Properties

Label 570.2.bh.b
Level $570$
Weight $2$
Character orbit 570.bh
Analytic conductor $4.551$
Analytic rank $0$
Dimension $120$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 570 = 2 \cdot 3 \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 570.bh (of order \(36\), degree \(12\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(4.55147291521\)
Analytic rank: \(0\)
Dimension: \(120\)
Relative dimension: \(10\) over \(\Q(\zeta_{36})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{36}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 120q + 12q^{5} + 12q^{7} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 120q + 12q^{5} + 12q^{7} + 12q^{10} - 48q^{13} - 12q^{17} + 60q^{21} + 12q^{22} + 48q^{23} + 12q^{25} + 12q^{26} - 12q^{30} + 24q^{33} - 24q^{38} - 36q^{41} + 24q^{43} + 96q^{47} + 24q^{52} - 36q^{55} + 12q^{57} - 24q^{58} - 12q^{60} - 24q^{61} + 24q^{62} - 24q^{66} + 72q^{67} + 12q^{68} + 96q^{70} + 36q^{73} - 12q^{76} - 24q^{78} - 12q^{80} - 24q^{82} - 60q^{83} - 36q^{85} - 72q^{86} + 12q^{87} - 144q^{91} - 12q^{92} + 48q^{93} + 156q^{95} - 120q^{96} - 216q^{97} - 96q^{98} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
13.1 −0.996195 0.0871557i 0.906308 0.422618i 0.984808 + 0.173648i −2.21625 0.297081i −0.939693 + 0.342020i 0.959162 + 3.57964i −0.965926 0.258819i 0.642788 0.766044i 2.18192 + 0.489109i
13.2 −0.996195 0.0871557i 0.906308 0.422618i 0.984808 + 0.173648i −1.80134 + 1.32483i −0.939693 + 0.342020i −0.863423 3.22234i −0.965926 0.258819i 0.642788 0.766044i 1.90995 1.16279i
13.3 −0.996195 0.0871557i 0.906308 0.422618i 0.984808 + 0.173648i 0.0670163 + 2.23506i −0.939693 + 0.342020i 0.772638 + 2.88353i −0.965926 0.258819i 0.642788 0.766044i 0.128037 2.23240i
13.4 −0.996195 0.0871557i 0.906308 0.422618i 0.984808 + 0.173648i 1.19781 + 1.88819i −0.939693 + 0.342020i −0.291238 1.08692i −0.965926 0.258819i 0.642788 0.766044i −1.02868 1.98540i
13.5 −0.996195 0.0871557i 0.906308 0.422618i 0.984808 + 0.173648i 2.12303 0.701946i −0.939693 + 0.342020i 0.232073 + 0.866107i −0.965926 0.258819i 0.642788 0.766044i −2.17613 + 0.514240i
13.6 0.996195 + 0.0871557i −0.906308 + 0.422618i 0.984808 + 0.173648i −2.00382 0.992325i −0.939693 + 0.342020i −0.441919 1.64926i 0.965926 + 0.258819i 0.642788 0.766044i −1.90971 1.16319i
13.7 0.996195 + 0.0871557i −0.906308 + 0.422618i 0.984808 + 0.173648i −1.60997 + 1.55177i −0.939693 + 0.342020i −0.201530 0.752119i 0.965926 + 0.258819i 0.642788 0.766044i −1.73909 + 1.40555i
13.8 0.996195 + 0.0871557i −0.906308 + 0.422618i 0.984808 + 0.173648i 1.47291 1.68242i −0.939693 + 0.342020i −0.298344 1.11343i 0.965926 + 0.258819i 0.642788 0.766044i 1.61394 1.54765i
13.9 0.996195 + 0.0871557i −0.906308 + 0.422618i 0.984808 + 0.173648i 1.74492 + 1.39830i −0.939693 + 0.342020i 0.637980 + 2.38097i 0.965926 + 0.258819i 0.642788 0.766044i 1.61641 + 1.54506i
13.10 0.996195 + 0.0871557i −0.906308 + 0.422618i 0.984808 + 0.173648i 2.02568 + 0.946899i −0.939693 + 0.342020i −1.23745 4.61822i 0.965926 + 0.258819i 0.642788 0.766044i 1.93544 + 1.11985i
