Properties

 Label 5054.2.a.bb.1.5 Level $5054$ Weight $2$ Character 5054.1 Self dual yes Analytic conductor $40.356$ Analytic rank $1$ Dimension $6$ CM no Inner twists $1$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$5054 = 2 \cdot 7 \cdot 19^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 5054.a (trivial)

Newform invariants

 Self dual: yes Analytic conductor: $$40.3563931816$$ Analytic rank: $$1$$ Dimension: $$6$$ Coefficient field: 6.6.1528713.1 Defining polynomial: $$x^{6} - 3x^{5} - 3x^{4} + 7x^{3} + 3x^{2} - 3x - 1$$ x^6 - 3*x^5 - 3*x^4 + 7*x^3 + 3*x^2 - 3*x - 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 266) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

 Embedding label 1.5 Root $$-0.725554$$ of defining polynomial Character $$\chi$$ $$=$$ 5054.1

$q$-expansion

 $$f(q)$$ $$=$$ $$q-1.00000 q^{2} +2.11161 q^{3} +1.00000 q^{4} -3.59028 q^{5} -2.11161 q^{6} -1.00000 q^{7} -1.00000 q^{8} +1.45891 q^{9} +O(q^{10})$$ $$q-1.00000 q^{2} +2.11161 q^{3} +1.00000 q^{4} -3.59028 q^{5} -2.11161 q^{6} -1.00000 q^{7} -1.00000 q^{8} +1.45891 q^{9} +3.59028 q^{10} +3.24298 q^{11} +2.11161 q^{12} +0.613941 q^{13} +1.00000 q^{14} -7.58127 q^{15} +1.00000 q^{16} -4.69713 q^{17} -1.45891 q^{18} -3.59028 q^{20} -2.11161 q^{21} -3.24298 q^{22} +4.53089 q^{23} -2.11161 q^{24} +7.89009 q^{25} -0.613941 q^{26} -3.25419 q^{27} -1.00000 q^{28} +4.45024 q^{29} +7.58127 q^{30} -6.48937 q^{31} -1.00000 q^{32} +6.84792 q^{33} +4.69713 q^{34} +3.59028 q^{35} +1.45891 q^{36} +5.34592 q^{37} +1.29641 q^{39} +3.59028 q^{40} -1.08664 q^{41} +2.11161 q^{42} +5.44103 q^{43} +3.24298 q^{44} -5.23789 q^{45} -4.53089 q^{46} +11.6016 q^{47} +2.11161 q^{48} +1.00000 q^{49} -7.89009 q^{50} -9.91851 q^{51} +0.613941 q^{52} -6.12638 q^{53} +3.25419 q^{54} -11.6432 q^{55} +1.00000 q^{56} -4.45024 q^{58} -2.89177 q^{59} -7.58127 q^{60} -13.2269 q^{61} +6.48937 q^{62} -1.45891 q^{63} +1.00000 q^{64} -2.20422 q^{65} -6.84792 q^{66} -6.41158 q^{67} -4.69713 q^{68} +9.56748 q^{69} -3.59028 q^{70} +11.5793 q^{71} -1.45891 q^{72} -16.3204 q^{73} -5.34592 q^{74} +16.6608 q^{75} -3.24298 q^{77} -1.29641 q^{78} -6.43324 q^{79} -3.59028 q^{80} -11.2483 q^{81} +1.08664 q^{82} -8.52076 q^{83} -2.11161 q^{84} +16.8640 q^{85} -5.44103 q^{86} +9.39718 q^{87} -3.24298 q^{88} +1.49287 q^{89} +5.23789 q^{90} -0.613941 q^{91} +4.53089 q^{92} -13.7030 q^{93} -11.6016 q^{94} -2.11161 q^{96} +1.67786 q^{97} -1.00000 q^{98} +4.73121 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q - 6 q^{2} + 3 q^{3} + 6 q^{4} - 3 q^{5} - 3 q^{6} - 6 q^{7} - 6 q^{8} - 3 q^{9}+O(q^{10})$$ 6 * q - 6 * q^2 + 3 * q^3 + 6 * q^4 - 3 * q^5 - 3 * q^6 - 6 * q^7 - 6 * q^8 - 3 * q^9 $$6 q - 6 q^{2} + 3 q^{3} + 6 q^{4} - 3 q^{5} - 3 q^{6} - 6 q^{7} - 6 q^{8} - 3 q^{9} + 3 q^{10} + 3 q^{11} + 3 q^{12} + 6 q^{13} + 6 q^{14} - 6 q^{15} + 6 q^{16} - 9 q^{17} + 3 q^{18} - 3 q^{20} - 3 q^{21} - 3 q^{22} - 3 q^{24} + 3 q^{25} - 6 q^{26} + 3 q^{27} - 6 q^{28} + 6 q^{29} + 6 q^{30} + 3 q^{31} - 6 q^{32} + 6 q^{33} + 9 q^{34} + 3 q^{35} - 3 q^{36} - 3 q^{37} - 9 q^{39} + 3 q^{40} + 9 q^{41} + 3 q^{42} - 6 q^{43} + 3 q^{44} - 12 q^{45} + 9 q^{47} + 3 q^{48} + 6 q^{49} - 3 q^{50} - 27 q^{51} + 6 q^{52} + 6 q^{53} - 3 q^{54} - 24 q^{55} + 6 q^{56} - 6 q^{58} + 15 q^{59} - 6 q^{60} - 30 q^{61} - 3 q^{62} + 3 q^{63} + 6 q^{64} + 3 q^{65} - 6 q^{66} + 15 q^{67} - 9 q^{68} + 24 q^{69} - 3 q^{70} + 21 q^{71} + 3 q^{72} - 33 q^{73} + 3 q^{74} + 33 q^{75} - 3 q^{77} + 9 q^{78} - 30 q^{79} - 3 q^{80} - 18 q^{81} - 9 q^{82} - 33 q^{83} - 3 q^{84} - 18 q^{85} + 6 q^{86} + 15 q^{87} - 3 q^{88} - 12 q^{89} + 12 q^{90} - 6 q^{91} - 21 q^{93} - 9 q^{94} - 3 q^{96} - 18 q^{97} - 6 q^{98}+O(q^{100})$$ 6 * q - 6 * q^2 + 3 * q^3 + 6 * q^4 - 3 * q^5 - 3 * q^6 - 6 * q^7 - 6 * q^8 - 3 * q^9 + 3 * q^10 + 3 * q^11 + 3 * q^12 + 6 * q^13 + 6 * q^14 - 6 * q^15 + 6 * q^16 - 9 * q^17 + 3 * q^18 - 3 * q^20 - 3 * q^21 - 3 * q^22 - 3 * q^24 + 3 * q^25 - 6 * q^26 + 3 * q^27 - 6 * q^28 + 6 * q^29 + 6 * q^30 + 3 * q^31 - 6 * q^32 + 6 * q^33 + 9 * q^34 + 3 * q^35 - 3 * q^36 - 3 * q^37 - 9 * q^39 + 3 * q^40 + 9 * q^41 + 3 * q^42 - 6 * q^43 + 3 * q^44 - 12 * q^45 + 9 * q^47 + 3 * q^48 + 6 * q^49 - 3 * q^50 - 27 * q^51 + 6 * q^52 + 6 * q^53 - 3 * q^54 - 24 * q^55 + 6 * q^56 - 6 * q^58 + 15 * q^59 - 6 * q^60 - 30 * q^61 - 3 * q^62 + 3 * q^63 + 6 * q^64 + 3 * q^65 - 6 * q^66 + 15 * q^67 - 9 * q^68 + 24 * q^69 - 3 * q^70 + 21 * q^71 + 3 * q^72 - 33 * q^73 + 3 * q^74 + 33 * q^75 - 3 * q^77 + 9 * q^78 - 30 * q^79 - 3 * q^80 - 18 * q^81 - 9 * q^82 - 33 * q^83 - 3 * q^84 - 18 * q^85 + 6 * q^86 + 15 * q^87 - 3 * q^88 - 12 * q^89 + 12 * q^90 - 6 * q^91 - 21 * q^93 - 9 * q^94 - 3 * q^96 - 18 * q^97 - 6 * q^98

Coefficient data

For each $$n$$ we display the coefficients of the $$q$$-expansion $$a_n$$, the Satake parameters $$\alpha_p$$, and the Satake angles $$\theta_p = \textrm{Arg}(\alpha_p)$$.

