Newspace parameters
Level: | \( N \) | \(=\) | \( 1183 = 7 \cdot 13^{2} \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 1183.c (of order \(2\), degree \(1\), not minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(9.44630255912\) |
Analytic rank: | \(0\) |
Dimension: | \(4\) |
Coefficient field: | \(\Q(\zeta_{12})\) |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
Defining polynomial: |
\( x^{4} - x^{2} + 1 \)
|
Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
Coefficient ring index: | \( 2^{2} \) |
Twist minimal: | no (minimal twist has level 91) |
Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.
Basis of coefficient ring
\(\beta_{1}\) | \(=\) |
\( \zeta_{12}^{3} \)
|
\(\beta_{2}\) | \(=\) |
\( 2\zeta_{12}^{2} - 1 \)
|
\(\beta_{3}\) | \(=\) |
\( -\zeta_{12}^{3} + 2\zeta_{12} \)
|
\(\zeta_{12}\) | \(=\) |
\( ( \beta_{3} + \beta_1 ) / 2 \)
|
\(\zeta_{12}^{2}\) | \(=\) |
\( ( \beta_{2} + 1 ) / 2 \)
|
\(\zeta_{12}^{3}\) | \(=\) |
\( \beta_1 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1183\mathbb{Z}\right)^\times\).
\(n\) | \(339\) | \(1016\) |
\(\chi(n)\) | \(1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
337.1 |
|
− | 1.73205i | −0.732051 | −1.00000 | − | 1.73205i | 1.26795i | − | 1.00000i | − | 1.73205i | −2.46410 | −3.00000 | ||||||||||||||||||||||||||
337.2 | − | 1.73205i | 2.73205 | −1.00000 | − | 1.73205i | − | 4.73205i | 1.00000i | − | 1.73205i | 4.46410 | −3.00000 | |||||||||||||||||||||||||||
337.3 | 1.73205i | −0.732051 | −1.00000 | 1.73205i | − | 1.26795i | 1.00000i | 1.73205i | −2.46410 | −3.00000 | ||||||||||||||||||||||||||||||
337.4 | 1.73205i | 2.73205 | −1.00000 | 1.73205i | 4.73205i | − | 1.00000i | 1.73205i | 4.46410 | −3.00000 | ||||||||||||||||||||||||||||||
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 | 1183.2.c.e | 4 | |
13.b | even | 2 | 1 | inner | 1183.2.c.e | 4 | |
13.d | odd | 4 | 1 | 1183.2.a.e | 2 | ||
13.d | odd | 4 | 1 | 1183.2.a.f | 2 | ||
13.f | odd | 12 | 2 | 91.2.f.b | ✓ | 4 | |
39.k | even | 12 | 2 | 819.2.o.b | 4 | ||
52.l | even | 12 | 2 | 1456.2.s.o | 4 | ||
91.i | even | 4 | 1 | 8281.2.a.r | 2 | ||
91.i | even | 4 | 1 | 8281.2.a.t | 2 | ||
91.w | even | 12 | 2 | 637.2.g.d | 4 | ||
91.x | odd | 12 | 2 | 637.2.h.d | 4 | ||
91.ba | even | 12 | 2 | 637.2.h.e | 4 | ||
91.bc | even | 12 | 2 | 637.2.f.d | 4 | ||
91.bd | odd | 12 | 2 | 637.2.g.e | 4 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
91.2.f.b | ✓ | 4 | 13.f | odd | 12 | 2 | |
637.2.f.d | 4 | 91.bc | even | 12 | 2 | ||
637.2.g.d | 4 | 91.w | even | 12 | 2 | ||
637.2.g.e | 4 | 91.bd | odd | 12 | 2 | ||
637.2.h.d | 4 | 91.x | odd | 12 | 2 | ||
637.2.h.e | 4 | 91.ba | even | 12 | 2 | ||
819.2.o.b | 4 | 39.k | even | 12 | 2 | ||
1183.2.a.e | 2 | 13.d | odd | 4 | 1 | ||
1183.2.a.f | 2 | 13.d | odd | 4 | 1 | ||
1183.2.c.e | 4 | 1.a | even | 1 | 1 | trivial | |
1183.2.c.e | 4 | 13.b | even | 2 | 1 | inner | |
1456.2.s.o | 4 | 52.l | even | 12 | 2 | ||
8281.2.a.r | 2 | 91.i | even | 4 | 1 | ||
8281.2.a.t | 2 | 91.i | even | 4 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{2} + 3 \)
acting on \(S_{2}^{\mathrm{new}}(1183, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( (T^{2} + 3)^{2} \)
$3$
\( (T^{2} - 2 T - 2)^{2} \)
$5$
\( (T^{2} + 3)^{2} \)
$7$
\( (T^{2} + 1)^{2} \)
$11$
\( T^{4} + 24T^{2} + 36 \)
$13$
\( T^{4} \)
$17$
\( (T^{2} - 12 T + 33)^{2} \)
$19$
\( (T^{2} + 4)^{2} \)
$23$
\( (T^{2} + 6 T + 6)^{2} \)
$29$
\( (T + 3)^{4} \)
$31$
\( T^{4} + 56T^{2} + 676 \)
$37$
\( (T^{2} + 49)^{2} \)
$41$
\( (T^{2} + 27)^{2} \)
$43$
\( (T^{2} + 10 T - 2)^{2} \)
$47$
\( T^{4} + 168T^{2} + 144 \)
$53$
\( (T^{2} + 6 T - 39)^{2} \)
$59$
\( T^{4} + 168T^{2} + 6084 \)
$61$
\( (T^{2} + 20 T + 73)^{2} \)
$67$
\( T^{4} + 56T^{2} + 676 \)
$71$
\( (T^{2} + 36)^{2} \)
$73$
\( T^{4} + 62T^{2} + 529 \)
$79$
\( (T^{2} - 22 T + 94)^{2} \)
$83$
\( T^{4} + 72T^{2} + 324 \)
$89$
\( T^{4} + 168T^{2} + 144 \)
$97$
\( T^{4} + 248T^{2} + 8464 \)
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