# Properties

 Label 380.2.n.b Level $380$ Weight $2$ Character orbit 380.n Analytic conductor $3.034$ Analytic rank $0$ Dimension $40$ CM no Inner twists $4$

# Learn more

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [380,2,Mod(31,380)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(380, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([3, 0, 5]))

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

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

chi := DirichletCharacter("380.31");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$380 = 2^{2} \cdot 5 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 380.n (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: $$3.03431527681$$ Analytic rank: $$0$$ Dimension: $$40$$ Relative dimension: $$20$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$40 q + 3 q^{2} + q^{4} - 20 q^{5} - 5 q^{6} - 22 q^{9}+O(q^{10})$$ 40 * q + 3 * q^2 + q^4 - 20 * q^5 - 5 * q^6 - 22 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$40 q + 3 q^{2} + q^{4} - 20 q^{5} - 5 q^{6} - 22 q^{9} - 3 q^{10} + 12 q^{13} - 18 q^{14} + 9 q^{16} - 12 q^{17} - 2 q^{20} + 12 q^{21} + 20 q^{24} - 20 q^{25} + 34 q^{26} + 12 q^{28} + 10 q^{30} - 12 q^{32} - 6 q^{33} - 27 q^{34} - 6 q^{36} + 12 q^{38} + 36 q^{41} - 21 q^{42} + 28 q^{44} + 44 q^{45} + 36 q^{48} - 60 q^{49} + 15 q^{52} - 66 q^{53} - 31 q^{54} + 28 q^{57} + 2 q^{58} + 3 q^{60} + 4 q^{61} - 57 q^{62} - 80 q^{64} + 18 q^{66} - 20 q^{68} + 18 q^{70} + 42 q^{72} - 18 q^{73} + 6 q^{74} + 30 q^{76} + 12 q^{77} + 144 q^{78} + 9 q^{80} + 16 q^{81} - 41 q^{82} - 12 q^{85} + 18 q^{86} - 18 q^{89} + 39 q^{90} + 40 q^{92} - 8 q^{93} - 160 q^{96} - 18 q^{97} - 66 q^{98}+O(q^{100})$$ 40 * q + 3 * q^2 + q^4 - 20 * q^5 - 5 * q^6 - 22 * q^9 - 3 * q^10 + 12 * q^13 - 18 * q^14 + 9 * q^16 - 12 * q^17 - 2 * q^20 + 12 * q^21 + 20 * q^24 - 20 * q^25 + 34 * q^26 + 12 * q^28 + 10 * q^30 - 12 * q^32 - 6 * q^33 - 27 * q^34 - 6 * q^36 + 12 * q^38 + 36 * q^41 - 21 * q^42 + 28 * q^44 + 44 * q^45 + 36 * q^48 - 60 * q^49 + 15 * q^52 - 66 * q^53 - 31 * q^54 + 28 * q^57 + 2 * q^58 + 3 * q^60 + 4 * q^61 - 57 * q^62 - 80 * q^64 + 18 * q^66 - 20 * q^68 + 18 * q^70 + 42 * q^72 - 18 * q^73 + 6 * q^74 + 30 * q^76 + 12 * q^77 + 144 * q^78 + 9 * q^80 + 16 * q^81 - 41 * q^82 - 12 * q^85 + 18 * q^86 - 18 * q^89 + 39 * q^90 + 40 * q^92 - 8 * q^93 - 160 * q^96 - 18 * q^97 - 66 * q^98

