Properties

Label 2850.2.d.p.799.2
Level $2850$
Weight $2$
Character 2850.799
Analytic conductor $22.757$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 2850 = 2 \cdot 3 \cdot 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2850.d (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(22.7573645761\)
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 114)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 799.2
Root \(-1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 2850.799
Dual form 2850.2.d.p.799.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} -4.00000i q^{7} -1.00000i q^{8} -1.00000 q^{9} +O(q^{10})\) \(q+1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} -4.00000i q^{7} -1.00000i q^{8} -1.00000 q^{9} +1.00000i q^{12} +4.00000i q^{13} +4.00000 q^{14} +1.00000 q^{16} +6.00000i q^{17} -1.00000i q^{18} -1.00000 q^{19} -4.00000 q^{21} +6.00000i q^{23} -1.00000 q^{24} -4.00000 q^{26} +1.00000i q^{27} +4.00000i q^{28} -6.00000 q^{29} +2.00000 q^{31} +1.00000i q^{32} -6.00000 q^{34} +1.00000 q^{36} -4.00000i q^{37} -1.00000i q^{38} +4.00000 q^{39} +6.00000 q^{41} -4.00000i q^{42} +4.00000i q^{43} -6.00000 q^{46} +6.00000i q^{47} -1.00000i q^{48} -9.00000 q^{49} +6.00000 q^{51} -4.00000i q^{52} -6.00000i q^{53} -1.00000 q^{54} -4.00000 q^{56} +1.00000i q^{57} -6.00000i q^{58} +12.0000 q^{59} +14.0000 q^{61} +2.00000i q^{62} +4.00000i q^{63} -1.00000 q^{64} +8.00000i q^{67} -6.00000i q^{68} +6.00000 q^{69} +1.00000i q^{72} -14.0000i q^{73} +4.00000 q^{74} +1.00000 q^{76} +4.00000i q^{78} +10.0000 q^{79} +1.00000 q^{81} +6.00000i q^{82} +12.0000i q^{83} +4.00000 q^{84} -4.00000 q^{86} +6.00000i q^{87} +6.00000 q^{89} +16.0000 q^{91} -6.00000i q^{92} -2.00000i q^{93} -6.00000 q^{94} +1.00000 q^{96} -10.0000i q^{97} -9.00000i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{4} + 2q^{6} - 2q^{9} + O(q^{10}) \) \( 2q - 2q^{4} + 2q^{6} - 2q^{9} + 8q^{14} + 2q^{16} - 2q^{19} - 8q^{21} - 2q^{24} - 8q^{26} - 12q^{29} + 4q^{31} - 12q^{34} + 2q^{36} + 8q^{39} + 12q^{41} - 12q^{46} - 18q^{49} + 12q^{51} - 2q^{54} - 8q^{56} + 24q^{59} + 28q^{61} - 2q^{64} + 12q^{69} + 8q^{74} + 2q^{76} + 20q^{79} + 2q^{81} + 8q^{84} - 8q^{86} + 12q^{89} + 32q^{91} - 12q^{94} + 2q^{96} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2850\mathbb{Z}\right)^\times\).

\(n\) \(1027\) \(1351\) \(1901\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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
\(4\) −1.00000 −0.500000
\(5\) 0 0
\(6\) 1.00000 0.408248
\(7\) − 4.00000i − 1.51186i −0.654654 0.755929i \(-0.727186\pi\)
0.654654 0.755929i \(-0.272814\pi\)
\(8\) − 1.00000i − 0.353553i
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) 1.00000i 0.288675i
\(13\) 4.00000i 1.10940i 0.832050 + 0.554700i \(0.187167\pi\)
−0.832050 + 0.554700i \(0.812833\pi\)
\(14\) 4.00000 1.06904
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 6.00000i 1.45521i 0.685994 + 0.727607i \(0.259367\pi\)
−0.685994 + 0.727607i \(0.740633\pi\)
\(18\) − 1.00000i − 0.235702i
\(19\) −1.00000 −0.229416
\(20\) 0 0
\(21\) −4.00000 −0.872872
\(22\) 0 0
\(23\) 6.00000i 1.25109i 0.780189 + 0.625543i \(0.215123\pi\)
−0.780189 + 0.625543i \(0.784877\pi\)
\(24\) −1.00000 −0.204124
\(25\) 0 0
\(26\) −4.00000 −0.784465
\(27\) 1.00000i 0.192450i
\(28\) 4.00000i 0.755929i
\(29\) −6.00000 −1.11417 −0.557086 0.830455i \(-0.688081\pi\)
−0.557086 + 0.830455i \(0.688081\pi\)
\(30\) 0 0
\(31\) 2.00000 0.359211 0.179605 0.983739i \(-0.442518\pi\)
0.179605 + 0.983739i \(0.442518\pi\)
\(32\) 1.00000i 0.176777i
\(33\) 0 0
\(34\) −6.00000 −1.02899
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) − 4.00000i − 0.657596i −0.944400 0.328798i \(-0.893356\pi\)
0.944400 0.328798i \(-0.106644\pi\)
\(38\) − 1.00000i − 0.162221i
\(39\) 4.00000 0.640513
\(40\) 0 0
\(41\) 6.00000 0.937043 0.468521 0.883452i \(-0.344787\pi\)
0.468521 + 0.883452i \(0.344787\pi\)
\(42\) − 4.00000i − 0.617213i
\(43\) 4.00000i 0.609994i 0.952353 + 0.304997i \(0.0986555\pi\)
−0.952353 + 0.304997i \(0.901344\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) −6.00000 −0.884652
\(47\) 6.00000i 0.875190i 0.899172 + 0.437595i \(0.144170\pi\)
−0.899172 + 0.437595i \(0.855830\pi\)
\(48\) − 1.00000i − 0.144338i
\(49\) −9.00000 −1.28571
\(50\) 0 0
\(51\) 6.00000 0.840168
\(52\) − 4.00000i − 0.554700i
\(53\) − 6.00000i − 0.824163i −0.911147 0.412082i \(-0.864802\pi\)
0.911147 0.412082i \(-0.135198\pi\)
\(54\) −1.00000 −0.136083
\(55\) 0 0
\(56\) −4.00000 −0.534522
\(57\) 1.00000i 0.132453i
\(58\) − 6.00000i − 0.787839i
\(59\) 12.0000 1.56227 0.781133 0.624364i \(-0.214642\pi\)
0.781133 + 0.624364i \(0.214642\pi\)
\(60\) 0 0
\(61\) 14.0000 1.79252 0.896258 0.443533i \(-0.146275\pi\)
0.896258 + 0.443533i \(0.146275\pi\)
