Properties

Label 390.2.b.c.181.2
Level $390$
Weight $2$
Character 390.181
Analytic conductor $3.114$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 390 = 2 \cdot 3 \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 390.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(3.11416567883\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{13})\)
Defining polynomial: \(x^{4} + 7 x^{2} + 9\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 181.2
Root \(2.30278i\) of defining polynomial
Character \(\chi\) \(=\) 390.181
Dual form 390.2.b.c.181.3

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000i q^{2} -1.00000 q^{3} -1.00000 q^{4} -1.00000i q^{5} +1.00000i q^{6} +4.60555i q^{7} +1.00000i q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000i q^{2} -1.00000 q^{3} -1.00000 q^{4} -1.00000i q^{5} +1.00000i q^{6} +4.60555i q^{7} +1.00000i q^{8} +1.00000 q^{9} -1.00000 q^{10} +1.00000 q^{12} +3.60555 q^{13} +4.60555 q^{14} +1.00000i q^{15} +1.00000 q^{16} +4.60555 q^{17} -1.00000i q^{18} +4.60555i q^{19} +1.00000i q^{20} -4.60555i q^{21} -1.39445 q^{23} -1.00000i q^{24} -1.00000 q^{25} -3.60555i q^{26} -1.00000 q^{27} -4.60555i q^{28} +4.60555 q^{29} +1.00000 q^{30} -6.00000i q^{31} -1.00000i q^{32} -4.60555i q^{34} +4.60555 q^{35} -1.00000 q^{36} +9.21110i q^{37} +4.60555 q^{38} -3.60555 q^{39} +1.00000 q^{40} -3.21110i q^{41} -4.60555 q^{42} +8.00000 q^{43} -1.00000i q^{45} +1.39445i q^{46} +9.21110i q^{47} -1.00000 q^{48} -14.2111 q^{49} +1.00000i q^{50} -4.60555 q^{51} -3.60555 q^{52} +6.00000 q^{53} +1.00000i q^{54} -4.60555 q^{56} -4.60555i q^{57} -4.60555i q^{58} +9.21110i q^{59} -1.00000i q^{60} -11.2111 q^{61} -6.00000 q^{62} +4.60555i q^{63} -1.00000 q^{64} -3.60555i q^{65} +3.21110i q^{67} -4.60555 q^{68} +1.39445 q^{69} -4.60555i q^{70} -9.21110i q^{71} +1.00000i q^{72} +1.39445i q^{73} +9.21110 q^{74} +1.00000 q^{75} -4.60555i q^{76} +3.60555i q^{78} -14.4222 q^{79} -1.00000i q^{80} +1.00000 q^{81} -3.21110 q^{82} -2.78890i q^{83} +4.60555i q^{84} -4.60555i q^{85} -8.00000i q^{86} -4.60555 q^{87} -15.2111i q^{89} -1.00000 q^{90} +16.6056i q^{91} +1.39445 q^{92} +6.00000i q^{93} +9.21110 q^{94} +4.60555 q^{95} +1.00000i q^{96} -1.39445i q^{97} +14.2111i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 4q^{3} - 4q^{4} + 4q^{9} + O(q^{10}) \) \( 4q - 4q^{3} - 4q^{4} + 4q^{9} - 4q^{10} + 4q^{12} + 4q^{14} + 4q^{16} + 4q^{17} - 20q^{23} - 4q^{25} - 4q^{27} + 4q^{29} + 4q^{30} + 4q^{35} - 4q^{36} + 4q^{38} + 4q^{40} - 4q^{42} + 32q^{43} - 4q^{48} - 28q^{49} - 4q^{51} + 24q^{53} - 4q^{56} - 16q^{61} - 24q^{62} - 4q^{64} - 4q^{68} + 20q^{69} + 8q^{74} + 4q^{75} + 4q^{81} + 16q^{82} - 4q^{87} - 4q^{90} + 20q^{92} + 8q^{94} + 4q^{95} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/390\mathbb{Z}\right)^\times\).

