Properties

Label 5070.2.b.n.1351.2
Level $5070$
Weight $2$
Character 5070.1351
Analytic conductor $40.484$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 5070 = 2 \cdot 3 \cdot 5 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5070.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(40.4841538248\)
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 390)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1351.2
Root \(-1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 5070.1351
Dual form 5070.2.b.n.1351.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000i q^{2} +1.00000 q^{3} -1.00000 q^{4} -1.00000i q^{5} +1.00000i q^{6} -2.00000i q^{7} -1.00000i q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000i q^{2} +1.00000 q^{3} -1.00000 q^{4} -1.00000i q^{5} +1.00000i q^{6} -2.00000i q^{7} -1.00000i q^{8} +1.00000 q^{9} +1.00000 q^{10} -1.00000 q^{12} +2.00000 q^{14} -1.00000i q^{15} +1.00000 q^{16} +1.00000i q^{18} +2.00000i q^{19} +1.00000i q^{20} -2.00000i q^{21} +6.00000 q^{23} -1.00000i q^{24} -1.00000 q^{25} +1.00000 q^{27} +2.00000i q^{28} +1.00000 q^{30} -4.00000i q^{31} +1.00000i q^{32} -2.00000 q^{35} -1.00000 q^{36} -2.00000i q^{37} -2.00000 q^{38} -1.00000 q^{40} -6.00000i q^{41} +2.00000 q^{42} +4.00000 q^{43} -1.00000i q^{45} +6.00000i q^{46} +1.00000 q^{48} +3.00000 q^{49} -1.00000i q^{50} -6.00000 q^{53} +1.00000i q^{54} -2.00000 q^{56} +2.00000i q^{57} +1.00000i q^{60} -10.0000 q^{61} +4.00000 q^{62} -2.00000i q^{63} -1.00000 q^{64} +8.00000i q^{67} +6.00000 q^{69} -2.00000i q^{70} -1.00000i q^{72} -8.00000i q^{73} +2.00000 q^{74} -1.00000 q^{75} -2.00000i q^{76} +8.00000 q^{79} -1.00000i q^{80} +1.00000 q^{81} +6.00000 q^{82} -12.0000i q^{83} +2.00000i q^{84} +4.00000i q^{86} -6.00000i q^{89} +1.00000 q^{90} -6.00000 q^{92} -4.00000i q^{93} +2.00000 q^{95} +1.00000i q^{96} +8.00000i q^{97} +3.00000i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{3} - 2q^{4} + 2q^{9} + O(q^{10}) \) \( 2q + 2q^{3} - 2q^{4} + 2q^{9} + 2q^{10} - 2q^{12} + 4q^{14} + 2q^{16} + 12q^{23} - 2q^{25} + 2q^{27} + 2q^{30} - 4q^{35} - 2q^{36} - 4q^{38} - 2q^{40} + 4q^{42} + 8q^{43} + 2q^{48} + 6q^{49} - 12q^{53} - 4q^{56} - 20q^{61} + 8q^{62} - 2q^{64} + 12q^{69} + 4q^{74} - 2q^{75} + 16q^{79} + 2q^{81} + 12q^{82} + 2q^{90} - 12q^{92} + 4q^{95} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/5070\mathbb{Z}\right)^\times\).

\(n\) \(1691\) \(1861\) \(4057\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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
\(3\) 1.00000 0.577350
\(4\) −1.00000 −0.500000
\(5\) − 1.00000i − 0.447214i
\(6\) 1.00000i 0.408248i
\(7\) − 2.00000i − 0.755929i −0.925820 0.377964i \(-0.876624\pi\)
0.925820 0.377964i \(-0.123376\pi\)
\(8\) − 1.00000i − 0.353553i
\(9\) 1.00000 0.333333
\(10\) 1.00000 0.316228
\(11\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(12\) −1.00000 −0.288675
\(13\) 0 0
\(14\) 2.00000 0.534522
\(15\) − 1.00000i − 0.258199i
\(16\) 1.00000 0.250000
\(17\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(18\) 1.00000i 0.235702i
\(19\) 2.00000i 0.458831i 0.973329 + 0.229416i \(0.0736815\pi\)
−0.973329 + 0.229416i \(0.926318\pi\)
\(20\) 1.00000i 0.223607i
\(21\) − 2.00000i − 0.436436i
\(22\) 0 0
\(23\) 6.00000 1.25109 0.625543 0.780189i \(-0.284877\pi\)
0.625543 + 0.780189i \(0.284877\pi\)
\(24\) − 1.00000i − 0.204124i
\(25\) −1.00000 −0.200000
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 2.00000i 0.377964i
\(29\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(30\) 1.00000 0.182574
\(31\) − 4.00000i − 0.718421i −0.933257 0.359211i \(-0.883046\pi\)
0.933257 0.359211i \(-0.116954\pi\)
\(32\) 1.00000i 0.176777i
\(33\) 0 0
\(34\) 0 0
\(35\) −2.00000 −0.338062
\(36\) −1.00000 −0.166667
\(37\) − 2.00000i − 0.328798i −0.986394 0.164399i \(-0.947432\pi\)
0.986394 0.164399i \(-0.0525685\pi\)
\(38\) −2.00000 −0.324443
\(39\) 0 0
\(40\) −1.00000 −0.158114
\(41\) − 6.00000i − 0.937043i −0.883452 0.468521i \(-0.844787\pi\)
0.883452 0.468521i \(-0.155213\pi\)
\(42\) 2.00000 0.308607
\(43\) 4.00000 0.609994 0.304997 0.952353i \(-0.401344\pi\)
0.304997 + 0.952353i \(0.401344\pi\)
\(44\) 0 0
\(45\) − 1.00000i − 0.149071i
\(46\) 6.00000i 0.884652i
\(47\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(48\) 1.00000 0.144338
\(49\) 3.00000 0.428571
\(50\) − 1.00000i − 0.141421i
\(51\) 0 0
\(52\) 0 0
\(53\) −6.00000 −0.824163 −0.412082 0.911147i \(-0.635198\pi\)
−0.412082 + 0.911147i \(0.635198\pi\)
\(54\) 1.00000i 0.136083i
\(55\) 0 0
\(56\) −2.00000 −0.267261
\(57\) 2.00000i 0.264906i
\(58\) 0 0
\(59\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(60\) 1.00000i 0.129099i
\(61\) −10.0000 −1.28037 −0.640184 0.768221i \(-0.721142\pi\)
−0.640184 + 0.768221i \(0.721142\pi\)
\(62\) 4.00000 0.508001
\(63\) − 2.00000i − 0.251976i
