Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [7581,2,Mod(1,7581)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(7581, 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("7581.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 7581 = 3 \cdot 7 \cdot 19^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 7581.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: | \(60.5345897723\) |
| Analytic rank: | \(1\) |
| Dimension: | \(9\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{9} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{9} - 3x^{8} - 9x^{7} + 34x^{6} + 6x^{5} - 99x^{4} + 75x^{3} + 15x^{2} - 18x - 1 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 399) |
| 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 \(1,\beta_1,\ldots,\beta_{8}\) 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
\( x^{9} - 3x^{8} - 9x^{7} + 34x^{6} + 6x^{5} - 99x^{4} + 75x^{3} + 15x^{2} - 18x - 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{7} - 11\nu^{5} + 32\nu^{3} - 2\nu^{2} - 17\nu - 2 ) / 2 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{7} - \nu^{6} - 11\nu^{5} + 11\nu^{4} + 30\nu^{3} - 32\nu^{2} - 3\nu + 3 ) / 2 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{7} - \nu^{6} - 11\nu^{5} + 13\nu^{4} + 28\nu^{3} - 44\nu^{2} + 9\nu + 7 ) / 2 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( \nu^{8} - \nu^{7} - 11\nu^{6} + 11\nu^{5} + 30\nu^{4} - 32\nu^{3} - 3\nu^{2} + 3\nu + 2 ) / 2 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( -\nu^{8} + 2\nu^{7} + 11\nu^{6} - 24\nu^{5} - 28\nu^{4} + 78\nu^{3} - 11\nu^{2} - 32\nu + 2 ) / 2 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 2\nu^{8} - 3\nu^{7} - 22\nu^{6} + 35\nu^{5} + 58\nu^{4} - 108\nu^{3} + 8\nu^{2} + 25\nu + 2 ) / 2 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 3\nu^{8} - 4\nu^{7} - 33\nu^{6} + 46\nu^{5} + 88\nu^{4} - 140\nu^{3} + 7\nu^{2} + 28\nu - 2 ) / 2 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{8} - \beta_{7} - \beta_{5} + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{7} + \beta_{6} - \beta_{5} + 5\beta _1 - 1 \)
|
| \(\nu^{4}\) | \(=\) |
\( 6\beta_{8} - 5\beta_{7} + \beta_{6} - 7\beta_{5} + \beta_{4} - \beta_{3} - \beta _1 + 15 \)
|
| \(\nu^{5}\) | \(=\) |
\( 8\beta_{7} + 7\beta_{6} - 9\beta_{5} + \beta_{4} - \beta_{3} + \beta_{2} + 28\beta _1 - 7 \)
|
| \(\nu^{6}\) | \(=\) |
\( 36\beta_{8} - 27\beta_{7} + 9\beta_{6} - 45\beta_{5} + 11\beta_{4} - 13\beta_{3} + 2\beta_{2} - 7\beta _1 + 82 \)
|
| \(\nu^{7}\) | \(=\) |
\( 2\beta_{8} + 54\beta_{7} + 45\beta_{6} - 69\beta_{5} + 11\beta_{4} - 11\beta_{3} + 13\beta_{2} + 165\beta _1 - 37 \)
|
| \(\nu^{8}\) | \(=\) |
\( 221 \beta_{8} - 152 \beta_{7} + 69 \beta_{6} - 288 \beta_{5} + 91 \beta_{4} - 113 \beta_{3} + 24 \beta_{2} + \cdots + 467 \)
|
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 |
|
