Properties

Label 4650.2.d.bd.3349.4
Level $4650$
Weight $2$
Character 4650.3349
Analytic conductor $37.130$
Analytic rank $0$
Dimension $4$
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: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{17})\)
Defining polynomial: \( x^{4} + 9x^{2} + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 930)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 3349.4
Root \(2.56155i\) of defining polynomial
Character \(\chi\) \(=\) 4650.3349
Dual form 4650.2.d.bd.3349.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{6} +2.56155i q^{7} -1.00000i q^{8} -1.00000 q^{9} +O(q^{10})\) \(q+1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{6} +2.56155i q^{7} -1.00000i q^{8} -1.00000 q^{9} +2.56155 q^{11} -1.00000i q^{12} -2.00000i q^{13} -2.56155 q^{14} +1.00000 q^{16} -3.12311i q^{17} -1.00000i q^{18} +7.68466 q^{19} -2.56155 q^{21} +2.56155i q^{22} -1.43845i q^{23} +1.00000 q^{24} +2.00000 q^{26} -1.00000i q^{27} -2.56155i q^{28} -7.12311 q^{29} +1.00000 q^{31} +1.00000i q^{32} +2.56155i q^{33} +3.12311 q^{34} +1.00000 q^{36} -3.12311i q^{37} +7.68466i q^{38} +2.00000 q^{39} +7.12311 q^{41} -2.56155i q^{42} -12.8078i q^{43} -2.56155 q^{44} +1.43845 q^{46} +5.12311i q^{47} +1.00000i q^{48} +0.438447 q^{49} +3.12311 q^{51} +2.00000i q^{52} -7.43845i q^{53} +1.00000 q^{54} +2.56155 q^{56} +7.68466i q^{57} -7.12311i q^{58} +13.1231 q^{59} +6.00000 q^{61} +1.00000i q^{62} -2.56155i q^{63} -1.00000 q^{64} -2.56155 q^{66} -15.3693i q^{67} +3.12311i q^{68} +1.43845 q^{69} -7.68466 q^{71} +1.00000i q^{72} +10.8078i q^{73} +3.12311 q^{74} -7.68466 q^{76} +6.56155i q^{77} +2.00000i q^{78} +4.31534 q^{79} +1.00000 q^{81} +7.12311i q^{82} -14.2462i q^{83} +2.56155 q^{84} +12.8078 q^{86} -7.12311i q^{87} -2.56155i q^{88} +13.6847 q^{89} +5.12311 q^{91} +1.43845i q^{92} +1.00000i q^{93} -5.12311 q^{94} -1.00000 q^{96} -6.00000i q^{97} +0.438447i q^{98} -2.56155 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{4} - 4 q^{6} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{4} - 4 q^{6} - 4 q^{9} + 2 q^{11} - 2 q^{14} + 4 q^{16} + 6 q^{19} - 2 q^{21} + 4 q^{24} + 8 q^{26} - 12 q^{29} + 4 q^{31} - 4 q^{34} + 4 q^{36} + 8 q^{39} + 12 q^{41} - 2 q^{44} + 14 q^{46} + 10 q^{49} - 4 q^{51} + 4 q^{54} + 2 q^{56} + 36 q^{59} + 24 q^{61} - 4 q^{64} - 2 q^{66} + 14 q^{69} - 6 q^{71} - 4 q^{74} - 6 q^{76} + 42 q^{79} + 4 q^{81} + 2 q^{84} + 10 q^{86} + 30 q^{89} + 4 q^{91} - 4 q^{94} - 4 q^{96} - 2 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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
\(7\) 2.56155i 0.968176i 0.875019 + 0.484088i \(0.160849\pi\)
−0.875019 + 0.484088i \(0.839151\pi\)
\(8\) − 1.00000i − 0.353553i
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) 2.56155 0.772337 0.386169 0.922428i \(-0.373798\pi\)
0.386169 + 0.922428i \(0.373798\pi\)
\(12\) − 1.00000i − 0.288675i
\(13\) − 2.00000i − 0.554700i −0.960769 0.277350i \(-0.910544\pi\)
0.960769 0.277350i \(-0.0894562\pi\)
\(14\) −2.56155 −0.684604
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) − 3.12311i − 0.757464i −0.925506 0.378732i \(-0.876360\pi\)
0.925506 0.378732i \(-0.123640\pi\)
\(18\) − 1.00000i − 0.235702i
\(19\) 7.68466 1.76298 0.881491 0.472201i \(-0.156540\pi\)
0.881491 + 0.472201i \(0.156540\pi\)
\(20\) 0 0
\(21\) −2.56155 −0.558977
\(22\) 2.56155i 0.546125i
\(23\) − 1.43845i − 0.299937i −0.988691 0.149968i \(-0.952083\pi\)
0.988691 0.149968i \(-0.0479172\pi\)
\(24\) 1.00000 0.204124
\(25\) 0 0
\(26\) 2.00000 0.392232
\(27\) − 1.00000i − 0.192450i
\(28\) − 2.56155i − 0.484088i
\(29\) −7.12311 −1.32273 −0.661364 0.750065i \(-0.730022\pi\)
−0.661364 + 0.750065i \(0.730022\pi\)
\(30\) 0 0
\(31\) 1.00000 0.179605
\(32\) 1.00000i 0.176777i
\(33\) 2.56155i 0.445909i
\(34\) 3.12311 0.535608
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) − 3.12311i − 0.513435i −0.966486 0.256718i \(-0.917359\pi\)
0.966486 0.256718i \(-0.0826411\pi\)
\(38\) 7.68466i 1.24662i
\(39\) 2.00000 0.320256
\(40\) 0 0
\(41\) 7.12311 1.11244 0.556221 0.831034i \(-0.312251\pi\)
0.556221 + 0.831034i \(0.312251\pi\)
\(42\) − 2.56155i − 0.395256i
\(43\) − 12.8078i − 1.95317i −0.215142 0.976583i \(-0.569021\pi\)
0.215142 0.976583i \(-0.430979\pi\)
\(44\) −2.56155 −0.386169
\(45\) 0 0
\(46\) 1.43845 0.212087
\(47\) 5.12311i 0.747282i 0.927573 + 0.373641i \(0.121891\pi\)
−0.927573 + 0.373641i \(0.878109\pi\)
\(48\) 1.00000i 0.144338i
\(49\) 0.438447 0.0626353
\(50\) 0 0
\(51\) 3.12311 0.437322
\(52\) 2.00000i 0.277350i
\(53\) − 7.43845i − 1.02175i −0.859655 0.510875i \(-0.829322\pi\)
0.859655 0.510875i \(-0.170678\pi\)
\(54\) 1.00000 0.136083
\(55\) 0 0
\(56\) 2.56155 0.342302
\(57\) 7.68466i 1.01786i
\(58\) − 7.12311i − 0.935310i
\(59\) 13.1231 1.70848 0.854241 0.519877i \(-0.174022\pi\)
0.854241 + 0.519877i \(0.174022\pi\)
\(60\) 0 0
\(61\) 6.00000 0.768221 0.384111 0.923287i \(-0.374508\pi\)
