# Properties

 Label 370.2.o.c Level $370$ Weight $2$ Character orbit 370.o Analytic conductor $2.954$ Analytic rank $0$ Dimension $24$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$370 = 2 \cdot 5 \cdot 37$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 370.o (of order $$9$$, degree $$6$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$2.95446487479$$ Analytic rank: $$0$$ Dimension: $$24$$ Relative dimension: $$4$$ over $$\Q(\zeta_{9})$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$24q - 6q^{6} - 3q^{7} + 12q^{8} + 12q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$24q - 6q^{6} - 3q^{7} + 12q^{8} + 12q^{9} + 12q^{10} - 15q^{11} + 3q^{14} - 6q^{17} + 6q^{18} - 6q^{19} + 36q^{21} + 9q^{22} + 21q^{23} - 6q^{26} - 12q^{27} - 3q^{28} + 12q^{29} + 30q^{31} - 39q^{33} - 21q^{34} + 6q^{35} + 54q^{36} - 12q^{37} - 36q^{38} - 18q^{39} + 27q^{41} - 36q^{42} - 24q^{43} - 9q^{44} - 27q^{45} + 3q^{46} - 6q^{47} - 3q^{48} - 27q^{49} - 6q^{51} - 9q^{52} + 6q^{53} + 9q^{54} + 9q^{55} - 6q^{56} - 27q^{57} + 27q^{58} + 51q^{59} - 3q^{60} - 3q^{62} - 27q^{63} - 12q^{64} + 15q^{66} - 18q^{69} + 3q^{70} - 66q^{71} + 6q^{72} + 42q^{73} - 42q^{74} + 6q^{75} - 6q^{76} + 69q^{77} + 36q^{78} - 30q^{79} + 24q^{80} - 90q^{81} + 24q^{82} + 57q^{83} + 6q^{84} - 6q^{86} + 6q^{87} + 15q^{88} - 57q^{89} + 6q^{90} + 3q^{91} + 15q^{92} - 72q^{93} + 3q^{94} - 15q^{95} - 3q^{97} - 72q^{98} + 63q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
71.1 −0.766044 0.642788i −2.54754 + 2.13764i 0.173648 + 0.984808i −0.939693 0.342020i 3.32558 −3.31719 1.20736i 0.500000 0.866025i 1.39951 7.93703i 0.500000 + 0.866025i
71.2 −0.766044 0.642788i 0.204920 0.171948i 0.173648 + 0.984808i −0.939693 0.342020i −0.267503 −3.03750 1.10556i 0.500000 0.866025i −0.508519 + 2.88395i 0.500000 + 0.866025i
71.3 −0.766044 0.642788i 0.727578 0.610510i 0.173648 + 0.984808i −0.939693 0.342020i −0.949785 4.16326 + 1.51530i 0.500000 0.866025i −0.364298 + 2.06604i 0.500000 + 0.866025i
71.4 −0.766044 0.642788i 2.38109 1.99797i 0.173648 + 0.984808i −0.939693 0.342020i −3.10829 −0.187965 0.0684136i 0.500000 0.866025i 1.15675 6.56026i 0.500000 + 0.866025i
81.1 0.939693 0.342020i −2.67866 0.974954i 0.766044 0.642788i 0.173648 + 0.984808i −2.85057 −0.751904 4.26426i 0.500000 0.866025i 3.92657 + 3.29478i 0.500000 + 0.866025i
81.2 0.939693 0.342020i −2.36335 0.860188i 0.766044 0.642788i 0.173648 + 0.984808i −2.51502 0.832620 + 4.72202i 0.500000 0.866025i 2.54735 + 2.13748i 0.500000 + 0.866025i
81.3 0.939693 0.342020i 1.85873 + 0.676523i 0.766044 0.642788i 0.173648 + 0.984808i 1.97802 0.241589 + 1.37012i 0.500000 0.866025i 0.699071 + 0.586590i 0.500000 + 0.866025i
