# Properties

 Label 546.2.bn.e 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.

 $$\operatorname{Tr}(f)(q) =$$ $$34q - 17q^{2} + 3q^{3} - 17q^{4} + 9q^{5} - 6q^{6} + 5q^{7} + 34q^{8} + 7q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$34q - 17q^{2} + 3q^{3} - 17q^{4} + 9q^{5} - 6q^{6} + 5q^{7} + 34q^{8} + 7q^{9} - 18q^{11} + 3q^{12} - 8q^{13} - 4q^{14} - 17q^{15} - 17q^{16} + 6q^{17} - 11q^{18} - 10q^{19} - 9q^{20} - 4q^{21} + 9q^{22} + 6q^{23} + 3q^{24} + 16q^{25} + 13q^{26} + 18q^{27} - q^{28} + 27q^{29} + 13q^{30} + q^{31} - 17q^{32} + 21q^{33} - 12q^{34} - 3q^{35} + 4q^{36} + 6q^{37} + 5q^{38} + 20q^{39} + 9q^{40} + 3q^{41} + 20q^{42} - 3q^{43} + 9q^{44} - 6q^{46} - 27q^{47} - 6q^{48} - 5q^{49} + 16q^{50} + 24q^{51} - 5q^{52} + 21q^{53} - 18q^{54} + 57q^{55} + 5q^{56} - 17q^{57} - 6q^{59} + 4q^{60} + q^{62} - 21q^{63} + 34q^{64} + 33q^{65} - 21q^{66} + 6q^{68} - 30q^{69} + 3q^{70} - 15q^{71} + 7q^{72} + 19q^{73} - 6q^{74} - 63q^{75} + 5q^{76} - 9q^{77} - 10q^{78} - 9q^{79} - 5q^{81} - 16q^{84} - 42q^{85} - 3q^{86} - 75q^{87} - 18q^{88} - 18q^{89} - 9q^{90} - 27q^{91} + 25q^{93} - 3q^{95} + 3q^{96} - 19q^{97} + 7q^{98} - 27q^{99} + O(q^{100})$$

## 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 −1.73183 0.0276700i −0.500000 0.866025i −2.84717 + 1.64381i 0.889878 1.48597i −0.683149 + 2.55603i 1.00000 2.99847 + 0.0958395i 3.28763i
101.2 −0.500000 + 0.866025i −1.72973 0.0896134i −0.500000 0.866025i 3.27919 1.89324i 0.942473 1.45318i 2.49160 0.889895i 1.00000 2.98394 + 0.310014i 3.78649i
101.3 −0.500000 + 0.866025i −1.25705 + 1.19156i −0.500000 0.866025i −1.80315 + 1.04105i −0.403394 1.68442i 1.78353 1.95424i 1.00000 0.160371 2.99571i 2.08210i
101.4 −0.500000 + 0.866025i −1.23327 1.21616i −0.500000 0.866025i 0.567570 0.327687i 1.66986 0.459961i −2.19987 1.46989i 1.00000 0.0419009 + 2.99971i 0.655374i
101.5 −0.500000 + 0.866025i −1.17271 + 1.27466i −0.500000 0.866025i 0.870413 0.502533i −0.517534 1.65292i 1.33597 + 2.28368i 1.00000 −0.249516 2.98961i 1.00507i
101.6 −0.500000 + 0.866025i −0.991815 1.41997i −0.500000 0.866025i 3.72094 2.14828i 1.72563 0.148954i −1.08595 + 2.41262i 1.00000 −1.03261 + 2.81669i 4.29657i
101.7 −0.500000 + 0.866025i −0.786858 + 1.54300i −0.500000 0.866025i 1.98183 1.14421i −0.942850 1.45294i −2.60041 + 0.487738i 1.00000 −1.76171 2.42825i 2.28842i
101.8 −0.500000 + 0.866025i −0.320103 1.70221i −0.500000 0.866025i −1.62172 + 0.936303i 1.63421 + 0.573890i 2.47749 + 0.928453i 1.00000 −2.79507 + 1.08977i 1.87261i
101.9 −0.500000 + 0.866025i 0.428009 1.67834i −0.500000 0.866025i 1.58996 0.917964i 1.23948 + 1.20983i 2.08732 1.62576i 1.00000 −2.63362 1.43668i 1.83593i
101.10 −0.500000 + 0.866025i 0.458130 + 1.67036i −0.500000 0.866025i 1.57344 0.908426i −1.67564 0.438430i −0.414054 2.61315i 1.00000 −2.58023 + 1.53049i 1.81685i
101.11 −0.500000 + 0.866025i 0.801045 + 1.53568i −0.500000 0.866025i −1.09866 + 0.634311i −1.73046 0.0741172i −2.21179 + 1.45190i 1.00000 −1.71665 + 2.46030i 1.26862i
101.12 −0.500000 + 0.866025i 0.925467 + 1.46407i −0.500000 0.866025i −2.64914 + 1.52948i −1.73066 + 0.0694407i 2.61668 0.391106i 1.00000 −1.28702 + 2.70990i 3.05896i
101.13 −0.500000 + 0.866025i 1.51226 + 0.844435i −0.500000 0.866025i 0.511132 0.295102i −1.48743 + 0.887438i 1.57814 + 2.12355i 1.00000 1.57386 + 2.55401i 0.590205i
101.14 −0.500000 + 0.866025i 1.57643 0.717543i −0.500000 0.866025i 1.41302 0.815806i −0.166804 + 1.72400i 1.06124 + 2.42359i 1.00000 1.97026 2.26231i 1.63161i
101.15 −0.500000 + 0.866025i 1.60671 0.646913i −0.500000 0.866025i −2.60318 + 1.50295i −0.243110 + 1.71490i −0.916975 2.48177i 1.00000 2.16301 2.07880i 3.00590i
101.16 −0.500000 + 0.866025i 1.70046 + 0.329307i −0.500000 0.866025i −1.26448 + 0.730045i −1.13542 + 1.30799i −2.63261 + 0.263392i 1.00000 2.78311 + 1.11994i 1.46009i
101.17 −0.500000 + 0.866025i 1.71486 + 0.243407i −0.500000 0.866025i 2.88000 1.66277i −1.06823 + 1.36341i −0.187179 2.63912i 1.00000 2.88151 + 0.834819i 3.32554i
173.1 −0.500000 0.866025i −1.73183 + 0.0276700i −0.500000 + 0.866025i −2.84717 1.64381i 0.889878 + 1.48597i −0.683149 2.55603i 1.00000 2.99847 0.0958395i 3.28763i
173.2 −0.500000 0.866025i −1.72973 + 0.0896134i −0.500000 + 0.866025i 3.27919 + 1.89324i 0.942473 + 1.45318i 2.49160 + 0.889895i 1.00000 2.98394 0.310014i 3.78649i
173.3 −0.500000 0.866025i −1.25705 1.19156i −0.500000 + 0.866025i −1.80315 1.04105i −0.403394 + 1.68442i 1.78353 + 1.95424i 1.00000 0.160371 + 2.99571i 2.08210i
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.e yes 34
3.b odd 2 1 546.2.bn.f yes 34
7.d odd 6 1 546.2.bi.f yes 34
13.e even 6 1 546.2.bi.e 34
21.g even 6 1 546.2.bi.e 34
39.h odd 6 1 546.2.bi.f yes 34
91.p odd 6 1 546.2.bn.f yes 34
273.y even 6 1 inner 546.2.bn.e yes 34

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

## Hecke kernels

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