# Properties

 Label 98.4.a.g.1.2 Level $98$ Weight $4$ Character 98.1 Self dual yes Analytic conductor $5.782$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [98,4,Mod(1,98)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(98, 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("98.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$98 = 2 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 98.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: $$5.78218718056$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\zeta_{8})^+$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - 2$$ x^2 - 2 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$1.41421$$ of defining polynomial Character $$\chi$$ $$=$$ 98.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-2.00000 q^{2} +7.07107 q^{3} +4.00000 q^{4} +19.7990 q^{5} -14.1421 q^{6} -8.00000 q^{8} +23.0000 q^{9} +O(q^{10})$$ $$q-2.00000 q^{2} +7.07107 q^{3} +4.00000 q^{4} +19.7990 q^{5} -14.1421 q^{6} -8.00000 q^{8} +23.0000 q^{9} -39.5980 q^{10} -14.0000 q^{11} +28.2843 q^{12} -50.9117 q^{13} +140.000 q^{15} +16.0000 q^{16} -1.41421 q^{17} -46.0000 q^{18} +1.41421 q^{19} +79.1960 q^{20} +28.0000 q^{22} +140.000 q^{23} -56.5685 q^{24} +267.000 q^{25} +101.823 q^{26} -28.2843 q^{27} -286.000 q^{29} -280.000 q^{30} +93.3381 q^{31} -32.0000 q^{32} -98.9949 q^{33} +2.82843 q^{34} +92.0000 q^{36} -38.0000 q^{37} -2.82843 q^{38} -360.000 q^{39} -158.392 q^{40} +125.865 q^{41} -34.0000 q^{43} -56.0000 q^{44} +455.377 q^{45} -280.000 q^{46} -523.259 q^{47} +113.137 q^{48} -534.000 q^{50} -10.0000 q^{51} -203.647 q^{52} -74.0000 q^{53} +56.5685 q^{54} -277.186 q^{55} +10.0000 q^{57} +572.000 q^{58} -434.164 q^{59} +560.000 q^{60} -14.1421 q^{61} -186.676 q^{62} +64.0000 q^{64} -1008.00 q^{65} +197.990 q^{66} +684.000 q^{67} -5.65685 q^{68} +989.949 q^{69} +588.000 q^{71} -184.000 q^{72} +270.115 q^{73} +76.0000 q^{74} +1887.98 q^{75} +5.65685 q^{76} +720.000 q^{78} +1220.00 q^{79} +316.784 q^{80} -821.000 q^{81} -251.730 q^{82} -422.850 q^{83} -28.0000 q^{85} +68.0000 q^{86} -2022.33 q^{87} +112.000 q^{88} -618.011 q^{89} -910.754 q^{90} +560.000 q^{92} +660.000 q^{93} +1046.52 q^{94} +28.0000 q^{95} -226.274 q^{96} -1483.51 q^{97} -322.000 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 4 q^{2} + 8 q^{4} - 16 q^{8} + 46 q^{9}+O(q^{10})$$ 2 * q - 4 * q^2 + 8 * q^4 - 16 * q^8 + 46 * q^9 $$2 q - 4 q^{2} + 8 q^{4} - 16 q^{8} + 46 q^{9} - 28 q^{11} + 280 q^{15} + 32 q^{16} - 92 q^{18} + 56 q^{22} + 280 q^{23} + 534 q^{25} - 572 q^{29} - 560 q^{30} - 64 q^{32} + 184 q^{36} - 76 q^{37} - 720 q^{39} - 68 q^{43} - 112 q^{44} - 560 q^{46} - 1068 q^{50} - 20 q^{51} - 148 q^{53} + 20 q^{57} + 1144 q^{58} + 1120 q^{60} + 128 q^{64} - 2016 q^{65} + 1368 q^{67} + 1176 q^{71} - 368 q^{72} + 152 q^{74} + 1440 q^{78} + 2440 q^{79} - 1642 q^{81} - 56 q^{85} + 136 q^{86} + 224 q^{88} + 1120 q^{92} + 1320 q^{93} + 56 q^{95} - 644 q^{99}+O(q^{100})$$ 2 * q - 4 * q^2 + 8 * q^4 - 16 * q^8 + 46 * q^9 - 28 * q^11 + 280 * q^15 + 32 * q^16 - 92 * q^18 + 56 * q^22 + 280 * q^23 + 534 * q^25 - 572 * q^29 - 560 * q^30 - 64 * q^32 + 184 * q^36 - 76 * q^37 - 720 * q^39 - 68 * q^43 - 112 * q^44 - 560 * q^46 - 1068 * q^50 - 20 * q^51 - 148 * q^53 + 20 * q^57 + 1144 * q^58 + 1120 * q^60 + 128 * q^64 - 2016 * q^65 + 1368 * q^67 + 1176 * q^71 - 368 * q^72 + 152 * q^74 + 1440 * q^78 + 2440 * q^79 - 1642 * q^81 - 56 * q^85 + 136 * q^86 + 224 * q^88 + 1120 * q^92 + 1320 * q^93 + 56 * q^95 - 644 * q^99

## 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
$$3$$ 7.07107 1.36083 0.680414 0.732828i $$-0.261800\pi$$
0.680414 + 0.732828i $$0.261800\pi$$
$$4$$ 4.00000 0.500000
$$5$$ 19.7990 1.77088 0.885438 0.464758i $$-0.153859\pi$$
0.885438 + 0.464758i $$0.153859\pi$$
$$6$$ −14.1421 −0.962250
$$7$$ 0 0
$$8$$ −8.00000 −0.353553
$$9$$ 23.0000 0.851852
$$10$$ −39.5980 −1.25220
$$11$$ −14.0000 −0.383742 −0.191871 0.981420i $$-0.561455\pi$$
−0.191871 + 0.981420i $$0.561455\pi$$
$$12$$ 28.2843 0.680414
$$13$$ −50.9117 −1.08618 −0.543091 0.839674i $$-0.682746\pi$$
−0.543091 + 0.839674i $$0.682746\pi$$
$$14$$ 0 0
$$15$$ 140.000 2.40986
$$16$$ 16.0000 0.250000
$$17$$ −1.41421 −0.0201763 −0.0100882 0.999949i $$-0.503211\pi$$
−0.0100882 + 0.999949i $$0.503211\pi$$
$$18$$ −46.0000 −0.602350
$$19$$ 1.41421 0.0170759 0.00853797 0.999964i $$-0.497282\pi$$
0.00853797 + 0.999964i $$0.497282\pi$$
$$20$$ 79.1960 0.885438
$$21$$ 0 0
$$22$$ 28.0000 0.271346
$$23$$ 140.000 1.26922 0.634609 0.772833i $$-0.281161\pi$$
0.634609 + 0.772833i $$0.281161\pi$$
$$24$$ −56.5685 −0.481125
$$25$$ 267.000 2.13600
$$26$$ 101.823 0.768046
$$27$$ −28.2843 −0.201604
$$28$$ 0 0
$$29$$ −286.000 −1.83134 −0.915670 0.401931i $$-0.868339\pi$$
−0.915670 + 0.401931i $$0.868339\pi$$
$$30$$ −280.000 −1.70403
$$31$$ 93.3381 0.540775 0.270387 0.962752i $$-0.412848\pi$$
0.270387 + 0.962752i $$0.412848\pi$$
$$32$$ −32.0000 −0.176777
$$33$$ −98.9949 −0.522206
$$34$$ 2.82843 0.0142668
$$35$$ 0 0
$$36$$ 92.0000 0.425926
$$37$$ −38.0000 −0.168842 −0.0844211 0.996430i $$-0.526904\pi$$
−0.0844211 + 0.996430i $$0.526904\pi$$
