# Properties

 Label 6825.2.a.bg.1.2 Level $6825$ Weight $2$ Character 6825.1 Self dual yes Analytic conductor $54.498$ Analytic rank $1$ 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] = [6825,2,Mod(1,6825)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6825, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("6825.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$6825 = 3 \cdot 5^{2} \cdot 7 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 6825.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: $$54.4978993795$$ Analytic rank: $$1$$ 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 $$\chi$$ $$=$$ 6825.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-1.43986 q^{2} -1.00000 q^{3} +0.0731828 q^{4} +1.43986 q^{6} -1.00000 q^{7} +2.77434 q^{8} +1.00000 q^{9} +O(q^{10})$$ $$q-1.43986 q^{2} -1.00000 q^{3} +0.0731828 q^{4} +1.43986 q^{6} -1.00000 q^{7} +2.77434 q^{8} +1.00000 q^{9} +4.21419 q^{11} -0.0731828 q^{12} -1.00000 q^{13} +1.43986 q^{14} -4.14101 q^{16} +2.87971 q^{17} -1.43986 q^{18} -1.28738 q^{19} +1.00000 q^{21} -6.06783 q^{22} +8.02072 q^{23} -2.77434 q^{24} +1.43986 q^{26} -1.00000 q^{27} -0.0731828 q^{28} +3.28738 q^{29} -7.04680 q^{31} +0.413779 q^{32} -4.21419 q^{33} -4.14637 q^{34} +0.0731828 q^{36} -8.57475 q^{37} +1.85363 q^{38} +1.00000 q^{39} -12.0678 q^{41} -1.43986 q^{42} -7.14101 q^{43} +0.308407 q^{44} -11.5487 q^{46} -1.95289 q^{47} +4.14101 q^{48} +1.00000 q^{49} -2.87971 q^{51} -0.0731828 q^{52} +5.14101 q^{53} +1.43986 q^{54} -2.77434 q^{56} +1.28738 q^{57} -4.73334 q^{58} -7.33448 q^{59} +7.75942 q^{61} +10.1464 q^{62} -1.00000 q^{63} +7.68624 q^{64} +6.06783 q^{66} -12.0414 q^{67} +0.210745 q^{68} -8.02072 q^{69} +10.7889 q^{71} +2.77434 q^{72} +8.32568 q^{73} +12.3464 q^{74} -0.0942138 q^{76} -4.21419 q^{77} -1.43986 q^{78} +4.47204 q^{79} +1.00000 q^{81} +17.3759 q^{82} +3.80653 q^{83} +0.0731828 q^{84} +10.2820 q^{86} -3.28738 q^{87} +11.6916 q^{88} -5.64793 q^{89} +1.00000 q^{91} +0.586979 q^{92} +7.04680 q^{93} +2.81188 q^{94} -0.413779 q^{96} -6.90043 q^{97} -1.43986 q^{98} +4.21419 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - q^{2} - 4 q^{3} + 7 q^{4} + q^{6} - 4 q^{7} - 3 q^{8} + 4 q^{9}+O(q^{10})$$ 4 * q - q^2 - 4 * q^3 + 7 * q^4 + q^6 - 4 * q^7 - 3 * q^8 + 4 * q^9 $$4 q - q^{2} - 4 q^{3} + 7 q^{4} + q^{6} - 4 q^{7} - 3 q^{8} + 4 q^{9} - 2 q^{11} - 7 q^{12} - 4 q^{13} + q^{14} + 9 q^{16} + 2 q^{17} - q^{18} + 7 q^{19} + 4 q^{21} + 8 q^{22} - 3 q^{23} + 3 q^{24} + q^{26} - 4 q^{27} - 7 q^{28} + q^{29} + 3 q^{31} - 7 q^{32} + 2 q^{33} - 30 q^{34} + 7 q^{36} - 10 q^{37} - 6 q^{38} + 4 q^{39} - 16 q^{41} - q^{42} - 3 q^{43} - 12 q^{44} - 18 q^{46} - 5 q^{47} - 9 q^{48} + 4 q^{49} - 2 q^{51} - 7 q^{52} - 5 q^{53} + q^{54} + 3 q^{56} - 7 q^{57} + 4 q^{58} - 20 q^{59} + 12 q^{61} + 54 q^{62} - 4 q^{63} + 5 q^{64} - 8 q^{66} + 22 q^{67} + 10 q^{68} + 3 q^{69} - 3 q^{72} + 13 q^{73} - 6 q^{74} - 6 q^{76} + 2 q^{77} - q^{78} + 11 q^{79} + 4 q^{81} - 10 q^{82} - q^{83} + 7 q^{84} - 10 q^{86} - q^{87} + 60 q^{88} - 5 q^{89} + 4 q^{91} - 34 q^{92} - 3 q^{93} + 34 q^{94} + 7 q^{96} + 17 q^{97} - q^{98} - 2 q^{99}+O(q^{100})$$ 4 * q - q^2 - 4 * q^3 + 7 * q^4 + q^6 - 4 * q^7 - 3 * q^8 + 4 * q^9 - 2 * q^11 - 7 * q^12 - 4 * q^13 + q^14 + 9 * q^16 + 2 * q^17 - q^18 + 7 * q^19 + 4 * q^21 + 8 * q^22 - 3 * q^23 + 3 * q^24 + q^26 - 4 * q^27 - 7 * q^28 + q^29 + 3 * q^31 - 7 * q^32 + 2 * q^33 - 30 * q^34 + 7 * q^36 - 10 * q^37 - 6 * q^38 + 4 * q^39 - 16 * q^41 - q^42 - 3 * q^43 - 12 * q^44 - 18 * q^46 - 5 * q^47 - 9 * q^48 + 4 * q^49 - 2 * q^51 - 7 * q^52 - 5 * q^53 + q^54 + 3 * q^56 - 7 * q^57 + 4 * q^58 - 20 * q^59 + 12 * q^61 + 54 * q^62 - 4 * q^63 + 5 * q^64 - 8 * q^66 + 22 * q^67 + 10 * q^68 + 3 * q^69 - 3 * q^72 + 13 * q^73 - 6 * q^74 - 6 * q^76 + 2 * q^77 - q^78 + 11 * q^79 + 4 * q^81 - 10 * q^82 - q^83 + 7 * q^84 - 10 * q^86 - q^87 + 60 * q^88 - 5 * q^89 + 4 * q^91 - 34 * q^92 - 3 * q^93 + 34 * q^94 + 7 * q^96 + 17 * q^97 - q^98 - 2 * 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −1.43986 −1.01813 −0.509066 0.860728i $$-0.670009\pi$$
−0.509066 + 0.860728i $$0.670009\pi$$
$$3$$ −1.00000 −0.577350
$$4$$ 0.0731828 0.0365914
$$5$$ 0 0
$$6$$ 1.43986 0.587818
$$7$$ −1.00000 −0.377964
$$8$$ 2.77434 0.980876
$$9$$ 1.00000 0.333333
$$10$$ 0 0
$$11$$ 4.21419 1.27063 0.635313 0.772254i $$-0.280871\pi$$
0.635313 + 0.772254i $$0.280871\pi$$
$$12$$ −0.0731828 −0.0211261
$$13$$ −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 $$0.386448\pi$$
$$18$$ −1.43986 −0.339377
