Properties

Label 5070.2.a.bf.1.2
Level $5070$
Weight $2$
Character 5070.1
Self dual yes
Analytic conductor $40.484$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 5070 = 2 \cdot 3 \cdot 5 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5070.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(40.4841538248\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{13}) \)
Defining polynomial: \(x^{2} - x - 3\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 390)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(2.30278\) of defining polynomial
Character \(\chi\) \(=\) 5070.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{6} +4.60555 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{6} +4.60555 q^{7} +1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{10} -1.00000 q^{12} +4.60555 q^{14} -1.00000 q^{15} +1.00000 q^{16} -4.60555 q^{17} +1.00000 q^{18} -4.60555 q^{19} +1.00000 q^{20} -4.60555 q^{21} +1.39445 q^{23} -1.00000 q^{24} +1.00000 q^{25} -1.00000 q^{27} +4.60555 q^{28} +4.60555 q^{29} -1.00000 q^{30} +6.00000 q^{31} +1.00000 q^{32} -4.60555 q^{34} +4.60555 q^{35} +1.00000 q^{36} +9.21110 q^{37} -4.60555 q^{38} +1.00000 q^{40} +3.21110 q^{41} -4.60555 q^{42} -8.00000 q^{43} +1.00000 q^{45} +1.39445 q^{46} +9.21110 q^{47} -1.00000 q^{48} +14.2111 q^{49} +1.00000 q^{50} +4.60555 q^{51} +6.00000 q^{53} -1.00000 q^{54} +4.60555 q^{56} +4.60555 q^{57} +4.60555 q^{58} +9.21110 q^{59} -1.00000 q^{60} -11.2111 q^{61} +6.00000 q^{62} +4.60555 q^{63} +1.00000 q^{64} -3.21110 q^{67} -4.60555 q^{68} -1.39445 q^{69} +4.60555 q^{70} +9.21110 q^{71} +1.00000 q^{72} +1.39445 q^{73} +9.21110 q^{74} -1.00000 q^{75} -4.60555 q^{76} -14.4222 q^{79} +1.00000 q^{80} +1.00000 q^{81} +3.21110 q^{82} +2.78890 q^{83} -4.60555 q^{84} -4.60555 q^{85} -8.00000 q^{86} -4.60555 q^{87} -15.2111 q^{89} +1.00000 q^{90} +1.39445 q^{92} -6.00000 q^{93} +9.21110 q^{94} -4.60555 q^{95} -1.00000 q^{96} +1.39445 q^{97} +14.2111 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{2} - 2q^{3} + 2q^{4} + 2q^{5} - 2q^{6} + 2q^{7} + 2q^{8} + 2q^{9} + O(q^{10}) \) \( 2q + 2q^{2} - 2q^{3} + 2q^{4} + 2q^{5} - 2q^{6} + 2q^{7} + 2q^{8} + 2q^{9} + 2q^{10} - 2q^{12} + 2q^{14} - 2q^{15} + 2q^{16} - 2q^{17} + 2q^{18} - 2q^{19} + 2q^{20} - 2q^{21} + 10q^{23} - 2q^{24} + 2q^{25} - 2q^{27} + 2q^{28} + 2q^{29} - 2q^{30} + 12q^{31} + 2q^{32} - 2q^{34} + 2q^{35} + 2q^{36} + 4q^{37} - 2q^{38} + 2q^{40} - 8q^{41} - 2q^{42} - 16q^{43} + 2q^{45} + 10q^{46} + 4q^{47} - 2q^{48} + 14q^{49} + 2q^{50} + 2q^{51} + 12q^{53} - 2q^{54} + 2q^{56} + 2q^{57} + 2q^{58} + 4q^{59} - 2q^{60} - 8q^{61} + 12q^{62} + 2q^{63} + 2q^{64} + 8q^{67} - 2q^{68} - 10q^{69} + 2q^{70} + 4q^{71} + 2q^{72} + 10q^{73} + 4q^{74} - 2q^{75} - 2q^{76} + 2q^{80} + 2q^{81} - 8q^{82} + 20q^{83} - 2q^{84} - 2q^{85} - 16q^{86} - 2q^{87} - 16q^{89} + 2q^{90} + 10q^{92} - 12q^{93} + 4q^{94} - 2q^{95} - 2q^{96} + 10q^{97} + 14q^{98} + O(q^{100}) \)

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
\(5\) 1.00000 0.447214
\(6\) −1.00000 −0.408248
\(7\) 4.60555 1.74073 0.870367 0.492403i \(-0.163881\pi\)
0.870367 + 0.492403i \(0.163881\pi\)
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) 1.00000 0.316228
\(11\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) −1.00000 −0.288675
\(13\) 0 0
\(14\) 4.60555 1.23089
\(15\) −1.00000 −0.258199
\(16\) 1.00000 0.250000
\(17\) −4.60555 −1.11701 −0.558505 0.829501i \(-0.688625\pi\)
−0.558505 + 0.829501i \(0.688625\pi\)
\(18\) 1.00000 0.235702
\(19\) −4.60555 −1.05659 −0.528293 0.849062i \(-0.677168\pi\)
−0.528293 + 0.849062i \(0.677168\pi\)
\(20\) 1.00000 0.223607
\(21\) −4.60555 −1.00501
\(22\) 0 0
\(23\) 1.39445 0.290763 0.145381 0.989376i \(-0.453559\pi\)
0.145381 + 0.989376i \(0.453559\pi\)
\(24\) −1.00000 −0.204124
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 4.60555 0.870367
\(29\) 4.60555 0.855229 0.427615 0.903961i \(-0.359354\pi\)
0.427615 + 0.903961i \(0.359354\pi\)
\(30\) −1.00000 −0.182574
\(31\) 6.00000 1.07763 0.538816 0.842424i \(-0.318872\pi\)
0.538816 + 0.842424i \(0.318872\pi\)
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) −4.60555 −0.789846
\(35\) 4.60555 0.778480
\(36\) 1.00000 0.166667
\(37\) 9.21110 1.51430 0.757148 0.653243i \(-0.226592\pi\)
0.757148 + 0.653243i \(0.226592\pi\)
\(38\) −4.60555 −0.747119
\(39\) 0 0
\(40\) 1.00000 0.158114
\(41\) 3.21110 0.501490 0.250745 0.968053i \(-0.419324\pi\)
0.250745 + 0.968053i \(0.419324\pi\)
\(42\) −4.60555 −0.710652
\(43\) −8.00000 −1.21999 −0.609994 0.792406i \(-0.708828\pi\)
−0.609994 + 0.792406i \(0.708828\pi\)
\(44\) 0 0
\(45\) 1.00000 0.149071
\(46\) 1.39445 0.205600
\(47\) 9.21110 1.34358 0.671789 0.740743i \(-0.265526\pi\)
0.671789 + 0.740743i \(0.265526\pi\)
\(48\) −1.00000 −0.144338
\(49\) 14.2111 2.03016
\(50\) 1.00000 0.141421
\(51\) 4.60555 0.644906
\(52\) 0 0
\(53\) 6.00000 0.824163 0.412082 0.911147i \(-0.364802\pi\)
0.412082 + 0.911147i \(0.364802\pi\)
