Newspace parameters
Level: | \( N \) | \(=\) | \( 663 = 3 \cdot 13 \cdot 17 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 663.b (of order \(2\), degree \(1\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(5.29408165401\) |
Analytic rank: | \(0\) |
Dimension: | \(8\) |
Coefficient field: | \(\mathbb{Q}[x]/(x^{8} + \cdots)\) |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
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Defining polynomial: | \( x^{8} + 10x^{6} + 28x^{4} + 24x^{2} + 1 \) |
Coefficient ring: | \(\Z[a_1, a_2]\) |
Coefficient ring index: | \( 2^{3} \) |
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 \( x^{8} + 10x^{6} + 28x^{4} + 24x^{2} + 1 \) :
\(\beta_{1}\) | \(=\) | \( \nu \) |
\(\beta_{2}\) | \(=\) | \( \nu^{2} + 3 \) |
\(\beta_{3}\) | \(=\) | \( \nu^{3} + 4\nu \) |
\(\beta_{4}\) | \(=\) | \( \nu^{5} + 8\nu^{3} + 12\nu \) |
\(\beta_{5}\) | \(=\) | \( \nu^{6} + 8\nu^{4} + 12\nu^{2} \) |
\(\beta_{6}\) | \(=\) | \( \nu^{6} + 9\nu^{4} + 19\nu^{2} + 7 \) |
\(\beta_{7}\) | \(=\) | \( \nu^{7} + 9\nu^{5} + 20\nu^{3} + 12\nu \) |
\(\nu\) | \(=\) | \( \beta_1 \) |
\(\nu^{2}\) | \(=\) | \( \beta_{2} - 3 \) |
\(\nu^{3}\) | \(=\) | \( \beta_{3} - 4\beta_1 \) |
\(\nu^{4}\) | \(=\) | \( \beta_{6} - \beta_{5} - 7\beta_{2} + 14 \) |
\(\nu^{5}\) | \(=\) | \( \beta_{4} - 8\beta_{3} + 20\beta_1 \) |
\(\nu^{6}\) | \(=\) | \( -8\beta_{6} + 9\beta_{5} + 44\beta_{2} - 76 \) |
\(\nu^{7}\) | \(=\) | \( \beta_{7} - 9\beta_{4} + 52\beta_{3} - 112\beta_1 \) |
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/663\mathbb{Z}\right)^\times\).
\(n\) | \(443\) | \(547\) | \(613\) |
\(\chi(n)\) | \(1\) | \(1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
103.1 |
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− | 2.44735i | 1.00000 | −3.98951 | 2.54977i | − | 2.44735i | 0.511026i | 4.86902i | 1.00000 | 6.24017 | ||||||||||||||||||||||||||||||||||||||||
103.2 | − | 1.53025i | 1.00000 | −0.341661 | 3.44293i | − | 1.53025i | 2.56617i | − | 2.53767i | 1.00000 | 5.26854 | ||||||||||||||||||||||||||||||||||||||||
103.3 | − | 1.27474i | 1.00000 | 0.375049 | − | 0.816926i | − | 1.27474i | − | 1.30719i | − | 3.02756i | 1.00000 | −1.04137 | ||||||||||||||||||||||||||||||||||||||
103.4 | − | 0.209470i | 1.00000 | 1.95612 | − | 2.23105i | − | 0.209470i | 2.33343i | − | 0.828690i | 1.00000 | −0.467338 | |||||||||||||||||||||||||||||||||||||||
103.5 | 0.209470i | 1.00000 | 1.95612 | 2.23105i | 0.209470i | − | 2.33343i | 0.828690i | 1.00000 | −0.467338 | ||||||||||||||||||||||||||||||||||||||||||