$$19$$ −1.28738 −0.295344 −0.147672 0.989036i $$-0.547178\pi$$
−0.147672 + 0.989036i $$0.547178\pi$$
$$20$$ 0 0
$$21$$ 1.00000 0.218218
$$22$$ −6.06783 −1.29367
$$23$$ 8.02072 1.67244 0.836218 0.548397i $$-0.184762\pi$$
0.836218 + 0.548397i $$0.184762\pi$$
$$24$$ −2.77434 −0.566309
$$25$$ 0 0
$$26$$ 1.43986 0.282379
$$27$$ −1.00000 −0.192450
$$28$$ −0.0731828 −0.0138303
$$29$$ 3.28738 0.610450 0.305225 0.952280i $$-0.401268\pi$$
0.305225 + 0.952280i $$0.401268\pi$$
$$30$$ 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
$$33$$ −4.21419 −0.733597
$$34$$ −4.14637 −0.711096
$$35$$ 0 0
$$36$$ 0.0731828 0.0121971
$$37$$ −8.57475 −1.40968 −0.704840 0.709366i $$-0.748981\pi$$
−0.704840 + 0.709366i $$0.748981\pi$$
$$38$$ 1.85363 0.300699
$$39$$ 1.00000 0.160128
$$40$$ 0 0
$$41$$ −12.0678 −1.88468 −0.942339 0.334660i $$-0.891379\pi$$
−0.942339 + 0.334660i $$0.891379\pi$$
$$42$$ −1.43986 −0.222174
$$43$$ −7.14101 −1.08899 −0.544497 0.838763i $$-0.683279\pi$$
−0.544497 + 0.838763i $$0.683279\pi$$
$$44$$ 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 $$0.545491\pi$$
$$48$$ 4.14101 0.597703
$$49$$ 1.00000 0.142857
$$50$$ 0 0
$$51$$ −2.87971 −0.403240
$$52$$ −0.0731828 −0.0101486
$$53$$ 5.14101 0.706172 0.353086 0.935591i $$-0.385132\pi$$
0.353086 + 0.935591i $$0.385132\pi$$
$$54$$ 1.43986 0.195939
$$55$$ 0 0
$$56$$ −2.77434 −0.370736
$$57$$ 1.28738 0.170517
$$58$$ −4.73334 −0.621519
$$59$$ −7.33448 −0.954868 −0.477434 0.878668i $$-0.658433\pi$$
−0.477434 + 0.878668i $$0.658433\pi$$
$$60$$ 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
$$63$$ −1.00000 −0.125988
$$64$$ 7.68624 0.960780
$$65$$ 0 0
$$66$$ 6.06783 0.746898
$$67$$ −12.0414 −1.47110 −0.735548 0.677473i $$-0.763075\pi$$
−0.735548 + 0.677473i $$0.763075\pi$$
$$68$$ 0.210745 0.0255566
$$69$$ −8.02072 −0.965581
$$70$$ 0 0
$$71$$ 10.7889 1.28041 0.640206 0.768203i $$-0.278849\pi$$
0.640206 + 0.768203i $$0.278849\pi$$
$$72$$ 2.77434 0.326959
$$73$$ 8.32568 0.974447 0.487224 0.873277i $$-0.338010\pi$$
0.487224 + 0.873277i $$0.338010\pi$$
$$74$$ 12.3464 1.43524
$$75$$ 0 0
$$76$$ −0.0942138 −0.0108071
$$77$$ −4.21419 −0.480252
$$78$$ −1.43986 −0.163031
$$79$$ 4.47204 0.503144 0.251572 0.967839i $$-0.419052\pi$$
0.251572 + 0.967839i $$0.419052\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 17.3759 1.91885
$$83$$ 3.80653 0.417821 0.208910 0.977935i $$-0.433008\pi$$
0.208910 + 0.977935i $$0.433008\pi$$
$$84$$ 0.0731828 0.00798490
$$85$$ 0 0
$$86$$ 10.2820 1.10874
$$87$$ −3.28738 −0.352444
$$88$$ 11.6916 1.24633
$$89$$ −5.64793 −0.598680 −0.299340 0.954147i $$-0.596766\pi$$
−0.299340 + 0.954147i $$0.596766\pi$$
$$90$$ 0 0
$$91$$ 1.00000 0.104828
$$92$$ 0.586979 0.0611968
$$93$$ 7.04680 0.730719
$$94$$ 2.81188 0.290024
$$95$$ 0 0
$$96$$ −0.413779 −0.0422312
$$97$$ −6.90043 −0.700633 −0.350316 0.936631i $$-0.613926\pi$$
−0.350316 + 0.936631i $$0.613926\pi$$
$$98$$ −1.43986 −0.145447
$$99$$ 4.21419 0.423542
$$100$$ 0 0
$$101$$ −10.9739 −1.09195 −0.545973 0.837803i $$-0.683840\pi$$
−0.545973 + 0.837803i $$0.683840\pi$$
$$102$$ 4.14637 0.410551
$$103$$ −16.4284 −1.61874 −0.809368 0.587301i $$-0.800190\pi$$
−0.809368 + 0.587301i $$0.800190\pi$$
$$104$$ −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 $$0.446599\pi$$
$$108$$ −0.0731828 −0.00704202
$$109$$ 4.66896 0.447206 0.223603 0.974680i $$-0.428218\pi$$
0.223603 + 0.974680i $$0.428218\pi$$
$$110$$ 0 0
$$111$$ 8.57475 0.813879
$$112$$ 4.14101 0.391289
$$113$$ 3.28738 0.309250 0.154625 0.987973i $$-0.450583\pi$$
0.154625 + 0.987973i $$0.450583\pi$$
$$114$$ −1.85363 −0.173609
$$115$$ 0 0
$$116$$ 0.240579 0.0223372
$$117$$ −1.00000 −0.0924500
$$118$$ 10.5606 0.972181
$$119$$ −2.87971 −0.263983
$$120$$ 0 0
$$121$$ 6.75942 0.614493
$$122$$ −11.1724 −1.01151
$$123$$ 12.0678 1.08812
$$124$$ −0.515704 −0.0463116
$$125$$ 0 0
$$126$$ 1.43986 0.128272
$$127$$ 0.428386 0.0380131 0.0190065 0.999819i $$-0.493950\pi$$
0.0190065 + 0.999819i $$0.493950\pi$$
$$128$$ −11.8946 −1.05135
$$129$$ 7.14101 0.628731
$$130$$ 0 0
$$131$$ 14.1878 1.23959 0.619797 0.784762i $$-0.287215\pi$$
0.619797 + 0.784762i $$0.287215\pi$$
$$132$$ −0.308407 −0.0268433
$$133$$ 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 $$0.716229\pi$$
$$138$$ 11.5487 0.983089
$$139$$ 9.85363 0.835774 0.417887 0.908499i $$-0.362771\pi$$
0.417887 + 0.908499i $$0.362771\pi$$
$$140$$ 0 0
$$141$$ 1.95289 0.164463
$$142$$ −15.5345 −1.30363
$$143$$ −4.21419 −0.352409
$$144$$ −4.14101 −0.345084
$$145$$ 0 0
$$146$$ −11.9878 −0.992115
$$147$$ −1.00000 −0.0824786
