Newspace parameters
| Level: | \( N \) | \(=\) | \( 350 = 2 \cdot 5^{2} \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 350.j (of order \(6\), degree \(2\), not minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(20.6506685020\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Relative dimension: | \(2\) over \(\Q(\zeta_{6})\) |
| Coefficient field: | \(\Q(\zeta_{12})\) |
|
|
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| Defining polynomial: |
\( x^{4} - x^{2} + 1 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{6}]$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{12}\). We also show the integral \(q\)-expansion of the trace form.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/350\mathbb{Z}\right)^\times\).
| \(n\) | \(101\) | \(127\) |
| \(\chi(n)\) | \(-\zeta_{12}^{2}\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 149.1 |
|
−1.73205 | − | 1.00000i | 3.46410 | − | 2.00000i | 2.00000 | + | 3.46410i | 0 | −8.00000 | −6.06218 | − | 17.5000i | − | 8.00000i | −5.50000 | + | 9.52628i | 0 | |||||||||||||||||||
| 149.2 | 1.73205 | + | 1.00000i | −3.46410 | + | 2.00000i | 2.00000 | + | 3.46410i | 0 | −8.00000 | 6.06218 | + | 17.5000i | 8.00000i | −5.50000 | + | 9.52628i | 0 | |||||||||||||||||||||
| 249.1 | −1.73205 | + | 1.00000i | 3.46410 | + | 2.00000i | 2.00000 | − | 3.46410i | 0 | −8.00000 | −6.06218 | + | 17.5000i | 8.00000i | −5.50000 | − | 9.52628i | 0 | |||||||||||||||||||||
| 249.2 | 1.73205 | − | 1.00000i | −3.46410 | − | 2.00000i | 2.00000 | − | 3.46410i | 0 | −8.00000 | 6.06218 | − | 17.5000i | − | 8.00000i | −5.50000 | − | 9.52628i | 0 | ||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 5.b | even | 2 | 1 | inner |
| 7.c | even | 3 | 1 | inner |
| 35.j | even | 6 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 350.4.j.c | 4 | |
| 5.b | even | 2 | 1 | inner | 350.4.j.c | 4 | |
| 5.c | odd | 4 | 1 | 350.4.e.c | ✓ | 2 | |
| 5.c | odd | 4 | 1 | 350.4.e.f | yes | 2 | |
| 7.c | even | 3 | 1 | inner | 350.4.j.c | 4 | |
| 35.j | even | 6 | 1 | inner | 350.4.j.c | 4 | |
| 35.k | even | 12 | 1 | 2450.4.a.f | 1 | ||
| 35.k | even | 12 | 1 | 2450.4.a.bl | 1 | ||
| 35.l | odd | 12 | 1 | 350.4.e.c | ✓ | 2 | |
| 35.l | odd | 12 | 1 | 350.4.e.f | yes | 2 | |
| 35.l | odd | 12 | 1 | 2450.4.a.p | 1 | ||
| 35.l | odd | 12 | 1 | 2450.4.a.z | 1 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 350.4.e.c | ✓ | 2 | 5.c | odd | 4 | 1 | |
| 350.4.e.c | ✓ | 2 | 35.l | odd | 12 | 1 | |
| 350.4.e.f | yes | 2 | 5.c | odd | 4 | 1 | |
| 350.4.e.f | yes | 2 | 35.l | odd | 12 | 1 | |
| 350.4.j.c | 4 | 1.a | even | 1 | 1 | trivial | |
| 350.4.j.c | 4 | 5.b | even | 2 | 1 | inner | |
| 350.4.j.c | 4 | 7.c | even | 3 | 1 | inner | |
| 350.4.j.c | 4 | 35.j | even | 6 | 1 | inner | |
| 2450.4.a.f | 1 | 35.k | even | 12 | 1 | ||
| 2450.4.a.p | 1 | 35.l | odd | 12 | 1 | ||
| 2450.4.a.z | 1 | 35.l | odd | 12 | 1 | ||
| 2450.4.a.bl | 1 | 35.k | even | 12 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(350, [\chi])\):
|
\( T_{3}^{4} - 16T_{3}^{2} + 256 \)
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\( T_{11}^{2} + 30T_{11} + 900 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} - 4T^{2} + 16 \)
|
| $3$ |
\( T^{4} - 16T^{2} + 256 \)
|
| $5$ |
\( T^{4} \)
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| $7$ |
\( T^{4} + 539 T^{2} + 117649 \)
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| $11$ |
\( (T^{2} + 30 T + 900)^{2} \)
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| $13$ |
\( (T^{2} + 16)^{2} \)
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| $17$ |
\( T^{4} - 81T^{2} + 6561 \)
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| $19$ |
\( (T^{2} + 88 T + 7744)^{2} \)
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| $23$ |
\( T^{4} - 1089 T^{2} + 1185921 \)
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| $29$ |
\( (T + 126)^{4} \)
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| $31$ |
\( (T^{2} + 155 T + 24025)^{2} \)
|
| $37$ |
\( T^{4} - 13456 T^{2} + 181063936 \)
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| $41$ |
\( (T + 423)^{4} \)
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| $43$ |
\( (T^{2} + 115600)^{2} \)
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| $47$ |
\( T^{4} + \cdots + 13206836241 \)
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| $53$ |
\( T^{4} + \cdots + 9475854336 \)
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| $59$ |
\( (T^{2} + 462 T + 213444)^{2} \)
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| $61$ |
\( (T^{2} + 326 T + 106276)^{2} \)
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| $67$ |
\( T^{4} + \cdots + 245635219456 \)
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| $71$ |
\( (T - 621)^{4} \)
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| $73$ |
\( T^{4} + \cdots + 3906250000 \)
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| $79$ |
\( (T^{2} + 1105 T + 1221025)^{2} \)
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| $83$ |
\( (T^{2} + 39204)^{2} \)
|
| $89$ |
\( (T^{2} + 873 T + 762129)^{2} \)
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| $97$ |
\( (T^{2} + 819025)^{2} \)
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