Properties

Label 2646.2.a.be.1.1
Level $2646$
Weight $2$
Character 2646.1
Self dual yes
Analytic conductor $21.128$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 2646 = 2 \cdot 3^{3} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2646.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(21.1284163748\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{8})^+\)
Defining polynomial: \(x^{2} - 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.41421\) of defining polynomial
Character \(\chi\) \(=\) 2646.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{4} -4.41421 q^{5} -1.00000 q^{8} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{4} -4.41421 q^{5} -1.00000 q^{8} +4.41421 q^{10} -1.00000 q^{11} -4.24264 q^{13} +1.00000 q^{16} +2.82843 q^{17} +5.82843 q^{19} -4.41421 q^{20} +1.00000 q^{22} +5.24264 q^{23} +14.4853 q^{25} +4.24264 q^{26} -0.242641 q^{29} +7.24264 q^{31} -1.00000 q^{32} -2.82843 q^{34} +5.24264 q^{37} -5.82843 q^{38} +4.41421 q^{40} -6.17157 q^{41} -6.48528 q^{43} -1.00000 q^{44} -5.24264 q^{46} -11.6569 q^{47} -14.4853 q^{50} -4.24264 q^{52} +8.00000 q^{53} +4.41421 q^{55} +0.242641 q^{58} -11.6569 q^{59} -14.8284 q^{61} -7.24264 q^{62} +1.00000 q^{64} +18.7279 q^{65} -1.75736 q^{67} +2.82843 q^{68} -1.24264 q^{71} +3.17157 q^{73} -5.24264 q^{74} +5.82843 q^{76} -6.48528 q^{79} -4.41421 q^{80} +6.17157 q^{82} +3.89949 q^{83} -12.4853 q^{85} +6.48528 q^{86} +1.00000 q^{88} -3.34315 q^{89} +5.24264 q^{92} +11.6569 q^{94} -25.7279 q^{95} +11.3137 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} + 2 q^{4} - 6 q^{5} - 2 q^{8} + O(q^{10}) \) \( 2 q - 2 q^{2} + 2 q^{4} - 6 q^{5} - 2 q^{8} + 6 q^{10} - 2 q^{11} + 2 q^{16} + 6 q^{19} - 6 q^{20} + 2 q^{22} + 2 q^{23} + 12 q^{25} + 8 q^{29} + 6 q^{31} - 2 q^{32} + 2 q^{37} - 6 q^{38} + 6 q^{40} - 18 q^{41} + 4 q^{43} - 2 q^{44} - 2 q^{46} - 12 q^{47} - 12 q^{50} + 16 q^{53} + 6 q^{55} - 8 q^{58} - 12 q^{59} - 24 q^{61} - 6 q^{62} + 2 q^{64} + 12 q^{65} - 12 q^{67} + 6 q^{71} + 12 q^{73} - 2 q^{74} + 6 q^{76} + 4 q^{79} - 6 q^{80} + 18 q^{82} - 12 q^{83} - 8 q^{85} - 4 q^{86} + 2 q^{88} - 18 q^{89} + 2 q^{92} + 12 q^{94} - 26 q^{95} + O(q^{100}) \)

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 0
\(4\) 1.00000 0.500000
\(5\) −4.41421 −1.97410 −0.987048 0.160424i \(-0.948714\pi\)
−0.987048 + 0.160424i \(0.948714\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) −1.00000 −0.353553
\(9\) 0 0
\(10\) 4.41421 1.39590
\(11\) −1.00000 −0.301511 −0.150756 0.988571i \(-0.548171\pi\)
−0.150756 + 0.988571i \(0.548171\pi\)
\(12\) 0 0
\(13\) −4.24264 −1.17670 −0.588348 0.808608i \(-0.700222\pi\)
−0.588348 + 0.808608i \(0.700222\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 2.82843 0.685994 0.342997 0.939336i \(-0.388558\pi\)
0.342997 + 0.939336i \(0.388558\pi\)
\(18\) 0 0
\(19\) 5.82843 1.33713 0.668566 0.743652i \(-0.266908\pi\)
0.668566 + 0.743652i \(0.266908\pi\)
\(20\) −4.41421 −0.987048
\(21\) 0 0
\(22\) 1.00000 0.213201
\(23\) 5.24264 1.09317 0.546583 0.837405i \(-0.315928\pi\)
0.546583 + 0.837405i \(0.315928\pi\)
\(24\) 0 0
\(25\) 14.4853 2.89706
\(26\) 4.24264 0.832050
\(27\) 0 0
\(28\) 0 0
\(29\) −0.242641 −0.0450572 −0.0225286 0.999746i \(-0.507172\pi\)
−0.0225286 + 0.999746i \(0.507172\pi\)
\(30\) 0 0
\(31\) 7.24264 1.30082 0.650408 0.759585i \(-0.274598\pi\)
0.650408 + 0.759585i \(0.274598\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0 0
\(34\) −2.82843 −0.485071
\(35\) 0 0
\(36\) 0 0
\(37\) 5.24264 0.861885 0.430942 0.902379i \(-0.358181\pi\)
0.430942 + 0.902379i \(0.358181\pi\)
\(38\) −5.82843 −0.945496
\(39\) 0 0
\(40\) 4.41421 0.697948
\(41\) −6.17157 −0.963838 −0.481919 0.876216i \(-0.660060\pi\)
−0.481919 + 0.876216i \(0.660060\pi\)
\(42\) 0 0
\(43\) −6.48528 −0.988996 −0.494498 0.869179i \(-0.664648\pi\)
−0.494498 + 0.869179i \(0.664648\pi\)
\(44\) −1.00000 −0.150756
\(45\) 0 0
\(46\) −5.24264 −0.772985
\(47\) −11.6569 −1.70033 −0.850163 0.526519i \(-0.823497\pi\)
−0.850163 + 0.526519i \(0.823497\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) −14.4853 −2.04853
\(51\) 0 0
\(52\) −4.24264 −0.588348
\(53\) 8.00000 1.09888 0.549442 0.835532i \(-0.314840\pi\)
0.549442 + 0.835532i \(0.314840\pi\)
\(54\) 0 0
\(55\) 4.41421 0.595212
\(56\) 0 0
\(57\) 0 0
\(58\) 0.242641 0.0318603
\(59\) −11.6569 −1.51759 −0.758797 0.651328i \(-0.774212\pi\)
−0.758797 + 0.651328i \(0.774212\pi\)
\(60\) 0 0
