# Properties

 Label 546.2.ch.a Level $546$ Weight $2$ Character orbit 546.ch Analytic conductor $4.360$ Analytic rank $0$ Dimension $152$ CM no Inner twists $4$

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$4.35983195036$$ Analytic rank: $$0$$ Dimension: $$152$$ Relative dimension: $$38$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$152q + 8q^{7} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$152q + 8q^{7} - 152q^{16} - 8q^{18} + 36q^{19} + 8q^{21} + 24q^{27} - 12q^{28} - 24q^{30} - 36q^{31} + 12q^{33} + 24q^{36} - 12q^{37} - 8q^{39} - 36q^{42} + 84q^{43} - 40q^{45} + 8q^{46} + 4q^{49} - 20q^{52} - 12q^{54} + 16q^{55} - 8q^{57} + 8q^{58} + 24q^{60} - 32q^{61} - 28q^{63} + 16q^{66} + 92q^{67} + 84q^{69} - 8q^{72} + 84q^{73} - 48q^{76} - 8q^{78} + 32q^{79} - 24q^{81} - 48q^{82} - 8q^{84} - 152q^{85} - 96q^{87} - 88q^{91} - 48q^{93} + 32q^{94} - 100q^{97} + 16q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
137.1 −0.707107 + 0.707107i −1.72681 0.134626i 1.00000i −1.91810 0.513954i 1.31623 1.12584i 1.87225 1.86941i 0.707107 + 0.707107i 2.96375 + 0.464948i 1.71972 0.992884i
137.2 −0.707107 + 0.707107i −1.72614 + 0.143006i 1.00000i −1.09645 0.293793i 1.11944 1.32168i −0.0234379 + 2.64565i 0.707107 + 0.707107i 2.95910 0.493697i 0.983052 0.567565i
137.3 −0.707107 + 0.707107i −1.47304 0.911131i 1.00000i 1.94682 + 0.521648i 1.68586 0.397328i −2.63140 0.275187i 0.707107 + 0.707107i 1.33968 + 2.68426i −1.74547 + 1.00775i
137.4 −0.707107 + 0.707107i −1.45428 0.940775i 1.00000i 3.45354 + 0.925372i 1.69356 0.363106i 2.64433 + 0.0866153i 0.707107 + 0.707107i 1.22989 + 2.73631i −3.09636 + 1.78768i
137.5 −0.707107 + 0.707107i −1.33512 + 1.10338i 1.00000i 2.13400 + 0.571804i 0.163866 1.72428i −2.34532 1.22452i 0.707107 + 0.707107i 0.565101 2.94630i −1.91329 + 1.10464i
137.6 −0.707107 + 0.707107i −0.975072 + 1.43151i 1.00000i −0.438011 0.117365i −0.322754 1.70171i 2.35961 + 1.19677i 0.707107 + 0.707107i −1.09847 2.79166i 0.392710 0.226731i
137.7 −0.707107 + 0.707107i −0.965783 1.43780i 1.00000i −4.19664 1.12449i 1.69959 + 0.333765i −2.15090 1.54066i 0.707107 + 0.707107i −1.13453 + 2.77720i 3.76260 2.17234i
137.8 −0.707107 + 0.707107i −0.649813 + 1.60554i 1.00000i −1.93876 0.519490i −0.675797 1.59477i −1.14094 2.38710i 0.707107 + 0.707107i −2.15549 2.08660i 1.73825 1.00358i
137.9 −0.707107 + 0.707107i −0.238917 1.71549i 1.00000i 0.465027 + 0.124604i 1.38198 + 1.04410i −1.66275 + 2.05797i 0.707107 + 0.707107i −2.88584 + 0.819723i −0.416932 + 0.240716i
137.10 −0.707107 + 0.707107i 0.116244 1.72815i 1.00000i −1.15194 0.308662i 1.13979 + 1.30418i 1.79371 + 1.94489i 0.707107 + 0.707107i −2.97297 0.401773i 1.03280 0.596290i
