Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [728,2,Mod(293,728)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(728, base_ring=CyclotomicField(12))
chi = DirichletCharacter(H, H._module([0, 6, 6, 11]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("728.293");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 728 = 2^{3} \cdot 7 \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 728.ds (of order \(12\), degree \(4\), 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: | \(5.81310926715\) |
| Analytic rank: | \(0\) |
| Dimension: | \(16\) |
| Relative dimension: | \(4\) over \(\Q(\zeta_{12})\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{16} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{16} - 4x^{14} + 8x^{12} + 40x^{10} - 161x^{8} + 360x^{6} + 648x^{4} - 2916x^{2} + 6561 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| Coefficient ring index: | \( 13 \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{U}(1)[D_{12}]$ |
$q$-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
\( x^{16} - 4x^{14} + 8x^{12} + 40x^{10} - 161x^{8} + 360x^{6} + 648x^{4} - 2916x^{2} + 6561 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( - 56 \nu^{14} + 13823 \nu^{12} - 7 \nu^{10} - 1988 \nu^{8} + 520738 \nu^{6} - 2268 \nu^{4} + \cdots + 9893259 ) / 521235 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -2\nu^{14} - 63\nu^{12} - 36\nu^{10} - 71\nu^{8} - 2556\nu^{6} - 81\nu^{4} + 2621\nu^{2} - 52488 ) / 6435 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 304 \nu^{14} + 577 \nu^{12} - 38 \nu^{10} - 10792 \nu^{8} - 2698 \nu^{6} - 12312 \nu^{4} + \cdots - 55404 ) / 521235 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -\nu^{15} + 4\nu^{13} - 8\nu^{11} - 40\nu^{9} + 161\nu^{7} - 360\nu^{5} - 648\nu^{3} + 729\nu ) / 2187 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 71\nu^{15} - 266\nu^{13} - 152\nu^{11} + 473\nu^{9} - 10792\nu^{7} - 342\nu^{5} + 46737\nu^{3} - 79461\nu ) / 142155 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 76 \nu^{15} - 367 \nu^{13} + 455 \nu^{11} + 2698 \nu^{9} - 7790 \nu^{7} + 3078 \nu^{5} + \cdots - 138510 \nu ) / 120285 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 17\nu^{15} + 112\nu^{13} + 64\nu^{11} + 257\nu^{9} + 4544\nu^{7} + 144\nu^{5} + 9396\nu^{3} + 69255\nu ) / 24057 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 284 \nu^{14} - 1064 \nu^{12} - 608 \nu^{10} + 11369 \nu^{8} - 43168 \nu^{6} - 1368 \nu^{4} + \cdots - 886464 ) / 104247 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 1582 \nu^{14} - 217 \nu^{12} - 124 \nu^{10} + 62596 \nu^{8} - 8804 \nu^{6} - 279 \nu^{4} + \cdots - 180792 ) / 521235 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( 2363 \nu^{14} - 3143 \nu^{12} - 1796 \nu^{10} + 67799 \nu^{8} - 127516 \nu^{6} - 4041 \nu^{4} + \cdots - 2618568 ) / 521235 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( - 3413 \nu^{14} + 7820 \nu^{12} - 2518 \nu^{10} - 137249 \nu^{8} + 342457 \nu^{6} - 237969 \nu^{4} + \cdots + 6232221 ) / 521235 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( - 346 \nu^{15} + 1069 \nu^{13} + 31 \nu^{11} - 13768 \nu^{9} + 42296 \nu^{7} + 837 \nu^{5} + \cdots + 847098 \nu ) / 120285 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( - 4879 \nu^{15} + 23989 \nu^{13} + 13708 \nu^{11} - 169987 \nu^{9} + 973268 \nu^{7} + \cdots + 18422559 \nu ) / 1563705 \)
