Properties

Label 8470.2.a.dh.1.7
Level $8470$
Weight $2$
Character 8470.1
Self dual yes
Analytic conductor $67.633$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 8470 = 2 \cdot 5 \cdot 7 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8470.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(67.6332905120\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Defining polynomial: \(x^{8} - 16 x^{6} + 69 x^{4} - 10 x^{3} - 70 x^{2} + 10 x + 5\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 770)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.7
Root \(2.00431\) of defining polynomial
Character \(\chi\) \(=\) 8470.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} +2.00431 q^{3} +1.00000 q^{4} -1.00000 q^{5} +2.00431 q^{6} -1.00000 q^{7} +1.00000 q^{8} +1.01724 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} +2.00431 q^{3} +1.00000 q^{4} -1.00000 q^{5} +2.00431 q^{6} -1.00000 q^{7} +1.00000 q^{8} +1.01724 q^{9} -1.00000 q^{10} +2.00431 q^{12} -4.02930 q^{13} -1.00000 q^{14} -2.00431 q^{15} +1.00000 q^{16} +0.300566 q^{17} +1.01724 q^{18} +7.73531 q^{19} -1.00000 q^{20} -2.00431 q^{21} -6.48607 q^{23} +2.00431 q^{24} +1.00000 q^{25} -4.02930 q^{26} -3.97405 q^{27} -1.00000 q^{28} +1.81673 q^{29} -2.00431 q^{30} +7.53480 q^{31} +1.00000 q^{32} +0.300566 q^{34} +1.00000 q^{35} +1.01724 q^{36} +9.11210 q^{37} +7.73531 q^{38} -8.07596 q^{39} -1.00000 q^{40} +9.68330 q^{41} -2.00431 q^{42} +3.42569 q^{43} -1.01724 q^{45} -6.48607 q^{46} -6.21202 q^{47} +2.00431 q^{48} +1.00000 q^{49} +1.00000 q^{50} +0.602426 q^{51} -4.02930 q^{52} +6.39502 q^{53} -3.97405 q^{54} -1.00000 q^{56} +15.5039 q^{57} +1.81673 q^{58} +5.51945 q^{59} -2.00431 q^{60} -3.62844 q^{61} +7.53480 q^{62} -1.01724 q^{63} +1.00000 q^{64} +4.02930 q^{65} +0.505448 q^{67} +0.300566 q^{68} -13.0001 q^{69} +1.00000 q^{70} +8.37915 q^{71} +1.01724 q^{72} +1.40274 q^{73} +9.11210 q^{74} +2.00431 q^{75} +7.73531 q^{76} -8.07596 q^{78} +6.20005 q^{79} -1.00000 q^{80} -11.0169 q^{81} +9.68330 q^{82} +14.5768 q^{83} -2.00431 q^{84} -0.300566 q^{85} +3.42569 q^{86} +3.64127 q^{87} +1.92815 q^{89} -1.01724 q^{90} +4.02930 q^{91} -6.48607 q^{92} +15.1020 q^{93} -6.21202 q^{94} -7.73531 q^{95} +2.00431 q^{96} -16.4944 q^{97} +1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + 8q^{2} + 8q^{4} - 8q^{5} - 8q^{7} + 8q^{8} + 8q^{9} + O(q^{10}) \) \( 8q + 8q^{2} + 8q^{4} - 8q^{5} - 8q^{7} + 8q^{8} + 8q^{9} - 8q^{10} - q^{13} - 8q^{14} + 8q^{16} + 6q^{17} + 8q^{18} + 5q^{19} - 8q^{20} + 10q^{23} + 8q^{25} - q^{26} - 8q^{28} + 3q^{29} - 8q^{31} + 8q^{32} + 6q^{34} + 8q^{35} + 8q^{36} - 6q^{37} + 5q^{38} + 35q^{39} - 8q^{40} + 11q^{41} - 5q^{43} - 8q^{45} + 10q^{46} - 15q^{47} + 8q^{49} + 8q^{50} - 6q^{51} - q^{52} - 16q^{53} - 8q^{56} + 38q^{57} + 3q^{58} - 9q^{59} + 32q^{61} - 8q^{62} - 8q^{63} + 8q^{64} + q^{65} + 33q^{67} + 6q^{68} - 22q^{69} + 8q^{70} + 11q^{71} + 8q^{72} - 34q^{73} - 6q^{74} + 5q^{76} + 35q^{78} + 31q^{79} - 8q^{80} + 20q^{81} + 11q^{82} + 50q^{83} - 6q^{85} - 5q^{86} - 12q^{87} + q^{89} - 8q^{90} + q^{91} + 10q^{92} + 26q^{93} - 15q^{94} - 5q^{95} - 4q^{97} + 8q^{98} + O(q^{100}) \)

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
\(3\) 2.00431 1.15719 0.578593 0.815616i \(-0.303602\pi\)
0.578593 + 0.815616i \(0.303602\pi\)
\(4\) 1.00000 0.500000
\(5\) −1.00000 −0.447214
\(6\) 2.00431 0.818254
\(7\) −1.00000 −0.377964
\(8\) 1.00000 0.353553
\(9\) 1.01724 0.339080
\(10\) −1.00000 −0.316228
\(11\) 0 0
\(12\) 2.00431 0.578593
\(13\) −4.02930 −1.11753 −0.558764 0.829327i \(-0.688724\pi\)
−0.558764 + 0.829327i \(0.688724\pi\)
\(14\) −1.00000 −0.267261
\(15\) −2.00431 −0.517509
\(16\) 1.00000 0.250000
\(17\) 0.300566 0.0728979 0.0364490 0.999336i \(-0.488395\pi\)
0.0364490 + 0.999336i \(0.488395\pi\)
\(18\) 1.01724 0.239766
\(19\) 7.73531 1.77460 0.887301 0.461191i \(-0.152578\pi\)
0.887301 + 0.461191i \(0.152578\pi\)
\(20\) −1.00000 −0.223607
\(21\) −2.00431 −0.437375
\(22\) 0 0
\(23\) −6.48607 −1.35244 −0.676219 0.736700i \(-0.736383\pi\)
−0.676219 + 0.736700i \(0.736383\pi\)
\(24\) 2.00431 0.409127
\(25\) 1.00000 0.200000
\(26\) −4.02930 −0.790211
\(27\) −3.97405 −0.764807
\(28\) −1.00000 −0.188982
\(29\) 1.81673 0.337358 0.168679 0.985671i \(-0.446050\pi\)
0.168679 + 0.985671i \(0.446050\pi\)
\(30\) −2.00431 −0.365934
\(31\) 7.53480 1.35329 0.676645 0.736310i \(-0.263433\pi\)
0.676645 + 0.736310i \(0.263433\pi\)
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) 0.300566 0.0515466
\(35\) 1.00000 0.169031
\(36\) 1.01724 0.169540
\(37\) 9.11210 1.49802 0.749010 0.662559i \(-0.230530\pi\)
0.749010 + 0.662559i \(0.230530\pi\)
\(38\) 7.73531 1.25483
\(39\) −8.07596 −1.29319
\(40\) −1.00000 −0.158114
\(41\) 9.68330 1.51228 0.756139 0.654411i \(-0.227083\pi\)
0.756139 + 0.654411i \(0.227083\pi\)
\(42\) −2.00431 −0.309271
\(43\) 3.42569 0.522412 0.261206 0.965283i \(-0.415880\pi\)
0.261206 + 0.965283i \(0.415880\pi\)
\(44\) 0 0
\(45\) −1.01724 −0.151641
\(46\) −6.48607 −0.956319
\(47\) −6.21202 −0.906116 −0.453058 0.891481i \(-0.649667\pi\)
