Properties

Label 4650.2.a.ce.1.1
Level $4650$
Weight $2$
Character 4650.1
Self dual yes
Analytic conductor $37.130$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 4650 = 2 \cdot 3 \cdot 5^{2} \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4650.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(37.1304369399\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{17}) \)
Defining polynomial: \( x^{2} - x - 4 \) 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)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(2.56155\) of defining polynomial
Character \(\chi\) \(=\) 4650.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} -2.56155 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} -2.56155 q^{7} -1.00000 q^{8} +1.00000 q^{9} +2.56155 q^{11} +1.00000 q^{12} -2.00000 q^{13} +2.56155 q^{14} +1.00000 q^{16} +3.12311 q^{17} -1.00000 q^{18} -7.68466 q^{19} -2.56155 q^{21} -2.56155 q^{22} -1.43845 q^{23} -1.00000 q^{24} +2.00000 q^{26} +1.00000 q^{27} -2.56155 q^{28} +7.12311 q^{29} +1.00000 q^{31} -1.00000 q^{32} +2.56155 q^{33} -3.12311 q^{34} +1.00000 q^{36} +3.12311 q^{37} +7.68466 q^{38} -2.00000 q^{39} +7.12311 q^{41} +2.56155 q^{42} -12.8078 q^{43} +2.56155 q^{44} +1.43845 q^{46} -5.12311 q^{47} +1.00000 q^{48} -0.438447 q^{49} +3.12311 q^{51} -2.00000 q^{52} -7.43845 q^{53} -1.00000 q^{54} +2.56155 q^{56} -7.68466 q^{57} -7.12311 q^{58} -13.1231 q^{59} +6.00000 q^{61} -1.00000 q^{62} -2.56155 q^{63} +1.00000 q^{64} -2.56155 q^{66} +15.3693 q^{67} +3.12311 q^{68} -1.43845 q^{69} -7.68466 q^{71} -1.00000 q^{72} +10.8078 q^{73} -3.12311 q^{74} -7.68466 q^{76} -6.56155 q^{77} +2.00000 q^{78} -4.31534 q^{79} +1.00000 q^{81} -7.12311 q^{82} -14.2462 q^{83} -2.56155 q^{84} +12.8078 q^{86} +7.12311 q^{87} -2.56155 q^{88} -13.6847 q^{89} +5.12311 q^{91} -1.43845 q^{92} +1.00000 q^{93} +5.12311 q^{94} -1.00000 q^{96} +6.00000 q^{97} +0.438447 q^{98} +2.56155 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} + 2 q^{3} + 2 q^{4} - 2 q^{6} - q^{7} - 2 q^{8} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{2} + 2 q^{3} + 2 q^{4} - 2 q^{6} - q^{7} - 2 q^{8} + 2 q^{9} + q^{11} + 2 q^{12} - 4 q^{13} + q^{14} + 2 q^{16} - 2 q^{17} - 2 q^{18} - 3 q^{19} - q^{21} - q^{22} - 7 q^{23} - 2 q^{24} + 4 q^{26} + 2 q^{27} - q^{28} + 6 q^{29} + 2 q^{31} - 2 q^{32} + q^{33} + 2 q^{34} + 2 q^{36} - 2 q^{37} + 3 q^{38} - 4 q^{39} + 6 q^{41} + q^{42} - 5 q^{43} + q^{44} + 7 q^{46} - 2 q^{47} + 2 q^{48} - 5 q^{49} - 2 q^{51} - 4 q^{52} - 19 q^{53} - 2 q^{54} + q^{56} - 3 q^{57} - 6 q^{58} - 18 q^{59} + 12 q^{61} - 2 q^{62} - q^{63} + 2 q^{64} - q^{66} + 6 q^{67} - 2 q^{68} - 7 q^{69} - 3 q^{71} - 2 q^{72} + q^{73} + 2 q^{74} - 3 q^{76} - 9 q^{77} + 4 q^{78} - 21 q^{79} + 2 q^{81} - 6 q^{82} - 12 q^{83} - q^{84} + 5 q^{86} + 6 q^{87} - q^{88} - 15 q^{89} + 2 q^{91} - 7 q^{92} + 2 q^{93} + 2 q^{94} - 2 q^{96} + 12 q^{97} + 5 q^{98} + q^{99}+O(q^{100}) \) Copy content Toggle raw display

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) 1.00000 0.577350
\(4\) 1.00000 0.500000
\(5\) 0 0
\(6\) −1.00000 −0.408248
\(7\) −2.56155 −0.968176 −0.484088 0.875019i \(-0.660849\pi\)
−0.484088 + 0.875019i \(0.660849\pi\)
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 2.56155 0.772337 0.386169 0.922428i \(-0.373798\pi\)
0.386169 + 0.922428i \(0.373798\pi\)
\(12\) 1.00000 0.288675
\(13\) −2.00000 −0.554700 −0.277350 0.960769i \(-0.589456\pi\)
−0.277350 + 0.960769i \(0.589456\pi\)
\(14\) 2.56155 0.684604
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 3.12311 0.757464 0.378732 0.925506i \(-0.376360\pi\)
0.378732 + 0.925506i \(0.376360\pi\)
\(18\) −1.00000 −0.235702
\(19\) −7.68466 −1.76298 −0.881491 0.472201i \(-0.843460\pi\)
−0.881491 + 0.472201i \(0.843460\pi\)
\(20\) 0 0
\(21\) −2.56155 −0.558977
\(22\) −2.56155 −0.546125
\(23\) −1.43845 −0.299937 −0.149968 0.988691i \(-0.547917\pi\)
−0.149968 + 0.988691i \(0.547917\pi\)
\(24\) −1.00000 −0.204124
\(25\) 0 0
\(26\) 2.00000 0.392232
\(27\) 1.00000 0.192450
\(28\) −2.56155 −0.484088
\(29\) 7.12311 1.32273 0.661364 0.750065i \(-0.269978\pi\)
0.661364 + 0.750065i \(0.269978\pi\)
\(30\) 0 0
\(31\) 1.00000 0.179605
\(32\) −1.00000 −0.176777
\(33\) 2.56155 0.445909
\(34\) −3.12311 −0.535608
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) 3.12311 0.513435 0.256718 0.966486i \(-0.417359\pi\)
0.256718 + 0.966486i \(0.417359\pi\)
\(38\) 7.68466 1.24662
\(39\) −2.00000 −0.320256
\(40\) 0 0
\(41\) 7.12311 1.11244 0.556221 0.831034i \(-0.312251\pi\)
0.556221 + 0.831034i \(0.312251\pi\)
\(42\) 2.56155 0.395256
\(43\) −12.8078 −1.95317 −0.976583 0.215142i \(-0.930979\pi\)
−0.976583 + 0.215142i \(0.930979\pi\)
\(44\) 2.56155 0.386169
\(45\) 0 0
\(46\) 1.43845 0.212087
\(47\) −5.12311 −0.747282 −0.373641 0.927573i \(-0.621891\pi\)
−0.373641 + 0.927573i \(0.621891\pi\)
