Properties

Label 1650.2.c.d.199.1
Level $1650$
Weight $2$
Character 1650.199
Analytic conductor $13.175$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1650 = 2 \cdot 3 \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1650.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(13.1753163335\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 66)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 199.1
Root \(1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 1650.199
Dual form 1650.2.c.d.199.2

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{6} +2.00000i q^{7} +1.00000i q^{8} -1.00000 q^{9} +O(q^{10})\) \(q-1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{6} +2.00000i q^{7} +1.00000i q^{8} -1.00000 q^{9} -1.00000 q^{11} +1.00000i q^{12} +4.00000i q^{13} +2.00000 q^{14} +1.00000 q^{16} -6.00000i q^{17} +1.00000i q^{18} +4.00000 q^{19} +2.00000 q^{21} +1.00000i q^{22} -6.00000i q^{23} +1.00000 q^{24} +4.00000 q^{26} +1.00000i q^{27} -2.00000i q^{28} -6.00000 q^{29} +8.00000 q^{31} -1.00000i q^{32} +1.00000i q^{33} -6.00000 q^{34} +1.00000 q^{36} -10.0000i q^{37} -4.00000i q^{38} +4.00000 q^{39} +6.00000 q^{41} -2.00000i q^{42} -8.00000i q^{43} +1.00000 q^{44} -6.00000 q^{46} -6.00000i q^{47} -1.00000i q^{48} +3.00000 q^{49} -6.00000 q^{51} -4.00000i q^{52} +1.00000 q^{54} -2.00000 q^{56} -4.00000i q^{57} +6.00000i q^{58} +8.00000 q^{61} -8.00000i q^{62} -2.00000i q^{63} -1.00000 q^{64} +1.00000 q^{66} -4.00000i q^{67} +6.00000i q^{68} -6.00000 q^{69} +6.00000 q^{71} -1.00000i q^{72} -2.00000i q^{73} -10.0000 q^{74} -4.00000 q^{76} -2.00000i q^{77} -4.00000i q^{78} -14.0000 q^{79} +1.00000 q^{81} -6.00000i q^{82} +12.0000i q^{83} -2.00000 q^{84} -8.00000 q^{86} +6.00000i q^{87} -1.00000i q^{88} +6.00000 q^{89} -8.00000 q^{91} +6.00000i q^{92} -8.00000i q^{93} -6.00000 q^{94} -1.00000 q^{96} +14.0000i q^{97} -3.00000i q^{98} +1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{4} - 2 q^{6} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{4} - 2 q^{6} - 2 q^{9} - 2 q^{11} + 4 q^{14} + 2 q^{16} + 8 q^{19} + 4 q^{21} + 2 q^{24} + 8 q^{26} - 12 q^{29} + 16 q^{31} - 12 q^{34} + 2 q^{36} + 8 q^{39} + 12 q^{41} + 2 q^{44} - 12 q^{46} + 6 q^{49} - 12 q^{51} + 2 q^{54} - 4 q^{56} + 16 q^{61} - 2 q^{64} + 2 q^{66} - 12 q^{69} + 12 q^{71} - 20 q^{74} - 8 q^{76} - 28 q^{79} + 2 q^{81} - 4 q^{84} - 16 q^{86} + 12 q^{89} - 16 q^{91} - 12 q^{94} - 2 q^{96} + 2 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1650\mathbb{Z}\right)^\times\).

\(n\) \(551\) \(727\) \(1201\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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
\(11\) −1.00000 −0.301511
\(12\) 1.00000i 0.288675i
\(13\) 4.00000i 1.10940i 0.832050 + 0.554700i \(0.187167\pi\)
−0.832050 + 0.554700i \(0.812833\pi\)
\(14\) 2.00000 0.534522
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) − 6.00000i − 1.45521i −0.685994 0.727607i \(-0.740633\pi\)
0.685994 0.727607i \(-0.259367\pi\)
\(18\) 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
\(21\) 2.00000 0.436436
\(22\) 1.00000i 0.213201i
\(23\) − 6.00000i − 1.25109i −0.780189 0.625543i \(-0.784877\pi\)
0.780189 0.625543i \(-0.215123\pi\)
\(24\) 1.00000 0.204124
\(25\) 0 0
\(26\) 4.00000 0.784465
\(27\) 1.00000i 0.192450i
\(28\) − 2.00000i − 0.377964i
\(29\) −6.00000 −1.11417 −0.557086 0.830455i \(-0.688081\pi\)
−0.557086 + 0.830455i \(0.688081\pi\)
\(30\) 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
\(33\) 1.00000i 0.174078i
\(34\) −6.00000 −1.02899
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) − 10.0000i − 1.64399i −0.569495 0.821995i \(-0.692861\pi\)
0.569495 0.821995i \(-0.307139\pi\)
\(38\) − 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 \(0.344787\pi\)
\(42\) − 2.00000i − 0.308607i
\(43\) − 8.00000i − 1.21999i −0.792406 0.609994i \(-0.791172\pi\)
0.792406 0.609994i \(-0.208828\pi\)
\(44\) 1.00000 0.150756
\(45\) 0 0
\(46\) −6.00000 −0.884652
\(47\) − 6.00000i − 0.875190i −0.899172 0.437595i \(-0.855830\pi\)
0.899172 0.437595i \(-0.144170\pi\)
\(48\) − 1.00000i − 0.144338i
\(49\) 3.00000 0.428571
\(50\) 0 0
\(51\) −6.00000 −0.840168
\(52\) − 4.00000i − 0.554700i
\(53\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(54\) 1.00000 0.136083
\(55\) 0 0
\(56\) −2.00000 −0.267261
\(57\) − 4.00000i − 0.529813i
\(58\) 6.00000i 0.787839i
\(59\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 0 0
\(61\) 8.00000 1.02430 0.512148 0.858898i \(-0.328850\pi\)
0.512148 + 0.858898i \(0.328850\pi\)
\(62\) − 8.00000i − 1.01600i
\(63\) − 2.00000i − 0.251976i
\(64\) −1.00000 −0.125000
\(65\) 0 0
\(66\) 1.00000 0.123091
\(67\) − 4.00000i − 0.488678i −0.969690 0.244339i \(-0.921429\pi\)
0.969690 0.244339i \(-0.0785709\pi\)
\(68\) 6.00000i 0.727607i
\(69\) −6.00000 −0.722315
\(70\) 0 0
\(71\) 6.00000 0.712069 0.356034 0.934473i \(-0.384129\pi\)
0.356034 + 0.934473i \(0.384129\pi\)
\(72\) − 1.00000i − 0.117851i
