# Properties

 Label 819.2.a.k.1.2 Level $819$ Weight $2$ Character 819.1 Self dual yes Analytic conductor $6.540$ Analytic rank $0$ Dimension $4$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [819,2,Mod(1,819)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(819, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("819.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$819 = 3^{2} \cdot 7 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 819.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: $$6.53974792554$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: 4.4.17428.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - x^{3} - 6x^{2} + 4x + 6$$ x^4 - x^3 - 6*x^2 + 4*x + 6 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 273) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$-2.10710$$ of defining polynomial Character $$\chi$$ $$=$$ 819.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-1.43986 q^{2} +0.0731828 q^{4} -0.926817 q^{5} +1.00000 q^{7} +2.77434 q^{8} +O(q^{10})$$ $$q-1.43986 q^{2} +0.0731828 q^{4} -0.926817 q^{5} +1.00000 q^{7} +2.77434 q^{8} +1.33448 q^{10} -4.21419 q^{11} +1.00000 q^{13} -1.43986 q^{14} -4.14101 q^{16} +2.87971 q^{17} -1.28738 q^{19} -0.0678271 q^{20} +6.06783 q^{22} +8.02072 q^{23} -4.14101 q^{25} -1.43986 q^{26} +0.0731828 q^{28} -3.28738 q^{29} -7.04680 q^{31} +0.413779 q^{32} -4.14637 q^{34} -0.926817 q^{35} +8.57475 q^{37} +1.85363 q^{38} -2.57130 q^{40} +12.0678 q^{41} +7.14101 q^{43} -0.308407 q^{44} -11.5487 q^{46} -1.95289 q^{47} +1.00000 q^{49} +5.96245 q^{50} +0.0731828 q^{52} +5.14101 q^{53} +3.90579 q^{55} +2.77434 q^{56} +4.73334 q^{58} +7.33448 q^{59} +7.75942 q^{61} +10.1464 q^{62} +7.68624 q^{64} -0.926817 q^{65} +12.0414 q^{67} +0.210745 q^{68} +1.33448 q^{70} -10.7889 q^{71} -8.32568 q^{73} -12.3464 q^{74} -0.0942138 q^{76} -4.21419 q^{77} +4.47204 q^{79} +3.83796 q^{80} -17.3759 q^{82} +3.80653 q^{83} -2.66896 q^{85} -10.2820 q^{86} -11.6916 q^{88} +5.64793 q^{89} +1.00000 q^{91} +0.586979 q^{92} +2.81188 q^{94} +1.19316 q^{95} +6.90043 q^{97} -1.43986 q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - q^{2} + 7 q^{4} + 3 q^{5} + 4 q^{7} - 3 q^{8}+O(q^{10})$$ 4 * q - q^2 + 7 * q^4 + 3 * q^5 + 4 * q^7 - 3 * q^8 $$4 q - q^{2} + 7 q^{4} + 3 q^{5} + 4 q^{7} - 3 q^{8} - 4 q^{10} + 2 q^{11} + 4 q^{13} - q^{14} + 9 q^{16} + 2 q^{17} + 7 q^{19} + 32 q^{20} - 8 q^{22} - 3 q^{23} + 9 q^{25} - q^{26} + 7 q^{28} - q^{29} + 3 q^{31} - 7 q^{32} - 30 q^{34} + 3 q^{35} + 10 q^{37} - 6 q^{38} - 14 q^{40} + 16 q^{41} + 3 q^{43} + 12 q^{44} - 18 q^{46} - 5 q^{47} + 4 q^{49} - 13 q^{50} + 7 q^{52} - 5 q^{53} + 10 q^{55} - 3 q^{56} - 4 q^{58} + 20 q^{59} + 12 q^{61} + 54 q^{62} + 5 q^{64} + 3 q^{65} - 22 q^{67} + 10 q^{68} - 4 q^{70} - 13 q^{73} + 6 q^{74} - 6 q^{76} + 2 q^{77} + 11 q^{79} + 42 q^{80} + 10 q^{82} - q^{83} + 8 q^{85} + 10 q^{86} - 60 q^{88} + 5 q^{89} + 4 q^{91} - 34 q^{92} + 34 q^{94} - 13 q^{95} - 17 q^{97} - q^{98}+O(q^{100})$$ 4 * q - q^2 + 7 * q^4 + 3 * q^5 + 4 * q^7 - 3 * q^8 - 4 * q^10 + 2 * q^11 + 4 * q^13 - q^14 + 9 * q^16 + 2 * q^17 + 7 * q^19 + 32 * q^20 - 8 * q^22 - 3 * q^23 + 9 * q^25 - q^26 + 7 * q^28 - q^29 + 3 * q^31 - 7 * q^32 - 30 * q^34 + 3 * q^35 + 10 * q^37 - 6 * q^38 - 14 * q^40 + 16 * q^41 + 3 * q^43 + 12 * q^44 - 18 * q^46 - 5 * q^47 + 4 * q^49 - 13 * q^50 + 7 * q^52 - 5 * q^53 + 10 * q^55 - 3 * q^56 - 4 * q^58 + 20 * q^59 + 12 * q^61 + 54 * q^62 + 5 * q^64 + 3 * q^65 - 22 * q^67 + 10 * q^68 - 4 * q^70 - 13 * q^73 + 6 * q^74 - 6 * q^76 + 2 * q^77 + 11 * q^79 + 42 * q^80 + 10 * q^82 - q^83 + 8 * q^85 + 10 * q^86 - 60 * q^88 + 5 * q^89 + 4 * q^91 - 34 * q^92 + 34 * q^94 - 13 * q^95 - 17 * q^97 - q^98

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −1.43986 −1.01813 −0.509066 0.860728i $$-0.670009\pi$$
−0.509066 + 0.860728i $$0.670009\pi$$
$$3$$ 0 0
$$4$$ 0.0731828 0.0365914
$$5$$ −0.926817 −0.414485 −0.207243 0.978290i $$-0.566449\pi$$
−0.207243 + 0.978290i $$0.566449\pi$$
$$6$$ 0 0
$$7$$ 1.00000 0.377964
$$8$$ 2.77434 0.980876
$$9$$ 0 0
$$10$$ 1.33448 0.422000
$$11$$ −4.21419 −1.27063 −0.635313 0.772254i $$-0.719129\pi$$
−0.635313 + 0.772254i $$0.719129\pi$$
$$12$$ 0 0
$$13$$ 1.00000 0.277350
$$14$$ −1.43986 −0.384817
$$15$$ 0 0
$$16$$ −4.14101 −1.03525
$$17$$ 2.87971 0.698432 0.349216 0.937042i $$-0.386448\pi$$
0.349216 + 0.937042i $$0.386448\pi$$
$$18$$ 0 0
$$19$$ −1.28738 −0.295344 −0.147672 0.989036i $$-0.547178\pi$$
−0.147672 + 0.989036i $$0.547178\pi$$
$$20$$ −0.0678271 −0.0151666
$$21$$ 0 0
$$22$$ 6.06783 1.29367
$$23$$ 8.02072 1.67244 0.836218 0.548397i $$-0.184762\pi$$
0.836218 + 0.548397i $$0.184762\pi$$
$$24$$ 0 0
$$25$$ −4.14101 −0.828202
$$26$$ −1.43986 −0.282379
$$27$$ 0 0
$$28$$ 0.0731828 0.0138303
