Properties

Label 950.2.b.g.799.6
Level $950$
Weight $2$
Character 950.799
Analytic conductor $7.586$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 950 = 2 \cdot 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 950.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(7.58578819202\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.4227136.2
Defining polynomial: \(x^{6} + 9 x^{4} + 22 x^{2} + 9\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 799.6
Root \(2.19869i\) of defining polynomial
Character \(\chi\) \(=\) 950.799
Dual form 950.2.b.g.799.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000i q^{2} +3.03293i q^{3} -1.00000 q^{4} -3.03293 q^{6} +2.46980i q^{7} -1.00000i q^{8} -6.19869 q^{9} +O(q^{10})\) \(q+1.00000i q^{2} +3.03293i q^{3} -1.00000 q^{4} -3.03293 q^{6} +2.46980i q^{7} -1.00000i q^{8} -6.19869 q^{9} +0.728896 q^{11} -3.03293i q^{12} +6.23163i q^{13} -2.46980 q^{14} +1.00000 q^{16} +0.563139i q^{17} -6.19869i q^{18} +1.00000 q^{19} -7.49073 q^{21} +0.728896i q^{22} -4.63555i q^{23} +3.03293 q^{24} -6.23163 q^{26} -9.70142i q^{27} -2.46980i q^{28} -10.2316 q^{29} +6.06587 q^{31} +1.00000i q^{32} +2.21069i q^{33} -0.563139 q^{34} +6.19869 q^{36} -5.72890i q^{37} +1.00000i q^{38} -18.9001 q^{39} +4.79476 q^{41} -7.49073i q^{42} -8.06587i q^{43} -0.728896 q^{44} +4.63555 q^{46} +8.12628i q^{47} +3.03293i q^{48} +0.900112 q^{49} -1.70796 q^{51} -6.23163i q^{52} +1.53020i q^{53} +9.70142 q^{54} +2.46980 q^{56} +3.03293i q^{57} -10.2316i q^{58} +5.76183 q^{59} +10.9396 q^{61} +6.06587i q^{62} -15.3095i q^{63} -1.00000 q^{64} -2.21069 q^{66} +12.9330i q^{67} -0.563139i q^{68} +14.0593 q^{69} -4.39738 q^{71} +6.19869i q^{72} +4.09334i q^{73} +5.72890 q^{74} -1.00000 q^{76} +1.80022i q^{77} -18.9001i q^{78} -15.3370 q^{79} +10.8277 q^{81} +4.79476i q^{82} -7.85517i q^{83} +7.49073 q^{84} +8.06587 q^{86} -31.0318i q^{87} -0.728896i q^{88} +10.0000 q^{89} -15.3908 q^{91} +4.63555i q^{92} +18.3974i q^{93} -8.12628 q^{94} -3.03293 q^{96} +11.0055i q^{97} +0.900112i q^{98} -4.51820 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q - 6q^{4} + 4q^{6} - 26q^{9} + O(q^{10}) \) \( 6q - 6q^{4} + 4q^{6} - 26q^{9} + 4q^{11} - 4q^{14} + 6q^{16} + 6q^{19} - 22q^{21} - 4q^{24} - 4q^{26} - 28q^{29} - 8q^{31} + 8q^{34} + 26q^{36} - 58q^{39} - 16q^{41} - 4q^{44} + 28q^{46} - 50q^{49} - 22q^{51} + 14q^{54} + 4q^{56} + 12q^{59} + 44q^{61} - 6q^{64} + 8q^{66} - 16q^{69} - 4q^{71} + 34q^{74} - 6q^{76} - 48q^{79} - 2q^{81} + 22q^{84} + 4q^{86} + 60q^{89} - 14q^{91} - 26q^{94} + 4q^{96} - 48q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/950\mathbb{Z}\right)^\times\).

\(n\) \(77\) \(401\)
\(\chi(n)\) \(-1\) \(1\)

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
\(3\) 3.03293i 1.75107i 0.483159 + 0.875533i \(0.339489\pi\)
−0.483159 + 0.875533i \(0.660511\pi\)
\(4\) −1.00000 −0.500000
\(5\) 0 0
\(6\) −3.03293 −1.23819
\(7\) 2.46980i 0.933495i 0.884391 + 0.466747i \(0.154574\pi\)
−0.884391 + 0.466747i \(0.845426\pi\)
\(8\) − 1.00000i − 0.353553i
\(9\) −6.19869 −2.06623
\(10\) 0 0
\(11\) 0.728896 0.219770 0.109885 0.993944i \(-0.464952\pi\)
0.109885 + 0.993944i \(0.464952\pi\)
\(12\) − 3.03293i − 0.875533i
\(13\) 6.23163i 1.72834i 0.503198 + 0.864171i \(0.332157\pi\)
−0.503198 + 0.864171i \(0.667843\pi\)
\(14\) −2.46980 −0.660081
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 0.563139i 0.136581i 0.997665 + 0.0682907i \(0.0217545\pi\)
−0.997665 + 0.0682907i \(0.978245\pi\)
\(18\) − 6.19869i − 1.46105i
\(19\) 1.00000 0.229416
\(20\) 0 0
\(21\) −7.49073 −1.63461
\(22\) 0.728896i 0.155401i
\(23\) − 4.63555i − 0.966579i −0.875460 0.483290i \(-0.839442\pi\)
0.875460 0.483290i \(-0.160558\pi\)
\(24\) 3.03293 0.619095
\(25\) 0 0
\(26\) −6.23163 −1.22212
\(27\) − 9.70142i − 1.86704i
\(28\) − 2.46980i − 0.466747i
\(29\) −10.2316 −1.89997 −0.949983 0.312303i \(-0.898900\pi\)
−0.949983 + 0.312303i \(0.898900\pi\)
\(30\) 0 0
\(31\) 6.06587 1.08946 0.544731 0.838611i \(-0.316632\pi\)
0.544731 + 0.838611i \(0.316632\pi\)
\(32\) 1.00000i 0.176777i
\(33\) 2.21069i 0.384832i
\(34\) −0.563139 −0.0965776
\(35\) 0 0
\(36\) 6.19869 1.03312
\(37\) − 5.72890i − 0.941825i −0.882180 0.470912i \(-0.843925\pi\)
0.882180 0.470912i \(-0.156075\pi\)
\(38\) 1.00000i 0.162221i
\(39\) −18.9001 −3.02644
\(40\) 0 0
\(41\) 4.79476 0.748816 0.374408 0.927264i \(-0.377846\pi\)
0.374408 + 0.927264i \(0.377846\pi\)
\(42\) − 7.49073i − 1.15584i
\(43\) − 8.06587i − 1.23003i −0.788514 0.615017i \(-0.789149\pi\)
0.788514 0.615017i \(-0.210851\pi\)
\(44\) −0.728896 −0.109885
\(45\) 0 0
\(46\) 4.63555 0.683475
\(47\) 8.12628i 1.18534i 0.805446 + 0.592670i \(0.201926\pi\)
−0.805446 + 0.592670i \(0.798074\pi\)
\(48\) 3.03293i 0.437766i
\(49\) 0.900112 0.128587
\(50\) 0 0
\(51\) −1.70796 −0.239163
\(52\) − 6.23163i − 0.864171i
\(53\) 1.53020i 0.210190i 0.994462 + 0.105095i \(0.0335146\pi\)
−0.994462 + 0.105095i \(0.966485\pi\)
\(54\) 9.70142 1.32020
\(55\) 0 0
\(56\) 2.46980 0.330040
\(57\) 3.03293i 0.401722i