\(n\) \(131\) \(157\) \(301\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 1.00000i − 0.707107i
\(3\) −1.00000 −0.577350
\(4\) −1.00000 −0.500000
\(5\) − 1.00000i − 0.447214i
\(6\) 1.00000i 0.408248i
\(7\) 4.60555i 1.74073i 0.492403 + 0.870367i \(0.336119\pi\)
−0.492403 + 0.870367i \(0.663881\pi\)
\(8\) 1.00000i 0.353553i
\(9\) 1.00000 0.333333
\(10\) −1.00000 −0.316228
\(11\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(12\) 1.00000 0.288675
\(13\) 3.60555 1.00000
\(14\) 4.60555 1.23089
\(15\) 1.00000i 0.258199i
\(16\) 1.00000 0.250000
\(17\) 4.60555 1.11701 0.558505 0.829501i \(-0.311375\pi\)
0.558505 + 0.829501i \(0.311375\pi\)
\(18\) − 1.00000i − 0.235702i
\(19\) 4.60555i 1.05659i 0.849062 + 0.528293i \(0.177168\pi\)
−0.849062 + 0.528293i \(0.822832\pi\)
\(20\) 1.00000i 0.223607i
\(21\) − 4.60555i − 1.00501i
\(22\) 0 0
\(23\) −1.39445 −0.290763 −0.145381 0.989376i \(-0.546441\pi\)
−0.145381 + 0.989376i \(0.546441\pi\)
\(24\) − 1.00000i − 0.204124i
\(25\) −1.00000 −0.200000
\(26\) − 3.60555i − 0.707107i
\(27\) −1.00000 −0.192450
\(28\) − 4.60555i − 0.870367i
\(29\) 4.60555 0.855229 0.427615 0.903961i \(-0.359354\pi\)
0.427615 + 0.903961i \(0.359354\pi\)
\(30\) 1.00000 0.182574
\(31\) − 6.00000i − 1.07763i −0.842424 0.538816i \(-0.818872\pi\)
0.842424 0.538816i \(-0.181128\pi\)
\(32\) − 1.00000i − 0.176777i
\(33\) 0 0
\(34\) − 4.60555i − 0.789846i
\(35\) 4.60555 0.778480
\(36\) −1.00000 −0.166667
\(37\) 9.21110i 1.51430i 0.653243 + 0.757148i \(0.273408\pi\)
−0.653243 + 0.757148i \(0.726592\pi\)
\(38\) 4.60555 0.747119
\(39\) −3.60555 −0.577350
\(40\) 1.00000 0.158114
\(41\) − 3.21110i − 0.501490i −0.968053 0.250745i \(-0.919324\pi\)
0.968053 0.250745i \(-0.0806756\pi\)
\(42\) −4.60555 −0.710652
\(43\) 8.00000 1.21999 0.609994 0.792406i \(-0.291172\pi\)
0.609994 + 0.792406i \(0.291172\pi\)
\(44\) 0 0
\(45\) − 1.00000i − 0.149071i
\(46\) 1.39445i 0.205600i
\(47\) 9.21110i 1.34358i 0.740743 + 0.671789i \(0.234474\pi\)
−0.740743 + 0.671789i \(0.765526\pi\)
\(48\) −1.00000 −0.144338
\(49\) −14.2111 −2.03016
\(50\) 1.00000i 0.141421i
\(51\) −4.60555 −0.644906
\(52\) −3.60555 −0.500000
\(53\) 6.00000 0.824163 0.412082 0.911147i \(-0.364802\pi\)
0.412082 + 0.911147i \(0.364802\pi\)
\(54\) 1.00000i 0.136083i
\(55\) 0 0
\(56\) −4.60555 −0.615443
\(57\) − 4.60555i − 0.610020i
\(58\) − 4.60555i − 0.604739i
\(59\) 9.21110i 1.19918i 0.800306 + 0.599592i \(0.204670\pi\)
−0.800306 + 0.599592i \(0.795330\pi\)
\(60\) − 1.00000i − 0.129099i
\(61\) −11.2111 −1.43543 −0.717717 0.696335i \(-0.754813\pi\)
−0.717717 + 0.696335i \(0.754813\pi\)
\(62\) −6.00000 −0.762001
\(63\) 4.60555i 0.580245i
\(64\) −1.00000 −0.125000
\(65\) − 3.60555i − 0.447214i
\(66\) 0 0
\(67\) 3.21110i 0.392299i 0.980574 + 0.196149i \(0.0628437\pi\)
−0.980574 + 0.196149i \(0.937156\pi\)
\(68\) −4.60555 −0.558505
\(69\) 1.39445 0.167872
\(70\) − 4.60555i − 0.550469i
\(71\) − 9.21110i − 1.09316i −0.837408 0.546578i \(-0.815930\pi\)
0.837408 0.546578i \(-0.184070\pi\)
\(72\) 1.00000i 0.117851i
\(73\) 1.39445i 0.163208i 0.996665 + 0.0816039i \(0.0260043\pi\)
−0.996665 + 0.0816039i \(0.973996\pi\)
\(74\) 9.21110 1.07077
\(75\) 1.00000 0.115470
\(76\) − 4.60555i − 0.528293i
\(77\) 0 0
\(78\) 3.60555i 0.408248i
