Properties

Label 2850.2.a.bd.1.2
Level $2850$
Weight $2$
Character 2850.1
Self dual yes
Analytic conductor $22.757$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 2850 = 2 \cdot 3 \cdot 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2850.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(22.7573645761\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{12})^+\)
Defining polynomial: \(x^{2} - 3\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.73205\) of defining polynomial
Character \(\chi\) \(=\) 2850.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} +0.732051 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} +0.732051 q^{7} -1.00000 q^{8} +1.00000 q^{9} +3.73205 q^{11} -1.00000 q^{12} -2.73205 q^{13} -0.732051 q^{14} +1.00000 q^{16} -6.19615 q^{17} -1.00000 q^{18} -1.00000 q^{19} -0.732051 q^{21} -3.73205 q^{22} +5.92820 q^{23} +1.00000 q^{24} +2.73205 q^{26} -1.00000 q^{27} +0.732051 q^{28} -1.73205 q^{29} -2.46410 q^{31} -1.00000 q^{32} -3.73205 q^{33} +6.19615 q^{34} +1.00000 q^{36} -2.00000 q^{37} +1.00000 q^{38} +2.73205 q^{39} -2.92820 q^{41} +0.732051 q^{42} -8.19615 q^{43} +3.73205 q^{44} -5.92820 q^{46} +3.46410 q^{47} -1.00000 q^{48} -6.46410 q^{49} +6.19615 q^{51} -2.73205 q^{52} -1.73205 q^{53} +1.00000 q^{54} -0.732051 q^{56} +1.00000 q^{57} +1.73205 q^{58} -8.19615 q^{59} +10.6603 q^{61} +2.46410 q^{62} +0.732051 q^{63} +1.00000 q^{64} +3.73205 q^{66} -0.267949 q^{67} -6.19615 q^{68} -5.92820 q^{69} +12.1962 q^{71} -1.00000 q^{72} -2.46410 q^{73} +2.00000 q^{74} -1.00000 q^{76} +2.73205 q^{77} -2.73205 q^{78} -5.53590 q^{79} +1.00000 q^{81} +2.92820 q^{82} +3.73205 q^{83} -0.732051 q^{84} +8.19615 q^{86} +1.73205 q^{87} -3.73205 q^{88} +10.8564 q^{89} -2.00000 q^{91} +5.92820 q^{92} +2.46410 q^{93} -3.46410 q^{94} +1.00000 q^{96} -15.1244 q^{97} +6.46410 q^{98} +3.73205 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{2} - 2q^{3} + 2q^{4} + 2q^{6} - 2q^{7} - 2q^{8} + 2q^{9} + O(q^{10}) \) \( 2q - 2q^{2} - 2q^{3} + 2q^{4} + 2q^{6} - 2q^{7} - 2q^{8} + 2q^{9} + 4q^{11} - 2q^{12} - 2q^{13} + 2q^{14} + 2q^{16} - 2q^{17} - 2q^{18} - 2q^{19} + 2q^{21} - 4q^{22} - 2q^{23} + 2q^{24} + 2q^{26} - 2q^{27} - 2q^{28} + 2q^{31} - 2q^{32} - 4q^{33} + 2q^{34} + 2q^{36} - 4q^{37} + 2q^{38} + 2q^{39} + 8q^{41} - 2q^{42} - 6q^{43} + 4q^{44} + 2q^{46} - 2q^{48} - 6q^{49} + 2q^{51} - 2q^{52} + 2q^{54} + 2q^{56} + 2q^{57} - 6q^{59} + 4q^{61} - 2q^{62} - 2q^{63} + 2q^{64} + 4q^{66} - 4q^{67} - 2q^{68} + 2q^{69} + 14q^{71} - 2q^{72} + 2q^{73} + 4q^{74} - 2q^{76} + 2q^{77} - 2q^{78} - 18q^{79} + 2q^{81} - 8q^{82} + 4q^{83} + 2q^{84} + 6q^{86} - 4q^{88} - 6q^{89} - 4q^{91} - 2q^{92} - 2q^{93} + 2q^{96} - 6q^{97} + 6q^{98} + 4q^{99} + O(q^{100}) \)

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
\(7\) 0.732051 0.276689 0.138345 0.990384i \(-0.455822\pi\)
0.138345 + 0.990384i \(0.455822\pi\)
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 3.73205 1.12526 0.562628 0.826710i \(-0.309790\pi\)
0.562628 + 0.826710i \(0.309790\pi\)
\(12\) −1.00000 −0.288675
\(13\) −2.73205 −0.757735 −0.378867 0.925451i \(-0.623686\pi\)
−0.378867 + 0.925451i \(0.623686\pi\)
\(14\) −0.732051 −0.195649
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −6.19615 −1.50279 −0.751394 0.659854i \(-0.770618\pi\)
−0.751394 + 0.659854i \(0.770618\pi\)
\(18\) −1.00000 −0.235702
\(19\) −1.00000 −0.229416
\(20\) 0 0
\(21\) −0.732051 −0.159747
\(22\) −3.73205 −0.795676
\(23\) 5.92820 1.23612 0.618058 0.786133i \(-0.287920\pi\)
0.618058 + 0.786133i \(0.287920\pi\)
\(24\) 1.00000 0.204124
\(25\) 0 0
\(26\) 2.73205 0.535799
\(27\) −1.00000 −0.192450
\(28\) 0.732051 0.138345
\(29\) −1.73205 −0.321634 −0.160817 0.986984i \(-0.551413\pi\)
−0.160817 + 0.986984i \(0.551413\pi\)
\(30\) 0 0
\(31\) −2.46410 −0.442566 −0.221283 0.975210i \(-0.571024\pi\)
−0.221283 + 0.975210i \(0.571024\pi\)
\(32\) −1.00000 −0.176777
\(33\) −3.73205 −0.649667
\(34\) 6.19615 1.06263
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) −2.00000 −0.328798 −0.164399 0.986394i \(-0.552568\pi\)
−0.164399 + 0.986394i \(0.552568\pi\)
\(38\) 1.00000 0.162221
\(39\) 2.73205 0.437478
\(40\) 0 0
\(41\) −2.92820 −0.457309 −0.228654 0.973508i \(-0.573433\pi\)
−0.228654 + 0.973508i \(0.573433\pi\)
\(42\) 0.732051 0.112958
\(43\) −8.19615 −1.24990 −0.624951 0.780664i \(-0.714881\pi\)
−0.624951 + 0.780664i \(0.714881\pi\)
\(44\) 3.73205 0.562628
\(45\) 0 0
\(46\) −5.92820 −0.874066
\(47\) 3.46410 0.505291 0.252646 0.967559i \(-0.418699\pi\)
0.252646 + 0.967559i \(0.418699\pi\)
\(48\) −1.00000 −0.144338
\(49\) −6.46410 −0.923443
\(50\) 0 0
\(51\) 6.19615 0.867635
\(52\) −2.73205 −0.378867
\(53\) −1.73205 −0.237915 −0.118958 0.992899i \(-0.537955\pi\)
−0.118958 + 0.992899i \(0.537955\pi\)
\(54\) 1.00000 0.136083
\(55\) 0 0
\(56\) −0.732051 −0.0978244
\(57\) 1.00000 0.132453