\(58\) − 10.2316i − 1.34348i
\(59\) 5.76183 0.750126 0.375063 0.926999i \(-0.377621\pi\)
0.375063 + 0.926999i \(0.377621\pi\)
\(60\) 0 0
\(61\) 10.9396 1.40067 0.700336 0.713814i \(-0.253034\pi\)
0.700336 + 0.713814i \(0.253034\pi\)
\(62\) 6.06587i 0.770366i
\(63\) − 15.3095i − 1.92882i
\(64\) −1.00000 −0.125000
\(65\) 0 0
\(66\) −2.21069 −0.272118
\(67\) 12.9330i 1.58002i 0.613092 + 0.790012i \(0.289926\pi\)
−0.613092 + 0.790012i \(0.710074\pi\)
\(68\) − 0.563139i − 0.0682907i
\(69\) 14.0593 1.69254
\(70\) 0 0
\(71\) −4.39738 −0.521873 −0.260937 0.965356i \(-0.584031\pi\)
−0.260937 + 0.965356i \(0.584031\pi\)
\(72\) 6.19869i 0.730523i
\(73\) 4.09334i 0.479090i 0.970885 + 0.239545i \(0.0769982\pi\)
−0.970885 + 0.239545i \(0.923002\pi\)
\(74\) 5.72890 0.665971
\(75\) 0 0
\(76\) −1.00000 −0.114708
\(77\) 1.80022i 0.205155i
\(78\) − 18.9001i − 2.14002i
\(79\) −15.3370 −1.72554 −0.862772 0.505593i \(-0.831274\pi\)
−0.862772 + 0.505593i \(0.831274\pi\)
\(80\) 0 0
\(81\) 10.8277 1.20308
\(82\) 4.79476i 0.529493i
\(83\) − 7.85517i − 0.862217i −0.902300 0.431109i \(-0.858123\pi\)
0.902300 0.431109i \(-0.141877\pi\)
\(84\) 7.49073 0.817305
\(85\) 0 0
\(86\) 8.06587 0.869765
\(87\) − 31.0318i − 3.32696i
\(88\) − 0.728896i − 0.0777006i
\(89\) 10.0000 1.06000 0.529999 0.847998i \(-0.322192\pi\)
0.529999 + 0.847998i \(0.322192\pi\)
\(90\) 0 0
\(91\) −15.3908 −1.61340
\(92\) 4.63555i 0.483290i
\(93\) 18.3974i 1.90772i
\(94\) −8.12628 −0.838162
\(95\) 0 0
\(96\) −3.03293 −0.309548
\(97\) 11.0055i 1.11744i 0.829358 + 0.558718i \(0.188706\pi\)
−0.829358 + 0.558718i \(0.811294\pi\)
\(98\) 0.900112i 0.0909250i
\(99\) −4.51820 −0.454096
\(100\) 0 0
\(101\) −9.73436 −0.968605 −0.484302 0.874901i \(-0.660926\pi\)
−0.484302 + 0.874901i \(0.660926\pi\)
\(102\) − 1.70796i − 0.169114i
\(103\) 8.33151i 0.820928i 0.911877 + 0.410464i \(0.134633\pi\)
−0.911877 + 0.410464i \(0.865367\pi\)
\(104\) 6.23163 0.611061
\(105\) 0 0
\(106\) −1.53020 −0.148627
\(107\) − 15.0264i − 1.45266i −0.687348 0.726328i \(-0.741225\pi\)
0.687348 0.726328i \(-0.258775\pi\)
\(108\) 9.70142i 0.933520i
\(109\) 13.6619 1.30858 0.654288 0.756245i \(-0.272968\pi\)
0.654288 + 0.756245i \(0.272968\pi\)
\(110\) 0 0
\(111\) 17.3754 1.64920
\(112\) 2.46980i 0.233374i
\(113\) − 1.20524i − 0.113379i −0.998392 0.0566895i \(-0.981945\pi\)
0.998392 0.0566895i \(-0.0180545\pi\)
\(114\) −3.03293 −0.284060
\(115\) 0 0
\(116\) 10.2316 0.949983
\(117\) − 38.6279i − 3.57115i
\(118\) 5.76183i 0.530419i
\(119\) −1.39084 −0.127498
\(120\) 0 0
\(121\) −10.4687 −0.951701
\(122\) 10.9396i 0.990424i
\(123\) 14.5422i 1.31123i
\(124\) −6.06587 −0.544731
\(125\) 0 0
\(126\) 15.3095 1.36388
\(127\) 9.52366i 0.845088i 0.906342 + 0.422544i \(0.138863\pi\)
−0.906342 + 0.422544i \(0.861137\pi\)
\(128\) − 1.00000i − 0.0883883i
\(129\) 24.4633 2.15387
\(130\) 0 0
\(131\) −17.4028 −1.52049 −0.760247 0.649635i \(-0.774922\pi\)
−0.760247 + 0.649635i \(0.774922\pi\)
\(132\) − 2.21069i − 0.192416i
\(133\) 2.46980i 0.214158i
\(134\) −12.9330 −1.11725
\(135\) 0 0
\(136\) 0.563139 0.0482888
\(137\) 2.25910i 0.193008i 0.995333 + 0.0965040i \(0.0307661\pi\)
−0.995333 + 0.0965040i \(0.969234\pi\)
\(138\) 14.0593i 1.19681i
\(139\) −16.8606 −1.43010 −0.715050 0.699073i \(-0.753596\pi\)
−0.715050 + 0.699073i \(0.753596\pi\)
\(140\) 0 0
\(141\) −24.6465 −2.07561
\(142\) − 4.39738i − 0.369020i
\(143\) 4.54221i 0.379838i
\(144\) −6.19869 −0.516558
\(145\) 0 0
\(146\) −4.09334 −0.338768
\(147\) 2.72998i 0.225165i
\(148\) 5.72890i 0.470912i
\(149\) 13.0055 1.06545 0.532724 0.846289i \(-0.321168\pi\)
0.532724 + 0.846289i \(0.321168\pi\)
\(150\) 0 0
\(151\) 17.5895 1.43142 0.715708 0.698400i \(-0.246104\pi\)
0.715708 + 0.698400i \(0.246104\pi\)
\(152\) − 1.00000i − 0.0811107i
\(153\) − 3.49073i − 0.282209i
\(154\) −1.80022 −0.145066
\(155\) 0 0
\(156\) 18.9001 1.51322
\(157\) 9.52366i 0.760071i 0.924972 + 0.380035i \(0.124088\pi\)
−0.924972 + 0.380035i \(0.875912\pi\)
\(158\) − 15.3370i − 1.22014i
\(159\) −4.64101 −0.368056
\(160\) 0 0
\(161\) 11.4489 0.902297
\(162\) 10.8277i 0.850704i
\(163\) − 13.1921i − 1.03329i −0.856200 0.516644i \(-0.827181\pi\)
0.856200 0.516644i \(-0.172819\pi\)
\(164\) −4.79476 −0.374408
\(165\) 0 0
\(166\) 7.85517 0.609680
\(167\) 1.81331i 0.140318i 0.997536 + 0.0701591i \(0.0223507\pi\)
−0.997536 + 0.0701591i \(0.977649\pi\)
\(168\) 7.49073i 0.577922i
\(169\) −25.8332 −1.98717
\(170\) 0 0
\(171\) −6.19869 −0.474026
\(172\) 8.06587i 0.615017i
\(173\) 12.5237i 0.952156i 0.879403 + 0.476078i \(0.157942\pi\)
−0.879403 + 0.476078i \(0.842058\pi\)
\(174\) 31.0318 2.35252
\(175\) 0 0
\(176\) 0.728896 0.0549426
\(177\) 17.4753i 1.31352i
\(178\) 10.0000i 0.749532i
\(179\) −1.39738 −0.104445 −0.0522226 0.998635i \(-0.516631\pi\)
−0.0522226 + 0.998635i \(0.516631\pi\)
\(180\) 0 0
\(181\) 5.72890 0.425825 0.212913 0.977071i \(-0.431705\pi\)
0.212913 + 0.977071i \(0.431705\pi\)
\(182\) − 15.3908i − 1.14084i
\(183\) 33.1791i 2.45267i
\(184\) −4.63555 −0.341737
