Properties

Label 8085.2.a.bk.1.1
Level $8085$
Weight $2$
Character 8085.1
Self dual yes
Analytic conductor $64.559$
Analytic rank $1$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 8085 = 3 \cdot 5 \cdot 7^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8085.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(64.5590500342\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.148.1
Defining polynomial: \( x^{3} - x^{2} - 3x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 165)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(2.17009\) of defining polynomial
Character \(\chi\) \(=\) 8085.1

$q$-expansion

\(f(q)\) \(=\) \(q-2.70928 q^{2} -1.00000 q^{3} +5.34017 q^{4} -1.00000 q^{5} +2.70928 q^{6} -9.04945 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-2.70928 q^{2} -1.00000 q^{3} +5.34017 q^{4} -1.00000 q^{5} +2.70928 q^{6} -9.04945 q^{8} +1.00000 q^{9} +2.70928 q^{10} +1.00000 q^{11} -5.34017 q^{12} +4.34017 q^{13} +1.00000 q^{15} +13.8371 q^{16} -7.75872 q^{17} -2.70928 q^{18} -5.26180 q^{19} -5.34017 q^{20} -2.70928 q^{22} -2.15676 q^{23} +9.04945 q^{24} +1.00000 q^{25} -11.7587 q^{26} -1.00000 q^{27} +1.41855 q^{29} -2.70928 q^{30} +4.68035 q^{31} -19.3896 q^{32} -1.00000 q^{33} +21.0205 q^{34} +5.34017 q^{36} -2.00000 q^{37} +14.2557 q^{38} -4.34017 q^{39} +9.04945 q^{40} +9.41855 q^{41} +7.60197 q^{43} +5.34017 q^{44} -1.00000 q^{45} +5.84324 q^{46} -4.68035 q^{47} -13.8371 q^{48} -2.70928 q^{50} +7.75872 q^{51} +23.1773 q^{52} +0.156755 q^{53} +2.70928 q^{54} -1.00000 q^{55} +5.26180 q^{57} -3.84324 q^{58} -6.15676 q^{59} +5.34017 q^{60} +4.15676 q^{61} -12.6803 q^{62} +24.8576 q^{64} -4.34017 q^{65} +2.70928 q^{66} -8.68035 q^{67} -41.4329 q^{68} +2.15676 q^{69} -4.68035 q^{71} -9.04945 q^{72} +10.4969 q^{73} +5.41855 q^{74} -1.00000 q^{75} -28.0989 q^{76} +11.7587 q^{78} -8.09890 q^{79} -13.8371 q^{80} +1.00000 q^{81} -25.5174 q^{82} +11.0205 q^{83} +7.75872 q^{85} -20.5958 q^{86} -1.41855 q^{87} -9.04945 q^{88} +12.8371 q^{89} +2.70928 q^{90} -11.5174 q^{92} -4.68035 q^{93} +12.6803 q^{94} +5.26180 q^{95} +19.3896 q^{96} -14.6803 q^{97} +1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - q^{2} - 3 q^{3} + 5 q^{4} - 3 q^{5} + q^{6} - 9 q^{8} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - q^{2} - 3 q^{3} + 5 q^{4} - 3 q^{5} + q^{6} - 9 q^{8} + 3 q^{9} + q^{10} + 3 q^{11} - 5 q^{12} + 2 q^{13} + 3 q^{15} + 13 q^{16} + 2 q^{17} - q^{18} - 8 q^{19} - 5 q^{20} - q^{22} + 9 q^{24} + 3 q^{25} - 10 q^{26} - 3 q^{27} - 10 q^{29} - q^{30} - 8 q^{31} - 29 q^{32} - 3 q^{33} + 30 q^{34} + 5 q^{36} - 6 q^{37} - 2 q^{39} + 9 q^{40} + 14 q^{41} + 4 q^{43} + 5 q^{44} - 3 q^{45} + 24 q^{46} + 8 q^{47} - 13 q^{48} - q^{50} - 2 q^{51} + 30 q^{52} - 6 q^{53} + q^{54} - 3 q^{55} + 8 q^{57} - 18 q^{58} - 12 q^{59} + 5 q^{60} + 6 q^{61} - 16 q^{62} + 13 q^{64} - 2 q^{65} + q^{66} - 4 q^{67} - 42 q^{68} + 8 q^{71} - 9 q^{72} + 14 q^{73} + 2 q^{74} - 3 q^{75} - 48 q^{76} + 10 q^{78} + 12 q^{79} - 13 q^{80} + 3 q^{81} - 26 q^{82} - 2 q^{85} - 8 q^{86} + 10 q^{87} - 9 q^{88} + 10 q^{89} + q^{90} + 16 q^{92} + 8 q^{93} + 16 q^{94} + 8 q^{95} + 29 q^{96} - 22 q^{97} + 3 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.70928 −1.91575 −0.957873 0.287190i \(-0.907279\pi\)
−0.957873 + 0.287190i \(0.907279\pi\)
\(3\) −1.00000 −0.577350
\(4\) 5.34017 2.67009
\(5\) −1.00000 −0.447214
\(6\) 2.70928 1.10606
\(7\) 0 0
\(8\) −9.04945 −3.19946
\(9\) 1.00000 0.333333
\(10\) 2.70928 0.856748
\(11\) 1.00000 0.301511
\(12\) −5.34017 −1.54158
\(13\) 4.34017 1.20375 0.601874 0.798591i \(-0.294421\pi\)
0.601874 + 0.798591i \(0.294421\pi\)
\(14\) 0 0
\(15\) 1.00000 0.258199
\(16\) 13.8371 3.45928
\(17\) −7.75872 −1.88177 −0.940883 0.338730i \(-0.890003\pi\)
−0.940883 + 0.338730i \(0.890003\pi\)
\(18\) −2.70928 −0.638582
\(19\) −5.26180 −1.20714 −0.603569 0.797311i \(-0.706255\pi\)
−0.603569 + 0.797311i \(0.706255\pi\)
\(20\) −5.34017 −1.19410
\(21\) 0 0
\(22\) −2.70928 −0.577619
\(23\) −2.15676 −0.449715 −0.224857 0.974392i \(-0.572192\pi\)
−0.224857 + 0.974392i \(0.572192\pi\)
\(24\) 9.04945 1.84721
\(25\) 1.00000 0.200000
\(26\) −11.7587 −2.30608
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) 1.41855 0.263418 0.131709 0.991288i \(-0.457954\pi\)
0.131709 + 0.991288i \(0.457954\pi\)
\(30\) −2.70928 −0.494644
\(31\) 4.68035 0.840615 0.420307 0.907382i \(-0.361922\pi\)
0.420307 + 0.907382i \(0.361922\pi\)
\(32\) −19.3896 −3.42763
\(33\) −1.00000 −0.174078
\(34\) 21.0205 3.60499
\(35\) 0 0
\(36\) 5.34017 0.890029
\(37\) −2.00000 −0.328798 −0.164399 0.986394i \(-0.552568\pi\)
−0.164399 + 0.986394i \(0.552568\pi\)
\(38\) 14.2557 2.31257
\(39\) −4.34017 −0.694984
\(40\) 9.04945 1.43084
\(41\) 9.41855 1.47093 0.735465 0.677562i \(-0.236964\pi\)
0.735465 + 0.677562i \(0.236964\pi\)
\(42\) 0 0
\(43\) 7.60197 1.15929 0.579645 0.814869i \(-0.303191\pi\)
0.579645 + 0.814869i \(0.303191\pi\)
\(44\) 5.34017 0.805061
\(45\) −1.00000 −0.149071
\(46\) 5.84324 0.861539
\(47\) −4.68035 −0.682699 −0.341349 0.939937i \(-0.610884\pi\)