\(48\) 1.00000 0.144338
\(49\) −0.438447 −0.0626353
\(50\) 0 0
\(51\) 3.12311 0.437322
\(52\) −2.00000 −0.277350
\(53\) −7.43845 −1.02175 −0.510875 0.859655i \(-0.670678\pi\)
−0.510875 + 0.859655i \(0.670678\pi\)
\(54\) −1.00000 −0.136083
\(55\) 0 0
\(56\) 2.56155 0.342302
\(57\) −7.68466 −1.01786
\(58\) −7.12311 −0.935310
\(59\) −13.1231 −1.70848 −0.854241 0.519877i \(-0.825978\pi\)
−0.854241 + 0.519877i \(0.825978\pi\)
\(60\) 0 0
\(61\) 6.00000 0.768221 0.384111 0.923287i \(-0.374508\pi\)
0.384111 + 0.923287i \(0.374508\pi\)
\(62\) −1.00000 −0.127000
\(63\) −2.56155 −0.322725
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) −2.56155 −0.315305
\(67\) 15.3693 1.87766 0.938830 0.344380i \(-0.111911\pi\)
0.938830 + 0.344380i \(0.111911\pi\)
\(68\) 3.12311 0.378732
\(69\) −1.43845 −0.173169
\(70\) 0 0
\(71\) −7.68466 −0.912001 −0.456001 0.889979i \(-0.650719\pi\)
−0.456001 + 0.889979i \(0.650719\pi\)
\(72\) −1.00000 −0.117851
\(73\) 10.8078 1.26495 0.632477 0.774580i \(-0.282038\pi\)
0.632477 + 0.774580i \(0.282038\pi\)
\(74\) −3.12311 −0.363054
\(75\) 0 0
\(76\) −7.68466 −0.881491
\(77\) −6.56155 −0.747758
\(78\) 2.00000 0.226455
\(79\) −4.31534 −0.485514 −0.242757 0.970087i \(-0.578052\pi\)
−0.242757 + 0.970087i \(0.578052\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) −7.12311 −0.786615
\(83\) −14.2462 −1.56372 −0.781862 0.623451i \(-0.785730\pi\)
−0.781862 + 0.623451i \(0.785730\pi\)
\(84\) −2.56155 −0.279488
\(85\) 0 0
\(86\) 12.8078 1.38110
\(87\) 7.12311 0.763677
\(88\) −2.56155 −0.273062
\(89\) −13.6847 −1.45057 −0.725285 0.688448i \(-0.758292\pi\)
−0.725285 + 0.688448i \(0.758292\pi\)
\(90\) 0 0
\(91\) 5.12311 0.537047
\(92\) −1.43845 −0.149968
\(93\) 1.00000 0.103695
\(94\) 5.12311 0.528408
\(95\) 0 0
\(96\) −1.00000 −0.102062
\(97\) 6.00000 0.609208 0.304604 0.952479i \(-0.401476\pi\)
0.304604 + 0.952479i \(0.401476\pi\)
\(98\) 0.438447 0.0442899
\(99\) 2.56155 0.257446
\(100\) 0 0
\(101\) −0.561553 −0.0558766 −0.0279383 0.999610i \(-0.508894\pi\)
−0.0279383 + 0.999610i \(0.508894\pi\)
\(102\) −3.12311 −0.309234
\(103\) 1.75379 0.172806 0.0864030 0.996260i \(-0.472463\pi\)
0.0864030 + 0.996260i \(0.472463\pi\)
\(104\) 2.00000 0.196116
\(105\) 0 0
\(106\) 7.43845 0.722486
\(107\) 2.56155 0.247635 0.123817 0.992305i \(-0.460486\pi\)
0.123817 + 0.992305i \(0.460486\pi\)
\(108\) 1.00000 0.0962250
\(109\) 5.36932 0.514287 0.257144 0.966373i \(-0.417219\pi\)
0.257144 + 0.966373i \(0.417219\pi\)
\(110\) 0 0
\(111\) 3.12311 0.296432
\(112\) −2.56155 −0.242044
\(113\) 1.68466 0.158479 0.0792397 0.996856i \(-0.474751\pi\)
0.0792397 + 0.996856i \(0.474751\pi\)
\(114\) 7.68466 0.719734
\(115\) 0 0
\(116\) 7.12311 0.661364
\(117\) −2.00000 −0.184900
\(118\) 13.1231 1.20808
\(119\) −8.00000 −0.733359
\(120\) 0 0
\(121\) −4.43845 −0.403495
\(122\) −6.00000 −0.543214
\(123\) 7.12311 0.642269
\(124\) 1.00000 0.0898027
\(125\) 0 0
\(126\) 2.56155 0.228201
\(127\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(128\) −1.00000 −0.0883883
\(129\) −12.8078 −1.12766
\(130\) 0 0
\(131\) −15.3693 −1.34282 −0.671412 0.741085i \(-0.734312\pi\)
−0.671412 + 0.741085i \(0.734312\pi\)
\(132\) 2.56155 0.222955
\(133\) 19.6847 1.70688
\(134\) −15.3693 −1.32771
\(135\) 0 0
\(136\) −3.12311 −0.267804
\(137\) −20.2462 −1.72975 −0.864875 0.501987i \(-0.832603\pi\)
−0.864875 + 0.501987i \(0.832603\pi\)
\(138\) 1.43845 0.122449
\(139\) −17.1231 −1.45236 −0.726181 0.687503i \(-0.758707\pi\)
−0.726181 + 0.687503i \(0.758707\pi\)
\(140\) 0 0
\(141\) −5.12311 −0.431443
\(142\) 7.68466 0.644882
\(143\) −5.12311 −0.428416
\(144\) 1.00000 0.0833333
\(145\) 0 0
\(146\) −10.8078 −0.894457
\(147\) −0.438447 −0.0361625
\(148\) 3.12311 0.256718
\(149\) 17.0540 1.39712 0.698558 0.715553i \(-0.253825\pi\)
0.698558 + 0.715553i \(0.253825\pi\)
\(150\) 0 0
\(151\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(152\) 7.68466 0.623308
\(153\) 3.12311 0.252488
\(154\) 6.56155 0.528745
\(155\) 0 0
\(156\) −2.00000 −0.160128
\(157\) −17.0540 −1.36106 −0.680528 0.732722i \(-0.738249\pi\)
−0.680528 + 0.732722i \(0.738249\pi\)
\(158\) 4.31534 0.343310
\(159\) −7.43845 −0.589907
\(160\) 0 0
\(161\) 3.68466 0.290392
\(162\) −1.00000 −0.0785674
\(163\) 15.3693 1.20382 0.601909 0.798565i \(-0.294407\pi\)
0.601909 + 0.798565i \(0.294407\pi\)
\(164\) 7.12311 0.556221
\(165\) 0 0
\(166\) 14.2462 1.10572
\(167\) 6.56155 0.507748 0.253874 0.967237i \(-0.418295\pi\)
0.253874 + 0.967237i \(0.418295\pi\)
\(168\) 2.56155 0.197628
\(169\) −9.00000 −0.692308
\(170\) 0 0
\(171\) −7.68466 −0.587661
\(172\) −12.8078 −0.976583
\(173\) −8.24621 −0.626948 −0.313474 0.949597i \(-0.601493\pi\)
−0.313474 + 0.949597i \(0.601493\pi\)
\(174\) −7.12311 −0.540001
\(175\) 0 0
\(176\) 2.56155 0.193084
\(177\) −13.1231 −0.986393
\(178\) 13.6847 1.02571
