# Properties

 Label 1014.2.a.o.1.1 Level $1014$ Weight $2$ Character 1014.1 Self dual yes Analytic conductor $8.097$ Analytic rank $0$ Dimension $3$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1014 = 2 \cdot 3 \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1014.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$8.09683076496$$ Analytic rank: $$0$$ Dimension: $$3$$ Coefficient field: $$\Q(\zeta_{14})^+$$ Defining polynomial: $$x^{3} - x^{2} - 2x + 1$$ x^3 - x^2 - 2*x + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$1.80194$$ of defining polynomial Character $$\chi$$ $$=$$ 1014.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -4.04892 q^{5} +1.00000 q^{6} -0.692021 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$$ $$q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -4.04892 q^{5} +1.00000 q^{6} -0.692021 q^{7} +1.00000 q^{8} +1.00000 q^{9} -4.04892 q^{10} +4.85086 q^{11} +1.00000 q^{12} -0.692021 q^{14} -4.04892 q^{15} +1.00000 q^{16} +7.38404 q^{17} +1.00000 q^{18} +1.78017 q^{19} -4.04892 q^{20} -0.692021 q^{21} +4.85086 q^{22} +5.10992 q^{23} +1.00000 q^{24} +11.3937 q^{25} +1.00000 q^{27} -0.692021 q^{28} -3.34481 q^{29} -4.04892 q^{30} -0.972853 q^{31} +1.00000 q^{32} +4.85086 q^{33} +7.38404 q^{34} +2.80194 q^{35} +1.00000 q^{36} -1.28621 q^{37} +1.78017 q^{38} -4.04892 q^{40} -1.50604 q^{41} -0.692021 q^{42} -8.31767 q^{43} +4.85086 q^{44} -4.04892 q^{45} +5.10992 q^{46} +7.20775 q^{47} +1.00000 q^{48} -6.52111 q^{49} +11.3937 q^{50} +7.38404 q^{51} +13.4765 q^{53} +1.00000 q^{54} -19.6407 q^{55} -0.692021 q^{56} +1.78017 q^{57} -3.34481 q^{58} -1.30798 q^{59} -4.04892 q^{60} -0.396125 q^{61} -0.972853 q^{62} -0.692021 q^{63} +1.00000 q^{64} +4.85086 q^{66} -6.05429 q^{67} +7.38404 q^{68} +5.10992 q^{69} +2.80194 q^{70} +1.32975 q^{71} +1.00000 q^{72} +7.65279 q^{73} -1.28621 q^{74} +11.3937 q^{75} +1.78017 q^{76} -3.35690 q^{77} -8.33944 q^{79} -4.04892 q^{80} +1.00000 q^{81} -1.50604 q^{82} -15.3274 q^{83} -0.692021 q^{84} -29.8974 q^{85} -8.31767 q^{86} -3.34481 q^{87} +4.85086 q^{88} -3.10992 q^{89} -4.04892 q^{90} +5.10992 q^{92} -0.972853 q^{93} +7.20775 q^{94} -7.20775 q^{95} +1.00000 q^{96} +8.54288 q^{97} -6.52111 q^{98} +4.85086 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q + 3 q^{2} + 3 q^{3} + 3 q^{4} - 3 q^{5} + 3 q^{6} + 3 q^{7} + 3 q^{8} + 3 q^{9}+O(q^{10})$$ 3 * q + 3 * q^2 + 3 * q^3 + 3 * q^4 - 3 * q^5 + 3 * q^6 + 3 * q^7 + 3 * q^8 + 3 * q^9 $$3 q + 3 q^{2} + 3 q^{3} + 3 q^{4} - 3 q^{5} + 3 q^{6} + 3 q^{7} + 3 q^{8} + 3 q^{9} - 3 q^{10} + q^{11} + 3 q^{12} + 3 q^{14} - 3 q^{15} + 3 q^{16} + 12 q^{17} + 3 q^{18} + 4 q^{19} - 3 q^{20} + 3 q^{21} + q^{22} + 16 q^{23} + 3 q^{24} + 2 q^{25} + 3 q^{27} + 3 q^{28} + 13 q^{29} - 3 q^{30} - 9 q^{31} + 3 q^{32} + q^{33} + 12 q^{34} + 4 q^{35} + 3 q^{36} - 12 q^{37} + 4 q^{38} - 3 q^{40} - 14 q^{41} + 3 q^{42} - 8 q^{43} + q^{44} - 3 q^{45} + 16 q^{46} + 4 q^{47} + 3 q^{48} - 4 q^{49} + 2 q^{50} + 12 q^{51} + 15 q^{53} + 3 q^{54} - 22 q^{55} + 3 q^{56} + 4 q^{57} + 13 q^{58} - 9 q^{59} - 3 q^{60} - 10 q^{61} - 9 q^{62} + 3 q^{63} + 3 q^{64} + q^{66} - 6 q^{67} + 12 q^{68} + 16 q^{69} + 4 q^{70} + 6 q^{71} + 3 q^{72} + 5 q^{73} - 12 q^{74} + 2 q^{75} + 4 q^{76} - 6 q^{77} - 5 q^{79} - 3 q^{80} + 3 q^{81} - 14 q^{82} - 7 q^{83} + 3 q^{84} - 26 q^{85} - 8 q^{86} + 13 q^{87} + q^{88} - 10 q^{89} - 3 q^{90} + 16 q^{92} - 9 q^{93} + 4 q^{94} - 4 q^{95} + 3 q^{96} + 7 q^{97} - 4 q^{98} + q^{99}+O(q^{100})$$ 3 * q + 3 * q^2 + 3 * q^3 + 3 * q^4 - 3 * q^5 + 3 * q^6 + 3 * q^7 + 3 * q^8 + 3 * q^9 - 3 * q^10 + q^11 + 3 * q^12 + 3 * q^14 - 3 * q^15 + 3 * q^16 + 12 * q^17 + 3 * q^18 + 4 * q^19 - 3 * q^20 + 3 * q^21 + q^22 + 16 * q^23 + 3 * q^24 + 2 * q^25 + 3 * q^27 + 3 * q^28 + 13 * q^29 - 3 * q^30 - 9 * q^31 + 3 * q^32 + q^33 + 12 * q^34 + 4 * q^35 + 3 * q^36 - 12 * q^37 + 4 * q^38 - 3 * q^40 - 14 * q^41 + 3 * q^42 - 8 * q^43 + q^44 - 3 * q^45 + 16 * q^46 + 4 * q^47 + 3 * q^48 - 4 * q^49 + 2 * q^50 + 12 * q^51 + 15 * q^53 + 3 * q^54 - 22 * q^55 + 3 * q^56 + 4 * q^57 + 13 * q^58 - 9 * q^59 - 3 * q^60 - 10 * q^61 - 9 * q^62 + 3 * q^63 + 3 * q^64 + q^66 - 6 * q^67 + 12 * q^68 + 16 * q^69 + 4 * q^70 + 6 * q^71 + 3 * q^72 + 5 * q^73 - 12 * q^74 + 2 * q^75 + 4 * q^76 - 6 * q^77 - 5 * q^79 - 3 * q^80 + 3 * q^81 - 14 * q^82 - 7 * q^83 + 3 * q^84 - 26 * q^85 - 8 * q^86 + 13 * q^87 + q^88 - 10 * q^89 - 3 * q^90 + 16 * q^92 - 9 * q^93 + 4 * q^94 - 4 * q^95 + 3 * q^96 + 7 * q^97 - 4 * q^98 + 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 1.00000 0.707107
$$3$$ 1.00000 0.577350
$$4$$ 1.00000 0.500000
$$5$$ −4.04892 −1.81073 −0.905365 0.424633i $$-0.860403\pi$$
−0.905365 + 0.424633i $$0.860403\pi$$
$$6$$ 1.00000 0.408248
$$7$$ −0.692021 −0.261560 −0.130780 0.991411i $$-0.541748\pi$$
−0.130780 + 0.991411i $$0.541748\pi$$
$$8$$ 1.00000 0.353553
$$9$$ 1.00000 0.333333
$$10$$ −4.04892 −1.28038
$$11$$ 4.85086 1.46259 0.731294 0.682062i $$-0.238917\pi$$
0.731294 + 0.682062i $$0.238917\pi$$
$$12$$ 1.00000 0.288675
$$13$$ 0 0
$$14$$ −0.692021 −0.184951
$$15$$ −4.04892 −1.04543
$$16$$ 1.00000 0.250000
$$17$$ 7.38404 1.79089 0.895447 0.445169i $$-0.146856\pi$$
0.895447 + 0.445169i $$0.146856\pi$$
$$18$$ 1.00000 0.235702
$$19$$ 1.78017 0.408398 0.204199 0.978929i $$-0.434541\pi$$
0.204199 + 0.978929i $$0.434541\pi$$
$$20$$ −4.04892 −0.905365
$$21$$ −0.692021 −0.151011