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −1.00000 −0.707107
$$3$$ 2.11161 1.21914 0.609570 0.792732i $$-0.291342\pi$$
0.609570 + 0.792732i $$0.291342\pi$$
$$4$$ 1.00000 0.500000
$$5$$ −3.59028 −1.60562 −0.802810 0.596235i $$-0.796663\pi$$
−0.802810 + 0.596235i $$0.796663\pi$$
$$6$$ −2.11161 −0.862062
$$7$$ −1.00000 −0.377964
$$8$$ −1.00000 −0.353553
$$9$$ 1.45891 0.486303
$$10$$ 3.59028 1.13535
$$11$$ 3.24298 0.977795 0.488898 0.872341i $$-0.337399\pi$$
0.488898 + 0.872341i $$0.337399\pi$$
$$12$$ 2.11161 0.609570
$$13$$ 0.613941 0.170277 0.0851383 0.996369i $$-0.472867\pi$$
0.0851383 + 0.996369i $$0.472867\pi$$
$$14$$ 1.00000 0.267261
$$15$$ −7.58127 −1.95748
$$16$$ 1.00000 0.250000
$$17$$ −4.69713 −1.13922 −0.569610 0.821915i $$-0.692906\pi$$
−0.569610 + 0.821915i $$0.692906\pi$$
$$18$$ −1.45891 −0.343868
$$19$$ 0 0
$$20$$ −3.59028 −0.802810
$$21$$ −2.11161 −0.460792
$$22$$ −3.24298 −0.691406
$$23$$ 4.53089 0.944756 0.472378 0.881396i $$-0.343396\pi$$
0.472378 + 0.881396i $$0.343396\pi$$
$$24$$ −2.11161 −0.431031
$$25$$ 7.89009 1.57802
$$26$$ −0.613941 −0.120404
$$27$$ −3.25419 −0.626269
$$28$$ −1.00000 −0.188982
$$29$$ 4.45024 0.826388 0.413194 0.910643i $$-0.364413\pi$$
0.413194 + 0.910643i $$0.364413\pi$$
$$30$$ 7.58127 1.38415
$$31$$ −6.48937 −1.16552 −0.582762 0.812643i $$-0.698028\pi$$
−0.582762 + 0.812643i $$0.698028\pi$$
$$32$$ −1.00000 −0.176777
$$33$$ 6.84792 1.19207
$$34$$ 4.69713 0.805551
$$35$$ 3.59028 0.606868
$$36$$ 1.45891 0.243152
$$37$$ 5.34592 0.878864 0.439432 0.898276i $$-0.355180\pi$$
0.439432 + 0.898276i $$0.355180\pi$$
$$38$$ 0 0
$$39$$ 1.29641 0.207591
$$40$$ 3.59028 0.567673
$$41$$ −1.08664 −0.169705 −0.0848525 0.996394i $$-0.527042\pi$$
−0.0848525 + 0.996394i $$0.527042\pi$$
$$42$$ 2.11161 0.325829
$$43$$ 5.44103 0.829750 0.414875 0.909878i $$-0.363825\pi$$
0.414875 + 0.909878i $$0.363825\pi$$
$$44$$ 3.24298 0.488898
$$45$$ −5.23789 −0.780818
$$46$$ −4.53089 −0.668043
$$47$$ 11.6016 1.69227 0.846133 0.532971i $$-0.178925\pi$$
0.846133 + 0.532971i $$0.178925\pi$$
$$48$$ 2.11161 0.304785
$$49$$ 1.00000 0.142857
$$50$$ −7.89009 −1.11583
$$51$$ −9.91851 −1.38887
$$52$$ 0.613941 0.0851383
$$53$$ −6.12638 −0.841523 −0.420762 0.907171i $$-0.638237\pi$$
−0.420762 + 0.907171i $$0.638237\pi$$
$$54$$ 3.25419 0.442839
$$55$$ −11.6432 −1.56997
$$56$$ 1.00000 0.133631
$$57$$ 0 0
$$58$$ −4.45024 −0.584345
$$59$$ −2.89177 −0.376477 −0.188238 0.982123i $$-0.560278\pi$$
−0.188238 + 0.982123i $$0.560278\pi$$
$$60$$ −7.58127 −0.978738
$$61$$ −13.2269 −1.69353 −0.846764 0.531969i $$-0.821452\pi$$
−0.846764 + 0.531969i $$0.821452\pi$$
$$62$$ 6.48937 0.824150
$$63$$ −1.45891 −0.183805
$$64$$ 1.00000 0.125000
$$65$$ −2.20422 −0.273400
$$66$$ −6.84792 −0.842921
$$67$$ −6.41158 −0.783299 −0.391650 0.920114i $$-0.628095\pi$$
−0.391650 + 0.920114i $$0.628095\pi$$
$$68$$ −4.69713 −0.569610
$$69$$ 9.56748 1.15179
$$70$$ −3.59028 −0.429120
$$71$$ 11.5793 1.37421 0.687103 0.726560i $$-0.258882\pi$$
0.687103 + 0.726560i $$0.258882\pi$$
$$72$$ −1.45891 −0.171934
$$73$$ −16.3204 −1.91015 −0.955077 0.296357i $$-0.904228\pi$$
−0.955077 + 0.296357i $$0.904228\pi$$
$$74$$ −5.34592 −0.621451
$$75$$ 16.6608 1.92382
$$76$$ 0 0
$$77$$ −3.24298 −0.369572
$$78$$ −1.29641 −0.146789
$$79$$ −6.43324 −0.723796 −0.361898 0.932218i $$-0.617871\pi$$
−0.361898 + 0.932218i $$0.617871\pi$$
$$80$$ −3.59028 −0.401405
$$81$$ −11.2483 −1.24981
$$82$$ 1.08664 0.120000
$$83$$ −8.52076 −0.935274 −0.467637 0.883921i $$-0.654895\pi$$
−0.467637 + 0.883921i $$0.654895\pi$$
$$84$$ −2.11161 −0.230396
$$85$$ 16.8640 1.82916
$$86$$ −5.44103 −0.586722
$$87$$ 9.39718 1.00748
$$88$$ −3.24298 −0.345703
$$89$$ 1.49287 0.158244 0.0791221 0.996865i $$-0.474788\pi$$
0.0791221 + 0.996865i $$0.474788\pi$$
$$90$$ 5.23789 0.552122
$$91$$ −0.613941 −0.0643585
$$92$$ 4.53089 0.472378
$$93$$ −13.7030 −1.42094
$$94$$ −11.6016 −1.19661
$$95$$ 0 0
$$96$$ −2.11161 −0.215516
$$97$$ 1.67786 0.170361 0.0851804 0.996366i $$-0.472853\pi$$
0.0851804 + 0.996366i $$0.472853\pi$$
$$98$$ −1.00000 −0.101015
$$99$$ 4.73121 0.475505
$$100$$ 7.89009 0.789009
$$101$$ 1.86854 0.185927 0.0929636 0.995670i $$-0.470366\pi$$
0.0929636 + 0.995670i $$0.470366\pi$$
$$102$$ 9.91851 0.982079
$$103$$ −20.1263 −1.98310 −0.991550 0.129727i $$-0.958590\pi$$
−0.991550 + 0.129727i $$0.958590\pi$$
$$104$$ −0.613941 −0.0602019
$$105$$ 7.58127 0.739857
$$106$$ 6.12638 0.595047
$$107$$ 9.37414 0.906233 0.453116 0.891451i $$-0.350312\pi$$
0.453116 + 0.891451i $$0.350312\pi$$
$$108$$ −3.25419 −0.313134
$$109$$ −9.28080 −0.888940 −0.444470 0.895794i $$-0.646608\pi$$
−0.444470 + 0.895794i $$0.646608\pi$$
$$110$$ 11.6432 1.11014
$$111$$ 11.2885 1.07146
$$112$$ −1.00000 −0.0944911
$$113$$ −8.67289 −0.815877 −0.407938 0.913009i $$-0.633752\pi$$
−0.407938 + 0.913009i $$0.633752\pi$$
$$114$$ 0 0
$$115$$ −16.2671 −1.51692
$$116$$ 4.45024 0.413194
$$117$$ 0.895684 0.0828060
$$118$$ 2.89177 0.266209
$$119$$ 4.69713 0.430585
$$120$$ 7.58127 0.692073
$$121$$ −0.483078 −0.0439162
$$122$$ 13.2269 1.19750
$$123$$ −2.29457 −0.206894
$$124$$ −6.48937 −0.582762
$$125$$ −10.3762 −0.928077
$$126$$ 1.45891 0.129970
$$127$$ 18.2693 1.62113 0.810567 0.585645i $$-0.199159\pi$$
0.810567 + 0.585645i $$0.199159\pi$$
$$128$$ −1.00000 −0.0883883
$$129$$ 11.4894 1.01158
$$130$$ 2.20422 0.193323
$$131$$ −8.09149 −0.706957 −0.353478 0.935443i $$-0.615001\pi$$
−0.353478 + 0.935443i $$0.615001\pi$$
