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: | \(3\) |
| Coefficient field: | 3.3.7057.1 |
|
|
|
| Defining polynomial: |
\( x^{3} - x^{2} - 22x + 32 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2 \) |
| 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 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
\( x^{3} - x^{2} - 22x + 32 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{2} + 3\nu - 16 ) / 2 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{2} - \nu - 14 ) / 2 \)
|
| \(\nu\) | \(=\) |
\( ( -\beta_{2} + \beta _1 + 1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 3\beta_{2} + \beta _1 + 29 ) / 2 \)
|
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 | −8.28632 | 0 | 2.38427 | 0 | −5.83529 | 0 | 41.6631 | 0 | |||||||||||||||||||||||||||
| 1.2 | 0 | −1.33180 | 0 | 18.4427 | 0 | −10.3872 | 0 | −25.2263 | 0 | ||||||||||||||||||||||||||||
| 1.3 | 0 | 5.61812 | 0 | −13.8269 | 0 | 9.22252 | 0 | 4.56325 | 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.g | 3 | |
| 4.b | odd | 2 | 1 | 152.4.a.c | ✓ | 3 | |
| 8.b | even | 2 | 1 | 1216.4.a.w | 3 | ||
| 8.d | odd | 2 | 1 | 1216.4.a.r | 3 | ||
| 12.b | even | 2 | 1 | 1368.4.a.d | 3 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 152.4.a.c | ✓ | 3 | 4.b | odd | 2 | 1 | |
| 304.4.a.g | 3 | 1.a | even | 1 | 1 | trivial | |
| 1216.4.a.r | 3 | 8.d | odd | 2 | 1 | ||
| 1216.4.a.w | 3 | 8.b | even | 2 | 1 | ||
| 1368.4.a.d | 3 | 12.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{3} + 4T_{3}^{2} - 43T_{3} - 62 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(304))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{3} \)
|
| $3$ |
\( T^{3} + 4 T^{2} + \cdots - 62 \)
|
| $5$ |
\( T^{3} - 7 T^{2} + \cdots + 608 \)
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| $7$ |
\( T^{3} + 7 T^{2} + \cdots - 559 \)
|
| $11$ |
\( T^{3} + 103 T^{2} + \cdots - 22156 \)
|
| $13$ |
\( T^{3} - 32 T^{2} + \cdots + 131420 \)
|
| $17$ |
\( T^{3} - 11 T^{2} + \cdots - 32173 \)
|
| $19$ |
\( (T - 19)^{3} \)
|
| $23$ |
\( T^{3} + 316 T^{2} + \cdots - 242336 \)
|
| $29$ |
\( T^{3} + 138 T^{2} + \cdots - 345766 \)
|
| $31$ |
\( T^{3} + 420 T^{2} + \cdots + 2336256 \)
|
| $37$ |
\( T^{3} - 102 T^{2} + \cdots + 15565400 \)
|
| $41$ |
\( T^{3} + 370 T^{2} + \cdots + 707456 \)
|
| $43$ |
\( T^{3} + 431 T^{2} + \cdots - 5420480 \)
|
| $47$ |
\( T^{3} - 199 T^{2} + \cdots + 45306112 \)
|
| $53$ |
\( T^{3} + 308 T^{2} + \cdots + 6630640 \)
|
| $59$ |
\( T^{3} - 188 T^{2} + \cdots + 1077242 \)
|
| $61$ |
\( T^{3} + 609 T^{2} + \cdots - 50618548 \)
|
| $67$ |
\( T^{3} - 246 T^{2} + \cdots + 96246632 \)
|
| $71$ |
\( T^{3} - 954 T^{2} + \cdots + 237273448 \)
|
| $73$ |
\( T^{3} + 629 T^{2} + \cdots - 417051529 \)
|
| $79$ |
\( T^{3} + 452 T^{2} + \cdots - 601611824 \)
|
| $83$ |
\( T^{3} + \cdots + 1048786960 \)
|
| $89$ |
\( T^{3} + 1356 T^{2} + \cdots + 71672320 \)
|
| $97$ |
\( T^{3} + \cdots - 2035482752 \)
|
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