Properties

Label 3450.2.d.n.2899.2
Level $3450$
Weight $2$
Character 3450.2899
Analytic conductor $27.548$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

Related objects

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Newspace parameters

Level: \( N \) \(=\) \( 3450 = 2 \cdot 3 \cdot 5^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3450.d (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(27.5483886973\)
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 690)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 2899.2
Root \(-1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 3450.2899
Dual form 3450.2.d.n.2899.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} -4.00000i q^{7} -1.00000i q^{8} -1.00000 q^{9} +O(q^{10})\) \(q+1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} -4.00000i q^{7} -1.00000i q^{8} -1.00000 q^{9} -2.00000 q^{11} +1.00000i q^{12} +4.00000 q^{14} +1.00000 q^{16} -2.00000i q^{17} -1.00000i q^{18} -4.00000 q^{21} -2.00000i q^{22} +1.00000i q^{23} -1.00000 q^{24} +1.00000i q^{27} +4.00000i q^{28} +4.00000 q^{29} +1.00000i q^{32} +2.00000i q^{33} +2.00000 q^{34} +1.00000 q^{36} -10.0000i q^{37} +6.00000 q^{41} -4.00000i q^{42} +2.00000i q^{43} +2.00000 q^{44} -1.00000 q^{46} -12.0000i q^{47} -1.00000i q^{48} -9.00000 q^{49} -2.00000 q^{51} +6.00000i q^{53} -1.00000 q^{54} -4.00000 q^{56} +4.00000i q^{58} -12.0000 q^{59} -14.0000 q^{61} +4.00000i q^{63} -1.00000 q^{64} -2.00000 q^{66} -2.00000i q^{67} +2.00000i q^{68} +1.00000 q^{69} -2.00000 q^{71} +1.00000i q^{72} +6.00000i q^{73} +10.0000 q^{74} +8.00000i q^{77} -8.00000 q^{79} +1.00000 q^{81} +6.00000i q^{82} +8.00000i q^{83} +4.00000 q^{84} -2.00000 q^{86} -4.00000i q^{87} +2.00000i q^{88} +8.00000 q^{89} -1.00000i q^{92} +12.0000 q^{94} +1.00000 q^{96} -9.00000i q^{98} +2.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{4} + 2q^{6} - 2q^{9} + O(q^{10}) \) \( 2q - 2q^{4} + 2q^{6} - 2q^{9} - 4q^{11} + 8q^{14} + 2q^{16} - 8q^{21} - 2q^{24} + 8q^{29} + 4q^{34} + 2q^{36} + 12q^{41} + 4q^{44} - 2q^{46} - 18q^{49} - 4q^{51} - 2q^{54} - 8q^{56} - 24q^{59} - 28q^{61} - 2q^{64} - 4q^{66} + 2q^{69} - 4q^{71} + 20q^{74} - 16q^{79} + 2q^{81} + 8q^{84} - 4q^{86} + 16q^{89} + 24q^{94} + 2q^{96} + 4q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3450\mathbb{Z}\right)^\times\).

\(n\) \(277\) \(1151\) \(1201\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000i 0.707107i
\(3\) − 1.00000i − 0.577350i
\(4\) −1.00000 −0.500000
\(5\) 0 0
\(6\) 1.00000 0.408248
\(7\) − 4.00000i − 1.51186i −0.654654 0.755929i \(-0.727186\pi\)
0.654654 0.755929i \(-0.272814\pi\)
\(8\) − 1.00000i − 0.353553i
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) −2.00000 −0.603023 −0.301511 0.953463i \(-0.597491\pi\)
−0.301511 + 0.953463i \(0.597491\pi\)
\(12\) 1.00000i 0.288675i
\(13\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(14\) 4.00000 1.06904
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) − 2.00000i − 0.485071i −0.970143 0.242536i \(-0.922021\pi\)
0.970143 0.242536i \(-0.0779791\pi\)
\(18\) − 1.00000i − 0.235702i
\(19\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(20\) 0 0
\(21\) −4.00000 −0.872872
\(22\) − 2.00000i − 0.426401i
\(23\) 1.00000i 0.208514i
\(24\) −1.00000 −0.204124
\(25\) 0 0
\(26\) 0 0
\(27\) 1.00000i 0.192450i
\(28\) 4.00000i 0.755929i
\(29\) 4.00000 0.742781 0.371391 0.928477i \(-0.378881\pi\)
0.371391 + 0.928477i \(0.378881\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) 1.00000i 0.176777i
\(33\) 2.00000i 0.348155i
\(34\) 2.00000 0.342997
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) − 10.0000i − 1.64399i −0.569495 0.821995i \(-0.692861\pi\)
0.569495 0.821995i \(-0.307139\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 6.00000 0.937043 0.468521 0.883452i \(-0.344787\pi\)
0.468521 + 0.883452i \(0.344787\pi\)
\(42\) − 4.00000i − 0.617213i
\(43\) 2.00000i 0.304997i 0.988304 + 0.152499i \(0.0487319\pi\)
−0.988304 + 0.152499i \(0.951268\pi\)
\(44\) 2.00000 0.301511
\(45\) 0 0
\(46\) −1.00000 −0.147442
\(47\) − 12.0000i − 1.75038i −0.483779 0.875190i \(-0.660736\pi\)
0.483779 0.875190i \(-0.339264\pi\)
\(48\) − 1.00000i − 0.144338i
\(49\) −9.00000 −1.28571
\(50\) 0 0
\(51\) −2.00000 −0.280056
\(52\) 0 0
\(53\) 6.00000i 0.824163i 0.911147 + 0.412082i \(0.135198\pi\)
−0.911147 + 0.412082i \(0.864802\pi\)
\(54\) −1.00000 −0.136083
\(55\) 0 0
\(56\) −4.00000 −0.534522
\(57\) 0 0
\(58\) 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
\(61\) −14.0000 −1.79252 −0.896258 0.443533i \(-0.853725\pi\)
−0.896258 + 0.443533i \(0.853725\pi\)
\(62\) 0 0
\(63\) 4.00000i 0.503953i
\(64\) −1.00000 −0.125000