\(73\) − 2.00000i − 0.234082i −0.993127 0.117041i \(-0.962659\pi\)
0.993127 0.117041i \(-0.0373409\pi\)
\(74\) −10.0000 −1.16248
\(75\) 0 0
\(76\) −4.00000 −0.458831
\(77\) − 2.00000i − 0.227921i
\(78\) − 4.00000i − 0.452911i
\(79\) −14.0000 −1.57512 −0.787562 0.616236i \(-0.788657\pi\)
−0.787562 + 0.616236i \(0.788657\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) − 6.00000i − 0.662589i
\(83\) 12.0000i 1.31717i 0.752506 + 0.658586i \(0.228845\pi\)
−0.752506 + 0.658586i \(0.771155\pi\)
\(84\) −2.00000 −0.218218
\(85\) 0 0
\(86\) −8.00000 −0.862662
\(87\) 6.00000i 0.643268i
\(88\) − 1.00000i − 0.106600i
\(89\) 6.00000 0.635999 0.317999 0.948091i \(-0.396989\pi\)
0.317999 + 0.948091i \(0.396989\pi\)
\(90\) 0 0
\(91\) −8.00000 −0.838628
\(92\) 6.00000i 0.625543i
\(93\) − 8.00000i − 0.829561i
\(94\) −6.00000 −0.618853
\(95\) 0 0
\(96\) −1.00000 −0.102062
\(97\) 14.0000i 1.42148i 0.703452 + 0.710742i \(0.251641\pi\)
−0.703452 + 0.710742i \(0.748359\pi\)
\(98\) − 3.00000i − 0.303046i
\(99\) 1.00000 0.100504
\(100\) 0 0
\(101\) 6.00000 0.597022 0.298511 0.954406i \(-0.403510\pi\)
0.298511 + 0.954406i \(0.403510\pi\)
\(102\) 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
\(106\) 0 0
\(107\) − 12.0000i − 1.16008i −0.814587 0.580042i \(-0.803036\pi\)
0.814587 0.580042i \(-0.196964\pi\)
\(108\) − 1.00000i − 0.0962250i
\(109\) 4.00000 0.383131 0.191565 0.981480i \(-0.438644\pi\)
0.191565 + 0.981480i \(0.438644\pi\)
\(110\) 0 0
\(111\) −10.0000 −0.949158
\(112\) 2.00000i 0.188982i
\(113\) − 18.0000i − 1.69330i −0.532152 0.846649i \(-0.678617\pi\)
0.532152 0.846649i \(-0.321383\pi\)
\(114\) −4.00000 −0.374634
\(115\) 0 0
\(116\) 6.00000 0.557086
\(117\) − 4.00000i − 0.369800i
\(118\) 0 0
\(119\) 12.0000 1.10004
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) − 8.00000i − 0.724286i
\(123\) − 6.00000i − 0.541002i
\(124\) −8.00000 −0.718421
\(125\) 0 0
\(126\) −2.00000 −0.178174
\(127\) 14.0000i 1.24230i 0.783692 + 0.621150i \(0.213334\pi\)
−0.783692 + 0.621150i \(0.786666\pi\)
\(128\) 1.00000i 0.0883883i
\(129\) −8.00000 −0.704361
\(130\) 0 0
\(131\) −12.0000 −1.04844 −0.524222 0.851581i \(-0.675644\pi\)
−0.524222 + 0.851581i \(0.675644\pi\)
\(132\) − 1.00000i − 0.0870388i
\(133\) 8.00000i 0.693688i
\(134\) −4.00000 −0.345547
\(135\) 0 0
\(136\) 6.00000 0.514496
\(137\) − 18.0000i − 1.53784i −0.639343 0.768922i \(-0.720793\pi\)
0.639343 0.768922i \(-0.279207\pi\)
\(138\) 6.00000i 0.510754i
\(139\) 4.00000 0.339276 0.169638 0.985506i \(-0.445740\pi\)
0.169638 + 0.985506i \(0.445740\pi\)
\(140\) 0 0
\(141\) −6.00000 −0.505291
\(142\) − 6.00000i − 0.503509i
\(143\) − 4.00000i − 0.334497i
\(144\) −1.00000 −0.0833333
\(145\) 0 0
\(146\) −2.00000 −0.165521
\(147\) − 3.00000i − 0.247436i
\(148\) 10.0000i 0.821995i
\(149\) 6.00000 0.491539 0.245770 0.969328i \(-0.420959\pi\)
0.245770 + 0.969328i \(0.420959\pi\)
\(150\) 0 0
\(151\) −10.0000 −0.813788 −0.406894 0.913475i \(-0.633388\pi\)
−0.406894 + 0.913475i \(0.633388\pi\)
\(152\) 4.00000i 0.324443i
\(153\) 6.00000i 0.485071i
\(154\) −2.00000 −0.161165
\(155\) 0 0
\(156\) −4.00000 −0.320256
\(157\) 2.00000i 0.159617i 0.996810 + 0.0798087i \(0.0254309\pi\)
−0.996810 + 0.0798087i \(0.974569\pi\)
\(158\) 14.0000i 1.11378i
\(159\) 0 0
\(160\) 0 0
\(161\) 12.0000 0.945732
\(162\) − 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
\(166\) 12.0000 0.931381
\(167\) 12.0000i 0.928588i 0.885681 + 0.464294i \(0.153692\pi\)
−0.885681 + 0.464294i \(0.846308\pi\)
\(168\) 2.00000i 0.154303i
\(169\) −3.00000 −0.230769
\(170\) 0 0
\(171\) −4.00000 −0.305888
\(172\) 8.00000i 0.609994i
\(173\) − 6.00000i − 0.456172i −0.973641 0.228086i \(-0.926753\pi\)
0.973641 0.228086i \(-0.0732467\pi\)
\(174\) 6.00000 0.454859
\(175\) 0 0
\(176\) −1.00000 −0.0753778
\(177\) 0 0
\(178\) − 6.00000i − 0.449719i
\(179\) −24.0000 −1.79384 −0.896922 0.442189i \(-0.854202\pi\)
−0.896922 + 0.442189i \(0.854202\pi\)
\(180\) 0 0
\(181\) −22.0000 −1.63525 −0.817624 0.575753i \(-0.804709\pi\)
−0.817624 + 0.575753i \(0.804709\pi\)
\(182\) 8.00000i 0.592999i
\(183\) − 8.00000i − 0.591377i
\(184\) 6.00000 0.442326
\(185\) 0 0
\(186\) −8.00000 −0.586588
\(187\) 6.00000i 0.438763i
\(188\) 6.00000i 0.437595i
\(189\) −2.00000 −0.145479
\(190\) 0 0
\(191\) 18.0000 1.30243 0.651217 0.758891i \(-0.274259\pi\)
0.651217 + 0.758891i \(0.274259\pi\)
\(192\) 1.00000i 0.0721688i
\(193\) − 14.0000i − 1.00774i −0.863779 0.503871i \(-0.831909\pi\)
0.863779 0.503871i \(-0.168091\pi\)
\(194\) 14.0000 1.00514
\(195\) 0 0
\(196\) −3.00000 −0.214286
\(197\) 6.00000i 0.427482i 0.976890 + 0.213741i \(0.0685649\pi\)
−0.976890 + 0.213741i \(0.931435\pi\)
\(198\) − 1.00000i − 0.0710669i
\(199\) 4.00000 0.283552 0.141776 0.989899i \(-0.454719\pi\)
0.141776 + 0.989899i \(0.454719\pi\)
\(200\) 0 0
\(201\) −4.00000 −0.282138
\(202\) − 6.00000i − 0.422159i
\(203\) − 12.0000i − 0.842235i
\(204\) 6.00000 0.420084
\(205\) 0 0
\(206\) 4.00000 0.278693