0.384111 + 0.923287i \(0.374508\pi\)
\(62\) 1.00000i 0.127000i
\(63\) − 2.56155i − 0.322725i
\(64\) −1.00000 −0.125000
\(65\) 0 0
\(66\) −2.56155 −0.315305
\(67\) − 15.3693i − 1.87766i −0.344380 0.938830i \(-0.611911\pi\)
0.344380 0.938830i \(-0.388089\pi\)
\(68\) 3.12311i 0.378732i
\(69\) 1.43845 0.173169
\(70\) 0 0
\(71\) −7.68466 −0.912001 −0.456001 0.889979i \(-0.650719\pi\)
−0.456001 + 0.889979i \(0.650719\pi\)
\(72\) 1.00000i 0.117851i
\(73\) 10.8078i 1.26495i 0.774580 + 0.632477i \(0.217962\pi\)
−0.774580 + 0.632477i \(0.782038\pi\)
\(74\) 3.12311 0.363054
\(75\) 0 0
\(76\) −7.68466 −0.881491
\(77\) 6.56155i 0.747758i
\(78\) 2.00000i 0.226455i
\(79\) 4.31534 0.485514 0.242757 0.970087i \(-0.421948\pi\)
0.242757 + 0.970087i \(0.421948\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 7.12311i 0.786615i
\(83\) − 14.2462i − 1.56372i −0.623451 0.781862i \(-0.714270\pi\)
0.623451 0.781862i \(-0.285730\pi\)
\(84\) 2.56155 0.279488
\(85\) 0 0
\(86\) 12.8078 1.38110
\(87\) − 7.12311i − 0.763677i
\(88\) − 2.56155i − 0.273062i
\(89\) 13.6847 1.45057 0.725285 0.688448i \(-0.241708\pi\)
0.725285 + 0.688448i \(0.241708\pi\)
\(90\) 0 0
\(91\) 5.12311 0.537047
\(92\) 1.43845i 0.149968i
\(93\) 1.00000i 0.103695i
\(94\) −5.12311 −0.528408
\(95\) 0 0
\(96\) −1.00000 −0.102062
\(97\) − 6.00000i − 0.609208i −0.952479 0.304604i \(-0.901476\pi\)
0.952479 0.304604i \(-0.0985241\pi\)
\(98\) 0.438447i 0.0442899i
\(99\) −2.56155 −0.257446
\(100\) 0 0
\(101\) −0.561553 −0.0558766 −0.0279383 0.999610i \(-0.508894\pi\)
−0.0279383 + 0.999610i \(0.508894\pi\)
\(102\) 3.12311i 0.309234i
\(103\) 1.75379i 0.172806i 0.996260 + 0.0864030i \(0.0275373\pi\)
−0.996260 + 0.0864030i \(0.972463\pi\)
\(104\) −2.00000 −0.196116
\(105\) 0 0
\(106\) 7.43845 0.722486
\(107\) − 2.56155i − 0.247635i −0.992305 0.123817i \(-0.960486\pi\)
0.992305 0.123817i \(-0.0395137\pi\)
\(108\) 1.00000i 0.0962250i
\(109\) −5.36932 −0.514287 −0.257144 0.966373i \(-0.582781\pi\)
−0.257144 + 0.966373i \(0.582781\pi\)
\(110\) 0 0
\(111\) 3.12311 0.296432
\(112\) 2.56155i 0.242044i
\(113\) 1.68466i 0.158479i 0.996856 + 0.0792397i \(0.0252492\pi\)
−0.996856 + 0.0792397i \(0.974751\pi\)
\(114\) −7.68466 −0.719734
\(115\) 0 0
\(116\) 7.12311 0.661364
\(117\) 2.00000i 0.184900i
\(118\) 13.1231i 1.20808i
\(119\) 8.00000 0.733359
\(120\) 0 0
\(121\) −4.43845 −0.403495
\(122\) 6.00000i 0.543214i
\(123\) 7.12311i 0.642269i
\(124\) −1.00000 −0.0898027
\(125\) 0 0
\(126\) 2.56155 0.228201
\(127\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(128\) − 1.00000i − 0.0883883i
\(129\) 12.8078 1.12766
\(130\) 0 0
\(131\) −15.3693 −1.34282 −0.671412 0.741085i \(-0.734312\pi\)
−0.671412 + 0.741085i \(0.734312\pi\)
\(132\) − 2.56155i − 0.222955i
\(133\) 19.6847i 1.70688i
\(134\) 15.3693 1.32771
\(135\) 0 0
\(136\) −3.12311 −0.267804
\(137\) 20.2462i 1.72975i 0.501987 + 0.864875i \(0.332603\pi\)
−0.501987 + 0.864875i \(0.667397\pi\)
\(138\) 1.43845i 0.122449i
\(139\) 17.1231 1.45236 0.726181 0.687503i \(-0.241293\pi\)
0.726181 + 0.687503i \(0.241293\pi\)
\(140\) 0 0
\(141\) −5.12311 −0.431443
\(142\) − 7.68466i − 0.644882i
\(143\) − 5.12311i − 0.428416i
\(144\) −1.00000 −0.0833333
\(145\) 0 0
\(146\) −10.8078 −0.894457
\(147\) 0.438447i 0.0361625i
\(148\) 3.12311i 0.256718i
\(149\) −17.0540 −1.39712 −0.698558 0.715553i \(-0.746175\pi\)
−0.698558 + 0.715553i \(0.746175\pi\)
\(150\) 0 0
\(151\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(152\) − 7.68466i − 0.623308i
\(153\) 3.12311i 0.252488i
\(154\) −6.56155 −0.528745
\(155\) 0 0
\(156\) −2.00000 −0.160128
\(157\) 17.0540i 1.36106i 0.732722 + 0.680528i \(0.238249\pi\)
−0.732722 + 0.680528i \(0.761751\pi\)
\(158\) 4.31534i 0.343310i
\(159\) 7.43845 0.589907
\(160\) 0 0
\(161\) 3.68466 0.290392
\(162\) 1.00000i 0.0785674i
\(163\) 15.3693i 1.20382i 0.798565 + 0.601909i \(0.205593\pi\)
−0.798565 + 0.601909i \(0.794407\pi\)
\(164\) −7.12311 −0.556221
\(165\) 0 0
\(166\) 14.2462 1.10572
\(167\) − 6.56155i − 0.507748i −0.967237 0.253874i \(-0.918295\pi\)
0.967237 0.253874i \(-0.0817049\pi\)
\(168\) 2.56155i 0.197628i
\(169\) 9.00000 0.692308
\(170\) 0 0
\(171\) −7.68466 −0.587661
\(172\) 12.8078i 0.976583i
\(173\) − 8.24621i − 0.626948i −0.949597 0.313474i \(-0.898507\pi\)
0.949597 0.313474i \(-0.101493\pi\)
\(174\) 7.12311 0.540001
\(175\) 0 0
\(176\) 2.56155 0.193084
\(177\) 13.1231i 0.986393i
\(178\) 13.6847i 1.02571i
\(179\) 12.0000 0.896922 0.448461 0.893802i \(-0.351972\pi\)
0.448461 + 0.893802i \(0.351972\pi\)
\(180\) 0 0
\(181\) −13.0540 −0.970294 −0.485147 0.874433i \(-0.661234\pi\)
−0.485147 + 0.874433i \(0.661234\pi\)
\(182\) 5.12311i 0.379750i
\(183\) 6.00000i 0.443533i
\(184\) −1.43845 −0.106044
\(185\) 0 0
\(186\) −1.00000 −0.0733236
\(187\) − 8.00000i − 0.585018i
\(188\) − 5.12311i − 0.373641i
\(189\) 2.56155 0.186326
\(190\) 0 0
\(191\) −14.2462 −1.03082 −0.515410 0.856944i \(-0.672360\pi\)
−0.515410 + 0.856944i \(0.672360\pi\)