$$22$$ 4.85086 1.03421
$$23$$ 5.10992 1.06549 0.532746 0.846275i $$-0.321160\pi$$
0.532746 + 0.846275i $$0.321160\pi$$
$$24$$ 1.00000 0.204124
$$25$$ 11.3937 2.27875
$$26$$ 0 0
$$27$$ 1.00000 0.192450
$$28$$ −0.692021 −0.130780
$$29$$ −3.34481 −0.621116 −0.310558 0.950554i $$-0.600516\pi$$
−0.310558 + 0.950554i $$0.600516\pi$$
$$30$$ −4.04892 −0.739228
$$31$$ −0.972853 −0.174730 −0.0873648 0.996176i $$-0.527845\pi$$
−0.0873648 + 0.996176i $$0.527845\pi$$
$$32$$ 1.00000 0.176777
$$33$$ 4.85086 0.844425
$$34$$ 7.38404 1.26635
$$35$$ 2.80194 0.473614
$$36$$ 1.00000 0.166667
$$37$$ −1.28621 −0.211451 −0.105726 0.994395i $$-0.533717\pi$$
−0.105726 + 0.994395i $$0.533717\pi$$
$$38$$ 1.78017 0.288781
$$39$$ 0 0
$$40$$ −4.04892 −0.640190
$$41$$ −1.50604 −0.235204 −0.117602 0.993061i $$-0.537521\pi$$
−0.117602 + 0.993061i $$0.537521\pi$$
$$42$$ −0.692021 −0.106781
$$43$$ −8.31767 −1.26843 −0.634216 0.773156i $$-0.718677\pi$$
−0.634216 + 0.773156i $$0.718677\pi$$
$$44$$ 4.85086 0.731294
$$45$$ −4.04892 −0.603577
$$46$$ 5.10992 0.753416
$$47$$ 7.20775 1.05136 0.525679 0.850683i $$-0.323811\pi$$
0.525679 + 0.850683i $$0.323811\pi$$
$$48$$ 1.00000 0.144338
$$49$$ −6.52111 −0.931587
$$50$$ 11.3937 1.61132
$$51$$ 7.38404 1.03397
$$52$$ 0 0
$$53$$ 13.4765 1.85114 0.925570 0.378577i $$-0.123586\pi$$
0.925570 + 0.378577i $$0.123586\pi$$
$$54$$ 1.00000 0.136083
$$55$$ −19.6407 −2.64835
$$56$$ −0.692021 −0.0924753
$$57$$ 1.78017 0.235789
$$58$$ −3.34481 −0.439196
$$59$$ −1.30798 −0.170284 −0.0851422 0.996369i $$-0.527134\pi$$
−0.0851422 + 0.996369i $$0.527134\pi$$
$$60$$ −4.04892 −0.522713
$$61$$ −0.396125 −0.0507185 −0.0253593 0.999678i $$-0.508073\pi$$
−0.0253593 + 0.999678i $$0.508073\pi$$
$$62$$ −0.972853 −0.123552
$$63$$ −0.692021 −0.0871865
$$64$$ 1.00000 0.125000
$$65$$ 0 0
$$66$$ 4.85086 0.597099
$$67$$ −6.05429 −0.739650 −0.369825 0.929101i $$-0.620582\pi$$
−0.369825 + 0.929101i $$0.620582\pi$$
$$68$$ 7.38404 0.895447
$$69$$ 5.10992 0.615162
$$70$$ 2.80194 0.334896
$$71$$ 1.32975 0.157812 0.0789061 0.996882i $$-0.474857\pi$$
0.0789061 + 0.996882i $$0.474857\pi$$
$$72$$ 1.00000 0.117851
$$73$$ 7.65279 0.895692 0.447846 0.894111i $$-0.352191\pi$$
0.447846 + 0.894111i $$0.352191\pi$$
$$74$$ −1.28621 −0.149519
$$75$$ 11.3937 1.31563
$$76$$ 1.78017 0.204199
$$77$$ −3.35690 −0.382554
$$78$$ 0 0
$$79$$ −8.33944 −0.938260 −0.469130 0.883129i $$-0.655432\pi$$
−0.469130 + 0.883129i $$0.655432\pi$$
$$80$$ −4.04892 −0.452683
$$81$$ 1.00000 0.111111
$$82$$ −1.50604 −0.166314
$$83$$ −15.3274 −1.68240 −0.841198 0.540727i $$-0.818149\pi$$
−0.841198 + 0.540727i $$0.818149\pi$$
$$84$$ −0.692021 −0.0755057
$$85$$ −29.8974 −3.24283
$$86$$ −8.31767 −0.896917
$$87$$ −3.34481 −0.358602
$$88$$ 4.85086 0.517103
$$89$$ −3.10992 −0.329650 −0.164825 0.986323i $$-0.552706\pi$$
−0.164825 + 0.986323i $$0.552706\pi$$
$$90$$ −4.04892 −0.426793
$$91$$ 0 0
$$92$$ 5.10992 0.532746
$$93$$ −0.972853 −0.100880
$$94$$ 7.20775 0.743423
$$95$$ −7.20775 −0.739500
$$96$$ 1.00000 0.102062
$$97$$ 8.54288 0.867398 0.433699 0.901058i $$-0.357208\pi$$
0.433699 + 0.901058i $$0.357208\pi$$
$$98$$ −6.52111 −0.658731
$$99$$ 4.85086 0.487529
$$100$$ 11.3937 1.13937
$$101$$ −11.9976 −1.19381 −0.596903 0.802313i $$-0.703602\pi$$
−0.596903 + 0.802313i $$0.703602\pi$$
$$102$$ 7.38404 0.731129
$$103$$ −12.3230 −1.21423 −0.607113 0.794616i $$-0.707672\pi$$
−0.607113 + 0.794616i $$0.707672\pi$$
$$104$$ 0 0
$$105$$ 2.80194 0.273441
$$106$$ 13.4765 1.30895
$$107$$ 5.89977 0.570353 0.285176 0.958475i $$-0.407948\pi$$
0.285176 + 0.958475i $$0.407948\pi$$
$$108$$ 1.00000 0.0962250
$$109$$ −0.792249 −0.0758837 −0.0379418 0.999280i $$-0.512080\pi$$
−0.0379418 + 0.999280i $$0.512080\pi$$
$$110$$ −19.6407 −1.87267
$$111$$ −1.28621 −0.122081
$$112$$ −0.692021 −0.0653899
$$113$$ 6.21983 0.585113 0.292556 0.956248i $$-0.405494\pi$$
0.292556 + 0.956248i $$0.405494\pi$$
$$114$$ 1.78017 0.166728
$$115$$ −20.6896 −1.92932
$$116$$ −3.34481 −0.310558
$$117$$ 0 0
$$118$$ −1.30798 −0.120409
$$119$$ −5.10992 −0.468425
$$120$$ −4.04892 −0.369614
$$121$$ 12.5308 1.13916
$$122$$ −0.396125 −0.0358634
$$123$$ −1.50604 −0.135795
$$124$$ −0.972853 −0.0873648
$$125$$ −25.8877 −2.31547
$$126$$ −0.692021 −0.0616502
$$127$$ −6.00538 −0.532891 −0.266446 0.963850i $$-0.585849\pi$$
−0.266446 + 0.963850i $$0.585849\pi$$
$$128$$ 1.00000 0.0883883
$$129$$ −8.31767 −0.732330
$$130$$ 0 0
$$131$$ 8.81700 0.770345 0.385173 0.922845i $$-0.374142\pi$$
0.385173 + 0.922845i $$0.374142\pi$$
$$132$$ 4.85086 0.422213
$$133$$ −1.23191 −0.106821
$$134$$ −6.05429 −0.523011
$$135$$ −4.04892 −0.348475
$$136$$ 7.38404 0.633176
$$137$$ 15.7560 1.34613 0.673063 0.739585i $$-0.264978\pi$$
0.673063 + 0.739585i $$0.264978\pi$$
$$138$$ 5.10992 0.434985
$$139$$ −6.09783 −0.517212 −0.258606 0.965983i $$-0.583263\pi$$
−0.258606 + 0.965983i $$0.583263\pi$$
$$140$$ 2.80194 0.236807
$$141$$ 7.20775 0.607002
$$142$$ 1.32975 0.111590
$$143$$ 0 0
$$144$$ 1.00000 0.0833333
$$145$$ 13.5429 1.12467
$$146$$ 7.65279 0.633350
$$147$$ −6.52111 −0.537852
$$148$$ −1.28621 −0.105726
$$149$$ 2.55257 0.209114 0.104557 0.994519i $$-0.466657\pi$$
0.104557 + 0.994519i $$0.466657\pi$$
$$150$$ 11.3937 0.930294
$$151$$ 17.7168 1.44177 0.720885 0.693054i $$-0.243735\pi$$
0.720885 + 0.693054i $$0.243735\pi$$
$$152$$ 1.78017 0.144391
$$153$$ 7.38404 0.596964
$$154$$ −3.35690 −0.270506
$$155$$ 3.93900 0.316388
