Properties

Label 8470.2.a.dc.1.6
Level $8470$
Weight $2$
Character 8470.1
Self dual yes
Analytic conductor $67.633$
Analytic rank $0$
Dimension $6$
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: \(6\)
Coefficient field: 6.6.19898000.1
Defining polynomial: \(x^{6} - x^{5} - 10 x^{4} + 7 x^{3} + 24 x^{2} - 15 x - 5\)
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 1.6
Root \(-2.18328\) of defining polynomial
Character \(\chi\) \(=\) 8470.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} +2.18328 q^{3} +1.00000 q^{4} +1.00000 q^{5} +2.18328 q^{6} -1.00000 q^{7} +1.00000 q^{8} +1.76673 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} +2.18328 q^{3} +1.00000 q^{4} +1.00000 q^{5} +2.18328 q^{6} -1.00000 q^{7} +1.00000 q^{8} +1.76673 q^{9} +1.00000 q^{10} +2.18328 q^{12} -2.53263 q^{13} -1.00000 q^{14} +2.18328 q^{15} +1.00000 q^{16} +5.73813 q^{17} +1.76673 q^{18} +2.24060 q^{19} +1.00000 q^{20} -2.18328 q^{21} -2.51489 q^{23} +2.18328 q^{24} +1.00000 q^{25} -2.53263 q^{26} -2.69257 q^{27} -1.00000 q^{28} +8.62000 q^{29} +2.18328 q^{30} +1.84089 q^{31} +1.00000 q^{32} +5.73813 q^{34} -1.00000 q^{35} +1.76673 q^{36} -4.63124 q^{37} +2.24060 q^{38} -5.52945 q^{39} +1.00000 q^{40} +10.5523 q^{41} -2.18328 q^{42} +2.18804 q^{43} +1.76673 q^{45} -2.51489 q^{46} +3.22482 q^{47} +2.18328 q^{48} +1.00000 q^{49} +1.00000 q^{50} +12.5280 q^{51} -2.53263 q^{52} -4.01746 q^{53} -2.69257 q^{54} -1.00000 q^{56} +4.89187 q^{57} +8.62000 q^{58} +2.01405 q^{59} +2.18328 q^{60} +0.268235 q^{61} +1.84089 q^{62} -1.76673 q^{63} +1.00000 q^{64} -2.53263 q^{65} +3.86233 q^{67} +5.73813 q^{68} -5.49071 q^{69} -1.00000 q^{70} -9.55514 q^{71} +1.76673 q^{72} +11.4258 q^{73} -4.63124 q^{74} +2.18328 q^{75} +2.24060 q^{76} -5.52945 q^{78} +11.7534 q^{79} +1.00000 q^{80} -11.1789 q^{81} +10.5523 q^{82} +7.34851 q^{83} -2.18328 q^{84} +5.73813 q^{85} +2.18804 q^{86} +18.8199 q^{87} -2.65397 q^{89} +1.76673 q^{90} +2.53263 q^{91} -2.51489 q^{92} +4.01919 q^{93} +3.22482 q^{94} +2.24060 q^{95} +2.18328 q^{96} +16.5045 q^{97} +1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q + 6q^{2} - q^{3} + 6q^{4} + 6q^{5} - q^{6} - 6q^{7} + 6q^{8} + 3q^{9} + O(q^{10}) \) \( 6q + 6q^{2} - q^{3} + 6q^{4} + 6q^{5} - q^{6} - 6q^{7} + 6q^{8} + 3q^{9} + 6q^{10} - q^{12} + 9q^{13} - 6q^{14} - q^{15} + 6q^{16} + 9q^{17} + 3q^{18} + 12q^{19} + 6q^{20} + q^{21} + 4q^{23} - q^{24} + 6q^{25} + 9q^{26} - 4q^{27} - 6q^{28} + 15q^{29} - q^{30} + 8q^{31} + 6q^{32} + 9q^{34} - 6q^{35} + 3q^{36} - 4q^{37} + 12q^{38} - 19q^{39} + 6q^{40} + 4q^{41} + q^{42} + 30q^{43} + 3q^{45} + 4q^{46} - 7q^{47} - q^{48} + 6q^{49} + 6q^{50} + 16q^{51} + 9q^{52} - 6q^{53} - 4q^{54} - 6q^{56} - 14q^{57} + 15q^{58} + 4q^{59} - q^{60} - 14q^{61} + 8q^{62} - 3q^{63} + 6q^{64} + 9q^{65} + 18q^{67} + 9q^{68} - 10q^{69} - 6q^{70} + 23q^{71} + 3q^{72} + 23q^{73} - 4q^{74} - q^{75} + 12q^{76} - 19q^{78} + 21q^{79} + 6q^{80} - 18q^{81} + 4q^{82} + 25q^{83} + q^{84} + 9q^{85} + 30q^{86} + 14q^{87} - 18q^{89} + 3q^{90} - 9q^{91} + 4q^{92} - 24q^{93} - 7q^{94} + 12q^{95} - q^{96} + 7q^{97} + 6q^{98} + O(q^{100}) \)

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
\(3\) 2.18328 1.26052 0.630260 0.776384i \(-0.282948\pi\)
0.630260 + 0.776384i \(0.282948\pi\)
\(4\) 1.00000 0.500000
\(5\) 1.00000 0.447214
\(6\) 2.18328 0.891322
\(7\) −1.00000 −0.377964
\(8\) 1.00000 0.353553
\(9\) 1.76673 0.588911
\(10\) 1.00000 0.316228
\(11\) 0 0
\(12\) 2.18328 0.630260
\(13\) −2.53263 −0.702425 −0.351212 0.936296i \(-0.614231\pi\)
−0.351212 + 0.936296i \(0.614231\pi\)
\(14\) −1.00000 −0.267261
\(15\) 2.18328 0.563722
\(16\) 1.00000 0.250000
\(17\) 5.73813 1.39170 0.695850 0.718187i \(-0.255028\pi\)
0.695850 + 0.718187i \(0.255028\pi\)
\(18\) 1.76673 0.416423
\(19\) 2.24060 0.514029 0.257014 0.966408i \(-0.417261\pi\)
0.257014 + 0.966408i \(0.417261\pi\)
\(20\) 1.00000 0.223607
\(21\) −2.18328 −0.476432
\(22\) 0 0
\(23\) −2.51489 −0.524390 −0.262195 0.965015i \(-0.584446\pi\)
−0.262195 + 0.965015i \(0.584446\pi\)
\(24\) 2.18328 0.445661
\(25\) 1.00000 0.200000
\(26\) −2.53263 −0.496689
\(27\) −2.69257 −0.518186
\(28\) −1.00000 −0.188982
\(29\) 8.62000 1.60069 0.800347 0.599537i \(-0.204649\pi\)
0.800347 + 0.599537i \(0.204649\pi\)
\(30\) 2.18328 0.398611
\(31\) 1.84089 0.330634 0.165317 0.986240i \(-0.447135\pi\)
0.165317 + 0.986240i \(0.447135\pi\)
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) 5.73813 0.984080
\(35\) −1.00000 −0.169031
\(36\) 1.76673 0.294455
\(37\) −4.63124 −0.761372 −0.380686 0.924704i \(-0.624312\pi\)
−0.380686 + 0.924704i \(0.624312\pi\)
\(38\) 2.24060 0.363473
\(39\) −5.52945 −0.885421
\(40\) 1.00000 0.158114
\(41\) 10.5523 1.64800 0.823999 0.566591i \(-0.191738\pi\)
0.823999 + 0.566591i \(0.191738\pi\)
\(42\) −2.18328 −0.336888
\(43\) 2.18804 0.333672 0.166836 0.985985i \(-0.446645\pi\)
0.166836 + 0.985985i \(0.446645\pi\)
\(44\) 0 0
\(45\) 1.76673 0.263369
\(46\) −2.51489 −0.370800
\(47\) 3.22482 0.470389 0.235194 0.971948i \(-0.424427\pi\)
0.235194 + 0.971948i \(0.424427\pi\)
