Properties

Label 4650.2.d.u.3349.2
Level $4650$
Weight $2$
Character 4650.3349
Analytic conductor $37.130$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 4650 = 2 \cdot 3 \cdot 5^{2} \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4650.d (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(37.1304369399\)
Analytic rank: \(1\)
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 930)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 3349.2
Root \(1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 4650.3349
Dual form 4650.2.d.u.3349.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} +2.00000i q^{7} -1.00000i q^{8} -1.00000 q^{9} +O(q^{10})\) \(q+1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} +2.00000i q^{7} -1.00000i q^{8} -1.00000 q^{9} +1.00000i q^{12} +4.00000i q^{13} -2.00000 q^{14} +1.00000 q^{16} +6.00000i q^{17} -1.00000i q^{18} -8.00000 q^{19} +2.00000 q^{21} -1.00000 q^{24} -4.00000 q^{26} +1.00000i q^{27} -2.00000i q^{28} +1.00000 q^{31} +1.00000i q^{32} -6.00000 q^{34} +1.00000 q^{36} -4.00000i q^{37} -8.00000i q^{38} +4.00000 q^{39} -6.00000 q^{41} +2.00000i q^{42} -8.00000i q^{43} -12.0000i q^{47} -1.00000i q^{48} +3.00000 q^{49} +6.00000 q^{51} -4.00000i q^{52} +6.00000i q^{53} -1.00000 q^{54} +2.00000 q^{56} +8.00000i q^{57} +6.00000 q^{59} +2.00000 q^{61} +1.00000i q^{62} -2.00000i q^{63} -1.00000 q^{64} +2.00000i q^{67} -6.00000i q^{68} -6.00000 q^{71} +1.00000i q^{72} -8.00000i q^{73} +4.00000 q^{74} +8.00000 q^{76} +4.00000i q^{78} -8.00000 q^{79} +1.00000 q^{81} -6.00000i q^{82} -12.0000i q^{83} -2.00000 q^{84} +8.00000 q^{86} -8.00000 q^{91} -1.00000i q^{93} +12.0000 q^{94} +1.00000 q^{96} -10.0000i q^{97} +3.00000i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{4} + 2 q^{6} - 2 q^{9} + O(q^{10}) \) \( 2 q - 2 q^{4} + 2 q^{6} - 2 q^{9} - 4 q^{14} + 2 q^{16} - 16 q^{19} + 4 q^{21} - 2 q^{24} - 8 q^{26} + 2 q^{31} - 12 q^{34} + 2 q^{36} + 8 q^{39} - 12 q^{41} + 6 q^{49} + 12 q^{51} - 2 q^{54} + 4 q^{56} + 12 q^{59} + 4 q^{61} - 2 q^{64} - 12 q^{71} + 8 q^{74} + 16 q^{76} - 16 q^{79} + 2 q^{81} - 4 q^{84} + 16 q^{86} - 16 q^{91} + 24 q^{94} + 2 q^{96} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4650\mathbb{Z}\right)^\times\).

\(n\) \(1801\) \(2977\) \(3101\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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
\(11\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) 1.00000i 0.288675i
\(13\) 4.00000i 1.10940i 0.832050 + 0.554700i \(0.187167\pi\)
−0.832050 + 0.554700i \(0.812833\pi\)
\(14\) −2.00000 −0.534522
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 6.00000i 1.45521i 0.685994 + 0.727607i \(0.259367\pi\)
−0.685994 + 0.727607i \(0.740633\pi\)
\(18\) − 1.00000i − 0.235702i
\(19\) −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
\(22\) 0 0
\(23\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(24\) −1.00000 −0.204124
\(25\) 0 0
\(26\) −4.00000 −0.784465
\(27\) 1.00000i 0.192450i
\(28\) − 2.00000i − 0.377964i
\(29\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(30\) 0 0
\(31\) 1.00000 0.179605
\(32\) 1.00000i 0.176777i
\(33\) 0 0
\(34\) −6.00000 −1.02899
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) − 4.00000i − 0.657596i −0.944400 0.328798i \(-0.893356\pi\)
0.944400 0.328798i \(-0.106644\pi\)
\(38\) − 8.00000i − 1.29777i
\(39\) 4.00000 0.640513
\(40\) 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
\(43\) − 8.00000i − 1.21999i −0.792406 0.609994i \(-0.791172\pi\)
0.792406 0.609994i \(-0.208828\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) − 12.0000i − 1.75038i −0.483779 0.875190i \(-0.660736\pi\)
0.483779 0.875190i \(-0.339264\pi\)
\(48\) − 1.00000i − 0.144338i
\(49\) 3.00000 0.428571
\(50\) 0 0
\(51\) 6.00000 0.840168
\(52\) − 4.00000i − 0.554700i
\(53\) 6.00000i 0.824163i 0.911147 + 0.412082i \(0.135198\pi\)
−0.911147 + 0.412082i \(0.864802\pi\)
\(54\) −1.00000 −0.136083
\(55\) 0 0
\(56\) 2.00000 0.267261
\(57\) 8.00000i 1.05963i
\(58\) 0 0
\(59\) 6.00000 0.781133 0.390567 0.920575i \(-0.372279\pi\)
0.390567 + 0.920575i \(0.372279\pi\)
\(60\) 0 0
\(61\) 2.00000 0.256074 0.128037 0.991769i \(-0.459132\pi\)
0.128037 + 0.991769i \(0.459132\pi\)
\(62\) 1.00000i 0.127000i
\(63\) − 2.00000i − 0.251976i
\(64\) −1.00000 −0.125000
\(65\) 0 0
\(66\) 0 0
\(67\) 2.00000i 0.244339i 0.992509 + 0.122169i \(0.0389851\pi\)
−0.992509 + 0.122169i \(0.961015\pi\)
