Properties

Label 3450.2.d.q.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

Downloads

Learn more about

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.q.2899.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} +2.00000i q^{7} -1.00000i q^{8} -1.00000 q^{9} +O(q^{10})\) \(q+1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} +2.00000i q^{7} -1.00000i q^{8} -1.00000 q^{9} +2.00000 q^{11} +1.00000i q^{12} -2.00000i q^{13} -2.00000 q^{14} +1.00000 q^{16} -1.00000i q^{18} -8.00000 q^{19} +2.00000 q^{21} +2.00000i q^{22} -1.00000i q^{23} -1.00000 q^{24} +2.00000 q^{26} +1.00000i q^{27} -2.00000i q^{28} +10.0000 q^{29} +8.00000 q^{31} +1.00000i q^{32} -2.00000i q^{33} +1.00000 q^{36} -8.00000i q^{37} -8.00000i q^{38} -2.00000 q^{39} -6.00000 q^{41} +2.00000i q^{42} +12.0000i q^{43} -2.00000 q^{44} +1.00000 q^{46} -8.00000i q^{47} -1.00000i q^{48} +3.00000 q^{49} +2.00000i q^{52} +10.0000i q^{53} -1.00000 q^{54} +2.00000 q^{56} +8.00000i q^{57} +10.0000i q^{58} -4.00000 q^{59} +12.0000 q^{61} +8.00000i q^{62} -2.00000i q^{63} -1.00000 q^{64} +2.00000 q^{66} +4.00000i q^{67} -1.00000 q^{69} +16.0000 q^{71} +1.00000i q^{72} -10.0000i q^{73} +8.00000 q^{74} +8.00000 q^{76} +4.00000i q^{77} -2.00000i q^{78} -10.0000 q^{79} +1.00000 q^{81} -6.00000i q^{82} -10.0000i q^{83} -2.00000 q^{84} -12.0000 q^{86} -10.0000i q^{87} -2.00000i q^{88} +4.00000 q^{91} +1.00000i q^{92} -8.00000i q^{93} +8.00000 q^{94} +1.00000 q^{96} -10.0000i q^{97} +3.00000i q^{98} -2.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{4} + 2q^{6} - 2q^{9} + O(q^{10}) \) \( 2q - 2q^{4} + 2q^{6} - 2q^{9} + 4q^{11} - 4q^{14} + 2q^{16} - 16q^{19} + 4q^{21} - 2q^{24} + 4q^{26} + 20q^{29} + 16q^{31} + 2q^{36} - 4q^{39} - 12q^{41} - 4q^{44} + 2q^{46} + 6q^{49} - 2q^{54} + 4q^{56} - 8q^{59} + 24q^{61} - 2q^{64} + 4q^{66} - 2q^{69} + 32q^{71} + 16q^{74} + 16q^{76} - 20q^{79} + 2q^{81} - 4q^{84} - 24q^{86} + 8q^{91} + 16q^{94} + 2q^{96} - 4q^{99} + O(q^{100}) \)

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
\(7\) 2.00000i 0.755929i 0.925820 + 0.377964i \(0.123376\pi\)
−0.925820 + 0.377964i \(0.876624\pi\)
\(8\) − 1.00000i − 0.353553i
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) 2.00000 0.603023 0.301511 0.953463i \(-0.402509\pi\)
0.301511 + 0.953463i \(0.402509\pi\)
\(12\) 1.00000i 0.288675i
\(13\) − 2.00000i − 0.554700i −0.960769 0.277350i \(-0.910544\pi\)
0.960769 0.277350i \(-0.0894562\pi\)
\(14\) −2.00000 −0.534522
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(18\) − 1.00000i − 0.235702i
\(19\) −8.00000 −1.83533 −0.917663 0.397360i \(-0.869927\pi\)
−0.917663 + 0.397360i \(0.869927\pi\)
\(20\) 0 0
\(21\) 2.00000 0.436436
\(22\) 2.00000i 0.426401i
\(23\) − 1.00000i − 0.208514i
\(24\) −1.00000 −0.204124
\(25\) 0 0
\(26\) 2.00000 0.392232
\(27\) 1.00000i 0.192450i
\(28\) − 2.00000i − 0.377964i
\(29\) 10.0000 1.85695 0.928477 0.371391i \(-0.121119\pi\)
0.928477 + 0.371391i \(0.121119\pi\)
\(30\) 0 0
\(31\) 8.00000 1.43684 0.718421 0.695608i \(-0.244865\pi\)
0.718421 + 0.695608i \(0.244865\pi\)
\(32\) 1.00000i 0.176777i
\(33\) − 2.00000i − 0.348155i
\(34\) 0 0
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) − 8.00000i − 1.31519i −0.753371 0.657596i \(-0.771573\pi\)
0.753371 0.657596i \(-0.228427\pi\)
\(38\) − 8.00000i − 1.29777i
\(39\) −2.00000 −0.320256
\(40\) 0 0
\(41\) −6.00000 −0.937043 −0.468521 0.883452i \(-0.655213\pi\)
−0.468521 + 0.883452i \(0.655213\pi\)
\(42\) 2.00000i 0.308607i
\(43\) 12.0000i 1.82998i 0.403473 + 0.914991i \(0.367803\pi\)
−0.403473 + 0.914991i \(0.632197\pi\)
\(44\) −2.00000 −0.301511
\(45\) 0 0
\(46\) 1.00000 0.147442
\(47\) − 8.00000i − 1.16692i −0.812142 0.583460i \(-0.801699\pi\)
0.812142 0.583460i \(-0.198301\pi\)
\(48\) − 1.00000i − 0.144338i
\(49\) 3.00000 0.428571
\(50\) 0 0
\(51\) 0 0
\(52\) 2.00000i 0.277350i
\(53\) 10.0000i 1.37361i 0.726844 + 0.686803i \(0.240986\pi\)
−0.726844 + 0.686803i \(0.759014\pi\)
\(54\) −1.00000 −0.136083
\(55\) 0 0
\(56\) 2.00000 0.267261
\(57\) 8.00000i 1.05963i
\(58\) 10.0000i 1.31306i
\(59\) −4.00000 −0.520756 −0.260378 0.965507i \(-0.583847\pi\)
−0.260378 + 0.965507i \(0.583847\pi\)
\(60\) 0 0
\(61\) 12.0000 1.53644 0.768221 0.640184i \(-0.221142\pi\)
0.768221 + 0.640184i \(0.221142\pi\)
\(62\) 8.00000i 1.01600i
\(63\) − 2.00000i − 0.251976i
\(64\) −1.00000 −0.125000
\(65\) 0 0
\(66\) 2.00000 0.246183
\(67\) 4.00000i 0.488678i 0.969690 + 0.244339i \(0.0785709\pi\)
−0.969690 + 0.244339i \(0.921429\pi\)
\(68\) 0 0
\(69\) −1.00000 −0.120386
\(70\) 0 0
\(71\) 16.0000 1.89885 0.949425 0.313993i \(-0.101667\pi\)
0.949425 + 0.313993i \(0.101667\pi\)
\(72\) 1.00000i 0.117851i
\(73\) − 10.0000i − 1.17041i −0.810885 0.585206i \(-0.801014\pi\)
