# Properties

 Label 546.2.by.a Level $546$ Weight $2$ Character orbit 546.by Analytic conductor $4.360$ Analytic rank $0$ Dimension $32$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$546 = 2 \cdot 3 \cdot 7 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 546.by (of order $$12$$, degree $$4$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.35983195036$$ Analytic rank: $$0$$ Dimension: $$32$$ Relative dimension: $$8$$ over $$\Q(\zeta_{12})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$32q + 8q^{7} - 32q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$32q + 8q^{7} - 32q^{9} - 4q^{11} - 16q^{12} + 4q^{14} + 16q^{16} - 8q^{17} - 12q^{19} - 8q^{21} - 4q^{22} - 24q^{23} - 24q^{25} + 8q^{26} - 4q^{28} - 12q^{29} - 28q^{31} + 4q^{33} - 8q^{34} - 44q^{35} - 36q^{37} + 8q^{38} - 20q^{39} - 28q^{41} + 12q^{42} + 84q^{43} + 8q^{44} - 4q^{46} + 16q^{47} + 24q^{49} - 16q^{50} + 8q^{52} - 4q^{53} + 48q^{55} + 12q^{56} + 12q^{57} + 24q^{58} + 12q^{59} - 40q^{62} - 8q^{63} - 4q^{65} + 8q^{67} - 8q^{69} + 60q^{70} + 20q^{71} + 4q^{73} + 20q^{74} + 8q^{75} + 36q^{76} - 12q^{77} - 20q^{78} + 32q^{81} - 48q^{82} + 12q^{83} - 16q^{84} - 4q^{85} + 52q^{86} - 36q^{87} - 36q^{89} - 16q^{92} + 4q^{93} - 48q^{95} + 76q^{97} + 8q^{98} + 4q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
19.1 −0.965926 0.258819i 1.00000i 0.866025 + 0.500000i −2.19856 + 0.589103i 0.258819 0.965926i 1.83682 1.90423i −0.707107 0.707107i −1.00000 2.27612
19.2 −0.965926 0.258819i 1.00000i 0.866025 + 0.500000i −1.43558 + 0.384662i 0.258819 0.965926i 1.74877 + 1.98539i −0.707107 0.707107i −1.00000 1.48622
19.3 −0.965926 0.258819i 1.00000i 0.866025 + 0.500000i −0.106603 + 0.0285643i 0.258819 0.965926i −1.53649 + 2.15388i −0.707107 0.707107i −1.00000 0.110364
19.4 −0.965926 0.258819i 1.00000i 0.866025 + 0.500000i 3.74075 1.00233i 0.258819 0.965926i −2.20451 + 1.46293i −0.707107 0.707107i −1.00000 −3.87271
19.5 0.965926 + 0.258819i 1.00000i 0.866025 + 0.500000i −2.85793 + 0.765779i −0.258819 + 0.965926i 0.565013 + 2.58472i 0.707107 + 0.707107i −1.00000 −2.95874
19.6 0.965926 + 0.258819i 1.00000i 0.866025 + 0.500000i 0.128673 0.0344778i −0.258819 + 0.965926i −2.64307 + 0.119073i 0.707107 + 0.707107i −1.00000 0.133212
19.7 0.965926 + 0.258819i 1.00000i 0.866025 + 0.500000i 0.421063 0.112824i −0.258819 + 0.965926i 2.44466 + 1.01175i 0.707107 + 0.707107i −1.00000 0.435917
19.8 0.965926 + 0.258819i 1.00000i 0.866025 + 0.500000i 2.30819 0.618478i −0.258819 + 0.965926i 1.78879 1.94942i 0.707107 + 0.707107i −1.00000 2.38962
115.1 −0.965926 + 0.258819i 1.00000i 0.866025 0.500000i −2.19856 0.589103i 0.258819 + 0.965926i 1.83682 + 1.90423i −0.707107 + 0.707107i −1.00000 2.27612
115.2 −0.965926 + 0.258819i 1.00000i 0.866025 0.500000i −1.43558 0.384662i 0.258819 + 0.965926i 1.74877 1.98539i −0.707107 + 0.707107i −1.00000 1.48622
115.3 −0.965926 + 0.258819i 1.00000i 0.866025 0.500000i −0.106603 0.0285643i 0.258819 + 0.965926i −1.53649 2.15388i −0.707107 + 0.707107i −1.00000 0.110364
115.4 −0.965926 + 0.258819i 1.00000i 0.866025 0.500000i 3.74075 + 1.00233i 0.258819 + 0.965926i −2.20451 1.46293i −0.707107 + 0.707107i −1.00000 −3.87271
115.5 0.965926 0.258819i 1.00000i 0.866025 0.500000i −2.85793 0.765779i −0.258819 0.965926i 0.565013 2.58472i 0.707107 0.707107i −1.00000 −2.95874
115.6 0.965926 0.258819i 1.00000i 0.866025 0.500000i 0.128673 + 0.0344778i −0.258819 0.965926i −2.64307 0.119073i 0.707107 0.707107i −1.00000 0.133212
115.7 0.965926 0.258819i 1.00000i 0.866025 0.500000i 0.421063 + 0.112824i −0.258819 0.965926i 2.44466 1.01175i 0.707107 0.707107i −1.00000 0.435917
115.8 0.965926 0.258819i 1.00000i 0.866025 0.500000i 2.30819 + 0.618478i −0.258819 0.965926i 1.78879 + 1.94942i 0.707107 0.707107i −1.00000 2.38962
397.1 −0.258819 + 0.965926i 1.00000i −0.866025 0.500000i −0.995376 3.71479i 0.965926 + 0.258819i 2.54668 0.717240i 0.707107 0.707107i −1.00000 3.84584
397.2 −0.258819 + 0.965926i 1.00000i −0.866025 0.500000i −0.166844 0.622671i 0.965926 + 0.258819i −2.38407 + 1.14726i 0.707107 0.707107i −1.00000 0.644637
397.3 −0.258819 + 0.965926i 1.00000i −0.866025 0.500000i 0.379698 + 1.41705i 0.965926 + 0.258819i 1.63697 2.07854i 0.707107 0.707107i −1.00000 −1.46704
397.4 −0.258819 + 0.965926i 1.00000i −0.866025 0.500000i 0.782522 + 2.92041i 0.965926 + 0.258819i 1.58056 + 2.12175i 0.707107 0.707107i −1.00000 −3.02343
See all 32 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 535.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
91.w even 12 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 546.2.by.a 32
7.d odd 6 1 546.2.cg.a yes 32
13.f odd 12 1 546.2.cg.a yes 32
91.w even 12 1 inner 546.2.by.a 32

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
546.2.by.a 32 1.a even 1 1 trivial
546.2.by.a 32 91.w even 12 1 inner
546.2.cg.a yes 32 7.d odd 6 1
546.2.cg.a yes 32 13.f odd 12 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{32} + \cdots$$ acting on $$S_{2}^{\mathrm{new}}(546, [\chi])$$.