Properties

Label 570.2.bb.a
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.

\(\operatorname{Tr}(f)(q) = \) \( 84q - 6q^{3} - 42q^{8} + 6q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 84q - 6q^{3} - 42q^{8} + 6q^{9} + 24q^{13} + 24q^{14} - 12q^{17} - 12q^{19} + 36q^{22} - 6q^{24} - 6q^{27} - 12q^{28} + 12q^{29} - 6q^{33} + 12q^{34} - 18q^{38} + 12q^{39} - 6q^{41} + 24q^{43} - 36q^{44} - 12q^{47} - 54q^{49} + 42q^{50} - 54q^{51} + 12q^{52} + 60q^{53} - 54q^{54} - 60q^{57} - 24q^{58} + 18q^{59} - 48q^{61} + 12q^{62} + 18q^{63} - 42q^{64} + 54q^{66} + 6q^{67} + 54q^{68} - 60q^{69} - 48q^{71} - 12q^{72} + 84q^{73} - 24q^{74} + 36q^{78} - 12q^{79} + 114q^{81} - 6q^{82} - 36q^{83} - 18q^{84} - 12q^{86} + 6q^{87} + 12q^{89} + 24q^{91} + 6q^{93} + 12q^{95} - 42q^{97} - 36q^{98} + 102q^{99} + O(q^{100}) \)

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 −1.71569 + 0.237506i 0.173648 + 0.984808i −0.984808 0.173648i −1.46696 0.920884i −2.15391 3.73068i −0.500000 + 0.866025i 2.88718 0.814973i −0.642788 0.766044i
41.2 0.766044 + 0.642788i −1.65947 + 0.496132i 0.173648 + 0.984808i 0.984808 + 0.173648i −1.59014 0.686630i 1.07093 + 1.85491i −0.500000 + 0.866025i 2.50771 1.64664i 0.642788 + 0.766044i
41.3 0.766044 + 0.642788i −1.64147 0.552780i 0.173648 + 0.984808i 0.984808 + 0.173648i −0.902121 1.47857i −1.68197 2.91326i −0.500000 + 0.866025i 2.38887 + 1.81475i 0.642788 + 0.766044i
41.4 0.766044 + 0.642788i −1.63627 + 0.567997i 0.173648 + 0.984808i −0.984808 0.173648i −1.61856 0.616663i 1.59792 + 2.76767i −0.500000 + 0.866025i 2.35476 1.85879i −0.642788 0.766044i
41.5 0.766044 + 0.642788i −0.978339 1.42928i 0.173648 + 0.984808i 0.984808 + 0.173648i 0.169275 1.72376i 1.36690 + 2.36755i −0.500000 + 0.866025i −1.08571 + 2.79665i 0.642788 + 0.766044i
41.6 0.766044 + 0.642788i −0.565043 + 1.63729i 0.173648 + 0.984808i −0.984808 0.173648i −1.48528 + 0.891036i −2.04901 3.54899i −0.500000 + 0.866025i −2.36145 1.85028i −0.642788 0.766044i
41.7 0.766044 + 0.642788i 0.0562681 1.73114i 0.173648 + 0.984808i −0.984808 0.173648i 1.15586 1.28996i −0.561471 0.972497i −0.500000 + 0.866025i −2.99367 0.194816i −0.642788 0.766044i
41.8 0.766044 + 0.642788i 0.185551 + 1.72208i 0.173648 + 0.984808i 0.984808 + 0.173648i −0.964794 + 1.43846i −1.20034 2.07905i −0.500000 + 0.866025i −2.93114 + 0.639068i 0.642788 + 0.766044i
41.9 0.766044 + 0.642788i 0.605893 1.62262i 0.173648 + 0.984808i −0.984808 0.173648i 1.50714 0.853538i 1.94656 + 3.37154i −0.500000 + 0.866025i −2.26579 1.96627i −0.642788 0.766044i
41.10 0.766044 + 0.642788i 0.733933 + 1.56887i 0.173648 + 0.984808i 0.984808 + 0.173648i −0.446223 + 1.67358i 2.13041 + 3.68998i −0.500000 + 0.866025i −1.92269 + 2.30289i 0.642788 + 0.766044i
41.11 0.766044 + 0.642788i 1.07402 + 1.35885i 0.173648 + 0.984808i −0.984808 0.173648i −0.0507095 + 1.73131i 0.847833 + 1.46849i −0.500000 + 0.866025i −0.692971 + 2.91887i −0.642788 0.766044i
41.12 0.766044 + 0.642788i 1.16246 1.28401i 0.173648 + 0.984808i 0.984808 + 0.173648i 1.71584 0.236392i −1.76528 3.05755i −0.500000 + 0.866025i −0.297363 2.98523i 0.642788 + 0.766044i
41.13 0.766044 + 0.642788i 1.68082 + 0.418131i 0.173648 + 0.984808i −0.984808 0.173648i 1.01882 + 1.40072i 0.0247885 + 0.0429350i −0.500000 + 0.866025i 2.65033 + 1.40561i −0.642788 0.766044i
41.14 0.766044 + 0.642788i 1.69734 + 0.345017i 0.173648 + 0.984808i 0.984808 + 0.173648i 1.07847 + 1.35533i −0.267952 0.464106i −0.500000 + 0.866025i 2.76193 + 1.17122i 0.642788 + 0.766044i
71.1 −0.939693 + 0.342020i −1.72889 0.104526i 0.766044 0.642788i −0.642788 + 0.766044i 1.66038 0.493094i 0.222334 + 0.385093i −0.500000 + 0.866025i 2.97815 + 0.361430i 0.342020 0.939693i
71.2 −0.939693 + 0.342020i −1.70290 + 0.316445i 0.766044 0.642788i 0.642788 0.766044i 1.49197 0.879786i −1.40312 2.43028i −0.500000 + 0.866025i 2.79973 1.07775i −0.342020 + 0.939693i
71.3 −0.939693 + 0.342020i −1.50842 + 0.851282i 0.766044 0.642788i 0.642788 0.766044i 1.12629 1.31585i 2.21271 + 3.83252i −0.500000 + 0.866025i 1.55064 2.56818i −0.342020 + 0.939693i
71.4 −0.939693 + 0.342020i −1.33410 1.10462i 0.766044 0.642788i 0.642788 0.766044i 1.63144 + 0.581714i −0.223800 0.387633i −0.500000 + 0.866025i 0.559632 + 2.94734i −0.342020 + 0.939693i
71.5 −0.939693 + 0.342020i −0.939713 + 1.45497i 0.766044 0.642788i −0.642788 + 0.766044i 0.385412 1.68863i −1.27478 2.20798i −0.500000 + 0.866025i −1.23388 2.73451i 0.342020 0.939693i
71.6 −0.939693 + 0.342020i −0.398282 1.68564i 0.766044 0.642788i −0.642788 + 0.766044i 0.950784 + 1.44776i −2.26841 3.92899i −0.500000 + 0.866025i −2.68274 + 1.34272i 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.a 84
3.b odd 2 1 570.2.bb.b yes 84
19.f odd 18 1 570.2.bb.b yes 84
57.j even 18 1 inner 570.2.bb.a 84
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
570.2.bb.a 84 1.a even 1 1 trivial
570.2.bb.a 84 57.j even 18 1 inner
570.2.bb.b yes 84 3.b odd 2 1
570.2.bb.b yes 84 19.f odd 18 1

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