Properties

Label 1150.2.b.j.599.5
Level $1150$
Weight $2$
Character 1150.599
Analytic conductor $9.183$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1150 = 2 \cdot 5^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1150.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(9.18279623245\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.77580864.1
Defining polynomial: \(x^{6} + 19 x^{4} + 105 x^{2} + 144\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 230)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 599.5
Root \(-1.43163i\) of defining polynomial
Character \(\chi\) \(=\) 1150.599
Dual form 1150.2.b.j.599.2

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000i q^{2} -1.43163i q^{3} -1.00000 q^{4} +1.43163 q^{6} +3.08719i q^{7} -1.00000i q^{8} +0.950444 q^{9} +O(q^{10})\) \(q+1.00000i q^{2} -1.43163i q^{3} -1.00000 q^{4} +1.43163 q^{6} +3.08719i q^{7} -1.00000i q^{8} +0.950444 q^{9} -6.46926 q^{11} +1.43163i q^{12} -3.95044i q^{13} -3.08719 q^{14} +1.00000 q^{16} -3.43163i q^{17} +0.950444i q^{18} -3.08719 q^{19} +4.41970 q^{21} -6.46926i q^{22} +1.00000i q^{23} -1.43163 q^{24} +3.95044 q^{26} -5.65556i q^{27} -3.08719i q^{28} -0.863254 q^{29} -5.95044 q^{31} +1.00000i q^{32} +9.26157i q^{33} +3.43163 q^{34} -0.950444 q^{36} -7.03763i q^{37} -3.08719i q^{38} -5.65556 q^{39} +5.60601 q^{41} +4.41970i q^{42} -8.00000i q^{43} +6.46926 q^{44} -1.00000 q^{46} +3.90089i q^{47} -1.43163i q^{48} -2.53074 q^{49} -4.91281 q^{51} +3.95044i q^{52} +6.00000i q^{53} +5.65556 q^{54} +3.08719 q^{56} +4.41970i q^{57} -0.863254i q^{58} -6.86325 q^{59} -13.5069 q^{61} -5.95044i q^{62} +2.93420i q^{63} -1.00000 q^{64} -9.26157 q^{66} -10.0753i q^{67} +3.43163i q^{68} +1.43163 q^{69} +2.56837 q^{71} -0.950444i q^{72} -5.90089i q^{73} +7.03763 q^{74} +3.08719 q^{76} -19.9718i q^{77} -5.65556i q^{78} -15.8018 q^{79} -5.24533 q^{81} +5.60601i q^{82} -9.03763i q^{83} -4.41970 q^{84} +8.00000 q^{86} +1.23586i q^{87} +6.46926i q^{88} -16.7641 q^{89} +12.1958 q^{91} -1.00000i q^{92} +8.51882i q^{93} -3.90089 q^{94} +1.43163 q^{96} -14.2949i q^{97} -2.53074i q^{98} -6.14867 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q - 6q^{4} + 2q^{6} - 20q^{9} + O(q^{10}) \) \( 6q - 6q^{4} + 2q^{6} - 20q^{9} + 6q^{11} - 6q^{14} + 6q^{16} - 6q^{19} - 44q^{21} - 2q^{24} - 2q^{26} + 8q^{29} - 10q^{31} + 14q^{34} + 20q^{36} - 28q^{39} + 2q^{41} - 6q^{44} - 6q^{46} - 60q^{49} - 42q^{51} + 28q^{54} + 6q^{56} - 28q^{59} + 2q^{61} - 6q^{64} - 18q^{66} + 2q^{69} + 22q^{71} + 4q^{74} + 6q^{76} + 8q^{79} + 14q^{81} + 44q^{84} + 48q^{86} - 36q^{89} + 2q^{91} + 28q^{94} + 2q^{96} - 114q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1150\mathbb{Z}\right)^\times\).

\(n\) \(51\) \(277\)
\(\chi(n)\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000i 0.707107i
\(3\) − 1.43163i − 0.826550i −0.910606 0.413275i \(-0.864385\pi\)
0.910606 0.413275i \(-0.135615\pi\)
\(4\) −1.00000 −0.500000
\(5\) 0 0
\(6\) 1.43163 0.584459
\(7\) 3.08719i 1.16685i 0.812168 + 0.583424i \(0.198287\pi\)
−0.812168 + 0.583424i \(0.801713\pi\)
\(8\) − 1.00000i − 0.353553i
\(9\) 0.950444 0.316815
\(10\) 0 0
\(11\) −6.46926 −1.95056 −0.975278 0.220983i \(-0.929074\pi\)
−0.975278 + 0.220983i \(0.929074\pi\)
\(12\) 1.43163i 0.413275i
\(13\) − 3.95044i − 1.09566i −0.836591 0.547828i \(-0.815455\pi\)
0.836591 0.547828i \(-0.184545\pi\)
\(14\) −3.08719 −0.825086
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) − 3.43163i − 0.832292i −0.909298 0.416146i \(-0.863381\pi\)
0.909298 0.416146i \(-0.136619\pi\)
\(18\) 0.950444i 0.224022i
\(19\) −3.08719 −0.708250 −0.354125 0.935198i \(-0.615221\pi\)
−0.354125 + 0.935198i \(0.615221\pi\)
\(20\) 0 0
\(21\) 4.41970 0.964459
\(22\) − 6.46926i − 1.37925i
\(23\) 1.00000i 0.208514i
\(24\) −1.43163 −0.292230
\(25\) 0 0
\(26\) 3.95044 0.774746
\(27\) − 5.65556i − 1.08841i
\(28\) − 3.08719i − 0.583424i
\(29\) −0.863254 −0.160302 −0.0801511 0.996783i \(-0.525540\pi\)
−0.0801511 + 0.996783i \(0.525540\pi\)
\(30\) 0 0
\(31\) −5.95044 −1.06873 −0.534366 0.845253i \(-0.679449\pi\)
−0.534366 + 0.845253i \(0.679449\pi\)
\(32\) 1.00000i 0.176777i
\(33\) 9.26157i 1.61223i
\(34\) 3.43163 0.588519
\(35\) 0 0
\(36\) −0.950444 −0.158407
\(37\) − 7.03763i − 1.15698i −0.815690 0.578490i \(-0.803642\pi\)
0.815690 0.578490i \(-0.196358\pi\)
\(38\) − 3.08719i − 0.500808i
\(39\) −5.65556 −0.905615
\(40\) 0 0
\(41\) 5.60601 0.875511 0.437756 0.899094i \(-0.355774\pi\)
0.437756 + 0.899094i \(0.355774\pi\)
\(42\) 4.41970i 0.681975i
\(43\) − 8.00000i − 1.21999i −0.792406 0.609994i \(-0.791172\pi\)
0.792406 0.609994i \(-0.208828\pi\)
\(44\) 6.46926 0.975278
\(45\) 0 0
\(46\) −1.00000 −0.147442
\(47\) 3.90089i 0.569003i 0.958676 + 0.284501i \(0.0918281\pi\)
−0.958676 + 0.284501i \(0.908172\pi\)
\(48\) − 1.43163i − 0.206638i
\(49\) −2.53074 −0.361534
\(50\) 0 0
\(51\) −4.91281 −0.687931
\(52\) 3.95044i 0.547828i
\(53\) 6.00000i 0.824163i 0.911147 + 0.412082i \(0.135198\pi\)
−0.911147 + 0.412082i \(0.864802\pi\)
\(54\) 5.65556 0.769625