\(61\) −14.8284 −1.89859 −0.949293 0.314393i \(-0.898199\pi\)
−0.949293 + 0.314393i \(0.898199\pi\)
\(62\) −7.24264 −0.919816
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 18.7279 2.32291
\(66\) 0 0
\(67\) −1.75736 −0.214696 −0.107348 0.994222i \(-0.534236\pi\)
−0.107348 + 0.994222i \(0.534236\pi\)
\(68\) 2.82843 0.342997
\(69\) 0 0
\(70\) 0 0
\(71\) −1.24264 −0.147474 −0.0737372 0.997278i \(-0.523493\pi\)
−0.0737372 + 0.997278i \(0.523493\pi\)
\(72\) 0 0
\(73\) 3.17157 0.371205 0.185602 0.982625i \(-0.440576\pi\)
0.185602 + 0.982625i \(0.440576\pi\)
\(74\) −5.24264 −0.609445
\(75\) 0 0
\(76\) 5.82843 0.668566
\(77\) 0 0
\(78\) 0 0
\(79\) −6.48528 −0.729651 −0.364826 0.931076i \(-0.618871\pi\)
−0.364826 + 0.931076i \(0.618871\pi\)
\(80\) −4.41421 −0.493524
\(81\) 0 0
\(82\) 6.17157 0.681536
\(83\) 3.89949 0.428025 0.214013 0.976831i \(-0.431347\pi\)
0.214013 + 0.976831i \(0.431347\pi\)
\(84\) 0 0
\(85\) −12.4853 −1.35422
\(86\) 6.48528 0.699326
\(87\) 0 0
\(88\) 1.00000 0.106600
\(89\) −3.34315 −0.354373 −0.177186 0.984177i \(-0.556700\pi\)
−0.177186 + 0.984177i \(0.556700\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 5.24264 0.546583
\(93\) 0 0
\(94\) 11.6569 1.20231
\(95\) −25.7279 −2.63963
\(96\) 0 0
\(97\) 11.3137 1.14873 0.574367 0.818598i \(-0.305248\pi\)
0.574367 + 0.818598i \(0.305248\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 14.4853 1.44853
\(101\) 14.4853 1.44134 0.720670 0.693279i \(-0.243834\pi\)
0.720670 + 0.693279i \(0.243834\pi\)
\(102\) 0 0
\(103\) −10.0711 −0.992332 −0.496166 0.868228i \(-0.665259\pi\)
−0.496166 + 0.868228i \(0.665259\pi\)
\(104\) 4.24264 0.416025
\(105\) 0 0
\(106\) −8.00000 −0.777029
\(107\) 14.9706 1.44726 0.723629 0.690189i \(-0.242472\pi\)
0.723629 + 0.690189i \(0.242472\pi\)
\(108\) 0 0
\(109\) −15.2426 −1.45998 −0.729990 0.683458i \(-0.760475\pi\)
−0.729990 + 0.683458i \(0.760475\pi\)
\(110\) −4.41421 −0.420879
\(111\) 0 0
\(112\) 0 0
\(113\) 4.24264 0.399114 0.199557 0.979886i \(-0.436050\pi\)
0.199557 + 0.979886i \(0.436050\pi\)
\(114\) 0 0
\(115\) −23.1421 −2.15802
\(116\) −0.242641 −0.0225286
\(117\) 0 0
\(118\) 11.6569 1.07310
\(119\) 0 0
\(120\) 0 0
\(121\) −10.0000 −0.909091
\(122\) 14.8284 1.34250
\(123\) 0 0
\(124\) 7.24264 0.650408
\(125\) −41.8701 −3.74497
\(126\) 0 0
\(127\) −12.2426 −1.08636 −0.543179 0.839617i \(-0.682780\pi\)
−0.543179 + 0.839617i \(0.682780\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 0 0
\(130\) −18.7279 −1.64255
\(131\) −14.4853 −1.26558 −0.632792 0.774321i \(-0.718091\pi\)
−0.632792 + 0.774321i \(0.718091\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 1.75736 0.151813
\(135\) 0 0
\(136\) −2.82843 −0.242536
\(137\) 1.75736 0.150141 0.0750707 0.997178i \(-0.476082\pi\)
0.0750707 + 0.997178i \(0.476082\pi\)
\(138\) 0 0
\(139\) 8.82843 0.748817 0.374409 0.927264i \(-0.377846\pi\)
0.374409 + 0.927264i \(0.377846\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 1.24264 0.104280
\(143\) 4.24264 0.354787
\(144\) 0 0
\(145\) 1.07107 0.0889473
\(146\) −3.17157 −0.262481
\(147\) 0 0
\(148\) 5.24264 0.430942
\(149\) −5.75736 −0.471661 −0.235831 0.971794i \(-0.575781\pi\)
−0.235831 + 0.971794i \(0.575781\pi\)
\(150\) 0 0
\(151\) 14.7279 1.19854 0.599271 0.800546i \(-0.295457\pi\)
0.599271 + 0.800546i \(0.295457\pi\)
\(152\) −5.82843 −0.472748
\(153\) 0 0
\(154\) 0 0
\(155\) −31.9706 −2.56794
\(156\) 0 0
\(157\) −4.92893 −0.393372 −0.196686 0.980467i \(-0.563018\pi\)
−0.196686 + 0.980467i \(0.563018\pi\)
\(158\) 6.48528 0.515941
\(159\) 0 0
\(160\) 4.41421 0.348974
\(161\) 0 0
\(162\) 0 0
\(163\) −14.7279 −1.15358 −0.576790 0.816893i \(-0.695695\pi\)
−0.576790 + 0.816893i \(0.695695\pi\)
\(164\) −6.17157 −0.481919
\(165\) 0 0
\(166\) −3.89949 −0.302660
\(167\) 12.3848 0.958363 0.479181 0.877716i \(-0.340934\pi\)
0.479181 + 0.877716i \(0.340934\pi\)
\(168\) 0 0
\(169\) 5.00000 0.384615
\(170\) 12.4853 0.957577
\(171\) 0 0
\(172\) −6.48528 −0.494498
\(173\) 13.2426 1.00682 0.503410 0.864048i \(-0.332079\pi\)
0.503410 + 0.864048i \(0.332079\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −1.00000 −0.0753778
\(177\) 0 0
\(178\) 3.34315 0.250579
\(179\) −10.9706 −0.819978 −0.409989 0.912090i \(-0.634468\pi\)
−0.409989 + 0.912090i \(0.634468\pi\)
\(180\) 0 0
\(181\) −10.2426 −0.761329 −0.380665 0.924713i \(-0.624305\pi\)
−0.380665 + 0.924713i \(0.624305\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) −5.24264 −0.386493
