# Properties

 Label 546.2.z.a 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.

 $$\operatorname{Tr}(f)(q) =$$ $$32q + 16q^{4} + 2q^{7} + 2q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$32q + 16q^{4} + 2q^{7} + 2q^{9} + 6q^{10} - 24q^{11} + 4q^{14} - 12q^{15} - 16q^{16} + 4q^{17} + 24q^{18} - 12q^{21} - 12q^{22} - 6q^{24} - 18q^{25} + 16q^{26} - 6q^{27} - 2q^{28} + 10q^{30} - 6q^{31} - 14q^{33} + 48q^{35} + 4q^{36} + 16q^{38} + 6q^{39} + 6q^{40} + 16q^{41} - 18q^{42} + 32q^{43} - 24q^{44} - 20q^{45} + 16q^{46} + 20q^{47} - 26q^{49} + 44q^{51} - 60q^{53} - 6q^{54} + 8q^{56} - 8q^{57} - 10q^{58} + 8q^{59} - 6q^{60} + 36q^{61} + 36q^{62} - 28q^{63} - 32q^{64} - 6q^{65} + 12q^{66} - 4q^{68} - 36q^{69} - 18q^{70} + 24q^{72} - 48q^{73} - 84q^{74} + 14q^{75} + 52q^{77} + 18q^{79} + 70q^{81} + 24q^{83} - 24q^{84} - 32q^{85} - 24q^{86} - 26q^{87} - 6q^{88} - 20q^{89} - 6q^{90} - 8q^{91} + 92q^{93} - 72q^{95} - 6q^{96} + 32q^{98} + O(q^{100})$$

## 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 −1.70193 + 0.321635i 0.500000 + 0.866025i 1.63964 2.83994i 1.63473 + 0.572419i 1.76715 + 1.96906i 1.00000i 2.79310 1.09480i −2.83994 + 1.63964i
131.2 −0.866025 0.500000i −1.54393 0.785030i 0.500000 + 0.866025i −0.890016 + 1.54155i 0.944570 + 1.45182i −1.51727 + 2.16746i 1.00000i 1.76746 + 2.42407i 1.54155 0.890016i
131.3 −0.866025 0.500000i −0.631675 1.61276i 0.500000 + 0.866025i −1.49759 + 2.59391i −0.259332 + 1.71253i −0.0366585 2.64550i 1.00000i −2.20197 + 2.03748i 2.59391 1.49759i
131.4 −0.866025 0.500000i −0.526573 1.65007i 0.500000 + 0.866025i 1.35360 2.34450i −0.369008 + 1.69229i −2.61407 + 0.408207i 1.00000i −2.44544 + 1.73776i −2.34450 + 1.35360i
131.5 −0.866025 0.500000i 0.115738 + 1.72818i 0.500000 + 0.866025i −1.85995 + 3.22152i 0.763858 1.55452i −1.49982 2.17958i 1.00000i −2.97321 + 0.400031i 3.22152 1.85995i
131.6 −0.866025 0.500000i 0.587248 + 1.62946i 0.500000 + 0.866025i 0.386271 0.669041i 0.306158 1.70478i 1.82832 + 1.91239i 1.00000i −2.31028 + 1.91379i −0.669041 + 0.386271i
131.7 −0.866025 0.500000i 1.11404 1.32624i 0.500000 + 0.866025i 0.737837 1.27797i −1.62791 + 0.591542i 2.24993 1.39206i 1.00000i −0.517844 2.95497i −1.27797 + 0.737837i
131.8 −0.866025 0.500000i 1.72106 + 0.194824i 0.500000 + 0.866025i −0.735814 + 1.27447i −1.39307 1.02925i 0.322414 + 2.62603i 1.00000i 2.92409 + 0.670607i 1.27447 0.735814i
131.9 0.866025 + 0.500000i −1.73120 0.0542481i 0.500000 + 0.866025i −0.401573 + 0.695544i −1.47214 0.912581i 1.69379 2.03250i 1.00000i 2.99411 + 0.187829i −0.695544 + 0.401573i
131.10 0.866025 + 0.500000i −1.61832 + 0.617297i 0.500000 + 0.866025i −0.166488 + 0.288366i −1.71015 0.274563i −2.56503 + 0.648546i 1.00000i 2.23789 1.99796i −0.288366 + 0.166488i
131.11 0.866025 + 0.500000i −0.913445 + 1.47160i 0.500000 + 0.866025i 1.72310 2.98449i −1.52687 + 0.817724i 2.50296 + 0.857422i 1.00000i −1.33124 2.68846i 2.98449 1.72310i
131.12 0.866025 + 0.500000i 0.110315 + 1.72853i 0.500000 + 0.866025i −1.50901 + 2.61368i −0.768732 + 1.55211i 0.237102 2.63511i 1.00000i −2.97566 + 0.381366i −2.61368 + 1.50901i
131.13 0.866025 + 0.500000i 0.305007 1.70498i 0.500000 + 0.866025i 1.04730 1.81398i 1.11664 1.32406i 1.84957 1.89185i 1.00000i −2.81394 1.04007i 1.81398 1.04730i
131.14 0.866025 + 0.500000i 1.48032 0.899243i 0.500000 + 0.866025i −1.85844 + 3.21892i 1.73162 0.0386050i −1.15681 + 2.37945i 1.00000i 1.38272 2.66234i −3.21892 + 1.85844i
131.15 0.866025 + 0.500000i 1.55397 + 0.764969i 0.500000 + 0.866025i 0.698212 1.20934i 0.963293 + 1.43947i 0.352771 + 2.62213i 1.00000i 1.82964 + 2.37748i 1.20934 0.698212i
131.16 0.866025 + 0.500000i 1.67937 0.423929i 0.500000 + 0.866025i 1.33293 2.30870i 1.66634 + 0.472552i −2.41436 1.08207i 1.00000i 2.64057 1.42387i 2.30870 1.33293i
521.1 −0.866025 + 0.500000i −1.70193 0.321635i 0.500000 0.866025i 1.63964 + 2.83994i 1.63473 0.572419i 1.76715 1.96906i 1.00000i 2.79310 + 1.09480i −2.83994 1.63964i
521.2 −0.866025 + 0.500000i −1.54393 + 0.785030i 0.500000 0.866025i −0.890016 1.54155i 0.944570 1.45182i −1.51727 2.16746i 1.00000i 1.76746 2.42407i 1.54155 + 0.890016i
521.3 −0.866025 + 0.500000i −0.631675 + 1.61276i 0.500000 0.866025i −1.49759 2.59391i −0.259332 1.71253i −0.0366585 + 2.64550i 1.00000i −2.20197 2.03748i 2.59391 + 1.49759i
521.4 −0.866025 + 0.500000i −0.526573 + 1.65007i 0.500000 0.866025i 1.35360 + 2.34450i −0.369008 1.69229i −2.61407 0.408207i 1.00000i −2.44544 1.73776i −2.34450 1.35360i
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.a 32
3.b odd 2 1 546.2.z.b yes 32
7.d odd 6 1 546.2.z.b yes 32
21.g even 6 1 inner 546.2.z.a 32

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

## Hecke kernels

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