Properties

Label 91.2.a.c.1.2
Level $91$
Weight $2$
Character 91.1
Self dual yes
Analytic conductor $0.727$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 91 = 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 91.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(0.726638658394\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{8})^+\)
Defining polynomial: \(x^{2} - 2\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.41421\) of defining polynomial
Character \(\chi\) \(=\) 91.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.41421 q^{2} -1.41421 q^{3} +4.41421 q^{5} -2.00000 q^{6} +1.00000 q^{7} -2.82843 q^{8} -1.00000 q^{9} +O(q^{10})\) \(q+1.41421 q^{2} -1.41421 q^{3} +4.41421 q^{5} -2.00000 q^{6} +1.00000 q^{7} -2.82843 q^{8} -1.00000 q^{9} +6.24264 q^{10} -4.24264 q^{11} -1.00000 q^{13} +1.41421 q^{14} -6.24264 q^{15} -4.00000 q^{16} -1.41421 q^{17} -1.41421 q^{18} +1.24264 q^{19} -1.41421 q^{21} -6.00000 q^{22} -0.171573 q^{23} +4.00000 q^{24} +14.4853 q^{25} -1.41421 q^{26} +5.65685 q^{27} +5.82843 q^{29} -8.82843 q^{30} -5.24264 q^{31} +6.00000 q^{33} -2.00000 q^{34} +4.41421 q^{35} -6.24264 q^{37} +1.75736 q^{38} +1.41421 q^{39} -12.4853 q^{40} +3.17157 q^{41} -2.00000 q^{42} -5.00000 q^{43} -4.41421 q^{45} -0.242641 q^{46} +4.41421 q^{47} +5.65685 q^{48} +1.00000 q^{49} +20.4853 q^{50} +2.00000 q^{51} -5.82843 q^{53} +8.00000 q^{54} -18.7279 q^{55} -2.82843 q^{56} -1.75736 q^{57} +8.24264 q^{58} +11.6569 q^{59} +6.00000 q^{61} -7.41421 q^{62} -1.00000 q^{63} +8.00000 q^{64} -4.41421 q^{65} +8.48528 q^{66} +2.48528 q^{67} +0.242641 q^{69} +6.24264 q^{70} +1.07107 q^{71} +2.82843 q^{72} -0.757359 q^{73} -8.82843 q^{74} -20.4853 q^{75} -4.24264 q^{77} +2.00000 q^{78} -1.48528 q^{79} -17.6569 q^{80} -5.00000 q^{81} +4.48528 q^{82} +4.75736 q^{83} -6.24264 q^{85} -7.07107 q^{86} -8.24264 q^{87} +12.0000 q^{88} +4.41421 q^{89} -6.24264 q^{90} -1.00000 q^{91} +7.41421 q^{93} +6.24264 q^{94} +5.48528 q^{95} -13.7279 q^{97} +1.41421 q^{98} +4.24264 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 6q^{5} - 4q^{6} + 2q^{7} - 2q^{9} + O(q^{10}) \) \( 2q + 6q^{5} - 4q^{6} + 2q^{7} - 2q^{9} + 4q^{10} - 2q^{13} - 4q^{15} - 8q^{16} - 6q^{19} - 12q^{22} - 6q^{23} + 8q^{24} + 12q^{25} + 6q^{29} - 12q^{30} - 2q^{31} + 12q^{33} - 4q^{34} + 6q^{35} - 4q^{37} + 12q^{38} - 8q^{40} + 12q^{41} - 4q^{42} - 10q^{43} - 6q^{45} + 8q^{46} + 6q^{47} + 2q^{49} + 24q^{50} + 4q^{51} - 6q^{53} + 16q^{54} - 12q^{55} - 12q^{57} + 8q^{58} + 12q^{59} + 12q^{61} - 12q^{62} - 2q^{63} + 16q^{64} - 6q^{65} - 12q^{67} - 8q^{69} + 4q^{70} - 12q^{71} - 10q^{73} - 12q^{74} - 24q^{75} + 4q^{78} + 14q^{79} - 24q^{80} - 10q^{81} - 8q^{82} + 18q^{83} - 4q^{85} - 8q^{87} + 24q^{88} + 6q^{89} - 4q^{90} - 2q^{91} + 12q^{93} + 4q^{94} - 6q^{95} - 2q^{97} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.41421 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(3\) −1.41421 −0.816497 −0.408248 0.912871i \(-0.633860\pi\)
−0.408248 + 0.912871i \(0.633860\pi\)
\(4\) 0 0
\(5\) 4.41421 1.97410 0.987048 0.160424i \(-0.0512862\pi\)
0.987048 + 0.160424i \(0.0512862\pi\)
\(6\) −2.00000 −0.816497
\(7\) 1.00000 0.377964
\(8\) −2.82843 −1.00000
\(9\) −1.00000 −0.333333
\(10\) 6.24264 1.97410
\(11\) −4.24264 −1.27920 −0.639602 0.768706i \(-0.720901\pi\)
−0.639602 + 0.768706i \(0.720901\pi\)
\(12\) 0 0
\(13\) −1.00000 −0.277350
\(14\) 1.41421 0.377964
\(15\) −6.24264 −1.61184
\(16\) −4.00000 −1.00000
\(17\) −1.41421 −0.342997 −0.171499 0.985184i \(-0.554861\pi\)
−0.171499 + 0.985184i \(0.554861\pi\)
\(18\) −1.41421 −0.333333
\(19\) 1.24264 0.285081 0.142541 0.989789i \(-0.454473\pi\)
0.142541 + 0.989789i \(0.454473\pi\)
\(20\) 0 0
\(21\) −1.41421 −0.308607
\(22\) −6.00000 −1.27920
\(23\) −0.171573 −0.0357754 −0.0178877 0.999840i \(-0.505694\pi\)
−0.0178877 + 0.999840i \(0.505694\pi\)
\(24\) 4.00000 0.816497
\(25\) 14.4853 2.89706
\(26\) −1.41421 −0.277350
\(27\) 5.65685 1.08866
\(28\) 0 0
\(29\) 5.82843 1.08231 0.541156 0.840922i \(-0.317987\pi\)
0.541156 + 0.840922i \(0.317987\pi\)
\(30\) −8.82843 −1.61184
\(31\) −5.24264 −0.941606 −0.470803 0.882238i \(-0.656036\pi\)
−0.470803 + 0.882238i \(0.656036\pi\)
\(32\) 0 0
\(33\) 6.00000 1.04447
\(34\) −2.00000 −0.342997
\(35\) 4.41421 0.746138
\(36\) 0 0
\(37\) −6.24264 −1.02628 −0.513142 0.858304i \(-0.671519\pi\)
−0.513142 + 0.858304i \(0.671519\pi\)
\(38\) 1.75736 0.285081
\(39\) 1.41421 0.226455
\(40\) −12.4853 −1.97410
\(41\) 3.17157 0.495316 0.247658 0.968847i \(-0.420339\pi\)
0.247658 + 0.968847i \(0.420339\pi\)
\(42\) −2.00000 −0.308607
\(43\) −5.00000 −0.762493 −0.381246 0.924473i \(-0.624505\pi\)
−0.381246 + 0.924473i \(0.624505\pi\)
\(44\) 0 0
\(45\) −4.41421 −0.658032
\(46\) −0.242641 −0.0357754
\(47\) 4.41421 0.643879 0.321940 0.946760i \(-0.395665\pi\)
0.321940 + 0.946760i \(0.395665\pi\)
\(48\) 5.65685 0.816497
\(49\) 1.00000 0.142857
\(50\) 20.4853 2.89706
\(51\) 2.00000 0.280056
\(52\) 0 0
\(53\) −5.82843 −0.800596 −0.400298 0.916385i \(-0.631093\pi\)
