Properties

Label 966.2.a.m.1.1
Level $966$
Weight $2$
Character 966.1
Self dual yes
Analytic conductor $7.714$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 966 = 2 \cdot 3 \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 966.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(7.71354883526\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{41}) \)
Defining polynomial: \(x^{2} - x - 10\)
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 \(3.70156\) of defining polynomial
Character \(\chi\) \(=\) 966.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -3.70156 q^{5} -1.00000 q^{6} -1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -3.70156 q^{5} -1.00000 q^{6} -1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +3.70156 q^{10} +1.00000 q^{12} +5.70156 q^{13} +1.00000 q^{14} -3.70156 q^{15} +1.00000 q^{16} -1.00000 q^{18} -5.40312 q^{19} -3.70156 q^{20} -1.00000 q^{21} -1.00000 q^{23} -1.00000 q^{24} +8.70156 q^{25} -5.70156 q^{26} +1.00000 q^{27} -1.00000 q^{28} +0.298438 q^{29} +3.70156 q^{30} +6.00000 q^{31} -1.00000 q^{32} +3.70156 q^{35} +1.00000 q^{36} +3.70156 q^{37} +5.40312 q^{38} +5.70156 q^{39} +3.70156 q^{40} +7.70156 q^{41} +1.00000 q^{42} +5.70156 q^{43} -3.70156 q^{45} +1.00000 q^{46} +3.70156 q^{47} +1.00000 q^{48} +1.00000 q^{49} -8.70156 q^{50} +5.70156 q^{52} +9.40312 q^{53} -1.00000 q^{54} +1.00000 q^{56} -5.40312 q^{57} -0.298438 q^{58} -0.596876 q^{59} -3.70156 q^{60} +10.0000 q^{61} -6.00000 q^{62} -1.00000 q^{63} +1.00000 q^{64} -21.1047 q^{65} +4.00000 q^{67} -1.00000 q^{69} -3.70156 q^{70} +7.40312 q^{71} -1.00000 q^{72} -5.40312 q^{73} -3.70156 q^{74} +8.70156 q^{75} -5.40312 q^{76} -5.70156 q^{78} -3.70156 q^{80} +1.00000 q^{81} -7.70156 q^{82} -2.59688 q^{83} -1.00000 q^{84} -5.70156 q^{86} +0.298438 q^{87} -15.4031 q^{89} +3.70156 q^{90} -5.70156 q^{91} -1.00000 q^{92} +6.00000 q^{93} -3.70156 q^{94} +20.0000 q^{95} -1.00000 q^{96} -1.10469 q^{97} -1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{2} + 2q^{3} + 2q^{4} - q^{5} - 2q^{6} - 2q^{7} - 2q^{8} + 2q^{9} + O(q^{10}) \) \( 2q - 2q^{2} + 2q^{3} + 2q^{4} - q^{5} - 2q^{6} - 2q^{7} - 2q^{8} + 2q^{9} + q^{10} + 2q^{12} + 5q^{13} + 2q^{14} - q^{15} + 2q^{16} - 2q^{18} + 2q^{19} - q^{20} - 2q^{21} - 2q^{23} - 2q^{24} + 11q^{25} - 5q^{26} + 2q^{27} - 2q^{28} + 7q^{29} + q^{30} + 12q^{31} - 2q^{32} + q^{35} + 2q^{36} + q^{37} - 2q^{38} + 5q^{39} + q^{40} + 9q^{41} + 2q^{42} + 5q^{43} - q^{45} + 2q^{46} + q^{47} + 2q^{48} + 2q^{49} - 11q^{50} + 5q^{52} + 6q^{53} - 2q^{54} + 2q^{56} + 2q^{57} - 7q^{58} - 14q^{59} - q^{60} + 20q^{61} - 12q^{62} - 2q^{63} + 2q^{64} - 23q^{65} + 8q^{67} - 2q^{69} - q^{70} + 2q^{71} - 2q^{72} + 2q^{73} - q^{74} + 11q^{75} + 2q^{76} - 5q^{78} - q^{80} + 2q^{81} - 9q^{82} - 18q^{83} - 2q^{84} - 5q^{86} + 7q^{87} - 18q^{89} + q^{90} - 5q^{91} - 2q^{92} + 12q^{93} - q^{94} + 40q^{95} - 2q^{96} + 17q^{97} - 2q^{98} + O(q^{100}) \)

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
\(3\) 1.00000 0.577350
\(4\) 1.00000 0.500000
\(5\) −3.70156 −1.65539 −0.827694 0.561179i \(-0.810348\pi\)
−0.827694 + 0.561179i \(0.810348\pi\)
\(6\) −1.00000 −0.408248
\(7\) −1.00000 −0.377964
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) 3.70156 1.17054
\(11\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) 1.00000 0.288675
\(13\) 5.70156 1.58133 0.790664 0.612250i \(-0.209735\pi\)
0.790664 + 0.612250i \(0.209735\pi\)
\(14\) 1.00000 0.267261
\(15\) −3.70156 −0.955739
\(16\) 1.00000 0.250000
\(17\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(18\) −1.00000 −0.235702
\(19\) −5.40312 −1.23956 −0.619781 0.784775i \(-0.712779\pi\)
−0.619781 + 0.784775i \(0.712779\pi\)
\(20\) −3.70156 −0.827694
\(21\) −1.00000 −0.218218
\(22\) 0 0
\(23\) −1.00000 −0.208514
\(24\) −1.00000 −0.204124
\(25\) 8.70156 1.74031
\(26\) −5.70156 −1.11817
\(27\) 1.00000 0.192450
\(28\) −1.00000 −0.188982
\(29\) 0.298438 0.0554185 0.0277093 0.999616i \(-0.491179\pi\)
0.0277093 + 0.999616i \(0.491179\pi\)
\(30\) 3.70156 0.675810
\(31\) 6.00000 1.07763 0.538816 0.842424i \(-0.318872\pi\)
0.538816 + 0.842424i \(0.318872\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0 0
\(34\) 0 0
\(35\) 3.70156 0.625678
\(36\) 1.00000 0.166667
\(37\) 3.70156 0.608533 0.304267 0.952587i \(-0.401589\pi\)
0.304267 + 0.952587i \(0.401589\pi\)
\(38\) 5.40312 0.876502
\(39\) 5.70156 0.912981
\(40\) 3.70156 0.585268
\(41\) 7.70156 1.20278 0.601391 0.798955i \(-0.294613\pi\)
0.601391 + 0.798955i \(0.294613\pi\)
\(42\) 1.00000 0.154303
\(43\) 5.70156 0.869480 0.434740 0.900556i \(-0.356840\pi\)
0.434740 + 0.900556i \(0.356840\pi\)
\(44\) 0 0
\(45\) −3.70156 −0.551796
\(46\) 1.00000 0.147442
\(47\) 3.70156 0.539928 0.269964 0.962870i \(-0.412988\pi\)
0.269964 + 0.962870i \(0.412988\pi\)
\(48\) 1.00000 0.144338
\(49\) 1.00000 0.142857
\(50\) −8.70156 −1.23059
\(51\) 0 0
\(52\) 5.70156 0.790664
\(53\) 9.40312 1.29162 0.645809 0.763499i \(-0.276520\pi\)
