Properties

Label 507.2.b.f.337.2
Level $507$
Weight $2$
Character 507.337
Analytic conductor $4.048$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 507 = 3 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 507.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(4.04841538248\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.153664.1
Defining polynomial: \(x^{6} + 5 x^{4} + 6 x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 337.2
Root \(-1.80194i\) of defining polynomial
Character \(\chi\) \(=\) 507.337
Dual form 507.2.b.f.337.5

$q$-expansion

\(f(q)\) \(=\) \(q-2.35690i q^{2} -1.00000 q^{3} -3.55496 q^{4} -3.69202i q^{5} +2.35690i q^{6} -0.801938i q^{7} +3.66487i q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-2.35690i q^{2} -1.00000 q^{3} -3.55496 q^{4} -3.69202i q^{5} +2.35690i q^{6} -0.801938i q^{7} +3.66487i q^{8} +1.00000 q^{9} -8.70171 q^{10} -2.85086i q^{11} +3.55496 q^{12} -1.89008 q^{14} +3.69202i q^{15} +1.52781 q^{16} -2.93900 q^{17} -2.35690i q^{18} +2.44504i q^{19} +13.1250i q^{20} +0.801938i q^{21} -6.71917 q^{22} +7.78986 q^{23} -3.66487i q^{24} -8.63102 q^{25} -1.00000 q^{27} +2.85086i q^{28} +3.85086 q^{29} +8.70171 q^{30} -2.34481i q^{31} +3.72886i q^{32} +2.85086i q^{33} +6.92692i q^{34} -2.96077 q^{35} -3.55496 q^{36} +7.44504i q^{37} +5.76271 q^{38} +13.5308 q^{40} +0.850855i q^{41} +1.89008 q^{42} +1.61596 q^{43} +10.1347i q^{44} -3.69202i q^{45} -18.3599i q^{46} +2.44504i q^{47} -1.52781 q^{48} +6.35690 q^{49} +20.3424i q^{50} +2.93900 q^{51} -9.96077 q^{53} +2.35690i q^{54} -10.5254 q^{55} +2.93900 q^{56} -2.44504i q^{57} -9.07606i q^{58} -5.38404i q^{59} -13.1250i q^{60} -13.2567 q^{61} -5.52648 q^{62} -0.801938i q^{63} +11.8442 q^{64} +6.71917 q^{66} -14.3937i q^{67} +10.4480 q^{68} -7.78986 q^{69} +6.97823i q^{70} -8.12498i q^{71} +3.66487i q^{72} -11.8877i q^{73} +17.5472 q^{74} +8.63102 q^{75} -8.69202i q^{76} -2.28621 q^{77} +5.40581 q^{79} -5.64071i q^{80} +1.00000 q^{81} +2.00538 q^{82} -7.04892i q^{83} -2.85086i q^{84} +10.8509i q^{85} -3.80864i q^{86} -3.85086 q^{87} +10.4480 q^{88} +1.13169i q^{89} -8.70171 q^{90} -27.6926 q^{92} +2.34481i q^{93} +5.76271 q^{94} +9.02715 q^{95} -3.72886i q^{96} +5.94438i q^{97} -14.9825i q^{98} -2.85086i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q - 6q^{3} - 22q^{4} + 6q^{9} + O(q^{10}) \) \( 6q - 6q^{3} - 22q^{4} + 6q^{9} + 2q^{10} + 22q^{12} - 10q^{14} + 22q^{16} + 2q^{17} - 18q^{22} - 22q^{25} - 6q^{27} - 4q^{29} - 2q^{30} + 8q^{35} - 22q^{36} + 6q^{40} + 10q^{42} + 30q^{43} - 22q^{48} + 30q^{49} - 2q^{51} - 34q^{53} + 6q^{55} - 2q^{56} - 26q^{61} + 4q^{62} + 18q^{66} - 26q^{68} + 30q^{74} + 22q^{75} - 30q^{77} + 6q^{79} + 6q^{81} + 6q^{82} + 4q^{87} - 26q^{88} + 2q^{90} + 14q^{92} + 42q^{95} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/507\mathbb{Z}\right)^\times\).

\(n\) \(170\) \(340\)
\(\chi(n)\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 2.35690i − 1.66658i −0.552838 0.833289i \(-0.686455\pi\)
0.552838 0.833289i \(-0.313545\pi\)
\(3\) −1.00000 −0.577350
\(4\) −3.55496 −1.77748
\(5\) − 3.69202i − 1.65112i −0.564313 0.825561i \(-0.690859\pi\)
0.564313 0.825561i \(-0.309141\pi\)
\(6\) 2.35690i 0.962199i
\(7\) − 0.801938i − 0.303104i −0.988449 0.151552i \(-0.951573\pi\)
0.988449 0.151552i \(-0.0484271\pi\)
\(8\) 3.66487i 1.29573i
\(9\) 1.00000 0.333333
\(10\) −8.70171 −2.75172
\(11\) − 2.85086i − 0.859565i −0.902932 0.429783i \(-0.858590\pi\)
0.902932 0.429783i \(-0.141410\pi\)
\(12\) 3.55496 1.02623
\(13\) 0 0
\(14\) −1.89008 −0.505146
\(15\) 3.69202i 0.953276i
\(16\) 1.52781 0.381953
\(17\) −2.93900 −0.712812 −0.356406 0.934331i \(-0.615998\pi\)
−0.356406 + 0.934331i \(0.615998\pi\)
\(18\) − 2.35690i − 0.555526i
\(19\) 2.44504i 0.560931i 0.959864 + 0.280466i \(0.0904888\pi\)
−0.959864 + 0.280466i \(0.909511\pi\)
\(20\) 13.1250i 2.93484i
\(21\) 0.801938i 0.174997i
\(22\) −6.71917 −1.43253
\(23\) 7.78986 1.62430 0.812149 0.583451i \(-0.198298\pi\)
0.812149 + 0.583451i \(0.198298\pi\)
\(24\) − 3.66487i − 0.748089i
\(25\) −8.63102 −1.72620
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 2.85086i 0.538761i
\(29\) 3.85086 0.715086 0.357543 0.933897i \(-0.383615\pi\)
0.357543 + 0.933897i \(0.383615\pi\)
\(30\) 8.70171 1.58871
\(31\) − 2.34481i − 0.421141i −0.977579 0.210571i \(-0.932468\pi\)
0.977579 0.210571i \(-0.0675322\pi\)
\(32\) 3.72886i 0.659175i
\(33\) 2.85086i 0.496270i
\(34\) 6.92692i 1.18796i
\(35\) −2.96077 −0.500462
\(36\) −3.55496 −0.592493
\(37\) 7.44504i 1.22396i 0.790874 + 0.611979i \(0.209626\pi\)
−0.790874 + 0.611979i \(0.790374\pi\)
\(38\) 5.76271 0.934835
\(39\) 0 0
\(40\) 13.5308 2.13941
\(41\) 0.850855i 0.132881i 0.997790 + 0.0664406i \(0.0211643\pi\)
−0.997790 + 0.0664406i \(0.978836\pi\)
\(42\) 1.89008 0.291646
\(43\) 1.61596 0.246431 0.123216 0.992380i \(-0.460679\pi\)
0.123216 + 0.992380i \(0.460679\pi\)
\(44\) 10.1347i 1.52786i
\(45\) − 3.69202i − 0.550374i
\(46\) − 18.3599i − 2.70702i
\(47\) 2.44504i 0.356646i 0.983972 + 0.178323i \(0.0570672\pi\)
−0.983972 + 0.178323i \(0.942933\pi\)
\(48\) −1.52781 −0.220521
\(49\) 6.35690 0.908128
\(50\) 20.3424i 2.87685i
\(51\) 2.93900 0.411542
\(52\) 0 0
\(53\) −9.96077 −1.36822 −0.684109 0.729380i \(-0.739809\pi\)
−0.684109 + 0.729380i \(0.739809\pi\)
\(54\) 2.35690i 0.320733i
\(55\) −10.5254 −1.41925
