Properties

Label 570.2.bh.a
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.

\(\operatorname{Tr}(f)(q) = \) \( 120q - 12q^{5} + 12q^{7} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 120q - 12q^{5} + 12q^{7} - 12q^{10} + 48q^{13} + 12q^{17} + 36q^{21} - 36q^{22} - 96q^{23} + 12q^{25} + 12q^{26} + 12q^{30} + 24q^{38} + 60q^{41} + 96q^{43} - 48q^{47} - 24q^{52} - 72q^{53} + 108q^{55} + 12q^{57} + 24q^{58} - 12q^{60} - 24q^{61} + 24q^{62} + 24q^{66} + 72q^{67} + 12q^{68} - 48q^{70} + 36q^{73} - 12q^{76} - 24q^{78} + 12q^{80} - 72q^{82} + 12q^{83} - 108q^{85} - 24q^{86} + 12q^{87} - 48q^{91} - 84q^{92} - 48q^{93} - 204q^{95} + 120q^{96} + 24q^{97} - 96q^{98} + O(q^{100}) \)

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 −1.79770 1.32977i 0.939693 0.342020i −1.12737 4.20740i −0.965926 0.258819i 0.642788 0.766044i 1.67496 + 1.48138i
13.2 −0.996195 0.0871557i −0.906308 + 0.422618i 0.984808 + 0.173648i −0.538816 2.17018i 0.939693 0.342020i 0.793037 + 2.95965i −0.965926 0.258819i 0.642788 0.766044i 0.347622 + 2.20888i
13.3 −0.996195 0.0871557i −0.906308 + 0.422618i 0.984808 + 0.173648i −0.393795 + 2.20112i 0.939693 0.342020i 0.524562 + 1.95769i −0.965926 0.258819i 0.642788 0.766044i 0.584136 2.15842i
13.4 −0.996195 0.0871557i −0.906308 + 0.422618i 0.984808 + 0.173648i 1.51913 1.64081i 0.939693 0.342020i 0.148294 + 0.553442i −0.965926 0.258819i 0.642788 0.766044i −1.65636 + 1.50216i
13.5 −0.996195 0.0871557i −0.906308 + 0.422618i 0.984808 + 0.173648i 2.18820 + 0.460190i 0.939693 0.342020i −0.860238 3.21045i −0.965926 0.258819i 0.642788 0.766044i −2.13977 0.649154i
13.6 0.996195 + 0.0871557i 0.906308 0.422618i 0.984808 + 0.173648i −2.19145 + 0.444477i 0.939693 0.342020i −1.33986 5.00042i 0.965926 + 0.258819i 0.642788 0.766044i −2.22185 + 0.251788i
13.7 0.996195 + 0.0871557i 0.906308 0.422618i 0.984808 + 0.173648i −2.02193 + 0.954879i 0.939693 0.342020i 0.864180 + 3.22516i 0.965926 + 0.258819i 0.642788 0.766044i −2.09746 + 0.775022i
13.8 0.996195 + 0.0871557i 0.906308 0.422618i 0.984808 + 0.173648i −0.261022 2.22078i 0.939693 0.342020i 0.151755 + 0.566356i 0.965926 + 0.258819i 0.642788 0.766044i −0.0664750 2.23508i
13.9 0.996195 + 0.0871557i 0.906308 0.422618i 0.984808 + 0.173648i 0.998061 + 2.00097i 0.939693 0.342020i 0.394140 + 1.47095i 0.965926 + 0.258819i 0.642788 0.766044i 0.819868 + 2.08034i
13.10 0.996195 + 0.0871557i 0.906308 0.422618i 0.984808 + 0.173648i 2.19391 0.432154i 0.939693 0.342020i −0.280552 1.04704i 0.965926 + 0.258819i 0.642788 0.766044i 2.22323 0.239298i
67.1 −0.819152 0.573576i 0.996195 + 0.0871557i 0.342020 + 0.939693i −2.23514 0.0643515i −0.766044 0.642788i 2.48134 0.664873i 0.258819 0.965926i 0.984808 + 0.173648i 1.79401 + 1.33474i
67.2 −0.819152 0.573576i 0.996195 + 0.0871557i 0.342020 + 0.939693i −1.90079 1.17771i −0.766044 0.642788i −2.42358 + 0.649396i 0.258819 0.965926i 0.984808 + 0.173648i 0.881533 + 2.05497i
