Newspace parameters
Level: | \( N \) | \(=\) | \( 117 = 3^{2} \cdot 13 \) |
Weight: | \( k \) | \(=\) | \( 6 \) |
Character orbit: | \([\chi]\) | \(=\) | 117.g (of order \(3\), degree \(2\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(18.7649069181\) |
Analytic rank: | \(0\) |
Dimension: | \(2\) |
Coefficient field: | \(\Q(\sqrt{-3}) \) |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
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Defining polynomial: | \( x^{2} - x + 1 \) |
Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
Coefficient ring index: | \( 1 \) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{U}(1)[D_{3}]$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/117\mathbb{Z}\right)^\times\).
\(n\) | \(28\) | \(92\) |
\(\chi(n)\) | \(-\zeta_{6}\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
55.1 |
|
0 | 0 | 16.0000 | − | 27.7128i | 0 | 0 | 105.500 | − | 182.731i | 0 | 0 | 0 | ||||||||||||||||||||
100.1 | 0 | 0 | 16.0000 | + | 27.7128i | 0 | 0 | 105.500 | + | 182.731i | 0 | 0 | 0 | |||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
3.b | odd | 2 | 1 | CM by \(\Q(\sqrt{-3}) \) |
13.c | even | 3 | 1 | inner |
39.i | odd | 6 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 117.6.g.a | ✓ | 2 |
3.b | odd | 2 | 1 | CM | 117.6.g.a | ✓ | 2 |
13.c | even | 3 | 1 | inner | 117.6.g.a | ✓ | 2 |
39.i | odd | 6 | 1 | inner | 117.6.g.a | ✓ | 2 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
117.6.g.a | ✓ | 2 | 1.a | even | 1 | 1 | trivial |
117.6.g.a | ✓ | 2 | 3.b | odd | 2 | 1 | CM |
117.6.g.a | ✓ | 2 | 13.c | even | 3 | 1 | inner |
117.6.g.a | ✓ | 2 | 39.i | odd | 6 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2} \)
acting on \(S_{6}^{\mathrm{new}}(117, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{2} \)
$3$
\( T^{2} \)
$5$
\( T^{2} \)
$7$
\( T^{2} - 211T + 44521 \)
$11$
\( T^{2} \)
$13$
\( T^{2} + 775T + 371293 \)
$17$
\( T^{2} \)
$19$
\( T^{2} - 1432 T + 2050624 \)
$23$
\( T^{2} \)
$29$
\( T^{2} \)
$31$
\( (T - 2723)^{2} \)
$37$
\( T^{2} + 16550 T + 273902500 \)
$41$
\( T^{2} \)
$43$
\( T^{2} - 19123 T + 365689129 \)
$47$
\( T^{2} \)
$53$
\( T^{2} \)
$59$
\( T^{2} \)
$61$
\( T^{2} + 56927 T + 3240683329 \)
$67$
\( T^{2} - 37939 T + 1439367721 \)
$71$
\( T^{2} \)
$73$
\( (T - 79577)^{2} \)
$79$
\( (T - 90857)^{2} \)
$83$
\( T^{2} \)
$89$
\( T^{2} \)
$97$
\( T^{2} + 177725 T + 31586175625 \)
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