Properties

Label 370.2.a.g.1.2
Level $370$
Weight $2$
Character 370.1
Self dual yes
Analytic conductor $2.954$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 370 = 2 \cdot 5 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 370.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(2.95446487479\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.892.1
Defining polynomial: \( x^{3} - x^{2} - 8x + 10 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-2.91729\) of defining polynomial
Character \(\chi\) \(=\) 370.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} +0.406728 q^{3} +1.00000 q^{4} -1.00000 q^{5} +0.406728 q^{6} +2.91729 q^{7} +1.00000 q^{8} -2.83457 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} +0.406728 q^{3} +1.00000 q^{4} -1.00000 q^{5} +0.406728 q^{6} +2.91729 q^{7} +1.00000 q^{8} -2.83457 q^{9} -1.00000 q^{10} +6.51056 q^{11} +0.406728 q^{12} -0.813457 q^{13} +2.91729 q^{14} -0.406728 q^{15} +1.00000 q^{16} -2.51056 q^{17} -2.83457 q^{18} +0.406728 q^{19} -1.00000 q^{20} +1.18654 q^{21} +6.51056 q^{22} +5.02112 q^{23} +0.406728 q^{24} +1.00000 q^{25} -0.813457 q^{26} -2.37309 q^{27} +2.91729 q^{28} -5.32401 q^{29} -0.406728 q^{30} -8.75186 q^{31} +1.00000 q^{32} +2.64803 q^{33} -2.51056 q^{34} -2.91729 q^{35} -2.83457 q^{36} -1.00000 q^{37} +0.406728 q^{38} -0.330856 q^{39} -1.00000 q^{40} +6.34513 q^{41} +1.18654 q^{42} -7.32401 q^{43} +6.51056 q^{44} +2.83457 q^{45} +5.02112 q^{46} -5.42784 q^{47} +0.406728 q^{48} +1.51056 q^{49} +1.00000 q^{50} -1.02112 q^{51} -0.813457 q^{52} -2.34513 q^{53} -2.37309 q^{54} -6.51056 q^{55} +2.91729 q^{56} +0.165428 q^{57} -5.32401 q^{58} -1.42784 q^{59} -0.406728 q^{60} -1.32401 q^{61} -8.75186 q^{62} -8.26926 q^{63} +1.00000 q^{64} +0.813457 q^{65} +2.64803 q^{66} -5.42784 q^{67} -2.51056 q^{68} +2.04223 q^{69} -2.91729 q^{70} +14.6480 q^{71} -2.83457 q^{72} -11.0211 q^{73} -1.00000 q^{74} +0.406728 q^{75} +0.406728 q^{76} +18.9932 q^{77} -0.330856 q^{78} -1.75870 q^{79} -1.00000 q^{80} +7.53851 q^{81} +6.34513 q^{82} -7.05476 q^{83} +1.18654 q^{84} +2.51056 q^{85} -7.32401 q^{86} -2.16543 q^{87} +6.51056 q^{88} +6.00000 q^{89} +2.83457 q^{90} -2.37309 q^{91} +5.02112 q^{92} -3.55963 q^{93} -5.42784 q^{94} -0.406728 q^{95} +0.406728 q^{96} +2.34513 q^{97} +1.51056 q^{98} -18.4546 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 3 q^{2} + 3 q^{4} - 3 q^{5} - q^{7} + 3 q^{8} + 11 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 3 q^{2} + 3 q^{4} - 3 q^{5} - q^{7} + 3 q^{8} + 11 q^{9} - 3 q^{10} + 11 q^{11} - q^{14} + 3 q^{16} + q^{17} + 11 q^{18} - 3 q^{20} + 6 q^{21} + 11 q^{22} - 2 q^{23} + 3 q^{25} - 12 q^{27} - q^{28} - 5 q^{29} + 3 q^{31} + 3 q^{32} - 14 q^{33} + q^{34} + q^{35} + 11 q^{36} - 3 q^{37} - 40 q^{39} - 3 q^{40} - 9 q^{41} + 6 q^{42} - 11 q^{43} + 11 q^{44} - 11 q^{45} - 2 q^{46} + 2 q^{47} - 4 q^{49} + 3 q^{50} + 14 q^{51} + 21 q^{53} - 12 q^{54} - 11 q^{55} - q^{56} + 20 q^{57} - 5 q^{58} + 14 q^{59} + 7 q^{61} + 3 q^{62} - 37 q^{63} + 3 q^{64} - 14 q^{66} + 2 q^{67} + q^{68} - 28 q^{69} + q^{70} + 22 q^{71} + 11 q^{72} - 16 q^{73} - 3 q^{74} + 7 q^{77} - 40 q^{78} - 26 q^{79} - 3 q^{80} + 47 q^{81} - 9 q^{82} + 2 q^{83} + 6 q^{84} - q^{85} - 11 q^{86} - 26 q^{87} + 11 q^{88} + 18 q^{89} - 11 q^{90} - 12 q^{91} - 2 q^{92} - 18 q^{93} + 2 q^{94} - 21 q^{97} - 4 q^{98} + 19 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) 0.406728 0.234825 0.117412 0.993083i \(-0.462540\pi\)
0.117412 + 0.993083i \(0.462540\pi\)
\(4\) 1.00000 0.500000
\(5\) −1.00000 −0.447214
\(6\) 0.406728 0.166046
\(7\) 2.91729 1.10263 0.551315 0.834297i \(-0.314126\pi\)
0.551315 + 0.834297i \(0.314126\pi\)
\(8\) 1.00000 0.353553
\(9\) −2.83457 −0.944857
\(10\) −1.00000 −0.316228
\(11\) 6.51056 1.96301 0.981503 0.191444i \(-0.0613171\pi\)
0.981503 + 0.191444i \(0.0613171\pi\)
\(12\) 0.406728 0.117412
\(13\) −0.813457 −0.225612 −0.112806 0.993617i \(-0.535984\pi\)
−0.112806 + 0.993617i \(0.535984\pi\)
\(14\) 2.91729 0.779677
\(15\) −0.406728 −0.105017
\(16\) 1.00000 0.250000
\(17\) −2.51056 −0.608900 −0.304450 0.952528i \(-0.598473\pi\)
−0.304450 + 0.952528i \(0.598473\pi\)
\(18\) −2.83457 −0.668115
\(19\) 0.406728 0.0933099 0.0466550 0.998911i \(-0.485144\pi\)
0.0466550 + 0.998911i \(0.485144\pi\)
\(20\) −1.00000 −0.223607
\(21\) 1.18654 0.258925
\(22\) 6.51056 1.38806
\(23\) 5.02112 1.04697 0.523487 0.852033i \(-0.324631\pi\)
0.523487 + 0.852033i \(0.324631\pi\)
\(24\) 0.406728 0.0830231
\(25\) 1.00000 0.200000
\(26\) −0.813457 −0.159532
\(27\) −2.37309 −0.456701
\(28\) 2.91729 0.551315
\(29\) −5.32401 −0.988645 −0.494322 0.869279i \(-0.664584\pi\)
−0.494322 + 0.869279i \(0.664584\pi\)
\(30\) −0.406728 −0.0742581
\(31\) −8.75186 −1.57188 −0.785940 0.618303i \(-0.787821\pi\)
−0.785940 + 0.618303i \(0.787821\pi\)
\(32\) 1.00000 0.176777
\(33\) 2.64803 0.460963
\(34\) −2.51056 −0.430557
\(35\) −2.91729 −0.493111
\(36\) −2.83457 −0.472429
\(37\) −1.00000 −0.164399
\(38\) 0.406728 0.0659801
\(39\) −0.330856 −0.0529794
\(40\) −1.00000 −0.158114
\(41\) 6.34513 0.990943 0.495471 0.868624i \(-0.334995\pi\)
0.495471 + 0.868624i \(0.334995\pi\)
\(42\) 1.18654 0.183088
\(43\) −7.32401 −1.11690 −0.558451 0.829538i \(-0.688604\pi\)
−0.558451 + 0.829538i \(0.688604\pi\)
\(44\) 6.51056 0.981503
\(45\) 2.83457 0.422553