$$132$$ 6.84792 0.596035
$$133$$ 0 0
$$134$$ 6.41158 0.553876
$$135$$ 11.6834 1.00555
$$136$$ 4.69713 0.402775
$$137$$ 4.55482 0.389145 0.194572 0.980888i $$-0.437668\pi$$
0.194572 + 0.980888i $$0.437668\pi$$
$$138$$ −9.56748 −0.814438
$$139$$ −21.6945 −1.84010 −0.920051 0.391799i $$-0.871853\pi$$
−0.920051 + 0.391799i $$0.871853\pi$$
$$140$$ 3.59028 0.303434
$$141$$ 24.4981 2.06311
$$142$$ −11.5793 −0.971710
$$143$$ 1.99100 0.166496
$$144$$ 1.45891 0.121576
$$145$$ −15.9776 −1.32687
$$146$$ 16.3204 1.35068
$$147$$ 2.11161 0.174163
$$148$$ 5.34592 0.439432
$$149$$ 21.4072 1.75375 0.876874 0.480721i $$-0.159625\pi$$
0.876874 + 0.480721i $$0.159625\pi$$
$$150$$ −16.6608 −1.36035
$$151$$ 5.52022 0.449229 0.224615 0.974448i $$-0.427888\pi$$
0.224615 + 0.974448i $$0.427888\pi$$
$$152$$ 0 0
$$153$$ −6.85268 −0.554007
$$154$$ 3.24298 0.261327
$$155$$ 23.2986 1.87139
$$156$$ 1.29641 0.103796
$$157$$ −13.5838 −1.08410 −0.542052 0.840345i $$-0.682352\pi$$
−0.542052 + 0.840345i $$0.682352\pi$$
$$158$$ 6.43324 0.511801
$$159$$ −12.9365 −1.02594
$$160$$ 3.59028 0.283836
$$161$$ −4.53089 −0.357084
$$162$$ 11.2483 0.883751
$$163$$ −6.32223 −0.495195 −0.247598 0.968863i $$-0.579641\pi$$
−0.247598 + 0.968863i $$0.579641\pi$$
$$164$$ −1.08664 −0.0848525
$$165$$ −24.5859 −1.91401
$$166$$ 8.52076 0.661339
$$167$$ −14.1989 −1.09875 −0.549373 0.835577i $$-0.685133\pi$$
−0.549373 + 0.835577i $$0.685133\pi$$
$$168$$ 2.11161 0.162914
$$169$$ −12.6231 −0.971006
$$170$$ −16.8640 −1.29341
$$171$$ 0 0
$$172$$ 5.44103 0.414875
$$173$$ 7.38609 0.561554 0.280777 0.959773i $$-0.409408\pi$$
0.280777 + 0.959773i $$0.409408\pi$$
$$174$$ −9.39718 −0.712398
$$175$$ −7.89009 −0.596435
$$176$$ 3.24298 0.244449
$$177$$ −6.10631 −0.458978
$$178$$ −1.49287 −0.111895
$$179$$ −6.37886 −0.476778 −0.238389 0.971170i $$-0.576619\pi$$
−0.238389 + 0.971170i $$0.576619\pi$$
$$180$$ −5.23789 −0.390409
$$181$$ 22.2731 1.65555 0.827774 0.561062i $$-0.189607\pi$$
0.827774 + 0.561062i $$0.189607\pi$$
$$182$$ 0.613941 0.0455083
$$183$$ −27.9300 −2.06465
$$184$$ −4.53089 −0.334022
$$185$$ −19.1933 −1.41112
$$186$$ 13.7030 1.00475
$$187$$ −15.2327 −1.11392
$$188$$ 11.6016 0.846133
$$189$$ 3.25419 0.236707
$$190$$ 0 0
$$191$$ −23.4129 −1.69410 −0.847048 0.531516i $$-0.821623\pi$$
−0.847048 + 0.531516i $$0.821623\pi$$
$$192$$ 2.11161 0.152393
$$193$$ −15.1577 −1.09107 −0.545537 0.838087i $$-0.683674\pi$$
−0.545537 + 0.838087i $$0.683674\pi$$
$$194$$ −1.67786 −0.120463
$$195$$ −4.65445 −0.333312
$$196$$ 1.00000 0.0714286
$$197$$ −14.5729 −1.03828 −0.519138 0.854691i $$-0.673747\pi$$
−0.519138 + 0.854691i $$0.673747\pi$$
$$198$$ −4.73121 −0.336233
$$199$$ 7.05899 0.500398 0.250199 0.968194i $$-0.419504\pi$$
0.250199 + 0.968194i $$0.419504\pi$$
$$200$$ −7.89009 −0.557913
$$201$$ −13.5388 −0.954952
$$202$$ −1.86854 −0.131470
$$203$$ −4.45024 −0.312345
$$204$$ −9.91851 −0.694435
$$205$$ 3.90134 0.272482
$$206$$ 20.1263 1.40226
$$207$$ 6.61016 0.459438
$$208$$ 0.613941 0.0425691
$$209$$ 0 0
$$210$$ −7.58127 −0.523158
$$211$$ −5.69557 −0.392099 −0.196050 0.980594i $$-0.562811\pi$$
−0.196050 + 0.980594i $$0.562811\pi$$
$$212$$ −6.12638 −0.420762
$$213$$ 24.4509 1.67535
$$214$$ −9.37414 −0.640803
$$215$$ −19.5348 −1.33226
$$216$$ 3.25419 0.221419
$$217$$ 6.48937 0.440527
$$218$$ 9.28080 0.628575
$$219$$ −34.4623 −2.32875
$$220$$ −11.6432 −0.784984
$$221$$ −2.88376 −0.193983
$$222$$ −11.2885 −0.757636
$$223$$ 18.5836 1.24445 0.622226 0.782838i $$-0.286228\pi$$
0.622226 + 0.782838i $$0.286228\pi$$
$$224$$ 1.00000 0.0668153
$$225$$ 11.5109 0.767395
$$226$$ 8.67289 0.576912
$$227$$ 0.527433 0.0350070 0.0175035 0.999847i $$-0.494428\pi$$
0.0175035 + 0.999847i $$0.494428\pi$$
$$228$$ 0 0
$$229$$ −11.9310 −0.788426 −0.394213 0.919019i $$-0.628983\pi$$
−0.394213 + 0.919019i $$0.628983\pi$$
$$230$$ 16.2671 1.07262
$$231$$ −6.84792 −0.450560
$$232$$ −4.45024 −0.292172
$$233$$ −6.00187 −0.393195 −0.196598 0.980484i $$-0.562989\pi$$
−0.196598 + 0.980484i $$0.562989\pi$$
$$234$$ −0.895684 −0.0585527
$$235$$ −41.6529 −2.71714
$$236$$ −2.89177 −0.188238
$$237$$ −13.5845 −0.882409
$$238$$ −4.69713 −0.304470
$$239$$ −22.2435 −1.43882 −0.719408 0.694588i $$-0.755587\pi$$
−0.719408 + 0.694588i $$0.755587\pi$$
$$240$$ −7.58127 −0.489369
$$241$$ 19.8067 1.27586 0.637932 0.770093i $$-0.279790\pi$$
0.637932 + 0.770093i $$0.279790\pi$$
$$242$$ 0.483078 0.0310534
$$243$$ −13.9895 −0.897428
$$244$$ −13.2269 −0.846764
$$245$$ −3.59028 −0.229374
$$246$$ 2.29457 0.146296
$$247$$ 0 0
$$248$$ 6.48937 0.412075
$$249$$ −17.9925 −1.14023
$$250$$ 10.3762 0.656249
$$251$$ 6.27968 0.396370 0.198185 0.980165i $$-0.436495\pi$$
0.198185 + 0.980165i $$0.436495\pi$$
$$252$$ −1.45891 −0.0919026
$$253$$ 14.6936 0.923778
$$254$$ −18.2693 −1.14632
$$255$$ 35.6102 2.23000
$$256$$ 1.00000 0.0625000
$$257$$ −11.4959 −0.717096 −0.358548 0.933511i $$-0.616728\pi$$
−0.358548 + 0.933511i $$0.616728\pi$$
$$258$$ −11.4894 −0.715296
$$259$$ −5.34592 −0.332179
$$260$$ −2.20422 −0.136700
$$261$$ 6.49249 0.401875
$$262$$ 8.09149 0.499894
$$263$$ 4.04284 0.249292 0.124646 0.992201i $$-0.460220\pi$$
0.124646 + 0.992201i $$0.460220\pi$$
$$264$$ −6.84792 −0.421460
$$265$$ 21.9954 1.35117
$$266$$ 0 0
$$267$$ 3.15237 0.192922
$$268$$ −6.41158 −0.391650
$$269$$ −15.2503 −0.929826 −0.464913 0.885356i $$-0.653914\pi$$
−0.464913 + 0.885356i $$0.653914\pi$$
$$270$$ −11.6834 −0.711031
$$271$$ −17.9249 −1.08886 −0.544429 0.838807i $$-0.683254\pi$$