−0.341349 + 0.939937i \(0.610884\pi\)
\(48\) −13.8371 −1.99721
\(49\) 0 0
\(50\) −2.70928 −0.383149
\(51\) 7.75872 1.08644
\(52\) 23.1773 3.21411
\(53\) 0.156755 0.0215320 0.0107660 0.999942i \(-0.496573\pi\)
0.0107660 + 0.999942i \(0.496573\pi\)
\(54\) 2.70928 0.368686
\(55\) −1.00000 −0.134840
\(56\) 0 0
\(57\) 5.26180 0.696942
\(58\) −3.84324 −0.504643
\(59\) −6.15676 −0.801541 −0.400771 0.916178i \(-0.631258\pi\)
−0.400771 + 0.916178i \(0.631258\pi\)
\(60\) 5.34017 0.689413
\(61\) 4.15676 0.532218 0.266109 0.963943i \(-0.414262\pi\)
0.266109 + 0.963943i \(0.414262\pi\)
\(62\) −12.6803 −1.61041
\(63\) 0 0
\(64\) 24.8576 3.10720
\(65\) −4.34017 −0.538332
\(66\) 2.70928 0.333489
\(67\) −8.68035 −1.06047 −0.530237 0.847850i \(-0.677897\pi\)
−0.530237 + 0.847850i \(0.677897\pi\)
\(68\) −41.4329 −5.02448
\(69\) 2.15676 0.259643
\(70\) 0 0
\(71\) −4.68035 −0.555455 −0.277727 0.960660i \(-0.589581\pi\)
−0.277727 + 0.960660i \(0.589581\pi\)
\(72\) −9.04945 −1.06649
\(73\) 10.4969 1.22857 0.614286 0.789083i \(-0.289444\pi\)
0.614286 + 0.789083i \(0.289444\pi\)
\(74\) 5.41855 0.629894
\(75\) −1.00000 −0.115470
\(76\) −28.0989 −3.22316
\(77\) 0 0
\(78\) 11.7587 1.33141
\(79\) −8.09890 −0.911197 −0.455599 0.890185i \(-0.650575\pi\)
−0.455599 + 0.890185i \(0.650575\pi\)
\(80\) −13.8371 −1.54703
\(81\) 1.00000 0.111111
\(82\) −25.5174 −2.81793
\(83\) 11.0205 1.20966 0.604830 0.796355i \(-0.293241\pi\)
0.604830 + 0.796355i \(0.293241\pi\)
\(84\) 0 0
\(85\) 7.75872 0.841552
\(86\) −20.5958 −2.22090
\(87\) −1.41855 −0.152085
\(88\) −9.04945 −0.964674
\(89\) 12.8371 1.36073 0.680365 0.732873i \(-0.261821\pi\)
0.680365 + 0.732873i \(0.261821\pi\)
\(90\) 2.70928 0.285583
\(91\) 0 0
\(92\) −11.5174 −1.20078
\(93\) −4.68035 −0.485329
\(94\) 12.6803 1.30788
\(95\) 5.26180 0.539849
\(96\) 19.3896 1.97894
\(97\) −14.6803 −1.49056 −0.745282 0.666750i \(-0.767685\pi\)
−0.745282 + 0.666750i \(0.767685\pi\)
\(98\) 0 0
\(99\) 1.00000 0.100504
\(100\) 5.34017 0.534017
\(101\) 15.5753 1.54980 0.774900 0.632083i \(-0.217800\pi\)
0.774900 + 0.632083i \(0.217800\pi\)
\(102\) −21.0205 −2.08134
\(103\) −6.83710 −0.673680 −0.336840 0.941562i \(-0.609358\pi\)
−0.336840 + 0.941562i \(0.609358\pi\)
\(104\) −39.2762 −3.85135
\(105\) 0 0
\(106\) −0.424694 −0.0412499
\(107\) 6.34017 0.612928 0.306464 0.951882i \(-0.400854\pi\)
0.306464 + 0.951882i \(0.400854\pi\)
\(108\) −5.34017 −0.513858
\(109\) 2.31351 0.221594 0.110797 0.993843i \(-0.464660\pi\)
0.110797 + 0.993843i \(0.464660\pi\)
\(110\) 2.70928 0.258319
\(111\) 2.00000 0.189832
\(112\) 0 0
\(113\) −6.00000 −0.564433 −0.282216 0.959351i \(-0.591070\pi\)
−0.282216 + 0.959351i \(0.591070\pi\)
\(114\) −14.2557 −1.33516
\(115\) 2.15676 0.201118
\(116\) 7.57531 0.703350
\(117\) 4.34017 0.401249
\(118\) 16.6803 1.53555
\(119\) 0 0
\(120\) −9.04945 −0.826098
\(121\) 1.00000 0.0909091
\(122\) −11.2618 −1.01960
\(123\) −9.41855 −0.849242
\(124\) 24.9939 2.24451
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) −2.24128 −0.198881 −0.0994406 0.995044i \(-0.531705\pi\)
−0.0994406 + 0.995044i \(0.531705\pi\)
\(128\) −28.5669 −2.52498
\(129\) −7.60197 −0.669316
\(130\) 11.7587 1.03131
\(131\) −8.68035 −0.758405 −0.379203 0.925314i \(-0.623802\pi\)
−0.379203 + 0.925314i \(0.623802\pi\)
\(132\) −5.34017 −0.464802
\(133\) 0 0
\(134\) 23.5174 2.03160
\(135\) 1.00000 0.0860663
\(136\) 70.2122 6.02064
\(137\) −15.3607 −1.31235 −0.656176 0.754608i \(-0.727827\pi\)
−0.656176 + 0.754608i \(0.727827\pi\)
\(138\) −5.84324 −0.497410
\(139\) −8.58145 −0.727869 −0.363935 0.931425i \(-0.618567\pi\)
−0.363935 + 0.931425i \(0.618567\pi\)
\(140\) 0 0
\(141\) 4.68035 0.394156
\(142\) 12.6803 1.06411
\(143\) 4.34017 0.362943
\(144\) 13.8371 1.15309
\(145\) −1.41855 −0.117804
\(146\) −28.4391 −2.35363
\(147\) 0 0
\(148\) −10.6803 −0.877919
\(149\) −18.0989 −1.48272 −0.741360 0.671108i \(-0.765819\pi\)
−0.741360 + 0.671108i \(0.765819\pi\)
\(150\) 2.70928 0.221211
\(151\) 22.9360 1.86651 0.933253 0.359221i \(-0.116958\pi\)
0.933253 + 0.359221i \(0.116958\pi\)
\(152\) 47.6163 3.86220
\(153\) −7.75872 −0.627256
\(154\) 0 0
\(155\) −4.68035 −0.375934
\(156\) −23.1773 −1.85567
\(157\) 10.9939 0.877405 0.438703 0.898632i \(-0.355438\pi\)
0.438703 + 0.898632i \(0.355438\pi\)
\(158\) 21.9421 1.74562
\(159\) −0.156755 −0.0124315
\(160\) 19.3896 1.53288
\(161\) 0 0
\(162\) −2.70928 −0.212861
\(163\) −6.52359 −0.510967 −0.255484 0.966813i \(-0.582235\pi\)
−0.255484 + 0.966813i \(0.582235\pi\)
\(164\) 50.2967 3.92751
\(165\) 1.00000 0.0778499
\(166\) −29.8576 −2.31740
\(167\) −1.97334 −0.152701 −0.0763507 0.997081i \(-0.524327\pi\)
−0.0763507 + 0.997081i \(0.524327\pi\)
\(168\) 0 0
\(169\) 5.83710 0.449008
\(170\) −21.0205 −1.61220
\(171\) −5.26180 −0.402380
\(172\) 40.5958 3.09540
\(173\) −3.75872 −0.285770 −0.142885 0.989739i \(-0.545638\pi\)
−0.142885 + 0.989739i \(0.545638\pi\)
\(174\) 3.84324 0.291356
\(175\) 0 0
\(176\) 13.8371 1.04301
\(177\) 6.15676 0.462770
\(178\) −34.7792 −2.60681
