Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [260,2,Mod(33,260)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(260, base_ring=CyclotomicField(12))
chi = DirichletCharacter(H, H._module([0, 9, 11]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("260.33");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 260 = 2^{2} \cdot 5 \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 260.bk (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: | \(2.07611045255\) |
| Analytic rank: | \(0\) |
| Dimension: | \(20\) |
| Relative dimension: | \(5\) over \(\Q(\zeta_{12})\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{20} + \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{20} + 30 x^{18} + 371 x^{16} + 2460 x^{14} + 9517 x^{12} + 21870 x^{10} + 29001 x^{8} + 20400 x^{6} + \cdots + 9 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2^{2} \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{12}]$ |
$q$-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 \(1,\beta_1,\ldots,\beta_{19}\) 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
\( x^{20} + 30 x^{18} + 371 x^{16} + 2460 x^{14} + 9517 x^{12} + 21870 x^{10} + 29001 x^{8} + 20400 x^{6} + \cdots + 9 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 33785 \nu^{19} + 39285 \nu^{18} + 894975 \nu^{17} + 1362663 \nu^{16} + 9328060 \nu^{15} + \cdots - 30772215 ) / 13619376 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 33785 \nu^{19} - 39285 \nu^{18} + 894975 \nu^{17} - 1362663 \nu^{16} + 9328060 \nu^{15} + \cdots + 30772215 ) / 13619376 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 13302 \nu^{18} - 387493 \nu^{16} - 4601880 \nu^{14} - 28817129 \nu^{12} - 102464589 \nu^{10} + \cdots + 1823586 ) / 1134948 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 7095 \nu^{19} + 54336 \nu^{18} + 236351 \nu^{17} + 1624980 \nu^{16} + 3193514 \nu^{15} + \cdots + 14385840 ) / 4539792 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 15705 \nu^{19} - 7449 \nu^{18} + 402397 \nu^{17} - 223669 \nu^{16} + 3930438 \nu^{15} + \cdots - 661521 ) / 4539792 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 66482 \nu^{19} + 47115 \nu^{18} + 2012742 \nu^{17} + 1207191 \nu^{16} + 25148026 \nu^{15} + \cdots - 15821217 ) / 13619376 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 66482 \nu^{19} - 47115 \nu^{18} + 2012742 \nu^{17} - 1207191 \nu^{16} + 25148026 \nu^{15} + \cdots + 15821217 ) / 13619376 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 65481 \nu^{19} + 34053 \nu^{18} + 1937459 \nu^{17} + 998655 \nu^{16} + 23623520 \nu^{15} + \cdots - 715755 ) / 4539792 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( - 201140 \nu^{19} + 14217 \nu^{18} - 5787312 \nu^{17} + 295401 \nu^{16} - 67780378 \nu^{15} + \cdots - 9990927 ) / 13619376 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( 65481 \nu^{19} - 34053 \nu^{18} + 1937459 \nu^{17} - 998655 \nu^{16} + 23623520 \nu^{15} + \cdots + 715755 ) / 4539792 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( 72585 \nu^{19} - 59235 \nu^{18} + 2051657 \nu^{17} - 1779699 \nu^{16} + 23339538 \nu^{15} + \cdots + 8463585 ) / 4539792 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( - 72576 \nu^{19} + 88389 \nu^{18} - 2173810 \nu^{17} + 2623635 \nu^{16} - 26817034 \nu^{15} + \cdots + 13670085 ) / 4539792 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( 89663 \nu^{19} - 8196 \nu^{18} + 2650665 \nu^{17} - 299502 \nu^{16} + 32159852 \nu^{15} + \cdots - 12592020 ) / 4539792 \)
|
| \(\beta_{14}\) | \(=\) |
\( ( - 353 \nu^{19} - 10585 \nu^{17} - 130732 \nu^{15} - 864458 \nu^{13} - 3326801 \nu^{11} + \cdots - 8376 ) / 16752 \)
|
| \(\beta_{15}\) | \(=\) |
\( ( 210017 \nu^{19} + 54111 \nu^{18} + 6333330 \nu^{17} + 1652277 \nu^{16} + 78814861 \nu^{15} + \cdots + 22008474 ) / 6809688 \)