−0.544429 + 0.838807i $$0.683254\pi$$
$$272$$ −4.69713 −0.284805
$$273$$ −1.29641 −0.0784620
$$274$$ −4.55482 −0.275167
$$275$$ 25.5874 1.54298
$$276$$ 9.56748 0.575895
$$277$$ −3.58168 −0.215202 −0.107601 0.994194i $$-0.534317\pi$$
−0.107601 + 0.994194i $$0.534317\pi$$
$$278$$ 21.6945 1.30115
$$279$$ −9.46740 −0.566798
$$280$$ −3.59028 −0.214560
$$281$$ −7.09159 −0.423049 −0.211524 0.977373i $$-0.567843\pi$$
−0.211524 + 0.977373i $$0.567843\pi$$
$$282$$ −24.4981 −1.45884
$$283$$ −14.0809 −0.837021 −0.418511 0.908212i $$-0.637448\pi$$
−0.418511 + 0.908212i $$0.637448\pi$$
$$284$$ 11.5793 0.687103
$$285$$ 0 0
$$286$$ −1.99100 −0.117730
$$287$$ 1.08664 0.0641424
$$288$$ −1.45891 −0.0859671
$$289$$ 5.06300 0.297824
$$290$$ 15.9776 0.938236
$$291$$ 3.54299 0.207694
$$292$$ −16.3204 −0.955077
$$293$$ 22.8960 1.33760 0.668798 0.743444i $$-0.266809\pi$$
0.668798 + 0.743444i $$0.266809\pi$$
$$294$$ −2.11161 −0.123152
$$295$$ 10.3823 0.604479
$$296$$ −5.34592 −0.310725
$$297$$ −10.5533 −0.612363
$$298$$ −21.4072 −1.24009
$$299$$ 2.78170 0.160870
$$300$$ 16.6608 0.961912
$$301$$ −5.44103 −0.313616
$$302$$ −5.52022 −0.317653
$$303$$ 3.94564 0.226671
$$304$$ 0 0
$$305$$ 47.4881 2.71916
$$306$$ 6.85268 0.391742
$$307$$ −32.8057 −1.87232 −0.936161 0.351572i $$-0.885647\pi$$
−0.936161 + 0.351572i $$0.885647\pi$$
$$308$$ −3.24298 −0.184786
$$309$$ −42.4989 −2.41768
$$310$$ −23.2986 −1.32327
$$311$$ 13.6967 0.776671 0.388336 0.921518i $$-0.373050\pi$$
0.388336 + 0.921518i $$0.373050\pi$$
$$312$$ −1.29641 −0.0733945
$$313$$ 7.88847 0.445883 0.222941 0.974832i $$-0.428434\pi$$
0.222941 + 0.974832i $$0.428434\pi$$
$$314$$ 13.5838 0.766577
$$315$$ 5.23789 0.295122
$$316$$ −6.43324 −0.361898
$$317$$ 28.0164 1.57356 0.786780 0.617234i $$-0.211747\pi$$
0.786780 + 0.617234i $$0.211747\pi$$
$$318$$ 12.9365 0.725446
$$319$$ 14.4320 0.808039
$$320$$ −3.59028 −0.200703
$$321$$ 19.7946 1.10482
$$322$$ 4.53089 0.252497
$$323$$ 0 0
$$324$$ −11.2483 −0.624906
$$325$$ 4.84405 0.268699
$$326$$ 6.32223 0.350156
$$327$$ −19.5975 −1.08374
$$328$$ 1.08664 0.0599998
$$329$$ −11.6016 −0.639617
$$330$$ 24.5859 1.35341
$$331$$ −13.6501 −0.750278 −0.375139 0.926969i $$-0.622405\pi$$
−0.375139 + 0.926969i $$0.622405\pi$$
$$332$$ −8.52076 −0.467637
$$333$$ 7.79921 0.427394
$$334$$ 14.1989 0.776931
$$335$$ 23.0194 1.25768
$$336$$ −2.11161 −0.115198
$$337$$ −17.2621 −0.940324 −0.470162 0.882580i $$-0.655805\pi$$
−0.470162 + 0.882580i $$0.655805\pi$$
$$338$$ 12.6231 0.686605
$$339$$ −18.3138 −0.994668
$$340$$ 16.8640 0.914578
$$341$$ −21.0449 −1.13964
$$342$$ 0 0
$$343$$ −1.00000 −0.0539949
$$344$$ −5.44103 −0.293361
$$345$$ −34.3499 −1.84934
$$346$$ −7.38609 −0.397078
$$347$$ −2.69423 −0.144634 −0.0723170 0.997382i $$-0.523039\pi$$
−0.0723170 + 0.997382i $$0.523039\pi$$
$$348$$ 9.39718 0.503742
$$349$$ −5.06542 −0.271145 −0.135573 0.990767i $$-0.543287\pi$$
−0.135573 + 0.990767i $$0.543287\pi$$
$$350$$ 7.89009 0.421743
$$351$$ −1.99788 −0.106639
$$352$$ −3.24298 −0.172851
$$353$$ −21.7080 −1.15540 −0.577701 0.816249i $$-0.696050\pi$$
−0.577701 + 0.816249i $$0.696050\pi$$
$$354$$ 6.10631 0.324547
$$355$$ −41.5728 −2.20645
$$356$$ 1.49287 0.0791221
$$357$$ 9.91851 0.524943
$$358$$ 6.37886 0.337133
$$359$$ 5.02083 0.264989 0.132495 0.991184i $$-0.457701\pi$$
0.132495 + 0.991184i $$0.457701\pi$$
$$360$$ 5.23789 0.276061
$$361$$ 0 0
$$362$$ −22.2731 −1.17065
$$363$$ −1.02007 −0.0535400
$$364$$ −0.613941 −0.0321792
$$365$$ 58.5946 3.06698
$$366$$ 27.9300 1.45993
$$367$$ −7.51570 −0.392316 −0.196158 0.980572i $$-0.562847\pi$$
−0.196158 + 0.980572i $$0.562847\pi$$
$$368$$ 4.53089 0.236189
$$369$$ −1.58531 −0.0825280
$$370$$ 19.1933 0.997814
$$371$$ 6.12638 0.318066
$$372$$ −13.7030 −0.710469
$$373$$ 14.0820 0.729137 0.364569 0.931177i $$-0.381216\pi$$
0.364569 + 0.931177i $$0.381216\pi$$
$$374$$ 15.2327 0.787664
$$375$$ −21.9105 −1.13146
$$376$$ −11.6016 −0.598307
$$377$$ 2.73218 0.140715
$$378$$ −3.25419 −0.167377
$$379$$ 6.46372 0.332019 0.166010 0.986124i $$-0.446912\pi$$
0.166010 + 0.986124i $$0.446912\pi$$
$$380$$ 0 0
$$381$$ 38.5776 1.97639
$$382$$ 23.4129 1.19791
$$383$$ −18.5181 −0.946229 −0.473115 0.881001i $$-0.656870\pi$$
−0.473115 + 0.881001i $$0.656870\pi$$
$$384$$ −2.11161 −0.107758
$$385$$ 11.6432 0.593392
$$386$$ 15.1577 0.771506
$$387$$ 7.93797 0.403510
$$388$$ 1.67786 0.0851804
$$389$$ −37.2070 −1.88647 −0.943234 0.332128i $$-0.892233\pi$$
−0.943234 + 0.332128i $$0.892233\pi$$
$$390$$ 4.65445 0.235687
$$391$$ −21.2822 −1.07629
$$392$$ −1.00000 −0.0505076
$$393$$ −17.0861 −0.861880
$$394$$ 14.5729 0.734172
$$395$$ 23.0971 1.16214
$$396$$ 4.73121 0.237752
$$397$$ 34.3856 1.72576 0.862881 0.505407i $$-0.168658\pi$$
0.862881 + 0.505407i $$0.168658\pi$$
$$398$$ −7.05899 −0.353835
$$399$$ 0 0
$$400$$ 7.89009 0.394504
$$401$$ 24.7614 1.23653 0.618263 0.785971i $$-0.287837\pi$$
0.618263 + 0.785971i $$0.287837\pi$$
$$402$$ 13.5388 0.675253
$$403$$ −3.98409 −0.198462
$$404$$ 1.86854 0.0929636
$$405$$ 40.3846 2.00672
$$406$$ 4.45024 0.220862
$$407$$ 17.3367 0.859349
$$408$$ 9.91851 0.491040
$$409$$ 8.46451 0.418543 0.209271 0.977858i $$-0.432891\pi$$
0.209271 + 0.977858i $$0.432891\pi$$
$$410$$ −3.90134 −0.192674
$$411$$ 9.61802 0.474422
$$412$$ −20.1263 −0.991550
$$413$$ 2.89177 0.142295
$$414$$ −6.61016 −0.324871
$$415$$ 30.5919 1.50170
$$416$$ −0.613941 −0.0301009
$$417$$ −45.8103 −2.24334
$$418$$ 0 0