\(185\) −23.1421 −1.70144
\(186\) 0 0
\(187\) −2.82843 −0.206835
\(188\) −11.6569 −0.850163
\(189\) 0 0
\(190\) 25.7279 1.86650
\(191\) −0.757359 −0.0548006 −0.0274003 0.999625i \(-0.508723\pi\)
−0.0274003 + 0.999625i \(0.508723\pi\)
\(192\) 0 0
\(193\) 4.48528 0.322858 0.161429 0.986884i \(-0.448390\pi\)
0.161429 + 0.986884i \(0.448390\pi\)
\(194\) −11.3137 −0.812277
\(195\) 0 0
\(196\) 0 0
\(197\) −2.72792 −0.194356 −0.0971782 0.995267i \(-0.530982\pi\)
−0.0971782 + 0.995267i \(0.530982\pi\)
\(198\) 0 0
\(199\) −4.75736 −0.337240 −0.168620 0.985681i \(-0.553931\pi\)
−0.168620 + 0.985681i \(0.553931\pi\)
\(200\) −14.4853 −1.02426
\(201\) 0 0
\(202\) −14.4853 −1.01918
\(203\) 0 0
\(204\) 0 0
\(205\) 27.2426 1.90271
\(206\) 10.0711 0.701685
\(207\) 0 0
\(208\) −4.24264 −0.294174
\(209\) −5.82843 −0.403161
\(210\) 0 0
\(211\) 12.7279 0.876226 0.438113 0.898920i \(-0.355647\pi\)
0.438113 + 0.898920i \(0.355647\pi\)
\(212\) 8.00000 0.549442
\(213\) 0 0
\(214\) −14.9706 −1.02337
\(215\) 28.6274 1.95237
\(216\) 0 0
\(217\) 0 0
\(218\) 15.2426 1.03236
\(219\) 0 0
\(220\) 4.41421 0.297606
\(221\) −12.0000 −0.807207
\(222\) 0 0
\(223\) −7.24264 −0.485003 −0.242502 0.970151i \(-0.577968\pi\)
−0.242502 + 0.970151i \(0.577968\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) −4.24264 −0.282216
\(227\) −12.7279 −0.844782 −0.422391 0.906414i \(-0.638809\pi\)
−0.422391 + 0.906414i \(0.638809\pi\)
\(228\) 0 0
\(229\) −12.0000 −0.792982 −0.396491 0.918039i \(-0.629772\pi\)
−0.396491 + 0.918039i \(0.629772\pi\)
\(230\) 23.1421 1.52595
\(231\) 0 0
\(232\) 0.242641 0.0159301
\(233\) 16.2426 1.06409 0.532045 0.846716i \(-0.321424\pi\)
0.532045 + 0.846716i \(0.321424\pi\)
\(234\) 0 0
\(235\) 51.4558 3.35661
\(236\) −11.6569 −0.758797
\(237\) 0 0
\(238\) 0 0
\(239\) −18.0000 −1.16432 −0.582162 0.813073i \(-0.697793\pi\)
−0.582162 + 0.813073i \(0.697793\pi\)
\(240\) 0 0
\(241\) 2.48528 0.160091 0.0800455 0.996791i \(-0.474493\pi\)
0.0800455 + 0.996791i \(0.474493\pi\)
\(242\) 10.0000 0.642824
\(243\) 0 0
\(244\) −14.8284 −0.949293
\(245\) 0 0
\(246\) 0 0
\(247\) −24.7279 −1.57340
\(248\) −7.24264 −0.459908
\(249\) 0 0
\(250\) 41.8701 2.64809
\(251\) −11.3137 −0.714115 −0.357057 0.934082i \(-0.616220\pi\)
−0.357057 + 0.934082i \(0.616220\pi\)
\(252\) 0 0
\(253\) −5.24264 −0.329602
\(254\) 12.2426 0.768172
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 11.4853 0.716432 0.358216 0.933639i \(-0.383385\pi\)
0.358216 + 0.933639i \(0.383385\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 18.7279 1.16146
\(261\) 0 0
\(262\) 14.4853 0.894904
\(263\) 3.72792 0.229874 0.114937 0.993373i \(-0.463333\pi\)
0.114937 + 0.993373i \(0.463333\pi\)
\(264\) 0 0
\(265\) −35.3137 −2.16930
\(266\) 0 0
\(267\) 0 0
\(268\) −1.75736 −0.107348
\(269\) 15.3848 0.938026 0.469013 0.883191i \(-0.344610\pi\)
0.469013 + 0.883191i \(0.344610\pi\)
\(270\) 0 0
\(271\) −6.68629 −0.406163 −0.203082 0.979162i \(-0.565096\pi\)
−0.203082 + 0.979162i \(0.565096\pi\)
\(272\) 2.82843 0.171499
\(273\) 0 0
\(274\) −1.75736 −0.106166
\(275\) −14.4853 −0.873495
\(276\) 0 0
\(277\) −14.2132 −0.853989 −0.426994 0.904254i \(-0.640428\pi\)
−0.426994 + 0.904254i \(0.640428\pi\)
\(278\) −8.82843 −0.529494
\(279\) 0 0
\(280\) 0 0
\(281\) 10.4853 0.625499 0.312750 0.949836i \(-0.398750\pi\)
0.312750 + 0.949836i \(0.398750\pi\)
\(282\) 0 0
\(283\) 8.14214 0.484000 0.242000 0.970276i \(-0.422197\pi\)
0.242000 + 0.970276i \(0.422197\pi\)
\(284\) −1.24264 −0.0737372
\(285\) 0 0
\(286\) −4.24264 −0.250873
\(287\) 0 0
\(288\) 0 0
\(289\) −9.00000 −0.529412
\(290\) −1.07107 −0.0628953
\(291\) 0 0
\(292\) 3.17157 0.185602
\(293\) −0.343146 −0.0200468 −0.0100234 0.999950i \(-0.503191\pi\)
−0.0100234 + 0.999950i \(0.503191\pi\)
\(294\) 0 0
\(295\) 51.4558 2.99588
\(296\) −5.24264 −0.304722
\(297\) 0 0
\(298\) 5.75736 0.333515
\(299\) −22.2426 −1.28633
\(300\) 0 0
\(301\) 0 0
\(302\) −14.7279 −0.847497
\(303\) 0 0
\(304\) 5.82843 0.334283
\(305\) 65.4558 3.74799
\(306\) 0 0
\(307\) −22.7990 −1.30121 −0.650604 0.759418i \(-0.725484\pi\)
−0.650604 + 0.759418i \(0.725484\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 31.9706 1.81581
\(311\) −15.5563 −0.882120 −0.441060 0.897478i \(-0.645397\pi\)
−0.441060 + 0.897478i \(0.645397\pi\)
\(312\) 0 0
\(313\) 1.75736 0.0993318 0.0496659 0.998766i \(-0.484184\pi\)
0.0496659 + 0.998766i \(0.484184\pi\)