\(179\) 15.1506 1.13241 0.566205 0.824264i \(-0.308411\pi\)
0.566205 + 0.824264i \(0.308411\pi\)
\(180\) −5.34017 −0.398033
\(181\) −4.83710 −0.359539 −0.179769 0.983709i \(-0.557535\pi\)
−0.179769 + 0.983709i \(0.557535\pi\)
\(182\) 0 0
\(183\) −4.15676 −0.307276
\(184\) 19.5174 1.43885
\(185\) 2.00000 0.147043
\(186\) 12.6803 0.929768
\(187\) −7.75872 −0.567374
\(188\) −24.9939 −1.82286
\(189\) 0 0
\(190\) −14.2557 −1.03421
\(191\) 2.52359 0.182601 0.0913003 0.995823i \(-0.470898\pi\)
0.0913003 + 0.995823i \(0.470898\pi\)
\(192\) −24.8576 −1.79394
\(193\) 0.0266620 0.00191917 0.000959586 1.00000i \(-0.499695\pi\)
0.000959586 1.00000i \(0.499695\pi\)
\(194\) 39.7731 2.85554
\(195\) 4.34017 0.310806
\(196\) 0 0
\(197\) 21.1194 1.50470 0.752348 0.658766i \(-0.228921\pi\)
0.752348 + 0.658766i \(0.228921\pi\)
\(198\) −2.70928 −0.192540
\(199\) −10.5236 −0.745998 −0.372999 0.927832i \(-0.621670\pi\)
−0.372999 + 0.927832i \(0.621670\pi\)
\(200\) −9.04945 −0.639893
\(201\) 8.68035 0.612264
\(202\) −42.1978 −2.96903
\(203\) 0 0
\(204\) 41.4329 2.90089
\(205\) −9.41855 −0.657820
\(206\) 18.5236 1.29060
\(207\) −2.15676 −0.149905
\(208\) 60.0554 4.16409
\(209\) −5.26180 −0.363966
\(210\) 0 0
\(211\) 9.57531 0.659191 0.329596 0.944122i \(-0.393088\pi\)
0.329596 + 0.944122i \(0.393088\pi\)
\(212\) 0.837101 0.0574924
\(213\) 4.68035 0.320692
\(214\) −17.1773 −1.17421
\(215\) −7.60197 −0.518450
\(216\) 9.04945 0.615737
\(217\) 0 0
\(218\) −6.26794 −0.424518
\(219\) −10.4969 −0.709317
\(220\) −5.34017 −0.360034
\(221\) −33.6742 −2.26517
\(222\) −5.41855 −0.363669
\(223\) 2.15676 0.144427 0.0722135 0.997389i \(-0.476994\pi\)
0.0722135 + 0.997389i \(0.476994\pi\)
\(224\) 0 0
\(225\) 1.00000 0.0666667
\(226\) 16.2557 1.08131
\(227\) −9.65983 −0.641145 −0.320573 0.947224i \(-0.603875\pi\)
−0.320573 + 0.947224i \(0.603875\pi\)
\(228\) 28.0989 1.86089
\(229\) 3.36069 0.222081 0.111040 0.993816i \(-0.464582\pi\)
0.111040 + 0.993816i \(0.464582\pi\)
\(230\) −5.84324 −0.385292
\(231\) 0 0
\(232\) −12.8371 −0.842797
\(233\) −2.39803 −0.157100 −0.0785501 0.996910i \(-0.525029\pi\)
−0.0785501 + 0.996910i \(0.525029\pi\)
\(234\) −11.7587 −0.768692
\(235\) 4.68035 0.305312
\(236\) −32.8781 −2.14018
\(237\) 8.09890 0.526080
\(238\) 0 0
\(239\) −7.20394 −0.465984 −0.232992 0.972479i \(-0.574852\pi\)
−0.232992 + 0.972479i \(0.574852\pi\)
\(240\) 13.8371 0.893181
\(241\) 5.20394 0.335215 0.167608 0.985854i \(-0.446396\pi\)
0.167608 + 0.985854i \(0.446396\pi\)
\(242\) −2.70928 −0.174159
\(243\) −1.00000 −0.0641500
\(244\) 22.1978 1.42107
\(245\) 0 0
\(246\) 25.5174 1.62693
\(247\) −22.8371 −1.45309
\(248\) −42.3545 −2.68952
\(249\) −11.0205 −0.698397
\(250\) 2.70928 0.171350
\(251\) −15.3197 −0.966968 −0.483484 0.875353i \(-0.660629\pi\)
−0.483484 + 0.875353i \(0.660629\pi\)
\(252\) 0 0
\(253\) −2.15676 −0.135594
\(254\) 6.07223 0.381006
\(255\) −7.75872 −0.485870
\(256\) 27.6803 1.73002
\(257\) −4.15676 −0.259291 −0.129646 0.991560i \(-0.541384\pi\)
−0.129646 + 0.991560i \(0.541384\pi\)
\(258\) 20.5958 1.28224
\(259\) 0 0
\(260\) −23.1773 −1.43739
\(261\) 1.41855 0.0878061
\(262\) 23.5174 1.45291
\(263\) −18.7070 −1.15352 −0.576762 0.816912i \(-0.695684\pi\)
−0.576762 + 0.816912i \(0.695684\pi\)
\(264\) 9.04945 0.556955
\(265\) −0.156755 −0.00962941
\(266\) 0 0
\(267\) −12.8371 −0.785618
\(268\) −46.3545 −2.83155
\(269\) −23.3607 −1.42433 −0.712163 0.702014i \(-0.752284\pi\)
−0.712163 + 0.702014i \(0.752284\pi\)
\(270\) −2.70928 −0.164881
\(271\) 5.57531 0.338676 0.169338 0.985558i \(-0.445837\pi\)
0.169338 + 0.985558i \(0.445837\pi\)
\(272\) −107.358 −6.50955
\(273\) 0 0
\(274\) 41.6163 2.51414
\(275\) 1.00000 0.0603023
\(276\) 11.5174 0.693269
\(277\) −26.0144 −1.56305 −0.781526 0.623872i \(-0.785558\pi\)
−0.781526 + 0.623872i \(0.785558\pi\)
\(278\) 23.2495 1.39441
\(279\) 4.68035 0.280205
\(280\) 0 0
\(281\) −9.41855 −0.561864 −0.280932 0.959728i \(-0.590643\pi\)
−0.280932 + 0.959728i \(0.590643\pi\)
\(282\) −12.6803 −0.755104
\(283\) −14.2413 −0.846556 −0.423278 0.906000i \(-0.639121\pi\)
−0.423278 + 0.906000i \(0.639121\pi\)
\(284\) −24.9939 −1.48311
\(285\) −5.26180 −0.311682
\(286\) −11.7587 −0.695308
\(287\) 0 0
\(288\) −19.3896 −1.14254
\(289\) 43.1978 2.54105
\(290\) 3.84324 0.225683
\(291\) 14.6803 0.860577
\(292\) 56.0554 3.28039
\(293\) 15.7587 0.920634 0.460317 0.887754i \(-0.347736\pi\)
0.460317 + 0.887754i \(0.347736\pi\)
\(294\) 0 0
\(295\) 6.15676 0.358460
\(296\) 18.0989 1.05198
\(297\) −1.00000 −0.0580259
\(298\) 49.0349 2.84052
\(299\) −9.36069 −0.541343
\(300\) −5.34017 −0.308315
\(301\) 0 0
\(302\) −62.1399 −3.57575
\(303\) −15.5753 −0.894778
\(304\) −72.8080 −4.17582
\(305\) −4.15676 −0.238015
\(306\) 21.0205 1.20166
\(307\) 18.9216 1.07991 0.539957 0.841693i \(-0.318440\pi\)
0.539957 + 0.841693i \(0.318440\pi\)
\(308\) 0 0
\(309\) 6.83710 0.388949
\(310\) 12.6803 0.720195
\(311\) 20.8781 1.18389 0.591945 0.805978i \(-0.298360\pi\)
0.591945 + 0.805978i \(0.298360\pi\)
\(312\) 39.2762 2.22358
\(313\) −6.31351 −0.356861 −0.178430 0.983953i \(-0.557102\pi\)
