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: | \(8\) |
Relative dimension: | \(4\) over \(\Q(\zeta_{6})\) |
Coefficient field: | 8.0.1731891456.1 |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
Defining polynomial: | \( x^{8} - 9x^{6} + 65x^{4} - 144x^{2} + 256 \) |
Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
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 basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.
Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 9x^{6} + 65x^{4} - 144x^{2} + 256 \) :
\(\beta_{1}\) | \(=\) | \( \nu \) |
\(\beta_{2}\) | \(=\) | \( ( 9\nu^{6} - 65\nu^{4} + 585\nu^{2} - 1296 ) / 1040 \) |
\(\beta_{3}\) | \(=\) | \( ( -\nu^{7} - 181\nu ) / 260 \) |
\(\beta_{4}\) | \(=\) | \( ( \nu^{6} + 116 ) / 65 \) |
\(\beta_{5}\) | \(=\) | \( ( -29\nu^{6} + 325\nu^{4} - 1885\nu^{2} + 4176 ) / 1040 \) |
\(\beta_{6}\) | \(=\) | \( ( 9\nu^{7} - 65\nu^{5} + 585\nu^{3} - 256\nu ) / 1040 \) |
\(\beta_{7}\) | \(=\) | \( ( 9\nu^{7} - 65\nu^{5} + 377\nu^{3} - 256\nu ) / 832 \) |
\(\nu\) | \(=\) | \( \beta_1 \) |
\(\nu^{2}\) | \(=\) | \( \beta_{5} - \beta_{4} + 5\beta_{2} + 4 \) |
\(\nu^{3}\) | \(=\) | \( -4\beta_{7} + 5\beta_{6} \) |
\(\nu^{4}\) | \(=\) | \( 9\beta_{5} + 29\beta_{2} \) |
\(\nu^{5}\) | \(=\) | \( -36\beta_{7} + 29\beta_{6} - 36\beta_{3} - 29\beta_1 \) |
\(\nu^{6}\) | \(=\) | \( 65\beta_{4} - 116 \) |
\(\nu^{7}\) | \(=\) | \( -260\beta_{3} - 181\beta_1 \) |
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/169\mathbb{Z}\right)^\times\).
\(n\) | \(2\) |
\(\chi(n)\) | \(-\beta_{2}\) |
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 |
|
−3.95042 | + | 2.28078i | −4.34233 | − | 7.52113i | 6.40388 | − | 11.0918i | 2.80776i | 34.3081 | + | 19.8078i | 8.28055 | + | 4.78078i | 21.9309i | −24.2116 | + | 41.9358i | −6.40388 | − | 11.0918i | ||||||||||||||||||||||||||||
23.2 | −0.379706 | + | 0.219224i | 1.84233 | + | 3.19101i | −3.90388 | + | 6.76172i | − | 17.8078i | −1.39909 | − | 0.807764i | 4.70983 | + | 2.71922i | − | 6.93087i | 6.71165 | − | 11.6249i | 3.90388 | + | 6.76172i | |||||||||||||||||||||||||||
23.3 | 0.379706 | − | 0.219224i | 1.84233 | + | 3.19101i | −3.90388 | + | 6.76172i | 17.8078i | 1.39909 | + | 0.807764i | −4.70983 | − | 2.71922i | 6.93087i | 6.71165 | − | 11.6249i | 3.90388 | + | 6.76172i | |||||||||||||||||||||||||||||
23.4 | 3.95042 | − | 2.28078i | −4.34233 | − | 7.52113i | 6.40388 | − | 11.0918i | − | 2.80776i | −34.3081 | − | 19.8078i | −8.28055 | − | 4.78078i | − | 21.9309i | −24.2116 | + | 41.9358i | −6.40388 | − | 11.0918i | |||||||||||||||||||||||||||
147.1 | −3.95042 | − | 2.28078i | −4.34233 | + | 7.52113i | 6.40388 | + | 11.0918i | − | 2.80776i | 34.3081 | − | 19.8078i | 8.28055 | − | 4.78078i | − | 21.9309i | −24.2116 | − | 41.9358i | −6.40388 | + | 11.0918i | |||||||||||||||||||||||||||
147.2 | −0.379706 | − | 0.219224i | 1.84233 | − | 3.19101i | −3.90388 | − | 6.76172i | 17.8078i | −1.39909 | + | 0.807764i | 4.70983 | − | 2.71922i | 6.93087i | 6.71165 | + | 11.6249i | 3.90388 | − | 6.76172i | |||||||||||||||||||||||||||||
147.3 | 0.379706 | + | 0.219224i | 1.84233 | − | 3.19101i | −3.90388 | − | 6.76172i | − | 17.8078i | 1.39909 | − | 0.807764i | −4.70983 | + | 2.71922i | − | 6.93087i | 6.71165 | + | 11.6249i | 3.90388 | − | 6.76172i | |||||||||||||||||||||||||||