\(48\) 2.18328 0.315130
\(49\) 1.00000 0.142857
\(50\) 1.00000 0.141421
\(51\) 12.5280 1.75427
\(52\) −2.53263 −0.351212
\(53\) −4.01746 −0.551841 −0.275920 0.961180i \(-0.588983\pi\)
−0.275920 + 0.961180i \(0.588983\pi\)
\(54\) −2.69257 −0.366413
\(55\) 0 0
\(56\) −1.00000 −0.133631
\(57\) 4.89187 0.647944
\(58\) 8.62000 1.13186
\(59\) 2.01405 0.262207 0.131103 0.991369i \(-0.458148\pi\)
0.131103 + 0.991369i \(0.458148\pi\)
\(60\) 2.18328 0.281861
\(61\) 0.268235 0.0343440 0.0171720 0.999853i \(-0.494534\pi\)
0.0171720 + 0.999853i \(0.494534\pi\)
\(62\) 1.84089 0.233793
\(63\) −1.76673 −0.222587
\(64\) 1.00000 0.125000
\(65\) −2.53263 −0.314134
\(66\) 0 0
\(67\) 3.86233 0.471859 0.235929 0.971770i \(-0.424187\pi\)
0.235929 + 0.971770i \(0.424187\pi\)
\(68\) 5.73813 0.695850
\(69\) −5.49071 −0.661004
\(70\) −1.00000 −0.119523
\(71\) −9.55514 −1.13399 −0.566993 0.823723i \(-0.691893\pi\)
−0.566993 + 0.823723i \(0.691893\pi\)
\(72\) 1.76673 0.208211
\(73\) 11.4258 1.33729 0.668647 0.743580i \(-0.266874\pi\)
0.668647 + 0.743580i \(0.266874\pi\)
\(74\) −4.63124 −0.538371
\(75\) 2.18328 0.252104
\(76\) 2.24060 0.257014
\(77\) 0 0
\(78\) −5.52945 −0.626087
\(79\) 11.7534 1.32236 0.661182 0.750225i \(-0.270055\pi\)
0.661182 + 0.750225i \(0.270055\pi\)
\(80\) 1.00000 0.111803
\(81\) −11.1789 −1.24209
\(82\) 10.5523 1.16531
\(83\) 7.34851 0.806604 0.403302 0.915067i \(-0.367862\pi\)
0.403302 + 0.915067i \(0.367862\pi\)
\(84\) −2.18328 −0.238216
\(85\) 5.73813 0.622387
\(86\) 2.18804 0.235942
\(87\) 18.8199 2.01771
\(88\) 0 0
\(89\) −2.65397 −0.281321 −0.140660 0.990058i \(-0.544923\pi\)
−0.140660 + 0.990058i \(0.544923\pi\)
\(90\) 1.76673 0.186230
\(91\) 2.53263 0.265492
\(92\) −2.51489 −0.262195
\(93\) 4.01919 0.416771
\(94\) 3.22482 0.332615
\(95\) 2.24060 0.229881
\(96\) 2.18328 0.222831
\(97\) 16.5045 1.67578 0.837891 0.545838i \(-0.183789\pi\)
0.837891 + 0.545838i \(0.183789\pi\)
\(98\) 1.00000 0.101015
\(99\) 0 0
\(100\) 1.00000 0.100000
\(101\) −16.0478 −1.59682 −0.798408 0.602116i \(-0.794324\pi\)
−0.798408 + 0.602116i \(0.794324\pi\)
\(102\) 12.5280 1.24045
\(103\) −12.9029 −1.27136 −0.635680 0.771953i \(-0.719280\pi\)
−0.635680 + 0.771953i \(0.719280\pi\)
\(104\) −2.53263 −0.248345
\(105\) −2.18328 −0.213067
\(106\) −4.01746 −0.390210
\(107\) 6.74567 0.652129 0.326064 0.945348i \(-0.394277\pi\)
0.326064 + 0.945348i \(0.394277\pi\)
\(108\) −2.69257 −0.259093
\(109\) −18.6864 −1.78983 −0.894916 0.446235i \(-0.852765\pi\)
−0.894916 + 0.446235i \(0.852765\pi\)
\(110\) 0 0
\(111\) −10.1113 −0.959725
\(112\) −1.00000 −0.0944911
\(113\) 7.56433 0.711593 0.355796 0.934564i \(-0.384210\pi\)
0.355796 + 0.934564i \(0.384210\pi\)
\(114\) 4.89187 0.458165
\(115\) −2.51489 −0.234514
\(116\) 8.62000 0.800347
\(117\) −4.47448 −0.413666
\(118\) 2.01405 0.185408
\(119\) −5.73813 −0.526013
\(120\) 2.18328 0.199306
\(121\) 0 0
\(122\) 0.268235 0.0242849
\(123\) 23.0388 2.07733
\(124\) 1.84089 0.165317
\(125\) 1.00000 0.0894427
\(126\) −1.76673 −0.157393
\(127\) 10.7426 0.953249 0.476625 0.879107i \(-0.341860\pi\)
0.476625 + 0.879107i \(0.341860\pi\)
\(128\) 1.00000 0.0883883
\(129\) 4.77710 0.420601
\(130\) −2.53263 −0.222126
\(131\) 17.0221 1.48723 0.743613 0.668610i \(-0.233110\pi\)
0.743613 + 0.668610i \(0.233110\pi\)
\(132\) 0 0
\(133\) −2.24060 −0.194285
\(134\) 3.86233 0.333654
\(135\) −2.69257 −0.231740
\(136\) 5.73813 0.492040
\(137\) −12.4777 −1.06604 −0.533019 0.846103i \(-0.678943\pi\)
−0.533019 + 0.846103i \(0.678943\pi\)
\(138\) −5.49071 −0.467401
\(139\) −17.3812 −1.47426 −0.737129 0.675752i \(-0.763819\pi\)
−0.737129 + 0.675752i \(0.763819\pi\)
\(140\) −1.00000 −0.0845154
\(141\) 7.04070 0.592934
\(142\) −9.55514 −0.801849
\(143\) 0 0
\(144\) 1.76673 0.147228
\(145\) 8.62000 0.715852
\(146\) 11.4258 0.945610
\(147\) 2.18328 0.180074
\(148\) −4.63124 −0.380686
\(149\) 1.49206 0.122235 0.0611173 0.998131i \(-0.480534\pi\)
0.0611173 + 0.998131i \(0.480534\pi\)
\(150\) 2.18328 0.178264
\(151\) 7.99555 0.650668 0.325334 0.945599i \(-0.394523\pi\)
0.325334 + 0.945599i \(0.394523\pi\)
\(152\) 2.24060 0.181737
\(153\) 10.1377 0.819587
\(154\) 0 0
\(155\) 1.84089 0.147864
\(156\) −5.52945 −0.442710
\(157\) 2.05173 0.163746 0.0818729 0.996643i \(-0.473910\pi\)
0.0818729 + 0.996643i \(0.473910\pi\)
\(158\) 11.7534 0.935053
\(159\) −8.77126 −0.695606
\(160\) 1.00000 0.0790569
\(161\) 2.51489 0.198201
\(162\) −11.1789 −0.878294
\(163\) 20.6976 1.62116 0.810579 0.585629i \(-0.199152\pi\)
0.810579 + 0.585629i \(0.199152\pi\)
\(164\) 10.5523 0.823999
\(165\) 0 0
\(166\) 7.34851 0.570355
\(167\) −1.32050 −0.102183 −0.0510917 0.998694i \(-0.516270\pi\)
−0.0510917 + 0.998694i \(0.516270\pi\)
\(168\) −2.18328 −0.168444
\(169\) −6.58579 −0.506599
\(170\) 5.73813 0.440094
\(171\) 3.95854 0.302717
\(172\) 2.18804 0.166836
\(173\) −15.7029 −1.19387 −0.596933 0.802291i \(-0.703614\pi\)
−0.596933 + 0.802291i \(0.703614\pi\)
\(174\) 18.8199 1.42673
\(175\) −1.00000 −0.0755929
\(176\) 0 0
\(177\) 4.39724 0.330517
\(178\) −2.65397 −0.198924
