Properties

 Label 50.4.a.d.1.1 Level $50$ Weight $4$ Character 50.1 Self dual yes Analytic conductor $2.950$ Analytic rank $0$ 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: $$0$$ 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 $$-0.438357\pi$$
0.192450 + 0.981307i $$0.438357\pi$$
$$4$$ 4.00000 0.500000
$$5$$ 0 0
$$6$$ 4.00000 0.272166
$$7$$ 26.0000 1.40387 0.701934 0.712242i $$-0.252320\pi$$
0.701934 + 0.712242i $$0.252320\pi$$
$$8$$ 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 $$-0.459142\pi$$
0.128008 + 0.991773i $$0.459142\pi$$
$$14$$ 52.0000 0.992685
$$15$$ 0 0
$$16$$ 16.0000 0.250000
$$17$$ −64.0000 −0.913075 −0.456538 0.889704i $$-0.650911\pi$$
−0.456538 + 0.889704i $$0.650911\pi$$
$$18$$ −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 $$-0.584682\pi$$
−0.262909 + 0.964821i $$0.584682\pi$$
$$24$$ 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 $$-0.324327\pi$$
0.524299 + 0.851534i $$0.324327\pi$$
$$38$$ −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 $$-0.278141\pi$$
0.641913 + 0.766778i $$0.278141\pi$$
$$44$$ −112.000 −0.383742
$$45$$ 0 0
$$46$$ −116.000 −0.371810
$$47$$ 226.000 0.701393 0.350697 0.936489i $$-0.385945\pi$$
0.350697 + 0.936489i $$0.385945\pi$$
$$48$$ 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 $$-0.544695\pi$$
−0.139952 + 0.990158i $$0.544695\pi$$
$$54$$ −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 $$-0.629492\pi$$
−0.395683 + 0.918387i $$0.629492\pi$$
$$68$$ −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 $$-0.330886\pi$$
0.506644 + 0.862155i $$0.330886\pi$$
$$74$$ 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 $$-0.602363\pi$$
−0.316068 + 0.948736i $$0.602363\pi$$
$$84$$ 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 $$-0.224201\pi$$
0.762033 + 0.647538i $$0.224201\pi$$
$$98$$ 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 $$-0.253497\pi$$
0.699297 + 0.714831i $$0.253497\pi$$
$$104$$ 96.0000 0.0905151
$$105$$ 0 0
$$106$$ −216.000 −0.197922
$$107$$ 966.000 0.872773 0.436387 0.899759i $$-0.356258\pi$$
0.436387 + 0.899759i $$0.356258\pi$$
$$108$$ −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 $$-0.570534\pi$$
−0.219779 + 0.975550i $$0.570534\pi$$
$$114$$ −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 $$-0.680030\pi$$
−0.535907 + 0.844277i $$0.680030\pi$$
$$128$$ 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 $$-0.624646\pi$$
−0.381655 + 0.924305i $$0.624646\pi$$
$$138$$ −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 $$-0.542520\pi$$
−0.133184 + 0.991091i $$0.542520\pi$$
$$158$$ −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 $$-0.820548\pi$$
−0.845249 + 0.534373i $$0.820548\pi$$
$$164$$ 968.000 0.460903
$$165$$ 0 0
$$166$$ −956.000 −0.446988
$$167$$ −534.000 −0.247438 −0.123719 0.992317i $$-0.539482\pi$$
−0.123719 + 0.992317i $$0.539482\pi$$
$$168$$ 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 $$-0.116011\pi$$
0.934317 + 0.356444i $$0.116011\pi$$
$$174$$ 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 $$-0.661669\pi$$
−0.486342 + 0.873769i $$0.661669\pi$$
$$194$$ 2912.00 1.07768
$$195$$ 0 0
$$196$$ 1332.00 0.485423
$$197$$ 5116.00 1.85025 0.925127 0.379659i $$-0.123959\pi$$
0.925127 + 0.379659i $$0.123959\pi$$
$$198$$ 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 $$-0.497037\pi$$
0.00930903 + 0.999957i $$0.497037\pi$$
$$224$$ 832.000 0.248171
$$225$$ 0 0
$$226$$ −1056.00 −0.310814
$$227$$ −5314.00 −1.55376 −0.776878 0.629651i $$-0.783198\pi$$
−0.776878 + 0.629651i $$0.783198\pi$$
$$228$$ −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 $$-0.609928\pi$$
−0.338526 + 0.940957i $$0.609928\pi$$
$$234$$ −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 $$-0.440316\pi$$
0.186407 + 0.982473i $$0.440316\pi$$
$$258$$ 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 $$-0.692863\pi$$
−0.569500 + 0.821991i $$0.692863\pi$$
$$264$$ −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 $$-0.566913\pi$$
−0.208668 + 0.977987i $$0.566913\pi$$