103.6 | 1.27474i | 1.00000 | 0.375049 | 0.816926i | 1.27474i | 1.30719i | 3.02756i | 1.00000 | −1.04137 | |||||||||||||||||||||||||||||||||||||||||||
103.7 | 1.53025i | 1.00000 | −0.341661 | − | 3.44293i | 1.53025i | − | 2.56617i | 2.53767i | 1.00000 | 5.26854 | |||||||||||||||||||||||||||||||||||||||||
103.8 | 2.44735i | 1.00000 | −3.98951 | − | 2.54977i | 2.44735i | − | 0.511026i | − | 4.86902i | 1.00000 | 6.24017 | ||||||||||||||||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
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 | 663.2.b.e | ✓ | 8 |
3.b | odd | 2 | 1 | 1989.2.b.h | 8 | ||
13.b | even | 2 | 1 | inner | 663.2.b.e | ✓ | 8 |
13.d | odd | 4 | 2 | 8619.2.a.bg | 8 | ||
39.d | odd | 2 | 1 | 1989.2.b.h | 8 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
663.2.b.e | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
663.2.b.e | ✓ | 8 | 13.b | even | 2 | 1 | inner |
1989.2.b.h | 8 | 3.b | odd | 2 | 1 | ||
1989.2.b.h | 8 | 39.d | odd | 2 | 1 | ||
8619.2.a.bg | 8 | 13.d | odd | 4 | 2 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{8} + 10T_{2}^{6} + 28T_{2}^{4} + 24T_{2}^{2} + 1 \)
acting on \(S_{2}^{\mathrm{new}}(663, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{8} + 10 T^{6} + 28 T^{4} + 24 T^{2} + \cdots + 1 \)
$3$
\( (T - 1)^{8} \)
$5$
\( T^{8} + 24 T^{6} + 184 T^{4} + \cdots + 256 \)
$7$
\( T^{8} + 14 T^{6} + 60 T^{4} + 76 T^{2} + \cdots + 16 \)
$11$
\( T^{8} + 52 T^{6} + 824 T^{4} + \cdots + 7744 \)
$13$
\( T^{8} + 4 T^{7} + 4 T^{6} + \cdots + 28561 \)
$17$
\( (T - 1)^{8} \)
$19$
\( T^{8} + 68 T^{6} + 1120 T^{4} + \cdots + 1024 \)
$23$
\( (T^{4} - 12 T^{3} + 4 T^{2} + 356 T - 992)^{2} \)
$29$
\( (T^{4} + 4 T^{3} - 68 T^{2} - 468 T - 776)^{2} \)
$31$
\( T^{8} + 74 T^{6} + 1436 T^{4} + \cdots + 8464 \)
$37$
\( T^{8} + 266 T^{6} + 21940 T^{4} + \cdots + 4227136 \)
$41$
\( T^{8} + 232 T^{6} + 18168 T^{4} + \cdots + 4596736 \)
$43$
\( (T^{4} - 6 T^{3} - 128 T^{2} + 1244 T - 2768)^{2} \)
$47$
\( T^{8} + 118 T^{6} + 4288 T^{4} + \cdots + 204304 \)
$53$
\( (T^{4} + 14 T^{3} + 24 T^{2} - 144 T - 352)^{2} \)
$59$
\( T^{8} + 230 T^{6} + 9616 T^{4} + \cdots + 258064 \)
$61$
\( (T^{4} + 10 T^{3} - 80 T^{2} - 832 T - 256)^{2} \)
$67$
\( T^{8} + 196 T^{6} + 9504 T^{4} + \cdots + 541696 \)
$71$
\( T^{8} + 356 T^{6} + 36136 T^{4} + \cdots + 7441984 \)
$73$
\( T^{8} + 106 T^{6} + 3732 T^{4} + \cdots + 30976 \)
$79$
\( (T^{4} + 8 T^{3} - 128 T^{2} - 1152 T - 2048)^{2} \)
$83$
\( T^{8} + 522 T^{6} + 72832 T^{4} + \cdots + 234256 \)
$89$
\( T^{8} + 434 T^{6} + \cdots + 69488896 \)
$97$
\( T^{8} + 294 T^{6} + 7396 T^{4} + \cdots + 123904 \)
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