Properties

Label 950.2.b.c.799.1
Level $950$
Weight $2$
Character 950.799
Analytic conductor $7.586$
Analytic rank $0$
Dimension $2$
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: \(2\)
Coefficient field: \(\Q(i)\)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 38)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 799.1
Root \(-1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 950.799
Dual form 950.2.b.c.799.2

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{6} -3.00000i q^{7} +1.00000i q^{8} +2.00000 q^{9} +O(q^{10})\) \(q-1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{6} -3.00000i q^{7} +1.00000i q^{8} +2.00000 q^{9} +2.00000 q^{11} +1.00000i q^{12} -1.00000i q^{13} -3.00000 q^{14} +1.00000 q^{16} -3.00000i q^{17} -2.00000i q^{18} +1.00000 q^{19} -3.00000 q^{21} -2.00000i q^{22} -1.00000i q^{23} +1.00000 q^{24} -1.00000 q^{26} -5.00000i q^{27} +3.00000i q^{28} +5.00000 q^{29} -8.00000 q^{31} -1.00000i q^{32} -2.00000i q^{33} -3.00000 q^{34} -2.00000 q^{36} +2.00000i q^{37} -1.00000i q^{38} -1.00000 q^{39} -8.00000 q^{41} +3.00000i q^{42} +4.00000i q^{43} -2.00000 q^{44} -1.00000 q^{46} -8.00000i q^{47} -1.00000i q^{48} -2.00000 q^{49} -3.00000 q^{51} +1.00000i q^{52} -1.00000i q^{53} -5.00000 q^{54} +3.00000 q^{56} -1.00000i q^{57} -5.00000i q^{58} -15.0000 q^{59} +2.00000 q^{61} +8.00000i q^{62} -6.00000i q^{63} -1.00000 q^{64} -2.00000 q^{66} -3.00000i q^{67} +3.00000i q^{68} -1.00000 q^{69} +2.00000 q^{71} +2.00000i q^{72} +9.00000i q^{73} +2.00000 q^{74} -1.00000 q^{76} -6.00000i q^{77} +1.00000i q^{78} +10.0000 q^{79} +1.00000 q^{81} +8.00000i q^{82} -6.00000i q^{83} +3.00000 q^{84} +4.00000 q^{86} -5.00000i q^{87} +2.00000i q^{88} -3.00000 q^{91} +1.00000i q^{92} +8.00000i q^{93} -8.00000 q^{94} -1.00000 q^{96} +2.00000i q^{97} +2.00000i q^{98} +4.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{4} - 2q^{6} + 4q^{9} + O(q^{10}) \) \( 2q - 2q^{4} - 2q^{6} + 4q^{9} + 4q^{11} - 6q^{14} + 2q^{16} + 2q^{19} - 6q^{21} + 2q^{24} - 2q^{26} + 10q^{29} - 16q^{31} - 6q^{34} - 4q^{36} - 2q^{39} - 16q^{41} - 4q^{44} - 2q^{46} - 4q^{49} - 6q^{51} - 10q^{54} + 6q^{56} - 30q^{59} + 4q^{61} - 2q^{64} - 4q^{66} - 2q^{69} + 4q^{71} + 4q^{74} - 2q^{76} + 20q^{79} + 2q^{81} + 6q^{84} + 8q^{86} - 6q^{91} - 16q^{94} - 2q^{96} + 8q^{99} + O(q^{100}) \)

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
\(3\) − 1.00000i − 0.577350i −0.957427 0.288675i \(-0.906785\pi\)
0.957427 0.288675i \(-0.0932147\pi\)
\(4\) −1.00000 −0.500000
\(5\) 0 0
\(6\) −1.00000 −0.408248
\(7\) − 3.00000i − 1.13389i −0.823754 0.566947i \(-0.808125\pi\)
0.823754 0.566947i \(-0.191875\pi\)
\(8\) 1.00000i 0.353553i
\(9\) 2.00000 0.666667
\(10\) 0 0
\(11\) 2.00000 0.603023 0.301511 0.953463i \(-0.402509\pi\)
0.301511 + 0.953463i \(0.402509\pi\)
\(12\) 1.00000i 0.288675i
\(13\) − 1.00000i − 0.277350i −0.990338 0.138675i \(-0.955716\pi\)
0.990338 0.138675i \(-0.0442844\pi\)
\(14\) −3.00000 −0.801784
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) − 3.00000i − 0.727607i −0.931476 0.363803i \(-0.881478\pi\)
0.931476 0.363803i \(-0.118522\pi\)
\(18\) − 2.00000i − 0.471405i
\(19\) 1.00000 0.229416
\(20\) 0 0
\(21\) −3.00000 −0.654654
\(22\) − 2.00000i − 0.426401i
\(23\) − 1.00000i − 0.208514i −0.994550 0.104257i \(-0.966753\pi\)
0.994550 0.104257i \(-0.0332465\pi\)
\(24\) 1.00000 0.204124
\(25\) 0 0
\(26\) −1.00000 −0.196116
\(27\) − 5.00000i − 0.962250i
\(28\) 3.00000i 0.566947i
\(29\) 5.00000 0.928477 0.464238 0.885710i \(-0.346328\pi\)
0.464238 + 0.885710i \(0.346328\pi\)
\(30\) 0 0
\(31\) −8.00000 −1.43684 −0.718421 0.695608i \(-0.755135\pi\)
−0.718421 + 0.695608i \(0.755135\pi\)
\(32\) − 1.00000i − 0.176777i
\(33\) − 2.00000i − 0.348155i
\(34\) −3.00000 −0.514496
\(35\) 0 0
\(36\) −2.00000 −0.333333
\(37\) 2.00000i 0.328798i 0.986394 + 0.164399i \(0.0525685\pi\)
−0.986394 + 0.164399i \(0.947432\pi\)
\(38\) − 1.00000i − 0.162221i
\(39\) −1.00000 −0.160128
\(40\) 0 0
\(41\) −8.00000 −1.24939 −0.624695 0.780869i \(-0.714777\pi\)
−0.624695 + 0.780869i \(0.714777\pi\)
\(42\) 3.00000i 0.462910i
\(43\) 4.00000i 0.609994i 0.952353 + 0.304997i \(0.0986555\pi\)
−0.952353 + 0.304997i \(0.901344\pi\)
\(44\) −2.00000 −0.301511
\(45\) 0 0
\(46\) −1.00000 −0.147442
\(47\) − 8.00000i − 1.16692i −0.812142 0.583460i \(-0.801699\pi\)
0.812142 0.583460i \(-0.198301\pi\)
\(48\) − 1.00000i − 0.144338i
\(49\) −2.00000 −0.285714
\(50\) 0 0
\(51\) −3.00000 −0.420084
\(52\) 1.00000i 0.138675i
\(53\) − 1.00000i − 0.137361i −0.997639 0.0686803i \(-0.978121\pi\)
0.997639 0.0686803i \(-0.0218788\pi\)
\(54\) −5.00000 −0.680414
\(55\) 0 0
\(56\) 3.00000 0.400892
\(57\) − 1.00000i − 0.132453i
\(58\) − 5.00000i − 0.656532i
\(59\) −15.0000 −1.95283 −0.976417 0.215894i \(-0.930733\pi\)
−0.976417 + 0.215894i \(0.930733\pi\)
\(60\) 0 0
\(61\) 2.00000 0.256074 0.128037 0.991769i \(-0.459132\pi\)
0.128037 + 0.991769i \(0.459132\pi\)
\(62\) 8.00000i 1.01600i
\(63\) − 6.00000i − 0.755929i
\(64\) −1.00000 −0.125000
\(65\) 0 0
\(66\) −2.00000 −0.246183
\(67\) − 3.00000i − 0.366508i −0.983066 0.183254i \(-0.941337\pi\)
