# Properties

 Label 99.8.a.a.1.1 Level $99$ Weight $8$ Character 99.1 Self dual yes Analytic conductor $30.926$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Learn more

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [99,8,Mod(1,99)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(99, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("99.1");

S:= CuspForms(chi, 8);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$99 = 3^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$8$$ Character orbit: $$[\chi]$$ $$=$$ 99.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: $$30.9261175229$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 33) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 99.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-10.0000 q^{2} -28.0000 q^{4} +410.000 q^{5} -1028.00 q^{7} +1560.00 q^{8} +O(q^{10})$$ $$q-10.0000 q^{2} -28.0000 q^{4} +410.000 q^{5} -1028.00 q^{7} +1560.00 q^{8} -4100.00 q^{10} +1331.00 q^{11} +12958.0 q^{13} +10280.0 q^{14} -12016.0 q^{16} -17062.0 q^{17} -54168.0 q^{19} -11480.0 q^{20} -13310.0 q^{22} +11488.0 q^{23} +89975.0 q^{25} -129580. q^{26} +28784.0 q^{28} +186654. q^{29} -188672. q^{31} -79520.0 q^{32} +170620. q^{34} -421480. q^{35} +395886. q^{37} +541680. q^{38} +639600. q^{40} +47546.0 q^{41} +602088. q^{43} -37268.0 q^{44} -114880. q^{46} +647200. q^{47} +233241. q^{49} -899750. q^{50} -362824. q^{52} +1.31272e6 q^{53} +545710. q^{55} -1.60368e6 q^{56} -1.86654e6 q^{58} +2.68114e6 q^{59} +551190. q^{61} +1.88672e6 q^{62} +2.33325e6 q^{64} +5.31278e6 q^{65} +459260. q^{67} +477736. q^{68} +4.21480e6 q^{70} +18072.0 q^{71} -426062. q^{73} -3.95886e6 q^{74} +1.51670e6 q^{76} -1.36827e6 q^{77} +297764. q^{79} -4.92656e6 q^{80} -475460. q^{82} -5.68403e6 q^{83} -6.99542e6 q^{85} -6.02088e6 q^{86} +2.07636e6 q^{88} +6.34297e6 q^{89} -1.33208e7 q^{91} -321664. q^{92} -6.47200e6 q^{94} -2.22089e7 q^{95} +1.66516e7 q^{97} -2.33241e6 q^{98} +O(q^{100})$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −10.0000 −0.883883 −0.441942 0.897044i $$-0.645710\pi$$
−0.441942 + 0.897044i $$0.645710\pi$$
$$3$$ 0 0
$$4$$ −28.0000 −0.218750
$$5$$ 410.000 1.46686 0.733430 0.679765i $$-0.237918\pi$$
0.733430 + 0.679765i $$0.237918\pi$$
$$6$$ 0 0
$$7$$ −1028.00 −1.13279 −0.566396 0.824133i $$-0.691663\pi$$
−0.566396 + 0.824133i $$0.691663\pi$$
$$8$$ 1560.00 1.07723
$$9$$ 0 0
$$10$$ −4100.00 −1.29653
$$11$$ 1331.00 0.301511
$$12$$ 0 0
$$13$$ 12958.0 1.63582 0.817911 0.575344i $$-0.195132\pi$$
0.817911 + 0.575344i $$0.195132\pi$$
$$14$$ 10280.0 1.00126
$$15$$ 0 0
$$16$$ −12016.0 −0.733398
$$17$$ −17062.0 −0.842284 −0.421142 0.906995i $$-0.638371\pi$$
−0.421142 + 0.906995i $$0.638371\pi$$
$$18$$ 0 0
$$19$$ −54168.0 −1.81178 −0.905889 0.423514i $$-0.860796\pi$$
−0.905889 + 0.423514i $$0.860796\pi$$
$$20$$ −11480.0 −0.320876
$$21$$ 0 0
$$22$$ −13310.0 −0.266501
$$23$$ 11488.0 0.196878 0.0984390 0.995143i $$-0.468615\pi$$
0.0984390 + 0.995143i $$0.468615\pi$$
$$24$$ 0 0
$$25$$ 89975.0 1.15168
$$26$$ −129580. −1.44588
$$27$$ 0 0
$$28$$ 28784.0 0.247798
$$29$$ 186654. 1.42116 0.710582 0.703614i $$-0.248432\pi$$
0.710582 + 0.703614i $$0.248432\pi$$
$$30$$ 0 0
$$31$$ −188672. −1.13747 −0.568737 0.822519i $$-0.692568\pi$$
−0.568737 + 0.822519i $$0.692568\pi$$
$$32$$ −79520.0 −0.428994
$$33$$ 0 0
$$34$$ 170620. 0.744481
$$35$$ −421480. −1.66165
$$36$$ 0 0
$$37$$ 395886. 1.28488 0.642442 0.766334i $$-0.277921\pi$$
0.642442 + 0.766334i $$0.277921\pi$$
$$38$$ 541680. 1.60140
$$39$$ 0 0
$$40$$ 639600. 1.58015
$$41$$ 47546.0 0.107738 0.0538692 0.998548i $$-0.482845\pi$$
0.0538692 + 0.998548i $$0.482845\pi$$
$$42$$ 0 0
$$43$$ 602088. 1.15484 0.577418 0.816449i $$-0.304060\pi$$
0.577418 + 0.816449i $$0.304060\pi$$
$$44$$ −37268.0 −0.0659556
$$45$$ 0 0
$$46$$ −114880. −0.174017
$$47$$ 647200. 0.909277 0.454638 0.890676i $$-0.349769\pi$$
0.454638 + 0.890676i $$0.349769\pi$$
$$48$$ 0 0
$$49$$ 233241. 0.283217
$$50$$ −899750. −1.01795
$$51$$ 0 0
$$52$$ −362824. −0.357836
$$53$$ 1.31272e6 1.21118 0.605588 0.795778i $$-0.292938\pi$$
0.605588 + 0.795778i $$0.292938\pi$$
$$54$$ 0 0
$$55$$ 545710. 0.442275
$$56$$ −1.60368e6 −1.22028
$$57$$ 0 0
$$58$$ −1.86654e6 −1.25614
$$59$$ 2.68114e6 1.69956 0.849782 0.527135i $$-0.176734\pi$$
0.849782 + 0.527135i $$0.176734\pi$$
$$60$$ 0 0
$$61$$ 551190. 0.310919 0.155459 0.987842i $$-0.450314\pi$$
0.155459 + 0.987842i $$0.450314\pi$$
$$62$$ 1.88672e6 1.00539
$$63$$ 0 0
$$64$$ 2.33325e6 1.11258
$$65$$ 5.31278e6 2.39952
$$66$$ 0 0
$$67$$ 459260. 0.186551 0.0932753 0.995640i $$-0.470266\pi$$
0.0932753 + 0.995640i $$0.470266\pi$$
$$68$$ 477736. 0.184250
$$69$$ 0 0
$$70$$ 4.21480e6 1.46870