\(192\) − 1.00000i − 0.0721688i
\(193\) − 14.4924i − 1.04319i −0.853194 0.521594i \(-0.825338\pi\)
0.853194 0.521594i \(-0.174662\pi\)
\(194\) 6.00000 0.430775
\(195\) 0 0
\(196\) −0.438447 −0.0313177
\(197\) 6.00000i 0.427482i 0.976890 + 0.213741i \(0.0685649\pi\)
−0.976890 + 0.213741i \(0.931435\pi\)
\(198\) − 2.56155i − 0.182042i
\(199\) 16.8078 1.19147 0.595735 0.803181i \(-0.296861\pi\)
0.595735 + 0.803181i \(0.296861\pi\)
\(200\) 0 0
\(201\) 15.3693 1.08407
\(202\) − 0.561553i − 0.0395107i
\(203\) − 18.2462i − 1.28063i
\(204\) −3.12311 −0.218661
\(205\) 0 0
\(206\) −1.75379 −0.122192
\(207\) 1.43845i 0.0999790i
\(208\) − 2.00000i − 0.138675i
\(209\) 19.6847 1.36162
\(210\) 0 0
\(211\) −23.6847 −1.63052 −0.815260 0.579096i \(-0.803406\pi\)
−0.815260 + 0.579096i \(0.803406\pi\)
\(212\) 7.43845i 0.510875i
\(213\) − 7.68466i − 0.526544i
\(214\) 2.56155 0.175104
\(215\) 0 0
\(216\) −1.00000 −0.0680414
\(217\) 2.56155i 0.173890i
\(218\) − 5.36932i − 0.363656i
\(219\) −10.8078 −0.730321
\(220\) 0 0
\(221\) −6.24621 −0.420166
\(222\) 3.12311i 0.209609i
\(223\) 21.1231i 1.41451i 0.706960 + 0.707254i \(0.250066\pi\)
−0.706960 + 0.707254i \(0.749934\pi\)
\(224\) −2.56155 −0.171151
\(225\) 0 0
\(226\) −1.68466 −0.112062
\(227\) 7.68466i 0.510049i 0.966935 + 0.255024i \(0.0820835\pi\)
−0.966935 + 0.255024i \(0.917917\pi\)
\(228\) − 7.68466i − 0.508929i
\(229\) 16.5616 1.09442 0.547209 0.836996i \(-0.315690\pi\)
0.547209 + 0.836996i \(0.315690\pi\)
\(230\) 0 0
\(231\) −6.56155 −0.431718
\(232\) 7.12311i 0.467655i
\(233\) 17.0540i 1.11724i 0.829423 + 0.558622i \(0.188670\pi\)
−0.829423 + 0.558622i \(0.811330\pi\)
\(234\) −2.00000 −0.130744
\(235\) 0 0
\(236\) −13.1231 −0.854241
\(237\) 4.31534i 0.280312i
\(238\) 8.00000i 0.518563i
\(239\) −24.0000 −1.55243 −0.776215 0.630468i \(-0.782863\pi\)
−0.776215 + 0.630468i \(0.782863\pi\)
\(240\) 0 0
\(241\) 12.2462 0.788848 0.394424 0.918929i \(-0.370944\pi\)
0.394424 + 0.918929i \(0.370944\pi\)
\(242\) − 4.43845i − 0.285314i
\(243\) 1.00000i 0.0641500i
\(244\) −6.00000 −0.384111
\(245\) 0 0
\(246\) −7.12311 −0.454153
\(247\) − 15.3693i − 0.977926i
\(248\) − 1.00000i − 0.0635001i
\(249\) 14.2462 0.902817
\(250\) 0 0
\(251\) 16.4924 1.04099 0.520496 0.853864i \(-0.325747\pi\)
0.520496 + 0.853864i \(0.325747\pi\)
\(252\) 2.56155i 0.161363i
\(253\) − 3.68466i − 0.231652i
\(254\) 0 0
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) − 1.68466i − 0.105086i −0.998619 0.0525431i \(-0.983267\pi\)
0.998619 0.0525431i \(-0.0167327\pi\)
\(258\) 12.8078i 0.797377i
\(259\) 8.00000 0.497096
\(260\) 0 0
\(261\) 7.12311 0.440909
\(262\) − 15.3693i − 0.949520i
\(263\) 20.4924i 1.26362i 0.775125 + 0.631808i \(0.217687\pi\)
−0.775125 + 0.631808i \(0.782313\pi\)
\(264\) 2.56155 0.157653
\(265\) 0 0
\(266\) −19.6847 −1.20694
\(267\) 13.6847i 0.837487i
\(268\) 15.3693i 0.938830i
\(269\) 0.246211 0.0150118 0.00750588 0.999972i \(-0.497611\pi\)
0.00750588 + 0.999972i \(0.497611\pi\)
\(270\) 0 0
\(271\) 27.0540 1.64341 0.821706 0.569912i \(-0.193023\pi\)
0.821706 + 0.569912i \(0.193023\pi\)
\(272\) − 3.12311i − 0.189366i
\(273\) 5.12311i 0.310064i
\(274\) −20.2462 −1.22312
\(275\) 0 0
\(276\) −1.43845 −0.0865843
\(277\) − 5.36932i − 0.322611i −0.986905 0.161305i \(-0.948430\pi\)
0.986905 0.161305i \(-0.0515704\pi\)
\(278\) 17.1231i 1.02698i
\(279\) −1.00000 −0.0598684
\(280\) 0 0
\(281\) 4.87689 0.290931 0.145466 0.989363i \(-0.453532\pi\)
0.145466 + 0.989363i \(0.453532\pi\)
\(282\) − 5.12311i − 0.305077i
\(283\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(284\) 7.68466 0.456001
\(285\) 0 0
\(286\) 5.12311 0.302936
\(287\) 18.2462i 1.07704i
\(288\) − 1.00000i − 0.0589256i
\(289\) 7.24621 0.426248
\(290\) 0 0
\(291\) 6.00000 0.351726
\(292\) − 10.8078i − 0.632477i
\(293\) − 26.4924i − 1.54770i −0.633367 0.773852i \(-0.718327\pi\)
0.633367 0.773852i \(-0.281673\pi\)
\(294\) −0.438447 −0.0255708
\(295\) 0 0
\(296\) −3.12311 −0.181527
\(297\) − 2.56155i − 0.148636i
\(298\) − 17.0540i − 0.987910i
\(299\) −2.87689 −0.166375
\(300\) 0 0
\(301\) 32.8078 1.89101
\(302\) 0 0
\(303\) − 0.561553i − 0.0322604i
\(304\) 7.68466 0.440745
\(305\) 0 0
\(306\) −3.12311 −0.178536
\(307\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(308\) − 6.56155i − 0.373879i
\(309\) −1.75379 −0.0997696
\(310\) 0 0
\(311\) −1.75379 −0.0994482 −0.0497241 0.998763i \(-0.515834\pi\)
−0.0497241 + 0.998763i \(0.515834\pi\)
\(312\) − 2.00000i − 0.113228i
\(313\) 10.0000i 0.565233i 0.959233 + 0.282617i \(0.0912024\pi\)
−0.959233 + 0.282617i \(0.908798\pi\)
\(314\) −17.0540 −0.962412
\(315\) 0 0
\(316\) −4.31534 −0.242757
\(317\) 16.8769i 0.947901i 0.880552 + 0.473950i \(0.157172\pi\)
−0.880552 + 0.473950i \(0.842828\pi\)
\(318\) 7.43845i 0.417127i
\(319\) −18.2462 −1.02159
\(320\) 0 0
\(321\) 2.56155 0.142972
\(322\) 3.68466i 0.205338i
\(323\) − 24.0000i − 1.33540i
\(324\) −1.00000 −0.0555556
\(325\) 0 0