−0.178430 + 0.983953i \(0.557102\pi\)
\(314\) −29.7854 −1.68089
\(315\) 0 0
\(316\) −43.2495 −2.43297
\(317\) 31.3607 1.76139 0.880696 0.473682i \(-0.157075\pi\)
0.880696 + 0.473682i \(0.157075\pi\)
\(318\) 0.424694 0.0238156
\(319\) 1.41855 0.0794236
\(320\) −24.8576 −1.38958
\(321\) −6.34017 −0.353874
\(322\) 0 0
\(323\) 40.8248 2.27155
\(324\) 5.34017 0.296676
\(325\) 4.34017 0.240749
\(326\) 17.6742 0.978884
\(327\) −2.31351 −0.127937
\(328\) −85.2327 −4.70619
\(329\) 0 0
\(330\) −2.70928 −0.149141
\(331\) 19.2039 1.05554 0.527772 0.849386i \(-0.323028\pi\)
0.527772 + 0.849386i \(0.323028\pi\)
\(332\) 58.8515 3.22989
\(333\) −2.00000 −0.109599
\(334\) 5.34632 0.292537
\(335\) 8.68035 0.474258
\(336\) 0 0
\(337\) 13.5031 0.735559 0.367780 0.929913i \(-0.380118\pi\)
0.367780 + 0.929913i \(0.380118\pi\)
\(338\) −15.8143 −0.860185
\(339\) 6.00000 0.325875
\(340\) 41.4329 2.24702
\(341\) 4.68035 0.253455
\(342\) 14.2557 0.770857
\(343\) 0 0
\(344\) −68.7936 −3.70910
\(345\) −2.15676 −0.116116
\(346\) 10.1834 0.547464
\(347\) 6.34017 0.340358 0.170179 0.985413i \(-0.445565\pi\)
0.170179 + 0.985413i \(0.445565\pi\)
\(348\) −7.57531 −0.406079
\(349\) −16.1568 −0.864851 −0.432426 0.901670i \(-0.642342\pi\)
−0.432426 + 0.901670i \(0.642342\pi\)
\(350\) 0 0
\(351\) −4.34017 −0.231661
\(352\) −19.3896 −1.03347
\(353\) 13.2039 0.702775 0.351387 0.936230i \(-0.385710\pi\)
0.351387 + 0.936230i \(0.385710\pi\)
\(354\) −16.6803 −0.886550
\(355\) 4.68035 0.248407
\(356\) 68.5523 3.63327
\(357\) 0 0
\(358\) −41.0472 −2.16941
\(359\) 3.31965 0.175205 0.0876023 0.996156i \(-0.472080\pi\)
0.0876023 + 0.996156i \(0.472080\pi\)
\(360\) 9.04945 0.476948
\(361\) 8.68649 0.457184
\(362\) 13.1050 0.688786
\(363\) −1.00000 −0.0524864
\(364\) 0 0
\(365\) −10.4969 −0.549434
\(366\) 11.2618 0.588663
\(367\) 36.1445 1.88673 0.943363 0.331762i \(-0.107643\pi\)
0.943363 + 0.331762i \(0.107643\pi\)
\(368\) −29.8432 −1.55569
\(369\) 9.41855 0.490310
\(370\) −5.41855 −0.281697
\(371\) 0 0
\(372\) −24.9939 −1.29587
\(373\) −2.81044 −0.145519 −0.0727595 0.997350i \(-0.523181\pi\)
−0.0727595 + 0.997350i \(0.523181\pi\)
\(374\) 21.0205 1.08695
\(375\) 1.00000 0.0516398
\(376\) 42.3545 2.18427
\(377\) 6.15676 0.317089
\(378\) 0 0
\(379\) −20.0000 −1.02733 −0.513665 0.857991i \(-0.671713\pi\)
−0.513665 + 0.857991i \(0.671713\pi\)
\(380\) 28.0989 1.44144
\(381\) 2.24128 0.114824
\(382\) −6.83710 −0.349817
\(383\) −33.5585 −1.71476 −0.857379 0.514685i \(-0.827909\pi\)
−0.857379 + 0.514685i \(0.827909\pi\)
\(384\) 28.5669 1.45780
\(385\) 0 0
\(386\) −0.0722347 −0.00367665
\(387\) 7.60197 0.386430
\(388\) −78.3956 −3.97993
\(389\) 12.8371 0.650867 0.325433 0.945565i \(-0.394490\pi\)
0.325433 + 0.945565i \(0.394490\pi\)
\(390\) −11.7587 −0.595426
\(391\) 16.7337 0.846258
\(392\) 0 0
\(393\) 8.68035 0.437866
\(394\) −57.2183 −2.88262
\(395\) 8.09890 0.407500
\(396\) 5.34017 0.268354
\(397\) 5.31965 0.266986 0.133493 0.991050i \(-0.457381\pi\)
0.133493 + 0.991050i \(0.457381\pi\)
\(398\) 28.5113 1.42914
\(399\) 0 0
\(400\) 13.8371 0.691855
\(401\) 2.00000 0.0998752 0.0499376 0.998752i \(-0.484098\pi\)
0.0499376 + 0.998752i \(0.484098\pi\)
\(402\) −23.5174 −1.17294
\(403\) 20.3135 1.01189
\(404\) 83.1748 4.13810
\(405\) −1.00000 −0.0496904
\(406\) 0 0
\(407\) −2.00000 −0.0991363
\(408\) −70.2122 −3.47602
\(409\) −26.1978 −1.29540 −0.647699 0.761897i \(-0.724268\pi\)
−0.647699 + 0.761897i \(0.724268\pi\)
\(410\) 25.5174 1.26022
\(411\) 15.3607 0.757687
\(412\) −36.5113 −1.79878
\(413\) 0 0
\(414\) 5.84324 0.287180
\(415\) −11.0205 −0.540976
\(416\) −84.1543 −4.12600
\(417\) 8.58145 0.420235
\(418\) 14.2557 0.697267
\(419\) 2.83710 0.138601 0.0693007 0.997596i \(-0.477923\pi\)
0.0693007 + 0.997596i \(0.477923\pi\)
\(420\) 0 0
\(421\) 11.4764 0.559326 0.279663 0.960098i \(-0.409777\pi\)
0.279663 + 0.960098i \(0.409777\pi\)
\(422\) −25.9421 −1.26284
\(423\) −4.68035 −0.227566
\(424\) −1.41855 −0.0688909
\(425\) −7.75872 −0.376353
\(426\) −12.6803 −0.614365
\(427\) 0 0
\(428\) 33.8576 1.63657
\(429\) −4.34017 −0.209546
\(430\) 20.5958 0.993219
\(431\) 23.5708 1.13536 0.567682 0.823248i \(-0.307840\pi\)
0.567682 + 0.823248i \(0.307840\pi\)
\(432\) −13.8371 −0.665738
\(433\) 14.9939 0.720559 0.360279 0.932844i \(-0.382681\pi\)
0.360279 + 0.932844i \(0.382681\pi\)
\(434\) 0 0
\(435\) 1.41855 0.0680143
\(436\) 12.3545 0.591676
\(437\) 11.3484 0.542868
\(438\) 28.4391 1.35887
\(439\) −4.77924 −0.228101 −0.114050 0.993475i \(-0.536383\pi\)
−0.114050 + 0.993475i \(0.536383\pi\)
\(440\) 9.04945 0.431416
\(441\) 0 0
\(442\) 91.2327 4.33950
\(443\) 20.1978 0.959626 0.479813 0.877371i \(-0.340704\pi\)
0.479813 + 0.877371i \(0.340704\pi\)
\(444\) 10.6803 0.506867
\(445\) −12.8371 −0.608537
\(446\) −5.84324 −0.276686
\(447\) 18.0989 0.856048
\(448\) 0 0
\(449\) −21.5708 −1.01799 −0.508994 0.860770i \(-0.669982\pi\)
−0.508994 + 0.860770i \(0.669982\pi\)
\(450\) −2.70928 −0.127716
\(451\) 9.41855 0.443502
\(452\) −32.0410 −1.50708
\(453\) −22.9360 −1.07763