\(65\) 0 0
\(66\) −2.00000 −0.246183
\(67\) − 2.00000i − 0.244339i −0.992509 0.122169i \(-0.961015\pi\)
0.992509 0.122169i \(-0.0389851\pi\)
\(68\) 2.00000i 0.242536i
\(69\) 1.00000 0.120386
\(70\) 0 0
\(71\) −2.00000 −0.237356 −0.118678 0.992933i \(-0.537866\pi\)
−0.118678 + 0.992933i \(0.537866\pi\)
\(72\) 1.00000i 0.117851i
\(73\) 6.00000i 0.702247i 0.936329 + 0.351123i \(0.114200\pi\)
−0.936329 + 0.351123i \(0.885800\pi\)
\(74\) 10.0000 1.16248
\(75\) 0 0
\(76\) 0 0
\(77\) 8.00000i 0.911685i
\(78\) 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
\(82\) 6.00000i 0.662589i
\(83\) 8.00000i 0.878114i 0.898459 + 0.439057i \(0.144687\pi\)
−0.898459 + 0.439057i \(0.855313\pi\)
\(84\) 4.00000 0.436436
\(85\) 0 0
\(86\) −2.00000 −0.215666
\(87\) − 4.00000i − 0.428845i
\(88\) 2.00000i 0.213201i
\(89\) 8.00000 0.847998 0.423999 0.905663i \(-0.360626\pi\)
0.423999 + 0.905663i \(0.360626\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) − 1.00000i − 0.104257i
\(93\) 0 0
\(94\) 12.0000 1.23771
\(95\) 0 0
\(96\) 1.00000 0.102062
\(97\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(98\) − 9.00000i − 0.909137i
\(99\) 2.00000 0.201008
\(100\) 0 0
\(101\) −12.0000 −1.19404 −0.597022 0.802225i \(-0.703650\pi\)
−0.597022 + 0.802225i \(0.703650\pi\)
\(102\) − 2.00000i − 0.198030i
\(103\) 4.00000i 0.394132i 0.980390 + 0.197066i \(0.0631413\pi\)
−0.980390 + 0.197066i \(0.936859\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) −6.00000 −0.582772
\(107\) 12.0000i 1.16008i 0.814587 + 0.580042i \(0.196964\pi\)
−0.814587 + 0.580042i \(0.803036\pi\)
\(108\) − 1.00000i − 0.0962250i
\(109\) −14.0000 −1.34096 −0.670478 0.741929i \(-0.733911\pi\)
−0.670478 + 0.741929i \(0.733911\pi\)
\(110\) 0 0
\(111\) −10.0000 −0.949158
\(112\) − 4.00000i − 0.377964i
\(113\) 6.00000i 0.564433i 0.959351 + 0.282216i \(0.0910696\pi\)
−0.959351 + 0.282216i \(0.908930\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) −4.00000 −0.371391
\(117\) 0 0
\(118\) − 12.0000i − 1.10469i
\(119\) −8.00000 −0.733359
\(120\) 0 0
\(121\) −7.00000 −0.636364
\(122\) − 14.0000i − 1.26750i
\(123\) − 6.00000i − 0.541002i
\(124\) 0 0
\(125\) 0 0
\(126\) −4.00000 −0.356348
\(127\) − 6.00000i − 0.532414i −0.963916 0.266207i \(-0.914230\pi\)
0.963916 0.266207i \(-0.0857705\pi\)
\(128\) − 1.00000i − 0.0883883i
\(129\) 2.00000 0.176090
\(130\) 0 0
\(131\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(132\) − 2.00000i − 0.174078i
\(133\) 0 0
\(134\) 2.00000 0.172774
\(135\) 0 0
\(136\) −2.00000 −0.171499
\(137\) 2.00000i 0.170872i 0.996344 + 0.0854358i \(0.0272282\pi\)
−0.996344 + 0.0854358i \(0.972772\pi\)
\(138\) 1.00000i 0.0851257i
\(139\) −20.0000 −1.69638 −0.848189 0.529694i \(-0.822307\pi\)
−0.848189 + 0.529694i \(0.822307\pi\)
\(140\) 0 0
\(141\) −12.0000 −1.01058
\(142\) − 2.00000i − 0.167836i
\(143\) 0 0
\(144\) −1.00000 −0.0833333
\(145\) 0 0
\(146\) −6.00000 −0.496564
\(147\) 9.00000i 0.742307i
\(148\) 10.0000i 0.821995i
\(149\) 14.0000 1.14692 0.573462 0.819232i \(-0.305600\pi\)
0.573462 + 0.819232i \(0.305600\pi\)
\(150\) 0 0
\(151\) −12.0000 −0.976546 −0.488273 0.872691i \(-0.662373\pi\)
−0.488273 + 0.872691i \(0.662373\pi\)
\(152\) 0 0
\(153\) 2.00000i 0.161690i
\(154\) −8.00000 −0.644658
\(155\) 0 0
\(156\) 0 0
\(157\) 22.0000i 1.75579i 0.478852 + 0.877896i \(0.341053\pi\)
−0.478852 + 0.877896i \(0.658947\pi\)
\(158\) − 8.00000i − 0.636446i
\(159\) 6.00000 0.475831
\(160\) 0 0
\(161\) 4.00000 0.315244
\(162\) 1.00000i 0.0785674i
\(163\) − 16.0000i − 1.25322i −0.779334 0.626608i \(-0.784443\pi\)
0.779334 0.626608i \(-0.215557\pi\)
\(164\) −6.00000 −0.468521
\(165\) 0 0
\(166\) −8.00000 −0.620920
\(167\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(168\) 4.00000i 0.308607i
\(169\) 13.0000 1.00000
\(170\) 0 0
\(171\) 0 0
\(172\) − 2.00000i − 0.152499i
\(173\) − 6.00000i − 0.456172i −0.973641 0.228086i \(-0.926753\pi\)
0.973641 0.228086i \(-0.0732467\pi\)
\(174\) 4.00000 0.303239
\(175\) 0 0
\(176\) −2.00000 −0.150756
\(177\) 12.0000i 0.901975i
\(178\) 8.00000i 0.599625i
\(179\) −4.00000 −0.298974 −0.149487 0.988764i \(-0.547762\pi\)
−0.149487 + 0.988764i \(0.547762\pi\)
\(180\) 0 0
\(181\) −6.00000 −0.445976 −0.222988 0.974821i \(-0.571581\pi\)
−0.222988 + 0.974821i \(0.571581\pi\)
\(182\) 0 0
\(183\) 14.0000i 1.03491i
\(184\) 1.00000 0.0737210
\(185\) 0 0
\(186\) 0 0
\(187\) 4.00000i 0.292509i
\(188\) 12.0000i 0.875190i
\(189\) 4.00000 0.290957
\(190\) 0 0
\(191\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(192\) 1.00000i 0.0721688i
\(193\) − 14.0000i − 1.00774i −0.863779 0.503871i \(-0.831909\pi\)
