Newspace parameters
| Level: | \( N \) | \(=\) | \( 169 = 13^{2} \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 169.c (of order \(3\), degree \(2\), not minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(9.97132279097\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Relative dimension: | \(2\) over \(\Q(\zeta_{3})\) |
| Coefficient field: | \(\Q(\zeta_{12})\) |
| Defining polynomial: |
\( x^{4} - x^{2} + 1 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 3 \) |
| Twist minimal: | no (minimal twist has level 13) |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{3}]$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.
Basis of coefficient ring
| \(\beta_{1}\) | \(=\) |
\( \zeta_{12}^{2} \)
|
| \(\beta_{2}\) | \(=\) |
\( \zeta_{12}^{3} + \zeta_{12} \)
|
| \(\beta_{3}\) | \(=\) |
\( -\zeta_{12}^{3} + 2\zeta_{12} \)
|
| \(\zeta_{12}\) | \(=\) |
\( ( \beta_{3} + \beta_{2} ) / 3 \)
|
| \(\zeta_{12}^{2}\) | \(=\) |
\( \beta_1 \)
|
| \(\zeta_{12}^{3}\) | \(=\) |
\( ( -\beta_{3} + 2\beta_{2} ) / 3 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/169\mathbb{Z}\right)^\times\).
| \(n\) | \(2\) |
| \(\chi(n)\) | \(-1 + \beta_{1}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 22.1 |
|
−0.866025 | + | 1.50000i | −1.00000 | + | 1.73205i | 2.50000 | + | 4.33013i | 1.73205 | −1.73205 | − | 3.00000i | 6.92820 | + | 12.0000i | −22.5167 | 11.5000 | + | 19.9186i | −1.50000 | + | 2.59808i | ||||||||||||||||
| 22.2 | 0.866025 | − | 1.50000i | −1.00000 | + | 1.73205i | 2.50000 | + | 4.33013i | −1.73205 | 1.73205 | + | 3.00000i | −6.92820 | − | 12.0000i | 22.5167 | 11.5000 | + | 19.9186i | −1.50000 | + | 2.59808i | |||||||||||||||||
| 146.1 | −0.866025 | − | 1.50000i | −1.00000 | − | 1.73205i | 2.50000 | − | 4.33013i | 1.73205 | −1.73205 | + | 3.00000i | 6.92820 | − | 12.0000i | −22.5167 | 11.5000 | − | 19.9186i | −1.50000 | − | 2.59808i | |||||||||||||||||
| 146.2 | 0.866025 | + | 1.50000i | −1.00000 | − | 1.73205i | 2.50000 | − | 4.33013i | −1.73205 | 1.73205 | − | 3.00000i | −6.92820 | + | 12.0000i | 22.5167 | 11.5000 | − | 19.9186i | −1.50000 | − | 2.59808i | |||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 13.b | even | 2 | 1 | inner |
| 13.c | even | 3 | 1 | inner |
| 13.e | even | 6 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 169.4.c.h | 4 | |
| 13.b | even | 2 | 1 | inner | 169.4.c.h | 4 | |
| 13.c | even | 3 | 1 | 169.4.a.i | 2 | ||
| 13.c | even | 3 | 1 | inner | 169.4.c.h | 4 | |
| 13.d | odd | 4 | 1 | 13.4.e.b | ✓ | 2 | |
| 13.d | odd | 4 | 1 | 169.4.e.a | 2 | ||
| 13.e | even | 6 | 1 | 169.4.a.i | 2 | ||
| 13.e | even | 6 | 1 | inner | 169.4.c.h | 4 | |
| 13.f | odd | 12 | 1 | 13.4.e.b | ✓ | 2 | |
| 13.f | odd | 12 | 2 | 169.4.b.d | 2 | ||
| 13.f | odd | 12 | 1 | 169.4.e.a | 2 | ||
| 39.f | even | 4 | 1 | 117.4.q.a | 2 | ||
| 39.h | odd | 6 | 1 | 1521.4.a.o | 2 | ||
| 39.i | odd | 6 | 1 | 1521.4.a.o | 2 | ||
| 39.k | even | 12 | 1 | 117.4.q.a | 2 | ||
| 52.f | even | 4 | 1 | 208.4.w.b | 2 | ||
| 52.l | even | 12 | 1 | 208.4.w.b | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 13.4.e.b | ✓ | 2 | 13.d | odd | 4 | 1 | |
| 13.4.e.b | ✓ | 2 | 13.f | odd | 12 | 1 | |
| 117.4.q.a | 2 | 39.f | even | 4 | 1 | ||
| 117.4.q.a | 2 | 39.k | even | 12 | 1 | ||
| 169.4.a.i | 2 | 13.c | even | 3 | 1 | ||
| 169.4.a.i | 2 | 13.e | even | 6 | 1 | ||
| 169.4.b.d | 2 | 13.f | odd | 12 | 2 | ||
| 169.4.c.h | 4 | 1.a | even | 1 | 1 | trivial | |
| 169.4.c.h | 4 | 13.b | even | 2 | 1 | inner | |
| 169.4.c.h | 4 | 13.c | even | 3 | 1 | inner | |
| 169.4.c.h | 4 | 13.e | even | 6 | 1 | inner | |
| 169.4.e.a | 2 | 13.d | odd | 4 | 1 | ||
| 169.4.e.a | 2 | 13.f | odd | 12 | 1 | ||
| 208.4.w.b | 2 | 52.f | even | 4 | 1 | ||
| 208.4.w.b | 2 | 52.l | even | 12 | 1 | ||
| 1521.4.a.o | 2 | 39.h | odd | 6 | 1 | ||
| 1521.4.a.o | 2 | 39.i | odd | 6 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{4} + 3T_{2}^{2} + 9 \)
acting on \(S_{4}^{\mathrm{new}}(169, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{4} + 3T^{2} + 9 \)
$3$
\( (T^{2} + 2 T + 4)^{2} \)
$5$
\( (T^{2} - 3)^{2} \)
$7$
\( T^{4} + 192 T^{2} + 36864 \)
$11$
\( T^{4} + 192 T^{2} + 36864 \)
$13$
\( T^{4} \)
$17$
\( (T^{2} - 117 T + 13689)^{2} \)
$19$
\( T^{4} + 13068 T^{2} + \cdots + 170772624 \)
$23$
\( (T^{2} + 78 T + 6084)^{2} \)
$29$
\( (T^{2} - 141 T + 19881)^{2} \)
$31$
\( (T^{2} - 24300)^{2} \)
$37$
\( T^{4} + 20667 T^{2} + \cdots + 427124889 \)
$41$
\( T^{4} + 73947 T^{2} + \cdots + 5468158809 \)
$43$
\( (T^{2} - 104 T + 10816)^{2} \)
$47$
\( (T^{2} - 90828)^{2} \)
$53$
\( (T - 93)^{4} \)
$59$
\( T^{4} + 80688 T^{2} + \cdots + 6510553344 \)
$61$
\( (T^{2} + 145 T + 21025)^{2} \)
$67$
\( T^{4} + 618348 T^{2} + \cdots + 382354249104 \)
$71$
\( T^{4} + 1116300 T^{2} + \cdots + 1246125690000 \)
$73$
\( (T^{2} - 210675)^{2} \)
$79$
\( (T - 1276)^{4} \)
$83$
\( (T^{2} - 623808)^{2} \)
$89$
\( T^{4} + 954288 T^{2} + \cdots + 910665586944 \)
$97$
\( T^{4} + 40368 T^{2} + \cdots + 1629575424 \)
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