$$38$$ −2.82843 −0.0120745
$$39$$ −360.000 −1.47811
$$40$$ −158.392 −0.626099
$$41$$ 125.865 0.479434 0.239717 0.970843i $$-0.422945\pi$$
0.239717 + 0.970843i $$0.422945\pi$$
$$42$$ 0 0
$$43$$ −34.0000 −0.120580 −0.0602901 0.998181i $$-0.519203\pi$$
−0.0602901 + 0.998181i $$0.519203\pi$$
$$44$$ −56.0000 −0.191871
$$45$$ 455.377 1.50852
$$46$$ −280.000 −0.897473
$$47$$ −523.259 −1.62394 −0.811970 0.583699i $$-0.801605\pi$$
−0.811970 + 0.583699i $$0.801605\pi$$
$$48$$ 113.137 0.340207
$$49$$ 0 0
$$50$$ −534.000 −1.51038
$$51$$ −10.0000 −0.0274565
$$52$$ −203.647 −0.543091
$$53$$ −74.0000 −0.191786 −0.0958932 0.995392i $$-0.530571\pi$$
−0.0958932 + 0.995392i $$0.530571\pi$$
$$54$$ 56.5685 0.142556
$$55$$ −277.186 −0.679559
$$56$$ 0 0
$$57$$ 10.0000 0.0232374
$$58$$ 572.000 1.29495
$$59$$ −434.164 −0.958022 −0.479011 0.877809i $$-0.659005\pi$$
−0.479011 + 0.877809i $$0.659005\pi$$
$$60$$ 560.000 1.20493
$$61$$ −14.1421 −0.0296839 −0.0148419 0.999890i $$-0.504725\pi$$
−0.0148419 + 0.999890i $$0.504725\pi$$
$$62$$ −186.676 −0.382385
$$63$$ 0 0
$$64$$ 64.0000 0.125000
$$65$$ −1008.00 −1.92349
$$66$$ 197.990 0.369256
$$67$$ 684.000 1.24722 0.623611 0.781735i $$-0.285665\pi$$
0.623611 + 0.781735i $$0.285665\pi$$
$$68$$ −5.65685 −0.0100882
$$69$$ 989.949 1.72719
$$70$$ 0 0
$$71$$ 588.000 0.982856 0.491428 0.870918i $$-0.336475\pi$$
0.491428 + 0.870918i $$0.336475\pi$$
$$72$$ −184.000 −0.301175
$$73$$ 270.115 0.433076 0.216538 0.976274i $$-0.430523\pi$$
0.216538 + 0.976274i $$0.430523\pi$$
$$74$$ 76.0000 0.119389
$$75$$ 1887.98 2.90673
$$76$$ 5.65685 0.00853797
$$77$$ 0 0
$$78$$ 720.000 1.04518
$$79$$ 1220.00 1.73748 0.868739 0.495271i $$-0.164931\pi$$
0.868739 + 0.495271i $$0.164931\pi$$
$$80$$ 316.784 0.442719
$$81$$ −821.000 −1.12620
$$82$$ −251.730 −0.339011
$$83$$ −422.850 −0.559202 −0.279601 0.960116i $$-0.590202\pi$$
−0.279601 + 0.960116i $$0.590202\pi$$
$$84$$ 0 0
$$85$$ −28.0000 −0.0357297
$$86$$ 68.0000 0.0852631
$$87$$ −2022.33 −2.49214
$$88$$ 112.000 0.135673
$$89$$ −618.011 −0.736057 −0.368028 0.929815i $$-0.619967\pi$$
−0.368028 + 0.929815i $$0.619967\pi$$
$$90$$ −910.754 −1.06669
$$91$$ 0 0
$$92$$ 560.000 0.634609
$$93$$ 660.000 0.735901
$$94$$ 1046.52 1.14830
$$95$$ 28.0000 0.0302394
$$96$$ −226.274 −0.240563
$$97$$ −1483.51 −1.55286 −0.776431 0.630202i $$-0.782972\pi$$
−0.776431 + 0.630202i $$0.782972\pi$$
$$98$$ 0 0
$$99$$ −322.000 −0.326891
$$100$$ 1068.00 1.06800
$$101$$ −1128.54 −1.11182 −0.555912 0.831241i $$-0.687631\pi$$
−0.555912 + 0.831241i $$0.687631\pi$$
$$102$$ 20.0000 0.0194147
$$103$$ 868.327 0.830668 0.415334 0.909669i $$-0.363665\pi$$
0.415334 + 0.909669i $$0.363665\pi$$
$$104$$ 407.294 0.384023
$$105$$ 0 0
$$106$$ 148.000 0.135613
$$107$$ −1684.00 −1.52148 −0.760740 0.649056i $$-0.775164\pi$$
−0.760740 + 0.649056i $$0.775164\pi$$
$$108$$ −113.137 −0.100802
$$109$$ −818.000 −0.718809 −0.359405 0.933182i $$-0.617020\pi$$
−0.359405 + 0.933182i $$0.617020\pi$$
$$110$$ 554.372 0.480521
$$111$$ −268.701 −0.229765
$$112$$ 0 0
$$113$$ −540.000 −0.449548 −0.224774 0.974411i $$-0.572164\pi$$
−0.224774 + 0.974411i $$0.572164\pi$$
$$114$$ −20.0000 −0.0164313
$$115$$ 2771.86 2.24763
$$116$$ −1144.00 −0.915670
$$117$$ −1170.97 −0.925266
$$118$$ 868.327 0.677424
$$119$$ 0 0
$$120$$ −1120.00 −0.852013
$$121$$ −1135.00 −0.852742
$$122$$ 28.2843 0.0209897
$$123$$ 890.000 0.652428
$$124$$ 373.352 0.270387
$$125$$ 2811.46 2.01171
$$126$$ 0 0
$$127$$ 1720.00 1.20177 0.600887 0.799334i $$-0.294814\pi$$
0.600887 + 0.799334i $$0.294814\pi$$
$$128$$ −128.000 −0.0883883
$$129$$ −240.416 −0.164089
$$130$$ 2016.00 1.36011
$$131$$ 1735.24 1.15732 0.578659 0.815570i $$-0.303576\pi$$
0.578659 + 0.815570i $$0.303576\pi$$
$$132$$ −395.980 −0.261103
$$133$$ 0 0
$$134$$ −1368.00 −0.881919
$$135$$ −560.000 −0.357016
$$136$$ 11.3137 0.00713340
$$137$$ 828.000 0.516356 0.258178 0.966097i $$-0.416878\pi$$
0.258178 + 0.966097i $$0.416878\pi$$
$$138$$ −1979.90 −1.22131
$$139$$ 425.678 0.259752 0.129876 0.991530i $$-0.458542\pi$$
0.129876 + 0.991530i $$0.458542\pi$$
$$140$$ 0 0
$$141$$ −3700.00 −2.20990
$$142$$ −1176.00 −0.694984
$$143$$ 712.764 0.416813
$$144$$ 368.000 0.212963
$$145$$ −5662.51 −3.24308
$$146$$ −540.230 −0.306231
$$147$$ 0 0
$$148$$ −152.000 −0.0844211
$$149$$ 2050.00 1.12713 0.563566 0.826071i $$-0.309429\pi$$
0.563566 + 0.826071i $$0.309429\pi$$
$$150$$ −3775.95 −2.05537
$$151$$ −472.000 −0.254376 −0.127188 0.991879i $$-0.540595\pi$$
−0.127188 + 0.991879i $$0.540595\pi$$
$$152$$ −11.3137 −0.00603726
$$153$$ −32.5269 −0.0171872
$$154$$ 0 0
$$155$$ 1848.00 0.957645
$$156$$ −1440.00 −0.739053
$$157$$ 2211.83 1.12435 0.562176 0.827018i $$-0.309964\pi$$
0.562176 + 0.827018i $$0.309964\pi$$
$$158$$ −2440.00 −1.22858
$$159$$ −523.259 −0.260988
$$160$$ −633.568 −0.313050
$$161$$ 0 0
$$162$$ 1642.00 0.796344
$$163$$ 3286.00 1.57901 0.789507 0.613741i $$-0.210336\pi$$
0.789507 + 0.613741i $$0.210336\pi$$
$$164$$ 503.460 0.239717
$$165$$ −1960.00 −0.924762
$$166$$ 845.700 0.395416
$$167$$ 1490.58 0.690686 0.345343 0.938476i $$-0.387763\pi$$
0.345343 + 0.938476i $$0.387763\pi$$
$$168$$ 0 0
$$169$$ 395.000 0.179791