\(326\) −15.3693 −0.851228
\(327\) − 5.36932i − 0.296924i
\(328\) − 7.12311i − 0.393308i
\(329\) −13.1231 −0.723500
\(330\) 0 0
\(331\) −6.24621 −0.343323 −0.171661 0.985156i \(-0.554914\pi\)
−0.171661 + 0.985156i \(0.554914\pi\)
\(332\) 14.2462i 0.781862i
\(333\) 3.12311i 0.171145i
\(334\) 6.56155 0.359032
\(335\) 0 0
\(336\) −2.56155 −0.139744
\(337\) − 10.0000i − 0.544735i −0.962193 0.272367i \(-0.912193\pi\)
0.962193 0.272367i \(-0.0878066\pi\)
\(338\) 9.00000i 0.489535i
\(339\) −1.68466 −0.0914981
\(340\) 0 0
\(341\) 2.56155 0.138716
\(342\) − 7.68466i − 0.415539i
\(343\) 19.0540i 1.02882i
\(344\) −12.8078 −0.690548
\(345\) 0 0
\(346\) 8.24621 0.443319
\(347\) 14.2462i 0.764777i 0.924002 + 0.382388i \(0.124898\pi\)
−0.924002 + 0.382388i \(0.875102\pi\)
\(348\) 7.12311i 0.381839i
\(349\) −5.36932 −0.287413 −0.143706 0.989620i \(-0.545902\pi\)
−0.143706 + 0.989620i \(0.545902\pi\)
\(350\) 0 0
\(351\) −2.00000 −0.106752
\(352\) 2.56155i 0.136531i
\(353\) 0.246211i 0.0131045i 0.999979 + 0.00655225i \(0.00208566\pi\)
−0.999979 + 0.00655225i \(0.997914\pi\)
\(354\) −13.1231 −0.697485
\(355\) 0 0
\(356\) −13.6847 −0.725285
\(357\) 8.00000i 0.423405i
\(358\) 12.0000i 0.634220i
\(359\) 31.6847 1.67225 0.836126 0.548537i \(-0.184815\pi\)
0.836126 + 0.548537i \(0.184815\pi\)
\(360\) 0 0
\(361\) 40.0540 2.10810
\(362\) − 13.0540i − 0.686102i
\(363\) − 4.43845i − 0.232958i
\(364\) −5.12311 −0.268524
\(365\) 0 0
\(366\) −6.00000 −0.313625
\(367\) 15.3693i 0.802272i 0.916019 + 0.401136i \(0.131384\pi\)
−0.916019 + 0.401136i \(0.868616\pi\)
\(368\) − 1.43845i − 0.0749842i
\(369\) −7.12311 −0.370814
\(370\) 0 0
\(371\) 19.0540 0.989233
\(372\) − 1.00000i − 0.0518476i
\(373\) 5.68466i 0.294340i 0.989111 + 0.147170i \(0.0470165\pi\)
−0.989111 + 0.147170i \(0.952983\pi\)
\(374\) 8.00000 0.413670
\(375\) 0 0
\(376\) 5.12311 0.264204
\(377\) 14.2462i 0.733717i
\(378\) 2.56155i 0.131752i
\(379\) 7.05398 0.362338 0.181169 0.983452i \(-0.442012\pi\)
0.181169 + 0.983452i \(0.442012\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) − 14.2462i − 0.728900i
\(383\) 10.2462i 0.523557i 0.965128 + 0.261778i \(0.0843090\pi\)
−0.965128 + 0.261778i \(0.915691\pi\)
\(384\) 1.00000 0.0510310
\(385\) 0 0
\(386\) 14.4924 0.737645
\(387\) 12.8078i 0.651055i
\(388\) 6.00000i 0.304604i
\(389\) −7.75379 −0.393133 −0.196566 0.980491i \(-0.562979\pi\)
−0.196566 + 0.980491i \(0.562979\pi\)
\(390\) 0 0
\(391\) −4.49242 −0.227192
\(392\) − 0.438447i − 0.0221449i
\(393\) − 15.3693i − 0.775279i
\(394\) −6.00000 −0.302276
\(395\) 0 0
\(396\) 2.56155 0.128723
\(397\) 9.05398i 0.454406i 0.973847 + 0.227203i \(0.0729581\pi\)
−0.973847 + 0.227203i \(0.927042\pi\)
\(398\) 16.8078i 0.842497i
\(399\) −19.6847 −0.985466
\(400\) 0 0
\(401\) −13.6847 −0.683379 −0.341690 0.939813i \(-0.610999\pi\)
−0.341690 + 0.939813i \(0.610999\pi\)
\(402\) 15.3693i 0.766552i
\(403\) − 2.00000i − 0.0996271i
\(404\) 0.561553 0.0279383
\(405\) 0 0
\(406\) 18.2462 0.905544
\(407\) − 8.00000i − 0.396545i
\(408\) − 3.12311i − 0.154617i
\(409\) 37.3693 1.84779 0.923897 0.382642i \(-0.124986\pi\)
0.923897 + 0.382642i \(0.124986\pi\)
\(410\) 0 0
\(411\) −20.2462 −0.998672
\(412\) − 1.75379i − 0.0864030i
\(413\) 33.6155i 1.65411i
\(414\) −1.43845 −0.0706958
\(415\) 0 0
\(416\) 2.00000 0.0980581
\(417\) 17.1231i 0.838522i
\(418\) 19.6847i 0.962808i
\(419\) −5.75379 −0.281091 −0.140545 0.990074i \(-0.544886\pi\)
−0.140545 + 0.990074i \(0.544886\pi\)
\(420\) 0 0
\(421\) −14.4924 −0.706317 −0.353159 0.935563i \(-0.614892\pi\)
−0.353159 + 0.935563i \(0.614892\pi\)
\(422\) − 23.6847i − 1.15295i
\(423\) − 5.12311i − 0.249094i
\(424\) −7.43845 −0.361243
\(425\) 0 0
\(426\) 7.68466 0.372323
\(427\) 15.3693i 0.743773i
\(428\) 2.56155i 0.123817i
\(429\) 5.12311 0.247346
\(430\) 0 0
\(431\) −30.2462 −1.45691 −0.728454 0.685094i \(-0.759761\pi\)
−0.728454 + 0.685094i \(0.759761\pi\)
\(432\) − 1.00000i − 0.0481125i
\(433\) − 35.3002i − 1.69642i −0.529661 0.848209i \(-0.677681\pi\)
0.529661 0.848209i \(-0.322319\pi\)
\(434\) −2.56155 −0.122958
\(435\) 0 0
\(436\) 5.36932 0.257144
\(437\) − 11.0540i − 0.528783i
\(438\) − 10.8078i − 0.516415i
\(439\) −9.61553 −0.458924 −0.229462 0.973318i \(-0.573697\pi\)
−0.229462 + 0.973318i \(0.573697\pi\)
\(440\) 0 0
\(441\) −0.438447 −0.0208784
\(442\) − 6.24621i − 0.297102i
\(443\) − 23.0540i − 1.09533i −0.836699 0.547664i \(-0.815517\pi\)
0.836699 0.547664i \(-0.184483\pi\)
\(444\) −3.12311 −0.148216
\(445\) 0 0
\(446\) −21.1231 −1.00021
\(447\) − 17.0540i − 0.806625i
\(448\) − 2.56155i − 0.121022i
\(449\) −10.4924 −0.495168 −0.247584 0.968866i \(-0.579637\pi\)
−0.247584 + 0.968866i \(0.579637\pi\)
\(450\) 0 0
\(451\) 18.2462 0.859181
\(452\) − 1.68466i − 0.0792397i
\(453\) 0 0
\(454\) −7.68466 −0.360659
\(455\) 0 0
\(456\) 7.68466 0.359867
\(457\) − 24.7386i − 1.15722i −0.815603 0.578612i \(-0.803594\pi\)
0.815603 0.578612i \(-0.196406\pi\)
\(458\) 16.5616i 0.773871i