\(46\) 5.02112 0.740323
\(47\) −5.42784 −0.791732 −0.395866 0.918308i \(-0.629556\pi\)
−0.395866 + 0.918308i \(0.629556\pi\)
\(48\) 0.406728 0.0587062
\(49\) 1.51056 0.215794
\(50\) 1.00000 0.141421
\(51\) −1.02112 −0.142985
\(52\) −0.813457 −0.112806
\(53\) −2.34513 −0.322128 −0.161064 0.986944i \(-0.551493\pi\)
−0.161064 + 0.986944i \(0.551493\pi\)
\(54\) −2.37309 −0.322936
\(55\) −6.51056 −0.877883
\(56\) 2.91729 0.389839
\(57\) 0.165428 0.0219115
\(58\) −5.32401 −0.699077
\(59\) −1.42784 −0.185889 −0.0929447 0.995671i \(-0.529628\pi\)
−0.0929447 + 0.995671i \(0.529628\pi\)
\(60\) −0.406728 −0.0525084
\(61\) −1.32401 −0.169523 −0.0847613 0.996401i \(-0.527013\pi\)
−0.0847613 + 0.996401i \(0.527013\pi\)
\(62\) −8.75186 −1.11149
\(63\) −8.26926 −1.04183
\(64\) 1.00000 0.125000
\(65\) 0.813457 0.100897
\(66\) 2.64803 0.325950
\(67\) −5.42784 −0.663117 −0.331558 0.943435i \(-0.607574\pi\)
−0.331558 + 0.943435i \(0.607574\pi\)
\(68\) −2.51056 −0.304450
\(69\) 2.04223 0.245856
\(70\) −2.91729 −0.348682
\(71\) 14.6480 1.73840 0.869201 0.494460i \(-0.164634\pi\)
0.869201 + 0.494460i \(0.164634\pi\)
\(72\) −2.83457 −0.334058
\(73\) −11.0211 −1.28992 −0.644962 0.764215i \(-0.723127\pi\)
−0.644962 + 0.764215i \(0.723127\pi\)
\(74\) −1.00000 −0.116248
\(75\) 0.406728 0.0469650
\(76\) 0.406728 0.0466550
\(77\) 18.9932 2.16447
\(78\) −0.330856 −0.0374621
\(79\) −1.75870 −0.197869 −0.0989346 0.995094i \(-0.531543\pi\)
−0.0989346 + 0.995094i \(0.531543\pi\)
\(80\) −1.00000 −0.111803
\(81\) 7.53851 0.837613
\(82\) 6.34513 0.700702
\(83\) −7.05476 −0.774360 −0.387180 0.922004i \(-0.626551\pi\)
−0.387180 + 0.922004i \(0.626551\pi\)
\(84\) 1.18654 0.129462
\(85\) 2.51056 0.272308
\(86\) −7.32401 −0.789769
\(87\) −2.16543 −0.232158
\(88\) 6.51056 0.694028
\(89\) 6.00000 0.635999 0.317999 0.948091i \(-0.396989\pi\)
0.317999 + 0.948091i \(0.396989\pi\)
\(90\) 2.83457 0.298790
\(91\) −2.37309 −0.248767
\(92\) 5.02112 0.523487
\(93\) −3.55963 −0.369116
\(94\) −5.42784 −0.559839
\(95\) −0.406728 −0.0417295
\(96\) 0.406728 0.0415115
\(97\) 2.34513 0.238112 0.119056 0.992888i \(-0.462013\pi\)
0.119056 + 0.992888i \(0.462013\pi\)
\(98\) 1.51056 0.152589
\(99\) −18.4546 −1.85476
\(100\) 1.00000 0.100000
\(101\) −8.20766 −0.816693 −0.408346 0.912827i \(-0.633894\pi\)
−0.408346 + 0.912827i \(0.633894\pi\)
\(102\) −1.02112 −0.101105
\(103\) −8.81346 −0.868416 −0.434208 0.900813i \(-0.642972\pi\)
−0.434208 + 0.900813i \(0.642972\pi\)
\(104\) −0.813457 −0.0797660
\(105\) −1.18654 −0.115795
\(106\) −2.34513 −0.227779
\(107\) 15.2624 1.47547 0.737737 0.675089i \(-0.235895\pi\)
0.737737 + 0.675089i \(0.235895\pi\)
\(108\) −2.37309 −0.228350
\(109\) 4.30290 0.412143 0.206072 0.978537i \(-0.433932\pi\)
0.206072 + 0.978537i \(0.433932\pi\)
\(110\) −6.51056 −0.620757
\(111\) −0.406728 −0.0386050
\(112\) 2.91729 0.275658
\(113\) −9.15859 −0.861567 −0.430784 0.902455i \(-0.641763\pi\)
−0.430784 + 0.902455i \(0.641763\pi\)
\(114\) 0.165428 0.0154938
\(115\) −5.02112 −0.468221
\(116\) −5.32401 −0.494322
\(117\) 2.30580 0.213171
\(118\) −1.42784 −0.131444
\(119\) −7.32401 −0.671391
\(120\) −0.406728 −0.0371291
\(121\) 31.3874 2.85340
\(122\) −1.32401 −0.119871
\(123\) 2.58074 0.232698
\(124\) −8.75186 −0.785940
\(125\) −1.00000 −0.0894427
\(126\) −8.26926 −0.736684
\(127\) −8.61439 −0.764403 −0.382202 0.924079i \(-0.624834\pi\)
−0.382202 + 0.924079i \(0.624834\pi\)
\(128\) 1.00000 0.0883883
\(129\) −2.97888 −0.262276
\(130\) 0.813457 0.0713449
\(131\) 13.4278 1.17320 0.586598 0.809878i \(-0.300467\pi\)
0.586598 + 0.809878i \(0.300467\pi\)
\(132\) 2.64803 0.230481
\(133\) 1.18654 0.102886
\(134\) −5.42784 −0.468894
\(135\) 2.37309 0.204243
\(136\) −2.51056 −0.215279
\(137\) −14.6903 −1.25507 −0.627537 0.778587i \(-0.715937\pi\)
−0.627537 + 0.778587i \(0.715937\pi\)
\(138\) 2.04223 0.173846
\(139\) 17.3662 1.47299 0.736493 0.676445i \(-0.236481\pi\)
0.736493 + 0.676445i \(0.236481\pi\)
\(140\) −2.91729 −0.246556
\(141\) −2.20766 −0.185918
\(142\) 14.6480 1.22924
\(143\) −5.29606 −0.442879
\(144\) −2.83457 −0.236214
\(145\) 5.32401 0.442135
\(146\) −11.0211 −0.912114
\(147\) 0.614387 0.0506738
\(148\) −1.00000 −0.0821995
\(149\) −0.207658 −0.0170120 −0.00850601 0.999964i \(-0.502708\pi\)
−0.00850601 + 0.999964i \(0.502708\pi\)
\(150\) 0.406728 0.0332092
\(151\) 0.813457 0.0661982 0.0330991 0.999452i \(-0.489462\pi\)
0.0330991 + 0.999452i \(0.489462\pi\)
\(152\) 0.406728 0.0329900
\(153\) 7.11636 0.575323
\(154\) 18.9932 1.53051
\(155\) 8.75186 0.702966
\(156\) −0.330856 −0.0264897
\(157\) 5.32401 0.424903 0.212451 0.977172i \(-0.431855\pi\)
0.212451 + 0.977172i \(0.431855\pi\)
\(158\) −1.75870 −0.139915
\(159\) −0.953831 −0.0756437
\(160\) −1.00000 −0.0790569
\(161\) 14.6480 1.15443
\(162\) 7.53851 0.592282
\(163\) 15.2008 1.19062 0.595310 0.803496i \(-0.297029\pi\)
0.595310 + 0.803496i \(0.297029\pi\)
\(164\) 6.34513 0.495471
\(165\) −2.64803 −0.206149
\(166\) −7.05476 −0.547555
\(167\) −12.4826 −0.965933 −0.482966 0.875639i \(-0.660441\pi\)
−0.482966 + 0.875639i \(0.660441\pi\)
\(168\) 1.18654 0.0915438
\(169\) −12.3383 −0.949099
\(170\) 2.51056 0.192551
\(171\) −1.15290 −0.0881645
\(172\) −7.32401 −0.558451
\(173\) 23.0354 1.75135 0.875674 0.482903i \(-0.160417\pi\)
0.875674 + 0.482903i \(0.160417\pi\)
