Newspace parameters
| Level: | \( N \) | \(=\) | \( 880 = 2^{4} \cdot 5 \cdot 11 \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 880.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(141.137761435\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{6} - \cdots)\) |
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| Defining polynomial: |
\( x^{6} - 3x^{5} - 149x^{4} + 297x^{3} + 4052x^{2} - 6522x - 20692 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| Coefficient ring index: | \( 2^{9} \) |
| Twist minimal: | no (minimal twist has level 55) |
| Fricke sign: | \(-1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{5}\) 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
\( x^{6} - 3x^{5} - 149x^{4} + 297x^{3} + 4052x^{2} - 6522x - 20692 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -5\nu^{5} - 16\nu^{4} + 789\nu^{3} + 2250\nu^{2} - 17938\nu - 13100 ) / 312 \)
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| \(\beta_{2}\) | \(=\) |
\( ( \nu^{5} + 8\nu^{4} - 153\nu^{3} - 1218\nu^{2} + 3242\nu + 17668 ) / 312 \)
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| \(\beta_{3}\) | \(=\) |
\( ( 5\nu^{5} - 701\nu^{3} - 626\nu^{2} + 13266\nu + 5684 ) / 104 \)
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| \(\beta_{4}\) | \(=\) |
\( ( -11\nu^{5} + 1563\nu^{3} + 2022\nu^{2} - 29518\nu - 46908 ) / 312 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 13\nu^{5} + 8\nu^{4} - 1773\nu^{3} - 3282\nu^{2} + 26594\nu + 59140 ) / 312 \)
|
| \(\nu\) | \(=\) |
\( ( -\beta_{5} + \beta_{3} + 3\beta_{2} + \beta _1 + 7 ) / 16 \)
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| \(\nu^{2}\) | \(=\) |
\( ( \beta_{5} + 6\beta_{4} + 3\beta_{3} - 7\beta_{2} - 3\beta _1 + 819 ) / 16 \)
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| \(\nu^{3}\) | \(=\) |
\( ( -47\beta_{5} + 54\beta_{4} + 99\beta_{3} + 265\beta_{2} + 109\beta _1 + 1187 ) / 16 \)
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| \(\nu^{4}\) | \(=\) |
\( ( 135\beta_{5} + 906\beta_{4} + 453\beta_{3} - 129\beta_{2} - 309\beta _1 + 80197 ) / 16 \)
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| \(\nu^{5}\) | \(=\) |
\( ( -3811\beta_{5} + 8322\beta_{4} + 11935\beta_{3} + 28317\beta_{2} + 12253\beta _1 + 232195 ) / 16 \)
|
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 | −28.4424 | 0 | −25.0000 | 0 | −115.579 | 0 | 565.969 | 0 | ||||||||||||||||||||||||||||||||||||
| 1.2 | 0 | −13.5338 | 0 | −25.0000 | 0 | −134.766 | 0 | −59.8351 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.3 | 0 | −11.4706 | 0 | −25.0000 | 0 | 132.518 | 0 | −111.426 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 0 | 10.1483 | 0 | −25.0000 | 0 | 228.119 | 0 | −140.013 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 0 | 13.7145 | 0 | −25.0000 | 0 | 131.565 | 0 | −54.9135 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 0 | 29.5841 | 0 | −25.0000 | 0 | −175.856 | 0 | 632.218 | 0 | |||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( -1 \) |
| \(5\) | \( +1 \) |
| \(11\) | \( +1 \) |
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 | 880.6.a.u | 6 | |
| 4.b | odd | 2 | 1 | 55.6.a.d | ✓ | 6 | |
| 12.b | even | 2 | 1 | 495.6.a.l | 6 | ||
| 20.d | odd | 2 | 1 | 275.6.a.f | 6 | ||
| 20.e | even | 4 | 2 | 275.6.b.f | 12 | ||
| 44.c | even | 2 | 1 | 605.6.a.e | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 55.6.a.d | ✓ | 6 | 4.b | odd | 2 | 1 | |
| 275.6.a.f | 6 | 20.d | odd | 2 | 1 | ||
| 275.6.b.f | 12 | 20.e | even | 4 | 2 | ||
| 495.6.a.l | 6 | 12.b | even | 2 | 1 | ||
| 605.6.a.e | 6 | 44.c | even | 2 | 1 | ||
| 880.6.a.u | 6 | 1.a | even | 1 | 1 | trivial | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{6} - 1145T_{3}^{4} - 840T_{3}^{3} + 276192T_{3}^{2} + 164160T_{3} - 18180288 \)
acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(880))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} \)
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| $3$ |
\( T^{6} - 1145 T^{4} + \cdots - 18180288 \)
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| $5$ |
\( (T + 25)^{6} \)
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| $7$ |
\( T^{6} + \cdots - 10894071216816 \)
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| $11$ |
\( (T + 121)^{6} \)
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| $13$ |
\( T^{6} + \cdots - 24\!\cdots\!32 \)
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| $17$ |
\( T^{6} + \cdots - 20\!\cdots\!92 \)
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| $19$ |
\( T^{6} + \cdots - 13\!\cdots\!00 \)
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| $23$ |
\( T^{6} + \cdots - 28\!\cdots\!68 \)
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| $29$ |
\( T^{6} + \cdots + 94\!\cdots\!00 \)
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| $31$ |
\( T^{6} + \cdots + 11\!\cdots\!00 \)
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| $37$ |
\( T^{6} + \cdots + 22\!\cdots\!12 \)
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| $41$ |
\( T^{6} + \cdots + 80\!\cdots\!56 \)
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| $43$ |
\( T^{6} + \cdots + 11\!\cdots\!76 \)
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| $47$ |
\( T^{6} + \cdots - 70\!\cdots\!68 \)
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| $53$ |
\( T^{6} + \cdots - 18\!\cdots\!56 \)
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| $59$ |
\( T^{6} + \cdots - 84\!\cdots\!00 \)
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| $61$ |
\( T^{6} + \cdots - 24\!\cdots\!64 \)
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| $67$ |
\( T^{6} + \cdots - 26\!\cdots\!16 \)
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| $71$ |
\( T^{6} + \cdots - 10\!\cdots\!28 \)
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| $73$ |
\( T^{6} + \cdots - 23\!\cdots\!72 \)
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| $79$ |
\( T^{6} + \cdots + 29\!\cdots\!00 \)
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| $83$ |
\( T^{6} + \cdots - 30\!\cdots\!84 \)
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| $89$ |
\( T^{6} + \cdots + 30\!\cdots\!00 \)
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| $97$ |
\( T^{6} + \cdots - 41\!\cdots\!28 \)
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