# Properties

 Label 546.2.bi.e Level $546$ Weight $2$ Character orbit 546.bi Analytic conductor $4.360$ Analytic rank $0$ Dimension $34$ CM no Inner twists $2$

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## Newspace parameters

 Level: $$N$$ $$=$$ $$546 = 2 \cdot 3 \cdot 7 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 546.bi (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.

 $$\operatorname{Tr}(f)(q) =$$ $$34q - 34q^{2} + 3q^{3} + 34q^{4} - 9q^{5} - 3q^{6} + 4q^{7} - 34q^{8} - 11q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$34q - 34q^{2} + 3q^{3} + 34q^{4} - 9q^{5} - 3q^{6} + 4q^{7} - 34q^{8} - 11q^{9} + 9q^{10} - 9q^{11} + 3q^{12} + 8q^{13} - 4q^{14} - 4q^{15} + 34q^{16} - 12q^{17} + 11q^{18} - 5q^{19} - 9q^{20} + 4q^{21} + 9q^{22} - 3q^{24} + 16q^{25} - 8q^{26} + 18q^{27} + 4q^{28} - 27q^{29} + 4q^{30} - q^{31} - 34q^{32} + 21q^{33} + 12q^{34} + 3q^{35} - 11q^{36} + 5q^{38} + 7q^{39} + 9q^{40} + 3q^{41} - 4q^{42} - 3q^{43} - 9q^{44} + 9q^{45} + 27q^{47} + 3q^{48} - 2q^{49} - 16q^{50} + 24q^{51} + 8q^{52} + 21q^{53} - 18q^{54} - 57q^{55} - 4q^{56} + 17q^{57} + 27q^{58} - 4q^{60} - 51q^{61} + q^{62} + 3q^{63} + 34q^{64} + 21q^{65} - 21q^{66} - 21q^{67} - 12q^{68} + 42q^{69} - 3q^{70} + 15q^{71} + 11q^{72} - 19q^{73} + 54q^{75} - 5q^{76} - 9q^{77} - 7q^{78} - 9q^{79} - 9q^{80} - 23q^{81} - 3q^{82} + 4q^{84} - 42q^{85} + 3q^{86} + 81q^{87} + 9q^{88} - 9q^{90} - 72q^{91} + 17q^{93} - 27q^{94} - 3q^{96} + 19q^{97} + 2q^{98} + 27q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
17.1 −1.00000 −1.66749 + 0.468501i 1.00000 −1.58996 + 0.917964i 1.66749 0.468501i −0.364289 2.62055i −1.00000 2.56101 1.56244i 1.58996 0.917964i
17.2 −1.00000 −1.40963 1.00646i 1.00000 −1.41302 + 0.815806i 1.40963 + 1.00646i 2.62951 + 0.292738i −1.00000 0.974089 + 2.83745i 1.41302 0.815806i
17.3 −1.00000 −1.36360 1.06799i 1.00000 2.60318 1.50295i 1.36360 + 1.06799i −2.60776 0.446759i −1.00000 0.718787 + 2.91262i −2.60318 + 1.50295i
17.4 −1.00000 −1.31411 + 1.12832i 1.00000 1.62172 0.936303i 1.31411 1.12832i 2.04281 1.68135i −1.00000 0.453768 2.96548i −1.62172 + 0.936303i
17.5 −1.00000 −0.733819 + 1.56892i 1.00000 −3.72094 + 2.14828i 0.733819 1.56892i 1.54641 + 2.14677i −1.00000 −1.92302 2.30261i 3.72094 2.14828i
17.6 −1.00000 −0.646635 1.60682i 1.00000 −2.88000 + 1.66277i 0.646635 + 1.60682i −2.37914 1.15746i −1.00000 −2.16373 + 2.07805i 2.88000 1.66277i
17.7 −1.00000 −0.565041 1.63729i 1.00000 1.26448 0.730045i 0.565041 + 1.63729i −1.08820 + 2.41160i −1.00000 −2.36146 + 1.85028i −1.26448 + 0.730045i
17.8 −1.00000 −0.436593 + 1.67612i 1.00000 −0.567570 + 0.327687i 0.436593 1.67612i −2.37289 + 1.17020i −1.00000 −2.61877 1.46357i 0.567570 0.327687i
17.9 −1.00000 −0.0248275 1.73187i 1.00000 −0.511132 + 0.295102i 0.0248275 + 1.73187i 2.62812 0.304939i −1.00000 −2.99877 + 0.0859963i 0.511132 0.295102i
17.10 −1.00000 0.787258 + 1.54280i 1.00000 −3.27919 + 1.89324i −0.787258 1.54280i 0.475130 2.60274i −1.00000 −1.76045 + 2.42916i 3.27919 1.89324i
17.11 −1.00000 0.805192 1.53351i 1.00000 2.64914 1.52948i −0.805192 + 1.53351i 0.969634 2.46167i −1.00000 −1.70333 2.46955i −2.64914 + 1.52948i
17.12 −1.00000 0.841952 + 1.51364i 1.00000 2.84717 1.64381i −0.841952 1.51364i 1.87202 + 1.86964i −1.00000 −1.58223 + 2.54883i −2.84717 + 1.64381i
17.13 −1.00000 0.929420 1.46157i 1.00000 1.09866 0.634311i −0.929420 + 1.46157i 0.151485 + 2.64141i −1.00000 −1.27236 2.71682i −1.09866 + 0.634311i
17.14 −1.00000 1.21751 1.23193i 1.00000 −1.57344 + 0.908426i −1.21751 + 1.23193i −2.47008 0.947995i −1.00000 −0.0353243 2.99979i 1.57344 0.908426i
17.15 −1.00000 1.66045 + 0.492861i 1.00000 1.80315 1.04105i −1.66045 0.492861i −0.800654 2.52170i −1.00000 2.51418 + 1.63674i −1.80315 + 1.04105i
17.16 −1.00000 1.69024 + 0.378264i 1.00000 −0.870413 + 0.502533i −1.69024 0.378264i 2.64571 0.0151415i −1.00000 2.71383 + 1.27872i 0.870413 0.502533i
17.17 −1.00000 1.72971 0.0900624i 1.00000 −1.98183 + 1.14421i −1.72971 + 0.0900624i −0.877809 + 2.49589i −1.00000 2.98378 0.311563i 1.98183 1.14421i
257.1 −1.00000 −1.66749 0.468501i 1.00000 −1.58996 0.917964i 1.66749 + 0.468501i −0.364289 + 2.62055i −1.00000 2.56101 + 1.56244i 1.58996 + 0.917964i
257.2 −1.00000 −1.40963 + 1.00646i 1.00000 −1.41302 0.815806i 1.40963 1.00646i 2.62951 0.292738i −1.00000 0.974089 2.83745i 1.41302 + 0.815806i
257.3 −1.00000 −1.36360 + 1.06799i 1.00000 2.60318 + 1.50295i 1.36360 1.06799i −2.60776 + 0.446759i −1.00000 0.718787 2.91262i −2.60318 1.50295i
See all 34 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 257.17 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.br 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.bi.e 34
3.b odd 2 1 546.2.bi.f yes 34
7.d odd 6 1 546.2.bn.f yes 34
13.e even 6 1 546.2.bn.e yes 34
21.g even 6 1 546.2.bn.e yes 34
39.h odd 6 1 546.2.bn.f yes 34
91.l odd 6 1 546.2.bi.f yes 34
273.br even 6 1 inner 546.2.bi.e 34

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

## Hecke kernels

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