\(174\) −2.16543 −0.164161
\(175\) 2.91729 0.220526
\(176\) 6.51056 0.490752
\(177\) −0.580745 −0.0436514
\(178\) 6.00000 0.449719
\(179\) 24.2835 1.81504 0.907518 0.420013i \(-0.137974\pi\)
0.907518 + 0.420013i \(0.137974\pi\)
\(180\) 2.83457 0.211277
\(181\) 2.16543 0.160955 0.0804775 0.996756i \(-0.474355\pi\)
0.0804775 + 0.996756i \(0.474355\pi\)
\(182\) −2.37309 −0.175905
\(183\) −0.538514 −0.0398081
\(184\) 5.02112 0.370162
\(185\) 1.00000 0.0735215
\(186\) −3.55963 −0.261005
\(187\) −16.3451 −1.19527
\(188\) −5.42784 −0.395866
\(189\) −6.92297 −0.503572
\(190\) −0.406728 −0.0295072
\(191\) −19.3326 −1.39886 −0.699429 0.714702i \(-0.746562\pi\)
−0.699429 + 0.714702i \(0.746562\pi\)
\(192\) 0.406728 0.0293531
\(193\) −20.0422 −1.44267 −0.721336 0.692586i \(-0.756471\pi\)
−0.721336 + 0.692586i \(0.756471\pi\)
\(194\) 2.34513 0.168370
\(195\) 0.330856 0.0236931
\(196\) 1.51056 0.107897
\(197\) 15.6269 1.11337 0.556686 0.830723i \(-0.312073\pi\)
0.556686 + 0.830723i \(0.312073\pi\)
\(198\) −18.4546 −1.31151
\(199\) 20.2835 1.43786 0.718931 0.695082i \(-0.244632\pi\)
0.718931 + 0.695082i \(0.244632\pi\)
\(200\) 1.00000 0.0707107
\(201\) −2.20766 −0.155716
\(202\) −8.20766 −0.577489
\(203\) −15.5317 −1.09011
\(204\) −1.02112 −0.0714924
\(205\) −6.34513 −0.443163
\(206\) −8.81346 −0.614063
\(207\) −14.2327 −0.989242
\(208\) −0.813457 −0.0564031
\(209\) 2.64803 0.183168
\(210\) −1.18654 −0.0818793
\(211\) 18.7182 1.28862 0.644308 0.764766i \(-0.277146\pi\)
0.644308 + 0.764766i \(0.277146\pi\)
\(212\) −2.34513 −0.161064
\(213\) 5.95777 0.408220
\(214\) 15.2624 1.04332
\(215\) 7.32401 0.499494
\(216\) −2.37309 −0.161468
\(217\) −25.5317 −1.73320
\(218\) 4.30290 0.291429
\(219\) −4.48260 −0.302906
\(220\) −6.51056 −0.438942
\(221\) 2.04223 0.137375
\(222\) −0.406728 −0.0272978
\(223\) −15.7307 −1.05341 −0.526704 0.850049i \(-0.676572\pi\)
−0.526704 + 0.850049i \(0.676572\pi\)
\(224\) 2.91729 0.194919
\(225\) −2.83457 −0.188971
\(226\) −9.15859 −0.609220
\(227\) 12.8836 0.855117 0.427559 0.903988i \(-0.359374\pi\)
0.427559 + 0.903988i \(0.359374\pi\)
\(228\) 0.165428 0.0109557
\(229\) 5.25383 0.347183 0.173591 0.984818i \(-0.444463\pi\)
0.173591 + 0.984818i \(0.444463\pi\)
\(230\) −5.02112 −0.331083
\(231\) 7.72506 0.508271
\(232\) −5.32401 −0.349539
\(233\) 28.3172 1.85512 0.927560 0.373675i \(-0.121902\pi\)
0.927560 + 0.373675i \(0.121902\pi\)
\(234\) 2.30580 0.150735
\(235\) 5.42784 0.354073
\(236\) −1.42784 −0.0929447
\(237\) −0.715313 −0.0464646
\(238\) −7.32401 −0.474745
\(239\) 14.3788 0.930085 0.465043 0.885288i \(-0.346039\pi\)
0.465043 + 0.885288i \(0.346039\pi\)
\(240\) −0.406728 −0.0262542
\(241\) 18.2749 1.17719 0.588596 0.808427i \(-0.299681\pi\)
0.588596 + 0.808427i \(0.299681\pi\)
\(242\) 31.3874 2.01766
\(243\) 10.1854 0.653393
\(244\) −1.32401 −0.0847613
\(245\) −1.51056 −0.0965060
\(246\) 2.58074 0.164542
\(247\) −0.330856 −0.0210519
\(248\) −8.75186 −0.555744
\(249\) −2.86937 −0.181839
\(250\) −1.00000 −0.0632456
\(251\) −1.22019 −0.0770174 −0.0385087 0.999258i \(-0.512261\pi\)
−0.0385087 + 0.999258i \(0.512261\pi\)
\(252\) −8.26926 −0.520914
\(253\) 32.6903 2.05522
\(254\) −8.61439 −0.540515
\(255\) 1.02112 0.0639447
\(256\) 1.00000 0.0625000
\(257\) 27.1306 1.69236 0.846181 0.532895i \(-0.178896\pi\)
0.846181 + 0.532895i \(0.178896\pi\)
\(258\) −2.97888 −0.185457
\(259\) −2.91729 −0.181271
\(260\) 0.813457 0.0504485
\(261\) 15.0913 0.934128
\(262\) 13.4278 0.829575
\(263\) −20.2133 −1.24641 −0.623204 0.782059i \(-0.714169\pi\)
−0.623204 + 0.782059i \(0.714169\pi\)
\(264\) 2.64803 0.162975
\(265\) 2.34513 0.144060
\(266\) 1.18654 0.0727516
\(267\) 2.44037 0.149348
\(268\) −5.42784 −0.331558
\(269\) −0.207658 −0.0126611 −0.00633057 0.999980i \(-0.502015\pi\)
−0.00633057 + 0.999980i \(0.502015\pi\)
\(270\) 2.37309 0.144421
\(271\) −30.6480 −1.86174 −0.930868 0.365357i \(-0.880947\pi\)
−0.930868 + 0.365357i \(0.880947\pi\)
\(272\) −2.51056 −0.152225
\(273\) −0.965202 −0.0584167
\(274\) −14.6903 −0.887471
\(275\) 6.51056 0.392601
\(276\) 2.04223 0.122928
\(277\) −16.8135 −1.01022 −0.505111 0.863054i \(-0.668549\pi\)
−0.505111 + 0.863054i \(0.668549\pi\)
\(278\) 17.3662 1.04156
\(279\) 24.8078 1.48520
\(280\) −2.91729 −0.174341
\(281\) −14.2749 −0.851572 −0.425786 0.904824i \(-0.640002\pi\)
−0.425786 + 0.904824i \(0.640002\pi\)
\(282\) −2.20766 −0.131464
\(283\) −13.2288 −0.786369 −0.393184 0.919460i \(-0.628627\pi\)
−0.393184 + 0.919460i \(0.628627\pi\)
\(284\) 14.6480 0.869201
\(285\) −0.165428 −0.00979911
\(286\) −5.29606 −0.313162
\(287\) 18.5106 1.09264
\(288\) −2.83457 −0.167029
\(289\) −10.6971 −0.629241
\(290\) 5.32401 0.312637
\(291\) 0.953831 0.0559146
\(292\) −11.0211 −0.644962
\(293\) 2.34513 0.137004 0.0685020 0.997651i \(-0.478178\pi\)
0.0685020 + 0.997651i \(0.478178\pi\)
\(294\) 0.614387 0.0358318
\(295\) 1.42784 0.0831323
\(296\) −1.00000 −0.0581238
\(297\) −15.4501 −0.896507
\(298\) −0.207658 −0.0120293
\(299\) −4.08446 −0.236210
\(300\) 0.406728 0.0234825
\(301\) −21.3662 −1.23153
\(302\) 0.813457 0.0468092
\(303\) −3.33829 −0.191780
\(304\) 0.406728 0.0233275
\(305\) 1.32401 0.0758128
\(306\) 7.11636 0.406815
\(307\) −23.2624 −1.32766 −0.663828 0.747885i \(-0.731069\pi\)
−0.663828 + 0.747885i \(0.731069\pi\)
\(308\) 18.9932 1.08224
\(309\) −3.58468 −0.203926
