Newspace parameters
| Level: | \( N \) | \(=\) | \( 304 = 2^{4} \cdot 19 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 304.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(17.9365806417\) |
| Analytic rank: | \(1\) |
| Dimension: | \(2\) |
| Coefficient field: | \(\Q(\sqrt{57}) \) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{2} - x - 14 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 152) |
| Fricke sign: | \(-1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \frac{1}{2}(1 + \sqrt{57})\). We also show the integral \(q\)-expansion of the trace form.
Embeddings
For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.
For more information on an embedded modular form you can click on its label.
| Label | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1.1 |
|
0 | −3.27492 | 0 | −6.27492 | 0 | 18.0997 | 0 | −16.2749 | 0 | ||||||||||||||||||||||||
| 1.2 | 0 | 4.27492 | 0 | 1.27492 | 0 | −12.0997 | 0 | −8.72508 | 0 | |||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(1\) |
| \(19\) | \(-1\) |
Inner twists
This newform does not admit any (nontrivial) inner twists.
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 304.4.a.e | 2 | |
| 4.b | odd | 2 | 1 | 152.4.a.a | ✓ | 2 | |
| 8.b | even | 2 | 1 | 1216.4.a.k | 2 | ||
| 8.d | odd | 2 | 1 | 1216.4.a.m | 2 | ||
| 12.b | even | 2 | 1 | 1368.4.a.a | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 152.4.a.a | ✓ | 2 | 4.b | odd | 2 | 1 | |
| 304.4.a.e | 2 | 1.a | even | 1 | 1 | trivial | |
| 1216.4.a.k | 2 | 8.b | even | 2 | 1 | ||
| 1216.4.a.m | 2 | 8.d | odd | 2 | 1 | ||
| 1368.4.a.a | 2 | 12.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{2} - T_{3} - 14 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(304))\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{2} \)
$3$
\( T^{2} - T - 14 \)
$5$
\( T^{2} + 5T - 8 \)
$7$
\( T^{2} - 6T - 219 \)
$11$
\( T^{2} - 15T - 642 \)
$13$
\( T^{2} + 59T + 172 \)
$17$
\( T^{2} + 104T + 1279 \)
$19$
\( (T - 19)^{2} \)
$23$
\( T^{2} - 21T - 15408 \)
$29$
\( T^{2} + 137T + 3538 \)
$31$
\( T^{2} + 4T - 11168 \)
$37$
\( T^{2} + 152T - 5396 \)
$41$
\( T^{2} + 210T - 104400 \)
$43$
\( T^{2} + 67T - 6416 \)
$47$
\( T^{2} + 273T - 10224 \)
$53$
\( T^{2} - 209T + 10792 \)
$59$
\( T^{2} + 799T + 103042 \)
$61$
\( T^{2} - 149T - 34478 \)
$67$
\( T^{2} + 201T + 9744 \)
$71$
\( T^{2} + 792T + 14316 \)
$73$
\( T^{2} + 246T - 58743 \)
$79$
\( T^{2} - 254T - 169064 \)
$83$
\( T^{2} + 374T - 302984 \)
$89$
\( T^{2} + 564T - 467904 \)
$97$
\( T^{2} + 178 T - 1985312 \)
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