0.863779 0.503871i \(-0.168091\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 9.00000 0.642857
\(197\) 6.00000i 0.427482i 0.976890 + 0.213741i \(0.0685649\pi\)
−0.976890 + 0.213741i \(0.931435\pi\)
\(198\) 2.00000i 0.142134i
\(199\) −16.0000 −1.13421 −0.567105 0.823646i \(-0.691937\pi\)
−0.567105 + 0.823646i \(0.691937\pi\)
\(200\) 0 0
\(201\) −2.00000 −0.141069
\(202\) − 12.0000i − 0.844317i
\(203\) − 16.0000i − 1.12298i
\(204\) 2.00000 0.140028
\(205\) 0 0
\(206\) −4.00000 −0.278693
\(207\) − 1.00000i − 0.0695048i
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 4.00000 0.275371 0.137686 0.990476i \(-0.456034\pi\)
0.137686 + 0.990476i \(0.456034\pi\)
\(212\) − 6.00000i − 0.412082i
\(213\) 2.00000i 0.137038i
\(214\) −12.0000 −0.820303
\(215\) 0 0
\(216\) 1.00000 0.0680414
\(217\) 0 0
\(218\) − 14.0000i − 0.948200i
\(219\) 6.00000 0.405442
\(220\) 0 0
\(221\) 0 0
\(222\) − 10.0000i − 0.671156i
\(223\) 2.00000i 0.133930i 0.997755 + 0.0669650i \(0.0213316\pi\)
−0.997755 + 0.0669650i \(0.978668\pi\)
\(224\) 4.00000 0.267261
\(225\) 0 0
\(226\) −6.00000 −0.399114
\(227\) − 12.0000i − 0.796468i −0.917284 0.398234i \(-0.869623\pi\)
0.917284 0.398234i \(-0.130377\pi\)
\(228\) 0 0
\(229\) 26.0000 1.71813 0.859064 0.511868i \(-0.171046\pi\)
0.859064 + 0.511868i \(0.171046\pi\)
\(230\) 0 0
\(231\) 8.00000 0.526361
\(232\) − 4.00000i − 0.262613i
\(233\) 10.0000i 0.655122i 0.944830 + 0.327561i \(0.106227\pi\)
−0.944830 + 0.327561i \(0.893773\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 12.0000 0.781133
\(237\) 8.00000i 0.519656i
\(238\) − 8.00000i − 0.518563i
\(239\) −22.0000 −1.42306 −0.711531 0.702655i \(-0.751998\pi\)
−0.711531 + 0.702655i \(0.751998\pi\)
\(240\) 0 0
\(241\) −14.0000 −0.901819 −0.450910 0.892570i \(-0.648900\pi\)
−0.450910 + 0.892570i \(0.648900\pi\)
\(242\) − 7.00000i − 0.449977i
\(243\) − 1.00000i − 0.0641500i
\(244\) 14.0000 0.896258
\(245\) 0 0
\(246\) 6.00000 0.382546
\(247\) 0 0
\(248\) 0 0
\(249\) 8.00000 0.506979
\(250\) 0 0
\(251\) 18.0000 1.13615 0.568075 0.822977i \(-0.307688\pi\)
0.568075 + 0.822977i \(0.307688\pi\)
\(252\) − 4.00000i − 0.251976i
\(253\) − 2.00000i − 0.125739i
\(254\) 6.00000 0.376473
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) − 2.00000i − 0.124757i −0.998053 0.0623783i \(-0.980131\pi\)
0.998053 0.0623783i \(-0.0198685\pi\)
\(258\) 2.00000i 0.124515i
\(259\) −40.0000 −2.48548
\(260\) 0 0
\(261\) −4.00000 −0.247594
\(262\) 0 0
\(263\) − 32.0000i − 1.97320i −0.163144 0.986602i \(-0.552164\pi\)
0.163144 0.986602i \(-0.447836\pi\)
\(264\) 2.00000 0.123091
\(265\) 0 0
\(266\) 0 0
\(267\) − 8.00000i − 0.489592i
\(268\) 2.00000i 0.122169i
\(269\) −24.0000 −1.46331 −0.731653 0.681677i \(-0.761251\pi\)
−0.731653 + 0.681677i \(0.761251\pi\)
\(270\) 0 0
\(271\) 8.00000 0.485965 0.242983 0.970031i \(-0.421874\pi\)
0.242983 + 0.970031i \(0.421874\pi\)
\(272\) − 2.00000i − 0.121268i
\(273\) 0 0
\(274\) −2.00000 −0.120824
\(275\) 0 0
\(276\) −1.00000 −0.0601929
\(277\) − 8.00000i − 0.480673i −0.970690 0.240337i \(-0.922742\pi\)
0.970690 0.240337i \(-0.0772579\pi\)
\(278\) − 20.0000i − 1.19952i
\(279\) 0 0
\(280\) 0 0
\(281\) 32.0000 1.90896 0.954480 0.298275i \(-0.0964112\pi\)
0.954480 + 0.298275i \(0.0964112\pi\)
\(282\) − 12.0000i − 0.714590i
\(283\) − 10.0000i − 0.594438i −0.954809 0.297219i \(-0.903941\pi\)
0.954809 0.297219i \(-0.0960592\pi\)
\(284\) 2.00000 0.118678
\(285\) 0 0
\(286\) 0 0
\(287\) − 24.0000i − 1.41668i
\(288\) − 1.00000i − 0.0589256i
\(289\) 13.0000 0.764706
\(290\) 0 0
\(291\) 0 0
\(292\) − 6.00000i − 0.351123i
\(293\) 14.0000i 0.817889i 0.912559 + 0.408944i \(0.134103\pi\)
−0.912559 + 0.408944i \(0.865897\pi\)
\(294\) −9.00000 −0.524891
\(295\) 0 0
\(296\) −10.0000 −0.581238
\(297\) − 2.00000i − 0.116052i
\(298\) 14.0000i 0.810998i
\(299\) 0 0
\(300\) 0 0
\(301\) 8.00000 0.461112
\(302\) − 12.0000i − 0.690522i
\(303\) 12.0000i 0.689382i
\(304\) 0 0
\(305\) 0 0
\(306\) −2.00000 −0.114332
\(307\) − 4.00000i − 0.228292i −0.993464 0.114146i \(-0.963587\pi\)
0.993464 0.114146i \(-0.0364132\pi\)
\(308\) − 8.00000i − 0.455842i
\(309\) 4.00000 0.227552
\(310\) 0 0
\(311\) −2.00000 −0.113410 −0.0567048 0.998391i \(-0.518059\pi\)
−0.0567048 + 0.998391i \(0.518059\pi\)
\(312\) 0 0
\(313\) 20.0000i 1.13047i 0.824931 + 0.565233i \(0.191214\pi\)
−0.824931 + 0.565233i \(0.808786\pi\)
\(314\) −22.0000 −1.24153
\(315\) 0 0
\(316\) 8.00000 0.450035
\(317\) 10.0000i 0.561656i 0.959758 + 0.280828i \(0.0906090\pi\)
−0.959758 + 0.280828i \(0.909391\pi\)
\(318\) 6.00000i 0.336463i
\(319\) −8.00000 −0.447914
\(320\) 0 0
\(321\) 12.0000 0.669775
\(322\) 4.00000i 0.222911i
\(323\) 0 0
\(324\) −1.00000 −0.0555556
\(325\) 0 0
