Newspace parameters
| Level: | \( N \) | \(=\) | \( 115 = 5 \cdot 23 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 115.b (of order \(2\), degree \(1\), minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(0.918279623245\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Coefficient field: | 8.0.527896576.2 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} - 2x^{7} + 2x^{6} + 2x^{5} + 7x^{4} - 10x^{3} + 8x^{2} + 4x + 1 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) 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
\( x^{8} - 2x^{7} + 2x^{6} + 2x^{5} + 7x^{4} - 10x^{3} + 8x^{2} + 4x + 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -27\nu^{7} + 31\nu^{6} + 7\nu^{5} - 221\nu^{4} - 187\nu^{3} + 128\nu^{2} + 66\nu - 882 ) / 467 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 52\nu^{7} - 77\nu^{6} + 73\nu^{5} + 97\nu^{4} + 585\nu^{3} - 333\nu^{2} + 288\nu + 142 ) / 467 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 138\nu^{7} - 366\nu^{6} + 535\nu^{5} - 12\nu^{4} + 852\nu^{3} - 1692\nu^{2} + 2309\nu - 162 ) / 467 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -142\nu^{7} + 336\nu^{6} - 361\nu^{5} - 211\nu^{4} - 897\nu^{3} + 2005\nu^{2} - 1469\nu - 280 ) / 467 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 273\nu^{7} - 521\nu^{6} + 500\nu^{5} + 626\nu^{4} + 1787\nu^{3} - 2332\nu^{2} + 1979\nu + 979 ) / 467 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( -284\nu^{7} + 672\nu^{6} - 722\nu^{5} - 422\nu^{4} - 1794\nu^{3} + 3543\nu^{2} - 2938\nu - 560 ) / 467 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( -\beta_{7} + 2\beta_{5} \)
|
| \(\nu^{3}\) | \(=\) |
\( -\beta_{7} - \beta_{6} + \beta_{5} + 2\beta_{3} - \beta_{2} - 1 \)
|
| \(\nu^{4}\) | \(=\) |
\( -\beta_{6} + \beta_{4} - 5\beta_{2} - 7 \)
|
| \(\nu^{5}\) | \(=\) |
\( 6\beta_{7} - 6\beta_{5} + 5\beta_{4} - 6\beta_{2} - 5\beta _1 - 6 \)
|
| \(\nu^{6}\) | \(=\) |
\( 22\beta_{7} + 6\beta_{6} - 27\beta_{5} + 6\beta_{4} - \beta_{3} - \beta_1 \)
|
| \(\nu^{7}\) | \(=\) |
\( 29\beta_{7} + 22\beta_{6} - 30\beta_{5} - 15\beta_{3} + 29\beta_{2} + 30 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/115\mathbb{Z}\right)^\times\).
| \(n\) | \(47\) | \(51\) |
| \(\chi(n)\) | \(-1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 24.1 |
|
− | 2.37988i | − | 1.95969i | −3.66382 | 0.479844 | + | 2.18398i | −4.66382 | − | 2.28394i | 3.95969i | −0.840379 | 5.19760 | − | 1.14197i | |||||||||||||||||||||||||||||||||||
| 24.2 | − | 1.92022i | − | 1.39945i | −1.68725 | −1.19972 | − | 1.88697i | −2.68725 | 4.60747i | − | 0.600553i | 1.04155 | −3.62340 | + | 2.30373i | ||||||||||||||||||||||||||||||||||||
| 24.3 | − | 0.751024i | 0.580491i | 1.43596 | −0.209755 | + | 2.22621i | 0.435963 | 0.315061i | − | 2.58049i | 2.66303 | 1.67194 | + | 0.157531i | |||||||||||||||||||||||||||||||||||||
| 24.4 | − | 0.291367i | 3.14073i | 1.91511 | −2.07037 | − | 0.844739i | 0.915105 | 1.20647i | − | 1.14073i | −6.86420 | −0.246129 | + | 0.603236i | |||||||||||||||||||||||||||||||||||||
| 24.5 | 0.291367i | − | 3.14073i | 1.91511 | −2.07037 | + | 0.844739i | 0.915105 | − | 1.20647i | 1.14073i | −6.86420 | −0.246129 | − | 0.603236i | |||||||||||||||||||||||||||||||||||||