\(185\) 0 0
\(186\) −18.3974 −1.34896
\(187\) 0.410470i 0.0300165i
\(188\) − 8.12628i − 0.592670i
\(189\) 23.9605 1.74287
\(190\) 0 0
\(191\) −27.0198 −1.95509 −0.977544 0.210733i \(-0.932415\pi\)
−0.977544 + 0.210733i \(0.932415\pi\)
\(192\) − 3.03293i − 0.218883i
\(193\) 23.0713i 1.66071i 0.557234 + 0.830355i \(0.311862\pi\)
−0.557234 + 0.830355i \(0.688138\pi\)
\(194\) −11.0055 −0.790146
\(195\) 0 0
\(196\) −0.900112 −0.0642937
\(197\) 0.794765i 0.0566247i 0.999599 + 0.0283123i \(0.00901330\pi\)
−0.999599 + 0.0283123i \(0.990987\pi\)
\(198\) − 4.51820i − 0.321095i
\(199\) −8.07241 −0.572238 −0.286119 0.958194i \(-0.592365\pi\)
−0.286119 + 0.958194i \(0.592365\pi\)
\(200\) 0 0
\(201\) −39.2251 −2.76672
\(202\) − 9.73436i − 0.684907i
\(203\) − 25.2700i − 1.77361i
\(204\) 1.70796 0.119581
\(205\) 0 0
\(206\) −8.33151 −0.580484
\(207\) 28.7344i 1.99718i
\(208\) 6.23163i 0.432085i
\(209\) 0.728896 0.0504188
\(210\) 0 0
\(211\) −11.6410 −0.801400 −0.400700 0.916209i \(-0.631233\pi\)
−0.400700 + 0.916209i \(0.631233\pi\)
\(212\) − 1.53020i − 0.105095i
\(213\) − 13.3370i − 0.913834i
\(214\) 15.0264 1.02718
\(215\) 0 0
\(216\) −9.70142 −0.660098
\(217\) 14.9815i 1.01701i
\(218\) 13.6619i 0.925304i
\(219\) −12.4148 −0.838917
\(220\) 0 0
\(221\) −3.50927 −0.236059
\(222\) 17.3754i 1.16616i
\(223\) 15.6554i 1.04836i 0.851607 + 0.524182i \(0.175629\pi\)
−0.851607 + 0.524182i \(0.824371\pi\)
\(224\) −2.46980 −0.165020
\(225\) 0 0
\(226\) 1.20524 0.0801710
\(227\) 4.80131i 0.318674i 0.987224 + 0.159337i \(0.0509357\pi\)
−0.987224 + 0.159337i \(0.949064\pi\)
\(228\) − 3.03293i − 0.200861i
\(229\) −2.79476 −0.184683 −0.0923416 0.995727i \(-0.529435\pi\)
−0.0923416 + 0.995727i \(0.529435\pi\)
\(230\) 0 0
\(231\) −5.45996 −0.359239
\(232\) 10.2316i 0.671739i
\(233\) − 11.5422i − 0.756155i −0.925774 0.378078i \(-0.876585\pi\)
0.925774 0.378078i \(-0.123415\pi\)
\(234\) 38.6279 2.52519
\(235\) 0 0
\(236\) −5.76183 −0.375063
\(237\) − 46.5160i − 3.02154i
\(238\) − 1.39084i − 0.0901547i
\(239\) 26.2251 1.69636 0.848180 0.529708i \(-0.177699\pi\)
0.848180 + 0.529708i \(0.177699\pi\)
\(240\) 0 0
\(241\) −12.0659 −0.777231 −0.388615 0.921400i \(-0.627047\pi\)
−0.388615 + 0.921400i \(0.627047\pi\)
\(242\) − 10.4687i − 0.672954i
\(243\) 3.73544i 0.239629i
\(244\) −10.9396 −0.700336
\(245\) 0 0
\(246\) −14.5422 −0.927177
\(247\) 6.23163i 0.396509i
\(248\) − 6.06587i − 0.385183i
\(249\) 23.8242 1.50980
\(250\) 0 0
\(251\) 13.5237 0.853606 0.426803 0.904345i \(-0.359640\pi\)
0.426803 + 0.904345i \(0.359640\pi\)
\(252\) 15.3095i 0.964408i
\(253\) − 3.37884i − 0.212426i
\(254\) −9.52366 −0.597568
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 22.7398i 1.41847i 0.704972 + 0.709235i \(0.250960\pi\)
−0.704972 + 0.709235i \(0.749040\pi\)
\(258\) 24.4633i 1.52302i
\(259\) 14.1492 0.879188
\(260\) 0 0
\(261\) 63.4227 3.92577
\(262\) − 17.4028i − 1.07515i
\(263\) 6.67395i 0.411533i 0.978601 + 0.205767i \(0.0659688\pi\)
−0.978601 + 0.205767i \(0.934031\pi\)
\(264\) 2.21069 0.136059
\(265\) 0 0
\(266\) −2.46980 −0.151433
\(267\) 30.3293i 1.85613i
\(268\) − 12.9330i − 0.790012i
\(269\) 29.7398 1.81327 0.906634 0.421917i \(-0.138643\pi\)
0.906634 + 0.421917i \(0.138643\pi\)
\(270\) 0 0
\(271\) 1.11189 0.0675426 0.0337713 0.999430i \(-0.489248\pi\)
0.0337713 + 0.999430i \(0.489248\pi\)
\(272\) 0.563139i 0.0341453i
\(273\) − 46.6794i − 2.82517i
\(274\) −2.25910 −0.136477
\(275\) 0 0
\(276\) −14.0593 −0.846272
\(277\) 13.1263i 0.788682i 0.918964 + 0.394341i \(0.129027\pi\)
−0.918964 + 0.394341i \(0.870973\pi\)
\(278\) − 16.8606i − 1.01123i
\(279\) −37.6004 −2.25108
\(280\) 0 0
\(281\) 22.9265 1.36768 0.683840 0.729632i \(-0.260309\pi\)
0.683840 + 0.729632i \(0.260309\pi\)
\(282\) − 24.6465i − 1.46768i
\(283\) − 0.860634i − 0.0511594i −0.999673 0.0255797i \(-0.991857\pi\)
0.999673 0.0255797i \(-0.00814315\pi\)
\(284\) 4.39738 0.260937
\(285\) 0 0
\(286\) −4.54221 −0.268586
\(287\) 11.8421i 0.699016i
\(288\) − 6.19869i − 0.365261i
\(289\) 16.6829 0.981346
\(290\) 0 0
\(291\) −33.3788 −1.95670
\(292\) − 4.09334i − 0.239545i
\(293\) − 19.8212i − 1.15796i −0.815340 0.578982i \(-0.803450\pi\)
0.815340 0.578982i \(-0.196550\pi\)
\(294\) −2.72998 −0.159216
\(295\) 0 0
\(296\) −5.72890 −0.332985
\(297\) − 7.07133i − 0.410320i
\(298\) 13.0055i 0.753386i
\(299\) 28.8870 1.67058
\(300\) 0 0
\(301\) 19.9210 1.14823
\(302\) 17.5895i 1.01216i
\(303\) − 29.5237i − 1.69609i
\(304\) 1.00000 0.0573539
\(305\) 0 0
\(306\) 3.49073 0.199552
\(307\) − 16.8661i − 0.962599i −0.876556 0.481299i \(-0.840165\pi\)
0.876556 0.481299i \(-0.159835\pi\)
\(308\) − 1.80022i − 0.102577i
\(309\) −25.2689 −1.43750
\(310\) 0 0
\(311\) 10.3250 0.585475 0.292738 0.956193i \(-0.405434\pi\)
0.292738 + 0.956193i \(0.405434\pi\)
\(312\) 18.9001i 1.07001i
\(313\) 15.7684i 0.891281i 0.895212 + 0.445641i \(0.147024\pi\)
−0.895212 + 0.445641i \(0.852976\pi\)
\(314\) −9.52366 −0.537451
\(315\) 0 0
\(316\) 15.3370 0.862772
\(317\) − 2.17230i − 0.122009i −0.998138 0.0610043i \(-0.980570\pi\)