0.645809 + 0.763499i \(0.276520\pi\)
\(54\) −1.00000 −0.136083
\(55\) 0 0
\(56\) 1.00000 0.133631
\(57\) −5.40312 −0.715661
\(58\) −0.298438 −0.0391868
\(59\) −0.596876 −0.0777066 −0.0388533 0.999245i \(-0.512371\pi\)
−0.0388533 + 0.999245i \(0.512371\pi\)
\(60\) −3.70156 −0.477870
\(61\) 10.0000 1.28037 0.640184 0.768221i \(-0.278858\pi\)
0.640184 + 0.768221i \(0.278858\pi\)
\(62\) −6.00000 −0.762001
\(63\) −1.00000 −0.125988
\(64\) 1.00000 0.125000
\(65\) −21.1047 −2.61771
\(66\) 0 0
\(67\) 4.00000 0.488678 0.244339 0.969690i \(-0.421429\pi\)
0.244339 + 0.969690i \(0.421429\pi\)
\(68\) 0 0
\(69\) −1.00000 −0.120386
\(70\) −3.70156 −0.442421
\(71\) 7.40312 0.878589 0.439295 0.898343i \(-0.355228\pi\)
0.439295 + 0.898343i \(0.355228\pi\)
\(72\) −1.00000 −0.117851
\(73\) −5.40312 −0.632388 −0.316194 0.948695i \(-0.602405\pi\)
−0.316194 + 0.948695i \(0.602405\pi\)
\(74\) −3.70156 −0.430298
\(75\) 8.70156 1.00477
\(76\) −5.40312 −0.619781
\(77\) 0 0
\(78\) −5.70156 −0.645575
\(79\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(80\) −3.70156 −0.413847
\(81\) 1.00000 0.111111
\(82\) −7.70156 −0.850495
\(83\) −2.59688 −0.285044 −0.142522 0.989792i \(-0.545521\pi\)
−0.142522 + 0.989792i \(0.545521\pi\)
\(84\) −1.00000 −0.109109
\(85\) 0 0
\(86\) −5.70156 −0.614815
\(87\) 0.298438 0.0319959
\(88\) 0 0
\(89\) −15.4031 −1.63273 −0.816364 0.577538i \(-0.804014\pi\)
−0.816364 + 0.577538i \(0.804014\pi\)
\(90\) 3.70156 0.390179
\(91\) −5.70156 −0.597686
\(92\) −1.00000 −0.104257
\(93\) 6.00000 0.622171
\(94\) −3.70156 −0.381787
\(95\) 20.0000 2.05196
\(96\) −1.00000 −0.102062
\(97\) −1.10469 −0.112164 −0.0560820 0.998426i \(-0.517861\pi\)
−0.0560820 + 0.998426i \(0.517861\pi\)
\(98\) −1.00000 −0.101015
\(99\) 0 0
\(100\) 8.70156 0.870156
\(101\) 4.59688 0.457406 0.228703 0.973496i \(-0.426551\pi\)
0.228703 + 0.973496i \(0.426551\pi\)
\(102\) 0 0
\(103\) −17.1047 −1.68537 −0.842687 0.538403i \(-0.819028\pi\)
−0.842687 + 0.538403i \(0.819028\pi\)
\(104\) −5.70156 −0.559084
\(105\) 3.70156 0.361235
\(106\) −9.40312 −0.913312
\(107\) 14.8062 1.43137 0.715687 0.698421i \(-0.246114\pi\)
0.715687 + 0.698421i \(0.246114\pi\)
\(108\) 1.00000 0.0962250
\(109\) 15.7016 1.50394 0.751968 0.659199i \(-0.229105\pi\)
0.751968 + 0.659199i \(0.229105\pi\)
\(110\) 0 0
\(111\) 3.70156 0.351337
\(112\) −1.00000 −0.0944911
\(113\) −4.29844 −0.404363 −0.202182 0.979348i \(-0.564803\pi\)
−0.202182 + 0.979348i \(0.564803\pi\)
\(114\) 5.40312 0.506049
\(115\) 3.70156 0.345172
\(116\) 0.298438 0.0277093
\(117\) 5.70156 0.527110
\(118\) 0.596876 0.0549469
\(119\) 0 0
\(120\) 3.70156 0.337905
\(121\) −11.0000 −1.00000
\(122\) −10.0000 −0.905357
\(123\) 7.70156 0.694426
\(124\) 6.00000 0.538816
\(125\) −13.7016 −1.22550
\(126\) 1.00000 0.0890871
\(127\) 17.1047 1.51780 0.758898 0.651210i \(-0.225738\pi\)
0.758898 + 0.651210i \(0.225738\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 5.70156 0.501995
\(130\) 21.1047 1.85100
\(131\) 15.4031 1.34578 0.672889 0.739744i \(-0.265053\pi\)
0.672889 + 0.739744i \(0.265053\pi\)
\(132\) 0 0
\(133\) 5.40312 0.468510
\(134\) −4.00000 −0.345547
\(135\) −3.70156 −0.318580
\(136\) 0 0
\(137\) 0.895314 0.0764918 0.0382459 0.999268i \(-0.487823\pi\)
0.0382459 + 0.999268i \(0.487823\pi\)
\(138\) 1.00000 0.0851257
\(139\) −21.1047 −1.79008 −0.895038 0.445990i \(-0.852852\pi\)
−0.895038 + 0.445990i \(0.852852\pi\)
\(140\) 3.70156 0.312839
\(141\) 3.70156 0.311728
\(142\) −7.40312 −0.621256
\(143\) 0 0
\(144\) 1.00000 0.0833333
\(145\) −1.10469 −0.0917392
\(146\) 5.40312 0.447166
\(147\) 1.00000 0.0824786
\(148\) 3.70156 0.304267
\(149\) −17.4031 −1.42572 −0.712860 0.701307i \(-0.752600\pi\)
−0.712860 + 0.701307i \(0.752600\pi\)
\(150\) −8.70156 −0.710480
\(151\) 9.10469 0.740929 0.370464 0.928847i \(-0.379199\pi\)
0.370464 + 0.928847i \(0.379199\pi\)
\(152\) 5.40312 0.438251
\(153\) 0 0
\(154\) 0 0
\(155\) −22.2094 −1.78390
\(156\) 5.70156 0.456490
\(157\) −13.4031 −1.06969 −0.534843 0.844952i \(-0.679629\pi\)
−0.534843 + 0.844952i \(0.679629\pi\)
\(158\) 0 0
\(159\) 9.40312 0.745716
\(160\) 3.70156 0.292634
\(161\) 1.00000 0.0788110
\(162\) −1.00000 −0.0785674
\(163\) −12.0000 −0.939913 −0.469956 0.882690i \(-0.655730\pi\)
−0.469956 + 0.882690i \(0.655730\pi\)
\(164\) 7.70156 0.601391
\(165\) 0 0
\(166\) 2.59688 0.201557
\(167\) −21.4031 −1.65622 −0.828112 0.560563i \(-0.810585\pi\)
−0.828112 + 0.560563i \(0.810585\pi\)
\(168\) 1.00000 0.0771517
\(169\) 19.5078 1.50060
\(170\) 0 0
\(171\) −5.40312 −0.413187
\(172\) 5.70156 0.434740
\(173\) −3.40312 −0.258735 −0.129367 0.991597i \(-0.541295\pi\)
−0.129367 + 0.991597i \(0.541295\pi\)
\(174\) −0.298438 −0.0226245
\(175\) −8.70156 −0.657776
\(176\) 0 0
\(177\) −0.596876 −0.0448639
\(178\) 15.4031 1.15451
\(179\) 5.70156 0.426155 0.213077 0.977035i \(-0.431651\pi\)
0.213077 + 0.977035i \(0.431651\pi\)
\(180\) −3.70156 −0.275898
