Newspace parameters
| Level: | \( N \) | \(=\) | \( 930 = 2 \cdot 3 \cdot 5 \cdot 31 \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 930.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(149.156952426\) |
| Analytic rank: | \(1\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{4} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - x^{3} - 106x^{2} - 4x + 2040 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2 \) |
| 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
\( x^{4} - x^{3} - 106x^{2} - 4x + 2040 \)
:
| \(\beta_{1}\) | \(=\) |
\( 2\nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{2} - \nu - 54 ) / 2 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{3} - 3\nu^{2} - 68\nu + 92 ) / 4 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 4\beta_{2} + \beta _1 + 108 ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 8\beta_{3} + 12\beta_{2} + 71\beta _1 + 140 ) / 2 \)
|
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 |
|
4.00000 | 9.00000 | 16.0000 | 25.0000 | 36.0000 | −101.640 | 64.0000 | 81.0000 | 100.000 | ||||||||||||||||||||||||||||||
| 1.2 | 4.00000 | 9.00000 | 16.0000 | 25.0000 | 36.0000 | −87.4260 | 64.0000 | 81.0000 | 100.000 | |||||||||||||||||||||||||||||||
| 1.3 | 4.00000 | 9.00000 | 16.0000 | 25.0000 | 36.0000 | −64.7610 | 64.0000 | 81.0000 | 100.000 | |||||||||||||||||||||||||||||||
| 1.4 | 4.00000 | 9.00000 | 16.0000 | 25.0000 | 36.0000 | 103.827 | 64.0000 | 81.0000 | 100.000 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(3\) | \(-1\) |
| \(5\) | \(-1\) |
| \(31\) | \(-1\) |
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 | 930.6.a.a | ✓ | 4 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 930.6.a.a | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{7}^{4} + 150T_{7}^{3} - 5224T_{7}^{2} - 1618406T_{7} - 59748708 \)
acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(930))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 4)^{4} \)
|
| $3$ |
\( (T - 9)^{4} \)
|
| $5$ |
\( (T - 25)^{4} \)
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| $7$ |
\( T^{4} + 150 T^{3} + \cdots - 59748708 \)
|
| $11$ |
\( T^{4} + \cdots - 38606444640 \)
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| $13$ |
\( T^{4} + \cdots + 7640067536 \)
|
| $17$ |
\( T^{4} + \cdots - 19639003840 \)
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| $19$ |
\( T^{4} + \cdots - 187396358400 \)
|
| $23$ |
\( T^{4} + \cdots + 49886182898240 \)
|
| $29$ |
\( T^{4} + \cdots + 7008543901780 \)
|
| $31$ |
\( (T - 961)^{4} \)
|
| $37$ |
\( T^{4} + \cdots + 80530921718760 \)
|
| $41$ |
\( T^{4} + \cdots - 14\!\cdots\!48 \)
|
| $43$ |
\( T^{4} + \cdots - 91\!\cdots\!16 \)
|
| $47$ |
\( T^{4} + \cdots - 25\!\cdots\!60 \)
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| $53$ |
\( T^{4} + \cdots + 11\!\cdots\!92 \)
|
| $59$ |
\( T^{4} + \cdots + 53\!\cdots\!00 \)
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| $61$ |
\( T^{4} + \cdots + 52\!\cdots\!48 \)
|
| $67$ |
\( T^{4} + \cdots + 81\!\cdots\!88 \)
|
| $71$ |
\( T^{4} + \cdots + 20\!\cdots\!28 \)
|
| $73$ |
\( T^{4} + \cdots + 49\!\cdots\!40 \)
|
| $79$ |
\( T^{4} + \cdots + 10\!\cdots\!00 \)
|
| $83$ |
\( T^{4} + \cdots - 45\!\cdots\!24 \)
|
| $89$ |
\( T^{4} + \cdots + 36\!\cdots\!60 \)
|
| $97$ |
\( T^{4} + \cdots + 48\!\cdots\!40 \)
|
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