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 $4$

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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:

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 \(S_{2}^{\mathrm{new}}(684, [\chi])\):

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