$$156$$ 0 0
$$157$$ −6.31767 −0.504205 −0.252102 0.967701i $$-0.581122\pi$$
−0.252102 + 0.967701i $$0.581122\pi$$
$$158$$ −8.33944 −0.663450
$$159$$ 13.4765 1.06876
$$160$$ −4.04892 −0.320095
$$161$$ −3.53617 −0.278689
$$162$$ 1.00000 0.0785674
$$163$$ 14.5918 1.14292 0.571459 0.820631i $$-0.306378\pi$$
0.571459 + 0.820631i $$0.306378\pi$$
$$164$$ −1.50604 −0.117602
$$165$$ −19.6407 −1.52903
$$166$$ −15.3274 −1.18963
$$167$$ −19.5013 −1.50905 −0.754526 0.656270i $$-0.772133\pi$$
−0.754526 + 0.656270i $$0.772133\pi$$
$$168$$ −0.692021 −0.0533906
$$169$$ 0 0
$$170$$ −29.8974 −2.29302
$$171$$ 1.78017 0.136133
$$172$$ −8.31767 −0.634216
$$173$$ −9.29052 −0.706345 −0.353173 0.935558i $$-0.614897\pi$$
−0.353173 + 0.935558i $$0.614897\pi$$
$$174$$ −3.34481 −0.253570
$$175$$ −7.88471 −0.596028
$$176$$ 4.85086 0.365647
$$177$$ −1.30798 −0.0983137
$$178$$ −3.10992 −0.233098
$$179$$ 22.7928 1.70362 0.851808 0.523853i $$-0.175506\pi$$
0.851808 + 0.523853i $$0.175506\pi$$
$$180$$ −4.04892 −0.301788
$$181$$ 0.537500 0.0399520 0.0199760 0.999800i $$-0.493641\pi$$
0.0199760 + 0.999800i $$0.493641\pi$$
$$182$$ 0 0
$$183$$ −0.396125 −0.0292824
$$184$$ 5.10992 0.376708
$$185$$ 5.20775 0.382881
$$186$$ −0.972853 −0.0713330
$$187$$ 35.8189 2.61934
$$188$$ 7.20775 0.525679
$$189$$ −0.692021 −0.0503372
$$190$$ −7.20775 −0.522905
$$191$$ −9.79954 −0.709070 −0.354535 0.935043i $$-0.615361\pi$$
−0.354535 + 0.935043i $$0.615361\pi$$
$$192$$ 1.00000 0.0721688
$$193$$ −14.1957 −1.02183 −0.510913 0.859632i $$-0.670693\pi$$
−0.510913 + 0.859632i $$0.670693\pi$$
$$194$$ 8.54288 0.613343
$$195$$ 0 0
$$196$$ −6.52111 −0.465793
$$197$$ −3.00969 −0.214431 −0.107216 0.994236i $$-0.534194\pi$$
−0.107216 + 0.994236i $$0.534194\pi$$
$$198$$ 4.85086 0.344735
$$199$$ 12.8944 0.914059 0.457030 0.889451i $$-0.348913\pi$$
0.457030 + 0.889451i $$0.348913\pi$$
$$200$$ 11.3937 0.805658
$$201$$ −6.05429 −0.427037
$$202$$ −11.9976 −0.844149
$$203$$ 2.31468 0.162459
$$204$$ 7.38404 0.516986
$$205$$ 6.09783 0.425891
$$206$$ −12.3230 −0.858587
$$207$$ 5.10992 0.355164
$$208$$ 0 0
$$209$$ 8.63533 0.597319
$$210$$ 2.80194 0.193352
$$211$$ 7.79954 0.536943 0.268471 0.963288i $$-0.413482\pi$$
0.268471 + 0.963288i $$0.413482\pi$$
$$212$$ 13.4765 0.925570
$$213$$ 1.32975 0.0911129
$$214$$ 5.89977 0.403300
$$215$$ 33.6775 2.29679
$$216$$ 1.00000 0.0680414
$$217$$ 0.673235 0.0457022
$$218$$ −0.792249 −0.0536579
$$219$$ 7.65279 0.517128
$$220$$ −19.6407 −1.32418
$$221$$ 0 0
$$222$$ −1.28621 −0.0863246
$$223$$ −12.1957 −0.816682 −0.408341 0.912829i $$-0.633893\pi$$
−0.408341 + 0.912829i $$0.633893\pi$$
$$224$$ −0.692021 −0.0462376
$$225$$ 11.3937 0.759582
$$226$$ 6.21983 0.413737
$$227$$ 6.74333 0.447571 0.223785 0.974638i $$-0.428159\pi$$
0.223785 + 0.974638i $$0.428159\pi$$
$$228$$ 1.78017 0.117894
$$229$$ −19.8237 −1.30999 −0.654994 0.755634i $$-0.727329\pi$$
−0.654994 + 0.755634i $$0.727329\pi$$
$$230$$ −20.6896 −1.36423
$$231$$ −3.35690 −0.220868
$$232$$ −3.34481 −0.219598
$$233$$ −30.0301 −1.96734 −0.983670 0.179983i $$-0.942396\pi$$
−0.983670 + 0.179983i $$0.942396\pi$$
$$234$$ 0 0
$$235$$ −29.1836 −1.90373
$$236$$ −1.30798 −0.0851422
$$237$$ −8.33944 −0.541705
$$238$$ −5.10992 −0.331227
$$239$$ 22.0978 1.42939 0.714695 0.699436i $$-0.246565\pi$$
0.714695 + 0.699436i $$0.246565\pi$$
$$240$$ −4.04892 −0.261356
$$241$$ 10.1274 0.652362 0.326181 0.945307i $$-0.394238\pi$$
0.326181 + 0.945307i $$0.394238\pi$$
$$242$$ 12.5308 0.805510
$$243$$ 1.00000 0.0641500
$$244$$ −0.396125 −0.0253593
$$245$$ 26.4034 1.68685
$$246$$ −1.50604 −0.0960217
$$247$$ 0 0
$$248$$ −0.972853 −0.0617762
$$249$$ −15.3274 −0.971332
$$250$$ −25.8877 −1.63728
$$251$$ 5.54719 0.350135 0.175068 0.984556i $$-0.443986\pi$$
0.175068 + 0.984556i $$0.443986\pi$$
$$252$$ −0.692021 −0.0435933
$$253$$ 24.7875 1.55837
$$254$$ −6.00538 −0.376811
$$255$$ −29.8974 −1.87225
$$256$$ 1.00000 0.0625000
$$257$$ −13.7995 −0.860792 −0.430396 0.902640i $$-0.641626\pi$$
−0.430396 + 0.902640i $$0.641626\pi$$
$$258$$ −8.31767 −0.517835
$$259$$ 0.890084 0.0553071
$$260$$ 0 0
$$261$$ −3.34481 −0.207039
$$262$$ 8.81700 0.544716
$$263$$ −22.4698 −1.38555 −0.692773 0.721155i $$-0.743611\pi$$
−0.692773 + 0.721155i $$0.743611\pi$$
$$264$$ 4.85086 0.298549
$$265$$ −54.5652 −3.35192
$$266$$ −1.23191 −0.0755335
$$267$$ −3.10992 −0.190324
$$268$$ −6.05429 −0.369825
$$269$$ −26.0140 −1.58610 −0.793051 0.609156i $$-0.791509\pi$$
−0.793051 + 0.609156i $$0.791509\pi$$
$$270$$ −4.04892 −0.246409
$$271$$ 2.88471 0.175233 0.0876167 0.996154i $$-0.472075\pi$$
0.0876167 + 0.996154i $$0.472075\pi$$
$$272$$ 7.38404 0.447723
$$273$$ 0 0
$$274$$ 15.7560 0.951855
$$275$$ 55.2693 3.33287
$$276$$ 5.10992 0.307581
$$277$$ −1.46250 −0.0878731 −0.0439366 0.999034i $$-0.513990\pi$$
−0.0439366 + 0.999034i $$0.513990\pi$$
$$278$$ −6.09783 −0.365724
$$279$$ −0.972853 −0.0582432
$$280$$ 2.80194 0.167448
$$281$$ 5.68233 0.338980 0.169490 0.985532i $$-0.445788\pi$$
0.169490 + 0.985532i $$0.445788\pi$$
$$282$$ 7.20775 0.429215
$$283$$ −25.2078 −1.49845 −0.749223 0.662318i $$-0.769573\pi$$
−0.749223 + 0.662318i $$0.769573\pi$$
$$284$$ 1.32975 0.0789061
$$285$$ −7.20775 −0.426950
$$286$$ 0 0
$$287$$ 1.04221 0.0615199
$$288$$ 1.00000 0.0589256
$$289$$ 37.5241 2.20730
$$290$$ 13.5429 0.795265
$$291$$ 8.54288 0.500792
$$292$$ 7.65279 0.447846
$$293$$ −7.14914 −0.417658 −0.208829 0.977952i $$-0.566965\pi$$
