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: | \(11\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{11} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{11} - 3 x^{10} - 19 x^{9} + 55 x^{8} + 128 x^{7} - 361 x^{6} - 343 x^{5} + 1012 x^{4} + 215 x^{3} + \cdots + 160 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{17}]\) |
| Coefficient ring index: | \( 2 \) |
| 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 \(1,\beta_1,\ldots,\beta_{10}\) 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
\( x^{11} - 3 x^{10} - 19 x^{9} + 55 x^{8} + 128 x^{7} - 361 x^{6} - 343 x^{5} + 1012 x^{4} + 215 x^{3} + \cdots + 160 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 4 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 225 \nu^{10} - 279 \nu^{9} - 6263 \nu^{8} + 5843 \nu^{7} + 64532 \nu^{6} - 49233 \nu^{5} + \cdots - 66384 ) / 18712 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 195 \nu^{10} + 226 \nu^{9} - 5272 \nu^{8} - 4448 \nu^{7} + 48287 \nu^{6} + 28437 \nu^{5} + \cdots - 36014 ) / 4678 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 1275 \nu^{10} + 1581 \nu^{9} + 29253 \nu^{8} - 26873 \nu^{7} - 247172 \nu^{6} + 166715 \nu^{5} + \cdots + 394888 ) / 18712 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 687 \nu^{10} - 1039 \nu^{9} - 15443 \nu^{8} + 17903 \nu^{7} + 125246 \nu^{6} - 103919 \nu^{5} + \cdots - 127096 ) / 9356 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 393 \nu^{10} - 768 \nu^{9} - 8538 \nu^{8} + 13418 \nu^{7} + 68961 \nu^{6} - 81877 \nu^{5} + \cdots - 68048 ) / 4678 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 1671 \nu^{10} - 3569 \nu^{9} - 35785 \nu^{8} + 62605 \nu^{7} + 279164 \nu^{6} - 368631 \nu^{5} + \cdots - 262480 ) / 18712 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 3391 \nu^{10} - 7573 \nu^{9} - 67653 \nu^{8} + 122449 \nu^{7} + 499488 \nu^{6} - 667647 \nu^{5} + \cdots - 444360 ) / 18712 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( 1773 \nu^{10} - 2947 \nu^{9} - 37751 \nu^{8} + 49411 \nu^{7} + 289956 \nu^{6} - 286537 \nu^{5} + \cdots - 173940 ) / 9356 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( -\beta_{8} + \beta_{7} + \beta_{6} + \beta_{5} + 2\beta_{2} + 5\beta _1 + 1 \)
|
| \(\nu^{4}\) | \(=\) |
\( 2\beta_{7} - \beta_{6} + 2\beta_{5} + \beta_{4} + 11\beta_{2} + \beta _1 + 25 \)
|
| \(\nu^{5}\) | \(=\) |
\( - \beta_{10} - 10 \beta_{8} + 15 \beta_{7} + 9 \beta_{6} + 13 \beta_{5} + 2 \beta_{4} - 3 \beta_{3} + \cdots + 20 \)
|
| \(\nu^{6}\) | \(=\) |
\( - 2 \beta_{10} - 3 \beta_{8} + 38 \beta_{7} - 14 \beta_{6} + 31 \beta_{5} + 16 \beta_{4} - 6 \beta_{3} + \cdots + 187 \)
|
| \(\nu^{7}\) | \(=\) |
\( - 14 \beta_{10} - 3 \beta_{9} - 87 \beta_{8} + 186 \beta_{7} + 58 \beta_{6} + 144 \beta_{5} + \cdots + 260 \)
|
| \(\nu^{8}\) | \(=\) |
\( - 34 \beta_{10} - 3 \beta_{9} - 69 \beta_{8} + 529 \beta_{7} - 158 \beta_{6} + 387 \beta_{5} + \cdots + 1566 \)
|
| \(\nu^{9}\) | \(=\) |
\( - 154 \beta_{10} - 60 \beta_{9} - 768 \beta_{8} + 2168 \beta_{7} + 264 \beta_{6} + 1551 \beta_{5} + \cdots + 3013 \)
|
| \(\nu^{10}\) | \(=\) |
\( - 419 \beta_{10} - 80 \beta_{9} - 1060 \beta_{8} + 6566 \beta_{7} - 1714 \beta_{6} + 4509 \beta_{5} + \cdots + 14264 \)