−0.400298 + 0.916385i \(0.631093\pi\)
\(54\) 8.00000 1.08866
\(55\) −18.7279 −2.52527
\(56\) −2.82843 −0.377964
\(57\) −1.75736 −0.232768
\(58\) 8.24264 1.08231
\(59\) 11.6569 1.51759 0.758797 0.651328i \(-0.225788\pi\)
0.758797 + 0.651328i \(0.225788\pi\)
\(60\) 0 0
\(61\) 6.00000 0.768221 0.384111 0.923287i \(-0.374508\pi\)
0.384111 + 0.923287i \(0.374508\pi\)
\(62\) −7.41421 −0.941606
\(63\) −1.00000 −0.125988
\(64\) 8.00000 1.00000
\(65\) −4.41421 −0.547516
\(66\) 8.48528 1.04447
\(67\) 2.48528 0.303625 0.151813 0.988409i \(-0.451489\pi\)
0.151813 + 0.988409i \(0.451489\pi\)
\(68\) 0 0
\(69\) 0.242641 0.0292105
\(70\) 6.24264 0.746138
\(71\) 1.07107 0.127112 0.0635562 0.997978i \(-0.479756\pi\)
0.0635562 + 0.997978i \(0.479756\pi\)
\(72\) 2.82843 0.333333
\(73\) −0.757359 −0.0886422 −0.0443211 0.999017i \(-0.514112\pi\)
−0.0443211 + 0.999017i \(0.514112\pi\)
\(74\) −8.82843 −1.02628
\(75\) −20.4853 −2.36544
\(76\) 0 0
\(77\) −4.24264 −0.483494
\(78\) 2.00000 0.226455
\(79\) −1.48528 −0.167107 −0.0835536 0.996503i \(-0.526627\pi\)
−0.0835536 + 0.996503i \(0.526627\pi\)
\(80\) −17.6569 −1.97410
\(81\) −5.00000 −0.555556
\(82\) 4.48528 0.495316
\(83\) 4.75736 0.522188 0.261094 0.965313i \(-0.415917\pi\)
0.261094 + 0.965313i \(0.415917\pi\)
\(84\) 0 0
\(85\) −6.24264 −0.677109
\(86\) −7.07107 −0.762493
\(87\) −8.24264 −0.883704
\(88\) 12.0000 1.27920
\(89\) 4.41421 0.467906 0.233953 0.972248i \(-0.424834\pi\)
0.233953 + 0.972248i \(0.424834\pi\)
\(90\) −6.24264 −0.658032
\(91\) −1.00000 −0.104828
\(92\) 0 0
\(93\) 7.41421 0.768818
\(94\) 6.24264 0.643879
\(95\) 5.48528 0.562778
\(96\) 0 0
\(97\) −13.7279 −1.39386 −0.696930 0.717139i \(-0.745451\pi\)
−0.696930 + 0.717139i \(0.745451\pi\)
\(98\) 1.41421 0.142857
\(99\) 4.24264 0.426401
\(100\) 0 0
\(101\) 1.75736 0.174864 0.0874319 0.996170i \(-0.472134\pi\)
0.0874319 + 0.996170i \(0.472134\pi\)
\(102\) 2.82843 0.280056
\(103\) −8.00000 −0.788263 −0.394132 0.919054i \(-0.628955\pi\)
−0.394132 + 0.919054i \(0.628955\pi\)
\(104\) 2.82843 0.277350
\(105\) −6.24264 −0.609219
\(106\) −8.24264 −0.800596
\(107\) 8.14214 0.787130 0.393565 0.919297i \(-0.371242\pi\)
0.393565 + 0.919297i \(0.371242\pi\)
\(108\) 0 0
\(109\) −8.72792 −0.835983 −0.417992 0.908451i \(-0.637266\pi\)
−0.417992 + 0.908451i \(0.637266\pi\)
\(110\) −26.4853 −2.52527
\(111\) 8.82843 0.837957
\(112\) −4.00000 −0.377964
\(113\) −20.3137 −1.91095 −0.955476 0.295067i \(-0.904658\pi\)
−0.955476 + 0.295067i \(0.904658\pi\)
\(114\) −2.48528 −0.232768
\(115\) −0.757359 −0.0706241
\(116\) 0 0
\(117\) 1.00000 0.0924500
\(118\) 16.4853 1.51759
\(119\) −1.41421 −0.129641
\(120\) 17.6569 1.61184
\(121\) 7.00000 0.636364
\(122\) 8.48528 0.768221
\(123\) −4.48528 −0.404424
\(124\) 0 0
\(125\) 41.8701 3.74497
\(126\) −1.41421 −0.125988
\(127\) 2.00000 0.177471 0.0887357 0.996055i \(-0.471717\pi\)
0.0887357 + 0.996055i \(0.471717\pi\)
\(128\) 11.3137 1.00000
\(129\) 7.07107 0.622573
\(130\) −6.24264 −0.547516
\(131\) 2.82843 0.247121 0.123560 0.992337i \(-0.460569\pi\)
0.123560 + 0.992337i \(0.460569\pi\)
\(132\) 0 0
\(133\) 1.24264 0.107751
\(134\) 3.51472 0.303625
\(135\) 24.9706 2.14912
\(136\) 4.00000 0.342997
\(137\) −7.41421 −0.633439 −0.316720 0.948519i \(-0.602581\pi\)
−0.316720 + 0.948519i \(0.602581\pi\)
\(138\) 0.343146 0.0292105
\(139\) −2.24264 −0.190218 −0.0951092 0.995467i \(-0.530320\pi\)
−0.0951092 + 0.995467i \(0.530320\pi\)
\(140\) 0 0
\(141\) −6.24264 −0.525725
\(142\) 1.51472 0.127112
\(143\) 4.24264 0.354787
\(144\) 4.00000 0.333333
\(145\) 25.7279 2.13659
\(146\) −1.07107 −0.0886422
\(147\) −1.41421 −0.116642
\(148\) 0 0
\(149\) 7.75736 0.635508 0.317754 0.948173i \(-0.397071\pi\)
0.317754 + 0.948173i \(0.397071\pi\)
\(150\) −28.9706 −2.36544
\(151\) −18.2426 −1.48457 −0.742283 0.670087i \(-0.766257\pi\)
−0.742283 + 0.670087i \(0.766257\pi\)
\(152\) −3.51472 −0.285081
\(153\) 1.41421 0.114332
\(154\) −6.00000 −0.483494
\(155\) −23.1421 −1.85882
\(156\) 0 0
\(157\) −12.2426 −0.977069 −0.488535 0.872545i \(-0.662468\pi\)
−0.488535 + 0.872545i \(0.662468\pi\)
\(158\) −2.10051 −0.167107
\(159\) 8.24264 0.653684
\(160\) 0 0
\(161\) −0.171573 −0.0135218
\(162\) −7.07107 −0.555556
\(163\) −8.48528 −0.664619 −0.332309 0.943170i \(-0.607828\pi\)
−0.332309 + 0.943170i \(0.607828\pi\)
\(164\) 0 0
\(165\) 26.4853 2.06188
\(166\) 6.72792 0.522188
\(167\) −21.3848 −1.65480 −0.827402 0.561610i \(-0.810182\pi\)
−0.827402 + 0.561610i \(0.810182\pi\)
\(168\) 4.00000 0.308607
\(169\) 1.00000 0.0769231
\(170\) −8.82843 −0.677109
\(171\) −1.24264 −0.0950271
\(172\) 0 0
\(173\) −0.727922 −0.0553429 −0.0276714 0.999617i \(-0.508809\pi\)
−0.0276714 + 0.999617i \(0.508809\pi\)
\(174\) −11.6569 −0.883704
\(175\) 14.4853 1.09498
\(176\) 16.9706 1.27920
\(177\) −16.4853 −1.23911
\(178\) 6.24264 0.467906
\(179\) −9.00000 −0.672692 −0.336346 0.941739i \(-0.609191\pi\)
−0.336346 + 0.941739i \(0.609191\pi\)
\(180\) 0 0
\(181\) 6.72792 0.500083 0.250041 0.968235i \(-0.419556\pi\)
0.250041 + 0.968235i \(0.419556\pi\)