\(58\) 1.73205 0.227429
\(59\) −8.19615 −1.06705 −0.533524 0.845785i \(-0.679133\pi\)
−0.533524 + 0.845785i \(0.679133\pi\)
\(60\) 0 0
\(61\) 10.6603 1.36491 0.682453 0.730930i \(-0.260913\pi\)
0.682453 + 0.730930i \(0.260913\pi\)
\(62\) 2.46410 0.312941
\(63\) 0.732051 0.0922297
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 3.73205 0.459384
\(67\) −0.267949 −0.0327352 −0.0163676 0.999866i \(-0.505210\pi\)
−0.0163676 + 0.999866i \(0.505210\pi\)
\(68\) −6.19615 −0.751394
\(69\) −5.92820 −0.713672
\(70\) 0 0
\(71\) 12.1962 1.44742 0.723708 0.690106i \(-0.242436\pi\)
0.723708 + 0.690106i \(0.242436\pi\)
\(72\) −1.00000 −0.117851
\(73\) −2.46410 −0.288401 −0.144201 0.989548i \(-0.546061\pi\)
−0.144201 + 0.989548i \(0.546061\pi\)
\(74\) 2.00000 0.232495
\(75\) 0 0
\(76\) −1.00000 −0.114708
\(77\) 2.73205 0.311346
\(78\) −2.73205 −0.309344
\(79\) −5.53590 −0.622837 −0.311419 0.950273i \(-0.600804\pi\)
−0.311419 + 0.950273i \(0.600804\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 2.92820 0.323366
\(83\) 3.73205 0.409646 0.204823 0.978799i \(-0.434338\pi\)
0.204823 + 0.978799i \(0.434338\pi\)
\(84\) −0.732051 −0.0798733
\(85\) 0 0
\(86\) 8.19615 0.883814
\(87\) 1.73205 0.185695
\(88\) −3.73205 −0.397838
\(89\) 10.8564 1.15078 0.575388 0.817880i \(-0.304851\pi\)
0.575388 + 0.817880i \(0.304851\pi\)
\(90\) 0 0
\(91\) −2.00000 −0.209657
\(92\) 5.92820 0.618058
\(93\) 2.46410 0.255515
\(94\) −3.46410 −0.357295
\(95\) 0 0
\(96\) 1.00000 0.102062
\(97\) −15.1244 −1.53565 −0.767823 0.640662i \(-0.778660\pi\)
−0.767823 + 0.640662i \(0.778660\pi\)
\(98\) 6.46410 0.652973
\(99\) 3.73205 0.375085
\(100\) 0 0
\(101\) 9.26795 0.922195 0.461098 0.887349i \(-0.347456\pi\)
0.461098 + 0.887349i \(0.347456\pi\)
\(102\) −6.19615 −0.613511
\(103\) −3.53590 −0.348402 −0.174201 0.984710i \(-0.555734\pi\)
−0.174201 + 0.984710i \(0.555734\pi\)
\(104\) 2.73205 0.267900
\(105\) 0 0
\(106\) 1.73205 0.168232
\(107\) 2.19615 0.212310 0.106155 0.994350i \(-0.466146\pi\)
0.106155 + 0.994350i \(0.466146\pi\)
\(108\) −1.00000 −0.0962250
\(109\) 2.39230 0.229141 0.114571 0.993415i \(-0.463451\pi\)
0.114571 + 0.993415i \(0.463451\pi\)
\(110\) 0 0
\(111\) 2.00000 0.189832
\(112\) 0.732051 0.0691723
\(113\) −13.3923 −1.25984 −0.629921 0.776659i \(-0.716913\pi\)
−0.629921 + 0.776659i \(0.716913\pi\)
\(114\) −1.00000 −0.0936586
\(115\) 0 0
\(116\) −1.73205 −0.160817
\(117\) −2.73205 −0.252578
\(118\) 8.19615 0.754517
\(119\) −4.53590 −0.415805
\(120\) 0 0
\(121\) 2.92820 0.266200
\(122\) −10.6603 −0.965134
\(123\) 2.92820 0.264027
\(124\) −2.46410 −0.221283
\(125\) 0 0
\(126\) −0.732051 −0.0652163
\(127\) −5.00000 −0.443678 −0.221839 0.975083i \(-0.571206\pi\)
−0.221839 + 0.975083i \(0.571206\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 8.19615 0.721631
\(130\) 0 0
\(131\) 10.2679 0.897115 0.448557 0.893754i \(-0.351938\pi\)
0.448557 + 0.893754i \(0.351938\pi\)
\(132\) −3.73205 −0.324833
\(133\) −0.732051 −0.0634769
\(134\) 0.267949 0.0231473
\(135\) 0 0
\(136\) 6.19615 0.531316
\(137\) −12.3923 −1.05875 −0.529373 0.848389i \(-0.677573\pi\)
−0.529373 + 0.848389i \(0.677573\pi\)
\(138\) 5.92820 0.504642
\(139\) −21.1244 −1.79174 −0.895872 0.444312i \(-0.853448\pi\)
−0.895872 + 0.444312i \(0.853448\pi\)
\(140\) 0 0
\(141\) −3.46410 −0.291730
\(142\) −12.1962 −1.02348
\(143\) −10.1962 −0.852645
\(144\) 1.00000 0.0833333
\(145\) 0 0
\(146\) 2.46410 0.203931
\(147\) 6.46410 0.533150
\(148\) −2.00000 −0.164399
\(149\) −21.8564 −1.79055 −0.895273 0.445517i \(-0.853020\pi\)
−0.895273 + 0.445517i \(0.853020\pi\)
\(150\) 0 0
\(151\) −11.4641 −0.932935 −0.466468 0.884538i \(-0.654474\pi\)
−0.466468 + 0.884538i \(0.654474\pi\)
\(152\) 1.00000 0.0811107
\(153\) −6.19615 −0.500929
\(154\) −2.73205 −0.220155
\(155\) 0 0
\(156\) 2.73205 0.218739
\(157\) 13.8564 1.10586 0.552931 0.833227i \(-0.313509\pi\)
0.552931 + 0.833227i \(0.313509\pi\)
\(158\) 5.53590 0.440412
\(159\) 1.73205 0.137361
\(160\) 0 0
\(161\) 4.33975 0.342020
\(162\) −1.00000 −0.0785674
\(163\) −2.00000 −0.156652 −0.0783260 0.996928i \(-0.524958\pi\)
−0.0783260 + 0.996928i \(0.524958\pi\)
\(164\) −2.92820 −0.228654
\(165\) 0 0
\(166\) −3.73205 −0.289663
\(167\) −8.92820 −0.690885 −0.345443 0.938440i \(-0.612271\pi\)
−0.345443 + 0.938440i \(0.612271\pi\)
\(168\) 0.732051 0.0564789
\(169\) −5.53590 −0.425838
\(170\) 0 0
\(171\) −1.00000 −0.0764719
\(172\) −8.19615 −0.624951
\(173\) 13.5885 1.03311 0.516556 0.856254i \(-0.327214\pi\)
0.516556 + 0.856254i \(0.327214\pi\)
\(174\) −1.73205 −0.131306
\(175\) 0 0
\(176\) 3.73205 0.281314
\(177\) 8.19615 0.616061
\(178\) −10.8564 −0.813722
\(179\) −12.0000 −0.896922 −0.448461 0.893802i \(-0.648028\pi\)
−0.448461 + 0.893802i \(0.648028\pi\)
\(180\) 0 0
\(181\) −16.5885 −1.23301 −0.616505 0.787351i \(-0.711452\pi\)
−0.616505 + 0.787351i \(0.711452\pi\)
