Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [7504,2,Mod(1,7504)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(7504, 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("7504.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 7504 = 2^{4} \cdot 7 \cdot 67 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 7504.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: | \(59.9197416768\) |
| Analytic rank: | \(1\) |
| 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: |
\( x^{7} - 10x^{5} - x^{4} + 28x^{3} + 3x^{2} - 17x + 4 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 2 \) |
| Twist minimal: | no (minimal twist has level 1876) |
| 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
\( x^{7} - 10x^{5} - x^{4} + 28x^{3} + 3x^{2} - 17x + 4 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - 4\nu \)
|
| \(\beta_{4}\) | \(=\) |
\( \nu^{4} - 6\nu^{2} - \nu + 5 \)
|
| \(\beta_{5}\) | \(=\) |
\( \nu^{5} - \nu^{4} - 7\nu^{3} + 5\nu^{2} + 11\nu - 4 \)
|
| \(\beta_{6}\) | \(=\) |
\( \nu^{6} - 9\nu^{4} - 2\nu^{3} + 21\nu^{2} + 8\nu - 7 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + 4\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{4} + 6\beta_{2} + \beta _1 + 13 \)
|
| \(\nu^{5}\) | \(=\) |
\( \beta_{5} + \beta_{4} + 7\beta_{3} + \beta_{2} + 18\beta _1 + 2 \)
|
| \(\nu^{6}\) | \(=\) |
\( \beta_{6} + 9\beta_{4} + 2\beta_{3} + 33\beta_{2} + 9\beta _1 + 61 \)
|
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 | −2.20649 | 0 | −2.86858 | 0 | 1.00000 | 0 | 1.86858 | 0 | |||||||||||||||||||||||||||||||||||||||
| 1.2 | 0 | −1.88667 | 0 | −1.55952 | 0 | 1.00000 | 0 | 0.559517 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 1.3 | 0 | −1.17933 | 0 | 0.609183 | 0 | 1.00000 | 0 | −1.60918 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 1.4 | 0 | 0.287447 | 0 | 1.91737 | 0 | 1.00000 | 0 | −2.91737 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 1.5 | 0 | 0.594951 | 0 | 1.64603 | 0 | 1.00000 | 0 | −2.64603 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 1.6 | 0 | 1.96259 | 0 | −1.85178 | 0 | 1.00000 | 0 | 0.851777 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 1.7 | 0 | 2.42749 | 0 | −3.89271 | 0 | 1.00000 | 0 | 2.89271 | 0 | ||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(7\) | \(-1\) |
| \(67\) | \(1\) |
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 | 7504.2.a.bq | 7 | |
| 4.b | odd | 2 | 1 | 1876.2.a.e | ✓ | 7 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1876.2.a.e | ✓ | 7 | 4.b | odd | 2 | 1 | |
| 7504.2.a.bq | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(7504))\):
|
\( T_{3}^{7} - 10T_{3}^{5} - T_{3}^{4} + 28T_{3}^{3} + 3T_{3}^{2} - 17T_{3} + 4 \)
|
|
\( T_{5}^{7} + 6T_{5}^{6} - 45T_{5}^{4} - 30T_{5}^{3} + 101T_{5}^{2} + 61T_{5} - 62 \)
|
|
\( T_{11}^{7} + 2T_{11}^{6} - 27T_{11}^{5} - 87T_{11}^{4} + 28T_{11}^{3} + 344T_{11}^{2} + 384T_{11} + 128 \)
|
|
\( T_{13}^{7} - 2T_{13}^{6} - 52T_{13}^{5} + 67T_{13}^{4} + 670T_{13}^{3} - 453T_{13}^{2} - 1223T_{13} + 808 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{7} \)
|
| $3$ |
\( T^{7} - 10 T^{5} + \cdots + 4 \)
|
| $5$ |
\( T^{7} + 6 T^{6} + \cdots - 62 \)
|
| $7$ |
\( (T - 1)^{7} \)
|
| $11$ |
\( T^{7} + 2 T^{6} + \cdots + 128 \)
|
| $13$ |
\( T^{7} - 2 T^{6} + \cdots + 808 \)
|
| $17$ |
\( T^{7} + 8 T^{6} + \cdots + 1096 \)
|
| $19$ |
\( T^{7} - 2 T^{6} + \cdots - 32 \)
|
| $23$ |
\( T^{7} - 96 T^{5} + \cdots - 10328 \)
|
| $29$ |
\( T^{7} + 10 T^{6} + \cdots + 15203 \)
|
| $31$ |
\( T^{7} - 12 T^{6} + \cdots + 109202 \)
|
| $37$ |
\( T^{7} + 9 T^{6} + \cdots - 2347 \)
|
| $41$ |
\( T^{7} + 20 T^{6} + \cdots - 4364 \)
|
| $43$ |
\( T^{7} + 3 T^{6} + \cdots + 12368 \)
|
| $47$ |
\( T^{7} - 11 T^{6} + \cdots + 23896 \)
|
| $53$ |
\( T^{7} + 30 T^{6} + \cdots + 6592 \)
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| $59$ |
\( T^{7} - 17 T^{6} + \cdots + 97676 \)
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| $61$ |
\( T^{7} + 26 T^{6} + \cdots + 421888 \)
|
| $67$ |
\( (T + 1)^{7} \)
|
| $71$ |
\( T^{7} - 4 T^{6} + \cdots + 4616 \)
|
| $73$ |
\( T^{7} + 32 T^{6} + \cdots - 7088 \)
|
| $79$ |
\( T^{7} - 22 T^{6} + \cdots - 215104 \)
|
| $83$ |
\( T^{7} - 6 T^{6} + \cdots + 681472 \)
|
| $89$ |
\( T^{7} + 23 T^{6} + \cdots - 283072 \)
|
| $97$ |
\( T^{7} - 29 T^{6} + \cdots + 7457024 \)
|
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