\(207\) 6.00000i 0.417029i
\(208\) 4.00000i 0.277350i
\(209\) −4.00000 −0.276686
\(210\) 0 0
\(211\) 8.00000 0.550743 0.275371 0.961338i \(-0.411199\pi\)
0.275371 + 0.961338i \(0.411199\pi\)
\(212\) 0 0
\(213\) − 6.00000i − 0.411113i
\(214\) −12.0000 −0.820303
\(215\) 0 0
\(216\) −1.00000 −0.0680414
\(217\) 16.0000i 1.08615i
\(218\) − 4.00000i − 0.270914i
\(219\) −2.00000 −0.135147
\(220\) 0 0
\(221\) 24.0000 1.61441
\(222\) 10.0000i 0.671156i
\(223\) 16.0000i 1.07144i 0.844396 + 0.535720i \(0.179960\pi\)
−0.844396 + 0.535720i \(0.820040\pi\)
\(224\) 2.00000 0.133631
\(225\) 0 0
\(226\) −18.0000 −1.19734
\(227\) − 12.0000i − 0.796468i −0.917284 0.398234i \(-0.869623\pi\)
0.917284 0.398234i \(-0.130377\pi\)
\(228\) 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
\(231\) −2.00000 −0.131590
\(232\) − 6.00000i − 0.393919i
\(233\) 18.0000i 1.17922i 0.807688 + 0.589610i \(0.200718\pi\)
−0.807688 + 0.589610i \(0.799282\pi\)
\(234\) −4.00000 −0.261488
\(235\) 0 0
\(236\) 0 0
\(237\) 14.0000i 0.909398i
\(238\) − 12.0000i − 0.777844i
\(239\) 12.0000 0.776215 0.388108 0.921614i \(-0.373129\pi\)
0.388108 + 0.921614i \(0.373129\pi\)
\(240\) 0 0
\(241\) −10.0000 −0.644157 −0.322078 0.946713i \(-0.604381\pi\)
−0.322078 + 0.946713i \(0.604381\pi\)
\(242\) − 1.00000i − 0.0642824i
\(243\) − 1.00000i − 0.0641500i
\(244\) −8.00000 −0.512148
\(245\) 0 0
\(246\) −6.00000 −0.382546
\(247\) 16.0000i 1.01806i
\(248\) 8.00000i 0.508001i
\(249\) 12.0000 0.760469
\(250\) 0 0
\(251\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(252\) 2.00000i 0.125988i
\(253\) 6.00000i 0.377217i
\(254\) 14.0000 0.878438
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) − 30.0000i − 1.87135i −0.352865 0.935674i \(-0.614792\pi\)
0.352865 0.935674i \(-0.385208\pi\)
\(258\) 8.00000i 0.498058i
\(259\) 20.0000 1.24274
\(260\) 0 0
\(261\) 6.00000 0.371391
\(262\) 12.0000i 0.741362i
\(263\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(264\) −1.00000 −0.0615457
\(265\) 0 0
\(266\) 8.00000 0.490511
\(267\) − 6.00000i − 0.367194i
\(268\) 4.00000i 0.244339i
\(269\) 24.0000 1.46331 0.731653 0.681677i \(-0.238749\pi\)
0.731653 + 0.681677i \(0.238749\pi\)
\(270\) 0 0
\(271\) 2.00000 0.121491 0.0607457 0.998153i \(-0.480652\pi\)
0.0607457 + 0.998153i \(0.480652\pi\)
\(272\) − 6.00000i − 0.363803i
\(273\) 8.00000i 0.484182i
\(274\) −18.0000 −1.08742
\(275\) 0 0
\(276\) 6.00000 0.361158
\(277\) − 16.0000i − 0.961347i −0.876900 0.480673i \(-0.840392\pi\)
0.876900 0.480673i \(-0.159608\pi\)
\(278\) − 4.00000i − 0.239904i
\(279\) −8.00000 −0.478947
\(280\) 0 0
\(281\) −6.00000 −0.357930 −0.178965 0.983855i \(-0.557275\pi\)
−0.178965 + 0.983855i \(0.557275\pi\)
\(282\) 6.00000i 0.357295i
\(283\) − 8.00000i − 0.475551i −0.971320 0.237775i \(-0.923582\pi\)
0.971320 0.237775i \(-0.0764182\pi\)
\(284\) −6.00000 −0.356034
\(285\) 0 0
\(286\) −4.00000 −0.236525
\(287\) 12.0000i 0.708338i
\(288\) 1.00000i 0.0589256i
\(289\) −19.0000 −1.11765
\(290\) 0 0
\(291\) 14.0000 0.820695
\(292\) 2.00000i 0.117041i
\(293\) 6.00000i 0.350524i 0.984522 + 0.175262i \(0.0560772\pi\)
−0.984522 + 0.175262i \(0.943923\pi\)
\(294\) −3.00000 −0.174964
\(295\) 0 0
\(296\) 10.0000 0.581238
\(297\) − 1.00000i − 0.0580259i
\(298\) − 6.00000i − 0.347571i
\(299\) 24.0000 1.38796
\(300\) 0 0
\(301\) 16.0000 0.922225
\(302\) 10.0000i 0.575435i
\(303\) − 6.00000i − 0.344691i
\(304\) 4.00000 0.229416
\(305\) 0 0
\(306\) 6.00000 0.342997
\(307\) 20.0000i 1.14146i 0.821138 + 0.570730i \(0.193340\pi\)
−0.821138 + 0.570730i \(0.806660\pi\)
\(308\) 2.00000i 0.113961i
\(309\) 4.00000 0.227552
\(310\) 0 0
\(311\) −18.0000 −1.02069 −0.510343 0.859971i \(-0.670482\pi\)
−0.510343 + 0.859971i \(0.670482\pi\)
\(312\) 4.00000i 0.226455i
\(313\) − 26.0000i − 1.46961i −0.678280 0.734803i \(-0.737274\pi\)
0.678280 0.734803i \(-0.262726\pi\)
\(314\) 2.00000 0.112867
\(315\) 0 0
\(316\) 14.0000 0.787562
\(317\) 12.0000i 0.673987i 0.941507 + 0.336994i \(0.109410\pi\)
−0.941507 + 0.336994i \(0.890590\pi\)
\(318\) 0 0
\(319\) 6.00000 0.335936
\(320\) 0 0
\(321\) −12.0000 −0.669775
\(322\) − 12.0000i − 0.668734i
\(323\) − 24.0000i − 1.33540i
\(324\) −1.00000 −0.0555556
\(325\) 0 0
\(326\) 4.00000 0.221540
\(327\) − 4.00000i − 0.221201i
\(328\) 6.00000i 0.331295i
\(329\) 12.0000 0.661581
\(330\) 0 0
\(331\) −4.00000 −0.219860 −0.109930 0.993939i \(-0.535063\pi\)
−0.109930 + 0.993939i \(0.535063\pi\)
\(332\) − 12.0000i − 0.658586i
\(333\) 10.0000i 0.547997i
\(334\) 12.0000 0.656611
\(335\) 0 0
\(336\) 2.00000 0.109109
\(337\) 2.00000i 0.108947i 0.998515 + 0.0544735i \(0.0173480\pi\)
−0.998515 + 0.0544735i \(0.982652\pi\)
\(338\) 3.00000i 0.163178i
\(339\) −18.0000 −0.977626
\(340\) 0 0
\(341\) −8.00000 −0.433224
\(342\) 4.00000i 0.216295i
\(343\) 20.0000i 1.07990i
\(344\) 8.00000 0.431331
\(345\) 0 0
\(346\) −6.00000 −0.322562
\(347\) 36.0000i 1.93258i 0.257454 + 0.966291i \(0.417117\pi\)
−0.257454 + 0.966291i \(0.582883\pi\)