$$71$$ 18072.0 0.00599242 0.00299621 0.999996i $$-0.499046\pi$$
0.00299621 + 0.999996i $$0.499046\pi$$
$$72$$ 0 0
$$73$$ −426062. −0.128187 −0.0640933 0.997944i $$-0.520416\pi$$
−0.0640933 + 0.997944i $$0.520416\pi$$
$$74$$ −3.95886e6 −1.13569
$$75$$ 0 0
$$76$$ 1.51670e6 0.396327
$$77$$ −1.36827e6 −0.341549
$$78$$ 0 0
$$79$$ 297764. 0.0679481 0.0339741 0.999423i $$-0.489184\pi$$
0.0339741 + 0.999423i $$0.489184\pi$$
$$80$$ −4.92656e6 −1.07579
$$81$$ 0 0
$$82$$ −475460. −0.0952282
$$83$$ −5.68403e6 −1.09115 −0.545573 0.838063i $$-0.683688\pi$$
−0.545573 + 0.838063i $$0.683688\pi$$
$$84$$ 0 0
$$85$$ −6.99542e6 −1.23551
$$86$$ −6.02088e6 −1.02074
$$87$$ 0 0
$$88$$ 2.07636e6 0.324798
$$89$$ 6.34297e6 0.953734 0.476867 0.878975i $$-0.341772\pi$$
0.476867 + 0.878975i $$0.341772\pi$$
$$90$$ 0 0
$$91$$ −1.33208e7 −1.85305
$$92$$ −321664. −0.0430670
$$93$$ 0 0
$$94$$ −6.47200e6 −0.803695
$$95$$ −2.22089e7 −2.65763
$$96$$ 0 0
$$97$$ 1.66516e7 1.85248 0.926242 0.376929i $$-0.123020\pi$$
0.926242 + 0.376929i $$0.123020\pi$$
$$98$$ −2.33241e6 −0.250330
$$99$$ 0 0
$$100$$ −2.51930e6 −0.251930
$$101$$ 2.08327e6 0.201197 0.100598 0.994927i $$-0.467924\pi$$
0.100598 + 0.994927i $$0.467924\pi$$
$$102$$ 0 0
$$103$$ −2.39046e6 −0.215552 −0.107776 0.994175i $$-0.534373\pi$$
−0.107776 + 0.994175i $$0.534373\pi$$
$$104$$ 2.02145e7 1.76216
$$105$$ 0 0
$$106$$ −1.31272e7 −1.07054
$$107$$ 1.40615e7 1.10965 0.554827 0.831966i $$-0.312785\pi$$
0.554827 + 0.831966i $$0.312785\pi$$
$$108$$ 0 0
$$109$$ −1.11321e7 −0.823347 −0.411674 0.911331i $$-0.635056\pi$$
−0.411674 + 0.911331i $$0.635056\pi$$
$$110$$ −5.45710e6 −0.390920
$$111$$ 0 0
$$112$$ 1.23524e7 0.830788
$$113$$ −5.66903e6 −0.369602 −0.184801 0.982776i $$-0.559164\pi$$
−0.184801 + 0.982776i $$0.559164\pi$$
$$114$$ 0 0
$$115$$ 4.71008e6 0.288792
$$116$$ −5.22631e6 −0.310880
$$117$$ 0 0
$$118$$ −2.68114e7 −1.50222
$$119$$ 1.75397e7 0.954132
$$120$$ 0 0
$$121$$ 1.77156e6 0.0909091
$$122$$ −5.51190e6 −0.274816
$$123$$ 0 0
$$124$$ 5.28282e6 0.248822
$$125$$ 4.85850e6 0.222493
$$126$$ 0 0
$$127$$ −2.09170e7 −0.906123 −0.453061 0.891479i $$-0.649668\pi$$
−0.453061 + 0.891479i $$0.649668\pi$$
$$128$$ −1.31539e7 −0.554396
$$129$$ 0 0
$$130$$ −5.31278e7 −2.12090
$$131$$ 1.12649e7 0.437802 0.218901 0.975747i $$-0.429753\pi$$
0.218901 + 0.975747i $$0.429753\pi$$
$$132$$ 0 0
$$133$$ 5.56847e7 2.05237
$$134$$ −4.59260e6 −0.164889
$$135$$ 0 0
$$136$$ −2.66167e7 −0.907336
$$137$$ −444290. −0.0147620 −0.00738099 0.999973i $$-0.502349\pi$$
−0.00738099 + 0.999973i $$0.502349\pi$$
$$138$$ 0 0
$$139$$ 3.42613e7 1.08206 0.541030 0.841003i $$-0.318034\pi$$
0.541030 + 0.841003i $$0.318034\pi$$
$$140$$ 1.18014e7 0.363485
$$141$$ 0 0
$$142$$ −180720. −0.00529660
$$143$$ 1.72471e7 0.493219
$$144$$ 0 0
$$145$$ 7.65281e7 2.08465
$$146$$ 4.26062e6 0.113302
$$147$$ 0 0
$$148$$ −1.10848e7 −0.281068
$$149$$ 4.82211e7 1.19422 0.597112 0.802158i $$-0.296315\pi$$
0.597112 + 0.802158i $$0.296315\pi$$
$$150$$ 0 0
$$151$$ −4.48693e7 −1.06055 −0.530273 0.847827i $$-0.677911\pi$$
−0.530273 + 0.847827i $$0.677911\pi$$
$$152$$ −8.45021e7 −1.95171
$$153$$ 0 0
$$154$$ 1.36827e7 0.301890
$$155$$ −7.73555e7 −1.66852
$$156$$ 0 0
$$157$$ −5.38907e6 −0.111139 −0.0555693 0.998455i $$-0.517697\pi$$
−0.0555693 + 0.998455i $$0.517697\pi$$
$$158$$ −2.97764e6 −0.0600582
$$159$$ 0 0
$$160$$ −3.26032e7 −0.629275
$$161$$ −1.18097e7 −0.223022
$$162$$ 0 0
$$163$$ 9.81674e7 1.77546 0.887730 0.460365i $$-0.152281\pi$$
0.887730 + 0.460365i $$0.152281\pi$$
$$164$$ −1.33129e6 −0.0235678
$$165$$ 0 0
$$166$$ 5.68403e7 0.964446
$$167$$ 4.40611e7 0.732062 0.366031 0.930603i $$-0.380716\pi$$
0.366031 + 0.930603i $$0.380716\pi$$
$$168$$ 0 0
$$169$$ 1.05161e8 1.67592
$$170$$ 6.99542e7 1.09205
$$171$$ 0 0
$$172$$ −1.68585e7 −0.252620
$$173$$ −6.71087e7 −0.985411 −0.492706 0.870196i $$-0.663992\pi$$
−0.492706 + 0.870196i $$0.663992\pi$$
$$174$$ 0 0
$$175$$ −9.24943e7 −1.30461
$$176$$ −1.59933e7 −0.221128
$$177$$ 0 0
$$178$$ −6.34297e7 −0.842990
$$179$$ −4.34929e6 −0.0566804 −0.0283402 0.999598i $$-0.509022\pi$$
−0.0283402 + 0.999598i $$0.509022\pi$$
$$180$$ 0 0
$$181$$ −1.20238e7 −0.150719 −0.0753593 0.997156i $$-0.524010\pi$$
−0.0753593 + 0.997156i $$0.524010\pi$$
$$182$$ 1.33208e8 1.63788
$$183$$ 0 0
$$184$$ 1.79213e7 0.212083
$$185$$ 1.62313e8 1.88475
$$186$$ 0 0
$$187$$ −2.27095e7 −0.253958
$$188$$ −1.81216e7 −0.198904
$$189$$ 0 0
$$190$$ 2.22089e8 2.34903
$$191$$ −5.96399e7 −0.619327 −0.309664 0.950846i $$-0.600216\pi$$
−0.309664 + 0.950846i $$0.600216\pi$$
$$192$$ 0 0
$$193$$ −9.81036e7 −0.982278 −0.491139 0.871081i $$-0.663419\pi$$
−0.491139 + 0.871081i $$0.663419\pi$$