$$148$$ −0.627525 −0.0515822
$$149$$ −3.38663 −0.277444 −0.138722 0.990331i $$-0.544299\pi$$
−0.138722 + 0.990331i $$0.544299\pi$$
$$150$$ 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
$$153$$ 2.87971 0.232811
$$154$$ 6.06783 0.488959
$$155$$ 0 0
$$156$$ 0.0731828 0.00585932
$$157$$ 0.867482 0.0692326 0.0346163 0.999401i $$-0.488979\pi$$
0.0346163 + 0.999401i $$0.488979\pi$$
$$158$$ −6.43910 −0.512267
$$159$$ −5.14101 −0.407709
$$160$$ 0 0
$$161$$ −8.02072 −0.632121
$$162$$ −1.43986 −0.113126
$$163$$ −4.42839 −0.346858 −0.173429 0.984846i $$-0.555485\pi$$
−0.173429 + 0.984846i $$0.555485\pi$$
$$164$$ −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 $$0.792130\pi$$
$$168$$ 2.77434 0.214045
$$169$$ 1.00000 0.0769231
$$170$$ 0 0
$$171$$ −1.28738 −0.0984481
$$172$$ −0.522599 −0.0398478
$$173$$ −15.0154 −1.14160 −0.570799 0.821090i $$-0.693366\pi$$
−0.570799 + 0.821090i $$0.693366\pi$$
$$174$$ 4.73334 0.358834
$$175$$ 0 0
$$176$$ −17.4510 −1.31542
$$177$$ 7.33448 0.551293
$$178$$ 8.13221 0.609535
$$179$$ −5.35176 −0.400009 −0.200004 0.979795i $$-0.564096\pi$$
−0.200004 + 0.979795i $$0.564096\pi$$
$$180$$ 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
$$183$$ −7.75942 −0.573593
$$184$$ 22.2522 1.64045
$$185$$ 0 0
$$186$$ −10.1464 −0.743968
$$187$$ 12.1357 0.887447
$$188$$ −0.142918 −0.0104234
$$189$$ 1.00000 0.0727393
$$190$$ 0 0
$$191$$ 20.1756 1.45985 0.729927 0.683525i $$-0.239554\pi$$
0.729927 + 0.683525i $$0.239554\pi$$
$$192$$ −7.68624 −0.554706
$$193$$ 26.3756 1.89856 0.949279 0.314435i $$-0.101815\pi$$
0.949279 + 0.314435i $$0.101815\pi$$
$$194$$ 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 $$0.245432\pi$$
$$198$$ −6.06783 −0.431222
$$199$$ 1.75942 0.124722 0.0623610 0.998054i $$-0.480137\pi$$
0.0623610 + 0.998054i $$0.480137\pi$$
$$200$$ 0 0
$$201$$ 12.0414 0.849338
$$202$$ 15.8009 1.11174
$$203$$ −3.28738 −0.230729
$$204$$ −0.210745 −0.0147551
$$205$$ 0 0
$$206$$ 23.6545 1.64809
$$207$$ 8.02072 0.557479
$$208$$ 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
$$213$$ −10.7889 −0.739246
$$214$$ −4.97392 −0.340010
$$215$$ 0 0
$$216$$ −2.77434 −0.188770
$$217$$ 7.04680 0.478368
$$218$$ −6.72263 −0.455314
$$219$$ −8.32568 −0.562597
$$220$$ 0 0
$$221$$ −2.87971 −0.193710
$$222$$ −12.3464 −0.828636
$$223$$ 11.5694 0.774744 0.387372 0.921923i $$-0.373383\pi$$
0.387372 + 0.921923i $$0.373383\pi$$
$$224$$ −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 $$0.380579\pi$$
$$228$$ 0.0942138 0.00623946
$$229$$ −13.7073 −0.905802 −0.452901 0.891561i $$-0.649611\pi$$
−0.452901 + 0.891561i $$0.649611\pi$$
$$230$$ 0 0
$$231$$ 4.21419 0.277274
$$232$$ 9.12029 0.598776
$$233$$ −1.23522 −0.0809222 −0.0404611 0.999181i $$-0.512883\pi$$
−0.0404611 + 0.999181i $$0.512883\pi$$
$$234$$ 1.43986 0.0941263
$$235$$ 0 0
$$236$$ −0.536758 −0.0349400
$$237$$ −4.47204 −0.290491
$$238$$ 4.14637 0.268769
$$239$$ 19.1061 1.23587 0.617936 0.786228i $$-0.287969\pi$$
0.617936 + 0.786228i $$0.287969\pi$$
$$240$$ 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
$$243$$ −1.00000 −0.0641500
$$244$$ 0.567856 0.0363533
$$245$$ 0 0
$$246$$ −17.3759 −1.10785
$$247$$ 1.28738 0.0819137
$$248$$ −19.5502 −1.24144
$$249$$ −3.80653 −0.241229
$$250$$ 0 0
$$251$$ −15.0031 −0.946990 −0.473495 0.880797i $$-0.657008\pi$$
−0.473495 + 0.880797i $$0.657008\pi$$
$$252$$ −0.0731828 −0.00461008
$$253$$ 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 $$0.366828\pi$$
$$258$$ −10.2820 −0.640131
$$259$$ 8.57475 0.532809
$$260$$ 0 0
$$261$$ 3.28738 0.203483
$$262$$ −20.4284 −1.26207
$$263$$ 2.59547 0.160044 0.0800218 0.996793i $$-0.474501\pi$$
0.0800218 + 0.996793i $$0.474501\pi$$
$$264$$ −11.6916 −0.719568
$$265$$ 0 0
$$266$$ −1.85363 −0.113654
$$267$$ 5.64793 0.345648
$$268$$ −0.881227 −0.0538295
$$269$$ −23.3081 −1.42112 −0.710560 0.703637i $$-0.751558\pi$$
−0.710560 + 0.703637i $$0.751558\pi$$
$$270$$ 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
$$273$$ −1.00000 −0.0605228
$$274$$ 21.1759 1.27928
$$275$$ 0 0
$$276$$ −0.586979 −0.0353320
$$277$$ −15.1932 −0.912869 −0.456434 0.889757i $$-0.650874\pi$$
−0.456434 + 0.889757i $$0.650874\pi$$
$$278$$ −14.1878 −0.850928
$$279$$ −7.04680 −0.421881
$$280$$ 0 0
$$281$$ −7.57506 −0.451890 −0.225945 0.974140i $$-0.572547\pi$$
−0.225945 + 0.974140i $$0.572547\pi$$
$$282$$ −2.81188 −0.167445
$$283$$ −19.0974 −1.13522 −0.567610 0.823298i $$-0.692132\pi$$
−0.567610 + 0.823298i $$0.692132\pi$$
$$284$$ 0.789565 0.0468521
$$285$$ 0 0
$$286$$ 6.06783 0.358798
$$287$$ 12.0678 0.712341
$$288$$ 0.413779 0.0243822