\(68\) − 6.00000i − 0.727607i
\(69\) 0 0
\(70\) 0 0
\(71\) −6.00000 −0.712069 −0.356034 0.934473i \(-0.615871\pi\)
−0.356034 + 0.934473i \(0.615871\pi\)
\(72\) 1.00000i 0.117851i
\(73\) − 8.00000i − 0.936329i −0.883641 0.468165i \(-0.844915\pi\)
0.883641 0.468165i \(-0.155085\pi\)
\(74\) 4.00000 0.464991
\(75\) 0 0
\(76\) 8.00000 0.917663
\(77\) 0 0
\(78\) 4.00000i 0.452911i
\(79\) −8.00000 −0.900070 −0.450035 0.893011i \(-0.648589\pi\)
−0.450035 + 0.893011i \(0.648589\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) − 6.00000i − 0.662589i
\(83\) − 12.0000i − 1.31717i −0.752506 0.658586i \(-0.771155\pi\)
0.752506 0.658586i \(-0.228845\pi\)
\(84\) −2.00000 −0.218218
\(85\) 0 0
\(86\) 8.00000 0.862662
\(87\) 0 0
\(88\) 0 0
\(89\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(90\) 0 0
\(91\) −8.00000 −0.838628
\(92\) 0 0
\(93\) − 1.00000i − 0.103695i
\(94\) 12.0000 1.23771
\(95\) 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
\(99\) 0 0
\(100\) 0 0
\(101\) −6.00000 −0.597022 −0.298511 0.954406i \(-0.596490\pi\)
−0.298511 + 0.954406i \(0.596490\pi\)
\(102\) 6.00000i 0.594089i
\(103\) 10.0000i 0.985329i 0.870219 + 0.492665i \(0.163977\pi\)
−0.870219 + 0.492665i \(0.836023\pi\)
\(104\) 4.00000 0.392232
\(105\) 0 0
\(106\) −6.00000 −0.582772
\(107\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(108\) − 1.00000i − 0.0962250i
\(109\) −2.00000 −0.191565 −0.0957826 0.995402i \(-0.530535\pi\)
−0.0957826 + 0.995402i \(0.530535\pi\)
\(110\) 0 0
\(111\) −4.00000 −0.379663
\(112\) 2.00000i 0.188982i
\(113\) 18.0000i 1.69330i 0.532152 + 0.846649i \(0.321383\pi\)
−0.532152 + 0.846649i \(0.678617\pi\)
\(114\) −8.00000 −0.749269
\(115\) 0 0
\(116\) 0 0
\(117\) − 4.00000i − 0.369800i
\(118\) 6.00000i 0.552345i
\(119\) −12.0000 −1.10004
\(120\) 0 0
\(121\) −11.0000 −1.00000
\(122\) 2.00000i 0.181071i
\(123\) 6.00000i 0.541002i
\(124\) −1.00000 −0.0898027
\(125\) 0 0
\(126\) 2.00000 0.178174
\(127\) − 16.0000i − 1.41977i −0.704317 0.709885i \(-0.748747\pi\)
0.704317 0.709885i \(-0.251253\pi\)
\(128\) − 1.00000i − 0.0883883i
\(129\) −8.00000 −0.704361
\(130\) 0 0
\(131\) −6.00000 −0.524222 −0.262111 0.965038i \(-0.584419\pi\)
−0.262111 + 0.965038i \(0.584419\pi\)
\(132\) 0 0
\(133\) − 16.0000i − 1.38738i
\(134\) −2.00000 −0.172774
\(135\) 0 0
\(136\) 6.00000 0.514496
\(137\) − 6.00000i − 0.512615i −0.966595 0.256307i \(-0.917494\pi\)
0.966595 0.256307i \(-0.0825059\pi\)
\(138\) 0 0
\(139\) 4.00000 0.339276 0.169638 0.985506i \(-0.445740\pi\)
0.169638 + 0.985506i \(0.445740\pi\)
\(140\) 0 0
\(141\) −12.0000 −1.01058
\(142\) − 6.00000i − 0.503509i
\(143\) 0 0
\(144\) −1.00000 −0.0833333
\(145\) 0 0
\(146\) 8.00000 0.662085
\(147\) − 3.00000i − 0.247436i
\(148\) 4.00000i 0.328798i
\(149\) −18.0000 −1.47462 −0.737309 0.675556i \(-0.763904\pi\)
−0.737309 + 0.675556i \(0.763904\pi\)
\(150\) 0 0
\(151\) 8.00000 0.651031 0.325515 0.945537i \(-0.394462\pi\)
0.325515 + 0.945537i \(0.394462\pi\)
\(152\) 8.00000i 0.648886i
\(153\) − 6.00000i − 0.485071i
\(154\) 0 0
\(155\) 0 0
\(156\) −4.00000 −0.320256
\(157\) 14.0000i 1.11732i 0.829396 + 0.558661i \(0.188685\pi\)
−0.829396 + 0.558661i \(0.811315\pi\)
\(158\) − 8.00000i − 0.636446i
\(159\) 6.00000 0.475831
\(160\) 0 0
\(161\) 0 0
\(162\) 1.00000i 0.0785674i
\(163\) − 2.00000i − 0.156652i −0.996928 0.0783260i \(-0.975042\pi\)
0.996928 0.0783260i \(-0.0249575\pi\)
\(164\) 6.00000 0.468521
\(165\) 0 0
\(166\) 12.0000 0.931381
\(167\) − 24.0000i − 1.85718i −0.371113 0.928588i \(-0.621024\pi\)
0.371113 0.928588i \(-0.378976\pi\)
\(168\) − 2.00000i − 0.154303i
\(169\) −3.00000 −0.230769
\(170\) 0 0
\(171\) 8.00000 0.611775
\(172\) 8.00000i 0.609994i
\(173\) 6.00000i 0.456172i 0.973641 + 0.228086i \(0.0732467\pi\)
−0.973641 + 0.228086i \(0.926753\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) − 6.00000i − 0.450988i
\(178\) 0 0
\(179\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(180\) 0 0
\(181\) 2.00000 0.148659 0.0743294 0.997234i \(-0.476318\pi\)
0.0743294 + 0.997234i \(0.476318\pi\)
\(182\) − 8.00000i − 0.592999i
\(183\) − 2.00000i − 0.147844i
\(184\) 0 0
\(185\) 0 0
\(186\) 1.00000 0.0733236
\(187\) 0 0
\(188\) 12.0000i 0.875190i
\(189\) −2.00000 −0.145479
\(190\) 0 0
\(191\) −18.0000 −1.30243 −0.651217 0.758891i \(-0.725741\pi\)
−0.651217 + 0.758891i \(0.725741\pi\)
\(192\) 1.00000i 0.0721688i
\(193\) − 14.0000i − 1.00774i −0.863779 0.503871i \(-0.831909\pi\)
0.863779 0.503871i \(-0.168091\pi\)
