Newspace parameters
| Level: | \( N \) | \(=\) | \( 390 = 2 \cdot 3 \cdot 5 \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 8 \) |
| Character orbit: | \([\chi]\) | \(=\) | 390.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(121.830159939\) |
| Analytic rank: | \(0\) |
| 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: |
\( x^{4} - 2x^{3} - 121275x^{2} - 12923350x + 102608000 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2^{3}\cdot 3\cdot 5 \) |
| 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
\( x^{4} - 2x^{3} - 121275x^{2} - 12923350x + 102608000 \)
:
| \(\beta_{1}\) | \(=\) |
\( 4\nu - 2 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -\nu^{3} + 112\nu^{2} + 108955\nu + 3027200 ) / 4950 \)
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| \(\beta_{3}\) | \(=\) |
\( ( \nu^{3} + 1868\nu^{2} - 429715\nu - 122936000 ) / 14850 \)
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| \(\nu\) | \(=\) |
\( ( \beta _1 + 2 ) / 4 \)
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| \(\nu^{2}\) | \(=\) |
\( ( 15\beta_{3} + 5\beta_{2} + 81\beta _1 + 121282 ) / 2 \)
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| \(\nu^{3}\) | \(=\) |
\( ( 3360\beta_{3} - 18680\beta_{2} + 127099\beta _1 + 39493878 ) / 4 \)
|
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 | 27.0000 | 64.0000 | −125.000 | −216.000 | −476.524 | −512.000 | 729.000 | 1000.00 | ||||||||||||||||||||||||||||||
| 1.2 | −8.00000 | 27.0000 | 64.0000 | −125.000 | −216.000 | −317.559 | −512.000 | 729.000 | 1000.00 | |||||||||||||||||||||||||||||||
| 1.3 | −8.00000 | 27.0000 | 64.0000 | −125.000 | −216.000 | 1336.41 | −512.000 | 729.000 | 1000.00 | |||||||||||||||||||||||||||||||
| 1.4 | −8.00000 | 27.0000 | 64.0000 | −125.000 | −216.000 | 1808.67 | −512.000 | 729.000 | 1000.00 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(1\) |
| \(3\) | \(-1\) |
| \(5\) | \(1\) |
| \(13\) | \(-1\) |
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 | 390.8.a.p | ✓ | 4 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 390.8.a.p | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{7}^{4} - 2351T_{7}^{3} + 70998T_{7}^{2} + 1443473856T_{7} + 365770934496 \)
acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(390))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T + 8)^{4} \)
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| $3$ |
\( (T - 27)^{4} \)
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| $5$ |
\( (T + 125)^{4} \)
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| $7$ |
\( T^{4} + \cdots + 365770934496 \)
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| $11$ |
\( T^{4} + \cdots + 699584558400000 \)
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| $13$ |
\( (T - 2197)^{4} \)
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| $17$ |
\( T^{4} + \cdots - 10\!\cdots\!00 \)
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| $19$ |
\( T^{4} + \cdots - 11\!\cdots\!00 \)
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| $23$ |
\( T^{4} + \cdots - 95\!\cdots\!00 \)
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| $29$ |
\( T^{4} + \cdots + 22\!\cdots\!40 \)
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| $31$ |
\( T^{4} + \cdots + 51\!\cdots\!20 \)
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| $37$ |
\( T^{4} + \cdots - 22\!\cdots\!44 \)
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| $41$ |
\( T^{4} + \cdots + 93\!\cdots\!12 \)
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| $43$ |
\( T^{4} + \cdots - 14\!\cdots\!84 \)
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| $47$ |
\( T^{4} + \cdots - 41\!\cdots\!00 \)
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| $53$ |
\( T^{4} + \cdots - 11\!\cdots\!40 \)
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| $59$ |
\( T^{4} + \cdots + 13\!\cdots\!80 \)
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| $61$ |
\( T^{4} + \cdots + 41\!\cdots\!40 \)
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| $67$ |
\( T^{4} + \cdots - 14\!\cdots\!00 \)
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| $71$ |
\( T^{4} + \cdots + 19\!\cdots\!40 \)
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| $73$ |
\( T^{4} + \cdots - 35\!\cdots\!32 \)
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| $79$ |
\( T^{4} + \cdots + 17\!\cdots\!12 \)
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| $83$ |
\( T^{4} + \cdots + 48\!\cdots\!04 \)
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| $89$ |
\( T^{4} + \cdots - 16\!\cdots\!56 \)
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| $97$ |
\( T^{4} + \cdots - 14\!\cdots\!56 \)
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