$$170$$ 56.0000 0.0252647
$$171$$ 32.5269 0.0145462
$$172$$ −136.000 −0.0602901
$$173$$ 2070.41 0.909886 0.454943 0.890521i $$-0.349660\pi$$
0.454943 + 0.890521i $$0.349660\pi$$
$$174$$ 4044.65 1.76221
$$175$$ 0 0
$$176$$ −224.000 −0.0959354
$$177$$ −3070.00 −1.30370
$$178$$ 1236.02 0.520471
$$179$$ 540.000 0.225483 0.112742 0.993624i $$-0.464037\pi$$
0.112742 + 0.993624i $$0.464037\pi$$
$$180$$ 1821.51 0.754262
$$181$$ −3784.44 −1.55412 −0.777058 0.629429i $$-0.783289\pi$$
−0.777058 + 0.629429i $$0.783289\pi$$
$$182$$ 0 0
$$183$$ −100.000 −0.0403946
$$184$$ −1120.00 −0.448736
$$185$$ −752.362 −0.298999
$$186$$ −1320.00 −0.520361
$$187$$ 19.7990 0.00774249
$$188$$ −2093.04 −0.811970
$$189$$ 0 0
$$190$$ −56.0000 −0.0213825
$$191$$ 1028.00 0.389442 0.194721 0.980859i $$-0.437620\pi$$
0.194721 + 0.980859i $$0.437620\pi$$
$$192$$ 452.548 0.170103
$$193$$ 4592.00 1.71264 0.856320 0.516446i $$-0.172745\pi$$
0.856320 + 0.516446i $$0.172745\pi$$
$$194$$ 2967.02 1.09804
$$195$$ −7127.64 −2.61754
$$196$$ 0 0
$$197$$ 794.000 0.287158 0.143579 0.989639i $$-0.454139\pi$$
0.143579 + 0.989639i $$0.454139\pi$$
$$198$$ 644.000 0.231147
$$199$$ −2486.19 −0.885634 −0.442817 0.896612i $$-0.646021\pi$$
−0.442817 + 0.896612i $$0.646021\pi$$
$$200$$ −2136.00 −0.755190
$$201$$ 4836.61 1.69725
$$202$$ 2257.08 0.786178
$$203$$ 0 0
$$204$$ −40.0000 −0.0137282
$$205$$ 2492.00 0.849019
$$206$$ −1736.65 −0.587371
$$207$$ 3220.00 1.08119
$$208$$ −814.587 −0.271545
$$209$$ −19.7990 −0.00655275
$$210$$ 0 0
$$211$$ −2748.00 −0.896588 −0.448294 0.893886i $$-0.647968\pi$$
−0.448294 + 0.893886i $$0.647968\pi$$
$$212$$ −296.000 −0.0958932
$$213$$ 4157.79 1.33750
$$214$$ 3368.00 1.07585
$$215$$ −673.166 −0.213533
$$216$$ 226.274 0.0712778
$$217$$ 0 0
$$218$$ 1636.00 0.508275
$$219$$ 1910.00 0.589342
$$220$$ −1108.74 −0.339779
$$221$$ 72.0000 0.0219151
$$222$$ 537.401 0.162468
$$223$$ 3428.05 1.02941 0.514707 0.857366i $$-0.327901\pi$$
0.514707 + 0.857366i $$0.327901\pi$$
$$224$$ 0 0
$$225$$ 6141.00 1.81956
$$226$$ 1080.00 0.317878
$$227$$ −5290.57 −1.54691 −0.773453 0.633854i $$-0.781472\pi$$
−0.773453 + 0.633854i $$0.781472\pi$$
$$228$$ 40.0000 0.0116187
$$229$$ 2749.23 0.793338 0.396669 0.917962i $$-0.370166\pi$$
0.396669 + 0.917962i $$0.370166\pi$$
$$230$$ −5543.72 −1.58931
$$231$$ 0 0
$$232$$ 2288.00 0.647477
$$233$$ 72.0000 0.0202441 0.0101221 0.999949i $$-0.496778\pi$$
0.0101221 + 0.999949i $$0.496778\pi$$
$$234$$ 2341.94 0.654262
$$235$$ −10360.0 −2.87580
$$236$$ −1736.65 −0.479011
$$237$$ 8626.70 2.36441
$$238$$ 0 0
$$239$$ 4308.00 1.16595 0.582974 0.812491i $$-0.301889\pi$$
0.582974 + 0.812491i $$0.301889\pi$$
$$240$$ 2240.00 0.602464
$$241$$ 1540.08 0.411640 0.205820 0.978590i $$-0.434014\pi$$
0.205820 + 0.978590i $$0.434014\pi$$
$$242$$ 2270.00 0.602980
$$243$$ −5041.67 −1.33096
$$244$$ −56.5685 −0.0148419
$$245$$ 0 0
$$246$$ −1780.00 −0.461336
$$247$$ −72.0000 −0.0185476
$$248$$ −746.705 −0.191193
$$249$$ −2990.00 −0.760978
$$250$$ −5622.91 −1.42250
$$251$$ −931.967 −0.234363 −0.117182 0.993110i $$-0.537386\pi$$
−0.117182 + 0.993110i $$0.537386\pi$$
$$252$$ 0 0
$$253$$ −1960.00 −0.487052
$$254$$ −3440.00 −0.849783
$$255$$ −197.990 −0.0486220
$$256$$ 256.000 0.0625000
$$257$$ −937.624 −0.227577 −0.113789 0.993505i $$-0.536299\pi$$
−0.113789 + 0.993505i $$0.536299\pi$$
$$258$$ 480.833 0.116028
$$259$$ 0 0
$$260$$ −4032.00 −0.961746
$$261$$ −6578.00 −1.56003
$$262$$ −3470.48 −0.818347
$$263$$ 7140.00 1.67404 0.837018 0.547176i $$-0.184297\pi$$
0.837018 + 0.547176i $$0.184297\pi$$
$$264$$ 791.960 0.184628
$$265$$ −1465.13 −0.339630
$$266$$ 0 0
$$267$$ −4370.00 −1.00165
$$268$$ 2736.00 0.623611
$$269$$ −4610.34 −1.04497 −0.522485 0.852648i $$-0.674995\pi$$
−0.522485 + 0.852648i $$0.674995\pi$$
$$270$$ 1120.00 0.252448
$$271$$ 2364.57 0.530026 0.265013 0.964245i $$-0.414624\pi$$
0.265013 + 0.964245i $$0.414624\pi$$
$$272$$ −22.6274 −0.00504408
$$273$$ 0 0
$$274$$ −1656.00 −0.365119
$$275$$ −3738.00 −0.819672
$$276$$ 3959.80 0.863594
$$277$$ 4006.00 0.868943 0.434472 0.900686i $$-0.356935\pi$$
0.434472 + 0.900686i $$0.356935\pi$$
$$278$$ −851.357 −0.183673
$$279$$ 2146.78 0.460660
$$280$$ 0 0
$$281$$ −5984.00 −1.27038 −0.635188 0.772358i $$-0.719077\pi$$
−0.635188 + 0.772358i $$0.719077\pi$$
$$282$$ 7400.00 1.56264
$$283$$ 4928.53 1.03523 0.517617 0.855613i $$-0.326819\pi$$
0.517617 + 0.855613i $$0.326819\pi$$
$$284$$ 2352.00 0.491428
$$285$$ 197.990 0.0411506
$$286$$ −1425.53 −0.294731
$$287$$ 0 0
$$288$$ −736.000 −0.150588
$$289$$ −4911.00 −0.999593
$$290$$ 11325.0 2.29320
$$291$$ −10490.0 −2.11318
$$292$$ 1080.46 0.216538
$$293$$ −1971.41 −0.393076 −0.196538 0.980496i $$-0.562970\pi$$
−0.196538 + 0.980496i $$0.562970\pi$$
$$294$$ 0 0
$$295$$ −8596.00 −1.69654
$$296$$ 304.000 0.0596947
$$297$$ 395.980 0.0773639
$$298$$ −4100.00 −0.797002
$$299$$ −7127.64 −1.37860
$$300$$ 7551.90 1.45336
$$301$$ 0 0
$$302$$ 944.000 0.179871
$$303$$ −7980.00 −1.51300
$$304$$ 22.6274 0.00426898
$$305$$ −280.000 −0.0525664
$$306$$ 65.0538 0.0121532
$$307$$ 4767.31 0.886270 0.443135 0.896455i $$-0.353866\pi$$
0.443135 + 0.896455i $$0.353866\pi$$