0.998138 0.0610043i \(-0.0194303\pi\)
\(318\) − 4.64101i − 0.260255i
\(319\) −7.45779 −0.417556
\(320\) 0 0
\(321\) 45.5741 2.54370
\(322\) 11.4489i 0.638020i
\(323\) 0.563139i 0.0313339i
\(324\) −10.8277 −0.601539
\(325\) 0 0
\(326\) 13.1921 0.730645
\(327\) 41.4358i 2.29140i
\(328\) − 4.79476i − 0.264747i
\(329\) −20.0702 −1.10651
\(330\) 0 0
\(331\) 11.9791 0.658429 0.329215 0.944255i \(-0.393216\pi\)
0.329215 + 0.944255i \(0.393216\pi\)
\(332\) 7.85517i 0.431109i
\(333\) 35.5117i 1.94603i
\(334\) −1.81331 −0.0992200
\(335\) 0 0
\(336\) −7.49073 −0.408653
\(337\) − 11.1921i − 0.609675i −0.952404 0.304838i \(-0.901398\pi\)
0.952404 0.304838i \(-0.0986022\pi\)
\(338\) − 25.8332i − 1.40514i
\(339\) 3.65540 0.198534
\(340\) 0 0
\(341\) 4.42139 0.239432
\(342\) − 6.19869i − 0.335187i
\(343\) 19.5117i 1.05353i
\(344\) −8.06587 −0.434883
\(345\) 0 0
\(346\) −12.5237 −0.673276
\(347\) − 20.3843i − 1.09429i −0.837039 0.547143i \(-0.815715\pi\)
0.837039 0.547143i \(-0.184285\pi\)
\(348\) 31.0318i 1.66348i
\(349\) 0.252557 0.0135191 0.00675954 0.999977i \(-0.497848\pi\)
0.00675954 + 0.999977i \(0.497848\pi\)
\(350\) 0 0
\(351\) 60.4556 3.22688
\(352\) 0.728896i 0.0388503i
\(353\) 28.6434i 1.52453i 0.647263 + 0.762267i \(0.275914\pi\)
−0.647263 + 0.762267i \(0.724086\pi\)
\(354\) −17.4753 −0.928799
\(355\) 0 0
\(356\) −10.0000 −0.529999
\(357\) − 4.21832i − 0.223257i
\(358\) − 1.39738i − 0.0738540i
\(359\) 18.5741 0.980301 0.490151 0.871638i \(-0.336942\pi\)
0.490151 + 0.871638i \(0.336942\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 5.72890i 0.301104i
\(363\) − 31.7509i − 1.66649i
\(364\) 15.3908 0.806699
\(365\) 0 0
\(366\) −33.1791 −1.73430
\(367\) − 4.79476i − 0.250285i −0.992139 0.125142i \(-0.960061\pi\)
0.992139 0.125142i \(-0.0399388\pi\)
\(368\) − 4.63555i − 0.241645i
\(369\) −29.7213 −1.54723
\(370\) 0 0
\(371\) −3.77929 −0.196211
\(372\) − 18.3974i − 0.953860i
\(373\) − 15.9485i − 0.825783i −0.910780 0.412892i \(-0.864519\pi\)
0.910780 0.412892i \(-0.135481\pi\)
\(374\) −0.410470 −0.0212249
\(375\) 0 0
\(376\) 8.12628 0.419081
\(377\) − 63.7597i − 3.28379i
\(378\) 23.9605i 1.23240i
\(379\) 16.8013 0.863025 0.431513 0.902107i \(-0.357980\pi\)
0.431513 + 0.902107i \(0.357980\pi\)
\(380\) 0 0
\(381\) −28.8846 −1.47980
\(382\) − 27.0198i − 1.38246i
\(383\) − 26.7948i − 1.36915i −0.728943 0.684574i \(-0.759988\pi\)
0.728943 0.684574i \(-0.240012\pi\)
\(384\) 3.03293 0.154774
\(385\) 0 0
\(386\) −23.0713 −1.17430
\(387\) 49.9978i 2.54153i
\(388\) − 11.0055i − 0.558718i
\(389\) −18.3424 −0.929998 −0.464999 0.885311i \(-0.653945\pi\)
−0.464999 + 0.885311i \(0.653945\pi\)
\(390\) 0 0
\(391\) 2.61046 0.132017
\(392\) − 0.900112i − 0.0454625i
\(393\) − 52.7817i − 2.66248i
\(394\) −0.794765 −0.0400397
\(395\) 0 0
\(396\) 4.51820 0.227048
\(397\) 21.2162i 1.06481i 0.846490 + 0.532404i \(0.178711\pi\)
−0.846490 + 0.532404i \(0.821289\pi\)
\(398\) − 8.07241i − 0.404633i
\(399\) −7.49073 −0.375005
\(400\) 0 0
\(401\) −27.8661 −1.39157 −0.695783 0.718252i \(-0.744943\pi\)
−0.695783 + 0.718252i \(0.744943\pi\)
\(402\) − 39.2251i − 1.95637i
\(403\) 37.8002i 1.88296i
\(404\) 9.73436 0.484302
\(405\) 0 0
\(406\) 25.2700 1.25413
\(407\) − 4.17577i − 0.206985i
\(408\) 1.70796i 0.0845568i
\(409\) 18.0899 0.894487 0.447243 0.894412i \(-0.352406\pi\)
0.447243 + 0.894412i \(0.352406\pi\)
\(410\) 0 0
\(411\) −6.85171 −0.337970
\(412\) − 8.33151i − 0.410464i
\(413\) 14.2305i 0.700239i
\(414\) −28.7344 −1.41222
\(415\) 0 0
\(416\) −6.23163 −0.305531
\(417\) − 51.1372i − 2.50420i
\(418\) 0.728896i 0.0356515i
\(419\) 15.3370 0.749260 0.374630 0.927174i \(-0.377770\pi\)
0.374630 + 0.927174i \(0.377770\pi\)
\(420\) 0 0
\(421\) −8.42486 −0.410602 −0.205301 0.978699i \(-0.565817\pi\)
−0.205301 + 0.978699i \(0.565817\pi\)
\(422\) − 11.6410i − 0.566676i
\(423\) − 50.3723i − 2.44918i
\(424\) 1.53020 0.0743133
\(425\) 0 0
\(426\) 13.3370 0.646178
\(427\) 27.0185i 1.30752i
\(428\) 15.0264i 0.726328i
\(429\) −13.7762 −0.665122
\(430\) 0 0
\(431\) 16.2766 0.784014 0.392007 0.919962i \(-0.371781\pi\)
0.392007 + 0.919962i \(0.371781\pi\)
\(432\) − 9.70142i − 0.466760i
\(433\) 18.1976i 0.874521i 0.899335 + 0.437261i \(0.144051\pi\)
−0.899335 + 0.437261i \(0.855949\pi\)
\(434\) −14.9815 −0.719133
\(435\) 0 0
\(436\) −13.6619 −0.654288
\(437\) − 4.63555i − 0.221749i
\(438\) − 12.4148i − 0.593204i
\(439\) −20.4214 −0.974660 −0.487330 0.873218i \(-0.662029\pi\)
−0.487330 + 0.873218i \(0.662029\pi\)
\(440\) 0 0
\(441\) −5.57952 −0.265691
\(442\) − 3.50927i − 0.166919i
\(443\) − 21.6685i − 1.02950i −0.857340 0.514750i \(-0.827885\pi\)
0.857340 0.514750i \(-0.172115\pi\)
\(444\) −17.3754 −0.824598
\(445\) 0 0
\(446\) −15.6554 −0.741305
\(447\) 39.4447i 1.86567i
\(448\) − 2.46980i − 0.116687i
\(449\) −23.8133 −1.12382 −0.561910 0.827198i \(-0.689933\pi\)
−0.561910 + 0.827198i \(0.689933\pi\)
\(450\) 0 0
\(451\) 3.49489 0.164568
\(452\) 1.20524i 0.0566895i
\(453\) 53.3479i 2.50650i