\(62\) 2.00000i 0.254000i
\(63\) 4.00000i 0.503953i
\(64\) −1.00000 −0.125000
\(65\) 0 0
\(66\) 0 0
\(67\) 8.00000i 0.977356i 0.872464 + 0.488678i \(0.162521\pi\)
−0.872464 + 0.488678i \(0.837479\pi\)
\(68\) − 6.00000i − 0.727607i
\(69\) 6.00000 0.722315
\(70\) 0 0
\(71\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 1.00000i 0.117851i
\(73\) − 14.0000i − 1.63858i −0.573382 0.819288i \(-0.694369\pi\)
0.573382 0.819288i \(-0.305631\pi\)
\(74\) 4.00000 0.464991
\(75\) 0 0
\(76\) 1.00000 0.114708
\(77\) 0 0
\(78\) 4.00000i 0.452911i
\(79\) 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
\(82\) 6.00000i 0.662589i
\(83\) 12.0000i 1.31717i 0.752506 + 0.658586i \(0.228845\pi\)
−0.752506 + 0.658586i \(0.771155\pi\)
\(84\) 4.00000 0.436436
\(85\) 0 0
\(86\) −4.00000 −0.431331
\(87\) 6.00000i 0.643268i
\(88\) 0 0
\(89\) 6.00000 0.635999 0.317999 0.948091i \(-0.396989\pi\)
0.317999 + 0.948091i \(0.396989\pi\)
\(90\) 0 0
\(91\) 16.0000 1.67726
\(92\) − 6.00000i − 0.625543i
\(93\) − 2.00000i − 0.207390i
\(94\) −6.00000 −0.618853
\(95\) 0 0
\(96\) 1.00000 0.102062
\(97\) − 10.0000i − 1.01535i −0.861550 0.507673i \(-0.830506\pi\)
0.861550 0.507673i \(-0.169494\pi\)
\(98\) − 9.00000i − 0.909137i
\(99\) 0 0
\(100\) 0 0
\(101\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(102\) 6.00000i 0.594089i
\(103\) 10.0000i 0.985329i 0.870219 + 0.492665i \(0.163977\pi\)
−0.870219 + 0.492665i \(0.836023\pi\)
\(104\) 4.00000 0.392232
\(105\) 0 0
\(106\) 6.00000 0.582772
\(107\) 12.0000i 1.16008i 0.814587 + 0.580042i \(0.196964\pi\)
−0.814587 + 0.580042i \(0.803036\pi\)
\(108\) − 1.00000i − 0.0962250i
\(109\) 16.0000 1.53252 0.766261 0.642529i \(-0.222115\pi\)
0.766261 + 0.642529i \(0.222115\pi\)
\(110\) 0 0
\(111\) −4.00000 −0.379663
\(112\) − 4.00000i − 0.377964i
\(113\) 18.0000i 1.69330i 0.532152 + 0.846649i \(0.321383\pi\)
−0.532152 + 0.846649i \(0.678617\pi\)
\(114\) −1.00000 −0.0936586
\(115\) 0 0
\(116\) 6.00000 0.557086
\(117\) − 4.00000i − 0.369800i
\(118\) 12.0000i 1.10469i
\(119\) 24.0000 2.20008
\(120\) 0 0
\(121\) −11.0000 −1.00000
\(122\) 14.0000i 1.26750i
\(123\) − 6.00000i − 0.541002i
\(124\) −2.00000 −0.179605
\(125\) 0 0
\(126\) −4.00000 −0.356348
\(127\) 2.00000i 0.177471i 0.996055 + 0.0887357i \(0.0282826\pi\)
−0.996055 + 0.0887357i \(0.971717\pi\)
\(128\) − 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 \(0.324356\pi\)
\(132\) 0 0
\(133\) 4.00000i 0.346844i
\(134\) −8.00000 −0.691095
\(135\) 0 0
\(136\) 6.00000 0.514496
\(137\) − 18.0000i − 1.53784i −0.639343 0.768922i \(-0.720793\pi\)
0.639343 0.768922i \(-0.279207\pi\)
\(138\) 6.00000i 0.510754i
\(139\) −20.0000 −1.69638 −0.848189 0.529694i \(-0.822307\pi\)
−0.848189 + 0.529694i \(0.822307\pi\)
\(140\) 0 0
\(141\) 6.00000 0.505291
\(142\) 0 0
\(143\) 0 0
\(144\) −1.00000 −0.0833333
\(145\) 0 0
\(146\) 14.0000 1.15865
\(147\) 9.00000i 0.742307i
\(148\) 4.00000i 0.328798i
\(149\) −12.0000 −0.983078 −0.491539 0.870855i \(-0.663566\pi\)
−0.491539 + 0.870855i \(0.663566\pi\)
\(150\) 0 0
\(151\) −10.0000 −0.813788 −0.406894 0.913475i \(-0.633388\pi\)
−0.406894 + 0.913475i \(0.633388\pi\)
\(152\) 1.00000i 0.0811107i
\(153\) − 6.00000i − 0.485071i
\(154\) 0 0
\(155\) 0 0
\(156\) −4.00000 −0.320256
\(157\) 2.00000i 0.159617i 0.996810 + 0.0798087i \(0.0254309\pi\)
−0.996810 + 0.0798087i \(0.974569\pi\)
\(158\) 10.0000i 0.795557i
\(159\) −6.00000 −0.475831
\(160\) 0 0
\(161\) 24.0000 1.89146
\(162\) 1.00000i 0.0785674i
\(163\) 4.00000i 0.313304i 0.987654 + 0.156652i \(0.0500701\pi\)
−0.987654 + 0.156652i \(0.949930\pi\)
\(164\) −6.00000 −0.468521
\(165\) 0 0
\(166\) −12.0000 −0.931381
\(167\) − 12.0000i − 0.928588i −0.885681 0.464294i \(-0.846308\pi\)
0.885681 0.464294i \(-0.153692\pi\)
\(168\) 4.00000i 0.308607i
\(169\) −3.00000 −0.230769
\(170\) 0 0
\(171\) 1.00000 0.0764719
\(172\) − 4.00000i − 0.304997i
\(173\) − 18.0000i − 1.36851i −0.729241 0.684257i \(-0.760127\pi\)
0.729241 0.684257i \(-0.239873\pi\)
\(174\) −6.00000 −0.454859
\(175\) 0 0
\(176\) 0 0
\(177\) − 12.0000i − 0.901975i
\(178\) 6.00000i 0.449719i
\(179\) −12.0000 −0.896922 −0.448461 0.893802i \(-0.648028\pi\)
−0.448461 + 0.893802i \(0.648028\pi\)
\(180\) 0 0
\(181\) −16.0000 −1.18927 −0.594635 0.803996i \(-0.702704\pi\)
−0.594635 + 0.803996i \(0.702704\pi\)
\(182\) 16.0000i 1.18600i
\(183\) − 14.0000i − 1.03491i
\(184\) 6.00000 0.442326
\(185\) 0 0
\(186\) 2.00000 0.146647
\(187\) 0 0
\(188\) − 6.00000i − 0.437595i
\(189\) 4.00000 0.290957
\(190\) 0 0
\(191\) 18.0000 1.30243 0.651217 0.758891i \(-0.274259\pi\)
0.651217 + 0.758891i \(0.274259\pi\)
\(192\) 1.00000i 0.0721688i
\(193\) 10.0000i 0.719816i 0.932988 + 0.359908i \(0.117192\pi\)
−0.932988 + 0.359908i \(0.882808\pi\)
\(194\) 10.0000 0.717958
\(195\) 0 0
