Properties

Label 76.5.b.a
Level $76$
Weight $5$
Character orbit 76.b
Analytic conductor $7.856$
Analytic rank $0$
Dimension $36$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 76 = 2^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 76.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(7.85611719437\)
Analytic rank: \(0\)
Dimension: \(36\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 36q + 6q^{2} - 6q^{4} + 24q^{5} + 66q^{6} + 216q^{8} - 972q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 36q + 6q^{2} - 6q^{4} + 24q^{5} + 66q^{6} + 216q^{8} - 972q^{9} + 152q^{10} + 160q^{12} + 120q^{13} - 60q^{14} - 38q^{16} - 600q^{17} + 286q^{18} - 600q^{20} + 608q^{21} + 1080q^{22} + 958q^{24} + 4604q^{25} - 2766q^{26} - 2250q^{28} - 168q^{29} - 1380q^{30} + 3576q^{32} + 1440q^{33} + 908q^{34} - 5836q^{36} - 2248q^{37} - 1716q^{40} + 1800q^{41} - 5006q^{42} - 2520q^{44} + 88q^{45} + 6404q^{46} + 1064q^{48} - 12188q^{49} + 3354q^{50} + 15492q^{52} - 6600q^{53} + 1654q^{54} + 12924q^{56} + 5450q^{58} - 11188q^{60} + 2200q^{61} - 9972q^{62} + 12570q^{64} - 15792q^{65} + 10500q^{66} - 22614q^{68} + 19904q^{69} + 900q^{70} - 11376q^{72} + 11560q^{73} + 17304q^{74} + 1680q^{77} - 24740q^{78} + 12900q^{80} + 13604q^{81} - 18420q^{82} + 5644q^{84} - 11552q^{85} + 24564q^{86} - 15304q^{88} + 13800q^{89} - 60212q^{90} - 2142q^{92} + 34592q^{93} - 23096q^{94} - 35770q^{96} + 8200q^{97} + 25566q^{98} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
39.1 −3.88689 0.944484i 15.4306i 14.2159 + 7.34222i −33.7971 −14.5740 + 59.9772i 52.2303i −48.3211 41.9651i −157.104 131.366 + 31.9208i
39.2 −3.88689 + 0.944484i 15.4306i 14.2159 7.34222i −33.7971 −14.5740 59.9772i 52.2303i −48.3211 + 41.9651i −157.104 131.366 31.9208i
39.3 −3.82653 1.16519i 9.98677i 13.2847 + 8.91726i 35.6225 11.6365 38.2147i 14.7837i −40.4439 49.6013i −18.7357 −136.310 41.5069i
39.4 −3.82653 + 1.16519i 9.98677i 13.2847 8.91726i 35.6225 11.6365 + 38.2147i 14.7837i −40.4439 + 49.6013i −18.7357 −136.310 + 41.5069i
39.5 −3.24361 2.34072i 7.00580i 5.04202 + 15.1848i 9.28704 −16.3986 + 22.7241i 54.2259i 19.1890 61.0556i 31.9187 −30.1236 21.7384i
39.6 −3.24361 + 2.34072i 7.00580i 5.04202 15.1848i 9.28704 −16.3986 22.7241i 54.2259i 19.1890 + 61.0556i 31.9187 −30.1236 + 21.7384i
39.7 −3.11720 2.50661i 12.3170i 3.43384 + 15.6272i −18.3676 30.8738 38.3944i 80.8078i 28.4673 57.3203i −70.7078 57.2555 + 46.0405i
39.8 −3.11720 + 2.50661i 12.3170i 3.43384 15.6272i −18.3676 30.8738 + 38.3944i 80.8078i 28.4673 + 57.3203i −70.7078 57.2555 46.0405i
39.9 −2.45315 3.15944i 2.52727i −3.96408 + 15.5012i −40.2815 7.98474 6.19977i 43.5781i 58.6994 25.5025i 74.6129 98.8167 + 127.267i
39.10 −2.45315 + 3.15944i 2.52727i −3.96408 15.5012i −40.2815 7.98474 + 6.19977i 43.5781i 58.6994 + 25.5025i 74.6129 98.8167 127.267i
39.11 −2.32469 3.25512i 7.75287i −5.19165 + 15.1343i 27.0702 −25.2365 + 18.0230i 95.4821i 61.3329 18.2830i 20.8930 −62.9297 88.1167i
39.12 −2.32469 + 3.25512i 7.75287i −5.19165 15.1343i 27.0702 −25.2365 18.0230i 95.4821i 61.3329 + 18.2830i 20.8930 −62.9297 + 88.1167i
39.13 −1.32688 3.77351i 14.8593i −12.4788 + 10.0140i 13.5934 56.0716 19.7164i 39.3950i 54.3456 + 33.8016i −139.798 −18.0368 51.2950i
39.14 −1.32688 + 3.77351i 14.8593i −12.4788 10.0140i 13.5934 56.0716 + 19.7164i 39.3950i 54.3456 33.8016i −139.798 −18.0368 + 51.2950i
39.15 −0.672122 3.94313i 15.5341i −15.0965 + 5.30053i −5.44730 −61.2528 + 10.4408i 31.1799i 31.0473 + 55.9648i −160.307 3.66125 + 21.4794i
39.16 −0.672122 + 3.94313i 15.5341i −15.0965 5.30053i −5.44730 −61.2528 10.4408i 31.1799i 31.0473 55.9648i −160.307 3.66125 21.4794i
39.17 −0.591318 3.95605i 4.43430i −15.3007 + 4.67857i 13.1035 17.5423 2.62208i 8.96798i 27.5562 + 57.7638i 61.3370 −7.74835 51.8382i
39.18 −0.591318 + 3.95605i 4.43430i −15.3007 4.67857i 13.1035 17.5423 + 2.62208i 8.96798i 27.5562 57.7638i 61.3370 −7.74835 + 51.8382i
39.19 0.515277 3.96667i 1.66205i −15.4690 4.08787i −27.3711 −6.59279 0.856415i 39.8073i −24.1861 + 59.2540i 78.2376 −14.1037 + 108.572i
39.20 0.515277 + 3.96667i 1.66205i −15.4690 + 4.08787i −27.3711 −6.59279 + 0.856415i 39.8073i −24.1861 59.2540i 78.2376 −14.1037 108.572i
See all 36 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 39.36
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 76.5.b.a 36
4.b odd 2 1 inner 76.5.b.a 36
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
76.5.b.a 36 1.a even 1 1 trivial
76.5.b.a 36 4.b odd 2 1 inner

Hecke kernels

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