Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1980,2,Mod(661,1980)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1980, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([0, 2, 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("1980.661");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1980 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 11 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1980.q (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: | \(15.8103796002\) |
| Analytic rank: | \(0\) |
| Dimension: | \(14\) |
| Relative dimension: | \(7\) over \(\Q(\zeta_{3})\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{14} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{14} - 5 x^{12} - 6 x^{11} + 10 x^{10} + 21 x^{9} - 9 x^{8} - 27 x^{7} - 27 x^{6} + 189 x^{5} + \cdots + 2187 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 3^{2} \) |
| 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 \(1,\beta_1,\ldots,\beta_{13}\) 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
\( x^{14} - 5 x^{12} - 6 x^{11} + 10 x^{10} + 21 x^{9} - 9 x^{8} - 27 x^{7} - 27 x^{6} + 189 x^{5} + \cdots + 2187 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 2 \nu^{13} + 21 \nu^{12} + 143 \nu^{11} - 171 \nu^{10} - 466 \nu^{9} + 90 \nu^{8} + 252 \nu^{7} + \cdots - 61965 ) / 2187 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( - 7 \nu^{13} + 129 \nu^{12} - 136 \nu^{11} - 414 \nu^{10} + 173 \nu^{9} + 252 \nu^{8} + 576 \nu^{7} + \cdots + 10206 ) / 2187 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 13 \nu^{13} + 102 \nu^{12} - 241 \nu^{11} - 162 \nu^{10} + 707 \nu^{9} + 18 \nu^{8} + 171 \nu^{7} + \cdots + 86751 ) / 2187 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 13 \nu^{13} - 102 \nu^{12} + 241 \nu^{11} + 162 \nu^{10} - 707 \nu^{9} - 18 \nu^{8} - 171 \nu^{7} + \cdots - 86751 ) / 2187 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 47 \nu^{13} + 114 \nu^{12} + 190 \nu^{11} - 396 \nu^{10} - 362 \nu^{9} - 9 \nu^{8} + 909 \nu^{7} + \cdots - 47385 ) / 2187 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 19 \nu^{13} - 57 \nu^{12} + 311 \nu^{11} + 75 \nu^{10} - 685 \nu^{9} - 78 \nu^{8} + 162 \nu^{7} + \cdots - 81648 ) / 729 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 11 \nu^{13} + 8 \nu^{12} + 43 \nu^{11} - 10 \nu^{10} - 44 \nu^{9} - 34 \nu^{8} + 93 \nu^{7} + \cdots - 3645 ) / 243 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 32 \nu^{13} + 15 \nu^{12} + 232 \nu^{11} - 72 \nu^{10} - 446 \nu^{9} - 90 \nu^{8} + 351 \nu^{7} + \cdots - 49572 ) / 729 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 34 \nu^{13} - 102 \nu^{12} - 80 \nu^{11} + 279 \nu^{10} + 97 \nu^{9} + 18 \nu^{8} - 576 \nu^{7} + \cdots + 9477 ) / 729 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( 46 \nu^{13} - 69 \nu^{12} - 185 \nu^{11} + 177 \nu^{10} + 244 \nu^{9} + 114 \nu^{8} - 522 \nu^{7} + \cdots + 24057 ) / 729 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( - 49 \nu^{13} - 6 \nu^{12} + 371 \nu^{11} - 27 \nu^{10} - 679 \nu^{9} - 171 \nu^{8} + 441 \nu^{7} + \cdots - 75816 ) / 729 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( - 191 \nu^{13} + 42 \nu^{12} + 1450 \nu^{11} - 387 \nu^{10} - 2855 \nu^{9} - 432 \nu^{8} + \cdots - 331695 ) / 2187 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( - 26 \nu^{13} + 27 \nu^{12} + 160 \nu^{11} - 105 \nu^{10} - 302 \nu^{9} - 42 \nu^{8} + 345 \nu^{7} + \cdots - 35478 ) / 243 \)