\(459\) −3.12311 −0.145774
\(460\) 0 0
\(461\) 9.36932 0.436373 0.218186 0.975907i \(-0.429986\pi\)
0.218186 + 0.975907i \(0.429986\pi\)
\(462\) − 6.56155i − 0.305271i
\(463\) − 30.7386i − 1.42855i −0.699867 0.714273i \(-0.746758\pi\)
0.699867 0.714273i \(-0.253242\pi\)
\(464\) −7.12311 −0.330682
\(465\) 0 0
\(466\) −17.0540 −0.790010
\(467\) − 9.75379i − 0.451352i −0.974202 0.225676i \(-0.927541\pi\)
0.974202 0.225676i \(-0.0724590\pi\)
\(468\) − 2.00000i − 0.0924500i
\(469\) 39.3693 1.81791
\(470\) 0 0
\(471\) −17.0540 −0.785806
\(472\) − 13.1231i − 0.604040i
\(473\) − 32.8078i − 1.50850i
\(474\) −4.31534 −0.198210
\(475\) 0 0
\(476\) −8.00000 −0.366679
\(477\) 7.43845i 0.340583i
\(478\) − 24.0000i − 1.09773i
\(479\) −10.5616 −0.482570 −0.241285 0.970454i \(-0.577569\pi\)
−0.241285 + 0.970454i \(0.577569\pi\)
\(480\) 0 0
\(481\) −6.24621 −0.284803
\(482\) 12.2462i 0.557800i
\(483\) 3.68466i 0.167658i
\(484\) 4.43845 0.201748
\(485\) 0 0
\(486\) −1.00000 −0.0453609
\(487\) 24.0000i 1.08754i 0.839233 + 0.543772i \(0.183004\pi\)
−0.839233 + 0.543772i \(0.816996\pi\)
\(488\) − 6.00000i − 0.271607i
\(489\) −15.3693 −0.695025
\(490\) 0 0
\(491\) −25.9309 −1.17024 −0.585122 0.810945i \(-0.698953\pi\)
−0.585122 + 0.810945i \(0.698953\pi\)
\(492\) − 7.12311i − 0.321134i
\(493\) 22.2462i 1.00192i
\(494\) 15.3693 0.691498
\(495\) 0 0
\(496\) 1.00000 0.0449013
\(497\) − 19.6847i − 0.882978i
\(498\) 14.2462i 0.638388i
\(499\) −20.0000 −0.895323 −0.447661 0.894203i \(-0.647743\pi\)
−0.447661 + 0.894203i \(0.647743\pi\)
\(500\) 0 0
\(501\) 6.56155 0.293149
\(502\) 16.4924i 0.736093i
\(503\) − 26.2462i − 1.17026i −0.810939 0.585130i \(-0.801043\pi\)
0.810939 0.585130i \(-0.198957\pi\)
\(504\) −2.56155 −0.114101
\(505\) 0 0
\(506\) 3.68466 0.163803
\(507\) 9.00000i 0.399704i
\(508\) 0 0
\(509\) −19.6155 −0.869443 −0.434721 0.900565i \(-0.643153\pi\)
−0.434721 + 0.900565i \(0.643153\pi\)
\(510\) 0 0
\(511\) −27.6847 −1.22470
\(512\) 1.00000i 0.0441942i
\(513\) − 7.68466i − 0.339286i
\(514\) 1.68466 0.0743071
\(515\) 0 0
\(516\) −12.8078 −0.563830
\(517\) 13.1231i 0.577154i
\(518\) 8.00000i 0.351500i
\(519\) 8.24621 0.361968
\(520\) 0 0
\(521\) −32.2462 −1.41273 −0.706366 0.707847i \(-0.749667\pi\)
−0.706366 + 0.707847i \(0.749667\pi\)
\(522\) 7.12311i 0.311770i
\(523\) 22.4233i 0.980502i 0.871581 + 0.490251i \(0.163095\pi\)
−0.871581 + 0.490251i \(0.836905\pi\)
\(524\) 15.3693 0.671412
\(525\) 0 0
\(526\) −20.4924 −0.893512
\(527\) − 3.12311i − 0.136045i
\(528\) 2.56155i 0.111477i
\(529\) 20.9309 0.910038
\(530\) 0 0
\(531\) −13.1231 −0.569494
\(532\) − 19.6847i − 0.853438i
\(533\) − 14.2462i − 0.617072i
\(534\) −13.6847 −0.592193
\(535\) 0 0
\(536\) −15.3693 −0.663853
\(537\) 12.0000i 0.517838i
\(538\) 0.246211i 0.0106149i
\(539\) 1.12311 0.0483756
\(540\) 0 0
\(541\) 19.7538 0.849282 0.424641 0.905362i \(-0.360400\pi\)
0.424641 + 0.905362i \(0.360400\pi\)
\(542\) 27.0540i 1.16207i
\(543\) − 13.0540i − 0.560200i
\(544\) 3.12311 0.133902
\(545\) 0 0
\(546\) −5.12311 −0.219249
\(547\) 26.8769i 1.14917i 0.818444 + 0.574587i \(0.194837\pi\)
−0.818444 + 0.574587i \(0.805163\pi\)
\(548\) − 20.2462i − 0.864875i
\(549\) −6.00000 −0.256074
\(550\) 0 0
\(551\) −54.7386 −2.33194
\(552\) − 1.43845i − 0.0612244i
\(553\) 11.0540i 0.470063i
\(554\) 5.36932 0.228120
\(555\) 0 0
\(556\) −17.1231 −0.726181
\(557\) − 26.8078i − 1.13588i −0.823069 0.567941i \(-0.807740\pi\)
0.823069 0.567941i \(-0.192260\pi\)
\(558\) − 1.00000i − 0.0423334i
\(559\) −25.6155 −1.08342
\(560\) 0 0
\(561\) 8.00000 0.337760
\(562\) 4.87689i 0.205719i
\(563\) − 16.4924i − 0.695073i −0.937667 0.347536i \(-0.887018\pi\)
0.937667 0.347536i \(-0.112982\pi\)
\(564\) 5.12311 0.215722
\(565\) 0 0
\(566\) 0 0
\(567\) 2.56155i 0.107575i
\(568\) 7.68466i 0.322441i
\(569\) −30.8078 −1.29153 −0.645764 0.763537i \(-0.723461\pi\)
−0.645764 + 0.763537i \(0.723461\pi\)
\(570\) 0 0
\(571\) 41.1231 1.72095 0.860474 0.509494i \(-0.170167\pi\)
0.860474 + 0.509494i \(0.170167\pi\)
\(572\) 5.12311i 0.214208i
\(573\) − 14.2462i − 0.595144i
\(574\) −18.2462 −0.761582
\(575\) 0 0
\(576\) 1.00000 0.0416667
\(577\) 15.1231i 0.629583i 0.949161 + 0.314792i \(0.101935\pi\)
−0.949161 + 0.314792i \(0.898065\pi\)
\(578\) 7.24621i 0.301403i
\(579\) 14.4924 0.602285
\(580\) 0 0
\(581\) 36.4924 1.51396
\(582\) 6.00000i 0.248708i
\(583\) − 19.0540i − 0.789135i
\(584\) 10.8078 0.447228
\(585\) 0 0
\(586\) 26.4924 1.09439
\(587\) − 0.492423i − 0.0203245i −0.999948 0.0101622i \(-0.996765\pi\)
0.999948 0.0101622i \(-0.00323479\pi\)
\(588\) − 0.438447i − 0.0180813i
\(589\) 7.68466 0.316641
\(590\) 0 0
\(591\) −6.00000 −0.246807
\(592\) − 3.12311i − 0.128359i
\(593\) 30.0000i 1.23195i 0.787765 + 0.615976i \(0.211238\pi\)
−0.787765 + 0.615976i \(0.788762\pi\)
\(594\) 2.56155 0.105102
\(595\) 0 0
\(596\) 17.0540 0.698558
\(597\) 16.8078i 0.687896i
\(598\) − 2.87689i − 0.117645i
