Properties

Label 3850.2.c.m.1849.2
Level $3850$
Weight $2$
Character 3850.1849
Analytic conductor $30.742$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 3850 = 2 \cdot 5^{2} \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3850.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(30.7424047782\)
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 770)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1849.2
Root \(1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 3850.1849
Dual form 3850.2.c.m.1849.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000i q^{2} -1.00000 q^{4} -1.00000i q^{7} -1.00000i q^{8} +3.00000 q^{9} +O(q^{10})\) \(q+1.00000i q^{2} -1.00000 q^{4} -1.00000i q^{7} -1.00000i q^{8} +3.00000 q^{9} -1.00000 q^{11} -6.00000i q^{13} +1.00000 q^{14} +1.00000 q^{16} +2.00000i q^{17} +3.00000i q^{18} +4.00000 q^{19} -1.00000i q^{22} -4.00000i q^{23} +6.00000 q^{26} +1.00000i q^{28} -6.00000 q^{29} +1.00000i q^{32} -2.00000 q^{34} -3.00000 q^{36} +2.00000i q^{37} +4.00000i q^{38} -6.00000 q^{41} -4.00000i q^{43} +1.00000 q^{44} +4.00000 q^{46} -4.00000i q^{47} -1.00000 q^{49} +6.00000i q^{52} -2.00000i q^{53} -1.00000 q^{56} -6.00000i q^{58} -12.0000 q^{59} -2.00000 q^{61} -3.00000i q^{63} -1.00000 q^{64} +8.00000i q^{67} -2.00000i q^{68} -8.00000 q^{71} -3.00000i q^{72} -10.0000i q^{73} -2.00000 q^{74} -4.00000 q^{76} +1.00000i q^{77} +8.00000 q^{79} +9.00000 q^{81} -6.00000i q^{82} -12.0000i q^{83} +4.00000 q^{86} +1.00000i q^{88} -10.0000 q^{89} -6.00000 q^{91} +4.00000i q^{92} +4.00000 q^{94} +6.00000i q^{97} -1.00000i q^{98} -3.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{4} + 6 q^{9} + O(q^{10}) \) \( 2 q - 2 q^{4} + 6 q^{9} - 2 q^{11} + 2 q^{14} + 2 q^{16} + 8 q^{19} + 12 q^{26} - 12 q^{29} - 4 q^{34} - 6 q^{36} - 12 q^{41} + 2 q^{44} + 8 q^{46} - 2 q^{49} - 2 q^{56} - 24 q^{59} - 4 q^{61} - 2 q^{64} - 16 q^{71} - 4 q^{74} - 8 q^{76} + 16 q^{79} + 18 q^{81} + 8 q^{86} - 20 q^{89} - 12 q^{91} + 8 q^{94} - 6 q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3850\mathbb{Z}\right)^\times\).

\(n\) \(1751\) \(2201\) \(2927\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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
\(3\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(4\) −1.00000 −0.500000
\(5\) 0 0
\(6\) 0 0
\(7\) − 1.00000i − 0.377964i
\(8\) − 1.00000i − 0.353553i
\(9\) 3.00000 1.00000
\(10\) 0 0
\(11\) −1.00000 −0.301511
\(12\) 0 0
\(13\) − 6.00000i − 1.66410i −0.554700 0.832050i \(-0.687167\pi\)
0.554700 0.832050i \(-0.312833\pi\)
\(14\) 1.00000 0.267261
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 2.00000i 0.485071i 0.970143 + 0.242536i \(0.0779791\pi\)
−0.970143 + 0.242536i \(0.922021\pi\)
\(18\) 3.00000i 0.707107i
\(19\) 4.00000 0.917663 0.458831 0.888523i \(-0.348268\pi\)
0.458831 + 0.888523i \(0.348268\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) − 1.00000i − 0.213201i
\(23\) − 4.00000i − 0.834058i −0.908893 0.417029i \(-0.863071\pi\)
0.908893 0.417029i \(-0.136929\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 6.00000 1.17670
\(27\) 0 0
\(28\) 1.00000i 0.188982i
\(29\) −6.00000 −1.11417 −0.557086 0.830455i \(-0.688081\pi\)
−0.557086 + 0.830455i \(0.688081\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) 1.00000i 0.176777i
\(33\) 0 0
\(34\) −2.00000 −0.342997
\(35\) 0 0
\(36\) −3.00000 −0.500000
\(37\) 2.00000i 0.328798i 0.986394 + 0.164399i \(0.0525685\pi\)
−0.986394 + 0.164399i \(0.947432\pi\)
\(38\) 4.00000i 0.648886i
\(39\) 0 0
\(40\) 0 0
\(41\) −6.00000 −0.937043 −0.468521 0.883452i \(-0.655213\pi\)
−0.468521 + 0.883452i \(0.655213\pi\)
\(42\) 0 0
\(43\) − 4.00000i − 0.609994i −0.952353 0.304997i \(-0.901344\pi\)
0.952353 0.304997i \(-0.0986555\pi\)
\(44\) 1.00000 0.150756
\(45\) 0 0
\(46\) 4.00000 0.589768
\(47\) − 4.00000i − 0.583460i −0.956501 0.291730i \(-0.905769\pi\)
0.956501 0.291730i \(-0.0942309\pi\)
\(48\) 0 0
\(49\) −1.00000 −0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) 6.00000i 0.832050i
\(53\) − 2.00000i − 0.274721i −0.990521 0.137361i \(-0.956138\pi\)
0.990521 0.137361i \(-0.0438619\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −1.00000 −0.133631
\(57\) 0 0
\(58\) − 6.00000i − 0.787839i
\(59\) −12.0000 −1.56227 −0.781133 0.624364i \(-0.785358\pi\)
−0.781133 + 0.624364i \(0.785358\pi\)
\(60\) 0 0
\(61\) −2.00000 −0.256074 −0.128037 0.991769i \(-0.540868\pi\)
−0.128037 + 0.991769i \(0.540868\pi\)
\(62\) 0 0
\(63\) − 3.00000i − 0.377964i