67.1 −0.819152 0.573576i −0.996195 0.0871557i 0.342020 + 0.939693i −2.16408 0.562797i 0.766044 + 0.642788i −2.70522 + 0.724862i 0.258819 0.965926i 0.984808 + 0.173648i 1.44991 + 1.70228i
67.2 −0.819152 0.573576i −0.996195 0.0871557i 0.342020 + 0.939693i −1.51860 + 1.64130i 0.766044 + 0.642788i 0.131281 0.0351766i 0.258819 0.965926i 0.984808 + 0.173648i 2.18537 0.473443i
67.3 −0.819152 0.573576i −0.996195 0.0871557i 0.342020 + 0.939693i 0.430308 + 2.19427i 0.766044 + 0.642788i 0.0612713 0.0164176i 0.258819 0.965926i 0.984808 + 0.173648i 0.906096 2.04426i
67.4 −0.819152 0.573576i −0.996195 0.0871557i 0.342020 + 0.939693i 1.05649 1.97075i 0.766044 + 0.642788i 3.32682 0.891419i 0.258819 0.965926i 0.984808 + 0.173648i −1.99580 + 1.00836i
67.5 −0.819152 0.573576i −0.996195 0.0871557i 0.342020 + 0.939693i 2.07594 0.830958i 0.766044 + 0.642788i −3.91386 + 1.04872i 0.258819 0.965926i 0.984808 + 0.173648i −2.17712 0.510027i
67.6 0.819152 + 0.573576i 0.996195 + 0.0871557i 0.342020 + 0.939693i −2.05776 0.874999i 0.766044 + 0.642788i 1.82281 0.488422i −0.258819 + 0.965926i 0.984808 + 0.173648i −1.18374 1.89704i
67.7 0.819152 + 0.573576i 0.996195 + 0.0871557i 0.342020 + 0.939693i −0.137857 2.23181i 0.766044 + 0.642788i 3.36188 0.900812i −0.258819 + 0.965926i 0.984808 + 0.173648i 1.16719 1.90727i
67.8 0.819152 + 0.573576i 0.996195 + 0.0871557i 0.342020 + 0.939693i 0.285903 + 2.21771i 0.766044 + 0.642788i −1.89491 + 0.507740i −0.258819 + 0.965926i 0.984808 + 0.173648i −1.03783 + 1.98063i
67.9 0.819152 + 0.573576i 0.996195 + 0.0871557i 0.342020 + 0.939693i 1.11555 + 1.93792i 0.766044 + 0.642788i 3.25218 0.871419i −0.258819 + 0.965926i 0.984808 + 0.173648i −0.197741 + 2.22731i
67.10 0.819152 + 0.573576i 0.996195 + 0.0871557i 0.342020 + 0.939693i 1.91411 1.15593i 0.766044 + 0.642788i −0.710195 + 0.190296i −0.258819 + 0.965926i 0.984808 + 0.173648i 2.23096 + 0.151011i
See next 80 embeddings (of 120 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 553.10
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner
19.f odd 18 1 inner
95.r even 36 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 570.2.bh.b 120
5.c odd 4 1 inner 570.2.bh.b 120
19.f odd 18 1 inner 570.2.bh.b 120
95.r even 36 1 inner 570.2.bh.b 120
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
570.2.bh.b 120 1.a even 1 1 trivial
570.2.bh.b 120 5.c odd 4 1 inner
570.2.bh.b 120 19.f odd 18 1 inner
570.2.bh.b 120 95.r even 36 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(79\!\cdots\!88\)\( T_{7}^{102} - \)\(49\!\cdots\!32\)\( T_{7}^{101} + \)\(16\!\cdots\!37\)\( T_{7}^{100} - \)\(99\!\cdots\!48\)\( T_{7}^{99} - \)\(18\!\cdots\!60\)\( T_{7}^{98} + \)\(12\!\cdots\!16\)\( T_{7}^{97} - \)\(46\!\cdots\!06\)\( T_{7}^{96} + \)\(53\!\cdots\!92\)\( T_{7}^{95} + \)\(31\!\cdots\!44\)\( T_{7}^{94} - \)\(25\!\cdots\!80\)\( T_{7}^{93} + \)\(97\!\cdots\!34\)\( T_{7}^{92} - \)\(15\!\cdots\!84\)\( T_{7}^{91} - \)\(36\!\cdots\!84\)\( T_{7}^{90} + \)\(38\!\cdots\!56\)\( T_{7}^{89} - \)\(16\!\cdots\!88\)\( T_{7}^{88} + \)\(30\!\cdots\!20\)\( T_{7}^{87} + \)\(23\!\cdots\!16\)\( T_{7}^{86} - \)\(46\!\cdots\!36\)\( T_{7}^{85} + \)\(20\!