\(79\) −14.4222 −1.62262 −0.811312 0.584613i \(-0.801246\pi\)
−0.811312 + 0.584613i \(0.801246\pi\)
\(80\) − 1.00000i − 0.111803i
\(81\) 1.00000 0.111111
\(82\) −3.21110 −0.354607
\(83\) − 2.78890i − 0.306121i −0.988217 0.153061i \(-0.951087\pi\)
0.988217 0.153061i \(-0.0489130\pi\)
\(84\) 4.60555i 0.502507i
\(85\) − 4.60555i − 0.499542i
\(86\) − 8.00000i − 0.862662i
\(87\) −4.60555 −0.493767
\(88\) 0 0
\(89\) − 15.2111i − 1.61237i −0.591661 0.806187i \(-0.701528\pi\)
0.591661 0.806187i \(-0.298472\pi\)
\(90\) −1.00000 −0.105409
\(91\) 16.6056i 1.74073i
\(92\) 1.39445 0.145381
\(93\) 6.00000i 0.622171i
\(94\) 9.21110 0.950053
\(95\) 4.60555 0.472520
\(96\) 1.00000i 0.102062i
\(97\) − 1.39445i − 0.141585i −0.997491 0.0707924i \(-0.977447\pi\)
0.997491 0.0707924i \(-0.0225528\pi\)
\(98\) 14.2111i 1.43554i
\(99\) 0 0
\(100\) 1.00000 0.100000
\(101\) −7.39445 −0.735775 −0.367888 0.929870i \(-0.619919\pi\)
−0.367888 + 0.929870i \(0.619919\pi\)
\(102\) 4.60555i 0.456018i
\(103\) −4.00000 −0.394132 −0.197066 0.980390i \(-0.563141\pi\)
−0.197066 + 0.980390i \(0.563141\pi\)
\(104\) 3.60555i 0.353553i
\(105\) −4.60555 −0.449456
\(106\) − 6.00000i − 0.582772i
\(107\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(108\) 1.00000 0.0962250
\(109\) − 1.39445i − 0.133564i −0.997768 0.0667820i \(-0.978727\pi\)
0.997768 0.0667820i \(-0.0212732\pi\)
\(110\) 0 0
\(111\) − 9.21110i − 0.874279i
\(112\) 4.60555i 0.435184i
\(113\) −13.8167 −1.29976 −0.649881 0.760036i \(-0.725181\pi\)
−0.649881 + 0.760036i \(0.725181\pi\)
\(114\) −4.60555 −0.431349
\(115\) 1.39445i 0.130033i
\(116\) −4.60555 −0.427615
\(117\) 3.60555 0.333333
\(118\) 9.21110 0.847951
\(119\) 21.2111i 1.94442i
\(120\) −1.00000 −0.0912871
\(121\) 11.0000 1.00000
\(122\) 11.2111i 1.01501i
\(123\) 3.21110i 0.289535i
\(124\) 6.00000i 0.538816i
\(125\) 1.00000i 0.0894427i
\(126\) 4.60555 0.410295
\(127\) 1.21110 0.107468 0.0537340 0.998555i \(-0.482888\pi\)
0.0537340 + 0.998555i \(0.482888\pi\)
\(128\) 1.00000i 0.0883883i
\(129\) −8.00000 −0.704361
\(130\) −3.60555 −0.316228
\(131\) 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
\(135\) 1.00000i 0.0860663i
\(136\) 4.60555i 0.394923i
\(137\) − 3.21110i − 0.274343i −0.990547 0.137172i \(-0.956199\pi\)
0.990547 0.137172i \(-0.0438011\pi\)
\(138\) − 1.39445i − 0.118703i
\(139\) 17.2111 1.45983 0.729913 0.683540i \(-0.239560\pi\)
0.729913 + 0.683540i \(0.239560\pi\)
\(140\) −4.60555 −0.389240
\(141\) − 9.21110i − 0.775715i
\(142\) −9.21110 −0.772979
\(143\) 0 0
\(144\) 1.00000 0.0833333
\(145\) − 4.60555i − 0.382470i
\(146\) 1.39445 0.115405
\(147\) 14.2111 1.17211
\(148\) − 9.21110i − 0.757148i
\(149\) − 15.2111i − 1.24614i −0.782165 0.623071i \(-0.785885\pi\)
0.782165 0.623071i \(-0.214115\pi\)
\(150\) − 1.00000i − 0.0816497i
\(151\) − 6.00000i − 0.488273i −0.969741 0.244137i \(-0.921495\pi\)
0.969741 0.244137i \(-0.0785045\pi\)
\(152\) −4.60555 −0.373560
\(153\) 4.60555 0.372337
\(154\) 0 0
\(155\) −6.00000 −0.481932
\(156\) 3.60555 0.288675
\(157\) 20.4222 1.62987 0.814935 0.579553i \(-0.196773\pi\)
0.814935 + 0.579553i \(0.196773\pi\)
\(158\) 14.4222i 1.14737i
\(159\) −6.00000 −0.475831
\(160\) −1.00000 −0.0790569
\(161\) − 6.42221i − 0.506141i
\(162\) − 1.00000i − 0.0785674i
\(163\) − 24.4222i − 1.91289i −0.291905 0.956447i \(-0.594289\pi\)