\(64\) −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 \(0.837479\pi\)
\(68\) − 2.00000i − 0.242536i
\(69\) 0 0
\(70\) 0 0
\(71\) −8.00000 −0.949425 −0.474713 0.880141i \(-0.657448\pi\)
−0.474713 + 0.880141i \(0.657448\pi\)
\(72\) − 3.00000i − 0.353553i
\(73\) − 10.0000i − 1.17041i −0.810885 0.585206i \(-0.801014\pi\)
0.810885 0.585206i \(-0.198986\pi\)
\(74\) −2.00000 −0.232495
\(75\) 0 0
\(76\) −4.00000 −0.458831
\(77\) 1.00000i 0.113961i
\(78\) 0 0
\(79\) 8.00000 0.900070 0.450035 0.893011i \(-0.351411\pi\)
0.450035 + 0.893011i \(0.351411\pi\)
\(80\) 0 0
\(81\) 9.00000 1.00000
\(82\) − 6.00000i − 0.662589i
\(83\) − 12.0000i − 1.31717i −0.752506 0.658586i \(-0.771155\pi\)
0.752506 0.658586i \(-0.228845\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 4.00000 0.431331
\(87\) 0 0
\(88\) 1.00000i 0.106600i
\(89\) −10.0000 −1.06000 −0.529999 0.847998i \(-0.677808\pi\)
−0.529999 + 0.847998i \(0.677808\pi\)
\(90\) 0 0
\(91\) −6.00000 −0.628971
\(92\) 4.00000i 0.417029i
\(93\) 0 0
\(94\) 4.00000 0.412568
\(95\) 0 0
\(96\) 0 0
\(97\) 6.00000i 0.609208i 0.952479 + 0.304604i \(0.0985241\pi\)
−0.952479 + 0.304604i \(0.901476\pi\)
\(98\) − 1.00000i − 0.101015i
\(99\) −3.00000 −0.301511
\(100\) 0 0
\(101\) 14.0000 1.39305 0.696526 0.717532i \(-0.254728\pi\)
0.696526 + 0.717532i \(0.254728\pi\)
\(102\) 0 0
\(103\) − 4.00000i − 0.394132i −0.980390 0.197066i \(-0.936859\pi\)
0.980390 0.197066i \(-0.0631413\pi\)
\(104\) −6.00000 −0.588348
\(105\) 0 0
\(106\) 2.00000 0.194257
\(107\) − 4.00000i − 0.386695i −0.981130 0.193347i \(-0.938066\pi\)
0.981130 0.193347i \(-0.0619344\pi\)
\(108\) 0 0
\(109\) −14.0000 −1.34096 −0.670478 0.741929i \(-0.733911\pi\)
−0.670478 + 0.741929i \(0.733911\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) − 1.00000i − 0.0944911i
\(113\) − 6.00000i − 0.564433i −0.959351 0.282216i \(-0.908930\pi\)
0.959351 0.282216i \(-0.0910696\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 6.00000 0.557086
\(117\) − 18.0000i − 1.66410i
\(118\) − 12.0000i − 1.10469i
\(119\) 2.00000 0.183340
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) − 2.00000i − 0.181071i
\(123\) 0 0
\(124\) 0 0
\(125\) 0 0
\(126\) 3.00000 0.267261
\(127\) 8.00000i 0.709885i 0.934888 + 0.354943i \(0.115500\pi\)
−0.934888 + 0.354943i \(0.884500\pi\)
\(128\) − 1.00000i − 0.0883883i
\(129\) 0 0
\(130\) 0 0
\(131\) −12.0000 −1.04844 −0.524222 0.851581i \(-0.675644\pi\)
−0.524222 + 0.851581i \(0.675644\pi\)
\(132\) 0 0
\(133\) − 4.00000i − 0.346844i
\(134\) −8.00000 −0.691095
\(135\) 0 0
\(136\) 2.00000 0.171499
\(137\) − 18.0000i − 1.53784i −0.639343 0.768922i \(-0.720793\pi\)
0.639343 0.768922i \(-0.279207\pi\)
\(138\) 0 0
\(139\) 20.0000 1.69638 0.848189 0.529694i \(-0.177693\pi\)
0.848189 + 0.529694i \(0.177693\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) − 8.00000i − 0.671345i
\(143\) 6.00000i 0.501745i
\(144\) 3.00000 0.250000
\(145\) 0 0
\(146\) 10.0000 0.827606
\(147\) 0 0
\(148\) − 2.00000i − 0.164399i
\(149\) −6.00000 −0.491539 −0.245770 0.969328i \(-0.579041\pi\)
−0.245770 + 0.969328i \(0.579041\pi\)
\(150\) 0 0
\(151\) 8.00000 0.651031 0.325515 0.945537i \(-0.394462\pi\)
0.325515 + 0.945537i \(0.394462\pi\)
\(152\) − 4.00000i − 0.324443i
\(153\) 6.00000i 0.485071i
\(154\) −1.00000 −0.0805823
\(155\) 0 0
\(156\) 0 0
\(157\) 2.00000i 0.159617i 0.996810 + 0.0798087i \(0.0254309\pi\)
−0.996810 + 0.0798087i \(0.974569\pi\)
\(158\) 8.00000i 0.636446i
\(159\) 0 0
\(160\) 0 0
\(161\) −4.00000 −0.315244
\(162\) 9.00000i 0.707107i
\(163\) 8.00000i 0.626608i 0.949653 + 0.313304i \(0.101436\pi\)
−0.949653 + 0.313304i \(0.898564\pi\)
\(164\) 6.00000 0.468521
\(165\) 0 0
\(166\) 12.0000 0.931381
\(167\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(168\) 0 0
\(169\) −23.0000 −1.76923
\(170\) 0 0
\(171\) 12.0000 0.917663
\(172\) 4.00000i 0.304997i
\(173\) − 14.0000i − 1.06440i −0.846619 0.532200i \(-0.821365\pi\)
0.846619 0.532200i \(-0.178635\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −1.00000 −0.0753778
\(177\) 0 0
\(178\) − 10.0000i − 0.749532i
\(179\) −20.0000 −1.49487 −0.747435 0.664335i \(-0.768715\pi\)
−0.747435 + 0.664335i \(0.768715\pi\)
\(180\) 0 0
\(181\) 6.00000 0.445976 0.222988 0.974821i \(-0.428419\pi\)
0.222988 + 0.974821i \(0.428419\pi\)
\(182\) − 6.00000i − 0.444750i
\(183\) 0 0
\(184\) −4.00000 −0.294884
\(185\) 0 0
\(186\) 0 0
\(187\) − 2.00000i − 0.146254i
\(188\) 4.00000i 0.291730i
\(189\) 0 0
\(190\) 0 0
\(191\) 8.00000 0.578860 0.289430 0.957199i \(-0.406534\pi\)
