# Properties

 Label 378.2.w.a Level 378 Weight 2 Character orbit 378.w Analytic conductor 3.018 Analytic rank 0 Dimension 72 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.w (of order $$9$$, degree $$6$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$3.01834519640$$ Analytic rank: $$0$$ Dimension: $$72$$ Relative dimension: $$12$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$72q + 3q^{6} - 3q^{7} - 36q^{8} - 12q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$72q + 3q^{6} - 3q^{7} - 36q^{8} - 12q^{9} + 12q^{10} - 6q^{11} - 12q^{13} - 3q^{14} - 6q^{15} + 24q^{17} + 36q^{19} - 12q^{21} - 6q^{22} + 12q^{23} - 30q^{25} - 18q^{26} - 6q^{27} + 3q^{29} + 3q^{30} + 9q^{31} - 18q^{33} + 9q^{34} + 9q^{35} + 9q^{36} + 3q^{39} - 6q^{41} + 3q^{42} + 24q^{43} - 30q^{45} - 9q^{47} - 6q^{48} + 51q^{49} + 6q^{50} + 12q^{51} + 6q^{52} - 15q^{53} - 27q^{54} + 72q^{55} + 6q^{56} - 63q^{57} + 3q^{58} + 15q^{59} - 3q^{60} - 18q^{61} - 24q^{62} - 48q^{63} - 36q^{64} - 18q^{65} - 36q^{66} + 66q^{67} - 18q^{68} - 21q^{69} - 6q^{70} + 12q^{71} - 12q^{72} - 66q^{73} + 9q^{74} + 15q^{75} - 15q^{77} + 30q^{78} + 9q^{79} - 6q^{80} - 33q^{82} - 18q^{83} - 12q^{84} + 21q^{85} - 12q^{86} - 48q^{87} + 12q^{88} + 72q^{89} + 69q^{90} + 12q^{91} + 30q^{92} + 60q^{93} - 36q^{94} + 93q^{95} - 48q^{97} + 6q^{98} - 54q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
25.1 −0.939693 + 0.342020i −1.70460 + 0.307144i 0.766044 0.642788i −2.61773 0.952774i 1.49675 0.871628i −1.47144 2.19883i −0.500000 + 0.866025i 2.81133 1.04711i 2.78573
25.2 −0.939693 + 0.342020i −1.58205 0.705062i 0.766044 0.642788i 1.56427 + 0.569347i 1.72779 + 0.121449i 0.319985 2.62633i −0.500000 + 0.866025i 2.00577 + 2.23089i −1.66466
25.3 −0.939693 + 0.342020i −1.25749 + 1.19110i 0.766044 0.642788i 1.98579 + 0.722770i 0.774279 1.54935i 2.63247 + 0.264792i −0.500000 + 0.866025i 0.162584 2.99559i −2.11324
25.4 −0.939693 + 0.342020i −1.03071 1.39199i 0.766044 0.642788i −3.69961 1.34655i 1.44464 + 0.955517i 0.644767 + 2.56598i −0.500000 + 0.866025i −0.875266 + 2.86948i 3.93704
25.5 −0.939693 + 0.342020i −0.649954 1.60548i 0.766044 0.642788i 3.27017 + 1.19024i 1.15986 + 1.28636i 1.64949 + 2.06862i −0.500000 + 0.866025i −2.15512 + 2.08697i −3.48004
25.6 −0.939693 + 0.342020i −0.195246 1.72101i 0.766044 0.642788i 0.182581 + 0.0664541i 0.772092 + 1.55044i −2.63939 0.183356i −0.500000 + 0.866025i −2.92376 + 0.672041i −0.194299
25.7 −0.939693 + 0.342020i −0.135903 + 1.72671i 0.766044 0.642788i −1.85969 0.676871i −0.462863 1.66906i 2.63192 + 0.270209i −0.500000 + 0.866025i −2.96306 0.469329i 1.97904
25.8 −0.939693 + 0.342020i 0.807797 + 1.53214i 0.766044 0.642788i 1.59297 + 0.579794i −1.28310 1.16346i −2.02988 + 1.69693i −0.500000 + 0.866025i −1.69493 + 2.47532i −1.69520
