Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1300,2,Mod(193,1300)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1300, base_ring=CyclotomicField(12))
chi = DirichletCharacter(H, H._module([0, 9, 7]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1300.193");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1300 = 2^{2} \cdot 5^{2} \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1300.bs (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: | \(10.3805522628\) |
| 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: | \(\Z[a_1, \ldots, a_{13}]\) |
| Coefficient ring index: | \( 2^{4} \) |
| Twist minimal: | no (minimal twist has level 260) |
| 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
\( x^{20} + 30 x^{18} + 371 x^{16} + 2460 x^{14} + 9517 x^{12} + 21870 x^{10} + 29001 x^{8} + 20400 x^{6} + \cdots + 9 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( - 3047 \nu^{18} - 80534 \nu^{16} - 834504 \nu^{14} - 4296019 \nu^{12} - 11423658 \nu^{10} + \cdots + 99723 ) / 1134948 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 19367 \nu^{19} + 69462 \nu^{18} + 805551 \nu^{17} + 1878198 \nu^{16} + 13356712 \nu^{15} + \cdots - 217278 ) / 13619376 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 15705 \nu^{19} + 7449 \nu^{18} + 402397 \nu^{17} + 223669 \nu^{16} + 3930438 \nu^{15} + \cdots + 661521 ) / 4539792 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 66482 \nu^{19} - 47115 \nu^{18} - 2012742 \nu^{17} - 1207191 \nu^{16} - 25148026 \nu^{15} + \cdots + 15821217 ) / 13619376 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 66482 \nu^{19} + 47115 \nu^{18} - 2012742 \nu^{17} + 1207191 \nu^{16} - 25148026 \nu^{15} + \cdots - 15821217 ) / 13619376 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 66482 \nu^{19} - 83679 \nu^{18} + 2012742 \nu^{17} - 2173599 \nu^{16} + 25148026 \nu^{15} + \cdots + 17017893 ) / 13619376 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 165961 \nu^{19} - 181245 \nu^{18} - 5106855 \nu^{17} - 5590593 \nu^{16} - 65333654 \nu^{15} + \cdots - 45388899 ) / 13619376 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 27888 \nu^{19} + 17400 \nu^{18} + 876466 \nu^{17} + 537244 \nu^{16} + 11480801 \nu^{15} + \cdots + 4472424 ) / 2269896 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( - 84623 \nu^{19} - 64035 \nu^{18} - 2581149 \nu^{17} - 1764837 \nu^{16} - 32554318 \nu^{15} + \cdots + 12376215 ) / 6809688 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( 27888 \nu^{19} - 17400 \nu^{18} + 876466 \nu^{17} - 537244 \nu^{16} + 11480801 \nu^{15} + \cdots - 4472424 ) / 2269896 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( - 84623 \nu^{19} + 64035 \nu^{18} - 2581149 \nu^{17} + 1764837 \nu^{16} - 32554318 \nu^{15} + \cdots - 12376215 ) / 6809688 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( 165961 \nu^{19} - 320169 \nu^{18} + 5106855 \nu^{17} - 9346989 \nu^{16} + 65333654 \nu^{15} + \cdots - 31334967 ) / 13619376 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( 282319 \nu^{19} + 174918 \nu^{18} + 8264211 \nu^{17} + 5018946 \nu^{16} + 98942810 \nu^{15} + \cdots + 42156486 ) / 13619376 \)