\(64\) −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 \(0.837479\pi\)
\(68\) 0 0
\(69\) 6.00000 0.722315
\(70\) − 2.00000i − 0.239046i
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) − 1.00000i − 0.117851i
\(73\) − 8.00000i − 0.936329i −0.883641 0.468165i \(-0.844915\pi\)
0.883641 0.468165i \(-0.155085\pi\)
\(74\) 2.00000 0.232495
\(75\) −1.00000 −0.115470
\(76\) − 2.00000i − 0.229416i
\(77\) 0 0
\(78\) 0 0
\(79\) 8.00000 0.900070 0.450035 0.893011i \(-0.351411\pi\)
0.450035 + 0.893011i \(0.351411\pi\)
\(80\) − 1.00000i − 0.111803i
\(81\) 1.00000 0.111111
\(82\) 6.00000 0.662589
\(83\) − 12.0000i − 1.31717i −0.752506 0.658586i \(-0.771155\pi\)
0.752506 0.658586i \(-0.228845\pi\)
\(84\) 2.00000i 0.218218i
\(85\) 0 0
\(86\) 4.00000i 0.431331i
\(87\) 0 0
\(88\) 0 0
\(89\) − 6.00000i − 0.635999i −0.948091 0.317999i \(-0.896989\pi\)
0.948091 0.317999i \(-0.103011\pi\)
\(90\) 1.00000 0.105409
\(91\) 0 0
\(92\) −6.00000 −0.625543
\(93\) − 4.00000i − 0.414781i
\(94\) 0 0
\(95\) 2.00000 0.205196
\(96\) 1.00000i 0.102062i
\(97\) 8.00000i 0.812277i 0.913812 + 0.406138i \(0.133125\pi\)
−0.913812 + 0.406138i \(0.866875\pi\)
\(98\) 3.00000i 0.303046i
\(99\) 0 0
\(100\) 1.00000 0.100000
\(101\) 12.0000 1.19404 0.597022 0.802225i \(-0.296350\pi\)
0.597022 + 0.802225i \(0.296350\pi\)
\(102\) 0 0
\(103\) −8.00000 −0.788263 −0.394132 0.919054i \(-0.628955\pi\)
−0.394132 + 0.919054i \(0.628955\pi\)
\(104\) 0 0
\(105\) −2.00000 −0.195180
\(106\) − 6.00000i − 0.582772i
\(107\) 12.0000 1.16008 0.580042 0.814587i \(-0.303036\pi\)
0.580042 + 0.814587i \(0.303036\pi\)
\(108\) −1.00000 −0.0962250
\(109\) − 16.0000i − 1.53252i −0.642529 0.766261i \(-0.722115\pi\)
0.642529 0.766261i \(-0.277885\pi\)
\(110\) 0 0
\(111\) − 2.00000i − 0.189832i
\(112\) − 2.00000i − 0.188982i
\(113\) −12.0000 −1.12887 −0.564433 0.825479i \(-0.690905\pi\)
−0.564433 + 0.825479i \(0.690905\pi\)
\(114\) −2.00000 −0.187317
\(115\) − 6.00000i − 0.559503i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) −1.00000 −0.0912871
\(121\) 11.0000 1.00000
\(122\) − 10.0000i − 0.905357i
\(123\) − 6.00000i − 0.541002i
\(124\) 4.00000i 0.359211i
\(125\) 1.00000i 0.0894427i
\(126\) 2.00000 0.178174
\(127\) −8.00000 −0.709885 −0.354943 0.934888i \(-0.615500\pi\)
−0.354943 + 0.934888i \(0.615500\pi\)
\(128\) − 1.00000i − 0.0883883i
\(129\) 4.00000 0.352180
\(130\) 0 0
\(131\) −18.0000 −1.57267 −0.786334 0.617802i \(-0.788023\pi\)
−0.786334 + 0.617802i \(0.788023\pi\)
\(132\) 0 0
\(133\) 4.00000 0.346844
\(134\) −8.00000 −0.691095
\(135\) − 1.00000i − 0.0860663i
\(136\) 0 0
\(137\) − 18.0000i − 1.53784i −0.639343 0.768922i \(-0.720793\pi\)
0.639343 0.768922i \(-0.279207\pi\)
\(138\) 6.00000i 0.510754i
\(139\) 8.00000 0.678551 0.339276 0.940687i \(-0.389818\pi\)
0.339276 + 0.940687i \(0.389818\pi\)
\(140\) 2.00000 0.169031
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 1.00000 0.0833333
\(145\) 0 0
\(146\) 8.00000 0.662085
\(147\) 3.00000 0.247436
\(148\) 2.00000i 0.164399i
\(149\) 6.00000i 0.491539i 0.969328 + 0.245770i \(0.0790407\pi\)
−0.969328 + 0.245770i \(0.920959\pi\)
\(150\) − 1.00000i − 0.0816497i
\(151\) − 20.0000i − 1.62758i −0.581161 0.813788i \(-0.697401\pi\)
0.581161 0.813788i \(-0.302599\pi\)
\(152\) 2.00000 0.162221
\(153\) 0 0
\(154\) 0 0
\(155\) −4.00000 −0.321288
\(156\) 0 0
\(157\) 2.00000 0.159617 0.0798087 0.996810i \(-0.474569\pi\)
0.0798087 + 0.996810i \(0.474569\pi\)
\(158\) 8.00000i 0.636446i
\(159\) −6.00000 −0.475831
\(160\) 1.00000 0.0790569
\(161\) − 12.0000i − 0.945732i
\(162\) 1.00000i 0.0785674i
\(163\) − 20.0000i − 1.56652i −0.621694 0.783260i \(-0.713555\pi\)
0.621694 0.783260i \(-0.286445\pi\)
\(164\) 6.00000i 0.468521i
\(165\) 0 0
\(166\) 12.0000 0.931381
\(167\) − 12.0000i − 0.928588i −0.885681 0.464294i \(-0.846308\pi\)
0.885681 0.464294i \(-0.153692\pi\)
\(168\) −2.00000 −0.154303
\(169\) 0 0
\(170\) 0 0
\(171\) 2.00000i 0.152944i
\(172\) −4.00000 −0.304997
\(173\) 6.00000 0.456172 0.228086 0.973641i \(-0.426753\pi\)
0.228086 + 0.973641i \(0.426753\pi\)
\(174\) 0 0
\(175\) 2.00000i 0.151186i
\(176\) 0 0
\(177\) 0 0
\(178\) 6.00000 0.449719
\(179\) −18.0000 −1.34538 −0.672692 0.739923i \(-0.734862\pi\)
−0.672692 + 0.739923i \(0.734862\pi\)
\(180\) 1.00000i 0.0745356i
\(181\) 22.0000 1.63525 0.817624 0.575753i \(-0.195291\pi\)
0.817624 + 0.575753i \(0.195291\pi\)
\(182\) 0 0
\(183\) −10.0000 −0.739221
\(184\) − 6.00000i − 0.442326i
\(185\) −2.00000 −0.147043
\(186\) 4.00000 0.293294
\(187\) 0 0
\(188\) 0 0
\(189\) − 2.00000i − 0.145479i
\(190\) 2.00000i 0.145095i
\(191\) 24.0000 1.73658 0.868290 0.496058i \(-0.165220\pi\)
0.868290 + 0.496058i \(0.165220\pi\)
\(192\) −1.00000 −0.0721688
\(193\) − 20.0000i − 1.43963i −0.694165 0.719816i \(-0.744226\pi\)
0.694165 0.719816i \(-0.255774\pi\)
\(194\) −8.00000 −0.574367
\(195\) 0 0
\(196\) −3.00000 −0.214286