\(348\) − 6.00000i − 0.321634i
\(349\) 4.00000 0.214115 0.107058 0.994253i \(-0.465857\pi\)
0.107058 + 0.994253i \(0.465857\pi\)
\(350\) 0 0
\(351\) −4.00000 −0.213504
\(352\) 1.00000i 0.0533002i
\(353\) 6.00000i 0.319348i 0.987170 + 0.159674i \(0.0510443\pi\)
−0.987170 + 0.159674i \(0.948956\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) −6.00000 −0.317999
\(357\) − 12.0000i − 0.635107i
\(358\) 24.0000i 1.26844i
\(359\) 12.0000 0.633336 0.316668 0.948536i \(-0.397436\pi\)
0.316668 + 0.948536i \(0.397436\pi\)
\(360\) 0 0
\(361\) −3.00000 −0.157895
\(362\) 22.0000i 1.15629i
\(363\) − 1.00000i − 0.0524864i
\(364\) 8.00000 0.419314
\(365\) 0 0
\(366\) −8.00000 −0.418167
\(367\) 8.00000i 0.417597i 0.977959 + 0.208798i \(0.0669552\pi\)
−0.977959 + 0.208798i \(0.933045\pi\)
\(368\) − 6.00000i − 0.312772i
\(369\) −6.00000 −0.312348
\(370\) 0 0
\(371\) 0 0
\(372\) 8.00000i 0.414781i
\(373\) − 20.0000i − 1.03556i −0.855514 0.517780i \(-0.826758\pi\)
0.855514 0.517780i \(-0.173242\pi\)
\(374\) 6.00000 0.310253
\(375\) 0 0
\(376\) 6.00000 0.309426
\(377\) − 24.0000i − 1.23606i
\(378\) 2.00000i 0.102869i
\(379\) −20.0000 −1.02733 −0.513665 0.857991i \(-0.671713\pi\)
−0.513665 + 0.857991i \(0.671713\pi\)
\(380\) 0 0
\(381\) 14.0000 0.717242
\(382\) − 18.0000i − 0.920960i
\(383\) − 6.00000i − 0.306586i −0.988181 0.153293i \(-0.951012\pi\)
0.988181 0.153293i \(-0.0489878\pi\)
\(384\) 1.00000 0.0510310
\(385\) 0 0
\(386\) −14.0000 −0.712581
\(387\) 8.00000i 0.406663i
\(388\) − 14.0000i − 0.710742i
\(389\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(390\) 0 0
\(391\) −36.0000 −1.82060
\(392\) 3.00000i 0.151523i
\(393\) 12.0000i 0.605320i
\(394\) 6.00000 0.302276
\(395\) 0 0
\(396\) −1.00000 −0.0502519
\(397\) 26.0000i 1.30490i 0.757831 + 0.652451i \(0.226259\pi\)
−0.757831 + 0.652451i \(0.773741\pi\)
\(398\) − 4.00000i − 0.200502i
\(399\) 8.00000 0.400501
\(400\) 0 0
\(401\) 30.0000 1.49813 0.749064 0.662497i \(-0.230503\pi\)
0.749064 + 0.662497i \(0.230503\pi\)
\(402\) 4.00000i 0.199502i
\(403\) 32.0000i 1.59403i
\(404\) −6.00000 −0.298511
\(405\) 0 0
\(406\) −12.0000 −0.595550
\(407\) 10.0000i 0.495682i
\(408\) − 6.00000i − 0.297044i
\(409\) 34.0000 1.68119 0.840596 0.541663i \(-0.182205\pi\)
0.840596 + 0.541663i \(0.182205\pi\)
\(410\) 0 0
\(411\) −18.0000 −0.887875
\(412\) − 4.00000i − 0.197066i
\(413\) 0 0
\(414\) 6.00000 0.294884
\(415\) 0 0
\(416\) 4.00000 0.196116
\(417\) − 4.00000i − 0.195881i
\(418\) 4.00000i 0.195646i
\(419\) −24.0000 −1.17248 −0.586238 0.810139i \(-0.699392\pi\)
−0.586238 + 0.810139i \(0.699392\pi\)
\(420\) 0 0
\(421\) −10.0000 −0.487370 −0.243685 0.969854i \(-0.578356\pi\)
−0.243685 + 0.969854i \(0.578356\pi\)
\(422\) − 8.00000i − 0.389434i
\(423\) 6.00000i 0.291730i
\(424\) 0 0
\(425\) 0 0
\(426\) −6.00000 −0.290701
\(427\) 16.0000i 0.774294i
\(428\) 12.0000i 0.580042i
\(429\) −4.00000 −0.193122
\(430\) 0 0
\(431\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(432\) 1.00000i 0.0481125i
\(433\) − 26.0000i − 1.24948i −0.780833 0.624740i \(-0.785205\pi\)
0.780833 0.624740i \(-0.214795\pi\)
\(434\) 16.0000 0.768025
\(435\) 0 0
\(436\) −4.00000 −0.191565
\(437\) − 24.0000i − 1.14808i
\(438\) 2.00000i 0.0955637i
\(439\) 10.0000 0.477274 0.238637 0.971109i \(-0.423299\pi\)
0.238637 + 0.971109i \(0.423299\pi\)
\(440\) 0 0
\(441\) −3.00000 −0.142857
\(442\) − 24.0000i − 1.14156i
\(443\) 24.0000i 1.14027i 0.821549 + 0.570137i \(0.193110\pi\)
−0.821549 + 0.570137i \(0.806890\pi\)
\(444\) 10.0000 0.474579
\(445\) 0 0
\(446\) 16.0000 0.757622
\(447\) − 6.00000i − 0.283790i
\(448\) − 2.00000i − 0.0944911i
\(449\) −6.00000 −0.283158 −0.141579 0.989927i \(-0.545218\pi\)
−0.141579 + 0.989927i \(0.545218\pi\)
\(450\) 0 0
\(451\) −6.00000 −0.282529
\(452\) 18.0000i 0.846649i
\(453\) 10.0000i 0.469841i
\(454\) −12.0000 −0.563188
\(455\) 0 0
\(456\) 4.00000 0.187317
\(457\) − 10.0000i − 0.467780i −0.972263 0.233890i \(-0.924854\pi\)
0.972263 0.233890i \(-0.0751456\pi\)
\(458\) − 22.0000i − 1.02799i
\(459\) 6.00000 0.280056
\(460\) 0 0
\(461\) 42.0000 1.95614 0.978068 0.208288i \(-0.0667892\pi\)
0.978068 + 0.208288i \(0.0667892\pi\)
\(462\) 2.00000i 0.0930484i
\(463\) 4.00000i 0.185896i 0.995671 + 0.0929479i \(0.0296290\pi\)
−0.995671 + 0.0929479i \(0.970371\pi\)
\(464\) −6.00000 −0.278543
\(465\) 0 0
\(466\) 18.0000 0.833834
\(467\) − 12.0000i − 0.555294i −0.960683 0.277647i \(-0.910445\pi\)
0.960683 0.277647i \(-0.0895545\pi\)
\(468\) 4.00000i 0.184900i
\(469\) 8.00000 0.369406
\(470\) 0 0
\(471\) 2.00000 0.0921551
\(472\) 0 0
\(473\) 8.00000i 0.367840i
\(474\) 14.0000 0.643041
\(475\) 0 0
\(476\) −12.0000 −0.550019
\(477\) 0 0
\(478\) − 12.0000i − 0.548867i
\(479\) 24.0000 1.09659 0.548294 0.836286i \(-0.315277\pi\)
0.548294 + 0.836286i \(0.315277\pi\)
\(480\) 0 0
\(481\) 40.0000 1.82384
\(482\) 10.0000i 0.455488i
\(483\) − 12.0000i − 0.546019i
\(484\) −1.00000 −0.0454545
\(485\) 0 0