\cdots\!31\)\( T_{7}^{84} - \)\(45\!\cdots\!12\)\( T_{7}^{83} + \)\(82\!\cdots\!08\)\( T_{7}^{82} + \)\(41\!\cdots\!80\)\( T_{7}^{81} - \)\(20\!\cdots\!48\)\( T_{7}^{80} + \)\(51\!\cdots\!32\)\( T_{7}^{79} - \)\(39\!\cdots\!64\)\( T_{7}^{78} - \)\(28\!\cdots\!88\)\( T_{7}^{77} + \)\(16\!\cdots\!74\)\( T_{7}^{76} - \)\(44\!\cdots\!48\)\( T_{7}^{75} + \)\(54\!\cdots\!44\)\( T_{7}^{74} + \)\(12\!\cdots\!88\)\( T_{7}^{73} - \)\(99\!\cdots\!35\)\( T_{7}^{72} + \)\(29\!\cdots\!12\)\( T_{7}^{71} - \)\(47\!\cdots\!48\)\( T_{7}^{70} - \)\(24\!\cdots\!60\)\( T_{7}^{69} + \)\(44\!\cdots\!57\)\( T_{7}^{68} - \)\(15\!\cdots\!68\)\( T_{7}^{67} + \)\(29\!\cdots\!76\)\( T_{7}^{66} - \)\(14\!\cdots\!80\)\( T_{7}^{65} - \)\(13\!\cdots\!49\)\( T_{7}^{64} + \)\(58\!\cdots\!48\)\( T_{7}^{63} - \)\(13\!\cdots\!16\)\( T_{7}^{62} + \)\(14\!\cdots\!48\)\( T_{7}^{61} + \)\(22\!\cdots\!38\)\( T_{7}^{60} - \)\(15\!\cdots\!68\)\( T_{7}^{59} + \)\(43\!\cdots\!28\)\( T_{7}^{58} - \)\(71\!\cdots\!60\)\( T_{7}^{57} + \)\(29\!\cdots\!51\)\( T_{7}^{56} + \)\(24\!\cdots\!92\)\( T_{7}^{55} - \)\(91\!\cdots\!36\)\( T_{7}^{54} + \)\(19\!\cdots\!00\)\( T_{7}^{53} - \)\(24\!\cdots\!06\)\( T_{7}^{52} - \)\(57\!\cdots\!08\)\( T_{7}^{51} + \)\(11\!\cdots\!08\)\( T_{7}^{50} - \)\(34\!\cdots\!44\)\( T_{7}^{49} + \)\(66\!\cdots\!63\)\( T_{7}^{48} - \)\(80\!\cdots\!60\)\( T_{7}^{47} + \)\(21\!\cdots\!60\)\( T_{7}^{46} + \)\(18\!\cdots\!60\)\( T_{7}^{45} - \)\(60\!\cdots\!86\)\( T_{7}^{44} + \)\(10\!\cdots\!56\)\( T_{7}^{43} - \)\(12\!\cdots\!24\)\( T_{7}^{42} + \)\(18\!\cdots\!96\)\( T_{7}^{41} + \)\(33\!\cdots\!25\)\( T_{7}^{40} - \)\(10\!\cdots\!88\)\( T_{7}^{39} + \)\(21\!\cdots\!84\)\( T_{7}^{38} - \)\(34\!\cdots\!92\)\( T_{7}^{37} + \)\(46\!\cdots\!93\)\( T_{7}^{36} - \)\(52\!\cdots\!44\)\( T_{7}^{35} + \)\(48\!\cdots\!00\)\( T_{7}^{34} - \)\(33\!\cdots\!52\)\( T_{7}^{33} + \)\(10\!\cdots\!37\)\( T_{7}^{32} + \)\(12\!\cdots\!68\)\( T_{7}^{31} - \)\(27\!\cdots\!28\)\( T_{7}^{30} + \)\(29\!\cdots\!52\)\( T_{7}^{29} - \)\(17\!\cdots\!00\)\( T_{7}^{28} - \)\(36\!\cdots\!24\)\( T_{7}^{27} + \)\(21\!\cdots\!80\)\( T_{7}^{26} - \)\(31\!\cdots\!76\)\( T_{7}^{25} + \)\(29\!\cdots\!71\)\( T_{7}^{24} - \)\(17\!\cdots\!72\)\( T_{7}^{23} + \)\(46\!\cdots\!60\)\( T_{7}^{22} + \)\(46\!\cdots\!32\)\( T_{7}^{21} - \)\(79\!\cdots\!12\)\( T_{7}^{20} + \)\(56\!\cdots\!64\)\( T_{7}^{19} - \)\(19\!\cdots\!08\)\( T_{7}^{18} - \)\(80\!\cdots\!24\)\( T_{7}^{17} + \)\(20\!\cdots\!49\)\( T_{7}^{16} - \)\(16\!\cdots\!60\)\( T_{7}^{15} + \)\(89\!\cdots\!04\)\( T_{7}^{14} - \)\(28\!\cdots\!56\)\( T_{7}^{13} - \)\(21\!\cdots\!52\)\( T_{7}^{12} + \)\(50\!\cdots\!20\)\( T_{7}^{11} - \)\(12\!\cdots\!56\)\( T_{7}^{10} + \)\(15\!\cdots\!36\)\( T_{7}^{9} + \)\(12\!\cdots\!75\)\( T_{7}^{8} - \)\(38\!\cdots\!76\)\( T_{7}^{7} + \)\(57\!\cdots\!04\)\( T_{7}^{6} - \)\(93\!\cdots\!84\)\( T_{7}^{5} + \)\(12\!\cdots\!43\)\( T_{7}^{4} - \)\(80\!\cdots\!96\)\( T_{7}^{3} + \)\(43\!\cdots\!92\)\( T_{7}^{2} - \)\(26\!\cdots\!28\)\( T_{7} + \)\(80\!\cdots\!01\)\( \)">\(T_{7}^{120} - \cdots\) acting on \(S_{2}^{\mathrm{new}}(570, [\chi])\).