\(56\) 2.93900 0.392741
\(57\) − 2.44504i − 0.323854i
\(58\) − 9.07606i − 1.19175i
\(59\) − 5.38404i − 0.700943i −0.936573 0.350471i \(-0.886021\pi\)
0.936573 0.350471i \(-0.113979\pi\)
\(60\) − 13.1250i − 1.69443i
\(61\) −13.2567 −1.69734 −0.848671 0.528921i \(-0.822597\pi\)
−0.848671 + 0.528921i \(0.822597\pi\)
\(62\) −5.52648 −0.701864
\(63\) − 0.801938i − 0.101035i
\(64\) 11.8442 1.48052
\(65\) 0 0
\(66\) 6.71917 0.827072
\(67\) − 14.3937i − 1.75847i −0.476384 0.879237i \(-0.658053\pi\)
0.476384 0.879237i \(-0.341947\pi\)
\(68\) 10.4480 1.26701
\(69\) −7.78986 −0.937788
\(70\) 6.97823i 0.834058i
\(71\) − 8.12498i − 0.964258i −0.876100 0.482129i \(-0.839864\pi\)
0.876100 0.482129i \(-0.160136\pi\)
\(72\) 3.66487i 0.431910i
\(73\) − 11.8877i − 1.39135i −0.718357 0.695674i \(-0.755106\pi\)
0.718357 0.695674i \(-0.244894\pi\)
\(74\) 17.5472 2.03982
\(75\) 8.63102 0.996625
\(76\) − 8.69202i − 0.997043i
\(77\) −2.28621 −0.260538
\(78\) 0 0
\(79\) 5.40581 0.608202 0.304101 0.952640i \(-0.401644\pi\)
0.304101 + 0.952640i \(0.401644\pi\)
\(80\) − 5.64071i − 0.630651i
\(81\) 1.00000 0.111111
\(82\) 2.00538 0.221457
\(83\) − 7.04892i − 0.773719i −0.922139 0.386860i \(-0.873560\pi\)
0.922139 0.386860i \(-0.126440\pi\)
\(84\) − 2.85086i − 0.311054i
\(85\) 10.8509i 1.17694i
\(86\) − 3.80864i − 0.410696i
\(87\) −3.85086 −0.412855
\(88\) 10.4480 1.11376
\(89\) 1.13169i 0.119959i 0.998200 + 0.0599793i \(0.0191035\pi\)
−0.998200 + 0.0599793i \(0.980897\pi\)
\(90\) −8.70171 −0.917241
\(91\) 0 0
\(92\) −27.6926 −2.88715
\(93\) 2.34481i 0.243146i
\(94\) 5.76271 0.594378
\(95\) 9.02715 0.926166
\(96\) − 3.72886i − 0.380575i
\(97\) 5.94438i 0.603560i 0.953378 + 0.301780i \(0.0975808\pi\)
−0.953378 + 0.301780i \(0.902419\pi\)
\(98\) − 14.9825i − 1.51347i
\(99\) − 2.85086i − 0.286522i
\(100\) 30.6829 3.06829
\(101\) −4.62565 −0.460269 −0.230134 0.973159i \(-0.573917\pi\)
−0.230134 + 0.973159i \(0.573917\pi\)
\(102\) − 6.92692i − 0.685867i
\(103\) −1.20775 −0.119003 −0.0595016 0.998228i \(-0.518951\pi\)
−0.0595016 + 0.998228i \(0.518951\pi\)
\(104\) 0 0
\(105\) 2.96077 0.288942
\(106\) 23.4765i 2.28024i
\(107\) −9.52111 −0.920440 −0.460220 0.887805i \(-0.652229\pi\)
−0.460220 + 0.887805i \(0.652229\pi\)
\(108\) 3.55496 0.342076
\(109\) 1.78448i 0.170922i 0.996342 + 0.0854611i \(0.0272363\pi\)
−0.996342 + 0.0854611i \(0.972764\pi\)
\(110\) 24.8073i 2.36528i
\(111\) − 7.44504i − 0.706652i
\(112\) − 1.22521i − 0.115771i
\(113\) 4.95108 0.465759 0.232879 0.972506i \(-0.425185\pi\)
0.232879 + 0.972506i \(0.425185\pi\)
\(114\) −5.76271 −0.539727
\(115\) − 28.7603i − 2.68191i
\(116\) −13.6896 −1.27105
\(117\) 0 0
\(118\) −12.6896 −1.16817
\(119\) 2.35690i 0.216056i
\(120\) −13.5308 −1.23519
\(121\) 2.87263 0.261148
\(122\) 31.2446i 2.82875i
\(123\) − 0.850855i − 0.0767190i
\(124\) 8.33572i 0.748569i
\(125\) 13.4058i 1.19905i
\(126\) −1.89008 −0.168382
\(127\) −5.67025 −0.503153 −0.251577 0.967837i \(-0.580949\pi\)
−0.251577 + 0.967837i \(0.580949\pi\)
\(128\) − 20.4577i − 1.80822i
\(129\) −1.61596 −0.142277
\(130\) 0 0
\(131\) 18.2228 1.59213 0.796067 0.605208i \(-0.206910\pi\)
0.796067 + 0.605208i \(0.206910\pi\)
\(132\) − 10.1347i − 0.882110i
\(133\) 1.96077 0.170020
\(134\) −33.9245 −2.93063
\(135\) 3.69202i 0.317759i
\(136\) − 10.7711i − 0.923612i
\(137\) 9.45042i 0.807404i 0.914891 + 0.403702i \(0.132277\pi\)
−0.914891 + 0.403702i \(0.867723\pi\)
\(138\) 18.3599i 1.56290i
\(139\) 4.01507 0.340553 0.170277 0.985396i \(-0.445534\pi\)
0.170277 + 0.985396i \(0.445534\pi\)
\(140\) 10.5254 0.889560
\(141\) − 2.44504i − 0.205910i
\(142\) −19.1497 −1.60701
\(143\) 0 0
\(144\) 1.52781 0.127318
\(145\) − 14.2174i − 1.18069i
\(146\) −28.0180 −2.31879
\(147\) −6.35690 −0.524308
\(148\) − 26.4668i − 2.17556i
\(149\) − 19.4058i − 1.58979i −0.606750 0.794893i \(-0.707527\pi\)
0.606750 0.794893i \(-0.292473\pi\)
\(150\) − 20.3424i − 1.66095i
\(151\) − 12.3623i − 1.00603i −0.864278 0.503014i \(-0.832225\pi\)
0.864278 0.503014i \(-0.167775\pi\)
\(152\) −8.96077 −0.726815
\(153\) −2.93900 −0.237604
\(154\) 5.38835i 0.434206i
\(155\) −8.65710 −0.695355
\(156\) 0 0
\(157\) −18.6775 −1.49063 −0.745315 0.666712i \(-0.767701\pi\)
−0.745315 + 0.666712i \(0.767701\pi\)
\(158\) − 12.7409i − 1.01361i
\(159\) 9.96077 0.789941
\(160\) 13.7670 1.08838
\(161\) − 6.24698i − 0.492331i
\(162\) − 2.35690i − 0.185175i
\(163\) 12.3394i 0.966499i 0.875483 + 0.483250i \(0.160544\pi\)
−0.875483 + 0.483250i \(0.839456\pi\)
\(164\) − 3.02475i − 0.236194i
\(165\) 10.5254 0.819403
\(166\) −16.6136 −1.28946
\(167\) 11.4940i 0.889429i 0.895672 + 0.444715i \(0.146695\pi\)
−0.895672 + 0.444715i \(0.853305\pi\)
\(168\) −2.93900 −0.226749
\(169\) 0 0
\(170\) 25.5743 1.96146
\(171\) 2.44504i 0.186977i
\(172\) −5.74466 −0.438026
\(173\) −12.1142 −0.921028 −0.460514 0.887653i \(-0.652335\pi\)
−0.460514 + 0.887653i \(0.652335\pi\)
\(174\) 9.07606i 0.688055i
\(175\) 6.92154i 0.523219i
\(176\) − 4.35557i − 0.328313i
\(177\) 5.38404i 0.404689i
\(178\) 2.66727 0.199920
\(179\) 0.538565 0.0402542 0.0201271 0.999797i \(-0.493593\pi\)
0.0201271 + 0.999797i \(0.493593\pi\)
\(180\) 13.1250i 0.978278i
\(181\) 23.2838 1.73067 0.865336 0.501192i \(-0.167105\pi\)
0.865336 + 0.501192i \(0.167105\pi\)
\(182\) 0 0
\(183\) 13.2567 0.979961
\(184\) 28.5488i 2.10465i
\(185\) 27.4873 2.02090
\(186\) 5.52648 0.405221