\(182\) 2.00000 0.148250
\(183\) −10.6603 −0.788029
\(184\) −5.92820 −0.437033
\(185\) 0 0
\(186\) −2.46410 −0.180677
\(187\) −23.1244 −1.69102
\(188\) 3.46410 0.252646
\(189\) −0.732051 −0.0532489
\(190\) 0 0
\(191\) 3.00000 0.217072 0.108536 0.994092i \(-0.465384\pi\)
0.108536 + 0.994092i \(0.465384\pi\)
\(192\) −1.00000 −0.0721688
\(193\) 22.0526 1.58738 0.793689 0.608324i \(-0.208158\pi\)
0.793689 + 0.608324i \(0.208158\pi\)
\(194\) 15.1244 1.08587
\(195\) 0 0
\(196\) −6.46410 −0.461722
\(197\) 24.5885 1.75186 0.875928 0.482443i \(-0.160250\pi\)
0.875928 + 0.482443i \(0.160250\pi\)
\(198\) −3.73205 −0.265225
\(199\) −14.0000 −0.992434 −0.496217 0.868199i \(-0.665278\pi\)
−0.496217 + 0.868199i \(0.665278\pi\)
\(200\) 0 0
\(201\) 0.267949 0.0188997
\(202\) −9.26795 −0.652091
\(203\) −1.26795 −0.0889926
\(204\) 6.19615 0.433817
\(205\) 0 0
\(206\) 3.53590 0.246358
\(207\) 5.92820 0.412039
\(208\) −2.73205 −0.189434
\(209\) −3.73205 −0.258151
\(210\) 0 0
\(211\) −10.2679 −0.706875 −0.353437 0.935458i \(-0.614987\pi\)
−0.353437 + 0.935458i \(0.614987\pi\)
\(212\) −1.73205 −0.118958
\(213\) −12.1962 −0.835667
\(214\) −2.19615 −0.150126
\(215\) 0 0
\(216\) 1.00000 0.0680414
\(217\) −1.80385 −0.122453
\(218\) −2.39230 −0.162027
\(219\) 2.46410 0.166509
\(220\) 0 0
\(221\) 16.9282 1.13871
\(222\) −2.00000 −0.134231
\(223\) −3.53590 −0.236781 −0.118391 0.992967i \(-0.537774\pi\)
−0.118391 + 0.992967i \(0.537774\pi\)
\(224\) −0.732051 −0.0489122
\(225\) 0 0
\(226\) 13.3923 0.890843
\(227\) −6.00000 −0.398234 −0.199117 0.979976i \(-0.563807\pi\)
−0.199117 + 0.979976i \(0.563807\pi\)
\(228\) 1.00000 0.0662266
\(229\) −6.26795 −0.414198 −0.207099 0.978320i \(-0.566402\pi\)
−0.207099 + 0.978320i \(0.566402\pi\)
\(230\) 0 0
\(231\) −2.73205 −0.179756
\(232\) 1.73205 0.113715
\(233\) −21.1244 −1.38390 −0.691951 0.721944i \(-0.743249\pi\)
−0.691951 + 0.721944i \(0.743249\pi\)
\(234\) 2.73205 0.178600
\(235\) 0 0
\(236\) −8.19615 −0.533524
\(237\) 5.53590 0.359595
\(238\) 4.53590 0.294019
\(239\) −14.0000 −0.905585 −0.452792 0.891616i \(-0.649572\pi\)
−0.452792 + 0.891616i \(0.649572\pi\)
\(240\) 0 0
\(241\) −30.1962 −1.94511 −0.972553 0.232683i \(-0.925249\pi\)
−0.972553 + 0.232683i \(0.925249\pi\)
\(242\) −2.92820 −0.188232
\(243\) −1.00000 −0.0641500
\(244\) 10.6603 0.682453
\(245\) 0 0
\(246\) −2.92820 −0.186695
\(247\) 2.73205 0.173836
\(248\) 2.46410 0.156471
\(249\) −3.73205 −0.236509
\(250\) 0 0
\(251\) 11.4641 0.723608 0.361804 0.932254i \(-0.382161\pi\)
0.361804 + 0.932254i \(0.382161\pi\)
\(252\) 0.732051 0.0461149
\(253\) 22.1244 1.39095
\(254\) 5.00000 0.313728
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 9.00000 0.561405 0.280702 0.959795i \(-0.409433\pi\)
0.280702 + 0.959795i \(0.409433\pi\)
\(258\) −8.19615 −0.510270
\(259\) −1.46410 −0.0909748
\(260\) 0 0
\(261\) −1.73205 −0.107211
\(262\) −10.2679 −0.634356
\(263\) 15.0000 0.924940 0.462470 0.886635i \(-0.346963\pi\)
0.462470 + 0.886635i \(0.346963\pi\)
\(264\) 3.73205 0.229692
\(265\) 0 0
\(266\) 0.732051 0.0448849
\(267\) −10.8564 −0.664401
\(268\) −0.267949 −0.0163676
\(269\) −20.5359 −1.25210 −0.626048 0.779785i \(-0.715329\pi\)
−0.626048 + 0.779785i \(0.715329\pi\)
\(270\) 0 0
\(271\) −15.8038 −0.960015 −0.480008 0.877264i \(-0.659366\pi\)
−0.480008 + 0.877264i \(0.659366\pi\)
\(272\) −6.19615 −0.375697
\(273\) 2.00000 0.121046
\(274\) 12.3923 0.748647
\(275\) 0 0
\(276\) −5.92820 −0.356836
\(277\) 11.1962 0.672712 0.336356 0.941735i \(-0.390806\pi\)
0.336356 + 0.941735i \(0.390806\pi\)
\(278\) 21.1244 1.26695
\(279\) −2.46410 −0.147522
\(280\) 0 0
\(281\) 17.7846 1.06094 0.530470 0.847703i \(-0.322015\pi\)
0.530470 + 0.847703i \(0.322015\pi\)
\(282\) 3.46410 0.206284
\(283\) 4.00000 0.237775 0.118888 0.992908i \(-0.462067\pi\)
0.118888 + 0.992908i \(0.462067\pi\)
\(284\) 12.1962 0.723708
\(285\) 0 0
\(286\) 10.1962 0.602911
\(287\) −2.14359 −0.126532
\(288\) −1.00000 −0.0589256
\(289\) 21.3923 1.25837
\(290\) 0 0
\(291\) 15.1244 0.886605
\(292\) −2.46410 −0.144201
\(293\) −27.7321 −1.62012 −0.810062 0.586344i \(-0.800567\pi\)
−0.810062 + 0.586344i \(0.800567\pi\)
\(294\) −6.46410 −0.376994
\(295\) 0 0
\(296\) 2.00000 0.116248
\(297\) −3.73205 −0.216556
\(298\) 21.8564 1.26611
\(299\) −16.1962 −0.936648
\(300\) 0 0
\(301\) −6.00000 −0.345834
\(302\) 11.4641 0.659685
\(303\) −9.26795 −0.532430
\(304\) −1.00000 −0.0573539
\(305\) 0 0
\(306\) 6.19615 0.354210
\(307\) −20.2679 −1.15675 −0.578376 0.815770i \(-0.696314\pi\)
−0.578376 + 0.815770i \(0.696314\pi\)
\(308\) 2.73205 0.155673
\(309\) 3.53590 0.201150
\(310\) 0 0
\(311\) −29.3205 −1.66261 −0.831307 0.555814i \(-0.812407\pi\)
−0.831307 + 0.555814i \(0.812407\pi\)
\(312\) −2.73205 −0.154672
\(313\) −3.92820 −0.222035 −0.111018 0.993818i \(-0.535411\pi\)
−0.111018 + 0.993818i \(0.535411\pi\)
\(314\) −13.8564 −0.781962