\(454\) −4.80131 −0.225337
\(455\) 0 0
\(456\) 3.03293 0.142030
\(457\) 9.15813i 0.428399i 0.976790 + 0.214200i \(0.0687143\pi\)
−0.976790 + 0.214200i \(0.931286\pi\)
\(458\) − 2.79476i − 0.130591i
\(459\) 5.46325 0.255003
\(460\) 0 0
\(461\) −7.78931 −0.362784 −0.181392 0.983411i \(-0.558060\pi\)
−0.181392 + 0.983411i \(0.558060\pi\)
\(462\) − 5.45996i − 0.254020i
\(463\) − 0.405011i − 0.0188225i −0.999956 0.00941123i \(-0.997004\pi\)
0.999956 0.00941123i \(-0.00299573\pi\)
\(464\) −10.2316 −0.474991
\(465\) 0 0
\(466\) 11.5422 0.534682
\(467\) − 1.95814i − 0.0906118i −0.998973 0.0453059i \(-0.985574\pi\)
0.998973 0.0453059i \(-0.0144262\pi\)
\(468\) 38.6279i 1.78558i
\(469\) −31.9420 −1.47494
\(470\) 0 0
\(471\) −28.8846 −1.33093
\(472\) − 5.76183i − 0.265210i
\(473\) − 5.87918i − 0.270325i
\(474\) 46.5160 2.13655
\(475\) 0 0
\(476\) 1.39084 0.0637490
\(477\) − 9.48527i − 0.434301i
\(478\) 26.2251i 1.19951i
\(479\) 22.9210 1.04729 0.523645 0.851937i \(-0.324572\pi\)
0.523645 + 0.851937i \(0.324572\pi\)
\(480\) 0 0
\(481\) 35.7003 1.62780
\(482\) − 12.0659i − 0.549585i
\(483\) 34.7237i 1.57998i
\(484\) 10.4687 0.475850
\(485\) 0 0
\(486\) −3.73544 −0.169443
\(487\) − 23.9450i − 1.08505i −0.840038 0.542527i \(-0.817468\pi\)
0.840038 0.542527i \(-0.182532\pi\)
\(488\) − 10.9396i − 0.495212i
\(489\) 40.0109 1.80936
\(490\) 0 0
\(491\) 3.86826 0.174572 0.0872861 0.996183i \(-0.472181\pi\)
0.0872861 + 0.996183i \(0.472181\pi\)
\(492\) − 14.5422i − 0.655613i
\(493\) − 5.76183i − 0.259500i
\(494\) −6.23163 −0.280374
\(495\) 0 0
\(496\) 6.06587 0.272366
\(497\) − 10.8606i − 0.487166i
\(498\) 23.8242i 1.06759i
\(499\) 7.92104 0.354595 0.177297 0.984157i \(-0.443265\pi\)
0.177297 + 0.984157i \(0.443265\pi\)
\(500\) 0 0
\(501\) −5.49966 −0.245706
\(502\) 13.5237i 0.603591i
\(503\) 16.6949i 0.744388i 0.928155 + 0.372194i \(0.121394\pi\)
−0.928155 + 0.372194i \(0.878606\pi\)
\(504\) −15.3095 −0.681939
\(505\) 0 0
\(506\) 3.37884 0.150208
\(507\) − 78.3503i − 3.47966i
\(508\) − 9.52366i − 0.422544i
\(509\) −24.4028 −1.08164 −0.540818 0.841139i \(-0.681885\pi\)
−0.540818 + 0.841139i \(0.681885\pi\)
\(510\) 0 0
\(511\) −10.1097 −0.447228
\(512\) 1.00000i 0.0441942i
\(513\) − 9.70142i − 0.428328i
\(514\) −22.7398 −1.00301
\(515\) 0 0
\(516\) −24.4633 −1.07693
\(517\) 5.92321i 0.260503i
\(518\) 14.1492i 0.621680i
\(519\) −37.9834 −1.66729
\(520\) 0 0
\(521\) 26.0659 1.14197 0.570983 0.820962i \(-0.306562\pi\)
0.570983 + 0.820962i \(0.306562\pi\)
\(522\) 63.4227i 2.77594i
\(523\) − 17.8726i − 0.781516i −0.920493 0.390758i \(-0.872213\pi\)
0.920493 0.390758i \(-0.127787\pi\)
\(524\) 17.4028 0.760247
\(525\) 0 0
\(526\) −6.67395 −0.290998
\(527\) 3.41593i 0.148800i
\(528\) 2.21069i 0.0962081i
\(529\) 1.51166 0.0657243
\(530\) 0 0
\(531\) −35.7158 −1.54993
\(532\) − 2.46980i − 0.107079i
\(533\) 29.8792i 1.29421i
\(534\) −30.3293 −1.31248
\(535\) 0 0
\(536\) 12.9330 0.558623
\(537\) − 4.23817i − 0.182891i
\(538\) 29.7398i 1.28217i
\(539\) 0.656088 0.0282597
\(540\) 0 0
\(541\) 23.0604 0.991444 0.495722 0.868481i \(-0.334903\pi\)
0.495722 + 0.868481i \(0.334903\pi\)
\(542\) 1.11189i 0.0477598i
\(543\) 17.3754i 0.745648i
\(544\) −0.563139 −0.0241444
\(545\) 0 0
\(546\) 46.6794 1.99769
\(547\) − 27.7584i − 1.18686i −0.804885 0.593431i \(-0.797773\pi\)
0.804885 0.593431i \(-0.202227\pi\)
\(548\) − 2.25910i − 0.0965040i
\(549\) −67.8111 −2.89411
\(550\) 0 0
\(551\) −10.2316 −0.435882
\(552\) − 14.0593i − 0.598405i
\(553\) − 37.8792i − 1.61079i
\(554\) −13.1263 −0.557682
\(555\) 0 0
\(556\) 16.8606 0.715050
\(557\) − 8.60808i − 0.364736i −0.983230 0.182368i \(-0.941624\pi\)
0.983230 0.182368i \(-0.0583762\pi\)
\(558\) − 37.6004i − 1.59175i
\(559\) 50.2635 2.12592
\(560\) 0 0
\(561\) −1.24493 −0.0525609
\(562\) 22.9265i 0.967096i
\(563\) − 7.93959i − 0.334614i −0.985905 0.167307i \(-0.946493\pi\)
0.985905 0.167307i \(-0.0535071\pi\)
\(564\) 24.6465 1.03780
\(565\) 0 0
\(566\) 0.860634 0.0361751
\(567\) 26.7422i 1.12307i
\(568\) 4.39738i 0.184510i
\(569\) 1.65757 0.0694889 0.0347444 0.999396i \(-0.488938\pi\)
0.0347444 + 0.999396i \(0.488938\pi\)
\(570\) 0 0
\(571\) −14.0528 −0.588091 −0.294045 0.955791i \(-0.595002\pi\)
−0.294045 + 0.955791i \(0.595002\pi\)
\(572\) − 4.54221i − 0.189919i
\(573\) − 81.9494i − 3.42349i
\(574\) −11.8421 −0.494279
\(575\) 0 0
\(576\) 6.19869 0.258279
\(577\) − 1.36445i − 0.0568027i −0.999597 0.0284014i \(-0.990958\pi\)
0.999597 0.0284014i \(-0.00904165\pi\)
\(578\) 16.6829i 0.693916i
\(579\) −69.9738 −2.90801
\(580\) 0 0
\(581\) 19.4007 0.804876
\(582\) − 33.3788i − 1.38360i
\(583\) 1.11536i 0.0461935i
\(584\) 4.09334 0.169384
\(585\) 0 0
\(586\) 19.8212 0.818804
\(587\) − 24.6609i − 1.01786i −0.860807 0.508931i \(-0.830041\pi\)
0.860807 0.508931i \(-0.169959\pi\)
\(588\) − 2.72998i − 0.112583i
\(589\) 6.06587 0.249940
\(590\) 0 0
\(591\) −2.41047 −0.0991535
\(592\) − 5.72890i − 0.235456i
\(593\) 0.747443i 0.0306938i 0.999882 + 0.0153469i \(0.00488526\pi\)
−0.999882 + 0.0153469i \(0.995115\pi\)
