Properties

Label 1425.2.c.c.799.2
Level $1425$
Weight $2$
Character 1425.799
Analytic conductor $11.379$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1425 = 3 \cdot 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1425.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(11.3786822880\)
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 285)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 799.2
Root \(1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 1425.799
Dual form 1425.2.c.c.799.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000i q^{2} +1.00000i q^{3} +1.00000 q^{4} -1.00000 q^{6} +2.00000i q^{7} +3.00000i q^{8} -1.00000 q^{9} +O(q^{10})\) \(q+1.00000i q^{2} +1.00000i q^{3} +1.00000 q^{4} -1.00000 q^{6} +2.00000i q^{7} +3.00000i q^{8} -1.00000 q^{9} -6.00000 q^{11} +1.00000i q^{12} -2.00000 q^{14} -1.00000 q^{16} +6.00000i q^{17} -1.00000i q^{18} -1.00000 q^{19} -2.00000 q^{21} -6.00000i q^{22} -8.00000i q^{23} -3.00000 q^{24} -1.00000i q^{27} +2.00000i q^{28} -4.00000 q^{29} +5.00000i q^{32} -6.00000i q^{33} -6.00000 q^{34} -1.00000 q^{36} -4.00000i q^{37} -1.00000i q^{38} -2.00000i q^{42} -2.00000i q^{43} -6.00000 q^{44} +8.00000 q^{46} +8.00000i q^{47} -1.00000i q^{48} +3.00000 q^{49} -6.00000 q^{51} +2.00000i q^{53} +1.00000 q^{54} -6.00000 q^{56} -1.00000i q^{57} -4.00000i q^{58} -12.0000 q^{59} +2.00000 q^{61} -2.00000i q^{63} -7.00000 q^{64} +6.00000 q^{66} +8.00000i q^{67} +6.00000i q^{68} +8.00000 q^{69} +16.0000 q^{71} -3.00000i q^{72} +14.0000i q^{73} +4.00000 q^{74} -1.00000 q^{76} -12.0000i q^{77} -8.00000 q^{79} +1.00000 q^{81} -2.00000 q^{84} +2.00000 q^{86} -4.00000i q^{87} -18.0000i q^{88} -8.00000i q^{92} -8.00000 q^{94} -5.00000 q^{96} +12.0000i q^{97} +3.00000i q^{98} +6.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{4} - 2 q^{6} - 2 q^{9} + O(q^{10}) \) \( 2 q + 2 q^{4} - 2 q^{6} - 2 q^{9} - 12 q^{11} - 4 q^{14} - 2 q^{16} - 2 q^{19} - 4 q^{21} - 6 q^{24} - 8 q^{29} - 12 q^{34} - 2 q^{36} - 12 q^{44} + 16 q^{46} + 6 q^{49} - 12 q^{51} + 2 q^{54} - 12 q^{56} - 24 q^{59} + 4 q^{61} - 14 q^{64} + 12 q^{66} + 16 q^{69} + 32 q^{71} + 8 q^{74} - 2 q^{76} - 16 q^{79} + 2 q^{81} - 4 q^{84} + 4 q^{86} - 16 q^{94} - 10 q^{96} + 12 q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1425\mathbb{Z}\right)^\times\).

\(n\) \(476\) \(1027\) \(1351\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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 0.935414 + 0.353553i \(0.115027\pi\)
−0.935414 + 0.353553i \(0.884973\pi\)
\(3\) 1.00000i 0.577350i
\(4\) 1.00000 0.500000
\(5\) 0 0
\(6\) −1.00000 −0.408248
\(7\) 2.00000i 0.755929i 0.925820 + 0.377964i \(0.123376\pi\)
−0.925820 + 0.377964i \(0.876624\pi\)
\(8\) 3.00000i 1.06066i
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) −6.00000 −1.80907 −0.904534 0.426401i \(-0.859781\pi\)
−0.904534 + 0.426401i \(0.859781\pi\)
\(12\) 1.00000i 0.288675i
\(13\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(14\) −2.00000 −0.534522
\(15\) 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
\(21\) −2.00000 −0.436436
\(22\) − 6.00000i − 1.27920i
\(23\) − 8.00000i − 1.66812i −0.551677 0.834058i \(-0.686012\pi\)
0.551677 0.834058i \(-0.313988\pi\)
\(24\) −3.00000 −0.612372
\(25\) 0 0
\(26\) 0 0
\(27\) − 1.00000i − 0.192450i
\(28\) 2.00000i 0.377964i
\(29\) −4.00000 −0.742781 −0.371391 0.928477i \(-0.621119\pi\)
−0.371391 + 0.928477i \(0.621119\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) 5.00000i 0.883883i
\(33\) − 6.00000i − 1.04447i
\(34\) −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
\(39\) 0 0
\(40\) 0 0
\(41\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(42\) − 2.00000i − 0.308607i
\(43\) − 2.00000i − 0.304997i −0.988304 0.152499i \(-0.951268\pi\)
0.988304 0.152499i \(-0.0487319\pi\)
\(44\) −6.00000 −0.904534
\(45\) 0 0
\(46\) 8.00000 1.17954
\(47\) 8.00000i 1.16692i 0.812142 + 0.583460i \(0.198301\pi\)
−0.812142 + 0.583460i \(0.801699\pi\)
\(48\) − 1.00000i − 0.144338i
\(49\) 3.00000 0.428571
\(50\) 0 0
\(51\) −6.00000 −0.840168
\(52\) 0 0
\(53\) 2.00000i 0.274721i 0.990521 + 0.137361i \(0.0438619\pi\)
−0.990521 + 0.137361i \(0.956138\pi\)
\(54\) 1.00000 0.136083
\(55\) 0 0
\(56\) −6.00000 −0.801784
\(57\) − 1.00000i − 0.132453i
\(58\) − 4.00000i − 0.525226i
\(59\) −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 \(-0.459132\pi\)
0.128037 + 0.991769i \(0.459132\pi\)
\(62\) 0 0
\(63\) − 2.00000i − 0.251976i
\(64\) −7.00000 −0.875000
\(65\) 0 0
\(66\) 6.00000 0.738549
\(67\) 8.00000i 0.977356i 0.872464 + 0.488678i \(0.162521\pi\)
−0.872464 + 0.488678i \(0.837479\pi\)
\(68\) 6.00000i 0.727607i