$$419$$ −10.9702 −0.535929 −0.267964 0.963429i $$-0.586351\pi$$
−0.267964 + 0.963429i $$0.586351\pi$$
$$420$$ 7.58127 0.369928
$$421$$ 32.1234 1.56560 0.782800 0.622273i $$-0.213791\pi$$
0.782800 + 0.622273i $$0.213791\pi$$
$$422$$ 5.69557 0.277256
$$423$$ 16.9257 0.822955
$$424$$ 6.12638 0.297523
$$425$$ −37.0607 −1.79771
$$426$$ −24.4509 −1.18465
$$427$$ 13.2269 0.640093
$$428$$ 9.37414 0.453116
$$429$$ 4.20422 0.202982
$$430$$ 19.5348 0.942052
$$431$$ −7.26547 −0.349966 −0.174983 0.984571i $$-0.555987\pi$$
−0.174983 + 0.984571i $$0.555987\pi$$
$$432$$ −3.25419 −0.156567
$$433$$ 18.2307 0.876111 0.438056 0.898948i $$-0.355667\pi$$
0.438056 + 0.898948i $$0.355667\pi$$
$$434$$ −6.48937 −0.311500
$$435$$ −33.7385 −1.61764
$$436$$ −9.28080 −0.444470
$$437$$ 0 0
$$438$$ 34.4623 1.64667
$$439$$ −32.5406 −1.55308 −0.776539 0.630069i $$-0.783027\pi$$
−0.776539 + 0.630069i $$0.783027\pi$$
$$440$$ 11.6432 0.555068
$$441$$ 1.45891 0.0694719
$$442$$ 2.88376 0.137166
$$443$$ −0.0189181 −0.000898825 0 −0.000449412 1.00000i $$-0.500143\pi$$
−0.000449412 1.00000i $$0.500143\pi$$
$$444$$ 11.2885 0.535729
$$445$$ −5.35982 −0.254080
$$446$$ −18.5836 −0.879961
$$447$$ 45.2037 2.13806
$$448$$ −1.00000 −0.0472456
$$449$$ 4.85684 0.229209 0.114604 0.993411i $$-0.463440\pi$$
0.114604 + 0.993411i $$0.463440\pi$$
$$450$$ −11.5109 −0.542630
$$451$$ −3.52396 −0.165937
$$452$$ −8.67289 −0.407938
$$453$$ 11.6566 0.547673
$$454$$ −0.527433 −0.0247537
$$455$$ 2.20422 0.103335
$$456$$ 0 0
$$457$$ 30.2842 1.41663 0.708317 0.705895i $$-0.249455\pi$$
0.708317 + 0.705895i $$0.249455\pi$$
$$458$$ 11.9310 0.557501
$$459$$ 15.2853 0.713458
$$460$$ −16.2671 −0.758460
$$461$$ 20.6428 0.961432 0.480716 0.876876i $$-0.340377\pi$$
0.480716 + 0.876876i $$0.340377\pi$$
$$462$$ 6.84792 0.318594
$$463$$ 21.7134 1.00911 0.504554 0.863380i $$-0.331657\pi$$
0.504554 + 0.863380i $$0.331657\pi$$
$$464$$ 4.45024 0.206597
$$465$$ 49.1977 2.28149
$$466$$ 6.00187 0.278031
$$467$$ 18.9511 0.876952 0.438476 0.898743i $$-0.355518\pi$$
0.438476 + 0.898743i $$0.355518\pi$$
$$468$$ 0.895684 0.0414030
$$469$$ 6.41158 0.296059
$$470$$ 41.6529 1.92131
$$471$$ −28.6837 −1.32167
$$472$$ 2.89177 0.133105
$$473$$ 17.6452 0.811325
$$474$$ 13.5845 0.623958
$$475$$ 0 0
$$476$$ 4.69713 0.215292
$$477$$ −8.93784 −0.409235
$$478$$ 22.2435 1.01740
$$479$$ 20.3473 0.929690 0.464845 0.885392i $$-0.346110\pi$$
0.464845 + 0.885392i $$0.346110\pi$$
$$480$$ 7.58127 0.346036
$$481$$ 3.28208 0.149650
$$482$$ −19.8067 −0.902171
$$483$$ −9.56748 −0.435336
$$484$$ −0.483078 −0.0219581
$$485$$ −6.02398 −0.273535
$$486$$ 13.9895 0.634577
$$487$$ 0.582139 0.0263793 0.0131896 0.999913i $$-0.495801\pi$$
0.0131896 + 0.999913i $$0.495801\pi$$
$$488$$ 13.2269 0.598752
$$489$$ −13.3501 −0.603713
$$490$$ 3.59028 0.162192
$$491$$ −33.5615 −1.51461 −0.757304 0.653063i $$-0.773484\pi$$
−0.757304 + 0.653063i $$0.773484\pi$$
$$492$$ −2.29457 −0.103447
$$493$$ −20.9033 −0.941439
$$494$$ 0 0
$$495$$ −16.9864 −0.763481
$$496$$ −6.48937 −0.291381
$$497$$ −11.5793 −0.519401
$$498$$ 17.9925 0.806265
$$499$$ −7.71505 −0.345373 −0.172687 0.984977i $$-0.555245\pi$$
−0.172687 + 0.984977i $$0.555245\pi$$
$$500$$ −10.3762 −0.464038
$$501$$ −29.9826 −1.33953
$$502$$ −6.27968 −0.280276
$$503$$ 17.1560 0.764949 0.382475 0.923966i $$-0.375072\pi$$
0.382475 + 0.923966i $$0.375072\pi$$
$$504$$ 1.45891 0.0649850
$$505$$ −6.70859 −0.298528
$$506$$ −14.6936 −0.653209
$$507$$ −26.6551 −1.18379
$$508$$ 18.2693 0.810567
$$509$$ −25.9509 −1.15025 −0.575126 0.818065i $$-0.695047\pi$$
−0.575126 + 0.818065i $$0.695047\pi$$
$$510$$ −35.6102 −1.57685
$$511$$ 16.3204 0.721971
$$512$$ −1.00000 −0.0441942
$$513$$ 0 0
$$514$$ 11.4959 0.507064
$$515$$ 72.2588 3.18411
$$516$$ 11.4894 0.505791
$$517$$ 37.6238 1.65469
$$518$$ 5.34592 0.234886
$$519$$ 15.5966 0.684613
$$520$$ 2.20422 0.0966613
$$521$$ −19.9715 −0.874969 −0.437485 0.899226i $$-0.644131\pi$$
−0.437485 + 0.899226i $$0.644131\pi$$
$$522$$ −6.49249 −0.284169
$$523$$ 8.44182 0.369135 0.184568 0.982820i $$-0.440912\pi$$
0.184568 + 0.982820i $$0.440912\pi$$
$$524$$ −8.09149 −0.353478
$$525$$ −16.6608 −0.727137
$$526$$ −4.04284 −0.176276
$$527$$ 30.4814 1.32779
$$528$$ 6.84792 0.298017
$$529$$ −2.47105 −0.107437
$$530$$ −21.9954 −0.955419
$$531$$ −4.21884 −0.183082
$$532$$ 0 0
$$533$$ −0.667134 −0.0288968
$$534$$ −3.15237 −0.136416
$$535$$ −33.6558 −1.45507
$$536$$ 6.41158 0.276938
$$537$$ −13.4697 −0.581260
$$538$$ 15.2503 0.657486
$$539$$ 3.24298 0.139685
$$540$$ 11.6834 0.502775
$$541$$ −39.3246 −1.69070 −0.845349 0.534214i $$-0.820608\pi$$
−0.845349 + 0.534214i $$0.820608\pi$$
$$542$$ 17.9249 0.769939
$$543$$ 47.0322 2.01834
$$544$$ 4.69713 0.201388
$$545$$ 33.3206 1.42730
$$546$$ 1.29641 0.0554810
$$547$$ −20.7275 −0.886245 −0.443122 0.896461i $$-0.646129\pi$$
−0.443122 + 0.896461i $$0.646129\pi$$
$$548$$ 4.55482 0.194572
$$549$$ −19.2968 −0.823568
$$550$$ −25.5874 −1.09105
$$551$$ 0 0
$$552$$ −9.56748 −0.407219
$$553$$ 6.43324 0.273569
$$554$$ 3.58168 0.152171
$$555$$ −40.5289 −1.72036
$$556$$ −21.6945 −0.920051
$$557$$ −30.6400 −1.29826 −0.649130 0.760677i $$-0.724867\pi$$
−0.649130 + 0.760677i $$0.724867\pi$$
$$558$$ 9.46740 0.400787
$$559$$ 3.34047 0.141287
$$560$$ 3.59028 0.151717
$$561$$ −32.1655 −1.35803
$$562$$ 7.09159 0.299141
$$563$$ −19.3305 −0.814682 −0.407341 0.913276i $$-0.633544\pi$$
−0.407341 + 0.913276i $$0.633544\pi$$
$$564$$ 24.4981 1.03156
