Newspace parameters
| Level: | \( N \) | \(=\) | \( 115 = 5 \cdot 23 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 115.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(6.78521965066\) |
| Analytic rank: | \(0\) |
| Dimension: | \(5\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{5} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{5} - x^{4} - 27x^{3} + 7x^{2} + 168x + 92 \)
|
| 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 \(1,\beta_1,\beta_2,\beta_3,\beta_4\) 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
\( x^{5} - x^{4} - 27x^{3} + 7x^{2} + 168x + 92 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{4} + \nu^{3} - 25\nu^{2} - 11\nu + 98 ) / 16 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -\nu^{4} + 3\nu^{3} + 17\nu^{2} - 41\nu - 42 ) / 8 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -\nu^{4} + 7\nu^{3} + 25\nu^{2} - 109\nu - 162 ) / 16 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{4} - \beta_{3} - \beta_{2} + \beta _1 + 11 \)
|
| \(\nu^{3}\) | \(=\) |
\( 2\beta_{4} + 2\beta_{2} + 15\beta _1 + 8 \)
|
| \(\nu^{4}\) | \(=\) |
\( 23\beta_{4} - 25\beta_{3} - 11\beta_{2} + 21\beta _1 + 169 \)
|
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 |
|
−2.93900 | −3.85751 | 0.637693 | −5.00000 | 11.3372 | −23.5932 | 21.6378 | −12.1196 | 14.6950 | |||||||||||||||||||||||||||||||||
| 1.2 | −1.49214 | 9.02447 | −5.77352 | −5.00000 | −13.4658 | 4.33445 | 20.5520 | 54.4411 | 7.46070 | ||||||||||||||||||||||||||||||||||
| 1.3 | 0.404957 | −7.11323 | −7.83601 | −5.00000 | −2.88055 | 13.7888 | −6.41290 | 23.5981 | −2.02479 | ||||||||||||||||||||||||||||||||||
| 1.4 | 4.41740 | 7.84147 | 11.5134 | −5.00000 | 34.6389 | −8.97260 | 15.5200 | 34.4886 | −22.0870 | ||||||||||||||||||||||||||||||||||
| 1.5 | 5.60878 | −1.89520 | 23.4584 | −5.00000 | −10.6297 | 11.4426 | 86.7031 | −23.4082 | −28.0439 | ||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(5\) | \(1\) |
| \(23\) | \(1\) |
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 | 115.4.a.e | ✓ | 5 |
| 3.b | odd | 2 | 1 | 1035.4.a.k | 5 | ||
| 4.b | odd | 2 | 1 | 1840.4.a.n | 5 | ||
| 5.b | even | 2 | 1 | 575.4.a.j | 5 | ||
| 5.c | odd | 4 | 2 | 575.4.b.i | 10 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 115.4.a.e | ✓ | 5 | 1.a | even | 1 | 1 | trivial |
| 575.4.a.j | 5 | 5.b | even | 2 | 1 | ||
| 575.4.b.i | 10 | 5.c | odd | 4 | 2 | ||
| 1035.4.a.k | 5 | 3.b | odd | 2 | 1 | ||
| 1840.4.a.n | 5 | 4.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{5} - 6T_{2}^{4} - 13T_{2}^{3} + 72T_{2}^{2} + 82T_{2} - 44 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(115))\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{5} - 6 T^{4} - 13 T^{3} + 72 T^{2} + \cdots - 44 \)
$3$
\( T^{5} - 4 T^{4} - 98 T^{3} + \cdots + 3680 \)
$5$
\( (T + 5)^{5} \)
$7$
\( T^{5} + 3 T^{4} - 484 T^{3} + \cdots - 144774 \)
$11$
\( T^{5} - 23 T^{4} - 4015 T^{3} + \cdots - 74136848 \)
$13$
\( T^{5} - 132 T^{4} + 4786 T^{3} + \cdots + 1550116 \)
$17$
\( T^{5} - 23 T^{4} + \cdots - 1039045340 \)
$19$
\( T^{5} + 161 T^{4} + 3999 T^{3} + \cdots - 801280 \)
$23$
\( (T + 23)^{5} \)
$29$
\( T^{5} - 401 T^{4} + \cdots - 6149898500 \)
$31$
\( T^{5} - 32 T^{4} + \cdots - 438072447 \)
$37$
\( T^{5} + 38 T^{4} + \cdots + 1590700778176 \)
$41$
\( T^{5} + 12 T^{4} + \cdots + 114116030755 \)
$43$
\( T^{5} + 566 T^{4} + \cdots + 504784881664 \)
$47$
\( T^{5} - 919 T^{4} + \cdots + 117787714816 \)
$53$
\( T^{5} - 1156 T^{4} + \cdots + 5720332226904 \)
$59$
\( T^{5} - 1324 T^{4} + \cdots + 24279649927232 \)
$61$
\( T^{5} + 1673 T^{4} + \cdots - 34095834816896 \)
$67$
\( T^{5} - 558 T^{4} + \cdots + 5644442112 \)
$71$
\( T^{5} + 108 T^{4} + \cdots + 15638892903635 \)
$73$
\( T^{5} + \cdots - 100895881632176 \)
$79$
\( T^{5} - 656 T^{4} + \cdots - 90481602379776 \)
$83$
\( T^{5} + 82 T^{4} + \cdots + 18307318870176 \)
$89$
\( T^{5} + \cdots - 115104799418880 \)
$97$
\( T^{5} + \cdots - 480989167569272 \)
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