Properties

Label 8470.2.a.dg.1.1
Level $8470$
Weight $2$
Character 8470.1
Self dual yes
Analytic conductor $67.633$
Analytic rank $1$
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: \(1\)
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 1.1
Root \(-2.90474\) of defining polynomial
Character \(\chi\) \(=\) 8470.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} -2.90474 q^{3} +1.00000 q^{4} -1.00000 q^{5} +2.90474 q^{6} +1.00000 q^{7} -1.00000 q^{8} +5.43751 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -2.90474 q^{3} +1.00000 q^{4} -1.00000 q^{5} +2.90474 q^{6} +1.00000 q^{7} -1.00000 q^{8} +5.43751 q^{9} +1.00000 q^{10} -2.90474 q^{12} +6.51020 q^{13} -1.00000 q^{14} +2.90474 q^{15} +1.00000 q^{16} -7.07831 q^{17} -5.43751 q^{18} -3.97479 q^{19} -1.00000 q^{20} -2.90474 q^{21} +9.39994 q^{23} +2.90474 q^{24} +1.00000 q^{25} -6.51020 q^{26} -7.08034 q^{27} +1.00000 q^{28} -1.17923 q^{29} -2.90474 q^{30} -1.28349 q^{31} -1.00000 q^{32} +7.07831 q^{34} -1.00000 q^{35} +5.43751 q^{36} -1.00807 q^{37} +3.97479 q^{38} -18.9104 q^{39} +1.00000 q^{40} +5.43144 q^{41} +2.90474 q^{42} +11.2003 q^{43} -5.43751 q^{45} -9.39994 q^{46} -11.3915 q^{47} -2.90474 q^{48} +1.00000 q^{49} -1.00000 q^{50} +20.5606 q^{51} +6.51020 q^{52} -7.75606 q^{53} +7.08034 q^{54} -1.00000 q^{56} +11.5457 q^{57} +1.17923 q^{58} -5.64597 q^{59} +2.90474 q^{60} -6.33662 q^{61} +1.28349 q^{62} +5.43751 q^{63} +1.00000 q^{64} -6.51020 q^{65} +3.88122 q^{67} -7.07831 q^{68} -27.3044 q^{69} +1.00000 q^{70} -0.259926 q^{71} -5.43751 q^{72} -4.82486 q^{73} +1.00807 q^{74} -2.90474 q^{75} -3.97479 q^{76} +18.9104 q^{78} -14.3428 q^{79} -1.00000 q^{80} +4.25401 q^{81} -5.43144 q^{82} -4.59502 q^{83} -2.90474 q^{84} +7.07831 q^{85} -11.2003 q^{86} +3.42535 q^{87} -1.43088 q^{89} +5.43751 q^{90} +6.51020 q^{91} +9.39994 q^{92} +3.72822 q^{93} +11.3915 q^{94} +3.97479 q^{95} +2.90474 q^{96} -8.60695 q^{97} -1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q - 8q^{2} + 8q^{4} - 8q^{5} + 8q^{7} - 8q^{8} + 8q^{9} + O(q^{10}) \) \( 8q - 8q^{2} + 8q^{4} - 8q^{5} + 8q^{7} - 8q^{8} + 8q^{9} + 8q^{10} + q^{13} - 8q^{14} + 8q^{16} - 6q^{17} - 8q^{18} - 5q^{19} - 8q^{20} + 10q^{23} + 8q^{25} - q^{26} + 8q^{28} - 3q^{29} - 8q^{31} - 8q^{32} + 6q^{34} - 8q^{35} + 8q^{36} - 6q^{37} + 5q^{38} - 35q^{39} + 8q^{40} - 11q^{41} + 5q^{43} - 8q^{45} - 10q^{46} - 15q^{47} + 8q^{49} - 8q^{50} + 6q^{51} + q^{52} - 16q^{53} - 8q^{56} - 38q^{57} + 3q^{58} - 9q^{59} - 32q^{61} + 8q^{62} + 8q^{63} + 8q^{64} - q^{65} + 33q^{67} - 6q^{68} - 22q^{69} + 8q^{70} + 11q^{71} - 8q^{72} + 34q^{73} + 6q^{74} - 5q^{76} + 35q^{78} - 31q^{79} - 8q^{80} + 20q^{81} + 11q^{82} - 50q^{83} + 6q^{85} - 5q^{86} + 12q^{87} + q^{89} + 8q^{90} + q^{91} + 10q^{92} + 26q^{93} + 15q^{94} + 5q^{95} - 4q^{97} - 8q^{98} + O(q^{100}) \)

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
\(3\) −2.90474 −1.67705 −0.838526 0.544861i \(-0.816582\pi\)
−0.838526 + 0.544861i \(0.816582\pi\)
\(4\) 1.00000 0.500000
\(5\) −1.00000 −0.447214
\(6\) 2.90474 1.18586
\(7\) 1.00000 0.377964
\(8\) −1.00000 −0.353553
\(9\) 5.43751 1.81250
\(10\) 1.00000 0.316228
\(11\) 0 0
\(12\) −2.90474 −0.838526
\(13\) 6.51020 1.80560 0.902802 0.430056i \(-0.141506\pi\)
0.902802 + 0.430056i \(0.141506\pi\)
\(14\) −1.00000 −0.267261
\(15\) 2.90474 0.750001
\(16\) 1.00000 0.250000
\(17\) −7.07831 −1.71674 −0.858371 0.513030i \(-0.828523\pi\)
−0.858371 + 0.513030i \(0.828523\pi\)
\(18\) −5.43751 −1.28163
\(19\) −3.97479 −0.911879 −0.455940 0.890011i \(-0.650697\pi\)
−0.455940 + 0.890011i \(0.650697\pi\)
\(20\) −1.00000 −0.223607
\(21\) −2.90474 −0.633866
\(22\) 0 0
\(23\) 9.39994 1.96002 0.980011 0.198943i \(-0.0637509\pi\)
0.980011 + 0.198943i \(0.0637509\pi\)
\(24\) 2.90474 0.592928
\(25\) 1.00000 0.200000
\(26\) −6.51020 −1.27676
\(27\) −7.08034 −1.36261
\(28\) 1.00000 0.188982
\(29\) −1.17923 −0.218977 −0.109489 0.993988i \(-0.534921\pi\)
−0.109489 + 0.993988i \(0.534921\pi\)
\(30\) −2.90474 −0.530331
\(31\) −1.28349 −0.230522 −0.115261 0.993335i \(-0.536770\pi\)
−0.115261 + 0.993335i \(0.536770\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0 0
\(34\) 7.07831 1.21392
\(35\) −1.00000 −0.169031
\(36\) 5.43751 0.906252
\(37\) −1.00807 −0.165726 −0.0828630 0.996561i \(-0.526406\pi\)
−0.0828630 + 0.996561i \(0.526406\pi\)
\(38\) 3.97479 0.644796
\(39\) −18.9104 −3.02809
\(40\) 1.00000 0.158114
\(41\) 5.43144 0.848248 0.424124 0.905604i \(-0.360582\pi\)
0.424124 + 0.905604i \(0.360582\pi\)
\(42\) 2.90474 0.448211
\(43\) 11.2003 1.70803 0.854013 0.520251i \(-0.174162\pi\)
0.854013 + 0.520251i \(0.174162\pi\)
\(44\) 0 0
\(45\) −5.43751 −0.810577
\(46\) −9.39994 −1.38594
\(47\) −11.3915 −1.66161 −0.830807 0.556561i \(-0.812121\pi\)
−0.830807 + 0.556561i \(0.812121\pi\)
\(48\) −2.90474 −0.419263
\(49\) 1.00000 0.142857
\(50\) −1.00000 −0.141421
\(51\) 20.5606 2.87907
\(52\) 6.51020 0.902802
\(53\) −7.75606 −1.06538 −0.532688 0.846311i \(-0.678818\pi\)
−0.532688 + 0.846311i \(0.678818\pi\)
\(54\) 7.08034 0.963513