$$29$$ −3.28738 −0.610450 −0.305225 0.952280i $$-0.598732\pi$$
−0.305225 + 0.952280i $$0.598732\pi$$
$$30$$ 0 0
$$31$$ −7.04680 −1.26564 −0.632821 0.774298i $$-0.718103\pi$$
−0.632821 + 0.774298i $$0.718103\pi$$
$$32$$ 0.413779 0.0731465
$$33$$ 0 0
$$34$$ −4.14637 −0.711096
$$35$$ −0.926817 −0.156661
$$36$$ 0 0
$$37$$ 8.57475 1.40968 0.704840 0.709366i $$-0.251019\pi$$
0.704840 + 0.709366i $$0.251019\pi$$
$$38$$ 1.85363 0.300699
$$39$$ 0 0
$$40$$ −2.57130 −0.406559
$$41$$ 12.0678 1.88468 0.942339 0.334660i $$-0.108621\pi$$
0.942339 + 0.334660i $$0.108621\pi$$
$$42$$ 0 0
$$43$$ 7.14101 1.08899 0.544497 0.838763i $$-0.316721\pi$$
0.544497 + 0.838763i $$0.316721\pi$$
$$44$$ −0.308407 −0.0464940
$$45$$ 0 0
$$46$$ −11.5487 −1.70276
$$47$$ −1.95289 −0.284859 −0.142429 0.989805i $$-0.545491\pi$$
−0.142429 + 0.989805i $$0.545491\pi$$
$$48$$ 0 0
$$49$$ 1.00000 0.142857
$$50$$ 5.96245 0.843218
$$51$$ 0 0
$$52$$ 0.0731828 0.0101486
$$53$$ 5.14101 0.706172 0.353086 0.935591i $$-0.385132\pi$$
0.353086 + 0.935591i $$0.385132\pi$$
$$54$$ 0 0
$$55$$ 3.90579 0.526656
$$56$$ 2.77434 0.370736
$$57$$ 0 0
$$58$$ 4.73334 0.621519
$$59$$ 7.33448 0.954868 0.477434 0.878668i $$-0.341567\pi$$
0.477434 + 0.878668i $$0.341567\pi$$
$$60$$ 0 0
$$61$$ 7.75942 0.993492 0.496746 0.867896i $$-0.334528\pi$$
0.496746 + 0.867896i $$0.334528\pi$$
$$62$$ 10.1464 1.28859
$$63$$ 0 0
$$64$$ 7.68624 0.960780
$$65$$ −0.926817 −0.114958
$$66$$ 0 0
$$67$$ 12.0414 1.47110 0.735548 0.677473i $$-0.236925\pi$$
0.735548 + 0.677473i $$0.236925\pi$$
$$68$$ 0.210745 0.0255566
$$69$$ 0 0
$$70$$ 1.33448 0.159501
$$71$$ −10.7889 −1.28041 −0.640206 0.768203i $$-0.721151\pi$$
−0.640206 + 0.768203i $$0.721151\pi$$
$$72$$ 0 0
$$73$$ −8.32568 −0.974447 −0.487224 0.873277i $$-0.661990\pi$$
−0.487224 + 0.873277i $$0.661990\pi$$
$$74$$ −12.3464 −1.43524
$$75$$ 0 0
$$76$$ −0.0942138 −0.0108071
$$77$$ −4.21419 −0.480252
$$78$$ 0 0
$$79$$ 4.47204 0.503144 0.251572 0.967839i $$-0.419052\pi$$
0.251572 + 0.967839i $$0.419052\pi$$
$$80$$ 3.83796 0.429097
$$81$$ 0 0
$$82$$ −17.3759 −1.91885
$$83$$ 3.80653 0.417821 0.208910 0.977935i $$-0.433008\pi$$
0.208910 + 0.977935i $$0.433008\pi$$
$$84$$ 0 0
$$85$$ −2.66896 −0.289490
$$86$$ −10.2820 −1.10874
$$87$$ 0 0
$$88$$ −11.6916 −1.24633
$$89$$ 5.64793 0.598680 0.299340 0.954147i $$-0.403234\pi$$
0.299340 + 0.954147i $$0.403234\pi$$
$$90$$ 0 0
$$91$$ 1.00000 0.104828
$$92$$ 0.586979 0.0611968
$$93$$ 0 0
$$94$$ 2.81188 0.290024
$$95$$ 1.19316 0.122416
$$96$$ 0 0
$$97$$ 6.90043 0.700633 0.350316 0.936631i $$-0.386074\pi$$
0.350316 + 0.936631i $$0.386074\pi$$
$$98$$ −1.43986 −0.145447
$$99$$ 0 0
$$100$$ −0.303051 −0.0303051
$$101$$ 10.9739 1.09195 0.545973 0.837803i $$-0.316160\pi$$
0.545973 + 0.837803i $$0.316160\pi$$
$$102$$ 0 0
$$103$$ 16.4284 1.61874 0.809368 0.587301i $$-0.199810\pi$$
0.809368 + 0.587301i $$0.199810\pi$$
$$104$$ 2.77434 0.272046
$$105$$ 0 0
$$106$$ −7.40231 −0.718976
$$107$$ 3.45446 0.333955 0.166978 0.985961i $$-0.446599\pi$$
0.166978 + 0.985961i $$0.446599\pi$$
$$108$$ 0 0
$$109$$ 4.66896 0.447206 0.223603 0.974680i $$-0.428218\pi$$
0.223603 + 0.974680i $$0.428218\pi$$
$$110$$ −5.62377 −0.536205
$$111$$ 0 0
$$112$$ −4.14101 −0.391289
$$113$$ 3.28738 0.309250 0.154625 0.987973i $$-0.450583\pi$$
0.154625 + 0.987973i $$0.450583\pi$$
$$114$$ 0 0
$$115$$ −7.43374 −0.693200
$$116$$ −0.240579 −0.0223372
$$117$$ 0 0
$$118$$ −10.5606 −0.972181
$$119$$ 2.87971 0.263983
$$120$$ 0 0
$$121$$ 6.75942 0.614493
$$122$$ −11.1724 −1.01151
$$123$$ 0 0
$$124$$ −0.515704 −0.0463116
$$125$$ 8.47204 0.757763
$$126$$ 0 0
$$127$$ −0.428386 −0.0380131 −0.0190065 0.999819i $$-0.506050\pi$$
−0.0190065 + 0.999819i $$0.506050\pi$$
$$128$$ −11.8946 −1.05135
$$129$$ 0 0
$$130$$ 1.33448 0.117042
$$131$$ −14.1878 −1.23959 −0.619797 0.784762i $$-0.712785\pi$$
−0.619797 + 0.784762i $$0.712785\pi$$
$$132$$ 0 0
$$133$$ −1.28738 −0.111630
$$134$$ −17.3379 −1.49777
$$135$$ 0 0
$$136$$ 7.98929 0.685076
$$137$$ −14.7070 −1.25650 −0.628250 0.778011i $$-0.716229\pi$$
−0.628250 + 0.778011i $$0.716229\pi$$
$$138$$ 0 0
$$139$$ 9.85363 0.835774 0.417887 0.908499i $$-0.362771\pi$$
0.417887 + 0.908499i $$0.362771\pi$$
$$140$$ −0.0678271 −0.00573244
$$141$$ 0 0
$$142$$ 15.5345 1.30363
$$143$$ −4.21419 −0.352409
$$144$$ 0 0
$$145$$ 3.04680 0.253023
$$146$$ 11.9878 0.992115
$$147$$ 0 0
$$148$$ 0.627525 0.0515822
$$149$$ 3.38663 0.277444 0.138722 0.990331i $$-0.455701\pi$$
0.138722 + 0.990331i $$0.455701\pi$$
$$150$$ 0 0
$$151$$ −21.8951 −1.78180 −0.890898 0.454204i $$-0.849924\pi$$
−0.890898 + 0.454204i $$0.849924\pi$$
$$152$$ −3.57161 −0.289696
$$153$$ 0 0
$$154$$ 6.06783 0.488959
$$155$$ 6.53109 0.524590
$$156$$ 0 0
$$157$$ −0.867482 −0.0692326 −0.0346163 0.999401i $$-0.511021\pi$$