0.810885 0.585206i \(-0.198986\pi\)
\(74\) 8.00000 0.929981
\(75\) 0 0
\(76\) 8.00000 0.917663
\(77\) 4.00000i 0.455842i
\(78\) − 2.00000i − 0.226455i
\(79\) −10.0000 −1.12509 −0.562544 0.826767i \(-0.690177\pi\)
−0.562544 + 0.826767i \(0.690177\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) − 6.00000i − 0.662589i
\(83\) − 10.0000i − 1.09764i −0.835940 0.548821i \(-0.815077\pi\)
0.835940 0.548821i \(-0.184923\pi\)
\(84\) −2.00000 −0.218218
\(85\) 0 0
\(86\) −12.0000 −1.29399
\(87\) − 10.0000i − 1.07211i
\(88\) − 2.00000i − 0.213201i
\(89\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(90\) 0 0
\(91\) 4.00000 0.419314
\(92\) 1.00000i 0.104257i
\(93\) − 8.00000i − 0.829561i
\(94\) 8.00000 0.825137
\(95\) 0 0
\(96\) 1.00000 0.102062
\(97\) − 10.0000i − 1.01535i −0.861550 0.507673i \(-0.830506\pi\)
0.861550 0.507673i \(-0.169494\pi\)
\(98\) 3.00000i 0.303046i
\(99\) −2.00000 −0.201008
\(100\) 0 0
\(101\) 10.0000 0.995037 0.497519 0.867453i \(-0.334245\pi\)
0.497519 + 0.867453i \(0.334245\pi\)
\(102\) 0 0
\(103\) − 2.00000i − 0.197066i −0.995134 0.0985329i \(-0.968585\pi\)
0.995134 0.0985329i \(-0.0314150\pi\)
\(104\) −2.00000 −0.196116
\(105\) 0 0
\(106\) −10.0000 −0.971286
\(107\) 10.0000i 0.966736i 0.875417 + 0.483368i \(0.160587\pi\)
−0.875417 + 0.483368i \(0.839413\pi\)
\(108\) − 1.00000i − 0.0962250i
\(109\) 8.00000 0.766261 0.383131 0.923694i \(-0.374846\pi\)
0.383131 + 0.923694i \(0.374846\pi\)
\(110\) 0 0
\(111\) −8.00000 −0.759326
\(112\) 2.00000i 0.188982i
\(113\) 8.00000i 0.752577i 0.926503 + 0.376288i \(0.122800\pi\)
−0.926503 + 0.376288i \(0.877200\pi\)
\(114\) −8.00000 −0.749269
\(115\) 0 0
\(116\) −10.0000 −0.928477
\(117\) 2.00000i 0.184900i
\(118\) − 4.00000i − 0.368230i
\(119\) 0 0
\(120\) 0 0
\(121\) −7.00000 −0.636364
\(122\) 12.0000i 1.08643i
\(123\) 6.00000i 0.541002i
\(124\) −8.00000 −0.718421
\(125\) 0 0
\(126\) 2.00000 0.178174
\(127\) 4.00000i 0.354943i 0.984126 + 0.177471i \(0.0567917\pi\)
−0.984126 + 0.177471i \(0.943208\pi\)
\(128\) − 1.00000i − 0.0883883i
\(129\) 12.0000 1.05654
\(130\) 0 0
\(131\) −8.00000 −0.698963 −0.349482 0.936943i \(-0.613642\pi\)
−0.349482 + 0.936943i \(0.613642\pi\)
\(132\) 2.00000i 0.174078i
\(133\) − 16.0000i − 1.38738i
\(134\) −4.00000 −0.345547
\(135\) 0 0
\(136\) 0 0
\(137\) 12.0000i 1.02523i 0.858619 + 0.512615i \(0.171323\pi\)
−0.858619 + 0.512615i \(0.828677\pi\)
\(138\) − 1.00000i − 0.0851257i
\(139\) 20.0000 1.69638 0.848189 0.529694i \(-0.177693\pi\)
0.848189 + 0.529694i \(0.177693\pi\)
\(140\) 0 0
\(141\) −8.00000 −0.673722
\(142\) 16.0000i 1.34269i
\(143\) − 4.00000i − 0.334497i
\(144\) −1.00000 −0.0833333
\(145\) 0 0
\(146\) 10.0000 0.827606
\(147\) − 3.00000i − 0.247436i
\(148\) 8.00000i 0.657596i
\(149\) 10.0000 0.819232 0.409616 0.912258i \(-0.365663\pi\)
0.409616 + 0.912258i \(0.365663\pi\)
\(150\) 0 0
\(151\) 20.0000 1.62758 0.813788 0.581161i \(-0.197401\pi\)
0.813788 + 0.581161i \(0.197401\pi\)
\(152\) 8.00000i 0.648886i
\(153\) 0 0
\(154\) −4.00000 −0.322329
\(155\) 0 0
\(156\) 2.00000 0.160128
\(157\) 8.00000i 0.638470i 0.947676 + 0.319235i \(0.103426\pi\)
−0.947676 + 0.319235i \(0.896574\pi\)
\(158\) − 10.0000i − 0.795557i
\(159\) 10.0000 0.793052
\(160\) 0 0
\(161\) 2.00000 0.157622
\(162\) 1.00000i 0.0785674i
\(163\) 4.00000i 0.313304i 0.987654 + 0.156652i \(0.0500701\pi\)
−0.987654 + 0.156652i \(0.949930\pi\)
\(164\) 6.00000 0.468521
\(165\) 0 0
\(166\) 10.0000 0.776151
\(167\) 8.00000i 0.619059i 0.950890 + 0.309529i \(0.100171\pi\)
−0.950890 + 0.309529i \(0.899829\pi\)
\(168\) − 2.00000i − 0.154303i
\(169\) 9.00000 0.692308
\(170\) 0 0
\(171\) 8.00000 0.611775
\(172\) − 12.0000i − 0.914991i
\(173\) − 2.00000i − 0.152057i −0.997106 0.0760286i \(-0.975776\pi\)
0.997106 0.0760286i \(-0.0242240\pi\)
\(174\) 10.0000 0.758098
\(175\) 0 0
\(176\) 2.00000 0.150756
\(177\) 4.00000i 0.300658i
\(178\) 0 0
\(179\) 16.0000 1.19590 0.597948 0.801535i \(-0.295983\pi\)
0.597948 + 0.801535i \(0.295983\pi\)
\(180\) 0 0
\(181\) 16.0000 1.18927 0.594635 0.803996i \(-0.297296\pi\)
0.594635 + 0.803996i \(0.297296\pi\)
\(182\) 4.00000i 0.296500i
\(183\) − 12.0000i − 0.887066i
\(184\) −1.00000 −0.0737210
\(185\) 0 0
\(186\) 8.00000 0.586588
\(187\) 0 0
\(188\) 8.00000i 0.583460i
\(189\) −2.00000 −0.145479
\(190\) 0 0
\(191\) −16.0000 −1.15772 −0.578860 0.815427i \(-0.696502\pi\)
−0.578860 + 0.815427i \(0.696502\pi\)
\(192\) 1.00000i 0.0721688i
\(193\) 14.0000i 1.00774i 0.863779 + 0.503871i \(0.168091\pi\)
−0.863779 + 0.503871i \(0.831909\pi\)
\(194\) 10.0000 0.717958
\(195\) 0 0
\(196\) −3.00000 −0.214286
\(197\) − 26.0000i − 1.85242i −0.377004 0.926212i \(-0.623046\pi\)
0.377004 0.926212i \(-0.376954\pi\)
\(198\) − 2.00000i − 0.142134i