$$194$$ −1.66516e8 −1.63738
$$195$$ 0 0
$$196$$ −6.53075e6 −0.0619536
$$197$$ 1.09317e8 1.01872 0.509361 0.860553i $$-0.329882\pi$$
0.509361 + 0.860553i $$0.329882\pi$$
$$198$$ 0 0
$$199$$ −3.64317e7 −0.327713 −0.163857 0.986484i $$-0.552393\pi$$
−0.163857 + 0.986484i $$0.552393\pi$$
$$200$$ 1.40361e8 1.24063
$$201$$ 0 0
$$202$$ −2.08327e7 −0.177834
$$203$$ −1.91880e8 −1.60988
$$204$$ 0 0
$$205$$ 1.94939e7 0.158037
$$206$$ 2.39046e7 0.190523
$$207$$ 0 0
$$208$$ −1.55703e8 −1.19971
$$209$$ −7.20976e7 −0.546272
$$210$$ 0 0
$$211$$ 1.38637e7 0.101599 0.0507997 0.998709i $$-0.483823\pi$$
0.0507997 + 0.998709i $$0.483823\pi$$
$$212$$ −3.67562e7 −0.264945
$$213$$ 0 0
$$214$$ −1.40615e8 −0.980805
$$215$$ 2.46856e8 1.69398
$$216$$ 0 0
$$217$$ 1.93955e8 1.28852
$$218$$ 1.11321e8 0.727743
$$219$$ 0 0
$$220$$ −1.52799e7 −0.0967477
$$221$$ −2.21089e8 −1.37783
$$222$$ 0 0
$$223$$ 1.35935e8 0.820850 0.410425 0.911894i $$-0.365380\pi$$
0.410425 + 0.911894i $$0.365380\pi$$
$$224$$ 8.17466e7 0.485961
$$225$$ 0 0
$$226$$ 5.66903e7 0.326685
$$227$$ −2.82203e7 −0.160129 −0.0800646 0.996790i $$-0.525513\pi$$
−0.0800646 + 0.996790i $$0.525513\pi$$
$$228$$ 0 0
$$229$$ −5.31215e7 −0.292312 −0.146156 0.989262i $$-0.546690\pi$$
−0.146156 + 0.989262i $$0.546690\pi$$
$$230$$ −4.71008e7 −0.255259
$$231$$ 0 0
$$232$$ 2.91180e8 1.53093
$$233$$ −1.54589e8 −0.800631 −0.400316 0.916377i $$-0.631099\pi$$
−0.400316 + 0.916377i $$0.631099\pi$$
$$234$$ 0 0
$$235$$ 2.65352e8 1.33378
$$236$$ −7.50719e7 −0.371780
$$237$$ 0 0
$$238$$ −1.75397e8 −0.843342
$$239$$ 1.86143e8 0.881972 0.440986 0.897514i $$-0.354629\pi$$
0.440986 + 0.897514i $$0.354629\pi$$
$$240$$ 0 0
$$241$$ 2.62107e8 1.20620 0.603100 0.797666i $$-0.293932\pi$$
0.603100 + 0.797666i $$0.293932\pi$$
$$242$$ −1.77156e7 −0.0803530
$$243$$ 0 0
$$244$$ −1.54333e7 −0.0680135
$$245$$ 9.56288e7 0.415439
$$246$$ 0 0
$$247$$ −7.01909e8 −2.96375
$$248$$ −2.94328e8 −1.22532
$$249$$ 0 0
$$250$$ −4.85850e7 −0.196658
$$251$$ 2.75827e8 1.10098 0.550489 0.834842i $$-0.314441\pi$$
0.550489 + 0.834842i $$0.314441\pi$$
$$252$$ 0 0
$$253$$ 1.52905e7 0.0593609
$$254$$ 2.09170e8 0.800907
$$255$$ 0 0
$$256$$ −1.67117e8 −0.622558
$$257$$ −1.06856e6 −0.00392675 −0.00196338 0.999998i $$-0.500625\pi$$
−0.00196338 + 0.999998i $$0.500625\pi$$
$$258$$ 0 0
$$259$$ −4.06971e8 −1.45551
$$260$$ −1.48758e8 −0.524896
$$261$$ 0 0
$$262$$ −1.12649e8 −0.386966
$$263$$ 7.92924e7 0.268774 0.134387 0.990929i $$-0.457094\pi$$
0.134387 + 0.990929i $$0.457094\pi$$
$$264$$ 0 0
$$265$$ 5.38216e8 1.77663
$$266$$ −5.56847e8 −1.81405
$$267$$ 0 0
$$268$$ −1.28593e7 −0.0408080
$$269$$ −2.10170e8 −0.658321 −0.329160 0.944274i $$-0.606766\pi$$
−0.329160 + 0.944274i $$0.606766\pi$$
$$270$$ 0 0
$$271$$ −2.65510e8 −0.810378 −0.405189 0.914233i $$-0.632794\pi$$
−0.405189 + 0.914233i $$0.632794\pi$$
$$272$$ 2.05017e8 0.617730
$$273$$ 0 0
$$274$$ 4.44290e6 0.0130479
$$275$$ 1.19757e8 0.347245
$$276$$ 0 0
$$277$$ −6.23529e8 −1.76270 −0.881349 0.472466i $$-0.843364\pi$$
−0.881349 + 0.472466i $$0.843364\pi$$
$$278$$ −3.42613e8 −0.956415
$$279$$ 0 0
$$280$$ −6.57509e8 −1.78998
$$281$$ −1.30611e8 −0.351162 −0.175581 0.984465i $$-0.556180\pi$$
−0.175581 + 0.984465i $$0.556180\pi$$
$$282$$ 0 0
$$283$$ −2.20874e7 −0.0579283 −0.0289642 0.999580i $$-0.509221\pi$$
−0.0289642 + 0.999580i $$0.509221\pi$$
$$284$$ −506016. −0.00131084
$$285$$ 0 0
$$286$$ −1.72471e8 −0.435948
$$287$$ −4.88773e7 −0.122045
$$288$$ 0 0
$$289$$ −1.19227e8 −0.290557
$$290$$ −7.65281e8 −1.84259
$$291$$ 0 0
$$292$$ 1.19297e7 0.0280408
$$293$$ −2.00188e8 −0.464944 −0.232472 0.972603i $$-0.574681\pi$$
−0.232472 + 0.972603i $$0.574681\pi$$
$$294$$ 0 0
$$295$$ 1.09927e9 2.49302
$$296$$ 6.17582e8 1.38412
$$297$$ 0 0
$$298$$ −4.82211e8 −1.05555
$$299$$ 1.48862e8 0.322057
$$300$$ 0 0
$$301$$ −6.18946e8 −1.30819
$$302$$ 4.48693e8 0.937399
$$303$$ 0 0
$$304$$ 6.50883e8 1.32876
$$305$$ 2.25988e8 0.456074
$$306$$ 0 0
$$307$$ −4.79736e8 −0.946276 −0.473138 0.880988i $$-0.656879\pi$$
−0.473138 + 0.880988i $$0.656879\pi$$
$$308$$ 3.83115e7 0.0747139
$$309$$ 0 0
$$310$$ 7.73555e8 1.47477
$$311$$ 5.19734e8 0.979761 0.489880 0.871790i $$-0.337040\pi$$
0.489880 + 0.871790i $$0.337040\pi$$
$$312$$ 0 0
$$313$$ 9.69759e8 1.78755 0.893776 0.448514i $$-0.148047\pi$$
0.893776 + 0.448514i $$0.148047\pi$$
$$314$$ 5.38907e7 0.0982335
$$315$$ 0 0
$$316$$ −8.33739e6 −0.0148636
$$317$$ −7.56875e8 −1.33450 −0.667248 0.744836i $$-0.732528\pi$$
−0.667248 + 0.744836i $$0.732528\pi$$
$$318$$ 0 0
$$319$$ 2.48436e8 0.428497
$$320$$ 9.56632e8 1.63200
$$321$$ 0 0
$$322$$ 1.18097e8 0.197125
$$323$$ 9.24214e8 1.52603
$$324$$ 0 0
$$325$$ 1.16590e9 1.88394
$$326$$ −9.81674e8 −1.56930