−0.0346163 + 0.999401i $$0.511021\pi$$
$$158$$ −6.43910 −0.512267
$$159$$ 0 0
$$160$$ −0.383498 −0.0303182
$$161$$ 8.02072 0.632121
$$162$$ 0 0
$$163$$ 4.42839 0.346858 0.173429 0.984846i $$-0.444515\pi$$
0.173429 + 0.984846i $$0.444515\pi$$
$$164$$ 0.883158 0.0689630
$$165$$ 0 0
$$166$$ −5.48085 −0.425396
$$167$$ −20.5276 −1.58848 −0.794238 0.607606i $$-0.792130\pi$$
−0.794238 + 0.607606i $$0.792130\pi$$
$$168$$ 0 0
$$169$$ 1.00000 0.0769231
$$170$$ 3.84292 0.294739
$$171$$ 0 0
$$172$$ 0.522599 0.0398478
$$173$$ −15.0154 −1.14160 −0.570799 0.821090i $$-0.693366\pi$$
−0.570799 + 0.821090i $$0.693366\pi$$
$$174$$ 0 0
$$175$$ −4.14101 −0.313031
$$176$$ 17.4510 1.31542
$$177$$ 0 0
$$178$$ −8.13221 −0.609535
$$179$$ 5.35176 0.400009 0.200004 0.979795i $$-0.435904\pi$$
0.200004 + 0.979795i $$0.435904\pi$$
$$180$$ 0 0
$$181$$ 11.4667 0.852312 0.426156 0.904650i $$-0.359867\pi$$
0.426156 + 0.904650i $$0.359867\pi$$
$$182$$ −1.43986 −0.106729
$$183$$ 0 0
$$184$$ 22.2522 1.64045
$$185$$ −7.94723 −0.584292
$$186$$ 0 0
$$187$$ −12.1357 −0.887447
$$188$$ −0.142918 −0.0104234
$$189$$ 0 0
$$190$$ −1.71798 −0.124635
$$191$$ −20.1756 −1.45985 −0.729927 0.683525i $$-0.760446\pi$$
−0.729927 + 0.683525i $$0.760446\pi$$
$$192$$ 0 0
$$193$$ −26.3756 −1.89856 −0.949279 0.314435i $$-0.898185\pi$$
−0.949279 + 0.314435i $$0.898185\pi$$
$$194$$ −9.93562 −0.713336
$$195$$ 0 0
$$196$$ 0.0731828 0.00522734
$$197$$ 20.1322 1.43436 0.717180 0.696888i $$-0.245432\pi$$
0.717180 + 0.696888i $$0.245432\pi$$
$$198$$ 0 0
$$199$$ 1.75942 0.124722 0.0623610 0.998054i $$-0.480137\pi$$
0.0623610 + 0.998054i $$0.480137\pi$$
$$200$$ −11.4886 −0.812364
$$201$$ 0 0
$$202$$ −15.8009 −1.11174
$$203$$ −3.28738 −0.230729
$$204$$ 0 0
$$205$$ −11.1847 −0.781171
$$206$$ −23.6545 −1.64809
$$207$$ 0 0
$$208$$ −4.14101 −0.287127
$$209$$ 5.42525 0.375272
$$210$$ 0 0
$$211$$ 15.5694 1.07184 0.535921 0.844268i $$-0.319965\pi$$
0.535921 + 0.844268i $$0.319965\pi$$
$$212$$ 0.376234 0.0258398
$$213$$ 0 0
$$214$$ −4.97392 −0.340010
$$215$$ −6.61841 −0.451372
$$216$$ 0 0
$$217$$ −7.04680 −0.478368
$$218$$ −6.72263 −0.455314
$$219$$ 0 0
$$220$$ 0.285836 0.0192711
$$221$$ 2.87971 0.193710
$$222$$ 0 0
$$223$$ −11.5694 −0.774744 −0.387372 0.921923i $$-0.626617\pi$$
−0.387372 + 0.921923i $$0.626617\pi$$
$$224$$ 0.413779 0.0276468
$$225$$ 0 0
$$226$$ −4.73334 −0.314857
$$227$$ 11.0418 0.732867 0.366433 0.930444i $$-0.380579\pi$$
0.366433 + 0.930444i $$0.380579\pi$$
$$228$$ 0 0
$$229$$ −13.7073 −0.905802 −0.452901 0.891561i $$-0.649611\pi$$
−0.452901 + 0.891561i $$0.649611\pi$$
$$230$$ 10.7035 0.705769
$$231$$ 0 0
$$232$$ −9.12029 −0.598776
$$233$$ −1.23522 −0.0809222 −0.0404611 0.999181i $$-0.512883\pi$$
−0.0404611 + 0.999181i $$0.512883\pi$$
$$234$$ 0 0
$$235$$ 1.80997 0.118070
$$236$$ 0.536758 0.0349400
$$237$$ 0 0
$$238$$ −4.14637 −0.268769
$$239$$ −19.1061 −1.23587 −0.617936 0.786228i $$-0.712031\pi$$
−0.617936 + 0.786228i $$0.712031\pi$$
$$240$$ 0 0
$$241$$ −12.0329 −0.775110 −0.387555 0.921847i $$-0.626680\pi$$
−0.387555 + 0.921847i $$0.626680\pi$$
$$242$$ −9.73259 −0.625634
$$243$$ 0 0
$$244$$ 0.567856 0.0363533
$$245$$ −0.926817 −0.0592122
$$246$$ 0 0
$$247$$ −1.28738 −0.0819137
$$248$$ −19.5502 −1.24144
$$249$$ 0 0
$$250$$ −12.1985 −0.771502
$$251$$ 15.0031 0.946990 0.473495 0.880797i $$-0.342992\pi$$
0.473495 + 0.880797i $$0.342992\pi$$
$$252$$ 0 0
$$253$$ −33.8009 −2.12504
$$254$$ 0.616813 0.0387023
$$255$$ 0 0
$$256$$ 1.75406 0.109629
$$257$$ 13.0261 0.812544 0.406272 0.913752i $$-0.366828\pi$$
0.406272 + 0.913752i $$0.366828\pi$$
$$258$$ 0 0
$$259$$ 8.57475 0.532809
$$260$$ −0.0678271 −0.00420646
$$261$$ 0 0
$$262$$ 20.4284 1.26207
$$263$$ 2.59547 0.160044 0.0800218 0.996793i $$-0.474501\pi$$
0.0800218 + 0.996793i $$0.474501\pi$$
$$264$$ 0 0
$$265$$ −4.76478 −0.292698
$$266$$ 1.85363 0.113654
$$267$$ 0 0
$$268$$ 0.881227 0.0538295
$$269$$ 23.3081 1.42112 0.710560 0.703637i $$-0.248442\pi$$
0.710560 + 0.703637i $$0.248442\pi$$
$$270$$ 0 0
$$271$$ −6.05215 −0.367642 −0.183821 0.982960i $$-0.558847\pi$$
−0.183821 + 0.982960i $$0.558847\pi$$
$$272$$ −11.9249 −0.723054
$$273$$ 0 0
$$274$$ 21.1759 1.27928
$$275$$ 17.4510 1.05234
$$276$$ 0 0
$$277$$ 15.1932 0.912869 0.456434 0.889757i $$-0.349126\pi$$
0.456434 + 0.889757i $$0.349126\pi$$
$$278$$ −14.1878 −0.850928
$$279$$ 0 0
$$280$$ −2.57130 −0.153665
$$281$$ 7.57506 0.451890 0.225945 0.974140i $$-0.427453\pi$$
0.225945 + 0.974140i $$0.427453\pi$$
$$282$$ 0 0
$$283$$ 19.0974 1.13522 0.567610 0.823298i $$-0.307868\pi$$
0.567610 + 0.823298i $$0.307868\pi$$
$$284$$ −0.789565 −0.0468521
$$285$$ 0 0
$$286$$ 6.06783 0.358798
$$287$$ 12.0678 0.712341
$$288$$ 0 0
$$289$$ −8.70727 −0.512192
$$290$$ −4.38695 −0.257610
$$291$$ 0 0
$$292$$ −0.609297 −0.0356564