\(594\) 7.07133 0.290140
\(595\) 0 0
\(596\) −13.0055 −0.532724
\(597\) − 24.4831i − 1.00203i
\(598\) 28.8870i 1.18128i
\(599\) −1.40501 −0.0574072 −0.0287036 0.999588i \(-0.509138\pi\)
−0.0287036 + 0.999588i \(0.509138\pi\)
\(600\) 0 0
\(601\) 29.7453 1.21334 0.606668 0.794956i \(-0.292506\pi\)
0.606668 + 0.794956i \(0.292506\pi\)
\(602\) 19.9210i 0.811921i
\(603\) − 80.1680i − 3.26469i
\(604\) −17.5895 −0.715708
\(605\) 0 0
\(606\) 29.5237 1.19932
\(607\) 5.66849i 0.230077i 0.993361 + 0.115038i \(0.0366991\pi\)
−0.993361 + 0.115038i \(0.963301\pi\)
\(608\) 1.00000i 0.0405554i
\(609\) 76.6423 3.10570
\(610\) 0 0
\(611\) −50.6399 −2.04867
\(612\) 3.49073i 0.141104i
\(613\) 2.99454i 0.120948i 0.998170 + 0.0604742i \(0.0192613\pi\)
−0.998170 + 0.0604742i \(0.980739\pi\)
\(614\) 16.8661 0.680660
\(615\) 0 0
\(616\) 1.80022 0.0725331
\(617\) − 32.3370i − 1.30184i −0.759147 0.650919i \(-0.774384\pi\)
0.759147 0.650919i \(-0.225616\pi\)
\(618\) − 25.2689i − 1.01647i
\(619\) 20.9265 0.841107 0.420554 0.907268i \(-0.361836\pi\)
0.420554 + 0.907268i \(0.361836\pi\)
\(620\) 0 0
\(621\) −44.9714 −1.80464
\(622\) 10.3250i 0.413994i
\(623\) 24.6980i 0.989503i
\(624\) −18.9001 −0.756610
\(625\) 0 0
\(626\) −15.7684 −0.630231
\(627\) 2.21069i 0.0882866i
\(628\) − 9.52366i − 0.380035i
\(629\) 3.22617 0.128636
\(630\) 0 0
\(631\) 22.8002 0.907663 0.453831 0.891088i \(-0.350057\pi\)
0.453831 + 0.891088i \(0.350057\pi\)
\(632\) 15.3370i 0.610072i
\(633\) − 35.3064i − 1.40330i
\(634\) 2.17230 0.0862731
\(635\) 0 0
\(636\) 4.64101 0.184028
\(637\) 5.60916i 0.222243i
\(638\) − 7.45779i − 0.295257i
\(639\) 27.2580 1.07831
\(640\) 0 0
\(641\) 36.3184 1.43449 0.717246 0.696820i \(-0.245402\pi\)
0.717246 + 0.696820i \(0.245402\pi\)
\(642\) 45.5741i 1.79866i
\(643\) − 13.6135i − 0.536865i −0.963298 0.268433i \(-0.913494\pi\)
0.963298 0.268433i \(-0.0865057\pi\)
\(644\) −11.4489 −0.451148
\(645\) 0 0
\(646\) −0.563139 −0.0221564
\(647\) − 0.0724126i − 0.00284683i −0.999999 0.00142342i \(-0.999547\pi\)
0.999999 0.00142342i \(-0.000453088\pi\)
\(648\) − 10.8277i − 0.425352i
\(649\) 4.19978 0.164856
\(650\) 0 0
\(651\) −45.4378 −1.78085
\(652\) 13.1921i 0.516644i
\(653\) 43.5346i 1.70364i 0.523835 + 0.851820i \(0.324501\pi\)
−0.523835 + 0.851820i \(0.675499\pi\)
\(654\) −41.4358 −1.62027
\(655\) 0 0
\(656\) 4.79476 0.187204
\(657\) − 25.3734i − 0.989910i
\(658\) − 20.0702i − 0.782420i
\(659\) 33.4512 1.30308 0.651538 0.758616i \(-0.274124\pi\)
0.651538 + 0.758616i \(0.274124\pi\)
\(660\) 0 0
\(661\) −2.89465 −0.112589 −0.0562945 0.998414i \(-0.517929\pi\)
−0.0562945 + 0.998414i \(0.517929\pi\)
\(662\) 11.9791i 0.465580i
\(663\) − 10.6434i − 0.413355i
\(664\) −7.85517 −0.304840
\(665\) 0 0
\(666\) −35.5117 −1.37605
\(667\) 47.4292i 1.83647i
\(668\) − 1.81331i − 0.0701591i
\(669\) −47.4818 −1.83575
\(670\) 0 0
\(671\) 7.97382 0.307826
\(672\) − 7.49073i − 0.288961i
\(673\) − 42.8475i − 1.65165i −0.563925 0.825826i \(-0.690709\pi\)
0.563925 0.825826i \(-0.309291\pi\)
\(674\) 11.1921 0.431105
\(675\) 0 0
\(676\) 25.8332 0.993583
\(677\) 34.4567i 1.32428i 0.749381 + 0.662139i \(0.230351\pi\)
−0.749381 + 0.662139i \(0.769649\pi\)
\(678\) 3.65540i 0.140385i
\(679\) −27.1812 −1.04312
\(680\) 0 0
\(681\) −14.5621 −0.558019
\(682\) 4.42139i 0.169304i
\(683\) 2.73436i 0.104627i 0.998631 + 0.0523136i \(0.0166595\pi\)
−0.998631 + 0.0523136i \(0.983340\pi\)
\(684\) 6.19869 0.237013
\(685\) 0 0
\(686\) −19.5117 −0.744959
\(687\) − 8.47634i − 0.323393i
\(688\) − 8.06587i − 0.307508i
\(689\) −9.53566 −0.363280
\(690\) 0 0
\(691\) −4.74198 −0.180394 −0.0901968 0.995924i \(-0.528750\pi\)
−0.0901968 + 0.995924i \(0.528750\pi\)
\(692\) − 12.5237i − 0.476078i
\(693\) − 11.1590i − 0.423897i
\(694\) 20.3843 0.773777
\(695\) 0 0
\(696\) −31.0318 −1.17626
\(697\) 2.70012i 0.102274i
\(698\) 0.252557i 0.00955943i
\(699\) 35.0068 1.32408
\(700\) 0 0
\(701\) 33.1372 1.25157 0.625787 0.779994i \(-0.284778\pi\)
0.625787 + 0.779994i \(0.284778\pi\)
\(702\) 60.4556i 2.28175i
\(703\) − 5.72890i − 0.216069i
\(704\) −0.728896 −0.0274713
\(705\) 0 0
\(706\) −28.6434 −1.07801
\(707\) − 24.0419i − 0.904187i
\(708\) − 17.4753i − 0.656760i
\(709\) −5.37884 −0.202006 −0.101003 0.994886i \(-0.532205\pi\)
−0.101003 + 0.994886i \(0.532205\pi\)
\(710\) 0 0
\(711\) 95.0692 3.56537
\(712\) − 10.0000i − 0.374766i
\(713\) − 28.1187i − 1.05305i
\(714\) 4.21832 0.157867
\(715\) 0 0
\(716\) 1.39738 0.0522226
\(717\) 79.5390i 2.97044i
\(718\) 18.5741i 0.693178i
\(719\) −33.0857 −1.23389 −0.616944 0.787007i \(-0.711630\pi\)
−0.616944 + 0.787007i \(0.711630\pi\)
\(720\) 0 0
\(721\) −20.5771 −0.766332
\(722\) 1.00000i 0.0372161i
\(723\) − 36.5950i − 1.36098i
\(724\) −5.72890 −0.212913
\(725\) 0 0
\(726\) 31.7509 1.17839
\(727\) 2.62463i 0.0973423i 0.998815 + 0.0486711i \(0.0154986\pi\)
−0.998815 + 0.0486711i \(0.984501\pi\)
\(728\) 15.3908i 0.570422i
\(729\) 21.1538 0.783472
\(730\) 0 0
\(731\) 4.54221 0.168000
\(732\) − 33.1791i − 1.22633i
