# Properties

 Label 546.2.bi.f Level $546$ Weight $2$ Character orbit 546.bi 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.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.

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

## 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 −1.72563 0.148954i 1.00000 3.72094 2.14828i −1.72563 0.148954i 1.54641 + 2.14677i 1.00000 2.95563 + 0.514080i 3.72094 2.14828i
17.2 1.00000 −1.66986 0.459961i 1.00000 0.567570 0.327687i −1.66986 0.459961i −2.37289 + 1.17020i 1.00000 2.57687 + 1.53614i 0.567570 0.327687i
17.3 1.00000 −1.63421 + 0.573890i 1.00000 −1.62172 + 0.936303i −1.63421 + 0.573890i 2.04281 1.68135i 1.00000 2.34130 1.87572i −1.62172 + 0.936303i
17.4 1.00000 −1.23948 + 1.20983i 1.00000 1.58996 0.917964i −1.23948 + 1.20983i −0.364289 2.62055i 1.00000 0.0726040 2.99912i 1.58996 0.917964i
17.5 1.00000 −0.942473 1.45318i 1.00000 3.27919 1.89324i −0.942473 1.45318i 0.475130 2.60274i 1.00000 −1.22349 + 2.73917i 3.27919 1.89324i
17.6 1.00000 −0.889878 1.48597i 1.00000 −2.84717 + 1.64381i −0.889878 1.48597i 1.87202 + 1.86964i 1.00000 −1.41623 + 2.64467i −2.84717 + 1.64381i
17.7 1.00000 0.166804 + 1.72400i 1.00000 1.41302 0.815806i 0.166804 + 1.72400i 2.62951 + 0.292738i 1.00000 −2.94435 + 0.575141i 1.41302 0.815806i
17.8 1.00000 0.243110 + 1.71490i 1.00000 −2.60318 + 1.50295i 0.243110 + 1.71490i −2.60776 0.446759i 1.00000 −2.88179 + 0.833822i −2.60318 + 1.50295i
17.9 1.00000 0.403394 1.68442i 1.00000 −1.80315 + 1.04105i 0.403394 1.68442i −0.800654 2.52170i 1.00000 −2.67455 1.35897i −1.80315 + 1.04105i
17.10 1.00000 0.517534 1.65292i 1.00000 0.870413 0.502533i 0.517534 1.65292i 2.64571 0.0151415i 1.00000 −2.46432 1.71089i 0.870413 0.502533i
17.11 1.00000 0.942850 1.45294i 1.00000 1.98183 1.14421i 0.942850 1.45294i −0.877809 + 2.49589i 1.00000 −1.22207 2.73981i 1.98183 1.14421i
17.12 1.00000 1.06823 + 1.36341i 1.00000 2.88000 1.66277i 1.06823 + 1.36341i −2.37914 1.15746i 1.00000 −0.717779 + 2.91287i 2.88000 1.66277i
17.13 1.00000 1.13542 + 1.30799i 1.00000 −1.26448 + 0.730045i 1.13542 + 1.30799i −1.08820 + 2.41160i 1.00000 −0.421657 + 2.97022i −1.26448 + 0.730045i
17.14 1.00000 1.48743 + 0.887438i 1.00000 0.511132 0.295102i 1.48743 + 0.887438i 2.62812 0.304939i 1.00000 1.42491 + 2.64001i 0.511132 0.295102i
17.15 1.00000 1.67564 0.438430i 1.00000 1.57344 0.908426i 1.67564 0.438430i −2.47008 0.947995i 1.00000 2.61556 1.46930i 1.57344 0.908426i
17.16 1.00000 1.73046 0.0741172i 1.00000 −1.09866 + 0.634311i 1.73046 0.0741172i 0.151485 + 2.64141i 1.00000 2.98901 0.256514i −1.09866 + 0.634311i
17.17 1.00000 1.73066 + 0.0694407i 1.00000 −2.64914 + 1.52948i 1.73066 + 0.0694407i 0.969634 2.46167i 1.00000 2.99036 + 0.240356i −2.64914 + 1.52948i
257.1 1.00000 −1.72563 + 0.148954i 1.00000 3.72094 + 2.14828i −1.72563 + 0.148954i 1.54641 2.14677i 1.00000 2.95563 0.514080i 3.72094 + 2.14828i
257.2 1.00000 −1.66986 + 0.459961i 1.00000 0.567570 + 0.327687i −1.66986 + 0.459961i −2.37289 1.17020i 1.00000 2.57687 1.53614i 0.567570 + 0.327687i
257.3 1.00000 −1.63421 0.573890i 1.00000 −1.62172 0.936303i −1.63421 0.573890i 2.04281 + 1.68135i 1.00000 2.34130 + 1.87572i −1.62172 0.936303i
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.f yes 34
3.b odd 2 1 546.2.bi.e 34
7.d odd 6 1 546.2.bn.e yes 34
13.e even 6 1 546.2.bn.f yes 34
21.g even 6 1 546.2.bn.f yes 34
39.h odd 6 1 546.2.bn.e yes 34
91.l odd 6 1 546.2.bi.e 34
273.br even 6 1 inner 546.2.bi.f yes 34

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

## Hecke kernels

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