\(197\) 6.00000i 0.427482i 0.976890 + 0.213741i \(0.0685649\pi\)
−0.976890 + 0.213741i \(0.931435\pi\)
\(198\) 0 0
\(199\) 16.0000 1.13421 0.567105 0.823646i \(-0.308063\pi\)
0.567105 + 0.823646i \(0.308063\pi\)
\(200\) 1.00000i 0.0707107i
\(201\) 8.00000i 0.564276i
\(202\) 12.0000i 0.844317i
\(203\) 0 0
\(204\) 0 0
\(205\) −6.00000 −0.419058
\(206\) − 8.00000i − 0.557386i
\(207\) 6.00000 0.417029
\(208\) 0 0
\(209\) 0 0
\(210\) − 2.00000i − 0.138013i
\(211\) −4.00000 −0.275371 −0.137686 0.990476i \(-0.543966\pi\)
−0.137686 + 0.990476i \(0.543966\pi\)
\(212\) 6.00000 0.412082
\(213\) 0 0
\(214\) 12.0000i 0.820303i
\(215\) − 4.00000i − 0.272798i
\(216\) − 1.00000i − 0.0680414i
\(217\) −8.00000 −0.543075
\(218\) 16.0000 1.08366
\(219\) − 8.00000i − 0.540590i
\(220\) 0 0
\(221\) 0 0
\(222\) 2.00000 0.134231
\(223\) 14.0000i 0.937509i 0.883328 + 0.468755i \(0.155297\pi\)
−0.883328 + 0.468755i \(0.844703\pi\)
\(224\) 2.00000 0.133631
\(225\) −1.00000 −0.0666667
\(226\) − 12.0000i − 0.798228i
\(227\) 12.0000i 0.796468i 0.917284 + 0.398234i \(0.130377\pi\)
−0.917284 + 0.398234i \(0.869623\pi\)
\(228\) − 2.00000i − 0.132453i
\(229\) 16.0000i 1.05731i 0.848837 + 0.528655i \(0.177303\pi\)
−0.848837 + 0.528655i \(0.822697\pi\)
\(230\) 6.00000 0.395628
\(231\) 0 0
\(232\) 0 0
\(233\) −24.0000 −1.57229 −0.786146 0.618041i \(-0.787927\pi\)
−0.786146 + 0.618041i \(0.787927\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 8.00000 0.519656
\(238\) 0 0
\(239\) − 24.0000i − 1.55243i −0.630468 0.776215i \(-0.717137\pi\)
0.630468 0.776215i \(-0.282863\pi\)
\(240\) − 1.00000i − 0.0645497i
\(241\) − 2.00000i − 0.128831i −0.997923 0.0644157i \(-0.979482\pi\)
0.997923 0.0644157i \(-0.0205183\pi\)
\(242\) 11.0000i 0.707107i
\(243\) 1.00000 0.0641500
\(244\) 10.0000 0.640184
\(245\) − 3.00000i − 0.191663i
\(246\) 6.00000 0.382546
\(247\) 0 0
\(248\) −4.00000 −0.254000
\(249\) − 12.0000i − 0.760469i
\(250\) −1.00000 −0.0632456
\(251\) −18.0000 −1.13615 −0.568075 0.822977i \(-0.692312\pi\)
−0.568075 + 0.822977i \(0.692312\pi\)
\(252\) 2.00000i 0.125988i
\(253\) 0 0
\(254\) − 8.00000i − 0.501965i
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −12.0000 −0.748539 −0.374270 0.927320i \(-0.622107\pi\)
−0.374270 + 0.927320i \(0.622107\pi\)
\(258\) 4.00000i 0.249029i
\(259\) −4.00000 −0.248548
\(260\) 0 0
\(261\) 0 0
\(262\) − 18.0000i − 1.11204i
\(263\) 6.00000 0.369976 0.184988 0.982741i \(-0.440775\pi\)
0.184988 + 0.982741i \(0.440775\pi\)
\(264\) 0 0
\(265\) 6.00000i 0.368577i
\(266\) 4.00000i 0.245256i
\(267\) − 6.00000i − 0.367194i
\(268\) − 8.00000i − 0.488678i
\(269\) 12.0000 0.731653 0.365826 0.930683i \(-0.380786\pi\)
0.365826 + 0.930683i \(0.380786\pi\)
\(270\) 1.00000 0.0608581
\(271\) 4.00000i 0.242983i 0.992592 + 0.121491i \(0.0387677\pi\)
−0.992592 + 0.121491i \(0.961232\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 18.0000 1.08742
\(275\) 0 0
\(276\) −6.00000 −0.361158
\(277\) 22.0000 1.32185 0.660926 0.750451i \(-0.270164\pi\)
0.660926 + 0.750451i \(0.270164\pi\)
\(278\) 8.00000i 0.479808i
\(279\) − 4.00000i − 0.239474i
\(280\) 2.00000i 0.119523i
\(281\) − 6.00000i − 0.357930i −0.983855 0.178965i \(-0.942725\pi\)
0.983855 0.178965i \(-0.0572749\pi\)
\(282\) 0 0
\(283\) 4.00000 0.237775 0.118888 0.992908i \(-0.462067\pi\)
0.118888 + 0.992908i \(0.462067\pi\)
\(284\) 0 0
\(285\) 2.00000 0.118470
\(286\) 0 0
\(287\) −12.0000 −0.708338
\(288\) 1.00000i 0.0589256i
\(289\) −17.0000 −1.00000
\(290\) 0 0
\(291\) 8.00000i 0.468968i
\(292\) 8.00000i 0.468165i
\(293\) − 6.00000i − 0.350524i −0.984522 0.175262i \(-0.943923\pi\)
0.984522 0.175262i \(-0.0560772\pi\)
\(294\) 3.00000i 0.174964i
\(295\) 0 0
\(296\) −2.00000 −0.116248
\(297\) 0 0
\(298\) −6.00000 −0.347571
\(299\) 0 0
\(300\) 1.00000 0.0577350
\(301\) − 8.00000i − 0.461112i
\(302\) 20.0000 1.15087
\(303\) 12.0000 0.689382
\(304\) 2.00000i 0.114708i
\(305\) 10.0000i 0.572598i
\(306\) 0 0
\(307\) 16.0000i 0.913168i 0.889680 + 0.456584i \(0.150927\pi\)
−0.889680 + 0.456584i \(0.849073\pi\)
\(308\) 0 0
\(309\) −8.00000 −0.455104
\(310\) − 4.00000i − 0.227185i
\(311\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(312\) 0 0
\(313\) 14.0000 0.791327 0.395663 0.918396i \(-0.370515\pi\)
0.395663 + 0.918396i \(0.370515\pi\)
\(314\) 2.00000i 0.112867i
\(315\) −2.00000 −0.112687
\(316\) −8.00000 −0.450035
\(317\) − 30.0000i − 1.68497i −0.538721 0.842484i \(-0.681092\pi\)
0.538721 0.842484i \(-0.318908\pi\)
\(318\) − 6.00000i − 0.336463i
\(319\) 0 0
\(320\) 1.00000i 0.0559017i
\(321\) 12.0000 0.669775
\(322\) 12.0000 0.668734
\(323\) 0 0
\(324\) −1.00000 −0.0555556
\(325\) 0 0
\(326\) 20.0000 1.10770
\(327\) − 16.0000i − 0.884802i
\(328\) −6.00000 −0.331295
\(329\) 0 0
\(330\) 0 0
\(331\) 26.0000i 1.42909i 0.699590 + 0.714545i \(0.253366\pi\)
−0.699590 + 0.714545i \(0.746634\pi\)