\(486\) −1.00000 −0.0453609
\(487\) 20.0000i 0.906287i 0.891438 + 0.453143i \(0.149697\pi\)
−0.891438 + 0.453143i \(0.850303\pi\)
\(488\) 8.00000i 0.362143i
\(489\) 4.00000 0.180886
\(490\) 0 0
\(491\) 12.0000 0.541552 0.270776 0.962642i \(-0.412720\pi\)
0.270776 + 0.962642i \(0.412720\pi\)
\(492\) 6.00000i 0.270501i
\(493\) 36.0000i 1.62136i
\(494\) 16.0000 0.719874
\(495\) 0 0
\(496\) 8.00000 0.359211
\(497\) 12.0000i 0.538274i
\(498\) − 12.0000i − 0.537733i
\(499\) 4.00000 0.179065 0.0895323 0.995984i \(-0.471463\pi\)
0.0895323 + 0.995984i \(0.471463\pi\)
\(500\) 0 0
\(501\) 12.0000 0.536120
\(502\) 0 0
\(503\) − 12.0000i − 0.535054i −0.963550 0.267527i \(-0.913794\pi\)
0.963550 0.267527i \(-0.0862064\pi\)
\(504\) 2.00000 0.0890871
\(505\) 0 0
\(506\) 6.00000 0.266733
\(507\) 3.00000i 0.133235i
\(508\) − 14.0000i − 0.621150i
\(509\) −24.0000 −1.06378 −0.531891 0.846813i \(-0.678518\pi\)
−0.531891 + 0.846813i \(0.678518\pi\)
\(510\) 0 0
\(511\) 4.00000 0.176950
\(512\) − 1.00000i − 0.0441942i
\(513\) 4.00000i 0.176604i
\(514\) −30.0000 −1.32324
\(515\) 0 0
\(516\) 8.00000 0.352180
\(517\) 6.00000i 0.263880i
\(518\) − 20.0000i − 0.878750i
\(519\) −6.00000 −0.263371
\(520\) 0 0
\(521\) −18.0000 −0.788594 −0.394297 0.918983i \(-0.629012\pi\)
−0.394297 + 0.918983i \(0.629012\pi\)
\(522\) − 6.00000i − 0.262613i
\(523\) 16.0000i 0.699631i 0.936819 + 0.349816i \(0.113756\pi\)
−0.936819 + 0.349816i \(0.886244\pi\)
\(524\) 12.0000 0.524222
\(525\) 0 0
\(526\) 0 0
\(527\) − 48.0000i − 2.09091i
\(528\) 1.00000i 0.0435194i
\(529\) −13.0000 −0.565217
\(530\) 0 0
\(531\) 0 0
\(532\) − 8.00000i − 0.346844i
\(533\) 24.0000i 1.03956i
\(534\) −6.00000 −0.259645
\(535\) 0 0
\(536\) 4.00000 0.172774
\(537\) 24.0000i 1.03568i
\(538\) − 24.0000i − 1.03471i
\(539\) −3.00000 −0.129219
\(540\) 0 0
\(541\) 20.0000 0.859867 0.429934 0.902861i \(-0.358537\pi\)
0.429934 + 0.902861i \(0.358537\pi\)
\(542\) − 2.00000i − 0.0859074i
\(543\) 22.0000i 0.944110i
\(544\) −6.00000 −0.257248
\(545\) 0 0
\(546\) 8.00000 0.342368
\(547\) − 28.0000i − 1.19719i −0.801050 0.598597i \(-0.795725\pi\)
0.801050 0.598597i \(-0.204275\pi\)
\(548\) 18.0000i 0.768922i
\(549\) −8.00000 −0.341432
\(550\) 0 0
\(551\) −24.0000 −1.02243
\(552\) − 6.00000i − 0.255377i
\(553\) − 28.0000i − 1.19068i
\(554\) −16.0000 −0.679775
\(555\) 0 0
\(556\) −4.00000 −0.169638
\(557\) 18.0000i 0.762684i 0.924434 + 0.381342i \(0.124538\pi\)
−0.924434 + 0.381342i \(0.875462\pi\)
\(558\) 8.00000i 0.338667i
\(559\) 32.0000 1.35346
\(560\) 0 0
\(561\) 6.00000 0.253320
\(562\) 6.00000i 0.253095i
\(563\) 12.0000i 0.505740i 0.967500 + 0.252870i \(0.0813744\pi\)
−0.967500 + 0.252870i \(0.918626\pi\)
\(564\) 6.00000 0.252646
\(565\) 0 0
\(566\) −8.00000 −0.336265
\(567\) 2.00000i 0.0839921i
\(568\) 6.00000i 0.251754i
\(569\) −18.0000 −0.754599 −0.377300 0.926091i \(-0.623147\pi\)
−0.377300 + 0.926091i \(0.623147\pi\)
\(570\) 0 0
\(571\) −28.0000 −1.17176 −0.585882 0.810397i \(-0.699252\pi\)
−0.585882 + 0.810397i \(0.699252\pi\)
\(572\) 4.00000i 0.167248i
\(573\) − 18.0000i − 0.751961i
\(574\) 12.0000 0.500870
\(575\) 0 0
\(576\) 1.00000 0.0416667
\(577\) − 34.0000i − 1.41544i −0.706494 0.707719i \(-0.749724\pi\)
0.706494 0.707719i \(-0.250276\pi\)
\(578\) 19.0000i 0.790296i
\(579\) −14.0000 −0.581820
\(580\) 0 0
\(581\) −24.0000 −0.995688
\(582\) − 14.0000i − 0.580319i
\(583\) 0 0
\(584\) 2.00000 0.0827606
\(585\) 0 0
\(586\) 6.00000 0.247858
\(587\) 24.0000i 0.990586i 0.868726 + 0.495293i \(0.164939\pi\)
−0.868726 + 0.495293i \(0.835061\pi\)
\(588\) 3.00000i 0.123718i
\(589\) 32.0000 1.31854
\(590\) 0 0
\(591\) 6.00000 0.246807
\(592\) − 10.0000i − 0.410997i
\(593\) 30.0000i 1.23195i 0.787765 + 0.615976i \(0.211238\pi\)
−0.787765 + 0.615976i \(0.788762\pi\)
\(594\) −1.00000 −0.0410305
\(595\) 0 0
\(596\) −6.00000 −0.245770
\(597\) − 4.00000i − 0.163709i
\(598\) − 24.0000i − 0.981433i
\(599\) −30.0000 −1.22577 −0.612883 0.790173i \(-0.709990\pi\)
−0.612883 + 0.790173i \(0.709990\pi\)
\(600\) 0 0
\(601\) −22.0000 −0.897399 −0.448699 0.893683i \(-0.648113\pi\)
−0.448699 + 0.893683i \(0.648113\pi\)
\(602\) − 16.0000i − 0.652111i
\(603\) 4.00000i 0.162893i
\(604\) 10.0000 0.406894
\(605\) 0 0
\(606\) −6.00000 −0.243733
\(607\) 14.0000i 0.568242i 0.958788 + 0.284121i \(0.0917018\pi\)
−0.958788 + 0.284121i \(0.908298\pi\)
\(608\) − 4.00000i − 0.162221i
\(609\) −12.0000 −0.486265
\(610\) 0 0
\(611\) 24.0000 0.970936
\(612\) − 6.00000i − 0.242536i
\(613\) 16.0000i 0.646234i 0.946359 + 0.323117i \(0.104731\pi\)
−0.946359 + 0.323117i \(0.895269\pi\)
\(614\) 20.0000 0.807134
\(615\) 0 0
\(616\) 2.00000 0.0805823
\(617\) − 30.0000i − 1.20775i −0.797077 0.603877i \(-0.793622\pi\)
0.797077 0.603877i \(-0.206378\pi\)
\(618\) − 4.00000i − 0.160904i
\(619\) −44.0000 −1.76851 −0.884255 0.467005i \(-0.845333\pi\)
−0.884255 + 0.467005i \(0.845333\pi\)
\(620\) 0 0
\(621\) 6.00000 0.240772
\(622\) 18.0000i 0.721734i
