Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [4508,2,Mod(1,4508)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4508, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([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("4508.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 4508 = 2^{2} \cdot 7^{2} \cdot 23 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 4508.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: | \(35.9965612312\) |
| 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: |
\( x^{7} - 13x^{5} - 4x^{4} + 39x^{3} + 26x^{2} - 9x - 1 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 644) |
| 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
\( x^{7} - 13x^{5} - 4x^{4} + 39x^{3} + 26x^{2} - 9x - 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -\nu^{5} + \nu^{4} + 13\nu^{3} - 5\nu^{2} - 37\nu - 9 ) / 5 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{6} - 14\nu^{4} - 3\nu^{3} + 47\nu^{2} + 21\nu - 11 ) / 5 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 2\nu^{5} - 2\nu^{4} - 21\nu^{3} + 15\nu^{2} + 39\nu - 7 ) / 5 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -3\nu^{6} + 4\nu^{5} + 33\nu^{4} - 33\nu^{3} - 66\nu^{2} + 20\nu - 6 ) / 5 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 4\nu^{6} - 3\nu^{5} - 48\nu^{4} + 17\nu^{3} + 123\nu^{2} + 33\nu - 11 ) / 5 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{6} + \beta_{5} - \beta_{3} + \beta_{2} + \beta _1 + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( -\beta_{6} - \beta_{5} + \beta_{4} + \beta_{3} + \beta_{2} + 6\beta _1 + 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( 9\beta_{6} + 8\beta_{5} + 2\beta_{4} - 12\beta_{3} + 9\beta_{2} + 10\beta _1 + 22 \)
|
| \(\nu^{5}\) | \(=\) |
\( -9\beta_{6} - 10\beta_{5} + 15\beta_{4} + 6\beta_{3} + 12\beta_{2} + 46\beta _1 + 24 \)
|
| \(\nu^{6}\) | \(=\) |
\( 76\beta_{6} + 62\beta_{5} + 31\beta_{4} - 113\beta_{3} + 82\beta_{2} + 90\beta _1 + 184 \)
|
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 | −2.18899 | 0 | −0.0719668 | 0 | 0 | 0 | 1.79167 | 0 | |||||||||||||||||||||||||||||||||||||||
| 1.2 | 0 | −2.08484 | 0 | 3.88251 | 0 | 0 | 0 | 1.34656 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 1.3 | 0 | −1.27401 | 0 | −1.42423 | 0 | 0 | 0 | −1.37690 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 1.4 | 0 | 0.662318 | 0 | −2.99194 | 0 | 0 | 0 | −2.56133 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 1.5 | 0 | 2.07917 | 0 | −1.66559 | 0 | 0 | 0 | 1.32296 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 1.6 | 0 | 2.34631 | 0 | 1.98567 | 0 | 0 | 0 | 2.50518 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 1.7 | 0 | 3.46004 | 0 | 2.28554 | 0 | 0 | 0 | 8.97186 | 0 | ||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(7\) | \(1\) |
| \(23\) | \(1\) |
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 | 4508.2.a.m | 7 | |
| 7.b | odd | 2 | 1 | 4508.2.a.l | 7 | ||
| 7.c | even | 3 | 2 | 644.2.i.a | ✓ | 14 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 644.2.i.a | ✓ | 14 | 7.c | even | 3 | 2 | |
| 4508.2.a.l | 7 | 7.b | odd | 2 | 1 | ||
| 4508.2.a.m | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4508))\):
|
\( T_{3}^{7} - 3T_{3}^{6} - 12T_{3}^{5} + 31T_{3}^{4} + 48T_{3}^{3} - 93T_{3}^{2} - 64T_{3} + 65 \)
|
|
\( T_{5}^{7} - 2T_{5}^{6} - 17T_{5}^{5} + 22T_{5}^{4} + 82T_{5}^{3} - 49T_{5}^{2} - 129T_{5} - 9 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{7} \)
|
| $3$ |
\( T^{7} - 3 T^{6} + \cdots + 65 \)
|
| $5$ |
\( T^{7} - 2 T^{6} + \cdots - 9 \)
|
| $7$ |
\( T^{7} \)
|
| $11$ |
\( T^{7} - 67 T^{5} + \cdots + 10233 \)
|
| $13$ |
\( T^{7} - 6 T^{6} + \cdots + 933 \)
|
| $17$ |
\( T^{7} - 4 T^{6} + \cdots - 117 \)
|
| $19$ |
\( T^{7} - 11 T^{6} + \cdots - 7659 \)
|
| $23$ |
\( (T + 1)^{7} \)
|
| $29$ |
\( T^{7} + T^{6} + \cdots - 975 \)
|
| $31$ |
\( T^{7} - 24 T^{6} + \cdots + 52897 \)
|
| $37$ |
\( T^{7} - 11 T^{6} + \cdots - 729 \)
|
| $41$ |
\( T^{7} + 9 T^{6} + \cdots - 1893 \)
|
| $43$ |
\( T^{7} - 5 T^{6} + \cdots + 87797 \)
|
| $47$ |
\( T^{7} - 8 T^{6} + \cdots + 702081 \)
|
| $53$ |
\( T^{7} + 20 T^{6} + \cdots - 81 \)
|
| $59$ |
\( T^{7} - 13 T^{6} + \cdots - 329751 \)
|
| $61$ |
\( T^{7} + 2 T^{6} + \cdots + 74913 \)
|
| $67$ |
\( T^{7} + 4 T^{6} + \cdots - 471 \)
|
| $71$ |
\( T^{7} + 8 T^{6} + \cdots + 21195 \)
|
| $73$ |
\( T^{7} - 11 T^{6} + \cdots + 22815 \)
|
| $79$ |
\( T^{7} - 28 T^{6} + \cdots - 118171 \)
|
| $83$ |
\( T^{7} - 21 T^{6} + \cdots - 452721 \)
|
| $89$ |
\( T^{7} + 9 T^{6} + \cdots + 42591 \)
|
| $97$ |
\( T^{7} - T^{6} + \cdots + 2066909 \)
|
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