$$278$$ 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 $$-0.557750\pi$$
−0.180432 + 0.983587i $$0.557750\pi$$
$$284$$ −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 $$-0.426618\pi$$
0.228498 + 0.973544i $$0.426618\pi$$
$$294$$ 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 $$-0.332413\pi$$
0.502503 + 0.864576i $$0.332413\pi$$
$$308$$ −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 $$-0.268928\pi$$
0.663833 + 0.747881i $$0.268928\pi$$
$$314$$ −1048.00 −0.188351
$$315$$ 0 0
$$316$$ −2880.00 −0.512698
$$317$$ −3484.00 −0.617290 −0.308645 0.951177i $$-0.599876\pi$$
−0.308645 + 0.951177i $$0.599876\pi$$
$$318$$ −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 $$-0.483116\pi$$
0.0530187 + 0.998594i $$0.483116\pi$$
$$338$$ −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 $$-0.646828\pi$$
−0.445088 + 0.895487i $$0.646828\pi$$
$$348$$ 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 $$-0.700335\pi$$
−0.588637 + 0.808397i $$0.700335\pi$$
$$354$$ −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 $$-0.573405\pi$$
−0.228569 + 0.973528i $$0.573405\pi$$
$$368$$ −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 $$-0.507689\pi$$
−0.0241538 + 0.999708i $$0.507689\pi$$
$$374$$ 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 $$-0.365628\pi$$
0.409715 + 0.912214i $$0.365628\pi$$
$$384$$ 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 $$-0.389207\pi$$
0.341080 + 0.940034i $$0.389207\pi$$
$$398$$ −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 $$-0.243699\pi$$
0.720965 + 0.692971i $$0.243699\pi$$
$$434$$ −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 $$-0.661838\pi$$
−0.486805 + 0.873511i $$0.661838\pi$$
$$444$$ 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 $$-0.538278\pi$$
−0.119965 + 0.992778i $$0.538278\pi$$
$$458$$ −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 $$-0.201565\pi$$
0.806117 + 0.591756i $$0.201565\pi$$
$$464$$ 1440.00 0.144074
$$465$$ 0 0
$$466$$ −4816.00 −0.478749
$$467$$ 17166.0 1.70096 0.850479 0.526008i $$-0.176312\pi$$
0.850479 + 0.526008i $$0.176312\pi$$
$$468$$ −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 $$-0.685233\pi$$
−0.549634 + 0.835406i $$0.685233\pi$$
$$488$$ 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 $$-0.525676\pi$$
−0.0805772 + 0.996748i $$0.525676\pi$$
$$504$$ −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 $$-0.584529\pi$$
−0.262445 + 0.964947i $$0.584529\pi$$
$$524$$ 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 $$-0.662670\pi$$
−0.489086 + 0.872236i $$0.662670\pi$$
$$548$$ −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 $$-0.367960\pi$$
0.403022 + 0.915190i $$0.367960\pi$$
$$558$$ 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 $$-0.324409\pi$$
0.524080 + 0.851669i $$0.324409\pi$$
$$564$$ 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 $$-0.293672\pi$$
0.603751 + 0.797173i $$0.293672\pi$$
$$578$$ −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 $$-0.584170\pi$$
−0.261358 + 0.965242i $$0.584170\pi$$
$$588$$ 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 $$-0.146593\pi$$
0.895814 + 0.444429i $$0.146593\pi$$
$$594$$ 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 $$-0.577538\pi$$
−0.241192 + 0.970477i $$0.577538\pi$$
$$608$$ −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 $$-0.550845\pi$$
−0.159055 + 0.987270i $$0.550845\pi$$
$$614$$ 10812.0 0.710646
$$615$$ 0 0
$$616$$ −5824.00 −0.380934
$$617$$ 27656.0 1.80452 0.902260 0.431193i $$-0.141907\pi$$
0.902260 + 0.431193i $$0.141907\pi$$
$$618$$ 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 $$-0.581294\pi$$
−0.252624 + 0.967564i $$0.581294\pi$$
$$644$$ −6032.00 −0.369090
$$645$$ 0 0
$$646$$ 7680.00 0.467749
$$647$$ 6426.00 0.390467 0.195233 0.980757i $$-0.437454\pi$$
0.195233 + 0.980757i $$0.437454\pi$$
$$648$$ 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 $$-0.556648\pi$$
−0.177027 + 0.984206i $$0.556648\pi$$
$$654$$ 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 $$-0.809478\pi$$