0.983066 0.183254i \(-0.0586631\pi\)
\(68\) 3.00000i 0.363803i
\(69\) −1.00000 −0.120386
\(70\) 0 0
\(71\) 2.00000 0.237356 0.118678 0.992933i \(-0.462134\pi\)
0.118678 + 0.992933i \(0.462134\pi\)
\(72\) 2.00000i 0.235702i
\(73\) 9.00000i 1.05337i 0.850060 + 0.526685i \(0.176565\pi\)
−0.850060 + 0.526685i \(0.823435\pi\)
\(74\) 2.00000 0.232495
\(75\) 0 0
\(76\) −1.00000 −0.114708
\(77\) − 6.00000i − 0.683763i
\(78\) 1.00000i 0.113228i
\(79\) 10.0000 1.12509 0.562544 0.826767i \(-0.309823\pi\)
0.562544 + 0.826767i \(0.309823\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 8.00000i 0.883452i
\(83\) − 6.00000i − 0.658586i −0.944228 0.329293i \(-0.893190\pi\)
0.944228 0.329293i \(-0.106810\pi\)
\(84\) 3.00000 0.327327
\(85\) 0 0
\(86\) 4.00000 0.431331
\(87\) − 5.00000i − 0.536056i
\(88\) 2.00000i 0.213201i
\(89\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(90\) 0 0
\(91\) −3.00000 −0.314485
\(92\) 1.00000i 0.104257i
\(93\) 8.00000i 0.829561i
\(94\) −8.00000 −0.825137
\(95\) 0 0
\(96\) −1.00000 −0.102062
\(97\) 2.00000i 0.203069i 0.994832 + 0.101535i \(0.0323753\pi\)
−0.994832 + 0.101535i \(0.967625\pi\)
\(98\) 2.00000i 0.202031i
\(99\) 4.00000 0.402015
\(100\) 0 0
\(101\) 2.00000 0.199007 0.0995037 0.995037i \(-0.468274\pi\)
0.0995037 + 0.995037i \(0.468274\pi\)
\(102\) 3.00000i 0.297044i
\(103\) − 6.00000i − 0.591198i −0.955312 0.295599i \(-0.904481\pi\)
0.955312 0.295599i \(-0.0955191\pi\)
\(104\) 1.00000 0.0980581
\(105\) 0 0
\(106\) −1.00000 −0.0971286
\(107\) 7.00000i 0.676716i 0.941018 + 0.338358i \(0.109871\pi\)
−0.941018 + 0.338358i \(0.890129\pi\)
\(108\) 5.00000i 0.481125i
\(109\) 15.0000 1.43674 0.718370 0.695662i \(-0.244889\pi\)
0.718370 + 0.695662i \(0.244889\pi\)
\(110\) 0 0
\(111\) 2.00000 0.189832
\(112\) − 3.00000i − 0.283473i
\(113\) 14.0000i 1.31701i 0.752577 + 0.658505i \(0.228811\pi\)
−0.752577 + 0.658505i \(0.771189\pi\)
\(114\) −1.00000 −0.0936586
\(115\) 0 0
\(116\) −5.00000 −0.464238
\(117\) − 2.00000i − 0.184900i
\(118\) 15.0000i 1.38086i
\(119\) −9.00000 −0.825029
\(120\) 0 0
\(121\) −7.00000 −0.636364
\(122\) − 2.00000i − 0.181071i
\(123\) 8.00000i 0.721336i
\(124\) 8.00000 0.718421
\(125\) 0 0
\(126\) −6.00000 −0.534522
\(127\) − 18.0000i − 1.59724i −0.601834 0.798621i \(-0.705563\pi\)
0.601834 0.798621i \(-0.294437\pi\)
\(128\) 1.00000i 0.0883883i
\(129\) 4.00000 0.352180
\(130\) 0 0
\(131\) 12.0000 1.04844 0.524222 0.851581i \(-0.324356\pi\)
0.524222 + 0.851581i \(0.324356\pi\)
\(132\) 2.00000i 0.174078i
\(133\) − 3.00000i − 0.260133i
\(134\) −3.00000 −0.259161
\(135\) 0 0
\(136\) 3.00000 0.257248
\(137\) 17.0000i 1.45241i 0.687479 + 0.726204i \(0.258717\pi\)
−0.687479 + 0.726204i \(0.741283\pi\)
\(138\) 1.00000i 0.0851257i
\(139\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(140\) 0 0
\(141\) −8.00000 −0.673722
\(142\) − 2.00000i − 0.167836i
\(143\) − 2.00000i − 0.167248i
\(144\) 2.00000 0.166667
\(145\) 0 0
\(146\) 9.00000 0.744845
\(147\) 2.00000i 0.164957i
\(148\) − 2.00000i − 0.164399i
\(149\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(150\) 0 0
\(151\) 2.00000 0.162758 0.0813788 0.996683i \(-0.474068\pi\)
0.0813788 + 0.996683i \(0.474068\pi\)
\(152\) 1.00000i 0.0811107i
\(153\) − 6.00000i − 0.485071i
\(154\) −6.00000 −0.483494
\(155\) 0 0
\(156\) 1.00000 0.0800641
\(157\) 2.00000i 0.159617i 0.996810 + 0.0798087i \(0.0254309\pi\)
−0.996810 + 0.0798087i \(0.974569\pi\)
\(158\) − 10.0000i − 0.795557i
\(159\) −1.00000 −0.0793052
\(160\) 0 0
\(161\) −3.00000 −0.236433
\(162\) − 1.00000i − 0.0785674i
\(163\) − 16.0000i − 1.25322i −0.779334 0.626608i \(-0.784443\pi\)
0.779334 0.626608i \(-0.215557\pi\)
\(164\) 8.00000 0.624695
\(165\) 0 0
\(166\) −6.00000 −0.465690
\(167\) 12.0000i 0.928588i 0.885681 + 0.464294i \(0.153692\pi\)
−0.885681 + 0.464294i \(0.846308\pi\)
\(168\) − 3.00000i − 0.231455i
\(169\) 12.0000 0.923077
\(170\) 0 0
\(171\) 2.00000 0.152944
\(172\) − 4.00000i − 0.304997i
\(173\) − 6.00000i − 0.456172i −0.973641 0.228086i \(-0.926753\pi\)
0.973641 0.228086i \(-0.0732467\pi\)
\(174\) −5.00000 −0.379049
\(175\) 0 0
\(176\) 2.00000 0.150756
\(177\) 15.0000i 1.12747i
\(178\) 0 0
\(179\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(180\) 0 0
\(181\) 22.0000 1.63525 0.817624 0.575753i \(-0.195291\pi\)
0.817624 + 0.575753i \(0.195291\pi\)
\(182\) 3.00000i 0.222375i
\(183\) − 2.00000i − 0.147844i
\(184\) 1.00000 0.0737210
\(185\) 0 0
\(186\) 8.00000 0.586588
\(187\) − 6.00000i − 0.438763i
\(188\) 8.00000i 0.583460i
\(189\) −15.0000 −1.09109
\(190\) 0 0
\(191\) 7.00000 0.506502 0.253251 0.967401i \(-0.418500\pi\)
0.253251 + 0.967401i \(0.418500\pi\)
\(192\) 1.00000i 0.0721688i
\(193\) − 6.00000i − 0.431889i −0.976406 0.215945i \(-0.930717\pi\)
0.976406 0.215945i \(-0.0692831\pi\)
\(194\) 2.00000 0.143592
\(195\) 0 0
\(196\) 2.00000 0.142857
\(197\) − 8.00000i − 0.569976i −0.958531 0.284988i \(-0.908010\pi\)
0.958531 0.284988i \(-0.0919897\pi\)
\(198\) − 4.00000i − 0.284268i
\(199\) 25.0000 1.77220 0.886102 0.463491i \(-0.153403\pi\)
0.886102 + 0.463491i \(0.153403\pi\)
