Newspace parameters
Level: | \( N \) | \(=\) | \( 1183 = 7 \cdot 13^{2} \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 1183.c (of order \(2\), degree \(1\), not minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(9.44630255912\) |
Analytic rank: | \(0\) |
Dimension: | \(6\) |
Coefficient field: | 6.0.399424.1 |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
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Defining polynomial: |
\( x^{6} - 2x^{5} + 3x^{4} - 6x^{3} + 6x^{2} - 8x + 8 \)
|
Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
Coefficient ring index: | \( 2^{2} \) |
Twist minimal: | no (minimal twist has level 91) |
Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{5}\) 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
\( x^{6} - 2x^{5} + 3x^{4} - 6x^{3} + 6x^{2} - 8x + 8 \)
:
\(\beta_{1}\) | \(=\) |
\( ( -\nu^{4} + 2\nu^{3} - \nu^{2} + 2\nu - 2 ) / 2 \)
|
\(\beta_{2}\) | \(=\) |
\( ( -\nu^{5} - 3\nu^{3} + 4\nu^{2} - 2\nu + 8 ) / 4 \)
|
\(\beta_{3}\) | \(=\) |
\( ( -\nu^{5} + 2\nu^{4} - 3\nu^{3} + 6\nu^{2} - 2\nu + 4 ) / 4 \)
|
\(\beta_{4}\) | \(=\) |
\( ( \nu^{5} - 2\nu^{4} + 3\nu^{3} - 6\nu^{2} + 10\nu - 8 ) / 4 \)
|
\(\beta_{5}\) | \(=\) |
\( ( 3\nu^{5} - 2\nu^{4} + 5\nu^{3} - 6\nu^{2} + 2\nu - 12 ) / 4 \)
|
\(\nu\) | \(=\) |
\( ( \beta_{4} + \beta_{3} + 1 ) / 2 \)
|
\(\nu^{2}\) | \(=\) |
\( ( \beta_{5} + 2\beta_{3} + \beta_{2} + \beta_1 ) / 2 \)
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\(\nu^{3}\) | \(=\) |
\( ( -\beta_{4} + \beta_{3} - 2\beta_{2} + 2\beta _1 + 3 ) / 2 \)
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\(\nu^{4}\) | \(=\) |
\( ( -\beta_{5} + 2\beta_{3} - 5\beta_{2} - \beta _1 + 4 ) / 2 \)
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\(\nu^{5}\) | \(=\) |
\( ( 4\beta_{5} + \beta_{4} + 3\beta_{3} + 2\beta_{2} - 2\beta _1 + 5 ) / 2 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1183\mathbb{Z}\right)^\times\).
\(n\) | \(339\) | \(1016\) |
\(\chi(n)\) | \(1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
337.1 |
|
− | 2.34292i | −1.14637 | −3.48929 | 1.34292i | 2.68585i | − | 1.00000i | 3.48929i | −1.68585 | 3.14637 | ||||||||||||||||||||||||||||||||||
337.2 | − | 1.81361i | −3.10278 | −1.28917 | 2.81361i | 5.62721i | 1.00000i | − | 1.28917i | 6.62721 | 5.10278 | |||||||||||||||||||||||||||||||||||
337.3 | − | 0.470683i | 2.24914 | 1.77846 | − | 0.529317i | − | 1.05863i | − | 1.00000i | − | 1.77846i | 2.05863 | −0.249141 | ||||||||||||||||||||||||||||||||
337.4 | 0.470683i | 2.24914 | 1.77846 | 0.529317i | 1.05863i | 1.00000i | 1.77846i | 2.05863 | −0.249141 | |||||||||||||||||||||||||||||||||||||
337.5 | 1.81361i | −3.10278 | −1.28917 | − | 2.81361i | − | 5.62721i | − | 1.00000i | 1.28917i | 6.62721 | 5.10278 | ||||||||||||||||||||||||||||||||||