\(454\) 26.1711 1.22827
\(455\) 0 0
\(456\) −47.6163 −2.22984
\(457\) 28.1711 1.31779 0.658895 0.752235i \(-0.271024\pi\)
0.658895 + 0.752235i \(0.271024\pi\)
\(458\) −9.10504 −0.425451
\(459\) 7.75872 0.362146
\(460\) 11.5174 0.537004
\(461\) 1.47187 0.0685520 0.0342760 0.999412i \(-0.489087\pi\)
0.0342760 + 0.999412i \(0.489087\pi\)
\(462\) 0 0
\(463\) −23.2039 −1.07838 −0.539189 0.842185i \(-0.681269\pi\)
−0.539189 + 0.842185i \(0.681269\pi\)
\(464\) 19.6286 0.911236
\(465\) 4.68035 0.217046
\(466\) 6.49693 0.300964
\(467\) −14.1568 −0.655097 −0.327548 0.944834i \(-0.606222\pi\)
−0.327548 + 0.944834i \(0.606222\pi\)
\(468\) 23.1773 1.07137
\(469\) 0 0
\(470\) −12.6803 −0.584901
\(471\) −10.9939 −0.506570
\(472\) 55.7152 2.56450
\(473\) 7.60197 0.349539
\(474\) −21.9421 −1.00784
\(475\) −5.26180 −0.241428
\(476\) 0 0
\(477\) 0.156755 0.00717734
\(478\) 19.5174 0.892707
\(479\) 13.8432 0.632514 0.316257 0.948674i \(-0.397574\pi\)
0.316257 + 0.948674i \(0.397574\pi\)
\(480\) −19.3896 −0.885011
\(481\) −8.68035 −0.395790
\(482\) −14.0989 −0.642187
\(483\) 0 0
\(484\) 5.34017 0.242735
\(485\) 14.6803 0.666600
\(486\) 2.70928 0.122895
\(487\) −40.9939 −1.85761 −0.928804 0.370570i \(-0.879162\pi\)
−0.928804 + 0.370570i \(0.879162\pi\)
\(488\) −37.6163 −1.70281
\(489\) 6.52359 0.295007
\(490\) 0 0
\(491\) 34.8371 1.57218 0.786088 0.618114i \(-0.212103\pi\)
0.786088 + 0.618114i \(0.212103\pi\)
\(492\) −50.2967 −2.26755
\(493\) −11.0061 −0.495692
\(494\) 61.8720 2.78375
\(495\) −1.00000 −0.0449467
\(496\) 64.7624 2.90792
\(497\) 0 0
\(498\) 29.8576 1.33795
\(499\) 15.1506 0.678235 0.339117 0.940744i \(-0.389872\pi\)
0.339117 + 0.940744i \(0.389872\pi\)
\(500\) −5.34017 −0.238820
\(501\) 1.97334 0.0881622
\(502\) 41.5052 1.85247
\(503\) 6.65368 0.296673 0.148337 0.988937i \(-0.452608\pi\)
0.148337 + 0.988937i \(0.452608\pi\)
\(504\) 0 0
\(505\) −15.5753 −0.693092
\(506\) 5.84324 0.259764
\(507\) −5.83710 −0.259235
\(508\) −11.9688 −0.531030
\(509\) −41.3484 −1.83274 −0.916368 0.400337i \(-0.868893\pi\)
−0.916368 + 0.400337i \(0.868893\pi\)
\(510\) 21.0205 0.930804
\(511\) 0 0
\(512\) −17.8599 −0.789303
\(513\) 5.26180 0.232314
\(514\) 11.2618 0.496736
\(515\) 6.83710 0.301279
\(516\) −40.5958 −1.78713
\(517\) −4.68035 −0.205841
\(518\) 0 0
\(519\) 3.75872 0.164990
\(520\) 39.2762 1.72237
\(521\) −7.67420 −0.336213 −0.168106 0.985769i \(-0.553765\pi\)
−0.168106 + 0.985769i \(0.553765\pi\)
\(522\) −3.84324 −0.168214
\(523\) −23.2351 −1.01600 −0.508001 0.861357i \(-0.669615\pi\)
−0.508001 + 0.861357i \(0.669615\pi\)
\(524\) −46.3545 −2.02501
\(525\) 0 0
\(526\) 50.6824 2.20986
\(527\) −36.3135 −1.58184
\(528\) −13.8371 −0.602183
\(529\) −18.3484 −0.797757
\(530\) 0.424694 0.0184475
\(531\) −6.15676 −0.267180
\(532\) 0 0
\(533\) 40.8781 1.77063
\(534\) 34.7792 1.50505
\(535\) −6.34017 −0.274110
\(536\) 78.5523 3.39294
\(537\) −15.1506 −0.653797
\(538\) 63.2905 2.72865
\(539\) 0 0
\(540\) 5.34017 0.229804
\(541\) −18.0000 −0.773880 −0.386940 0.922105i \(-0.626468\pi\)
−0.386940 + 0.922105i \(0.626468\pi\)
\(542\) −15.1050 −0.648817
\(543\) 4.83710 0.207580
\(544\) 150.439 6.45001
\(545\) −2.31351 −0.0990999
\(546\) 0 0
\(547\) −23.0661 −0.986235 −0.493117 0.869963i \(-0.664143\pi\)
−0.493117 + 0.869963i \(0.664143\pi\)
\(548\) −82.0288 −3.50409
\(549\) 4.15676 0.177406
\(550\) −2.70928 −0.115524
\(551\) −7.46412 −0.317982
\(552\) −19.5174 −0.830718
\(553\) 0 0
\(554\) 70.4801 2.99441
\(555\) −2.00000 −0.0848953
\(556\) −45.8264 −1.94347
\(557\) 10.5958 0.448960 0.224480 0.974479i \(-0.427932\pi\)
0.224480 + 0.974479i \(0.427932\pi\)
\(558\) −12.6803 −0.536802
\(559\) 32.9939 1.39549
\(560\) 0 0
\(561\) 7.75872 0.327574
\(562\) 25.5174 1.07639
\(563\) 36.2122 1.52616 0.763080 0.646303i \(-0.223686\pi\)
0.763080 + 0.646303i \(0.223686\pi\)
\(564\) 24.9939 1.05243
\(565\) 6.00000 0.252422
\(566\) 38.5835 1.62179
\(567\) 0 0
\(568\) 42.3545 1.77716
\(569\) −27.5753 −1.15602 −0.578008 0.816031i \(-0.696170\pi\)
−0.578008 + 0.816031i \(0.696170\pi\)
\(570\) 14.2557 0.597104
\(571\) −27.9299 −1.16883 −0.584414 0.811456i \(-0.698676\pi\)
−0.584414 + 0.811456i \(0.698676\pi\)
\(572\) 23.1773 0.969091
\(573\) −2.52359 −0.105425
\(574\) 0 0
\(575\) −2.15676 −0.0899429
\(576\) 24.8576 1.03573
\(577\) −41.4017 −1.72358 −0.861788 0.507268i \(-0.830655\pi\)
−0.861788 + 0.507268i \(0.830655\pi\)
\(578\) −117.035 −4.86800
\(579\) −0.0266620 −0.00110803
\(580\) −7.57531 −0.314547
\(581\) 0 0
\(582\) −39.7731 −1.64865
\(583\) 0.156755 0.00649215
\(584\) −94.9914 −3.93077
\(585\) −4.34017 −0.179444
\(586\) −42.6947 −1.76370
\(587\) −8.48255 −0.350112 −0.175056 0.984558i \(-0.556011\pi\)
−0.175056 + 0.984558i \(0.556011\pi\)
\(588\) 0 0
\(589\) −24.6270 −1.01474
\(590\) −16.6803 −0.686719
\(591\) −21.1194 −0.868737
\(592\) −27.6742 −1.13740
\(593\) −7.56093 −0.310490 −0.155245 0.987876i \(-0.549617\pi\)
−0.155245 + 0.987876i \(0.549617\pi\)
\(594\) 2.70928 0.111163
\(595\) 0 0
\(596\) −96.6512 −3.95899
\(597\) 10.5236 0.430702
\(598\) 25.3607 1.03708
