Properties

Label 76.4.f.a
Level $76$
Weight $4$
Character orbit 76.f
Analytic conductor $4.484$
Analytic rank $0$
Dimension $56$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 76 = 2^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 76.f (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(4.48414516044\)
Analytic rank: \(0\)
Dimension: \(56\)
Relative dimension: \(28\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 56q - 3q^{2} + 5q^{4} - 2q^{5} + 21q^{6} - 228q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 56q - 3q^{2} + 5q^{4} - 2q^{5} + 21q^{6} - 228q^{9} + 96q^{10} + 102q^{13} - 78q^{14} - 67q^{16} + 74q^{17} - 276q^{20} - 24q^{21} + 21q^{22} - 79q^{24} - 502q^{25} + 492q^{26} + 412q^{28} - 6q^{29} + 928q^{30} + 147q^{32} + 558q^{33} - 1170q^{34} + 70q^{36} - 1066q^{38} + 336q^{40} + 588q^{41} - 368q^{42} + 443q^{44} + 600q^{45} + 1353q^{48} - 2552q^{49} - 1086q^{52} - 594q^{53} + 21q^{54} + 574q^{57} + 1564q^{58} - 2826q^{60} + 2262q^{61} - 456q^{62} - 2098q^{64} - 2609q^{66} - 1612q^{68} + 3402q^{70} + 7350q^{72} - 92q^{73} - 62q^{74} + 667q^{76} + 1168q^{77} - 666q^{78} - 1558q^{80} - 2144q^{81} - 2113q^{82} + 1974q^{85} + 1590q^{86} + 258q^{89} + 294q^{90} - 3016q^{92} + 1780q^{93} + 158q^{96} - 792q^{97} + 3819q^{98} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
27.1 −2.82835 0.0213793i 0.662470 1.14743i 7.99909 + 0.120936i 0.369055 0.639222i −1.89823 + 3.23117i 4.50637i −22.6216 0.513064i 12.6223 + 21.8624i −1.05748 + 1.80005i
27.2 −2.76333 0.603342i −2.62655 + 4.54932i 7.27196 + 3.33446i 6.06129 10.4985i 10.0028 10.9866i 11.2918i −18.0830 13.6017i −0.297553 0.515377i −23.0835 + 25.3537i
27.3 −2.68912 + 0.876719i 4.88508 8.46120i 6.46273 4.71521i 6.65041 11.5188i −5.71846 + 27.0360i 4.05088i −13.2451 + 18.3457i −34.2280 59.2846i −7.78495 + 36.8061i
27.4 −2.49901 1.32474i 3.47531 6.01941i 4.49010 + 6.62110i −9.47803 + 16.4164i −16.6590 + 10.4387i 33.3441i −2.44954 22.4944i −10.6555 18.4559i 45.4333 28.4689i
27.5 −2.46181 + 1.39265i −4.46255 + 7.72937i 4.12104 6.85689i −6.17128 + 10.6890i 0.221653 25.2430i 29.2047i −0.595974 + 22.6196i −26.3288 45.6028i 0.306525 34.9087i
27.6 −2.45296 + 1.40819i 0.676811 1.17227i 4.03401 6.90846i −5.49309 + 9.51431i −0.00941024 + 3.82861i 11.1169i −0.166844 + 22.6268i 12.5839 + 21.7959i 0.0763748 31.0735i
27.7 −2.39677 1.50183i −3.47531 + 6.01941i 3.48899 + 7.19909i −9.47803 + 16.4164i 17.3697 9.20780i 33.3441i 2.44954 22.4944i −10.6555 18.4559i 47.3714 25.1119i
27.8 −1.90417 2.09144i 2.62655 4.54932i −0.748247 + 7.96493i 6.06129 10.4985i −14.5160 + 3.16942i 11.2918i 18.0830 13.6017i −0.297553 0.515377i −33.4987 + 7.31407i
27.9 −1.80915 + 2.17416i −2.57005 + 4.45145i −1.45393 7.86677i 7.41606 12.8450i −5.02855 13.6410i 20.1038i 19.7340 + 11.0711i 0.289720 + 0.501809i 14.5103 + 39.3623i
27.10 −1.43269 2.43873i −0.662470 + 1.14743i −3.89481 + 6.98788i 0.369055 0.639222i 3.74739 0.0283263i 4.50637i 22.6216 0.513064i 12.6223 + 21.8624i −2.08763 + 0.0157803i
27.11 −1.06422 + 2.62058i 1.31663 2.28047i −5.73488 5.57774i 1.30962 2.26832i 4.57498 + 5.87726i 35.2043i 20.7201 9.09277i 10.0330 + 17.3776i 4.55060 + 5.84595i
27.12 −1.03532 + 2.63213i 3.69266 6.39588i −5.85623 5.45019i −3.68797 + 6.38776i 13.0117 + 16.3413i 17.6395i 20.4087 9.77170i −13.7715 23.8529i −12.9952 16.3206i
27.13 −0.585298 2.76721i −4.88508 + 8.46120i −7.31485 + 3.23928i 6.65041 11.5188i 26.2731 + 8.56569i 4.05088i 13.2451 + 18.3457i −34.2280 59.2846i −35.7675 11.6611i
27.14 −0.0248348 2.82832i 4.46255 7.72937i −7.99877 + 0.140481i −6.17128 + 10.6890i −21.9719 12.4296i 29.2047i 0.595974 + 22.6196i −26.3288 45.6028i 30.3851 + 17.1889i
27.15 −0.00695190 2.82842i −0.676811 + 1.17227i −7.99990 + 0.0393258i −5.49309 + 9.51431i 3.32038 + 1.90615i 11.1169i 0.166844 + 22.6268i 12.5839 + 21.7959i 26.9486 + 15.4706i
27.16 0.147992 + 2.82455i −2.98979 + 5.17847i −7.95620 + 0.836024i −1.90510 + 3.29973i −15.0693 7.67845i 2.74987i −3.53885 22.3490i −4.37772 7.58243i −9.60221 4.89273i
27.17 0.750206 + 2.72712i 2.20715 3.82289i −6.87438 + 4.09180i 9.79476 16.9650i 12.0813 + 3.15120i 17.1729i −16.3160 15.6776i 3.75702 + 6.50734i 53.6137 + 13.9842i
27.18 0.978299 2.65385i 2.57005 4.45145i −6.08586 5.19252i 7.41606 12.8450i −9.29922 11.1754i 20.1038i −19.7340 + 11.0711i 0.289720 + 0.501809i −26.8336 32.2474i
27.19 1.37769 + 2.47022i 1.48658 2.57483i −4.20394 + 6.80639i −8.90564 + 15.4250i 8.40843 + 0.124854i 5.78146i −22.6050 1.00755i 9.08017 + 15.7273i −50.3724 0.747961i
27.20 1.73738 2.23193i −1.31663 + 2.28047i −1.96303 7.75542i 1.30962 2.26832i 2.80237 + 6.90067i 35.2043i −20.7201 9.09277i 10.0330 + 17.3776i −2.78744 6.86391i
See all 56 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 31.28
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
19.d odd 6 1 inner
76.f even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 76.4.f.a 56
4.b odd 2 1 inner 76.4.f.a 56
19.d odd 6 1 inner 76.4.f.a 56
76.f even 6 1 inner 76.4.f.a 56
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
76.4.f.a 56 1.a even 1 1 trivial
76.4.f.a 56 4.b odd 2 1 inner
76.4.f.a 56 19.d odd 6 1 inner
76.4.f.a 56 76.f even 6 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(76, [\chi])\).