|
| \(\beta_{14}\) | \(=\) |
\( ( - 454 \nu^{15} - 1145 \nu^{13} + 31 \nu^{11} - 17872 \nu^{9} - 45184 \nu^{7} - 837 \nu^{5} + \cdots - 760347 \nu ) / 142155 \)
|
| \(\beta_{15}\) | \(=\) |
\( ( - 2806 \nu^{15} + 1558 \nu^{13} + 1258 \nu^{11} - 112483 \nu^{9} + 89318 \nu^{7} + \cdots + 1834164 \nu ) / 521235 \)
|
| \(\nu\) | \(=\) |
\( ( -4\beta_{15} + \beta_{14} + 2\beta_{13} + 5\beta_{12} - 6\beta_{7} + 4\beta_{6} + 8\beta_{5} ) / 13 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{9} - \beta_{8} + \beta_{2} \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 14\beta_{15} - 10\beta_{14} + 6\beta_{13} - 24\beta_{12} - 5\beta_{7} - 14\beta_{6} + 11\beta_{5} ) / 13 \)
|
| \(\nu^{4}\) | \(=\) |
\( -2\beta_{11} - 5\beta_{9} - 5\beta_{3} + 2\beta_{2} + 2\beta_1 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 68 \beta_{15} - 56 \beta_{14} - 34 \beta_{13} - 33 \beta_{12} + 11 \beta_{7} - 55 \beta_{6} + \cdots - 78 \beta_{4} ) / 13 \)
|
| \(\nu^{6}\) | \(=\) |
\( \beta_{11} - \beta_{10} - 19\beta_{3} - 19 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( 191 \beta_{15} - 168 \beta_{14} - 76 \beta_{13} - 99 \beta_{12} + 7 \beta_{7} + 69 \beta_{6} + \cdots + 39 \beta_{4} ) / 13 \)
|
| \(\nu^{8}\) | \(=\) |
\( -20\beta_{10} + 31\beta_{8} - 20\beta_{2} \)
|
| \(\nu^{9}\) | \(=\) |
\( ( -207\beta_{15} + 16\beta_{14} + 32\beta_{13} + 223\beta_{12} - 382\beta_{7} + 207\beta_{6} - 366\beta_{5} ) / 13 \)
|
| \(\nu^{10}\) | \(=\) |
\( 109\beta_{9} - 109\beta_{8} + 109\beta_{3} - 71\beta_{2} - 71\beta _1 - 109 \)
|
| \(\nu^{11}\) | \(=\) |
\( ( - 1544 \beta_{15} + 1894 \beta_{14} + 1019 \beta_{13} + 175 \beta_{12} + 1194 \beta_{7} + \cdots - 175 \beta_{5} ) / 13 \)
|
| \(\nu^{12}\) | \(=\) |
\( -38\beta_{11} + 38\beta_{10} + 715\beta_{3} + 38\beta_1 \)
|
| \(\nu^{13}\) | \(=\) |
\( ( - 4632 \beta_{15} + 5188 \beta_{14} + 1575 \beta_{13} + 1019 \beta_{12} + 4076 \beta_{7} + \cdots - 1482 \beta_{4} ) / 13 \)
|
| \(\nu^{14}\) | \(=\) |
\( 791\beta_{10} - 449\beta_{9} - 449\beta_{8} \)
|
| \(\nu^{15}\) | \(=\) |
\( ( - 3613 \beta_{15} + 5281 \beta_{14} - 5558 \beta_{13} + 8894 \beta_{12} + 18065 \beta_{7} + \cdots + 7226 \beta_{5} ) / 13 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/728\mathbb{Z}\right)^\times\).
| \(n\) | \(183\) | \(365\) | \(521\) | \(561\) |
| \(\chi(n)\) | \(1\) | \(-1\) | \(-1\) | \(\beta_{3} + \beta_{9}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 293.1 |
|
1.36603 | − | 0.366025i | −1.07674 | + | 1.86497i | 1.73205 | − | 1.00000i | 2.94171 | + | 2.94171i | −0.788230 | + | 2.94171i | 2.55560 | + | 0.684771i | 2.00000 | − | 2.00000i | −0.818745 | − | 1.41811i | 5.09520 | + | 2.94171i | ||||||||||||||||||||||||||||||||||||||||||||||||
| 293.2 | 1.36603 | − | 0.366025i | −0.688914 | + | 1.19323i | 1.73205 | − | 1.00000i | 1.88215 | + | 1.88215i | −0.504320 | + | 1.88215i | −2.55560 | − | 0.684771i | 2.00000 | − | 2.00000i | 0.550796 | + | 0.954007i | 3.25997 | + | 1.88215i | |||||||||||||||||||||||||||||||||||||||||||||||||