\(182\) −1.41421 −0.104828
\(183\) −8.48528 −0.627250
\(184\) 0.485281 0.0357754
\(185\) −27.5563 −2.02598
\(186\) 10.4853 0.768818
\(187\) 6.00000 0.438763
\(188\) 0 0
\(189\) 5.65685 0.411476
\(190\) 7.75736 0.562778
\(191\) 20.8284 1.50709 0.753546 0.657395i \(-0.228342\pi\)
0.753546 + 0.657395i \(0.228342\pi\)
\(192\) −11.3137 −0.816497
\(193\) 2.48528 0.178894 0.0894472 0.995992i \(-0.471490\pi\)
0.0894472 + 0.995992i \(0.471490\pi\)
\(194\) −19.4142 −1.39386
\(195\) 6.24264 0.447045
\(196\) 0 0
\(197\) 23.6569 1.68548 0.842741 0.538320i \(-0.180941\pi\)
0.842741 + 0.538320i \(0.180941\pi\)
\(198\) 6.00000 0.426401
\(199\) −12.2426 −0.867858 −0.433929 0.900947i \(-0.642873\pi\)
−0.433929 + 0.900947i \(0.642873\pi\)
\(200\) −40.9706 −2.89706
\(201\) −3.51472 −0.247909
\(202\) 2.48528 0.174864
\(203\) 5.82843 0.409075
\(204\) 0 0
\(205\) 14.0000 0.977802
\(206\) −11.3137 −0.788263
\(207\) 0.171573 0.0119251
\(208\) 4.00000 0.277350
\(209\) −5.27208 −0.364677
\(210\) −8.82843 −0.609219
\(211\) 17.9706 1.23714 0.618572 0.785728i \(-0.287711\pi\)
0.618572 + 0.785728i \(0.287711\pi\)
\(212\) 0 0
\(213\) −1.51472 −0.103787
\(214\) 11.5147 0.787130
\(215\) −22.0711 −1.50523
\(216\) −16.0000 −1.08866
\(217\) −5.24264 −0.355894
\(218\) −12.3431 −0.835983
\(219\) 1.07107 0.0723761
\(220\) 0 0
\(221\) 1.41421 0.0951303
\(222\) 12.4853 0.837957
\(223\) 9.24264 0.618933 0.309466 0.950910i \(-0.399850\pi\)
0.309466 + 0.950910i \(0.399850\pi\)
\(224\) 0 0
\(225\) −14.4853 −0.965685
\(226\) −28.7279 −1.91095
\(227\) 21.1716 1.40521 0.702603 0.711582i \(-0.252021\pi\)
0.702603 + 0.711582i \(0.252021\pi\)
\(228\) 0 0
\(229\) −21.4558 −1.41784 −0.708921 0.705288i \(-0.750818\pi\)
−0.708921 + 0.705288i \(0.750818\pi\)
\(230\) −1.07107 −0.0706241
\(231\) 6.00000 0.394771
\(232\) −16.4853 −1.08231
\(233\) −3.34315 −0.219017 −0.109508 0.993986i \(-0.534928\pi\)
−0.109508 + 0.993986i \(0.534928\pi\)
\(234\) 1.41421 0.0924500
\(235\) 19.4853 1.27108
\(236\) 0 0
\(237\) 2.10051 0.136442
\(238\) −2.00000 −0.129641
\(239\) 20.4853 1.32508 0.662541 0.749025i \(-0.269478\pi\)
0.662541 + 0.749025i \(0.269478\pi\)
\(240\) 24.9706 1.61184
\(241\) 22.2132 1.43088 0.715439 0.698675i \(-0.246227\pi\)
0.715439 + 0.698675i \(0.246227\pi\)
\(242\) 9.89949 0.636364
\(243\) −9.89949 −0.635053
\(244\) 0 0
\(245\) 4.41421 0.282014
\(246\) −6.34315 −0.404424
\(247\) −1.24264 −0.0790673
\(248\) 14.8284 0.941606
\(249\) −6.72792 −0.426365
\(250\) 59.2132 3.74497
\(251\) 19.4142 1.22541 0.612707 0.790310i \(-0.290081\pi\)
0.612707 + 0.790310i \(0.290081\pi\)
\(252\) 0 0
\(253\) 0.727922 0.0457641
\(254\) 2.82843 0.177471
\(255\) 8.82843 0.552858
\(256\) 0 0
\(257\) −16.5858 −1.03459 −0.517296 0.855806i \(-0.673062\pi\)
−0.517296 + 0.855806i \(0.673062\pi\)
\(258\) 10.0000 0.622573
\(259\) −6.24264 −0.387899
\(260\) 0 0
\(261\) −5.82843 −0.360771
\(262\) 4.00000 0.247121
\(263\) −31.9706 −1.97139 −0.985695 0.168541i \(-0.946095\pi\)
−0.985695 + 0.168541i \(0.946095\pi\)
\(264\) −16.9706 −1.04447
\(265\) −25.7279 −1.58045
\(266\) 1.75736 0.107751
\(267\) −6.24264 −0.382043
\(268\) 0 0
\(269\) 14.8284 0.904105 0.452053 0.891991i \(-0.350692\pi\)
0.452053 + 0.891991i \(0.350692\pi\)
\(270\) 35.3137 2.14912
\(271\) 20.0000 1.21491 0.607457 0.794353i \(-0.292190\pi\)
0.607457 + 0.794353i \(0.292190\pi\)
\(272\) 5.65685 0.342997
\(273\) 1.41421 0.0855921
\(274\) −10.4853 −0.633439
\(275\) −61.4558 −3.70593
\(276\) 0 0
\(277\) 9.48528 0.569915 0.284958 0.958540i \(-0.408020\pi\)
0.284958 + 0.958540i \(0.408020\pi\)
\(278\) −3.17157 −0.190218
\(279\) 5.24264 0.313869
\(280\) −12.4853 −0.746138
\(281\) −15.5563 −0.928014 −0.464007 0.885832i \(-0.653589\pi\)
−0.464007 + 0.885832i \(0.653589\pi\)
\(282\) −8.82843 −0.525725
\(283\) 8.48528 0.504398 0.252199 0.967675i \(-0.418846\pi\)
0.252199 + 0.967675i \(0.418846\pi\)
\(284\) 0 0
\(285\) −7.75736 −0.459506
\(286\) 6.00000 0.354787
\(287\) 3.17157 0.187212
\(288\) 0 0
\(289\) −15.0000 −0.882353
\(290\) 36.3848 2.13659
\(291\) 19.4142 1.13808
\(292\) 0 0
\(293\) −21.3848 −1.24931 −0.624656 0.780900i \(-0.714761\pi\)
−0.624656 + 0.780900i \(0.714761\pi\)
\(294\) −2.00000 −0.116642
\(295\) 51.4558 2.99588
\(296\) 17.6569 1.02628
\(297\) −24.0000 −1.39262
\(298\) 10.9706 0.635508
\(299\) 0.171573 0.00992232
\(300\) 0 0
\(301\) −5.00000 −0.288195
\(302\) −25.7990 −1.48457
\(303\) −2.48528 −0.142776
\(304\) −4.97056 −0.285081
\(305\) 26.4853 1.51654
\(306\) 2.00000 0.114332
\(307\) 4.75736 0.271517 0.135758 0.990742i \(-0.456653\pi\)
0.135758 + 0.990742i \(0.456653\pi\)
\(308\) 0 0
\(309\) 11.3137 0.643614
\(310\) −32.7279 −1.85882
\(311\) −4.58579 −0.260036 −0.130018 0.991512i \(-0.541504\pi\)
−0.130018 + 0.991512i \(0.541504\pi\)
\(312\) −4.00000 −0.226455
\(313\) −19.2132 −1.08599 −0.542997 0.839734i \(-0.682711\pi\)
−0.542997 + 0.839734i \(0.682711\pi\)
\(314\) −17.3137 −0.977069
\(315\) −4.41421 −0.248713
\(316\) 0 0
\(317\) 11.3137 0.635441 0.317721 0.948184i \(-0.397083\pi\)