\(179\) 4.65497 0.347929 0.173964 0.984752i \(-0.444342\pi\)
0.173964 + 0.984752i \(0.444342\pi\)
\(180\) 1.76673 0.131684
\(181\) −5.90450 −0.438878 −0.219439 0.975626i \(-0.570423\pi\)
−0.219439 + 0.975626i \(0.570423\pi\)
\(182\) 2.53263 0.187731
\(183\) 0.585633 0.0432913
\(184\) −2.51489 −0.185400
\(185\) −4.63124 −0.340496
\(186\) 4.01919 0.294701
\(187\) 0 0
\(188\) 3.22482 0.235194
\(189\) 2.69257 0.195856
\(190\) 2.24060 0.162550
\(191\) −5.93400 −0.429369 −0.214685 0.976683i \(-0.568872\pi\)
−0.214685 + 0.976683i \(0.568872\pi\)
\(192\) 2.18328 0.157565
\(193\) −10.8377 −0.780113 −0.390056 0.920791i \(-0.627544\pi\)
−0.390056 + 0.920791i \(0.627544\pi\)
\(194\) 16.5045 1.18496
\(195\) −5.52945 −0.395972
\(196\) 1.00000 0.0714286
\(197\) −12.0176 −0.856216 −0.428108 0.903728i \(-0.640820\pi\)
−0.428108 + 0.903728i \(0.640820\pi\)
\(198\) 0 0
\(199\) 18.4014 1.30444 0.652220 0.758030i \(-0.273838\pi\)
0.652220 + 0.758030i \(0.273838\pi\)
\(200\) 1.00000 0.0707107
\(201\) 8.43256 0.594787
\(202\) −16.0478 −1.12912
\(203\) −8.62000 −0.605005
\(204\) 12.5280 0.877133
\(205\) 10.5523 0.737007
\(206\) −12.9029 −0.898987
\(207\) −4.44313 −0.308819
\(208\) −2.53263 −0.175606
\(209\) 0 0
\(210\) −2.18328 −0.150661
\(211\) −20.0422 −1.37976 −0.689880 0.723924i \(-0.742337\pi\)
−0.689880 + 0.723924i \(0.742337\pi\)
\(212\) −4.01746 −0.275920
\(213\) −20.8616 −1.42941
\(214\) 6.74567 0.461125
\(215\) 2.18804 0.149223
\(216\) −2.69257 −0.183206
\(217\) −1.84089 −0.124968
\(218\) −18.6864 −1.26560
\(219\) 24.9459 1.68569
\(220\) 0 0
\(221\) −14.5325 −0.977565
\(222\) −10.1113 −0.678628
\(223\) 11.8000 0.790185 0.395092 0.918641i \(-0.370713\pi\)
0.395092 + 0.918641i \(0.370713\pi\)
\(224\) −1.00000 −0.0668153
\(225\) 1.76673 0.117782
\(226\) 7.56433 0.503172
\(227\) −0.990121 −0.0657166 −0.0328583 0.999460i \(-0.510461\pi\)
−0.0328583 + 0.999460i \(0.510461\pi\)
\(228\) 4.89187 0.323972
\(229\) 18.3559 1.21299 0.606495 0.795087i \(-0.292575\pi\)
0.606495 + 0.795087i \(0.292575\pi\)
\(230\) −2.51489 −0.165827
\(231\) 0 0
\(232\) 8.62000 0.565931
\(233\) 17.3650 1.13762 0.568809 0.822470i \(-0.307404\pi\)
0.568809 + 0.822470i \(0.307404\pi\)
\(234\) −4.47448 −0.292506
\(235\) 3.22482 0.210364
\(236\) 2.01405 0.131103
\(237\) 25.6611 1.66687
\(238\) −5.73813 −0.371947
\(239\) −8.92113 −0.577060 −0.288530 0.957471i \(-0.593166\pi\)
−0.288530 + 0.957471i \(0.593166\pi\)
\(240\) 2.18328 0.140930
\(241\) 0.896696 0.0577613 0.0288806 0.999583i \(-0.490806\pi\)
0.0288806 + 0.999583i \(0.490806\pi\)
\(242\) 0 0
\(243\) −16.3289 −1.04750
\(244\) 0.268235 0.0171720
\(245\) 1.00000 0.0638877
\(246\) 23.0388 1.46890
\(247\) −5.67461 −0.361067
\(248\) 1.84089 0.116897
\(249\) 16.0439 1.01674
\(250\) 1.00000 0.0632456
\(251\) −22.4978 −1.42005 −0.710026 0.704176i \(-0.751317\pi\)
−0.710026 + 0.704176i \(0.751317\pi\)
\(252\) −1.76673 −0.111294
\(253\) 0 0
\(254\) 10.7426 0.674049
\(255\) 12.5280 0.784531
\(256\) 1.00000 0.0625000
\(257\) −24.9449 −1.55602 −0.778009 0.628254i \(-0.783770\pi\)
−0.778009 + 0.628254i \(0.783770\pi\)
\(258\) 4.77710 0.297410
\(259\) 4.63124 0.287772
\(260\) −2.53263 −0.157067
\(261\) 15.2292 0.942666
\(262\) 17.0221 1.05163
\(263\) −6.25594 −0.385758 −0.192879 0.981223i \(-0.561782\pi\)
−0.192879 + 0.981223i \(0.561782\pi\)
\(264\) 0 0
\(265\) −4.01746 −0.246791
\(266\) −2.24060 −0.137380
\(267\) −5.79438 −0.354610
\(268\) 3.86233 0.235929
\(269\) −13.0673 −0.796727 −0.398364 0.917228i \(-0.630422\pi\)
−0.398364 + 0.917228i \(0.630422\pi\)
\(270\) −2.69257 −0.163865
\(271\) 24.3157 1.47707 0.738535 0.674215i \(-0.235518\pi\)
0.738535 + 0.674215i \(0.235518\pi\)
\(272\) 5.73813 0.347925
\(273\) 5.52945 0.334658
\(274\) −12.4777 −0.753803
\(275\) 0 0
\(276\) −5.49071 −0.330502
\(277\) 13.4094 0.805696 0.402848 0.915267i \(-0.368020\pi\)
0.402848 + 0.915267i \(0.368020\pi\)
\(278\) −17.3812 −1.04246
\(279\) 3.25236 0.194714
\(280\) −1.00000 −0.0597614
\(281\) −12.7398 −0.759993 −0.379997 0.924988i \(-0.624075\pi\)
−0.379997 + 0.924988i \(0.624075\pi\)
\(282\) 7.04070 0.419268
\(283\) 13.7495 0.817325 0.408662 0.912686i \(-0.365995\pi\)
0.408662 + 0.912686i \(0.365995\pi\)
\(284\) −9.55514 −0.566993
\(285\) 4.89187 0.289769
\(286\) 0 0
\(287\) −10.5523 −0.622885
\(288\) 1.76673 0.104106
\(289\) 15.9261 0.936828
\(290\) 8.62000 0.506184
\(291\) 36.0341 2.11236
\(292\) 11.4258 0.668647
\(293\) −16.6376 −0.971982 −0.485991 0.873964i \(-0.661541\pi\)
−0.485991 + 0.873964i \(0.661541\pi\)
\(294\) 2.18328 0.127332
\(295\) 2.01405 0.117262
\(296\) −4.63124 −0.269186
\(297\) 0 0
\(298\) 1.49206 0.0864329
\(299\) 6.36928 0.368345
\(300\) 2.18328 0.126052
\(301\) −2.18804 −0.126116
\(302\) 7.99555 0.460092
\(303\) −35.0369 −2.01282
\(304\) 2.24060 0.128507
\(305\) 0.268235 0.0153591
\(306\) 10.1377 0.579536
\(307\) −5.56226 −0.317455 −0.158728 0.987322i \(-0.550739\pi\)
−0.158728 + 0.987322i \(0.550739\pi\)
\(308\) 0 0
\(309\) −28.1707 −1.60258
\(310\) 1.84089 0.104556
\(311\) 17.9179 1.01603 0.508014 0.861349i \(-0.330380\pi\)
0.508014 + 0.861349i \(0.330380\pi\)