| 293.3 | 1.36603 | − | 0.366025i | 0.688914 | − | 1.19323i | 1.73205 | − | 1.00000i | −1.88215 | − | 1.88215i | 0.504320 | − | 1.88215i | −2.55560 | − | 0.684771i | 2.00000 | − | 2.00000i | 0.550796 | + | 0.954007i | −3.25997 | − | 1.88215i | |||||||||||||||||||||||||||||||||||||||||||||||||
| 293.4 | 1.36603 | − | 0.366025i | 1.07674 | − | 1.86497i | 1.73205 | − | 1.00000i | −2.94171 | − | 2.94171i | 0.788230 | − | 2.94171i | 2.55560 | + | 0.684771i | 2.00000 | − | 2.00000i | −0.818745 | − | 1.41811i | −5.09520 | − | 2.94171i | |||||||||||||||||||||||||||||||||||||||||||||||||
| 349.1 | −0.366025 | + | 1.36603i | −1.72070 | − | 2.98034i | −1.73205 | − | 1.00000i | −1.25964 | − | 1.25964i | 4.70104 | − | 1.25964i | 0.684771 | + | 2.55560i | 2.00000 | − | 2.00000i | −4.42162 | + | 7.65848i | 2.18176 | − | 1.25964i | |||||||||||||||||||||||||||||||||||||||||||||||||
| 349.2 | −0.366025 | + | 1.36603i | −0.636563 | − | 1.10256i | −1.73205 | − | 1.00000i | −0.465997 | − | 0.465997i | 1.73912 | − | 0.465997i | −0.684771 | − | 2.55560i | 2.00000 | − | 2.00000i | 0.689574 | − | 1.19438i | 0.807130 | − | 0.465997i | |||||||||||||||||||||||||||||||||||||||||||||||||
| 349.3 | −0.366025 | + | 1.36603i | 0.636563 | + | 1.10256i | −1.73205 | − | 1.00000i | 0.465997 | + | 0.465997i | −1.73912 | + | 0.465997i | −0.684771 | − | 2.55560i | 2.00000 | − | 2.00000i | 0.689574 | − | 1.19438i | −0.807130 | + | 0.465997i | |||||||||||||||||||||||||||||||||||||||||||||||||
| 349.4 | −0.366025 | + | 1.36603i | 1.72070 | + | 2.98034i | −1.73205 | − | 1.00000i | 1.25964 | + | 1.25964i | −4.70104 | + | 1.25964i | 0.684771 | + | 2.55560i | 2.00000 | − | 2.00000i | −4.42162 | + | 7.65848i | −2.18176 | + | 1.25964i | |||||||||||||||||||||||||||||||||||||||||||||||||
| 405.1 | 1.36603 | + | 0.366025i | −1.07674 | − | 1.86497i | 1.73205 | + | 1.00000i | 2.94171 | − | 2.94171i | −0.788230 | − | 2.94171i | 2.55560 | − | 0.684771i | 2.00000 | + | 2.00000i | −0.818745 | + | 1.41811i | 5.09520 | − | 2.94171i | |||||||||||||||||||||||||||||||||||||||||||||||||
| 405.2 | 1.36603 | + | 0.366025i | −0.688914 | − | 1.19323i | 1.73205 | + | 1.00000i | 1.88215 | − | 1.88215i | −0.504320 | − | 1.88215i | −2.55560 | + | 0.684771i | 2.00000 | + | 2.00000i | 0.550796 | − | 0.954007i | 3.25997 | − | 1.88215i | |||||||||||||||||||||||||||||||||||||||||||||||||
| 405.3 | 1.36603 | + | 0.366025i | 0.688914 | + | 1.19323i | 1.73205 | + | 1.00000i | −1.88215 | + | 1.88215i | 0.504320 | + | 1.88215i | −2.55560 | + | 0.684771i | 2.00000 | + | 2.00000i | 0.550796 | − | 0.954007i | −3.25997 | + | 1.88215i | |||||||||||||||||||||||||||||||||||||||||||||||||
| 405.4 | 1.36603 | + | 0.366025i | 1.07674 | + | 1.86497i | 1.73205 | + | 1.00000i | −2.94171 | + | 2.94171i | 0.788230 | + | 2.94171i | 2.55560 | − | 0.684771i | 2.00000 | + | 2.00000i | −0.818745 | + | 1.41811i | −5.09520 | + | 2.94171i | |||||||||||||||||||||||||||||||||||||||||||||||||
| 461.1 | −0.366025 | − | 1.36603i | −1.72070 | + | 2.98034i | −1.73205 | + | 1.00000i | −1.25964 | + | 1.25964i | 4.70104 | + | 1.25964i | 0.684771 | − | 2.55560i | 2.00000 | + | 2.00000i | −4.42162 | − | 7.65848i | 2.18176 | + | 1.25964i | |||||||||||||||||||||||||||||||||||||||||||||||||
