Newspace parameters
| Level: | \( N \) | \(=\) | \( 5070 = 2 \cdot 3 \cdot 5 \cdot 13^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 5070.b (of order \(2\), degree \(1\), not minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(40.4841538248\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\Q(\zeta_{8})\) |
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| Defining polynomial: |
\( x^{4} + 1 \)
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| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{3} \) |
| Twist minimal: | no (minimal twist has level 390) |
| 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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| \(\zeta_{8}^{3}\) | \(=\) |
\( ( -\beta_{3} + \beta_{2} ) / 4 \)
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Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/5070\mathbb{Z}\right)^\times\).
| \(n\) | \(1691\) | \(1861\) | \(4057\) |
| \(\chi(n)\) | \(1\) | \(-1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1351.1 |
|
− | 1.00000i | 1.00000 | −1.00000 | − | 1.00000i | − | 1.00000i | − | 2.82843i | 1.00000i | 1.00000 | −1.00000 | ||||||||||||||||||||||||||
| 1351.2 | − | 1.00000i | 1.00000 | −1.00000 | − | 1.00000i | − | 1.00000i | 2.82843i | 1.00000i | 1.00000 | −1.00000 | ||||||||||||||||||||||||||||
| 1351.3 | 1.00000i | 1.00000 | −1.00000 | 1.00000i | 1.00000i | − | 2.82843i | − | 1.00000i | 1.00000 | −1.00000 | |||||||||||||||||||||||||||||
| 1351.4 | 1.00000i | 1.00000 | −1.00000 | 1.00000i | 1.00000i | 2.82843i | − | 1.00000i | 1.00000 | −1.00000 | ||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 13.b | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 5070.2.b.q | 4 | |
| 13.b | even | 2 | 1 | inner | 5070.2.b.q | 4 | |
| 13.d | odd | 4 | 1 | 390.2.a.h | ✓ | 2 | |
| 13.d | odd | 4 | 1 | 5070.2.a.bc | 2 | ||
| 39.f | even | 4 | 1 | 1170.2.a.o | 2 | ||
| 52.f | even | 4 | 1 | 3120.2.a.bc | 2 | ||
| 65.f | even | 4 | 1 | 1950.2.e.o | 4 | ||
| 65.g | odd | 4 | 1 | 1950.2.a.bd | 2 | ||
| 65.k | even | 4 | 1 | 1950.2.e.o | 4 | ||
| 156.l | odd | 4 | 1 | 9360.2.a.ch | 2 | ||
| 195.j | odd | 4 | 1 | 5850.2.e.bk | 4 | ||
| 195.n | even | 4 | 1 | 5850.2.a.cl | 2 | ||
| 195.u | odd | 4 | 1 | 5850.2.e.bk | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 390.2.a.h | ✓ | 2 | 13.d | odd | 4 | 1 | |
| 1170.2.a.o | 2 | 39.f | even | 4 | 1 | ||
| 1950.2.a.bd | 2 | 65.g | odd | 4 | 1 | ||
| 1950.2.e.o | 4 | 65.f | even | 4 | 1 | ||
| 1950.2.e.o | 4 | 65.k | even | 4 | 1 | ||
| 3120.2.a.bc | 2 | 52.f | even | 4 | 1 | ||
| 5070.2.a.bc | 2 | 13.d | odd | 4 | 1 | ||
| 5070.2.b.q | 4 | 1.a | even | 1 | 1 | trivial | |
| 5070.2.b.q | 4 | 13.b | even | 2 | 1 | inner | |
| 5850.2.a.cl | 2 | 195.n | even | 4 | 1 | ||
| 5850.2.e.bk | 4 | 195.j | odd | 4 | 1 | ||
| 5850.2.e.bk | 4 | 195.u | odd | 4 | 1 | ||
| 9360.2.a.ch | 2 | 156.l | odd | 4 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(5070, [\chi])\):
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\( T_{7}^{2} + 8 \)
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\( T_{11}^{2} + 32 \)
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\( T_{17}^{2} - 4T_{17} - 4 \)
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\( T_{31}^{2} + 16 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} + 1)^{2} \)
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| $3$ |
\( (T - 1)^{4} \)
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| $5$ |
\( (T^{2} + 1)^{2} \)
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| $7$ |
\( (T^{2} + 8)^{2} \)
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| $11$ |
\( (T^{2} + 32)^{2} \)
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| $13$ |
\( T^{4} \)
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| $17$ |
\( (T^{2} - 4 T - 4)^{2} \)
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| $19$ |
\( (T^{2} + 8)^{2} \)
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| $23$ |
\( (T^{2} - 72)^{2} \)
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| $29$ |
\( (T^{2} + 12 T + 28)^{2} \)
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| $31$ |
\( (T^{2} + 16)^{2} \)
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| $37$ |
\( T^{4} + 136T^{2} + 16 \)
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| $41$ |
\( T^{4} + 72T^{2} + 784 \)
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| $43$ |
\( (T^{2} + 8 T - 16)^{2} \)
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| $47$ |
\( (T^{2} + 64)^{2} \)
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| $53$ |
\( (T^{2} - 4 T - 124)^{2} \)
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| $59$ |
\( T^{4} + 192T^{2} + 1024 \)
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| $61$ |
\( (T - 6)^{4} \)
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| $67$ |
\( (T^{2} + 32)^{2} \)
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| $71$ |
\( (T^{2} + 32)^{2} \)
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| $73$ |
\( T^{4} + 216T^{2} + 1296 \)
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| $79$ |
\( (T^{2} - 16 T + 32)^{2} \)
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| $83$ |
\( T^{4} + 352 T^{2} + 12544 \)
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| $89$ |
\( T^{4} + 264T^{2} + 4624 \)
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| $97$ |
\( T^{4} + 88T^{2} + 784 \)
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