Newspace parameters
| Level: | \( N \) | \(=\) | \( 3555 = 3^{2} \cdot 5 \cdot 79 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 3555.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(28.3868179186\) |
| Analytic rank: | \(0\) |
| Dimension: | \(7\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{7} - \cdots)\) |
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| Defining polynomial: |
\( x^{7} - 16x^{5} - 2x^{4} + 80x^{3} + 21x^{2} - 119x - 53 \)
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| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 2 \) |
| Twist minimal: | no (minimal twist has level 1185) |
| 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
\( x^{7} - 16x^{5} - 2x^{4} + 80x^{3} + 21x^{2} - 119x - 53 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
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| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 5 \)
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| \(\beta_{3}\) | \(=\) |
\( \nu^{6} - 2\nu^{5} - 10\nu^{4} + 17\nu^{3} + 26\nu^{2} - 27\nu - 20 \)
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| \(\beta_{4}\) | \(=\) |
\( \nu^{6} - 2\nu^{5} - 10\nu^{4} + 18\nu^{3} + 26\nu^{2} - 33\nu - 21 \)
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| \(\beta_{5}\) | \(=\) |
\( -\nu^{6} + 3\nu^{5} + 10\nu^{4} - 27\nu^{3} - 28\nu^{2} + 49\nu + 30 \)
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| \(\beta_{6}\) | \(=\) |
\( 2\nu^{6} - 5\nu^{5} - 19\nu^{4} + 44\nu^{3} + 46\nu^{2} - 77\nu - 41 \)
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| \(\nu\) | \(=\) |
\( \beta_1 \)
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| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 5 \)
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| \(\nu^{3}\) | \(=\) |
\( \beta_{4} - \beta_{3} + 6\beta _1 + 1 \)
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| \(\nu^{4}\) | \(=\) |
\( \beta_{6} + \beta_{5} - \beta_{3} + 8\beta_{2} + \beta _1 + 31 \)
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| \(\nu^{5}\) | \(=\) |
\( \beta_{5} + 10\beta_{4} - 9\beta_{3} + 2\beta_{2} + 38\beta _1 + 10 \)
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| \(\nu^{6}\) | \(=\) |
\( 10\beta_{6} + 12\beta_{5} + 3\beta_{4} - 10\beta_{3} + 58\beta_{2} + 11\beta _1 + 203 \)
|
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 |
|
−2.54339 | 0 | 4.46884 | −1.00000 | 0 | −1.26030 | −6.27922 | 0 | 2.54339 | |||||||||||||||||||||||||||||||||||||||
| 1.2 | −2.52207 | 0 | 4.36086 | −1.00000 | 0 | 3.39797 | −5.95425 | 0 | 2.52207 | ||||||||||||||||||||||||||||||||||||||||
| 1.3 | −1.45084 | 0 | 0.104934 | −1.00000 | 0 | 1.14015 | 2.74944 | 0 | 1.45084 | ||||||||||||||||||||||||||||||||||||||||
| 1.4 | −0.475353 | 0 | −1.77404 | −1.00000 | 0 | 4.56700 | 1.79400 | 0 | 0.475353 | ||||||||||||||||||||||||||||||||||||||||
| 1.5 | 1.74276 | 0 | 1.03722 | −1.00000 | 0 | −4.51420 | −1.67789 | 0 | −1.74276 | ||||||||||||||||||||||||||||||||||||||||
| 1.6 | 2.50880 | 0 | 4.29410 | −1.00000 | 0 | 2.23766 | 5.75544 | 0 | −2.50880 | ||||||||||||||||||||||||||||||||||||||||
| 1.7 | 2.74009 | 0 | 5.50809 | −1.00000 | 0 | −0.568263 | 9.61249 | 0 | −2.74009 | ||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \( -1 \) |
| \(5\) | \( +1 \) |
| \(79\) | \( +1 \) |
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 | 3555.2.a.u | 7 | |
| 3.b | odd | 2 | 1 | 1185.2.a.n | ✓ | 7 | |
| 15.d | odd | 2 | 1 | 5925.2.a.u | 7 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1185.2.a.n | ✓ | 7 | 3.b | odd | 2 | 1 | |
| 3555.2.a.u | 7 | 1.a | even | 1 | 1 | trivial | |
| 5925.2.a.u | 7 | 15.d | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(3555))\):
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\( T_{2}^{7} - 16T_{2}^{5} - 2T_{2}^{4} + 80T_{2}^{3} + 21T_{2}^{2} - 119T_{2} - 53 \)
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\( T_{7}^{7} - 5T_{7}^{6} - 18T_{7}^{5} + 114T_{7}^{4} - 55T_{7}^{3} - 256T_{7}^{2} + 120T_{7} + 128 \)
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\( T_{11}^{7} + 4T_{11}^{6} - 33T_{11}^{5} - 160T_{11}^{4} + 119T_{11}^{3} + 1460T_{11}^{2} + 2302T_{11} + 1096 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{7} - 16 T^{5} + \cdots - 53 \)
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| $3$ |
\( T^{7} \)
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| $5$ |
\( (T + 1)^{7} \)
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| $7$ |
\( T^{7} - 5 T^{6} + \cdots + 128 \)
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| $11$ |
\( T^{7} + 4 T^{6} + \cdots + 1096 \)
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| $13$ |
\( T^{7} - 6 T^{6} + \cdots + 760 \)
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| $17$ |
\( T^{7} - 8 T^{6} + \cdots + 1408 \)
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| $19$ |
\( T^{7} - 10 T^{6} + \cdots - 3200 \)
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| $23$ |
\( T^{7} - 2 T^{6} + \cdots - 8608 \)
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| $29$ |
\( T^{7} - 15 T^{6} + \cdots - 346636 \)
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| $31$ |
\( T^{7} - 40 T^{6} + \cdots + 78080 \)
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| $37$ |
\( T^{7} - 6 T^{6} + \cdots - 153724 \)
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| $41$ |
\( T^{7} + 9 T^{6} + \cdots - 597148 \)
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| $43$ |
\( T^{7} - 21 T^{6} + \cdots - 704 \)
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| $47$ |
\( T^{7} + 5 T^{6} + \cdots - 6688 \)
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| $53$ |
\( T^{7} - 3 T^{6} + \cdots + 608 \)
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| $59$ |
\( T^{7} + T^{6} + \cdots + 135808 \)
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| $61$ |
\( T^{7} - 6 T^{6} + \cdots - 172840 \)
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| $67$ |
\( T^{7} - 13 T^{6} + \cdots + 160040 \)
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| $71$ |
\( T^{7} + 8 T^{6} + \cdots - 334016 \)
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| $73$ |
\( T^{7} + 24 T^{6} + \cdots - 2120 \)
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| $79$ |
\( (T + 1)^{7} \)
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| $83$ |
\( T^{7} - 5 T^{6} + \cdots - 2512 \)
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| $89$ |
\( T^{7} - 8 T^{6} + \cdots + 8262568 \)
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| $97$ |
\( T^{7} + 5 T^{6} + \cdots + 1847656 \)
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