# Properties

 Label 50.4.a.b.1.1 Level $50$ Weight $4$ Character 50.1 Self dual yes Analytic conductor $2.950$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [50,4,Mod(1,50)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(50, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("50.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$50 = 2 \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 50.a (trivial)

## 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: yes Analytic conductor: $$2.95009550029$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 10) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 50.1

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-2.00000 q^{2} -2.00000 q^{3} +4.00000 q^{4} +4.00000 q^{6} -26.0000 q^{7} -8.00000 q^{8} -23.0000 q^{9} +O(q^{10})$$ $$q-2.00000 q^{2} -2.00000 q^{3} +4.00000 q^{4} +4.00000 q^{6} -26.0000 q^{7} -8.00000 q^{8} -23.0000 q^{9} -28.0000 q^{11} -8.00000 q^{12} -12.0000 q^{13} +52.0000 q^{14} +16.0000 q^{16} +64.0000 q^{17} +46.0000 q^{18} -60.0000 q^{19} +52.0000 q^{21} +56.0000 q^{22} +58.0000 q^{23} +16.0000 q^{24} +24.0000 q^{26} +100.000 q^{27} -104.000 q^{28} +90.0000 q^{29} -128.000 q^{31} -32.0000 q^{32} +56.0000 q^{33} -128.000 q^{34} -92.0000 q^{36} -236.000 q^{37} +120.000 q^{38} +24.0000 q^{39} +242.000 q^{41} -104.000 q^{42} -362.000 q^{43} -112.000 q^{44} -116.000 q^{46} -226.000 q^{47} -32.0000 q^{48} +333.000 q^{49} -128.000 q^{51} -48.0000 q^{52} +108.000 q^{53} -200.000 q^{54} +208.000 q^{56} +120.000 q^{57} -180.000 q^{58} -20.0000 q^{59} +542.000 q^{61} +256.000 q^{62} +598.000 q^{63} +64.0000 q^{64} -112.000 q^{66} +434.000 q^{67} +256.000 q^{68} -116.000 q^{69} -1128.00 q^{71} +184.000 q^{72} -632.000 q^{73} +472.000 q^{74} -240.000 q^{76} +728.000 q^{77} -48.0000 q^{78} -720.000 q^{79} +421.000 q^{81} -484.000 q^{82} +478.000 q^{83} +208.000 q^{84} +724.000 q^{86} -180.000 q^{87} +224.000 q^{88} -490.000 q^{89} +312.000 q^{91} +232.000 q^{92} +256.000 q^{93} +452.000 q^{94} +64.0000 q^{96} -1456.00 q^{97} -666.000 q^{98} +644.000 q^{99} +O(q^{100})$$

## Coefficient data

For each $$n$$ we display the coefficients of the $$q$$-expansion $$a_n$$, the Satake parameters $$\alpha_p$$, and the Satake angles $$\theta_p = \textrm{Arg}(\alpha_p)$$.

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −2.00000 −0.707107
$$3$$ −2.00000 −0.384900 −0.192450 0.981307i $$-0.561643\pi$$
−0.192450 + 0.981307i $$0.561643\pi$$
$$4$$ 4.00000 0.500000
$$5$$ 0 0
$$6$$ 4.00000 0.272166
$$7$$ −26.0000 −1.40387 −0.701934 0.712242i $$-0.747680\pi$$
−0.701934 + 0.712242i $$0.747680\pi$$
$$8$$ −8.00000 −0.353553
$$9$$ −23.0000 −0.851852
$$10$$ 0 0
$$11$$ −28.0000 −0.767483 −0.383742 0.923440i $$-0.625365\pi$$
−0.383742 + 0.923440i $$0.625365\pi$$
$$12$$ −8.00000 −0.192450
$$13$$ −12.0000 −0.256015 −0.128008 0.991773i $$-0.540858\pi$$
−0.128008 + 0.991773i $$0.540858\pi$$
$$14$$ 52.0000 0.992685
$$15$$ 0 0
$$16$$ 16.0000 0.250000
$$17$$ 64.0000 0.913075 0.456538 0.889704i $$-0.349089\pi$$
0.456538 + 0.889704i $$0.349089\pi$$
$$18$$ 46.0000 0.602350
$$19$$ −60.0000 −0.724471 −0.362235 0.932087i $$-0.617986\pi$$
−0.362235 + 0.932087i $$0.617986\pi$$
$$20$$ 0 0
$$21$$ 52.0000 0.540349
$$22$$ 56.0000 0.542693
$$23$$ 58.0000 0.525819 0.262909 0.964821i $$-0.415318\pi$$
0.262909 + 0.964821i $$0.415318\pi$$
$$24$$ 16.0000 0.136083
$$25$$ 0 0
$$26$$ 24.0000 0.181030
$$27$$ 100.000 0.712778
$$28$$ −104.000 −0.701934
$$29$$ 90.0000 0.576296 0.288148 0.957586i $$-0.406961\pi$$
0.288148 + 0.957586i $$0.406961\pi$$
$$30$$ 0 0
$$31$$ −128.000 −0.741596 −0.370798 0.928714i $$-0.620916\pi$$
−0.370798 + 0.928714i $$0.620916\pi$$
$$32$$ −32.0000 −0.176777
$$33$$ 56.0000 0.295405
$$34$$ −128.000 −0.645642
$$35$$ 0 0
$$36$$ −92.0000 −0.425926
$$37$$ −236.000 −1.04860 −0.524299 0.851534i $$-0.675673\pi$$
−0.524299 + 0.851534i $$0.675673\pi$$
$$38$$ 120.000 0.512278
$$39$$ 24.0000 0.0985404
$$40$$ 0 0
$$41$$ 242.000 0.921806 0.460903 0.887450i $$-0.347526\pi$$
0.460903 + 0.887450i $$0.347526\pi$$
$$42$$ −104.000 −0.382084
$$43$$ −362.000 −1.28383 −0.641913 0.766778i $$-0.721859\pi$$
−0.641913 + 0.766778i $$0.721859\pi$$
$$44$$ −112.000 −0.383742
$$45$$ 0 0
$$46$$ −116.000 −0.371810
$$47$$ −226.000 −0.701393 −0.350697 0.936489i $$-0.614055\pi$$
−0.350697 + 0.936489i $$0.614055\pi$$
$$48$$ −32.0000 −0.0962250
$$49$$ 333.000 0.970845
$$50$$ 0 0
$$51$$ −128.000 −0.351443
$$52$$ −48.0000 −0.128008
$$53$$ 108.000 0.279905 0.139952 0.990158i $$-0.455305\pi$$
0.139952 + 0.990158i $$0.455305\pi$$
$$54$$ −200.000 −0.504010
$$55$$ 0 0
$$56$$ 208.000 0.496342
$$57$$ 120.000 0.278849
$$58$$ −180.000 −0.407503
$$59$$ −20.0000 −0.0441318 −0.0220659 0.999757i $$-0.507024\pi$$
−0.0220659 + 0.999757i $$0.507024\pi$$
$$60$$ 0 0
$$61$$ 542.000 1.13764 0.568820 0.822462i $$-0.307400\pi$$
0.568820 + 0.822462i $$0.307400\pi$$
$$62$$ 256.000 0.524388
$$63$$ 598.000 1.19589
$$64$$ 64.0000 0.125000
$$65$$ 0 0
