# Properties

 Label 378.2.u.d Level $378$ Weight $2$ Character orbit 378.u Analytic conductor $3.018$ Analytic rank $0$ Dimension $30$ 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: $$30$$ Relative dimension: $$5$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$30 q - 3 q^{3} + 3 q^{5} - 6 q^{6} - 15 q^{8} + 3 q^{9}+O(q^{10})$$ 30 * q - 3 * q^3 + 3 * q^5 - 6 * q^6 - 15 * q^8 + 3 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$30 q - 3 q^{3} + 3 q^{5} - 6 q^{6} - 15 q^{8} + 3 q^{9} - 3 q^{10} + 9 q^{11} + 3 q^{12} - 9 q^{13} + 12 q^{15} - 12 q^{18} - 15 q^{19} - 6 q^{20} - 3 q^{21} + 9 q^{22} - 12 q^{23} - 3 q^{24} + 21 q^{25} + 18 q^{26} + 3 q^{27} - 30 q^{28} - 30 q^{29} - 15 q^{30} + 18 q^{31} + 9 q^{33} + 12 q^{34} + 3 q^{35} - 12 q^{36} + 3 q^{37} - 3 q^{38} + 72 q^{39} + 3 q^{40} + 36 q^{41} - 3 q^{42} - 9 q^{43} - 3 q^{44} - 48 q^{45} - 15 q^{46} + 3 q^{48} + 3 q^{50} - 6 q^{51} - 9 q^{52} + 6 q^{53} + 45 q^{54} + 66 q^{55} - 51 q^{57} - 3 q^{58} - 48 q^{59} + 24 q^{60} + 57 q^{61} - 18 q^{62} - 15 q^{63} - 15 q^{64} + 24 q^{65} + 54 q^{66} + 6 q^{67} - 15 q^{68} - 48 q^{69} - 3 q^{70} - 24 q^{71} - 6 q^{72} - 36 q^{73} - 48 q^{74} - 129 q^{75} - 3 q^{76} + 9 q^{77} + 24 q^{78} - 21 q^{79} + 6 q^{80} + 27 q^{81} - 15 q^{83} + 3 q^{84} - 72 q^{85} - 9 q^{86} + 42 q^{87} - 9 q^{88} - 30 q^{89} - 30 q^{90} + 9 q^{91} + 6 q^{92} + 111 q^{93} - 45 q^{94} + 81 q^{95} - 6 q^{96} - 15 q^{97} - 15 q^{98} + 45 q^{99}+O(q^{100})$$ 30 * q - 3 * q^3 + 3 * q^5 - 6 * q^6 - 15 * q^8 + 3 * q^9 - 3 * q^10 + 9 * q^11 + 3 * q^12 - 9 * q^13 + 12 * q^15 - 12 * q^18 - 15 * q^19 - 6 * q^20 - 3 * q^21 + 9 * q^22 - 12 * q^23 - 3 * q^24 + 21 * q^25 + 18 * q^26 + 3 * q^27 - 30 * q^28 - 30 * q^29 - 15 * q^30 + 18 * q^31 + 9 * q^33 + 12 * q^34 + 3 * q^35 - 12 * q^36 + 3 * q^37 - 3 * q^38 + 72 * q^39 + 3 * q^40 + 36 * q^41 - 3 * q^42 - 9 * q^43 - 3 * q^44 - 48 * q^45 - 15 * q^46 + 3 * q^48 + 3 * q^50 - 6 * q^51 - 9 * q^52 + 6 * q^53 + 45 * q^54 + 66 * q^55 - 51 * q^57 - 3 * q^58 - 48 * q^59 + 24 * q^60 + 57 * q^61 - 18 * q^62 - 15 * q^63 - 15 * q^64 + 24 * q^65 + 54 * q^66 + 6 * q^67 - 15 * q^68 - 48 * q^69 - 3 * q^70 - 24 * q^71 - 6 * q^72 - 36 * q^73 - 48 * q^74 - 129 * q^75 - 3 * q^76 + 9 * q^77 + 24 * q^78 - 21 * q^79 + 6 * q^80 + 27 * q^81 - 15 * q^83 + 3 * q^84 - 72 * q^85 - 9 * q^86 + 42 * q^87 - 9 * q^88 - 30 * q^89 - 30 * q^90 + 9 * q^91 + 6 * q^92 + 111 * q^93 - 45 * q^94 + 81 * q^95 - 6 * q^96 - 15 * q^97 - 15 * q^98 + 45 * 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 −1.62854 0.589807i 0.173648 + 0.984808i −1.36620 0.497255i −0.868410 1.49862i −0.173648 + 0.984808i −0.500000 + 0.866025i 2.30426 + 1.92104i −0.726937 1.25909i
43.2 0.766044 + 0.642788i −0.764934 1.55399i 0.173648 + 0.984808i 3.01809 + 1.09849i 0.412911 1.68211i −0.173648 + 0.984808i −0.500000 + 0.866025i −1.82975 + 2.37739i 1.60589 + 2.78148i
43.3 0.766044 + 0.642788i −0.167318 + 1.72395i 0.173648 + 0.984808i 2.66798 + 0.971065i −1.23631 + 1.21307i −0.173648 + 0.984808i −0.500000 + 0.866025i −2.94401 0.576895i 1.41960 + 2.45882i
43.4 0.766044 + 0.642788i 0.503821 + 1.65716i 0.173648 + 0.984808i −4.01520 1.46142i −0.679250 + 1.59330i −0.173648 + 0.984808i −0.500000 + 0.866025i −2.49233 + 1.66982i −2.13645 3.70043i
