Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [570,2,Mod(83,570)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(570, base_ring=CyclotomicField(12))
chi = DirichletCharacter(H, H._module([6, 9, 4]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("570.83");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 570 = 2 \cdot 3 \cdot 5 \cdot 19 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 570.v (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: | \(4.55147291521\) |
Analytic rank: | \(0\) |
Dimension: | \(144\) |
Relative dimension: | \(36\) over \(\Q(\zeta_{12})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{12}]$ |
$q$-expansion
The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.
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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
83.1 | −0.258819 | + | 0.965926i | −1.72915 | + | 0.100231i | −0.866025 | − | 0.500000i | 0.941726 | + | 2.02809i | 0.350721 | − | 1.69617i | 0.473527 | − | 0.473527i | 0.707107 | − | 0.707107i | 2.97991 | − | 0.346627i | −2.20272 | + | 0.384729i |
83.2 | −0.258819 | + | 0.965926i | −1.72667 | + | 0.136400i | −0.866025 | − | 0.500000i | 0.777252 | − | 2.09664i | 0.315143 | − | 1.70314i | −2.77147 | + | 2.77147i | 0.707107 | − | 0.707107i | 2.96279 | − | 0.471038i | 1.82403 | + | 1.29342i |
83.3 | −0.258819 | + | 0.965926i | −1.71974 | − | 0.206162i | −0.866025 | − | 0.500000i | −2.08833 | − | 0.799309i | 0.644238 | − | 1.60778i | 2.22018 | − | 2.22018i | 0.707107 | − | 0.707107i | 2.91499 | + | 0.709088i | 1.31257 | − | 1.81029i |
83.4 | −0.258819 | + | 0.965926i | −1.42499 | + | 0.984579i | −0.866025 | − | 0.500000i | −2.13559 | + | 0.662776i | −0.582215 | − | 1.63127i | −2.79338 | + | 2.79338i | 0.707107 | − | 0.707107i | 1.06121 | − | 2.80604i | −0.0874624 | − | 2.23436i |
83.5 | −0.258819 | + | 0.965926i | −1.23645 | + | 1.21293i | −0.866025 | − | 0.500000i | 2.20180 | + | 0.389966i | −0.851583 | − | 1.50825i | 0.853440 | − | 0.853440i | 0.707107 | − | 0.707107i | 0.0576060 | − | 2.99945i | −0.946546 | + | 2.02585i |
83.6 | −0.258819 | + | 0.965926i | −0.986467 | − | 1.42369i | −0.866025 | − | 0.500000i | −2.21751 | − | 0.287509i | 1.63049 | − | 0.584377i | −1.76892 | + | 1.76892i | 0.707107 | − | 0.707107i | −1.05376 | + | 2.80884i | 0.851646 | − | 2.06753i |
83.7 | −0.258819 | + | 0.965926i | −0.822951 | − | 1.52406i | −0.866025 | − | 0.500000i | −0.462693 | + | 2.18767i | 1.68512 | − | 0.400455i | 2.22340 | − | 2.22340i | 0.707107 | − | 0.707107i | −1.64550 | + | 2.50845i | −1.99338 | − | 1.01314i |
83.8 | −0.258819 | + | 0.965926i | −0.681821 | + | 1.59221i | −0.866025 | − | 0.500000i | 1.65405 | − | 1.50470i | −1.36148 | − | 1.07068i | 0.292109 | − | 0.292109i | 0.707107 | − | 0.707107i | −2.07024 | − | 2.17120i | 1.02533 | + | 1.98714i |
83.9 | −0.258819 | + | 0.965926i | −0.396548 | − | 1.68605i | −0.866025 | − | 0.500000i | 2.12625 | + | 0.692155i | 1.73123 | + | 0.0533452i | −3.45963 | + | 3.45963i | 0.707107 | − | 0.707107i | −2.68550 | + | 1.33719i | −1.21888 | + | 1.87465i |
83.10 | −0.258819 | + | 0.965926i | −0.366145 | + | 1.69291i | −0.866025 | − | 0.500000i | −0.619415 | + | 2.14856i | −1.54046 | − | 0.791825i | −1.07462 | + | 1.07462i | 0.707107 | − | 0.707107i | −2.73188 | − | 1.23970i | −1.91504 | − | 1.15440i |
83.11 | −0.258819 | + | 0.965926i | 0.00494183 | − | 1.73204i | −0.866025 | − | 0.500000i | 0.471665 | − | 2.18576i | 1.67175 | + | 0.453059i | 2.57943 | − | 2.57943i | 0.707107 | − | 0.707107i | −2.99995 | − | 0.0171189i | 1.98920 | + | 1.02131i |