\(181\) 16.8062 1.24920 0.624599 0.780945i \(-0.285262\pi\)
0.624599 + 0.780945i \(0.285262\pi\)
\(182\) 5.70156 0.422628
\(183\) 10.0000 0.739221
\(184\) 1.00000 0.0737210
\(185\) −13.7016 −1.00736
\(186\) −6.00000 −0.439941
\(187\) 0 0
\(188\) 3.70156 0.269964
\(189\) −1.00000 −0.0727393
\(190\) −20.0000 −1.45095
\(191\) 22.8062 1.65020 0.825101 0.564985i \(-0.191118\pi\)
0.825101 + 0.564985i \(0.191118\pi\)
\(192\) 1.00000 0.0721688
\(193\) −7.10469 −0.511407 −0.255703 0.966755i \(-0.582307\pi\)
−0.255703 + 0.966755i \(0.582307\pi\)
\(194\) 1.10469 0.0793119
\(195\) −21.1047 −1.51134
\(196\) 1.00000 0.0714286
\(197\) 6.50781 0.463662 0.231831 0.972756i \(-0.425528\pi\)
0.231831 + 0.972756i \(0.425528\pi\)
\(198\) 0 0
\(199\) 1.10469 0.0783091 0.0391546 0.999233i \(-0.487534\pi\)
0.0391546 + 0.999233i \(0.487534\pi\)
\(200\) −8.70156 −0.615293
\(201\) 4.00000 0.282138
\(202\) −4.59688 −0.323435
\(203\) −0.298438 −0.0209462
\(204\) 0 0
\(205\) −28.5078 −1.99107
\(206\) 17.1047 1.19174
\(207\) −1.00000 −0.0695048
\(208\) 5.70156 0.395332
\(209\) 0 0
\(210\) −3.70156 −0.255432
\(211\) −12.0000 −0.826114 −0.413057 0.910705i \(-0.635539\pi\)
−0.413057 + 0.910705i \(0.635539\pi\)
\(212\) 9.40312 0.645809
\(213\) 7.40312 0.507254
\(214\) −14.8062 −1.01213
\(215\) −21.1047 −1.43933
\(216\) −1.00000 −0.0680414
\(217\) −6.00000 −0.407307
\(218\) −15.7016 −1.06344
\(219\) −5.40312 −0.365109
\(220\) 0 0
\(221\) 0 0
\(222\) −3.70156 −0.248433
\(223\) −20.8062 −1.39329 −0.696645 0.717416i \(-0.745325\pi\)
−0.696645 + 0.717416i \(0.745325\pi\)
\(224\) 1.00000 0.0668153
\(225\) 8.70156 0.580104
\(226\) 4.29844 0.285928
\(227\) −26.5078 −1.75939 −0.879693 0.475543i \(-0.842252\pi\)
−0.879693 + 0.475543i \(0.842252\pi\)
\(228\) −5.40312 −0.357831
\(229\) 28.2094 1.86413 0.932064 0.362294i \(-0.118006\pi\)
0.932064 + 0.362294i \(0.118006\pi\)
\(230\) −3.70156 −0.244074
\(231\) 0 0
\(232\) −0.298438 −0.0195934
\(233\) 28.8062 1.88716 0.943580 0.331145i \(-0.107435\pi\)
0.943580 + 0.331145i \(0.107435\pi\)
\(234\) −5.70156 −0.372723
\(235\) −13.7016 −0.893791
\(236\) −0.596876 −0.0388533
\(237\) 0 0
\(238\) 0 0
\(239\) −18.8062 −1.21648 −0.608238 0.793755i \(-0.708123\pi\)
−0.608238 + 0.793755i \(0.708123\pi\)
\(240\) −3.70156 −0.238935
\(241\) 5.10469 0.328822 0.164411 0.986392i \(-0.447428\pi\)
0.164411 + 0.986392i \(0.447428\pi\)
\(242\) 11.0000 0.707107
\(243\) 1.00000 0.0641500
\(244\) 10.0000 0.640184
\(245\) −3.70156 −0.236484
\(246\) −7.70156 −0.491034
\(247\) −30.8062 −1.96015
\(248\) −6.00000 −0.381000
\(249\) −2.59688 −0.164570
\(250\) 13.7016 0.866563
\(251\) 13.9109 0.878050 0.439025 0.898475i \(-0.355324\pi\)
0.439025 + 0.898475i \(0.355324\pi\)
\(252\) −1.00000 −0.0629941
\(253\) 0 0
\(254\) −17.1047 −1.07324
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 23.6125 1.47291 0.736454 0.676488i \(-0.236499\pi\)
0.736454 + 0.676488i \(0.236499\pi\)
\(258\) −5.70156 −0.354964
\(259\) −3.70156 −0.230004
\(260\) −21.1047 −1.30886
\(261\) 0.298438 0.0184728
\(262\) −15.4031 −0.951608
\(263\) −16.5078 −1.01792 −0.508958 0.860792i \(-0.669969\pi\)
−0.508958 + 0.860792i \(0.669969\pi\)
\(264\) 0 0
\(265\) −34.8062 −2.13813
\(266\) −5.40312 −0.331287
\(267\) −15.4031 −0.942656
\(268\) 4.00000 0.244339
\(269\) −3.40312 −0.207492 −0.103746 0.994604i \(-0.533083\pi\)
−0.103746 + 0.994604i \(0.533083\pi\)
\(270\) 3.70156 0.225270
\(271\) 17.4031 1.05716 0.528582 0.848882i \(-0.322724\pi\)
0.528582 + 0.848882i \(0.322724\pi\)
\(272\) 0 0
\(273\) −5.70156 −0.345074
\(274\) −0.895314 −0.0540879
\(275\) 0 0
\(276\) −1.00000 −0.0601929
\(277\) −13.4031 −0.805316 −0.402658 0.915351i \(-0.631914\pi\)
−0.402658 + 0.915351i \(0.631914\pi\)
\(278\) 21.1047 1.26577
\(279\) 6.00000 0.359211
\(280\) −3.70156 −0.221211
\(281\) 4.29844 0.256423 0.128212 0.991747i \(-0.459076\pi\)
0.128212 + 0.991747i \(0.459076\pi\)
\(282\) −3.70156 −0.220425
\(283\) −10.0000 −0.594438 −0.297219 0.954809i \(-0.596059\pi\)
−0.297219 + 0.954809i \(0.596059\pi\)
\(284\) 7.40312 0.439295
\(285\) 20.0000 1.18470
\(286\) 0 0
\(287\) −7.70156 −0.454609
\(288\) −1.00000 −0.0589256
\(289\) −17.0000 −1.00000
\(290\) 1.10469 0.0648694
\(291\) −1.10469 −0.0647579
\(292\) −5.40312 −0.316194
\(293\) −3.19375 −0.186581 −0.0932905 0.995639i \(-0.529739\pi\)
−0.0932905 + 0.995639i \(0.529739\pi\)
\(294\) −1.00000 −0.0583212
\(295\) 2.20937 0.128635
\(296\) −3.70156 −0.215149
\(297\) 0 0
\(298\) 17.4031 1.00814
\(299\) −5.70156 −0.329730
\(300\) 8.70156 0.502385
\(301\) −5.70156 −0.328633
\(302\) −9.10469 −0.523916
\(303\) 4.59688 0.264084
\(304\) −5.40312 −0.309890
\(305\) −37.0156 −2.11951
\(306\) 0 0
\(307\) 5.10469 0.291340 0.145670 0.989333i \(-0.453466\pi\)
0.145670 + 0.989333i \(0.453466\pi\)
\(308\) 0 0
\(309\) −17.1047 −0.973052
\(310\) 22.2094 1.26141
\(311\) −16.2094 −0.919149 −0.459575 0.888139i \(-0.651998\pi\)
−0.459575 + 0.888139i \(0.651998\pi\)
\(312\) −5.70156 −0.322787