0.289430 + 0.957199i \(0.406534\pi\)
\(192\) 0 0
\(193\) 6.00000i 0.431889i 0.976406 + 0.215945i \(0.0692831\pi\)
−0.976406 + 0.215945i \(0.930717\pi\)
\(194\) −6.00000 −0.430775
\(195\) 0 0
\(196\) 1.00000 0.0714286
\(197\) − 18.0000i − 1.28245i −0.767354 0.641223i \(-0.778427\pi\)
0.767354 0.641223i \(-0.221573\pi\)
\(198\) − 3.00000i − 0.213201i
\(199\) 24.0000 1.70131 0.850657 0.525720i \(-0.176204\pi\)
0.850657 + 0.525720i \(0.176204\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 14.0000i 0.985037i
\(203\) 6.00000i 0.421117i
\(204\) 0 0
\(205\) 0 0
\(206\) 4.00000 0.278693
\(207\) − 12.0000i − 0.834058i
\(208\) − 6.00000i − 0.416025i
\(209\) −4.00000 −0.276686
\(210\) 0 0
\(211\) 4.00000 0.275371 0.137686 0.990476i \(-0.456034\pi\)
0.137686 + 0.990476i \(0.456034\pi\)
\(212\) 2.00000i 0.137361i
\(213\) 0 0
\(214\) 4.00000 0.273434
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) − 14.0000i − 0.948200i
\(219\) 0 0
\(220\) 0 0
\(221\) 12.0000 0.807207
\(222\) 0 0
\(223\) − 4.00000i − 0.267860i −0.990991 0.133930i \(-0.957240\pi\)
0.990991 0.133930i \(-0.0427597\pi\)
\(224\) 1.00000 0.0668153
\(225\) 0 0
\(226\) 6.00000 0.399114
\(227\) − 4.00000i − 0.265489i −0.991150 0.132745i \(-0.957621\pi\)
0.991150 0.132745i \(-0.0423790\pi\)
\(228\) 0 0
\(229\) −6.00000 −0.396491 −0.198246 0.980152i \(-0.563524\pi\)
−0.198246 + 0.980152i \(0.563524\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 6.00000i 0.393919i
\(233\) − 10.0000i − 0.655122i −0.944830 0.327561i \(-0.893773\pi\)
0.944830 0.327561i \(-0.106227\pi\)
\(234\) 18.0000 1.17670
\(235\) 0 0
\(236\) 12.0000 0.781133
\(237\) 0 0
\(238\) 2.00000i 0.129641i
\(239\) 24.0000 1.55243 0.776215 0.630468i \(-0.217137\pi\)
0.776215 + 0.630468i \(0.217137\pi\)
\(240\) 0 0
\(241\) 2.00000 0.128831 0.0644157 0.997923i \(-0.479482\pi\)
0.0644157 + 0.997923i \(0.479482\pi\)
\(242\) 1.00000i 0.0642824i
\(243\) 0 0
\(244\) 2.00000 0.128037
\(245\) 0 0
\(246\) 0 0
\(247\) − 24.0000i − 1.52708i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 20.0000 1.26239 0.631194 0.775625i \(-0.282565\pi\)
0.631194 + 0.775625i \(0.282565\pi\)
\(252\) 3.00000i 0.188982i
\(253\) 4.00000i 0.251478i
\(254\) −8.00000 −0.501965
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) − 18.0000i − 1.12281i −0.827541 0.561405i \(-0.810261\pi\)
0.827541 0.561405i \(-0.189739\pi\)
\(258\) 0 0
\(259\) 2.00000 0.124274
\(260\) 0 0
\(261\) −18.0000 −1.11417
\(262\) − 12.0000i − 0.741362i
\(263\) 16.0000i 0.986602i 0.869859 + 0.493301i \(0.164210\pi\)
−0.869859 + 0.493301i \(0.835790\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 4.00000 0.245256
\(267\) 0 0
\(268\) − 8.00000i − 0.488678i
\(269\) 2.00000 0.121942 0.0609711 0.998140i \(-0.480580\pi\)
0.0609711 + 0.998140i \(0.480580\pi\)
\(270\) 0 0
\(271\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(272\) 2.00000i 0.121268i
\(273\) 0 0
\(274\) 18.0000 1.08742
\(275\) 0 0
\(276\) 0 0
\(277\) − 26.0000i − 1.56219i −0.624413 0.781094i \(-0.714662\pi\)
0.624413 0.781094i \(-0.285338\pi\)
\(278\) 20.0000i 1.19952i
\(279\) 0 0
\(280\) 0 0
\(281\) −30.0000 −1.78965 −0.894825 0.446417i \(-0.852700\pi\)
−0.894825 + 0.446417i \(0.852700\pi\)
\(282\) 0 0
\(283\) − 20.0000i − 1.18888i −0.804141 0.594438i \(-0.797374\pi\)
0.804141 0.594438i \(-0.202626\pi\)
\(284\) 8.00000 0.474713
\(285\) 0 0
\(286\) −6.00000 −0.354787
\(287\) 6.00000i 0.354169i
\(288\) 3.00000i 0.176777i
\(289\) 13.0000 0.764706
\(290\) 0 0
\(291\) 0 0
\(292\) 10.0000i 0.585206i
\(293\) − 14.0000i − 0.817889i −0.912559 0.408944i \(-0.865897\pi\)
0.912559 0.408944i \(-0.134103\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 2.00000 0.116248
\(297\) 0 0
\(298\) − 6.00000i − 0.347571i
\(299\) −24.0000 −1.38796
\(300\) 0 0
\(301\) −4.00000 −0.230556
\(302\) 8.00000i 0.460348i
\(303\) 0 0
\(304\) 4.00000 0.229416
\(305\) 0 0
\(306\) −6.00000 −0.342997
\(307\) 20.0000i 1.14146i 0.821138 + 0.570730i \(0.193340\pi\)
−0.821138 + 0.570730i \(0.806660\pi\)
\(308\) − 1.00000i − 0.0569803i
\(309\) 0 0
\(310\) 0 0
\(311\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(312\) 0 0
\(313\) 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
\(316\) −8.00000 −0.450035
\(317\) − 14.0000i − 0.786318i −0.919470 0.393159i \(-0.871382\pi\)
0.919470 0.393159i \(-0.128618\pi\)
\(318\) 0 0
\(319\) 6.00000 0.335936
\(320\) 0 0
\(321\) 0 0
\(322\) − 4.00000i − 0.222911i
\(323\) 8.00000i 0.445132i
\(324\) −9.00000 −0.500000
\(325\) 0 0
\(326\) −8.00000 −0.443079
\(327\) 0 0
