# Properties

 Label 546.2.u.a Level $546$ Weight $2$ Character orbit 546.u Analytic conductor $4.360$ Analytic rank $0$ Dimension $76$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$546 = 2 \cdot 3 \cdot 7 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 546.u (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.35983195036$$ Analytic rank: $$0$$ Dimension: $$76$$ Relative dimension: $$38$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$76 q + 38 q^{4} + 10 q^{7} - 8 q^{9}+O(q^{10})$$ 76 * q + 38 * q^4 + 10 * q^7 - 8 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$76 q + 38 q^{4} + 10 q^{7} - 8 q^{9} + 2 q^{13} + 12 q^{15} - 38 q^{16} - 8 q^{21} - 42 q^{25} + 20 q^{28} + 20 q^{30} + 12 q^{31} - 4 q^{36} - 6 q^{37} + 12 q^{39} + 8 q^{42} - 2 q^{43} - 4 q^{46} + 2 q^{49} + 10 q^{51} - 14 q^{52} + 18 q^{54} + 24 q^{55} + 8 q^{57} + 16 q^{58} - 12 q^{60} + 32 q^{63} - 76 q^{64} - 24 q^{66} + 96 q^{67} - 30 q^{69} - 54 q^{73} - 12 q^{75} + 18 q^{76} - 12 q^{78} - 60 q^{79} + 8 q^{81} + 8 q^{84} + 8 q^{85} - 24 q^{87} + 48 q^{91} + 16 q^{93} - 66 q^{97}+O(q^{100})$$ 76 * q + 38 * q^4 + 10 * q^7 - 8 * q^9 + 2 * q^13 + 12 * q^15 - 38 * q^16 - 8 * q^21 - 42 * q^25 + 20 * q^28 + 20 * q^30 + 12 * q^31 - 4 * q^36 - 6 * q^37 + 12 * q^39 + 8 * q^42 - 2 * q^43 - 4 * q^46 + 2 * q^49 + 10 * q^51 - 14 * q^52 + 18 * q^54 + 24 * q^55 + 8 * q^57 + 16 * q^58 - 12 * q^60 + 32 * q^63 - 76 * q^64 - 24 * q^66 + 96 * q^67 - 30 * q^69 - 54 * q^73 - 12 * q^75 + 18 * q^76 - 12 * q^78 - 60 * q^79 + 8 * q^81 + 8 * q^84 + 8 * q^85 - 24 * q^87 + 48 * q^91 + 16 * q^93 - 66 * q^97