| 461.2 | −0.366025 | − | 1.36603i | −0.636563 | + | 1.10256i | −1.73205 | + | 1.00000i | −0.465997 | + | 0.465997i | 1.73912 | + | 0.465997i | −0.684771 | + | 2.55560i | 2.00000 | + | 2.00000i | 0.689574 | + | 1.19438i | 0.807130 | + | 0.465997i | |||||||||||||||||||||||||||||||||||||||||||||||||
| 461.3 | −0.366025 | − | 1.36603i | 0.636563 | − | 1.10256i | −1.73205 | + | 1.00000i | 0.465997 | − | 0.465997i | −1.73912 | − | 0.465997i | −0.684771 | + | 2.55560i | 2.00000 | + | 2.00000i | 0.689574 | + | 1.19438i | −0.807130 | − | 0.465997i | |||||||||||||||||||||||||||||||||||||||||||||||||
| 461.4 | −0.366025 | − | 1.36603i | 1.72070 | − | 2.98034i | −1.73205 | + | 1.00000i | 1.25964 | − | 1.25964i | −4.70104 | − | 1.25964i | 0.684771 | − | 2.55560i | 2.00000 | + | 2.00000i | −4.42162 | − | 7.65848i | −2.18176 | − | 1.25964i | |||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 56.h | odd | 2 | 1 | CM by \(\Q(\sqrt{-14}) \) |
| 7.b | odd | 2 | 1 | inner |
| 8.b | even | 2 | 1 | inner |
| 13.f | odd | 12 | 1 | inner |
| 91.bc | even | 12 | 1 | inner |
| 104.x | odd | 12 | 1 | inner |
| 728.ds | even | 12 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 728.2.ds.b | ✓ | 16 |
| 7.b | odd | 2 | 1 | inner | 728.2.ds.b | ✓ | 16 |
| 8.b | even | 2 | 1 | inner | 728.2.ds.b | ✓ | 16 |
| 13.f | odd | 12 | 1 | inner | 728.2.ds.b | ✓ | 16 |
| 56.h | odd | 2 | 1 | CM | 728.2.ds.b | ✓ | 16 |
| 91.bc | even | 12 | 1 | inner | 728.2.ds.b | ✓ | 16 |
| 104.x | odd | 12 | 1 | inner | 728.2.ds.b | ✓ | 16 |
| 728.ds | even | 12 | 1 | inner | 728.2.ds.b | ✓ | 16 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 728.2.ds.b | ✓ | 16 | 1.a | even | 1 | 1 | trivial |
| 728.2.ds.b | ✓ | 16 | 7.b | odd | 2 | 1 | inner |
| 728.2.ds.b | ✓ | 16 | 8.b | even | 2 | 1 | inner |
| 728.2.ds.b | ✓ | 16 | 13.f | odd | 12 | 1 | inner |
| 728.2.ds.b | ✓ | 16 | 56.h | odd | 2 | 1 | CM |
| 728.2.ds.b | ✓ | 16 | 91.bc | even | 12 | 1 | inner |
| 728.2.ds.b | ✓ | 16 | 104.x | odd | 12 | 1 | inner |
| 728.2.ds.b | ✓ | 16 | 728.ds | even | 12 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{16} + 20T_{3}^{14} + 284T_{3}^{12} + 1832T_{3}^{10} + 8407T_{3}^{8} + 21544T_{3}^{6} + 39932T_{3}^{4} + 41236T_{3}^{2} + 28561 \)
acting on \(S_{2}^{\mathrm{new}}(728, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{4} - 2 T^{3} + 2 T^{2} + \cdots + 4)^{4} \)
|
| $3$ |
\( T^{16} + 20 T^{14} + \cdots + 28561 \)
|
| $5$ |
\( T^{16} + 360 T^{12} + \cdots + 28561 \)
|
| $7$ |
\( (T^{8} - 49 T^{4} + 2401)^{2} \)
|
| $11$ |
\( T^{16} \)
|
| $13$ |
\( T^{16} + \cdots + 815730721 \)
|
| $17$ |
\( T^{16} \)
|
| $19$ |
\( T^{16} + \cdots + 13051691536 \)
|
| $23$ |
\( (T^{8} - 102 T^{6} + \cdots + 1185921)^{2} \)
|
| $29$ |
\( T^{16} \)
|
| $31$ |
\( T^{16} \)
|
| $37$ |
\( T^{16} \)
|
| $41$ |
\( T^{16} \)
|
| $43$ |
\( T^{16} \)
|
| $47$ |
\( T^{16} \)
|
| $53$ |
\( T^{16} \)
|
| $59$ |
\( T^{16} + \cdots + 57\!\cdots\!41 \)
|
| $61$ |
\( T^{16} + \cdots + 368377429171921 \)
|
| $67$ |
\( T^{16} \)
|
| $71$ |
\( (T^{8} + 8 T^{7} + \cdots + 1073283121)^{2} \)
|
| $73$ |
\( T^{16} \)
|
| $79$ |
\( (T^{2} - 24 T + 130)^{8} \)
|
| $83$ |
\( (T^{8} + 70716 T^{4} + 71402500)^{2} \)
|
| $89$ |
\( T^{16} \)
|
| $97$ |
\( T^{16} \)
|
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