\(54\) −1.00000 −0.136083
\(55\) 0 0
\(56\) 4.60555 0.615443
\(57\) 4.60555 0.610020
\(58\) 4.60555 0.604739
\(59\) 9.21110 1.19918 0.599592 0.800306i \(-0.295330\pi\)
0.599592 + 0.800306i \(0.295330\pi\)
\(60\) −1.00000 −0.129099
\(61\) −11.2111 −1.43543 −0.717717 0.696335i \(-0.754813\pi\)
−0.717717 + 0.696335i \(0.754813\pi\)
\(62\) 6.00000 0.762001
\(63\) 4.60555 0.580245
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 0 0
\(67\) −3.21110 −0.392299 −0.196149 0.980574i \(-0.562844\pi\)
−0.196149 + 0.980574i \(0.562844\pi\)
\(68\) −4.60555 −0.558505
\(69\) −1.39445 −0.167872
\(70\) 4.60555 0.550469
\(71\) 9.21110 1.09316 0.546578 0.837408i \(-0.315930\pi\)
0.546578 + 0.837408i \(0.315930\pi\)
\(72\) 1.00000 0.117851
\(73\) 1.39445 0.163208 0.0816039 0.996665i \(-0.473996\pi\)
0.0816039 + 0.996665i \(0.473996\pi\)
\(74\) 9.21110 1.07077
\(75\) −1.00000 −0.115470
\(76\) −4.60555 −0.528293
\(77\) 0 0
\(78\) 0 0
\(79\) −14.4222 −1.62262 −0.811312 0.584613i \(-0.801246\pi\)
−0.811312 + 0.584613i \(0.801246\pi\)
\(80\) 1.00000 0.111803
\(81\) 1.00000 0.111111
\(82\) 3.21110 0.354607
\(83\) 2.78890 0.306121 0.153061 0.988217i \(-0.451087\pi\)
0.153061 + 0.988217i \(0.451087\pi\)
\(84\) −4.60555 −0.502507
\(85\) −4.60555 −0.499542
\(86\) −8.00000 −0.862662
\(87\) −4.60555 −0.493767
\(88\) 0 0
\(89\) −15.2111 −1.61237 −0.806187 0.591661i \(-0.798472\pi\)
−0.806187 + 0.591661i \(0.798472\pi\)
\(90\) 1.00000 0.105409
\(91\) 0 0
\(92\) 1.39445 0.145381
\(93\) −6.00000 −0.622171
\(94\) 9.21110 0.950053
\(95\) −4.60555 −0.472520
\(96\) −1.00000 −0.102062
\(97\) 1.39445 0.141585 0.0707924 0.997491i \(-0.477447\pi\)
0.0707924 + 0.997491i \(0.477447\pi\)
\(98\) 14.2111 1.43554
\(99\) 0 0
\(100\) 1.00000 0.100000
\(101\) 7.39445 0.735775 0.367888 0.929870i \(-0.380081\pi\)
0.367888 + 0.929870i \(0.380081\pi\)
\(102\) 4.60555 0.456018
\(103\) 4.00000 0.394132 0.197066 0.980390i \(-0.436859\pi\)
0.197066 + 0.980390i \(0.436859\pi\)
\(104\) 0 0
\(105\) −4.60555 −0.449456
\(106\) 6.00000 0.582772
\(107\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(108\) −1.00000 −0.0962250
\(109\) 1.39445 0.133564 0.0667820 0.997768i \(-0.478727\pi\)
0.0667820 + 0.997768i \(0.478727\pi\)
\(110\) 0 0
\(111\) −9.21110 −0.874279
\(112\) 4.60555 0.435184
\(113\) −13.8167 −1.29976 −0.649881 0.760036i \(-0.725181\pi\)
−0.649881 + 0.760036i \(0.725181\pi\)
\(114\) 4.60555 0.431349
\(115\) 1.39445 0.130033
\(116\) 4.60555 0.427615
\(117\) 0 0
\(118\) 9.21110 0.847951
\(119\) −21.2111 −1.94442
\(120\) −1.00000 −0.0912871
\(121\) −11.0000 −1.00000
\(122\) −11.2111 −1.01501
\(123\) −3.21110 −0.289535
\(124\) 6.00000 0.538816
\(125\) 1.00000 0.0894427
\(126\) 4.60555 0.410295
\(127\) −1.21110 −0.107468 −0.0537340 0.998555i \(-0.517112\pi\)
−0.0537340 + 0.998555i \(0.517112\pi\)
\(128\) 1.00000 0.0883883
\(129\) 8.00000 0.704361
\(130\) 0 0
\(131\) 22.6056 1.97506 0.987528 0.157443i \(-0.0503250\pi\)
0.987528 + 0.157443i \(0.0503250\pi\)
\(132\) 0 0
\(133\) −21.2111 −1.83924
\(134\) −3.21110 −0.277397
\(135\) −1.00000 −0.0860663
\(136\) −4.60555 −0.394923
\(137\) −3.21110 −0.274343 −0.137172 0.990547i \(-0.543801\pi\)
−0.137172 + 0.990547i \(0.543801\pi\)
\(138\) −1.39445 −0.118703
\(139\) 17.2111 1.45983 0.729913 0.683540i \(-0.239560\pi\)
0.729913 + 0.683540i \(0.239560\pi\)
\(140\) 4.60555 0.389240
\(141\) −9.21110 −0.775715
\(142\) 9.21110 0.772979
\(143\) 0 0
\(144\) 1.00000 0.0833333
\(145\) 4.60555 0.382470
\(146\) 1.39445 0.115405
\(147\) −14.2111 −1.17211
\(148\) 9.21110 0.757148
\(149\) 15.2111 1.24614 0.623071 0.782165i \(-0.285885\pi\)
0.623071 + 0.782165i \(0.285885\pi\)
\(150\) −1.00000 −0.0816497
\(151\) −6.00000 −0.488273 −0.244137 0.969741i \(-0.578505\pi\)
−0.244137 + 0.969741i \(0.578505\pi\)
\(152\) −4.60555 −0.373560
\(153\) −4.60555 −0.372337
\(154\) 0 0
\(155\) 6.00000 0.481932
\(156\) 0 0
\(157\) 20.4222 1.62987 0.814935 0.579553i \(-0.196773\pi\)
0.814935 + 0.579553i \(0.196773\pi\)
\(158\) −14.4222 −1.14737
\(159\) −6.00000 −0.475831
\(160\) 1.00000 0.0790569
\(161\) 6.42221 0.506141
\(162\) 1.00000 0.0785674
\(163\) −24.4222 −1.91289 −0.956447 0.291905i \(-0.905711\pi\)
−0.956447 + 0.291905i \(0.905711\pi\)
\(164\) 3.21110 0.250745
\(165\) 0 0
\(166\) 2.78890 0.216460
\(167\) −9.21110 −0.712777 −0.356388 0.934338i \(-0.615992\pi\)
−0.356388 + 0.934338i \(0.615992\pi\)
\(168\) −4.60555 −0.355326
\(169\) 0 0
\(170\) −4.60555 −0.353230
\(171\) −4.60555 −0.352195
\(172\) −8.00000 −0.609994
\(173\) 12.4222 0.944443 0.472221 0.881480i \(-0.343452\pi\)
0.472221 + 0.881480i \(0.343452\pi\)
\(174\) −4.60555 −0.349146
\(175\) 4.60555 0.348147
\(176\) 0 0
\(177\) −9.21110 −0.692349
\(178\) −15.2111 −1.14012
\(179\) 19.8167 1.48117 0.740583 0.671965i \(-0.234549\pi\)
0.740583 + 0.671965i \(0.234549\pi\)
\(180\) 1.00000 0.0745356
\(181\) −8.42221 −0.626018 −0.313009 0.949750i \(-0.601337\pi\)
−0.313009 + 0.949750i \(0.601337\pi\)
\(182\) 0 0
\(183\) 11.2111 0.828749