$$66$$ −112.000 −0.208883
$$67$$ 434.000 0.791366 0.395683 0.918387i $$-0.370508\pi$$
0.395683 + 0.918387i $$0.370508\pi$$
$$68$$ 256.000 0.456538
$$69$$ −116.000 −0.202388
$$70$$ 0 0
$$71$$ −1128.00 −1.88548 −0.942739 0.333531i $$-0.891760\pi$$
−0.942739 + 0.333531i $$0.891760\pi$$
$$72$$ 184.000 0.301175
$$73$$ −632.000 −1.01329 −0.506644 0.862155i $$-0.669114\pi$$
−0.506644 + 0.862155i $$0.669114\pi$$
$$74$$ 472.000 0.741471
$$75$$ 0 0
$$76$$ −240.000 −0.362235
$$77$$ 728.000 1.07745
$$78$$ −48.0000 −0.0696786
$$79$$ −720.000 −1.02540 −0.512698 0.858569i $$-0.671354\pi$$
−0.512698 + 0.858569i $$0.671354\pi$$
$$80$$ 0 0
$$81$$ 421.000 0.577503
$$82$$ −484.000 −0.651815
$$83$$ 478.000 0.632136 0.316068 0.948736i $$-0.397637\pi$$
0.316068 + 0.948736i $$0.397637\pi$$
$$84$$ 208.000 0.270175
$$85$$ 0 0
$$86$$ 724.000 0.907801
$$87$$ −180.000 −0.221816
$$88$$ 224.000 0.271346
$$89$$ −490.000 −0.583594 −0.291797 0.956480i $$-0.594253\pi$$
−0.291797 + 0.956480i $$0.594253\pi$$
$$90$$ 0 0
$$91$$ 312.000 0.359412
$$92$$ 232.000 0.262909
$$93$$ 256.000 0.285440
$$94$$ 452.000 0.495960
$$95$$ 0 0
$$96$$ 64.0000 0.0680414
$$97$$ −1456.00 −1.52407 −0.762033 0.647538i $$-0.775799\pi$$
−0.762033 + 0.647538i $$0.775799\pi$$
$$98$$ −666.000 −0.686491
$$99$$ 644.000 0.653782
$$100$$ 0 0
$$101$$ −578.000 −0.569437 −0.284719 0.958611i $$-0.591900\pi$$
−0.284719 + 0.958611i $$0.591900\pi$$
$$102$$ 256.000 0.248508
$$103$$ −1462.00 −1.39859 −0.699297 0.714831i $$-0.746503\pi$$
−0.699297 + 0.714831i $$0.746503\pi$$
$$104$$ 96.0000 0.0905151
$$105$$ 0 0
$$106$$ −216.000 −0.197922
$$107$$ −966.000 −0.872773 −0.436387 0.899759i $$-0.643742\pi$$
−0.436387 + 0.899759i $$0.643742\pi$$
$$108$$ 400.000 0.356389
$$109$$ 370.000 0.325134 0.162567 0.986698i $$-0.448023\pi$$
0.162567 + 0.986698i $$0.448023\pi$$
$$110$$ 0 0
$$111$$ 472.000 0.403606
$$112$$ −416.000 −0.350967
$$113$$ 528.000 0.439558 0.219779 0.975550i $$-0.429466\pi$$
0.219779 + 0.975550i $$0.429466\pi$$
$$114$$ −240.000 −0.197176
$$115$$ 0 0
$$116$$ 360.000 0.288148
$$117$$ 276.000 0.218087
$$118$$ 40.0000 0.0312059
$$119$$ −1664.00 −1.28184
$$120$$ 0 0
$$121$$ −547.000 −0.410969
$$122$$ −1084.00 −0.804432
$$123$$ −484.000 −0.354803
$$124$$ −512.000 −0.370798
$$125$$ 0 0
$$126$$ −1196.00 −0.845620
$$127$$ 1534.00 1.07181 0.535907 0.844277i $$-0.319970\pi$$
0.535907 + 0.844277i $$0.319970\pi$$
$$128$$ −128.000 −0.0883883
$$129$$ 724.000 0.494145
$$130$$ 0 0
$$131$$ 12.0000 0.00800340 0.00400170 0.999992i $$-0.498726\pi$$
0.00400170 + 0.999992i $$0.498726\pi$$
$$132$$ 224.000 0.147702
$$133$$ 1560.00 1.01706
$$134$$ −868.000 −0.559580
$$135$$ 0 0
$$136$$ −512.000 −0.322821
$$137$$ 1224.00 0.763309 0.381655 0.924305i $$-0.375354\pi$$
0.381655 + 0.924305i $$0.375354\pi$$
$$138$$ 232.000 0.143110
$$139$$ 3100.00 1.89164 0.945822 0.324685i $$-0.105258\pi$$
0.945822 + 0.324685i $$0.105258\pi$$
$$140$$ 0 0
$$141$$ 452.000 0.269966
$$142$$ 2256.00 1.33323
$$143$$ 336.000 0.196488
$$144$$ −368.000 −0.212963
$$145$$ 0 0
$$146$$ 1264.00 0.716503
$$147$$ −666.000 −0.373679
$$148$$ −944.000 −0.524299
$$149$$ 250.000 0.137455 0.0687275 0.997635i $$-0.478106\pi$$
0.0687275 + 0.997635i $$0.478106\pi$$
$$150$$ 0 0
$$151$$ 2152.00 1.15978 0.579892 0.814694i $$-0.303095\pi$$
0.579892 + 0.814694i $$0.303095\pi$$
$$152$$ 480.000 0.256139
$$153$$ −1472.00 −0.777805
$$154$$ −1456.00 −0.761869
$$155$$ 0 0
$$156$$ 96.0000 0.0492702
$$157$$ 524.000 0.266368 0.133184 0.991091i $$-0.457480\pi$$
0.133184 + 0.991091i $$0.457480\pi$$
$$158$$ 1440.00 0.725065
$$159$$ −216.000 −0.107735
$$160$$ 0 0
$$161$$ −1508.00 −0.738180
$$162$$ −842.000 −0.408357
$$163$$ 3518.00 1.69050 0.845249 0.534373i $$-0.179452\pi$$
0.845249 + 0.534373i $$0.179452\pi$$
$$164$$ 968.000 0.460903
$$165$$ 0 0
$$166$$ −956.000 −0.446988
$$167$$ 534.000 0.247438 0.123719 0.992317i $$-0.460518\pi$$
0.123719 + 0.992317i $$0.460518\pi$$
$$168$$ −416.000 −0.191042
$$169$$ −2053.00 −0.934456
$$170$$ 0 0
$$171$$ 1380.00 0.617142
$$172$$ −1448.00 −0.641913
$$173$$ −4252.00 −1.86863 −0.934317 0.356444i $$-0.883989\pi$$
−0.934317 + 0.356444i $$0.883989\pi$$
$$174$$ 360.000 0.156848
$$175$$ 0 0
$$176$$ −448.000 −0.191871
$$177$$ 40.0000 0.0169864
$$178$$ 980.000 0.412664
$$179$$ 2500.00 1.04390 0.521952 0.852975i $$-0.325204\pi$$
0.521952 + 0.852975i $$0.325204\pi$$
$$180$$ 0 0
$$181$$ −2578.00 −1.05868 −0.529340 0.848410i $$-0.677561\pi$$
−0.529340 + 0.848410i $$0.677561\pi$$
$$182$$ −624.000 −0.254143
$$183$$ −1084.00 −0.437878
$$184$$ −464.000 −0.185905
$$185$$ 0 0
$$186$$ −512.000 −0.201837
$$187$$ −1792.00 −0.700770
$$188$$ −904.000 −0.350697
$$189$$ −2600.00 −1.00065
$$190$$ 0 0
$$191$$ −768.000 −0.290945 −0.145473 0.989362i $$-0.546470\pi$$
−0.145473 + 0.989362i $$0.546470\pi$$
$$192$$ −128.000 −0.0481125
$$193$$ 2608.00 0.972684 0.486342 0.873769i $$-0.338331\pi$$
0.486342 + 0.873769i $$0.338331\pi$$
$$194$$ 2912.00 1.07768
$$195$$ 0 0
$$196$$ 1332.00 0.485423