|
| \(\beta_{14}\) | \(=\) |
\( ( - 353 \nu^{19} - 10585 \nu^{17} - 130732 \nu^{15} - 864458 \nu^{13} - 3326801 \nu^{11} + \cdots - 8376 ) / 16752 \)
|
| \(\beta_{15}\) | \(=\) |
\( ( 397382 \nu^{19} + 50979 \nu^{18} + 11805666 \nu^{17} + 1600971 \nu^{16} + 144498028 \nu^{15} + \cdots + 24710427 ) / 13619376 \)
|
| \(\beta_{16}\) | \(=\) |
\( ( 210017 \nu^{19} + 54111 \nu^{18} + 6333330 \nu^{17} + 1652277 \nu^{16} + 78814861 \nu^{15} + \cdots + 22008474 ) / 6809688 \)
|
| \(\beta_{17}\) | \(=\) |
\( ( 67147 \nu^{19} - 27215 \nu^{18} + 2045559 \nu^{17} - 784089 \nu^{16} + 25759345 \nu^{15} + \cdots + 7414695 ) / 2269896 \)
|
| \(\beta_{18}\) | \(=\) |
\( ( 938081 \nu^{19} + 181527 \nu^{18} + 28105911 \nu^{17} + 5420187 \nu^{16} + 346829194 \nu^{15} + \cdots - 35447103 ) / 13619376 \)
|
| \(\beta_{19}\) | \(=\) |
\( ( - 1080005 \nu^{19} - 98094 \nu^{18} - 32065821 \nu^{17} - 2808162 \nu^{16} - 391463830 \nu^{15} + \cdots - 15698898 ) / 13619376 \)
|
| \(\nu\) | \(=\) |
\( ( -\beta_{6} - \beta_{5} + \beta_1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( - \beta_{19} - \beta_{18} + \beta_{16} - \beta_{15} - \beta_{14} - \beta_{13} - \beta_{12} + \beta_{10} + \cdots - 6 ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( \beta_{19} + \beta_{18} + 2 \beta_{17} - \beta_{16} - \beta_{15} - \beta_{14} - \beta_{13} - 3 \beta_{12} + \cdots + 4 ) / 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 9 \beta_{19} + 7 \beta_{18} - 13 \beta_{16} + 15 \beta_{15} + 9 \beta_{14} + 11 \beta_{13} + 5 \beta_{12} + \cdots + 34 ) / 2 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( - 13 \beta_{19} - 11 \beta_{18} - 22 \beta_{17} + 11 \beta_{16} + 9 \beta_{15} + 17 \beta_{14} + \cdots - 40 ) / 2 \)
|
| \(\nu^{6}\) | \(=\) |
\( - 38 \beta_{19} - 29 \beta_{18} + 58 \beta_{16} - 67 \beta_{15} - 38 \beta_{14} - 47 \beta_{13} + \cdots - 119 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( 118 \beta_{19} + 106 \beta_{18} + 212 \beta_{17} - 106 \beta_{16} - 94 \beta_{15} - 166 \beta_{14} + \cdots + 352 ) / 2 \)
|
| \(\nu^{8}\) | \(=\) |
\( ( 637 \beta_{19} + 499 \beta_{18} - 951 \beta_{16} + 1089 \beta_{15} + 637 \beta_{14} + 775 \beta_{13} + \cdots + 1830 ) / 2 \)
|
| \(\nu^{9}\) | \(=\) |
\( ( - 975 \beta_{19} - 971 \beta_{18} - 1942 \beta_{17} + 971 \beta_{16} + 967 \beta_{15} + 1443 \beta_{14} + \cdots - 2992 ) / 2 \)
|
| \(\nu^{10}\) | \(=\) |
\( ( - 5317 \beta_{19} - 4287 \beta_{18} + 7635 \beta_{16} - 8665 \beta_{15} - 5317 \beta_{14} + \cdots - 14638 ) / 2 \)
|
| \(\nu^{11}\) | \(=\) |
\( ( 7845 \beta_{19} + 8595 \beta_{18} + 17190 \beta_{17} - 8615 \beta_{16} - 9365 \beta_{15} - 12173 \beta_{14} + \cdots + 25044 ) / 2 \)
|
| \(\nu^{12}\) | \(=\) |
\( 22087 \beta_{19} + 18229 \beta_{18} - 30559 \beta_{16} + 34417 \beta_{15} + 22087 \beta_{14} + \cdots + 59485 \)
|
| \(\nu^{13}\) | \(=\) |
\( ( - 62818 \beta_{19} - 74282 \beta_{18} - 148564 \beta_{17} + 74742 \beta_{16} + 86206 \beta_{15} + \cdots - 207904 ) / 2 \)
|
| \(\nu^{14}\) | \(=\) |
\( ( - 365529 \beta_{19} - 307025 \beta_{18} + 490501 \beta_{16} - 549005 \beta_{15} - 365529 \beta_{14} + \cdots - 973110 ) / 2 \)