\(55\) 0 0
\(56\) 3.08719 0.412543
\(57\) 4.41970i 0.585404i
\(58\) − 0.863254i − 0.113351i
\(59\) −6.86325 −0.893520 −0.446760 0.894654i \(-0.647422\pi\)
−0.446760 + 0.894654i \(0.647422\pi\)
\(60\) 0 0
\(61\) −13.5069 −1.72938 −0.864690 0.502305i \(-0.832485\pi\)
−0.864690 + 0.502305i \(0.832485\pi\)
\(62\) − 5.95044i − 0.755707i
\(63\) 2.93420i 0.369674i
\(64\) −1.00000 −0.125000
\(65\) 0 0
\(66\) −9.26157 −1.14002
\(67\) − 10.0753i − 1.23089i −0.788180 0.615445i \(-0.788976\pi\)
0.788180 0.615445i \(-0.211024\pi\)
\(68\) 3.43163i 0.416146i
\(69\) 1.43163 0.172348
\(70\) 0 0
\(71\) 2.56837 0.304810 0.152405 0.988318i \(-0.451298\pi\)
0.152405 + 0.988318i \(0.451298\pi\)
\(72\) − 0.950444i − 0.112011i
\(73\) − 5.90089i − 0.690647i −0.938484 0.345323i \(-0.887769\pi\)
0.938484 0.345323i \(-0.112231\pi\)
\(74\) 7.03763 0.818108
\(75\) 0 0
\(76\) 3.08719 0.354125
\(77\) − 19.9718i − 2.27600i
\(78\) − 5.65556i − 0.640366i
\(79\) −15.8018 −1.77784 −0.888919 0.458064i \(-0.848543\pi\)
−0.888919 + 0.458064i \(0.848543\pi\)
\(80\) 0 0
\(81\) −5.24533 −0.582814
\(82\) 5.60601i 0.619080i
\(83\) − 9.03763i − 0.992009i −0.868320 0.496005i \(-0.834800\pi\)
0.868320 0.496005i \(-0.165200\pi\)
\(84\) −4.41970 −0.482229
\(85\) 0 0
\(86\) 8.00000 0.862662
\(87\) 1.23586i 0.132498i
\(88\) 6.46926i 0.689625i
\(89\) −16.7641 −1.77700 −0.888498 0.458881i \(-0.848250\pi\)
−0.888498 + 0.458881i \(0.848250\pi\)
\(90\) 0 0
\(91\) 12.1958 1.27846
\(92\) − 1.00000i − 0.104257i
\(93\) 8.51882i 0.883360i
\(94\) −3.90089 −0.402346
\(95\) 0 0
\(96\) 1.43163 0.146115
\(97\) − 14.2949i − 1.45143i −0.687998 0.725713i \(-0.741510\pi\)
0.687998 0.725713i \(-0.258490\pi\)
\(98\) − 2.53074i − 0.255643i
\(99\) −6.14867 −0.617964
\(100\) 0 0
\(101\) 0.863254 0.0858970 0.0429485 0.999077i \(-0.486325\pi\)
0.0429485 + 0.999077i \(0.486325\pi\)
\(102\) − 4.91281i − 0.486441i
\(103\) − 1.53074i − 0.150828i −0.997152 0.0754141i \(-0.975972\pi\)
0.997152 0.0754141i \(-0.0240279\pi\)
\(104\) −3.95044 −0.387373
\(105\) 0 0
\(106\) −6.00000 −0.582772
\(107\) 17.6274i 1.70410i 0.523457 + 0.852052i \(0.324642\pi\)
−0.523457 + 0.852052i \(0.675358\pi\)
\(108\) 5.65556i 0.544207i
\(109\) 2.91281 0.278997 0.139498 0.990222i \(-0.455451\pi\)
0.139498 + 0.990222i \(0.455451\pi\)
\(110\) 0 0
\(111\) −10.0753 −0.956302
\(112\) 3.08719i 0.291712i
\(113\) 6.00000i 0.564433i 0.959351 + 0.282216i \(0.0910696\pi\)
−0.959351 + 0.282216i \(0.908930\pi\)
\(114\) −4.41970 −0.413943
\(115\) 0 0
\(116\) 0.863254 0.0801511
\(117\) − 3.75467i − 0.347120i
\(118\) − 6.86325i − 0.631814i
\(119\) 10.5941 0.971158
\(120\) 0 0
\(121\) 30.8513 2.80467
\(122\) − 13.5069i − 1.22286i
\(123\) − 8.02571i − 0.723654i
\(124\) 5.95044 0.534366
\(125\) 0 0
\(126\) −2.93420 −0.261399
\(127\) 20.9385i 1.85799i 0.370088 + 0.928997i \(0.379327\pi\)
−0.370088 + 0.928997i \(0.620673\pi\)
\(128\) − 1.00000i − 0.0883883i
\(129\) −11.4530 −1.00838
\(130\) 0 0
\(131\) 10.7641 0.940467 0.470234 0.882542i \(-0.344170\pi\)
0.470234 + 0.882542i \(0.344170\pi\)
\(132\) − 9.26157i − 0.806116i
\(133\) − 9.53074i − 0.826420i
\(134\) 10.0753 0.870370
\(135\) 0 0
\(136\) −3.43163 −0.294260
\(137\) 3.26157i 0.278655i 0.990246 + 0.139327i \(0.0444940\pi\)
−0.990246 + 0.139327i \(0.955506\pi\)
\(138\) 1.43163i 0.121868i
\(139\) 16.9385 1.43671 0.718353 0.695678i \(-0.244896\pi\)
0.718353 + 0.695678i \(0.244896\pi\)
\(140\) 0 0
\(141\) 5.58462 0.470310
\(142\) 2.56837i 0.215533i
\(143\) 25.5565i 2.13714i
\(144\) 0.950444 0.0792037
\(145\) 0 0
\(146\) 5.90089 0.488361
\(147\) 3.62308i 0.298826i
\(148\) 7.03763i 0.578490i
\(149\) −3.26157 −0.267198 −0.133599 0.991035i \(-0.542653\pi\)
−0.133599 + 0.991035i \(0.542653\pi\)
\(150\) 0 0
\(151\) 0.294881 0.0239971 0.0119986 0.999928i \(-0.496181\pi\)
0.0119986 + 0.999928i \(0.496181\pi\)
\(152\) 3.08719i 0.250404i
\(153\) − 3.26157i − 0.263682i
\(154\) 19.9718 1.60938
\(155\) 0 0
\(156\) 5.65556 0.452807
\(157\) 7.13675i 0.569574i 0.958591 + 0.284787i \(0.0919229\pi\)
−0.958591 + 0.284787i \(0.908077\pi\)
\(158\) − 15.8018i − 1.25712i
\(159\) 8.58976 0.681212
\(160\) 0 0
\(161\) −3.08719 −0.243305
\(162\) − 5.24533i − 0.412112i
\(163\) 8.12482i 0.636385i 0.948026 + 0.318193i \(0.103076\pi\)
−0.948026 + 0.318193i \(0.896924\pi\)
\(164\) −5.60601 −0.437756
\(165\) 0 0
\(166\) 9.03763 0.701456
\(167\) − 7.80178i − 0.603719i −0.953352 0.301860i \(-0.902393\pi\)
0.953352 0.301860i \(-0.0976074\pi\)
\(168\) − 4.41970i − 0.340988i
\(169\) −2.60601 −0.200462
\(170\) 0 0
\(171\) −2.93420 −0.224384
\(172\) 8.00000i 0.609994i
\(173\) 19.3325i 1.46982i 0.678163 + 0.734912i \(0.262777\pi\)
−0.678163 + 0.734912i \(0.737223\pi\)
\(174\) −1.23586 −0.0936902
\(175\) 0 0
\(176\) −6.46926 −0.487639
\(177\) 9.82562i 0.738539i
\(178\) − 16.7641i − 1.25653i
\(179\) −2.17438 −0.162521 −0.0812604 0.996693i \(-0.525895\pi\)
−0.0812604 + 0.996693i \(0.525895\pi\)
\(180\) 0 0
\(181\) −7.26157 −0.539748 −0.269874 0.962896i \(-0.586982\pi\)
−0.269874 + 0.962896i \(0.586982\pi\)