\(623\) 12.0000i 0.480770i
\(624\) 4.00000 0.160128
\(625\) 0 0
\(626\) −26.0000 −1.03917
\(627\) 4.00000i 0.159745i
\(628\) − 2.00000i − 0.0798087i
\(629\) −60.0000 −2.39236
\(630\) 0 0
\(631\) −16.0000 −0.636950 −0.318475 0.947931i \(-0.603171\pi\)
−0.318475 + 0.947931i \(0.603171\pi\)
\(632\) − 14.0000i − 0.556890i
\(633\) − 8.00000i − 0.317971i
\(634\) 12.0000 0.476581
\(635\) 0 0
\(636\) 0 0
\(637\) 12.0000i 0.475457i
\(638\) − 6.00000i − 0.237542i
\(639\) −6.00000 −0.237356
\(640\) 0 0
\(641\) 6.00000 0.236986 0.118493 0.992955i \(-0.462194\pi\)
0.118493 + 0.992955i \(0.462194\pi\)
\(642\) 12.0000i 0.473602i
\(643\) 4.00000i 0.157745i 0.996885 + 0.0788723i \(0.0251319\pi\)
−0.996885 + 0.0788723i \(0.974868\pi\)
\(644\) −12.0000 −0.472866
\(645\) 0 0
\(646\) −24.0000 −0.944267
\(647\) 6.00000i 0.235884i 0.993020 + 0.117942i \(0.0376297\pi\)
−0.993020 + 0.117942i \(0.962370\pi\)
\(648\) 1.00000i 0.0392837i
\(649\) 0 0
\(650\) 0 0
\(651\) 16.0000 0.627089
\(652\) − 4.00000i − 0.156652i
\(653\) 36.0000i 1.40879i 0.709809 + 0.704394i \(0.248781\pi\)
−0.709809 + 0.704394i \(0.751219\pi\)
\(654\) −4.00000 −0.156412
\(655\) 0 0
\(656\) 6.00000 0.234261
\(657\) 2.00000i 0.0780274i
\(658\) − 12.0000i − 0.467809i
\(659\) −36.0000 −1.40236 −0.701180 0.712984i \(-0.747343\pi\)
−0.701180 + 0.712984i \(0.747343\pi\)
\(660\) 0 0
\(661\) −22.0000 −0.855701 −0.427850 0.903850i \(-0.640729\pi\)
−0.427850 + 0.903850i \(0.640729\pi\)
\(662\) 4.00000i 0.155464i
\(663\) − 24.0000i − 0.932083i
\(664\) −12.0000 −0.465690
\(665\) 0 0
\(666\) 10.0000 0.387492
\(667\) 36.0000i 1.39393i
\(668\) − 12.0000i − 0.464294i
\(669\) 16.0000 0.618596
\(670\) 0 0
\(671\) −8.00000 −0.308837
\(672\) − 2.00000i − 0.0771517i
\(673\) − 14.0000i − 0.539660i −0.962908 0.269830i \(-0.913032\pi\)
0.962908 0.269830i \(-0.0869676\pi\)
\(674\) 2.00000 0.0770371
\(675\) 0 0
\(676\) 3.00000 0.115385
\(677\) 30.0000i 1.15299i 0.817099 + 0.576497i \(0.195581\pi\)
−0.817099 + 0.576497i \(0.804419\pi\)
\(678\) 18.0000i 0.691286i
\(679\) −28.0000 −1.07454
\(680\) 0 0
\(681\) −12.0000 −0.459841
\(682\) 8.00000i 0.306336i
\(683\) 24.0000i 0.918334i 0.888350 + 0.459167i \(0.151852\pi\)
−0.888350 + 0.459167i \(0.848148\pi\)
\(684\) 4.00000 0.152944
\(685\) 0 0
\(686\) 20.0000 0.763604
\(687\) − 22.0000i − 0.839352i
\(688\) − 8.00000i − 0.304997i
\(689\) 0 0
\(690\) 0 0
\(691\) 20.0000 0.760836 0.380418 0.924815i \(-0.375780\pi\)
0.380418 + 0.924815i \(0.375780\pi\)
\(692\) 6.00000i 0.228086i
\(693\) 2.00000i 0.0759737i
\(694\) 36.0000 1.36654
\(695\) 0 0
\(696\) −6.00000 −0.227429
\(697\) − 36.0000i − 1.36360i
\(698\) − 4.00000i − 0.151402i
\(699\) 18.0000 0.680823
\(700\) 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
\(704\) 1.00000 0.0376889
\(705\) 0 0
\(706\) 6.00000 0.225813
\(707\) 12.0000i 0.451306i
\(708\) 0 0
\(709\) −26.0000 −0.976450 −0.488225 0.872718i \(-0.662356\pi\)
−0.488225 + 0.872718i \(0.662356\pi\)
\(710\) 0 0
\(711\) 14.0000 0.525041
\(712\) 6.00000i 0.224860i
\(713\) − 48.0000i − 1.79761i
\(714\) −12.0000 −0.449089
\(715\) 0 0
\(716\) 24.0000 0.896922
\(717\) − 12.0000i − 0.448148i
\(718\) − 12.0000i − 0.447836i
\(719\) −30.0000 −1.11881 −0.559406 0.828894i \(-0.688971\pi\)
−0.559406 + 0.828894i \(0.688971\pi\)
\(720\) 0 0
\(721\) −8.00000 −0.297936
\(722\) 3.00000i 0.111648i
\(723\) 10.0000i 0.371904i
\(724\) 22.0000 0.817624
\(725\) 0 0
\(726\) −1.00000 −0.0371135
\(727\) − 28.0000i − 1.03846i −0.854634 0.519231i \(-0.826218\pi\)
0.854634 0.519231i \(-0.173782\pi\)
\(728\) − 8.00000i − 0.296500i
\(729\) −1.00000 −0.0370370
\(730\) 0 0
\(731\) −48.0000 −1.77534
\(732\) 8.00000i 0.295689i
\(733\) 4.00000i 0.147743i 0.997268 + 0.0738717i \(0.0235355\pi\)
−0.997268 + 0.0738717i \(0.976464\pi\)
\(734\) 8.00000 0.295285
\(735\) 0 0
\(736\) −6.00000 −0.221163
\(737\) 4.00000i 0.147342i
\(738\) 6.00000i 0.220863i
\(739\) −8.00000 −0.294285 −0.147142 0.989115i \(-0.547008\pi\)
−0.147142 + 0.989115i \(0.547008\pi\)
\(740\) 0 0
\(741\) 16.0000 0.587775
\(742\) 0 0
\(743\) − 36.0000i − 1.32071i −0.750953 0.660356i \(-0.770405\pi\)
0.750953 0.660356i \(-0.229595\pi\)
\(744\) 8.00000 0.293294
\(745\) 0 0
\(746\) −20.0000 −0.732252
\(747\) − 12.0000i − 0.439057i
\(748\) − 6.00000i − 0.219382i
\(749\) 24.0000 0.876941
\(750\) 0 0
\(751\) 8.00000 0.291924 0.145962 0.989290i \(-0.453372\pi\)
0.145962 + 0.989290i \(0.453372\pi\)
\(752\) − 6.00000i − 0.218797i
\(753\) 0 0
\(754\) −24.0000 −0.874028
\(755\) 0 0
\(756\) 2.00000 0.0727393
\(757\) − 34.0000i − 1.23575i −0.786276 0.617876i \(-0.787994\pi\)
0.786276 0.617876i \(-0.212006\pi\)
\(758\) 20.0000i 0.726433i
\(759\) 6.00000 0.217786
\(760\) 0 0
\(761\) 18.0000 0.652499 0.326250 0.945284i \(-0.394215\pi\)
0.326250 + 0.945284i \(0.394215\pi\)
\(762\) − 14.0000i − 0.507166i
\(763\) 8.00000i 0.289619i
\(764\) −18.0000 −0.651217
\(765\) 0 0
\(766\) −6.00000 −0.216789
\(767\) 0 0
\(768\) − 1.00000i − 0.0360844i