0.317721 + 0.948184i \(0.397083\pi\)
\(318\) 11.6569 0.653684
\(319\) −24.7279 −1.38450
\(320\) 35.3137 1.97410
\(321\) −11.5147 −0.642689
\(322\) −0.242641 −0.0135218
\(323\) −1.75736 −0.0977821
\(324\) 0 0
\(325\) −14.4853 −0.803499
\(326\) −12.0000 −0.664619
\(327\) 12.3431 0.682578
\(328\) −8.97056 −0.495316
\(329\) 4.41421 0.243363
\(330\) 37.4558 2.06188
\(331\) −18.0000 −0.989369 −0.494685 0.869072i \(-0.664716\pi\)
−0.494685 + 0.869072i \(0.664716\pi\)
\(332\) 0 0
\(333\) 6.24264 0.342095
\(334\) −30.2426 −1.65480
\(335\) 10.9706 0.599386
\(336\) 5.65685 0.308607
\(337\) −33.0000 −1.79762 −0.898812 0.438334i \(-0.855569\pi\)
−0.898812 + 0.438334i \(0.855569\pi\)
\(338\) 1.41421 0.0769231
\(339\) 28.7279 1.56029
\(340\) 0 0
\(341\) 22.2426 1.20451
\(342\) −1.75736 −0.0950271
\(343\) 1.00000 0.0539949
\(344\) 14.1421 0.762493
\(345\) 1.07107 0.0576644
\(346\) −1.02944 −0.0553429
\(347\) 5.65685 0.303676 0.151838 0.988405i \(-0.451481\pi\)
0.151838 + 0.988405i \(0.451481\pi\)
\(348\) 0 0
\(349\) −0.272078 −0.0145640 −0.00728200 0.999973i \(-0.502318\pi\)
−0.00728200 + 0.999973i \(0.502318\pi\)
\(350\) 20.4853 1.09498
\(351\) −5.65685 −0.301941
\(352\) 0 0
\(353\) 8.48528 0.451626 0.225813 0.974171i \(-0.427496\pi\)
0.225813 + 0.974171i \(0.427496\pi\)
\(354\) −23.3137 −1.23911
\(355\) 4.72792 0.250932
\(356\) 0 0
\(357\) 2.00000 0.105851
\(358\) −12.7279 −0.672692
\(359\) −8.10051 −0.427528 −0.213764 0.976885i \(-0.568572\pi\)
−0.213764 + 0.976885i \(0.568572\pi\)
\(360\) 12.4853 0.658032
\(361\) −17.4558 −0.918729
\(362\) 9.51472 0.500083
\(363\) −9.89949 −0.519589
\(364\) 0 0
\(365\) −3.34315 −0.174988
\(366\) −12.0000 −0.627250
\(367\) 1.75736 0.0917334 0.0458667 0.998948i \(-0.485395\pi\)
0.0458667 + 0.998948i \(0.485395\pi\)
\(368\) 0.686292 0.0357754
\(369\) −3.17157 −0.165105
\(370\) −38.9706 −2.02598
\(371\) −5.82843 −0.302597
\(372\) 0 0
\(373\) −8.48528 −0.439351 −0.219676 0.975573i \(-0.570500\pi\)
−0.219676 + 0.975573i \(0.570500\pi\)
\(374\) 8.48528 0.438763
\(375\) −59.2132 −3.05776
\(376\) −12.4853 −0.643879
\(377\) −5.82843 −0.300179
\(378\) 8.00000 0.411476
\(379\) 32.2426 1.65619 0.828097 0.560585i \(-0.189424\pi\)
0.828097 + 0.560585i \(0.189424\pi\)
\(380\) 0 0
\(381\) −2.82843 −0.144905
\(382\) 29.4558 1.50709
\(383\) −3.51472 −0.179594 −0.0897969 0.995960i \(-0.528622\pi\)
−0.0897969 + 0.995960i \(0.528622\pi\)
\(384\) −16.0000 −0.816497
\(385\) −18.7279 −0.954463
\(386\) 3.51472 0.178894
\(387\) 5.00000 0.254164
\(388\) 0 0
\(389\) 18.3431 0.930034 0.465017 0.885302i \(-0.346048\pi\)
0.465017 + 0.885302i \(0.346048\pi\)
\(390\) 8.82843 0.447045
\(391\) 0.242641 0.0122709
\(392\) −2.82843 −0.142857
\(393\) −4.00000 −0.201773
\(394\) 33.4558 1.68548
\(395\) −6.55635 −0.329886
\(396\) 0 0
\(397\) 24.2132 1.21523 0.607613 0.794233i \(-0.292127\pi\)
0.607613 + 0.794233i \(0.292127\pi\)
\(398\) −17.3137 −0.867858
\(399\) −1.75736 −0.0879780
\(400\) −57.9411 −2.89706
\(401\) 17.6569 0.881741 0.440871 0.897571i \(-0.354670\pi\)
0.440871 + 0.897571i \(0.354670\pi\)
\(402\) −4.97056 −0.247909
\(403\) 5.24264 0.261155
\(404\) 0 0
\(405\) −22.0711 −1.09672
\(406\) 8.24264 0.409075
\(407\) 26.4853 1.31283
\(408\) −5.65685 −0.280056
\(409\) −5.24264 −0.259232 −0.129616 0.991564i \(-0.541374\pi\)
−0.129616 + 0.991564i \(0.541374\pi\)
\(410\) 19.7990 0.977802
\(411\) 10.4853 0.517201
\(412\) 0 0
\(413\) 11.6569 0.573596
\(414\) 0.242641 0.0119251
\(415\) 21.0000 1.03085
\(416\) 0 0
\(417\) 3.17157 0.155313
\(418\) −7.45584 −0.364677
\(419\) 32.8701 1.60581 0.802904 0.596109i \(-0.203287\pi\)
0.802904 + 0.596109i \(0.203287\pi\)
\(420\) 0 0
\(421\) −18.7279 −0.912743 −0.456372 0.889789i \(-0.650851\pi\)
−0.456372 + 0.889789i \(0.650851\pi\)
\(422\) 25.4142 1.23714
\(423\) −4.41421 −0.214626
\(424\) 16.4853 0.800596
\(425\) −20.4853 −0.993682
\(426\) −2.14214 −0.103787
\(427\) 6.00000 0.290360
\(428\) 0 0
\(429\) −6.00000 −0.289683
\(430\) −31.2132 −1.50523
\(431\) −23.6569 −1.13951 −0.569755 0.821814i \(-0.692962\pi\)
−0.569755 + 0.821814i \(0.692962\pi\)
\(432\) −22.6274 −1.08866
\(433\) 8.97056 0.431098 0.215549 0.976493i \(-0.430846\pi\)
0.215549 + 0.976493i \(0.430846\pi\)
\(434\) −7.41421 −0.355894
\(435\) −36.3848 −1.74452
\(436\) 0 0
\(437\) −0.213203 −0.0101989
\(438\) 1.51472 0.0723761
\(439\) −17.5147 −0.835932 −0.417966 0.908463i \(-0.637257\pi\)
−0.417966 + 0.908463i \(0.637257\pi\)
\(440\) 52.9706 2.52527
\(441\) −1.00000 −0.0476190
\(442\) 2.00000 0.0951303
\(443\) −26.3137 −1.25020 −0.625101 0.780544i \(-0.714942\pi\)
−0.625101 + 0.780544i \(0.714942\pi\)
\(444\) 0 0
\(445\) 19.4853 0.923691
\(446\) 13.0711 0.618933
\(447\) −10.9706 −0.518890
\(448\) 8.00000 0.377964
\(449\) 26.8284 1.26611 0.633056 0.774106i \(-0.281800\pi\)
0.633056 + 0.774106i \(0.281800\pi\)
\(450\) −20.4853 −0.965685
\(451\) −13.4558 −0.633611
\(452\) 0 0
\(453\) 25.7990 1.21214
\(454\) 29.9411 1.40521
\(455\) −4.41421 −0.206942
\(456\) 4.97056 0.232768
\(457\) 35.2132 1.64720 0.823602 0.567168i \(-0.191961\pi\)
