# Properties

 Label 585.2.bt.b Level $585$ Weight $2$ Character orbit 585.bt Analytic conductor $4.671$ Analytic rank $0$ Dimension $108$ CM no Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [585,2,Mod(376,585)]

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, 3]))

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

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

chi := DirichletCharacter("585.376");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$585 = 3^{2} \cdot 5 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 585.bt (of order $$6$$, 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: $$108$$ Relative dimension: $$54$$ over $$\Q(\zeta_{6})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$108 q - 6 q^{3} + 52 q^{4} - 14 q^{9}+O(q^{10})$$ 108 * q - 6 * q^3 + 52 * q^4 - 14 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$108 q - 6 q^{3} + 52 q^{4} - 14 q^{9} - 8 q^{10} - 36 q^{12} - 4 q^{13} - 8 q^{14} - 64 q^{16} + 36 q^{17} + 24 q^{22} - 22 q^{23} + 54 q^{25} + 40 q^{26} + 48 q^{27} - 16 q^{29} + 20 q^{30} - 40 q^{35} - 76 q^{36} - 12 q^{38} + 26 q^{39} - 52 q^{42} - 6 q^{43} - 68 q^{48} + 58 q^{49} + 34 q^{51} + 22 q^{52} - 116 q^{53} + 24 q^{55} + 40 q^{56} + 18 q^{61} + 152 q^{62} - 216 q^{64} + 2 q^{65} - 72 q^{66} - 134 q^{69} + 32 q^{74} + 40 q^{77} - 34 q^{78} + 18 q^{79} + 34 q^{81} + 88 q^{82} + 60 q^{87} + 8 q^{90} + 16 q^{91} + 176 q^{92} - 76 q^{94} - 56 q^{95}+O(q^{100})$$ 108 * q - 6 * q^3 + 52 * q^4 - 14 * q^9 - 8 * q^10 - 36 * q^12 - 4 * q^13 - 8 * q^14 - 64 * q^16 + 36 * q^17 + 24 * q^22 - 22 * q^23 + 54 * q^25 + 40 * q^26 + 48 * q^27 - 16 * q^29 + 20 * q^30 - 40 * q^35 - 76 * q^36 - 12 * q^38 + 26 * q^39 - 52 * q^42 - 6 * q^43 - 68 * q^48 + 58 * q^49 + 34 * q^51 + 22 * q^52 - 116 * q^53 + 24 * q^55 + 40 * q^56 + 18 * q^61 + 152 * q^62 - 216 * q^64 + 2 * q^65 - 72 * q^66 - 134 * q^69 + 32 * q^74 + 40 * q^77 - 34 * q^78 + 18 * q^79 + 34 * q^81 + 88 * q^82 + 60 * q^87 + 8 * q^90 + 16 * q^91 + 176 * q^92 - 76 * q^94 - 56 * q^95