\(179\) −12.0000 −0.896922 −0.448461 0.893802i \(-0.648028\pi\)
−0.448461 + 0.893802i \(0.648028\pi\)
\(180\) 0 0
\(181\) −13.0540 −0.970294 −0.485147 0.874433i \(-0.661234\pi\)
−0.485147 + 0.874433i \(0.661234\pi\)
\(182\) −5.12311 −0.379750
\(183\) 6.00000 0.443533
\(184\) 1.43845 0.106044
\(185\) 0 0
\(186\) −1.00000 −0.0733236
\(187\) 8.00000 0.585018
\(188\) −5.12311 −0.373641
\(189\) −2.56155 −0.186326
\(190\) 0 0
\(191\) −14.2462 −1.03082 −0.515410 0.856944i \(-0.672360\pi\)
−0.515410 + 0.856944i \(0.672360\pi\)
\(192\) 1.00000 0.0721688
\(193\) −14.4924 −1.04319 −0.521594 0.853194i \(-0.674662\pi\)
−0.521594 + 0.853194i \(0.674662\pi\)
\(194\) −6.00000 −0.430775
\(195\) 0 0
\(196\) −0.438447 −0.0313177
\(197\) −6.00000 −0.427482 −0.213741 0.976890i \(-0.568565\pi\)
−0.213741 + 0.976890i \(0.568565\pi\)
\(198\) −2.56155 −0.182042
\(199\) −16.8078 −1.19147 −0.595735 0.803181i \(-0.703139\pi\)
−0.595735 + 0.803181i \(0.703139\pi\)
\(200\) 0 0
\(201\) 15.3693 1.08407
\(202\) 0.561553 0.0395107
\(203\) −18.2462 −1.28063
\(204\) 3.12311 0.218661
\(205\) 0 0
\(206\) −1.75379 −0.122192
\(207\) −1.43845 −0.0999790
\(208\) −2.00000 −0.138675
\(209\) −19.6847 −1.36162
\(210\) 0 0
\(211\) −23.6847 −1.63052 −0.815260 0.579096i \(-0.803406\pi\)
−0.815260 + 0.579096i \(0.803406\pi\)
\(212\) −7.43845 −0.510875
\(213\) −7.68466 −0.526544
\(214\) −2.56155 −0.175104
\(215\) 0 0
\(216\) −1.00000 −0.0680414
\(217\) −2.56155 −0.173890
\(218\) −5.36932 −0.363656
\(219\) 10.8078 0.730321
\(220\) 0 0
\(221\) −6.24621 −0.420166
\(222\) −3.12311 −0.209609
\(223\) 21.1231 1.41451 0.707254 0.706960i \(-0.249934\pi\)
0.707254 + 0.706960i \(0.249934\pi\)
\(224\) 2.56155 0.171151
\(225\) 0 0
\(226\) −1.68466 −0.112062
\(227\) −7.68466 −0.510049 −0.255024 0.966935i \(-0.582083\pi\)
−0.255024 + 0.966935i \(0.582083\pi\)
\(228\) −7.68466 −0.508929
\(229\) −16.5616 −1.09442 −0.547209 0.836996i \(-0.684310\pi\)
−0.547209 + 0.836996i \(0.684310\pi\)
\(230\) 0 0
\(231\) −6.56155 −0.431718
\(232\) −7.12311 −0.467655
\(233\) 17.0540 1.11724 0.558622 0.829423i \(-0.311330\pi\)
0.558622 + 0.829423i \(0.311330\pi\)
\(234\) 2.00000 0.130744
\(235\) 0 0
\(236\) −13.1231 −0.854241
\(237\) −4.31534 −0.280312
\(238\) 8.00000 0.518563
\(239\) 24.0000 1.55243 0.776215 0.630468i \(-0.217137\pi\)
0.776215 + 0.630468i \(0.217137\pi\)
\(240\) 0 0
\(241\) 12.2462 0.788848 0.394424 0.918929i \(-0.370944\pi\)
0.394424 + 0.918929i \(0.370944\pi\)
\(242\) 4.43845 0.285314
\(243\) 1.00000 0.0641500
\(244\) 6.00000 0.384111
\(245\) 0 0
\(246\) −7.12311 −0.454153
\(247\) 15.3693 0.977926
\(248\) −1.00000 −0.0635001
\(249\) −14.2462 −0.902817
\(250\) 0 0
\(251\) 16.4924 1.04099 0.520496 0.853864i \(-0.325747\pi\)
0.520496 + 0.853864i \(0.325747\pi\)
\(252\) −2.56155 −0.161363
\(253\) −3.68466 −0.231652
\(254\) 0 0
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 1.68466 0.105086 0.0525431 0.998619i \(-0.483267\pi\)
0.0525431 + 0.998619i \(0.483267\pi\)
\(258\) 12.8078 0.797377
\(259\) −8.00000 −0.497096
\(260\) 0 0
\(261\) 7.12311 0.440909
\(262\) 15.3693 0.949520
\(263\) 20.4924 1.26362 0.631808 0.775125i \(-0.282313\pi\)
0.631808 + 0.775125i \(0.282313\pi\)
\(264\) −2.56155 −0.157653
\(265\) 0 0
\(266\) −19.6847 −1.20694
\(267\) −13.6847 −0.837487
\(268\) 15.3693 0.938830
\(269\) −0.246211 −0.0150118 −0.00750588 0.999972i \(-0.502389\pi\)
−0.00750588 + 0.999972i \(0.502389\pi\)
\(270\) 0 0
\(271\) 27.0540 1.64341 0.821706 0.569912i \(-0.193023\pi\)
0.821706 + 0.569912i \(0.193023\pi\)
\(272\) 3.12311 0.189366
\(273\) 5.12311 0.310064
\(274\) 20.2462 1.22312
\(275\) 0 0
\(276\) −1.43845 −0.0865843
\(277\) 5.36932 0.322611 0.161305 0.986905i \(-0.448430\pi\)
0.161305 + 0.986905i \(0.448430\pi\)
\(278\) 17.1231 1.02698
\(279\) 1.00000 0.0598684
\(280\) 0 0
\(281\) 4.87689 0.290931 0.145466 0.989363i \(-0.453532\pi\)
0.145466 + 0.989363i \(0.453532\pi\)
\(282\) 5.12311 0.305077
\(283\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(284\) −7.68466 −0.456001
\(285\) 0 0
\(286\) 5.12311 0.302936
\(287\) −18.2462 −1.07704
\(288\) −1.00000 −0.0589256
\(289\) −7.24621 −0.426248
\(290\) 0 0
\(291\) 6.00000 0.351726
\(292\) 10.8078 0.632477
\(293\) −26.4924 −1.54770 −0.773852 0.633367i \(-0.781673\pi\)
−0.773852 + 0.633367i \(0.781673\pi\)
\(294\) 0.438447 0.0255708
\(295\) 0 0
\(296\) −3.12311 −0.181527
\(297\) 2.56155 0.148636
\(298\) −17.0540 −0.987910
\(299\) 2.87689 0.166375
\(300\) 0 0
\(301\) 32.8078 1.89101
\(302\) 0 0
\(303\) −0.561553 −0.0322604
\(304\) −7.68466 −0.440745
\(305\) 0 0
\(306\) −3.12311 −0.178536
\(307\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(308\) −6.56155 −0.373879
\(309\) 1.75379 0.0997696
\(310\) 0 0
\(311\) −1.75379 −0.0994482 −0.0497241 0.998763i \(-0.515834\pi\)
−0.0497241 + 0.998763i \(0.515834\pi\)