\(314\) 4.92893 0.278156
\(315\) 0 0
\(316\) −6.48528 −0.364826
\(317\) −10.0000 −0.561656 −0.280828 0.959758i \(-0.590609\pi\)
−0.280828 + 0.959758i \(0.590609\pi\)
\(318\) 0 0
\(319\) 0.242641 0.0135853
\(320\) −4.41421 −0.246762
\(321\) 0 0
\(322\) 0 0
\(323\) 16.4853 0.917266
\(324\) 0 0
\(325\) −61.4558 −3.40896
\(326\) 14.7279 0.815704
\(327\) 0 0
\(328\) 6.17157 0.340768
\(329\) 0 0
\(330\) 0 0
\(331\) 7.51472 0.413046 0.206523 0.978442i \(-0.433785\pi\)
0.206523 + 0.978442i \(0.433785\pi\)
\(332\) 3.89949 0.214013
\(333\) 0 0
\(334\) −12.3848 −0.677665
\(335\) 7.75736 0.423830
\(336\) 0 0
\(337\) −3.48528 −0.189855 −0.0949277 0.995484i \(-0.530262\pi\)
−0.0949277 + 0.995484i \(0.530262\pi\)
\(338\) −5.00000 −0.271964
\(339\) 0 0
\(340\) −12.4853 −0.677109
\(341\) −7.24264 −0.392211
\(342\) 0 0
\(343\) 0 0
\(344\) 6.48528 0.349663
\(345\) 0 0
\(346\) −13.2426 −0.711929
\(347\) −18.5147 −0.993922 −0.496961 0.867773i \(-0.665551\pi\)
−0.496961 + 0.867773i \(0.665551\pi\)
\(348\) 0 0
\(349\) −13.7574 −0.736415 −0.368207 0.929744i \(-0.620028\pi\)
−0.368207 + 0.929744i \(0.620028\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 1.00000 0.0533002
\(353\) −2.65685 −0.141410 −0.0707050 0.997497i \(-0.522525\pi\)
−0.0707050 + 0.997497i \(0.522525\pi\)
\(354\) 0 0
\(355\) 5.48528 0.291129
\(356\) −3.34315 −0.177186
\(357\) 0 0
\(358\) 10.9706 0.579812
\(359\) −17.4558 −0.921284 −0.460642 0.887586i \(-0.652381\pi\)
−0.460642 + 0.887586i \(0.652381\pi\)
\(360\) 0 0
\(361\) 14.9706 0.787924
\(362\) 10.2426 0.538341
\(363\) 0 0
\(364\) 0 0
\(365\) −14.0000 −0.732793
\(366\) 0 0
\(367\) 30.8995 1.61294 0.806470 0.591275i \(-0.201375\pi\)
0.806470 + 0.591275i \(0.201375\pi\)
\(368\) 5.24264 0.273292
\(369\) 0 0
\(370\) 23.1421 1.20310
\(371\) 0 0
\(372\) 0 0
\(373\) −15.2426 −0.789234 −0.394617 0.918846i \(-0.629123\pi\)
−0.394617 + 0.918846i \(0.629123\pi\)
\(374\) 2.82843 0.146254
\(375\) 0 0
\(376\) 11.6569 0.601156
\(377\) 1.02944 0.0530187
\(378\) 0 0
\(379\) 2.97056 0.152588 0.0762938 0.997085i \(-0.475691\pi\)
0.0762938 + 0.997085i \(0.475691\pi\)
\(380\) −25.7279 −1.31981
\(381\) 0 0
\(382\) 0.757359 0.0387499
\(383\) −18.0416 −0.921884 −0.460942 0.887430i \(-0.652488\pi\)
−0.460942 + 0.887430i \(0.652488\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −4.48528 −0.228295
\(387\) 0 0
\(388\) 11.3137 0.574367
\(389\) −36.9706 −1.87448 −0.937241 0.348682i \(-0.886629\pi\)
−0.937241 + 0.348682i \(0.886629\pi\)
\(390\) 0 0
\(391\) 14.8284 0.749906
\(392\) 0 0
\(393\) 0 0
\(394\) 2.72792 0.137431
\(395\) 28.6274 1.44040
\(396\) 0 0
\(397\) −19.4142 −0.974371 −0.487186 0.873298i \(-0.661977\pi\)
−0.487186 + 0.873298i \(0.661977\pi\)
\(398\) 4.75736 0.238465
\(399\) 0 0
\(400\) 14.4853 0.724264
\(401\) 1.75736 0.0877583 0.0438792 0.999037i \(-0.486028\pi\)
0.0438792 + 0.999037i \(0.486028\pi\)
\(402\) 0 0
\(403\) −30.7279 −1.53067
\(404\) 14.4853 0.720670
\(405\) 0 0
\(406\) 0 0
\(407\) −5.24264 −0.259868
\(408\) 0 0
\(409\) 6.00000 0.296681 0.148340 0.988936i \(-0.452607\pi\)
0.148340 + 0.988936i \(0.452607\pi\)
\(410\) −27.2426 −1.34542
\(411\) 0 0
\(412\) −10.0711 −0.496166
\(413\) 0 0
\(414\) 0 0
\(415\) −17.2132 −0.844963
\(416\) 4.24264 0.208013
\(417\) 0 0
\(418\) 5.82843 0.285078
\(419\) 36.0416 1.76075 0.880374 0.474279i \(-0.157291\pi\)
0.880374 + 0.474279i \(0.157291\pi\)
\(420\) 0 0
\(421\) −24.2132 −1.18008 −0.590040 0.807374i \(-0.700888\pi\)
−0.590040 + 0.807374i \(0.700888\pi\)
\(422\) −12.7279 −0.619586
\(423\) 0 0
\(424\) −8.00000 −0.388514
\(425\) 40.9706 1.98736
\(426\) 0 0
\(427\) 0 0
\(428\) 14.9706 0.723629
\(429\) 0 0
\(430\) −28.6274 −1.38054
\(431\) 10.7574 0.518164 0.259082 0.965855i \(-0.416580\pi\)
0.259082 + 0.965855i \(0.416580\pi\)
\(432\) 0 0
\(433\) −5.27208 −0.253360 −0.126680 0.991944i \(-0.540432\pi\)
−0.126680 + 0.991944i \(0.540432\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −15.2426 −0.729990
\(437\) 30.5563 1.46171
\(438\) 0 0
\(439\) −4.97056 −0.237232 −0.118616 0.992940i \(-0.537846\pi\)
−0.118616 + 0.992940i \(0.537846\pi\)
\(440\) −4.41421 −0.210439
\(441\) 0 0
\(442\) 12.0000 0.570782
\(443\) −38.9411 −1.85015 −0.925074 0.379786i \(-0.875998\pi\)
−0.925074 + 0.379786i \(0.875998\pi\)
\(444\) 0 0
\(445\) 14.7574 0.699566
\(446\) 7.24264 0.342949
\(447\) 0 0
\(448\) 0 0
\(449\) 36.7279 1.73330 0.866649 0.498919i \(-0.166269\pi\)
0.866649 + 0.498919i \(0.166269\pi\)