\(55\) 0 0
\(56\) −1.00000 −0.133631
\(57\) 11.5457 1.52927
\(58\) 1.17923 0.154840
\(59\) −5.64597 −0.735042 −0.367521 0.930015i \(-0.619793\pi\)
−0.367521 + 0.930015i \(0.619793\pi\)
\(60\) 2.90474 0.375000
\(61\) −6.33662 −0.811321 −0.405660 0.914024i \(-0.632958\pi\)
−0.405660 + 0.914024i \(0.632958\pi\)
\(62\) 1.28349 0.163004
\(63\) 5.43751 0.685062
\(64\) 1.00000 0.125000
\(65\) −6.51020 −0.807491
\(66\) 0 0
\(67\) 3.88122 0.474166 0.237083 0.971489i \(-0.423809\pi\)
0.237083 + 0.971489i \(0.423809\pi\)
\(68\) −7.07831 −0.858371
\(69\) −27.3044 −3.28706
\(70\) 1.00000 0.119523
\(71\) −0.259926 −0.0308475 −0.0154238 0.999881i \(-0.504910\pi\)
−0.0154238 + 0.999881i \(0.504910\pi\)
\(72\) −5.43751 −0.640817
\(73\) −4.82486 −0.564707 −0.282354 0.959310i \(-0.591115\pi\)
−0.282354 + 0.959310i \(0.591115\pi\)
\(74\) 1.00807 0.117186
\(75\) −2.90474 −0.335410
\(76\) −3.97479 −0.455940
\(77\) 0 0
\(78\) 18.9104 2.14119
\(79\) −14.3428 −1.61369 −0.806843 0.590765i \(-0.798826\pi\)
−0.806843 + 0.590765i \(0.798826\pi\)
\(80\) −1.00000 −0.111803
\(81\) 4.25401 0.472668
\(82\) −5.43144 −0.599802
\(83\) −4.59502 −0.504368 −0.252184 0.967679i \(-0.581149\pi\)
−0.252184 + 0.967679i \(0.581149\pi\)
\(84\) −2.90474 −0.316933
\(85\) 7.07831 0.767750
\(86\) −11.2003 −1.20776
\(87\) 3.42535 0.367236
\(88\) 0 0
\(89\) −1.43088 −0.151673 −0.0758366 0.997120i \(-0.524163\pi\)
−0.0758366 + 0.997120i \(0.524163\pi\)
\(90\) 5.43751 0.573164
\(91\) 6.51020 0.682454
\(92\) 9.39994 0.980011
\(93\) 3.72822 0.386598
\(94\) 11.3915 1.17494
\(95\) 3.97479 0.407805
\(96\) 2.90474 0.296464
\(97\) −8.60695 −0.873903 −0.436952 0.899485i \(-0.643942\pi\)
−0.436952 + 0.899485i \(0.643942\pi\)
\(98\) −1.00000 −0.101015
\(99\) 0 0
\(100\) 1.00000 0.100000
\(101\) 7.14255 0.710710 0.355355 0.934731i \(-0.384360\pi\)
0.355355 + 0.934731i \(0.384360\pi\)
\(102\) −20.5606 −2.03581
\(103\) 3.12129 0.307550 0.153775 0.988106i \(-0.450857\pi\)
0.153775 + 0.988106i \(0.450857\pi\)
\(104\) −6.51020 −0.638378
\(105\) 2.90474 0.283474
\(106\) 7.75606 0.753335
\(107\) 3.29188 0.318238 0.159119 0.987259i \(-0.449135\pi\)
0.159119 + 0.987259i \(0.449135\pi\)
\(108\) −7.08034 −0.681306
\(109\) −12.5855 −1.20547 −0.602736 0.797941i \(-0.705923\pi\)
−0.602736 + 0.797941i \(0.705923\pi\)
\(110\) 0 0
\(111\) 2.92819 0.277931
\(112\) 1.00000 0.0944911
\(113\) 1.20490 0.113347 0.0566737 0.998393i \(-0.481951\pi\)
0.0566737 + 0.998393i \(0.481951\pi\)
\(114\) −11.5457 −1.08136
\(115\) −9.39994 −0.876549
\(116\) −1.17923 −0.109489
\(117\) 35.3993 3.27267
\(118\) 5.64597 0.519753
\(119\) −7.07831 −0.648867
\(120\) −2.90474 −0.265165
\(121\) 0 0
\(122\) 6.33662 0.573690
\(123\) −15.7769 −1.42256
\(124\) −1.28349 −0.115261
\(125\) −1.00000 −0.0894427
\(126\) −5.43751 −0.484412
\(127\) 16.0026 1.42000 0.710000 0.704202i \(-0.248695\pi\)
0.710000 + 0.704202i \(0.248695\pi\)
\(128\) −1.00000 −0.0883883
\(129\) −32.5339 −2.86445
\(130\) 6.51020 0.570982
\(131\) −14.6019 −1.27577 −0.637887 0.770130i \(-0.720191\pi\)
−0.637887 + 0.770130i \(0.720191\pi\)
\(132\) 0 0
\(133\) −3.97479 −0.344658
\(134\) −3.88122 −0.335286
\(135\) 7.08034 0.609379
\(136\) 7.07831 0.606960
\(137\) 10.0996 0.862871 0.431435 0.902144i \(-0.358007\pi\)
0.431435 + 0.902144i \(0.358007\pi\)
\(138\) 27.3044 2.32430
\(139\) 18.5110 1.57009 0.785043 0.619442i \(-0.212641\pi\)
0.785043 + 0.619442i \(0.212641\pi\)
\(140\) −1.00000 −0.0845154
\(141\) 33.0892 2.78661
\(142\) 0.259926 0.0218125
\(143\) 0 0
\(144\) 5.43751 0.453126
\(145\) 1.17923 0.0979296
\(146\) 4.82486 0.399308
\(147\) −2.90474 −0.239579
\(148\) −1.00807 −0.0828630
\(149\) −11.5126 −0.943152 −0.471576 0.881826i \(-0.656315\pi\)
−0.471576 + 0.881826i \(0.656315\pi\)
\(150\) 2.90474 0.237171
\(151\) −5.43407 −0.442218 −0.221109 0.975249i \(-0.570968\pi\)
−0.221109 + 0.975249i \(0.570968\pi\)
\(152\) 3.97479 0.322398
\(153\) −38.4884 −3.11160
\(154\) 0 0
\(155\) 1.28349 0.103093
\(156\) −18.9104 −1.51405
\(157\) 7.97651 0.636595 0.318297 0.947991i \(-0.396889\pi\)
0.318297 + 0.947991i \(0.396889\pi\)
\(158\) 14.3428 1.14105
\(159\) 22.5293 1.78669
\(160\) 1.00000 0.0790569
\(161\) 9.39994 0.740819
\(162\) −4.25401 −0.334227
\(163\) 14.3432 1.12345 0.561723 0.827326i \(-0.310139\pi\)
0.561723 + 0.827326i \(0.310139\pi\)
\(164\) 5.43144 0.424124
\(165\) 0 0
\(166\) 4.59502 0.356642
\(167\) 1.91040 0.147831 0.0739155 0.997265i \(-0.476450\pi\)
0.0739155 + 0.997265i \(0.476450\pi\)
\(168\) 2.90474 0.224106
\(169\) 29.3827 2.26021
\(170\) −7.07831 −0.542881
\(171\) −21.6130 −1.65279
\(172\) 11.2003 0.854013
\(173\) 14.5926 1.10945 0.554726 0.832033i \(-0.312823\pi\)
0.554726 + 0.832033i \(0.312823\pi\)
\(174\) −3.42535 −0.259675
\(175\) 1.00000 0.0755929
\(176\) 0 0
\(177\) 16.4001 1.23270
\(178\) 1.43088 0.107249
\(179\) −3.15693 −0.235960 −0.117980 0.993016i \(-0.537642\pi\)
−0.117980 + 0.993016i \(0.537642\pi\)
\(180\) −5.43751 −0.405288
\(181\) −7.09045 −0.527029 −0.263514 0.964655i \(-0.584882\pi\)
−0.263514 + 0.964655i \(0.584882\pi\)