\(315\) 0 0
\(316\) −5.53590 −0.311419
\(317\) −1.73205 −0.0972817 −0.0486408 0.998816i \(-0.515489\pi\)
−0.0486408 + 0.998816i \(0.515489\pi\)
\(318\) −1.73205 −0.0971286
\(319\) −6.46410 −0.361920
\(320\) 0 0
\(321\) −2.19615 −0.122577
\(322\) −4.33975 −0.241845
\(323\) 6.19615 0.344763
\(324\) 1.00000 0.0555556
\(325\) 0 0
\(326\) 2.00000 0.110770
\(327\) −2.39230 −0.132295
\(328\) 2.92820 0.161683
\(329\) 2.53590 0.139809
\(330\) 0 0
\(331\) −22.2679 −1.22396 −0.611979 0.790874i \(-0.709626\pi\)
−0.611979 + 0.790874i \(0.709626\pi\)
\(332\) 3.73205 0.204823
\(333\) −2.00000 −0.109599
\(334\) 8.92820 0.488530
\(335\) 0 0
\(336\) −0.732051 −0.0399366
\(337\) 10.5359 0.573927 0.286963 0.957942i \(-0.407354\pi\)
0.286963 + 0.957942i \(0.407354\pi\)
\(338\) 5.53590 0.301113
\(339\) 13.3923 0.727370
\(340\) 0 0
\(341\) −9.19615 −0.498000
\(342\) 1.00000 0.0540738
\(343\) −9.85641 −0.532196
\(344\) 8.19615 0.441907
\(345\) 0 0
\(346\) −13.5885 −0.730520
\(347\) −20.7846 −1.11578 −0.557888 0.829916i \(-0.688388\pi\)
−0.557888 + 0.829916i \(0.688388\pi\)
\(348\) 1.73205 0.0928477
\(349\) −17.0526 −0.912803 −0.456401 0.889774i \(-0.650862\pi\)
−0.456401 + 0.889774i \(0.650862\pi\)
\(350\) 0 0
\(351\) 2.73205 0.145826
\(352\) −3.73205 −0.198919
\(353\) 16.0526 0.854391 0.427196 0.904159i \(-0.359502\pi\)
0.427196 + 0.904159i \(0.359502\pi\)
\(354\) −8.19615 −0.435621
\(355\) 0 0
\(356\) 10.8564 0.575388
\(357\) 4.53590 0.240065
\(358\) 12.0000 0.634220
\(359\) −16.9282 −0.893436 −0.446718 0.894675i \(-0.647407\pi\)
−0.446718 + 0.894675i \(0.647407\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 16.5885 0.871870
\(363\) −2.92820 −0.153691
\(364\) −2.00000 −0.104828
\(365\) 0 0
\(366\) 10.6603 0.557220
\(367\) −13.2679 −0.692581 −0.346291 0.938127i \(-0.612559\pi\)
−0.346291 + 0.938127i \(0.612559\pi\)
\(368\) 5.92820 0.309029
\(369\) −2.92820 −0.152436
\(370\) 0 0
\(371\) −1.26795 −0.0658286
\(372\) 2.46410 0.127758
\(373\) −31.5167 −1.63187 −0.815935 0.578143i \(-0.803778\pi\)
−0.815935 + 0.578143i \(0.803778\pi\)
\(374\) 23.1244 1.19573
\(375\) 0 0
\(376\) −3.46410 −0.178647
\(377\) 4.73205 0.243713
\(378\) 0.732051 0.0376526
\(379\) −0.392305 −0.0201513 −0.0100757 0.999949i \(-0.503207\pi\)
−0.0100757 + 0.999949i \(0.503207\pi\)
\(380\) 0 0
\(381\) 5.00000 0.256158
\(382\) −3.00000 −0.153493
\(383\) −19.5167 −0.997255 −0.498627 0.866816i \(-0.666162\pi\)
−0.498627 + 0.866816i \(0.666162\pi\)
\(384\) 1.00000 0.0510310
\(385\) 0 0
\(386\) −22.0526 −1.12245
\(387\) −8.19615 −0.416634
\(388\) −15.1244 −0.767823
\(389\) 21.7128 1.10088 0.550442 0.834874i \(-0.314459\pi\)
0.550442 + 0.834874i \(0.314459\pi\)
\(390\) 0 0
\(391\) −36.7321 −1.85762
\(392\) 6.46410 0.326486
\(393\) −10.2679 −0.517950
\(394\) −24.5885 −1.23875
\(395\) 0 0
\(396\) 3.73205 0.187543
\(397\) 11.7321 0.588815 0.294407 0.955680i \(-0.404878\pi\)
0.294407 + 0.955680i \(0.404878\pi\)
\(398\) 14.0000 0.701757
\(399\) 0.732051 0.0366484
\(400\) 0 0
\(401\) 16.8564 0.841769 0.420884 0.907114i \(-0.361720\pi\)
0.420884 + 0.907114i \(0.361720\pi\)
\(402\) −0.267949 −0.0133641
\(403\) 6.73205 0.335347
\(404\) 9.26795 0.461098
\(405\) 0 0
\(406\) 1.26795 0.0629273
\(407\) −7.46410 −0.369982
\(408\) −6.19615 −0.306755
\(409\) 1.60770 0.0794954 0.0397477 0.999210i \(-0.487345\pi\)
0.0397477 + 0.999210i \(0.487345\pi\)
\(410\) 0 0
\(411\) 12.3923 0.611267
\(412\) −3.53590 −0.174201
\(413\) −6.00000 −0.295241
\(414\) −5.92820 −0.291355
\(415\) 0 0
\(416\) 2.73205 0.133950
\(417\) 21.1244 1.03446
\(418\) 3.73205 0.182541
\(419\) 13.8564 0.676930 0.338465 0.940979i \(-0.390092\pi\)
0.338465 + 0.940979i \(0.390092\pi\)
\(420\) 0 0
\(421\) 6.78461 0.330662 0.165331 0.986238i \(-0.447131\pi\)
0.165331 + 0.986238i \(0.447131\pi\)
\(422\) 10.2679 0.499836
\(423\) 3.46410 0.168430
\(424\) 1.73205 0.0841158
\(425\) 0 0
\(426\) 12.1962 0.590906
\(427\) 7.80385 0.377655
\(428\) 2.19615 0.106155
\(429\) 10.1962 0.492275
\(430\) 0 0
\(431\) −29.3205 −1.41232 −0.706160 0.708053i \(-0.749574\pi\)
−0.706160 + 0.708053i \(0.749574\pi\)
\(432\) −1.00000 −0.0481125
\(433\) 24.1962 1.16279 0.581396 0.813620i \(-0.302507\pi\)
0.581396 + 0.813620i \(0.302507\pi\)
\(434\) 1.80385 0.0865875
\(435\) 0 0
\(436\) 2.39230 0.114571
\(437\) −5.92820 −0.283584
\(438\) −2.46410 −0.117739
\(439\) 25.7846 1.23063 0.615316 0.788280i \(-0.289028\pi\)
0.615316 + 0.788280i \(0.289028\pi\)
\(440\) 0 0
\(441\) −6.46410 −0.307814
\(442\) −16.9282 −0.805193
\(443\) 3.33975 0.158676 0.0793381 0.996848i \(-0.474719\pi\)
0.0793381 + 0.996848i \(0.474719\pi\)
\(444\) 2.00000 0.0949158
\(445\) 0 0
\(446\) 3.53590 0.167430
\(447\) 21.8564 1.03377
\(448\) 0.732051 0.0345861
\(449\) 24.4641 1.15453 0.577266 0.816556i \(-0.304120\pi\)
0.577266 + 0.816556i \(0.304120\pi\)
\(450\) 0 0
\(451\) −10.9282 −0.514589