\(313\) −4.59688 −0.259831 −0.129915 0.991525i \(-0.541471\pi\)
−0.129915 + 0.991525i \(0.541471\pi\)
\(314\) 13.4031 0.756382
\(315\) 3.70156 0.208559
\(316\) 0 0
\(317\) 7.70156 0.432563 0.216281 0.976331i \(-0.430607\pi\)
0.216281 + 0.976331i \(0.430607\pi\)
\(318\) −9.40312 −0.527301
\(319\) 0 0
\(320\) −3.70156 −0.206924
\(321\) 14.8062 0.826404
\(322\) −1.00000 −0.0557278
\(323\) 0 0
\(324\) 1.00000 0.0555556
\(325\) 49.6125 2.75201
\(326\) 12.0000 0.664619
\(327\) 15.7016 0.868298
\(328\) −7.70156 −0.425248
\(329\) −3.70156 −0.204074
\(330\) 0 0
\(331\) −17.6125 −0.968070 −0.484035 0.875049i \(-0.660829\pi\)
−0.484035 + 0.875049i \(0.660829\pi\)
\(332\) −2.59688 −0.142522
\(333\) 3.70156 0.202844
\(334\) 21.4031 1.17113
\(335\) −14.8062 −0.808952
\(336\) −1.00000 −0.0545545
\(337\) 17.4031 0.948009 0.474004 0.880523i \(-0.342808\pi\)
0.474004 + 0.880523i \(0.342808\pi\)
\(338\) −19.5078 −1.06109
\(339\) −4.29844 −0.233459
\(340\) 0 0
\(341\) 0 0
\(342\) 5.40312 0.292167
\(343\) −1.00000 −0.0539949
\(344\) −5.70156 −0.307408
\(345\) 3.70156 0.199285
\(346\) 3.40312 0.182953
\(347\) 20.5078 1.10092 0.550458 0.834863i \(-0.314453\pi\)
0.550458 + 0.834863i \(0.314453\pi\)
\(348\) 0.298438 0.0159979
\(349\) 34.2094 1.83119 0.915593 0.402107i \(-0.131722\pi\)
0.915593 + 0.402107i \(0.131722\pi\)
\(350\) 8.70156 0.465118
\(351\) 5.70156 0.304327
\(352\) 0 0
\(353\) −33.3141 −1.77313 −0.886564 0.462606i \(-0.846915\pi\)
−0.886564 + 0.462606i \(0.846915\pi\)
\(354\) 0.596876 0.0317236
\(355\) −27.4031 −1.45441
\(356\) −15.4031 −0.816364
\(357\) 0 0
\(358\) −5.70156 −0.301337
\(359\) 9.70156 0.512029 0.256014 0.966673i \(-0.417591\pi\)
0.256014 + 0.966673i \(0.417591\pi\)
\(360\) 3.70156 0.195089
\(361\) 10.1938 0.536513
\(362\) −16.8062 −0.883317
\(363\) −11.0000 −0.577350
\(364\) −5.70156 −0.298843
\(365\) 20.0000 1.04685
\(366\) −10.0000 −0.522708
\(367\) 15.9109 0.830544 0.415272 0.909697i \(-0.363686\pi\)
0.415272 + 0.909697i \(0.363686\pi\)
\(368\) −1.00000 −0.0521286
\(369\) 7.70156 0.400927
\(370\) 13.7016 0.712310
\(371\) −9.40312 −0.488186
\(372\) 6.00000 0.311086
\(373\) −4.80625 −0.248858 −0.124429 0.992229i \(-0.539710\pi\)
−0.124429 + 0.992229i \(0.539710\pi\)
\(374\) 0 0
\(375\) −13.7016 −0.707546
\(376\) −3.70156 −0.190893
\(377\) 1.70156 0.0876349
\(378\) 1.00000 0.0514344
\(379\) −17.7016 −0.909268 −0.454634 0.890678i \(-0.650230\pi\)
−0.454634 + 0.890678i \(0.650230\pi\)
\(380\) 20.0000 1.02598
\(381\) 17.1047 0.876300
\(382\) −22.8062 −1.16687
\(383\) 6.80625 0.347783 0.173892 0.984765i \(-0.444366\pi\)
0.173892 + 0.984765i \(0.444366\pi\)
\(384\) −1.00000 −0.0510310
\(385\) 0 0
\(386\) 7.10469 0.361619
\(387\) 5.70156 0.289827
\(388\) −1.10469 −0.0560820
\(389\) 36.8062 1.86615 0.933075 0.359681i \(-0.117114\pi\)
0.933075 + 0.359681i \(0.117114\pi\)
\(390\) 21.1047 1.06868
\(391\) 0 0
\(392\) −1.00000 −0.0505076
\(393\) 15.4031 0.776985
\(394\) −6.50781 −0.327859
\(395\) 0 0
\(396\) 0 0
\(397\) 6.20937 0.311639 0.155820 0.987786i \(-0.450198\pi\)
0.155820 + 0.987786i \(0.450198\pi\)
\(398\) −1.10469 −0.0553729
\(399\) 5.40312 0.270495
\(400\) 8.70156 0.435078
\(401\) 28.8062 1.43852 0.719258 0.694743i \(-0.244482\pi\)
0.719258 + 0.694743i \(0.244482\pi\)
\(402\) −4.00000 −0.199502
\(403\) 34.2094 1.70409
\(404\) 4.59688 0.228703
\(405\) −3.70156 −0.183932
\(406\) 0.298438 0.0148112
\(407\) 0 0
\(408\) 0 0
\(409\) −32.2094 −1.59265 −0.796325 0.604868i \(-0.793226\pi\)
−0.796325 + 0.604868i \(0.793226\pi\)
\(410\) 28.5078 1.40790
\(411\) 0.895314 0.0441626
\(412\) −17.1047 −0.842687
\(413\) 0.596876 0.0293703
\(414\) 1.00000 0.0491473
\(415\) 9.61250 0.471859
\(416\) −5.70156 −0.279542
\(417\) −21.1047 −1.03350
\(418\) 0 0
\(419\) −2.59688 −0.126866 −0.0634328 0.997986i \(-0.520205\pi\)
−0.0634328 + 0.997986i \(0.520205\pi\)
\(420\) 3.70156 0.180618
\(421\) 15.7016 0.765247 0.382624 0.923904i \(-0.375021\pi\)
0.382624 + 0.923904i \(0.375021\pi\)
\(422\) 12.0000 0.584151
\(423\) 3.70156 0.179976
\(424\) −9.40312 −0.456656
\(425\) 0 0
\(426\) −7.40312 −0.358683
\(427\) −10.0000 −0.483934
\(428\) 14.8062 0.715687
\(429\) 0 0
\(430\) 21.1047 1.01776
\(431\) 5.10469 0.245884 0.122942 0.992414i \(-0.460767\pi\)
0.122942 + 0.992414i \(0.460767\pi\)
\(432\) 1.00000 0.0481125
\(433\) 17.1047 0.821999 0.410999 0.911636i \(-0.365180\pi\)
0.410999 + 0.911636i \(0.365180\pi\)
\(434\) 6.00000 0.288009
\(435\) −1.10469 −0.0529657
\(436\) 15.7016 0.751968
\(437\) 5.40312 0.258466
\(438\) 5.40312 0.258171
\(439\) 3.19375 0.152429 0.0762147 0.997091i \(-0.475717\pi\)
0.0762147 + 0.997091i \(0.475717\pi\)
\(440\) 0 0
\(441\) 1.00000 0.0476190
\(442\) 0 0
\(443\) 17.1047 0.812668 0.406334 0.913725i \(-0.366807\pi\)
0.406334 + 0.913725i \(0.366807\pi\)
\(444\) 3.70156 0.175668
\(445\) 57.0156 2.70280
\(446\) 20.8062 0.985204
\(447\) −17.4031 −0.823140
\(448\) −1.00000 −0.0472456
\(449\) 27.6125 1.30311 0.651557 0.758600i \(-0.274116\pi\)