\(326\) 16.0000 0.886158
\(327\) 14.0000i 0.774202i
\(328\) − 6.00000i − 0.331295i
\(329\) −48.0000 −2.64633
\(330\) 0 0
\(331\) −28.0000 −1.53902 −0.769510 0.638635i \(-0.779499\pi\)
−0.769510 + 0.638635i \(0.779499\pi\)
\(332\) − 8.00000i − 0.439057i
\(333\) 10.0000i 0.547997i
\(334\) 0 0
\(335\) 0 0
\(336\) −4.00000 −0.218218
\(337\) 4.00000i 0.217894i 0.994048 + 0.108947i \(0.0347479\pi\)
−0.994048 + 0.108947i \(0.965252\pi\)
\(338\) 13.0000i 0.707107i
\(339\) 6.00000 0.325875
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 8.00000i 0.431959i
\(344\) 2.00000 0.107833
\(345\) 0 0
\(346\) 6.00000 0.322562
\(347\) − 4.00000i − 0.214731i −0.994220 0.107366i \(-0.965758\pi\)
0.994220 0.107366i \(-0.0342415\pi\)
\(348\) 4.00000i 0.214423i
\(349\) 30.0000 1.60586 0.802932 0.596071i \(-0.203272\pi\)
0.802932 + 0.596071i \(0.203272\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) − 2.00000i − 0.106600i
\(353\) − 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
\(356\) −8.00000 −0.423999
\(357\) 8.00000i 0.423405i
\(358\) − 4.00000i − 0.211407i
\(359\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(360\) 0 0
\(361\) −19.0000 −1.00000
\(362\) − 6.00000i − 0.315353i
\(363\) 7.00000i 0.367405i
\(364\) 0 0
\(365\) 0 0
\(366\) −14.0000 −0.731792
\(367\) − 4.00000i − 0.208798i −0.994535 0.104399i \(-0.966708\pi\)
0.994535 0.104399i \(-0.0332919\pi\)
\(368\) 1.00000i 0.0521286i
\(369\) −6.00000 −0.312348
\(370\) 0 0
\(371\) 24.0000 1.24602
\(372\) 0 0
\(373\) − 22.0000i − 1.13912i −0.821951 0.569558i \(-0.807114\pi\)
0.821951 0.569558i \(-0.192886\pi\)
\(374\) −4.00000 −0.206835
\(375\) 0 0
\(376\) −12.0000 −0.618853
\(377\) 0 0
\(378\) 4.00000i 0.205738i
\(379\) −4.00000 −0.205466 −0.102733 0.994709i \(-0.532759\pi\)
−0.102733 + 0.994709i \(0.532759\pi\)
\(380\) 0 0
\(381\) −6.00000 −0.307389
\(382\) 0 0
\(383\) − 24.0000i − 1.22634i −0.789950 0.613171i \(-0.789894\pi\)
0.789950 0.613171i \(-0.210106\pi\)
\(384\) −1.00000 −0.0510310
\(385\) 0 0
\(386\) 14.0000 0.712581
\(387\) − 2.00000i − 0.101666i
\(388\) 0 0
\(389\) −22.0000 −1.11544 −0.557722 0.830028i \(-0.688325\pi\)
−0.557722 + 0.830028i \(0.688325\pi\)
\(390\) 0 0
\(391\) 2.00000 0.101144
\(392\) 9.00000i 0.454569i
\(393\) 0 0
\(394\) −6.00000 −0.302276
\(395\) 0 0
\(396\) −2.00000 −0.100504
\(397\) − 32.0000i − 1.60603i −0.595956 0.803017i \(-0.703227\pi\)
0.595956 0.803017i \(-0.296773\pi\)
\(398\) − 16.0000i − 0.802008i
\(399\) 0 0
\(400\) 0 0
\(401\) −12.0000 −0.599251 −0.299626 0.954057i \(-0.596862\pi\)
−0.299626 + 0.954057i \(0.596862\pi\)
\(402\) − 2.00000i − 0.0997509i
\(403\) 0 0
\(404\) 12.0000 0.597022
\(405\) 0 0
\(406\) 16.0000 0.794067
\(407\) 20.0000i 0.991363i
\(408\) 2.00000i 0.0990148i
\(409\) 10.0000 0.494468 0.247234 0.968956i \(-0.420478\pi\)
0.247234 + 0.968956i \(0.420478\pi\)
\(410\) 0 0
\(411\) 2.00000 0.0986527
\(412\) − 4.00000i − 0.197066i
\(413\) 48.0000i 2.36193i
\(414\) 1.00000 0.0491473
\(415\) 0 0
\(416\) 0 0
\(417\) 20.0000i 0.979404i
\(418\) 0 0
\(419\) 26.0000 1.27018 0.635092 0.772437i \(-0.280962\pi\)
0.635092 + 0.772437i \(0.280962\pi\)
\(420\) 0 0
\(421\) −6.00000 −0.292422 −0.146211 0.989253i \(-0.546708\pi\)
−0.146211 + 0.989253i \(0.546708\pi\)
\(422\) 4.00000i 0.194717i
\(423\) 12.0000i 0.583460i
\(424\) 6.00000 0.291386
\(425\) 0 0
\(426\) −2.00000 −0.0969003
\(427\) 56.0000i 2.71003i
\(428\) − 12.0000i − 0.580042i
\(429\) 0 0
\(430\) 0 0
\(431\) 24.0000 1.15604 0.578020 0.816023i \(-0.303826\pi\)
0.578020 + 0.816023i \(0.303826\pi\)
\(432\) 1.00000i 0.0481125i
\(433\) 24.0000i 1.15337i 0.816968 + 0.576683i \(0.195653\pi\)
−0.816968 + 0.576683i \(0.804347\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 14.0000 0.670478
\(437\) 0 0
\(438\) 6.00000i 0.286691i
\(439\) −20.0000 −0.954548 −0.477274 0.878755i \(-0.658375\pi\)
−0.477274 + 0.878755i \(0.658375\pi\)
\(440\) 0 0
\(441\) 9.00000 0.428571
\(442\) 0 0
\(443\) − 20.0000i − 0.950229i −0.879924 0.475114i \(-0.842407\pi\)
0.879924 0.475114i \(-0.157593\pi\)
\(444\) 10.0000 0.474579
\(445\) 0 0
\(446\) −2.00000 −0.0947027
\(447\) − 14.0000i − 0.662177i
\(448\) 4.00000i 0.188982i
\(449\) 34.0000 1.60456 0.802280 0.596948i \(-0.203620\pi\)
0.802280 + 0.596948i \(0.203620\pi\)
\(450\) 0 0
\(451\) −12.0000 −0.565058
\(452\) − 6.00000i − 0.282216i
\(453\) 12.0000i 0.563809i
\(454\) 12.0000 0.563188
\(455\) 0 0
\(456\) 0 0
\(457\) 24.0000i 1.12267i 0.827588 + 0.561336i \(0.189713\pi\)
−0.827588 + 0.561336i \(0.810287\pi\)
\(458\) 26.0000i 1.21490i
\(459\) 2.00000 0.0933520
\(460\) 0 0
\(461\) 20.0000 0.931493 0.465746 0.884918i \(-0.345786\pi\)
0.465746 + 0.884918i \(0.345786\pi\)