\(310\) 8.75186 0.497072
\(311\) −31.3999 −1.78052 −0.890262 0.455449i \(-0.849479\pi\)
−0.890262 + 0.455449i \(0.849479\pi\)
\(312\) −0.330856 −0.0187310
\(313\) −8.04223 −0.454574 −0.227287 0.973828i \(-0.572986\pi\)
−0.227287 + 0.973828i \(0.572986\pi\)
\(314\) 5.32401 0.300452
\(315\) 8.26926 0.465920
\(316\) −1.75870 −0.0989346
\(317\) −12.3029 −0.691000 −0.345500 0.938419i \(-0.612291\pi\)
−0.345500 + 0.938419i \(0.612291\pi\)
\(318\) −0.953831 −0.0534882
\(319\) −34.6623 −1.94072
\(320\) −1.00000 −0.0559017
\(321\) 6.20766 0.346478
\(322\) 14.6480 0.816303
\(323\) −1.02112 −0.0568164
\(324\) 7.53851 0.418806
\(325\) −0.813457 −0.0451225
\(326\) 15.2008 0.841895
\(327\) 1.75011 0.0967814
\(328\) 6.34513 0.350351
\(329\) −15.8346 −0.872988
\(330\) −2.64803 −0.145769
\(331\) −6.77981 −0.372652 −0.186326 0.982488i \(-0.559658\pi\)
−0.186326 + 0.982488i \(0.559658\pi\)
\(332\) −7.05476 −0.387180
\(333\) 2.83457 0.155334
\(334\) −12.4826 −0.683018
\(335\) 5.42784 0.296555
\(336\) 1.18654 0.0647312
\(337\) 10.6903 0.582336 0.291168 0.956672i \(-0.405956\pi\)
0.291168 + 0.956672i \(0.405956\pi\)
\(338\) −12.3383 −0.671114
\(339\) −3.72506 −0.202317
\(340\) 2.51056 0.136154
\(341\) −56.9795 −3.08561
\(342\) −1.15290 −0.0623417
\(343\) −16.0143 −0.864689
\(344\) −7.32401 −0.394884
\(345\) −2.04223 −0.109950
\(346\) 23.0354 1.23839
\(347\) 5.62691 0.302069 0.151034 0.988529i \(-0.451740\pi\)
0.151034 + 0.988529i \(0.451740\pi\)
\(348\) −2.16543 −0.116079
\(349\) 15.1306 0.809924 0.404962 0.914334i \(-0.367285\pi\)
0.404962 + 0.914334i \(0.367285\pi\)
\(350\) 2.91729 0.155935
\(351\) 1.93040 0.103037
\(352\) 6.51056 0.347014
\(353\) 35.7816 1.90446 0.952230 0.305381i \(-0.0987839\pi\)
0.952230 + 0.305381i \(0.0987839\pi\)
\(354\) −0.580745 −0.0308662
\(355\) −14.6480 −0.777437
\(356\) 6.00000 0.317999
\(357\) −2.97888 −0.157659
\(358\) 24.2835 1.28342
\(359\) 4.48260 0.236583 0.118291 0.992979i \(-0.462258\pi\)
0.118291 + 0.992979i \(0.462258\pi\)
\(360\) 2.83457 0.149395
\(361\) −18.8346 −0.991293
\(362\) 2.16543 0.113812
\(363\) 12.7661 0.670048
\(364\) −2.37309 −0.124384
\(365\) 11.0211 0.576872
\(366\) −0.538514 −0.0281486
\(367\) −21.5653 −1.12570 −0.562850 0.826559i \(-0.690295\pi\)
−0.562850 + 0.826559i \(0.690295\pi\)
\(368\) 5.02112 0.261744
\(369\) −17.9857 −0.936300
\(370\) 1.00000 0.0519875
\(371\) −6.84141 −0.355188
\(372\) −3.55963 −0.184558
\(373\) 16.9230 0.876238 0.438119 0.898917i \(-0.355645\pi\)
0.438119 + 0.898917i \(0.355645\pi\)
\(374\) −16.3451 −0.845187
\(375\) −0.406728 −0.0210034
\(376\) −5.42784 −0.279920
\(377\) 4.33086 0.223050
\(378\) −6.92297 −0.356079
\(379\) 31.2710 1.60628 0.803142 0.595788i \(-0.203160\pi\)
0.803142 + 0.595788i \(0.203160\pi\)
\(380\) −0.406728 −0.0208647
\(381\) −3.50372 −0.179501
\(382\) −19.3326 −0.989142
\(383\) 1.62691 0.0831314 0.0415657 0.999136i \(-0.486765\pi\)
0.0415657 + 0.999136i \(0.486765\pi\)
\(384\) 0.406728 0.0207558
\(385\) −18.9932 −0.967981
\(386\) −20.0422 −1.02012
\(387\) 20.7604 1.05531
\(388\) 2.34513 0.119056
\(389\) −31.6412 −1.60427 −0.802136 0.597141i \(-0.796303\pi\)
−0.802136 + 0.597141i \(0.796303\pi\)
\(390\) 0.330856 0.0167535
\(391\) −12.6058 −0.637503
\(392\) 1.51056 0.0762947
\(393\) 5.46149 0.275496
\(394\) 15.6269 0.787273
\(395\) 1.75870 0.0884898
\(396\) −18.4546 −0.927381
\(397\) −39.2961 −1.97221 −0.986106 0.166116i \(-0.946878\pi\)
−0.986106 + 0.166116i \(0.946878\pi\)
\(398\) 20.2835 1.01672
\(399\) 0.482601 0.0241603
\(400\) 1.00000 0.0500000
\(401\) −9.66914 −0.482854 −0.241427 0.970419i \(-0.577615\pi\)
−0.241427 + 0.970419i \(0.577615\pi\)
\(402\) −2.20766 −0.110108
\(403\) 7.11926 0.354636
\(404\) −8.20766 −0.408346
\(405\) −7.53851 −0.374592
\(406\) −15.5317 −0.770824
\(407\) −6.51056 −0.322716
\(408\) −1.02112 −0.0505527
\(409\) −26.2749 −1.29921 −0.649606 0.760271i \(-0.725066\pi\)
−0.649606 + 0.760271i \(0.725066\pi\)
\(410\) −6.34513 −0.313364
\(411\) −5.97495 −0.294722
\(412\) −8.81346 −0.434208
\(413\) −4.16543 −0.204967
\(414\) −14.2327 −0.699500
\(415\) 7.05476 0.346304
\(416\) −0.813457 −0.0398830
\(417\) 7.06335 0.345894
\(418\) 2.64803 0.129519
\(419\) −30.1940 −1.47507 −0.737536 0.675308i \(-0.764011\pi\)
−0.737536 + 0.675308i \(0.764011\pi\)
\(420\) −1.18654 −0.0578974
\(421\) 22.0000 1.07221 0.536107 0.844150i \(-0.319894\pi\)
0.536107 + 0.844150i \(0.319894\pi\)
\(422\) 18.7182 0.911188
\(423\) 15.3856 0.748074
\(424\) −2.34513 −0.113890
\(425\) −2.51056 −0.121780
\(426\) 5.95777 0.288655
\(427\) −3.86253 −0.186921
\(428\) 15.2624 0.737737
\(429\) −2.15406 −0.103999
\(430\) 7.32401 0.353195
\(431\) 25.5653 1.23144 0.615719 0.787966i \(-0.288866\pi\)
0.615719 + 0.787966i \(0.288866\pi\)
\(432\) −2.37309 −0.114175
\(433\) 32.0422 1.53985 0.769926 0.638134i \(-0.220293\pi\)
0.769926 + 0.638134i \(0.220293\pi\)
\(434\) −25.5317 −1.22556
\(435\) 2.16543 0.103824
\(436\) 4.30290 0.206072
\(437\) 2.04223 0.0976931
\(438\) −4.48260 −0.214187
\(439\) 3.93840 0.187970 0.0939848 0.995574i \(-0.470039\pi\)
0.0939848 + 0.995574i \(0.470039\pi\)
\(440\) −6.51056 −0.310379
\(441\) −4.28178 −0.203894
\(442\) 2.04223 0.0971390
\(443\) −0.0758724 −0.00360481 −0.00180240 0.999998i \(-0.500574\pi\)
−0.00180240 + 0.999998i \(0.500574\pi\)
\(444\) −0.406728 −0.0193025
\(445\) −6.00000 −0.284427