\(200\) 0 0
\(201\) −3.00000 −0.211604
\(202\) − 2.00000i − 0.140720i
\(203\) − 15.0000i − 1.05279i
\(204\) 3.00000 0.210042
\(205\) 0 0
\(206\) −6.00000 −0.418040
\(207\) − 2.00000i − 0.139010i
\(208\) − 1.00000i − 0.0693375i
\(209\) 2.00000 0.138343
\(210\) 0 0
\(211\) 27.0000 1.85876 0.929378 0.369129i \(-0.120344\pi\)
0.929378 + 0.369129i \(0.120344\pi\)
\(212\) 1.00000i 0.0686803i
\(213\) − 2.00000i − 0.137038i
\(214\) 7.00000 0.478510
\(215\) 0 0
\(216\) 5.00000 0.340207
\(217\) 24.0000i 1.62923i
\(218\) − 15.0000i − 1.01593i
\(219\) 9.00000 0.608164
\(220\) 0 0
\(221\) −3.00000 −0.201802
\(222\) − 2.00000i − 0.134231i
\(223\) 14.0000i 0.937509i 0.883328 + 0.468755i \(0.155297\pi\)
−0.883328 + 0.468755i \(0.844703\pi\)
\(224\) −3.00000 −0.200446
\(225\) 0 0
\(226\) 14.0000 0.931266
\(227\) 17.0000i 1.12833i 0.825662 + 0.564165i \(0.190802\pi\)
−0.825662 + 0.564165i \(0.809198\pi\)
\(228\) 1.00000i 0.0662266i
\(229\) 10.0000 0.660819 0.330409 0.943838i \(-0.392813\pi\)
0.330409 + 0.943838i \(0.392813\pi\)
\(230\) 0 0
\(231\) −6.00000 −0.394771
\(232\) 5.00000i 0.328266i
\(233\) − 6.00000i − 0.393073i −0.980497 0.196537i \(-0.937031\pi\)
0.980497 0.196537i \(-0.0629694\pi\)
\(234\) −2.00000 −0.130744
\(235\) 0 0
\(236\) 15.0000 0.976417
\(237\) − 10.0000i − 0.649570i
\(238\) 9.00000i 0.583383i
\(239\) −15.0000 −0.970269 −0.485135 0.874439i \(-0.661229\pi\)
−0.485135 + 0.874439i \(0.661229\pi\)
\(240\) 0 0
\(241\) −8.00000 −0.515325 −0.257663 0.966235i \(-0.582952\pi\)
−0.257663 + 0.966235i \(0.582952\pi\)
\(242\) 7.00000i 0.449977i
\(243\) − 16.0000i − 1.02640i
\(244\) −2.00000 −0.128037
\(245\) 0 0
\(246\) 8.00000 0.510061
\(247\) − 1.00000i − 0.0636285i
\(248\) − 8.00000i − 0.508001i
\(249\) −6.00000 −0.380235
\(250\) 0 0
\(251\) 2.00000 0.126239 0.0631194 0.998006i \(-0.479895\pi\)
0.0631194 + 0.998006i \(0.479895\pi\)
\(252\) 6.00000i 0.377964i
\(253\) − 2.00000i − 0.125739i
\(254\) −18.0000 −1.12942
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) − 8.00000i − 0.499026i −0.968371 0.249513i \(-0.919729\pi\)
0.968371 0.249513i \(-0.0802706\pi\)
\(258\) − 4.00000i − 0.249029i
\(259\) 6.00000 0.372822
\(260\) 0 0
\(261\) 10.0000 0.618984
\(262\) − 12.0000i − 0.741362i
\(263\) 24.0000i 1.47990i 0.672660 + 0.739952i \(0.265152\pi\)
−0.672660 + 0.739952i \(0.734848\pi\)
\(264\) 2.00000 0.123091
\(265\) 0 0
\(266\) −3.00000 −0.183942
\(267\) 0 0
\(268\) 3.00000i 0.183254i
\(269\) −30.0000 −1.82913 −0.914566 0.404436i \(-0.867468\pi\)
−0.914566 + 0.404436i \(0.867468\pi\)
\(270\) 0 0
\(271\) 7.00000 0.425220 0.212610 0.977137i \(-0.431804\pi\)
0.212610 + 0.977137i \(0.431804\pi\)
\(272\) − 3.00000i − 0.181902i
\(273\) 3.00000i 0.181568i
\(274\) 17.0000 1.02701
\(275\) 0 0
\(276\) 1.00000 0.0601929
\(277\) − 28.0000i − 1.68236i −0.540758 0.841178i \(-0.681862\pi\)
0.540758 0.841178i \(-0.318138\pi\)
\(278\) 0 0
\(279\) −16.0000 −0.957895
\(280\) 0 0
\(281\) −8.00000 −0.477240 −0.238620 0.971113i \(-0.576695\pi\)
−0.238620 + 0.971113i \(0.576695\pi\)
\(282\) 8.00000i 0.476393i
\(283\) − 6.00000i − 0.356663i −0.983970 0.178331i \(-0.942930\pi\)
0.983970 0.178331i \(-0.0570699\pi\)
\(284\) −2.00000 −0.118678
\(285\) 0 0
\(286\) −2.00000 −0.118262
\(287\) 24.0000i 1.41668i
\(288\) − 2.00000i − 0.117851i
\(289\) 8.00000 0.470588
\(290\) 0 0
\(291\) 2.00000 0.117242
\(292\) − 9.00000i − 0.526685i
\(293\) 9.00000i 0.525786i 0.964825 + 0.262893i \(0.0846766\pi\)
−0.964825 + 0.262893i \(0.915323\pi\)
\(294\) 2.00000 0.116642
\(295\) 0 0
\(296\) −2.00000 −0.116248
\(297\) − 10.0000i − 0.580259i
\(298\) 0 0
\(299\) −1.00000 −0.0578315
\(300\) 0 0
\(301\) 12.0000 0.691669
\(302\) − 2.00000i − 0.115087i
\(303\) − 2.00000i − 0.114897i
\(304\) 1.00000 0.0573539
\(305\) 0 0
\(306\) −6.00000 −0.342997
\(307\) 12.0000i 0.684876i 0.939540 + 0.342438i \(0.111253\pi\)
−0.939540 + 0.342438i \(0.888747\pi\)
\(308\) 6.00000i 0.341882i
\(309\) −6.00000 −0.341328
\(310\) 0 0
\(311\) 7.00000 0.396934 0.198467 0.980108i \(-0.436404\pi\)
0.198467 + 0.980108i \(0.436404\pi\)
\(312\) − 1.00000i − 0.0566139i
\(313\) 29.0000i 1.63918i 0.572953 + 0.819588i \(0.305798\pi\)
−0.572953 + 0.819588i \(0.694202\pi\)
\(314\) 2.00000 0.112867
\(315\) 0 0
\(316\) −10.0000 −0.562544
\(317\) 27.0000i 1.51647i 0.651981 + 0.758236i \(0.273938\pi\)
−0.651981 + 0.758236i \(0.726062\pi\)
\(318\) 1.00000i 0.0560772i
\(319\) 10.0000 0.559893
\(320\) 0 0
\(321\) 7.00000 0.390702
\(322\) 3.00000i 0.167183i
\(323\) − 3.00000i − 0.166924i
\(324\) −1.00000 −0.0555556
\(325\) 0 0
\(326\) −16.0000 −0.886158
\(327\) − 15.0000i − 0.829502i
\(328\) − 8.00000i − 0.441726i
\(329\) −24.0000 −1.32316
\(330\) 0 0
\(331\) 17.0000 0.934405 0.467202 0.884150i \(-0.345262\pi\)
0.467202 + 0.884150i \(0.345262\pi\)
\(332\) 6.00000i 0.329293i
\(333\) 4.00000i 0.219199i
\(334\) 12.0000 0.656611
\(335\) 0 0
\(336\) −3.00000 −0.163663
\(337\) 32.0000i 1.74315i 0.490261 + 0.871576i \(0.336901\pi\)
−0.490261 + 0.871576i \(0.663099\pi\)
\(338\) − 12.0000i − 0.652714i
\(339\) 14.0000 0.760376
\(340\) 0 0
\(341\) −16.0000 −0.866449
