Newspace parameters
| Level: | \( N \) | \(=\) | \( 117 = 3^{2} \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 117.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(6.90322347067\) |
| Analytic rank: | \(1\) |
| Dimension: | \(2\) |
| Coefficient field: | \(\Q(\sqrt{14}) \) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{2} - 14 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 39) |
| Fricke sign: | \(-1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{14}\). We also show the integral \(q\)-expansion of the trace form.
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1.1 |
|
−4.74166 | 0 | 14.4833 | −4.51669 | 0 | −7.48331 | −30.7417 | 0 | 21.4166 | ||||||||||||||||||||||||
| 1.2 | 2.74166 | 0 | −0.483315 | −19.4833 | 0 | 7.48331 | −23.2583 | 0 | −53.4166 | |||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(-1\) |
| \(13\) | \(1\) |
Inner twists
This newform does not admit any (nontrivial) inner twists.
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 117.4.a.c | 2 | |
| 3.b | odd | 2 | 1 | 39.4.a.b | ✓ | 2 | |
| 4.b | odd | 2 | 1 | 1872.4.a.t | 2 | ||
| 12.b | even | 2 | 1 | 624.4.a.r | 2 | ||
| 13.b | even | 2 | 1 | 1521.4.a.s | 2 | ||
| 15.d | odd | 2 | 1 | 975.4.a.j | 2 | ||
| 21.c | even | 2 | 1 | 1911.4.a.h | 2 | ||
| 24.f | even | 2 | 1 | 2496.4.a.s | 2 | ||
| 24.h | odd | 2 | 1 | 2496.4.a.bc | 2 | ||
| 39.d | odd | 2 | 1 | 507.4.a.f | 2 | ||
| 39.f | even | 4 | 2 | 507.4.b.f | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 39.4.a.b | ✓ | 2 | 3.b | odd | 2 | 1 | |
| 117.4.a.c | 2 | 1.a | even | 1 | 1 | trivial | |
| 507.4.a.f | 2 | 39.d | odd | 2 | 1 | ||
| 507.4.b.f | 4 | 39.f | even | 4 | 2 | ||
| 624.4.a.r | 2 | 12.b | even | 2 | 1 | ||
| 975.4.a.j | 2 | 15.d | odd | 2 | 1 | ||
| 1521.4.a.s | 2 | 13.b | even | 2 | 1 | ||
| 1872.4.a.t | 2 | 4.b | odd | 2 | 1 | ||
| 1911.4.a.h | 2 | 21.c | even | 2 | 1 | ||
| 2496.4.a.s | 2 | 24.f | even | 2 | 1 | ||
| 2496.4.a.bc | 2 | 24.h | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{2} + 2T_{2} - 13 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(117))\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{2} + 2T - 13 \)
$3$
\( T^{2} \)
$5$
\( T^{2} + 24T + 88 \)
$7$
\( T^{2} - 56 \)
$11$
\( T^{2} - 44T - 1532 \)
$13$
\( (T + 13)^{2} \)
$17$
\( T^{2} + 164T + 6500 \)
$19$
\( T^{2} - 48T + 520 \)
$23$
\( T^{2} + 8T - 32240 \)
$29$
\( T^{2} + 404T + 32740 \)
$31$
\( T^{2} - 40T - 9064 \)
$37$
\( T^{2} + 100T - 8476 \)
$41$
\( T^{2} + 200T - 113704 \)
$43$
\( T^{2} + 616T + 57008 \)
$47$
\( T^{2} - 324T + 11908 \)
$53$
\( T^{2} - 164T - 194876 \)
$59$
\( T^{2} + 140T - 17500 \)
$61$
\( T^{2} - 628T - 160348 \)
$67$
\( T^{2} + 472T - 348904 \)
$71$
\( T^{2} + 428T - 52988 \)
$73$
\( T^{2} + 900T + 121636 \)
$79$
\( T^{2} + 432T - 61760 \)
$83$
\( T^{2} - 1388 T + 424292 \)
$89$
\( T^{2} + 960T - 275000 \)
$97$
\( T^{2} + 532T - 606844 \)
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