\(312\) 2.00000 0.113228
\(313\) 10.0000 0.565233 0.282617 0.959233i \(-0.408798\pi\)
0.282617 + 0.959233i \(0.408798\pi\)
\(314\) 17.0540 0.962412
\(315\) 0 0
\(316\) −4.31534 −0.242757
\(317\) −16.8769 −0.947901 −0.473950 0.880552i \(-0.657172\pi\)
−0.473950 + 0.880552i \(0.657172\pi\)
\(318\) 7.43845 0.417127
\(319\) 18.2462 1.02159
\(320\) 0 0
\(321\) 2.56155 0.142972
\(322\) −3.68466 −0.205338
\(323\) −24.0000 −1.33540
\(324\) 1.00000 0.0555556
\(325\) 0 0
\(326\) −15.3693 −0.851228
\(327\) 5.36932 0.296924
\(328\) −7.12311 −0.393308
\(329\) 13.1231 0.723500
\(330\) 0 0
\(331\) −6.24621 −0.343323 −0.171661 0.985156i \(-0.554914\pi\)
−0.171661 + 0.985156i \(0.554914\pi\)
\(332\) −14.2462 −0.781862
\(333\) 3.12311 0.171145
\(334\) −6.56155 −0.359032
\(335\) 0 0
\(336\) −2.56155 −0.139744
\(337\) 10.0000 0.544735 0.272367 0.962193i \(-0.412193\pi\)
0.272367 + 0.962193i \(0.412193\pi\)
\(338\) 9.00000 0.489535
\(339\) 1.68466 0.0914981
\(340\) 0 0
\(341\) 2.56155 0.138716
\(342\) 7.68466 0.415539
\(343\) 19.0540 1.02882
\(344\) 12.8078 0.690548
\(345\) 0 0
\(346\) 8.24621 0.443319
\(347\) −14.2462 −0.764777 −0.382388 0.924002i \(-0.624898\pi\)
−0.382388 + 0.924002i \(0.624898\pi\)
\(348\) 7.12311 0.381839
\(349\) 5.36932 0.287413 0.143706 0.989620i \(-0.454098\pi\)
0.143706 + 0.989620i \(0.454098\pi\)
\(350\) 0 0
\(351\) −2.00000 −0.106752
\(352\) −2.56155 −0.136531
\(353\) 0.246211 0.0131045 0.00655225 0.999979i \(-0.497914\pi\)
0.00655225 + 0.999979i \(0.497914\pi\)
\(354\) 13.1231 0.697485
\(355\) 0 0
\(356\) −13.6847 −0.725285
\(357\) −8.00000 −0.423405
\(358\) 12.0000 0.634220
\(359\) −31.6847 −1.67225 −0.836126 0.548537i \(-0.815185\pi\)
−0.836126 + 0.548537i \(0.815185\pi\)
\(360\) 0 0
\(361\) 40.0540 2.10810
\(362\) 13.0540 0.686102
\(363\) −4.43845 −0.232958
\(364\) 5.12311 0.268524
\(365\) 0 0
\(366\) −6.00000 −0.313625
\(367\) −15.3693 −0.802272 −0.401136 0.916019i \(-0.631384\pi\)
−0.401136 + 0.916019i \(0.631384\pi\)
\(368\) −1.43845 −0.0749842
\(369\) 7.12311 0.370814
\(370\) 0 0
\(371\) 19.0540 0.989233
\(372\) 1.00000 0.0518476
\(373\) 5.68466 0.294340 0.147170 0.989111i \(-0.452983\pi\)
0.147170 + 0.989111i \(0.452983\pi\)
\(374\) −8.00000 −0.413670
\(375\) 0 0
\(376\) 5.12311 0.264204
\(377\) −14.2462 −0.733717
\(378\) 2.56155 0.131752
\(379\) −7.05398 −0.362338 −0.181169 0.983452i \(-0.557988\pi\)
−0.181169 + 0.983452i \(0.557988\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 14.2462 0.728900
\(383\) 10.2462 0.523557 0.261778 0.965128i \(-0.415691\pi\)
0.261778 + 0.965128i \(0.415691\pi\)
\(384\) −1.00000 −0.0510310
\(385\) 0 0
\(386\) 14.4924 0.737645
\(387\) −12.8078 −0.651055
\(388\) 6.00000 0.304604
\(389\) 7.75379 0.393133 0.196566 0.980491i \(-0.437021\pi\)
0.196566 + 0.980491i \(0.437021\pi\)
\(390\) 0 0
\(391\) −4.49242 −0.227192
\(392\) 0.438447 0.0221449
\(393\) −15.3693 −0.775279
\(394\) 6.00000 0.302276
\(395\) 0 0
\(396\) 2.56155 0.128723
\(397\) −9.05398 −0.454406 −0.227203 0.973847i \(-0.572958\pi\)
−0.227203 + 0.973847i \(0.572958\pi\)
\(398\) 16.8078 0.842497
\(399\) 19.6847 0.985466
\(400\) 0 0
\(401\) −13.6847 −0.683379 −0.341690 0.939813i \(-0.610999\pi\)
−0.341690 + 0.939813i \(0.610999\pi\)
\(402\) −15.3693 −0.766552
\(403\) −2.00000 −0.0996271
\(404\) −0.561553 −0.0279383
\(405\) 0 0
\(406\) 18.2462 0.905544
\(407\) 8.00000 0.396545
\(408\) −3.12311 −0.154617
\(409\) −37.3693 −1.84779 −0.923897 0.382642i \(-0.875014\pi\)
−0.923897 + 0.382642i \(0.875014\pi\)
\(410\) 0 0
\(411\) −20.2462 −0.998672
\(412\) 1.75379 0.0864030
\(413\) 33.6155 1.65411
\(414\) 1.43845 0.0706958
\(415\) 0 0
\(416\) 2.00000 0.0980581
\(417\) −17.1231 −0.838522
\(418\) 19.6847 0.962808
\(419\) 5.75379 0.281091 0.140545 0.990074i \(-0.455114\pi\)
0.140545 + 0.990074i \(0.455114\pi\)
\(420\) 0 0
\(421\) −14.4924 −0.706317 −0.353159 0.935563i \(-0.614892\pi\)
−0.353159 + 0.935563i \(0.614892\pi\)
\(422\) 23.6847 1.15295
\(423\) −5.12311 −0.249094
\(424\) 7.43845 0.361243
\(425\) 0 0
\(426\) 7.68466 0.372323
\(427\) −15.3693 −0.743773
\(428\) 2.56155 0.123817
\(429\) −5.12311 −0.247346
\(430\) 0 0
\(431\) −30.2462 −1.45691 −0.728454 0.685094i \(-0.759761\pi\)
−0.728454 + 0.685094i \(0.759761\pi\)
\(432\) 1.00000 0.0481125
\(433\) −35.3002 −1.69642 −0.848209 0.529661i \(-0.822319\pi\)
−0.848209 + 0.529661i \(0.822319\pi\)
\(434\) 2.56155 0.122958
\(435\) 0 0
\(436\) 5.36932 0.257144
\(437\) 11.0540 0.528783
\(438\) −10.8078 −0.516415
\(439\) 9.61553 0.458924 0.229462 0.973318i \(-0.426303\pi\)
0.229462 + 0.973318i \(0.426303\pi\)
\(440\) 0 0
\(441\) −0.438447 −0.0208784
\(442\) 6.24621 0.297102
\(443\) −23.0540 −1.09533 −0.547664 0.836699i \(-0.684483\pi\)
−0.547664 + 0.836699i \(0.684483\pi\)
\(444\) 3.12311 0.148216
\(445\) 0 0
\(446\) −21.1231 −1.00021
\(447\) 17.0540 0.806625
