# Properties

 Label 546.2.z.b Level $546$ Weight $2$ Character orbit 546.z Analytic conductor $4.360$ Analytic rank $0$ Dimension $32$ 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.z (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.35983195036$$ Analytic rank: $$0$$ Dimension: $$32$$ Relative dimension: $$16$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$32q + 16q^{4} + 2q^{7} + 2q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$32q + 16q^{4} + 2q^{7} + 2q^{9} + 6q^{10} + 24q^{11} - 4q^{14} - 12q^{15} - 16q^{16} - 4q^{17} - 24q^{18} + 4q^{21} - 12q^{22} - 6q^{24} - 18q^{25} - 16q^{26} + 6q^{27} - 2q^{28} - 8q^{30} - 6q^{31} + 14q^{33} - 48q^{35} + 4q^{36} - 16q^{38} - 6q^{39} + 6q^{40} - 16q^{41} + 14q^{42} + 32q^{43} + 24q^{44} + 20q^{45} + 16q^{46} - 20q^{47} - 26q^{49} - 46q^{51} + 60q^{53} + 6q^{54} - 8q^{56} - 8q^{57} - 10q^{58} - 8q^{59} - 6q^{60} + 36q^{61} - 36q^{62} + 84q^{63} - 32q^{64} + 6q^{65} + 36q^{66} + 4q^{68} + 36q^{69} - 18q^{70} - 24q^{72} - 48q^{73} + 84q^{74} - 2q^{75} - 52q^{77} + 18q^{79} - 74q^{81} - 24q^{83} + 8q^{84} - 32q^{85} + 24q^{86} + 14q^{87} - 6q^{88} + 20q^{89} + 6q^{90} - 8q^{91} - 40q^{93} + 72q^{95} - 6q^{96} - 32q^{98} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
131.1 −0.866025 0.500000i −1.73117 + 0.0552650i 0.500000 + 0.866025i −1.72310 + 2.98449i 1.52687 + 0.817724i 2.50296 + 0.857422i 1.00000i 2.99389 0.191346i 2.98449 1.72310i
131.2 −0.866025 0.500000i −1.44180 0.959803i 0.500000 + 0.866025i 1.50901 2.61368i 0.768732 + 1.55211i 0.237102 2.63511i 1.00000i 1.15756 + 2.76768i −2.61368 + 1.50901i
131.3 −0.866025 0.500000i −1.34375 + 1.09285i 0.500000 + 0.866025i 0.166488 0.288366i 1.71015 0.274563i −2.56503 + 0.648546i 1.00000i 0.611342 2.93705i −0.288366 + 0.166488i
131.4 −0.866025 0.500000i −0.818620 + 1.52639i 0.500000 + 0.866025i 0.401573 0.695544i 1.47214 0.912581i 1.69379 2.03250i 1.00000i −1.65972 2.49906i −0.695544 + 0.401573i
131.5 −0.866025 0.500000i 0.114502 1.72826i 0.500000 + 0.866025i −0.698212 + 1.20934i −0.963293 + 1.43947i 0.352771 + 2.62213i 1.00000i −2.97378 0.395779i 1.20934 0.698212i
131.6 −0.866025 0.500000i 1.20682 1.24241i 0.500000 + 0.866025i −1.33293 + 2.30870i −1.66634 + 0.472552i −2.41436 1.08207i 1.00000i −0.0871789 2.99873i 2.30870 1.33293i
131.7 −0.866025 0.500000i 1.51893 0.832377i 0.500000 + 0.866025i 1.85844 3.21892i −1.73162 0.0386050i −1.15681 + 2.37945i 1.00000i 1.61430 2.52865i −3.21892 + 1.85844i
131.8 −0.866025 0.500000i 1.62906 + 0.588348i 0.500000 + 0.866025i −1.04730 + 1.81398i −1.11664 1.32406i 1.84957 1.89185i 1.00000i 2.30769 + 1.91691i 1.81398 1.04730i