$$197$$ −5116.00 −1.85025 −0.925127 0.379659i $$-0.876041\pi$$
−0.925127 + 0.379659i $$0.876041\pi$$
$$198$$ −1288.00 −0.462294
$$199$$ −3480.00 −1.23965 −0.619826 0.784739i $$-0.712797\pi$$
−0.619826 + 0.784739i $$0.712797\pi$$
$$200$$ 0 0
$$201$$ −868.000 −0.304597
$$202$$ 1156.00 0.402653
$$203$$ −2340.00 −0.809043
$$204$$ −512.000 −0.175721
$$205$$ 0 0
$$206$$ 2924.00 0.988955
$$207$$ −1334.00 −0.447920
$$208$$ −192.000 −0.0640039
$$209$$ 1680.00 0.556019
$$210$$ 0 0
$$211$$ 3132.00 1.02188 0.510938 0.859618i $$-0.329298\pi$$
0.510938 + 0.859618i $$0.329298\pi$$
$$212$$ 432.000 0.139952
$$213$$ 2256.00 0.725721
$$214$$ 1932.00 0.617144
$$215$$ 0 0
$$216$$ −800.000 −0.252005
$$217$$ 3328.00 1.04110
$$218$$ −740.000 −0.229904
$$219$$ 1264.00 0.390015
$$220$$ 0 0
$$221$$ −768.000 −0.233761
$$222$$ −944.000 −0.285392
$$223$$ −62.0000 −0.0186181 −0.00930903 0.999957i $$-0.502963\pi$$
−0.00930903 + 0.999957i $$0.502963\pi$$
$$224$$ 832.000 0.248171
$$225$$ 0 0
$$226$$ −1056.00 −0.310814
$$227$$ 5314.00 1.55376 0.776878 0.629651i $$-0.216802\pi$$
0.776878 + 0.629651i $$0.216802\pi$$
$$228$$ 480.000 0.139424
$$229$$ −190.000 −0.0548277 −0.0274139 0.999624i $$-0.508727\pi$$
−0.0274139 + 0.999624i $$0.508727\pi$$
$$230$$ 0 0
$$231$$ −1456.00 −0.414709
$$232$$ −720.000 −0.203751
$$233$$ 2408.00 0.677053 0.338526 0.940957i $$-0.390072\pi$$
0.338526 + 0.940957i $$0.390072\pi$$
$$234$$ −552.000 −0.154211
$$235$$ 0 0
$$236$$ −80.0000 −0.0220659
$$237$$ 1440.00 0.394675
$$238$$ 3328.00 0.906396
$$239$$ −5680.00 −1.53727 −0.768637 0.639685i $$-0.779065\pi$$
−0.768637 + 0.639685i $$0.779065\pi$$
$$240$$ 0 0
$$241$$ −278.000 −0.0743052 −0.0371526 0.999310i $$-0.511829\pi$$
−0.0371526 + 0.999310i $$0.511829\pi$$
$$242$$ 1094.00 0.290599
$$243$$ −3542.00 −0.935059
$$244$$ 2168.00 0.568820
$$245$$ 0 0
$$246$$ 968.000 0.250884
$$247$$ 720.000 0.185476
$$248$$ 1024.00 0.262194
$$249$$ −956.000 −0.243309
$$250$$ 0 0
$$251$$ 3252.00 0.817787 0.408893 0.912582i $$-0.365915\pi$$
0.408893 + 0.912582i $$0.365915\pi$$
$$252$$ 2392.00 0.597944
$$253$$ −1624.00 −0.403557
$$254$$ −3068.00 −0.757888
$$255$$ 0 0
$$256$$ 256.000 0.0625000
$$257$$ −1536.00 −0.372813 −0.186407 0.982473i $$-0.559684\pi$$
−0.186407 + 0.982473i $$0.559684\pi$$
$$258$$ −1448.00 −0.349413
$$259$$ 6136.00 1.47209
$$260$$ 0 0
$$261$$ −2070.00 −0.490919
$$262$$ −24.0000 −0.00565926
$$263$$ 4858.00 1.13900 0.569500 0.821991i $$-0.307137\pi$$
0.569500 + 0.821991i $$0.307137\pi$$
$$264$$ −448.000 −0.104441
$$265$$ 0 0
$$266$$ −3120.00 −0.719171
$$267$$ 980.000 0.224626
$$268$$ 1736.00 0.395683
$$269$$ 2610.00 0.591578 0.295789 0.955253i $$-0.404417\pi$$
0.295789 + 0.955253i $$0.404417\pi$$
$$270$$ 0 0
$$271$$ −5168.00 −1.15843 −0.579213 0.815176i $$-0.696640\pi$$
−0.579213 + 0.815176i $$0.696640\pi$$
$$272$$ 1024.00 0.228269
$$273$$ −624.000 −0.138338
$$274$$ −2448.00 −0.539741
$$275$$ 0 0
$$276$$ −464.000 −0.101194
$$277$$ 1924.00 0.417336 0.208668 0.977987i $$-0.433087\pi$$
0.208668 + 0.977987i $$0.433087\pi$$
$$278$$ −6200.00 −1.33759
$$279$$ 2944.00 0.631730
$$280$$ 0 0
$$281$$ 3042.00 0.645803 0.322901 0.946433i $$-0.395342\pi$$
0.322901 + 0.946433i $$0.395342\pi$$
$$282$$ −904.000 −0.190895
$$283$$ 1718.00 0.360864 0.180432 0.983587i $$-0.442250\pi$$
0.180432 + 0.983587i $$0.442250\pi$$
$$284$$ −4512.00 −0.942739
$$285$$ 0 0
$$286$$ −672.000 −0.138938
$$287$$ −6292.00 −1.29409
$$288$$ 736.000 0.150588
$$289$$ −817.000 −0.166294
$$290$$ 0 0
$$291$$ 2912.00 0.586613
$$292$$ −2528.00 −0.506644
$$293$$ −2292.00 −0.456997 −0.228498 0.973544i $$-0.573382\pi$$
−0.228498 + 0.973544i $$0.573382\pi$$
$$294$$ 1332.00 0.264231
$$295$$ 0 0
$$296$$ 1888.00 0.370736
$$297$$ −2800.00 −0.547045
$$298$$ −500.000 −0.0971954
$$299$$ −696.000 −0.134618
$$300$$ 0 0
$$301$$ 9412.00 1.80232
$$302$$ −4304.00 −0.820091
$$303$$ 1156.00 0.219176
$$304$$ −960.000 −0.181118
$$305$$ 0 0
$$306$$ 2944.00 0.549991
$$307$$ −5406.00 −1.00501 −0.502503 0.864576i $$-0.667587\pi$$
−0.502503 + 0.864576i $$0.667587\pi$$
$$308$$ 2912.00 0.538723
$$309$$ 2924.00 0.538319
$$310$$ 0 0
$$311$$ −5688.00 −1.03710 −0.518548 0.855048i $$-0.673527\pi$$
−0.518548 + 0.855048i $$0.673527\pi$$
$$312$$ −192.000 −0.0348393
$$313$$ −7352.00 −1.32767 −0.663833 0.747881i $$-0.731072\pi$$
−0.663833 + 0.747881i $$0.731072\pi$$
$$314$$ −1048.00 −0.188351
$$315$$ 0 0
$$316$$ −2880.00 −0.512698
$$317$$ 3484.00 0.617290 0.308645 0.951177i $$-0.400124\pi$$
0.308645 + 0.951177i $$0.400124\pi$$
$$318$$ 432.000 0.0761804
$$319$$ −2520.00 −0.442298
$$320$$ 0 0
$$321$$ 1932.00 0.335931
$$322$$ 3016.00 0.521972
$$323$$ −3840.00 −0.661496
$$324$$ 1684.00 0.288752
$$325$$ 0 0
$$326$$ −7036.00 −1.19536
$$327$$ −740.000 −0.125144
$$328$$ −1936.00 −0.325908
$$329$$ 5876.00 0.984664
$$330$$ 0 0
$$331$$ −7868.00 −1.30654 −0.653269 0.757125i $$-0.726603\pi$$
−0.653269 + 0.757125i $$0.726603\pi$$
$$332$$ 1912.00 0.316068
$$333$$ 5428.00 0.893251
$$334$$ −1068.00 −0.174965
