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: |
\( x^{4} - x^{2} + 1 \)
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| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 70) |
| 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 | −0.866025 | + | 0.500000i | 2.00000 | + | 3.46410i | 0 | 2.00000 | −16.4545 | − | 8.50000i | − | 8.00000i | −13.0000 | + | 22.5167i | 0 | |||||||||||||||||||
| 149.2 | 1.73205 | + | 1.00000i | 0.866025 | − | 0.500000i | 2.00000 | + | 3.46410i | 0 | 2.00000 | 16.4545 | + | 8.50000i | 8.00000i | −13.0000 | + | 22.5167i | 0 | |||||||||||||||||||||
| 249.1 | −1.73205 | + | 1.00000i | −0.866025 | − | 0.500000i | 2.00000 | − | 3.46410i | 0 | 2.00000 | −16.4545 | + | 8.50000i | 8.00000i | −13.0000 | − | 22.5167i | 0 | |||||||||||||||||||||
| 249.2 | 1.73205 | − | 1.00000i | 0.866025 | + | 0.500000i | 2.00000 | − | 3.46410i | 0 | 2.00000 | 16.4545 | − | 8.50000i | − | 8.00000i | −13.0000 | − | 22.5167i | 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.e | 4 | |
| 5.b | even | 2 | 1 | inner | 350.4.j.e | 4 | |
| 5.c | odd | 4 | 1 | 70.4.e.c | ✓ | 2 | |
| 5.c | odd | 4 | 1 | 350.4.e.a | 2 | ||
| 7.c | even | 3 | 1 | inner | 350.4.j.e | 4 | |
| 15.e | even | 4 | 1 | 630.4.k.b | 2 | ||
| 20.e | even | 4 | 1 | 560.4.q.d | 2 | ||
| 35.f | even | 4 | 1 | 490.4.e.m | 2 | ||
| 35.j | even | 6 | 1 | inner | 350.4.j.e | 4 | |
| 35.k | even | 12 | 1 | 490.4.a.e | 1 | ||
| 35.k | even | 12 | 1 | 490.4.e.m | 2 | ||
| 35.k | even | 12 | 1 | 2450.4.a.be | 1 | ||
| 35.l | odd | 12 | 1 | 70.4.e.c | ✓ | 2 | |
| 35.l | odd | 12 | 1 | 350.4.e.a | 2 | ||
| 35.l | odd | 12 | 1 | 490.4.a.c | 1 | ||
| 35.l | odd | 12 | 1 | 2450.4.a.bg | 1 | ||
| 105.x | even | 12 | 1 | 630.4.k.b | 2 | ||
| 140.w | even | 12 | 1 | 560.4.q.d | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 70.4.e.c | ✓ | 2 | 5.c | odd | 4 | 1 | |
| 70.4.e.c | ✓ | 2 | 35.l | odd | 12 | 1 | |
| 350.4.e.a | 2 | 5.c | odd | 4 | 1 | ||
| 350.4.e.a | 2 | 35.l | odd | 12 | 1 | ||
| 350.4.j.e | 4 | 1.a | even | 1 | 1 | trivial | |
| 350.4.j.e | 4 | 5.b | even | 2 | 1 | inner | |
| 350.4.j.e | 4 | 7.c | even | 3 | 1 | inner | |
| 350.4.j.e | 4 | 35.j | even | 6 | 1 | inner | |
| 490.4.a.c | 1 | 35.l | odd | 12 | 1 | ||
| 490.4.a.e | 1 | 35.k | even | 12 | 1 | ||
| 490.4.e.m | 2 | 35.f | even | 4 | 1 | ||
| 490.4.e.m | 2 | 35.k | even | 12 | 1 | ||
| 560.4.q.d | 2 | 20.e | even | 4 | 1 | ||
| 560.4.q.d | 2 | 140.w | even | 12 | 1 | ||
| 630.4.k.b | 2 | 15.e | even | 4 | 1 | ||
| 630.4.k.b | 2 | 105.x | even | 12 | 1 | ||
| 2450.4.a.be | 1 | 35.k | even | 12 | 1 | ||
| 2450.4.a.bg | 1 | 35.l | odd | 12 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(350, [\chi])\):
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\( T_{3}^{4} - T_{3}^{2} + 1 \)
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\( T_{11}^{2} - 2T_{11} + 4 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} - 4T^{2} + 16 \)
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| $3$ |
\( T^{4} - T^{2} + 1 \)
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| $5$ |
\( T^{4} \)
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| $7$ |
\( T^{4} - 397 T^{2} + 117649 \)
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| $11$ |
\( (T^{2} - 2 T + 4)^{2} \)
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| $13$ |
\( (T^{2} + 64)^{2} \)
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| $17$ |
\( T^{4} - 2704 T^{2} + 7311616 \)
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| $19$ |
\( (T^{2} - 26 T + 676)^{2} \)
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| $23$ |
\( T^{4} - 4489 T^{2} + 20151121 \)
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| $29$ |
\( (T + 69)^{4} \)
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| $31$ |
\( (T^{2} - 332 T + 110224)^{2} \)
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| $37$ |
\( T^{4} + \cdots + 1475789056 \)
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| $41$ |
\( (T - 353)^{4} \)
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| $43$ |
\( (T^{2} + 136161)^{2} \)
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| $47$ |
\( T^{4} - 7744 T^{2} + 59969536 \)
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| $53$ |
\( T^{4} + \cdots + 114733948176 \)
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| $59$ |
\( (T^{2} + 350 T + 122500)^{2} \)
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| $61$ |
\( (T^{2} - 467 T + 218089)^{2} \)
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| $67$ |
\( T^{4} + \cdots + 7170871761 \)
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| $71$ |
\( (T - 770)^{4} \)
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| $73$ |
\( T^{4} + \cdots + 155538739456 \)
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| $79$ |
\( (T^{2} - 1170 T + 1368900)^{2} \)
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| $83$ |
\( (T^{2} + 275625)^{2} \)
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| $89$ |
\( (T^{2} - 89 T + 7921)^{2} \)
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| $97$ |
\( (T^{2} + 84100)^{2} \)
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