Properties

Label 76.5.l.a
Level $76$
Weight $5$
Character orbit 76.l
Analytic conductor $7.856$
Analytic rank $0$
Dimension $228$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 76 = 2^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 76.l (of order \(18\), degree \(6\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(7.85611719437\)
Analytic rank: \(0\)
Dimension: \(228\)
Relative dimension: \(38\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 228q - 6q^{2} - 48q^{4} - 12q^{5} - 60q^{6} - 3q^{8} - 180q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 228q - 6q^{2} - 48q^{4} - 12q^{5} - 60q^{6} - 3q^{8} - 180q^{9} - 453q^{10} - 3q^{12} + 516q^{13} + 81q^{14} - 720q^{16} - 12q^{17} - 12q^{18} - 1182q^{20} - 930q^{21} - 54q^{22} - 2664q^{24} - 12q^{25} - 3099q^{26} - 4824q^{28} - 12q^{29} + 5814q^{30} + 7419q^{32} + 1542q^{33} + 6912q^{34} + 8547q^{36} - 24q^{37} - 1536q^{38} - 6942q^{40} + 6396q^{41} - 18855q^{42} - 1143q^{44} - 6q^{45} - 1488q^{46} + 6093q^{48} + 26748q^{49} + 8070q^{50} - 12591q^{52} + 2436q^{53} - 7365q^{54} - 14418q^{56} - 12q^{57} + 8268q^{58} + 34632q^{60} - 31068q^{61} + 27396q^{62} + 4575q^{64} + 3594q^{65} - 29556q^{66} - 22782q^{68} + 29370q^{69} - 71541q^{70} - 30174q^{72} - 6348q^{73} - 417q^{74} + 43194q^{76} - 50436q^{77} + 45537q^{78} + 30081q^{80} - 41994q^{81} + 72219q^{82} + 13779q^{84} - 59244q^{85} + 19200q^{86} - 22395q^{88} + 1860q^{89} + 45594q^{90} - 64140q^{92} + 59118q^{93} - 55590q^{94} - 98646q^{96} - 5556q^{97} - 52215q^{98} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
23.1 −3.99987 0.0325330i −1.55169 + 0.273604i 15.9979 + 0.260256i 3.23088 2.71103i 6.21545 1.04390i 4.97171 + 2.87042i −63.9809 1.56145i −73.7822 + 26.8545i −13.0113 + 10.7387i
23.2 −3.99725 0.148413i 8.88596 1.56683i 15.9559 + 1.18649i −25.4406 + 21.3472i −35.7519 + 4.94423i −18.4296 10.6403i −63.6038 7.11074i 0.390232 0.142033i 104.860 81.5542i
23.3 −3.94825 + 0.641375i −13.6721 + 2.41076i 15.1773 5.06461i 3.65652 3.06818i 52.4346 18.2872i 76.9302 + 44.4157i −56.6753 + 29.7306i 105.000 38.2168i −12.4690 + 14.4591i
23.4 −3.86299 1.03792i 13.6381 2.40477i 13.8454 + 8.01895i 29.8632 25.0582i −55.1799 4.86566i 54.4581 + 31.4414i −45.1618 45.3476i 104.100 37.8894i −141.370 + 65.8041i
23.5 −3.62639 1.68799i −13.6381 + 2.40477i 10.3014 + 12.2426i 29.8632 25.0582i 53.5163 + 14.3004i −54.4581 31.4414i −16.6913 61.7851i 104.100 37.8894i −150.594 + 40.4619i
23.6 −3.55482 + 1.83391i −12.3954 + 2.18564i 9.27354 13.0385i −26.3005 + 22.0687i 40.0551 30.5016i −82.9404 47.8857i −9.05443 + 63.3563i 72.7534 26.4801i 53.0215 126.683i
23.7 −3.53287 + 1.87585i 3.32006 0.585416i 8.96237 13.2543i 31.7438 26.6362i −10.6312 + 8.29614i −51.6354 29.8117i −6.79985 + 63.6377i −65.4350 + 23.8164i −62.1812 + 153.649i