$$293$$ 3.79430 0.221665 0.110833 0.993839i $$-0.464648\pi$$
0.110833 + 0.993839i $$0.464648\pi$$
$$294$$ 0 0
$$295$$ −6.79772 −0.395779
$$296$$ 23.7893 1.38272
$$297$$ 0 0
$$298$$ −4.87626 −0.282474
$$299$$ 8.02072 0.463850
$$300$$ 0 0
$$301$$ 7.14101 0.411601
$$302$$ 31.5257 1.81410
$$303$$ 0 0
$$304$$ 5.33104 0.305756
$$305$$ −7.19156 −0.411788
$$306$$ 0 0
$$307$$ −5.19316 −0.296389 −0.148195 0.988958i $$-0.547346\pi$$
−0.148195 + 0.988958i $$0.547346\pi$$
$$308$$ −0.308407 −0.0175731
$$309$$ 0 0
$$310$$ −9.40383 −0.534101
$$311$$ 13.8536 0.785568 0.392784 0.919631i $$-0.371512\pi$$
0.392784 + 0.919631i $$0.371512\pi$$
$$312$$ 0 0
$$313$$ −32.6162 −1.84358 −0.921788 0.387694i $$-0.873272\pi$$
−0.921788 + 0.387694i $$0.873272\pi$$
$$314$$ 1.24905 0.0704879
$$315$$ 0 0
$$316$$ 0.327277 0.0184108
$$317$$ −12.0380 −0.676121 −0.338061 0.941124i $$-0.609771\pi$$
−0.338061 + 0.941124i $$0.609771\pi$$
$$318$$ 0 0
$$319$$ 13.8536 0.775655
$$320$$ −7.12374 −0.398229
$$321$$ 0 0
$$322$$ −11.5487 −0.643583
$$323$$ −3.70727 −0.206278
$$324$$ 0 0
$$325$$ −4.14101 −0.229702
$$326$$ −6.37623 −0.353147
$$327$$ 0 0
$$328$$ 33.4802 1.84864
$$329$$ −1.95289 −0.107666
$$330$$ 0 0
$$331$$ −26.2820 −1.44459 −0.722295 0.691585i $$-0.756913\pi$$
−0.722295 + 0.691585i $$0.756913\pi$$
$$332$$ 0.278572 0.0152886
$$333$$ 0 0
$$334$$ 29.5568 1.61728
$$335$$ −11.1602 −0.609748
$$336$$ 0 0
$$337$$ 26.2905 1.43214 0.716068 0.698031i $$-0.245940\pi$$
0.716068 + 0.698031i $$0.245940\pi$$
$$338$$ −1.43986 −0.0783178
$$339$$ 0 0
$$340$$ −0.195322 −0.0105928
$$341$$ 29.6966 1.60816
$$342$$ 0 0
$$343$$ 1.00000 0.0539949
$$344$$ 19.8116 1.06817
$$345$$ 0 0
$$346$$ 21.6199 1.16230
$$347$$ −16.4928 −0.885378 −0.442689 0.896675i $$-0.645975\pi$$
−0.442689 + 0.896675i $$0.645975\pi$$
$$348$$ 0 0
$$349$$ 14.1793 0.759001 0.379501 0.925191i $$-0.376096\pi$$
0.379501 + 0.925191i $$0.376096\pi$$
$$350$$ 5.96245 0.318707
$$351$$ 0 0
$$352$$ −1.74375 −0.0929419
$$353$$ 29.8272 1.58754 0.793772 0.608215i $$-0.208114\pi$$
0.793772 + 0.608215i $$0.208114\pi$$
$$354$$ 0 0
$$355$$ 9.99938 0.530712
$$356$$ 0.413332 0.0219065
$$357$$ 0 0
$$358$$ −7.70575 −0.407262
$$359$$ −1.34671 −0.0710767 −0.0355383 0.999368i $$-0.511315\pi$$
−0.0355383 + 0.999368i $$0.511315\pi$$
$$360$$ 0 0
$$361$$ −17.3427 −0.912772
$$362$$ −16.5104 −0.867766
$$363$$ 0 0
$$364$$ 0.0731828 0.00383582
$$365$$ 7.71638 0.403894
$$366$$ 0 0
$$367$$ 36.1771 1.88843 0.944214 0.329331i $$-0.106823\pi$$
0.944214 + 0.329331i $$0.106823\pi$$
$$368$$ −33.2139 −1.73139
$$369$$ 0 0
$$370$$ 11.4429 0.594886
$$371$$ 5.14101 0.266908
$$372$$ 0 0
$$373$$ −16.9089 −0.875511 −0.437755 0.899094i $$-0.644226\pi$$
−0.437755 + 0.899094i $$0.644226\pi$$
$$374$$ 17.4736 0.903538
$$375$$ 0 0
$$376$$ −5.41798 −0.279411
$$377$$ −3.28738 −0.169308
$$378$$ 0 0
$$379$$ −11.6131 −0.596523 −0.298261 0.954484i $$-0.596407\pi$$
−0.298261 + 0.954484i $$0.596407\pi$$
$$380$$ 0.0873190 0.00447937
$$381$$ 0 0
$$382$$ 29.0499 1.48632
$$383$$ −14.6134 −0.746708 −0.373354 0.927689i $$-0.621792\pi$$
−0.373354 + 0.927689i $$0.621792\pi$$
$$384$$ 0 0
$$385$$ 3.90579 0.199057
$$386$$ 37.9771 1.93298
$$387$$ 0 0
$$388$$ 0.504993 0.0256371
$$389$$ −6.00000 −0.304212 −0.152106 0.988364i $$-0.548606\pi$$
−0.152106 + 0.988364i $$0.548606\pi$$
$$390$$ 0 0
$$391$$ 23.0974 1.16808
$$392$$ 2.77434 0.140125
$$393$$ 0 0
$$394$$ −28.9875 −1.46037
$$395$$ −4.14477 −0.208546
$$396$$ 0 0
$$397$$ 25.6982 1.28975 0.644877 0.764287i $$-0.276909\pi$$
0.644877 + 0.764287i $$0.276909\pi$$
$$398$$ −2.53331 −0.126983
$$399$$ 0 0
$$400$$ 17.1480 0.857398
$$401$$ 19.5637 0.976966 0.488483 0.872573i $$-0.337550\pi$$
0.488483 + 0.872573i $$0.337550\pi$$
$$402$$ 0 0
$$403$$ −7.04680 −0.351026
$$404$$ 0.803103 0.0399559
$$405$$ 0 0
$$406$$ 4.73334 0.234912
$$407$$ −36.1357 −1.79118
$$408$$ 0 0
$$409$$ 3.19316 0.157892 0.0789458 0.996879i $$-0.474845\pi$$
0.0789458 + 0.996879i $$0.474845\pi$$
$$410$$ 16.1043 0.795335
$$411$$ 0 0
$$412$$ 1.20228 0.0592319
$$413$$ 7.33448 0.360906
$$414$$ 0 0
$$415$$ −3.52796 −0.173181
$$416$$ 0.413779 0.0202872
$$417$$ 0 0
$$418$$ −7.81157 −0.382076
$$419$$ −30.2292 −1.47680 −0.738398 0.674366i $$-0.764417\pi$$
−0.738398 + 0.674366i $$0.764417\pi$$
$$420$$ 0 0
$$421$$ 26.9510 1.31351 0.656755 0.754104i $$-0.271928\pi$$
0.656755 + 0.754104i $$0.271928\pi$$
$$422$$ −22.4177 −1.09128
$$423$$ 0 0
$$424$$ 14.2629 0.692668
$$425$$ −11.9249 −0.578443
$$426$$ 0 0
$$427$$ 7.75942 0.375505
$$428$$ 0.252807 0.0122199
$$429$$ 0 0
$$430$$ 9.52955 0.459556
$$431$$ −16.7713 −0.807847 −0.403923 0.914793i $$-0.632354\pi$$
−0.403923 + 0.914793i $$0.632354\pi$$
$$432$$ 0 0
$$433$$ 25.8530 1.24242 0.621208 0.783646i $$-0.286642\pi$$
0.621208 + 0.783646i $$0.286642\pi$$