|
| \(\nu^{15}\) | \(=\) |
\( ( 504149 \beta_{19} + 631589 \beta_{18} + 1263178 \beta_{17} - 638233 \beta_{16} - 765673 \beta_{15} + \cdots + 1717868 ) / 2 \)
|
| \(\nu^{16}\) | \(=\) |
\( ( 3015847 \beta_{19} + 2566165 \beta_{18} - 3952079 \beta_{16} + 4401761 \beta_{15} + 3015847 \beta_{14} + \cdots + 7979906 ) / 2 \)
|
| \(\nu^{17}\) | \(=\) |
\( ( - 4062775 \beta_{19} - 5311317 \beta_{18} - 10622634 \beta_{17} + 5389093 \beta_{16} + 6637635 \beta_{15} + \cdots - 14156328 ) / 2 \)
|
| \(\nu^{18}\) | \(=\) |
\( - 12417331 \beta_{19} - 10666792 \beta_{18} + 15983777 \beta_{16} - 17734316 \beta_{15} + \cdots - 32752987 \)
|
| \(\nu^{19}\) | \(=\) |
\( ( 32875304 \beta_{19} + 44334764 \beta_{18} + 88669528 \beta_{17} - 45144368 \beta_{16} - 56603828 \beta_{15} + \cdots + 116477984 ) / 2 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1300\mathbb{Z}\right)^\times\).
| \(n\) | \(301\) | \(651\) | \(677\) |
| \(\chi(n)\) | \(\beta_{5}\) | \(1\) | \(\beta_{4} + \beta_{5}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 193.1 |
|
0 | −0.877081 | + | 3.27331i | 0 | 0 | 0 | −2.27273 | − | 1.31216i | 0 | −7.34722 | − | 4.24192i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 193.2 | 0 | −0.387600 | + | 1.44654i | 0 | 0 | 0 | −2.10545 | − | 1.21558i | 0 | 0.655821 | + | 0.378639i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 193.3 | 0 | −0.355132 | + | 1.32537i | 0 | 0 | 0 | 3.40883 | + | 1.96809i | 0 | 0.967585 | + | 0.558635i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 193.4 | 0 | 0.563138 | − | 2.10166i | 0 | 0 | 0 | −1.25530 | − | 0.724750i | 0 | −1.50178 | − | 0.867051i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 193.5 | 0 | 0.690650 | − | 2.57754i | 0 | 0 | 0 | −1.87342 | − | 1.08162i | 0 | −3.56864 | − | 2.06036i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 293.1 | 0 | −2.49316 | + | 0.668040i | 0 | 0 | 0 | 0.749297 | − | 0.432607i | 0 | 3.17149 | − | 1.83106i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 293.2 | 0 | −0.834228 | + | 0.223531i | 0 | 0 | 0 | −2.07295 | + | 1.19682i | 0 | −1.95211 | + | 1.12705i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 293.3 | 0 | −0.243028 | + | 0.0651192i | 0 | 0 | 0 | 4.30824 | − | 2.48736i | 0 | −2.54325 | + | 1.46835i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 293.4 | 0 | 2.41732 | − | 0.647720i | 0 | 0 | 0 | −3.34088 | + | 1.92886i | 0 | 2.82584 | − | 1.63150i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 293.5 | 0 | 2.51912 | − | 0.674996i | 0 | 0 | 0 | 1.45437 | − | 0.839682i | 0 | 3.29226 | − | 1.90079i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 357.1 | 0 | −0.877081 | − | 3.27331i | 0 | 0 | 0 | −2.27273 | + | 1.31216i | 0 | −7.34722 | + | 4.24192i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 357.2 | 0 | −0.387600 | − | 1.44654i | 0 | 0 | 0 | −2.10545 | + | 1.21558i | 0 | 0.655821 | − | 0.378639i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 357.3 | 0 | −0.355132 | − | 1.32537i | 0 | 0 | 0 | 3.40883 | − | 1.96809i | 0 | 0.967585 | − | 0.558635i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 357.4 | 0 | 0.563138 | + | 2.10166i | 0 | 0 | 0 | −1.25530 | + | 0.724750i | 0 | −1.50178 | + | 0.867051i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 357.5 | 0 | 0.690650 | + | 2.57754i | 0 | 0 | 0 | −1.87342 | + | 1.08162i | 0 | −3.56864 | + | 2.06036i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 457.1 | 0 | −2.49316 | − | 0.668040i | 0 | 0 | 0 | 0.749297 | + | 0.432607i | 0 | 3.17149 | + | 1.83106i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 457.2 | 0 | −0.834228 | − | 0.223531i | 0 | 0 | 0 | −2.07295 | − | 1.19682i | 0 | −1.95211 | − | 1.12705i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 457.3 | 0 | −0.243028 | − | 0.0651192i | 0 | 0 | 0 | 4.30824 | + | 2.48736i | 0 | −2.54325 | − | 1.46835i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 457.4 | 0 | 2.41732 | + | 0.647720i | 0 | 0 | 0 | −3.34088 | − | 1.92886i | 0 | 2.82584 | + | 1.63150i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 457.5 | 0 | 2.51912 | + | 0.674996i | 0 | 0 | 0 | 1.45437 | + | 0.839682i | 0 | 3.29226 | + | 1.90079i | 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 | 1300.2.bs.d | 20 | |
| 5.b | even | 2 | 1 | 260.2.bk.c | yes | 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 | 1300.2.bn.d | 20 | ||
| 65.o | even | 12 | 1 | inner | 1300.2.bs.d | 20 | |
| 65.s | odd | 12 | 1 | 260.2.bf.c | ✓ | 20 | |
| 65.t | even | 12 | 1 | 260.2.bk.c | yes | 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 | 65.s | odd | 12 | 1 | |
| 260.2.bk.c | yes | 20 | 5.b | even | 2 | 1 | |
| 260.2.bk.c | yes | 20 | 65.t | even | 12 | 1 | |
| 1300.2.bn.d | 20 | 5.c | odd | 4 | 1 | ||
| 1300.2.bn.d | 20 | 13.f | odd | 12 | 1 | ||
| 1300.2.bs.d | 20 | 1.a | even | 1 | 1 | trivial | |
| 1300.2.bs.d | 20 | 65.o | even | 12 | 1 | inner | |
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 \(S_{2}^{\mathrm{new}}(1300, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{20} \)
|
| $3$ |
\( T^{20} - 2 T^{19} + \cdots + 21904 \)
|
| $5$ |
\( T^{20} \)
|
| $7$ |
\( T^{20} + 6 T^{19} + \cdots + 27625536 \)
|
| $11$ |
\( T^{20} - 12 T^{18} + \cdots + 8202496 \)
|
| $13$ |
\( T^{20} + \cdots + 137858491849 \)
|
| $17$ |
\( T^{20} + \cdots + 109980446689 \)
|
| $19$ |
\( T^{20} + \cdots + 472279824 \)
|
| $23$ |
\( T^{20} + 6 T^{19} + \cdots + 3139984 \)
|
| $29$ |
\( T^{20} + \cdots + 341103961 \)
|
| $31$ |
\( T^{20} + \cdots + 54838398976 \)
|
| $37$ |
\( T^{20} + \cdots + 52\!\cdots\!69 \)
|
| $41$ |
\( T^{20} + \cdots + 10017978284161 \)
|
| $43$ |
\( T^{20} + \cdots + 319577657344 \)
|
| $47$ |
\( T^{20} + \cdots + 5858983936 \)
|
| $53$ |
\( T^{20} + \cdots + 90\!\cdots\!64 \)
|
| $59$ |
\( T^{20} + \cdots + 26\!\cdots\!96 \)
|
| $61$ |
\( T^{20} + \cdots + 43\!\cdots\!41 \)
|
| $67$ |
\( T^{20} + \cdots + 11546104977936 \)
|
| $71$ |
\( T^{20} + \cdots + 27\!\cdots\!76 \)
|
| $73$ |
\( (T^{10} - 22 T^{9} + \cdots - 119575728)^{2} \)
|
| $79$ |
\( T^{20} + \cdots + 207459544743936 \)
|
| $83$ |
\( T^{20} + \cdots + 10\!\cdots\!36 \)
|
| $89$ |
\( T^{20} + \cdots + 237919334033956 \)
|
| $97$ |
\( T^{20} + \cdots + 963004643584 \)
|
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