\(187\) 8.37867i 0.612709i
\(188\) − 8.69202i − 0.633931i
\(189\) 0.801938i 0.0583324i
\(190\) − 21.2760i − 1.54353i
\(191\) 16.7657 1.21312 0.606561 0.795037i \(-0.292548\pi\)
0.606561 + 0.795037i \(0.292548\pi\)
\(192\) −11.8442 −0.854778
\(193\) 25.7439i 1.85309i 0.376186 + 0.926544i \(0.377235\pi\)
−0.376186 + 0.926544i \(0.622765\pi\)
\(194\) 14.0103 1.00588
\(195\) 0 0
\(196\) −22.5985 −1.61418
\(197\) − 21.4209i − 1.52617i −0.646296 0.763087i \(-0.723683\pi\)
0.646296 0.763087i \(-0.276317\pi\)
\(198\) −6.71917 −0.477511
\(199\) −3.52781 −0.250080 −0.125040 0.992152i \(-0.539906\pi\)
−0.125040 + 0.992152i \(0.539906\pi\)
\(200\) − 31.6316i − 2.23669i
\(201\) 14.3937i 1.01526i
\(202\) 10.9022i 0.767074i
\(203\) − 3.08815i − 0.216745i
\(204\) −10.4480 −0.731508
\(205\) 3.14138 0.219403
\(206\) 2.84654i 0.198328i
\(207\) 7.78986 0.541432
\(208\) 0 0
\(209\) 6.97046 0.482157
\(210\) − 6.97823i − 0.481544i
\(211\) 1.21552 0.0836799 0.0418399 0.999124i \(-0.486678\pi\)
0.0418399 + 0.999124i \(0.486678\pi\)
\(212\) 35.4101 2.43198
\(213\) 8.12498i 0.556715i
\(214\) 22.4403i 1.53398i
\(215\) − 5.96615i − 0.406888i
\(216\) − 3.66487i − 0.249363i
\(217\) −1.88040 −0.127650
\(218\) 4.20583 0.284855
\(219\) 11.8877i 0.803296i
\(220\) 37.4174 2.52268
\(221\) 0 0
\(222\) −17.5472 −1.17769
\(223\) 17.3884i 1.16441i 0.813042 + 0.582205i \(0.197810\pi\)
−0.813042 + 0.582205i \(0.802190\pi\)
\(224\) 2.99031 0.199799
\(225\) −8.63102 −0.575402
\(226\) − 11.6692i − 0.776223i
\(227\) − 17.4155i − 1.15591i −0.816070 0.577954i \(-0.803851\pi\)
0.816070 0.577954i \(-0.196149\pi\)
\(228\) 8.69202i 0.575643i
\(229\) 18.7603i 1.23972i 0.784714 + 0.619858i \(0.212810\pi\)
−0.784714 + 0.619858i \(0.787190\pi\)
\(230\) −67.7851 −4.46962
\(231\) 2.28621 0.150421
\(232\) 14.1129i 0.926557i
\(233\) −3.95108 −0.258844 −0.129422 0.991590i \(-0.541312\pi\)
−0.129422 + 0.991590i \(0.541312\pi\)
\(234\) 0 0
\(235\) 9.02715 0.588866
\(236\) 19.1400i 1.24591i
\(237\) −5.40581 −0.351145
\(238\) 5.55496 0.360074
\(239\) − 0.818331i − 0.0529334i −0.999650 0.0264667i \(-0.991574\pi\)
0.999650 0.0264667i \(-0.00842560\pi\)
\(240\) 5.64071i 0.364106i
\(241\) 6.03252i 0.388589i 0.980943 + 0.194295i \(0.0622417\pi\)
−0.980943 + 0.194295i \(0.937758\pi\)
\(242\) − 6.77048i − 0.435223i
\(243\) −1.00000 −0.0641500
\(244\) 47.1269 3.01699
\(245\) − 23.4698i − 1.49943i
\(246\) −2.00538 −0.127858
\(247\) 0 0
\(248\) 8.59345 0.545685
\(249\) 7.04892i 0.446707i
\(250\) 31.5961 1.99831
\(251\) 26.8799 1.69665 0.848323 0.529479i \(-0.177613\pi\)
0.848323 + 0.529479i \(0.177613\pi\)
\(252\) 2.85086i 0.179587i
\(253\) − 22.2078i − 1.39619i
\(254\) 13.3642i 0.838544i
\(255\) − 10.8509i − 0.679507i
\(256\) −24.5284 −1.53303
\(257\) 9.05323 0.564725 0.282362 0.959308i \(-0.408882\pi\)
0.282362 + 0.959308i \(0.408882\pi\)
\(258\) 3.80864i 0.237116i
\(259\) 5.97046 0.370986
\(260\) 0 0
\(261\) 3.85086 0.238362
\(262\) − 42.9493i − 2.65342i
\(263\) 23.1511 1.42756 0.713778 0.700372i \(-0.246983\pi\)
0.713778 + 0.700372i \(0.246983\pi\)
\(264\) −10.4480 −0.643032
\(265\) 36.7754i 2.25909i
\(266\) − 4.62133i − 0.283352i
\(267\) − 1.13169i − 0.0692581i
\(268\) 51.1691i 3.12565i
\(269\) −2.42088 −0.147604 −0.0738018 0.997273i \(-0.523513\pi\)
−0.0738018 + 0.997273i \(0.523513\pi\)
\(270\) 8.70171 0.529569
\(271\) − 21.4450i − 1.30269i −0.758780 0.651347i \(-0.774204\pi\)
0.758780 0.651347i \(-0.225796\pi\)
\(272\) −4.49024 −0.272261
\(273\) 0 0
\(274\) 22.2737 1.34560
\(275\) 24.6058i 1.48379i
\(276\) 27.6926 1.66690
\(277\) 14.8073 0.889685 0.444843 0.895609i \(-0.353260\pi\)
0.444843 + 0.895609i \(0.353260\pi\)
\(278\) − 9.46309i − 0.567558i
\(279\) − 2.34481i − 0.140380i
\(280\) − 10.8509i − 0.648463i
\(281\) 14.5036i 0.865215i 0.901582 + 0.432608i \(0.142406\pi\)
−0.901582 + 0.432608i \(0.857594\pi\)
\(282\) −5.76271 −0.343164
\(283\) −25.6722 −1.52605 −0.763026 0.646368i \(-0.776287\pi\)
−0.763026 + 0.646368i \(0.776287\pi\)
\(284\) 28.8840i 1.71395i
\(285\) −9.02715 −0.534722
\(286\) 0 0
\(287\) 0.682333 0.0402768
\(288\) 3.72886i 0.219725i
\(289\) −8.36227 −0.491898
\(290\) −33.5090 −1.96772
\(291\) − 5.94438i − 0.348466i
\(292\) 42.2602i 2.47309i
\(293\) − 26.5230i − 1.54949i −0.632273 0.774746i \(-0.717878\pi\)
0.632273 0.774746i \(-0.282122\pi\)
\(294\) 14.9825i 0.873800i
\(295\) −19.8780 −1.15734
\(296\) −27.2851 −1.58592
\(297\) 2.85086i 0.165423i
\(298\) −45.7375 −2.64950
\(299\) 0 0
\(300\) −30.6829 −1.77148
\(301\) − 1.29590i − 0.0746943i
\(302\) −29.1366 −1.67662
\(303\) 4.62565 0.265736
\(304\) 3.73556i 0.214249i
\(305\) 48.9439i 2.80252i
\(306\) 6.92692i 0.395986i
\(307\) − 8.24698i − 0.470680i −0.971913 0.235340i \(-0.924380\pi\)
0.971913 0.235340i \(-0.0756204\pi\)
\(308\) 8.12737 0.463100
\(309\) 1.20775 0.0687066
\(310\) 20.4039i 1.15886i
\(311\) 14.4179 0.817564 0.408782 0.912632i \(-0.365954\pi\)
0.408782 + 0.912632i \(0.365954\pi\)
\(312\) 0 0
\(313\) 14.2338 0.804544 0.402272 0.915520i \(-0.368221\pi\)
0.402272 + 0.915520i \(0.368221\pi\)
\(314\) 44.0210i 2.48425i
\(315\) −2.96077 −0.166821
\(316\) −19.2174 −1.08107
\(317\) 6.84415i 0.384406i 0.981355 + 0.192203i \(0.0615632\pi\)
−0.981355 + 0.192203i \(0.938437\pi\)
\(318\) − 23.4765i − 1.31650i
\(319\) − 10.9782i − 0.614663i
\(320\) − 43.7289i − 2.44452i
\(321\) 9.52111 0.531416
