Properties

Label 3450.2.d.s.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.s.2899.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} -4.00000i q^{7} -1.00000i q^{8} -1.00000 q^{9} +O(q^{10})\) \(q+1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} -4.00000i q^{7} -1.00000i q^{8} -1.00000 q^{9} +2.00000 q^{11} +1.00000i q^{12} +4.00000i q^{13} +4.00000 q^{14} +1.00000 q^{16} +6.00000i q^{17} -1.00000i q^{18} +4.00000 q^{19} -4.00000 q^{21} +2.00000i q^{22} -1.00000i q^{23} -1.00000 q^{24} -4.00000 q^{26} +1.00000i q^{27} +4.00000i q^{28} -8.00000 q^{29} +8.00000 q^{31} +1.00000i q^{32} -2.00000i q^{33} -6.00000 q^{34} +1.00000 q^{36} +10.0000i q^{37} +4.00000i q^{38} +4.00000 q^{39} +6.00000 q^{41} -4.00000i q^{42} +6.00000i q^{43} -2.00000 q^{44} +1.00000 q^{46} +4.00000i q^{47} -1.00000i q^{48} -9.00000 q^{49} +6.00000 q^{51} -4.00000i q^{52} -14.0000i q^{53} -1.00000 q^{54} -4.00000 q^{56} -4.00000i q^{57} -8.00000i q^{58} -4.00000 q^{59} +6.00000 q^{61} +8.00000i q^{62} +4.00000i q^{63} -1.00000 q^{64} +2.00000 q^{66} -14.0000i q^{67} -6.00000i q^{68} -1.00000 q^{69} +10.0000 q^{71} +1.00000i q^{72} +14.0000i q^{73} -10.0000 q^{74} -4.00000 q^{76} -8.00000i q^{77} +4.00000i q^{78} +8.00000 q^{79} +1.00000 q^{81} +6.00000i q^{82} -4.00000i q^{83} +4.00000 q^{84} -6.00000 q^{86} +8.00000i q^{87} -2.00000i q^{88} +16.0000 q^{91} +1.00000i q^{92} -8.00000i q^{93} -4.00000 q^{94} +1.00000 q^{96} +8.00000i q^{97} -9.00000i q^{98} -2.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{4} + 2q^{6} - 2q^{9} + O(q^{10}) \) \( 2q - 2q^{4} + 2q^{6} - 2q^{9} + 4q^{11} + 8q^{14} + 2q^{16} + 8q^{19} - 8q^{21} - 2q^{24} - 8q^{26} - 16q^{29} + 16q^{31} - 12q^{34} + 2q^{36} + 8q^{39} + 12q^{41} - 4q^{44} + 2q^{46} - 18q^{49} + 12q^{51} - 2q^{54} - 8q^{56} - 8q^{59} + 12q^{61} - 2q^{64} + 4q^{66} - 2q^{69} + 20q^{71} - 20q^{74} - 8q^{76} + 16q^{79} + 2q^{81} + 8q^{84} - 12q^{86} + 32q^{91} - 8q^{94} + 2q^{96} - 4q^{99} + O(q^{100}) \)

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 \(-0.402509\pi\)
0.301511 + 0.953463i \(0.402509\pi\)
\(12\) 1.00000i 0.288675i
\(13\) 4.00000i 1.10940i 0.832050 + 0.554700i \(0.187167\pi\)
−0.832050 + 0.554700i \(0.812833\pi\)
\(14\) 4.00000 1.06904
\(15\) 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
\(19\) 4.00000 0.917663 0.458831 0.888523i \(-0.348268\pi\)
0.458831 + 0.888523i \(0.348268\pi\)
\(20\) 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
\(26\) −4.00000 −0.784465
\(27\) 1.00000i 0.192450i
\(28\) 4.00000i 0.755929i
\(29\) −8.00000 −1.48556 −0.742781 0.669534i \(-0.766494\pi\)
−0.742781 + 0.669534i \(0.766494\pi\)
\(30\) 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
\(34\) −6.00000 −1.02899
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) 10.0000i 1.64399i 0.569495 + 0.821995i \(0.307139\pi\)
−0.569495 + 0.821995i \(0.692861\pi\)
\(38\) 4.00000i 0.648886i
\(39\) 4.00000 0.640513
\(40\) 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
\(43\) 6.00000i 0.914991i 0.889212 + 0.457496i \(0.151253\pi\)
−0.889212 + 0.457496i \(0.848747\pi\)
\(44\) −2.00000 −0.301511
\(45\) 0 0
\(46\) 1.00000 0.147442
\(47\) 4.00000i 0.583460i 0.956501 + 0.291730i \(0.0942309\pi\)
−0.956501 + 0.291730i \(0.905769\pi\)
\(48\) − 1.00000i − 0.144338i
\(49\) −9.00000 −1.28571
\(50\) 0 0
\(51\) 6.00000 0.840168
\(52\) − 4.00000i − 0.554700i
\(53\) − 14.0000i − 1.92305i −0.274721 0.961524i \(-0.588586\pi\)
0.274721 0.961524i \(-0.411414\pi\)
\(54\) −1.00000 −0.136083
\(55\) 0 0
\(56\) −4.00000 −0.534522
\(57\) − 4.00000i − 0.529813i
\(58\) − 8.00000i − 1.05045i
\(59\) −4.00000 −0.520756 −0.260378 0.965507i \(-0.583847\pi\)
−0.260378 + 0.965507i \(0.583847\pi\)
\(60\) 0 0
\(61\) 6.00000 0.768221 0.384111 0.923287i \(-0.374508\pi\)
0.384111 + 0.923287i \(0.374508\pi\)
\(62\) 8.00000i 1.01600i
\(63\) 4.00000i 0.503953i
\(64\) −1.00000 −0.125000
\(65\) 0 0
\(66\) 2.00000 0.246183
\(67\) − 14.0000i − 1.71037i −0.518321 0.855186i \(-0.673443\pi\)
0.518321 0.855186i \(-0.326557\pi\)
\(68\) − 6.00000i − 0.727607i
\(69\) −1.00000 −0.120386
\(70\) 0 0
\(71\) 10.0000 1.18678 0.593391 0.804914i \(-0.297789\pi\)
0.593391 + 0.804914i \(0.297789\pi\)
\(72\) 1.00000i 0.117851i
\(73\) 14.0000i 1.63858i 0.573382 + 0.819288i \(0.305631\pi\)
−0.573382 + 0.819288i \(0.694369\pi\)
\(74\) −10.0000 −1.16248
\(75\) 0 0
\(76\) −4.00000 −0.458831
\(77\) − 8.00000i − 0.911685i
\(78\) 4.00000i 0.452911i
\(79\) 8.00000 0.900070 0.450035 0.893011i \(-0.351411\pi\)
0.450035 + 0.893011i \(0.351411\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 6.00000i 0.662589i
