# Properties

 Label 1911.2.a.s.1.3 Level $1911$ Weight $2$ Character 1911.1 Self dual yes Analytic conductor $15.259$ 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] = [1911,2,Mod(1,1911)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1911, 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("1911.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1911 = 3 \cdot 7^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1911.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: $$15.2594118263$$ 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 1.3 Root $$-2.10710$$ of defining polynomial Character $$\chi$$ $$=$$ 1911.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+1.43986 q^{2} -1.00000 q^{3} +0.0731828 q^{4} -0.926817 q^{5} -1.43986 q^{6} -2.77434 q^{8} +1.00000 q^{9} +O(q^{10})$$ $$q+1.43986 q^{2} -1.00000 q^{3} +0.0731828 q^{4} -0.926817 q^{5} -1.43986 q^{6} -2.77434 q^{8} +1.00000 q^{9} -1.33448 q^{10} +4.21419 q^{11} -0.0731828 q^{12} -1.00000 q^{13} +0.926817 q^{15} -4.14101 q^{16} +2.87971 q^{17} +1.43986 q^{18} +1.28738 q^{19} -0.0678271 q^{20} +6.06783 q^{22} -8.02072 q^{23} +2.77434 q^{24} -4.14101 q^{25} -1.43986 q^{26} -1.00000 q^{27} +3.28738 q^{29} +1.33448 q^{30} +7.04680 q^{31} -0.413779 q^{32} -4.21419 q^{33} +4.14637 q^{34} +0.0731828 q^{36} +8.57475 q^{37} +1.85363 q^{38} +1.00000 q^{39} +2.57130 q^{40} +12.0678 q^{41} +7.14101 q^{43} +0.308407 q^{44} -0.926817 q^{45} -11.5487 q^{46} -1.95289 q^{47} +4.14101 q^{48} -5.96245 q^{50} -2.87971 q^{51} -0.0731828 q^{52} -5.14101 q^{53} -1.43986 q^{54} -3.90579 q^{55} -1.28738 q^{57} +4.73334 q^{58} +7.33448 q^{59} +0.0678271 q^{60} -7.75942 q^{61} +10.1464 q^{62} +7.68624 q^{64} +0.926817 q^{65} -6.06783 q^{66} +12.0414 q^{67} +0.210745 q^{68} +8.02072 q^{69} +10.7889 q^{71} -2.77434 q^{72} +8.32568 q^{73} +12.3464 q^{74} +4.14101 q^{75} +0.0942138 q^{76} +1.43986 q^{78} +4.47204 q^{79} +3.83796 q^{80} +1.00000 q^{81} +17.3759 q^{82} +3.80653 q^{83} -2.66896 q^{85} +10.2820 q^{86} -3.28738 q^{87} -11.6916 q^{88} +5.64793 q^{89} -1.33448 q^{90} -0.586979 q^{92} -7.04680 q^{93} -2.81188 q^{94} -1.19316 q^{95} +0.413779 q^{96} -6.90043 q^{97} +4.21419 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + q^{2} - 4 q^{3} + 7 q^{4} + 3 q^{5} - q^{6} + 3 q^{8} + 4 q^{9}+O(q^{10})$$ 4 * q + q^2 - 4 * q^3 + 7 * q^4 + 3 * q^5 - q^6 + 3 * q^8 + 4 * q^9 $$4 q + q^{2} - 4 q^{3} + 7 q^{4} + 3 q^{5} - q^{6} + 3 q^{8} + 4 q^{9} + 4 q^{10} - 2 q^{11} - 7 q^{12} - 4 q^{13} - 3 q^{15} + 9 q^{16} + 2 q^{17} + q^{18} - 7 q^{19} + 32 q^{20} - 8 q^{22} + 3 q^{23} - 3 q^{24} + 9 q^{25} - q^{26} - 4 q^{27} + q^{29} - 4 q^{30} - 3 q^{31} + 7 q^{32} + 2 q^{33} + 30 q^{34} + 7 q^{36} + 10 q^{37} - 6 q^{38} + 4 q^{39} + 14 q^{40} + 16 q^{41} + 3 q^{43} - 12 q^{44} + 3 q^{45} - 18 q^{46} - 5 q^{47} - 9 q^{48} + 13 q^{50} - 2 q^{51} - 7 q^{52} + 5 q^{53} - q^{54} - 10 q^{55} + 7 q^{57} - 4 q^{58} + 20 q^{59} - 32 q^{60} - 12 q^{61} + 54 q^{62} + 5 q^{64} - 3 q^{65} + 8 q^{66} - 22 q^{67} + 10 q^{68} - 3 q^{69} + 3 q^{72} + 13 q^{73} - 6 q^{74} - 9 q^{75} + 6 q^{76} + q^{78} + 11 q^{79} + 42 q^{80} + 4 q^{81} - 10 q^{82} - q^{83} + 8 q^{85} - 10 q^{86} - q^{87} - 60 q^{88} + 5 q^{89} + 4 q^{90} + 34 q^{92} + 3 q^{93} - 34 q^{94} + 13 q^{95} - 7 q^{96} + 17 q^{97} - 2 q^{99}+O(q^{100})$$ 4 * q + q^2 - 4 * q^3 + 7 * q^4 + 3 * q^5 - q^6 + 3 * q^8 + 4 * q^9 + 4 * q^10 - 2 * q^11 - 7 * q^12 - 4 * q^13 - 3 * q^15 + 9 * q^16 + 2 * q^17 + q^18 - 7 * q^19 + 32 * q^20 - 8 * q^22 + 3 * q^23 - 3 * q^24 + 9 * q^25 - q^26 - 4 * q^27 + q^29 - 4 * q^30 - 3 * q^31 + 7 * q^32 + 2 * q^33 + 30 * q^34 + 7 * q^36 + 10 * q^37 - 6 * q^38 + 4 * q^39 + 14 * q^40 + 16 * q^41 + 3 * q^43 - 12 * q^44 + 3 * q^45 - 18 * q^46 - 5 * q^47 - 9 * q^48 + 13 * q^50 - 2 * q^51 - 7 * q^52 + 5 * q^53 - q^54 - 10 * q^55 + 7 * q^57 - 4 * q^58 + 20 * q^59 - 32 * q^60 - 12 * q^61 + 54 * q^62 + 5 * q^64 - 3 * q^65 + 8 * q^66 - 22 * q^67 + 10 * q^68 - 3 * q^69 + 3 * q^72 + 13 * q^73 - 6 * q^74 - 9 * q^75 + 6 * q^76 + q^78 + 11 * q^79 + 42 * q^80 + 4 * q^81 - 10 * q^82 - q^83 + 8 * q^85 - 10 * q^86 - q^87 - 60 * q^88 + 5 * q^89 + 4 * q^90 + 34 * q^92 + 3 * q^93 - 34 * q^94 + 13 * q^95 - 7 * q^96 + 17 * q^97 - 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 $$-0.329991\pi$$