\(328\) 6.00000i 0.331295i
\(329\) −4.00000 −0.220527
\(330\) 0 0
\(331\) −28.0000 −1.53902 −0.769510 0.638635i \(-0.779499\pi\)
−0.769510 + 0.638635i \(0.779499\pi\)
\(332\) 12.0000i 0.658586i
\(333\) 6.00000i 0.328798i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) − 22.0000i − 1.19842i −0.800593 0.599208i \(-0.795482\pi\)
0.800593 0.599208i \(-0.204518\pi\)
\(338\) − 23.0000i − 1.25104i
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 12.0000i 0.648886i
\(343\) 1.00000i 0.0539949i
\(344\) −4.00000 −0.215666
\(345\) 0 0
\(346\) 14.0000 0.752645
\(347\) 28.0000i 1.50312i 0.659665 + 0.751559i \(0.270698\pi\)
−0.659665 + 0.751559i \(0.729302\pi\)
\(348\) 0 0
\(349\) 34.0000 1.81998 0.909989 0.414632i \(-0.136090\pi\)
0.909989 + 0.414632i \(0.136090\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) − 1.00000i − 0.0533002i
\(353\) − 14.0000i − 0.745145i −0.928003 0.372572i \(-0.878476\pi\)
0.928003 0.372572i \(-0.121524\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 10.0000 0.529999
\(357\) 0 0
\(358\) − 20.0000i − 1.05703i
\(359\) 16.0000 0.844448 0.422224 0.906492i \(-0.361250\pi\)
0.422224 + 0.906492i \(0.361250\pi\)
\(360\) 0 0
\(361\) −3.00000 −0.157895
\(362\) 6.00000i 0.315353i
\(363\) 0 0
\(364\) 6.00000 0.314485
\(365\) 0 0
\(366\) 0 0
\(367\) − 20.0000i − 1.04399i −0.852948 0.521996i \(-0.825188\pi\)
0.852948 0.521996i \(-0.174812\pi\)
\(368\) − 4.00000i − 0.208514i
\(369\) −18.0000 −0.937043
\(370\) 0 0
\(371\) −2.00000 −0.103835
\(372\) 0 0
\(373\) 34.0000i 1.76045i 0.474554 + 0.880227i \(0.342610\pi\)
−0.474554 + 0.880227i \(0.657390\pi\)
\(374\) 2.00000 0.103418
\(375\) 0 0
\(376\) −4.00000 −0.206284
\(377\) 36.0000i 1.85409i
\(378\) 0 0
\(379\) 12.0000 0.616399 0.308199 0.951322i \(-0.400274\pi\)
0.308199 + 0.951322i \(0.400274\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 8.00000i 0.409316i
\(383\) − 28.0000i − 1.43073i −0.698749 0.715367i \(-0.746260\pi\)
0.698749 0.715367i \(-0.253740\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −6.00000 −0.305392
\(387\) − 12.0000i − 0.609994i
\(388\) − 6.00000i − 0.304604i
\(389\) −6.00000 −0.304212 −0.152106 0.988364i \(-0.548606\pi\)
−0.152106 + 0.988364i \(0.548606\pi\)
\(390\) 0 0
\(391\) 8.00000 0.404577
\(392\) 1.00000i 0.0505076i
\(393\) 0 0
\(394\) 18.0000 0.906827
\(395\) 0 0
\(396\) 3.00000 0.150756
\(397\) 34.0000i 1.70641i 0.521575 + 0.853206i \(0.325345\pi\)
−0.521575 + 0.853206i \(0.674655\pi\)
\(398\) 24.0000i 1.20301i
\(399\) 0 0
\(400\) 0 0
\(401\) 34.0000 1.69788 0.848939 0.528490i \(-0.177242\pi\)
0.848939 + 0.528490i \(0.177242\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) −14.0000 −0.696526
\(405\) 0 0
\(406\) −6.00000 −0.297775
\(407\) − 2.00000i − 0.0991363i
\(408\) 0 0
\(409\) −34.0000 −1.68119 −0.840596 0.541663i \(-0.817795\pi\)
−0.840596 + 0.541663i \(0.817795\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 4.00000i 0.197066i
\(413\) 12.0000i 0.590481i
\(414\) 12.0000 0.589768
\(415\) 0 0
\(416\) 6.00000 0.294174
\(417\) 0 0
\(418\) − 4.00000i − 0.195646i
\(419\) −12.0000 −0.586238 −0.293119 0.956076i \(-0.594693\pi\)
−0.293119 + 0.956076i \(0.594693\pi\)
\(420\) 0 0
\(421\) −26.0000 −1.26716 −0.633581 0.773676i \(-0.718416\pi\)
−0.633581 + 0.773676i \(0.718416\pi\)
\(422\) 4.00000i 0.194717i
\(423\) − 12.0000i − 0.583460i
\(424\) −2.00000 −0.0971286
\(425\) 0 0
\(426\) 0 0
\(427\) 2.00000i 0.0967868i
\(428\) 4.00000i 0.193347i
\(429\) 0 0
\(430\) 0 0
\(431\) 8.00000 0.385346 0.192673 0.981263i \(-0.438284\pi\)
0.192673 + 0.981263i \(0.438284\pi\)
\(432\) 0 0
\(433\) − 6.00000i − 0.288342i −0.989553 0.144171i \(-0.953949\pi\)
0.989553 0.144171i \(-0.0460515\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 14.0000 0.670478
\(437\) − 16.0000i − 0.765384i
\(438\) 0 0
\(439\) 24.0000 1.14546 0.572729 0.819745i \(-0.305885\pi\)
0.572729 + 0.819745i \(0.305885\pi\)
\(440\) 0 0
\(441\) −3.00000 −0.142857
\(442\) 12.0000i 0.570782i
\(443\) 16.0000i 0.760183i 0.924949 + 0.380091i \(0.124107\pi\)
−0.924949 + 0.380091i \(0.875893\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 4.00000 0.189405
\(447\) 0 0
\(448\) 1.00000i 0.0472456i
\(449\) −2.00000 −0.0943858 −0.0471929 0.998886i \(-0.515028\pi\)
−0.0471929 + 0.998886i \(0.515028\pi\)
\(450\) 0 0
\(451\) 6.00000 0.282529
\(452\) 6.00000i 0.282216i
\(453\) 0 0
\(454\) 4.00000 0.187729
\(455\) 0 0
\(456\) 0 0
\(457\) 26.0000i 1.21623i 0.793849 + 0.608114i \(0.208074\pi\)
−0.793849 + 0.608114i \(0.791926\pi\)