\(769\) 34.0000 1.22607 0.613036 0.790055i \(-0.289948\pi\)
0.613036 + 0.790055i \(0.289948\pi\)
\(770\) 0 0
\(771\) −30.0000 −1.08042
\(772\) 14.0000i 0.503871i
\(773\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(774\) 8.00000 0.287554
\(775\) 0 0
\(776\) −14.0000 −0.502571
\(777\) − 20.0000i − 0.717496i
\(778\) 0 0
\(779\) 24.0000 0.859889
\(780\) 0 0
\(781\) −6.00000 −0.214697
\(782\) 36.0000i 1.28736i
\(783\) − 6.00000i − 0.214423i
\(784\) 3.00000 0.107143
\(785\) 0 0
\(786\) 12.0000 0.428026
\(787\) 32.0000i 1.14068i 0.821410 + 0.570338i \(0.193188\pi\)
−0.821410 + 0.570338i \(0.806812\pi\)
\(788\) − 6.00000i − 0.213741i
\(789\) 0 0
\(790\) 0 0
\(791\) 36.0000 1.28001
\(792\) 1.00000i 0.0355335i
\(793\) 32.0000i 1.13635i
\(794\) 26.0000 0.922705
\(795\) 0 0
\(796\) −4.00000 −0.141776
\(797\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(798\) − 8.00000i − 0.283197i
\(799\) −36.0000 −1.27359
\(800\) 0 0
\(801\) −6.00000 −0.212000
\(802\) − 30.0000i − 1.05934i
\(803\) 2.00000i 0.0705785i
\(804\) 4.00000 0.141069
\(805\) 0 0
\(806\) 32.0000 1.12715
\(807\) − 24.0000i − 0.844840i
\(808\) 6.00000i 0.211079i
\(809\) 42.0000 1.47664 0.738321 0.674450i \(-0.235619\pi\)
0.738321 + 0.674450i \(0.235619\pi\)
\(810\) 0 0
\(811\) 8.00000 0.280918 0.140459 0.990086i \(-0.455142\pi\)
0.140459 + 0.990086i \(0.455142\pi\)
\(812\) 12.0000i 0.421117i
\(813\) − 2.00000i − 0.0701431i
\(814\) 10.0000 0.350500
\(815\) 0 0
\(816\) −6.00000 −0.210042
\(817\) − 32.0000i − 1.11954i
\(818\) − 34.0000i − 1.18878i
\(819\) 8.00000 0.279543
\(820\) 0 0
\(821\) −18.0000 −0.628204 −0.314102 0.949389i \(-0.601703\pi\)
−0.314102 + 0.949389i \(0.601703\pi\)
\(822\) 18.0000i 0.627822i
\(823\) − 8.00000i − 0.278862i −0.990232 0.139431i \(-0.955473\pi\)
0.990232 0.139431i \(-0.0445274\pi\)
\(824\) −4.00000 −0.139347
\(825\) 0 0
\(826\) 0 0
\(827\) − 12.0000i − 0.417281i −0.977992 0.208640i \(-0.933096\pi\)
0.977992 0.208640i \(-0.0669038\pi\)
\(828\) − 6.00000i − 0.208514i
\(829\) −14.0000 −0.486240 −0.243120 0.969996i \(-0.578171\pi\)
−0.243120 + 0.969996i \(0.578171\pi\)
\(830\) 0 0
\(831\) −16.0000 −0.555034
\(832\) − 4.00000i − 0.138675i
\(833\) − 18.0000i − 0.623663i
\(834\) −4.00000 −0.138509
\(835\) 0 0
\(836\) 4.00000 0.138343
\(837\) 8.00000i 0.276520i
\(838\) 24.0000i 0.829066i
\(839\) −18.0000 −0.621429 −0.310715 0.950503i \(-0.600568\pi\)
−0.310715 + 0.950503i \(0.600568\pi\)
\(840\) 0 0
\(841\) 7.00000 0.241379
\(842\) 10.0000i 0.344623i
\(843\) 6.00000i 0.206651i
\(844\) −8.00000 −0.275371
\(845\) 0 0
\(846\) 6.00000 0.206284
\(847\) 2.00000i 0.0687208i
\(848\) 0 0
\(849\) −8.00000 −0.274559
\(850\) 0 0
\(851\) −60.0000 −2.05677
\(852\) 6.00000i 0.205557i
\(853\) − 8.00000i − 0.273915i −0.990577 0.136957i \(-0.956268\pi\)
0.990577 0.136957i \(-0.0437323\pi\)
\(854\) 16.0000 0.547509
\(855\) 0 0
\(856\) 12.0000 0.410152
\(857\) 54.0000i 1.84460i 0.386469 + 0.922302i \(0.373695\pi\)
−0.386469 + 0.922302i \(0.626305\pi\)
\(858\) 4.00000i 0.136558i
\(859\) −20.0000 −0.682391 −0.341196 0.939992i \(-0.610832\pi\)
−0.341196 + 0.939992i \(0.610832\pi\)
\(860\) 0 0
\(861\) 12.0000 0.408959
\(862\) 0 0
\(863\) − 42.0000i − 1.42970i −0.699280 0.714848i \(-0.746496\pi\)
0.699280 0.714848i \(-0.253504\pi\)
\(864\) 1.00000 0.0340207
\(865\) 0 0
\(866\) −26.0000 −0.883516
\(867\) 19.0000i 0.645274i
\(868\) − 16.0000i − 0.543075i
\(869\) 14.0000 0.474917
\(870\) 0 0
\(871\) 16.0000 0.542139
\(872\) 4.00000i 0.135457i
\(873\) − 14.0000i − 0.473828i
\(874\) −24.0000 −0.811812
\(875\) 0 0
\(876\) 2.00000 0.0675737
\(877\) − 52.0000i − 1.75592i −0.478738 0.877958i \(-0.658906\pi\)
0.478738 0.877958i \(-0.341094\pi\)
\(878\) − 10.0000i − 0.337484i
\(879\) 6.00000 0.202375
\(880\) 0 0
\(881\) −30.0000 −1.01073 −0.505363 0.862907i \(-0.668641\pi\)
−0.505363 + 0.862907i \(0.668641\pi\)
\(882\) 3.00000i 0.101015i
\(883\) 28.0000i 0.942275i 0.882060 + 0.471138i \(0.156156\pi\)
−0.882060 + 0.471138i \(0.843844\pi\)
\(884\) −24.0000 −0.807207
\(885\) 0 0
\(886\) 24.0000 0.806296
\(887\) − 36.0000i − 1.20876i −0.796696 0.604381i \(-0.793421\pi\)
0.796696 0.604381i \(-0.206579\pi\)
\(888\) − 10.0000i − 0.335578i
\(889\) −28.0000 −0.939090
\(890\) 0 0
\(891\) −1.00000 −0.0335013
\(892\) − 16.0000i − 0.535720i
\(893\) − 24.0000i − 0.803129i
\(894\) −6.00000 −0.200670
\(895\) 0 0
\(896\) −2.00000 −0.0668153
\(897\) − 24.0000i − 0.801337i
\(898\) 6.00000i 0.200223i
\(899\) −48.0000 −1.60089
\(900\) 0 0
\(901\) 0 0
\(902\) 6.00000i 0.199778i
\(903\) − 16.0000i − 0.532447i
\(904\) 18.0000 0.598671
\(905\) 0 0
\(906\) 10.0000 0.332228
\(907\) 20.0000i 0.664089i 0.943264 + 0.332045i \(0.107738\pi\)
−0.943264 + 0.332045i \(0.892262\pi\)
\(908\) 12.0000i 0.398234i
\(909\) −6.00000 −0.199007
\(910\) 0 0
\(911\) −42.0000 −1.39152 −0.695761 0.718273i \(-0.744933\pi\)
−0.695761 + 0.718273i \(0.744933\pi\)
\(912\) − 4.00000i − 0.132453i
\(913\) − 12.0000i − 0.397142i