0.509066 + 0.860728i $$0.329991\pi$$
$$3$$ −1.00000 −0.577350
$$4$$ 0.0731828 0.0365914
$$5$$ −0.926817 −0.414485 −0.207243 0.978290i $$-0.566449\pi$$
−0.207243 + 0.978290i $$0.566449\pi$$
$$6$$ −1.43986 −0.587818
$$7$$ 0 0
$$8$$ −2.77434 −0.980876
$$9$$ 1.00000 0.333333
$$10$$ −1.33448 −0.422000
$$11$$ 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
$$14$$ 0 0
$$15$$ 0.926817 0.239303
$$16$$ −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 $$-0.452822\pi$$
0.147672 + 0.989036i $$0.452822\pi$$
$$20$$ −0.0678271 −0.0151666
$$21$$ 0 0
$$22$$ 6.06783 1.29367
$$23$$ −8.02072 −1.67244 −0.836218 0.548397i $$-0.815238\pi$$
−0.836218 + 0.548397i $$0.815238\pi$$
$$24$$ 2.77434 0.566309
$$25$$ −4.14101 −0.828202
$$26$$ −1.43986 −0.282379
$$27$$ −1.00000 −0.192450
$$28$$ 0 0
$$29$$ 3.28738 0.610450 0.305225 0.952280i $$-0.401268\pi$$
0.305225 + 0.952280i $$0.401268\pi$$
$$30$$ 1.33448 0.243642
$$31$$ 7.04680 1.26564 0.632821 0.774298i $$-0.281897\pi$$
0.632821 + 0.774298i $$0.281897\pi$$
$$32$$ −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 $$-0.251019\pi$$
0.704840 + 0.709366i $$0.251019\pi$$
$$38$$ 1.85363 0.300699
$$39$$ 1.00000 0.160128
$$40$$ 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
$$45$$ −0.926817 −0.138162
$$46$$ −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
$$49$$ 0 0
$$50$$ −5.96245 −0.843218
$$51$$ −2.87971 −0.403240
$$52$$ −0.0731828 −0.0101486
$$53$$ −5.14101 −0.706172 −0.353086 0.935591i $$-0.614868\pi$$
−0.353086 + 0.935591i $$0.614868\pi$$
$$54$$ −1.43986 −0.195939
$$55$$ −3.90579 −0.526656
$$56$$ 0 0
$$57$$ −1.28738 −0.170517
$$58$$ 4.73334 0.621519
$$59$$ 7.33448 0.954868 0.477434 0.878668i $$-0.341567\pi$$
0.477434 + 0.878668i $$0.341567\pi$$
$$60$$ 0.0678271 0.00875644
$$61$$ −7.75942 −0.993492 −0.496746 0.867896i $$-0.665472\pi$$
−0.496746 + 0.867896i $$0.665472\pi$$
$$62$$ 10.1464 1.28859
$$63$$ 0 0
$$64$$ 7.68624 0.960780
$$65$$ 0.926817 0.114958
$$66$$ −6.06783 −0.746898
$$67$$ 12.0414 1.47110 0.735548 0.677473i $$-0.236925\pi$$
0.735548 + 0.677473i $$0.236925\pi$$
$$68$$ 0.210745 0.0255566
$$69$$ 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
$$75$$ 4.14101 0.478163
$$76$$ 0.0942138 0.0108071
$$77$$ 0 0
$$78$$ 1.43986 0.163031
$$79$$ 4.47204 0.503144 0.251572 0.967839i $$-0.419052\pi$$
0.251572 + 0.967839i $$0.419052\pi$$
$$80$$ 3.83796 0.429097
$$81$$ 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 0
$$85$$ −2.66896 −0.289490
$$86$$ 10.2820 1.10874
$$87$$ −3.28738 −0.352444
$$88$$ −11.6916 −1.24633
$$89$$ 5.64793 0.598680 0.299340 0.954147i $$-0.403234\pi$$
0.299340 + 0.954147i $$0.403234\pi$$
$$90$$ −1.33448 −0.140667
$$91$$ 0 0
$$92$$ −0.586979 −0.0611968
$$93$$ −7.04680 −0.730719
$$94$$ −2.81188 −0.290024
$$95$$ −1.19316 −0.122416
$$96$$ 0.413779 0.0422312
$$97$$ −6.90043 −0.700633 −0.350316 0.936631i $$-0.613926\pi$$
−0.350316 + 0.936631i $$0.613926\pi$$
$$98$$ 0 0
$$99$$ 4.21419 0.423542
$$100$$ −0.303051 −0.0303051
$$101$$ 10.9739 1.09195 0.545973 0.837803i $$-0.316160\pi$$
0.545973 + 0.837803i $$0.316160\pi$$
$$102$$ −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 $$-0.553401\pi$$
−0.166978 + 0.985961i $$0.553401\pi$$
$$108$$ −0.0731828 −0.00704202
$$109$$ 4.66896 0.447206 0.223603 0.974680i $$-0.428218\pi$$
0.223603 + 0.974680i $$0.428218\pi$$
$$110$$ −5.62377 −0.536205
$$111$$ −8.57475 −0.813879
