# Properties

 Label 740.2.k.c Level $740$ Weight $2$ Character orbit 740.k Analytic conductor $5.909$ Analytic rank $0$ Dimension $216$ Inner twists $8$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [740,2,Mod(179,740)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(740, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([2, 2, 1]))

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

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

chi := DirichletCharacter("740.179");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$740 = 2^{2} \cdot 5 \cdot 37$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 740.k (of order $$4$$, 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: $$5.90892974957$$ Analytic rank: $$0$$ Dimension: $$216$$ Relative dimension: $$108$$ over $$\Q(i)$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-expansion

The algebraic $$q$$-expansion of this newform has not been computed, but we have computed the trace expansion.

 $$\operatorname{Tr}(f)(q) =$$ $$216 q - 8 q^{5} + 4 q^{6} - 232 q^{9}+O(q^{10})$$ 216 * q - 8 * q^5 + 4 * q^6 - 232 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$216 q - 8 q^{5} + 4 q^{6} - 232 q^{9} - 16 q^{10} - 12 q^{14} + 8 q^{16} - 4 q^{20} - 36 q^{24} - 48 q^{26} + 16 q^{29} - 16 q^{34} - 72 q^{44} + 16 q^{45} - 136 q^{46} + 184 q^{49} - 32 q^{50} - 20 q^{54} + 20 q^{56} + 36 q^{60} + 16 q^{61} + 28 q^{66} - 32 q^{69} - 60 q^{70} - 92 q^{74} - 28 q^{76} - 12 q^{80} + 152 q^{81} - 32 q^{84} + 72 q^{86} - 40 q^{89} - 16 q^{90} - 48 q^{94} + 16 q^{96}+O(q^{100})$$ 216 * q - 8 * q^5 + 4 * q^6 - 232 * q^9 - 16 * q^10 - 12 * q^14 + 8 * q^16 - 4 * q^20 - 36 * q^24 - 48 * q^26 + 16 * q^29 - 16 * q^34 - 72 * q^44 + 16 * q^45 - 136 * q^46 + 184 * q^49 - 32 * q^50 - 20 * q^54 + 20 * q^56 + 36 * q^60 + 16 * q^61 + 28 * q^66 - 32 * q^69 - 60 * q^70 - 92 * q^74 - 28 * q^76 - 12 * q^80 + 152 * q^81 - 32 * q^84 + 72 * q^86 - 40 * q^89 - 16 * q^90 - 48 * q^94 + 16 * q^96

## 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   $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
179.1 −1.41417 + 0.0112112i 1.10399i 1.99975 0.0317091i −0.0702723 2.23496i 0.0123771 + 1.56123i 0.423249 −2.82763 + 0.0672617i 1.78120 0.124434 + 3.15983i
179.2 −1.41411 + 0.0169178i 1.43960i 1.99943 0.0478472i −1.78993 + 1.34020i −0.0243548 2.03576i −0.0804759 −2.82661 + 0.101487i 0.927550 2.50849 1.92548i
179.3 −1.41220 + 0.0754091i 1.91116i 1.98863 0.212986i −2.20851 0.349987i 0.144119 + 2.69894i 3.09756 −2.79228 + 0.450740i −0.652519 3.14525 + 0.327711i
179.4 −1.39238 0.247533i 3.24286i 1.87745 + 0.689322i −0.455272 + 2.18923i −0.802717 + 4.51530i −2.08644 −2.44350 1.42453i −7.51616 1.17582 2.93555i
179.5 −1.39007 0.260196i 2.69556i 1.86460 + 0.723381i 1.08099 + 1.95741i 0.701373 3.74702i 2.34449 −2.40370 1.49071i −4.26605 −0.993338 3.00221i
179.6 −1.38535 0.284249i 3.33433i 1.83841 + 0.787570i −1.65612 1.50242i 0.947781 4.61923i −2.00442 −2.32297 1.61363i −8.11779 1.86725 + 2.55213i
179.7 −1.38410 0.290269i 0.837020i 1.83149 + 0.803524i 1.69797 1.45495i −0.242961 + 1.15852i −0.279655 −2.30173 1.64379i 2.29940 −2.77250 + 1.52094i
179.8 −1.37717 + 0.321579i 1.34163i 1.79317 0.885734i 1.95385 + 1.08741i 0.431440 + 1.84765i −1.32565 −2.18467 + 1.79645i 1.20002 −3.04047 0.869228i
179.9 −1.37515 + 0.330097i 2.62444i 1.78207 0.907866i 1.29187 1.82512i −0.866322 3.60900i 3.53783 −2.15093 + 1.83671i −3.88771 −1.17405 + 2.93626i
179.10 −1.37317 0.338251i 0.235460i 1.77117 + 0.928950i −1.34759 + 1.78438i −0.0796445 + 0.323325i −4.45596 −2.11790 1.87470i 2.94456 2.45403 1.99443i
179.11 −1.36131 0.383186i 0.145516i 1.70634 + 1.04327i 0.799638 + 2.08820i −0.0557598 + 0.198093i 3.05986 −1.92309 2.07406i 2.97883 −0.288388 3.14910i
179.12 −1.36028 + 0.386818i 1.72633i 1.70074 1.05236i 1.60779 + 1.55403i −0.667777 2.34830i −3.84183 −1.90642 + 2.08939i 0.0197698 −2.78818 1.49200i
179.13 −1.31376 + 0.523479i 3.06130i 1.45194 1.37545i 1.66991 1.48707i 1.60252 + 4.02182i 0.959145 −1.18748 + 2.56708i −6.37155 −1.41542 + 2.82783i
179.14 −1.30358 + 0.548342i 1.78936i 1.39864 1.42962i −0.621142 2.14806i −0.981180 2.33257i −0.447293 −1.03932 + 2.63055i −0.201798 1.98758 + 2.45958i
179.15 −1.28091 + 0.599393i 0.924956i 1.28146 1.53554i −2.23380 + 0.100625i −0.554412 1.18479i 0.438499 −0.721042 + 2.73498i 2.14446 2.80098 1.46782i
179.16 −1.27262 0.616804i 2.09980i 1.23910 + 1.56991i 2.13596 0.661580i 1.29516 2.67223i −2.52168 −0.608578 2.76218i −1.40914 −3.12632 0.475530i
179.17 −1.25262 + 0.656462i 0.0160284i 1.13811 1.64460i −0.177395 + 2.22902i −0.0105220 0.0200775i 3.72929 −0.346011 + 2.80718i 2.99974 −1.24106 2.90857i
179.18 −1.24820 + 0.664834i 2.39692i 1.11599 1.65969i −1.98754 1.02454i 1.59355 + 2.99183i −4.93364 −0.289561 + 2.81357i −2.74521 3.16199 0.0425512i
179.19 −1.22029 0.714774i 2.58416i 0.978197 + 1.74446i 2.17123 + 0.534580i −1.84709 + 3.15342i 4.55780 0.0532110 2.82793i −3.67790 −2.26741 2.20428i
179.20 −1.21943 0.716239i 0.846684i 0.974003 + 1.74680i −0.116235 2.23304i 0.606428 1.03247i 3.61247 0.0634021 2.82772i 2.28313 −1.45765 + 2.80629i
See next 80 embeddings (of 216 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 179.108 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
5.b even 2 1 inner
20.d odd 2 1 inner
37.d odd 4 1 inner
148.g even 4 1 inner
185.j odd 4 1 inner
740.k even 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 740.2.k.c 216
4.b odd 2 1 inner 740.2.k.c 216
5.b even 2 1 inner 740.2.k.c 216
20.d odd 2 1 inner 740.2.k.c 216
37.d odd 4 1 inner 740.2.k.c 216
148.g even 4 1 inner 740.2.k.c 216
185.j odd 4 1 inner 740.2.k.c 216
740.k even 4 1 inner 740.2.k.c 216