0.651557 + 0.758600i \(0.274116\pi\)
\(450\) −8.70156 −0.410196
\(451\) 0 0
\(452\) −4.29844 −0.202182
\(453\) 9.10469 0.427775
\(454\) 26.5078 1.24407
\(455\) 21.1047 0.989403
\(456\) 5.40312 0.253024
\(457\) −16.2094 −0.758242 −0.379121 0.925347i \(-0.623774\pi\)
−0.379121 + 0.925347i \(0.623774\pi\)
\(458\) −28.2094 −1.31814
\(459\) 0 0
\(460\) 3.70156 0.172586
\(461\) 27.4031 1.27629 0.638145 0.769916i \(-0.279702\pi\)
0.638145 + 0.769916i \(0.279702\pi\)
\(462\) 0 0
\(463\) −13.1047 −0.609026 −0.304513 0.952508i \(-0.598494\pi\)
−0.304513 + 0.952508i \(0.598494\pi\)
\(464\) 0.298438 0.0138546
\(465\) −22.2094 −1.02993
\(466\) −28.8062 −1.33442
\(467\) −8.29844 −0.384006 −0.192003 0.981394i \(-0.561498\pi\)
−0.192003 + 0.981394i \(0.561498\pi\)
\(468\) 5.70156 0.263555
\(469\) −4.00000 −0.184703
\(470\) 13.7016 0.632006
\(471\) −13.4031 −0.617583
\(472\) 0.596876 0.0274734
\(473\) 0 0
\(474\) 0 0
\(475\) −47.0156 −2.15722
\(476\) 0 0
\(477\) 9.40312 0.430539
\(478\) 18.8062 0.860178
\(479\) 7.40312 0.338257 0.169129 0.985594i \(-0.445905\pi\)
0.169129 + 0.985594i \(0.445905\pi\)
\(480\) 3.70156 0.168952
\(481\) 21.1047 0.962291
\(482\) −5.10469 −0.232512
\(483\) 1.00000 0.0455016
\(484\) −11.0000 −0.500000
\(485\) 4.08907 0.185675
\(486\) −1.00000 −0.0453609
\(487\) −13.1047 −0.593830 −0.296915 0.954904i \(-0.595958\pi\)
−0.296915 + 0.954904i \(0.595958\pi\)
\(488\) −10.0000 −0.452679
\(489\) −12.0000 −0.542659
\(490\) 3.70156 0.167220
\(491\) −17.6125 −0.794841 −0.397420 0.917637i \(-0.630095\pi\)
−0.397420 + 0.917637i \(0.630095\pi\)
\(492\) 7.70156 0.347213
\(493\) 0 0
\(494\) 30.8062 1.38604
\(495\) 0 0
\(496\) 6.00000 0.269408
\(497\) −7.40312 −0.332076
\(498\) 2.59688 0.116369
\(499\) 37.0156 1.65705 0.828523 0.559954i \(-0.189181\pi\)
0.828523 + 0.559954i \(0.189181\pi\)
\(500\) −13.7016 −0.612752
\(501\) −21.4031 −0.956221
\(502\) −13.9109 −0.620875
\(503\) −11.4031 −0.508440 −0.254220 0.967146i \(-0.581819\pi\)
−0.254220 + 0.967146i \(0.581819\pi\)
\(504\) 1.00000 0.0445435
\(505\) −17.0156 −0.757185
\(506\) 0 0
\(507\) 19.5078 0.866372
\(508\) 17.1047 0.758898
\(509\) −14.8062 −0.656275 −0.328138 0.944630i \(-0.606421\pi\)
−0.328138 + 0.944630i \(0.606421\pi\)
\(510\) 0 0
\(511\) 5.40312 0.239020
\(512\) −1.00000 −0.0441942
\(513\) −5.40312 −0.238554
\(514\) −23.6125 −1.04150
\(515\) 63.3141 2.78995
\(516\) 5.70156 0.250997
\(517\) 0 0
\(518\) 3.70156 0.162637
\(519\) −3.40312 −0.149381
\(520\) 21.1047 0.925502
\(521\) 15.4031 0.674823 0.337412 0.941357i \(-0.390449\pi\)
0.337412 + 0.941357i \(0.390449\pi\)
\(522\) −0.298438 −0.0130623
\(523\) −38.4187 −1.67993 −0.839967 0.542637i \(-0.817426\pi\)
−0.839967 + 0.542637i \(0.817426\pi\)
\(524\) 15.4031 0.672889
\(525\) −8.70156 −0.379767
\(526\) 16.5078 0.719775
\(527\) 0 0
\(528\) 0 0
\(529\) 1.00000 0.0434783
\(530\) 34.8062 1.51189
\(531\) −0.596876 −0.0259022
\(532\) 5.40312 0.234255
\(533\) 43.9109 1.90199
\(534\) 15.4031 0.666558
\(535\) −54.8062 −2.36948
\(536\) −4.00000 −0.172774
\(537\) 5.70156 0.246041
\(538\) 3.40312 0.146719
\(539\) 0 0
\(540\) −3.70156 −0.159290
\(541\) −20.2094 −0.868869 −0.434434 0.900703i \(-0.643052\pi\)
−0.434434 + 0.900703i \(0.643052\pi\)
\(542\) −17.4031 −0.747528
\(543\) 16.8062 0.721225
\(544\) 0 0
\(545\) −58.1203 −2.48960
\(546\) 5.70156 0.244004
\(547\) −14.2094 −0.607549 −0.303774 0.952744i \(-0.598247\pi\)
−0.303774 + 0.952744i \(0.598247\pi\)
\(548\) 0.895314 0.0382459
\(549\) 10.0000 0.426790
\(550\) 0 0
\(551\) −1.61250 −0.0686947
\(552\) 1.00000 0.0425628
\(553\) 0 0
\(554\) 13.4031 0.569444
\(555\) −13.7016 −0.581599
\(556\) −21.1047 −0.895038
\(557\) 21.4031 0.906879 0.453440 0.891287i \(-0.350197\pi\)
0.453440 + 0.891287i \(0.350197\pi\)
\(558\) −6.00000 −0.254000
\(559\) 32.5078 1.37493
\(560\) 3.70156 0.156420
\(561\) 0 0
\(562\) −4.29844 −0.181319
\(563\) 44.7172 1.88460 0.942302 0.334763i \(-0.108656\pi\)
0.942302 + 0.334763i \(0.108656\pi\)
\(564\) 3.70156 0.155864
\(565\) 15.9109 0.669378
\(566\) 10.0000 0.420331
\(567\) −1.00000 −0.0419961
\(568\) −7.40312 −0.310628
\(569\) 11.1047 0.465533 0.232766 0.972533i \(-0.425222\pi\)
0.232766 + 0.972533i \(0.425222\pi\)
\(570\) −20.0000 −0.837708
\(571\) −12.0000 −0.502184 −0.251092 0.967963i \(-0.580790\pi\)
−0.251092 + 0.967963i \(0.580790\pi\)
\(572\) 0 0
\(573\) 22.8062 0.952745
\(574\) 7.70156 0.321457
\(575\) −8.70156 −0.362880
\(576\) 1.00000 0.0416667
\(577\) −23.6125 −0.983001 −0.491501 0.870877i \(-0.663551\pi\)
−0.491501 + 0.870877i \(0.663551\pi\)
\(578\) 17.0000 0.707107
\(579\) −7.10469 −0.295261
\(580\) −1.10469 −0.0458696
\(581\) 2.59688 0.107737
\(582\) 1.10469 0.0457907
\(583\) 0 0
\(584\) 5.40312 0.223583
\(585\) −21.1047 −0.872571
\(586\) 3.19375 0.131933
\(587\) −1.19375 −0.0492714 −0.0246357 0.999696i \(-0.507843\pi\)
−0.0246357 + 0.999696i \(0.507843\pi\)
\(588\) 1.00000 0.0412393
\(589\) −32.4187 −1.33579
\(590\) −2.20937 −0.0909584
