Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [7524,2,Mod(1,7524)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(7524, 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("7524.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 7524 = 2^{2} \cdot 3^{2} \cdot 11 \cdot 19 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 7524.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: | \(60.0794424808\) |
| 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: |
\( x^{7} - 3x^{6} - 20x^{5} + 16x^{4} + 124x^{3} + 100x^{2} - 24x - 24 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| 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
\( x^{7} - 3x^{6} - 20x^{5} + 16x^{4} + 124x^{3} + 100x^{2} - 24x - 24 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{5} - 5\nu^{4} - 10\nu^{3} + 36\nu^{2} + 52\nu - 4 ) / 4 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -3\nu^{6} + 14\nu^{5} + 37\nu^{4} - 112\nu^{3} - 188\nu^{2} + 36\nu + 40 ) / 4 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 2\nu^{6} - 9\nu^{5} - 26\nu^{4} + 69\nu^{3} + 138\nu^{2} + 6\nu - 28 ) / 2 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -3\nu^{6} + 14\nu^{5} + 37\nu^{4} - 111\nu^{3} - 191\nu^{2} + 26\nu + 40 ) / 2 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( -6\nu^{6} + 27\nu^{5} + 79\nu^{4} - 212\nu^{3} - 422\nu^{2} + 16\nu + 108 ) / 4 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( -\beta_{6} + \beta_{5} - \beta_{2} + 4\beta _1 + 6 \)
|
| \(\nu^{3}\) | \(=\) |
\( -3\beta_{6} + 5\beta_{5} - 4\beta_{3} - 3\beta_{2} + 22\beta _1 + 18 \)
|
| \(\nu^{4}\) | \(=\) |
\( -19\beta_{6} + 33\beta_{5} + 6\beta_{4} - 20\beta_{3} - 23\beta_{2} + 108\beta _1 + 114 \)
|
| \(\nu^{5}\) | \(=\) |
\( -89\beta_{6} + 179\beta_{5} + 30\beta_{4} - 140\beta_{3} - 105\beta_{2} + 564\beta _1 + 538 \)
|
| \(\nu^{6}\) | \(=\) |
\( -475\beta_{6} + 993\beta_{5} + 214\beta_{4} - 752\beta_{3} - 599\beta_{2} + 2904\beta _1 + 2882 \)
|
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 | 0 | 0 | −3.96346 | 0 | −4.92304 | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||
| 1.2 | 0 | 0 | 0 | −3.26966 | 0 | 4.63854 | 0 | 0 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 1.3 | 0 | 0 | 0 | −3.26027 | 0 | 1.07116 | 0 | 0 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 1.4 | 0 | 0 | 0 | 0.457443 | 0 | −1.15819 | 0 | 0 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 1.5 | 0 | 0 | 0 | 0.876239 | 0 | 3.13754 | 0 | 0 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 1.6 | 0 | 0 | 0 | 1.42172 | 0 | −2.91084 | 0 | 0 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 1.7 | 0 | 0 | 0 | 3.73799 | 0 | −1.85517 | 0 | 0 | 0 | ||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(3\) | \(1\) |
| \(11\) | \(-1\) |
| \(19\) | \(1\) |
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 | 7524.2.a.s | ✓ | 7 |
| 3.b | odd | 2 | 1 | 7524.2.a.v | yes | 7 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 7524.2.a.s | ✓ | 7 | 1.a | even | 1 | 1 | trivial |
| 7524.2.a.v | yes | 7 | 3.b | odd | 2 | 1 | |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(7524))\):
|
\( T_{5}^{7} + 4T_{5}^{6} - 19T_{5}^{5} - 72T_{5}^{4} + 92T_{5}^{3} + 220T_{5}^{2} - 309T_{5} + 90 \)
|
|
\( T_{7}^{7} + 2T_{7}^{6} - 33T_{7}^{5} - 62T_{7}^{4} + 248T_{7}^{3} + 476T_{7}^{2} - 231T_{7} - 480 \)
|
|
\( T_{17}^{7} + 3T_{17}^{6} - 44T_{17}^{5} - 174T_{17}^{4} + 31T_{17}^{3} + 341T_{17}^{2} - 155T_{17} + 17 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{7} \)
|
| $3$ |
\( T^{7} \)
|
| $5$ |
\( T^{7} + 4 T^{6} + \cdots + 90 \)
|
| $7$ |
\( T^{7} + 2 T^{6} + \cdots - 480 \)
|
| $11$ |
\( (T - 1)^{7} \)
|
| $13$ |
\( T^{7} + 3 T^{6} + \cdots - 7573 \)
|
| $17$ |
\( T^{7} + 3 T^{6} + \cdots + 17 \)
|
| $19$ |
\( (T + 1)^{7} \)
|
| $23$ |
\( T^{7} + 2 T^{6} + \cdots + 412 \)
|
| $29$ |
\( T^{7} + 18 T^{6} + \cdots - 1770 \)
|
| $31$ |
\( T^{7} + 4 T^{6} + \cdots + 434756 \)
|
| $37$ |
\( T^{7} + 10 T^{6} + \cdots + 888 \)
|
| $41$ |
\( T^{7} + 6 T^{6} + \cdots + 1052 \)
|
| $43$ |
\( T^{7} + 12 T^{6} + \cdots + 2412 \)
|
| $47$ |
\( T^{7} - 22 T^{6} + \cdots + 622254 \)
|
| $53$ |
\( T^{7} + 13 T^{6} + \cdots + 399041 \)
|
| $59$ |
\( T^{7} + 23 T^{6} + \cdots - 53019 \)
|
| $61$ |
\( T^{7} + 4 T^{6} + \cdots + 42 \)
|
| $67$ |
\( T^{7} + 14 T^{6} + \cdots + 18360 \)
|
| $71$ |
\( T^{7} - 5 T^{6} + \cdots + 278307 \)
|
| $73$ |
\( T^{7} - 154 T^{5} + \cdots + 6792 \)
|
| $79$ |
\( T^{7} - 13 T^{6} + \cdots + 62957 \)
|
| $83$ |
\( T^{7} + T^{6} + \cdots - 243147 \)
|
| $89$ |
\( T^{7} + 11 T^{6} + \cdots - 10161 \)
|
| $97$ |
\( T^{7} + 4 T^{6} + \cdots + 415828 \)
|
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