\(452\) −13.3923 −0.629921
\(453\) 11.4641 0.538630
\(454\) 6.00000 0.281594
\(455\) 0 0
\(456\) −1.00000 −0.0468293
\(457\) 5.07180 0.237249 0.118624 0.992939i \(-0.462152\pi\)
0.118624 + 0.992939i \(0.462152\pi\)
\(458\) 6.26795 0.292882
\(459\) 6.19615 0.289212
\(460\) 0 0
\(461\) −17.8564 −0.831656 −0.415828 0.909443i \(-0.636508\pi\)
−0.415828 + 0.909443i \(0.636508\pi\)
\(462\) 2.73205 0.127107
\(463\) 35.3205 1.64148 0.820742 0.571300i \(-0.193561\pi\)
0.820742 + 0.571300i \(0.193561\pi\)
\(464\) −1.73205 −0.0804084
\(465\) 0 0
\(466\) 21.1244 0.978567
\(467\) −7.19615 −0.332998 −0.166499 0.986042i \(-0.553246\pi\)
−0.166499 + 0.986042i \(0.553246\pi\)
\(468\) −2.73205 −0.126289
\(469\) −0.196152 −0.00905748
\(470\) 0 0
\(471\) −13.8564 −0.638470
\(472\) 8.19615 0.377258
\(473\) −30.5885 −1.40646
\(474\) −5.53590 −0.254272
\(475\) 0 0
\(476\) −4.53590 −0.207903
\(477\) −1.73205 −0.0793052
\(478\) 14.0000 0.640345
\(479\) 3.14359 0.143634 0.0718172 0.997418i \(-0.477120\pi\)
0.0718172 + 0.997418i \(0.477120\pi\)
\(480\) 0 0
\(481\) 5.46410 0.249142
\(482\) 30.1962 1.37540
\(483\) −4.33975 −0.197465
\(484\) 2.92820 0.133100
\(485\) 0 0
\(486\) 1.00000 0.0453609
\(487\) −2.14359 −0.0971355 −0.0485677 0.998820i \(-0.515466\pi\)
−0.0485677 + 0.998820i \(0.515466\pi\)
\(488\) −10.6603 −0.482567
\(489\) 2.00000 0.0904431
\(490\) 0 0
\(491\) −2.92820 −0.132148 −0.0660740 0.997815i \(-0.521047\pi\)
−0.0660740 + 0.997815i \(0.521047\pi\)
\(492\) 2.92820 0.132014
\(493\) 10.7321 0.483347
\(494\) −2.73205 −0.122921
\(495\) 0 0
\(496\) −2.46410 −0.110641
\(497\) 8.92820 0.400485
\(498\) 3.73205 0.167237
\(499\) 41.3731 1.85211 0.926056 0.377385i \(-0.123177\pi\)
0.926056 + 0.377385i \(0.123177\pi\)
\(500\) 0 0
\(501\) 8.92820 0.398883
\(502\) −11.4641 −0.511668
\(503\) 17.8564 0.796178 0.398089 0.917347i \(-0.369674\pi\)
0.398089 + 0.917347i \(0.369674\pi\)
\(504\) −0.732051 −0.0326081
\(505\) 0 0
\(506\) −22.1244 −0.983548
\(507\) 5.53590 0.245858
\(508\) −5.00000 −0.221839
\(509\) 13.8756 0.615027 0.307514 0.951544i \(-0.400503\pi\)
0.307514 + 0.951544i \(0.400503\pi\)
\(510\) 0 0
\(511\) −1.80385 −0.0797975
\(512\) −1.00000 −0.0441942
\(513\) 1.00000 0.0441511
\(514\) −9.00000 −0.396973
\(515\) 0 0
\(516\) 8.19615 0.360815
\(517\) 12.9282 0.568582
\(518\) 1.46410 0.0643289
\(519\) −13.5885 −0.596467
\(520\) 0 0
\(521\) 24.1769 1.05921 0.529605 0.848244i \(-0.322340\pi\)
0.529605 + 0.848244i \(0.322340\pi\)
\(522\) 1.73205 0.0758098
\(523\) −4.00000 −0.174908 −0.0874539 0.996169i \(-0.527873\pi\)
−0.0874539 + 0.996169i \(0.527873\pi\)
\(524\) 10.2679 0.448557
\(525\) 0 0
\(526\) −15.0000 −0.654031
\(527\) 15.2679 0.665082
\(528\) −3.73205 −0.162417
\(529\) 12.1436 0.527982
\(530\) 0 0
\(531\) −8.19615 −0.355683
\(532\) −0.732051 −0.0317384
\(533\) 8.00000 0.346518
\(534\) 10.8564 0.469803
\(535\) 0 0
\(536\) 0.267949 0.0115736
\(537\) 12.0000 0.517838
\(538\) 20.5359 0.885365
\(539\) −24.1244 −1.03911
\(540\) 0 0
\(541\) 17.1962 0.739320 0.369660 0.929167i \(-0.379474\pi\)
0.369660 + 0.929167i \(0.379474\pi\)
\(542\) 15.8038 0.678833
\(543\) 16.5885 0.711879
\(544\) 6.19615 0.265658
\(545\) 0 0
\(546\) −2.00000 −0.0855921
\(547\) 43.4449 1.85757 0.928784 0.370621i \(-0.120855\pi\)
0.928784 + 0.370621i \(0.120855\pi\)
\(548\) −12.3923 −0.529373
\(549\) 10.6603 0.454969
\(550\) 0 0
\(551\) 1.73205 0.0737878
\(552\) 5.92820 0.252321
\(553\) −4.05256 −0.172332
\(554\) −11.1962 −0.475679
\(555\) 0 0
\(556\) −21.1244 −0.895872
\(557\) 3.80385 0.161174 0.0805871 0.996748i \(-0.474320\pi\)
0.0805871 + 0.996748i \(0.474320\pi\)
\(558\) 2.46410 0.104314
\(559\) 22.3923 0.947094
\(560\) 0 0
\(561\) 23.1244 0.976311
\(562\) −17.7846 −0.750198
\(563\) 7.66025 0.322841 0.161421 0.986886i \(-0.448392\pi\)
0.161421 + 0.986886i \(0.448392\pi\)
\(564\) −3.46410 −0.145865
\(565\) 0 0
\(566\) −4.00000 −0.168133
\(567\) 0.732051 0.0307432
\(568\) −12.1962 −0.511739
\(569\) 18.0000 0.754599 0.377300 0.926091i \(-0.376853\pi\)
0.377300 + 0.926091i \(0.376853\pi\)
\(570\) 0 0
\(571\) −6.19615 −0.259301 −0.129650 0.991560i \(-0.541386\pi\)
−0.129650 + 0.991560i \(0.541386\pi\)
\(572\) −10.1962 −0.426323
\(573\) −3.00000 −0.125327
\(574\) 2.14359 0.0894719
\(575\) 0 0
\(576\) 1.00000 0.0416667
\(577\) −17.2487 −0.718073 −0.359037 0.933323i \(-0.616895\pi\)
−0.359037 + 0.933323i \(0.616895\pi\)
\(578\) −21.3923 −0.889803
\(579\) −22.0526 −0.916473
\(580\) 0 0
\(581\) 2.73205 0.113345
\(582\) −15.1244 −0.626925
\(583\) −6.46410 −0.267716
\(584\) 2.46410 0.101965
\(585\) 0 0
\(586\) 27.7321 1.14560
\(587\) −12.2679 −0.506352 −0.253176 0.967420i \(-0.581475\pi\)
−0.253176 + 0.967420i \(0.581475\pi\)
\(588\) 6.46410 0.266575
\(589\) 2.46410 0.101532
\(590\) 0 0
\(591\) −24.5885 −1.01143
\(592\) −2.00000 −0.0821995
\(593\) −27.6077 −1.13371 −0.566856 0.823817i \(-0.691840\pi\)