\(194\) 10.0000 0.717958
\(195\) 0 0
\(196\) −3.00000 −0.214286
\(197\) 6.00000i 0.427482i 0.976890 + 0.213741i \(0.0685649\pi\)
−0.976890 + 0.213741i \(0.931435\pi\)
\(198\) 0 0
\(199\) −8.00000 −0.567105 −0.283552 0.958957i \(-0.591513\pi\)
−0.283552 + 0.958957i \(0.591513\pi\)
\(200\) 0 0
\(201\) 2.00000 0.141069
\(202\) − 6.00000i − 0.422159i
\(203\) 0 0
\(204\) −6.00000 −0.420084
\(205\) 0 0
\(206\) −10.0000 −0.696733
\(207\) 0 0
\(208\) 4.00000i 0.277350i
\(209\) 0 0
\(210\) 0 0
\(211\) 8.00000 0.550743 0.275371 0.961338i \(-0.411199\pi\)
0.275371 + 0.961338i \(0.411199\pi\)
\(212\) − 6.00000i − 0.412082i
\(213\) 6.00000i 0.411113i
\(214\) 0 0
\(215\) 0 0
\(216\) 1.00000 0.0680414
\(217\) 2.00000i 0.135769i
\(218\) − 2.00000i − 0.135457i
\(219\) −8.00000 −0.540590
\(220\) 0 0
\(221\) −24.0000 −1.61441
\(222\) − 4.00000i − 0.268462i
\(223\) 28.0000i 1.87502i 0.347960 + 0.937509i \(0.386874\pi\)
−0.347960 + 0.937509i \(0.613126\pi\)
\(224\) −2.00000 −0.133631
\(225\) 0 0
\(226\) −18.0000 −1.19734
\(227\) 12.0000i 0.796468i 0.917284 + 0.398234i \(0.130377\pi\)
−0.917284 + 0.398234i \(0.869623\pi\)
\(228\) − 8.00000i − 0.529813i
\(229\) −14.0000 −0.925146 −0.462573 0.886581i \(-0.653074\pi\)
−0.462573 + 0.886581i \(0.653074\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) − 18.0000i − 1.17922i −0.807688 0.589610i \(-0.799282\pi\)
0.807688 0.589610i \(-0.200718\pi\)
\(234\) 4.00000 0.261488
\(235\) 0 0
\(236\) −6.00000 −0.390567
\(237\) 8.00000i 0.519656i
\(238\) − 12.0000i − 0.777844i
\(239\) −12.0000 −0.776215 −0.388108 0.921614i \(-0.626871\pi\)
−0.388108 + 0.921614i \(0.626871\pi\)
\(240\) 0 0
\(241\) 2.00000 0.128831 0.0644157 0.997923i \(-0.479482\pi\)
0.0644157 + 0.997923i \(0.479482\pi\)
\(242\) − 11.0000i − 0.707107i
\(243\) − 1.00000i − 0.0641500i
\(244\) −2.00000 −0.128037
\(245\) 0 0
\(246\) −6.00000 −0.382546
\(247\) − 32.0000i − 2.03611i
\(248\) − 1.00000i − 0.0635001i
\(249\) −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
\(253\) 0 0
\(254\) 16.0000 1.00393
\(255\) 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
\(259\) 8.00000 0.497096
\(260\) 0 0
\(261\) 0 0
\(262\) − 6.00000i − 0.370681i
\(263\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 16.0000 0.981023
\(267\) 0 0
\(268\) − 2.00000i − 0.122169i
\(269\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(270\) 0 0
\(271\) 32.0000 1.94386 0.971931 0.235267i \(-0.0755965\pi\)
0.971931 + 0.235267i \(0.0755965\pi\)
\(272\) 6.00000i 0.363803i
\(273\) 8.00000i 0.484182i
\(274\) 6.00000 0.362473
\(275\) 0 0
\(276\) 0 0
\(277\) 8.00000i 0.480673i 0.970690 + 0.240337i \(0.0772579\pi\)
−0.970690 + 0.240337i \(0.922742\pi\)
\(278\) 4.00000i 0.239904i
\(279\) −1.00000 −0.0598684
\(280\) 0 0
\(281\) −18.0000 −1.07379 −0.536895 0.843649i \(-0.680403\pi\)
−0.536895 + 0.843649i \(0.680403\pi\)
\(282\) − 12.0000i − 0.714590i
\(283\) 22.0000i 1.30776i 0.756596 + 0.653882i \(0.226861\pi\)
−0.756596 + 0.653882i \(0.773139\pi\)
\(284\) 6.00000 0.356034
\(285\) 0 0
\(286\) 0 0
\(287\) − 12.0000i − 0.708338i
\(288\) − 1.00000i − 0.0589256i
\(289\) −19.0000 −1.11765
\(290\) 0 0
\(291\) −10.0000 −0.586210
\(292\) 8.00000i 0.468165i
\(293\) − 6.00000i − 0.350524i −0.984522 0.175262i \(-0.943923\pi\)
0.984522 0.175262i \(-0.0560772\pi\)
\(294\) 3.00000 0.174964
\(295\) 0 0
\(296\) −4.00000 −0.232495
\(297\) 0 0
\(298\) − 18.0000i − 1.04271i
\(299\) 0 0
\(300\) 0 0
\(301\) 16.0000 0.922225
\(302\) 8.00000i 0.460348i
\(303\) 6.00000i 0.344691i
\(304\) −8.00000 −0.458831
\(305\) 0 0
\(306\) 6.00000 0.342997
\(307\) − 34.0000i − 1.94048i −0.242140 0.970241i \(-0.577849\pi\)
0.242140 0.970241i \(-0.422151\pi\)
\(308\) 0 0
\(309\) 10.0000 0.568880
\(310\) 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
\(313\) 16.0000i 0.904373i 0.891923 + 0.452187i \(0.149356\pi\)
−0.891923 + 0.452187i \(0.850644\pi\)
\(314\) −14.0000 −0.790066
\(315\) 0 0
\(316\) 8.00000 0.450035
\(317\) 6.00000i 0.336994i 0.985702 + 0.168497i \(0.0538913\pi\)
−0.985702 + 0.168497i \(0.946109\pi\)
\(318\) 6.00000i 0.336463i
\(319\) 0 0
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) − 48.0000i − 2.67079i
\(324\) −1.00000 −0.0555556
\(325\) 0 0
\(326\) 2.00000 0.110770
\(327\) 2.00000i 0.110600i
\(328\) 6.00000i 0.331295i
\(329\) 24.0000 1.32316
\(330\) 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
\(333\) 4.00000i 0.219199i
\(334\) 24.0000 1.31322
\(335\) 0 0
