# Properties

 Label 37.5.i.a Level $37$ Weight $5$ Character orbit 37.i Analytic conductor $3.825$ Analytic rank $0$ Dimension $144$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [37,5,Mod(2,37)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(37, base_ring=CyclotomicField(36))

chi = DirichletCharacter(H, H._module([1]))

N = Newforms(chi, 5, names="a")

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

chi := DirichletCharacter("37.2");

S:= CuspForms(chi, 5);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$37$$ Weight: $$k$$ $$=$$ $$5$$ Character orbit: $$[\chi]$$ $$=$$ 37.i (of order $$36$$, degree $$12$$, 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: $$3.82468863410$$ Analytic rank: $$0$$ Dimension: $$144$$ Relative dimension: $$12$$ over $$\Q(\zeta_{36})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{36}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$144 q - 12 q^{2} - 12 q^{3} + 6 q^{4} - 120 q^{5} - 12 q^{6} - 12 q^{7} - 282 q^{8} + 198 q^{9}+O(q^{10})$$ 144 * q - 12 * q^2 - 12 * q^3 + 6 * q^4 - 120 * q^5 - 12 * q^6 - 12 * q^7 - 282 * q^8 + 198 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$144 q - 12 q^{2} - 12 q^{3} + 6 q^{4} - 120 q^{5} - 12 q^{6} - 12 q^{7} - 282 q^{8} + 198 q^{9} - 6 q^{10} - 18 q^{11} + 948 q^{12} - 12 q^{13} - 588 q^{14} - 810 q^{15} - 942 q^{16} + 708 q^{17} + 2418 q^{18} + 132 q^{19} - 1932 q^{20} - 750 q^{21} - 108 q^{22} + 1986 q^{23} - 4632 q^{24} + 1752 q^{25} + 1434 q^{26} + 5814 q^{27} + 9396 q^{28} - 768 q^{29} - 3204 q^{30} - 11364 q^{31} - 13818 q^{32} - 4602 q^{33} - 4044 q^{34} - 7140 q^{35} + 3096 q^{37} + 9732 q^{38} - 174 q^{39} + 24564 q^{40} + 7152 q^{41} + 11892 q^{42} + 14556 q^{43} - 522 q^{44} + 24912 q^{45} - 23952 q^{46} - 1788 q^{47} - 34578 q^{48} - 10050 q^{49} - 11238 q^{50} - 20652 q^{51} + 29394 q^{52} - 1902 q^{53} - 5916 q^{54} + 1692 q^{55} + 13236 q^{56} - 5484 q^{57} + 28128 q^{58} + 26538 q^{59} + 85248 q^{60} - 15996 q^{61} + 17016 q^{62} - 9366 q^{63} - 53316 q^{64} - 27984 q^{65} - 128484 q^{66} + 8346 q^{67} - 76026 q^{68} - 27294 q^{69} - 56298 q^{70} - 31692 q^{71} - 16314 q^{72} + 13614 q^{74} + 49320 q^{75} + 17598 q^{76} + 26190 q^{77} + 119382 q^{78} + 72852 q^{79} + 115608 q^{80} + 34026 q^{81} + 109680 q^{82} - 12792 q^{83} + 84462 q^{84} + 25254 q^{85} + 34890 q^{86} - 86928 q^{87} - 82800 q^{88} + 4992 q^{89} - 175584 q^{90} - 118062 q^{91} + 112794 q^{92} + 119424 q^{93} + 4692 q^{94} + 23802 q^{95} + 109008 q^{96} + 18186 q^{97} - 155478 q^{98} - 4032 q^{99}+O(q^{100})$$ 144 * q - 12 * q^2 - 12 * q^3 + 6 * q^4 - 120 * q^5 - 12 * q^6 - 12 * q^7 - 282 * q^8 + 198 * q^9 - 6 * q^10 - 18 * q^11 + 948 * q^12 - 12 * q^13 - 588 * q^14 - 810 * q^15 - 942 * q^16 + 708 * q^17 + 2418 * q^18 + 132 * q^19 - 1932 * q^20 - 750 * q^21 - 108 * q^22 + 1986 * q^23 - 4632 * q^24 + 1752 * q^25 + 1434 * q^26 + 5814 * q^27 + 9396 * q^28 - 768 * q^29 - 3204 * q^30 - 11364 * q^31 - 13818 * q^32 - 4602 * q^33 - 4044 * q^34 - 7140 * q^35 + 3096 * q^37 + 9732 * q^38 - 174 * q^39 + 24564 * q^40 + 7152 * q^41 + 11892 * q^42 + 14556 * q^43 - 522 * q^44 + 24912 * q^45 - 23952 * q^46 - 1788 * q^47 - 34578 * q^48 - 10050 * q^49 - 11238 * q^50 - 20652 * q^51 + 29394 * q^52 - 1902 * q^53 - 5916 * q^54 + 1692 * q^55 + 13236 * q^56 - 5484 * q^57 + 28128 * q^58 + 26538 * q^59 + 85248 * q^60 - 15996 * q^61 + 17016 * q^62 - 9366 * q^63 - 53316 * q^64 - 27984 * q^65 - 128484 * q^66 + 8346 * q^67 - 76026 * q^68 - 27294 * q^69 - 56298 * q^70 - 31692 * q^71 - 16314 * q^72 + 13614 * q^74 + 49320 * q^75 + 17598 * q^76 + 26190 * q^77 + 119382 * q^78 + 72852 * q^79 + 115608 * q^80 + 34026 * q^81 + 109680 * q^82 - 12792 * q^83 + 84462 * q^84 + 25254 * q^85 + 34890 * q^86 - 86928 * q^87 - 82800 * q^88 + 4992 * q^89 - 175584 * q^90 - 118062 * q^91 + 112794 * q^92 + 119424 * q^93 + 4692 * q^94 + 23802 * q^95 + 109008 * q^96 + 18186 * q^97 - 155478 * q^98 - 4032 * 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
2.1 −7.43299 0.650302i −8.90517 + 10.6128i 39.0695 + 6.88901i −18.8246 + 40.3695i 73.0935 73.0935i −26.2057 + 9.53809i −170.609 45.7146i −19.2633 109.247i 166.176 287.824i
2.2 −7.07168 0.618692i 10.4867 12.4976i 33.8689 + 5.97201i −4.92870 + 10.5696i −81.8911 + 81.8911i 75.0961 27.3328i −126.106 33.7901i −32.1531 182.349i 41.3935 71.6956i
2.3 −5.82370 0.509508i 0.698255 0.832148i 17.8989 + 3.15607i 10.8573 23.2835i −4.49041 + 4.49041i −65.7606 + 23.9349i −12.2821 3.29097i 13.8606 + 78.6073i −75.0928 + 130.064i
2.4 −4.38998 0.384074i 0.247322 0.294747i 3.36752 + 0.593785i −2.39998 + 5.14676i −1.19894 + 1.19894i 28.5706 10.3989i 53.5502 + 14.3487i 14.0398 + 79.6236i 12.5126 21.6724i
2.5 −2.93366 0.256662i −9.35942 + 11.1541i −7.21645 1.27246i 9.43269 20.2285i 30.3202 30.3202i 39.1303 14.2423i 66.3563 + 17.7801i −22.7502 129.023i −32.8641 + 56.9224i
2.6 −1.13246 0.0990776i 9.68265 11.5393i −14.4843 2.55397i 6.08797 13.0557i −12.1085 + 12.1085i −66.6341 + 24.2528i 33.7187 + 9.03489i −25.3370 143.693i −8.18792 + 14.1819i
2.7 −0.818396 0.0716004i 2.52396 3.00793i −15.0923 2.66118i −18.6068 + 39.9023i −2.28096 + 2.28096i 8.45114 3.07597i 24.8574 + 6.66052i 11.3882 + 64.5856i 18.0847 31.3237i
2.8 2.03026 + 0.177625i −6.00370 + 7.15494i −11.6665 2.05712i −1.02327 + 2.19440i −13.4600 + 13.4600i −47.7624 + 17.3841i −54.8178 14.6884i −1.08314 6.14280i −2.46728 + 4.27345i