\(182\) −6.51020 −0.482568
\(183\) 18.4062 1.36063
\(184\) −9.39994 −0.692972
\(185\) 1.00807 0.0741149
\(186\) −3.72822 −0.273366
\(187\) 0 0
\(188\) −11.3915 −0.830807
\(189\) −7.08034 −0.515019
\(190\) −3.97479 −0.288362
\(191\) 7.61496 0.550999 0.275500 0.961301i \(-0.411157\pi\)
0.275500 + 0.961301i \(0.411157\pi\)
\(192\) −2.90474 −0.209632
\(193\) 8.55503 0.615804 0.307902 0.951418i \(-0.400373\pi\)
0.307902 + 0.951418i \(0.400373\pi\)
\(194\) 8.60695 0.617943
\(195\) 18.9104 1.35420
\(196\) 1.00000 0.0714286
\(197\) −5.30318 −0.377836 −0.188918 0.981993i \(-0.560498\pi\)
−0.188918 + 0.981993i \(0.560498\pi\)
\(198\) 0 0
\(199\) 27.8463 1.97397 0.986987 0.160798i \(-0.0514068\pi\)
0.986987 + 0.160798i \(0.0514068\pi\)
\(200\) −1.00000 −0.0707107
\(201\) −11.2739 −0.795201
\(202\) −7.14255 −0.502548
\(203\) −1.17923 −0.0827656
\(204\) 20.5606 1.43953
\(205\) −5.43144 −0.379348
\(206\) −3.12129 −0.217470
\(207\) 51.1123 3.55255
\(208\) 6.51020 0.451401
\(209\) 0 0
\(210\) −2.90474 −0.200446
\(211\) −9.26733 −0.637989 −0.318995 0.947757i \(-0.603345\pi\)
−0.318995 + 0.947757i \(0.603345\pi\)
\(212\) −7.75606 −0.532688
\(213\) 0.755017 0.0517329
\(214\) −3.29188 −0.225028
\(215\) −11.2003 −0.763853
\(216\) 7.08034 0.481756
\(217\) −1.28349 −0.0871293
\(218\) 12.5855 0.852397
\(219\) 14.0150 0.947044
\(220\) 0 0
\(221\) −46.0812 −3.09976
\(222\) −2.92819 −0.196527
\(223\) 26.0134 1.74198 0.870992 0.491297i \(-0.163477\pi\)
0.870992 + 0.491297i \(0.163477\pi\)
\(224\) −1.00000 −0.0668153
\(225\) 5.43751 0.362501
\(226\) −1.20490 −0.0801487
\(227\) 0.293119 0.0194550 0.00972750 0.999953i \(-0.496904\pi\)
0.00972750 + 0.999953i \(0.496904\pi\)
\(228\) 11.5457 0.764635
\(229\) −19.9552 −1.31867 −0.659337 0.751847i \(-0.729163\pi\)
−0.659337 + 0.751847i \(0.729163\pi\)
\(230\) 9.39994 0.619813
\(231\) 0 0
\(232\) 1.17923 0.0774202
\(233\) −0.201558 −0.0132045 −0.00660225 0.999978i \(-0.502102\pi\)
−0.00660225 + 0.999978i \(0.502102\pi\)
\(234\) −35.3993 −2.31412
\(235\) 11.3915 0.743096
\(236\) −5.64597 −0.367521
\(237\) 41.6620 2.70624
\(238\) 7.07831 0.458819
\(239\) −23.9193 −1.54721 −0.773604 0.633669i \(-0.781548\pi\)
−0.773604 + 0.633669i \(0.781548\pi\)
\(240\) 2.90474 0.187500
\(241\) 18.5851 1.19717 0.598586 0.801059i \(-0.295730\pi\)
0.598586 + 0.801059i \(0.295730\pi\)
\(242\) 0 0
\(243\) 8.88423 0.569924
\(244\) −6.33662 −0.405660
\(245\) −1.00000 −0.0638877
\(246\) 15.7769 1.00590
\(247\) −25.8767 −1.64649
\(248\) 1.28349 0.0815020
\(249\) 13.3473 0.845852
\(250\) 1.00000 0.0632456
\(251\) −10.0947 −0.637174 −0.318587 0.947894i \(-0.603208\pi\)
−0.318587 + 0.947894i \(0.603208\pi\)
\(252\) 5.43751 0.342531
\(253\) 0 0
\(254\) −16.0026 −1.00409
\(255\) −20.5606 −1.28756
\(256\) 1.00000 0.0625000
\(257\) −5.47320 −0.341409 −0.170704 0.985322i \(-0.554604\pi\)
−0.170704 + 0.985322i \(0.554604\pi\)
\(258\) 32.5339 2.02547
\(259\) −1.00807 −0.0626385
\(260\) −6.51020 −0.403745
\(261\) −6.41207 −0.396897
\(262\) 14.6019 0.902109
\(263\) 12.8440 0.791998 0.395999 0.918251i \(-0.370398\pi\)
0.395999 + 0.918251i \(0.370398\pi\)
\(264\) 0 0
\(265\) 7.75606 0.476451
\(266\) 3.97479 0.243710
\(267\) 4.15634 0.254364
\(268\) 3.88122 0.237083
\(269\) 7.87541 0.480172 0.240086 0.970752i \(-0.422824\pi\)
0.240086 + 0.970752i \(0.422824\pi\)
\(270\) −7.08034 −0.430896
\(271\) −9.87947 −0.600135 −0.300068 0.953918i \(-0.597009\pi\)
−0.300068 + 0.953918i \(0.597009\pi\)
\(272\) −7.07831 −0.429185
\(273\) −18.9104 −1.14451
\(274\) −10.0996 −0.610142
\(275\) 0 0
\(276\) −27.3044 −1.64353
\(277\) −27.0289 −1.62401 −0.812006 0.583649i \(-0.801624\pi\)
−0.812006 + 0.583649i \(0.801624\pi\)
\(278\) −18.5110 −1.11022
\(279\) −6.97902 −0.417823
\(280\) 1.00000 0.0597614
\(281\) −13.6419 −0.813806 −0.406903 0.913471i \(-0.633391\pi\)
−0.406903 + 0.913471i \(0.633391\pi\)
\(282\) −33.0892 −1.97043
\(283\) −18.3850 −1.09287 −0.546437 0.837500i \(-0.684016\pi\)
−0.546437 + 0.837500i \(0.684016\pi\)
\(284\) −0.259926 −0.0154238
\(285\) −11.5457 −0.683910
\(286\) 0 0
\(287\) 5.43144 0.320608
\(288\) −5.43751 −0.320409
\(289\) 33.1024 1.94720
\(290\) −1.17923 −0.0692467
\(291\) 25.0009 1.46558
\(292\) −4.82486 −0.282354
\(293\) −9.87445 −0.576871 −0.288436 0.957499i \(-0.593135\pi\)
−0.288436 + 0.957499i \(0.593135\pi\)
\(294\) 2.90474 0.169408
\(295\) 5.64597 0.328721
\(296\) 1.00807 0.0585930
\(297\) 0 0
\(298\) 11.5126 0.666909
\(299\) 61.1955 3.53902
\(300\) −2.90474 −0.167705
\(301\) 11.2003 0.645573
\(302\) 5.43407 0.312696
\(303\) −20.7473 −1.19190
\(304\) −3.97479 −0.227970
\(305\) 6.33662 0.362834
\(306\) 38.4884 2.20023
\(307\) −6.50344 −0.371171 −0.185585 0.982628i \(-0.559418\pi\)
−0.185585 + 0.982628i \(0.559418\pi\)
\(308\) 0 0
\(309\) −9.06653 −0.515777
\(310\) −1.28349 −0.0728976
\(311\) 3.36996 0.191093 0.0955464 0.995425i \(-0.469540\pi\)
0.0955464 + 0.995425i \(0.469540\pi\)
\(312\) 18.9104 1.07059
\(313\) −5.42220 −0.306481 −0.153240 0.988189i \(-0.548971\pi\)
−0.153240 + 0.988189i \(0.548971\pi\)
