Newspace parameters
Level: | \( N \) | \(=\) | \( 117 = 3^{2} \cdot 13 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 117.r (of order \(6\), degree \(2\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(0.934249703649\) |
Analytic rank: | \(1\) |
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: | \(\Z[a_1, a_2]\) |
Coefficient ring index: | \( 1 \) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{6}]$ |
$q$-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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
43.1 |
|
−1.50000 | + | 0.866025i | −1.50000 | + | 0.866025i | 0.500000 | − | 0.866025i | −1.50000 | + | 0.866025i | 1.50000 | − | 2.59808i | − | 1.73205i | − | 1.73205i | 1.50000 | − | 2.59808i | 1.50000 | − | 2.59808i | ||||||||
49.1 | −1.50000 | − | 0.866025i | −1.50000 | − | 0.866025i | 0.500000 | + | 0.866025i | −1.50000 | − | 0.866025i | 1.50000 | + | 2.59808i | 1.73205i | 1.73205i | 1.50000 | + | 2.59808i | 1.50000 | + | 2.59808i | |||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
117.r | even | 6 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 117.2.r.a | yes | 2 |
3.b | odd | 2 | 1 | 351.2.r.a | 2 | ||
9.c | even | 3 | 1 | 117.2.l.a | ✓ | 2 | |
9.d | odd | 6 | 1 | 351.2.l.a | 2 | ||
13.e | even | 6 | 1 | 117.2.l.a | ✓ | 2 | |
39.h | odd | 6 | 1 | 351.2.l.a | 2 | ||
117.m | odd | 6 | 1 | 351.2.r.a | 2 | ||
117.r | even | 6 | 1 | inner | 117.2.r.a | yes | 2 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
117.2.l.a | ✓ | 2 | 9.c | even | 3 | 1 | |
117.2.l.a | ✓ | 2 | 13.e | even | 6 | 1 | |
117.2.r.a | yes | 2 | 1.a | even | 1 | 1 | trivial |
117.2.r.a | yes | 2 | 117.r | even | 6 | 1 | inner |
351.2.l.a | 2 | 9.d | odd | 6 | 1 | ||
351.2.l.a | 2 | 39.h | odd | 6 | 1 | ||
351.2.r.a | 2 | 3.b | odd | 2 | 1 | ||
351.2.r.a | 2 | 117.m | odd | 6 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{2} + 3T_{2} + 3 \)
acting on \(S_{2}^{\mathrm{new}}(117, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{2} + 3T + 3 \)
$3$
\( T^{2} + 3T + 3 \)
$5$
\( T^{2} + 3T + 3 \)
$7$
\( T^{2} + 3 \)
$11$
\( T^{2} + 6T + 12 \)
$13$
\( T^{2} + 5T + 13 \)
$17$
\( T^{2} + 3T + 9 \)
$19$
\( T^{2} + 3T + 3 \)
$23$
\( (T + 3)^{2} \)
$29$
\( T^{2} - 6T + 36 \)
$31$
\( T^{2} + 15T + 75 \)
$37$
\( T^{2} + 9T + 27 \)
$41$
\( T^{2} + 147 \)
$43$
\( (T + 1)^{2} \)
$47$
\( T^{2} + 9T + 27 \)
$53$
\( (T - 6)^{2} \)
$59$
\( T^{2} - 6T + 12 \)
$61$
\( (T + 5)^{2} \)
$67$
\( T^{2} + 147 \)
$71$
\( T^{2} + 15T + 75 \)
$73$
\( T^{2} + 48 \)
$79$
\( T^{2} - 11T + 121 \)
$83$
\( T^{2} + 9T + 27 \)
$89$
\( T^{2} - 27T + 243 \)
$97$
\( T^{2} + 243 \)
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