\(336\) 2.00000 0.109109
\(337\) 20.0000i 1.08947i 0.838608 + 0.544735i \(0.183370\pi\)
−0.838608 + 0.544735i \(0.816630\pi\)
\(338\) − 3.00000i − 0.163178i
\(339\) 18.0000 0.977626
\(340\) 0 0
\(341\) 0 0
\(342\) 8.00000i 0.432590i
\(343\) 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 \(-0.582883\pi\)
0.257454 0.966291i \(-0.417117\pi\)
\(348\) 0 0
\(349\) −2.00000 −0.107058 −0.0535288 0.998566i \(-0.517047\pi\)
−0.0535288 + 0.998566i \(0.517047\pi\)
\(350\) 0 0
\(351\) −4.00000 −0.213504
\(352\) 0 0
\(353\) − 30.0000i − 1.59674i −0.602168 0.798369i \(-0.705696\pi\)
0.602168 0.798369i \(-0.294304\pi\)
\(354\) 6.00000 0.318896
\(355\) 0 0
\(356\) 0 0
\(357\) 12.0000i 0.635107i
\(358\) 0 0
\(359\) −30.0000 −1.58334 −0.791670 0.610949i \(-0.790788\pi\)
−0.791670 + 0.610949i \(0.790788\pi\)
\(360\) 0 0
\(361\) 45.0000 2.36842
\(362\) 2.00000i 0.105118i
\(363\) 11.0000i 0.577350i
\(364\) 8.00000 0.419314
\(365\) 0 0
\(366\) 2.00000 0.104542
\(367\) 20.0000i 1.04399i 0.852948 + 0.521996i \(0.174812\pi\)
−0.852948 + 0.521996i \(0.825188\pi\)
\(368\) 0 0
\(369\) 6.00000 0.312348
\(370\) 0 0
\(371\) −12.0000 −0.623009
\(372\) 1.00000i 0.0518476i
\(373\) − 38.0000i − 1.96757i −0.179364 0.983783i \(-0.557404\pi\)
0.179364 0.983783i \(-0.442596\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) −12.0000 −0.618853
\(377\) 0 0
\(378\) − 2.00000i − 0.102869i
\(379\) −32.0000 −1.64373 −0.821865 0.569683i \(-0.807066\pi\)
−0.821865 + 0.569683i \(0.807066\pi\)
\(380\) 0 0
\(381\) −16.0000 −0.819705
\(382\) − 18.0000i − 0.920960i
\(383\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(384\) −1.00000 −0.0510310
\(385\) 0 0
\(386\) 14.0000 0.712581
\(387\) 8.00000i 0.406663i
\(388\) 10.0000i 0.507673i
\(389\) −24.0000 −1.21685 −0.608424 0.793612i \(-0.708198\pi\)
−0.608424 + 0.793612i \(0.708198\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) − 3.00000i − 0.151523i
\(393\) 6.00000i 0.302660i
\(394\) −6.00000 −0.302276
\(395\) 0 0
\(396\) 0 0
\(397\) 2.00000i 0.100377i 0.998740 + 0.0501886i \(0.0159822\pi\)
−0.998740 + 0.0501886i \(0.984018\pi\)
\(398\) − 8.00000i − 0.401004i
\(399\) −16.0000 −0.801002
\(400\) 0 0
\(401\) 36.0000 1.79775 0.898877 0.438201i \(-0.144384\pi\)
0.898877 + 0.438201i \(0.144384\pi\)
\(402\) 2.00000i 0.0997509i
\(403\) 4.00000i 0.199254i
\(404\) 6.00000 0.298511
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) − 6.00000i − 0.297044i
\(409\) −14.0000 −0.692255 −0.346128 0.938187i \(-0.612504\pi\)
−0.346128 + 0.938187i \(0.612504\pi\)
\(410\) 0 0
\(411\) −6.00000 −0.295958
\(412\) − 10.0000i − 0.492665i
\(413\) 12.0000i 0.590481i
\(414\) 0 0
\(415\) 0 0
\(416\) −4.00000 −0.196116
\(417\) − 4.00000i − 0.195881i
\(418\) 0 0
\(419\) −30.0000 −1.46560 −0.732798 0.680446i \(-0.761786\pi\)
−0.732798 + 0.680446i \(0.761786\pi\)
\(420\) 0 0
\(421\) 26.0000 1.26716 0.633581 0.773676i \(-0.281584\pi\)
0.633581 + 0.773676i \(0.281584\pi\)
\(422\) 8.00000i 0.389434i
\(423\) 12.0000i 0.583460i
\(424\) 6.00000 0.291386
\(425\) 0 0
\(426\) −6.00000 −0.290701
\(427\) 4.00000i 0.193574i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 6.00000 0.289010 0.144505 0.989504i \(-0.453841\pi\)
0.144505 + 0.989504i \(0.453841\pi\)
\(432\) 1.00000i 0.0481125i
\(433\) 16.0000i 0.768911i 0.923144 + 0.384455i \(0.125611\pi\)
−0.923144 + 0.384455i \(0.874389\pi\)
\(434\) −2.00000 −0.0960031
\(435\) 0 0
\(436\) 2.00000 0.0957826
\(437\) 0 0
\(438\) − 8.00000i − 0.382255i
\(439\) −8.00000 −0.381819 −0.190910 0.981608i \(-0.561144\pi\)
−0.190910 + 0.981608i \(0.561144\pi\)
\(440\) 0 0
\(441\) −3.00000 −0.142857
\(442\) − 24.0000i − 1.14156i
\(443\) 36.0000i 1.71041i 0.518289 + 0.855206i \(0.326569\pi\)
−0.518289 + 0.855206i \(0.673431\pi\)
\(444\) 4.00000 0.189832
\(445\) 0 0
\(446\) −28.0000 −1.32584
\(447\) 18.0000i 0.851371i
\(448\) − 2.00000i − 0.0944911i
\(449\) −12.0000 −0.566315 −0.283158 0.959073i \(-0.591382\pi\)
−0.283158 + 0.959073i \(0.591382\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) − 18.0000i − 0.846649i
\(453\) − 8.00000i − 0.375873i
\(454\) −12.0000 −0.563188
\(455\) 0 0
\(456\) 8.00000 0.374634
\(457\) 8.00000i 0.374224i 0.982339 + 0.187112i \(0.0599128\pi\)
−0.982339 + 0.187112i \(0.940087\pi\)
\(458\) − 14.0000i − 0.654177i
\(459\) −6.00000 −0.280056
\(460\) 0 0
\(461\) −36.0000 −1.67669 −0.838344 0.545142i \(-0.816476\pi\)
−0.838344 + 0.545142i \(0.816476\pi\)
\(462\) 0 0