$$335$$ 0 0
$$336$$ 832.000 0.135087
$$337$$ −656.000 −0.106037 −0.0530187 0.998594i $$-0.516884\pi$$
−0.0530187 + 0.998594i $$0.516884\pi$$
$$338$$ 4106.00 0.660760
$$339$$ −1056.00 −0.169186
$$340$$ 0 0
$$341$$ 3584.00 0.569163
$$342$$ −2760.00 −0.436385
$$343$$ 260.000 0.0409291
$$344$$ 2896.00 0.453901
$$345$$ 0 0
$$346$$ 8504.00 1.32132
$$347$$ 5754.00 0.890176 0.445088 0.895487i $$-0.353172\pi$$
0.445088 + 0.895487i $$0.353172\pi$$
$$348$$ −720.000 −0.110908
$$349$$ −3110.00 −0.477004 −0.238502 0.971142i $$-0.576656\pi$$
−0.238502 + 0.971142i $$0.576656\pi$$
$$350$$ 0 0
$$351$$ −1200.00 −0.182482
$$352$$ 896.000 0.135673
$$353$$ 7808.00 1.17727 0.588637 0.808397i $$-0.299665\pi$$
0.588637 + 0.808397i $$0.299665\pi$$
$$354$$ −80.0000 −0.0120112
$$355$$ 0 0
$$356$$ −1960.00 −0.291797
$$357$$ 3328.00 0.493379
$$358$$ −5000.00 −0.738151
$$359$$ −9240.00 −1.35841 −0.679204 0.733949i $$-0.737675\pi$$
−0.679204 + 0.733949i $$0.737675\pi$$
$$360$$ 0 0
$$361$$ −3259.00 −0.475142
$$362$$ 5156.00 0.748600
$$363$$ 1094.00 0.158182
$$364$$ 1248.00 0.179706
$$365$$ 0 0
$$366$$ 2168.00 0.309626
$$367$$ 3214.00 0.457137 0.228569 0.973528i $$-0.426595\pi$$
0.228569 + 0.973528i $$0.426595\pi$$
$$368$$ 928.000 0.131455
$$369$$ −5566.00 −0.785242
$$370$$ 0 0
$$371$$ −2808.00 −0.392949
$$372$$ 1024.00 0.142720
$$373$$ 348.000 0.0483077 0.0241538 0.999708i $$-0.492311\pi$$
0.0241538 + 0.999708i $$0.492311\pi$$
$$374$$ 3584.00 0.495519
$$375$$ 0 0
$$376$$ 1808.00 0.247980
$$377$$ −1080.00 −0.147541
$$378$$ 5200.00 0.707564
$$379$$ 4940.00 0.669527 0.334764 0.942302i $$-0.391344\pi$$
0.334764 + 0.942302i $$0.391344\pi$$
$$380$$ 0 0
$$381$$ −3068.00 −0.412542
$$382$$ 1536.00 0.205729
$$383$$ −6142.00 −0.819430 −0.409715 0.912214i $$-0.634372\pi$$
−0.409715 + 0.912214i $$0.634372\pi$$
$$384$$ 256.000 0.0340207
$$385$$ 0 0
$$386$$ −5216.00 −0.687791
$$387$$ 8326.00 1.09363
$$388$$ −5824.00 −0.762033
$$389$$ 3050.00 0.397535 0.198768 0.980047i $$-0.436306\pi$$
0.198768 + 0.980047i $$0.436306\pi$$
$$390$$ 0 0
$$391$$ 3712.00 0.480112
$$392$$ −2664.00 −0.343246
$$393$$ −24.0000 −0.00308051
$$394$$ 10232.0 1.30833
$$395$$ 0 0
$$396$$ 2576.00 0.326891
$$397$$ −5396.00 −0.682160 −0.341080 0.940034i $$-0.610793\pi$$
−0.341080 + 0.940034i $$0.610793\pi$$
$$398$$ 6960.00 0.876566
$$399$$ −3120.00 −0.391467
$$400$$ 0 0
$$401$$ 14482.0 1.80348 0.901741 0.432276i $$-0.142289\pi$$
0.901741 + 0.432276i $$0.142289\pi$$
$$402$$ 1736.00 0.215383
$$403$$ 1536.00 0.189860
$$404$$ −2312.00 −0.284719
$$405$$ 0 0
$$406$$ 4680.00 0.572080
$$407$$ 6608.00 0.804782
$$408$$ 1024.00 0.124254
$$409$$ −1090.00 −0.131778 −0.0658888 0.997827i $$-0.520988\pi$$
−0.0658888 + 0.997827i $$0.520988\pi$$
$$410$$ 0 0
$$411$$ −2448.00 −0.293798
$$412$$ −5848.00 −0.699297
$$413$$ 520.000 0.0619553
$$414$$ 2668.00 0.316727
$$415$$ 0 0
$$416$$ 384.000 0.0452576
$$417$$ −6200.00 −0.728094
$$418$$ −3360.00 −0.393165
$$419$$ −7180.00 −0.837150 −0.418575 0.908182i $$-0.637470\pi$$
−0.418575 + 0.908182i $$0.637470\pi$$
$$420$$ 0 0
$$421$$ −8138.00 −0.942095 −0.471047 0.882108i $$-0.656124\pi$$
−0.471047 + 0.882108i $$0.656124\pi$$
$$422$$ −6264.00 −0.722575
$$423$$ 5198.00 0.597483
$$424$$ −864.000 −0.0989612
$$425$$ 0 0
$$426$$ −4512.00 −0.513162
$$427$$ −14092.0 −1.59710
$$428$$ −3864.00 −0.436387
$$429$$ −672.000 −0.0756281
$$430$$ 0 0
$$431$$ −208.000 −0.0232460 −0.0116230 0.999932i $$-0.503700\pi$$
−0.0116230 + 0.999932i $$0.503700\pi$$
$$432$$ 1600.00 0.178195
$$433$$ −12992.0 −1.44193 −0.720965 0.692971i $$-0.756301\pi$$
−0.720965 + 0.692971i $$0.756301\pi$$
$$434$$ −6656.00 −0.736171
$$435$$ 0 0
$$436$$ 1480.00 0.162567
$$437$$ −3480.00 −0.380940
$$438$$ −2528.00 −0.275782
$$439$$ 1080.00 0.117416 0.0587080 0.998275i $$-0.481302\pi$$
0.0587080 + 0.998275i $$0.481302\pi$$
$$440$$ 0 0
$$441$$ −7659.00 −0.827017
$$442$$ 1536.00 0.165294
$$443$$ 9078.00 0.973609 0.486805 0.873511i $$-0.338162\pi$$
0.486805 + 0.873511i $$0.338162\pi$$
$$444$$ 1888.00 0.201803
$$445$$ 0 0
$$446$$ 124.000 0.0131650
$$447$$ −500.000 −0.0529065
$$448$$ −1664.00 −0.175484
$$449$$ 14310.0 1.50408 0.752039 0.659119i $$-0.229071\pi$$
0.752039 + 0.659119i $$0.229071\pi$$
$$450$$ 0 0
$$451$$ −6776.00 −0.707471
$$452$$ 2112.00 0.219779
$$453$$ −4304.00 −0.446401
$$454$$ −10628.0 −1.09867
$$455$$ 0 0
$$456$$ −960.000 −0.0985880
$$457$$ 2344.00 0.239929 0.119965 0.992778i $$-0.461722\pi$$
0.119965 + 0.992778i $$0.461722\pi$$
$$458$$ 380.000 0.0387691
$$459$$ 6400.00 0.650820
$$460$$ 0 0
$$461$$ 11382.0 1.14992 0.574959 0.818182i $$-0.305018\pi$$
0.574959 + 0.818182i $$0.305018\pi$$
$$462$$ 2912.00 0.293244
$$463$$ −16062.0 −1.61223 −0.806117 0.591756i $$-0.798435\pi$$
−0.806117 + 0.591756i $$0.798435\pi$$
$$464$$ 1440.00 0.144074
$$465$$ 0 0
$$466$$ −4816.00 −0.478749
$$467$$ −17166.0 −1.70096 −0.850479 0.526008i $$-0.823688\pi$$
−0.850479 + 0.526008i $$0.823688\pi$$
$$468$$ 1104.00 0.109044
$$469$$ −11284.0 −1.11097
