# Properties

 Label 546.2.bw.a Level $546$ Weight $2$ Character orbit 546.bw Analytic conductor $4.360$ Analytic rank $0$ Dimension $152$ 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.bw (of order $$12$$, degree $$4$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.35983195036$$ Analytic rank: $$0$$ Dimension: $$152$$ Relative dimension: $$38$$ over $$\Q(\zeta_{12})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

## $q$-expansion

The dimension is sufficiently large that we do not compute an algebraic $$q$$-expansion, but we have computed the trace expansion.

 $$\operatorname{Tr}(f)(q) =$$ $$152 q + 12 q^{7} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$152 q + 12 q^{7} + 76 q^{16} - 8 q^{18} + 12 q^{19} + 8 q^{21} + 24 q^{27} + 16 q^{28} - 12 q^{31} - 48 q^{33} + 24 q^{36} - 36 q^{37} - 8 q^{39} + 84 q^{43} - 40 q^{45} - 16 q^{46} - 28 q^{49} + 4 q^{52} + 24 q^{54} + 16 q^{55} - 8 q^{57} - 16 q^{58} + 64 q^{61} - 100 q^{63} + 16 q^{66} - 112 q^{67} + 84 q^{69} + 16 q^{72} - 120 q^{73} - 48 q^{76} - 8 q^{78} + 32 q^{79} + 48 q^{81} - 8 q^{84} - 152 q^{85} + 48 q^{87} + 16 q^{91} - 12 q^{93} - 64 q^{94} - 100 q^{97} + 16 q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
11.1 −0.258819 0.965926i −1.73001 0.0839781i −0.866025 + 0.500000i −2.83225 0.758898i 0.366644 + 1.69280i −2.04314 + 1.68095i 0.707107 + 0.707107i 2.98590 + 0.290567i 2.93216i
11.2 −0.258819 0.965926i −1.61660 0.621783i −0.866025 + 0.500000i −0.475031 0.127284i −0.182190 + 1.72244i 2.42388 1.06056i 0.707107 + 0.707107i 2.22677 + 2.01034i 0.491788i
11.3 −0.258819 0.965926i −1.45291 0.942896i −0.866025 + 0.500000i 0.0616403 + 0.0165165i −0.534727 + 1.64744i −1.95035 1.78777i 0.707107 + 0.707107i 1.22189 + 2.73989i 0.0638147i
11.4 −0.258819 0.965926i −1.41158 + 1.00372i −0.866025 + 0.500000i −0.854914 0.229074i 1.33486 + 1.10370i 0.275303 2.63139i 0.707107 + 0.707107i 0.985106 2.83365i 0.885072i
11.5 −0.258819 0.965926i −1.37659 + 1.05119i −0.866025 + 0.500000i 3.72584 + 0.998336i 1.37166 + 1.05761i −1.75384 1.98092i 0.707107 + 0.707107i 0.789991 2.89412i 3.85727i
11.6 −0.258819 0.965926i −0.980845 + 1.42757i −0.866025 + 0.500000i −3.87106 1.03725i 1.63278 + 0.577943i 2.44614 + 1.00817i 0.707107 + 0.707107i −1.07589 2.80044i 4.00762i
11.7 −0.258819 0.965926i −0.773747 1.54962i −0.866025 + 0.500000i 2.25164 + 0.603326i −1.29656 + 1.14845i −1.19264 + 2.36170i 0.707107 + 0.707107i −1.80263 + 2.39802i 2.33107i
11.8 −0.258819 0.965926i −0.328251 + 1.70066i −0.866025 + 0.500000i 1.75073 + 0.469106i 1.72767 0.123098i 1.22850 + 2.34324i 0.707107 + 0.707107i −2.78450 1.11649i 1.81249i
11.9 −0.258819 0.965926i −0.229196 + 1.71682i −0.866025 + 0.500000i 0.0852751 + 0.0228494i 1.71764 0.222959i −2.48393 + 0.911101i 0.707107 + 0.707107i −2.89494 0.786976i 0.0882833i
