Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [6024,2,Mod(1,6024)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6024, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([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("6024.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 6024 = 2^{3} \cdot 3 \cdot 251 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 6024.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: | \(48.1018821776\) |
| 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: |
\( x^{11} - 5x^{10} - 10x^{9} + 80x^{8} - 78x^{7} - 150x^{6} + 296x^{5} - 116x^{4} - 45x^{3} + 31x^{2} - x - 1 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| 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
\( x^{11} - 5x^{10} - 10x^{9} + 80x^{8} - 78x^{7} - 150x^{6} + 296x^{5} - 116x^{4} - 45x^{3} + 31x^{2} - x - 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( - 13 \nu^{10} + 66 \nu^{9} + 127 \nu^{8} - 1056 \nu^{7} + 1062 \nu^{6} + 1968 \nu^{5} - 3962 \nu^{4} + \cdots + 34 ) / 27 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( - 13 \nu^{10} + 66 \nu^{9} + 127 \nu^{8} - 1056 \nu^{7} + 1062 \nu^{6} + 1968 \nu^{5} - 3962 \nu^{4} + \cdots + 7 ) / 27 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 14 \nu^{10} - 69 \nu^{9} - 143 \nu^{8} + 1104 \nu^{7} - 1044 \nu^{6} - 2082 \nu^{5} + 4030 \nu^{4} + \cdots - 20 ) / 27 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 9 \nu^{10} - 34 \nu^{9} - 143 \nu^{8} + 597 \nu^{7} + 165 \nu^{6} - 1992 \nu^{5} + 747 \nu^{4} + \cdots + 91 ) / 18 \)
|
| \(\beta_{5}\) | \(=\) |
\( -\nu^{10} + 4\nu^{9} + 14\nu^{8} - 66\nu^{7} + 12\nu^{6} + 162\nu^{5} - 134\nu^{4} - 18\nu^{3} + 27\nu^{2} - 4\nu - 2 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 41 \nu^{10} + 185 \nu^{9} + 495 \nu^{8} - 3021 \nu^{7} + 1806 \nu^{6} + 6747 \nu^{5} - 8914 \nu^{4} + \cdots - 99 ) / 18 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 21 \nu^{10} + 91 \nu^{9} + 271 \nu^{8} - 1501 \nu^{7} + 634 \nu^{6} + 3599 \nu^{5} - 3842 \nu^{4} + \cdots - 27 ) / 6 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 62 \nu^{10} - 275 \nu^{9} - 771 \nu^{8} + 4512 \nu^{7} - 2361 \nu^{6} - 10422 \nu^{5} + 12622 \nu^{4} + \cdots + 117 ) / 18 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 99 \nu^{10} - 454 \nu^{9} - 1175 \nu^{8} + 7425 \nu^{7} - 4701 \nu^{6} - 16656 \nu^{5} + 22557 \nu^{4} + \cdots + 253 ) / 18 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( 135 \nu^{10} - 613 \nu^{9} - 1625 \nu^{8} + 10029 \nu^{7} - 6018 \nu^{6} - 22611 \nu^{5} + 29538 \nu^{4} + \cdots + 301 ) / 18 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{2} - \beta _1 + 1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -2\beta_{10} - 2\beta_{7} - 4\beta_{6} + 2\beta_{5} - 2\beta_{3} - 3\beta_{2} - \beta _1 + 7 ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( -\beta_{10} + 2\beta_{9} - 4\beta_{8} - 2\beta_{7} - 3\beta_{6} - 2\beta_{4} - 4\beta_{3} + 7\beta_{2} - 6\beta _1 + 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( - 32 \beta_{10} + 2 \beta_{9} - 12 \beta_{8} - 32 \beta_{7} - 72 \beta_{6} + 26 \beta_{5} + 6 \beta_{4} + \cdots + 85 ) / 2 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( - 50 \beta_{10} + 78 \beta_{9} - 160 \beta_{8} - 88 \beta_{7} - 134 \beta_{6} + 4 \beta_{5} - 66 \beta_{4} + \cdots + 65 ) / 2 \)
|
| \(\nu^{6}\) | \(=\) |
\( - 258 \beta_{10} + 34 \beta_{9} - 150 \beta_{8} - 269 \beta_{7} - 599 \beta_{6} + 193 \beta_{5} + \cdots + 631 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( - 1008 \beta_{10} + 1290 \beta_{9} - 2736 \beta_{8} - 1638 \beta_{7} - 2622 \beta_{6} + 196 \beta_{5} + \cdots + 1393 ) / 2 \)