$$327$$ 0 0
$$328$$ 7.41718e7 0.116059
$$329$$ −6.65322e8 −1.03002
$$330$$ 0 0
$$331$$ −1.79867e8 −0.272618 −0.136309 0.990666i $$-0.543524\pi$$
−0.136309 + 0.990666i $$0.543524\pi$$
$$332$$ 1.59153e8 0.238688
$$333$$ 0 0
$$334$$ −4.40611e8 −0.647058
$$335$$ 1.88297e8 0.273644
$$336$$ 0 0
$$337$$ −1.38092e9 −1.96546 −0.982728 0.185054i $$-0.940754\pi$$
−0.982728 + 0.185054i $$0.940754\pi$$
$$338$$ −1.05161e9 −1.48131
$$339$$ 0 0
$$340$$ 1.95872e8 0.270269
$$341$$ −2.51122e8 −0.342961
$$342$$ 0 0
$$343$$ 6.06830e8 0.811966
$$344$$ 9.39257e8 1.24403
$$345$$ 0 0
$$346$$ 6.71087e8 0.870989
$$347$$ −7.66253e8 −0.984507 −0.492254 0.870452i $$-0.663827\pi$$
−0.492254 + 0.870452i $$0.663827\pi$$
$$348$$ 0 0
$$349$$ −2.68852e8 −0.338552 −0.169276 0.985569i $$-0.554143\pi$$
−0.169276 + 0.985569i $$0.554143\pi$$
$$350$$ 9.24943e8 1.15313
$$351$$ 0 0
$$352$$ −1.05841e8 −0.129347
$$353$$ 3.95002e8 0.477956 0.238978 0.971025i $$-0.423188\pi$$
0.238978 + 0.971025i $$0.423188\pi$$
$$354$$ 0 0
$$355$$ 7.40952e6 0.00879004
$$356$$ −1.77603e8 −0.208629
$$357$$ 0 0
$$358$$ 4.34929e7 0.0500989
$$359$$ 4.25768e7 0.0485671 0.0242836 0.999705i $$-0.492270\pi$$
0.0242836 + 0.999705i $$0.492270\pi$$
$$360$$ 0 0
$$361$$ 2.04030e9 2.28254
$$362$$ 1.20238e8 0.133218
$$363$$ 0 0
$$364$$ 3.72983e8 0.405354
$$365$$ −1.74685e8 −0.188032
$$366$$ 0 0
$$367$$ 1.85295e9 1.95673 0.978366 0.206882i $$-0.0663315\pi$$
0.978366 + 0.206882i $$0.0663315\pi$$
$$368$$ −1.38040e8 −0.144390
$$369$$ 0 0
$$370$$ −1.62313e9 −1.66590
$$371$$ −1.34948e9 −1.37201
$$372$$ 0 0
$$373$$ −4.83602e7 −0.0482511 −0.0241256 0.999709i $$-0.507680\pi$$
−0.0241256 + 0.999709i $$0.507680\pi$$
$$374$$ 2.27095e8 0.224470
$$375$$ 0 0
$$376$$ 1.00963e9 0.979503
$$377$$ 2.41866e9 2.32477
$$378$$ 0 0
$$379$$ −2.26078e8 −0.213315 −0.106658 0.994296i $$-0.534015\pi$$
−0.106658 + 0.994296i $$0.534015\pi$$
$$380$$ 6.21849e8 0.581356
$$381$$ 0 0
$$382$$ 5.96399e8 0.547413
$$383$$ 1.35198e9 1.22963 0.614815 0.788671i $$-0.289231\pi$$
0.614815 + 0.788671i $$0.289231\pi$$
$$384$$ 0 0
$$385$$ −5.60990e8 −0.501005
$$386$$ 9.81036e8 0.868219
$$387$$ 0 0
$$388$$ −4.66244e8 −0.405231
$$389$$ 1.09107e9 0.939789 0.469894 0.882723i $$-0.344292\pi$$
0.469894 + 0.882723i $$0.344292\pi$$
$$390$$ 0 0
$$391$$ −1.96008e8 −0.165827
$$392$$ 3.63856e8 0.305090
$$393$$ 0 0
$$394$$ −1.09317e9 −0.900431
$$395$$ 1.22083e8 0.0996704
$$396$$ 0 0
$$397$$ −6.97868e8 −0.559766 −0.279883 0.960034i $$-0.590296\pi$$
−0.279883 + 0.960034i $$0.590296\pi$$
$$398$$ 3.64317e8 0.289660
$$399$$ 0 0
$$400$$ −1.08114e9 −0.844640
$$401$$ −1.74689e9 −1.35288 −0.676441 0.736497i $$-0.736479\pi$$
−0.676441 + 0.736497i $$0.736479\pi$$
$$402$$ 0 0
$$403$$ −2.44481e9 −1.86071
$$404$$ −5.83316e7 −0.0440118
$$405$$ 0 0
$$406$$ 1.91880e9 1.42295
$$407$$ 5.26924e8 0.387407
$$408$$ 0 0
$$409$$ −1.30304e9 −0.941729 −0.470865 0.882205i $$-0.656058\pi$$
−0.470865 + 0.882205i $$0.656058\pi$$
$$410$$ −1.94939e8 −0.139686
$$411$$ 0 0
$$412$$ 6.69330e7 0.0471520
$$413$$ −2.75621e9 −1.92525
$$414$$ 0 0
$$415$$ −2.33045e9 −1.60056
$$416$$ −1.03042e9 −0.701759
$$417$$ 0 0
$$418$$ 7.20976e8 0.482841
$$419$$ −2.87139e9 −1.90697 −0.953484 0.301443i $$-0.902532\pi$$
−0.953484 + 0.301443i $$0.902532\pi$$
$$420$$ 0 0
$$421$$ 1.15946e9 0.757299 0.378650 0.925540i $$-0.376389\pi$$
0.378650 + 0.925540i $$0.376389\pi$$
$$422$$ −1.38637e8 −0.0898020
$$423$$ 0 0
$$424$$ 2.04785e9 1.30472
$$425$$ −1.53515e9 −0.970042
$$426$$ 0 0
$$427$$ −5.66623e8 −0.352206
$$428$$ −3.93721e8 −0.242737
$$429$$ 0 0
$$430$$ −2.46856e9 −1.49728
$$431$$ 1.66703e9 1.00294 0.501468 0.865176i $$-0.332793\pi$$
0.501468 + 0.865176i $$0.332793\pi$$
$$432$$ 0 0
$$433$$ 6.34094e8 0.375358 0.187679 0.982230i $$-0.439903\pi$$
0.187679 + 0.982230i $$0.439903\pi$$
$$434$$ −1.93955e9 −1.13890
$$435$$ 0 0
$$436$$ 3.11698e8 0.180107
$$437$$ −6.22282e8 −0.356699
$$438$$ 0 0
$$439$$ −1.22368e9 −0.690307 −0.345154 0.938546i $$-0.612173\pi$$
−0.345154 + 0.938546i $$0.612173\pi$$
$$440$$ 8.51308e8 0.476433
$$441$$ 0 0
$$442$$ 2.21089e9 1.21784
$$443$$ 1.23213e9 0.673355 0.336677 0.941620i $$-0.390697\pi$$
0.336677 + 0.941620i $$0.390697\pi$$
$$444$$ 0 0
$$445$$ 2.60062e9 1.39900
$$446$$ −1.35935e9 −0.725536
$$447$$ 0 0
$$448$$ −2.39858e9 −1.26032
$$449$$ 3.07511e9 1.60324 0.801621 0.597833i $$-0.203971\pi$$
0.801621 + 0.597833i $$0.203971\pi$$
$$450$$ 0 0
$$451$$ 6.32837e7 0.0324843
$$452$$ 1.58733e8 0.0808505
$$453$$ 0 0
$$454$$ 2.82203e8 0.141536
$$455$$ −5.46154e9 −2.71816
$$456$$ 0 0
$$457$$ −2.44730e9 −1.19945 −0.599723 0.800207i $$-0.704723\pi$$
−0.599723 + 0.800207i $$0.704723\pi$$
$$458$$ 5.31215e8 0.258370
$$459$$ 0 0