−0.208829 + 0.977952i $$0.566965\pi$$
$$294$$ −6.52111 −0.380319
$$295$$ 5.29590 0.308339
$$296$$ −1.28621 −0.0747593
$$297$$ 4.85086 0.281475
$$298$$ 2.55257 0.147866
$$299$$ 0 0
$$300$$ 11.3937 0.657817
$$301$$ 5.75600 0.331771
$$302$$ 17.7168 1.01949
$$303$$ −11.9976 −0.689245
$$304$$ 1.78017 0.102100
$$305$$ 1.60388 0.0918376
$$306$$ 7.38404 0.422118
$$307$$ −17.9952 −1.02704 −0.513521 0.858077i $$-0.671659\pi$$
−0.513521 + 0.858077i $$0.671659\pi$$
$$308$$ −3.35690 −0.191277
$$309$$ −12.3230 −0.701033
$$310$$ 3.93900 0.223720
$$311$$ 3.32975 0.188813 0.0944064 0.995534i $$-0.469905\pi$$
0.0944064 + 0.995534i $$0.469905\pi$$
$$312$$ 0 0
$$313$$ −17.8834 −1.01083 −0.505414 0.862877i $$-0.668660\pi$$
−0.505414 + 0.862877i $$0.668660\pi$$
$$314$$ −6.31767 −0.356527
$$315$$ 2.80194 0.157871
$$316$$ −8.33944 −0.469130
$$317$$ 4.39373 0.246777 0.123388 0.992358i $$-0.460624\pi$$
0.123388 + 0.992358i $$0.460624\pi$$
$$318$$ 13.4765 0.755725
$$319$$ −16.2252 −0.908437
$$320$$ −4.04892 −0.226341
$$321$$ 5.89977 0.329293
$$322$$ −3.53617 −0.197063
$$323$$ 13.1448 0.731398
$$324$$ 1.00000 0.0555556
$$325$$ 0 0
$$326$$ 14.5918 0.808165
$$327$$ −0.792249 −0.0438115
$$328$$ −1.50604 −0.0831572
$$329$$ −4.98792 −0.274993
$$330$$ −19.6407 −1.08119
$$331$$ −25.6775 −1.41137 −0.705683 0.708528i $$-0.749360\pi$$
−0.705683 + 0.708528i $$0.749360\pi$$
$$332$$ −15.3274 −0.841198
$$333$$ −1.28621 −0.0704838
$$334$$ −19.5013 −1.06706
$$335$$ 24.5133 1.33931
$$336$$ −0.692021 −0.0377529
$$337$$ −24.6504 −1.34279 −0.671396 0.741098i $$-0.734305\pi$$
−0.671396 + 0.741098i $$0.734305\pi$$
$$338$$ 0 0
$$339$$ 6.21983 0.337815
$$340$$ −29.8974 −1.62141
$$341$$ −4.71917 −0.255557
$$342$$ 1.78017 0.0962604
$$343$$ 9.35690 0.505225
$$344$$ −8.31767 −0.448459
$$345$$ −20.6896 −1.11389
$$346$$ −9.29052 −0.499461
$$347$$ 14.2959 0.767444 0.383722 0.923449i $$-0.374642\pi$$
0.383722 + 0.923449i $$0.374642\pi$$
$$348$$ −3.34481 −0.179301
$$349$$ 11.0616 0.592113 0.296057 0.955170i $$-0.404328\pi$$
0.296057 + 0.955170i $$0.404328\pi$$
$$350$$ −7.88471 −0.421455
$$351$$ 0 0
$$352$$ 4.85086 0.258551
$$353$$ −10.5047 −0.559109 −0.279555 0.960130i $$-0.590187\pi$$
−0.279555 + 0.960130i $$0.590187\pi$$
$$354$$ −1.30798 −0.0695183
$$355$$ −5.38404 −0.285755
$$356$$ −3.10992 −0.164825
$$357$$ −5.10992 −0.270445
$$358$$ 22.7928 1.20464
$$359$$ 5.10992 0.269691 0.134846 0.990867i $$-0.456946\pi$$
0.134846 + 0.990867i $$0.456946\pi$$
$$360$$ −4.04892 −0.213397
$$361$$ −15.8310 −0.833211
$$362$$ 0.537500 0.0282504
$$363$$ 12.5308 0.657696
$$364$$ 0 0
$$365$$ −30.9855 −1.62186
$$366$$ −0.396125 −0.0207058
$$367$$ −8.44803 −0.440983 −0.220492 0.975389i $$-0.570766\pi$$
−0.220492 + 0.975389i $$0.570766\pi$$
$$368$$ 5.10992 0.266373
$$369$$ −1.50604 −0.0784014
$$370$$ 5.20775 0.270738
$$371$$ −9.32603 −0.484183
$$372$$ −0.972853 −0.0504401
$$373$$ 7.69096 0.398223 0.199111 0.979977i $$-0.436194\pi$$
0.199111 + 0.979977i $$0.436194\pi$$
$$374$$ 35.8189 1.85215
$$375$$ −25.8877 −1.33683
$$376$$ 7.20775 0.371711
$$377$$ 0 0
$$378$$ −0.692021 −0.0355937
$$379$$ −11.4034 −0.585754 −0.292877 0.956150i $$-0.594613\pi$$
−0.292877 + 0.956150i $$0.594613\pi$$
$$380$$ −7.20775 −0.369750
$$381$$ −6.00538 −0.307665
$$382$$ −9.79954 −0.501388
$$383$$ 20.6703 1.05620 0.528100 0.849182i $$-0.322905\pi$$
0.528100 + 0.849182i $$0.322905\pi$$
$$384$$ 1.00000 0.0510310
$$385$$ 13.5918 0.692702
$$386$$ −14.1957 −0.722541
$$387$$ −8.31767 −0.422811
$$388$$ 8.54288 0.433699
$$389$$ 17.4776 0.886148 0.443074 0.896485i $$-0.353888\pi$$
0.443074 + 0.896485i $$0.353888\pi$$
$$390$$ 0 0
$$391$$ 37.7318 1.90818
$$392$$ −6.52111 −0.329366
$$393$$ 8.81700 0.444759
$$394$$ −3.00969 −0.151626
$$395$$ 33.7657 1.69894
$$396$$ 4.85086 0.243765
$$397$$ −19.3599 −0.971645 −0.485822 0.874058i $$-0.661480\pi$$
−0.485822 + 0.874058i $$0.661480\pi$$
$$398$$ 12.8944 0.646338
$$399$$ −1.23191 −0.0616728
$$400$$ 11.3937 0.569687
$$401$$ −14.4832 −0.723257 −0.361628 0.932322i $$-0.617779\pi$$
−0.361628 + 0.932322i $$0.617779\pi$$
$$402$$ −6.05429 −0.301961
$$403$$ 0 0
$$404$$ −11.9976 −0.596903
$$405$$ −4.04892 −0.201192
$$406$$ 2.31468 0.114876
$$407$$ −6.23921 −0.309266
$$408$$ 7.38404 0.365565
$$409$$ −18.8984 −0.934468 −0.467234 0.884134i $$-0.654749\pi$$
−0.467234 + 0.884134i $$0.654749\pi$$
$$410$$ 6.09783 0.301151
$$411$$ 15.7560 0.777186
$$412$$ −12.3230 −0.607113
$$413$$ 0.905149 0.0445395
$$414$$ 5.10992 0.251139
$$415$$ 62.0592 3.04637
$$416$$ 0 0
$$417$$ −6.09783 −0.298612
$$418$$ 8.63533 0.422368
$$419$$ −21.7603 −1.06306 −0.531531 0.847039i $$-0.678383\pi$$
−0.531531 + 0.847039i $$0.678383\pi$$
$$420$$ 2.80194 0.136721
$$421$$ 20.5918 1.00358 0.501791 0.864989i $$-0.332675\pi$$
0.501791 + 0.864989i $$0.332675\pi$$
$$422$$ 7.79954 0.379676
$$423$$ 7.20775 0.350453
$$424$$ 13.4765 0.654477
$$425$$ 84.1318 4.08099
$$426$$ 1.32975 0.0644265
$$427$$ 0.274127 0.0132659
$$428$$ 5.89977 0.285176
$$429$$ 0 0
$$430$$ 33.6775 1.62408
$$431$$ 34.5133 1.66245 0.831224 0.555937i $$-0.187640\pi$$
0.831224 + 0.555937i $$0.187640\pi$$
$$432$$ 1.00000 0.0481125
$$433$$ −2.12631 −0.102184 −0.0510920 0.998694i $$-0.516270\pi$$
−0.0510920 + 0.998694i $$0.516270\pi$$
$$434$$ 0.673235 0.0323163
$$435$$ 13.5429 0.649331
$$436$$ −0.792249 −0.0379418
$$437$$ 9.09651 0.435145
$$438$$ 7.65279 0.365665
$$439$$ 21.8321 1.04199 0.520994 0.853560i $$-0.325561\pi$$