0.823602 + 0.567168i \(0.191961\pi\)
\(458\) −30.3431 −1.41784
\(459\) −8.00000 −0.373408
\(460\) 0 0
\(461\) 3.17157 0.147715 0.0738574 0.997269i \(-0.476469\pi\)
0.0738574 + 0.997269i \(0.476469\pi\)
\(462\) 8.48528 0.394771
\(463\) 4.24264 0.197172 0.0985861 0.995129i \(-0.468568\pi\)
0.0985861 + 0.995129i \(0.468568\pi\)
\(464\) −23.3137 −1.08231
\(465\) 32.7279 1.51772
\(466\) −4.72792 −0.219017
\(467\) 15.8995 0.735741 0.367870 0.929877i \(-0.380087\pi\)
0.367870 + 0.929877i \(0.380087\pi\)
\(468\) 0 0
\(469\) 2.48528 0.114760
\(470\) 27.5563 1.27108
\(471\) 17.3137 0.797774
\(472\) −32.9706 −1.51759
\(473\) 21.2132 0.975384
\(474\) 2.97056 0.136442
\(475\) 18.0000 0.825897
\(476\) 0 0
\(477\) 5.82843 0.266865
\(478\) 28.9706 1.32508
\(479\) 36.2132 1.65462 0.827312 0.561743i \(-0.189869\pi\)
0.827312 + 0.561743i \(0.189869\pi\)
\(480\) 0 0
\(481\) 6.24264 0.284640
\(482\) 31.4142 1.43088
\(483\) 0.242641 0.0110405
\(484\) 0 0
\(485\) −60.5980 −2.75161
\(486\) −14.0000 −0.635053
\(487\) 39.4558 1.78791 0.893957 0.448152i \(-0.147918\pi\)
0.893957 + 0.448152i \(0.147918\pi\)
\(488\) −16.9706 −0.768221
\(489\) 12.0000 0.542659
\(490\) 6.24264 0.282014
\(491\) −28.6274 −1.29194 −0.645969 0.763364i \(-0.723546\pi\)
−0.645969 + 0.763364i \(0.723546\pi\)
\(492\) 0 0
\(493\) −8.24264 −0.371230
\(494\) −1.75736 −0.0790673
\(495\) 18.7279 0.841757
\(496\) 20.9706 0.941606
\(497\) 1.07107 0.0480440
\(498\) −9.51472 −0.426365
\(499\) −38.7279 −1.73370 −0.866850 0.498569i \(-0.833859\pi\)
−0.866850 + 0.498569i \(0.833859\pi\)
\(500\) 0 0
\(501\) 30.2426 1.35114
\(502\) 27.4558 1.22541
\(503\) 16.6274 0.741380 0.370690 0.928757i \(-0.379121\pi\)
0.370690 + 0.928757i \(0.379121\pi\)
\(504\) 2.82843 0.125988
\(505\) 7.75736 0.345198
\(506\) 1.02944 0.0457641
\(507\) −1.41421 −0.0628074
\(508\) 0 0
\(509\) −24.8995 −1.10365 −0.551825 0.833960i \(-0.686069\pi\)
−0.551825 + 0.833960i \(0.686069\pi\)
\(510\) 12.4853 0.552858
\(511\) −0.757359 −0.0335036
\(512\) −22.6274 −1.00000
\(513\) 7.02944 0.310357
\(514\) −23.4558 −1.03459
\(515\) −35.3137 −1.55611
\(516\) 0 0
\(517\) −18.7279 −0.823653
\(518\) −8.82843 −0.387899
\(519\) 1.02944 0.0451873
\(520\) 12.4853 0.547516
\(521\) −17.6569 −0.773561 −0.386780 0.922172i \(-0.626413\pi\)
−0.386780 + 0.922172i \(0.626413\pi\)
\(522\) −8.24264 −0.360771
\(523\) −2.97056 −0.129894 −0.0649468 0.997889i \(-0.520688\pi\)
−0.0649468 + 0.997889i \(0.520688\pi\)
\(524\) 0 0
\(525\) −20.4853 −0.894051
\(526\) −45.2132 −1.97139
\(527\) 7.41421 0.322968
\(528\) −24.0000 −1.04447
\(529\) −22.9706 −0.998720
\(530\) −36.3848 −1.58045
\(531\) −11.6569 −0.505864
\(532\) 0 0
\(533\) −3.17157 −0.137376
\(534\) −8.82843 −0.382043
\(535\) 35.9411 1.55387
\(536\) −7.02944 −0.303625
\(537\) 12.7279 0.549250
\(538\) 20.9706 0.904105
\(539\) −4.24264 −0.182743
\(540\) 0 0
\(541\) 35.2132 1.51393 0.756967 0.653453i \(-0.226680\pi\)
0.756967 + 0.653453i \(0.226680\pi\)
\(542\) 28.2843 1.21491
\(543\) −9.51472 −0.408316
\(544\) 0 0
\(545\) −38.5269 −1.65031
\(546\) 2.00000 0.0855921
\(547\) 18.5147 0.791632 0.395816 0.918330i \(-0.370462\pi\)
0.395816 + 0.918330i \(0.370462\pi\)
\(548\) 0 0
\(549\) −6.00000 −0.256074
\(550\) −86.9117 −3.70593
\(551\) 7.24264 0.308547
\(552\) −0.686292 −0.0292105
\(553\) −1.48528 −0.0631606
\(554\) 13.4142 0.569915
\(555\) 38.9706 1.65421
\(556\) 0 0
\(557\) 12.0000 0.508456 0.254228 0.967144i \(-0.418179\pi\)
0.254228 + 0.967144i \(0.418179\pi\)
\(558\) 7.41421 0.313869
\(559\) 5.00000 0.211477
\(560\) −17.6569 −0.746138
\(561\) −8.48528 −0.358249
\(562\) −22.0000 −0.928014
\(563\) −17.6569 −0.744148 −0.372074 0.928203i \(-0.621353\pi\)
−0.372074 + 0.928203i \(0.621353\pi\)
\(564\) 0 0
\(565\) −89.6690 −3.77241
\(566\) 12.0000 0.504398
\(567\) −5.00000 −0.209980
\(568\) −3.02944 −0.127112
\(569\) −23.1421 −0.970169 −0.485084 0.874467i \(-0.661211\pi\)
−0.485084 + 0.874467i \(0.661211\pi\)
\(570\) −10.9706 −0.459506
\(571\) −8.45584 −0.353866 −0.176933 0.984223i \(-0.556618\pi\)
−0.176933 + 0.984223i \(0.556618\pi\)
\(572\) 0 0
\(573\) −29.4558 −1.23054
\(574\) 4.48528 0.187212
\(575\) −2.48528 −0.103643
\(576\) −8.00000 −0.333333
\(577\) 26.9706 1.12280 0.561400 0.827545i \(-0.310263\pi\)
0.561400 + 0.827545i \(0.310263\pi\)
\(578\) −21.2132 −0.882353
\(579\) −3.51472 −0.146067
\(580\) 0 0
\(581\) 4.75736 0.197369
\(582\) 27.4558 1.13808
\(583\) 24.7279 1.02413
\(584\) 2.14214 0.0886422
\(585\) 4.41421 0.182505
\(586\) −30.2426 −1.24931
\(587\) 24.5563 1.01355 0.506775 0.862079i \(-0.330838\pi\)
0.506775 + 0.862079i \(0.330838\pi\)
\(588\) 0 0
\(589\) −6.51472 −0.268434
\(590\) 72.7696 2.99588
\(591\) −33.4558 −1.37619
\(592\) 24.9706 1.02628
\(593\) 30.5563 1.25480 0.627399 0.778698i \(-0.284119\pi\)
0.627399 + 0.778698i \(0.284119\pi\)
\(594\) −33.9411 −1.39262
\(595\) −6.24264 −0.255923
\(596\) 0 0
\(597\) 17.3137 0.708603
\(598\) 0.242641 0.00992232
\(599\) 10.7990 0.441235 0.220617 0.975360i \(-0.429193\pi\)
0.220617 + 0.975360i \(0.429193\pi\)