\(196\) 9.00000 0.642857
\(197\) 12.0000i 0.854965i 0.904024 + 0.427482i \(0.140599\pi\)
−0.904024 + 0.427482i \(0.859401\pi\)
\(198\) 0 0
\(199\) −20.0000 −1.41776 −0.708881 0.705328i \(-0.750800\pi\)
−0.708881 + 0.705328i \(0.750800\pi\)
\(200\) 0 0
\(201\) 8.00000 0.564276
\(202\) 0 0
\(203\) 24.0000i 1.68447i
\(204\) −6.00000 −0.420084
\(205\) 0 0
\(206\) −10.0000 −0.696733
\(207\) − 6.00000i − 0.417029i
\(208\) 4.00000i 0.277350i
\(209\) 0 0
\(210\) 0 0
\(211\) −4.00000 −0.275371 −0.137686 0.990476i \(-0.543966\pi\)
−0.137686 + 0.990476i \(0.543966\pi\)
\(212\) 6.00000i 0.412082i
\(213\) 0 0
\(214\) −12.0000 −0.820303
\(215\) 0 0
\(216\) 1.00000 0.0680414
\(217\) − 8.00000i − 0.543075i
\(218\) 16.0000i 1.08366i
\(219\) −14.0000 −0.946032
\(220\) 0 0
\(221\) −24.0000 −1.61441
\(222\) − 4.00000i − 0.268462i
\(223\) 22.0000i 1.47323i 0.676313 + 0.736614i \(0.263577\pi\)
−0.676313 + 0.736614i \(0.736423\pi\)
\(224\) 4.00000 0.267261
\(225\) 0 0
\(226\) −18.0000 −1.19734
\(227\) 12.0000i 0.796468i 0.917284 + 0.398234i \(0.130377\pi\)
−0.917284 + 0.398234i \(0.869623\pi\)
\(228\) − 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
\(231\) 0 0
\(232\) 6.00000i 0.393919i
\(233\) 6.00000i 0.393073i 0.980497 + 0.196537i \(0.0629694\pi\)
−0.980497 + 0.196537i \(0.937031\pi\)
\(234\) 4.00000 0.261488
\(235\) 0 0
\(236\) −12.0000 −0.781133
\(237\) − 10.0000i − 0.649570i
\(238\) 24.0000i 1.55569i
\(239\) 18.0000 1.16432 0.582162 0.813073i \(-0.302207\pi\)
0.582162 + 0.813073i \(0.302207\pi\)
\(240\) 0 0
\(241\) 14.0000 0.901819 0.450910 0.892570i \(-0.351100\pi\)
0.450910 + 0.892570i \(0.351100\pi\)
\(242\) − 11.0000i − 0.707107i
\(243\) − 1.00000i − 0.0641500i
\(244\) −14.0000 −0.896258
\(245\) 0 0
\(246\) 6.00000 0.382546
\(247\) − 4.00000i − 0.254514i
\(248\) − 2.00000i − 0.127000i
\(249\) 12.0000 0.760469
\(250\) 0 0
\(251\) 12.0000 0.757433 0.378717 0.925513i \(-0.376365\pi\)
0.378717 + 0.925513i \(0.376365\pi\)
\(252\) − 4.00000i − 0.251976i
\(253\) 0 0
\(254\) −2.00000 −0.125491
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 18.0000i 1.12281i 0.827541 + 0.561405i \(0.189739\pi\)
−0.827541 + 0.561405i \(0.810261\pi\)
\(258\) 4.00000i 0.249029i
\(259\) −16.0000 −0.994192
\(260\) 0 0
\(261\) 6.00000 0.371391
\(262\) 12.0000i 0.741362i
\(263\) 18.0000i 1.10993i 0.831875 + 0.554964i \(0.187268\pi\)
−0.831875 + 0.554964i \(0.812732\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) −4.00000 −0.245256
\(267\) − 6.00000i − 0.367194i
\(268\) − 8.00000i − 0.488678i
\(269\) 6.00000 0.365826 0.182913 0.983129i \(-0.441447\pi\)
0.182913 + 0.983129i \(0.441447\pi\)
\(270\) 0 0
\(271\) 20.0000 1.21491 0.607457 0.794353i \(-0.292190\pi\)
0.607457 + 0.794353i \(0.292190\pi\)
\(272\) 6.00000i 0.363803i
\(273\) − 16.0000i − 0.968364i
\(274\) 18.0000 1.08742
\(275\) 0 0
\(276\) −6.00000 −0.361158
\(277\) − 10.0000i − 0.600842i −0.953807 0.300421i \(-0.902873\pi\)
0.953807 0.300421i \(-0.0971271\pi\)
\(278\) − 20.0000i − 1.19952i
\(279\) −2.00000 −0.119737
\(280\) 0 0
\(281\) −18.0000 −1.07379 −0.536895 0.843649i \(-0.680403\pi\)
−0.536895 + 0.843649i \(0.680403\pi\)
\(282\) 6.00000i 0.357295i
\(283\) 4.00000i 0.237775i 0.992908 + 0.118888i \(0.0379328\pi\)
−0.992908 + 0.118888i \(0.962067\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) − 24.0000i − 1.41668i
\(288\) − 1.00000i − 0.0589256i
\(289\) −19.0000 −1.11765
\(290\) 0 0
\(291\) −10.0000 −0.586210
\(292\) 14.0000i 0.819288i
\(293\) 6.00000i 0.350524i 0.984522 + 0.175262i \(0.0560772\pi\)
−0.984522 + 0.175262i \(0.943923\pi\)
\(294\) −9.00000 −0.524891
\(295\) 0 0
\(296\) −4.00000 −0.232495
\(297\) 0 0
\(298\) − 12.0000i − 0.695141i
\(299\) −24.0000 −1.38796
\(300\) 0 0
\(301\) 16.0000 0.922225
\(302\) − 10.0000i − 0.575435i
\(303\) 0 0
\(304\) −1.00000 −0.0573539
\(305\) 0 0
\(306\) 6.00000 0.342997
\(307\) − 16.0000i − 0.913168i −0.889680 0.456584i \(-0.849073\pi\)
0.889680 0.456584i \(-0.150927\pi\)
\(308\) 0 0
\(309\) 10.0000 0.568880
\(310\) 0 0
\(311\) 30.0000 1.70114 0.850572 0.525859i \(-0.176256\pi\)
0.850572 + 0.525859i \(0.176256\pi\)
\(312\) − 4.00000i − 0.226455i
\(313\) 10.0000i 0.565233i 0.959233 + 0.282617i \(0.0912024\pi\)
−0.959233 + 0.282617i \(0.908798\pi\)
\(314\) −2.00000 −0.112867
\(315\) 0 0
\(316\) −10.0000 −0.562544
\(317\) 18.0000i 1.01098i 0.862832 + 0.505490i \(0.168688\pi\)
−0.862832 + 0.505490i \(0.831312\pi\)
\(318\) − 6.00000i − 0.336463i
\(319\) 0 0
\(320\) 0 0
\(321\) 12.0000 0.669775
\(322\) 24.0000i 1.33747i
\(323\) − 6.00000i − 0.333849i
\(324\) −1.00000 −0.0555556
\(325\) 0 0
\(326\) −4.00000 −0.221540
\(327\) − 16.0000i − 0.884802i
\(328\) − 6.00000i − 0.331295i
\(329\) 24.0000 1.32316
\(330\) 0 0
\(331\) 8.00000 0.439720 0.219860 0.975531i \(-0.429440\pi\)
