# Properties

 Label 684.3.bl.a Level $684$ Weight $3$ Character orbit 684.bl 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.bl (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.

 $$\operatorname{Tr}(f)(q) =$$ $$80 q - 6 q^{3} - q^{7} - 2 q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$80 q - 6 q^{3} - q^{7} - 2 q^{9} - 6 q^{11} - 15 q^{13} + 24 q^{15} - 21 q^{17} - 20 q^{19} + 24 q^{23} + 400 q^{25} + 63 q^{27} + 24 q^{31} + 30 q^{33} - 54 q^{35} - 81 q^{39} + 76 q^{43} + 188 q^{45} + 24 q^{47} - 267 q^{49} - 243 q^{51} - 36 q^{53} + 72 q^{57} + 14 q^{61} + 284 q^{63} + 288 q^{65} - 21 q^{67} - 48 q^{69} - 81 q^{71} + 55 q^{73} - 165 q^{75} + 30 q^{77} - 51 q^{79} - 110 q^{81} - 93 q^{83} + 306 q^{87} + 216 q^{89} + 96 q^{91} + 204 q^{93} - 432 q^{95} + 90 q^{97} + 260 q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
373.1 0 −2.98864 0.260813i 0 −2.43552 0 −1.46858 2.54365i 0 8.86395 + 1.55895i 0
373.2 0 −2.95765 + 0.502306i 0 5.96221 0 6.83689 + 11.8418i 0 8.49538 2.97129i 0
373.3 0 −2.87278 + 0.864385i 0 −6.92087 0 −3.78976 6.56406i 0 7.50568 4.96637i 0
373.4 0 −2.85833 + 0.911006i 0 4.10591 0 −6.48935 11.2399i 0 7.34014 5.20792i 0
373.5 0 −2.81963 1.02453i 0 6.99525 0 0.133800 + 0.231748i 0 6.90066 + 5.77762i 0
373.6 0 −2.72800 1.24821i 0 −3.30662 0 0.469266 + 0.812792i 0 5.88393 + 6.81024i 0
373.7 0 −2.70298 1.30149i 0 −2.67889 0 3.94899 + 6.83985i 0 5.61225 + 7.03581i 0
373.8 0 −2.63299 + 1.43783i 0 3.77174 0 −0.501515 0.868649i 0 4.86530 7.57158i 0
373.9 0 −2.51806 + 1.63076i 0 −7.88879 0 6.38121 + 11.0526i 0 3.68127 8.21269i 0
373.10 0 −2.00289 2.23348i 0 4.14557 0 −5.99339 10.3809i 0 −0.976899 + 8.94682i 0
373.11 0 −1.97980 2.25397i 0 8.35221 0 0.352343 + 0.610277i 0 −1.16076 + 8.92483i 0
373.12 0 −1.89697 + 2.32411i 0 1.84061 0 3.31490 + 5.74157i 0 −1.80300 8.81755i 0
373.13 0 −1.68084 + 2.48491i 0 −5.55679 0 0.506731 + 0.877684i 0 −3.34954 8.35348i 0
373.14 0 −1.54730 2.57019i 0 −9.33980 0 0.278827 + 0.482942i 0 −4.21173 + 7.95370i 0
373.15 0 −1.43106 + 2.63668i 0 5.52048 0 0.838417 + 1.45218i 0 −4.90416 7.54647i 0
373.16 0 −1.11019 2.78702i 0 −7.12097 0 −5.30926 9.19591i 0 −6.53494 + 6.18826i 0
373.17 0 −0.981973 2.83474i 0 0.638534 0 −1.11619 1.93329i 0 −7.07146 + 5.56727i 0
373.18 0 −0.927485 + 2.85303i 0 −7.29666 0 −4.80976 8.33075i 0 −7.27954 5.29228i 0
373.19 0 −0.522039 2.95423i 0 4.76547 0 3.51868 + 6.09454i 0 −8.45495 + 3.08444i 0
373.20 0 −0.334639 2.98128i 0 0.387901 0 5.33508 + 9.24062i 0 −8.77603 + 1.99530i 0
See all 80 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 673.40 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.s 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.bl.a yes 80
3.b odd 2 1 2052.3.bl.a 80
9.c even 3 1 684.3.s.a 80
9.d odd 6 1 2052.3.s.a 80
19.d odd 6 1 684.3.s.a 80
57.f even 6 1 2052.3.s.a 80
171.k even 6 1 2052.3.bl.a 80
171.s odd 6 1 inner 684.3.bl.a yes 80

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
684.3.s.a 80 9.c even 3 1
684.3.s.a 80 19.d odd 6 1
684.3.bl.a yes 80 1.a even 1 1 trivial
684.3.bl.a yes 80 171.s odd 6 1 inner
2052.3.s.a 80 9.d odd 6 1
2052.3.s.a 80 57.f even 6 1
2052.3.bl.a 80 3.b odd 2 1
2052.3.bl.a 80 171.k even 6 1

## Hecke kernels

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