$$460$$ −1.31882e8 −0.0631733
$$461$$ −9.52419e8 −0.452767 −0.226383 0.974038i $$-0.572690\pi$$
−0.226383 + 0.974038i $$0.572690\pi$$
$$462$$ 0 0
$$463$$ −6.05200e8 −0.283378 −0.141689 0.989911i $$-0.545253\pi$$
−0.141689 + 0.989911i $$0.545253\pi$$
$$464$$ −2.24283e9 −1.04228
$$465$$ 0 0
$$466$$ 1.54589e9 0.707665
$$467$$ 1.37708e9 0.625676 0.312838 0.949806i $$-0.398720\pi$$
0.312838 + 0.949806i $$0.398720\pi$$
$$468$$ 0 0
$$469$$ −4.72119e8 −0.211323
$$470$$ −2.65352e9 −1.17891
$$471$$ 0 0
$$472$$ 4.18258e9 1.83083
$$473$$ 8.01379e8 0.348196
$$474$$ 0 0
$$475$$ −4.87377e9 −2.08659
$$476$$ −4.91113e8 −0.208716
$$477$$ 0 0
$$478$$ −1.86143e9 −0.779561
$$479$$ 4.00222e9 1.66390 0.831949 0.554851i $$-0.187225\pi$$
0.831949 + 0.554851i $$0.187225\pi$$
$$480$$ 0 0
$$481$$ 5.12989e9 2.10184
$$482$$ −2.62107e9 −1.06614
$$483$$ 0 0
$$484$$ −4.96037e7 −0.0198864
$$485$$ 6.82715e9 2.71734
$$486$$ 0 0
$$487$$ −2.88677e9 −1.13256 −0.566279 0.824214i $$-0.691617\pi$$
−0.566279 + 0.824214i $$0.691617\pi$$
$$488$$ 8.59856e8 0.334932
$$489$$ 0 0
$$490$$ −9.56288e8 −0.367200
$$491$$ −1.19743e8 −0.0456525 −0.0228262 0.999739i $$-0.507266\pi$$
−0.0228262 + 0.999739i $$0.507266\pi$$
$$492$$ 0 0
$$493$$ −3.18469e9 −1.19702
$$494$$ 7.01909e9 2.61961
$$495$$ 0 0
$$496$$ 2.26708e9 0.834222
$$497$$ −1.85780e7 −0.00678816
$$498$$ 0 0
$$499$$ −4.78950e9 −1.72559 −0.862796 0.505552i $$-0.831289\pi$$
−0.862796 + 0.505552i $$0.831289\pi$$
$$500$$ −1.36038e8 −0.0486704
$$501$$ 0 0
$$502$$ −2.75827e9 −0.973137
$$503$$ −3.83047e9 −1.34203 −0.671017 0.741442i $$-0.734142\pi$$
−0.671017 + 0.741442i $$0.734142\pi$$
$$504$$ 0 0
$$505$$ 8.54141e8 0.295127
$$506$$ −1.52905e8 −0.0524681
$$507$$ 0 0
$$508$$ 5.85677e8 0.198214
$$509$$ 2.34385e9 0.787803 0.393902 0.919153i $$-0.371125\pi$$
0.393902 + 0.919153i $$0.371125\pi$$
$$510$$ 0 0
$$511$$ 4.37992e8 0.145209
$$512$$ 3.35487e9 1.10466
$$513$$ 0 0
$$514$$ 1.06856e7 0.00347079
$$515$$ −9.80090e8 −0.316185
$$516$$ 0 0
$$517$$ 8.61423e8 0.274157
$$518$$ 4.06971e9 1.28650
$$519$$ 0 0
$$520$$ 8.28794e9 2.58485
$$521$$ −5.77085e9 −1.78775 −0.893877 0.448313i $$-0.852025\pi$$
−0.893877 + 0.448313i $$0.852025\pi$$
$$522$$ 0 0
$$523$$ −3.49411e8 −0.106802 −0.0534012 0.998573i $$-0.517006\pi$$
−0.0534012 + 0.998573i $$0.517006\pi$$
$$524$$ −3.15417e8 −0.0957691
$$525$$ 0 0
$$526$$ −7.92924e8 −0.237565
$$527$$ 3.21912e9 0.958077
$$528$$ 0 0
$$529$$ −3.27285e9 −0.961239
$$530$$ −5.38216e9 −1.57033
$$531$$ 0 0
$$532$$ −1.55917e9 −0.448955
$$533$$ 6.16101e8 0.176241
$$534$$ 0 0
$$535$$ 5.76520e9 1.62771
$$536$$ 7.16446e8 0.200959
$$537$$ 0 0
$$538$$ 2.10170e9 0.581879
$$539$$ 3.10444e8 0.0853930
$$540$$ 0 0
$$541$$ −5.10025e9 −1.38484 −0.692422 0.721493i $$-0.743456\pi$$
−0.692422 + 0.721493i $$0.743456\pi$$
$$542$$ 2.65510e9 0.716280
$$543$$ 0 0
$$544$$ 1.35677e9 0.361335
$$545$$ −4.56415e9 −1.20774
$$546$$ 0 0
$$547$$ 4.96217e9 1.29633 0.648166 0.761499i $$-0.275536\pi$$
0.648166 + 0.761499i $$0.275536\pi$$
$$548$$ 1.24401e7 0.00322918
$$549$$ 0 0
$$550$$ −1.19757e9 −0.306924
$$551$$ −1.01107e10 −2.57484
$$552$$ 0 0
$$553$$ −3.06101e8 −0.0769710
$$554$$ 6.23529e9 1.55802
$$555$$ 0 0
$$556$$ −9.59315e8 −0.236701
$$557$$ −1.42590e9 −0.349620 −0.174810 0.984602i $$-0.555931\pi$$
−0.174810 + 0.984602i $$0.555931\pi$$
$$558$$ 0 0
$$559$$ 7.80186e9 1.88911
$$560$$ 5.06450e9 1.21865
$$561$$ 0 0
$$562$$ 1.30611e9 0.310386
$$563$$ −5.96929e9 −1.40975 −0.704876 0.709330i $$-0.748998\pi$$
−0.704876 + 0.709330i $$0.748998\pi$$
$$564$$ 0 0
$$565$$ −2.32430e9 −0.542155
$$566$$ 2.20874e8 0.0512019
$$567$$ 0 0
$$568$$ 2.81923e7 0.00645523
$$569$$ 3.51616e9 0.800158 0.400079 0.916481i $$-0.368983\pi$$
0.400079 + 0.916481i $$0.368983\pi$$
$$570$$ 0 0
$$571$$ 6.44706e8 0.144922 0.0724611 0.997371i $$-0.476915\pi$$
0.0724611 + 0.997371i $$0.476915\pi$$
$$572$$ −4.82919e8 −0.107892
$$573$$ 0 0
$$574$$ 4.88773e8 0.107874
$$575$$ 1.03363e9 0.226740
$$576$$ 0 0
$$577$$ −2.63322e9 −0.570652 −0.285326 0.958430i $$-0.592102\pi$$
−0.285326 + 0.958430i $$0.592102\pi$$
$$578$$ 1.19227e9 0.256819
$$579$$ 0 0
$$580$$ −2.14279e9 −0.456017
$$581$$ 5.84318e9 1.23604
$$582$$ 0 0
$$583$$ 1.74723e9 0.365183
$$584$$ −6.64657e8 −0.138087
$$585$$ 0 0
$$586$$ 2.00188e9 0.410956
$$587$$ −6.76347e9 −1.38018 −0.690090 0.723723i $$-0.742429\pi$$
−0.690090 + 0.723723i $$0.742429\pi$$
$$588$$ 0 0
$$589$$ 1.02200e10 2.06085
$$590$$ −1.09927e10 −2.20354
$$591$$ 0 0
$$592$$ −4.75697e9 −0.942332
$$593$$ 4.22718e9 0.832452 0.416226 0.909261i $$-0.363352\pi$$
0.416226 + 0.909261i $$0.363352\pi$$
$$594$$ 0 0
$$595$$ 7.19129e9 1.39958
$$596$$ −1.35019e9 −0.261236
$$597$$ 0 0
$$598$$ −1.48862e9 −0.284661
