Newspace parameters
| Level: | \( N \) | \(=\) | \( 7232 = 2^{6} \cdot 113 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 7232.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(57.7478107418\) |
| Analytic rank: | \(0\) |
| Dimension: | \(5\) |
| Coefficient field: | 5.5.138136.1 |
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| Defining polynomial: |
\( x^{5} - x^{4} - 6x^{3} + 3x^{2} + 4x - 2 \)
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| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 904) |
| 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
\( x^{5} - x^{4} - 6x^{3} + 3x^{2} + 4x - 2 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{4} - 6\nu^{2} - 3\nu + 2 \)
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| \(\beta_{3}\) | \(=\) |
\( \nu^{4} - 7\nu^{2} - 2\nu + 5 \)
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| \(\beta_{4}\) | \(=\) |
\( 2\nu^{4} - \nu^{3} - 12\nu^{2} - \nu + 6 \)
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| \(\nu\) | \(=\) |
\( \beta_1 \)
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| \(\nu^{2}\) | \(=\) |
\( -\beta_{3} + \beta_{2} + \beta _1 + 3 \)
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| \(\nu^{3}\) | \(=\) |
\( -\beta_{4} + 2\beta_{2} + 5\beta _1 + 2 \)
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| \(\nu^{4}\) | \(=\) |
\( -6\beta_{3} + 7\beta_{2} + 9\beta _1 + 16 \)
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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 | −2.05592 | 0 | 1.55403 | 0 | −0.300036 | 0 | 1.22679 | 0 | |||||||||||||||||||||||||||||||||
| 1.2 | 0 | −0.904340 | 0 | −1.65705 | 0 | −1.73667 | 0 | −2.18217 | 0 | ||||||||||||||||||||||||||||||||||
| 1.3 | 0 | 0.514054 | 0 | −0.677907 | 0 | 2.83280 | 0 | −2.73575 | 0 | ||||||||||||||||||||||||||||||||||
| 1.4 | 0 | 0.786893 | 0 | 2.31167 | 0 | −1.15083 | 0 | −2.38080 | 0 | ||||||||||||||||||||||||||||||||||
| 1.5 | 0 | 2.65931 | 0 | 3.46927 | 0 | 2.35474 | 0 | 4.07193 | 0 | ||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( -1 \) |
| \(113\) | \( +1 \) |
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 | 7232.2.a.x | 5 | |
| 4.b | odd | 2 | 1 | 7232.2.a.w | 5 | ||
| 8.b | even | 2 | 1 | 1808.2.a.k | 5 | ||
| 8.d | odd | 2 | 1 | 904.2.a.b | ✓ | 5 | |
| 24.f | even | 2 | 1 | 8136.2.a.r | 5 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 904.2.a.b | ✓ | 5 | 8.d | odd | 2 | 1 | |
| 1808.2.a.k | 5 | 8.b | even | 2 | 1 | ||
| 7232.2.a.w | 5 | 4.b | odd | 2 | 1 | ||
| 7232.2.a.x | 5 | 1.a | even | 1 | 1 | trivial | |
| 8136.2.a.r | 5 | 24.f | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(7232))\):
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\( T_{3}^{5} - T_{3}^{4} - 6T_{3}^{3} + 3T_{3}^{2} + 4T_{3} - 2 \)
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\( T_{5}^{5} - 5T_{5}^{4} + T_{5}^{3} + 19T_{5}^{2} - 10T_{5} - 14 \)
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\( T_{7}^{5} - 2T_{7}^{4} - 7T_{7}^{3} + 7T_{7}^{2} + 16T_{7} + 4 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{5} \)
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| $3$ |
\( T^{5} - T^{4} - 6 T^{3} + \cdots - 2 \)
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| $5$ |
\( T^{5} - 5 T^{4} + \cdots - 14 \)
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| $7$ |
\( T^{5} - 2 T^{4} + \cdots + 4 \)
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| $11$ |
\( T^{5} + 4 T^{4} + \cdots + 256 \)
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| $13$ |
\( T^{5} - 12 T^{4} + \cdots + 956 \)
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| $17$ |
\( T^{5} + 10 T^{4} + \cdots - 4 \)
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| $19$ |
\( T^{5} - 4 T^{4} + \cdots - 62 \)
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| $23$ |
\( T^{5} - 12 T^{4} + \cdots - 3464 \)
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| $29$ |
\( T^{5} - 11 T^{4} + \cdots + 4262 \)
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| $31$ |
\( T^{5} - 11 T^{4} + \cdots - 28 \)
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| $37$ |
\( T^{5} - 6 T^{4} + \cdots - 2014 \)
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| $41$ |
\( T^{5} + 19 T^{4} + \cdots + 4036 \)
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| $43$ |
\( T^{5} - 20 T^{4} + \cdots + 8246 \)
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| $47$ |
\( T^{5} - 5 T^{4} + \cdots - 472 \)
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| $53$ |
\( T^{5} - T^{4} + \cdots - 76 \)
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| $59$ |
\( T^{5} + 15 T^{4} + \cdots - 25826 \)
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| $61$ |
\( T^{5} - 9 T^{4} + \cdots - 18508 \)
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| $67$ |
\( T^{5} - 23 T^{4} + \cdots - 71602 \)
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| $71$ |
\( T^{5} - 22 T^{4} + \cdots + 28544 \)
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| $73$ |
\( T^{5} + 11 T^{4} + \cdots - 76 \)
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| $79$ |
\( T^{5} + 11 T^{4} + \cdots + 1696 \)
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| $83$ |
\( T^{5} - 20 T^{4} + \cdots + 4408 \)
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| $89$ |
\( T^{5} + 24 T^{4} + \cdots + 2764 \)
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| $97$ |
\( T^{5} - 357 T^{3} + \cdots - 29792 \)
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