$$470$$ 0 0
$$471$$ −1048.00 −0.102525
$$472$$ 160.000 0.0156030
$$473$$ 10136.0 0.985315
$$474$$ −2880.00 −0.279078
$$475$$ 0 0
$$476$$ −6656.00 −0.640919
$$477$$ −2484.00 −0.238437
$$478$$ 11360.0 1.08702
$$479$$ 7520.00 0.717323 0.358661 0.933468i $$-0.383233\pi$$
0.358661 + 0.933468i $$0.383233\pi$$
$$480$$ 0 0
$$481$$ 2832.00 0.268458
$$482$$ 556.000 0.0525417
$$483$$ 3016.00 0.284126
$$484$$ −2188.00 −0.205485
$$485$$ 0 0
$$486$$ 7084.00 0.661187
$$487$$ 11814.0 1.09927 0.549634 0.835406i $$-0.314767\pi$$
0.549634 + 0.835406i $$0.314767\pi$$
$$488$$ −4336.00 −0.402216
$$489$$ −7036.00 −0.650673
$$490$$ 0 0
$$491$$ 14052.0 1.29156 0.645782 0.763522i $$-0.276532\pi$$
0.645782 + 0.763522i $$0.276532\pi$$
$$492$$ −1936.00 −0.177402
$$493$$ 5760.00 0.526202
$$494$$ −1440.00 −0.131151
$$495$$ 0 0
$$496$$ −2048.00 −0.185399
$$497$$ 29328.0 2.64696
$$498$$ 1912.00 0.172046
$$499$$ 7620.00 0.683603 0.341802 0.939772i $$-0.388963\pi$$
0.341802 + 0.939772i $$0.388963\pi$$
$$500$$ 0 0
$$501$$ −1068.00 −0.0952390
$$502$$ −6504.00 −0.578262
$$503$$ 1818.00 0.161154 0.0805772 0.996748i $$-0.474324\pi$$
0.0805772 + 0.996748i $$0.474324\pi$$
$$504$$ −4784.00 −0.422810
$$505$$ 0 0
$$506$$ 3248.00 0.285358
$$507$$ 4106.00 0.359672
$$508$$ 6136.00 0.535907
$$509$$ 17850.0 1.55440 0.777198 0.629256i $$-0.216640\pi$$
0.777198 + 0.629256i $$0.216640\pi$$
$$510$$ 0 0
$$511$$ 16432.0 1.42252
$$512$$ −512.000 −0.0441942
$$513$$ −6000.00 −0.516387
$$514$$ 3072.00 0.263619
$$515$$ 0 0
$$516$$ 2896.00 0.247072
$$517$$ 6328.00 0.538308
$$518$$ −12272.0 −1.04093
$$519$$ 8504.00 0.719237
$$520$$ 0 0
$$521$$ −19238.0 −1.61772 −0.808860 0.588001i $$-0.799915\pi$$
−0.808860 + 0.588001i $$0.799915\pi$$
$$522$$ 4140.00 0.347132
$$523$$ 6278.00 0.524891 0.262445 0.964947i $$-0.415471\pi$$
0.262445 + 0.964947i $$0.415471\pi$$
$$524$$ 48.0000 0.00400170
$$525$$ 0 0
$$526$$ −9716.00 −0.805395
$$527$$ −8192.00 −0.677133
$$528$$ 896.000 0.0738511
$$529$$ −8803.00 −0.723514
$$530$$ 0 0
$$531$$ 460.000 0.0375938
$$532$$ 6240.00 0.508531
$$533$$ −2904.00 −0.235997
$$534$$ −1960.00 −0.158834
$$535$$ 0 0
$$536$$ −3472.00 −0.279790
$$537$$ −5000.00 −0.401799
$$538$$ −5220.00 −0.418309
$$539$$ −9324.00 −0.745108
$$540$$ 0 0
$$541$$ −9818.00 −0.780238 −0.390119 0.920764i $$-0.627566\pi$$
−0.390119 + 0.920764i $$0.627566\pi$$
$$542$$ 10336.0 0.819131
$$543$$ 5156.00 0.407486
$$544$$ −2048.00 −0.161410
$$545$$ 0 0
$$546$$ 1248.00 0.0978195
$$547$$ 12514.0 0.978172 0.489086 0.872236i $$-0.337330\pi$$
0.489086 + 0.872236i $$0.337330\pi$$
$$548$$ 4896.00 0.381655
$$549$$ −12466.0 −0.969100
$$550$$ 0 0
$$551$$ −5400.00 −0.417509
$$552$$ 928.000 0.0715549
$$553$$ 18720.0 1.43952
$$554$$ −3848.00 −0.295101
$$555$$ 0 0
$$556$$ 12400.0 0.945822
$$557$$ −10596.0 −0.806045 −0.403022 0.915190i $$-0.632040\pi$$
−0.403022 + 0.915190i $$0.632040\pi$$
$$558$$ −5888.00 −0.446701
$$559$$ 4344.00 0.328679
$$560$$ 0 0
$$561$$ 3584.00 0.269727
$$562$$ −6084.00 −0.456651
$$563$$ −14002.0 −1.04816 −0.524080 0.851669i $$-0.675591\pi$$
−0.524080 + 0.851669i $$0.675591\pi$$
$$564$$ 1808.00 0.134983
$$565$$ 0 0
$$566$$ −3436.00 −0.255169
$$567$$ −10946.0 −0.810739
$$568$$ 9024.00 0.666617
$$569$$ −7330.00 −0.540052 −0.270026 0.962853i $$-0.587032\pi$$
−0.270026 + 0.962853i $$0.587032\pi$$
$$570$$ 0 0
$$571$$ 5812.00 0.425963 0.212981 0.977056i $$-0.431683\pi$$
0.212981 + 0.977056i $$0.431683\pi$$
$$572$$ 1344.00 0.0982438
$$573$$ 1536.00 0.111985
$$574$$ 12584.0 0.915063
$$575$$ 0 0
$$576$$ −1472.00 −0.106481
$$577$$ −16736.0 −1.20750 −0.603751 0.797173i $$-0.706328\pi$$
−0.603751 + 0.797173i $$0.706328\pi$$
$$578$$ 1634.00 0.117587
$$579$$ −5216.00 −0.374386
$$580$$ 0 0
$$581$$ −12428.0 −0.887436
$$582$$ −5824.00 −0.414798
$$583$$ −3024.00 −0.214822
$$584$$ 5056.00 0.358251
$$585$$ 0 0
$$586$$ 4584.00 0.323146
$$587$$ 7434.00 0.522716 0.261358 0.965242i $$-0.415830\pi$$
0.261358 + 0.965242i $$0.415830\pi$$
$$588$$ −2664.00 −0.186839
$$589$$ 7680.00 0.537265
$$590$$ 0 0
$$591$$ 10232.0 0.712163
$$592$$ −3776.00 −0.262150
$$593$$ −25872.0 −1.79163 −0.895814 0.444429i $$-0.853407\pi$$
−0.895814 + 0.444429i $$0.853407\pi$$
$$594$$ 5600.00 0.386820
$$595$$ 0 0
$$596$$ 1000.00 0.0687275
$$597$$ 6960.00 0.477142
$$598$$ 1392.00 0.0951892
$$599$$ −3720.00 −0.253748 −0.126874 0.991919i $$-0.540494\pi$$
−0.126874 + 0.991919i $$0.540494\pi$$
$$600$$ 0 0
$$601$$ −12958.0 −0.879481 −0.439740 0.898125i $$-0.644930\pi$$
−0.439740 + 0.898125i $$0.644930\pi$$
$$602$$ −18824.0 −1.27443
$$603$$ −9982.00 −0.674127
$$604$$ 8608.00 0.579892
$$605$$ 0 0
$$606$$ −2312.00 −0.154981
$$607$$ 7214.00 0.482384 0.241192 0.970477i $$-0.422462\pi$$
0.241192 + 0.970477i $$0.422462\pi$$
$$608$$ 1920.00 0.128070
$$609$$ 4680.00 0.311401
$$610$$ 0 0
$$611$$ 2712.00 0.179568
$$612$$ −5888.00 −0.388902
$$613$$ 4828.00 0.318109 0.159055 0.987270i $$-0.449155\pi$$
0.159055 + 0.987270i $$0.449155\pi$$
$$614$$ 10812.0 0.710646
$$615$$ 0 0
