Newspace parameters
| Level: | \( N \) | \(=\) | \( 400 = 2^{4} \cdot 5^{2} \) |
| Weight: | \( k \) | \(=\) | \( 8 \) |
| Character orbit: | \([\chi]\) | \(=\) | 400.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(124.954010194\) |
| Analytic rank: | \(1\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{4} - \cdots)\) |
|
|
|
| Defining polynomial: |
\( x^{4} - 2x^{3} - 96x^{2} - 318x - 135 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2^{8}\cdot 5^{2} \) |
| Twist minimal: | no (minimal twist has level 40) |
| 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
\( x^{4} - 2x^{3} - 96x^{2} - 318x - 135 \)
:
| \(\beta_{1}\) | \(=\) |
\( 2\nu^{2} - 12\nu - 92 \)
|
| \(\beta_{2}\) | \(=\) |
\( -6\nu^{2} + 116\nu + 236 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 40\nu^{3} - 236\nu^{2} - 2784\nu - 2424 ) / 3 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{2} + 3\beta _1 + 40 ) / 80 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 3\beta_{2} + 29\beta _1 + 1960 ) / 40 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 6\beta_{3} + 105\beta_{2} + 551\beta _1 + 30760 ) / 80 \)
|
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 | −65.0953 | 0 | 0 | 0 | −1567.21 | 0 | 2050.39 | 0 | ||||||||||||||||||||||||||||||
| 1.2 | 0 | −52.5319 | 0 | 0 | 0 | 338.895 | 0 | 572.598 | 0 | |||||||||||||||||||||||||||||||
| 1.3 | 0 | 8.10762 | 0 | 0 | 0 | 980.714 | 0 | −2121.27 | 0 | |||||||||||||||||||||||||||||||
| 1.4 | 0 | 77.5195 | 0 | 0 | 0 | −1384.40 | 0 | 3822.28 | 0 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( +1 \) |
| \(5\) | \( -1 \) |
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 | 400.8.a.bj | 4 | |
| 4.b | odd | 2 | 1 | 200.8.a.r | 4 | ||
| 5.b | even | 2 | 1 | 400.8.a.bl | 4 | ||
| 5.c | odd | 4 | 2 | 80.8.c.e | 8 | ||
| 20.d | odd | 2 | 1 | 200.8.a.q | 4 | ||
| 20.e | even | 4 | 2 | 40.8.c.b | ✓ | 8 | |
| 40.i | odd | 4 | 2 | 320.8.c.l | 8 | ||
| 40.k | even | 4 | 2 | 320.8.c.k | 8 | ||
| 60.l | odd | 4 | 2 | 360.8.f.b | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 40.8.c.b | ✓ | 8 | 20.e | even | 4 | 2 | |
| 80.8.c.e | 8 | 5.c | odd | 4 | 2 | ||
| 200.8.a.q | 4 | 20.d | odd | 2 | 1 | ||
| 200.8.a.r | 4 | 4.b | odd | 2 | 1 | ||
| 320.8.c.k | 8 | 40.k | even | 4 | 2 | ||
| 320.8.c.l | 8 | 40.i | odd | 4 | 2 | ||
| 360.8.f.b | 8 | 60.l | odd | 4 | 2 | ||
| 400.8.a.bj | 4 | 1.a | even | 1 | 1 | trivial | |
| 400.8.a.bl | 4 | 5.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{4} + 32T_{3}^{3} - 6024T_{3}^{2} - 218880T_{3} + 2149200 \)
acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(400))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} \)
|
| $3$ |
\( T^{4} + 32 T^{3} + \cdots + 2149200 \)
|
| $5$ |
\( T^{4} \)
|
| $7$ |
\( T^{4} + \cdots + 721102628944 \)
|
| $11$ |
\( T^{4} + \cdots + 6076238622976 \)
|
| $13$ |
\( T^{4} + \cdots - 697938363343616 \)
|
| $17$ |
\( T^{4} + \cdots + 26\!\cdots\!00 \)
|
| $19$ |
\( T^{4} + \cdots - 50\!\cdots\!84 \)
|
| $23$ |
\( T^{4} + \cdots - 47\!\cdots\!96 \)
|
| $29$ |
\( T^{4} + \cdots + 35\!\cdots\!96 \)
|
| $31$ |
\( T^{4} + \cdots - 11\!\cdots\!00 \)
|
| $37$ |
\( T^{4} + \cdots - 16\!\cdots\!84 \)
|
| $41$ |
\( T^{4} + \cdots + 36\!\cdots\!64 \)
|
| $43$ |
\( T^{4} + \cdots + 34\!\cdots\!24 \)
|
| $47$ |
\( T^{4} + \cdots - 12\!\cdots\!64 \)
|
| $53$ |
\( T^{4} + \cdots - 47\!\cdots\!84 \)
|
| $59$ |
\( T^{4} + \cdots - 15\!\cdots\!96 \)
|
| $61$ |
\( T^{4} + \cdots - 10\!\cdots\!00 \)
|
| $67$ |
\( T^{4} + \cdots - 33\!\cdots\!76 \)
|
| $71$ |
\( T^{4} + \cdots - 19\!\cdots\!00 \)
|
| $73$ |
\( T^{4} + \cdots + 28\!\cdots\!04 \)
|
| $79$ |
\( T^{4} + \cdots - 17\!\cdots\!00 \)
|
| $83$ |
\( T^{4} + \cdots + 26\!\cdots\!84 \)
|
| $89$ |
\( T^{4} + \cdots - 25\!\cdots\!84 \)
|
| $97$ |
\( T^{4} + \cdots - 74\!\cdots\!44 \)
|
| show more | |
| show less |