\(312\) −5.52945 −0.313043
\(313\) 5.15478 0.291365 0.145683 0.989331i \(-0.453462\pi\)
0.145683 + 0.989331i \(0.453462\pi\)
\(314\) 2.05173 0.115786
\(315\) −1.76673 −0.0995441
\(316\) 11.7534 0.661182
\(317\) 28.4614 1.59855 0.799275 0.600965i \(-0.205217\pi\)
0.799275 + 0.600965i \(0.205217\pi\)
\(318\) −8.77126 −0.491868
\(319\) 0 0
\(320\) 1.00000 0.0559017
\(321\) 14.7277 0.822021
\(322\) 2.51489 0.140149
\(323\) 12.8568 0.715374
\(324\) −11.1789 −0.621047
\(325\) −2.53263 −0.140485
\(326\) 20.6976 1.14633
\(327\) −40.7977 −2.25612
\(328\) 10.5523 0.582655
\(329\) −3.22482 −0.177790
\(330\) 0 0
\(331\) −24.8572 −1.36627 −0.683136 0.730291i \(-0.739385\pi\)
−0.683136 + 0.730291i \(0.739385\pi\)
\(332\) 7.34851 0.403302
\(333\) −8.18217 −0.448380
\(334\) −1.32050 −0.0722546
\(335\) 3.86233 0.211022
\(336\) −2.18328 −0.119108
\(337\) 15.6474 0.852369 0.426184 0.904636i \(-0.359858\pi\)
0.426184 + 0.904636i \(0.359858\pi\)
\(338\) −6.58579 −0.358220
\(339\) 16.5151 0.896977
\(340\) 5.73813 0.311194
\(341\) 0 0
\(342\) 3.95854 0.214053
\(343\) −1.00000 −0.0539949
\(344\) 2.18804 0.117971
\(345\) −5.49071 −0.295610
\(346\) −15.7029 −0.844191
\(347\) −14.6646 −0.787239 −0.393620 0.919273i \(-0.628777\pi\)
−0.393620 + 0.919273i \(0.628777\pi\)
\(348\) 18.8199 1.00885
\(349\) −26.9763 −1.44401 −0.722004 0.691889i \(-0.756779\pi\)
−0.722004 + 0.691889i \(0.756779\pi\)
\(350\) −1.00000 −0.0534522
\(351\) 6.81929 0.363987
\(352\) 0 0
\(353\) 4.77463 0.254128 0.127064 0.991895i \(-0.459445\pi\)
0.127064 + 0.991895i \(0.459445\pi\)
\(354\) 4.39724 0.233711
\(355\) −9.55514 −0.507134
\(356\) −2.65397 −0.140660
\(357\) −12.5280 −0.663050
\(358\) 4.65497 0.246023
\(359\) −34.0982 −1.79963 −0.899816 0.436269i \(-0.856300\pi\)
−0.899816 + 0.436269i \(0.856300\pi\)
\(360\) 1.76673 0.0931150
\(361\) −13.9797 −0.735774
\(362\) −5.90450 −0.310334
\(363\) 0 0
\(364\) 2.53263 0.132746
\(365\) 11.4258 0.598056
\(366\) 0.585633 0.0306115
\(367\) −9.98965 −0.521455 −0.260728 0.965412i \(-0.583962\pi\)
−0.260728 + 0.965412i \(0.583962\pi\)
\(368\) −2.51489 −0.131098
\(369\) 18.6432 0.970524
\(370\) −4.63124 −0.240767
\(371\) 4.01746 0.208576
\(372\) 4.01919 0.208385
\(373\) 0.510916 0.0264542 0.0132271 0.999913i \(-0.495790\pi\)
0.0132271 + 0.999913i \(0.495790\pi\)
\(374\) 0 0
\(375\) 2.18328 0.112744
\(376\) 3.22482 0.166307
\(377\) −21.8313 −1.12437
\(378\) 2.69257 0.138491
\(379\) 5.31376 0.272949 0.136475 0.990644i \(-0.456423\pi\)
0.136475 + 0.990644i \(0.456423\pi\)
\(380\) 2.24060 0.114940
\(381\) 23.4541 1.20159
\(382\) −5.93400 −0.303610
\(383\) 23.5202 1.20183 0.600914 0.799314i \(-0.294803\pi\)
0.600914 + 0.799314i \(0.294803\pi\)
\(384\) 2.18328 0.111415
\(385\) 0 0
\(386\) −10.8377 −0.551623
\(387\) 3.86567 0.196503
\(388\) 16.5045 0.837891
\(389\) 9.18397 0.465646 0.232823 0.972519i \(-0.425204\pi\)
0.232823 + 0.972519i \(0.425204\pi\)
\(390\) −5.52945 −0.279995
\(391\) −14.4307 −0.729794
\(392\) 1.00000 0.0505076
\(393\) 37.1641 1.87468
\(394\) −12.0176 −0.605436
\(395\) 11.7534 0.591380
\(396\) 0 0
\(397\) −31.5709 −1.58450 −0.792248 0.610199i \(-0.791089\pi\)
−0.792248 + 0.610199i \(0.791089\pi\)
\(398\) 18.4014 0.922378
\(399\) −4.89187 −0.244900
\(400\) 1.00000 0.0500000
\(401\) −10.6804 −0.533354 −0.266677 0.963786i \(-0.585926\pi\)
−0.266677 + 0.963786i \(0.585926\pi\)
\(402\) 8.43256 0.420578
\(403\) −4.66229 −0.232245
\(404\) −16.0478 −0.798408
\(405\) −11.1789 −0.555482
\(406\) −8.62000 −0.427803
\(407\) 0 0
\(408\) 12.5280 0.620227
\(409\) 23.5770 1.16581 0.582903 0.812542i \(-0.301917\pi\)
0.582903 + 0.812542i \(0.301917\pi\)
\(410\) 10.5523 0.521143
\(411\) −27.2423 −1.34376
\(412\) −12.9029 −0.635680
\(413\) −2.01405 −0.0991048
\(414\) −4.44313 −0.218368
\(415\) 7.34851 0.360724
\(416\) −2.53263 −0.124172
\(417\) −37.9482 −1.85833
\(418\) 0 0
\(419\) −15.9235 −0.777915 −0.388958 0.921256i \(-0.627165\pi\)
−0.388958 + 0.921256i \(0.627165\pi\)
\(420\) −2.18328 −0.106533
\(421\) −39.8602 −1.94267 −0.971333 0.237722i \(-0.923599\pi\)
−0.971333 + 0.237722i \(0.923599\pi\)
\(422\) −20.0422 −0.975638
\(423\) 5.69740 0.277017
\(424\) −4.01746 −0.195105
\(425\) 5.73813 0.278340
\(426\) −20.8616 −1.01075
\(427\) −0.268235 −0.0129808
\(428\) 6.74567 0.326064
\(429\) 0 0
\(430\) 2.18804 0.105516
\(431\) 17.5909 0.847325 0.423662 0.905820i \(-0.360744\pi\)
0.423662 + 0.905820i \(0.360744\pi\)
\(432\) −2.69257 −0.129547
\(433\) 20.5443 0.987296 0.493648 0.869662i \(-0.335663\pi\)
0.493648 + 0.869662i \(0.335663\pi\)
\(434\) −1.84089 −0.0883656
\(435\) 18.8199 0.902346
\(436\) −18.6864 −0.894916
\(437\) −5.63485 −0.269552
\(438\) 24.9459 1.19196
\(439\) −22.4200 −1.07005 −0.535023 0.844837i \(-0.679697\pi\)
−0.535023 + 0.844837i \(0.679697\pi\)
\(440\) 0 0
\(441\) 1.76673 0.0841301
\(442\) −14.5325 −0.691243
\(443\) 7.63516 0.362757 0.181379 0.983413i \(-0.441944\pi\)
0.181379 + 0.983413i \(0.441944\pi\)
\(444\) −10.1113 −0.479862
\(445\) −2.65397 −0.125810
\(446\) 11.8000 0.558745
\(447\) 3.25760 0.154079
\(448\) −1.00000 −0.0472456