23.8 −3.15747 2.45569i −8.88596 + 1.56683i 3.93918 + 15.5075i −25.4406 + 21.3472i 31.9048 + 16.8739i 18.4296 + 10.6403i 25.6438 58.6378i 0.390232 0.142033i 132.750 4.92883i
23.9 −3.08499 2.54614i 1.55169 0.273604i 3.03431 + 15.7096i 3.23088 2.71103i −5.48357 3.10675i −4.97171 2.87042i 30.6382 56.1899i −73.7822 + 26.8545i −16.8699 + 0.137212i
23.10 −3.06815 + 2.56640i 14.2932 2.52028i 2.82714 15.7482i −7.84979 + 6.58676i −37.3857 + 44.4148i 31.3486 + 18.0991i 31.7423 + 55.5736i 121.829 44.3421i 7.18009 40.3549i
23.11 −2.96694 + 2.68277i −1.05794 + 0.186543i 1.60551 15.9192i 1.47493 1.23761i 2.63839 3.39166i 26.6122 + 15.3646i 37.9442 + 51.5387i −75.0307 + 27.3089i −1.05580 + 7.62880i
23.12 −2.61226 3.02920i 13.6721 2.41076i −2.35216 + 15.8262i 3.65652 3.06818i −43.0179 35.1181i −76.9302 44.4157i 54.0851 34.2169i 105.000 38.2168i −18.8459 3.06144i
23.13 −1.54434 3.68985i 12.3954 2.18564i −11.2300 + 11.3968i −26.3005 + 22.0687i −27.2074 42.3618i 82.9404 + 47.8857i 59.3954 + 23.8368i 72.7534 26.4801i 122.047 + 62.9634i
23.14 −1.50056 3.70787i −3.32006 + 0.585416i −11.4966 + 11.1278i 31.7438 26.6362i 7.15261 + 11.4319i 51.6354 + 29.8117i 58.5118 + 25.9300i −65.4350 + 23.8164i −146.397 77.7326i
23.15 −1.47024 + 3.72000i −11.9688 + 2.11041i −11.6768 10.9386i 19.6524 16.4903i 9.74622 47.6266i 0.422999 + 0.244219i 57.8592 27.3552i 62.6822 22.8144i 32.4501 + 97.3515i
23.16 −1.33273 + 3.77145i −4.88717 + 0.861739i −12.4477 10.0526i −29.7827 + 24.9906i 3.26326 19.5802i 20.5177 + 11.8459i 54.5024 33.5483i −52.9733 + 19.2807i −54.5587 145.630i
23.17 −1.14588 + 3.83236i 9.44677 1.66572i −13.3739 8.78286i −12.8360 + 10.7707i −4.44126 + 38.1121i −66.0612 38.1405i 48.9840 41.1894i 10.3517 3.76773i −26.5686 61.5342i
23.18 −0.700689 3.93815i −14.2932 + 2.52028i −15.0181 + 5.51884i −7.84979 + 6.58676i 19.9403 + 54.5229i −31.3486 18.0991i 32.2570 + 55.2764i 121.829 44.3421i 31.4399 + 26.2984i
23.19 −0.548361 3.96223i 1.05794 0.186543i −15.3986 + 4.34547i 1.47493 1.23761i −1.31926 4.08950i −26.6122 15.3646i 25.6618 + 58.6300i −75.0307 + 27.3089i −5.71249 5.16535i
23.20 −0.324370 + 3.98683i 10.8623 1.91531i −15.7896 2.58641i 23.2093 19.4749i 4.11262 + 43.9272i 10.4065 + 6.00818i 15.4332 62.1113i 38.2053 13.9056i 70.1147 + 98.8485i
See next 80 embeddings (of 228 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 63.38
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
19.e even 9 1 inner
76.l odd 18 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 76.5.l.a 228
4.b odd 2 1 inner 76.5.l.a 228
19.e even 9 1 inner 76.5.l.a 228
76.l odd 18 1 inner 76.5.l.a 228
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
76.5.l.a 228 1.a even 1 1 trivial
76.5.l.a 228 4.b odd 2 1 inner
76.5.l.a 228 19.e even 9 1 inner
76.5.l.a 228 76.l odd 18 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{5}^{\mathrm{new}}(76, [\chi])\).