Properties

Label 5290.2.a.j.1.1
Level $5290$
Weight $2$
Character 5290.1
Self dual yes
Analytic conductor $42.241$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 5290 = 2 \cdot 5 \cdot 23^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5290.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(42.2408626693\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{13}) \)
Defining polynomial: \(x^{2} - x - 3\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 230)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.30278\) of defining polynomial
Character \(\chi\) \(=\) 5290.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} -0.302776 q^{3} +1.00000 q^{4} -1.00000 q^{5} +0.302776 q^{6} -3.30278 q^{7} -1.00000 q^{8} -2.90833 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -0.302776 q^{3} +1.00000 q^{4} -1.00000 q^{5} +0.302776 q^{6} -3.30278 q^{7} -1.00000 q^{8} -2.90833 q^{9} +1.00000 q^{10} +1.69722 q^{11} -0.302776 q^{12} +3.30278 q^{13} +3.30278 q^{14} +0.302776 q^{15} +1.00000 q^{16} -6.90833 q^{17} +2.90833 q^{18} -5.90833 q^{19} -1.00000 q^{20} +1.00000 q^{21} -1.69722 q^{22} +0.302776 q^{24} +1.00000 q^{25} -3.30278 q^{26} +1.78890 q^{27} -3.30278 q^{28} -2.60555 q^{29} -0.302776 q^{30} -7.90833 q^{31} -1.00000 q^{32} -0.513878 q^{33} +6.90833 q^{34} +3.30278 q^{35} -2.90833 q^{36} -8.00000 q^{37} +5.90833 q^{38} -1.00000 q^{39} +1.00000 q^{40} +0.908327 q^{41} -1.00000 q^{42} +9.21110 q^{43} +1.69722 q^{44} +2.90833 q^{45} -2.60555 q^{47} -0.302776 q^{48} +3.90833 q^{49} -1.00000 q^{50} +2.09167 q^{51} +3.30278 q^{52} +11.2111 q^{53} -1.78890 q^{54} -1.69722 q^{55} +3.30278 q^{56} +1.78890 q^{57} +2.60555 q^{58} -3.39445 q^{59} +0.302776 q^{60} -11.5139 q^{61} +7.90833 q^{62} +9.60555 q^{63} +1.00000 q^{64} -3.30278 q^{65} +0.513878 q^{66} +4.00000 q^{67} -6.90833 q^{68} -3.30278 q^{70} -16.3028 q^{71} +2.90833 q^{72} -5.81665 q^{73} +8.00000 q^{74} -0.302776 q^{75} -5.90833 q^{76} -5.60555 q^{77} +1.00000 q^{78} +14.4222 q^{79} -1.00000 q^{80} +8.18335 q^{81} -0.908327 q^{82} -11.2111 q^{83} +1.00000 q^{84} +6.90833 q^{85} -9.21110 q^{86} +0.788897 q^{87} -1.69722 q^{88} -2.90833 q^{90} -10.9083 q^{91} +2.39445 q^{93} +2.60555 q^{94} +5.90833 q^{95} +0.302776 q^{96} -6.30278 q^{97} -3.90833 q^{98} -4.93608 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{2} + 3q^{3} + 2q^{4} - 2q^{5} - 3q^{6} - 3q^{7} - 2q^{8} + 5q^{9} + O(q^{10}) \) \( 2q - 2q^{2} + 3q^{3} + 2q^{4} - 2q^{5} - 3q^{6} - 3q^{7} - 2q^{8} + 5q^{9} + 2q^{10} + 7q^{11} + 3q^{12} + 3q^{13} + 3q^{14} - 3q^{15} + 2q^{16} - 3q^{17} - 5q^{18} - q^{19} - 2q^{20} + 2q^{21} - 7q^{22} - 3q^{24} + 2q^{25} - 3q^{26} + 18q^{27} - 3q^{28} + 2q^{29} + 3q^{30} - 5q^{31} - 2q^{32} + 17q^{33} + 3q^{34} + 3q^{35} + 5q^{36} - 16q^{37} + q^{38} - 2q^{39} + 2q^{40} - 9q^{41} - 2q^{42} + 4q^{43} + 7q^{44} - 5q^{45} + 2q^{47} + 3q^{48} - 3q^{49} - 2q^{50} + 15q^{51} + 3q^{52} + 8q^{53} - 18q^{54} - 7q^{55} + 3q^{56} + 18q^{57} - 2q^{58} - 14q^{59} - 3q^{60} - 5q^{61} + 5q^{62} + 12q^{63} + 2q^{64} - 3q^{65} - 17q^{66} + 8q^{67} - 3q^{68} - 3q^{70} - 29q^{71} - 5q^{72} + 10q^{73} + 16q^{74} + 3q^{75} - q^{76} - 4q^{77} + 2q^{78} - 2q^{80} + 38q^{81} + 9q^{82} - 8q^{83} + 2q^{84} + 3q^{85} - 4q^{86} + 16q^{87} - 7q^{88} + 5q^{90} - 11q^{91} + 12q^{93} - 2q^{94} + q^{95} - 3q^{96} - 9q^{97} + 3q^{98} + 37q^{99} + O(q^{100}) \)

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 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) −0.302776 −0.174808 −0.0874038 0.996173i \(-0.527857\pi\)
−0.0874038 + 0.996173i \(0.527857\pi\)
\(4\) 1.00000 0.500000
\(5\) −1.00000 −0.447214
\(6\) 0.302776 0.123608
\(7\) −3.30278 −1.24833 −0.624166 0.781292i \(-0.714561\pi\)
−0.624166 + 0.781292i \(0.714561\pi\)
\(8\) −1.00000 −0.353553
\(9\) −2.90833 −0.969442
\(10\) 1.00000 0.316228
\(11\) 1.69722 0.511732 0.255866 0.966712i \(-0.417639\pi\)
0.255866 + 0.966712i \(0.417639\pi\)
\(12\) −0.302776 −0.0874038
\(13\) 3.30278 0.916025 0.458013 0.888946i \(-0.348561\pi\)
0.458013 + 0.888946i \(0.348561\pi\)
\(14\) 3.30278 0.882704
\(15\) 0.302776 0.0781763
\(16\) 1.00000 0.250000
\(17\) −6.90833 −1.67552 −0.837758 0.546042i \(-0.816134\pi\)
−0.837758 + 0.546042i \(0.816134\pi\)
\(18\) 2.90833 0.685499
\(19\) −5.90833 −1.35546 −0.677732 0.735309i \(-0.737037\pi\)
−0.677732 + 0.735309i \(0.737037\pi\)
\(20\) −1.00000 −0.223607
\(21\) 1.00000 0.218218
\(22\) −1.69722 −0.361849
\(23\) 0 0
\(24\) 0.302776 0.0618038
\(25\) 1.00000 0.200000
\(26\) −3.30278 −0.647728
\(27\) 1.78890 0.344273
\(28\) −3.30278 −0.624166
\(29\) −2.60555 −0.483839 −0.241919 0.970296i \(-0.577777\pi\)
−0.241919 + 0.970296i \(0.577777\pi\)
\(30\) −0.302776 −0.0552790
\(31\) −7.90833 −1.42038 −0.710189 0.704011i \(-0.751390\pi\)
−0.710189 + 0.704011i \(0.751390\pi\)
\(32\) −1.00000 −0.176777
\(33\) −0.513878 −0.0894547
\(34\) 6.90833 1.18477
\(35\) 3.30278 0.558271
\(36\) −2.90833 −0.484721
\(37\) −8.00000 −1.31519 −0.657596 0.753371i \(-0.728427\pi\)
−0.657596 + 0.753371i \(0.728427\pi\)
\(38\) 5.90833 0.958457
\(39\) −1.00000 −0.160128
\(40\) 1.00000 0.158114
\(41\) 0.908327 0.141857 0.0709284 0.997481i \(-0.477404\pi\)
0.0709284 + 0.997481i \(0.477404\pi\)
\(42\) −1.00000 −0.154303