$$565$$ 31.1381 1.30999
$$566$$ 14.0809 0.591863
$$567$$ 11.2483 0.472385
$$568$$ −11.5793 −0.485855
$$569$$ −18.0853 −0.758174 −0.379087 0.925361i $$-0.623762\pi$$
−0.379087 + 0.925361i $$0.623762\pi$$
$$570$$ 0 0
$$571$$ 1.49157 0.0624203 0.0312102 0.999513i $$-0.490064\pi$$
0.0312102 + 0.999513i $$0.490064\pi$$
$$572$$ 1.99100 0.0832478
$$573$$ −49.4390 −2.06534
$$574$$ −1.08664 −0.0453555
$$575$$ 35.7491 1.49084
$$576$$ 1.45891 0.0607879
$$577$$ 26.2740 1.09380 0.546900 0.837198i $$-0.315808\pi$$
0.546900 + 0.837198i $$0.315808\pi$$
$$578$$ −5.06300 −0.210593
$$579$$ −32.0072 −1.33017
$$580$$ −15.9776 −0.663433
$$581$$ 8.52076 0.353500
$$582$$ −3.54299 −0.146862
$$583$$ −19.8677 −0.822838
$$584$$ 16.3204 0.675342
$$585$$ −3.21575 −0.132955
$$586$$ −22.8960 −0.945823
$$587$$ 14.2560 0.588408 0.294204 0.955743i $$-0.404946\pi$$
0.294204 + 0.955743i $$0.404946\pi$$
$$588$$ 2.11161 0.0870815
$$589$$ 0 0
$$590$$ −10.3823 −0.427431
$$591$$ −30.7723 −1.26580
$$592$$ 5.34592 0.219716
$$593$$ 26.0203 1.06852 0.534262 0.845319i $$-0.320589\pi$$
0.534262 + 0.845319i $$0.320589\pi$$
$$594$$ 10.5533 0.433006
$$595$$ −16.8640 −0.691356
$$596$$ 21.4072 0.876874
$$597$$ 14.9058 0.610056
$$598$$ −2.78170 −0.113752
$$599$$ −26.0530 −1.06450 −0.532248 0.846589i $$-0.678653\pi$$
−0.532248 + 0.846589i $$0.678653\pi$$
$$600$$ −16.6608 −0.680175
$$601$$ 26.5798 1.08421 0.542106 0.840310i $$-0.317627\pi$$
0.542106 + 0.840310i $$0.317627\pi$$
$$602$$ 5.44103 0.221760
$$603$$ −9.35392 −0.380921
$$604$$ 5.52022 0.224615
$$605$$ 1.73438 0.0705127
$$606$$ −3.94564 −0.160281
$$607$$ 10.3567 0.420367 0.210184 0.977662i $$-0.432594\pi$$
0.210184 + 0.977662i $$0.432594\pi$$
$$608$$ 0 0
$$609$$ −9.39718 −0.380793
$$610$$ −47.4881 −1.92274
$$611$$ 7.12270 0.288153
$$612$$ −6.85268 −0.277003
$$613$$ −13.1694 −0.531908 −0.265954 0.963986i $$-0.585687\pi$$
−0.265954 + 0.963986i $$0.585687\pi$$
$$614$$ 32.8057 1.32393
$$615$$ 8.23813 0.332193
$$616$$ 3.24298 0.130663
$$617$$ 6.28250 0.252924 0.126462 0.991971i $$-0.459638\pi$$
0.126462 + 0.991971i $$0.459638\pi$$
$$618$$ 42.4989 1.70956
$$619$$ −6.86393 −0.275885 −0.137942 0.990440i $$-0.544049\pi$$
−0.137942 + 0.990440i $$0.544049\pi$$
$$620$$ 23.2986 0.935695
$$621$$ −14.7444 −0.591671
$$622$$ −13.6967 −0.549190
$$623$$ −1.49287 −0.0598107
$$624$$ 1.29641 0.0518978
$$625$$ −2.19696 −0.0878784
$$626$$ −7.88847 −0.315287
$$627$$ 0 0
$$628$$ −13.5838 −0.542052
$$629$$ −25.1105 −1.00122
$$630$$ −5.23789 −0.208682
$$631$$ −34.6486 −1.37934 −0.689669 0.724125i $$-0.742244\pi$$
−0.689669 + 0.724125i $$0.742244\pi$$
$$632$$ 6.43324 0.255901
$$633$$ −12.0268 −0.478024
$$634$$ −28.0164 −1.11267
$$635$$ −65.5917 −2.60293
$$636$$ −12.9365 −0.512968
$$637$$ 0.613941 0.0243252
$$638$$ −14.4320 −0.571370
$$639$$ 16.8931 0.668280
$$640$$ 3.59028 0.141918
$$641$$ −2.33296 −0.0921463 −0.0460732 0.998938i $$-0.514671\pi$$
−0.0460732 + 0.998938i $$0.514671\pi$$
$$642$$ −19.7946 −0.781229
$$643$$ −42.1217 −1.66112 −0.830559 0.556931i $$-0.811979\pi$$
−0.830559 + 0.556931i $$0.811979\pi$$
$$644$$ −4.53089 −0.178542
$$645$$ −41.2500 −1.62422
$$646$$ 0 0
$$647$$ 18.7499 0.737133 0.368566 0.929601i $$-0.379849\pi$$
0.368566 + 0.929601i $$0.379849\pi$$
$$648$$ 11.2483 0.441875
$$649$$ −9.37797 −0.368117
$$650$$ −4.84405 −0.189999
$$651$$ 13.7030 0.537064
$$652$$ −6.32223 −0.247598
$$653$$ 12.8937 0.504569 0.252285 0.967653i $$-0.418818\pi$$
0.252285 + 0.967653i $$0.418818\pi$$
$$654$$ 19.5975 0.766321
$$655$$ 29.0507 1.13510
$$656$$ −1.08664 −0.0424262
$$657$$ −23.8099 −0.928914
$$658$$ 11.6016 0.452277
$$659$$ 18.6966 0.728318 0.364159 0.931337i $$-0.381357\pi$$
0.364159 + 0.931337i $$0.381357\pi$$
$$660$$ −24.5859 −0.957006
$$661$$ 34.7384 1.35117 0.675584 0.737283i $$-0.263892\pi$$
0.675584 + 0.737283i $$0.263892\pi$$
$$662$$ 13.6501 0.530526
$$663$$ −6.08938 −0.236492
$$664$$ 8.52076 0.330669
$$665$$ 0 0
$$666$$ −7.79921 −0.302213
$$667$$ 20.1635 0.780735
$$668$$ −14.1989 −0.549373
$$669$$ 39.2414 1.51716
$$670$$ −23.0194 −0.889315
$$671$$ −42.8945 −1.65592
$$672$$ 2.11161 0.0814572
$$673$$ −32.2228 −1.24210 −0.621049 0.783771i $$-0.713293\pi$$
−0.621049 + 0.783771i $$0.713293\pi$$
$$674$$ 17.2621 0.664910
$$675$$ −25.6758 −0.988263
$$676$$ −12.6231 −0.485503
$$677$$ −25.6438 −0.985570 −0.492785 0.870151i $$-0.664021\pi$$
−0.492785 + 0.870151i $$0.664021\pi$$
$$678$$ 18.3138 0.703337
$$679$$ −1.67786 −0.0643903
$$680$$ −16.8640 −0.646704
$$681$$ 1.11373 0.0426784
$$682$$ 21.0449 0.805850
$$683$$ −3.96087 −0.151558 −0.0757792 0.997125i $$-0.524144\pi$$
−0.0757792 + 0.997125i $$0.524144\pi$$
$$684$$ 0 0
$$685$$ −16.3531 −0.624819
$$686$$ 1.00000 0.0381802
$$687$$ −25.1938 −0.961202
$$688$$ 5.44103 0.207437
$$689$$ −3.76124 −0.143292
$$690$$ 34.3499 1.30768
$$691$$ 16.9238 0.643811 0.321905 0.946772i $$-0.395677\pi$$
0.321905 + 0.946772i $$0.395677\pi$$
$$692$$ 7.38609 0.280777
$$693$$ −4.73121 −0.179724
$$694$$ 2.69423 0.102272
$$695$$ 77.8892 2.95450
$$696$$ −9.39718 −0.356199
$$697$$ 5.10409 0.193331
$$698$$ 5.06542 0.191729
$$699$$ −12.6736 −0.479360
$$700$$ −7.89009 −0.298217
$$701$$ 16.8434 0.636166 0.318083 0.948063i $$-0.396961\pi$$
0.318083 + 0.948063i $$0.396961\pi$$
$$702$$ 1.99788 0.0754051
$$703$$ 0 0
$$704$$ 3.24298 0.122224
$$705$$ −87.9549 −3.31257
$$706$$ 21.7080 0.816992
$$707$$ −1.86854 −0.0702739
$$708$$ −6.10631 −0.229489
$$709$$ −28.7267 −1.07885 −0.539426 0.842033i $$-0.681359\pi$$
