Newspace parameters
Level: | \( N \) | \(=\) | \( 117 = 3^{2} \cdot 13 \) |
Weight: | \( k \) | \(=\) | \( 4 \) |
Character orbit: | \([\chi]\) | \(=\) | 117.g (of order \(3\), degree \(2\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(6.90322347067\) |
Analytic rank: | \(0\) |
Dimension: | \(2\) |
Coefficient field: | \(\Q(\sqrt{-3}) \) |
Defining polynomial: |
\( x^{2} - x + 1 \)
|
Coefficient ring: | \(\Z[a_1, \ldots, a_{4}]\) |
Coefficient ring index: | \( 1 \) |
Twist minimal: | no (minimal twist has level 39) |
Sato-Tate group: | $\mathrm{SU}(2)[C_{3}]$ |
$q$-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 |
|
1.50000 | + | 2.59808i | 0 | −0.500000 | + | 0.866025i | 9.00000 | 0 | −1.00000 | + | 1.73205i | 21.0000 | 0 | 13.5000 | + | 23.3827i | ||||||||||||||||
100.1 | 1.50000 | − | 2.59808i | 0 | −0.500000 | − | 0.866025i | 9.00000 | 0 | −1.00000 | − | 1.73205i | 21.0000 | 0 | 13.5000 | − | 23.3827i | |||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
13.c | even | 3 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 117.4.g.b | 2 | |
3.b | odd | 2 | 1 | 39.4.e.a | ✓ | 2 | |
12.b | even | 2 | 1 | 624.4.q.b | 2 | ||
13.c | even | 3 | 1 | inner | 117.4.g.b | 2 | |
13.c | even | 3 | 1 | 1521.4.a.c | 1 | ||
13.e | even | 6 | 1 | 1521.4.a.j | 1 | ||
39.h | odd | 6 | 1 | 507.4.a.a | 1 | ||
39.i | odd | 6 | 1 | 39.4.e.a | ✓ | 2 | |
39.i | odd | 6 | 1 | 507.4.a.e | 1 | ||
39.k | even | 12 | 2 | 507.4.b.c | 2 | ||
156.p | even | 6 | 1 | 624.4.q.b | 2 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
39.4.e.a | ✓ | 2 | 3.b | odd | 2 | 1 | |
39.4.e.a | ✓ | 2 | 39.i | odd | 6 | 1 | |
117.4.g.b | 2 | 1.a | even | 1 | 1 | trivial | |
117.4.g.b | 2 | 13.c | even | 3 | 1 | inner | |
507.4.a.a | 1 | 39.h | odd | 6 | 1 | ||
507.4.a.e | 1 | 39.i | odd | 6 | 1 | ||
507.4.b.c | 2 | 39.k | even | 12 | 2 | ||
624.4.q.b | 2 | 12.b | even | 2 | 1 | ||
624.4.q.b | 2 | 156.p | even | 6 | 1 | ||
1521.4.a.c | 1 | 13.c | even | 3 | 1 | ||
1521.4.a.j | 1 | 13.e | even | 6 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{2} - 3T_{2} + 9 \)
acting on \(S_{4}^{\mathrm{new}}(117, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{2} - 3T + 9 \)
$3$
\( T^{2} \)
$5$
\( (T - 9)^{2} \)
$7$
\( T^{2} + 2T + 4 \)
$11$
\( T^{2} - 30T + 900 \)
$13$
\( T^{2} - 65T + 2197 \)
$17$
\( T^{2} + 111T + 12321 \)
$19$
\( T^{2} - 46T + 2116 \)
$23$
\( T^{2} + 6T + 36 \)
$29$
\( T^{2} + 105T + 11025 \)
$31$
\( (T + 100)^{2} \)
$37$
\( T^{2} + 17T + 289 \)
$41$
\( T^{2} + 231T + 53361 \)
$43$
\( T^{2} - 514T + 264196 \)
$47$
\( (T - 162)^{2} \)
$53$
\( (T + 639)^{2} \)
$59$
\( T^{2} - 600T + 360000 \)
$61$
\( T^{2} + 233T + 54289 \)
$67$
\( T^{2} + 926T + 857476 \)
$71$
\( T^{2} + 930T + 864900 \)
$73$
\( (T + 253)^{2} \)
$79$
\( (T + 1324)^{2} \)
$83$
\( (T + 810)^{2} \)
$89$
\( T^{2} - 498T + 248004 \)
$97$
\( T^{2} + 1358 T + 1844164 \)
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