\(182\) 12.1958i 0.904011i
\(183\) 19.3368i 1.42942i
\(184\) 1.00000 0.0737210
\(185\) 0 0
\(186\) −8.51882 −0.624630
\(187\) 22.2001i 1.62343i
\(188\) − 3.90089i − 0.284501i
\(189\) 17.4598 1.27001
\(190\) 0 0
\(191\) 8.58976 0.621533 0.310767 0.950486i \(-0.399414\pi\)
0.310767 + 0.950486i \(0.399414\pi\)
\(192\) 1.43163i 0.103319i
\(193\) − 2.44787i − 0.176202i −0.996112 0.0881008i \(-0.971920\pi\)
0.996112 0.0881008i \(-0.0280798\pi\)
\(194\) 14.2949 1.02631
\(195\) 0 0
\(196\) 2.53074 0.180767
\(197\) − 10.7428i − 0.765389i −0.923875 0.382695i \(-0.874996\pi\)
0.923875 0.382695i \(-0.125004\pi\)
\(198\) − 6.14867i − 0.436967i
\(199\) 11.3111 0.801824 0.400912 0.916116i \(-0.368693\pi\)
0.400912 + 0.916116i \(0.368693\pi\)
\(200\) 0 0
\(201\) −14.4240 −1.01739
\(202\) 0.863254i 0.0607384i
\(203\) − 2.66503i − 0.187048i
\(204\) 4.91281 0.343966
\(205\) 0 0
\(206\) 1.53074 0.106652
\(207\) 0.950444i 0.0660604i
\(208\) − 3.95044i − 0.273914i
\(209\) 19.9718 1.38148
\(210\) 0 0
\(211\) 23.1129 1.59116 0.795579 0.605850i \(-0.207167\pi\)
0.795579 + 0.605850i \(0.207167\pi\)
\(212\) − 6.00000i − 0.412082i
\(213\) − 3.67695i − 0.251941i
\(214\) −17.6274 −1.20498
\(215\) 0 0
\(216\) −5.65556 −0.384812
\(217\) − 18.3701i − 1.24705i
\(218\) 2.91281i 0.197280i
\(219\) −8.44787 −0.570854
\(220\) 0 0
\(221\) −13.5565 −0.911906
\(222\) − 10.0753i − 0.676208i
\(223\) 5.72651i 0.383475i 0.981446 + 0.191738i \(0.0614123\pi\)
−0.981446 + 0.191738i \(0.938588\pi\)
\(224\) −3.08719 −0.206272
\(225\) 0 0
\(226\) −6.00000 −0.399114
\(227\) − 17.6274i − 1.16997i −0.811044 0.584986i \(-0.801100\pi\)
0.811044 0.584986i \(-0.198900\pi\)
\(228\) − 4.41970i − 0.292702i
\(229\) 16.0753 1.06228 0.531142 0.847283i \(-0.321763\pi\)
0.531142 + 0.847283i \(0.321763\pi\)
\(230\) 0 0
\(231\) −28.5922 −1.88123
\(232\) 0.863254i 0.0566754i
\(233\) 6.44787i 0.422414i 0.977441 + 0.211207i \(0.0677394\pi\)
−0.977441 + 0.211207i \(0.932261\pi\)
\(234\) 3.75467 0.245451
\(235\) 0 0
\(236\) 6.86325 0.446760
\(237\) 22.6222i 1.46947i
\(238\) 10.5941i 0.686712i
\(239\) −4.34876 −0.281298 −0.140649 0.990060i \(-0.544919\pi\)
−0.140649 + 0.990060i \(0.544919\pi\)
\(240\) 0 0
\(241\) 0.764142 0.0492227 0.0246114 0.999697i \(-0.492165\pi\)
0.0246114 + 0.999697i \(0.492165\pi\)
\(242\) 30.8513i 1.98320i
\(243\) − 9.45734i − 0.606688i
\(244\) 13.5069 0.864690
\(245\) 0 0
\(246\) 8.02571 0.511701
\(247\) 12.1958i 0.775998i
\(248\) 5.95044i 0.377854i
\(249\) −12.9385 −0.819945
\(250\) 0 0
\(251\) −22.5402 −1.42273 −0.711363 0.702825i \(-0.751922\pi\)
−0.711363 + 0.702825i \(0.751922\pi\)
\(252\) − 2.93420i − 0.184837i
\(253\) − 6.46926i − 0.406719i
\(254\) −20.9385 −1.31380
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) − 9.90089i − 0.617600i −0.951127 0.308800i \(-0.900073\pi\)
0.951127 0.308800i \(-0.0999274\pi\)
\(258\) − 11.4530i − 0.713034i
\(259\) 21.7265 1.35002
\(260\) 0 0
\(261\) −0.820475 −0.0507861
\(262\) 10.7641i 0.665011i
\(263\) 7.25725i 0.447501i 0.974646 + 0.223751i \(0.0718301\pi\)
−0.974646 + 0.223751i \(0.928170\pi\)
\(264\) 9.26157 0.570010
\(265\) 0 0
\(266\) 9.53074 0.584367
\(267\) 24.0000i 1.46878i
\(268\) 10.0753i 0.615445i
\(269\) 6.93852 0.423049 0.211525 0.977373i \(-0.432157\pi\)
0.211525 + 0.977373i \(0.432157\pi\)
\(270\) 0 0
\(271\) −10.2992 −0.625632 −0.312816 0.949814i \(-0.601272\pi\)
−0.312816 + 0.949814i \(0.601272\pi\)
\(272\) − 3.43163i − 0.208073i
\(273\) − 17.4598i − 1.05671i
\(274\) −3.26157 −0.197039
\(275\) 0 0
\(276\) −1.43163 −0.0861738
\(277\) − 17.8018i − 1.06961i −0.844977 0.534803i \(-0.820386\pi\)
0.844977 0.534803i \(-0.179614\pi\)
\(278\) 16.9385i 1.01590i
\(279\) −5.65556 −0.338590
\(280\) 0 0
\(281\) −18.9385 −1.12978 −0.564889 0.825167i \(-0.691081\pi\)
−0.564889 + 0.825167i \(0.691081\pi\)
\(282\) 5.58462i 0.332559i
\(283\) − 12.6889i − 0.754275i −0.926157 0.377138i \(-0.876908\pi\)
0.926157 0.377138i \(-0.123092\pi\)
\(284\) −2.56837 −0.152405
\(285\) 0 0
\(286\) −25.5565 −1.51118
\(287\) 17.3068i 1.02159i
\(288\) 0.950444i 0.0560054i
\(289\) 5.22394 0.307290
\(290\) 0 0
\(291\) −20.4649 −1.19968
\(292\) 5.90089i 0.345323i
\(293\) 6.00000i 0.350524i 0.984522 + 0.175262i \(0.0560772\pi\)
−0.984522 + 0.175262i \(0.943923\pi\)
\(294\) −3.62308 −0.211302
\(295\) 0 0
\(296\) −7.03763 −0.409054
\(297\) 36.5873i 2.12301i
\(298\) − 3.26157i − 0.188938i
\(299\) 3.95044 0.228460
\(300\) 0 0
\(301\) 24.6975 1.42354
\(302\) 0.294881i 0.0169685i
\(303\) − 1.23586i − 0.0709982i
\(304\) −3.08719 −0.177062
\(305\) 0 0
\(306\) 3.26157 0.186451
\(307\) − 22.9599i − 1.31039i −0.755459 0.655196i \(-0.772586\pi\)
0.755459 0.655196i \(-0.227414\pi\)
\(308\) 19.9718i 1.13800i
\(309\) −2.19145 −0.124667
\(310\) 0 0
\(311\) 18.0753 1.02495 0.512477 0.858701i \(-0.328728\pi\)
0.512477 + 0.858701i \(0.328728\pi\)
\(312\) 5.65556i 0.320183i
\(313\) − 15.1625i − 0.857033i −0.903534 0.428516i \(-0.859036\pi\)
0.903534 0.428516i \(-0.140964\pi\)
\(314\) −7.13675 −0.402750
\(315\) 0 0
\(316\) 15.8018 0.888919
