Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [756,2,Mod(341,756)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(756, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([0, 5, 5]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("756.341");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 756 = 2^{2} \cdot 3^{3} \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 756.w (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: | \(6.03669039281\) |
| 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: | \( 3^{6} \) |
| 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
\( x^{16} - 2 x^{15} + 5 x^{14} - 17 x^{13} + 22 x^{12} - 31 x^{11} + 62 x^{10} - 52 x^{9} + 52 x^{8} + \cdots + 6561 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 19 \nu^{15} + 139 \nu^{14} + 1928 \nu^{13} + 8221 \nu^{12} + 10009 \nu^{11} + 14762 \nu^{10} + \cdots + 4788801 ) / 103518 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 1307 \nu^{15} - 5068 \nu^{14} - 824 \nu^{13} - 49267 \nu^{12} - 2716 \nu^{11} - 77018 \nu^{10} + \cdots - 19166868 ) / 621108 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 3695 \nu^{15} + 20725 \nu^{14} - 51544 \nu^{13} + 99223 \nu^{12} - 215537 \nu^{11} + \cdots + 46300977 ) / 1242216 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 6677 \nu^{15} + 21577 \nu^{14} - 16612 \nu^{13} + 91129 \nu^{12} - 145673 \nu^{11} + \cdots - 5872095 ) / 1242216 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 1669 \nu^{15} + 1930 \nu^{14} + 23477 \nu^{13} + 5248 \nu^{12} + 44857 \nu^{11} - 82192 \nu^{10} + \cdots - 4533651 ) / 310554 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 2739 \nu^{15} - 9229 \nu^{14} + 1724 \nu^{13} - 88319 \nu^{12} - 8827 \nu^{11} - 147394 \nu^{10} + \cdots - 35170605 ) / 414072 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 8405 \nu^{15} - 16535 \nu^{14} - 52096 \nu^{13} - 27863 \nu^{12} - 20561 \nu^{11} + \cdots - 11234619 ) / 621108 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 4934 \nu^{15} + 6616 \nu^{14} - 21253 \nu^{13} + 86401 \nu^{12} - 24581 \nu^{11} + \cdots + 34596153 ) / 310554 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 13862 \nu^{15} - 1333 \nu^{14} + 45760 \nu^{13} - 164782 \nu^{12} - 15775 \nu^{11} + \cdots - 61443765 ) / 621108 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( 27959 \nu^{15} + 45263 \nu^{14} + 129088 \nu^{13} - 58111 \nu^{12} - 95755 \nu^{11} + \cdots - 47062053 ) / 1242216 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( - 16952 \nu^{15} + 9175 \nu^{14} - 73804 \nu^{13} + 123904 \nu^{12} - 128807 \nu^{11} + \cdots + 23648031 ) / 621108 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( - 19793 \nu^{15} - 1430 \nu^{14} - 108688 \nu^{13} + 147133 \nu^{12} - 121322 \nu^{11} + \cdots + 66410442 ) / 621108 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( 3656 \nu^{15} - 865 \nu^{14} + 15076 \nu^{13} - 31234 \nu^{12} + 13505 \nu^{11} - 75428 \nu^{10} + \cdots - 10890531 ) / 103518 \)
