# 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 $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [37,8,Mod(10,37)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(37, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([4]))

N = Newforms(chi, 8, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("37.10");

S:= CuspForms(chi, 8);

N := Newforms(S);

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

## Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 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.

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

## 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.

comment: embeddings in the coefficient field

gp: mfembed(f)

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: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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 $$S_{8}^{\mathrm{new}}(37, [\chi])$$.