\(446\) −15.7307 −0.744872
\(447\) −0.0844605 −0.00399485
\(448\) 2.91729 0.137829
\(449\) 19.2961 0.910637 0.455319 0.890329i \(-0.349525\pi\)
0.455319 + 0.890329i \(0.349525\pi\)
\(450\) −2.83457 −0.133623
\(451\) 41.3103 1.94523
\(452\) −9.15859 −0.430784
\(453\) 0.330856 0.0155450
\(454\) 12.8836 0.604659
\(455\) 2.37309 0.111252
\(456\) 0.165428 0.00774688
\(457\) 3.00684 0.140654 0.0703271 0.997524i \(-0.477596\pi\)
0.0703271 + 0.997524i \(0.477596\pi\)
\(458\) 5.25383 0.245495
\(459\) 5.95777 0.278085
\(460\) −5.02112 −0.234111
\(461\) −0.633755 −0.0295169 −0.0147585 0.999891i \(-0.504698\pi\)
−0.0147585 + 0.999891i \(0.504698\pi\)
\(462\) 7.72506 0.359402
\(463\) 18.0422 0.838494 0.419247 0.907872i \(-0.362294\pi\)
0.419247 + 0.907872i \(0.362294\pi\)
\(464\) −5.32401 −0.247161
\(465\) 3.55963 0.165074
\(466\) 28.3172 1.31177
\(467\) −23.4758 −1.08633 −0.543164 0.839626i \(-0.682774\pi\)
−0.543164 + 0.839626i \(0.682774\pi\)
\(468\) 2.30580 0.106586
\(469\) −15.8346 −0.731173
\(470\) 5.42784 0.250368
\(471\) 2.16543 0.0997777
\(472\) −1.42784 −0.0657218
\(473\) −47.6834 −2.19249
\(474\) −0.715313 −0.0328554
\(475\) 0.406728 0.0186620
\(476\) −7.32401 −0.335696
\(477\) 6.64744 0.304365
\(478\) 14.3788 0.657670
\(479\) −8.19907 −0.374625 −0.187313 0.982300i \(-0.559978\pi\)
−0.187313 + 0.982300i \(0.559978\pi\)
\(480\) −0.406728 −0.0185645
\(481\) 0.813457 0.0370904
\(482\) 18.2749 0.832401
\(483\) 5.95777 0.271088
\(484\) 31.3874 1.42670
\(485\) −2.34513 −0.106487
\(486\) 10.1854 0.462019
\(487\) 0.953831 0.0432222 0.0216111 0.999766i \(-0.493120\pi\)
0.0216111 + 0.999766i \(0.493120\pi\)
\(488\) −1.32401 −0.0599353
\(489\) 6.18260 0.279587
\(490\) −1.51056 −0.0682400
\(491\) 19.3383 0.872725 0.436362 0.899771i \(-0.356267\pi\)
0.436362 + 0.899771i \(0.356267\pi\)
\(492\) 2.58074 0.116349
\(493\) 13.3662 0.601985
\(494\) −0.330856 −0.0148859
\(495\) 18.4546 0.829475
\(496\) −8.75186 −0.392970
\(497\) 42.7325 1.91681
\(498\) −2.86937 −0.128580
\(499\) −5.63550 −0.252280 −0.126140 0.992012i \(-0.540259\pi\)
−0.126140 + 0.992012i \(0.540259\pi\)
\(500\) −1.00000 −0.0447214
\(501\) −5.07703 −0.226825
\(502\) −1.22019 −0.0544595
\(503\) −33.6269 −1.49935 −0.749675 0.661806i \(-0.769790\pi\)
−0.749675 + 0.661806i \(0.769790\pi\)
\(504\) −8.26926 −0.368342
\(505\) 8.20766 0.365236
\(506\) 32.6903 1.45326
\(507\) −5.01833 −0.222872
\(508\) −8.61439 −0.382202
\(509\) −18.6903 −0.828431 −0.414216 0.910179i \(-0.635944\pi\)
−0.414216 + 0.910179i \(0.635944\pi\)
\(510\) 1.02112 0.0452157
\(511\) −32.1517 −1.42231
\(512\) 1.00000 0.0441942
\(513\) −0.965202 −0.0426147
\(514\) 27.1306 1.19668
\(515\) 8.81346 0.388367
\(516\) −2.97888 −0.131138
\(517\) −35.3383 −1.55418
\(518\) −2.91729 −0.128178
\(519\) 9.36915 0.411260
\(520\) 0.813457 0.0356724
\(521\) 25.1586 1.10222 0.551109 0.834433i \(-0.314205\pi\)
0.551109 + 0.834433i \(0.314205\pi\)
\(522\) 15.0913 0.660528
\(523\) −7.25383 −0.317188 −0.158594 0.987344i \(-0.550696\pi\)
−0.158594 + 0.987344i \(0.550696\pi\)
\(524\) 13.4278 0.586598
\(525\) 1.18654 0.0517850
\(526\) −20.2133 −0.881344
\(527\) 21.9720 0.957117
\(528\) 2.64803 0.115241
\(529\) 2.21160 0.0961564
\(530\) 2.34513 0.101866
\(531\) 4.04733 0.175639
\(532\) 1.18654 0.0514432
\(533\) −5.16149 −0.223569
\(534\) 2.44037 0.105605
\(535\) −15.2624 −0.659852
\(536\) −5.42784 −0.234447
\(537\) 9.87680 0.426215
\(538\) −0.207658 −0.00895278
\(539\) 9.83457 0.423605
\(540\) 2.37309 0.102121
\(541\) 16.0422 0.689709 0.344855 0.938656i \(-0.387928\pi\)
0.344855 + 0.938656i \(0.387928\pi\)
\(542\) −30.6480 −1.31645
\(543\) 0.880741 0.0377962
\(544\) −2.51056 −0.107639
\(545\) −4.30290 −0.184316
\(546\) −0.965202 −0.0413068
\(547\) 16.0143 0.684721 0.342360 0.939569i \(-0.388774\pi\)
0.342360 + 0.939569i \(0.388774\pi\)
\(548\) −14.6903 −0.627537
\(549\) 3.75301 0.160175
\(550\) 6.51056 0.277611
\(551\) −2.16543 −0.0922503
\(552\) 2.04223 0.0869231
\(553\) −5.13063 −0.218177
\(554\) −16.8135 −0.714335
\(555\) 0.406728 0.0172647
\(556\) 17.3662 0.736493
\(557\) −10.8557 −0.459970 −0.229985 0.973194i \(-0.573868\pi\)
−0.229985 + 0.973194i \(0.573868\pi\)
\(558\) 24.8078 1.05020
\(559\) 5.95777 0.251987
\(560\) −2.91729 −0.123278
\(561\) −6.64803 −0.280680
\(562\) −14.2749 −0.602152
\(563\) 31.0776 1.30977 0.654883 0.755731i \(-0.272718\pi\)
0.654883 + 0.755731i \(0.272718\pi\)
\(564\) −2.20766 −0.0929592
\(565\) 9.15859 0.385305
\(566\) −13.2288 −0.556047
\(567\) 21.9920 0.923577
\(568\) 14.6480 0.614618
\(569\) 30.0845 1.26121 0.630603 0.776105i \(-0.282808\pi\)
0.630603 + 0.776105i \(0.282808\pi\)
\(570\) −0.165428 −0.00692902
\(571\) 5.55673 0.232542 0.116271 0.993218i \(-0.462906\pi\)
0.116271 + 0.993218i \(0.462906\pi\)
\(572\) −5.29606 −0.221439
\(573\) −7.86312 −0.328487
\(574\) 18.5106 0.772616
\(575\) 5.02112 0.209395
\(576\) −2.83457 −0.118107
\(577\) −22.2499 −0.926275 −0.463137 0.886286i \(-0.653276\pi\)
−0.463137 + 0.886286i \(0.653276\pi\)
\(578\) −10.6971 −0.444941
\(579\) −8.15174 −0.338775
\(580\) 5.32401 0.221068
\(581\) −20.5807 −0.853833
\(582\) 0.953831 0.0395376
\(583\) −15.2681 −0.632340
\(584\) −11.0211 −0.456057
\(585\) −2.30580 −0.0953332
\(586\) 2.34513 0.0968764
\(587\) 45.3103 1.87016 0.935079 0.354440i \(-0.115328\pi\)
0.935079 + 0.354440i \(0.115328\pi\)