−0.453058 + 0.891481i \(0.649667\pi\)
\(48\) 2.00431 0.289297
\(49\) 1.00000 0.142857
\(50\) 1.00000 0.141421
\(51\) 0.602426 0.0843565
\(52\) −4.02930 −0.558764
\(53\) 6.39502 0.878423 0.439212 0.898384i \(-0.355258\pi\)
0.439212 + 0.898384i \(0.355258\pi\)
\(54\) −3.97405 −0.540800
\(55\) 0 0
\(56\) −1.00000 −0.133631
\(57\) 15.5039 2.05355
\(58\) 1.81673 0.238548
\(59\) 5.51945 0.718571 0.359286 0.933228i \(-0.383020\pi\)
0.359286 + 0.933228i \(0.383020\pi\)
\(60\) −2.00431 −0.258755
\(61\) −3.62844 −0.464574 −0.232287 0.972647i \(-0.574621\pi\)
−0.232287 + 0.972647i \(0.574621\pi\)
\(62\) 7.53480 0.956920
\(63\) −1.01724 −0.128160
\(64\) 1.00000 0.125000
\(65\) 4.02930 0.499774
\(66\) 0 0
\(67\) 0.505448 0.0617503 0.0308751 0.999523i \(-0.490171\pi\)
0.0308751 + 0.999523i \(0.490171\pi\)
\(68\) 0.300566 0.0364490
\(69\) −13.0001 −1.56502
\(70\) 1.00000 0.119523
\(71\) 8.37915 0.994422 0.497211 0.867630i \(-0.334357\pi\)
0.497211 + 0.867630i \(0.334357\pi\)
\(72\) 1.01724 0.119883
\(73\) 1.40274 0.164178 0.0820892 0.996625i \(-0.473841\pi\)
0.0820892 + 0.996625i \(0.473841\pi\)
\(74\) 9.11210 1.05926
\(75\) 2.00431 0.231437
\(76\) 7.73531 0.887301
\(77\) 0 0
\(78\) −8.07596 −0.914422
\(79\) 6.20005 0.697560 0.348780 0.937205i \(-0.386596\pi\)
0.348780 + 0.937205i \(0.386596\pi\)
\(80\) −1.00000 −0.111803
\(81\) −11.0169 −1.22410
\(82\) 9.68330 1.06934
\(83\) 14.5768 1.60001 0.800006 0.599992i \(-0.204830\pi\)
0.800006 + 0.599992i \(0.204830\pi\)
\(84\) −2.00431 −0.218688
\(85\) −0.300566 −0.0326009
\(86\) 3.42569 0.369401
\(87\) 3.64127 0.390386
\(88\) 0 0
\(89\) 1.92815 0.204384 0.102192 0.994765i \(-0.467414\pi\)
0.102192 + 0.994765i \(0.467414\pi\)
\(90\) −1.01724 −0.107227
\(91\) 4.02930 0.422386
\(92\) −6.48607 −0.676219
\(93\) 15.1020 1.56601
\(94\) −6.21202 −0.640721
\(95\) −7.73531 −0.793626
\(96\) 2.00431 0.204564
\(97\) −16.4944 −1.67475 −0.837374 0.546630i \(-0.815910\pi\)
−0.837374 + 0.546630i \(0.815910\pi\)
\(98\) 1.00000 0.101015
\(99\) 0 0
\(100\) 1.00000 0.100000
\(101\) 9.44404 0.939717 0.469859 0.882742i \(-0.344305\pi\)
0.469859 + 0.882742i \(0.344305\pi\)
\(102\) 0.602426 0.0596491
\(103\) −3.51983 −0.346820 −0.173410 0.984850i \(-0.555479\pi\)
−0.173410 + 0.984850i \(0.555479\pi\)
\(104\) −4.02930 −0.395106
\(105\) 2.00431 0.195600
\(106\) 6.39502 0.621139
\(107\) 10.4251 1.00783 0.503917 0.863752i \(-0.331892\pi\)
0.503917 + 0.863752i \(0.331892\pi\)
\(108\) −3.97405 −0.382404
\(109\) 12.1571 1.16444 0.582221 0.813030i \(-0.302184\pi\)
0.582221 + 0.813030i \(0.302184\pi\)
\(110\) 0 0
\(111\) 18.2634 1.73349
\(112\) −1.00000 −0.0944911
\(113\) −6.67216 −0.627664 −0.313832 0.949478i \(-0.601613\pi\)
−0.313832 + 0.949478i \(0.601613\pi\)
\(114\) 15.5039 1.45208
\(115\) 6.48607 0.604829
\(116\) 1.81673 0.168679
\(117\) −4.09877 −0.378932
\(118\) 5.51945 0.508106
\(119\) −0.300566 −0.0275528
\(120\) −2.00431 −0.182967
\(121\) 0 0
\(122\) −3.62844 −0.328503
\(123\) 19.4083 1.74999
\(124\) 7.53480 0.676645
\(125\) −1.00000 −0.0894427
\(126\) −1.01724 −0.0906231
\(127\) −12.8919 −1.14397 −0.571984 0.820265i \(-0.693826\pi\)
−0.571984 + 0.820265i \(0.693826\pi\)
\(128\) 1.00000 0.0883883
\(129\) 6.86612 0.604528
\(130\) 4.02930 0.353393
\(131\) 20.3862 1.78115 0.890577 0.454833i \(-0.150301\pi\)
0.890577 + 0.454833i \(0.150301\pi\)
\(132\) 0 0
\(133\) −7.73531 −0.670737
\(134\) 0.505448 0.0436640
\(135\) 3.97405 0.342032
\(136\) 0.300566 0.0257733
\(137\) −17.6954 −1.51182 −0.755911 0.654674i \(-0.772806\pi\)
−0.755911 + 0.654674i \(0.772806\pi\)
\(138\) −13.0001 −1.10664
\(139\) 13.4774 1.14314 0.571570 0.820553i \(-0.306335\pi\)
0.571570 + 0.820553i \(0.306335\pi\)
\(140\) 1.00000 0.0845154
\(141\) −12.4508 −1.04855
\(142\) 8.37915 0.703163
\(143\) 0 0
\(144\) 1.01724 0.0847701
\(145\) −1.81673 −0.150871
\(146\) 1.40274 0.116092
\(147\) 2.00431 0.165312
\(148\) 9.11210 0.749010
\(149\) −1.72417 −0.141249 −0.0706246 0.997503i \(-0.522499\pi\)
−0.0706246 + 0.997503i \(0.522499\pi\)
\(150\) 2.00431 0.163651
\(151\) −9.14895 −0.744531 −0.372266 0.928126i \(-0.621419\pi\)
−0.372266 + 0.928126i \(0.621419\pi\)
\(152\) 7.73531 0.627417
\(153\) 0.305748 0.0247183
\(154\) 0 0
\(155\) −7.53480 −0.605209
\(156\) −8.07596 −0.646594
\(157\) −0.876025 −0.0699144 −0.0349572 0.999389i \(-0.511129\pi\)
−0.0349572 + 0.999389i \(0.511129\pi\)
\(158\) 6.20005 0.493250
\(159\) 12.8176 1.01650
\(160\) −1.00000 −0.0790569
\(161\) 6.48607 0.511174
\(162\) −11.0169 −0.865573
\(163\) −6.49269 −0.508547 −0.254274 0.967132i \(-0.581836\pi\)
−0.254274 + 0.967132i \(0.581836\pi\)
\(164\) 9.68330 0.756139
\(165\) 0 0
\(166\) 14.5768 1.13138
\(167\) −20.0322 −1.55014 −0.775070 0.631875i \(-0.782286\pi\)
−0.775070 + 0.631875i \(0.782286\pi\)
\(168\) −2.00431 −0.154636
\(169\) 3.23529 0.248868
\(170\) −0.300566 −0.0230524
\(171\) 7.86868 0.601733
\(172\) 3.42569 0.261206
\(173\) 1.52088 0.115630 0.0578151 0.998327i \(-0.481587\pi\)
0.0578151 + 0.998327i \(0.481587\pi\)
\(174\) 3.64127 0.276044
\(175\) −1.00000 −0.0755929
\(176\) 0 0
\(177\) 11.0627 0.831521