\(69\) 8.00000 0.963087
\(70\) 0 0
\(71\) 16.0000 1.89885 0.949425 0.313993i \(-0.101667\pi\)
0.949425 + 0.313993i \(0.101667\pi\)
\(72\) − 3.00000i − 0.353553i
\(73\) 14.0000i 1.63858i 0.573382 + 0.819288i \(0.305631\pi\)
−0.573382 + 0.819288i \(0.694369\pi\)
\(74\) 4.00000 0.464991
\(75\) 0 0
\(76\) −1.00000 −0.114708
\(77\) − 12.0000i − 1.36753i
\(78\) 0 0
\(79\) −8.00000 −0.900070 −0.450035 0.893011i \(-0.648589\pi\)
−0.450035 + 0.893011i \(0.648589\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(84\) −2.00000 −0.218218
\(85\) 0 0
\(86\) 2.00000 0.215666
\(87\) − 4.00000i − 0.428845i
\(88\) − 18.0000i − 1.91881i
\(89\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) − 8.00000i − 0.834058i
\(93\) 0 0
\(94\) −8.00000 −0.825137
\(95\) 0 0
\(96\) −5.00000 −0.510310
\(97\) 12.0000i 1.21842i 0.793011 + 0.609208i \(0.208512\pi\)
−0.793011 + 0.609208i \(0.791488\pi\)
\(98\) 3.00000i 0.303046i
\(99\) 6.00000 0.603023
\(100\) 0 0
\(101\) −18.0000 −1.79107 −0.895533 0.444994i \(-0.853206\pi\)
−0.895533 + 0.444994i \(0.853206\pi\)
\(102\) − 6.00000i − 0.594089i
\(103\) − 8.00000i − 0.788263i −0.919054 0.394132i \(-0.871045\pi\)
0.919054 0.394132i \(-0.128955\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) −2.00000 −0.194257
\(107\) 12.0000i 1.16008i 0.814587 + 0.580042i \(0.196964\pi\)
−0.814587 + 0.580042i \(0.803036\pi\)
\(108\) − 1.00000i − 0.0962250i
\(109\) 6.00000 0.574696 0.287348 0.957826i \(-0.407226\pi\)
0.287348 + 0.957826i \(0.407226\pi\)
\(110\) 0 0
\(111\) 4.00000 0.379663
\(112\) − 2.00000i − 0.188982i
\(113\) − 6.00000i − 0.564433i −0.959351 0.282216i \(-0.908930\pi\)
0.959351 0.282216i \(-0.0910696\pi\)
\(114\) 1.00000 0.0936586
\(115\) 0 0
\(116\) −4.00000 −0.371391
\(117\) 0 0
\(118\) − 12.0000i − 1.10469i
\(119\) −12.0000 −1.10004
\(120\) 0 0
\(121\) 25.0000 2.27273
\(122\) 2.00000i 0.181071i
\(123\) 0 0
\(124\) 0 0
\(125\) 0 0
\(126\) 2.00000 0.178174
\(127\) − 4.00000i − 0.354943i −0.984126 0.177471i \(-0.943208\pi\)
0.984126 0.177471i \(-0.0567917\pi\)
\(128\) 3.00000i 0.265165i
\(129\) 2.00000 0.176090
\(130\) 0 0
\(131\) −2.00000 −0.174741 −0.0873704 0.996176i \(-0.527846\pi\)
−0.0873704 + 0.996176i \(0.527846\pi\)
\(132\) − 6.00000i − 0.522233i
\(133\) − 2.00000i − 0.173422i
\(134\) −8.00000 −0.691095
\(135\) 0 0
\(136\) −18.0000 −1.54349
\(137\) 10.0000i 0.854358i 0.904167 + 0.427179i \(0.140493\pi\)
−0.904167 + 0.427179i \(0.859507\pi\)
\(138\) 8.00000i 0.681005i
\(139\) 16.0000 1.35710 0.678551 0.734553i \(-0.262608\pi\)
0.678551 + 0.734553i \(0.262608\pi\)
\(140\) 0 0
\(141\) −8.00000 −0.673722
\(142\) 16.0000i 1.34269i
\(143\) 0 0
\(144\) 1.00000 0.0833333
\(145\) 0 0
\(146\) −14.0000 −1.15865
\(147\) 3.00000i 0.247436i
\(148\) − 4.00000i − 0.328798i
\(149\) −6.00000 −0.491539 −0.245770 0.969328i \(-0.579041\pi\)
−0.245770 + 0.969328i \(0.579041\pi\)
\(150\) 0 0
\(151\) −16.0000 −1.30206 −0.651031 0.759051i \(-0.725663\pi\)
−0.651031 + 0.759051i \(0.725663\pi\)
\(152\) − 3.00000i − 0.243332i
\(153\) − 6.00000i − 0.485071i
\(154\) 12.0000 0.966988
\(155\) 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
\(159\) −2.00000 −0.158610
\(160\) 0 0
\(161\) 16.0000 1.26098
\(162\) 1.00000i 0.0785674i
\(163\) − 22.0000i − 1.72317i −0.507611 0.861586i \(-0.669471\pi\)
0.507611 0.861586i \(-0.330529\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) − 16.0000i − 1.23812i −0.785345 0.619059i \(-0.787514\pi\)
0.785345 0.619059i \(-0.212486\pi\)
\(168\) − 6.00000i − 0.462910i
\(169\) 13.0000 1.00000
\(170\) 0 0
\(171\) 1.00000 0.0764719
\(172\) − 2.00000i − 0.152499i
\(173\) 6.00000i 0.456172i 0.973641 + 0.228086i \(0.0732467\pi\)
−0.973641 + 0.228086i \(0.926753\pi\)
\(174\) 4.00000 0.303239
\(175\) 0 0
\(176\) 6.00000 0.452267
\(177\) − 12.0000i − 0.901975i
\(178\) 0 0
\(179\) 20.0000 1.49487 0.747435 0.664335i \(-0.231285\pi\)
0.747435 + 0.664335i \(0.231285\pi\)
\(180\) 0 0
\(181\) 10.0000 0.743294 0.371647 0.928374i \(-0.378793\pi\)
0.371647 + 0.928374i \(0.378793\pi\)
\(182\) 0 0
\(183\) 2.00000i 0.147844i
\(184\) 24.0000 1.76930
\(185\) 0 0
\(186\) 0 0
\(187\) − 36.0000i − 2.63258i
\(188\) 8.00000i 0.583460i
\(189\) 2.00000 0.145479
\(190\) 0 0
\(191\) 10.0000 0.723575 0.361787 0.932261i \(-0.382167\pi\)
0.361787 + 0.932261i \(0.382167\pi\)
\(192\) − 7.00000i − 0.505181i
\(193\) 24.0000i 1.72756i 0.503871 + 0.863779i \(0.331909\pi\)
−0.503871 + 0.863779i \(0.668091\pi\)
\(194\) −12.0000 −0.861550
\(195\) 0 0
\(196\) 3.00000 0.214286
\(197\) − 6.00000i − 0.427482i −0.976890 0.213741i \(-0.931435\pi\)
0.976890 0.213741i \(-0.0685649\pi\)
\(198\) 6.00000i 0.426401i
