Newspace parameters
| Level: | \( N \) | \(=\) | \( 356 = 2^{2} \cdot 89 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 356.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(2.84267431196\) |
| Analytic rank: | \(0\) |
| Dimension: | \(7\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{7} - \cdots)\) |
|
|
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| Defining polynomial: |
\( x^{7} - x^{6} - 18x^{5} + 18x^{4} + 93x^{3} - 95x^{2} - 126x + 134 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| 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 \(1,\beta_1,\ldots,\beta_{6}\) 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
\( x^{7} - x^{6} - 18x^{5} + 18x^{4} + 93x^{3} - 95x^{2} - 126x + 134 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -4\nu^{6} + 9\nu^{5} + 79\nu^{4} - 116\nu^{3} - 446\nu^{2} + 317\nu + 564 ) / 73 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 19\nu^{6} + 12\nu^{5} - 357\nu^{4} - 179\nu^{3} + 1936\nu^{2} + 593\nu - 2679 ) / 73 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 28\nu^{6} + 10\nu^{5} - 480\nu^{4} - 137\nu^{3} + 2319\nu^{2} + 409\nu - 2853 ) / 73 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 32\nu^{6} + \nu^{5} - 559\nu^{4} - 21\nu^{3} + 2692\nu^{2} + 92\nu - 3052 ) / 73 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 46\nu^{6} + 6\nu^{5} - 799\nu^{4} - 126\nu^{3} + 3888\nu^{2} + 552\nu - 4734 ) / 73 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( -\beta_{5} + \beta_{4} - \beta_{2} + 5 \)
|
| \(\nu^{3}\) | \(=\) |
\( -2\beta_{6} + \beta_{5} + 2\beta_{4} - \beta_{2} + 7\beta _1 - 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{6} - 12\beta_{5} + 12\beta_{4} - 2\beta_{3} - 10\beta_{2} + 36 \)
|
| \(\nu^{5}\) | \(=\) |
\( -27\beta_{6} + 14\beta_{5} + 26\beta_{4} + 2\beta_{3} - 7\beta_{2} + 55\beta _1 - 22 \)
|
| \(\nu^{6}\) | \(=\) |
\( 17\beta_{6} - 123\beta_{5} + 126\beta_{4} - 35\beta_{3} - 91\beta_{2} + 303 \)
|
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 | −3.12667 | 0 | −3.07771 | 0 | −4.43694 | 0 | 6.77606 | 0 | |||||||||||||||||||||||||||||||||||||||
| 1.2 | 0 | −2.46578 | 0 | 4.26598 | 0 | −2.04350 | 0 | 3.08006 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 1.3 | 0 | −1.47549 | 0 | 0.459570 | 0 | 2.49680 | 0 | −0.822924 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 1.4 | 0 | 1.18392 | 0 | 3.68551 | 0 | 1.01833 | 0 | −1.59834 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 1.5 | 0 | 1.30035 | 0 | −3.30434 | 0 | 4.66487 | 0 | −1.30909 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 1.6 | 0 | 2.41399 | 0 | 0.754566 | 0 | −3.82143 | 0 | 2.82735 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 1.7 | 0 | 3.16968 | 0 | 0.216423 | 0 | 2.12188 | 0 | 7.04689 | 0 | ||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( -1 \) |
| \(89\) | \( +1 \) |
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 | 356.2.a.b | ✓ | 7 |
| 3.b | odd | 2 | 1 | 3204.2.a.j | 7 | ||
| 4.b | odd | 2 | 1 | 1424.2.a.n | 7 | ||
| 5.b | even | 2 | 1 | 8900.2.a.p | 7 | ||
| 8.b | even | 2 | 1 | 5696.2.a.bj | 7 | ||
| 8.d | odd | 2 | 1 | 5696.2.a.bk | 7 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 356.2.a.b | ✓ | 7 | 1.a | even | 1 | 1 | trivial |
| 1424.2.a.n | 7 | 4.b | odd | 2 | 1 | ||
| 3204.2.a.j | 7 | 3.b | odd | 2 | 1 | ||
| 5696.2.a.bj | 7 | 8.b | even | 2 | 1 | ||
| 5696.2.a.bk | 7 | 8.d | odd | 2 | 1 | ||
| 8900.2.a.p | 7 | 5.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{7} - T_{3}^{6} - 18T_{3}^{5} + 18T_{3}^{4} + 93T_{3}^{3} - 95T_{3}^{2} - 126T_{3} + 134 \)
acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(356))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{7} \)
|
| $3$ |
\( T^{7} - T^{6} + \cdots + 134 \)
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| $5$ |
\( T^{7} - 3 T^{6} + \cdots - 12 \)
|
| $7$ |
\( T^{7} - 36 T^{5} + \cdots + 872 \)
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| $11$ |
\( T^{7} - 8 T^{6} + \cdots + 6912 \)
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| $13$ |
\( T^{7} - 10 T^{6} + \cdots - 26048 \)
|
| $17$ |
\( T^{7} - 7 T^{6} + \cdots - 19536 \)
|
| $19$ |
\( T^{7} - 13 T^{6} + \cdots - 10826 \)
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| $23$ |
\( T^{7} + 3 T^{6} + \cdots - 55962 \)
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| $29$ |
\( T^{7} - 4 T^{6} + \cdots - 7104 \)
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| $31$ |
\( T^{7} + 9 T^{6} + \cdots - 79138 \)
|
| $37$ |
\( T^{7} - 2 T^{6} + \cdots - 5248 \)
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| $41$ |
\( T^{7} - 8 T^{6} + \cdots + 199488 \)
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| $43$ |
\( T^{7} - 13 T^{6} + \cdots - 433502 \)
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| $47$ |
\( T^{7} + 30 T^{6} + \cdots + 342144 \)
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| $53$ |
\( T^{7} + 7 T^{6} + \cdots + 432 \)
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| $59$ |
\( T^{7} - 260 T^{5} + \cdots + 31944 \)
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| $61$ |
\( T^{7} - 18 T^{6} + \cdots + 22208 \)
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| $67$ |
\( T^{7} + 2 T^{6} + \cdots - 41728 \)
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| $71$ |
\( T^{7} + 30 T^{6} + \cdots + 28416 \)
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| $73$ |
\( T^{7} - 19 T^{6} + \cdots + 11430388 \)
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| $79$ |
\( T^{7} + 18 T^{6} + \cdots + 256 \)
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| $83$ |
\( T^{7} + 20 T^{6} + \cdots + 600 \)
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| $89$ |
\( (T + 1)^{7} \)
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| $97$ |
\( T^{7} - 13 T^{6} + \cdots - 3488 \)
|
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