Newspace parameters
Level: | \( N \) | \(=\) | \( 117 = 3^{2} \cdot 13 \) |
Weight: | \( k \) | \(=\) | \( 4 \) |
Character orbit: | \([\chi]\) | \(=\) | 117.f (of order \(3\), degree \(2\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(6.90322347067\) |
Analytic rank: | \(0\) |
Dimension: | \(2\) |
Coefficient field: | \(\Q(\sqrt{-3}) \) |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
Defining polynomial: | \( x^{2} - x + 1 \) |
Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
Coefficient ring index: | \( 1 \) |
Twist minimal: | yes |
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 \(\left(\mathbb{Z}/117\mathbb{Z}\right)^\times\).
\(n\) | \(28\) | \(92\) |
\(\chi(n)\) | \(-\zeta_{6}\) | \(-\zeta_{6}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
61.1 |
|
2.50000 | + | 4.33013i | −4.50000 | + | 2.59808i | −8.50000 | + | 14.7224i | 9.50000 | + | 16.4545i | −22.5000 | − | 12.9904i | 11.0000 | −45.0000 | 13.5000 | − | 23.3827i | −47.5000 | + | 82.2724i | ||||||||||
94.1 | 2.50000 | − | 4.33013i | −4.50000 | − | 2.59808i | −8.50000 | − | 14.7224i | 9.50000 | − | 16.4545i | −22.5000 | + | 12.9904i | 11.0000 | −45.0000 | 13.5000 | + | 23.3827i | −47.5000 | − | 82.2724i | |||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
117.f | 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.f.a | ✓ | 2 |
3.b | odd | 2 | 1 | 351.4.f.a | 2 | ||
9.c | even | 3 | 1 | 117.4.h.a | yes | 2 | |
9.d | odd | 6 | 1 | 351.4.h.a | 2 | ||
13.c | even | 3 | 1 | 117.4.h.a | yes | 2 | |
39.i | odd | 6 | 1 | 351.4.h.a | 2 | ||
117.f | even | 3 | 1 | inner | 117.4.f.a | ✓ | 2 |
117.u | odd | 6 | 1 | 351.4.f.a | 2 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
117.4.f.a | ✓ | 2 | 1.a | even | 1 | 1 | trivial |
117.4.f.a | ✓ | 2 | 117.f | even | 3 | 1 | inner |
117.4.h.a | yes | 2 | 9.c | even | 3 | 1 | |
117.4.h.a | yes | 2 | 13.c | even | 3 | 1 | |
351.4.f.a | 2 | 3.b | odd | 2 | 1 | ||
351.4.f.a | 2 | 117.u | odd | 6 | 1 | ||
351.4.h.a | 2 | 9.d | odd | 6 | 1 | ||
351.4.h.a | 2 | 39.i | odd | 6 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{2} - 5T_{2} + 25 \)
acting on \(S_{4}^{\mathrm{new}}(117, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{2} - 5T + 25 \)
$3$
\( T^{2} + 9T + 27 \)
$5$
\( T^{2} - 19T + 361 \)
$7$
\( (T - 11)^{2} \)
$11$
\( T^{2} + 16T + 256 \)
$13$
\( T^{2} - 89T + 2197 \)
$17$
\( T^{2} + 131T + 17161 \)
$19$
\( T^{2} - 93T + 8649 \)
$23$
\( (T - 69)^{2} \)
$29$
\( T^{2} + 86T + 7396 \)
$31$
\( T^{2} - 173T + 29929 \)
$37$
\( T^{2} - 323T + 104329 \)
$41$
\( (T + 9)^{2} \)
$43$
\( (T - 69)^{2} \)
$47$
\( T^{2} + 369T + 136161 \)
$53$
\( (T + 306)^{2} \)
$59$
\( T^{2} - 420T + 176400 \)
$61$
\( (T + 383)^{2} \)
$67$
\( (T - 139)^{2} \)
$71$
\( T^{2} + 237T + 56169 \)
$73$
\( (T - 518)^{2} \)
$79$
\( T^{2} + 41T + 1681 \)
$83$
\( T^{2} - 905T + 819025 \)
$89$
\( T^{2} - 929T + 863041 \)
$97$
\( (T + 5)^{2} \)
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