\(184\) 1.39445 0.102800
\(185\) 9.21110 0.677214
\(186\) −6.00000 −0.439941
\(187\) 0 0
\(188\) 9.21110 0.671789
\(189\) −4.60555 −0.335005
\(190\) −4.60555 −0.334122
\(191\) −12.0000 −0.868290 −0.434145 0.900843i \(-0.642949\pi\)
−0.434145 + 0.900843i \(0.642949\pi\)
\(192\) −1.00000 −0.0721688
\(193\) −7.81665 −0.562655 −0.281328 0.959612i \(-0.590775\pi\)
−0.281328 + 0.959612i \(0.590775\pi\)
\(194\) 1.39445 0.100116
\(195\) 0 0
\(196\) 14.2111 1.01508
\(197\) −6.00000 −0.427482 −0.213741 0.976890i \(-0.568565\pi\)
−0.213741 + 0.976890i \(0.568565\pi\)
\(198\) 0 0
\(199\) −22.4222 −1.58947 −0.794734 0.606958i \(-0.792390\pi\)
−0.794734 + 0.606958i \(0.792390\pi\)
\(200\) 1.00000 0.0707107
\(201\) 3.21110 0.226494
\(202\) 7.39445 0.520272
\(203\) 21.2111 1.48873
\(204\) 4.60555 0.322453
\(205\) 3.21110 0.224273
\(206\) 4.00000 0.278693
\(207\) 1.39445 0.0969209
\(208\) 0 0
\(209\) 0 0
\(210\) −4.60555 −0.317813
\(211\) −17.2111 −1.18486 −0.592431 0.805622i \(-0.701832\pi\)
−0.592431 + 0.805622i \(0.701832\pi\)
\(212\) 6.00000 0.412082
\(213\) −9.21110 −0.631134
\(214\) 0 0
\(215\) −8.00000 −0.545595
\(216\) −1.00000 −0.0680414
\(217\) 27.6333 1.87587
\(218\) 1.39445 0.0944440
\(219\) −1.39445 −0.0942281
\(220\) 0 0
\(221\) 0 0
\(222\) −9.21110 −0.618209
\(223\) 1.81665 0.121652 0.0608261 0.998148i \(-0.480627\pi\)
0.0608261 + 0.998148i \(0.480627\pi\)
\(224\) 4.60555 0.307721
\(225\) 1.00000 0.0666667
\(226\) −13.8167 −0.919070
\(227\) 24.0000 1.59294 0.796468 0.604681i \(-0.206699\pi\)
0.796468 + 0.604681i \(0.206699\pi\)
\(228\) 4.60555 0.305010
\(229\) 19.8167 1.30952 0.654761 0.755836i \(-0.272769\pi\)
0.654761 + 0.755836i \(0.272769\pi\)
\(230\) 1.39445 0.0919472
\(231\) 0 0
\(232\) 4.60555 0.302369
\(233\) 1.81665 0.119013 0.0595065 0.998228i \(-0.481047\pi\)
0.0595065 + 0.998228i \(0.481047\pi\)
\(234\) 0 0
\(235\) 9.21110 0.600866
\(236\) 9.21110 0.599592
\(237\) 14.4222 0.936823
\(238\) −21.2111 −1.37491
\(239\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(240\) −1.00000 −0.0645497
\(241\) 6.42221 0.413691 0.206845 0.978374i \(-0.433680\pi\)
0.206845 + 0.978374i \(0.433680\pi\)
\(242\) −11.0000 −0.707107
\(243\) −1.00000 −0.0641500
\(244\) −11.2111 −0.717717
\(245\) 14.2111 0.907914
\(246\) −3.21110 −0.204732
\(247\) 0 0
\(248\) 6.00000 0.381000
\(249\) −2.78890 −0.176739
\(250\) 1.00000 0.0632456
\(251\) 13.3944 0.845450 0.422725 0.906258i \(-0.361074\pi\)
0.422725 + 0.906258i \(0.361074\pi\)
\(252\) 4.60555 0.290122
\(253\) 0 0
\(254\) −1.21110 −0.0759913
\(255\) 4.60555 0.288411
\(256\) 1.00000 0.0625000
\(257\) 28.6056 1.78437 0.892183 0.451675i \(-0.149173\pi\)
0.892183 + 0.451675i \(0.149173\pi\)
\(258\) 8.00000 0.498058
\(259\) 42.4222 2.63599
\(260\) 0 0
\(261\) 4.60555 0.285076
\(262\) 22.6056 1.39658
\(263\) −7.81665 −0.481996 −0.240998 0.970526i \(-0.577475\pi\)
−0.240998 + 0.970526i \(0.577475\pi\)
\(264\) 0 0
\(265\) 6.00000 0.368577
\(266\) −21.2111 −1.30054
\(267\) 15.2111 0.930904
\(268\) −3.21110 −0.196149
\(269\) 25.8167 1.57407 0.787035 0.616909i \(-0.211615\pi\)
0.787035 + 0.616909i \(0.211615\pi\)
\(270\) −1.00000 −0.0608581
\(271\) 0.422205 0.0256471 0.0128236 0.999918i \(-0.495918\pi\)
0.0128236 + 0.999918i \(0.495918\pi\)
\(272\) −4.60555 −0.279253
\(273\) 0 0
\(274\) −3.21110 −0.193990
\(275\) 0 0
\(276\) −1.39445 −0.0839359
\(277\) 16.4222 0.986715 0.493357 0.869827i \(-0.335770\pi\)
0.493357 + 0.869827i \(0.335770\pi\)
\(278\) 17.2111 1.03225
\(279\) 6.00000 0.359211
\(280\) 4.60555 0.275234
\(281\) −27.2111 −1.62328 −0.811639 0.584159i \(-0.801424\pi\)
−0.811639 + 0.584159i \(0.801424\pi\)
\(282\) −9.21110 −0.548513
\(283\) −10.4222 −0.619536 −0.309768 0.950812i \(-0.600251\pi\)
−0.309768 + 0.950812i \(0.600251\pi\)
\(284\) 9.21110 0.546578
\(285\) 4.60555 0.272809
\(286\) 0 0
\(287\) 14.7889 0.872961
\(288\) 1.00000 0.0589256
\(289\) 4.21110 0.247712
\(290\) 4.60555 0.270447
\(291\) −1.39445 −0.0817440
\(292\) 1.39445 0.0816039
\(293\) −18.0000 −1.05157 −0.525786 0.850617i \(-0.676229\pi\)
−0.525786 + 0.850617i \(0.676229\pi\)
\(294\) −14.2111 −0.828808
\(295\) 9.21110 0.536291
\(296\) 9.21110 0.535384
\(297\) 0 0
\(298\) 15.2111 0.881156
\(299\) 0 0
\(300\) −1.00000 −0.0577350
\(301\) −36.8444 −2.12368
\(302\) −6.00000 −0.345261
\(303\) −7.39445 −0.424800
\(304\) −4.60555 −0.264146
\(305\) −11.2111 −0.641946
\(306\) −4.60555 −0.263282
\(307\) 8.78890 0.501609 0.250804 0.968038i \(-0.419305\pi\)
0.250804 + 0.968038i \(0.419305\pi\)
\(308\) 0 0
\(309\) −4.00000 −0.227552
\(310\) 6.00000 0.340777
\(311\) −12.0000 −0.680458 −0.340229 0.940343i \(-0.610505\pi\)
−0.340229 + 0.940343i \(0.610505\pi\)
\(312\) 0 0
\(313\) 3.57779 0.202229 0.101114 0.994875i \(-0.467759\pi\)
0.101114 + 0.994875i \(0.467759\pi\)
\(314\) 20.4222 1.15249
\(315\) 4.60555 0.259493
\(316\) −14.4222 −0.811312
\(317\) −18.0000 −1.01098 −0.505490 0.862832i \(-0.668688\pi\)
−0.505490 + 0.862832i \(0.668688\pi\)
\(318\) −6.00000 −0.336463