\(591\) 6.50781 0.267696
\(592\) 3.70156 0.152133
\(593\) −41.9109 −1.72108 −0.860538 0.509386i \(-0.829872\pi\)
−0.860538 + 0.509386i \(0.829872\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −17.4031 −0.712860
\(597\) 1.10469 0.0452118
\(598\) 5.70156 0.233154
\(599\) −29.6125 −1.20993 −0.604967 0.796251i \(-0.706814\pi\)
−0.604967 + 0.796251i \(0.706814\pi\)
\(600\) −8.70156 −0.355240
\(601\) 25.4031 1.03622 0.518108 0.855315i \(-0.326637\pi\)
0.518108 + 0.855315i \(0.326637\pi\)
\(602\) 5.70156 0.232378
\(603\) 4.00000 0.162893
\(604\) 9.10469 0.370464
\(605\) 40.7172 1.65539
\(606\) −4.59688 −0.186735
\(607\) −2.00000 −0.0811775 −0.0405887 0.999176i \(-0.512923\pi\)
−0.0405887 + 0.999176i \(0.512923\pi\)
\(608\) 5.40312 0.219126
\(609\) −0.298438 −0.0120933
\(610\) 37.0156 1.49872
\(611\) 21.1047 0.853804
\(612\) 0 0
\(613\) −25.3141 −1.02243 −0.511213 0.859454i \(-0.670804\pi\)
−0.511213 + 0.859454i \(0.670804\pi\)
\(614\) −5.10469 −0.206008
\(615\) −28.5078 −1.14955
\(616\) 0 0
\(617\) −42.0000 −1.69086 −0.845428 0.534089i \(-0.820655\pi\)
−0.845428 + 0.534089i \(0.820655\pi\)
\(618\) 17.1047 0.688051
\(619\) −30.0000 −1.20580 −0.602901 0.797816i \(-0.705989\pi\)
−0.602901 + 0.797816i \(0.705989\pi\)
\(620\) −22.2094 −0.891950
\(621\) −1.00000 −0.0401286
\(622\) 16.2094 0.649937
\(623\) 15.4031 0.617113
\(624\) 5.70156 0.228245
\(625\) 7.20937 0.288375
\(626\) 4.59688 0.183728
\(627\) 0 0
\(628\) −13.4031 −0.534843
\(629\) 0 0
\(630\) −3.70156 −0.147474
\(631\) 9.19375 0.365997 0.182999 0.983113i \(-0.441420\pi\)
0.182999 + 0.983113i \(0.441420\pi\)
\(632\) 0 0
\(633\) −12.0000 −0.476957
\(634\) −7.70156 −0.305868
\(635\) −63.3141 −2.51254
\(636\) 9.40312 0.372858
\(637\) 5.70156 0.225904
\(638\) 0 0
\(639\) 7.40312 0.292863
\(640\) 3.70156 0.146317
\(641\) −11.7016 −0.462184 −0.231092 0.972932i \(-0.574230\pi\)
−0.231092 + 0.972932i \(0.574230\pi\)
\(642\) −14.8062 −0.584356
\(643\) 3.19375 0.125949 0.0629746 0.998015i \(-0.479941\pi\)
0.0629746 + 0.998015i \(0.479941\pi\)
\(644\) 1.00000 0.0394055
\(645\) −21.1047 −0.830996
\(646\) 0 0
\(647\) −4.20937 −0.165488 −0.0827438 0.996571i \(-0.526368\pi\)
−0.0827438 + 0.996571i \(0.526368\pi\)
\(648\) −1.00000 −0.0392837
\(649\) 0 0
\(650\) −49.6125 −1.94596
\(651\) −6.00000 −0.235159
\(652\) −12.0000 −0.469956
\(653\) 47.1047 1.84335 0.921674 0.387964i \(-0.126822\pi\)
0.921674 + 0.387964i \(0.126822\pi\)
\(654\) −15.7016 −0.613980
\(655\) −57.0156 −2.22778
\(656\) 7.70156 0.300695
\(657\) −5.40312 −0.210796
\(658\) 3.70156 0.144302
\(659\) 25.6125 0.997721 0.498861 0.866682i \(-0.333752\pi\)
0.498861 + 0.866682i \(0.333752\pi\)
\(660\) 0 0
\(661\) −20.8062 −0.809269 −0.404635 0.914478i \(-0.632601\pi\)
−0.404635 + 0.914478i \(0.632601\pi\)
\(662\) 17.6125 0.684529
\(663\) 0 0
\(664\) 2.59688 0.100778
\(665\) −20.0000 −0.775567
\(666\) −3.70156 −0.143433
\(667\) −0.298438 −0.0115556
\(668\) −21.4031 −0.828112
\(669\) −20.8062 −0.804416
\(670\) 14.8062 0.572015
\(671\) 0 0
\(672\) 1.00000 0.0385758
\(673\) −8.29844 −0.319881 −0.159941 0.987127i \(-0.551130\pi\)
−0.159941 + 0.987127i \(0.551130\pi\)
\(674\) −17.4031 −0.670343
\(675\) 8.70156 0.334923
\(676\) 19.5078 0.750300
\(677\) −3.19375 −0.122746 −0.0613729 0.998115i \(-0.519548\pi\)
−0.0613729 + 0.998115i \(0.519548\pi\)
\(678\) 4.29844 0.165081
\(679\) 1.10469 0.0423940
\(680\) 0 0
\(681\) −26.5078 −1.01578
\(682\) 0 0
\(683\) 36.0000 1.37750 0.688751 0.724998i \(-0.258159\pi\)
0.688751 + 0.724998i \(0.258159\pi\)
\(684\) −5.40312 −0.206594
\(685\) −3.31406 −0.126624
\(686\) 1.00000 0.0381802
\(687\) 28.2094 1.07625
\(688\) 5.70156 0.217370
\(689\) 53.6125 2.04247
\(690\) −3.70156 −0.140916
\(691\) −41.7016 −1.58640 −0.793201 0.608960i \(-0.791587\pi\)
−0.793201 + 0.608960i \(0.791587\pi\)
\(692\) −3.40312 −0.129367
\(693\) 0 0
\(694\) −20.5078 −0.778466
\(695\) 78.1203 2.96327
\(696\) −0.298438 −0.0113123
\(697\) 0 0
\(698\) −34.2094 −1.29484
\(699\) 28.8062 1.08955
\(700\) −8.70156 −0.328888
\(701\) 14.5969 0.551316 0.275658 0.961256i \(-0.411104\pi\)
0.275658 + 0.961256i \(0.411104\pi\)
\(702\) −5.70156 −0.215192
\(703\) −20.0000 −0.754314
\(704\) 0 0
\(705\) −13.7016 −0.516031
\(706\) 33.3141 1.25379
\(707\) −4.59688 −0.172883
\(708\) −0.596876 −0.0224320
\(709\) −15.1938 −0.570613 −0.285307 0.958436i \(-0.592095\pi\)
−0.285307 + 0.958436i \(0.592095\pi\)
\(710\) 27.4031 1.02842
\(711\) 0 0
\(712\) 15.4031 0.577256
\(713\) −6.00000 −0.224702
\(714\) 0 0
\(715\) 0 0
\(716\) 5.70156 0.213077
\(717\) −18.8062 −0.702332
\(718\) −9.70156 −0.362059
\(719\) 14.5078 0.541050 0.270525 0.962713i \(-0.412803\pi\)
0.270525 + 0.962713i \(0.412803\pi\)
\(720\) −3.70156 −0.137949
\(721\) 17.1047 0.637012
\(722\) −10.1938 −0.379372
\(723\) 5.10469 0.189845
\(724\) 16.8062 0.624599
\(725\) 2.59688 0.0964455
\(726\) 11.0000 0.408248
\(727\) −44.0000 −1.63187 −0.815935 0.578144i \(-0.803777\pi\)
−0.815935 + 0.578144i \(0.803777\pi\)
