Properties

Label 570.2.bb.b
Level $570$
Weight $2$
Character orbit 570.bb
Analytic conductor $4.551$
Analytic rank $0$
Dimension $84$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 570 = 2 \cdot 3 \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 570.bb (of order \(18\), degree \(6\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(4.55147291521\)
Analytic rank: \(0\)
Dimension: \(84\)
Relative dimension: \(14\) over \(\Q(\zeta_{18})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{18}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 84q - 6q^{6} + 42q^{8} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 84q - 6q^{6} + 42q^{8} + 24q^{13} - 24q^{14} + 12q^{17} - 12q^{19} + 36q^{22} + 24q^{27} - 12q^{28} - 12q^{29} + 36q^{33} + 12q^{34} + 6q^{36} + 18q^{38} + 12q^{39} + 6q^{41} + 24q^{43} + 36q^{44} + 12q^{47} - 12q^{48} - 54q^{49} - 42q^{50} + 96q^{51} + 12q^{52} - 60q^{53} - 18q^{54} - 96q^{57} - 24q^{58} - 18q^{59} - 48q^{61} - 12q^{62} - 114q^{63} - 42q^{64} - 24q^{66} + 6q^{67} - 54q^{68} - 48q^{69} + 48q^{71} + 84q^{73} + 24q^{74} - 12q^{79} - 36q^{81} - 6q^{82} + 36q^{83} + 18q^{84} + 12q^{86} + 6q^{87} - 12q^{89} - 24q^{90} + 24q^{91} - 6q^{93} - 12q^{95} - 42q^{97} + 36q^{98} - 6q^{99} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
41.1 −0.766044 0.642788i −1.71584 0.236392i 0.173648 + 0.984808i −0.984808 0.173648i 1.16246 + 1.28401i −1.76528 3.05755i 0.500000 0.866025i 2.88824 + 0.811224i 0.642788 + 0.766044i
41.2 −0.766044 0.642788i −1.50714 0.853538i 0.173648 + 0.984808i 0.984808 + 0.173648i 0.605893 + 1.62262i 1.94656 + 3.37154i 0.500000 0.866025i 1.54295 + 2.57280i −0.642788 0.766044i
41.3 −0.766044 0.642788i −1.15586 1.28996i 0.173648 + 0.984808i 0.984808 + 0.173648i 0.0562681 + 1.73114i −0.561471 0.972497i 0.500000 0.866025i −0.327989 + 2.98202i −0.642788 0.766044i
41.4 −0.766044 0.642788i −1.07847 + 1.35533i 0.173648 + 0.984808i −0.984808 0.173648i 1.69734 0.345017i −0.267952 0.464106i 0.500000 0.866025i −0.673825 2.92335i 0.642788 + 0.766044i
41.5 −0.766044 0.642788i −1.01882 + 1.40072i 0.173648 + 0.984808i 0.984808 + 0.173648i 1.68082 0.418131i 0.0247885 + 0.0429350i 0.500000 0.866025i −0.924029 2.85415i −0.642788 0.766044i
41.6 −0.766044 0.642788i −0.169275 1.72376i 0.173648 + 0.984808i −0.984808 0.173648i −0.978339 + 1.42928i 1.36690 + 2.36755i 0.500000 0.866025i −2.94269 + 0.583578i 0.642788 + 0.766044i
41.7 −0.766044 0.642788i 0.0507095 + 1.73131i 0.173648 + 0.984808i 0.984808 + 0.173648i 1.07402 1.35885i 0.847833 + 1.46849i 0.500000 0.866025i −2.99486 + 0.175587i −0.642788 0.766044i
41.8 −0.766044 0.642788i 0.446223 + 1.67358i 0.173648 + 0.984808i −0.984808 0.173648i 0.733933 1.56887i 2.13041 + 3.68998i 0.500000 0.866025i −2.60177 + 1.49358i 0.642788 + 0.766044i