0.520994 + 0.853560i $$0.325561\pi$$
$$440$$ −19.6407 −0.936334
$$441$$ −6.52111 −0.310529
$$442$$ 0 0
$$443$$ −7.54048 −0.358259 −0.179130 0.983825i $$-0.557328\pi$$
−0.179130 + 0.983825i $$0.557328\pi$$
$$444$$ −1.28621 −0.0610407
$$445$$ 12.5918 0.596908
$$446$$ −12.1957 −0.577482
$$447$$ 2.55257 0.120732
$$448$$ −0.692021 −0.0326949
$$449$$ 19.7560 0.932343 0.466172 0.884694i $$-0.345633\pi$$
0.466172 + 0.884694i $$0.345633\pi$$
$$450$$ 11.3937 0.537106
$$451$$ −7.30559 −0.344007
$$452$$ 6.21983 0.292556
$$453$$ 17.7168 0.832407
$$454$$ 6.74333 0.316480
$$455$$ 0 0
$$456$$ 1.78017 0.0833640
$$457$$ 23.8582 1.11604 0.558019 0.829828i $$-0.311562\pi$$
0.558019 + 0.829828i $$0.311562\pi$$
$$458$$ −19.8237 −0.926301
$$459$$ 7.38404 0.344658
$$460$$ −20.6896 −0.964659
$$461$$ −17.5773 −0.818657 −0.409329 0.912387i $$-0.634237\pi$$
−0.409329 + 0.912387i $$0.634237\pi$$
$$462$$ −3.35690 −0.156177
$$463$$ 23.8431 1.10808 0.554041 0.832489i $$-0.313085\pi$$
0.554041 + 0.832489i $$0.313085\pi$$
$$464$$ −3.34481 −0.155279
$$465$$ 3.93900 0.182667
$$466$$ −30.0301 −1.39112
$$467$$ 8.61058 0.398450 0.199225 0.979954i $$-0.436158\pi$$
0.199225 + 0.979954i $$0.436158\pi$$
$$468$$ 0 0
$$469$$ 4.18970 0.193462
$$470$$ −29.1836 −1.34614
$$471$$ −6.31767 −0.291103
$$472$$ −1.30798 −0.0602046
$$473$$ −40.3478 −1.85519
$$474$$ −8.33944 −0.383043
$$475$$ 20.2828 0.930636
$$476$$ −5.10992 −0.234213
$$477$$ 13.4765 0.617047
$$478$$ 22.0978 1.01073
$$479$$ 6.58104 0.300695 0.150348 0.988633i $$-0.451961\pi$$
0.150348 + 0.988633i $$0.451961\pi$$
$$480$$ −4.04892 −0.184807
$$481$$ 0 0
$$482$$ 10.1274 0.461289
$$483$$ −3.53617 −0.160901
$$484$$ 12.5308 0.569582
$$485$$ −34.5894 −1.57062
$$486$$ 1.00000 0.0453609
$$487$$ −23.2760 −1.05474 −0.527369 0.849636i $$-0.676822\pi$$
−0.527369 + 0.849636i $$0.676822\pi$$
$$488$$ −0.396125 −0.0179317
$$489$$ 14.5918 0.659864
$$490$$ 26.4034 1.19278
$$491$$ −15.5405 −0.701332 −0.350666 0.936501i $$-0.614045\pi$$
−0.350666 + 0.936501i $$0.614045\pi$$
$$492$$ −1.50604 −0.0678976
$$493$$ −24.6983 −1.11235
$$494$$ 0 0
$$495$$ −19.6407 −0.882784
$$496$$ −0.972853 −0.0436824
$$497$$ −0.920215 −0.0412773
$$498$$ −15.3274 −0.686835
$$499$$ 9.53617 0.426898 0.213449 0.976954i $$-0.431530\pi$$
0.213449 + 0.976954i $$0.431530\pi$$
$$500$$ −25.8877 −1.15773
$$501$$ −19.5013 −0.871252
$$502$$ 5.54719 0.247583
$$503$$ −13.8345 −0.616848 −0.308424 0.951249i $$-0.599802\pi$$
−0.308424 + 0.951249i $$0.599802\pi$$
$$504$$ −0.692021 −0.0308251
$$505$$ 48.5773 2.16166
$$506$$ 24.7875 1.10194
$$507$$ 0 0
$$508$$ −6.00538 −0.266446
$$509$$ −40.9638 −1.81569 −0.907843 0.419310i $$-0.862272\pi$$
−0.907843 + 0.419310i $$0.862272\pi$$
$$510$$ −29.8974 −1.32388
$$511$$ −5.29590 −0.234277
$$512$$ 1.00000 0.0441942
$$513$$ 1.78017 0.0785963
$$514$$ −13.7995 −0.608672
$$515$$ 49.8950 2.19864
$$516$$ −8.31767 −0.366165
$$517$$ 34.9638 1.53770
$$518$$ 0.890084 0.0391080
$$519$$ −9.29052 −0.407809
$$520$$ 0 0
$$521$$ 36.3672 1.59327 0.796637 0.604457i $$-0.206610\pi$$
0.796637 + 0.604457i $$0.206610\pi$$
$$522$$ −3.34481 −0.146399
$$523$$ −6.03013 −0.263679 −0.131840 0.991271i $$-0.542088\pi$$
−0.131840 + 0.991271i $$0.542088\pi$$
$$524$$ 8.81700 0.385173
$$525$$ −7.88471 −0.344117
$$526$$ −22.4698 −0.979730
$$527$$ −7.18359 −0.312922
$$528$$ 4.85086 0.211106
$$529$$ 3.11124 0.135271
$$530$$ −54.5652 −2.37016
$$531$$ −1.30798 −0.0567614
$$532$$ −1.23191 −0.0534103
$$533$$ 0 0
$$534$$ −3.10992 −0.134579
$$535$$ −23.8877 −1.03275
$$536$$ −6.05429 −0.261506
$$537$$ 22.7928 0.983584
$$538$$ −26.0140 −1.12154
$$539$$ −31.6329 −1.36253
$$540$$ −4.04892 −0.174238
$$541$$ −7.92154 −0.340574 −0.170287 0.985395i $$-0.554469\pi$$
−0.170287 + 0.985395i $$0.554469\pi$$
$$542$$ 2.88471 0.123909
$$543$$ 0.537500 0.0230663
$$544$$ 7.38404 0.316588
$$545$$ 3.20775 0.137405
$$546$$ 0 0
$$547$$ 18.4155 0.787390 0.393695 0.919241i $$-0.371197\pi$$
0.393695 + 0.919241i $$0.371197\pi$$
$$548$$ 15.7560 0.673063
$$549$$ −0.396125 −0.0169062
$$550$$ 55.2693 2.35669
$$551$$ −5.95433 −0.253663
$$552$$ 5.10992 0.217492
$$553$$ 5.77107 0.245411
$$554$$ −1.46250 −0.0621357
$$555$$ 5.20775 0.221057
$$556$$ −6.09783 −0.258606
$$557$$ −23.9758 −1.01589 −0.507944 0.861390i $$-0.669594\pi$$
−0.507944 + 0.861390i $$0.669594\pi$$
$$558$$ −0.972853 −0.0411841
$$559$$ 0 0
$$560$$ 2.80194 0.118403
$$561$$ 35.8189 1.51228
$$562$$ 5.68233 0.239695
$$563$$ −2.29291 −0.0966348 −0.0483174 0.998832i $$-0.515386\pi$$
−0.0483174 + 0.998832i $$0.515386\pi$$
$$564$$ 7.20775 0.303501
$$565$$ −25.1836 −1.05948
$$566$$ −25.2078 −1.05956
$$567$$ −0.692021 −0.0290622
$$568$$ 1.32975 0.0557950
$$569$$ −44.3430 −1.85896 −0.929478 0.368878i $$-0.879742\pi$$
−0.929478 + 0.368878i $$0.879742\pi$$
$$570$$ −7.20775 −0.301899
$$571$$ −15.2707 −0.639058 −0.319529 0.947577i $$-0.603525\pi$$
−0.319529 + 0.947577i $$0.603525\pi$$
$$572$$ 0 0
$$573$$ −9.79954 −0.409382
$$574$$ 1.04221 0.0435011
$$575$$ 58.2210 2.42798
$$576$$ 1.00000 0.0416667
$$577$$ 8.77048 0.365120 0.182560 0.983195i $$-0.441562\pi$$
0.182560 + 0.983195i $$0.441562\pi$$
$$578$$ 37.5241 1.56080
$$579$$ −14.1957 −0.589952
$$580$$ 13.5429 0.562337
$$581$$ 10.6069 0.440047
$$582$$ 8.54288 0.354114
$$583$$ 65.3726 2.70745
$$584$$ 7.65279 0.316675
$$585$$ 0 0
$$586$$ −7.14914 −0.295328
$$587$$ −38.1430 −1.57433 −0.787166 0.616742i $$-0.788452\pi$$
