Newspace parameters
| Level: | \( N \) | \(=\) | \( 1372 = 2^{2} \cdot 7^{3} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1372.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(10.9554751573\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | 6.6.2624293.1 |
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| Defining polynomial: |
\( x^{6} - x^{5} - 9x^{4} + 8x^{3} + 21x^{2} - 14x - 7 \)
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| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Fricke sign: | \(-1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{5}\) 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
\( x^{6} - x^{5} - 9x^{4} + 8x^{3} + 21x^{2} - 14x - 7 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
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| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 3 \)
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| \(\beta_{3}\) | \(=\) |
\( \nu^{5} + \nu^{4} - 6\nu^{3} - 5\nu^{2} + 6\nu + 2 \)
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| \(\beta_{4}\) | \(=\) |
\( \nu^{5} + \nu^{4} - 7\nu^{3} - 5\nu^{2} + 10\nu + 2 \)
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| \(\beta_{5}\) | \(=\) |
\( 2\nu^{5} + 3\nu^{4} - 13\nu^{3} - 16\nu^{2} + 16\nu + 10 \)
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| \(\nu\) | \(=\) |
\( \beta_1 \)
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| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 3 \)
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| \(\nu^{3}\) | \(=\) |
\( -\beta_{4} + \beta_{3} + 4\beta_1 \)
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| \(\nu^{4}\) | \(=\) |
\( \beta_{5} - \beta_{4} - \beta_{3} + 6\beta_{2} + 12 \)
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| \(\nu^{5}\) | \(=\) |
\( -\beta_{5} - 5\beta_{4} + 8\beta_{3} - \beta_{2} + 18\beta _1 + 1 \)
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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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1.1 |
|
0 | −1.30687 | 0 | 0.966761 | 0 | 0 | 0 | −1.29208 | 0 | ||||||||||||||||||||||||||||||||||||
| 1.2 | 0 | −0.739991 | 0 | 1.95032 | 0 | 0 | 0 | −2.45241 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.3 | 0 | 0.650063 | 0 | −1.58824 | 0 | 0 | 0 | −2.57742 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 0 | 2.05989 | 0 | 2.83518 | 0 | 0 | 0 | 1.24316 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 0 | 3.15187 | 0 | 4.03328 | 0 | 0 | 0 | 6.93431 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 0 | 3.18503 | 0 | −1.19730 | 0 | 0 | 0 | 7.14443 | 0 | |||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( -1 \) |
| \(7\) | \( +1 \) |
Inner twists
This newform does not admit any (nontrivial) inner twists.
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 1372.2.a.d | yes | 6 |
| 4.b | odd | 2 | 1 | 5488.2.a.g | 6 | ||
| 7.b | odd | 2 | 1 | 1372.2.a.a | ✓ | 6 | |
| 7.c | even | 3 | 2 | 1372.2.e.a | 12 | ||
| 7.d | odd | 6 | 2 | 1372.2.e.d | 12 | ||
| 28.d | even | 2 | 1 | 5488.2.a.q | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1372.2.a.a | ✓ | 6 | 7.b | odd | 2 | 1 | |
| 1372.2.a.d | yes | 6 | 1.a | even | 1 | 1 | trivial |
| 1372.2.e.a | 12 | 7.c | even | 3 | 2 | ||
| 1372.2.e.d | 12 | 7.d | odd | 6 | 2 | ||
| 5488.2.a.g | 6 | 4.b | odd | 2 | 1 | ||
| 5488.2.a.q | 6 | 28.d | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1372))\):
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\( T_{3}^{6} - 7T_{3}^{5} + 11T_{3}^{4} + 14T_{3}^{3} - 32T_{3}^{2} - 7T_{3} + 13 \)
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\( T_{11}^{6} - 3T_{11}^{5} - 40T_{11}^{4} + 36T_{11}^{3} + 459T_{11}^{2} + 331T_{11} - 433 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} \)
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| $3$ |
\( T^{6} - 7 T^{5} + \cdots + 13 \)
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| $5$ |
\( T^{6} - 7 T^{5} + \cdots + 41 \)
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| $7$ |
\( T^{6} \)
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| $11$ |
\( T^{6} - 3 T^{5} + \cdots - 433 \)
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| $13$ |
\( T^{6} - 7 T^{5} + \cdots + 349 \)
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| $17$ |
\( T^{6} - 34 T^{4} + \cdots + 349 \)
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| $19$ |
\( T^{6} - 7 T^{5} + \cdots + 419 \)
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| $23$ |
\( T^{6} - 11 T^{5} + \cdots + 1597 \)
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| $29$ |
\( T^{6} - 6 T^{5} + \cdots + 351 \)
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| $31$ |
\( T^{6} - 14 T^{5} + \cdots + 31507 \)
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| $37$ |
\( T^{6} + 6 T^{5} + \cdots - 31401 \)
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| $41$ |
\( T^{6} - 136 T^{4} + \cdots + 2113 \)
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| $43$ |
\( T^{6} + 3 T^{5} + \cdots + 7631 \)
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| $47$ |
\( T^{6} - 42 T^{5} + \cdots + 37729 \)
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| $53$ |
\( T^{6} + 17 T^{5} + \cdots - 11367 \)
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| $59$ |
\( T^{6} - 35 T^{5} + \cdots + 19837 \)
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| $61$ |
\( T^{6} - 7 T^{5} + \cdots - 144257 \)
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| $67$ |
\( T^{6} + 14 T^{5} + \cdots + 20531 \)
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| $71$ |
\( (T^{3} + 14 T^{2} + 49 T + 7)^{2} \)
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| $73$ |
\( T^{6} - 269 T^{4} + \cdots - 138629 \)
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| $79$ |
\( T^{6} - 6 T^{5} + \cdots + 491 \)
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| $83$ |
\( T^{6} - 49 T^{5} + \cdots + 20327 \)
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| $89$ |
\( T^{6} + 7 T^{5} + \cdots + 339499 \)
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| $97$ |
\( T^{6} + 21 T^{5} + \cdots + 297317 \)
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