\(332\) 12.0000i 0.658586i
\(333\) − 2.00000i − 0.109599i
\(334\) 12.0000 0.656611
\(335\) 8.00000 0.437087
\(336\) − 2.00000i − 0.109109i
\(337\) 34.0000 1.85210 0.926049 0.377403i \(-0.123183\pi\)
0.926049 + 0.377403i \(0.123183\pi\)
\(338\) 0 0
\(339\) −12.0000 −0.651751
\(340\) 0 0
\(341\) 0 0
\(342\) −2.00000 −0.108148
\(343\) − 20.0000i − 1.07990i
\(344\) − 4.00000i − 0.215666i
\(345\) − 6.00000i − 0.323029i
\(346\) 6.00000i 0.322562i
\(347\) 24.0000 1.28839 0.644194 0.764862i \(-0.277193\pi\)
0.644194 + 0.764862i \(0.277193\pi\)
\(348\) 0 0
\(349\) 16.0000i 0.856460i 0.903670 + 0.428230i \(0.140863\pi\)
−0.903670 + 0.428230i \(0.859137\pi\)
\(350\) −2.00000 −0.106904
\(351\) 0 0
\(352\) 0 0
\(353\) 18.0000i 0.958043i 0.877803 + 0.479022i \(0.159008\pi\)
−0.877803 + 0.479022i \(0.840992\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 6.00000i 0.317999i
\(357\) 0 0
\(358\) − 18.0000i − 0.951330i
\(359\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(360\) −1.00000 −0.0527046
\(361\) 15.0000 0.789474
\(362\) 22.0000i 1.15629i
\(363\) 11.0000 0.577350
\(364\) 0 0
\(365\) −8.00000 −0.418739
\(366\) − 10.0000i − 0.522708i
\(367\) −16.0000 −0.835193 −0.417597 0.908633i \(-0.637127\pi\)
−0.417597 + 0.908633i \(0.637127\pi\)
\(368\) 6.00000 0.312772
\(369\) − 6.00000i − 0.312348i
\(370\) − 2.00000i − 0.103975i
\(371\) 12.0000i 0.623009i
\(372\) 4.00000i 0.207390i
\(373\) −22.0000 −1.13912 −0.569558 0.821951i \(-0.692886\pi\)
−0.569558 + 0.821951i \(0.692886\pi\)
\(374\) 0 0
\(375\) 1.00000i 0.0516398i
\(376\) 0 0
\(377\) 0 0
\(378\) 2.00000 0.102869
\(379\) − 10.0000i − 0.513665i −0.966456 0.256833i \(-0.917321\pi\)
0.966456 0.256833i \(-0.0826790\pi\)
\(380\) −2.00000 −0.102598
\(381\) −8.00000 −0.409852
\(382\) 24.0000i 1.22795i
\(383\) − 36.0000i − 1.83951i −0.392488 0.919757i \(-0.628386\pi\)
0.392488 0.919757i \(-0.371614\pi\)
\(384\) − 1.00000i − 0.0510310i
\(385\) 0 0
\(386\) 20.0000 1.01797
\(387\) 4.00000 0.203331
\(388\) − 8.00000i − 0.406138i
\(389\) −12.0000 −0.608424 −0.304212 0.952604i \(-0.598393\pi\)
−0.304212 + 0.952604i \(0.598393\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) − 3.00000i − 0.151523i
\(393\) −18.0000 −0.907980
\(394\) −6.00000 −0.302276
\(395\) − 8.00000i − 0.402524i
\(396\) 0 0
\(397\) 22.0000i 1.10415i 0.833795 + 0.552074i \(0.186163\pi\)
−0.833795 + 0.552074i \(0.813837\pi\)
\(398\) 16.0000i 0.802008i
\(399\) 4.00000 0.200250
\(400\) −1.00000 −0.0500000
\(401\) 6.00000i 0.299626i 0.988714 + 0.149813i \(0.0478671\pi\)
−0.988714 + 0.149813i \(0.952133\pi\)
\(402\) −8.00000 −0.399004
\(403\) 0 0
\(404\) −12.0000 −0.597022
\(405\) − 1.00000i − 0.0496904i
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 14.0000i 0.692255i 0.938187 + 0.346128i \(0.112504\pi\)
−0.938187 + 0.346128i \(0.887496\pi\)
\(410\) − 6.00000i − 0.296319i
\(411\) − 18.0000i − 0.887875i
\(412\) 8.00000 0.394132
\(413\) 0 0
\(414\) 6.00000i 0.294884i
\(415\) −12.0000 −0.589057
\(416\) 0 0
\(417\) 8.00000 0.391762
\(418\) 0 0
\(419\) −18.0000 −0.879358 −0.439679 0.898155i \(-0.644908\pi\)
−0.439679 + 0.898155i \(0.644908\pi\)
\(420\) 2.00000 0.0975900
\(421\) − 4.00000i − 0.194948i −0.995238 0.0974740i \(-0.968924\pi\)
0.995238 0.0974740i \(-0.0310763\pi\)
\(422\) − 4.00000i − 0.194717i
\(423\) 0 0
\(424\) 6.00000i 0.291386i
\(425\) 0 0
\(426\) 0 0
\(427\) 20.0000i 0.967868i
\(428\) −12.0000 −0.580042
\(429\) 0 0
\(430\) 4.00000 0.192897
\(431\) 24.0000i 1.15604i 0.816023 + 0.578020i \(0.196174\pi\)
−0.816023 + 0.578020i \(0.803826\pi\)
\(432\) 1.00000 0.0481125
\(433\) 34.0000 1.63394 0.816968 0.576683i \(-0.195653\pi\)
0.816968 + 0.576683i \(0.195653\pi\)
\(434\) − 8.00000i − 0.384012i
\(435\) 0 0
\(436\) 16.0000i 0.766261i
\(437\) 12.0000i 0.574038i
\(438\) 8.00000 0.382255
\(439\) −8.00000 −0.381819 −0.190910 0.981608i \(-0.561144\pi\)
−0.190910 + 0.981608i \(0.561144\pi\)
\(440\) 0 0
\(441\) 3.00000 0.142857
\(442\) 0 0
\(443\) −24.0000 −1.14027 −0.570137 0.821549i \(-0.693110\pi\)
−0.570137 + 0.821549i \(0.693110\pi\)
\(444\) 2.00000i 0.0949158i
\(445\) −6.00000 −0.284427
\(446\) −14.0000 −0.662919
\(447\) 6.00000i 0.283790i
\(448\) 2.00000i 0.0944911i
\(449\) 18.0000i 0.849473i 0.905317 + 0.424736i \(0.139633\pi\)
−0.905317 + 0.424736i \(0.860367\pi\)
\(450\) − 1.00000i − 0.0471405i
\(451\) 0 0
\(452\) 12.0000 0.564433
\(453\) − 20.0000i − 0.939682i
\(454\) −12.0000 −0.563188
\(455\) 0 0
\(456\) 2.00000 0.0936586
\(457\) 32.0000i 1.49690i 0.663193 + 0.748448i \(0.269201\pi\)
−0.663193 + 0.748448i \(0.730799\pi\)
\(458\) −16.0000 −0.747631
\(459\) 0 0
\(460\) 6.00000i 0.279751i
\(461\) 18.0000i 0.838344i 0.907907 + 0.419172i \(0.137680\pi\)
−0.907907 + 0.419172i \(0.862320\pi\)
\(462\) 0 0
\(463\) − 14.0000i − 0.650635i −0.945605 0.325318i \(-0.894529\pi\)
0.945605 0.325318i \(-0.105471\pi\)
\(464\) 0 0