\(342\) − 2.00000i − 0.108148i
\(343\) − 15.0000i − 0.809924i
\(344\) −4.00000 −0.215666
\(345\) 0 0
\(346\) −6.00000 −0.322562
\(347\) 2.00000i 0.107366i 0.998558 + 0.0536828i \(0.0170960\pi\)
−0.998558 + 0.0536828i \(0.982904\pi\)
\(348\) 5.00000i 0.268028i
\(349\) −10.0000 −0.535288 −0.267644 0.963518i \(-0.586245\pi\)
−0.267644 + 0.963518i \(0.586245\pi\)
\(350\) 0 0
\(351\) −5.00000 −0.266880
\(352\) − 2.00000i − 0.106600i
\(353\) 9.00000i 0.479022i 0.970894 + 0.239511i \(0.0769871\pi\)
−0.970894 + 0.239511i \(0.923013\pi\)
\(354\) 15.0000 0.797241
\(355\) 0 0
\(356\) 0 0
\(357\) 9.00000i 0.476331i
\(358\) 0 0
\(359\) 15.0000 0.791670 0.395835 0.918322i \(-0.370455\pi\)
0.395835 + 0.918322i \(0.370455\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) − 22.0000i − 1.15629i
\(363\) 7.00000i 0.367405i
\(364\) 3.00000 0.157243
\(365\) 0 0
\(366\) −2.00000 −0.104542
\(367\) − 28.0000i − 1.46159i −0.682598 0.730794i \(-0.739150\pi\)
0.682598 0.730794i \(-0.260850\pi\)
\(368\) − 1.00000i − 0.0521286i
\(369\) −16.0000 −0.832927
\(370\) 0 0
\(371\) −3.00000 −0.155752
\(372\) − 8.00000i − 0.414781i
\(373\) 29.0000i 1.50156i 0.660551 + 0.750782i \(0.270323\pi\)
−0.660551 + 0.750782i \(0.729677\pi\)
\(374\) −6.00000 −0.310253
\(375\) 0 0
\(376\) 8.00000 0.412568
\(377\) − 5.00000i − 0.257513i
\(378\) 15.0000i 0.771517i
\(379\) −15.0000 −0.770498 −0.385249 0.922813i \(-0.625884\pi\)
−0.385249 + 0.922813i \(0.625884\pi\)
\(380\) 0 0
\(381\) −18.0000 −0.922168
\(382\) − 7.00000i − 0.358151i
\(383\) − 26.0000i − 1.32854i −0.747494 0.664269i \(-0.768743\pi\)
0.747494 0.664269i \(-0.231257\pi\)
\(384\) 1.00000 0.0510310
\(385\) 0 0
\(386\) −6.00000 −0.305392
\(387\) 8.00000i 0.406663i
\(388\) − 2.00000i − 0.101535i
\(389\) 30.0000 1.52106 0.760530 0.649303i \(-0.224939\pi\)
0.760530 + 0.649303i \(0.224939\pi\)
\(390\) 0 0
\(391\) −3.00000 −0.151717
\(392\) − 2.00000i − 0.101015i
\(393\) − 12.0000i − 0.605320i
\(394\) −8.00000 −0.403034
\(395\) 0 0
\(396\) −4.00000 −0.201008
\(397\) − 8.00000i − 0.401508i −0.979642 0.200754i \(-0.935661\pi\)
0.979642 0.200754i \(-0.0643393\pi\)
\(398\) − 25.0000i − 1.25314i
\(399\) −3.00000 −0.150188
\(400\) 0 0
\(401\) −8.00000 −0.399501 −0.199750 0.979847i \(-0.564013\pi\)
−0.199750 + 0.979847i \(0.564013\pi\)
\(402\) 3.00000i 0.149626i
\(403\) 8.00000i 0.398508i
\(404\) −2.00000 −0.0995037
\(405\) 0 0
\(406\) −15.0000 −0.744438
\(407\) 4.00000i 0.198273i
\(408\) − 3.00000i − 0.148522i
\(409\) 20.0000 0.988936 0.494468 0.869196i \(-0.335363\pi\)
0.494468 + 0.869196i \(0.335363\pi\)
\(410\) 0 0
\(411\) 17.0000 0.838548
\(412\) 6.00000i 0.295599i
\(413\) 45.0000i 2.21431i
\(414\) −2.00000 −0.0982946
\(415\) 0 0
\(416\) −1.00000 −0.0490290
\(417\) 0 0
\(418\) − 2.00000i − 0.0978232i
\(419\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(420\) 0 0
\(421\) −13.0000 −0.633581 −0.316791 0.948495i \(-0.602605\pi\)
−0.316791 + 0.948495i \(0.602605\pi\)
\(422\) − 27.0000i − 1.31434i
\(423\) − 16.0000i − 0.777947i
\(424\) 1.00000 0.0485643
\(425\) 0 0
\(426\) −2.00000 −0.0969003
\(427\) − 6.00000i − 0.290360i
\(428\) − 7.00000i − 0.338358i
\(429\) −2.00000 −0.0965609
\(430\) 0 0
\(431\) −18.0000 −0.867029 −0.433515 0.901146i \(-0.642727\pi\)
−0.433515 + 0.901146i \(0.642727\pi\)
\(432\) − 5.00000i − 0.240563i
\(433\) 14.0000i 0.672797i 0.941720 + 0.336399i \(0.109209\pi\)
−0.941720 + 0.336399i \(0.890791\pi\)
\(434\) 24.0000 1.15204
\(435\) 0 0
\(436\) −15.0000 −0.718370
\(437\) − 1.00000i − 0.0478365i
\(438\) − 9.00000i − 0.430037i
\(439\) −20.0000 −0.954548 −0.477274 0.878755i \(-0.658375\pi\)
−0.477274 + 0.878755i \(0.658375\pi\)
\(440\) 0 0
\(441\) −4.00000 −0.190476
\(442\) 3.00000i 0.142695i
\(443\) − 26.0000i − 1.23530i −0.786454 0.617649i \(-0.788085\pi\)
0.786454 0.617649i \(-0.211915\pi\)
\(444\) −2.00000 −0.0949158
\(445\) 0 0
\(446\) 14.0000 0.662919
\(447\) 0 0
\(448\) 3.00000i 0.141737i
\(449\) −10.0000 −0.471929 −0.235965 0.971762i \(-0.575825\pi\)
−0.235965 + 0.971762i \(0.575825\pi\)
\(450\) 0 0
\(451\) −16.0000 −0.753411
\(452\) − 14.0000i − 0.658505i
\(453\) − 2.00000i − 0.0939682i
\(454\) 17.0000 0.797850
\(455\) 0 0
\(456\) 1.00000 0.0468293
\(457\) 7.00000i 0.327446i 0.986506 + 0.163723i \(0.0523504\pi\)
−0.986506 + 0.163723i \(0.947650\pi\)
\(458\) − 10.0000i − 0.467269i
\(459\) −15.0000 −0.700140
\(460\) 0 0
\(461\) −28.0000 −1.30409 −0.652045 0.758180i \(-0.726089\pi\)
−0.652045 + 0.758180i \(0.726089\pi\)
\(462\) 6.00000i 0.279145i
\(463\) 4.00000i 0.185896i 0.995671 + 0.0929479i \(0.0296290\pi\)
−0.995671 + 0.0929479i \(0.970371\pi\)
\(464\) 5.00000 0.232119
\(465\) 0 0
\(466\) −6.00000 −0.277945
\(467\) 2.00000i 0.0925490i 0.998929 + 0.0462745i \(0.0147349\pi\)
−0.998929 + 0.0462745i \(0.985265\pi\)
\(468\) 2.00000i 0.0924500i
\(469\) −9.00000 −0.415581
\(470\) 0 0
\(471\) 2.00000 0.0921551
\(472\) − 15.0000i − 0.690431i
\(473\) 8.00000i 0.367840i
\(474\) −10.0000 −0.459315
\(475\) 0 0
\(476\) 9.00000 0.412514
\(477\) − 2.00000i − 0.0915737i
\(478\) 15.0000i 0.686084i
\(479\) 20.0000 0.913823 0.456912 0.889512i \(-0.348956\pi\)