\(43\) 9.21110 1.40468 0.702340 0.711842i \(-0.252139\pi\)
0.702340 + 0.711842i \(0.252139\pi\)
\(44\) 1.69722 0.255866
\(45\) 2.90833 0.433548
\(46\) 0 0
\(47\) −2.60555 −0.380059 −0.190029 0.981778i \(-0.560858\pi\)
−0.190029 + 0.981778i \(0.560858\pi\)
\(48\) −0.302776 −0.0437019
\(49\) 3.90833 0.558332
\(50\) −1.00000 −0.141421
\(51\) 2.09167 0.292893
\(52\) 3.30278 0.458013
\(53\) 11.2111 1.53996 0.769982 0.638066i \(-0.220265\pi\)
0.769982 + 0.638066i \(0.220265\pi\)
\(54\) −1.78890 −0.243438
\(55\) −1.69722 −0.228854
\(56\) 3.30278 0.441352
\(57\) 1.78890 0.236945
\(58\) 2.60555 0.342126
\(59\) −3.39445 −0.441920 −0.220960 0.975283i \(-0.570919\pi\)
−0.220960 + 0.975283i \(0.570919\pi\)
\(60\) 0.302776 0.0390882
\(61\) −11.5139 −1.47420 −0.737101 0.675783i \(-0.763806\pi\)
−0.737101 + 0.675783i \(0.763806\pi\)
\(62\) 7.90833 1.00436
\(63\) 9.60555 1.21019
\(64\) 1.00000 0.125000
\(65\) −3.30278 −0.409659
\(66\) 0.513878 0.0632540
\(67\) 4.00000 0.488678 0.244339 0.969690i \(-0.421429\pi\)
0.244339 + 0.969690i \(0.421429\pi\)
\(68\) −6.90833 −0.837758
\(69\) 0 0
\(70\) −3.30278 −0.394757
\(71\) −16.3028 −1.93478 −0.967392 0.253285i \(-0.918489\pi\)
−0.967392 + 0.253285i \(0.918489\pi\)
\(72\) 2.90833 0.342750
\(73\) −5.81665 −0.680788 −0.340394 0.940283i \(-0.610560\pi\)
−0.340394 + 0.940283i \(0.610560\pi\)
\(74\) 8.00000 0.929981
\(75\) −0.302776 −0.0349615
\(76\) −5.90833 −0.677732
\(77\) −5.60555 −0.638812
\(78\) 1.00000 0.113228
\(79\) 14.4222 1.62262 0.811312 0.584613i \(-0.198754\pi\)
0.811312 + 0.584613i \(0.198754\pi\)
\(80\) −1.00000 −0.111803
\(81\) 8.18335 0.909261
\(82\) −0.908327 −0.100308
\(83\) −11.2111 −1.23058 −0.615289 0.788301i \(-0.710961\pi\)
−0.615289 + 0.788301i \(0.710961\pi\)
\(84\) 1.00000 0.109109
\(85\) 6.90833 0.749313
\(86\) −9.21110 −0.993259
\(87\) 0.788897 0.0845787
\(88\) −1.69722 −0.180925
\(89\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(90\) −2.90833 −0.306565
\(91\) −10.9083 −1.14350
\(92\) 0 0
\(93\) 2.39445 0.248293
\(94\) 2.60555 0.268742
\(95\) 5.90833 0.606182
\(96\) 0.302776 0.0309019
\(97\) −6.30278 −0.639950 −0.319975 0.947426i \(-0.603675\pi\)
−0.319975 + 0.947426i \(0.603675\pi\)
\(98\) −3.90833 −0.394801
\(99\) −4.93608 −0.496095
\(100\) 1.00000 0.100000
\(101\) 2.60555 0.259262 0.129631 0.991562i \(-0.458621\pi\)
0.129631 + 0.991562i \(0.458621\pi\)
\(102\) −2.09167 −0.207106
\(103\) −8.11943 −0.800031 −0.400016 0.916508i \(-0.630995\pi\)
−0.400016 + 0.916508i \(0.630995\pi\)
\(104\) −3.30278 −0.323864
\(105\) −1.00000 −0.0975900
\(106\) −11.2111 −1.08892
\(107\) 2.60555 0.251888 0.125944 0.992037i \(-0.459804\pi\)
0.125944 + 0.992037i \(0.459804\pi\)
\(108\) 1.78890 0.172137
\(109\) −1.48612 −0.142345 −0.0711723 0.997464i \(-0.522674\pi\)
−0.0711723 + 0.997464i \(0.522674\pi\)
\(110\) 1.69722 0.161824
\(111\) 2.42221 0.229906
\(112\) −3.30278 −0.312083
\(113\) 16.4222 1.54487 0.772436 0.635093i \(-0.219038\pi\)
0.772436 + 0.635093i \(0.219038\pi\)
\(114\) −1.78890 −0.167546
\(115\) 0 0
\(116\) −2.60555 −0.241919
\(117\) −9.60555 −0.888034
\(118\) 3.39445 0.312484
\(119\) 22.8167 2.09160
\(120\) −0.302776 −0.0276395
\(121\) −8.11943 −0.738130
\(122\) 11.5139 1.04242
\(123\) −0.275019 −0.0247977
\(124\) −7.90833 −0.710189
\(125\) −1.00000 −0.0894427
\(126\) −9.60555 −0.855731
\(127\) 9.81665 0.871087 0.435544 0.900168i \(-0.356556\pi\)
0.435544 + 0.900168i \(0.356556\pi\)
\(128\) −1.00000 −0.0883883
\(129\) −2.78890 −0.245549
\(130\) 3.30278 0.289673
\(131\) −11.2111 −0.979519 −0.489759 0.871858i \(-0.662915\pi\)
−0.489759 + 0.871858i \(0.662915\pi\)
\(132\) −0.513878 −0.0447274
\(133\) 19.5139 1.69207
\(134\) −4.00000 −0.345547
\(135\) −1.78890 −0.153964
\(136\) 6.90833 0.592384
\(137\) −3.90833 −0.333911 −0.166955 0.985964i \(-0.553394\pi\)
−0.166955 + 0.985964i \(0.553394\pi\)
\(138\) 0 0
\(139\) −12.6056 −1.06919 −0.534594 0.845109i \(-0.679536\pi\)
−0.534594 + 0.845109i \(0.679536\pi\)
\(140\) 3.30278 0.279135
\(141\) 0.788897 0.0664372
\(142\) 16.3028 1.36810
\(143\) 5.60555 0.468760
\(144\) −2.90833 −0.242361
\(145\) 2.60555 0.216379
\(146\) 5.81665 0.481390
\(147\) −1.18335 −0.0976007
\(148\) −8.00000 −0.657596
\(149\) −13.3028 −1.08981 −0.544903 0.838499i \(-0.683433\pi\)
−0.544903 + 0.838499i \(0.683433\pi\)
\(150\) 0.302776 0.0247215
\(151\) 8.90833 0.724949 0.362475 0.931994i \(-0.381932\pi\)
0.362475 + 0.931994i \(0.381932\pi\)
\(152\) 5.90833 0.479229
\(153\) 20.0917 1.62432
\(154\) 5.60555 0.451708
\(155\) 7.90833 0.635212
\(156\) −1.00000 −0.0800641
\(157\) 18.6056 1.48488 0.742442 0.669910i \(-0.233667\pi\)
0.742442 + 0.669910i \(0.233667\pi\)
\(158\) −14.4222 −1.14737
\(159\) −3.39445 −0.269197
\(160\) 1.00000 0.0790569
\(161\) 0 0
\(162\) −8.18335 −0.642944
\(163\) 9.30278 0.728650 0.364325 0.931272i \(-0.381300\pi\)
0.364325 + 0.931272i \(0.381300\pi\)
\(164\) 0.908327 0.0709284
\(165\) 0.513878 0.0400054
\(166\) 11.2111 0.870150
\(167\) 6.78890 0.525341 0.262670 0.964886i \(-0.415397\pi\)
0.262670 + 0.964886i \(0.415397\pi\)
\(168\) −1.00000 −0.0771517
\(169\) −2.09167 −0.160898
\(170\) −6.90833 −0.529844
\(171\) 17.1833 1.31404
\(172\) 9.21110 0.702340
\(173\) 19.6972 1.49755 0.748776 0.662823i \(-0.230642\pi\)