\(588\) 0.614387 0.0253369
\(589\) −3.55963 −0.146672
\(590\) 1.42784 0.0587834
\(591\) 6.35591 0.261447
\(592\) −1.00000 −0.0410997
\(593\) −0.232712 −0.00955635 −0.00477817 0.999989i \(-0.501521\pi\)
−0.00477817 + 0.999989i \(0.501521\pi\)
\(594\) −15.4501 −0.633926
\(595\) 7.32401 0.300255
\(596\) −0.207658 −0.00850601
\(597\) 8.24989 0.337645
\(598\) −4.08446 −0.167026
\(599\) 43.4056 1.77350 0.886752 0.462246i \(-0.152956\pi\)
0.886752 + 0.462246i \(0.152956\pi\)
\(600\) 0.406728 0.0166046
\(601\) 35.5317 1.44937 0.724684 0.689082i \(-0.241986\pi\)
0.724684 + 0.689082i \(0.241986\pi\)
\(602\) −21.3662 −0.870823
\(603\) 15.3856 0.626551
\(604\) 0.813457 0.0330991
\(605\) −31.3874 −1.27608
\(606\) −3.33829 −0.135609
\(607\) 26.7325 1.08504 0.542519 0.840043i \(-0.317471\pi\)
0.542519 + 0.840043i \(0.317471\pi\)
\(608\) 0.406728 0.0164950
\(609\) −6.31717 −0.255985
\(610\) 1.32401 0.0536078
\(611\) 4.41532 0.178625
\(612\) 7.11636 0.287662
\(613\) 48.0565 1.94098 0.970492 0.241134i \(-0.0775192\pi\)
0.970492 + 0.241134i \(0.0775192\pi\)
\(614\) −23.2624 −0.938795
\(615\) −2.58074 −0.104066
\(616\) 18.9932 0.765256
\(617\) 12.9789 0.522510 0.261255 0.965270i \(-0.415864\pi\)
0.261255 + 0.965270i \(0.415864\pi\)
\(618\) −3.58468 −0.144197
\(619\) 25.3662 1.01956 0.509778 0.860306i \(-0.329728\pi\)
0.509778 + 0.860306i \(0.329728\pi\)
\(620\) 8.75186 0.351483
\(621\) −11.9155 −0.478154
\(622\) −31.3999 −1.25902
\(623\) 17.5037 0.701272
\(624\) −0.330856 −0.0132448
\(625\) 1.00000 0.0400000
\(626\) −8.04223 −0.321432
\(627\) 1.07703 0.0430124
\(628\) 5.32401 0.212451
\(629\) 2.51056 0.100102
\(630\) 8.26926 0.329455
\(631\) −31.6075 −1.25828 −0.629138 0.777293i \(-0.716592\pi\)
−0.629138 + 0.777293i \(0.716592\pi\)
\(632\) −1.75870 −0.0699573
\(633\) 7.61323 0.302599
\(634\) −12.3029 −0.488611
\(635\) 8.61439 0.341852
\(636\) −0.953831 −0.0378219
\(637\) −1.22877 −0.0486858
\(638\) −34.6623 −1.37229
\(639\) −41.5209 −1.64254
\(640\) −1.00000 −0.0395285
\(641\) −23.8625 −0.942513 −0.471257 0.881996i \(-0.656199\pi\)
−0.471257 + 0.881996i \(0.656199\pi\)
\(642\) 6.20766 0.244997
\(643\) 32.8277 1.29460 0.647300 0.762236i \(-0.275898\pi\)
0.647300 + 0.762236i \(0.275898\pi\)
\(644\) 14.6480 0.577213
\(645\) 2.97888 0.117293
\(646\) −1.02112 −0.0401752
\(647\) −13.9578 −0.548737 −0.274368 0.961625i \(-0.588469\pi\)
−0.274368 + 0.961625i \(0.588469\pi\)
\(648\) 7.53851 0.296141
\(649\) −9.29606 −0.364902
\(650\) −0.813457 −0.0319064
\(651\) −10.3845 −0.406999
\(652\) 15.2008 0.595310
\(653\) 28.5921 1.11890 0.559448 0.828865i \(-0.311013\pi\)
0.559448 + 0.828865i \(0.311013\pi\)
\(654\) 1.75011 0.0684348
\(655\) −13.4278 −0.524669
\(656\) 6.34513 0.247736
\(657\) 31.2401 1.21879
\(658\) −15.8346 −0.617296
\(659\) −5.62691 −0.219193 −0.109597 0.993976i \(-0.534956\pi\)
−0.109597 + 0.993976i \(0.534956\pi\)
\(660\) −2.64803 −0.103074
\(661\) 13.3240 0.518244 0.259122 0.965845i \(-0.416567\pi\)
0.259122 + 0.965845i \(0.416567\pi\)
\(662\) −6.77981 −0.263505
\(663\) 0.830633 0.0322591
\(664\) −7.05476 −0.273778
\(665\) −1.18654 −0.0460122
\(666\) 2.83457 0.109837
\(667\) −26.7325 −1.03509
\(668\) −12.4826 −0.482966
\(669\) −6.39814 −0.247366
\(670\) 5.42784 0.209696
\(671\) −8.62007 −0.332774
\(672\) 1.18654 0.0457719
\(673\) 29.6691 1.14366 0.571831 0.820372i \(-0.306233\pi\)
0.571831 + 0.820372i \(0.306233\pi\)
\(674\) 10.6903 0.411773
\(675\) −2.37309 −0.0913401
\(676\) −12.3383 −0.474550
\(677\) −1.25383 −0.0481885 −0.0240943 0.999710i \(-0.507670\pi\)
−0.0240943 + 0.999710i \(0.507670\pi\)
\(678\) −3.72506 −0.143060
\(679\) 6.84141 0.262549
\(680\) 2.51056 0.0962755
\(681\) 5.24014 0.200803
\(682\) −56.9795 −2.18186
\(683\) −4.76045 −0.182153 −0.0910767 0.995844i \(-0.529031\pi\)
−0.0910767 + 0.995844i \(0.529031\pi\)
\(684\) −1.15290 −0.0440823
\(685\) 14.6903 0.561286
\(686\) −16.0143 −0.611428
\(687\) 2.13688 0.0815271
\(688\) −7.32401 −0.279225
\(689\) 1.90766 0.0726761
\(690\) −2.04223 −0.0777464
\(691\) −20.2219 −0.769279 −0.384639 0.923067i \(-0.625674\pi\)
−0.384639 + 0.923067i \(0.625674\pi\)
\(692\) 23.0354 0.875674
\(693\) −53.8375 −2.04512
\(694\) 5.62691 0.213595
\(695\) −17.3662 −0.658739
\(696\) −2.16543 −0.0820803
\(697\) −15.9298 −0.603385
\(698\) 15.1306 0.572703
\(699\) 11.5174 0.435628
\(700\) 2.91729 0.110263
\(701\) −22.0845 −0.834119 −0.417059 0.908879i \(-0.636939\pi\)
−0.417059 + 0.908879i \(0.636939\pi\)
\(702\) 1.93040 0.0728584
\(703\) −0.406728 −0.0153401
\(704\) 6.51056 0.245376
\(705\) 2.20766 0.0831452
\(706\) 35.7816 1.34666
\(707\) −23.9441 −0.900510
\(708\) −0.580745 −0.0218257
\(709\) 5.59896 0.210273 0.105137 0.994458i \(-0.466472\pi\)
0.105137 + 0.994458i \(0.466472\pi\)
\(710\) −14.6480 −0.549731
\(711\) 4.98516 0.186958
\(712\) 6.00000 0.224860
\(713\) −43.9441 −1.64572
\(714\) −2.97888 −0.111482
\(715\) 5.29606 0.198061
\(716\) 24.2835 0.907518
\(717\) 5.84826 0.218407
\(718\) 4.48260 0.167289
\(719\) 40.2921 1.50264 0.751321 0.659937i \(-0.229417\pi\)
0.751321 + 0.659937i \(0.229417\pi\)
\(720\) 2.83457 0.105638
\(721\) −25.7114 −0.957542
\(722\) −18.8346 −0.700950
\(723\) 7.43294 0.276434
\(724\) 2.16543 0.0804775
\(725\) −5.32401 −0.197729
\(726\) 12.7661 0.473796
\(727\) 11.2538 0.417381 0.208691 0.977982i \(-0.433080\pi\)
