# Properties

 Label 546.2.bz.a 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.

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

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

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$26\!\cdots\!44$$$$T_{5}^{13} +$$$$52\!\cdots\!56$$$$T_{5}^{12} +$$$$18\!\cdots\!28$$$$T_{5}^{11} + 864463721312 T_{5}^{10} -$$$$10\!\cdots\!24$$$$T_{5}^{9} -$$$$57\!\cdots\!08$$$$T_{5}^{8} +$$$$12\!\cdots\!52$$$$T_{5}^{7} +$$$$13\!\cdots\!00$$$$T_{5}^{6} -$$$$56\!\cdots\!52$$$$T_{5}^{5} +$$$$18\!\cdots\!32$$$$T_{5}^{4} - 217890207360 T_{5}^{3} + 61552141056 T_{5}^{2} - 7105335552 T_{5} + 252047376$$">$$T_{5}^{40} + \cdots$$ acting on $$S_{2}^{\mathrm{new}}(546, [\chi])$$.