Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [4655,2,Mod(1,4655)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4655, 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("4655.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 4655 = 5 \cdot 7^{2} \cdot 19 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 4655.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: | \(37.1703621409\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | 6.6.79799552.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - x^{5} - 8x^{4} + 6x^{3} + 15x^{2} - 7x - 2 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 665) |
| 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
\( x^{6} - x^{5} - 8x^{4} + 6x^{3} + 15x^{2} - 7x - 2 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - 5\nu \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{5} - 8\nu^{3} + 13\nu - 2 ) / 2 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -\nu^{5} + 2\nu^{4} + 6\nu^{3} - 10\nu^{2} - 7\nu + 6 ) / 2 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + 5\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{5} + \beta_{4} + \beta_{3} + 5\beta_{2} + 2\beta _1 + 13 \)
|
| \(\nu^{5}\) | \(=\) |
\( 2\beta_{4} + 8\beta_{3} + 27\beta _1 + 2 \)
|
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 |
|
−2.46972 | 1.73873 | 4.09953 | 1.00000 | −4.29419 | 0 | −5.18527 | 0.0231906 | −2.46972 | ||||||||||||||||||||||||||||||||||||
| 1.2 | −1.80624 | −2.21815 | 1.26250 | 1.00000 | 4.00651 | 0 | 1.33210 | 1.92019 | −1.80624 | |||||||||||||||||||||||||||||||||||||
| 1.3 | −0.652999 | 3.19008 | −1.57359 | 1.00000 | −2.08312 | 0 | 2.33355 | 7.17664 | −0.652999 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 0.205001 | −1.29822 | −1.95797 | 1.00000 | −0.266137 | 0 | −0.811388 | −1.31461 | 0.205001 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 1.51877 | 0.100701 | 0.306663 | 1.00000 | 0.152942 | 0 | −2.57179 | −2.98986 | 1.51877 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 2.20519 | 2.48686 | 2.86287 | 1.00000 | 5.48400 | 0 | 1.90279 | 3.18446 | 2.20519 | |||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(5\) | \( -1 \) |
| \(7\) | \( -1 \) |
| \(19\) | \( -1 \) |
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 | 4655.2.a.bc | 6 | |
| 7.b | odd | 2 | 1 | 665.2.a.i | ✓ | 6 | |
| 21.c | even | 2 | 1 | 5985.2.a.bn | 6 | ||
| 35.c | odd | 2 | 1 | 3325.2.a.u | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 665.2.a.i | ✓ | 6 | 7.b | odd | 2 | 1 | |
| 3325.2.a.u | 6 | 35.c | odd | 2 | 1 | ||
| 4655.2.a.bc | 6 | 1.a | even | 1 | 1 | trivial | |
| 5985.2.a.bn | 6 | 21.c | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4655))\):
|
\( T_{2}^{6} + T_{2}^{5} - 8T_{2}^{4} - 6T_{2}^{3} + 15T_{2}^{2} + 7T_{2} - 2 \)
|
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\( T_{3}^{6} - 4T_{3}^{5} - 5T_{3}^{4} + 28T_{3}^{3} - 40T_{3} + 4 \)
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\( T_{11}^{6} - 5T_{11}^{5} - 32T_{11}^{4} + 136T_{11}^{3} + 304T_{11}^{2} - 848T_{11} - 512 \)
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\( T_{13}^{6} - 15T_{13}^{5} + 52T_{13}^{4} + 172T_{13}^{3} - 1068T_{13}^{2} - 116T_{13} + 4528 \)
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\( T_{17}^{6} - 8T_{17}^{5} - 33T_{17}^{4} + 336T_{17}^{3} - 8T_{17}^{2} - 2048T_{17} - 1616 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} + T^{5} - 8 T^{4} + \cdots - 2 \)
|
| $3$ |
\( T^{6} - 4 T^{5} + \cdots + 4 \)
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| $5$ |
\( (T - 1)^{6} \)
|
| $7$ |
\( T^{6} \)
|
| $11$ |
\( T^{6} - 5 T^{5} + \cdots - 512 \)
|
| $13$ |
\( T^{6} - 15 T^{5} + \cdots + 4528 \)
|
| $17$ |
\( T^{6} - 8 T^{5} + \cdots - 1616 \)
|
| $19$ |
\( (T - 1)^{6} \)
|
| $23$ |
\( T^{6} + 5 T^{5} + \cdots - 31232 \)
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| $29$ |
\( T^{6} - 3 T^{5} + \cdots - 608 \)
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| $31$ |
\( (T - 4)^{6} \)
|
| $37$ |
\( T^{6} + T^{5} + \cdots - 32 \)
|
| $41$ |
\( T^{6} - T^{5} + \cdots + 3008 \)
|
| $43$ |
\( T^{6} + 5 T^{5} + \cdots + 512 \)
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| $47$ |
\( T^{6} - 13 T^{5} + \cdots - 4096 \)
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| $53$ |
\( T^{6} - 7 T^{5} + \cdots - 22112 \)
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| $59$ |
\( T^{6} + 16 T^{5} + \cdots + 256 \)
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| $61$ |
\( T^{6} + 2 T^{5} + \cdots - 168448 \)
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| $67$ |
\( T^{6} + 22 T^{5} + \cdots - 6272 \)
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| $71$ |
\( T^{6} - T^{5} + \cdots - 1178624 \)
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| $73$ |
\( T^{6} - 29 T^{5} + \cdots + 2396128 \)
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| $79$ |
\( T^{6} + 7 T^{5} + \cdots - 109184 \)
|
| $83$ |
\( T^{6} - 36 T^{5} + \cdots + 434176 \)
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| $89$ |
\( T^{6} + 4 T^{5} + \cdots + 10688 \)
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| $97$ |
\( T^{6} - 3 T^{5} + \cdots - 706712 \)
|
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