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: | \(2\) |
Coefficient field: | \(\Q(\sqrt{-3}) \) |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
Defining polynomial: |
\( x^{2} - x + 1 \)
|
Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
Coefficient ring index: | \( 1 \) |
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 primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/169\mathbb{Z}\right)^\times\).
\(n\) | \(2\) |
\(\chi(n)\) | \(-\zeta_{6}\) |
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 |
|
1.50000 | − | 2.59808i | 0.500000 | − | 0.866025i | −0.500000 | − | 0.866025i | 9.00000 | −1.50000 | − | 2.59808i | 7.50000 | + | 12.9904i | 21.0000 | 13.0000 | + | 22.5167i | 13.5000 | − | 23.3827i | ||||||||||
146.1 | 1.50000 | + | 2.59808i | 0.500000 | + | 0.866025i | −0.500000 | + | 0.866025i | 9.00000 | −1.50000 | + | 2.59808i | 7.50000 | − | 12.9904i | 21.0000 | 13.0000 | − | 22.5167i | 13.5000 | + | 23.3827i | |||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
13.c | even | 3 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 169.4.c.c | 2 | |
13.b | even | 2 | 1 | 169.4.c.b | 2 | ||
13.c | even | 3 | 1 | 169.4.a.b | 1 | ||
13.c | even | 3 | 1 | inner | 169.4.c.c | 2 | |
13.d | odd | 4 | 2 | 169.4.e.d | 4 | ||
13.e | even | 6 | 1 | 169.4.a.c | 1 | ||
13.e | even | 6 | 1 | 169.4.c.b | 2 | ||
13.f | odd | 12 | 2 | 13.4.b.a | ✓ | 2 | |
13.f | odd | 12 | 2 | 169.4.e.d | 4 | ||
39.h | odd | 6 | 1 | 1521.4.a.d | 1 | ||
39.i | odd | 6 | 1 | 1521.4.a.i | 1 | ||
39.k | even | 12 | 2 | 117.4.b.a | 2 | ||
52.l | even | 12 | 2 | 208.4.f.b | 2 | ||
65.o | even | 12 | 2 | 325.4.d.a | 2 | ||
65.s | odd | 12 | 2 | 325.4.c.b | 2 | ||
65.t | even | 12 | 2 | 325.4.d.b | 2 | ||
104.u | even | 12 | 2 | 832.4.f.c | 2 | ||
104.x | odd | 12 | 2 | 832.4.f.e | 2 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
13.4.b.a | ✓ | 2 | 13.f | odd | 12 | 2 | |
117.4.b.a | 2 | 39.k | even | 12 | 2 | ||
169.4.a.b | 1 | 13.c | even | 3 | 1 | ||
169.4.a.c | 1 | 13.e | even | 6 | 1 | ||
169.4.c.b | 2 | 13.b | even | 2 | 1 | ||
169.4.c.b | 2 | 13.e | even | 6 | 1 | ||
169.4.c.c | 2 | 1.a | even | 1 | 1 | trivial | |
169.4.c.c | 2 | 13.c | even | 3 | 1 | inner | |
169.4.e.d | 4 | 13.d | odd | 4 | 2 | ||
169.4.e.d | 4 | 13.f | odd | 12 | 2 | ||
208.4.f.b | 2 | 52.l | even | 12 | 2 | ||
325.4.c.b | 2 | 65.s | odd | 12 | 2 | ||
325.4.d.a | 2 | 65.o | even | 12 | 2 | ||
325.4.d.b | 2 | 65.t | even | 12 | 2 | ||
832.4.f.c | 2 | 104.u | even | 12 | 2 | ||
832.4.f.e | 2 | 104.x | odd | 12 | 2 | ||
1521.4.a.d | 1 | 39.h | odd | 6 | 1 | ||
1521.4.a.i | 1 | 39.i | odd | 6 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{2} - 3T_{2} + 9 \)
acting on \(S_{4}^{\mathrm{new}}(169, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{2} - 3T + 9 \)
$3$
\( T^{2} - T + 1 \)
$5$
\( (T - 9)^{2} \)
$7$
\( T^{2} - 15T + 225 \)
$11$
\( T^{2} + 48T + 2304 \)
$13$
\( T^{2} \)
$17$
\( T^{2} + 45T + 2025 \)
$19$
\( T^{2} - 6T + 36 \)
$23$
\( T^{2} - 162T + 26244 \)
$29$
\( T^{2} - 144T + 20736 \)
$31$
\( (T + 264)^{2} \)
$37$
\( T^{2} - 303T + 91809 \)
$41$
\( T^{2} + 192T + 36864 \)
$43$
\( T^{2} + 97T + 9409 \)
$47$
\( (T + 111)^{2} \)
$53$
\( (T + 414)^{2} \)
$59$
\( T^{2} - 522T + 272484 \)
$61$
\( T^{2} + 376T + 141376 \)
$67$
\( T^{2} + 36T + 1296 \)
$71$
\( T^{2} - 357T + 127449 \)
$73$
\( (T - 1098)^{2} \)
$79$
\( (T + 830)^{2} \)
$83$
\( (T - 438)^{2} \)
$89$
\( T^{2} + 438T + 191844 \)
$97$
\( T^{2} + 852T + 725904 \)
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