# Properties

 Label 684.2.cf.a Level $684$ Weight $2$ Character orbit 684.cf Analytic conductor $5.462$ Analytic rank $0$ Dimension $48$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$684 = 2^{2} \cdot 3^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 684.cf (of order $$18$$, degree $$6$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$5.46176749826$$ Analytic rank: $$0$$ Dimension: $$48$$ Relative dimension: $$8$$ over $$\Q(\zeta_{18})$$ Twist minimal: no (minimal twist has level 76) Sato-Tate group: $\mathrm{SU}(2)[C_{18}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$48q + 6q^{2} + 12q^{5} + 9q^{8} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$48q + 6q^{2} + 12q^{5} + 9q^{8} - 3q^{10} + 3q^{14} + 12q^{17} + 42q^{20} - 12q^{22} - 12q^{25} - 21q^{26} + 12q^{29} - 9q^{32} - 60q^{38} + 6q^{40} - 30q^{41} - 45q^{44} + 36q^{46} - 18q^{49} - 18q^{50} - 15q^{52} + 24q^{53} + 60q^{58} + 66q^{62} - 45q^{64} - 18q^{65} + 42q^{68} - 63q^{70} - 12q^{73} + 105q^{74} - 126q^{76} + 36q^{77} + 3q^{80} - 111q^{82} + 108q^{85} + 24q^{86} - 81q^{88} - 36q^{92} - 6q^{97} - 39q^{98} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
91.1 −1.41400 + 0.0247662i 0 1.99877 0.0700386i 2.99965 2.51700i 0 −0.0108543 0.00626673i −2.82452 + 0.148536i 0 −4.17916 + 3.63332i
91.2 −0.927206 + 1.06784i 0 −0.280577 1.98022i 0.220151 0.184728i 0 −0.588321 0.339668i 2.37472 + 1.53646i 0 −0.00686428 + 0.406368i
91.3 −0.854626 1.12677i 0 −0.539229 + 1.92594i −0.579816 + 0.486524i 0 −2.62687 1.51662i 2.63093 1.03837i 0 1.04373 + 0.237525i
91.4 0.0238852 + 1.41401i 0 −1.99886 + 0.0675479i 0.220151 0.184728i 0 0.588321 + 0.339668i −0.143257 2.82480i 0 0.266467 + 0.306884i
91.5 0.312491 1.37926i 0 −1.80470 0.862010i −1.46633 + 1.23040i 0 1.58907 + 0.917452i −1.75289 + 2.21977i 0 1.23882 + 2.40694i
91.6 0.647187 1.25744i 0 −1.16230 1.62759i −1.46633 + 1.23040i 0 −1.58907 0.917452i −2.79882 + 0.408157i 0 0.598159 + 2.64012i
91.7 1.06726 + 0.927872i 0 0.278109 + 1.98057i 2.99965 2.51700i 0 0.0108543 + 0.00626673i −1.54090 + 2.37184i 0 5.53688 + 0.0969785i
91.8 1.37896 0.313814i 0 1.80304 0.865472i −0.579816 + 0.486524i 0 2.62687 + 1.51662i 2.21472 1.75927i 0 −0.646863 + 0.852849i
127.1 −1.33647 0.462424i 0 1.57233 + 1.23604i 0.00805719 + 0.0456946i 0 −1.20959 0.698356i −1.52980 2.37901i 0 0.0103621 0.0647955i
127.2 −0.947064 + 1.05027i 0 −0.206140 1.98935i 0.165124 + 0.936467i 0 2.67390 + 1.54378i 2.28458 + 1.66754i 0 −1.13993 0.713469i
127.3 −0.719007 + 1.21780i 0 −0.966058 1.75121i −0.615920 3.49306i 0 −3.53555 2.04125i 2.82722 + 0.0826699i 0 4.69669 + 1.76147i
127.4 −0.223323 1.39647i 0 −1.90025 + 0.623726i 0.00805719 + 0.0456946i 0 1.20959 + 0.698356i 1.29538 + 2.51435i 0 0.0620117 0.0214562i
127.5 0.178792 + 1.40287i 0 −1.93607 + 0.501643i 0.503046 + 2.85292i 0 2.71118 + 1.56530i −1.04989 2.62635i 0 −3.91232 + 1.21579i
127.6 1.19877 0.750298i 0 0.874105 1.79887i 0.165124 + 0.936467i 0 −2.67390 1.54378i −0.301838 2.81228i 0 0.900576 + 0.998718i
127.7 1.32415 0.496616i 0 1.50675 1.31519i −0.615920 3.49306i 0 3.53555 + 2.04125i 1.34202 2.48978i 0 −2.55028 4.31946i
127.8 1.35051 + 0.419681i 0 1.64774 + 1.13356i 0.503046 + 2.85292i 0 −2.71118 1.56530i 1.74954 + 2.22241i 0 −0.517948 + 4.06400i
307.1 −1.33647 + 0.462424i 0 1.57233 1.23604i 0.00805719 0.0456946i 0 −1.20959 + 0.698356i −1.52980 + 2.37901i 0 0.0103621 + 0.0647955i
307.2 −0.947064 1.05027i 0 −0.206140 + 1.98935i 0.165124 0.936467i 0 2.67390 1.54378i 2.28458 1.66754i 0 −1.13993 + 0.713469i
307.3 −0.719007 1.21780i 0 −0.966058 + 1.75121i −0.615920 + 3.49306i 0 −3.53555 + 2.04125i 2.82722 0.0826699i 0 4.69669 1.76147i
307.4 −0.223323 + 1.39647i 0 −1.90025 0.623726i 0.00805719 0.0456946i 0 1.20959 0.698356i 1.29538 2.51435i 0 0.0620117 + 0.0214562i
See all 48 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 667.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
19.f odd 18 1 inner
76.k even 18 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 684.2.cf.a 48
3.b odd 2 1 76.2.k.a 48
4.b odd 2 1 inner 684.2.cf.a 48
12.b even 2 1 76.2.k.a 48
19.f odd 18 1 inner 684.2.cf.a 48
57.j even 18 1 76.2.k.a 48
76.k even 18 1 inner 684.2.cf.a 48
228.u odd 18 1 76.2.k.a 48