\(199\) 14.0000 0.992434 0.496217 0.868199i \(-0.334722\pi\)
0.496217 + 0.868199i \(0.334722\pi\)
\(200\) 0 0
\(201\) 4.00000 0.282138
\(202\) 10.0000i 0.703598i
\(203\) 20.0000i 1.40372i
\(204\) 0 0
\(205\) 0 0
\(206\) 2.00000 0.139347
\(207\) 1.00000i 0.0695048i
\(208\) − 2.00000i − 0.138675i
\(209\) −16.0000 −1.10674
\(210\) 0 0
\(211\) 4.00000 0.275371 0.137686 0.990476i \(-0.456034\pi\)
0.137686 + 0.990476i \(0.456034\pi\)
\(212\) − 10.0000i − 0.686803i
\(213\) − 16.0000i − 1.09630i
\(214\) −10.0000 −0.683586
\(215\) 0 0
\(216\) 1.00000 0.0680414
\(217\) 16.0000i 1.08615i
\(218\) 8.00000i 0.541828i
\(219\) −10.0000 −0.675737
\(220\) 0 0
\(221\) 0 0
\(222\) − 8.00000i − 0.536925i
\(223\) − 4.00000i − 0.267860i −0.990991 0.133930i \(-0.957240\pi\)
0.990991 0.133930i \(-0.0427597\pi\)
\(224\) −2.00000 −0.133631
\(225\) 0 0
\(226\) −8.00000 −0.532152
\(227\) − 14.0000i − 0.929213i −0.885517 0.464606i \(-0.846196\pi\)
0.885517 0.464606i \(-0.153804\pi\)
\(228\) − 8.00000i − 0.529813i
\(229\) 16.0000 1.05731 0.528655 0.848837i \(-0.322697\pi\)
0.528655 + 0.848837i \(0.322697\pi\)
\(230\) 0 0
\(231\) 4.00000 0.263181
\(232\) − 10.0000i − 0.656532i
\(233\) − 6.00000i − 0.393073i −0.980497 0.196537i \(-0.937031\pi\)
0.980497 0.196537i \(-0.0629694\pi\)
\(234\) −2.00000 −0.130744
\(235\) 0 0
\(236\) 4.00000 0.260378
\(237\) 10.0000i 0.649570i
\(238\) 0 0
\(239\) 16.0000 1.03495 0.517477 0.855697i \(-0.326871\pi\)
0.517477 + 0.855697i \(0.326871\pi\)
\(240\) 0 0
\(241\) 10.0000 0.644157 0.322078 0.946713i \(-0.395619\pi\)
0.322078 + 0.946713i \(0.395619\pi\)
\(242\) − 7.00000i − 0.449977i
\(243\) − 1.00000i − 0.0641500i
\(244\) −12.0000 −0.768221
\(245\) 0 0
\(246\) −6.00000 −0.382546
\(247\) 16.0000i 1.01806i
\(248\) − 8.00000i − 0.508001i
\(249\) −10.0000 −0.633724
\(250\) 0 0
\(251\) −14.0000 −0.883672 −0.441836 0.897096i \(-0.645673\pi\)
−0.441836 + 0.897096i \(0.645673\pi\)
\(252\) 2.00000i 0.125988i
\(253\) − 2.00000i − 0.125739i
\(254\) −4.00000 −0.250982
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 6.00000i 0.374270i 0.982334 + 0.187135i \(0.0599201\pi\)
−0.982334 + 0.187135i \(0.940080\pi\)
\(258\) 12.0000i 0.747087i
\(259\) 16.0000 0.994192
\(260\) 0 0
\(261\) −10.0000 −0.618984
\(262\) − 8.00000i − 0.494242i
\(263\) 4.00000i 0.246651i 0.992366 + 0.123325i \(0.0393559\pi\)
−0.992366 + 0.123325i \(0.960644\pi\)
\(264\) −2.00000 −0.123091
\(265\) 0 0
\(266\) 16.0000 0.981023
\(267\) 0 0
\(268\) − 4.00000i − 0.244339i
\(269\) −18.0000 −1.09748 −0.548740 0.835993i \(-0.684892\pi\)
−0.548740 + 0.835993i \(0.684892\pi\)
\(270\) 0 0
\(271\) 28.0000 1.70088 0.850439 0.526073i \(-0.176336\pi\)
0.850439 + 0.526073i \(0.176336\pi\)
\(272\) 0 0
\(273\) − 4.00000i − 0.242091i
\(274\) −12.0000 −0.724947
\(275\) 0 0
\(276\) 1.00000 0.0601929
\(277\) 6.00000i 0.360505i 0.983620 + 0.180253i \(0.0576915\pi\)
−0.983620 + 0.180253i \(0.942309\pi\)
\(278\) 20.0000i 1.19952i
\(279\) −8.00000 −0.478947
\(280\) 0 0
\(281\) −16.0000 −0.954480 −0.477240 0.878773i \(-0.658363\pi\)
−0.477240 + 0.878773i \(0.658363\pi\)
\(282\) − 8.00000i − 0.476393i
\(283\) − 16.0000i − 0.951101i −0.879688 0.475551i \(-0.842249\pi\)
0.879688 0.475551i \(-0.157751\pi\)
\(284\) −16.0000 −0.949425
\(285\) 0 0
\(286\) 4.00000 0.236525
\(287\) − 12.0000i − 0.708338i
\(288\) − 1.00000i − 0.0589256i
\(289\) 17.0000 1.00000
\(290\) 0 0
\(291\) −10.0000 −0.586210
\(292\) 10.0000i 0.585206i
\(293\) 2.00000i 0.116841i 0.998292 + 0.0584206i \(0.0186065\pi\)
−0.998292 + 0.0584206i \(0.981394\pi\)
\(294\) 3.00000 0.174964
\(295\) 0 0
\(296\) −8.00000 −0.464991
\(297\) 2.00000i 0.116052i
\(298\) 10.0000i 0.579284i
\(299\) −2.00000 −0.115663
\(300\) 0 0
\(301\) −24.0000 −1.38334
\(302\) 20.0000i 1.15087i
\(303\) − 10.0000i − 0.574485i
\(304\) −8.00000 −0.458831
\(305\) 0 0
\(306\) 0 0
\(307\) − 28.0000i − 1.59804i −0.601302 0.799022i \(-0.705351\pi\)
0.601302 0.799022i \(-0.294649\pi\)
\(308\) − 4.00000i − 0.227921i
\(309\) −2.00000 −0.113776
\(310\) 0 0
\(311\) 16.0000 0.907277 0.453638 0.891186i \(-0.350126\pi\)
0.453638 + 0.891186i \(0.350126\pi\)
\(312\) 2.00000i 0.113228i
\(313\) − 10.0000i − 0.565233i −0.959233 0.282617i \(-0.908798\pi\)
0.959233 0.282617i \(-0.0912024\pi\)
\(314\) −8.00000 −0.451466
\(315\) 0 0
\(316\) 10.0000 0.562544
\(317\) − 6.00000i − 0.336994i −0.985702 0.168497i \(-0.946109\pi\)
0.985702 0.168497i \(-0.0538913\pi\)
\(318\) 10.0000i 0.560772i
\(319\) 20.0000 1.11979
\(320\) 0 0
\(321\) 10.0000 0.558146
\(322\) 2.00000i 0.111456i
\(323\) 0 0
\(324\) −1.00000 −0.0555556
\(325\) 0 0
\(326\) −4.00000 −0.221540
\(327\) − 8.00000i − 0.442401i
\(328\) 6.00000i 0.331295i
\(329\) 16.0000 0.882109
\(330\) 0 0
