Newspace parameters
| Level: | \( N \) | \(=\) | \( 1296 = 2^{4} \cdot 3^{4} \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1296.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(76.4664753674\) |
| Analytic rank: | \(0\) |
| Dimension: | \(5\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{5} - \cdots)\) |
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|
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| Defining polynomial: |
\( x^{5} - x^{4} - 19x^{3} + 4x^{2} + 81x + 12 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2^{4}\cdot 3^{4} \) |
| Twist minimal: | no (minimal twist has level 72) |
| 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
\( x^{5} - x^{4} - 19x^{3} + 4x^{2} + 81x + 12 \)
:
| \(\beta_{1}\) | \(=\) |
\( 2\nu^{4} - 7\nu^{3} - 14\nu^{2} + 43\nu - 19 \)
|
| \(\beta_{2}\) | \(=\) |
\( 2\nu^{4} - 7\nu^{3} - 20\nu^{2} + 49\nu + 27 \)
|
| \(\beta_{3}\) | \(=\) |
\( 6\nu^{4} - 18\nu^{3} - 60\nu^{2} + 126\nu + 58 \)
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| \(\beta_{4}\) | \(=\) |
\( 10\nu^{4} - 38\nu^{3} - 88\nu^{2} + 290\nu + 60 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{4} + \beta_{3} - 6\beta_{2} - 2\beta _1 + 6 ) / 36 \)
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| \(\nu^{2}\) | \(=\) |
\( ( \beta_{4} + \beta_{3} - 12\beta_{2} + 4\beta _1 + 282 ) / 36 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 7\beta_{4} + 19\beta_{3} - 78\beta_{2} - 14\beta _1 + 318 ) / 36 \)
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| \(\nu^{4}\) | \(=\) |
\( ( 5\beta_{4} + 26\beta_{3} - 114\beta_{2} + 20\beta _1 + 1650 ) / 18 \)
|
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 | 0 | 0 | −18.3184 | 0 | −14.9787 | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||
| 1.2 | 0 | 0 | 0 | −6.31871 | 0 | 29.5796 | 0 | 0 | 0 | ||||||||||||||||||||||||||||||||||
| 1.3 | 0 | 0 | 0 | −2.98196 | 0 | −11.7109 | 0 | 0 | 0 | ||||||||||||||||||||||||||||||||||
| 1.4 | 0 | 0 | 0 | 12.3969 | 0 | −30.6325 | 0 | 0 | 0 | ||||||||||||||||||||||||||||||||||
| 1.5 | 0 | 0 | 0 | 20.2222 | 0 | 24.7425 | 0 | 0 | 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 | 1296.4.a.bd | 5 | |
| 3.b | odd | 2 | 1 | 1296.4.a.bc | 5 | ||
| 4.b | odd | 2 | 1 | 648.4.a.l | 5 | ||
| 9.c | even | 3 | 2 | 144.4.i.f | 10 | ||
| 9.d | odd | 6 | 2 | 432.4.i.f | 10 | ||
| 12.b | even | 2 | 1 | 648.4.a.k | 5 | ||
| 36.f | odd | 6 | 2 | 72.4.i.b | ✓ | 10 | |
| 36.h | even | 6 | 2 | 216.4.i.b | 10 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 72.4.i.b | ✓ | 10 | 36.f | odd | 6 | 2 | |
| 144.4.i.f | 10 | 9.c | even | 3 | 2 | ||
| 216.4.i.b | 10 | 36.h | even | 6 | 2 | ||
| 432.4.i.f | 10 | 9.d | odd | 6 | 2 | ||
| 648.4.a.k | 5 | 12.b | even | 2 | 1 | ||
| 648.4.a.l | 5 | 4.b | odd | 2 | 1 | ||
| 1296.4.a.bc | 5 | 3.b | odd | 2 | 1 | ||
| 1296.4.a.bd | 5 | 1.a | even | 1 | 1 | trivial | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{5} - 5T_{5}^{4} - 461T_{5}^{3} + 1097T_{5}^{2} + 36176T_{5} + 86528 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1296))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{5} \)
|
| $3$ |
\( T^{5} \)
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| $5$ |
\( T^{5} - 5 T^{4} + \cdots + 86528 \)
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| $7$ |
\( T^{5} + 3 T^{4} + \cdots + 3932604 \)
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| $11$ |
\( T^{5} - 25 T^{4} + \cdots + 2450383 \)
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| $13$ |
\( T^{5} - 29 T^{4} + \cdots + 11723252 \)
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| $17$ |
\( T^{5} - 28 T^{4} + \cdots - 465026264 \)
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| $19$ |
\( T^{5} + 64 T^{4} + \cdots + 219796928 \)
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| $23$ |
\( T^{5} + \cdots + 54657912052 \)
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| $29$ |
\( T^{5} + \cdots + 2405376324 \)
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| $31$ |
\( T^{5} + \cdots + 2225150096 \)
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| $37$ |
\( T^{5} + \cdots - 525481299072 \)
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| $41$ |
\( T^{5} - 171 T^{4} + \cdots + 816147765 \)
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| $43$ |
\( T^{5} + \cdots - 236167016117 \)
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| $47$ |
\( T^{5} + \cdots + 134565454332 \)
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| $53$ |
\( T^{5} + \cdots - 758189779072 \)
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| $59$ |
\( T^{5} + \cdots + 804795801121 \)
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| $61$ |
\( T^{5} + \cdots - 4452174344512 \)
|
| $67$ |
\( T^{5} + \cdots + 6065728349525 \)
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| $71$ |
\( T^{5} + \cdots - 24420797440 \)
|
| $73$ |
\( T^{5} + \cdots + 28702127905256 \)
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| $79$ |
\( T^{5} + \cdots - 85346901126224 \)
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| $83$ |
\( T^{5} + \cdots - 70154840960560 \)
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| $89$ |
\( T^{5} + \cdots + 887800297007520 \)
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| $97$ |
\( T^{5} + \cdots + 27675431349397 \)
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