\(462\) 8.00000i 0.372194i
\(463\) 18.0000i 0.836531i 0.908325 + 0.418265i \(0.137362\pi\)
−0.908325 + 0.418265i \(0.862638\pi\)
\(464\) 4.00000 0.185695
\(465\) 0 0
\(466\) −10.0000 −0.463241
\(467\) − 24.0000i − 1.11059i −0.831654 0.555294i \(-0.812606\pi\)
0.831654 0.555294i \(-0.187394\pi\)
\(468\) 0 0
\(469\) −8.00000 −0.369406
\(470\) 0 0
\(471\) 22.0000 1.01371
\(472\) 12.0000i 0.552345i
\(473\) − 4.00000i − 0.183920i
\(474\) −8.00000 −0.367452
\(475\) 0 0
\(476\) 8.00000 0.366679
\(477\) − 6.00000i − 0.274721i
\(478\) − 22.0000i − 1.00626i
\(479\) 40.0000 1.82765 0.913823 0.406112i \(-0.133116\pi\)
0.913823 + 0.406112i \(0.133116\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) − 14.0000i − 0.637683i
\(483\) − 4.00000i − 0.182006i
\(484\) 7.00000 0.318182
\(485\) 0 0
\(486\) 1.00000 0.0453609
\(487\) − 18.0000i − 0.815658i −0.913058 0.407829i \(-0.866286\pi\)
0.913058 0.407829i \(-0.133714\pi\)
\(488\) 14.0000i 0.633750i
\(489\) −16.0000 −0.723545
\(490\) 0 0
\(491\) −24.0000 −1.08310 −0.541552 0.840667i \(-0.682163\pi\)
−0.541552 + 0.840667i \(0.682163\pi\)
\(492\) 6.00000i 0.270501i
\(493\) − 8.00000i − 0.360302i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 8.00000i 0.358849i
\(498\) 8.00000i 0.358489i
\(499\) 28.0000 1.25345 0.626726 0.779240i \(-0.284395\pi\)
0.626726 + 0.779240i \(0.284395\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 18.0000i 0.803379i
\(503\) − 8.00000i − 0.356702i −0.983967 0.178351i \(-0.942924\pi\)
0.983967 0.178351i \(-0.0570763\pi\)
\(504\) 4.00000 0.178174
\(505\) 0 0
\(506\) 2.00000 0.0889108
\(507\) − 13.0000i − 0.577350i
\(508\) 6.00000i 0.266207i
\(509\) −28.0000 −1.24108 −0.620539 0.784176i \(-0.713086\pi\)
−0.620539 + 0.784176i \(0.713086\pi\)
\(510\) 0 0
\(511\) 24.0000 1.06170
\(512\) 1.00000i 0.0441942i
\(513\) 0 0
\(514\) 2.00000 0.0882162
\(515\) 0 0
\(516\) −2.00000 −0.0880451
\(517\) 24.0000i 1.05552i
\(518\) − 40.0000i − 1.75750i
\(519\) −6.00000 −0.263371
\(520\) 0 0
\(521\) 32.0000 1.40195 0.700973 0.713188i \(-0.252749\pi\)
0.700973 + 0.713188i \(0.252749\pi\)
\(522\) − 4.00000i − 0.175075i
\(523\) 22.0000i 0.961993i 0.876723 + 0.480996i \(0.159725\pi\)
−0.876723 + 0.480996i \(0.840275\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 32.0000 1.39527
\(527\) 0 0
\(528\) 2.00000i 0.0870388i
\(529\) −1.00000 −0.0434783
\(530\) 0 0
\(531\) 12.0000 0.520756
\(532\) 0 0
\(533\) 0 0
\(534\) 8.00000 0.346194
\(535\) 0 0
\(536\) −2.00000 −0.0863868
\(537\) 4.00000i 0.172613i
\(538\) − 24.0000i − 1.03471i
\(539\) 18.0000 0.775315
\(540\) 0 0
\(541\) −22.0000 −0.945854 −0.472927 0.881102i \(-0.656803\pi\)
−0.472927 + 0.881102i \(0.656803\pi\)
\(542\) 8.00000i 0.343629i
\(543\) 6.00000i 0.257485i
\(544\) 2.00000 0.0857493
\(545\) 0 0
\(546\) 0 0
\(547\) − 12.0000i − 0.513083i −0.966533 0.256541i \(-0.917417\pi\)
0.966533 0.256541i \(-0.0825830\pi\)
\(548\) − 2.00000i − 0.0854358i
\(549\) 14.0000 0.597505
\(550\) 0 0
\(551\) 0 0
\(552\) − 1.00000i − 0.0425628i
\(553\) 32.0000i 1.36078i
\(554\) 8.00000 0.339887
\(555\) 0 0
\(556\) 20.0000 0.848189
\(557\) − 26.0000i − 1.10166i −0.834619 0.550828i \(-0.814312\pi\)
0.834619 0.550828i \(-0.185688\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 4.00000 0.168880
\(562\) 32.0000i 1.34984i
\(563\) 8.00000i 0.337160i 0.985688 + 0.168580i \(0.0539181\pi\)
−0.985688 + 0.168580i \(0.946082\pi\)
\(564\) 12.0000 0.505291
\(565\) 0 0
\(566\) 10.0000 0.420331
\(567\) − 4.00000i − 0.167984i
\(568\) 2.00000i 0.0839181i
\(569\) 12.0000 0.503066 0.251533 0.967849i \(-0.419065\pi\)
0.251533 + 0.967849i \(0.419065\pi\)
\(570\) 0 0
\(571\) −36.0000 −1.50655 −0.753277 0.657704i \(-0.771528\pi\)
−0.753277 + 0.657704i \(0.771528\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 24.0000 1.00174
\(575\) 0 0
\(576\) 1.00000 0.0416667
\(577\) 10.0000i 0.416305i 0.978096 + 0.208153i \(0.0667451\pi\)
−0.978096 + 0.208153i \(0.933255\pi\)
\(578\) 13.0000i 0.540729i
\(579\) −14.0000 −0.581820
\(580\) 0 0
\(581\) 32.0000 1.32758
\(582\) 0 0
\(583\) − 12.0000i − 0.496989i
\(584\) 6.00000 0.248282
\(585\) 0 0
\(586\) −14.0000 −0.578335
\(587\) 4.00000i 0.165098i 0.996587 + 0.0825488i \(0.0263060\pi\)
−0.996587 + 0.0825488i \(0.973694\pi\)
\(588\) − 9.00000i − 0.371154i
\(589\) 0 0
\(590\) 0 0
\(591\) 6.00000 0.246807
\(592\) − 10.0000i − 0.410997i
\(593\) − 30.0000i − 1.23195i −0.787765 0.615976i \(-0.788762\pi\)
0.787765 0.615976i \(-0.211238\pi\)
\(594\) 2.00000 0.0820610
\(595\) 0 0
\(596\) −14.0000 −0.573462
\(597\) 16.0000i 0.654836i
\(598\) 0 0
\(599\) 30.0000 1.22577 0.612883 0.790173i \(-0.290010\pi\)
0.612883 + 0.790173i \(0.290010\pi\)
