# Properties

 Label 740.2.m.c Level $740$ Weight $2$ Character orbit 740.m 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(147,740)]

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, 1, 2]))

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

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

chi := DirichletCharacter("740.147");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$740 = 2^{2} \cdot 5 \cdot 37$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 740.m (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.

 $$\operatorname{Tr}(f)(q) =$$ $$216 q+O(q^{10})$$ 216 * q $$\operatorname{Tr}(f)(q) =$$ $$216 q - 16 q^{10} - 16 q^{12} + 8 q^{16} - 16 q^{21} - 32 q^{26} - 36 q^{28} + 16 q^{30} + 16 q^{33} + 16 q^{36} + 16 q^{37} + 4 q^{38} + 44 q^{40} + 32 q^{41} + 40 q^{46} + 52 q^{48} - 32 q^{53} - 144 q^{58} - 12 q^{62} + 16 q^{65} + 16 q^{70} + 64 q^{73} + 48 q^{77} - 8 q^{78} - 136 q^{81} - 64 q^{85} - 88 q^{86} + 96 q^{90}+O(q^{100})$$ 216 * q - 16 * q^10 - 16 * q^12 + 8 * q^16 - 16 * q^21 - 32 * q^26 - 36 * q^28 + 16 * q^30 + 16 * q^33 + 16 * q^36 + 16 * q^37 + 4 * q^38 + 44 * q^40 + 32 * q^41 + 40 * q^46 + 52 * q^48 - 32 * q^53 - 144 * q^58 - 12 * q^62 + 16 * q^65 + 16 * q^70 + 64 * q^73 + 48 * q^77 - 8 * q^78 - 136 * q^81 - 64 * q^85 - 88 * q^86 + 96 * q^90

## 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   $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
147.1 −1.41400 0.0244005i 0.843248 + 0.843248i 1.99881 + 0.0690048i −2.22853 + 0.183419i −1.17178 1.21293i −1.85966 + 1.85966i −2.82464 0.146345i 1.57787i 3.15563 0.204978i
147.2 −1.41365 0.0397689i 0.222918 + 0.222918i 1.99684 + 0.112439i −0.291733 + 2.21696i −0.306263 0.323994i −0.407595 + 0.407595i −2.81837 0.238362i 2.90062i 0.500576 3.12241i
147.3 −1.41352 + 0.0444054i −1.40518 1.40518i 1.99606 0.125535i 2.22590 0.213026i 2.04865 + 1.92385i −2.97474 + 2.97474i −2.81588 + 0.266082i 0.949079i −3.13688 + 0.399957i
147.4 −1.40656 + 0.146921i −0.239392 0.239392i 1.95683 0.413307i −0.944979 2.02658i 0.371891 + 0.301547i 2.73816 2.73816i −2.69167 + 0.868842i 2.88538i 1.62692 + 2.71167i
147.5 −1.40450 0.165503i 1.54210 + 1.54210i 1.94522 + 0.464898i 1.05021 1.97410i −1.91066 2.42110i 1.25747 1.25747i −2.65511 0.974887i 1.75618i −1.80173 + 2.59880i
147.6 −1.39702 + 0.219882i −2.06451 2.06451i 1.90330 0.614357i −1.04865 1.97493i 3.33810 + 2.43020i −1.52612 + 1.52612i −2.52386 + 1.27677i 5.52441i 1.89923 + 2.52842i
147.7 −1.39372 0.239886i 2.25982 + 2.25982i 1.88491 + 0.668667i 1.90207 + 1.17564i −2.60745 3.69165i −2.89652 + 2.89652i −2.46663 1.38410i 7.21355i −2.36893 2.09479i
147.8 −1.39360 0.240561i −1.88344 1.88344i 1.88426 + 0.670494i 1.40145 + 1.74239i 2.17168 + 3.07785i 1.74017 1.74017i −2.46462 1.38768i 4.09468i −1.53391 2.76534i
147.9 −1.37564 + 0.328030i 0.683771 + 0.683771i 1.78479 0.902504i 1.49080 + 1.66659i −1.16492 0.716328i 0.871879 0.871879i −2.15919 + 1.82699i 2.06491i −2.59750 1.80361i
147.10 −1.35540 + 0.403607i −1.73649 1.73649i 1.67420 1.09410i −2.21752 + 0.287427i 3.05449 + 1.65277i 2.51648 2.51648i −1.82762 + 2.15865i 3.03080i 2.88961 1.28458i
147.11 −1.33091 0.478214i −0.667697 0.667697i 1.54262 + 1.27292i −1.40325 1.74095i 0.569340 + 1.20794i −1.52970 + 1.52970i −1.44436 2.43184i 2.10836i 1.03504 + 2.98809i
147.12 −1.32750 + 0.487585i −0.405252 0.405252i 1.52452 1.29454i 1.43771 1.71260i 0.735568 + 0.340378i 0.390006 0.390006i −1.39261 + 2.46184i 2.67154i −1.07352 + 2.97448i
147.13 −1.31027 + 0.532153i 1.34504 + 1.34504i 1.43363 1.39453i 0.234074 2.22378i −2.47813 1.04660i −3.59854 + 3.59854i −1.13634 + 2.59012i 0.618246i 0.876691 + 3.03832i
147.14 −1.29368 + 0.571318i 2.14770 + 2.14770i 1.34719 1.47820i −1.42073 + 1.72671i −4.00544 1.55140i −0.502479 + 0.502479i −0.898303 + 2.68199i 6.22520i 0.851457 3.04549i
147.15 −1.29229 0.574434i −0.283957 0.283957i 1.34005 + 1.48468i 2.23379 + 0.100889i 0.203841 + 0.530070i 2.24373 2.24373i −0.878890 2.68841i 2.83874i −2.82876 1.41354i
147.16 −1.24863 0.664020i −0.800527 0.800527i 1.11815 + 1.65823i −1.95987 + 1.07653i 0.467996 + 1.53113i 2.28993 2.28993i −0.295064 2.81299i 1.71831i 3.16199 0.0428021i
147.17 −1.24773 + 0.665716i −1.47275 1.47275i 1.11364 1.66126i 0.432079 + 2.19393i 2.81803 + 0.857157i 0.497315 0.497315i −0.283592 + 2.81417i 1.33800i −1.99965 2.44978i
147.18 −1.23300 0.692606i −2.39205 2.39205i 1.04059 + 1.70797i −1.76278 + 1.37572i 1.29266 + 4.60615i −2.24066 + 2.24066i −0.100107 2.82666i 8.44377i 3.12635 0.475351i
147.19 −1.22095 + 0.713647i 1.73173 + 1.73173i 0.981417 1.74265i 2.14628 + 0.627282i −3.35020 0.878506i 2.36798 2.36798i 0.0453786 + 2.82806i 2.99779i −3.06815 + 0.765808i
147.20 −1.19937 0.749346i 1.06956 + 1.06956i 0.876962 + 1.79748i 2.21974 + 0.269739i −0.481325 2.08427i −0.686867 + 0.686867i 0.295135 2.81299i 0.712080i −2.46015 1.98687i
See next 80 embeddings (of 216 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 147.108 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.c odd 4 1 inner
20.e even 4 1 inner
37.b even 2 1 inner
148.b odd 2 1 inner
185.h odd 4 1 inner
740.m 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.m.c 216
4.b odd 2 1 inner 740.2.m.c 216
5.c odd 4 1 inner 740.2.m.c 216
20.e even 4 1 inner 740.2.m.c 216
37.b even 2 1 inner 740.2.m.c 216
148.b odd 2 1 inner 740.2.m.c 216
185.h odd 4 1 inner 740.2.m.c 216
740.m even 4 1 inner 740.2.m.c 216

