Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [9264,2,Mod(1,9264)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(9264, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("9264.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 9264 = 2^{4} \cdot 3 \cdot 193 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 9264.a (trivial) |
Newform invariants
comment: select newform
sage: f = N[0] # Warning: the index may be different
gp: f = lf[1] \\ Warning: the index may be different
| Self dual: | yes |
| Analytic conductor: | \(73.9734124325\) |
| Analytic rank: | \(0\) |
| Dimension: | \(7\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{7} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{7} - x^{6} - 10x^{5} + 11x^{4} + 25x^{3} - 31x^{2} - 6x + 9 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2\cdot 3 \) |
| Twist minimal: | no (minimal twist has level 2316) |
| Fricke sign: | \(-1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-expansion
comment: q-expansion
sage: f.q_expansion() # note that sage often uses an isomorphic number field
gp: mfcoefs(f, 20)
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{6}\) 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
\( x^{7} - x^{6} - 10x^{5} + 11x^{4} + 25x^{3} - 31x^{2} - 6x + 9 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu^{2} - 3 \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{3} - 5\nu + 1 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{6} + \nu^{5} - 9\nu^{4} - 7\nu^{3} + 19\nu^{2} + 8\nu - 5 \)
|
| \(\beta_{4}\) | \(=\) |
\( -2\nu^{6} - \nu^{5} + 18\nu^{4} + 6\nu^{3} - 38\nu^{2} - \nu + 9 \)
|
| \(\beta_{5}\) | \(=\) |
\( 2\nu^{6} + \nu^{5} - 19\nu^{4} - 6\nu^{3} + 45\nu^{2} + \nu - 16 \)
|
| \(\beta_{6}\) | \(=\) |
\( 3\nu^{6} + \nu^{5} - 28\nu^{4} - 5\nu^{3} + 64\nu^{2} - 4\nu - 20 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{6} - \beta_{5} + \beta_{4} + \beta_{3} ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta _1 + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 5\beta_{6} - 5\beta_{5} + 5\beta_{4} + 5\beta_{3} + 2\beta_{2} - 2 ) / 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( -\beta_{5} - \beta_{4} + 7\beta _1 + 14 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 25\beta_{6} - 25\beta_{5} + 27\beta_{4} + 29\beta_{3} + 16\beta_{2} - 14 ) / 2 \)
|
| \(\nu^{6}\) | \(=\) |
\( \beta_{6} - 10\beta_{5} - 9\beta_{4} - \beta_{2} + 44\beta _1 + 74 \)
|
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.
comment: embeddings in the coefficient field
gp: mfembed(f)
| Label | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | |||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1.1 |
|
0 | 1.00000 | 0 | −3.07315 | 0 | 3.31771 | 0 | 1.00000 | 0 | |||||||||||||||||||||||||||||||||||||||
| 1.2 | 0 | 1.00000 | 0 | −2.86897 | 0 | −0.114890 | 0 | 1.00000 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 1.3 | 0 | 1.00000 | 0 | −0.957685 | 0 | −1.47571 | 0 | 1.00000 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 1.4 | 0 | 1.00000 | 0 | 0.625786 | 0 | −4.20092 | 0 | 1.00000 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 1.5 | 0 | 1.00000 | 0 | 0.968278 | 0 | 1.61011 | 0 | 1.00000 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 1.6 | 0 | 1.00000 | 0 | 2.61467 | 0 | 3.36209 | 0 | 1.00000 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 1.7 | 0 | 1.00000 | 0 | 2.69108 | 0 | 2.50161 | 0 | 1.00000 | 0 | ||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( -1 \) |
| \(3\) | \( -1 \) |
| \(193\) | \( -1 \) |
Inner twists
This newform does not admit any (nontrivial) inner twists.
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 9264.2.a.bi | 7 | |
| 4.b | odd | 2 | 1 | 2316.2.a.d | ✓ | 7 | |
| 12.b | even | 2 | 1 | 6948.2.a.h | 7 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 2316.2.a.d | ✓ | 7 | 4.b | odd | 2 | 1 | |
| 6948.2.a.h | 7 | 12.b | even | 2 | 1 | ||
| 9264.2.a.bi | 7 | 1.a | even | 1 | 1 | trivial | |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(9264))\):
|
\( T_{5}^{7} - 17T_{5}^{5} + 5T_{5}^{4} + 80T_{5}^{3} - 44T_{5}^{2} - 60T_{5} + 36 \)
|
|
\( T_{7}^{7} - 5T_{7}^{6} - 13T_{7}^{5} + 101T_{7}^{4} - 92T_{7}^{3} - 208T_{7}^{2} + 256T_{7} + 32 \)
|
|
\( T_{11}^{7} - 7T_{11}^{6} - 5T_{11}^{5} + 127T_{11}^{4} - 260T_{11}^{3} + 64T_{11}^{2} + 144T_{11} - 72 \)
|
|
\( T_{13}^{7} - 5T_{13}^{6} - 32T_{13}^{5} + 85T_{13}^{4} + 352T_{13}^{3} + 344T_{13}^{2} + 128T_{13} + 16 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{7} \)
|
| $3$ |
\( (T - 1)^{7} \)
|
| $5$ |
\( T^{7} - 17 T^{5} + \cdots + 36 \)
|
| $7$ |
\( T^{7} - 5 T^{6} + \cdots + 32 \)
|
| $11$ |
\( T^{7} - 7 T^{6} + \cdots - 72 \)
|
| $13$ |
\( T^{7} - 5 T^{6} + \cdots + 16 \)
|
| $17$ |
\( T^{7} + 2 T^{6} + \cdots - 4964 \)
|
| $19$ |
\( T^{7} - T^{6} + \cdots + 40948 \)
|
| $23$ |
\( T^{7} - 6 T^{6} + \cdots + 144 \)
|
| $29$ |
\( T^{7} + T^{6} + \cdots + 892 \)
|
| $31$ |
\( T^{7} - 15 T^{6} + \cdots + 288 \)
|
| $37$ |
\( T^{7} - 15 T^{6} + \cdots - 2608 \)
|
| $41$ |
\( T^{7} + 9 T^{6} + \cdots + 12924 \)
|
| $43$ |
\( T^{7} - 8 T^{6} + \cdots + 32 \)
|
| $47$ |
\( T^{7} - 16 T^{6} + \cdots - 5832 \)
|
| $53$ |
\( T^{7} + 8 T^{6} + \cdots + 17644 \)
|
| $59$ |
\( T^{7} - 26 T^{6} + \cdots + 896 \)
|
| $61$ |
\( T^{7} - 6 T^{6} + \cdots + 306576 \)
|
| $67$ |
\( T^{7} - 23 T^{6} + \cdots - 64 \)
|
| $71$ |
\( T^{7} - 44 T^{6} + \cdots - 420248 \)
|
| $73$ |
\( T^{7} - 12 T^{6} + \cdots - 22608 \)
|
| $79$ |
\( T^{7} - 25 T^{6} + \cdots - 497044 \)
|
| $83$ |
\( T^{7} - 22 T^{6} + \cdots - 67248 \)
|
| $89$ |
\( T^{7} + 23 T^{6} + \cdots + 3492 \)
|
| $97$ |
\( T^{7} - 2 T^{6} + \cdots - 53321328 \)
|
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