0.748776 + 0.662823i \(0.230642\pi\)
\(174\) −0.788897 −0.0598062
\(175\) −3.30278 −0.249666
\(176\) 1.69722 0.127933
\(177\) 1.02776 0.0772509
\(178\) 0 0
\(179\) 9.39445 0.702174 0.351087 0.936343i \(-0.385812\pi\)
0.351087 + 0.936343i \(0.385812\pi\)
\(180\) 2.90833 0.216774
\(181\) −17.1194 −1.27248 −0.636239 0.771492i \(-0.719511\pi\)
−0.636239 + 0.771492i \(0.719511\pi\)
\(182\) 10.9083 0.808579
\(183\) 3.48612 0.257702
\(184\) 0 0
\(185\) 8.00000 0.588172
\(186\) −2.39445 −0.175569
\(187\) −11.7250 −0.857416
\(188\) −2.60555 −0.190029
\(189\) −5.90833 −0.429768
\(190\) −5.90833 −0.428635
\(191\) 8.60555 0.622676 0.311338 0.950299i \(-0.399223\pi\)
0.311338 + 0.950299i \(0.399223\pi\)
\(192\) −0.302776 −0.0218509
\(193\) −17.8167 −1.28247 −0.641235 0.767344i \(-0.721578\pi\)
−0.641235 + 0.767344i \(0.721578\pi\)
\(194\) 6.30278 0.452513
\(195\) 1.00000 0.0716115
\(196\) 3.90833 0.279166
\(197\) 4.30278 0.306560 0.153280 0.988183i \(-0.451016\pi\)
0.153280 + 0.988183i \(0.451016\pi\)
\(198\) 4.93608 0.350792
\(199\) 20.4222 1.44769 0.723846 0.689962i \(-0.242373\pi\)
0.723846 + 0.689962i \(0.242373\pi\)
\(200\) −1.00000 −0.0707107
\(201\) −1.21110 −0.0854246
\(202\) −2.60555 −0.183326
\(203\) 8.60555 0.603991
\(204\) 2.09167 0.146446
\(205\) −0.908327 −0.0634403
\(206\) 8.11943 0.565707
\(207\) 0 0
\(208\) 3.30278 0.229006
\(209\) −10.0278 −0.693634
\(210\) 1.00000 0.0690066
\(211\) 7.21110 0.496433 0.248216 0.968705i \(-0.420156\pi\)
0.248216 + 0.968705i \(0.420156\pi\)
\(212\) 11.2111 0.769982
\(213\) 4.93608 0.338215
\(214\) −2.60555 −0.178112
\(215\) −9.21110 −0.628192
\(216\) −1.78890 −0.121719
\(217\) 26.1194 1.77310
\(218\) 1.48612 0.100653
\(219\) 1.76114 0.119007
\(220\) −1.69722 −0.114427
\(221\) −22.8167 −1.53481
\(222\) −2.42221 −0.162568
\(223\) −4.00000 −0.267860 −0.133930 0.990991i \(-0.542760\pi\)
−0.133930 + 0.990991i \(0.542760\pi\)
\(224\) 3.30278 0.220676
\(225\) −2.90833 −0.193888
\(226\) −16.4222 −1.09239
\(227\) 14.6056 0.969404 0.484702 0.874679i \(-0.338928\pi\)
0.484702 + 0.874679i \(0.338928\pi\)
\(228\) 1.78890 0.118473
\(229\) −2.00000 −0.132164 −0.0660819 0.997814i \(-0.521050\pi\)
−0.0660819 + 0.997814i \(0.521050\pi\)
\(230\) 0 0
\(231\) 1.69722 0.111669
\(232\) 2.60555 0.171063
\(233\) 25.8167 1.69131 0.845653 0.533734i \(-0.179212\pi\)
0.845653 + 0.533734i \(0.179212\pi\)
\(234\) 9.60555 0.627935
\(235\) 2.60555 0.169967
\(236\) −3.39445 −0.220960
\(237\) −4.36669 −0.283647
\(238\) −22.8167 −1.47898
\(239\) 5.21110 0.337078 0.168539 0.985695i \(-0.446095\pi\)
0.168539 + 0.985695i \(0.446095\pi\)
\(240\) 0.302776 0.0195441
\(241\) 14.4222 0.929016 0.464508 0.885569i \(-0.346231\pi\)
0.464508 + 0.885569i \(0.346231\pi\)
\(242\) 8.11943 0.521937
\(243\) −7.84441 −0.503219
\(244\) −11.5139 −0.737101
\(245\) −3.90833 −0.249694
\(246\) 0.275019 0.0175346
\(247\) −19.5139 −1.24164
\(248\) 7.90833 0.502179
\(249\) 3.39445 0.215114
\(250\) 1.00000 0.0632456
\(251\) −12.5139 −0.789869 −0.394934 0.918709i \(-0.629233\pi\)
−0.394934 + 0.918709i \(0.629233\pi\)
\(252\) 9.60555 0.605093
\(253\) 0 0
\(254\) −9.81665 −0.615952
\(255\) −2.09167 −0.130986
\(256\) 1.00000 0.0625000
\(257\) −1.81665 −0.113320 −0.0566599 0.998394i \(-0.518045\pi\)
−0.0566599 + 0.998394i \(0.518045\pi\)
\(258\) 2.78890 0.173629
\(259\) 26.4222 1.64180
\(260\) −3.30278 −0.204829
\(261\) 7.57779 0.469054
\(262\) 11.2111 0.692624
\(263\) −3.51388 −0.216675 −0.108338 0.994114i \(-0.534553\pi\)
−0.108338 + 0.994114i \(0.534553\pi\)
\(264\) 0.513878 0.0316270
\(265\) −11.2111 −0.688693
\(266\) −19.5139 −1.19647
\(267\) 0 0
\(268\) 4.00000 0.244339
\(269\) 4.18335 0.255063 0.127532 0.991835i \(-0.459295\pi\)
0.127532 + 0.991835i \(0.459295\pi\)
\(270\) 1.78890 0.108869
\(271\) −2.69722 −0.163845 −0.0819224 0.996639i \(-0.526106\pi\)
−0.0819224 + 0.996639i \(0.526106\pi\)
\(272\) −6.90833 −0.418879
\(273\) 3.30278 0.199893
\(274\) 3.90833 0.236111
\(275\) 1.69722 0.102346
\(276\) 0 0
\(277\) −27.2111 −1.63496 −0.817478 0.575959i \(-0.804629\pi\)
−0.817478 + 0.575959i \(0.804629\pi\)
\(278\) 12.6056 0.756031
\(279\) 23.0000 1.37697
\(280\) −3.30278 −0.197379
\(281\) 26.6056 1.58715 0.793577 0.608470i \(-0.208216\pi\)
0.793577 + 0.608470i \(0.208216\pi\)
\(282\) −0.788897 −0.0469782
\(283\) −2.00000 −0.118888 −0.0594438 0.998232i \(-0.518933\pi\)
−0.0594438 + 0.998232i \(0.518933\pi\)
\(284\) −16.3028 −0.967392
\(285\) −1.78890 −0.105965
\(286\) −5.60555 −0.331463
\(287\) −3.00000 −0.177084
\(288\) 2.90833 0.171375
\(289\) 30.7250 1.80735
\(290\) −2.60555 −0.153003
\(291\) 1.90833 0.111868
\(292\) −5.81665 −0.340394
\(293\) −23.2111 −1.35601 −0.678004 0.735059i \(-0.737155\pi\)
−0.678004 + 0.735059i \(0.737155\pi\)
\(294\) 1.18335 0.0690142
\(295\) 3.39445 0.197632
\(296\) 8.00000 0.464991
\(297\) 3.03616 0.176176
\(298\) 13.3028 0.770609
\(299\) 0 0
\(300\) −0.302776 −0.0174808
\(301\) −30.4222 −1.75351
\(302\) −8.90833 −0.512617
\(303\) −0.788897 −0.0453210
\(304\) −5.90833 −0.338866
\(305\) 11.5139 0.659283
\(306\) −20.0917 −1.14856
\(307\) −11.6972 −0.667596 −0.333798 0.942645i \(-0.608330\pi\)
−0.333798 + 0.942645i \(0.608330\pi\)
\(308\) −5.60555 −0.319406
\(309\) 2.45837 0.139852