\(450\) 0 0
\(451\) 6.17157 0.290608
\(452\) 4.24264 0.199557
\(453\) 0 0
\(454\) 12.7279 0.597351
\(455\) 0 0
\(456\) 0 0
\(457\) 19.9706 0.934184 0.467092 0.884209i \(-0.345302\pi\)
0.467092 + 0.884209i \(0.345302\pi\)
\(458\) 12.0000 0.560723
\(459\) 0 0
\(460\) −23.1421 −1.07901
\(461\) 12.8995 0.600789 0.300395 0.953815i \(-0.402882\pi\)
0.300395 + 0.953815i \(0.402882\pi\)
\(462\) 0 0
\(463\) −38.7279 −1.79984 −0.899920 0.436056i \(-0.856375\pi\)
−0.899920 + 0.436056i \(0.856375\pi\)
\(464\) −0.242641 −0.0112643
\(465\) 0 0
\(466\) −16.2426 −0.752426
\(467\) 12.3431 0.571173 0.285586 0.958353i \(-0.407812\pi\)
0.285586 + 0.958353i \(0.407812\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) −51.4558 −2.37348
\(471\) 0 0
\(472\) 11.6569 0.536550
\(473\) 6.48528 0.298194
\(474\) 0 0
\(475\) 84.4264 3.87375
\(476\) 0 0
\(477\) 0 0
\(478\) 18.0000 0.823301
\(479\) −29.6569 −1.35506 −0.677528 0.735497i \(-0.736949\pi\)
−0.677528 + 0.735497i \(0.736949\pi\)
\(480\) 0 0
\(481\) −22.2426 −1.01418
\(482\) −2.48528 −0.113201
\(483\) 0 0
\(484\) −10.0000 −0.454545
\(485\) −49.9411 −2.26771
\(486\) 0 0
\(487\) −38.4853 −1.74393 −0.871967 0.489564i \(-0.837156\pi\)
−0.871967 + 0.489564i \(0.837156\pi\)
\(488\) 14.8284 0.671251
\(489\) 0 0
\(490\) 0 0
\(491\) 17.4853 0.789100 0.394550 0.918875i \(-0.370901\pi\)
0.394550 + 0.918875i \(0.370901\pi\)
\(492\) 0 0
\(493\) −0.686292 −0.0309090
\(494\) 24.7279 1.11256
\(495\) 0 0
\(496\) 7.24264 0.325204
\(497\) 0 0
\(498\) 0 0
\(499\) −14.9706 −0.670174 −0.335087 0.942187i \(-0.608766\pi\)
−0.335087 + 0.942187i \(0.608766\pi\)
\(500\) −41.8701 −1.87249
\(501\) 0 0
\(502\) 11.3137 0.504956
\(503\) 8.44365 0.376484 0.188242 0.982123i \(-0.439721\pi\)
0.188242 + 0.982123i \(0.439721\pi\)
\(504\) 0 0
\(505\) −63.9411 −2.84534
\(506\) 5.24264 0.233064
\(507\) 0 0
\(508\) −12.2426 −0.543179
\(509\) −34.2843 −1.51962 −0.759812 0.650143i \(-0.774709\pi\)
−0.759812 + 0.650143i \(0.774709\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) −1.00000 −0.0441942
\(513\) 0 0
\(514\) −11.4853 −0.506594
\(515\) 44.4558 1.95896
\(516\) 0 0
\(517\) 11.6569 0.512668
\(518\) 0 0
\(519\) 0 0
\(520\) −18.7279 −0.821274
\(521\) −23.1421 −1.01388 −0.506938 0.861983i \(-0.669223\pi\)
−0.506938 + 0.861983i \(0.669223\pi\)
\(522\) 0 0
\(523\) 17.4853 0.764578 0.382289 0.924043i \(-0.375136\pi\)
0.382289 + 0.924043i \(0.375136\pi\)
\(524\) −14.4853 −0.632792
\(525\) 0 0
\(526\) −3.72792 −0.162545
\(527\) 20.4853 0.892353
\(528\) 0 0
\(529\) 4.48528 0.195012
\(530\) 35.3137 1.53393
\(531\) 0 0
\(532\) 0 0
\(533\) 26.1838 1.13414
\(534\) 0 0
\(535\) −66.0833 −2.85703
\(536\) 1.75736 0.0759064
\(537\) 0 0
\(538\) −15.3848 −0.663285
\(539\) 0 0
\(540\) 0 0
\(541\) −24.7574 −1.06440 −0.532201 0.846618i \(-0.678635\pi\)
−0.532201 + 0.846618i \(0.678635\pi\)
\(542\) 6.68629 0.287201
\(543\) 0 0
\(544\) −2.82843 −0.121268
\(545\) 67.2843 2.88214
\(546\) 0 0
\(547\) 24.4853 1.04692 0.523458 0.852052i \(-0.324642\pi\)
0.523458 + 0.852052i \(0.324642\pi\)
\(548\) 1.75736 0.0750707
\(549\) 0 0
\(550\) 14.4853 0.617654
\(551\) −1.41421 −0.0602475
\(552\) 0 0
\(553\) 0 0
\(554\) 14.2132 0.603861
\(555\) 0 0
\(556\) 8.82843 0.374409
\(557\) 18.7279 0.793528 0.396764 0.917921i \(-0.370133\pi\)
0.396764 + 0.917921i \(0.370133\pi\)
\(558\) 0 0
\(559\) 27.5147 1.16375
\(560\) 0 0
\(561\) 0 0
\(562\) −10.4853 −0.442295
\(563\) 16.5858 0.699008 0.349504 0.936935i \(-0.386350\pi\)
0.349504 + 0.936935i \(0.386350\pi\)
\(564\) 0 0
\(565\) −18.7279 −0.787890
\(566\) −8.14214 −0.342239
\(567\) 0 0
\(568\) 1.24264 0.0521400
\(569\) 27.7574 1.16365 0.581825 0.813314i \(-0.302339\pi\)
0.581825 + 0.813314i \(0.302339\pi\)
\(570\) 0 0
\(571\) 1.21320 0.0507710 0.0253855 0.999678i \(-0.491919\pi\)
0.0253855 + 0.999678i \(0.491919\pi\)
\(572\) 4.24264 0.177394
\(573\) 0 0
\(574\) 0 0
\(575\) 75.9411 3.16696
\(576\) 0 0
\(577\) 19.7574 0.822510 0.411255 0.911520i \(-0.365091\pi\)
0.411255 + 0.911520i \(0.365091\pi\)
\(578\) 9.00000 0.374351
\(579\) 0 0
\(580\) 1.07107 0.0444737
\(581\) 0 0
\(582\) 0 0
\(583\) −8.00000 −0.331326
\(584\) −3.17157 −0.131241
\(585\) 0 0
\(586\) 0.343146 0.0141752
\(587\) 7.75736 0.320180 0.160090 0.987102i \(-0.448821\pi\)
0.160090 + 0.987102i \(0.448821\pi\)
\(588\) 0 0
\(589\) 42.2132 1.73936
\(590\) −51.4558 −2.11840
\(591\) 0 0
\(592\) 5.24264 0.215471
\(593\) 2.65685 0.109104 0.0545520 0.998511i \(-0.482627\pi\)