$$112$$ 0 0
$$113$$ −3.28738 −0.309250 −0.154625 0.987973i $$-0.549417\pi$$
−0.154625 + 0.987973i $$0.549417\pi$$
$$114$$ −1.85363 −0.173609
$$115$$ 7.43374 0.693200
$$116$$ 0.240579 0.0223372
$$117$$ −1.00000 −0.0924500
$$118$$ 10.5606 0.972181
$$119$$ 0 0
$$120$$ −2.57130 −0.234727
$$121$$ 6.75942 0.614493
$$122$$ −11.1724 −1.01151
$$123$$ −12.0678 −1.08812
$$124$$ 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
$$129$$ −7.14101 −0.628731
$$130$$ 1.33448 0.117042
$$131$$ −14.1878 −1.23959 −0.619797 0.784762i $$-0.712785\pi$$
−0.619797 + 0.784762i $$0.712785\pi$$
$$132$$ −0.308407 −0.0268433
$$133$$ 0 0
$$134$$ 17.3379 1.49777
$$135$$ 0.926817 0.0797677
$$136$$ −7.98929 −0.685076
$$137$$ 14.7070 1.25650 0.628250 0.778011i $$-0.283771\pi$$
0.628250 + 0.778011i $$0.283771\pi$$
$$138$$ 11.5487 0.983089
$$139$$ −9.85363 −0.835774 −0.417887 0.908499i $$-0.637229\pi$$
−0.417887 + 0.908499i $$0.637229\pi$$
$$140$$ 0 0
$$141$$ 1.95289 0.164463
$$142$$ 15.5345 1.30363
$$143$$ −4.21419 −0.352409
$$144$$ −4.14101 −0.345084
$$145$$ −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 $$-0.544299\pi$$
−0.138722 + 0.990331i $$0.544299\pi$$
$$150$$ 5.96245 0.486832
$$151$$ −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
$$154$$ 0 0
$$155$$ −6.53109 −0.524590
$$156$$ 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
$$160$$ 0.383498 0.0303182
$$161$$ 0 0
$$162$$ 1.43986 0.113126
$$163$$ 4.42839 0.346858 0.173429 0.984846i $$-0.444515\pi$$
0.173429 + 0.984846i $$0.444515\pi$$
$$164$$ 0.883158 0.0689630
$$165$$ 3.90579 0.304065
$$166$$ 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
$$171$$ 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 $$0.564096\pi$$
$$180$$ −0.0678271 −0.00505553
$$181$$ −11.4667 −0.852312 −0.426156 0.904650i $$-0.640133\pi$$
−0.426156 + 0.904650i $$0.640133\pi$$
$$182$$ 0 0
$$183$$ 7.75942 0.573593
$$184$$ 22.2522 1.64045
$$185$$ −7.94723 −0.584292
$$186$$ −10.1464 −0.743968
$$187$$ 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 $$-0.239554\pi$$
0.729927 + 0.683525i $$0.239554\pi$$
$$192$$ −7.68624 −0.554706
$$193$$ −26.3756 −1.89856 −0.949279 0.314435i $$-0.898185\pi$$
−0.949279 + 0.314435i $$0.898185\pi$$
$$194$$ −9.93562 −0.713336
$$195$$ −0.926817 −0.0663708
$$196$$ 0 0
$$197$$ −20.1322 −1.43436 −0.717180 0.696888i $$-0.754568\pi$$
−0.717180 + 0.696888i $$0.754568\pi$$
$$198$$ 6.06783 0.431222
$$199$$ −1.75942 −0.124722 −0.0623610 0.998054i $$-0.519863\pi$$
−0.0623610 + 0.998054i $$0.519863\pi$$
$$200$$ 11.4886 0.812364
$$201$$ −12.0414 −0.849338
$$202$$ 15.8009 1.11174
$$203$$ 0 0
$$204$$ −0.210745 −0.0147551
$$205$$ −11.1847 −0.781171
$$206$$ −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
$$215$$ −6.61841 −0.451372
$$216$$ 2.77434 0.188770
$$217$$ 0 0
$$218$$ 6.72263 0.455314
$$219$$ −8.32568 −0.562597
$$220$$ −0.285836 −0.0192711
$$221$$ −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 0
$$225$$ −4.14101 −0.276067
$$226$$ −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 $$-0.350389\pi$$
0.452901 + 0.891561i $$0.350389\pi$$
$$230$$ 10.7035 0.705769
$$231$$ 0 0
$$232$$ −9.12029 −0.598776
$$233$$ 1.23522 0.0809222 0.0404611 0.999181i $$-0.487117\pi$$
0.0404611 + 0.999181i $$0.487117\pi$$
$$234$$ −1.43986 −0.0941263
$$235$$ 1.80997 0.118070
$$236$$ 0.536758 0.0349400
$$237$$ −4.47204 −0.290491
$$238$$ 0 0
$$239$$ 19.1061 1.23587 0.617936 0.786228i $$-0.287969\pi$$
0.617936 + 0.786228i $$0.287969\pi$$
$$240$$ −3.83796 −0.247739
$$241$$ 12.0329 0.775110 0.387555 0.921847i $$-0.373320\pi$$
0.387555 + 0.921847i $$0.373320\pi$$
$$242$$ 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
$$250$$ 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
$$255$$ 2.66896 0.167137
$$256$$ 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
$$259$$ 0 0
$$260$$ 0.0678271 0.00420646