\(310\) −7.90833 −0.449163
\(311\) 22.4222 1.27145 0.635723 0.771917i \(-0.280702\pi\)
0.635723 + 0.771917i \(0.280702\pi\)
\(312\) 1.00000 0.0566139
\(313\) −19.7250 −1.11492 −0.557461 0.830203i \(-0.688224\pi\)
−0.557461 + 0.830203i \(0.688224\pi\)
\(314\) −18.6056 −1.04997
\(315\) −9.60555 −0.541212
\(316\) 14.4222 0.811312
\(317\) 17.7250 0.995534 0.497767 0.867311i \(-0.334153\pi\)
0.497767 + 0.867311i \(0.334153\pi\)
\(318\) 3.39445 0.190351
\(319\) −4.42221 −0.247596
\(320\) −1.00000 −0.0559017
\(321\) −0.788897 −0.0440320
\(322\) 0 0
\(323\) 40.8167 2.27110
\(324\) 8.18335 0.454630
\(325\) 3.30278 0.183205
\(326\) −9.30278 −0.515233
\(327\) 0.449961 0.0248829
\(328\) −0.908327 −0.0501540
\(329\) 8.60555 0.474439
\(330\) −0.513878 −0.0282881
\(331\) 16.6056 0.912724 0.456362 0.889794i \(-0.349152\pi\)
0.456362 + 0.889794i \(0.349152\pi\)
\(332\) −11.2111 −0.615289
\(333\) 23.2666 1.27500
\(334\) −6.78890 −0.371472
\(335\) −4.00000 −0.218543
\(336\) 1.00000 0.0545545
\(337\) 22.5139 1.22641 0.613205 0.789924i \(-0.289880\pi\)
0.613205 + 0.789924i \(0.289880\pi\)
\(338\) 2.09167 0.113772
\(339\) −4.97224 −0.270055
\(340\) 6.90833 0.374657
\(341\) −13.4222 −0.726853
\(342\) −17.1833 −0.929169
\(343\) 10.2111 0.551348
\(344\) −9.21110 −0.496629
\(345\) 0 0
\(346\) −19.6972 −1.05893
\(347\) 28.5416 1.53220 0.766098 0.642724i \(-0.222196\pi\)
0.766098 + 0.642724i \(0.222196\pi\)
\(348\) 0.788897 0.0422893
\(349\) −27.2111 −1.45658 −0.728288 0.685271i \(-0.759684\pi\)
−0.728288 + 0.685271i \(0.759684\pi\)
\(350\) 3.30278 0.176541
\(351\) 5.90833 0.315363
\(352\) −1.69722 −0.0904624
\(353\) −10.4222 −0.554718 −0.277359 0.960766i \(-0.589459\pi\)
−0.277359 + 0.960766i \(0.589459\pi\)
\(354\) −1.02776 −0.0546246
\(355\) 16.3028 0.865261
\(356\) 0 0
\(357\) −6.90833 −0.365627
\(358\) −9.39445 −0.496512
\(359\) −11.2111 −0.591699 −0.295850 0.955235i \(-0.595603\pi\)
−0.295850 + 0.955235i \(0.595603\pi\)
\(360\) −2.90833 −0.153282
\(361\) 15.9083 0.837280
\(362\) 17.1194 0.899777
\(363\) 2.45837 0.129031
\(364\) −10.9083 −0.571752
\(365\) 5.81665 0.304458
\(366\) −3.48612 −0.182223
\(367\) −14.7889 −0.771974 −0.385987 0.922504i \(-0.626139\pi\)
−0.385987 + 0.922504i \(0.626139\pi\)
\(368\) 0 0
\(369\) −2.64171 −0.137522
\(370\) −8.00000 −0.415900
\(371\) −37.0278 −1.92239
\(372\) 2.39445 0.124146
\(373\) −4.60555 −0.238466 −0.119233 0.992866i \(-0.538044\pi\)
−0.119233 + 0.992866i \(0.538044\pi\)
\(374\) 11.7250 0.606284
\(375\) 0.302776 0.0156353
\(376\) 2.60555 0.134371
\(377\) −8.60555 −0.443208
\(378\) 5.90833 0.303892
\(379\) −14.9083 −0.765789 −0.382895 0.923792i \(-0.625073\pi\)
−0.382895 + 0.923792i \(0.625073\pi\)
\(380\) 5.90833 0.303091
\(381\) −2.97224 −0.152273
\(382\) −8.60555 −0.440298
\(383\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(384\) 0.302776 0.0154510
\(385\) 5.60555 0.285685
\(386\) 17.8167 0.906844
\(387\) −26.7889 −1.36176
\(388\) −6.30278 −0.319975
\(389\) 25.9361 1.31501 0.657506 0.753449i \(-0.271612\pi\)
0.657506 + 0.753449i \(0.271612\pi\)
\(390\) −1.00000 −0.0506370
\(391\) 0 0
\(392\) −3.90833 −0.197400
\(393\) 3.39445 0.171227
\(394\) −4.30278 −0.216771
\(395\) −14.4222 −0.725660
\(396\) −4.93608 −0.248048
\(397\) 10.7250 0.538271 0.269136 0.963102i \(-0.413262\pi\)
0.269136 + 0.963102i \(0.413262\pi\)
\(398\) −20.4222 −1.02367
\(399\) −5.90833 −0.295786
\(400\) 1.00000 0.0500000
\(401\) 8.60555 0.429741 0.214870 0.976643i \(-0.431067\pi\)
0.214870 + 0.976643i \(0.431067\pi\)
\(402\) 1.21110 0.0604043
\(403\) −26.1194 −1.30110
\(404\) 2.60555 0.129631
\(405\) −8.18335 −0.406634
\(406\) −8.60555 −0.427086
\(407\) −13.5778 −0.673026
\(408\) −2.09167 −0.103553
\(409\) −25.9083 −1.28108 −0.640542 0.767923i \(-0.721290\pi\)
−0.640542 + 0.767923i \(0.721290\pi\)
\(410\) 0.908327 0.0448591
\(411\) 1.18335 0.0583702
\(412\) −8.11943 −0.400016
\(413\) 11.2111 0.551662
\(414\) 0 0
\(415\) 11.2111 0.550331
\(416\) −3.30278 −0.161932
\(417\) 3.81665 0.186902
\(418\) 10.0278 0.490474
\(419\) −3.63331 −0.177499 −0.0887493 0.996054i \(-0.528287\pi\)
−0.0887493 + 0.996054i \(0.528287\pi\)
\(420\) −1.00000 −0.0487950
\(421\) −30.6972 −1.49609 −0.748046 0.663647i \(-0.769008\pi\)
−0.748046 + 0.663647i \(0.769008\pi\)
\(422\) −7.21110 −0.351031
\(423\) 7.57779 0.368445
\(424\) −11.2111 −0.544459
\(425\) −6.90833 −0.335103
\(426\) −4.93608 −0.239154
\(427\) 38.0278 1.84029
\(428\) 2.60555 0.125944
\(429\) −1.69722 −0.0819428
\(430\) 9.21110 0.444199
\(431\) 30.2389 1.45655 0.728277 0.685283i \(-0.240321\pi\)
0.728277 + 0.685283i \(0.240321\pi\)
\(432\) 1.78890 0.0860684
\(433\) 24.0917 1.15777 0.578886 0.815409i \(-0.303488\pi\)
0.578886 + 0.815409i \(0.303488\pi\)
\(434\) −26.1194 −1.25377
\(435\) −0.788897 −0.0378247
\(436\) −1.48612 −0.0711723
\(437\) 0 0
\(438\) −1.76114 −0.0841506
\(439\) −14.6972 −0.701460 −0.350730 0.936477i \(-0.614067\pi\)
−0.350730 + 0.936477i \(0.614067\pi\)
\(440\) 1.69722 0.0809120
\(441\) −11.3667 −0.541271
\(442\) 22.8167 1.08528
\(443\) 17.4861 0.830791 0.415395 0.909641i \(-0.363643\pi\)
0.415395 + 0.909641i \(0.363643\pi\)
\(444\) 2.42221 0.114953
\(445\) 0 0
\(446\) 4.00000 0.189405
\(447\) 4.02776 0.190506
\(448\) −3.30278 −0.156041