$$434$$ 10.1464 0.487041
$$435$$ 0 0
$$436$$ 0.341688 0.0163639
$$437$$ −10.3257 −0.493944
$$438$$ 0 0
$$439$$ 25.0553 1.19582 0.597912 0.801562i $$-0.295997\pi$$
0.597912 + 0.801562i $$0.295997\pi$$
$$440$$ 10.8360 0.516585
$$441$$ 0 0
$$442$$ −4.14637 −0.197223
$$443$$ −4.64762 −0.220815 −0.110408 0.993886i $$-0.535216\pi$$
−0.110408 + 0.993886i $$0.535216\pi$$
$$444$$ 0 0
$$445$$ −5.23460 −0.248144
$$446$$ 16.6583 0.788791
$$447$$ 0 0
$$448$$ 7.68624 0.363141
$$449$$ −0.494696 −0.0233462 −0.0116731 0.999932i $$-0.503716\pi$$
−0.0116731 + 0.999932i $$0.503716\pi$$
$$450$$ 0 0
$$451$$ −50.8561 −2.39472
$$452$$ 0.240579 0.0113159
$$453$$ 0 0
$$454$$ −15.8985 −0.746155
$$455$$ −0.926817 −0.0434499
$$456$$ 0 0
$$457$$ 26.4698 1.23821 0.619103 0.785310i $$-0.287496\pi$$
0.619103 + 0.785310i $$0.287496\pi$$
$$458$$ 19.7365 0.922225
$$459$$ 0 0
$$460$$ −0.544022 −0.0253652
$$461$$ 1.08168 0.0503786 0.0251893 0.999683i $$-0.491981\pi$$
0.0251893 + 0.999683i $$0.491981\pi$$
$$462$$ 0 0
$$463$$ −32.7098 −1.52015 −0.760076 0.649834i $$-0.774838\pi$$
−0.760076 + 0.649834i $$0.774838\pi$$
$$464$$ 13.6131 0.631970
$$465$$ 0 0
$$466$$ 1.77854 0.0823894
$$467$$ 17.0797 0.790356 0.395178 0.918605i $$-0.370683\pi$$
0.395178 + 0.918605i $$0.370683\pi$$
$$468$$ 0 0
$$469$$ 12.0414 0.556022
$$470$$ −2.60610 −0.120211
$$471$$ 0 0
$$472$$ 20.3483 0.936608
$$473$$ −30.0936 −1.38370
$$474$$ 0 0
$$475$$ 5.33104 0.244605
$$476$$ 0.210745 0.00965950
$$477$$ 0 0
$$478$$ 27.5101 1.25828
$$479$$ 17.1790 0.784929 0.392464 0.919767i $$-0.371623\pi$$
0.392464 + 0.919767i $$0.371623\pi$$
$$480$$ 0 0
$$481$$ 8.57475 0.390975
$$482$$ 17.3257 0.789164
$$483$$ 0 0
$$484$$ 0.494674 0.0224852
$$485$$ −6.39544 −0.290402
$$486$$ 0 0
$$487$$ −4.28264 −0.194065 −0.0970325 0.995281i $$-0.530935\pi$$
−0.0970325 + 0.995281i $$0.530935\pi$$
$$488$$ 21.5273 0.974493
$$489$$ 0 0
$$490$$ 1.33448 0.0602858
$$491$$ −27.4545 −1.23900 −0.619501 0.784996i $$-0.712665\pi$$
−0.619501 + 0.784996i $$0.712665\pi$$
$$492$$ 0 0
$$493$$ −9.46669 −0.426358
$$494$$ 1.85363 0.0833990
$$495$$ 0 0
$$496$$ 29.1809 1.31026
$$497$$ −10.7889 −0.483950
$$498$$ 0 0
$$499$$ −37.0452 −1.65837 −0.829185 0.558974i $$-0.811195\pi$$
−0.829185 + 0.558974i $$0.811195\pi$$
$$500$$ 0.620008 0.0277276
$$501$$ 0 0
$$502$$ −21.6023 −0.964160
$$503$$ −3.23744 −0.144350 −0.0721752 0.997392i $$-0.522994\pi$$
−0.0721752 + 0.997392i $$0.522994\pi$$
$$504$$ 0 0
$$505$$ −10.1708 −0.452596
$$506$$ 48.6683 2.16357
$$507$$ 0 0
$$508$$ −0.0313505 −0.00139095
$$509$$ 30.4877 1.35134 0.675672 0.737202i $$-0.263853\pi$$
0.675672 + 0.737202i $$0.263853\pi$$
$$510$$ 0 0
$$511$$ −8.32568 −0.368306
$$512$$ 21.2637 0.939730
$$513$$ 0 0
$$514$$ −18.7557 −0.827277
$$515$$ −15.2261 −0.670943
$$516$$ 0 0
$$517$$ 8.22987 0.361949
$$518$$ −12.3464 −0.542470
$$519$$ 0 0
$$520$$ −2.57130 −0.112759
$$521$$ 25.3602 1.11105 0.555526 0.831499i $$-0.312517\pi$$
0.555526 + 0.831499i $$0.312517\pi$$
$$522$$ 0 0
$$523$$ −7.00314 −0.306226 −0.153113 0.988209i $$-0.548930\pi$$
−0.153113 + 0.988209i $$0.548930\pi$$
$$524$$ −1.03830 −0.0453585
$$525$$ 0 0
$$526$$ −3.73710 −0.162945
$$527$$ −20.2927 −0.883965
$$528$$ 0 0
$$529$$ 41.3320 1.79704
$$530$$ 6.86059 0.298005
$$531$$ 0 0
$$532$$ −0.0942138 −0.00408469
$$533$$ 12.0678 0.522716
$$534$$ 0 0
$$535$$ −3.20165 −0.138420
$$536$$ 33.4070 1.44296
$$537$$ 0 0
$$538$$ −33.5603 −1.44689
$$539$$ −4.21419 −0.181518
$$540$$ 0 0
$$541$$ −1.42463 −0.0612495 −0.0306248 0.999531i $$-0.509750\pi$$
−0.0306248 + 0.999531i $$0.509750\pi$$
$$542$$ 8.71422 0.374308
$$543$$ 0 0
$$544$$ 1.19156 0.0510879
$$545$$ −4.32728 −0.185360
$$546$$ 0 0
$$547$$ −22.0499 −0.942787 −0.471394 0.881923i $$-0.656249\pi$$
−0.471394 + 0.881923i $$0.656249\pi$$
$$548$$ −1.07630 −0.0459771
$$549$$ 0 0
$$550$$ −25.1269 −1.07142
$$551$$ 4.23209 0.180293
$$552$$ 0 0
$$553$$ 4.47204 0.190171
$$554$$ −21.8760 −0.929420
$$555$$ 0 0
$$556$$ 0.721117 0.0305822
$$557$$ 9.08319 0.384867 0.192434 0.981310i $$-0.438362\pi$$
0.192434 + 0.981310i $$0.438362\pi$$
$$558$$ 0 0
$$559$$ 7.14101 0.302033
$$560$$ 3.83796 0.162183
$$561$$ 0 0
$$562$$ −10.9070 −0.460084
$$563$$ −34.1350 −1.43862 −0.719310 0.694689i $$-0.755542\pi$$
−0.719310 + 0.694689i $$0.755542\pi$$
$$564$$ 0 0
$$565$$ −3.04680 −0.128180
$$566$$ −27.4974 −1.15580
$$567$$ 0 0
$$568$$ −29.9322 −1.25593
$$569$$ −1.23522 −0.0517833 −0.0258916 0.999665i $$-0.508242\pi$$
−0.0258916 + 0.999665i $$0.508242\pi$$
$$570$$ 0 0
$$571$$ −26.4613 −1.10737 −0.553686 0.832725i $$-0.686779\pi$$
−0.553686 + 0.832725i $$0.686779\pi$$
$$572$$ −0.308407 −0.0128951
$$573$$ 0 0
$$574$$ −17.3759 −0.725257
$$575$$ −33.2139 −1.38511
$$576$$ 0 0
$$577$$ −23.2852 −0.969374 −0.484687 0.874688i $$-0.661066\pi$$