$$616$$ −5824.00 −0.380934
$$617$$ −27656.0 −1.80452 −0.902260 0.431193i $$-0.858093\pi$$
−0.902260 + 0.431193i $$0.858093\pi$$
$$618$$ −5848.00 −0.380649
$$619$$ −21220.0 −1.37787 −0.688937 0.724821i $$-0.741922\pi$$
−0.688937 + 0.724821i $$0.741922\pi$$
$$620$$ 0 0
$$621$$ 5800.00 0.374792
$$622$$ 11376.0 0.733338
$$623$$ 12740.0 0.819289
$$624$$ 384.000 0.0246351
$$625$$ 0 0
$$626$$ 14704.0 0.938802
$$627$$ −3360.00 −0.214012
$$628$$ 2096.00 0.133184
$$629$$ −15104.0 −0.957450
$$630$$ 0 0
$$631$$ 17672.0 1.11491 0.557457 0.830206i $$-0.311777\pi$$
0.557457 + 0.830206i $$0.311777\pi$$
$$632$$ 5760.00 0.362532
$$633$$ −6264.00 −0.393320
$$634$$ −6968.00 −0.436490
$$635$$ 0 0
$$636$$ −864.000 −0.0538677
$$637$$ −3996.00 −0.248551
$$638$$ 5040.00 0.312752
$$639$$ 25944.0 1.60615
$$640$$ 0 0
$$641$$ 7322.00 0.451173 0.225586 0.974223i $$-0.427570\pi$$
0.225586 + 0.974223i $$0.427570\pi$$
$$642$$ −3864.00 −0.237539
$$643$$ 8238.00 0.505249 0.252624 0.967564i $$-0.418706\pi$$
0.252624 + 0.967564i $$0.418706\pi$$
$$644$$ −6032.00 −0.369090
$$645$$ 0 0
$$646$$ 7680.00 0.467749
$$647$$ −6426.00 −0.390467 −0.195233 0.980757i $$-0.562546\pi$$
−0.195233 + 0.980757i $$0.562546\pi$$
$$648$$ −3368.00 −0.204178
$$649$$ 560.000 0.0338705
$$650$$ 0 0
$$651$$ −6656.00 −0.400721
$$652$$ 14072.0 0.845249
$$653$$ 5908.00 0.354055 0.177027 0.984206i $$-0.443352\pi$$
0.177027 + 0.984206i $$0.443352\pi$$
$$654$$ 1480.00 0.0884902
$$655$$ 0 0
$$656$$ 3872.00 0.230452
$$657$$ 14536.0 0.863171
$$658$$ −11752.0 −0.696262
$$659$$ −26780.0 −1.58301 −0.791503 0.611166i $$-0.790701\pi$$
−0.791503 + 0.611166i $$0.790701\pi$$
$$660$$ 0 0
$$661$$ −24538.0 −1.44390 −0.721950 0.691945i $$-0.756754\pi$$
−0.721950 + 0.691945i $$0.756754\pi$$
$$662$$ 15736.0 0.923863
$$663$$ 1536.00 0.0899748
$$664$$ −3824.00 −0.223494
$$665$$ 0 0
$$666$$ −10856.0 −0.631624
$$667$$ 5220.00 0.303027
$$668$$ 2136.00 0.123719
$$669$$ 124.000 0.00716609
$$670$$ 0 0
$$671$$ −15176.0 −0.873119
$$672$$ −1664.00 −0.0955211
$$673$$ 28848.0 1.65232 0.826158 0.563439i $$-0.190522\pi$$
0.826158 + 0.563439i $$0.190522\pi$$
$$674$$ 1312.00 0.0749798
$$675$$ 0 0
$$676$$ −8212.00 −0.467228
$$677$$ 26884.0 1.52620 0.763099 0.646282i $$-0.223677\pi$$
0.763099 + 0.646282i $$0.223677\pi$$
$$678$$ 2112.00 0.119633
$$679$$ 37856.0 2.13959
$$680$$ 0 0
$$681$$ −10628.0 −0.598041
$$682$$ −7168.00 −0.402459
$$683$$ −14282.0 −0.800125 −0.400063 0.916488i $$-0.631012\pi$$
−0.400063 + 0.916488i $$0.631012\pi$$
$$684$$ 5520.00 0.308571
$$685$$ 0 0
$$686$$ −520.000 −0.0289412
$$687$$ 380.000 0.0211032
$$688$$ −5792.00 −0.320956
$$689$$ −1296.00 −0.0716599
$$690$$ 0 0
$$691$$ −3428.00 −0.188723 −0.0943613 0.995538i $$-0.530081\pi$$
−0.0943613 + 0.995538i $$0.530081\pi$$
$$692$$ −17008.0 −0.934317
$$693$$ −16744.0 −0.917824
$$694$$ −11508.0 −0.629449
$$695$$ 0 0
$$696$$ 1440.00 0.0784239
$$697$$ 15488.0 0.841678
$$698$$ 6220.00 0.337293
$$699$$ −4816.00 −0.260598
$$700$$ 0 0
$$701$$ 26942.0 1.45162 0.725810 0.687895i $$-0.241465\pi$$
0.725810 + 0.687895i $$0.241465\pi$$
$$702$$ 2400.00 0.129034
$$703$$ 14160.0 0.759679
$$704$$ −1792.00 −0.0959354
$$705$$ 0 0
$$706$$ −15616.0 −0.832459
$$707$$ 15028.0 0.799415
$$708$$ 160.000 0.00849318
$$709$$ −1950.00 −0.103292 −0.0516458 0.998665i $$-0.516447\pi$$
−0.0516458 + 0.998665i $$0.516447\pi$$
$$710$$ 0 0
$$711$$ 16560.0 0.873486
$$712$$ 3920.00 0.206332
$$713$$ −7424.00 −0.389945
$$714$$ −6656.00 −0.348872
$$715$$ 0 0
$$716$$ 10000.0 0.521952
$$717$$ 11360.0 0.591697
$$718$$ 18480.0 0.960540
$$719$$ 12080.0 0.626576 0.313288 0.949658i $$-0.398570\pi$$
0.313288 + 0.949658i $$0.398570\pi$$
$$720$$ 0 0
$$721$$ 38012.0 1.96344
$$722$$ 6518.00 0.335976
$$723$$ 556.000 0.0286001
$$724$$ −10312.0 −0.529340
$$725$$ 0 0
$$726$$ −2188.00 −0.111852
$$727$$ −17226.0 −0.878785 −0.439393 0.898295i $$-0.644806\pi$$
−0.439393 + 0.898295i $$0.644806\pi$$
$$728$$ −2496.00 −0.127071
$$729$$ −4283.00 −0.217599
$$730$$ 0 0
$$731$$ −23168.0 −1.17223
$$732$$ −4336.00 −0.218939
$$733$$ 788.000 0.0397073 0.0198536 0.999803i $$-0.493680\pi$$
0.0198536 + 0.999803i $$0.493680\pi$$
$$734$$ −6428.00 −0.323245
$$735$$ 0 0
$$736$$ −1856.00 −0.0929525
$$737$$ −12152.0 −0.607360
$$738$$ 11132.0 0.555250
$$739$$ −2060.00 −0.102542 −0.0512709 0.998685i $$-0.516327\pi$$
−0.0512709 + 0.998685i $$0.516327\pi$$
$$740$$ 0 0
$$741$$ −1440.00 −0.0713896
$$742$$ 5616.00 0.277857
$$743$$ 3258.00 0.160867 0.0804337 0.996760i $$-0.474369\pi$$
0.0804337 + 0.996760i $$0.474369\pi$$
$$744$$ −2048.00 −0.100918
$$745$$ 0 0
$$746$$ −696.000 −0.0341587
$$747$$ −10994.0 −0.538487
$$748$$ −7168.00 −0.350385
$$749$$ 25116.0 1.22526
$$750$$ 0 0
$$751$$ −4528.00 −0.220012 −0.110006 0.993931i $$-0.535087\pi$$
−0.110006 + 0.993931i $$0.535087\pi$$
$$752$$ −3616.00 −0.175348
$$753$$ −6504.00 −0.314766
$$754$$ 2160.00 0.104327
$$755$$ 0 0
$$756$$ −10400.0 −0.500323
$$757$$ −18236.0 −0.875560 −0.437780 0.899082i $$-0.644235\pi$$