\(317\) 14.0257i 0.787762i 0.919161 + 0.393881i \(0.128868\pi\)
−0.919161 + 0.393881i \(0.871132\pi\)
\(318\) 8.58976i 0.481690i
\(319\) 5.58462 0.312678
\(320\) 0 0
\(321\) 25.2359 1.40853
\(322\) − 3.08719i − 0.172042i
\(323\) 10.5941i 0.589471i
\(324\) 5.24533 0.291407
\(325\) 0 0
\(326\) −8.12482 −0.449992
\(327\) − 4.17006i − 0.230605i
\(328\) − 5.60601i − 0.309540i
\(329\) −12.0428 −0.663940
\(330\) 0 0
\(331\) −24.7403 −1.35985 −0.679925 0.733282i \(-0.737988\pi\)
−0.679925 + 0.733282i \(0.737988\pi\)
\(332\) 9.03763i 0.496005i
\(333\) − 6.68888i − 0.366548i
\(334\) 7.80178 0.426894
\(335\) 0 0
\(336\) 4.41970 0.241115
\(337\) 14.7146i 0.801555i 0.916176 + 0.400777i \(0.131260\pi\)
−0.916176 + 0.400777i \(0.868740\pi\)
\(338\) − 2.60601i − 0.141748i
\(339\) 8.58976 0.466532
\(340\) 0 0
\(341\) 38.4950 2.08462
\(342\) − 2.93420i − 0.158663i
\(343\) 13.7975i 0.744993i
\(344\) −8.00000 −0.431331
\(345\) 0 0
\(346\) −19.3325 −1.03932
\(347\) 9.43163i 0.506316i 0.967425 + 0.253158i \(0.0814693\pi\)
−0.967425 + 0.253158i \(0.918531\pi\)
\(348\) − 1.23586i − 0.0662490i
\(349\) −18.3488 −0.982187 −0.491093 0.871107i \(-0.663403\pi\)
−0.491093 + 0.871107i \(0.663403\pi\)
\(350\) 0 0
\(351\) −22.3420 −1.19253
\(352\) − 6.46926i − 0.344813i
\(353\) 33.1129i 1.76242i 0.472723 + 0.881211i \(0.343271\pi\)
−0.472723 + 0.881211i \(0.656729\pi\)
\(354\) −9.82562 −0.522226
\(355\) 0 0
\(356\) 16.7641 0.888498
\(357\) − 15.1668i − 0.802711i
\(358\) − 2.17438i − 0.114920i
\(359\) −33.0376 −1.74366 −0.871830 0.489809i \(-0.837067\pi\)
−0.871830 + 0.489809i \(0.837067\pi\)
\(360\) 0 0
\(361\) −9.46926 −0.498382
\(362\) − 7.26157i − 0.381660i
\(363\) − 44.1676i − 2.31820i
\(364\) −12.1958 −0.639232
\(365\) 0 0
\(366\) −19.3368 −1.01075
\(367\) − 2.27349i − 0.118675i −0.998238 0.0593376i \(-0.981101\pi\)
0.998238 0.0593376i \(-0.0188989\pi\)
\(368\) 1.00000i 0.0521286i
\(369\) 5.32819 0.277375
\(370\) 0 0
\(371\) −18.5231 −0.961673
\(372\) − 8.51882i − 0.441680i
\(373\) − 23.9762i − 1.24144i −0.784033 0.620719i \(-0.786841\pi\)
0.784033 0.620719i \(-0.213159\pi\)
\(374\) −22.2001 −1.14794
\(375\) 0 0
\(376\) 3.90089 0.201173
\(377\) 3.41024i 0.175636i
\(378\) 17.4598i 0.898035i
\(379\) 13.8795 0.712942 0.356471 0.934306i \(-0.383980\pi\)
0.356471 + 0.934306i \(0.383980\pi\)
\(380\) 0 0
\(381\) 29.9762 1.53572
\(382\) 8.58976i 0.439490i
\(383\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(384\) −1.43163 −0.0730574
\(385\) 0 0
\(386\) 2.44787 0.124593
\(387\) − 7.60355i − 0.386510i
\(388\) 14.2949i 0.725713i
\(389\) 26.6436 1.35089 0.675443 0.737412i \(-0.263952\pi\)
0.675443 + 0.737412i \(0.263952\pi\)
\(390\) 0 0
\(391\) 3.43163 0.173545
\(392\) 2.53074i 0.127822i
\(393\) − 15.4102i − 0.777344i
\(394\) 10.7428 0.541212
\(395\) 0 0
\(396\) 6.14867 0.308982
\(397\) − 8.66749i − 0.435009i −0.976059 0.217504i \(-0.930208\pi\)
0.976059 0.217504i \(-0.0697916\pi\)
\(398\) 11.3111i 0.566975i
\(399\) −13.6445 −0.683078
\(400\) 0 0
\(401\) 39.9762 1.99631 0.998157 0.0606854i \(-0.0193286\pi\)
0.998157 + 0.0606854i \(0.0193286\pi\)
\(402\) − 14.4240i − 0.719405i
\(403\) 23.5069i 1.17096i
\(404\) −0.863254 −0.0429485
\(405\) 0 0
\(406\) 2.66503 0.132263
\(407\) 45.5283i 2.25675i
\(408\) 4.91281i 0.243220i
\(409\) −30.1248 −1.48958 −0.744788 0.667301i \(-0.767450\pi\)
−0.744788 + 0.667301i \(0.767450\pi\)
\(410\) 0 0
\(411\) 4.66935 0.230322
\(412\) 1.53074i 0.0754141i
\(413\) − 21.1882i − 1.04260i
\(414\) −0.950444 −0.0467118
\(415\) 0 0
\(416\) 3.95044 0.193686
\(417\) − 24.2496i − 1.18751i
\(418\) 19.9718i 0.976854i
\(419\) −25.8770 −1.26418 −0.632088 0.774897i \(-0.717802\pi\)
−0.632088 + 0.774897i \(0.717802\pi\)
\(420\) 0 0
\(421\) −10.7146 −0.522197 −0.261098 0.965312i \(-0.584085\pi\)
−0.261098 + 0.965312i \(0.584085\pi\)
\(422\) 23.1129i 1.12512i
\(423\) 3.70757i 0.180268i
\(424\) 6.00000 0.291386
\(425\) 0 0
\(426\) 3.67695 0.178149
\(427\) − 41.6983i − 2.01792i
\(428\) − 17.6274i − 0.852052i
\(429\) 36.5873 1.76645
\(430\) 0 0
\(431\) 32.2496 1.55341 0.776705 0.629864i \(-0.216889\pi\)
0.776705 + 0.629864i \(0.216889\pi\)
\(432\) − 5.65556i − 0.272103i
\(433\) − 20.6393i − 0.991862i −0.868362 0.495931i \(-0.834827\pi\)
0.868362 0.495931i \(-0.165173\pi\)
\(434\) 18.3701 0.881795
\(435\) 0 0
\(436\) −2.91281 −0.139498
\(437\) − 3.08719i − 0.147680i
\(438\) − 8.44787i − 0.403655i
\(439\) 16.2239 0.774326 0.387163 0.922011i \(-0.373455\pi\)
0.387163 + 0.922011i \(0.373455\pi\)
\(440\) 0 0
\(441\) −2.40533 −0.114539
\(442\) − 13.5565i − 0.644815i
\(443\) 15.6770i 0.744834i 0.928065 + 0.372417i \(0.121471\pi\)
−0.928065 + 0.372417i \(0.878529\pi\)
\(444\) 10.0753 0.478151
\(445\) 0 0
\(446\) −5.72651 −0.271158
\(447\) 4.66935i 0.220853i
\(448\) − 3.08719i − 0.145856i
\(449\) −22.5727 −1.06527 −0.532636 0.846345i \(-0.678798\pi\)
−0.532636 + 0.846345i \(0.678798\pi\)
\(450\) 0 0
\(451\) −36.2667 −1.70773
\(452\) − 6.00000i − 0.282216i
\(453\) − 0.422160i − 0.0198348i
\(454\) 17.6274 0.827295
\(455\) 0 0
