Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1764,2,Mod(1685,1764)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1764, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([0, 1, 5]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1764.1685");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1764 = 2^{2} \cdot 3^{2} \cdot 7^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1764.bm (of order \(6\), degree \(2\), not 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: | \(14.0856109166\) |
| Analytic rank: | \(0\) |
| Dimension: | \(16\) |
| Relative dimension: | \(8\) over \(\Q(\zeta_{6})\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{16} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{16} - 2 x^{15} + 5 x^{14} - 17 x^{13} + 22 x^{12} - 31 x^{11} + 62 x^{10} - 52 x^{9} + 52 x^{8} + \cdots + 6561 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| Coefficient ring index: | \( 3^{4} \) |
| Twist minimal: | no (minimal twist has level 252) |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{6}]$ |
$q$-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 \(1,\beta_1,\ldots,\beta_{15}\) 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
\( x^{16} - 2 x^{15} + 5 x^{14} - 17 x^{13} + 22 x^{12} - 31 x^{11} + 62 x^{10} - 52 x^{9} + 52 x^{8} + \cdots + 6561 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( - 1307 \nu^{15} + 5068 \nu^{14} + 824 \nu^{13} + 49267 \nu^{12} + 2716 \nu^{11} + 77018 \nu^{10} + \cdots + 19787976 ) / 621108 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( - 3695 \nu^{15} + 20725 \nu^{14} - 51544 \nu^{13} + 99223 \nu^{12} - 215537 \nu^{11} + \cdots + 46300977 ) / 1242216 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 6613 \nu^{15} - 4165 \nu^{14} - 58592 \nu^{13} + 79853 \nu^{12} - 42655 \nu^{11} + \cdots + 63188991 ) / 1242216 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 1730 \nu^{15} + 4365 \nu^{14} - 8834 \nu^{13} + 21044 \nu^{12} - 39051 \nu^{11} + 29320 \nu^{10} + \cdots + 1850931 ) / 207036 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 8480 \nu^{15} + 13039 \nu^{14} - 27196 \nu^{13} + 146632 \nu^{12} - 38759 \nu^{11} + \cdots + 50810571 ) / 621108 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 9197 \nu^{15} + 929 \nu^{14} - 62076 \nu^{13} + 49233 \nu^{12} - 112105 \nu^{11} + \cdots + 20080305 ) / 414072 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 13862 \nu^{15} - 1333 \nu^{14} + 45760 \nu^{13} - 164782 \nu^{12} - 15775 \nu^{11} + \cdots - 61443765 ) / 621108 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 10991 \nu^{15} + 435 \nu^{14} - 52928 \nu^{13} + 71471 \nu^{12} - 66663 \nu^{11} + \cdots + 23210631 ) / 414072 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( - 16952 \nu^{15} + 9175 \nu^{14} - 73804 \nu^{13} + 123904 \nu^{12} - 128807 \nu^{11} + \cdots + 23648031 ) / 621108 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( 17318 \nu^{15} + 11843 \nu^{14} + 72850 \nu^{13} - 74620 \nu^{12} + 24959 \nu^{11} + \cdots - 20594979 ) / 621108 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( 8906 \nu^{15} - 13891 \nu^{14} + 29326 \nu^{13} - 153874 \nu^{12} + 48131 \nu^{11} + \cdots - 52673895 ) / 310554 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( - 20668 \nu^{15} + 11675 \nu^{14} - 91538 \nu^{13} + 200162 \nu^{12} - 134761 \nu^{11} + \cdots + 65179161 ) / 621108 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( - 7981 \nu^{15} + 3790 \nu^{14} - 34392 \nu^{13} + 96837 \nu^{12} - 29234 \nu^{11} + \cdots + 39766950 ) / 207036 \)