\(458\) − 6.00000i − 0.280362i
\(459\) 0 0
\(460\) 0 0
\(461\) 6.00000 0.279448 0.139724 0.990190i \(-0.455378\pi\)
0.139724 + 0.990190i \(0.455378\pi\)
\(462\) 0 0
\(463\) 20.0000i 0.929479i 0.885448 + 0.464739i \(0.153852\pi\)
−0.885448 + 0.464739i \(0.846148\pi\)
\(464\) −6.00000 −0.278543
\(465\) 0 0
\(466\) 10.0000 0.463241
\(467\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(468\) 18.0000i 0.832050i
\(469\) 8.00000 0.369406
\(470\) 0 0
\(471\) 0 0
\(472\) 12.0000i 0.552345i
\(473\) 4.00000i 0.183920i
\(474\) 0 0
\(475\) 0 0
\(476\) −2.00000 −0.0916698
\(477\) − 6.00000i − 0.274721i
\(478\) 24.0000i 1.09773i
\(479\) 24.0000 1.09659 0.548294 0.836286i \(-0.315277\pi\)
0.548294 + 0.836286i \(0.315277\pi\)
\(480\) 0 0
\(481\) 12.0000 0.547153
\(482\) 2.00000i 0.0910975i
\(483\) 0 0
\(484\) −1.00000 −0.0454545
\(485\) 0 0
\(486\) 0 0
\(487\) 28.0000i 1.26880i 0.773004 + 0.634401i \(0.218753\pi\)
−0.773004 + 0.634401i \(0.781247\pi\)
\(488\) 2.00000i 0.0905357i
\(489\) 0 0
\(490\) 0 0
\(491\) 20.0000 0.902587 0.451294 0.892375i \(-0.350963\pi\)
0.451294 + 0.892375i \(0.350963\pi\)
\(492\) 0 0
\(493\) − 12.0000i − 0.540453i
\(494\) 24.0000 1.07981
\(495\) 0 0
\(496\) 0 0
\(497\) 8.00000i 0.358849i
\(498\) 0 0
\(499\) −28.0000 −1.25345 −0.626726 0.779240i \(-0.715605\pi\)
−0.626726 + 0.779240i \(0.715605\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 20.0000i 0.892644i
\(503\) − 8.00000i − 0.356702i −0.983967 0.178351i \(-0.942924\pi\)
0.983967 0.178351i \(-0.0570763\pi\)
\(504\) −3.00000 −0.133631
\(505\) 0 0
\(506\) −4.00000 −0.177822
\(507\) 0 0
\(508\) − 8.00000i − 0.354943i
\(509\) 2.00000 0.0886484 0.0443242 0.999017i \(-0.485887\pi\)
0.0443242 + 0.999017i \(0.485887\pi\)
\(510\) 0 0
\(511\) −10.0000 −0.442374
\(512\) 1.00000i 0.0441942i
\(513\) 0 0
\(514\) 18.0000 0.793946
\(515\) 0 0
\(516\) 0 0
\(517\) 4.00000i 0.175920i
\(518\) 2.00000i 0.0878750i
\(519\) 0 0
\(520\) 0 0
\(521\) −22.0000 −0.963837 −0.481919 0.876216i \(-0.660060\pi\)
−0.481919 + 0.876216i \(0.660060\pi\)
\(522\) − 18.0000i − 0.787839i
\(523\) − 12.0000i − 0.524723i −0.964970 0.262362i \(-0.915499\pi\)
0.964970 0.262362i \(-0.0845013\pi\)
\(524\) 12.0000 0.524222
\(525\) 0 0
\(526\) −16.0000 −0.697633
\(527\) 0 0
\(528\) 0 0
\(529\) 7.00000 0.304348
\(530\) 0 0
\(531\) −36.0000 −1.56227
\(532\) 4.00000i 0.173422i
\(533\) 36.0000i 1.55933i
\(534\) 0 0
\(535\) 0 0
\(536\) 8.00000 0.345547
\(537\) 0 0
\(538\) 2.00000i 0.0862261i
\(539\) 1.00000 0.0430730
\(540\) 0 0
\(541\) −18.0000 −0.773880 −0.386940 0.922105i \(-0.626468\pi\)
−0.386940 + 0.922105i \(0.626468\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) −2.00000 −0.0857493
\(545\) 0 0
\(546\) 0 0
\(547\) − 36.0000i − 1.53925i −0.638497 0.769624i \(-0.720443\pi\)
0.638497 0.769624i \(-0.279557\pi\)
\(548\) 18.0000i 0.768922i
\(549\) −6.00000 −0.256074
\(550\) 0 0
\(551\) −24.0000 −1.02243
\(552\) 0 0
\(553\) − 8.00000i − 0.340195i
\(554\) 26.0000 1.10463
\(555\) 0 0
\(556\) −20.0000 −0.848189
\(557\) 6.00000i 0.254228i 0.991888 + 0.127114i \(0.0405714\pi\)
−0.991888 + 0.127114i \(0.959429\pi\)
\(558\) 0 0
\(559\) −24.0000 −1.01509
\(560\) 0 0
\(561\) 0 0
\(562\) − 30.0000i − 1.26547i
\(563\) − 4.00000i − 0.168580i −0.996441 0.0842900i \(-0.973138\pi\)
0.996441 0.0842900i \(-0.0268622\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 20.0000 0.840663
\(567\) − 9.00000i − 0.377964i
\(568\) 8.00000i 0.335673i
\(569\) −42.0000 −1.76073 −0.880366 0.474295i \(-0.842703\pi\)
−0.880366 + 0.474295i \(0.842703\pi\)
\(570\) 0 0
\(571\) 4.00000 0.167395 0.0836974 0.996491i \(-0.473327\pi\)
0.0836974 + 0.996491i \(0.473327\pi\)
\(572\) − 6.00000i − 0.250873i
\(573\) 0 0
\(574\) −6.00000 −0.250435
\(575\) 0 0
\(576\) −3.00000 −0.125000
\(577\) 38.0000i 1.58196i 0.611842 + 0.790980i \(0.290429\pi\)
−0.611842 + 0.790980i \(0.709571\pi\)
\(578\) 13.0000i 0.540729i
\(579\) 0 0
\(580\) 0 0
\(581\) −12.0000 −0.497844
\(582\) 0 0
\(583\) 2.00000i 0.0828315i
\(584\) −10.0000 −0.413803
\(585\) 0 0
\(586\) 14.0000 0.578335
\(587\) − 8.00000i − 0.330195i −0.986277 0.165098i \(-0.947206\pi\)
0.986277 0.165098i \(-0.0527939\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) 2.00000i 0.0821995i
\(593\) 30.0000i 1.23195i 0.787765 + 0.615976i \(0.211238\pi\)
−0.787765 + 0.615976i \(0.788762\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 6.00000 0.245770
\(597\) 0 0
\(598\) − 24.0000i − 0.981433i
\(599\) 40.0000 1.63436 0.817178 0.576386i \(-0.195537\pi\)
0.817178 + 0.576386i \(0.195537\pi\)