0.219860 + 0.975531i \(0.429440\pi\)
\(332\) − 12.0000i − 0.658586i
\(333\) 4.00000i 0.219199i
\(334\) 12.0000 0.656611
\(335\) 0 0
\(336\) −4.00000 −0.218218
\(337\) 2.00000i 0.108947i 0.998515 + 0.0544735i \(0.0173480\pi\)
−0.998515 + 0.0544735i \(0.982652\pi\)
\(338\) − 3.00000i − 0.163178i
\(339\) 18.0000 0.977626
\(340\) 0 0
\(341\) 0 0
\(342\) 1.00000i 0.0540738i
\(343\) 8.00000i 0.431959i
\(344\) 4.00000 0.215666
\(345\) 0 0
\(346\) 18.0000 0.967686
\(347\) 24.0000i 1.28839i 0.764862 + 0.644194i \(0.222807\pi\)
−0.764862 + 0.644194i \(0.777193\pi\)
\(348\) − 6.00000i − 0.321634i
\(349\) −2.00000 −0.107058 −0.0535288 0.998566i \(-0.517047\pi\)
−0.0535288 + 0.998566i \(0.517047\pi\)
\(350\) 0 0
\(351\) −4.00000 −0.213504
\(352\) 0 0
\(353\) 6.00000i 0.319348i 0.987170 + 0.159674i \(0.0510443\pi\)
−0.987170 + 0.159674i \(0.948956\pi\)
\(354\) 12.0000 0.637793
\(355\) 0 0
\(356\) −6.00000 −0.317999
\(357\) − 24.0000i − 1.27021i
\(358\) − 12.0000i − 0.634220i
\(359\) 6.00000 0.316668 0.158334 0.987386i \(-0.449388\pi\)
0.158334 + 0.987386i \(0.449388\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) − 16.0000i − 0.840941i
\(363\) 11.0000i 0.577350i
\(364\) −16.0000 −0.838628
\(365\) 0 0
\(366\) 14.0000 0.731792
\(367\) − 4.00000i − 0.208798i −0.994535 0.104399i \(-0.966708\pi\)
0.994535 0.104399i \(-0.0332919\pi\)
\(368\) 6.00000i 0.312772i
\(369\) −6.00000 −0.312348
\(370\) 0 0
\(371\) −24.0000 −1.24602
\(372\) 2.00000i 0.103695i
\(373\) 4.00000i 0.207112i 0.994624 + 0.103556i \(0.0330221\pi\)
−0.994624 + 0.103556i \(0.966978\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 6.00000 0.309426
\(377\) − 24.0000i − 1.23606i
\(378\) 4.00000i 0.205738i
\(379\) −8.00000 −0.410932 −0.205466 0.978664i \(-0.565871\pi\)
−0.205466 + 0.978664i \(0.565871\pi\)
\(380\) 0 0
\(381\) 2.00000 0.102463
\(382\) 18.0000i 0.920960i
\(383\) − 24.0000i − 1.22634i −0.789950 0.613171i \(-0.789894\pi\)
0.789950 0.613171i \(-0.210106\pi\)
\(384\) −1.00000 −0.0510310
\(385\) 0 0
\(386\) −10.0000 −0.508987
\(387\) − 4.00000i − 0.203331i
\(388\) 10.0000i 0.507673i
\(389\) −24.0000 −1.21685 −0.608424 0.793612i \(-0.708198\pi\)
−0.608424 + 0.793612i \(0.708198\pi\)
\(390\) 0 0
\(391\) −36.0000 −1.82060
\(392\) 9.00000i 0.454569i
\(393\) − 12.0000i − 0.605320i
\(394\) −12.0000 −0.604551
\(395\) 0 0
\(396\) 0 0
\(397\) 26.0000i 1.30490i 0.757831 + 0.652451i \(0.226259\pi\)
−0.757831 + 0.652451i \(0.773741\pi\)
\(398\) − 20.0000i − 1.00251i
\(399\) 4.00000 0.200250
\(400\) 0 0
\(401\) 30.0000 1.49813 0.749064 0.662497i \(-0.230503\pi\)
0.749064 + 0.662497i \(0.230503\pi\)
\(402\) 8.00000i 0.399004i
\(403\) 8.00000i 0.398508i
\(404\) 0 0
\(405\) 0 0
\(406\) −24.0000 −1.19110
\(407\) 0 0
\(408\) − 6.00000i − 0.297044i
\(409\) −14.0000 −0.692255 −0.346128 0.938187i \(-0.612504\pi\)
−0.346128 + 0.938187i \(0.612504\pi\)
\(410\) 0 0
\(411\) −18.0000 −0.887875
\(412\) − 10.0000i − 0.492665i
\(413\) − 48.0000i − 2.36193i
\(414\) 6.00000 0.294884
\(415\) 0 0
\(416\) −4.00000 −0.196116
\(417\) 20.0000i 0.979404i
\(418\) 0 0
\(419\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(420\) 0 0
\(421\) −28.0000 −1.36464 −0.682318 0.731055i \(-0.739028\pi\)
−0.682318 + 0.731055i \(0.739028\pi\)
\(422\) − 4.00000i − 0.194717i
\(423\) − 6.00000i − 0.291730i
\(424\) −6.00000 −0.291386
\(425\) 0 0
\(426\) 0 0
\(427\) − 56.0000i − 2.71003i
\(428\) − 12.0000i − 0.580042i
\(429\) 0 0
\(430\) 0 0
\(431\) −12.0000 −0.578020 −0.289010 0.957326i \(-0.593326\pi\)
−0.289010 + 0.957326i \(0.593326\pi\)
\(432\) 1.00000i 0.0481125i
\(433\) 10.0000i 0.480569i 0.970702 + 0.240285i \(0.0772408\pi\)
−0.970702 + 0.240285i \(0.922759\pi\)
\(434\) 8.00000 0.384012
\(435\) 0 0
\(436\) −16.0000 −0.766261
\(437\) − 6.00000i − 0.287019i
\(438\) − 14.0000i − 0.668946i
\(439\) −26.0000 −1.24091 −0.620456 0.784241i \(-0.713053\pi\)
−0.620456 + 0.784241i \(0.713053\pi\)
\(440\) 0 0
\(441\) 9.00000 0.428571
\(442\) − 24.0000i − 1.14156i
\(443\) − 24.0000i − 1.14027i −0.821549 0.570137i \(-0.806890\pi\)
0.821549 0.570137i \(-0.193110\pi\)
\(444\) 4.00000 0.189832
\(445\) 0 0
\(446\) −22.0000 −1.04173
\(447\) 12.0000i 0.567581i
\(448\) 4.00000i 0.188982i
\(449\) 30.0000 1.41579 0.707894 0.706319i \(-0.249646\pi\)
0.707894 + 0.706319i \(0.249646\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) − 18.0000i − 0.846649i
\(453\) 10.0000i 0.469841i
\(454\) −12.0000 −0.563188
\(455\) 0 0
\(456\) 1.00000 0.0468293
\(457\) 2.00000i 0.0935561i 0.998905 + 0.0467780i \(0.0148953\pi\)
−0.998905 + 0.0467780i \(0.985105\pi\)
\(458\) 10.0000i 0.467269i
\(459\) −6.00000 −0.280056
\(460\) 0 0
\(461\) 12.0000 0.558896 0.279448 0.960161i \(-0.409849\pi\)
0.279448 + 0.960161i \(0.409849\pi\)
\(462\) 0 0