41.9 −0.766044 0.642788i 0.902121 1.47857i 0.173648 + 0.984808i −0.984808 0.173648i −1.64147 + 0.552780i −1.68197 2.91326i 0.500000 0.866025i −1.37235 2.66770i 0.642788 + 0.766044i
41.10 −0.766044 0.642788i 0.964794 + 1.43846i 0.173648 + 0.984808i −0.984808 0.173648i 0.185551 1.72208i −1.20034 2.07905i 0.500000 0.866025i −1.13835 + 2.77564i 0.642788 + 0.766044i
41.11 −0.766044 0.642788i 1.46696 0.920884i 0.173648 + 0.984808i 0.984808 + 0.173648i −1.71569 0.237506i −2.15391 3.73068i 0.500000 0.866025i 1.30395 2.70180i −0.642788 0.766044i
41.12 −0.766044 0.642788i 1.48528 + 0.891036i 0.173648 + 0.984808i 0.984808 + 0.173648i −0.565043 1.63729i −2.04901 3.54899i 0.500000 0.866025i 1.41211 + 2.64687i −0.642788 0.766044i
41.13 −0.766044 0.642788i 1.59014 0.686630i 0.173648 + 0.984808i −0.984808 0.173648i −1.65947 0.496132i 1.07093 + 1.85491i 0.500000 0.866025i 2.05708 2.18367i 0.642788 + 0.766044i
41.14 −0.766044 0.642788i 1.61856 0.616663i 0.173648 + 0.984808i 0.984808 + 0.173648i −1.63627 0.567997i 1.59792 + 2.76767i 0.500000 0.866025i 2.23945 1.99621i −0.642788 0.766044i
71.1 0.939693 0.342020i −1.66038 0.493094i 0.766044 0.642788i 0.642788 0.766044i −1.72889 + 0.104526i 0.222334 + 0.385093i 0.500000 0.866025i 2.51372 + 1.63745i 0.342020 0.939693i
71.2 0.939693 0.342020i −1.63144 + 0.581714i 0.766044 0.642788i −0.642788 + 0.766044i −1.33410 + 1.10462i −0.223800 0.387633i 0.500000 0.866025i 2.32322 1.89807i −0.342020 + 0.939693i
71.3 0.939693 0.342020i −1.49197 0.879786i 0.766044 0.642788i −0.642788 + 0.766044i −1.70290 0.316445i −1.40312 2.43028i 0.500000 0.866025i 1.45195 + 2.62523i −0.342020 + 0.939693i
71.4 0.939693 0.342020i −1.12629 1.31585i 0.766044 0.642788i −0.642788 + 0.766044i −1.50842 0.851282i 2.21271 + 3.83252i 0.500000 0.866025i −0.462935 + 2.96407i −0.342020 + 0.939693i
71.5 0.939693 0.342020i −0.950784 + 1.44776i 0.766044 0.642788i 0.642788 0.766044i −0.398282 + 1.68564i −2.26841 3.92899i 0.500000 0.866025i −1.19202 2.75301i 0.342020 0.939693i
71.6 0.939693 0.342020i −0.936498 + 1.45704i 0.766044 0.642788i 0.642788 0.766044i −0.381683 + 1.68947i 0.199417 + 0.345401i 0.500000 0.866025i −1.24594 2.72903i 0.342020 0.939693i
See all 84 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 431.14
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
57.j even 18 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 570.2.bb.b yes 84
3.b odd 2 1 570.2.bb.a 84
19.f odd 18 1 570.2.bb.a 84
57.j even 18 1 inner 570.2.bb.b yes 84
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
570.2.bb.a 84 3.b odd 2 1
570.2.bb.a 84 19.f odd 18 1
570.2.bb.b yes 84 1.a even 1 1 trivial