$$599$$ −4.00299e9 −0.761010 −0.380505 0.924779i $$-0.624250\pi$$
−0.380505 + 0.924779i $$0.624250\pi$$
$$600$$ 0 0
$$601$$ 6.67554e9 1.25437 0.627185 0.778870i $$-0.284207\pi$$
0.627185 + 0.778870i $$0.284207\pi$$
$$602$$ 6.18946e9 1.15629
$$603$$ 0 0
$$604$$ 1.25634e9 0.231994
$$605$$ 7.26340e8 0.133351
$$606$$ 0 0
$$607$$ −5.30634e9 −0.963018 −0.481509 0.876441i $$-0.659911\pi$$
−0.481509 + 0.876441i $$0.659911\pi$$
$$608$$ 4.30744e9 0.777243
$$609$$ 0 0
$$610$$ −2.25988e9 −0.403117
$$611$$ 8.38642e9 1.48742
$$612$$ 0 0
$$613$$ −8.65802e9 −1.51812 −0.759061 0.651019i $$-0.774342\pi$$
−0.759061 + 0.651019i $$0.774342\pi$$
$$614$$ 4.79736e9 0.836398
$$615$$ 0 0
$$616$$ −2.13450e9 −0.367928
$$617$$ 7.38891e9 1.26643 0.633217 0.773974i $$-0.281734\pi$$
0.633217 + 0.773974i $$0.281734\pi$$
$$618$$ 0 0
$$619$$ 9.99141e9 1.69321 0.846603 0.532225i $$-0.178644\pi$$
0.846603 + 0.532225i $$0.178644\pi$$
$$620$$ 2.16595e9 0.364988
$$621$$ 0 0
$$622$$ −5.19734e9 −0.865994
$$623$$ −6.52057e9 −1.08038
$$624$$ 0 0
$$625$$ −5.03731e9 −0.825313
$$626$$ −9.69759e9 −1.57999
$$627$$ 0 0
$$628$$ 1.50894e8 0.0243116
$$629$$ −6.75461e9 −1.08224
$$630$$ 0 0
$$631$$ −3.29834e9 −0.522628 −0.261314 0.965254i $$-0.584156\pi$$
−0.261314 + 0.965254i $$0.584156\pi$$
$$632$$ 4.64512e8 0.0731959
$$633$$ 0 0
$$634$$ 7.56875e9 1.17954
$$635$$ −8.57598e9 −1.32916
$$636$$ 0 0
$$637$$ 3.02234e9 0.463292
$$638$$ −2.48436e9 −0.378742
$$639$$ 0 0
$$640$$ −5.39311e9 −0.813222
$$641$$ 9.76971e9 1.46514 0.732569 0.680692i $$-0.238321\pi$$
0.732569 + 0.680692i $$0.238321\pi$$
$$642$$ 0 0
$$643$$ −4.18444e9 −0.620724 −0.310362 0.950618i $$-0.600450\pi$$
−0.310362 + 0.950618i $$0.600450\pi$$
$$644$$ 3.30671e8 0.0487860
$$645$$ 0 0
$$646$$ −9.24214e9 −1.34884
$$647$$ 6.96085e8 0.101041 0.0505204 0.998723i $$-0.483912\pi$$
0.0505204 + 0.998723i $$0.483912\pi$$
$$648$$ 0 0
$$649$$ 3.56860e9 0.512438
$$650$$ −1.16590e10 −1.66519
$$651$$ 0 0
$$652$$ −2.74869e9 −0.388382
$$653$$ −6.20046e9 −0.871420 −0.435710 0.900087i $$-0.643503\pi$$
−0.435710 + 0.900087i $$0.643503\pi$$
$$654$$ 0 0
$$655$$ 4.61861e9 0.642194
$$656$$ −5.71313e8 −0.0790152
$$657$$ 0 0
$$658$$ 6.65322e9 0.910418
$$659$$ 1.11404e10 1.51636 0.758178 0.652047i $$-0.226090\pi$$
0.758178 + 0.652047i $$0.226090\pi$$
$$660$$ 0 0
$$661$$ 4.56096e9 0.614258 0.307129 0.951668i $$-0.400632\pi$$
0.307129 + 0.951668i $$0.400632\pi$$
$$662$$ 1.79867e9 0.240963
$$663$$ 0 0
$$664$$ −8.86708e9 −1.17542
$$665$$ 2.28307e10 3.01054
$$666$$ 0 0
$$667$$ 2.14428e9 0.279796
$$668$$ −1.23371e9 −0.160139
$$669$$ 0 0
$$670$$ −1.88297e9 −0.241869
$$671$$ 7.33634e8 0.0937455
$$672$$ 0 0
$$673$$ 5.82879e9 0.737099 0.368550 0.929608i $$-0.379854\pi$$
0.368550 + 0.929608i $$0.379854\pi$$
$$674$$ 1.38092e10 1.73723
$$675$$ 0 0
$$676$$ −2.94451e9 −0.366607
$$677$$ −4.99624e9 −0.618846 −0.309423 0.950924i $$-0.600136\pi$$
−0.309423 + 0.950924i $$0.600136\pi$$
$$678$$ 0 0
$$679$$ −1.71178e10 −2.09848
$$680$$ −1.09129e10 −1.33094
$$681$$ 0 0
$$682$$ 2.51122e9 0.303138
$$683$$ −1.21371e10 −1.45762 −0.728808 0.684718i $$-0.759925\pi$$
−0.728808 + 0.684718i $$0.759925\pi$$
$$684$$ 0 0
$$685$$ −1.82159e8 −0.0216538
$$686$$ −6.06830e9 −0.717684
$$687$$ 0 0
$$688$$ −7.23469e9 −0.846955
$$689$$ 1.70103e10 1.98127
$$690$$ 0 0
$$691$$ −9.23403e9 −1.06468 −0.532339 0.846531i $$-0.678687\pi$$
−0.532339 + 0.846531i $$0.678687\pi$$
$$692$$ 1.87904e9 0.215559
$$693$$ 0 0
$$694$$ 7.66253e9 0.870190
$$695$$ 1.40471e10 1.58723
$$696$$ 0 0
$$697$$ −8.11230e8 −0.0907464
$$698$$ 2.68852e9 0.299240
$$699$$ 0 0
$$700$$ 2.58984e9 0.285384
$$701$$ −4.74530e9 −0.520296 −0.260148 0.965569i $$-0.583771\pi$$
−0.260148 + 0.965569i $$0.583771\pi$$
$$702$$ 0 0
$$703$$ −2.14444e10 −2.32793
$$704$$ 3.10555e9 0.335455
$$705$$ 0 0
$$706$$ −3.95002e9 −0.422458
$$707$$ −2.14160e9 −0.227914
$$708$$ 0 0
$$709$$ 1.34547e10 1.41779 0.708894 0.705315i $$-0.249195\pi$$
0.708894 + 0.705315i $$0.249195\pi$$
$$710$$ −7.40952e7 −0.00776937
$$711$$ 0 0
$$712$$ 9.89503e9 1.02739
$$713$$ −2.16746e9 −0.223944
$$714$$ 0 0
$$715$$ 7.07131e9 0.723484
$$716$$ 1.21780e8 0.0123988
$$717$$ 0 0
$$718$$ −4.25768e8 −0.0429277
$$719$$ −2.63976e9 −0.264858 −0.132429 0.991192i $$-0.542278\pi$$
−0.132429 + 0.991192i $$0.542278\pi$$
$$720$$ 0 0
$$721$$ 2.45740e9 0.244175
$$722$$ −2.04030e10 −2.01750
$$723$$ 0 0
$$724$$ 3.36667e8 0.0329697
$$725$$ 1.67942e10 1.63673
$$726$$ 0 0
$$727$$ −3.52707e9 −0.340442 −0.170221 0.985406i $$-0.554448\pi$$
−0.170221 + 0.985406i $$0.554448\pi$$
$$728$$ −2.07805e10 −1.99616
$$729$$ 0 0
$$730$$ 1.74685e9 0.166198
$$731$$ −1.02728e10 −0.972700
$$732$$ 0 0
$$733$$ −1.03828e10 −0.973760 −0.486880 0.873469i $$-0.661865\pi$$