|
| \(\beta_{14}\) | \(=\) |
\( ( - 25616 \nu^{15} - 1811 \nu^{14} - 72430 \nu^{13} + 225742 \nu^{12} + 48469 \nu^{11} + \cdots + 67869171 ) / 621108 \)
|
| \(\beta_{15}\) | \(=\) |
\( ( - 19601 \nu^{15} + 19537 \nu^{14} - 71044 \nu^{13} + 245365 \nu^{12} - 112385 \nu^{11} + \cdots + 77064777 ) / 414072 \)
|
| \(\nu\) | \(=\) |
\( ( 2\beta_{15} - \beta_{13} - \beta_{12} - \beta_{9} - \beta_{8} + 2\beta_{6} - \beta_{4} - \beta_{2} - \beta_1 ) / 3 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{15} - 2 \beta_{14} + \beta_{11} - \beta_{10} + 2 \beta_{9} - 2 \beta_{6} + \beta_{5} + 3 \beta_{3} + \cdots - 2 ) / 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( - 2 \beta_{15} + \beta_{14} - 3 \beta_{13} + 7 \beta_{12} - 2 \beta_{11} - \beta_{10} - 2 \beta_{9} + \cdots + 9 ) / 3 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( - 9 \beta_{13} + 2 \beta_{12} - 6 \beta_{11} - 5 \beta_{9} + 9 \beta_{8} + \beta_{7} - \beta_{6} + \cdots + 13 ) / 3 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( - 5 \beta_{15} - 3 \beta_{14} + 7 \beta_{13} - 11 \beta_{12} - 3 \beta_{11} - 12 \beta_{10} + 16 \beta_{9} + \cdots - 21 ) / 3 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( - 7 \beta_{15} + 7 \beta_{14} - 11 \beta_{13} + 2 \beta_{12} - 11 \beta_{11} + 5 \beta_{10} + 11 \beta_{9} + \cdots - 3 ) / 3 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( - 4 \beta_{15} + 6 \beta_{14} + 11 \beta_{13} - 39 \beta_{12} - 15 \beta_{11} - 6 \beta_{10} + \cdots + 116 ) / 3 \)
|
| \(\nu^{8}\) | \(=\) |
\( ( 51 \beta_{15} + 43 \beta_{14} - 43 \beta_{13} - 37 \beta_{12} + 10 \beta_{11} + 2 \beta_{10} + \cdots - 116 ) / 3 \)
|
| \(\nu^{9}\) | \(=\) |
\( ( - 4 \beta_{15} - 48 \beta_{14} - 4 \beta_{13} + 35 \beta_{12} + 126 \beta_{11} - 33 \beta_{10} + \cdots - 177 ) / 3 \)
|
| \(\nu^{10}\) | \(=\) |
\( ( - 41 \beta_{15} + 60 \beta_{14} - 71 \beta_{13} + 164 \beta_{12} - 57 \beta_{11} + 6 \beta_{10} + \cdots + 47 ) / 3 \)
|
| \(\nu^{11}\) | \(=\) |
\( ( - 275 \beta_{15} + 134 \beta_{14} - 31 \beta_{13} + 20 \beta_{12} - 181 \beta_{11} + 58 \beta_{10} + \cdots + 17 ) / 3 \)
|
| \(\nu^{12}\) | \(=\) |
\( ( - 438 \beta_{15} + 208 \beta_{14} + 101 \beta_{13} + 9 \beta_{12} + 133 \beta_{11} - 184 \beta_{10} + \cdots - 1389 ) / 3 \)
|
| \(\nu^{13}\) | \(=\) |
\( ( 4 \beta_{15} - 144 \beta_{14} + 283 \beta_{13} - 602 \beta_{12} + 243 \beta_{11} + 237 \beta_{10} + \cdots - 1269 ) / 3 \)
|
| \(\nu^{14}\) | \(=\) |
\( ( - 679 \beta_{15} + 815 \beta_{14} + 2796 \beta_{13} - 2187 \beta_{12} - 229 \beta_{11} + 331 \beta_{10} + \cdots + 1187 ) / 3 \)
|
| \(\nu^{15}\) | \(=\) |
\( ( 1166 \beta_{15} + 2420 \beta_{14} - 2349 \beta_{13} + 662 \beta_{12} + 1109 \beta_{11} + 2848 \beta_{10} + \cdots - 3948 ) / 3 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1764\mathbb{Z}\right)^\times\).
