Newspace parameters
| Level: | \( N \) | \(=\) | \( 8325 = 3^{2} \cdot 5^{2} \cdot 37 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 8325.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(66.4754596827\) |
| Analytic rank: | \(1\) |
| Dimension: | \(5\) |
| Coefficient field: | 5.5.457904.1 |
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| Defining polynomial: |
\( x^{5} - 2x^{4} - 5x^{3} + 8x^{2} + 5x - 6 \)
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| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 1665) |
| 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
\( x^{5} - 2x^{4} - 5x^{3} + 8x^{2} + 5x - 6 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
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| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - \nu - 2 \)
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| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - 2\nu^{2} - 3\nu + 4 \)
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| \(\beta_{4}\) | \(=\) |
\( \nu^{4} - \nu^{3} - 5\nu^{2} + 2\nu + 4 \)
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| \(\nu\) | \(=\) |
\( \beta_1 \)
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| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + \beta _1 + 2 \)
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| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + 2\beta_{2} + 5\beta_1 \)
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| \(\nu^{4}\) | \(=\) |
\( \beta_{4} + \beta_{3} + 7\beta_{2} + 8\beta _1 + 6 \)
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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 |
|
−2.58059 | 0 | 4.65942 | 0 | 0 | −0.947908 | −6.86286 | 0 | 0 | |||||||||||||||||||||||||||||||||
| 1.2 | −1.51432 | 0 | 0.293161 | 0 | 0 | 1.43000 | 2.58470 | 0 | 0 | ||||||||||||||||||||||||||||||||||
| 1.3 | −0.788997 | 0 | −1.37748 | 0 | 0 | −4.52826 | 2.66482 | 0 | 0 | ||||||||||||||||||||||||||||||||||
| 1.4 | 1.07694 | 0 | −0.840195 | 0 | 0 | 1.59547 | −3.05873 | 0 | 0 | ||||||||||||||||||||||||||||||||||
| 1.5 | 1.80696 | 0 | 1.26510 | 0 | 0 | 2.45070 | −1.32794 | 0 | 0 | ||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \( +1 \) |
| \(5\) | \( +1 \) |
| \(37\) | \( +1 \) |
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 | 8325.2.a.by | 5 | |
| 3.b | odd | 2 | 1 | 8325.2.a.cf | 5 | ||
| 5.b | even | 2 | 1 | 1665.2.a.r | yes | 5 | |
| 15.d | odd | 2 | 1 | 1665.2.a.o | ✓ | 5 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1665.2.a.o | ✓ | 5 | 15.d | odd | 2 | 1 | |
| 1665.2.a.r | yes | 5 | 5.b | even | 2 | 1 | |
| 8325.2.a.by | 5 | 1.a | even | 1 | 1 | trivial | |
| 8325.2.a.cf | 5 | 3.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8325))\):
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\( T_{2}^{5} + 2T_{2}^{4} - 5T_{2}^{3} - 8T_{2}^{2} + 5T_{2} + 6 \)
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\( T_{7}^{5} - 16T_{7}^{3} + 24T_{7}^{2} + 11T_{7} - 24 \)
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\( T_{11}^{5} - 12T_{11}^{4} + 33T_{11}^{3} + 76T_{11}^{2} - 389T_{11} + 292 \)
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\( T_{13}^{5} - 3T_{13}^{4} - 30T_{13}^{3} + 38T_{13}^{2} + 249T_{13} + 113 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{5} + 2 T^{4} + \cdots + 6 \)
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| $3$ |
\( T^{5} \)
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| $5$ |
\( T^{5} \)
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| $7$ |
\( T^{5} - 16 T^{3} + \cdots - 24 \)
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| $11$ |
\( T^{5} - 12 T^{4} + \cdots + 292 \)
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| $13$ |
\( T^{5} - 3 T^{4} + \cdots + 113 \)
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| $17$ |
\( T^{5} + 4 T^{4} + \cdots + 2048 \)
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| $19$ |
\( T^{5} + 8 T^{4} + \cdots - 4 \)
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| $23$ |
\( T^{5} + 18 T^{4} + \cdots + 128 \)
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| $29$ |
\( T^{5} - 13 T^{4} + \cdots - 4877 \)
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| $31$ |
\( T^{5} + 16 T^{4} + \cdots - 7938 \)
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| $37$ |
\( (T + 1)^{5} \)
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| $41$ |
\( T^{5} - 6 T^{4} + \cdots + 334 \)
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| $43$ |
\( T^{5} + 5 T^{4} + \cdots - 3 \)
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| $47$ |
\( T^{5} + 19 T^{4} + \cdots - 117 \)
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| $53$ |
\( T^{5} + 5 T^{4} + \cdots + 4176 \)
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| $59$ |
\( T^{5} + T^{4} + \cdots - 13 \)
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| $61$ |
\( T^{5} + 2 T^{4} + \cdots + 1664 \)
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| $67$ |
\( T^{5} + 12 T^{4} + \cdots + 292 \)
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| $71$ |
\( T^{5} - 30 T^{4} + \cdots + 19678 \)
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| $73$ |
\( T^{5} + 12 T^{4} + \cdots - 444 \)
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| $79$ |
\( T^{5} + 12 T^{4} + \cdots + 2126 \)
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| $83$ |
\( T^{5} + 19 T^{4} + \cdots - 3071 \)
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| $89$ |
\( T^{5} - 11 T^{4} + \cdots - 6019 \)
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| $97$ |
\( T^{5} - 456 T^{3} + \cdots + 245914 \)
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