## 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   $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
376.1 −2.39334 1.38179i −0.969577 1.43524i 2.81871 + 4.88215i 0.866025 0.500000i 0.337317 + 4.77477i −0.814209 0.470084i 10.0523i −1.11984 + 2.78316i −2.76359
376.2 −2.36351 1.36457i 1.68089 + 0.417855i 2.72413 + 4.71833i −0.866025 + 0.500000i −3.40262 3.28131i 3.99906 + 2.30886i 9.41082i 2.65080 + 1.40474i 2.72915
376.3 −2.26372 1.30696i −0.538321 + 1.64627i 2.41629 + 4.18514i 0.866025 0.500000i 3.37022 3.02314i 3.14827 + 1.81766i 7.40414i −2.42042 1.77245i −2.61392
376.4 −2.24228 1.29458i −0.927160 + 1.46300i 2.35187 + 4.07355i −0.866025 + 0.500000i 3.97292 2.08017i −1.32705 0.766174i 7.00039i −1.28075 2.71287i 2.58916
376.5 −2.20318 1.27200i 0.350451 1.69623i 2.23599 + 3.87285i −0.866025 + 0.500000i −2.92971 + 3.29131i −0.586224 0.338456i 6.28874i −2.75437 1.18889i 2.54401
376.6 −2.03176 1.17304i 0.447750 + 1.67318i 1.75203 + 3.03461i 0.866025 0.500000i 1.05298 3.92472i −2.60767 1.50554i 3.52864i −2.59904 + 1.49833i −2.34607
376.7 −1.91643 1.10645i 1.62047 + 0.611626i 1.44848 + 2.50883i 0.866025 0.500000i −2.42878 2.96511i 0.916933 + 0.529392i 1.98487i 2.25183 + 1.98224i −2.21291
376.8 −1.85413 1.07048i −1.70628 + 0.297682i 1.29187 + 2.23759i 0.866025 0.500000i 3.48233 + 1.27460i −4.18669 2.41719i 1.24978i 2.82277 1.01586i −2.14097
376.9 −1.71224 0.988562i −1.13208 1.31087i 0.954509 + 1.65326i −0.866025 + 0.500000i 0.642514 + 3.36366i −3.18919 1.84128i 0.179881i −0.436783 + 2.96803i 1.97712
376.10 −1.71151 0.988142i 1.03985 + 1.38517i 0.952848 + 1.65038i −0.866025 + 0.500000i −0.410976 3.39826i −1.22619 0.707941i 0.186373i −0.837406 + 2.88076i 1.97628
376.11 −1.64129 0.947602i −1.39724 1.02358i 0.795899 + 1.37854i 0.866025 0.500000i 1.32334 + 3.00402i 2.82539 + 1.63124i 0.773626i 0.904573 + 2.86038i −1.89520
376.12 −1.62679 0.939229i −1.72455 + 0.161066i 0.764302 + 1.32381i −0.866025 + 0.500000i 2.95676 + 1.35772i 1.80641 + 1.04293i 0.885499i 2.94812 0.555532i 1.87846
376.13 −1.39901 0.807720i 0.0517001 1.73128i 0.304825 + 0.527972i 0.866025 0.500000i −1.47072 + 2.38032i 1.67328 + 0.966071i 2.24603i −2.99465 0.179015i −1.61544
376.14 −1.23573 0.713450i −1.00032 + 1.41399i 0.0180219 + 0.0312149i −0.866025 + 0.500000i 2.24493 1.03364i 2.69830 + 1.55786i 2.80237i −0.998738 2.82887i 1.42690
376.15 −1.22618 0.707937i 1.65800 0.501035i 0.00235097 + 0.00407201i 0.866025 0.500000i −2.38771 0.559399i −0.00992873 0.00573236i 2.82509i 2.49793 1.66143i −1.41587
376.16 −1.16682 0.673666i 0.619932 1.61731i −0.0923480 0.159951i 0.866025 0.500000i −1.81288 + 1.46949i −4.07352 2.35185i 2.94351i −2.23137 2.00524i −1.34733
376.17 −1.09899 0.634500i 1.58392 0.700848i −0.194820 0.337439i −0.866025 + 0.500000i −2.18540 0.234776i 4.16228 + 2.40310i 3.03245i 2.01762 2.22018i 1.26900
376.18 −1.07284 0.619404i −1.48219 + 0.896168i −0.232678 0.403011i 0.866025 0.500000i 2.14524 0.0433707i 0.166629 + 0.0962033i 3.05410i 1.39377 2.65658i −1.23881
376.19 −0.883008 0.509805i −0.597446 + 1.62575i −0.480198 0.831727i −0.866025 + 0.500000i 1.35636 1.13097i −3.32453 1.91942i 3.01845i −2.28612 1.94260i 1.01961
376.20 −0.774258 0.447018i −0.513274 1.65425i −0.600350 1.03984i −0.866025 + 0.500000i −0.342074 + 1.51026i 1.76954 + 1.02164i 2.86154i −2.47310 + 1.69817i 0.894036
See next 80 embeddings (of 108 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 376.54 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
13.b even 2 1 inner
117.t even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 585.2.bt.b 108
9.c even 3 1 inner 585.2.bt.b 108
13.b even 2 1 inner 585.2.bt.b 108
117.t even 6 1 inner 585.2.bt.b 108

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
585.2.bt.b 108 1.a even 1 1 trivial
585.2.bt.b 108 9.c even 3 1 inner
585.2.bt.b 108 13.b even 2 1 inner
585.2.bt.b 108 117.t even 6 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{108} - 80 T_{2}^{106} + 3402 T_{2}^{104} - 99900 T_{2}^{102} + 2249817 T_{2}^{100} - 41081196 T_{2}^{98} + 629426852 T_{2}^{96} - 8280112066 T_{2}^{94} + 95063249334 T_{2}^{92} + \cdots + 1679616$$ acting on $$S_{2}^{\mathrm{new}}(585, [\chi])$$.