0.208691 + 0.977982i \(0.433080\pi\)
\(728\) −2.37309 −0.0879524
\(729\) −18.4729 −0.684180
\(730\) 11.0211 0.407910
\(731\) 18.3874 0.680081
\(732\) −0.538514 −0.0199041
\(733\) −42.2892 −1.56199 −0.780994 0.624539i \(-0.785287\pi\)
−0.780994 + 0.624539i \(0.785287\pi\)
\(734\) −21.5653 −0.795990
\(735\) −0.614387 −0.0226620
\(736\) 5.02112 0.185081
\(737\) −35.3383 −1.30170
\(738\) −17.9857 −0.662064
\(739\) −29.3103 −1.07820 −0.539099 0.842242i \(-0.681235\pi\)
−0.539099 + 0.842242i \(0.681235\pi\)
\(740\) 1.00000 0.0367607
\(741\) −0.134569 −0.00494350
\(742\) −6.84141 −0.251156
\(743\) −4.39245 −0.161144 −0.0805718 0.996749i \(-0.525675\pi\)
−0.0805718 + 0.996749i \(0.525675\pi\)
\(744\) −3.55963 −0.130502
\(745\) 0.207658 0.00760801
\(746\) 16.9230 0.619594
\(747\) 19.9972 0.731660
\(748\) −16.3451 −0.597637
\(749\) 44.5248 1.62690
\(750\) −0.406728 −0.0148516
\(751\) 2.58074 0.0941727 0.0470864 0.998891i \(-0.485006\pi\)
0.0470864 + 0.998891i \(0.485006\pi\)
\(752\) −5.42784 −0.197933
\(753\) −0.496284 −0.0180856
\(754\) 4.33086 0.157720
\(755\) −0.813457 −0.0296047
\(756\) −6.92297 −0.251786
\(757\) 27.3805 0.995162 0.497581 0.867418i \(-0.334222\pi\)
0.497581 + 0.867418i \(0.334222\pi\)
\(758\) 31.2710 1.13581
\(759\) 13.2961 0.482616
\(760\) −0.406728 −0.0147536
\(761\) −42.9509 −1.55697 −0.778485 0.627663i \(-0.784011\pi\)
−0.778485 + 0.627663i \(0.784011\pi\)
\(762\) −3.50372 −0.126926
\(763\) 12.5528 0.454441
\(764\) −19.3326 −0.699429
\(765\) −7.11636 −0.257292
\(766\) 1.62691 0.0587828
\(767\) 1.16149 0.0419389
\(768\) 0.406728 0.0146765
\(769\) −13.0633 −0.471076 −0.235538 0.971865i \(-0.575685\pi\)
−0.235538 + 0.971865i \(0.575685\pi\)
\(770\) −18.9932 −0.684466
\(771\) 11.0348 0.397409
\(772\) −20.0422 −0.721336
\(773\) −15.3662 −0.552685 −0.276343 0.961059i \(-0.589122\pi\)
−0.276343 + 0.961059i \(0.589122\pi\)
\(774\) 20.7604 0.746219
\(775\) −8.75186 −0.314376
\(776\) 2.34513 0.0841852
\(777\) −1.18654 −0.0425670
\(778\) −31.6412 −1.13439
\(779\) 2.58074 0.0924648
\(780\) 0.330856 0.0118465
\(781\) 95.3668 3.41249
\(782\) −12.6058 −0.450782
\(783\) 12.6343 0.451515
\(784\) 1.51056 0.0539485
\(785\) −5.32401 −0.190022
\(786\) 5.46149 0.194805
\(787\) 10.9316 0.389668 0.194834 0.980836i \(-0.437583\pi\)
0.194834 + 0.980836i \(0.437583\pi\)
\(788\) 15.6269 0.556686
\(789\) −8.22134 −0.292688
\(790\) 1.75870 0.0625717
\(791\) −26.7182 −0.949990
\(792\) −18.4546 −0.655757
\(793\) 1.07703 0.0382464
\(794\) −39.2961 −1.39456
\(795\) 0.953831 0.0338289
\(796\) 20.2835 0.718931
\(797\) −24.2921 −0.860471 −0.430235 0.902717i \(-0.641569\pi\)
−0.430235 + 0.902717i \(0.641569\pi\)
\(798\) 0.482601 0.0170839
\(799\) 13.6269 0.482086
\(800\) 1.00000 0.0353553
\(801\) −17.0074 −0.600928
\(802\) −9.66914 −0.341429
\(803\) −71.7536 −2.53213
\(804\) −2.20766 −0.0778581
\(805\) −14.6480 −0.516275
\(806\) 7.11926 0.250765
\(807\) −0.0844605 −0.00297315
\(808\) −8.20766 −0.288744
\(809\) −40.0422 −1.40781 −0.703905 0.710294i \(-0.748562\pi\)
−0.703905 + 0.710294i \(0.748562\pi\)
\(810\) −7.53851 −0.264876
\(811\) −40.6343 −1.42686 −0.713432 0.700724i \(-0.752860\pi\)
−0.713432 + 0.700724i \(0.752860\pi\)
\(812\) −15.5317 −0.545055
\(813\) −12.4654 −0.437182
\(814\) −6.51056 −0.228195
\(815\) −15.2008 −0.532461
\(816\) −1.02112 −0.0357462
\(817\) −2.97888 −0.104218
\(818\) −26.2749 −0.918682
\(819\) 6.72668 0.235049
\(820\) −6.34513 −0.221582
\(821\) −41.4787 −1.44762 −0.723808 0.690002i \(-0.757610\pi\)
−0.723808 + 0.690002i \(0.757610\pi\)
\(822\) −5.97495 −0.208400
\(823\) −5.27610 −0.183913 −0.0919566 0.995763i \(-0.529312\pi\)
−0.0919566 + 0.995763i \(0.529312\pi\)
\(824\) −8.81346 −0.307031
\(825\) 2.64803 0.0921925
\(826\) −4.16543 −0.144934
\(827\) 1.76438 0.0613537 0.0306768 0.999529i \(-0.490234\pi\)
0.0306768 + 0.999529i \(0.490234\pi\)
\(828\) −14.2327 −0.494621
\(829\) 16.3029 0.566223 0.283112 0.959087i \(-0.408633\pi\)
0.283112 + 0.959087i \(0.408633\pi\)
\(830\) 7.05476 0.244874
\(831\) −6.83851 −0.237225
\(832\) −0.813457 −0.0282015
\(833\) −3.79234 −0.131397
\(834\) 7.06335 0.244584
\(835\) 12.4826 0.431978
\(836\) 2.64803 0.0915840
\(837\) 20.7689 0.717879
\(838\) −30.1940 −1.04303
\(839\) 14.3731 0.496214 0.248107 0.968733i \(-0.420192\pi\)
0.248107 + 0.968733i \(0.420192\pi\)
\(840\) −1.18654 −0.0409396
\(841\) −0.654870 −0.0225817
\(842\) 22.0000 0.758170
\(843\) −5.80602 −0.199970
\(844\) 18.7182 0.644308
\(845\) 12.3383 0.424450
\(846\) 15.3856 0.528968
\(847\) 91.5659 3.14624
\(848\) −2.34513 −0.0805321
\(849\) −5.38052 −0.184659
\(850\) −2.51056 −0.0861114
\(851\) −5.02112 −0.172122
\(852\) 5.95777 0.204110
\(853\) 43.2961 1.48243 0.741214 0.671268i \(-0.234250\pi\)
0.741214 + 0.671268i \(0.234250\pi\)
\(854\) −3.86253 −0.132173
\(855\) 1.15290 0.0394284
\(856\) 15.2624 0.521659
\(857\) −49.0776 −1.67646 −0.838230 0.545317i \(-0.816409\pi\)
−0.838230 + 0.545317i \(0.816409\pi\)
\(858\) −2.15406 −0.0735383
\(859\) 15.5374 0.530128 0.265064 0.964231i \(-0.414607\pi\)
0.265064 + 0.964231i \(0.414607\pi\)
\(860\) 7.32401 0.249747
\(861\) 7.52877 0.256580
\(862\) 25.5653 0.870758
\(863\) −36.0901 −1.22852 −0.614261 0.789103i \(-0.710546\pi\)
−0.614261 + 0.789103i \(0.710546\pi\)
\(864\) −2.37309 −0.0807340
\(865\) −23.0354 −0.783227
\(866\) 32.0422 1.08884