\(733\) − 0.608077i − 0.0224598i −0.999937 0.0112299i \(-0.996425\pi\)
0.999937 0.0112299i \(-0.00357467\pi\)
\(734\) 4.79476 0.176978
\(735\) 0 0
\(736\) 4.63555 0.170869
\(737\) 9.42685i 0.347242i
\(738\) − 29.7213i − 1.09405i
\(739\) −36.3974 −1.33890 −0.669450 0.742857i \(-0.733470\pi\)
−0.669450 + 0.742857i \(0.733470\pi\)
\(740\) 0 0
\(741\) −18.9001 −0.694313
\(742\) − 3.77929i − 0.138742i
\(743\) 23.0713i 0.846405i 0.906035 + 0.423202i \(0.139094\pi\)
−0.906035 + 0.423202i \(0.860906\pi\)
\(744\) 18.3974 0.674481
\(745\) 0 0
\(746\) 15.9485 0.583917
\(747\) 48.6918i 1.78154i
\(748\) − 0.410470i − 0.0150083i
\(749\) 37.1121 1.35605
\(750\) 0 0
\(751\) 4.75290 0.173436 0.0867179 0.996233i \(-0.472362\pi\)
0.0867179 + 0.996233i \(0.472362\pi\)
\(752\) 8.12628i 0.296335i
\(753\) 41.0164i 1.49472i
\(754\) 63.7597 2.32199
\(755\) 0 0
\(756\) −23.9605 −0.871436
\(757\) 26.8057i 0.974269i 0.873327 + 0.487135i \(0.161958\pi\)
−0.873327 + 0.487135i \(0.838042\pi\)
\(758\) 16.8013i 0.610251i
\(759\) 10.2478 0.371971
\(760\) 0 0
\(761\) 25.4478 0.922481 0.461241 0.887275i \(-0.347404\pi\)
0.461241 + 0.887275i \(0.347404\pi\)
\(762\) − 28.8846i − 1.04638i
\(763\) 33.7422i 1.22155i
\(764\) 27.0198 0.977544
\(765\) 0 0
\(766\) 26.7948 0.968134
\(767\) 35.9056i 1.29648i
\(768\) 3.03293i 0.109442i
\(769\) 14.6421 0.528007 0.264004 0.964522i \(-0.414957\pi\)
0.264004 + 0.964522i \(0.414957\pi\)
\(770\) 0 0
\(771\) −68.9684 −2.48384
\(772\) − 23.0713i − 0.830355i
\(773\) 16.8462i 0.605917i 0.953004 + 0.302959i \(0.0979744\pi\)
−0.953004 + 0.302959i \(0.902026\pi\)
\(774\) −49.9978 −1.79713
\(775\) 0 0
\(776\) 11.0055 0.395073
\(777\) 42.9136i 1.53952i
\(778\) − 18.3424i − 0.657608i
\(779\) 4.79476 0.171790
\(780\) 0 0
\(781\) −3.20524 −0.114692
\(782\) 2.61046i 0.0933499i
\(783\) 99.2613i 3.54731i
\(784\) 0.900112 0.0321469
\(785\) 0 0
\(786\) 52.7817 1.88266
\(787\) 18.7367i 0.667893i 0.942592 + 0.333946i \(0.108380\pi\)
−0.942592 + 0.333946i \(0.891620\pi\)
\(788\) − 0.794765i − 0.0283123i
\(789\) −20.2416 −0.720621
\(790\) 0 0
\(791\) 2.97668 0.105839
\(792\) 4.51820i 0.160547i
\(793\) 68.1714i 2.42084i
\(794\) −21.2162 −0.752933
\(795\) 0 0
\(796\) 8.07241 0.286119
\(797\) − 37.9900i − 1.34567i −0.739791 0.672837i \(-0.765075\pi\)
0.739791 0.672837i \(-0.234925\pi\)
\(798\) − 7.49073i − 0.265169i
\(799\) −4.57623 −0.161895
\(800\) 0 0
\(801\) −61.9869 −2.19020
\(802\) − 27.8661i − 0.983986i
\(803\) 2.98362i 0.105290i
\(804\) 39.2251 1.38336
\(805\) 0 0
\(806\) −37.8002 −1.33146
\(807\) 90.1989i 3.17515i
\(808\) 9.73436i 0.342453i
\(809\) −21.8857 −0.769461 −0.384731 0.923029i \(-0.625706\pi\)
−0.384731 + 0.923029i \(0.625706\pi\)
\(810\) 0 0
\(811\) 37.8595 1.32943 0.664714 0.747098i \(-0.268553\pi\)
0.664714 + 0.747098i \(0.268553\pi\)
\(812\) 25.2700i 0.886804i
\(813\) 3.37229i 0.118271i
\(814\) 4.17577 0.146361
\(815\) 0 0
\(816\) −1.70796 −0.0597907
\(817\) − 8.06587i − 0.282189i
\(818\) 18.0899i 0.632498i
\(819\) 95.4031 3.33365
\(820\) 0 0
\(821\) −20.1976 −0.704901 −0.352451 0.935830i \(-0.614652\pi\)
−0.352451 + 0.935830i \(0.614652\pi\)
\(822\) − 6.85171i − 0.238981i
\(823\) 4.34590i 0.151489i 0.997127 + 0.0757443i \(0.0241333\pi\)
−0.997127 + 0.0757443i \(0.975867\pi\)
\(824\) 8.33151 0.290242
\(825\) 0 0
\(826\) −14.2305 −0.495144
\(827\) − 56.8375i − 1.97643i −0.153057 0.988217i \(-0.548912\pi\)
0.153057 0.988217i \(-0.451088\pi\)
\(828\) − 28.7344i − 0.998588i
\(829\) 17.4543 0.606214 0.303107 0.952957i \(-0.401976\pi\)
0.303107 + 0.952957i \(0.401976\pi\)
\(830\) 0 0
\(831\) −39.8111 −1.38103
\(832\) − 6.23163i − 0.216043i
\(833\) 0.506888i 0.0175626i
\(834\) 51.1372 1.77074
\(835\) 0 0
\(836\) −0.728896 −0.0252094
\(837\) − 58.8475i − 2.03407i
\(838\) 15.3370i 0.529807i
\(839\) −1.77622 −0.0613219 −0.0306609 0.999530i \(-0.509761\pi\)
−0.0306609 + 0.999530i \(0.509761\pi\)
\(840\) 0 0
\(841\) 75.6862 2.60987
\(842\) − 8.42486i − 0.290340i
\(843\) 69.5346i 2.39490i
\(844\) 11.6410 0.400700
\(845\) 0 0
\(846\) 50.3723 1.73184
\(847\) − 25.8556i − 0.888408i
\(848\) 1.53020i 0.0525475i
\(849\) 2.61025 0.0895834
\(850\) 0 0
\(851\) −26.5566 −0.910348
\(852\) 13.3370i 0.456917i
\(853\) 16.1187i 0.551892i 0.961173 + 0.275946i \(0.0889911\pi\)
−0.961173 + 0.275946i \(0.911009\pi\)
\(854\) −27.0185 −0.924556
\(855\) 0 0
\(856\) −15.0264 −0.513591
\(857\) − 47.2031i − 1.61243i −0.591625 0.806213i \(-0.701514\pi\)
0.591625 0.806213i \(-0.298486\pi\)
\(858\) − 13.7762i − 0.470312i
\(859\) −37.3239 −1.27347 −0.636737 0.771081i \(-0.719716\pi\)
−0.636737 + 0.771081i \(0.719716\pi\)
\(860\) 0 0
\(861\) −35.9163 −1.22402
\(862\) 16.2766i 0.554382i
\(863\) 1.33697i 0.0455111i 0.999741 + 0.0227555i \(0.00724394\pi\)
−0.999741 + 0.0227555i \(0.992756\pi\)
\(864\) 9.70142 0.330049
\(865\) 0 0
\(866\) −18.1976 −0.618380
\(867\) 50.5981i 1.71840i
\(868\) − 14.9815i − 0.508504i
\(869\) −11.1791 −0.379224
\(870\) 0 0
\(871\) −80.5939 −2.73082
\(872\) − 13.6619i − 0.462652i
\(873\) − 68.2194i − 2.30888i