−0.787166 + 0.616742i $$0.788452\pi$$
$$588$$ −6.52111 −0.268926
$$589$$ −1.73184 −0.0713593
$$590$$ 5.29590 0.218029
$$591$$ −3.00969 −0.123802
$$592$$ −1.28621 −0.0528628
$$593$$ −37.9517 −1.55849 −0.779244 0.626720i $$-0.784397\pi$$
−0.779244 + 0.626720i $$0.784397\pi$$
$$594$$ 4.85086 0.199033
$$595$$ 20.6896 0.848192
$$596$$ 2.55257 0.104557
$$597$$ 12.8944 0.527732
$$598$$ 0 0
$$599$$ −3.57971 −0.146263 −0.0731315 0.997322i $$-0.523299\pi$$
−0.0731315 + 0.997322i $$0.523299\pi$$
$$600$$ 11.3937 0.465147
$$601$$ −5.71678 −0.233192 −0.116596 0.993179i $$-0.537198\pi$$
−0.116596 + 0.993179i $$0.537198\pi$$
$$602$$ 5.75600 0.234597
$$603$$ −6.05429 −0.246550
$$604$$ 17.7168 0.720885
$$605$$ −50.7362 −2.06272
$$606$$ −11.9976 −0.487369
$$607$$ 22.4286 0.910351 0.455175 0.890402i $$-0.349577\pi$$
0.455175 + 0.890402i $$0.349577\pi$$
$$608$$ 1.78017 0.0721953
$$609$$ 2.31468 0.0937957
$$610$$ 1.60388 0.0649390
$$611$$ 0 0
$$612$$ 7.38404 0.298482
$$613$$ −39.9603 −1.61398 −0.806991 0.590564i $$-0.798905\pi$$
−0.806991 + 0.590564i $$0.798905\pi$$
$$614$$ −17.9952 −0.726228
$$615$$ 6.09783 0.245888
$$616$$ −3.35690 −0.135253
$$617$$ 31.4470 1.26601 0.633003 0.774149i $$-0.281822\pi$$
0.633003 + 0.774149i $$0.281822\pi$$
$$618$$ −12.3230 −0.495706
$$619$$ −29.3685 −1.18042 −0.590210 0.807250i $$-0.700955\pi$$
−0.590210 + 0.807250i $$0.700955\pi$$
$$620$$ 3.93900 0.158194
$$621$$ 5.10992 0.205054
$$622$$ 3.32975 0.133511
$$623$$ 2.15213 0.0862232
$$624$$ 0 0
$$625$$ 47.8485 1.91394
$$626$$ −17.8834 −0.714764
$$627$$ 8.63533 0.344862
$$628$$ −6.31767 −0.252102
$$629$$ −9.49742 −0.378687
$$630$$ 2.80194 0.111632
$$631$$ 21.6799 0.863065 0.431532 0.902097i $$-0.357973\pi$$
0.431532 + 0.902097i $$0.357973\pi$$
$$632$$ −8.33944 −0.331725
$$633$$ 7.79954 0.310004
$$634$$ 4.39373 0.174497
$$635$$ 24.3153 0.964922
$$636$$ 13.4765 0.534378
$$637$$ 0 0
$$638$$ −16.2252 −0.642362
$$639$$ 1.32975 0.0526040
$$640$$ −4.04892 −0.160048
$$641$$ 14.0108 0.553391 0.276696 0.960958i $$-0.410761\pi$$
0.276696 + 0.960958i $$0.410761\pi$$
$$642$$ 5.89977 0.232845
$$643$$ 30.7875 1.21414 0.607070 0.794649i $$-0.292345\pi$$
0.607070 + 0.794649i $$0.292345\pi$$
$$644$$ −3.53617 −0.139345
$$645$$ 33.6775 1.32605
$$646$$ 13.1448 0.517177
$$647$$ −16.6025 −0.652713 −0.326357 0.945247i $$-0.605821\pi$$
−0.326357 + 0.945247i $$0.605821\pi$$
$$648$$ 1.00000 0.0392837
$$649$$ −6.34481 −0.249056
$$650$$ 0 0
$$651$$ 0.673235 0.0263862
$$652$$ 14.5918 0.571459
$$653$$ 28.5459 1.11709 0.558543 0.829476i $$-0.311361\pi$$
0.558543 + 0.829476i $$0.311361\pi$$
$$654$$ −0.792249 −0.0309794
$$655$$ −35.6993 −1.39489
$$656$$ −1.50604 −0.0588010
$$657$$ 7.65279 0.298564
$$658$$ −4.98792 −0.194449
$$659$$ 27.7187 1.07977 0.539884 0.841740i $$-0.318468\pi$$
0.539884 + 0.841740i $$0.318468\pi$$
$$660$$ −19.6407 −0.764514
$$661$$ 10.8009 0.420105 0.210053 0.977690i $$-0.432636\pi$$
0.210053 + 0.977690i $$0.432636\pi$$
$$662$$ −25.6775 −0.997986
$$663$$ 0 0
$$664$$ −15.3274 −0.594817
$$665$$ 4.98792 0.193423
$$666$$ −1.28621 −0.0498396
$$667$$ −17.0917 −0.661794
$$668$$ −19.5013 −0.754526
$$669$$ −12.1957 −0.471512
$$670$$ 24.5133 0.947033
$$671$$ −1.92154 −0.0741803
$$672$$ −0.692021 −0.0266953
$$673$$ 16.3260 0.629322 0.314661 0.949204i $$-0.398109\pi$$
0.314661 + 0.949204i $$0.398109\pi$$
$$674$$ −24.6504 −0.949498
$$675$$ 11.3937 0.438545
$$676$$ 0 0
$$677$$ 41.4252 1.59210 0.796050 0.605231i $$-0.206919\pi$$
0.796050 + 0.605231i $$0.206919\pi$$
$$678$$ 6.21983 0.238871
$$679$$ −5.91185 −0.226876
$$680$$ −29.8974 −1.14651
$$681$$ 6.74333 0.258405
$$682$$ −4.71917 −0.180706
$$683$$ 31.2325 1.19508 0.597539 0.801840i $$-0.296145\pi$$
0.597539 + 0.801840i $$0.296145\pi$$
$$684$$ 1.78017 0.0680664
$$685$$ −63.7948 −2.43747
$$686$$ 9.35690 0.357248
$$687$$ −19.8237 −0.756322
$$688$$ −8.31767 −0.317108
$$689$$ 0 0
$$690$$ −20.6896 −0.787641
$$691$$ 24.9638 0.949666 0.474833 0.880076i $$-0.342508\pi$$
0.474833 + 0.880076i $$0.342508\pi$$
$$692$$ −9.29052 −0.353173
$$693$$ −3.35690 −0.127518
$$694$$ 14.2959 0.542665
$$695$$ 24.6896 0.936531
$$696$$ −3.34481 −0.126785
$$697$$ −11.1207 −0.421225
$$698$$ 11.0616 0.418687
$$699$$ −30.0301 −1.13584
$$700$$ −7.88471 −0.298014
$$701$$ 8.17151 0.308634 0.154317 0.988021i $$-0.450682\pi$$
0.154317 + 0.988021i $$0.450682\pi$$
$$702$$ 0 0
$$703$$ −2.28967 −0.0863564
$$704$$ 4.85086 0.182823
$$705$$ −29.1836 −1.09912
$$706$$ −10.5047 −0.395350
$$707$$ 8.30260 0.312251
$$708$$ −1.30798 −0.0491568
$$709$$ −36.7982 −1.38199 −0.690993 0.722861i $$-0.742826\pi$$
−0.690993 + 0.722861i $$0.742826\pi$$
$$710$$ −5.38404 −0.202060
$$711$$ −8.33944 −0.312753
$$712$$ −3.10992 −0.116549
$$713$$ −4.97120 −0.186173
$$714$$ −5.10992 −0.191234
$$715$$ 0 0
$$716$$ 22.7928 0.851808
$$717$$ 22.0978 0.825259
$$718$$ 5.10992 0.190700
$$719$$ 35.2223 1.31357 0.656786 0.754077i $$-0.271916\pi$$
0.656786 + 0.754077i $$0.271916\pi$$
$$720$$ −4.04892 −0.150894
$$721$$ 8.52781 0.317592
$$722$$ −15.8310 −0.589169
$$723$$ 10.1274 0.376641
$$724$$ 0.537500 0.0199760
$$725$$ −38.1099 −1.41537
$$726$$ 12.5308 0.465061
$$727$$ 40.6872 1.50901 0.754503 0.656297i $$-0.227878\pi$$
0.754503 + 0.656297i $$0.227878\pi$$
$$728$$ 0 0
$$729$$ 1.00000 0.0370370
$$730$$ −30.9855 −1.14683
$$731$$ −61.4180 −2.27163
$$732$$ −0.396125 −0.0146412
$$733$$ 27.1400 1.00244 0.501220 0.865320i $$-0.332885\pi$$
0.501220 + 0.865320i $$0.332885\pi$$