\(178\) 1.92815 0.144521
\(179\) 24.1512 1.80515 0.902573 0.430537i \(-0.141676\pi\)
0.902573 + 0.430537i \(0.141676\pi\)
\(180\) −1.01724 −0.0758207
\(181\) 16.6623 1.23850 0.619249 0.785195i \(-0.287437\pi\)
0.619249 + 0.785195i \(0.287437\pi\)
\(182\) 4.02930 0.298672
\(183\) −7.27250 −0.537598
\(184\) −6.48607 −0.478159
\(185\) −9.11210 −0.669935
\(186\) 15.1020 1.10733
\(187\) 0 0
\(188\) −6.21202 −0.453058
\(189\) 3.97405 0.289070
\(190\) −7.73531 −0.561178
\(191\) −2.97548 −0.215298 −0.107649 0.994189i \(-0.534332\pi\)
−0.107649 + 0.994189i \(0.534332\pi\)
\(192\) 2.00431 0.144648
\(193\) −11.5006 −0.827831 −0.413916 0.910315i \(-0.635839\pi\)
−0.413916 + 0.910315i \(0.635839\pi\)
\(194\) −16.4944 −1.18423
\(195\) 8.07596 0.578331
\(196\) 1.00000 0.0714286
\(197\) 16.0597 1.14420 0.572101 0.820183i \(-0.306128\pi\)
0.572101 + 0.820183i \(0.306128\pi\)
\(198\) 0 0
\(199\) 11.0899 0.786140 0.393070 0.919509i \(-0.371413\pi\)
0.393070 + 0.919509i \(0.371413\pi\)
\(200\) 1.00000 0.0707107
\(201\) 1.01307 0.0714566
\(202\) 9.44404 0.664480
\(203\) −1.81673 −0.127509
\(204\) 0.602426 0.0421783
\(205\) −9.68330 −0.676311
\(206\) −3.51983 −0.245238
\(207\) −6.59790 −0.458586
\(208\) −4.02930 −0.279382
\(209\) 0 0
\(210\) 2.00431 0.138310
\(211\) −25.5146 −1.75650 −0.878250 0.478203i \(-0.841288\pi\)
−0.878250 + 0.478203i \(0.841288\pi\)
\(212\) 6.39502 0.439212
\(213\) 16.7944 1.15073
\(214\) 10.4251 0.712646
\(215\) −3.42569 −0.233630
\(216\) −3.97405 −0.270400
\(217\) −7.53480 −0.511495
\(218\) 12.1571 0.823385
\(219\) 2.81152 0.189985
\(220\) 0 0
\(221\) −1.21107 −0.0814655
\(222\) 18.2634 1.22576
\(223\) −23.1843 −1.55254 −0.776268 0.630403i \(-0.782890\pi\)
−0.776268 + 0.630403i \(0.782890\pi\)
\(224\) −1.00000 −0.0668153
\(225\) 1.01724 0.0678161
\(226\) −6.67216 −0.443826
\(227\) −4.97362 −0.330111 −0.165055 0.986284i \(-0.552780\pi\)
−0.165055 + 0.986284i \(0.552780\pi\)
\(228\) 15.5039 1.02677
\(229\) −25.6627 −1.69584 −0.847921 0.530123i \(-0.822146\pi\)
−0.847921 + 0.530123i \(0.822146\pi\)
\(230\) 6.48607 0.427679
\(231\) 0 0
\(232\) 1.81673 0.119274
\(233\) 3.81437 0.249888 0.124944 0.992164i \(-0.460125\pi\)
0.124944 + 0.992164i \(0.460125\pi\)
\(234\) −4.09877 −0.267945
\(235\) 6.21202 0.405227
\(236\) 5.51945 0.359286
\(237\) 12.4268 0.807207
\(238\) −0.300566 −0.0194828
\(239\) 6.58100 0.425689 0.212845 0.977086i \(-0.431727\pi\)
0.212845 + 0.977086i \(0.431727\pi\)
\(240\) −2.00431 −0.129377
\(241\) 23.9679 1.54391 0.771955 0.635677i \(-0.219279\pi\)
0.771955 + 0.635677i \(0.219279\pi\)
\(242\) 0 0
\(243\) −10.1592 −0.651710
\(244\) −3.62844 −0.232287
\(245\) −1.00000 −0.0638877
\(246\) 19.4083 1.23743
\(247\) −31.1679 −1.98317
\(248\) 7.53480 0.478460
\(249\) 29.2164 1.85151
\(250\) −1.00000 −0.0632456
\(251\) 5.08595 0.321022 0.160511 0.987034i \(-0.448686\pi\)
0.160511 + 0.987034i \(0.448686\pi\)
\(252\) −1.01724 −0.0640802
\(253\) 0 0
\(254\) −12.8919 −0.808908
\(255\) −0.602426 −0.0377254
\(256\) 1.00000 0.0625000
\(257\) 11.0968 0.692198 0.346099 0.938198i \(-0.387506\pi\)
0.346099 + 0.938198i \(0.387506\pi\)
\(258\) 6.86612 0.427466
\(259\) −9.11210 −0.566198
\(260\) 4.02930 0.249887
\(261\) 1.84805 0.114391
\(262\) 20.3862 1.25947
\(263\) −2.64779 −0.163270 −0.0816348 0.996662i \(-0.526014\pi\)
−0.0816348 + 0.996662i \(0.526014\pi\)
\(264\) 0 0
\(265\) −6.39502 −0.392843
\(266\) −7.73531 −0.474282
\(267\) 3.86460 0.236510
\(268\) 0.505448 0.0308751
\(269\) −3.42199 −0.208642 −0.104321 0.994544i \(-0.533267\pi\)
−0.104321 + 0.994544i \(0.533267\pi\)
\(270\) 3.97405 0.241853
\(271\) −21.1788 −1.28652 −0.643259 0.765649i \(-0.722418\pi\)
−0.643259 + 0.765649i \(0.722418\pi\)
\(272\) 0.300566 0.0182245
\(273\) 8.07596 0.488779
\(274\) −17.6954 −1.06902
\(275\) 0 0
\(276\) −13.0001 −0.782512
\(277\) 27.3915 1.64580 0.822899 0.568188i \(-0.192355\pi\)
0.822899 + 0.568188i \(0.192355\pi\)
\(278\) 13.4774 0.808322
\(279\) 7.66470 0.458874
\(280\) 1.00000 0.0597614
\(281\) 11.2620 0.671832 0.335916 0.941892i \(-0.390954\pi\)
0.335916 + 0.941892i \(0.390954\pi\)
\(282\) −12.4508 −0.741433
\(283\) −20.2928 −1.20628 −0.603142 0.797634i \(-0.706085\pi\)
−0.603142 + 0.797634i \(0.706085\pi\)
\(284\) 8.37915 0.497211
\(285\) −15.5039 −0.918373
\(286\) 0 0
\(287\) −9.68330 −0.571587
\(288\) 1.01724 0.0599415
\(289\) −16.9097 −0.994686
\(290\) −1.81673 −0.106682
\(291\) −33.0597 −1.93800
\(292\) 1.40274 0.0820892
\(293\) −11.9548 −0.698405 −0.349202 0.937047i \(-0.613547\pi\)
−0.349202 + 0.937047i \(0.613547\pi\)
\(294\) 2.00431 0.116893
\(295\) −5.51945 −0.321355
\(296\) 9.11210 0.529630
\(297\) 0 0
\(298\) −1.72417 −0.0998783
\(299\) 26.1343 1.51139
\(300\) 2.00431 0.115719
\(301\) −3.42569 −0.197453
\(302\) −9.14895 −0.526463
\(303\) 18.9287 1.08743
\(304\) 7.73531 0.443651
\(305\) 3.62844 0.207764
\(306\) 0.305748 0.0174785
\(307\) −19.6625 −1.12220 −0.561100 0.827748i \(-0.689622\pi\)
−0.561100 + 0.827748i \(0.689622\pi\)
\(308\) 0 0
\(309\) −7.05482 −0.401335
\(310\) −7.53480 −0.427948
\(311\) −10.8025 −0.612552 −0.306276 0.951943i \(-0.599083\pi\)