\(874\) 4.63555 0.156800
\(875\) 0 0
\(876\) 12.4148 0.419459
\(877\) 13.1647i 0.444539i 0.974985 + 0.222270i \(0.0713465\pi\)
−0.974985 + 0.222270i \(0.928653\pi\)
\(878\) − 20.4214i − 0.689188i
\(879\) 60.1163 2.02767
\(880\) 0 0
\(881\) −10.2052 −0.343823 −0.171912 0.985112i \(-0.554994\pi\)
−0.171912 + 0.985112i \(0.554994\pi\)
\(882\) − 5.57952i − 0.187872i
\(883\) − 1.49966i − 0.0504674i −0.999682 0.0252337i \(-0.991967\pi\)
0.999682 0.0252337i \(-0.00803299\pi\)
\(884\) 3.50927 0.118030
\(885\) 0 0
\(886\) 21.6685 0.727967
\(887\) − 46.9505i − 1.57644i −0.615391 0.788222i \(-0.711002\pi\)
0.615391 0.788222i \(-0.288998\pi\)
\(888\) − 17.3754i − 0.583079i
\(889\) −23.5215 −0.788886
\(890\) 0 0
\(891\) 7.89227 0.264401
\(892\) − 15.6554i − 0.524182i
\(893\) 8.12628i 0.271936i
\(894\) −39.4447 −1.31923
\(895\) 0 0
\(896\) 2.46980 0.0825101
\(897\) 87.6125i 2.92529i
\(898\) − 23.8133i − 0.794661i
\(899\) −62.0637 −2.06994
\(900\) 0 0
\(901\) −0.861719 −0.0287080
\(902\) 3.49489i 0.116367i
\(903\) 60.4192i 2.01063i
\(904\) −1.20524 −0.0400855
\(905\) 0 0
\(906\) −53.3479 −1.77236
\(907\) 16.3668i 0.543452i 0.962375 + 0.271726i \(0.0875944\pi\)
−0.962375 + 0.271726i \(0.912406\pi\)
\(908\) − 4.80131i − 0.159337i
\(909\) 60.3403 2.00136
\(910\) 0 0
\(911\) 32.3293 1.07112 0.535559 0.844498i \(-0.320101\pi\)
0.535559 + 0.844498i \(0.320101\pi\)
\(912\) 3.03293i 0.100430i
\(913\) − 5.72561i − 0.189490i
\(914\) −9.15813 −0.302924
\(915\) 0 0
\(916\) 2.79476 0.0923416
\(917\) − 42.9815i − 1.41937i
\(918\) 5.46325i 0.180314i
\(919\) 22.8157 0.752620 0.376310 0.926494i \(-0.377193\pi\)
0.376310 + 0.926494i \(0.377193\pi\)
\(920\) 0 0
\(921\) 51.1538 1.68557
\(922\) − 7.78931i − 0.256527i
\(923\) − 27.4028i − 0.901976i
\(924\) 5.45996 0.179620
\(925\) 0 0
\(926\) 0.405011 0.0133095
\(927\) − 51.6445i − 1.69623i
\(928\) − 10.2316i − 0.335870i
\(929\) 27.2436 0.893834 0.446917 0.894575i \(-0.352522\pi\)
0.446917 + 0.894575i \(0.352522\pi\)
\(930\) 0 0
\(931\) 0.900112 0.0295000
\(932\) 11.5422i 0.378078i
\(933\) 31.3150i 1.02521i
\(934\) 1.95814 0.0640722
\(935\) 0 0
\(936\) −38.6279 −1.26259
\(937\) − 51.5006i − 1.68245i −0.540685 0.841225i \(-0.681835\pi\)
0.540685 0.841225i \(-0.318165\pi\)
\(938\) − 31.9420i − 1.04294i
\(939\) −47.8244 −1.56069
\(940\) 0 0
\(941\) −40.8541 −1.33181 −0.665903 0.746039i \(-0.731953\pi\)
−0.665903 + 0.746039i \(0.731953\pi\)
\(942\) − 28.8846i − 0.941112i
\(943\) − 22.2264i − 0.723791i
\(944\) 5.76183 0.187532
\(945\) 0 0
\(946\) 5.87918 0.191149
\(947\) 38.2526i 1.24304i 0.783398 + 0.621521i \(0.213485\pi\)
−0.783398 + 0.621521i \(0.786515\pi\)
\(948\) 46.5160i 1.51077i
\(949\) −25.5082 −0.828031
\(950\) 0 0
\(951\) 6.58845 0.213645
\(952\) 1.39084i 0.0450773i
\(953\) − 54.1187i − 1.75308i −0.481334 0.876538i \(-0.659847\pi\)
0.481334 0.876538i \(-0.340153\pi\)
\(954\) 9.48527 0.307097
\(955\) 0 0
\(956\) −26.2251 −0.848180
\(957\) − 22.6190i − 0.731168i
\(958\) 22.9210i 0.740545i
\(959\) −5.57952 −0.180172
\(960\) 0 0
\(961\) 5.79476 0.186928
\(962\) 35.7003i 1.15103i
\(963\) 93.1440i 3.00152i
\(964\) 12.0659 0.388615
\(965\) 0 0
\(966\) −34.7237 −1.11722
\(967\) 28.4214i 0.913970i 0.889474 + 0.456985i \(0.151071\pi\)
−0.889474 + 0.456985i \(0.848929\pi\)
\(968\) 10.4687i 0.336477i
\(969\) −1.70796 −0.0548677
\(970\) 0 0
\(971\) 30.8057 0.988601 0.494301 0.869291i \(-0.335424\pi\)
0.494301 + 0.869291i \(0.335424\pi\)
\(972\) − 3.73544i − 0.119814i
\(973\) − 41.6423i − 1.33499i
\(974\) 23.9450 0.767249
\(975\) 0 0
\(976\) 10.9396 0.350168
\(977\) 37.6554i 1.20470i 0.798231 + 0.602351i \(0.205769\pi\)
−0.798231 + 0.602351i \(0.794231\pi\)
\(978\) 40.0109i 1.27941i
\(979\) 7.28896 0.232956
\(980\) 0 0
\(981\) −84.6862 −2.70382
\(982\) 3.86826i 0.123441i
\(983\) 38.7948i 1.23736i 0.785643 + 0.618680i \(0.212332\pi\)
−0.785643 + 0.618680i \(0.787668\pi\)
\(984\) 14.5422 0.463589
\(985\) 0 0
\(986\) 5.76183 0.183494
\(987\) − 60.8717i − 1.93757i
\(988\) − 6.23163i − 0.198254i
\(989\) −37.3898 −1.18893
\(990\) 0 0
\(991\) −15.0295 −0.477427 −0.238713 0.971090i \(-0.576726\pi\)
−0.238713 + 0.971090i \(0.576726\pi\)
\(992\) 6.06587i 0.192592i
\(993\) 36.3317i 1.15295i
\(994\) 10.8606 0.344478
\(995\) 0 0
\(996\) −23.8242 −0.754900
\(997\) 1.18123i 0.0374099i 0.999825 + 0.0187050i \(0.00595432\pi\)
−0.999825 + 0.0187050i \(0.994046\pi\)
\(998\) 7.92104i 0.250736i
\(999\) −55.5784 −1.75842
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 950.2.b.g.799.6 6
5.2 odd 4 950.2.a.k.1.3 3
5.3 odd 4 950.2.a.m.1.1 yes 3
5.4 even 2 inner 950.2.b.g.799.1 6
15.2 even 4 8550.2.a.co.1.2 3
15.8 even 4 8550.2.a.cj.1.2 3
20.3 even 4 7600.2.a.bm.1.3 3
20.7 even 4 7600.2.a.cb.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
950.2.a.k.1.3 3 5.2 odd 4
950.2.a.m.1.1 yes 3 5.3 odd 4
950.2.b.g.799.1 6 5.4 even 2 inner
950.2.b.g.799.6 6 1.1 even 1 trivial
7600.2.a.bm.1.3 3 20.3 even 4
7600.2.a.cb.1.1 3 20.7 even 4
8550.2.a.cj.1.2 3 15.8 even 4
8550.2.a.co.1.2 3 15.2 even 4