Properties

Label 570.2.v.c
Level $570$
Weight $2$
Character orbit 570.v
Analytic conductor $4.551$
Analytic rank $0$
Dimension $144$
CM no
Inner twists $8$

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Newspace parameters

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

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.

\(\operatorname{Tr}(f)(q) = \) \( 144q - 4q^{3} + 8q^{6} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 144q - 4q^{3} + 8q^{6} - 16q^{10} - 8q^{12} - 32q^{13} + 4q^{15} + 72q^{16} + 32q^{18} - 24q^{21} + 8q^{22} + 40q^{25} - 16q^{27} - 8q^{30} - 16q^{33} - 16q^{37} + 8q^{40} - 44q^{42} - 16q^{43} - 56q^{45} - 48q^{46} + 4q^{48} + 32q^{51} + 32q^{52} + 8q^{55} - 64q^{57} + 32q^{58} - 24q^{60} + 76q^{63} + 16q^{66} - 56q^{67} - 8q^{70} + 16q^{72} + 88q^{73} + 32q^{75} + 48q^{76} - 56q^{81} - 152q^{87} - 16q^{88} - 4q^{90} + 80q^{91} - 52q^{93} + 16q^{96} + 32q^{97} + O(q^{100}) \)

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.

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)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 467.36
Significant digits:
Format:

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 \(S_{2}^{\mathrm{new}}(570, [\chi])\):

\(12\!\cdots\!80\)\( T_{7}^{9} + \)\(20\!\cdots\!04\)\( T_{7}^{8} - 326923301896 T_{7}^{7} + 520929772032 T_{7}^{6} - \)\(20\!\cdots\!52\)\( T_{7}^{5} + \)\(28\!\cdots\!44\)\( T_{7}^{4} - \)\(17\!\cdots\!84\)\( T_{7}^{3} + 660964830752 T_{7}^{2} - 133819635280 T_{7} + 13546632100 \)">\(T_{7}^{36} - \cdots\)
\(16\!\cdots\!32\)\( T_{17}^{128} - \)\(44\!\cdots\!44\)\( T_{17}^{124} + \)\(96\!\cdots\!80\)\( T_{17}^{120} - \)\(16\!\cdots\!68\)\( T_{17}^{116} + \)\(21\!\cdots\!24\)\( T_{17}^{112} - \)\(21\!\cdots\!28\)\( T_{17}^{108} + \)\(17\!\cdots\!72\)\( T_{17}^{104} - \)\(10\!\cdots\!40\)\( T_{17}^{100} + \)\(46\!\cdots\!32\)\( T_{17}^{96} - \)\(16\!\cdots\!72\)\( T_{17}^{92} + \)\(48\!\cdots\!56\)\( T_{17}^{88} - \)\(11\!\cdots\!96\)\( T_{17}^{84} + \)\(20\!\cdots\!80\)\( T_{17}^{80} - \)\(31\!\cdots\!48\)\( T_{17}^{76} + \)\(38\!\cdots\!68\)\( T_{17}^{72} - \)\(37\!\cdots\!80\)\( T_{17}^{68} + \)\(30\!\cdots\!28\)\( T_{17}^{64} - \)\(19\!\cdots\!04\)\( T_{17}^{60} + \)\(99\!\cdots\!12\)\( T_{17}^{56} - \)\(40\!\cdots\!80\)\( T_{17}^{52} + \)\(12\!\cdots\!96\)\( T_{17}^{48} - \)\(31\!\cdots\!60\)\( T_{17}^{44} + \)\(59\!\cdots\!24\)\( T_{17}^{40} - \)\(77\!\cdots\!00\)\( T_{17}^{36} + \)\(70\!\cdots\!76\)\( T_{17}^{32} - \)\(35\!\cdots\!40\)\( T_{17}^{28} + \)\(12\!\cdots\!48\)\( T_{17}^{24} - \)\(14\!\cdots\!52\)\( T_{17}^{20} + \)\(12\!\cdots\!08\)\( T_{17}^{16} - \)\(45\!\cdots\!88\)\( T_{17}^{12} + \)\(15\!\cdots\!20\)\( T_{17}^{8} - \)\(22\!\cdots\!04\)\( T_{17}^{4} + \)\(32\!\cdots\!56\)\( \)">\(T_{17}^{144} - \cdots\)