Newspace parameters
| Level: | \( N \) | \(=\) | \( 169 = 13^{2} \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 169.b (of order \(2\), degree \(1\), not minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(9.97132279097\) |
| Analytic rank: | \(0\) |
| Dimension: | \(2\) |
| Coefficient field: | \(\Q(\sqrt{-1}) \) |
| Defining polynomial: |
\( x^{2} + 1 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 13) |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/169\mathbb{Z}\right)^\times\).
| \(n\) | \(2\) |
| \(\chi(n)\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 168.1 |
|
− | 5.00000i | −7.00000 | −17.0000 | − | 7.00000i | 35.0000i | 13.0000i | 45.0000i | 22.0000 | −35.0000 | ||||||||||||||||||||||
| 168.2 | 5.00000i | −7.00000 | −17.0000 | 7.00000i | − | 35.0000i | − | 13.0000i | − | 45.0000i | 22.0000 | −35.0000 | ||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 13.b | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 169.4.b.a | 2 | |
| 13.b | even | 2 | 1 | inner | 169.4.b.a | 2 | |
| 13.c | even | 3 | 2 | 169.4.e.e | 4 | ||
| 13.d | odd | 4 | 1 | 13.4.a.a | ✓ | 1 | |
| 13.d | odd | 4 | 1 | 169.4.a.e | 1 | ||
| 13.e | even | 6 | 2 | 169.4.e.e | 4 | ||
| 13.f | odd | 12 | 2 | 169.4.c.a | 2 | ||
| 13.f | odd | 12 | 2 | 169.4.c.e | 2 | ||
| 39.f | even | 4 | 1 | 117.4.a.b | 1 | ||
| 39.f | even | 4 | 1 | 1521.4.a.a | 1 | ||
| 52.f | even | 4 | 1 | 208.4.a.g | 1 | ||
| 65.f | even | 4 | 1 | 325.4.b.b | 2 | ||
| 65.g | odd | 4 | 1 | 325.4.a.d | 1 | ||
| 65.k | even | 4 | 1 | 325.4.b.b | 2 | ||
| 91.i | even | 4 | 1 | 637.4.a.a | 1 | ||
| 104.j | odd | 4 | 1 | 832.4.a.r | 1 | ||
| 104.m | even | 4 | 1 | 832.4.a.a | 1 | ||
| 143.g | even | 4 | 1 | 1573.4.a.a | 1 | ||
| 156.l | odd | 4 | 1 | 1872.4.a.k | 1 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 13.4.a.a | ✓ | 1 | 13.d | odd | 4 | 1 | |
| 117.4.a.b | 1 | 39.f | even | 4 | 1 | ||
| 169.4.a.e | 1 | 13.d | odd | 4 | 1 | ||
| 169.4.b.a | 2 | 1.a | even | 1 | 1 | trivial | |
| 169.4.b.a | 2 | 13.b | even | 2 | 1 | inner | |
| 169.4.c.a | 2 | 13.f | odd | 12 | 2 | ||
| 169.4.c.e | 2 | 13.f | odd | 12 | 2 | ||
| 169.4.e.e | 4 | 13.c | even | 3 | 2 | ||
| 169.4.e.e | 4 | 13.e | even | 6 | 2 | ||
| 208.4.a.g | 1 | 52.f | even | 4 | 1 | ||
| 325.4.a.d | 1 | 65.g | odd | 4 | 1 | ||
| 325.4.b.b | 2 | 65.f | even | 4 | 1 | ||
| 325.4.b.b | 2 | 65.k | even | 4 | 1 | ||
| 637.4.a.a | 1 | 91.i | even | 4 | 1 | ||
| 832.4.a.a | 1 | 104.m | even | 4 | 1 | ||
| 832.4.a.r | 1 | 104.j | odd | 4 | 1 | ||
| 1521.4.a.a | 1 | 39.f | even | 4 | 1 | ||
| 1573.4.a.a | 1 | 143.g | even | 4 | 1 | ||
| 1872.4.a.k | 1 | 156.l | odd | 4 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{2} + 25 \)
acting on \(S_{4}^{\mathrm{new}}(169, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{2} + 25 \)
$3$
\( (T + 7)^{2} \)
$5$
\( T^{2} + 49 \)
$7$
\( T^{2} + 169 \)
$11$
\( T^{2} + 676 \)
$13$
\( T^{2} \)
$17$
\( (T + 77)^{2} \)
$19$
\( T^{2} + 15876 \)
$23$
\( (T - 96)^{2} \)
$29$
\( (T + 82)^{2} \)
$31$
\( T^{2} + 38416 \)
$37$
\( T^{2} + 17161 \)
$41$
\( T^{2} + 112896 \)
$43$
\( (T - 201)^{2} \)
$47$
\( T^{2} + 11025 \)
$53$
\( (T + 432)^{2} \)
$59$
\( T^{2} + 86436 \)
$61$
\( (T + 56)^{2} \)
$67$
\( T^{2} + 228484 \)
$71$
\( T^{2} + 81 \)
$73$
\( T^{2} + 9604 \)
$79$
\( (T - 1304)^{2} \)
$83$
\( T^{2} + 94864 \)
$89$
\( T^{2} + 1416100 \)
$97$
\( T^{2} + 4900 \)
show more
show less