\(199\) 20.0000 1.41776 0.708881 0.705328i \(-0.249200\pi\)
0.708881 + 0.705328i \(0.249200\pi\)
\(200\) 0 0
\(201\) −8.00000 −0.564276
\(202\) − 18.0000i − 1.26648i
\(203\) − 8.00000i − 0.561490i
\(204\) −6.00000 −0.420084
\(205\) 0 0
\(206\) 8.00000 0.557386
\(207\) 8.00000i 0.556038i
\(208\) 0 0
\(209\) 6.00000 0.415029
\(210\) 0 0
\(211\) −20.0000 −1.37686 −0.688428 0.725304i \(-0.741699\pi\)
−0.688428 + 0.725304i \(0.741699\pi\)
\(212\) 2.00000i 0.137361i
\(213\) 16.0000i 1.09630i
\(214\) −12.0000 −0.820303
\(215\) 0 0
\(216\) 3.00000 0.204124
\(217\) 0 0
\(218\) 6.00000i 0.406371i
\(219\) −14.0000 −0.946032
\(220\) 0 0
\(221\) 0 0
\(222\) 4.00000i 0.268462i
\(223\) 8.00000i 0.535720i 0.963458 + 0.267860i \(0.0863164\pi\)
−0.963458 + 0.267860i \(0.913684\pi\)
\(224\) −10.0000 −0.668153
\(225\) 0 0
\(226\) 6.00000 0.399114
\(227\) 28.0000i 1.85843i 0.369546 + 0.929213i \(0.379513\pi\)
−0.369546 + 0.929213i \(0.620487\pi\)
\(228\) − 1.00000i − 0.0662266i
\(229\) −2.00000 −0.132164 −0.0660819 0.997814i \(-0.521050\pi\)
−0.0660819 + 0.997814i \(0.521050\pi\)
\(230\) 0 0
\(231\) 12.0000 0.789542
\(232\) − 12.0000i − 0.787839i
\(233\) 22.0000i 1.44127i 0.693316 + 0.720634i \(0.256149\pi\)
−0.693316 + 0.720634i \(0.743851\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) −12.0000 −0.781133
\(237\) − 8.00000i − 0.519656i
\(238\) − 12.0000i − 0.777844i
\(239\) −6.00000 −0.388108 −0.194054 0.980991i \(-0.562164\pi\)
−0.194054 + 0.980991i \(0.562164\pi\)
\(240\) 0 0
\(241\) 18.0000 1.15948 0.579741 0.814801i \(-0.303154\pi\)
0.579741 + 0.814801i \(0.303154\pi\)
\(242\) 25.0000i 1.60706i
\(243\) 1.00000i 0.0641500i
\(244\) 2.00000 0.128037
\(245\) 0 0
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −6.00000 −0.378717 −0.189358 0.981908i \(-0.560641\pi\)
−0.189358 + 0.981908i \(0.560641\pi\)
\(252\) − 2.00000i − 0.125988i
\(253\) 48.0000i 3.01773i
\(254\) 4.00000 0.250982
\(255\) 0 0
\(256\) −17.0000 −1.06250
\(257\) 22.0000i 1.37232i 0.727450 + 0.686161i \(0.240706\pi\)
−0.727450 + 0.686161i \(0.759294\pi\)
\(258\) 2.00000i 0.124515i
\(259\) 8.00000 0.497096
\(260\) 0 0
\(261\) 4.00000 0.247594
\(262\) − 2.00000i − 0.123560i
\(263\) − 12.0000i − 0.739952i −0.929041 0.369976i \(-0.879366\pi\)
0.929041 0.369976i \(-0.120634\pi\)
\(264\) 18.0000 1.10782
\(265\) 0 0
\(266\) 2.00000 0.122628
\(267\) 0 0
\(268\) 8.00000i 0.488678i
\(269\) −12.0000 −0.731653 −0.365826 0.930683i \(-0.619214\pi\)
−0.365826 + 0.930683i \(0.619214\pi\)
\(270\) 0 0
\(271\) −20.0000 −1.21491 −0.607457 0.794353i \(-0.707810\pi\)
−0.607457 + 0.794353i \(0.707810\pi\)
\(272\) − 6.00000i − 0.363803i
\(273\) 0 0
\(274\) −10.0000 −0.604122
\(275\) 0 0
\(276\) 8.00000 0.481543
\(277\) 2.00000i 0.120168i 0.998193 + 0.0600842i \(0.0191369\pi\)
−0.998193 + 0.0600842i \(0.980863\pi\)
\(278\) 16.0000i 0.959616i
\(279\) 0 0
\(280\) 0 0
\(281\) −4.00000 −0.238620 −0.119310 0.992857i \(-0.538068\pi\)
−0.119310 + 0.992857i \(0.538068\pi\)
\(282\) − 8.00000i − 0.476393i
\(283\) − 14.0000i − 0.832214i −0.909316 0.416107i \(-0.863394\pi\)
0.909316 0.416107i \(-0.136606\pi\)
\(284\) 16.0000 0.949425
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) − 5.00000i − 0.294628i
\(289\) −19.0000 −1.11765
\(290\) 0 0
\(291\) −12.0000 −0.703452
\(292\) 14.0000i 0.819288i
\(293\) 30.0000i 1.75262i 0.481749 + 0.876309i \(0.340002\pi\)
−0.481749 + 0.876309i \(0.659998\pi\)
\(294\) −3.00000 −0.174964
\(295\) 0 0
\(296\) 12.0000 0.697486
\(297\) 6.00000i 0.348155i
\(298\) − 6.00000i − 0.347571i
\(299\) 0 0
\(300\) 0 0
\(301\) 4.00000 0.230556
\(302\) − 16.0000i − 0.920697i
\(303\) − 18.0000i − 1.03407i
\(304\) 1.00000 0.0573539
\(305\) 0 0
\(306\) 6.00000 0.342997
\(307\) 8.00000i 0.456584i 0.973593 + 0.228292i \(0.0733141\pi\)
−0.973593 + 0.228292i \(0.926686\pi\)
\(308\) − 12.0000i − 0.683763i
\(309\) 8.00000 0.455104
\(310\) 0 0
\(311\) 34.0000 1.92796 0.963982 0.265969i \(-0.0856919\pi\)
0.963982 + 0.265969i \(0.0856919\pi\)
\(312\) 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
\(317\) 6.00000i 0.336994i 0.985702 + 0.168497i \(0.0538913\pi\)
−0.985702 + 0.168497i \(0.946109\pi\)
\(318\) − 2.00000i − 0.112154i
\(319\) 24.0000 1.34374
\(320\) 0 0
\(321\) −12.0000 −0.669775
\(322\) 16.0000i 0.891645i
\(323\) − 6.00000i − 0.333849i
\(324\) 1.00000 0.0555556
\(325\) 0 0
\(326\) 22.0000 1.21847
\(327\) 6.00000i 0.331801i
\(328\) 0 0
\(329\) −16.0000 −0.882109
\(330\) 0 0
\(331\) 28.0000 1.53902 0.769510 0.638635i \(-0.220501\pi\)
0.769510 + 0.638635i \(0.220501\pi\)
\(332\) 0 0
\(333\) 4.00000i 0.219199i
\(334\) 16.0000 0.875481
\(335\) 0 0
\(336\) 2.00000 0.109109