\(465\) −4.00000 −0.185496
\(466\) − 24.0000i − 1.11178i
\(467\) 12.0000 0.555294 0.277647 0.960683i \(-0.410445\pi\)
0.277647 + 0.960683i \(0.410445\pi\)
\(468\) 0 0
\(469\) 16.0000 0.738811
\(470\) 0 0
\(471\) 2.00000 0.0921551
\(472\) 0 0
\(473\) 0 0
\(474\) 8.00000i 0.367452i
\(475\) − 2.00000i − 0.0917663i
\(476\) 0 0
\(477\) −6.00000 −0.274721
\(478\) 24.0000 1.09773
\(479\) 24.0000i 1.09659i 0.836286 + 0.548294i \(0.184723\pi\)
−0.836286 + 0.548294i \(0.815277\pi\)
\(480\) 1.00000 0.0456435
\(481\) 0 0
\(482\) 2.00000 0.0910975
\(483\) − 12.0000i − 0.546019i
\(484\) −11.0000 −0.500000
\(485\) 8.00000 0.363261
\(486\) 1.00000i 0.0453609i
\(487\) 2.00000i 0.0906287i 0.998973 + 0.0453143i \(0.0144289\pi\)
−0.998973 + 0.0453143i \(0.985571\pi\)
\(488\) 10.0000i 0.452679i
\(489\) − 20.0000i − 0.904431i
\(490\) 3.00000 0.135526
\(491\) −6.00000 −0.270776 −0.135388 0.990793i \(-0.543228\pi\)
−0.135388 + 0.990793i \(0.543228\pi\)
\(492\) 6.00000i 0.270501i
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) − 4.00000i − 0.179605i
\(497\) 0 0
\(498\) 12.0000 0.537733
\(499\) 14.0000i 0.626726i 0.949633 + 0.313363i \(0.101456\pi\)
−0.949633 + 0.313363i \(0.898544\pi\)
\(500\) − 1.00000i − 0.0447214i
\(501\) − 12.0000i − 0.536120i
\(502\) − 18.0000i − 0.803379i
\(503\) 6.00000 0.267527 0.133763 0.991013i \(-0.457294\pi\)
0.133763 + 0.991013i \(0.457294\pi\)
\(504\) −2.00000 −0.0890871
\(505\) − 12.0000i − 0.533993i
\(506\) 0 0
\(507\) 0 0
\(508\) 8.00000 0.354943
\(509\) 18.0000i 0.797836i 0.916987 + 0.398918i \(0.130614\pi\)
−0.916987 + 0.398918i \(0.869386\pi\)
\(510\) 0 0
\(511\) −16.0000 −0.707798
\(512\) 1.00000i 0.0441942i
\(513\) 2.00000i 0.0883022i
\(514\) − 12.0000i − 0.529297i
\(515\) 8.00000i 0.352522i
\(516\) −4.00000 −0.176090
\(517\) 0 0
\(518\) − 4.00000i − 0.175750i
\(519\) 6.00000 0.263371
\(520\) 0 0
\(521\) −18.0000 −0.788594 −0.394297 0.918983i \(-0.629012\pi\)
−0.394297 + 0.918983i \(0.629012\pi\)
\(522\) 0 0
\(523\) −4.00000 −0.174908 −0.0874539 0.996169i \(-0.527873\pi\)
−0.0874539 + 0.996169i \(0.527873\pi\)
\(524\) 18.0000 0.786334
\(525\) 2.00000i 0.0872872i
\(526\) 6.00000i 0.261612i
\(527\) 0 0
\(528\) 0 0
\(529\) 13.0000 0.565217
\(530\) −6.00000 −0.260623
\(531\) 0 0
\(532\) −4.00000 −0.173422
\(533\) 0 0
\(534\) 6.00000 0.259645
\(535\) − 12.0000i − 0.518805i
\(536\) 8.00000 0.345547
\(537\) −18.0000 −0.776757
\(538\) 12.0000i 0.517357i
\(539\) 0 0
\(540\) 1.00000i 0.0430331i
\(541\) − 20.0000i − 0.859867i −0.902861 0.429934i \(-0.858537\pi\)
0.902861 0.429934i \(-0.141463\pi\)
\(542\) −4.00000 −0.171815
\(543\) 22.0000 0.944110
\(544\) 0 0
\(545\) −16.0000 −0.685365
\(546\) 0 0
\(547\) −4.00000 −0.171028 −0.0855138 0.996337i \(-0.527253\pi\)
−0.0855138 + 0.996337i \(0.527253\pi\)
\(548\) 18.0000i 0.768922i
\(549\) −10.0000 −0.426790
\(550\) 0 0
\(551\) 0 0
\(552\) − 6.00000i − 0.255377i
\(553\) − 16.0000i − 0.680389i
\(554\) 22.0000i 0.934690i
\(555\) −2.00000 −0.0848953
\(556\) −8.00000 −0.339276
\(557\) 6.00000i 0.254228i 0.991888 + 0.127114i \(0.0405714\pi\)
−0.991888 + 0.127114i \(0.959429\pi\)
\(558\) 4.00000 0.169334
\(559\) 0 0
\(560\) −2.00000 −0.0845154
\(561\) 0 0
\(562\) 6.00000 0.253095
\(563\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(564\) 0 0
\(565\) 12.0000i 0.504844i
\(566\) 4.00000i 0.168133i
\(567\) − 2.00000i − 0.0839921i
\(568\) 0 0
\(569\) −30.0000 −1.25767 −0.628833 0.777541i \(-0.716467\pi\)
−0.628833 + 0.777541i \(0.716467\pi\)
\(570\) 2.00000i 0.0837708i
\(571\) 16.0000 0.669579 0.334790 0.942293i \(-0.391335\pi\)
0.334790 + 0.942293i \(0.391335\pi\)
\(572\) 0 0
\(573\) 24.0000 1.00261
\(574\) − 12.0000i − 0.500870i
\(575\) −6.00000 −0.250217
\(576\) −1.00000 −0.0416667
\(577\) − 4.00000i − 0.166522i −0.996528 0.0832611i \(-0.973466\pi\)
0.996528 0.0832611i \(-0.0265335\pi\)
\(578\) − 17.0000i − 0.707107i
\(579\) − 20.0000i − 0.831172i
\(580\) 0 0
\(581\) −24.0000 −0.995688
\(582\) −8.00000 −0.331611
\(583\) 0 0
\(584\) −8.00000 −0.331042
\(585\) 0 0
\(586\) 6.00000 0.247858
\(587\) 36.0000i 1.48588i 0.669359 + 0.742940i \(0.266569\pi\)
−0.669359 + 0.742940i \(0.733431\pi\)
\(588\) −3.00000 −0.123718
\(589\) 8.00000 0.329634
\(590\) 0 0
\(591\) 6.00000i 0.246807i
\(592\) − 2.00000i − 0.0821995i
\(593\) − 6.00000i − 0.246390i −0.992382 0.123195i \(-0.960686\pi\)
0.992382 0.123195i \(-0.0393141\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) − 6.00000i − 0.245770i
\(597\) 16.0000 0.654836
\(598\) 0 0
\(599\) −24.0000 −0.980613 −0.490307 0.871550i \(-0.663115\pi\)
−0.490307 + 0.871550i \(0.663115\pi\)
\(600\) 1.00000i 0.0408248i
\(601\) 38.0000 1.55005 0.775026 0.631929i \(-0.217737\pi\)
0.775026 + 0.631929i \(0.217737\pi\)
\(602\) 8.00000 0.326056
\(603\) 8.00000i 0.325785i
\(604\) 20.0000i 0.813788i
\(605\) − 11.0000i − 0.447214i
\(606\) 12.0000i 0.487467i
