# Properties

 Label 380.2.k.c Level $380$ Weight $2$ Character orbit 380.k Analytic conductor $3.034$ Analytic rank $0$ Dimension $52$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [380,2,Mod(267,380)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(380, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([2, 1, 0]))

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

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

chi := DirichletCharacter("380.267");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$380 = 2^{2} \cdot 5 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 380.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: $$3.03431527681$$ Analytic rank: $$0$$ Dimension: $$52$$ Relative dimension: $$26$$ over $$\Q(i)$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$52 q - 2 q^{2} - 2 q^{3} + 4 q^{5} - 4 q^{6} - 4 q^{7} + 4 q^{8}+O(q^{10})$$ 52 * q - 2 * q^2 - 2 * q^3 + 4 * q^5 - 4 * q^6 - 4 * q^7 + 4 * q^8 $$\operatorname{Tr}(f)(q) =$$ $$52 q - 2 q^{2} - 2 q^{3} + 4 q^{5} - 4 q^{6} - 4 q^{7} + 4 q^{8} - 18 q^{10} - 38 q^{12} - 2 q^{13} + 2 q^{15} + 16 q^{16} - 20 q^{17} + 2 q^{18} + 52 q^{19} + 20 q^{20} - 16 q^{21} + 20 q^{22} + 20 q^{23} - 16 q^{25} - 8 q^{27} - 8 q^{28} - 48 q^{30} - 42 q^{32} + 8 q^{33} - 20 q^{34} + 12 q^{35} - 4 q^{36} + 10 q^{37} - 2 q^{38} - 64 q^{39} + 60 q^{40} - 4 q^{41} + 60 q^{42} - 28 q^{43} + 8 q^{44} + 12 q^{45} - 8 q^{46} - 4 q^{47} - 18 q^{48} - 42 q^{50} - 54 q^{52} - 2 q^{53} - 24 q^{54} + 12 q^{56} - 2 q^{57} - 12 q^{58} + 28 q^{59} + 90 q^{60} - 4 q^{61} + 56 q^{62} + 44 q^{63} + 24 q^{64} + 10 q^{65} + 36 q^{66} + 6 q^{67} - 60 q^{68} - 32 q^{70} - 24 q^{72} + 8 q^{73} - 88 q^{74} + 2 q^{75} + 12 q^{77} + 64 q^{78} - 52 q^{79} - 8 q^{80} - 24 q^{81} + 56 q^{82} - 76 q^{83} + 40 q^{84} + 12 q^{85} - 8 q^{86} + 12 q^{87} - 40 q^{88} - 94 q^{90} - 48 q^{92} + 24 q^{93} - 32 q^{94} + 4 q^{95} - 4 q^{96} - 10 q^{97} + 58 q^{98} + 128 q^{99}+O(q^{100})$$ 52 * q - 2 * q^2 - 2 * q^3 + 4 * q^5 - 4 * q^6 - 4 * q^7 + 4 * q^8 - 18 * q^10 - 38 * q^12 - 2 * q^13 + 2 * q^15 + 16 * q^16 - 20 * q^17 + 2 * q^18 + 52 * q^19 + 20 * q^20 - 16 * q^21 + 20 * q^22 + 20 * q^23 - 16 * q^25 - 8 * q^27 - 8 * q^28 - 48 * q^30 - 42 * q^32 + 8 * q^33 - 20 * q^34 + 12 * q^35 - 4 * q^36 + 10 * q^37 - 2 * q^38 - 64 * q^39 + 60 * q^40 - 4 * q^41 + 60 * q^42 - 28 * q^43 + 8 * q^44 + 12 * q^45 - 8 * q^46 - 4 * q^47 - 18 * q^48 - 42 * q^50 - 54 * q^52 - 2 * q^53 - 24 * q^54 + 12 * q^56 - 2 * q^57 - 12 * q^58 + 28 * q^59 + 90 * q^60 - 4 * q^61 + 56 * q^62 + 44 * q^63 + 24 * q^64 + 10 * q^65 + 36 * q^66 + 6 * q^67 - 60 * q^68 - 32 * q^70 - 24 * q^72 + 8 * q^73 - 88 * q^74 + 2 * q^75 + 12 * q^77 + 64 * q^78 - 52 * q^79 - 8 * q^80 - 24 * q^81 + 56 * q^82 - 76 * q^83 + 40 * q^84 + 12 * q^85 - 8 * q^86 + 12 * q^87 - 40 * q^88 - 94 * q^90 - 48 * q^92 + 24 * q^93 - 32 * q^94 + 4 * q^95 - 4 * q^96 - 10 * q^97 + 58 * q^98 + 128 * 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   $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
267.1 −1.41374 + 0.0365245i −0.389264 0.389264i 1.99733 0.103273i −1.49421 + 1.66354i 0.564536 + 0.536101i −2.85353 + 2.85353i −2.81994 + 0.218952i 2.69695i 2.05166 2.40638i
267.2 −1.38561 0.282971i 0.321015 + 0.321015i 1.83985 + 0.784178i 1.77712 + 1.35715i −0.353965 0.535642i 1.91243 1.91243i −2.32743 1.60719i 2.79390i −2.07836 2.38336i
267.3 −1.38463 0.287770i 1.15541 + 1.15541i 1.83438 + 0.796907i −0.283261 2.21805i −1.26732 1.93230i −3.26325 + 3.26325i −2.31060 1.63130i 0.330062i −0.246078 + 3.15269i
