# Properties

 Label 378.2.u.e Level $378$ Weight $2$ Character orbit 378.u Analytic conductor $3.018$ Analytic rank $0$ Dimension $36$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [378,2,Mod(43,378)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(378, base_ring=CyclotomicField(18))

chi = DirichletCharacter(H, H._module([4, 0]))

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

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

chi := DirichletCharacter("378.43");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$378 = 2 \cdot 3^{3} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 378.u (of order $$9$$, degree $$6$$, 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: $$3.01834519640$$ Analytic rank: $$0$$ Dimension: $$36$$ Relative dimension: $$6$$ over $$\Q(\zeta_{9})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{9}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$36 q + 3 q^{3} + 3 q^{5} + 6 q^{6} + 18 q^{8} + 3 q^{9}+O(q^{10})$$ 36 * q + 3 * q^3 + 3 * q^5 + 6 * q^6 + 18 * q^8 + 3 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$36 q + 3 q^{3} + 3 q^{5} + 6 q^{6} + 18 q^{8} + 3 q^{9} - 3 q^{10} + 9 q^{13} - 6 q^{15} + 6 q^{18} - 15 q^{19} - 6 q^{20} + 3 q^{21} - 3 q^{24} - 3 q^{25} - 18 q^{26} - 30 q^{27} + 36 q^{28} - 6 q^{29} - 9 q^{30} + 60 q^{33} + 9 q^{34} + 3 q^{35} - 27 q^{37} + 15 q^{38} + 6 q^{39} - 3 q^{40} - 9 q^{41} - 6 q^{43} + 9 q^{44} + 42 q^{45} - 9 q^{46} + 36 q^{47} + 3 q^{48} - 15 q^{50} - 36 q^{51} + 9 q^{52} - 42 q^{53} - 27 q^{54} + 30 q^{55} - 18 q^{57} - 21 q^{58} + 12 q^{59} + 24 q^{60} + 3 q^{61} + 18 q^{62} + 12 q^{63} - 18 q^{64} - 84 q^{65} + 18 q^{66} - 69 q^{67} + 9 q^{68} - 48 q^{69} - 3 q^{70} + 12 q^{71} + 6 q^{72} - 12 q^{73} + 9 q^{75} - 15 q^{76} + 9 q^{77} - 48 q^{78} - 51 q^{79} - 6 q^{80} - 69 q^{81} + 15 q^{83} + 3 q^{84} - 12 q^{85} + 6 q^{86} + 84 q^{87} - 9 q^{88} - 3 q^{89} + 6 q^{90} - 9 q^{91} + 18 q^{92} - 21 q^{93} + 9 q^{94} - 75 q^{95} - 42 q^{97} + 18 q^{98} + 39 q^{99}+O(q^{100})$$ 36 * q + 3 * q^3 + 3 * q^5 + 6 * q^6 + 18 * q^8 + 3 * q^9 - 3 * q^10 + 9 * q^13 - 6 * q^15 + 6 * q^18 - 15 * q^19 - 6 * q^20 + 3 * q^21 - 3 * q^24 - 3 * q^25 - 18 * q^26 - 30 * q^27 + 36 * q^28 - 6 * q^29 - 9 * q^30 + 60 * q^33 + 9 * q^34 + 3 * q^35 - 27 * q^37 + 15 * q^38 + 6 * q^39 - 3 * q^40 - 9 * q^41 - 6 * q^43 + 9 * q^44 + 42 * q^45 - 9 * q^46 + 36 * q^47 + 3 * q^48 - 15 * q^50 - 36 * q^51 + 9 * q^52 - 42 * q^53 - 27 * q^54 + 30 * q^55 - 18 * q^57 - 21 * q^58 + 12 * q^59 + 24 * q^60 + 3 * q^61 + 18 * q^62 + 12 * q^63 - 18 * q^64 - 84 * q^65 + 18 * q^66 - 69 * q^67 + 9 * q^68 - 48 * q^69 - 3 * q^70 + 12 * q^71 + 6 * q^72 - 12 * q^73 + 9 * q^75 - 15 * q^76 + 9 * q^77 - 48 * q^78 - 51 * q^79 - 6 * q^80 - 69 * q^81 + 15 * q^83 + 3 * q^84 - 12 * q^85 + 6 * q^86 + 84 * q^87 - 9 * q^88 - 3 * q^89 + 6 * q^90 - 9 * q^91 + 18 * q^92 - 21 * q^93 + 9 * q^94 - 75 * q^95 - 42 * q^97 + 18 * q^98 + 39 * 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   $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
43.1 −0.766044 0.642788i −1.73154 0.0419083i 0.173648 + 0.984808i −0.354259 0.128940i 1.29950 + 1.14512i 0.173648 0.984808i 0.500000 0.866025i 2.99649 + 0.145132i 0.188497 + 0.326487i
43.2 −0.766044 0.642788i −1.15570 1.29010i 0.173648 + 0.984808i −2.06967 0.753297i 0.0560567 + 1.73114i 0.173648 0.984808i 0.500000 0.866025i −0.328718 + 2.98194i 1.10125 + 1.90741i
43.3 −0.766044 0.642788i −0.910669 + 1.47332i 0.173648 + 0.984808i 1.95320 + 0.710905i 1.64465 0.543265i 0.173648 0.984808i 0.500000 0.866025i −1.34137 2.68342i −1.03927 1.80007i
