# Properties

 Label 684.3.m.a Level $684$ Weight $3$ Character orbit 684.m Analytic conductor $18.638$ Analytic rank $0$ Dimension $80$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$684 = 2^{2} \cdot 3^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 684.m (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$18.6376500822$$ Analytic rank: $$0$$ Dimension: $$80$$ Relative dimension: $$40$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$80 q - 2 q^{3} + q^{7} - 2 q^{9}+O(q^{10})$$ 80 * q - 2 * q^3 + q^7 - 2 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$80 q - 2 q^{3} + q^{7} - 2 q^{9} + 18 q^{11} - 5 q^{13} - 2 q^{15} - 9 q^{17} + 20 q^{19} - 30 q^{21} + 72 q^{23} - 400 q^{25} + 25 q^{27} - 8 q^{31} - 64 q^{33} + 22 q^{37} + 39 q^{39} - 44 q^{43} - 196 q^{45} - 267 q^{49} - 47 q^{51} - 36 q^{53} + 84 q^{57} - 14 q^{61} - 260 q^{63} - 144 q^{65} - 77 q^{67} + 44 q^{69} - 135 q^{71} + 43 q^{73} + 69 q^{75} + 216 q^{77} - 17 q^{79} - 254 q^{81} - 171 q^{83} - 244 q^{87} + 216 q^{89} + 122 q^{91} + 292 q^{93} - 288 q^{95} - 8 q^{97} + 172 q^{99}+O(q^{100})$$ 80 * q - 2 * q^3 + q^7 - 2 * q^9 + 18 * q^11 - 5 * q^13 - 2 * q^15 - 9 * q^17 + 20 * q^19 - 30 * q^21 + 72 * q^23 - 400 * q^25 + 25 * q^27 - 8 * q^31 - 64 * q^33 + 22 * q^37 + 39 * q^39 - 44 * q^43 - 196 * q^45 - 267 * q^49 - 47 * q^51 - 36 * q^53 + 84 * q^57 - 14 * q^61 - 260 * q^63 - 144 * q^65 - 77 * q^67 + 44 * q^69 - 135 * q^71 + 43 * q^73 + 69 * q^75 + 216 * q^77 - 17 * q^79 - 254 * q^81 - 171 * q^83 - 244 * q^87 + 216 * q^89 + 122 * q^91 + 292 * q^93 - 288 * q^95 - 8 * q^97 + 172 * q^99

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
353.1 0 −2.99548 0.164631i 0 2.57048i 0 1.88687 3.26816i 0 8.94579 + 0.986299i 0
353.2 0 −2.99460 0.179992i 0 0.944063i 0 −5.01764 + 8.69081i 0 8.93521 + 1.07801i 0
353.3 0 −2.88193 + 0.833354i 0 7.26979i 0 3.75776 6.50863i 0 7.61104 4.80334i 0
353.4 0 −2.84492 0.952067i 0 6.24143i 0 4.14502 7.17938i 0 7.18714 + 5.41711i 0
353.5 0 −2.80552 + 1.06257i 0 5.14812i 0 2.38165 4.12514i 0 6.74191 5.96210i 0
353.6 0 −2.74110 1.21916i 0 1.38038i 0 −1.29258 + 2.23881i 0 6.02728 + 6.68370i 0
353.7 0 −2.64568 + 1.41435i 0 6.34957i 0 −3.73312 + 6.46596i 0 4.99921 7.48384i 0
353.8 0 −2.55262 1.57611i 0 9.08186i 0 3.71777 6.43936i 0 4.03172 + 8.04644i 0
353.9 0 −2.38029 + 1.82598i 0 5.89395i 0 −0.351537 + 0.608880i 0 2.33159 8.69274i 0
353.10 0 −2.11779 2.12485i 0 8.55202i 0 −5.68709 + 9.85033i 0 −0.0299448 + 8.99995i 0
353.11 0 −1.79577 + 2.40317i 0 4.55087i 0 5.85150 10.1351i 0 −2.55043 8.63107i 0
353.12 0 −1.70356 2.46939i 0 9.28650i 0 −2.28067 + 3.95023i 0 −3.19575 + 8.41351i 0
353.13 0 −1.61882 2.52575i 0 0.497329i 0 5.73504 9.93338i 0 −3.75887 + 8.17746i 0
353.14 0 −1.60029 2.53754i 0 1.56237i 0 −4.59594 + 7.96040i 0 −3.87817 + 8.12156i 0
353.15 0 −1.55115 + 2.56787i 0 5.60337i 0 −5.05578 + 8.75686i 0 −4.18789 7.96628i 0
353.16 0 −1.21870 + 2.74131i 0 2.92593i 0 −2.71227 + 4.69779i 0 −6.02956 6.68165i 0
353.17 0 −1.06149 + 2.80593i 0 1.84834i 0 2.46982 4.27785i 0 −6.74649 5.95692i 0
353.18 0 −1.03515 2.81575i 0 0.329507i 0 2.81673 4.87871i 0 −6.85692 + 5.82947i 0
353.19 0 −0.358884 2.97846i 0 0.408400i 0 −0.674294 + 1.16791i 0 −8.74240 + 2.13784i 0
353.20 0 −0.0106609 + 2.99998i 0 4.51007i 0 1.48435 2.57096i 0 −8.99977 0.0639652i 0
See all 80 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 653.40 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
171.n odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 684.3.m.a 80
3.b odd 2 1 2052.3.m.a 80
9.c even 3 1 2052.3.be.a 80
9.d odd 6 1 684.3.be.a yes 80
19.c even 3 1 684.3.be.a yes 80
57.h odd 6 1 2052.3.be.a 80
171.g even 3 1 2052.3.m.a 80
171.n odd 6 1 inner 684.3.m.a 80

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
684.3.m.a 80 1.a even 1 1 trivial
684.3.m.a 80 171.n odd 6 1 inner
684.3.be.a yes 80 9.d odd 6 1
684.3.be.a yes 80 19.c even 3 1
2052.3.m.a 80 3.b odd 2 1
2052.3.m.a 80 171.g even 3 1
2052.3.be.a 80 9.c even 3 1
2052.3.be.a 80 57.h odd 6 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{3}^{\mathrm{new}}(684, [\chi])$$.