\(449\) −0.652470 −0.0307920 −0.0153960 0.999881i \(-0.504901\pi\)
−0.0153960 + 0.999881i \(0.504901\pi\)
\(450\) 1.76673 0.0832846
\(451\) 0 0
\(452\) 7.56433 0.355796
\(453\) 17.4566 0.820181
\(454\) −0.990121 −0.0464687
\(455\) 2.53263 0.118731
\(456\) 4.89187 0.229083
\(457\) −30.5691 −1.42996 −0.714981 0.699144i \(-0.753565\pi\)
−0.714981 + 0.699144i \(0.753565\pi\)
\(458\) 18.3559 0.857714
\(459\) −15.4503 −0.721160
\(460\) −2.51489 −0.117257
\(461\) 3.55735 0.165682 0.0828412 0.996563i \(-0.473601\pi\)
0.0828412 + 0.996563i \(0.473601\pi\)
\(462\) 0 0
\(463\) 6.57677 0.305648 0.152824 0.988253i \(-0.451163\pi\)
0.152824 + 0.988253i \(0.451163\pi\)
\(464\) 8.62000 0.400173
\(465\) 4.01919 0.186385
\(466\) 17.3650 0.804418
\(467\) −18.3935 −0.851150 −0.425575 0.904923i \(-0.639928\pi\)
−0.425575 + 0.904923i \(0.639928\pi\)
\(468\) −4.47448 −0.206833
\(469\) −3.86233 −0.178346
\(470\) 3.22482 0.148750
\(471\) 4.47951 0.206405
\(472\) 2.01405 0.0927040
\(473\) 0 0
\(474\) 25.6611 1.17865
\(475\) 2.24060 0.102806
\(476\) −5.73813 −0.263007
\(477\) −7.09778 −0.324985
\(478\) −8.92113 −0.408043
\(479\) −33.1011 −1.51243 −0.756213 0.654325i \(-0.772953\pi\)
−0.756213 + 0.654325i \(0.772953\pi\)
\(480\) 2.18328 0.0996529
\(481\) 11.7292 0.534807
\(482\) 0.896696 0.0408434
\(483\) 5.49071 0.249836
\(484\) 0 0
\(485\) 16.5045 0.749432
\(486\) −16.3289 −0.740694
\(487\) −15.5981 −0.706819 −0.353410 0.935469i \(-0.614978\pi\)
−0.353410 + 0.935469i \(0.614978\pi\)
\(488\) 0.268235 0.0121424
\(489\) 45.1887 2.04350
\(490\) 1.00000 0.0451754
\(491\) 41.6636 1.88025 0.940127 0.340825i \(-0.110706\pi\)
0.940127 + 0.340825i \(0.110706\pi\)
\(492\) 23.0388 1.03867
\(493\) 49.4626 2.22768
\(494\) −5.67461 −0.255313
\(495\) 0 0
\(496\) 1.84089 0.0826585
\(497\) 9.55514 0.428606
\(498\) 16.0439 0.718944
\(499\) −2.45110 −0.109726 −0.0548631 0.998494i \(-0.517472\pi\)
−0.0548631 + 0.998494i \(0.517472\pi\)
\(500\) 1.00000 0.0447214
\(501\) −2.88303 −0.128804
\(502\) −22.4978 −1.00413
\(503\) 6.68855 0.298228 0.149114 0.988820i \(-0.452358\pi\)
0.149114 + 0.988820i \(0.452358\pi\)
\(504\) −1.76673 −0.0786965
\(505\) −16.0478 −0.714118
\(506\) 0 0
\(507\) −14.3787 −0.638579
\(508\) 10.7426 0.476625
\(509\) −14.0762 −0.623917 −0.311959 0.950096i \(-0.600985\pi\)
−0.311959 + 0.950096i \(0.600985\pi\)
\(510\) 12.5280 0.554747
\(511\) −11.4258 −0.505450
\(512\) 1.00000 0.0441942
\(513\) −6.03298 −0.266363
\(514\) −24.9449 −1.10027
\(515\) −12.9029 −0.568570
\(516\) 4.77710 0.210300
\(517\) 0 0
\(518\) 4.63124 0.203485
\(519\) −34.2838 −1.50489
\(520\) −2.53263 −0.111063
\(521\) 41.6989 1.82686 0.913431 0.406993i \(-0.133423\pi\)
0.913431 + 0.406993i \(0.133423\pi\)
\(522\) 15.2292 0.666565
\(523\) −10.6960 −0.467703 −0.233851 0.972272i \(-0.575133\pi\)
−0.233851 + 0.972272i \(0.575133\pi\)
\(524\) 17.0221 0.743613
\(525\) −2.18328 −0.0952864
\(526\) −6.25594 −0.272772
\(527\) 10.5633 0.460143
\(528\) 0 0
\(529\) −16.6753 −0.725015
\(530\) −4.01746 −0.174507
\(531\) 3.55828 0.154416
\(532\) −2.24060 −0.0971423
\(533\) −26.7252 −1.15759
\(534\) −5.79438 −0.250747
\(535\) 6.74567 0.291641
\(536\) 3.86233 0.166827
\(537\) 10.1631 0.438571
\(538\) −13.0673 −0.563371
\(539\) 0 0
\(540\) −2.69257 −0.115870
\(541\) −9.64911 −0.414847 −0.207424 0.978251i \(-0.566508\pi\)
−0.207424 + 0.978251i \(0.566508\pi\)
\(542\) 24.3157 1.04445
\(543\) −12.8912 −0.553215
\(544\) 5.73813 0.246020
\(545\) −18.6864 −0.800437
\(546\) 5.52945 0.236639
\(547\) −14.8545 −0.635133 −0.317567 0.948236i \(-0.602866\pi\)
−0.317567 + 0.948236i \(0.602866\pi\)
\(548\) −12.4777 −0.533019
\(549\) 0.473900 0.0202255
\(550\) 0 0
\(551\) 19.3140 0.822803
\(552\) −5.49071 −0.233700
\(553\) −11.7534 −0.499807
\(554\) 13.4094 0.569713
\(555\) −10.1113 −0.429202
\(556\) −17.3812 −0.737129
\(557\) −39.9349 −1.69210 −0.846048 0.533106i \(-0.821025\pi\)
−0.846048 + 0.533106i \(0.821025\pi\)
\(558\) 3.25236 0.137683
\(559\) −5.54148 −0.234380
\(560\) −1.00000 −0.0422577
\(561\) 0 0
\(562\) −12.7398 −0.537396
\(563\) −4.06409 −0.171281 −0.0856405 0.996326i \(-0.527294\pi\)
−0.0856405 + 0.996326i \(0.527294\pi\)
\(564\) 7.04070 0.296467
\(565\) 7.56433 0.318234
\(566\) 13.7495 0.577936
\(567\) 11.1789 0.469468
\(568\) −9.55514 −0.400925
\(569\) −31.8994 −1.33729 −0.668647 0.743580i \(-0.733126\pi\)
−0.668647 + 0.743580i \(0.733126\pi\)
\(570\) 4.89187 0.204898
\(571\) −20.3883 −0.853222 −0.426611 0.904435i \(-0.640293\pi\)
−0.426611 + 0.904435i \(0.640293\pi\)
\(572\) 0 0
\(573\) −12.9556 −0.541229
\(574\) −10.5523 −0.440446
\(575\) −2.51489 −0.104878
\(576\) 1.76673 0.0736139
\(577\) −26.0450 −1.08427 −0.542134 0.840292i \(-0.682383\pi\)
−0.542134 + 0.840292i \(0.682383\pi\)
\(578\) 15.9261 0.662438
\(579\) −23.6617 −0.983348
\(580\) 8.62000 0.357926
\(581\) −7.34851 −0.304867
\(582\) 36.0341 1.49366
\(583\) 0 0
\(584\) 11.4258 0.472805
\(585\) −4.47448 −0.184997
\(586\) −16.6376 −0.687295
\(587\) −38.9847 −1.60907 −0.804536 0.593904i \(-0.797586\pi\)
−0.804536 + 0.593904i \(0.797586\pi\)