147.4 | 3.95042 | + | 2.28078i | −4.34233 | + | 7.52113i | 6.40388 | + | 11.0918i | 2.80776i | −34.3081 | + | 19.8078i | −8.28055 | + | 4.78078i | 21.9309i | −24.2116 | − | 41.9358i | −6.40388 | + | 11.0918i | |||||||||||||||||||||||||||||
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.g | 8 | |
13.b | even | 2 | 1 | inner | 169.4.e.g | 8 | |
13.c | even | 3 | 1 | 169.4.b.e | 4 | ||
13.c | even | 3 | 1 | inner | 169.4.e.g | 8 | |
13.d | odd | 4 | 1 | 13.4.c.b | ✓ | 4 | |
13.d | odd | 4 | 1 | 169.4.c.f | 4 | ||
13.e | even | 6 | 1 | 169.4.b.e | 4 | ||
13.e | even | 6 | 1 | inner | 169.4.e.g | 8 | |
13.f | odd | 12 | 1 | 13.4.c.b | ✓ | 4 | |
13.f | odd | 12 | 1 | 169.4.a.f | 2 | ||
13.f | odd | 12 | 1 | 169.4.a.j | 2 | ||
13.f | odd | 12 | 1 | 169.4.c.f | 4 | ||
39.f | even | 4 | 1 | 117.4.g.d | 4 | ||
39.k | even | 12 | 1 | 117.4.g.d | 4 | ||
39.k | even | 12 | 1 | 1521.4.a.l | 2 | ||
39.k | even | 12 | 1 | 1521.4.a.t | 2 | ||
52.f | even | 4 | 1 | 208.4.i.e | 4 | ||
52.l | even | 12 | 1 | 208.4.i.e | 4 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
13.4.c.b | ✓ | 4 | 13.d | odd | 4 | 1 | |
13.4.c.b | ✓ | 4 | 13.f | odd | 12 | 1 | |
117.4.g.d | 4 | 39.f | even | 4 | 1 | ||
117.4.g.d | 4 | 39.k | even | 12 | 1 | ||
169.4.a.f | 2 | 13.f | odd | 12 | 1 | ||
169.4.a.j | 2 | 13.f | odd | 12 | 1 | ||
169.4.b.e | 4 | 13.c | even | 3 | 1 | ||
169.4.b.e | 4 | 13.e | even | 6 | 1 | ||
169.4.c.f | 4 | 13.d | odd | 4 | 1 | ||
169.4.c.f | 4 | 13.f | odd | 12 | 1 | ||
169.4.e.g | 8 | 1.a | even | 1 | 1 | trivial | |
169.4.e.g | 8 | 13.b | even | 2 | 1 | inner | |
169.4.e.g | 8 | 13.c | even | 3 | 1 | inner | |
169.4.e.g | 8 | 13.e | even | 6 | 1 | inner | |
208.4.i.e | 4 | 52.f | even | 4 | 1 | ||
208.4.i.e | 4 | 52.l | even | 12 | 1 | ||
1521.4.a.l | 2 | 39.k | even | 12 | 1 | ||
1521.4.a.t | 2 | 39.k | even | 12 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{8} - 21T_{2}^{6} + 437T_{2}^{4} - 84T_{2}^{2} + 16 \)
acting on \(S_{4}^{\mathrm{new}}(169, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{8} - 21 T^{6} + 437 T^{4} + \cdots + 16 \)
$3$
\( (T^{4} + 5 T^{3} + 57 T^{2} - 160 T + 1024)^{2} \)
$5$
\( (T^{4} + 325 T^{2} + 2500)^{2} \)
$7$
\( T^{8} - 121 T^{6} + 11937 T^{4} + \cdots + 7311616 \)
$11$
\( T^{8} - 2057 T^{6} + \cdots + 610673479936 \)
$13$
\( T^{8} \)
$17$
\( (T^{4} - 70 T^{3} + 4763 T^{2} + \cdots + 18769)^{2} \)
$19$
\( T^{8} + \cdots + 559724432982016 \)
$23$
\( (T^{4} - 145 T^{3} + 20397 T^{2} + \cdots + 394384)^{2} \)
$29$
\( (T^{4} + 34 T^{3} + 16167 T^{2} + \cdots + 225330121)^{2} \)
$31$
\( (T^{4} + 94800 T^{2} + \cdots + 1413760000)^{2} \)
$37$
\( T^{8} - 34506 T^{6} + \cdots + 403490473681 \)
$41$
\( T^{8} - 148122 T^{6} + \cdots + 24\!\cdots\!41 \)
$43$
\( (T^{4} - 455 T^{3} + 195257 T^{2} + \cdots + 138485824)^{2} \)
$47$
\( (T^{4} + 168400 T^{2} + \cdots + 6789760000)^{2} \)
$53$
\( (T^{2} - 545 T - 41450)^{4} \)
$59$
\( T^{8} - 352953 T^{6} + \cdots + 51\!\cdots\!16 \)
$61$
\( (T^{4} + 502 T^{3} + \cdots + 11448786001)^{2} \)
$67$
\( T^{8} - 183201 T^{6} + \cdots + 20\!\cdots\!36 \)
$71$
\( T^{8} - 101777 T^{6} + \cdots + 33\!\cdots\!76 \)
$73$
\( (T^{4} + 232525 T^{2} + \cdots + 3008522500)^{2} \)
$79$
\( (T^{2} - 240 T + 7600)^{4} \)
$83$
\( (T^{4} + 118800 T^{2} + \cdots + 655360000)^{2} \)
$89$
\( T^{8} - 556933 T^{6} + \cdots + 45\!\cdots\!56 \)
$97$
\( T^{8} - 1955781 T^{6} + \cdots + 63\!\cdots\!56 \)
show more
show less