# Properties

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

 $$\operatorname{Tr}(f)(q) =$$ $$80q - q^{7} + 4q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$80q - q^{7} + 4q^{9} - 6q^{11} + 33q^{15} - 21q^{17} - 20q^{19} - 48q^{23} - 200q^{25} - 63q^{27} - 27q^{29} - 24q^{31} + 27q^{33} - 54q^{35} - 81q^{39} - 18q^{41} - 152q^{43} + 188q^{45} - 12q^{47} - 267q^{49} - 126q^{51} - 36q^{53} + 126q^{57} - 135q^{59} - 7q^{61} - 190q^{63} - 288q^{65} + 48q^{69} - 81q^{71} + 55q^{73} + 165q^{75} + 30q^{77} + 28q^{81} - 93q^{83} + 306q^{87} + 216q^{89} + 96q^{91} + 24q^{93} + 288q^{95} - 241q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
445.1 0 −2.99896 + 0.0790095i 0 −2.76024 4.78088i 0 0.838417 + 1.45218i 0 8.98751 0.473893i 0
445.2 0 −2.99241 0.213197i 0 2.77840 + 4.81232i 0 0.506731 + 0.877684i 0 8.90909 + 1.27595i 0
445.3 0 −2.96123 0.480768i 0 −0.920304 1.59401i 0 3.31490 + 5.74157i 0 8.53772 + 2.84733i 0
445.4 0 −2.93454 + 0.623289i 0 3.64833 + 6.31909i 0 −4.80976 8.33075i 0 8.22302 3.65813i 0
445.5 0 −2.72361 + 1.25776i 0 0.340706 + 0.590120i 0 −2.79581 4.84248i 0 5.83609 6.85128i 0
445.6 0 −2.67140 + 1.36514i 0 −4.45215 7.71135i 0 −2.28143 3.95155i 0 5.27278 7.29368i 0
445.7 0 −2.67131 1.36533i 0 3.94439 + 6.83189i 0 6.38121 + 11.0526i 0 5.27176 + 7.29442i 0
445.8 0 −2.56169 1.56133i 0 −1.88587 3.26643i 0 −0.501515 0.868649i 0 4.12453 + 7.99927i 0
445.9 0 −2.34179 + 1.87511i 0 1.96284 + 3.39973i 0 2.56510 + 4.44288i 0 1.96795 8.78221i 0
445.10 0 −2.21812 2.01989i 0 −2.05295 3.55582i 0 −6.48935 11.2399i 0 0.840121 + 8.96070i 0
445.11 0 −2.18497 2.05570i 0 3.46044 + 5.99365i 0 −3.78976 6.56406i 0 0.548164 + 8.98329i 0
445.12 0 −1.99772 + 2.23810i 0 0.267948 + 0.464100i 0 4.62209 + 8.00569i 0 −1.01820 8.94222i 0
445.13 0 −1.91383 2.31025i 0 −2.98110 5.16342i 0 6.83689 + 11.8418i 0 −1.67447 + 8.84286i 0
445.14 0 −1.40415 + 2.65111i 0 −1.52868 2.64775i 0 −5.58622 9.67561i 0 −5.05674 7.44509i 0
445.15 0 −1.26845 2.71865i 0 1.21776 + 2.10923i 0 −1.46858 2.54365i 0 −5.78207 + 6.89693i 0
445.16 0 −0.719501 + 2.91244i 0 −1.98064 3.43056i 0 2.79341 + 4.83834i 0 −7.96464 4.19101i 0
445.17 0 −0.522544 2.95414i 0 −3.49763 6.05807i 0 0.133800 + 0.231748i 0 −8.45389 + 3.08734i 0
445.18 0 −0.455700 + 2.96519i 0 1.91242 + 3.31241i 0 −3.97487 6.88467i 0 −8.58467 2.70247i 0
445.19 0 −0.283015 2.98662i 0 1.65331 + 2.86362i 0 0.469266 + 0.812792i 0 −8.83980 + 1.69052i 0
445.20 0 −0.224370 2.99160i 0 1.33945 + 2.31999i 0 3.94899 + 6.83985i 0 −8.89932 + 1.34245i 0
See all 80 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 601.40 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.i 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.s.a 80
3.b odd 2 1 2052.3.s.a 80
9.c even 3 1 684.3.bl.a yes 80
9.d odd 6 1 2052.3.bl.a 80
19.d odd 6 1 684.3.bl.a yes 80
57.f even 6 1 2052.3.bl.a 80
171.i odd 6 1 inner 684.3.s.a 80
171.t even 6 1 2052.3.s.a 80

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

## Hecke kernels

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