\(314\) −7.97651 −0.450141
\(315\) −5.43751 −0.306369
\(316\) −14.3428 −0.806843
\(317\) −7.15998 −0.402145 −0.201072 0.979576i \(-0.564443\pi\)
−0.201072 + 0.979576i \(0.564443\pi\)
\(318\) −22.5293 −1.26338
\(319\) 0 0
\(320\) −1.00000 −0.0559017
\(321\) −9.56205 −0.533702
\(322\) −9.39994 −0.523838
\(323\) 28.1348 1.56546
\(324\) 4.25401 0.236334
\(325\) 6.51020 0.361121
\(326\) −14.3432 −0.794396
\(327\) 36.5576 2.02164
\(328\) −5.43144 −0.299901
\(329\) −11.3915 −0.628031
\(330\) 0 0
\(331\) 20.6983 1.13768 0.568842 0.822447i \(-0.307392\pi\)
0.568842 + 0.822447i \(0.307392\pi\)
\(332\) −4.59502 −0.252184
\(333\) −5.48140 −0.300379
\(334\) −1.91040 −0.104532
\(335\) −3.88122 −0.212053
\(336\) −2.90474 −0.158467
\(337\) 7.81093 0.425488 0.212744 0.977108i \(-0.431760\pi\)
0.212744 + 0.977108i \(0.431760\pi\)
\(338\) −29.3827 −1.59821
\(339\) −3.49992 −0.190090
\(340\) 7.07831 0.383875
\(341\) 0 0
\(342\) 21.6130 1.16870
\(343\) 1.00000 0.0539949
\(344\) −11.2003 −0.603879
\(345\) 27.3044 1.47002
\(346\) −14.5926 −0.784502
\(347\) 27.9220 1.49893 0.749465 0.662044i \(-0.230311\pi\)
0.749465 + 0.662044i \(0.230311\pi\)
\(348\) 3.42535 0.183618
\(349\) −20.5937 −1.10236 −0.551178 0.834387i \(-0.685822\pi\)
−0.551178 + 0.834387i \(0.685822\pi\)
\(350\) −1.00000 −0.0534522
\(351\) −46.0944 −2.46034
\(352\) 0 0
\(353\) −5.61133 −0.298661 −0.149330 0.988787i \(-0.547712\pi\)
−0.149330 + 0.988787i \(0.547712\pi\)
\(354\) −16.4001 −0.871653
\(355\) 0.259926 0.0137954
\(356\) −1.43088 −0.0758366
\(357\) 20.5606 1.08818
\(358\) 3.15693 0.166849
\(359\) −8.24264 −0.435030 −0.217515 0.976057i \(-0.569795\pi\)
−0.217515 + 0.976057i \(0.569795\pi\)
\(360\) 5.43751 0.286582
\(361\) −3.20104 −0.168476
\(362\) 7.09045 0.372666
\(363\) 0 0
\(364\) 6.51020 0.341227
\(365\) 4.82486 0.252545
\(366\) −18.4062 −0.962109
\(367\) −4.10818 −0.214445 −0.107223 0.994235i \(-0.534196\pi\)
−0.107223 + 0.994235i \(0.534196\pi\)
\(368\) 9.39994 0.490006
\(369\) 29.5335 1.53745
\(370\) −1.00807 −0.0524071
\(371\) −7.75606 −0.402675
\(372\) 3.72822 0.193299
\(373\) 26.2182 1.35753 0.678764 0.734356i \(-0.262516\pi\)
0.678764 + 0.734356i \(0.262516\pi\)
\(374\) 0 0
\(375\) 2.90474 0.150000
\(376\) 11.3915 0.587469
\(377\) −7.67702 −0.395386
\(378\) 7.08034 0.364174
\(379\) 19.7398 1.01397 0.506983 0.861956i \(-0.330761\pi\)
0.506983 + 0.861956i \(0.330761\pi\)
\(380\) 3.97479 0.203902
\(381\) −46.4834 −2.38141
\(382\) −7.61496 −0.389615
\(383\) 35.8040 1.82950 0.914749 0.404022i \(-0.132388\pi\)
0.914749 + 0.404022i \(0.132388\pi\)
\(384\) 2.90474 0.148232
\(385\) 0 0
\(386\) −8.55503 −0.435439
\(387\) 60.9017 3.09581
\(388\) −8.60695 −0.436952
\(389\) −28.8835 −1.46445 −0.732226 0.681062i \(-0.761518\pi\)
−0.732226 + 0.681062i \(0.761518\pi\)
\(390\) −18.9104 −0.957567
\(391\) −66.5356 −3.36485
\(392\) −1.00000 −0.0505076
\(393\) 42.4147 2.13954
\(394\) 5.30318 0.267171
\(395\) 14.3428 0.721663
\(396\) 0 0
\(397\) −31.7117 −1.59156 −0.795782 0.605583i \(-0.792940\pi\)
−0.795782 + 0.605583i \(0.792940\pi\)
\(398\) −27.8463 −1.39581
\(399\) 11.5457 0.578010
\(400\) 1.00000 0.0500000
\(401\) −24.8442 −1.24066 −0.620331 0.784340i \(-0.713002\pi\)
−0.620331 + 0.784340i \(0.713002\pi\)
\(402\) 11.2739 0.562292
\(403\) −8.35580 −0.416232
\(404\) 7.14255 0.355355
\(405\) −4.25401 −0.211384
\(406\) 1.17923 0.0585242
\(407\) 0 0
\(408\) −20.5606 −1.01790
\(409\) −17.6373 −0.872107 −0.436054 0.899921i \(-0.643624\pi\)
−0.436054 + 0.899921i \(0.643624\pi\)
\(410\) 5.43144 0.268240
\(411\) −29.3368 −1.44708
\(412\) 3.12129 0.153775
\(413\) −5.64597 −0.277820
\(414\) −51.1123 −2.51203
\(415\) 4.59502 0.225560
\(416\) −6.51020 −0.319189
\(417\) −53.7697 −2.63312
\(418\) 0 0
\(419\) 6.70771 0.327693 0.163847 0.986486i \(-0.447610\pi\)
0.163847 + 0.986486i \(0.447610\pi\)
\(420\) 2.90474 0.141737
\(421\) 16.0438 0.781925 0.390962 0.920407i \(-0.372142\pi\)
0.390962 + 0.920407i \(0.372142\pi\)
\(422\) 9.26733 0.451126
\(423\) −61.9412 −3.01168
\(424\) 7.75606 0.376668
\(425\) −7.07831 −0.343348
\(426\) −0.755017 −0.0365807
\(427\) −6.33662 −0.306650
\(428\) 3.29188 0.159119
\(429\) 0 0
\(430\) 11.2003 0.540125
\(431\) −31.5480 −1.51961 −0.759807 0.650148i \(-0.774707\pi\)
−0.759807 + 0.650148i \(0.774707\pi\)
\(432\) −7.08034 −0.340653
\(433\) −8.26911 −0.397388 −0.198694 0.980062i \(-0.563670\pi\)
−0.198694 + 0.980062i \(0.563670\pi\)
\(434\) 1.28349 0.0616097
\(435\) −3.42535 −0.164233
\(436\) −12.5855 −0.602736
\(437\) −37.3628 −1.78730
\(438\) −14.0150 −0.669661
\(439\) −11.1008 −0.529814 −0.264907 0.964274i \(-0.585341\pi\)
−0.264907 + 0.964274i \(0.585341\pi\)
\(440\) 0 0
\(441\) 5.43751 0.258929
\(442\) 46.0812 2.19186
\(443\) 4.53871 0.215641 0.107820 0.994170i \(-0.465613\pi\)
0.107820 + 0.994170i \(0.465613\pi\)
\(444\) 2.92819 0.138966
\(445\) 1.43088 0.0678303
\(446\) −26.0134 −1.23177
\(447\) 33.4412 1.58171
\(448\) 1.00000 0.0472456
\(449\) −9.71975 −0.458703 −0.229352 0.973344i \(-0.573661\pi\)
−0.229352 + 0.973344i \(0.573661\pi\)