43.4 −0.766044 0.642788i 0.391603 1.68720i 0.173648 + 0.984808i 2.08466 + 0.758756i −1.38450 + 1.04075i 0.173648 0.984808i 0.500000 0.866025i −2.69329 1.32143i −1.10923 1.92124i
43.5 −0.766044 0.642788i 1.44903 + 0.948848i 0.173648 + 0.984808i −3.88962 1.41571i −0.500114 1.65828i 0.173648 0.984808i 0.500000 0.866025i 1.19938 + 2.74982i 2.06962 + 3.58470i
43.6 −0.766044 0.642788i 1.69123 + 0.373800i 0.173648 + 0.984808i 3.54173 + 1.28909i −1.05529 1.37345i 0.173648 0.984808i 0.500000 0.866025i 2.72055 + 1.26437i −1.88452 3.26408i
85.1 0.939693 + 0.342020i −1.62983 0.586228i 0.766044 + 0.642788i 0.184482 1.04625i −1.33103 1.10831i 0.766044 0.642788i 0.500000 + 0.866025i 2.31267 + 1.91090i 0.531195 0.920056i
85.2 0.939693 + 0.342020i −1.33644 + 1.10178i 0.766044 + 0.642788i −0.619299 + 3.51222i −1.63268 + 0.578248i 0.766044 0.642788i 0.500000 + 0.866025i 0.572147 2.94494i −1.78320 + 3.08859i
85.3 0.939693 + 0.342020i 0.873169 1.49585i 0.766044 + 0.642788i −0.605889 + 3.43617i 1.33212 1.10700i 0.766044 0.642788i 0.500000 + 0.866025i −1.47515 2.61227i −1.74459 + 3.02172i
85.4 0.939693 + 0.342020i 0.927705 + 1.46266i 0.766044 + 0.642788i 0.587615 3.33253i 0.371500 + 1.69174i 0.766044 0.642788i 0.500000 + 0.866025i −1.27873 + 2.71383i 1.69197 2.93058i
85.5 0.939693 + 0.342020i 1.28430 + 1.16214i 0.766044 + 0.642788i −0.345781 + 1.96102i 0.809368 + 1.53131i 0.766044 0.642788i 0.500000 + 0.866025i 0.298838 + 2.98508i −0.995637 + 1.72449i
85.6 0.939693 + 0.342020i 1.32079 1.12050i 0.766044 + 0.642788i 0.359180 2.03701i 1.62437 0.601188i 0.766044 0.642788i 0.500000 + 0.866025i 0.488967 2.95988i 1.03422 1.79132i
169.1 0.939693 0.342020i −1.62983 + 0.586228i 0.766044 0.642788i 0.184482 + 1.04625i −1.33103 + 1.10831i 0.766044 + 0.642788i 0.500000 0.866025i 2.31267 1.91090i 0.531195 + 0.920056i
169.2 0.939693 0.342020i −1.33644 1.10178i 0.766044 0.642788i −0.619299 3.51222i −1.63268 0.578248i 0.766044 + 0.642788i 0.500000 0.866025i 0.572147 + 2.94494i −1.78320 3.08859i
169.3 0.939693 0.342020i 0.873169 + 1.49585i 0.766044 0.642788i −0.605889 3.43617i 1.33212 + 1.10700i 0.766044 + 0.642788i 0.500000 0.866025i −1.47515 + 2.61227i −1.74459 3.02172i
169.4 0.939693 0.342020i 0.927705 1.46266i 0.766044 0.642788i 0.587615 + 3.33253i 0.371500 1.69174i 0.766044 + 0.642788i 0.500000 0.866025i −1.27873 2.71383i 1.69197 + 2.93058i
169.5 0.939693 0.342020i 1.28430 1.16214i 0.766044 0.642788i −0.345781 1.96102i 0.809368 1.53131i 0.766044 + 0.642788i 0.500000 0.866025i 0.298838 2.98508i −0.995637 1.72449i
169.6 0.939693 0.342020i 1.32079 + 1.12050i 0.766044 0.642788i 0.359180 + 2.03701i 1.62437 + 0.601188i 0.766044 + 0.642788i 0.500000 0.866025i 0.488967 + 2.95988i 1.03422 + 1.79132i
211.1 −0.766044 + 0.642788i −1.73154 + 0.0419083i 0.173648 0.984808i −0.354259 + 0.128940i 1.29950 1.14512i 0.173648 + 0.984808i 0.500000 + 0.866025i 2.99649 0.145132i 0.188497 0.326487i
211.2 −0.766044 + 0.642788i −1.15570 + 1.29010i 0.173648 0.984808i −2.06967 + 0.753297i 0.0560567 1.73114i 0.173648 + 0.984808i 0.500000 + 0.866025i −0.328718 2.98194i 1.10125 1.90741i
See all 36 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 43.6 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
27.e even 9 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 378.2.u.e 36
27.e even 9 1 inner 378.2.u.e 36

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
378.2.u.e 36 1.a even 1 1 trivial
378.2.u.e 36 27.e even 9 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{36} - 3 T_{5}^{35} + 6 T_{5}^{34} - 15 T_{5}^{33} - 9 T_{5}^{32} + 297 T_{5}^{31} + \cdots + 7766544384$$ acting on $$S_{2}^{\mathrm{new}}(378, [\chi])$$.