0.291905 0.956447i \(-0.405711\pi\)
\(164\) 3.21110i 0.250745i
\(165\) 0 0
\(166\) −2.78890 −0.216460
\(167\) − 9.21110i − 0.712777i −0.934338 0.356388i \(-0.884008\pi\)
0.934338 0.356388i \(-0.115992\pi\)
\(168\) 4.60555 0.355326
\(169\) 13.0000 1.00000
\(170\) −4.60555 −0.353230
\(171\) 4.60555i 0.352195i
\(172\) −8.00000 −0.609994
\(173\) −12.4222 −0.944443 −0.472221 0.881480i \(-0.656548\pi\)
−0.472221 + 0.881480i \(0.656548\pi\)
\(174\) 4.60555i 0.349146i
\(175\) − 4.60555i − 0.348147i
\(176\) 0 0
\(177\) − 9.21110i − 0.692349i
\(178\) −15.2111 −1.14012
\(179\) −19.8167 −1.48117 −0.740583 0.671965i \(-0.765451\pi\)
−0.740583 + 0.671965i \(0.765451\pi\)
\(180\) 1.00000i 0.0745356i
\(181\) 8.42221 0.626018 0.313009 0.949750i \(-0.398663\pi\)
0.313009 + 0.949750i \(0.398663\pi\)
\(182\) 16.6056 1.23089
\(183\) 11.2111 0.828749
\(184\) − 1.39445i − 0.102800i
\(185\) 9.21110 0.677214
\(186\) 6.00000 0.439941
\(187\) 0 0
\(188\) − 9.21110i − 0.671789i
\(189\) − 4.60555i − 0.335005i
\(190\) − 4.60555i − 0.334122i
\(191\) −12.0000 −0.868290 −0.434145 0.900843i \(-0.642949\pi\)
−0.434145 + 0.900843i \(0.642949\pi\)
\(192\) 1.00000 0.0721688
\(193\) − 7.81665i − 0.562655i −0.959612 0.281328i \(-0.909225\pi\)
0.959612 0.281328i \(-0.0907747\pi\)
\(194\) −1.39445 −0.100116
\(195\) 3.60555i 0.258199i
\(196\) 14.2111 1.01508
\(197\) 6.00000i 0.427482i 0.976890 + 0.213741i \(0.0685649\pi\)
−0.976890 + 0.213741i \(0.931435\pi\)
\(198\) 0 0
\(199\) 22.4222 1.58947 0.794734 0.606958i \(-0.207610\pi\)
0.794734 + 0.606958i \(0.207610\pi\)
\(200\) − 1.00000i − 0.0707107i
\(201\) − 3.21110i − 0.226494i
\(202\) 7.39445i 0.520272i
\(203\) 21.2111i 1.48873i
\(204\) 4.60555 0.322453
\(205\) −3.21110 −0.224273
\(206\) 4.00000i 0.278693i
\(207\) −1.39445 −0.0969209
\(208\) 3.60555 0.250000
\(209\) 0 0
\(210\) 4.60555i 0.317813i
\(211\) −17.2111 −1.18486 −0.592431 0.805622i \(-0.701832\pi\)
−0.592431 + 0.805622i \(0.701832\pi\)
\(212\) −6.00000 −0.412082
\(213\) 9.21110i 0.631134i
\(214\) 0 0
\(215\) − 8.00000i − 0.545595i
\(216\) − 1.00000i − 0.0680414i
\(217\) 27.6333 1.87587
\(218\) −1.39445 −0.0944440
\(219\) − 1.39445i − 0.0942281i
\(220\) 0 0
\(221\) 16.6056 1.11701
\(222\) −9.21110 −0.618209
\(223\) − 1.81665i − 0.121652i −0.998148 0.0608261i \(-0.980627\pi\)
0.998148 0.0608261i \(-0.0193735\pi\)
\(224\) 4.60555 0.307721
\(225\) −1.00000 −0.0666667
\(226\) 13.8167i 0.919070i
\(227\) − 24.0000i − 1.59294i −0.604681 0.796468i \(-0.706699\pi\)
0.604681 0.796468i \(-0.293301\pi\)
\(228\) 4.60555i 0.305010i
\(229\) 19.8167i 1.30952i 0.755836 + 0.654761i \(0.227231\pi\)
−0.755836 + 0.654761i \(0.772769\pi\)
\(230\) 1.39445 0.0919472
\(231\) 0 0
\(232\) 4.60555i 0.302369i
\(233\) −1.81665 −0.119013 −0.0595065 0.998228i \(-0.518953\pi\)
−0.0595065 + 0.998228i \(0.518953\pi\)
\(234\) − 3.60555i − 0.235702i
\(235\) 9.21110 0.600866
\(236\) − 9.21110i − 0.599592i
\(237\) 14.4222 0.936823
\(238\) 21.2111 1.37491
\(239\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(240\) 1.00000i 0.0645497i
\(241\) 6.42221i 0.413691i 0.978374 + 0.206845i \(0.0663197\pi\)
−0.978374 + 0.206845i \(0.933680\pi\)
\(242\) − 11.0000i − 0.707107i
\(243\) −1.00000 −0.0641500
\(244\) 11.2111 0.717717
\(245\) 14.2111i 0.907914i