\(83\) − 4.00000i − 0.439057i −0.975606 0.219529i \(-0.929548\pi\)
0.975606 0.219529i \(-0.0704519\pi\)
\(84\) 4.00000 0.436436
\(85\) 0 0
\(86\) −6.00000 −0.646997
\(87\) 8.00000i 0.857690i
\(88\) − 2.00000i − 0.213201i
\(89\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(90\) 0 0
\(91\) 16.0000 1.67726
\(92\) 1.00000i 0.104257i
\(93\) − 8.00000i − 0.829561i
\(94\) −4.00000 −0.412568
\(95\) 0 0
\(96\) 1.00000 0.102062
\(97\) 8.00000i 0.812277i 0.913812 + 0.406138i \(0.133125\pi\)
−0.913812 + 0.406138i \(0.866875\pi\)
\(98\) − 9.00000i − 0.909137i
\(99\) −2.00000 −0.201008
\(100\) 0 0
\(101\) −8.00000 −0.796030 −0.398015 0.917379i \(-0.630301\pi\)
−0.398015 + 0.917379i \(0.630301\pi\)
\(102\) 6.00000i 0.594089i
\(103\) 4.00000i 0.394132i 0.980390 + 0.197066i \(0.0631413\pi\)
−0.980390 + 0.197066i \(0.936859\pi\)
\(104\) 4.00000 0.392232
\(105\) 0 0
\(106\) 14.0000 1.35980
\(107\) − 8.00000i − 0.773389i −0.922208 0.386695i \(-0.873617\pi\)
0.922208 0.386695i \(-0.126383\pi\)
\(108\) − 1.00000i − 0.0962250i
\(109\) 14.0000 1.34096 0.670478 0.741929i \(-0.266089\pi\)
0.670478 + 0.741929i \(0.266089\pi\)
\(110\) 0 0
\(111\) 10.0000 0.949158
\(112\) − 4.00000i − 0.377964i
\(113\) − 10.0000i − 0.940721i −0.882474 0.470360i \(-0.844124\pi\)
0.882474 0.470360i \(-0.155876\pi\)
\(114\) 4.00000 0.374634
\(115\) 0 0
\(116\) 8.00000 0.742781
\(117\) − 4.00000i − 0.369800i
\(118\) − 4.00000i − 0.368230i
\(119\) 24.0000 2.20008
\(120\) 0 0
\(121\) −7.00000 −0.636364
\(122\) 6.00000i 0.543214i
\(123\) − 6.00000i − 0.541002i
\(124\) −8.00000 −0.718421
\(125\) 0 0
\(126\) −4.00000 −0.356348
\(127\) − 2.00000i − 0.177471i −0.996055 0.0887357i \(-0.971717\pi\)
0.996055 0.0887357i \(-0.0282826\pi\)
\(128\) − 1.00000i − 0.0883883i
\(129\) 6.00000 0.528271
\(130\) 0 0
\(131\) 16.0000 1.39793 0.698963 0.715158i \(-0.253645\pi\)
0.698963 + 0.715158i \(0.253645\pi\)
\(132\) 2.00000i 0.174078i
\(133\) − 16.0000i − 1.38738i
\(134\) 14.0000 1.20942
\(135\) 0 0
\(136\) 6.00000 0.514496
\(137\) − 6.00000i − 0.512615i −0.966595 0.256307i \(-0.917494\pi\)
0.966595 0.256307i \(-0.0825059\pi\)
\(138\) − 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
\(141\) 4.00000 0.336861
\(142\) 10.0000i 0.839181i
\(143\) 8.00000i 0.668994i
\(144\) −1.00000 −0.0833333
\(145\) 0 0
\(146\) −14.0000 −1.15865
\(147\) 9.00000i 0.742307i
\(148\) − 10.0000i − 0.821995i
\(149\) 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 \(0.197401\pi\)
\(152\) − 4.00000i − 0.324443i
\(153\) − 6.00000i − 0.485071i
\(154\) 8.00000 0.644658
\(155\) 0 0
\(156\) −4.00000 −0.320256
\(157\) 2.00000i 0.159617i 0.996810 + 0.0798087i \(0.0254309\pi\)
−0.996810 + 0.0798087i \(0.974569\pi\)
\(158\) 8.00000i 0.636446i
\(159\) −14.0000 −1.11027
\(160\) 0 0
\(161\) −4.00000 −0.315244
\(162\) 1.00000i 0.0785674i
\(163\) − 8.00000i − 0.626608i −0.949653 0.313304i \(-0.898564\pi\)
0.949653 0.313304i \(-0.101436\pi\)
\(164\) −6.00000 −0.468521
\(165\) 0 0
\(166\) 4.00000 0.310460
\(167\) 8.00000i 0.619059i 0.950890 + 0.309529i \(0.100171\pi\)
−0.950890 + 0.309529i \(0.899829\pi\)
\(168\) 4.00000i 0.308607i
\(169\) −3.00000 −0.230769
\(170\) 0 0
\(171\) −4.00000 −0.305888
\(172\) − 6.00000i − 0.457496i
\(173\) 10.0000i 0.760286i 0.924928 + 0.380143i \(0.124125\pi\)
−0.924928 + 0.380143i \(0.875875\pi\)
\(174\) −8.00000 −0.606478
\(175\) 0 0
\(176\) 2.00000 0.150756
\(177\) 4.00000i 0.300658i
\(178\) 0 0
\(179\) 4.00000 0.298974 0.149487 0.988764i \(-0.452238\pi\)
0.149487 + 0.988764i \(0.452238\pi\)
\(180\) 0 0
\(181\) −2.00000 −0.148659 −0.0743294 0.997234i \(-0.523682\pi\)
−0.0743294 + 0.997234i \(0.523682\pi\)
\(182\) 16.0000i 1.18600i
\(183\) − 6.00000i − 0.443533i
\(184\) −1.00000 −0.0737210
\(185\) 0 0
\(186\) 8.00000 0.586588
\(187\) 12.0000i 0.877527i
\(188\) − 4.00000i − 0.291730i
\(189\) 4.00000 0.290957
\(190\) 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
\(193\) − 22.0000i − 1.58359i −0.610784 0.791797i \(-0.709146\pi\)
0.610784 0.791797i \(-0.290854\pi\)
\(194\) −8.00000 −0.574367
\(195\) 0 0
\(196\) 9.00000 0.642857
\(197\) 22.0000i 1.56744i 0.621117 + 0.783718i \(0.286679\pi\)
−0.621117 + 0.783718i \(0.713321\pi\)
\(198\) − 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
\(201\) −14.0000 −0.987484
\(202\) − 8.00000i − 0.562878i
\(203\) 32.0000i 2.24596i
\(204\) −6.00000 −0.420084
\(205\) 0 0
\(206\) −4.00000 −0.278693
\(207\) 1.00000i 0.0695048i
\(208\) 4.00000i 0.277350i
\(209\) 8.00000 0.553372
\(210\) 0 0
\(211\) −20.0000 −1.37686 −0.688428 0.725304i \(-0.741699\pi\)
−0.688428 + 0.725304i \(0.741699\pi\)