−0.486880 + 0.873469i $$0.661865\pi$$
$$734$$ −1.85295e10 −1.72952
$$735$$ 0 0
$$736$$ −9.13526e8 −0.0844595
$$737$$ 6.11275e8 0.0562471
$$738$$ 0 0
$$739$$ 2.05418e9 0.187233 0.0936164 0.995608i $$-0.470157\pi$$
0.0936164 + 0.995608i $$0.470157\pi$$
$$740$$ −4.54477e9 −0.412288
$$741$$ 0 0
$$742$$ 1.34948e10 1.21270
$$743$$ −4.87476e9 −0.436006 −0.218003 0.975948i $$-0.569954\pi$$
−0.218003 + 0.975948i $$0.569954\pi$$
$$744$$ 0 0
$$745$$ 1.97707e10 1.75176
$$746$$ 4.83602e8 0.0426484
$$747$$ 0 0
$$748$$ 6.35867e8 0.0555534
$$749$$ −1.44552e10 −1.25701
$$750$$ 0 0
$$751$$ 1.15809e10 0.997705 0.498853 0.866687i $$-0.333755\pi$$
0.498853 + 0.866687i $$0.333755\pi$$
$$752$$ −7.77676e9 −0.666862
$$753$$ 0 0
$$754$$ −2.41866e10 −2.05483
$$755$$ −1.83964e10 −1.55567
$$756$$ 0 0
$$757$$ 3.46735e9 0.290511 0.145255 0.989394i $$-0.453600\pi$$
0.145255 + 0.989394i $$0.453600\pi$$
$$758$$ 2.26078e9 0.188546
$$759$$ 0 0
$$760$$ −3.46459e10 −2.86288
$$761$$ 1.14023e10 0.937877 0.468938 0.883231i $$-0.344637\pi$$
0.468938 + 0.883231i $$0.344637\pi$$
$$762$$ 0 0
$$763$$ 1.14438e10 0.932681
$$764$$ 1.66992e9 0.135478
$$765$$ 0 0
$$766$$ −1.35198e10 −1.08685
$$767$$ 3.47422e10 2.78018
$$768$$ 0 0
$$769$$ 2.30715e10 1.82951 0.914754 0.404012i $$-0.132384\pi$$
0.914754 + 0.404012i $$0.132384\pi$$
$$770$$ 5.60990e9 0.442830
$$771$$ 0 0
$$772$$ 2.74690e9 0.214873
$$773$$ −2.15091e10 −1.67492 −0.837461 0.546497i $$-0.815961\pi$$
−0.837461 + 0.546497i $$0.815961\pi$$
$$774$$ 0 0
$$775$$ −1.69758e10 −1.31001
$$776$$ 2.59765e10 1.99556
$$777$$ 0 0
$$778$$ −1.09107e10 −0.830664
$$779$$ −2.57547e9 −0.195198
$$780$$ 0 0
$$781$$ 2.40538e7 0.00180678
$$782$$ 1.96008e9 0.146572
$$783$$ 0 0
$$784$$ −2.80262e9 −0.207711
$$785$$ −2.20952e9 −0.163025
$$786$$ 0 0
$$787$$ −4.46678e9 −0.326651 −0.163325 0.986572i $$-0.552222\pi$$
−0.163325 + 0.986572i $$0.552222\pi$$
$$788$$ −3.06087e9 −0.222845
$$789$$ 0 0
$$790$$ −1.22083e9 −0.0880970
$$791$$ 5.82777e9 0.418682
$$792$$ 0 0
$$793$$ 7.14232e9 0.508608
$$794$$ 6.97868e9 0.494768
$$795$$ 0 0
$$796$$ 1.02009e9 0.0716873
$$797$$ −2.43899e10 −1.70650 −0.853248 0.521505i $$-0.825371\pi$$
−0.853248 + 0.521505i $$0.825371\pi$$
$$798$$ 0 0
$$799$$ −1.10425e10 −0.765869
$$800$$ −7.15481e9 −0.494064
$$801$$ 0 0
$$802$$ 1.74689e10 1.19579
$$803$$ −5.67089e8 −0.0386497
$$804$$ 0 0
$$805$$ −4.84196e9 −0.327142
$$806$$ 2.44481e10 1.64465
$$807$$ 0 0
$$808$$ 3.24990e9 0.216736
$$809$$ −9.88857e9 −0.656620 −0.328310 0.944570i $$-0.606479\pi$$
−0.328310 + 0.944570i $$0.606479\pi$$
$$810$$ 0 0
$$811$$ −1.15204e10 −0.758395 −0.379198 0.925316i $$-0.623800\pi$$
−0.379198 + 0.925316i $$0.623800\pi$$
$$812$$ 5.37265e9 0.352162
$$813$$ 0 0
$$814$$ −5.26924e9 −0.342423
$$815$$ 4.02487e10 2.60435
$$816$$ 0 0
$$817$$ −3.26139e10 −2.09231
$$818$$ 1.30304e10 0.832379
$$819$$ 0 0
$$820$$ −5.45828e8 −0.0345706
$$821$$ −2.63516e9 −0.166191 −0.0830953 0.996542i $$-0.526481\pi$$
−0.0830953 + 0.996542i $$0.526481\pi$$
$$822$$ 0 0
$$823$$ −1.27039e10 −0.794400 −0.397200 0.917732i $$-0.630018\pi$$
−0.397200 + 0.917732i $$0.630018\pi$$
$$824$$ −3.72912e9 −0.232200
$$825$$ 0 0
$$826$$ 2.75621e10 1.70170
$$827$$ 1.11339e10 0.684504 0.342252 0.939608i $$-0.388810\pi$$
0.342252 + 0.939608i $$0.388810\pi$$
$$828$$ 0 0
$$829$$ 2.79852e10 1.70604 0.853018 0.521881i $$-0.174770\pi$$
0.853018 + 0.521881i $$0.174770\pi$$
$$830$$ 2.33045e10 1.41471
$$831$$ 0 0
$$832$$ 3.02342e10 1.81998
$$833$$ −3.97956e9 −0.238549
$$834$$ 0 0
$$835$$ 1.80651e10 1.07383
$$836$$ 2.01873e9 0.119497
$$837$$ 0 0
$$838$$ 2.87139e10 1.68554
$$839$$ −2.71170e9 −0.158517 −0.0792583 0.996854i $$-0.525255\pi$$
−0.0792583 + 0.996854i $$0.525255\pi$$
$$840$$ 0 0
$$841$$ 1.75898e10 1.01971
$$842$$ −1.15946e10 −0.669364
$$843$$ 0 0
$$844$$ −3.88184e8 −0.0222249
$$845$$ 4.31161e10 2.45834
$$846$$ 0 0
$$847$$ −1.82116e9 −0.102981
$$848$$ −1.57737e10 −0.888275
$$849$$ 0 0
$$850$$ 1.53515e10 0.857404
$$851$$ 4.54794e9 0.252965
$$852$$ 0 0
$$853$$ 1.97175e10 1.08775 0.543877 0.839165i $$-0.316956\pi$$
0.543877 + 0.839165i $$0.316956\pi$$
$$854$$ 5.66623e9 0.311309
$$855$$ 0 0
$$856$$ 2.19359e10 1.19536
$$857$$ −1.89411e10 −1.02795 −0.513976 0.857804i $$-0.671828\pi$$
−0.513976 + 0.857804i $$0.671828\pi$$
$$858$$ 0 0
$$859$$ −6.77637e9 −0.364772 −0.182386 0.983227i $$-0.558382\pi$$
−0.182386 + 0.983227i $$0.558382\pi$$
$$860$$ −6.91197e9 −0.370559
$$861$$ 0 0
$$862$$ −1.66703e10 −0.886479
$$863$$ −2.80635e10 −1.48629 −0.743146 0.669129i $$-0.766667\pi$$
−0.743146 + 0.669129i $$0.766667\pi$$
$$864$$ 0 0
$$865$$ −2.75146e10 −1.44546
$$866$$ −6.34094e9 −0.331773
$$867$$ 0 0
$$868$$ −5.43073e9 −0.281864