|
| \(\beta_{16}\) | \(=\) |
\( ( 430079 \nu^{19} + 137379 \nu^{18} + 12923433 \nu^{17} + 4170825 \nu^{16} + 160317994 \nu^{15} + \cdots - 21883005 ) / 13619376 \)
|
| \(\beta_{17}\) | \(=\) |
\( ( 191039 \nu^{19} - 26604 \nu^{18} + 5681093 \nu^{17} - 774986 \nu^{16} + 69369358 \nu^{15} + \cdots + 3647172 ) / 4539792 \)
|
| \(\beta_{18}\) | \(=\) |
\( ( - 600719 \nu^{19} + 267846 \nu^{18} - 17871213 \nu^{17} + 7954470 \nu^{16} - 218545294 \nu^{15} + \cdots + 28943604 ) / 13619376 \)
|
| \(\beta_{19}\) | \(=\) |
\( ( 870232 \nu^{19} - 57567 \nu^{18} + 25941228 \nu^{17} - 1845867 \nu^{16} + 318130016 \nu^{15} + \cdots + 10941489 ) / 13619376 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{17} + \beta_{14} + \beta_{12} + \beta_{10} - \beta_{8} - \beta_{5} - \beta_{4} - \beta_{3} + 1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( - 2 \beta_{19} + \beta_{17} + 2 \beta_{16} + \beta_{14} - \beta_{12} + \beta_{10} - 2 \beta_{9} + \cdots - 5 ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( - \beta_{19} - \beta_{18} - \beta_{17} - \beta_{16} + \beta_{15} - 3 \beta_{14} - \beta_{13} - 2 \beta_{12} + \cdots - 3 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 12 \beta_{19} - 5 \beta_{17} - 12 \beta_{16} - 11 \beta_{14} - 2 \beta_{13} + 7 \beta_{12} + 2 \beta_{11} + \cdots + 23 ) / 2 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 18 \beta_{19} + 18 \beta_{18} + 7 \beta_{17} + 18 \beta_{16} - 18 \beta_{15} + 47 \beta_{14} + 18 \beta_{13} + \cdots + 45 ) / 2 \)
|
| \(\nu^{6}\) | \(=\) |
\( - 39 \beta_{19} - 2 \beta_{18} + 6 \beta_{17} + 39 \beta_{16} - 2 \beta_{15} + 49 \beta_{14} + \cdots - 63 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( - 136 \beta_{19} - 136 \beta_{18} - 37 \beta_{17} - 136 \beta_{16} + 136 \beta_{15} - 369 \beta_{14} + \cdots - 357 ) / 2 \)
|
| \(\nu^{8}\) | \(=\) |
\( ( 554 \beta_{19} + 76 \beta_{18} + 87 \beta_{17} - 554 \beta_{16} + 76 \beta_{15} - 833 \beta_{14} + \cdots + 745 ) / 2 \)
|
| \(\nu^{9}\) | \(=\) |
\( 499 \beta_{19} + 499 \beta_{18} + 124 \beta_{17} + 499 \beta_{16} - 499 \beta_{15} + 1460 \beta_{14} + \cdots + 1448 \)
|
| \(\nu^{10}\) | \(=\) |
\( ( - 4154 \beta_{19} - 944 \beta_{18} - 1823 \beta_{17} + 4154 \beta_{16} - 944 \beta_{15} + 6965 \beta_{14} + \cdots - 4627 ) / 2 \)
|
| \(\nu^{11}\) | \(=\) |
\( ( - 7394 \beta_{19} - 7374 \beta_{18} - 1867 \beta_{17} - 7394 \beta_{16} + 7374 \beta_{15} + \cdots - 23737 ) / 2 \)
|
| \(\nu^{12}\) | \(=\) |
\( 16060 \beta_{19} + 4908 \beta_{18} + 10713 \beta_{17} - 16060 \beta_{16} + 4908 \beta_{15} + \cdots + 14982 \)
|
| \(\nu^{13}\) | \(=\) |
\( ( 55912 \beta_{19} + 55452 \beta_{18} + 14887 \beta_{17} + 55912 \beta_{16} - 55452 \beta_{15} + \cdots + 195423 ) / 2 \)
|
| \(\nu^{14}\) | \(=\) |
\( ( - 252878 \beta_{19} - 93112 \beta_{18} - 213107 \beta_{17} + 252878 \beta_{16} - 93112 \beta_{15} + \cdots - 201853 ) / 2 \)
|
| \(\nu^{15}\) | \(=\) |
\( - 215855 \beta_{19} - 212533 \beta_{18} - 61098 \beta_{17} - 215855 \beta_{16} + 212533 \beta_{15} + \cdots - 805634 \)
|
| \(\nu^{16}\) | \(=\) |
\( ( 2013708 \beta_{19} + 838112 \beta_{18} + 1962909 \beta_{17} - 2013708 \beta_{16} + 838112 \beta_{15} + \cdots + 1411421 ) / 2 \)
|
| \(\nu^{17}\) | \(=\) |
\( ( 3392222 \beta_{19} + 3314446 \beta_{18} + 1016735 \beta_{17} + 3392222 \beta_{16} - 3314446 \beta_{15} + \cdots + 13286633 ) / 2 \)
|
| \(\nu^{18}\) | \(=\) |
\( - 8080497 \beta_{19} - 3654094 \beta_{18} - 8669055 \beta_{17} + 8080497 \beta_{16} - 3654094 \beta_{15} + \cdots - 5104464 \)
|
| \(\nu^{19}\) | \(=\) |
\( ( - 27013592 \beta_{19} - 26203988 \beta_{18} - 8502479 \beta_{17} - 27013592 \beta_{16} + \cdots - 109516551 ) / 2 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/260\mathbb{Z}\right)^\times\).