\(463\) 4.00000i 0.185896i 0.995671 + 0.0929479i \(0.0296290\pi\)
−0.995671 + 0.0929479i \(0.970371\pi\)
\(464\) −6.00000 −0.278543
\(465\) 0 0
\(466\) −6.00000 −0.277945
\(467\) − 36.0000i − 1.66588i −0.553362 0.832941i \(-0.686655\pi\)
0.553362 0.832941i \(-0.313345\pi\)
\(468\) 4.00000i 0.184900i
\(469\) 32.0000 1.47762
\(470\) 0 0
\(471\) 2.00000 0.0921551
\(472\) − 12.0000i − 0.552345i
\(473\) 0 0
\(474\) 10.0000 0.459315
\(475\) 0 0
\(476\) −24.0000 −1.10004
\(477\) 6.00000i 0.274721i
\(478\) 18.0000i 0.823301i
\(479\) 6.00000 0.274147 0.137073 0.990561i \(-0.456230\pi\)
0.137073 + 0.990561i \(0.456230\pi\)
\(480\) 0 0
\(481\) 16.0000 0.729537
\(482\) 14.0000i 0.637683i
\(483\) − 24.0000i − 1.09204i
\(484\) 11.0000 0.500000
\(485\) 0 0
\(486\) 1.00000 0.0453609
\(487\) 38.0000i 1.72194i 0.508652 + 0.860972i \(0.330144\pi\)
−0.508652 + 0.860972i \(0.669856\pi\)
\(488\) − 14.0000i − 0.633750i
\(489\) 4.00000 0.180886
\(490\) 0 0
\(491\) 36.0000 1.62466 0.812329 0.583200i \(-0.198200\pi\)
0.812329 + 0.583200i \(0.198200\pi\)
\(492\) 6.00000i 0.270501i
\(493\) − 36.0000i − 1.62136i
\(494\) 4.00000 0.179969
\(495\) 0 0
\(496\) 2.00000 0.0898027
\(497\) 0 0
\(498\) 12.0000i 0.537733i
\(499\) 4.00000 0.179065 0.0895323 0.995984i \(-0.471463\pi\)
0.0895323 + 0.995984i \(0.471463\pi\)
\(500\) 0 0
\(501\) −12.0000 −0.536120
\(502\) 12.0000i 0.535586i
\(503\) − 6.00000i − 0.267527i −0.991013 0.133763i \(-0.957294\pi\)
0.991013 0.133763i \(-0.0427062\pi\)
\(504\) 4.00000 0.178174
\(505\) 0 0
\(506\) 0 0
\(507\) 3.00000i 0.133235i
\(508\) − 2.00000i − 0.0887357i
\(509\) 6.00000 0.265945 0.132973 0.991120i \(-0.457548\pi\)
0.132973 + 0.991120i \(0.457548\pi\)
\(510\) 0 0
\(511\) −56.0000 −2.47729
\(512\) 1.00000i 0.0441942i
\(513\) − 1.00000i − 0.0441511i
\(514\) −18.0000 −0.793946
\(515\) 0 0
\(516\) −4.00000 −0.176090
\(517\) 0 0
\(518\) − 16.0000i − 0.703000i
\(519\) −18.0000 −0.790112
\(520\) 0 0
\(521\) −30.0000 −1.31432 −0.657162 0.753749i \(-0.728243\pi\)
−0.657162 + 0.753749i \(0.728243\pi\)
\(522\) 6.00000i 0.262613i
\(523\) − 8.00000i − 0.349816i −0.984585 0.174908i \(-0.944037\pi\)
0.984585 0.174908i \(-0.0559627\pi\)
\(524\) −12.0000 −0.524222
\(525\) 0 0
\(526\) −18.0000 −0.784837
\(527\) 12.0000i 0.522728i
\(528\) 0 0
\(529\) −13.0000 −0.565217
\(530\) 0 0
\(531\) −12.0000 −0.520756
\(532\) − 4.00000i − 0.173422i
\(533\) 24.0000i 1.03956i
\(534\) 6.00000 0.259645
\(535\) 0 0
\(536\) 8.00000 0.345547
\(537\) 12.0000i 0.517838i
\(538\) 6.00000i 0.258678i
\(539\) 0 0
\(540\) 0 0
\(541\) −22.0000 −0.945854 −0.472927 0.881102i \(-0.656803\pi\)
−0.472927 + 0.881102i \(0.656803\pi\)
\(542\) 20.0000i 0.859074i
\(543\) 16.0000i 0.686626i
\(544\) −6.00000 −0.257248
\(545\) 0 0
\(546\) 16.0000 0.684737
\(547\) 32.0000i 1.36822i 0.729378 + 0.684111i \(0.239809\pi\)
−0.729378 + 0.684111i \(0.760191\pi\)
\(548\) 18.0000i 0.768922i
\(549\) −14.0000 −0.597505
\(550\) 0 0
\(551\) 6.00000 0.255609
\(552\) − 6.00000i − 0.255377i
\(553\) − 40.0000i − 1.70097i
\(554\) 10.0000 0.424859
\(555\) 0 0
\(556\) 20.0000 0.848189
\(557\) 12.0000i 0.508456i 0.967144 + 0.254228i \(0.0818214\pi\)
−0.967144 + 0.254228i \(0.918179\pi\)
\(558\) − 2.00000i − 0.0846668i
\(559\) −16.0000 −0.676728
\(560\) 0 0
\(561\) 0 0
\(562\) − 18.0000i − 0.759284i
\(563\) 36.0000i 1.51722i 0.651546 + 0.758610i \(0.274121\pi\)
−0.651546 + 0.758610i \(0.725879\pi\)
\(564\) −6.00000 −0.252646
\(565\) 0 0
\(566\) −4.00000 −0.168133
\(567\) − 4.00000i − 0.167984i
\(568\) 0 0
\(569\) −42.0000 −1.76073 −0.880366 0.474295i \(-0.842703\pi\)
−0.880366 + 0.474295i \(0.842703\pi\)
\(570\) 0 0
\(571\) −28.0000 −1.17176 −0.585882 0.810397i \(-0.699252\pi\)
−0.585882 + 0.810397i \(0.699252\pi\)
\(572\) 0 0
\(573\) − 18.0000i − 0.751961i
\(574\) 24.0000 1.00174
\(575\) 0 0
\(576\) 1.00000 0.0416667
\(577\) 2.00000i 0.0832611i 0.999133 + 0.0416305i \(0.0132552\pi\)
−0.999133 + 0.0416305i \(0.986745\pi\)
\(578\) − 19.0000i − 0.790296i
\(579\) 10.0000 0.415586
\(580\) 0 0
\(581\) 48.0000 1.99138
\(582\) − 10.0000i − 0.414513i
\(583\) 0 0
\(584\) −14.0000 −0.579324
\(585\) 0 0
\(586\) −6.00000 −0.247858
\(587\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(588\) − 9.00000i − 0.371154i
\(589\) −2.00000 −0.0824086
\(590\) 0 0
\(591\) 12.0000 0.493614
\(592\) − 4.00000i − 0.164399i
\(593\) 6.00000i 0.246390i 0.992382 + 0.123195i \(0.0393141\pi\)
−0.992382 + 0.123195i \(0.960686\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 12.0000 0.491539
\(597\) 20.0000i 0.818546i
\(598\) − 24.0000i − 0.981433i
\(599\) 12.0000 0.490307 0.245153 0.969484i \(-0.421162\pi\)
0.245153 + 0.969484i \(0.421162\pi\)
\(600\) 0 0
\(601\) −22.0000 −0.897399 −0.448699 0.893683i \(-0.648113\pi\)
−0.448699 + 0.893683i \(0.648113\pi\)
\(602\) 16.0000i 0.652111i