\(599\) 5.67420 0.231842 0.115921 0.993258i \(-0.463018\pi\)
0.115921 + 0.993258i \(0.463018\pi\)
\(600\) 9.04945 0.369442
\(601\) 1.31965 0.0538298 0.0269149 0.999638i \(-0.491432\pi\)
0.0269149 + 0.999638i \(0.491432\pi\)
\(602\) 0 0
\(603\) −8.68035 −0.353491
\(604\) 122.482 4.98373
\(605\) −1.00000 −0.0406558
\(606\) 42.1978 1.71417
\(607\) 2.24128 0.0909706 0.0454853 0.998965i \(-0.485517\pi\)
0.0454853 + 0.998965i \(0.485517\pi\)
\(608\) 102.024 4.13763
\(609\) 0 0
\(610\) 11.2618 0.455977
\(611\) −20.3135 −0.821797
\(612\) −41.4329 −1.67483
\(613\) 42.8638 1.73125 0.865626 0.500692i \(-0.166921\pi\)
0.865626 + 0.500692i \(0.166921\pi\)
\(614\) −51.2639 −2.06884
\(615\) 9.41855 0.379793
\(616\) 0 0
\(617\) 11.3607 0.457364 0.228682 0.973501i \(-0.426558\pi\)
0.228682 + 0.973501i \(0.426558\pi\)
\(618\) −18.5236 −0.745128
\(619\) 45.1917 1.81641 0.908203 0.418530i \(-0.137455\pi\)
0.908203 + 0.418530i \(0.137455\pi\)
\(620\) −24.9939 −1.00378
\(621\) 2.15676 0.0865476
\(622\) −56.5646 −2.26803
\(623\) 0 0
\(624\) −60.0554 −2.40414
\(625\) 1.00000 0.0400000
\(626\) 17.1050 0.683655
\(627\) 5.26180 0.210136
\(628\) 58.7091 2.34275
\(629\) 15.5174 0.618721
\(630\) 0 0
\(631\) −9.78992 −0.389731 −0.194865 0.980830i \(-0.562427\pi\)
−0.194865 + 0.980830i \(0.562427\pi\)
\(632\) 73.2905 2.91534
\(633\) −9.57531 −0.380584
\(634\) −84.9647 −3.37438
\(635\) 2.24128 0.0889423
\(636\) −0.837101 −0.0331932
\(637\) 0 0
\(638\) −3.84324 −0.152156
\(639\) −4.68035 −0.185152
\(640\) 28.5669 1.12921
\(641\) 0.210079 0.00829764 0.00414882 0.999991i \(-0.498679\pi\)
0.00414882 + 0.999991i \(0.498679\pi\)
\(642\) 17.1773 0.677933
\(643\) −14.5236 −0.572754 −0.286377 0.958117i \(-0.592451\pi\)
−0.286377 + 0.958117i \(0.592451\pi\)
\(644\) 0 0
\(645\) 7.60197 0.299327
\(646\) −110.606 −4.35172
\(647\) −15.4641 −0.607957 −0.303979 0.952679i \(-0.598315\pi\)
−0.303979 + 0.952679i \(0.598315\pi\)
\(648\) −9.04945 −0.355496
\(649\) −6.15676 −0.241674
\(650\) −11.7587 −0.461215
\(651\) 0 0
\(652\) −34.8371 −1.36433
\(653\) −17.8310 −0.697779 −0.348890 0.937164i \(-0.613441\pi\)
−0.348890 + 0.937164i \(0.613441\pi\)
\(654\) 6.26794 0.245096
\(655\) 8.68035 0.339169
\(656\) 130.325 5.08835
\(657\) 10.4969 0.409524
\(658\) 0 0
\(659\) −32.3135 −1.25876 −0.629378 0.777099i \(-0.716690\pi\)
−0.629378 + 0.777099i \(0.716690\pi\)
\(660\) 5.34017 0.207866
\(661\) 5.68649 0.221179 0.110589 0.993866i \(-0.464726\pi\)
0.110589 + 0.993866i \(0.464726\pi\)
\(662\) −52.0288 −2.02215
\(663\) 33.6742 1.30780
\(664\) −99.7296 −3.87026
\(665\) 0 0
\(666\) 5.41855 0.209965
\(667\) −3.05947 −0.118463
\(668\) −10.5380 −0.407726
\(669\) −2.15676 −0.0833850
\(670\) −23.5174 −0.908558
\(671\) 4.15676 0.160470
\(672\) 0 0
\(673\) −21.0205 −0.810281 −0.405141 0.914254i \(-0.632777\pi\)
−0.405141 + 0.914254i \(0.632777\pi\)
\(674\) −36.5835 −1.40915
\(675\) −1.00000 −0.0384900
\(676\) 31.1711 1.19889
\(677\) −36.7526 −1.41252 −0.706258 0.707954i \(-0.749618\pi\)
−0.706258 + 0.707954i \(0.749618\pi\)
\(678\) −16.2557 −0.624295
\(679\) 0 0
\(680\) −70.2122 −2.69251
\(681\) 9.65983 0.370165
\(682\) −12.6803 −0.485556
\(683\) −17.3074 −0.662248 −0.331124 0.943587i \(-0.607428\pi\)
−0.331124 + 0.943587i \(0.607428\pi\)
\(684\) −28.0989 −1.07439
\(685\) 15.3607 0.586902
\(686\) 0 0
\(687\) −3.36069 −0.128218
\(688\) 105.189 4.01030
\(689\) 0.680346 0.0259191
\(690\) 5.84324 0.222449
\(691\) −17.6742 −0.672358 −0.336179 0.941798i \(-0.609135\pi\)
−0.336179 + 0.941798i \(0.609135\pi\)
\(692\) −20.0722 −0.763032
\(693\) 0 0
\(694\) −17.1773 −0.652040
\(695\) 8.58145 0.325513
\(696\) 12.8371 0.486589
\(697\) −73.0759 −2.76795
\(698\) 43.7731 1.65684
\(699\) 2.39803 0.0907019
\(700\) 0 0
\(701\) −17.1050 −0.646048 −0.323024 0.946391i \(-0.604700\pi\)
−0.323024 + 0.946391i \(0.604700\pi\)
\(702\) 11.7587 0.443804
\(703\) 10.5236 0.396905
\(704\) 24.8576 0.936857
\(705\) −4.68035 −0.176272
\(706\) −35.7731 −1.34634
\(707\) 0 0
\(708\) 32.8781 1.23564
\(709\) 25.1506 0.944551 0.472276 0.881451i \(-0.343433\pi\)
0.472276 + 0.881451i \(0.343433\pi\)
\(710\) −12.6803 −0.475885
\(711\) −8.09890 −0.303732
\(712\) −116.169 −4.35361
\(713\) −10.0944 −0.378037
\(714\) 0 0
\(715\) −4.34017 −0.162313
\(716\) 80.9069 3.02363
\(717\) 7.20394 0.269036
\(718\) −8.99386 −0.335648
\(719\) 1.78992 0.0667528 0.0333764 0.999443i \(-0.489374\pi\)
0.0333764 + 0.999443i \(0.489374\pi\)
\(720\) −13.8371 −0.515678
\(721\) 0 0
\(722\) −23.5341 −0.875848
\(723\) −5.20394 −0.193536
\(724\) −25.8310 −0.960000
\(725\) 1.41855 0.0526837
\(726\) 2.70928 0.100551
\(727\) −25.9877 −0.963831 −0.481915 0.876218i \(-0.660059\pi\)
−0.481915 + 0.876218i \(0.660059\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 28.4391 1.05258
\(731\) −58.9816 −2.18151
\(732\) −22.1978 −0.820454
\(733\) 41.0205 1.51513 0.757564 0.652761i \(-0.226390\pi\)
0.757564 + 0.652761i \(0.226390\pi\)
\(734\) −97.9253 −3.61449
\(735\) 0 0
\(736\) 41.8187 1.54146
\(737\) −8.68035 −0.319745
\(738\) −25.5174 −0.939310