|
| \(\beta_{14}\) | \(=\) |
\( ( - 27554 \nu^{15} + 18943 \nu^{14} - 80794 \nu^{13} + 318436 \nu^{12} - 60809 \nu^{11} + \cdots + 92446677 ) / 621108 \)
|
| \(\beta_{15}\) | \(=\) |
\( ( - 23045 \nu^{15} + 19169 \nu^{14} - 116460 \nu^{13} + 283569 \nu^{12} - 185713 \nu^{11} + \cdots + 111152817 ) / 414072 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{15} - \beta_{12} - \beta_{11} - \beta_{10} + 2 \beta_{9} + \beta_{7} - \beta_{6} + 2 \beta_{4} + \cdots + 1 ) / 3 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{15} - \beta_{13} - 2\beta_{11} + \beta_{10} - \beta_{8} - \beta_{5} + \beta_{4} - 3\beta_{3} - \beta_1 ) / 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( - 4 \beta_{15} + 4 \beta_{14} + \beta_{13} + \beta_{12} + \beta_{10} - 4 \beta_{9} + \beta_{8} + \cdots - 4 ) / 3 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( - 2 \beta_{15} + 3 \beta_{14} + 4 \beta_{13} + 4 \beta_{12} - \beta_{11} + 5 \beta_{9} + 2 \beta_{8} + \cdots + 10 ) / 3 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 6 \beta_{15} - 14 \beta_{14} - 2 \beta_{13} - \beta_{12} + \beta_{11} - 10 \beta_{10} - 5 \beta_{8} + \cdots - 3 ) / 3 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( - 5 \beta_{15} + 9 \beta_{14} - 10 \beta_{13} - 13 \beta_{12} - 13 \beta_{11} + 3 \beta_{10} - 6 \beta_{9} + \cdots + 5 ) / 3 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( 12 \beta_{15} - 9 \beta_{14} + 2 \beta_{13} + 10 \beta_{12} - 15 \beta_{11} + 2 \beta_{10} - 15 \beta_{9} + \cdots + 60 ) / 3 \)
|
| \(\nu^{8}\) | \(=\) |
\( ( 61 \beta_{15} - 25 \beta_{14} + 25 \beta_{13} - 41 \beta_{12} + 6 \beta_{11} - 46 \beta_{10} + 54 \beta_{9} + \cdots + 15 ) / 3 \)
|
| \(\nu^{9}\) | \(=\) |
\( ( - 56 \beta_{15} + 13 \beta_{14} - 67 \beta_{13} + 14 \beta_{12} + 7 \beta_{11} + 118 \beta_{10} + \cdots - 81 ) / 3 \)
|
| \(\nu^{10}\) | \(=\) |
\( ( - 185 \beta_{15} + 207 \beta_{14} + 14 \beta_{13} + 99 \beta_{12} - 61 \beta_{11} + 112 \beta_{10} + \cdots - 632 ) / 3 \)
|
| \(\nu^{11}\) | \(=\) |
\( ( - 7 \beta_{15} - 106 \beta_{14} + 234 \beta_{13} + 286 \beta_{12} + 274 \beta_{11} + 48 \beta_{10} + \cdots - 63 ) / 3 \)
|
| \(\nu^{12}\) | \(=\) |
\( ( - 144 \beta_{15} - 439 \beta_{14} + 36 \beta_{13} - 147 \beta_{12} + 673 \beta_{11} - 220 \beta_{10} + \cdots - 502 ) / 3 \)
|
| \(\nu^{13}\) | \(=\) |
\( ( - 373 \beta_{15} + 399 \beta_{14} - 651 \beta_{13} - 44 \beta_{12} - 632 \beta_{11} + 544 \beta_{10} + \cdots - 517 ) / 3 \)
|
| \(\nu^{14}\) | \(=\) |
\( ( 1073 \beta_{15} - 1398 \beta_{14} - 11 \beta_{13} + 363 \beta_{12} + 530 \beta_{11} - 367 \beta_{10} + \cdots - 1014 ) / 3 \)
|
| \(\nu^{15}\) | \(=\) |
\( ( 1627 \beta_{15} - 544 \beta_{14} + 1472 \beta_{13} - 2368 \beta_{12} + 2547 \beta_{11} - 361 \beta_{10} + \cdots + 1741 ) / 3 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/756\mathbb{Z}\right)^\times\).
