Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [8008,2,Mod(1,8008)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8008, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 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("8008.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 8008 = 2^{3} \cdot 7 \cdot 11 \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 8008.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: | \(63.9442019386\) |
| Analytic rank: | \(0\) |
| Dimension: | \(12\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{12} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{12} - 4 x^{11} - 17 x^{10} + 79 x^{9} + 80 x^{8} - 536 x^{7} - 4 x^{6} + 1484 x^{5} - 682 x^{4} + \cdots - 64 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{17}]\) |
| Coefficient ring index: | \( 2^{6} \) |
| 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 \(1,\beta_1,\ldots,\beta_{11}\) 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
\( x^{12} - 4 x^{11} - 17 x^{10} + 79 x^{9} + 80 x^{8} - 536 x^{7} - 4 x^{6} + 1484 x^{5} - 682 x^{4} + \cdots - 64 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 4 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 77 \nu^{11} + 116 \nu^{10} - 2017 \nu^{9} - 1221 \nu^{8} + 18608 \nu^{7} - 2976 \nu^{6} + \cdots + 42444 ) / 6100 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 113 \nu^{11} - 416 \nu^{10} - 2453 \nu^{9} + 8491 \nu^{8} + 18752 \nu^{7} - 60424 \nu^{6} + \cdots + 31376 ) / 6100 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 46 \nu^{11} - 2 \nu^{10} - 896 \nu^{9} - 468 \nu^{8} + 6284 \nu^{7} + 8097 \nu^{6} - 22080 \nu^{5} + \cdots - 248 ) / 1525 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 163 \nu^{11} - 816 \nu^{10} - 1703 \nu^{9} + 15541 \nu^{8} - 5448 \nu^{7} - 100224 \nu^{6} + \cdots + 18376 ) / 6100 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 19 \nu^{11} + 91 \nu^{10} + 386 \nu^{9} - 1764 \nu^{8} - 2821 \nu^{7} + 11464 \nu^{6} + 9303 \nu^{5} + \cdots - 3234 ) / 610 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 124 \nu^{11} - 443 \nu^{10} - 2044 \nu^{9} + 7968 \nu^{8} + 9646 \nu^{7} - 46427 \nu^{6} + \cdots + 4848 ) / 3050 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 65 \nu^{11} - 154 \nu^{10} - 1221 \nu^{9} + 2943 \nu^{8} + 7580 \nu^{7} - 19166 \nu^{6} - 17414 \nu^{5} + \cdots + 9208 ) / 1220 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( - 103 \nu^{11} + 214 \nu^{10} + 2115 \nu^{9} - 4397 \nu^{8} - 15052 \nu^{7} + 31846 \nu^{6} + \cdots - 6892 ) / 1220 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( - 679 \nu^{11} + 1528 \nu^{10} + 13849 \nu^{9} - 30103 \nu^{8} - 97266 \nu^{7} + 203642 \nu^{6} + \cdots - 22808 ) / 6100 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( -\beta_{9} + \beta_{8} + \beta_{3} + 8\beta _1 - 1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{10} + \beta_{9} + \beta_{4} + \beta_{3} + 10\beta_{2} + \beta _1 + 26 \)
|
| \(\nu^{5}\) | \(=\) |
\( -10\beta_{9} + 12\beta_{8} - \beta_{6} - \beta_{5} - 2\beta_{4} + 11\beta_{3} + 65\beta _1 - 7 \)
|
| \(\nu^{6}\) | \(=\) |
\( - 2 \beta_{11} + 15 \beta_{10} + 15 \beta_{9} - 2 \beta_{8} - 2 \beta_{7} - \beta_{5} + 7 \beta_{4} + \cdots + 202 \)
|
| \(\nu^{7}\) | \(=\) |
\( 3 \beta_{11} - 3 \beta_{10} - 84 \beta_{9} + 118 \beta_{8} - 3 \beta_{7} - 17 \beta_{6} - 16 \beta_{5} + \cdots - 42 \)
|
| \(\nu^{8}\) | \(=\) |
\( - 32 \beta_{11} + 165 \beta_{10} + 168 \beta_{9} - 33 \beta_{8} - 36 \beta_{7} - 19 \beta_{5} + \cdots + 1688 \)
|
| \(\nu^{9}\) | \(=\) |
\( 53 \beta_{11} - 50 \beta_{10} - 681 \beta_{9} + 1094 \beta_{8} - 55 \beta_{7} - 201 \beta_{6} + \cdots - 229 \)
|