0.0545520 + 0.998511i \(0.482627\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −5.75736 −0.235831
\(597\) 0 0
\(598\) 22.2426 0.909569
\(599\) 19.7279 0.806061 0.403031 0.915187i \(-0.367957\pi\)
0.403031 + 0.915187i \(0.367957\pi\)
\(600\) 0 0
\(601\) −30.7279 −1.25342 −0.626709 0.779253i \(-0.715599\pi\)
−0.626709 + 0.779253i \(0.715599\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 14.7279 0.599271
\(605\) 44.1421 1.79463
\(606\) 0 0
\(607\) 30.0000 1.21766 0.608831 0.793300i \(-0.291639\pi\)
0.608831 + 0.793300i \(0.291639\pi\)
\(608\) −5.82843 −0.236374
\(609\) 0 0
\(610\) −65.4558 −2.65023
\(611\) 49.4558 2.00077
\(612\) 0 0
\(613\) −21.7279 −0.877583 −0.438791 0.898589i \(-0.644593\pi\)
−0.438791 + 0.898589i \(0.644593\pi\)
\(614\) 22.7990 0.920092
\(615\) 0 0
\(616\) 0 0
\(617\) −26.9706 −1.08579 −0.542897 0.839799i \(-0.682673\pi\)
−0.542897 + 0.839799i \(0.682673\pi\)
\(618\) 0 0
\(619\) 16.7990 0.675208 0.337604 0.941288i \(-0.390383\pi\)
0.337604 + 0.941288i \(0.390383\pi\)
\(620\) −31.9706 −1.28397
\(621\) 0 0
\(622\) 15.5563 0.623753
\(623\) 0 0
\(624\) 0 0
\(625\) 112.397 4.49588
\(626\) −1.75736 −0.0702382
\(627\) 0 0
\(628\) −4.92893 −0.196686
\(629\) 14.8284 0.591248
\(630\) 0 0
\(631\) 14.0000 0.557331 0.278666 0.960388i \(-0.410108\pi\)
0.278666 + 0.960388i \(0.410108\pi\)
\(632\) 6.48528 0.257971
\(633\) 0 0
\(634\) 10.0000 0.397151
\(635\) 54.0416 2.14458
\(636\) 0 0
\(637\) 0 0
\(638\) −0.242641 −0.00960624
\(639\) 0 0
\(640\) 4.41421 0.174487
\(641\) −41.2132 −1.62782 −0.813912 0.580988i \(-0.802666\pi\)
−0.813912 + 0.580988i \(0.802666\pi\)
\(642\) 0 0
\(643\) 5.82843 0.229851 0.114925 0.993374i \(-0.463337\pi\)
0.114925 + 0.993374i \(0.463337\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) −16.4853 −0.648605
\(647\) 38.4853 1.51301 0.756506 0.653986i \(-0.226905\pi\)
0.756506 + 0.653986i \(0.226905\pi\)
\(648\) 0 0
\(649\) 11.6569 0.457572
\(650\) 61.4558 2.41050
\(651\) 0 0
\(652\) −14.7279 −0.576790
\(653\) −39.9411 −1.56302 −0.781509 0.623895i \(-0.785549\pi\)
−0.781509 + 0.623895i \(0.785549\pi\)
\(654\) 0 0
\(655\) 63.9411 2.49839
\(656\) −6.17157 −0.240959
\(657\) 0 0
\(658\) 0 0
\(659\) 42.9411 1.67275 0.836374 0.548159i \(-0.184671\pi\)
0.836374 + 0.548159i \(0.184671\pi\)
\(660\) 0 0
\(661\) 24.7279 0.961805 0.480902 0.876774i \(-0.340309\pi\)
0.480902 + 0.876774i \(0.340309\pi\)
\(662\) −7.51472 −0.292068
\(663\) 0 0
\(664\) −3.89949 −0.151330
\(665\) 0 0
\(666\) 0 0
\(667\) −1.27208 −0.0492551
\(668\) 12.3848 0.479181
\(669\) 0 0
\(670\) −7.75736 −0.299693
\(671\) 14.8284 0.572445
\(672\) 0 0
\(673\) −1.45584 −0.0561187 −0.0280593 0.999606i \(-0.508933\pi\)
−0.0280593 + 0.999606i \(0.508933\pi\)
\(674\) 3.48528 0.134248
\(675\) 0 0
\(676\) 5.00000 0.192308
\(677\) −20.6985 −0.795507 −0.397754 0.917492i \(-0.630210\pi\)
−0.397754 + 0.917492i \(0.630210\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 12.4853 0.478789
\(681\) 0 0
\(682\) 7.24264 0.277335
\(683\) −19.9706 −0.764152 −0.382076 0.924131i \(-0.624791\pi\)
−0.382076 + 0.924131i \(0.624791\pi\)
\(684\) 0 0
\(685\) −7.75736 −0.296393
\(686\) 0 0
\(687\) 0 0
\(688\) −6.48528 −0.247249
\(689\) −33.9411 −1.29305
\(690\) 0 0
\(691\) −45.9411 −1.74768 −0.873841 0.486211i \(-0.838379\pi\)
−0.873841 + 0.486211i \(0.838379\pi\)
\(692\) 13.2426 0.503410
\(693\) 0 0
\(694\) 18.5147 0.702809
\(695\) −38.9706 −1.47824
\(696\) 0 0
\(697\) −17.4558 −0.661187
\(698\) 13.7574 0.520724
\(699\) 0 0
\(700\) 0 0
\(701\) −28.2426 −1.06671 −0.533355 0.845892i \(-0.679069\pi\)
−0.533355 + 0.845892i \(0.679069\pi\)
\(702\) 0 0
\(703\) 30.5563 1.15245
\(704\) −1.00000 −0.0376889
\(705\) 0 0
\(706\) 2.65685 0.0999920
\(707\) 0 0
\(708\) 0 0
\(709\) −18.7574 −0.704447 −0.352224 0.935916i \(-0.614574\pi\)
−0.352224 + 0.935916i \(0.614574\pi\)
\(710\) −5.48528 −0.205859
\(711\) 0 0
\(712\) 3.34315 0.125290
\(713\) 37.9706 1.42201
\(714\) 0 0
\(715\) −18.7279 −0.700385
\(716\) −10.9706 −0.409989
\(717\) 0 0
\(718\) 17.4558 0.651446
\(719\) 12.3431 0.460322 0.230161 0.973153i \(-0.426075\pi\)
0.230161 + 0.973153i \(0.426075\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) −14.9706 −0.557147
\(723\) 0 0
\(724\) −10.2426 −0.380665
\(725\) −3.51472 −0.130533
\(726\) 0 0
\(727\) −14.1421 −0.524503 −0.262251 0.965000i \(-0.584465\pi\)
−0.262251 + 0.965000i \(0.584465\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 14.0000 0.518163