$$734$$ −8.44803 −0.311822
$$735$$ 26.4034 0.973905
$$736$$ 5.10992 0.188354
$$737$$ −29.3685 −1.08180
$$738$$ −1.50604 −0.0554381
$$739$$ 3.72587 0.137058 0.0685292 0.997649i $$-0.478169\pi$$
0.0685292 + 0.997649i $$0.478169\pi$$
$$740$$ 5.20775 0.191441
$$741$$ 0 0
$$742$$ −9.32603 −0.342369
$$743$$ 21.8586 0.801915 0.400958 0.916097i $$-0.368677\pi$$
0.400958 + 0.916097i $$0.368677\pi$$
$$744$$ −0.972853 −0.0356665
$$745$$ −10.3351 −0.378650
$$746$$ 7.69096 0.281586
$$747$$ −15.3274 −0.560799
$$748$$ 35.8189 1.30967
$$749$$ −4.08277 −0.149181
$$750$$ −25.8877 −0.945285
$$751$$ 53.3642 1.94729 0.973644 0.228075i $$-0.0732432\pi$$
0.973644 + 0.228075i $$0.0732432\pi$$
$$752$$ 7.20775 0.262840
$$753$$ 5.54719 0.202151
$$754$$ 0 0
$$755$$ −71.7338 −2.61066
$$756$$ −0.692021 −0.0251686
$$757$$ −4.63533 −0.168474 −0.0842370 0.996446i $$-0.526845\pi$$
−0.0842370 + 0.996446i $$0.526845\pi$$
$$758$$ −11.4034 −0.414191
$$759$$ 24.7875 0.899728
$$760$$ −7.20775 −0.261453
$$761$$ 11.5603 0.419062 0.209531 0.977802i $$-0.432806\pi$$
0.209531 + 0.977802i $$0.432806\pi$$
$$762$$ −6.00538 −0.217552
$$763$$ 0.548253 0.0198481
$$764$$ −9.79954 −0.354535
$$765$$ −29.8974 −1.08094
$$766$$ 20.6703 0.746847
$$767$$ 0 0
$$768$$ 1.00000 0.0360844
$$769$$ 14.7439 0.531679 0.265840 0.964017i $$-0.414351\pi$$
0.265840 + 0.964017i $$0.414351\pi$$
$$770$$ 13.5918 0.489814
$$771$$ −13.7995 −0.496978
$$772$$ −14.1957 −0.510913
$$773$$ −6.42268 −0.231008 −0.115504 0.993307i $$-0.536848\pi$$
−0.115504 + 0.993307i $$0.536848\pi$$
$$774$$ −8.31767 −0.298972
$$775$$ −11.0844 −0.398164
$$776$$ 8.54288 0.306671
$$777$$ 0.890084 0.0319316
$$778$$ 17.4776 0.626601
$$779$$ −2.68100 −0.0960570
$$780$$ 0 0
$$781$$ 6.45042 0.230814
$$782$$ 37.7318 1.34929
$$783$$ −3.34481 −0.119534
$$784$$ −6.52111 −0.232897
$$785$$ 25.5797 0.912979
$$786$$ 8.81700 0.314492
$$787$$ 43.4336 1.54824 0.774119 0.633040i $$-0.218193\pi$$
0.774119 + 0.633040i $$0.218193\pi$$
$$788$$ −3.00969 −0.107216
$$789$$ −22.4698 −0.799946
$$790$$ 33.7657 1.20133
$$791$$ −4.30426 −0.153042
$$792$$ 4.85086 0.172368
$$793$$ 0 0
$$794$$ −19.3599 −0.687056
$$795$$ −54.5652 −1.93523
$$796$$ 12.8944 0.457030
$$797$$ 21.0164 0.744439 0.372219 0.928145i $$-0.378597\pi$$
0.372219 + 0.928145i $$0.378597\pi$$
$$798$$ −1.23191 −0.0436093
$$799$$ 53.2223 1.88287
$$800$$ 11.3937 0.402829
$$801$$ −3.10992 −0.109883
$$802$$ −14.4832 −0.511420
$$803$$ 37.1226 1.31003
$$804$$ −6.05429 −0.213518
$$805$$ 14.3177 0.504631
$$806$$ 0 0
$$807$$ −26.0140 −0.915736
$$808$$ −11.9976 −0.422074
$$809$$ −8.32245 −0.292602 −0.146301 0.989240i $$-0.546737\pi$$
−0.146301 + 0.989240i $$0.546737\pi$$
$$810$$ −4.04892 −0.142264
$$811$$ 14.4638 0.507894 0.253947 0.967218i $$-0.418271\pi$$
0.253947 + 0.967218i $$0.418271\pi$$
$$812$$ 2.31468 0.0812295
$$813$$ 2.88471 0.101171
$$814$$ −6.23921 −0.218684
$$815$$ −59.0810 −2.06952
$$816$$ 7.38404 0.258493
$$817$$ −14.8068 −0.518026
$$818$$ −18.8984 −0.660769
$$819$$ 0 0
$$820$$ 6.09783 0.212946
$$821$$ 38.4161 1.34073 0.670365 0.742031i $$-0.266137\pi$$
0.670365 + 0.742031i $$0.266137\pi$$
$$822$$ 15.7560 0.549554
$$823$$ −40.9748 −1.42829 −0.714145 0.699997i $$-0.753184\pi$$
−0.714145 + 0.699997i $$0.753184\pi$$
$$824$$ −12.3230 −0.429294
$$825$$ 55.2693 1.92423
$$826$$ 0.905149 0.0314942
$$827$$ −3.51035 −0.122067 −0.0610335 0.998136i $$-0.519440\pi$$
−0.0610335 + 0.998136i $$0.519440\pi$$
$$828$$ 5.10992 0.177582
$$829$$ −13.2185 −0.459098 −0.229549 0.973297i $$-0.573725\pi$$
−0.229549 + 0.973297i $$0.573725\pi$$
$$830$$ 62.0592 2.15411
$$831$$ −1.46250 −0.0507336
$$832$$ 0 0
$$833$$ −48.1521 −1.66837
$$834$$ −6.09783 −0.211151
$$835$$ 78.9590 2.73249
$$836$$ 8.63533 0.298659
$$837$$ −0.972853 −0.0336267
$$838$$ −21.7603 −0.751698
$$839$$ −55.8491 −1.92812 −0.964062 0.265678i $$-0.914404\pi$$
−0.964062 + 0.265678i $$0.914404\pi$$
$$840$$ 2.80194 0.0966760
$$841$$ −17.8122 −0.614214
$$842$$ 20.5918 0.709640
$$843$$ 5.68233 0.195710
$$844$$ 7.79954 0.268471
$$845$$ 0 0
$$846$$ 7.20775 0.247808
$$847$$ −8.67158 −0.297959
$$848$$ 13.4765 0.462785
$$849$$ −25.2078 −0.865128
$$850$$ 84.1318 2.88570
$$851$$ −6.57242 −0.225300
$$852$$ 1.32975 0.0455564
$$853$$ −21.8103 −0.746770 −0.373385 0.927676i $$-0.621803\pi$$
−0.373385 + 0.927676i $$0.621803\pi$$
$$854$$ 0.274127 0.00938042
$$855$$ −7.20775 −0.246500
$$856$$ 5.89977 0.201650
$$857$$ 28.8961 0.987070 0.493535 0.869726i $$-0.335704\pi$$
0.493535 + 0.869726i $$0.335704\pi$$
$$858$$ 0 0
$$859$$ −17.2755 −0.589431 −0.294715 0.955585i $$-0.595225\pi$$
−0.294715 + 0.955585i $$0.595225\pi$$
$$860$$ 33.6775 1.14839
$$861$$ 1.04221 0.0355185
$$862$$ 34.5133 1.17553
$$863$$ −44.7741 −1.52413 −0.762063 0.647503i $$-0.775813\pi$$
−0.762063 + 0.647503i $$0.775813\pi$$
$$864$$ 1.00000 0.0340207
$$865$$ 37.6165 1.27900
$$866$$ −2.12631 −0.0722549
$$867$$ 37.5241 1.27438
$$868$$ 0.673235 0.0228511
$$869$$ −40.4534 −1.37229
$$870$$ 13.5429 0.459147
$$871$$ 0 0
$$872$$ −0.792249 −0.0268289
$$873$$ 8.54288 0.289133
$$874$$ 9.09651 0.307694
$$875$$ 17.9148 0.605632
$$876$$ 7.65279 0.258564
$$877$$ −40.2741 −1.35996 −0.679980 0.733230i $$-0.738012\pi$$
−0.679980 + 0.733230i $$0.738012\pi$$
$$878$$ 21.8321 0.736797
$$879$$ −7.14914 −0.241135
$$880$$ −19.6407 −0.662088
$$881$$ 9.40821 0.316971 0.158485 0.987361i $$-0.449339\pi$$
0.158485 + 0.987361i $$0.449339\pi$$
