# Properties

 Label 99.8.e.b Level $99$ Weight $8$ Character orbit 99.e Analytic conductor $30.926$ Analytic rank $0$ Dimension $74$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [99,8,Mod(34,99)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(99, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([4, 0]))

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

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

chi := DirichletCharacter("99.34");

S:= CuspForms(chi, 8);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$99 = 3^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$8$$ Character orbit: $$[\chi]$$ $$=$$ 99.e (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: $$30.9261175229$$ Analytic rank: $$0$$ Dimension: $$74$$ Relative dimension: $$37$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$74 q + 35 q^{3} - 2560 q^{4} + 468 q^{5} + 2405 q^{6} - 1289 q^{7} - 2598 q^{8} + 2551 q^{9}+O(q^{10})$$ 74 * q + 35 * q^3 - 2560 * q^4 + 468 * q^5 + 2405 * q^6 - 1289 * q^7 - 2598 * q^8 + 2551 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$74 q + 35 q^{3} - 2560 q^{4} + 468 q^{5} + 2405 q^{6} - 1289 q^{7} - 2598 q^{8} + 2551 q^{9} + 12000 q^{10} + 49247 q^{11} + 16333 q^{12} - 20123 q^{13} + 16527 q^{14} - 46018 q^{15} - 188416 q^{16} - 27222 q^{17} + 100057 q^{18} + 114838 q^{19} + 147411 q^{20} + 35555 q^{21} + 141468 q^{23} - 750321 q^{24} - 771523 q^{25} - 337260 q^{26} - 24619 q^{27} + 903382 q^{28} + 96018 q^{29} + 1011452 q^{30} - 455402 q^{31} + 551667 q^{32} + 93170 q^{33} + 33054 q^{34} - 1270080 q^{35} - 1905247 q^{36} + 834262 q^{37} + 1588893 q^{38} + 1563728 q^{39} + 765 q^{40} + 261042 q^{41} - 159013 q^{42} + 65491 q^{43} - 6814720 q^{44} - 3598190 q^{45} + 927504 q^{46} - 91584 q^{47} + 3937657 q^{48} - 5321982 q^{49} - 200199 q^{50} + 4327679 q^{51} - 1325648 q^{52} - 2527182 q^{53} - 2071810 q^{54} + 1245816 q^{55} + 1817094 q^{56} + 6385113 q^{57} + 445839 q^{58} + 382386 q^{59} + 1314187 q^{60} - 3800531 q^{61} - 2418690 q^{62} - 14518613 q^{63} + 22914182 q^{64} + 4439373 q^{65} + 1426832 q^{66} - 5389556 q^{67} - 2056977 q^{68} + 3611203 q^{69} - 7917372 q^{70} + 8739834 q^{71} - 7210734 q^{72} + 23797384 q^{73} - 11488821 q^{74} + 3626101 q^{75} - 11416346 q^{76} + 1715659 q^{77} + 10651715 q^{78} - 11242367 q^{79} - 35830230 q^{80} - 15548357 q^{81} + 29393394 q^{82} - 11510640 q^{83} + 30982504 q^{84} - 23163939 q^{85} - 12696132 q^{86} - 7248951 q^{87} - 1728969 q^{88} - 12598836 q^{89} - 46478909 q^{90} + 42923150 q^{91} - 3050307 q^{92} + 19348778 q^{93} - 43129860 q^{94} - 5856174 q^{95} + 17580091 q^{96} - 46858874 q^{97} + 53876532 q^{98} - 3395381 q^{99}+O(q^{100})$$ 74 * q + 35 * q^3 - 2560 * q^4 + 468 * q^5 + 2405 * q^6 - 1289 * q^7 - 2598 * q^8 + 2551 * q^9 + 12000 * q^10 + 49247 * q^11 + 16333 * q^12 - 20123 * q^13 + 16527 * q^14 - 46018 * q^15 - 188416 * q^16 - 27222 * q^17 + 100057 * q^18 + 114838 * q^19 + 147411 * q^20 + 35555 * q^21 + 141468 * q^23 - 750321 * q^24 - 771523 * q^25 - 337260 * q^26 - 24619 * q^27 + 903382 * q^28 + 96018 * q^29 + 1011452 * q^30 - 455402 * q^31 + 551667 * q^32 + 93170 * q^33 + 33054 * q^34 - 1270080 * q^35 - 1905247 * q^36 + 834262 * q^37 + 1588893 * q^38 + 1563728 * q^39 + 765 * q^40 + 261042 * q^41 - 159013 * q^42 + 65491 * q^43 - 6814720 * q^44 - 3598190 * q^45 + 927504 * q^46 - 91584 * q^47 + 3937657 * q^48 - 5321982 * q^49 - 200199 * q^50 + 4327679 * q^51 - 1325648 * q^52 - 2527182 * q^53 - 2071810 * q^54 + 1245816 * q^55 + 1817094 * q^56 + 6385113 * q^57 + 445839 * q^58 + 382386 * q^59 + 1314187 * q^60 - 3800531 * q^61 - 2418690 * q^62 - 14518613 * q^63 + 22914182 * q^64 + 4439373 * q^65 + 1426832 * q^66 - 5389556 * q^67 - 2056977 * q^68 + 3611203 * q^69 - 7917372 * q^70 + 8739834 * q^71 - 7210734 * q^72 + 23797384 * q^73 - 11488821 * q^74 + 3626101 * q^75 - 11416346 * q^76 + 1715659 * q^77 + 10651715 * q^78 - 11242367 * q^79 - 35830230 * q^80 - 15548357 * q^81 + 29393394 * q^82 - 11510640 * q^83 + 30982504 * q^84 - 23163939 * q^85 - 12696132 * q^86 - 7248951 * q^87 - 1728969 * q^88 - 12598836 * q^89 - 46478909 * q^90 + 42923150 * q^91 - 3050307 * q^92 + 19348778 * q^93 - 43129860 * q^94 - 5856174 * q^95 + 17580091 * q^96 - 46858874 * q^97 + 53876532 * q^98 - 3395381 * 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   $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