−0.484687 + 0.874688i $$0.661066\pi$$
$$578$$ 12.5372 0.521479
$$579$$ 0 0
$$580$$ 0.222973 0.00925846
$$581$$ 3.80653 0.157921
$$582$$ 0 0
$$583$$ −21.6652 −0.897281
$$584$$ −23.0982 −0.955812
$$585$$ 0 0
$$586$$ −5.46324 −0.225684
$$587$$ 10.7155 0.442274 0.221137 0.975243i $$-0.429023\pi$$
0.221137 + 0.975243i $$0.429023\pi$$
$$588$$ 0 0
$$589$$ 9.07187 0.373800
$$590$$ 9.78774 0.402955
$$591$$ 0 0
$$592$$ −35.5081 −1.45938
$$593$$ −14.1008 −0.579049 −0.289525 0.957171i $$-0.593497\pi$$
−0.289525 + 0.957171i $$0.593497\pi$$
$$594$$ 0 0
$$595$$ −2.66896 −0.109417
$$596$$ 0.247843 0.0101521
$$597$$ 0 0
$$598$$ −11.5487 −0.472260
$$599$$ 38.9296 1.59062 0.795311 0.606202i $$-0.207308\pi$$
0.795311 + 0.606202i $$0.207308\pi$$
$$600$$ 0 0
$$601$$ −32.6162 −1.33044 −0.665221 0.746646i $$-0.731663\pi$$
−0.665221 + 0.746646i $$0.731663\pi$$
$$602$$ −10.2820 −0.419064
$$603$$ 0 0
$$604$$ −1.60234 −0.0651984
$$605$$ −6.26475 −0.254698
$$606$$ 0 0
$$607$$ 19.3203 0.784188 0.392094 0.919925i $$-0.371751\pi$$
0.392094 + 0.919925i $$0.371751\pi$$
$$608$$ −0.532689 −0.0216034
$$609$$ 0 0
$$610$$ 10.3548 0.419254
$$611$$ −1.95289 −0.0790056
$$612$$ 0 0
$$613$$ 35.3197 1.42655 0.713275 0.700885i $$-0.247211\pi$$
0.713275 + 0.700885i $$0.247211\pi$$
$$614$$ 7.47740 0.301763
$$615$$ 0 0
$$616$$ −11.6916 −0.471068
$$617$$ 24.4318 0.983589 0.491794 0.870711i $$-0.336341\pi$$
0.491794 + 0.870711i $$0.336341\pi$$
$$618$$ 0 0
$$619$$ 2.90892 0.116919 0.0584597 0.998290i $$-0.481381\pi$$
0.0584597 + 0.998290i $$0.481381\pi$$
$$620$$ 0.477964 0.0191955
$$621$$ 0 0
$$622$$ −19.9472 −0.799811
$$623$$ 5.64793 0.226280
$$624$$ 0 0
$$625$$ 12.8530 0.514121
$$626$$ 46.9626 1.87700
$$627$$ 0 0
$$628$$ −0.0634848 −0.00253332
$$629$$ 24.6928 0.984566
$$630$$ 0 0
$$631$$ 5.66521 0.225528 0.112764 0.993622i $$-0.464030\pi$$
0.112764 + 0.993622i $$0.464030\pi$$
$$632$$ 12.4070 0.493522
$$633$$ 0 0
$$634$$ 17.3330 0.688380
$$635$$ 0.397035 0.0157559
$$636$$ 0 0
$$637$$ 1.00000 0.0396214
$$638$$ −19.9472 −0.789718
$$639$$ 0 0
$$640$$ 11.0241 0.435768
$$641$$ −12.2842 −0.485198 −0.242599 0.970127i $$-0.578000\pi$$
−0.242599 + 0.970127i $$0.578000\pi$$
$$642$$ 0 0
$$643$$ 9.14950 0.360821 0.180411 0.983591i $$-0.442257\pi$$
0.180411 + 0.983591i $$0.442257\pi$$
$$644$$ 0.586979 0.0231302
$$645$$ 0 0
$$646$$ 5.33793 0.210018
$$647$$ −38.4108 −1.51008 −0.755042 0.655677i $$-0.772383\pi$$
−0.755042 + 0.655677i $$0.772383\pi$$
$$648$$ 0 0
$$649$$ −30.9089 −1.21328
$$650$$ 5.96245 0.233867
$$651$$ 0 0
$$652$$ 0.324082 0.0126920
$$653$$ −28.4805 −1.11453 −0.557265 0.830335i $$-0.688149\pi$$
−0.557265 + 0.830335i $$0.688149\pi$$
$$654$$ 0 0
$$655$$ 13.1495 0.513794
$$656$$ −49.9730 −1.95112
$$657$$ 0 0
$$658$$ 2.81188 0.109619
$$659$$ 30.4384 1.18571 0.592856 0.805309i $$-0.298000\pi$$
0.592856 + 0.805309i $$0.298000\pi$$
$$660$$ 0 0
$$661$$ 36.7710 1.43023 0.715114 0.699008i $$-0.246375\pi$$
0.715114 + 0.699008i $$0.246375\pi$$
$$662$$ 37.8423 1.47078
$$663$$ 0 0
$$664$$ 10.5606 0.409830
$$665$$ 1.19316 0.0462688
$$666$$ 0 0
$$667$$ −26.3671 −1.02094
$$668$$ −1.50227 −0.0581246
$$669$$ 0 0
$$670$$ 16.0691 0.620803
$$671$$ −32.6997 −1.26236
$$672$$ 0 0
$$673$$ 26.7961 1.03291 0.516457 0.856313i $$-0.327250\pi$$
0.516457 + 0.856313i $$0.327250\pi$$
$$674$$ −37.8545 −1.45810
$$675$$ 0 0
$$676$$ 0.0731828 0.00281472
$$677$$ 20.3050 0.780383 0.390191 0.920734i $$-0.372409\pi$$
0.390191 + 0.920734i $$0.372409\pi$$
$$678$$ 0 0
$$679$$ 6.90043 0.264814
$$680$$ −7.40461 −0.283954
$$681$$ 0 0
$$682$$ −42.7587 −1.63732
$$683$$ −44.9240 −1.71897 −0.859484 0.511162i $$-0.829215\pi$$
−0.859484 + 0.511162i $$0.829215\pi$$
$$684$$ 0 0
$$685$$ 13.6307 0.520801
$$686$$ −1.43986 −0.0549739
$$687$$ 0 0
$$688$$ −29.5710 −1.12738
$$689$$ 5.14101 0.195857
$$690$$ 0 0
$$691$$ 22.0085 0.837243 0.418621 0.908161i $$-0.362513\pi$$
0.418621 + 0.908161i $$0.362513\pi$$
$$692$$ −1.09887 −0.0417726
$$693$$ 0 0
$$694$$ 23.7472 0.901431
$$695$$ −9.13252 −0.346416
$$696$$ 0 0
$$697$$ 34.7518 1.31632
$$698$$ −20.4162 −0.772763
$$699$$ 0 0
$$700$$ −0.303051 −0.0114542
$$701$$ −31.4645 −1.18840 −0.594198 0.804319i $$-0.702531\pi$$
−0.594198 + 0.804319i $$0.702531\pi$$
$$702$$ 0 0
$$703$$ −11.0389 −0.416341
$$704$$ −32.3913 −1.22079
$$705$$ 0 0
$$706$$ −42.9469 −1.61633
$$707$$ 10.9739 0.412717
$$708$$ 0 0
$$709$$ −25.3373 −0.951563 −0.475781 0.879564i $$-0.657835\pi$$
−0.475781 + 0.879564i $$0.657835\pi$$
$$710$$ −14.3977 −0.540334
$$711$$ 0 0
$$712$$ 15.6693 0.587231
$$713$$ −56.5204 −2.11670
$$714$$ 0 0
$$715$$ 3.90579 0.146068
$$716$$ 0.391657 0.0146369
$$717$$ 0 0
$$718$$ 1.93907 0.0723654
$$719$$ 43.8002 1.63347 0.816737 0.577011i $$-0.195781\pi$$
0.816737 + 0.577011i $$0.195781\pi$$
$$720$$ 0 0
$$721$$ 16.4284 0.611825