\(600\) 57.9411 2.36544
\(601\) −5.02944 −0.205155 −0.102578 0.994725i \(-0.532709\pi\)
−0.102578 + 0.994725i \(0.532709\pi\)
\(602\) −7.07107 −0.288195
\(603\) −2.48528 −0.101208
\(604\) 0 0
\(605\) 30.8995 1.25624
\(606\) −3.51472 −0.142776
\(607\) −1.27208 −0.0516321 −0.0258160 0.999667i \(-0.508218\pi\)
−0.0258160 + 0.999667i \(0.508218\pi\)
\(608\) 0 0
\(609\) −8.24264 −0.334009
\(610\) 37.4558 1.51654
\(611\) −4.41421 −0.178580
\(612\) 0 0
\(613\) −16.0000 −0.646234 −0.323117 0.946359i \(-0.604731\pi\)
−0.323117 + 0.946359i \(0.604731\pi\)
\(614\) 6.72792 0.271517
\(615\) −19.7990 −0.798372
\(616\) 12.0000 0.483494
\(617\) 23.6569 0.952389 0.476195 0.879340i \(-0.342016\pi\)
0.476195 + 0.879340i \(0.342016\pi\)
\(618\) 16.0000 0.643614
\(619\) −32.9706 −1.32520 −0.662599 0.748974i \(-0.730547\pi\)
−0.662599 + 0.748974i \(0.730547\pi\)
\(620\) 0 0
\(621\) −0.970563 −0.0389473
\(622\) −6.48528 −0.260036
\(623\) 4.41421 0.176852
\(624\) −5.65685 −0.226455
\(625\) 112.397 4.49588
\(626\) −27.1716 −1.08599
\(627\) 7.45584 0.297758
\(628\) 0 0
\(629\) 8.82843 0.352012
\(630\) −6.24264 −0.248713
\(631\) 2.00000 0.0796187 0.0398094 0.999207i \(-0.487325\pi\)
0.0398094 + 0.999207i \(0.487325\pi\)
\(632\) 4.20101 0.167107
\(633\) −25.4142 −1.01012
\(634\) 16.0000 0.635441
\(635\) 8.82843 0.350345
\(636\) 0 0
\(637\) −1.00000 −0.0396214
\(638\) −34.9706 −1.38450
\(639\) −1.07107 −0.0423708
\(640\) 49.9411 1.97410
\(641\) 26.6569 1.05288 0.526441 0.850212i \(-0.323526\pi\)
0.526441 + 0.850212i \(0.323526\pi\)
\(642\) −16.2843 −0.642689
\(643\) 4.48528 0.176882 0.0884411 0.996081i \(-0.471811\pi\)
0.0884411 + 0.996081i \(0.471811\pi\)
\(644\) 0 0
\(645\) 31.2132 1.22902
\(646\) −2.48528 −0.0977821
\(647\) −50.5269 −1.98642 −0.993209 0.116344i \(-0.962882\pi\)
−0.993209 + 0.116344i \(0.962882\pi\)
\(648\) 14.1421 0.555556
\(649\) −49.4558 −1.94131
\(650\) −20.4853 −0.803499
\(651\) 7.41421 0.290586
\(652\) 0 0
\(653\) −5.31371 −0.207941 −0.103971 0.994580i \(-0.533155\pi\)
−0.103971 + 0.994580i \(0.533155\pi\)
\(654\) 17.4558 0.682578
\(655\) 12.4853 0.487840
\(656\) −12.6863 −0.495316
\(657\) 0.757359 0.0295474
\(658\) 6.24264 0.243363
\(659\) −26.6569 −1.03840 −0.519202 0.854652i \(-0.673771\pi\)
−0.519202 + 0.854652i \(0.673771\pi\)
\(660\) 0 0
\(661\) −19.2426 −0.748452 −0.374226 0.927338i \(-0.622092\pi\)
−0.374226 + 0.927338i \(0.622092\pi\)
\(662\) −25.4558 −0.989369
\(663\) −2.00000 −0.0776736
\(664\) −13.4558 −0.522188
\(665\) 5.48528 0.212710
\(666\) 8.82843 0.342095
\(667\) −1.00000 −0.0387202
\(668\) 0 0
\(669\) −13.0711 −0.505357
\(670\) 15.5147 0.599386
\(671\) −25.4558 −0.982712
\(672\) 0 0
\(673\) 40.9411 1.57816 0.789082 0.614288i \(-0.210557\pi\)
0.789082 + 0.614288i \(0.210557\pi\)
\(674\) −46.6690 −1.79762
\(675\) 81.9411 3.15392
\(676\) 0 0
\(677\) 29.3553 1.12822 0.564109 0.825701i \(-0.309220\pi\)
0.564109 + 0.825701i \(0.309220\pi\)
\(678\) 40.6274 1.56029
\(679\) −13.7279 −0.526829
\(680\) 17.6569 0.677109
\(681\) −29.9411 −1.14735
\(682\) 31.4558 1.20451
\(683\) −26.8284 −1.02656 −0.513281 0.858221i \(-0.671570\pi\)
−0.513281 + 0.858221i \(0.671570\pi\)
\(684\) 0 0
\(685\) −32.7279 −1.25047
\(686\) 1.41421 0.0539949
\(687\) 30.3431 1.15766
\(688\) 20.0000 0.762493
\(689\) 5.82843 0.222045
\(690\) 1.51472 0.0576644
\(691\) −30.6985 −1.16783 −0.583913 0.811816i \(-0.698479\pi\)
−0.583913 + 0.811816i \(0.698479\pi\)
\(692\) 0 0
\(693\) 4.24264 0.161165
\(694\) 8.00000 0.303676
\(695\) −9.89949 −0.375509
\(696\) 23.3137 0.883704
\(697\) −4.48528 −0.169892
\(698\) −0.384776 −0.0145640
\(699\) 4.72792 0.178826
\(700\) 0 0
\(701\) 28.7990 1.08772 0.543861 0.839175i \(-0.316962\pi\)
0.543861 + 0.839175i \(0.316962\pi\)
\(702\) −8.00000 −0.301941
\(703\) −7.75736 −0.292574
\(704\) −33.9411 −1.27920
\(705\) −27.5563 −1.03783
\(706\) 12.0000 0.451626
\(707\) 1.75736 0.0660923
\(708\) 0 0
\(709\) −24.7279 −0.928677 −0.464338 0.885658i \(-0.653708\pi\)
−0.464338 + 0.885658i \(0.653708\pi\)
\(710\) 6.68629 0.250932
\(711\) 1.48528 0.0557024
\(712\) −12.4853 −0.467906
\(713\) 0.899495 0.0336864
\(714\) 2.82843 0.105851
\(715\) 18.7279 0.700385
\(716\) 0 0
\(717\) −28.9706 −1.08193
\(718\) −11.4558 −0.427528
\(719\) −10.2426 −0.381986 −0.190993 0.981591i \(-0.561171\pi\)
−0.190993 + 0.981591i \(0.561171\pi\)
\(720\) 17.6569 0.658032
\(721\) −8.00000 −0.297936
\(722\) −24.6863 −0.918729
\(723\) −31.4142 −1.16831
\(724\) 0 0
\(725\) 84.4264 3.13552
\(726\) −14.0000 −0.519589
\(727\) 42.0000 1.55769 0.778847 0.627214i \(-0.215805\pi\)
0.778847 + 0.627214i \(0.215805\pi\)
\(728\) 2.82843 0.104828
\(729\) 29.0000 1.07407
\(730\) −4.72792 −0.174988
\(731\) 7.07107 0.261533
\(732\) 0 0
\(733\) 42.6985 1.57710 0.788552 0.614968i \(-0.210831\pi\)
0.788552 + 0.614968i \(0.210831\pi\)
\(734\) 2.48528 0.0917334
\(735\) −6.24264 −0.230263
\(736\) 0 0
\(737\) −10.5442 −0.388399
\(738\) −4.48528 −0.165105
\(739\) 41.6985 1.53390 0.766952 0.641705i \(-0.221773\pi\)
0.766952 + 0.641705i \(0.221773\pi\)