$$308$$ 0 0
$$309$$ 6140.00 1.13040
$$310$$ −3696.00 −0.677157
$$311$$ 6776.91 1.23564 0.617819 0.786320i $$-0.288016\pi$$
0.617819 + 0.786320i $$0.288016\pi$$
$$312$$ 2880.00 0.522589
$$313$$ 6190.01 1.11783 0.558914 0.829226i $$-0.311218\pi$$
0.558914 + 0.829226i $$0.311218\pi$$
$$314$$ −4423.66 −0.795037
$$315$$ 0 0
$$316$$ 4880.00 0.868739
$$317$$ 9826.00 1.74096 0.870478 0.492207i $$-0.163810\pi$$
0.870478 + 0.492207i $$0.163810\pi$$
$$318$$ 1046.52 0.184547
$$319$$ 4004.00 0.702762
$$320$$ 1267.14 0.221359
$$321$$ −11907.7 −2.07047
$$322$$ 0 0
$$323$$ −2.00000 −0.000344529 0
$$324$$ −3284.00 −0.563100
$$325$$ −13593.4 −2.32008
$$326$$ −6572.00 −1.11653
$$327$$ −5784.13 −0.978175
$$328$$ −1006.92 −0.169506
$$329$$ 0 0
$$330$$ 3920.00 0.653906
$$331$$ −5738.00 −0.952837 −0.476418 0.879219i $$-0.658065\pi$$
−0.476418 + 0.879219i $$0.658065\pi$$
$$332$$ −1691.40 −0.279601
$$333$$ −874.000 −0.143829
$$334$$ −2981.16 −0.488389
$$335$$ 13542.5 2.20868
$$336$$ 0 0
$$337$$ −2254.00 −0.364342 −0.182171 0.983267i $$-0.558312\pi$$
−0.182171 + 0.983267i $$0.558312\pi$$
$$338$$ −790.000 −0.127131
$$339$$ −3818.38 −0.611757
$$340$$ −112.000 −0.0178649
$$341$$ −1306.73 −0.207518
$$342$$ −65.0538 −0.0102857
$$343$$ 0 0
$$344$$ 272.000 0.0426316
$$345$$ 19600.0 3.05863
$$346$$ −4140.82 −0.643386
$$347$$ −1986.00 −0.307245 −0.153623 0.988130i $$-0.549094\pi$$
−0.153623 + 0.988130i $$0.549094\pi$$
$$348$$ −8089.30 −1.24607
$$349$$ −6771.25 −1.03856 −0.519279 0.854605i $$-0.673800\pi$$
−0.519279 + 0.854605i $$0.673800\pi$$
$$350$$ 0 0
$$351$$ 1440.00 0.218979
$$352$$ 448.000 0.0678366
$$353$$ −6993.29 −1.05443 −0.527217 0.849731i $$-0.676764\pi$$
−0.527217 + 0.849731i $$0.676764\pi$$
$$354$$ 6140.00 0.921857
$$355$$ 11641.8 1.74052
$$356$$ −2472.05 −0.368028
$$357$$ 0 0
$$358$$ −1080.00 −0.159441
$$359$$ 5944.00 0.873850 0.436925 0.899498i $$-0.356067\pi$$
0.436925 + 0.899498i $$0.356067\pi$$
$$360$$ −3643.01 −0.533344
$$361$$ −6857.00 −0.999708
$$362$$ 7568.87 1.09893
$$363$$ −8025.66 −1.16044
$$364$$ 0 0
$$365$$ 5348.00 0.766924
$$366$$ 200.000 0.0285633
$$367$$ 842.871 0.119884 0.0599421 0.998202i $$-0.480908\pi$$
0.0599421 + 0.998202i $$0.480908\pi$$
$$368$$ 2240.00 0.317305
$$369$$ 2894.90 0.408407
$$370$$ 1504.72 0.211424
$$371$$ 0 0
$$372$$ 2640.00 0.367951
$$373$$ −5726.00 −0.794855 −0.397428 0.917634i $$-0.630097\pi$$
−0.397428 + 0.917634i $$0.630097\pi$$
$$374$$ −39.5980 −0.00547477
$$375$$ 19880.0 2.73760
$$376$$ 4186.07 0.574149
$$377$$ 14560.7 1.98917
$$378$$ 0 0
$$379$$ 10330.0 1.40004 0.700022 0.714122i $$-0.253174\pi$$
0.700022 + 0.714122i $$0.253174\pi$$
$$380$$ 112.000 0.0151197
$$381$$ 12162.2 1.63541
$$382$$ −2056.00 −0.275377
$$383$$ −1004.09 −0.133960 −0.0669800 0.997754i $$-0.521336\pi$$
−0.0669800 + 0.997754i $$0.521336\pi$$
$$384$$ −905.097 −0.120281
$$385$$ 0 0
$$386$$ −9184.00 −1.21102
$$387$$ −782.000 −0.102717
$$388$$ −5934.04 −0.776431
$$389$$ 5210.00 0.679068 0.339534 0.940594i $$-0.389731\pi$$
0.339534 + 0.940594i $$0.389731\pi$$
$$390$$ 14255.3 1.85088
$$391$$ −197.990 −0.0256081
$$392$$ 0 0
$$393$$ 12270.0 1.57491
$$394$$ −1588.00 −0.203051
$$395$$ 24154.8 3.07686
$$396$$ −1288.00 −0.163446
$$397$$ 73.5391 0.00929678 0.00464839 0.999989i $$-0.498520\pi$$
0.00464839 + 0.999989i $$0.498520\pi$$
$$398$$ 4972.37 0.626238
$$399$$ 0 0
$$400$$ 4272.00 0.534000
$$401$$ −498.000 −0.0620173 −0.0310086 0.999519i $$-0.509872\pi$$
−0.0310086 + 0.999519i $$0.509872\pi$$
$$402$$ −9673.22 −1.20014
$$403$$ −4752.00 −0.587380
$$404$$ −4514.17 −0.555912
$$405$$ −16255.0 −1.99436
$$406$$ 0 0
$$407$$ 532.000 0.0647918
$$408$$ 80.0000 0.00970733
$$409$$ −3355.93 −0.405721 −0.202861 0.979208i $$-0.565024\pi$$
−0.202861 + 0.979208i $$0.565024\pi$$
$$410$$ −4984.00 −0.600347
$$411$$ 5854.84 0.702672
$$412$$ 3473.31 0.415334
$$413$$ 0 0
$$414$$ −6440.00 −0.764514
$$415$$ −8372.00 −0.990278
$$416$$ 1629.17 0.192012
$$417$$ 3010.00 0.353478
$$418$$ 39.5980 0.00463349
$$419$$ −14545.2 −1.69589 −0.847946 0.530082i $$-0.822161\pi$$
−0.847946 + 0.530082i $$0.822161\pi$$
$$420$$ 0 0
$$421$$ 10854.0 1.25651 0.628256 0.778007i $$-0.283769\pi$$
0.628256 + 0.778007i $$0.283769\pi$$
$$422$$ 5496.00 0.633984
$$423$$ −12035.0 −1.38336
$$424$$ 592.000 0.0678067
$$425$$ −377.595 −0.0430966
$$426$$ −8315.58 −0.945753
$$427$$ 0 0
$$428$$ −6736.00 −0.760740
$$429$$ 5040.00 0.567211
$$430$$ 1346.33 0.150990
$$431$$ −5364.00 −0.599477 −0.299739 0.954021i $$-0.596900\pi$$
−0.299739 + 0.954021i $$0.596900\pi$$
$$432$$ −452.548 −0.0504010
$$433$$ 6487.00 0.719966 0.359983 0.932959i $$-0.382783\pi$$
0.359983 + 0.932959i $$0.382783\pi$$
$$434$$ 0 0
$$435$$ −40040.0 −4.41327
$$436$$ −3272.00 −0.359405
$$437$$ 197.990 0.0216731
$$438$$ −3820.00 −0.416728
$$439$$ −13932.8 −1.51476 −0.757378 0.652977i $$-0.773520\pi$$
−0.757378 + 0.652977i $$0.773520\pi$$
$$440$$ 2217.49 0.240260
$$441$$ 0 0
$$442$$ −144.000 −0.0154963
$$443$$ 5996.00 0.643067 0.321533 0.946898i $$-0.395802\pi$$
0.321533 + 0.946898i $$0.395802\pi$$
$$444$$ −1074.80 −0.114883
$$445$$ −12236.0 −1.30347
$$446$$ −6856.11 −0.727906
$$447$$ 14495.7 1.53383
$$448$$ 0 0