\(337\) 12.0000i 0.653682i 0.945079 + 0.326841i \(0.105984\pi\)
−0.945079 + 0.326841i \(0.894016\pi\)
\(338\) 13.0000i 0.707107i
\(339\) 6.00000 0.325875
\(340\) 0 0
\(341\) 0 0
\(342\) 1.00000i 0.0540738i
\(343\) 20.0000i 1.07990i
\(344\) 6.00000 0.323498
\(345\) 0 0
\(346\) −6.00000 −0.322562
\(347\) − 8.00000i − 0.429463i −0.976673 0.214731i \(-0.931112\pi\)
0.976673 0.214731i \(-0.0688876\pi\)
\(348\) − 4.00000i − 0.214423i
\(349\) 2.00000 0.107058 0.0535288 0.998566i \(-0.482953\pi\)
0.0535288 + 0.998566i \(0.482953\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) − 30.0000i − 1.59901i
\(353\) − 34.0000i − 1.80964i −0.425797 0.904819i \(-0.640006\pi\)
0.425797 0.904819i \(-0.359994\pi\)
\(354\) 12.0000 0.637793
\(355\) 0 0
\(356\) 0 0
\(357\) − 12.0000i − 0.635107i
\(358\) 20.0000i 1.05703i
\(359\) −10.0000 −0.527780 −0.263890 0.964553i \(-0.585006\pi\)
−0.263890 + 0.964553i \(0.585006\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 10.0000i 0.525588i
\(363\) 25.0000i 1.31216i
\(364\) 0 0
\(365\) 0 0
\(366\) −2.00000 −0.104542
\(367\) 14.0000i 0.730794i 0.930852 + 0.365397i \(0.119067\pi\)
−0.930852 + 0.365397i \(0.880933\pi\)
\(368\) 8.00000i 0.417029i
\(369\) 0 0
\(370\) 0 0
\(371\) −4.00000 −0.207670
\(372\) 0 0
\(373\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(374\) 36.0000 1.86152
\(375\) 0 0
\(376\) −24.0000 −1.23771
\(377\) 0 0
\(378\) 2.00000i 0.102869i
\(379\) −20.0000 −1.02733 −0.513665 0.857991i \(-0.671713\pi\)
−0.513665 + 0.857991i \(0.671713\pi\)
\(380\) 0 0
\(381\) 4.00000 0.204926
\(382\) 10.0000i 0.511645i
\(383\) 8.00000i 0.408781i 0.978889 + 0.204390i \(0.0655212\pi\)
−0.978889 + 0.204390i \(0.934479\pi\)
\(384\) −3.00000 −0.153093
\(385\) 0 0
\(386\) −24.0000 −1.22157
\(387\) 2.00000i 0.101666i
\(388\) 12.0000i 0.609208i
\(389\) 14.0000 0.709828 0.354914 0.934899i \(-0.384510\pi\)
0.354914 + 0.934899i \(0.384510\pi\)
\(390\) 0 0
\(391\) 48.0000 2.42746
\(392\) 9.00000i 0.454569i
\(393\) − 2.00000i − 0.100887i
\(394\) 6.00000 0.302276
\(395\) 0 0
\(396\) 6.00000 0.301511
\(397\) − 6.00000i − 0.301131i −0.988600 0.150566i \(-0.951890\pi\)
0.988600 0.150566i \(-0.0481095\pi\)
\(398\) 20.0000i 1.00251i
\(399\) 2.00000 0.100125
\(400\) 0 0
\(401\) 4.00000 0.199750 0.0998752 0.995000i \(-0.468156\pi\)
0.0998752 + 0.995000i \(0.468156\pi\)
\(402\) − 8.00000i − 0.399004i
\(403\) 0 0
\(404\) −18.0000 −0.895533
\(405\) 0 0
\(406\) 8.00000 0.397033
\(407\) 24.0000i 1.18964i
\(408\) − 18.0000i − 0.891133i
\(409\) 10.0000 0.494468 0.247234 0.968956i \(-0.420478\pi\)
0.247234 + 0.968956i \(0.420478\pi\)
\(410\) 0 0
\(411\) −10.0000 −0.493264
\(412\) − 8.00000i − 0.394132i
\(413\) − 24.0000i − 1.18096i
\(414\) −8.00000 −0.393179
\(415\) 0 0
\(416\) 0 0
\(417\) 16.0000i 0.783523i
\(418\) 6.00000i 0.293470i
\(419\) 30.0000 1.46560 0.732798 0.680446i \(-0.238214\pi\)
0.732798 + 0.680446i \(0.238214\pi\)
\(420\) 0 0
\(421\) 6.00000 0.292422 0.146211 0.989253i \(-0.453292\pi\)
0.146211 + 0.989253i \(0.453292\pi\)
\(422\) − 20.0000i − 0.973585i
\(423\) − 8.00000i − 0.388973i
\(424\) −6.00000 −0.291386
\(425\) 0 0
\(426\) −16.0000 −0.775203
\(427\) 4.00000i 0.193574i
\(428\) 12.0000i 0.580042i
\(429\) 0 0
\(430\) 0 0
\(431\) 36.0000 1.73406 0.867029 0.498257i \(-0.166026\pi\)
0.867029 + 0.498257i \(0.166026\pi\)
\(432\) 1.00000i 0.0481125i
\(433\) − 16.0000i − 0.768911i −0.923144 0.384455i \(-0.874389\pi\)
0.923144 0.384455i \(-0.125611\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 6.00000 0.287348
\(437\) 8.00000i 0.382692i
\(438\) − 14.0000i − 0.668946i
\(439\) −24.0000 −1.14546 −0.572729 0.819745i \(-0.694115\pi\)
−0.572729 + 0.819745i \(0.694115\pi\)
\(440\) 0 0
\(441\) −3.00000 −0.142857
\(442\) 0 0
\(443\) − 12.0000i − 0.570137i −0.958507 0.285069i \(-0.907984\pi\)
0.958507 0.285069i \(-0.0920164\pi\)
\(444\) 4.00000 0.189832
\(445\) 0 0
\(446\) −8.00000 −0.378811
\(447\) − 6.00000i − 0.283790i
\(448\) − 14.0000i − 0.661438i
\(449\) 8.00000 0.377543 0.188772 0.982021i \(-0.439549\pi\)
0.188772 + 0.982021i \(0.439549\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) − 6.00000i − 0.282216i
\(453\) − 16.0000i − 0.751746i
\(454\) −28.0000 −1.31411
\(455\) 0 0
\(456\) 3.00000 0.140488
\(457\) 6.00000i 0.280668i 0.990104 + 0.140334i \(0.0448177\pi\)
−0.990104 + 0.140334i \(0.955182\pi\)
\(458\) − 2.00000i − 0.0934539i
\(459\) 6.00000 0.280056
\(460\) 0 0
\(461\) 30.0000 1.39724 0.698620 0.715493i \(-0.253798\pi\)
0.698620 + 0.715493i \(0.253798\pi\)
\(462\) 12.0000i 0.558291i
\(463\) − 14.0000i − 0.650635i −0.945605 0.325318i \(-0.894529\pi\)
0.945605 0.325318i \(-0.105471\pi\)
\(464\) 4.00000 0.185695
\(465\) 0 0