267.4 −1.34620 + 0.433304i −0.497882 0.497882i 1.62450 1.16662i −2.01079 0.978129i 0.885982 + 0.454514i 2.91849 2.91849i −1.68139 + 2.27441i 2.50423i 3.13074 + 0.445474i
267.5 −1.34536 + 0.435901i −1.82161 1.82161i 1.61998 1.17289i 2.11542 + 0.724569i 3.24476 + 1.65668i −1.36053 + 1.36053i −1.66819 + 2.28411i 3.63653i −3.16184 0.0526904i
267.6 −1.01972 + 0.979881i 0.582309 + 0.582309i 0.0796647 1.99841i 1.46731 + 1.68731i −1.16439 0.0231993i 0.972933 0.972933i 1.87697 + 2.11589i 2.32183i −3.14961 0.282794i
267.7 −0.923868 1.07073i 1.14885 + 1.14885i −0.292934 + 1.97843i 1.95382 1.08746i 0.168724 2.29149i 2.70883 2.70883i 2.38900 1.51416i 0.360306i −2.96946 1.08735i
267.8 −0.830627 1.14458i −1.85040 1.85040i −0.620117 + 1.90143i 1.13686 + 1.92550i −0.580936 + 3.65493i 1.96775 1.96775i 2.69143 0.869611i 3.84798i 1.25958 2.90060i
267.9 −0.780925 + 1.17905i −2.27237 2.27237i −0.780312 1.84150i 0.440177 2.19231i 4.45378 0.904682i 2.19634 2.19634i 2.78058 + 0.518047i 7.32729i 2.24110 + 2.23102i
267.10 −0.600761 + 1.28027i 1.93295 + 1.93295i −1.27817 1.53827i 2.18132 + 0.491792i −3.63593 + 1.31345i −3.06024 + 3.06024i 2.73727 0.712271i 4.47257i −1.94007 + 2.49722i
267.11 −0.424847 1.34889i 2.38733 + 2.38733i −1.63901 + 1.14614i −2.23504 0.0679064i 2.20600 4.23450i 0.160827 0.160827i 2.24235 + 1.72391i 8.39872i 0.857951 + 3.04367i
267.12 −0.378581 1.36260i −1.58108 1.58108i −1.71335 + 1.03171i −1.74843 1.39392i −1.55581 + 2.75294i −1.94368 + 1.94368i 2.05445 + 1.94403i 1.99961i −1.23743 + 2.91012i
267.13 −0.291109 1.38393i 1.18694 + 1.18694i −1.83051 + 0.805748i 1.27222 1.83887i 1.29711 1.98816i −1.53689 + 1.53689i 1.64798 + 2.29873i 0.182360i −2.91522 1.22536i
267.14 0.0237221 + 1.41401i −1.56852 1.56852i −1.99887 + 0.0670868i −1.80219 + 1.32368i 2.18070 2.25511i 2.36081 2.36081i −0.142279 2.82485i 1.92048i −1.91445 2.51692i
267.15 0.0413255 + 1.41361i 0.135093 + 0.135093i −1.99658 + 0.116836i −0.869041 2.06028i −0.185386 + 0.196551i −0.485902 + 0.485902i −0.247670 2.81756i 2.96350i 2.87652 1.31363i
267.16 0.413585 + 1.35239i 2.24210 + 2.24210i −1.65789 + 1.11865i −0.349685 + 2.20856i −2.10488 + 3.95948i 2.40464 2.40464i −2.19853 1.77945i 7.05400i −3.13144 + 0.440517i
267.17 0.436150 1.34528i −1.71881 1.71881i −1.61955 1.17349i −0.923046 + 2.03666i −3.06193 + 1.56262i −0.323011 + 0.323011i −2.28503 + 1.66692i 2.90860i 2.33729 + 2.13004i
267.18 0.655429 + 1.25316i −0.124098 0.124098i −1.14082 + 1.64272i −0.0211435 + 2.23597i 0.0741774 0.236853i −2.60801 + 2.60801i −2.80632 0.352951i 2.96920i −2.81589 + 1.43902i
267.19 0.683080 1.23831i −0.494605 0.494605i −1.06680 1.69172i 1.31079 1.81158i −0.950327 + 0.274617i 0.327577 0.327577i −2.82358 + 0.165443i 2.51073i −1.34791 2.86062i
267.20 1.08061 + 0.912296i 1.34016 + 1.34016i 0.335432 + 1.97167i −2.21942 0.272360i 0.225567 + 2.67081i −0.697742 + 0.697742i −1.43628 + 2.43662i 0.592060i −2.14985 2.31908i
See all 52 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 267.26 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
20.e even 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 380.2.k.c 52
4.b odd 2 1 380.2.k.d yes 52
5.c odd 4 1 380.2.k.d yes 52
20.e even 4 1 inner 380.2.k.c 52

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
380.2.k.c 52 1.a even 1 1 trivial
380.2.k.c 52 20.e even 4 1 inner
380.2.k.d yes 52 4.b odd 2 1
380.2.k.d yes 52 5.c odd 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{52} + 2 T_{3}^{51} + 2 T_{3}^{50} + 364 T_{3}^{48} + 768 T_{3}^{47} + 808 T_{3}^{46} + \cdots + 6553600$$ acting on $$S_{2}^{\mathrm{new}}(380, [\chi])$$.