\(914\) −10.0000 −0.330771
\(915\) 0 0
\(916\) −22.0000 −0.726900
\(917\) − 24.0000i − 0.792550i
\(918\) − 6.00000i − 0.198030i
\(919\) −2.00000 −0.0659739 −0.0329870 0.999456i \(-0.510502\pi\)
−0.0329870 + 0.999456i \(0.510502\pi\)
\(920\) 0 0
\(921\) 20.0000 0.659022
\(922\) − 42.0000i − 1.38320i
\(923\) 24.0000i 0.789970i
\(924\) 2.00000 0.0657952
\(925\) 0 0
\(926\) 4.00000 0.131448
\(927\) − 4.00000i − 0.131377i
\(928\) 6.00000i 0.196960i
\(929\) −30.0000 −0.984268 −0.492134 0.870519i \(-0.663783\pi\)
−0.492134 + 0.870519i \(0.663783\pi\)
\(930\) 0 0
\(931\) 12.0000 0.393284
\(932\) − 18.0000i − 0.589610i
\(933\) 18.0000i 0.589294i
\(934\) −12.0000 −0.392652
\(935\) 0 0
\(936\) 4.00000 0.130744
\(937\) − 22.0000i − 0.718709i −0.933201 0.359354i \(-0.882997\pi\)
0.933201 0.359354i \(-0.117003\pi\)
\(938\) − 8.00000i − 0.261209i
\(939\) −26.0000 −0.848478
\(940\) 0 0
\(941\) −18.0000 −0.586783 −0.293392 0.955992i \(-0.594784\pi\)
−0.293392 + 0.955992i \(0.594784\pi\)
\(942\) − 2.00000i − 0.0651635i
\(943\) − 36.0000i − 1.17232i
\(944\) 0 0
\(945\) 0 0
\(946\) 8.00000 0.260102
\(947\) 12.0000i 0.389948i 0.980808 + 0.194974i \(0.0624622\pi\)
−0.980808 + 0.194974i \(0.937538\pi\)
\(948\) − 14.0000i − 0.454699i
\(949\) 8.00000 0.259691
\(950\) 0 0
\(951\) 12.0000 0.389127
\(952\) 12.0000i 0.388922i
\(953\) 42.0000i 1.36051i 0.732974 + 0.680257i \(0.238132\pi\)
−0.732974 + 0.680257i \(0.761868\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) −12.0000 −0.388108
\(957\) − 6.00000i − 0.193952i
\(958\) − 24.0000i − 0.775405i
\(959\) 36.0000 1.16250
\(960\) 0 0
\(961\) 33.0000 1.06452
\(962\) − 40.0000i − 1.28965i
\(963\) 12.0000i 0.386695i
\(964\) 10.0000 0.322078
\(965\) 0 0
\(966\) −12.0000 −0.386094
\(967\) 14.0000i 0.450210i 0.974335 + 0.225105i \(0.0722725\pi\)
−0.974335 + 0.225105i \(0.927728\pi\)
\(968\) 1.00000i 0.0321412i
\(969\) −24.0000 −0.770991
\(970\) 0 0
\(971\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(972\) 1.00000i 0.0320750i
\(973\) 8.00000i 0.256468i
\(974\) 20.0000 0.640841
\(975\) 0 0
\(976\) 8.00000 0.256074
\(977\) 54.0000i 1.72761i 0.503824 + 0.863807i \(0.331926\pi\)
−0.503824 + 0.863807i \(0.668074\pi\)
\(978\) − 4.00000i − 0.127906i
\(979\) −6.00000 −0.191761
\(980\) 0 0
\(981\) −4.00000 −0.127710
\(982\) − 12.0000i − 0.382935i
\(983\) − 30.0000i − 0.956851i −0.878128 0.478426i \(-0.841208\pi\)
0.878128 0.478426i \(-0.158792\pi\)
\(984\) 6.00000 0.191273
\(985\) 0 0
\(986\) 36.0000 1.14647
\(987\) − 12.0000i − 0.381964i
\(988\) − 16.0000i − 0.509028i
\(989\) −48.0000 −1.52631
\(990\) 0 0
\(991\) 56.0000 1.77890 0.889449 0.457034i \(-0.151088\pi\)
0.889449 + 0.457034i \(0.151088\pi\)
\(992\) − 8.00000i − 0.254000i
\(993\) 4.00000i 0.126936i
\(994\) 12.0000 0.380617
\(995\) 0 0
\(996\) −12.0000 −0.380235
\(997\) 44.0000i 1.39349i 0.717317 + 0.696747i \(0.245370\pi\)
−0.717317 + 0.696747i \(0.754630\pi\)
\(998\) − 4.00000i − 0.126618i
\(999\) 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 1650.2.c.d.199.1 2
3.2 odd 2 4950.2.c.r.199.2 2
5.2 odd 4 1650.2.a.m.1.1 1
5.3 odd 4 66.2.a.a.1.1 1
5.4 even 2 inner 1650.2.c.d.199.2 2
15.2 even 4 4950.2.a.g.1.1 1
15.8 even 4 198.2.a.e.1.1 1
15.14 odd 2 4950.2.c.r.199.1 2
20.3 even 4 528.2.a.d.1.1 1
35.13 even 4 3234.2.a.d.1.1 1
40.3 even 4 2112.2.a.v.1.1 1
40.13 odd 4 2112.2.a.i.1.1 1
45.13 odd 12 1782.2.e.s.1189.1 2
45.23 even 12 1782.2.e.f.1189.1 2
45.38 even 12 1782.2.e.f.595.1 2
45.43 odd 12 1782.2.e.s.595.1 2
55.3 odd 20 726.2.e.k.493.1 4
55.8 even 20 726.2.e.b.493.1 4
55.13 even 20 726.2.e.b.565.1 4
55.18 even 20 726.2.e.b.511.1 4
55.28 even 20 726.2.e.b.487.1 4
55.38 odd 20 726.2.e.k.487.1 4
55.43 even 4 726.2.a.i.1.1 1
55.48 odd 20 726.2.e.k.511.1 4
55.53 odd 20 726.2.e.k.565.1 4
60.23 odd 4 1584.2.a.h.1.1 1
105.83 odd 4 9702.2.a.bu.1.1 1
120.53 even 4 6336.2.a.bj.1.1 1
120.83 odd 4 6336.2.a.bf.1.1 1
165.98 odd 4 2178.2.a.b.1.1 1
220.43 odd 4 5808.2.a.l.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
66.2.a.a.1.1 1 5.3 odd 4
198.2.a.e.1.1 1 15.8 even 4
528.2.a.d.1.1 1 20.3 even 4
726.2.a.i.1.1 1 55.43 even 4
726.2.e.b.487.1 4 55.28 even 20
726.2.e.b.493.1 4 55.8 even 20
726.2.e.b.511.1 4 55.18 even 20
726.2.e.b.565.1 4 55.13 even 20
726.2.e.k.487.1 4 55.38 odd 20
726.2.e.k.493.1 4 55.3 odd 20
726.2.e.k.511.1 4 55.48 odd 20
726.2.e.k.565.1 4 55.53 odd 20
1584.2.a.h.1.1 1 60.23 odd 4
1650.2.a.m.1.1 1 5.2 odd 4
1650.2.c.d.199.1 2 1.1 even 1 trivial
1650.2.c.d.199.2 2 5.4 even 2 inner
1782.2.e.f.595.1 2 45.38 even 12
1782.2.e.f.1189.1 2 45.23 even 12
1782.2.e.s.595.1 2 45.43 odd 12
1782.2.e.s.1189.1 2 45.13 odd 12
2112.2.a.i.1.1 1 40.13 odd 4
2112.2.a.v.1.1 1 40.3 even 4
2178.2.a.b.1.1 1 165.98 odd 4
3234.2.a.d.1.1 1 35.13 even 4
4950.2.a.g.1.1 1 15.2 even 4
4950.2.c.r.199.1 2 15.14 odd 2
4950.2.c.r.199.2 2 3.2 odd 2
5808.2.a.l.1.1 1 220.43 odd 4
6336.2.a.bf.1.1 1 120.83 odd 4
6336.2.a.bj.1.1 1 120.53 even 4
9702.2.a.bu.1.1 1 105.83 odd 4