\(322\) −14.7235 −0.820507
\(323\) − 7.18598i − 0.399839i
\(324\) −3.55496 −0.197498
\(325\) 0 0
\(326\) 29.0828 1.61075
\(327\) − 1.78448i − 0.0986819i
\(328\) −3.11828 −0.172178
\(329\) 1.96077 0.108101
\(330\) − 24.8073i − 1.36560i
\(331\) − 9.44265i − 0.519015i −0.965741 0.259507i \(-0.916440\pi\)
0.965741 0.259507i \(-0.0835602\pi\)
\(332\) 25.0586i 1.37527i
\(333\) 7.44504i 0.407986i
\(334\) 27.0901 1.48230
\(335\) −53.1420 −2.90346
\(336\) 1.22521i 0.0668406i
\(337\) −2.64310 −0.143979 −0.0719895 0.997405i \(-0.522935\pi\)
−0.0719895 + 0.997405i \(0.522935\pi\)
\(338\) 0 0
\(339\) −4.95108 −0.268906
\(340\) − 38.5743i − 2.09199i
\(341\) −6.68473 −0.361998
\(342\) 5.76271 0.311612
\(343\) − 10.7114i − 0.578361i
\(344\) 5.92228i 0.319308i
\(345\) 28.7603i 1.54840i
\(346\) 28.5520i 1.53496i
\(347\) −10.1588 −0.545355 −0.272677 0.962106i \(-0.587909\pi\)
−0.272677 + 0.962106i \(0.587909\pi\)
\(348\) 13.6896 0.733841
\(349\) 10.4397i 0.558822i 0.960172 + 0.279411i \(0.0901393\pi\)
−0.960172 + 0.279411i \(0.909861\pi\)
\(350\) 16.3134 0.871986
\(351\) 0 0
\(352\) 10.6304 0.566604
\(353\) − 18.2911i − 0.973538i −0.873531 0.486769i \(-0.838175\pi\)
0.873531 0.486769i \(-0.161825\pi\)
\(354\) 12.6896 0.674446
\(355\) −29.9976 −1.59211
\(356\) − 4.02310i − 0.213224i
\(357\) − 2.35690i − 0.124740i
\(358\) − 1.26934i − 0.0670867i
\(359\) − 15.2731i − 0.806081i −0.915182 0.403041i \(-0.867953\pi\)
0.915182 0.403041i \(-0.132047\pi\)
\(360\) 13.5308 0.713136
\(361\) 13.0218 0.685356
\(362\) − 54.8775i − 2.88430i
\(363\) −2.87263 −0.150774
\(364\) 0 0
\(365\) −43.8896 −2.29729
\(366\) − 31.2446i − 1.63318i
\(367\) −22.2717 −1.16258 −0.581288 0.813698i \(-0.697451\pi\)
−0.581288 + 0.813698i \(0.697451\pi\)
\(368\) 11.9014 0.620405
\(369\) 0.850855i 0.0442937i
\(370\) − 64.7846i − 3.36799i
\(371\) 7.98792i 0.414712i
\(372\) − 8.33572i − 0.432187i
\(373\) 4.12631 0.213652 0.106826 0.994278i \(-0.465931\pi\)
0.106826 + 0.994278i \(0.465931\pi\)
\(374\) 19.7476 1.02113
\(375\) − 13.4058i − 0.692273i
\(376\) −8.96077 −0.462116
\(377\) 0 0
\(378\) 1.89008 0.0972154
\(379\) 10.7071i 0.549986i 0.961446 + 0.274993i \(0.0886756\pi\)
−0.961446 + 0.274993i \(0.911324\pi\)
\(380\) −32.0911 −1.64624
\(381\) 5.67025 0.290496
\(382\) − 39.5150i − 2.02176i
\(383\) − 6.52648i − 0.333488i −0.986000 0.166744i \(-0.946675\pi\)
0.986000 0.166744i \(-0.0533253\pi\)
\(384\) 20.4577i 1.04398i
\(385\) 8.44073i 0.430179i
\(386\) 60.6757 3.08831
\(387\) 1.61596 0.0821437
\(388\) − 21.1320i − 1.07282i
\(389\) 11.7922 0.597891 0.298945 0.954270i \(-0.403365\pi\)
0.298945 + 0.954270i \(0.403365\pi\)
\(390\) 0 0
\(391\) −22.8944 −1.15782
\(392\) 23.2972i 1.17669i
\(393\) −18.2228 −0.919219
\(394\) −50.4868 −2.54349
\(395\) − 19.9584i − 1.00422i
\(396\) 10.1347i 0.509286i
\(397\) − 12.5429i − 0.629509i −0.949173 0.314754i \(-0.898078\pi\)
0.949173 0.314754i \(-0.101922\pi\)
\(398\) 8.31468i 0.416777i
\(399\) −1.96077 −0.0981613
\(400\) −13.1866 −0.659329
\(401\) 17.8702i 0.892397i 0.894934 + 0.446198i \(0.147222\pi\)
−0.894934 + 0.446198i \(0.852778\pi\)
\(402\) 33.9245 1.69200
\(403\) 0 0
\(404\) 16.4440 0.818118
\(405\) − 3.69202i − 0.183458i
\(406\) −7.27844 −0.361223
\(407\) 21.2247 1.05207
\(408\) 10.7711i 0.533247i
\(409\) 27.9119i 1.38015i 0.723737 + 0.690076i \(0.242423\pi\)
−0.723737 + 0.690076i \(0.757577\pi\)
\(410\) − 7.40389i − 0.365652i
\(411\) − 9.45042i − 0.466155i
\(412\) 4.29350 0.211526
\(413\) −4.31767 −0.212459
\(414\) − 18.3599i − 0.902339i
\(415\) −26.0248 −1.27750
\(416\) 0 0
\(417\) −4.01507 −0.196619
\(418\) − 16.4286i − 0.803551i
\(419\) −16.4034 −0.801360 −0.400680 0.916218i \(-0.631226\pi\)
−0.400680 + 0.916218i \(0.631226\pi\)
\(420\) −10.5254 −0.513588
\(421\) 3.03684i 0.148006i 0.997258 + 0.0740032i \(0.0235775\pi\)
−0.997258 + 0.0740032i \(0.976423\pi\)
\(422\) − 2.86486i − 0.139459i
\(423\) 2.44504i 0.118882i
\(424\) − 36.5050i − 1.77284i
\(425\) 25.3666 1.23046
\(426\) 19.1497 0.927808
\(427\) 10.6310i 0.514471i
\(428\) 33.8471 1.63606
\(429\) 0 0
\(430\) −14.0616 −0.678110
\(431\) 3.33811i 0.160791i 0.996763 + 0.0803955i \(0.0256183\pi\)
−0.996763 + 0.0803955i \(0.974382\pi\)
\(432\) −1.52781 −0.0735068
\(433\) −11.9028 −0.572010 −0.286005 0.958228i \(-0.592327\pi\)
−0.286005 + 0.958228i \(0.592327\pi\)
\(434\) 4.43190i 0.212738i
\(435\) 14.2174i 0.681674i
\(436\) − 6.34375i − 0.303810i
\(437\) 19.0465i 0.911119i
\(438\) 28.0180 1.33875
\(439\) −3.71810 −0.177455 −0.0887277 0.996056i \(-0.528280\pi\)
−0.0887277 + 0.996056i \(0.528280\pi\)
\(440\) − 38.5743i − 1.83896i
\(441\) 6.35690 0.302709
\(442\) 0 0
\(443\) 1.45712 0.0692300 0.0346150 0.999401i \(-0.488979\pi\)
0.0346150 + 0.999401i \(0.488979\pi\)
\(444\) 26.4668i 1.25606i
\(445\) 4.17821 0.198066
\(446\) 40.9825 1.94058
\(447\) 19.4058i 0.917863i
\(448\) − 9.49827i − 0.448751i
\(449\) − 12.1274i − 0.572326i −0.958181 0.286163i \(-0.907620\pi\)
0.958181 0.286163i \(-0.0923799\pi\)
\(450\) 20.3424i 0.958951i
\(451\) 2.42566 0.114220
\(452\) −17.6009 −0.827876
\(453\) 12.3623i 0.580830i
\(454\) −41.0465 −1.92641
\(455\) 0 0
\(456\) 8.96077 0.419627
\(457\) 3.44803i 0.161292i 0.996743 + 0.0806459i \(0.0256983\pi\)
−0.996743 + 0.0806459i \(0.974302\pi\)
\(458\) 44.2161 2.06608
\(459\) 2.93900 0.137181
\(460\) 102.242i 4.76704i