$$869$$ 3.96324e8 0.0204871
$$870$$ 0 0
$$871$$ 5.95109e9 0.305164
$$872$$ −1.73660e10 −0.886937
$$873$$ 0 0
$$874$$ 6.22282e9 0.315281
$$875$$ −4.99454e9 −0.252039
$$876$$ 0 0
$$877$$ −1.01559e10 −0.508418 −0.254209 0.967149i $$-0.581815\pi$$
−0.254209 + 0.967149i $$0.581815\pi$$
$$878$$ 1.22368e10 0.610151
$$879$$ 0 0
$$880$$ −6.55725e9 −0.324364
$$881$$ 2.78023e10 1.36982 0.684912 0.728626i $$-0.259841\pi$$
0.684912 + 0.728626i $$0.259841\pi$$
$$882$$ 0 0
$$883$$ 2.15199e10 1.05191 0.525954 0.850513i $$-0.323709\pi$$
0.525954 + 0.850513i $$0.323709\pi$$
$$884$$ 6.19050e9 0.301400
$$885$$ 0 0
$$886$$ −1.23213e10 −0.595167
$$887$$ 3.13376e10 1.50776 0.753881 0.657011i $$-0.228180\pi$$
0.753881 + 0.657011i $$0.228180\pi$$
$$888$$ 0 0
$$889$$ 2.15027e10 1.02645
$$890$$ −2.60062e10 −1.23655
$$891$$ 0 0
$$892$$ −3.80618e9 −0.179561
$$893$$ −3.50575e10 −1.64741
$$894$$ 0 0
$$895$$ −1.78321e9 −0.0831423
$$896$$ 1.35222e10 0.628015
$$897$$ 0 0
$$898$$ −3.07511e10 −1.41708
$$899$$ −3.52164e10 −1.61654
$$900$$ 0 0
$$901$$ −2.23977e10 −1.02015
$$902$$ −6.32837e8 −0.0287124
$$903$$ 0 0
$$904$$ −8.84369e9 −0.398148
$$905$$ −4.92976e9 −0.221083
$$906$$ 0 0
$$907$$ −1.62459e10 −0.722966 −0.361483 0.932379i $$-0.617730\pi$$
−0.361483 + 0.932379i $$0.617730\pi$$
$$908$$ 7.90168e8 0.0350283
$$909$$ 0 0
$$910$$ 5.46154e10 2.40254
$$911$$ 3.15726e9 0.138356 0.0691778 0.997604i $$-0.477962\pi$$
0.0691778 + 0.997604i $$0.477962\pi$$
$$912$$ 0 0
$$913$$ −7.56544e9 −0.328993
$$914$$ 2.44730e10 1.06017
$$915$$ 0 0
$$916$$ 1.48740e9 0.0639432
$$917$$ −1.15803e10 −0.495938
$$918$$ 0 0
$$919$$ 2.53655e9 0.107805 0.0539026 0.998546i $$-0.482834\pi$$
0.0539026 + 0.998546i $$0.482834\pi$$
$$920$$ 7.34772e9 0.311097
$$921$$ 0 0
$$922$$ 9.52419e9 0.400193
$$923$$ 2.34177e8 0.00980253
$$924$$ 0 0
$$925$$ 3.56198e10 1.47978
$$926$$ 6.05200e9 0.250473
$$927$$ 0 0
$$928$$ −1.48427e10 −0.609671
$$929$$ 2.10282e10 0.860494 0.430247 0.902711i $$-0.358426\pi$$
0.430247 + 0.902711i $$0.358426\pi$$
$$930$$ 0 0
$$931$$ −1.26342e10 −0.513126
$$932$$ 4.32849e9 0.175138
$$933$$ 0 0
$$934$$ −1.37708e10 −0.553025
$$935$$ −9.31090e9 −0.372521
$$936$$ 0 0
$$937$$ 3.12893e10 1.24253 0.621265 0.783601i $$-0.286619\pi$$
0.621265 + 0.783601i $$0.286619\pi$$
$$938$$ 4.72119e9 0.186785
$$939$$ 0 0
$$940$$ −7.42986e9 −0.291765
$$941$$ −7.91706e9 −0.309742 −0.154871 0.987935i $$-0.549496\pi$$
−0.154871 + 0.987935i $$0.549496\pi$$
$$942$$ 0 0
$$943$$ 5.46208e8 0.0212113
$$944$$ −3.22166e10 −1.24646
$$945$$ 0 0
$$946$$ −8.01379e9 −0.307765
$$947$$ −8.55849e9 −0.327471 −0.163735 0.986504i $$-0.552354\pi$$
−0.163735 + 0.986504i $$0.552354\pi$$
$$948$$ 0 0
$$949$$ −5.52091e9 −0.209691
$$950$$ 4.87377e10 1.84430
$$951$$ 0 0
$$952$$ 2.73620e10 1.02782
$$953$$ 8.49661e9 0.317995 0.158998 0.987279i $$-0.449174\pi$$
0.158998 + 0.987279i $$0.449174\pi$$
$$954$$ 0 0
$$955$$ −2.44524e10 −0.908466
$$956$$ −5.21202e9 −0.192931
$$957$$ 0 0
$$958$$ −4.00222e10 −1.47069
$$959$$ 4.56730e8 0.0167222
$$960$$ 0 0
$$961$$ 8.08451e9 0.293847
$$962$$ −5.12989e10 −1.85778
$$963$$ 0 0
$$964$$ −7.33900e9 −0.263856
$$965$$ −4.02225e10 −1.44086
$$966$$ 0 0
$$967$$ −1.47988e10 −0.526300 −0.263150 0.964755i $$-0.584761\pi$$
−0.263150 + 0.964755i $$0.584761\pi$$
$$968$$ 2.76364e9 0.0979303
$$969$$ 0 0
$$970$$ −6.82715e10 −2.40181
$$971$$ 2.86157e10 1.00308 0.501542 0.865133i $$-0.332766\pi$$
0.501542 + 0.865133i $$0.332766\pi$$
$$972$$ 0 0
$$973$$ −3.52206e10 −1.22575
$$974$$ 2.88677e10 1.00105
$$975$$ 0 0
$$976$$ −6.62310e9 −0.228027
$$977$$ −3.37991e10 −1.15951 −0.579755 0.814791i $$-0.696852\pi$$
−0.579755 + 0.814791i $$0.696852\pi$$
$$978$$ 0 0
$$979$$ 8.44249e9 0.287562
$$980$$ −2.67761e9 −0.0908773
$$981$$ 0 0
$$982$$ 1.19743e9 0.0403515
$$983$$ −1.03134e9 −0.0346308 −0.0173154 0.999850i $$-0.505512\pi$$
−0.0173154 + 0.999850i $$0.505512\pi$$
$$984$$ 0 0
$$985$$ 4.48199e10 1.49432
$$986$$ 3.18469e10 1.05803
$$987$$ 0 0
$$988$$ 1.96535e10 0.648320
$$989$$ 6.91679e9 0.227362
$$990$$ 0 0
$$991$$ −5.63139e10 −1.83805 −0.919027 0.394195i $$-0.871023\pi$$
−0.919027 + 0.394195i $$0.871023\pi$$
$$992$$ 1.50032e10 0.487970
$$993$$ 0 0
$$994$$ 1.85780e8 0.00599994
$$995$$ −1.49370e10 −0.480710
$$996$$ 0 0
$$997$$ 2.55531e10 0.816603 0.408301 0.912847i $$-0.366121\pi$$
0.408301 + 0.912847i $$0.366121\pi$$
$$998$$ 4.78950e10 1.52522
$$999$$ 0 0
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 99.8.a.a.1.1 1
3.2 odd 2 33.8.a.a.1.1 1
12.11 even 2 528.8.a.a.1.1 1
33.32 even 2 363.8.a.a.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
33.8.a.a.1.1 1 3.2 odd 2
99.8.a.a.1.1 1 1.1 even 1 trivial
363.8.a.a.1.1 1 33.32 even 2
528.8.a.a.1.1 1 12.11 even 2