337.6 | 2.34292i | −1.14637 | −3.48929 | − | 1.34292i | − | 2.68585i | 1.00000i | − | 3.48929i | −1.68585 | 3.14637 | ||||||||||||||||||||||||||||||||||
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 | 1183.2.c.f | 6 | |
13.b | even | 2 | 1 | inner | 1183.2.c.f | 6 | |
13.d | odd | 4 | 1 | 91.2.a.d | ✓ | 3 | |
13.d | odd | 4 | 1 | 1183.2.a.i | 3 | ||
39.f | even | 4 | 1 | 819.2.a.i | 3 | ||
52.f | even | 4 | 1 | 1456.2.a.t | 3 | ||
65.g | odd | 4 | 1 | 2275.2.a.m | 3 | ||
91.i | even | 4 | 1 | 637.2.a.j | 3 | ||
91.i | even | 4 | 1 | 8281.2.a.bg | 3 | ||
91.z | odd | 12 | 2 | 637.2.e.j | 6 | ||
91.bb | even | 12 | 2 | 637.2.e.i | 6 | ||
104.j | odd | 4 | 1 | 5824.2.a.by | 3 | ||
104.m | even | 4 | 1 | 5824.2.a.bs | 3 | ||
273.o | odd | 4 | 1 | 5733.2.a.x | 3 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
91.2.a.d | ✓ | 3 | 13.d | odd | 4 | 1 | |
637.2.a.j | 3 | 91.i | even | 4 | 1 | ||
637.2.e.i | 6 | 91.bb | even | 12 | 2 | ||
637.2.e.j | 6 | 91.z | odd | 12 | 2 | ||
819.2.a.i | 3 | 39.f | even | 4 | 1 | ||
1183.2.a.i | 3 | 13.d | odd | 4 | 1 | ||
1183.2.c.f | 6 | 1.a | even | 1 | 1 | trivial | |
1183.2.c.f | 6 | 13.b | even | 2 | 1 | inner | |
1456.2.a.t | 3 | 52.f | even | 4 | 1 | ||
2275.2.a.m | 3 | 65.g | odd | 4 | 1 | ||
5733.2.a.x | 3 | 273.o | odd | 4 | 1 | ||
5824.2.a.bs | 3 | 104.m | even | 4 | 1 | ||
5824.2.a.by | 3 | 104.j | odd | 4 | 1 | ||
8281.2.a.bg | 3 | 91.i | even | 4 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{6} + 9T_{2}^{4} + 20T_{2}^{2} + 4 \)
acting on \(S_{2}^{\mathrm{new}}(1183, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{6} + 9 T^{4} + 20 T^{2} + 4 \)
$3$
\( (T^{3} + 2 T^{2} - 6 T - 8)^{2} \)
$5$
\( T^{6} + 10 T^{4} + 17 T^{2} + 4 \)
$7$
\( (T^{2} + 1)^{3} \)
$11$
\( T^{6} + 16 T^{4} + 68 T^{2} + 64 \)
$13$
\( T^{6} \)
$17$
\( (T^{3} + 4 T^{2} - 10 T + 4)^{2} \)
$19$
\( T^{6} + 14 T^{4} + 33 T^{2} + 16 \)
$23$
\( (T^{3} + 10 T^{2} + T - 136)^{2} \)
$29$
\( (T^{3} - 24 T^{2} + 185 T - 454)^{2} \)
$31$
\( T^{6} + 54 T^{4} + 233 T^{2} + \cdots + 256 \)
$37$
\( T^{6} + 116 T^{4} + 3364 T^{2} + \cdots + 15376 \)
$41$
\( T^{6} + 60 T^{4} + 752 T^{2} + \cdots + 64 \)
$43$
\( (T^{3} + 10 T^{2} - 71 T - 628)^{2} \)
$47$
\( T^{6} + 222 T^{4} + 14945 T^{2} + \cdots + 295936 \)
$53$
\( (T^{3} - 8 T^{2} - 35 T - 22)^{2} \)
$59$
\( T^{6} + 328 T^{4} + 29840 T^{2} + \cdots + 473344 \)
$61$
\( (T + 2)^{6} \)
$67$
\( T^{6} + 392 T^{4} + 38800 T^{2} + \cdots + 952576 \)
$71$
\( T^{6} + 80 T^{4} + 292 T^{2} + \cdots + 256 \)
$73$
\( T^{6} + 298 T^{4} + 15281 T^{2} + \cdots + 75076 \)
$79$
\( (T^{3} + 14 T^{2} + 5 T - 16)^{2} \)
$83$
\( T^{6} + 686 T^{4} + \cdots + 10679824 \)
$89$
\( T^{6} + 194 T^{4} + 10713 T^{2} + \cdots + 178084 \)
$97$
\( T^{6} + 42 T^{4} + 401 T^{2} + \cdots + 484 \)
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