$$449$$ 2622.00 0.275590 0.137795 0.990461i $$-0.455999\pi$$
0.137795 + 0.990461i $$0.455999\pi$$
$$450$$ −12282.0 −1.28662
$$451$$ −1762.11 −0.183979
$$452$$ −2160.00 −0.224774
$$453$$ −3337.54 −0.346162
$$454$$ 10581.1 1.09383
$$455$$ 0 0
$$456$$ −80.0000 −0.00821567
$$457$$ 11208.0 1.14724 0.573619 0.819122i $$-0.305539\pi$$
0.573619 + 0.819122i $$0.305539\pi$$
$$458$$ −5498.46 −0.560974
$$459$$ 40.0000 0.00406763
$$460$$ 11087.4 1.12381
$$461$$ −9786.36 −0.988712 −0.494356 0.869260i $$-0.664596\pi$$
−0.494356 + 0.869260i $$0.664596\pi$$
$$462$$ 0 0
$$463$$ 3952.00 0.396685 0.198342 0.980133i $$-0.436444\pi$$
0.198342 + 0.980133i $$0.436444\pi$$
$$464$$ −4576.00 −0.457835
$$465$$ 13067.3 1.30319
$$466$$ −144.000 −0.0143147
$$467$$ −17506.5 −1.73470 −0.867352 0.497696i $$-0.834180\pi$$
−0.867352 + 0.497696i $$0.834180\pi$$
$$468$$ −4683.88 −0.462633
$$469$$ 0 0
$$470$$ 20720.0 2.03349
$$471$$ 15640.0 1.53005
$$472$$ 3473.31 0.338712
$$473$$ 476.000 0.0462717
$$474$$ −17253.4 −1.67189
$$475$$ 377.595 0.0364742
$$476$$ 0 0
$$477$$ −1702.00 −0.163374
$$478$$ −8616.00 −0.824449
$$479$$ −2288.20 −0.218268 −0.109134 0.994027i $$-0.534808\pi$$
−0.109134 + 0.994027i $$0.534808\pi$$
$$480$$ −4480.00 −0.426006
$$481$$ 1934.64 0.183393
$$482$$ −3080.16 −0.291073
$$483$$ 0 0
$$484$$ −4540.00 −0.426371
$$485$$ −29372.0 −2.74993
$$486$$ 10083.3 0.941131
$$487$$ 972.000 0.0904426 0.0452213 0.998977i $$-0.485601\pi$$
0.0452213 + 0.998977i $$0.485601\pi$$
$$488$$ 113.137 0.0104948
$$489$$ 23235.5 2.14877
$$490$$ 0 0
$$491$$ −7404.00 −0.680525 −0.340263 0.940330i $$-0.610516\pi$$
−0.340263 + 0.940330i $$0.610516\pi$$
$$492$$ 3560.00 0.326214
$$493$$ 404.465 0.0369497
$$494$$ 144.000 0.0131151
$$495$$ −6375.27 −0.578883
$$496$$ 1493.41 0.135194
$$497$$ 0 0
$$498$$ 5980.00 0.538093
$$499$$ −12244.0 −1.09843 −0.549215 0.835681i $$-0.685073\pi$$
−0.549215 + 0.835681i $$0.685073\pi$$
$$500$$ 11245.8 1.00586
$$501$$ 10540.0 0.939905
$$502$$ 1863.93 0.165720
$$503$$ 2415.48 0.214117 0.107058 0.994253i $$-0.465857\pi$$
0.107058 + 0.994253i $$0.465857\pi$$
$$504$$ 0 0
$$505$$ −22344.0 −1.96890
$$506$$ 3920.00 0.344398
$$507$$ 2793.07 0.244664
$$508$$ 6880.00 0.600887
$$509$$ 5707.77 0.497038 0.248519 0.968627i $$-0.420056\pi$$
0.248519 + 0.968627i $$0.420056\pi$$
$$510$$ 395.980 0.0343809
$$511$$ 0 0
$$512$$ −512.000 −0.0441942
$$513$$ −40.0000 −0.00344258
$$514$$ 1875.25 0.160921
$$515$$ 17192.0 1.47101
$$516$$ −961.665 −0.0820445
$$517$$ 7325.63 0.623173
$$518$$ 0 0
$$519$$ 14640.0 1.23820
$$520$$ 8064.00 0.680057
$$521$$ −1.41421 −0.000118921 0 −5.94605e−5 1.00000i $$-0.500019\pi$$
−5.94605e−5 1.00000i $$0.500019\pi$$
$$522$$ 13156.0 1.10311
$$523$$ 12257.0 1.02478 0.512391 0.858752i $$-0.328760\pi$$
0.512391 + 0.858752i $$0.328760\pi$$
$$524$$ 6940.96 0.578659
$$525$$ 0 0
$$526$$ −14280.0 −1.18372
$$527$$ −132.000 −0.0109108
$$528$$ −1583.92 −0.130552
$$529$$ 7433.00 0.610915
$$530$$ 2930.25 0.240155
$$531$$ −9985.76 −0.816093
$$532$$ 0 0
$$533$$ −6408.00 −0.520753
$$534$$ 8740.00 0.708271
$$535$$ −33341.5 −2.69435
$$536$$ −5472.00 −0.440960
$$537$$ 3818.38 0.306844
$$538$$ 9220.67 0.738906
$$539$$ 0 0
$$540$$ −2240.00 −0.178508
$$541$$ 2050.00 0.162914 0.0814569 0.996677i $$-0.474043\pi$$
0.0814569 + 0.996677i $$0.474043\pi$$
$$542$$ −4729.13 −0.374785
$$543$$ −26760.0 −2.11488
$$544$$ 45.2548 0.00356670
$$545$$ −16195.6 −1.27292
$$546$$ 0 0
$$547$$ 14554.0 1.13763 0.568815 0.822465i $$-0.307402\pi$$
0.568815 + 0.822465i $$0.307402\pi$$
$$548$$ 3312.00 0.258178
$$549$$ −325.269 −0.0252862
$$550$$ 7476.00 0.579596
$$551$$ −404.465 −0.0312719
$$552$$ −7919.60 −0.610653
$$553$$ 0 0
$$554$$ −8012.00 −0.614435
$$555$$ −5320.00 −0.406885
$$556$$ 1702.71 0.129876
$$557$$ 6954.00 0.528995 0.264498 0.964386i $$-0.414794\pi$$
0.264498 + 0.964386i $$0.414794\pi$$
$$558$$ −4293.55 −0.325736
$$559$$ 1731.00 0.130972
$$560$$ 0 0
$$561$$ 140.000 0.0105362
$$562$$ 11968.0 0.898291
$$563$$ 1636.25 0.122486 0.0612429 0.998123i $$-0.480494\pi$$
0.0612429 + 0.998123i $$0.480494\pi$$
$$564$$ −14800.0 −1.10495
$$565$$ −10691.5 −0.796094
$$566$$ −9857.07 −0.732020
$$567$$ 0 0
$$568$$ −4704.00 −0.347492
$$569$$ −7142.00 −0.526201 −0.263100 0.964768i $$-0.584745\pi$$
−0.263100 + 0.964768i $$0.584745\pi$$
$$570$$ −395.980 −0.0290978
$$571$$ −20606.0 −1.51022 −0.755109 0.655599i $$-0.772416\pi$$
−0.755109 + 0.655599i $$0.772416\pi$$
$$572$$ 2851.05 0.208407
$$573$$ 7269.06 0.529964
$$574$$ 0 0
$$575$$ 37380.0 2.71105
$$576$$ 1472.00 0.106481
$$577$$ 8803.48 0.635171 0.317585 0.948230i $$-0.397128\pi$$
0.317585 + 0.948230i $$0.397128\pi$$
$$578$$ 9822.00 0.706819
$$579$$ 32470.3 2.33061
$$580$$ −22650.0 −1.62154
$$581$$ 0 0
$$582$$ 20980.0 1.49424
$$583$$ 1036.00 0.0735965
$$584$$ −2160.92 −0.153115
$$585$$ −23184.0 −1.63853
$$586$$ 3942.83 0.277947
$$587$$ −6503.97 −0.457321 −0.228661 0.973506i $$-0.573435\pi$$
−0.228661 + 0.973506i $$0.573435\pi$$
$$588$$ 0 0
$$589$$ 132.000 0.00923424
$$590$$ 17192.0 1.19963
$$591$$ 5614.43 0.390773
$$592$$ −608.000 −0.0422106
$$593$$ 23140.8 1.60249 0.801246 0.598335i $$-0.204171\pi$$
