Newspace parameters
Level: | \( N \) | \(=\) | \( 363 = 3 \cdot 11^{2} \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 363.d (of order \(2\), degree \(1\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(2.89856959337\) |
Analytic rank: | \(0\) |
Dimension: | \(8\) |
Coefficient field: | 8.0.3588489216.5 |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
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Defining polynomial: | \( x^{8} - 4x^{7} + 10x^{6} - 8x^{5} + 8x^{4} + 4x^{3} + 16x^{2} + 32x + 22 \) |
Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
Coefficient ring index: | \( 2 \) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) 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 \( x^{8} - 4x^{7} + 10x^{6} - 8x^{5} + 8x^{4} + 4x^{3} + 16x^{2} + 32x + 22 \) :
\(\beta_{1}\) | \(=\) | \( ( \nu^{5} - 4\nu^{4} + 7\nu^{3} - \nu^{2} - 2\nu - 1 ) / 9 \) |
\(\beta_{2}\) | \(=\) | \( ( -\nu^{5} + \nu^{4} + 2\nu^{3} - 11\nu^{2} + 2\nu - 2 ) / 9 \) |
\(\beta_{3}\) | \(=\) | \( ( \nu^{7} - 3\nu^{6} + 7\nu^{5} - \nu^{4} + 7\nu^{3} + 11\nu^{2} + 59 ) / 27 \) |
\(\beta_{4}\) | \(=\) | \( ( \nu^{7} - 3\nu^{6} + 10\nu^{5} - 13\nu^{4} + 28\nu^{3} - 19\nu^{2} + 48\nu + 29 ) / 27 \) |
\(\beta_{5}\) | \(=\) | \( ( \nu^{6} - 4\nu^{5} + 10\nu^{4} - 10\nu^{3} + 10\nu^{2} + 8\nu + 3 ) / 9 \) |
\(\beta_{6}\) | \(=\) | \( ( -\nu^{7} + 6\nu^{6} - 16\nu^{5} + 19\nu^{4} - 16\nu^{3} + 16\nu^{2} - 36\nu - 26 ) / 27 \) |
\(\beta_{7}\) | \(=\) | \( ( -\nu^{7} + 4\nu^{6} - 10\nu^{5} + 10\nu^{4} - 10\nu^{3} + \nu^{2} - 12\nu - 18 ) / 9 \) |
\(\nu\) | \(=\) | \( ( -\beta_{6} + \beta_{5} - \beta_{3} + \beta _1 + 1 ) / 2 \) |
\(\nu^{2}\) | \(=\) | \( -\beta_{6} + \beta_{5} - \beta_{4} + 2\beta_1 \) |
\(\nu^{3}\) | \(=\) | \( \beta_{7} - 2\beta_{6} + \beta_{5} - \beta_{4} + 2\beta_{3} + \beta_{2} + 3\beta _1 - 3 \) |
\(\nu^{4}\) | \(=\) | \( 3\beta_{7} - 2\beta_{6} - \beta_{5} + \beta_{4} + 6\beta_{3} - 2\beta _1 - 10 \) |
\(\nu^{5}\) | \(=\) | \( 5\beta_{7} + 4\beta_{6} - 9\beta_{5} + 10\beta_{4} + 9\beta_{3} - 7\beta_{2} - 17\beta _1 - 17 \) |
\(\nu^{6}\) | \(=\) | \( 30\beta_{6} - 21\beta_{5} + 30\beta_{4} - 18\beta_{2} - 42\beta _1 - 5 \) |
\(\nu^{7}\) | \(=\) | \( -39\beta_{7} + 85\beta_{6} - 19\beta_{5} + 39\beta_{4} - 44\beta_{3} - 12\beta_{2} - 52\beta _1 + 56 \) |
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/363\mathbb{Z}\right)^\times\).