−2.42419 | 1.00000 | 3.87669 | 0.0945335 | −2.42419 | −1.00000 | −4.54945 | 1.00000 | −0.229167 | |||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −2.30497 | 1.00000 | 3.31290 | −0.817606 | −2.30497 | −1.00000 | −3.02619 | 1.00000 | 1.88456 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −0.437590 | 1.00000 | −1.80852 | −3.15346 | −0.437590 | −1.00000 | 1.66657 | 1.00000 | 1.37992 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −0.0538367 | 1.00000 | −1.99710 | 2.07987 | −0.0538367 | −1.00000 | 0.215191 | 1.00000 | −0.111973 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | 0.716345 | 1.00000 | −1.48685 | 0.830283 | 0.716345 | −1.00000 | −2.49779 | 1.00000 | 0.594769 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 1.17166 | 1.00000 | −0.627216 | −2.42717 | 1.17166 | −1.00000 | −3.07820 | 1.00000 | −2.84381 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 1.82672 | 1.00000 | 1.33691 | 1.80191 | 1.82672 | −1.00000 | −1.21129 | 1.00000 | 3.29159 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 1.90523 | 1.00000 | 1.62991 | 3.02723 | 1.90523 | −1.00000 | −0.705103 | 1.00000 | 5.76757 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 2.60063 | 1.00000 | 4.76328 | −1.43560 | 2.60063 | −1.00000 | 7.18626 | 1.00000 | −3.73346 | ||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(-1\) |
| \(7\) | \(1\) |
| \(19\) | \(-1\) |
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 | 7581.2.a.bh | 9 | |
| 19.b | odd | 2 | 1 | 7581.2.a.bg | 9 | ||
| 19.f | odd | 18 | 2 | 399.2.bo.b | ✓ | 18 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 399.2.bo.b | ✓ | 18 | 19.f | odd | 18 | 2 | |
| 7581.2.a.bg | 9 | 19.b | odd | 2 | 1 | ||
| 7581.2.a.bh | 9 | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(7581))\):
|
\( T_{2}^{9} - 3T_{2}^{8} - 9T_{2}^{7} + 34T_{2}^{6} + 6T_{2}^{5} - 99T_{2}^{4} + 75T_{2}^{3} + 15T_{2}^{2} - 18T_{2} - 1 \)
|
|
\( T_{5}^{9} - 18T_{5}^{7} + 2T_{5}^{6} + 99T_{5}^{5} - 18T_{5}^{4} - 183T_{5}^{3} + 24T_{5}^{2} + 84T_{5} - 8 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{9} - 3 T^{8} + \cdots - 1 \)
|
| $3$ |
\( (T - 1)^{9} \)
|
| $5$ |
\( T^{9} - 18 T^{7} + \cdots - 8 \)
|
| $7$ |
\( (T + 1)^{9} \)
|
| $11$ |
\( T^{9} + 12 T^{8} + \cdots + 53 \)
|
| $13$ |
\( T^{9} + 9 T^{8} + \cdots - 2591 \)
|
| $17$ |
\( T^{9} - 72 T^{7} + \cdots + 13661 \)
|
| $19$ |
\( T^{9} \)
|
| $23$ |
\( T^{9} + 21 T^{8} + \cdots - 3781 \)
|
| $29$ |
\( T^{9} - 15 T^{8} + \cdots - 429029 \)
|
| $31$ |
\( T^{9} + 27 T^{8} + \cdots + 9521 \)
|
| $37$ |
\( T^{9} - 108 T^{7} + \cdots + 937 \)
|
| $41$ |
\( T^{9} - 9 T^{8} + \cdots + 90199 \)
|
| $43$ |
\( T^{9} + 9 T^{8} + \cdots + 1877327 \)
|
| $47$ |
\( T^{9} + 15 T^{8} + \cdots + 199223 \)
|
| $53$ |
\( T^{9} - 15 T^{8} + \cdots - 1302317 \)
|
| $59$ |
\( T^{9} + 9 T^{8} + \cdots - 122201 \)
|
| $61$ |
\( T^{9} - 9 T^{8} + \cdots - 92555821 \)
|
| $67$ |
\( T^{9} + 18 T^{8} + \cdots + 33012136 \)
|
| $71$ |
\( T^{9} + 9 T^{8} + \cdots - 5146343 \)
|
| $73$ |
\( T^{9} + 39 T^{8} + \cdots - 8976493 \)
|
| $79$ |
\( T^{9} - 453 T^{7} + \cdots + 171288512 \)
|
| $83$ |
\( T^{9} - 30 T^{8} + \cdots + 415881863 \)
|
| $89$ |
\( T^{9} - 9 T^{8} + \cdots - 13260617 \)
|
| $97$ |
\( T^{9} + 3 T^{8} + \cdots - 6338737 \)
|
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