\(466\) −22.0000 −1.01913
\(467\) − 12.0000i − 0.555294i −0.960683 0.277647i \(-0.910445\pi\)
0.960683 0.277647i \(-0.0895545\pi\)
\(468\) 0 0
\(469\) −16.0000 −0.738811
\(470\) 0 0
\(471\) −2.00000 −0.0921551
\(472\) − 36.0000i − 1.65703i
\(473\) 12.0000i 0.551761i
\(474\) 8.00000 0.367452
\(475\) 0 0
\(476\) −12.0000 −0.550019
\(477\) − 2.00000i − 0.0915737i
\(478\) − 6.00000i − 0.274434i
\(479\) −18.0000 −0.822441 −0.411220 0.911536i \(-0.634897\pi\)
−0.411220 + 0.911536i \(0.634897\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 18.0000i 0.819878i
\(483\) 16.0000i 0.728025i
\(484\) 25.0000 1.13636
\(485\) 0 0
\(486\) −1.00000 −0.0453609
\(487\) − 8.00000i − 0.362515i −0.983436 0.181257i \(-0.941983\pi\)
0.983436 0.181257i \(-0.0580167\pi\)
\(488\) 6.00000i 0.271607i
\(489\) 22.0000 0.994874
\(490\) 0 0
\(491\) −26.0000 −1.17336 −0.586682 0.809818i \(-0.699566\pi\)
−0.586682 + 0.809818i \(0.699566\pi\)
\(492\) 0 0
\(493\) − 24.0000i − 1.08091i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 32.0000i 1.43540i
\(498\) 0 0
\(499\) 24.0000 1.07439 0.537194 0.843459i \(-0.319484\pi\)
0.537194 + 0.843459i \(0.319484\pi\)
\(500\) 0 0
\(501\) 16.0000 0.714827
\(502\) − 6.00000i − 0.267793i
\(503\) − 36.0000i − 1.60516i −0.596544 0.802580i \(-0.703460\pi\)
0.596544 0.802580i \(-0.296540\pi\)
\(504\) 6.00000 0.267261
\(505\) 0 0
\(506\) −48.0000 −2.13386
\(507\) 13.0000i 0.577350i
\(508\) − 4.00000i − 0.177471i
\(509\) 24.0000 1.06378 0.531891 0.846813i \(-0.321482\pi\)
0.531891 + 0.846813i \(0.321482\pi\)
\(510\) 0 0
\(511\) −28.0000 −1.23865
\(512\) − 11.0000i − 0.486136i
\(513\) 1.00000i 0.0441511i
\(514\) −22.0000 −0.970378
\(515\) 0 0
\(516\) 2.00000 0.0880451
\(517\) − 48.0000i − 2.11104i
\(518\) 8.00000i 0.351500i
\(519\) −6.00000 −0.263371
\(520\) 0 0
\(521\) −4.00000 −0.175243 −0.0876216 0.996154i \(-0.527927\pi\)
−0.0876216 + 0.996154i \(0.527927\pi\)
\(522\) 4.00000i 0.175075i
\(523\) − 4.00000i − 0.174908i −0.996169 0.0874539i \(-0.972127\pi\)
0.996169 0.0874539i \(-0.0278730\pi\)
\(524\) −2.00000 −0.0873704
\(525\) 0 0
\(526\) 12.0000 0.523225
\(527\) 0 0
\(528\) 6.00000i 0.261116i
\(529\) −41.0000 −1.78261
\(530\) 0 0
\(531\) 12.0000 0.520756
\(532\) − 2.00000i − 0.0867110i
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) −24.0000 −1.03664
\(537\) 20.0000i 0.863064i
\(538\) − 12.0000i − 0.517357i
\(539\) −18.0000 −0.775315
\(540\) 0 0
\(541\) −42.0000 −1.80572 −0.902861 0.429934i \(-0.858537\pi\)
−0.902861 + 0.429934i \(0.858537\pi\)
\(542\) − 20.0000i − 0.859074i
\(543\) 10.0000i 0.429141i
\(544\) −30.0000 −1.28624
\(545\) 0 0
\(546\) 0 0
\(547\) 8.00000i 0.342055i 0.985266 + 0.171028i \(0.0547087\pi\)
−0.985266 + 0.171028i \(0.945291\pi\)
\(548\) 10.0000i 0.427179i
\(549\) −2.00000 −0.0853579
\(550\) 0 0
\(551\) 4.00000 0.170406
\(552\) 24.0000i 1.02151i
\(553\) − 16.0000i − 0.680389i
\(554\) −2.00000 −0.0849719
\(555\) 0 0
\(556\) 16.0000 0.678551
\(557\) − 18.0000i − 0.762684i −0.924434 0.381342i \(-0.875462\pi\)
0.924434 0.381342i \(-0.124538\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 36.0000 1.51992
\(562\) − 4.00000i − 0.168730i
\(563\) 36.0000i 1.51722i 0.651546 + 0.758610i \(0.274121\pi\)
−0.651546 + 0.758610i \(0.725879\pi\)
\(564\) −8.00000 −0.336861
\(565\) 0 0
\(566\) 14.0000 0.588464
\(567\) 2.00000i 0.0839921i
\(568\) 48.0000i 2.01404i
\(569\) 8.00000 0.335377 0.167689 0.985840i \(-0.446370\pi\)
0.167689 + 0.985840i \(0.446370\pi\)
\(570\) 0 0
\(571\) −24.0000 −1.00437 −0.502184 0.864761i \(-0.667470\pi\)
−0.502184 + 0.864761i \(0.667470\pi\)
\(572\) 0 0
\(573\) 10.0000i 0.417756i
\(574\) 0 0
\(575\) 0 0
\(576\) 7.00000 0.291667
\(577\) − 2.00000i − 0.0832611i −0.999133 0.0416305i \(-0.986745\pi\)
0.999133 0.0416305i \(-0.0132552\pi\)
\(578\) − 19.0000i − 0.790296i
\(579\) −24.0000 −0.997406
\(580\) 0 0
\(581\) 0 0
\(582\) − 12.0000i − 0.497416i
\(583\) − 12.0000i − 0.496989i
\(584\) −42.0000 −1.73797
\(585\) 0 0
\(586\) −30.0000 −1.23929
\(587\) 20.0000i 0.825488i 0.910847 + 0.412744i \(0.135430\pi\)
−0.910847 + 0.412744i \(0.864570\pi\)
\(588\) 3.00000i 0.123718i
\(589\) 0 0
\(590\) 0 0
\(591\) 6.00000 0.246807
\(592\) 4.00000i 0.164399i
\(593\) − 10.0000i − 0.410651i −0.978694 0.205325i \(-0.934175\pi\)
0.978694 0.205325i \(-0.0658253\pi\)
\(594\) −6.00000 −0.246183
\(595\) 0 0
\(596\) −6.00000 −0.245770
\(597\) 20.0000i 0.818546i
\(598\) 0 0
\(599\) 20.0000 0.817178 0.408589 0.912719i \(-0.366021\pi\)
0.408589 + 0.912719i \(0.366021\pi\)
\(600\) 0 0
\(601\) 22.0000 0.897399 0.448699 0.893683i \(-0.351887\pi\)
0.448699 + 0.893683i \(0.351887\pi\)
\(602\) 4.00000i 0.163028i