| \(n\) | \(785\) | \(883\) | \(1081\) |
| \(\chi(n)\) | \(1 - \beta_{1}\) | \(1\) | \(\beta_{1}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1685.1 |
|
0 | −1.71501 | + | 0.242362i | 0 | −0.699656 | 0 | 0 | 0 | 2.88252 | − | 0.831308i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1685.2 | 0 | −1.42826 | − | 0.979841i | 0 | 1.68574 | 0 | 0 | 0 | 1.07983 | + | 2.79892i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1685.3 | 0 | −0.615921 | + | 1.61884i | 0 | 3.91482 | 0 | 0 | 0 | −2.24128 | − | 1.99416i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1685.4 | 0 | −0.478402 | + | 1.66467i | 0 | −2.74332 | 0 | 0 | 0 | −2.54226 | − | 1.59276i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1685.5 | 0 | −0.127742 | − | 1.72733i | 0 | −2.18300 | 0 | 0 | 0 | −2.96736 | + | 0.441307i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1685.6 | 0 | 1.31579 | − | 1.12637i | 0 | −0.0764245 | 0 | 0 | 0 | 0.462593 | − | 2.96412i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1685.7 | 0 | 1.44376 | + | 0.956855i | 0 | −2.86804 | 0 | 0 | 0 | 1.16886 | + | 2.76293i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1685.8 | 0 | 1.60579 | − | 0.649187i | 0 | 2.96988 | 0 | 0 | 0 | 2.15711 | − | 2.08491i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1697.1 | 0 | −1.71501 | − | 0.242362i | 0 | −0.699656 | 0 | 0 | 0 | 2.88252 | + | 0.831308i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1697.2 | 0 | −1.42826 | + | 0.979841i | 0 | 1.68574 | 0 | 0 | 0 | 1.07983 | − | 2.79892i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1697.3 | 0 | −0.615921 | − | 1.61884i | 0 | 3.91482 | 0 | 0 | 0 | −2.24128 | + | 1.99416i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1697.4 | 0 | −0.478402 | − | 1.66467i | 0 | −2.74332 | 0 | 0 | 0 | −2.54226 | + | 1.59276i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1697.5 | 0 | −0.127742 | + | 1.72733i | 0 | −2.18300 | 0 | 0 | 0 | −2.96736 | − | 0.441307i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1697.6 | 0 | 1.31579 | + | 1.12637i | 0 | −0.0764245 | 0 | 0 | 0 | 0.462593 | + | 2.96412i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1697.7 | 0 | 1.44376 | − | 0.956855i | 0 | −2.86804 | 0 | 0 | 0 | 1.16886 | − | 2.76293i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1697.8 | 0 | 1.60579 | + | 0.649187i | 0 | 2.96988 | 0 | 0 | 0 | 2.15711 | + | 2.08491i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 63.s | even | 6 | 1 | inner |
Twists
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{8} - 24T_{5}^{6} - 12T_{5}^{5} + 171T_{5}^{4} + 135T_{5}^{3} - 324T_{5}^{2} - 261T_{5} - 18 \)
acting on \(S_{2}^{\mathrm{new}}(1764, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{16} \)
|
| $3$ |
\( T^{16} - 3 T^{13} + \cdots + 6561 \)
|
| $5$ |
\( (T^{8} - 24 T^{6} + \cdots - 18)^{2} \)
|
| $7$ |
\( T^{16} \)
|
| $11$ |
\( T^{16} + 90 T^{14} + \cdots + 26244 \)
|
| $13$ |
\( T^{16} + 3 T^{15} + \cdots + 3337929 \)
|
| $17$ |
\( T^{16} - 9 T^{15} + \cdots + 13549761 \)
|
| $19$ |
\( T^{16} - 93 T^{14} + \cdots + 2099601 \)
|
| $23$ |
\( T^{16} + \cdots + 15198451524 \)
|
| $29$ |
\( T^{16} - 6 T^{15} + \cdots + 15752961 \)
|
| $31$ |
\( T^{16} + \cdots + 3910251024 \)
|
| $37$ |
\( T^{16} - T^{15} + \cdots + 52765696 \)
|
| $41$ |
\( T^{16} + \cdots + 91647269289 \)
|
| $43$ |
\( T^{16} + \cdots + 28009034881 \)
|
| $47$ |
\( T^{16} + \cdots + 1971620372736 \)
|
| $53$ |
\( T^{16} - 153 T^{14} + \cdots + 531441 \)
|
| $59$ |
\( T^{16} + \cdots + 165574120464 \)
|
| $61$ |
\( T^{16} + \cdots + 1475481744 \)
|
| $67$ |
\( T^{16} + \cdots + 2114953586944 \)
|
| $71$ |
\( T^{16} + \cdots + 780959242139904 \)
|
| $73$ |
\( T^{16} + \cdots + 7523023152969 \)
|
| $79$ |
\( T^{16} + T^{15} + \cdots + 10549504 \)
|
| $83$ |
\( T^{16} + \cdots + 669184533369 \)
|
| $89$ |
\( T^{16} + \cdots + 7161826993281 \)
|
| $97$ |
\( T^{16} + \cdots + 22864161681 \)
|
| show more | |
| show less |