\(331\) −12.0000 −0.659580 −0.329790 0.944054i \(-0.606978\pi\)
−0.329790 + 0.944054i \(0.606978\pi\)
\(332\) 10.0000i 0.548821i
\(333\) 8.00000i 0.438397i
\(334\) −8.00000 −0.437741
\(335\) 0 0
\(336\) 2.00000 0.109109
\(337\) − 22.0000i − 1.19842i −0.800593 0.599208i \(-0.795482\pi\)
0.800593 0.599208i \(-0.204518\pi\)
\(338\) 9.00000i 0.489535i
\(339\) 8.00000 0.434500
\(340\) 0 0
\(341\) 16.0000 0.866449
\(342\) 8.00000i 0.432590i
\(343\) 20.0000i 1.07990i
\(344\) 12.0000 0.646997
\(345\) 0 0
\(346\) 2.00000 0.107521
\(347\) 24.0000i 1.28839i 0.764862 + 0.644194i \(0.222807\pi\)
−0.764862 + 0.644194i \(0.777193\pi\)
\(348\) 10.0000i 0.536056i
\(349\) 6.00000 0.321173 0.160586 0.987022i \(-0.448662\pi\)
0.160586 + 0.987022i \(0.448662\pi\)
\(350\) 0 0
\(351\) 2.00000 0.106752
\(352\) 2.00000i 0.106600i
\(353\) 14.0000i 0.745145i 0.928003 + 0.372572i \(0.121524\pi\)
−0.928003 + 0.372572i \(0.878476\pi\)
\(354\) −4.00000 −0.212598
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 16.0000i 0.845626i
\(359\) −8.00000 −0.422224 −0.211112 0.977462i \(-0.567708\pi\)
−0.211112 + 0.977462i \(0.567708\pi\)
\(360\) 0 0
\(361\) 45.0000 2.36842
\(362\) 16.0000i 0.840941i
\(363\) 7.00000i 0.367405i
\(364\) −4.00000 −0.209657
\(365\) 0 0
\(366\) 12.0000 0.627250
\(367\) − 18.0000i − 0.939592i −0.882775 0.469796i \(-0.844327\pi\)
0.882775 0.469796i \(-0.155673\pi\)
\(368\) − 1.00000i − 0.0521286i
\(369\) 6.00000 0.312348
\(370\) 0 0
\(371\) −20.0000 −1.03835
\(372\) 8.00000i 0.414781i
\(373\) − 24.0000i − 1.24267i −0.783544 0.621336i \(-0.786590\pi\)
0.783544 0.621336i \(-0.213410\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) −8.00000 −0.412568
\(377\) − 20.0000i − 1.03005i
\(378\) − 2.00000i − 0.102869i
\(379\) −8.00000 −0.410932 −0.205466 0.978664i \(-0.565871\pi\)
−0.205466 + 0.978664i \(0.565871\pi\)
\(380\) 0 0
\(381\) 4.00000 0.204926
\(382\) − 16.0000i − 0.818631i
\(383\) 36.0000i 1.83951i 0.392488 + 0.919757i \(0.371614\pi\)
−0.392488 + 0.919757i \(0.628386\pi\)
\(384\) −1.00000 −0.0510310
\(385\) 0 0
\(386\) −14.0000 −0.712581
\(387\) − 12.0000i − 0.609994i
\(388\) 10.0000i 0.507673i
\(389\) 2.00000 0.101404 0.0507020 0.998714i \(-0.483854\pi\)
0.0507020 + 0.998714i \(0.483854\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) − 3.00000i − 0.151523i
\(393\) 8.00000i 0.403547i
\(394\) 26.0000 1.30986
\(395\) 0 0
\(396\) 2.00000 0.100504
\(397\) 10.0000i 0.501886i 0.968002 + 0.250943i \(0.0807406\pi\)
−0.968002 + 0.250943i \(0.919259\pi\)
\(398\) 14.0000i 0.701757i
\(399\) −16.0000 −0.801002
\(400\) 0 0
\(401\) −16.0000 −0.799002 −0.399501 0.916733i \(-0.630817\pi\)
−0.399501 + 0.916733i \(0.630817\pi\)
\(402\) 4.00000i 0.199502i
\(403\) − 16.0000i − 0.797017i
\(404\) −10.0000 −0.497519
\(405\) 0 0
\(406\) −20.0000 −0.992583
\(407\) − 16.0000i − 0.793091i
\(408\) 0 0
\(409\) 10.0000 0.494468 0.247234 0.968956i \(-0.420478\pi\)
0.247234 + 0.968956i \(0.420478\pi\)
\(410\) 0 0
\(411\) 12.0000 0.591916
\(412\) 2.00000i 0.0985329i
\(413\) − 8.00000i − 0.393654i
\(414\) −1.00000 −0.0491473
\(415\) 0 0
\(416\) 2.00000 0.0980581
\(417\) − 20.0000i − 0.979404i
\(418\) − 16.0000i − 0.782586i
\(419\) −30.0000 −1.46560 −0.732798 0.680446i \(-0.761786\pi\)
−0.732798 + 0.680446i \(0.761786\pi\)
\(420\) 0 0
\(421\) −36.0000 −1.75453 −0.877266 0.480004i \(-0.840635\pi\)
−0.877266 + 0.480004i \(0.840635\pi\)
\(422\) 4.00000i 0.194717i
\(423\) 8.00000i 0.388973i
\(424\) 10.0000 0.485643
\(425\) 0 0
\(426\) 16.0000 0.775203
\(427\) 24.0000i 1.16144i
\(428\) − 10.0000i − 0.483368i
\(429\) −4.00000 −0.193122
\(430\) 0 0
\(431\) −28.0000 −1.34871 −0.674356 0.738406i \(-0.735579\pi\)
−0.674356 + 0.738406i \(0.735579\pi\)
\(432\) 1.00000i 0.0481125i
\(433\) 14.0000i 0.672797i 0.941720 + 0.336399i \(0.109209\pi\)
−0.941720 + 0.336399i \(0.890791\pi\)
\(434\) −16.0000 −0.768025
\(435\) 0 0
\(436\) −8.00000 −0.383131
\(437\) 8.00000i 0.382692i
\(438\) − 10.0000i − 0.477818i
\(439\) −16.0000 −0.763638 −0.381819 0.924237i \(-0.624702\pi\)
−0.381819 + 0.924237i \(0.624702\pi\)
\(440\) 0 0
\(441\) −3.00000 −0.142857
\(442\) 0 0
\(443\) − 28.0000i − 1.33032i −0.746701 0.665160i \(-0.768363\pi\)
0.746701 0.665160i \(-0.231637\pi\)
\(444\) 8.00000 0.379663
\(445\) 0 0
\(446\) 4.00000 0.189405
\(447\) − 10.0000i − 0.472984i
\(448\) − 2.00000i − 0.0944911i
\(449\) −18.0000 −0.849473 −0.424736 0.905317i \(-0.639633\pi\)
−0.424736 + 0.905317i \(0.639633\pi\)
\(450\) 0 0
\(451\) −12.0000 −0.565058
\(452\) − 8.00000i − 0.376288i
\(453\) − 20.0000i − 0.939682i
\(454\) 14.0000 0.657053
\(455\) 0 0
\(456\) 8.00000 0.374634
\(457\) − 10.0000i − 0.467780i −0.972263 0.233890i \(-0.924854\pi\)
0.972263 0.233890i \(-0.0751456\pi\)
\(458\) 16.0000i 0.747631i