\(212\) 14.0000i 0.961524i
\(213\) − 10.0000i − 0.685189i
\(214\) 8.00000 0.546869
\(215\) 0 0
\(216\) 1.00000 0.0680414
\(217\) − 32.0000i − 2.17230i
\(218\) 14.0000i 0.948200i
\(219\) 14.0000 0.946032
\(220\) 0 0
\(221\) −24.0000 −1.61441
\(222\) 10.0000i 0.671156i
\(223\) 14.0000i 0.937509i 0.883328 + 0.468755i \(0.155297\pi\)
−0.883328 + 0.468755i \(0.844703\pi\)
\(224\) 4.00000 0.267261
\(225\) 0 0
\(226\) 10.0000 0.665190
\(227\) − 8.00000i − 0.530979i −0.964114 0.265489i \(-0.914466\pi\)
0.964114 0.265489i \(-0.0855335\pi\)
\(228\) 4.00000i 0.264906i
\(229\) 22.0000 1.45380 0.726900 0.686743i \(-0.240960\pi\)
0.726900 + 0.686743i \(0.240960\pi\)
\(230\) 0 0
\(231\) −8.00000 −0.526361
\(232\) 8.00000i 0.525226i
\(233\) − 6.00000i − 0.393073i −0.980497 0.196537i \(-0.937031\pi\)
0.980497 0.196537i \(-0.0629694\pi\)
\(234\) 4.00000 0.261488
\(235\) 0 0
\(236\) 4.00000 0.260378
\(237\) − 8.00000i − 0.519656i
\(238\) 24.0000i 1.55569i
\(239\) −26.0000 −1.68180 −0.840900 0.541190i \(-0.817974\pi\)
−0.840900 + 0.541190i \(0.817974\pi\)
\(240\) 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
\(244\) −6.00000 −0.384111
\(245\) 0 0
\(246\) 6.00000 0.382546
\(247\) 16.0000i 1.01806i
\(248\) − 8.00000i − 0.508001i
\(249\) −4.00000 −0.253490
\(250\) 0 0
\(251\) −2.00000 −0.126239 −0.0631194 0.998006i \(-0.520105\pi\)
−0.0631194 + 0.998006i \(0.520105\pi\)
\(252\) − 4.00000i − 0.251976i
\(253\) − 2.00000i − 0.125739i
\(254\) 2.00000 0.125491
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) − 18.0000i − 1.12281i −0.827541 0.561405i \(-0.810261\pi\)
0.827541 0.561405i \(-0.189739\pi\)
\(258\) 6.00000i 0.373544i
\(259\) 40.0000 2.48548
\(260\) 0 0
\(261\) 8.00000 0.495188
\(262\) 16.0000i 0.988483i
\(263\) 16.0000i 0.986602i 0.869859 + 0.493301i \(0.164210\pi\)
−0.869859 + 0.493301i \(0.835790\pi\)
\(264\) −2.00000 −0.123091
\(265\) 0 0
\(266\) 16.0000 0.981023
\(267\) 0 0
\(268\) 14.0000i 0.855186i
\(269\) −12.0000 −0.731653 −0.365826 0.930683i \(-0.619214\pi\)
−0.365826 + 0.930683i \(0.619214\pi\)
\(270\) 0 0
\(271\) −8.00000 −0.485965 −0.242983 0.970031i \(-0.578126\pi\)
−0.242983 + 0.970031i \(0.578126\pi\)
\(272\) 6.00000i 0.363803i
\(273\) − 16.0000i − 0.968364i
\(274\) 6.00000 0.362473
\(275\) 0 0
\(276\) 1.00000 0.0601929
\(277\) 12.0000i 0.721010i 0.932757 + 0.360505i \(0.117396\pi\)
−0.932757 + 0.360505i \(0.882604\pi\)
\(278\) 20.0000i 1.19952i
\(279\) −8.00000 −0.478947
\(280\) 0 0
\(281\) 8.00000 0.477240 0.238620 0.971113i \(-0.423305\pi\)
0.238620 + 0.971113i \(0.423305\pi\)
\(282\) 4.00000i 0.238197i
\(283\) − 22.0000i − 1.30776i −0.756596 0.653882i \(-0.773139\pi\)
0.756596 0.653882i \(-0.226861\pi\)
\(284\) −10.0000 −0.593391
\(285\) 0 0
\(286\) −8.00000 −0.473050
\(287\) − 24.0000i − 1.41668i
\(288\) − 1.00000i − 0.0589256i
\(289\) −19.0000 −1.11765
\(290\) 0 0
\(291\) 8.00000 0.468968
\(292\) − 14.0000i − 0.819288i
\(293\) 26.0000i 1.51894i 0.650545 + 0.759468i \(0.274541\pi\)
−0.650545 + 0.759468i \(0.725459\pi\)
\(294\) −9.00000 −0.524891
\(295\) 0 0
\(296\) 10.0000 0.581238
\(297\) 2.00000i 0.116052i
\(298\) 10.0000i 0.579284i
\(299\) 4.00000 0.231326
\(300\) 0 0
\(301\) 24.0000 1.38334
\(302\) 20.0000i 1.15087i
\(303\) 8.00000i 0.459588i
\(304\) 4.00000 0.229416
\(305\) 0 0
\(306\) 6.00000 0.342997
\(307\) − 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
\(311\) −14.0000 −0.793867 −0.396934 0.917847i \(-0.629926\pi\)
−0.396934 + 0.917847i \(0.629926\pi\)
\(312\) − 4.00000i − 0.226455i
\(313\) − 4.00000i − 0.226093i −0.993590 0.113047i \(-0.963939\pi\)
0.993590 0.113047i \(-0.0360610\pi\)
\(314\) −2.00000 −0.112867
\(315\) 0 0
\(316\) −8.00000 −0.450035
\(317\) 18.0000i 1.01098i 0.862832 + 0.505490i \(0.168688\pi\)
−0.862832 + 0.505490i \(0.831312\pi\)
\(318\) − 14.0000i − 0.785081i
\(319\) −16.0000 −0.895828
\(320\) 0 0
\(321\) −8.00000 −0.446516
\(322\) − 4.00000i − 0.222911i
\(323\) 24.0000i 1.33540i
\(324\) −1.00000 −0.0555556
\(325\) 0 0
\(326\) 8.00000 0.443079
\(327\) − 14.0000i − 0.774202i
\(328\) − 6.00000i − 0.331295i
\(329\) 16.0000 0.882109
\(330\) 0 0
\(331\) 12.0000 0.659580 0.329790 0.944054i \(-0.393022\pi\)
0.329790 + 0.944054i \(0.393022\pi\)
\(332\) 4.00000i 0.219529i
\(333\) − 10.0000i − 0.547997i
\(334\) −8.00000 −0.437741
\(335\) 0 0
\(336\) −4.00000 −0.218218
\(337\) 20.0000i 1.08947i 0.838608 + 0.544735i \(0.183370\pi\)
−0.838608 + 0.544735i \(0.816630\pi\)
\(338\) − 3.00000i − 0.163178i
\(339\) −10.0000 −0.543125
\(340\) 0 0
\(341\) 16.0000 0.866449
\(342\) − 4.00000i − 0.216295i
\(343\) 8.00000i 0.431959i