\(456\) 4.41970 0.206972
\(457\) − 1.45302i − 0.0679693i −0.999422 0.0339846i \(-0.989180\pi\)
0.999422 0.0339846i \(-0.0108197\pi\)
\(458\) 16.0753i 0.751148i
\(459\) −19.4078 −0.905878
\(460\) 0 0
\(461\) −29.7027 −1.38339 −0.691695 0.722189i \(-0.743136\pi\)
−0.691695 + 0.722189i \(0.743136\pi\)
\(462\) − 28.5922i − 1.33023i
\(463\) − 22.9624i − 1.06715i −0.845752 0.533576i \(-0.820848\pi\)
0.845752 0.533576i \(-0.179152\pi\)
\(464\) −0.863254 −0.0400756
\(465\) 0 0
\(466\) −6.44787 −0.298692
\(467\) 9.48550i 0.438937i 0.975620 + 0.219468i \(0.0704323\pi\)
−0.975620 + 0.219468i \(0.929568\pi\)
\(468\) 3.75467i 0.173560i
\(469\) 31.1043 1.43626
\(470\) 0 0
\(471\) 10.2172 0.470782
\(472\) 6.86325i 0.315907i
\(473\) 51.7541i 2.37966i
\(474\) −22.6222 −1.03907
\(475\) 0 0
\(476\) −10.5941 −0.485579
\(477\) 5.70266i 0.261107i
\(478\) − 4.34876i − 0.198908i
\(479\) 30.5659 1.39659 0.698296 0.715809i \(-0.253942\pi\)
0.698296 + 0.715809i \(0.253942\pi\)
\(480\) 0 0
\(481\) −27.8018 −1.26765
\(482\) 0.764142i 0.0348057i
\(483\) 4.41970i 0.201104i
\(484\) −30.8513 −1.40233
\(485\) 0 0
\(486\) 9.45734 0.428994
\(487\) − 21.1367i − 0.957797i −0.877870 0.478899i \(-0.841036\pi\)
0.877870 0.478899i \(-0.158964\pi\)
\(488\) 13.5069i 0.611428i
\(489\) 11.6317 0.526004
\(490\) 0 0
\(491\) −28.0514 −1.26594 −0.632971 0.774175i \(-0.718165\pi\)
−0.632971 + 0.774175i \(0.718165\pi\)
\(492\) 8.02571i 0.361827i
\(493\) 2.96237i 0.133418i
\(494\) −12.1958 −0.548714
\(495\) 0 0
\(496\) −5.95044 −0.267183
\(497\) 7.92905i 0.355667i
\(498\) − 12.9385i − 0.579789i
\(499\) −3.80178 −0.170191 −0.0850954 0.996373i \(-0.527120\pi\)
−0.0850954 + 0.996373i \(0.527120\pi\)
\(500\) 0 0
\(501\) −11.1692 −0.499005
\(502\) − 22.5402i − 1.00602i
\(503\) − 19.0872i − 0.851056i −0.904945 0.425528i \(-0.860088\pi\)
0.904945 0.425528i \(-0.139912\pi\)
\(504\) 2.93420 0.130700
\(505\) 0 0
\(506\) 6.46926 0.287594
\(507\) 3.73083i 0.165692i
\(508\) − 20.9385i − 0.928997i
\(509\) 9.41024 0.417101 0.208551 0.978012i \(-0.433125\pi\)
0.208551 + 0.978012i \(0.433125\pi\)
\(510\) 0 0
\(511\) 18.2172 0.805880
\(512\) 1.00000i 0.0441942i
\(513\) 17.4598i 0.770869i
\(514\) 9.90089 0.436709
\(515\) 0 0
\(516\) 11.4530 0.504191
\(517\) − 25.2359i − 1.10987i
\(518\) 21.7265i 0.954608i
\(519\) 27.6770 1.21488
\(520\) 0 0
\(521\) 25.8018 1.13040 0.565198 0.824955i \(-0.308800\pi\)
0.565198 + 0.824955i \(0.308800\pi\)
\(522\) − 0.820475i − 0.0359112i
\(523\) 19.1129i 0.835749i 0.908505 + 0.417874i \(0.137225\pi\)
−0.908505 + 0.417874i \(0.862775\pi\)
\(524\) −10.7641 −0.470234
\(525\) 0 0
\(526\) −7.25725 −0.316431
\(527\) 20.4197i 0.889496i
\(528\) 9.26157i 0.403058i
\(529\) −1.00000 −0.0434783
\(530\) 0 0
\(531\) −6.52314 −0.283080
\(532\) 9.53074i 0.413210i
\(533\) − 22.1462i − 0.959259i
\(534\) −24.0000 −1.03858
\(535\) 0 0
\(536\) −10.0753 −0.435185
\(537\) 3.11290i 0.134332i
\(538\) 6.93852i 0.299141i
\(539\) 16.3720 0.705193
\(540\) 0 0
\(541\) 30.8394 1.32589 0.662945 0.748668i \(-0.269306\pi\)
0.662945 + 0.748668i \(0.269306\pi\)
\(542\) − 10.2992i − 0.442389i
\(543\) 10.3959i 0.446129i
\(544\) 3.43163 0.147130
\(545\) 0 0
\(546\) 17.4598 0.747210
\(547\) 13.8513i 0.592240i 0.955151 + 0.296120i \(0.0956929\pi\)
−0.955151 + 0.296120i \(0.904307\pi\)
\(548\) − 3.26157i − 0.139327i
\(549\) −12.8375 −0.547893
\(550\) 0 0
\(551\) 2.66503 0.113534
\(552\) − 1.43163i − 0.0609341i
\(553\) − 48.7831i − 2.07447i
\(554\) 17.8018 0.756325
\(555\) 0 0
\(556\) −16.9385 −0.718353
\(557\) − 28.7641i − 1.21878i −0.792872 0.609388i \(-0.791415\pi\)
0.792872 0.609388i \(-0.208585\pi\)
\(558\) − 5.65556i − 0.239419i
\(559\) −31.6036 −1.33669
\(560\) 0 0
\(561\) 31.7823 1.34185
\(562\) − 18.9385i − 0.798873i
\(563\) − 36.1505i − 1.52356i −0.647834 0.761782i \(-0.724325\pi\)
0.647834 0.761782i \(-0.275675\pi\)
\(564\) −5.58462 −0.235155
\(565\) 0 0
\(566\) 12.6889 0.533353
\(567\) − 16.1933i − 0.680055i
\(568\) − 2.56837i − 0.107767i
\(569\) 10.3488 0.433843 0.216921 0.976189i \(-0.430399\pi\)
0.216921 + 0.976189i \(0.430399\pi\)
\(570\) 0 0
\(571\) 19.6060 0.820486 0.410243 0.911976i \(-0.365444\pi\)
0.410243 + 0.911976i \(0.365444\pi\)
\(572\) − 25.5565i − 1.06857i
\(573\) − 12.2973i − 0.513729i
\(574\) −17.3068 −0.722372
\(575\) 0 0
\(576\) −0.950444 −0.0396018
\(577\) − 34.1505i − 1.42171i −0.703341 0.710853i \(-0.748309\pi\)
0.703341 0.710853i \(-0.251691\pi\)
\(578\) 5.22394i 0.217287i
\(579\) −3.50444 −0.145639
\(580\) 0 0
\(581\) 27.9009 1.15752
\(582\) − 20.4649i − 0.848299i
\(583\) − 38.8156i − 1.60758i
\(584\) −5.90089 −0.244180
\(585\) 0 0
\(586\) −6.00000 −0.247858
\(587\) − 5.19062i − 0.214240i −0.994246 0.107120i \(-0.965837\pi\)
0.994246 0.107120i \(-0.0341629\pi\)
\(588\) − 3.62308i − 0.149413i
\(589\) 18.3701 0.756929
\(590\) 0 0
\(591\) −15.3796 −0.632633
\(592\) − 7.03763i − 0.289245i
\(593\) 2.09911i 0.0862002i 0.999071 + 0.0431001i \(0.0137234\pi\)
−0.999071 + 0.0431001i \(0.986277\pi\)
\(594\) −36.5873 −1.50120
\(595\) 0 0
\(596\) 3.26157 0.133599
