# Properties

 Label 546.2.cg.a Level $546$ Weight $2$ Character orbit 546.cg 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.cg (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.

 $$\operatorname{Tr}(f)(q) =$$ $$32q - 4q^{7} + 16q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$32q - 4q^{7} + 16q^{9} + 8q^{11} + 16q^{12} + 4q^{14} - 32q^{16} - 16q^{17} + 24q^{19} + 8q^{21} - 4q^{22} + 24q^{25} - 8q^{26} + 8q^{28} - 12q^{29} + 4q^{31} + 8q^{33} + 8q^{34} + 40q^{35} - 12q^{37} - 8q^{38} + 28q^{39} + 28q^{41} - 12q^{42} + 84q^{43} - 4q^{44} - 40q^{46} - 4q^{47} - 36q^{49} - 16q^{50} - 20q^{52} - 4q^{53} - 48q^{55} + 12q^{56} + 12q^{57} + 12q^{58} - 24q^{59} - 36q^{61} + 40q^{62} + 4q^{63} + 44q^{65} - 16q^{67} + 8q^{69} + 40q^{70} + 20q^{71} - 16q^{73} - 40q^{74} + 16q^{75} - 36q^{76} + 12q^{77} - 20q^{78} - 16q^{81} - 24q^{82} - 12q^{83} - 8q^{84} - 4q^{85} - 20q^{86} - 60q^{89} - 48q^{91} - 16q^{92} - 32q^{93} - 76q^{97} - 8q^{98} + 4q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
145.1 −0.707107 + 0.707107i 0.866025 + 0.500000i 1.00000i −3.71479 + 0.995376i −0.965926 + 0.258819i 1.84687 1.89449i 0.707107 + 0.707107i 0.500000 + 0.866025i 1.92292 3.33059i
145.2 −0.707107 + 0.707107i 0.866025 + 0.500000i 1.00000i −0.622671 + 0.166844i −0.965926 + 0.258819i −1.49104 + 2.18559i 0.707107 + 0.707107i 0.500000 + 0.866025i 0.322318 0.558272i
145.3 −0.707107 + 0.707107i 0.866025 + 0.500000i 1.00000i 1.41705 0.379698i −0.965926 + 0.258819i 0.378394 2.61855i 0.707107 + 0.707107i 0.500000 + 0.866025i −0.733521 + 1.27049i
145.4 −0.707107 + 0.707107i 0.866025 + 0.500000i 1.00000i 2.92041 0.782522i −0.965926 + 0.258819i 2.42968 + 1.04721i 0.707107 + 0.707107i 0.500000 + 0.866025i −1.51172 + 2.61837i
145.5 0.707107 0.707107i 0.866025 + 0.500000i 1.00000i −3.46134 + 0.927465i 0.965926 0.258819i −1.74330 + 1.99020i −0.707107 0.707107i 0.500000 + 0.866025i −1.79172 + 3.10336i
145.6 0.707107 0.707107i 0.866025 + 0.500000i 1.00000i −0.864363 + 0.231605i 0.965926 0.258819i −1.65489 2.06430i −0.707107 0.707107i 0.500000 + 0.866025i −0.447427 + 0.774967i
145.7 0.707107 0.707107i 0.866025 + 0.500000i 1.00000i 1.56767 0.420055i 0.965926 0.258819i 2.44485 1.01129i −0.707107 0.707107i 0.500000 + 0.866025i 0.811483 1.40553i
145.8 0.707107 0.707107i 0.866025 + 0.500000i 1.00000i 2.75804 0.739015i 0.965926 0.258819i 0.253540 + 2.63358i −0.707107 0.707107i 0.500000 + 0.866025i 1.42767 2.47279i
241.1 −0.707107 0.707107i 0.866025 0.500000i 1.00000i −3.71479 0.995376i −0.965926 0.258819i 1.84687 + 1.89449i 0.707107 0.707107i 0.500000 0.866025i 1.92292 + 3.33059i
241.2 −0.707107 0.707107i 0.866025 0.500000i 1.00000i −0.622671 0.166844i −0.965926 0.258819i −1.49104 2.18559i 0.707107 0.707107i 0.500000 0.866025i 0.322318 + 0.558272i
241.3 −0.707107 0.707107i 0.866025 0.500000i 1.00000i 1.41705 + 0.379698i −0.965926 0.258819i 0.378394 + 2.61855i 0.707107 0.707107i 0.500000 0.866025i −0.733521 1.27049i
241.4 −0.707107 0.707107i 0.866025 0.500000i 1.00000i 2.92041 + 0.782522i −0.965926 0.258819i 2.42968 1.04721i 0.707107 0.707107i 0.500000 0.866025i −1.51172 2.61837i
241.5 0.707107 + 0.707107i 0.866025 0.500000i 1.00000i −3.46134 0.927465i 0.965926 + 0.258819i −1.74330 1.99020i −0.707107 + 0.707107i 0.500000 0.866025i −1.79172 3.10336i
241.6 0.707107 + 0.707107i 0.866025 0.500000i 1.00000i −0.864363 0.231605i 0.965926 + 0.258819i −1.65489 + 2.06430i −0.707107 + 0.707107i 0.500000 0.866025i −0.447427 0.774967i
241.7 0.707107 + 0.707107i 0.866025 0.500000i 1.00000i 1.56767 + 0.420055i 0.965926 + 0.258819i 2.44485 + 1.01129i −0.707107 + 0.707107i 0.500000 0.866025i 0.811483 + 1.40553i
241.8 0.707107 + 0.707107i 0.866025 0.500000i 1.00000i 2.75804 + 0.739015i 0.965926 + 0.258819i 0.253540 2.63358i −0.707107 + 0.707107i 0.500000 0.866025i 1.42767 + 2.47279i
271.1 −0.707107 0.707107i −0.866025 0.500000i 1.00000i −0.765779 2.85793i 0.258819 + 0.965926i −1.78167 1.95592i 0.707107 0.707107i 0.500000 + 0.866025i −1.47937 + 2.56235i
271.2 −0.707107 0.707107i −0.866025 0.500000i 1.00000i 0.0344778 + 0.128673i 0.258819 + 0.965926i 2.22943 1.42466i 0.707107 0.707107i 0.500000 + 0.866025i 0.0666060 0.115365i
271.3 −0.707107 0.707107i −0.866025 0.500000i 1.00000i 0.112824 + 0.421063i 0.258819 + 0.965926i −2.62301 + 0.346128i 0.707107 0.707107i 0.500000 + 0.866025i 0.217958 0.377515i
271.4 −0.707107 0.707107i −0.866025 0.500000i 1.00000i 0.618478 + 2.30819i 0.258819 + 0.965926i −0.574432 + 2.58264i 0.707107 0.707107i 0.500000 + 0.866025i 1.19481 2.06947i
See all 32 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 409.8 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.ba 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.cg.a yes 32
7.d odd 6 1 546.2.by.a 32
13.f odd 12 1 546.2.by.a 32
91.ba even 12 1 inner 546.2.cg.a yes 32

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

## Hecke kernels

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