\(448\) −2.56155 −0.121022
\(449\) 10.4924 0.495168 0.247584 0.968866i \(-0.420363\pi\)
0.247584 + 0.968866i \(0.420363\pi\)
\(450\) 0 0
\(451\) 18.2462 0.859181
\(452\) 1.68466 0.0792397
\(453\) 0 0
\(454\) 7.68466 0.360659
\(455\) 0 0
\(456\) 7.68466 0.359867
\(457\) 24.7386 1.15722 0.578612 0.815603i \(-0.303594\pi\)
0.578612 + 0.815603i \(0.303594\pi\)
\(458\) 16.5616 0.773871
\(459\) 3.12311 0.145774
\(460\) 0 0
\(461\) 9.36932 0.436373 0.218186 0.975907i \(-0.429986\pi\)
0.218186 + 0.975907i \(0.429986\pi\)
\(462\) 6.56155 0.305271
\(463\) −30.7386 −1.42855 −0.714273 0.699867i \(-0.753242\pi\)
−0.714273 + 0.699867i \(0.753242\pi\)
\(464\) 7.12311 0.330682
\(465\) 0 0
\(466\) −17.0540 −0.790010
\(467\) 9.75379 0.451352 0.225676 0.974202i \(-0.427541\pi\)
0.225676 + 0.974202i \(0.427541\pi\)
\(468\) −2.00000 −0.0924500
\(469\) −39.3693 −1.81791
\(470\) 0 0
\(471\) −17.0540 −0.785806
\(472\) 13.1231 0.604040
\(473\) −32.8078 −1.50850
\(474\) 4.31534 0.198210
\(475\) 0 0
\(476\) −8.00000 −0.366679
\(477\) −7.43845 −0.340583
\(478\) −24.0000 −1.09773
\(479\) 10.5616 0.482570 0.241285 0.970454i \(-0.422431\pi\)
0.241285 + 0.970454i \(0.422431\pi\)
\(480\) 0 0
\(481\) −6.24621 −0.284803
\(482\) −12.2462 −0.557800
\(483\) 3.68466 0.167658
\(484\) −4.43845 −0.201748
\(485\) 0 0
\(486\) −1.00000 −0.0453609
\(487\) −24.0000 −1.08754 −0.543772 0.839233i \(-0.683004\pi\)
−0.543772 + 0.839233i \(0.683004\pi\)
\(488\) −6.00000 −0.271607
\(489\) 15.3693 0.695025
\(490\) 0 0
\(491\) −25.9309 −1.17024 −0.585122 0.810945i \(-0.698953\pi\)
−0.585122 + 0.810945i \(0.698953\pi\)
\(492\) 7.12311 0.321134
\(493\) 22.2462 1.00192
\(494\) −15.3693 −0.691498
\(495\) 0 0
\(496\) 1.00000 0.0449013
\(497\) 19.6847 0.882978
\(498\) 14.2462 0.638388
\(499\) 20.0000 0.895323 0.447661 0.894203i \(-0.352257\pi\)
0.447661 + 0.894203i \(0.352257\pi\)
\(500\) 0 0
\(501\) 6.56155 0.293149
\(502\) −16.4924 −0.736093
\(503\) −26.2462 −1.17026 −0.585130 0.810939i \(-0.698957\pi\)
−0.585130 + 0.810939i \(0.698957\pi\)
\(504\) 2.56155 0.114101
\(505\) 0 0
\(506\) 3.68466 0.163803
\(507\) −9.00000 −0.399704
\(508\) 0 0
\(509\) 19.6155 0.869443 0.434721 0.900565i \(-0.356847\pi\)
0.434721 + 0.900565i \(0.356847\pi\)
\(510\) 0 0
\(511\) −27.6847 −1.22470
\(512\) −1.00000 −0.0441942
\(513\) −7.68466 −0.339286
\(514\) −1.68466 −0.0743071
\(515\) 0 0
\(516\) −12.8078 −0.563830
\(517\) −13.1231 −0.577154
\(518\) 8.00000 0.351500
\(519\) −8.24621 −0.361968
\(520\) 0 0
\(521\) −32.2462 −1.41273 −0.706366 0.707847i \(-0.749667\pi\)
−0.706366 + 0.707847i \(0.749667\pi\)
\(522\) −7.12311 −0.311770
\(523\) 22.4233 0.980502 0.490251 0.871581i \(-0.336905\pi\)
0.490251 + 0.871581i \(0.336905\pi\)
\(524\) −15.3693 −0.671412
\(525\) 0 0
\(526\) −20.4924 −0.893512
\(527\) 3.12311 0.136045
\(528\) 2.56155 0.111477
\(529\) −20.9309 −0.910038
\(530\) 0 0
\(531\) −13.1231 −0.569494
\(532\) 19.6847 0.853438
\(533\) −14.2462 −0.617072
\(534\) 13.6847 0.592193
\(535\) 0 0
\(536\) −15.3693 −0.663853
\(537\) −12.0000 −0.517838
\(538\) 0.246211 0.0106149
\(539\) −1.12311 −0.0483756
\(540\) 0 0
\(541\) 19.7538 0.849282 0.424641 0.905362i \(-0.360400\pi\)
0.424641 + 0.905362i \(0.360400\pi\)
\(542\) −27.0540 −1.16207
\(543\) −13.0540 −0.560200
\(544\) −3.12311 −0.133902
\(545\) 0 0
\(546\) −5.12311 −0.219249
\(547\) −26.8769 −1.14917 −0.574587 0.818444i \(-0.694837\pi\)
−0.574587 + 0.818444i \(0.694837\pi\)
\(548\) −20.2462 −0.864875
\(549\) 6.00000 0.256074
\(550\) 0 0
\(551\) −54.7386 −2.33194
\(552\) 1.43845 0.0612244
\(553\) 11.0540 0.470063
\(554\) −5.36932 −0.228120
\(555\) 0 0
\(556\) −17.1231 −0.726181
\(557\) 26.8078 1.13588 0.567941 0.823069i \(-0.307740\pi\)
0.567941 + 0.823069i \(0.307740\pi\)
\(558\) −1.00000 −0.0423334
\(559\) 25.6155 1.08342
\(560\) 0 0
\(561\) 8.00000 0.337760
\(562\) −4.87689 −0.205719
\(563\) −16.4924 −0.695073 −0.347536 0.937667i \(-0.612982\pi\)
−0.347536 + 0.937667i \(0.612982\pi\)
\(564\) −5.12311 −0.215722
\(565\) 0 0
\(566\) 0 0
\(567\) −2.56155 −0.107575
\(568\) 7.68466 0.322441
\(569\) 30.8078 1.29153 0.645764 0.763537i \(-0.276539\pi\)
0.645764 + 0.763537i \(0.276539\pi\)
\(570\) 0 0
\(571\) 41.1231 1.72095 0.860474 0.509494i \(-0.170167\pi\)
0.860474 + 0.509494i \(0.170167\pi\)
\(572\) −5.12311 −0.214208
\(573\) −14.2462 −0.595144
\(574\) 18.2462 0.761582
\(575\) 0 0
\(576\) 1.00000 0.0416667
\(577\) −15.1231 −0.629583 −0.314792 0.949161i \(-0.601935\pi\)
−0.314792 + 0.949161i \(0.601935\pi\)
\(578\) 7.24621 0.301403
\(579\) −14.4924 −0.602285
\(580\) 0 0
\(581\) 36.4924 1.51396
\(582\) −6.00000 −0.248708
\(583\) −19.0540 −0.789135
\(584\) −10.8078 −0.447228
\(585\) 0 0
\(586\) 26.4924 1.09439
\(587\) 0.492423 0.0203245 0.0101622 0.999948i \(-0.496765\pi\)
0.0101622 + 0.999948i \(0.496765\pi\)
\(588\) −0.438447 −0.0180813