\(603\) − 8.00000i − 0.325785i
\(604\) −16.0000 −0.651031
\(605\) 0 0
\(606\) 18.0000 0.731200
\(607\) − 16.0000i − 0.649420i −0.945814 0.324710i \(-0.894733\pi\)
0.945814 0.324710i \(-0.105267\pi\)
\(608\) − 5.00000i − 0.202777i
\(609\) 8.00000 0.324176
\(610\) 0 0
\(611\) 0 0
\(612\) − 6.00000i − 0.242536i
\(613\) − 2.00000i − 0.0807792i −0.999184 0.0403896i \(-0.987140\pi\)
0.999184 0.0403896i \(-0.0128599\pi\)
\(614\) −8.00000 −0.322854
\(615\) 0 0
\(616\) 36.0000 1.45048
\(617\) − 34.0000i − 1.36879i −0.729112 0.684394i \(-0.760067\pi\)
0.729112 0.684394i \(-0.239933\pi\)
\(618\) 8.00000i 0.321807i
\(619\) −8.00000 −0.321547 −0.160774 0.986991i \(-0.551399\pi\)
−0.160774 + 0.986991i \(0.551399\pi\)
\(620\) 0 0
\(621\) −8.00000 −0.321029
\(622\) 34.0000i 1.36328i
\(623\) 0 0
\(624\) 0 0
\(625\) 0 0
\(626\) −10.0000 −0.399680
\(627\) 6.00000i 0.239617i
\(628\) 2.00000i 0.0798087i
\(629\) 24.0000 0.956943
\(630\) 0 0
\(631\) −40.0000 −1.59237 −0.796187 0.605050i \(-0.793153\pi\)
−0.796187 + 0.605050i \(0.793153\pi\)
\(632\) − 24.0000i − 0.954669i
\(633\) − 20.0000i − 0.794929i
\(634\) −6.00000 −0.238290
\(635\) 0 0
\(636\) −2.00000 −0.0793052
\(637\) 0 0
\(638\) 24.0000i 0.950169i
\(639\) −16.0000 −0.632950
\(640\) 0 0
\(641\) 40.0000 1.57991 0.789953 0.613168i \(-0.210105\pi\)
0.789953 + 0.613168i \(0.210105\pi\)
\(642\) − 12.0000i − 0.473602i
\(643\) 2.00000i 0.0788723i 0.999222 + 0.0394362i \(0.0125562\pi\)
−0.999222 + 0.0394362i \(0.987444\pi\)
\(644\) 16.0000 0.630488
\(645\) 0 0
\(646\) 6.00000 0.236067
\(647\) − 20.0000i − 0.786281i −0.919478 0.393141i \(-0.871389\pi\)
0.919478 0.393141i \(-0.128611\pi\)
\(648\) 3.00000i 0.117851i
\(649\) 72.0000 2.82625
\(650\) 0 0
\(651\) 0 0
\(652\) − 22.0000i − 0.861586i
\(653\) − 18.0000i − 0.704394i −0.935926 0.352197i \(-0.885435\pi\)
0.935926 0.352197i \(-0.114565\pi\)
\(654\) −6.00000 −0.234619
\(655\) 0 0
\(656\) 0 0
\(657\) − 14.0000i − 0.546192i
\(658\) − 16.0000i − 0.623745i
\(659\) −48.0000 −1.86981 −0.934907 0.354892i \(-0.884518\pi\)
−0.934907 + 0.354892i \(0.884518\pi\)
\(660\) 0 0
\(661\) −22.0000 −0.855701 −0.427850 0.903850i \(-0.640729\pi\)
−0.427850 + 0.903850i \(0.640729\pi\)
\(662\) 28.0000i 1.08825i
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) −4.00000 −0.154997
\(667\) 32.0000i 1.23904i
\(668\) − 16.0000i − 0.619059i
\(669\) −8.00000 −0.309298
\(670\) 0 0
\(671\) −12.0000 −0.463255
\(672\) − 10.0000i − 0.385758i
\(673\) 36.0000i 1.38770i 0.720121 + 0.693849i \(0.244086\pi\)
−0.720121 + 0.693849i \(0.755914\pi\)
\(674\) −12.0000 −0.462223
\(675\) 0 0
\(676\) 13.0000 0.500000
\(677\) − 18.0000i − 0.691796i −0.938272 0.345898i \(-0.887574\pi\)
0.938272 0.345898i \(-0.112426\pi\)
\(678\) 6.00000i 0.230429i
\(679\) −24.0000 −0.921035
\(680\) 0 0
\(681\) −28.0000 −1.07296
\(682\) 0 0
\(683\) − 36.0000i − 1.37750i −0.724998 0.688751i \(-0.758159\pi\)
0.724998 0.688751i \(-0.241841\pi\)
\(684\) 1.00000 0.0382360
\(685\) 0 0
\(686\) −20.0000 −0.763604
\(687\) − 2.00000i − 0.0763048i
\(688\) 2.00000i 0.0762493i
\(689\) 0 0
\(690\) 0 0
\(691\) 32.0000 1.21734 0.608669 0.793424i \(-0.291704\pi\)
0.608669 + 0.793424i \(0.291704\pi\)
\(692\) 6.00000i 0.228086i
\(693\) 12.0000i 0.455842i
\(694\) 8.00000 0.303676
\(695\) 0 0
\(696\) 12.0000 0.454859
\(697\) 0 0
\(698\) 2.00000i 0.0757011i
\(699\) −22.0000 −0.832116
\(700\) 0 0
\(701\) −42.0000 −1.58632 −0.793159 0.609015i \(-0.791565\pi\)
−0.793159 + 0.609015i \(0.791565\pi\)
\(702\) 0 0
\(703\) 4.00000i 0.150863i
\(704\) 42.0000 1.58293
\(705\) 0 0
\(706\) 34.0000 1.27961
\(707\) − 36.0000i − 1.35392i
\(708\) − 12.0000i − 0.450988i
\(709\) 14.0000 0.525781 0.262891 0.964826i \(-0.415324\pi\)
0.262891 + 0.964826i \(0.415324\pi\)
\(710\) 0 0
\(711\) 8.00000 0.300023
\(712\) 0 0
\(713\) 0 0
\(714\) 12.0000 0.449089
\(715\) 0 0
\(716\) 20.0000 0.747435
\(717\) − 6.00000i − 0.224074i
\(718\) − 10.0000i − 0.373197i
\(719\) 18.0000 0.671287 0.335643 0.941989i \(-0.391046\pi\)
0.335643 + 0.941989i \(0.391046\pi\)
\(720\) 0 0
\(721\) 16.0000 0.595871
\(722\) 1.00000i 0.0372161i
\(723\) 18.0000i 0.669427i
\(724\) 10.0000 0.371647
\(725\) 0 0
\(726\) −25.0000 −0.927837
\(727\) − 26.0000i − 0.964287i −0.876092 0.482143i \(-0.839858\pi\)
0.876092 0.482143i \(-0.160142\pi\)
\(728\) 0 0
\(729\) −1.00000 −0.0370370
\(730\) 0 0
\(731\) 12.0000 0.443836
\(732\) 2.00000i 0.0739221i
\(733\) − 22.0000i − 0.812589i −0.913742 0.406294i \(-0.866821\pi\)
0.913742 0.406294i \(-0.133179\pi\)
\(734\) −14.0000 −0.516749
\(735\) 0 0
\(736\) 40.0000 1.47442
\(737\) − 48.0000i − 1.76810i
\(738\) 0 0
\(739\) 20.0000 0.735712 0.367856 0.929883i \(-0.380092\pi\)
