Newspace parameters
Level: | \( N \) | \(=\) | \( 169 = 13^{2} \) |
Weight: | \( k \) | \(=\) | \( 4 \) |
Character orbit: | \([\chi]\) | \(=\) | 169.e (of order \(6\), degree \(2\), not minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(9.97132279097\) |
Analytic rank: | \(0\) |
Dimension: | \(4\) |
Relative dimension: | \(2\) over \(\Q(\zeta_{6})\) |
Coefficient field: | \(\Q(\zeta_{12})\) |
Defining polynomial: |
\( x^{4} - x^{2} + 1 \)
|
Coefficient ring: | \(\Z[a_1, \ldots, a_{9}]\) |
Coefficient ring index: | \( 1 \) |
Twist minimal: | no (minimal twist has level 13) |
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 \(\zeta_{12}\). We also show the integral \(q\)-expansion of the trace form.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/169\mathbb{Z}\right)^\times\).
\(n\) | \(2\) |
\(\chi(n)\) | \(\zeta_{12}^{2}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
23.1 |
|
−3.46410 | + | 2.00000i | −1.00000 | − | 1.73205i | 4.00000 | − | 6.92820i | − | 17.0000i | 6.92820 | + | 4.00000i | −17.3205 | − | 10.0000i | 0 | 11.5000 | − | 19.9186i | 34.0000 | + | 58.8897i | |||||||||||||||
23.2 | 3.46410 | − | 2.00000i | −1.00000 | − | 1.73205i | 4.00000 | − | 6.92820i | 17.0000i | −6.92820 | − | 4.00000i | 17.3205 | + | 10.0000i | 0 | 11.5000 | − | 19.9186i | 34.0000 | + | 58.8897i | |||||||||||||||||
147.1 | −3.46410 | − | 2.00000i | −1.00000 | + | 1.73205i | 4.00000 | + | 6.92820i | 17.0000i | 6.92820 | − | 4.00000i | −17.3205 | + | 10.0000i | 0 | 11.5000 | + | 19.9186i | 34.0000 | − | 58.8897i | |||||||||||||||||
147.2 | 3.46410 | + | 2.00000i | −1.00000 | + | 1.73205i | 4.00000 | + | 6.92820i | − | 17.0000i | −6.92820 | + | 4.00000i | 17.3205 | − | 10.0000i | 0 | 11.5000 | + | 19.9186i | 34.0000 | − | 58.8897i | ||||||||||||||||
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.e.c | 4 | |
13.b | even | 2 | 1 | inner | 169.4.e.c | 4 | |
13.c | even | 3 | 1 | 169.4.b.c | 2 | ||
13.c | even | 3 | 1 | inner | 169.4.e.c | 4 | |
13.d | odd | 4 | 1 | 13.4.c.a | ✓ | 2 | |
13.d | odd | 4 | 1 | 169.4.c.d | 2 | ||
13.e | even | 6 | 1 | 169.4.b.c | 2 | ||
13.e | even | 6 | 1 | inner | 169.4.e.c | 4 | |
13.f | odd | 12 | 1 | 13.4.c.a | ✓ | 2 | |
13.f | odd | 12 | 1 | 169.4.a.a | 1 | ||
13.f | odd | 12 | 1 | 169.4.a.d | 1 | ||
13.f | odd | 12 | 1 | 169.4.c.d | 2 | ||
39.f | even | 4 | 1 | 117.4.g.c | 2 | ||
39.k | even | 12 | 1 | 117.4.g.c | 2 | ||
39.k | even | 12 | 1 | 1521.4.a.b | 1 | ||
39.k | even | 12 | 1 | 1521.4.a.k | 1 | ||
52.f | even | 4 | 1 | 208.4.i.b | 2 | ||
52.l | even | 12 | 1 | 208.4.i.b | 2 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
13.4.c.a | ✓ | 2 | 13.d | odd | 4 | 1 | |
13.4.c.a | ✓ | 2 | 13.f | odd | 12 | 1 | |
117.4.g.c | 2 | 39.f | even | 4 | 1 | ||
117.4.g.c | 2 | 39.k | even | 12 | 1 | ||
169.4.a.a | 1 | 13.f | odd | 12 | 1 | ||
169.4.a.d | 1 | 13.f | odd | 12 | 1 | ||
169.4.b.c | 2 | 13.c | even | 3 | 1 | ||
169.4.b.c | 2 | 13.e | even | 6 | 1 | ||
169.4.c.d | 2 | 13.d | odd | 4 | 1 | ||
169.4.c.d | 2 | 13.f | odd | 12 | 1 | ||
169.4.e.c | 4 | 1.a | even | 1 | 1 | trivial | |
169.4.e.c | 4 | 13.b | even | 2 | 1 | inner | |
169.4.e.c | 4 | 13.c | even | 3 | 1 | inner | |
169.4.e.c | 4 | 13.e | even | 6 | 1 | inner | |
208.4.i.b | 2 | 52.f | even | 4 | 1 | ||
208.4.i.b | 2 | 52.l | even | 12 | 1 | ||
1521.4.a.b | 1 | 39.k | even | 12 | 1 | ||
1521.4.a.k | 1 | 39.k | even | 12 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{4} - 16T_{2}^{2} + 256 \)
acting on \(S_{4}^{\mathrm{new}}(169, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{4} - 16T^{2} + 256 \)
$3$
\( (T^{2} + 2 T + 4)^{2} \)
$5$
\( (T^{2} + 289)^{2} \)
$7$
\( T^{4} - 400 T^{2} + 160000 \)
$11$
\( T^{4} - 1024 T^{2} + \cdots + 1048576 \)
$13$
\( T^{4} \)
$17$
\( (T^{2} + 13 T + 169)^{2} \)
$19$
\( T^{4} - 900 T^{2} + 810000 \)
$23$
\( (T^{2} - 78 T + 6084)^{2} \)
$29$
\( (T^{2} + 197 T + 38809)^{2} \)
$31$
\( (T^{2} + 5476)^{2} \)
$37$
\( T^{4} - 51529 T^{2} + \cdots + 2655237841 \)
$41$
\( T^{4} - 27225 T^{2} + \cdots + 741200625 \)
$43$
\( (T^{2} + 156 T + 24336)^{2} \)
$47$
\( (T^{2} + 26244)^{2} \)
$53$
\( (T - 93)^{4} \)
$59$
\( T^{4} - 746496 T^{2} + \cdots + 557256278016 \)
$61$
\( (T^{2} + 145 T + 21025)^{2} \)
$67$
\( T^{4} - 743044 T^{2} + \cdots + 552114385936 \)
$71$
\( T^{4} - 427716 T^{2} + \cdots + 182940976656 \)
$73$
\( (T^{2} + 46225)^{2} \)
$79$
\( (T + 76)^{4} \)
$83$
\( (T^{2} + 394384)^{2} \)
$89$
\( T^{4} - 70756 T^{2} + \cdots + 5006411536 \)
$97$
\( T^{4} - 56644 T^{2} + \cdots + 3208542736 \)
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