570.2.bb.b yes 84 57.j even 18 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(37\!\cdots\!79\)\( T_{11}^{70} - \)\(32\!\cdots\!96\)\( T_{11}^{69} + \)\(13\!\cdots\!38\)\( T_{11}^{68} + \)\(14\!\cdots\!78\)\( T_{11}^{67} - \)\(41\!\cdots\!89\)\( T_{11}^{66} - \)\(54\!\cdots\!78\)\( T_{11}^{65} + \)\(11\!\cdots\!14\)\( T_{11}^{64} + \)\(17\!\cdots\!92\)\( T_{11}^{63} - \)\(25\!\cdots\!80\)\( T_{11}^{62} - \)\(46\!\cdots\!52\)\( T_{11}^{61} + \)\(52\!\cdots\!57\)\( T_{11}^{60} + \)\(10\!\cdots\!18\)\( T_{11}^{59} - \)\(92\!\cdots\!84\)\( T_{11}^{58} - \)\(22\!\cdots\!02\)\( T_{11}^{57} + \)\(14\!\cdots\!27\)\( T_{11}^{56} + \)\(39\!\cdots\!82\)\( T_{11}^{55} - \)\(18\!\cdots\!41\)\( T_{11}^{54} - \)\(60\!\cdots\!86\)\( T_{11}^{53} + \)\(21\!\cdots\!77\)\( T_{11}^{52} + \)\(81\!\cdots\!28\)\( T_{11}^{51} - \)\(21\!\cdots\!66\)\( T_{11}^{50} - \)\(95\!\cdots\!20\)\( T_{11}^{49} + \)\(16\!\cdots\!53\)\( T_{11}^{48} + \)\(98\!\cdots\!80\)\( T_{11}^{47} - \)\(10\!\cdots\!02\)\( T_{11}^{46} - \)\(86\!\cdots\!96\)\( T_{11}^{45} + \)\(43\!\cdots\!02\)\( T_{11}^{44} + \)\(66\!\cdots\!82\)\( T_{11}^{43} + \)\(10\!\cdots\!16\)\( T_{11}^{42} - \)\(43\!\cdots\!86\)\( T_{11}^{41} - \)\(21\!\cdots\!99\)\( T_{11}^{40} + \)\(24\!\cdots\!60\)\( T_{11}^{39} + \)\(23\!\cdots\!81\)\( T_{11}^{38} - \)\(11\!\cdots\!08\)\( T_{11}^{37} - \)\(16\!\cdots\!90\)\( T_{11}^{36} + \)\(40\!\cdots\!56\)\( T_{11}^{35} + \)\(91\!\cdots\!51\)\( T_{11}^{34} - \)\(11\!\cdots\!00\)\( T_{11}^{33} - \)\(39\!\cdots\!14\)\( T_{11}^{32} + \)\(17\!\cdots\!36\)\( T_{11}^{31} + \)\(13\!\cdots\!77\)\( T_{11}^{30} + \)\(20\!\cdots\!80\)\( T_{11}^{29} - \)\(35\!\cdots\!72\)\( T_{11}^{28} - \)\(24\!\cdots\!80\)\( T_{11}^{27} + \)\(69\!\cdots\!15\)\( T_{11}^{26} + \)\(88\!\cdots\!48\)\( T_{11}^{25} - \)\(88\!\cdots\!15\)\( T_{11}^{24} - \)\(20\!\cdots\!76\)\( T_{11}^{23} + \)\(37\!\cdots\!09\)\( T_{11}^{22} + \)\(33\!\cdots\!72\)\( T_{11}^{21} + \)\(12\!\cdots\!28\)\( T_{11}^{20} - \)\(33\!\cdots\!10\)\( T_{11}^{19} - \)\(29\!\cdots\!71\)\( T_{11}^{18} + \)\(20\!\cdots\!86\)\( T_{11}^{17} + \)\(35\!\cdots\!45\)\( T_{11}^{16} + \)\(90\!\cdots\!32\)\( T_{11}^{15} - \)\(23\!\cdots\!38\)\( T_{11}^{14} - \)\(86\!\cdots\!52\)\( T_{11}^{13} + \)\(97\!\cdots\!87\)\( T_{11}^{12} + \)\(77\!\cdots\!68\)\( T_{11}^{11} - \)\(11\!\cdots\!30\)\( T_{11}^{10} - \)\(28\!\cdots\!44\)\( T_{11}^{9} - \)\(34\!\cdots\!27\)\( T_{11}^{8} + \)\(72\!\cdots\!98\)\( T_{11}^{7} + \)\(30\!\cdots\!04\)\( T_{11}^{6} - \)\(34\!\cdots\!70\)\( T_{11}^{5} - \)\(41\!\cdots\!21\)\( T_{11}^{4} - \)\(11\!\cdots\!02\)\( T_{11}^{3} + \)\(47\!\cdots\!83\)\( T_{11}^{2} + \)\(13\!\cdots\!02\)\( T_{11} + \)\(12\!\cdots\!49\)\( \)">\(T_{11}^{84} - \cdots\) acting on \(S_{2}^{\mathrm{new}}(570, [\chi])\).