\(319\) 0 0
\(320\) 1.00000 0.0559017
\(321\) 0 0
\(322\) 6.42221 0.357895
\(323\) 21.2111 1.18022
\(324\) 1.00000 0.0555556
\(325\) 0 0
\(326\) −24.4222 −1.35262
\(327\) −1.39445 −0.0771132
\(328\) 3.21110 0.177303
\(329\) 42.4222 2.33881
\(330\) 0 0
\(331\) −16.6056 −0.912724 −0.456362 0.889794i \(-0.650848\pi\)
−0.456362 + 0.889794i \(0.650848\pi\)
\(332\) 2.78890 0.153061
\(333\) 9.21110 0.504765
\(334\) −9.21110 −0.504009
\(335\) −3.21110 −0.175441
\(336\) −4.60555 −0.251253
\(337\) −13.6333 −0.742654 −0.371327 0.928502i \(-0.621097\pi\)
−0.371327 + 0.928502i \(0.621097\pi\)
\(338\) 0 0
\(339\) 13.8167 0.750418
\(340\) −4.60555 −0.249771
\(341\) 0 0
\(342\) −4.60555 −0.249040
\(343\) 33.2111 1.79323
\(344\) −8.00000 −0.431331
\(345\) −1.39445 −0.0750746
\(346\) 12.4222 0.667822
\(347\) −27.6333 −1.48343 −0.741717 0.670713i \(-0.765988\pi\)
−0.741717 + 0.670713i \(0.765988\pi\)
\(348\) −4.60555 −0.246883
\(349\) 7.81665 0.418416 0.209208 0.977871i \(-0.432911\pi\)
0.209208 + 0.977871i \(0.432911\pi\)
\(350\) 4.60555 0.246177
\(351\) 0 0
\(352\) 0 0
\(353\) −8.78890 −0.467786 −0.233893 0.972262i \(-0.575147\pi\)
−0.233893 + 0.972262i \(0.575147\pi\)
\(354\) −9.21110 −0.489565
\(355\) 9.21110 0.488875
\(356\) −15.2111 −0.806187
\(357\) 21.2111 1.12261
\(358\) 19.8167 1.04734
\(359\) −15.6333 −0.825094 −0.412547 0.910936i \(-0.635361\pi\)
−0.412547 + 0.910936i \(0.635361\pi\)
\(360\) 1.00000 0.0527046
\(361\) 2.21110 0.116374
\(362\) −8.42221 −0.442661
\(363\) 11.0000 0.577350
\(364\) 0 0
\(365\) 1.39445 0.0729888
\(366\) 11.2111 0.586014
\(367\) −19.6333 −1.02485 −0.512425 0.858732i \(-0.671253\pi\)
−0.512425 + 0.858732i \(0.671253\pi\)
\(368\) 1.39445 0.0726907
\(369\) 3.21110 0.167163
\(370\) 9.21110 0.478862
\(371\) 27.6333 1.43465
\(372\) −6.00000 −0.311086
\(373\) −20.4222 −1.05742 −0.528711 0.848802i \(-0.677324\pi\)
−0.528711 + 0.848802i \(0.677324\pi\)
\(374\) 0 0
\(375\) −1.00000 −0.0516398
\(376\) 9.21110 0.475026
\(377\) 0 0
\(378\) −4.60555 −0.236884
\(379\) 35.0278 1.79925 0.899627 0.436658i \(-0.143838\pi\)
0.899627 + 0.436658i \(0.143838\pi\)
\(380\) −4.60555 −0.236260
\(381\) 1.21110 0.0620467
\(382\) −12.0000 −0.613973
\(383\) −27.6333 −1.41200 −0.705998 0.708214i \(-0.749501\pi\)
−0.705998 + 0.708214i \(0.749501\pi\)
\(384\) −1.00000 −0.0510310
\(385\) 0 0
\(386\) −7.81665 −0.397857
\(387\) −8.00000 −0.406663
\(388\) 1.39445 0.0707924
\(389\) −4.60555 −0.233511 −0.116755 0.993161i \(-0.537249\pi\)
−0.116755 + 0.993161i \(0.537249\pi\)
\(390\) 0 0
\(391\) −6.42221 −0.324785
\(392\) 14.2111 0.717769
\(393\) −22.6056 −1.14030
\(394\) −6.00000 −0.302276
\(395\) −14.4222 −0.725660
\(396\) 0 0
\(397\) 3.63331 0.182350 0.0911752 0.995835i \(-0.470938\pi\)
0.0911752 + 0.995835i \(0.470938\pi\)
\(398\) −22.4222 −1.12392
\(399\) 21.2111 1.06188
\(400\) 1.00000 0.0500000
\(401\) −8.78890 −0.438897 −0.219448 0.975624i \(-0.570426\pi\)
−0.219448 + 0.975624i \(0.570426\pi\)
\(402\) 3.21110 0.160155
\(403\) 0 0
\(404\) 7.39445 0.367888
\(405\) 1.00000 0.0496904
\(406\) 21.2111 1.05269
\(407\) 0 0
\(408\) 4.60555 0.228009
\(409\) −14.7889 −0.731264 −0.365632 0.930760i \(-0.619147\pi\)
−0.365632 + 0.930760i \(0.619147\pi\)
\(410\) 3.21110 0.158585
\(411\) 3.21110 0.158392
\(412\) 4.00000 0.197066
\(413\) 42.4222 2.08746
\(414\) 1.39445 0.0685334
\(415\) 2.78890 0.136902
\(416\) 0 0
\(417\) −17.2111 −0.842831
\(418\) 0 0
\(419\) 4.18335 0.204370 0.102185 0.994765i \(-0.467417\pi\)
0.102185 + 0.994765i \(0.467417\pi\)
\(420\) −4.60555 −0.224728
\(421\) −19.8167 −0.965805 −0.482902 0.875674i \(-0.660417\pi\)
−0.482902 + 0.875674i \(0.660417\pi\)
\(422\) −17.2111 −0.837823
\(423\) 9.21110 0.447859
\(424\) 6.00000 0.291386
\(425\) −4.60555 −0.223402
\(426\) −9.21110 −0.446279
\(427\) −51.6333 −2.49871
\(428\) 0 0
\(429\) 0 0
\(430\) −8.00000 −0.385794
\(431\) 12.0000 0.578020 0.289010 0.957326i \(-0.406674\pi\)
0.289010 + 0.957326i \(0.406674\pi\)
\(432\) −1.00000 −0.0481125
\(433\) −19.2111 −0.923227 −0.461613 0.887081i \(-0.652729\pi\)
−0.461613 + 0.887081i \(0.652729\pi\)
\(434\) 27.6333 1.32644
\(435\) −4.60555 −0.220819
\(436\) 1.39445 0.0667820
\(437\) −6.42221 −0.307216
\(438\) −1.39445 −0.0666293
\(439\) −8.00000 −0.381819 −0.190910 0.981608i \(-0.561144\pi\)
−0.190910 + 0.981608i \(0.561144\pi\)
\(440\) 0 0
\(441\) 14.2111 0.676719
\(442\) 0 0
\(443\) 15.6333 0.742761 0.371380 0.928481i \(-0.378885\pi\)
0.371380 + 0.928481i \(0.378885\pi\)
\(444\) −9.21110 −0.437140
\(445\) −15.2111 −0.721075
\(446\) 1.81665 0.0860211
\(447\) −15.2111 −0.719460
\(448\) 4.60555 0.217592
\(449\) −33.6333 −1.58725 −0.793627 0.608405i \(-0.791810\pi\)
−0.793627 + 0.608405i \(0.791810\pi\)
\(450\) 1.00000 0.0471405
\(451\) 0 0
\(452\) −13.8167 −0.649881
\(453\) 6.00000 0.281905
\(454\) 24.0000 1.12638
\(455\) 0 0
\(456\) 4.60555 0.215675
\(457\) 38.2389 1.78874 0.894369 0.447330i \(-0.147625\pi\)
0.894369 + 0.447330i \(0.147625\pi\)
\(458\) 19.8167 0.925971