\(344\) 6.00000 0.323498
\(345\) 0 0
\(346\) −10.0000 −0.537603
\(347\) − 12.0000i − 0.644194i −0.946707 0.322097i \(-0.895612\pi\)
0.946707 0.322097i \(-0.104388\pi\)
\(348\) − 8.00000i − 0.428845i
\(349\) 30.0000 1.60586 0.802932 0.596071i \(-0.203272\pi\)
0.802932 + 0.596071i \(0.203272\pi\)
\(350\) 0 0
\(351\) −4.00000 −0.213504
\(352\) 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
\(357\) − 24.0000i − 1.27021i
\(358\) 4.00000i 0.211407i
\(359\) 16.0000 0.844448 0.422224 0.906492i \(-0.361250\pi\)
0.422224 + 0.906492i \(0.361250\pi\)
\(360\) 0 0
\(361\) −3.00000 −0.157895
\(362\) − 2.00000i − 0.105118i
\(363\) 7.00000i 0.367405i
\(364\) −16.0000 −0.838628
\(365\) 0 0
\(366\) 6.00000 0.313625
\(367\) 12.0000i 0.626395i 0.949688 + 0.313197i \(0.101400\pi\)
−0.949688 + 0.313197i \(0.898600\pi\)
\(368\) − 1.00000i − 0.0521286i
\(369\) −6.00000 −0.312348
\(370\) 0 0
\(371\) −56.0000 −2.90738
\(372\) 8.00000i 0.414781i
\(373\) − 18.0000i − 0.932005i −0.884783 0.466002i \(-0.845694\pi\)
0.884783 0.466002i \(-0.154306\pi\)
\(374\) −12.0000 −0.620505
\(375\) 0 0
\(376\) 4.00000 0.206284
\(377\) − 32.0000i − 1.64808i
\(378\) 4.00000i 0.205738i
\(379\) 16.0000 0.821865 0.410932 0.911666i \(-0.365203\pi\)
0.410932 + 0.911666i \(0.365203\pi\)
\(380\) 0 0
\(381\) −2.00000 −0.102463
\(382\) − 16.0000i − 0.818631i
\(383\) − 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
\(386\) 22.0000 1.11977
\(387\) − 6.00000i − 0.304997i
\(388\) − 8.00000i − 0.406138i
\(389\) 14.0000 0.709828 0.354914 0.934899i \(-0.384510\pi\)
0.354914 + 0.934899i \(0.384510\pi\)
\(390\) 0 0
\(391\) 6.00000 0.303433
\(392\) 9.00000i 0.454569i
\(393\) − 16.0000i − 0.807093i
\(394\) −22.0000 −1.10834
\(395\) 0 0
\(396\) 2.00000 0.100504
\(397\) 28.0000i 1.40528i 0.711546 + 0.702640i \(0.247995\pi\)
−0.711546 + 0.702640i \(0.752005\pi\)
\(398\) − 16.0000i − 0.802008i
\(399\) −16.0000 −0.801002
\(400\) 0 0
\(401\) 20.0000 0.998752 0.499376 0.866385i \(-0.333563\pi\)
0.499376 + 0.866385i \(0.333563\pi\)
\(402\) − 14.0000i − 0.698257i
\(403\) 32.0000i 1.59403i
\(404\) 8.00000 0.398015
\(405\) 0 0
\(406\) −32.0000 −1.58813
\(407\) 20.0000i 0.991363i
\(408\) − 6.00000i − 0.297044i
\(409\) 10.0000 0.494468 0.247234 0.968956i \(-0.420478\pi\)
0.247234 + 0.968956i \(0.420478\pi\)
\(410\) 0 0
\(411\) −6.00000 −0.295958
\(412\) − 4.00000i − 0.197066i
\(413\) 16.0000i 0.787309i
\(414\) −1.00000 −0.0491473
\(415\) 0 0
\(416\) −4.00000 −0.196116
\(417\) − 20.0000i − 0.979404i
\(418\) 8.00000i 0.391293i
\(419\) 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
\(423\) − 4.00000i − 0.194487i
\(424\) −14.0000 −0.679900
\(425\) 0 0
\(426\) 10.0000 0.484502
\(427\) − 24.0000i − 1.16144i
\(428\) 8.00000i 0.386695i
\(429\) 8.00000 0.386244
\(430\) 0 0
\(431\) 8.00000 0.385346 0.192673 0.981263i \(-0.438284\pi\)
0.192673 + 0.981263i \(0.438284\pi\)
\(432\) 1.00000i 0.0481125i
\(433\) 32.0000i 1.53782i 0.639356 + 0.768911i \(0.279201\pi\)
−0.639356 + 0.768911i \(0.720799\pi\)
\(434\) 32.0000 1.53605
\(435\) 0 0
\(436\) −14.0000 −0.670478
\(437\) − 4.00000i − 0.191346i
\(438\) 14.0000i 0.668946i
\(439\) 20.0000 0.954548 0.477274 0.878755i \(-0.341625\pi\)
0.477274 + 0.878755i \(0.341625\pi\)
\(440\) 0 0
\(441\) 9.00000 0.428571
\(442\) − 24.0000i − 1.14156i
\(443\) − 28.0000i − 1.33032i −0.746701 0.665160i \(-0.768363\pi\)
0.746701 0.665160i \(-0.231637\pi\)
\(444\) −10.0000 −0.474579
\(445\) 0 0
\(446\) −14.0000 −0.662919
\(447\) − 10.0000i − 0.472984i
\(448\) 4.00000i 0.188982i
\(449\) −30.0000 −1.41579 −0.707894 0.706319i \(-0.750354\pi\)
−0.707894 + 0.706319i \(0.750354\pi\)
\(450\) 0 0
\(451\) 12.0000 0.565058
\(452\) 10.0000i 0.470360i
\(453\) − 20.0000i − 0.939682i
\(454\) 8.00000 0.375459
\(455\) 0 0
\(456\) −4.00000 −0.187317
\(457\) − 16.0000i − 0.748448i −0.927338 0.374224i \(-0.877909\pi\)
0.927338 0.374224i \(-0.122091\pi\)
\(458\) 22.0000i 1.02799i
\(459\) −6.00000 −0.280056
\(460\) 0 0
\(461\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(462\) − 8.00000i − 0.372194i
\(463\) − 10.0000i − 0.464739i −0.972628 0.232370i \(-0.925352\pi\)
0.972628 0.232370i \(-0.0746479\pi\)
\(464\) −8.00000 −0.371391
\(465\) 0 0
\(466\) 6.00000 0.277945
\(467\) 36.0000i 1.66588i 0.553362 + 0.832941i \(0.313345\pi\)
−0.553362 + 0.832941i \(0.686655\pi\)
\(468\) 4.00000i 0.184900i
\(469\) −56.0000 −2.58584
\(470\) 0 0
\(471\) 2.00000 0.0921551
\(472\) 4.00000i 0.184115i
\(473\) 12.0000i 0.551761i
\(474\) 8.00000 0.367452
\(475\) 0 0
\(476\) −24.0000 −1.10004
\(477\) 14.0000i 0.641016i
\(478\) − 26.0000i − 1.18921i