\(461\) 6.75600i 0.314658i 0.987546 + 0.157329i \(0.0502884\pi\)
−0.987546 + 0.157329i \(0.949712\pi\)
\(462\) − 5.38835i − 0.250689i
\(463\) − 7.45175i − 0.346312i −0.984894 0.173156i \(-0.944604\pi\)
0.984894 0.173156i \(-0.0553965\pi\)
\(464\) 5.88338 0.273129
\(465\) 8.65710 0.401464
\(466\) 9.31229i 0.431384i
\(467\) −32.6098 −1.50900 −0.754502 0.656298i \(-0.772121\pi\)
−0.754502 + 0.656298i \(0.772121\pi\)
\(468\) 0 0
\(469\) −11.5429 −0.533001
\(470\) − 21.2760i − 0.981391i
\(471\) 18.6775 0.860616
\(472\) 19.7318 0.908232
\(473\) − 4.60686i − 0.211824i
\(474\) 12.7409i 0.585211i
\(475\) − 21.1032i − 0.968282i
\(476\) − 8.37867i − 0.384036i
\(477\) −9.96077 −0.456072
\(478\) −1.92872 −0.0882177
\(479\) 2.82908i 0.129264i 0.997909 + 0.0646321i \(0.0205874\pi\)
−0.997909 + 0.0646321i \(0.979413\pi\)
\(480\) −13.7670 −0.628376
\(481\) 0 0
\(482\) 14.2180 0.647614
\(483\) 6.24698i 0.284247i
\(484\) −10.2121 −0.464185
\(485\) 21.9468 0.996552
\(486\) 2.35690i 0.106911i
\(487\) 41.2935i 1.87119i 0.353079 + 0.935594i \(0.385135\pi\)
−0.353079 + 0.935594i \(0.614865\pi\)
\(488\) − 48.5840i − 2.19930i
\(489\) − 12.3394i − 0.558009i
\(490\) −55.3159 −2.49892
\(491\) 34.6698 1.56463 0.782313 0.622886i \(-0.214040\pi\)
0.782313 + 0.622886i \(0.214040\pi\)
\(492\) 3.02475i 0.136366i
\(493\) −11.3177 −0.509722
\(494\) 0 0
\(495\) −10.5254 −0.473082
\(496\) − 3.58243i − 0.160856i
\(497\) −6.51573 −0.292270
\(498\) 16.6136 0.744472
\(499\) − 17.9409i − 0.803146i −0.915827 0.401573i \(-0.868464\pi\)
0.915827 0.401573i \(-0.131536\pi\)
\(500\) − 47.6571i − 2.13129i
\(501\) − 11.4940i − 0.513512i
\(502\) − 63.3532i − 2.82759i
\(503\) −26.1812 −1.16736 −0.583681 0.811983i \(-0.698388\pi\)
−0.583681 + 0.811983i \(0.698388\pi\)
\(504\) 2.93900 0.130914
\(505\) 17.0780i 0.759960i
\(506\) −52.3414 −2.32686
\(507\) 0 0
\(508\) 20.1575 0.894345
\(509\) 5.50604i 0.244051i 0.992527 + 0.122025i \(0.0389390\pi\)
−0.992527 + 0.122025i \(0.961061\pi\)
\(510\) −25.5743 −1.13245
\(511\) −9.53319 −0.421723
\(512\) 16.8955i 0.746681i
\(513\) − 2.44504i − 0.107951i
\(514\) − 21.3375i − 0.941158i
\(515\) 4.45904i 0.196489i
\(516\) 5.74466 0.252895
\(517\) 6.97046 0.306560
\(518\) − 14.0718i − 0.618277i
\(519\) 12.1142 0.531756
\(520\) 0 0
\(521\) −26.7211 −1.17067 −0.585336 0.810791i \(-0.699037\pi\)
−0.585336 + 0.810791i \(0.699037\pi\)
\(522\) − 9.07606i − 0.397249i
\(523\) 36.5230 1.59704 0.798520 0.601968i \(-0.205617\pi\)
0.798520 + 0.601968i \(0.205617\pi\)
\(524\) −64.7813 −2.82999
\(525\) − 6.92154i − 0.302081i
\(526\) − 54.5646i − 2.37913i
\(527\) 6.89141i 0.300195i
\(528\) 4.35557i 0.189552i
\(529\) 37.6819 1.63834
\(530\) 86.6757 3.76495
\(531\) − 5.38404i − 0.233648i
\(532\) −6.97046 −0.302208
\(533\) 0 0
\(534\) −2.66727 −0.115424
\(535\) 35.1521i 1.51976i
\(536\) 52.7512 2.27851
\(537\) −0.538565 −0.0232408
\(538\) 5.70576i 0.245993i
\(539\) − 18.1226i − 0.780595i
\(540\) − 13.1250i − 0.564809i
\(541\) 18.4655i 0.793893i 0.917842 + 0.396947i \(0.129930\pi\)
−0.917842 + 0.396947i \(0.870070\pi\)
\(542\) −50.5437 −2.17104
\(543\) −23.2838 −0.999204
\(544\) − 10.9591i − 0.469868i
\(545\) 6.58834 0.282213
\(546\) 0 0
\(547\) 39.8471 1.70374 0.851870 0.523753i \(-0.175468\pi\)
0.851870 + 0.523753i \(0.175468\pi\)
\(548\) − 33.5958i − 1.43514i
\(549\) −13.2567 −0.565781
\(550\) 57.9933 2.47284
\(551\) 9.41550i 0.401114i
\(552\) − 28.5488i − 1.21512i
\(553\) − 4.33513i − 0.184348i
\(554\) − 34.8993i − 1.48273i
\(555\) −27.4873 −1.16677
\(556\) −14.2734 −0.605327
\(557\) 9.20477i 0.390018i 0.980801 + 0.195009i \(0.0624737\pi\)
−0.980801 + 0.195009i \(0.937526\pi\)
\(558\) −5.52648 −0.233955
\(559\) 0 0
\(560\) −4.52350 −0.191153
\(561\) − 8.37867i − 0.353748i
\(562\) 34.1836 1.44195
\(563\) 0.975246 0.0411017 0.0205509 0.999789i \(-0.493458\pi\)
0.0205509 + 0.999789i \(0.493458\pi\)
\(564\) 8.69202i 0.366000i
\(565\) − 18.2795i − 0.769024i
\(566\) 60.5066i 2.54328i
\(567\) − 0.801938i − 0.0336782i
\(568\) 29.7770 1.24942
\(569\) 16.8944 0.708250 0.354125 0.935198i \(-0.384779\pi\)
0.354125 + 0.935198i \(0.384779\pi\)
\(570\) 21.2760i 0.891155i
\(571\) 44.3226 1.85484 0.927421 0.374019i \(-0.122021\pi\)
0.927421 + 0.374019i \(0.122021\pi\)
\(572\) 0 0
\(573\) −16.7657 −0.700397
\(574\) − 1.60819i − 0.0671244i
\(575\) −67.2344 −2.80387
\(576\) 11.8442 0.493506
\(577\) − 3.56704i − 0.148498i −0.997240 0.0742489i \(-0.976344\pi\)
0.997240 0.0742489i \(-0.0236559\pi\)
\(578\) 19.7090i 0.819787i
\(579\) − 25.7439i − 1.06988i
\(580\) 50.5424i 2.09866i
\(581\) −5.65279 −0.234517
\(582\) −14.0103 −0.580745
\(583\) 28.3967i 1.17607i
\(584\) 43.5669 1.80281
\(585\) 0 0
\(586\) −62.5120 −2.58235
\(587\) 16.1172i 0.665229i 0.943063 + 0.332614i \(0.107931\pi\)
−0.943063 + 0.332614i \(0.892069\pi\)
\(588\) 22.5985 0.931946
\(589\) 5.73317 0.236231
\(590\) 46.8504i 1.92880i
\(591\) 21.4209i 0.881137i
\(592\) 11.3746i 0.467494i
\(593\) − 42.8611i − 1.76010i −0.474885 0.880048i \(-0.657510\pi\)
0.474885 0.880048i \(-0.342490\pi\)
\(594\) 6.71917 0.275691
\(595\) 8.70171 0.356735
\(596\) 68.9869i 2.82581i
\(597\) 3.52781 0.144384
\(598\) 0 0
\(599\) 40.9420 1.67284 0.836422 0.548086i \(-0.184643\pi\)
0.836422 + 0.548086i \(0.184643\pi\)
\(600\) 31.6316i 1.29136i
\(601\) 1.18705 0.0484206 0.0242103 0.999707i \(-0.492293\pi\)
0.0242103 + 0.999707i \(0.492293\pi\)
\(602\) −3.05429 −0.124484
