# 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 $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [585,2,Mod(196,585)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(585, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([4, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("585.196");

S:= CuspForms(chi, 2);

N := Newforms(S);

 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

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 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.

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

## 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.

comment: embeddings in the coefficient field

gp: mfembed(f)

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 196.15 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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 $$S_{2}^{\mathrm{new}}(585, [\chi])$$:

 $$T_{2}^{30} - T_{2}^{29} + 26 T_{2}^{28} - 23 T_{2}^{27} + 405 T_{2}^{26} - 338 T_{2}^{25} + 4110 T_{2}^{24} - 3223 T_{2}^{23} + 30766 T_{2}^{22} - 23145 T_{2}^{21} + 170006 T_{2}^{20} - 122117 T_{2}^{19} + 718425 T_{2}^{18} + \cdots + 20736$$ T2^30 - T2^29 + 26*T2^28 - 23*T2^27 + 405*T2^26 - 338*T2^25 + 4110*T2^24 - 3223*T2^23 + 30766*T2^22 - 23145*T2^21 + 170006*T2^20 - 122117*T2^19 + 718425*T2^18 - 495520*T2^17 + 2257873*T2^16 - 1462802*T2^15 + 5330152*T2^14 - 3169368*T2^13 + 8856771*T2^12 - 4469319*T2^11 + 10446576*T2^10 - 4397121*T2^9 + 7968663*T2^8 - 2195721*T2^7 + 3781224*T2^6 - 911817*T2^5 + 1173906*T2^4 - 182061*T2^3 + 210033*T2^2 - 32400*T2 + 20736 $$T_{7}^{30} + 10 T_{7}^{29} + 121 T_{7}^{28} + 770 T_{7}^{27} + 5910 T_{7}^{26} + 30389 T_{7}^{25} + 183025 T_{7}^{24} + 787276 T_{7}^{23} + 3908218 T_{7}^{22} + 14334577 T_{7}^{21} + 61116299 T_{7}^{20} + \cdots + 11314151424$$ T7^30 + 10*T7^29 + 121*T7^28 + 770*T7^27 + 5910*T7^26 + 30389*T7^25 + 183025*T7^24 + 787276*T7^23 + 3908218*T7^22 + 14334577*T7^21 + 61116299*T7^20 + 192710575*T7^19 + 713387163*T7^18 + 1916734871*T7^17 + 6264272907*T7^16 + 14272507970*T7^15 + 41591788717*T7^14 + 77805400979*T7^13 + 204394693947*T7^12 + 302117024506*T7^11 + 737180340096*T7^10 + 773423798536*T7^9 + 1817194487888*T7^8 + 1095087849424*T7^7 + 3159073592032*T7^6 + 719631348864*T7^5 + 3089693970688*T7^4 - 791264495616*T7^3 + 1921993552128*T7^2 - 148416546816*T7 + 11314151424