\(739\) −47.6163 −1.75160 −0.875798 0.482678i \(-0.839664\pi\)
−0.875798 + 0.482678i \(0.839664\pi\)
\(740\) 10.6803 0.392617
\(741\) 22.8371 0.838942
\(742\) 0 0
\(743\) 0.550252 0.0201868 0.0100934 0.999949i \(-0.496787\pi\)
0.0100934 + 0.999949i \(0.496787\pi\)
\(744\) 42.3545 1.55279
\(745\) 18.0989 0.663092
\(746\) 7.61425 0.278778
\(747\) 11.0205 0.403220
\(748\) −41.4329 −1.51494
\(749\) 0 0
\(750\) −2.70928 −0.0989287
\(751\) 41.5585 1.51649 0.758245 0.651969i \(-0.226057\pi\)
0.758245 + 0.651969i \(0.226057\pi\)
\(752\) −64.7624 −2.36164
\(753\) 15.3197 0.558279
\(754\) −16.6803 −0.607462
\(755\) −22.9360 −0.834726
\(756\) 0 0
\(757\) 1.31965 0.0479636 0.0239818 0.999712i \(-0.492366\pi\)
0.0239818 + 0.999712i \(0.492366\pi\)
\(758\) 54.1855 1.96811
\(759\) 2.15676 0.0782853
\(760\) −47.6163 −1.72723
\(761\) 2.21461 0.0802797 0.0401399 0.999194i \(-0.487220\pi\)
0.0401399 + 0.999194i \(0.487220\pi\)
\(762\) −6.07223 −0.219974
\(763\) 0 0
\(764\) 13.4764 0.487559
\(765\) 7.75872 0.280517
\(766\) 90.9192 3.28504
\(767\) −26.7214 −0.964853
\(768\) −27.6803 −0.998828
\(769\) 14.3668 0.518081 0.259041 0.965866i \(-0.416594\pi\)
0.259041 + 0.965866i \(0.416594\pi\)
\(770\) 0 0
\(771\) 4.15676 0.149702
\(772\) 0.142380 0.00512436
\(773\) −40.1568 −1.44434 −0.722169 0.691717i \(-0.756855\pi\)
−0.722169 + 0.691717i \(0.756855\pi\)
\(774\) −20.5958 −0.740302
\(775\) 4.68035 0.168123
\(776\) 132.849 4.76900
\(777\) 0 0
\(778\) −34.7792 −1.24690
\(779\) −49.5585 −1.77562
\(780\) 23.1773 0.829880
\(781\) −4.68035 −0.167476
\(782\) −45.3361 −1.62122
\(783\) −1.41855 −0.0506949
\(784\) 0 0
\(785\) −10.9939 −0.392388
\(786\) −23.5174 −0.838840
\(787\) −49.5897 −1.76768 −0.883841 0.467788i \(-0.845051\pi\)
−0.883841 + 0.467788i \(0.845051\pi\)
\(788\) 112.781 4.01767
\(789\) 18.7070 0.665987
\(790\) −21.9421 −0.780666
\(791\) 0 0
\(792\) −9.04945 −0.321558
\(793\) 18.0410 0.640656
\(794\) −14.4124 −0.511477
\(795\) 0.156755 0.00555954
\(796\) −56.1978 −1.99188
\(797\) 46.7091 1.65452 0.827261 0.561818i \(-0.189898\pi\)
0.827261 + 0.561818i \(0.189898\pi\)
\(798\) 0 0
\(799\) 36.3135 1.28468
\(800\) −19.3896 −0.685527
\(801\) 12.8371 0.453577
\(802\) −5.41855 −0.191336
\(803\) 10.4969 0.370429
\(804\) 46.3545 1.63480
\(805\) 0 0
\(806\) −55.0349 −1.93852
\(807\) 23.3607 0.822335
\(808\) −140.948 −4.95853
\(809\) −18.5814 −0.653289 −0.326644 0.945147i \(-0.605918\pi\)
−0.326644 + 0.945147i \(0.605918\pi\)
\(810\) 2.70928 0.0951942
\(811\) −27.3028 −0.958732 −0.479366 0.877615i \(-0.659133\pi\)
−0.479366 + 0.877615i \(0.659133\pi\)
\(812\) 0 0
\(813\) −5.57531 −0.195535
\(814\) 5.41855 0.189920
\(815\) 6.52359 0.228511
\(816\) 107.358 3.75829
\(817\) −40.0000 −1.39942
\(818\) 70.9770 2.48165
\(819\) 0 0
\(820\) −50.2967 −1.75644
\(821\) −31.2085 −1.08918 −0.544592 0.838701i \(-0.683315\pi\)
−0.544592 + 0.838701i \(0.683315\pi\)
\(822\) −41.6163 −1.45154
\(823\) −50.1855 −1.74936 −0.874678 0.484704i \(-0.838927\pi\)
−0.874678 + 0.484704i \(0.838927\pi\)
\(824\) 61.8720 2.15541
\(825\) −1.00000 −0.0348155
\(826\) 0 0
\(827\) 27.3874 0.952352 0.476176 0.879350i \(-0.342023\pi\)
0.476176 + 0.879350i \(0.342023\pi\)
\(828\) −11.5174 −0.400259
\(829\) 26.1978 0.909887 0.454943 0.890520i \(-0.349659\pi\)
0.454943 + 0.890520i \(0.349659\pi\)
\(830\) 29.8576 1.03637
\(831\) 26.0144 0.902429
\(832\) 107.886 3.74029
\(833\) 0 0
\(834\) −23.2495 −0.805065
\(835\) 1.97334 0.0682902
\(836\) −28.0989 −0.971821
\(837\) −4.68035 −0.161776
\(838\) −7.68649 −0.265525
\(839\) −7.20394 −0.248708 −0.124354 0.992238i \(-0.539686\pi\)
−0.124354 + 0.992238i \(0.539686\pi\)
\(840\) 0 0
\(841\) −26.9877 −0.930611
\(842\) −31.0928 −1.07153
\(843\) 9.41855 0.324392
\(844\) 51.1338 1.76010
\(845\) −5.83710 −0.200802
\(846\) 12.6803 0.435959
\(847\) 0 0
\(848\) 2.16904 0.0744852
\(849\) 14.2413 0.488759
\(850\) 21.0205 0.720998
\(851\) 4.31351 0.147865
\(852\) 24.9939 0.856275
\(853\) −39.8043 −1.36287 −0.681437 0.731877i \(-0.738644\pi\)
−0.681437 + 0.731877i \(0.738644\pi\)
\(854\) 0 0
\(855\) 5.26180 0.179950
\(856\) −57.3751 −1.96104
\(857\) 36.9504 1.26220 0.631100 0.775701i \(-0.282604\pi\)
0.631100 + 0.775701i \(0.282604\pi\)
\(858\) 11.7587 0.401436
\(859\) −57.5052 −1.96205 −0.981025 0.193879i \(-0.937893\pi\)
−0.981025 + 0.193879i \(0.937893\pi\)
\(860\) −40.5958 −1.38431
\(861\) 0 0
\(862\) −63.8597 −2.17507
\(863\) −1.89657 −0.0645599 −0.0322800 0.999479i \(-0.510277\pi\)
−0.0322800 + 0.999479i \(0.510277\pi\)
\(864\) 19.3896 0.659648
\(865\) 3.75872 0.127800
\(866\) −40.6225 −1.38041
\(867\) −43.1978 −1.46707
\(868\) 0 0
\(869\) −8.09890 −0.274736
\(870\) −3.84324 −0.130298
\(871\) −37.6742 −1.27654
\(872\) −20.9360 −0.708982
\(873\) −14.6803 −0.496854
\(874\) −30.7460 −1.04000
\(875\) 0 0
\(876\) −56.0554 −1.89394
\(877\) 32.5380 1.09873 0.549365 0.835583i \(-0.314870\pi\)
0.549365 + 0.835583i \(0.314870\pi\)
\(878\) 12.9483 0.436983
\(879\) −15.7587 −0.531529
\(880\) −13.8371 −0.466449
\(881\) −18.1978 −0.613099 −0.306550 0.951855i \(-0.599175\pi\)