−0.306276 + 0.951943i \(0.599083\pi\)
\(312\) −8.07596 −0.457211
\(313\) −14.7449 −0.833431 −0.416715 0.909037i \(-0.636819\pi\)
−0.416715 + 0.909037i \(0.636819\pi\)
\(314\) −0.876025 −0.0494370
\(315\) 1.01724 0.0573151
\(316\) 6.20005 0.348780
\(317\) 17.4668 0.981035 0.490518 0.871431i \(-0.336808\pi\)
0.490518 + 0.871431i \(0.336808\pi\)
\(318\) 12.8176 0.718774
\(319\) 0 0
\(320\) −1.00000 −0.0559017
\(321\) 20.8951 1.16625
\(322\) 6.48607 0.361455
\(323\) 2.32497 0.129365
\(324\) −11.0169 −0.612052
\(325\) −4.02930 −0.223506
\(326\) −6.49269 −0.359597
\(327\) 24.3666 1.34748
\(328\) 9.68330 0.534671
\(329\) 6.21202 0.342480
\(330\) 0 0
\(331\) 22.3375 1.22778 0.613890 0.789391i \(-0.289604\pi\)
0.613890 + 0.789391i \(0.289604\pi\)
\(332\) 14.5768 0.800006
\(333\) 9.26920 0.507949
\(334\) −20.0322 −1.09611
\(335\) −0.505448 −0.0276156
\(336\) −2.00431 −0.109344
\(337\) 14.2386 0.775624 0.387812 0.921739i \(-0.373231\pi\)
0.387812 + 0.921739i \(0.373231\pi\)
\(338\) 3.23529 0.175976
\(339\) −13.3731 −0.726325
\(340\) −0.300566 −0.0163005
\(341\) 0 0
\(342\) 7.86868 0.425489
\(343\) −1.00000 −0.0539949
\(344\) 3.42569 0.184701
\(345\) 13.0001 0.699900
\(346\) 1.52088 0.0817629
\(347\) 20.6300 1.10748 0.553739 0.832690i \(-0.313200\pi\)
0.553739 + 0.832690i \(0.313200\pi\)
\(348\) 3.64127 0.195193
\(349\) −16.0047 −0.856711 −0.428355 0.903610i \(-0.640907\pi\)
−0.428355 + 0.903610i \(0.640907\pi\)
\(350\) −1.00000 −0.0534522
\(351\) 16.0127 0.854693
\(352\) 0 0
\(353\) −1.46881 −0.0781767 −0.0390883 0.999236i \(-0.512445\pi\)
−0.0390883 + 0.999236i \(0.512445\pi\)
\(354\) 11.0627 0.587974
\(355\) −8.37915 −0.444719
\(356\) 1.92815 0.102192
\(357\) −0.602426 −0.0318838
\(358\) 24.1512 1.27643
\(359\) −8.43193 −0.445020 −0.222510 0.974930i \(-0.571425\pi\)
−0.222510 + 0.974930i \(0.571425\pi\)
\(360\) −1.01724 −0.0536133
\(361\) 40.8350 2.14921
\(362\) 16.6623 0.875750
\(363\) 0 0
\(364\) 4.02930 0.211193
\(365\) −1.40274 −0.0734228
\(366\) −7.27250 −0.380139
\(367\) 27.2590 1.42291 0.711454 0.702732i \(-0.248037\pi\)
0.711454 + 0.702732i \(0.248037\pi\)
\(368\) −6.48607 −0.338110
\(369\) 9.85025 0.512784
\(370\) −9.11210 −0.473715
\(371\) −6.39502 −0.332013
\(372\) 15.1020 0.783004
\(373\) 14.7417 0.763299 0.381649 0.924307i \(-0.375356\pi\)
0.381649 + 0.924307i \(0.375356\pi\)
\(374\) 0 0
\(375\) −2.00431 −0.103502
\(376\) −6.21202 −0.320360
\(377\) −7.32014 −0.377006
\(378\) 3.97405 0.204403
\(379\) −26.2074 −1.34618 −0.673091 0.739560i \(-0.735034\pi\)
−0.673091 + 0.739560i \(0.735034\pi\)
\(380\) −7.73531 −0.396813
\(381\) −25.8392 −1.32378
\(382\) −2.97548 −0.152239
\(383\) −18.0535 −0.922492 −0.461246 0.887272i \(-0.652597\pi\)
−0.461246 + 0.887272i \(0.652597\pi\)
\(384\) 2.00431 0.102282
\(385\) 0 0
\(386\) −11.5006 −0.585365
\(387\) 3.48475 0.177140
\(388\) −16.4944 −0.837374
\(389\) −21.4054 −1.08530 −0.542648 0.839960i \(-0.682578\pi\)
−0.542648 + 0.839960i \(0.682578\pi\)
\(390\) 8.07596 0.408942
\(391\) −1.94949 −0.0985900
\(392\) 1.00000 0.0505076
\(393\) 40.8602 2.06113
\(394\) 16.0597 0.809074
\(395\) −6.20005 −0.311959
\(396\) 0 0
\(397\) 4.87615 0.244727 0.122364 0.992485i \(-0.460953\pi\)
0.122364 + 0.992485i \(0.460953\pi\)
\(398\) 11.0899 0.555885
\(399\) −15.5039 −0.776167
\(400\) 1.00000 0.0500000
\(401\) 15.6680 0.782423 0.391212 0.920301i \(-0.372056\pi\)
0.391212 + 0.920301i \(0.372056\pi\)
\(402\) 1.01307 0.0505274
\(403\) −30.3600 −1.51234
\(404\) 9.44404 0.469859
\(405\) 11.0169 0.547436
\(406\) −1.81673 −0.0901626
\(407\) 0 0
\(408\) 0.602426 0.0298245
\(409\) 37.1821 1.83854 0.919268 0.393633i \(-0.128782\pi\)
0.919268 + 0.393633i \(0.128782\pi\)
\(410\) −9.68330 −0.478224
\(411\) −35.4670 −1.74946
\(412\) −3.51983 −0.173410
\(413\) −5.51945 −0.271594
\(414\) −6.59790 −0.324269
\(415\) −14.5768 −0.715547
\(416\) −4.02930 −0.197553
\(417\) 27.0129 1.32283
\(418\) 0 0
\(419\) −27.2599 −1.33173 −0.665865 0.746072i \(-0.731938\pi\)
−0.665865 + 0.746072i \(0.731938\pi\)
\(420\) 2.00431 0.0978001
\(421\) −13.1997 −0.643315 −0.321658 0.946856i \(-0.604240\pi\)
−0.321658 + 0.946856i \(0.604240\pi\)
\(422\) −25.5146 −1.24203
\(423\) −6.31912 −0.307246
\(424\) 6.39502 0.310570
\(425\) 0.300566 0.0145796
\(426\) 16.7944 0.813690
\(427\) 3.62844 0.175592
\(428\) 10.4251 0.503917
\(429\) 0 0
\(430\) −3.42569 −0.165201
\(431\) −17.4534 −0.840699 −0.420350 0.907362i \(-0.638093\pi\)
−0.420350 + 0.907362i \(0.638093\pi\)
\(432\) −3.97405 −0.191202
\(433\) −28.2800 −1.35905 −0.679525 0.733652i \(-0.737814\pi\)
−0.679525 + 0.733652i \(0.737814\pi\)
\(434\) −7.53480 −0.361682
\(435\) −3.64127 −0.174586
\(436\) 12.1571 0.582221
\(437\) −50.1718 −2.40004
\(438\) 2.81152 0.134340
\(439\) 35.7484 1.70618 0.853090 0.521764i \(-0.174726\pi\)
0.853090 + 0.521764i \(0.174726\pi\)
\(440\) 0 0
\(441\) 1.01724 0.0484401
\(442\) −1.21107 −0.0576048
\(443\) 14.9441 0.710016 0.355008 0.934863i \(-0.384478\pi\)
0.355008 + 0.934863i \(0.384478\pi\)
\(444\) 18.2634 0.866744
\(445\) −1.92815 −0.0914031
\(446\) −23.1843 −1.09781
