Newspace parameters
| Level: | \( N \) | \(=\) | \( 3750 = 2 \cdot 3 \cdot 5^{4} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 3750.c (of order \(2\), degree \(1\), not minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(29.9439007580\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Coefficient field: | 8.0.324000000.1 |
|
|
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| Defining polynomial: |
\( x^{8} + 9x^{6} + 26x^{4} + 24x^{2} + 1 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) 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
\( x^{8} + 9x^{6} + 26x^{4} + 24x^{2} + 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} + 2 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} + 3\nu \)
|
| \(\beta_{4}\) | \(=\) |
\( \nu^{4} + 4\nu^{2} + 2 \)
|
| \(\beta_{5}\) | \(=\) |
\( \nu^{5} + 5\nu^{3} + 5\nu \)
|
| \(\beta_{6}\) | \(=\) |
\( \nu^{6} + 6\nu^{4} + 9\nu^{2} + 2 \)
|
| \(\beta_{7}\) | \(=\) |
\( \nu^{7} + 7\nu^{5} + 14\nu^{3} + 7\nu \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} - 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} - 3\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{4} - 4\beta_{2} + 6 \)
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| \(\nu^{5}\) | \(=\) |
\( \beta_{5} - 5\beta_{3} + 10\beta_1 \)
|
| \(\nu^{6}\) | \(=\) |
\( \beta_{6} - 6\beta_{4} + 15\beta_{2} - 20 \)
|
| \(\nu^{7}\) | \(=\) |
\( \beta_{7} - 7\beta_{5} + 21\beta_{3} - 35\beta_1 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3750\mathbb{Z}\right)^\times\).
| \(n\) | \(2501\) | \(3127\) |
| \(\chi(n)\) | \(1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1249.1 |
|
− | 1.00000i | 1.00000i | −1.00000 | 0 | 1.00000 | − | 3.36527i | 1.00000i | −1.00000 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 1249.2 | − | 1.00000i | 1.00000i | −1.00000 | 0 | 1.00000 | − | 2.10686i | 1.00000i | −1.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||
| 1249.3 | − | 1.00000i | 1.00000i | −1.00000 | 0 | 1.00000 | − | 0.929284i | 1.00000i | −1.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||
| 1249.4 | − | 1.00000i | 1.00000i | −1.00000 | 0 | 1.00000 | 4.40142i | 1.00000i | −1.00000 | 0 | ||||||||||||||||||||||||||||||||||||||||||
| 1249.5 | 1.00000i | − | 1.00000i | −1.00000 | 0 | 1.00000 | − | 4.40142i | − | 1.00000i | −1.00000 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 1249.6 | 1.00000i | − | 1.00000i | −1.00000 | 0 | 1.00000 | 0.929284i | − | 1.00000i | −1.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||
| 1249.7 | 1.00000i | − | 1.00000i | −1.00000 | 0 | 1.00000 | 2.10686i | − | 1.00000i | −1.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||
| 1249.8 | 1.00000i | − | 1.00000i | −1.00000 | 0 | 1.00000 | 3.36527i | − | 1.00000i | −1.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 5.b | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 3750.2.c.j | 8 | |
| 5.b | even | 2 | 1 | inner | 3750.2.c.j | 8 | |
| 5.c | odd | 4 | 1 | 3750.2.a.i | ✓ | 4 | |
| 5.c | odd | 4 | 1 | 3750.2.a.t | yes | 4 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 3750.2.a.i | ✓ | 4 | 5.c | odd | 4 | 1 | |
| 3750.2.a.t | yes | 4 | 5.c | odd | 4 | 1 | |
| 3750.2.c.j | 8 | 1.a | even | 1 | 1 | trivial | |
| 3750.2.c.j | 8 | 5.b | even | 2 | 1 | inner | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{7}^{8} + 36T_{7}^{6} + 386T_{7}^{4} + 1281T_{7}^{2} + 841 \)
acting on \(S_{2}^{\mathrm{new}}(3750, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} + 1)^{4} \)
|
| $3$ |
\( (T^{2} + 1)^{4} \)
|
| $5$ |
\( T^{8} \)
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| $7$ |
\( T^{8} + 36 T^{6} + \cdots + 841 \)
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| $11$ |
\( (T^{4} - 8 T^{3} + 9 T^{2} + \cdots + 1)^{2} \)
|
| $13$ |
\( T^{8} + 44 T^{6} + \cdots + 841 \)
|
| $17$ |
\( T^{8} + 11 T^{6} + \cdots + 1 \)
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| $19$ |
\( (T^{4} - 5 T^{3} + 10 T - 5)^{2} \)
|
| $23$ |
\( T^{8} + 129 T^{6} + \cdots + 7921 \)
|
| $29$ |
\( (T^{4} - 70 T^{2} + \cdots + 445)^{2} \)
|
| $31$ |
\( (T^{4} - 8 T^{3} + 9 T^{2} + \cdots + 1)^{2} \)
|
| $37$ |
\( T^{8} + 291 T^{6} + \cdots + 8637721 \)
|
| $41$ |
\( (T^{4} - 8 T^{3} + \cdots - 149)^{2} \)
|
| $43$ |
\( T^{8} + 264 T^{6} + \cdots + 9369721 \)
|
| $47$ |
\( T^{8} + 301 T^{6} + \cdots + 16168441 \)
|
| $53$ |
\( T^{8} + 329 T^{6} + \cdots + 15437041 \)
|
| $59$ |
\( (T^{4} - 5 T^{3} - 85 T^{2} + \cdots + 25)^{2} \)
|
| $61$ |
\( (T^{4} + 27 T^{3} + \cdots - 279)^{2} \)
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| $67$ |
\( T^{8} + 351 T^{6} + \cdots + 30791401 \)
|
| $71$ |
\( (T^{4} - 28 T^{3} + \cdots + 1801)^{2} \)
|
| $73$ |
\( T^{8} + 444 T^{6} + \cdots + 281961 \)
|
| $79$ |
\( (T^{4} + 15 T^{3} + \cdots - 1055)^{2} \)
|
| $83$ |
\( T^{8} + 404 T^{6} + \cdots + 292681 \)
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| $89$ |
\( (T^{4} + 15 T^{3} + \cdots + 2025)^{2} \)
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| $97$ |
\( T^{8} + 601 T^{6} + \cdots + 116618401 \)
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