| \(n\) | \(41\) | \(131\) | \(157\) |
| \(\chi(n)\) | \(\beta_{6}\) | \(1\) | \(\beta_{6} + \beta_{7}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 33.1 |
|
0 | −2.51912 | + | 0.674996i | 0 | 2.14030 | − | 0.647383i | 0 | −1.45437 | + | 0.839682i | 0 | 3.29226 | − | 1.90079i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 33.2 | 0 | −2.41732 | + | 0.647720i | 0 | −2.18528 | − | 0.473860i | 0 | 3.34088 | − | 1.92886i | 0 | 2.82584 | − | 1.63150i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 33.3 | 0 | 0.243028 | − | 0.0651192i | 0 | 0.356684 | + | 2.20744i | 0 | −4.30824 | + | 2.48736i | 0 | −2.54325 | + | 1.46835i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 33.4 | 0 | 0.834228 | − | 0.223531i | 0 | 2.20306 | − | 0.382788i | 0 | 2.07295 | − | 1.19682i | 0 | −1.95211 | + | 1.12705i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 33.5 | 0 | 2.49316 | − | 0.668040i | 0 | −0.380790 | − | 2.20341i | 0 | −0.749297 | + | 0.432607i | 0 | 3.17149 | − | 1.83106i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 97.1 | 0 | −0.690650 | − | 2.57754i | 0 | −0.0143596 | − | 2.23602i | 0 | 1.87342 | − | 1.08162i | 0 | −3.56864 | + | 2.06036i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 97.2 | 0 | −0.563138 | − | 2.10166i | 0 | 1.75876 | + | 1.38086i | 0 | 1.25530 | − | 0.724750i | 0 | −1.50178 | + | 0.867051i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 97.3 | 0 | 0.355132 | + | 1.32537i | 0 | 1.50965 | + | 1.64953i | 0 | −3.40883 | + | 1.96809i | 0 | 0.967585 | − | 0.558635i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 97.4 | 0 | 0.387600 | + | 1.44654i | 0 | 1.95799 | − | 1.07994i | 0 | 2.10545 | − | 1.21558i | 0 | 0.655821 | − | 0.378639i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 97.5 | 0 | 0.877081 | + | 3.27331i | 0 | −1.34601 | + | 1.78557i | 0 | 2.27273 | − | 1.31216i | 0 | −7.34722 | + | 4.24192i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 193.1 | 0 | −0.690650 | + | 2.57754i | 0 | −0.0143596 | + | 2.23602i | 0 | 1.87342 | + | 1.08162i | 0 | −3.56864 | − | 2.06036i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 193.2 | 0 | −0.563138 | + | 2.10166i | 0 | 1.75876 | − | 1.38086i | 0 | 1.25530 | + | 0.724750i | 0 | −1.50178 | − | 0.867051i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 193.3 | 0 | 0.355132 | − | 1.32537i | 0 | 1.50965 | − | 1.64953i | 0 | −3.40883 | − | 1.96809i | 0 | 0.967585 | + | 0.558635i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 193.4 | 0 | 0.387600 | − | 1.44654i | 0 | 1.95799 | + | 1.07994i | 0 | 2.10545 | + | 1.21558i | 0 | 0.655821 | + | 0.378639i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 193.5 | 0 | 0.877081 | − | 3.27331i | 0 | −1.34601 | − | 1.78557i | 0 | 2.27273 | + | 1.31216i | 0 | −7.34722 | − | 4.24192i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 197.1 | 0 | −2.51912 | − | 0.674996i | 0 | 2.14030 | + | 0.647383i | 0 | −1.45437 | − | 0.839682i | 0 | 3.29226 | + | 1.90079i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 197.2 | 0 | −2.41732 | − | 0.647720i | 0 | −2.18528 | + | 0.473860i | 0 | 3.34088 | + | 1.92886i | 0 | 2.82584 | + | 1.63150i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 197.3 | 0 | 0.243028 | + | 0.0651192i | 0 | 0.356684 | − | 2.20744i | 0 | −4.30824 | − | 2.48736i | 0 | −2.54325 | − | 1.46835i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 197.4 | 0 | 0.834228 | + | 0.223531i | 0 | 2.20306 | + | 0.382788i | 0 | 2.07295 | + | 1.19682i | 0 | −1.95211 | − | 1.12705i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 197.5 | 0 | 2.49316 | + | 0.668040i | 0 | −0.380790 | + | 2.20341i | 0 | −0.749297 | − | 0.432607i | 0 | 3.17149 | + | 1.83106i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 65.o | even | 12 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 260.2.bk.c | yes | 20 |
| 5.b | even | 2 | 1 | 1300.2.bs.d | 20 | ||
| 5.c | odd | 4 | 1 | 260.2.bf.c | ✓ | 20 | |
| 5.c | odd | 4 | 1 | 1300.2.bn.d | 20 | ||
| 13.f | odd | 12 | 1 | 260.2.bf.c | ✓ | 20 | |
| 65.o | even | 12 | 1 | inner | 260.2.bk.c | yes | 20 |
| 65.s | odd | 12 | 1 | 1300.2.bn.d | 20 | ||
| 65.t | even | 12 | 1 | 1300.2.bs.d | 20 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 260.2.bf.c | ✓ | 20 | 5.c | odd | 4 | 1 | |
| 260.2.bf.c | ✓ | 20 | 13.f | odd | 12 | 1 | |
| 260.2.bk.c | yes | 20 | 1.a | even | 1 | 1 | trivial |
| 260.2.bk.c | yes | 20 | 65.o | even | 12 | 1 | inner |
| 1300.2.bn.d | 20 | 5.c | odd | 4 | 1 | ||
| 1300.2.bn.d | 20 | 65.s | odd | 12 | 1 | ||
| 1300.2.bs.d | 20 | 5.b | even | 2 | 1 | ||
| 1300.2.bs.d | 20 | 65.t | even | 12 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{20} + 2 T_{3}^{19} + 8 T_{3}^{18} + 28 T_{3}^{17} - 26 T_{3}^{16} - 106 T_{3}^{15} - 500 T_{3}^{14} + \cdots + 21904 \)
acting on \(S_{2}^{\mathrm{new}}(260, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{20} \)
|
| $3$ |
\( T^{20} + 2 T^{19} + \cdots + 21904 \)
|
| $5$ |
\( T^{20} - 12 T^{19} + \cdots + 9765625 \)
|
| $7$ |
\( T^{20} - 6 T^{19} + \cdots + 27625536 \)
|
| $11$ |
\( T^{20} - 12 T^{18} + \cdots + 8202496 \)
|
| $13$ |
\( T^{20} + \cdots + 137858491849 \)
|
| $17$ |
\( T^{20} + \cdots + 109980446689 \)
|
| $19$ |
\( T^{20} + \cdots + 472279824 \)
|
| $23$ |
\( T^{20} - 6 T^{19} + \cdots + 3139984 \)
|
| $29$ |
\( T^{20} + \cdots + 341103961 \)
|
| $31$ |
\( T^{20} + \cdots + 54838398976 \)
|
| $37$ |
\( T^{20} + \cdots + 52\!\cdots\!69 \)
|
| $41$ |
\( T^{20} + \cdots + 10017978284161 \)
|
| $43$ |
\( T^{20} + \cdots + 319577657344 \)
|
| $47$ |
\( T^{20} + \cdots + 5858983936 \)
|
| $53$ |
\( T^{20} + \cdots + 90\!\cdots\!64 \)
|
| $59$ |
\( T^{20} + \cdots + 26\!\cdots\!96 \)
|
| $61$ |
\( T^{20} + \cdots + 43\!\cdots\!41 \)
|
| $67$ |
\( T^{20} + \cdots + 11546104977936 \)
|
| $71$ |
\( T^{20} + \cdots + 27\!\cdots\!76 \)
|
| $73$ |
\( (T^{10} + 22 T^{9} + \cdots - 119575728)^{2} \)
|
| $79$ |
\( T^{20} + \cdots + 207459544743936 \)
|
| $83$ |
\( T^{20} + \cdots + 10\!\cdots\!36 \)
|
| $89$ |
\( T^{20} + \cdots + 237919334033956 \)
|
| $97$ |
\( T^{20} + \cdots + 963004643584 \)
|
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