$$882$$ −6.52111 −0.219577
$$883$$ −51.2271 −1.72393 −0.861965 0.506968i $$-0.830766\pi$$
−0.861965 + 0.506968i $$0.830766\pi$$
$$884$$ 0 0
$$885$$ 5.29590 0.178020
$$886$$ −7.54048 −0.253328
$$887$$ −39.9215 −1.34043 −0.670217 0.742165i $$-0.733799\pi$$
−0.670217 + 0.742165i $$0.733799\pi$$
$$888$$ −1.28621 −0.0431623
$$889$$ 4.15585 0.139383
$$890$$ 12.5918 0.422078
$$891$$ 4.85086 0.162510
$$892$$ −12.1957 −0.408341
$$893$$ 12.8310 0.429373
$$894$$ 2.55257 0.0853706
$$895$$ −92.2863 −3.08479
$$896$$ −0.692021 −0.0231188
$$897$$ 0 0
$$898$$ 19.7560 0.659266
$$899$$ 3.25401 0.108527
$$900$$ 11.3937 0.379791
$$901$$ 99.5111 3.31519
$$902$$ −7.30559 −0.243249
$$903$$ 5.75600 0.191548
$$904$$ 6.21983 0.206869
$$905$$ −2.17629 −0.0723424
$$906$$ 17.7168 0.588600
$$907$$ −34.8310 −1.15654 −0.578272 0.815844i $$-0.696273\pi$$
−0.578272 + 0.815844i $$0.696273\pi$$
$$908$$ 6.74333 0.223785
$$909$$ −11.9976 −0.397936
$$910$$ 0 0
$$911$$ 13.2125 0.437751 0.218875 0.975753i $$-0.429761\pi$$
0.218875 + 0.975753i $$0.429761\pi$$
$$912$$ 1.78017 0.0589472
$$913$$ −74.3508 −2.46065
$$914$$ 23.8582 0.789157
$$915$$ 1.60388 0.0530225
$$916$$ −19.8237 −0.654994
$$917$$ −6.10156 −0.201491
$$918$$ 7.38404 0.243710
$$919$$ 21.0175 0.693302 0.346651 0.937994i $$-0.387319\pi$$
0.346651 + 0.937994i $$0.387319\pi$$
$$920$$ −20.6896 −0.682117
$$921$$ −17.9952 −0.592962
$$922$$ −17.5773 −0.578878
$$923$$ 0 0
$$924$$ −3.35690 −0.110434
$$925$$ −14.6547 −0.481844
$$926$$ 23.8431 0.783532
$$927$$ −12.3230 −0.404742
$$928$$ −3.34481 −0.109799
$$929$$ −26.9965 −0.885728 −0.442864 0.896589i $$-0.646038\pi$$
−0.442864 + 0.896589i $$0.646038\pi$$
$$930$$ 3.93900 0.129165
$$931$$ −11.6087 −0.380459
$$932$$ −30.0301 −0.983670
$$933$$ 3.32975 0.109011
$$934$$ 8.61058 0.281747
$$935$$ −145.028 −4.74292
$$936$$ 0 0
$$937$$ 23.1745 0.757078 0.378539 0.925585i $$-0.376427\pi$$
0.378539 + 0.925585i $$0.376427\pi$$
$$938$$ 4.18970 0.136799
$$939$$ −17.8834 −0.583602
$$940$$ −29.1836 −0.951864
$$941$$ 7.54048 0.245813 0.122906 0.992418i $$-0.460779\pi$$
0.122906 + 0.992418i $$0.460779\pi$$
$$942$$ −6.31767 −0.205841
$$943$$ −7.69574 −0.250608
$$944$$ −1.30798 −0.0425711
$$945$$ 2.80194 0.0911470
$$946$$ −40.3478 −1.31182
$$947$$ 20.6601 0.671363 0.335681 0.941976i $$-0.391033\pi$$
0.335681 + 0.941976i $$0.391033\pi$$
$$948$$ −8.33944 −0.270852
$$949$$ 0 0
$$950$$ 20.2828 0.658059
$$951$$ 4.39373 0.142477
$$952$$ −5.10992 −0.165613
$$953$$ −30.2935 −0.981303 −0.490651 0.871356i $$-0.663241\pi$$
−0.490651 + 0.871356i $$0.663241\pi$$
$$954$$ 13.4765 0.436318
$$955$$ 39.6775 1.28394
$$956$$ 22.0978 0.714695
$$957$$ −16.2252 −0.524487
$$958$$ 6.58104 0.212624
$$959$$ −10.9035 −0.352092
$$960$$ −4.04892 −0.130678
$$961$$ −30.0536 −0.969470
$$962$$ 0 0
$$963$$ 5.89977 0.190118
$$964$$ 10.1274 0.326181
$$965$$ 57.4771 1.85025
$$966$$ −3.53617 −0.113774
$$967$$ −20.7289 −0.666595 −0.333298 0.942822i $$-0.608161\pi$$
−0.333298 + 0.942822i $$0.608161\pi$$
$$968$$ 12.5308 0.402755
$$969$$ 13.1448 0.422273
$$970$$ −34.5894 −1.11060
$$971$$ −39.1094 −1.25508 −0.627541 0.778584i $$-0.715938\pi$$
−0.627541 + 0.778584i $$0.715938\pi$$
$$972$$ 1.00000 0.0320750
$$973$$ 4.21983 0.135282
$$974$$ −23.2760 −0.745813
$$975$$ 0 0
$$976$$ −0.396125 −0.0126796
$$977$$ 7.27545 0.232762 0.116381 0.993205i $$-0.462871\pi$$
0.116381 + 0.993205i $$0.462871\pi$$
$$978$$ 14.5918 0.466594
$$979$$ −15.0858 −0.482143
$$980$$ 26.4034 0.843426
$$981$$ −0.792249 −0.0252946
$$982$$ −15.5405 −0.495917
$$983$$ 53.4857 1.70593 0.852965 0.521969i $$-0.174802\pi$$
0.852965 + 0.521969i $$0.174802\pi$$
$$984$$ −1.50604 −0.0480108
$$985$$ 12.1860 0.388278
$$986$$ −24.6983 −0.786553
$$987$$ −4.98792 −0.158767
$$988$$ 0 0
$$989$$ −42.5026 −1.35150
$$990$$ −19.6407 −0.624223
$$991$$ 16.0575 0.510085 0.255042 0.966930i $$-0.417911\pi$$
0.255042 + 0.966930i $$0.417911\pi$$
$$992$$ −0.972853 −0.0308881
$$993$$ −25.6775 −0.814852
$$994$$ −0.920215 −0.0291874
$$995$$ −52.2083 −1.65512
$$996$$ −15.3274 −0.485666
$$997$$ 22.4263 0.710247 0.355123 0.934819i $$-0.384439\pi$$
0.355123 + 0.934819i $$0.384439\pi$$
$$998$$ 9.53617 0.301862
$$999$$ −1.28621 −0.0406938
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 1014.2.a.o.1.1 yes 3
3.2 odd 2 3042.2.a.bd.1.3 3
4.3 odd 2 8112.2.a.bz.1.1 3
13.2 odd 12 1014.2.i.g.823.6 12
13.3 even 3 1014.2.e.k.529.1 6
13.4 even 6 1014.2.e.m.991.3 6
13.5 odd 4 1014.2.b.g.337.3 6
13.6 odd 12 1014.2.i.g.361.3 12
13.7 odd 12 1014.2.i.g.361.4 12
13.8 odd 4 1014.2.b.g.337.4 6
13.9 even 3 1014.2.e.k.991.1 6
13.10 even 6 1014.2.e.m.529.3 6
13.11 odd 12 1014.2.i.g.823.1 12
13.12 even 2 1014.2.a.m.1.3 3
39.5 even 4 3042.2.b.r.1351.4 6
39.8 even 4 3042.2.b.r.1351.3 6
39.38 odd 2 3042.2.a.be.1.1 3
52.51 odd 2 8112.2.a.ce.1.3 3

By twisted newform
Twist Min Dim Char Parity Ord Type
1014.2.a.m.1.3 3 13.12 even 2
1014.2.a.o.1.1 yes 3 1.1 even 1 trivial
1014.2.b.g.337.3 6 13.5 odd 4
1014.2.b.g.337.4 6 13.8 odd 4
1014.2.e.k.529.1 6 13.3 even 3
1014.2.e.k.991.1 6 13.9 even 3
1014.2.e.m.529.3 6 13.10 even 6
1014.2.e.m.991.3 6 13.4 even 6
1014.2.i.g.361.3 12 13.6 odd 12
1014.2.i.g.361.4 12 13.7 odd 12
1014.2.i.g.823.1 12 13.11 odd 12
1014.2.i.g.823.6 12 13.2 odd 12
3042.2.a.bd.1.3 3 3.2 odd 2
3042.2.a.be.1.1 3 39.38 odd 2
3042.2.b.r.1351.3 6 39.8 even 4
3042.2.b.r.1351.4 6 39.5 even 4
8112.2.a.bz.1.1 3 4.3 odd 2
8112.2.a.ce.1.3 3 52.51 odd 2