\(459\) 4.60555 0.214969
\(460\) 1.39445 0.0650165
\(461\) −33.6333 −1.56646 −0.783230 0.621733i \(-0.786429\pi\)
−0.783230 + 0.621733i \(0.786429\pi\)
\(462\) 0 0
\(463\) 31.3944 1.45902 0.729512 0.683968i \(-0.239747\pi\)
0.729512 + 0.683968i \(0.239747\pi\)
\(464\) 4.60555 0.213807
\(465\) −6.00000 −0.278243
\(466\) 1.81665 0.0841549
\(467\) 30.4222 1.40777 0.703886 0.710313i \(-0.251447\pi\)
0.703886 + 0.710313i \(0.251447\pi\)
\(468\) 0 0
\(469\) −14.7889 −0.682888
\(470\) 9.21110 0.424876
\(471\) −20.4222 −0.941006
\(472\) 9.21110 0.423975
\(473\) 0 0
\(474\) 14.4222 0.662434
\(475\) −4.60555 −0.211317
\(476\) −21.2111 −0.972209
\(477\) 6.00000 0.274721
\(478\) 0 0
\(479\) −5.57779 −0.254856 −0.127428 0.991848i \(-0.540672\pi\)
−0.127428 + 0.991848i \(0.540672\pi\)
\(480\) −1.00000 −0.0456435
\(481\) 0 0
\(482\) 6.42221 0.292523
\(483\) −6.42221 −0.292220
\(484\) −11.0000 −0.500000
\(485\) 1.39445 0.0633187
\(486\) −1.00000 −0.0453609
\(487\) −0.972244 −0.0440566 −0.0220283 0.999757i \(-0.507012\pi\)
−0.0220283 + 0.999757i \(0.507012\pi\)
\(488\) −11.2111 −0.507503
\(489\) 24.4222 1.10441
\(490\) 14.2111 0.641992
\(491\) −7.81665 −0.352761 −0.176380 0.984322i \(-0.556439\pi\)
−0.176380 + 0.984322i \(0.556439\pi\)
\(492\) −3.21110 −0.144768
\(493\) −21.2111 −0.955300
\(494\) 0 0
\(495\) 0 0
\(496\) 6.00000 0.269408
\(497\) 42.4222 1.90290
\(498\) −2.78890 −0.124973
\(499\) 23.0278 1.03086 0.515432 0.856930i \(-0.327631\pi\)
0.515432 + 0.856930i \(0.327631\pi\)
\(500\) 1.00000 0.0447214
\(501\) 9.21110 0.411522
\(502\) 13.3944 0.597824
\(503\) 23.4500 1.04558 0.522791 0.852461i \(-0.324891\pi\)
0.522791 + 0.852461i \(0.324891\pi\)
\(504\) 4.60555 0.205148
\(505\) 7.39445 0.329049
\(506\) 0 0
\(507\) 0 0
\(508\) −1.21110 −0.0537340
\(509\) −33.6333 −1.49077 −0.745385 0.666634i \(-0.767734\pi\)
−0.745385 + 0.666634i \(0.767734\pi\)
\(510\) 4.60555 0.203937
\(511\) 6.42221 0.284102
\(512\) 1.00000 0.0441942
\(513\) 4.60555 0.203340
\(514\) 28.6056 1.26174
\(515\) 4.00000 0.176261
\(516\) 8.00000 0.352180
\(517\) 0 0
\(518\) 42.4222 1.86392
\(519\) −12.4222 −0.545274
\(520\) 0 0
\(521\) 21.6333 0.947772 0.473886 0.880586i \(-0.342851\pi\)
0.473886 + 0.880586i \(0.342851\pi\)
\(522\) 4.60555 0.201580
\(523\) 32.8444 1.43619 0.718093 0.695947i \(-0.245015\pi\)
0.718093 + 0.695947i \(0.245015\pi\)
\(524\) 22.6056 0.987528
\(525\) −4.60555 −0.201003
\(526\) −7.81665 −0.340822
\(527\) −27.6333 −1.20373
\(528\) 0 0
\(529\) −21.0555 −0.915457
\(530\) 6.00000 0.260623
\(531\) 9.21110 0.399728
\(532\) −21.2111 −0.919618
\(533\) 0 0
\(534\) 15.2111 0.658249
\(535\) 0 0
\(536\) −3.21110 −0.138699
\(537\) −19.8167 −0.855152
\(538\) 25.8167 1.11303
\(539\) 0 0
\(540\) −1.00000 −0.0430331
\(541\) −6.97224 −0.299760 −0.149880 0.988704i \(-0.547889\pi\)
−0.149880 + 0.988704i \(0.547889\pi\)
\(542\) 0.422205 0.0181353
\(543\) 8.42221 0.361431
\(544\) −4.60555 −0.197461
\(545\) 1.39445 0.0597316
\(546\) 0 0
\(547\) 14.4222 0.616649 0.308324 0.951281i \(-0.400232\pi\)
0.308324 + 0.951281i \(0.400232\pi\)
\(548\) −3.21110 −0.137172
\(549\) −11.2111 −0.478478
\(550\) 0 0
\(551\) −21.2111 −0.903623
\(552\) −1.39445 −0.0593517
\(553\) −66.4222 −2.82456
\(554\) 16.4222 0.697713
\(555\) −9.21110 −0.390990
\(556\) 17.2111 0.729913
\(557\) −11.5778 −0.490567 −0.245283 0.969451i \(-0.578881\pi\)
−0.245283 + 0.969451i \(0.578881\pi\)
\(558\) 6.00000 0.254000
\(559\) 0 0
\(560\) 4.60555 0.194620
\(561\) 0 0
\(562\) −27.2111 −1.14783
\(563\) −34.0555 −1.43527 −0.717634 0.696420i \(-0.754775\pi\)
−0.717634 + 0.696420i \(0.754775\pi\)
\(564\) −9.21110 −0.387857
\(565\) −13.8167 −0.581271
\(566\) −10.4222 −0.438078
\(567\) 4.60555 0.193415
\(568\) 9.21110 0.386489
\(569\) −33.6333 −1.40998 −0.704991 0.709216i \(-0.749049\pi\)
−0.704991 + 0.709216i \(0.749049\pi\)
\(570\) 4.60555 0.192905
\(571\) 30.0555 1.25778 0.628892 0.777493i \(-0.283509\pi\)
0.628892 + 0.777493i \(0.283509\pi\)
\(572\) 0 0
\(573\) 12.0000 0.501307
\(574\) 14.7889 0.617277
\(575\) 1.39445 0.0581525
\(576\) 1.00000 0.0416667
\(577\) −37.3944 −1.55675 −0.778376 0.627799i \(-0.783956\pi\)
−0.778376 + 0.627799i \(0.783956\pi\)
\(578\) 4.21110 0.175159
\(579\) 7.81665 0.324849
\(580\) 4.60555 0.191235
\(581\) 12.8444 0.532876
\(582\) −1.39445 −0.0578018
\(583\) 0 0
\(584\) 1.39445 0.0577027
\(585\) 0 0
\(586\) −18.0000 −0.743573
\(587\) −6.42221 −0.265073 −0.132536 0.991178i \(-0.542312\pi\)
−0.132536 + 0.991178i \(0.542312\pi\)
\(588\) −14.2111 −0.586056
\(589\) −27.6333 −1.13861
\(590\) 9.21110 0.379215
\(591\) 6.00000 0.246807
\(592\) 9.21110 0.378574
\(593\) 24.4222 1.00290 0.501450 0.865187i \(-0.332800\pi\)
0.501450 + 0.865187i \(0.332800\pi\)
\(594\) 0 0
\(595\) −21.2111 −0.869570
\(596\) 15.2111 0.623071
\(597\) 22.4222 0.917680
\(598\) 0 0
\(599\) −24.0000 −0.980613 −0.490307 0.871550i \(-0.663115\pi\)
−0.490307 + 0.871550i \(0.663115\pi\)
\(600\) −1.00000 −0.0408248