\(607\) 32.0000 1.29884 0.649420 0.760430i \(-0.275012\pi\)
0.649420 + 0.760430i \(0.275012\pi\)
\(608\) −2.00000 −0.0811107
\(609\) 0 0
\(610\) −10.0000 −0.404888
\(611\) 0 0
\(612\) 0 0
\(613\) 26.0000i 1.05013i 0.851062 + 0.525065i \(0.175959\pi\)
−0.851062 + 0.525065i \(0.824041\pi\)
\(614\) −16.0000 −0.645707
\(615\) −6.00000 −0.241943
\(616\) 0 0
\(617\) 18.0000i 0.724653i 0.932051 + 0.362326i \(0.118017\pi\)
−0.932051 + 0.362326i \(0.881983\pi\)
\(618\) − 8.00000i − 0.321807i
\(619\) − 14.0000i − 0.562708i −0.959604 0.281354i \(-0.909217\pi\)
0.959604 0.281354i \(-0.0907834\pi\)
\(620\) 4.00000 0.160644
\(621\) 6.00000 0.240772
\(622\) 0 0
\(623\) −12.0000 −0.480770
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 14.0000i 0.559553i
\(627\) 0 0
\(628\) −2.00000 −0.0798087
\(629\) 0 0
\(630\) − 2.00000i − 0.0796819i
\(631\) − 8.00000i − 0.318475i −0.987240 0.159237i \(-0.949096\pi\)
0.987240 0.159237i \(-0.0509036\pi\)
\(632\) − 8.00000i − 0.318223i
\(633\) −4.00000 −0.158986
\(634\) 30.0000 1.19145
\(635\) 8.00000i 0.317470i
\(636\) 6.00000 0.237915
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) −1.00000 −0.0395285
\(641\) −42.0000 −1.65890 −0.829450 0.558581i \(-0.811346\pi\)
−0.829450 + 0.558581i \(0.811346\pi\)
\(642\) 12.0000i 0.473602i
\(643\) − 40.0000i − 1.57745i −0.614749 0.788723i \(-0.710743\pi\)
0.614749 0.788723i \(-0.289257\pi\)
\(644\) 12.0000i 0.472866i
\(645\) − 4.00000i − 0.157500i
\(646\) 0 0
\(647\) 18.0000 0.707653 0.353827 0.935311i \(-0.384880\pi\)
0.353827 + 0.935311i \(0.384880\pi\)
\(648\) − 1.00000i − 0.0392837i
\(649\) 0 0
\(650\) 0 0
\(651\) −8.00000 −0.313545
\(652\) 20.0000i 0.783260i
\(653\) 6.00000 0.234798 0.117399 0.993085i \(-0.462544\pi\)
0.117399 + 0.993085i \(0.462544\pi\)
\(654\) 16.0000 0.625650
\(655\) 18.0000i 0.703318i
\(656\) − 6.00000i − 0.234261i
\(657\) − 8.00000i − 0.312110i
\(658\) 0 0
\(659\) −18.0000 −0.701180 −0.350590 0.936529i \(-0.614019\pi\)
−0.350590 + 0.936529i \(0.614019\pi\)
\(660\) 0 0
\(661\) 28.0000i 1.08907i 0.838737 + 0.544537i \(0.183295\pi\)
−0.838737 + 0.544537i \(0.816705\pi\)
\(662\) −26.0000 −1.01052
\(663\) 0 0
\(664\) −12.0000 −0.465690
\(665\) − 4.00000i − 0.155113i
\(666\) 2.00000 0.0774984
\(667\) 0 0
\(668\) 12.0000i 0.464294i
\(669\) 14.0000i 0.541271i
\(670\) 8.00000i 0.309067i
\(671\) 0 0
\(672\) 2.00000 0.0771517
\(673\) −26.0000 −1.00223 −0.501113 0.865382i \(-0.667076\pi\)
−0.501113 + 0.865382i \(0.667076\pi\)
\(674\) 34.0000i 1.30963i
\(675\) −1.00000 −0.0384900
\(676\) 0 0
\(677\) 6.00000 0.230599 0.115299 0.993331i \(-0.463217\pi\)
0.115299 + 0.993331i \(0.463217\pi\)
\(678\) − 12.0000i − 0.460857i
\(679\) 16.0000 0.614024
\(680\) 0 0
\(681\) 12.0000i 0.459841i
\(682\) 0 0
\(683\) − 36.0000i − 1.37750i −0.724998 0.688751i \(-0.758159\pi\)
0.724998 0.688751i \(-0.241841\pi\)
\(684\) − 2.00000i − 0.0764719i
\(685\) −18.0000 −0.687745
\(686\) 20.0000 0.763604
\(687\) 16.0000i 0.610438i
\(688\) 4.00000 0.152499
\(689\) 0 0
\(690\) 6.00000 0.228416
\(691\) 2.00000i 0.0760836i 0.999276 + 0.0380418i \(0.0121120\pi\)
−0.999276 + 0.0380418i \(0.987888\pi\)
\(692\) −6.00000 −0.228086
\(693\) 0 0
\(694\) 24.0000i 0.911028i
\(695\) − 8.00000i − 0.303457i
\(696\) 0 0
\(697\) 0 0
\(698\) −16.0000 −0.605609
\(699\) −24.0000 −0.907763
\(700\) − 2.00000i − 0.0755929i
\(701\) 36.0000 1.35970 0.679851 0.733351i \(-0.262045\pi\)
0.679851 + 0.733351i \(0.262045\pi\)
\(702\) 0 0
\(703\) 4.00000 0.150863
\(704\) 0 0
\(705\) 0 0
\(706\) −18.0000 −0.677439
\(707\) − 24.0000i − 0.902613i
\(708\) 0 0
\(709\) − 8.00000i − 0.300446i −0.988652 0.150223i \(-0.952001\pi\)
0.988652 0.150223i \(-0.0479992\pi\)
\(710\) 0 0
\(711\) 8.00000 0.300023
\(712\) −6.00000 −0.224860
\(713\) − 24.0000i − 0.898807i
\(714\) 0 0
\(715\) 0 0
\(716\) 18.0000 0.672692
\(717\) − 24.0000i − 0.896296i
\(718\) 0 0
\(719\) −36.0000 −1.34257 −0.671287 0.741198i \(-0.734258\pi\)
−0.671287 + 0.741198i \(0.734258\pi\)
\(720\) − 1.00000i − 0.0372678i
\(721\) 16.0000i 0.595871i
\(722\) 15.0000i 0.558242i
\(723\) − 2.00000i − 0.0743808i
\(724\) −22.0000 −0.817624
\(725\) 0 0
\(726\) 11.0000i 0.408248i
\(727\) 28.0000 1.03846 0.519231 0.854634i \(-0.326218\pi\)
0.519231 + 0.854634i \(0.326218\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) − 8.00000i − 0.296093i
\(731\) 0 0
\(732\) 10.0000 0.369611
\(733\) 26.0000i 0.960332i 0.877178 + 0.480166i \(0.159424\pi\)
−0.877178 + 0.480166i \(0.840576\pi\)
\(734\) − 16.0000i − 0.590571i
\(735\) − 3.00000i − 0.110657i
\(736\) 6.00000i 0.221163i
\(737\) 0 0
\(738\) 6.00000 0.220863
\(739\) 34.0000i 1.25071i 0.780340 + 0.625355i \(0.215046\pi\)
−0.780340 + 0.625355i \(0.784954\pi\)
\(740\) 2.00000 0.0735215
\(741\) 0 0
\(742\) −12.0000 −0.440534
\(743\) − 12.0000i − 0.440237i −0.975473 0.220119i \(-0.929356\pi\)
0.975473 0.220119i \(-0.0706445\pi\)
