Newspace parameters
| Level: | \( N \) | \(=\) | \( 9114 = 2 \cdot 3 \cdot 7^{2} \cdot 31 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 9114.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(72.7756564022\) |
| Analytic rank: | \(0\) |
| Dimension: | \(5\) |
| Coefficient field: | 5.5.11978672.1 |
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|
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| Defining polynomial: |
\( x^{5} - 15x^{3} + 48x - 16 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{19}]\) |
| Coefficient ring index: | \( 2^{2} \) |
| 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 \(1,\beta_1,\beta_2,\beta_3,\beta_4\) 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
\( x^{5} - 15x^{3} + 48x - 16 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 6 \)
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| \(\beta_{3}\) | \(=\) |
\( ( \nu^{4} - 11\nu^{2} - 2\nu + 14 ) / 2 \)
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| \(\beta_{4}\) | \(=\) |
\( \nu^{3} - \nu^{2} - 9\nu + 6 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
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| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 6 \)
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| \(\nu^{3}\) | \(=\) |
\( \beta_{4} + \beta_{2} + 9\beta_1 \)
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| \(\nu^{4}\) | \(=\) |
\( 2\beta_{3} + 11\beta_{2} + 2\beta _1 + 52 \)
|
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 |
|
1.00000 | −1.00000 | 1.00000 | −3.33353 | −1.00000 | 0 | 1.00000 | 1.00000 | −3.33353 | |||||||||||||||||||||||||||||||||
| 1.2 | 1.00000 | −1.00000 | 1.00000 | −1.83932 | −1.00000 | 0 | 1.00000 | 1.00000 | −1.83932 | ||||||||||||||||||||||||||||||||||
| 1.3 | 1.00000 | −1.00000 | 1.00000 | −0.346196 | −1.00000 | 0 | 1.00000 | 1.00000 | −0.346196 | ||||||||||||||||||||||||||||||||||
| 1.4 | 1.00000 | −1.00000 | 1.00000 | 2.48144 | −1.00000 | 0 | 1.00000 | 1.00000 | 2.48144 | ||||||||||||||||||||||||||||||||||
| 1.5 | 1.00000 | −1.00000 | 1.00000 | 3.03761 | −1.00000 | 0 | 1.00000 | 1.00000 | 3.03761 | ||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( -1 \) |
| \(3\) | \( +1 \) |
| \(7\) | \( -1 \) |
| \(31\) | \( -1 \) |
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 | 9114.2.a.bx | ✓ | 5 |
| 7.b | odd | 2 | 1 | 9114.2.a.cc | yes | 5 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 9114.2.a.bx | ✓ | 5 | 1.a | even | 1 | 1 | trivial |
| 9114.2.a.cc | yes | 5 | 7.b | odd | 2 | 1 | |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(9114))\):
|
\( T_{5}^{5} - 15T_{5}^{3} + 48T_{5} + 16 \)
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\( T_{11}^{5} - 10T_{11}^{4} + 25T_{11}^{3} + 10T_{11}^{2} - 52T_{11} - 24 \)
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\( T_{13}^{5} - 39T_{13}^{3} + 18T_{13}^{2} + 252T_{13} - 40 \)
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\( T_{17}^{5} - 39T_{17}^{3} + 18T_{17}^{2} + 252T_{17} - 40 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 1)^{5} \)
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| $3$ |
\( (T + 1)^{5} \)
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| $5$ |
\( T^{5} - 15 T^{3} + \cdots + 16 \)
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| $7$ |
\( T^{5} \)
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| $11$ |
\( T^{5} - 10 T^{4} + \cdots - 24 \)
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| $13$ |
\( T^{5} - 39 T^{3} + \cdots - 40 \)
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| $17$ |
\( T^{5} - 39 T^{3} + \cdots - 40 \)
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| $19$ |
\( T^{5} + 4 T^{4} + \cdots + 3152 \)
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| $23$ |
\( T^{5} - 14 T^{4} + \cdots + 96 \)
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| $29$ |
\( T^{5} - 10 T^{4} + \cdots - 10656 \)
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| $31$ |
\( (T - 1)^{5} \)
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| $37$ |
\( T^{5} - 6 T^{4} + \cdots - 640 \)
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| $41$ |
\( T^{5} - 4 T^{4} + \cdots + 384 \)
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| $43$ |
\( T^{5} - 6 T^{4} + \cdots - 8160 \)
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| $47$ |
\( T^{5} - 4 T^{4} + \cdots + 352 \)
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| $53$ |
\( T^{5} - 10 T^{4} + \cdots - 352 \)
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| $59$ |
\( T^{5} - 108 T^{3} + \cdots - 1088 \)
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| $61$ |
\( T^{5} - 303 T^{3} + \cdots + 27832 \)
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| $67$ |
\( T^{5} - 10 T^{4} + \cdots + 2176 \)
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| $71$ |
\( T^{5} - 10 T^{4} + \cdots - 121152 \)
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| $73$ |
\( T^{5} - 10 T^{4} + \cdots - 130784 \)
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| $79$ |
\( T^{5} - 10 T^{4} + \cdots + 344 \)
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| $83$ |
\( T^{5} + 10 T^{4} + \cdots + 704 \)
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| $89$ |
\( T^{5} - 10 T^{4} + \cdots - 6304 \)
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| $97$ |
\( T^{5} + 20 T^{4} + \cdots - 42048 \)
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