\(603\) − 14.3937i − 0.586158i
\(604\) 43.9474i 1.78819i
\(605\) − 10.6058i − 0.431187i
\(606\) − 10.9022i − 0.442870i
\(607\) 19.9922 0.811460 0.405730 0.913993i \(-0.367017\pi\)
0.405730 + 0.913993i \(0.367017\pi\)
\(608\) −9.11721 −0.369752
\(609\) 3.08815i 0.125138i
\(610\) 115.356 4.67062
\(611\) 0 0
\(612\) 10.4480 0.422336
\(613\) 33.3618i 1.34747i 0.738973 + 0.673735i \(0.235311\pi\)
−0.738973 + 0.673735i \(0.764689\pi\)
\(614\) −19.4373 −0.784424
\(615\) −3.14138 −0.126672
\(616\) − 8.37867i − 0.337586i
\(617\) − 11.6233i − 0.467935i −0.972244 0.233967i \(-0.924829\pi\)
0.972244 0.233967i \(-0.0751709\pi\)
\(618\) − 2.84654i − 0.114505i
\(619\) − 16.5381i − 0.664722i −0.943152 0.332361i \(-0.892155\pi\)
0.943152 0.332361i \(-0.107845\pi\)
\(620\) 30.7756 1.23598
\(621\) −7.78986 −0.312596
\(622\) − 33.9815i − 1.36253i
\(623\) 0.907542 0.0363599
\(624\) 0 0
\(625\) 6.33944 0.253577
\(626\) − 33.5477i − 1.34083i
\(627\) −6.97046 −0.278373
\(628\) 66.3979 2.64956
\(629\) − 21.8810i − 0.872452i
\(630\) 6.97823i 0.278019i
\(631\) − 36.4416i − 1.45072i −0.688372 0.725358i \(-0.741674\pi\)
0.688372 0.725358i \(-0.258326\pi\)
\(632\) 19.8116i 0.788064i
\(633\) −1.21552 −0.0483126
\(634\) 16.1309 0.640642
\(635\) 20.9347i 0.830768i
\(636\) −35.4101 −1.40410
\(637\) 0 0
\(638\) −25.8745 −1.02438
\(639\) − 8.12498i − 0.321419i
\(640\) −75.5303 −2.98560
\(641\) −27.2067 −1.07460 −0.537300 0.843391i \(-0.680556\pi\)
−0.537300 + 0.843391i \(0.680556\pi\)
\(642\) − 22.4403i − 0.885646i
\(643\) 5.06962i 0.199926i 0.994991 + 0.0999632i \(0.0318725\pi\)
−0.994991 + 0.0999632i \(0.968127\pi\)
\(644\) 22.2078i 0.875108i
\(645\) 5.96615i 0.234917i
\(646\) −16.9366 −0.666362
\(647\) −19.3207 −0.759573 −0.379787 0.925074i \(-0.624003\pi\)
−0.379787 + 0.925074i \(0.624003\pi\)
\(648\) 3.66487i 0.143970i
\(649\) −15.3491 −0.602506
\(650\) 0 0
\(651\) 1.88040 0.0736985
\(652\) − 43.8662i − 1.71793i
\(653\) 35.2355 1.37887 0.689436 0.724347i \(-0.257859\pi\)
0.689436 + 0.724347i \(0.257859\pi\)
\(654\) −4.20583 −0.164461
\(655\) − 67.2790i − 2.62881i
\(656\) 1.29995i 0.0507544i
\(657\) − 11.8877i − 0.463783i
\(658\) − 4.62133i − 0.180158i
\(659\) −4.36168 −0.169907 −0.0849535 0.996385i \(-0.527074\pi\)
−0.0849535 + 0.996385i \(0.527074\pi\)
\(660\) −37.4174 −1.45647
\(661\) − 15.4709i − 0.601747i −0.953664 0.300873i \(-0.902722\pi\)
0.953664 0.300873i \(-0.0972782\pi\)
\(662\) −22.2553 −0.864978
\(663\) 0 0
\(664\) 25.8334 1.00253
\(665\) − 7.23921i − 0.280725i
\(666\) 17.5472 0.679940
\(667\) 29.9976 1.16151
\(668\) − 40.8605i − 1.58094i
\(669\) − 17.3884i − 0.672273i
\(670\) 125.250i 4.83883i
\(671\) 37.7928i 1.45898i
\(672\) −2.99031 −0.115354
\(673\) 11.7409 0.452580 0.226290 0.974060i \(-0.427340\pi\)
0.226290 + 0.974060i \(0.427340\pi\)
\(674\) 6.22952i 0.239952i
\(675\) 8.63102 0.332208
\(676\) 0 0
\(677\) 3.44504 0.132404 0.0662019 0.997806i \(-0.478912\pi\)
0.0662019 + 0.997806i \(0.478912\pi\)
\(678\) 11.6692i 0.448152i
\(679\) 4.76702 0.182941
\(680\) −39.7670 −1.52500
\(681\) 17.4155i 0.667363i
\(682\) 15.7552i 0.603298i
\(683\) 20.4058i 0.780807i 0.920644 + 0.390403i \(0.127664\pi\)
−0.920644 + 0.390403i \(0.872336\pi\)
\(684\) − 8.69202i − 0.332348i
\(685\) 34.8911 1.33312
\(686\) −25.2457 −0.963883
\(687\) − 18.7603i − 0.715751i
\(688\) 2.46888 0.0941251
\(689\) 0 0
\(690\) 67.7851 2.58053
\(691\) − 27.8039i − 1.05771i −0.848713 0.528854i \(-0.822622\pi\)
0.848713 0.528854i \(-0.177378\pi\)
\(692\) 43.0656 1.63711
\(693\) −2.28621 −0.0868459
\(694\) 23.9433i 0.908876i
\(695\) − 14.8237i − 0.562295i
\(696\) − 14.1129i − 0.534948i
\(697\) − 2.50066i − 0.0947194i
\(698\) 24.6052 0.931321
\(699\) 3.95108 0.149444
\(700\) − 24.6058i − 0.930012i
\(701\) −11.9715 −0.452158 −0.226079 0.974109i \(-0.572591\pi\)
−0.226079 + 0.974109i \(0.572591\pi\)
\(702\) 0 0
\(703\) −18.2034 −0.686556
\(704\) − 33.7660i − 1.27260i
\(705\) −9.02715 −0.339982
\(706\) −43.1102 −1.62248
\(707\) 3.70948i 0.139509i
\(708\) − 19.1400i − 0.719327i
\(709\) 32.2664i 1.21179i 0.795545 + 0.605894i \(0.207185\pi\)
−0.795545 + 0.605894i \(0.792815\pi\)
\(710\) 70.7012i 2.65337i
\(711\) 5.40581 0.202734
\(712\) −4.14749 −0.155434
\(713\) − 18.2658i − 0.684058i
\(714\) −5.55496 −0.207889
\(715\) 0 0
\(716\) −1.91457 −0.0715510
\(717\) 0.818331i 0.0305611i
\(718\) −35.9970 −1.34340
\(719\) −12.1086 −0.451574 −0.225787 0.974177i \(-0.572495\pi\)
−0.225787 + 0.974177i \(0.572495\pi\)
\(720\) − 5.64071i − 0.210217i
\(721\) 0.968541i 0.0360704i
\(722\) − 30.6910i − 1.14220i
\(723\) − 6.03252i − 0.224352i
\(724\) −82.7730 −3.07623
\(725\) −33.2368 −1.23438
\(726\) 6.77048i 0.251276i
\(727\) 16.6200 0.616402 0.308201 0.951321i \(-0.400273\pi\)
0.308201 + 0.951321i \(0.400273\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 103.443i 3.82861i
\(731\) −4.74930 −0.175659
\(732\) −47.1269 −1.74186
\(733\) − 17.7912i − 0.657132i −0.944481 0.328566i \(-0.893435\pi\)
0.944481 0.328566i \(-0.106565\pi\)
\(734\) 52.4922i 1.93752i
\(735\) 23.4698i 0.865696i
\(736\) 29.0473i 1.07070i
\(737\) −41.0344 −1.51152
\(738\) 2.00538 0.0738189
\(739\) − 27.3618i − 1.00652i −0.864135 0.503260i \(-0.832134\pi\)
0.864135 0.503260i \(-0.167866\pi\)
\(740\) −97.7160 −3.59211
\(741\) 0 0
\(742\) 18.8267 0.691150
\(743\) − 8.38596i − 0.307651i −0.988098 0.153826i \(-0.950841\pi\)