\(740\) 0 0
\(741\) 1.75736 0.0645582
\(742\) −8.24264 −0.302597
\(743\) −18.3431 −0.672945 −0.336472 0.941693i \(-0.609234\pi\)
−0.336472 + 0.941693i \(0.609234\pi\)
\(744\) −20.9706 −0.768818
\(745\) 34.2426 1.25455
\(746\) −12.0000 −0.439351
\(747\) −4.75736 −0.174063
\(748\) 0 0
\(749\) 8.14214 0.297507
\(750\) −83.7401 −3.05776
\(751\) −1.48528 −0.0541987 −0.0270993 0.999633i \(-0.508627\pi\)
−0.0270993 + 0.999633i \(0.508627\pi\)
\(752\) −17.6569 −0.643879
\(753\) −27.4558 −1.00055
\(754\) −8.24264 −0.300179
\(755\) −80.5269 −2.93067
\(756\) 0 0
\(757\) 4.51472 0.164090 0.0820451 0.996629i \(-0.473855\pi\)
0.0820451 + 0.996629i \(0.473855\pi\)
\(758\) 45.5980 1.65619
\(759\) −1.02944 −0.0373662
\(760\) −15.5147 −0.562778
\(761\) 43.2426 1.56754 0.783772 0.621048i \(-0.213293\pi\)
0.783772 + 0.621048i \(0.213293\pi\)
\(762\) −4.00000 −0.144905
\(763\) −8.72792 −0.315972
\(764\) 0 0
\(765\) 6.24264 0.225703
\(766\) −4.97056 −0.179594
\(767\) −11.6569 −0.420905
\(768\) 0 0
\(769\) 9.78680 0.352921 0.176460 0.984308i \(-0.443535\pi\)
0.176460 + 0.984308i \(0.443535\pi\)
\(770\) −26.4853 −0.954463
\(771\) 23.4558 0.844742
\(772\) 0 0
\(773\) −27.1716 −0.977294 −0.488647 0.872482i \(-0.662509\pi\)
−0.488647 + 0.872482i \(0.662509\pi\)
\(774\) 7.07107 0.254164
\(775\) −75.9411 −2.72789
\(776\) 38.8284 1.39386
\(777\) 8.82843 0.316718
\(778\) 25.9411 0.930034
\(779\) 3.94113 0.141205
\(780\) 0 0
\(781\) −4.54416 −0.162603
\(782\) 0.343146 0.0122709
\(783\) 32.9706 1.17827
\(784\) −4.00000 −0.142857
\(785\) −54.0416 −1.92883
\(786\) −5.65685 −0.201773
\(787\) −33.2426 −1.18497 −0.592486 0.805581i \(-0.701853\pi\)
−0.592486 + 0.805581i \(0.701853\pi\)
\(788\) 0 0
\(789\) 45.2132 1.60963
\(790\) −9.27208 −0.329886
\(791\) −20.3137 −0.722272
\(792\) −12.0000 −0.426401
\(793\) −6.00000 −0.213066
\(794\) 34.2426 1.21523
\(795\) 36.3848 1.29044
\(796\) 0 0
\(797\) −35.6569 −1.26303 −0.631515 0.775363i \(-0.717567\pi\)
−0.631515 + 0.775363i \(0.717567\pi\)
\(798\) −2.48528 −0.0879780
\(799\) −6.24264 −0.220849
\(800\) 0 0
\(801\) −4.41421 −0.155969
\(802\) 24.9706 0.881741
\(803\) 3.21320 0.113391
\(804\) 0 0
\(805\) −0.757359 −0.0266934
\(806\) 7.41421 0.261155
\(807\) −20.9706 −0.738199
\(808\) −4.97056 −0.174864
\(809\) 0.514719 0.0180965 0.00904827 0.999959i \(-0.497120\pi\)
0.00904827 + 0.999959i \(0.497120\pi\)
\(810\) −31.2132 −1.09672
\(811\) 45.9411 1.61321 0.806606 0.591090i \(-0.201302\pi\)
0.806606 + 0.591090i \(0.201302\pi\)
\(812\) 0 0
\(813\) −28.2843 −0.991973
\(814\) 37.4558 1.31283
\(815\) −37.4558 −1.31202
\(816\) −8.00000 −0.280056
\(817\) −6.21320 −0.217372
\(818\) −7.41421 −0.259232
\(819\) 1.00000 0.0349428
\(820\) 0 0
\(821\) 8.14214 0.284162 0.142081 0.989855i \(-0.454621\pi\)
0.142081 + 0.989855i \(0.454621\pi\)
\(822\) 14.8284 0.517201
\(823\) −6.48528 −0.226063 −0.113031 0.993591i \(-0.536056\pi\)
−0.113031 + 0.993591i \(0.536056\pi\)
\(824\) 22.6274 0.788263
\(825\) 86.9117 3.02588
\(826\) 16.4853 0.573596
\(827\) 1.45584 0.0506247 0.0253123 0.999680i \(-0.491942\pi\)
0.0253123 + 0.999680i \(0.491942\pi\)
\(828\) 0 0
\(829\) 25.2721 0.877736 0.438868 0.898552i \(-0.355380\pi\)
0.438868 + 0.898552i \(0.355380\pi\)
\(830\) 29.6985 1.03085
\(831\) −13.4142 −0.465334
\(832\) −8.00000 −0.277350
\(833\) −1.41421 −0.0489996
\(834\) 4.48528 0.155313
\(835\) −94.3970 −3.26674
\(836\) 0 0
\(837\) −29.6569 −1.02509
\(838\) 46.4853 1.60581
\(839\) −38.8284 −1.34051 −0.670253 0.742133i \(-0.733814\pi\)
−0.670253 + 0.742133i \(0.733814\pi\)
\(840\) 17.6569 0.609219
\(841\) 4.97056 0.171399
\(842\) −26.4853 −0.912743
\(843\) 22.0000 0.757720
\(844\) 0 0
\(845\) 4.41421 0.151854
\(846\) −6.24264 −0.214626
\(847\) 7.00000 0.240523
\(848\) 23.3137 0.800596
\(849\) −12.0000 −0.411839
\(850\) −28.9706 −0.993682
\(851\) 1.07107 0.0367157
\(852\) 0 0
\(853\) 14.2721 0.488667 0.244333 0.969691i \(-0.421431\pi\)
0.244333 + 0.969691i \(0.421431\pi\)
\(854\) 8.48528 0.290360
\(855\) −5.48528 −0.187593
\(856\) −23.0294 −0.787130
\(857\) 18.7696 0.641156 0.320578 0.947222i \(-0.396123\pi\)
0.320578 + 0.947222i \(0.396123\pi\)
\(858\) −8.48528 −0.289683
\(859\) −2.97056 −0.101354 −0.0506771 0.998715i \(-0.516138\pi\)
−0.0506771 + 0.998715i \(0.516138\pi\)
\(860\) 0 0
\(861\) −4.48528 −0.152858
\(862\) −33.4558 −1.13951
\(863\) −28.2843 −0.962808 −0.481404 0.876499i \(-0.659873\pi\)
−0.481404 + 0.876499i \(0.659873\pi\)
\(864\) 0 0
\(865\) −3.21320 −0.109252
\(866\) 12.6863 0.431098
\(867\) 21.2132 0.720438
\(868\) 0 0
\(869\) 6.30152 0.213764
\(870\) −51.4558 −1.74452
\(871\) −2.48528 −0.0842105
\(872\) 24.6863 0.835983
\(873\) 13.7279 0.464620
\(874\) −0.301515 −0.0101989
\(875\) 41.8701 1.41547
\(876\) 0 0
\(877\) −1.75736 −0.0593418 −0.0296709 0.999560i \(-0.509446\pi\)
−0.0296709 + 0.999560i \(0.509446\pi\)
\(878\) −24.7696 −0.835932
\(879\) 30.2426 1.02006
\(880\) 74.9117 2.52527
\(881\) −49.1127 −1.65465 −0.827324 0.561724i \(-0.810138\pi\)
−0.827324 + 0.561724i \(0.810138\pi\)
\(882\) −1.41421 −0.0476190