−0.539426 + 0.842033i $$0.681359\pi$$
$$710$$ 41.5728 1.56020
$$711$$ −9.38552 −0.351984
$$712$$ −1.49287 −0.0559477
$$713$$ −29.4026 −1.10114
$$714$$ −9.91851 −0.371191
$$715$$ −7.14823 −0.267329
$$716$$ −6.37886 −0.238389
$$717$$ −46.9698 −1.75412
$$718$$ −5.02083 −0.187376
$$719$$ −32.2712 −1.20351 −0.601757 0.798679i $$-0.705532\pi$$
−0.601757 + 0.798679i $$0.705532\pi$$
$$720$$ −5.23789 −0.195205
$$721$$ 20.1263 0.749541
$$722$$ 0 0
$$723$$ 41.8241 1.55546
$$724$$ 22.2731 0.827774
$$725$$ 35.1128 1.30406
$$726$$ 1.02007 0.0378585
$$727$$ 6.48738 0.240604 0.120302 0.992737i $$-0.461614\pi$$
0.120302 + 0.992737i $$0.461614\pi$$
$$728$$ 0.613941 0.0227542
$$729$$ 4.20448 0.155722
$$730$$ −58.5946 −2.16868
$$731$$ −25.5572 −0.945268
$$732$$ −27.9300 −1.03232
$$733$$ −15.0737 −0.556758 −0.278379 0.960471i $$-0.589797\pi$$
−0.278379 + 0.960471i $$0.589797\pi$$
$$734$$ 7.51570 0.277410
$$735$$ −7.58127 −0.279640
$$736$$ −4.53089 −0.167011
$$737$$ −20.7926 −0.765906
$$738$$ 1.58531 0.0583561
$$739$$ −21.8502 −0.803772 −0.401886 0.915690i $$-0.631645\pi$$
−0.401886 + 0.915690i $$0.631645\pi$$
$$740$$ −19.1933 −0.705561
$$741$$ 0 0
$$742$$ −6.12638 −0.224907
$$743$$ 41.7004 1.52984 0.764919 0.644127i $$-0.222779\pi$$
0.764919 + 0.644127i $$0.222779\pi$$
$$744$$ 13.7030 0.502377
$$745$$ −76.8578 −2.81585
$$746$$ −14.0820 −0.515578
$$747$$ −12.4310 −0.454827
$$748$$ −15.2327 −0.556962
$$749$$ −9.37414 −0.342524
$$750$$ 21.9105 0.800060
$$751$$ −36.4048 −1.32843 −0.664215 0.747542i $$-0.731234\pi$$
−0.664215 + 0.747542i $$0.731234\pi$$
$$752$$ 11.6016 0.423067
$$753$$ 13.2603 0.483230
$$754$$ −2.73218 −0.0995002
$$755$$ −19.8191 −0.721292
$$756$$ 3.25419 0.118354
$$757$$ 4.98234 0.181086 0.0905431 0.995893i $$-0.471140\pi$$
0.0905431 + 0.995893i $$0.471140\pi$$
$$758$$ −6.46372 −0.234773
$$759$$ 31.0272 1.12621
$$760$$ 0 0
$$761$$ 16.1769 0.586412 0.293206 0.956049i $$-0.405278\pi$$
0.293206 + 0.956049i $$0.405278\pi$$
$$762$$ −38.5776 −1.39752
$$763$$ 9.28080 0.335988
$$764$$ −23.4129 −0.847048
$$765$$ 24.6030 0.889524
$$766$$ 18.5181 0.669085
$$767$$ −1.77538 −0.0641052
$$768$$ 2.11161 0.0761963
$$769$$ 29.6375 1.06876 0.534378 0.845246i $$-0.320546\pi$$
0.534378 + 0.845246i $$0.320546\pi$$
$$770$$ −11.6432 −0.419592
$$771$$ −24.2749 −0.874241
$$772$$ −15.1577 −0.545537
$$773$$ 35.2879 1.26922 0.634608 0.772834i $$-0.281161\pi$$
0.634608 + 0.772834i $$0.281161\pi$$
$$774$$ −7.93797 −0.285325
$$775$$ −51.2017 −1.83922
$$776$$ −1.67786 −0.0602316
$$777$$ −11.2885 −0.404973
$$778$$ 37.2070 1.33393
$$779$$ 0 0
$$780$$ −4.65445 −0.166656
$$781$$ 37.5513 1.34369
$$782$$ 21.2822 0.761048
$$783$$ −14.4819 −0.517541
$$784$$ 1.00000 0.0357143
$$785$$ 48.7695 1.74066
$$786$$ 17.0861 0.609441
$$787$$ 48.1868 1.71767 0.858836 0.512250i $$-0.171188\pi$$
0.858836 + 0.512250i $$0.171188\pi$$
$$788$$ −14.5729 −0.519138
$$789$$ 8.53691 0.303922
$$790$$ −23.0971 −0.821759
$$791$$ 8.67289 0.308372
$$792$$ −4.73121 −0.168116
$$793$$ −8.12052 −0.288368
$$794$$ −34.3856 −1.22030
$$795$$ 46.4458 1.64726
$$796$$ 7.05899 0.250199
$$797$$ 10.0767 0.356934 0.178467 0.983946i $$-0.442886\pi$$
0.178467 + 0.983946i $$0.442886\pi$$
$$798$$ 0 0
$$799$$ −54.4942 −1.92787
$$800$$ −7.89009 −0.278957
$$801$$ 2.17797 0.0769546
$$802$$ −24.7614 −0.874356
$$803$$ −52.9266 −1.86774
$$804$$ −13.5388 −0.477476
$$805$$ 16.2671 0.573342
$$806$$ 3.98409 0.140333
$$807$$ −32.2027 −1.13359
$$808$$ −1.86854 −0.0657352
$$809$$ 37.6152 1.32248 0.661240 0.750174i $$-0.270030\pi$$
0.661240 + 0.750174i $$0.270030\pi$$
$$810$$ −40.3846 −1.41897
$$811$$ −1.02897 −0.0361321 −0.0180661 0.999837i $$-0.505751\pi$$
−0.0180661 + 0.999837i $$0.505751\pi$$
$$812$$ −4.45024 −0.156173
$$813$$ −37.8504 −1.32747
$$814$$ −17.3367 −0.607652
$$815$$ 22.6986 0.795096
$$816$$ −9.91851 −0.347217
$$817$$ 0 0
$$818$$ −8.46451 −0.295954
$$819$$ −0.895684 −0.0312977
$$820$$ 3.90134 0.136241
$$821$$ −29.6164 −1.03362 −0.516809 0.856101i $$-0.672880\pi$$
−0.516809 + 0.856101i $$0.672880\pi$$
$$822$$ −9.61802 −0.335467
$$823$$ −29.6541 −1.03368 −0.516839 0.856083i $$-0.672891\pi$$
−0.516839 + 0.856083i $$0.672891\pi$$
$$824$$ 20.1263 0.701132
$$825$$ 54.0307 1.88111
$$826$$ −2.89177 −0.100618
$$827$$ 5.98622 0.208161 0.104081 0.994569i $$-0.466810\pi$$
0.104081 + 0.994569i $$0.466810\pi$$
$$828$$ 6.61016 0.229719
$$829$$ 3.97321 0.137995 0.0689976 0.997617i $$-0.478020\pi$$
0.0689976 + 0.997617i $$0.478020\pi$$
$$830$$ −30.5919 −1.06186
$$831$$ −7.56312 −0.262362
$$832$$ 0.613941 0.0212846
$$833$$ −4.69713 −0.162746
$$834$$ 45.8103 1.58628
$$835$$ 50.9781 1.76417
$$836$$ 0 0
$$837$$ 21.1176 0.729931
$$838$$ 10.9702 0.378959
$$839$$ 44.1324 1.52362 0.761810 0.647801i $$-0.224311\pi$$
0.761810 + 0.647801i $$0.224311\pi$$
$$840$$ −7.58127 −0.261579
$$841$$ −9.19539 −0.317082
$$842$$ −32.1234 −1.10705
$$843$$ −14.9747 −0.515756
$$844$$ −5.69557 −0.196050
$$845$$ 45.3203 1.55907
$$846$$ −16.9257 −0.581917
$$847$$ 0.483078 0.0165988
$$848$$ −6.12638 −0.210381
$$849$$ −29.7334 −1.02045
$$850$$ 37.0607 1.27117
$$851$$ 24.2218 0.830312
$$852$$ 24.4509 0.837675
$$853$$ 1.76629 0.0604767 0.0302384 0.999543i $$-0.490373\pi$$
0.0302384 + 0.999543i $$0.490373\pi$$
$$854$$ −13.2269 −0.452614
$$855$$ 0 0
$$856$$ −9.37414 −0.320402
$$857$$ −42.8844 −1.46490 −0.732451 0.680819i $$-0.761624\pi$$
−0.732451 + 0.680819i $$0.761624\pi$$
$$858$$ −4.20422 −0.143530