\(731\) −18.3431 −0.678446
\(732\) 0 0
\(733\) 39.9411 1.47526 0.737630 0.675206i \(-0.235945\pi\)
0.737630 + 0.675206i \(0.235945\pi\)
\(734\) −30.8995 −1.14052
\(735\) 0 0
\(736\) −5.24264 −0.193246
\(737\) 1.75736 0.0647332
\(738\) 0 0
\(739\) 23.2721 0.856077 0.428039 0.903760i \(-0.359205\pi\)
0.428039 + 0.903760i \(0.359205\pi\)
\(740\) −23.1421 −0.850722
\(741\) 0 0
\(742\) 0 0
\(743\) −23.2426 −0.852690 −0.426345 0.904561i \(-0.640199\pi\)
−0.426345 + 0.904561i \(0.640199\pi\)
\(744\) 0 0
\(745\) 25.4142 0.931105
\(746\) 15.2426 0.558073
\(747\) 0 0
\(748\) −2.82843 −0.103418
\(749\) 0 0
\(750\) 0 0
\(751\) −11.4558 −0.418030 −0.209015 0.977912i \(-0.567026\pi\)
−0.209015 + 0.977912i \(0.567026\pi\)
\(752\) −11.6569 −0.425082
\(753\) 0 0
\(754\) −1.02944 −0.0374899
\(755\) −65.0122 −2.36604
\(756\) 0 0
\(757\) 3.45584 0.125605 0.0628024 0.998026i \(-0.479996\pi\)
0.0628024 + 0.998026i \(0.479996\pi\)
\(758\) −2.97056 −0.107896
\(759\) 0 0
\(760\) 25.7279 0.933250
\(761\) 6.68629 0.242378 0.121189 0.992629i \(-0.461329\pi\)
0.121189 + 0.992629i \(0.461329\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) −0.757359 −0.0274003
\(765\) 0 0
\(766\) 18.0416 0.651871
\(767\) 49.4558 1.78575
\(768\) 0 0
\(769\) −32.1838 −1.16058 −0.580288 0.814411i \(-0.697060\pi\)
−0.580288 + 0.814411i \(0.697060\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 4.48528 0.161429
\(773\) −2.61522 −0.0940631 −0.0470315 0.998893i \(-0.514976\pi\)
−0.0470315 + 0.998893i \(0.514976\pi\)
\(774\) 0 0
\(775\) 104.912 3.76854
\(776\) −11.3137 −0.406138
\(777\) 0 0
\(778\) 36.9706 1.32546
\(779\) −35.9706 −1.28878
\(780\) 0 0
\(781\) 1.24264 0.0444652
\(782\) −14.8284 −0.530263
\(783\) 0 0
\(784\) 0 0
\(785\) 21.7574 0.776553
\(786\) 0 0
\(787\) 30.6863 1.09385 0.546924 0.837182i \(-0.315799\pi\)
0.546924 + 0.837182i \(0.315799\pi\)
\(788\) −2.72792 −0.0971782
\(789\) 0 0
\(790\) −28.6274 −1.01852
\(791\) 0 0
\(792\) 0 0
\(793\) 62.9117 2.23406
\(794\) 19.4142 0.688985
\(795\) 0 0
\(796\) −4.75736 −0.168620
\(797\) 16.7574 0.593576 0.296788 0.954943i \(-0.404085\pi\)
0.296788 + 0.954943i \(0.404085\pi\)
\(798\) 0 0
\(799\) −32.9706 −1.16641
\(800\) −14.4853 −0.512132
\(801\) 0 0
\(802\) −1.75736 −0.0620545
\(803\) −3.17157 −0.111922
\(804\) 0 0
\(805\) 0 0
\(806\) 30.7279 1.08234
\(807\) 0 0
\(808\) −14.4853 −0.509590
\(809\) −9.75736 −0.343050 −0.171525 0.985180i \(-0.554870\pi\)
−0.171525 + 0.985180i \(0.554870\pi\)
\(810\) 0 0
\(811\) −39.4264 −1.38445 −0.692224 0.721683i \(-0.743369\pi\)
−0.692224 + 0.721683i \(0.743369\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 5.24264 0.183754
\(815\) 65.0122 2.27728
\(816\) 0 0
\(817\) −37.7990 −1.32242
\(818\) −6.00000 −0.209785
\(819\) 0 0
\(820\) 27.2426 0.951354
\(821\) −46.9706 −1.63928 −0.819642 0.572876i \(-0.805828\pi\)
−0.819642 + 0.572876i \(0.805828\pi\)
\(822\) 0 0
\(823\) 38.2426 1.33305 0.666527 0.745481i \(-0.267780\pi\)
0.666527 + 0.745481i \(0.267780\pi\)
\(824\) 10.0711 0.350842
\(825\) 0 0
\(826\) 0 0
\(827\) −54.9411 −1.91049 −0.955245 0.295816i \(-0.904409\pi\)
−0.955245 + 0.295816i \(0.904409\pi\)
\(828\) 0 0
\(829\) 44.1421 1.53312 0.766560 0.642173i \(-0.221967\pi\)
0.766560 + 0.642173i \(0.221967\pi\)
\(830\) 17.2132 0.597479
\(831\) 0 0
\(832\) −4.24264 −0.147087
\(833\) 0 0
\(834\) 0 0
\(835\) −54.6690 −1.89190
\(836\) −5.82843 −0.201580
\(837\) 0 0
\(838\) −36.0416 −1.24504
\(839\) −54.3848 −1.87757 −0.938785 0.344502i \(-0.888048\pi\)
−0.938785 + 0.344502i \(0.888048\pi\)
\(840\) 0 0
\(841\) −28.9411 −0.997970
\(842\) 24.2132 0.834442
\(843\) 0 0
\(844\) 12.7279 0.438113
\(845\) −22.0711 −0.759268
\(846\) 0 0
\(847\) 0 0
\(848\) 8.00000 0.274721
\(849\) 0 0
\(850\) −40.9706 −1.40528
\(851\) 27.4853 0.942183
\(852\) 0 0
\(853\) −24.0000 −0.821744 −0.410872 0.911693i \(-0.634776\pi\)
−0.410872 + 0.911693i \(0.634776\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) −14.9706 −0.511683
\(857\) −39.7696 −1.35850 −0.679251 0.733906i \(-0.737695\pi\)
−0.679251 + 0.733906i \(0.737695\pi\)
\(858\) 0 0
\(859\) 24.1716 0.824723 0.412362 0.911020i \(-0.364704\pi\)
0.412362 + 0.911020i \(0.364704\pi\)
\(860\) 28.6274 0.976187
\(861\) 0 0
\(862\) −10.7574 −0.366397
\(863\) −30.9706 −1.05425 −0.527125 0.849788i \(-0.676730\pi\)
−0.527125 + 0.849788i \(0.676730\pi\)
\(864\) 0 0
\(865\) −58.4558 −1.98756
\(866\) 5.27208 0.179153