\(463\) 16.0000i 0.743583i 0.928316 + 0.371792i \(0.121256\pi\)
−0.928316 + 0.371792i \(0.878744\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 18.0000 0.833834
\(467\) 24.0000i 1.11059i 0.831654 + 0.555294i \(0.187394\pi\)
−0.831654 + 0.555294i \(0.812606\pi\)
\(468\) 4.00000i 0.184900i
\(469\) −4.00000 −0.184703
\(470\) 0 0
\(471\) 14.0000 0.645086
\(472\) − 6.00000i − 0.276172i
\(473\) 0 0
\(474\) −8.00000 −0.367452
\(475\) 0 0
\(476\) 12.0000 0.550019
\(477\) − 6.00000i − 0.274721i
\(478\) − 12.0000i − 0.548867i
\(479\) 6.00000 0.274147 0.137073 0.990561i \(-0.456230\pi\)
0.137073 + 0.990561i \(0.456230\pi\)
\(480\) 0 0
\(481\) 16.0000 0.729537
\(482\) 2.00000i 0.0910975i
\(483\) 0 0
\(484\) 11.0000 0.500000
\(485\) 0 0
\(486\) 1.00000 0.0453609
\(487\) − 40.0000i − 1.81257i −0.422664 0.906287i \(-0.638905\pi\)
0.422664 0.906287i \(-0.361095\pi\)
\(488\) − 2.00000i − 0.0905357i
\(489\) −2.00000 −0.0904431
\(490\) 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
\(493\) 0 0
\(494\) 32.0000 1.43975
\(495\) 0 0
\(496\) 1.00000 0.0449013
\(497\) − 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
\(501\) −24.0000 −1.07224
\(502\) 0 0
\(503\) − 36.0000i − 1.60516i −0.596544 0.802580i \(-0.703460\pi\)
0.596544 0.802580i \(-0.296540\pi\)
\(504\) −2.00000 −0.0890871
\(505\) 0 0
\(506\) 0 0
\(507\) 3.00000i 0.133235i
\(508\) 16.0000i 0.709885i
\(509\) −12.0000 −0.531891 −0.265945 0.963988i \(-0.585684\pi\)
−0.265945 + 0.963988i \(0.585684\pi\)
\(510\) 0 0
\(511\) 16.0000 0.707798
\(512\) 1.00000i 0.0441942i
\(513\) − 8.00000i − 0.353209i
\(514\) 30.0000 1.32324
\(515\) 0 0
\(516\) 8.00000 0.352180
\(517\) 0 0
\(518\) 8.00000i 0.351500i
\(519\) 6.00000 0.263371
\(520\) 0 0
\(521\) −6.00000 −0.262865 −0.131432 0.991325i \(-0.541958\pi\)
−0.131432 + 0.991325i \(0.541958\pi\)
\(522\) 0 0
\(523\) 40.0000i 1.74908i 0.484955 + 0.874539i \(0.338836\pi\)
−0.484955 + 0.874539i \(0.661164\pi\)
\(524\) 6.00000 0.262111
\(525\) 0 0
\(526\) 0 0
\(527\) 6.00000i 0.261364i
\(528\) 0 0
\(529\) 23.0000 1.00000
\(530\) 0 0
\(531\) −6.00000 −0.260378
\(532\) 16.0000i 0.693688i
\(533\) − 24.0000i − 1.03956i
\(534\) 0 0
\(535\) 0 0
\(536\) 2.00000 0.0863868
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 2.00000 0.0859867 0.0429934 0.999075i \(-0.486311\pi\)
0.0429934 + 0.999075i \(0.486311\pi\)
\(542\) 32.0000i 1.37452i
\(543\) − 2.00000i − 0.0858282i
\(544\) −6.00000 −0.257248
\(545\) 0 0
\(546\) −8.00000 −0.342368
\(547\) 2.00000i 0.0855138i 0.999086 + 0.0427569i \(0.0136141\pi\)
−0.999086 + 0.0427569i \(0.986386\pi\)
\(548\) 6.00000i 0.256307i
\(549\) −2.00000 −0.0853579
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) − 16.0000i − 0.680389i
\(554\) −8.00000 −0.339887
\(555\) 0 0
\(556\) −4.00000 −0.169638
\(557\) 42.0000i 1.77960i 0.456354 + 0.889799i \(0.349155\pi\)
−0.456354 + 0.889799i \(0.650845\pi\)
\(558\) − 1.00000i − 0.0423334i
\(559\) 32.0000 1.35346
\(560\) 0 0
\(561\) 0 0
\(562\) − 18.0000i − 0.759284i
\(563\) 24.0000i 1.01148i 0.862686 + 0.505740i \(0.168780\pi\)
−0.862686 + 0.505740i \(0.831220\pi\)
\(564\) 12.0000 0.505291
\(565\) 0 0
\(566\) −22.0000 −0.924729
\(567\) 2.00000i 0.0839921i
\(568\) 6.00000i 0.251754i
\(569\) −36.0000 −1.50920 −0.754599 0.656186i \(-0.772169\pi\)
−0.754599 + 0.656186i \(0.772169\pi\)
\(570\) 0 0
\(571\) 20.0000 0.836974 0.418487 0.908223i \(-0.362561\pi\)
0.418487 + 0.908223i \(0.362561\pi\)
\(572\) 0 0
\(573\) 18.0000i 0.751961i
\(574\) 12.0000 0.500870
\(575\) 0 0
\(576\) 1.00000 0.0416667
\(577\) − 22.0000i − 0.915872i −0.888985 0.457936i \(-0.848589\pi\)
0.888985 0.457936i \(-0.151411\pi\)
\(578\) − 19.0000i − 0.790296i
\(579\) −14.0000 −0.581820
\(580\) 0 0
\(581\) 24.0000 0.995688
\(582\) − 10.0000i − 0.414513i
\(583\) 0 0
\(584\) −8.00000 −0.331042
\(585\) 0 0
\(586\) 6.00000 0.247858
\(587\) − 12.0000i − 0.495293i −0.968850 0.247647i \(-0.920343\pi\)
0.968850 0.247647i \(-0.0796572\pi\)
\(588\) 3.00000i 0.123718i
\(589\) −8.00000 −0.329634
\(590\) 0 0
\(591\) 6.00000 0.246807
\(592\) − 4.00000i − 0.164399i
\(593\) − 6.00000i − 0.246390i −0.992382 0.123195i \(-0.960686\pi\)
0.992382 0.123195i \(-0.0393141\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 18.0000 0.737309
\(597\) 8.00000i 0.327418i
\(598\) 0 0
\(599\) −6.00000 −0.245153 −0.122577 0.992459i \(-0.539116\pi\)
−0.122577 + 0.992459i \(0.539116\pi\)
\(600\) 0 0
\(601\) −10.0000 −0.407909 −0.203954 0.978980i \(-0.565379\pi\)