\(867\) −4.35081 −0.147761
\(868\) −25.5317 −0.866601
\(869\) −11.4501 −0.388419
\(870\) 2.16543 0.0734149
\(871\) 4.41532 0.149607
\(872\) 4.30290 0.145715
\(873\) −6.64744 −0.224982
\(874\) 2.04223 0.0690795
\(875\) −2.91729 −0.0986223
\(876\) −4.48260 −0.151453
\(877\) 29.3240 0.990202 0.495101 0.868836i \(-0.335131\pi\)
0.495101 + 0.868836i \(0.335131\pi\)
\(878\) 3.93840 0.132915
\(879\) 0.953831 0.0321719
\(880\) −6.51056 −0.219471
\(881\) 23.8066 0.802065 0.401033 0.916064i \(-0.368651\pi\)
0.401033 + 0.916064i \(0.368651\pi\)
\(882\) −4.28178 −0.144175
\(883\) −3.94699 −0.132827 −0.0664134 0.997792i \(-0.521156\pi\)
−0.0664134 + 0.997792i \(0.521156\pi\)
\(884\) 2.04223 0.0686876
\(885\) 0.580745 0.0195215
\(886\) −0.0758724 −0.00254898
\(887\) 1.96346 0.0659264 0.0329632 0.999457i \(-0.489506\pi\)
0.0329632 + 0.999457i \(0.489506\pi\)
\(888\) −0.406728 −0.0136489
\(889\) −25.1306 −0.842854
\(890\) −6.00000 −0.201120
\(891\) 49.0799 1.64424
\(892\) −15.7307 −0.526704
\(893\) −2.20766 −0.0738765
\(894\) −0.0844605 −0.00282478
\(895\) −24.2835 −0.811709
\(896\) 2.91729 0.0974597
\(897\) −1.66127 −0.0554681
\(898\) 19.2961 0.643918
\(899\) 46.5950 1.55403
\(900\) −2.83457 −0.0944857
\(901\) 5.88758 0.196144
\(902\) 41.3103 1.37548
\(903\) −8.69026 −0.289194
\(904\) −9.15859 −0.304610
\(905\) −2.16543 −0.0719813
\(906\) 0.330856 0.0109920
\(907\) 5.89049 0.195590 0.0977952 0.995207i \(-0.468821\pi\)
0.0977952 + 0.995207i \(0.468821\pi\)
\(908\) 12.8836 0.427559
\(909\) 23.2652 0.771658
\(910\) 2.37309 0.0786670
\(911\) −27.9527 −0.926113 −0.463057 0.886329i \(-0.653247\pi\)
−0.463057 + 0.886329i \(0.653247\pi\)
\(912\) 0.165428 0.00547787
\(913\) −45.9304 −1.52007
\(914\) 3.00684 0.0994575
\(915\) 0.538514 0.0178027
\(916\) 5.25383 0.173591
\(917\) 39.1729 1.29360
\(918\) 5.95777 0.196636
\(919\) 17.7587 0.585805 0.292903 0.956142i \(-0.405379\pi\)
0.292903 + 0.956142i \(0.405379\pi\)
\(920\) −5.02112 −0.165541
\(921\) −9.46149 −0.311767
\(922\) −0.633755 −0.0208716
\(923\) −11.9155 −0.392205
\(924\) 7.72506 0.254136
\(925\) −1.00000 −0.0328798
\(926\) 18.0422 0.592904
\(927\) 24.9824 0.820529
\(928\) −5.32401 −0.174769
\(929\) 34.2892 1.12499 0.562496 0.826800i \(-0.309841\pi\)
0.562496 + 0.826800i \(0.309841\pi\)
\(930\) 3.55963 0.116725
\(931\) 0.614387 0.0201357
\(932\) 28.3172 0.927560
\(933\) −12.7712 −0.418111
\(934\) −23.4758 −0.768150
\(935\) 16.3451 0.534543
\(936\) 2.30580 0.0753675
\(937\) −14.0845 −0.460119 −0.230060 0.973177i \(-0.573892\pi\)
−0.230060 + 0.973177i \(0.573892\pi\)
\(938\) −15.8346 −0.517017
\(939\) −3.27100 −0.106745
\(940\) 5.42784 0.177037
\(941\) 14.2499 0.464533 0.232267 0.972652i \(-0.425386\pi\)
0.232267 + 0.972652i \(0.425386\pi\)
\(942\) 2.16543 0.0705535
\(943\) 31.8596 1.03749
\(944\) −1.42784 −0.0464723
\(945\) 6.92297 0.225204
\(946\) −47.6834 −1.55032
\(947\) −29.8203 −0.969029 −0.484515 0.874783i \(-0.661004\pi\)
−0.484515 + 0.874783i \(0.661004\pi\)
\(948\) −0.715313 −0.0232323
\(949\) 8.96520 0.291023
\(950\) 0.406728 0.0131960
\(951\) −5.00394 −0.162264
\(952\) −7.32401 −0.237373
\(953\) 8.64803 0.280137 0.140069 0.990142i \(-0.455268\pi\)
0.140069 + 0.990142i \(0.455268\pi\)
\(954\) 6.64744 0.215219
\(955\) 19.3326 0.625588
\(956\) 14.3788 0.465043
\(957\) −14.0981 −0.455728
\(958\) −8.19907 −0.264900
\(959\) −42.8557 −1.38388
\(960\) −0.406728 −0.0131271
\(961\) 45.5950 1.47081
\(962\) 0.813457 0.0262269
\(963\) −43.2624 −1.39411
\(964\) 18.2749 0.588596
\(965\) 20.0422 0.645182
\(966\) 5.95777 0.191688
\(967\) −45.0325 −1.44815 −0.724074 0.689723i \(-0.757732\pi\)
−0.724074 + 0.689723i \(0.757732\pi\)
\(968\) 31.3874 1.00883
\(969\) −0.415317 −0.0133419
\(970\) −2.34513 −0.0752976
\(971\) 26.3029 0.844100 0.422050 0.906573i \(-0.361311\pi\)
0.422050 + 0.906573i \(0.361311\pi\)
\(972\) 10.1854 0.326696
\(973\) 50.6623 1.62416
\(974\) 0.953831 0.0305627
\(975\) −0.330856 −0.0105959
\(976\) −1.32401 −0.0423807
\(977\) 25.6549 0.820772 0.410386 0.911912i \(-0.365394\pi\)
0.410386 + 0.911912i \(0.365394\pi\)
\(978\) 6.18260 0.197698
\(979\) 39.0633 1.24847
\(980\) −1.51056 −0.0482530
\(981\) −12.1969 −0.389416
\(982\) 19.3383 0.617110
\(983\) −55.3611 −1.76575 −0.882873 0.469611i \(-0.844394\pi\)
−0.882873 + 0.469611i \(0.844394\pi\)
\(984\) 2.58074 0.0822711
\(985\) −15.6269 −0.497915
\(986\) 13.3662 0.425668
\(987\) −6.44037 −0.204999
\(988\) −0.330856 −0.0105259
\(989\) −36.7747 −1.16937
\(990\) 18.4546 0.586527
\(991\) −3.24814 −0.103181 −0.0515903 0.998668i \(-0.516429\pi\)
−0.0515903 + 0.998668i \(0.516429\pi\)
\(992\) −8.75186 −0.277872
\(993\) −2.75754 −0.0875080
\(994\) 42.7325 1.35539
\(995\) −20.2835 −0.643031
\(996\) −2.86937 −0.0909195
\(997\) −12.6480 −0.400567 −0.200284 0.979738i \(-0.564186\pi\)
−0.200284 + 0.979738i \(0.564186\pi\)
\(998\) −5.63550 −0.178389
\(999\) 2.37309 0.0750811
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 370.2.a.g.1.2 3
3.2 odd 2 3330.2.a.bg.1.3 3
4.3 odd 2 2960.2.a.u.1.2 3
5.2 odd 4 1850.2.b.o.149.5 6
5.3 odd 4 1850.2.b.o.149.2 6
5.4 even 2 1850.2.a.z.1.2 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
370.2.a.g.1.2 3 1.1 even 1 trivial
1850.2.a.z.1.2 3 5.4 even 2
1850.2.b.o.149.2 6 5.3 odd 4
1850.2.b.o.149.5 6 5.2 odd 4
2960.2.a.u.1.2 3 4.3 odd 2
3330.2.a.bg.1.3 3 3.2 odd 2