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
76.2.k.a 48 3.b odd 2 1
76.2.k.a 48 12.b even 2 1
76.2.k.a 48 57.j even 18 1
76.2.k.a 48 228.u odd 18 1
684.2.cf.a 48 1.a even 1 1 trivial
684.2.cf.a 48 4.b odd 2 1 inner
684.2.cf.a 48 19.f odd 18 1 inner
684.2.cf.a 48 76.k even 18 1 inner

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(684, [\chi])$$:

 $$T_{5}^{24} - \cdots$$ $$20\!\cdots\!74$$$$T_{7}^{28} -$$$$10\!\cdots\!20$$$$T_{7}^{26} +$$$$41\!\cdots\!57$$$$T_{7}^{24} -$$$$13\!\cdots\!07$$$$T_{7}^{22} +$$$$37\!\cdots\!73$$$$T_{7}^{20} -$$$$81\!\cdots\!10$$$$T_{7}^{18} +$$$$14\!\cdots\!75$$$$T_{7}^{16} -$$$$19\!\cdots\!55$$$$T_{7}^{14} +$$$$21\!\cdots\!37$$$$T_{7}^{12} -$$$$16\!\cdots\!76$$$$T_{7}^{10} +$$$$91\!\cdots\!32$$$$T_{7}^{8} -$$$$29\!\cdots\!16$$$$T_{7}^{6} +$$$$59\!\cdots\!00$$$$T_{7}^{4} - 9420244992 T_{7}^{2} + 1478656$$">$$T_{7}^{48} - \cdots$$