Newspace parameters
| Level: | \( N \) | \(=\) | \( 525 = 3 \cdot 5^{2} \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 525.d (of order \(2\), degree \(1\), not minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(30.9760027530\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\Q(\zeta_{8})\) |
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|
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| Defining polynomial: |
\( x^{4} + 1 \)
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| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 2^{3} \) |
| Twist minimal: | no (minimal twist has level 105) |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.
Basis of coefficient ring
| \(\beta_{1}\) | \(=\) |
\( \zeta_{8}^{2} \)
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| \(\beta_{2}\) | \(=\) |
\( 2\zeta_{8}^{3} + 2\zeta_{8} \)
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| \(\beta_{3}\) | \(=\) |
\( -2\zeta_{8}^{3} + 2\zeta_{8} \)
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| \(\zeta_{8}\) | \(=\) |
\( ( \beta_{3} + \beta_{2} ) / 4 \)
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| \(\zeta_{8}^{2}\) | \(=\) |
\( \beta_1 \)
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| \(\zeta_{8}^{3}\) | \(=\) |
\( ( -\beta_{3} + \beta_{2} ) / 4 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/525\mathbb{Z}\right)^\times\).
| \(n\) | \(127\) | \(176\) | \(451\) |
| \(\chi(n)\) | \(-1\) | \(1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 274.1 |
|
− | 3.82843i | 3.00000i | −6.65685 | 0 | 11.4853 | − | 7.00000i | − | 5.14214i | −9.00000 | 0 | |||||||||||||||||||||||||||
| 274.2 | − | 1.82843i | − | 3.00000i | 4.65685 | 0 | −5.48528 | 7.00000i | − | 23.1421i | −9.00000 | 0 | ||||||||||||||||||||||||||||
| 274.3 | 1.82843i | 3.00000i | 4.65685 | 0 | −5.48528 | − | 7.00000i | 23.1421i | −9.00000 | 0 | ||||||||||||||||||||||||||||||
| 274.4 | 3.82843i | − | 3.00000i | −6.65685 | 0 | 11.4853 | 7.00000i | 5.14214i | −9.00000 | 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 | 525.4.d.l | 4 | |
| 5.b | even | 2 | 1 | inner | 525.4.d.l | 4 | |
| 5.c | odd | 4 | 1 | 105.4.a.e | ✓ | 2 | |
| 5.c | odd | 4 | 1 | 525.4.a.l | 2 | ||
| 15.e | even | 4 | 1 | 315.4.a.k | 2 | ||
| 15.e | even | 4 | 1 | 1575.4.a.q | 2 | ||
| 20.e | even | 4 | 1 | 1680.4.a.bo | 2 | ||
| 35.f | even | 4 | 1 | 735.4.a.o | 2 | ||
| 105.k | odd | 4 | 1 | 2205.4.a.bb | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 105.4.a.e | ✓ | 2 | 5.c | odd | 4 | 1 | |
| 315.4.a.k | 2 | 15.e | even | 4 | 1 | ||
| 525.4.a.l | 2 | 5.c | odd | 4 | 1 | ||
| 525.4.d.l | 4 | 1.a | even | 1 | 1 | trivial | |
| 525.4.d.l | 4 | 5.b | even | 2 | 1 | inner | |
| 735.4.a.o | 2 | 35.f | even | 4 | 1 | ||
| 1575.4.a.q | 2 | 15.e | even | 4 | 1 | ||
| 1680.4.a.bo | 2 | 20.e | even | 4 | 1 | ||
| 2205.4.a.bb | 2 | 105.k | odd | 4 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(525, [\chi])\):
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\( T_{2}^{4} + 18T_{2}^{2} + 49 \)
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\( T_{11}^{2} + 16T_{11} - 3136 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} + 18T^{2} + 49 \)
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| $3$ |
\( (T^{2} + 9)^{2} \)
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| $5$ |
\( T^{4} \)
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| $7$ |
\( (T^{2} + 49)^{2} \)
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| $11$ |
\( (T^{2} + 16 T - 3136)^{2} \)
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| $13$ |
\( T^{4} + 2952 T^{2} + 1993744 \)
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| $17$ |
\( T^{4} + 7752 T^{2} + 14531344 \)
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| $19$ |
\( (T^{2} - 96 T + 256)^{2} \)
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| $23$ |
\( T^{4} + 18624 T^{2} + 84345856 \)
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| $29$ |
\( (T^{2} + 188 T - 14492)^{2} \)
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| $31$ |
\( (T^{2} + 120 T + 1008)^{2} \)
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| $37$ |
\( T^{4} + 78408 T^{2} + 929762064 \)
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| $41$ |
\( (T^{2} - 100 T - 48700)^{2} \)
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| $43$ |
\( T^{4} + \cdots + 1686909184 \)
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| $47$ |
\( T^{4} + \cdots + 40475001856 \)
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| $53$ |
\( T^{4} + \cdots + 26437459216 \)
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| $59$ |
\( (T^{2} + 104 T - 330224)^{2} \)
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| $61$ |
\( (T^{2} + 468 T + 40644)^{2} \)
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| $67$ |
\( T^{4} + \cdots + 491098214656 \)
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| $71$ |
\( (T^{2} + 136 T - 175376)^{2} \)
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| $73$ |
\( T^{4} + \cdots + 585157681936 \)
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| $79$ |
\( (T^{2} - 432 T - 383936)^{2} \)
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| $83$ |
\( T^{4} + 283424 T^{2} + 830361856 \)
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| $89$ |
\( (T^{2} - 1404 T + 392452)^{2} \)
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| $97$ |
\( T^{4} + \cdots + 112834184464 \)
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