| \(\nu^{10}\) | \(=\) |
\( - 368 \beta_{11} + 1629 \beta_{10} + 1694 \beta_{9} - 381 \beta_{8} - 444 \beta_{7} - 5 \beta_{6} + \cdots + 14538 \)
|
| \(\nu^{11}\) | \(=\) |
\( 633 \beta_{11} - 569 \beta_{10} - 5511 \beta_{9} + 9899 \beta_{8} - 695 \beta_{7} - 2073 \beta_{6} + \cdots - 1019 \)
|
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 |
|
0 | −2.95543 | 0 | 3.71005 | 0 | 1.00000 | 0 | 5.73458 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | 0 | −2.33021 | 0 | −0.450718 | 0 | 1.00000 | 0 | 2.42987 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | 0 | −1.78254 | 0 | 0.474905 | 0 | 1.00000 | 0 | 0.177459 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | 0 | −1.54910 | 0 | −3.31877 | 0 | 1.00000 | 0 | −0.600281 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | 0 | −0.219004 | 0 | 2.16734 | 0 | 1.00000 | 0 | −2.95204 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 0 | 0.336827 | 0 | 3.95783 | 0 | 1.00000 | 0 | −2.88655 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 0 | 0.818045 | 0 | −0.570018 | 0 | 1.00000 | 0 | −2.33080 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 0 | 1.39079 | 0 | 0.208204 | 0 | 1.00000 | 0 | −1.06569 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 0 | 1.75763 | 0 | −3.24204 | 0 | 1.00000 | 0 | 0.0892594 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 0 | 2.54627 | 0 | 3.33743 | 0 | 1.00000 | 0 | 3.48349 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.11 | 0 | 2.98199 | 0 | 1.64834 | 0 | 1.00000 | 0 | 5.89225 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.12 | 0 | 3.00474 | 0 | −1.92256 | 0 | 1.00000 | 0 | 6.02845 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(1\) |
| \(7\) | \(-1\) |
| \(11\) | \(-1\) |
| \(13\) | \(-1\) |
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 | 8008.2.a.x | ✓ | 12 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 8008.2.a.x | ✓ | 12 | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8008))\):
|
\( T_{3}^{12} - 4 T_{3}^{11} - 17 T_{3}^{10} + 79 T_{3}^{9} + 80 T_{3}^{8} - 536 T_{3}^{7} - 4 T_{3}^{6} + \cdots - 64 \)
|
|
\( T_{5}^{12} - 6 T_{5}^{11} - 19 T_{5}^{10} + 157 T_{5}^{9} + 38 T_{5}^{8} - 1310 T_{5}^{7} + 816 T_{5}^{6} + \cdots - 92 \)
|
|
\( T_{17}^{12} - 16 T_{17}^{11} + 7 T_{17}^{10} + 1031 T_{17}^{9} - 4150 T_{17}^{8} - 13894 T_{17}^{7} + \cdots - 3460 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{12} \)
|
| $3$ |
\( T^{12} - 4 T^{11} + \cdots - 64 \)
|
| $5$ |
\( T^{12} - 6 T^{11} + \cdots - 92 \)
|
| $7$ |
\( (T - 1)^{12} \)
|
| $11$ |
\( (T - 1)^{12} \)
|
| $13$ |
\( (T - 1)^{12} \)
|
| $17$ |
\( T^{12} - 16 T^{11} + \cdots - 3460 \)
|
| $19$ |
\( T^{12} + 2 T^{11} + \cdots - 51984 \)
|
| $23$ |
\( T^{12} - 9 T^{11} + \cdots + 2293760 \)
|
| $29$ |
\( T^{12} - 15 T^{11} + \cdots + 877312 \)
|
| $31$ |
\( T^{12} - 10 T^{11} + \cdots + 179200 \)
|
| $37$ |
\( T^{12} - 18 T^{11} + \cdots + 32083712 \)
|
| $41$ |
\( T^{12} - 24 T^{11} + \cdots + 49408 \)
|
| $43$ |
\( T^{12} - 200 T^{10} + \cdots + 12467456 \)
|
| $47$ |
\( T^{12} - 5 T^{11} + \cdots - 1289216 \)
|
| $53$ |
\( T^{12} + \cdots - 10360994036 \)
|
| $59$ |
\( T^{12} - 15 T^{11} + \cdots - 4305920 \)
|
| $61$ |
\( T^{12} - 17 T^{11} + \cdots - 260820 \)
|
| $67$ |
\( T^{12} + \cdots - 309893103568 \)
|
| $71$ |
\( T^{12} - 9 T^{11} + \cdots - 5328640 \)
|
| $73$ |
\( T^{12} - 32 T^{11} + \cdots - 31928576 \)
|
| $79$ |
\( T^{12} + \cdots + 61219583440 \)
|
| $83$ |
\( T^{12} + 5 T^{11} + \cdots + 30750608 \)
|
| $89$ |
\( T^{12} - 16 T^{11} + \cdots - 322708 \)
|
| $97$ |
\( T^{12} - 10 T^{11} + \cdots - 66277120 \)
|
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