\(597\) − 16.1933i − 0.662748i
\(598\) 3.95044i 0.161546i
\(599\) −11.1581 −0.455909 −0.227955 0.973672i \(-0.573204\pi\)
−0.227955 + 0.973672i \(0.573204\pi\)
\(600\) 0 0
\(601\) 9.57784 0.390688 0.195344 0.980735i \(-0.437418\pi\)
0.195344 + 0.980735i \(0.437418\pi\)
\(602\) 24.6975i 1.00660i
\(603\) − 9.57597i − 0.389964i
\(604\) −0.294881 −0.0119986
\(605\) 0 0
\(606\) 1.23586 0.0502033
\(607\) − 24.6975i − 1.00244i −0.865320 0.501221i \(-0.832885\pi\)
0.865320 0.501221i \(-0.167115\pi\)
\(608\) − 3.08719i − 0.125202i
\(609\) −3.81533 −0.154605
\(610\) 0 0
\(611\) 15.4102 0.623431
\(612\) 3.26157i 0.131841i
\(613\) 26.3488i 1.06422i 0.846676 + 0.532108i \(0.178600\pi\)
−0.846676 + 0.532108i \(0.821400\pi\)
\(614\) 22.9599 0.926587
\(615\) 0 0
\(616\) −19.9718 −0.804688
\(617\) 5.94612i 0.239382i 0.992811 + 0.119691i \(0.0381904\pi\)
−0.992811 + 0.119691i \(0.961810\pi\)
\(618\) − 2.19145i − 0.0881530i
\(619\) −39.6856 −1.59510 −0.797549 0.603254i \(-0.793871\pi\)
−0.797549 + 0.603254i \(0.793871\pi\)
\(620\) 0 0
\(621\) 5.65556 0.226950
\(622\) 18.0753i 0.724752i
\(623\) − 51.7541i − 2.07348i
\(624\) −5.65556 −0.226404
\(625\) 0 0
\(626\) 15.1625 0.606014
\(627\) − 28.5922i − 1.14186i
\(628\) − 7.13675i − 0.284787i
\(629\) −24.1505 −0.962945
\(630\) 0 0
\(631\) −23.0138 −0.916164 −0.458082 0.888910i \(-0.651463\pi\)
−0.458082 + 0.888910i \(0.651463\pi\)
\(632\) 15.8018i 0.628561i
\(633\) − 33.0891i − 1.31517i
\(634\) −14.0257 −0.557032
\(635\) 0 0
\(636\) −8.58976 −0.340606
\(637\) 9.99754i 0.396117i
\(638\) 5.58462i 0.221097i
\(639\) 2.44109 0.0965682
\(640\) 0 0
\(641\) 25.4617 1.00568 0.502838 0.864381i \(-0.332289\pi\)
0.502838 + 0.864381i \(0.332289\pi\)
\(642\) 25.2359i 0.995980i
\(643\) 16.4479i 0.648641i 0.945947 + 0.324320i \(0.105136\pi\)
−0.945947 + 0.324320i \(0.894864\pi\)
\(644\) 3.08719 0.121652
\(645\) 0 0
\(646\) −10.5941 −0.416819
\(647\) − 34.9147i − 1.37264i −0.727301 0.686319i \(-0.759226\pi\)
0.727301 0.686319i \(-0.240774\pi\)
\(648\) 5.24533i 0.206056i
\(649\) 44.4002 1.74286
\(650\) 0 0
\(651\) −26.2992 −1.03075
\(652\) − 8.12482i − 0.318193i
\(653\) 34.2754i 1.34130i 0.741775 + 0.670649i \(0.233984\pi\)
−0.741775 + 0.670649i \(0.766016\pi\)
\(654\) 4.17006 0.163062
\(655\) 0 0
\(656\) 5.60601 0.218878
\(657\) − 5.60846i − 0.218807i
\(658\) − 12.0428i − 0.469476i
\(659\) −35.2548 −1.37333 −0.686666 0.726973i \(-0.740926\pi\)
−0.686666 + 0.726973i \(0.740926\pi\)
\(660\) 0 0
\(661\) 28.5684 1.11118 0.555590 0.831456i \(-0.312492\pi\)
0.555590 + 0.831456i \(0.312492\pi\)
\(662\) − 24.7403i − 0.961559i
\(663\) 19.4078i 0.753736i
\(664\) −9.03763 −0.350728
\(665\) 0 0
\(666\) 6.68888 0.259189
\(667\) − 0.863254i − 0.0334253i
\(668\) 7.80178i 0.301860i
\(669\) 8.19822 0.316962
\(670\) 0 0
\(671\) 87.3796 3.37325
\(672\) 4.41970i 0.170494i
\(673\) 23.8770i 0.920392i 0.887817 + 0.460196i \(0.152221\pi\)
−0.887817 + 0.460196i \(0.847779\pi\)
\(674\) −14.7146 −0.566785
\(675\) 0 0
\(676\) 2.60601 0.100231
\(677\) − 6.00000i − 0.230599i −0.993331 0.115299i \(-0.963217\pi\)
0.993331 0.115299i \(-0.0367827\pi\)
\(678\) 8.58976i 0.329888i
\(679\) 44.1310 1.69359
\(680\) 0 0
\(681\) −25.2359 −0.967040
\(682\) 38.4950i 1.47405i
\(683\) − 34.6480i − 1.32577i −0.748722 0.662884i \(-0.769332\pi\)
0.748722 0.662884i \(-0.230668\pi\)
\(684\) 2.93420 0.112192
\(685\) 0 0
\(686\) −13.7975 −0.526789
\(687\) − 23.0138i − 0.878031i
\(688\) − 8.00000i − 0.304997i
\(689\) 23.7027 0.903000
\(690\) 0 0
\(691\) −18.1744 −0.691386 −0.345693 0.938348i \(-0.612356\pi\)
−0.345693 + 0.938348i \(0.612356\pi\)
\(692\) − 19.3325i − 0.734912i
\(693\) − 18.9821i − 0.721071i
\(694\) −9.43163 −0.358020
\(695\) 0 0
\(696\) 1.23586 0.0468451
\(697\) − 19.2377i − 0.728681i
\(698\) − 18.3488i − 0.694511i
\(699\) 9.23095 0.349146
\(700\) 0 0
\(701\) −40.6907 −1.53687 −0.768434 0.639929i \(-0.778964\pi\)
−0.768434 + 0.639929i \(0.778964\pi\)
\(702\) − 22.3420i − 0.843244i
\(703\) 21.7265i 0.819431i
\(704\) 6.46926 0.243819
\(705\) 0 0
\(706\) −33.1129 −1.24622
\(707\) 2.66503i 0.100229i
\(708\) − 9.82562i − 0.369269i
\(709\) 26.4454 0.993178 0.496589 0.867986i \(-0.334586\pi\)
0.496589 + 0.867986i \(0.334586\pi\)
\(710\) 0 0
\(711\) −15.0187 −0.563245
\(712\) 16.7641i 0.628263i
\(713\) − 5.95044i − 0.222846i
\(714\) 15.1668 0.567602
\(715\) 0 0
\(716\) 2.17438 0.0812604
\(717\) 6.22580i 0.232507i
\(718\) − 33.0376i − 1.23295i
\(719\) 2.24778 0.0838281 0.0419140 0.999121i \(-0.486654\pi\)
0.0419140 + 0.999121i \(0.486654\pi\)
\(720\) 0 0
\(721\) 4.72568 0.175994
\(722\) − 9.46926i − 0.352409i
\(723\) − 1.09397i − 0.0406850i
\(724\) 7.26157 0.269874
\(725\) 0 0
\(726\) 44.1676 1.63921
\(727\) 21.4830i 0.796762i 0.917220 + 0.398381i \(0.130428\pi\)
−0.917220 + 0.398381i \(0.869572\pi\)
\(728\) − 12.1958i − 0.452005i
\(729\) −29.2754 −1.08427
\(730\) 0 0
\(731\) −27.4530 −1.01539
\(732\) − 19.3368i − 0.714710i
\(733\) − 11.3778i − 0.420247i −0.977675 0.210123i \(-0.932613\pi\)
0.977675 0.210123i \(-0.0673866\pi\)