$$289$$ −8.70727 −0.512192
$$290$$ 0 0
$$291$$ 6.90043 0.404510
$$292$$ 0.609297 0.0356564
$$293$$ 3.79430 0.221665 0.110833 0.993839i $$-0.464648\pi$$
0.110833 + 0.993839i $$0.464648\pi$$
$$294$$ 1.43986 0.0839741
$$295$$ 0 0
$$296$$ −23.7893 −1.38272
$$297$$ −4.21419 −0.244532
$$298$$ 4.87626 0.282474
$$299$$ −8.02072 −0.463850
$$300$$ 0 0
$$301$$ 7.14101 0.411601
$$302$$ 31.5257 1.81410
$$303$$ 10.9739 0.630435
$$304$$ 5.33104 0.305756
$$305$$ 0 0
$$306$$ −4.14637 −0.237032
$$307$$ 5.19316 0.296389 0.148195 0.988958i $$-0.452654\pi$$
0.148195 + 0.988958i $$0.452654\pi$$
$$308$$ −0.308407 −0.0175731
$$309$$ 16.4284 0.934578
$$310$$ 0 0
$$311$$ −13.8536 −0.785568 −0.392784 0.919631i $$-0.628488\pi$$
−0.392784 + 0.919631i $$0.628488\pi$$
$$312$$ 2.77434 0.157066
$$313$$ 32.6162 1.84358 0.921788 0.387694i $$-0.126728\pi$$
0.921788 + 0.387694i $$0.126728\pi$$
$$314$$ −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 $$0.609771\pi$$
$$318$$ 7.40231 0.415101
$$319$$ 13.8536 0.775655
$$320$$ 0 0
$$321$$ −3.45446 −0.192809
$$322$$ 11.5487 0.643583
$$323$$ −3.70727 −0.206278
$$324$$ 0.0731828 0.00406571
$$325$$ 0 0
$$326$$ 6.37623 0.353147
$$327$$ −4.66896 −0.258194
$$328$$ −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
$$333$$ −8.57475 −0.469893
$$334$$ 29.5568 1.61728
$$335$$ 0 0
$$336$$ −4.14101 −0.225911
$$337$$ −26.2905 −1.43214 −0.716068 0.698031i $$-0.754060\pi$$
−0.716068 + 0.698031i $$0.754060\pi$$
$$338$$ −1.43986 −0.0783178
$$339$$ −3.28738 −0.178546
$$340$$ 0 0
$$341$$ −29.6966 −1.60816
$$342$$ 1.85363 0.100233
$$343$$ −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 $$0.645975\pi$$
$$348$$ −0.240579 −0.0128964
$$349$$ 14.1793 0.759001 0.379501 0.925191i $$-0.376096\pi$$
0.379501 + 0.925191i $$0.376096\pi$$
$$350$$ 0 0
$$351$$ 1.00000 0.0533761
$$352$$ 1.74375 0.0929419
$$353$$ 29.8272 1.58754 0.793772 0.608215i $$-0.208114\pi$$
0.793772 + 0.608215i $$0.208114\pi$$
$$354$$ −10.5606 −0.561289
$$355$$ 0 0
$$356$$ −0.413332 −0.0219065
$$357$$ 2.87971 0.152410
$$358$$ 7.70575 0.407262
$$359$$ 1.34671 0.0710767 0.0355383 0.999368i $$-0.488685\pi$$
0.0355383 + 0.999368i $$0.488685\pi$$
$$360$$ 0 0
$$361$$ −17.3427 −0.912772
$$362$$ −16.5104 −0.867766
$$363$$ −6.75942 −0.354778
$$364$$ 0.0731828 0.00383582
$$365$$ 0 0
$$366$$ 11.1724 0.583993
$$367$$ −36.1771 −1.88843 −0.944214 0.329331i $$-0.893177\pi$$
−0.944214 + 0.329331i $$0.893177\pi$$
$$368$$ −33.2139 −1.73139
$$369$$ −12.0678 −0.628226
$$370$$ 0 0
$$371$$ −5.14101 −0.266908
$$372$$ 0.515704 0.0267380
$$373$$ 16.9089 0.875511 0.437755 0.899094i $$-0.355774\pi$$
0.437755 + 0.899094i $$0.355774\pi$$
$$374$$ −17.4736 −0.903538
$$375$$ 0 0
$$376$$ −5.41798 −0.279411
$$377$$ −3.28738 −0.169308
$$378$$ −1.43986 −0.0740582
$$379$$ −11.6131 −0.596523 −0.298261 0.954484i $$-0.596407\pi$$
−0.298261 + 0.954484i $$0.596407\pi$$
$$380$$ 0 0
$$381$$ −0.428386 −0.0219469
$$382$$ −29.0499 −1.48632
$$383$$ −14.6134 −0.746708 −0.373354 0.927689i $$-0.621792\pi$$
−0.373354 + 0.927689i $$0.621792\pi$$
$$384$$ 11.8946 0.606995
$$385$$ 0 0
$$386$$ −37.9771 −1.93298
$$387$$ −7.14101 −0.362998
$$388$$ −0.504993 −0.0256371
$$389$$ 6.00000 0.304212 0.152106 0.988364i $$-0.451394\pi$$
0.152106 + 0.988364i $$0.451394\pi$$
$$390$$ 0 0
$$391$$ 23.0974 1.16808
$$392$$ 2.77434 0.140125
$$393$$ −14.1878 −0.715680
$$394$$ −28.9875 −1.46037
$$395$$ 0 0
$$396$$ 0.308407 0.0154980
$$397$$ −25.6982 −1.28975 −0.644877 0.764287i $$-0.723091\pi$$
−0.644877 + 0.764287i $$0.723091\pi$$
$$398$$ −2.53331 −0.126983
$$399$$ −1.28738 −0.0644494
$$400$$ 0 0
$$401$$ −19.5637 −0.976966 −0.488483 0.872573i $$-0.662450\pi$$
−0.488483 + 0.872573i $$0.662450\pi$$
$$402$$ −17.3379 −0.864737
$$403$$ 7.04680 0.351026
$$404$$ −0.803103 −0.0399559
$$405$$ 0 0
$$406$$ 4.73334 0.234912
$$407$$ −36.1357 −1.79118
$$408$$ −7.98929 −0.395529
$$409$$ 3.19316 0.157892 0.0789458 0.996879i $$-0.474845\pi$$
0.0789458 + 0.996879i $$0.474845\pi$$
$$410$$ 0 0
$$411$$ 14.7070 0.725441
$$412$$ −1.20228 −0.0592319
$$413$$ 7.33448 0.360906
$$414$$ −11.5487 −0.567586
$$415$$ 0 0
$$416$$ −0.413779 −0.0202872
$$417$$ −9.85363 −0.482535
$$418$$ 7.81157 0.382076
$$419$$ 30.2292 1.47680 0.738398 0.674366i $$-0.235583\pi$$
0.738398 + 0.674366i $$0.235583\pi$$
$$420$$ 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
$$423$$ −1.95289 −0.0949529
$$424$$ 14.2629 0.692668
$$425$$ 0 0
$$426$$ 15.5345 0.752650
$$427$$ −7.75942 −0.375505
$$428$$ 0.252807 0.0122199
$$429$$ 4.21419 0.203463
$$430$$ 0 0
$$431$$ 16.7713 0.807847 0.403923 0.914793i $$-0.367646\pi$$
