Newspace parameters
| Level: | \( N \) | \(=\) | \( 660 = 2^{2} \cdot 3 \cdot 5 \cdot 11 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 660.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(38.9412606038\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{4} - \cdots)\) |
|
|
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| Defining polynomial: |
\( x^{4} - 248x^{2} - 1722x - 2900 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| Coefficient ring index: | \( 2^{5} \) |
| 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
\( x^{4} - 248x^{2} - 1722x - 2900 \)
:
| \(\beta_{1}\) | \(=\) |
\( 2\nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 2\nu^{3} - 22\nu^{2} - 314\nu + 145 ) / 15 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 8\nu^{3} - 28\nu^{2} - 1886\nu - 6860 ) / 15 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{3} - 4\beta_{2} + 21\beta _1 + 496 ) / 4 \)
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| \(\nu^{3}\) | \(=\) |
\( ( 11\beta_{3} - 14\beta_{2} + 545\beta _1 + 5166 ) / 4 \)
|
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 |
|
0 | 3.00000 | 0 | 5.00000 | 0 | −33.3393 | 0 | 9.00000 | 0 | ||||||||||||||||||||||||||||||
| 1.2 | 0 | 3.00000 | 0 | 5.00000 | 0 | 9.42342 | 0 | 9.00000 | 0 | |||||||||||||||||||||||||||||||
| 1.3 | 0 | 3.00000 | 0 | 5.00000 | 0 | 14.9055 | 0 | 9.00000 | 0 | |||||||||||||||||||||||||||||||
| 1.4 | 0 | 3.00000 | 0 | 5.00000 | 0 | 25.0104 | 0 | 9.00000 | 0 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( -1 \) |
| \(3\) | \( -1 \) |
| \(5\) | \( -1 \) |
| \(11\) | \( -1 \) |
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 | 660.4.a.h | ✓ | 4 |
| 3.b | odd | 2 | 1 | 1980.4.a.n | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 660.4.a.h | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 1980.4.a.n | 4 | 3.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{7}^{4} - 16T_{7}^{3} - 896T_{7}^{2} + 21456T_{7} - 117120 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(660))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} \)
|
| $3$ |
\( (T - 3)^{4} \)
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| $5$ |
\( (T - 5)^{4} \)
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| $7$ |
\( T^{4} - 16 T^{3} + \cdots - 117120 \)
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| $11$ |
\( (T - 11)^{4} \)
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| $13$ |
\( T^{4} - 12 T^{3} + \cdots + 6278448 \)
|
| $17$ |
\( T^{4} - 136 T^{3} + \cdots - 46290000 \)
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| $19$ |
\( T^{4} - 28 T^{3} + \cdots + 68987264 \)
|
| $23$ |
\( T^{4} - 168 T^{3} + \cdots - 250905600 \)
|
| $29$ |
\( T^{4} - 204 T^{3} + \cdots + 137097840 \)
|
| $31$ |
\( T^{4} - 216 T^{3} + \cdots + 77334528 \)
|
| $37$ |
\( T^{4} - 208 T^{3} + \cdots + 7244880 \)
|
| $41$ |
\( T^{4} + \cdots + 11183936880 \)
|
| $43$ |
\( T^{4} - 184 T^{3} + \cdots - 658040640 \)
|
| $47$ |
\( T^{4} + \cdots - 1864028160 \)
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| $53$ |
\( T^{4} + \cdots + 26277271632 \)
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| $59$ |
\( T^{4} + \cdots - 8545008384 \)
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| $61$ |
\( T^{4} + \cdots + 3753588880 \)
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| $67$ |
\( T^{4} + \cdots + 160379406592 \)
|
| $71$ |
\( T^{4} + \cdots - 7153689600 \)
|
| $73$ |
\( T^{4} + \cdots - 3873416656 \)
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| $79$ |
\( T^{4} + \cdots + 8163540480 \)
|
| $83$ |
\( T^{4} + \cdots + 230897300544 \)
|
| $89$ |
\( T^{4} + \cdots + 174517116624 \)
|
| $97$ |
\( T^{4} + \cdots + 16089701520 \)
|
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