Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [6675,2,Mod(1,6675)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6675, 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("6675.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 6675 = 3 \cdot 5^{2} \cdot 89 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 6675.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: | \(53.3001433492\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | 6.6.10407557.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - x^{5} - 8x^{4} + 2x^{3} + 18x^{2} + 7x - 2 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 2^{2} \) |
| Twist minimal: | no (minimal twist has level 1335) |
| 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
\( x^{6} - x^{5} - 8x^{4} + 2x^{3} + 18x^{2} + 7x - 2 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu^{3} - \nu^{2} - 5\nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - \nu - 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{4} - 2\nu^{3} - 4\nu^{2} + 5\nu + 3 \)
|
| \(\beta_{4}\) | \(=\) |
\( \nu^{5} - 2\nu^{4} - 6\nu^{3} + 7\nu^{2} + 10\nu \)
|
| \(\beta_{5}\) | \(=\) |
\( 2\nu^{5} - 4\nu^{4} - 11\nu^{3} + 14\nu^{2} + 18\nu - 3 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{5} - 2\beta_{4} - \beta_{2} - \beta_1 ) / 4 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{5} - 2\beta_{4} + 3\beta_{2} - \beta _1 + 12 ) / 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 3\beta_{5} - 6\beta_{4} - \beta_{2} - \beta _1 + 6 ) / 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 11\beta_{5} - 22\beta_{4} + 4\beta_{3} + 13\beta_{2} - 3\beta _1 + 60 ) / 4 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 41\beta_{5} - 78\beta_{4} + 8\beta_{3} + 3\beta_{2} - \beta _1 + 108 ) / 4 \)
|
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 |
|
−2.15466 | 1.00000 | 2.64258 | 0 | −2.15466 | 2.12396 | −1.38454 | 1.00000 | 0 | ||||||||||||||||||||||||||||||||||||
| 1.2 | −0.652386 | 1.00000 | −1.57439 | 0 | −0.652386 | −1.82035 | 2.33188 | 1.00000 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.3 | 0.305330 | 1.00000 | −1.90677 | 0 | 0.305330 | −1.72660 | −1.19286 | 1.00000 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 1.49828 | 1.00000 | 0.244835 | 0 | 1.49828 | 3.23110 | −2.62972 | 1.00000 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 2.30610 | 1.00000 | 3.31809 | 0 | 2.30610 | −4.21574 | 3.03965 | 1.00000 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 2.69734 | 1.00000 | 5.27566 | 0 | 2.69734 | 1.40763 | 8.83558 | 1.00000 | 0 | |||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(-1\) |
| \(5\) | \(1\) |
| \(89\) | \(1\) |
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 | 6675.2.a.u | 6 | |
| 5.b | even | 2 | 1 | 1335.2.a.g | ✓ | 6 | |
| 15.d | odd | 2 | 1 | 4005.2.a.n | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1335.2.a.g | ✓ | 6 | 5.b | even | 2 | 1 | |
| 4005.2.a.n | 6 | 15.d | odd | 2 | 1 | ||
| 6675.2.a.u | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6675))\):
|
\( T_{2}^{6} - 4T_{2}^{5} - 2T_{2}^{4} + 21T_{2}^{3} - 13T_{2}^{2} - 11T_{2} + 4 \)
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\( T_{7}^{6} + T_{7}^{5} - 20T_{7}^{4} - 7T_{7}^{3} + 96T_{7}^{2} + 16T_{7} - 128 \)
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\( T_{11} \)
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\( T_{13}^{6} - 3T_{13}^{5} - 20T_{13}^{4} + 17T_{13}^{3} + 70T_{13}^{2} - 32T_{13} - 32 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} - 4 T^{5} + \cdots + 4 \)
|
| $3$ |
\( (T - 1)^{6} \)
|
| $5$ |
\( T^{6} \)
|
| $7$ |
\( T^{6} + T^{5} + \cdots - 128 \)
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| $11$ |
\( T^{6} \)
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| $13$ |
\( T^{6} - 3 T^{5} + \cdots - 32 \)
|
| $17$ |
\( T^{6} - 13 T^{5} + \cdots - 128 \)
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| $19$ |
\( (T - 2)^{6} \)
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| $23$ |
\( T^{6} - 19 T^{5} + \cdots - 2048 \)
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| $29$ |
\( T^{6} - 55 T^{4} + \cdots + 652 \)
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| $31$ |
\( T^{6} - 10 T^{5} + \cdots - 3904 \)
|
| $37$ |
\( T^{6} - T^{5} + \cdots - 32 \)
|
| $41$ |
\( T^{6} + 4 T^{5} + \cdots - 4964 \)
|
| $43$ |
\( T^{6} - 7 T^{5} + \cdots + 1024 \)
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| $47$ |
\( T^{6} - 15 T^{5} + \cdots + 512 \)
|
| $53$ |
\( T^{6} - 27 T^{5} + \cdots - 34376 \)
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| $59$ |
\( T^{6} + 4 T^{5} + \cdots - 4 \)
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| $61$ |
\( T^{6} - 8 T^{5} + \cdots - 1024 \)
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| $67$ |
\( T^{6} - 11 T^{5} + \cdots + 53888 \)
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| $71$ |
\( T^{6} + 16 T^{5} + \cdots + 22016 \)
|
| $73$ |
\( T^{6} + T^{5} + \cdots - 2343136 \)
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| $79$ |
\( T^{6} - 121 T^{4} + \cdots - 10048 \)
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| $83$ |
\( T^{6} - 17 T^{5} + \cdots + 512 \)
|
| $89$ |
\( (T + 1)^{6} \)
|
| $97$ |
\( T^{6} - 29 T^{5} + \cdots - 30112 \)
|
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