| \(n\) | \(29\) | \(325\) | \(379\) |
| \(\chi(n)\) | \(1 - \beta_{2}\) | \(1 - \beta_{2}\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 341.1 |
|
0 | 0 | 0 | −1.43402 | − | 2.48379i | 0 | 2.56899 | − | 0.632668i | 0 | 0 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 341.2 | 0 | 0 | 0 | −1.37166 | − | 2.37578i | 0 | −2.60476 | + | 0.463945i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 341.3 | 0 | 0 | 0 | −1.09150 | − | 1.89054i | 0 | −1.25859 | + | 2.32722i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 341.4 | 0 | 0 | 0 | −0.349828 | − | 0.605920i | 0 | 2.48683 | + | 0.903137i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 341.5 | 0 | 0 | 0 | −0.0382122 | − | 0.0661855i | 0 | 0.232935 | − | 2.63548i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 341.6 | 0 | 0 | 0 | 0.842869 | + | 1.45989i | 0 | −2.27938 | − | 1.34329i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 341.7 | 0 | 0 | 0 | 1.48494 | + | 2.57199i | 0 | −0.200279 | + | 2.63816i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 341.8 | 0 | 0 | 0 | 1.95741 | + | 3.39033i | 0 | 0.554241 | − | 2.58705i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 521.1 | 0 | 0 | 0 | −1.43402 | + | 2.48379i | 0 | 2.56899 | + | 0.632668i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 521.2 | 0 | 0 | 0 | −1.37166 | + | 2.37578i | 0 | −2.60476 | − | 0.463945i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 521.3 | 0 | 0 | 0 | −1.09150 | + | 1.89054i | 0 | −1.25859 | − | 2.32722i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 521.4 | 0 | 0 | 0 | −0.349828 | + | 0.605920i | 0 | 2.48683 | − | 0.903137i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 521.5 | 0 | 0 | 0 | −0.0382122 | + | 0.0661855i | 0 | 0.232935 | + | 2.63548i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 521.6 | 0 | 0 | 0 | 0.842869 | − | 1.45989i | 0 | −2.27938 | + | 1.34329i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 521.7 | 0 | 0 | 0 | 1.48494 | − | 2.57199i | 0 | −0.200279 | − | 2.63816i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 521.8 | 0 | 0 | 0 | 1.95741 | − | 3.39033i | 0 | 0.554241 | + | 2.58705i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 63.i | even | 6 | 1 | inner |
Twists
Hecke kernels
This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(756, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{16} \)
|
| $3$ |
\( T^{16} \)
|
| $5$ |
\( T^{16} + 24 T^{14} + \cdots + 324 \)
|
| $7$ |
\( T^{16} + T^{15} + \cdots + 5764801 \)
|
| $11$ |
\( T^{16} - 6 T^{15} + \cdots + 26244 \)
|
| $13$ |
\( T^{16} + 3 T^{15} + \cdots + 3337929 \)
|
| $17$ |
\( T^{16} + 9 T^{15} + \cdots + 13549761 \)
|
| $19$ |
\( T^{16} - 93 T^{14} + \cdots + 2099601 \)
|
| $23$ |
\( T^{16} + \cdots + 15198451524 \)
|
| $29$ |
\( T^{16} + 6 T^{15} + \cdots + 15752961 \)
|
| $31$ |
\( T^{16} + \cdots + 3910251024 \)
|
| $37$ |
\( T^{16} - T^{15} + \cdots + 52765696 \)
|
| $41$ |
\( T^{16} + \cdots + 91647269289 \)
|
| $43$ |
\( T^{16} + \cdots + 28009034881 \)
|
| $47$ |
\( (T^{8} - 18 T^{7} + \cdots + 1404144)^{2} \)
|
| $53$ |
\( T^{16} - 153 T^{14} + \cdots + 531441 \)
|
| $59$ |
\( (T^{8} - 15 T^{7} + \cdots - 406908)^{2} \)
|
| $61$ |
\( T^{16} + \cdots + 1475481744 \)
|
| $67$ |
\( (T^{8} - 7 T^{7} + \cdots - 1454288)^{2} \)
|
| $71$ |
\( T^{16} + \cdots + 780959242139904 \)
|
| $73$ |
\( T^{16} + \cdots + 7523023152969 \)
|
| $79$ |
\( (T^{8} - T^{7} - 143 T^{6} + \cdots - 3248)^{2} \)
|
| $83$ |
\( T^{16} + \cdots + 669184533369 \)
|
| $89$ |
\( T^{16} + \cdots + 7161826993281 \)
|
| $97$ |
\( T^{16} + \cdots + 22864161681 \)
|
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