0.456912 + 0.889512i \(0.348956\pi\)
\(480\) 0 0
\(481\) 2.00000 0.0911922
\(482\) 8.00000i 0.364390i
\(483\) 3.00000i 0.136505i
\(484\) 7.00000 0.318182
\(485\) 0 0
\(486\) −16.0000 −0.725775
\(487\) 2.00000i 0.0906287i 0.998973 + 0.0453143i \(0.0144289\pi\)
−0.998973 + 0.0453143i \(0.985571\pi\)
\(488\) 2.00000i 0.0905357i
\(489\) −16.0000 −0.723545
\(490\) 0 0
\(491\) −28.0000 −1.26362 −0.631811 0.775122i \(-0.717688\pi\)
−0.631811 + 0.775122i \(0.717688\pi\)
\(492\) − 8.00000i − 0.360668i
\(493\) − 15.0000i − 0.675566i
\(494\) −1.00000 −0.0449921
\(495\) 0 0
\(496\) −8.00000 −0.359211
\(497\) − 6.00000i − 0.269137i
\(498\) 6.00000i 0.268866i
\(499\) −40.0000 −1.79065 −0.895323 0.445418i \(-0.853055\pi\)
−0.895323 + 0.445418i \(0.853055\pi\)
\(500\) 0 0
\(501\) 12.0000 0.536120
\(502\) − 2.00000i − 0.0892644i
\(503\) 39.0000i 1.73892i 0.494000 + 0.869462i \(0.335534\pi\)
−0.494000 + 0.869462i \(0.664466\pi\)
\(504\) 6.00000 0.267261
\(505\) 0 0
\(506\) −2.00000 −0.0889108
\(507\) − 12.0000i − 0.532939i
\(508\) 18.0000i 0.798621i
\(509\) 30.0000 1.32973 0.664863 0.746965i \(-0.268490\pi\)
0.664863 + 0.746965i \(0.268490\pi\)
\(510\) 0 0
\(511\) 27.0000 1.19441
\(512\) − 1.00000i − 0.0441942i
\(513\) − 5.00000i − 0.220755i
\(514\) −8.00000 −0.352865
\(515\) 0 0
\(516\) −4.00000 −0.176090
\(517\) − 16.0000i − 0.703679i
\(518\) − 6.00000i − 0.263625i
\(519\) −6.00000 −0.263371
\(520\) 0 0
\(521\) −28.0000 −1.22670 −0.613351 0.789810i \(-0.710179\pi\)
−0.613351 + 0.789810i \(0.710179\pi\)
\(522\) − 10.0000i − 0.437688i
\(523\) 29.0000i 1.26808i 0.773300 + 0.634041i \(0.218605\pi\)
−0.773300 + 0.634041i \(0.781395\pi\)
\(524\) −12.0000 −0.524222
\(525\) 0 0
\(526\) 24.0000 1.04645
\(527\) 24.0000i 1.04546i
\(528\) − 2.00000i − 0.0870388i
\(529\) 22.0000 0.956522
\(530\) 0 0
\(531\) −30.0000 −1.30189
\(532\) 3.00000i 0.130066i
\(533\) 8.00000i 0.346518i
\(534\) 0 0
\(535\) 0 0
\(536\) 3.00000 0.129580
\(537\) 0 0
\(538\) 30.0000i 1.29339i
\(539\) −4.00000 −0.172292
\(540\) 0 0
\(541\) 2.00000 0.0859867 0.0429934 0.999075i \(-0.486311\pi\)
0.0429934 + 0.999075i \(0.486311\pi\)
\(542\) − 7.00000i − 0.300676i
\(543\) − 22.0000i − 0.944110i
\(544\) −3.00000 −0.128624
\(545\) 0 0
\(546\) 3.00000 0.128388
\(547\) − 28.0000i − 1.19719i −0.801050 0.598597i \(-0.795725\pi\)
0.801050 0.598597i \(-0.204275\pi\)
\(548\) − 17.0000i − 0.726204i
\(549\) 4.00000 0.170716
\(550\) 0 0
\(551\) 5.00000 0.213007
\(552\) − 1.00000i − 0.0425628i
\(553\) − 30.0000i − 1.27573i
\(554\) −28.0000 −1.18961
\(555\) 0 0
\(556\) 0 0
\(557\) − 28.0000i − 1.18640i −0.805056 0.593199i \(-0.797865\pi\)
0.805056 0.593199i \(-0.202135\pi\)
\(558\) 16.0000i 0.677334i
\(559\) 4.00000 0.169182
\(560\) 0 0
\(561\) −6.00000 −0.253320
\(562\) 8.00000i 0.337460i
\(563\) − 36.0000i − 1.51722i −0.651546 0.758610i \(-0.725879\pi\)
0.651546 0.758610i \(-0.274121\pi\)
\(564\) 8.00000 0.336861
\(565\) 0 0
\(566\) −6.00000 −0.252199
\(567\) − 3.00000i − 0.125988i
\(568\) 2.00000i 0.0839181i
\(569\) −40.0000 −1.67689 −0.838444 0.544988i \(-0.816534\pi\)
−0.838444 + 0.544988i \(0.816534\pi\)
\(570\) 0 0
\(571\) −28.0000 −1.17176 −0.585882 0.810397i \(-0.699252\pi\)
−0.585882 + 0.810397i \(0.699252\pi\)
\(572\) 2.00000i 0.0836242i
\(573\) − 7.00000i − 0.292429i
\(574\) 24.0000 1.00174
\(575\) 0 0
\(576\) −2.00000 −0.0833333
\(577\) 37.0000i 1.54033i 0.637845 + 0.770165i \(0.279826\pi\)
−0.637845 + 0.770165i \(0.720174\pi\)
\(578\) − 8.00000i − 0.332756i
\(579\) −6.00000 −0.249351
\(580\) 0 0
\(581\) −18.0000 −0.746766
\(582\) − 2.00000i − 0.0829027i
\(583\) − 2.00000i − 0.0828315i
\(584\) −9.00000 −0.372423
\(585\) 0 0
\(586\) 9.00000 0.371787
\(587\) 12.0000i 0.495293i 0.968850 + 0.247647i \(0.0796572\pi\)
−0.968850 + 0.247647i \(0.920343\pi\)
\(588\) − 2.00000i − 0.0824786i
\(589\) −8.00000 −0.329634
\(590\) 0 0
\(591\) −8.00000 −0.329076
\(592\) 2.00000i 0.0821995i
\(593\) 34.0000i 1.39621i 0.715994 + 0.698106i \(0.245974\pi\)
−0.715994 + 0.698106i \(0.754026\pi\)
\(594\) −10.0000 −0.410305
\(595\) 0 0
\(596\) 0 0
\(597\) − 25.0000i − 1.02318i
\(598\) 1.00000i 0.0408930i
\(599\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(600\) 0 0
\(601\) −8.00000 −0.326327 −0.163163 0.986599i \(-0.552170\pi\)
−0.163163 + 0.986599i \(0.552170\pi\)
\(602\) − 12.0000i − 0.489083i
\(603\) − 6.00000i − 0.244339i
\(604\) −2.00000 −0.0813788
\(605\) 0 0
\(606\) −2.00000 −0.0812444
\(607\) 22.0000i 0.892952i 0.894795 + 0.446476i \(0.147321\pi\)
−0.894795 + 0.446476i \(0.852679\pi\)
\(608\) − 1.00000i − 0.0405554i
\(609\) −15.0000 −0.607831
\(610\) 0 0
\(611\) −8.00000 −0.323645
\(612\) 6.00000i 0.242536i
\(613\) 34.0000i 1.37325i 0.727013 + 0.686624i \(0.240908\pi\)
−0.727013 + 0.686624i \(0.759092\pi\)
\(614\) 12.0000 0.484281
\(615\) 0 0
\(616\) 6.00000 0.241747
\(617\) − 18.0000i − 0.724653i −0.932051 0.362326i \(-0.881983\pi\)
0.932051 0.362326i \(-0.118017\pi\)
\(618\) 6.00000i 0.241355i
\(619\) −10.0000 −0.401934 −0.200967 0.979598i \(-0.564408\pi\)
−0.200967 + 0.979598i \(0.564408\pi\)
\(620\) 0 0
\(621\) −5.00000 −0.200643