0.403923 + 0.914793i $$0.367646\pi$$
$$432$$ 4.14101 0.199234
$$433$$ −25.8530 −1.24242 −0.621208 0.783646i $$-0.713358\pi$$
−0.621208 + 0.783646i $$0.713358\pi$$
$$434$$ −10.1464 −0.487041
$$435$$ 0 0
$$436$$ 0.341688 0.0163639
$$437$$ −10.3257 −0.493944
$$438$$ 11.9878 0.572798
$$439$$ 25.0553 1.19582 0.597912 0.801562i $$-0.295997\pi$$
0.597912 + 0.801562i $$0.295997\pi$$
$$440$$ 0 0
$$441$$ 1.00000 0.0476190
$$442$$ 4.14637 0.197223
$$443$$ −4.64762 −0.220815 −0.110408 0.993886i $$-0.535216\pi$$
−0.110408 + 0.993886i $$0.535216\pi$$
$$444$$ 0.627525 0.0297810
$$445$$ 0 0
$$446$$ −16.6583 −0.788791
$$447$$ 3.38663 0.160182
$$448$$ −7.68624 −0.363141
$$449$$ 0.494696 0.0233462 0.0116731 0.999932i $$-0.496284\pi$$
0.0116731 + 0.999932i $$0.496284\pi$$
$$450$$ 0 0
$$451$$ −50.8561 −2.39472
$$452$$ 0.240579 0.0113159
$$453$$ 21.8951 1.02872
$$454$$ −15.8985 −0.746155
$$455$$ 0 0
$$456$$ 3.57161 0.167256
$$457$$ −26.4698 −1.23821 −0.619103 0.785310i $$-0.712504\pi$$
−0.619103 + 0.785310i $$0.712504\pi$$
$$458$$ 19.7365 0.922225
$$459$$ −2.87971 −0.134413
$$460$$ 0 0
$$461$$ −1.08168 −0.0503786 −0.0251893 0.999683i $$-0.508019\pi$$
−0.0251893 + 0.999683i $$0.508019\pi$$
$$462$$ −6.06783 −0.282301
$$463$$ 32.7098 1.52015 0.760076 0.649834i $$-0.225162\pi$$
0.760076 + 0.649834i $$0.225162\pi$$
$$464$$ −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 $$0.370683\pi$$
$$468$$ −0.0731828 −0.00338288
$$469$$ 12.0414 0.556022
$$470$$ 0 0
$$471$$ −0.867482 −0.0399715
$$472$$ −20.3483 −0.936608
$$473$$ −30.0936 −1.38370
$$474$$ 6.43910 0.295758
$$475$$ 0 0
$$476$$ −0.210745 −0.00965950
$$477$$ 5.14101 0.235391
$$478$$ −27.5101 −1.25828
$$479$$ −17.1790 −0.784929 −0.392464 0.919767i $$-0.628377\pi$$
−0.392464 + 0.919767i $$0.628377\pi$$
$$480$$ 0 0
$$481$$ 8.57475 0.390975
$$482$$ 17.3257 0.789164
$$483$$ 8.02072 0.364955
$$484$$ 0.494674 0.0224852
$$485$$ 0 0
$$486$$ 1.43986 0.0653132
$$487$$ 4.28264 0.194065 0.0970325 0.995281i $$-0.469065\pi$$
0.0970325 + 0.995281i $$0.469065\pi$$
$$488$$ 21.5273 0.974493
$$489$$ 4.42839 0.200259
$$490$$ 0 0
$$491$$ 27.4545 1.23900 0.619501 0.784996i $$-0.287335\pi$$
0.619501 + 0.784996i $$0.287335\pi$$
$$492$$ 0.883158 0.0398158
$$493$$ 9.46669 0.426358
$$494$$ −1.85363 −0.0833990
$$495$$ 0 0
$$496$$ 29.1809 1.31026
$$497$$ −10.7889 −0.483950
$$498$$ 5.48085 0.245603
$$499$$ −37.0452 −1.65837 −0.829185 0.558974i $$-0.811195\pi$$
−0.829185 + 0.558974i $$0.811195\pi$$
$$500$$ 0 0
$$501$$ 20.5276 0.917108
$$502$$ 21.6023 0.964160
$$503$$ −3.23744 −0.144350 −0.0721752 0.997392i $$-0.522994\pi$$
−0.0721752 + 0.997392i $$0.522994\pi$$
$$504$$ −2.77434 −0.123579
$$505$$ 0 0
$$506$$ −48.6683 −2.16357
$$507$$ −1.00000 −0.0444116
$$508$$ 0.0313505 0.00139095
$$509$$ −30.4877 −1.35134 −0.675672 0.737202i $$-0.736147\pi$$
−0.675672 + 0.737202i $$0.736147\pi$$
$$510$$ 0 0
$$511$$ −8.32568 −0.368306
$$512$$ 21.2637 0.939730
$$513$$ 1.28738 0.0568390
$$514$$ −18.7557 −0.827277
$$515$$ 0 0
$$516$$ 0.522599 0.0230062
$$517$$ −8.22987 −0.361949
$$518$$ −12.3464 −0.542470
$$519$$ 15.0154 0.659101
$$520$$ 0 0
$$521$$ −25.3602 −1.11105 −0.555526 0.831499i $$-0.687483\pi$$
−0.555526 + 0.831499i $$0.687483\pi$$
$$522$$ −4.73334 −0.207173
$$523$$ 7.00314 0.306226 0.153113 0.988209i $$-0.451070\pi$$
0.153113 + 0.988209i $$0.451070\pi$$
$$524$$ 1.03830 0.0453585
$$525$$ 0 0
$$526$$ −3.73710 −0.162945
$$527$$ −20.2927 −0.883965
$$528$$ 17.4510 0.759458
$$529$$ 41.3320 1.79704
$$530$$ 0 0
$$531$$ −7.33448 −0.318289
$$532$$ 0.0942138 0.00408469
$$533$$ 12.0678 0.522716
$$534$$ −8.13221 −0.351915
$$535$$ 0 0
$$536$$ −33.4070 −1.44296
$$537$$ 5.35176 0.230945
$$538$$ 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
$$543$$ −11.4667 −0.492083
$$544$$ 1.19156 0.0510879
$$545$$ 0 0
$$546$$ 1.43986 0.0616201
$$547$$ 22.0499 0.942787 0.471394 0.881923i $$-0.343751\pi$$
0.471394 + 0.881923i $$0.343751\pi$$
$$548$$ −1.07630 −0.0459771
$$549$$ 7.75942 0.331164
$$550$$ 0 0
$$551$$ −4.23209 −0.180293
$$552$$ −22.2522 −0.947116
$$553$$ −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 $$0.438362\pi$$
$$558$$ 10.1464 0.429530
$$559$$ 7.14101 0.302033
$$560$$ 0 0
$$561$$ −12.1357 −0.512368
$$562$$ 10.9070 0.460084
$$563$$ −34.1350 −1.43862 −0.719310 0.694689i $$-0.755542\pi$$
−0.719310 + 0.694689i $$0.755542\pi$$
$$564$$ 0.142918 0.00601794
$$565$$ 0 0
$$566$$ 27.4974 1.15580
$$567$$ −1.00000 −0.0419961
$$568$$ 29.9322 1.25593
$$569$$ 1.23522 0.0517833 0.0258916 0.999665i $$-0.491758\pi$$
0.0258916 + 0.999665i $$0.491758\pi$$