−0.566856 + 0.823817i \(0.691840\pi\)
\(594\) 3.73205 0.153128
\(595\) 0 0
\(596\) −21.8564 −0.895273
\(597\) 14.0000 0.572982
\(598\) 16.1962 0.662310
\(599\) −3.66025 −0.149554 −0.0747770 0.997200i \(-0.523824\pi\)
−0.0747770 + 0.997200i \(0.523824\pi\)
\(600\) 0 0
\(601\) −10.0526 −0.410052 −0.205026 0.978756i \(-0.565728\pi\)
−0.205026 + 0.978756i \(0.565728\pi\)
\(602\) 6.00000 0.244542
\(603\) −0.267949 −0.0109117
\(604\) −11.4641 −0.466468
\(605\) 0 0
\(606\) 9.26795 0.376485
\(607\) −11.7846 −0.478323 −0.239161 0.970980i \(-0.576872\pi\)
−0.239161 + 0.970980i \(0.576872\pi\)
\(608\) 1.00000 0.0405554
\(609\) 1.26795 0.0513799
\(610\) 0 0
\(611\) −9.46410 −0.382877
\(612\) −6.19615 −0.250465
\(613\) 38.7846 1.56650 0.783248 0.621710i \(-0.213562\pi\)
0.783248 + 0.621710i \(0.213562\pi\)
\(614\) 20.2679 0.817948
\(615\) 0 0
\(616\) −2.73205 −0.110077
\(617\) 46.3923 1.86768 0.933842 0.357686i \(-0.116434\pi\)
0.933842 + 0.357686i \(0.116434\pi\)
\(618\) −3.53590 −0.142235
\(619\) −0.196152 −0.00788403 −0.00394202 0.999992i \(-0.501255\pi\)
−0.00394202 + 0.999992i \(0.501255\pi\)
\(620\) 0 0
\(621\) −5.92820 −0.237891
\(622\) 29.3205 1.17565
\(623\) 7.94744 0.318408
\(624\) 2.73205 0.109370
\(625\) 0 0
\(626\) 3.92820 0.157003
\(627\) 3.73205 0.149044
\(628\) 13.8564 0.552931
\(629\) 12.3923 0.494114
\(630\) 0 0
\(631\) 2.92820 0.116570 0.0582850 0.998300i \(-0.481437\pi\)
0.0582850 + 0.998300i \(0.481437\pi\)
\(632\) 5.53590 0.220206
\(633\) 10.2679 0.408114
\(634\) 1.73205 0.0687885
\(635\) 0 0
\(636\) 1.73205 0.0686803
\(637\) 17.6603 0.699725
\(638\) 6.46410 0.255916
\(639\) 12.1962 0.482472
\(640\) 0 0
\(641\) 10.5359 0.416143 0.208071 0.978114i \(-0.433281\pi\)
0.208071 + 0.978114i \(0.433281\pi\)
\(642\) 2.19615 0.0866752
\(643\) 27.0718 1.06761 0.533804 0.845608i \(-0.320762\pi\)
0.533804 + 0.845608i \(0.320762\pi\)
\(644\) 4.33975 0.171010
\(645\) 0 0
\(646\) −6.19615 −0.243784
\(647\) 9.39230 0.369250 0.184625 0.982809i \(-0.440893\pi\)
0.184625 + 0.982809i \(0.440893\pi\)
\(648\) −1.00000 −0.0392837
\(649\) −30.5885 −1.20070
\(650\) 0 0
\(651\) 1.80385 0.0706984
\(652\) −2.00000 −0.0783260
\(653\) −17.4641 −0.683423 −0.341712 0.939805i \(-0.611007\pi\)
−0.341712 + 0.939805i \(0.611007\pi\)
\(654\) 2.39230 0.0935465
\(655\) 0 0
\(656\) −2.92820 −0.114327
\(657\) −2.46410 −0.0961338
\(658\) −2.53590 −0.0988596
\(659\) 7.41154 0.288713 0.144356 0.989526i \(-0.453889\pi\)
0.144356 + 0.989526i \(0.453889\pi\)
\(660\) 0 0
\(661\) −22.6410 −0.880633 −0.440317 0.897843i \(-0.645134\pi\)
−0.440317 + 0.897843i \(0.645134\pi\)
\(662\) 22.2679 0.865468
\(663\) −16.9282 −0.657437
\(664\) −3.73205 −0.144832
\(665\) 0 0
\(666\) 2.00000 0.0774984
\(667\) −10.2679 −0.397577
\(668\) −8.92820 −0.345443
\(669\) 3.53590 0.136706
\(670\) 0 0
\(671\) 39.7846 1.53587
\(672\) 0.732051 0.0282395
\(673\) 6.98076 0.269089 0.134544 0.990908i \(-0.457043\pi\)
0.134544 + 0.990908i \(0.457043\pi\)
\(674\) −10.5359 −0.405828
\(675\) 0 0
\(676\) −5.53590 −0.212919
\(677\) −8.66025 −0.332841 −0.166420 0.986055i \(-0.553221\pi\)
−0.166420 + 0.986055i \(0.553221\pi\)
\(678\) −13.3923 −0.514328
\(679\) −11.0718 −0.424897
\(680\) 0 0
\(681\) 6.00000 0.229920
\(682\) 9.19615 0.352139
\(683\) 46.9808 1.79767 0.898834 0.438288i \(-0.144415\pi\)
0.898834 + 0.438288i \(0.144415\pi\)
\(684\) −1.00000 −0.0382360
\(685\) 0 0
\(686\) 9.85641 0.376319
\(687\) 6.26795 0.239137
\(688\) −8.19615 −0.312475
\(689\) 4.73205 0.180277
\(690\) 0 0
\(691\) −22.1962 −0.844381 −0.422191 0.906507i \(-0.638739\pi\)
−0.422191 + 0.906507i \(0.638739\pi\)
\(692\) 13.5885 0.516556
\(693\) 2.73205 0.103782
\(694\) 20.7846 0.788973
\(695\) 0 0
\(696\) −1.73205 −0.0656532
\(697\) 18.1436 0.687238
\(698\) 17.0526 0.645449
\(699\) 21.1244 0.798997
\(700\) 0 0
\(701\) 3.60770 0.136261 0.0681304 0.997676i \(-0.478297\pi\)
0.0681304 + 0.997676i \(0.478297\pi\)
\(702\) −2.73205 −0.103115
\(703\) 2.00000 0.0754314
\(704\) 3.73205 0.140657
\(705\) 0 0
\(706\) −16.0526 −0.604146
\(707\) 6.78461 0.255162
\(708\) 8.19615 0.308030
\(709\) 47.3013 1.77644 0.888218 0.459422i \(-0.151943\pi\)
0.888218 + 0.459422i \(0.151943\pi\)
\(710\) 0 0
\(711\) −5.53590 −0.207612
\(712\) −10.8564 −0.406861
\(713\) −14.6077 −0.547062
\(714\) −4.53590 −0.169752
\(715\) 0 0
\(716\) −12.0000 −0.448461
\(717\) 14.0000 0.522840
\(718\) 16.9282 0.631755
\(719\) 12.1769 0.454122 0.227061 0.973881i \(-0.427088\pi\)
0.227061 + 0.973881i \(0.427088\pi\)
\(720\) 0 0
\(721\) −2.58846 −0.0963992
\(722\) −1.00000 −0.0372161
\(723\) 30.1962 1.12301
\(724\) −16.5885 −0.616505
\(725\) 0 0
\(726\) 2.92820 0.108676
\(727\) −38.1051 −1.41324 −0.706620 0.707593i \(-0.749781\pi\)
−0.706620 + 0.707593i \(0.749781\pi\)
\(728\) 2.00000 0.0741249
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 50.7846 1.87834