\(447\) −3.45576 −0.163452
\(448\) −1.00000 −0.0472456
\(449\) 13.9847 0.659981 0.329990 0.943984i \(-0.392954\pi\)
0.329990 + 0.943984i \(0.392954\pi\)
\(450\) 1.01724 0.0479532
\(451\) 0 0
\(452\) −6.67216 −0.313832
\(453\) −18.3373 −0.861562
\(454\) −4.97362 −0.233423
\(455\) −4.02930 −0.188897
\(456\) 15.5039 0.726038
\(457\) 10.2714 0.480477 0.240238 0.970714i \(-0.422774\pi\)
0.240238 + 0.970714i \(0.422774\pi\)
\(458\) −25.6627 −1.19914
\(459\) −1.19447 −0.0557529
\(460\) 6.48607 0.302415
\(461\) −17.4058 −0.810671 −0.405335 0.914168i \(-0.632845\pi\)
−0.405335 + 0.914168i \(0.632845\pi\)
\(462\) 0 0
\(463\) −30.3943 −1.41254 −0.706271 0.707942i \(-0.749624\pi\)
−0.706271 + 0.707942i \(0.749624\pi\)
\(464\) 1.81673 0.0843394
\(465\) −15.1020 −0.700340
\(466\) 3.81437 0.176697
\(467\) 23.8052 1.10158 0.550788 0.834646i \(-0.314327\pi\)
0.550788 + 0.834646i \(0.314327\pi\)
\(468\) −4.09877 −0.189466
\(469\) −0.505448 −0.0233394
\(470\) 6.21202 0.286539
\(471\) −1.75582 −0.0809040
\(472\) 5.51945 0.254053
\(473\) 0 0
\(474\) 12.4268 0.570782
\(475\) 7.73531 0.354920
\(476\) −0.300566 −0.0137764
\(477\) 6.50528 0.297856
\(478\) 6.58100 0.301008
\(479\) 7.79996 0.356389 0.178195 0.983995i \(-0.442974\pi\)
0.178195 + 0.983995i \(0.442974\pi\)
\(480\) −2.00431 −0.0914836
\(481\) −36.7154 −1.67408
\(482\) 23.9679 1.09171
\(483\) 13.0001 0.591523
\(484\) 0 0
\(485\) 16.4944 0.748970
\(486\) −10.1592 −0.460829
\(487\) −19.3109 −0.875060 −0.437530 0.899204i \(-0.644147\pi\)
−0.437530 + 0.899204i \(0.644147\pi\)
\(488\) −3.62844 −0.164252
\(489\) −13.0133 −0.588484
\(490\) −1.00000 −0.0451754
\(491\) 16.5180 0.745449 0.372724 0.927942i \(-0.378424\pi\)
0.372724 + 0.927942i \(0.378424\pi\)
\(492\) 19.4083 0.874994
\(493\) 0.546046 0.0245927
\(494\) −31.1679 −1.40231
\(495\) 0 0
\(496\) 7.53480 0.338322
\(497\) −8.37915 −0.375856
\(498\) 29.2164 1.30922
\(499\) 11.8692 0.531341 0.265670 0.964064i \(-0.414407\pi\)
0.265670 + 0.964064i \(0.414407\pi\)
\(500\) −1.00000 −0.0447214
\(501\) −40.1507 −1.79380
\(502\) 5.08595 0.226997
\(503\) −6.76151 −0.301481 −0.150741 0.988573i \(-0.548166\pi\)
−0.150741 + 0.988573i \(0.548166\pi\)
\(504\) −1.01724 −0.0453115
\(505\) −9.44404 −0.420254
\(506\) 0 0
\(507\) 6.48451 0.287987
\(508\) −12.8919 −0.571984
\(509\) −37.3337 −1.65479 −0.827393 0.561624i \(-0.810177\pi\)
−0.827393 + 0.561624i \(0.810177\pi\)
\(510\) −0.602426 −0.0266759
\(511\) −1.40274 −0.0620536
\(512\) 1.00000 0.0441942
\(513\) −30.7405 −1.35723
\(514\) 11.0968 0.489458
\(515\) 3.51983 0.155102
\(516\) 6.86612 0.302264
\(517\) 0 0
\(518\) −9.11210 −0.400363
\(519\) 3.04831 0.133806
\(520\) 4.02930 0.176697
\(521\) −13.6299 −0.597137 −0.298569 0.954388i \(-0.596509\pi\)
−0.298569 + 0.954388i \(0.596509\pi\)
\(522\) 1.84805 0.0808869
\(523\) −7.03010 −0.307405 −0.153702 0.988117i \(-0.549120\pi\)
−0.153702 + 0.988117i \(0.549120\pi\)
\(524\) 20.3862 0.890577
\(525\) −2.00431 −0.0874751
\(526\) −2.64779 −0.115449
\(527\) 2.26470 0.0986520
\(528\) 0 0
\(529\) 19.0691 0.829091
\(530\) −6.39502 −0.277782
\(531\) 5.61461 0.243653
\(532\) −7.73531 −0.335368
\(533\) −39.0170 −1.69001
\(534\) 3.86460 0.167238
\(535\) −10.4251 −0.450717
\(536\) 0.505448 0.0218320
\(537\) 48.4064 2.08889
\(538\) −3.42199 −0.147532
\(539\) 0 0
\(540\) 3.97405 0.171016
\(541\) 15.3397 0.659507 0.329753 0.944067i \(-0.393034\pi\)
0.329753 + 0.944067i \(0.393034\pi\)
\(542\) −21.1788 −0.909706
\(543\) 33.3963 1.43317
\(544\) 0.300566 0.0128867
\(545\) −12.1571 −0.520755
\(546\) 8.07596 0.345619
\(547\) 17.6527 0.754775 0.377388 0.926055i \(-0.376822\pi\)
0.377388 + 0.926055i \(0.376822\pi\)
\(548\) −17.6954 −0.755911
\(549\) −3.69100 −0.157528
\(550\) 0 0
\(551\) 14.0529 0.598675
\(552\) −13.0001 −0.553320
\(553\) −6.20005 −0.263653
\(554\) 27.3915 1.16375
\(555\) −18.2634 −0.775239
\(556\) 13.4774 0.571570
\(557\) 27.5742 1.16836 0.584178 0.811626i \(-0.301417\pi\)
0.584178 + 0.811626i \(0.301417\pi\)
\(558\) 7.66470 0.324473
\(559\) −13.8031 −0.583810
\(560\) 1.00000 0.0422577
\(561\) 0 0
\(562\) 11.2620 0.475057
\(563\) −14.3283 −0.603866 −0.301933 0.953329i \(-0.597632\pi\)
−0.301933 + 0.953329i \(0.597632\pi\)
\(564\) −12.4508 −0.524273
\(565\) 6.67216 0.280700
\(566\) −20.2928 −0.852971
\(567\) 11.0169 0.462668
\(568\) 8.37915 0.351581
\(569\) −29.2285 −1.22532 −0.612662 0.790345i \(-0.709901\pi\)
−0.612662 + 0.790345i \(0.709901\pi\)
\(570\) −15.5039 −0.649388
\(571\) −42.5911 −1.78238 −0.891190 0.453630i \(-0.850129\pi\)
−0.891190 + 0.453630i \(0.850129\pi\)
\(572\) 0 0
\(573\) −5.96378 −0.249140
\(574\) −9.68330 −0.404173
\(575\) −6.48607 −0.270488
\(576\) 1.01724 0.0423851
\(577\) 25.4138 1.05799 0.528994 0.848625i \(-0.322569\pi\)
0.528994 + 0.848625i \(0.322569\pi\)
\(578\) −16.9097 −0.703349
\(579\) −23.0507 −0.957955
\(580\) −1.81673 −0.0754354
\(581\) −14.5768 −0.604748
\(582\) −33.0597 −1.37037
\(583\) 0 0
\(584\) 1.40274 0.0580458
\(585\) 4.09877 0.169463
\(586\) −11.9548 −0.493847
\(587\) 48.0000 1.98117 0.990586 0.136892i \(-0.0437115\pi\)
0.990586 + 0.136892i \(0.0437115\pi\)
