Newspace parameters
| Level: | \( N \) | \(=\) | \( 130 = 2 \cdot 5 \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 8 \) |
| Character orbit: | \([\chi]\) | \(=\) | 130.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(40.6100533129\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{4} - \cdots)\) |
|
|
|
| Defining polynomial: |
\( x^{4} - 7066x^{2} - 15288x + 8343477 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{4} \) |
| 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 \(1,\beta_1,\beta_2,\beta_3\) 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
\( x^{4} - 7066x^{2} - 15288x + 8343477 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{3} - 21\nu^{2} - 4645\nu + 62727 ) / 90 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{2} - 3\nu - 3533 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{3} + 3\beta _1 + 3533 \)
|
| \(\nu^{3}\) | \(=\) |
\( 21\beta_{3} + 90\beta_{2} + 4708\beta _1 + 11466 \)
|
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 |
|
8.00000 | −57.6171 | 64.0000 | 125.000 | −460.937 | 1412.21 | 512.000 | 1132.73 | 1000.00 | ||||||||||||||||||||||||||||||
| 1.2 | 8.00000 | −25.7169 | 64.0000 | 125.000 | −205.735 | −1385.39 | 512.000 | −1525.64 | 1000.00 | |||||||||||||||||||||||||||||||
| 1.3 | 8.00000 | 51.9363 | 64.0000 | 125.000 | 415.491 | 1010.97 | 512.000 | 510.382 | 1000.00 | |||||||||||||||||||||||||||||||
| 1.4 | 8.00000 | 91.3977 | 64.0000 | 125.000 | 731.181 | −421.792 | 512.000 | 6166.53 | 1000.00 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( -1 \) |
| \(5\) | \( -1 \) |
| \(13\) | \( +1 \) |
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 | 130.8.a.j | ✓ | 4 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 130.8.a.j | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{4} - 60T_{3}^{3} - 5716T_{3}^{2} + 183192T_{3} + 7033572 \)
acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(130))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 8)^{4} \)
|
| $3$ |
\( T^{4} - 60 T^{3} + \cdots + 7033572 \)
|
| $5$ |
\( (T - 125)^{4} \)
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| $7$ |
\( T^{4} + \cdots + 834275027664 \)
|
| $11$ |
\( T^{4} + \cdots - 6547102671564 \)
|
| $13$ |
\( (T + 2197)^{4} \)
|
| $17$ |
\( T^{4} + \cdots + 24\!\cdots\!52 \)
|
| $19$ |
\( T^{4} + \cdots - 18\!\cdots\!96 \)
|
| $23$ |
\( T^{4} + \cdots + 13\!\cdots\!00 \)
|
| $29$ |
\( T^{4} + \cdots - 92\!\cdots\!24 \)
|
| $31$ |
\( T^{4} + \cdots + 71\!\cdots\!52 \)
|
| $37$ |
\( T^{4} + \cdots + 97\!\cdots\!44 \)
|
| $41$ |
\( T^{4} + \cdots - 64\!\cdots\!00 \)
|
| $43$ |
\( T^{4} + \cdots + 94\!\cdots\!80 \)
|
| $47$ |
\( T^{4} + \cdots - 44\!\cdots\!04 \)
|
| $53$ |
\( T^{4} + \cdots + 34\!\cdots\!00 \)
|
| $59$ |
\( T^{4} + \cdots + 30\!\cdots\!40 \)
|
| $61$ |
\( T^{4} + \cdots - 43\!\cdots\!56 \)
|
| $67$ |
\( T^{4} + \cdots - 14\!\cdots\!04 \)
|
| $71$ |
\( T^{4} + \cdots - 42\!\cdots\!80 \)
|
| $73$ |
\( T^{4} + \cdots + 17\!\cdots\!00 \)
|
| $79$ |
\( T^{4} + \cdots + 46\!\cdots\!68 \)
|
| $83$ |
\( T^{4} + \cdots + 95\!\cdots\!00 \)
|
| $89$ |
\( T^{4} + \cdots - 17\!\cdots\!44 \)
|
| $97$ |
\( T^{4} + \cdots - 10\!\cdots\!00 \)
|
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