Properties

Label 76.2.k.a
Level $76$
Weight $2$
Character orbit 76.k
Analytic conductor $0.607$
Analytic rank $0$
Dimension $48$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 76 = 2^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 76.k (of order \(18\), degree \(6\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.606863055362\)
Analytic rank: \(0\)
Dimension: \(48\)
Relative dimension: \(8\) over \(\Q(\zeta_{18})\)
Twist minimal: yes
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.

\(\operatorname{Tr}(f)(q) = \) \( 48q - 6q^{2} - 12q^{5} - 12q^{6} - 9q^{8} - 18q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 48q - 6q^{2} - 12q^{5} - 12q^{6} - 9q^{8} - 18q^{9} - 3q^{10} - 9q^{12} - 3q^{14} - 12q^{17} - 42q^{20} - 18q^{21} - 12q^{22} + 24q^{24} - 12q^{25} + 21q^{26} - 12q^{29} + 42q^{30} + 9q^{32} - 36q^{33} + 87q^{36} + 60q^{38} + 6q^{40} + 30q^{41} + 3q^{42} + 45q^{44} - 6q^{45} + 36q^{46} + 45q^{48} - 18q^{49} + 18q^{50} - 15q^{52} - 24q^{53} - 75q^{54} - 12q^{57} + 60q^{58} + 6q^{60} - 66q^{62} - 45q^{64} + 18q^{65} - 42q^{66} - 42q^{68} + 126q^{69} - 63q^{70} - 78q^{72} - 12q^{73} - 105q^{74} - 126q^{76} - 36q^{77} + 3q^{78} - 3q^{80} + 72q^{81} - 111q^{82} - 117q^{84} + 108q^{85} - 24q^{86} - 81q^{88} - 18q^{90} + 36q^{92} + 30q^{93} - 66q^{96} - 6q^{97} + 39q^{98} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
3.1 −1.35051 + 0.419681i −1.23656 + 0.450071i 1.64774 1.13356i −0.503046 + 2.85292i 1.48110 1.12678i −2.71118 + 1.56530i −1.74954 + 2.22241i −0.971618 + 0.815285i −0.517948 4.06400i
3.2 −1.32415 0.496616i −1.09089 + 0.397053i 1.50675 + 1.31519i 0.615920 3.49306i 1.64169 + 0.0159973i 3.53555 2.04125i −1.34202 2.48978i −1.26573 + 1.06208i −2.55028 + 4.31946i
3.3 −1.19877 0.750298i 2.80443 1.02073i 0.874105 + 1.79887i −0.165124 + 0.936467i −4.12772 0.880539i −2.67390 + 1.54378i 0.301838 2.81228i 4.52481 3.79677i 0.900576 0.998718i
3.4 −0.178792 + 1.40287i 1.23656 0.450071i −1.93607 0.501643i −0.503046 + 2.85292i 0.410302 + 1.81520i 2.71118 1.56530i 1.04989 2.62635i −0.971618 + 0.815285i −3.91232 1.21579i
3.5 0.223323 1.39647i 0.855656 0.311433i −1.90025 0.623726i −0.00805719 + 0.0456946i −0.243820 1.26445i 1.20959 0.698356i −1.29538 + 2.51435i −1.66298 + 1.39540i 0.0620117 + 0.0214562i
3.6 0.719007 + 1.21780i 1.09089 0.397053i −0.966058 + 1.75121i 0.615920 3.49306i 1.26789 + 1.04300i −3.53555 + 2.04125i −2.82722 + 0.0826699i −1.26573 + 1.06208i 4.69669 1.76147i
3.7 0.947064 + 1.05027i −2.80443 + 1.02073i −0.206140 + 1.98935i −0.165124 + 0.936467i −3.72802 1.97872i 2.67390 1.54378i −2.28458 + 1.66754i 4.52481 3.79677i −1.13993 + 0.713469i
3.8 1.33647 0.462424i −0.855656 + 0.311433i 1.57233 1.23604i −0.00805719 + 0.0456946i −0.999548 + 0.811899i −1.20959 + 0.698356i 1.52980 2.37901i −1.66298 + 1.39540i 0.0103621 + 0.0647955i
