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: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | 6.6.297869800.1 |
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| Defining polynomial: |
\( x^{6} - 2x^{5} - 9x^{4} + 7x^{3} + 23x^{2} + 2x - 6 \)
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| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 2775) |
| Fricke sign: | \(-1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{5}\) 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
\( x^{6} - 2x^{5} - 9x^{4} + 7x^{3} + 23x^{2} + 2x - 6 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
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| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 2\nu - 3 \)
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| \(\beta_{3}\) | \(=\) |
\( -\nu^{5} + 4\nu^{4} + 2\nu^{3} - 14\nu^{2} + 6 \)
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| \(\beta_{4}\) | \(=\) |
\( \nu^{5} - 3\nu^{4} - 5\nu^{3} + 10\nu^{2} + 8\nu - 3 \)
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| \(\beta_{5}\) | \(=\) |
\( -\nu^{5} + 3\nu^{4} + 6\nu^{3} - 12\nu^{2} - 12\nu + 5 \)
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| \(\nu\) | \(=\) |
\( \beta_1 \)
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| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 2\beta _1 + 3 \)
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| \(\nu^{3}\) | \(=\) |
\( \beta_{5} + \beta_{4} + 2\beta_{2} + 8\beta _1 + 4 \)
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| \(\nu^{4}\) | \(=\) |
\( 3\beta_{5} + 4\beta_{4} + \beta_{3} + 10\beta_{2} + 24\beta _1 + 21 \)
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| \(\nu^{5}\) | \(=\) |
\( 14\beta_{5} + 18\beta_{4} + 3\beta_{3} + 30\beta_{2} + 84\beta _1 + 56 \)
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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 |
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−2.69361 | 0 | 5.25551 | 0 | 0 | −2.97039 | −8.76907 | 0 | 0 | ||||||||||||||||||||||||||||||||||||
| 1.2 | −2.52727 | 0 | 4.38709 | 0 | 0 | 3.70785 | −6.03281 | 0 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.3 | −1.72206 | 0 | 0.965489 | 0 | 0 | 0.768516 | 1.78149 | 0 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.4 | −0.541409 | 0 | −1.70688 | 0 | 0 | −3.40524 | 2.00694 | 0 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 1.02507 | 0 | −0.949227 | 0 | 0 | 1.59014 | −3.02317 | 0 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 2.45927 | 0 | 4.04801 | 0 | 0 | 1.30912 | 5.03662 | 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.ci | 6 | |
| 3.b | odd | 2 | 1 | 2775.2.a.bf | yes | 6 | |
| 5.b | even | 2 | 1 | 8325.2.a.cl | 6 | ||
| 15.d | odd | 2 | 1 | 2775.2.a.be | ✓ | 6 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 2775.2.a.be | ✓ | 6 | 15.d | odd | 2 | 1 | |
| 2775.2.a.bf | yes | 6 | 3.b | odd | 2 | 1 | |
| 8325.2.a.ci | 6 | 1.a | even | 1 | 1 | trivial | |
| 8325.2.a.cl | 6 | 5.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8325))\):
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\( T_{2}^{6} + 4T_{2}^{5} - 4T_{2}^{4} - 29T_{2}^{3} - 15T_{2}^{2} + 29T_{2} + 16 \)
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\( T_{7}^{6} - T_{7}^{5} - 19T_{7}^{4} + 22T_{7}^{3} + 75T_{7}^{2} - 140T_{7} + 60 \)
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\( T_{11}^{6} - 5T_{11}^{5} - 30T_{11}^{4} + 91T_{11}^{3} + 284T_{11}^{2} + 200T_{11} + 32 \)
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\( T_{13}^{6} - T_{13}^{5} - 41T_{13}^{4} + 42T_{13}^{3} + 227T_{13}^{2} + 60T_{13} + 4 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} + 4 T^{5} + \cdots + 16 \)
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| $3$ |
\( T^{6} \)
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| $5$ |
\( T^{6} \)
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| $7$ |
\( T^{6} - T^{5} + \cdots + 60 \)
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| $11$ |
\( T^{6} - 5 T^{5} + \cdots + 32 \)
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| $13$ |
\( T^{6} - T^{5} - 41 T^{4} + \cdots + 4 \)
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| $17$ |
\( T^{6} + 15 T^{5} + \cdots - 520 \)
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| $19$ |
\( T^{6} - 11 T^{5} + \cdots - 4073 \)
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| $23$ |
\( T^{6} + 12 T^{5} + \cdots - 708 \)
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| $29$ |
\( T^{6} - 9 T^{5} + \cdots + 344 \)
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| $31$ |
\( T^{6} - 14 T^{5} + \cdots + 21036 \)
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| $37$ |
\( (T + 1)^{6} \)
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| $41$ |
\( T^{6} - 4 T^{5} + \cdots - 398 \)
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| $43$ |
\( T^{6} - 58 T^{4} + \cdots - 905 \)
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| $47$ |
\( T^{6} + 15 T^{5} + \cdots + 104 \)
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| $53$ |
\( T^{6} + 41 T^{5} + \cdots + 65524 \)
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| $59$ |
\( T^{6} - 3 T^{5} + \cdots + 908 \)
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| $61$ |
\( T^{6} - 5 T^{5} + \cdots + 35216 \)
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| $67$ |
\( T^{6} - 4 T^{5} + \cdots - 257820 \)
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| $71$ |
\( T^{6} + 3 T^{5} + \cdots + 1888 \)
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| $73$ |
\( T^{6} - 11 T^{5} + \cdots + 35570 \)
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| $79$ |
\( T^{6} - 4 T^{5} + \cdots - 84024 \)
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| $83$ |
\( T^{6} + 27 T^{5} + \cdots + 245568 \)
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| $89$ |
\( T^{6} - 15 T^{5} + \cdots + 127800 \)
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| $97$ |
\( T^{6} + 17 T^{5} + \cdots + 162368 \)
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