\(600\) 0 0
\(601\) −10.0000 −0.407909 −0.203954 0.978980i \(-0.565379\pi\)
−0.203954 + 0.978980i \(0.565379\pi\)
\(602\) 8.00000i 0.326056i
\(603\) 2.00000i 0.0814463i
\(604\) 12.0000 0.488273
\(605\) 0 0
\(606\) −12.0000 −0.487467
\(607\) 14.0000i 0.568242i 0.958788 + 0.284121i \(0.0917018\pi\)
−0.958788 + 0.284121i \(0.908298\pi\)
\(608\) 0 0
\(609\) −16.0000 −0.648353
\(610\) 0 0
\(611\) 0 0
\(612\) − 2.00000i − 0.0808452i
\(613\) 14.0000i 0.565455i 0.959200 + 0.282727i \(0.0912392\pi\)
−0.959200 + 0.282727i \(0.908761\pi\)
\(614\) 4.00000 0.161427
\(615\) 0 0
\(616\) 8.00000 0.322329
\(617\) 14.0000i 0.563619i 0.959470 + 0.281809i \(0.0909346\pi\)
−0.959470 + 0.281809i \(0.909065\pi\)
\(618\) 4.00000i 0.160904i
\(619\) 24.0000 0.964641 0.482321 0.875995i \(-0.339794\pi\)
0.482321 + 0.875995i \(0.339794\pi\)
\(620\) 0 0
\(621\) −1.00000 −0.0401286
\(622\) − 2.00000i − 0.0801927i
\(623\) − 32.0000i − 1.28205i
\(624\) 0 0
\(625\) 0 0
\(626\) −20.0000 −0.799361
\(627\) 0 0
\(628\) − 22.0000i − 0.877896i
\(629\) −20.0000 −0.797452
\(630\) 0 0
\(631\) −16.0000 −0.636950 −0.318475 0.947931i \(-0.603171\pi\)
−0.318475 + 0.947931i \(0.603171\pi\)
\(632\) 8.00000i 0.318223i
\(633\) − 4.00000i − 0.158986i
\(634\) −10.0000 −0.397151
\(635\) 0 0
\(636\) −6.00000 −0.237915
\(637\) 0 0
\(638\) − 8.00000i − 0.316723i
\(639\) 2.00000 0.0791188
\(640\) 0 0
\(641\) −8.00000 −0.315981 −0.157991 0.987441i \(-0.550502\pi\)
−0.157991 + 0.987441i \(0.550502\pi\)
\(642\) 12.0000i 0.473602i
\(643\) 14.0000i 0.552106i 0.961142 + 0.276053i \(0.0890266\pi\)
−0.961142 + 0.276053i \(0.910973\pi\)
\(644\) −4.00000 −0.157622
\(645\) 0 0
\(646\) 0 0
\(647\) − 12.0000i − 0.471769i −0.971781 0.235884i \(-0.924201\pi\)
0.971781 0.235884i \(-0.0757987\pi\)
\(648\) − 1.00000i − 0.0392837i
\(649\) 24.0000 0.942082
\(650\) 0 0
\(651\) 0 0
\(652\) 16.0000i 0.626608i
\(653\) − 26.0000i − 1.01746i −0.860927 0.508729i \(-0.830115\pi\)
0.860927 0.508729i \(-0.169885\pi\)
\(654\) −14.0000 −0.547443
\(655\) 0 0
\(656\) 6.00000 0.234261
\(657\) − 6.00000i − 0.234082i
\(658\) − 48.0000i − 1.87123i
\(659\) 18.0000 0.701180 0.350590 0.936529i \(-0.385981\pi\)
0.350590 + 0.936529i \(0.385981\pi\)
\(660\) 0 0
\(661\) 6.00000 0.233373 0.116686 0.993169i \(-0.462773\pi\)
0.116686 + 0.993169i \(0.462773\pi\)
\(662\) − 28.0000i − 1.08825i
\(663\) 0 0
\(664\) 8.00000 0.310460
\(665\) 0 0
\(666\) −10.0000 −0.387492
\(667\) 4.00000i 0.154881i
\(668\) 0 0
\(669\) 2.00000 0.0773245
\(670\) 0 0
\(671\) 28.0000 1.08093
\(672\) − 4.00000i − 0.154303i
\(673\) − 18.0000i − 0.693849i −0.937893 0.346925i \(-0.887226\pi\)
0.937893 0.346925i \(-0.112774\pi\)
\(674\) −4.00000 −0.154074
\(675\) 0 0
\(676\) −13.0000 −0.500000
\(677\) 22.0000i 0.845529i 0.906240 + 0.422764i \(0.138940\pi\)
−0.906240 + 0.422764i \(0.861060\pi\)
\(678\) 6.00000i 0.230429i
\(679\) 0 0
\(680\) 0 0
\(681\) −12.0000 −0.459841
\(682\) 0 0
\(683\) − 36.0000i − 1.37750i −0.724998 0.688751i \(-0.758159\pi\)
0.724998 0.688751i \(-0.241841\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) −8.00000 −0.305441
\(687\) − 26.0000i − 0.991962i
\(688\) 2.00000i 0.0762493i
\(689\) 0 0
\(690\) 0 0
\(691\) −20.0000 −0.760836 −0.380418 0.924815i \(-0.624220\pi\)
−0.380418 + 0.924815i \(0.624220\pi\)
\(692\) 6.00000i 0.228086i
\(693\) − 8.00000i − 0.303895i
\(694\) 4.00000 0.151838
\(695\) 0 0
\(696\) −4.00000 −0.151620
\(697\) − 12.0000i − 0.454532i
\(698\) 30.0000i 1.13552i
\(699\) 10.0000 0.378235
\(700\) 0 0
\(701\) 6.00000 0.226617 0.113308 0.993560i \(-0.463855\pi\)
0.113308 + 0.993560i \(0.463855\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 2.00000 0.0753778
\(705\) 0 0
\(706\) 34.0000 1.27961
\(707\) 48.0000i 1.80523i
\(708\) − 12.0000i − 0.450988i
\(709\) −22.0000 −0.826227 −0.413114 0.910679i \(-0.635559\pi\)
−0.413114 + 0.910679i \(0.635559\pi\)
\(710\) 0 0
\(711\) 8.00000 0.300023
\(712\) − 8.00000i − 0.299813i
\(713\) 0 0
\(714\) −8.00000 −0.299392
\(715\) 0 0
\(716\) 4.00000 0.149487
\(717\) 22.0000i 0.821605i
\(718\) 0 0
\(719\) −18.0000 −0.671287 −0.335643 0.941989i \(-0.608954\pi\)
−0.335643 + 0.941989i \(0.608954\pi\)
\(720\) 0 0
\(721\) 16.0000 0.595871
\(722\) − 19.0000i − 0.707107i
\(723\) 14.0000i 0.520666i
\(724\) 6.00000 0.222988
\(725\) 0 0
\(726\) −7.00000 −0.259794
\(727\) − 8.00000i − 0.296704i −0.988935 0.148352i \(-0.952603\pi\)
0.988935 0.148352i \(-0.0473968\pi\)
\(728\) 0 0
\(729\) −1.00000 −0.0370370
\(730\) 0 0
\(731\) 4.00000 0.147945
\(732\) − 14.0000i − 0.517455i
\(733\) − 30.0000i − 1.10808i −0.832492 0.554038i \(-0.813086\pi\)
0.832492 0.554038i \(-0.186914\pi\)
\(734\) 4.00000 0.147643