\(450\) −5.43751 −0.256327
\(451\) 0 0
\(452\) 1.20490 0.0566737
\(453\) 15.7846 0.741623
\(454\) −0.293119 −0.0137568
\(455\) −6.51020 −0.305203
\(456\) −11.5457 −0.540678
\(457\) −21.1478 −0.989251 −0.494626 0.869106i \(-0.664695\pi\)
−0.494626 + 0.869106i \(0.664695\pi\)
\(458\) 19.9552 0.932444
\(459\) 50.1168 2.33925
\(460\) −9.39994 −0.438274
\(461\) −20.5158 −0.955516 −0.477758 0.878492i \(-0.658550\pi\)
−0.477758 + 0.878492i \(0.658550\pi\)
\(462\) 0 0
\(463\) −5.06175 −0.235239 −0.117620 0.993059i \(-0.537526\pi\)
−0.117620 + 0.993059i \(0.537526\pi\)
\(464\) −1.17923 −0.0547443
\(465\) −3.72822 −0.172892
\(466\) 0.201558 0.00933699
\(467\) −7.48743 −0.346477 −0.173239 0.984880i \(-0.555423\pi\)
−0.173239 + 0.984880i \(0.555423\pi\)
\(468\) 35.3993 1.63633
\(469\) 3.88122 0.179218
\(470\) −11.3915 −0.525448
\(471\) −23.1697 −1.06760
\(472\) 5.64597 0.259877
\(473\) 0 0
\(474\) −41.6620 −1.91360
\(475\) −3.97479 −0.182376
\(476\) −7.07831 −0.324434
\(477\) −42.1737 −1.93100
\(478\) 23.9193 1.09404
\(479\) 0.414174 0.0189241 0.00946206 0.999955i \(-0.496988\pi\)
0.00946206 + 0.999955i \(0.496988\pi\)
\(480\) −2.90474 −0.132583
\(481\) −6.56275 −0.299235
\(482\) −18.5851 −0.846528
\(483\) −27.3044 −1.24239
\(484\) 0 0
\(485\) 8.60695 0.390821
\(486\) −8.88423 −0.402997
\(487\) −23.7328 −1.07544 −0.537718 0.843125i \(-0.680714\pi\)
−0.537718 + 0.843125i \(0.680714\pi\)
\(488\) 6.33662 0.286845
\(489\) −41.6632 −1.88408
\(490\) 1.00000 0.0451754
\(491\) −32.2162 −1.45390 −0.726949 0.686691i \(-0.759063\pi\)
−0.726949 + 0.686691i \(0.759063\pi\)
\(492\) −15.7769 −0.711278
\(493\) 8.34695 0.375928
\(494\) 25.8767 1.16425
\(495\) 0 0
\(496\) −1.28349 −0.0576306
\(497\) −0.259926 −0.0116593
\(498\) −13.3473 −0.598108
\(499\) −4.48988 −0.200995 −0.100497 0.994937i \(-0.532043\pi\)
−0.100497 + 0.994937i \(0.532043\pi\)
\(500\) −1.00000 −0.0447214
\(501\) −5.54921 −0.247920
\(502\) 10.0947 0.450550
\(503\) −30.8299 −1.37464 −0.687319 0.726355i \(-0.741213\pi\)
−0.687319 + 0.726355i \(0.741213\pi\)
\(504\) −5.43751 −0.242206
\(505\) −7.14255 −0.317839
\(506\) 0 0
\(507\) −85.3491 −3.79049
\(508\) 16.0026 0.710000
\(509\) −2.65682 −0.117762 −0.0588808 0.998265i \(-0.518753\pi\)
−0.0588808 + 0.998265i \(0.518753\pi\)
\(510\) 20.5606 0.910440
\(511\) −4.82486 −0.213439
\(512\) −1.00000 −0.0441942
\(513\) 28.1429 1.24254
\(514\) 5.47320 0.241412
\(515\) −3.12129 −0.137540
\(516\) −32.5339 −1.43222
\(517\) 0 0
\(518\) 1.00807 0.0442921
\(519\) −42.3876 −1.86061
\(520\) 6.51020 0.285491
\(521\) 5.98594 0.262249 0.131124 0.991366i \(-0.458141\pi\)
0.131124 + 0.991366i \(0.458141\pi\)
\(522\) 6.41207 0.280649
\(523\) −0.966746 −0.0422728 −0.0211364 0.999777i \(-0.506728\pi\)
−0.0211364 + 0.999777i \(0.506728\pi\)
\(524\) −14.6019 −0.637887
\(525\) −2.90474 −0.126773
\(526\) −12.8440 −0.560027
\(527\) 9.08497 0.395747
\(528\) 0 0
\(529\) 65.3588 2.84169
\(530\) −7.75606 −0.336902
\(531\) −30.7000 −1.33227
\(532\) −3.97479 −0.172329
\(533\) 35.3597 1.53160
\(534\) −4.15634 −0.179862
\(535\) −3.29188 −0.142320
\(536\) −3.88122 −0.167643
\(537\) 9.17005 0.395717
\(538\) −7.87541 −0.339533
\(539\) 0 0
\(540\) 7.08034 0.304689
\(541\) 1.56401 0.0672420 0.0336210 0.999435i \(-0.489296\pi\)
0.0336210 + 0.999435i \(0.489296\pi\)
\(542\) 9.87947 0.424360
\(543\) 20.5959 0.883855
\(544\) 7.07831 0.303480
\(545\) 12.5855 0.539103
\(546\) 18.9104 0.809292
\(547\) 14.5413 0.621741 0.310870 0.950452i \(-0.399380\pi\)
0.310870 + 0.950452i \(0.399380\pi\)
\(548\) 10.0996 0.431435
\(549\) −34.4554 −1.47052
\(550\) 0 0
\(551\) 4.68719 0.199681
\(552\) 27.3044 1.16215
\(553\) −14.3428 −0.609916
\(554\) 27.0289 1.14835
\(555\) −2.92819 −0.124295
\(556\) 18.5110 0.785043
\(557\) 37.0040 1.56791 0.783956 0.620817i \(-0.213199\pi\)
0.783956 + 0.620817i \(0.213199\pi\)
\(558\) 6.97902 0.295445
\(559\) 72.9160 3.08402
\(560\) −1.00000 −0.0422577
\(561\) 0 0
\(562\) 13.6419 0.575448
\(563\) −23.9389 −1.00890 −0.504452 0.863440i \(-0.668306\pi\)
−0.504452 + 0.863440i \(0.668306\pi\)
\(564\) 33.0892 1.39331
\(565\) −1.20490 −0.0506905
\(566\) 18.3850 0.772779
\(567\) 4.25401 0.178652
\(568\) 0.259926 0.0109063
\(569\) 26.6076 1.11545 0.557725 0.830026i \(-0.311674\pi\)
0.557725 + 0.830026i \(0.311674\pi\)
\(570\) 11.5457 0.483597
\(571\) −17.4823 −0.731610 −0.365805 0.930692i \(-0.619206\pi\)
−0.365805 + 0.930692i \(0.619206\pi\)
\(572\) 0 0
\(573\) −22.1195 −0.924055
\(574\) −5.43144 −0.226704
\(575\) 9.39994 0.392004
\(576\) 5.43751 0.226563
\(577\) 31.3194 1.30385 0.651923 0.758285i \(-0.273963\pi\)
0.651923 + 0.758285i \(0.273963\pi\)
\(578\) −33.1024 −1.37688
\(579\) −24.8501 −1.03274
\(580\) 1.17923 0.0489648
\(581\) −4.59502 −0.190633
\(582\) −25.0009 −1.03632
\(583\) 0 0
\(584\) 4.82486 0.199654
\(585\) −35.3993 −1.46358
\(586\) 9.87445 0.407910
\(587\) 4.04656 0.167019 0.0835097 0.996507i \(-0.473387\pi\)
0.0835097 + 0.996507i \(0.473387\pi\)
\(588\) −2.90474 −0.119789
\(589\) 5.10162 0.210209
\(590\) −5.64597 −0.232441