$$261$$ 3.28738 0.203483
$$262$$ −20.4284 −1.26207
$$263$$ −2.59547 −0.160044 −0.0800218 0.996793i $$-0.525499\pi$$
−0.0800218 + 0.996793i $$0.525499\pi$$
$$264$$ 11.6916 0.719568
$$265$$ 4.76478 0.292698
$$266$$ 0 0
$$267$$ −5.64793 −0.345648
$$268$$ 0.881227 0.0538295
$$269$$ 23.3081 1.42112 0.710560 0.703637i $$-0.248442\pi$$
0.710560 + 0.703637i $$0.248442\pi$$
$$270$$ 1.33448 0.0812140
$$271$$ 6.05215 0.367642 0.183821 0.982960i $$-0.441153\pi$$
0.183821 + 0.982960i $$0.441153\pi$$
$$272$$ −11.9249 −0.723054
$$273$$ 0 0
$$274$$ 21.1759 1.27928
$$275$$ −17.4510 −1.05234
$$276$$ 0.586979 0.0353320
$$277$$ 15.1932 0.912869 0.456434 0.889757i $$-0.349126\pi$$
0.456434 + 0.889757i $$0.349126\pi$$
$$278$$ −14.1878 −0.850928
$$279$$ 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
$$285$$ 1.19316 0.0706768
$$286$$ −6.06783 −0.358798
$$287$$ 0 0
$$288$$ −0.413779 −0.0243822
$$289$$ −8.70727 −0.512192
$$290$$ −4.38695 −0.257610
$$291$$ 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 $$0.464648\pi$$
$$294$$ 0 0
$$295$$ −6.79772 −0.395779
$$296$$ −23.7893 −1.38272
$$297$$ −4.21419 −0.244532
$$298$$ −4.87626 −0.282474
$$299$$ 8.02072 0.463850
$$300$$ 0.303051 0.0174966
$$301$$ 0 0
$$302$$ −31.5257 −1.81410
$$303$$ −10.9739 −0.630435
$$304$$ −5.33104 −0.305756
$$305$$ 7.19156 0.411788
$$306$$ 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 0
$$309$$ 16.4284 0.934578
$$310$$ −9.40383 −0.534101
$$311$$ 13.8536 0.785568 0.392784 0.919631i $$-0.371512\pi$$
0.392784 + 0.919631i $$0.371512\pi$$
$$312$$ −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 $$-0.390229\pi$$
0.338061 + 0.941124i $$0.390229\pi$$
$$318$$ 7.40231 0.415101
$$319$$ 13.8536 0.775655
$$320$$ −7.12374 −0.398229
$$321$$ 3.45446 0.192809
$$322$$ 0 0
$$323$$ 3.70727 0.206278
$$324$$ 0.0731828 0.00406571
$$325$$ 4.14101 0.229702
$$326$$ 6.37623 0.353147
$$327$$ −4.66896 −0.258194
$$328$$ −33.4802 −1.84864
$$329$$ 0 0
$$330$$ 5.62377 0.309578
$$331$$ −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
$$335$$ −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
$$339$$ 3.28738 0.178546
$$340$$ −0.195322 −0.0105928
$$341$$ 29.6966 1.60816
$$342$$ 1.85363 0.100233
$$343$$ 0 0
$$344$$ −19.8116 −1.06817
$$345$$ −7.43374 −0.400219
$$346$$ −21.6199 −1.16230
$$347$$ 16.4928 0.885378 0.442689 0.896675i $$-0.354025\pi$$
0.442689 + 0.896675i $$0.354025\pi$$
$$348$$ −0.240579 −0.0128964
$$349$$ −14.1793 −0.759001 −0.379501 0.925191i $$-0.623904\pi$$
−0.379501 + 0.925191i $$0.623904\pi$$
$$350$$ 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
$$355$$ −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 $$-0.488685\pi$$
0.0355383 + 0.999368i $$0.488685\pi$$
$$360$$ 2.57130 0.135520
$$361$$ −17.3427 −0.912772
$$362$$ −16.5104 −0.867766
$$363$$ −6.75942 −0.354778
$$364$$ 0 0
$$365$$ −7.71638 −0.403894
$$366$$ 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
$$370$$ −11.4429 −0.594886
$$371$$ 0 0
$$372$$ −0.515704 −0.0267380
$$373$$ −16.9089 −0.875511 −0.437755 0.899094i $$-0.644226\pi$$
−0.437755 + 0.899094i $$0.644226\pi$$
$$374$$ 17.4736 0.903538
$$375$$ −8.47204 −0.437495
$$376$$ 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
$$381$$ 0.428386 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 $$0.451394\pi$$
$$390$$ −1.33448 −0.0675741
$$391$$ −23.0974 −1.16808
$$392$$ 0 0
$$393$$ 14.1878 0.715680
$$394$$ −28.9875 −1.46037
$$395$$ −4.14477 −0.208546
$$396$$ 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
$$399$$ 0 0
$$400$$ 17.1480 0.857398
$$401$$ −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
$$405$$ −0.926817 −0.0460539
$$406$$ 0 0
$$407$$ 36.1357 1.79118
$$408$$ 7.98929 0.395529
$$409$$ −3.19316 −0.157892 −0.0789458 0.996879i $$-0.525155\pi$$
−0.0789458 + 0.996879i $$0.525155\pi$$
$$410$$ −16.1043 −0.795335