\(601\) 1.63331 0.0666240 0.0333120 0.999445i \(-0.489394\pi\)
0.0333120 + 0.999445i \(0.489394\pi\)
\(602\) −36.8444 −1.50167
\(603\) −3.21110 −0.130766
\(604\) −6.00000 −0.244137
\(605\) −11.0000 −0.447214
\(606\) −7.39445 −0.300379
\(607\) 17.2111 0.698577 0.349289 0.937015i \(-0.386423\pi\)
0.349289 + 0.937015i \(0.386423\pi\)
\(608\) −4.60555 −0.186780
\(609\) −21.2111 −0.859517
\(610\) −11.2111 −0.453924
\(611\) 0 0
\(612\) −4.60555 −0.186168
\(613\) −33.2111 −1.34138 −0.670692 0.741736i \(-0.734003\pi\)
−0.670692 + 0.741736i \(0.734003\pi\)
\(614\) 8.78890 0.354691
\(615\) −3.21110 −0.129484
\(616\) 0 0
\(617\) 12.4222 0.500099 0.250050 0.968233i \(-0.419553\pi\)
0.250050 + 0.968233i \(0.419553\pi\)
\(618\) −4.00000 −0.160904
\(619\) −25.8167 −1.03766 −0.518829 0.854878i \(-0.673632\pi\)
−0.518829 + 0.854878i \(0.673632\pi\)
\(620\) 6.00000 0.240966
\(621\) −1.39445 −0.0559573
\(622\) −12.0000 −0.481156
\(623\) −70.0555 −2.80671
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 3.57779 0.142997
\(627\) 0 0
\(628\) 20.4222 0.814935
\(629\) −42.4222 −1.69148
\(630\) 4.60555 0.183490
\(631\) −3.21110 −0.127832 −0.0639160 0.997955i \(-0.520359\pi\)
−0.0639160 + 0.997955i \(0.520359\pi\)
\(632\) −14.4222 −0.573685
\(633\) 17.2111 0.684080
\(634\) −18.0000 −0.714871
\(635\) −1.21110 −0.0480611
\(636\) −6.00000 −0.237915
\(637\) 0 0
\(638\) 0 0
\(639\) 9.21110 0.364386
\(640\) 1.00000 0.0395285
\(641\) −0.422205 −0.0166761 −0.00833805 0.999965i \(-0.502654\pi\)
−0.00833805 + 0.999965i \(0.502654\pi\)
\(642\) 0 0
\(643\) 9.63331 0.379901 0.189950 0.981794i \(-0.439167\pi\)
0.189950 + 0.981794i \(0.439167\pi\)
\(644\) 6.42221 0.253070
\(645\) 8.00000 0.315000
\(646\) 21.2111 0.834540
\(647\) 34.6056 1.36048 0.680242 0.732987i \(-0.261875\pi\)
0.680242 + 0.732987i \(0.261875\pi\)
\(648\) 1.00000 0.0392837
\(649\) 0 0
\(650\) 0 0
\(651\) −27.6333 −1.08303
\(652\) −24.4222 −0.956447
\(653\) 39.2111 1.53445 0.767225 0.641379i \(-0.221637\pi\)
0.767225 + 0.641379i \(0.221637\pi\)
\(654\) −1.39445 −0.0545273
\(655\) 22.6056 0.883272
\(656\) 3.21110 0.125372
\(657\) 1.39445 0.0544026
\(658\) 42.4222 1.65379
\(659\) −26.2389 −1.02212 −0.511060 0.859545i \(-0.670747\pi\)
−0.511060 + 0.859545i \(0.670747\pi\)
\(660\) 0 0
\(661\) −50.2389 −1.95407 −0.977033 0.213090i \(-0.931647\pi\)
−0.977033 + 0.213090i \(0.931647\pi\)
\(662\) −16.6056 −0.645393
\(663\) 0 0
\(664\) 2.78890 0.108230
\(665\) −21.2111 −0.822531
\(666\) 9.21110 0.356923
\(667\) 6.42221 0.248669
\(668\) −9.21110 −0.356388
\(669\) −1.81665 −0.0702359
\(670\) −3.21110 −0.124056
\(671\) 0 0
\(672\) −4.60555 −0.177663
\(673\) 37.6333 1.45066 0.725329 0.688403i \(-0.241688\pi\)
0.725329 + 0.688403i \(0.241688\pi\)
\(674\) −13.6333 −0.525135
\(675\) −1.00000 −0.0384900
\(676\) 0 0
\(677\) −28.0555 −1.07826 −0.539130 0.842222i \(-0.681247\pi\)
−0.539130 + 0.842222i \(0.681247\pi\)
\(678\) 13.8167 0.530625
\(679\) 6.42221 0.246462
\(680\) −4.60555 −0.176615
\(681\) −24.0000 −0.919682
\(682\) 0 0
\(683\) −9.21110 −0.352453 −0.176227 0.984350i \(-0.556389\pi\)
−0.176227 + 0.984350i \(0.556389\pi\)
\(684\) −4.60555 −0.176098
\(685\) −3.21110 −0.122690
\(686\) 33.2111 1.26801
\(687\) −19.8167 −0.756053
\(688\) −8.00000 −0.304997
\(689\) 0 0
\(690\) −1.39445 −0.0530858
\(691\) 20.2389 0.769922 0.384961 0.922933i \(-0.374215\pi\)
0.384961 + 0.922933i \(0.374215\pi\)
\(692\) 12.4222 0.472221
\(693\) 0 0
\(694\) −27.6333 −1.04895
\(695\) 17.2111 0.652854
\(696\) −4.60555 −0.174573
\(697\) −14.7889 −0.560169
\(698\) 7.81665 0.295865
\(699\) −1.81665 −0.0687122
\(700\) 4.60555 0.174073
\(701\) −47.0278 −1.77621 −0.888107 0.459637i \(-0.847980\pi\)
−0.888107 + 0.459637i \(0.847980\pi\)
\(702\) 0 0
\(703\) −42.4222 −1.59998
\(704\) 0 0
\(705\) −9.21110 −0.346910
\(706\) −8.78890 −0.330775
\(707\) 34.0555 1.28079
\(708\) −9.21110 −0.346174
\(709\) 1.39445 0.0523696 0.0261848 0.999657i \(-0.491664\pi\)
0.0261848 + 0.999657i \(0.491664\pi\)
\(710\) 9.21110 0.345687
\(711\) −14.4222 −0.540875
\(712\) −15.2111 −0.570060
\(713\) 8.36669 0.313335
\(714\) 21.2111 0.793806
\(715\) 0 0
\(716\) 19.8167 0.740583
\(717\) 0 0
\(718\) −15.6333 −0.583430
\(719\) 51.6333 1.92560 0.962799 0.270220i \(-0.0870963\pi\)
0.962799 + 0.270220i \(0.0870963\pi\)
\(720\) 1.00000 0.0372678
\(721\) 18.4222 0.686079
\(722\) 2.21110 0.0822887
\(723\) −6.42221 −0.238844
\(724\) −8.42221 −0.313009
\(725\) 4.60555 0.171046
\(726\) 11.0000 0.408248
\(727\) 14.4222 0.534890 0.267445 0.963573i \(-0.413821\pi\)
0.267445 + 0.963573i \(0.413821\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 1.39445 0.0516109
\(731\) 36.8444 1.36274
\(732\) 11.2111 0.414374
\(733\) −34.0555 −1.25787 −0.628935 0.777458i \(-0.716509\pi\)
−0.628935 + 0.777458i \(0.716509\pi\)
\(734\) −19.6333 −0.724679
\(735\) −14.2111 −0.524184
\(736\) 1.39445 0.0514001
\(737\) 0 0
\(738\) 3.21110 0.118202
\(739\) −20.2389 −0.744498 −0.372249 0.928133i \(-0.621413\pi\)
−0.372249 + 0.928133i \(0.621413\pi\)