\(449\) −2.09167 −0.0987122 −0.0493561 0.998781i \(-0.515717\pi\)
−0.0493561 + 0.998781i \(0.515717\pi\)
\(450\) 2.90833 0.137100
\(451\) 1.54163 0.0725927
\(452\) 16.4222 0.772436
\(453\) −2.69722 −0.126727
\(454\) −14.6056 −0.685472
\(455\) 10.9083 0.511390
\(456\) −1.78890 −0.0837728
\(457\) 32.4222 1.51665 0.758323 0.651879i \(-0.226019\pi\)
0.758323 + 0.651879i \(0.226019\pi\)
\(458\) 2.00000 0.0934539
\(459\) −12.3583 −0.576836
\(460\) 0 0
\(461\) 10.1833 0.474286 0.237143 0.971475i \(-0.423789\pi\)
0.237143 + 0.971475i \(0.423789\pi\)
\(462\) −1.69722 −0.0789620
\(463\) 17.6333 0.819489 0.409745 0.912200i \(-0.365618\pi\)
0.409745 + 0.912200i \(0.365618\pi\)
\(464\) −2.60555 −0.120960
\(465\) −2.39445 −0.111040
\(466\) −25.8167 −1.19593
\(467\) 1.81665 0.0840647 0.0420324 0.999116i \(-0.486617\pi\)
0.0420324 + 0.999116i \(0.486617\pi\)
\(468\) −9.60555 −0.444017
\(469\) −13.2111 −0.610032
\(470\) −2.60555 −0.120185
\(471\) −5.63331 −0.259569
\(472\) 3.39445 0.156242
\(473\) 15.6333 0.718820
\(474\) 4.36669 0.200569
\(475\) −5.90833 −0.271093
\(476\) 22.8167 1.04580
\(477\) −32.6056 −1.49291
\(478\) −5.21110 −0.238350
\(479\) 30.0000 1.37073 0.685367 0.728197i \(-0.259642\pi\)
0.685367 + 0.728197i \(0.259642\pi\)
\(480\) −0.302776 −0.0138198
\(481\) −26.4222 −1.20475
\(482\) −14.4222 −0.656913
\(483\) 0 0
\(484\) −8.11943 −0.369065
\(485\) 6.30278 0.286194
\(486\) 7.84441 0.355830
\(487\) 9.81665 0.444835 0.222418 0.974952i \(-0.428605\pi\)
0.222418 + 0.974952i \(0.428605\pi\)
\(488\) 11.5139 0.521209
\(489\) −2.81665 −0.127373
\(490\) 3.90833 0.176560
\(491\) −4.18335 −0.188792 −0.0943959 0.995535i \(-0.530092\pi\)
−0.0943959 + 0.995535i \(0.530092\pi\)
\(492\) −0.275019 −0.0123988
\(493\) 18.0000 0.810679
\(494\) 19.5139 0.877971
\(495\) 4.93608 0.221860
\(496\) −7.90833 −0.355094
\(497\) 53.8444 2.41525
\(498\) −3.39445 −0.152109
\(499\) −31.6333 −1.41610 −0.708051 0.706162i \(-0.750425\pi\)
−0.708051 + 0.706162i \(0.750425\pi\)
\(500\) −1.00000 −0.0447214
\(501\) −2.05551 −0.0918335
\(502\) 12.5139 0.558522
\(503\) −29.7250 −1.32537 −0.662686 0.748898i \(-0.730583\pi\)
−0.662686 + 0.748898i \(0.730583\pi\)
\(504\) −9.60555 −0.427865
\(505\) −2.60555 −0.115946
\(506\) 0 0
\(507\) 0.633308 0.0281262
\(508\) 9.81665 0.435544
\(509\) −35.4500 −1.57129 −0.785646 0.618676i \(-0.787669\pi\)
−0.785646 + 0.618676i \(0.787669\pi\)
\(510\) 2.09167 0.0926208
\(511\) 19.2111 0.849849
\(512\) −1.00000 −0.0441942
\(513\) −10.5694 −0.466650
\(514\) 1.81665 0.0801292
\(515\) 8.11943 0.357785
\(516\) −2.78890 −0.122774
\(517\) −4.42221 −0.194488
\(518\) −26.4222 −1.16093
\(519\) −5.96384 −0.261784
\(520\) 3.30278 0.144836
\(521\) −6.00000 −0.262865 −0.131432 0.991325i \(-0.541958\pi\)
−0.131432 + 0.991325i \(0.541958\pi\)
\(522\) −7.57779 −0.331671
\(523\) 20.4222 0.893001 0.446500 0.894783i \(-0.352670\pi\)
0.446500 + 0.894783i \(0.352670\pi\)
\(524\) −11.2111 −0.489759
\(525\) 1.00000 0.0436436
\(526\) 3.51388 0.153212
\(527\) 54.6333 2.37986
\(528\) −0.513878 −0.0223637
\(529\) 0 0
\(530\) 11.2111 0.486979
\(531\) 9.87217 0.428416
\(532\) 19.5139 0.846034
\(533\) 3.00000 0.129944
\(534\) 0 0
\(535\) −2.60555 −0.112648
\(536\) −4.00000 −0.172774
\(537\) −2.84441 −0.122745
\(538\) −4.18335 −0.180357
\(539\) 6.63331 0.285717
\(540\) −1.78890 −0.0769819
\(541\) 28.8444 1.24012 0.620059 0.784555i \(-0.287109\pi\)
0.620059 + 0.784555i \(0.287109\pi\)
\(542\) 2.69722 0.115856
\(543\) 5.18335 0.222439
\(544\) 6.90833 0.296192
\(545\) 1.48612 0.0636585
\(546\) −3.30278 −0.141346
\(547\) −10.5139 −0.449541 −0.224770 0.974412i \(-0.572163\pi\)
−0.224770 + 0.974412i \(0.572163\pi\)
\(548\) −3.90833 −0.166955
\(549\) 33.4861 1.42915
\(550\) −1.69722 −0.0723699
\(551\) 15.3944 0.655826
\(552\) 0 0
\(553\) −47.6333 −2.02557
\(554\) 27.2111 1.15609
\(555\) −2.42221 −0.102817
\(556\) −12.6056 −0.534594
\(557\) −22.4222 −0.950059 −0.475030 0.879970i \(-0.657563\pi\)
−0.475030 + 0.879970i \(0.657563\pi\)
\(558\) −23.0000 −0.973668
\(559\) 30.4222 1.28672
\(560\) 3.30278 0.139568
\(561\) 3.55004 0.149883
\(562\) −26.6056 −1.12229
\(563\) 3.63331 0.153126 0.0765628 0.997065i \(-0.475605\pi\)
0.0765628 + 0.997065i \(0.475605\pi\)
\(564\) 0.788897 0.0332186
\(565\) −16.4222 −0.690887
\(566\) 2.00000 0.0840663
\(567\) −27.0278 −1.13506
\(568\) 16.3028 0.684049
\(569\) −28.4222 −1.19152 −0.595760 0.803162i \(-0.703149\pi\)
−0.595760 + 0.803162i \(0.703149\pi\)
\(570\) 1.78890 0.0749287
\(571\) 16.1194 0.674577 0.337289 0.941401i \(-0.390490\pi\)
0.337289 + 0.941401i \(0.390490\pi\)
\(572\) 5.60555 0.234380
\(573\) −2.60555 −0.108848
\(574\) 3.00000 0.125218
\(575\) 0 0
\(576\) −2.90833 −0.121180
\(577\) 2.00000 0.0832611 0.0416305 0.999133i \(-0.486745\pi\)
0.0416305 + 0.999133i \(0.486745\pi\)
\(578\) −30.7250 −1.27799
\(579\) 5.39445 0.224186
\(580\) 2.60555 0.108190
\(581\) 37.0278 1.53617
\(582\) −1.90833 −0.0791027
\(583\) 19.0278 0.788049
\(584\) 5.81665 0.240695
\(585\) 9.60555 0.397141
\(586\) 23.2111 0.958842
\(587\) 16.5416 0.682746 0.341373 0.939928i \(-0.389108\pi\)
0.341373 + 0.939928i \(0.389108\pi\)
\(588\) −1.18335 −0.0488004
\(589\) 46.7250 1.92527
\(590\) −3.39445 −0.139747
\(591\) −1.30278 −0.0535890