\(735\) 0 0
\(736\) −1.00000 −0.0368605
\(737\) 4.00000i 0.147342i
\(738\) − 6.00000i − 0.220863i
\(739\) −36.0000 −1.32428 −0.662141 0.749380i \(-0.730352\pi\)
−0.662141 + 0.749380i \(0.730352\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 24.0000i 0.881068i
\(743\) 16.0000i 0.586983i 0.955962 + 0.293492i \(0.0948173\pi\)
−0.955962 + 0.293492i \(0.905183\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 22.0000 0.805477
\(747\) − 8.00000i − 0.292705i
\(748\) − 4.00000i − 0.146254i
\(749\) 48.0000 1.75388
\(750\) 0 0
\(751\) 40.0000 1.45962 0.729810 0.683650i \(-0.239608\pi\)
0.729810 + 0.683650i \(0.239608\pi\)
\(752\) − 12.0000i − 0.437595i
\(753\) − 18.0000i − 0.655956i
\(754\) 0 0
\(755\) 0 0
\(756\) −4.00000 −0.145479
\(757\) − 22.0000i − 0.799604i −0.916602 0.399802i \(-0.869079\pi\)
0.916602 0.399802i \(-0.130921\pi\)
\(758\) − 4.00000i − 0.145287i
\(759\) −2.00000 −0.0725954
\(760\) 0 0
\(761\) 2.00000 0.0724999 0.0362500 0.999343i \(-0.488459\pi\)
0.0362500 + 0.999343i \(0.488459\pi\)
\(762\) − 6.00000i − 0.217357i
\(763\) 56.0000i 2.02734i
\(764\) 0 0
\(765\) 0 0
\(766\) 24.0000 0.867155
\(767\) 0 0
\(768\) − 1.00000i − 0.0360844i
\(769\) −6.00000 −0.216366 −0.108183 0.994131i \(-0.534503\pi\)
−0.108183 + 0.994131i \(0.534503\pi\)
\(770\) 0 0
\(771\) −2.00000 −0.0720282
\(772\) 14.0000i 0.503871i
\(773\) 6.00000i 0.215805i 0.994161 + 0.107903i \(0.0344134\pi\)
−0.994161 + 0.107903i \(0.965587\pi\)
\(774\) 2.00000 0.0718885
\(775\) 0 0
\(776\) 0 0
\(777\) 40.0000i 1.43499i
\(778\) − 22.0000i − 0.788738i
\(779\) 0 0
\(780\) 0 0
\(781\) 4.00000 0.143131
\(782\) 2.00000i 0.0715199i
\(783\) 4.00000i 0.142948i
\(784\) −9.00000 −0.321429
\(785\) 0 0
\(786\) 0 0
\(787\) 34.0000i 1.21197i 0.795476 + 0.605985i \(0.207221\pi\)
−0.795476 + 0.605985i \(0.792779\pi\)
\(788\) − 6.00000i − 0.213741i
\(789\) −32.0000 −1.13923
\(790\) 0 0
\(791\) 24.0000 0.853342
\(792\) − 2.00000i − 0.0710669i
\(793\) 0 0
\(794\) 32.0000 1.13564
\(795\) 0 0
\(796\) 16.0000 0.567105
\(797\) − 50.0000i − 1.77109i −0.464553 0.885545i \(-0.653785\pi\)
0.464553 0.885545i \(-0.346215\pi\)
\(798\) 0 0
\(799\) −24.0000 −0.849059
\(800\) 0 0
\(801\) −8.00000 −0.282666
\(802\) − 12.0000i − 0.423735i
\(803\) − 12.0000i − 0.423471i
\(804\) 2.00000 0.0705346
\(805\) 0 0
\(806\) 0 0
\(807\) 24.0000i 0.844840i
\(808\) 12.0000i 0.422159i
\(809\) 30.0000 1.05474 0.527372 0.849635i \(-0.323177\pi\)
0.527372 + 0.849635i \(0.323177\pi\)
\(810\) 0 0
\(811\) 52.0000 1.82597 0.912983 0.407997i \(-0.133772\pi\)
0.912983 + 0.407997i \(0.133772\pi\)
\(812\) 16.0000i 0.561490i
\(813\) − 8.00000i − 0.280572i
\(814\) −20.0000 −0.701000
\(815\) 0 0
\(816\) −2.00000 −0.0700140
\(817\) 0 0
\(818\) 10.0000i 0.349642i
\(819\) 0 0
\(820\) 0 0
\(821\) −16.0000 −0.558404 −0.279202 0.960232i \(-0.590070\pi\)
−0.279202 + 0.960232i \(0.590070\pi\)
\(822\) 2.00000i 0.0697580i
\(823\) − 10.0000i − 0.348578i −0.984695 0.174289i \(-0.944237\pi\)
0.984695 0.174289i \(-0.0557627\pi\)
\(824\) 4.00000 0.139347
\(825\) 0 0
\(826\) −48.0000 −1.67013
\(827\) − 44.0000i − 1.53003i −0.644013 0.765015i \(-0.722732\pi\)
0.644013 0.765015i \(-0.277268\pi\)
\(828\) 1.00000i 0.0347524i
\(829\) −10.0000 −0.347314 −0.173657 0.984806i \(-0.555558\pi\)
−0.173657 + 0.984806i \(0.555558\pi\)
\(830\) 0 0
\(831\) −8.00000 −0.277517
\(832\) 0 0
\(833\) 18.0000i 0.623663i
\(834\) −20.0000 −0.692543
\(835\) 0 0
\(836\) 0 0
\(837\) 0 0
\(838\) 26.0000i 0.898155i
\(839\) 20.0000 0.690477 0.345238 0.938515i \(-0.387798\pi\)
0.345238 + 0.938515i \(0.387798\pi\)
\(840\) 0 0
\(841\) −13.0000 −0.448276
\(842\) − 6.00000i − 0.206774i
\(843\) − 32.0000i − 1.10214i
\(844\) −4.00000 −0.137686
\(845\) 0 0
\(846\) −12.0000 −0.412568
\(847\) 28.0000i 0.962091i
\(848\) 6.00000i 0.206041i
\(849\) −10.0000 −0.343199
\(850\) 0 0
\(851\) 10.0000 0.342796
\(852\) − 2.00000i − 0.0685189i
\(853\) 56.0000i 1.91740i 0.284413 + 0.958702i \(0.408201\pi\)
−0.284413 + 0.958702i \(0.591799\pi\)
\(854\) −56.0000 −1.91628
\(855\) 0 0
\(856\) 12.0000 0.410152
\(857\) 42.0000i 1.43469i 0.696717 + 0.717346i \(0.254643\pi\)
−0.696717 + 0.717346i \(0.745357\pi\)
\(858\) 0 0
\(859\) −36.0000 −1.22830 −0.614152 0.789188i \(-0.710502\pi\)
−0.614152 + 0.789188i \(0.710502\pi\)
\(860\) 0 0
\(861\) −24.0000 −0.817918
\(862\) 24.0000i 0.817443i
\(863\) 36.0000i 1.22545i 0.790295 + 0.612727i \(0.209928\pi\)
−0.790295 + 0.612727i \(0.790072\pi\)
\(864\) −1.00000 −0.0340207
\(865\) 0 0
\(866\) −24.0000 −0.815553
\(867\) − 13.0000i − 0.441503i
\(868\) 0 0
\(869\) 16.0000 0.542763
\(870\) 0 0
\(871\) 0 0
\(872\) 14.0000i 0.474100i
\(873\) 0 0
\(874\) 0 0
