Newspace parameters
| Level: | \( N \) | \(=\) | \( 464 = 2^{4} \cdot 29 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 464.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(27.3768862427\) |
| Analytic rank: | \(1\) |
| Dimension: | \(4\) |
| Coefficient field: | 4.4.225792.1 |
| Defining polynomial: |
\( x^{4} - 18x^{2} + 18 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{3} \) |
| Twist minimal: | no (minimal twist has level 232) |
| 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
\( x^{4} - 18x^{2} + 18 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{3} + \nu^{2} - 12\nu - 9 ) / 3 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -2\nu^{3} + 36\nu ) / 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 2\nu^{2} - 18 ) / 3 \)
|
| \(\nu\) | \(=\) |
\( ( -\beta_{3} + \beta_{2} + 2\beta_1 ) / 4 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 3\beta_{3} + 18 ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -9\beta_{3} + 6\beta_{2} + 18\beta_1 ) / 2 \)
|
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 | −6.40414 | 0 | −16.6404 | 0 | 3.16792 | 0 | 14.0130 | 0 | ||||||||||||||||||||||||||||||
| 1.2 | 0 | −4.12732 | 0 | −2.08419 | 0 | 13.1705 | 0 | −9.96522 | 0 | |||||||||||||||||||||||||||||||
| 1.3 | 0 | 1.11264 | 0 | 6.64036 | 0 | 11.4151 | 0 | −25.7620 | 0 | |||||||||||||||||||||||||||||||
| 1.4 | 0 | 9.41882 | 0 | −7.91581 | 0 | −19.7535 | 0 | 61.7142 | 0 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(1\) |
| \(29\) | \(-1\) |
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 | 464.4.a.k | 4 | |
| 4.b | odd | 2 | 1 | 232.4.a.c | ✓ | 4 | |
| 8.b | even | 2 | 1 | 1856.4.a.x | 4 | ||
| 8.d | odd | 2 | 1 | 1856.4.a.w | 4 | ||
| 12.b | even | 2 | 1 | 2088.4.a.e | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 232.4.a.c | ✓ | 4 | 4.b | odd | 2 | 1 | |
| 464.4.a.k | 4 | 1.a | even | 1 | 1 | trivial | |
| 1856.4.a.w | 4 | 8.d | odd | 2 | 1 | ||
| 1856.4.a.x | 4 | 8.b | even | 2 | 1 | ||
| 2088.4.a.e | 4 | 12.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{4} - 74T_{3}^{2} - 168T_{3} + 277 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(464))\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{4} \)
$3$
\( T^{4} - 74 T^{2} - 168 T + 277 \)
$5$
\( T^{4} + 20 T^{3} + 6 T^{2} + \cdots - 1823 \)
$7$
\( T^{4} - 8 T^{3} - 320 T^{2} + \cdots - 9408 \)
$11$
\( T^{4} + 8 T^{3} - 3746 T^{2} + \cdots + 2296893 \)
$13$
\( T^{4} + 60 T^{3} - 3698 T^{2} + \cdots - 4326887 \)
$17$
\( T^{4} + 224 T^{3} + \cdots - 55312368 \)
$19$
\( T^{4} - 200 T^{3} + \cdots - 18943728 \)
$23$
\( T^{4} - 104 T^{3} + \cdots + 26110672 \)
$29$
\( (T - 29)^{4} \)
$31$
\( T^{4} + 424 T^{3} + \cdots - 147822387 \)
$37$
\( T^{4} + 496 T^{3} + \cdots - 8389147392 \)
$41$
\( T^{4} + 264 T^{3} + \cdots + 796866624 \)
$43$
\( T^{4} + 376 T^{3} + \cdots - 6763463843 \)
$47$
\( T^{4} - 480 T^{3} + \cdots - 46277307 \)
$53$
\( T^{4} + 756 T^{3} + \cdots + 48201739177 \)
$59$
\( T^{4} - 440 T^{3} + \cdots + 27370709968 \)
$61$
\( T^{4} + 976 T^{3} + \cdots + 1049490576 \)
$67$
\( T^{4} - 80 T^{3} - 205760 T^{2} + \cdots - 715776 \)
$71$
\( T^{4} + 208 T^{3} + \cdots + 50595925968 \)
$73$
\( T^{4} - 528 T^{3} + \cdots + 4749558016 \)
$79$
\( T^{4} + 2512 T^{3} + \cdots + 75686955061 \)
$83$
\( T^{4} - 440 T^{3} + \cdots + 17011084752 \)
$89$
\( T^{4} + 1608 T^{3} + \cdots - 2795276736 \)
$97$
\( T^{4} - 584 T^{3} + \cdots + 10030146112 \)
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