0.367856 + 0.929883i \(0.380092\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) − 4.00000i − 0.146845i
\(743\) 24.0000i 0.880475i 0.897881 + 0.440237i \(0.145106\pi\)
−0.897881 + 0.440237i \(0.854894\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 0 0
\(748\) − 36.0000i − 1.31629i
\(749\) −24.0000 −0.876941
\(750\) 0 0
\(751\) −40.0000 −1.45962 −0.729810 0.683650i \(-0.760392\pi\)
−0.729810 + 0.683650i \(0.760392\pi\)
\(752\) − 8.00000i − 0.291730i
\(753\) − 6.00000i − 0.218652i
\(754\) 0 0
\(755\) 0 0
\(756\) 2.00000 0.0727393
\(757\) 38.0000i 1.38113i 0.723269 + 0.690567i \(0.242639\pi\)
−0.723269 + 0.690567i \(0.757361\pi\)
\(758\) − 20.0000i − 0.726433i
\(759\) −48.0000 −1.74229
\(760\) 0 0
\(761\) 30.0000 1.08750 0.543750 0.839248i \(-0.317004\pi\)
0.543750 + 0.839248i \(0.317004\pi\)
\(762\) 4.00000i 0.144905i
\(763\) 12.0000i 0.434429i
\(764\) 10.0000 0.361787
\(765\) 0 0
\(766\) −8.00000 −0.289052
\(767\) 0 0
\(768\) − 17.0000i − 0.613435i
\(769\) 14.0000 0.504853 0.252426 0.967616i \(-0.418771\pi\)
0.252426 + 0.967616i \(0.418771\pi\)
\(770\) 0 0
\(771\) −22.0000 −0.792311
\(772\) 24.0000i 0.863779i
\(773\) − 6.00000i − 0.215805i −0.994161 0.107903i \(-0.965587\pi\)
0.994161 0.107903i \(-0.0344134\pi\)
\(774\) −2.00000 −0.0718885
\(775\) 0 0
\(776\) −36.0000 −1.29232
\(777\) 8.00000i 0.286998i
\(778\) 14.0000i 0.501924i
\(779\) 0 0
\(780\) 0 0
\(781\) −96.0000 −3.43515
\(782\) 48.0000i 1.71648i
\(783\) 4.00000i 0.142948i
\(784\) −3.00000 −0.107143
\(785\) 0 0
\(786\) 2.00000 0.0713376
\(787\) 28.0000i 0.998092i 0.866575 + 0.499046i \(0.166316\pi\)
−0.866575 + 0.499046i \(0.833684\pi\)
\(788\) − 6.00000i − 0.213741i
\(789\) 12.0000 0.427211
\(790\) 0 0
\(791\) 12.0000 0.426671
\(792\) 18.0000i 0.639602i
\(793\) 0 0
\(794\) 6.00000 0.212932
\(795\) 0 0
\(796\) 20.0000 0.708881
\(797\) 26.0000i 0.920967i 0.887668 + 0.460484i \(0.152324\pi\)
−0.887668 + 0.460484i \(0.847676\pi\)
\(798\) 2.00000i 0.0707992i
\(799\) −48.0000 −1.69812
\(800\) 0 0
\(801\) 0 0
\(802\) 4.00000i 0.141245i
\(803\) − 84.0000i − 2.96430i
\(804\) −8.00000 −0.282138
\(805\) 0 0
\(806\) 0 0
\(807\) − 12.0000i − 0.422420i
\(808\) − 54.0000i − 1.89971i
\(809\) −2.00000 −0.0703163 −0.0351581 0.999382i \(-0.511193\pi\)
−0.0351581 + 0.999382i \(0.511193\pi\)
\(810\) 0 0
\(811\) 20.0000 0.702295 0.351147 0.936320i \(-0.385792\pi\)
0.351147 + 0.936320i \(0.385792\pi\)
\(812\) − 8.00000i − 0.280745i
\(813\) − 20.0000i − 0.701431i
\(814\) −24.0000 −0.841200
\(815\) 0 0
\(816\) 6.00000 0.210042
\(817\) 2.00000i 0.0699711i
\(818\) 10.0000i 0.349642i
\(819\) 0 0
\(820\) 0 0
\(821\) 6.00000 0.209401 0.104701 0.994504i \(-0.466612\pi\)
0.104701 + 0.994504i \(0.466612\pi\)
\(822\) − 10.0000i − 0.348790i
\(823\) − 14.0000i − 0.488009i −0.969774 0.244005i \(-0.921539\pi\)
0.969774 0.244005i \(-0.0784612\pi\)
\(824\) 24.0000 0.836080
\(825\) 0 0
\(826\) 24.0000 0.835067
\(827\) − 36.0000i − 1.25184i −0.779886 0.625921i \(-0.784723\pi\)
0.779886 0.625921i \(-0.215277\pi\)
\(828\) 8.00000i 0.278019i
\(829\) 6.00000 0.208389 0.104194 0.994557i \(-0.466774\pi\)
0.104194 + 0.994557i \(0.466774\pi\)
\(830\) 0 0
\(831\) −2.00000 −0.0693792
\(832\) 0 0
\(833\) 18.0000i 0.623663i
\(834\) −16.0000 −0.554035
\(835\) 0 0
\(836\) 6.00000 0.207514
\(837\) 0 0
\(838\) 30.0000i 1.03633i
\(839\) −8.00000 −0.276191 −0.138095 0.990419i \(-0.544098\pi\)
−0.138095 + 0.990419i \(0.544098\pi\)
\(840\) 0 0
\(841\) −13.0000 −0.448276
\(842\) 6.00000i 0.206774i
\(843\) − 4.00000i − 0.137767i
\(844\) −20.0000 −0.688428
\(845\) 0 0
\(846\) 8.00000 0.275046
\(847\) 50.0000i 1.71802i
\(848\) − 2.00000i − 0.0686803i
\(849\) 14.0000 0.480479
\(850\) 0 0
\(851\) −32.0000 −1.09695
\(852\) 16.0000i 0.548151i
\(853\) 26.0000i 0.890223i 0.895475 + 0.445112i \(0.146836\pi\)
−0.895475 + 0.445112i \(0.853164\pi\)
\(854\) −4.00000 −0.136877
\(855\) 0 0
\(856\) −36.0000 −1.23045
\(857\) − 46.0000i − 1.57133i −0.618652 0.785665i \(-0.712321\pi\)
0.618652 0.785665i \(-0.287679\pi\)
\(858\) 0 0
\(859\) −44.0000 −1.50126 −0.750630 0.660722i \(-0.770250\pi\)
−0.750630 + 0.660722i \(0.770250\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 36.0000i 1.22616i
\(863\) − 24.0000i − 0.816970i −0.912765 0.408485i \(-0.866057\pi\)
0.912765 0.408485i \(-0.133943\pi\)
\(864\) 5.00000 0.170103
\(865\) 0 0
\(866\) 16.0000 0.543702
\(867\) − 19.0000i − 0.645274i
\(868\) 0 0
\(869\) 48.0000 1.62829
\(870\) 0 0
\(871\) 0 0
\(872\) 18.0000i 0.609557i
\(873\) − 12.0000i − 0.406138i
\(874\) −8.00000 −0.270604
\(875\) 0 0
\(876\) −14.0000 −0.473016
\(877\) 24.0000i 0.810422i 0.914223 + 0.405211i \(0.132802\pi\)