\(734\) 2.27349 0.0839161
\(735\) 0 0
\(736\) −1.00000 −0.0368605
\(737\) 65.1795i 2.40092i
\(738\) 5.32819i 0.196134i
\(739\) −21.8770 −0.804760 −0.402380 0.915473i \(-0.631817\pi\)
−0.402380 + 0.915473i \(0.631817\pi\)
\(740\) 0 0
\(741\) 17.4598 0.641402
\(742\) − 18.5231i − 0.680006i
\(743\) − 5.36068i − 0.196664i −0.995154 0.0983322i \(-0.968649\pi\)
0.995154 0.0983322i \(-0.0313508\pi\)
\(744\) 8.51882 0.312315
\(745\) 0 0
\(746\) 23.9762 0.877829
\(747\) − 8.58976i − 0.314283i
\(748\) − 22.2001i − 0.811716i
\(749\) −54.4191 −1.98843
\(750\) 0 0
\(751\) 24.3915 0.890060 0.445030 0.895516i \(-0.353193\pi\)
0.445030 + 0.895516i \(0.353193\pi\)
\(752\) 3.90089i 0.142251i
\(753\) 32.2692i 1.17595i
\(754\) −3.41024 −0.124194
\(755\) 0 0
\(756\) −17.4598 −0.635007
\(757\) 41.9437i 1.52447i 0.647301 + 0.762234i \(0.275898\pi\)
−0.647301 + 0.762234i \(0.724102\pi\)
\(758\) 13.8795i 0.504126i
\(759\) −9.26157 −0.336174
\(760\) 0 0
\(761\) −18.3745 −0.666074 −0.333037 0.942914i \(-0.608073\pi\)
−0.333037 + 0.942914i \(0.608073\pi\)
\(762\) 29.9762i 1.08592i
\(763\) 8.99240i 0.325547i
\(764\) −8.58976 −0.310767
\(765\) 0 0
\(766\) 0 0
\(767\) 27.1129i 0.978990i
\(768\) − 1.43163i − 0.0516594i
\(769\) −42.4993 −1.53256 −0.766282 0.642505i \(-0.777895\pi\)
−0.766282 + 0.642505i \(0.777895\pi\)
\(770\) 0 0
\(771\) −14.1744 −0.510478
\(772\) 2.44787i 0.0881008i
\(773\) 15.0376i 0.540866i 0.962739 + 0.270433i \(0.0871669\pi\)
−0.962739 + 0.270433i \(0.912833\pi\)
\(774\) 7.60355 0.273304
\(775\) 0 0
\(776\) −14.2949 −0.513156
\(777\) − 31.1043i − 1.11586i
\(778\) 26.6436i 0.955221i
\(779\) −17.3068 −0.620081
\(780\) 0 0
\(781\) −16.6155 −0.594548
\(782\) 3.43163i 0.122715i
\(783\) 4.88219i 0.174475i
\(784\) −2.53074 −0.0903836
\(785\) 0 0
\(786\) 15.4102 0.549665
\(787\) − 8.39154i − 0.299126i −0.988752 0.149563i \(-0.952213\pi\)
0.988752 0.149563i \(-0.0477867\pi\)
\(788\) 10.7428i 0.382695i
\(789\) 10.3897 0.369882
\(790\) 0 0
\(791\) −18.5231 −0.658607
\(792\) 6.14867i 0.218483i
\(793\) 53.3582i 1.89481i
\(794\) 8.66749 0.307598
\(795\) 0 0
\(796\) −11.3111 −0.400912
\(797\) − 22.0514i − 0.781101i −0.920581 0.390551i \(-0.872285\pi\)
0.920581 0.390551i \(-0.127715\pi\)
\(798\) − 13.6445i − 0.483009i
\(799\) 13.3864 0.473576
\(800\) 0 0
\(801\) −15.9334 −0.562978
\(802\) 39.9762i 1.41161i
\(803\) 38.1744i 1.34714i
\(804\) 14.4240 0.508696
\(805\) 0 0
\(806\) −23.5069 −0.827995
\(807\) − 9.93337i − 0.349671i
\(808\) − 0.863254i − 0.0303692i
\(809\) −2.04524 −0.0719066 −0.0359533 0.999353i \(-0.511447\pi\)
−0.0359533 + 0.999353i \(0.511447\pi\)
\(810\) 0 0
\(811\) 4.24965 0.149225 0.0746126 0.997213i \(-0.476228\pi\)
0.0746126 + 0.997213i \(0.476228\pi\)
\(812\) 2.66503i 0.0935242i
\(813\) 14.7446i 0.517116i
\(814\) −45.5283 −1.59577
\(815\) 0 0
\(816\) −4.91281 −0.171983
\(817\) 24.6975i 0.864057i
\(818\) − 30.1248i − 1.05329i
\(819\) 11.5914 0.405036
\(820\) 0 0
\(821\) −29.2548 −1.02100 −0.510500 0.859878i \(-0.670540\pi\)
−0.510500 + 0.859878i \(0.670540\pi\)
\(822\) 4.66935i 0.162862i
\(823\) − 13.5846i − 0.473530i −0.971567 0.236765i \(-0.923913\pi\)
0.971567 0.236765i \(-0.0760871\pi\)
\(824\) −1.53074 −0.0533258
\(825\) 0 0
\(826\) 21.1882 0.737231
\(827\) 24.7880i 0.861963i 0.902361 + 0.430981i \(0.141833\pi\)
−0.902361 + 0.430981i \(0.858167\pi\)
\(828\) − 0.950444i − 0.0330302i
\(829\) −26.1505 −0.908246 −0.454123 0.890939i \(-0.650047\pi\)
−0.454123 + 0.890939i \(0.650047\pi\)
\(830\) 0 0
\(831\) −25.4855 −0.884082
\(832\) 3.95044i 0.136957i
\(833\) 8.68455i 0.300902i
\(834\) 24.2496 0.839697
\(835\) 0 0
\(836\) −19.9718 −0.690740
\(837\) 33.6531i 1.16322i
\(838\) − 25.8770i − 0.893908i
\(839\) −6.37260 −0.220007 −0.110003 0.993931i \(-0.535086\pi\)
−0.110003 + 0.993931i \(0.535086\pi\)
\(840\) 0 0
\(841\) −28.2548 −0.974303
\(842\) − 10.7146i − 0.369249i
\(843\) 27.1129i 0.933818i
\(844\) −23.1129 −0.795579
\(845\) 0 0
\(846\) −3.70757 −0.127469
\(847\) 95.2439i 3.27262i
\(848\) 6.00000i 0.206041i
\(849\) −18.1657 −0.623447
\(850\) 0 0
\(851\) 7.03763 0.241247
\(852\) 3.67695i 0.125970i
\(853\) 33.6293i 1.15144i 0.817646 + 0.575722i \(0.195279\pi\)
−0.817646 + 0.575722i \(0.804721\pi\)
\(854\) 41.6983 1.42689
\(855\) 0 0
\(856\) 17.6274 0.602492
\(857\) 11.1795i 0.381885i 0.981601 + 0.190943i \(0.0611545\pi\)
−0.981601 + 0.190943i \(0.938846\pi\)
\(858\) 36.5873i 1.24907i
\(859\) −47.9009 −1.63436 −0.817179 0.576385i \(-0.804463\pi\)
−0.817179 + 0.576385i \(0.804463\pi\)
\(860\) 0 0
\(861\) 24.7769 0.844394
\(862\) 32.2496i 1.09843i
\(863\) 30.1180i 1.02523i 0.858619 + 0.512615i \(0.171323\pi\)
−0.858619 + 0.512615i \(0.828677\pi\)
\(864\) 5.65556 0.192406
\(865\) 0 0
\(866\) 20.6393 0.701353
\(867\) − 7.47873i − 0.253991i
\(868\) 18.3701i 0.623523i
\(869\) 102.226 3.46777
\(870\) 0 0
\(871\) −39.8018 −1.34863
\(872\) − 2.91281i − 0.0986402i
\(873\) − 13.5865i − 0.459833i
\(874\) 3.08719 0.104426
\(875\) 0 0
\(876\) 8.44787 0.285427
\(877\) − 22.3940i − 0.756191i −0.925767 0.378096i \(-0.876579\pi\)