43.5 0.766044 + 0.642788i 1.73061 0.0705192i 0.173648 + 0.984808i −0.918005 0.334126i 1.37106 + 1.05840i −0.173648 + 0.984808i −0.500000 + 0.866025i 2.99005 0.244083i −0.488460 0.846038i
85.1 −0.939693 0.342020i −1.73005 + 0.0833302i 0.766044 + 0.642788i 0.111050 0.629795i 1.65421 + 0.513405i −0.766044 + 0.642788i −0.500000 0.866025i 2.98611 0.288330i −0.319755 + 0.553832i
85.2 −0.939693 0.342020i −1.21448 1.23493i 0.766044 + 0.642788i −0.386963 + 2.19458i 0.718864 + 1.57583i −0.766044 + 0.642788i −0.500000 0.866025i −0.0500961 + 2.99958i 1.11422 1.92988i
85.3 −0.939693 0.342020i 0.344942 1.69736i 0.766044 + 0.642788i −0.445239 + 2.52508i −0.904669 + 1.47702i −0.766044 + 0.642788i −0.500000 0.866025i −2.76203 1.17098i 1.28201 2.22051i
85.4 −0.939693 0.342020i 1.22833 + 1.22115i 0.766044 + 0.642788i −0.0262315 + 0.148766i −0.736599 1.56762i −0.766044 + 0.642788i −0.500000 0.866025i 0.0176026 + 2.99995i 0.0755306 0.130823i
85.5 −0.939693 0.342020i 1.63729 0.565046i 0.766044 + 0.642788i 0.654987 3.71462i −1.73181 0.0290168i −0.766044 + 0.642788i −0.500000 0.866025i 2.36145 1.85029i −1.88596 + 3.26658i
169.1 −0.939693 + 0.342020i −1.73005 0.0833302i 0.766044 0.642788i 0.111050 + 0.629795i 1.65421 0.513405i −0.766044 0.642788i −0.500000 + 0.866025i 2.98611 + 0.288330i −0.319755 0.553832i
169.2 −0.939693 + 0.342020i −1.21448 + 1.23493i 0.766044 0.642788i −0.386963 2.19458i 0.718864 1.57583i −0.766044 0.642788i −0.500000 + 0.866025i −0.0500961 2.99958i 1.11422 + 1.92988i
169.3 −0.939693 + 0.342020i 0.344942 + 1.69736i 0.766044 0.642788i −0.445239 2.52508i −0.904669 1.47702i −0.766044 0.642788i −0.500000 + 0.866025i −2.76203 + 1.17098i 1.28201 + 2.22051i
169.4 −0.939693 + 0.342020i 1.22833 1.22115i 0.766044 0.642788i −0.0262315 0.148766i −0.736599 + 1.56762i −0.766044 0.642788i −0.500000 + 0.866025i 0.0176026 2.99995i 0.0755306 + 0.130823i
169.5 −0.939693 + 0.342020i 1.63729 + 0.565046i 0.766044 0.642788i 0.654987 + 3.71462i −1.73181 + 0.0290168i −0.766044 0.642788i −0.500000 + 0.866025i 2.36145 + 1.85029i −1.88596 3.26658i
211.1 0.766044 0.642788i −1.62854 + 0.589807i 0.173648 0.984808i −1.36620 + 0.497255i −0.868410 + 1.49862i −0.173648 0.984808i −0.500000 0.866025i 2.30426 1.92104i −0.726937 + 1.25909i
211.2 0.766044 0.642788i −0.764934 + 1.55399i 0.173648 0.984808i 3.01809 1.09849i 0.412911 + 1.68211i −0.173648 0.984808i −0.500000 0.866025i −1.82975 2.37739i 1.60589 2.78148i
211.3 0.766044 0.642788i −0.167318 1.72395i 0.173648 0.984808i 2.66798 0.971065i −1.23631 1.21307i −0.173648 0.984808i −0.500000 0.866025i −2.94401 + 0.576895i 1.41960 2.45882i
211.4 0.766044 0.642788i 0.503821 1.65716i 0.173648 0.984808i −4.01520 + 1.46142i −0.679250 1.59330i −0.173648 0.984808i −0.500000 0.866025i −2.49233 1.66982i −2.13645 + 3.70043i
211.5 0.766044 0.642788i 1.73061 + 0.0705192i 0.173648 0.984808i −0.918005 + 0.334126i 1.37106 1.05840i −0.173648 0.984808i −0.500000 0.866025i 2.99005 + 0.244083i −0.488460 + 0.846038i
See all 30 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 43.5 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.d 30
27.e even 9 1 inner 378.2.u.d 30

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
378.2.u.d 30 1.a even 1 1 trivial
378.2.u.d 30 27.e even 9 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{30} - 3 T_{5}^{29} - 6 T_{5}^{28} + 53 T_{5}^{27} - 81 T_{5}^{26} - 333 T_{5}^{25} + \cdots + 185761$$ acting on $$S_{2}^{\mathrm{new}}(378, [\chi])$$.