\(588\) 2.00431 0.0826562
\(589\) 58.2840 2.40155
\(590\) −5.51945 −0.227232
\(591\) 32.1884 1.32406
\(592\) 9.11210 0.374505
\(593\) 3.34128 0.137210 0.0686050 0.997644i \(-0.478145\pi\)
0.0686050 + 0.997644i \(0.478145\pi\)
\(594\) 0 0
\(595\) 0.300566 0.0123220
\(596\) −1.72417 −0.0706246
\(597\) 22.2275 0.909711
\(598\) 26.1343 1.06871
\(599\) −43.1060 −1.76126 −0.880632 0.473800i \(-0.842882\pi\)
−0.880632 + 0.473800i \(0.842882\pi\)
\(600\) 2.00431 0.0818254
\(601\) −17.8432 −0.727840 −0.363920 0.931430i \(-0.618562\pi\)
−0.363920 + 0.931430i \(0.618562\pi\)
\(602\) −3.42569 −0.139621
\(603\) 0.514162 0.0209383
\(604\) −9.14895 −0.372266
\(605\) 0 0
\(606\) 18.9287 0.768928
\(607\) −11.3238 −0.459619 −0.229809 0.973236i \(-0.573810\pi\)
−0.229809 + 0.973236i \(0.573810\pi\)
\(608\) 7.73531 0.313708
\(609\) −3.64127 −0.147552
\(610\) 3.62844 0.146911
\(611\) 25.0301 1.01261
\(612\) 0.305748 0.0123591
\(613\) −36.2483 −1.46405 −0.732027 0.681275i \(-0.761426\pi\)
−0.732027 + 0.681275i \(0.761426\pi\)
\(614\) −19.6625 −0.793516
\(615\) −19.4083 −0.782618
\(616\) 0 0
\(617\) 40.1661 1.61702 0.808512 0.588479i \(-0.200273\pi\)
0.808512 + 0.588479i \(0.200273\pi\)
\(618\) −7.05482 −0.283787
\(619\) 8.86070 0.356141 0.178071 0.984018i \(-0.443014\pi\)
0.178071 + 0.984018i \(0.443014\pi\)
\(620\) −7.53480 −0.302605
\(621\) 25.7760 1.03436
\(622\) −10.8025 −0.433139
\(623\) −1.92815 −0.0772497
\(624\) −8.07596 −0.323297
\(625\) 1.00000 0.0400000
\(626\) −14.7449 −0.589325
\(627\) 0 0
\(628\) −0.876025 −0.0349572
\(629\) 2.73879 0.109203
\(630\) 1.01724 0.0405279
\(631\) −2.39090 −0.0951802 −0.0475901 0.998867i \(-0.515154\pi\)
−0.0475901 + 0.998867i \(0.515154\pi\)
\(632\) 6.20005 0.246625
\(633\) −51.1391 −2.03260
\(634\) 17.4668 0.693697
\(635\) 12.8919 0.511598
\(636\) 12.8176 0.508250
\(637\) −4.02930 −0.159647
\(638\) 0 0
\(639\) 8.52362 0.337189
\(640\) −1.00000 −0.0395285
\(641\) −7.27551 −0.287365 −0.143683 0.989624i \(-0.545894\pi\)
−0.143683 + 0.989624i \(0.545894\pi\)
\(642\) 20.8951 0.824664
\(643\) −9.04508 −0.356703 −0.178352 0.983967i \(-0.557076\pi\)
−0.178352 + 0.983967i \(0.557076\pi\)
\(644\) 6.48607 0.255587
\(645\) −6.86612 −0.270353
\(646\) 2.32497 0.0914747
\(647\) −31.0318 −1.21999 −0.609993 0.792407i \(-0.708828\pi\)
−0.609993 + 0.792407i \(0.708828\pi\)
\(648\) −11.0169 −0.432786
\(649\) 0 0
\(650\) −4.02930 −0.158042
\(651\) −15.1020 −0.591895
\(652\) −6.49269 −0.254274
\(653\) 42.3720 1.65814 0.829072 0.559142i \(-0.188870\pi\)
0.829072 + 0.559142i \(0.188870\pi\)
\(654\) 24.3666 0.952810
\(655\) −20.3862 −0.796556
\(656\) 9.68330 0.378069
\(657\) 1.42693 0.0556697
\(658\) 6.21202 0.242170
\(659\) 25.5490 0.995247 0.497624 0.867393i \(-0.334206\pi\)
0.497624 + 0.867393i \(0.334206\pi\)
\(660\) 0 0
\(661\) 1.55717 0.0605669 0.0302834 0.999541i \(-0.490359\pi\)
0.0302834 + 0.999541i \(0.490359\pi\)
\(662\) 22.3375 0.868172
\(663\) −2.42736 −0.0942707
\(664\) 14.5768 0.565690
\(665\) 7.73531 0.299962
\(666\) 9.26920 0.359174
\(667\) −11.7834 −0.456255
\(668\) −20.0322 −0.775070
\(669\) −46.4684 −1.79657
\(670\) −0.505448 −0.0195271
\(671\) 0 0
\(672\) −2.00431 −0.0773178
\(673\) 25.0244 0.964620 0.482310 0.876001i \(-0.339798\pi\)
0.482310 + 0.876001i \(0.339798\pi\)
\(674\) 14.2386 0.548449
\(675\) −3.97405 −0.152961
\(676\) 3.23529 0.124434
\(677\) 23.4532 0.901380 0.450690 0.892681i \(-0.351178\pi\)
0.450690 + 0.892681i \(0.351178\pi\)
\(678\) −13.3731 −0.513589
\(679\) 16.4944 0.632995
\(680\) −0.300566 −0.0115262
\(681\) −9.96865 −0.381999
\(682\) 0 0
\(683\) −36.6815 −1.40358 −0.701790 0.712384i \(-0.747616\pi\)
−0.701790 + 0.712384i \(0.747616\pi\)
\(684\) 7.86868 0.300866
\(685\) 17.6954 0.676107
\(686\) −1.00000 −0.0381802
\(687\) −51.4360 −1.96240
\(688\) 3.42569 0.130603
\(689\) −25.7675 −0.981662
\(690\) 13.0001 0.494904
\(691\) 27.6172 1.05061 0.525303 0.850915i \(-0.323952\pi\)
0.525303 + 0.850915i \(0.323952\pi\)
\(692\) 1.52088 0.0578151
\(693\) 0 0
\(694\) 20.6300 0.783106
\(695\) −13.4774 −0.511228
\(696\) 3.64127 0.138022
\(697\) 2.91047 0.110242
\(698\) −16.0047 −0.605786
\(699\) 7.64517 0.289167
\(700\) −1.00000 −0.0377964
\(701\) 29.9881 1.13263 0.566317 0.824188i \(-0.308368\pi\)
0.566317 + 0.824188i \(0.308368\pi\)
\(702\) 16.0127 0.604359
\(703\) 70.4849 2.65839
\(704\) 0 0
\(705\) 12.4508 0.468924
\(706\) −1.46881 −0.0552793
\(707\) −9.44404 −0.355180
\(708\) 11.0627 0.415760
\(709\) −5.18512 −0.194731 −0.0973656 0.995249i \(-0.531042\pi\)
−0.0973656 + 0.995249i \(0.531042\pi\)
\(710\) −8.37915 −0.314464
\(711\) 6.30695 0.236529
\(712\) 1.92815 0.0722605
\(713\) −48.8712 −1.83024
\(714\) −0.602426 −0.0225452
\(715\) 0 0
\(716\) 24.1512 0.902573
\(717\) 13.1903 0.492602
\(718\) −8.43193 −0.314677
\(719\) −16.7828 −0.625892 −0.312946 0.949771i \(-0.601316\pi\)
−0.312946 + 0.949771i \(0.601316\pi\)
\(720\) −1.01724 −0.0379103
\(721\) 3.51983 0.131085
\(722\) 40.8350 1.51972
\(723\) 48.0390 1.78659
\(724\) 16.6623 0.619249
\(725\) 1.81673 0.0674715
\(726\) 0 0
\(727\) 9.72794 0.360789 0.180395 0.983594i \(-0.442263\pi\)
