# Properties

 Label 378.2.v.b Level 378 Weight 2 Character orbit 378.v 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.v (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.

 $$\operatorname{Tr}(f)(q) =$$ $$72q + 3q^{6} + 6q^{7} - 36q^{8} + 24q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$72q + 3q^{6} + 6q^{7} - 36q^{8} + 24q^{9} - 6q^{10} + 12q^{11} - 12q^{13} - 3q^{14} - 6q^{15} - 12q^{17} - 18q^{19} - 30q^{21} - 6q^{22} + 30q^{23} + 6q^{25} - 18q^{26} - 6q^{27} + 3q^{29} + 3q^{30} + 9q^{31} - 18q^{33} + 9q^{34} - 18q^{35} + 9q^{36} - 24q^{39} - 6q^{41} + 12q^{42} + 24q^{43} + 51q^{45} + 45q^{47} - 6q^{48} - 12q^{49} + 6q^{50} - 51q^{51} + 6q^{52} - 15q^{53} + 27q^{54} + 72q^{55} - 3q^{56} - 63q^{57} + 3q^{58} - 30q^{59} - 3q^{60} + 9q^{61} - 24q^{62} - 39q^{63} - 36q^{64} + 9q^{65} - 6q^{67} + 9q^{68} - 21q^{69} - 33q^{70} + 12q^{71} - 12q^{72} + 132q^{73} - 18q^{74} - 30q^{75} - 33q^{77} + 30q^{78} - 9q^{79} - 6q^{80} + 36q^{81} - 33q^{82} - 18q^{83} + 15q^{84} + 21q^{85} - 12q^{86} - 39q^{87} - 6q^{88} - 36q^{89} + 69q^{90} + 39q^{91} + 30q^{92} - 66q^{93} - 9q^{94} - 33q^{95} - 48q^{97} - 12q^{98} - 54q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
67.1 0.766044 0.642788i −1.73077 + 0.0664993i 0.173648 0.984808i −0.294369 + 1.66945i −1.28310 + 1.16346i −2.64575 0.00486080i −0.500000 0.866025i 2.99116 0.230191i 0.847602 + 1.46809i
67.2 0.766044 0.642788i −1.69832 0.340165i 0.173648 0.984808i −0.294026 + 1.66751i −1.51964 + 0.831077i 1.80729 1.93228i −0.500000 0.866025i 2.76858 + 1.15542i 0.846615 + 1.46638i
67.3 0.766044 0.642788i −1.42742 + 0.981051i 0.173648 0.984808i 0.343656 1.94897i −0.462863 + 1.66906i 1.84248 + 1.89876i −0.500000 0.866025i 1.07508 2.80075i −0.989519 1.71390i
67.4 0.766044 0.642788i −1.07856 1.35526i 0.173648 0.984808i 0.620533 3.51922i −1.69736 0.344903i −0.670349 2.55942i −0.500000 0.866025i −0.673434 + 2.92344i −1.78675 3.09475i
67.5 0.766044 0.642788i −0.402772 + 1.68457i 0.173648 0.984808i −0.366960 + 2.08113i 0.774279 + 1.54935i 1.84638 + 1.89496i −0.500000 0.866025i −2.67555 1.35699i 1.05662 + 1.83012i
67.6 0.766044 0.642788i −0.389111 1.68778i 0.173648 0.984808i −0.0926928 + 0.525687i −1.38296 1.04280i −2.63702 + 0.214772i −0.500000 0.866025i −2.69718 + 1.31347i 0.266898 + 0.462282i
67.7 0.766044 0.642788i 0.586306 + 1.62980i 0.173648 0.984808i 0.483736 2.74340i 1.49675 + 0.871628i 0.286196 2.63023i −0.500000 0.866025i −2.31249 + 1.91112i −1.39286 2.41251i
67.8 0.766044 0.642788i 0.654437 1.60366i 0.173648 0.984808i 0.190866 1.08246i −0.529482 1.64914i 2.53672 + 0.751691i −0.500000 0.866025i −2.14342 2.09898i −0.549577 0.951896i
67.9 0.766044 0.642788i 1.40163 + 1.01757i 0.173648 0.984808i −0.289065 + 1.63937i 1.72779 0.121449i 1.93330 1.80620i −0.500000 0.866025i 0.929121 + 2.85250i 0.832330 + 1.44164i
67.10 0.766044 0.642788i 1.58806 0.691418i 0.173648 0.984808i −0.0337396 + 0.191347i 0.772092 1.55044i −1.90403 1.83703i −0.500000 0.866025i 2.04388 2.19603i 0.0971494 + 0.168268i
67.11 0.766044 0.642788i 1.71536 0.239862i 0.173648 0.984808i −0.604303 + 3.42717i 1.15986 1.28636i −0.0661038 + 2.64493i −0.500000 0.866025i 2.88493 0.822900i 1.74002 + 3.01380i
67.12 0.766044 0.642788i 1.72085 + 0.196628i 0.173648 0.984808i 0.683660 3.87723i 1.44464 0.955517i −1.15546 + 2.38011i −0.500000 0.866025i 2.92267 + 0.676737i −1.96852 3.40958i
79.1 0.766044 + 0.642788i −1.73077 0.0664993i 0.173648 + 0.984808i −0.294369 1.66945i −1.28310 1.16346i −2.64575 + 0.00486080i −0.500000 + 0.866025i 2.99116 + 0.230191i 0.847602 1.46809i
79.2 0.766044 + 0.642788i −1.69832 + 0.340165i 0.173648 + 0.984808i −0.294026 1.66751i −1.51964 0.831077i 1.80729 + 1.93228i −0.500000 + 0.866025i 2.76858 1.15542i 0.846615 1.46638i
79.3 0.766044 + 0.642788i −1.42742 0.981051i 0.173648 + 0.984808i 0.343656 + 1.94897i −0.462863 1.66906i 1.84248 1.89876i −0.500000 + 0.866025i 1.07508 + 2.80075i −0.989519 + 1.71390i
79.4 0.766044 + 0.642788i −1.07856 + 1.35526i 0.173648 + 0.984808i 0.620533 + 3.51922i −1.69736 + 0.344903i −0.670349 + 2.55942i −0.500000 + 0.866025i −0.673434 2.92344i −1.78675 + 3.09475i
79.5 0.766044 + 0.642788i −0.402772 1.68457i 0.173648 + 0.984808i −0.366960 2.08113i 0.774279 1.54935i 1.84638 1.89496i −0.500000 + 0.866025i −2.67555 + 1.35699i 1.05662 1.83012i
79.6 0.766044 + 0.642788i −0.389111 + 1.68778i 0.173648 + 0.984808i −0.0926928 0.525687i −1.38296 + 1.04280i −2.63702 0.214772i −0.500000 + 0.866025i −2.69718 1.31347i 0.266898 0.462282i
79.7 0.766044 + 0.642788i 0.586306 1.62980i 0.173648 + 0.984808i 0.483736 + 2.74340i 1.49675 0.871628i 0.286196 + 2.63023i −0.500000 + 0.866025i −2.31249 1.91112i −1.39286 + 2.41251i
79.8 0.766044 + 0.642788i 0.654437 + 1.60366i 0.173648 + 0.984808i 0.190866 + 1.08246i −0.529482 + 1.64914i 2.53672 0.751691i −0.500000 + 0.866025i −2.14342 + 2.09898i −0.549577 + 0.951896i
See all 72 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 331.12 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.u 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.v.b 72
7.c even 3 1 378.2.w.a yes 72
27.e even 9 1 378.2.w.a yes 72
189.u even 9 1 inner 378.2.v.b 72

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

## 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