\(603\) − 8.00000i − 0.325785i
\(604\) 10.0000 0.406894
\(605\) 0 0
\(606\) 0 0
\(607\) − 34.0000i − 1.38002i −0.723801 0.690009i \(-0.757607\pi\)
0.723801 0.690009i \(-0.242393\pi\)
\(608\) − 1.00000i − 0.0405554i
\(609\) 24.0000 0.972529
\(610\) 0 0
\(611\) −24.0000 −0.970936
\(612\) 6.00000i 0.242536i
\(613\) − 2.00000i − 0.0807792i −0.999184 0.0403896i \(-0.987140\pi\)
0.999184 0.0403896i \(-0.0128599\pi\)
\(614\) 16.0000 0.645707
\(615\) 0 0
\(616\) 0 0
\(617\) − 30.0000i − 1.20775i −0.797077 0.603877i \(-0.793622\pi\)
0.797077 0.603877i \(-0.206378\pi\)
\(618\) 10.0000i 0.402259i
\(619\) 4.00000 0.160774 0.0803868 0.996764i \(-0.474384\pi\)
0.0803868 + 0.996764i \(0.474384\pi\)
\(620\) 0 0
\(621\) −6.00000 −0.240772
\(622\) 30.0000i 1.20289i
\(623\) − 24.0000i − 0.961540i
\(624\) 4.00000 0.160128
\(625\) 0 0
\(626\) −10.0000 −0.399680
\(627\) 0 0
\(628\) − 2.00000i − 0.0798087i
\(629\) 24.0000 0.956943
\(630\) 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
\(633\) 4.00000i 0.158986i
\(634\) −18.0000 −0.714871
\(635\) 0 0
\(636\) 6.00000 0.237915
\(637\) − 36.0000i − 1.42637i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −6.00000 −0.236986 −0.118493 0.992955i \(-0.537806\pi\)
−0.118493 + 0.992955i \(0.537806\pi\)
\(642\) 12.0000i 0.473602i
\(643\) 4.00000i 0.157745i 0.996885 + 0.0788723i \(0.0251319\pi\)
−0.996885 + 0.0788723i \(0.974868\pi\)
\(644\) −24.0000 −0.945732
\(645\) 0 0
\(646\) 6.00000 0.236067
\(647\) − 6.00000i − 0.235884i −0.993020 0.117942i \(-0.962370\pi\)
0.993020 0.117942i \(-0.0376297\pi\)
\(648\) − 1.00000i − 0.0392837i
\(649\) 0 0
\(650\) 0 0
\(651\) −8.00000 −0.313545
\(652\) − 4.00000i − 0.156652i
\(653\) − 36.0000i − 1.40879i −0.709809 0.704394i \(-0.751219\pi\)
0.709809 0.704394i \(-0.248781\pi\)
\(654\) 16.0000 0.625650
\(655\) 0 0
\(656\) 6.00000 0.234261
\(657\) 14.0000i 0.546192i
\(658\) 24.0000i 0.935617i
\(659\) 12.0000 0.467454 0.233727 0.972302i \(-0.424908\pi\)
0.233727 + 0.972302i \(0.424908\pi\)
\(660\) 0 0
\(661\) −4.00000 −0.155582 −0.0777910 0.996970i \(-0.524787\pi\)
−0.0777910 + 0.996970i \(0.524787\pi\)
\(662\) 8.00000i 0.310929i
\(663\) 24.0000i 0.932083i
\(664\) 12.0000 0.465690
\(665\) 0 0
\(666\) −4.00000 −0.154997
\(667\) − 36.0000i − 1.39393i
\(668\) 12.0000i 0.464294i
\(669\) 22.0000 0.850569
\(670\) 0 0
\(671\) 0 0
\(672\) − 4.00000i − 0.154303i
\(673\) 10.0000i 0.385472i 0.981251 + 0.192736i \(0.0617360\pi\)
−0.981251 + 0.192736i \(0.938264\pi\)
\(674\) −2.00000 −0.0770371
\(675\) 0 0
\(676\) 3.00000 0.115385
\(677\) − 18.0000i − 0.691796i −0.938272 0.345898i \(-0.887574\pi\)
0.938272 0.345898i \(-0.112426\pi\)
\(678\) 18.0000i 0.691286i
\(679\) −40.0000 −1.53506
\(680\) 0 0
\(681\) 12.0000 0.459841
\(682\) 0 0
\(683\) 12.0000i 0.459167i 0.973289 + 0.229584i \(0.0737364\pi\)
−0.973289 + 0.229584i \(0.926264\pi\)
\(684\) −1.00000 −0.0382360
\(685\) 0 0
\(686\) −8.00000 −0.305441
\(687\) − 10.0000i − 0.381524i
\(688\) 4.00000i 0.152499i
\(689\) 24.0000 0.914327
\(690\) 0 0
\(691\) −28.0000 −1.06517 −0.532585 0.846376i \(-0.678779\pi\)
−0.532585 + 0.846376i \(0.678779\pi\)
\(692\) 18.0000i 0.684257i
\(693\) 0 0
\(694\) −24.0000 −0.911028
\(695\) 0 0
\(696\) 6.00000 0.227429
\(697\) 36.0000i 1.36360i
\(698\) − 2.00000i − 0.0757011i
\(699\) 6.00000 0.226941
\(700\) 0 0
\(701\) −48.0000 −1.81293 −0.906467 0.422276i \(-0.861231\pi\)
−0.906467 + 0.422276i \(0.861231\pi\)
\(702\) − 4.00000i − 0.150970i
\(703\) 4.00000i 0.150863i
\(704\) 0 0
\(705\) 0 0
\(706\) −6.00000 −0.225813
\(707\) 0 0
\(708\) 12.0000i 0.450988i
\(709\) 10.0000 0.375558 0.187779 0.982211i \(-0.439871\pi\)
0.187779 + 0.982211i \(0.439871\pi\)
\(710\) 0 0
\(711\) −10.0000 −0.375029
\(712\) − 6.00000i − 0.224860i
\(713\) 12.0000i 0.449404i
\(714\) 24.0000 0.898177
\(715\) 0 0
\(716\) 12.0000 0.448461
\(717\) − 18.0000i − 0.672222i
\(718\) 6.00000i 0.223918i
\(719\) 18.0000 0.671287 0.335643 0.941989i \(-0.391046\pi\)
0.335643 + 0.941989i \(0.391046\pi\)
\(720\) 0 0
\(721\) 40.0000 1.48968
\(722\) 1.00000i 0.0372161i
\(723\) − 14.0000i − 0.520666i
\(724\) 16.0000 0.594635
\(725\) 0 0
\(726\) −11.0000 −0.408248
\(727\) 8.00000i 0.296704i 0.988935 + 0.148352i \(0.0473968\pi\)
−0.988935 + 0.148352i \(0.952603\pi\)
\(728\) − 16.0000i − 0.592999i
\(729\) −1.00000 −0.0370370
\(730\) 0 0
\(731\) −24.0000 −0.887672
\(732\) 14.0000i 0.517455i
\(733\) − 50.0000i − 1.84679i −0.383849 0.923396i \(-0.625402\pi\)
0.383849 0.923396i \(-0.374598\pi\)
\(734\) 4.00000 0.147643
\(735\) 0 0
\(736\) −6.00000 −0.221163
\(737\) 0 0
\(738\) − 6.00000i − 0.220863i
\(739\) 4.00000 0.147142 0.0735712 0.997290i \(-0.476560\pi\)
0.0735712 + 0.997290i \(0.476560\pi\)
\(740\) 0 0
\(741\) −4.00000 −0.146944
\(742\) − 24.0000i − 0.881068i