\(599\) −38.4233 −1.56993 −0.784967 0.619538i \(-0.787320\pi\)
−0.784967 + 0.619538i \(0.787320\pi\)
\(600\) 0 0
\(601\) 2.00000 0.0815817 0.0407909 0.999168i \(-0.487012\pi\)
0.0407909 + 0.999168i \(0.487012\pi\)
\(602\) 32.8078i 1.33714i
\(603\) 15.3693i 0.625887i
\(604\) 0 0
\(605\) 0 0
\(606\) 0.561553 0.0228115
\(607\) 7.05398i 0.286312i 0.989700 + 0.143156i \(0.0457251\pi\)
−0.989700 + 0.143156i \(0.954275\pi\)
\(608\) 7.68466i 0.311654i
\(609\) 18.2462 0.739374
\(610\) 0 0
\(611\) 10.2462 0.414517
\(612\) − 3.12311i − 0.126244i
\(613\) − 2.00000i − 0.0807792i −0.999184 0.0403896i \(-0.987140\pi\)
0.999184 0.0403896i \(-0.0128599\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 6.56155 0.264372
\(617\) − 15.4384i − 0.621528i −0.950487 0.310764i \(-0.899415\pi\)
0.950487 0.310764i \(-0.100585\pi\)
\(618\) − 1.75379i − 0.0705477i
\(619\) −11.3693 −0.456971 −0.228486 0.973547i \(-0.573377\pi\)
−0.228486 + 0.973547i \(0.573377\pi\)
\(620\) 0 0
\(621\) −1.43845 −0.0577229
\(622\) − 1.75379i − 0.0703205i
\(623\) 35.0540i 1.40441i
\(624\) 2.00000 0.0800641
\(625\) 0 0
\(626\) −10.0000 −0.399680
\(627\) 19.6847i 0.786130i
\(628\) − 17.0540i − 0.680528i
\(629\) −9.75379 −0.388909
\(630\) 0 0
\(631\) 29.3002 1.16642 0.583211 0.812321i \(-0.301796\pi\)
0.583211 + 0.812321i \(0.301796\pi\)
\(632\) − 4.31534i − 0.171655i
\(633\) − 23.6847i − 0.941381i
\(634\) −16.8769 −0.670267
\(635\) 0 0
\(636\) −7.43845 −0.294954
\(637\) − 0.876894i − 0.0347438i
\(638\) − 18.2462i − 0.722374i
\(639\) 7.68466 0.304000
\(640\) 0 0
\(641\) −5.50758 −0.217536 −0.108768 0.994067i \(-0.534691\pi\)
−0.108768 + 0.994067i \(0.534691\pi\)
\(642\) 2.56155i 0.101096i
\(643\) 7.05398i 0.278182i 0.990280 + 0.139091i \(0.0444180\pi\)
−0.990280 + 0.139091i \(0.955582\pi\)
\(644\) −3.68466 −0.145196
\(645\) 0 0
\(646\) 24.0000 0.944267
\(647\) − 45.9309i − 1.80573i −0.429925 0.902864i \(-0.641460\pi\)
0.429925 0.902864i \(-0.358540\pi\)
\(648\) − 1.00000i − 0.0392837i
\(649\) 33.6155 1.31952
\(650\) 0 0
\(651\) −2.56155 −0.100395
\(652\) − 15.3693i − 0.601909i
\(653\) − 40.2462i − 1.57496i −0.616343 0.787478i \(-0.711386\pi\)
0.616343 0.787478i \(-0.288614\pi\)
\(654\) 5.36932 0.209957
\(655\) 0 0
\(656\) 7.12311 0.278111
\(657\) − 10.8078i − 0.421651i
\(658\) − 13.1231i − 0.511592i
\(659\) 30.7386 1.19741 0.598704 0.800971i \(-0.295683\pi\)
0.598704 + 0.800971i \(0.295683\pi\)
\(660\) 0 0
\(661\) 26.4924 1.03044 0.515218 0.857059i \(-0.327711\pi\)
0.515218 + 0.857059i \(0.327711\pi\)
\(662\) − 6.24621i − 0.242766i
\(663\) − 6.24621i − 0.242583i
\(664\) −14.2462 −0.552860
\(665\) 0 0
\(666\) −3.12311 −0.121018
\(667\) 10.2462i 0.396735i
\(668\) 6.56155i 0.253874i
\(669\) −21.1231 −0.816666
\(670\) 0 0
\(671\) 15.3693 0.593326
\(672\) − 2.56155i − 0.0988140i
\(673\) − 11.7538i − 0.453075i −0.974002 0.226538i \(-0.927259\pi\)
0.974002 0.226538i \(-0.0727406\pi\)
\(674\) 10.0000 0.385186
\(675\) 0 0
\(676\) −9.00000 −0.346154
\(677\) 32.4233i 1.24613i 0.782171 + 0.623064i \(0.214112\pi\)
−0.782171 + 0.623064i \(0.785888\pi\)
\(678\) − 1.68466i − 0.0646989i
\(679\) 15.3693 0.589820
\(680\) 0 0
\(681\) −7.68466 −0.294477
\(682\) 2.56155i 0.0980869i
\(683\) − 31.6847i − 1.21238i −0.795320 0.606190i \(-0.792697\pi\)
0.795320 0.606190i \(-0.207303\pi\)
\(684\) 7.68466 0.293830
\(685\) 0 0
\(686\) −19.0540 −0.727484
\(687\) 16.5616i 0.631863i
\(688\) − 12.8078i − 0.488291i
\(689\) −14.8769 −0.566765
\(690\) 0 0
\(691\) 15.0540 0.572680 0.286340 0.958128i \(-0.407561\pi\)
0.286340 + 0.958128i \(0.407561\pi\)
\(692\) 8.24621i 0.313474i
\(693\) − 6.56155i − 0.249253i
\(694\) −14.2462 −0.540779
\(695\) 0 0
\(696\) −7.12311 −0.270001
\(697\) − 22.2462i − 0.842635i
\(698\) − 5.36932i − 0.203232i
\(699\) −17.0540 −0.645041
\(700\) 0 0
\(701\) 5.19224 0.196108 0.0980540 0.995181i \(-0.468738\pi\)
0.0980540 + 0.995181i \(0.468738\pi\)
\(702\) − 2.00000i − 0.0754851i
\(703\) − 24.0000i − 0.905177i
\(704\) −2.56155 −0.0965422
\(705\) 0 0
\(706\) −0.246211 −0.00926628
\(707\) − 1.43845i − 0.0540984i
\(708\) − 13.1231i − 0.493197i
\(709\) −17.0540 −0.640475 −0.320238 0.947337i \(-0.603763\pi\)
−0.320238 + 0.947337i \(0.603763\pi\)
\(710\) 0 0
\(711\) −4.31534 −0.161838
\(712\) − 13.6847i − 0.512854i
\(713\) − 1.43845i − 0.0538703i
\(714\) −8.00000 −0.299392
\(715\) 0 0
\(716\) −12.0000 −0.448461
\(717\) − 24.0000i − 0.896296i
\(718\) 31.6847i 1.18246i
\(719\) 34.8769 1.30069 0.650344 0.759640i \(-0.274625\pi\)
0.650344 + 0.759640i \(0.274625\pi\)
\(720\) 0 0
\(721\) −4.49242 −0.167307
\(722\) 40.0540i 1.49065i
\(723\) 12.2462i 0.455441i
\(724\) 13.0540 0.485147
\(725\) 0 0
\(726\) 4.43845 0.164726
\(727\) 23.0540i 0.855025i 0.904010 + 0.427512i \(0.140610\pi\)
−0.904010 + 0.427512i \(0.859390\pi\)
\(728\) − 5.12311i − 0.189875i
\(729\) −1.00000 −0.0370370
\(730\) 0 0
\(731\) −40.0000 −1.47945
\(732\) − 6.00000i − 0.221766i
\(733\) 6.49242i 0.239803i 0.992786 + 0.119902i \(0.0382579\pi\)
