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: | \(6\) |
| Coefficient field: | 6.6.13231312.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - x^{5} - 8x^{4} + 9x^{3} + 8x^{2} - 9x + 1 \)
|
| 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 \(1,\beta_1,\ldots,\beta_{5}\) 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
\( x^{6} - x^{5} - 8x^{4} + 9x^{3} + 8x^{2} - 9x + 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{4} - 7\nu^{2} + 2\nu + 5 ) / 2 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{5} - 7\nu^{3} + 2\nu^{2} + 3\nu - 2 ) / 2 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{5} - \nu^{4} - 9\nu^{3} + 9\nu^{2} + 13\nu - 9 ) / 2 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 2\nu^{5} - \nu^{4} - 16\nu^{3} + 9\nu^{2} + 18\nu - 5 ) / 2 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( -\beta_{5} + \beta_{4} + \beta_{3} + \beta _1 + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( -\beta_{4} + \beta_{3} - \beta_{2} + 6\beta _1 - 1 \)
|
| \(\nu^{4}\) | \(=\) |
\( -7\beta_{5} + 7\beta_{4} + 7\beta_{3} + 2\beta_{2} + 5\beta _1 + 16 \)
|
| \(\nu^{5}\) | \(=\) |
\( 2\beta_{5} - 9\beta_{4} + 7\beta_{3} - 7\beta_{2} + 37\beta _1 - 11 \)
|
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 |
|
−2.57072 | −1.00000 | 4.60860 | 0 | 2.57072 | 4.28120 | −6.70597 | 1.00000 | 0 | ||||||||||||||||||||||||||||||||||||
| 1.2 | −2.12439 | −1.00000 | 2.51302 | 0 | 2.12439 | −2.71830 | −1.08985 | 1.00000 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.3 | −1.40654 | −1.00000 | −0.0216539 | 0 | 1.40654 | −1.09008 | 2.84353 | 1.00000 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.4 | −0.163212 | −1.00000 | −1.97336 | 0 | 0.163212 | −1.53872 | 0.648498 | 1.00000 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 1.11100 | −1.00000 | −0.765670 | 0 | −1.11100 | 0.814220 | −3.07267 | 1.00000 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 2.15385 | −1.00000 | 2.63907 | 0 | −2.15385 | 0.251675 | 1.37646 | 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.ba | 6 | |
| 5.b | even | 2 | 1 | 321.2.a.c | ✓ | 6 | |
| 15.d | odd | 2 | 1 | 963.2.a.d | 6 | ||
| 20.d | odd | 2 | 1 | 5136.2.a.bg | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 321.2.a.c | ✓ | 6 | 5.b | even | 2 | 1 | |
| 963.2.a.d | 6 | 15.d | odd | 2 | 1 | ||
| 5136.2.a.bg | 6 | 20.d | odd | 2 | 1 | ||
| 8025.2.a.ba | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8025))\):
|
\( T_{2}^{6} + 3T_{2}^{5} - 5T_{2}^{4} - 18T_{2}^{3} + T_{2}^{2} + 19T_{2} + 3 \)
|
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\( T_{7}^{6} - 15T_{7}^{4} - 18T_{7}^{3} + 13T_{7}^{2} + 14T_{7} - 4 \)
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\( T_{11}^{6} - 6T_{11}^{5} - 21T_{11}^{4} + 138T_{11}^{3} - 7T_{11}^{2} - 632T_{11} + 636 \)
|
|
\( T_{13}^{6} - 8T_{13}^{5} - 4T_{13}^{4} + 122T_{13}^{3} - 20T_{13}^{2} - 500T_{13} - 359 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} + 3 T^{5} + \cdots + 3 \)
|
| $3$ |
\( (T + 1)^{6} \)
|
| $5$ |
\( T^{6} \)
|
| $7$ |
\( T^{6} - 15 T^{4} + \cdots - 4 \)
|
| $11$ |
\( T^{6} - 6 T^{5} + \cdots + 636 \)
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| $13$ |
\( T^{6} - 8 T^{5} + \cdots - 359 \)
|
| $17$ |
\( T^{6} + 4 T^{5} + \cdots + 477 \)
|
| $19$ |
\( T^{6} + 4 T^{5} + \cdots - 16 \)
|
| $23$ |
\( T^{6} + 14 T^{5} + \cdots + 3264 \)
|
| $29$ |
\( T^{6} - 10 T^{5} + \cdots + 28608 \)
|
| $31$ |
\( T^{6} - 12 T^{5} + \cdots + 106468 \)
|
| $37$ |
\( T^{6} - 12 T^{5} + \cdots + 10721 \)
|
| $41$ |
\( T^{6} + 6 T^{5} + \cdots - 768 \)
|
| $43$ |
\( T^{6} - 12 T^{5} + \cdots + 58868 \)
|
| $47$ |
\( T^{6} + 16 T^{5} + \cdots + 27276 \)
|
| $53$ |
\( T^{6} + 12 T^{5} + \cdots + 2304 \)
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| $59$ |
\( T^{6} - 8 T^{5} + \cdots + 768 \)
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| $61$ |
\( T^{6} + 24 T^{5} + \cdots + 2054417 \)
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| $67$ |
\( T^{6} - 4 T^{5} + \cdots + 164608 \)
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| $71$ |
\( T^{6} - 36 T^{5} + \cdots - 58896 \)
|
| $73$ |
\( T^{6} - 26 T^{5} + \cdots + 53824 \)
|
| $79$ |
\( T^{6} - 8 T^{5} + \cdots + 4096 \)
|
| $83$ |
\( T^{6} - 8 T^{5} + \cdots + 22668 \)
|
| $89$ |
\( T^{6} + 8 T^{5} + \cdots - 36864 \)
|
| $97$ |
\( T^{6} - 24 T^{5} + \cdots - 25856 \)
|
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