$$411$$ −14.7070 −0.725441
$$412$$ −1.20228 −0.0592319
$$413$$ 0 0
$$414$$ −11.5487 −0.567586
$$415$$ −3.52796 −0.173181
$$416$$ 0.413779 0.0202872
$$417$$ 9.85363 0.482535
$$418$$ 7.81157 0.382076
$$419$$ −30.2292 −1.47680 −0.738398 0.674366i $$-0.764417\pi$$
−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
$$423$$ −1.95289 −0.0949529
$$424$$ 14.2629 0.692668
$$425$$ −11.9249 −0.578443
$$426$$ −15.5345 −0.752650
$$427$$ 0 0
$$428$$ −0.252807 −0.0122199
$$429$$ 4.21419 0.203463
$$430$$ −9.52955 −0.459556
$$431$$ 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 $$0.713358\pi$$
$$434$$ 0 0
$$435$$ 3.04680 0.146083
$$436$$ 0.341688 0.0163639
$$437$$ −10.3257 −0.493944
$$438$$ −11.9878 −0.572798
$$439$$ −25.0553 −1.19582 −0.597912 0.801562i $$-0.704003\pi$$
−0.597912 + 0.801562i $$0.704003\pi$$
$$440$$ 10.8360 0.516585
$$441$$ 0 0
$$442$$ −4.14637 −0.197223
$$443$$ 4.64762 0.220815 0.110408 0.993886i $$-0.464784\pi$$
0.110408 + 0.993886i $$0.464784\pi$$
$$444$$ −0.627525 −0.0297810
$$445$$ −5.23460 −0.248144
$$446$$ 16.6583 0.788791
$$447$$ 3.38663 0.160182
$$448$$ 0 0
$$449$$ 0.494696 0.0233462 0.0116731 0.999932i $$-0.496284\pi$$
0.0116731 + 0.999932i $$0.496284\pi$$
$$450$$ −5.96245 −0.281073
$$451$$ 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 $$-0.287496\pi$$
0.619103 + 0.785310i $$0.287496\pi$$
$$458$$ 19.7365 0.922225
$$459$$ −2.87971 −0.134413
$$460$$ 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
$$465$$ 6.53109 0.302872
$$466$$ 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
$$469$$ 0 0
$$470$$ 2.60610 0.120211
$$471$$ −0.867482 −0.0399715
$$472$$ −20.3483 −0.936608
$$473$$ 30.0936 1.38370
$$474$$ −6.43910 −0.295758
$$475$$ −5.33104 −0.244605
$$476$$ 0 0
$$477$$ −5.14101 −0.235391
$$478$$ 27.5101 1.25828
$$479$$ 17.1790 0.784929 0.392464 0.919767i $$-0.371623\pi$$
0.392464 + 0.919767i $$0.371623\pi$$
$$480$$ −0.383498 −0.0175042
$$481$$ −8.57475 −0.390975
$$482$$ 17.3257 0.789164
$$483$$ 0 0
$$484$$ 0.494674 0.0224852
$$485$$ 6.39544 0.290402
$$486$$ −1.43986 −0.0653132
$$487$$ −4.28264 −0.194065 −0.0970325 0.995281i $$-0.530935\pi$$
−0.0970325 + 0.995281i $$0.530935\pi$$
$$488$$ 21.5273 0.974493
$$489$$ −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
$$495$$ −3.90579 −0.175552
$$496$$ −29.1809 −1.31026
$$497$$ 0 0
$$498$$ −5.48085 −0.245603
$$499$$ −37.0452 −1.65837 −0.829185 0.558974i $$-0.811195\pi$$
−0.829185 + 0.558974i $$0.811195\pi$$
$$500$$ 0.620008 0.0277276
$$501$$ 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 $$0.522994\pi$$
$$504$$ 0 0
$$505$$ −10.1708 −0.452596
$$506$$ −48.6683 −2.16357
$$507$$ −1.00000 −0.0444116
$$508$$ −0.0313505 −0.00139095
$$509$$ 30.4877 1.35134 0.675672 0.737202i $$-0.263853\pi$$
0.675672 + 0.737202i $$0.263853\pi$$
$$510$$ 3.84292 0.170167
$$511$$ 0 0
$$512$$ −21.2637 −0.939730
$$513$$ −1.28738 −0.0568390
$$514$$ 18.7557 0.827277
$$515$$ 15.2261 0.670943
$$516$$ −0.522599 −0.0230062
$$517$$ −8.22987 −0.361949
$$518$$ 0 0
$$519$$ 15.0154 0.659101
$$520$$ −2.57130 −0.112759
$$521$$ 25.3602 1.11105 0.555526 0.831499i $$-0.312517\pi$$
0.555526 + 0.831499i $$0.312517\pi$$
$$522$$ 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
$$530$$ 6.86059 0.298005
$$531$$ 7.33448 0.318289
$$532$$ 0 0
$$533$$ −12.0678 −0.522716
$$534$$ −8.13221 −0.351915
$$535$$ 3.20165 0.138420
$$536$$ −33.4070 −1.44296
$$537$$ 5.35176 0.230945
$$538$$ 33.5603 1.44689
$$539$$ 0 0
$$540$$ 0.0678271 0.00291881
$$541$$ −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
$$545$$ −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
$$549$$ −7.75942 −0.331164
$$550$$ −25.1269 −1.07142
$$551$$ 4.23209 0.180293
$$552$$ −22.2522 −0.947116
$$553$$ 0 0
$$554$$ 21.8760 0.929420
$$555$$ 7.94723 0.337341
$$556$$ −0.721117 −0.0305822
$$557$$ −9.08319 −0.384867 −0.192434 0.981310i $$-0.561638\pi$$