$$570$$ 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
$$573$$ −20.1756 −0.842847
$$574$$ −17.3759 −0.725257
$$575$$ 0 0
$$576$$ 7.68624 0.320260
$$577$$ 23.2852 0.969374 0.484687 0.874688i $$-0.338934\pi$$
0.484687 + 0.874688i $$0.338934\pi$$
$$578$$ 12.5372 0.521479
$$579$$ −26.3756 −1.09613
$$580$$ 0 0
$$581$$ −3.80653 −0.157921
$$582$$ −9.93562 −0.411845
$$583$$ 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 $$0.429023\pi$$
$$588$$ −0.0731828 −0.00301801
$$589$$ 9.07187 0.373800
$$590$$ 0 0
$$591$$ −20.1322 −0.828128
$$592$$ 35.5081 1.45938
$$593$$ −14.1008 −0.579049 −0.289525 0.957171i $$-0.593497\pi$$
−0.289525 + 0.957171i $$0.593497\pi$$
$$594$$ 6.06783 0.248966
$$595$$ 0 0
$$596$$ −0.247843 −0.0101521
$$597$$ −1.75942 −0.0720083
$$598$$ 11.5487 0.472260
$$599$$ −38.9296 −1.59062 −0.795311 0.606202i $$-0.792692\pi$$
−0.795311 + 0.606202i $$0.792692\pi$$
$$600$$ 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
$$603$$ −12.0414 −0.490365
$$604$$ −1.60234 −0.0651984
$$605$$ 0 0
$$606$$ −15.8009 −0.641866
$$607$$ −19.3203 −0.784188 −0.392094 0.919925i $$-0.628249\pi$$
−0.392094 + 0.919925i $$0.628249\pi$$
$$608$$ −0.532689 −0.0216034
$$609$$ 3.28738 0.133211
$$610$$ 0 0
$$611$$ 1.95289 0.0790056
$$612$$ 0.210745 0.00851888
$$613$$ −35.3197 −1.42655 −0.713275 0.700885i $$-0.752789\pi$$
−0.713275 + 0.700885i $$0.752789\pi$$
$$614$$ −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 $$0.336341\pi$$
$$618$$ −23.6545 −0.951523
$$619$$ 2.90892 0.116919 0.0584597 0.998290i $$-0.481381\pi$$
0.0584597 + 0.998290i $$0.481381\pi$$
$$620$$ 0 0
$$621$$ −8.02072 −0.321860
$$622$$ 19.9472 0.799811
$$623$$ 5.64793 0.226280
$$624$$ −4.14101 −0.165773
$$625$$ 0 0
$$626$$ −46.9626 −1.87700
$$627$$ 5.42525 0.216664
$$628$$ 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
$$633$$ −15.5694 −0.618828
$$634$$ 17.3330 0.688380
$$635$$ 0 0
$$636$$ −0.376234 −0.0149186
$$637$$ −1.00000 −0.0396214
$$638$$ −19.9472 −0.789718
$$639$$ 10.7889 0.426804
$$640$$ 0 0
$$641$$ 12.2842 0.485198 0.242599 0.970127i $$-0.422000\pi$$
0.242599 + 0.970127i $$0.422000\pi$$
$$642$$ 4.97392 0.196305
$$643$$ −9.14950 −0.360821 −0.180411 0.983591i $$-0.557743\pi$$
−0.180411 + 0.983591i $$0.557743\pi$$
$$644$$ −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 $$0.772383\pi$$
$$648$$ 2.77434 0.108986
$$649$$ −30.9089 −1.21328
$$650$$ 0 0
$$651$$ −7.04680 −0.276186
$$652$$ −0.324082 −0.0126920
$$653$$ −28.4805 −1.11453 −0.557265 0.830335i $$-0.688149\pi$$
−0.557265 + 0.830335i $$0.688149\pi$$
$$654$$ 6.72263 0.262876
$$655$$ 0 0
$$656$$ 49.9730 1.95112
$$657$$ 8.32568 0.324816
$$658$$ −2.81188 −0.109619
$$659$$ −30.4384 −1.18571 −0.592856 0.805309i $$-0.702000\pi$$
−0.592856 + 0.805309i $$0.702000\pi$$
$$660$$ 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
$$663$$ 2.87971 0.111839
$$664$$ 10.5606 0.409830
$$665$$ 0 0
$$666$$ 12.3464 0.478413
$$667$$ 26.3671 1.02094
$$668$$ −1.50227 −0.0581246
$$669$$ −11.5694 −0.447299
$$670$$ 0 0
$$671$$ 32.6997 1.26236
$$672$$ 0.413779 0.0159619
$$673$$ −26.7961 −1.03291 −0.516457 0.856313i $$-0.672750\pi$$
−0.516457 + 0.856313i $$0.672750\pi$$
$$674$$ 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 $$0.372409\pi$$
$$678$$ 4.73334 0.181783
$$679$$ 6.90043 0.264814
$$680$$ 0 0
$$681$$ −11.0418 −0.423121
$$682$$ 42.7587 1.63732
$$683$$ −44.9240 −1.71897 −0.859484 0.511162i $$-0.829215\pi$$
−0.859484 + 0.511162i $$0.829215\pi$$
$$684$$ −0.0942138 −0.00360235
$$685$$ 0 0
$$686$$ 1.43986 0.0549739
$$687$$ 13.7073 0.522965
$$688$$ 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
$$693$$ −4.21419 −0.160084
$$694$$ 23.7472 0.901431
$$695$$ 0 0
$$696$$ −9.12029 −0.345704
$$697$$ −34.7518 −1.31632
$$698$$ −20.4162 −0.772763
$$699$$ 1.23522 0.0467205
$$700$$ 0 0
$$701$$ 31.4645 1.18840 0.594198 0.804319i $$-0.297469\pi$$
0.594198 + 0.804319i $$0.297469\pi$$
$$702$$ −1.43986 −0.0543438
$$703$$ 11.0389 0.416341
$$704$$ 32.3913 1.22079
$$705$$ 0 0
$$706$$ −42.9469 −1.61633
$$707$$ 10.9739 0.412717
$$708$$ 0.536758 0.0201726
$$709$$ −25.3373 −0.951563 −0.475781 0.879564i $$-0.657835\pi$$
−0.475781 + 0.879564i $$0.657835\pi$$
$$710$$ 0 0
$$711$$ 4.47204 0.167715
$$712$$ −15.6693 −0.587231
$$713$$ −56.5204 −2.11670
$$714$$ −4.14637 −0.155174
$$715$$ 0 0
$$716$$ −0.391657 −0.0146369
$$717$$ −19.1061 −0.713532
$$718$$ −1.93907 −0.0723654
$$719$$ −43.8002 −1.63347 −0.816737 0.577011i $$-0.804219\pi$$
−0.816737 + 0.577011i $$0.804219\pi$$
$$720$$ 0 0
$$721$$ 16.4284 0.611825
$$722$$ 24.9709 0.929322