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
185.1 −0.866025 0.500000i −1.67595 + 0.437249i 0.500000 + 0.866025i 1.40279 + 2.42970i 1.67004 + 0.459307i 1.42569 2.22877i 1.00000i 2.61763 1.46562i 2.80557i
185.2 −0.866025 0.500000i −1.59603 + 0.672816i 0.500000 + 0.866025i 1.07511 + 1.86215i 1.71861 + 0.215340i 0.447408 + 2.60765i 1.00000i 2.09464 2.14767i 2.15023i
185.3 −0.866025 0.500000i −1.56696 0.737995i 0.500000 + 0.866025i −1.41378 2.44875i 0.988029 + 1.42260i −2.45899 + 0.976416i 1.00000i 1.91073 + 2.31282i 2.82757i
185.4 −0.866025 0.500000i −1.51699 0.835899i 0.500000 + 0.866025i −0.699535 1.21163i 0.895807 + 1.48241i 1.97618 + 1.75918i 1.00000i 1.60255 + 2.53611i 1.39907i
185.5 −0.866025 0.500000i −1.38893 + 1.03484i 0.500000 + 0.866025i −1.75377 3.03762i 1.72026 0.201730i 1.37341 2.26136i 1.00000i 0.858232 2.87462i 3.50754i
185.6 −0.866025 0.500000i −1.18075 1.26722i 0.500000 + 0.866025i 0.386492 + 0.669424i 0.388948 + 1.68782i 0.662023 2.56159i 1.00000i −0.211678 + 2.99252i 0.772984i
185.7 −0.866025 0.500000i −0.685609 + 1.59058i 0.500000 + 0.866025i −0.587127 1.01693i 1.38904 1.03468i −1.73688 + 1.99580i 1.00000i −2.05988 2.18103i 1.17425i
185.8 −0.866025 0.500000i −0.266485 + 1.71143i 0.500000 + 0.866025i 0.279399 + 0.483934i 1.08650 1.34890i 0.253211 2.63361i 1.00000i −2.85797 0.912140i 0.558798i
185.9 −0.866025 0.500000i −0.219501 1.71809i 0.500000 + 0.866025i 0.356048 + 0.616693i −0.668950 + 1.59766i −2.23074 1.42260i 1.00000i −2.90364 + 0.754242i 0.712096i
185.10 −0.866025 0.500000i −0.0864870 1.72989i 0.500000 + 0.866025i −0.886718 1.53584i −0.790045 + 1.54137i 2.61154 + 0.424103i 1.00000i −2.98504 + 0.299226i 1.77344i
185.11 −0.866025 0.500000i 0.323515 + 1.70157i 0.500000 + 0.866025i 1.99252 + 3.45115i 0.570612 1.63536i −2.44529 1.01022i 1.00000i −2.79068 + 1.10097i 3.98504i
185.12 −0.866025 0.500000i 0.696855 + 1.58568i 0.500000 + 0.866025i −1.42622 2.47029i 0.189348 1.72167i 2.54641 + 0.718201i 1.00000i −2.02879 + 2.20998i 2.85245i
185.13 −0.866025 0.500000i 0.804592 1.53383i 0.500000 + 0.866025i 2.14723 + 3.71911i −1.46371 + 0.926039i 2.58099 0.581819i 1.00000i −1.70526 2.46821i 4.29446i
185.14 −0.866025 0.500000i 0.896713 1.48186i 0.500000 + 0.866025i −0.277881 0.481304i −1.51751 + 0.834971i −0.887101 + 2.49260i 1.00000i −1.39181 2.65760i 0.555761i
185.15 −0.866025 0.500000i 1.11365 1.32657i 0.500000 + 0.866025i −2.12998 3.68924i −1.62773 + 0.592023i −1.07162 2.41901i 1.00000i −0.519589 2.95466i 4.25996i
185.16 −0.866025 0.500000i 1.30626 + 1.13741i 0.500000 + 0.866025i −0.604752 1.04746i −0.562548 1.63815i −2.45540 + 0.985399i 1.00000i 0.412611 + 2.97149i 1.20950i
185.17 −0.866025 0.500000i 1.62717 + 0.593575i 0.500000 + 0.866025i 1.76347 + 3.05441i −1.11238 1.32763i 1.45770 + 2.20797i 1.00000i 2.29534 + 1.93169i 3.52693i
185.18 −0.866025 0.500000i 1.69393 + 0.361382i 0.500000 + 0.866025i 0.0272832 + 0.0472559i −1.28630 1.15993i 2.23165 1.42118i 1.00000i 2.73881 + 1.22431i 0.0545664i
185.19 −0.866025 0.500000i 1.72102 0.195177i 0.500000 + 0.866025i 0.349437 + 0.605242i −1.58803 0.691481i −1.78017 1.95729i 1.00000i 2.92381 0.671806i 0.698873i
185.20 0.866025 + 0.500000i −1.72102 0.195177i 0.500000 + 0.866025i −0.349437 0.605242i −1.39286 1.02954i −1.78017 1.95729i 1.00000i 2.92381 + 0.671806i 0.698873i
See all 76 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 425.38 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
91.m odd 6 1 inner
273.bf 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.u.a 76
3.b odd 2 1 inner 546.2.u.a 76
7.d odd 6 1 546.2.bb.a yes 76
13.c even 3 1 546.2.bb.a yes 76
21.g even 6 1 546.2.bb.a yes 76
39.i odd 6 1 546.2.bb.a yes 76
91.m odd 6 1 inner 546.2.u.a 76
273.bf even 6 1 inner 546.2.u.a 76

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
546.2.u.a 76 1.a even 1 1 trivial
546.2.u.a 76 3.b odd 2 1 inner
546.2.u.a 76 91.m odd 6 1 inner
546.2.u.a 76 273.bf even 6 1 inner
546.2.bb.a yes 76 7.d odd 6 1
546.2.bb.a yes 76 13.c even 3 1
546.2.bb.a yes 76 21.g even 6 1
546.2.bb.a yes 76 39.i odd 6 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(546, [\chi])$$.