2.9 2.08231 + 0.182179i 2.85756 3.40550i −11.4541 2.01967i 15.1773 32.5477i 6.57073 6.57073i 66.1524 24.0775i −55.7877 14.9483i 10.6337 + 60.3066i 37.5333 65.0095i
2.10 5.32165 + 0.465584i 8.08788 9.63876i 12.3463 + 2.17699i −8.12880 + 17.4323i 47.5286 47.5286i 2.51614 0.915801i −17.8701 4.78828i −13.4264 76.1449i −51.3748 + 88.9838i
2.11 5.93109 + 0.518903i −7.67181 + 9.14291i 19.1516 + 3.37695i −13.8164 + 29.6295i −50.2465 + 50.2465i 80.3866 29.2583i 19.8237 + 5.31175i −10.6706 60.5159i −97.3214 + 168.566i
2.12 6.83559 + 0.598037i −0.770402 + 0.918130i 30.6107 + 5.39749i 9.09446 19.5031i −5.81523 + 5.81523i −42.6753 + 15.5325i 99.9681 + 26.7864i 13.8161 + 78.3548i 73.8296 127.877i
5.1 −3.29658 + 7.06954i −3.57062 + 9.81021i −28.8263 34.3539i 0.215509 0.307778i −57.5828 57.5828i 11.8673 + 67.3028i 217.341 58.2363i −21.4413 17.9914i 1.46541 + 2.53816i
5.2 −2.62972 + 5.63946i 2.63021 7.22644i −14.6034 17.4037i 17.6934 25.2688i 33.8365 + 33.8365i −11.0584 62.7152i 40.3836 10.8208i 16.7462 + 14.0517i 95.9735 + 166.231i
5.3 −2.37818 + 5.10001i 4.82693 13.2619i −10.0698 12.0008i −25.5003 + 36.4182i 56.1566 + 56.1566i 13.1184 + 74.3980i −1.81618 + 0.486643i −90.5289 75.9628i −125.089 216.661i
5.4 −1.86456 + 3.99856i −1.37794 + 3.78585i −2.22731 2.65440i −13.4352 + 19.1875i −12.5687 12.5687i −8.06935 45.7636i −53.4187 + 14.3135i 49.6157 + 41.6325i −51.6715 89.4976i
5.5 −1.15872 + 2.48488i −5.54437 + 15.2330i 5.45259 + 6.49814i 6.68567 9.54813i −31.4280 31.4280i −2.18344 12.3829i −64.8386 + 17.3735i −139.256 116.849i 15.9792 + 27.6767i
5.6 −1.02938 + 2.20752i 0.471556 1.29559i 6.47109 + 7.71195i 15.7198 22.4501i 2.37463 + 2.37463i 13.1089 + 74.3445i −61.3292 + 16.4331i 60.5934 + 50.8439i 33.3774 + 57.8114i
5.7 0.240642 0.516058i 4.70675 12.9317i 10.0762 + 12.0083i 1.54727 2.20973i −5.54086 5.54086i −8.44071 47.8697i 17.4218 4.66816i −83.0255 69.6667i −0.768009 1.33023i
5.8 0.479059 1.02734i −0.611056 + 1.67886i 9.45866 + 11.2724i −16.2436 + 23.1983i 1.43204 + 1.43204i 0.371208 + 2.10523i 33.6307 9.01132i 59.6044 + 50.0140i 16.0510 + 27.8012i
See next 80 embeddings (of 144 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 35.12 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.i odd 36 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 37.5.i.a 144
37.i odd 36 1 inner 37.5.i.a 144

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
37.5.i.a 144 1.a even 1 1 trivial
37.5.i.a 144 37.i odd 36 1 inner

## Hecke kernels

This newform subspace is the entire newspace $$S_{5}^{\mathrm{new}}(37, [\chi])$$.