# Properties

 Label 546.2.bn.f Level $546$ Weight $2$ Character orbit 546.bn Analytic conductor $4.360$ Analytic rank $0$ Dimension $34$ 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.bn (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.35983195036$$ Analytic rank: $$0$$ Dimension: $$34$$ Relative dimension: $$17$$ over $$\Q(\zeta_{6})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$34q + 17q^{2} - 3q^{3} - 17q^{4} - 9q^{5} + 3q^{6} + 5q^{7} - 34q^{8} + 7q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$34q + 17q^{2} - 3q^{3} - 17q^{4} - 9q^{5} + 3q^{6} + 5q^{7} - 34q^{8} + 7q^{9} + 18q^{11} + 6q^{12} - 8q^{13} + 4q^{14} - 4q^{15} - 17q^{16} - 6q^{17} - 4q^{18} - 10q^{19} + 9q^{20} + 7q^{21} + 9q^{22} - 6q^{23} + 3q^{24} + 16q^{25} - 13q^{26} - 18q^{27} - q^{28} - 27q^{29} + 13q^{30} + q^{31} + 17q^{32} + 21q^{33} - 12q^{34} + 3q^{35} - 11q^{36} + 6q^{37} - 5q^{38} - 2q^{39} + 9q^{40} - 3q^{41} + 8q^{42} - 3q^{43} - 9q^{44} + 9q^{45} - 6q^{46} + 27q^{47} - 3q^{48} - 5q^{49} - 16q^{50} - 36q^{51} - 5q^{52} - 21q^{53} + 57q^{55} - 5q^{56} + 17q^{57} + 6q^{59} + 17q^{60} - q^{62} + 34q^{64} - 33q^{65} - 6q^{68} - 42q^{69} + 3q^{70} + 15q^{71} - 7q^{72} + 19q^{73} + 6q^{74} - 9q^{75} + 5q^{76} + 9q^{77} - 7q^{78} - 9q^{79} - 5q^{81} + q^{84} - 42q^{85} + 3q^{86} + 6q^{87} - 18q^{88} + 18q^{89} + 9q^{90} - 27q^{91} + 8q^{93} + 3q^{95} - 6q^{96} - 19q^{97} - 7q^{98} + 27q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
101.1 0.500000 0.866025i −1.71486 + 0.243407i −0.500000 0.866025i −2.88000 + 1.66277i −0.646635 + 1.60682i −0.187179 2.63912i −1.00000 2.88151 0.834819i 3.32554i
101.2 0.500000 0.866025i −1.70046 + 0.329307i −0.500000 0.866025i 1.26448 0.730045i −0.565041 + 1.63729i −2.63261 + 0.263392i −1.00000 2.78311 1.11994i 1.46009i
101.3 0.500000 0.866025i −1.60671 0.646913i −0.500000 0.866025i 2.60318 1.50295i −1.36360 + 1.06799i −0.916975 2.48177i −1.00000 2.16301 + 2.07880i 3.00590i
101.4 0.500000 0.866025i −1.57643 0.717543i −0.500000 0.866025i −1.41302 + 0.815806i −1.40963 + 1.00646i 1.06124 + 2.42359i −1.00000 1.97026 + 2.26231i 1.63161i
101.5 0.500000 0.866025i −1.51226 + 0.844435i −0.500000 0.866025i −0.511132 + 0.295102i −0.0248275 + 1.73187i 1.57814 + 2.12355i −1.00000 1.57386 2.55401i 0.590205i
101.6 0.500000 0.866025i −0.925467 + 1.46407i −0.500000 0.866025i 2.64914 1.52948i 0.805192 + 1.53351i 2.61668 0.391106i −1.00000 −1.28702 2.70990i 3.05896i
101.7 0.500000 0.866025i −0.801045 + 1.53568i −0.500000 0.866025i 1.09866 0.634311i 0.929420 + 1.46157i −2.21179 + 1.45190i −1.00000 −1.71665 2.46030i 1.26862i
101.8 0.500000 0.866025i −0.458130 + 1.67036i −0.500000 0.866025i −1.57344 + 0.908426i 1.21751 + 1.23193i −0.414054 2.61315i −1.00000 −2.58023 1.53049i 1.81685i