\(591\) 15.4044 0.633651
\(592\) −1.00807 −0.0414315
\(593\) −23.7125 −0.973756 −0.486878 0.873470i \(-0.661864\pi\)
−0.486878 + 0.873470i \(0.661864\pi\)
\(594\) 0 0
\(595\) 7.07831 0.290182
\(596\) −11.5126 −0.471576
\(597\) −80.8864 −3.31046
\(598\) −61.1955 −2.50247
\(599\) −10.4237 −0.425900 −0.212950 0.977063i \(-0.568307\pi\)
−0.212950 + 0.977063i \(0.568307\pi\)
\(600\) 2.90474 0.118586
\(601\) 15.5929 0.636049 0.318025 0.948082i \(-0.396981\pi\)
0.318025 + 0.948082i \(0.396981\pi\)
\(602\) −11.2003 −0.456489
\(603\) 21.1042 0.859428
\(604\) −5.43407 −0.221109
\(605\) 0 0
\(606\) 20.7473 0.842799
\(607\) 3.06966 0.124594 0.0622968 0.998058i \(-0.480157\pi\)
0.0622968 + 0.998058i \(0.480157\pi\)
\(608\) 3.97479 0.161199
\(609\) 3.42535 0.138802
\(610\) −6.33662 −0.256562
\(611\) −74.1606 −3.00022
\(612\) −38.4884 −1.55580
\(613\) −25.5273 −1.03104 −0.515520 0.856878i \(-0.672401\pi\)
−0.515520 + 0.856878i \(0.672401\pi\)
\(614\) 6.50344 0.262457
\(615\) 15.7769 0.636187
\(616\) 0 0
\(617\) −14.4437 −0.581483 −0.290741 0.956802i \(-0.593902\pi\)
−0.290741 + 0.956802i \(0.593902\pi\)
\(618\) 9.06653 0.364709
\(619\) 1.76790 0.0710578 0.0355289 0.999369i \(-0.488688\pi\)
0.0355289 + 0.999369i \(0.488688\pi\)
\(620\) 1.28349 0.0515464
\(621\) −66.5548 −2.67075
\(622\) −3.36996 −0.135123
\(623\) −1.43088 −0.0573270
\(624\) −18.9104 −0.757023
\(625\) 1.00000 0.0400000
\(626\) 5.42220 0.216715
\(627\) 0 0
\(628\) 7.97651 0.318297
\(629\) 7.13544 0.284509
\(630\) 5.43751 0.216636
\(631\) 21.3564 0.850185 0.425093 0.905150i \(-0.360242\pi\)
0.425093 + 0.905150i \(0.360242\pi\)
\(632\) 14.3428 0.570524
\(633\) 26.9192 1.06994
\(634\) 7.15998 0.284359
\(635\) −16.0026 −0.635043
\(636\) 22.5293 0.893346
\(637\) 6.51020 0.257943
\(638\) 0 0
\(639\) −1.41335 −0.0559113
\(640\) 1.00000 0.0395285
\(641\) 46.1398 1.82241 0.911205 0.411952i \(-0.135153\pi\)
0.911205 + 0.411952i \(0.135153\pi\)
\(642\) 9.56205 0.377384
\(643\) 18.4699 0.728383 0.364191 0.931324i \(-0.381345\pi\)
0.364191 + 0.931324i \(0.381345\pi\)
\(644\) 9.39994 0.370409
\(645\) 32.5339 1.28102
\(646\) −28.1348 −1.10695
\(647\) 36.3807 1.43027 0.715137 0.698984i \(-0.246364\pi\)
0.715137 + 0.698984i \(0.246364\pi\)
\(648\) −4.25401 −0.167113
\(649\) 0 0
\(650\) −6.51020 −0.255351
\(651\) 3.72822 0.146120
\(652\) 14.3432 0.561723
\(653\) −2.89414 −0.113256 −0.0566281 0.998395i \(-0.518035\pi\)
−0.0566281 + 0.998395i \(0.518035\pi\)
\(654\) −36.5576 −1.42951
\(655\) 14.6019 0.570544
\(656\) 5.43144 0.212062
\(657\) −26.2352 −1.02353
\(658\) 11.3915 0.444085
\(659\) −19.1764 −0.747008 −0.373504 0.927629i \(-0.621844\pi\)
−0.373504 + 0.927629i \(0.621844\pi\)
\(660\) 0 0
\(661\) −24.9234 −0.969406 −0.484703 0.874679i \(-0.661072\pi\)
−0.484703 + 0.874679i \(0.661072\pi\)
\(662\) −20.6983 −0.804463
\(663\) 133.854 5.19845
\(664\) 4.59502 0.178321
\(665\) 3.97479 0.154136
\(666\) 5.48140 0.212400
\(667\) −11.0847 −0.429200
\(668\) 1.91040 0.0739155
\(669\) −75.5621 −2.92140
\(670\) 3.88122 0.149944
\(671\) 0 0
\(672\) 2.90474 0.112053
\(673\) −13.3066 −0.512932 −0.256466 0.966553i \(-0.582558\pi\)
−0.256466 + 0.966553i \(0.582558\pi\)
\(674\) −7.81093 −0.300866
\(675\) −7.08034 −0.272523
\(676\) 29.3827 1.13010
\(677\) −28.2748 −1.08669 −0.543344 0.839510i \(-0.682842\pi\)
−0.543344 + 0.839510i \(0.682842\pi\)
\(678\) 3.49992 0.134414
\(679\) −8.60695 −0.330304
\(680\) −7.07831 −0.271441
\(681\) −0.851435 −0.0326271
\(682\) 0 0
\(683\) −20.1040 −0.769259 −0.384629 0.923071i \(-0.625671\pi\)
−0.384629 + 0.923071i \(0.625671\pi\)
\(684\) −21.6130 −0.826393
\(685\) −10.0996 −0.385887
\(686\) −1.00000 −0.0381802
\(687\) 57.9646 2.21149
\(688\) 11.2003 0.427007
\(689\) −50.4935 −1.92365
\(690\) −27.3044 −1.03946
\(691\) −1.68862 −0.0642382 −0.0321191 0.999484i \(-0.510226\pi\)
−0.0321191 + 0.999484i \(0.510226\pi\)
\(692\) 14.5926 0.554726
\(693\) 0 0
\(694\) −27.9220 −1.05990
\(695\) −18.5110 −0.702164
\(696\) −3.42535 −0.129838
\(697\) −38.4454 −1.45622
\(698\) 20.5937 0.779484
\(699\) 0.585473 0.0221446
\(700\) 1.00000 0.0377964
\(701\) −2.04142 −0.0771035 −0.0385517 0.999257i \(-0.512274\pi\)
−0.0385517 + 0.999257i \(0.512274\pi\)
\(702\) 46.0944 1.73972
\(703\) 4.00687 0.151122
\(704\) 0 0
\(705\) −33.0892 −1.24621
\(706\) 5.61133 0.211185
\(707\) 7.14255 0.268623
\(708\) 16.4001 0.616352
\(709\) 4.99388 0.187549 0.0937745 0.995593i \(-0.470107\pi\)
0.0937745 + 0.995593i \(0.470107\pi\)
\(710\) −0.259926 −0.00975485
\(711\) −77.9890 −2.92481
\(712\) 1.43088 0.0536245
\(713\) −12.0648 −0.451829
\(714\) −20.5606 −0.769463
\(715\) 0 0
\(716\) −3.15693 −0.117980
\(717\) 69.4792 2.59475
\(718\) 8.24264 0.307613
\(719\) 34.3305 1.28031 0.640156 0.768245i \(-0.278870\pi\)
0.640156 + 0.768245i \(0.278870\pi\)
\(720\) −5.43751 −0.202644
\(721\) 3.12129 0.116243
\(722\) 3.20104 0.119130
\(723\) −53.9849 −2.00772
\(724\) −7.09045 −0.263514
\(725\) −1.17923 −0.0437955
\(726\) 0 0
\(727\) 26.6447 0.988196 0.494098 0.869406i \(-0.335498\pi\)
0.494098 + 0.869406i \(0.335498\pi\)