−0.437780 + 0.899082i $$0.644235\pi$$
$$758$$ −9880.00 −0.473427
$$759$$ 3248.00 0.155329
$$760$$ 0 0
$$761$$ −18678.0 −0.889720 −0.444860 0.895600i $$-0.646747\pi$$
−0.444860 + 0.895600i $$0.646747\pi$$
$$762$$ 6136.00 0.291711
$$763$$ −9620.00 −0.456445
$$764$$ −3072.00 −0.145473
$$765$$ 0 0
$$766$$ 12284.0 0.579424
$$767$$ 240.000 0.0112984
$$768$$ −512.000 −0.0240563
$$769$$ 27390.0 1.28441 0.642203 0.766534i $$-0.278020\pi$$
0.642203 + 0.766534i $$0.278020\pi$$
$$770$$ 0 0
$$771$$ 3072.00 0.143496
$$772$$ 10432.0 0.486342
$$773$$ −9252.00 −0.430493 −0.215247 0.976560i $$-0.569056\pi$$
−0.215247 + 0.976560i $$0.569056\pi$$
$$774$$ −16652.0 −0.773312
$$775$$ 0 0
$$776$$ 11648.0 0.538839
$$777$$ −12272.0 −0.566609
$$778$$ −6100.00 −0.281100
$$779$$ −14520.0 −0.667822
$$780$$ 0 0
$$781$$ 31584.0 1.44707
$$782$$ −7424.00 −0.339491
$$783$$ 9000.00 0.410771
$$784$$ 5328.00 0.242711
$$785$$ 0 0
$$786$$ 48.0000 0.00217825
$$787$$ −5726.00 −0.259352 −0.129676 0.991556i $$-0.541394\pi$$
−0.129676 + 0.991556i $$0.541394\pi$$
$$788$$ −20464.0 −0.925127
$$789$$ −9716.00 −0.438401
$$790$$ 0 0
$$791$$ −13728.0 −0.617082
$$792$$ −5152.00 −0.231147
$$793$$ −6504.00 −0.291253
$$794$$ 10792.0 0.482360
$$795$$ 0 0
$$796$$ −13920.0 −0.619826
$$797$$ −27236.0 −1.21048 −0.605238 0.796045i $$-0.706922\pi$$
−0.605238 + 0.796045i $$0.706922\pi$$
$$798$$ 6240.00 0.276809
$$799$$ −14464.0 −0.640425
$$800$$ 0 0
$$801$$ 11270.0 0.497136
$$802$$ −28964.0 −1.27525
$$803$$ 17696.0 0.777682
$$804$$ −3472.00 −0.152299
$$805$$ 0 0
$$806$$ −3072.00 −0.134251
$$807$$ −5220.00 −0.227699
$$808$$ 4624.00 0.201326
$$809$$ 10950.0 0.475873 0.237937 0.971281i $$-0.423529\pi$$
0.237937 + 0.971281i $$0.423529\pi$$
$$810$$ 0 0
$$811$$ −8828.00 −0.382236 −0.191118 0.981567i $$-0.561211\pi$$
−0.191118 + 0.981567i $$0.561211\pi$$
$$812$$ −9360.00 −0.404522
$$813$$ 10336.0 0.445879
$$814$$ −13216.0 −0.569067
$$815$$ 0 0
$$816$$ −2048.00 −0.0878607
$$817$$ 21720.0 0.930094
$$818$$ 2180.00 0.0931808
$$819$$ −7176.00 −0.306166
$$820$$ 0 0
$$821$$ −16058.0 −0.682616 −0.341308 0.939951i $$-0.610870\pi$$
−0.341308 + 0.939951i $$0.610870\pi$$
$$822$$ 4896.00 0.207746
$$823$$ −41862.0 −1.77305 −0.886523 0.462684i $$-0.846887\pi$$
−0.886523 + 0.462684i $$0.846887\pi$$
$$824$$ 11696.0 0.494478
$$825$$ 0 0
$$826$$ −1040.00 −0.0438090
$$827$$ 12154.0 0.511047 0.255524 0.966803i $$-0.417752\pi$$
0.255524 + 0.966803i $$0.417752\pi$$
$$828$$ −5336.00 −0.223960
$$829$$ −15390.0 −0.644773 −0.322386 0.946608i $$-0.604485\pi$$
−0.322386 + 0.946608i $$0.604485\pi$$
$$830$$ 0 0
$$831$$ −3848.00 −0.160633
$$832$$ −768.000 −0.0320019
$$833$$ 21312.0 0.886455
$$834$$ 12400.0 0.514840
$$835$$ 0 0
$$836$$ 6720.00 0.278010
$$837$$ −12800.0 −0.528593
$$838$$ 14360.0 0.591955
$$839$$ −4280.00 −0.176117 −0.0880584 0.996115i $$-0.528066\pi$$
−0.0880584 + 0.996115i $$0.528066\pi$$
$$840$$ 0 0
$$841$$ −16289.0 −0.667883
$$842$$ 16276.0 0.666162
$$843$$ −6084.00 −0.248570
$$844$$ 12528.0 0.510938
$$845$$ 0 0
$$846$$ −10396.0 −0.422484
$$847$$ 14222.0 0.576947
$$848$$ 1728.00 0.0699761
$$849$$ −3436.00 −0.138897
$$850$$ 0 0
$$851$$ −13688.0 −0.551373
$$852$$ 9024.00 0.362860
$$853$$ −14452.0 −0.580102 −0.290051 0.957011i $$-0.593672\pi$$
−0.290051 + 0.957011i $$0.593672\pi$$
$$854$$ 28184.0 1.12932
$$855$$ 0 0
$$856$$ 7728.00 0.308572
$$857$$ 22584.0 0.900181 0.450090 0.892983i $$-0.351392\pi$$
0.450090 + 0.892983i $$0.351392\pi$$
$$858$$ 1344.00 0.0534772
$$859$$ −26740.0 −1.06212 −0.531058 0.847336i $$-0.678205\pi$$
−0.531058 + 0.847336i $$0.678205\pi$$
$$860$$ 0 0
$$861$$ 12584.0 0.498097
$$862$$ 416.000 0.0164374
$$863$$ 498.000 0.0196432 0.00982162 0.999952i $$-0.496874\pi$$
0.00982162 + 0.999952i $$0.496874\pi$$
$$864$$ −3200.00 −0.126003
$$865$$ 0 0
$$866$$ 25984.0 1.01960
$$867$$ 1634.00 0.0640064
$$868$$ 13312.0 0.520552
$$869$$ 20160.0 0.786975
$$870$$ 0 0
$$871$$ −5208.00 −0.202602
$$872$$ −2960.00 −0.114952
$$873$$ 33488.0 1.29828
$$874$$ 6960.00 0.269366
$$875$$ 0 0
$$876$$ 5056.00 0.195007
$$877$$ 13244.0 0.509941 0.254970 0.966949i $$-0.417934\pi$$
0.254970 + 0.966949i $$0.417934\pi$$
$$878$$ −2160.00 −0.0830256
$$879$$ 4584.00 0.175898
$$880$$ 0 0
$$881$$ 40842.0 1.56186 0.780932 0.624616i $$-0.214745\pi$$
0.780932 + 0.624616i $$0.214745\pi$$
$$882$$ 15318.0 0.584789
$$883$$ 12078.0 0.460314 0.230157 0.973154i $$-0.426076\pi$$
0.230157 + 0.973154i $$0.426076\pi$$
$$884$$ −3072.00 −0.116881
$$885$$ 0 0
$$886$$ −18156.0 −0.688446
$$887$$ 18294.0 0.692506 0.346253 0.938141i $$-0.387454\pi$$
0.346253 + 0.938141i $$0.387454\pi$$
$$888$$ −3776.00 −0.142696
$$889$$ −39884.0 −1.50469
$$890$$ 0 0
$$891$$ −11788.0 −0.443224
$$892$$ −248.000 −0.00930903
$$893$$ 13560.0 0.508139
$$894$$ 1000.00 0.0374105
$$895$$ 0 0
$$896$$ 3328.00 0.124086
$$897$$ 1392.00 0.0518144
$$898$$ −28620.0 −1.06354
$$899$$ −11520.0 −0.427379
$$900$$ 0 0
$$901$$ 6912.00 0.255574
$$902$$ 13552.0 0.500257
$$903$$ −18824.0 −0.693714
