Newspace parameters
| Level: | \( N \) | \(=\) | \( 351 = 3^{3} \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 351.b (of order \(2\), degree \(1\), minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(2.80274911095\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\Q(\sqrt{-10 +2 \sqrt{13}})\) |
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|
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| Defining polynomial: |
\( x^{4} + 5x^{2} + 3 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 3 \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{U}(1)[D_{2}]$ |
$q$-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 in terms of a root \(\nu\) of
\( x^{4} + 5x^{2} + 3 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu^{3} + 3\nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{3} + 6\nu \)
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| \(\beta_{3}\) | \(=\) |
\( \nu^{2} + 3 \)
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| \(\nu\) | \(=\) |
\( ( \beta_{2} - \beta_1 ) / 3 \)
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| \(\nu^{2}\) | \(=\) |
\( \beta_{3} - 3 \)
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| \(\nu^{3}\) | \(=\) |
\( -\beta_{2} + 2\beta_1 \)
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Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/351\mathbb{Z}\right)^\times\).
| \(n\) | \(28\) | \(326\) |
| \(\chi(n)\) | \(-1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 298.1 |
|
− | 2.70236i | 0 | −5.30278 | 3.52058i | 0 | 0 | 8.92530i | 0 | 9.51388 | |||||||||||||||||||||||||||||
| 298.2 | − | 1.92282i | 0 | −1.69722 | − | 4.42782i | 0 | 0 | − | 0.582182i | 0 | −8.51388 | ||||||||||||||||||||||||||||
| 298.3 | 1.92282i | 0 | −1.69722 | 4.42782i | 0 | 0 | 0.582182i | 0 | −8.51388 | |||||||||||||||||||||||||||||||
| 298.4 | 2.70236i | 0 | −5.30278 | − | 3.52058i | 0 | 0 | − | 8.92530i | 0 | 9.51388 | |||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 39.d | odd | 2 | 1 | CM by \(\Q(\sqrt{-39}) \) |
| 3.b | odd | 2 | 1 | inner |
| 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 | 351.2.b.b | ✓ | 4 |
| 3.b | odd | 2 | 1 | inner | 351.2.b.b | ✓ | 4 |
| 13.b | even | 2 | 1 | inner | 351.2.b.b | ✓ | 4 |
| 13.d | odd | 4 | 2 | 4563.2.a.bb | 4 | ||
| 39.d | odd | 2 | 1 | CM | 351.2.b.b | ✓ | 4 |
| 39.f | even | 4 | 2 | 4563.2.a.bb | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 351.2.b.b | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 351.2.b.b | ✓ | 4 | 3.b | odd | 2 | 1 | inner |
| 351.2.b.b | ✓ | 4 | 13.b | even | 2 | 1 | inner |
| 351.2.b.b | ✓ | 4 | 39.d | odd | 2 | 1 | CM |
| 4563.2.a.bb | 4 | 13.d | odd | 4 | 2 | ||
| 4563.2.a.bb | 4 | 39.f | even | 4 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(351, [\chi])\):
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\( T_{2}^{4} + 11T_{2}^{2} + 27 \)
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\( T_{5}^{4} + 32T_{5}^{2} + 243 \)
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\( T_{17} \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} + 11T^{2} + 27 \)
|
| $3$ |
\( T^{4} \)
|
| $5$ |
\( T^{4} + 32T^{2} + 243 \)
|
| $7$ |
\( T^{4} \)
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| $11$ |
\( T^{4} + 44T^{2} + 432 \)
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| $13$ |
\( (T^{2} - 13)^{2} \)
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| $17$ |
\( T^{4} \)
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| $19$ |
\( T^{4} \)
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| $23$ |
\( T^{4} \)
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| $29$ |
\( T^{4} \)
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| $31$ |
\( T^{4} \)
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| $37$ |
\( T^{4} \)
|
| $41$ |
\( T^{4} + 164T^{2} + 432 \)
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| $43$ |
\( (T^{2} - 4 T - 113)^{2} \)
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| $47$ |
\( T^{4} + 200T^{2} + 7803 \)
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| $53$ |
\( T^{4} \)
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| $59$ |
\( T^{4} + 32T^{2} + 243 \)
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| $61$ |
\( (T^{2} - 24 T + 131)^{2} \)
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| $67$ |
\( T^{4} \)
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| $71$ |
\( T^{4} + 80T^{2} + 27 \)
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| $73$ |
\( T^{4} \)
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| $79$ |
\( (T^{2} - 208)^{2} \)
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| $83$ |
\( T^{4} + 344 T^{2} + 22707 \)
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| $89$ |
\( T^{4} + 80T^{2} + 27 \)
|
| $97$ |
\( T^{4} \)
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