\(728\) 5.70156 0.211314
\(729\) 1.00000 0.0370370
\(730\) −20.0000 −0.740233
\(731\) 0 0
\(732\) 10.0000 0.369611
\(733\) 31.0156 1.14559 0.572794 0.819699i \(-0.305859\pi\)
0.572794 + 0.819699i \(0.305859\pi\)
\(734\) −15.9109 −0.587283
\(735\) −3.70156 −0.136534
\(736\) 1.00000 0.0368605
\(737\) 0 0
\(738\) −7.70156 −0.283498
\(739\) 38.2094 1.40555 0.702777 0.711410i \(-0.251943\pi\)
0.702777 + 0.711410i \(0.251943\pi\)
\(740\) −13.7016 −0.503679
\(741\) −30.8062 −1.13170
\(742\) 9.40312 0.345200
\(743\) −21.6125 −0.792886 −0.396443 0.918059i \(-0.629755\pi\)
−0.396443 + 0.918059i \(0.629755\pi\)
\(744\) −6.00000 −0.219971
\(745\) 64.4187 2.36012
\(746\) 4.80625 0.175969
\(747\) −2.59688 −0.0950147
\(748\) 0 0
\(749\) −14.8062 −0.541009
\(750\) 13.7016 0.500310
\(751\) 8.00000 0.291924 0.145962 0.989290i \(-0.453372\pi\)
0.145962 + 0.989290i \(0.453372\pi\)
\(752\) 3.70156 0.134982
\(753\) 13.9109 0.506943
\(754\) −1.70156 −0.0619672
\(755\) −33.7016 −1.22653
\(756\) −1.00000 −0.0363696
\(757\) 7.19375 0.261461 0.130731 0.991418i \(-0.458268\pi\)
0.130731 + 0.991418i \(0.458268\pi\)
\(758\) 17.7016 0.642950
\(759\) 0 0
\(760\) −20.0000 −0.725476
\(761\) 15.6125 0.565953 0.282976 0.959127i \(-0.408678\pi\)
0.282976 + 0.959127i \(0.408678\pi\)
\(762\) −17.1047 −0.619637
\(763\) −15.7016 −0.568435
\(764\) 22.8062 0.825101
\(765\) 0 0
\(766\) −6.80625 −0.245920
\(767\) −3.40312 −0.122880
\(768\) 1.00000 0.0360844
\(769\) −29.1047 −1.04954 −0.524771 0.851243i \(-0.675849\pi\)
−0.524771 + 0.851243i \(0.675849\pi\)
\(770\) 0 0
\(771\) 23.6125 0.850383
\(772\) −7.10469 −0.255703
\(773\) 44.7172 1.60837 0.804183 0.594382i \(-0.202603\pi\)
0.804183 + 0.594382i \(0.202603\pi\)
\(774\) −5.70156 −0.204938
\(775\) 52.2094 1.87542
\(776\) 1.10469 0.0396559
\(777\) −3.70156 −0.132793
\(778\) −36.8062 −1.31957
\(779\) −41.6125 −1.49092
\(780\) −21.1047 −0.755669
\(781\) 0 0
\(782\) 0 0
\(783\) 0.298438 0.0106653
\(784\) 1.00000 0.0357143
\(785\) 49.6125 1.77075
\(786\) −15.4031 −0.549411
\(787\) 49.8219 1.77596 0.887979 0.459884i \(-0.152109\pi\)
0.887979 + 0.459884i \(0.152109\pi\)
\(788\) 6.50781 0.231831
\(789\) −16.5078 −0.587694
\(790\) 0 0
\(791\) 4.29844 0.152835
\(792\) 0 0
\(793\) 57.0156 2.02468
\(794\) −6.20937 −0.220362
\(795\) −34.8062 −1.23445
\(796\) 1.10469 0.0391546
\(797\) −31.7016 −1.12293 −0.561463 0.827502i \(-0.689761\pi\)
−0.561463 + 0.827502i \(0.689761\pi\)
\(798\) −5.40312 −0.191269
\(799\) 0 0
\(800\) −8.70156 −0.307647
\(801\) −15.4031 −0.544243
\(802\) −28.8062 −1.01718
\(803\) 0 0
\(804\) 4.00000 0.141069
\(805\) −3.70156 −0.130463
\(806\) −34.2094 −1.20497
\(807\) −3.40312 −0.119796
\(808\) −4.59688 −0.161718
\(809\) −21.4031 −0.752494 −0.376247 0.926519i \(-0.622786\pi\)
−0.376247 + 0.926519i \(0.622786\pi\)
\(810\) 3.70156 0.130060
\(811\) 11.3141 0.397290 0.198645 0.980071i \(-0.436346\pi\)
0.198645 + 0.980071i \(0.436346\pi\)
\(812\) −0.298438 −0.0104731
\(813\) 17.4031 0.610354
\(814\) 0 0
\(815\) 44.4187 1.55592
\(816\) 0 0
\(817\) −30.8062 −1.07777
\(818\) 32.2094 1.12617
\(819\) −5.70156 −0.199229
\(820\) −28.5078 −0.995536
\(821\) 6.00000 0.209401 0.104701 0.994504i \(-0.466612\pi\)
0.104701 + 0.994504i \(0.466612\pi\)
\(822\) −0.895314 −0.0312276
\(823\) 46.7172 1.62846 0.814229 0.580543i \(-0.197160\pi\)
0.814229 + 0.580543i \(0.197160\pi\)
\(824\) 17.1047 0.595870
\(825\) 0 0
\(826\) −0.596876 −0.0207680
\(827\) −33.0156 −1.14807 −0.574033 0.818832i \(-0.694622\pi\)
−0.574033 + 0.818832i \(0.694622\pi\)
\(828\) −1.00000 −0.0347524
\(829\) −30.2094 −1.04921 −0.524607 0.851344i \(-0.675788\pi\)
−0.524607 + 0.851344i \(0.675788\pi\)
\(830\) −9.61250 −0.333655
\(831\) −13.4031 −0.464949
\(832\) 5.70156 0.197666
\(833\) 0 0
\(834\) 21.1047 0.730796
\(835\) 79.2250 2.74169
\(836\) 0 0
\(837\) 6.00000 0.207390
\(838\) 2.59688 0.0897076
\(839\) −19.4031 −0.669870 −0.334935 0.942241i \(-0.608714\pi\)
−0.334935 + 0.942241i \(0.608714\pi\)
\(840\) −3.70156 −0.127716
\(841\) −28.9109 −0.996929
\(842\) −15.7016 −0.541112
\(843\) 4.29844 0.148046
\(844\) −12.0000 −0.413057
\(845\) −72.2094 −2.48408
\(846\) −3.70156 −0.127262
\(847\) 11.0000 0.377964
\(848\) 9.40312 0.322905
\(849\) −10.0000 −0.343199
\(850\) 0 0
\(851\) −3.70156 −0.126888
\(852\) 7.40312 0.253627
\(853\) 41.1047 1.40740 0.703699 0.710498i \(-0.251530\pi\)
0.703699 + 0.710498i \(0.251530\pi\)
\(854\) 10.0000 0.342193
\(855\) 20.0000 0.683986
\(856\) −14.8062 −0.506067
\(857\) 2.08907 0.0713611 0.0356806 0.999363i \(-0.488640\pi\)
0.0356806 + 0.999363i \(0.488640\pi\)
\(858\) 0 0
\(859\) −3.91093 −0.133439 −0.0667197 0.997772i \(-0.521253\pi\)
−0.0667197 + 0.997772i \(0.521253\pi\)
\(860\) −21.1047 −0.719664
\(861\) −7.70156 −0.262469
\(862\) −5.10469 −0.173866
\(863\) −10.8062 −0.367849 −0.183924 0.982940i \(-0.558880\pi\)
−0.183924 + 0.982940i \(0.558880\pi\)
\(864\) −1.00000 −0.0340207
\(865\) 12.5969 0.428307
