Properties

Label 37.8.c.a
Level $37$
Weight $8$
Character orbit 37.c
Analytic conductor $11.558$
Analytic rank $0$
Dimension $42$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 37 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 37.c (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(11.5582459429\)
Analytic rank: \(0\)
Dimension: \(42\)
Relative dimension: \(21\) over \(\Q(\zeta_{3})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 42q - 16q^{2} - 68q^{3} - 1126q^{4} - 337q^{5} + 252q^{6} - 158q^{7} + 6264q^{8} - 12773q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 42q - 16q^{2} - 68q^{3} - 1126q^{4} - 337q^{5} + 252q^{6} - 158q^{7} + 6264q^{8} - 12773q^{9} + 10028q^{10} + 14876q^{11} - 4836q^{12} - 5922q^{13} + 20176q^{14} - 8460q^{15} - 36726q^{16} - 19095q^{17} - 7338q^{18} - 34334q^{19} - 75668q^{20} - 32764q^{21} + 18526q^{22} - 99896q^{23} - 203760q^{24} - 322660q^{25} + 101516q^{26} + 846616q^{27} - 73490q^{28} + 168186q^{29} - 88790q^{30} - 38052q^{31} - 172292q^{32} - 54536q^{33} + 90724q^{34} - 247046q^{35} + 352004q^{36} + 740185q^{37} + 1994976q^{38} - 591380q^{39} - 38962q^{40} - 292827q^{41} - 1499386q^{42} + 630980q^{43} - 1727390q^{44} - 994286q^{45} + 576988q^{46} + 3394668q^{47} - 79840q^{48} + 728851q^{49} - 69656q^{50} - 3970200q^{51} - 2043140q^{52} + 584438q^{53} - 2160774q^{54} + 2019874q^{55} - 658986q^{56} + 4295748q^{57} - 5684872q^{58} - 1773152q^{59} + 20208544q^{60} - 3596569q^{61} - 1538738q^{62} + 11795092q^{63} - 9622852q^{64} - 5027290q^{65} - 17464952q^{66} - 3249754q^{67} + 5608668q^{68} + 9599384q^{69} - 5102080q^{70} - 1162530q^{71} - 4730172q^{72} - 5266532q^{73} + 4052290q^{74} + 35543072q^{75} - 13781226q^{76} + 121180q^{77} - 7012890q^{78} - 13545718q^{79} - 17840888q^{80} - 16689077q^{81} - 43173544q^{82} + 7359048q^{83} + 41186680q^{84} + 21047382q^{85} + 6732838q^{86} + 7672800q^{87} + 42023140q^{88} + 2955725q^{89} - 14596808q^{90} - 19887752q^{91} + 13352900q^{92} + 9156784q^{93} + 690474q^{94} - 5667270q^{95} + 29986904q^{96} + 52065982q^{97} - 53664462q^{98} - 35534038q^{99} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
10.1 −11.0007 + 19.0538i 11.3160 + 19.5999i −178.031 308.359i 50.3125 + 87.1438i −497.937 −276.316 478.593i 5017.70 837.396 1450.41i −2213.89
10.2 −9.34255 + 16.1818i −43.2405 74.8948i −110.566 191.507i −171.836 297.629i 1615.91 −666.202 1153.90i 1740.20 −2645.99 + 4582.98i 6421.56
10.3 −8.90423 + 15.4226i 0.464514 + 0.804563i −94.5706 163.801i −251.123 434.958i −16.5446 772.514 + 1338.03i 1088.83 1093.07 1893.25i 8944.22
10.4 −8.43866 + 14.6162i −26.6641 46.1836i −78.4219 135.831i 196.505 + 340.357i 900.038 448.397 + 776.647i 486.805 −328.451 + 568.894i −6632.96
10.5 −7.57692 + 13.1236i 12.8050 + 22.1789i −50.8193 88.0217i 31.2763 + 54.1721i −388.090 −403.425 698.752i −399.475 765.563 1325.99i −947.911
10.6 −7.43537 + 12.8784i 41.2618 + 71.4676i −46.5696 80.6609i 62.4377 + 108.145i −1227.19 308.286 + 533.967i −518.407 −2311.58 + 4003.77i −1856.99
10.7 −5.02707 + 8.70714i −17.3307 30.0177i 13.4571 + 23.3084i 14.1829 + 24.5655i 348.491 −329.971 571.526i −1557.53 492.792 853.541i −285.194
10.8 −3.42160 + 5.92638i 28.0949 + 48.6617i 40.5853 + 70.2959i −242.165 419.442i −384.517 −642.579 1112.98i −1431.40 −485.141 + 840.289i 3314.36
10.9 −2.85295 + 4.94146i −22.7033 39.3233i 47.7213 + 82.6558i −48.1581 83.4123i 259.086 330.602 + 572.619i −1274.94 62.6168 108.455i 549.571
10.10 −1.26811 + 2.19642i 13.5559 + 23.4795i 60.7838 + 105.281i 253.342 + 438.801i −68.7612 −185.616 321.497i −632.957 725.976 1257.43i −1285.06
10.11 −1.18492 + 2.05234i 20.9206 + 36.2356i 61.1919 + 105.988i −19.7131 34.1440i −99.1570 666.527 + 1154.46i −593.369 218.155 377.855i 93.4336
10.12 0.746546 1.29305i −28.6079 49.5504i 62.8853 + 108.921i −130.514 226.056i −85.4285 −159.585 276.409i 378.903 −543.330 + 941.074i −389.737
10.13 2.32071 4.01959i −46.0982 79.8444i 53.2286 + 92.1946i 217.309 + 376.390i −427.923 19.1096 + 33.0988i 1088.22 −3156.59 + 5467.37i 2017.24
10.14 3.77156 6.53253i 9.73006 + 16.8530i 35.5507 + 61.5757i −147.558 255.578i 146.790 91.4021 + 158.313i 1501.84 904.152 1566.04i −2226.09
10.15 4.09845 7.09873i 41.0784 + 71.1498i 30.4054 + 52.6637i 21.4152 + 37.0921i 673.431 −128.692 222.901i 1547.66 −2281.37 + 3951.44i 351.076
10.16 4.93583 8.54911i −6.61210 11.4525i 15.2752 + 26.4574i 56.3382 + 97.5806i −130.545 −783.646 1357.32i 1565.16 1006.06 1742.55i 1112.30
10.17 6.29543 10.9040i −9.44168 16.3535i −15.2650 26.4397i 125.405 + 217.208i −237.758 817.371 + 1415.73i 1227.23 915.209 1585.19i 3157.91
10.18 7.43768 12.8824i −34.7930 60.2632i −46.6382 80.7798i −238.285 412.723i −1035.12 249.349 + 431.886i 516.525 −1327.60 + 2299.47i −7089.17
10.19 9.33278 16.1648i 28.0828 + 48.6408i −110.202 190.875i 166.579 + 288.523i 1048.36 1.55453 + 2.69252i −1724.76 −483.783 + 837.937i 6218.56
10.20 9.53293 16.5115i 19.4337 + 33.6601i −117.754 203.955i −179.660 311.180i 741.040 35.6214 + 61.6981i −2049.72 338.163 585.716i −6850.74
See all 42 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 26.21
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
37.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 37.8.c.a 42
37.c even 3 1 inner 37.8.c.a 42
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
37.8.c.a 42 1.a even 1 1 trivial
37.8.c.a 42 37.c even 3 1 inner

Hecke kernels

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