\(246\) 3.21110 0.204732
\(247\) 16.6056i 1.05659i
\(248\) 6.00000 0.381000
\(249\) 2.78890i 0.176739i
\(250\) 1.00000 0.0632456
\(251\) −13.3944 −0.845450 −0.422725 0.906258i \(-0.638926\pi\)
−0.422725 + 0.906258i \(0.638926\pi\)
\(252\) − 4.60555i − 0.290122i
\(253\) 0 0
\(254\) − 1.21110i − 0.0759913i
\(255\) 4.60555i 0.288411i
\(256\) 1.00000 0.0625000
\(257\) −28.6056 −1.78437 −0.892183 0.451675i \(-0.850827\pi\)
−0.892183 + 0.451675i \(0.850827\pi\)
\(258\) 8.00000i 0.498058i
\(259\) −42.4222 −2.63599
\(260\) 3.60555i 0.223607i
\(261\) 4.60555 0.285076
\(262\) − 22.6056i − 1.39658i
\(263\) −7.81665 −0.481996 −0.240998 0.970526i \(-0.577475\pi\)
−0.240998 + 0.970526i \(0.577475\pi\)
\(264\) 0 0
\(265\) − 6.00000i − 0.368577i
\(266\) 21.2111i 1.30054i
\(267\) 15.2111i 0.930904i
\(268\) − 3.21110i − 0.196149i
\(269\) 25.8167 1.57407 0.787035 0.616909i \(-0.211615\pi\)
0.787035 + 0.616909i \(0.211615\pi\)
\(270\) 1.00000 0.0608581
\(271\) 0.422205i 0.0256471i 0.999918 + 0.0128236i \(0.00408198\pi\)
−0.999918 + 0.0128236i \(0.995918\pi\)
\(272\) 4.60555 0.279253
\(273\) − 16.6056i − 1.00501i
\(274\) −3.21110 −0.193990
\(275\) 0 0
\(276\) −1.39445 −0.0839359
\(277\) −16.4222 −0.986715 −0.493357 0.869827i \(-0.664230\pi\)
−0.493357 + 0.869827i \(0.664230\pi\)
\(278\) − 17.2111i − 1.03225i
\(279\) − 6.00000i − 0.359211i
\(280\) 4.60555i 0.275234i
\(281\) − 27.2111i − 1.62328i −0.584159 0.811639i \(-0.698576\pi\)
0.584159 0.811639i \(-0.301424\pi\)
\(282\) −9.21110 −0.548513
\(283\) 10.4222 0.619536 0.309768 0.950812i \(-0.399749\pi\)
0.309768 + 0.950812i \(0.399749\pi\)
\(284\) 9.21110i 0.546578i
\(285\) −4.60555 −0.272809
\(286\) 0 0
\(287\) 14.7889 0.872961
\(288\) − 1.00000i − 0.0589256i
\(289\) 4.21110 0.247712
\(290\) −4.60555 −0.270447
\(291\) 1.39445i 0.0817440i
\(292\) − 1.39445i − 0.0816039i
\(293\) − 18.0000i − 1.05157i −0.850617 0.525786i \(-0.823771\pi\)
0.850617 0.525786i \(-0.176229\pi\)
\(294\) − 14.2111i − 0.828808i
\(295\) 9.21110 0.536291
\(296\) −9.21110 −0.535384
\(297\) 0 0
\(298\) −15.2111 −0.881156
\(299\) −5.02776 −0.290763
\(300\) −1.00000 −0.0577350
\(301\) 36.8444i 2.12368i
\(302\) −6.00000 −0.345261
\(303\) 7.39445 0.424800
\(304\) 4.60555i 0.264146i
\(305\) 11.2111i 0.641946i
\(306\) − 4.60555i − 0.263282i
\(307\) 8.78890i 0.501609i 0.968038 + 0.250804i \(0.0806951\pi\)
−0.968038 + 0.250804i \(0.919305\pi\)
\(308\) 0 0
\(309\) 4.00000 0.227552
\(310\) 6.00000i 0.340777i
\(311\) 12.0000 0.680458 0.340229 0.940343i \(-0.389495\pi\)
0.340229 + 0.940343i \(0.389495\pi\)
\(312\) − 3.60555i − 0.204124i
\(313\) 3.57779 0.202229 0.101114 0.994875i \(-0.467759\pi\)
0.101114 + 0.994875i \(0.467759\pi\)
\(314\) − 20.4222i − 1.15249i
\(315\) 4.60555 0.259493
\(316\) 14.4222 0.811312
\(317\) 18.0000i 1.01098i 0.862832 + 0.505490i \(0.168688\pi\)
−0.862832 + 0.505490i \(0.831312\pi\)
\(318\) 6.00000i 0.336463i
\(319\) 0 0
\(320\) 1.00000i 0.0559017i
\(321\) 0 0
\(322\) −6.42221 −0.357895
\(323\) 21.2111i 1.18022i
\(324\) −1.00000 −0.0555556
\(325\) −3.60555 −0.200000
\(326\) −24.4222 −1.35262
\(327\) 1.39445i 0.0771132i
\(328\) 3.21110 0.177303
\(329\) −42.4222 −2.33881
\(330\) 0 0
\(331\) 16.6056i 0.912724i 0.889794 + 0.456362i \(0.150848\pi\)
−0.889794 + 0.456362i \(0.849152\pi\)