$$722$$ 24.9709 0.929322
$$723$$ 0 0
$$724$$ 0.839165 0.0311873
$$725$$ 13.6131 0.505576
$$726$$ 0 0
$$727$$ 0.944090 0.0350144 0.0175072 0.999847i $$-0.494427\pi$$
0.0175072 + 0.999847i $$0.494427\pi$$
$$728$$ 2.77434 0.102824
$$729$$ 0 0
$$730$$ −11.1105 −0.411217
$$731$$ 20.5640 0.760588
$$732$$ 0 0
$$733$$ −5.09957 −0.188357 −0.0941784 0.995555i $$-0.530022\pi$$
−0.0941784 + 0.995555i $$0.530022\pi$$
$$734$$ −52.0898 −1.92267
$$735$$ 0 0
$$736$$ 3.31881 0.122333
$$737$$ −50.7450 −1.86921
$$738$$ 0 0
$$739$$ −8.25129 −0.303529 −0.151764 0.988417i $$-0.548495\pi$$
−0.151764 + 0.988417i $$0.548495\pi$$
$$740$$ −0.581601 −0.0213801
$$741$$ 0 0
$$742$$ −7.40231 −0.271747
$$743$$ −38.4962 −1.41229 −0.706145 0.708068i $$-0.749567\pi$$
−0.706145 + 0.708068i $$0.749567\pi$$
$$744$$ 0 0
$$745$$ −3.13879 −0.114996
$$746$$ 24.3464 0.891385
$$747$$ 0 0
$$748$$ −0.888121 −0.0324729
$$749$$ 3.45446 0.126223
$$750$$ 0 0
$$751$$ −41.3175 −1.50770 −0.753848 0.657049i $$-0.771805\pi$$
−0.753848 + 0.657049i $$0.771805\pi$$
$$752$$ 8.08695 0.294901
$$753$$ 0 0
$$754$$ 4.73334 0.172378
$$755$$ 20.2927 0.738528
$$756$$ 0 0
$$757$$ −39.7158 −1.44349 −0.721747 0.692157i $$-0.756661\pi$$
−0.721747 + 0.692157i $$0.756661\pi$$
$$758$$ 16.7211 0.607338
$$759$$ 0 0
$$760$$ 3.31023 0.120075
$$761$$ −10.7390 −0.389289 −0.194644 0.980874i $$-0.562355\pi$$
−0.194644 + 0.980874i $$0.562355\pi$$
$$762$$ 0 0
$$763$$ 4.66896 0.169028
$$764$$ −1.47651 −0.0534181
$$765$$ 0 0
$$766$$ 21.0411 0.760247
$$767$$ 7.33448 0.264833
$$768$$ 0 0
$$769$$ −50.5549 −1.82306 −0.911529 0.411237i $$-0.865097\pi$$
−0.911529 + 0.411237i $$0.865097\pi$$
$$770$$ −5.62377 −0.202666
$$771$$ 0 0
$$772$$ −1.93024 −0.0694709
$$773$$ −8.91770 −0.320747 −0.160374 0.987056i $$-0.551270\pi$$
−0.160374 + 0.987056i $$0.551270\pi$$
$$774$$ 0 0
$$775$$ 29.1809 1.04821
$$776$$ 19.1441 0.687234
$$777$$ 0 0
$$778$$ 8.63913 0.309728
$$779$$ −15.5358 −0.556629
$$780$$ 0 0
$$781$$ 45.4667 1.62693
$$782$$ −33.2568 −1.18926
$$783$$ 0 0
$$784$$ −4.14101 −0.147893
$$785$$ 0.803998 0.0286959
$$786$$ 0 0
$$787$$ 22.3671 0.797302 0.398651 0.917103i $$-0.369479\pi$$
0.398651 + 0.917103i $$0.369479\pi$$
$$788$$ 1.47333 0.0524853
$$789$$ 0 0
$$790$$ 5.96787 0.212327
$$791$$ 3.28738 0.116886
$$792$$ 0 0
$$793$$ 7.75942 0.275545
$$794$$ −37.0016 −1.31314
$$795$$ 0 0
$$796$$ 0.128759 0.00456375
$$797$$ −20.5518 −0.727983 −0.363991 0.931402i $$-0.618586\pi$$
−0.363991 + 0.931402i $$0.618586\pi$$
$$798$$ 0 0
$$799$$ −5.62377 −0.198955
$$800$$ −1.71346 −0.0605801
$$801$$ 0 0
$$802$$ −28.1689 −0.994680
$$803$$ 35.0860 1.23816
$$804$$ 0 0
$$805$$ −7.43374 −0.262005
$$806$$ 10.1464 0.357390
$$807$$ 0 0
$$808$$ 30.4454 1.07106
$$809$$ −7.83443 −0.275444 −0.137722 0.990471i $$-0.543978\pi$$
−0.137722 + 0.990471i $$0.543978\pi$$
$$810$$ 0 0
$$811$$ −47.5502 −1.66971 −0.834857 0.550468i $$-0.814449\pi$$
−0.834857 + 0.550468i $$0.814449\pi$$
$$812$$ −0.240579 −0.00844268
$$813$$ 0 0
$$814$$ 52.0301 1.82365
$$815$$ −4.10430 −0.143767
$$816$$ 0 0
$$817$$ −9.19316 −0.321628
$$818$$ −4.59769 −0.160754
$$819$$ 0 0
$$820$$ −0.818526 −0.0285842
$$821$$ −4.42494 −0.154431 −0.0772157 0.997014i $$-0.524603\pi$$
−0.0772157 + 0.997014i $$0.524603\pi$$
$$822$$ 0 0
$$823$$ −28.7525 −1.00225 −0.501124 0.865375i $$-0.667080\pi$$
−0.501124 + 0.865375i $$0.667080\pi$$
$$824$$ 45.5779 1.58778
$$825$$ 0 0
$$826$$ −10.5606 −0.367450
$$827$$ 5.58667 0.194267 0.0971337 0.995271i $$-0.469033\pi$$
0.0971337 + 0.995271i $$0.469033\pi$$
$$828$$ 0 0
$$829$$ 53.4974 1.85804 0.929021 0.370027i $$-0.120652\pi$$
0.929021 + 0.370027i $$0.120652\pi$$
$$830$$ 5.07974 0.176320
$$831$$ 0 0
$$832$$ 7.68624 0.266472
$$833$$ 2.87971 0.0997760
$$834$$ 0 0
$$835$$ 19.0254 0.658400
$$836$$ 0.397035 0.0137317
$$837$$ 0 0
$$838$$ 43.5257 1.50357
$$839$$ 43.0405 1.48592 0.742962 0.669334i $$-0.233420\pi$$
0.742962 + 0.669334i $$0.233420\pi$$
$$840$$ 0 0
$$841$$ −18.1932 −0.627350
$$842$$ −38.8055 −1.33733
$$843$$ 0 0
$$844$$ 1.13941 0.0392202
$$845$$ −0.926817 −0.0318835
$$846$$ 0 0
$$847$$ 6.75942 0.232256
$$848$$ −21.2890 −0.731066
$$849$$ 0 0
$$850$$ 17.1701 0.588931
$$851$$ 68.7757 2.35760
$$852$$ 0 0
$$853$$ −30.7434 −1.05263 −0.526316 0.850289i $$-0.676427\pi$$
−0.526316 + 0.850289i $$0.676427\pi$$
$$854$$ −11.1724 −0.382313
$$855$$ 0 0
$$856$$ 9.58384 0.327569
$$857$$ 24.5449 0.838438 0.419219 0.907885i $$-0.362304\pi$$
0.419219 + 0.907885i $$0.362304\pi$$
$$858$$ 0 0
$$859$$ −52.1243 −1.77846 −0.889229 0.457461i $$-0.848759\pi$$
−0.889229 + 0.457461i $$0.848759\pi$$
$$860$$ −0.484354 −0.0165163
$$861$$ 0 0
$$862$$ 24.1483 0.822494
$$863$$ −22.2563 −0.757612 −0.378806 0.925476i $$-0.623665\pi$$
−0.378806 + 0.925476i $$0.623665\pi$$
$$864$$ 0 0
$$865$$ 13.9165 0.473175