\(883\) 2.00000 0.0673054 0.0336527 0.999434i \(-0.489286\pi\)
0.0336527 + 0.999434i \(0.489286\pi\)
\(884\) 0 0
\(885\) −72.7696 −2.44612
\(886\) −37.2132 −1.25020
\(887\) −43.1127 −1.44758 −0.723791 0.690019i \(-0.757602\pi\)
−0.723791 + 0.690019i \(0.757602\pi\)
\(888\) −24.9706 −0.837957
\(889\) 2.00000 0.0670778
\(890\) 27.5563 0.923691
\(891\) 21.2132 0.710669
\(892\) 0 0
\(893\) 5.48528 0.183558
\(894\) −15.5147 −0.518890
\(895\) −39.7279 −1.32796
\(896\) 11.3137 0.377964
\(897\) −0.242641 −0.00810154
\(898\) 37.9411 1.26611
\(899\) −30.5563 −1.01911
\(900\) 0 0
\(901\) 8.24264 0.274602
\(902\) −19.0294 −0.633611
\(903\) 7.07107 0.235310
\(904\) 57.4558 1.91095
\(905\) 29.6985 0.987211
\(906\) 36.4853 1.21214
\(907\) −31.9706 −1.06157 −0.530783 0.847508i \(-0.678102\pi\)
−0.530783 + 0.847508i \(0.678102\pi\)
\(908\) 0 0
\(909\) −1.75736 −0.0582879
\(910\) −6.24264 −0.206942
\(911\) −10.0294 −0.332290 −0.166145 0.986101i \(-0.553132\pi\)
−0.166145 + 0.986101i \(0.553132\pi\)
\(912\) 7.02944 0.232768
\(913\) −20.1838 −0.667985
\(914\) 49.7990 1.64720
\(915\) −37.4558 −1.23825
\(916\) 0 0
\(917\) 2.82843 0.0934029
\(918\) −11.3137 −0.373408
\(919\) 24.0000 0.791687 0.395843 0.918318i \(-0.370452\pi\)
0.395843 + 0.918318i \(0.370452\pi\)
\(920\) 2.14214 0.0706241
\(921\) −6.72792 −0.221693
\(922\) 4.48528 0.147715
\(923\) −1.07107 −0.0352546
\(924\) 0 0
\(925\) −90.4264 −2.97320
\(926\) 6.00000 0.197172
\(927\) 8.00000 0.262754
\(928\) 0 0
\(929\) 17.1005 0.561049 0.280525 0.959847i \(-0.409492\pi\)
0.280525 + 0.959847i \(0.409492\pi\)
\(930\) 46.2843 1.51772
\(931\) 1.24264 0.0407259
\(932\) 0 0
\(933\) 6.48528 0.212319
\(934\) 22.4853 0.735741
\(935\) 26.4853 0.866161
\(936\) −2.82843 −0.0924500
\(937\) 2.78680 0.0910407 0.0455203 0.998963i \(-0.485505\pi\)
0.0455203 + 0.998963i \(0.485505\pi\)
\(938\) 3.51472 0.114760
\(939\) 27.1716 0.886711
\(940\) 0 0
\(941\) 25.9289 0.845259 0.422630 0.906303i \(-0.361107\pi\)
0.422630 + 0.906303i \(0.361107\pi\)
\(942\) 24.4853 0.797774
\(943\) −0.544156 −0.0177202
\(944\) −46.6274 −1.51759
\(945\) 24.9706 0.812292
\(946\) 30.0000 0.975384
\(947\) 20.8284 0.676833 0.338416 0.940996i \(-0.390109\pi\)
0.338416 + 0.940996i \(0.390109\pi\)
\(948\) 0 0
\(949\) 0.757359 0.0245849
\(950\) 25.4558 0.825897
\(951\) −16.0000 −0.518836
\(952\) 4.00000 0.129641
\(953\) −17.1421 −0.555288 −0.277644 0.960684i \(-0.589554\pi\)
−0.277644 + 0.960684i \(0.589554\pi\)
\(954\) 8.24264 0.266865
\(955\) 91.9411 2.97514
\(956\) 0 0
\(957\) 34.9706 1.13044
\(958\) 51.2132 1.65462
\(959\) −7.41421 −0.239417
\(960\) −49.9411 −1.61184
\(961\) −3.51472 −0.113378
\(962\) 8.82843 0.284640
\(963\) −8.14214 −0.262377
\(964\) 0 0
\(965\) 10.9706 0.353155
\(966\) 0.343146 0.0110405
\(967\) −41.6985 −1.34093 −0.670466 0.741940i \(-0.733906\pi\)
−0.670466 + 0.741940i \(0.733906\pi\)
\(968\) −19.7990 −0.636364
\(969\) 2.48528 0.0798387
\(970\) −85.6985 −2.75161
\(971\) −7.45584 −0.239269 −0.119635 0.992818i \(-0.538172\pi\)
−0.119635 + 0.992818i \(0.538172\pi\)
\(972\) 0 0
\(973\) −2.24264 −0.0718958
\(974\) 55.7990 1.78791
\(975\) 20.4853 0.656054
\(976\) −24.0000 −0.768221
\(977\) −24.0416 −0.769160 −0.384580 0.923092i \(-0.625654\pi\)
−0.384580 + 0.923092i \(0.625654\pi\)
\(978\) 16.9706 0.542659
\(979\) −18.7279 −0.598547
\(980\) 0 0
\(981\) 8.72792 0.278661
\(982\) −40.4853 −1.29194
\(983\) 33.0416 1.05386 0.526932 0.849907i \(-0.323342\pi\)
0.526932 + 0.849907i \(0.323342\pi\)
\(984\) 12.6863 0.404424
\(985\) 104.426 3.32730
\(986\) −11.6569 −0.371230
\(987\) −6.24264 −0.198705
\(988\) 0 0
\(989\) 0.857864 0.0272785
\(990\) 26.4853 0.841757
\(991\) 34.9706 1.11088 0.555438 0.831558i \(-0.312551\pi\)
0.555438 + 0.831558i \(0.312551\pi\)
\(992\) 0 0
\(993\) 25.4558 0.807817
\(994\) 1.51472 0.0480440
\(995\) −54.0416 −1.71323
\(996\) 0 0
\(997\) −28.4853 −0.902138 −0.451069 0.892489i \(-0.648957\pi\)
−0.451069 + 0.892489i \(0.648957\pi\)
\(998\) −54.7696 −1.73370
\(999\) −35.3137 −1.11728
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 91.2.a.c.1.2 2
3.2 odd 2 819.2.a.h.1.1 2
4.3 odd 2 1456.2.a.q.1.2 2
5.4 even 2 2275.2.a.j.1.1 2
7.2 even 3 637.2.e.f.508.1 4
7.3 odd 6 637.2.e.g.79.1 4
7.4 even 3 637.2.e.f.79.1 4
7.5 odd 6 637.2.e.g.508.1 4
7.6 odd 2 637.2.a.g.1.2 2
8.3 odd 2 5824.2.a.bk.1.1 2
8.5 even 2 5824.2.a.bl.1.2 2
13.5 odd 4 1183.2.c.d.337.1 4
13.8 odd 4 1183.2.c.d.337.3 4
13.12 even 2 1183.2.a.d.1.1 2
21.20 even 2 5733.2.a.s.1.1 2
91.90 odd 2 8281.2.a.v.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
91.2.a.c.1.2 2 1.1 even 1 trivial
637.2.a.g.1.2 2 7.6 odd 2
637.2.e.f.79.1 4 7.4 even 3
637.2.e.f.508.1 4 7.2 even 3
637.2.e.g.79.1 4 7.3 odd 6
637.2.e.g.508.1 4 7.5 odd 6
819.2.a.h.1.1 2 3.2 odd 2
1183.2.a.d.1.1 2 13.12 even 2
1183.2.c.d.337.1 4 13.5 odd 4
1183.2.c.d.337.3 4 13.8 odd 4
1456.2.a.q.1.2 2 4.3 odd 2
2275.2.a.j.1.1 2 5.4 even 2
5733.2.a.s.1.1 2 21.20 even 2
5824.2.a.bk.1.1 2 8.3 odd 2
5824.2.a.bl.1.2 2 8.5 even 2
8281.2.a.v.1.1 2 91.90 odd 2