83.12 | −0.258819 | + | 0.965926i | 0.881105 | − | 1.49119i | −0.866025 | − | 0.500000i | −1.25031 | + | 1.85384i | 1.21233 | + | 1.23703i | −1.01864 | + | 1.01864i | 0.707107 | − | 0.707107i | −1.44731 | − | 2.62779i | −1.46706 | − | 1.68752i |
83.13 | −0.258819 | + | 0.965926i | 1.06964 | + | 1.36231i | −0.866025 | − | 0.500000i | 1.30832 | + | 1.81337i | −1.59273 | + | 0.680599i | −1.22093 | + | 1.22093i | 0.707107 | − | 0.707107i | −0.711754 | + | 2.91434i | −2.09020 | + | 0.794403i |
83.14 | −0.258819 | + | 0.965926i | 1.25366 | − | 1.19513i | −0.866025 | − | 0.500000i | 1.74044 | − | 1.40387i | 0.829942 | + | 1.52026i | −1.56325 | + | 1.56325i | 0.707107 | − | 0.707107i | 0.143305 | − | 2.99658i | 0.905575 | + | 2.04449i |
83.15 | −0.258819 | + | 0.965926i | 1.38266 | + | 1.04319i | −0.866025 | − | 0.500000i | 1.72991 | − | 1.41683i | −1.36550 | + | 1.06555i | 0.867998 | − | 0.867998i | 0.707107 | − | 0.707107i | 0.823517 | + | 2.88476i | 0.920816 | + | 2.03767i |
83.16 | −0.258819 | + | 0.965926i | 1.59655 | − | 0.671583i | −0.866025 | − | 0.500000i | 2.02987 | + | 0.937881i | 0.235481 | + | 1.71597i | 2.68985 | − | 2.68985i | 0.707107 | − | 0.707107i | 2.09795 | − | 2.14443i | −1.43129 | + | 1.71796i |
83.17 | −0.258819 | + | 0.965926i | 1.65736 | − | 0.503149i | −0.866025 | − | 0.500000i | −1.39912 | − | 1.74426i | 0.0570481 | + | 1.73111i | 0.200069 | − | 0.200069i | 0.707107 | − | 0.707107i | 2.49368 | − | 1.66780i | 2.04695 | − | 0.899999i |
83.18 | −0.258819 | + | 0.965926i | 1.67919 | + | 0.424642i | −0.866025 | − | 0.500000i | −1.46226 | + | 1.69168i | −0.844779 | + | 1.51207i | 3.27081 | − | 3.27081i | 0.707107 | − | 0.707107i | 2.63936 | + | 1.42611i | −1.25558 | − | 1.85027i |
83.19 | 0.258819 | − | 0.965926i | −1.71902 | + | 0.212096i | −0.866025 | − | 0.500000i | −1.72991 | + | 1.41683i | −0.240045 | + | 1.71534i | 0.867998 | − | 0.867998i | −0.707107 | + | 0.707107i | 2.91003 | − | 0.729192i | 0.920816 | + | 2.03767i |
83.20 | 0.258819 | − | 0.965926i | −1.66654 | − | 0.471844i | −0.866025 | − | 0.500000i | 1.46226 | − | 1.69168i | −0.887099 | + | 1.48763i | 3.27081 | − | 3.27081i | −0.707107 | + | 0.707107i | 2.55473 | + | 1.57270i | −1.25558 | − | 1.85027i |
See next 80 embeddings (of 144 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
3.b | odd | 2 | 1 | inner |
5.c | odd | 4 | 1 | inner |
15.e | even | 4 | 1 | inner |
19.c | even | 3 | 1 | inner |
57.h | odd | 6 | 1 | inner |
95.m | odd | 12 | 1 | inner |
285.v | even | 12 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 570.2.v.c | ✓ | 144 |
3.b | odd | 2 | 1 | inner | 570.2.v.c | ✓ | 144 |
5.c | odd | 4 | 1 | inner | 570.2.v.c | ✓ | 144 |
15.e | even | 4 | 1 | inner | 570.2.v.c | ✓ | 144 |
19.c | even | 3 | 1 | inner | 570.2.v.c | ✓ | 144 |
57.h | odd | 6 | 1 | inner | 570.2.v.c | ✓ | 144 |
95.m | odd | 12 | 1 | inner | 570.2.v.c | ✓ | 144 |
285.v | even | 12 | 1 | inner | 570.2.v.c | ✓ | 144 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
570.2.v.c | ✓ | 144 | 1.a | even | 1 | 1 | trivial |
570.2.v.c | ✓ | 144 | 3.b | odd | 2 | 1 | inner |
570.2.v.c | ✓ | 144 | 5.c | odd | 4 | 1 | inner |
570.2.v.c | ✓ | 144 | 15.e | even | 4 | 1 | inner |
570.2.v.c | ✓ | 144 | 19.c | even | 3 | 1 | inner |
570.2.v.c | ✓ | 144 | 57.h | odd | 6 | 1 | inner |
570.2.v.c | ✓ | 144 | 95.m | odd | 12 | 1 | inner |
570.2.v.c | ✓ | 144 | 285.v | even | 12 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(570, [\chi])\):
\( T_{7}^{36} - 4 T_{7}^{33} + 1089 T_{7}^{32} - 196 T_{7}^{31} + 8 T_{7}^{30} + 856 T_{7}^{29} + \cdots + 13546632100 \) |
\( T_{17}^{144} - 12432 T_{17}^{140} + 91265720 T_{17}^{136} - 445652000624 T_{17}^{132} + \cdots + 32\!\cdots\!56 \) |