\(592\) −8.00000 −0.328798
\(593\) −1.81665 −0.0746010 −0.0373005 0.999304i \(-0.511876\pi\)
−0.0373005 + 0.999304i \(0.511876\pi\)
\(594\) −3.03616 −0.124575
\(595\) −22.8167 −0.935392
\(596\) −13.3028 −0.544903
\(597\) −6.18335 −0.253068
\(598\) 0 0
\(599\) −35.3305 −1.44357 −0.721783 0.692119i \(-0.756677\pi\)
−0.721783 + 0.692119i \(0.756677\pi\)
\(600\) 0.302776 0.0123608
\(601\) 42.9361 1.75140 0.875700 0.482856i \(-0.160401\pi\)
0.875700 + 0.482856i \(0.160401\pi\)
\(602\) 30.4222 1.23992
\(603\) −11.6333 −0.473745
\(604\) 8.90833 0.362475
\(605\) 8.11943 0.330102
\(606\) 0.788897 0.0320468
\(607\) 46.0555 1.86934 0.934668 0.355522i \(-0.115697\pi\)
0.934668 + 0.355522i \(0.115697\pi\)
\(608\) 5.90833 0.239614
\(609\) −2.60555 −0.105582
\(610\) −11.5139 −0.466183
\(611\) −8.60555 −0.348143
\(612\) 20.0917 0.812158
\(613\) −3.57779 −0.144506 −0.0722529 0.997386i \(-0.523019\pi\)
−0.0722529 + 0.997386i \(0.523019\pi\)
\(614\) 11.6972 0.472062
\(615\) 0.275019 0.0110898
\(616\) 5.60555 0.225854
\(617\) −18.9083 −0.761221 −0.380610 0.924736i \(-0.624286\pi\)
−0.380610 + 0.924736i \(0.624286\pi\)
\(618\) −2.45837 −0.0988900
\(619\) 12.3305 0.495606 0.247803 0.968810i \(-0.420291\pi\)
0.247803 + 0.968810i \(0.420291\pi\)
\(620\) 7.90833 0.317606
\(621\) 0 0
\(622\) −22.4222 −0.899049
\(623\) 0 0
\(624\) −1.00000 −0.0400320
\(625\) 1.00000 0.0400000
\(626\) 19.7250 0.788369
\(627\) 3.03616 0.121253
\(628\) 18.6056 0.742442
\(629\) 55.2666 2.20362
\(630\) 9.60555 0.382694
\(631\) −23.3944 −0.931318 −0.465659 0.884964i \(-0.654183\pi\)
−0.465659 + 0.884964i \(0.654183\pi\)
\(632\) −14.4222 −0.573685
\(633\) −2.18335 −0.0867802
\(634\) −17.7250 −0.703949
\(635\) −9.81665 −0.389562
\(636\) −3.39445 −0.134599
\(637\) 12.9083 0.511447
\(638\) 4.42221 0.175077
\(639\) 47.4138 1.87566
\(640\) 1.00000 0.0395285
\(641\) 36.0000 1.42191 0.710957 0.703235i \(-0.248262\pi\)
0.710957 + 0.703235i \(0.248262\pi\)
\(642\) 0.788897 0.0311353
\(643\) 34.2389 1.35025 0.675124 0.737704i \(-0.264090\pi\)
0.675124 + 0.737704i \(0.264090\pi\)
\(644\) 0 0
\(645\) 2.78890 0.109813
\(646\) −40.8167 −1.60591
\(647\) 26.8444 1.05536 0.527681 0.849442i \(-0.323062\pi\)
0.527681 + 0.849442i \(0.323062\pi\)
\(648\) −8.18335 −0.321472
\(649\) −5.76114 −0.226145
\(650\) −3.30278 −0.129546
\(651\) −7.90833 −0.309952
\(652\) 9.30278 0.364325
\(653\) 41.7250 1.63282 0.816412 0.577469i \(-0.195960\pi\)
0.816412 + 0.577469i \(0.195960\pi\)
\(654\) −0.449961 −0.0175949
\(655\) 11.2111 0.438054
\(656\) 0.908327 0.0354642
\(657\) 16.9167 0.659985
\(658\) −8.60555 −0.335479
\(659\) −15.6333 −0.608987 −0.304494 0.952514i \(-0.598487\pi\)
−0.304494 + 0.952514i \(0.598487\pi\)
\(660\) 0.513878 0.0200027
\(661\) 34.9083 1.35778 0.678888 0.734242i \(-0.262462\pi\)
0.678888 + 0.734242i \(0.262462\pi\)
\(662\) −16.6056 −0.645393
\(663\) 6.90833 0.268297
\(664\) 11.2111 0.435075
\(665\) −19.5139 −0.756716
\(666\) −23.2666 −0.901563
\(667\) 0 0
\(668\) 6.78890 0.262670
\(669\) 1.21110 0.0468239
\(670\) 4.00000 0.154533
\(671\) −19.5416 −0.754396
\(672\) −1.00000 −0.0385758
\(673\) −37.6333 −1.45066 −0.725329 0.688403i \(-0.758312\pi\)
−0.725329 + 0.688403i \(0.758312\pi\)
\(674\) −22.5139 −0.867202
\(675\) 1.78890 0.0688547
\(676\) −2.09167 −0.0804490
\(677\) −16.4222 −0.631157 −0.315578 0.948900i \(-0.602198\pi\)
−0.315578 + 0.948900i \(0.602198\pi\)
\(678\) 4.97224 0.190958
\(679\) 20.8167 0.798870
\(680\) −6.90833 −0.264922
\(681\) −4.42221 −0.169459
\(682\) 13.4222 0.513963
\(683\) −0.275019 −0.0105233 −0.00526166 0.999986i \(-0.501675\pi\)
−0.00526166 + 0.999986i \(0.501675\pi\)
\(684\) 17.1833 0.657022
\(685\) 3.90833 0.149329
\(686\) −10.2111 −0.389862
\(687\) 0.605551 0.0231032
\(688\) 9.21110 0.351170
\(689\) 37.0278 1.41065
\(690\) 0 0
\(691\) 51.8167 1.97120 0.985599 0.169098i \(-0.0540855\pi\)
0.985599 + 0.169098i \(0.0540855\pi\)
\(692\) 19.6972 0.748776
\(693\) 16.3028 0.619291
\(694\) −28.5416 −1.08343
\(695\) 12.6056 0.478156
\(696\) −0.788897 −0.0299031
\(697\) −6.27502 −0.237683
\(698\) 27.2111 1.02996
\(699\) −7.81665 −0.295653
\(700\) −3.30278 −0.124833
\(701\) −32.0917 −1.21209 −0.606043 0.795432i \(-0.707244\pi\)
−0.606043 + 0.795432i \(0.707244\pi\)
\(702\) −5.90833 −0.222995
\(703\) 47.2666 1.78269
\(704\) 1.69722 0.0639665
\(705\) −0.788897 −0.0297116
\(706\) 10.4222 0.392245
\(707\) −8.60555 −0.323645
\(708\) 1.02776 0.0386254
\(709\) 15.8806 0.596407 0.298204 0.954502i \(-0.403613\pi\)
0.298204 + 0.954502i \(0.403613\pi\)
\(710\) −16.3028 −0.611832
\(711\) −41.9445 −1.57304
\(712\) 0 0
\(713\) 0 0
\(714\) 6.90833 0.258538
\(715\) −5.60555 −0.209636
\(716\) 9.39445 0.351087
\(717\) −1.57779 −0.0589238
\(718\) 11.2111 0.418395
\(719\) 10.6972 0.398939 0.199470 0.979904i \(-0.436078\pi\)
0.199470 + 0.979904i \(0.436078\pi\)
\(720\) 2.90833 0.108387
\(721\) 26.8167 0.998704
\(722\) −15.9083 −0.592047
\(723\) −4.36669 −0.162399
\(724\) −17.1194 −0.636239
\(725\) −2.60555 −0.0967677
\(726\) −2.45837 −0.0912385
\(727\) −2.90833 −0.107864 −0.0539319 0.998545i \(-0.517175\pi\)
−0.0539319 + 0.998545i \(0.517175\pi\)
\(728\) 10.9083 0.404289
\(729\) −22.1749 −0.821294
\(730\) −5.81665 −0.215284
\(731\) −63.6333 −2.35356
\(732\) 3.48612 0.128851