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
740.2.k.c 216 1.a even 1 1 trivial
740.2.k.c 216 4.b odd 2 1 inner
740.2.k.c 216 5.b even 2 1 inner
740.2.k.c 216 20.d odd 2 1 inner
740.2.k.c 216 37.d odd 4 1 inner
740.2.k.c 216 148.g even 4 1 inner
740.2.k.c 216 185.j odd 4 1 inner
740.2.k.c 216 740.k even 4 1 inner

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(740, [\chi])$$:

 $$T_{3}^{54} + 110 T_{3}^{52} + 5667 T_{3}^{50} + 181782 T_{3}^{48} + 4071449 T_{3}^{46} + 67678388 T_{3}^{44} + \cdots + 19232$$ T3^54 + 110*T3^52 + 5667*T3^50 + 181782*T3^48 + 4071449*T3^46 + 67678388*T3^44 + 866360444*T3^42 + 8747070848*T3^40 + 70764850137*T3^38 + 463525873486*T3^36 + 2474017994647*T3^34 + 10794264666138*T3^32 + 38516686029151*T3^30 + 112171098009704*T3^28 + 265408207698066*T3^26 + 506548736107144*T3^24 + 771915291481950*T3^22 + 926323217877332*T3^20 + 859435111007668*T3^18 + 601518592973056*T3^16 + 307141214016496*T3^14 + 109151937150268*T3^12 + 25195200184236*T3^10 + 3402030876600*T3^8 + 231855503928*T3^6 + 7181435400*T3^4 + 76688768*T3^2 + 19232 $$T_{13}^{108} + 9598 T_{13}^{104} + 40254107 T_{13}^{100} + 97793463422 T_{13}^{96} + \cdots + 78\!\cdots\!00$$ T13^108 + 9598*T13^104 + 40254107*T13^100 + 97793463422*T13^96 + 153635518915929*T13^92 + 164925010771506608*T13^88 + 124635037573905762328*T13^84 + 67366859092182562677856*T13^80 + 26241003531721625297550192*T13^76 + 7385401516525633299134040416*T13^72 + 1501829772140479029987262972928*T13^68 + 220408706053596547701718046824320*T13^64 + 23309236445531271580099709648914432*T13^60 + 1772305680400442915054434776377042176*T13^56 + 96475243230201583500465945901647579392*T13^52 + 3728614548021630826499522333401877262336*T13^48 + 100828086524850469983548737781669794852864*T13^44 + 1864631687594808414617068719338466384351232*T13^40 + 22839665502812435800267309417104336540663808*T13^36 + 178085152608683670388275368250347575164665856*T13^32 + 845319330715974267753069742186668521861677056*T13^28 + 2311638933513934528063502650861943762971525120*T13^24 + 3361962529041440896109885418723374781052223488*T13^20 + 2282496271882786648651543541582665515869929472*T13^16 + 660743745078317077716821579750414703925395456*T13^12 + 86203669138339403193548663165480168494989312*T13^8 + 4744738116828919462366909788447891003539456*T13^4 + 78645253113434941199999357478491914240000