\(600\) 0 0
\(601\) 26.0000 1.06056 0.530281 0.847822i \(-0.322086\pi\)
0.530281 + 0.847822i \(0.322086\pi\)
\(602\) − 4.00000i − 0.163028i
\(603\) 24.0000i 0.977356i
\(604\) −8.00000 −0.325515
\(605\) 0 0
\(606\) 0 0
\(607\) − 40.0000i − 1.62355i −0.583970 0.811775i \(-0.698502\pi\)
0.583970 0.811775i \(-0.301498\pi\)
\(608\) 4.00000i 0.162221i
\(609\) 0 0
\(610\) 0 0
\(611\) −24.0000 −0.970936
\(612\) − 6.00000i − 0.242536i
\(613\) 34.0000i 1.37325i 0.727013 + 0.686624i \(0.240908\pi\)
−0.727013 + 0.686624i \(0.759092\pi\)
\(614\) −20.0000 −0.807134
\(615\) 0 0
\(616\) 1.00000 0.0402911
\(617\) − 34.0000i − 1.36879i −0.729112 0.684394i \(-0.760067\pi\)
0.729112 0.684394i \(-0.239933\pi\)
\(618\) 0 0
\(619\) −4.00000 −0.160774 −0.0803868 0.996764i \(-0.525616\pi\)
−0.0803868 + 0.996764i \(0.525616\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 10.0000i 0.400642i
\(624\) 0 0
\(625\) 0 0
\(626\) −10.0000 −0.399680
\(627\) 0 0
\(628\) − 2.00000i − 0.0798087i
\(629\) −4.00000 −0.159490
\(630\) 0 0
\(631\) 8.00000 0.318475 0.159237 0.987240i \(-0.449096\pi\)
0.159237 + 0.987240i \(0.449096\pi\)
\(632\) − 8.00000i − 0.318223i
\(633\) 0 0
\(634\) 14.0000 0.556011
\(635\) 0 0
\(636\) 0 0
\(637\) 6.00000i 0.237729i
\(638\) 6.00000i 0.237542i
\(639\) −24.0000 −0.949425
\(640\) 0 0
\(641\) −30.0000 −1.18493 −0.592464 0.805597i \(-0.701845\pi\)
−0.592464 + 0.805597i \(0.701845\pi\)
\(642\) 0 0
\(643\) 32.0000i 1.26196i 0.775800 + 0.630978i \(0.217346\pi\)
−0.775800 + 0.630978i \(0.782654\pi\)
\(644\) 4.00000 0.157622
\(645\) 0 0
\(646\) −8.00000 −0.314756
\(647\) 12.0000i 0.471769i 0.971781 + 0.235884i \(0.0757987\pi\)
−0.971781 + 0.235884i \(0.924201\pi\)
\(648\) − 9.00000i − 0.353553i
\(649\) 12.0000 0.471041
\(650\) 0 0
\(651\) 0 0
\(652\) − 8.00000i − 0.313304i
\(653\) − 42.0000i − 1.64359i −0.569785 0.821794i \(-0.692974\pi\)
0.569785 0.821794i \(-0.307026\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) −6.00000 −0.234261
\(657\) − 30.0000i − 1.17041i
\(658\) − 4.00000i − 0.155936i
\(659\) −36.0000 −1.40236 −0.701180 0.712984i \(-0.747343\pi\)
−0.701180 + 0.712984i \(0.747343\pi\)
\(660\) 0 0
\(661\) 22.0000 0.855701 0.427850 0.903850i \(-0.359271\pi\)
0.427850 + 0.903850i \(0.359271\pi\)
\(662\) − 28.0000i − 1.08825i
\(663\) 0 0
\(664\) −12.0000 −0.465690
\(665\) 0 0
\(666\) −6.00000 −0.232495
\(667\) 24.0000i 0.929284i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 2.00000 0.0772091
\(672\) 0 0
\(673\) − 10.0000i − 0.385472i −0.981251 0.192736i \(-0.938264\pi\)
0.981251 0.192736i \(-0.0617360\pi\)
\(674\) 22.0000 0.847408
\(675\) 0 0
\(676\) 23.0000 0.884615
\(677\) 14.0000i 0.538064i 0.963131 + 0.269032i \(0.0867037\pi\)
−0.963131 + 0.269032i \(0.913296\pi\)
\(678\) 0 0
\(679\) 6.00000 0.230259
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 24.0000i 0.918334i 0.888350 + 0.459167i \(0.151852\pi\)
−0.888350 + 0.459167i \(0.848148\pi\)
\(684\) −12.0000 −0.458831
\(685\) 0 0
\(686\) −1.00000 −0.0381802
\(687\) 0 0
\(688\) − 4.00000i − 0.152499i
\(689\) −12.0000 −0.457164
\(690\) 0 0
\(691\) −36.0000 −1.36950 −0.684752 0.728776i \(-0.740090\pi\)
−0.684752 + 0.728776i \(0.740090\pi\)
\(692\) 14.0000i 0.532200i
\(693\) 3.00000i 0.113961i
\(694\) −28.0000 −1.06287
\(695\) 0 0
\(696\) 0 0
\(697\) − 12.0000i − 0.454532i
\(698\) 34.0000i 1.28692i
\(699\) 0 0
\(700\) 0 0
\(701\) −42.0000 −1.58632 −0.793159 0.609015i \(-0.791565\pi\)
−0.793159 + 0.609015i \(0.791565\pi\)
\(702\) 0 0
\(703\) 8.00000i 0.301726i
\(704\) 1.00000 0.0376889
\(705\) 0 0
\(706\) 14.0000 0.526897
\(707\) − 14.0000i − 0.526524i
\(708\) 0 0
\(709\) −22.0000 −0.826227 −0.413114 0.910679i \(-0.635559\pi\)
−0.413114 + 0.910679i \(0.635559\pi\)
\(710\) 0 0
\(711\) 24.0000 0.900070
\(712\) 10.0000i 0.374766i
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 20.0000 0.747435
\(717\) 0 0
\(718\) 16.0000i 0.597115i
\(719\) 40.0000 1.49175 0.745874 0.666087i \(-0.232032\pi\)
0.745874 + 0.666087i \(0.232032\pi\)
\(720\) 0 0
\(721\) −4.00000 −0.148968
\(722\) − 3.00000i − 0.111648i
\(723\) 0 0
\(724\) −6.00000 −0.222988
\(725\) 0 0
\(726\) 0 0
\(727\) 12.0000i 0.445055i 0.974926 + 0.222528i \(0.0714308\pi\)
−0.974926 + 0.222528i \(0.928569\pi\)
\(728\) 6.00000i 0.222375i
\(729\) 27.0000 1.00000
\(730\) 0 0
\(731\) 8.00000 0.295891
\(732\) 0 0
\(733\) − 46.0000i − 1.69905i −0.527549 0.849524i \(-0.676889\pi\)
0.527549 0.849524i \(-0.323111\pi\)
\(734\) 20.0000 0.738213
\(735\) 0 0
\(736\) 4.00000 0.147442