0.988098 0.153826i \(-0.0491594\pi\)
\(744\) −8.59345 −0.315051
\(745\) −71.6467 −2.62493
\(746\) − 9.72528i − 0.356068i
\(747\) − 7.04892i − 0.257906i
\(748\) − 29.7858i − 1.08908i
\(749\) 7.63533i 0.278989i
\(750\) −31.5961 −1.15373
\(751\) 38.7778 1.41502 0.707511 0.706703i \(-0.249818\pi\)
0.707511 + 0.706703i \(0.249818\pi\)
\(752\) 3.73556i 0.136222i
\(753\) −26.8799 −0.979559
\(754\) 0 0
\(755\) −45.6418 −1.66107
\(756\) − 2.85086i − 0.103685i
\(757\) 12.9729 0.471506 0.235753 0.971813i \(-0.424244\pi\)
0.235753 + 0.971813i \(0.424244\pi\)
\(758\) 25.2355 0.916594
\(759\) 22.2078i 0.806090i
\(760\) 33.0834i 1.20006i
\(761\) − 5.15585i − 0.186899i −0.995624 0.0934497i \(-0.970211\pi\)
0.995624 0.0934497i \(-0.0297894\pi\)
\(762\) − 13.3642i − 0.484134i
\(763\) 1.43104 0.0518072
\(764\) −59.6013 −2.15630
\(765\) 10.8509i 0.392313i
\(766\) −15.3822 −0.555783
\(767\) 0 0
\(768\) 24.5284 0.885092
\(769\) 35.5013i 1.28021i 0.768288 + 0.640104i \(0.221109\pi\)
−0.768288 + 0.640104i \(0.778891\pi\)
\(770\) 19.8939 0.716927
\(771\) −9.05323 −0.326044
\(772\) − 91.5186i − 3.29383i
\(773\) 6.15585i 0.221411i 0.993853 + 0.110705i \(0.0353110\pi\)
−0.993853 + 0.110705i \(0.964689\pi\)
\(774\) − 3.80864i − 0.136899i
\(775\) 20.2381i 0.726976i
\(776\) −21.7854 −0.782050
\(777\) −5.97046 −0.214189
\(778\) − 27.7931i − 0.996431i
\(779\) −2.08038 −0.0745372
\(780\) 0 0
\(781\) −23.1631 −0.828843
\(782\) 53.9597i 1.92960i
\(783\) −3.85086 −0.137618
\(784\) 9.71214 0.346862
\(785\) 68.9579i 2.46121i
\(786\) 42.9493i 1.53195i
\(787\) 14.2107i 0.506558i 0.967393 + 0.253279i \(0.0815091\pi\)
−0.967393 + 0.253279i \(0.918491\pi\)
\(788\) 76.1503i 2.71274i
\(789\) −23.1511 −0.824200
\(790\) −47.0398 −1.67360
\(791\) − 3.97046i − 0.141173i
\(792\) 10.4480 0.371254
\(793\) 0 0
\(794\) −29.5623 −1.04913
\(795\) − 36.7754i − 1.30429i
\(796\) 12.5412 0.444512
\(797\) −11.9022 −0.421596 −0.210798 0.977530i \(-0.567606\pi\)
−0.210798 + 0.977530i \(0.567606\pi\)
\(798\) 4.62133i 0.163593i
\(799\) − 7.18598i − 0.254222i
\(800\) − 32.1839i − 1.13787i
\(801\) 1.13169i 0.0399862i
\(802\) 42.1183 1.48725
\(803\) −33.8901 −1.19596
\(804\) − 51.1691i − 1.80460i
\(805\) −23.0640 −0.812899
\(806\) 0 0
\(807\) 2.42088 0.0852190
\(808\) − 16.9524i − 0.596384i
\(809\) 1.61596 0.0568140 0.0284070 0.999596i \(-0.490957\pi\)
0.0284070 + 0.999596i \(0.490957\pi\)
\(810\) −8.70171 −0.305747
\(811\) 51.9657i 1.82476i 0.409342 + 0.912381i \(0.365758\pi\)
−0.409342 + 0.912381i \(0.634242\pi\)
\(812\) 10.9782i 0.385260i
\(813\) 21.4450i 0.752110i
\(814\) − 50.0245i − 1.75336i
\(815\) 45.5575 1.59581
\(816\) 4.49024 0.157190
\(817\) 3.95108i 0.138231i
\(818\) 65.7853 2.30013
\(819\) 0 0
\(820\) −11.1675 −0.389985
\(821\) − 54.8327i − 1.91367i −0.290627 0.956837i \(-0.593864\pi\)
0.290627 0.956837i \(-0.406136\pi\)
\(822\) −22.2737 −0.776883
\(823\) 46.8514 1.63314 0.816569 0.577247i \(-0.195873\pi\)
0.816569 + 0.577247i \(0.195873\pi\)
\(824\) − 4.42626i − 0.154196i
\(825\) − 24.6058i − 0.856664i
\(826\) 10.1763i 0.354078i
\(827\) 21.2021i 0.737270i 0.929574 + 0.368635i \(0.120175\pi\)
−0.929574 + 0.368635i \(0.879825\pi\)
\(828\) −27.6926 −0.962385
\(829\) −18.2972 −0.635489 −0.317744 0.948176i \(-0.602925\pi\)
−0.317744 + 0.948176i \(0.602925\pi\)
\(830\) 61.3376i 2.12906i
\(831\) −14.8073 −0.513660
\(832\) 0 0
\(833\) −18.6829 −0.647325
\(834\) 9.46309i 0.327680i
\(835\) 42.4359 1.46856
\(836\) −24.7797 −0.857024
\(837\) 2.34481i 0.0810486i
\(838\) 38.6612i 1.33553i
\(839\) 21.1414i 0.729881i 0.931031 + 0.364941i \(0.118911\pi\)
−0.931031 + 0.364941i \(0.881089\pi\)
\(840\) 10.8509i 0.374390i
\(841\) −14.1709 −0.488652
\(842\) 7.15751 0.246664
\(843\) − 14.5036i − 0.499532i
\(844\) −4.32113 −0.148739
\(845\) 0 0
\(846\) 5.76271 0.198126
\(847\) − 2.30367i − 0.0791549i
\(848\) −15.2182 −0.522594
\(849\) 25.6722 0.881067
\(850\) − 59.7864i − 2.05066i
\(851\) 57.9958i 1.98807i
\(852\) − 28.8840i − 0.989549i
\(853\) − 7.13036i − 0.244139i −0.992522 0.122069i \(-0.961047\pi\)
0.992522 0.122069i \(-0.0389531\pi\)
\(854\) 25.0562 0.857406
\(855\) 9.02715 0.308722
\(856\) − 34.8937i − 1.19264i
\(857\) 44.7741 1.52945 0.764726 0.644355i \(-0.222874\pi\)
0.764726 + 0.644355i \(0.222874\pi\)
\(858\) 0 0
\(859\) 57.3782 1.95772 0.978859 0.204535i \(-0.0655681\pi\)
0.978859 + 0.204535i \(0.0655681\pi\)
\(860\) 21.2094i 0.723235i
\(861\) −0.682333 −0.0232538
\(862\) 7.86758 0.267971
\(863\) 6.67563i 0.227241i 0.993524 + 0.113621i \(0.0362448\pi\)
−0.993524 + 0.113621i \(0.963755\pi\)
\(864\) − 3.72886i − 0.126858i
\(865\) 44.7260i 1.52073i
\(866\) 28.0536i 0.953299i
\(867\) 8.36227 0.283998
\(868\) 6.68473 0.226894
\(869\) − 15.4112i − 0.522789i
\(870\) 33.5090 1.13606
\(871\) 0 0
\(872\) −6.53989 −0.221469
\(873\) 5.94438i 0.201187i
\(874\) 44.8907 1.51845
\(875\) 10.7506 0.363438
\(876\) − 42.2602i − 1.42784i
\(877\) − 25.2983i − 0.854263i −0.904190 0.427131i \(-0.859524\pi\)
0.904190 0.427131i \(-0.140476\pi\)
\(878\) 8.76318i 0.295743i
\(879\) 26.5230i 0.894599i
\(880\) −16.0809 −0.542085
\(881\) 35.3787 1.19194 0.595969 0.803008i \(-0.296768\pi\)
0.595969 + 0.803008i \(0.296768\pi\)
\(882\) − 14.9825i − 0.504488i
\(883\) 11.5851 0.389869 0.194935 0.980816i \(-0.437551\pi\)
0.194935 + 0.980816i \(0.437551\pi\)
\(884\) 0 0
\(885\) 19.8780 0.668192