−0.192434 + 0.981310i $$0.561638\pi$$
$$558$$ 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
$$565$$ 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 $$-0.491758\pi$$
0.0258916 + 0.999665i $$0.491758\pi$$
$$570$$ 1.71798 0.0719583
$$571$$ −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
$$574$$ 0 0
$$575$$ 33.2139 1.38511
$$576$$ 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
$$580$$ −0.222973 −0.00925846
$$581$$ 0 0
$$582$$ 9.93562 0.411845
$$583$$ −21.6652 −0.897281
$$584$$ −23.0982 −0.955812
$$585$$ 0.926817 0.0383192
$$586$$ 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
$$591$$ 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 $$0.792692\pi$$
$$600$$ −11.4886 −0.469018
$$601$$ 32.6162 1.33044 0.665221 0.746646i $$-0.268337\pi$$
0.665221 + 0.746646i $$0.268337\pi$$
$$602$$ 0 0
$$603$$ 12.0414 0.490365
$$604$$ −1.60234 −0.0651984
$$605$$ −6.26475 −0.254698
$$606$$ −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
$$609$$ 0 0
$$610$$ 10.3548 0.419254
$$611$$ 1.95289 0.0790056
$$612$$ 0.210745 0.00851888
$$613$$ 35.3197 1.42655 0.713275 0.700885i $$-0.247211\pi$$
0.713275 + 0.700885i $$0.247211\pi$$
$$614$$ 7.47740 0.301763
$$615$$ 11.1847 0.451009
$$616$$ 0 0
$$617$$ −24.4318 −0.983589 −0.491794 0.870711i $$-0.663659\pi$$
−0.491794 + 0.870711i $$0.663659\pi$$
$$618$$ 23.6545 0.951523
$$619$$ −2.90892 −0.116919 −0.0584597 0.998290i $$-0.518619\pi$$
−0.0584597 + 0.998290i $$0.518619\pi$$
$$620$$ −0.477964 −0.0191955
$$621$$ 8.02072 0.321860
$$622$$ 19.9472 0.799811
$$623$$ 0 0
$$624$$ −4.14101 −0.165773
$$625$$ 12.8530 0.514121
$$626$$ 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
$$635$$ 0.397035 0.0157559
$$636$$ 0.376234 0.0149186
$$637$$ 0 0
$$638$$ 19.9472 0.789718
$$639$$ 10.7889 0.426804
$$640$$ −11.0241 −0.435768
$$641$$ 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 0
$$645$$ 6.61841 0.260600
$$646$$ 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
$$650$$ 5.96245 0.233867
$$651$$ 0 0
$$652$$ 0.324082 0.0126920
$$653$$ 28.4805 1.11453 0.557265 0.830335i $$-0.311851\pi$$
0.557265 + 0.830335i $$0.311851\pi$$
$$654$$ −6.72263 −0.262876
$$655$$ 13.1495 0.513794
$$656$$ −49.9730 −1.95112
$$657$$ 8.32568 0.324816
$$658$$ 0 0
$$659$$ −30.4384 −1.18571 −0.592856 0.805309i $$-0.702000\pi$$
−0.592856 + 0.805309i $$0.702000\pi$$
$$660$$ 0.285836 0.0111262
$$661$$ −36.7710 −1.43023 −0.715114 0.699008i $$-0.753625\pi$$
−0.715114 + 0.699008i $$0.753625\pi$$
$$662$$ −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
$$670$$ −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
$$675$$ 4.14101 0.159388
$$676$$ 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
$$679$$ 0 0
$$680$$ 7.40461 0.283954
$$681$$ −11.0418 −0.423121
$$682$$ 42.7587 1.63732
$$683$$ 44.9240 1.71897 0.859484 0.511162i $$-0.170785\pi$$
0.859484 + 0.511162i $$0.170785\pi$$
$$684$$ 0.0942138 0.00360235
$$685$$ −13.6307 −0.520801
$$686$$ 0 0
$$687$$ −13.7073 −0.522965
$$688$$ −29.5710 −1.12738
$$689$$ 5.14101 0.195857
$$690$$ −10.7035 −0.407476
$$691$$ −22.0085 −0.837243 −0.418621 0.908161i $$-0.637487\pi$$
−0.418621 + 0.908161i $$0.637487\pi$$
$$692$$ −1.09887 −0.0417726
$$693$$ 0 0
$$694$$ 23.7472 0.901431
$$695$$ 9.13252 0.346416
$$696$$ 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
$$705$$ −1.80997 −0.0681676
$$706$$ 42.9469 1.61633
$$707$$ 0 0
$$708$$ −0.536758 −0.0201726
$$709$$ −25.3373 −0.951563 −0.475781 0.879564i $$-0.657835\pi$$
−0.475781 + 0.879564i $$0.657835\pi$$
$$710$$ −14.3977 −0.540334
$$711$$ 4.47204 0.167715
$$712$$ −15.6693 −0.587231
$$713$$ −56.5204 −2.11670
$$714$$ 0 0
$$715$$ 3.90579 0.146068
$$716$$ −0.391657 −0.0146369
$$717$$ −19.1061 −0.713532
$$718$$ 1.93907 0.0723654
$$719$$ 43.8002 1.63347 0.816737 0.577011i $$-0.195781\pi$$
0.816737 + 0.577011i $$0.195781\pi$$