34.1 −10.7701 + 18.6544i −42.1814 20.1923i −167.991 290.970i 250.600 + 434.052i 830.974 569.396i −557.197 + 965.093i 4480.00 1371.55 + 1703.48i −10796.0
34.2 −10.5913 + 18.3447i −15.0079 44.2918i −160.353 277.740i −104.661 181.278i 971.475 + 193.793i 190.345 329.688i 4082.03 −1736.52 + 1329.46i 4434.00
34.3 −10.1903 + 17.6501i −27.0124 + 38.1750i −143.684 248.867i −224.899 389.537i −398.528 865.785i −85.8356 + 148.672i 3248.00 −727.659 2062.40i 9167.14
34.4 −9.73151 + 16.8555i 44.2084 15.2518i −125.405 217.207i −125.595 217.537i −173.138 + 893.576i −756.405 + 1310.13i 2390.23 1721.77 1348.51i 4888.92
34.5 −9.60619 + 16.6384i −28.4471 + 37.1183i −120.558 208.812i 199.370 + 345.318i −344.321 829.879i 787.256 1363.57i 2173.21 −568.530 2111.81i −7660.73
34.6 −8.23897 + 14.2703i 46.3941 + 5.88131i −71.7612 124.294i 48.3424 + 83.7314i −466.167 + 613.602i 453.585 785.633i 255.776 2117.82 + 545.716i −1593.16
34.7 −7.99874 + 13.8542i 21.9286 41.3054i −63.9596 110.781i 183.811 + 318.370i 396.853 + 634.195i −164.975 + 285.745i −1.29126 −1225.27 1811.54i −5881.03
34.8 −7.55397 + 13.0839i 25.2048 + 39.3919i −50.1248 86.8187i 55.0107 + 95.2813i −705.793 + 32.2111i 164.167 284.346i −419.252 −916.437 + 1985.73i −1662.20
34.9 −6.91175 + 11.9715i 20.7774 41.8963i −31.5445 54.6368i −194.870 337.526i 357.954 + 538.313i 839.163 1453.47i −897.296 −1323.60 1740.99i 5387.58
34.10 −6.80706 + 11.7902i −45.5659 + 10.5239i −28.6722 49.6618i 23.2959 + 40.3497i 186.091 608.866i −307.304 + 532.266i −961.913 1965.50 959.059i −634.307
34.11 −5.69140 + 9.85780i −35.9261 29.9386i −0.784104 1.35811i 43.1131 + 74.6742i 499.598 183.760i 193.141 334.530i −1439.15 394.365 + 2151.15i −981.497
34.12 −5.44144 + 9.42485i 16.5132 + 43.7529i 4.78148 + 8.28177i −137.296 237.804i −502.220 82.4445i −897.344 + 1554.24i −1497.08 −1641.63 + 1445.00i 2988.36
34.13 −4.07501 + 7.05812i −38.2445 + 26.9138i 30.7887 + 53.3275i −167.097 289.421i −34.1143 379.608i 440.578 763.104i −1545.06 738.290 2058.62i 2723.69
34.14 −3.25448 + 5.63693i 6.41613 46.3231i 42.8167 + 74.1606i 80.7900 + 139.932i 240.239 + 186.925i −775.330 + 1342.91i −1390.53 −2104.67 594.431i −1051.72
34.15 −3.21265 + 5.56447i −3.59885 + 46.6267i 43.3578 + 75.0979i 234.240 + 405.716i −247.891 169.821i −257.383 + 445.801i −1379.61 −2161.10 335.605i −3010.13
34.16 −2.31234 + 4.00510i 46.5868 4.08325i 53.3061 + 92.3289i 28.2378 + 48.9093i −91.3708 + 196.026i 106.272 184.069i −1085.01 2153.65 380.450i −261.182
34.17 −1.56778 + 2.71547i 39.0488 + 25.7331i 59.0841 + 102.337i −224.531 388.899i −131.097 + 65.6922i 165.561 286.759i −771.875 862.617 + 2009.69i 1408.06
34.18 −0.363127 + 0.628954i −11.5794 45.3091i 63.7363 + 110.394i 48.0698 + 83.2594i 32.7022 + 9.17009i 544.379 942.892i −185.538 −1918.84 + 1049.30i −69.8218
34.19 0.738283 1.27874i 1.80735 + 46.7304i 62.9099 + 108.963i −79.1166 137.034i 61.0906 + 32.1892i 538.464 932.647i 374.782 −2180.47 + 168.916i −233.642
34.20 0.834765 1.44586i 33.9730 32.1378i 62.6063 + 108.437i −240.191 416.023i −18.1071 75.9475i −368.038 + 637.460i 422.746 121.326 2183.63i −802.011
See all 74 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 34.37 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

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

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 99.8.e.b 74
9.c even 3 1 inner 99.8.e.b 74

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
99.8.e.b 74 1.a even 1 1 trivial
99.8.e.b 74 9.c even 3 1 inner