\(332\) 2.78890i 0.153061i
\(333\) 9.21110i 0.504765i
\(334\) −9.21110 −0.504009
\(335\) 3.21110 0.175441
\(336\) − 4.60555i − 0.251253i
\(337\) 13.6333 0.742654 0.371327 0.928502i \(-0.378903\pi\)
0.371327 + 0.928502i \(0.378903\pi\)
\(338\) − 13.0000i − 0.707107i
\(339\) 13.8167 0.750418
\(340\) 4.60555i 0.249771i
\(341\) 0 0
\(342\) 4.60555 0.249040
\(343\) − 33.2111i − 1.79323i
\(344\) 8.00000i 0.431331i
\(345\) − 1.39445i − 0.0750746i
\(346\) 12.4222i 0.667822i
\(347\) −27.6333 −1.48343 −0.741717 0.670713i \(-0.765988\pi\)
−0.741717 + 0.670713i \(0.765988\pi\)
\(348\) 4.60555 0.246883
\(349\) 7.81665i 0.418416i 0.977871 + 0.209208i \(0.0670886\pi\)
−0.977871 + 0.209208i \(0.932911\pi\)
\(350\) −4.60555 −0.246177
\(351\) −3.60555 −0.192450
\(352\) 0 0
\(353\) 8.78890i 0.467786i 0.972262 + 0.233893i \(0.0751465\pi\)
−0.972262 + 0.233893i \(0.924853\pi\)
\(354\) −9.21110 −0.489565
\(355\) −9.21110 −0.488875
\(356\) 15.2111i 0.806187i
\(357\) − 21.2111i − 1.12261i
\(358\) 19.8167i 1.04734i
\(359\) − 15.6333i − 0.825094i −0.910936 0.412547i \(-0.864639\pi\)
0.910936 0.412547i \(-0.135361\pi\)
\(360\) 1.00000 0.0527046
\(361\) −2.21110 −0.116374
\(362\) − 8.42221i − 0.442661i
\(363\) −11.0000 −0.577350
\(364\) − 16.6056i − 0.870367i
\(365\) 1.39445 0.0729888
\(366\) − 11.2111i − 0.586014i
\(367\) −19.6333 −1.02485 −0.512425 0.858732i \(-0.671253\pi\)
−0.512425 + 0.858732i \(0.671253\pi\)
\(368\) −1.39445 −0.0726907
\(369\) − 3.21110i − 0.167163i
\(370\) − 9.21110i − 0.478862i
\(371\) 27.6333i 1.43465i
\(372\) − 6.00000i − 0.311086i
\(373\) −20.4222 −1.05742 −0.528711 0.848802i \(-0.677324\pi\)
−0.528711 + 0.848802i \(0.677324\pi\)
\(374\) 0 0
\(375\) − 1.00000i − 0.0516398i
\(376\) −9.21110 −0.475026
\(377\) 16.6056 0.855229
\(378\) −4.60555 −0.236884
\(379\) − 35.0278i − 1.79925i −0.436658 0.899627i \(-0.643838\pi\)
0.436658 0.899627i \(-0.356162\pi\)
\(380\) −4.60555 −0.236260
\(381\) −1.21110 −0.0620467
\(382\) 12.0000i 0.613973i
\(383\) 27.6333i 1.41200i 0.708214 + 0.705998i \(0.249501\pi\)
−0.708214 + 0.705998i \(0.750499\pi\)
\(384\) − 1.00000i − 0.0510310i
\(385\) 0 0
\(386\) −7.81665 −0.397857
\(387\) 8.00000 0.406663
\(388\) 1.39445i 0.0707924i
\(389\) 4.60555 0.233511 0.116755 0.993161i \(-0.462751\pi\)
0.116755 + 0.993161i \(0.462751\pi\)
\(390\) 3.60555 0.182574
\(391\) −6.42221 −0.324785
\(392\) − 14.2111i − 0.717769i
\(393\) −22.6056 −1.14030
\(394\) 6.00000 0.302276
\(395\) 14.4222i 0.725660i
\(396\) 0 0
\(397\) 3.63331i 0.182350i 0.995835 + 0.0911752i \(0.0290623\pi\)
−0.995835 + 0.0911752i \(0.970938\pi\)
\(398\) − 22.4222i − 1.12392i
\(399\) 21.2111 1.06188
\(400\) −1.00000 −0.0500000
\(401\) − 8.78890i − 0.438897i −0.975624 0.219448i \(-0.929574\pi\)
0.975624 0.219448i \(-0.0704257\pi\)
\(402\) −3.21110 −0.160155
\(403\) − 21.6333i − 1.07763i
\(404\) 7.39445 0.367888
\(405\) − 1.00000i − 0.0496904i
\(406\) 21.2111 1.05269
\(407\) 0 0
\(408\) − 4.60555i − 0.228009i
\(409\) 14.7889i 0.731264i 0.930760 + 0.365632i \(0.119147\pi\)
−0.930760 + 0.365632i \(0.880853\pi\)
\(410\) 3.21110i 0.158585i
\(411\) 3.21110i 0.158392i
\(412\) 4.00000 0.197066
\(413\) −42.4222 −2.08746
\(414\) 1.39445i 0.0685334i
\(415\) −2.78890 −0.136902
\(416\) − 3.60555i − 0.176777i