−0.914223 + 0.405211i \(0.867198\pi\)
\(878\) − 24.0000i − 0.809961i
\(879\) −30.0000 −1.01187
\(880\) 0 0
\(881\) −42.0000 −1.41502 −0.707508 0.706705i \(-0.750181\pi\)
−0.707508 + 0.706705i \(0.750181\pi\)
\(882\) − 3.00000i − 0.101015i
\(883\) 2.00000i 0.0673054i 0.999434 + 0.0336527i \(0.0107140\pi\)
−0.999434 + 0.0336527i \(0.989286\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 12.0000 0.403148
\(887\) − 48.0000i − 1.61168i −0.592132 0.805841i \(-0.701714\pi\)
0.592132 0.805841i \(-0.298286\pi\)
\(888\) 12.0000i 0.402694i
\(889\) 8.00000 0.268311
\(890\) 0 0
\(891\) −6.00000 −0.201008
\(892\) 8.00000i 0.267860i
\(893\) − 8.00000i − 0.267710i
\(894\) 6.00000 0.200670
\(895\) 0 0
\(896\) −6.00000 −0.200446
\(897\) 0 0
\(898\) 8.00000i 0.266963i
\(899\) 0 0
\(900\) 0 0
\(901\) −12.0000 −0.399778
\(902\) 0 0
\(903\) 4.00000i 0.133112i
\(904\) 18.0000 0.598671
\(905\) 0 0
\(906\) 16.0000 0.531564
\(907\) 52.0000i 1.72663i 0.504664 + 0.863316i \(0.331616\pi\)
−0.504664 + 0.863316i \(0.668384\pi\)
\(908\) 28.0000i 0.929213i
\(909\) 18.0000 0.597022
\(910\) 0 0
\(911\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(912\) 1.00000i 0.0331133i
\(913\) 0 0
\(914\) −6.00000 −0.198462
\(915\) 0 0
\(916\) −2.00000 −0.0660819
\(917\) − 4.00000i − 0.132092i
\(918\) 6.00000i 0.198030i
\(919\) 40.0000 1.31948 0.659739 0.751495i \(-0.270667\pi\)
0.659739 + 0.751495i \(0.270667\pi\)
\(920\) 0 0
\(921\) −8.00000 −0.263609
\(922\) 30.0000i 0.987997i
\(923\) 0 0
\(924\) 12.0000 0.394771
\(925\) 0 0
\(926\) 14.0000 0.460069
\(927\) 8.00000i 0.262754i
\(928\) − 20.0000i − 0.656532i
\(929\) −18.0000 −0.590561 −0.295280 0.955411i \(-0.595413\pi\)
−0.295280 + 0.955411i \(0.595413\pi\)
\(930\) 0 0
\(931\) −3.00000 −0.0983210
\(932\) 22.0000i 0.720634i
\(933\) 34.0000i 1.11311i
\(934\) 12.0000 0.392652
\(935\) 0 0
\(936\) 0 0
\(937\) − 30.0000i − 0.980057i −0.871706 0.490029i \(-0.836986\pi\)
0.871706 0.490029i \(-0.163014\pi\)
\(938\) − 16.0000i − 0.522419i
\(939\) −10.0000 −0.326338
\(940\) 0 0
\(941\) 44.0000 1.43436 0.717180 0.696888i \(-0.245433\pi\)
0.717180 + 0.696888i \(0.245433\pi\)
\(942\) − 2.00000i − 0.0651635i
\(943\) 0 0
\(944\) 12.0000 0.390567
\(945\) 0 0
\(946\) −12.0000 −0.390154
\(947\) − 48.0000i − 1.55979i −0.625910 0.779895i \(-0.715272\pi\)
0.625910 0.779895i \(-0.284728\pi\)
\(948\) − 8.00000i − 0.259828i
\(949\) 0 0
\(950\) 0 0
\(951\) −6.00000 −0.194563
\(952\) − 36.0000i − 1.16677i
\(953\) − 30.0000i − 0.971795i −0.874016 0.485898i \(-0.838493\pi\)
0.874016 0.485898i \(-0.161507\pi\)
\(954\) 2.00000 0.0647524
\(955\) 0 0
\(956\) −6.00000 −0.194054
\(957\) 24.0000i 0.775810i
\(958\) − 18.0000i − 0.581554i
\(959\) −20.0000 −0.645834
\(960\) 0 0
\(961\) −31.0000 −1.00000
\(962\) 0 0
\(963\) − 12.0000i − 0.386695i
\(964\) 18.0000 0.579741
\(965\) 0 0
\(966\) −16.0000 −0.514792
\(967\) 14.0000i 0.450210i 0.974335 + 0.225105i \(0.0722725\pi\)
−0.974335 + 0.225105i \(0.927728\pi\)
\(968\) 75.0000i 2.41059i
\(969\) 6.00000 0.192748
\(970\) 0 0
\(971\) −8.00000 −0.256732 −0.128366 0.991727i \(-0.540973\pi\)
−0.128366 + 0.991727i \(0.540973\pi\)
\(972\) 1.00000i 0.0320750i
\(973\) 32.0000i 1.02587i
\(974\) 8.00000 0.256337
\(975\) 0 0
\(976\) −2.00000 −0.0640184
\(977\) 42.0000i 1.34370i 0.740688 + 0.671850i \(0.234500\pi\)
−0.740688 + 0.671850i \(0.765500\pi\)
\(978\) 22.0000i 0.703482i
\(979\) 0 0
\(980\) 0 0
\(981\) −6.00000 −0.191565
\(982\) − 26.0000i − 0.829693i
\(983\) 8.00000i 0.255160i 0.991828 + 0.127580i \(0.0407210\pi\)
−0.991828 + 0.127580i \(0.959279\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 24.0000 0.764316
\(987\) − 16.0000i − 0.509286i
\(988\) 0 0
\(989\) −16.0000 −0.508770
\(990\) 0 0
\(991\) −16.0000 −0.508257 −0.254128 0.967170i \(-0.581789\pi\)
−0.254128 + 0.967170i \(0.581789\pi\)
\(992\) 0 0
\(993\) 28.0000i 0.888553i
\(994\) −32.0000 −1.01498
\(995\) 0 0
\(996\) 0 0
\(997\) − 14.0000i − 0.443384i −0.975117 0.221692i \(-0.928842\pi\)
0.975117 0.221692i \(-0.0711580\pi\)
\(998\) 24.0000i 0.759707i
\(999\) −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 1425.2.c.c.799.2 2
5.2 odd 4 285.2.a.a.1.1 1
5.3 odd 4 1425.2.a.g.1.1 1
5.4 even 2 inner 1425.2.c.c.799.1 2
15.2 even 4 855.2.a.c.1.1 1
15.8 even 4 4275.2.a.h.1.1 1
20.7 even 4 4560.2.a.h.1.1 1
95.37 even 4 5415.2.a.h.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
285.2.a.a.1.1 1 5.2 odd 4
855.2.a.c.1.1 1 15.2 even 4
1425.2.a.g.1.1 1 5.3 odd 4
1425.2.c.c.799.1 2 5.4 even 2 inner
1425.2.c.c.799.2 2 1.1 even 1 trivial
4275.2.a.h.1.1 1 15.8 even 4
4560.2.a.h.1.1 1 20.7 even 4
5415.2.a.h.1.1 1 95.37 even 4