0.925767 0.378096i \(-0.123421\pi\)
\(878\) 16.2239i 0.547531i
\(879\) 8.58976 0.289726
\(880\) 0 0
\(881\) 21.1129 0.711312 0.355656 0.934617i \(-0.384258\pi\)
0.355656 + 0.934617i \(0.384258\pi\)
\(882\) − 2.40533i − 0.0809915i
\(883\) − 49.1061i − 1.65255i −0.563265 0.826276i \(-0.690455\pi\)
0.563265 0.826276i \(-0.309545\pi\)
\(884\) 13.5565 0.455953
\(885\) 0 0
\(886\) −15.6770 −0.526678
\(887\) − 43.0566i − 1.44570i −0.691006 0.722849i \(-0.742832\pi\)
0.691006 0.722849i \(-0.257168\pi\)
\(888\) 10.0753i 0.338104i
\(889\) −64.6412 −2.16800
\(890\) 0 0
\(891\) 33.9334 1.13681
\(892\) − 5.72651i − 0.191738i
\(893\) − 12.0428i − 0.402996i
\(894\) −4.66935 −0.156166
\(895\) 0 0
\(896\) 3.08719 0.103136
\(897\) − 5.65556i − 0.188834i
\(898\) − 22.5727i − 0.753261i
\(899\) 5.13675 0.171320
\(900\) 0 0
\(901\) 20.5898 0.685944
\(902\) − 36.2667i − 1.20755i
\(903\) − 35.3576i − 1.17663i
\(904\) 6.00000 0.199557
\(905\) 0 0
\(906\) 0.422160 0.0140253
\(907\) − 37.9762i − 1.26098i −0.776198 0.630489i \(-0.782854\pi\)
0.776198 0.630489i \(-0.217146\pi\)
\(908\) 17.6274i 0.584986i
\(909\) 0.820475 0.0272134
\(910\) 0 0
\(911\) −15.7504 −0.521833 −0.260916 0.965361i \(-0.584025\pi\)
−0.260916 + 0.965361i \(0.584025\pi\)
\(912\) 4.41970i 0.146351i
\(913\) 58.4668i 1.93497i
\(914\) 1.45302 0.0480615
\(915\) 0 0
\(916\) −16.0753 −0.531142
\(917\) 33.2309i 1.09738i
\(918\) − 19.4078i − 0.640552i
\(919\) −30.4240 −1.00360 −0.501798 0.864985i \(-0.667328\pi\)
−0.501798 + 0.864985i \(0.667328\pi\)
\(920\) 0 0
\(921\) −32.8700 −1.08310
\(922\) − 29.7027i − 0.978205i
\(923\) − 10.1462i − 0.333967i
\(924\) 28.5922 0.940615
\(925\) 0 0
\(926\) 22.9624 0.754590
\(927\) − 1.45488i − 0.0477846i
\(928\) − 0.863254i − 0.0283377i
\(929\) 37.8018 1.24024 0.620118 0.784509i \(-0.287085\pi\)
0.620118 + 0.784509i \(0.287085\pi\)
\(930\) 0 0
\(931\) 7.81287 0.256057
\(932\) − 6.44787i − 0.211207i
\(933\) − 25.8770i − 0.847176i
\(934\) −9.48550 −0.310375
\(935\) 0 0
\(936\) −3.75467 −0.122725
\(937\) − 32.6907i − 1.06796i −0.845497 0.533980i \(-0.820696\pi\)
0.845497 0.533980i \(-0.179304\pi\)
\(938\) 31.1043i 1.01559i
\(939\) −21.7070 −0.708381
\(940\) 0 0
\(941\) −46.4882 −1.51547 −0.757736 0.652561i \(-0.773694\pi\)
−0.757736 + 0.652561i \(0.773694\pi\)
\(942\) 10.2172i 0.332893i
\(943\) 5.60601i 0.182557i
\(944\) −6.86325 −0.223380
\(945\) 0 0
\(946\) −51.7541 −1.68267
\(947\) − 37.2052i − 1.20901i −0.796602 0.604504i \(-0.793371\pi\)
0.796602 0.604504i \(-0.206629\pi\)
\(948\) − 22.6222i − 0.734737i
\(949\) −23.3111 −0.756711
\(950\) 0 0
\(951\) 20.0796 0.651125
\(952\) − 10.5941i − 0.343356i
\(953\) − 49.1104i − 1.59084i −0.606056 0.795422i \(-0.707249\pi\)
0.606056 0.795422i \(-0.292751\pi\)
\(954\) −5.70266 −0.184631
\(955\) 0 0
\(956\) 4.34876 0.140649
\(957\) − 7.99509i − 0.258445i
\(958\) 30.5659i 0.987540i
\(959\) −10.0691 −0.325148
\(960\) 0 0
\(961\) 4.40778 0.142187
\(962\) − 27.8018i − 0.896365i
\(963\) 16.7538i 0.539885i
\(964\) −0.764142 −0.0246114
\(965\) 0 0
\(966\) −4.41970 −0.142202
\(967\) 1.96751i 0.0632709i 0.999499 + 0.0316355i \(0.0100716\pi\)
−0.999499 + 0.0316355i \(0.989928\pi\)
\(968\) − 30.8513i − 0.991599i
\(969\) 15.1668 0.487227
\(970\) 0 0
\(971\) 26.1205 0.838247 0.419123 0.907929i \(-0.362337\pi\)
0.419123 + 0.907929i \(0.362337\pi\)
\(972\) 9.45734i 0.303344i
\(973\) 52.2924i 1.67642i
\(974\) 21.1367 0.677265
\(975\) 0 0
\(976\) −13.5069 −0.432345
\(977\) − 0.639319i − 0.0204536i −0.999948 0.0102268i \(-0.996745\pi\)
0.999948 0.0102268i \(-0.00325535\pi\)
\(978\) 11.6317i 0.371941i
\(979\) 108.452 3.46613
\(980\) 0 0
\(981\) 2.76846 0.0883902
\(982\) − 28.0514i − 0.895157i
\(983\) 6.46926i 0.206337i 0.994664 + 0.103169i \(0.0328981\pi\)
−0.994664 + 0.103169i \(0.967102\pi\)
\(984\) −8.02571 −0.255850
\(985\) 0 0
\(986\) −2.96237 −0.0943410
\(987\) 17.2408i 0.548780i
\(988\) − 12.1958i − 0.387999i
\(989\) 8.00000 0.254385
\(990\) 0 0
\(991\) 37.2334 1.18276 0.591379 0.806394i \(-0.298584\pi\)
0.591379 + 0.806394i \(0.298584\pi\)
\(992\) − 5.95044i − 0.188927i
\(993\) 35.4189i 1.12398i
\(994\) −7.92905 −0.251494
\(995\) 0 0
\(996\) 12.9385 0.409973
\(997\) 9.65124i 0.305658i 0.988253 + 0.152829i \(0.0488384\pi\)
−0.988253 + 0.152829i \(0.951162\pi\)
\(998\) − 3.80178i − 0.120343i
\(999\) −39.8018 −1.25927
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 1150.2.b.j.599.5 6
5.2 odd 4 1150.2.a.q.1.2 3
5.3 odd 4 230.2.a.d.1.2 3
5.4 even 2 inner 1150.2.b.j.599.2 6
15.8 even 4 2070.2.a.z.1.2 3
20.3 even 4 1840.2.a.r.1.2 3
20.7 even 4 9200.2.a.cf.1.2 3
40.3 even 4 7360.2.a.ce.1.2 3
40.13 odd 4 7360.2.a.bz.1.2 3
115.68 even 4 5290.2.a.r.1.2 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
230.2.a.d.1.2 3 5.3 odd 4
1150.2.a.q.1.2 3 5.2 odd 4
1150.2.b.j.599.2 6 5.4 even 2 inner
1150.2.b.j.599.5 6 1.1 even 1 trivial
1840.2.a.r.1.2 3 20.3 even 4
2070.2.a.z.1.2 3 15.8 even 4
5290.2.a.r.1.2 3 115.68 even 4
7360.2.a.bz.1.2 3 40.13 odd 4
7360.2.a.ce.1.2 3 40.3 even 4
9200.2.a.cf.1.2 3 20.7 even 4