# Properties

 Label 546.2.bz.b Level $546$ Weight $2$ Character orbit 546.bz Analytic conductor $4.360$ Analytic rank $0$ Dimension $40$ 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.bz (of order $$12$$, degree $$4$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.35983195036$$ Analytic rank: $$0$$ Dimension: $$40$$ Relative dimension: $$10$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$40q + 4q^{7} + 20q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$40q + 4q^{7} + 20q^{9} - 4q^{11} - 20q^{12} + 4q^{14} + 20q^{16} - 8q^{17} + 16q^{19} + 4q^{21} + 8q^{22} + 24q^{23} + 48q^{25} + 8q^{26} - 4q^{28} + 24q^{29} + 4q^{33} + 16q^{34} - 8q^{35} - 32q^{37} - 16q^{38} - 4q^{39} + 8q^{41} - 4q^{44} - 28q^{46} - 20q^{47} - 16q^{49} + 32q^{50} + 12q^{51} + 4q^{52} - 4q^{53} - 16q^{57} + 12q^{58} + 24q^{59} - 12q^{61} - 16q^{62} + 8q^{63} + 8q^{65} - 24q^{67} - 12q^{68} + 16q^{69} + 12q^{70} + 8q^{71} - 36q^{73} - 40q^{74} + 36q^{75} + 16q^{76} - 48q^{77} - 8q^{78} - 20q^{81} - 24q^{83} - 8q^{84} - 40q^{85} - 56q^{86} - 72q^{87} - 72q^{89} - 24q^{91} - 16q^{92} - 36q^{94} + 32q^{98} - 8q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
31.1 −0.965926 0.258819i −0.866025 0.500000i 0.866025 + 0.500000i −0.953102 + 3.55703i 0.707107 + 0.707107i −2.47914 + 0.924041i −0.707107 0.707107i 0.500000 + 0.866025i 1.84125 3.18914i
31.2 −0.965926 0.258819i −0.866025 0.500000i 0.866025 + 0.500000i −0.326078 + 1.21694i 0.707107 + 0.707107i 0.699081 2.55172i −0.707107 0.707107i 0.500000 + 0.866025i 0.629934 1.09108i
31.3 −0.965926 0.258819i −0.866025 0.500000i 0.866025 + 0.500000i 0.0181845 0.0678656i 0.707107 + 0.707107i 2.49748 + 0.873279i −0.707107 0.707107i 0.500000 + 0.866025i −0.0351298 + 0.0608466i
31.4 −0.965926 0.258819i −0.866025 0.500000i 0.866025 + 0.500000i 0.466002 1.73914i 0.707107 + 0.707107i −0.0195352 + 2.64568i −0.707107 0.707107i 0.500000 + 0.866025i −0.900247 + 1.55927i
31.5 −0.965926 0.258819i −0.866025 0.500000i 0.866025 + 0.500000i 0.794994 2.96696i 0.707107 + 0.707107i −1.32272 2.29138i −0.707107 0.707107i 0.500000 + 0.866025i −1.53581 + 2.66010i
31.6 0.965926 + 0.258819i −0.866025 0.500000i 0.866025 + 0.500000i −0.930654 + 3.47325i −0.707107 0.707107i −2.03957 1.68528i 0.707107 + 0.707107i 0.500000 + 0.866025i −1.79789 + 3.11403i
31.7 0.965926 + 0.258819i −0.866025 0.500000i 0.866025 + 0.500000i −0.272675 + 1.01764i −0.707107 0.707107i 2.64349 + 0.109436i 0.707107 + 0.707107i 0.500000 + 0.866025i −0.526768 + 0.912388i
31.8 0.965926 + 0.258819i −0.866025 0.500000i 0.866025 + 0.500000i 0.140104 0.522875i −0.707107 0.707107i −2.24755 1.39589i 0.707107 + 0.707107i 0.500000 + 0.866025i 0.270660 0.468797i
31.9 0.965926 + 0.258819i −0.866025 0.500000i 0.866025 + 0.500000i 0.187100 0.698266i −0.707107 0.707107i −0.197219 + 2.63839i 0.707107 + 0.707107i 0.500000 + 0.866025i 0.361449 0.626048i
31.10 0.965926 + 0.258819i −0.866025 0.500000i 0.866025 + 0.500000i 0.876125 3.26974i −0.707107 0.707107i 1.73365 1.99861i 0.707107 + 0.707107i 0.500000 + 0.866025i 1.69254 2.93157i
