Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [8025,2,Mod(1,8025)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8025, 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("8025.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 8025 = 3 \cdot 5^{2} \cdot 107 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 8025.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: | \(64.0799476221\) |
| 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: |
\( x^{7} - 3x^{6} - 9x^{5} + 24x^{4} + 13x^{3} - 47x^{2} + 19x - 2 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 321) |
| 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
\( x^{7} - 3x^{6} - 9x^{5} + 24x^{4} + 13x^{3} - 47x^{2} + 19x - 2 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -\nu^{5} + \nu^{4} + 12\nu^{3} - \nu^{2} - 27\nu ) / 2 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{6} - 2\nu^{5} - 11\nu^{4} + 13\nu^{3} + 26\nu^{2} - 23\nu - 2 ) / 2 \)
|
| \(\beta_{4}\) | \(=\) |
\( \nu^{6} - 2\nu^{5} - 11\nu^{4} + 14\nu^{3} + 24\nu^{2} - 29\nu + 3 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 3\nu^{6} - 8\nu^{5} - 29\nu^{4} + 61\nu^{3} + 54\nu^{2} - 119\nu + 24 ) / 2 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( -3\nu^{6} + 9\nu^{5} + 28\nu^{4} - 73\nu^{3} - 51\nu^{2} + 144\nu - 32 ) / 2 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{6} + \beta_{5} + \beta_{2} + \beta _1 + 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( 2\beta_{6} + 2\beta_{5} + \beta_{4} - 2\beta_{3} + 2\beta_{2} + 8\beta _1 + 3 \)
|
| \(\nu^{4}\) | \(=\) |
\( 13\beta_{6} + 14\beta_{5} + \beta_{4} - 5\beta_{3} + 11\beta_{2} + 17\beta _1 + 32 \)
|
| \(\nu^{5}\) | \(=\) |
\( 36\beta_{6} + 37\beta_{5} + 13\beta_{4} - 29\beta_{3} + 32\beta_{2} + 85\beta _1 + 64 \)
|
| \(\nu^{6}\) | \(=\) |
\( 163\beta_{6} + 176\beta_{5} + 24\beta_{4} - 85\beta_{3} + 133\beta_{2} + 250\beta _1 + 339 \)
|
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 |
|
−2.79760 | 1.00000 | 5.82655 | 0 | −2.79760 | −3.22568 | −10.7051 | 1.00000 | 0 | |||||||||||||||||||||||||||||||||||||||
| 1.2 | −2.28249 | 1.00000 | 3.20977 | 0 | −2.28249 | 1.42306 | −2.76128 | 1.00000 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 1.3 | −1.21213 | 1.00000 | −0.530744 | 0 | −1.21213 | −4.47154 | 3.06759 | 1.00000 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 1.4 | 0.535263 | 1.00000 | −1.71349 | 0 | 0.535263 | 3.02020 | −1.98769 | 1.00000 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 1.5 | 0.726480 | 1.00000 | −1.47223 | 0 | 0.726480 | −3.04775 | −2.52250 | 1.00000 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 1.6 | 2.39839 | 1.00000 | 3.75227 | 0 | 2.39839 | −2.59863 | 4.20262 | 1.00000 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 1.7 | 2.63209 | 1.00000 | 4.92788 | 0 | 2.63209 | 2.90033 | 7.70643 | 1.00000 | 0 | ||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(-1\) |
| \(5\) | \(1\) |
| \(107\) | \(1\) |
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 | 8025.2.a.bb | 7 | |
| 5.b | even | 2 | 1 | 321.2.a.d | ✓ | 7 | |
| 15.d | odd | 2 | 1 | 963.2.a.e | 7 | ||
| 20.d | odd | 2 | 1 | 5136.2.a.bi | 7 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 321.2.a.d | ✓ | 7 | 5.b | even | 2 | 1 | |
| 963.2.a.e | 7 | 15.d | odd | 2 | 1 | ||
| 5136.2.a.bi | 7 | 20.d | odd | 2 | 1 | ||
| 8025.2.a.bb | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8025))\):
|
\( T_{2}^{7} - 14T_{2}^{5} + T_{2}^{4} + 55T_{2}^{3} - 8T_{2}^{2} - 46T_{2} + 19 \)
|
|
\( T_{7}^{7} + 6T_{7}^{6} - 15T_{7}^{5} - 124T_{7}^{4} + 33T_{7}^{3} + 788T_{7}^{2} + 188T_{7} - 1424 \)
|
|
\( T_{11}^{7} - 4T_{11}^{6} - 33T_{11}^{5} + 112T_{11}^{4} + 277T_{11}^{3} - 610T_{11}^{2} - 556T_{11} + 976 \)
|
|
\( T_{13}^{7} + 6T_{13}^{6} - 20T_{13}^{5} - 94T_{13}^{4} + 152T_{13}^{3} + 276T_{13}^{2} - 351T_{13} + 94 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{7} - 14 T^{5} + \cdots + 19 \)
|
| $3$ |
\( (T - 1)^{7} \)
|
| $5$ |
\( T^{7} \)
|
| $7$ |
\( T^{7} + 6 T^{6} + \cdots - 1424 \)
|
| $11$ |
\( T^{7} - 4 T^{6} + \cdots + 976 \)
|
| $13$ |
\( T^{7} + 6 T^{6} + \cdots + 94 \)
|
| $17$ |
\( T^{7} - 10 T^{6} + \cdots - 8762 \)
|
| $19$ |
\( T^{7} - 8 T^{6} + \cdots - 6208 \)
|
| $23$ |
\( T^{7} + 6 T^{6} + \cdots - 1664 \)
|
| $29$ |
\( T^{7} - 56 T^{5} + \cdots + 2432 \)
|
| $31$ |
\( T^{7} - 16 T^{6} + \cdots + 26512 \)
|
| $37$ |
\( T^{7} + 10 T^{6} + \cdots - 13642 \)
|
| $41$ |
\( T^{7} + 2 T^{6} + \cdots + 2048 \)
|
| $43$ |
\( T^{7} + 2 T^{6} + \cdots + 15424 \)
|
| $47$ |
\( T^{7} + 16 T^{6} + \cdots + 127808 \)
|
| $53$ |
\( T^{7} - 16 T^{6} + \cdots - 512 \)
|
| $59$ |
\( T^{7} - 20 T^{6} + \cdots + 806912 \)
|
| $61$ |
\( T^{7} - 2 T^{6} + \cdots - 2294 \)
|
| $67$ |
\( T^{7} + 30 T^{6} + \cdots - 607744 \)
|
| $71$ |
\( T^{7} - 32 T^{6} + \cdots + 192256 \)
|
| $73$ |
\( T^{7} - 12 T^{6} + \cdots - 23680 \)
|
| $79$ |
\( T^{7} - 36 T^{6} + \cdots + 8192 \)
|
| $83$ |
\( T^{7} - 10 T^{6} + \cdots + 3919616 \)
|
| $89$ |
\( T^{7} + 4 T^{6} + \cdots - 8192 \)
|
| $97$ |
\( T^{7} + 24 T^{6} + \cdots + 3247616 \)
|
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