# Properties

 Label 546.2.bb.a Level $546$ Weight $2$ Character orbit 546.bb 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.bb (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.

 $$\operatorname{Tr}(f)(q) =$$ $$76q - 76q^{4} + 10q^{7} + 4q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$76q - 76q^{4} + 10q^{7} + 4q^{9} - 2q^{13} + 12q^{15} + 76q^{16} + 18q^{19} - 8q^{21} - 42q^{25} - 10q^{28} - 10q^{30} + 12q^{31} + 72q^{33} - 4q^{36} + 12q^{37} - 42q^{39} + 2q^{42} - 2q^{43} + 8q^{46} - 10q^{49} + 10q^{51} + 2q^{52} - 24q^{55} + 8q^{57} - 8q^{58} - 12q^{60} + 78q^{61} + 2q^{63} - 76q^{64} - 24q^{66} - 48q^{67} + 30q^{69} - 54q^{73} - 18q^{76} - 12q^{78} - 60q^{79} - 4q^{81} - 24q^{82} + 8q^{84} + 8q^{85} - 8q^{91} - 32q^{93} - 24q^{94} + 66q^{97} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
269.1 1.00000i −1.68782 0.388948i −1.00000 −0.386492 0.669424i −0.388948 + 1.68782i 1.88739 + 1.85412i 1.00000i 2.69744 + 1.31294i −0.669424 + 0.386492i
269.2 1.00000i −1.59766 + 0.668950i −1.00000 −0.356048 0.616693i 0.668950 + 1.59766i 2.34738 1.22057i 1.00000i 2.10501 2.13750i −0.616693 + 0.356048i
269.3 1.00000i −1.54137 + 0.790045i −1.00000 0.886718 + 1.53584i 0.790045 + 1.54137i −1.67305 + 2.04961i 1.00000i 1.75166 2.43551i 1.53584 0.886718i
269.4 1.00000i −1.48241 0.895807i −1.00000 0.699535 + 1.21163i −0.895807 + 1.48241i −2.51159 + 0.831830i 1.00000i 1.39506 + 2.65590i 1.21163 0.699535i
269.5 1.00000i −1.42260 0.988029i −1.00000 1.41378 + 2.44875i −0.988029 + 1.42260i 0.383892 2.61775i 1.00000i 1.04760 + 2.81115i 2.44875 1.41378i
269.6 1.00000i −0.926039 + 1.46371i −1.00000 −2.14723 3.71911i 1.46371 + 0.926039i −0.786623 + 2.52611i 1.00000i −1.28490 2.71091i −3.71911 + 2.14723i
269.7 1.00000i −0.834971 + 1.51751i −1.00000 0.277881 + 0.481304i 1.51751 + 0.834971i −1.71510 2.01455i 1.00000i −1.60565 2.53415i 0.481304 0.277881i
269.8 1.00000i −0.592023 + 1.62773i −1.00000 2.12998 + 3.68924i 1.62773 + 0.592023i 2.63074 + 0.281454i 1.00000i −2.29902 1.92731i 3.68924 2.12998i
269.9 1.00000i −0.459307 1.67004i −1.00000 −1.40279 2.42970i −1.67004 + 0.459307i 1.21733 + 2.34907i 1.00000i −2.57807 + 1.53412i −2.42970 + 1.40279i
269.10 1.00000i −0.215340 1.71861i −1.00000 −1.07511 1.86215i −1.71861 + 0.215340i −2.48199 0.916357i 1.00000i −2.90726 + 0.740172i −1.86215 + 1.07511i
269.11 1.00000i 0.201730 1.72026i −1.00000 1.75377 + 3.03762i −1.72026 0.201730i 1.27169 + 2.32009i 1.00000i −2.91861 0.694059i 3.03762 1.75377i
269.12 1.00000i 0.691481 + 1.58803i −1.00000 −0.349437 0.605242i 1.58803 0.691481i 2.58515 0.563029i 1.00000i −2.04371 + 2.19619i −0.605242 + 0.349437i
269.13 1.00000i 1.03468 1.38904i −1.00000 0.587127 + 1.01693i −1.38904 1.03468i −0.859975 2.50209i 1.00000i −0.858887 2.87442i 1.01693 0.587127i
269.14 1.00000i 1.15993 + 1.28630i −1.00000 −0.0272832 0.0472559i 1.28630 1.15993i 0.114956 + 2.64325i 1.00000i −0.309116 + 2.98403i −0.0472559 + 0.0272832i
269.15 1.00000i 1.32763 + 1.11238i −1.00000 −1.76347 3.05441i 1.11238 1.32763i −2.64100 + 0.158422i 1.00000i 0.525225 + 2.95367i −3.05441 + 1.76347i
269.16 1.00000i 1.34890 1.08650i −1.00000 −0.279399 0.483934i −1.08650 1.34890i 2.15416 + 1.53609i 1.00000i 0.639049 2.93115i −0.483934 + 0.279399i
269.17 1.00000i 1.63536 0.570612i −1.00000 −1.99252 3.45115i −0.570612 1.63536i 2.09752 1.61258i 1.00000i 2.34880 1.86631i −3.45115 + 1.99252i
269.18 1.00000i 1.63815 + 0.562548i −1.00000 0.604752 + 1.04746i 0.562548 1.63815i 0.374319 2.61914i 1.00000i 2.36708 + 1.84308i 1.04746 0.604752i
269.19 1.00000i 1.72167 0.189348i −1.00000 1.42622 + 2.47029i −0.189348 1.72167i −1.89518 + 1.84615i 1.00000i 2.92829 0.651988i 2.47029 1.42622i
269.20 1.00000i −1.73168 0.0356471i −1.00000 −0.277881 0.481304i 0.0356471 1.73168i −1.71510 2.01455i 1.00000i 2.99746 + 0.123459i 0.481304 0.277881i
See all 76 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 341.38 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.v odd 6 1 inner
273.r 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.bb.a yes 76
3.b odd 2 1 inner 546.2.bb.a yes 76
7.d odd 6 1 546.2.u.a 76
13.c even 3 1 546.2.u.a 76
21.g even 6 1 546.2.u.a 76
39.i odd 6 1 546.2.u.a 76
91.v odd 6 1 inner 546.2.bb.a yes 76
273.r even 6 1 inner 546.2.bb.a yes 76

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

## Hecke kernels

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