Internal Server Error With Minimal PHP Contact Form - php

Question

This site is hosted on shared hosting, unfortunately, the only thing the host seems to make available is the access log, not the PHP error log, so I'm having trouble figuring out the specific error here.
I'm trying to do a very minimal, simple contact form in PHP, without much validation or complexity. Getting a 500 error, though I'm sure I'm overlooking something basic.
Any assistance greatly appreciated.
HTML:
<form name="htmlform" method="post" action="collins_contact.php">
<table width="450px" cellpadding="20px">
<tr><td> </td></tr>
<tr>
<td align="left">
<label for="first_name">First Name</label>
</td>
<td valign="top">
<input type="text" name="first_name" maxlength="50" size="30">
</td>
</tr>
<tr><td> </td></tr>
<tr>
<td align="left">
<label for="last_name">Last Name</label>
</td>
<td valign="top">
<input type="text" name="last_name" maxlength="50" size="30">
</td>
</tr>
<tr><td> </td></tr>
<tr>
<td align="left">
<label for="email">Email Address</label>
</td>
<td valign="top">
<input type="text" name="email" maxlength="80" size="30">
</td>
</tr>
<tr><td> </td></tr>
<tr>
<td align="left">
<label for="telephone">Telephone</label>
</td>
<td valign="top">
<input type="text" name="phone" maxlength="30" size="30">
</td>
</tr>
<tr><td> </td></tr>
<tr>
<td align="left">
<label for="comments">Questions / Comments</label>
</td>
<td valign="top">
<textarea name="message" maxlength="1000" cols="25" rows="6"></textarea>
</td>
<tr><td> </td></tr>
</tr>
<tr>
<td colspan="2" style="text-align:center">
<input type="submit" value="Submit">
</td>
</tr>
<tr><td> </td></tr>
</table>
</form>
PHP
<?php
if (isset($_POST["first_name"]) && isset($_POST["email"]) && isset($_POST["message"])) {
$name = $_POST["first_name"];
$email = $_POST["email"];
$subject = "Contact Form Submission";
$message = $_POST["message"];
$header = "from: $name <$email>";
$to = "contact#domain.com";
if (mail($to, $subject, $message, $header)) {
header("Location: index.html");
} else {
echo("Error With Form Submission");
}
}
?>

Answer 1

Permissions needed to be changed to 0644 on the php file, then it worked.