|
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 | −2.60864 | 0 | −4.23324 | 0 | −1.00000 | 0 | 3.80501 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | 0 | −2.37601 | 0 | 2.73467 | 0 | −1.00000 | 0 | 2.64540 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | 0 | −1.93952 | 0 | −0.283518 | 0 | −1.00000 | 0 | 0.761736 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | 0 | −1.59424 | 0 | 0.804472 | 0 | −1.00000 | 0 | −0.458395 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | 0 | −0.291357 | 0 | −2.53284 | 0 | −1.00000 | 0 | −2.91511 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 0 | 0.846033 | 0 | −0.529248 | 0 | −1.00000 | 0 | −2.28423 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 0 | 1.12948 | 0 | 2.32569 | 0 | −1.00000 | 0 | −1.72428 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 0 | 1.36523 | 0 | −2.50606 | 0 | −1.00000 | 0 | −1.13614 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 0 | 2.48144 | 0 | 1.56116 | 0 | −1.00000 | 0 | 3.15756 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 0 | 2.65994 | 0 | 3.55706 | 0 | −1.00000 | 0 | 4.07531 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.11 | 0 | 3.32763 | 0 | −2.89815 | 0 | −1.00000 | 0 | 8.07314 | 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.w | ✓ | 11 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 8008.2.a.w | ✓ | 11 | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8008))\):
|
\( T_{3}^{11} - 3 T_{3}^{10} - 19 T_{3}^{9} + 55 T_{3}^{8} + 128 T_{3}^{7} - 361 T_{3}^{6} - 343 T_{3}^{5} + \cdots + 160 \)
|
|
\( T_{5}^{11} + 2 T_{5}^{10} - 32 T_{5}^{9} - 50 T_{5}^{8} + 363 T_{5}^{7} + 409 T_{5}^{6} - 1790 T_{5}^{5} + \cdots - 332 \)
|
|
\( T_{17}^{11} - 9 T_{17}^{10} - 53 T_{17}^{9} + 679 T_{17}^{8} - 576 T_{17}^{7} - 11041 T_{17}^{6} + \cdots - 400 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{11} \)
|
| $3$ |
\( T^{11} - 3 T^{10} + \cdots + 160 \)
|
| $5$ |
\( T^{11} + 2 T^{10} + \cdots - 332 \)
|
| $7$ |
\( (T + 1)^{11} \)
|
| $11$ |
\( (T - 1)^{11} \)
|
| $13$ |
\( (T - 1)^{11} \)
|
| $17$ |
\( T^{11} - 9 T^{10} + \cdots - 400 \)
|
| $19$ |
\( T^{11} - 20 T^{10} + \cdots + 97312 \)
|
| $23$ |
\( T^{11} - 12 T^{10} + \cdots + 373376 \)
|
| $29$ |
\( T^{11} - 8 T^{10} + \cdots - 2076192 \)
|
| $31$ |
\( T^{11} - 7 T^{10} + \cdots + 70400 \)
|
| $37$ |
\( T^{11} + 10 T^{10} + \cdots - 128 \)
|
| $41$ |
\( T^{11} + 2 T^{10} + \cdots - 4000 \)
|
| $43$ |
\( T^{11} - 24 T^{10} + \cdots + 13955680 \)
|
| $47$ |
\( T^{11} - 2 T^{10} + \cdots - 126656 \)
|
| $53$ |
\( T^{11} - 3 T^{10} + \cdots - 9827116 \)
|
| $59$ |
\( T^{11} + \cdots + 11457505280 \)
|
| $61$ |
\( T^{11} + 22 T^{10} + \cdots - 12857624 \)
|
| $67$ |
\( T^{11} + \cdots + 549577120 \)
|
| $71$ |
\( T^{11} - 6 T^{10} + \cdots + 25600 \)
|
| $73$ |
\( T^{11} - 3 T^{10} + \cdots - 3439280 \)
|
| $79$ |
\( T^{11} - 8 T^{10} + \cdots - 54675632 \)
|
| $83$ |
\( T^{11} - 29 T^{10} + \cdots + 2916640 \)
|
| $89$ |
\( T^{11} - 20 T^{10} + \cdots - 26698136 \)
|
| $97$ |
\( T^{11} + \cdots - 943989120 \)
|
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