81.4 0.939693 0.342020i 2.24358 + 0.816598i 0.766044 0.642788i 0.173648 + 0.984808i 2.38757 −0.475008 2.69391i 0.500000 0.866025i 2.06871 + 1.73585i 0.500000 + 0.866025i
181.1 −0.173648 + 0.984808i −0.420953 2.38734i −0.939693 0.342020i 0.766044 0.642788i 2.42417 0.480222 0.402954i 0.500000 0.866025i −2.70313 + 0.983860i 0.500000 + 0.866025i
181.2 −0.173648 + 0.984808i −0.116582 0.661169i −0.939693 0.342020i 0.766044 0.642788i 0.671368 −1.79162 + 1.50334i 0.500000 0.866025i 2.39553 0.871900i 0.500000 + 0.866025i
181.3 −0.173648 + 0.984808i 0.160396 + 0.909649i −0.939693 0.342020i 0.766044 0.642788i −0.923681 1.63147 1.36896i 0.500000 0.866025i 2.01734 0.734253i 0.500000 + 0.866025i
181.4 −0.173648 + 0.984808i 0.550788 + 3.12367i −0.939693 0.342020i 0.766044 0.642788i −3.17186 0.712016 0.597452i 0.500000 0.866025i −6.63488 + 2.41490i 0.500000 + 0.866025i
201.1 0.939693 + 0.342020i −2.67866 + 0.974954i 0.766044 + 0.642788i 0.173648 0.984808i −2.85057 −0.751904 + 4.26426i 0.500000 + 0.866025i 3.92657 3.29478i 0.500000 0.866025i
201.2 0.939693 + 0.342020i −2.36335 + 0.860188i 0.766044 + 0.642788i 0.173648 0.984808i −2.51502 0.832620 4.72202i 0.500000 + 0.866025i 2.54735 2.13748i 0.500000 0.866025i
201.3 0.939693 + 0.342020i 1.85873 0.676523i 0.766044 + 0.642788i 0.173648 0.984808i 1.97802 0.241589 1.37012i 0.500000 + 0.866025i 0.699071 0.586590i 0.500000 0.866025i
201.4 0.939693 + 0.342020i 2.24358 0.816598i 0.766044 + 0.642788i 0.173648 0.984808i 2.38757 −0.475008 + 2.69391i 0.500000 + 0.866025i 2.06871 1.73585i 0.500000 0.866025i
231.1 −0.173648 0.984808i −0.420953 + 2.38734i −0.939693 + 0.342020i 0.766044 + 0.642788i 2.42417 0.480222 + 0.402954i 0.500000 + 0.866025i −2.70313 0.983860i 0.500000 0.866025i
231.2 −0.173648 0.984808i −0.116582 + 0.661169i −0.939693 + 0.342020i 0.766044 + 0.642788i 0.671368 −1.79162 1.50334i 0.500000 + 0.866025i 2.39553 + 0.871900i 0.500000 0.866025i
231.3 −0.173648 0.984808i 0.160396 0.909649i −0.939693 + 0.342020i 0.766044 + 0.642788i −0.923681 1.63147 + 1.36896i 0.500000 + 0.866025i 2.01734 + 0.734253i 0.500000 0.866025i
231.4 −0.173648 0.984808i 0.550788 3.12367i −0.939693 + 0.342020i 0.766044 + 0.642788i −3.17186 0.712016 + 0.597452i 0.500000 + 0.866025i −6.63488 2.41490i 0.500000 0.866025i
See all 24 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 271.4 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
37.f even 9 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 370.2.o.c 24
37.f even 9 1 inner 370.2.o.c 24

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
370.2.o.c 24 1.a even 1 1 trivial
370.2.o.c 24 37.f even 9 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{24} - \cdots$$ acting on $$S_{2}^{\mathrm{new}}(370, [\chi])$$.