Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [2280,2,Mod(121,2280)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2280, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([0, 0, 0, 0, 2]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("2280.121");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 2280 = 2^{3} \cdot 3 \cdot 5 \cdot 19 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 2280.bg (of order \(3\), degree \(2\), minimal) |
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: | no |
| Analytic conductor: | \(18.2058916609\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Relative dimension: | \(4\) over \(\Q(\zeta_{3})\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{8} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} - x^{7} + 13x^{6} + 126x^{4} - 24x^{3} + 324x^{2} + 144x + 576 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{19}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{3}]$ |
$q$-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
\( x^{8} - x^{7} + 13x^{6} + 126x^{4} - 24x^{3} + 324x^{2} + 144x + 576 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -8\nu^{7} - 97\nu^{6} - 84\nu^{5} - 68\nu^{4} - 2935\nu^{3} - 312\nu^{2} - 480\nu + 42204 ) / 12414 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 337 \nu^{7} + 1345 \nu^{6} - 4573 \nu^{5} + 10584 \nu^{4} - 33894 \nu^{3} + 142032 \nu^{2} + \cdots + 309888 ) / 297936 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 641 \nu^{7} - 1797 \nu^{6} + 16041 \nu^{5} - 16276 \nu^{4} + 124734 \nu^{3} - 80520 \nu^{2} + \cdots - 175680 ) / 297936 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -42\nu^{7} + 8\nu^{6} - 441\nu^{5} - 357\nu^{4} - 5581\nu^{3} - 1638\nu^{2} - 2520\nu - 8088 ) / 12414 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 201\nu^{7} - 925\nu^{6} + 3145\nu^{5} - 11740\nu^{4} + 23310\nu^{3} - 97680\nu^{2} + 12060\nu - 213120 ) / 49656 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 1227 \nu^{7} + 653 \nu^{6} + 7711 \nu^{5} + 28016 \nu^{4} + 62994 \nu^{3} + 178200 \nu^{2} + \cdots + 388800 ) / 297936 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{6} - \beta_{5} + 6\beta_{3} + \beta_{2} + \beta _1 - 6 \)
|
| \(\nu^{3}\) | \(=\) |
\( -4\beta_{7} - 7\beta_{5} - 2\beta_{4} + 2\beta_{3} + \beta_{2} - 2\beta _1 - 6 \)
|
| \(\nu^{4}\) | \(=\) |
\( -2\beta_{7} - 13\beta_{6} + 2\beta_{4} - 50\beta_{3} - 17\beta_1 \)
|
| \(\nu^{5}\) | \(=\) |
\( 26\beta_{7} - 19\beta_{6} + 67\beta_{5} + 52\beta_{4} - 64\beta_{3} - 19\beta_{2} - 93\beta _1 + 90 \)
|
| \(\nu^{6}\) | \(=\) |
\( 76\beta_{7} + 157\beta_{5} + 38\beta_{4} - 38\beta_{3} - 145\beta_{2} + 38\beta _1 + 558 \)
|
| \(\nu^{7}\) | \(=\) |
\( 290\beta_{7} + 271\beta_{6} - 290\beta_{4} + 590\beta_{3} + 1295\beta_1 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2280\mathbb{Z}\right)^\times\).