−0.203954 + 0.978980i \(0.565379\pi\)
\(602\) 16.0000i 0.652111i
\(603\) − 2.00000i − 0.0814463i
\(604\) −8.00000 −0.325515
\(605\) 0 0
\(606\) −6.00000 −0.243733
\(607\) − 34.0000i − 1.38002i −0.723801 0.690009i \(-0.757607\pi\)
0.723801 0.690009i \(-0.242393\pi\)
\(608\) − 8.00000i − 0.324443i
\(609\) 0 0
\(610\) 0 0
\(611\) 48.0000 1.94187
\(612\) 6.00000i 0.242536i
\(613\) 40.0000i 1.61558i 0.589467 + 0.807792i \(0.299338\pi\)
−0.589467 + 0.807792i \(0.700662\pi\)
\(614\) 34.0000 1.37213
\(615\) 0 0
\(616\) 0 0
\(617\) 6.00000i 0.241551i 0.992680 + 0.120775i \(0.0385381\pi\)
−0.992680 + 0.120775i \(0.961462\pi\)
\(618\) 10.0000i 0.402259i
\(619\) 4.00000 0.160774 0.0803868 0.996764i \(-0.474384\pi\)
0.0803868 + 0.996764i \(0.474384\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) − 18.0000i − 0.721734i
\(623\) 0 0
\(624\) 4.00000 0.160128
\(625\) 0 0
\(626\) −16.0000 −0.639489
\(627\) 0 0
\(628\) − 14.0000i − 0.558661i
\(629\) 24.0000 0.956943
\(630\) 0 0
\(631\) 8.00000 0.318475 0.159237 0.987240i \(-0.449096\pi\)
0.159237 + 0.987240i \(0.449096\pi\)
\(632\) 8.00000i 0.318223i
\(633\) − 8.00000i − 0.317971i
\(634\) −6.00000 −0.238290
\(635\) 0 0
\(636\) −6.00000 −0.237915
\(637\) 12.0000i 0.475457i
\(638\) 0 0
\(639\) 6.00000 0.237356
\(640\) 0 0
\(641\) −12.0000 −0.473972 −0.236986 0.971513i \(-0.576159\pi\)
−0.236986 + 0.971513i \(0.576159\pi\)
\(642\) 0 0
\(643\) − 32.0000i − 1.26196i −0.775800 0.630978i \(-0.782654\pi\)
0.775800 0.630978i \(-0.217346\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 48.0000 1.88853
\(647\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(648\) − 1.00000i − 0.0392837i
\(649\) 0 0
\(650\) 0 0
\(651\) 2.00000 0.0783862
\(652\) 2.00000i 0.0783260i
\(653\) − 6.00000i − 0.234798i −0.993085 0.117399i \(-0.962544\pi\)
0.993085 0.117399i \(-0.0374557\pi\)
\(654\) −2.00000 −0.0782062
\(655\) 0 0
\(656\) −6.00000 −0.234261
\(657\) 8.00000i 0.312110i
\(658\) 24.0000i 0.935617i
\(659\) −6.00000 −0.233727 −0.116863 0.993148i \(-0.537284\pi\)
−0.116863 + 0.993148i \(0.537284\pi\)
\(660\) 0 0
\(661\) −10.0000 −0.388955 −0.194477 0.980907i \(-0.562301\pi\)
−0.194477 + 0.980907i \(0.562301\pi\)
\(662\) − 4.00000i − 0.155464i
\(663\) 24.0000i 0.932083i
\(664\) −12.0000 −0.465690
\(665\) 0 0
\(666\) −4.00000 −0.154997
\(667\) 0 0
\(668\) 24.0000i 0.928588i
\(669\) 28.0000 1.08254
\(670\) 0 0
\(671\) 0 0
\(672\) 2.00000i 0.0771517i
\(673\) − 44.0000i − 1.69608i −0.529936 0.848038i \(-0.677784\pi\)
0.529936 0.848038i \(-0.322216\pi\)
\(674\) −20.0000 −0.770371
\(675\) 0 0
\(676\) 3.00000 0.115385
\(677\) 18.0000i 0.691796i 0.938272 + 0.345898i \(0.112426\pi\)
−0.938272 + 0.345898i \(0.887574\pi\)
\(678\) 18.0000i 0.691286i
\(679\) 20.0000 0.767530
\(680\) 0 0
\(681\) 12.0000 0.459841
\(682\) 0 0
\(683\) 36.0000i 1.37750i 0.724998 + 0.688751i \(0.241841\pi\)
−0.724998 + 0.688751i \(0.758159\pi\)
\(684\) −8.00000 −0.305888
\(685\) 0 0
\(686\) −20.0000 −0.763604
\(687\) 14.0000i 0.534133i
\(688\) − 8.00000i − 0.304997i
\(689\) −24.0000 −0.914327
\(690\) 0 0
\(691\) 44.0000 1.67384 0.836919 0.547326i \(-0.184354\pi\)
0.836919 + 0.547326i \(0.184354\pi\)
\(692\) − 6.00000i − 0.228086i
\(693\) 0 0
\(694\) 36.0000 1.36654
\(695\) 0 0
\(696\) 0 0
\(697\) − 36.0000i − 1.36360i
\(698\) − 2.00000i − 0.0757011i
\(699\) −18.0000 −0.680823
\(700\) 0 0
\(701\) 30.0000 1.13308 0.566542 0.824033i \(-0.308281\pi\)
0.566542 + 0.824033i \(0.308281\pi\)
\(702\) − 4.00000i − 0.150970i
\(703\) 32.0000i 1.20690i
\(704\) 0 0
\(705\) 0 0
\(706\) 30.0000 1.12906
\(707\) − 12.0000i − 0.451306i
\(708\) 6.00000i 0.225494i
\(709\) 10.0000 0.375558 0.187779 0.982211i \(-0.439871\pi\)
0.187779 + 0.982211i \(0.439871\pi\)
\(710\) 0 0
\(711\) 8.00000 0.300023
\(712\) 0 0
\(713\) 0 0
\(714\) −12.0000 −0.449089
\(715\) 0 0
\(716\) 0 0
\(717\) 12.0000i 0.448148i
\(718\) − 30.0000i − 1.11959i
\(719\) 24.0000 0.895049 0.447524 0.894272i \(-0.352306\pi\)
0.447524 + 0.894272i \(0.352306\pi\)
\(720\) 0 0
\(721\) −20.0000 −0.744839
\(722\) 45.0000i 1.67473i
\(723\) − 2.00000i − 0.0743808i
\(724\) −2.00000 −0.0743294
\(725\) 0 0
\(726\) −11.0000 −0.408248
\(727\) 26.0000i 0.964287i 0.876092 + 0.482143i \(0.160142\pi\)
−0.876092 + 0.482143i \(0.839858\pi\)
\(728\) 8.00000i 0.296500i
\(729\) −1.00000 −0.0370370
\(730\) 0 0
\(731\) 48.0000 1.77534
\(732\) 2.00000i 0.0739221i
\(733\) 22.0000i 0.812589i 0.913742 + 0.406294i \(0.133179\pi\)