−0.992786 + 0.119902i \(0.961742\pi\)
\(734\) −15.3693 −0.567292
\(735\) 0 0
\(736\) 1.43845 0.0530219
\(737\) − 39.3693i − 1.45019i
\(738\) − 7.12311i − 0.262205i
\(739\) −52.9848 −1.94908 −0.974540 0.224216i \(-0.928018\pi\)
−0.974540 + 0.224216i \(0.928018\pi\)
\(740\) 0 0
\(741\) 15.3693 0.564606
\(742\) 19.0540i 0.699493i
\(743\) 50.4233i 1.84985i 0.380148 + 0.924926i \(0.375873\pi\)
−0.380148 + 0.924926i \(0.624127\pi\)
\(744\) 1.00000 0.0366618
\(745\) 0 0
\(746\) −5.68466 −0.208130
\(747\) 14.2462i 0.521242i
\(748\) 8.00000i 0.292509i
\(749\) 6.56155 0.239754
\(750\) 0 0
\(751\) 45.1231 1.64657 0.823283 0.567631i \(-0.192140\pi\)
0.823283 + 0.567631i \(0.192140\pi\)
\(752\) 5.12311i 0.186820i
\(753\) 16.4924i 0.601017i
\(754\) −14.2462 −0.518816
\(755\) 0 0
\(756\) −2.56155 −0.0931628
\(757\) 10.0000i 0.363456i 0.983349 + 0.181728i \(0.0581691\pi\)
−0.983349 + 0.181728i \(0.941831\pi\)
\(758\) 7.05398i 0.256212i
\(759\) 3.68466 0.133745
\(760\) 0 0
\(761\) −20.4233 −0.740344 −0.370172 0.928963i \(-0.620701\pi\)
−0.370172 + 0.928963i \(0.620701\pi\)
\(762\) 0 0
\(763\) − 13.7538i − 0.497921i
\(764\) 14.2462 0.515410
\(765\) 0 0
\(766\) −10.2462 −0.370211
\(767\) − 26.2462i − 0.947696i
\(768\) 1.00000i 0.0360844i
\(769\) 20.5616 0.741469 0.370734 0.928739i \(-0.379106\pi\)
0.370734 + 0.928739i \(0.379106\pi\)
\(770\) 0 0
\(771\) 1.68466 0.0606715
\(772\) 14.4924i 0.521594i
\(773\) 11.4384i 0.411412i 0.978614 + 0.205706i \(0.0659491\pi\)
−0.978614 + 0.205706i \(0.934051\pi\)
\(774\) −12.8078 −0.460366
\(775\) 0 0
\(776\) −6.00000 −0.215387
\(777\) 8.00000i 0.286998i
\(778\) − 7.75379i − 0.277987i
\(779\) 54.7386 1.96122
\(780\) 0 0
\(781\) −19.6847 −0.704372
\(782\) − 4.49242i − 0.160649i
\(783\) 7.12311i 0.254559i
\(784\) 0.438447 0.0156588
\(785\) 0 0
\(786\) 15.3693 0.548205
\(787\) − 17.9309i − 0.639166i −0.947558 0.319583i \(-0.896457\pi\)
0.947558 0.319583i \(-0.103543\pi\)
\(788\) − 6.00000i − 0.213741i
\(789\) −20.4924 −0.729550
\(790\) 0 0
\(791\) −4.31534 −0.153436
\(792\) 2.56155i 0.0910208i
\(793\) − 12.0000i − 0.426132i
\(794\) −9.05398 −0.321314
\(795\) 0 0
\(796\) −16.8078 −0.595735
\(797\) − 26.9848i − 0.955852i −0.878400 0.477926i \(-0.841389\pi\)
0.878400 0.477926i \(-0.158611\pi\)
\(798\) − 19.6847i − 0.696829i
\(799\) 16.0000 0.566039
\(800\) 0 0
\(801\) −13.6847 −0.483524
\(802\) − 13.6847i − 0.483222i
\(803\) 27.6847i 0.976970i
\(804\) −15.3693 −0.542034
\(805\) 0 0
\(806\) 2.00000 0.0704470
\(807\) 0.246211i 0.00866705i
\(808\) 0.561553i 0.0197554i
\(809\) 37.6847 1.32492 0.662461 0.749096i \(-0.269512\pi\)
0.662461 + 0.749096i \(0.269512\pi\)
\(810\) 0 0
\(811\) −24.6695 −0.866263 −0.433132 0.901331i \(-0.642592\pi\)
−0.433132 + 0.901331i \(0.642592\pi\)
\(812\) 18.2462i 0.640316i
\(813\) 27.0540i 0.948824i
\(814\) 8.00000 0.280400
\(815\) 0 0
\(816\) 3.12311 0.109331
\(817\) − 98.4233i − 3.44340i
\(818\) 37.3693i 1.30659i
\(819\) −5.12311 −0.179016
\(820\) 0 0
\(821\) 19.6155 0.684587 0.342293 0.939593i \(-0.388796\pi\)
0.342293 + 0.939593i \(0.388796\pi\)
\(822\) − 20.2462i − 0.706168i
\(823\) 25.6155i 0.892901i 0.894808 + 0.446451i \(0.147312\pi\)
−0.894808 + 0.446451i \(0.852688\pi\)
\(824\) 1.75379 0.0610961
\(825\) 0 0
\(826\) −33.6155 −1.16963
\(827\) 1.75379i 0.0609852i 0.999535 + 0.0304926i \(0.00970760\pi\)
−0.999535 + 0.0304926i \(0.990292\pi\)
\(828\) − 1.43845i − 0.0499895i
\(829\) −24.4233 −0.848256 −0.424128 0.905602i \(-0.639419\pi\)
−0.424128 + 0.905602i \(0.639419\pi\)
\(830\) 0 0
\(831\) 5.36932 0.186260
\(832\) 2.00000i 0.0693375i
\(833\) − 1.36932i − 0.0474440i
\(834\) −17.1231 −0.592925
\(835\) 0 0
\(836\) −19.6847 −0.680808
\(837\) − 1.00000i − 0.0345651i
\(838\) − 5.75379i − 0.198761i
\(839\) −6.06913 −0.209530 −0.104765 0.994497i \(-0.533409\pi\)
−0.104765 + 0.994497i \(0.533409\pi\)
\(840\) 0 0
\(841\) 21.7386 0.749608
\(842\) − 14.4924i − 0.499442i
\(843\) 4.87689i 0.167969i
\(844\) 23.6847 0.815260
\(845\) 0 0
\(846\) 5.12311 0.176136
\(847\) − 11.3693i − 0.390654i
\(848\) − 7.43845i − 0.255437i
\(849\) 0 0
\(850\) 0 0
\(851\) −4.49242 −0.153998
\(852\) 7.68466i 0.263272i
\(853\) − 47.7926i − 1.63639i −0.574942 0.818194i \(-0.694976\pi\)
0.574942 0.818194i \(-0.305024\pi\)
\(854\) −15.3693 −0.525927
\(855\) 0 0
\(856\) −2.56155 −0.0875521
\(857\) 29.2311i 0.998514i 0.866454 + 0.499257i \(0.166394\pi\)
−0.866454 + 0.499257i \(0.833606\pi\)
\(858\) 5.12311i 0.174900i
\(859\) −26.7386 −0.912310 −0.456155 0.889900i \(-0.650774\pi\)
−0.456155 + 0.889900i \(0.650774\pi\)
\(860\) 0 0
\(861\) −18.2462 −0.621829
\(862\) − 30.2462i − 1.03019i
\(863\) − 39.5464i − 1.34618i −0.739563 0.673088i \(-0.764968\pi\)
0.739563 0.673088i \(-0.235032\pi\)
\(864\) 1.00000 0.0340207
\(865\) 0 0
\(866\) 35.3002 1.19955
\(867\) 7.24621i 0.246094i
\(868\) − 2.56155i − 0.0869448i
\(869\) 11.0540 0.374980
\(870\) 0 0
\(871\) −30.7386 −1.04154
\(872\) 5.36932i 0.181828i