$$720$$ 3.83796 0.143032
$$721$$ 0 0
$$722$$ −24.9709 −0.929322
$$723$$ −12.0329 −0.447510
$$724$$ −0.839165 −0.0311873
$$725$$ −13.6131 −0.505576
$$726$$ −9.73259 −0.361210
$$727$$ −0.944090 −0.0350144 −0.0175072 0.999847i $$-0.505573\pi$$
−0.0175072 + 0.999847i $$0.505573\pi$$
$$728$$ 0 0
$$729$$ 1.00000 0.0370370
$$730$$ −11.1105 −0.411217
$$731$$ 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 $$0.548495\pi$$
$$740$$ −0.581601 −0.0213801
$$741$$ 1.28738 0.0472929
$$742$$ 0 0
$$743$$ 38.4962 1.41229 0.706145 0.708068i $$-0.250433\pi$$
0.706145 + 0.708068i $$0.250433\pi$$
$$744$$ 19.5502 0.716745
$$745$$ 3.13879 0.114996
$$746$$ −24.3464 −0.891385
$$747$$ 3.80653 0.139274
$$748$$ 0.888121 0.0324729
$$749$$ 0 0
$$750$$ −12.1985 −0.445427
$$751$$ −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
$$755$$ 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
$$759$$ 33.8009 1.22689
$$760$$ 3.31023 0.120075
$$761$$ −10.7390 −0.389289 −0.194644 0.980874i $$-0.562355\pi$$
−0.194644 + 0.980874i $$0.562355\pi$$
$$762$$ 0.616813 0.0223448
$$763$$ 0 0
$$764$$ 1.47651 0.0534181
$$765$$ −2.66896 −0.0964966
$$766$$ −21.0411 −0.760247
$$767$$ −7.33448 −0.264833
$$768$$ −1.75406 −0.0632944
$$769$$ 50.5549 1.82306 0.911529 0.411237i $$-0.134903\pi$$
0.911529 + 0.411237i $$0.134903\pi$$
$$770$$ 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
$$775$$ −29.1809 −1.04821
$$776$$ 19.1441 0.687234
$$777$$ 0 0
$$778$$ 8.63913 0.309728
$$779$$ 15.5358 0.556629
$$780$$ −0.0678271 −0.00242860
$$781$$ 45.4667 1.62693
$$782$$ −33.2568 −1.18926
$$783$$ −3.28738 −0.117481
$$784$$ 0 0
$$785$$ −0.803998 −0.0286959
$$786$$ 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
$$790$$ −5.96787 −0.212327
$$791$$ 0 0
$$792$$ −11.6916 −0.415443
$$793$$ 7.75942 0.275545
$$794$$ −37.0016 −1.31314
$$795$$ −4.76478 −0.168989
$$796$$ −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
$$801$$ 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 $$0.456022\pi$$
$$810$$ −1.33448 −0.0468889
$$811$$ 47.5502 1.66971 0.834857 0.550468i $$-0.185551\pi$$
0.834857 + 0.550468i $$0.185551\pi$$
$$812$$ 0 0
$$813$$ −6.05215 −0.212258
$$814$$ 52.0301 1.82365
$$815$$ −4.10430 −0.143767
$$816$$ 11.9249 0.417455
$$817$$ 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 $$-0.475397\pi$$
0.0772157 + 0.997014i $$0.475397\pi$$
$$822$$ −21.1759 −0.738594
$$823$$ −28.7525 −1.00225 −0.501124 0.865375i $$-0.667080\pi$$
−0.501124 + 0.865375i $$0.667080\pi$$
$$824$$ 45.5779 1.58778
$$825$$ 17.4510 0.607566
$$826$$ 0 0
$$827$$ −5.58667 −0.194267 −0.0971337 0.995271i $$-0.530967\pi$$
−0.0971337 + 0.995271i $$0.530967\pi$$
$$828$$ −0.586979 −0.0203989
$$829$$ −53.4974 −1.85804 −0.929021 0.370027i $$-0.879348\pi$$
−0.929021 + 0.370027i $$0.879348\pi$$
$$830$$ −5.07974 −0.176320
$$831$$ −15.1932 −0.527045
$$832$$ −7.68624 −0.266472
$$833$$ 0 0
$$834$$ 14.1878 0.491284
$$835$$ 19.0254 0.658400
$$836$$ 0.397035 0.0137317
$$837$$ −7.04680 −0.243573
$$838$$ −43.5257 −1.50357
$$839$$ 43.0405 1.48592 0.742962 0.669334i $$-0.233420\pi$$
0.742962 + 0.669334i $$0.233420\pi$$
$$840$$ 0 0
$$841$$ −18.1932 −0.627350
$$842$$ 38.8055 1.33733
$$843$$ 7.57506 0.260899
$$844$$ 1.13941 0.0392202
$$845$$ −0.926817 −0.0318835
$$846$$ −2.81188 −0.0966745
$$847$$ 0 0
$$848$$ 21.2890 0.731066
$$849$$ 19.0974 0.655419
$$850$$ −17.1701 −0.588931
$$851$$ −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 $$0.323573\pi$$
$$854$$ 0 0
$$855$$ −1.19316 −0.0408053
$$856$$ 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 $$-0.151241\pi$$
0.889229 + 0.457461i $$0.151241\pi$$
$$860$$ −0.484354 −0.0165163
$$861$$ 0 0
$$862$$ 24.1483 0.822494
$$863$$ 22.2563 0.757612 0.378806 0.925476i $$-0.376335\pi$$
0.378806 + 0.925476i $$0.376335\pi$$
$$864$$ 0.413779 0.0140771
$$865$$ 13.9165 0.473175