\(588\) 2.18328 0.0900371
\(589\) 4.12470 0.169955
\(590\) 2.01405 0.0829170
\(591\) −26.2378 −1.07928
\(592\) −4.63124 −0.190343
\(593\) 0.526179 0.0216076 0.0108038 0.999942i \(-0.496561\pi\)
0.0108038 + 0.999942i \(0.496561\pi\)
\(594\) 0 0
\(595\) −5.73813 −0.235240
\(596\) 1.49206 0.0611173
\(597\) 40.1755 1.64427
\(598\) 6.36928 0.260459
\(599\) −12.3903 −0.506256 −0.253128 0.967433i \(-0.581459\pi\)
−0.253128 + 0.967433i \(0.581459\pi\)
\(600\) 2.18328 0.0891322
\(601\) −26.8732 −1.09618 −0.548091 0.836418i \(-0.684645\pi\)
−0.548091 + 0.836418i \(0.684645\pi\)
\(602\) −2.18804 −0.0891777
\(603\) 6.82370 0.277883
\(604\) 7.99555 0.325334
\(605\) 0 0
\(606\) −35.0369 −1.42328
\(607\) 36.1325 1.46657 0.733286 0.679920i \(-0.237986\pi\)
0.733286 + 0.679920i \(0.237986\pi\)
\(608\) 2.24060 0.0908683
\(609\) −18.8199 −0.762621
\(610\) 0.268235 0.0108605
\(611\) −8.16728 −0.330413
\(612\) 10.1377 0.409794
\(613\) 33.2590 1.34332 0.671659 0.740860i \(-0.265582\pi\)
0.671659 + 0.740860i \(0.265582\pi\)
\(614\) −5.56226 −0.224475
\(615\) 23.0388 0.929012
\(616\) 0 0
\(617\) −27.1415 −1.09267 −0.546337 0.837565i \(-0.683978\pi\)
−0.546337 + 0.837565i \(0.683978\pi\)
\(618\) −28.1707 −1.13319
\(619\) 2.93234 0.117861 0.0589303 0.998262i \(-0.481231\pi\)
0.0589303 + 0.998262i \(0.481231\pi\)
\(620\) 1.84089 0.0739320
\(621\) 6.77152 0.271732
\(622\) 17.9179 0.718441
\(623\) 2.65397 0.106329
\(624\) −5.52945 −0.221355
\(625\) 1.00000 0.0400000
\(626\) 5.15478 0.206026
\(627\) 0 0
\(628\) 2.05173 0.0818729
\(629\) −26.5747 −1.05960
\(630\) −1.76673 −0.0703883
\(631\) −7.10213 −0.282731 −0.141366 0.989957i \(-0.545149\pi\)
−0.141366 + 0.989957i \(0.545149\pi\)
\(632\) 11.7534 0.467527
\(633\) −43.7578 −1.73922
\(634\) 28.4614 1.13035
\(635\) 10.7426 0.426306
\(636\) −8.77126 −0.347803
\(637\) −2.53263 −0.100346
\(638\) 0 0
\(639\) −16.8814 −0.667817
\(640\) 1.00000 0.0395285
\(641\) 1.60982 0.0635842 0.0317921 0.999495i \(-0.489879\pi\)
0.0317921 + 0.999495i \(0.489879\pi\)
\(642\) 14.7277 0.581257
\(643\) −17.9131 −0.706422 −0.353211 0.935544i \(-0.614910\pi\)
−0.353211 + 0.935544i \(0.614910\pi\)
\(644\) 2.51489 0.0991004
\(645\) 4.77710 0.188098
\(646\) 12.8568 0.505846
\(647\) −5.38711 −0.211789 −0.105895 0.994377i \(-0.533771\pi\)
−0.105895 + 0.994377i \(0.533771\pi\)
\(648\) −11.1789 −0.439147
\(649\) 0 0
\(650\) −2.53263 −0.0993379
\(651\) −4.01919 −0.157524
\(652\) 20.6976 0.810579
\(653\) 48.6527 1.90393 0.951963 0.306212i \(-0.0990616\pi\)
0.951963 + 0.306212i \(0.0990616\pi\)
\(654\) −40.7977 −1.59532
\(655\) 17.0221 0.665108
\(656\) 10.5523 0.411999
\(657\) 20.1864 0.787547
\(658\) −3.22482 −0.125717
\(659\) 29.0793 1.13277 0.566385 0.824141i \(-0.308342\pi\)
0.566385 + 0.824141i \(0.308342\pi\)
\(660\) 0 0
\(661\) −13.9160 −0.541270 −0.270635 0.962682i \(-0.587234\pi\)
−0.270635 + 0.962682i \(0.587234\pi\)
\(662\) −24.8572 −0.966101
\(663\) −31.7287 −1.23224
\(664\) 7.34851 0.285177
\(665\) −2.24060 −0.0868867
\(666\) −8.18217 −0.317053
\(667\) −21.6783 −0.839388
\(668\) −1.32050 −0.0510917
\(669\) 25.7627 0.996043
\(670\) 3.86233 0.149215
\(671\) 0 0
\(672\) −2.18328 −0.0842220
\(673\) 32.6201 1.25741 0.628707 0.777642i \(-0.283585\pi\)
0.628707 + 0.777642i \(0.283585\pi\)
\(674\) 15.6474 0.602716
\(675\) −2.69257 −0.103637
\(676\) −6.58579 −0.253300
\(677\) −37.5887 −1.44465 −0.722325 0.691554i \(-0.756926\pi\)
−0.722325 + 0.691554i \(0.756926\pi\)
\(678\) 16.5151 0.634258
\(679\) −16.5045 −0.633386
\(680\) 5.73813 0.220047
\(681\) −2.16172 −0.0828372
\(682\) 0 0
\(683\) −17.5199 −0.670380 −0.335190 0.942151i \(-0.608800\pi\)
−0.335190 + 0.942151i \(0.608800\pi\)
\(684\) 3.95854 0.151359
\(685\) −12.4777 −0.476747
\(686\) −1.00000 −0.0381802
\(687\) 40.0761 1.52900
\(688\) 2.18804 0.0834181
\(689\) 10.1747 0.387627
\(690\) −5.49071 −0.209028
\(691\) 7.14352 0.271752 0.135876 0.990726i \(-0.456615\pi\)
0.135876 + 0.990726i \(0.456615\pi\)
\(692\) −15.7029 −0.596933
\(693\) 0 0
\(694\) −14.6646 −0.556662
\(695\) −17.3812 −0.659308
\(696\) 18.8199 0.713367
\(697\) 60.5506 2.29352
\(698\) −26.9763 −1.02107
\(699\) 37.9127 1.43399
\(700\) −1.00000 −0.0377964
\(701\) 38.7733 1.46445 0.732224 0.681064i \(-0.238482\pi\)
0.732224 + 0.681064i \(0.238482\pi\)
\(702\) 6.81929 0.257378
\(703\) −10.3768 −0.391367
\(704\) 0 0
\(705\) 7.04070 0.265168
\(706\) 4.77463 0.179696
\(707\) 16.0478 0.603540
\(708\) 4.39724 0.165258
\(709\) 30.3511 1.13986 0.569930 0.821693i \(-0.306970\pi\)
0.569930 + 0.821693i \(0.306970\pi\)
\(710\) −9.55514 −0.358598
\(711\) 20.7652 0.778755
\(712\) −2.65397 −0.0994618
\(713\) −4.62963 −0.173381
\(714\) −12.5280 −0.468847
\(715\) 0 0
\(716\) 4.65497 0.173964
\(717\) −19.4774 −0.727396
\(718\) −34.0982 −1.27253
\(719\) 36.0357 1.34391 0.671953 0.740594i \(-0.265456\pi\)
0.671953 + 0.740594i \(0.265456\pi\)
\(720\) 1.76673 0.0658422
\(721\) 12.9029 0.480529
\(722\) −13.9797 −0.520271
\(723\) 1.95774 0.0728092
\(724\) −5.90450 −0.219439
\(725\) 8.62000 0.320139
\(726\) 0 0
\(727\) 31.1833 1.15653 0.578263 0.815850i \(-0.303731\pi\)