73.1 −0.258819 + 0.965926i 0.866025 + 0.500000i −0.866025 0.500000i −2.23807 0.599690i −0.707107 + 0.707107i 1.69400 2.03233i 0.707107 0.707107i 0.500000 + 0.866025i 1.15851 2.00660i
73.2 −0.258819 + 0.965926i 0.866025 + 0.500000i −0.866025 0.500000i −1.99820 0.535417i −0.707107 + 0.707107i 0.718636 + 2.54628i 0.707107 0.707107i 0.500000 + 0.866025i 1.03435 1.79154i
73.3 −0.258819 + 0.965926i 0.866025 + 0.500000i −0.866025 0.500000i −1.56603 0.419616i −0.707107 + 0.707107i −2.13245 1.56610i 0.707107 0.707107i 0.500000 + 0.866025i 0.810635 1.40406i
73.4 −0.258819 + 0.965926i 0.866025 + 0.500000i −0.866025 0.500000i 2.74268 + 0.734900i −0.707107 + 0.707107i 2.59833 0.498684i 0.707107 0.707107i 0.500000 + 0.866025i −1.41972 + 2.45902i
73.5 −0.258819 + 0.965926i 0.866025 + 0.500000i −0.866025 0.500000i 3.05962 + 0.819823i −0.707107 + 0.707107i −2.47842 + 0.925986i 0.707107 0.707107i 0.500000 + 0.866025i −1.58378 + 2.74318i
73.6 0.258819 0.965926i 0.866025 + 0.500000i −0.866025 0.500000i −4.17716 1.11927i 0.707107 0.707107i −0.344270 2.62326i −0.707107 + 0.707107i 0.500000 + 0.866025i −2.16226 + 3.74514i
73.7 0.258819 0.965926i 0.866025 + 0.500000i −0.866025 0.500000i −2.53696 0.679776i 0.707107 0.707107i 2.51480 + 0.822050i −0.707107 + 0.707107i 0.500000 + 0.866025i −1.31323 + 2.27458i
73.8 0.258819 0.965926i 0.866025 + 0.500000i −0.866025 0.500000i −0.171119 0.0458512i 0.707107 0.707107i −1.18892 + 2.36357i −0.707107 + 0.707107i 0.500000 + 0.866025i −0.0885777 + 0.153421i
73.9 0.258819 0.965926i 0.866025 + 0.500000i −0.866025 0.500000i 2.98554 + 0.799972i 0.707107 0.707107i 1.80183 + 1.93737i −0.707107 + 0.707107i 0.500000 + 0.866025i 1.54543 2.67676i
73.10 0.258819 0.965926i 0.866025 + 0.500000i −0.866025 0.500000i 3.89970 + 1.04492i 0.707107 0.707107i −0.451486 2.60694i −0.707107 + 0.707107i 0.500000 + 0.866025i 2.01864 3.49638i
See all 40 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 229.10 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.bb 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.bz.b yes 40
7.d odd 6 1 546.2.bz.a 40
13.d odd 4 1 546.2.bz.a 40
91.bb even 12 1 inner 546.2.bz.b yes 40

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
546.2.bz.a 40 7.d odd 6 1
546.2.bz.a 40 13.d odd 4 1
546.2.bz.b yes 40 1.a even 1 1 trivial
546.2.bz.b yes 40 91.bb even 12 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$11\!\cdots\!00$$$$T_{5}^{16} -$$$$20\!\cdots\!80$$$$T_{5}^{15} - 57879661440 T_{5}^{14} +$$$$63\!\cdots\!44$$$$T_{5}^{13} +$$$$18\!\cdots\!88$$$$T_{5}^{12} +$$$$33\!\cdots\!08$$$$T_{5}^{11} +$$$$39\!\cdots\!84$$$$T_{5}^{10} +$$$$39\!\cdots\!28$$$$T_{5}^{9} +$$$$31\!\cdots\!56$$$$T_{5}^{8} +$$$$19\!\cdots\!16$$$$T_{5}^{7} +$$$$13\!\cdots\!32$$$$T_{5}^{6} +$$$$41\!\cdots\!24$$$$T_{5}^{5} +$$$$21\!\cdots\!00$$$$T_{5}^{4} + 529756827264 T_{5}^{3} + 40812242688 T_{5}^{2} + 1056198528 T_{5} + 252047376$$">$$T_{5}^{40} - \cdots$$ acting on $$S_{2}^{\mathrm{new}}(546, [\chi])$$.