\(622\) − 7.00000i − 0.280674i
\(623\) 0 0
\(624\) −1.00000 −0.0400320
\(625\) 0 0
\(626\) 29.0000 1.15907
\(627\) − 2.00000i − 0.0798723i
\(628\) − 2.00000i − 0.0798087i
\(629\) 6.00000 0.239236
\(630\) 0 0
\(631\) 32.0000 1.27390 0.636950 0.770905i \(-0.280196\pi\)
0.636950 + 0.770905i \(0.280196\pi\)
\(632\) 10.0000i 0.397779i
\(633\) − 27.0000i − 1.07315i
\(634\) 27.0000 1.07231
\(635\) 0 0
\(636\) 1.00000 0.0396526
\(637\) 2.00000i 0.0792429i
\(638\) − 10.0000i − 0.395904i
\(639\) 4.00000 0.158238
\(640\) 0 0
\(641\) 42.0000 1.65890 0.829450 0.558581i \(-0.188654\pi\)
0.829450 + 0.558581i \(0.188654\pi\)
\(642\) − 7.00000i − 0.276268i
\(643\) − 26.0000i − 1.02534i −0.858586 0.512670i \(-0.828656\pi\)
0.858586 0.512670i \(-0.171344\pi\)
\(644\) 3.00000 0.118217
\(645\) 0 0
\(646\) −3.00000 −0.118033
\(647\) − 23.0000i − 0.904223i −0.891961 0.452112i \(-0.850671\pi\)
0.891961 0.452112i \(-0.149329\pi\)
\(648\) 1.00000i 0.0392837i
\(649\) −30.0000 −1.17760
\(650\) 0 0
\(651\) 24.0000 0.940634
\(652\) 16.0000i 0.626608i
\(653\) − 36.0000i − 1.40879i −0.709809 0.704394i \(-0.751219\pi\)
0.709809 0.704394i \(-0.248781\pi\)
\(654\) −15.0000 −0.586546
\(655\) 0 0
\(656\) −8.00000 −0.312348
\(657\) 18.0000i 0.702247i
\(658\) 24.0000i 0.935617i
\(659\) −5.00000 −0.194772 −0.0973862 0.995247i \(-0.531048\pi\)
−0.0973862 + 0.995247i \(0.531048\pi\)
\(660\) 0 0
\(661\) −23.0000 −0.894596 −0.447298 0.894385i \(-0.647614\pi\)
−0.447298 + 0.894385i \(0.647614\pi\)
\(662\) − 17.0000i − 0.660724i
\(663\) 3.00000i 0.116510i
\(664\) 6.00000 0.232845
\(665\) 0 0
\(666\) 4.00000 0.154997
\(667\) − 5.00000i − 0.193601i
\(668\) − 12.0000i − 0.464294i
\(669\) 14.0000 0.541271
\(670\) 0 0
\(671\) 4.00000 0.154418
\(672\) 3.00000i 0.115728i
\(673\) 44.0000i 1.69608i 0.529936 + 0.848038i \(0.322216\pi\)
−0.529936 + 0.848038i \(0.677784\pi\)
\(674\) 32.0000 1.23259
\(675\) 0 0
\(676\) −12.0000 −0.461538
\(677\) − 13.0000i − 0.499631i −0.968294 0.249815i \(-0.919630\pi\)
0.968294 0.249815i \(-0.0803699\pi\)
\(678\) − 14.0000i − 0.537667i
\(679\) 6.00000 0.230259
\(680\) 0 0
\(681\) 17.0000 0.651441
\(682\) 16.0000i 0.612672i
\(683\) 4.00000i 0.153056i 0.997067 + 0.0765279i \(0.0243834\pi\)
−0.997067 + 0.0765279i \(0.975617\pi\)
\(684\) −2.00000 −0.0764719
\(685\) 0 0
\(686\) −15.0000 −0.572703
\(687\) − 10.0000i − 0.381524i
\(688\) 4.00000i 0.152499i
\(689\) −1.00000 −0.0380970
\(690\) 0 0
\(691\) 42.0000 1.59776 0.798878 0.601494i \(-0.205427\pi\)
0.798878 + 0.601494i \(0.205427\pi\)
\(692\) 6.00000i 0.228086i
\(693\) − 12.0000i − 0.455842i
\(694\) 2.00000 0.0759190
\(695\) 0 0
\(696\) 5.00000 0.189525
\(697\) 24.0000i 0.909065i
\(698\) 10.0000i 0.378506i
\(699\) −6.00000 −0.226941
\(700\) 0 0
\(701\) −28.0000 −1.05755 −0.528773 0.848763i \(-0.677348\pi\)
−0.528773 + 0.848763i \(0.677348\pi\)
\(702\) 5.00000i 0.188713i
\(703\) 2.00000i 0.0754314i
\(704\) −2.00000 −0.0753778
\(705\) 0 0
\(706\) 9.00000 0.338719
\(707\) − 6.00000i − 0.225653i
\(708\) − 15.0000i − 0.563735i
\(709\) 30.0000 1.12667 0.563337 0.826227i \(-0.309517\pi\)
0.563337 + 0.826227i \(0.309517\pi\)
\(710\) 0 0
\(711\) 20.0000 0.750059
\(712\) 0 0
\(713\) 8.00000i 0.299602i
\(714\) 9.00000 0.336817
\(715\) 0 0
\(716\) 0 0
\(717\) 15.0000i 0.560185i
\(718\) − 15.0000i − 0.559795i
\(719\) 5.00000 0.186469 0.0932343 0.995644i \(-0.470279\pi\)
0.0932343 + 0.995644i \(0.470279\pi\)
\(720\) 0 0
\(721\) −18.0000 −0.670355
\(722\) − 1.00000i − 0.0372161i
\(723\) 8.00000i 0.297523i
\(724\) −22.0000 −0.817624
\(725\) 0 0
\(726\) 7.00000 0.259794
\(727\) 17.0000i 0.630495i 0.949009 + 0.315248i \(0.102088\pi\)
−0.949009 + 0.315248i \(0.897912\pi\)
\(728\) − 3.00000i − 0.111187i
\(729\) −13.0000 −0.481481
\(730\) 0 0
\(731\) 12.0000 0.443836
\(732\) 2.00000i 0.0739221i
\(733\) − 36.0000i − 1.32969i −0.746981 0.664845i \(-0.768498\pi\)
0.746981 0.664845i \(-0.231502\pi\)
\(734\) −28.0000 −1.03350
\(735\) 0 0
\(736\) −1.00000 −0.0368605
\(737\) − 6.00000i − 0.221013i
\(738\) 16.0000i 0.588968i
\(739\) 40.0000 1.47142 0.735712 0.677295i \(-0.236848\pi\)
0.735712 + 0.677295i \(0.236848\pi\)
\(740\) 0 0
\(741\) −1.00000 −0.0367359
\(742\) 3.00000i 0.110133i
\(743\) − 16.0000i − 0.586983i −0.955962 0.293492i \(-0.905183\pi\)
0.955962 0.293492i \(-0.0948173\pi\)
\(744\) −8.00000 −0.293294
\(745\) 0 0
\(746\) 29.0000 1.06177
\(747\) − 12.0000i − 0.439057i
\(748\) 6.00000i 0.219382i
\(749\) 21.0000 0.767323
\(750\) 0 0
\(751\) 32.0000 1.16770 0.583848 0.811863i \(-0.301546\pi\)
0.583848 + 0.811863i \(0.301546\pi\)
\(752\) − 8.00000i − 0.291730i
\(753\) − 2.00000i − 0.0728841i
\(754\) −5.00000 −0.182089
\(755\) 0 0
\(756\) 15.0000 0.545545
\(757\) 2.00000i 0.0726912i 0.999339 + 0.0363456i \(0.0115717\pi\)
−0.999339 + 0.0363456i \(0.988428\pi\)
\(758\) 15.0000i 0.544825i
\(759\) −2.00000 −0.0725954
\(760\) 0 0
\(761\) 27.0000 0.978749 0.489375 0.872074i \(-0.337225\pi\)
0.489375 + 0.872074i \(0.337225\pi\)
\(762\) 18.0000i 0.652071i
\(763\) − 45.0000i − 1.62911i
\(764\) −7.00000 −0.253251
\(765\) 0 0
\(766\) −26.0000 −0.939418
\(767\) 15.0000i 0.541619i