15.1 −1.37896 + 0.313814i −0.00846870 0.0480284i 1.80304 0.865472i 0.579816 0.486524i 0.0267499 + 0.0635714i 2.62687 + 1.51662i −2.21472 + 1.75927i 2.81684 1.02525i −0.646863 + 0.852849i
15.2 −1.06726 0.927872i 0.361434 + 2.04979i 0.278109 + 1.98057i −2.99965 + 2.51700i 1.51620 2.52304i 0.0108543 + 0.00626673i 1.54090 2.37184i −1.25194 + 0.455671i 5.53688 + 0.0969785i
15.3 −0.647187 + 1.25744i 0.547993 + 3.10782i −1.16230 1.62759i 1.46633 1.23040i −4.26255 1.32228i −1.58907 0.917452i 2.79882 0.408157i −6.53918 + 2.38007i 0.598159 + 2.64012i
15.4 −0.312491 + 1.37926i −0.547993 3.10782i −1.80470 0.862010i 1.46633 1.23040i 4.45773 + 0.215343i 1.58907 + 0.917452i 1.75289 2.21977i −6.53918 + 2.38007i 1.23882 + 2.40694i
15.5 −0.0238852 1.41401i −0.306623 1.73895i −1.99886 + 0.0675479i −0.220151 + 0.184728i −2.45157 + 0.475104i 0.588321 + 0.339668i 0.143257 + 2.82480i −0.110838 + 0.0403418i 0.266467 + 0.306884i
15.6 0.854626 + 1.12677i 0.00846870 + 0.0480284i −0.539229 + 1.92594i 0.579816 0.486524i −0.0468794 + 0.0505886i −2.62687 1.51662i −2.63093 + 1.03837i 2.81684 1.02525i 1.04373 + 0.237525i
15.7 0.927206 1.06784i 0.306623 + 1.73895i −0.280577 1.98022i −0.220151 + 0.184728i 2.14122 + 1.28494i −0.588321 0.339668i −2.37472 1.53646i −0.110838 + 0.0403418i −0.00686428 + 0.406368i
15.8 1.41400 0.0247662i −0.361434 2.04979i 1.99877 0.0700386i −2.99965 + 2.51700i −0.561832 2.88945i −0.0108543 0.00626673i 2.82452 0.148536i −1.25194 + 0.455671i −4.17916 + 3.63332i
51.1 −1.35051 0.419681i −1.23656 0.450071i 1.64774 + 1.13356i −0.503046 2.85292i 1.48110 + 1.12678i −2.71118 1.56530i −1.74954 2.22241i −0.971618 0.815285i −0.517948 + 4.06400i
51.2 −1.32415 + 0.496616i −1.09089 0.397053i 1.50675 1.31519i 0.615920 + 3.49306i 1.64169 0.0159973i 3.53555 + 2.04125i −1.34202 + 2.48978i −1.26573 1.06208i −2.55028 4.31946i
51.3 −1.19877 + 0.750298i 2.80443 + 1.02073i 0.874105 1.79887i −0.165124 0.936467i −4.12772 + 0.880539i −2.67390 1.54378i 0.301838 + 2.81228i 4.52481 + 3.79677i 0.900576 + 0.998718i
51.4 −0.178792 1.40287i 1.23656 + 0.450071i −1.93607 + 0.501643i −0.503046 2.85292i 0.410302 1.81520i 2.71118 + 1.56530i 1.04989 + 2.62635i −0.971618 0.815285i −3.91232 + 1.21579i
See all 48 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 71.8
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 76.2.k.a 48
3.b odd 2 1 684.2.cf.a 48
4.b odd 2 1 inner 76.2.k.a 48
12.b even 2 1 684.2.cf.a 48
19.f odd 18 1 inner 76.2.k.a 48
57.j even 18 1 684.2.cf.a 48
76.k even 18 1 inner 76.2.k.a 48
228.u odd 18 1 684.2.cf.a 48
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
76.2.k.a 48 1.a even 1 1 trivial
76.2.k.a 48 4.b odd 2 1 inner
76.2.k.a 48 19.f odd 18 1 inner
76.2.k.a 48 76.k even 18 1 inner
684.2.cf.a 48 3.b odd 2 1
684.2.cf.a 48 12.b even 2 1
684.2.cf.a 48 57.j even 18 1
684.2.cf.a 48 228.u odd 18 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(76, [\chi])\).