\(589\) −7.68466 −0.316641
\(590\) 0 0
\(591\) −6.00000 −0.246807
\(592\) 3.12311 0.128359
\(593\) 30.0000 1.23195 0.615976 0.787765i \(-0.288762\pi\)
0.615976 + 0.787765i \(0.288762\pi\)
\(594\) −2.56155 −0.105102
\(595\) 0 0
\(596\) 17.0540 0.698558
\(597\) −16.8078 −0.687896
\(598\) −2.87689 −0.117645
\(599\) 38.4233 1.56993 0.784967 0.619538i \(-0.212680\pi\)
0.784967 + 0.619538i \(0.212680\pi\)
\(600\) 0 0
\(601\) 2.00000 0.0815817 0.0407909 0.999168i \(-0.487012\pi\)
0.0407909 + 0.999168i \(0.487012\pi\)
\(602\) −32.8078 −1.33714
\(603\) 15.3693 0.625887
\(604\) 0 0
\(605\) 0 0
\(606\) 0.561553 0.0228115
\(607\) −7.05398 −0.286312 −0.143156 0.989700i \(-0.545725\pi\)
−0.143156 + 0.989700i \(0.545725\pi\)
\(608\) 7.68466 0.311654
\(609\) −18.2462 −0.739374
\(610\) 0 0
\(611\) 10.2462 0.414517
\(612\) 3.12311 0.126244
\(613\) −2.00000 −0.0807792 −0.0403896 0.999184i \(-0.512860\pi\)
−0.0403896 + 0.999184i \(0.512860\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 6.56155 0.264372
\(617\) 15.4384 0.621528 0.310764 0.950487i \(-0.399415\pi\)
0.310764 + 0.950487i \(0.399415\pi\)
\(618\) −1.75379 −0.0705477
\(619\) 11.3693 0.456971 0.228486 0.973547i \(-0.426623\pi\)
0.228486 + 0.973547i \(0.426623\pi\)
\(620\) 0 0
\(621\) −1.43845 −0.0577229
\(622\) 1.75379 0.0703205
\(623\) 35.0540 1.40441
\(624\) −2.00000 −0.0800641
\(625\) 0 0
\(626\) −10.0000 −0.399680
\(627\) −19.6847 −0.786130
\(628\) −17.0540 −0.680528
\(629\) 9.75379 0.388909
\(630\) 0 0
\(631\) 29.3002 1.16642 0.583211 0.812321i \(-0.301796\pi\)
0.583211 + 0.812321i \(0.301796\pi\)
\(632\) 4.31534 0.171655
\(633\) −23.6847 −0.941381
\(634\) 16.8769 0.670267
\(635\) 0 0
\(636\) −7.43845 −0.294954
\(637\) 0.876894 0.0347438
\(638\) −18.2462 −0.722374
\(639\) −7.68466 −0.304000
\(640\) 0 0
\(641\) −5.50758 −0.217536 −0.108768 0.994067i \(-0.534691\pi\)
−0.108768 + 0.994067i \(0.534691\pi\)
\(642\) −2.56155 −0.101096
\(643\) 7.05398 0.278182 0.139091 0.990280i \(-0.455582\pi\)
0.139091 + 0.990280i \(0.455582\pi\)
\(644\) 3.68466 0.145196
\(645\) 0 0
\(646\) 24.0000 0.944267
\(647\) 45.9309 1.80573 0.902864 0.429925i \(-0.141460\pi\)
0.902864 + 0.429925i \(0.141460\pi\)
\(648\) −1.00000 −0.0392837
\(649\) −33.6155 −1.31952
\(650\) 0 0
\(651\) −2.56155 −0.100395
\(652\) 15.3693 0.601909
\(653\) −40.2462 −1.57496 −0.787478 0.616343i \(-0.788614\pi\)
−0.787478 + 0.616343i \(0.788614\pi\)
\(654\) −5.36932 −0.209957
\(655\) 0 0
\(656\) 7.12311 0.278111
\(657\) 10.8078 0.421651
\(658\) −13.1231 −0.511592
\(659\) −30.7386 −1.19741 −0.598704 0.800971i \(-0.704317\pi\)
−0.598704 + 0.800971i \(0.704317\pi\)
\(660\) 0 0
\(661\) 26.4924 1.03044 0.515218 0.857059i \(-0.327711\pi\)
0.515218 + 0.857059i \(0.327711\pi\)
\(662\) 6.24621 0.242766
\(663\) −6.24621 −0.242583
\(664\) 14.2462 0.552860
\(665\) 0 0
\(666\) −3.12311 −0.121018
\(667\) −10.2462 −0.396735
\(668\) 6.56155 0.253874
\(669\) 21.1231 0.816666
\(670\) 0 0
\(671\) 15.3693 0.593326
\(672\) 2.56155 0.0988140
\(673\) −11.7538 −0.453075 −0.226538 0.974002i \(-0.572741\pi\)
−0.226538 + 0.974002i \(0.572741\pi\)
\(674\) −10.0000 −0.385186
\(675\) 0 0
\(676\) −9.00000 −0.346154
\(677\) −32.4233 −1.24613 −0.623064 0.782171i \(-0.714112\pi\)
−0.623064 + 0.782171i \(0.714112\pi\)
\(678\) −1.68466 −0.0646989
\(679\) −15.3693 −0.589820
\(680\) 0 0
\(681\) −7.68466 −0.294477
\(682\) −2.56155 −0.0980869
\(683\) −31.6847 −1.21238 −0.606190 0.795320i \(-0.707303\pi\)
−0.606190 + 0.795320i \(0.707303\pi\)
\(684\) −7.68466 −0.293830
\(685\) 0 0
\(686\) −19.0540 −0.727484
\(687\) −16.5616 −0.631863
\(688\) −12.8078 −0.488291
\(689\) 14.8769 0.566765
\(690\) 0 0
\(691\) 15.0540 0.572680 0.286340 0.958128i \(-0.407561\pi\)
0.286340 + 0.958128i \(0.407561\pi\)
\(692\) −8.24621 −0.313474
\(693\) −6.56155 −0.249253
\(694\) 14.2462 0.540779
\(695\) 0 0
\(696\) −7.12311 −0.270001
\(697\) 22.2462 0.842635
\(698\) −5.36932 −0.203232
\(699\) 17.0540 0.645041
\(700\) 0 0
\(701\) 5.19224 0.196108 0.0980540 0.995181i \(-0.468738\pi\)
0.0980540 + 0.995181i \(0.468738\pi\)
\(702\) 2.00000 0.0754851
\(703\) −24.0000 −0.905177
\(704\) 2.56155 0.0965422
\(705\) 0 0
\(706\) −0.246211 −0.00926628
\(707\) 1.43845 0.0540984
\(708\) −13.1231 −0.493197
\(709\) 17.0540 0.640475 0.320238 0.947337i \(-0.396237\pi\)
0.320238 + 0.947337i \(0.396237\pi\)
\(710\) 0 0
\(711\) −4.31534 −0.161838
\(712\) 13.6847 0.512854
\(713\) −1.43845 −0.0538703
\(714\) 8.00000 0.299392
\(715\) 0 0
\(716\) −12.0000 −0.448461
\(717\) 24.0000 0.896296
\(718\) 31.6847 1.18246
\(719\) −34.8769 −1.30069 −0.650344 0.759640i \(-0.725375\pi\)
−0.650344 + 0.759640i \(0.725375\pi\)
\(720\) 0 0
\(721\) −4.49242 −0.167307
\(722\) −40.0540 −1.49065
\(723\) 12.2462 0.455441
\(724\) −13.0540 −0.485147
\(725\) 0 0
\(726\) 4.43845 0.164726
\(727\) −23.0540 −0.855025 −0.427512 0.904010i \(-0.640610\pi\)