\(768\) − 1.00000i − 0.0360844i
\(769\) 35.0000 1.26213 0.631066 0.775729i \(-0.282618\pi\)
0.631066 + 0.775729i \(0.282618\pi\)
\(770\) 0 0
\(771\) −8.00000 −0.288113
\(772\) 6.00000i 0.215945i
\(773\) 9.00000i 0.323708i 0.986815 + 0.161854i \(0.0517473\pi\)
−0.986815 + 0.161854i \(0.948253\pi\)
\(774\) 8.00000 0.287554
\(775\) 0 0
\(776\) −2.00000 −0.0717958
\(777\) − 6.00000i − 0.215249i
\(778\) − 30.0000i − 1.07555i
\(779\) −8.00000 −0.286630
\(780\) 0 0
\(781\) 4.00000 0.143131
\(782\) 3.00000i 0.107280i
\(783\) − 25.0000i − 0.893427i
\(784\) −2.00000 −0.0714286
\(785\) 0 0
\(786\) −12.0000 −0.428026
\(787\) 17.0000i 0.605985i 0.952993 + 0.302992i \(0.0979856\pi\)
−0.952993 + 0.302992i \(0.902014\pi\)
\(788\) 8.00000i 0.284988i
\(789\) 24.0000 0.854423
\(790\) 0 0
\(791\) 42.0000 1.49335
\(792\) 4.00000i 0.142134i
\(793\) − 2.00000i − 0.0710221i
\(794\) −8.00000 −0.283909
\(795\) 0 0
\(796\) −25.0000 −0.886102
\(797\) − 3.00000i − 0.106265i −0.998587 0.0531327i \(-0.983079\pi\)
0.998587 0.0531327i \(-0.0169206\pi\)
\(798\) 3.00000i 0.106199i
\(799\) −24.0000 −0.849059
\(800\) 0 0
\(801\) 0 0
\(802\) 8.00000i 0.282490i
\(803\) 18.0000i 0.635206i
\(804\) 3.00000 0.105802
\(805\) 0 0
\(806\) 8.00000 0.281788
\(807\) 30.0000i 1.05605i
\(808\) 2.00000i 0.0703598i
\(809\) 15.0000 0.527372 0.263686 0.964609i \(-0.415062\pi\)
0.263686 + 0.964609i \(0.415062\pi\)
\(810\) 0 0
\(811\) −3.00000 −0.105344 −0.0526721 0.998612i \(-0.516774\pi\)
−0.0526721 + 0.998612i \(0.516774\pi\)
\(812\) 15.0000i 0.526397i
\(813\) − 7.00000i − 0.245501i
\(814\) 4.00000 0.140200
\(815\) 0 0
\(816\) −3.00000 −0.105021
\(817\) 4.00000i 0.139942i
\(818\) − 20.0000i − 0.699284i
\(819\) −6.00000 −0.209657
\(820\) 0 0
\(821\) 12.0000 0.418803 0.209401 0.977830i \(-0.432848\pi\)
0.209401 + 0.977830i \(0.432848\pi\)
\(822\) − 17.0000i − 0.592943i
\(823\) 29.0000i 1.01088i 0.862863 + 0.505438i \(0.168669\pi\)
−0.862863 + 0.505438i \(0.831331\pi\)
\(824\) 6.00000 0.209020
\(825\) 0 0
\(826\) 45.0000 1.56575
\(827\) − 23.0000i − 0.799788i −0.916561 0.399894i \(-0.869047\pi\)
0.916561 0.399894i \(-0.130953\pi\)
\(828\) 2.00000i 0.0695048i
\(829\) 15.0000 0.520972 0.260486 0.965478i \(-0.416117\pi\)
0.260486 + 0.965478i \(0.416117\pi\)
\(830\) 0 0
\(831\) −28.0000 −0.971309
\(832\) 1.00000i 0.0346688i
\(833\) 6.00000i 0.207888i
\(834\) 0 0
\(835\) 0 0
\(836\) −2.00000 −0.0691714
\(837\) 40.0000i 1.38260i
\(838\) 0 0
\(839\) −20.0000 −0.690477 −0.345238 0.938515i \(-0.612202\pi\)
−0.345238 + 0.938515i \(0.612202\pi\)
\(840\) 0 0
\(841\) −4.00000 −0.137931
\(842\) 13.0000i 0.448010i
\(843\) 8.00000i 0.275535i
\(844\) −27.0000 −0.929378
\(845\) 0 0
\(846\) −16.0000 −0.550091
\(847\) 21.0000i 0.721569i
\(848\) − 1.00000i − 0.0343401i
\(849\) −6.00000 −0.205919
\(850\) 0 0
\(851\) 2.00000 0.0685591
\(852\) 2.00000i 0.0685189i
\(853\) − 6.00000i − 0.205436i −0.994711 0.102718i \(-0.967246\pi\)
0.994711 0.102718i \(-0.0327539\pi\)
\(854\) −6.00000 −0.205316
\(855\) 0 0
\(856\) −7.00000 −0.239255
\(857\) 12.0000i 0.409912i 0.978771 + 0.204956i \(0.0657052\pi\)
−0.978771 + 0.204956i \(0.934295\pi\)
\(858\) 2.00000i 0.0682789i
\(859\) 50.0000 1.70598 0.852989 0.521929i \(-0.174787\pi\)
0.852989 + 0.521929i \(0.174787\pi\)
\(860\) 0 0
\(861\) 24.0000 0.817918
\(862\) 18.0000i 0.613082i
\(863\) 54.0000i 1.83818i 0.394046 + 0.919091i \(0.371075\pi\)
−0.394046 + 0.919091i \(0.628925\pi\)
\(864\) −5.00000 −0.170103
\(865\) 0 0
\(866\) 14.0000 0.475739
\(867\) − 8.00000i − 0.271694i
\(868\) − 24.0000i − 0.814613i
\(869\) 20.0000 0.678454
\(870\) 0 0
\(871\) −3.00000 −0.101651
\(872\) 15.0000i 0.507964i
\(873\) 4.00000i 0.135379i
\(874\) −1.00000 −0.0338255
\(875\) 0 0
\(876\) −9.00000 −0.304082
\(877\) − 13.0000i − 0.438979i −0.975615 0.219489i \(-0.929561\pi\)
0.975615 0.219489i \(-0.0704391\pi\)
\(878\) 20.0000i 0.674967i
\(879\) 9.00000 0.303562
\(880\) 0 0
\(881\) −18.0000 −0.606435 −0.303218 0.952921i \(-0.598061\pi\)
−0.303218 + 0.952921i \(0.598061\pi\)
\(882\) 4.00000i 0.134687i
\(883\) 34.0000i 1.14419i 0.820187 + 0.572096i \(0.193869\pi\)
−0.820187 + 0.572096i \(0.806131\pi\)
\(884\) 3.00000 0.100901
\(885\) 0 0
\(886\) −26.0000 −0.873487
\(887\) 2.00000i 0.0671534i 0.999436 + 0.0335767i \(0.0106898\pi\)
−0.999436 + 0.0335767i \(0.989310\pi\)
\(888\) 2.00000i 0.0671156i
\(889\) −54.0000 −1.81110
\(890\) 0 0
\(891\) 2.00000 0.0670025
\(892\) − 14.0000i − 0.468755i
\(893\) − 8.00000i − 0.267710i
\(894\) 0 0
\(895\) 0 0
\(896\) 3.00000 0.100223
\(897\) 1.00000i 0.0333890i
\(898\) 10.0000i 0.333704i
\(899\) −40.0000 −1.33407
\(900\) 0 0
\(901\) −3.00000 −0.0999445
\(902\) 16.0000i 0.532742i
\(903\) − 12.0000i − 0.399335i
\(904\) −14.0000 −0.465633
\(905\) 0 0
\(906\) −2.00000 −0.0664455
\(907\) − 53.0000i − 1.75984i −0.475125 0.879918i \(-0.657597\pi\)
0.475125 0.879918i \(-0.342403\pi\)
\(908\) − 17.0000i − 0.564165i
\(909\) 4.00000 0.132672
\(910\) 0 0
\(911\) 12.0000 0.397578 0.198789 0.980042i \(-0.436299\pi\)
0.198789 + 0.980042i \(0.436299\pi\)
\(912\) − 1.00000i − 0.0331133i
\(913\) − 12.0000i − 0.397142i
