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: | \(4\) |
| Coefficient field: | 4.4.6224.1 |
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| Defining polynomial: |
\( x^{4} - 6x^{2} - 2x + 5 \)
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| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 111) |
| Fricke sign: | \(+1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) 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
\( x^{4} - 6x^{2} - 2x + 5 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
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| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - \nu - 3 \)
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| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - \nu^{2} - 4\nu + 2 \)
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| \(\nu\) | \(=\) |
\( \beta_1 \)
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| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + \beta _1 + 3 \)
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| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + \beta_{2} + 5\beta _1 + 1 \)
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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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−2.44579 | 0 | 3.98187 | 0 | 0 | −0.269264 | −4.84724 | 0 | 0 | ||||||||||||||||||||||||||||||
| 1.2 | −0.796815 | 0 | −1.36509 | 0 | 0 | −4.63253 | 2.68135 | 0 | 0 | |||||||||||||||||||||||||||||||
| 1.3 | 1.37033 | 0 | −0.122207 | 0 | 0 | 4.06064 | −2.90812 | 0 | 0 | |||||||||||||||||||||||||||||||
| 1.4 | 1.87228 | 0 | 1.50542 | 0 | 0 | −3.15885 | −0.925994 | 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.bv | 4 | |
| 3.b | odd | 2 | 1 | 2775.2.a.x | 4 | ||
| 5.b | even | 2 | 1 | 333.2.a.g | 4 | ||
| 15.d | odd | 2 | 1 | 111.2.a.b | ✓ | 4 | |
| 20.d | odd | 2 | 1 | 5328.2.a.bs | 4 | ||
| 60.h | even | 2 | 1 | 1776.2.a.u | 4 | ||
| 105.g | even | 2 | 1 | 5439.2.a.u | 4 | ||
| 120.i | odd | 2 | 1 | 7104.2.a.cc | 4 | ||
| 120.m | even | 2 | 1 | 7104.2.a.cf | 4 | ||
| 555.b | odd | 2 | 1 | 4107.2.a.i | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 111.2.a.b | ✓ | 4 | 15.d | odd | 2 | 1 | |
| 333.2.a.g | 4 | 5.b | even | 2 | 1 | ||
| 1776.2.a.u | 4 | 60.h | even | 2 | 1 | ||
| 2775.2.a.x | 4 | 3.b | odd | 2 | 1 | ||
| 4107.2.a.i | 4 | 555.b | odd | 2 | 1 | ||
| 5328.2.a.bs | 4 | 20.d | odd | 2 | 1 | ||
| 5439.2.a.u | 4 | 105.g | even | 2 | 1 | ||
| 7104.2.a.cc | 4 | 120.i | odd | 2 | 1 | ||
| 7104.2.a.cf | 4 | 120.m | even | 2 | 1 | ||
| 8325.2.a.bv | 4 | 1.a | even | 1 | 1 | trivial | |
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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\( T_{2}^{4} - 6T_{2}^{2} + 2T_{2} + 5 \)
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\( T_{7}^{4} + 4T_{7}^{3} - 16T_{7}^{2} - 64T_{7} - 16 \)
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\( T_{11}^{4} - 32T_{11}^{2} + 32T_{11} + 64 \)
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\( T_{13}^{4} + 4T_{13}^{3} - 32T_{13}^{2} - 144T_{13} - 80 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} - 6 T^{2} + \cdots + 5 \)
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| $3$ |
\( T^{4} \)
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| $5$ |
\( T^{4} \)
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| $7$ |
\( T^{4} + 4 T^{3} + \cdots - 16 \)
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| $11$ |
\( T^{4} - 32 T^{2} + \cdots + 64 \)
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| $13$ |
\( T^{4} + 4 T^{3} + \cdots - 80 \)
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| $17$ |
\( T^{4} + 2 T^{3} + \cdots - 28 \)
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| $19$ |
\( T^{4} - 8 T^{3} + \cdots - 224 \)
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| $23$ |
\( T^{4} + 10 T^{3} + \cdots + 652 \)
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| $29$ |
\( T^{4} - 2 T^{3} + \cdots + 724 \)
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| $31$ |
\( T^{4} - 4 T^{3} + \cdots + 32 \)
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| $37$ |
\( (T - 1)^{4} \)
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| $41$ |
\( T^{4} + 12 T^{3} + \cdots - 400 \)
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| $43$ |
\( T^{4} + 4 T^{3} + \cdots + 3424 \)
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| $47$ |
\( T^{4} + 12 T^{3} + \cdots - 128 \)
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| $53$ |
\( T^{4} - 8 T^{3} + \cdots + 464 \)
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| $59$ |
\( T^{4} - 10 T^{3} + \cdots - 7156 \)
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| $61$ |
\( T^{4} + 8 T^{3} + \cdots + 656 \)
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| $67$ |
\( T^{4} - 4 T^{3} + \cdots - 16 \)
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| $71$ |
\( T^{4} - 12 T^{3} + \cdots + 1664 \)
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| $73$ |
\( T^{4} + 12 T^{3} + \cdots - 32 \)
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| $79$ |
\( T^{4} + 8 T^{3} + \cdots - 1504 \)
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| $83$ |
\( T^{4} + 20 T^{3} + \cdots + 64 \)
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| $89$ |
\( T^{4} + 26 T^{3} + \cdots - 5452 \)
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| $97$ |
\( T^{4} - 4 T^{3} + \cdots + 17008 \)
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