$$859$$ 3.86858 0.131994 0.0659971 0.997820i $$-0.478977\pi$$
0.0659971 + 0.997820i $$0.478977\pi$$
$$860$$ −19.5348 −0.666132
$$861$$ 2.29457 0.0781986
$$862$$ 7.26547 0.247463
$$863$$ 32.7259 1.11400 0.557001 0.830512i $$-0.311952\pi$$
0.557001 + 0.830512i $$0.311952\pi$$
$$864$$ 3.25419 0.110710
$$865$$ −26.5181 −0.901642
$$866$$ −18.2307 −0.619504
$$867$$ 10.6911 0.363089
$$868$$ 6.48937 0.220263
$$869$$ −20.8629 −0.707725
$$870$$ 33.7385 1.14384
$$871$$ −3.93633 −0.133378
$$872$$ 9.28080 0.314288
$$873$$ 2.44784 0.0828470
$$874$$ 0 0
$$875$$ 10.3762 0.350780
$$876$$ −34.4623 −1.16437
$$877$$ −11.6549 −0.393559 −0.196779 0.980448i $$-0.563048\pi$$
−0.196779 + 0.980448i $$0.563048\pi$$
$$878$$ 32.5406 1.09819
$$879$$ 48.3474 1.63072
$$880$$ −11.6432 −0.392492
$$881$$ 52.9550 1.78410 0.892050 0.451937i $$-0.149267\pi$$
0.892050 + 0.451937i $$0.149267\pi$$
$$882$$ −1.45891 −0.0491240
$$883$$ −0.838696 −0.0282244 −0.0141122 0.999900i $$-0.504492\pi$$
−0.0141122 + 0.999900i $$0.504492\pi$$
$$884$$ −2.88376 −0.0969913
$$885$$ 21.9233 0.736945
$$886$$ 0.0189181 0.000635565 0
$$887$$ 48.3051 1.62193 0.810963 0.585097i $$-0.198944\pi$$
0.810963 + 0.585097i $$0.198944\pi$$
$$888$$ −11.2885 −0.378818
$$889$$ −18.2693 −0.612731
$$890$$ 5.35982 0.179662
$$891$$ −36.4781 −1.22206
$$892$$ 18.5836 0.622226
$$893$$ 0 0
$$894$$ −45.2037 −1.51184
$$895$$ 22.9019 0.765525
$$896$$ 1.00000 0.0334077
$$897$$ 5.87387 0.196123
$$898$$ −4.85684 −0.162075
$$899$$ −28.8792 −0.963176
$$900$$ 11.5109 0.383697
$$901$$ 28.7764 0.958681
$$902$$ 3.52396 0.117335
$$903$$ −11.4894 −0.382342
$$904$$ 8.67289 0.288456
$$905$$ −79.9667 −2.65818
$$906$$ −11.6566 −0.387264
$$907$$ −39.8456 −1.32305 −0.661526 0.749922i $$-0.730091\pi$$
−0.661526 + 0.749922i $$0.730091\pi$$
$$908$$ 0.527433 0.0175035
$$909$$ 2.72604 0.0904170
$$910$$ −2.20422 −0.0730691
$$911$$ 33.9742 1.12562 0.562808 0.826587i $$-0.309721\pi$$
0.562808 + 0.826587i $$0.309721\pi$$
$$912$$ 0 0
$$913$$ −27.6326 −0.914507
$$914$$ −30.2842 −1.00171
$$915$$ 100.277 3.31504
$$916$$ −11.9310 −0.394213
$$917$$ 8.09149 0.267205
$$918$$ −15.2853 −0.504491
$$919$$ 23.7930 0.784858 0.392429 0.919782i $$-0.371635\pi$$
0.392429 + 0.919782i $$0.371635\pi$$
$$920$$ 16.2671 0.536312
$$921$$ −69.2730 −2.28262
$$922$$ −20.6428 −0.679835
$$923$$ 7.10898 0.233995
$$924$$ −6.84792 −0.225280
$$925$$ 42.1798 1.38686
$$926$$ −21.7134 −0.713547
$$927$$ −29.3624 −0.964387
$$928$$ −4.45024 −0.146086
$$929$$ −6.90591 −0.226575 −0.113288 0.993562i $$-0.536138\pi$$
−0.113288 + 0.993562i $$0.536138\pi$$
$$930$$ −49.1977 −1.61326
$$931$$ 0 0
$$932$$ −6.00187 −0.196598
$$933$$ 28.9222 0.946871
$$934$$ −18.9511 −0.620099
$$935$$ 54.6896 1.78854
$$936$$ −0.895684 −0.0292763
$$937$$ 0.663778 0.0216847 0.0108423 0.999941i $$-0.496549\pi$$
0.0108423 + 0.999941i $$0.496549\pi$$
$$938$$ −6.41158 −0.209346
$$939$$ 16.6574 0.543593
$$940$$ −41.6529 −1.35857
$$941$$ 1.97654 0.0644334 0.0322167 0.999481i $$-0.489743\pi$$
0.0322167 + 0.999481i $$0.489743\pi$$
$$942$$ 28.6837 0.934564
$$943$$ −4.92345 −0.160330
$$944$$ −2.89177 −0.0941192
$$945$$ −11.6834 −0.380062
$$946$$ −17.6452 −0.573694
$$947$$ 42.1283 1.36899 0.684493 0.729020i $$-0.260024\pi$$
0.684493 + 0.729020i $$0.260024\pi$$
$$948$$ −13.5845 −0.441205
$$949$$ −10.0197 −0.325255
$$950$$ 0 0
$$951$$ 59.1599 1.91839
$$952$$ −4.69713 −0.152235
$$953$$ 50.5277 1.63675 0.818376 0.574683i $$-0.194875\pi$$
0.818376 + 0.574683i $$0.194875\pi$$
$$954$$ 8.93784 0.289373
$$955$$ 84.0587 2.72008
$$956$$ −22.2435 −0.719408
$$957$$ 30.4749 0.985112
$$958$$ −20.3473 −0.657390
$$959$$ −4.55482 −0.147083
$$960$$ −7.58127 −0.244685
$$961$$ 11.1119 0.358447
$$962$$ −3.28208 −0.105818
$$963$$ 13.6760 0.440704
$$964$$ 19.8067 0.637932
$$965$$ 54.4203 1.75185
$$966$$ 9.56748 0.307829
$$967$$ −38.8196 −1.24835 −0.624176 0.781284i $$-0.714565\pi$$
−0.624176 + 0.781284i $$0.714565\pi$$
$$968$$ 0.483078 0.0155267
$$969$$ 0 0
$$970$$ 6.02398 0.193418
$$971$$ 1.81105 0.0581194 0.0290597 0.999578i $$-0.490749\pi$$
0.0290597 + 0.999578i $$0.490749\pi$$
$$972$$ −13.9895 −0.448714
$$973$$ 21.6945 0.695493
$$974$$ −0.582139 −0.0186530
$$975$$ 10.2288 0.327582
$$976$$ −13.2269 −0.423382
$$977$$ −19.1940 −0.614069 −0.307035 0.951698i $$-0.599337\pi$$
−0.307035 + 0.951698i $$0.599337\pi$$
$$978$$ 13.3501 0.426889
$$979$$ 4.84136 0.154730
$$980$$ −3.59028 −0.114687
$$981$$ −13.5398 −0.432294
$$982$$ 33.5615 1.07099
$$983$$ 4.19927 0.133936 0.0669680 0.997755i $$-0.478667\pi$$
0.0669680 + 0.997755i $$0.478667\pi$$
$$984$$ 2.29457 0.0731481
$$985$$ 52.3207 1.66708
$$986$$ 20.9033 0.665698
$$987$$ −24.4981 −0.779783
$$988$$ 0 0
$$989$$ 24.6527 0.783911
$$990$$ 16.9864 0.539862
$$991$$ 9.13292 0.290117 0.145058 0.989423i $$-0.453663\pi$$
0.145058 + 0.989423i $$0.453663\pi$$
$$992$$ 6.48937 0.206038
$$993$$ −28.8237 −0.914694
$$994$$ 11.5793 0.367272
$$995$$ −25.3437 −0.803450
$$996$$ −17.9925 −0.570115
$$997$$ −23.2862 −0.737482 −0.368741 0.929532i $$-0.620211\pi$$
−0.368741 + 0.929532i $$0.620211\pi$$
$$998$$ 7.71505 0.244216
$$999$$ −17.3966 −0.550405
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 5054.2.a.bb.1.5 6
19.9 even 9 266.2.u.b.43.2 12
19.17 even 9 266.2.u.b.99.2 yes 12
19.18 odd 2 5054.2.a.bc.1.2 6

By twisted newform
Twist Min Dim Char Parity Ord Type
266.2.u.b.43.2 12 19.9 even 9
266.2.u.b.99.2 yes 12 19.17 even 9
5054.2.a.bb.1.5 6 1.1 even 1 trivial
5054.2.a.bc.1.2 6 19.18 odd 2