Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [4002,2,Mod(1,4002)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4002, 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("4002.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 4002 = 2 \cdot 3 \cdot 23 \cdot 29 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 4002.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: | \(31.9561308889\) |
| Analytic rank: | \(1\) |
| 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: |
\( x^{7} - x^{6} - 13x^{5} + 16x^{4} + 19x^{3} - 8x^{2} - 10x - 2 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{2} \) |
| Twist minimal: | yes |
| 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
\( x^{7} - x^{6} - 13x^{5} + 16x^{4} + 19x^{3} - 8x^{2} - 10x - 2 \)
:
| \(\beta_{1}\) | \(=\) |
\( 4\nu^{6} - 7\nu^{5} - 47\nu^{4} + 100\nu^{3} + 4\nu^{2} - 44\nu - 6 \)
|
| \(\beta_{2}\) | \(=\) |
\( -5\nu^{6} + 8\nu^{5} + 60\nu^{4} - 116\nu^{3} - 23\nu^{2} + 54\nu + 14 \)
|
| \(\beta_{3}\) | \(=\) |
\( 7\nu^{6} - 10\nu^{5} - 86\nu^{4} + 149\nu^{3} + 61\nu^{2} - 79\nu - 29 \)
|
| \(\beta_{4}\) | \(=\) |
\( -7\nu^{6} + 11\nu^{5} + 85\nu^{4} - 161\nu^{3} - 45\nu^{2} + 87\nu + 26 \)
|
| \(\beta_{5}\) | \(=\) |
\( 8\nu^{6} - 12\nu^{5} - 98\nu^{4} + 177\nu^{3} + 64\nu^{2} - 95\nu - 35 \)
|
| \(\beta_{6}\) | \(=\) |
\( -11\nu^{6} + 17\nu^{5} + 134\nu^{4} - 249\nu^{3} - 76\nu^{2} + 130\nu + 41 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{5} + \beta_{4} + \beta_{2} + \beta _1 + 1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 2\beta_{6} + 3\beta_{5} - \beta_{3} - \beta_{2} + 8 ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 5\beta_{5} + 5\beta_{4} + 2\beta_{3} + 11\beta_{2} + 9\beta _1 + 3 ) / 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 24\beta_{6} + 26\beta_{5} - 7\beta_{4} - 9\beta_{3} - 18\beta_{2} - 5\beta _1 + 69 ) / 2 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( -8\beta_{6} + 30\beta_{5} + 47\beta_{4} + 33\beta_{3} + 122\beta_{2} + 95\beta _1 - 25 ) / 2 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( 266\beta_{6} + 241\beta_{5} - 114\beta_{4} - 97\beta_{3} - 261\beta_{2} - 106\beta _1 + 698 ) / 2 \)
|
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 |
|
−1.00000 | −1.00000 | 1.00000 | −3.83000 | 1.00000 | −4.91072 | −1.00000 | 1.00000 | 3.83000 | |||||||||||||||||||||||||||||||||||||||
| 1.2 | −1.00000 | −1.00000 | 1.00000 | −0.694170 | 1.00000 | 1.42319 | −1.00000 | 1.00000 | 0.694170 | ||||||||||||||||||||||||||||||||||||||||
| 1.3 | −1.00000 | −1.00000 | 1.00000 | −0.662522 | 1.00000 | −0.537195 | −1.00000 | 1.00000 | 0.662522 | ||||||||||||||||||||||||||||||||||||||||
| 1.4 | −1.00000 | −1.00000 | 1.00000 | −0.324686 | 1.00000 | 2.40824 | −1.00000 | 1.00000 | 0.324686 | ||||||||||||||||||||||||||||||||||||||||
| 1.5 | −1.00000 | −1.00000 | 1.00000 | 1.78524 | 1.00000 | 0.541489 | −1.00000 | 1.00000 | −1.78524 | ||||||||||||||||||||||||||||||||||||||||
| 1.6 | −1.00000 | −1.00000 | 1.00000 | 2.26164 | 1.00000 | −5.18548 | −1.00000 | 1.00000 | −2.26164 | ||||||||||||||||||||||||||||||||||||||||
| 1.7 | −1.00000 | −1.00000 | 1.00000 | 3.46450 | 1.00000 | 1.26047 | −1.00000 | 1.00000 | −3.46450 | ||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(1\) |
| \(3\) | \(1\) |
| \(23\) | \(-1\) |
| \(29\) | \(-1\) |
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 | 4002.2.a.bg | ✓ | 7 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 4002.2.a.bg | ✓ | 7 | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4002))\):
|
\( T_{5}^{7} - 2T_{5}^{6} - 16T_{5}^{5} + 34T_{5}^{4} + 29T_{5}^{3} - 42T_{5}^{2} - 40T_{5} - 8 \)
|
|
\( T_{7}^{7} + 5T_{7}^{6} - 18T_{7}^{5} - 52T_{7}^{4} + 172T_{7}^{3} - 96T_{7}^{2} - 48T_{7} + 32 \)
|
|
\( T_{11}^{7} + T_{11}^{6} - 37T_{11}^{5} - 47T_{11}^{4} + 248T_{11}^{3} + 204T_{11}^{2} - 456T_{11} - 176 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T + 1)^{7} \)
|
| $3$ |
\( (T + 1)^{7} \)
|
| $5$ |
\( T^{7} - 2 T^{6} + \cdots - 8 \)
|
| $7$ |
\( T^{7} + 5 T^{6} + \cdots + 32 \)
|
| $11$ |
\( T^{7} + T^{6} + \cdots - 176 \)
|
| $13$ |
\( T^{7} - T^{6} + \cdots - 6016 \)
|
| $17$ |
\( T^{7} + T^{6} + \cdots - 256 \)
|
| $19$ |
\( T^{7} + 9 T^{6} + \cdots - 1312 \)
|
| $23$ |
\( (T - 1)^{7} \)
|
| $29$ |
\( (T - 1)^{7} \)
|
| $31$ |
\( T^{7} + 19 T^{6} + \cdots + 589312 \)
|
| $37$ |
\( T^{7} + 18 T^{6} + \cdots + 3112 \)
|
| $41$ |
\( T^{7} - 4 T^{6} + \cdots + 134408 \)
|
| $43$ |
\( T^{7} + 9 T^{6} + \cdots - 1364288 \)
|
| $47$ |
\( T^{7} + 11 T^{6} + \cdots + 88832 \)
|
| $53$ |
\( T^{7} - 8 T^{6} + \cdots - 536128 \)
|
| $59$ |
\( T^{7} - 12 T^{6} + \cdots - 95296 \)
|
| $61$ |
\( T^{7} + 5 T^{6} + \cdots - 22768 \)
|
| $67$ |
\( T^{7} + 13 T^{6} + \cdots + 203776 \)
|
| $71$ |
\( T^{7} + T^{6} + \cdots + 183296 \)
|
| $73$ |
\( T^{7} + 26 T^{6} + \cdots - 836032 \)
|
| $79$ |
\( T^{7} + 38 T^{6} + \cdots + 86848 \)
|
| $83$ |
\( T^{7} + 6 T^{6} + \cdots - 2048 \)
|
| $89$ |
\( T^{7} - 20 T^{6} + \cdots + 512 \)
|
| $97$ |
\( T^{7} + 8 T^{6} + \cdots - 407552 \)
|
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