|
| \(\nu\) | \(=\) |
\( \beta_{4} + \beta_{3} \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{12} + \beta_{10} - \beta_{6} + \beta_{3} - \beta_{2} + \beta_1 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{13} + \beta_{11} + 2\beta_{10} - \beta_{9} - \beta_{8} - \beta_{6} - \beta_{5} + 2\beta_{4} + 2\beta_{3} - \beta_{2} \)
|
| \(\nu^{4}\) | \(=\) |
\( -3\beta_{13} - \beta_{12} + 4\beta_{11} + 2\beta_{7} + 2\beta_{5} + 3\beta_{4} + 3\beta_{3} + 3\beta_{2} + 4\beta _1 - 1 \)
|
| \(\nu^{5}\) | \(=\) |
\( 2 \beta_{13} - 3 \beta_{12} + \beta_{11} + 3 \beta_{10} + \beta_{9} + 3 \beta_{8} + 2 \beta_{7} + \cdots + 1 \)
|
| \(\nu^{6}\) | \(=\) |
\( 4 \beta_{13} - 4 \beta_{12} + \beta_{11} - 2 \beta_{10} + 2 \beta_{9} - 8 \beta_{8} + 7 \beta_{7} + \cdots + 15 \)
|
| \(\nu^{7}\) | \(=\) |
\( - 6 \beta_{13} + 6 \beta_{12} - 5 \beta_{11} + 10 \beta_{10} - 12 \beta_{9} + \beta_{8} + \beta_{7} + \cdots - 4 \)
|
| \(\nu^{8}\) | \(=\) |
\( 8 \beta_{13} + 14 \beta_{12} - 8 \beta_{11} + 11 \beta_{10} + 4 \beta_{9} - 7 \beta_{8} + 3 \beta_{7} + \cdots + 43 \)
|
| \(\nu^{9}\) | \(=\) |
\( \beta_{13} + 3 \beta_{12} + 13 \beta_{11} + 17 \beta_{10} - 4 \beta_{9} - 10 \beta_{8} - 6 \beta_{7} + \cdots - 90 \)
|
| \(\nu^{10}\) | \(=\) |
\( 3 \beta_{13} + 65 \beta_{12} + 16 \beta_{11} + 108 \beta_{10} - 48 \beta_{9} - 39 \beta_{8} + 44 \beta_{7} + \cdots - 70 \)
|
| \(\nu^{11}\) | \(=\) |
\( - 7 \beta_{13} - 120 \beta_{12} + 109 \beta_{11} + 12 \beta_{10} - 44 \beta_{9} - 15 \beta_{8} + \cdots + 1 \)
|
| \(\nu^{12}\) | \(=\) |
\( - 113 \beta_{13} - 94 \beta_{12} + 55 \beta_{11} - 101 \beta_{10} + 20 \beta_{9} - 8 \beta_{8} + \cdots - 129 \)
|
| \(\nu^{13}\) | \(=\) |
\( 210 \beta_{13} + 6 \beta_{12} - 338 \beta_{11} + 127 \beta_{10} - 66 \beta_{9} + 82 \beta_{8} + \cdots + 167 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1980\mathbb{Z}\right)^\times\).
| \(n\) | \(397\) | \(541\) | \(991\) | \(1541\) |
| \(\chi(n)\) | \(1\) | \(1\) | \(1\) | \(-1 + \beta_{3}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 661.1 |
|
0 | −1.71925 | + | 0.210188i | 0 | 0.500000 | − | 0.866025i | 0 | 1.50392 | + | 2.60487i | 0 | 2.91164 | − | 0.722733i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 661.2 | 0 | −1.23770 | − | 1.21165i | 0 | 0.500000 | − | 0.866025i | 0 | −0.378341 | − | 0.655305i | 0 | 0.0637891 | + | 2.99932i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 661.3 | 0 | −1.23559 | + | 1.21380i | 0 | 0.500000 | − | 0.866025i | 0 | −1.25403 | − | 2.17204i | 0 | 0.0533611 | − | 2.99953i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 661.4 | 0 | −0.258003 | + | 1.71273i | 0 | 0.500000 | − | 0.866025i | 0 | 0.937480 | + | 1.62376i | 0 | −2.86687 | − | 0.883779i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 661.5 | 0 | 1.02424 | + | 1.39676i | 0 | 0.500000 | − | 0.866025i | 0 | −1.86264 | − | 3.22618i | 0 | −0.901859 | + | 2.86123i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 661.6 | 0 | 1.70298 | + | 0.315997i | 0 | 0.500000 | − | 0.866025i | 0 | −0.664196 | − | 1.15042i | 0 | 2.80029 | + | 1.07628i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 661.7 | 0 | 1.72332 | − | 0.173718i | 0 | 0.500000 | − | 0.866025i | 0 | 1.21780 | + | 2.10930i | 