$$866$$ −37.2246 −1.26494
$$867$$ 8.70727 0.295714
$$868$$ 0 0
$$869$$ 18.8461 0.639309
$$870$$ 4.38695 0.148731
$$871$$ −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 $$-0.256218\pi$$
0.693159 + 0.720785i $$0.256218\pi$$
$$878$$ −36.0760 −1.21751
$$879$$ −3.79430 −0.127979
$$880$$ 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
$$885$$ 6.79772 0.228503
$$886$$ 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 0
$$890$$ −7.53707 −0.252643
$$891$$ 4.21419 0.141181
$$892$$ 0.846681 0.0283490
$$893$$ −2.51411 −0.0841314
$$894$$ 4.87626 0.163087
$$895$$ 4.96010 0.165798
$$896$$ 0 0
$$897$$ −8.02072 −0.267804
$$898$$ 0.712291 0.0237695
$$899$$ 23.1655 0.772612
$$900$$ −0.303051 −0.0101017
$$901$$ −14.8046 −0.493213
$$902$$ 73.2255 2.43814
$$903$$ 0 0
$$904$$ 9.12029 0.303336
$$905$$ 10.6275 0.353271
$$906$$ 31.5257 1.04737
$$907$$ −31.2936 −1.03909 −0.519544 0.854444i $$-0.673898\pi$$
−0.519544 + 0.854444i $$0.673898\pi$$
$$908$$ 0.808067 0.0268166
$$909$$ 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
$$915$$ −7.19156 −0.237746
$$916$$ 1.00314 0.0331446
$$917$$ 0 0
$$918$$ −4.14637 −0.136850
$$919$$ 21.7066 0.716036 0.358018 0.933715i $$-0.383453\pi$$
0.358018 + 0.933715i $$0.383453\pi$$
$$920$$ −20.6237 −0.679944
$$921$$ −5.19316 −0.171120
$$922$$ 1.55746 0.0512921
$$923$$ −10.7889 −0.355122
$$924$$ 0 0
$$925$$ −35.5081 −1.16750
$$926$$ −47.0974 −1.54771
$$927$$ −16.4284 −0.539579
$$928$$ −1.36025 −0.0446523
$$929$$ −35.6787 −1.17058 −0.585289 0.810824i $$-0.699019\pi$$
−0.585289 + 0.810824i $$0.699019\pi$$
$$930$$ 9.40383 0.308364
$$931$$ 0 0
$$932$$ 0.0903972 0.00296106
$$933$$ −13.8536 −0.453548
$$934$$ 24.5924 0.804687
$$935$$ −11.2475 −0.367834
$$936$$ 2.77434 0.0906821
$$937$$ 47.0866 1.53825 0.769127 0.639096i $$-0.220691\pi$$
0.769127 + 0.639096i $$0.220691\pi$$
$$938$$ 0 0
$$939$$ −32.6162 −1.06439
$$940$$ 0.132459 0.00432034
$$941$$ −45.2923 −1.47649 −0.738244 0.674534i $$-0.764345\pi$$
−0.738244 + 0.674534i $$0.764345\pi$$
$$942$$ −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 $$-0.475299\pi$$
0.0775238 + 0.996991i $$0.475299\pi$$
$$948$$ −0.327277 −0.0106295
$$949$$ −8.32568 −0.270263
$$950$$ −7.67592 −0.249040
$$951$$ −12.0380 −0.390359
$$952$$ 0 0
$$953$$ −3.95572 −0.128138 −0.0640692 0.997945i $$-0.520408\pi$$
−0.0640692 + 0.997945i $$0.520408\pi$$
$$954$$ −7.40231 −0.239659
$$955$$ −18.6991 −0.605088
$$956$$ 1.39824 0.0452223
$$957$$ −13.8536 −0.447824
$$958$$ 24.7353 0.799160
$$959$$ 0 0
$$960$$ 7.12374 0.229918
$$961$$ 18.6573 0.601850
$$962$$ −12.3464 −0.398064
$$963$$ −3.45446 −0.111318
$$964$$ 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
$$969$$ −3.70727 −0.119095
$$970$$ 9.20850 0.295667
$$971$$ −38.6576 −1.24058 −0.620291 0.784372i $$-0.712986\pi$$
−0.620291 + 0.784372i $$0.712986\pi$$
$$972$$ −0.0731828 −0.00234734
$$973$$ 0 0
$$974$$ −6.16638 −0.197584
$$975$$ −4.14101 −0.132618
$$976$$ 32.1318 1.02852
$$977$$ −20.6134 −0.659480 −0.329740 0.944072i $$-0.606961\pi$$
−0.329740 + 0.944072i $$0.606961\pi$$
$$978$$ −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
$$985$$ 18.6589 0.594521
$$986$$ 13.6307 0.434089
$$987$$ 0 0
$$988$$ −0.0942138 −0.00299734
$$989$$ −57.2760 −1.82127
$$990$$ −5.62377 −0.178735
$$991$$ −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
$$994$$ 0 0
$$995$$ 1.63066 0.0516954
$$996$$ −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 1911.2.a.s.1.3 4
3.2 odd 2 5733.2.a.bf.1.2 4
7.6 odd 2 273.2.a.e.1.3 4
21.20 even 2 819.2.a.k.1.2 4
28.27 even 2 4368.2.a.br.1.3 4
35.34 odd 2 6825.2.a.bg.1.2 4
91.90 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 7.6 odd 2
819.2.a.k.1.2 4 21.20 even 2
1911.2.a.s.1.3 4 1.1 even 1 trivial
3549.2.a.w.1.2 4 91.90 odd 2
4368.2.a.br.1.3 4 28.27 even 2
5733.2.a.bf.1.2 4 3.2 odd 2
6825.2.a.bg.1.2 4 35.34 odd 2