\(743\) 12.0000i 0.440237i 0.975473 + 0.220119i \(0.0706445\pi\)
−0.975473 + 0.220119i \(0.929356\pi\)
\(744\) −2.00000 −0.0733236
\(745\) 0 0
\(746\) −4.00000 −0.146450
\(747\) − 12.0000i − 0.439057i
\(748\) 0 0
\(749\) 48.0000 1.75388
\(750\) 0 0
\(751\) −10.0000 −0.364905 −0.182453 0.983215i \(-0.558404\pi\)
−0.182453 + 0.983215i \(0.558404\pi\)
\(752\) 6.00000i 0.218797i
\(753\) − 12.0000i − 0.437304i
\(754\) 24.0000 0.874028
\(755\) 0 0
\(756\) −4.00000 −0.145479
\(757\) − 46.0000i − 1.67190i −0.548807 0.835949i \(-0.684918\pi\)
0.548807 0.835949i \(-0.315082\pi\)
\(758\) − 8.00000i − 0.290573i
\(759\) 0 0
\(760\) 0 0
\(761\) 6.00000 0.217500 0.108750 0.994069i \(-0.465315\pi\)
0.108750 + 0.994069i \(0.465315\pi\)
\(762\) 2.00000i 0.0724524i
\(763\) − 64.0000i − 2.31696i
\(764\) −18.0000 −0.651217
\(765\) 0 0
\(766\) 24.0000 0.867155
\(767\) 48.0000i 1.73318i
\(768\) − 1.00000i − 0.0360844i
\(769\) −14.0000 −0.504853 −0.252426 0.967616i \(-0.581229\pi\)
−0.252426 + 0.967616i \(0.581229\pi\)
\(770\) 0 0
\(771\) 18.0000 0.648254
\(772\) − 10.0000i − 0.359908i
\(773\) 6.00000i 0.215805i 0.994161 + 0.107903i \(0.0344134\pi\)
−0.994161 + 0.107903i \(0.965587\pi\)
\(774\) 4.00000 0.143777
\(775\) 0 0
\(776\) −10.0000 −0.358979
\(777\) 16.0000i 0.573997i
\(778\) − 24.0000i − 0.860442i
\(779\) −6.00000 −0.214972
\(780\) 0 0
\(781\) 0 0
\(782\) − 36.0000i − 1.28736i
\(783\) − 6.00000i − 0.214423i
\(784\) −9.00000 −0.321429
\(785\) 0 0
\(786\) 12.0000 0.428026
\(787\) 20.0000i 0.712923i 0.934310 + 0.356462i \(0.116017\pi\)
−0.934310 + 0.356462i \(0.883983\pi\)
\(788\) − 12.0000i − 0.427482i
\(789\) 18.0000 0.640817
\(790\) 0 0
\(791\) 72.0000 2.56003
\(792\) 0 0
\(793\) 56.0000i 1.98862i
\(794\) −26.0000 −0.922705
\(795\) 0 0
\(796\) 20.0000 0.708881
\(797\) 18.0000i 0.637593i 0.947823 + 0.318796i \(0.103279\pi\)
−0.947823 + 0.318796i \(0.896721\pi\)
\(798\) 4.00000i 0.141598i
\(799\) −36.0000 −1.27359
\(800\) 0 0
\(801\) −6.00000 −0.212000
\(802\) 30.0000i 1.05934i
\(803\) 0 0
\(804\) −8.00000 −0.282138
\(805\) 0 0
\(806\) −8.00000 −0.281788
\(807\) − 6.00000i − 0.211210i
\(808\) 0 0
\(809\) 42.0000 1.47664 0.738321 0.674450i \(-0.235619\pi\)
0.738321 + 0.674450i \(0.235619\pi\)
\(810\) 0 0
\(811\) −16.0000 −0.561836 −0.280918 0.959732i \(-0.590639\pi\)
−0.280918 + 0.959732i \(0.590639\pi\)
\(812\) − 24.0000i − 0.842235i
\(813\) − 20.0000i − 0.701431i
\(814\) 0 0
\(815\) 0 0
\(816\) 6.00000 0.210042
\(817\) − 4.00000i − 0.139942i
\(818\) − 14.0000i − 0.489499i
\(819\) −16.0000 −0.559085
\(820\) 0 0
\(821\) −24.0000 −0.837606 −0.418803 0.908077i \(-0.637550\pi\)
−0.418803 + 0.908077i \(0.637550\pi\)
\(822\) − 18.0000i − 0.627822i
\(823\) − 32.0000i − 1.11545i −0.830026 0.557725i \(-0.811674\pi\)
0.830026 0.557725i \(-0.188326\pi\)
\(824\) 10.0000 0.348367
\(825\) 0 0
\(826\) 48.0000 1.67013
\(827\) − 12.0000i − 0.417281i −0.977992 0.208640i \(-0.933096\pi\)
0.977992 0.208640i \(-0.0669038\pi\)
\(828\) 6.00000i 0.208514i
\(829\) 16.0000 0.555703 0.277851 0.960624i \(-0.410378\pi\)
0.277851 + 0.960624i \(0.410378\pi\)
\(830\) 0 0
\(831\) −10.0000 −0.346896
\(832\) − 4.00000i − 0.138675i
\(833\) − 54.0000i − 1.87099i
\(834\) −20.0000 −0.692543
\(835\) 0 0
\(836\) 0 0
\(837\) 2.00000i 0.0691301i
\(838\) 0 0
\(839\) −36.0000 −1.24286 −0.621429 0.783470i \(-0.713448\pi\)
−0.621429 + 0.783470i \(0.713448\pi\)
\(840\) 0 0
\(841\) 7.00000 0.241379
\(842\) − 28.0000i − 0.964944i
\(843\) 18.0000i 0.619953i
\(844\) 4.00000 0.137686
\(845\) 0 0
\(846\) 6.00000 0.206284
\(847\) 44.0000i 1.51186i
\(848\) − 6.00000i − 0.206041i
\(849\) 4.00000 0.137280
\(850\) 0 0
\(851\) 24.0000 0.822709
\(852\) 0 0
\(853\) − 38.0000i − 1.30110i −0.759465 0.650548i \(-0.774539\pi\)
0.759465 0.650548i \(-0.225461\pi\)
\(854\) 56.0000 1.91628
\(855\) 0 0
\(856\) 12.0000 0.410152
\(857\) 18.0000i 0.614868i 0.951569 + 0.307434i \(0.0994704\pi\)
−0.951569 + 0.307434i \(0.900530\pi\)
\(858\) 0 0
\(859\) 28.0000 0.955348 0.477674 0.878537i \(-0.341480\pi\)
0.477674 + 0.878537i \(0.341480\pi\)
\(860\) 0 0
\(861\) −24.0000 −0.817918
\(862\) − 12.0000i − 0.408722i
\(863\) − 12.0000i − 0.408485i −0.978920 0.204242i \(-0.934527\pi\)
0.978920 0.204242i \(-0.0654731\pi\)
\(864\) −1.00000 −0.0340207
\(865\) 0 0
\(866\) −10.0000 −0.339814
\(867\) 19.0000i 0.645274i
\(868\) 8.00000i 0.271538i
\(869\) 0 0
\(870\) 0 0
\(871\) −32.0000 −1.08428
\(872\) − 16.0000i − 0.541828i
\(873\) 10.0000i 0.338449i
\(874\) 6.00000 0.202953
\(875\) 0 0
\(876\) 14.0000 0.473016
\(877\) 44.0000i 1.48577i 0.669417 + 0.742887i \(0.266544\pi\)
−0.669417 + 0.742887i \(0.733456\pi\)
\(878\) − 26.0000i − 0.877457i
\(879\) 6.00000 0.202375
\(880\) 0 0
\(881\) −30.0000 −1.01073 −0.505363 0.862907i \(-0.668641\pi\)
