Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1512,2,Mod(865,1512)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1512, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([0, 0, 0, 4]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1512.865");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1512 = 2^{3} \cdot 3^{3} \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1512.s (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: | \(12.0733807856\) |
| 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: |
\( x^{8} - 3x^{7} + 4x^{6} + 28x^{5} + 14x^{4} - 52x^{3} + 306x^{2} + 1052x + 1051 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| 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
\( x^{8} - 3x^{7} + 4x^{6} + 28x^{5} + 14x^{4} - 52x^{3} + 306x^{2} + 1052x + 1051 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{7} - 16\nu^{6} + 212\nu^{5} - 2728\nu^{4} + 7695\nu^{3} - 16738\nu^{2} - 4364\nu - 25565 ) / 27783 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -29\nu^{7} + 86\nu^{6} + 845\nu^{5} - 4615\nu^{4} + 3267\nu^{3} + 11201\nu^{2} - 15194\nu - 88514 ) / 27783 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -31\nu^{7} + 307\nu^{6} - 1091\nu^{5} + 1030\nu^{4} + 1674\nu^{3} - 2006\nu^{2} - 14971\nu + 14402 ) / 27783 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -83\nu^{7} + 383\nu^{6} - 2098\nu^{5} + 3215\nu^{4} - 5157\nu^{3} - 12559\nu^{2} - 23915\nu + 21349 ) / 27783 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( -214\nu^{7} + 967\nu^{6} - 1898\nu^{5} - 2108\nu^{4} + 3618\nu^{3} + 5485\nu^{2} - 47014\nu - 32581 ) / 27783 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( -58\nu^{7} + 226\nu^{6} - 443\nu^{5} - 1238\nu^{4} + 837\nu^{3} + 2260\nu^{2} - 20344\nu - 33523 ) / 3969 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{7} - \beta_{6} - 2\beta_{5} + \beta_{4} - 2\beta_{3} - \beta_{2} + \beta _1 + 1 \)
|
| \(\nu^{3}\) | \(=\) |
\( 2\beta_{7} - 7\beta_{5} + \beta_{4} - 9\beta_{3} + \beta_{2} - 6 \)
|
| \(\nu^{4}\) | \(=\) |
\( -2\beta_{7} + 9\beta_{6} - 7\beta_{5} - 6\beta_{4} - 12\beta_{3} - \beta_{2} - 11\beta _1 - 37 \)
|
| \(\nu^{5}\) | \(=\) |
\( -27\beta_{7} + 46\beta_{6} + 15\beta_{5} - 33\beta_{4} + 31\beta_{3} + 3\beta_{2} - 48\beta _1 - 67 \)
|
| \(\nu^{6}\) | \(=\) |
\( -113\beta_{7} + 112\beta_{6} + 144\beta_{5} + 63\beta_{4} + 276\beta_{3} - \beta_{2} - 81\beta _1 - 88 \)
|
| \(\nu^{7}\) | \(=\) |
\( -192\beta_{7} - 146\beta_{6} + 417\beta_{5} + 679\beta_{4} + 887\beta_{3} - 30\beta_{2} - 26\beta _1 + 333 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1512\mathbb{Z}\right)^\times\).