0 | 2.93964 | − | 0.598741i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1321.1 | 0 | −1.71925 | − | 0.210188i | 0 | 0.500000 | + | 0.866025i | 0 | 1.50392 | − | 2.60487i | 0 | 2.91164 | + | 0.722733i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1321.2 | 0 | −1.23770 | + | 1.21165i | 0 | 0.500000 | + | 0.866025i | 0 | −0.378341 | + | 0.655305i | 0 | 0.0637891 | − | 2.99932i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1321.3 | 0 | −1.23559 | − | 1.21380i | 0 | 0.500000 | + | 0.866025i | 0 | −1.25403 | + | 2.17204i | 0 | 0.0533611 | + | 2.99953i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1321.4 | 0 | −0.258003 | − | 1.71273i | 0 | 0.500000 | + | 0.866025i | 0 | 0.937480 | − | 1.62376i | 0 | −2.86687 | + | 0.883779i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1321.5 | 0 | 1.02424 | − | 1.39676i | 0 | 0.500000 | + | 0.866025i | 0 | −1.86264 | + | 3.22618i | 0 | −0.901859 | − | 2.86123i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1321.6 | 0 | 1.70298 | − | 0.315997i | 0 | 0.500000 | + | 0.866025i | 0 | −0.664196 | + | 1.15042i | 0 | 2.80029 | − | 1.07628i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1321.7 | 0 | 1.72332 | + | 0.173718i | 0 | 0.500000 | + | 0.866025i | 0 | 1.21780 | − | 2.10930i | 0 | 2.93964 | + | 0.598741i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 9.c | even | 3 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 1980.2.q.e | ✓ | 14 |
| 3.b | odd | 2 | 1 | 5940.2.q.e | 14 | ||
| 9.c | even | 3 | 1 | inner | 1980.2.q.e | ✓ | 14 |
| 9.d | odd | 6 | 1 | 5940.2.q.e | 14 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1980.2.q.e | ✓ | 14 | 1.a | even | 1 | 1 | trivial |
| 1980.2.q.e | ✓ | 14 | 9.c | even | 3 | 1 | inner |
| 5940.2.q.e | 14 | 3.b | odd | 2 | 1 | ||
| 5940.2.q.e | 14 | 9.d | odd | 6 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{7}^{14} + T_{7}^{13} + 21 T_{7}^{12} + 6 T_{7}^{11} + 297 T_{7}^{10} + 90 T_{7}^{9} + 2049 T_{7}^{8} + \cdots + 16641 \)
acting on \(S_{2}^{\mathrm{new}}(1980, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{14} \)
|
| $3$ |
\( T^{14} - 5 T^{12} + \cdots + 2187 \)
|
| $5$ |
\( (T^{2} - T + 1)^{7} \)
|
| $7$ |
\( T^{14} + T^{13} + \cdots + 16641 \)
|
| $11$ |
\( (T^{2} + T + 1)^{7} \)
|
| $13$ |
\( T^{14} + 5 T^{13} + \cdots + 2401 \)
|
| $17$ |
\( (T^{7} - 2 T^{6} + \cdots - 2133)^{2} \)
|
| $19$ |
\( (T^{7} - 5 T^{6} + \cdots - 797)^{2} \)
|
| $23$ |
\( T^{14} - 4 T^{13} + \cdots + 7634169 \)
|
| $29$ |
\( T^{14} - 8 T^{13} + \cdots + 23707161 \)
|
| $31$ |
\( T^{14} - 4 T^{13} + \cdots + 6889 \)
|
| $37$ |
\( (T^{7} + 6 T^{6} + \cdots + 76669)^{2} \)
|
| $41$ |
\( T^{14} + \cdots + 2359336329 \)
|
| $43$ |
\( T^{14} + \cdots + 2797246321 \)
|
| $47$ |
\( T^{14} + \cdots + 2952074889 \)
|
| $53$ |
\( (T^{7} + 8 T^{6} + \cdots - 256689)^{2} \)
|
| $59$ |
\( T^{14} + \cdots + 2162529009 \)
|
| $61$ |
\( T^{14} + \cdots + 287404209 \)
|
| $67$ |
\( T^{14} + \cdots + 8558285121 \)
|
| $71$ |
\( (T^{7} + 18 T^{6} + \cdots - 383211)^{2} \)
|
| $73$ |
\( (T^{7} - 5 T^{6} + \cdots - 602867)^{2} \)
|
| $79$ |
\( T^{14} - 5 T^{13} + \cdots + 25130169 \)
|
| $83$ |
\( T^{14} + \cdots + 844541721 \)
|
| $89$ |
\( (T^{7} + 6 T^{6} + \cdots + 32319)^{2} \)
|
| $97$ |
\( T^{14} + \cdots + 1139306468689 \)
|
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