|
| \(\nu^{8}\) | \(=\) |
\( ( - 8396 \beta_{10} + 1632 \beta_{9} - 6024 \beta_{8} - 9044 \beta_{7} - 19720 \beta_{6} + 5970 \beta_{5} + \cdots + 19705 ) / 2 \)
|
| \(\nu^{9}\) | \(=\) |
\( - 9560 \beta_{10} + 10412 \beta_{9} - 22696 \beta_{8} - 14634 \beta_{7} - 24458 \beta_{6} + \cdots + 14469 \)
|
| \(\nu^{10}\) | \(=\) |
\( ( - 137518 \beta_{10} + 34116 \beta_{9} - 113096 \beta_{8} - 151986 \beta_{7} - 325128 \beta_{6} + \cdots + 314077 ) / 2 \)
|
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 |
|
0 | −1.00000 | 0 | −3.58783 | 0 | −3.54697 | 0 | 1.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | 0 | −1.00000 | 0 | −2.47965 | 0 | −2.94490 | 0 | 1.00000 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | 0 | −1.00000 | 0 | −0.580810 | 0 | 0.289648 | 0 | 1.00000 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | 0 | −1.00000 | 0 | −0.558544 | 0 | −4.82539 | 0 | 1.00000 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | 0 | −1.00000 | 0 | 0.110431 | 0 | 0.221258 | 0 | 1.00000 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 0 | −1.00000 | 0 | 0.127771 | 0 | 1.49959 | 0 | 1.00000 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 0 | −1.00000 | 0 | 1.86694 | 0 | −2.48455 | 0 | 1.00000 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 0 | −1.00000 | 0 | 2.04572 | 0 | 4.85647 | 0 | 1.00000 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 0 | −1.00000 | 0 | 2.40158 | 0 | −3.19474 | 0 | 1.00000 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 0 | −1.00000 | 0 | 2.72612 | 0 | 3.78126 | 0 | 1.00000 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.11 | 0 | −1.00000 | 0 | 3.92826 | 0 | 1.34833 | 0 | 1.00000 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(3\) | \(1\) |
| \(251\) | \(1\) |
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 | 6024.2.a.l | ✓ | 11 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 6024.2.a.l | ✓ | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6024))\):
|
\( T_{5}^{11} - 6 T_{5}^{10} - 10 T_{5}^{9} + 116 T_{5}^{8} - 102 T_{5}^{7} - 516 T_{5}^{6} + 883 T_{5}^{5} + \cdots - 4 \)
|
|
\( T_{7}^{11} + 5 T_{7}^{10} - 39 T_{7}^{9} - 218 T_{7}^{8} + 380 T_{7}^{7} + 2815 T_{7}^{6} - 401 T_{7}^{5} + \cdots + 952 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{11} \)
|
| $3$ |
\( (T + 1)^{11} \)
|
| $5$ |
\( T^{11} - 6 T^{10} + \cdots - 4 \)
|
| $7$ |
\( T^{11} + 5 T^{10} + \cdots + 952 \)
|
| $11$ |
\( T^{11} - 11 T^{10} + \cdots - 424 \)
|
| $13$ |
\( T^{11} + 4 T^{10} + \cdots + 2068 \)
|
| $17$ |
\( T^{11} + 20 T^{10} + \cdots - 3816 \)
|
| $19$ |
\( T^{11} - 4 T^{10} + \cdots + 113168 \)
|
| $23$ |
\( T^{11} - 6 T^{10} + \cdots + 88272 \)
|
| $29$ |
\( T^{11} + 6 T^{10} + \cdots + 20812 \)
|
| $31$ |
\( T^{11} - 19 T^{10} + \cdots - 16672 \)
|
| $37$ |
\( T^{11} + 24 T^{10} + \cdots + 3915072 \)
|
| $41$ |
\( T^{11} + 18 T^{10} + \cdots + 26871808 \)
|
| $43$ |
\( T^{11} + \cdots - 264723248 \)
|
| $47$ |
\( T^{11} - 27 T^{10} + \cdots - 8568 \)
|
| $53$ |
\( T^{11} + \cdots - 414667712 \)
|
| $59$ |
\( T^{11} + \cdots - 3951519158 \)
|
| $61$ |
\( T^{11} + \cdots - 20688705332 \)
|
| $67$ |
\( T^{11} + \cdots + 5051264102 \)
|
| $71$ |
\( T^{11} - 53 T^{10} + \cdots - 46932608 \)
|
| $73$ |
\( T^{11} + \cdots - 3757697863 \)
|
| $79$ |
\( T^{11} - 27 T^{10} + \cdots - 128 \)
|
| $83$ |
\( T^{11} - 25 T^{10} + \cdots - 2760128 \)
|
| $89$ |
\( T^{11} + \cdots + 14188740688 \)
|
| $97$ |
\( T^{11} + \cdots - 72428695048 \)
|
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