101.9 0.500000 0.866025i −0.428009 1.67834i −0.500000 0.866025i −1.58996 + 0.917964i −1.66749 0.468501i 2.08732 1.62576i −1.00000 −2.63362 + 1.43668i 1.83593i
101.10 0.500000 0.866025i 0.320103 1.70221i −0.500000 0.866025i 1.62172 0.936303i −1.31411 1.12832i 2.47749 + 0.928453i −1.00000 −2.79507 1.08977i 1.87261i
101.11 0.500000 0.866025i 0.786858 + 1.54300i −0.500000 0.866025i −1.98183 + 1.14421i 1.72971 + 0.0900624i −2.60041 + 0.487738i −1.00000 −1.76171 + 2.42825i 2.28842i
101.12 0.500000 0.866025i 0.991815 1.41997i −0.500000 0.866025i −3.72094 + 2.14828i −0.733819 1.56892i −1.08595 + 2.41262i −1.00000 −1.03261 2.81669i 4.29657i
101.13 0.500000 0.866025i 1.17271 + 1.27466i −0.500000 0.866025i −0.870413 + 0.502533i 1.69024 0.378264i 1.33597 + 2.28368i −1.00000 −0.249516 + 2.98961i 1.00507i
101.14 0.500000 0.866025i 1.23327 1.21616i −0.500000 0.866025i −0.567570 + 0.327687i −0.436593 1.67612i −2.19987 1.46989i −1.00000 0.0419009 2.99971i 0.655374i
101.15 0.500000 0.866025i 1.25705 + 1.19156i −0.500000 0.866025i 1.80315 1.04105i 1.66045 0.492861i 1.78353 1.95424i −1.00000 0.160371 + 2.99571i 2.08210i
101.16 0.500000 0.866025i 1.72973 0.0896134i −0.500000 0.866025i −3.27919 + 1.89324i 0.787258 1.54280i 2.49160 0.889895i −1.00000 2.98394 0.310014i 3.78649i
101.17 0.500000 0.866025i 1.73183 0.0276700i −0.500000 0.866025i 2.84717 1.64381i 0.841952 1.51364i −0.683149 + 2.55603i −1.00000 2.99847 0.0958395i 3.28763i
173.1 0.500000 + 0.866025i −1.71486 0.243407i −0.500000 + 0.866025i −2.88000 1.66277i −0.646635 1.60682i −0.187179 + 2.63912i −1.00000 2.88151 + 0.834819i 3.32554i
173.2 0.500000 + 0.866025i −1.70046 0.329307i −0.500000 + 0.866025i 1.26448 + 0.730045i −0.565041 1.63729i −2.63261 0.263392i −1.00000 2.78311 + 1.11994i 1.46009i
173.3 0.500000 + 0.866025i −1.60671 + 0.646913i −0.500000 + 0.866025i 2.60318 + 1.50295i −1.36360 1.06799i −0.916975 + 2.48177i −1.00000 2.16301 2.07880i 3.00590i
See all 34 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 173.17 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
273.y even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 546.2.bn.f yes 34
3.b odd 2 1 546.2.bn.e yes 34
7.d odd 6 1 546.2.bi.e 34
13.e even 6 1 546.2.bi.f yes 34
21.g even 6 1 546.2.bi.f yes 34
39.h odd 6 1 546.2.bi.e 34
91.p odd 6 1 546.2.bn.e yes 34
273.y even 6 1 inner 546.2.bn.f yes 34

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
546.2.bi.e 34 7.d odd 6 1
546.2.bi.e 34 39.h odd 6 1
546.2.bi.f yes 34 13.e even 6 1
546.2.bi.f yes 34 21.g even 6 1
546.2.bn.e yes 34 3.b odd 2 1
546.2.bn.e yes 34 91.p odd 6 1
546.2.bn.f yes 34 1.a even 1 1 trivial
546.2.bn.f yes 34 273.y even 6 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{34} + \cdots$$ acting on $$S_{2}^{\mathrm{new}}(546, [\chi])$$.