67.3 −0.819152 0.573576i 0.996195 + 0.0871557i 0.342020 + 0.939693i −0.0386474 + 2.23573i −0.766044 0.642788i 2.25738 0.604862i 0.258819 0.965926i 0.984808 + 0.173648i 1.31402 1.80924i
67.4 −0.819152 0.573576i 0.996195 + 0.0871557i 0.342020 + 0.939693i 0.180509 2.22877i −0.766044 0.642788i −2.19282 + 0.587565i 0.258819 0.965926i 0.984808 + 0.173648i −1.42623 + 1.72217i
67.5 −0.819152 0.573576i 0.996195 + 0.0871557i 0.342020 + 0.939693i 2.23464 + 0.0799806i −0.766044 0.642788i 4.38798 1.17576i 0.258819 0.965926i 0.984808 + 0.173648i −1.78463 1.34725i
67.6 0.819152 + 0.573576i −0.996195 0.0871557i 0.342020 + 0.939693i −2.19503 + 0.426412i −0.766044 0.642788i 2.31403 0.620043i −0.258819 + 0.965926i 0.984808 + 0.173648i −2.04265 0.909723i
67.7 0.819152 + 0.573576i −0.996195 0.0871557i 0.342020 + 0.939693i −1.42376 + 1.72421i −0.766044 0.642788i −2.79653 + 0.749327i −0.258819 + 0.965926i 0.984808 + 0.173648i −2.15524 + 0.595756i
67.8 0.819152 + 0.573576i −0.996195 0.0871557i 0.342020 + 0.939693i −0.598083 2.15460i −0.766044 0.642788i −0.0495570 + 0.0132788i −0.258819 + 0.965926i 0.984808 + 0.173648i 0.745906 2.10799i
67.9 0.819152 + 0.573576i −0.996195 0.0871557i 0.342020 + 0.939693i −0.299221 2.21596i −0.766044 0.642788i −3.99996 + 1.07179i −0.258819 + 0.965926i 0.984808 + 0.173648i 1.02591 1.98683i
67.10 0.819152 + 0.573576i −0.996195 0.0871557i 0.342020 + 0.939693i 1.51676 + 1.64300i −0.766044 0.642788i 2.75377 0.737870i −0.258819 + 0.965926i 0.984808 + 0.173648i 0.300075 + 2.21584i
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.a 120
5.c odd 4 1 inner 570.2.bh.a 120
19.f odd 18 1 inner 570.2.bh.a 120
95.r even 36 1 inner 570.2.bh.a 120
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
570.2.bh.a 120 1.a even 1 1 trivial
570.2.bh.a 120 5.c odd 4 1 inner
570.2.bh.a 120 19.f odd 18 1 inner
570.2.bh.a 120 95.r even 36 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(28\!\cdots\!40\)\( T_{7}^{103} - \)\(52\!\cdots\!28\)\( T_{7}^{102} + \)\(47\!\cdots\!08\)\( T_{7}^{101} + \)\(21\!\cdots\!81\)\( T_{7}^{100} - \)\(10\!\cdots\!12\)\( T_{7}^{99} + \)\(12\!\cdots\!16\)\( T_{7}^{98} - \)\(17\!\cdots\!44\)\( T_{7}^{97} - \)\(45\!\cdots\!26\)\( T_{7}^{96} + \)\(29\!\cdots\!84\)\( T_{7}^{95} - \)\(16\!\cdots\!84\)\( T_{7}^{94} + \)\(46\!\cdots\!12\)\( T_{7}^{93} + \)\(84\!\cdots\!98\)\( T_{7}^{92} - \)\(66\!\cdots\!28\)\( T_{7}^{91} + \)\(30\!\cdots\!56\)\( T_{7}^{90} - \)\(95\!\cdots\!88\)\( T_{7}^{89} - \)\(14\!\cdots\!00\)\( T_{7}^{88} + \)\(12\!\cdots\!72\)\( T_{7}^{87} + \)\(48\!\cdots\!00\)\( T_{7}^{86} + \)\(15\!\cdots\!92\)\( T_{7}^{85} + \)\(23\!\cdots\!31\)\( T_{7}^{84} - \)\(18\!\cdots\!36\)\( T_{7}^{83} - \)\(14\!\cdots\!48\)\( T_{7}^{82} - \)\(21\!\cdots\!08\)\( T_{7}^{81} - \)\(36\!\cdots\!24\)\( T_{7}^{80} + \)\(21\!\cdots\!68\)\( T_{7}^{79} + \)\(24\!