25.9 −0.939693 + 0.342020i 1.06159 1.36859i 0.766044 0.642788i −1.03287 0.375933i −0.529482 + 1.64914i 2.42642 1.05474i −0.500000 + 0.866025i −0.746062 2.90575i 1.09915
25.10 −0.939693 + 0.342020i 1.14375 + 1.30071i 0.766044 0.642788i 1.59112 + 0.579119i −1.51964 0.831077i 0.142417 2.64192i −0.500000 + 0.866025i −0.383668 + 2.97537i −1.69323
25.11 −0.939693 + 0.342020i 1.65621 0.506908i 0.766044 0.642788i 0.501605 + 0.182569i −1.38296 + 1.04280i −1.88202 + 1.85957i −0.500000 + 0.866025i 2.48609 1.67910i −0.533797
25.12 −0.939693 + 0.342020i 1.71296 + 0.256429i 0.766044 0.642788i −3.35800 1.22221i −1.69736 + 0.344903i −2.15868 1.52974i −0.500000 + 0.866025i 2.86849 + 0.878508i 3.57351
121.1 −0.939693 0.342020i −1.70460 0.307144i 0.766044 + 0.642788i −2.61773 + 0.952774i 1.49675 + 0.871628i −1.47144 + 2.19883i −0.500000 0.866025i 2.81133 + 1.04711i 2.78573
121.2 −0.939693 0.342020i −1.58205 + 0.705062i 0.766044 + 0.642788i 1.56427 0.569347i 1.72779 0.121449i 0.319985 + 2.62633i −0.500000 0.866025i 2.00577 2.23089i −1.66466
121.3 −0.939693 0.342020i −1.25749 1.19110i 0.766044 + 0.642788i 1.98579 0.722770i 0.774279 + 1.54935i 2.63247 0.264792i −0.500000 0.866025i 0.162584 + 2.99559i −2.11324
121.4 −0.939693 0.342020i −1.03071 + 1.39199i 0.766044 + 0.642788i −3.69961 + 1.34655i 1.44464 0.955517i 0.644767 2.56598i −0.500000 0.866025i −0.875266 2.86948i 3.93704
121.5 −0.939693 0.342020i −0.649954 + 1.60548i 0.766044 + 0.642788i 3.27017 1.19024i 1.15986 1.28636i 1.64949 2.06862i −0.500000 0.866025i −2.15512 2.08697i −3.48004
121.6 −0.939693 0.342020i −0.195246 + 1.72101i 0.766044 + 0.642788i 0.182581 0.0664541i 0.772092 1.55044i −2.63939 + 0.183356i −0.500000 0.866025i −2.92376 0.672041i −0.194299
121.7 −0.939693 0.342020i −0.135903 1.72671i 0.766044 + 0.642788i −1.85969 + 0.676871i −0.462863 + 1.66906i 2.63192 0.270209i −0.500000 0.866025i −2.96306 + 0.469329i 1.97904
121.8 −0.939693 0.342020i 0.807797 1.53214i 0.766044 + 0.642788i 1.59297 0.579794i −1.28310 + 1.16346i −2.02988 1.69693i −0.500000 0.866025i −1.69493 2.47532i −1.69520
See all 72 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 373.12 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
189.w 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.w.a yes 72
7.c even 3 1 378.2.v.b 72
27.e even 9 1 378.2.v.b 72
189.w even 9 1 inner 378.2.w.a yes 72

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
378.2.v.b 72 7.c even 3 1
378.2.v.b 72 27.e even 9 1
378.2.w.a yes 72 1.a even 1 1 trivial
378.2.w.a yes 72 189.w even 9 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{72} + \cdots$$ acting on $$S_{2}^{\mathrm{new}}(378, [\chi])$$.

## Hecke Characteristic Polynomials

There are no characteristic polynomials of Hecke operators in the database