\(417\) −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
\(421\) 19.8167i 0.965805i 0.875674 + 0.482902i \(0.160417\pi\)
−0.875674 + 0.482902i \(0.839583\pi\)
\(422\) 17.2111i 0.837823i
\(423\) 9.21110i 0.447859i
\(424\) 6.00000i 0.291386i
\(425\) −4.60555 −0.223402
\(426\) 9.21110 0.446279
\(427\) − 51.6333i − 2.49871i
\(428\) 0 0
\(429\) 0 0
\(430\) −8.00000 −0.385794
\(431\) − 12.0000i − 0.578020i −0.957326 0.289010i \(-0.906674\pi\)
0.957326 0.289010i \(-0.0933260\pi\)
\(432\) −1.00000 −0.0481125
\(433\) 19.2111 0.923227 0.461613 0.887081i \(-0.347271\pi\)
0.461613 + 0.887081i \(0.347271\pi\)
\(434\) − 27.6333i − 1.32644i
\(435\) 4.60555i 0.220819i
\(436\) 1.39445i 0.0667820i
\(437\) − 6.42221i − 0.307216i
\(438\) −1.39445 −0.0666293
\(439\) 8.00000 0.381819 0.190910 0.981608i \(-0.438856\pi\)
0.190910 + 0.981608i \(0.438856\pi\)
\(440\) 0 0
\(441\) −14.2111 −0.676719
\(442\) − 16.6056i − 0.789846i
\(443\) 15.6333 0.742761 0.371380 0.928481i \(-0.378885\pi\)
0.371380 + 0.928481i \(0.378885\pi\)
\(444\) 9.21110i 0.437140i
\(445\) −15.2111 −0.721075
\(446\) −1.81665 −0.0860211
\(447\) 15.2111i 0.719460i
\(448\) − 4.60555i − 0.217592i
\(449\) − 33.6333i − 1.58725i −0.608405 0.793627i \(-0.708190\pi\)
0.608405 0.793627i \(-0.291810\pi\)
\(450\) 1.00000i 0.0471405i
\(451\) 0 0
\(452\) 13.8167 0.649881
\(453\) 6.00000i 0.281905i
\(454\) −24.0000 −1.12638
\(455\) 16.6056 0.778480
\(456\) 4.60555 0.215675
\(457\) − 38.2389i − 1.78874i −0.447330 0.894369i \(-0.647625\pi\)
0.447330 0.894369i \(-0.352375\pi\)
\(458\) 19.8167 0.925971
\(459\) −4.60555 −0.214969
\(460\) − 1.39445i − 0.0650165i
\(461\) 33.6333i 1.56646i 0.621733 + 0.783230i \(0.286429\pi\)
−0.621733 + 0.783230i \(0.713571\pi\)
\(462\) 0 0
\(463\) 31.3944i 1.45902i 0.683968 + 0.729512i \(0.260253\pi\)
−0.683968 + 0.729512i \(0.739747\pi\)
\(464\) 4.60555 0.213807
\(465\) 6.00000 0.278243
\(466\) 1.81665i 0.0841549i
\(467\) −30.4222 −1.40777 −0.703886 0.710313i \(-0.748553\pi\)
−0.703886 + 0.710313i \(0.748553\pi\)
\(468\) −3.60555 −0.166667
\(469\) −14.7889 −0.682888
\(470\) − 9.21110i − 0.424876i
\(471\) −20.4222 −0.941006
\(472\) −9.21110 −0.423975
\(473\) 0 0
\(474\) − 14.4222i − 0.662434i
\(475\) − 4.60555i − 0.211317i
\(476\) − 21.2111i − 0.972209i
\(477\) 6.00000 0.274721
\(478\) 0 0
\(479\) − 5.57779i − 0.254856i −0.991848 0.127428i \(-0.959328\pi\)
0.991848 0.127428i \(-0.0406722\pi\)
\(480\) 1.00000 0.0456435
\(481\) 33.2111i 1.51430i
\(482\) 6.42221 0.292523
\(483\) 6.42221i 0.292220i
\(484\) −11.0000 −0.500000
\(485\) −1.39445 −0.0633187
\(486\) 1.00000i 0.0453609i
\(487\) 0.972244i 0.0440566i 0.999757 + 0.0220283i \(0.00701239\pi\)
−0.999757 + 0.0220283i \(0.992988\pi\)
\(488\) − 11.2111i − 0.507503i
\(489\) 24.4222i 1.10441i
\(490\) 14.2111 0.641992
\(491\) 7.81665 0.352761 0.176380 0.984322i \(-0.443561\pi\)
0.176380 + 0.984322i \(0.443561\pi\)
\(492\) − 3.21110i − 0.144768i
\(493\) 21.2111 0.955300
\(494\) 16.6056 0.747119
\(495\) 0 0
\(496\) − 6.00000i − 0.269408i
\(497\) 42.4222 1.90290
\(498\) 2.78890 0.124973
\(499\) − 23.0278i − 1.03086i −0.856930 0.515432i \(-0.827631\pi\)
0.856930 0.515432i \(-0.172369\pi\)
\(500\) − 1.00000i − 0.0447214i
\(501\) 9.21110i 0.411522i
\(502\) 13.3944i 0.597824i
\(503\) 23.4500 1.04558 0.522791 0.852461i \(-0.324891\pi\)