\(886\) − 3.43429i − 0.115377i
\(887\) −27.2892 −0.916281 −0.458141 0.888880i \(-0.651484\pi\)
−0.458141 + 0.888880i \(0.651484\pi\)
\(888\) 27.2851 0.915629
\(889\) 4.54719i 0.152508i
\(890\) − 9.84761i − 0.330093i
\(891\) − 2.85086i − 0.0955072i
\(892\) − 61.8149i − 2.06972i
\(893\) −5.97823 −0.200054
\(894\) 45.7375 1.52969
\(895\) − 1.98839i − 0.0664646i
\(896\) −16.4058 −0.548080
\(897\) 0 0
\(898\) −28.5830 −0.953826
\(899\) − 9.02954i − 0.301152i
\(900\) 30.6829 1.02276
\(901\) 29.2747 0.975282
\(902\) − 5.71704i − 0.190357i
\(903\) 1.29590i 0.0431247i
\(904\) 18.1451i 0.603497i
\(905\) − 85.9643i − 2.85755i
\(906\) 29.1366 0.967998
\(907\) 30.4219 1.01014 0.505072 0.863077i \(-0.331466\pi\)
0.505072 + 0.863077i \(0.331466\pi\)
\(908\) 61.9114i 2.05460i
\(909\) −4.62565 −0.153423
\(910\) 0 0
\(911\) 53.5719 1.77492 0.887459 0.460887i \(-0.152469\pi\)
0.887459 + 0.460887i \(0.152469\pi\)
\(912\) − 3.73556i − 0.123697i
\(913\) −20.0954 −0.665062
\(914\) 8.12664 0.268805
\(915\) − 48.9439i − 1.61804i
\(916\) − 66.6921i − 2.20357i
\(917\) − 14.6136i − 0.482582i
\(918\) − 6.92692i − 0.228622i
\(919\) −36.1672 −1.19305 −0.596523 0.802596i \(-0.703451\pi\)
−0.596523 + 0.802596i \(0.703451\pi\)
\(920\) 105.403 3.47503
\(921\) 8.24698i 0.271747i
\(922\) 15.9232 0.524403
\(923\) 0 0
\(924\) −8.12737 −0.267371
\(925\) − 64.2583i − 2.11280i
\(926\) −17.5630 −0.577156
\(927\) −1.20775 −0.0396677
\(928\) 14.3593i 0.471367i
\(929\) 13.3478i 0.437927i 0.975733 + 0.218964i \(0.0702676\pi\)
−0.975733 + 0.218964i \(0.929732\pi\)
\(930\) − 20.4039i − 0.669070i
\(931\) 15.5429i 0.509397i
\(932\) 14.0459 0.460090
\(933\) −14.4179 −0.472021
\(934\) 76.8580i 2.51487i
\(935\) 30.9342 1.01166
\(936\) 0 0
\(937\) 38.6872 1.26386 0.631928 0.775027i \(-0.282264\pi\)
0.631928 + 0.775027i \(0.282264\pi\)
\(938\) 27.2054i 0.888286i
\(939\) −14.2338 −0.464504
\(940\) −32.0911 −1.04670
\(941\) − 35.7275i − 1.16468i −0.812944 0.582342i \(-0.802136\pi\)
0.812944 0.582342i \(-0.197864\pi\)
\(942\) − 44.0210i − 1.43428i
\(943\) 6.62804i 0.215839i
\(944\) − 8.22580i − 0.267727i
\(945\) 2.96077 0.0963139
\(946\) −10.8579 −0.353020
\(947\) − 17.8436i − 0.579838i −0.957051 0.289919i \(-0.906372\pi\)
0.957051 0.289919i \(-0.0936283\pi\)
\(948\) 19.2174 0.624153
\(949\) 0 0
\(950\) −49.7381 −1.61372
\(951\) − 6.84415i − 0.221937i
\(952\) −8.63773 −0.279950
\(953\) −20.1691 −0.653342 −0.326671 0.945138i \(-0.605927\pi\)
−0.326671 + 0.945138i \(0.605927\pi\)
\(954\) 23.4765i 0.760080i
\(955\) − 61.8993i − 2.00301i
\(956\) 2.90913i 0.0940881i
\(957\) 10.9782i 0.354876i
\(958\) 6.66786 0.215429
\(959\) 7.57865 0.244727
\(960\) 43.7289i 1.41134i
\(961\) 25.5018 0.822640
\(962\) 0 0
\(963\) −9.52111 −0.306813
\(964\) − 21.4454i − 0.690709i
\(965\) 95.0471 3.05967
\(966\) 14.7235 0.473720
\(967\) 32.7894i 1.05444i 0.849730 + 0.527218i \(0.176765\pi\)
−0.849730 + 0.527218i \(0.823235\pi\)
\(968\) 10.5278i 0.338377i
\(969\) 7.18598i 0.230847i
\(970\) − 51.7263i − 1.66083i
\(971\) −26.9845 −0.865973 −0.432986 0.901401i \(-0.642540\pi\)
−0.432986 + 0.901401i \(0.642540\pi\)
\(972\) 3.55496 0.114025
\(973\) − 3.21983i − 0.103223i
\(974\) 97.3245 3.11848
\(975\) 0 0
\(976\) −20.2537 −0.648305
\(977\) 16.9571i 0.542504i 0.962508 + 0.271252i \(0.0874377\pi\)
−0.962508 + 0.271252i \(0.912562\pi\)
\(978\) −29.0828 −0.929964
\(979\) 3.22627 0.103112
\(980\) 83.4341i 2.66521i
\(981\) 1.78448i 0.0569740i
\(982\) − 81.7131i − 2.60757i
\(983\) − 32.6631i − 1.04179i −0.853621 0.520895i \(-0.825598\pi\)
0.853621 0.520895i \(-0.174402\pi\)
\(984\) 3.11828 0.0994070
\(985\) −79.0863 −2.51990
\(986\) 26.6746i 0.849491i
\(987\) −1.96077 −0.0624120
\(988\) 0 0
\(989\) 12.5881 0.400277
\(990\) 24.8073i 0.788428i
\(991\) 7.64310 0.242791 0.121396 0.992604i \(-0.461263\pi\)
0.121396 + 0.992604i \(0.461263\pi\)
\(992\) 8.74348 0.277606
\(993\) 9.44265i 0.299653i
\(994\) 15.3569i 0.487091i
\(995\) 13.0248i 0.412912i
\(996\) − 25.0586i − 0.794012i
\(997\) 36.1256 1.14411 0.572054 0.820216i \(-0.306147\pi\)
0.572054 + 0.820216i \(0.306147\pi\)
\(998\) −42.2849 −1.33850
\(999\) − 7.44504i − 0.235551i
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 507.2.b.f.337.2 6
3.2 odd 2 1521.2.b.k.1351.5 6
13.2 odd 12 507.2.e.l.22.2 6
13.3 even 3 507.2.j.i.316.5 12
13.4 even 6 507.2.j.i.361.5 12
13.5 odd 4 507.2.a.i.1.2 3
13.6 odd 12 507.2.e.l.484.2 6
13.7 odd 12 507.2.e.i.484.2 6
13.8 odd 4 507.2.a.l.1.2 yes 3
13.9 even 3 507.2.j.i.361.2 12
13.10 even 6 507.2.j.i.316.2 12
13.11 odd 12 507.2.e.i.22.2 6
13.12 even 2 inner 507.2.b.f.337.5 6
39.5 even 4 1521.2.a.s.1.2 3
39.8 even 4 1521.2.a.n.1.2 3
39.38 odd 2 1521.2.b.k.1351.2 6
52.31 even 4 8112.2.a.cg.1.1 3
52.47 even 4 8112.2.a.cp.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
507.2.a.i.1.2 3 13.5 odd 4
507.2.a.l.1.2 yes 3 13.8 odd 4
507.2.b.f.337.2 6 1.1 even 1 trivial
507.2.b.f.337.5 6 13.12 even 2 inner
507.2.e.i.22.2 6 13.11 odd 12
507.2.e.i.484.2 6 13.7 odd 12
507.2.e.l.22.2 6 13.2 odd 12
507.2.e.l.484.2 6 13.6 odd 12
507.2.j.i.316.2 12 13.10 even 6
507.2.j.i.316.5 12 13.3 even 3
507.2.j.i.361.2 12 13.9 even 3
507.2.j.i.361.5 12 13.4 even 6
1521.2.a.n.1.2 3 39.8 even 4
1521.2.a.s.1.2 3 39.5 even 4
1521.2.b.k.1351.2 6 39.38 odd 2
1521.2.b.k.1351.5 6 3.2 odd 2
8112.2.a.cg.1.1 3 52.31 even 4
8112.2.a.cp.1.3 3 52.47 even 4