\(744\) −4.00000 −0.146647
\(745\) 6.00000 0.219823
\(746\) − 22.0000i − 0.805477i
\(747\) − 12.0000i − 0.439057i
\(748\) 0 0
\(749\) − 24.0000i − 0.876941i
\(750\) −1.00000 −0.0365148
\(751\) −32.0000 −1.16770 −0.583848 0.811863i \(-0.698454\pi\)
−0.583848 + 0.811863i \(0.698454\pi\)
\(752\) 0 0
\(753\) −18.0000 −0.655956
\(754\) 0 0
\(755\) −20.0000 −0.727875
\(756\) 2.00000i 0.0727393i
\(757\) 14.0000 0.508839 0.254419 0.967094i \(-0.418116\pi\)
0.254419 + 0.967094i \(0.418116\pi\)
\(758\) 10.0000 0.363216
\(759\) 0 0
\(760\) − 2.00000i − 0.0725476i
\(761\) 54.0000i 1.95750i 0.205061 + 0.978749i \(0.434261\pi\)
−0.205061 + 0.978749i \(0.565739\pi\)
\(762\) − 8.00000i − 0.289809i
\(763\) −32.0000 −1.15848
\(764\) −24.0000 −0.868290
\(765\) 0 0
\(766\) 36.0000 1.30073
\(767\) 0 0
\(768\) 1.00000 0.0360844
\(769\) − 22.0000i − 0.793340i −0.917961 0.396670i \(-0.870166\pi\)
0.917961 0.396670i \(-0.129834\pi\)
\(770\) 0 0
\(771\) −12.0000 −0.432169
\(772\) 20.0000i 0.719816i
\(773\) − 42.0000i − 1.51064i −0.655359 0.755318i \(-0.727483\pi\)
0.655359 0.755318i \(-0.272517\pi\)
\(774\) 4.00000i 0.143777i
\(775\) 4.00000i 0.143684i
\(776\) 8.00000 0.287183
\(777\) −4.00000 −0.143499
\(778\) − 12.0000i − 0.430221i
\(779\) 12.0000 0.429945
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 3.00000 0.107143
\(785\) − 2.00000i − 0.0713831i
\(786\) − 18.0000i − 0.642039i
\(787\) 4.00000i 0.142585i 0.997455 + 0.0712923i \(0.0227123\pi\)
−0.997455 + 0.0712923i \(0.977288\pi\)
\(788\) − 6.00000i − 0.213741i
\(789\) 6.00000 0.213606
\(790\) 8.00000 0.284627
\(791\) 24.0000i 0.853342i
\(792\) 0 0
\(793\) 0 0
\(794\) −22.0000 −0.780751
\(795\) 6.00000i 0.212798i
\(796\) −16.0000 −0.567105
\(797\) −18.0000 −0.637593 −0.318796 0.947823i \(-0.603279\pi\)
−0.318796 + 0.947823i \(0.603279\pi\)
\(798\) 4.00000i 0.141598i
\(799\) 0 0
\(800\) − 1.00000i − 0.0353553i
\(801\) − 6.00000i − 0.212000i
\(802\) −6.00000 −0.211867
\(803\) 0 0
\(804\) − 8.00000i − 0.282138i
\(805\) −12.0000 −0.422944
\(806\) 0 0
\(807\) 12.0000 0.422420
\(808\) − 12.0000i − 0.422159i
\(809\) 18.0000 0.632846 0.316423 0.948618i \(-0.397518\pi\)
0.316423 + 0.948618i \(0.397518\pi\)
\(810\) 1.00000 0.0351364
\(811\) − 22.0000i − 0.772524i −0.922389 0.386262i \(-0.873766\pi\)
0.922389 0.386262i \(-0.126234\pi\)
\(812\) 0 0
\(813\) 4.00000i 0.140286i
\(814\) 0 0
\(815\) −20.0000 −0.700569
\(816\) 0 0
\(817\) 8.00000i 0.279885i
\(818\) −14.0000 −0.489499
\(819\) 0 0
\(820\) 6.00000 0.209529
\(821\) 30.0000i 1.04701i 0.852023 + 0.523504i \(0.175375\pi\)
−0.852023 + 0.523504i \(0.824625\pi\)
\(822\) 18.0000 0.627822
\(823\) 40.0000 1.39431 0.697156 0.716919i \(-0.254448\pi\)
0.697156 + 0.716919i \(0.254448\pi\)
\(824\) 8.00000i 0.278693i
\(825\) 0 0
\(826\) 0 0
\(827\) 36.0000i 1.25184i 0.779886 + 0.625921i \(0.215277\pi\)
−0.779886 + 0.625921i \(0.784723\pi\)
\(828\) −6.00000 −0.208514
\(829\) 34.0000 1.18087 0.590434 0.807086i \(-0.298956\pi\)
0.590434 + 0.807086i \(0.298956\pi\)
\(830\) − 12.0000i − 0.416526i
\(831\) 22.0000 0.763172
\(832\) 0 0
\(833\) 0 0
\(834\) 8.00000i 0.277017i
\(835\) −12.0000 −0.415277
\(836\) 0 0
\(837\) − 4.00000i − 0.138260i
\(838\) − 18.0000i − 0.621800i
\(839\) 24.0000i 0.828572i 0.910147 + 0.414286i \(0.135969\pi\)
−0.910147 + 0.414286i \(0.864031\pi\)
\(840\) 2.00000i 0.0690066i
\(841\) −29.0000 −1.00000
\(842\) 4.00000 0.137849
\(843\) − 6.00000i − 0.206651i
\(844\) 4.00000 0.137686
\(845\) 0 0
\(846\) 0 0
\(847\) − 22.0000i − 0.755929i
\(848\) −6.00000 −0.206041
\(849\) 4.00000 0.137280
\(850\) 0 0
\(851\) − 12.0000i − 0.411355i
\(852\) 0 0
\(853\) − 26.0000i − 0.890223i −0.895475 0.445112i \(-0.853164\pi\)
0.895475 0.445112i \(-0.146836\pi\)
\(854\) −20.0000 −0.684386
\(855\) 2.00000 0.0683986
\(856\) − 12.0000i − 0.410152i
\(857\) −12.0000 −0.409912 −0.204956 0.978771i \(-0.565705\pi\)
−0.204956 + 0.978771i \(0.565705\pi\)
\(858\) 0 0
\(859\) 20.0000 0.682391 0.341196 0.939992i \(-0.389168\pi\)
0.341196 + 0.939992i \(0.389168\pi\)
\(860\) 4.00000i 0.136399i
\(861\) −12.0000 −0.408959
\(862\) −24.0000 −0.817443
\(863\) 36.0000i 1.22545i 0.790295 + 0.612727i \(0.209928\pi\)
−0.790295 + 0.612727i \(0.790072\pi\)
\(864\) 1.00000i 0.0340207i
\(865\) − 6.00000i − 0.204006i
\(866\) 34.0000i 1.15537i
\(867\) −17.0000 −0.577350
\(868\) 8.00000 0.271538
\(869\) 0 0
\(870\) 0 0
\(871\) 0 0
\(872\) −16.0000 −0.541828
\(873\) 8.00000i 0.270759i
\(874\) −12.0000 −0.405906
\(875\) 2.00000 0.0676123
\(876\) 8.00000i 0.270295i
\(877\) − 34.0000i − 1.14810i −0.818821 0.574049i \(-0.805372\pi\)
0.818821 0.574049i \(-0.194628\pi\)
\(878\) − 8.00000i − 0.269987i
\(879\) − 6.00000i − 0.202375i
\(880\) 0 0
\(881\) 18.0000 0.606435 0.303218 0.952921i \(-0.401939\pi\)
0.303218 + 0.952921i \(0.401939\pi\)
\(882\) 3.00000i 0.101015i
\(883\) 52.0000 1.74994 0.874970 0.484178i \(-0.160881\pi\)