\(479\) −24.0000 −1.09659 −0.548294 0.836286i \(-0.684723\pi\)
−0.548294 + 0.836286i \(0.684723\pi\)
\(480\) 0 0
\(481\) −40.0000 −1.82384
\(482\) 10.0000i 0.455488i
\(483\) 4.00000i 0.182006i
\(484\) 7.00000 0.318182
\(485\) 0 0
\(486\) 1.00000 0.0453609
\(487\) 10.0000i 0.453143i 0.973995 + 0.226572i \(0.0727517\pi\)
−0.973995 + 0.226572i \(0.927248\pi\)
\(488\) − 6.00000i − 0.271607i
\(489\) −8.00000 −0.361773
\(490\) 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
\(493\) − 48.0000i − 2.16181i
\(494\) −16.0000 −0.719874
\(495\) 0 0
\(496\) 8.00000 0.359211
\(497\) − 40.0000i − 1.79425i
\(498\) − 4.00000i − 0.179244i
\(499\) −20.0000 −0.895323 −0.447661 0.894203i \(-0.647743\pi\)
−0.447661 + 0.894203i \(0.647743\pi\)
\(500\) 0 0
\(501\) 8.00000 0.357414
\(502\) − 2.00000i − 0.0892644i
\(503\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(504\) 4.00000 0.178174
\(505\) 0 0
\(506\) 2.00000 0.0889108
\(507\) 3.00000i 0.133235i
\(508\) 2.00000i 0.0887357i
\(509\) −24.0000 −1.06378 −0.531891 0.846813i \(-0.678518\pi\)
−0.531891 + 0.846813i \(0.678518\pi\)
\(510\) 0 0
\(511\) 56.0000 2.47729
\(512\) 1.00000i 0.0441942i
\(513\) 4.00000i 0.176604i
\(514\) 18.0000 0.793946
\(515\) 0 0
\(516\) −6.00000 −0.264135
\(517\) 8.00000i 0.351840i
\(518\) 40.0000i 1.75750i
\(519\) 10.0000 0.438951
\(520\) 0 0
\(521\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(522\) 8.00000i 0.350150i
\(523\) − 14.0000i − 0.612177i −0.952003 0.306089i \(-0.900980\pi\)
0.952003 0.306089i \(-0.0990204\pi\)
\(524\) −16.0000 −0.698963
\(525\) 0 0
\(526\) −16.0000 −0.697633
\(527\) 48.0000i 2.09091i
\(528\) − 2.00000i − 0.0870388i
\(529\) −1.00000 −0.0434783
\(530\) 0 0
\(531\) 4.00000 0.173585
\(532\) 16.0000i 0.693688i
\(533\) 24.0000i 1.03956i
\(534\) 0 0
\(535\) 0 0
\(536\) −14.0000 −0.604708
\(537\) − 4.00000i − 0.172613i
\(538\) − 12.0000i − 0.517357i
\(539\) −18.0000 −0.775315
\(540\) 0 0
\(541\) 10.0000 0.429934 0.214967 0.976621i \(-0.431036\pi\)
0.214967 + 0.976621i \(0.431036\pi\)
\(542\) − 8.00000i − 0.343629i
\(543\) 2.00000i 0.0858282i
\(544\) −6.00000 −0.257248
\(545\) 0 0
\(546\) 16.0000 0.684737
\(547\) − 28.0000i − 1.19719i −0.801050 0.598597i \(-0.795725\pi\)
0.801050 0.598597i \(-0.204275\pi\)
\(548\) 6.00000i 0.256307i
\(549\) −6.00000 −0.256074
\(550\) 0 0
\(551\) −32.0000 −1.36325
\(552\) 1.00000i 0.0425628i
\(553\) − 32.0000i − 1.36078i
\(554\) −12.0000 −0.509831
\(555\) 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
\(561\) 12.0000 0.506640
\(562\) 8.00000i 0.337460i
\(563\) − 12.0000i − 0.505740i −0.967500 0.252870i \(-0.918626\pi\)
0.967500 0.252870i \(-0.0813744\pi\)
\(564\) −4.00000 −0.168430
\(565\) 0 0
\(566\) 22.0000 0.924729
\(567\) − 4.00000i − 0.167984i
\(568\) − 10.0000i − 0.419591i
\(569\) 44.0000 1.84458 0.922288 0.386503i \(-0.126317\pi\)
0.922288 + 0.386503i \(0.126317\pi\)
\(570\) 0 0
\(571\) −24.0000 −1.00437 −0.502184 0.864761i \(-0.667470\pi\)
−0.502184 + 0.864761i \(0.667470\pi\)
\(572\) − 8.00000i − 0.334497i
\(573\) 16.0000i 0.668410i
\(574\) 24.0000 1.00174
\(575\) 0 0
\(576\) 1.00000 0.0416667
\(577\) 34.0000i 1.41544i 0.706494 + 0.707719i \(0.250276\pi\)
−0.706494 + 0.707719i \(0.749724\pi\)
\(578\) − 19.0000i − 0.790296i
\(579\) −22.0000 −0.914289
\(580\) 0 0
\(581\) −16.0000 −0.663792
\(582\) 8.00000i 0.331611i
\(583\) − 28.0000i − 1.15964i
\(584\) 14.0000 0.579324
\(585\) 0 0
\(586\) −26.0000 −1.07405
\(587\) − 12.0000i − 0.495293i −0.968850 0.247647i \(-0.920343\pi\)
0.968850 0.247647i \(-0.0796572\pi\)
\(588\) − 9.00000i − 0.371154i
\(589\) 32.0000 1.31854
\(590\) 0 0
\(591\) 22.0000 0.904959
\(592\) 10.0000i 0.410997i
\(593\) 18.0000i 0.739171i 0.929197 + 0.369586i \(0.120500\pi\)
−0.929197 + 0.369586i \(0.879500\pi\)
\(594\) −2.00000 −0.0820610
\(595\) 0 0
\(596\) −10.0000 −0.409616
\(597\) 16.0000i 0.654836i
\(598\) 4.00000i 0.163572i
\(599\) −14.0000 −0.572024 −0.286012 0.958226i \(-0.592330\pi\)
−0.286012 + 0.958226i \(0.592330\pi\)
\(600\) 0 0
\(601\) −26.0000 −1.06056 −0.530281 0.847822i \(-0.677914\pi\)
−0.530281 + 0.847822i \(0.677914\pi\)
\(602\) 24.0000i 0.978167i
\(603\) 14.0000i 0.570124i
\(604\) −20.0000 −0.813788
\(605\) 0 0
\(606\) −8.00000 −0.324978
\(607\) − 30.0000i − 1.21766i −0.793300 0.608831i \(-0.791639\pi\)
0.793300 0.608831i \(-0.208361\pi\)
\(608\) 4.00000i 0.162221i
\(609\) 32.0000 1.29671
\(610\) 0 0
\(611\) −16.0000 −0.647291
\(612\) 6.00000i 0.242536i
\(613\) − 14.0000i − 0.565455i −0.959200 0.282727i \(-0.908761\pi\)
0.959200 0.282727i \(-0.0912392\pi\)
\(614\) 4.00000 0.161427
\(615\) 0 0