\(728\) −6.51020 −0.241284
\(729\) −38.5684 −1.42846
\(730\) −4.82486 −0.178576
\(731\) −79.2790 −2.93224
\(732\) 18.4062 0.680314
\(733\) −14.1335 −0.522032 −0.261016 0.965334i \(-0.584058\pi\)
−0.261016 + 0.965334i \(0.584058\pi\)
\(734\) 4.10818 0.151636
\(735\) 2.90474 0.107143
\(736\) −9.39994 −0.346486
\(737\) 0 0
\(738\) −29.5335 −1.08714
\(739\) −36.0918 −1.32766 −0.663829 0.747884i \(-0.731070\pi\)
−0.663829 + 0.747884i \(0.731070\pi\)
\(740\) 1.00807 0.0370574
\(741\) 75.1650 2.76126
\(742\) 7.75606 0.284734
\(743\) 29.3508 1.07678 0.538389 0.842697i \(-0.319033\pi\)
0.538389 + 0.842697i \(0.319033\pi\)
\(744\) −3.72822 −0.136683
\(745\) 11.5126 0.421790
\(746\) −26.2182 −0.959917
\(747\) −24.9855 −0.914170
\(748\) 0 0
\(749\) 3.29188 0.120283
\(750\) −2.90474 −0.106066
\(751\) 36.8418 1.34438 0.672189 0.740380i \(-0.265354\pi\)
0.672189 + 0.740380i \(0.265354\pi\)
\(752\) −11.3915 −0.415404
\(753\) 29.3226 1.06857
\(754\) 7.67702 0.279580
\(755\) 5.43407 0.197766
\(756\) −7.08034 −0.257510
\(757\) −36.7671 −1.33632 −0.668161 0.744016i \(-0.732918\pi\)
−0.668161 + 0.744016i \(0.732918\pi\)
\(758\) −19.7398 −0.716982
\(759\) 0 0
\(760\) −3.97479 −0.144181
\(761\) −35.6456 −1.29215 −0.646075 0.763274i \(-0.723591\pi\)
−0.646075 + 0.763274i \(0.723591\pi\)
\(762\) 46.4834 1.68391
\(763\) −12.5855 −0.455625
\(764\) 7.61496 0.275500
\(765\) 38.4884 1.39155
\(766\) −35.8040 −1.29365
\(767\) −36.7564 −1.32720
\(768\) −2.90474 −0.104816
\(769\) −30.6264 −1.10442 −0.552209 0.833706i \(-0.686215\pi\)
−0.552209 + 0.833706i \(0.686215\pi\)
\(770\) 0 0
\(771\) 15.8982 0.572560
\(772\) 8.55503 0.307902
\(773\) −10.7328 −0.386033 −0.193017 0.981195i \(-0.561827\pi\)
−0.193017 + 0.981195i \(0.561827\pi\)
\(774\) −60.9017 −2.18907
\(775\) −1.28349 −0.0461045
\(776\) 8.60695 0.308971
\(777\) 2.92819 0.105048
\(778\) 28.8835 1.03552
\(779\) −21.5888 −0.773500
\(780\) 18.9104 0.677102
\(781\) 0 0
\(782\) 66.5356 2.37931
\(783\) 8.34934 0.298381
\(784\) 1.00000 0.0357143
\(785\) −7.97651 −0.284694
\(786\) −42.4147 −1.51288
\(787\) 18.2605 0.650917 0.325459 0.945556i \(-0.394481\pi\)
0.325459 + 0.945556i \(0.394481\pi\)
\(788\) −5.30318 −0.188918
\(789\) −37.3086 −1.32822
\(790\) −14.3428 −0.510293
\(791\) 1.20490 0.0428413
\(792\) 0 0
\(793\) −41.2526 −1.46492
\(794\) 31.7117 1.12541
\(795\) −22.5293 −0.799033
\(796\) 27.8463 0.986987
\(797\) −29.4786 −1.04418 −0.522092 0.852889i \(-0.674848\pi\)
−0.522092 + 0.852889i \(0.674848\pi\)
\(798\) −11.5457 −0.408714
\(799\) 80.6322 2.85256
\(800\) −1.00000 −0.0353553
\(801\) −7.78044 −0.274908
\(802\) 24.8442 0.877281
\(803\) 0 0
\(804\) −11.2739 −0.397601
\(805\) −9.39994 −0.331304
\(806\) 8.35580 0.294321
\(807\) −22.8760 −0.805274
\(808\) −7.14255 −0.251274
\(809\) 22.3819 0.786904 0.393452 0.919345i \(-0.371281\pi\)
0.393452 + 0.919345i \(0.371281\pi\)
\(810\) 4.25401 0.149471
\(811\) 27.0318 0.949214 0.474607 0.880198i \(-0.342590\pi\)
0.474607 + 0.880198i \(0.342590\pi\)
\(812\) −1.17923 −0.0413828
\(813\) 28.6973 1.00646
\(814\) 0 0
\(815\) −14.3432 −0.502420
\(816\) 20.5606 0.719766
\(817\) −44.5188 −1.55751
\(818\) 17.6373 0.616673
\(819\) 35.3993 1.23695
\(820\) −5.43144 −0.189674
\(821\) 14.4241 0.503404 0.251702 0.967805i \(-0.419010\pi\)
0.251702 + 0.967805i \(0.419010\pi\)
\(822\) 29.3368 1.02324
\(823\) −15.3358 −0.534572 −0.267286 0.963617i \(-0.586127\pi\)
−0.267286 + 0.963617i \(0.586127\pi\)
\(824\) −3.12129 −0.108735
\(825\) 0 0
\(826\) 5.64597 0.196448
\(827\) −44.2084 −1.53728 −0.768638 0.639685i \(-0.779065\pi\)
−0.768638 + 0.639685i \(0.779065\pi\)
\(828\) 51.1123 1.77627
\(829\) −27.9599 −0.971087 −0.485544 0.874212i \(-0.661378\pi\)
−0.485544 + 0.874212i \(0.661378\pi\)
\(830\) −4.59502 −0.159495
\(831\) 78.5120 2.72355
\(832\) 6.51020 0.225701
\(833\) −7.07831 −0.245249
\(834\) 53.7697 1.86189
\(835\) −1.91040 −0.0661120
\(836\) 0 0
\(837\) 9.08758 0.314113
\(838\) −6.70771 −0.231714
\(839\) −4.45002 −0.153632 −0.0768159 0.997045i \(-0.524475\pi\)
−0.0768159 + 0.997045i \(0.524475\pi\)
\(840\) −2.90474 −0.100223
\(841\) −27.6094 −0.952049
\(842\) −16.0438 −0.552904
\(843\) 39.6261 1.36480
\(844\) −9.26733 −0.318995
\(845\) −29.3827 −1.01080
\(846\) 61.9412 2.12958
\(847\) 0 0
\(848\) −7.75606 −0.266344
\(849\) 53.4036 1.83281
\(850\) 7.07831 0.242784
\(851\) −9.47581 −0.324826
\(852\) 0.755017 0.0258665
\(853\) −2.31141 −0.0791411 −0.0395705 0.999217i \(-0.512599\pi\)
−0.0395705 + 0.999217i \(0.512599\pi\)
\(854\) 6.33662 0.216835
\(855\) 21.6130 0.739148
\(856\) −3.29188 −0.112514
\(857\) −2.10357 −0.0718566 −0.0359283 0.999354i \(-0.511439\pi\)
−0.0359283 + 0.999354i \(0.511439\pi\)
\(858\) 0 0
\(859\) −6.91941 −0.236087 −0.118044 0.993008i \(-0.537662\pi\)
−0.118044 + 0.993008i \(0.537662\pi\)
\(860\) −11.2003 −0.381926
\(861\) −15.7769 −0.537676
\(862\) 31.5480 1.07453
\(863\) −37.5821 −1.27931 −0.639655 0.768662i \(-0.720923\pi\)
−0.639655 + 0.768662i \(0.720923\pi\)
\(864\) 7.08034 0.240878
\(865\) −14.5926 −0.496162
