Newspace parameters
| Level: | \( N \) | \(=\) | \( 768 = 2^{8} \cdot 3 \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 768.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(123.174773616\) |
| Analytic rank: | \(0\) |
| Dimension: | \(5\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{5} - \cdots)\) |
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| Defining polynomial: |
\( x^{5} - x^{4} - 65x^{3} + 85x^{2} + 856x - 1692 \)
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| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| Coefficient ring index: | \( 2^{18} \) |
| Twist minimal: | no (minimal twist has level 24) |
| Fricke sign: | \(-1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3,\beta_4\) 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
\( x^{5} - x^{4} - 65x^{3} + 85x^{2} + 856x - 1692 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{4} - 2\nu^{3} - 43\nu^{2} + 188\nu + 78 ) / 5 \)
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| \(\beta_{2}\) | \(=\) |
\( ( -\nu^{4} + 2\nu^{3} + 63\nu^{2} - 48\nu - 628 ) / 5 \)
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| \(\beta_{3}\) | \(=\) |
\( ( -2\nu^{4} + 4\nu^{3} + 46\nu^{2} - 16\nu + 824 ) / 5 \)
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| \(\beta_{4}\) | \(=\) |
\( 4\nu^{4} + 8\nu^{3} - 232\nu^{2} - 356\nu + 2286 \)
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| \(\nu\) | \(=\) |
\( ( \beta_{3} + 2\beta_{2} + 4\beta _1 + 24 ) / 128 \)
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| \(\nu^{2}\) | \(=\) |
\( ( -7\beta_{3} + 18\beta_{2} + 4\beta _1 + 3352 ) / 128 \)
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| \(\nu^{3}\) | \(=\) |
\( ( 8\beta_{4} + 43\beta_{3} + 206\beta_{2} + 132\beta _1 - 1560 ) / 128 \)
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| \(\nu^{4}\) | \(=\) |
\( ( 16\beta_{4} - 403\beta_{3} + 810\beta_{2} + 324\beta _1 + 126520 ) / 128 \)
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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 | 9.00000 | 0 | −67.1492 | 0 | −37.4101 | 0 | 81.0000 | 0 | |||||||||||||||||||||||||||||||||
| 1.2 | 0 | 9.00000 | 0 | −29.4414 | 0 | 94.2792 | 0 | 81.0000 | 0 | ||||||||||||||||||||||||||||||||||
| 1.3 | 0 | 9.00000 | 0 | −20.4846 | 0 | −35.4345 | 0 | 81.0000 | 0 | ||||||||||||||||||||||||||||||||||
| 1.4 | 0 | 9.00000 | 0 | 66.8717 | 0 | 230.960 | 0 | 81.0000 | 0 | ||||||||||||||||||||||||||||||||||
| 1.5 | 0 | 9.00000 | 0 | 100.204 | 0 | −154.395 | 0 | 81.0000 | 0 | ||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( +1 \) |
| \(3\) | \( -1 \) |
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 | 768.6.a.bd | 5 | |
| 4.b | odd | 2 | 1 | 768.6.a.bb | 5 | ||
| 8.b | even | 2 | 1 | 768.6.a.ba | 5 | ||
| 8.d | odd | 2 | 1 | 768.6.a.bc | 5 | ||
| 16.e | even | 4 | 2 | 24.6.d.a | ✓ | 10 | |
| 16.f | odd | 4 | 2 | 96.6.d.a | 10 | ||
| 48.i | odd | 4 | 2 | 72.6.d.d | 10 | ||
| 48.k | even | 4 | 2 | 288.6.d.d | 10 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 24.6.d.a | ✓ | 10 | 16.e | even | 4 | 2 | |
| 72.6.d.d | 10 | 48.i | odd | 4 | 2 | ||
| 96.6.d.a | 10 | 16.f | odd | 4 | 2 | ||
| 288.6.d.d | 10 | 48.k | even | 4 | 2 | ||
| 768.6.a.ba | 5 | 8.b | even | 2 | 1 | ||
| 768.6.a.bb | 5 | 4.b | odd | 2 | 1 | ||
| 768.6.a.bc | 5 | 8.d | odd | 2 | 1 | ||
| 768.6.a.bd | 5 | 1.a | even | 1 | 1 | trivial | |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(768))\):
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\( T_{5}^{5} - 50T_{5}^{4} - 8904T_{5}^{3} + 164112T_{5}^{2} + 19739408T_{5} + 271363808 \)
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\( T_{7}^{5} - 98T_{7}^{4} - 39560T_{7}^{3} + 1063696T_{7}^{2} + 207195664T_{7} + 4456567264 \)
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\( T_{11}^{5} - 392960T_{11}^{3} + 29048832T_{11}^{2} + 20197933056T_{11} + 1270622453760 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{5} \)
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| $3$ |
\( (T - 9)^{5} \)
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| $5$ |
\( T^{5} - 50 T^{4} + \cdots + 271363808 \)
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| $7$ |
\( T^{5} + \cdots + 4456567264 \)
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| $11$ |
\( T^{5} + \cdots + 1270622453760 \)
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| $13$ |
\( T^{5} + \cdots + 19009776061440 \)
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| $17$ |
\( T^{5} + \cdots + 25\!\cdots\!24 \)
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| $19$ |
\( T^{5} + \cdots - 174679081196544 \)
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| $23$ |
\( T^{5} + \cdots - 28\!\cdots\!56 \)
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| $29$ |
\( T^{5} + \cdots + 11\!\cdots\!28 \)
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| $31$ |
\( T^{5} + \cdots + 17\!\cdots\!80 \)
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| $37$ |
\( T^{5} + \cdots - 35\!\cdots\!88 \)
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| $41$ |
\( T^{5} + \cdots - 12\!\cdots\!20 \)
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| $43$ |
\( T^{5} + \cdots + 12\!\cdots\!48 \)
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| $47$ |
\( T^{5} + \cdots - 65\!\cdots\!12 \)
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| $53$ |
\( T^{5} + \cdots - 14\!\cdots\!60 \)
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| $59$ |
\( T^{5} + \cdots - 39\!\cdots\!96 \)
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| $61$ |
\( T^{5} + \cdots + 27\!\cdots\!60 \)
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| $67$ |
\( T^{5} + \cdots - 40\!\cdots\!24 \)
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| $71$ |
\( T^{5} + \cdots + 11\!\cdots\!32 \)
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| $73$ |
\( T^{5} + \cdots - 17\!\cdots\!60 \)
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| $79$ |
\( T^{5} + \cdots - 37\!\cdots\!60 \)
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| $83$ |
\( T^{5} + \cdots + 13\!\cdots\!00 \)
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| $89$ |
\( T^{5} + \cdots - 26\!\cdots\!00 \)
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| $97$ |
\( T^{5} + \cdots - 85\!\cdots\!20 \)
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