Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [9025,2,Mod(1,9025)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(9025, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([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("9025.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 9025 = 5^{2} \cdot 19^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 9025.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: | \(72.0649878242\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | 6.6.41289040.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - 3x^{5} - 8x^{4} + 21x^{3} + 18x^{2} - 25x - 5 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 475) |
| 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 \(1,\beta_1,\ldots,\beta_{5}\) 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
\( x^{6} - 3x^{5} - 8x^{4} + 21x^{3} + 18x^{2} - 25x - 5 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - \nu - 4 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{5} - \nu^{4} - 10\nu^{3} + 6\nu^{2} + 20\nu - 5 ) / 5 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 2\nu^{5} - 2\nu^{4} - 15\nu^{3} + 7\nu^{2} + 15\nu - 5 ) / 5 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 2\nu^{5} + 3\nu^{4} - 20\nu^{3} - 28\nu^{2} + 25\nu + 20 ) / 5 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + \beta _1 + 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{4} - 2\beta_{3} + \beta_{2} + 6\beta _1 + 3 \)
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| \(\nu^{4}\) | \(=\) |
\( \beta_{5} - 2\beta_{3} + 8\beta_{2} + 11\beta _1 + 26 \)
|
| \(\nu^{5}\) | \(=\) |
\( \beta_{5} + 10\beta_{4} - 17\beta_{3} + 12\beta_{2} + 45\beta _1 + 37 \)
|
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 |
|
−1.48119 | −0.181608 | 0.193937 | 0 | 0.268996 | −1.30422 | 2.67513 | −2.96702 | 0 | ||||||||||||||||||||||||||||||||||||
| 1.2 | −1.48119 | 2.85674 | 0.193937 | 0 | −4.23138 | 3.78541 | 2.67513 | 5.16096 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.3 | 0.311108 | −2.24415 | −1.90321 | 0 | −0.698174 | 3.96928 | −1.21432 | 2.03623 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 0.311108 | 1.02983 | −1.90321 | 0 | 0.320390 | −3.28038 | −1.21432 | −1.93944 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 2.17009 | −1.41269 | 2.70928 | 0 | −3.06566 | −1.76171 | 1.53919 | −1.00431 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 2.17009 | 2.95188 | 2.70928 | 0 | 6.40583 | 0.591620 | 1.53919 | 5.71358 | 0 | |||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(5\) | \(-1\) |
| \(19\) | \(1\) |
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 | 9025.2.a.bz | 6 | |
| 5.b | even | 2 | 1 | 9025.2.a.br | 6 | ||
| 19.b | odd | 2 | 1 | 9025.2.a.bs | 6 | ||
| 19.c | even | 3 | 2 | 475.2.e.f | ✓ | 12 | |
| 95.d | odd | 2 | 1 | 9025.2.a.by | 6 | ||
| 95.i | even | 6 | 2 | 475.2.e.h | yes | 12 | |
| 95.m | odd | 12 | 4 | 475.2.j.d | 24 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 475.2.e.f | ✓ | 12 | 19.c | even | 3 | 2 | |
| 475.2.e.h | yes | 12 | 95.i | even | 6 | 2 | |
| 475.2.j.d | 24 | 95.m | odd | 12 | 4 | ||
| 9025.2.a.br | 6 | 5.b | even | 2 | 1 | ||
| 9025.2.a.bs | 6 | 19.b | odd | 2 | 1 | ||
| 9025.2.a.by | 6 | 95.d | odd | 2 | 1 | ||
| 9025.2.a.bz | 6 | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(9025))\):
|
\( T_{2}^{3} - T_{2}^{2} - 3T_{2} + 1 \)
|
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\( T_{3}^{6} - 3T_{3}^{5} - 8T_{3}^{4} + 21T_{3}^{3} + 18T_{3}^{2} - 25T_{3} - 5 \)
|
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\( T_{7}^{6} - 2T_{7}^{5} - 21T_{7}^{4} + 20T_{7}^{3} + 123T_{7}^{2} + 38T_{7} - 67 \)
|
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\( T_{11}^{6} + T_{11}^{5} - 46T_{11}^{4} + 9T_{11}^{3} + 522T_{11}^{2} - 871T_{11} + 247 \)
|
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\( T_{29}^{6} - 3T_{29}^{5} - 108T_{29}^{4} + 97T_{29}^{3} + 3140T_{29}^{2} + 2693T_{29} - 12781 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{3} - T^{2} - 3 T + 1)^{2} \)
|
| $3$ |
\( T^{6} - 3 T^{5} + \cdots - 5 \)
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| $5$ |
\( T^{6} \)
|
| $7$ |
\( T^{6} - 2 T^{5} + \cdots - 67 \)
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| $11$ |
\( T^{6} + T^{5} + \cdots + 247 \)
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| $13$ |
\( T^{6} - 5 T^{5} + \cdots - 317 \)
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| $17$ |
\( T^{6} + 3 T^{5} + \cdots - 1 \)
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| $19$ |
\( T^{6} \)
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| $23$ |
\( T^{6} + 6 T^{5} + \cdots + 1073 \)
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| $29$ |
\( T^{6} - 3 T^{5} + \cdots - 12781 \)
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| $31$ |
\( T^{6} + 3 T^{5} + \cdots - 631 \)
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| $37$ |
\( T^{6} + 6 T^{5} + \cdots - 64 \)
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| $41$ |
\( T^{6} - 11 T^{5} + \cdots + 1319 \)
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| $43$ |
\( T^{6} - 13 T^{5} + \cdots - 911 \)
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| $47$ |
\( T^{6} + 6 T^{5} + \cdots + 23179 \)
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| $53$ |
\( T^{6} - 18 T^{5} + \cdots + 341861 \)
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| $59$ |
\( T^{6} - 4 T^{5} + \cdots + 11593 \)
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| $61$ |
\( T^{6} - 25 T^{5} + \cdots - 17491 \)
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| $67$ |
\( T^{6} - 6 T^{5} + \cdots - 12080 \)
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| $71$ |
\( T^{6} - 18 T^{5} + \cdots - 11657 \)
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| $73$ |
\( T^{6} - T^{5} + \cdots - 68555 \)
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| $79$ |
\( T^{6} - 3 T^{5} + \cdots + 326351 \)
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| $83$ |
\( T^{6} + 23 T^{5} + \cdots + 378053 \)
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| $89$ |
\( T^{6} - 12 T^{5} + \cdots - 349609 \)
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| $97$ |
\( T^{6} - 3 T^{5} + \cdots - 159631 \)
|
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