0.180395 + 0.983594i \(0.442263\pi\)
\(728\) 4.02930 0.149336
\(729\) 12.6888 0.469954
\(730\) −1.40274 −0.0519178
\(731\) 1.02964 0.0380828
\(732\) −7.27250 −0.268799
\(733\) 10.5873 0.391050 0.195525 0.980699i \(-0.437359\pi\)
0.195525 + 0.980699i \(0.437359\pi\)
\(734\) 27.2590 1.00615
\(735\) −2.00431 −0.0739299
\(736\) −6.48607 −0.239080
\(737\) 0 0
\(738\) 9.85025 0.362593
\(739\) −11.6909 −0.430058 −0.215029 0.976608i \(-0.568985\pi\)
−0.215029 + 0.976608i \(0.568985\pi\)
\(740\) −9.11210 −0.334967
\(741\) −62.4700 −2.29489
\(742\) −6.39502 −0.234768
\(743\) 2.32433 0.0852713 0.0426356 0.999091i \(-0.486425\pi\)
0.0426356 + 0.999091i \(0.486425\pi\)
\(744\) 15.1020 0.553667
\(745\) 1.72417 0.0631686
\(746\) 14.7417 0.539734
\(747\) 14.8281 0.542533
\(748\) 0 0
\(749\) −10.4251 −0.380925
\(750\) −2.00431 −0.0731869
\(751\) −13.6597 −0.498450 −0.249225 0.968446i \(-0.580176\pi\)
−0.249225 + 0.968446i \(0.580176\pi\)
\(752\) −6.21202 −0.226529
\(753\) 10.1938 0.371483
\(754\) −7.32014 −0.266584
\(755\) 9.14895 0.332965
\(756\) 3.97405 0.144535
\(757\) 20.4404 0.742919 0.371460 0.928449i \(-0.378857\pi\)
0.371460 + 0.928449i \(0.378857\pi\)
\(758\) −26.2074 −0.951894
\(759\) 0 0
\(760\) −7.73531 −0.280589
\(761\) 5.54838 0.201129 0.100564 0.994931i \(-0.467935\pi\)
0.100564 + 0.994931i \(0.467935\pi\)
\(762\) −25.8392 −0.936057
\(763\) −12.1571 −0.440118
\(764\) −2.97548 −0.107649
\(765\) −0.305748 −0.0110543
\(766\) −18.0535 −0.652300
\(767\) −22.2395 −0.803023
\(768\) 2.00431 0.0723242
\(769\) 8.18880 0.295296 0.147648 0.989040i \(-0.452830\pi\)
0.147648 + 0.989040i \(0.452830\pi\)
\(770\) 0 0
\(771\) 22.2413 0.801002
\(772\) −11.5006 −0.413916
\(773\) 10.5045 0.377819 0.188909 0.981995i \(-0.439505\pi\)
0.188909 + 0.981995i \(0.439505\pi\)
\(774\) 3.48475 0.125257
\(775\) 7.53480 0.270658
\(776\) −16.4944 −0.592113
\(777\) −18.2634 −0.655197
\(778\) −21.4054 −0.767420
\(779\) 74.9033 2.68369
\(780\) 8.07596 0.289166
\(781\) 0 0
\(782\) −1.94949 −0.0697137
\(783\) −7.21977 −0.258013
\(784\) 1.00000 0.0357143
\(785\) 0.876025 0.0312667
\(786\) 40.8602 1.45744
\(787\) 11.8432 0.422166 0.211083 0.977468i \(-0.432301\pi\)
0.211083 + 0.977468i \(0.432301\pi\)
\(788\) 16.0597 0.572101
\(789\) −5.30697 −0.188933
\(790\) −6.20005 −0.220588
\(791\) 6.67216 0.237235
\(792\) 0 0
\(793\) 14.6201 0.519174
\(794\) 4.87615 0.173048
\(795\) −12.8176 −0.454592
\(796\) 11.0899 0.393070
\(797\) −22.3237 −0.790747 −0.395374 0.918520i \(-0.629385\pi\)
−0.395374 + 0.918520i \(0.629385\pi\)
\(798\) −15.5039 −0.548833
\(799\) −1.86712 −0.0660540
\(800\) 1.00000 0.0353553
\(801\) 1.96139 0.0693025
\(802\) 15.6680 0.553257
\(803\) 0 0
\(804\) 1.01307 0.0357283
\(805\) −6.48607 −0.228604
\(806\) −30.3600 −1.06938
\(807\) −6.85871 −0.241438
\(808\) 9.44404 0.332240
\(809\) −32.0996 −1.12856 −0.564282 0.825582i \(-0.690847\pi\)
−0.564282 + 0.825582i \(0.690847\pi\)
\(810\) 11.0169 0.387096
\(811\) 42.4498 1.49061 0.745306 0.666722i \(-0.232303\pi\)
0.745306 + 0.666722i \(0.232303\pi\)
\(812\) −1.81673 −0.0637546
\(813\) −42.4487 −1.48874
\(814\) 0 0
\(815\) 6.49269 0.227429
\(816\) 0.602426 0.0210891
\(817\) 26.4987 0.927074
\(818\) 37.1821 1.30004
\(819\) 4.09877 0.143223
\(820\) −9.68330 −0.338156
\(821\) −17.7352 −0.618962 −0.309481 0.950906i \(-0.600155\pi\)
−0.309481 + 0.950906i \(0.600155\pi\)
\(822\) −35.4670 −1.23706
\(823\) 37.9393 1.32248 0.661241 0.750173i \(-0.270030\pi\)
0.661241 + 0.750173i \(0.270030\pi\)
\(824\) −3.51983 −0.122619
\(825\) 0 0
\(826\) −5.51945 −0.192046
\(827\) 40.9256 1.42312 0.711561 0.702624i \(-0.247988\pi\)
0.711561 + 0.702624i \(0.247988\pi\)
\(828\) −6.59790 −0.229293
\(829\) 2.81991 0.0979395 0.0489698 0.998800i \(-0.484406\pi\)
0.0489698 + 0.998800i \(0.484406\pi\)
\(830\) −14.5768 −0.505968
\(831\) 54.9010 1.90449
\(832\) −4.02930 −0.139691
\(833\) 0.300566 0.0104140
\(834\) 27.0129 0.935379
\(835\) 20.0322 0.693244
\(836\) 0 0
\(837\) −29.9437 −1.03501
\(838\) −27.2599 −0.941676
\(839\) −55.3938 −1.91241 −0.956204 0.292701i \(-0.905446\pi\)
−0.956204 + 0.292701i \(0.905446\pi\)
\(840\) 2.00431 0.0691551
\(841\) −25.6995 −0.886190
\(842\) −13.1997 −0.454892
\(843\) 22.5724 0.777435
\(844\) −25.5146 −0.878250
\(845\) −3.23529 −0.111297
\(846\) −6.31912 −0.217256
\(847\) 0 0
\(848\) 6.39502 0.219606
\(849\) −40.6730 −1.39589
\(850\) 0.300566 0.0103093
\(851\) −59.1017 −2.02598
\(852\) 16.7944 0.575366
\(853\) −49.6786 −1.70096 −0.850481 0.526006i \(-0.823689\pi\)
−0.850481 + 0.526006i \(0.823689\pi\)
\(854\) 3.62844 0.124163
\(855\) −7.86868 −0.269103
\(856\) 10.4251 0.356323
\(857\) −20.7002 −0.707104 −0.353552 0.935415i \(-0.615026\pi\)
−0.353552 + 0.935415i \(0.615026\pi\)
\(858\) 0 0
\(859\) −15.2630 −0.520766 −0.260383 0.965505i \(-0.583849\pi\)
−0.260383 + 0.965505i \(0.583849\pi\)
\(860\) −3.42569 −0.116815
\(861\) −19.4083 −0.661433
\(862\) −17.4534 −0.594464
\(863\) 8.86338 0.301713 0.150857 0.988556i \(-0.451797\pi\)
0.150857 + 0.988556i \(0.451797\pi\)
\(864\) −3.97405 −0.135200