\(740\) 9.21110 0.338607
\(741\) 0 0
\(742\) 27.6333 1.01445
\(743\) 36.8444 1.35169 0.675845 0.737044i \(-0.263779\pi\)
0.675845 + 0.737044i \(0.263779\pi\)
\(744\) −6.00000 −0.219971
\(745\) 15.2111 0.557292
\(746\) −20.4222 −0.747710
\(747\) 2.78890 0.102040
\(748\) 0 0
\(749\) 0 0
\(750\) −1.00000 −0.0365148
\(751\) 10.4222 0.380312 0.190156 0.981754i \(-0.439101\pi\)
0.190156 + 0.981754i \(0.439101\pi\)
\(752\) 9.21110 0.335894
\(753\) −13.3944 −0.488121
\(754\) 0 0
\(755\) −6.00000 −0.218362
\(756\) −4.60555 −0.167502
\(757\) 12.7889 0.464820 0.232410 0.972618i \(-0.425339\pi\)
0.232410 + 0.972618i \(0.425339\pi\)
\(758\) 35.0278 1.27227
\(759\) 0 0
\(760\) −4.60555 −0.167061
\(761\) −33.6333 −1.21921 −0.609603 0.792707i \(-0.708671\pi\)
−0.609603 + 0.792707i \(0.708671\pi\)
\(762\) 1.21110 0.0438736
\(763\) 6.42221 0.232499
\(764\) −12.0000 −0.434145
\(765\) −4.60555 −0.166514
\(766\) −27.6333 −0.998432
\(767\) 0 0
\(768\) −1.00000 −0.0360844
\(769\) −12.8444 −0.463181 −0.231591 0.972813i \(-0.574393\pi\)
−0.231591 + 0.972813i \(0.574393\pi\)
\(770\) 0 0
\(771\) −28.6056 −1.03020
\(772\) −7.81665 −0.281328
\(773\) 30.0000 1.07903 0.539513 0.841978i \(-0.318609\pi\)
0.539513 + 0.841978i \(0.318609\pi\)
\(774\) −8.00000 −0.287554
\(775\) 6.00000 0.215526
\(776\) 1.39445 0.0500578
\(777\) −42.4222 −1.52189
\(778\) −4.60555 −0.165117
\(779\) −14.7889 −0.529867
\(780\) 0 0
\(781\) 0 0
\(782\) −6.42221 −0.229658
\(783\) −4.60555 −0.164589
\(784\) 14.2111 0.507539
\(785\) 20.4222 0.728900
\(786\) −22.6056 −0.806313
\(787\) −49.2666 −1.75617 −0.878083 0.478509i \(-0.841177\pi\)
−0.878083 + 0.478509i \(0.841177\pi\)
\(788\) −6.00000 −0.213741
\(789\) 7.81665 0.278280
\(790\) −14.4222 −0.513119
\(791\) −63.6333 −2.26254
\(792\) 0 0
\(793\) 0 0
\(794\) 3.63331 0.128941
\(795\) −6.00000 −0.212798
\(796\) −22.4222 −0.794734
\(797\) 6.00000 0.212531 0.106265 0.994338i \(-0.466111\pi\)
0.106265 + 0.994338i \(0.466111\pi\)
\(798\) 21.2111 0.750865
\(799\) −42.4222 −1.50079
\(800\) 1.00000 0.0353553
\(801\) −15.2111 −0.537458
\(802\) −8.78890 −0.310347
\(803\) 0 0
\(804\) 3.21110 0.113247
\(805\) 6.42221 0.226353
\(806\) 0 0
\(807\) −25.8167 −0.908789
\(808\) 7.39445 0.260136
\(809\) −6.84441 −0.240637 −0.120318 0.992735i \(-0.538392\pi\)
−0.120318 + 0.992735i \(0.538392\pi\)
\(810\) 1.00000 0.0351364
\(811\) 32.2389 1.13206 0.566030 0.824385i \(-0.308479\pi\)
0.566030 + 0.824385i \(0.308479\pi\)
\(812\) 21.2111 0.744364
\(813\) −0.422205 −0.0148074
\(814\) 0 0
\(815\) −24.4222 −0.855473
\(816\) 4.60555 0.161227
\(817\) 36.8444 1.28902
\(818\) −14.7889 −0.517082
\(819\) 0 0
\(820\) 3.21110 0.112137
\(821\) −3.21110 −0.112068 −0.0560341 0.998429i \(-0.517846\pi\)
−0.0560341 + 0.998429i \(0.517846\pi\)
\(822\) 3.21110 0.112000
\(823\) 4.00000 0.139431 0.0697156 0.997567i \(-0.477791\pi\)
0.0697156 + 0.997567i \(0.477791\pi\)
\(824\) 4.00000 0.139347
\(825\) 0 0
\(826\) 42.4222 1.47606
\(827\) 27.6333 0.960904 0.480452 0.877021i \(-0.340473\pi\)
0.480452 + 0.877021i \(0.340473\pi\)
\(828\) 1.39445 0.0484604
\(829\) −46.8444 −1.62697 −0.813487 0.581583i \(-0.802433\pi\)
−0.813487 + 0.581583i \(0.802433\pi\)
\(830\) 2.78890 0.0968040
\(831\) −16.4222 −0.569680
\(832\) 0 0
\(833\) −65.4500 −2.26771
\(834\) −17.2111 −0.595972
\(835\) −9.21110 −0.318763
\(836\) 0 0
\(837\) −6.00000 −0.207390
\(838\) 4.18335 0.144511
\(839\) 18.4222 0.636005 0.318003 0.948090i \(-0.396988\pi\)
0.318003 + 0.948090i \(0.396988\pi\)
\(840\) −4.60555 −0.158907
\(841\) −7.78890 −0.268583
\(842\) −19.8167 −0.682927
\(843\) 27.2111 0.937200
\(844\) −17.2111 −0.592431
\(845\) 0 0
\(846\) 9.21110 0.316684
\(847\) −50.6611 −1.74073
\(848\) 6.00000 0.206041
\(849\) 10.4222 0.357689
\(850\) −4.60555 −0.157969
\(851\) 12.8444 0.440301
\(852\) −9.21110 −0.315567
\(853\) 14.7889 0.506362 0.253181 0.967419i \(-0.418523\pi\)
0.253181 + 0.967419i \(0.418523\pi\)
\(854\) −51.6333 −1.76686
\(855\) −4.60555 −0.157507
\(856\) 0 0
\(857\) 23.0278 0.786613 0.393307 0.919407i \(-0.371331\pi\)
0.393307 + 0.919407i \(0.371331\pi\)
\(858\) 0 0
\(859\) 25.2111 0.860192 0.430096 0.902783i \(-0.358480\pi\)
0.430096 + 0.902783i \(0.358480\pi\)
\(860\) −8.00000 −0.272798
\(861\) −14.7889 −0.504004
\(862\) 12.0000 0.408722
\(863\) 51.6333 1.75762 0.878809 0.477173i \(-0.158339\pi\)
0.878809 + 0.477173i \(0.158339\pi\)
\(864\) −1.00000 −0.0340207
\(865\) 12.4222 0.422368
\(866\) −19.2111 −0.652820
\(867\) −4.21110 −0.143017
\(868\) 27.6333 0.937936
\(869\) 0 0
\(870\) −4.60555 −0.156143
\(871\) 0 0
\(872\) 1.39445 0.0472220
\(873\) 1.39445 0.0471949
\(874\) −6.42221 −0.217234
\(875\) 4.60555 0.155696
\(876\) −1.39445 −0.0471141
\(877\) −24.8444 −0.838936 −0.419468 0.907770i \(-0.637783\pi\)
−0.419468 + 0.907770i \(0.637783\pi\)
\(878\) −8.00000 −0.269987
\(879\) 18.0000 0.607125
\(880\) 0 0
\(881\) −39.2111 −1.32106 −0.660528 0.750802i \(-0.729667\pi\)
−0.660528 + 0.750802i \(0.729667\pi\)
\(882\) 14.2111 0.478513