## 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   $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
31.1 −1.41089 0.0968610i 0.717455 + 1.24267i 1.98124 + 0.273321i −0.500000 0.866025i −0.891886 1.82277i 2.27676i −2.76884 0.577531i 0.470516 0.814958i 0.621562 + 1.27030i
31.2 −1.33235 0.474182i −0.435478 0.754271i 1.55030 + 1.26355i −0.500000 0.866025i 0.222547 + 1.21145i 3.15791i −1.46639 2.41862i 1.12072 1.94114i 0.255520 + 1.39094i
31.3 −1.31582 + 0.518293i −0.736860 1.27628i 1.46274 1.36396i −0.500000 0.866025i 1.63106 + 1.29744i 0.456236i −1.21777 + 2.55285i 0.414074 0.717197i 1.10676 + 0.880384i
31.4 −1.18980 0.764449i −1.38246 2.39450i 0.831236 + 1.81908i −0.500000 0.866025i −0.185619 + 3.90579i 2.18272i 0.401590 2.79977i −2.32241 + 4.02254i −0.0671334 + 1.41262i
31.5 −1.10676 + 0.880384i 0.736860 + 1.27628i 0.449849 1.94875i −0.500000 0.866025i −1.93915 0.763819i 0.456236i 1.21777 + 2.55285i 0.414074 0.717197i 1.31582 + 0.518293i
31.6 −0.785164 1.17623i 1.08269 + 1.87527i −0.767036 + 1.84707i −0.500000 0.866025i 1.35566 2.74588i 3.94804i 2.77483 0.548038i −0.844418 + 1.46258i −0.626064 + 1.26809i
31.7 −0.621562 + 1.27030i −0.717455 1.24267i −1.22732 1.57914i −0.500000 0.866025i 2.02450 0.138987i 2.27676i 2.76884 0.577531i 0.470516 0.814958i 1.41089 0.0968610i
31.8 −0.255520 + 1.39094i 0.435478 + 0.754271i −1.86942 0.710826i −0.500000 0.866025i −1.16042 + 0.412992i 3.15791i 1.46639 2.41862i 1.12072 1.94114i 1.33235 0.474182i
31.9 −0.0867802 1.41155i 0.980171 + 1.69771i −1.98494 + 0.244989i −0.500000 0.866025i 2.31133 1.53089i 1.97181i 0.518067 + 2.78058i −0.421469 + 0.730006i −1.17905 + 0.780928i
31.10 0.0669309 1.41263i −0.665205 1.15217i −1.99104 0.189097i −0.500000 0.866025i −1.67211 + 0.862572i 4.68878i −0.400386 + 2.79994i 0.615006 1.06522i −1.25684 + 0.648351i
31.11 0.0671334 + 1.41262i 1.38246 + 2.39450i −1.99099 + 0.189668i −0.500000 0.866025i −3.28970 + 2.11365i 2.18272i −0.401590 2.79977i −2.32241 + 4.02254i 1.18980 0.764449i
31.12 0.337532 1.37334i −1.50046 2.59888i −1.77214 0.927096i −0.500000 0.866025i −4.07560 + 1.18344i 4.06445i −1.87138 + 2.12084i −3.00277 + 5.20095i −1.35812 + 0.394360i
31.13 0.626064 + 1.26809i −1.08269 1.87527i −1.21609 + 1.58781i −0.500000 0.866025i 1.70017 2.54698i 3.94804i −2.77483 0.548038i −0.844418 + 1.46258i 0.785164 1.17623i
31.14 0.864116 1.11951i 0.0154003 + 0.0266742i −0.506608 1.93477i −0.500000 0.866025i 0.0431697 + 0.00580873i 0.370928i −2.60377 1.10472i 1.49953 2.59725i −1.40158 0.188591i
31.15 1.17905 + 0.780928i −0.980171 1.69771i 0.780303 + 1.84150i −0.500000 0.866025i 0.170119 2.76712i 1.97181i −0.518067 + 2.78058i −0.421469 + 0.730006i 0.0867802 1.41155i
31.16 1.19364 0.758433i 1.50479 + 2.60637i 0.849559 1.81059i −0.500000 0.866025i 3.77293 + 1.96979i 2.59896i −0.359145 2.80553i −3.02877 + 5.24598i −1.25364 0.654507i
31.17 1.25364 0.654507i −1.50479 2.60637i 1.14324 1.64104i −0.500000 0.866025i −3.59235 2.28256i 2.59896i 0.359145 2.80553i −3.02877 + 5.24598i −1.19364 0.758433i
31.18 1.25684 + 0.648351i 0.665205 + 1.15217i 1.15928 + 1.62974i −0.500000 0.866025i 0.0890455 + 1.87937i 4.68878i 0.400386 + 2.79994i 0.615006 1.06522i −0.0669309 1.41263i
31.19 1.35812 + 0.394360i 1.50046 + 2.59888i 1.68896 + 1.07117i −0.500000 0.866025i 1.01291 + 4.12130i 4.06445i 1.87138 + 2.12084i −3.00277 + 5.20095i −0.337532 1.37334i
31.20 1.40158 0.188591i −0.0154003 0.0266742i 1.92887 0.528652i −0.500000 0.866025i −0.0266153 0.0344817i 0.370928i 2.60377 1.10472i 1.49953 2.59725i −0.864116 1.11951i
See all 40 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 31.20 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
19.d odd 6 1 inner
76.f even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 380.2.n.b 40
4.b odd 2 1 inner 380.2.n.b 40
19.d odd 6 1 inner 380.2.n.b 40
76.f even 6 1 inner 380.2.n.b 40

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
380.2.n.b 40 1.a even 1 1 trivial
380.2.n.b 40 4.b odd 2 1 inner
380.2.n.b 40 19.d odd 6 1 inner
380.2.n.b 40 76.f even 6 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{40} + 41 T_{3}^{38} + 976 T_{3}^{36} + 15639 T_{3}^{34} + 187640 T_{3}^{32} + 1737687 T_{3}^{30} + 12807081 T_{3}^{28} + 75729868 T_{3}^{26} + 363589113 T_{3}^{24} + 1417213175 T_{3}^{22} + \cdots + 4096$$ acting on $$S_{2}^{\mathrm{new}}(380, [\chi])$$.