$$866$$ −37.2246 −1.26494
$$867$$ 0 0
$$868$$ −0.515704 −0.0175041
$$869$$ −18.8461 −0.639309
$$870$$ 0 0
$$871$$ 12.0414 0.408009
$$872$$ 12.9533 0.438654
$$873$$ 0 0
$$874$$ 14.8675 0.502900
$$875$$ 8.47204 0.286407
$$876$$ 0 0
$$877$$ 41.0547 1.38632 0.693159 0.720785i $$-0.256218\pi$$
0.693159 + 0.720785i $$0.256218\pi$$
$$878$$ −36.0760 −1.21751
$$879$$ 0 0
$$880$$ −16.1739 −0.545222
$$881$$ 31.0568 1.04633 0.523165 0.852231i $$-0.324751\pi$$
0.523165 + 0.852231i $$0.324751\pi$$
$$882$$ 0 0
$$883$$ −56.5357 −1.90258 −0.951289 0.308300i $$-0.900240\pi$$
−0.951289 + 0.308300i $$0.900240\pi$$
$$884$$ 0.210745 0.00708813
$$885$$ 0 0
$$886$$ 6.69190 0.224819
$$887$$ −9.94785 −0.334016 −0.167008 0.985956i $$-0.553411\pi$$
−0.167008 + 0.985956i $$0.553411\pi$$
$$888$$ 0 0
$$889$$ −0.428386 −0.0143676
$$890$$ 7.53707 0.252643
$$891$$ 0 0
$$892$$ −0.846681 −0.0283490
$$893$$ 2.51411 0.0841314
$$894$$ 0 0
$$895$$ −4.96010 −0.165798
$$896$$ −11.8946 −0.397372
$$897$$ 0 0
$$898$$ 0.712291 0.0237695
$$899$$ 23.1655 0.772612
$$900$$ 0 0
$$901$$ 14.8046 0.493213
$$902$$ 73.2255 2.43814
$$903$$ 0 0
$$904$$ 9.12029 0.303336
$$905$$ −10.6275 −0.353271
$$906$$ 0 0
$$907$$ −31.2936 −1.03909 −0.519544 0.854444i $$-0.673898\pi$$
−0.519544 + 0.854444i $$0.673898\pi$$
$$908$$ 0.808067 0.0268166
$$909$$ 0 0
$$910$$ 1.33448 0.0442377
$$911$$ 9.39320 0.311210 0.155605 0.987819i $$-0.450267\pi$$
0.155605 + 0.987819i $$0.450267\pi$$
$$912$$ 0 0
$$913$$ −16.0414 −0.530894
$$914$$ −38.1127 −1.26066
$$915$$ 0 0
$$916$$ −1.00314 −0.0331446
$$917$$ −14.1878 −0.468523
$$918$$ 0 0
$$919$$ 21.7066 0.716036 0.358018 0.933715i $$-0.383453\pi$$
0.358018 + 0.933715i $$0.383453\pi$$
$$920$$ −20.6237 −0.679944
$$921$$ 0 0
$$922$$ −1.55746 −0.0512921
$$923$$ −10.7889 −0.355122
$$924$$ 0 0
$$925$$ −35.5081 −1.16750
$$926$$ 47.0974 1.54771
$$927$$ 0 0
$$928$$ −1.36025 −0.0446523
$$929$$ −35.6787 −1.17058 −0.585289 0.810824i $$-0.699019\pi$$
−0.585289 + 0.810824i $$0.699019\pi$$
$$930$$ 0 0
$$931$$ −1.28738 −0.0421920
$$932$$ −0.0903972 −0.00296106
$$933$$ 0 0
$$934$$ −24.5924 −0.804687
$$935$$ 11.2475 0.367834
$$936$$ 0 0
$$937$$ −47.0866 −1.53825 −0.769127 0.639096i $$-0.779309\pi$$
−0.769127 + 0.639096i $$0.779309\pi$$
$$938$$ −17.3379 −0.566103
$$939$$ 0 0
$$940$$ 0.132459 0.00432034
$$941$$ −45.2923 −1.47649 −0.738244 0.674534i $$-0.764345\pi$$
−0.738244 + 0.674534i $$0.764345\pi$$
$$942$$ 0 0
$$943$$ 96.7927 3.15200
$$944$$ −30.3722 −0.988530
$$945$$ 0 0
$$946$$ 43.3304 1.40879
$$947$$ −4.77134 −0.155048 −0.0775238 0.996991i $$-0.524701\pi$$
−0.0775238 + 0.996991i $$0.524701\pi$$
$$948$$ 0 0
$$949$$ −8.32568 −0.270263
$$950$$ −7.67592 −0.249040
$$951$$ 0 0
$$952$$ 7.98929 0.258934
$$953$$ 3.95572 0.128138 0.0640692 0.997945i $$-0.479592\pi$$
0.0640692 + 0.997945i $$0.479592\pi$$
$$954$$ 0 0
$$955$$ 18.6991 0.605088
$$956$$ −1.39824 −0.0452223
$$957$$ 0 0
$$958$$ −24.7353 −0.799160
$$959$$ −14.7070 −0.474912
$$960$$ 0 0
$$961$$ 18.6573 0.601850
$$962$$ −12.3464 −0.398064
$$963$$ 0 0
$$964$$ −0.880605 −0.0283624
$$965$$ 24.4454 0.786924
$$966$$ 0 0
$$967$$ 50.2292 1.61526 0.807632 0.589687i $$-0.200749\pi$$
0.807632 + 0.589687i $$0.200749\pi$$
$$968$$ 18.7529 0.602742
$$969$$ 0 0
$$970$$ 9.20850 0.295667
$$971$$ −38.6576 −1.24058 −0.620291 0.784372i $$-0.712986\pi$$
−0.620291 + 0.784372i $$0.712986\pi$$
$$972$$ 0 0
$$973$$ 9.85363 0.315893
$$974$$ 6.16638 0.197584
$$975$$ 0 0
$$976$$ −32.1318 −1.02852
$$977$$ 20.6134 0.659480 0.329740 0.944072i $$-0.393039\pi$$
0.329740 + 0.944072i $$0.393039\pi$$
$$978$$ 0 0
$$979$$ −23.8015 −0.760699
$$980$$ −0.0678271 −0.00216666
$$981$$ 0 0
$$982$$ 39.5304 1.26147
$$983$$ 4.48620 0.143088 0.0715438 0.997437i $$-0.477207\pi$$
0.0715438 + 0.997437i $$0.477207\pi$$
$$984$$ 0 0
$$985$$ −18.6589 −0.594521
$$986$$ 13.6307 0.434089
$$987$$ 0 0
$$988$$ −0.0942138 −0.00299734
$$989$$ 57.2760 1.82127
$$990$$ 0 0
$$991$$ −32.7971 −1.04183 −0.520917 0.853607i $$-0.674410\pi$$
−0.520917 + 0.853607i $$0.674410\pi$$
$$992$$ −2.91582 −0.0925773
$$993$$ 0 0
$$994$$ 15.5345 0.492725
$$995$$ −1.63066 −0.0516954
$$996$$ 0 0
$$997$$ 0.491870 0.0155777 0.00778884 0.999970i $$-0.497521\pi$$
0.00778884 + 0.999970i $$0.497521\pi$$
$$998$$ 53.3397 1.68844
$$999$$ 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 819.2.a.k.1.2 4
3.2 odd 2 273.2.a.e.1.3 4
7.6 odd 2 5733.2.a.bf.1.2 4
12.11 even 2 4368.2.a.br.1.3 4
15.14 odd 2 6825.2.a.bg.1.2 4
21.20 even 2 1911.2.a.s.1.3 4
39.38 odd 2 3549.2.a.w.1.2 4

By twisted newform
Twist Min Dim Char Parity Ord Type
273.2.a.e.1.3 4 3.2 odd 2
819.2.a.k.1.2 4 1.1 even 1 trivial
1911.2.a.s.1.3 4 21.20 even 2
3549.2.a.w.1.2 4 39.38 odd 2
4368.2.a.br.1.3 4 12.11 even 2
5733.2.a.bf.1.2 4 7.6 odd 2
6825.2.a.bg.1.2 4 15.14 odd 2