−0.913742 + 0.406294i \(0.866821\pi\)
\(734\) −20.0000 −0.738213
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 6.00000i 0.220863i
\(739\) 52.0000 1.91285 0.956425 0.291977i \(-0.0943129\pi\)
0.956425 + 0.291977i \(0.0943129\pi\)
\(740\) 0 0
\(741\) −32.0000 −1.17555
\(742\) − 12.0000i − 0.440534i
\(743\) − 24.0000i − 0.880475i −0.897881 0.440237i \(-0.854894\pi\)
0.897881 0.440237i \(-0.145106\pi\)
\(744\) −1.00000 −0.0366618
\(745\) 0 0
\(746\) 38.0000 1.39128
\(747\) 12.0000i 0.439057i
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) −16.0000 −0.583848 −0.291924 0.956441i \(-0.594295\pi\)
−0.291924 + 0.956441i \(0.594295\pi\)
\(752\) − 12.0000i − 0.437595i
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 2.00000 0.0727393
\(757\) − 4.00000i − 0.145382i −0.997354 0.0726912i \(-0.976841\pi\)
0.997354 0.0726912i \(-0.0231588\pi\)
\(758\) − 32.0000i − 1.16229i
\(759\) 0 0
\(760\) 0 0
\(761\) 12.0000 0.435000 0.217500 0.976060i \(-0.430210\pi\)
0.217500 + 0.976060i \(0.430210\pi\)
\(762\) − 16.0000i − 0.579619i
\(763\) − 4.00000i − 0.144810i
\(764\) 18.0000 0.651217
\(765\) 0 0
\(766\) 0 0
\(767\) 24.0000i 0.866590i
\(768\) − 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
\(773\) 54.0000i 1.94225i 0.238581 + 0.971123i \(0.423318\pi\)
−0.238581 + 0.971123i \(0.576682\pi\)
\(774\) −8.00000 −0.287554
\(775\) 0 0
\(776\) −10.0000 −0.358979
\(777\) − 8.00000i − 0.286998i
\(778\) − 24.0000i − 0.860442i
\(779\) 48.0000 1.71978
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 3.00000 0.107143
\(785\) 0 0
\(786\) −6.00000 −0.214013
\(787\) 20.0000i 0.712923i 0.934310 + 0.356462i \(0.116017\pi\)
−0.934310 + 0.356462i \(0.883983\pi\)
\(788\) − 6.00000i − 0.213741i
\(789\) 0 0
\(790\) 0 0
\(791\) −36.0000 −1.28001
\(792\) 0 0
\(793\) 8.00000i 0.284088i
\(794\) −2.00000 −0.0709773
\(795\) 0 0
\(796\) 8.00000 0.283552
\(797\) 6.00000i 0.212531i 0.994338 + 0.106265i \(0.0338893\pi\)
−0.994338 + 0.106265i \(0.966111\pi\)
\(798\) − 16.0000i − 0.566394i
\(799\) 72.0000 2.54718
\(800\) 0 0
\(801\) 0 0
\(802\) 36.0000i 1.27120i
\(803\) 0 0
\(804\) −2.00000 −0.0705346
\(805\) 0 0
\(806\) −4.00000 −0.140894
\(807\) 0 0
\(808\) 6.00000i 0.211079i
\(809\) −36.0000 −1.26569 −0.632846 0.774277i \(-0.718114\pi\)
−0.632846 + 0.774277i \(0.718114\pi\)
\(810\) 0 0
\(811\) −16.0000 −0.561836 −0.280918 0.959732i \(-0.590639\pi\)
−0.280918 + 0.959732i \(0.590639\pi\)
\(812\) 0 0
\(813\) − 32.0000i − 1.12229i
\(814\) 0 0
\(815\) 0 0
\(816\) 6.00000 0.210042
\(817\) 64.0000i 2.23908i
\(818\) − 14.0000i − 0.489499i
\(819\) 8.00000 0.279543
\(820\) 0 0
\(821\) −24.0000 −0.837606 −0.418803 0.908077i \(-0.637550\pi\)
−0.418803 + 0.908077i \(0.637550\pi\)
\(822\) − 6.00000i − 0.209274i
\(823\) 16.0000i 0.557725i 0.960331 + 0.278862i \(0.0899574\pi\)
−0.960331 + 0.278862i \(0.910043\pi\)
\(824\) 10.0000 0.348367
\(825\) 0 0
\(826\) −12.0000 −0.417533
\(827\) − 12.0000i − 0.417281i −0.977992 0.208640i \(-0.933096\pi\)
0.977992 0.208640i \(-0.0669038\pi\)
\(828\) 0 0
\(829\) 22.0000 0.764092 0.382046 0.924143i \(-0.375220\pi\)
0.382046 + 0.924143i \(0.375220\pi\)
\(830\) 0 0
\(831\) 8.00000 0.277517
\(832\) − 4.00000i − 0.138675i
\(833\) 18.0000i 0.623663i
\(834\) 4.00000 0.138509
\(835\) 0 0
\(836\) 0 0
\(837\) 1.00000i 0.0345651i
\(838\) − 30.0000i − 1.03633i
\(839\) 18.0000 0.621429 0.310715 0.950503i \(-0.399432\pi\)
0.310715 + 0.950503i \(0.399432\pi\)
\(840\) 0 0
\(841\) −29.0000 −1.00000
\(842\) 26.0000i 0.896019i
\(843\) 18.0000i 0.619953i
\(844\) −8.00000 −0.275371
\(845\) 0 0
\(846\) −12.0000 −0.412568
\(847\) − 22.0000i − 0.755929i
\(848\) 6.00000i 0.206041i
\(849\) 22.0000 0.755038
\(850\) 0 0
\(851\) 0 0
\(852\) − 6.00000i − 0.205557i
\(853\) − 2.00000i − 0.0684787i −0.999414 0.0342393i \(-0.989099\pi\)
0.999414 0.0342393i \(-0.0109009\pi\)
\(854\) −4.00000 −0.136877
\(855\) 0 0
\(856\) 0 0
\(857\) 54.0000i 1.84460i 0.386469 + 0.922302i \(0.373695\pi\)
−0.386469 + 0.922302i \(0.626305\pi\)
\(858\) 0 0
\(859\) −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
\(862\) 6.00000i 0.204361i
\(863\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(864\) −1.00000 −0.0340207
\(865\) 0 0
\(866\) −16.0000 −0.543702
\(867\) 19.0000i 0.645274i
\(868\) − 2.00000i − 0.0678844i
\(869\) 0 0
\(870\) 0 0
\(871\) −8.00000 −0.271070
\(872\) 2.00000i 0.0677285i
\(873\) 10.0000i 0.338449i
\(874\) 0 0
\(875\) 0 0
\(876\) 8.00000 0.270295