\(616\) −8.00000 −0.322329
\(617\) − 18.0000i − 0.724653i −0.932051 0.362326i \(-0.881983\pi\)
0.932051 0.362326i \(-0.118017\pi\)
\(618\) 4.00000i 0.160904i
\(619\) 20.0000 0.803868 0.401934 0.915669i \(-0.368338\pi\)
0.401934 + 0.915669i \(0.368338\pi\)
\(620\) 0 0
\(621\) 1.00000 0.0401286
\(622\) − 14.0000i − 0.561349i
\(623\) 0 0
\(624\) 4.00000 0.160128
\(625\) 0 0
\(626\) 4.00000 0.159872
\(627\) − 8.00000i − 0.319489i
\(628\) − 2.00000i − 0.0798087i
\(629\) −60.0000 −2.39236
\(630\) 0 0
\(631\) −8.00000 −0.318475 −0.159237 0.987240i \(-0.550904\pi\)
−0.159237 + 0.987240i \(0.550904\pi\)
\(632\) − 8.00000i − 0.318223i
\(633\) 20.0000i 0.794929i
\(634\) −18.0000 −0.714871
\(635\) 0 0
\(636\) 14.0000 0.555136
\(637\) − 36.0000i − 1.42637i
\(638\) − 16.0000i − 0.633446i
\(639\) −10.0000 −0.395594
\(640\) 0 0
\(641\) −24.0000 −0.947943 −0.473972 0.880540i \(-0.657180\pi\)
−0.473972 + 0.880540i \(0.657180\pi\)
\(642\) − 8.00000i − 0.315735i
\(643\) 34.0000i 1.34083i 0.741987 + 0.670415i \(0.233884\pi\)
−0.741987 + 0.670415i \(0.766116\pi\)
\(644\) 4.00000 0.157622
\(645\) 0 0
\(646\) −24.0000 −0.944267
\(647\) 28.0000i 1.10079i 0.834903 + 0.550397i \(0.185524\pi\)
−0.834903 + 0.550397i \(0.814476\pi\)
\(648\) − 1.00000i − 0.0392837i
\(649\) −8.00000 −0.314027
\(650\) 0 0
\(651\) −32.0000 −1.25418
\(652\) 8.00000i 0.313304i
\(653\) − 42.0000i − 1.64359i −0.569785 0.821794i \(-0.692974\pi\)
0.569785 0.821794i \(-0.307026\pi\)
\(654\) 14.0000 0.547443
\(655\) 0 0
\(656\) 6.00000 0.234261
\(657\) − 14.0000i − 0.546192i
\(658\) 16.0000i 0.623745i
\(659\) 6.00000 0.233727 0.116863 0.993148i \(-0.462716\pi\)
0.116863 + 0.993148i \(0.462716\pi\)
\(660\) 0 0
\(661\) −22.0000 −0.855701 −0.427850 0.903850i \(-0.640729\pi\)
−0.427850 + 0.903850i \(0.640729\pi\)
\(662\) 12.0000i 0.466393i
\(663\) 24.0000i 0.932083i
\(664\) −4.00000 −0.155230
\(665\) 0 0
\(666\) 10.0000 0.387492
\(667\) 8.00000i 0.309761i
\(668\) − 8.00000i − 0.309529i
\(669\) 14.0000 0.541271
\(670\) 0 0
\(671\) 12.0000 0.463255
\(672\) − 4.00000i − 0.154303i
\(673\) − 34.0000i − 1.31060i −0.755367 0.655302i \(-0.772541\pi\)
0.755367 0.655302i \(-0.227459\pi\)
\(674\) −20.0000 −0.770371
\(675\) 0 0
\(676\) 3.00000 0.115385
\(677\) 26.0000i 0.999261i 0.866239 + 0.499631i \(0.166531\pi\)
−0.866239 + 0.499631i \(0.833469\pi\)
\(678\) − 10.0000i − 0.384048i
\(679\) 32.0000 1.22805
\(680\) 0 0
\(681\) −8.00000 −0.306561
\(682\) 16.0000i 0.612672i
\(683\) 12.0000i 0.459167i 0.973289 + 0.229584i \(0.0737364\pi\)
−0.973289 + 0.229584i \(0.926264\pi\)
\(684\) 4.00000 0.152944
\(685\) 0 0
\(686\) −8.00000 −0.305441
\(687\) − 22.0000i − 0.839352i
\(688\) 6.00000i 0.228748i
\(689\) 56.0000 2.13343
\(690\) 0 0
\(691\) −4.00000 −0.152167 −0.0760836 0.997101i \(-0.524242\pi\)
−0.0760836 + 0.997101i \(0.524242\pi\)
\(692\) − 10.0000i − 0.380143i
\(693\) 8.00000i 0.303895i
\(694\) 12.0000 0.455514
\(695\) 0 0
\(696\) 8.00000 0.303239
\(697\) 36.0000i 1.36360i
\(698\) 30.0000i 1.13552i
\(699\) −6.00000 −0.226941
\(700\) 0 0
\(701\) −6.00000 −0.226617 −0.113308 0.993560i \(-0.536145\pi\)
−0.113308 + 0.993560i \(0.536145\pi\)
\(702\) − 4.00000i − 0.150970i
\(703\) 40.0000i 1.50863i
\(704\) −2.00000 −0.0753778
\(705\) 0 0
\(706\) −14.0000 −0.526897
\(707\) 32.0000i 1.20348i
\(708\) − 4.00000i − 0.150329i
\(709\) −34.0000 −1.27690 −0.638448 0.769665i \(-0.720423\pi\)
−0.638448 + 0.769665i \(0.720423\pi\)
\(710\) 0 0
\(711\) −8.00000 −0.300023
\(712\) 0 0
\(713\) − 8.00000i − 0.299602i
\(714\) 24.0000 0.898177
\(715\) 0 0
\(716\) −4.00000 −0.149487
\(717\) 26.0000i 0.970988i
\(718\) 16.0000i 0.597115i
\(719\) −46.0000 −1.71551 −0.857755 0.514058i \(-0.828142\pi\)
−0.857755 + 0.514058i \(0.828142\pi\)
\(720\) 0 0
\(721\) 16.0000 0.595871
\(722\) − 3.00000i − 0.111648i
\(723\) − 10.0000i − 0.371904i
\(724\) 2.00000 0.0743294
\(725\) 0 0
\(726\) −7.00000 −0.259794
\(727\) − 32.0000i − 1.18681i −0.804902 0.593407i \(-0.797782\pi\)
0.804902 0.593407i \(-0.202218\pi\)
\(728\) − 16.0000i − 0.592999i
\(729\) −1.00000 −0.0370370
\(730\) 0 0
\(731\) −36.0000 −1.33151
\(732\) 6.00000i 0.221766i
\(733\) 14.0000i 0.517102i 0.965998 + 0.258551i \(0.0832450\pi\)
−0.965998 + 0.258551i \(0.916755\pi\)
\(734\) −12.0000 −0.442928
\(735\) 0 0
\(736\) 1.00000 0.0368605
\(737\) − 28.0000i − 1.03139i
\(738\) − 6.00000i − 0.220863i
\(739\) 4.00000 0.147142 0.0735712 0.997290i \(-0.476560\pi\)
0.0735712 + 0.997290i \(0.476560\pi\)
\(740\) 0 0
\(741\) 16.0000 0.587775
\(742\) − 56.0000i − 2.05582i
\(743\) 32.0000i 1.17397i 0.809599 + 0.586983i \(0.199684\pi\)