0.801246 + 0.598335i $$0.204171\pi$$
$$594$$ −791.960 −0.0547045
$$595$$ 0 0
$$596$$ 8200.00 0.563566
$$597$$ −17580.0 −1.20520
$$598$$ 14255.3 0.974818
$$599$$ −11296.0 −0.770521 −0.385260 0.922808i $$-0.625888\pi$$
−0.385260 + 0.922808i $$0.625888\pi$$
$$600$$ −15103.8 −1.02768
$$601$$ −8727.11 −0.592323 −0.296162 0.955138i $$-0.595707\pi$$
−0.296162 + 0.955138i $$0.595707\pi$$
$$602$$ 0 0
$$603$$ 15732.0 1.06245
$$604$$ −1888.00 −0.127188
$$605$$ −22471.9 −1.51010
$$606$$ 15960.0 1.06985
$$607$$ −19736.8 −1.31975 −0.659877 0.751374i $$-0.729392\pi$$
−0.659877 + 0.751374i $$0.729392\pi$$
$$608$$ −45.2548 −0.00301863
$$609$$ 0 0
$$610$$ 560.000 0.0371701
$$611$$ 26640.0 1.76389
$$612$$ −130.108 −0.00859361
$$613$$ 16962.0 1.11760 0.558800 0.829302i $$-0.311262\pi$$
0.558800 + 0.829302i $$0.311262\pi$$
$$614$$ −9534.63 −0.626688
$$615$$ 17621.1 1.15537
$$616$$ 0 0
$$617$$ −19034.0 −1.24194 −0.620972 0.783832i $$-0.713262\pi$$
−0.620972 + 0.783832i $$0.713262\pi$$
$$618$$ −12280.0 −0.799311
$$619$$ 18677.5 1.21278 0.606392 0.795166i $$-0.292616\pi$$
0.606392 + 0.795166i $$0.292616\pi$$
$$620$$ 7392.00 0.478822
$$621$$ −3959.80 −0.255880
$$622$$ −13553.8 −0.873728
$$623$$ 0 0
$$624$$ −5760.00 −0.369527
$$625$$ 22289.0 1.42650
$$626$$ −12380.0 −0.790424
$$627$$ −140.000 −0.00891716
$$628$$ 8847.32 0.562176
$$629$$ 53.7401 0.00340661
$$630$$ 0 0
$$631$$ −14716.0 −0.928423 −0.464211 0.885724i $$-0.653662\pi$$
−0.464211 + 0.885724i $$0.653662\pi$$
$$632$$ −9760.00 −0.614291
$$633$$ −19431.3 −1.22010
$$634$$ −19652.0 −1.23104
$$635$$ 34054.3 2.12819
$$636$$ −2093.04 −0.130494
$$637$$ 0 0
$$638$$ −8008.00 −0.496928
$$639$$ 13524.0 0.837248
$$640$$ −2534.27 −0.156525
$$641$$ 4730.00 0.291457 0.145728 0.989325i $$-0.453447\pi$$
0.145728 + 0.989325i $$0.453447\pi$$
$$642$$ 23815.4 1.46405
$$643$$ −19056.5 −1.16877 −0.584383 0.811478i $$-0.698663\pi$$
−0.584383 + 0.811478i $$0.698663\pi$$
$$644$$ 0 0
$$645$$ −4760.00 −0.290581
$$646$$ 4.00000 0.000243619 0
$$647$$ −9342.29 −0.567672 −0.283836 0.958873i $$-0.591607\pi$$
−0.283836 + 0.958873i $$0.591607\pi$$
$$648$$ 6568.00 0.398172
$$649$$ 6078.29 0.367633
$$650$$ 27186.8 1.64055
$$651$$ 0 0
$$652$$ 13144.0 0.789507
$$653$$ 3774.00 0.226169 0.113084 0.993585i $$-0.463927\pi$$
0.113084 + 0.993585i $$0.463927\pi$$
$$654$$ 11568.3 0.691674
$$655$$ 34356.0 2.04947
$$656$$ 2013.84 0.119859
$$657$$ 6212.64 0.368917
$$658$$ 0 0
$$659$$ −21150.0 −1.25021 −0.625104 0.780541i $$-0.714943\pi$$
−0.625104 + 0.780541i $$0.714943\pi$$
$$660$$ −7840.00 −0.462381
$$661$$ 10377.5 0.610647 0.305324 0.952249i $$-0.401235\pi$$
0.305324 + 0.952249i $$0.401235\pi$$
$$662$$ 11476.0 0.673757
$$663$$ 509.117 0.0298227
$$664$$ 3382.80 0.197708
$$665$$ 0 0
$$666$$ 1748.00 0.101702
$$667$$ −40040.0 −2.32437
$$668$$ 5962.32 0.345343
$$669$$ 24240.0 1.40086
$$670$$ −27085.0 −1.56177
$$671$$ 197.990 0.0113909
$$672$$ 0 0
$$673$$ −1164.00 −0.0666700 −0.0333350 0.999444i $$-0.510613\pi$$
−0.0333350 + 0.999444i $$0.510613\pi$$
$$674$$ 4508.00 0.257629
$$675$$ −7551.90 −0.430626
$$676$$ 1580.00 0.0898953
$$677$$ −27152.9 −1.54146 −0.770732 0.637160i $$-0.780109\pi$$
−0.770732 + 0.637160i $$0.780109\pi$$
$$678$$ 7636.75 0.432578
$$679$$ 0 0
$$680$$ 224.000 0.0126324
$$681$$ −37410.0 −2.10507
$$682$$ 2613.47 0.146737
$$683$$ −16596.0 −0.929763 −0.464882 0.885373i $$-0.653903\pi$$
−0.464882 + 0.885373i $$0.653903\pi$$
$$684$$ 130.108 0.00727309
$$685$$ 16393.6 0.914403
$$686$$ 0 0
$$687$$ 19440.0 1.07960
$$688$$ −544.000 −0.0301451
$$689$$ 3767.46 0.208315
$$690$$ −39200.0 −2.16278
$$691$$ −11298.2 −0.622000 −0.311000 0.950410i $$-0.600664\pi$$
−0.311000 + 0.950410i $$0.600664\pi$$
$$692$$ 8281.63 0.454943
$$693$$ 0 0
$$694$$ 3972.00 0.217255
$$695$$ 8428.00 0.459989
$$696$$ 16178.6 0.881104
$$697$$ −178.000 −0.00967321
$$698$$ 13542.5 0.734372
$$699$$ 509.117 0.0275487
$$700$$ 0 0
$$701$$ −2754.00 −0.148384 −0.0741920 0.997244i $$-0.523638\pi$$
−0.0741920 + 0.997244i $$0.523638\pi$$
$$702$$ −2880.00 −0.154841
$$703$$ −53.7401 −0.00288314
$$704$$ −896.000 −0.0479677
$$705$$ −73256.3 −3.91346
$$706$$ 13986.6 0.745597
$$707$$ 0 0
$$708$$ −12280.0 −0.651851
$$709$$ 29434.0 1.55912 0.779561 0.626327i $$-0.215442\pi$$
0.779561 + 0.626327i $$0.215442\pi$$
$$710$$ −23283.6 −1.23073
$$711$$ 28060.0 1.48007
$$712$$ 4944.09 0.260235
$$713$$ 13067.3 0.686361
$$714$$ 0 0
$$715$$ 14112.0 0.738124
$$716$$ 2160.00 0.112742
$$717$$ 30462.2 1.58665
$$718$$ −11888.0 −0.617906
$$719$$ 17669.2 0.916480 0.458240 0.888828i $$-0.348480\pi$$
0.458240 + 0.888828i $$0.348480\pi$$
$$720$$ 7286.03 0.377131
$$721$$ 0 0
$$722$$ 13714.0 0.706901
$$723$$ 10890.0 0.560171
$$724$$ −15137.7 −0.777058
$$725$$ −76362.0 −3.91174
$$726$$ 16051.3 0.820552
$$727$$ −28445.5 −1.45115 −0.725574 0.688144i $$-0.758426\pi$$
−0.725574 + 0.688144i $$0.758426\pi$$
$$728$$ 0 0
$$729$$ −13483.0 −0.685007
$$730$$ −10696.0 −0.542297
$$731$$ 48.0833 0.00243286
$$732$$ −400.000 −0.0201973
$$733$$ −22341.7 −1.12580 −0.562900 0.826525i $$-0.690314\pi$$
−0.562900 + 0.826525i $$0.690314\pi$$
$$734$$ −1685.74 −0.0847710
$$735$$ 0 0
$$736$$ −4480.00 −0.224368