−0.427512 + 0.904010i \(0.640610\pi\)
\(728\) −5.12311 −0.189875
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −40.0000 −1.47945
\(732\) 6.00000 0.221766
\(733\) 6.49242 0.239803 0.119902 0.992786i \(-0.461742\pi\)
0.119902 + 0.992786i \(0.461742\pi\)
\(734\) 15.3693 0.567292
\(735\) 0 0
\(736\) 1.43845 0.0530219
\(737\) 39.3693 1.45019
\(738\) −7.12311 −0.262205
\(739\) 52.9848 1.94908 0.974540 0.224216i \(-0.0719821\pi\)
0.974540 + 0.224216i \(0.0719821\pi\)
\(740\) 0 0
\(741\) 15.3693 0.564606
\(742\) −19.0540 −0.699493
\(743\) 50.4233 1.84985 0.924926 0.380148i \(-0.124127\pi\)
0.924926 + 0.380148i \(0.124127\pi\)
\(744\) −1.00000 −0.0366618
\(745\) 0 0
\(746\) −5.68466 −0.208130
\(747\) −14.2462 −0.521242
\(748\) 8.00000 0.292509
\(749\) −6.56155 −0.239754
\(750\) 0 0
\(751\) 45.1231 1.64657 0.823283 0.567631i \(-0.192140\pi\)
0.823283 + 0.567631i \(0.192140\pi\)
\(752\) −5.12311 −0.186820
\(753\) 16.4924 0.601017
\(754\) 14.2462 0.518816
\(755\) 0 0
\(756\) −2.56155 −0.0931628
\(757\) −10.0000 −0.363456 −0.181728 0.983349i \(-0.558169\pi\)
−0.181728 + 0.983349i \(0.558169\pi\)
\(758\) 7.05398 0.256212
\(759\) −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
\(763\) −13.7538 −0.497921
\(764\) −14.2462 −0.515410
\(765\) 0 0
\(766\) −10.2462 −0.370211
\(767\) 26.2462 0.947696
\(768\) 1.00000 0.0360844
\(769\) −20.5616 −0.741469 −0.370734 0.928739i \(-0.620894\pi\)
−0.370734 + 0.928739i \(0.620894\pi\)
\(770\) 0 0
\(771\) 1.68466 0.0606715
\(772\) −14.4924 −0.521594
\(773\) 11.4384 0.411412 0.205706 0.978614i \(-0.434051\pi\)
0.205706 + 0.978614i \(0.434051\pi\)
\(774\) 12.8078 0.460366
\(775\) 0 0
\(776\) −6.00000 −0.215387
\(777\) −8.00000 −0.286998
\(778\) −7.75379 −0.277987
\(779\) −54.7386 −1.96122
\(780\) 0 0
\(781\) −19.6847 −0.704372
\(782\) 4.49242 0.160649
\(783\) 7.12311 0.254559
\(784\) −0.438447 −0.0156588
\(785\) 0 0
\(786\) 15.3693 0.548205
\(787\) 17.9309 0.639166 0.319583 0.947558i \(-0.396457\pi\)
0.319583 + 0.947558i \(0.396457\pi\)
\(788\) −6.00000 −0.213741
\(789\) 20.4924 0.729550
\(790\) 0 0
\(791\) −4.31534 −0.153436
\(792\) −2.56155 −0.0910208
\(793\) −12.0000 −0.426132
\(794\) 9.05398 0.321314
\(795\) 0 0
\(796\) −16.8078 −0.595735
\(797\) 26.9848 0.955852 0.477926 0.878400i \(-0.341389\pi\)
0.477926 + 0.878400i \(0.341389\pi\)
\(798\) −19.6847 −0.696829
\(799\) −16.0000 −0.566039
\(800\) 0 0
\(801\) −13.6847 −0.483524
\(802\) 13.6847 0.483222
\(803\) 27.6847 0.976970
\(804\) 15.3693 0.542034
\(805\) 0 0
\(806\) 2.00000 0.0704470
\(807\) −0.246211 −0.00866705
\(808\) 0.561553 0.0197554
\(809\) −37.6847 −1.32492 −0.662461 0.749096i \(-0.730488\pi\)
−0.662461 + 0.749096i \(0.730488\pi\)
\(810\) 0 0
\(811\) −24.6695 −0.866263 −0.433132 0.901331i \(-0.642592\pi\)
−0.433132 + 0.901331i \(0.642592\pi\)
\(812\) −18.2462 −0.640316
\(813\) 27.0540 0.948824
\(814\) −8.00000 −0.280400
\(815\) 0 0
\(816\) 3.12311 0.109331
\(817\) 98.4233 3.44340
\(818\) 37.3693 1.30659
\(819\) 5.12311 0.179016
\(820\) 0 0
\(821\) 19.6155 0.684587 0.342293 0.939593i \(-0.388796\pi\)
0.342293 + 0.939593i \(0.388796\pi\)
\(822\) 20.2462 0.706168
\(823\) 25.6155 0.892901 0.446451 0.894808i \(-0.352688\pi\)
0.446451 + 0.894808i \(0.352688\pi\)
\(824\) −1.75379 −0.0610961
\(825\) 0 0
\(826\) −33.6155 −1.16963
\(827\) −1.75379 −0.0609852 −0.0304926 0.999535i \(-0.509708\pi\)
−0.0304926 + 0.999535i \(0.509708\pi\)
\(828\) −1.43845 −0.0499895
\(829\) 24.4233 0.848256 0.424128 0.905602i \(-0.360581\pi\)
0.424128 + 0.905602i \(0.360581\pi\)
\(830\) 0 0
\(831\) 5.36932 0.186260
\(832\) −2.00000 −0.0693375
\(833\) −1.36932 −0.0474440
\(834\) 17.1231 0.592925
\(835\) 0 0
\(836\) −19.6847 −0.680808
\(837\) 1.00000 0.0345651
\(838\) −5.75379 −0.198761
\(839\) 6.06913 0.209530 0.104765 0.994497i \(-0.466591\pi\)
0.104765 + 0.994497i \(0.466591\pi\)
\(840\) 0 0
\(841\) 21.7386 0.749608
\(842\) 14.4924 0.499442
\(843\) 4.87689 0.167969
\(844\) −23.6847 −0.815260
\(845\) 0 0
\(846\) 5.12311 0.176136
\(847\) 11.3693 0.390654
\(848\) −7.43845 −0.255437
\(849\) 0 0
\(850\) 0 0
\(851\) −4.49242 −0.153998
\(852\) −7.68466 −0.263272
\(853\) −47.7926 −1.63639 −0.818194 0.574942i \(-0.805024\pi\)
−0.818194 + 0.574942i \(0.805024\pi\)
\(854\) 15.3693 0.525927
\(855\) 0 0
\(856\) −2.56155 −0.0875521
\(857\) −29.2311 −0.998514 −0.499257 0.866454i \(-0.666394\pi\)
−0.499257 + 0.866454i \(0.666394\pi\)
\(858\) 5.12311 0.174900
\(859\) 26.7386 0.912310 0.456155 0.889900i \(-0.349226\pi\)
0.456155 + 0.889900i \(0.349226\pi\)
\(860\) 0 0
\(861\) −18.2462 −0.621829
\(862\) 30.2462 1.03019
\(863\) −39.5464 −1.34618 −0.673088 0.739563i \(-0.735032\pi\)
−0.673088 + 0.739563i \(0.735032\pi\)
\(864\) −1.00000 −0.0340207
\(865\) 0 0
\(866\) 35.3002 1.19955
\(867\) −7.24621 −0.246094
\(868\) −2.56155 −0.0869448