137.11 −0.707107 + 0.707107i 0.396140 1.68614i 1.00000i 0.940755 + 0.252074i 0.912169 + 1.47240i 1.13831 2.38836i 0.707107 + 0.707107i −2.68615 1.33590i −0.843457 + 0.486970i
137.12 −0.707107 + 0.707107i 0.480095 + 1.66418i 1.00000i 3.07620 + 0.824266i −1.51623 0.837278i 1.73277 1.99938i 0.707107 + 0.707107i −2.53902 + 1.59793i −2.75805 + 1.59236i
137.13 −0.707107 + 0.707107i 0.642070 + 1.60865i 1.00000i −0.758733 0.203302i −1.59150 0.683474i −1.58539 + 2.11814i 0.707107 + 0.707107i −2.17549 + 2.06573i 0.680261 0.392749i
137.14 −0.707107 + 0.707107i 0.922039 + 1.46623i 1.00000i −3.25307 0.871657i −1.68876 0.384805i 2.64559 0.0294574i 0.707107 + 0.707107i −1.29969 + 2.70385i 2.91662 1.68391i
137.15 −0.707107 + 0.707107i 1.47583 0.906598i 1.00000i 3.71236 + 0.994723i −0.402510 + 1.68463i 1.97669 + 1.75860i 0.707107 + 0.707107i 1.35616 2.67597i −3.32841 + 1.92166i
137.16 −0.707107 + 0.707107i 1.52325 + 0.824446i 1.00000i 2.39134 + 0.640759i −1.66007 + 0.494128i −0.703385 2.55054i 0.707107 + 0.707107i 1.64058 + 2.51167i −2.14402 + 1.23785i
137.17 −0.707107 + 0.707107i 1.59861 0.666660i 1.00000i −1.41769 0.379869i −0.658990 + 1.60179i 1.80686 1.93268i 0.707107 + 0.707107i 2.11113 2.13146i 1.27107 0.733851i
137.18 −0.707107 + 0.707107i 1.66850 + 0.464889i 1.00000i 1.76812 + 0.473766i −1.50853 + 0.851078i −1.72527 + 2.00586i 0.707107 + 0.707107i 2.56776 + 1.55133i −1.58525 + 0.915246i
137.19 −0.707107 + 0.707107i 1.72220 0.184470i 1.00000i −3.71876 0.996438i −1.08734 + 1.34822i −0.403232 + 2.61484i 0.707107 + 0.707107i 2.93194 0.635390i 3.33415 1.92497i
137.20 0.707107 0.707107i −1.65831 0.500003i 1.00000i −0.940755 0.252074i −1.52616 + 0.819048i 1.13831 2.38836i −0.707107 0.707107i 2.49999 + 1.65832i −0.843457 + 0.486970i
See next 80 embeddings (of 152 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 401.38 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
91.x odd 12 1 inner
273.bv 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.ch.a yes 152
3.b odd 2 1 inner 546.2.ch.a yes 152
7.c even 3 1 546.2.bw.a 152
13.f odd 12 1 546.2.bw.a 152
21.h odd 6 1 546.2.bw.a 152
39.k even 12 1 546.2.bw.a 152
91.x odd 12 1 inner 546.2.ch.a yes 152
273.bv even 12 1 inner 546.2.ch.a yes 152

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
546.2.bw.a 152 7.c even 3 1
546.2.bw.a 152 13.f odd 12 1
546.2.bw.a 152 21.h odd 6 1
546.2.bw.a 152 39.k even 12 1
546.2.ch.a yes 152 1.a even 1 1 trivial
546.2.ch.a yes 152 3.b odd 2 1 inner
546.2.ch.a yes 152 91.x odd 12 1 inner
546.2.ch.a yes 152 273.bv even 12 1 inner

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(546, [\chi])$$.