\(867\) 0 0
\(868\) 0 0
\(869\) 6.48528 0.219998
\(870\) 0 0
\(871\) 7.45584 0.252632
\(872\) 15.2426 0.516181
\(873\) 0 0
\(874\) −30.5563 −1.03358
\(875\) 0 0
\(876\) 0 0
\(877\) 18.9706 0.640590 0.320295 0.947318i \(-0.396218\pi\)
0.320295 + 0.947318i \(0.396218\pi\)
\(878\) 4.97056 0.167748
\(879\) 0 0
\(880\) 4.41421 0.148803
\(881\) 12.9411 0.435998 0.217999 0.975949i \(-0.430047\pi\)
0.217999 + 0.975949i \(0.430047\pi\)
\(882\) 0 0
\(883\) 28.4853 0.958606 0.479303 0.877649i \(-0.340889\pi\)
0.479303 + 0.877649i \(0.340889\pi\)
\(884\) −12.0000 −0.403604
\(885\) 0 0
\(886\) 38.9411 1.30825
\(887\) 23.3553 0.784196 0.392098 0.919924i \(-0.371749\pi\)
0.392098 + 0.919924i \(0.371749\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) −14.7574 −0.494668
\(891\) 0 0
\(892\) −7.24264 −0.242502
\(893\) −67.9411 −2.27356
\(894\) 0 0
\(895\) 48.4264 1.61872
\(896\) 0 0
\(897\) 0 0
\(898\) −36.7279 −1.22563
\(899\) −1.75736 −0.0586112
\(900\) 0 0
\(901\) 22.6274 0.753829
\(902\) −6.17157 −0.205491
\(903\) 0 0
\(904\) −4.24264 −0.141108
\(905\) 45.2132 1.50294
\(906\) 0 0
\(907\) 16.7279 0.555442 0.277721 0.960662i \(-0.410421\pi\)
0.277721 + 0.960662i \(0.410421\pi\)
\(908\) −12.7279 −0.422391
\(909\) 0 0
\(910\) 0 0
\(911\) −4.48528 −0.148604 −0.0743020 0.997236i \(-0.523673\pi\)
−0.0743020 + 0.997236i \(0.523673\pi\)
\(912\) 0 0
\(913\) −3.89949 −0.129054
\(914\) −19.9706 −0.660568
\(915\) 0 0
\(916\) −12.0000 −0.396491
\(917\) 0 0
\(918\) 0 0
\(919\) −4.48528 −0.147956 −0.0739779 0.997260i \(-0.523569\pi\)
−0.0739779 + 0.997260i \(0.523569\pi\)
\(920\) 23.1421 0.762974
\(921\) 0 0
\(922\) −12.8995 −0.424822
\(923\) 5.27208 0.173533
\(924\) 0 0
\(925\) 75.9411 2.49693
\(926\) 38.7279 1.27268
\(927\) 0 0
\(928\) 0.242641 0.00796507
\(929\) 40.2843 1.32168 0.660842 0.750525i \(-0.270199\pi\)
0.660842 + 0.750525i \(0.270199\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 16.2426 0.532045
\(933\) 0 0
\(934\) −12.3431 −0.403880
\(935\) 12.4853 0.408312
\(936\) 0 0
\(937\) 23.3137 0.761626 0.380813 0.924652i \(-0.375644\pi\)
0.380813 + 0.924652i \(0.375644\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 51.4558 1.67830
\(941\) 4.07107 0.132713 0.0663565 0.997796i \(-0.478863\pi\)
0.0663565 + 0.997796i \(0.478863\pi\)
\(942\) 0 0
\(943\) −32.3553 −1.05363
\(944\) −11.6569 −0.379398
\(945\) 0 0
\(946\) −6.48528 −0.210855
\(947\) 49.9706 1.62383 0.811913 0.583779i \(-0.198427\pi\)
0.811913 + 0.583779i \(0.198427\pi\)
\(948\) 0 0
\(949\) −13.4558 −0.436795
\(950\) −84.4264 −2.73915
\(951\) 0 0
\(952\) 0 0
\(953\) −26.2426 −0.850083 −0.425041 0.905174i \(-0.639740\pi\)
−0.425041 + 0.905174i \(0.639740\pi\)
\(954\) 0 0
\(955\) 3.34315 0.108182
\(956\) −18.0000 −0.582162
\(957\) 0 0
\(958\) 29.6569 0.958169
\(959\) 0 0
\(960\) 0 0
\(961\) 21.4558 0.692124
\(962\) 22.2426 0.717132
\(963\) 0 0
\(964\) 2.48528 0.0800455
\(965\) −19.7990 −0.637352
\(966\) 0 0
\(967\) 40.6690 1.30783 0.653914 0.756569i \(-0.273126\pi\)
0.653914 + 0.756569i \(0.273126\pi\)
\(968\) 10.0000 0.321412
\(969\) 0 0
\(970\) 49.9411 1.60351
\(971\) 43.4975 1.39590 0.697950 0.716146i \(-0.254096\pi\)
0.697950 + 0.716146i \(0.254096\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 38.4853 1.23315
\(975\) 0 0
\(976\) −14.8284 −0.474646
\(977\) 16.9706 0.542936 0.271468 0.962447i \(-0.412491\pi\)
0.271468 + 0.962447i \(0.412491\pi\)
\(978\) 0 0
\(979\) 3.34315 0.106847
\(980\) 0 0
\(981\) 0 0
\(982\) −17.4853 −0.557978
\(983\) −37.7990 −1.20560 −0.602800 0.797892i \(-0.705948\pi\)
−0.602800 + 0.797892i \(0.705948\pi\)
\(984\) 0 0
\(985\) 12.0416 0.383678
\(986\) 0.686292 0.0218560
\(987\) 0 0
\(988\) −24.7279 −0.786700
\(989\) −34.0000 −1.08114
\(990\) 0 0
\(991\) −8.24264 −0.261836 −0.130918 0.991393i \(-0.541792\pi\)
−0.130918 + 0.991393i \(0.541792\pi\)
\(992\) −7.24264 −0.229954
\(993\) 0 0
\(994\) 0 0
\(995\) 21.0000 0.665745
\(996\) 0 0
\(997\) −20.1005 −0.636589 −0.318295 0.947992i \(-0.603110\pi\)
−0.318295 + 0.947992i \(0.603110\pi\)
\(998\) 14.9706 0.473885
\(999\) 0 0
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 2646.2.a.be.1.1 2
3.2 odd 2 2646.2.a.bp.1.2 yes 2
7.6 odd 2 2646.2.a.bj.1.2 yes 2
21.20 even 2 2646.2.a.bk.1.1 yes 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2646.2.a.be.1.1 2 1.1 even 1 trivial
2646.2.a.bj.1.2 yes 2 7.6 odd 2
2646.2.a.bk.1.1 yes 2 21.20 even 2
2646.2.a.bp.1.2 yes 2 3.2 odd 2