\(914\) 7.00000 0.231539
\(915\) 0 0
\(916\) −10.0000 −0.330409
\(917\) − 36.0000i − 1.18882i
\(918\) 15.0000i 0.495074i
\(919\) −5.00000 −0.164935 −0.0824674 0.996594i \(-0.526280\pi\)
−0.0824674 + 0.996594i \(0.526280\pi\)
\(920\) 0 0
\(921\) 12.0000 0.395413
\(922\) 28.0000i 0.922131i
\(923\) − 2.00000i − 0.0658308i
\(924\) 6.00000 0.197386
\(925\) 0 0
\(926\) 4.00000 0.131448
\(927\) − 12.0000i − 0.394132i
\(928\) − 5.00000i − 0.164133i
\(929\) 55.0000 1.80449 0.902246 0.431222i \(-0.141918\pi\)
0.902246 + 0.431222i \(0.141918\pi\)
\(930\) 0 0
\(931\) −2.00000 −0.0655474
\(932\) 6.00000i 0.196537i
\(933\) − 7.00000i − 0.229170i
\(934\) 2.00000 0.0654420
\(935\) 0 0
\(936\) 2.00000 0.0653720
\(937\) 7.00000i 0.228680i 0.993442 + 0.114340i \(0.0364753\pi\)
−0.993442 + 0.114340i \(0.963525\pi\)
\(938\) 9.00000i 0.293860i
\(939\) 29.0000 0.946379
\(940\) 0 0
\(941\) 7.00000 0.228193 0.114097 0.993470i \(-0.463603\pi\)
0.114097 + 0.993470i \(0.463603\pi\)
\(942\) − 2.00000i − 0.0651635i
\(943\) 8.00000i 0.260516i
\(944\) −15.0000 −0.488208
\(945\) 0 0
\(946\) 8.00000 0.260102
\(947\) 12.0000i 0.389948i 0.980808 + 0.194974i \(0.0624622\pi\)
−0.980808 + 0.194974i \(0.937538\pi\)
\(948\) 10.0000i 0.324785i
\(949\) 9.00000 0.292152
\(950\) 0 0
\(951\) 27.0000 0.875535
\(952\) − 9.00000i − 0.291692i
\(953\) − 46.0000i − 1.49009i −0.667016 0.745043i \(-0.732429\pi\)
0.667016 0.745043i \(-0.267571\pi\)
\(954\) −2.00000 −0.0647524
\(955\) 0 0
\(956\) 15.0000 0.485135
\(957\) − 10.0000i − 0.323254i
\(958\) − 20.0000i − 0.646171i
\(959\) 51.0000 1.64688
\(960\) 0 0
\(961\) 33.0000 1.06452
\(962\) − 2.00000i − 0.0644826i
\(963\) 14.0000i 0.451144i
\(964\) 8.00000 0.257663
\(965\) 0 0
\(966\) 3.00000 0.0965234
\(967\) − 48.0000i − 1.54358i −0.635880 0.771788i \(-0.719363\pi\)
0.635880 0.771788i \(-0.280637\pi\)
\(968\) − 7.00000i − 0.224989i
\(969\) −3.00000 −0.0963739
\(970\) 0 0
\(971\) −28.0000 −0.898563 −0.449281 0.893390i \(-0.648320\pi\)
−0.449281 + 0.893390i \(0.648320\pi\)
\(972\) 16.0000i 0.513200i
\(973\) 0 0
\(974\) 2.00000 0.0640841
\(975\) 0 0
\(976\) 2.00000 0.0640184
\(977\) − 8.00000i − 0.255943i −0.991778 0.127971i \(-0.959153\pi\)
0.991778 0.127971i \(-0.0408466\pi\)
\(978\) 16.0000i 0.511624i
\(979\) 0 0
\(980\) 0 0
\(981\) 30.0000 0.957826
\(982\) 28.0000i 0.893516i
\(983\) − 6.00000i − 0.191370i −0.995412 0.0956851i \(-0.969496\pi\)
0.995412 0.0956851i \(-0.0305042\pi\)
\(984\) −8.00000 −0.255031
\(985\) 0 0
\(986\) −15.0000 −0.477697
\(987\) 24.0000i 0.763928i
\(988\) 1.00000i 0.0318142i
\(989\) 4.00000 0.127193
\(990\) 0 0
\(991\) −8.00000 −0.254128 −0.127064 0.991894i \(-0.540555\pi\)
−0.127064 + 0.991894i \(0.540555\pi\)
\(992\) 8.00000i 0.254000i
\(993\) − 17.0000i − 0.539479i
\(994\) −6.00000 −0.190308
\(995\) 0 0
\(996\) 6.00000 0.190117
\(997\) − 28.0000i − 0.886769i −0.896332 0.443384i \(-0.853778\pi\)
0.896332 0.443384i \(-0.146222\pi\)
\(998\) 40.0000i 1.26618i
\(999\) 10.0000 0.316386
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.c.799.1 2
5.2 odd 4 38.2.a.b.1.1 1
5.3 odd 4 950.2.a.b.1.1 1
5.4 even 2 inner 950.2.b.c.799.2 2
15.2 even 4 342.2.a.d.1.1 1
15.8 even 4 8550.2.a.u.1.1 1
20.3 even 4 7600.2.a.h.1.1 1
20.7 even 4 304.2.a.d.1.1 1
35.27 even 4 1862.2.a.f.1.1 1
40.27 even 4 1216.2.a.g.1.1 1
40.37 odd 4 1216.2.a.n.1.1 1
55.32 even 4 4598.2.a.a.1.1 1
60.47 odd 4 2736.2.a.w.1.1 1
65.12 odd 4 6422.2.a.b.1.1 1
95.2 even 36 722.2.e.d.99.1 6
95.7 odd 12 722.2.c.d.429.1 2
95.12 even 12 722.2.c.f.429.1 2
95.17 odd 36 722.2.e.c.99.1 6
95.22 even 36 722.2.e.d.389.1 6
95.27 even 12 722.2.c.f.653.1 2
95.32 even 36 722.2.e.d.245.1 6
95.37 even 4 722.2.a.b.1.1 1
95.42 odd 36 722.2.e.c.415.1 6
95.47 odd 36 722.2.e.c.423.1 6
95.52 even 36 722.2.e.d.595.1 6
95.62 odd 36 722.2.e.c.595.1 6
95.67 even 36 722.2.e.d.423.1 6
95.72 even 36 722.2.e.d.415.1 6
95.82 odd 36 722.2.e.c.245.1 6
95.87 odd 12 722.2.c.d.653.1 2
95.92 odd 36 722.2.e.c.389.1 6
285.227 odd 4 6498.2.a.y.1.1 1
380.227 odd 4 5776.2.a.d.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
38.2.a.b.1.1 1 5.2 odd 4
304.2.a.d.1.1 1 20.7 even 4
342.2.a.d.1.1 1 15.2 even 4
722.2.a.b.1.1 1 95.37 even 4
722.2.c.d.429.1 2 95.7 odd 12
722.2.c.d.653.1 2 95.87 odd 12
722.2.c.f.429.1 2 95.12 even 12
722.2.c.f.653.1 2 95.27 even 12
722.2.e.c.99.1 6 95.17 odd 36
722.2.e.c.245.1 6 95.82 odd 36
722.2.e.c.389.1 6 95.92 odd 36
722.2.e.c.415.1 6 95.42 odd 36
722.2.e.c.423.1 6 95.47 odd 36
722.2.e.c.595.1 6 95.62 odd 36
722.2.e.d.99.1 6 95.2 even 36
722.2.e.d.245.1 6 95.32 even 36
722.2.e.d.389.1 6 95.22 even 36
722.2.e.d.415.1 6 95.72 even 36
722.2.e.d.423.1 6 95.67 even 36
722.2.e.d.595.1 6 95.52 even 36
950.2.a.b.1.1 1 5.3 odd 4
950.2.b.c.799.1 2 1.1 even 1 trivial
950.2.b.c.799.2 2 5.4 even 2 inner
1216.2.a.g.1.1 1 40.27 even 4
1216.2.a.n.1.1 1 40.37 odd 4
1862.2.a.f.1.1 1 35.27 even 4
2736.2.a.w.1.1 1 60.47 odd 4
4598.2.a.a.1.1 1 55.32 even 4
5776.2.a.d.1.1 1 380.227 odd 4
6422.2.a.b.1.1 1 65.12 odd 4
6498.2.a.y.1.1 1 285.227 odd 4
7600.2.a.h.1.1 1 20.3 even 4
8550.2.a.u.1.1 1 15.8 even 4