\(459\) 0 0
\(460\) 0 0
\(461\) −30.0000 −1.39724 −0.698620 0.715493i \(-0.746202\pi\)
−0.698620 + 0.715493i \(0.746202\pi\)
\(462\) 4.00000i 0.186097i
\(463\) 8.00000i 0.371792i 0.982569 + 0.185896i \(0.0595187\pi\)
−0.982569 + 0.185896i \(0.940481\pi\)
\(464\) 10.0000 0.464238
\(465\) 0 0
\(466\) 6.00000 0.277945
\(467\) 6.00000i 0.277647i 0.990317 + 0.138823i \(0.0443321\pi\)
−0.990317 + 0.138823i \(0.955668\pi\)
\(468\) − 2.00000i − 0.0924500i
\(469\) −8.00000 −0.369406
\(470\) 0 0
\(471\) 8.00000 0.368621
\(472\) 4.00000i 0.184115i
\(473\) 24.0000i 1.10352i
\(474\) −10.0000 −0.459315
\(475\) 0 0
\(476\) 0 0
\(477\) − 10.0000i − 0.457869i
\(478\) 16.0000i 0.731823i
\(479\) −24.0000 −1.09659 −0.548294 0.836286i \(-0.684723\pi\)
−0.548294 + 0.836286i \(0.684723\pi\)
\(480\) 0 0
\(481\) −16.0000 −0.729537
\(482\) 10.0000i 0.455488i
\(483\) − 2.00000i − 0.0910032i
\(484\) 7.00000 0.318182
\(485\) 0 0
\(486\) 1.00000 0.0453609
\(487\) 40.0000i 1.81257i 0.422664 + 0.906287i \(0.361095\pi\)
−0.422664 + 0.906287i \(0.638905\pi\)
\(488\) − 12.0000i − 0.543214i
\(489\) 4.00000 0.180886
\(490\) 0 0
\(491\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(492\) − 6.00000i − 0.270501i
\(493\) 0 0
\(494\) −16.0000 −0.719874
\(495\) 0 0
\(496\) 8.00000 0.359211
\(497\) 32.0000i 1.43540i
\(498\) − 10.0000i − 0.448111i
\(499\) 4.00000 0.179065 0.0895323 0.995984i \(-0.471463\pi\)
0.0895323 + 0.995984i \(0.471463\pi\)
\(500\) 0 0
\(501\) 8.00000 0.357414
\(502\) − 14.0000i − 0.624851i
\(503\) 36.0000i 1.60516i 0.596544 + 0.802580i \(0.296540\pi\)
−0.596544 + 0.802580i \(0.703460\pi\)
\(504\) −2.00000 −0.0890871
\(505\) 0 0
\(506\) 2.00000 0.0889108
\(507\) − 9.00000i − 0.399704i
\(508\) − 4.00000i − 0.177471i
\(509\) −6.00000 −0.265945 −0.132973 0.991120i \(-0.542452\pi\)
−0.132973 + 0.991120i \(0.542452\pi\)
\(510\) 0 0
\(511\) 20.0000 0.884748
\(512\) 1.00000i 0.0441942i
\(513\) − 8.00000i − 0.353209i
\(514\) −6.00000 −0.264649
\(515\) 0 0
\(516\) −12.0000 −0.528271
\(517\) − 16.0000i − 0.703679i
\(518\) 16.0000i 0.703000i
\(519\) −2.00000 −0.0877903
\(520\) 0 0
\(521\) −12.0000 −0.525730 −0.262865 0.964833i \(-0.584667\pi\)
−0.262865 + 0.964833i \(0.584667\pi\)
\(522\) − 10.0000i − 0.437688i
\(523\) 4.00000i 0.174908i 0.996169 + 0.0874539i \(0.0278730\pi\)
−0.996169 + 0.0874539i \(0.972127\pi\)
\(524\) 8.00000 0.349482
\(525\) 0 0
\(526\) −4.00000 −0.174408
\(527\) 0 0
\(528\) − 2.00000i − 0.0870388i
\(529\) −1.00000 −0.0434783
\(530\) 0 0
\(531\) 4.00000 0.173585
\(532\) 16.0000i 0.693688i
\(533\) 12.0000i 0.519778i
\(534\) 0 0
\(535\) 0 0
\(536\) 4.00000 0.172774
\(537\) − 16.0000i − 0.690451i
\(538\) − 18.0000i − 0.776035i
\(539\) 6.00000 0.258438
\(540\) 0 0
\(541\) −38.0000 −1.63375 −0.816874 0.576816i \(-0.804295\pi\)
−0.816874 + 0.576816i \(0.804295\pi\)
\(542\) 28.0000i 1.20270i
\(543\) − 16.0000i − 0.686626i
\(544\) 0 0
\(545\) 0 0
\(546\) 4.00000 0.171184
\(547\) 20.0000i 0.855138i 0.903983 + 0.427569i \(0.140630\pi\)
−0.903983 + 0.427569i \(0.859370\pi\)
\(548\) − 12.0000i − 0.512615i
\(549\) −12.0000 −0.512148
\(550\) 0 0
\(551\) −80.0000 −3.40811
\(552\) 1.00000i 0.0425628i
\(553\) − 20.0000i − 0.850487i
\(554\) −6.00000 −0.254916
\(555\) 0 0
\(556\) −20.0000 −0.848189
\(557\) 2.00000i 0.0847427i 0.999102 + 0.0423714i \(0.0134913\pi\)
−0.999102 + 0.0423714i \(0.986509\pi\)
\(558\) − 8.00000i − 0.338667i
\(559\) 24.0000 1.01509
\(560\) 0 0
\(561\) 0 0
\(562\) − 16.0000i − 0.674919i
\(563\) 6.00000i 0.252870i 0.991975 + 0.126435i \(0.0403535\pi\)
−0.991975 + 0.126435i \(0.959647\pi\)
\(564\) 8.00000 0.336861
\(565\) 0 0
\(566\) 16.0000 0.672530
\(567\) 2.00000i 0.0839921i
\(568\) − 16.0000i − 0.671345i
\(569\) 32.0000 1.34151 0.670755 0.741679i \(-0.265970\pi\)
0.670755 + 0.741679i \(0.265970\pi\)
\(570\) 0 0
\(571\) 12.0000 0.502184 0.251092 0.967963i \(-0.419210\pi\)
0.251092 + 0.967963i \(0.419210\pi\)
\(572\) 4.00000i 0.167248i
\(573\) 16.0000i 0.668410i
\(574\) 12.0000 0.500870
\(575\) 0 0
\(576\) 1.00000 0.0416667
\(577\) − 2.00000i − 0.0832611i −0.999133 0.0416305i \(-0.986745\pi\)
0.999133 0.0416305i \(-0.0132552\pi\)
\(578\) 17.0000i 0.707107i
\(579\) 14.0000 0.581820
\(580\) 0 0
\(581\) 20.0000 0.829740
\(582\) − 10.0000i − 0.414513i
\(583\) 20.0000i 0.828315i
\(584\) −10.0000 −0.413803
\(585\) 0 0
\(586\) −2.00000 −0.0826192
\(587\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(588\) 3.00000i 0.123718i
\(589\) −64.0000 −2.63707
\(590\) 0 0
\(591\) −26.0000 −1.06950
\(592\) − 8.00000i − 0.328798i
\(593\) − 6.00000i − 0.246390i −0.992382 0.123195i \(-0.960686\pi\)
0.992382 0.123195i \(-0.0393141\pi\)
\(594\) −2.00000 −0.0820610
\(595\) 0 0
\(596\) −10.0000 −0.409616
\(597\) − 14.0000i − 0.572982i