\(733\) −29.6333 −1.09453 −0.547266 0.836959i \(-0.684331\pi\)
−0.547266 + 0.836959i \(0.684331\pi\)
\(734\) 14.7889 0.545868
\(735\) 1.18335 0.0436484
\(736\) 0 0
\(737\) 6.78890 0.250072
\(738\) 2.64171 0.0972427
\(739\) 35.6333 1.31079 0.655396 0.755285i \(-0.272502\pi\)
0.655396 + 0.755285i \(0.272502\pi\)
\(740\) 8.00000 0.294086
\(741\) 5.90833 0.217048
\(742\) 37.0278 1.35933
\(743\) 32.3305 1.18609 0.593046 0.805169i \(-0.297925\pi\)
0.593046 + 0.805169i \(0.297925\pi\)
\(744\) −2.39445 −0.0877847
\(745\) 13.3028 0.487376
\(746\) 4.60555 0.168621
\(747\) 32.6056 1.19297
\(748\) −11.7250 −0.428708
\(749\) −8.60555 −0.314440
\(750\) −0.302776 −0.0110558
\(751\) −21.8167 −0.796101 −0.398051 0.917364i \(-0.630313\pi\)
−0.398051 + 0.917364i \(0.630313\pi\)
\(752\) −2.60555 −0.0950147
\(753\) 3.78890 0.138075
\(754\) 8.60555 0.313396
\(755\) −8.90833 −0.324207
\(756\) −5.90833 −0.214884
\(757\) −13.2111 −0.480166 −0.240083 0.970752i \(-0.577175\pi\)
−0.240083 + 0.970752i \(0.577175\pi\)
\(758\) 14.9083 0.541495
\(759\) 0 0
\(760\) −5.90833 −0.214318
\(761\) −49.5416 −1.79588 −0.897941 0.440115i \(-0.854938\pi\)
−0.897941 + 0.440115i \(0.854938\pi\)
\(762\) 2.97224 0.107673
\(763\) 4.90833 0.177693
\(764\) 8.60555 0.311338
\(765\) −20.0917 −0.726416
\(766\) 0 0
\(767\) −11.2111 −0.404809
\(768\) −0.302776 −0.0109255
\(769\) −45.2666 −1.63236 −0.816178 0.577801i \(-0.803911\pi\)
−0.816178 + 0.577801i \(0.803911\pi\)
\(770\) −5.60555 −0.202010
\(771\) 0.550039 0.0198092
\(772\) −17.8167 −0.641235
\(773\) −12.0000 −0.431610 −0.215805 0.976436i \(-0.569238\pi\)
−0.215805 + 0.976436i \(0.569238\pi\)
\(774\) 26.7889 0.962907
\(775\) −7.90833 −0.284075
\(776\) 6.30278 0.226256
\(777\) −8.00000 −0.286998
\(778\) −25.9361 −0.929854
\(779\) −5.36669 −0.192282
\(780\) 1.00000 0.0358057
\(781\) −27.6695 −0.990091
\(782\) 0 0
\(783\) −4.66106 −0.166573
\(784\) 3.90833 0.139583
\(785\) −18.6056 −0.664061
\(786\) −3.39445 −0.121076
\(787\) −37.4500 −1.33495 −0.667473 0.744634i \(-0.732624\pi\)
−0.667473 + 0.744634i \(0.732624\pi\)
\(788\) 4.30278 0.153280
\(789\) 1.06392 0.0378764
\(790\) 14.4222 0.513119
\(791\) −54.2389 −1.92851
\(792\) 4.93608 0.175396
\(793\) −38.0278 −1.35041
\(794\) −10.7250 −0.380615
\(795\) 3.39445 0.120389
\(796\) 20.4222 0.723846
\(797\) −1.81665 −0.0643492 −0.0321746 0.999482i \(-0.510243\pi\)
−0.0321746 + 0.999482i \(0.510243\pi\)
\(798\) 5.90833 0.209153
\(799\) 18.0000 0.636794
\(800\) −1.00000 −0.0353553
\(801\) 0 0
\(802\) −8.60555 −0.303873
\(803\) −9.87217 −0.348381
\(804\) −1.21110 −0.0427123
\(805\) 0 0
\(806\) 26.1194 0.920018
\(807\) −1.26662 −0.0445870
\(808\) −2.60555 −0.0916630
\(809\) 50.7250 1.78340 0.891698 0.452631i \(-0.149515\pi\)
0.891698 + 0.452631i \(0.149515\pi\)
\(810\) 8.18335 0.287533
\(811\) −41.0278 −1.44068 −0.720340 0.693621i \(-0.756014\pi\)
−0.720340 + 0.693621i \(0.756014\pi\)
\(812\) 8.60555 0.301996
\(813\) 0.816654 0.0286413
\(814\) 13.5778 0.475901
\(815\) −9.30278 −0.325862
\(816\) 2.09167 0.0732232
\(817\) −54.4222 −1.90399
\(818\) 25.9083 0.905863
\(819\) 31.7250 1.10856
\(820\) −0.908327 −0.0317202
\(821\) −30.0000 −1.04701 −0.523504 0.852023i \(-0.675375\pi\)
−0.523504 + 0.852023i \(0.675375\pi\)
\(822\) −1.18335 −0.0412739
\(823\) −15.2111 −0.530226 −0.265113 0.964217i \(-0.585409\pi\)
−0.265113 + 0.964217i \(0.585409\pi\)
\(824\) 8.11943 0.282854
\(825\) −0.513878 −0.0178909
\(826\) −11.2111 −0.390084
\(827\) 29.4500 1.02408 0.512038 0.858963i \(-0.328891\pi\)
0.512038 + 0.858963i \(0.328891\pi\)
\(828\) 0 0
\(829\) 31.2111 1.08401 0.542003 0.840376i \(-0.317666\pi\)
0.542003 + 0.840376i \(0.317666\pi\)
\(830\) −11.2111 −0.389143
\(831\) 8.23886 0.285803
\(832\) 3.30278 0.114503
\(833\) −27.0000 −0.935495
\(834\) −3.81665 −0.132160
\(835\) −6.78890 −0.234939
\(836\) −10.0278 −0.346817
\(837\) −14.1472 −0.488998
\(838\) 3.63331 0.125511
\(839\) 43.8167 1.51272 0.756359 0.654156i \(-0.226976\pi\)
0.756359 + 0.654156i \(0.226976\pi\)
\(840\) 1.00000 0.0345033
\(841\) −22.2111 −0.765900
\(842\) 30.6972 1.05790
\(843\) −8.05551 −0.277447
\(844\) 7.21110 0.248216
\(845\) 2.09167 0.0719557
\(846\) −7.57779 −0.260530
\(847\) 26.8167 0.921431
\(848\) 11.2111 0.384991
\(849\) 0.605551 0.0207825
\(850\) 6.90833 0.236954
\(851\) 0 0
\(852\) 4.93608 0.169107
\(853\) −21.7250 −0.743849 −0.371925 0.928263i \(-0.621302\pi\)
−0.371925 + 0.928263i \(0.621302\pi\)
\(854\) −38.0278 −1.30128
\(855\) −17.1833 −0.587658
\(856\) −2.60555 −0.0890559
\(857\) 9.63331 0.329068 0.164534 0.986371i \(-0.447388\pi\)
0.164534 + 0.986371i \(0.447388\pi\)
\(858\) 1.69722 0.0579423
\(859\) −35.8167 −1.22205 −0.611024 0.791612i \(-0.709242\pi\)
−0.611024 + 0.791612i \(0.709242\pi\)
\(860\) −9.21110 −0.314096
\(861\) 0.908327 0.0309557
\(862\) −30.2389 −1.02994
\(863\) −41.4500 −1.41097 −0.705487 0.708723i \(-0.749271\pi\)
−0.705487 + 0.708723i \(0.749271\pi\)
\(864\) −1.78890 −0.0608595
\(865\) −19.6972 −0.669726
\(866\) −24.0917 −0.818668
\(867\) −9.30278 −0.315939
\(868\) 26.1194 0.886551
\(869\) 24.4777 0.830350
\(870\) 0.788897 0.0267461
\(871\) 13.2111 0.447641
\(872\) 1.48612 0.0503264
\(873\) 18.3305 0.620395
\(874\) 0 0
\(875\) 3.30278 0.111654
\(876\) 1.76114 0.0595034
