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: | \(\Z[a_1, \ldots, a_{7}]\) |
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 |
|
−1.73205 | + | 3.00000i | 3.50000 | − | 6.06218i | −2.00000 | − | 3.46410i | 13.8564 | 12.1244 | + | 21.0000i | 11.2583 | + | 19.5000i | −13.8564 | −11.0000 | − | 19.0526i | −24.0000 | + | 41.5692i | ||||||||||||||||
22.2 | 1.73205 | − | 3.00000i | 3.50000 | − | 6.06218i | −2.00000 | − | 3.46410i | −13.8564 | −12.1244 | − | 21.0000i | −11.2583 | − | 19.5000i | 13.8564 | −11.0000 | − | 19.0526i | −24.0000 | + | 41.5692i | |||||||||||||||||
146.1 | −1.73205 | − | 3.00000i | 3.50000 | + | 6.06218i | −2.00000 | + | 3.46410i | 13.8564 | 12.1244 | − | 21.0000i | 11.2583 | − | 19.5000i | −13.8564 | −11.0000 | + | 19.0526i | −24.0000 | − | 41.5692i | |||||||||||||||||
146.2 | 1.73205 | + | 3.00000i | 3.50000 | + | 6.06218i | −2.00000 | + | 3.46410i | −13.8564 | −12.1244 | + | 21.0000i | −11.2583 | + | 19.5000i | 13.8564 | −11.0000 | + | 19.0526i | −24.0000 | − | 41.5692i | |||||||||||||||||
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.i | 4 | |
13.b | even | 2 | 1 | inner | 169.4.c.i | 4 | |
13.c | even | 3 | 1 | 169.4.a.h | 2 | ||
13.c | even | 3 | 1 | inner | 169.4.c.i | 4 | |
13.d | odd | 4 | 1 | 13.4.e.a | ✓ | 2 | |
13.d | odd | 4 | 1 | 169.4.e.b | 2 | ||
13.e | even | 6 | 1 | 169.4.a.h | 2 | ||
13.e | even | 6 | 1 | inner | 169.4.c.i | 4 | |
13.f | odd | 12 | 1 | 13.4.e.a | ✓ | 2 | |
13.f | odd | 12 | 2 | 169.4.b.b | 2 | ||
13.f | odd | 12 | 1 | 169.4.e.b | 2 | ||
39.f | even | 4 | 1 | 117.4.q.c | 2 | ||
39.h | odd | 6 | 1 | 1521.4.a.q | 2 | ||
39.i | odd | 6 | 1 | 1521.4.a.q | 2 | ||
39.k | even | 12 | 1 | 117.4.q.c | 2 | ||
52.f | even | 4 | 1 | 208.4.w.a | 2 | ||
52.l | even | 12 | 1 | 208.4.w.a | 2 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
13.4.e.a | ✓ | 2 | 13.d | odd | 4 | 1 | |
13.4.e.a | ✓ | 2 | 13.f | odd | 12 | 1 | |
117.4.q.c | 2 | 39.f | even | 4 | 1 | ||
117.4.q.c | 2 | 39.k | even | 12 | 1 | ||
169.4.a.h | 2 | 13.c | even | 3 | 1 | ||
169.4.a.h | 2 | 13.e | even | 6 | 1 | ||
169.4.b.b | 2 | 13.f | odd | 12 | 2 | ||
169.4.c.i | 4 | 1.a | even | 1 | 1 | trivial | |
169.4.c.i | 4 | 13.b | even | 2 | 1 | inner | |
169.4.c.i | 4 | 13.c | even | 3 | 1 | inner | |
169.4.c.i | 4 | 13.e | even | 6 | 1 | inner | |
169.4.e.b | 2 | 13.d | odd | 4 | 1 | ||
169.4.e.b | 2 | 13.f | odd | 12 | 1 | ||
208.4.w.a | 2 | 52.f | even | 4 | 1 | ||
208.4.w.a | 2 | 52.l | even | 12 | 1 | ||
1521.4.a.q | 2 | 39.h | odd | 6 | 1 | ||
1521.4.a.q | 2 | 39.i | odd | 6 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{4} + 12T_{2}^{2} + 144 \)
acting on \(S_{4}^{\mathrm{new}}(169, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{4} + 12T^{2} + 144 \)
$3$
\( (T^{2} - 7 T + 49)^{2} \)
$5$
\( (T^{2} - 192)^{2} \)
$7$
\( T^{4} + 507 T^{2} + 257049 \)
$11$
\( T^{4} + 507 T^{2} + 257049 \)
$13$
\( T^{4} \)
$17$
\( (T^{2} - 27 T + 729)^{2} \)
$19$
\( T^{4} + 7803 T^{2} + \cdots + 60886809 \)
$23$
\( (T^{2} - 57 T + 3249)^{2} \)
$29$
\( (T^{2} - 69 T + 4761)^{2} \)
$31$
\( (T^{2} - 5292)^{2} \)
$37$
\( T^{4} + 1587 T^{2} + \cdots + 2518569 \)
$41$
\( T^{4} + 154587 T^{2} + \cdots + 23897140569 \)
$43$
\( (T^{2} + 85 T + 7225)^{2} \)
$47$
\( (T^{2} - 117612)^{2} \)
$53$
\( (T - 426)^{4} \)
$59$
\( T^{4} + 363 T^{2} + 131769 \)
$61$
\( (T^{2} - 17 T + 289)^{2} \)
$67$
\( T^{4} + 27075 T^{2} + \cdots + 733055625 \)
$71$
\( T^{4} + 340707 T^{2} + \cdots + 116081259849 \)
$73$
\( (T^{2} - 1009200)^{2} \)
$79$
\( (T + 1244)^{4} \)
$83$
\( (T^{2} - 181548)^{2} \)
$89$
\( T^{4} + 93987 T^{2} + \cdots + 8833556169 \)
$97$
\( T^{4} + 1525107 T^{2} + \cdots + 2325951361449 \)
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