−0.809599 + 0.586983i \(0.800316\pi\)
\(744\) −8.00000 −0.293294
\(745\) 0 0
\(746\) 18.0000 0.659027
\(747\) 4.00000i 0.146352i
\(748\) − 12.0000i − 0.438763i
\(749\) −32.0000 −1.16925
\(750\) 0 0
\(751\) −16.0000 −0.583848 −0.291924 0.956441i \(-0.594295\pi\)
−0.291924 + 0.956441i \(0.594295\pi\)
\(752\) 4.00000i 0.145865i
\(753\) 2.00000i 0.0728841i
\(754\) 32.0000 1.16537
\(755\) 0 0
\(756\) −4.00000 −0.145479
\(757\) − 34.0000i − 1.23575i −0.786276 0.617876i \(-0.787994\pi\)
0.786276 0.617876i \(-0.212006\pi\)
\(758\) 16.0000i 0.581146i
\(759\) −2.00000 −0.0725954
\(760\) 0 0
\(761\) −6.00000 −0.217500 −0.108750 0.994069i \(-0.534685\pi\)
−0.108750 + 0.994069i \(0.534685\pi\)
\(762\) − 2.00000i − 0.0724524i
\(763\) − 56.0000i − 2.02734i
\(764\) 16.0000 0.578860
\(765\) 0 0
\(766\) 24.0000 0.867155
\(767\) − 16.0000i − 0.577727i
\(768\) − 1.00000i − 0.0360844i
\(769\) −14.0000 −0.504853 −0.252426 0.967616i \(-0.581229\pi\)
−0.252426 + 0.967616i \(0.581229\pi\)
\(770\) 0 0
\(771\) −18.0000 −0.648254
\(772\) 22.0000i 0.791797i
\(773\) 42.0000i 1.51064i 0.655359 + 0.755318i \(0.272517\pi\)
−0.655359 + 0.755318i \(0.727483\pi\)
\(774\) 6.00000 0.215666
\(775\) 0 0
\(776\) 8.00000 0.287183
\(777\) − 40.0000i − 1.43499i
\(778\) 14.0000i 0.501924i
\(779\) 24.0000 0.859889
\(780\) 0 0
\(781\) 20.0000 0.715656
\(782\) 6.00000i 0.214560i
\(783\) − 8.00000i − 0.285897i
\(784\) −9.00000 −0.321429
\(785\) 0 0
\(786\) 16.0000 0.570701
\(787\) − 50.0000i − 1.78231i −0.453701 0.891154i \(-0.649897\pi\)
0.453701 0.891154i \(-0.350103\pi\)
\(788\) − 22.0000i − 0.783718i
\(789\) 16.0000 0.569615
\(790\) 0 0
\(791\) −40.0000 −1.42224
\(792\) 2.00000i 0.0710669i
\(793\) 24.0000i 0.852265i
\(794\) −28.0000 −0.993683
\(795\) 0 0
\(796\) 16.0000 0.567105
\(797\) − 54.0000i − 1.91278i −0.292096 0.956389i \(-0.594353\pi\)
0.292096 0.956389i \(-0.405647\pi\)
\(798\) − 16.0000i − 0.566394i
\(799\) −24.0000 −0.849059
\(800\) 0 0
\(801\) 0 0
\(802\) 20.0000i 0.706225i
\(803\) 28.0000i 0.988099i
\(804\) 14.0000 0.493742
\(805\) 0 0
\(806\) −32.0000 −1.12715
\(807\) 12.0000i 0.422420i
\(808\) 8.00000i 0.281439i
\(809\) 6.00000 0.210949 0.105474 0.994422i \(-0.466364\pi\)
0.105474 + 0.994422i \(0.466364\pi\)
\(810\) 0 0
\(811\) −12.0000 −0.421377 −0.210688 0.977553i \(-0.567571\pi\)
−0.210688 + 0.977553i \(0.567571\pi\)
\(812\) − 32.0000i − 1.12298i
\(813\) 8.00000i 0.280572i
\(814\) −20.0000 −0.701000
\(815\) 0 0
\(816\) 6.00000 0.210042
\(817\) 24.0000i 0.839654i
\(818\) 10.0000i 0.349642i
\(819\) −16.0000 −0.559085
\(820\) 0 0
\(821\) 12.0000 0.418803 0.209401 0.977830i \(-0.432848\pi\)
0.209401 + 0.977830i \(0.432848\pi\)
\(822\) − 6.00000i − 0.209274i
\(823\) 42.0000i 1.46403i 0.681290 + 0.732014i \(0.261419\pi\)
−0.681290 + 0.732014i \(0.738581\pi\)
\(824\) 4.00000 0.139347
\(825\) 0 0
\(826\) −16.0000 −0.556711
\(827\) 48.0000i 1.66912i 0.550914 + 0.834562i \(0.314279\pi\)
−0.550914 + 0.834562i \(0.685721\pi\)
\(828\) − 1.00000i − 0.0347524i
\(829\) 38.0000 1.31979 0.659897 0.751356i \(-0.270600\pi\)
0.659897 + 0.751356i \(0.270600\pi\)
\(830\) 0 0
\(831\) 12.0000 0.416275
\(832\) − 4.00000i − 0.138675i
\(833\) − 54.0000i − 1.87099i
\(834\) 20.0000 0.692543
\(835\) 0 0
\(836\) −8.00000 −0.276686
\(837\) 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
\(841\) 35.0000 1.20690
\(842\) 6.00000i 0.206774i
\(843\) − 8.00000i − 0.275535i
\(844\) 20.0000 0.688428
\(845\) 0 0
\(846\) 4.00000 0.137523
\(847\) 28.0000i 0.962091i
\(848\) − 14.0000i − 0.480762i
\(849\) −22.0000 −0.755038
\(850\) 0 0
\(851\) 10.0000 0.342796
\(852\) 10.0000i 0.342594i
\(853\) − 4.00000i − 0.136957i −0.997653 0.0684787i \(-0.978185\pi\)
0.997653 0.0684787i \(-0.0218145\pi\)
\(854\) 24.0000 0.821263
\(855\) 0 0
\(856\) −8.00000 −0.273434
\(857\) − 6.00000i − 0.204956i −0.994735 0.102478i \(-0.967323\pi\)
0.994735 0.102478i \(-0.0326771\pi\)
\(858\) 8.00000i 0.273115i
\(859\) 4.00000 0.136478 0.0682391 0.997669i \(-0.478262\pi\)
0.0682391 + 0.997669i \(0.478262\pi\)
\(860\) 0 0
\(861\) −24.0000 −0.817918
\(862\) 8.00000i 0.272481i
\(863\) − 36.0000i − 1.22545i −0.790295 0.612727i \(-0.790072\pi\)
0.790295 0.612727i \(-0.209928\pi\)
\(864\) −1.00000 −0.0340207
\(865\) 0 0
\(866\) −32.0000 −1.08740
\(867\) 19.0000i 0.645274i
\(868\) 32.0000i 1.08615i
\(869\) 16.0000 0.542763
\(870\) 0 0
\(871\) 56.0000 1.89749
\(872\) − 14.0000i − 0.474100i
\(873\) − 8.00000i − 0.270759i
\(874\) 4.00000 0.135302
\(875\) 0 0
\(876\) −14.0000 −0.473016
\(877\) − 20.0000i − 0.675352i −0.941262 0.337676i \(-0.890359\pi\)
0.941262 0.337676i \(-0.109641\pi\)