\(869\) −11.0540 −0.374980
\(870\) 0 0
\(871\) −30.7386 −1.04154
\(872\) −5.36932 −0.181828
\(873\) 6.00000 0.203069
\(874\) −11.0540 −0.373906
\(875\) 0 0
\(876\) 10.8078 0.365161
\(877\) 50.0000 1.68838 0.844190 0.536044i \(-0.180082\pi\)
0.844190 + 0.536044i \(0.180082\pi\)
\(878\) −9.61553 −0.324508
\(879\) −26.4924 −0.893567
\(880\) 0 0
\(881\) 9.50758 0.320318 0.160159 0.987091i \(-0.448799\pi\)
0.160159 + 0.987091i \(0.448799\pi\)
\(882\) 0.438447 0.0147633
\(883\) −30.4233 −1.02383 −0.511913 0.859038i \(-0.671063\pi\)
−0.511913 + 0.859038i \(0.671063\pi\)
\(884\) −6.24621 −0.210083
\(885\) 0 0
\(886\) 23.0540 0.774513
\(887\) 32.6307 1.09563 0.547816 0.836599i \(-0.315460\pi\)
0.547816 + 0.836599i \(0.315460\pi\)
\(888\) −3.12311 −0.104805
\(889\) 0 0
\(890\) 0 0
\(891\) 2.56155 0.0858152
\(892\) 21.1231 0.707254
\(893\) 39.3693 1.31744
\(894\) −17.0540 −0.570370
\(895\) 0 0
\(896\) 2.56155 0.0855755
\(897\) 2.87689 0.0960567
\(898\) −10.4924 −0.350137
\(899\) 7.12311 0.237569
\(900\) 0 0
\(901\) −23.2311 −0.773939
\(902\) −18.2462 −0.607532
\(903\) 32.8078 1.09177
\(904\) −1.68466 −0.0560309
\(905\) 0 0
\(906\) 0 0
\(907\) −24.0000 −0.796907 −0.398453 0.917189i \(-0.630453\pi\)
−0.398453 + 0.917189i \(0.630453\pi\)
\(908\) −7.68466 −0.255024
\(909\) −0.561553 −0.0186255
\(910\) 0 0
\(911\) −11.5076 −0.381263 −0.190632 0.981662i \(-0.561054\pi\)
−0.190632 + 0.981662i \(0.561054\pi\)
\(912\) −7.68466 −0.254464
\(913\) −36.4924 −1.20772
\(914\) −24.7386 −0.818281
\(915\) 0 0
\(916\) −16.5616 −0.547209
\(917\) 39.3693 1.30009
\(918\) −3.12311 −0.103078
\(919\) −1.61553 −0.0532914 −0.0266457 0.999645i \(-0.508483\pi\)
−0.0266457 + 0.999645i \(0.508483\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) −9.36932 −0.308562
\(923\) 15.3693 0.505887
\(924\) −6.56155 −0.215859
\(925\) 0 0
\(926\) 30.7386 1.01013
\(927\) 1.75379 0.0576020
\(928\) −7.12311 −0.233827
\(929\) 7.43845 0.244048 0.122024 0.992527i \(-0.461062\pi\)
0.122024 + 0.992527i \(0.461062\pi\)
\(930\) 0 0
\(931\) 3.36932 0.110425
\(932\) 17.0540 0.558622
\(933\) −1.75379 −0.0574165
\(934\) −9.75379 −0.319154
\(935\) 0 0
\(936\) 2.00000 0.0653720
\(937\) −41.3693 −1.35148 −0.675738 0.737142i \(-0.736175\pi\)
−0.675738 + 0.737142i \(0.736175\pi\)
\(938\) 39.3693 1.28545
\(939\) 10.0000 0.326338
\(940\) 0 0
\(941\) 2.63068 0.0857578 0.0428789 0.999080i \(-0.486347\pi\)
0.0428789 + 0.999080i \(0.486347\pi\)
\(942\) 17.0540 0.555649
\(943\) −10.2462 −0.333663
\(944\) −13.1231 −0.427121
\(945\) 0 0
\(946\) 32.8078 1.06667
\(947\) 9.12311 0.296461 0.148231 0.988953i \(-0.452642\pi\)
0.148231 + 0.988953i \(0.452642\pi\)
\(948\) −4.31534 −0.140156
\(949\) −21.6155 −0.701670
\(950\) 0 0
\(951\) −16.8769 −0.547271
\(952\) 8.00000 0.259281
\(953\) −16.1080 −0.521788 −0.260894 0.965367i \(-0.584017\pi\)
−0.260894 + 0.965367i \(0.584017\pi\)
\(954\) 7.43845 0.240829
\(955\) 0 0
\(956\) 24.0000 0.776215
\(957\) 18.2462 0.589816
\(958\) −10.5616 −0.341228
\(959\) 51.8617 1.67470
\(960\) 0 0
\(961\) 1.00000 0.0322581
\(962\) 6.24621 0.201386
\(963\) 2.56155 0.0825449
\(964\) 12.2462 0.394424
\(965\) 0 0
\(966\) −3.68466 −0.118552
\(967\) 42.2462 1.35855 0.679273 0.733885i \(-0.262295\pi\)
0.679273 + 0.733885i \(0.262295\pi\)
\(968\) 4.43845 0.142657
\(969\) −24.0000 −0.770991
\(970\) 0 0
\(971\) −37.1231 −1.19134 −0.595669 0.803230i \(-0.703113\pi\)
−0.595669 + 0.803230i \(0.703113\pi\)
\(972\) 1.00000 0.0320750
\(973\) 43.8617 1.40614
\(974\) 24.0000 0.769010
\(975\) 0 0
\(976\) 6.00000 0.192055
\(977\) −42.9848 −1.37521 −0.687604 0.726086i \(-0.741337\pi\)
−0.687604 + 0.726086i \(0.741337\pi\)
\(978\) −15.3693 −0.491457
\(979\) −35.0540 −1.12033
\(980\) 0 0
\(981\) 5.36932 0.171429
\(982\) 25.9309 0.827487
\(983\) −40.9848 −1.30721 −0.653607 0.756834i \(-0.726745\pi\)
−0.653607 + 0.756834i \(0.726745\pi\)
\(984\) −7.12311 −0.227076
\(985\) 0 0
\(986\) −22.2462 −0.708464
\(987\) 13.1231 0.417713
\(988\) 15.3693 0.488963
\(989\) 18.4233 0.585827
\(990\) 0 0
\(991\) 35.6847 1.13356 0.566780 0.823869i \(-0.308189\pi\)
0.566780 + 0.823869i \(0.308189\pi\)
\(992\) −1.00000 −0.0317500
\(993\) −6.24621 −0.198218
\(994\) −19.6847 −0.624359
\(995\) 0 0
\(996\) −14.2462 −0.451408
\(997\) 29.5076 0.934514 0.467257 0.884121i \(-0.345242\pi\)
0.467257 + 0.884121i \(0.345242\pi\)
\(998\) −20.0000 −0.633089
\(999\) 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.a.ce.1.1 2
5.2 odd 4 4650.2.d.bd.3349.1 4
5.3 odd 4 4650.2.d.bd.3349.4 4
5.4 even 2 930.2.a.p.1.2 2
15.14 odd 2 2790.2.a.be.1.2 2
20.19 odd 2 7440.2.a.bl.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
930.2.a.p.1.2 2 5.4 even 2
2790.2.a.be.1.2 2 15.14 odd 2
4650.2.a.ce.1.1 2 1.1 even 1 trivial
4650.2.d.bd.3349.1 4 5.2 odd 4
4650.2.d.bd.3349.4 4 5.3 odd 4
7440.2.a.bl.1.1 2 20.19 odd 2