Newspace parameters
| Level: | \( N \) | \(=\) | \( 304 = 2^{4} \cdot 19 \) |
| Weight: | \( k \) | \(=\) | \( 10 \) |
| Character orbit: | \([\chi]\) | \(=\) | 304.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(156.570894194\) |
| Analytic rank: | \(1\) |
| Dimension: | \(3\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{3} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{3} - x^{2} - 4552x + 85948 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{3}\cdot 3 \) |
| Twist minimal: | no (minimal twist has level 38) |
| Fricke sign: | \(1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.
Basis of coefficient ring in terms of a root \(\nu\) of
\( x^{3} - x^{2} - 4552x + 85948 \)
:
| \(\beta_{1}\) | \(=\) |
\( 3\nu - 1 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{2} + 24\nu - 3043 ) / 3 \)
|
| \(\nu\) | \(=\) |
\( ( \beta _1 + 1 ) / 3 \)
|
| \(\nu^{2}\) | \(=\) |
\( 3\beta_{2} - 8\beta _1 + 3035 \)
|
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 | −165.716 | 0 | 936.105 | 0 | 11191.8 | 0 | 7778.66 | 0 | |||||||||||||||||||||||||||
| 1.2 | 0 | −62.2478 | 0 | −420.333 | 0 | −1751.81 | 0 | −15808.2 | 0 | ||||||||||||||||||||||||||||
| 1.3 | 0 | 224.963 | 0 | −29.7721 | 0 | 3877.06 | 0 | 30925.5 | 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 | 304.10.a.c | 3 | |
| 4.b | odd | 2 | 1 | 38.10.a.c | ✓ | 3 | |
| 12.b | even | 2 | 1 | 342.10.a.e | 3 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 38.10.a.c | ✓ | 3 | 4.b | odd | 2 | 1 | |
| 304.10.a.c | 3 | 1.a | even | 1 | 1 | trivial | |
| 342.10.a.e | 3 | 12.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{3} + 3T_{3}^{2} - 40968T_{3} - 2320596 \)
acting on \(S_{10}^{\mathrm{new}}(\Gamma_0(304))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{3} \)
|
| $3$ |
\( T^{3} + 3 T^{2} + \cdots - 2320596 \)
|
| $5$ |
\( T^{3} - 486 T^{2} + \cdots - 11714584 \)
|
| $7$ |
\( T^{3} + \cdots + 76013110089 \)
|
| $11$ |
\( T^{3} + \cdots - 153565439505290 \)
|
| $13$ |
\( T^{3} + \cdots + 39757599966256 \)
|
| $17$ |
\( T^{3} + \cdots - 42\!\cdots\!43 \)
|
| $19$ |
\( (T - 130321)^{3} \)
|
| $23$ |
\( T^{3} + \cdots + 28\!\cdots\!60 \)
|
| $29$ |
\( T^{3} + \cdots + 12\!\cdots\!40 \)
|
| $31$ |
\( T^{3} + \cdots + 36\!\cdots\!60 \)
|
| $37$ |
\( T^{3} + \cdots + 15\!\cdots\!48 \)
|
| $41$ |
\( T^{3} + \cdots - 74\!\cdots\!76 \)
|
| $43$ |
\( T^{3} + \cdots - 11\!\cdots\!48 \)
|
| $47$ |
\( T^{3} + \cdots - 12\!\cdots\!80 \)
|
| $53$ |
\( T^{3} + \cdots + 23\!\cdots\!92 \)
|
| $59$ |
\( T^{3} + \cdots - 25\!\cdots\!32 \)
|
| $61$ |
\( T^{3} + \cdots + 41\!\cdots\!58 \)
|
| $67$ |
\( T^{3} + \cdots - 82\!\cdots\!08 \)
|
| $71$ |
\( T^{3} + \cdots + 16\!\cdots\!60 \)
|
| $73$ |
\( T^{3} + \cdots - 22\!\cdots\!17 \)
|
| $79$ |
\( T^{3} + \cdots + 19\!\cdots\!00 \)
|
| $83$ |
\( T^{3} + \cdots - 62\!\cdots\!44 \)
|
| $89$ |
\( T^{3} + \cdots - 27\!\cdots\!80 \)
|
| $97$ |
\( T^{3} + \cdots - 40\!\cdots\!92 \)
|
| show more | |
| show less |