11.10 −0.258819 0.965926i 0.0687143 1.73069i −0.866025 + 0.500000i 0.682162 + 0.182785i −1.68950 + 0.381562i 1.42099 + 2.23177i 0.707107 + 0.707107i −2.99056 0.237846i 0.706226i
11.11 −0.258819 0.965926i 0.131905 1.72702i −0.866025 + 0.500000i −3.76426 1.00863i −1.70231 + 0.319575i 1.07453 2.41772i 0.707107 + 0.707107i −2.96520 0.455606i 3.89704i
11.12 −0.258819 0.965926i 0.472866 1.66625i −0.866025 + 0.500000i 2.76510 + 0.740906i −1.73186 0.0254954i 0.700808 2.55125i 0.707107 + 0.707107i −2.55280 1.57583i 2.86264i
11.13 −0.258819 0.965926i 0.770566 + 1.55120i −0.866025 + 0.500000i −1.57124 0.421014i 1.29891 1.14579i 0.222219 2.63640i 0.707107 + 0.707107i −1.81246 + 2.39061i 1.62667i
11.14 −0.258819 0.965926i 0.868227 1.49873i −0.866025 + 0.500000i −3.18146 0.852470i −1.67237 0.450743i −1.87409 + 1.86756i 0.707107 + 0.707107i −1.49237 2.60247i 3.29369i
11.15 −0.258819 0.965926i 1.00636 + 1.40969i −0.866025 + 0.500000i 3.29330 + 0.882438i 1.10119 1.33693i 2.64105 0.157578i 0.707107 + 0.707107i −0.974462 + 2.83733i 3.40948i
11.16 −0.258819 0.965926i 1.57960 + 0.710547i −0.866025 + 0.500000i 2.90525 + 0.778459i 0.277506 1.70968i −2.63535 0.234373i 0.707107 + 0.707107i 1.99025 + 2.24475i 3.00773i
11.17 −0.258819 0.965926i 1.63257 0.578534i −0.866025 + 0.500000i −0.965749 0.258772i −0.981362 1.42721i 1.68839 + 2.03700i 0.707107 + 0.707107i 2.33060 1.88900i 0.999817i
11.18 −0.258819 0.965926i 1.66438 0.479415i −0.866025 + 0.500000i 1.88802 + 0.505894i −0.893853 1.48359i 2.05975 1.66055i 0.707107 + 0.707107i 2.54032 1.59586i 1.95463i
11.19 −0.258819 0.965926i 1.70453 + 0.307515i −0.866025 + 0.500000i −1.89300 0.507228i −0.144129 1.72604i −2.48026 0.921033i 0.707107 + 0.707107i 2.81087 + 1.04834i 1.95978i
11.20 0.258819 + 0.965926i −1.73001 + 0.0839781i −0.866025 + 0.500000i 2.83225 + 0.758898i −0.528877 1.64933i −2.04314 + 1.68095i −0.707107 0.707107i 2.98590 0.290567i 2.93216i
See next 80 embeddings (of 152 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 527.38 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.bd odd 12 1 inner
273.bw even 12 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 546.2.bw.a 152
3.b odd 2 1 inner 546.2.bw.a 152
7.c even 3 1 546.2.ch.a yes 152
13.f odd 12 1 546.2.ch.a yes 152
21.h odd 6 1 546.2.ch.a yes 152
39.k even 12 1 546.2.ch.a yes 152
91.bd odd 12 1 inner 546.2.bw.a 152
273.bw even 12 1 inner 546.2.bw.a 152

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
546.2.bw.a 152 1.a even 1 1 trivial
546.2.bw.a 152 3.b odd 2 1 inner
546.2.bw.a 152 91.bd odd 12 1 inner
546.2.bw.a 152 273.bw even 12 1 inner
546.2.ch.a yes 152 7.c even 3 1
546.2.ch.a yes 152 13.f odd 12 1
546.2.ch.a yes 152 21.h odd 6 1
546.2.ch.a yes 152 39.k even 12 1

## Hecke kernels

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