131.9 0.866025 + 0.500000i −1.43878 0.964322i 0.500000 + 0.866025i 1.85995 3.22152i −0.763858 1.55452i −1.49982 2.17958i 1.00000i 1.14017 + 2.77489i 3.22152 1.85995i
131.10 0.866025 + 0.500000i −1.12951 + 1.31309i 0.500000 + 0.866025i −1.63964 + 2.83994i −1.63473 + 0.572419i 1.76715 + 1.96906i 1.00000i −0.448430 2.96630i −2.83994 + 1.63964i
131.11 0.866025 + 0.500000i −1.11753 1.32330i 0.500000 + 0.866025i −0.386271 + 0.669041i −0.306158 1.70478i 1.82832 + 1.91239i 1.00000i −0.502255 + 2.95766i −0.669041 + 0.386271i
131.12 0.866025 + 0.500000i −0.0921101 + 1.72960i 0.500000 + 0.866025i 0.890016 1.54155i −0.944570 + 1.45182i −1.51727 + 2.16746i 1.00000i −2.98303 0.318627i 1.54155 0.890016i
131.13 0.866025 + 0.500000i 0.691807 1.58789i 0.500000 + 0.866025i 0.735814 1.27447i 1.39307 1.02925i 0.322414 + 2.62603i 1.00000i −2.04281 2.19703i 1.27447 0.735814i
131.14 0.866025 + 0.500000i 1.08085 + 1.35343i 0.500000 + 0.866025i 1.49759 2.59391i 0.259332 + 1.71253i −0.0366585 2.64550i 1.00000i −0.663521 + 2.92570i 2.59391 1.49759i
131.15 0.866025 + 0.500000i 1.16571 + 1.28106i 0.500000 + 0.866025i −1.35360 + 2.34450i 0.369008 + 1.69229i −2.61407 + 0.408207i 1.00000i −0.282226 + 2.98670i −2.34450 + 1.35360i
131.16 0.866025 + 0.500000i 1.70558 0.301663i 0.500000 + 0.866025i −0.737837 + 1.27797i 1.62791 + 0.591542i 2.24993 1.39206i 1.00000i 2.81800 1.02902i −1.27797 + 0.737837i
521.1 −0.866025 + 0.500000i −1.73117 0.0552650i 0.500000 0.866025i −1.72310 2.98449i 1.52687 0.817724i 2.50296 0.857422i 1.00000i 2.99389 + 0.191346i 2.98449 + 1.72310i
521.2 −0.866025 + 0.500000i −1.44180 + 0.959803i 0.500000 0.866025i 1.50901 + 2.61368i 0.768732 1.55211i 0.237102 + 2.63511i 1.00000i 1.15756 2.76768i −2.61368 1.50901i
521.3 −0.866025 + 0.500000i −1.34375 1.09285i 0.500000 0.866025i 0.166488 + 0.288366i 1.71015 + 0.274563i −2.56503 0.648546i 1.00000i 0.611342 + 2.93705i −0.288366 0.166488i
521.4 −0.866025 + 0.500000i −0.818620 1.52639i 0.500000 0.866025i 0.401573 + 0.695544i 1.47214 + 0.912581i 1.69379 + 2.03250i 1.00000i −1.65972 + 2.49906i −0.695544 0.401573i
See all 32 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 521.16 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
21.g 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.z.b yes 32
3.b odd 2 1 546.2.z.a 32
7.d odd 6 1 546.2.z.a 32
21.g even 6 1 inner 546.2.z.b yes 32

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
546.2.z.a 32 3.b odd 2 1
546.2.z.a 32 7.d odd 6 1
546.2.z.b yes 32 1.a even 1 1 trivial
546.2.z.b yes 32 21.g even 6 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{32} + \cdots$$ acting on $$S_{2}^{\mathrm{new}}(546, [\chi])$$.