\(598\) − 2.00000i − 0.0817861i
\(599\) −32.0000 −1.30748 −0.653742 0.756717i \(-0.726802\pi\)
−0.653742 + 0.756717i \(0.726802\pi\)
\(600\) 0 0
\(601\) 10.0000 0.407909 0.203954 0.978980i \(-0.434621\pi\)
0.203954 + 0.978980i \(0.434621\pi\)
\(602\) − 24.0000i − 0.978167i
\(603\) − 4.00000i − 0.162893i
\(604\) −20.0000 −0.813788
\(605\) 0 0
\(606\) 10.0000 0.406222
\(607\) − 12.0000i − 0.487065i −0.969893 0.243532i \(-0.921694\pi\)
0.969893 0.243532i \(-0.0783062\pi\)
\(608\) − 8.00000i − 0.324443i
\(609\) 20.0000 0.810441
\(610\) 0 0
\(611\) −16.0000 −0.647291
\(612\) 0 0
\(613\) − 44.0000i − 1.77714i −0.458738 0.888572i \(-0.651698\pi\)
0.458738 0.888572i \(-0.348302\pi\)
\(614\) 28.0000 1.12999
\(615\) 0 0
\(616\) 4.00000 0.161165
\(617\) − 48.0000i − 1.93241i −0.257780 0.966204i \(-0.582991\pi\)
0.257780 0.966204i \(-0.417009\pi\)
\(618\) − 2.00000i − 0.0804518i
\(619\) 8.00000 0.321547 0.160774 0.986991i \(-0.448601\pi\)
0.160774 + 0.986991i \(0.448601\pi\)
\(620\) 0 0
\(621\) 1.00000 0.0401286
\(622\) 16.0000i 0.641542i
\(623\) 0 0
\(624\) −2.00000 −0.0800641
\(625\) 0 0
\(626\) 10.0000 0.399680
\(627\) 16.0000i 0.638978i
\(628\) − 8.00000i − 0.319235i
\(629\) 0 0
\(630\) 0 0
\(631\) −2.00000 −0.0796187 −0.0398094 0.999207i \(-0.512675\pi\)
−0.0398094 + 0.999207i \(0.512675\pi\)
\(632\) 10.0000i 0.397779i
\(633\) − 4.00000i − 0.158986i
\(634\) 6.00000 0.238290
\(635\) 0 0
\(636\) −10.0000 −0.396526
\(637\) − 6.00000i − 0.237729i
\(638\) 20.0000i 0.791808i
\(639\) −16.0000 −0.632950
\(640\) 0 0
\(641\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(642\) 10.0000i 0.394669i
\(643\) 4.00000i 0.157745i 0.996885 + 0.0788723i \(0.0251319\pi\)
−0.996885 + 0.0788723i \(0.974868\pi\)
\(644\) −2.00000 −0.0788110
\(645\) 0 0
\(646\) 0 0
\(647\) 40.0000i 1.57256i 0.617869 + 0.786281i \(0.287996\pi\)
−0.617869 + 0.786281i \(0.712004\pi\)
\(648\) − 1.00000i − 0.0392837i
\(649\) −8.00000 −0.314027
\(650\) 0 0
\(651\) 16.0000 0.627089
\(652\) − 4.00000i − 0.156652i
\(653\) − 6.00000i − 0.234798i −0.993085 0.117399i \(-0.962544\pi\)
0.993085 0.117399i \(-0.0374557\pi\)
\(654\) 8.00000 0.312825
\(655\) 0 0
\(656\) −6.00000 −0.234261
\(657\) 10.0000i 0.390137i
\(658\) 16.0000i 0.623745i
\(659\) 30.0000 1.16863 0.584317 0.811525i \(-0.301362\pi\)
0.584317 + 0.811525i \(0.301362\pi\)
\(660\) 0 0
\(661\) −28.0000 −1.08907 −0.544537 0.838737i \(-0.683295\pi\)
−0.544537 + 0.838737i \(0.683295\pi\)
\(662\) − 12.0000i − 0.466393i
\(663\) 0 0
\(664\) −10.0000 −0.388075
\(665\) 0 0
\(666\) −8.00000 −0.309994
\(667\) − 10.0000i − 0.387202i
\(668\) − 8.00000i − 0.309529i
\(669\) −4.00000 −0.154649
\(670\) 0 0
\(671\) 24.0000 0.926510
\(672\) 2.00000i 0.0771517i
\(673\) − 46.0000i − 1.77317i −0.462566 0.886585i \(-0.653071\pi\)
0.462566 0.886585i \(-0.346929\pi\)
\(674\) 22.0000 0.847408
\(675\) 0 0
\(676\) −9.00000 −0.346154
\(677\) − 46.0000i − 1.76792i −0.467559 0.883962i \(-0.654866\pi\)
0.467559 0.883962i \(-0.345134\pi\)
\(678\) 8.00000i 0.307238i
\(679\) 20.0000 0.767530
\(680\) 0 0
\(681\) −14.0000 −0.536481
\(682\) 16.0000i 0.612672i
\(683\) 24.0000i 0.918334i 0.888350 + 0.459167i \(0.151852\pi\)
−0.888350 + 0.459167i \(0.848148\pi\)
\(684\) −8.00000 −0.305888
\(685\) 0 0
\(686\) −20.0000 −0.763604
\(687\) − 16.0000i − 0.610438i
\(688\) 12.0000i 0.457496i
\(689\) 20.0000 0.761939
\(690\) 0 0
\(691\) −52.0000 −1.97817 −0.989087 0.147335i \(-0.952930\pi\)
−0.989087 + 0.147335i \(0.952930\pi\)
\(692\) 2.00000i 0.0760286i
\(693\) − 4.00000i − 0.151947i
\(694\) −24.0000 −0.911028
\(695\) 0 0
\(696\) −10.0000 −0.379049
\(697\) 0 0
\(698\) 6.00000i 0.227103i
\(699\) −6.00000 −0.226941
\(700\) 0 0
\(701\) −6.00000 −0.226617 −0.113308 0.993560i \(-0.536145\pi\)
−0.113308 + 0.993560i \(0.536145\pi\)
\(702\) 2.00000i 0.0754851i
\(703\) 64.0000i 2.41381i
\(704\) −2.00000 −0.0753778
\(705\) 0 0
\(706\) −14.0000 −0.526897
\(707\) 20.0000i 0.752177i
\(708\) − 4.00000i − 0.150329i
\(709\) −28.0000 −1.05156 −0.525781 0.850620i \(-0.676227\pi\)
−0.525781 + 0.850620i \(0.676227\pi\)
\(710\) 0 0
\(711\) 10.0000 0.375029
\(712\) 0 0
\(713\) − 8.00000i − 0.299602i
\(714\) 0 0
\(715\) 0 0
\(716\) −16.0000 −0.597948
\(717\) − 16.0000i − 0.597531i
\(718\) − 8.00000i − 0.298557i
\(719\) −16.0000 −0.596699 −0.298350 0.954457i \(-0.596436\pi\)
−0.298350 + 0.954457i \(0.596436\pi\)
\(720\) 0 0
\(721\) 4.00000 0.148968
\(722\) 45.0000i 1.67473i
\(723\) − 10.0000i − 0.371904i
\(724\) −16.0000 −0.594635
\(725\) 0 0
\(726\) −7.00000 −0.259794
\(727\) − 14.0000i − 0.519231i −0.965712 0.259616i \(-0.916404\pi\)
0.965712 0.259616i \(-0.0835959\pi\)
\(728\) − 4.00000i − 0.148250i
\(729\) −1.00000 −0.0370370
\(730\) 0 0
\(731\) 0 0