\(n\) | \(122\) | \(244\) |
\(\chi(n)\) | \(-1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
362.1 |
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−2.39417 | 0.366025 | − | 1.69293i | 3.73205 | − | 1.23931i | −0.876327 | + | 4.05317i | − | 2.82843i | −4.14682 | −2.73205 | − | 1.23931i | 2.96713i | ||||||||||||||||||||||||||||||||||
362.2 | −2.39417 | 0.366025 | + | 1.69293i | 3.73205 | 1.23931i | −0.876327 | − | 4.05317i | 2.82843i | −4.14682 | −2.73205 | + | 1.23931i | − | 2.96713i | ||||||||||||||||||||||||||||||||||||
362.3 | −1.50597 | −1.36603 | − | 1.06488i | 0.267949 | 2.90931i | 2.05719 | + | 1.60368i | − | 2.82843i | 2.60842 | 0.732051 | + | 2.90931i | − | 4.38134i | |||||||||||||||||||||||||||||||||||
362.4 | −1.50597 | −1.36603 | + | 1.06488i | 0.267949 | − | 2.90931i | 2.05719 | − | 1.60368i | 2.82843i | 2.60842 | 0.732051 | − | 2.90931i | 4.38134i | ||||||||||||||||||||||||||||||||||||
362.5 | 1.50597 | −1.36603 | − | 1.06488i | 0.267949 | 2.90931i | −2.05719 | − | 1.60368i | 2.82843i | −2.60842 | 0.732051 | + | 2.90931i | 4.38134i | |||||||||||||||||||||||||||||||||||||
362.6 | 1.50597 | −1.36603 | + | 1.06488i | 0.267949 | − | 2.90931i | −2.05719 | + | 1.60368i | − | 2.82843i | −2.60842 | 0.732051 | − | 2.90931i | − | 4.38134i | ||||||||||||||||||||||||||||||||||
362.7 | 2.39417 | 0.366025 | − | 1.69293i | 3.73205 | − | 1.23931i | 0.876327 | − | 4.05317i | 2.82843i | 4.14682 | −2.73205 | − | 1.23931i | − | 2.96713i | |||||||||||||||||||||||||||||||||||
362.8 | 2.39417 | 0.366025 | + | 1.69293i | 3.73205 | 1.23931i | 0.876327 | + | 4.05317i | − | 2.82843i | 4.14682 | −2.73205 | + | 1.23931i | 2.96713i | ||||||||||||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
3.b | odd | 2 | 1 | inner |
11.b | odd | 2 | 1 | inner |
33.d | even | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 363.2.d.e | ✓ | 8 |
3.b | odd | 2 | 1 | inner | 363.2.d.e | ✓ | 8 |
11.b | odd | 2 | 1 | inner | 363.2.d.e | ✓ | 8 |
11.c | even | 5 | 4 | 363.2.f.i | 32 | ||
11.d | odd | 10 | 4 | 363.2.f.i | 32 | ||
33.d | even | 2 | 1 | inner | 363.2.d.e | ✓ | 8 |
33.f | even | 10 | 4 | 363.2.f.i | 32 | ||
33.h | odd | 10 | 4 | 363.2.f.i | 32 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
363.2.d.e | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
363.2.d.e | ✓ | 8 | 3.b | odd | 2 | 1 | inner |
363.2.d.e | ✓ | 8 | 11.b | odd | 2 | 1 | inner |
363.2.d.e | ✓ | 8 | 33.d | even | 2 | 1 | inner |
363.2.f.i | 32 | 11.c | even | 5 | 4 | ||
363.2.f.i | 32 | 11.d | odd | 10 | 4 | ||
363.2.f.i | 32 | 33.f | even | 10 | 4 | ||
363.2.f.i | 32 | 33.h | odd | 10 | 4 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{4} - 8T_{2}^{2} + 13 \)
acting on \(S_{2}^{\mathrm{new}}(363, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( (T^{4} - 8 T^{2} + 13)^{2} \)
$3$
\( (T^{4} + 2 T^{3} + 4 T^{2} + 6 T + 9)^{2} \)
$5$
\( (T^{4} + 10 T^{2} + 13)^{2} \)
$7$
\( (T^{2} + 8)^{4} \)
$11$
\( T^{8} \)
$13$
\( (T^{4} + 12 T^{2} + 9)^{2} \)
$17$
\( (T^{4} - 24 T^{2} + 117)^{2} \)
$19$
\( (T^{2} + 6)^{4} \)
$23$
\( T^{8} \)
$29$
\( (T^{4} - 104 T^{2} + 2197)^{2} \)
$31$
\( (T^{2} + 10 T - 2)^{4} \)
$37$
\( (T^{2} + 14 T + 37)^{4} \)
$41$
\( (T^{4} - 72 T^{2} + 1053)^{2} \)
$43$
\( (T^{2} + 18)^{4} \)
$47$
\( (T^{4} + 16 T^{2} + 52)^{2} \)
$53$
\( (T^{4} + 94 T^{2} + 2197)^{2} \)
$59$
\( (T^{4} + 160 T^{2} + 52)^{2} \)
$61$
\( (T^{4} + 124 T^{2} + 2116)^{2} \)
$67$
\( (T - 2)^{8} \)
$71$
\( (T^{4} + 160 T^{2} + 6292)^{2} \)
$73$
\( (T^{4} + 76 T^{2} + 676)^{2} \)
$79$
\( (T^{2} + 54)^{4} \)
$83$
\( (T^{4} - 260 T^{2} + 8788)^{2} \)
$89$
\( (T^{4} + 246 T^{2} + 14157)^{2} \)
$97$
\( (T^{2} - 8 T + 13)^{4} \)
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