\(732\) −10.6603 −0.394014
\(733\) −20.5167 −0.757800 −0.378900 0.925438i \(-0.623698\pi\)
−0.378900 + 0.925438i \(0.623698\pi\)
\(734\) 13.2679 0.489729
\(735\) 0 0
\(736\) −5.92820 −0.218516
\(737\) −1.00000 −0.0368355
\(738\) 2.92820 0.107789
\(739\) −53.1769 −1.95614 −0.978072 0.208266i \(-0.933218\pi\)
−0.978072 + 0.208266i \(0.933218\pi\)
\(740\) 0 0
\(741\) −2.73205 −0.100364
\(742\) 1.26795 0.0465479
\(743\) −5.07180 −0.186066 −0.0930331 0.995663i \(-0.529656\pi\)
−0.0930331 + 0.995663i \(0.529656\pi\)
\(744\) −2.46410 −0.0903383
\(745\) 0 0
\(746\) 31.5167 1.15391
\(747\) 3.73205 0.136549
\(748\) −23.1244 −0.845510
\(749\) 1.60770 0.0587439
\(750\) 0 0
\(751\) 7.46410 0.272369 0.136184 0.990683i \(-0.456516\pi\)
0.136184 + 0.990683i \(0.456516\pi\)
\(752\) 3.46410 0.126323
\(753\) −11.4641 −0.417775
\(754\) −4.73205 −0.172331
\(755\) 0 0
\(756\) −0.732051 −0.0266244
\(757\) −13.5885 −0.493881 −0.246940 0.969031i \(-0.579425\pi\)
−0.246940 + 0.969031i \(0.579425\pi\)
\(758\) 0.392305 0.0142492
\(759\) −22.1244 −0.803063
\(760\) 0 0
\(761\) 1.26795 0.0459631 0.0229816 0.999736i \(-0.492684\pi\)
0.0229816 + 0.999736i \(0.492684\pi\)
\(762\) −5.00000 −0.181131
\(763\) 1.75129 0.0634009
\(764\) 3.00000 0.108536
\(765\) 0 0
\(766\) 19.5167 0.705166
\(767\) 22.3923 0.808539
\(768\) −1.00000 −0.0360844
\(769\) −34.7128 −1.25178 −0.625888 0.779913i \(-0.715263\pi\)
−0.625888 + 0.779913i \(0.715263\pi\)
\(770\) 0 0
\(771\) −9.00000 −0.324127
\(772\) 22.0526 0.793689
\(773\) −45.3205 −1.63007 −0.815033 0.579415i \(-0.803281\pi\)
−0.815033 + 0.579415i \(0.803281\pi\)
\(774\) 8.19615 0.294605
\(775\) 0 0
\(776\) 15.1244 0.542933
\(777\) 1.46410 0.0525244
\(778\) −21.7128 −0.778442
\(779\) 2.92820 0.104914
\(780\) 0 0
\(781\) 45.5167 1.62871
\(782\) 36.7321 1.31354
\(783\) 1.73205 0.0618984
\(784\) −6.46410 −0.230861
\(785\) 0 0
\(786\) 10.2679 0.366246
\(787\) 2.12436 0.0757251 0.0378626 0.999283i \(-0.487945\pi\)
0.0378626 + 0.999283i \(0.487945\pi\)
\(788\) 24.5885 0.875928
\(789\) −15.0000 −0.534014
\(790\) 0 0
\(791\) −9.80385 −0.348585
\(792\) −3.73205 −0.132613
\(793\) −29.1244 −1.03424
\(794\) −11.7321 −0.416355
\(795\) 0 0
\(796\) −14.0000 −0.496217
\(797\) 20.0000 0.708436 0.354218 0.935163i \(-0.384747\pi\)
0.354218 + 0.935163i \(0.384747\pi\)
\(798\) −0.732051 −0.0259143
\(799\) −21.4641 −0.759345
\(800\) 0 0
\(801\) 10.8564 0.383592
\(802\) −16.8564 −0.595220
\(803\) −9.19615 −0.324525
\(804\) 0.267949 0.00944984
\(805\) 0 0
\(806\) −6.73205 −0.237126
\(807\) 20.5359 0.722898
\(808\) −9.26795 −0.326045
\(809\) 37.5167 1.31902 0.659508 0.751698i \(-0.270765\pi\)
0.659508 + 0.751698i \(0.270765\pi\)
\(810\) 0 0
\(811\) 18.3731 0.645166 0.322583 0.946541i \(-0.395449\pi\)
0.322583 + 0.946541i \(0.395449\pi\)
\(812\) −1.26795 −0.0444963
\(813\) 15.8038 0.554265
\(814\) 7.46410 0.261617
\(815\) 0 0
\(816\) 6.19615 0.216909
\(817\) 8.19615 0.286747
\(818\) −1.60770 −0.0562117
\(819\) −2.00000 −0.0698857
\(820\) 0 0
\(821\) 2.48334 0.0866691 0.0433346 0.999061i \(-0.486202\pi\)
0.0433346 + 0.999061i \(0.486202\pi\)
\(822\) −12.3923 −0.432231
\(823\) −6.78461 −0.236497 −0.118248 0.992984i \(-0.537728\pi\)
−0.118248 + 0.992984i \(0.537728\pi\)
\(824\) 3.53590 0.123179
\(825\) 0 0
\(826\) 6.00000 0.208767
\(827\) 6.73205 0.234096 0.117048 0.993126i \(-0.462657\pi\)
0.117048 + 0.993126i \(0.462657\pi\)
\(828\) 5.92820 0.206019
\(829\) 6.19615 0.215201 0.107601 0.994194i \(-0.465683\pi\)
0.107601 + 0.994194i \(0.465683\pi\)
\(830\) 0 0
\(831\) −11.1962 −0.388390
\(832\) −2.73205 −0.0947168
\(833\) 40.0526 1.38774
\(834\) −21.1244 −0.731477
\(835\) 0 0
\(836\) −3.73205 −0.129076
\(837\) 2.46410 0.0851718
\(838\) −13.8564 −0.478662
\(839\) −46.5885 −1.60841 −0.804206 0.594351i \(-0.797409\pi\)
−0.804206 + 0.594351i \(0.797409\pi\)
\(840\) 0 0
\(841\) −26.0000 −0.896552
\(842\) −6.78461 −0.233813
\(843\) −17.7846 −0.612534
\(844\) −10.2679 −0.353437
\(845\) 0 0
\(846\) −3.46410 −0.119098
\(847\) 2.14359 0.0736547
\(848\) −1.73205 −0.0594789
\(849\) −4.00000 −0.137280
\(850\) 0 0
\(851\) −11.8564 −0.406432
\(852\) −12.1962 −0.417833
\(853\) −4.53590 −0.155306 −0.0776531 0.996980i \(-0.524743\pi\)
−0.0776531 + 0.996980i \(0.524743\pi\)
\(854\) −7.80385 −0.267042
\(855\) 0 0
\(856\) −2.19615 −0.0750629
\(857\) −42.9282 −1.46640 −0.733200 0.680013i \(-0.761974\pi\)
−0.733200 + 0.680013i \(0.761974\pi\)
\(858\) −10.1962 −0.348091
\(859\) 56.9282 1.94237 0.971183 0.238337i \(-0.0766021\pi\)
0.971183 + 0.238337i \(0.0766021\pi\)
\(860\) 0 0
\(861\) 2.14359 0.0730535
\(862\) 29.3205 0.998660
\(863\) −55.3205 −1.88313 −0.941566 0.336829i \(-0.890646\pi\)
−0.941566 + 0.336829i \(0.890646\pi\)
\(864\) 1.00000 0.0340207
\(865\) 0 0
\(866\) −24.1962 −0.822219
\(867\) −21.3923 −0.726521
\(868\) −1.80385 −0.0612266
\(869\) −20.6603 −0.700851
\(870\) 0 0