0.874970 + 0.484178i \(0.160881\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) − 24.0000i − 0.806296i
\(887\) 18.0000 0.604381 0.302190 0.953248i \(-0.402282\pi\)
0.302190 + 0.953248i \(0.402282\pi\)
\(888\) −2.00000 −0.0671156
\(889\) 16.0000i 0.536623i
\(890\) − 6.00000i − 0.201120i
\(891\) 0 0
\(892\) − 14.0000i − 0.468755i
\(893\) 0 0
\(894\) −6.00000 −0.200670
\(895\) 18.0000i 0.601674i
\(896\) −2.00000 −0.0668153
\(897\) 0 0
\(898\) −18.0000 −0.600668
\(899\) 0 0
\(900\) 1.00000 0.0333333
\(901\) 0 0
\(902\) 0 0
\(903\) − 8.00000i − 0.266223i
\(904\) 12.0000i 0.399114i
\(905\) − 22.0000i − 0.731305i
\(906\) 20.0000 0.664455
\(907\) −44.0000 −1.46100 −0.730498 0.682915i \(-0.760712\pi\)
−0.730498 + 0.682915i \(0.760712\pi\)
\(908\) − 12.0000i − 0.398234i
\(909\) 12.0000 0.398015
\(910\) 0 0
\(911\) −36.0000 −1.19273 −0.596367 0.802712i \(-0.703390\pi\)
−0.596367 + 0.802712i \(0.703390\pi\)
\(912\) 2.00000i 0.0662266i
\(913\) 0 0
\(914\) −32.0000 −1.05847
\(915\) 10.0000i 0.330590i
\(916\) − 16.0000i − 0.528655i
\(917\) 36.0000i 1.18882i
\(918\) 0 0
\(919\) 56.0000 1.84727 0.923635 0.383274i \(-0.125203\pi\)
0.923635 + 0.383274i \(0.125203\pi\)
\(920\) −6.00000 −0.197814
\(921\) 16.0000i 0.527218i
\(922\) −18.0000 −0.592798
\(923\) 0 0
\(924\) 0 0
\(925\) 2.00000i 0.0657596i
\(926\) 14.0000 0.460069
\(927\) −8.00000 −0.262754
\(928\) 0 0
\(929\) 30.0000i 0.984268i 0.870519 + 0.492134i \(0.163783\pi\)
−0.870519 + 0.492134i \(0.836217\pi\)
\(930\) − 4.00000i − 0.131165i
\(931\) 6.00000i 0.196642i
\(932\) 24.0000 0.786146
\(933\) 0 0
\(934\) 12.0000i 0.392652i
\(935\) 0 0
\(936\) 0 0
\(937\) 26.0000 0.849383 0.424691 0.905338i \(-0.360383\pi\)
0.424691 + 0.905338i \(0.360383\pi\)
\(938\) 16.0000i 0.522419i
\(939\) 14.0000 0.456873
\(940\) 0 0
\(941\) 6.00000i 0.195594i 0.995206 + 0.0977972i \(0.0311797\pi\)
−0.995206 + 0.0977972i \(0.968820\pi\)
\(942\) 2.00000i 0.0651635i
\(943\) − 36.0000i − 1.17232i
\(944\) 0 0
\(945\) −2.00000 −0.0650600
\(946\) 0 0
\(947\) − 36.0000i − 1.16984i −0.811090 0.584921i \(-0.801125\pi\)
0.811090 0.584921i \(-0.198875\pi\)
\(948\) −8.00000 −0.259828
\(949\) 0 0
\(950\) 2.00000 0.0648886
\(951\) − 30.0000i − 0.972817i
\(952\) 0 0
\(953\) −48.0000 −1.55487 −0.777436 0.628962i \(-0.783480\pi\)
−0.777436 + 0.628962i \(0.783480\pi\)
\(954\) − 6.00000i − 0.194257i
\(955\) − 24.0000i − 0.776622i
\(956\) 24.0000i 0.776215i
\(957\) 0 0
\(958\) −24.0000 −0.775405
\(959\) −36.0000 −1.16250
\(960\) 1.00000i 0.0322749i
\(961\) 15.0000 0.483871
\(962\) 0 0
\(963\) 12.0000 0.386695
\(964\) 2.00000i 0.0644157i
\(965\) −20.0000 −0.643823
\(966\) 12.0000 0.386094
\(967\) 50.0000i 1.60789i 0.594703 + 0.803946i \(0.297270\pi\)
−0.594703 + 0.803946i \(0.702730\pi\)
\(968\) − 11.0000i − 0.353553i
\(969\) 0 0
\(970\) 8.00000i 0.256865i
\(971\) 18.0000 0.577647 0.288824 0.957382i \(-0.406736\pi\)
0.288824 + 0.957382i \(0.406736\pi\)
\(972\) −1.00000 −0.0320750
\(973\) − 16.0000i − 0.512936i
\(974\) −2.00000 −0.0640841
\(975\) 0 0
\(976\) −10.0000 −0.320092
\(977\) 6.00000i 0.191957i 0.995383 + 0.0959785i \(0.0305980\pi\)
−0.995383 + 0.0959785i \(0.969402\pi\)
\(978\) 20.0000 0.639529
\(979\) 0 0
\(980\) 3.00000i 0.0958315i
\(981\) − 16.0000i − 0.510841i
\(982\) − 6.00000i − 0.191468i
\(983\) 12.0000i 0.382741i 0.981518 + 0.191370i \(0.0612931\pi\)
−0.981518 + 0.191370i \(0.938707\pi\)
\(984\) −6.00000 −0.191273
\(985\) 6.00000 0.191176
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 24.0000 0.763156
\(990\) 0 0
\(991\) 32.0000 1.01651 0.508257 0.861206i \(-0.330290\pi\)
0.508257 + 0.861206i \(0.330290\pi\)
\(992\) 4.00000 0.127000
\(993\) 26.0000i 0.825085i
\(994\) 0 0
\(995\) − 16.0000i − 0.507234i
\(996\) 12.0000i 0.380235i
\(997\) −46.0000 −1.45683 −0.728417 0.685134i \(-0.759744\pi\)
−0.728417 + 0.685134i \(0.759744\pi\)
\(998\) −14.0000 −0.443162
\(999\) − 2.00000i − 0.0632772i
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 5070.2.b.n.1351.2 2
13.5 odd 4 390.2.a.g.1.1 1
13.8 odd 4 5070.2.a.k.1.1 1
13.12 even 2 inner 5070.2.b.n.1351.1 2
39.5 even 4 1170.2.a.g.1.1 1
52.31 even 4 3120.2.a.b.1.1 1
65.18 even 4 1950.2.e.k.1249.1 2
65.44 odd 4 1950.2.a.b.1.1 1
65.57 even 4 1950.2.e.k.1249.2 2
156.83 odd 4 9360.2.a.bg.1.1 1
195.44 even 4 5850.2.a.bk.1.1 1
195.83 odd 4 5850.2.e.r.5149.2 2
195.122 odd 4 5850.2.e.r.5149.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
390.2.a.g.1.1 1 13.5 odd 4
1170.2.a.g.1.1 1 39.5 even 4
1950.2.a.b.1.1 1 65.44 odd 4
1950.2.e.k.1249.1 2 65.18 even 4
1950.2.e.k.1249.2 2 65.57 even 4
3120.2.a.b.1.1 1 52.31 even 4
5070.2.a.k.1.1 1 13.8 odd 4
5070.2.b.n.1351.1 2 13.12 even 2 inner
5070.2.b.n.1351.2 2 1.1 even 1 trivial
5850.2.a.bk.1.1 1 195.44 even 4
5850.2.e.r.5149.1 2 195.122 odd 4
5850.2.e.r.5149.2 2 195.83 odd 4
9360.2.a.bg.1.1 1 156.83 odd 4