\(732\) 12.0000i 0.443533i
\(733\) − 52.0000i − 1.92066i −0.278859 0.960332i \(-0.589956\pi\)
0.278859 0.960332i \(-0.410044\pi\)
\(734\) 18.0000 0.664392
\(735\) 0 0
\(736\) 1.00000 0.0368605
\(737\) 8.00000i 0.294684i
\(738\) 6.00000i 0.220863i
\(739\) −44.0000 −1.61857 −0.809283 0.587419i \(-0.800144\pi\)
−0.809283 + 0.587419i \(0.800144\pi\)
\(740\) 0 0
\(741\) 16.0000 0.587775
\(742\) − 20.0000i − 0.734223i
\(743\) − 28.0000i − 1.02722i −0.858024 0.513610i \(-0.828308\pi\)
0.858024 0.513610i \(-0.171692\pi\)
\(744\) −8.00000 −0.293294
\(745\) 0 0
\(746\) 24.0000 0.878702
\(747\) 10.0000i 0.365881i
\(748\) 0 0
\(749\) −20.0000 −0.730784
\(750\) 0 0
\(751\) 26.0000 0.948753 0.474377 0.880322i \(-0.342673\pi\)
0.474377 + 0.880322i \(0.342673\pi\)
\(752\) − 8.00000i − 0.291730i
\(753\) 14.0000i 0.510188i
\(754\) 20.0000 0.728357
\(755\) 0 0
\(756\) 2.00000 0.0727393
\(757\) 8.00000i 0.290765i 0.989376 + 0.145382i \(0.0464413\pi\)
−0.989376 + 0.145382i \(0.953559\pi\)
\(758\) − 8.00000i − 0.290573i
\(759\) −2.00000 −0.0725954
\(760\) 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
\(763\) 16.0000i 0.579239i
\(764\) 16.0000 0.578860
\(765\) 0 0
\(766\) −36.0000 −1.30073
\(767\) 8.00000i 0.288863i
\(768\) − 1.00000i − 0.0360844i
\(769\) 10.0000 0.360609 0.180305 0.983611i \(-0.442292\pi\)
0.180305 + 0.983611i \(0.442292\pi\)
\(770\) 0 0
\(771\) 6.00000 0.216085
\(772\) − 14.0000i − 0.503871i
\(773\) − 6.00000i − 0.215805i −0.994161 0.107903i \(-0.965587\pi\)
0.994161 0.107903i \(-0.0344134\pi\)
\(774\) 12.0000 0.431331
\(775\) 0 0
\(776\) −10.0000 −0.358979
\(777\) − 16.0000i − 0.573997i
\(778\) 2.00000i 0.0717035i
\(779\) 48.0000 1.71978
\(780\) 0 0
\(781\) 32.0000 1.14505
\(782\) 0 0
\(783\) 10.0000i 0.357371i
\(784\) 3.00000 0.107143
\(785\) 0 0
\(786\) −8.00000 −0.285351
\(787\) 4.00000i 0.142585i 0.997455 + 0.0712923i \(0.0227123\pi\)
−0.997455 + 0.0712923i \(0.977288\pi\)
\(788\) 26.0000i 0.926212i
\(789\) 4.00000 0.142404
\(790\) 0 0
\(791\) −16.0000 −0.568895
\(792\) 2.00000i 0.0710669i
\(793\) − 24.0000i − 0.852265i
\(794\) −10.0000 −0.354887
\(795\) 0 0
\(796\) −14.0000 −0.496217
\(797\) − 6.00000i − 0.212531i −0.994338 0.106265i \(-0.966111\pi\)
0.994338 0.106265i \(-0.0338893\pi\)
\(798\) − 16.0000i − 0.566394i
\(799\) 0 0
\(800\) 0 0
\(801\) 0 0
\(802\) − 16.0000i − 0.564980i
\(803\) − 20.0000i − 0.705785i
\(804\) −4.00000 −0.141069
\(805\) 0 0
\(806\) 16.0000 0.563576
\(807\) 18.0000i 0.633630i
\(808\) − 10.0000i − 0.351799i
\(809\) −30.0000 −1.05474 −0.527372 0.849635i \(-0.676823\pi\)
−0.527372 + 0.849635i \(0.676823\pi\)
\(810\) 0 0
\(811\) 12.0000 0.421377 0.210688 0.977553i \(-0.432429\pi\)
0.210688 + 0.977553i \(0.432429\pi\)
\(812\) − 20.0000i − 0.701862i
\(813\) − 28.0000i − 0.982003i
\(814\) 16.0000 0.560800
\(815\) 0 0
\(816\) 0 0
\(817\) − 96.0000i − 3.35861i
\(818\) 10.0000i 0.349642i
\(819\) −4.00000 −0.139771
\(820\) 0 0
\(821\) 42.0000 1.46581 0.732905 0.680331i \(-0.238164\pi\)
0.732905 + 0.680331i \(0.238164\pi\)
\(822\) 12.0000i 0.418548i
\(823\) 24.0000i 0.836587i 0.908312 + 0.418294i \(0.137372\pi\)
−0.908312 + 0.418294i \(0.862628\pi\)
\(824\) −2.00000 −0.0696733
\(825\) 0 0
\(826\) 8.00000 0.278356
\(827\) 6.00000i 0.208640i 0.994544 + 0.104320i \(0.0332667\pi\)
−0.994544 + 0.104320i \(0.966733\pi\)
\(828\) − 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
\(831\) 6.00000 0.208138
\(832\) 2.00000i 0.0693375i
\(833\) 0 0
\(834\) 20.0000 0.692543
\(835\) 0 0
\(836\) 16.0000 0.553372
\(837\) 8.00000i 0.276520i
\(838\) − 30.0000i − 1.03633i
\(839\) 12.0000 0.414286 0.207143 0.978311i \(-0.433583\pi\)
0.207143 + 0.978311i \(0.433583\pi\)
\(840\) 0 0
\(841\) 71.0000 2.44828
\(842\) − 36.0000i − 1.24064i
\(843\) 16.0000i 0.551069i
\(844\) −4.00000 −0.137686
\(845\) 0 0
\(846\) −8.00000 −0.275046
\(847\) − 14.0000i − 0.481046i
\(848\) 10.0000i 0.343401i
\(849\) −16.0000 −0.549119
\(850\) 0 0
\(851\) −8.00000 −0.274236
\(852\) 16.0000i 0.548151i
\(853\) − 10.0000i − 0.342393i −0.985237 0.171197i \(-0.945237\pi\)
0.985237 0.171197i \(-0.0547634\pi\)
\(854\) −24.0000 −0.821263
\(855\) 0 0
\(856\) 10.0000 0.341793
\(857\) − 18.0000i − 0.614868i −0.951569 0.307434i \(-0.900530\pi\)
0.951569 0.307434i \(-0.0994704\pi\)
\(858\) − 4.00000i − 0.136558i
\(859\) 4.00000 0.136478 0.0682391 0.997669i \(-0.478262\pi\)
0.0682391 + 0.997669i \(0.478262\pi\)
\(860\) 0 0
\(861\) −12.0000 −0.408959
\(862\) − 28.0000i − 0.953684i
\(863\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(864\) −1.00000 −0.0340207
\(865\) 0 0
\(866\) −14.0000 −0.475739
\(867\) − 17.0000i − 0.577350i
\(868\) − 16.0000i − 0.543075i
\(869\) −20.0000 −0.678454
\(870\) 0 0
\(871\) 8.00000 0.271070
\(872\) − 8.00000i − 0.270914i