\(877\) −48.1749 −1.62675 −0.813376 0.581738i \(-0.802373\pi\)
−0.813376 + 0.581738i \(0.802373\pi\)
\(878\) 14.6972 0.496007
\(879\) 7.02776 0.237040
\(880\) −1.69722 −0.0572134
\(881\) −55.2666 −1.86198 −0.930990 0.365045i \(-0.881054\pi\)
−0.930990 + 0.365045i \(0.881054\pi\)
\(882\) 11.3667 0.382736
\(883\) 8.27502 0.278477 0.139238 0.990259i \(-0.455535\pi\)
0.139238 + 0.990259i \(0.455535\pi\)
\(884\) −22.8167 −0.767407
\(885\) −1.02776 −0.0345477
\(886\) −17.4861 −0.587458
\(887\) 27.6333 0.927836 0.463918 0.885878i \(-0.346443\pi\)
0.463918 + 0.885878i \(0.346443\pi\)
\(888\) −2.42221 −0.0812839
\(889\) −32.4222 −1.08741
\(890\) 0 0
\(891\) 13.8890 0.465298
\(892\) −4.00000 −0.133930
\(893\) 15.3944 0.515156
\(894\) −4.02776 −0.134708
\(895\) −9.39445 −0.314022
\(896\) 3.30278 0.110338
\(897\) 0 0
\(898\) 2.09167 0.0698000
\(899\) 20.6056 0.687234
\(900\) −2.90833 −0.0969442
\(901\) −77.4500 −2.58023
\(902\) −1.54163 −0.0513308
\(903\) 9.21110 0.306526
\(904\) −16.4222 −0.546194
\(905\) 17.1194 0.569069
\(906\) 2.69722 0.0896093
\(907\) −48.6611 −1.61576 −0.807882 0.589344i \(-0.799386\pi\)
−0.807882 + 0.589344i \(0.799386\pi\)
\(908\) 14.6056 0.484702
\(909\) −7.57779 −0.251340
\(910\) −10.9083 −0.361608
\(911\) 4.18335 0.138600 0.0693002 0.997596i \(-0.477923\pi\)
0.0693002 + 0.997596i \(0.477923\pi\)
\(912\) 1.78890 0.0592363
\(913\) −19.0278 −0.629727
\(914\) −32.4222 −1.07243
\(915\) −3.48612 −0.115248
\(916\) −2.00000 −0.0660819
\(917\) 37.0278 1.22276
\(918\) 12.3583 0.407884
\(919\) −44.0000 −1.45143 −0.725713 0.687998i \(-0.758490\pi\)
−0.725713 + 0.687998i \(0.758490\pi\)
\(920\) 0 0
\(921\) 3.54163 0.116701
\(922\) −10.1833 −0.335371
\(923\) −53.8444 −1.77231
\(924\) 1.69722 0.0558346
\(925\) −8.00000 −0.263038
\(926\) −17.6333 −0.579466
\(927\) 23.6140 0.775584
\(928\) 2.60555 0.0855314
\(929\) −14.3667 −0.471356 −0.235678 0.971831i \(-0.575731\pi\)
−0.235678 + 0.971831i \(0.575731\pi\)
\(930\) 2.39445 0.0785171
\(931\) −23.0917 −0.756799
\(932\) 25.8167 0.845653
\(933\) −6.78890 −0.222259
\(934\) −1.81665 −0.0594427
\(935\) 11.7250 0.383448
\(936\) 9.60555 0.313967
\(937\) −37.9638 −1.24022 −0.620112 0.784513i \(-0.712913\pi\)
−0.620112 + 0.784513i \(0.712913\pi\)
\(938\) 13.2111 0.431358
\(939\) 5.97224 0.194897
\(940\) 2.60555 0.0849837
\(941\) −25.9361 −0.845492 −0.422746 0.906248i \(-0.638934\pi\)
−0.422746 + 0.906248i \(0.638934\pi\)
\(942\) 5.63331 0.183543
\(943\) 0 0
\(944\) −3.39445 −0.110480
\(945\) 5.90833 0.192198
\(946\) −15.6333 −0.508283
\(947\) −4.93608 −0.160401 −0.0802006 0.996779i \(-0.525556\pi\)
−0.0802006 + 0.996779i \(0.525556\pi\)
\(948\) −4.36669 −0.141824
\(949\) −19.2111 −0.623619
\(950\) 5.90833 0.191691
\(951\) −5.36669 −0.174027
\(952\) −22.8167 −0.739492
\(953\) 41.3305 1.33883 0.669414 0.742890i \(-0.266545\pi\)
0.669414 + 0.742890i \(0.266545\pi\)
\(954\) 32.6056 1.05564
\(955\) −8.60555 −0.278469
\(956\) 5.21110 0.168539
\(957\) 1.33894 0.0432817
\(958\) −30.0000 −0.969256
\(959\) 12.9083 0.416832
\(960\) 0.302776 0.00977204
\(961\) 31.5416 1.01747
\(962\) 26.4222 0.851886
\(963\) −7.57779 −0.244191
\(964\) 14.4222 0.464508
\(965\) 17.8167 0.573538
\(966\) 0 0
\(967\) −12.6056 −0.405367 −0.202684 0.979244i \(-0.564966\pi\)
−0.202684 + 0.979244i \(0.564966\pi\)
\(968\) 8.11943 0.260968
\(969\) −12.3583 −0.397005
\(970\) −6.30278 −0.202370
\(971\) 17.0917 0.548498 0.274249 0.961659i \(-0.411571\pi\)
0.274249 + 0.961659i \(0.411571\pi\)
\(972\) −7.84441 −0.251610
\(973\) 41.6333 1.33470
\(974\) −9.81665 −0.314546
\(975\) −1.00000 −0.0320256
\(976\) −11.5139 −0.368550
\(977\) −6.51388 −0.208397 −0.104199 0.994556i \(-0.533228\pi\)
−0.104199 + 0.994556i \(0.533228\pi\)
\(978\) 2.81665 0.0900667
\(979\) 0 0
\(980\) −3.90833 −0.124847
\(981\) 4.32213 0.137995
\(982\) 4.18335 0.133496
\(983\) −34.5416 −1.10171 −0.550854 0.834602i \(-0.685698\pi\)
−0.550854 + 0.834602i \(0.685698\pi\)
\(984\) 0.275019 0.00876729
\(985\) −4.30278 −0.137098
\(986\) −18.0000 −0.573237
\(987\) −2.60555 −0.0829356
\(988\) −19.5139 −0.620819
\(989\) 0 0
\(990\) −4.93608 −0.156879
\(991\) −15.3305 −0.486990 −0.243495 0.969902i \(-0.578294\pi\)
−0.243495 + 0.969902i \(0.578294\pi\)
\(992\) 7.90833 0.251090
\(993\) −5.02776 −0.159551
\(994\) −53.8444 −1.70784
\(995\) −20.4222 −0.647427
\(996\) 3.39445 0.107557
\(997\) −16.7889 −0.531710 −0.265855 0.964013i \(-0.585654\pi\)
−0.265855 + 0.964013i \(0.585654\pi\)
\(998\) 31.6333 1.00133
\(999\) −14.3112 −0.452786
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 5290.2.a.j.1.1 2
23.22 odd 2 230.2.a.b.1.1 2
69.68 even 2 2070.2.a.w.1.2 2
92.91 even 2 1840.2.a.j.1.2 2
115.22 even 4 1150.2.b.f.599.2 4
115.68 even 4 1150.2.b.f.599.3 4
115.114 odd 2 1150.2.a.m.1.2 2
184.45 odd 2 7360.2.a.bc.1.2 2
184.91 even 2 7360.2.a.bu.1.1 2
460.459 even 2 9200.2.a.ca.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
230.2.a.b.1.1 2 23.22 odd 2
1150.2.a.m.1.2 2 115.114 odd 2
1150.2.b.f.599.2 4 115.22 even 4
1150.2.b.f.599.3 4 115.68 even 4
1840.2.a.j.1.2 2 92.91 even 2
2070.2.a.w.1.2 2 69.68 even 2
5290.2.a.j.1.1 2 1.1 even 1 trivial
7360.2.a.bc.1.2 2 184.45 odd 2
7360.2.a.bu.1.1 2 184.91 even 2
9200.2.a.ca.1.1 2 460.459 even 2