$$737$$ −9576.00 −0.478611
$$738$$ −5789.79 −0.288787
$$739$$ 20670.0 1.02890 0.514451 0.857520i $$-0.327996\pi$$
0.514451 + 0.857520i $$0.327996\pi$$
$$740$$ −3009.45 −0.149499
$$741$$ −509.117 −0.0252400
$$742$$ 0 0
$$743$$ −25400.0 −1.25415 −0.627076 0.778958i $$-0.715749\pi$$
−0.627076 + 0.778958i $$0.715749\pi$$
$$744$$ −5280.00 −0.260180
$$745$$ 40587.9 1.99601
$$746$$ 11452.0 0.562048
$$747$$ −9725.55 −0.476358
$$748$$ 79.1960 0.00387124
$$749$$ 0 0
$$750$$ −39760.0 −1.93577
$$751$$ 29180.0 1.41783 0.708917 0.705292i $$-0.249184\pi$$
0.708917 + 0.705292i $$0.249184\pi$$
$$752$$ −8372.14 −0.405985
$$753$$ −6590.00 −0.318928
$$754$$ −29121.5 −1.40655
$$755$$ −9345.12 −0.450469
$$756$$ 0 0
$$757$$ −26206.0 −1.25822 −0.629110 0.777316i $$-0.716581\pi$$
−0.629110 + 0.777316i $$0.716581\pi$$
$$758$$ −20660.0 −0.989980
$$759$$ −13859.3 −0.662794
$$760$$ −224.000 −0.0106912
$$761$$ −6863.18 −0.326925 −0.163463 0.986550i $$-0.552266\pi$$
−0.163463 + 0.986550i $$0.552266\pi$$
$$762$$ −24324.5 −1.15641
$$763$$ 0 0
$$764$$ 4112.00 0.194721
$$765$$ −644.000 −0.0304364
$$766$$ 2008.18 0.0947240
$$767$$ 22104.0 1.04059
$$768$$ 1810.19 0.0850517
$$769$$ 9058.04 0.424761 0.212380 0.977187i $$-0.431878\pi$$
0.212380 + 0.977187i $$0.431878\pi$$
$$770$$ 0 0
$$771$$ −6630.00 −0.309693
$$772$$ 18368.0 0.856320
$$773$$ 132.936 0.00618548 0.00309274 0.999995i $$-0.499016\pi$$
0.00309274 + 0.999995i $$0.499016\pi$$
$$774$$ 1564.00 0.0726315
$$775$$ 24921.3 1.15509
$$776$$ 11868.1 0.549020
$$777$$ 0 0
$$778$$ −10420.0 −0.480174
$$779$$ 178.000 0.00818679
$$780$$ −28510.5 −1.30877
$$781$$ −8232.00 −0.377163
$$782$$ 395.980 0.0181077
$$783$$ 8089.30 0.369206
$$784$$ 0 0
$$785$$ 43792.0 1.99109
$$786$$ −24540.0 −1.11363
$$787$$ 8729.94 0.395411 0.197706 0.980261i $$-0.436651\pi$$
0.197706 + 0.980261i $$0.436651\pi$$
$$788$$ 3176.00 0.143579
$$789$$ 50487.4 2.27807
$$790$$ −48309.5 −2.17567
$$791$$ 0 0
$$792$$ 2576.00 0.115573
$$793$$ 720.000 0.0322421
$$794$$ −147.078 −0.00657382
$$795$$ −10360.0 −0.462178
$$796$$ −9944.75 −0.442817
$$797$$ −7517.96 −0.334128 −0.167064 0.985946i $$-0.553429\pi$$
−0.167064 + 0.985946i $$0.553429\pi$$
$$798$$ 0 0
$$799$$ 740.000 0.0327651
$$800$$ −8544.00 −0.377595
$$801$$ −14214.3 −0.627011
$$802$$ 996.000 0.0438528
$$803$$ −3781.61 −0.166189
$$804$$ 19346.4 0.848627
$$805$$ 0 0
$$806$$ 9504.00 0.415340
$$807$$ −32600.0 −1.42203
$$808$$ 9028.34 0.393089
$$809$$ 3776.00 0.164100 0.0820501 0.996628i $$-0.473853\pi$$
0.0820501 + 0.996628i $$0.473853\pi$$
$$810$$ 32509.9 1.41023
$$811$$ 36227.9 1.56860 0.784300 0.620382i $$-0.213023\pi$$
0.784300 + 0.620382i $$0.213023\pi$$
$$812$$ 0 0
$$813$$ 16720.0 0.721274
$$814$$ −1064.00 −0.0458147
$$815$$ 65059.5 2.79624
$$816$$ −160.000 −0.00686412
$$817$$ −48.0833 −0.00205902
$$818$$ 6711.86 0.286888
$$819$$ 0 0
$$820$$ 9968.00 0.424509
$$821$$ −16410.0 −0.697580 −0.348790 0.937201i $$-0.613407\pi$$
−0.348790 + 0.937201i $$0.613407\pi$$
$$822$$ −11709.7 −0.496864
$$823$$ 22072.0 0.934850 0.467425 0.884033i $$-0.345182\pi$$
0.467425 + 0.884033i $$0.345182\pi$$
$$824$$ −6946.62 −0.293686
$$825$$ −26431.7 −1.11543
$$826$$ 0 0
$$827$$ −11628.0 −0.488930 −0.244465 0.969658i $$-0.578612\pi$$
−0.244465 + 0.969658i $$0.578612\pi$$
$$828$$ 12880.0 0.540593
$$829$$ 30906.2 1.29483 0.647417 0.762136i $$-0.275849\pi$$
0.647417 + 0.762136i $$0.275849\pi$$
$$830$$ 16744.0 0.700232
$$831$$ 28326.7 1.18248
$$832$$ −3258.35 −0.135773
$$833$$ 0 0
$$834$$ −6020.00 −0.249947
$$835$$ 29512.0 1.22312
$$836$$ −79.1960 −0.00327638
$$837$$ −2640.00 −0.109022
$$838$$ 29090.4 1.19918
$$839$$ −17884.1 −0.735911 −0.367955 0.929843i $$-0.619942\pi$$
−0.367955 + 0.929843i $$0.619942\pi$$
$$840$$ 0 0
$$841$$ 57407.0 2.35381
$$842$$ −21708.0 −0.888488
$$843$$ −42313.3 −1.72876
$$844$$ −10992.0 −0.448294
$$845$$ 7820.60 0.318387
$$846$$ 24069.9 0.978181
$$847$$ 0 0
$$848$$ −1184.00 −0.0479466
$$849$$ 34850.0 1.40877
$$850$$ 755.190 0.0304739
$$851$$ −5320.00 −0.214298
$$852$$ 16631.2 0.668749
$$853$$ 20755.0 0.833104 0.416552 0.909112i $$-0.363238\pi$$
0.416552 + 0.909112i $$0.363238\pi$$
$$854$$ 0 0
$$855$$ 644.000 0.0257595
$$856$$ 13472.0 0.537925
$$857$$ −44919.7 −1.79046 −0.895231 0.445602i $$-0.852990\pi$$
−0.895231 + 0.445602i $$0.852990\pi$$
$$858$$ −10080.0 −0.401079
$$859$$ −69.2965 −0.00275246 −0.00137623 0.999999i $$-0.500438\pi$$
−0.00137623 + 0.999999i $$0.500438\pi$$
$$860$$ −2692.66 −0.106766
$$861$$ 0 0
$$862$$ 10728.0 0.423895
$$863$$ −5452.00 −0.215050 −0.107525 0.994202i $$-0.534293\pi$$
−0.107525 + 0.994202i $$0.534293\pi$$
$$864$$ 905.097 0.0356389
$$865$$ 40992.0 1.61129
$$866$$ −12974.0 −0.509093
$$867$$ −34726.0 −1.36027
$$868$$ 0 0
$$869$$ −17080.0 −0.666743
$$870$$ 80080.0 3.12065
$$871$$ −34823.6 −1.35471
$$872$$ 6544.00 0.254137
$$873$$ −34120.7 −1.32281
$$874$$ −395.980 −0.0153252
$$875$$ 0 0
$$876$$ 7640.00 0.294671
$$877$$ 31106.0 1.19769 0.598845 0.800865i $$-0.295626\pi$$
0.598845 + 0.800865i $$0.295626\pi$$
$$878$$ 27865.7 1.07109
$$879$$ −13940.0 −0.534908
$$880$$ −4434.97 −0.169890
$$881$$ −5943.94 −0.227306 −0.113653 0.993521i $$-0.536255\pi$$