\(877\) − 22.0000i − 0.742887i −0.928456 0.371444i \(-0.878863\pi\)
0.928456 0.371444i \(-0.121137\pi\)
\(878\) − 8.00000i − 0.269987i
\(879\) −6.00000 −0.202375
\(880\) 0 0
\(881\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(882\) − 3.00000i − 0.101015i
\(883\) − 44.0000i − 1.48072i −0.672212 0.740359i \(-0.734656\pi\)
0.672212 0.740359i \(-0.265344\pi\)
\(884\) 24.0000 0.807207
\(885\) 0 0
\(886\) −36.0000 −1.20944
\(887\) 24.0000i 0.805841i 0.915235 + 0.402921i \(0.132005\pi\)
−0.915235 + 0.402921i \(0.867995\pi\)
\(888\) 4.00000i 0.134231i
\(889\) 32.0000 1.07325
\(890\) 0 0
\(891\) 0 0
\(892\) − 28.0000i − 0.937509i
\(893\) 96.0000i 3.21252i
\(894\) −18.0000 −0.602010
\(895\) 0 0
\(896\) 2.00000 0.0668153
\(897\) 0 0
\(898\) − 12.0000i − 0.400445i
\(899\) 0 0
\(900\) 0 0
\(901\) −36.0000 −1.19933
\(902\) 0 0
\(903\) − 16.0000i − 0.532447i
\(904\) 18.0000 0.598671
\(905\) 0 0
\(906\) 8.00000 0.265782
\(907\) − 10.0000i − 0.332045i −0.986122 0.166022i \(-0.946908\pi\)
0.986122 0.166022i \(-0.0530924\pi\)
\(908\) − 12.0000i − 0.398234i
\(909\) 6.00000 0.199007
\(910\) 0 0
\(911\) 24.0000 0.795155 0.397578 0.917568i \(-0.369851\pi\)
0.397578 + 0.917568i \(0.369851\pi\)
\(912\) 8.00000i 0.264906i
\(913\) 0 0
\(914\) −8.00000 −0.264616
\(915\) 0 0
\(916\) 14.0000 0.462573
\(917\) − 12.0000i − 0.396275i
\(918\) − 6.00000i − 0.198030i
\(919\) 52.0000 1.71532 0.857661 0.514216i \(-0.171917\pi\)
0.857661 + 0.514216i \(0.171917\pi\)
\(920\) 0 0
\(921\) −34.0000 −1.12034
\(922\) − 36.0000i − 1.18560i
\(923\) − 24.0000i − 0.789970i
\(924\) 0 0
\(925\) 0 0
\(926\) −16.0000 −0.525793
\(927\) − 10.0000i − 0.328443i
\(928\) 0 0
\(929\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(930\) 0 0
\(931\) −24.0000 −0.786568
\(932\) 18.0000i 0.589610i
\(933\) 18.0000i 0.589294i
\(934\) −24.0000 −0.785304
\(935\) 0 0
\(936\) −4.00000 −0.130744
\(937\) 38.0000i 1.24141i 0.784046 + 0.620703i \(0.213153\pi\)
−0.784046 + 0.620703i \(0.786847\pi\)
\(938\) − 4.00000i − 0.130605i
\(939\) 16.0000 0.522140
\(940\) 0 0
\(941\) −48.0000 −1.56476 −0.782378 0.622804i \(-0.785993\pi\)
−0.782378 + 0.622804i \(0.785993\pi\)
\(942\) 14.0000i 0.456145i
\(943\) 0 0
\(944\) 6.00000 0.195283
\(945\) 0 0
\(946\) 0 0
\(947\) 12.0000i 0.389948i 0.980808 + 0.194974i \(0.0624622\pi\)
−0.980808 + 0.194974i \(0.937538\pi\)
\(948\) − 8.00000i − 0.259828i
\(949\) 32.0000 1.03876
\(950\) 0 0
\(951\) 6.00000 0.194563
\(952\) 12.0000i 0.388922i
\(953\) − 18.0000i − 0.583077i −0.956559 0.291539i \(-0.905833\pi\)
0.956559 0.291539i \(-0.0941672\pi\)
\(954\) 6.00000 0.194257
\(955\) 0 0
\(956\) 12.0000 0.388108
\(957\) 0 0
\(958\) 6.00000i 0.193851i
\(959\) 12.0000 0.387500
\(960\) 0 0
\(961\) 1.00000 0.0322581
\(962\) 16.0000i 0.515861i
\(963\) 0 0
\(964\) −2.00000 −0.0644157
\(965\) 0 0
\(966\) 0 0
\(967\) − 4.00000i − 0.128631i −0.997930 0.0643157i \(-0.979514\pi\)
0.997930 0.0643157i \(-0.0204865\pi\)
\(968\) 11.0000i 0.353553i
\(969\) −48.0000 −1.54198
\(970\) 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
\(973\) 8.00000i 0.256468i
\(974\) 40.0000 1.28168
\(975\) 0 0
\(976\) 2.00000 0.0640184
\(977\) − 18.0000i − 0.575871i −0.957650 0.287936i \(-0.907031\pi\)
0.957650 0.287936i \(-0.0929689\pi\)
\(978\) − 2.00000i − 0.0639529i
\(979\) 0 0
\(980\) 0 0
\(981\) 2.00000 0.0638551
\(982\) 12.0000i 0.382935i
\(983\) − 24.0000i − 0.765481i −0.923856 0.382741i \(-0.874980\pi\)
0.923856 0.382741i \(-0.125020\pi\)
\(984\) 6.00000 0.191273
\(985\) 0 0
\(986\) 0 0
\(987\) − 24.0000i − 0.763928i
\(988\) 32.0000i 1.01806i
\(989\) 0 0
\(990\) 0 0
\(991\) −16.0000 −0.508257 −0.254128 0.967170i \(-0.581789\pi\)
−0.254128 + 0.967170i \(0.581789\pi\)
\(992\) 1.00000i 0.0317500i
\(993\) 4.00000i 0.126936i
\(994\) 12.0000 0.380617
\(995\) 0 0
\(996\) 12.0000 0.380235
\(997\) − 10.0000i − 0.316703i −0.987383 0.158352i \(-0.949382\pi\)
0.987383 0.158352i \(-0.0506179\pi\)
\(998\) 4.00000i 0.126618i
\(999\) 4.00000 0.126554
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 4650.2.d.u.3349.2 2
5.2 odd 4 4650.2.a.d.1.1 1
5.3 odd 4 930.2.a.n.1.1 1
5.4 even 2 inner 4650.2.d.u.3349.1 2
15.8 even 4 2790.2.a.k.1.1 1
20.3 even 4 7440.2.a.b.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
930.2.a.n.1.1 1 5.3 odd 4
2790.2.a.k.1.1 1 15.8 even 4
4650.2.a.d.1.1 1 5.2 odd 4
4650.2.d.u.3349.1 2 5.4 even 2 inner
4650.2.d.u.3349.2 2 1.1 even 1 trivial
7440.2.a.b.1.1 1 20.3 even 4