Properties

Label 37.8.e.a
Level $37$
Weight $8$
Character orbit 37.e
Analytic conductor $11.558$
Analytic rank $0$
Dimension $40$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 37 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 37.e (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(11.5582459429\)
Analytic rank: \(0\)
Dimension: \(40\)
Relative dimension: \(20\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 40q + 18q^{2} - 42q^{3} + 1114q^{4} - 672q^{5} + 156q^{7} - 9128q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 40q + 18q^{2} - 42q^{3} + 1114q^{4} - 672q^{5} + 156q^{7} - 9128q^{9} - 5124q^{10} + 12468q^{11} + 6926q^{12} + 10512q^{13} + 36402q^{15} - 31094q^{16} - 22440q^{17} - 153504q^{18} + 49086q^{19} + 28662q^{20} + 7736q^{21} + 132018q^{22} + 2754q^{24} + 72328q^{25} - 680244q^{26} + 42984q^{27} + 105488q^{28} + 382084q^{30} + 523656q^{32} + 9228q^{33} + 276914q^{34} - 904452q^{35} - 204828q^{36} + 1333776q^{37} + 50208q^{38} + 122040q^{39} - 1205236q^{40} - 871440q^{41} + 1436094q^{42} - 994326q^{44} - 987942q^{46} - 586932q^{47} + 3984388q^{48} - 1766072q^{49} + 1818708q^{50} - 2745228q^{52} + 1661988q^{53} + 1102554q^{54} - 2733258q^{55} - 5943924q^{56} + 3920592q^{57} + 5021672q^{58} + 5296632q^{59} - 4588524q^{61} - 8143398q^{62} - 2503648q^{63} + 10662964q^{64} - 1652736q^{65} + 1940622q^{67} - 10266552q^{69} - 5038546q^{70} + 6420528q^{71} - 6576036q^{72} - 3313032q^{73} + 16192386q^{74} - 5168536q^{75} + 10328022q^{76} - 8599404q^{77} + 18210222q^{78} - 15094374q^{79} + 1733956q^{81} - 191670q^{83} + 1175224q^{84} - 18053592q^{85} + 18575940q^{86} - 3762762q^{87} - 14873004q^{89} + 37293212q^{90} - 76496700q^{91} + 27846618q^{92} + 26687652q^{93} + 35421402q^{94} + 14310774q^{95} - 30894534q^{96} - 13793868q^{98} - 37141758q^{99} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
11.1 −18.1421 + 10.4744i −15.3322 + 26.5561i 155.424 269.203i 181.047 + 104.528i 642.380i −800.230 + 1386.04i 3830.44i 623.347 + 1079.67i −4379.44
11.2 −16.8690 + 9.73931i 5.24451 9.08376i 125.708 217.733i −269.659 155.688i 204.312i 385.556 667.802i 2403.99i 1038.49 + 1798.72i 6065.16
11.3 −14.9331 + 8.62160i 30.8914 53.5054i 84.6641 146.643i 237.219 + 136.958i 1065.33i 106.642 184.710i 712.630i −815.054 1411.72i −4723.20
11.4 −13.4758 + 7.78027i −38.3502 + 66.4244i 57.0652 98.8398i 4.26713 + 2.46363i 1193.50i 243.028 420.937i 215.818i −1847.97 3200.78i −76.6708
11.5 −10.0545 + 5.80498i 31.9798 55.3907i 3.39554 5.88124i −274.161 158.287i 742.569i −610.528 + 1057.47i 1407.23i −951.919 1648.77i 3675.40
11.6 −8.60443 + 4.96777i −5.90461 + 10.2271i −14.6425 + 25.3615i 369.746 + 213.473i 117.331i 428.306 741.848i 1562.71i 1023.77 + 1773.22i −4241.93
11.7 −8.00216 + 4.62005i −13.9119 + 24.0960i −21.3103 + 36.9106i −360.513 208.143i 257.094i 67.3669 116.683i 1576.55i 706.421 + 1223.56i 3846.51
11.8 −6.09195 + 3.51719i −7.67088 + 13.2864i −39.2587 + 67.9981i 66.8903 + 38.6191i 107.920i −602.681 + 1043.87i 1452.72i 975.815 + 1690.16i −543.323
11.9 −2.67688 + 1.54550i 32.7398 56.7070i −59.2229 + 102.577i −63.8099 36.8407i 202.397i 457.071 791.671i 761.764i −1050.29 1819.15i 227.749
11.10 0.806849 0.465835i −38.5559 + 66.7808i −63.5660 + 110.100i −299.174 172.728i 71.8427i −372.891 + 645.866i 237.699i −1879.62 3255.59i −321.851
11.11 2.05422 1.18601i 3.39795 5.88542i −61.1868 + 105.979i −176.269 101.769i 16.1199i 721.166 1249.10i 593.889i 1070.41 + 1854.00i −482.794
11.12 3.57062 2.06150i −32.2488 + 55.8566i −55.5005 + 96.1296i 228.187 + 131.744i 265.923i 124.613 215.836i 985.399i −986.473 1708.62i 1086.36
11.13 3.63260 2.09728i 24.6553 42.7042i −55.2028 + 95.6141i 292.816 + 169.058i 206.836i −480.087 + 831.535i 1000.01i −122.265 211.770i 1418.24
11.14 9.14535 5.28007i 2.08216 3.60640i −8.24170 + 14.2750i −276.472 159.621i 43.9758i −330.781 + 572.930i 1525.77i 1084.83 + 1878.98i −3371.24
11.15 12.1223 6.99882i 43.2891 74.9789i 33.9669 58.8324i −221.481 127.872i 1211.89i −32.7203 + 56.6733i 840.785i −2654.39 4597.54i −3579.82
11.16 12.1985 7.04279i −13.3616 + 23.1430i 35.2017 60.9712i 158.100 + 91.2791i 376.413i 380.940 659.807i 811.281i 736.434 + 1275.54i 2571.44
11.17 14.2692 8.23834i 23.0376 39.9023i 71.7405 124.258i 363.263 + 209.730i 759.167i 415.828 720.236i 255.076i 32.0370 + 55.4897i 6911.31
11.18 15.6973 9.06286i −28.3079 + 49.0307i 100.271 173.674i 169.896 + 98.0895i 1026.20i −758.009 + 1312.91i 1314.87i −509.171 881.910i 3555.89
11.19 15.8575 9.15532i −37.9133 + 65.6677i 103.640 179.510i −366.586 211.649i 1388.43i 808.308 1400.03i 1451.67i −1781.33 3085.36i −7750.85
11.20 18.4955 10.6784i 13.2396 22.9317i 164.055 284.151i −99.3053 57.3339i 565.509i −72.8986 + 126.264i 4273.68i 742.926 + 1286.79i −2448.93
See all 40 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 27.20
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
37.e even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 37.8.e.a 40
37.e even 6 1 inner 37.8.e.a 40
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
37.8.e.a 40 1.a even 1 1 trivial
37.8.e.a 40 37.e even 6 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{8}^{\mathrm{new}}(37, [\chi])\).