Newspace parameters
| Level: | \( N \) | \(=\) | \( 232 = 2^{3} \cdot 29 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 232.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(13.6884431213\) |
| Analytic rank: | \(0\) |
| Dimension: | \(5\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{5} - \cdots)\) |
| Defining polynomial: |
\( x^{5} - x^{4} - 34x^{3} + 74x^{2} + 94x - 198 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{3} \) |
| 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 \(1,\beta_1,\beta_2,\beta_3,\beta_4\) 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
\( x^{5} - x^{4} - 34x^{3} + 74x^{2} + 94x - 198 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -\nu^{4} - 5\nu^{3} + 23\nu^{2} + 64\nu - 90 ) / 19 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -\nu^{4} - 5\nu^{3} + 23\nu^{2} + 140\nu - 109 ) / 19 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -5\nu^{4} - 6\nu^{3} + 134\nu^{2} - 60\nu - 203 ) / 19 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -9\nu^{4} - 7\nu^{3} + 283\nu^{2} - 184\nu - 829 ) / 19 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{2} - \beta _1 + 1 ) / 4 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{4} - 2\beta_{3} + \beta _1 + 27 ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -\beta_{4} + 4\beta_{3} + 10\beta_{2} - 21\beta _1 - 43 ) / 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( 14\beta_{4} - 33\beta_{3} - 9\beta_{2} + 29\beta _1 + 344 \)
|
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 | −5.90002 | 0 | −14.9988 | 0 | −6.82211 | 0 | 7.81028 | 0 | |||||||||||||||||||||||||||||||||
| 1.2 | 0 | −4.03215 | 0 | 15.2584 | 0 | 33.3511 | 0 | −10.7418 | 0 | ||||||||||||||||||||||||||||||||||
| 1.3 | 0 | −1.01127 | 0 | 0.397400 | 0 | −14.4223 | 0 | −25.9773 | 0 | ||||||||||||||||||||||||||||||||||
| 1.4 | 0 | 7.11424 | 0 | 16.3850 | 0 | 5.74609 | 0 | 23.6124 | 0 | ||||||||||||||||||||||||||||||||||
| 1.5 | 0 | 7.82921 | 0 | −7.04208 | 0 | 14.1473 | 0 | 34.2965 | 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 | 232.4.a.d | ✓ | 5 |
| 3.b | odd | 2 | 1 | 2088.4.a.f | 5 | ||
| 4.b | odd | 2 | 1 | 464.4.a.m | 5 | ||
| 8.b | even | 2 | 1 | 1856.4.a.z | 5 | ||
| 8.d | odd | 2 | 1 | 1856.4.a.ba | 5 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 232.4.a.d | ✓ | 5 | 1.a | even | 1 | 1 | trivial |
| 464.4.a.m | 5 | 4.b | odd | 2 | 1 | ||
| 1856.4.a.z | 5 | 8.b | even | 2 | 1 | ||
| 1856.4.a.ba | 5 | 8.d | odd | 2 | 1 | ||
| 2088.4.a.f | 5 | 3.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{5} - 4T_{3}^{4} - 74T_{3}^{3} + 128T_{3}^{2} + 1525T_{3} + 1340 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(232))\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{5} \)
$3$
\( T^{5} - 4 T^{4} - 74 T^{3} + \cdots + 1340 \)
$5$
\( T^{5} - 10 T^{4} - 338 T^{3} + \cdots - 10494 \)
$7$
\( T^{5} - 32 T^{4} - 288 T^{3} + \cdots - 266752 \)
$11$
\( T^{5} - 36 T^{4} - 1346 T^{3} + \cdots + 1741860 \)
$13$
\( T^{5} - 26 T^{4} + \cdots + 105404410 \)
$17$
\( T^{5} - 82 T^{4} - 2776 T^{3} + \cdots - 49184 \)
$19$
\( T^{5} - 156 T^{4} + 4216 T^{3} + \cdots - 1820736 \)
$23$
\( T^{5} - 336 T^{4} + \cdots + 7489438848 \)
$29$
\( (T - 29)^{5} \)
$31$
\( T^{5} - 432 T^{4} + \cdots + 445071048 \)
$37$
\( T^{5} + 18 T^{4} + \cdots - 15294686720 \)
$41$
\( T^{5} - 82 T^{4} + \cdots - 731491061376 \)
$43$
\( T^{5} - 340 T^{4} + \cdots - 571309913052 \)
$47$
\( T^{5} - 680 T^{4} + \cdots + 2559413417896 \)
$53$
\( T^{5} + 102 T^{4} + \cdots + 103910584482 \)
$59$
\( T^{5} - 924 T^{4} + \cdots + 16799541984192 \)
$61$
\( T^{5} + 618 T^{4} + \cdots + 2366067286944 \)
$67$
\( T^{5} - 44 T^{4} + \cdots + 29804817076224 \)
$71$
\( T^{5} - 1032 T^{4} + \cdots - 50004302698368 \)
$73$
\( T^{5} + 1078 T^{4} + \cdots - 4755790305792 \)
$79$
\( T^{5} - 200 T^{4} + \cdots - 13404287016 \)
$83$
\( T^{5} + \cdots - 181825403631808 \)
$89$
\( T^{5} + 1790 T^{4} + \cdots - 79946879886976 \)
$97$
\( T^{5} + 2518 T^{4} + \cdots - 41073987679360 \)
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