−0.505363 + 0.862907i \(0.668641\pi\)
\(882\) 9.00000i 0.303046i
\(883\) − 20.0000i − 0.673054i −0.941674 0.336527i \(-0.890748\pi\)
0.941674 0.336527i \(-0.109252\pi\)
\(884\) 24.0000 0.807207
\(885\) 0 0
\(886\) 24.0000 0.806296
\(887\) 12.0000i 0.402921i 0.979497 + 0.201460i \(0.0645687\pi\)
−0.979497 + 0.201460i \(0.935431\pi\)
\(888\) 4.00000i 0.134231i
\(889\) 8.00000 0.268311
\(890\) 0 0
\(891\) 0 0
\(892\) − 22.0000i − 0.736614i
\(893\) − 6.00000i − 0.200782i
\(894\) −12.0000 −0.401340
\(895\) 0 0
\(896\) −4.00000 −0.133631
\(897\) 24.0000i 0.801337i
\(898\) 30.0000i 1.00111i
\(899\) −12.0000 −0.400222
\(900\) 0 0
\(901\) 36.0000 1.19933
\(902\) 0 0
\(903\) − 16.0000i − 0.532447i
\(904\) 18.0000 0.598671
\(905\) 0 0
\(906\) −10.0000 −0.332228
\(907\) 8.00000i 0.265636i 0.991140 + 0.132818i \(0.0424025\pi\)
−0.991140 + 0.132818i \(0.957597\pi\)
\(908\) − 12.0000i − 0.398234i
\(909\) 0 0
\(910\) 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
\(913\) 0 0
\(914\) −2.00000 −0.0661541
\(915\) 0 0
\(916\) −10.0000 −0.330409
\(917\) − 48.0000i − 1.58510i
\(918\) − 6.00000i − 0.198030i
\(919\) 16.0000 0.527791 0.263896 0.964551i \(-0.414993\pi\)
0.263896 + 0.964551i \(0.414993\pi\)
\(920\) 0 0
\(921\) −16.0000 −0.527218
\(922\) 12.0000i 0.395199i
\(923\) 0 0
\(924\) 0 0
\(925\) 0 0
\(926\) −4.00000 −0.131448
\(927\) − 10.0000i − 0.328443i
\(928\) − 6.00000i − 0.196960i
\(929\) −6.00000 −0.196854 −0.0984268 0.995144i \(-0.531381\pi\)
−0.0984268 + 0.995144i \(0.531381\pi\)
\(930\) 0 0
\(931\) 9.00000 0.294963
\(932\) − 6.00000i − 0.196537i
\(933\) − 30.0000i − 0.982156i
\(934\) 36.0000 1.17796
\(935\) 0 0
\(936\) −4.00000 −0.130744
\(937\) − 58.0000i − 1.89478i −0.320085 0.947389i \(-0.603712\pi\)
0.320085 0.947389i \(-0.396288\pi\)
\(938\) 32.0000i 1.04484i
\(939\) 10.0000 0.326338
\(940\) 0 0
\(941\) 18.0000 0.586783 0.293392 0.955992i \(-0.405216\pi\)
0.293392 + 0.955992i \(0.405216\pi\)
\(942\) 2.00000i 0.0651635i
\(943\) 36.0000i 1.17232i
\(944\) 12.0000 0.390567
\(945\) 0 0
\(946\) 0 0
\(947\) − 12.0000i − 0.389948i −0.980808 0.194974i \(-0.937538\pi\)
0.980808 0.194974i \(-0.0624622\pi\)
\(948\) 10.0000i 0.324785i
\(949\) 56.0000 1.81784
\(950\) 0 0
\(951\) 18.0000 0.583690
\(952\) − 24.0000i − 0.777844i
\(953\) − 54.0000i − 1.74923i −0.484817 0.874616i \(-0.661114\pi\)
0.484817 0.874616i \(-0.338886\pi\)
\(954\) −6.00000 −0.194257
\(955\) 0 0
\(956\) −18.0000 −0.582162
\(957\) 0 0
\(958\) 6.00000i 0.193851i
\(959\) −72.0000 −2.32500
\(960\) 0 0
\(961\) −27.0000 −0.870968
\(962\) 16.0000i 0.515861i
\(963\) − 12.0000i − 0.386695i
\(964\) −14.0000 −0.450910
\(965\) 0 0
\(966\) 24.0000 0.772187
\(967\) − 28.0000i − 0.900419i −0.892923 0.450210i \(-0.851349\pi\)
0.892923 0.450210i \(-0.148651\pi\)
\(968\) 11.0000i 0.353553i
\(969\) −6.00000 −0.192748
\(970\) 0 0
\(971\) −12.0000 −0.385098 −0.192549 0.981287i \(-0.561675\pi\)
−0.192549 + 0.981287i \(0.561675\pi\)
\(972\) 1.00000i 0.0320750i
\(973\) 80.0000i 2.56468i
\(974\) −38.0000 −1.21760
\(975\) 0 0
\(976\) 14.0000 0.448129
\(977\) − 18.0000i − 0.575871i −0.957650 0.287936i \(-0.907031\pi\)
0.957650 0.287936i \(-0.0929689\pi\)
\(978\) 4.00000i 0.127906i
\(979\) 0 0
\(980\) 0 0
\(981\) −16.0000 −0.510841
\(982\) 36.0000i 1.14881i
\(983\) 48.0000i 1.53096i 0.643458 + 0.765481i \(0.277499\pi\)
−0.643458 + 0.765481i \(0.722501\pi\)
\(984\) −6.00000 −0.191273
\(985\) 0 0
\(986\) 36.0000 1.14647
\(987\) − 24.0000i − 0.763928i
\(988\) 4.00000i 0.127257i
\(989\) −24.0000 −0.763156
\(990\) 0 0
\(991\) 26.0000 0.825917 0.412959 0.910750i \(-0.364495\pi\)
0.412959 + 0.910750i \(0.364495\pi\)
\(992\) 2.00000i 0.0635001i
\(993\) − 8.00000i − 0.253872i
\(994\) 0 0
\(995\) 0 0
\(996\) −12.0000 −0.380235
\(997\) 26.0000i 0.823428i 0.911313 + 0.411714i \(0.135070\pi\)
−0.911313 + 0.411714i \(0.864930\pi\)
\(998\) 4.00000i 0.126618i
\(999\) 4.00000 0.126554
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 2850.2.d.p.799.2 2
5.2 odd 4 2850.2.a.g.1.1 1
5.3 odd 4 114.2.a.c.1.1 1
5.4 even 2 inner 2850.2.d.p.799.1 2
15.2 even 4 8550.2.a.bj.1.1 1
15.8 even 4 342.2.a.c.1.1 1
20.3 even 4 912.2.a.c.1.1 1
35.13 even 4 5586.2.a.u.1.1 1
40.3 even 4 3648.2.a.bc.1.1 1
40.13 odd 4 3648.2.a.i.1.1 1
60.23 odd 4 2736.2.a.o.1.1 1
95.18 even 4 2166.2.a.a.1.1 1
285.113 odd 4 6498.2.a.t.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
114.2.a.c.1.1 1 5.3 odd 4
342.2.a.c.1.1 1 15.8 even 4
912.2.a.c.1.1 1 20.3 even 4
2166.2.a.a.1.1 1 95.18 even 4
2736.2.a.o.1.1 1 60.23 odd 4
2850.2.a.g.1.1 1 5.2 odd 4
2850.2.d.p.799.1 2 5.4 even 2 inner
2850.2.d.p.799.2 2 1.1 even 1 trivial
3648.2.a.i.1.1 1 40.13 odd 4
3648.2.a.bc.1.1 1 40.3 even 4
5586.2.a.u.1.1 1 35.13 even 4
6498.2.a.t.1.1 1 285.113 odd 4
8550.2.a.bj.1.1 1 15.2 even 4