\(871\) 0.732051 0.0248046
\(872\) −2.39230 −0.0810137
\(873\) −15.1244 −0.511882
\(874\) 5.92820 0.200524
\(875\) 0 0
\(876\) 2.46410 0.0832543
\(877\) 31.1769 1.05277 0.526385 0.850246i \(-0.323547\pi\)
0.526385 + 0.850246i \(0.323547\pi\)
\(878\) −25.7846 −0.870188
\(879\) 27.7321 0.935379
\(880\) 0 0
\(881\) −26.5359 −0.894017 −0.447009 0.894530i \(-0.647511\pi\)
−0.447009 + 0.894530i \(0.647511\pi\)
\(882\) 6.46410 0.217658
\(883\) 27.2679 0.917640 0.458820 0.888529i \(-0.348272\pi\)
0.458820 + 0.888529i \(0.348272\pi\)
\(884\) 16.9282 0.569357
\(885\) 0 0
\(886\) −3.33975 −0.112201
\(887\) −7.60770 −0.255441 −0.127721 0.991810i \(-0.540766\pi\)
−0.127721 + 0.991810i \(0.540766\pi\)
\(888\) −2.00000 −0.0671156
\(889\) −3.66025 −0.122761
\(890\) 0 0
\(891\) 3.73205 0.125028
\(892\) −3.53590 −0.118391
\(893\) −3.46410 −0.115922
\(894\) −21.8564 −0.730988
\(895\) 0 0
\(896\) −0.732051 −0.0244561
\(897\) 16.1962 0.540774
\(898\) −24.4641 −0.816378
\(899\) 4.26795 0.142344
\(900\) 0 0
\(901\) 10.7321 0.357536
\(902\) 10.9282 0.363869
\(903\) 6.00000 0.199667
\(904\) 13.3923 0.445421
\(905\) 0 0
\(906\) −11.4641 −0.380869
\(907\) 49.3205 1.63766 0.818830 0.574036i \(-0.194623\pi\)
0.818830 + 0.574036i \(0.194623\pi\)
\(908\) −6.00000 −0.199117
\(909\) 9.26795 0.307398
\(910\) 0 0
\(911\) 34.1051 1.12995 0.564976 0.825107i \(-0.308885\pi\)
0.564976 + 0.825107i \(0.308885\pi\)
\(912\) 1.00000 0.0331133
\(913\) 13.9282 0.460956
\(914\) −5.07180 −0.167760
\(915\) 0 0
\(916\) −6.26795 −0.207099
\(917\) 7.51666 0.248222
\(918\) −6.19615 −0.204504
\(919\) −4.48334 −0.147892 −0.0739459 0.997262i \(-0.523559\pi\)
−0.0739459 + 0.997262i \(0.523559\pi\)
\(920\) 0 0
\(921\) 20.2679 0.667852
\(922\) 17.8564 0.588069
\(923\) −33.3205 −1.09676
\(924\) −2.73205 −0.0898779
\(925\) 0 0
\(926\) −35.3205 −1.16070
\(927\) −3.53590 −0.116134
\(928\) 1.73205 0.0568574
\(929\) 42.9282 1.40843 0.704214 0.709987i \(-0.251299\pi\)
0.704214 + 0.709987i \(0.251299\pi\)
\(930\) 0 0
\(931\) 6.46410 0.211852
\(932\) −21.1244 −0.691951
\(933\) 29.3205 0.959910
\(934\) 7.19615 0.235465
\(935\) 0 0
\(936\) 2.73205 0.0892999
\(937\) −13.2154 −0.431728 −0.215864 0.976423i \(-0.569257\pi\)
−0.215864 + 0.976423i \(0.569257\pi\)
\(938\) 0.196152 0.00640460
\(939\) 3.92820 0.128192
\(940\) 0 0
\(941\) −12.2679 −0.399924 −0.199962 0.979804i \(-0.564082\pi\)
−0.199962 + 0.979804i \(0.564082\pi\)
\(942\) 13.8564 0.451466
\(943\) −17.3590 −0.565286
\(944\) −8.19615 −0.266762
\(945\) 0 0
\(946\) 30.5885 0.994517
\(947\) 5.85641 0.190308 0.0951538 0.995463i \(-0.469666\pi\)
0.0951538 + 0.995463i \(0.469666\pi\)
\(948\) 5.53590 0.179798
\(949\) 6.73205 0.218532
\(950\) 0 0
\(951\) 1.73205 0.0561656
\(952\) 4.53590 0.147009
\(953\) 5.24871 0.170022 0.0850112 0.996380i \(-0.472907\pi\)
0.0850112 + 0.996380i \(0.472907\pi\)
\(954\) 1.73205 0.0560772
\(955\) 0 0
\(956\) −14.0000 −0.452792
\(957\) 6.46410 0.208955
\(958\) −3.14359 −0.101565
\(959\) −9.07180 −0.292944
\(960\) 0 0
\(961\) −24.9282 −0.804136
\(962\) −5.46410 −0.176170
\(963\) 2.19615 0.0707700
\(964\) −30.1962 −0.972553
\(965\) 0 0
\(966\) 4.33975 0.139629
\(967\) −18.1051 −0.582221 −0.291111 0.956689i \(-0.594025\pi\)
−0.291111 + 0.956689i \(0.594025\pi\)
\(968\) −2.92820 −0.0941160
\(969\) −6.19615 −0.199049
\(970\) 0 0
\(971\) −37.7654 −1.21195 −0.605974 0.795484i \(-0.707217\pi\)
−0.605974 + 0.795484i \(0.707217\pi\)
\(972\) −1.00000 −0.0320750
\(973\) −15.4641 −0.495756
\(974\) 2.14359 0.0686852
\(975\) 0 0
\(976\) 10.6603 0.341226
\(977\) 1.46410 0.0468408 0.0234204 0.999726i \(-0.492544\pi\)
0.0234204 + 0.999726i \(0.492544\pi\)
\(978\) −2.00000 −0.0639529
\(979\) 40.5167 1.29492
\(980\) 0 0
\(981\) 2.39230 0.0763804
\(982\) 2.92820 0.0934427
\(983\) 30.3397 0.967688 0.483844 0.875154i \(-0.339240\pi\)
0.483844 + 0.875154i \(0.339240\pi\)
\(984\) −2.92820 −0.0933477
\(985\) 0 0
\(986\) −10.7321 −0.341778
\(987\) −2.53590 −0.0807185
\(988\) 2.73205 0.0869181
\(989\) −48.5885 −1.54502
\(990\) 0 0
\(991\) −40.3205 −1.28082 −0.640412 0.768032i \(-0.721236\pi\)
−0.640412 + 0.768032i \(0.721236\pi\)
\(992\) 2.46410 0.0782353
\(993\) 22.2679 0.706652
\(994\) −8.92820 −0.283185
\(995\) 0 0
\(996\) −3.73205 −0.118255
\(997\) 63.0526 1.99689 0.998447 0.0557047i \(-0.0177405\pi\)
0.998447 + 0.0557047i \(0.0177405\pi\)
\(998\) −41.3731 −1.30964
\(999\) 2.00000 0.0632772
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 2850.2.a.bd.1.2 2
3.2 odd 2 8550.2.a.by.1.2 2
5.2 odd 4 2850.2.d.x.799.2 4
5.3 odd 4 2850.2.d.x.799.3 4
5.4 even 2 2850.2.a.bi.1.1 yes 2
15.14 odd 2 8550.2.a.bs.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2850.2.a.bd.1.2 2 1.1 even 1 trivial
2850.2.a.bi.1.1 yes 2 5.4 even 2
2850.2.d.x.799.2 4 5.2 odd 4
2850.2.d.x.799.3 4 5.3 odd 4
8550.2.a.bs.1.1 2 15.14 odd 2
8550.2.a.by.1.2 2 3.2 odd 2