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: | \(7\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{7} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{7} - 3x^{6} - 9x^{5} + 29x^{4} + 17x^{3} - 67x^{2} - 5x + 33 \)
|
| 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 \(1,\beta_1,\ldots,\beta_{6}\) 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
\( x^{7} - 3x^{6} - 9x^{5} + 29x^{4} + 17x^{3} - 67x^{2} - 5x + 33 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 4 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{4} - 8\nu^{2} + 9 ) / 2 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{5} - \nu^{4} - 8\nu^{3} + 6\nu^{2} + 11\nu - 3 ) / 2 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -\nu^{4} + 2\nu^{3} + 8\nu^{2} - 10\nu - 11 ) / 2 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( \nu^{6} - \nu^{5} - 10\nu^{4} + 8\nu^{3} + 23\nu^{2} - 11\nu - 8 ) / 2 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{5} + \beta_{3} + 5\beta _1 + 1 \)
|
| \(\nu^{4}\) | \(=\) |
\( 2\beta_{3} + 8\beta_{2} + 23 \)
|
| \(\nu^{5}\) | \(=\) |
\( 8\beta_{5} + 2\beta_{4} + 10\beta_{3} + 2\beta_{2} + 29\beta _1 + 10 \)
|
| \(\nu^{6}\) | \(=\) |
\( 2\beta_{6} + 2\beta_{4} + 22\beta_{3} + 59\beta_{2} + 148 \)
|
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 |
|
−2.54836 | −0.389583 | 4.49416 | 1.00000 | 0.992800 | 0 | −6.35603 | −2.84823 | −2.54836 | |||||||||||||||||||||||||||||||||||||||
| 1.2 | −1.53790 | −2.16359 | 0.365123 | 1.00000 | 3.32737 | 0 | 2.51427 | 1.68112 | −1.53790 | ||||||||||||||||||||||||||||||||||||||||
| 1.3 | −0.788425 | 2.20675 | −1.37839 | 1.00000 | −1.73985 | 0 | 2.66360 | 1.86974 | −0.788425 | ||||||||||||||||||||||||||||||||||||||||
| 1.4 | 0.878865 | 1.70869 | −1.22760 | 1.00000 | 1.50171 | 0 | −2.83662 | −0.0803809 | 0.878865 | ||||||||||||||||||||||||||||||||||||||||
| 1.5 | 1.81826 | −3.25922 | 1.30606 | 1.00000 | −5.92610 | 0 | −1.26176 | 7.62253 | 1.81826 | ||||||||||||||||||||||||||||||||||||||||
| 1.6 | 2.45265 | −1.46894 | 4.01547 | 1.00000 | −3.60278 | 0 | 4.94324 | −0.842228 | 2.45265 | ||||||||||||||||||||||||||||||||||||||||
| 1.7 | 2.72492 | 2.36589 | 5.42517 | 1.00000 | 6.44686 | 0 | 9.33330 | 2.59745 | 2.72492 | ||||||||||||||||||||||||||||||||||||||||
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.bd | 7 | |
| 7.b | odd | 2 | 1 | 665.2.a.j | ✓ | 7 | |
| 21.c | even | 2 | 1 | 5985.2.a.bo | 7 | ||
| 35.c | odd | 2 | 1 | 3325.2.a.v | 7 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 665.2.a.j | ✓ | 7 | 7.b | odd | 2 | 1 | |
| 3325.2.a.v | 7 | 35.c | odd | 2 | 1 | ||
| 4655.2.a.bd | 7 | 1.a | even | 1 | 1 | trivial | |
| 5985.2.a.bo | 7 | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4655))\):
|
\( T_{2}^{7} - 3T_{2}^{6} - 9T_{2}^{5} + 29T_{2}^{4} + 17T_{2}^{3} - 67T_{2}^{2} - 5T_{2} + 33 \)
|
|
\( T_{3}^{7} + T_{3}^{6} - 15T_{3}^{5} - 9T_{3}^{4} + 68T_{3}^{3} + 28T_{3}^{2} - 92T_{3} - 36 \)
|
|
\( T_{11}^{7} - 48T_{11}^{5} + 44T_{11}^{4} + 592T_{11}^{3} - 672T_{11}^{2} - 2176T_{11} + 2496 \)
|
|
\( T_{13}^{7} + 4T_{13}^{6} - 64T_{13}^{5} - 296T_{13}^{4} + 836T_{13}^{3} + 5024T_{13}^{2} + 3008T_{13} - 4544 \)
|
|
\( T_{17}^{7} - 3T_{17}^{6} - 29T_{17}^{5} + 19T_{17}^{4} + 104T_{17}^{3} - 56T_{17}^{2} - 80T_{17} + 48 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{7} - 3 T^{6} + \cdots + 33 \)
|
| $3$ |
\( T^{7} + T^{6} + \cdots - 36 \)
|
| $5$ |
\( (T - 1)^{7} \)
|
| $7$ |
\( T^{7} \)
|
| $11$ |
\( T^{7} - 48 T^{5} + \cdots + 2496 \)
|
| $13$ |
\( T^{7} + 4 T^{6} + \cdots - 4544 \)
|
| $17$ |
\( T^{7} - 3 T^{6} + \cdots + 48 \)
|
| $19$ |
\( (T - 1)^{7} \)
|
| $23$ |
\( T^{7} - 11 T^{6} + \cdots + 192 \)
|
| $29$ |
\( T^{7} + 12 T^{6} + \cdots + 48704 \)
|
| $31$ |
\( T^{7} + 18 T^{6} + \cdots - 1024 \)
|
| $37$ |
\( T^{7} - 31 T^{6} + \cdots - 27396 \)
|
| $41$ |
\( T^{7} - 5 T^{6} + \cdots + 16336 \)
|
| $43$ |
\( T^{7} - 23 T^{6} + \cdots - 128896 \)
|
| $47$ |
\( T^{7} - 4 T^{6} + \cdots - 18432 \)
|
| $53$ |
\( T^{7} - 9 T^{6} + \cdots - 292804 \)
|
| $59$ |
\( T^{7} - 8 T^{6} + \cdots + 36608 \)
|
| $61$ |
\( T^{7} - 12 T^{6} + \cdots + 33536 \)
|
| $67$ |
\( T^{7} - 12 T^{6} + \cdots - 47088 \)
|
| $71$ |
\( T^{7} + 21 T^{6} + \cdots - 267872 \)
|
| $73$ |
\( T^{7} - T^{6} + \cdots + 326928 \)
|
| $79$ |
\( T^{7} - 30 T^{6} + \cdots + 1268352 \)
|
| $83$ |
\( T^{7} - 12 T^{6} + \cdots - 768 \)
|
| $89$ |
\( T^{7} - 4 T^{6} + \cdots - 249024 \)
|
| $97$ |
\( T^{7} + 4 T^{6} + \cdots + 9666128 \)
|
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