$$723$$ 12.0329 0.447510
$$724$$ 0.839165 0.0311873
$$725$$ 0 0
$$726$$ 9.73259 0.361210
$$727$$ −0.944090 −0.0350144 −0.0175072 0.999847i $$-0.505573\pi$$
−0.0175072 + 0.999847i $$0.505573\pi$$
$$728$$ 2.77434 0.102824
$$729$$ 1.00000 0.0370370
$$730$$ 0 0
$$731$$ −20.5640 −0.760588
$$732$$ −0.567856 −0.0209886
$$733$$ 5.09957 0.188357 0.0941784 0.995555i $$-0.469978\pi$$
0.0941784 + 0.995555i $$0.469978\pi$$
$$734$$ 52.0898 1.92267
$$735$$ 0 0
$$736$$ 3.31881 0.122333
$$737$$ −50.7450 −1.86921
$$738$$ 17.3759 0.639617
$$739$$ −8.25129 −0.303529 −0.151764 0.988417i $$-0.548495\pi$$
−0.151764 + 0.988417i $$0.548495\pi$$
$$740$$ 0 0
$$741$$ −1.28738 −0.0472929
$$742$$ 7.40231 0.271747
$$743$$ −38.4962 −1.41229 −0.706145 0.708068i $$-0.749567\pi$$
−0.706145 + 0.708068i $$0.749567\pi$$
$$744$$ 19.5502 0.716745
$$745$$ 0 0
$$746$$ −24.3464 −0.891385
$$747$$ 3.80653 0.139274
$$748$$ 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
$$753$$ 15.0031 0.546745
$$754$$ 4.73334 0.172378
$$755$$ 0 0
$$756$$ 0.0731828 0.00266163
$$757$$ 39.7158 1.44349 0.721747 0.692157i $$-0.243339\pi$$
0.721747 + 0.692157i $$0.243339\pi$$
$$758$$ 16.7211 0.607338
$$759$$ −33.8009 −1.22689
$$760$$ 0 0
$$761$$ 10.7390 0.389289 0.194644 0.980874i $$-0.437645\pi$$
0.194644 + 0.980874i $$0.437645\pi$$
$$762$$ 0.616813 0.0223448
$$763$$ −4.66896 −0.169028
$$764$$ 1.47651 0.0534181
$$765$$ 0 0
$$766$$ 21.0411 0.760247
$$767$$ 7.33448 0.264833
$$768$$ −1.75406 −0.0632944
$$769$$ −50.5549 −1.82306 −0.911529 0.411237i $$-0.865097\pi$$
−0.911529 + 0.411237i $$0.865097\pi$$
$$770$$ 0 0
$$771$$ −13.0261 −0.469123
$$772$$ 1.93024 0.0694709
$$773$$ −8.91770 −0.320747 −0.160374 0.987056i $$-0.551270\pi$$
−0.160374 + 0.987056i $$0.551270\pi$$
$$774$$ 10.2820 0.369580
$$775$$ 0 0
$$776$$ −19.1441 −0.687234
$$777$$ −8.57475 −0.307617
$$778$$ −8.63913 −0.309728
$$779$$ 15.5358 0.556629
$$780$$ 0 0
$$781$$ 45.4667 1.62693
$$782$$ −33.2568 −1.18926
$$783$$ −3.28738 −0.117481
$$784$$ −4.14101 −0.147893
$$785$$ 0 0
$$786$$ 20.4284 0.728656
$$787$$ −22.3671 −0.797302 −0.398651 0.917103i $$-0.630521\pi$$
−0.398651 + 0.917103i $$0.630521\pi$$
$$788$$ 1.47333 0.0524853
$$789$$ −2.59547 −0.0924012
$$790$$ 0 0
$$791$$ −3.28738 −0.116886
$$792$$ 11.6916 0.415443
$$793$$ −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 $$0.618586\pi$$
$$798$$ 1.85363 0.0656179
$$799$$ −5.62377 −0.198955
$$800$$ 0 0
$$801$$ −5.64793 −0.199560
$$802$$ 28.1689 0.994680
$$803$$ 35.0860 1.23816
$$804$$ 0.881227 0.0310785
$$805$$ 0 0
$$806$$ −10.1464 −0.357390
$$807$$ 23.3081 0.820484
$$808$$ −30.4454 −1.07106
$$809$$ 7.83443 0.275444 0.137722 0.990471i $$-0.456022\pi$$
0.137722 + 0.990471i $$0.456022\pi$$
$$810$$ 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
$$813$$ 6.05215 0.212258
$$814$$ 52.0301 1.82365
$$815$$ 0 0
$$816$$ 11.9249 0.417455
$$817$$ 9.19316 0.321628
$$818$$ −4.59769 −0.160754
$$819$$ 1.00000 0.0349428
$$820$$ 0 0
$$821$$ 4.42494 0.154431 0.0772157 0.997014i $$-0.475397\pi$$
0.0772157 + 0.997014i $$0.475397\pi$$
$$822$$ −21.1759 −0.738594
$$823$$ 28.7525 1.00225 0.501124 0.865375i $$-0.332920\pi$$
0.501124 + 0.865375i $$0.332920\pi$$
$$824$$ −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 $$0.469033\pi$$
$$828$$ 0.586979 0.0203989
$$829$$ 53.4974 1.85804 0.929021 0.370027i $$-0.120652\pi$$
0.929021 + 0.370027i $$0.120652\pi$$
$$830$$ 0 0
$$831$$ 15.1932 0.527045
$$832$$ −7.68624 −0.266472
$$833$$ 2.87971 0.0997760
$$834$$ 14.1878 0.491284
$$835$$ 0 0
$$836$$ −0.397035 −0.0137317
$$837$$ 7.04680 0.243573
$$838$$ −43.5257 −1.50357
$$839$$ −43.0405 −1.48592 −0.742962 0.669334i $$-0.766580\pi$$
−0.742962 + 0.669334i $$0.766580\pi$$
$$840$$ 0 0
$$841$$ −18.1932 −0.627350
$$842$$ −38.8055 −1.33733
$$843$$ 7.57506 0.260899
$$844$$ 1.13941 0.0392202
$$845$$ 0 0
$$846$$ 2.81188 0.0966745
$$847$$ −6.75942 −0.232256
$$848$$ −21.2890 −0.731066
$$849$$ 19.0974 0.655419
$$850$$ 0 0
$$851$$ −68.7757 −2.35760
$$852$$ −0.789565 −0.0270501
$$853$$ 30.7434 1.05263 0.526316 0.850289i $$-0.323573\pi$$
0.526316 + 0.850289i $$0.323573\pi$$
$$854$$ 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 $$0.362304\pi$$
$$858$$ −6.06783 −0.207152
$$859$$ −52.1243 −1.77846 −0.889229 0.457461i $$-0.848759\pi$$
−0.889229 + 0.457461i $$0.848759\pi$$
$$860$$ 0 0
$$861$$ −12.0678 −0.411270
$$862$$ −24.1483 −0.822494
$$863$$ −22.2563 −0.757612 −0.378806 0.925476i $$-0.623665\pi$$
−0.378806 + 0.925476i $$0.623665\pi$$
$$864$$ −0.413779 −0.0140771
$$865$$ 0 0
$$866$$ 37.2246 1.26494
$$867$$ 8.70727 0.295714