0.578263 + 0.815850i \(0.303731\pi\)
\(728\) 2.53263 0.0938655
\(729\) −2.11408 −0.0782992
\(730\) 11.4258 0.422890
\(731\) 12.5552 0.464372
\(732\) 0.585633 0.0216456
\(733\) 26.0329 0.961548 0.480774 0.876844i \(-0.340356\pi\)
0.480774 + 0.876844i \(0.340356\pi\)
\(734\) −9.98965 −0.368725
\(735\) 2.18328 0.0805317
\(736\) −2.51489 −0.0927000
\(737\) 0 0
\(738\) 18.6432 0.686264
\(739\) −22.4140 −0.824514 −0.412257 0.911068i \(-0.635259\pi\)
−0.412257 + 0.911068i \(0.635259\pi\)
\(740\) −4.63124 −0.170248
\(741\) −12.3893 −0.455132
\(742\) 4.01746 0.147486
\(743\) 44.3080 1.62550 0.812752 0.582610i \(-0.197969\pi\)
0.812752 + 0.582610i \(0.197969\pi\)
\(744\) 4.01919 0.147351
\(745\) 1.49206 0.0546650
\(746\) 0.510916 0.0187060
\(747\) 12.9828 0.475018
\(748\) 0 0
\(749\) −6.74567 −0.246482
\(750\) 2.18328 0.0797223
\(751\) 39.3682 1.43657 0.718283 0.695751i \(-0.244928\pi\)
0.718283 + 0.695751i \(0.244928\pi\)
\(752\) 3.22482 0.117597
\(753\) −49.1192 −1.79000
\(754\) −21.8313 −0.795048
\(755\) 7.99555 0.290988
\(756\) 2.69257 0.0979280
\(757\) −33.4097 −1.21430 −0.607149 0.794588i \(-0.707687\pi\)
−0.607149 + 0.794588i \(0.707687\pi\)
\(758\) 5.31376 0.193004
\(759\) 0 0
\(760\) 2.24060 0.0812751
\(761\) −38.9480 −1.41186 −0.705932 0.708279i \(-0.749472\pi\)
−0.705932 + 0.708279i \(0.749472\pi\)
\(762\) 23.4541 0.849652
\(763\) 18.6864 0.676493
\(764\) −5.93400 −0.214685
\(765\) 10.1377 0.366531
\(766\) 23.5202 0.849821
\(767\) −5.10083 −0.184180
\(768\) 2.18328 0.0787825
\(769\) 0.875099 0.0315569 0.0157784 0.999876i \(-0.494977\pi\)
0.0157784 + 0.999876i \(0.494977\pi\)
\(770\) 0 0
\(771\) −54.4617 −1.96139
\(772\) −10.8377 −0.390056
\(773\) −22.9131 −0.824125 −0.412063 0.911155i \(-0.635192\pi\)
−0.412063 + 0.911155i \(0.635192\pi\)
\(774\) 3.86567 0.138949
\(775\) 1.84089 0.0661268
\(776\) 16.5045 0.592478
\(777\) 10.1113 0.362742
\(778\) 9.18397 0.329261
\(779\) 23.6436 0.847118
\(780\) −5.52945 −0.197986
\(781\) 0 0
\(782\) −14.4307 −0.516042
\(783\) −23.2100 −0.829457
\(784\) 1.00000 0.0357143
\(785\) 2.05173 0.0732293
\(786\) 37.1641 1.32560
\(787\) 33.4092 1.19091 0.595455 0.803388i \(-0.296972\pi\)
0.595455 + 0.803388i \(0.296972\pi\)
\(788\) −12.0176 −0.428108
\(789\) −13.6585 −0.486255
\(790\) 11.7534 0.418168
\(791\) −7.56433 −0.268957
\(792\) 0 0
\(793\) −0.679340 −0.0241241
\(794\) −31.5709 −1.12041
\(795\) −8.77126 −0.311085
\(796\) 18.4014 0.652220
\(797\) −3.43820 −0.121787 −0.0608936 0.998144i \(-0.519395\pi\)
−0.0608936 + 0.998144i \(0.519395\pi\)
\(798\) −4.89187 −0.173170
\(799\) 18.5044 0.654640
\(800\) 1.00000 0.0353553
\(801\) −4.68886 −0.165673
\(802\) −10.6804 −0.377138
\(803\) 0 0
\(804\) 8.43256 0.297394
\(805\) 2.51489 0.0886381
\(806\) −4.66229 −0.164222
\(807\) −28.5296 −1.00429
\(808\) −16.0478 −0.564560
\(809\) 35.3314 1.24219 0.621093 0.783737i \(-0.286689\pi\)
0.621093 + 0.783737i \(0.286689\pi\)
\(810\) −11.1789 −0.392785
\(811\) −48.6796 −1.70937 −0.854686 0.519145i \(-0.826251\pi\)
−0.854686 + 0.519145i \(0.826251\pi\)
\(812\) −8.62000 −0.302503
\(813\) 53.0880 1.86188
\(814\) 0 0
\(815\) 20.6976 0.725004
\(816\) 12.5280 0.438566
\(817\) 4.90251 0.171517
\(818\) 23.5770 0.824349
\(819\) 4.47448 0.156351
\(820\) 10.5523 0.368504
\(821\) −17.1917 −0.599995 −0.299997 0.953940i \(-0.596986\pi\)
−0.299997 + 0.953940i \(0.596986\pi\)
\(822\) −27.2423 −0.950184
\(823\) −45.8799 −1.59927 −0.799636 0.600485i \(-0.794974\pi\)
−0.799636 + 0.600485i \(0.794974\pi\)
\(824\) −12.9029 −0.449494
\(825\) 0 0
\(826\) −2.01405 −0.0700777
\(827\) 27.9645 0.972420 0.486210 0.873842i \(-0.338379\pi\)
0.486210 + 0.873842i \(0.338379\pi\)
\(828\) −4.44313 −0.154410
\(829\) −18.8815 −0.655782 −0.327891 0.944715i \(-0.606338\pi\)
−0.327891 + 0.944715i \(0.606338\pi\)
\(830\) 7.34851 0.255070
\(831\) 29.2766 1.01560
\(832\) −2.53263 −0.0878031
\(833\) 5.73813 0.198814
\(834\) −37.9482 −1.31404
\(835\) −1.32050 −0.0456978
\(836\) 0 0
\(837\) −4.95674 −0.171330
\(838\) −15.9235 −0.550069
\(839\) −19.7116 −0.680521 −0.340261 0.940331i \(-0.610515\pi\)
−0.340261 + 0.940331i \(0.610515\pi\)
\(840\) −2.18328 −0.0753305
\(841\) 45.3044 1.56222
\(842\) −39.8602 −1.37367
\(843\) −27.8146 −0.957987
\(844\) −20.0422 −0.689880
\(845\) −6.58579 −0.226558
\(846\) 5.69740 0.195881
\(847\) 0 0
\(848\) −4.01746 −0.137960
\(849\) 30.0191 1.03025
\(850\) 5.73813 0.196816
\(851\) 11.6471 0.399256
\(852\) −20.8616 −0.714706
\(853\) 30.8936 1.05778 0.528888 0.848692i \(-0.322609\pi\)
0.528888 + 0.848692i \(0.322609\pi\)
\(854\) −0.268235 −0.00917881
\(855\) 3.95854 0.135379
\(856\) 6.74567 0.230562
\(857\) −24.2847 −0.829550 −0.414775 0.909924i \(-0.636140\pi\)
−0.414775 + 0.909924i \(0.636140\pi\)
\(858\) 0 0
\(859\) 22.0877 0.753621 0.376811 0.926290i \(-0.377021\pi\)
0.376811 + 0.926290i \(0.377021\pi\)
\(860\) 2.18804 0.0746114
\(861\) −23.0388 −0.785159
\(862\) 17.5909 0.599149
\(863\) −14.1052 −0.480146 −0.240073 0.970755i \(-0.577171\pi\)
−0.240073 + 0.970755i \(0.577171\pi\)
\(864\) −2.69257 −0.0916032