\(737\) − 8.00000i − 0.294684i
\(738\) − 18.0000i − 0.662589i
\(739\) 12.0000 0.441427 0.220714 0.975339i \(-0.429161\pi\)
0.220714 + 0.975339i \(0.429161\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) − 2.00000i − 0.0734223i
\(743\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) −34.0000 −1.24483
\(747\) − 36.0000i − 1.31717i
\(748\) 2.00000i 0.0731272i
\(749\) −4.00000 −0.146157
\(750\) 0 0
\(751\) −32.0000 −1.16770 −0.583848 0.811863i \(-0.698454\pi\)
−0.583848 + 0.811863i \(0.698454\pi\)
\(752\) − 4.00000i − 0.145865i
\(753\) 0 0
\(754\) −36.0000 −1.31104
\(755\) 0 0
\(756\) 0 0
\(757\) − 6.00000i − 0.218074i −0.994038 0.109037i \(-0.965223\pi\)
0.994038 0.109037i \(-0.0347767\pi\)
\(758\) 12.0000i 0.435860i
\(759\) 0 0
\(760\) 0 0
\(761\) −14.0000 −0.507500 −0.253750 0.967270i \(-0.581664\pi\)
−0.253750 + 0.967270i \(0.581664\pi\)
\(762\) 0 0
\(763\) 14.0000i 0.506834i
\(764\) −8.00000 −0.289430
\(765\) 0 0
\(766\) 28.0000 1.01168
\(767\) 72.0000i 2.59977i
\(768\) 0 0
\(769\) 14.0000 0.504853 0.252426 0.967616i \(-0.418771\pi\)
0.252426 + 0.967616i \(0.418771\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) − 6.00000i − 0.215945i
\(773\) 30.0000i 1.07903i 0.841978 + 0.539513i \(0.181391\pi\)
−0.841978 + 0.539513i \(0.818609\pi\)
\(774\) 12.0000 0.431331
\(775\) 0 0
\(776\) 6.00000 0.215387
\(777\) 0 0
\(778\) − 6.00000i − 0.215110i
\(779\) −24.0000 −0.859889
\(780\) 0 0
\(781\) 8.00000 0.286263
\(782\) 8.00000i 0.286079i
\(783\) 0 0
\(784\) −1.00000 −0.0357143
\(785\) 0 0
\(786\) 0 0
\(787\) − 28.0000i − 0.998092i −0.866575 0.499046i \(-0.833684\pi\)
0.866575 0.499046i \(-0.166316\pi\)
\(788\) 18.0000i 0.641223i
\(789\) 0 0
\(790\) 0 0
\(791\) −6.00000 −0.213335
\(792\) 3.00000i 0.106600i
\(793\) 12.0000i 0.426132i
\(794\) −34.0000 −1.20661
\(795\) 0 0
\(796\) −24.0000 −0.850657
\(797\) 34.0000i 1.20434i 0.798367 + 0.602171i \(0.205697\pi\)
−0.798367 + 0.602171i \(0.794303\pi\)
\(798\) 0 0
\(799\) 8.00000 0.283020
\(800\) 0 0
\(801\) −30.0000 −1.06000
\(802\) 34.0000i 1.20058i
\(803\) 10.0000i 0.352892i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) − 14.0000i − 0.492518i
\(809\) 46.0000 1.61727 0.808637 0.588308i \(-0.200206\pi\)
0.808637 + 0.588308i \(0.200206\pi\)
\(810\) 0 0
\(811\) 52.0000 1.82597 0.912983 0.407997i \(-0.133772\pi\)
0.912983 + 0.407997i \(0.133772\pi\)
\(812\) − 6.00000i − 0.210559i
\(813\) 0 0
\(814\) 2.00000 0.0701000
\(815\) 0 0
\(816\) 0 0
\(817\) − 16.0000i − 0.559769i
\(818\) − 34.0000i − 1.18878i
\(819\) −18.0000 −0.628971
\(820\) 0 0
\(821\) 30.0000 1.04701 0.523504 0.852023i \(-0.324625\pi\)
0.523504 + 0.852023i \(0.324625\pi\)
\(822\) 0 0
\(823\) 4.00000i 0.139431i 0.997567 + 0.0697156i \(0.0222092\pi\)
−0.997567 + 0.0697156i \(0.977791\pi\)
\(824\) −4.00000 −0.139347
\(825\) 0 0
\(826\) −12.0000 −0.417533
\(827\) 12.0000i 0.417281i 0.977992 + 0.208640i \(0.0669038\pi\)
−0.977992 + 0.208640i \(0.933096\pi\)
\(828\) 12.0000i 0.417029i
\(829\) −30.0000 −1.04194 −0.520972 0.853574i \(-0.674430\pi\)
−0.520972 + 0.853574i \(0.674430\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 6.00000i 0.208013i
\(833\) − 2.00000i − 0.0692959i
\(834\) 0 0
\(835\) 0 0
\(836\) 4.00000 0.138343
\(837\) 0 0
\(838\) − 12.0000i − 0.414533i
\(839\) 40.0000 1.38095 0.690477 0.723355i \(-0.257401\pi\)
0.690477 + 0.723355i \(0.257401\pi\)
\(840\) 0 0
\(841\) 7.00000 0.241379
\(842\) − 26.0000i − 0.896019i
\(843\) 0 0
\(844\) −4.00000 −0.137686
\(845\) 0 0
\(846\) 12.0000 0.412568
\(847\) − 1.00000i − 0.0343604i
\(848\) − 2.00000i − 0.0686803i
\(849\) 0 0
\(850\) 0 0
\(851\) 8.00000 0.274236
\(852\) 0 0
\(853\) − 6.00000i − 0.205436i −0.994711 0.102718i \(-0.967246\pi\)
0.994711 0.102718i \(-0.0327539\pi\)
\(854\) −2.00000 −0.0684386
\(855\) 0 0
\(856\) −4.00000 −0.136717
\(857\) 42.0000i 1.43469i 0.696717 + 0.717346i \(0.254643\pi\)
−0.696717 + 0.717346i \(0.745357\pi\)
\(858\) 0 0
\(859\) 44.0000 1.50126 0.750630 0.660722i \(-0.229750\pi\)
0.750630 + 0.660722i \(0.229750\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 8.00000i 0.272481i
\(863\) − 52.0000i − 1.77010i −0.465495 0.885050i \(-0.654124\pi\)
0.465495 0.885050i \(-0.345876\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 6.00000 0.203888
\(867\) 0 0
\(868\) 0 0
\(869\) −8.00000 −0.271381
\(870\) 0 0
\(871\) 48.0000 1.62642
\(872\) 14.0000i 0.474100i
\(873\) 18.0000i 0.609208i
\(874\) 16.0000 0.541208
\(875\) 0 0
\(876\) 0 0
\(877\) − 10.0000i − 0.337676i −0.985644 0.168838i \(-0.945999\pi\)