$$868$$ 0.515704 0.0175041
$$869$$ 18.8461 0.639309
$$870$$ 0 0
$$871$$ 12.0414 0.408009
$$872$$ 12.9533 0.438654
$$873$$ −6.90043 −0.233544
$$874$$ 14.8675 0.502900
$$875$$ 0 0
$$876$$ −0.609297 −0.0205862
$$877$$ −41.0547 −1.38632 −0.693159 0.720785i $$-0.743782\pi$$
−0.693159 + 0.720785i $$0.743782\pi$$
$$878$$ −36.0760 −1.21751
$$879$$ −3.79430 −0.127979
$$880$$ 0 0
$$881$$ −31.0568 −1.04633 −0.523165 0.852231i $$-0.675249\pi$$
−0.523165 + 0.852231i $$0.675249\pi$$
$$882$$ −1.43986 −0.0484824
$$883$$ 56.5357 1.90258 0.951289 0.308300i $$-0.0997600\pi$$
0.951289 + 0.308300i $$0.0997600\pi$$
$$884$$ −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 $$0.553411\pi$$
$$888$$ 23.7893 0.798315
$$889$$ −0.428386 −0.0143676
$$890$$ 0 0
$$891$$ 4.21419 0.141181
$$892$$ 0.846681 0.0283490
$$893$$ 2.51411 0.0841314
$$894$$ −4.87626 −0.163087
$$895$$ 0 0
$$896$$ 11.8946 0.397372
$$897$$ 8.02072 0.267804
$$898$$ −0.712291 −0.0237695
$$899$$ −23.1655 −0.772612
$$900$$ 0 0
$$901$$ 14.8046 0.493213
$$902$$ 73.2255 2.43814
$$903$$ −7.14101 −0.237638
$$904$$ 9.12029 0.303336
$$905$$ 0 0
$$906$$ −31.5257 −1.04737
$$907$$ 31.2936 1.03909 0.519544 0.854444i $$-0.326102\pi$$
0.519544 + 0.854444i $$0.326102\pi$$
$$908$$ 0.808067 0.0268166
$$909$$ −10.9739 −0.363982
$$910$$ 0 0
$$911$$ −9.39320 −0.311210 −0.155605 0.987819i $$-0.549733\pi$$
−0.155605 + 0.987819i $$0.549733\pi$$
$$912$$ −5.33104 −0.176528
$$913$$ 16.0414 0.530894
$$914$$ 38.1127 1.26066
$$915$$ 0 0
$$916$$ −1.00314 −0.0331446
$$917$$ −14.1878 −0.468523
$$918$$ 4.14637 0.136850
$$919$$ 21.7066 0.716036 0.358018 0.933715i $$-0.383453\pi$$
0.358018 + 0.933715i $$0.383453\pi$$
$$920$$ 0 0
$$921$$ −5.19316 −0.171120
$$922$$ 1.55746 0.0512921
$$923$$ −10.7889 −0.355122
$$924$$ 0.308407 0.0101458
$$925$$ 0 0
$$926$$ −47.0974 −1.54771
$$927$$ −16.4284 −0.539579
$$928$$ 1.36025 0.0446523
$$929$$ 35.6787 1.17058 0.585289 0.810824i $$-0.300981\pi$$
0.585289 + 0.810824i $$0.300981\pi$$
$$930$$ 0 0
$$931$$ −1.28738 −0.0421920
$$932$$ −0.0903972 −0.00296106
$$933$$ 13.8536 0.453548
$$934$$ −24.5924 −0.804687
$$935$$ 0 0
$$936$$ −2.77434 −0.0906821
$$937$$ 47.0866 1.53825 0.769127 0.639096i $$-0.220691\pi$$
0.769127 + 0.639096i $$0.220691\pi$$
$$938$$ −17.3379 −0.566103
$$939$$ −32.6162 −1.06439
$$940$$ 0 0
$$941$$ 45.2923 1.47649 0.738244 0.674534i $$-0.235655\pi$$
0.738244 + 0.674534i $$0.235655\pi$$
$$942$$ 1.24905 0.0406962
$$943$$ −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 $$0.524701\pi$$
$$948$$ −0.327277 −0.0106295
$$949$$ −8.32568 −0.270263
$$950$$ 0 0
$$951$$ 12.0380 0.390359
$$952$$ −7.98929 −0.258934
$$953$$ 3.95572 0.128138 0.0640692 0.997945i $$-0.479592\pi$$
0.0640692 + 0.997945i $$0.479592\pi$$
$$954$$ −7.40231 −0.239659
$$955$$ 0 0
$$956$$ 1.39824 0.0452223
$$957$$ −13.8536 −0.447824
$$958$$ 24.7353 0.799160
$$959$$ 14.7070 0.474912
$$960$$ 0 0
$$961$$ 18.6573 0.601850
$$962$$ −12.3464 −0.398064
$$963$$ 3.45446 0.111318
$$964$$ −0.880605 −0.0283624
$$965$$ 0 0
$$966$$ −11.5487 −0.371573
$$967$$ −50.2292 −1.61526 −0.807632 0.589687i $$-0.799251\pi$$
−0.807632 + 0.589687i $$0.799251\pi$$
$$968$$ 18.7529 0.602742
$$969$$ 3.70727 0.119095
$$970$$ 0 0
$$971$$ 38.6576 1.24058 0.620291 0.784372i $$-0.287014\pi$$
0.620291 + 0.784372i $$0.287014\pi$$
$$972$$ −0.0731828 −0.00234734
$$973$$ −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 $$0.393039\pi$$
$$978$$ −6.37623 −0.203889
$$979$$ −23.8015 −0.760699
$$980$$ 0 0
$$981$$ 4.66896 0.149069
$$982$$ −39.5304 −1.26147
$$983$$ 4.48620 0.143088 0.0715438 0.997437i $$-0.477207\pi$$
0.0715438 + 0.997437i $$0.477207\pi$$
$$984$$ 33.4802 1.06731
$$985$$ 0 0
$$986$$ −13.6307 −0.434089
$$987$$ −1.95289 −0.0621613
$$988$$ 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
$$993$$ 26.2820 0.834035
$$994$$ 15.5345 0.492725
$$995$$ 0 0
$$996$$ −0.278572 −0.00882691
$$997$$ −0.491870 −0.0155777 −0.00778884 0.999970i $$-0.502479\pi$$
−0.00778884 + 0.999970i $$0.502479\pi$$
$$998$$ 53.3397 1.68844
$$999$$ 8.57475 0.271293
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 6825.2.a.bg.1.2 4
5.4 even 2 273.2.a.e.1.3 4
15.14 odd 2 819.2.a.k.1.2 4
20.19 odd 2 4368.2.a.br.1.3 4
35.34 odd 2 1911.2.a.s.1.3 4
65.64 even 2 3549.2.a.w.1.2 4
105.104 even 2 5733.2.a.bf.1.2 4

By twisted newform
Twist Min Dim Char Parity Ord Type
273.2.a.e.1.3 4 5.4 even 2
819.2.a.k.1.2 4 15.14 odd 2
1911.2.a.s.1.3 4 35.34 odd 2
3549.2.a.w.1.2 4 65.64 even 2
4368.2.a.br.1.3 4 20.19 odd 2
5733.2.a.bf.1.2 4 105.104 even 2
6825.2.a.bg.1.2 4 1.1 even 1 trivial