\(873\) 6.00000i 0.203069i
\(874\) 11.0540 0.373906
\(875\) 0 0
\(876\) 10.8078 0.365161
\(877\) − 50.0000i − 1.68838i −0.536044 0.844190i \(-0.680082\pi\)
0.536044 0.844190i \(-0.319918\pi\)
\(878\) − 9.61553i − 0.324508i
\(879\) 26.4924 0.893567
\(880\) 0 0
\(881\) 9.50758 0.320318 0.160159 0.987091i \(-0.448799\pi\)
0.160159 + 0.987091i \(0.448799\pi\)
\(882\) − 0.438447i − 0.0147633i
\(883\) − 30.4233i − 1.02383i −0.859038 0.511913i \(-0.828937\pi\)
0.859038 0.511913i \(-0.171063\pi\)
\(884\) 6.24621 0.210083
\(885\) 0 0
\(886\) 23.0540 0.774513
\(887\) − 32.6307i − 1.09563i −0.836599 0.547816i \(-0.815460\pi\)
0.836599 0.547816i \(-0.184540\pi\)
\(888\) − 3.12311i − 0.104805i
\(889\) 0 0
\(890\) 0 0
\(891\) 2.56155 0.0858152
\(892\) − 21.1231i − 0.707254i
\(893\) 39.3693i 1.31744i
\(894\) 17.0540 0.570370
\(895\) 0 0
\(896\) 2.56155 0.0855755
\(897\) − 2.87689i − 0.0960567i
\(898\) − 10.4924i − 0.350137i
\(899\) −7.12311 −0.237569
\(900\) 0 0
\(901\) −23.2311 −0.773939
\(902\) 18.2462i 0.607532i
\(903\) 32.8078i 1.09177i
\(904\) 1.68466 0.0560309
\(905\) 0 0
\(906\) 0 0
\(907\) 24.0000i 0.796907i 0.917189 + 0.398453i \(0.130453\pi\)
−0.917189 + 0.398453i \(0.869547\pi\)
\(908\) − 7.68466i − 0.255024i
\(909\) 0.561553 0.0186255
\(910\) 0 0
\(911\) −11.5076 −0.381263 −0.190632 0.981662i \(-0.561054\pi\)
−0.190632 + 0.981662i \(0.561054\pi\)
\(912\) 7.68466i 0.254464i
\(913\) − 36.4924i − 1.20772i
\(914\) 24.7386 0.818281
\(915\) 0 0
\(916\) −16.5616 −0.547209
\(917\) − 39.3693i − 1.30009i
\(918\) − 3.12311i − 0.103078i
\(919\) 1.61553 0.0532914 0.0266457 0.999645i \(-0.491517\pi\)
0.0266457 + 0.999645i \(0.491517\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 9.36932i 0.308562i
\(923\) 15.3693i 0.505887i
\(924\) 6.56155 0.215859
\(925\) 0 0
\(926\) 30.7386 1.01013
\(927\) − 1.75379i − 0.0576020i
\(928\) − 7.12311i − 0.233827i
\(929\) −7.43845 −0.244048 −0.122024 0.992527i \(-0.538938\pi\)
−0.122024 + 0.992527i \(0.538938\pi\)
\(930\) 0 0
\(931\) 3.36932 0.110425
\(932\) − 17.0540i − 0.558622i
\(933\) − 1.75379i − 0.0574165i
\(934\) 9.75379 0.319154
\(935\) 0 0
\(936\) 2.00000 0.0653720
\(937\) 41.3693i 1.35148i 0.737142 + 0.675738i \(0.236175\pi\)
−0.737142 + 0.675738i \(0.763825\pi\)
\(938\) 39.3693i 1.28545i
\(939\) −10.0000 −0.326338
\(940\) 0 0
\(941\) 2.63068 0.0857578 0.0428789 0.999080i \(-0.486347\pi\)
0.0428789 + 0.999080i \(0.486347\pi\)
\(942\) − 17.0540i − 0.555649i
\(943\) − 10.2462i − 0.333663i
\(944\) 13.1231 0.427121
\(945\) 0 0
\(946\) 32.8078 1.06667
\(947\) − 9.12311i − 0.296461i −0.988953 0.148231i \(-0.952642\pi\)
0.988953 0.148231i \(-0.0473578\pi\)
\(948\) − 4.31534i − 0.140156i
\(949\) 21.6155 0.701670
\(950\) 0 0
\(951\) −16.8769 −0.547271
\(952\) − 8.00000i − 0.259281i
\(953\) − 16.1080i − 0.521788i −0.965367 0.260894i \(-0.915983\pi\)
0.965367 0.260894i \(-0.0840173\pi\)
\(954\) −7.43845 −0.240829
\(955\) 0 0
\(956\) 24.0000 0.776215
\(957\) − 18.2462i − 0.589816i
\(958\) − 10.5616i − 0.341228i
\(959\) −51.8617 −1.67470
\(960\) 0 0
\(961\) 1.00000 0.0322581
\(962\) − 6.24621i − 0.201386i
\(963\) 2.56155i 0.0825449i
\(964\) −12.2462 −0.394424
\(965\) 0 0
\(966\) −3.68466 −0.118552
\(967\) − 42.2462i − 1.35855i −0.733885 0.679273i \(-0.762295\pi\)
0.733885 0.679273i \(-0.237705\pi\)
\(968\) 4.43845i 0.142657i
\(969\) 24.0000 0.770991
\(970\) 0 0
\(971\) −37.1231 −1.19134 −0.595669 0.803230i \(-0.703113\pi\)
−0.595669 + 0.803230i \(0.703113\pi\)
\(972\) − 1.00000i − 0.0320750i
\(973\) 43.8617i 1.40614i
\(974\) −24.0000 −0.769010
\(975\) 0 0
\(976\) 6.00000 0.192055
\(977\) 42.9848i 1.37521i 0.726086 + 0.687604i \(0.241337\pi\)
−0.726086 + 0.687604i \(0.758663\pi\)
\(978\) − 15.3693i − 0.491457i
\(979\) 35.0540 1.12033
\(980\) 0 0
\(981\) 5.36932 0.171429
\(982\) − 25.9309i − 0.827487i
\(983\) − 40.9848i − 1.30721i −0.756834 0.653607i \(-0.773255\pi\)
0.756834 0.653607i \(-0.226745\pi\)
\(984\) 7.12311 0.227076
\(985\) 0 0
\(986\) −22.2462 −0.708464
\(987\) − 13.1231i − 0.417713i
\(988\) 15.3693i 0.488963i
\(989\) −18.4233 −0.585827
\(990\) 0 0
\(991\) 35.6847 1.13356 0.566780 0.823869i \(-0.308189\pi\)
0.566780 + 0.823869i \(0.308189\pi\)
\(992\) 1.00000i 0.0317500i
\(993\) − 6.24621i − 0.198218i
\(994\) 19.6847 0.624359
\(995\) 0 0
\(996\) −14.2462 −0.451408
\(997\) − 29.5076i − 0.934514i −0.884121 0.467257i \(-0.845242\pi\)
0.884121 0.467257i \(-0.154758\pi\)
\(998\) − 20.0000i − 0.633089i
\(999\) −3.12311 −0.0988107
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.bd.3349.4 4
5.2 odd 4 4650.2.a.ce.1.1 2
5.3 odd 4 930.2.a.p.1.2 2
5.4 even 2 inner 4650.2.d.bd.3349.1 4
15.8 even 4 2790.2.a.be.1.2 2
20.3 even 4 7440.2.a.bl.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
930.2.a.p.1.2 2 5.3 odd 4
2790.2.a.be.1.2 2 15.8 even 4
4650.2.a.ce.1.1 2 5.2 odd 4
4650.2.d.bd.3349.1 4 5.4 even 2 inner
4650.2.d.bd.3349.4 4 1.1 even 1 trivial
7440.2.a.bl.1.1 2 20.3 even 4