$$904$$ −4224.00 −0.155407
$$905$$ 0 0
$$906$$ 8608.00 0.315653
$$907$$ −22566.0 −0.826121 −0.413060 0.910704i $$-0.635540\pi$$
−0.413060 + 0.910704i $$0.635540\pi$$
$$908$$ 21256.0 0.776878
$$909$$ 13294.0 0.485076
$$910$$ 0 0
$$911$$ −6768.00 −0.246140 −0.123070 0.992398i $$-0.539274\pi$$
−0.123070 + 0.992398i $$0.539274\pi$$
$$912$$ 1920.00 0.0697122
$$913$$ −13384.0 −0.485154
$$914$$ −4688.00 −0.169656
$$915$$ 0 0
$$916$$ −760.000 −0.0274139
$$917$$ −312.000 −0.0112357
$$918$$ −12800.0 −0.460199
$$919$$ 22200.0 0.796856 0.398428 0.917200i $$-0.369556\pi$$
0.398428 + 0.917200i $$0.369556\pi$$
$$920$$ 0 0
$$921$$ 10812.0 0.386827
$$922$$ −22764.0 −0.813115
$$923$$ 13536.0 0.482712
$$924$$ −5824.00 −0.207354
$$925$$ 0 0
$$926$$ 32124.0 1.14002
$$927$$ 33626.0 1.19139
$$928$$ −2880.00 −0.101876
$$929$$ −6330.00 −0.223553 −0.111776 0.993733i $$-0.535654\pi$$
−0.111776 + 0.993733i $$0.535654\pi$$
$$930$$ 0 0
$$931$$ −19980.0 −0.703349
$$932$$ 9632.00 0.338526
$$933$$ 11376.0 0.399178
$$934$$ 34332.0 1.20276
$$935$$ 0 0
$$936$$ −2208.00 −0.0771055
$$937$$ 19544.0 0.681403 0.340702 0.940172i $$-0.389335\pi$$
0.340702 + 0.940172i $$0.389335\pi$$
$$938$$ 22568.0 0.785577
$$939$$ 14704.0 0.511019
$$940$$ 0 0
$$941$$ −9898.00 −0.342896 −0.171448 0.985193i $$-0.554845\pi$$
−0.171448 + 0.985193i $$0.554845\pi$$
$$942$$ 2096.00 0.0724961
$$943$$ 14036.0 0.484703
$$944$$ −320.000 −0.0110330
$$945$$ 0 0
$$946$$ −20272.0 −0.696723
$$947$$ −41406.0 −1.42082 −0.710409 0.703789i $$-0.751490\pi$$
−0.710409 + 0.703789i $$0.751490\pi$$
$$948$$ 5760.00 0.197338
$$949$$ 7584.00 0.259417
$$950$$ 0 0
$$951$$ −6968.00 −0.237595
$$952$$ 13312.0 0.453198
$$953$$ −25432.0 −0.864453 −0.432226 0.901765i $$-0.642272\pi$$
−0.432226 + 0.901765i $$0.642272\pi$$
$$954$$ 4968.00 0.168601
$$955$$ 0 0
$$956$$ −22720.0 −0.768637
$$957$$ 5040.00 0.170240
$$958$$ −15040.0 −0.507224
$$959$$ −31824.0 −1.07159
$$960$$ 0 0
$$961$$ −13407.0 −0.450035
$$962$$ −5664.00 −0.189828
$$963$$ 22218.0 0.743474
$$964$$ −1112.00 −0.0371526
$$965$$ 0 0
$$966$$ −6032.00 −0.200907
$$967$$ −12106.0 −0.402588 −0.201294 0.979531i $$-0.564515\pi$$
−0.201294 + 0.979531i $$0.564515\pi$$
$$968$$ 4376.00 0.145300
$$969$$ 7680.00 0.254610
$$970$$ 0 0
$$971$$ 7812.00 0.258186 0.129093 0.991632i $$-0.458793\pi$$
0.129093 + 0.991632i $$0.458793\pi$$
$$972$$ −14168.0 −0.467530
$$973$$ −80600.0 −2.65562
$$974$$ −23628.0 −0.777300
$$975$$ 0 0
$$976$$ 8672.00 0.284410
$$977$$ −12576.0 −0.411814 −0.205907 0.978572i $$-0.566014\pi$$
−0.205907 + 0.978572i $$0.566014\pi$$
$$978$$ 14072.0 0.460095
$$979$$ 13720.0 0.447899
$$980$$ 0 0
$$981$$ −8510.00 −0.276966
$$982$$ −28104.0 −0.913274
$$983$$ −4342.00 −0.140883 −0.0704417 0.997516i $$-0.522441\pi$$
−0.0704417 + 0.997516i $$0.522441\pi$$
$$984$$ 3872.00 0.125442
$$985$$ 0 0
$$986$$ −11520.0 −0.372081
$$987$$ −11752.0 −0.378997
$$988$$ 2880.00 0.0927379
$$989$$ −20996.0 −0.675060
$$990$$ 0 0
$$991$$ 26272.0 0.842137 0.421068 0.907029i $$-0.361655\pi$$
0.421068 + 0.907029i $$0.361655\pi$$
$$992$$ 4096.00 0.131097
$$993$$ 15736.0 0.502887
$$994$$ −58656.0 −1.87169
$$995$$ 0 0
$$996$$ −3824.00 −0.121655
$$997$$ −44796.0 −1.42297 −0.711486 0.702700i $$-0.751978\pi$$
−0.711486 + 0.702700i $$0.751978\pi$$
$$998$$ −15240.0 −0.483381
$$999$$ −23600.0 −0.747418
Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000

## Twists

By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 50.4.a.b.1.1 1
3.2 odd 2 450.4.a.k.1.1 1
4.3 odd 2 400.4.a.n.1.1 1
5.2 odd 4 10.4.b.a.9.1 2
5.3 odd 4 10.4.b.a.9.2 yes 2
5.4 even 2 50.4.a.d.1.1 1
7.6 odd 2 2450.4.a.o.1.1 1
8.3 odd 2 1600.4.a.t.1.1 1
8.5 even 2 1600.4.a.bh.1.1 1
15.2 even 4 90.4.c.b.19.2 2
15.8 even 4 90.4.c.b.19.1 2
15.14 odd 2 450.4.a.j.1.1 1
20.3 even 4 80.4.c.a.49.2 2
20.7 even 4 80.4.c.a.49.1 2
20.19 odd 2 400.4.a.h.1.1 1
35.13 even 4 490.4.c.b.99.2 2
35.27 even 4 490.4.c.b.99.1 2
35.34 odd 2 2450.4.a.bb.1.1 1
40.3 even 4 320.4.c.c.129.1 2
40.13 odd 4 320.4.c.d.129.2 2
40.19 odd 2 1600.4.a.bg.1.1 1
40.27 even 4 320.4.c.c.129.2 2
40.29 even 2 1600.4.a.u.1.1 1
40.37 odd 4 320.4.c.d.129.1 2
60.23 odd 4 720.4.f.f.289.2 2
60.47 odd 4 720.4.f.f.289.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
10.4.b.a.9.1 2 5.2 odd 4
10.4.b.a.9.2 yes 2 5.3 odd 4
50.4.a.b.1.1 1 1.1 even 1 trivial
50.4.a.d.1.1 1 5.4 even 2
80.4.c.a.49.1 2 20.7 even 4
80.4.c.a.49.2 2 20.3 even 4
90.4.c.b.19.1 2 15.8 even 4
90.4.c.b.19.2 2 15.2 even 4
320.4.c.c.129.1 2 40.3 even 4
320.4.c.c.129.2 2 40.27 even 4
320.4.c.d.129.1 2 40.37 odd 4
320.4.c.d.129.2 2 40.13 odd 4
400.4.a.h.1.1 1 20.19 odd 2
400.4.a.n.1.1 1 4.3 odd 2
450.4.a.j.1.1 1 15.14 odd 2
450.4.a.k.1.1 1 3.2 odd 2
490.4.c.b.99.1 2 35.27 even 4
490.4.c.b.99.2 2 35.13 even 4
720.4.f.f.289.1 2 60.47 odd 4
720.4.f.f.289.2 2 60.23 odd 4
1600.4.a.t.1.1 1 8.3 odd 2
1600.4.a.u.1.1 1 40.29 even 2
1600.4.a.bg.1.1 1 40.19 odd 2
1600.4.a.bh.1.1 1 8.5 even 2
2450.4.a.o.1.1 1 7.6 odd 2
2450.4.a.bb.1.1 1 35.34 odd 2