| 24.6 | 0.751024i | − | 0.580491i | 1.43596 | −0.209755 | − | 2.22621i | 0.435963 | − | 0.315061i | 2.58049i | 2.66303 | 1.67194 | − | 0.157531i | |||||||||||||||||||||||||||||||||||||
| 24.7 | 1.92022i | 1.39945i | −1.68725 | −1.19972 | + | 1.88697i | −2.68725 | − | 4.60747i | 0.600553i | 1.04155 | −3.62340 | − | 2.30373i | ||||||||||||||||||||||||||||||||||||||
| 24.8 | 2.37988i | 1.95969i | −3.66382 | 0.479844 | − | 2.18398i | −4.66382 | 2.28394i | − | 3.95969i | −0.840379 | 5.19760 | + | 1.14197i | ||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 5.b | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 115.2.b.b | ✓ | 8 |
| 3.b | odd | 2 | 1 | 1035.2.b.e | 8 | ||
| 4.b | odd | 2 | 1 | 1840.2.e.d | 8 | ||
| 5.b | even | 2 | 1 | inner | 115.2.b.b | ✓ | 8 |
| 5.c | odd | 4 | 1 | 575.2.a.i | 4 | ||
| 5.c | odd | 4 | 1 | 575.2.a.j | 4 | ||
| 15.d | odd | 2 | 1 | 1035.2.b.e | 8 | ||
| 15.e | even | 4 | 1 | 5175.2.a.bv | 4 | ||
| 15.e | even | 4 | 1 | 5175.2.a.bw | 4 | ||
| 20.d | odd | 2 | 1 | 1840.2.e.d | 8 | ||
| 20.e | even | 4 | 1 | 9200.2.a.ck | 4 | ||
| 20.e | even | 4 | 1 | 9200.2.a.cq | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 115.2.b.b | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 115.2.b.b | ✓ | 8 | 5.b | even | 2 | 1 | inner |
| 575.2.a.i | 4 | 5.c | odd | 4 | 1 | ||
| 575.2.a.j | 4 | 5.c | odd | 4 | 1 | ||
| 1035.2.b.e | 8 | 3.b | odd | 2 | 1 | ||
| 1035.2.b.e | 8 | 15.d | odd | 2 | 1 | ||
| 1840.2.e.d | 8 | 4.b | odd | 2 | 1 | ||
| 1840.2.e.d | 8 | 20.d | odd | 2 | 1 | ||
| 5175.2.a.bv | 4 | 15.e | even | 4 | 1 | ||
| 5175.2.a.bw | 4 | 15.e | even | 4 | 1 | ||
| 9200.2.a.ck | 4 | 20.e | even | 4 | 1 | ||
| 9200.2.a.cq | 4 | 20.e | even | 4 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{8} + 10T_{2}^{6} + 27T_{2}^{4} + 14T_{2}^{2} + 1 \)
acting on \(S_{2}^{\mathrm{new}}(115, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{8} + 10 T^{6} + 27 T^{4} + 14 T^{2} + \cdots + 1 \)
$3$
\( T^{8} + 16 T^{6} + 70 T^{4} + 96 T^{2} + \cdots + 25 \)
$5$
\( T^{8} + 6 T^{7} + 26 T^{6} + 82 T^{5} + \cdots + 625 \)
$7$
\( T^{8} + 28 T^{6} + 152 T^{4} + \cdots + 16 \)
$11$
\( (T^{4} - 2 T^{3} - 16 T^{2} + 44 T - 28)^{2} \)
$13$
\( T^{8} + 64 T^{6} + 1174 T^{4} + \cdots + 3721 \)
$17$
\( T^{8} + 80 T^{6} + 1612 T^{4} + \cdots + 400 \)
$19$
\( (T^{4} - 4 T^{3} - 6 T^{2} + 28 T - 20)^{2} \)
$23$
\( (T^{2} + 1)^{4} \)
$29$
\( (T^{4} + 4 T^{3} - 22 T^{2} - 4 T + 5)^{2} \)
$31$
\( (T^{4} - 74 T^{2} - 256 T - 167)^{2} \)
$37$
\( T^{8} + 148 T^{6} + 5752 T^{4} + \cdots + 226576 \)
$41$
\( (T^{4} + 8 T^{3} - 94 T^{2} - 368 T + 2485)^{2} \)
$43$
\( T^{8} + 252 T^{6} + 21340 T^{4} + \cdots + 3857296 \)
$47$
\( T^{8} + 280 T^{6} + \cdots + 20367169 \)
$53$
\( T^{8} + 224 T^{6} + 16096 T^{4} + \cdots + 4804864 \)
$59$
\( (T^{4} - 20 T^{2} - 16 T + 16)^{2} \)
$61$
\( (T^{4} + 8 T^{3} - 102 T^{2} - 1148 T - 2756)^{2} \)
$67$
\( T^{8} + 20 T^{6} + 120 T^{4} + \cdots + 16 \)
$71$
\( (T^{4} + 24 T^{3} + 66 T^{2} - 1568 T - 7435)^{2} \)
$73$
\( T^{8} + 368 T^{6} + \cdots + 69538921 \)
$79$
\( (T^{4} + 24 T^{3} + 178 T^{2} + 412 T + 28)^{2} \)
$83$
\( T^{8} + 576 T^{6} + \cdots + 223442704 \)
$89$
\( (T^{4} - 8 T^{3} - 86 T^{2} + 372 T + 2380)^{2} \)
$97$
\( T^{8} + 460 T^{6} + \cdots + 21864976 \)
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