\cdots\!68\)\( T_{7}^{78} + \)\(24\!\cdots\!80\)\( T_{7}^{77} + \)\(49\!\cdots\!54\)\( T_{7}^{76} - \)\(20\!\cdots\!16\)\( T_{7}^{75} - \)\(31\!\cdots\!68\)\( T_{7}^{74} - \)\(23\!\cdots\!76\)\( T_{7}^{73} - \)\(55\!\cdots\!87\)\( T_{7}^{72} + \)\(15\!\cdots\!96\)\( T_{7}^{71} + \)\(29\!\cdots\!72\)\( T_{7}^{70} + \)\(18\!\cdots\!84\)\( T_{7}^{69} + \)\(50\!\cdots\!77\)\( T_{7}^{68} - \)\(82\!\cdots\!00\)\( T_{7}^{67} - \)\(21\!\cdots\!56\)\( T_{7}^{66} - \)\(12\!\cdots\!64\)\( T_{7}^{65} - \)\(37\!\cdots\!57\)\( T_{7}^{64} + \)\(30\!\cdots\!28\)\( T_{7}^{63} + \)\(12\!\cdots\!16\)\( T_{7}^{62} + \)\(66\!\cdots\!12\)\( T_{7}^{61} + \)\(22\!\cdots\!18\)\( T_{7}^{60} - \)\(34\!\cdots\!88\)\( T_{7}^{59} - \)\(49\!\cdots\!52\)\( T_{7}^{58} - \)\(28\!\cdots\!52\)\( T_{7}^{57} - \)\(10\!\cdots\!41\)\( T_{7}^{56} - \)\(38\!\cdots\!36\)\( T_{7}^{55} + \)\(14\!\cdots\!52\)\( T_{7}^{54} + \)\(98\!\cdots\!36\)\( T_{7}^{53} + \)\(42\!\cdots\!78\)\( T_{7}^{52} + \)\(32\!\cdots\!60\)\( T_{7}^{51} - \)\(21\!\cdots\!76\)\( T_{7}^{50} - \)\(25\!\cdots\!36\)\( T_{7}^{49} - \)\(12\!\cdots\!41\)\( T_{7}^{48} - \)\(14\!\cdots\!20\)\( T_{7}^{47} - \)\(29\!\cdots\!80\)\( T_{7}^{46} + \)\(46\!\cdots\!64\)\( T_{7}^{45} + \)\(30\!\cdots\!14\)\( T_{7}^{44} + \)\(44\!\cdots\!08\)\( T_{7}^{43} + \)\(29\!\cdots\!32\)\( T_{7}^{42} - \)\(46\!\cdots\!64\)\( T_{7}^{41} - \)\(53\!\cdots\!43\)\( T_{7}^{40} - \)\(97\!\cdots\!16\)\( T_{7}^{39} - \)\(91\!\cdots\!16\)\( T_{7}^{38} - \)\(20\!\cdots\!20\)\( T_{7}^{37} + \)\(67\!\cdots\!53\)\( T_{7}^{36} + \)\(15\!\cdots\!40\)\( T_{7}^{35} + \)\(18\!\cdots\!40\)\( T_{7}^{34} + \)\(17\!\cdots\!32\)\( T_{7}^{33} - \)\(46\!\cdots\!99\)\( T_{7}^{32} - \)\(15\!\cdots\!68\)\( T_{7}^{31} - \)\(21\!\cdots\!04\)\( T_{7}^{30} - \)\(29\!\cdots\!72\)\( T_{7}^{29} + \)\(69\!\cdots\!64\)\( T_{7}^{28} + \)\(93\!\cdots\!84\)\( T_{7}^{27} + \)\(16\!\cdots\!60\)\( T_{7}^{26} + \)\(26\!\cdots\!32\)\( T_{7}^{25} + \)\(25\!\cdots\!43\)\( T_{7}^{24} - \)\(13\!\cdots\!76\)\( T_{7}^{23} - \)\(22\!\cdots\!72\)\( T_{7}^{22} - \)\(48\!\cdots\!52\)\( T_{7}^{21} - \)\(57\!\cdots\!32\)\( T_{7}^{20} - \)\(10\!\cdots\!64\)\( T_{7}^{19} + \)\(32\!\cdots\!84\)\( T_{7}^{18} + \)\(77\!\cdots\!88\)\( T_{7}^{17} + \)\(87\!\cdots\!65\)\( T_{7}^{16} + \)\(23\!\cdots\!40\)\( T_{7}^{15} - \)\(17\!\cdots\!84\)\( T_{7}^{14} - \)\(38\!\cdots\!88\)\( T_{7}^{13} - \)\(34\!\cdots\!00\)\( T_{7}^{12} - \)\(76\!\cdots\!04\)\( T_{7}^{11} + \)\(42\!\cdots\!68\)\( T_{7}^{10} + \)\(95\!\cdots\!20\)\( T_{7}^{9} + \)\(11\!\cdots\!27\)\( T_{7}^{8} + \)\(69\!\cdots\!68\)\( T_{7}^{7} + \)\(31\!\cdots\!88\)\( T_{7}^{6} + \)\(14\!\cdots\!20\)\( T_{7}^{5} + \)\(40\!\cdots\!43\)\( T_{7}^{4} + \)\(25\!\cdots\!04\)\( T_{7}^{3} + \)\(10\!\cdots\!00\)\( T_{7}^{2} + \)\(66\!\cdots\!80\)\( T_{7} + \)\(20\!\cdots\!41\)\( \)">\(T_{7}^{120} - \cdots\) acting on \(S_{2}^{\mathrm{new}}(570, [\chi])\).