\(865\) −1.52088 −0.0517114
\(866\) −28.2800 −0.960994
\(867\) −33.8921 −1.15104
\(868\) −7.53480 −0.255748
\(869\) 0 0
\(870\) −3.64127 −0.123451
\(871\) −2.03660 −0.0690076
\(872\) 12.1571 0.411693
\(873\) −16.7787 −0.567874
\(874\) −50.1718 −1.69709
\(875\) 1.00000 0.0338062
\(876\) 2.81152 0.0949925
\(877\) 16.7377 0.565192 0.282596 0.959239i \(-0.408804\pi\)
0.282596 + 0.959239i \(0.408804\pi\)
\(878\) 35.7484 1.20645
\(879\) −23.9610 −0.808185
\(880\) 0 0
\(881\) −4.10239 −0.138213 −0.0691065 0.997609i \(-0.522015\pi\)
−0.0691065 + 0.997609i \(0.522015\pi\)
\(882\) 1.01724 0.0342523
\(883\) 40.7341 1.37081 0.685406 0.728161i \(-0.259625\pi\)
0.685406 + 0.728161i \(0.259625\pi\)
\(884\) −1.21107 −0.0407327
\(885\) −11.0627 −0.371867
\(886\) 14.9441 0.502057
\(887\) −17.7893 −0.597305 −0.298652 0.954362i \(-0.596537\pi\)
−0.298652 + 0.954362i \(0.596537\pi\)
\(888\) 18.2634 0.612881
\(889\) 12.8919 0.432379
\(890\) −1.92815 −0.0646318
\(891\) 0 0
\(892\) −23.1843 −0.776268
\(893\) −48.0519 −1.60800
\(894\) −3.45576 −0.115578
\(895\) −24.1512 −0.807286
\(896\) −1.00000 −0.0334077
\(897\) 52.3812 1.74896
\(898\) 13.9847 0.466677
\(899\) 13.6887 0.456542
\(900\) 1.01724 0.0339080
\(901\) 1.92212 0.0640352
\(902\) 0 0
\(903\) −6.86612 −0.228490
\(904\) −6.67216 −0.221913
\(905\) −16.6623 −0.553873
\(906\) −18.3373 −0.609216
\(907\) −3.77888 −0.125476 −0.0627379 0.998030i \(-0.519983\pi\)
−0.0627379 + 0.998030i \(0.519983\pi\)
\(908\) −4.97362 −0.165055
\(909\) 9.60687 0.318640
\(910\) −4.02930 −0.133570
\(911\) 36.8482 1.22084 0.610418 0.792080i \(-0.291002\pi\)
0.610418 + 0.792080i \(0.291002\pi\)
\(912\) 15.5039 0.513386
\(913\) 0 0
\(914\) 10.2714 0.339748
\(915\) 7.27250 0.240421
\(916\) −25.6627 −0.847921
\(917\) −20.3862 −0.673213
\(918\) −1.19447 −0.0394232
\(919\) −56.7946 −1.87348 −0.936740 0.350025i \(-0.886173\pi\)
−0.936740 + 0.350025i \(0.886173\pi\)
\(920\) 6.48607 0.213839
\(921\) −39.4098 −1.29860
\(922\) −17.4058 −0.573231
\(923\) −33.7621 −1.11129
\(924\) 0 0
\(925\) 9.11210 0.299604
\(926\) −30.3943 −0.998818
\(927\) −3.58052 −0.117600
\(928\) 1.81673 0.0596369
\(929\) −18.0311 −0.591580 −0.295790 0.955253i \(-0.595583\pi\)
−0.295790 + 0.955253i \(0.595583\pi\)
\(930\) −15.1020 −0.495215
\(931\) 7.73531 0.253515
\(932\) 3.81437 0.124944
\(933\) −21.6514 −0.708836
\(934\) 23.8052 0.778931
\(935\) 0 0
\(936\) −4.09877 −0.133973
\(937\) −20.7850 −0.679016 −0.339508 0.940603i \(-0.610261\pi\)
−0.339508 + 0.940603i \(0.610261\pi\)
\(938\) −0.505448 −0.0165035
\(939\) −29.5533 −0.964435
\(940\) 6.21202 0.202614
\(941\) 4.11596 0.134176 0.0670882 0.997747i \(-0.478629\pi\)
0.0670882 + 0.997747i \(0.478629\pi\)
\(942\) −1.75582 −0.0572078
\(943\) −62.8066 −2.04526
\(944\) 5.51945 0.179643
\(945\) −3.97405 −0.129276
\(946\) 0 0
\(947\) −31.0569 −1.00921 −0.504607 0.863349i \(-0.668363\pi\)
−0.504607 + 0.863349i \(0.668363\pi\)
\(948\) 12.4268 0.403604
\(949\) −5.65207 −0.183474
\(950\) 7.73531 0.250967
\(951\) 35.0089 1.13524
\(952\) −0.300566 −0.00974140
\(953\) 17.8888 0.579474 0.289737 0.957106i \(-0.406432\pi\)
0.289737 + 0.957106i \(0.406432\pi\)
\(954\) 6.50528 0.210616
\(955\) 2.97548 0.0962843
\(956\) 6.58100 0.212845
\(957\) 0 0
\(958\) 7.79996 0.252005
\(959\) 17.6954 0.571415
\(960\) −2.00431 −0.0646887
\(961\) 25.7731 0.831392
\(962\) −36.7154 −1.18375
\(963\) 10.6049 0.341737
\(964\) 23.9679 0.771955
\(965\) 11.5006 0.370217
\(966\) 13.0001 0.418270
\(967\) −59.5313 −1.91440 −0.957199 0.289431i \(-0.906534\pi\)
−0.957199 + 0.289431i \(0.906534\pi\)
\(968\) 0 0
\(969\) 4.65995 0.149699
\(970\) 16.4944 0.529602
\(971\) −43.1529 −1.38484 −0.692421 0.721494i \(-0.743456\pi\)
−0.692421 + 0.721494i \(0.743456\pi\)
\(972\) −10.1592 −0.325855
\(973\) −13.4774 −0.432066
\(974\) −19.3109 −0.618761
\(975\) −8.07596 −0.258638
\(976\) −3.62844 −0.116143
\(977\) −46.0411 −1.47299 −0.736493 0.676445i \(-0.763520\pi\)
−0.736493 + 0.676445i \(0.763520\pi\)
\(978\) −13.0133 −0.416121
\(979\) 0 0
\(980\) −1.00000 −0.0319438
\(981\) 12.3667 0.394840
\(982\) 16.5180 0.527112
\(983\) 45.1941 1.44147 0.720734 0.693212i \(-0.243805\pi\)
0.720734 + 0.693212i \(0.243805\pi\)
\(984\) 19.4083 0.618714
\(985\) −16.0597 −0.511703
\(986\) 0.546046 0.0173896
\(987\) 12.4508 0.396313
\(988\) −31.1679 −0.991584
\(989\) −22.2192 −0.706531
\(990\) 0 0
\(991\) −32.5442 −1.03380 −0.516900 0.856046i \(-0.672914\pi\)
−0.516900 + 0.856046i \(0.672914\pi\)
\(992\) 7.53480 0.239230
\(993\) 44.7712 1.42077
\(994\) −8.37915 −0.265770
\(995\) −11.0899 −0.351573
\(996\) 29.2164 0.925756
\(997\) −44.8088 −1.41911 −0.709555 0.704650i \(-0.751104\pi\)
−0.709555 + 0.704650i \(0.751104\pi\)
\(998\) 11.8692 0.375715
\(999\) −36.2120 −1.14570
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 8470.2.a.dh.1.7 8
11.5 even 5 770.2.n.k.421.4 16
11.9 even 5 770.2.n.k.631.4 yes 16
11.10 odd 2 8470.2.a.dg.1.7 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
770.2.n.k.421.4 16 11.5 even 5
770.2.n.k.631.4 yes 16 11.9 even 5
8470.2.a.dg.1.7 8 11.10 odd 2
8470.2.a.dh.1.7 8 1.1 even 1 trivial