\(875\) 0 0
\(876\) −6.00000 −0.202721
\(877\) 56.0000i 1.89099i 0.325643 + 0.945493i \(0.394419\pi\)
−0.325643 + 0.945493i \(0.605581\pi\)
\(878\) − 20.0000i − 0.674967i
\(879\) 14.0000 0.472208
\(880\) 0 0
\(881\) 40.0000 1.34763 0.673817 0.738898i \(-0.264654\pi\)
0.673817 + 0.738898i \(0.264654\pi\)
\(882\) 9.00000i 0.303046i
\(883\) − 48.0000i − 1.61533i −0.589643 0.807664i \(-0.700731\pi\)
0.589643 0.807664i \(-0.299269\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 20.0000 0.671913
\(887\) 36.0000i 1.20876i 0.796696 + 0.604381i \(0.206579\pi\)
−0.796696 + 0.604381i \(0.793421\pi\)
\(888\) 10.0000i 0.335578i
\(889\) −24.0000 −0.804934
\(890\) 0 0
\(891\) −2.00000 −0.0670025
\(892\) − 2.00000i − 0.0669650i
\(893\) 0 0
\(894\) 14.0000 0.468230
\(895\) 0 0
\(896\) −4.00000 −0.133631
\(897\) 0 0
\(898\) 34.0000i 1.13459i
\(899\) 0 0
\(900\) 0 0
\(901\) 12.0000 0.399778
\(902\) − 12.0000i − 0.399556i
\(903\) − 8.00000i − 0.266223i
\(904\) 6.00000 0.199557
\(905\) 0 0
\(906\) −12.0000 −0.398673
\(907\) − 42.0000i − 1.39459i −0.716786 0.697294i \(-0.754387\pi\)
0.716786 0.697294i \(-0.245613\pi\)
\(908\) 12.0000i 0.398234i
\(909\) 12.0000 0.398015
\(910\) 0 0
\(911\) −12.0000 −0.397578 −0.198789 0.980042i \(-0.563701\pi\)
−0.198789 + 0.980042i \(0.563701\pi\)
\(912\) 0 0
\(913\) − 16.0000i − 0.529523i
\(914\) −24.0000 −0.793849
\(915\) 0 0
\(916\) −26.0000 −0.859064
\(917\) 0 0
\(918\) 2.00000i 0.0660098i
\(919\) 32.0000 1.05558 0.527791 0.849374i \(-0.323020\pi\)
0.527791 + 0.849374i \(0.323020\pi\)
\(920\) 0 0
\(921\) −4.00000 −0.131804
\(922\) 20.0000i 0.658665i
\(923\) 0 0
\(924\) −8.00000 −0.263181
\(925\) 0 0
\(926\) −18.0000 −0.591517
\(927\) − 4.00000i − 0.131377i
\(928\) 4.00000i 0.131306i
\(929\) −30.0000 −0.984268 −0.492134 0.870519i \(-0.663783\pi\)
−0.492134 + 0.870519i \(0.663783\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) − 10.0000i − 0.327561i
\(933\) 2.00000i 0.0654771i
\(934\) 24.0000 0.785304
\(935\) 0 0
\(936\) 0 0
\(937\) − 48.0000i − 1.56809i −0.620703 0.784046i \(-0.713153\pi\)
0.620703 0.784046i \(-0.286847\pi\)
\(938\) − 8.00000i − 0.261209i
\(939\) 20.0000 0.652675
\(940\) 0 0
\(941\) 26.0000 0.847576 0.423788 0.905761i \(-0.360700\pi\)
0.423788 + 0.905761i \(0.360700\pi\)
\(942\) 22.0000i 0.716799i
\(943\) 6.00000i 0.195387i
\(944\) −12.0000 −0.390567
\(945\) 0 0
\(946\) 4.00000 0.130051
\(947\) 52.0000i 1.68977i 0.534946 + 0.844886i \(0.320332\pi\)
−0.534946 + 0.844886i \(0.679668\pi\)
\(948\) − 8.00000i − 0.259828i
\(949\) 0 0
\(950\) 0 0
\(951\) 10.0000 0.324272
\(952\) 8.00000i 0.259281i
\(953\) 14.0000i 0.453504i 0.973952 + 0.226752i \(0.0728108\pi\)
−0.973952 + 0.226752i \(0.927189\pi\)
\(954\) 6.00000 0.194257
\(955\) 0 0
\(956\) 22.0000 0.711531
\(957\) 8.00000i 0.258603i
\(958\) 40.0000i 1.29234i
\(959\) 8.00000 0.258333
\(960\) 0 0
\(961\) −31.0000 −1.00000
\(962\) 0 0
\(963\) − 12.0000i − 0.386695i
\(964\) 14.0000 0.450910
\(965\) 0 0
\(966\) 4.00000 0.128698
\(967\) − 58.0000i − 1.86515i −0.360971 0.932577i \(-0.617555\pi\)
0.360971 0.932577i \(-0.382445\pi\)
\(968\) 7.00000i 0.224989i
\(969\) 0 0
\(970\) 0 0
\(971\) 50.0000 1.60458 0.802288 0.596937i \(-0.203616\pi\)
0.802288 + 0.596937i \(0.203616\pi\)
\(972\) 1.00000i 0.0320750i
\(973\) 80.0000i 2.56468i
\(974\) 18.0000 0.576757
\(975\) 0 0
\(976\) −14.0000 −0.448129
\(977\) − 18.0000i − 0.575871i −0.957650 0.287936i \(-0.907031\pi\)
0.957650 0.287936i \(-0.0929689\pi\)
\(978\) − 16.0000i − 0.511624i
\(979\) −16.0000 −0.511362
\(980\) 0 0
\(981\) 14.0000 0.446986
\(982\) − 24.0000i − 0.765871i
\(983\) 24.0000i 0.765481i 0.923856 + 0.382741i \(0.125020\pi\)
−0.923856 + 0.382741i \(0.874980\pi\)
\(984\) −6.00000 −0.191273
\(985\) 0 0
\(986\) 8.00000 0.254772
\(987\) 48.0000i 1.52786i
\(988\) 0 0
\(989\) −2.00000 −0.0635963
\(990\) 0 0
\(991\) 52.0000 1.65183 0.825917 0.563791i \(-0.190658\pi\)
0.825917 + 0.563791i \(0.190658\pi\)
\(992\) 0 0
\(993\) 28.0000i 0.888553i
\(994\) −8.00000 −0.253745
\(995\) 0 0
\(996\) −8.00000 −0.253490
\(997\) − 40.0000i − 1.26681i −0.773819 0.633406i \(-0.781656\pi\)
0.773819 0.633406i \(-0.218344\pi\)
\(998\) 28.0000i 0.886325i
\(999\) 10.0000 0.316386
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 3450.2.d.n.2899.2 2
5.2 odd 4 690.2.a.b.1.1 1
5.3 odd 4 3450.2.a.t.1.1 1
5.4 even 2 inner 3450.2.d.n.2899.1 2
15.2 even 4 2070.2.a.s.1.1 1
20.7 even 4 5520.2.a.r.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
690.2.a.b.1.1 1 5.2 odd 4
2070.2.a.s.1.1 1 15.2 even 4
3450.2.a.t.1.1 1 5.3 odd 4
3450.2.d.n.2899.1 2 5.4 even 2 inner
3450.2.d.n.2899.2 2 1.1 even 1 trivial
5520.2.a.r.1.1 1 20.7 even 4