\(866\) 8.26911 0.280996
\(867\) −96.1540 −3.26556
\(868\) −1.28349 −0.0435646
\(869\) 0 0
\(870\) 3.42535 0.116130
\(871\) 25.2675 0.856156
\(872\) 12.5855 0.426199
\(873\) −46.8004 −1.58395
\(874\) 37.3628 1.26381
\(875\) −1.00000 −0.0338062
\(876\) 14.0150 0.473522
\(877\) −34.1431 −1.15293 −0.576465 0.817122i \(-0.695568\pi\)
−0.576465 + 0.817122i \(0.695568\pi\)
\(878\) 11.1008 0.374635
\(879\) 28.6827 0.967443
\(880\) 0 0
\(881\) 38.7897 1.30686 0.653429 0.756987i \(-0.273330\pi\)
0.653429 + 0.756987i \(0.273330\pi\)
\(882\) −5.43751 −0.183091
\(883\) 1.29604 0.0436152 0.0218076 0.999762i \(-0.493058\pi\)
0.0218076 + 0.999762i \(0.493058\pi\)
\(884\) −46.0812 −1.54988
\(885\) −16.4001 −0.551282
\(886\) −4.53871 −0.152481
\(887\) −6.44111 −0.216271 −0.108136 0.994136i \(-0.534488\pi\)
−0.108136 + 0.994136i \(0.534488\pi\)
\(888\) −2.92819 −0.0982635
\(889\) 16.0026 0.536710
\(890\) −1.43088 −0.0479633
\(891\) 0 0
\(892\) 26.0134 0.870992
\(893\) 45.2786 1.51519
\(894\) −33.4412 −1.11844
\(895\) 3.15693 0.105524
\(896\) −1.00000 −0.0334077
\(897\) −177.757 −5.93513
\(898\) 9.71975 0.324352
\(899\) 1.51353 0.0504792
\(900\) 5.43751 0.181250
\(901\) 54.8998 1.82898
\(902\) 0 0
\(903\) −32.5339 −1.08266
\(904\) −1.20490 −0.0400744
\(905\) 7.09045 0.235694
\(906\) −15.7846 −0.524407
\(907\) 35.2975 1.17203 0.586017 0.810299i \(-0.300695\pi\)
0.586017 + 0.810299i \(0.300695\pi\)
\(908\) 0.293119 0.00972750
\(909\) 38.8377 1.28817
\(910\) 6.51020 0.215811
\(911\) −36.9287 −1.22350 −0.611752 0.791050i \(-0.709535\pi\)
−0.611752 + 0.791050i \(0.709535\pi\)
\(912\) 11.5457 0.382317
\(913\) 0 0
\(914\) 21.1478 0.699506
\(915\) −18.4062 −0.608491
\(916\) −19.9552 −0.659337
\(917\) −14.6019 −0.482197
\(918\) −50.1168 −1.65410
\(919\) 35.4760 1.17025 0.585123 0.810945i \(-0.301046\pi\)
0.585123 + 0.810945i \(0.301046\pi\)
\(920\) 9.39994 0.309907
\(921\) 18.8908 0.622473
\(922\) 20.5158 0.675652
\(923\) −1.69217 −0.0556984
\(924\) 0 0
\(925\) −1.00807 −0.0331452
\(926\) 5.06175 0.166339
\(927\) 16.9720 0.557435
\(928\) 1.17923 0.0387101
\(929\) −17.7706 −0.583034 −0.291517 0.956566i \(-0.594160\pi\)
−0.291517 + 0.956566i \(0.594160\pi\)
\(930\) 3.72822 0.122253
\(931\) −3.97479 −0.130268
\(932\) −0.201558 −0.00660225
\(933\) −9.78885 −0.320473
\(934\) 7.48743 0.244996
\(935\) 0 0
\(936\) −35.3993 −1.15706
\(937\) −6.01550 −0.196518 −0.0982589 0.995161i \(-0.531327\pi\)
−0.0982589 + 0.995161i \(0.531327\pi\)
\(938\) −3.88122 −0.126726
\(939\) 15.7501 0.513985
\(940\) 11.3915 0.371548
\(941\) −30.6883 −1.00041 −0.500205 0.865907i \(-0.666742\pi\)
−0.500205 + 0.865907i \(0.666742\pi\)
\(942\) 23.1697 0.754909
\(943\) 51.0552 1.66258
\(944\) −5.64597 −0.183761
\(945\) 7.08034 0.230324
\(946\) 0 0
\(947\) 32.3767 1.05210 0.526051 0.850453i \(-0.323672\pi\)
0.526051 + 0.850453i \(0.323672\pi\)
\(948\) 41.6620 1.35312
\(949\) −31.4108 −1.01964
\(950\) 3.97479 0.128959
\(951\) 20.7979 0.674417
\(952\) 7.07831 0.229409
\(953\) −11.8836 −0.384948 −0.192474 0.981302i \(-0.561651\pi\)
−0.192474 + 0.981302i \(0.561651\pi\)
\(954\) 42.1737 1.36542
\(955\) −7.61496 −0.246414
\(956\) −23.9193 −0.773604
\(957\) 0 0
\(958\) −0.414174 −0.0133814
\(959\) 10.0996 0.326134
\(960\) 2.90474 0.0937501
\(961\) −29.3526 −0.946859
\(962\) 6.56275 0.211591
\(963\) 17.8996 0.576808
\(964\) 18.5851 0.598586
\(965\) −8.55503 −0.275396
\(966\) 27.3044 0.878504
\(967\) −24.0988 −0.774965 −0.387483 0.921877i \(-0.626655\pi\)
−0.387483 + 0.921877i \(0.626655\pi\)
\(968\) 0 0
\(969\) −81.7242 −2.62536
\(970\) −8.60695 −0.276352
\(971\) −2.52446 −0.0810139 −0.0405069 0.999179i \(-0.512897\pi\)
−0.0405069 + 0.999179i \(0.512897\pi\)
\(972\) 8.88423 0.284962
\(973\) 18.5110 0.593437
\(974\) 23.7328 0.760448
\(975\) −18.9104 −0.605619
\(976\) −6.33662 −0.202830
\(977\) −25.8513 −0.827056 −0.413528 0.910491i \(-0.635704\pi\)
−0.413528 + 0.910491i \(0.635704\pi\)
\(978\) 41.6632 1.33224
\(979\) 0 0
\(980\) −1.00000 −0.0319438
\(981\) −68.4338 −2.18492
\(982\) 32.2162 1.02806
\(983\) −0.975472 −0.0311127 −0.0155564 0.999879i \(-0.504952\pi\)
−0.0155564 + 0.999879i \(0.504952\pi\)
\(984\) 15.7769 0.502950
\(985\) 5.30318 0.168973
\(986\) −8.34695 −0.265821
\(987\) 33.0892 1.05324
\(988\) −25.8767 −0.823247
\(989\) 105.282 3.34777
\(990\) 0 0
\(991\) 3.48309 0.110644 0.0553220 0.998469i \(-0.482381\pi\)
0.0553220 + 0.998469i \(0.482381\pi\)
\(992\) 1.28349 0.0407510
\(993\) −60.1233 −1.90795
\(994\) 0.259926 0.00824435
\(995\) −27.8463 −0.882788
\(996\) 13.3473 0.422926
\(997\) −43.0019 −1.36188 −0.680942 0.732337i \(-0.738430\pi\)
−0.680942 + 0.732337i \(0.738430\pi\)
\(998\) 4.48988 0.142125
\(999\) 7.13749 0.225820
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.dg.1.1 8
11.2 odd 10 770.2.n.k.631.1 yes 16
11.6 odd 10 770.2.n.k.421.1 16
11.10 odd 2 8470.2.a.dh.1.1 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
770.2.n.k.421.1 16 11.6 odd 10
770.2.n.k.631.1 yes 16 11.2 odd 10
8470.2.a.dg.1.1 8 1.1 even 1 trivial
8470.2.a.dh.1.1 8 11.10 odd 2