# Properties

 Label 378.2.u.d Level 378 Weight 2 Character orbit 378.u Analytic conductor 3.018 Analytic rank 0 Dimension 30 CM no Inner twists 2

# Related objects

## Newspace parameters

 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

 Self dual: no Analytic conductor: $$3.01834519640$$ Analytic rank: $$0$$ Dimension: $$30$$ Relative dimension: $$5$$ over $$\Q(\zeta_{9})$$ Coefficient ring index: multiple of None Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{9}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$30q - 3q^{3} + 3q^{5} - 6q^{6} - 15q^{8} + 3q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$30q - 3q^{3} + 3q^{5} - 6q^{6} - 15q^{8} + 3q^{9} - 3q^{10} + 9q^{11} + 3q^{12} - 9q^{13} + 12q^{15} - 12q^{18} - 15q^{19} - 6q^{20} - 3q^{21} + 9q^{22} - 12q^{23} - 3q^{24} + 21q^{25} + 18q^{26} + 3q^{27} - 30q^{28} - 30q^{29} - 15q^{30} + 18q^{31} + 9q^{33} + 12q^{34} + 3q^{35} - 12q^{36} + 3q^{37} - 3q^{38} + 72q^{39} + 3q^{40} + 36q^{41} - 3q^{42} - 9q^{43} - 3q^{44} - 48q^{45} - 15q^{46} + 3q^{48} + 3q^{50} - 6q^{51} - 9q^{52} + 6q^{53} + 45q^{54} + 66q^{55} - 51q^{57} - 3q^{58} - 48q^{59} + 24q^{60} + 57q^{61} - 18q^{62} - 15q^{63} - 15q^{64} + 24q^{65} + 54q^{66} + 6q^{67} - 15q^{68} - 48q^{69} - 3q^{70} - 24q^{71} - 6q^{72} - 36q^{73} - 48q^{74} - 129q^{75} - 3q^{76} + 9q^{77} + 24q^{78} - 21q^{79} + 6q^{80} + 27q^{81} - 15q^{83} + 3q^{84} - 72q^{85} - 9q^{86} + 42q^{87} - 9q^{88} - 30q^{89} - 30q^{90} + 9q^{91} + 6q^{92} + 111q^{93} - 45q^{94} + 81q^{95} - 6q^{96} - 15q^{97} - 15q^{98} + 45q^{99} + O(q^{100})$$

## 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.

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 337.5 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 $$T_{5}^{30} - \cdots$$ acting on $$S_{2}^{\mathrm{new}}(378, [\chi])$$.

## Hecke Characteristic Polynomials

There are no characteristic polynomials of Hecke operators in the database