−0.826158 + 0.563439i $$0.809478\pi$$
$$674$$ 1312.00 0.0749798
$$675$$ 0 0
$$676$$ −8212.00 −0.467228
$$677$$ −26884.0 −1.52620 −0.763099 0.646282i $$-0.776323\pi$$
−0.763099 + 0.646282i $$0.776323\pi$$
$$678$$ −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 $$-0.368988\pi$$
0.400063 + 0.916488i $$0.368988\pi$$
$$684$$ 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 $$-0.355194\pi$$
0.439393 + 0.898295i $$0.355194\pi$$
$$728$$ 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 $$-0.506320\pi$$
−0.0198536 + 0.999803i $$0.506320\pi$$
$$734$$ −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 $$-0.525631\pi$$
−0.0804337 + 0.996760i $$0.525631\pi$$
$$744$$ −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 $$-0.355765\pi$$
0.437780 + 0.899082i $$0.355765\pi$$
$$758$$ 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 $$-0.430944\pi$$
0.215247 + 0.976560i $$0.430944\pi$$
$$774$$ −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 $$-0.458606\pi$$
0.129676 + 0.991556i $$0.458606\pi$$
$$788$$ 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 $$-0.293078\pi$$
0.605238 + 0.796045i $$0.293078\pi$$
$$798$$ −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 $$-0.153113\pi$$
0.886523 + 0.462684i $$0.153113\pi$$
$$824$$ 11696.0 0.494478
$$825$$ 0 0
$$826$$ −1040.00 −0.0438090
$$827$$ −12154.0 −0.511047 −0.255524 0.966803i $$-0.582248\pi$$
−0.255524 + 0.966803i $$0.582248\pi$$
$$828$$ 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 $$-0.406328\pi$$
0.290051 + 0.957011i $$0.406328\pi$$
$$854$$ 28184.0 1.12932
$$855$$ 0 0
$$856$$ 7728.00 0.308572
$$857$$ −22584.0 −0.900181 −0.450090 0.892983i $$-0.648608\pi$$
−0.450090 + 0.892983i $$0.648608\pi$$
$$858$$ −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 $$-0.503126\pi$$
−0.00982162 + 0.999952i $$0.503126\pi$$
$$864$$ −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 $$-0.582066\pi$$
−0.254970 + 0.966949i $$0.582066\pi$$
$$878$$ 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 $$-0.573924\pi$$
−0.230157 + 0.973154i $$0.573924\pi$$
$$884$$ −3072.00 −0.116881
$$885$$ 0 0
$$886$$ −18156.0 −0.688446
$$887$$ −18294.0 −0.692506 −0.346253 0.938141i $$-0.612546\pi$$
−0.346253 + 0.938141i $$0.612546\pi$$
$$888$$ 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 $$-0.364460\pi$$
0.413060 + 0.910704i $$0.364460\pi$$
$$908$$ −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 $$-0.610665\pi$$
−0.340702 + 0.940172i $$0.610665\pi$$
$$938$$ −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 $$-0.248510\pi$$
0.710409 + 0.703789i $$0.248510\pi$$
$$948$$ −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 $$-0.357728\pi$$
0.432226 + 0.901765i $$0.357728\pi$$
$$954$$ 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 $$-0.435485\pi$$
0.201294 + 0.979531i $$0.435485\pi$$
$$968$$ −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 $$-0.433986\pi$$
0.205907 + 0.978572i $$0.433986\pi$$
$$978$$ −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 $$-0.477559\pi$$
0.0704417 + 0.997516i $$0.477559\pi$$
$$984$$ 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 $$-0.248022\pi$$
0.711486 + 0.702700i $$0.248022\pi$$
$$998$$ 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.d.1.1 1
3.2 odd 2 450.4.a.j.1.1 1
4.3 odd 2 400.4.a.h.1.1 1
5.2 odd 4 10.4.b.a.9.2 yes 2
5.3 odd 4 10.4.b.a.9.1 2
5.4 even 2 50.4.a.b.1.1 1
7.6 odd 2 2450.4.a.bb.1.1 1
8.3 odd 2 1600.4.a.bg.1.1 1
8.5 even 2 1600.4.a.u.1.1 1
15.2 even 4 90.4.c.b.19.1 2
15.8 even 4 90.4.c.b.19.2 2
15.14 odd 2 450.4.a.k.1.1 1
20.3 even 4 80.4.c.a.49.1 2
20.7 even 4 80.4.c.a.49.2 2
20.19 odd 2 400.4.a.n.1.1 1
35.13 even 4 490.4.c.b.99.1 2
35.27 even 4 490.4.c.b.99.2 2
35.34 odd 2 2450.4.a.o.1.1 1
40.3 even 4 320.4.c.c.129.2 2
40.13 odd 4 320.4.c.d.129.1 2
40.19 odd 2 1600.4.a.t.1.1 1
40.27 even 4 320.4.c.c.129.1 2
40.29 even 2 1600.4.a.bh.1.1 1
40.37 odd 4 320.4.c.d.129.2 2
60.23 odd 4 720.4.f.f.289.1 2
60.47 odd 4 720.4.f.f.289.2 2

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