| \(n\) | \(457\) | \(761\) | \(1141\) | \(1711\) | \(1921\) |
| \(\chi(n)\) | \(1\) | \(1\) | \(1\) | \(1\) | \(-\beta_{3}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 121.1 |
|
0 | −0.500000 | − | 0.866025i | 0 | 0.500000 | + | 0.866025i | 0 | −2.39980 | 0 | −0.500000 | + | 0.866025i | 0 | ||||||||||||||||||||||||||||||||||||
| 121.2 | 0 | −0.500000 | − | 0.866025i | 0 | 0.500000 | + | 0.866025i | 0 | −0.824880 | 0 | −0.500000 | + | 0.866025i | 0 | |||||||||||||||||||||||||||||||||||||
| 121.3 | 0 | −0.500000 | − | 0.866025i | 0 | 0.500000 | + | 0.866025i | 0 | 2.34186 | 0 | −0.500000 | + | 0.866025i | 0 | |||||||||||||||||||||||||||||||||||||
| 121.4 | 0 | −0.500000 | − | 0.866025i | 0 | 0.500000 | + | 0.866025i | 0 | 3.88281 | 0 | −0.500000 | + | 0.866025i | 0 | |||||||||||||||||||||||||||||||||||||
| 961.1 | 0 | −0.500000 | + | 0.866025i | 0 | 0.500000 | − | 0.866025i | 0 | −2.39980 | 0 | −0.500000 | − | 0.866025i | 0 | |||||||||||||||||||||||||||||||||||||
| 961.2 | 0 | −0.500000 | + | 0.866025i | 0 | 0.500000 | − | 0.866025i | 0 | −0.824880 | 0 | −0.500000 | − | 0.866025i | 0 | |||||||||||||||||||||||||||||||||||||
| 961.3 | 0 | −0.500000 | + | 0.866025i | 0 | 0.500000 | − | 0.866025i | 0 | 2.34186 | 0 | −0.500000 | − | 0.866025i | 0 | |||||||||||||||||||||||||||||||||||||
| 961.4 | 0 | −0.500000 | + | 0.866025i | 0 | 0.500000 | − | 0.866025i | 0 | 3.88281 | 0 | −0.500000 | − | 0.866025i | 0 | |||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 19.c | even | 3 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 2280.2.bg.u | ✓ | 8 |
| 19.c | even | 3 | 1 | inner | 2280.2.bg.u | ✓ | 8 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 2280.2.bg.u | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 2280.2.bg.u | ✓ | 8 | 19.c | even | 3 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(2280, [\chi])\):
|
\( T_{7}^{4} - 3T_{7}^{3} - 9T_{7}^{2} + 17T_{7} + 18 \)
|
|
\( T_{11}^{4} + 7T_{11}^{3} - 9T_{11}^{2} - 87T_{11} + 24 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} \)
|
| $3$ |
\( (T^{2} + T + 1)^{4} \)
|
| $5$ |
\( (T^{2} - T + 1)^{4} \)
|
| $7$ |
\( (T^{4} - 3 T^{3} - 9 T^{2} + \cdots + 18)^{2} \)
|
| $11$ |
\( (T^{4} + 7 T^{3} - 9 T^{2} + \cdots + 24)^{2} \)
|
| $13$ |
\( T^{8} - 5 T^{7} + \cdots + 9216 \)
|
| $17$ |
\( T^{8} - 8 T^{7} + \cdots + 256 \)
|
| $19$ |
\( T^{8} + T^{7} + \cdots + 130321 \)
|
| $23$ |
\( T^{8} + 8 T^{7} + \cdots + 1149184 \)
|
| $29$ |
\( T^{8} - 3 T^{7} + \cdots + 3779136 \)
|
| $31$ |
\( (T^{4} - 4 T^{3} - 57 T^{2} + \cdots - 24)^{2} \)
|
| $37$ |
\( (T^{4} + 13 T^{3} + \cdots - 906)^{2} \)
|
| $41$ |
\( T^{8} + 22 T^{7} + \cdots + 4096 \)
|
| $43$ |
\( T^{8} + T^{7} + \cdots + 287296 \)
|
| $47$ |
\( T^{8} + 10 T^{7} + \cdots + 3444736 \)
|
| $53$ |
\( T^{8} - 12 T^{7} + \cdots + 36433296 \)
|
| $59$ |
\( T^{8} + 25 T^{7} + \cdots + 331776 \)
|
| $61$ |
\( T^{8} + 2 T^{7} + \cdots + 17791524 \)
|
| $67$ |
\( T^{8} + 13 T^{7} + \cdots + 1478656 \)
|
| $71$ |
\( T^{8} - 23 T^{7} + \cdots + 501264 \)
|
| $73$ |
\( T^{8} - 15 T^{7} + \cdots + 40602384 \)
|
| $79$ |
\( T^{8} - 16 T^{7} + \cdots + 131698576 \)
|
| $83$ |
\( (T^{4} + 14 T^{3} + \cdots + 384)^{2} \)
|
| $89$ |
\( T^{8} + 3 T^{7} + \cdots + 4613904 \)
|
| $97$ |
\( T^{8} + 20 T^{7} + \cdots + 102252544 \)
|
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