\(865\) −15.7029 −0.533913
\(866\) 20.5443 0.698124
\(867\) 34.7712 1.18089
\(868\) −1.84089 −0.0624839
\(869\) 0 0
\(870\) 18.8199 0.638055
\(871\) −9.78185 −0.331445
\(872\) −18.6864 −0.632801
\(873\) 29.1591 0.986886
\(874\) −5.63485 −0.190602
\(875\) −1.00000 −0.0338062
\(876\) 24.9459 0.842843
\(877\) −55.9227 −1.88838 −0.944188 0.329406i \(-0.893151\pi\)
−0.944188 + 0.329406i \(0.893151\pi\)
\(878\) −22.4200 −0.756637
\(879\) −36.3247 −1.22520
\(880\) 0 0
\(881\) −46.2057 −1.55671 −0.778354 0.627826i \(-0.783945\pi\)
−0.778354 + 0.627826i \(0.783945\pi\)
\(882\) 1.76673 0.0594890
\(883\) −14.1307 −0.475536 −0.237768 0.971322i \(-0.576416\pi\)
−0.237768 + 0.971322i \(0.576416\pi\)
\(884\) −14.5325 −0.488782
\(885\) 4.39724 0.147812
\(886\) 7.63516 0.256508
\(887\) 37.6208 1.26318 0.631591 0.775302i \(-0.282402\pi\)
0.631591 + 0.775302i \(0.282402\pi\)
\(888\) −10.1113 −0.339314
\(889\) −10.7426 −0.360294
\(890\) −2.65397 −0.0889614
\(891\) 0 0
\(892\) 11.8000 0.395092
\(893\) 7.22553 0.241793
\(894\) 3.25760 0.108950
\(895\) 4.65497 0.155598
\(896\) −1.00000 −0.0334077
\(897\) 13.9059 0.464306
\(898\) −0.652470 −0.0217732
\(899\) 15.8685 0.529243
\(900\) 1.76673 0.0588911
\(901\) −23.0527 −0.767997
\(902\) 0 0
\(903\) −4.77710 −0.158972
\(904\) 7.56433 0.251586
\(905\) −5.90450 −0.196272
\(906\) 17.4566 0.579955
\(907\) −2.97518 −0.0987893 −0.0493947 0.998779i \(-0.515729\pi\)
−0.0493947 + 0.998779i \(0.515729\pi\)
\(908\) −0.990121 −0.0328583
\(909\) −28.3522 −0.940383
\(910\) 2.53263 0.0839558
\(911\) −53.8707 −1.78481 −0.892407 0.451231i \(-0.850985\pi\)
−0.892407 + 0.451231i \(0.850985\pi\)
\(912\) 4.89187 0.161986
\(913\) 0 0
\(914\) −30.5691 −1.01114
\(915\) 0.585633 0.0193604
\(916\) 18.3559 0.606495
\(917\) −17.0221 −0.562119
\(918\) −15.4503 −0.509937
\(919\) 16.9667 0.559678 0.279839 0.960047i \(-0.409719\pi\)
0.279839 + 0.960047i \(0.409719\pi\)
\(920\) −2.51489 −0.0829134
\(921\) −12.1440 −0.400159
\(922\) 3.55735 0.117155
\(923\) 24.1996 0.796540
\(924\) 0 0
\(925\) −4.63124 −0.152274
\(926\) 6.57677 0.216126
\(927\) −22.7960 −0.748718
\(928\) 8.62000 0.282965
\(929\) 32.3876 1.06260 0.531301 0.847183i \(-0.321703\pi\)
0.531301 + 0.847183i \(0.321703\pi\)
\(930\) 4.01919 0.131794
\(931\) 2.24060 0.0734327
\(932\) 17.3650 0.568809
\(933\) 39.1198 1.28072
\(934\) −18.3935 −0.601854
\(935\) 0 0
\(936\) −4.47448 −0.146253
\(937\) 24.1450 0.788784 0.394392 0.918942i \(-0.370955\pi\)
0.394392 + 0.918942i \(0.370955\pi\)
\(938\) −3.86233 −0.126110
\(939\) 11.2543 0.367272
\(940\) 3.22482 0.105182
\(941\) −40.6046 −1.32367 −0.661836 0.749648i \(-0.730223\pi\)
−0.661836 + 0.749648i \(0.730223\pi\)
\(942\) 4.47951 0.145950
\(943\) −26.5379 −0.864194
\(944\) 2.01405 0.0655516
\(945\) 2.69257 0.0875894
\(946\) 0 0
\(947\) 0.111103 0.00361036 0.00180518 0.999998i \(-0.499425\pi\)
0.00180518 + 0.999998i \(0.499425\pi\)
\(948\) 25.6611 0.833434
\(949\) −28.9374 −0.939349
\(950\) 2.24060 0.0726946
\(951\) 62.1393 2.01500
\(952\) −5.73813 −0.185974
\(953\) 7.70395 0.249555 0.124778 0.992185i \(-0.460178\pi\)
0.124778 + 0.992185i \(0.460178\pi\)
\(954\) −7.09778 −0.229799
\(955\) −5.93400 −0.192020
\(956\) −8.92113 −0.288530
\(957\) 0 0
\(958\) −33.1011 −1.06945
\(959\) 12.4777 0.402925
\(960\) 2.18328 0.0704652
\(961\) −27.6111 −0.890681
\(962\) 11.7292 0.378165
\(963\) 11.9178 0.384046
\(964\) 0.896696 0.0288806
\(965\) −10.8377 −0.348877
\(966\) 5.49071 0.176661
\(967\) −41.8250 −1.34500 −0.672501 0.740096i \(-0.734780\pi\)
−0.672501 + 0.740096i \(0.734780\pi\)
\(968\) 0 0
\(969\) 28.0701 0.901743
\(970\) 16.5045 0.529929
\(971\) 37.2897 1.19668 0.598342 0.801241i \(-0.295826\pi\)
0.598342 + 0.801241i \(0.295826\pi\)
\(972\) −16.3289 −0.523750
\(973\) 17.3812 0.557217
\(974\) −15.5981 −0.499797
\(975\) −5.52945 −0.177084
\(976\) 0.268235 0.00858599
\(977\) 2.41139 0.0771471 0.0385736 0.999256i \(-0.487719\pi\)
0.0385736 + 0.999256i \(0.487719\pi\)
\(978\) 45.1887 1.44498
\(979\) 0 0
\(980\) 1.00000 0.0319438
\(981\) −33.0139 −1.05405
\(982\) 41.6636 1.32954
\(983\) −40.4385 −1.28979 −0.644894 0.764272i \(-0.723098\pi\)
−0.644894 + 0.764272i \(0.723098\pi\)
\(984\) 23.0388 0.734449
\(985\) −12.0176 −0.382911
\(986\) 49.4626 1.57521
\(987\) −7.04070 −0.224108
\(988\) −5.67461 −0.180533
\(989\) −5.50266 −0.174974
\(990\) 0 0
\(991\) 26.7075 0.848392 0.424196 0.905570i \(-0.360557\pi\)
0.424196 + 0.905570i \(0.360557\pi\)
\(992\) 1.84089 0.0584484
\(993\) −54.2702 −1.72221
\(994\) 9.55514 0.303071
\(995\) 18.4014 0.583363
\(996\) 16.0439 0.508370
\(997\) 27.3325 0.865630 0.432815 0.901483i \(-0.357520\pi\)
0.432815 + 0.901483i \(0.357520\pi\)
\(998\) −2.45110 −0.0775881
\(999\) 12.4700 0.394532
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.dc.1.6 6
11.7 odd 10 770.2.n.j.71.1 12
11.8 odd 10 770.2.n.j.141.1 yes 12
11.10 odd 2 8470.2.a.cw.1.6 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
770.2.n.j.71.1 12 11.7 odd 10
770.2.n.j.141.1 yes 12 11.8 odd 10
8470.2.a.cw.1.6 6 11.10 odd 2
8470.2.a.dc.1.6 6 1.1 even 1 trivial