\(878\) 20.0000i 0.674967i
\(879\) 26.0000 0.876958
\(880\) 0 0
\(881\) −48.0000 −1.61716 −0.808581 0.588386i \(-0.799764\pi\)
−0.808581 + 0.588386i \(0.799764\pi\)
\(882\) 9.00000i 0.303046i
\(883\) 8.00000i 0.269221i 0.990899 + 0.134611i \(0.0429784\pi\)
−0.990899 + 0.134611i \(0.957022\pi\)
\(884\) 24.0000 0.807207
\(885\) 0 0
\(886\) 28.0000 0.940678
\(887\) − 12.0000i − 0.402921i −0.979497 0.201460i \(-0.935431\pi\)
0.979497 0.201460i \(-0.0645687\pi\)
\(888\) − 10.0000i − 0.335578i
\(889\) −8.00000 −0.268311
\(890\) 0 0
\(891\) 2.00000 0.0670025
\(892\) − 14.0000i − 0.468755i
\(893\) 16.0000i 0.535420i
\(894\) 10.0000 0.334450
\(895\) 0 0
\(896\) −4.00000 −0.133631
\(897\) − 4.00000i − 0.133556i
\(898\) − 30.0000i − 1.00111i
\(899\) −64.0000 −2.13452
\(900\) 0 0
\(901\) 84.0000 2.79845
\(902\) 12.0000i 0.399556i
\(903\) − 24.0000i − 0.798670i
\(904\) −10.0000 −0.332595
\(905\) 0 0
\(906\) 20.0000 0.664455
\(907\) − 46.0000i − 1.52740i −0.645568 0.763702i \(-0.723379\pi\)
0.645568 0.763702i \(-0.276621\pi\)
\(908\) 8.00000i 0.265489i
\(909\) 8.00000 0.265343
\(910\) 0 0
\(911\) −36.0000 −1.19273 −0.596367 0.802712i \(-0.703390\pi\)
−0.596367 + 0.802712i \(0.703390\pi\)
\(912\) − 4.00000i − 0.132453i
\(913\) − 8.00000i − 0.264761i
\(914\) 16.0000 0.529233
\(915\) 0 0
\(916\) −22.0000 −0.726900
\(917\) − 64.0000i − 2.11347i
\(918\) − 6.00000i − 0.198030i
\(919\) 16.0000 0.527791 0.263896 0.964551i \(-0.414993\pi\)
0.263896 + 0.964551i \(0.414993\pi\)
\(920\) 0 0
\(921\) −4.00000 −0.131804
\(922\) 0 0
\(923\) 40.0000i 1.31662i
\(924\) 8.00000 0.263181
\(925\) 0 0
\(926\) 10.0000 0.328620
\(927\) − 4.00000i − 0.131377i
\(928\) − 8.00000i − 0.262613i
\(929\) 18.0000 0.590561 0.295280 0.955411i \(-0.404587\pi\)
0.295280 + 0.955411i \(0.404587\pi\)
\(930\) 0 0
\(931\) −36.0000 −1.17985
\(932\) 6.00000i 0.196537i
\(933\) 14.0000i 0.458339i
\(934\) −36.0000 −1.17796
\(935\) 0 0
\(936\) −4.00000 −0.130744
\(937\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(938\) − 56.0000i − 1.82846i
\(939\) −4.00000 −0.130535
\(940\) 0 0
\(941\) 38.0000 1.23876 0.619382 0.785090i \(-0.287383\pi\)
0.619382 + 0.785090i \(0.287383\pi\)
\(942\) 2.00000i 0.0651635i
\(943\) − 6.00000i − 0.195387i
\(944\) −4.00000 −0.130189
\(945\) 0 0
\(946\) −12.0000 −0.390154
\(947\) 20.0000i 0.649913i 0.945729 + 0.324956i \(0.105350\pi\)
−0.945729 + 0.324956i \(0.894650\pi\)
\(948\) 8.00000i 0.259828i
\(949\) −56.0000 −1.81784
\(950\) 0 0
\(951\) 18.0000 0.583690
\(952\) − 24.0000i − 0.777844i
\(953\) 38.0000i 1.23094i 0.788160 + 0.615470i \(0.211034\pi\)
−0.788160 + 0.615470i \(0.788966\pi\)
\(954\) −14.0000 −0.453267
\(955\) 0 0
\(956\) 26.0000 0.840900
\(957\) 16.0000i 0.517207i
\(958\) − 24.0000i − 0.775405i
\(959\) −24.0000 −0.775000
\(960\) 0 0
\(961\) 33.0000 1.06452
\(962\) − 40.0000i − 1.28965i
\(963\) 8.00000i 0.257796i
\(964\) −10.0000 −0.322078
\(965\) 0 0
\(966\) −4.00000 −0.128698
\(967\) − 22.0000i − 0.707472i −0.935345 0.353736i \(-0.884911\pi\)
0.935345 0.353736i \(-0.115089\pi\)
\(968\) 7.00000i 0.224989i
\(969\) 24.0000 0.770991
\(970\) 0 0
\(971\) 6.00000 0.192549 0.0962746 0.995355i \(-0.469307\pi\)
0.0962746 + 0.995355i \(0.469307\pi\)
\(972\) 1.00000i 0.0320750i
\(973\) − 80.0000i − 2.56468i
\(974\) −10.0000 −0.320421
\(975\) 0 0
\(976\) 6.00000 0.192055
\(977\) − 10.0000i − 0.319928i −0.987123 0.159964i \(-0.948862\pi\)
0.987123 0.159964i \(-0.0511379\pi\)
\(978\) − 8.00000i − 0.255812i
\(979\) 0 0
\(980\) 0 0
\(981\) −14.0000 −0.446986
\(982\) − 24.0000i − 0.765871i
\(983\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(984\) −6.00000 −0.191273
\(985\) 0 0
\(986\) 48.0000 1.52863
\(987\) − 16.0000i − 0.509286i
\(988\) − 16.0000i − 0.509028i
\(989\) 6.00000 0.190789
\(990\) 0 0
\(991\) 4.00000 0.127064 0.0635321 0.997980i \(-0.479763\pi\)
0.0635321 + 0.997980i \(0.479763\pi\)
\(992\) 8.00000i 0.254000i
\(993\) − 12.0000i − 0.380808i
\(994\) 40.0000 1.26872
\(995\) 0 0
\(996\) 4.00000 0.126745
\(997\) − 4.00000i − 0.126681i −0.997992 0.0633406i \(-0.979825\pi\)
0.997992 0.0633406i \(-0.0201755\pi\)
\(998\) − 20.0000i − 0.633089i
\(999\) −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.s.2899.2 2
5.2 odd 4 690.2.a.d.1.1 1
5.3 odd 4 3450.2.a.u.1.1 1
5.4 even 2 inner 3450.2.d.s.2899.1 2
15.2 even 4 2070.2.a.o.1.1 1
20.7 even 4 5520.2.a.bb.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
690.2.a.d.1.1 1 5.2 odd 4
2070.2.a.o.1.1 1 15.2 even 4
3450.2.a.u.1.1 1 5.3 odd 4
3450.2.d.s.2899.1 2 5.4 even 2 inner
3450.2.d.s.2899.2 2 1.1 even 1 trivial
5520.2.a.bb.1.1 1 20.7 even 4