−0.113653 + 0.993521i $$0.536255\pi$$
$$882$$ 0 0
$$883$$ 34796.0 1.32614 0.663068 0.748559i $$-0.269254\pi$$
0.663068 + 0.748559i $$0.269254\pi$$
$$884$$ 288.000 0.0109576
$$885$$ −60782.9 −2.30869
$$886$$ −11992.0 −0.454717
$$887$$ −9964.55 −0.377200 −0.188600 0.982054i $$-0.560395\pi$$
−0.188600 + 0.982054i $$0.560395\pi$$
$$888$$ 2149.60 0.0812342
$$889$$ 0 0
$$890$$ 24472.0 0.921689
$$891$$ 11494.0 0.432170
$$892$$ 13712.2 0.514707
$$893$$ −740.000 −0.0277303
$$894$$ −28991.4 −1.08458
$$895$$ 10691.5 0.399303
$$896$$ 0 0
$$897$$ −50400.0 −1.87604
$$898$$ −5244.00 −0.194871
$$899$$ −26694.7 −0.990343
$$900$$ 24564.0 0.909778
$$901$$ 104.652 0.00386954
$$902$$ 3524.22 0.130093
$$903$$ 0 0
$$904$$ 4320.00 0.158939
$$905$$ −74928.0 −2.75214
$$906$$ 6675.09 0.244774
$$907$$ −29756.0 −1.08934 −0.544670 0.838650i $$-0.683345\pi$$
−0.544670 + 0.838650i $$0.683345\pi$$
$$908$$ −21162.3 −0.773453
$$909$$ −25956.5 −0.947109
$$910$$ 0 0
$$911$$ 21440.0 0.779735 0.389868 0.920871i $$-0.372521\pi$$
0.389868 + 0.920871i $$0.372521\pi$$
$$912$$ 160.000 0.00580935
$$913$$ 5919.90 0.214589
$$914$$ −22416.0 −0.811220
$$915$$ −1979.90 −0.0715338
$$916$$ 10996.9 0.396669
$$917$$ 0 0
$$918$$ −80.0000 −0.00287625
$$919$$ −8288.00 −0.297493 −0.148746 0.988875i $$-0.547524\pi$$
−0.148746 + 0.988875i $$0.547524\pi$$
$$920$$ −22174.9 −0.794656
$$921$$ 33710.0 1.20606
$$922$$ 19572.7 0.699125
$$923$$ −29936.1 −1.06756
$$924$$ 0 0
$$925$$ −10146.0 −0.360647
$$926$$ −7904.00 −0.280498
$$927$$ 19971.5 0.707606
$$928$$ 9152.00 0.323738
$$929$$ 45581.5 1.60978 0.804888 0.593427i $$-0.202226\pi$$
0.804888 + 0.593427i $$0.202226\pi$$
$$930$$ −26134.7 −0.921494
$$931$$ 0 0
$$932$$ 288.000 0.0101221
$$933$$ 47920.0 1.68149
$$934$$ 35013.1 1.22662
$$935$$ 392.000 0.0137110
$$936$$ 9367.75 0.327131
$$937$$ −11665.8 −0.406731 −0.203365 0.979103i $$-0.565188\pi$$
−0.203365 + 0.979103i $$0.565188\pi$$
$$938$$ 0 0
$$939$$ 43770.0 1.52117
$$940$$ −41440.0 −1.43790
$$941$$ −14.1421 −0.000489926 0 −0.000244963 1.00000i $$-0.500078\pi$$
−0.000244963 1.00000i $$0.500078\pi$$
$$942$$ −31280.0 −1.08191
$$943$$ 17621.1 0.608507
$$944$$ −6946.62 −0.239505
$$945$$ 0 0
$$946$$ −952.000 −0.0327190
$$947$$ 14034.0 0.481567 0.240783 0.970579i $$-0.422596\pi$$
0.240783 + 0.970579i $$0.422596\pi$$
$$948$$ 34506.8 1.18220
$$949$$ −13752.0 −0.470399
$$950$$ −755.190 −0.0257912
$$951$$ 69480.3 2.36914
$$952$$ 0 0
$$953$$ −42698.0 −1.45134 −0.725668 0.688045i $$-0.758469\pi$$
−0.725668 + 0.688045i $$0.758469\pi$$
$$954$$ 3404.00 0.115523
$$955$$ 20353.4 0.689654
$$956$$ 17232.0 0.582974
$$957$$ 28312.6 0.956337
$$958$$ 4576.40 0.154339
$$959$$ 0 0
$$960$$ 8960.00 0.301232
$$961$$ −21079.0 −0.707563
$$962$$ −3869.29 −0.129679
$$963$$ −38732.0 −1.29608
$$964$$ 6160.31 0.205820
$$965$$ 90917.0 3.03287
$$966$$ 0 0
$$967$$ −48492.0 −1.61261 −0.806307 0.591497i $$-0.798537\pi$$
−0.806307 + 0.591497i $$0.798537\pi$$
$$968$$ 9080.00 0.301490
$$969$$ −14.1421 −0.000468845 0
$$970$$ 58744.0 1.94449
$$971$$ −52669.6 −1.74073 −0.870364 0.492409i $$-0.836116\pi$$
−0.870364 + 0.492409i $$0.836116\pi$$
$$972$$ −20166.7 −0.665480
$$973$$ 0 0
$$974$$ −1944.00 −0.0639525
$$975$$ −96120.0 −3.15723
$$976$$ −226.274 −0.00742096
$$977$$ −55380.0 −1.81347 −0.906737 0.421698i $$-0.861434\pi$$
−0.906737 + 0.421698i $$0.861434\pi$$
$$978$$ −46471.1 −1.51941
$$979$$ 8652.16 0.282456
$$980$$ 0 0
$$981$$ −18814.0 −0.612319
$$982$$ 14808.0 0.481204
$$983$$ 50535.5 1.63971 0.819854 0.572573i $$-0.194055\pi$$
0.819854 + 0.572573i $$0.194055\pi$$
$$984$$ −7120.00 −0.230668
$$985$$ 15720.4 0.508521
$$986$$ −808.930 −0.0261274
$$987$$ 0 0
$$988$$ −288.000 −0.00927379
$$989$$ −4760.00 −0.153043
$$990$$ 12750.5 0.409332
$$991$$ −39712.0 −1.27295 −0.636475 0.771297i $$-0.719608\pi$$
−0.636475 + 0.771297i $$0.719608\pi$$
$$992$$ −2986.82 −0.0955964
$$993$$ −40573.8 −1.29665
$$994$$ 0 0
$$995$$ −49224.0 −1.56835
$$996$$ −11960.0 −0.380489
$$997$$ −2186.37 −0.0694515 −0.0347258 0.999397i $$-0.511056\pi$$
−0.0347258 + 0.999397i $$0.511056\pi$$
$$998$$ 24488.0 0.776708
$$999$$ 1074.80 0.0340393
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 98.4.a.g.1.2 yes 2
3.2 odd 2 882.4.a.bg.1.1 2
4.3 odd 2 784.4.a.y.1.1 2
5.4 even 2 2450.4.a.bx.1.1 2
7.2 even 3 98.4.c.h.67.1 4
7.3 odd 6 98.4.c.h.79.2 4
7.4 even 3 98.4.c.h.79.1 4
7.5 odd 6 98.4.c.h.67.2 4
7.6 odd 2 inner 98.4.a.g.1.1 2
21.2 odd 6 882.4.g.ba.361.2 4
21.5 even 6 882.4.g.ba.361.1 4
21.11 odd 6 882.4.g.ba.667.2 4
21.17 even 6 882.4.g.ba.667.1 4
21.20 even 2 882.4.a.bg.1.2 2
28.27 even 2 784.4.a.y.1.2 2
35.34 odd 2 2450.4.a.bx.1.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
98.4.a.g.1.1 2 7.6 odd 2 inner
98.4.a.g.1.2 yes 2 1.1 even 1 trivial
98.4.c.h.67.1 4 7.2 even 3
98.4.c.h.67.2 4 7.5 odd 6
98.4.c.h.79.1 4 7.4 even 3
98.4.c.h.79.2 4 7.3 odd 6
784.4.a.y.1.1 2 4.3 odd 2
784.4.a.y.1.2 2 28.27 even 2
882.4.a.bg.1.1 2 3.2 odd 2
882.4.a.bg.1.2 2 21.20 even 2
882.4.g.ba.361.1 4 21.5 even 6
882.4.g.ba.361.2 4 21.2 odd 6
882.4.g.ba.667.1 4 21.17 even 6
882.4.g.ba.667.2 4 21.11 odd 6
2450.4.a.bx.1.1 2 5.4 even 2
2450.4.a.bx.1.2 2 35.34 odd 2