Properties

Label 585.2.i.h
Level $585$
Weight $2$
Character orbit 585.i
Analytic conductor $4.671$
Analytic rank $0$
Dimension $30$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 585 = 3^{2} \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 585.i (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(4.67124851824\)
Analytic rank: \(0\)
Dimension: \(30\)
Relative dimension: \(15\) over \(\Q(\zeta_{3})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 30 q + q^{2} + q^{3} - 21 q^{4} + 15 q^{5} - 9 q^{6} - 10 q^{7} + q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 30 q + q^{2} + q^{3} - 21 q^{4} + 15 q^{5} - 9 q^{6} - 10 q^{7} + q^{9} + 2 q^{10} + 9 q^{11} + 18 q^{12} - 15 q^{13} + 3 q^{14} + 2 q^{15} - 33 q^{16} + 6 q^{17} + 9 q^{18} + 30 q^{19} + 21 q^{20} + 9 q^{21} - 10 q^{22} - 6 q^{23} + 24 q^{24} - 15 q^{25} - 2 q^{26} - 2 q^{27} + 70 q^{28} + 8 q^{29} - 6 q^{30} - 22 q^{31} + 21 q^{32} - 20 q^{33} - 9 q^{34} - 20 q^{35} - 7 q^{36} + 8 q^{37} - 14 q^{38} - 2 q^{39} + 13 q^{41} + 21 q^{42} - 24 q^{43} + 10 q^{44} - 7 q^{45} - 6 q^{46} - q^{47} - 27 q^{48} - 37 q^{49} + q^{50} - q^{51} - 21 q^{52} + 14 q^{53} - 24 q^{54} + 18 q^{55} + 17 q^{56} - 55 q^{57} - 22 q^{58} + 19 q^{59} + 9 q^{60} - 16 q^{61} + 26 q^{62} + 4 q^{63} + 72 q^{64} + 15 q^{65} + 24 q^{66} - 11 q^{67} - 28 q^{68} + 44 q^{69} - 3 q^{70} - 56 q^{71} - 18 q^{72} + 52 q^{73} + 8 q^{74} + q^{75} - 18 q^{76} - 24 q^{77} + 6 q^{78} - 44 q^{79} - 66 q^{80} + 37 q^{81} + 70 q^{82} - 3 q^{83} - 139 q^{84} + 3 q^{85} + 40 q^{86} + 60 q^{87} - 37 q^{88} - 8 q^{89} - 12 q^{90} + 20 q^{91} - 74 q^{92} - 55 q^{93} - 2 q^{94} + 15 q^{95} + 55 q^{96} - 33 q^{97} + 6 q^{98} + 26 q^{99} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
196.1 −1.33115 + 2.30562i 0.397495 + 1.68582i −2.54392 4.40620i 0.500000 + 0.866025i −4.41599 1.32761i −0.723369 + 1.25291i 8.22077 −2.68400 + 1.34021i −2.66230
196.2 −1.26000 + 2.18239i −1.67966 + 0.422782i −2.17520 3.76756i 0.500000 + 0.866025i 1.19370 4.19837i −1.50610 + 2.60865i 5.92303 2.64251 1.42026i −2.52000
196.3 −1.09333 + 1.89370i 1.72789 + 0.119945i −1.39073 2.40882i 0.500000 + 0.866025i −2.11629 + 3.14097i 2.12668 3.68352i 1.70880 2.97123 + 0.414503i −2.18666
196.4 −1.01209 + 1.75298i 0.826646 1.52206i −1.04864 1.81629i 0.500000 + 0.866025i 1.83150 + 2.98955i −1.83320 + 3.17519i 0.196897 −1.63331 2.51640i −2.02417
196.5 −0.644173 + 1.11574i −0.971694 1.43381i 0.170081 + 0.294590i 0.500000 + 0.866025i 2.22570 0.160537i 1.40888 2.44026i −3.01494 −1.11162 + 2.78645i −1.28835
196.6 −0.332241 + 0.575458i −0.727463 + 1.57188i 0.779232 + 1.34967i 0.500000 + 0.866025i −0.662856 0.940866i −1.40460 + 2.43283i −2.36453 −1.94159 2.28697i −0.664482
196.7 −0.264400 + 0.457954i 1.72073 0.197726i 0.860185 + 1.48988i 0.500000 + 0.866025i −0.364411 + 0.840293i −1.64961 + 2.85721i −1.96733 2.92181 0.680465i −0.528800
