Newspace parameters
| Level: | \( N \) | \(=\) | \( 374 = 2 \cdot 11 \cdot 17 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 374.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(2.98640503560\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | 4.4.55585.1 |
|
|
|
| Defining polynomial: |
\( x^{4} - x^{3} - 10x^{2} + 9x + 16 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Fricke sign: | \(-1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) 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
\( x^{4} - x^{3} - 10x^{2} + 9x + 16 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 5 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} + \nu^{2} - 7\nu - 5 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 5 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} - \beta_{2} + 7\beta_1 \)
|
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 |
|
−1.00000 | −2.88332 | 1.00000 | 4.31351 | 2.88332 | −2.83975 | −1.00000 | 5.31351 | −4.31351 | ||||||||||||||||||||||||||||||
| 1.2 | −1.00000 | −0.954430 | 1.00000 | −3.08906 | 0.954430 | 2.36654 | −1.00000 | −2.08906 | 3.08906 | |||||||||||||||||||||||||||||||
| 1.3 | −1.00000 | 2.22696 | 1.00000 | 0.959363 | −2.22696 | 4.62570 | −1.00000 | 1.95936 | −0.959363 | |||||||||||||||||||||||||||||||
| 1.4 | −1.00000 | 2.61078 | 1.00000 | 2.81619 | −2.61078 | −3.15249 | −1.00000 | 3.81619 | −2.81619 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( +1 \) |
| \(11\) | \( -1 \) |
| \(17\) | \( +1 \) |
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 | 374.2.a.d | ✓ | 4 |
| 3.b | odd | 2 | 1 | 3366.2.a.bg | 4 | ||
| 4.b | odd | 2 | 1 | 2992.2.a.w | 4 | ||
| 5.b | even | 2 | 1 | 9350.2.a.cl | 4 | ||
| 11.b | odd | 2 | 1 | 4114.2.a.bc | 4 | ||
| 17.b | even | 2 | 1 | 6358.2.a.t | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 374.2.a.d | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 2992.2.a.w | 4 | 4.b | odd | 2 | 1 | ||
| 3366.2.a.bg | 4 | 3.b | odd | 2 | 1 | ||
| 4114.2.a.bc | 4 | 11.b | odd | 2 | 1 | ||
| 6358.2.a.t | 4 | 17.b | even | 2 | 1 | ||
| 9350.2.a.cl | 4 | 5.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{4} - T_{3}^{3} - 10T_{3}^{2} + 9T_{3} + 16 \)
acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(374))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T + 1)^{4} \)
|
| $3$ |
\( T^{4} - T^{3} + \cdots + 16 \)
|
| $5$ |
\( T^{4} - 5 T^{3} + \cdots - 36 \)
|
| $7$ |
\( T^{4} - T^{3} + \cdots + 98 \)
|
| $11$ |
\( (T - 1)^{4} \)
|
| $13$ |
\( T^{4} + 3 T^{3} + \cdots + 350 \)
|
| $17$ |
\( (T + 1)^{4} \)
|
| $19$ |
\( T^{4} - 7 T^{3} + \cdots - 428 \)
|
| $23$ |
\( T^{4} - 6 T^{3} + \cdots + 240 \)
|
| $29$ |
\( T^{4} - 4 T^{3} + \cdots - 192 \)
|
| $31$ |
\( T^{4} - 4 T^{3} + \cdots + 80 \)
|
| $37$ |
\( T^{4} - 25 T^{3} + \cdots + 800 \)
|
| $41$ |
\( T^{4} - 5 T^{3} + \cdots + 810 \)
|
| $43$ |
\( T^{4} - 11 T^{3} + \cdots + 4 \)
|
| $47$ |
\( T^{4} + 17 T^{3} + \cdots - 192 \)
|
| $53$ |
\( T^{4} - 2 T^{3} + \cdots + 384 \)
|
| $59$ |
\( T^{4} - 4 T^{3} + \cdots + 480 \)
|
| $61$ |
\( T^{4} - 14 T^{3} + \cdots + 32 \)
|
| $67$ |
\( T^{4} - 14 T^{3} + \cdots + 32 \)
|
| $71$ |
\( T^{4} + 8 T^{3} + \cdots + 18000 \)
|
| $73$ |
\( T^{4} + 19 T^{3} + \cdots + 490 \)
|
| $79$ |
\( T^{4} - 17 T^{3} + \cdots + 6806 \)
|
| $83$ |
\( T^{4} + 19 T^{3} + \cdots + 2628 \)
|
| $89$ |
\( T^{4} + 5 T^{3} + \cdots + 5298 \)
|
| $97$ |
\( T^{4} + 6 T^{3} + \cdots + 896 \)
|
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