\(883\) −9.57779 −0.322318 −0.161159 0.986928i \(-0.551523\pi\)
−0.161159 + 0.986928i \(0.551523\pi\)
\(884\) 0 0
\(885\) −9.21110 −0.309628
\(886\) 15.6333 0.525211
\(887\) 6.97224 0.234105 0.117053 0.993126i \(-0.462655\pi\)
0.117053 + 0.993126i \(0.462655\pi\)
\(888\) −9.21110 −0.309104
\(889\) −5.57779 −0.187073
\(890\) −15.2111 −0.509877
\(891\) 0 0
\(892\) 1.81665 0.0608261
\(893\) −42.4222 −1.41960
\(894\) −15.2111 −0.508735
\(895\) 19.8167 0.662398
\(896\) 4.60555 0.153861
\(897\) 0 0
\(898\) −33.6333 −1.12236
\(899\) 27.6333 0.921622
\(900\) 1.00000 0.0333333
\(901\) −27.6333 −0.920599
\(902\) 0 0
\(903\) 36.8444 1.22611
\(904\) −13.8167 −0.459535
\(905\) −8.42221 −0.279964
\(906\) 6.00000 0.199337
\(907\) −21.5778 −0.716479 −0.358239 0.933630i \(-0.616623\pi\)
−0.358239 + 0.933630i \(0.616623\pi\)
\(908\) 24.0000 0.796468
\(909\) 7.39445 0.245258
\(910\) 0 0
\(911\) −27.6333 −0.915532 −0.457766 0.889073i \(-0.651350\pi\)
−0.457766 + 0.889073i \(0.651350\pi\)
\(912\) 4.60555 0.152505
\(913\) 0 0
\(914\) 38.2389 1.26483
\(915\) 11.2111 0.370628
\(916\) 19.8167 0.654761
\(917\) 104.111 3.43805
\(918\) 4.60555 0.152006
\(919\) 16.0000 0.527791 0.263896 0.964551i \(-0.414993\pi\)
0.263896 + 0.964551i \(0.414993\pi\)
\(920\) 1.39445 0.0459736
\(921\) −8.78890 −0.289604
\(922\) −33.6333 −1.10765
\(923\) 0 0
\(924\) 0 0
\(925\) 9.21110 0.302859
\(926\) 31.3944 1.03169
\(927\) 4.00000 0.131377
\(928\) 4.60555 0.151185
\(929\) 39.2111 1.28647 0.643237 0.765667i \(-0.277591\pi\)
0.643237 + 0.765667i \(0.277591\pi\)
\(930\) −6.00000 −0.196748
\(931\) −65.4500 −2.14504
\(932\) 1.81665 0.0595065
\(933\) 12.0000 0.392862
\(934\) 30.4222 0.995445
\(935\) 0 0
\(936\) 0 0
\(937\) −10.3667 −0.338665 −0.169333 0.985559i \(-0.554161\pi\)
−0.169333 + 0.985559i \(0.554161\pi\)
\(938\) −14.7889 −0.482875
\(939\) −3.57779 −0.116757
\(940\) 9.21110 0.300433
\(941\) −54.0000 −1.76035 −0.880175 0.474650i \(-0.842575\pi\)
−0.880175 + 0.474650i \(0.842575\pi\)
\(942\) −20.4222 −0.665391
\(943\) 4.47772 0.145815
\(944\) 9.21110 0.299796
\(945\) −4.60555 −0.149819
\(946\) 0 0
\(947\) 15.6333 0.508014 0.254007 0.967202i \(-0.418251\pi\)
0.254007 + 0.967202i \(0.418251\pi\)
\(948\) 14.4222 0.468411
\(949\) 0 0
\(950\) −4.60555 −0.149424
\(951\) 18.0000 0.583690
\(952\) −21.2111 −0.687456
\(953\) 20.2389 0.655601 0.327800 0.944747i \(-0.393693\pi\)
0.327800 + 0.944747i \(0.393693\pi\)
\(954\) 6.00000 0.194257
\(955\) −12.0000 −0.388311
\(956\) 0 0
\(957\) 0 0
\(958\) −5.57779 −0.180210
\(959\) −14.7889 −0.477558
\(960\) −1.00000 −0.0322749
\(961\) 5.00000 0.161290
\(962\) 0 0
\(963\) 0 0
\(964\) 6.42221 0.206845
\(965\) −7.81665 −0.251627
\(966\) −6.42221 −0.206631
\(967\) −8.23886 −0.264944 −0.132472 0.991187i \(-0.542291\pi\)
−0.132472 + 0.991187i \(0.542291\pi\)
\(968\) −11.0000 −0.353553
\(969\) −21.2111 −0.681399
\(970\) 1.39445 0.0447731
\(971\) −53.0278 −1.70174 −0.850871 0.525375i \(-0.823925\pi\)
−0.850871 + 0.525375i \(0.823925\pi\)
\(972\) −1.00000 −0.0320750
\(973\) 79.2666 2.54117
\(974\) −0.972244 −0.0311527
\(975\) 0 0
\(976\) −11.2111 −0.358859
\(977\) 18.8444 0.602886 0.301443 0.953484i \(-0.402532\pi\)
0.301443 + 0.953484i \(0.402532\pi\)
\(978\) 24.4222 0.780936
\(979\) 0 0
\(980\) 14.2111 0.453957
\(981\) 1.39445 0.0445213
\(982\) −7.81665 −0.249439
\(983\) 42.4222 1.35306 0.676529 0.736416i \(-0.263483\pi\)
0.676529 + 0.736416i \(0.263483\pi\)
\(984\) −3.21110 −0.102366
\(985\) −6.00000 −0.191176
\(986\) −21.2111 −0.675499
\(987\) −42.4222 −1.35031
\(988\) 0 0
\(989\) −11.1556 −0.354727
\(990\) 0 0
\(991\) −22.4222 −0.712265 −0.356132 0.934436i \(-0.615905\pi\)
−0.356132 + 0.934436i \(0.615905\pi\)
\(992\) 6.00000 0.190500
\(993\) 16.6056 0.526961
\(994\) 42.4222 1.34555
\(995\) −22.4222 −0.710832
\(996\) −2.78890 −0.0883696
\(997\) −16.4222 −0.520096 −0.260048 0.965596i \(-0.583738\pi\)
−0.260048 + 0.965596i \(0.583738\pi\)
\(998\) 23.0278 0.728931
\(999\) −9.21110 −0.291426
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 5070.2.a.bf.1.2 2
13.5 odd 4 390.2.b.c.181.2 4
13.8 odd 4 390.2.b.c.181.3 yes 4
13.12 even 2 5070.2.a.z.1.1 2
39.5 even 4 1170.2.b.d.181.4 4
39.8 even 4 1170.2.b.d.181.1 4
52.31 even 4 3120.2.g.q.961.1 4
52.47 even 4 3120.2.g.q.961.4 4
65.8 even 4 1950.2.f.n.649.1 4
65.18 even 4 1950.2.f.m.649.2 4
65.34 odd 4 1950.2.b.k.1351.2 4
65.44 odd 4 1950.2.b.k.1351.3 4
65.47 even 4 1950.2.f.m.649.4 4
65.57 even 4 1950.2.f.n.649.3 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
390.2.b.c.181.2 4 13.5 odd 4
390.2.b.c.181.3 yes 4 13.8 odd 4
1170.2.b.d.181.1 4 39.8 even 4
1170.2.b.d.181.4 4 39.5 even 4
1950.2.b.k.1351.2 4 65.34 odd 4
1950.2.b.k.1351.3 4 65.44 odd 4
1950.2.f.m.649.2 4 65.18 even 4
1950.2.f.m.649.4 4 65.47 even 4
1950.2.f.n.649.1 4 65.8 even 4
1950.2.f.n.649.3 4 65.57 even 4
3120.2.g.q.961.1 4 52.31 even 4
3120.2.g.q.961.4 4 52.47 even 4
5070.2.a.z.1.1 2 13.12 even 2
5070.2.a.bf.1.2 2 1.1 even 1 trivial