\(873\) 10.0000i 0.338449i
\(874\) −8.00000 −0.270604
\(875\) 0 0
\(876\) 10.0000 0.337869
\(877\) 22.0000i 0.742887i 0.928456 + 0.371444i \(0.121137\pi\)
−0.928456 + 0.371444i \(0.878863\pi\)
\(878\) − 16.0000i − 0.539974i
\(879\) 2.00000 0.0674583
\(880\) 0 0
\(881\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(882\) − 3.00000i − 0.101015i
\(883\) 20.0000i 0.673054i 0.941674 + 0.336527i \(0.109252\pi\)
−0.941674 + 0.336527i \(0.890748\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 28.0000 0.940678
\(887\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(888\) 8.00000i 0.268462i
\(889\) −8.00000 −0.268311
\(890\) 0 0
\(891\) 2.00000 0.0670025
\(892\) 4.00000i 0.133930i
\(893\) 64.0000i 2.14168i
\(894\) 10.0000 0.334450
\(895\) 0 0
\(896\) 2.00000 0.0668153
\(897\) 2.00000i 0.0667781i
\(898\) − 18.0000i − 0.600668i
\(899\) 80.0000 2.66815
\(900\) 0 0
\(901\) 0 0
\(902\) − 12.0000i − 0.399556i
\(903\) 24.0000i 0.798670i
\(904\) 8.00000 0.266076
\(905\) 0 0
\(906\) 20.0000 0.664455
\(907\) − 40.0000i − 1.32818i −0.747653 0.664089i \(-0.768820\pi\)
0.747653 0.664089i \(-0.231180\pi\)
\(908\) 14.0000i 0.464606i
\(909\) −10.0000 −0.331679
\(910\) 0 0
\(911\) −36.0000 −1.19273 −0.596367 0.802712i \(-0.703390\pi\)
−0.596367 + 0.802712i \(0.703390\pi\)
\(912\) 8.00000i 0.264906i
\(913\) − 20.0000i − 0.661903i
\(914\) 10.0000 0.330771
\(915\) 0 0
\(916\) −16.0000 −0.528655
\(917\) − 16.0000i − 0.528367i
\(918\) 0 0
\(919\) −26.0000 −0.857661 −0.428830 0.903385i \(-0.641074\pi\)
−0.428830 + 0.903385i \(0.641074\pi\)
\(920\) 0 0
\(921\) −28.0000 −0.922631
\(922\) − 30.0000i − 0.987997i
\(923\) − 32.0000i − 1.05329i
\(924\) −4.00000 −0.131590
\(925\) 0 0
\(926\) −8.00000 −0.262896
\(927\) 2.00000i 0.0656886i
\(928\) 10.0000i 0.328266i
\(929\) 6.00000 0.196854 0.0984268 0.995144i \(-0.468619\pi\)
0.0984268 + 0.995144i \(0.468619\pi\)
\(930\) 0 0
\(931\) −24.0000 −0.786568
\(932\) 6.00000i 0.196537i
\(933\) − 16.0000i − 0.523816i
\(934\) −6.00000 −0.196326
\(935\) 0 0
\(936\) 2.00000 0.0653720
\(937\) 30.0000i 0.980057i 0.871706 + 0.490029i \(0.163014\pi\)
−0.871706 + 0.490029i \(0.836986\pi\)
\(938\) − 8.00000i − 0.261209i
\(939\) −10.0000 −0.326338
\(940\) 0 0
\(941\) −34.0000 −1.10837 −0.554184 0.832394i \(-0.686970\pi\)
−0.554184 + 0.832394i \(0.686970\pi\)
\(942\) 8.00000i 0.260654i
\(943\) 6.00000i 0.195387i
\(944\) −4.00000 −0.130189
\(945\) 0 0
\(946\) −24.0000 −0.780307
\(947\) 8.00000i 0.259965i 0.991516 + 0.129983i \(0.0414921\pi\)
−0.991516 + 0.129983i \(0.958508\pi\)
\(948\) − 10.0000i − 0.324785i
\(949\) −20.0000 −0.649227
\(950\) 0 0
\(951\) −6.00000 −0.194563
\(952\) 0 0
\(953\) 8.00000i 0.259145i 0.991570 + 0.129573i \(0.0413606\pi\)
−0.991570 + 0.129573i \(0.958639\pi\)
\(954\) 10.0000 0.323762
\(955\) 0 0
\(956\) −16.0000 −0.517477
\(957\) − 20.0000i − 0.646508i
\(958\) − 24.0000i − 0.775405i
\(959\) −24.0000 −0.775000
\(960\) 0 0
\(961\) 33.0000 1.06452
\(962\) − 16.0000i − 0.515861i
\(963\) − 10.0000i − 0.322245i
\(964\) −10.0000 −0.322078
\(965\) 0 0
\(966\) 2.00000 0.0643489
\(967\) 56.0000i 1.80084i 0.435023 + 0.900419i \(0.356740\pi\)
−0.435023 + 0.900419i \(0.643260\pi\)
\(968\) 7.00000i 0.224989i
\(969\) 0 0
\(970\) 0 0
\(971\) 30.0000 0.962746 0.481373 0.876516i \(-0.340138\pi\)
0.481373 + 0.876516i \(0.340138\pi\)
\(972\) 1.00000i 0.0320750i
\(973\) 40.0000i 1.28234i
\(974\) −40.0000 −1.28168
\(975\) 0 0
\(976\) 12.0000 0.384111
\(977\) 20.0000i 0.639857i 0.947442 + 0.319928i \(0.103659\pi\)
−0.947442 + 0.319928i \(0.896341\pi\)
\(978\) 4.00000i 0.127906i
\(979\) 0 0
\(980\) 0 0
\(981\) −8.00000 −0.255420
\(982\) 0 0
\(983\) 36.0000i 1.14822i 0.818778 + 0.574111i \(0.194652\pi\)
−0.818778 + 0.574111i \(0.805348\pi\)
\(984\) 6.00000 0.191273
\(985\) 0 0
\(986\) 0 0
\(987\) − 16.0000i − 0.509286i
\(988\) − 16.0000i − 0.509028i
\(989\) 12.0000 0.381578
\(990\) 0 0
\(991\) 16.0000 0.508257 0.254128 0.967170i \(-0.418211\pi\)
0.254128 + 0.967170i \(0.418211\pi\)
\(992\) 8.00000i 0.254000i
\(993\) 12.0000i 0.380808i
\(994\) −32.0000 −1.01498
\(995\) 0 0
\(996\) 10.0000 0.316862
\(997\) 38.0000i 1.20347i 0.798695 + 0.601736i \(0.205524\pi\)
−0.798695 + 0.601736i \(0.794476\pi\)
\(998\) 4.00000i 0.126618i
\(999\) 8.00000 0.253109
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.q.2899.2 2
5.2 odd 4 690.2.a.c.1.1 1
5.3 odd 4 3450.2.a.z.1.1 1
5.4 even 2 inner 3450.2.d.q.2899.1 2
15.2 even 4 2070.2.a.k.1.1 1
20.7 even 4 5520.2.a.bg.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
690.2.a.c.1.1 1 5.2 odd 4
2070.2.a.k.1.1 1 15.2 even 4
3450.2.a.z.1.1 1 5.3 odd 4
3450.2.d.q.2899.1 2 5.4 even 2 inner
3450.2.d.q.2899.2 2 1.1 even 1 trivial
5520.2.a.bg.1.1 1 20.7 even 4