196.8 0.210096 0.363896i −1.72749 0.125594i 0.911720 + 1.57914i 0.500000 + 0.866025i −0.408641 + 0.602241i 0.421896 0.730745i 1.60658 2.96845 + 0.433924i 0.420191
196.9 0.300757 0.520926i 1.48936 + 0.884196i 0.819091 + 1.41871i 0.500000 + 0.866025i 0.908536 0.509919i 1.29950 2.25080i 2.18842 1.43640 + 2.63377i 0.601514
196.10 0.562020 0.973448i −0.0186726 1.73195i 0.368266 + 0.637856i 0.500000 + 0.866025i −1.69646 0.955214i 1.91433 3.31571i 3.07597 −2.99930 + 0.0646801i 1.12404
196.11 0.628641 1.08884i −0.933669 1.45886i 0.209620 + 0.363073i 0.500000 + 0.866025i −2.17540 + 0.0995174i −2.28445 + 3.95678i 3.04167 −1.25652 + 2.72418i 1.25728
196.12 0.909537 1.57536i 1.33257 + 1.10646i −0.654515 1.13365i 0.500000 + 0.866025i 2.95510 1.09292i −1.45386 + 2.51816i 1.25693 0.551488 + 2.94887i 1.81907
196.13 1.17680 2.03828i 0.988775 1.42208i −1.76973 3.06526i 0.500000 + 0.866025i −1.73501 3.68891i 0.0389983 0.0675470i −3.62327 −1.04465 2.81224i 2.35360
196.14 1.26258 2.18685i −0.259881 + 1.71244i −2.18820 3.79007i 0.500000 + 0.866025i 3.41673 + 2.73041i 0.799357 1.38453i −6.00077 −2.86492 0.890062i 2.52515
196.15 1.38695 2.40227i −1.66494 0.477475i −2.84726 4.93159i 0.500000 + 0.866025i −3.45621 + 3.33739i −2.15445 + 3.73162i −10.2482 2.54404 + 1.58993i 2.77390
391.1 −1.33115 2.30562i 0.397495 1.68582i −2.54392 + 4.40620i 0.500000 0.866025i −4.41599 + 1.32761i −0.723369 1.25291i 8.22077 −2.68400 1.34021i −2.66230
391.2 −1.26000 2.18239i −1.67966 0.422782i −2.17520 + 3.76756i 0.500000 0.866025i 1.19370 + 4.19837i −1.50610 2.60865i 5.92303 2.64251 + 1.42026i −2.52000
391.3 −1.09333 1.89370i 1.72789 0.119945i −1.39073 + 2.40882i 0.500000 0.866025i −2.11629 3.14097i 2.12668 + 3.68352i 1.70880 2.97123 0.414503i −2.18666
391.4 −1.01209 1.75298i 0.826646 + 1.52206i −1.04864 + 1.81629i 0.500000 0.866025i 1.83150 2.98955i −1.83320 3.17519i 0.196897 −1.63331 + 2.51640i −2.02417
391.5 −0.644173 1.11574i −0.971694 + 1.43381i 0.170081 0.294590i 0.500000 0.866025i 2.22570 + 0.160537i 1.40888 + 2.44026i −3.01494 −1.11162 2.78645i −1.28835
See all 30 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 391.15
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 585.2.i.h 30
3.b odd 2 1 1755.2.i.h 30
9.c even 3 1 inner 585.2.i.h 30
9.c even 3 1 5265.2.a.bk 15
9.d odd 6 1 1755.2.i.h 30
9.d odd 6 1 5265.2.a.bl 15
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
585.2.i.h 30 1.a even 1 1 trivial
585.2.i.h 30 9.c even 3 1 inner
1755.2.i.h 30 3.b odd 2 1
1755.2.i.h 30 9.d odd 6 1
5265.2.a.bk 15 9.c even 3 1
5265.2.a.bl 15 9.d odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(585, [\chi])\):

\(T_{2}^{30} - \cdots\)
\(18\!\cdots\!88\)\( T_{7}^{8} + \)\(10\!\cdots\!24\)\( T_{7}^{7} + \)\(31\!\cdots\!32\)\( T_{7}^{6} + 719631348864 T_{7}^{5} + \)\(30\!\cdots\!88\)\( T_{7}^{4} - 791264495616 T_{7}^{3} + \)\(19\!\cdots\!28\)\( T_{7}^{2} - 148416546816 T_{7} + 11314151424 \)">\(T_{7}^{30} + \cdots\)