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: | \(\Z[a_1, \ldots, a_{5}]\) |
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 |
|
−4.33013 | + | 2.50000i | 3.50000 | + | 6.06218i | 8.50000 | − | 14.7224i | − | 7.00000i | −30.3109 | − | 17.5000i | −11.2583 | − | 6.50000i | 45.0000i | −11.0000 | + | 19.0526i | 17.5000 | + | 30.3109i | |||||||||||||||
23.2 | 4.33013 | − | 2.50000i | 3.50000 | + | 6.06218i | 8.50000 | − | 14.7224i | 7.00000i | 30.3109 | + | 17.5000i | 11.2583 | + | 6.50000i | − | 45.0000i | −11.0000 | + | 19.0526i | 17.5000 | + | 30.3109i | ||||||||||||||||
147.1 | −4.33013 | − | 2.50000i | 3.50000 | − | 6.06218i | 8.50000 | + | 14.7224i | 7.00000i | −30.3109 | + | 17.5000i | −11.2583 | + | 6.50000i | − | 45.0000i | −11.0000 | − | 19.0526i | 17.5000 | − | 30.3109i | ||||||||||||||||
147.2 | 4.33013 | + | 2.50000i | 3.50000 | − | 6.06218i | 8.50000 | + | 14.7224i | − | 7.00000i | 30.3109 | − | 17.5000i | 11.2583 | − | 6.50000i | 45.0000i | −11.0000 | − | 19.0526i | 17.5000 | − | 30.3109i | ||||||||||||||||
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.e | 4 | |
13.b | even | 2 | 1 | inner | 169.4.e.e | 4 | |
13.c | even | 3 | 1 | 169.4.b.a | 2 | ||
13.c | even | 3 | 1 | inner | 169.4.e.e | 4 | |
13.d | odd | 4 | 1 | 169.4.c.a | 2 | ||
13.d | odd | 4 | 1 | 169.4.c.e | 2 | ||
13.e | even | 6 | 1 | 169.4.b.a | 2 | ||
13.e | even | 6 | 1 | inner | 169.4.e.e | 4 | |
13.f | odd | 12 | 1 | 13.4.a.a | ✓ | 1 | |
13.f | odd | 12 | 1 | 169.4.a.e | 1 | ||
13.f | odd | 12 | 1 | 169.4.c.a | 2 | ||
13.f | odd | 12 | 1 | 169.4.c.e | 2 | ||
39.k | even | 12 | 1 | 117.4.a.b | 1 | ||
39.k | even | 12 | 1 | 1521.4.a.a | 1 | ||
52.l | even | 12 | 1 | 208.4.a.g | 1 | ||
65.o | even | 12 | 1 | 325.4.b.b | 2 | ||
65.s | odd | 12 | 1 | 325.4.a.d | 1 | ||
65.t | even | 12 | 1 | 325.4.b.b | 2 | ||
91.bc | even | 12 | 1 | 637.4.a.a | 1 | ||
104.u | even | 12 | 1 | 832.4.a.a | 1 | ||
104.x | odd | 12 | 1 | 832.4.a.r | 1 | ||
143.o | even | 12 | 1 | 1573.4.a.a | 1 | ||
156.v | odd | 12 | 1 | 1872.4.a.k | 1 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
13.4.a.a | ✓ | 1 | 13.f | odd | 12 | 1 | |
117.4.a.b | 1 | 39.k | even | 12 | 1 | ||
169.4.a.e | 1 | 13.f | odd | 12 | 1 | ||
169.4.b.a | 2 | 13.c | even | 3 | 1 | ||
169.4.b.a | 2 | 13.e | even | 6 | 1 | ||
169.4.c.a | 2 | 13.d | odd | 4 | 1 | ||
169.4.c.a | 2 | 13.f | odd | 12 | 1 | ||
169.4.c.e | 2 | 13.d | odd | 4 | 1 | ||
169.4.c.e | 2 | 13.f | odd | 12 | 1 | ||
169.4.e.e | 4 | 1.a | even | 1 | 1 | trivial | |
169.4.e.e | 4 | 13.b | even | 2 | 1 | inner | |
169.4.e.e | 4 | 13.c | even | 3 | 1 | inner | |
169.4.e.e | 4 | 13.e | even | 6 | 1 | inner | |
208.4.a.g | 1 | 52.l | even | 12 | 1 | ||
325.4.a.d | 1 | 65.s | odd | 12 | 1 | ||
325.4.b.b | 2 | 65.o | even | 12 | 1 | ||
325.4.b.b | 2 | 65.t | even | 12 | 1 | ||
637.4.a.a | 1 | 91.bc | even | 12 | 1 | ||
832.4.a.a | 1 | 104.u | even | 12 | 1 | ||
832.4.a.r | 1 | 104.x | odd | 12 | 1 | ||
1521.4.a.a | 1 | 39.k | even | 12 | 1 | ||
1573.4.a.a | 1 | 143.o | even | 12 | 1 | ||
1872.4.a.k | 1 | 156.v | odd | 12 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{4} - 25T_{2}^{2} + 625 \)
acting on \(S_{4}^{\mathrm{new}}(169, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{4} - 25T^{2} + 625 \)
$3$
\( (T^{2} - 7 T + 49)^{2} \)
$5$
\( (T^{2} + 49)^{2} \)
$7$
\( T^{4} - 169 T^{2} + 28561 \)
$11$
\( T^{4} - 676 T^{2} + 456976 \)
$13$
\( T^{4} \)
$17$
\( (T^{2} - 77 T + 5929)^{2} \)
$19$
\( T^{4} - 15876 T^{2} + \cdots + 252047376 \)
$23$
\( (T^{2} + 96 T + 9216)^{2} \)
$29$
\( (T^{2} - 82 T + 6724)^{2} \)
$31$
\( (T^{2} + 38416)^{2} \)
$37$
\( T^{4} - 17161 T^{2} + \cdots + 294499921 \)
$41$
\( T^{4} - 112896 T^{2} + \cdots + 12745506816 \)
$43$
\( (T^{2} + 201 T + 40401)^{2} \)
$47$
\( (T^{2} + 11025)^{2} \)
$53$
\( (T + 432)^{4} \)
$59$
\( T^{4} - 86436 T^{2} + \cdots + 7471182096 \)
$61$
\( (T^{2} - 56 T + 3136)^{2} \)
$67$
\( T^{4} - 228484 T^{2} + \cdots + 52204938256 \)
$71$
\( T^{4} - 81T^{2} + 6561 \)
$73$
\( (T^{2} + 9604)^{2} \)
$79$
\( (T - 1304)^{4} \)
$83$
\( (T^{2} + 94864)^{2} \)
$89$
\( T^{4} - 1416100 T^{2} + \cdots + 2005339210000 \)
$97$
\( T^{4} - 4900 T^{2} + \cdots + 24010000 \)
show more
show less