\(866\) −17.1047 −0.581241
\(867\) −17.0000 −0.577350
\(868\) −6.00000 −0.203653
\(869\) 0 0
\(870\) 1.10469 0.0374524
\(871\) 22.8062 0.772760
\(872\) −15.7016 −0.531722
\(873\) −1.10469 −0.0373880
\(874\) −5.40312 −0.182763
\(875\) 13.7016 0.463197
\(876\) −5.40312 −0.182555
\(877\) −48.2094 −1.62791 −0.813957 0.580925i \(-0.802691\pi\)
−0.813957 + 0.580925i \(0.802691\pi\)
\(878\) −3.19375 −0.107784
\(879\) −3.19375 −0.107723
\(880\) 0 0
\(881\) −10.2094 −0.343963 −0.171981 0.985100i \(-0.555017\pi\)
−0.171981 + 0.985100i \(0.555017\pi\)
\(882\) −1.00000 −0.0336718
\(883\) 16.5969 0.558529 0.279265 0.960214i \(-0.409909\pi\)
0.279265 + 0.960214i \(0.409909\pi\)
\(884\) 0 0
\(885\) 2.20937 0.0742673
\(886\) −17.1047 −0.574643
\(887\) −52.2094 −1.75302 −0.876510 0.481384i \(-0.840134\pi\)
−0.876510 + 0.481384i \(0.840134\pi\)
\(888\) −3.70156 −0.124216
\(889\) −17.1047 −0.573673
\(890\) −57.0156 −1.91117
\(891\) 0 0
\(892\) −20.8062 −0.696645
\(893\) −20.0000 −0.669274
\(894\) 17.4031 0.582048
\(895\) −21.1047 −0.705452
\(896\) 1.00000 0.0334077
\(897\) −5.70156 −0.190370
\(898\) −27.6125 −0.921441
\(899\) 1.79063 0.0597208
\(900\) 8.70156 0.290052
\(901\) 0 0
\(902\) 0 0
\(903\) −5.70156 −0.189736
\(904\) 4.29844 0.142964
\(905\) −62.2094 −2.06791
\(906\) −9.10469 −0.302483
\(907\) −2.89531 −0.0961373 −0.0480687 0.998844i \(-0.515307\pi\)
−0.0480687 + 0.998844i \(0.515307\pi\)
\(908\) −26.5078 −0.879693
\(909\) 4.59688 0.152469
\(910\) −21.1047 −0.699614
\(911\) 1.70156 0.0563753 0.0281876 0.999603i \(-0.491026\pi\)
0.0281876 + 0.999603i \(0.491026\pi\)
\(912\) −5.40312 −0.178915
\(913\) 0 0
\(914\) 16.2094 0.536158
\(915\) −37.0156 −1.22370
\(916\) 28.2094 0.932064
\(917\) −15.4031 −0.508656
\(918\) 0 0
\(919\) 1.19375 0.0393782 0.0196891 0.999806i \(-0.493732\pi\)
0.0196891 + 0.999806i \(0.493732\pi\)
\(920\) −3.70156 −0.122037
\(921\) 5.10469 0.168205
\(922\) −27.4031 −0.902474
\(923\) 42.2094 1.38934
\(924\) 0 0
\(925\) 32.2094 1.05904
\(926\) 13.1047 0.430647
\(927\) −17.1047 −0.561792
\(928\) −0.298438 −0.00979670
\(929\) 23.7016 0.777623 0.388812 0.921317i \(-0.372886\pi\)
0.388812 + 0.921317i \(0.372886\pi\)
\(930\) 22.2094 0.728274
\(931\) −5.40312 −0.177080
\(932\) 28.8062 0.943580
\(933\) −16.2094 −0.530671
\(934\) 8.29844 0.271533
\(935\) 0 0
\(936\) −5.70156 −0.186361
\(937\) 47.3141 1.54568 0.772841 0.634599i \(-0.218835\pi\)
0.772841 + 0.634599i \(0.218835\pi\)
\(938\) 4.00000 0.130605
\(939\) −4.59688 −0.150013
\(940\) −13.7016 −0.446896
\(941\) −9.31406 −0.303630 −0.151815 0.988409i \(-0.548512\pi\)
−0.151815 + 0.988409i \(0.548512\pi\)
\(942\) 13.4031 0.436697
\(943\) −7.70156 −0.250797
\(944\) −0.596876 −0.0194267
\(945\) 3.70156 0.120412
\(946\) 0 0
\(947\) 3.49219 0.113481 0.0567405 0.998389i \(-0.481929\pi\)
0.0567405 + 0.998389i \(0.481929\pi\)
\(948\) 0 0
\(949\) −30.8062 −1.00001
\(950\) 47.0156 1.52539
\(951\) 7.70156 0.249740
\(952\) 0 0
\(953\) −20.8062 −0.673980 −0.336990 0.941508i \(-0.609409\pi\)
−0.336990 + 0.941508i \(0.609409\pi\)
\(954\) −9.40312 −0.304437
\(955\) −84.4187 −2.73173
\(956\) −18.8062 −0.608238
\(957\) 0 0
\(958\) −7.40312 −0.239184
\(959\) −0.895314 −0.0289112
\(960\) −3.70156 −0.119467
\(961\) 5.00000 0.161290
\(962\) −21.1047 −0.680442
\(963\) 14.8062 0.477125
\(964\) 5.10469 0.164411
\(965\) 26.2984 0.846577
\(966\) −1.00000 −0.0321745
\(967\) 20.4187 0.656623 0.328311 0.944570i \(-0.393521\pi\)
0.328311 + 0.944570i \(0.393521\pi\)
\(968\) 11.0000 0.353553
\(969\) 0 0
\(970\) −4.08907 −0.131292
\(971\) −33.4031 −1.07196 −0.535979 0.844232i \(-0.680057\pi\)
−0.535979 + 0.844232i \(0.680057\pi\)
\(972\) 1.00000 0.0320750
\(973\) 21.1047 0.676585
\(974\) 13.1047 0.419901
\(975\) 49.6125 1.58887
\(976\) 10.0000 0.320092
\(977\) −40.2984 −1.28926 −0.644631 0.764494i \(-0.722989\pi\)
−0.644631 + 0.764494i \(0.722989\pi\)
\(978\) 12.0000 0.383718
\(979\) 0 0
\(980\) −3.70156 −0.118242
\(981\) 15.7016 0.501312
\(982\) 17.6125 0.562037
\(983\) −55.8219 −1.78044 −0.890221 0.455530i \(-0.849450\pi\)
−0.890221 + 0.455530i \(0.849450\pi\)
\(984\) −7.70156 −0.245517
\(985\) −24.0891 −0.767541
\(986\) 0 0
\(987\) −3.70156 −0.117822
\(988\) −30.8062 −0.980077
\(989\) −5.70156 −0.181299
\(990\) 0 0
\(991\) 20.0000 0.635321 0.317660 0.948205i \(-0.397103\pi\)
0.317660 + 0.948205i \(0.397103\pi\)
\(992\) −6.00000 −0.190500
\(993\) −17.6125 −0.558916
\(994\) 7.40312 0.234813
\(995\) −4.08907 −0.129632
\(996\) −2.59688 −0.0822852
\(997\) 48.5969 1.53908 0.769539 0.638600i \(-0.220486\pi\)
0.769539 + 0.638600i \(0.220486\pi\)
\(998\) −37.0156 −1.17171
\(999\) 3.70156 0.117112
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 966.2.a.m.1.1 2
3.2 odd 2 2898.2.a.bc.1.2 2
4.3 odd 2 7728.2.a.z.1.1 2
7.6 odd 2 6762.2.a.bq.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
966.2.a.m.1.1 2 1.1 even 1 trivial
2898.2.a.bc.1.2 2 3.2 odd 2
6762.2.a.bq.1.2 2 7.6 odd 2
7728.2.a.z.1.1 2 4.3 odd 2