0.985644 0.168838i \(-0.0540015\pi\)
\(878\) 24.0000i 0.809961i
\(879\) 0 0
\(880\) 0 0
\(881\) 2.00000 0.0673817 0.0336909 0.999432i \(-0.489274\pi\)
0.0336909 + 0.999432i \(0.489274\pi\)
\(882\) − 3.00000i − 0.101015i
\(883\) 56.0000i 1.88455i 0.334840 + 0.942275i \(0.391318\pi\)
−0.334840 + 0.942275i \(0.608682\pi\)
\(884\) −12.0000 −0.403604
\(885\) 0 0
\(886\) −16.0000 −0.537531
\(887\) 48.0000i 1.61168i 0.592132 + 0.805841i \(0.298286\pi\)
−0.592132 + 0.805841i \(0.701714\pi\)
\(888\) 0 0
\(889\) 8.00000 0.268311
\(890\) 0 0
\(891\) −9.00000 −0.301511
\(892\) 4.00000i 0.133930i
\(893\) − 16.0000i − 0.535420i
\(894\) 0 0
\(895\) 0 0
\(896\) −1.00000 −0.0334077
\(897\) 0 0
\(898\) − 2.00000i − 0.0667409i
\(899\) 0 0
\(900\) 0 0
\(901\) 4.00000 0.133259
\(902\) 6.00000i 0.199778i
\(903\) 0 0
\(904\) −6.00000 −0.199557
\(905\) 0 0
\(906\) 0 0
\(907\) − 32.0000i − 1.06254i −0.847202 0.531271i \(-0.821714\pi\)
0.847202 0.531271i \(-0.178286\pi\)
\(908\) 4.00000i 0.132745i
\(909\) 42.0000 1.39305
\(910\) 0 0
\(911\) −40.0000 −1.32526 −0.662630 0.748947i \(-0.730560\pi\)
−0.662630 + 0.748947i \(0.730560\pi\)
\(912\) 0 0
\(913\) 12.0000i 0.397142i
\(914\) −26.0000 −0.860004
\(915\) 0 0
\(916\) 6.00000 0.198246
\(917\) 12.0000i 0.396275i
\(918\) 0 0
\(919\) 40.0000 1.31948 0.659739 0.751495i \(-0.270667\pi\)
0.659739 + 0.751495i \(0.270667\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 6.00000i 0.197599i
\(923\) 48.0000i 1.57994i
\(924\) 0 0
\(925\) 0 0
\(926\) −20.0000 −0.657241
\(927\) − 12.0000i − 0.394132i
\(928\) − 6.00000i − 0.196960i
\(929\) 30.0000 0.984268 0.492134 0.870519i \(-0.336217\pi\)
0.492134 + 0.870519i \(0.336217\pi\)
\(930\) 0 0
\(931\) −4.00000 −0.131095
\(932\) 10.0000i 0.327561i
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) −18.0000 −0.588348
\(937\) 50.0000i 1.63343i 0.577042 + 0.816714i \(0.304207\pi\)
−0.577042 + 0.816714i \(0.695793\pi\)
\(938\) 8.00000i 0.261209i
\(939\) 0 0
\(940\) 0 0
\(941\) 38.0000 1.23876 0.619382 0.785090i \(-0.287383\pi\)
0.619382 + 0.785090i \(0.287383\pi\)
\(942\) 0 0
\(943\) 24.0000i 0.781548i
\(944\) −12.0000 −0.390567
\(945\) 0 0
\(946\) −4.00000 −0.130051
\(947\) − 24.0000i − 0.779895i −0.920837 0.389948i \(-0.872493\pi\)
0.920837 0.389948i \(-0.127507\pi\)
\(948\) 0 0
\(949\) −60.0000 −1.94768
\(950\) 0 0
\(951\) 0 0
\(952\) − 2.00000i − 0.0648204i
\(953\) 22.0000i 0.712650i 0.934362 + 0.356325i \(0.115970\pi\)
−0.934362 + 0.356325i \(0.884030\pi\)
\(954\) 6.00000 0.194257
\(955\) 0 0
\(956\) −24.0000 −0.776215
\(957\) 0 0
\(958\) 24.0000i 0.775405i
\(959\) −18.0000 −0.581250
\(960\) 0 0
\(961\) −31.0000 −1.00000
\(962\) 12.0000i 0.386896i
\(963\) − 12.0000i − 0.386695i
\(964\) −2.00000 −0.0644157
\(965\) 0 0
\(966\) 0 0
\(967\) 32.0000i 1.02905i 0.857475 + 0.514525i \(0.172032\pi\)
−0.857475 + 0.514525i \(0.827968\pi\)
\(968\) − 1.00000i − 0.0321412i
\(969\) 0 0
\(970\) 0 0
\(971\) −52.0000 −1.66876 −0.834380 0.551190i \(-0.814174\pi\)
−0.834380 + 0.551190i \(0.814174\pi\)
\(972\) 0 0
\(973\) − 20.0000i − 0.641171i
\(974\) −28.0000 −0.897178
\(975\) 0 0
\(976\) −2.00000 −0.0640184
\(977\) 54.0000i 1.72761i 0.503824 + 0.863807i \(0.331926\pi\)
−0.503824 + 0.863807i \(0.668074\pi\)
\(978\) 0 0
\(979\) 10.0000 0.319601
\(980\) 0 0
\(981\) −42.0000 −1.34096
\(982\) 20.0000i 0.638226i
\(983\) − 4.00000i − 0.127580i −0.997963 0.0637901i \(-0.979681\pi\)
0.997963 0.0637901i \(-0.0203188\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 12.0000 0.382158
\(987\) 0 0
\(988\) 24.0000i 0.763542i
\(989\) −16.0000 −0.508770
\(990\) 0 0
\(991\) −40.0000 −1.27064 −0.635321 0.772248i \(-0.719132\pi\)
−0.635321 + 0.772248i \(0.719132\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) −8.00000 −0.253745
\(995\) 0 0
\(996\) 0 0
\(997\) − 10.0000i − 0.316703i −0.987383 0.158352i \(-0.949382\pi\)
0.987383 0.158352i \(-0.0506179\pi\)
\(998\) − 28.0000i − 0.886325i
\(999\) 0 0
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 3850.2.c.m.1849.2 2
5.2 odd 4 770.2.a.d.1.1 1
5.3 odd 4 3850.2.a.s.1.1 1
5.4 even 2 inner 3850.2.c.m.1849.1 2
15.2 even 4 6930.2.a.x.1.1 1
20.7 even 4 6160.2.a.e.1.1 1
35.27 even 4 5390.2.a.j.1.1 1
55.32 even 4 8470.2.a.z.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
770.2.a.d.1.1 1 5.2 odd 4
3850.2.a.s.1.1 1 5.3 odd 4
3850.2.c.m.1849.1 2 5.4 even 2 inner
3850.2.c.m.1849.2 2 1.1 even 1 trivial
5390.2.a.j.1.1 1 35.27 even 4
6160.2.a.e.1.1 1 20.7 even 4
6930.2.a.x.1.1 1 15.2 even 4
8470.2.a.z.1.1 1 55.32 even 4