| \(n\) | \(757\) | \(785\) | \(1081\) | \(1135\) |
| \(\chi(n)\) | \(1\) | \(1\) | \(-\beta_{4}\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 865.1 |
|
0 | 0 | 0 | −1.70873 | − | 2.95960i | 0 | −2.27499 | − | 1.35071i | 0 | 0 | 0 | ||||||||||||||||||||||||||||||||||||||
| 865.2 | 0 | 0 | 0 | −1.26913 | − | 2.19820i | 0 | 2.64086 | + | 0.160857i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||
| 865.3 | 0 | 0 | 0 | −0.0144658 | − | 0.0250554i | 0 | −1.70811 | + | 2.02048i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||
| 865.4 | 0 | 0 | 0 | 1.99232 | + | 3.45080i | 0 | −0.657753 | − | 2.56269i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||
| 1297.1 | 0 | 0 | 0 | −1.70873 | + | 2.95960i | 0 | −2.27499 | + | 1.35071i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||
| 1297.2 | 0 | 0 | 0 | −1.26913 | + | 2.19820i | 0 | 2.64086 | − | 0.160857i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||
| 1297.3 | 0 | 0 | 0 | −0.0144658 | + | 0.0250554i | 0 | −1.70811 | − | 2.02048i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||
| 1297.4 | 0 | 0 | 0 | 1.99232 | − | 3.45080i | 0 | −0.657753 | + | 2.56269i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 7.c | even | 3 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 1512.2.s.n | ✓ | 8 |
| 3.b | odd | 2 | 1 | 1512.2.s.o | yes | 8 | |
| 7.c | even | 3 | 1 | inner | 1512.2.s.n | ✓ | 8 |
| 21.h | odd | 6 | 1 | 1512.2.s.o | yes | 8 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1512.2.s.n | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 1512.2.s.n | ✓ | 8 | 7.c | even | 3 | 1 | inner |
| 1512.2.s.o | yes | 8 | 3.b | odd | 2 | 1 | |
| 1512.2.s.o | yes | 8 | 21.h | odd | 6 | 1 | |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1512, [\chi])\):
|
\( T_{5}^{8} + 2T_{5}^{7} + 19T_{5}^{6} + 40T_{5}^{5} + 296T_{5}^{4} + 529T_{5}^{3} + 1210T_{5}^{2} + 35T_{5} + 1 \)
|
|
\( T_{11}^{8} + T_{11}^{7} + 29T_{11}^{6} + 130T_{11}^{5} + 911T_{11}^{4} + 2308T_{11}^{3} + 4897T_{11}^{2} + 3792T_{11} + 2304 \)
|
|
\( T_{13}^{4} + 10T_{13}^{3} + 21T_{13}^{2} - T_{13} - 4 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} \)
|
| $3$ |
\( T^{8} \)
|
| $5$ |
\( T^{8} + 2 T^{7} + \cdots + 1 \)
|
| $7$ |
\( T^{8} + 4 T^{7} + \cdots + 2401 \)
|
| $11$ |
\( T^{8} + T^{7} + \cdots + 2304 \)
|
| $13$ |
\( (T^{4} + 10 T^{3} + 21 T^{2} + \cdots - 4)^{2} \)
|
| $17$ |
\( T^{8} + 4 T^{7} + \cdots + 324 \)
|
| $19$ |
\( T^{8} - T^{7} + \cdots + 633616 \)
|
| $23$ |
\( T^{8} - 12 T^{7} + \cdots + 910116 \)
|
| $29$ |
\( (T^{4} - 6 T^{3} + \cdots + 351)^{2} \)
|
| $31$ |
\( T^{8} - 8 T^{7} + \cdots + 430336 \)
|
| $37$ |
\( T^{8} + 85 T^{6} + \cdots + 69696 \)
|
| $41$ |
\( (T^{4} + 6 T^{3} + \cdots - 402)^{2} \)
|
| $43$ |
\( (T^{4} - T^{3} - 28 T^{2} + \cdots - 48)^{2} \)
|
| $47$ |
\( T^{8} + 9 T^{7} + \cdots + 777924 \)
|
| $53$ |
\( T^{8} - 7 T^{7} + \cdots + 81 \)
|
| $59$ |
\( T^{8} + 4 T^{7} + \cdots + 68906601 \)
|
| $61$ |
\( T^{8} + 25 T^{7} + \cdots + 1313316 \)
|
| $67$ |
\( T^{8} + 30 T^{7} + \cdots + 617796 \)
|
| $71$ |
\( (T^{4} + 11 T^{3} + \cdots - 622)^{2} \)
|
| $73$ |
\( T^{8} - 4 T^{7} + \cdots + 290521 \)
|
| $79$ |
\( T^{8} - 7 T^{7} + \cdots + 147889921 \)
|
| $83$ |
\( (T^{4} - 29 T^{3} + \cdots - 3428)^{2} \)
|
| $89$ |
\( T^{8} - 9 T^{7} + \cdots + 19131876 \)
|
| $97$ |
\( (T^{4} + 4 T^{3} + \cdots - 2544)^{2} \)
|
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