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: | \(0\) |
| Dimension: | \(8\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{8} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} - 2x^{7} - 30x^{6} + 52x^{5} + 267x^{4} - 352x^{3} - 632x^{2} + 240x + 288 \)
|
| 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 \(1,\beta_1,\ldots,\beta_{7}\) 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
\( x^{8} - 2x^{7} - 30x^{6} + 52x^{5} + 267x^{4} - 352x^{3} - 632x^{2} + 240x + 288 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 73\nu^{7} - 1010\nu^{6} - 2490\nu^{5} + 23896\nu^{4} + 27735\nu^{3} - 142756\nu^{2} - 87080\nu + 110232 ) / 21048 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 45\nu^{7} - 94\nu^{6} - 718\nu^{5} + 2092\nu^{4} + 1383\nu^{3} - 14092\nu^{2} + 1968\nu + 26528 ) / 7016 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 101\nu^{7} - 172\nu^{6} - 2508\nu^{5} + 1850\nu^{4} + 17253\nu^{3} + 9220\nu^{2} - 28792\nu - 30576 ) / 10524 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -68\nu^{7} + 220\nu^{6} + 1923\nu^{5} - 6221\nu^{4} - 13257\nu^{3} + 47741\nu^{2} - 986\nu - 51936 ) / 5262 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 407\nu^{7} - 1162\nu^{6} - 9846\nu^{5} + 31160\nu^{4} + 57177\nu^{3} - 212192\nu^{2} + 9848\nu + 97896 ) / 21048 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( -269\nu^{7} + 406\nu^{6} + 7878\nu^{5} - 11648\nu^{4} - 64863\nu^{3} + 89228\nu^{2} + 110548\nu - 71736 ) / 10524 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{6} + \beta_{5} - \beta_{3} + 9 \)
|
| \(\nu^{3}\) | \(=\) |
\( -\beta_{7} + 2\beta_{5} - \beta_{4} + \beta_{3} + \beta_{2} + 12\beta _1 + 1 \)
|
| \(\nu^{4}\) | \(=\) |
\( -\beta_{7} + 16\beta_{6} + 16\beta_{5} - 4\beta_{4} - 14\beta_{3} - \beta _1 + 118 \)
|
| \(\nu^{5}\) | \(=\) |
\( -14\beta_{7} + 5\beta_{6} + 37\beta_{5} - 19\beta_{4} + 24\beta_{3} + 15\beta_{2} + 155\beta _1 + 22 \)
|
| \(\nu^{6}\) | \(=\) |
\( -32\beta_{7} + 235\beta_{6} + 245\beta_{5} - 96\beta_{4} - 189\beta_{3} - 18\beta_{2} - 12\beta _1 + 1637 \)
|
| \(\nu^{7}\) | \(=\) |
\( -213\beta_{7} + 140\beta_{6} + 610\beta_{5} - 287\beta_{4} + 451\beta_{3} + 171\beta_{2} + 2082\beta _1 + 483 \)
|
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 | −3.78725 | −1.00000 | 3.57193 | 1.00000 | 1.00000 | −3.78725 | ||||||||||||||||||||||||||||||||||||||||||
| 1.2 | 1.00000 | −1.00000 | 1.00000 | −3.42323 | −1.00000 | −1.25167 | 1.00000 | 1.00000 | −3.42323 | |||||||||||||||||||||||||||||||||||||||||||
| 1.3 | 1.00000 | −1.00000 | 1.00000 | −1.12580 | −1.00000 | −0.350474 | 1.00000 | 1.00000 | −1.12580 | |||||||||||||||||||||||||||||||||||||||||||
| 1.4 | 1.00000 | −1.00000 | 1.00000 | −0.652958 | −1.00000 | −3.89658 | 1.00000 | 1.00000 | −0.652958 | |||||||||||||||||||||||||||||||||||||||||||
| 1.5 | 1.00000 | −1.00000 | 1.00000 | 0.815784 | −1.00000 | 2.48945 | 1.00000 | 1.00000 | 0.815784 | |||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 1.00000 | −1.00000 | 1.00000 | 2.56063 | −1.00000 | 3.94138 | 1.00000 | 1.00000 | 2.56063 | |||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 1.00000 | −1.00000 | 1.00000 | 3.65713 | −1.00000 | −4.08293 | 1.00000 | 1.00000 | 3.65713 | |||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 1.00000 | −1.00000 | 1.00000 | 3.95569 | −1.00000 | 4.57890 | 1.00000 | 1.00000 | 3.95569 | |||||||||||||||||||||||||||||||||||||||||||
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.bk | ✓ | 8 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 4002.2.a.bk | ✓ | 8 | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4002))\):
|
\( T_{5}^{8} - 2T_{5}^{7} - 30T_{5}^{6} + 52T_{5}^{5} + 267T_{5}^{4} - 352T_{5}^{3} - 632T_{5}^{2} + 240T_{5} + 288 \)
|
|
\( T_{7}^{8} - 5T_{7}^{7} - 32T_{7}^{6} + 172T_{7}^{5} + 260T_{7}^{4} - 1680T_{7}^{3} - 144T_{7}^{2} + 3360T_{7} + 1120 \)
|
|
\( T_{11}^{8} - 3T_{11}^{7} - 49T_{11}^{6} + 77T_{11}^{5} + 820T_{11}^{4} - 208T_{11}^{3} - 4336T_{11}^{2} - 672T_{11} + 6272 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 1)^{8} \)
|
| $3$ |
\( (T + 1)^{8} \)
|
| $5$ |
\( T^{8} - 2 T^{7} + \cdots + 288 \)
|
| $7$ |
\( T^{8} - 5 T^{7} + \cdots + 1120 \)
|
| $11$ |
\( T^{8} - 3 T^{7} + \cdots + 6272 \)
|
| $13$ |
\( T^{8} - 13 T^{7} + \cdots + 84640 \)
|
| $17$ |
\( T^{8} - 5 T^{7} + \cdots + 1792 \)
|
| $19$ |
\( T^{8} - 7 T^{7} + \cdots - 2592 \)
|
| $23$ |
\( (T + 1)^{8} \)
|
| $29$ |
\( (T + 1)^{8} \)
|
| $31$ |
\( T^{8} - 15 T^{7} + \cdots - 896 \)
|
| $37$ |
\( T^{8} - 22 T^{7} + \cdots - 244944 \)
|
| $41$ |
\( T^{8} + 8 T^{7} + \cdots + 75856 \)
|
| $43$ |
\( T^{8} + T^{7} + \cdots - 282944 \)
|
| $47$ |
\( T^{8} + 9 T^{7} + \cdots + 5120 \)
|
| $53$ |
\( T^{8} - 14 T^{7} + \cdots + 83712 \)
|
| $59$ |
\( T^{8} + 4 T^{7} + \cdots + 343680 \)
|
| $61$ |
\( T^{8} - 13 T^{7} + \cdots - 1760 \)
|
| $67$ |
\( T^{8} + 3 T^{7} + \cdots - 1323392 \)
|
| $71$ |
\( T^{8} - 7 T^{7} + \cdots + 224896 \)
|
| $73$ |
\( T^{8} - 16 T^{7} + \cdots - 833920 \)
|
| $79$ |
\( T^{8} - 14 T^{7} + \cdots + 881408 \)
|
| $83$ |
\( T^{8} - 36 T^{7} + \cdots - 36864 \)
|
| $89$ |
\( T^{8} + 12 T^{7} + \cdots - 1208832 \)
|
| $97$ |
\( T^{8} - 10 T^{7} + \cdots + 38656 \)
|
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