−0.306550 + 0.951855i \(0.599175\pi\)
\(882\) 0 0
\(883\) 36.3956 1.22481 0.612405 0.790545i \(-0.290202\pi\)
0.612405 + 0.790545i \(0.290202\pi\)
\(884\) −179.826 −6.04821
\(885\) −6.15676 −0.206957
\(886\) −54.7214 −1.83840
\(887\) −27.8699 −0.935780 −0.467890 0.883787i \(-0.654986\pi\)
−0.467890 + 0.883787i \(0.654986\pi\)
\(888\) −18.0989 −0.607359
\(889\) 0 0
\(890\) 34.7792 1.16580
\(891\) 1.00000 0.0335013
\(892\) 11.5174 0.385633
\(893\) 24.6270 0.824112
\(894\) −49.0349 −1.63997
\(895\) −15.1506 −0.506429
\(896\) 0 0
\(897\) 9.36069 0.312544
\(898\) 58.4412 1.95021
\(899\) 6.63931 0.221433
\(900\) 5.34017 0.178006
\(901\) −1.21622 −0.0405182
\(902\) −25.5174 −0.849638
\(903\) 0 0
\(904\) 54.2967 1.80588
\(905\) 4.83710 0.160791
\(906\) 62.1399 2.06446
\(907\) −27.9376 −0.927653 −0.463826 0.885926i \(-0.653524\pi\)
−0.463826 + 0.885926i \(0.653524\pi\)
\(908\) −51.5851 −1.71191
\(909\) 15.5753 0.516600
\(910\) 0 0
\(911\) −11.8843 −0.393744 −0.196872 0.980429i \(-0.563078\pi\)
−0.196872 + 0.980429i \(0.563078\pi\)
\(912\) 72.8080 2.41091
\(913\) 11.0205 0.364726
\(914\) −76.3234 −2.52455
\(915\) 4.15676 0.137418
\(916\) 17.9467 0.592975
\(917\) 0 0
\(918\) −21.0205 −0.693781
\(919\) −45.6041 −1.50434 −0.752170 0.658970i \(-0.770993\pi\)
−0.752170 + 0.658970i \(0.770993\pi\)
\(920\) −19.5174 −0.643471
\(921\) −18.9216 −0.623489
\(922\) −3.98771 −0.131328
\(923\) −20.3135 −0.668627
\(924\) 0 0
\(925\) −2.00000 −0.0657596
\(926\) 62.8659 2.06590
\(927\) −6.83710 −0.224560
\(928\) −27.5052 −0.902901
\(929\) 25.1506 0.825165 0.412582 0.910920i \(-0.364627\pi\)
0.412582 + 0.910920i \(0.364627\pi\)
\(930\) −12.6803 −0.415805
\(931\) 0 0
\(932\) −12.8059 −0.419471
\(933\) −20.8781 −0.683520
\(934\) 38.3545 1.25500
\(935\) 7.75872 0.253737
\(936\) −39.2762 −1.28378
\(937\) −5.33403 −0.174255 −0.0871276 0.996197i \(-0.527769\pi\)
−0.0871276 + 0.996197i \(0.527769\pi\)
\(938\) 0 0
\(939\) 6.31351 0.206034
\(940\) 24.9939 0.815210
\(941\) −56.8203 −1.85229 −0.926144 0.377170i \(-0.876897\pi\)
−0.926144 + 0.377170i \(0.876897\pi\)
\(942\) 29.7854 0.970460
\(943\) −20.3135 −0.661499
\(944\) −85.1917 −2.77275
\(945\) 0 0
\(946\) −20.5958 −0.669628
\(947\) −20.9939 −0.682209 −0.341104 0.940025i \(-0.610801\pi\)
−0.341104 + 0.940025i \(0.610801\pi\)
\(948\) 43.2495 1.40468
\(949\) 45.5585 1.47889
\(950\) 14.2557 0.462514
\(951\) −31.3607 −1.01694
\(952\) 0 0
\(953\) −25.2351 −0.817446 −0.408723 0.912658i \(-0.634026\pi\)
−0.408723 + 0.912658i \(0.634026\pi\)
\(954\) −0.424694 −0.0137500
\(955\) −2.52359 −0.0816615
\(956\) −38.4703 −1.24422
\(957\) −1.41855 −0.0458552
\(958\) −37.5052 −1.21174
\(959\) 0 0
\(960\) 24.8576 0.802276
\(961\) −9.09436 −0.293367
\(962\) 23.5174 0.758233
\(963\) 6.34017 0.204309
\(964\) 27.7899 0.895053
\(965\) −0.0266620 −0.000858280 0
\(966\) 0 0
\(967\) 13.1317 0.422287 0.211144 0.977455i \(-0.432281\pi\)
0.211144 + 0.977455i \(0.432281\pi\)
\(968\) −9.04945 −0.290860
\(969\) −40.8248 −1.31148
\(970\) −39.7731 −1.27704
\(971\) −8.94053 −0.286915 −0.143458 0.989656i \(-0.545822\pi\)
−0.143458 + 0.989656i \(0.545822\pi\)
\(972\) −5.34017 −0.171286
\(973\) 0 0
\(974\) 111.064 3.55871
\(975\) −4.34017 −0.138997
\(976\) 57.5174 1.84109
\(977\) 50.3956 1.61230 0.806149 0.591713i \(-0.201548\pi\)
0.806149 + 0.591713i \(0.201548\pi\)
\(978\) −17.6742 −0.565159
\(979\) 12.8371 0.410276
\(980\) 0 0
\(981\) 2.31351 0.0738647
\(982\) −94.3833 −3.01189
\(983\) 32.1978 1.02695 0.513475 0.858105i \(-0.328358\pi\)
0.513475 + 0.858105i \(0.328358\pi\)
\(984\) 85.2327 2.71712
\(985\) −21.1194 −0.672921
\(986\) 29.8187 0.949620
\(987\) 0 0
\(988\) −121.954 −3.87988
\(989\) −16.3956 −0.521349
\(990\) 2.70928 0.0861064
\(991\) −46.7747 −1.48585 −0.742924 0.669376i \(-0.766562\pi\)
−0.742924 + 0.669376i \(0.766562\pi\)
\(992\) −90.7501 −2.88132
\(993\) −19.2039 −0.609419
\(994\) 0 0
\(995\) 10.5236 0.333620
\(996\) −58.8515 −1.86478
\(997\) −38.2122 −1.21019 −0.605096 0.796153i \(-0.706865\pi\)
−0.605096 + 0.796153i \(0.706865\pi\)
\(998\) −41.0472 −1.29933
\(999\) 2.00000 0.0632772
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 8085.2.a.bk.1.1 3
7.6 odd 2 165.2.a.c.1.1 3
21.20 even 2 495.2.a.e.1.3 3
28.27 even 2 2640.2.a.be.1.2 3
35.13 even 4 825.2.c.g.199.6 6
35.27 even 4 825.2.c.g.199.1 6
35.34 odd 2 825.2.a.k.1.3 3
77.76 even 2 1815.2.a.m.1.3 3
84.83 odd 2 7920.2.a.cj.1.2 3
105.62 odd 4 2475.2.c.r.199.6 6
105.83 odd 4 2475.2.c.r.199.1 6
105.104 even 2 2475.2.a.bb.1.1 3
231.230 odd 2 5445.2.a.z.1.1 3
385.384 even 2 9075.2.a.cf.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
165.2.a.c.1.1 3 7.6 odd 2
495.2.a.e.1.3 3 21.20 even 2
825.2.a.k.1.3 3 35.34 odd 2
825.2.c.g.199.1 6 35.27 even 4
825.2.c.g.199.6 6 35.13 even 4
1815.2.a.m.1.3 3 77.76 even 2
2475.2.a.bb.1.1 3 105.104 even 2
2475.2.c.r.199.1 6 105.83 odd 4
2475.2.c.r.199.6 6 105.62 odd 4
2640.2.a.be.1.2 3 28.27 even 2
5445.2.a.z.1.1 3 231.230 odd 2
7920.2.a.cj.1.2 3 84.83 odd 2
8085.2.a.bk.1.1 3 1.1 even 1 trivial
9075.2.a.cf.1.1 3 385.384 even 2