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
740.2.m.c 216 1.a even 1 1 trivial
740.2.m.c 216 4.b odd 2 1 inner
740.2.m.c 216 5.c odd 4 1 inner
740.2.m.c 216 20.e even 4 1 inner
740.2.m.c 216 37.b even 2 1 inner
740.2.m.c 216 148.b odd 2 1 inner
740.2.m.c 216 185.h odd 4 1 inner
740.2.m.c 216 740.m 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 $$S_{2}^{\mathrm{new}}(740, [\chi])$$:

 $$T_{3}^{108} + 746 T_{3}^{104} + 251139 T_{3}^{100} + 50631122 T_{3}^{96} + 6839977299 T_{3}^{92} + \cdots + 43\!\cdots\!00$$ T3^108 + 746*T3^104 + 251139*T3^100 + 50631122*T3^96 + 6839977299*T3^92 + 656376372812*T3^88 + 46272266380274*T3^84 + 2444422045922044*T3^80 + 97829630521761967*T3^76 + 2979517193685884466*T3^72 + 69006531145852950283*T3^68 + 1209031302374217650978*T3^64 + 15877163647896828818905*T3^60 + 154310035064637772898688*T3^56 + 1093565160069249478522904*T3^52 + 5568321952588299595957264*T3^48 + 20134018316886819238944316*T3^44 + 51260047816075806184809768*T3^40 + 91161569663132762033557696*T3^36 + 111923986234963716678209056*T3^32 + 92946290552172639537836224*T3^28 + 50382261210977094821664336*T3^24 + 16757959926975804678274448*T3^20 + 3066031472322633180653376*T3^16 + 254682612385441919038400*T3^12 + 8117443998127113000000*T3^8 + 102566883435000000000*T3^4 + 438906250000000000 $$T_{13}^{108} + 10774 T_{13}^{104} + 51870747 T_{13}^{100} + 147687314502 T_{13}^{96} + \cdots + 35\!\cdots\!00$$ T13^108 + 10774*T13^104 + 51870747*T13^100 + 147687314502*T13^96 + 277161386648681*T13^92 + 361545407153398808*T13^88 + 336828004471258964728*T13^84 + 226805182926799225190208*T13^80 + 110575319609460550167452880*T13^76 + 38817123679581548078984758880*T13^72 + 9710509179006439486655050885632*T13^68 + 1709693667650615160494701408001920*T13^64 + 209477750843662338510522205493911296*T13^60 + 17711112579508188942530394932096698624*T13^56 + 1028135742060900629652736336618113689856*T13^52 + 40836863049656519366137354821030109771776*T13^48 + 1104971551654214561352532315135764695785472*T13^44 + 20187375380695292767376402731897202849951744*T13^40 + 244718078449547827183506435510489836196126720*T13^36 + 1911721064686281287794048402223477591482630144*T13^32 + 9237489953341794908812773868867324752892657664*T13^28 + 26464344988735058639472177933766158956152815616*T13^24 + 43936895135183812923817835037418867819554078720*T13^20 + 41373172534449118958910627829989284439901863936*T13^16 + 21390153874082622341241640854322210021605113856*T13^12 + 5779117812545396616903257564740543442662195200*T13^8 + 746399161308687625307429024545078782197760000*T13^4 + 35531559531908358160217936182640640000000000