0.522791 + 0.852461i \(0.324891\pi\)
\(504\) −4.60555 −0.205148
\(505\) 7.39445i 0.329049i
\(506\) 0 0
\(507\) −13.0000 −0.577350
\(508\) −1.21110 −0.0537340
\(509\) 33.6333i 1.49077i 0.666634 + 0.745385i \(0.267734\pi\)
−0.666634 + 0.745385i \(0.732266\pi\)
\(510\) 4.60555 0.203937
\(511\) −6.42221 −0.284102
\(512\) − 1.00000i − 0.0441942i
\(513\) − 4.60555i − 0.203340i
\(514\) 28.6056i 1.26174i
\(515\) 4.00000i 0.176261i
\(516\) 8.00000 0.352180
\(517\) 0 0
\(518\) 42.4222i 1.86392i
\(519\) 12.4222 0.545274
\(520\) 3.60555 0.158114
\(521\) 21.6333 0.947772 0.473886 0.880586i \(-0.342851\pi\)
0.473886 + 0.880586i \(0.342851\pi\)
\(522\) − 4.60555i − 0.201580i
\(523\) 32.8444 1.43619 0.718093 0.695947i \(-0.245015\pi\)
0.718093 + 0.695947i \(0.245015\pi\)
\(524\) −22.6056 −0.987528
\(525\) 4.60555i 0.201003i
\(526\) 7.81665i 0.340822i
\(527\) − 27.6333i − 1.20373i
\(528\) 0 0
\(529\) −21.0555 −0.915457
\(530\) −6.00000 −0.260623
\(531\) 9.21110i 0.399728i
\(532\) 21.2111 0.919618
\(533\) − 11.5778i − 0.501490i
\(534\) 15.2111 0.658249
\(535\) 0 0
\(536\) −3.21110 −0.138699
\(537\) 19.8167 0.855152
\(538\) − 25.8167i − 1.11303i
\(539\) 0 0
\(540\) − 1.00000i − 0.0430331i
\(541\) − 6.97224i − 0.299760i −0.988704 0.149880i \(-0.952111\pi\)
0.988704 0.149880i \(-0.0478888\pi\)
\(542\) 0.422205 0.0181353
\(543\) −8.42221 −0.361431
\(544\) − 4.60555i − 0.197461i
\(545\) −1.39445 −0.0597316
\(546\) −16.6056 −0.710652
\(547\) 14.4222 0.616649 0.308324 0.951281i \(-0.400232\pi\)
0.308324 + 0.951281i \(0.400232\pi\)
\(548\) 3.21110i 0.137172i
\(549\) −11.2111 −0.478478
\(550\) 0 0
\(551\) 21.2111i 0.903623i
\(552\) 1.39445i 0.0593517i
\(553\) − 66.4222i − 2.82456i
\(554\) 16.4222i 0.697713i
\(555\) −9.21110 −0.390990
\(556\) −17.2111 −0.729913
\(557\) − 11.5778i − 0.490567i −0.969451 0.245283i \(-0.921119\pi\)
0.969451 0.245283i \(-0.0788810\pi\)
\(558\) −6.00000 −0.254000
\(559\) 28.8444 1.21999
\(560\) 4.60555 0.194620
\(561\) 0 0
\(562\) −27.2111 −1.14783
\(563\) 34.0555 1.43527 0.717634 0.696420i \(-0.245225\pi\)
0.717634 + 0.696420i \(0.245225\pi\)
\(564\) 9.21110i 0.387857i
\(565\) 13.8167i 0.581271i
\(566\) − 10.4222i − 0.438078i
\(567\) 4.60555i 0.193415i
\(568\) 9.21110 0.386489
\(569\) 33.6333 1.40998 0.704991 0.709216i \(-0.250951\pi\)
0.704991 + 0.709216i \(0.250951\pi\)
\(570\) 4.60555i 0.192905i
\(571\) −30.0555 −1.25778 −0.628892 0.777493i \(-0.716491\pi\)
−0.628892 + 0.777493i \(0.716491\pi\)
\(572\) 0 0
\(573\) 12.0000 0.501307
\(574\) − 14.7889i − 0.617277i
\(575\) 1.39445 0.0581525
\(576\) −1.00000 −0.0416667
\(577\) 37.3944i 1.55675i 0.627799 + 0.778376i \(0.283956\pi\)
−0.627799 + 0.778376i \(0.716044\pi\)
\(578\) − 4.21110i − 0.175159i
\(579\) 7.81665i 0.324849i
\(580\) 4.60555i 0.191235i
\(581\) 12.8444 0.532876
\(582\) 1.39445 0.0578018
\(583\) 0 0
\(584\) −1.39445 −0.0577027
\(585\) − 3.60555i − 0.149071i
\(586\) −18.0000 −0.743573
\(587\) 6.42221i 0.265073i 0.991178 + 0.132536i \(0.0423121\pi\)
−0.991178 + 0.132536i \(0.957688\pi\)
\(588\) −14.2111 −0.586056
\(589\) 27.6333 1.13861
\(590\) − 9.21110i − 0.379215i
\(591\) − 6.00000i − 0.246807i
\(592\) 9.21110i 0.378574i
\(593\) 24.4222i 1.00290i 0.865187 + 0.501450i \(0.167200\pi\)
−0.865187 + 0.501450i \(0.832800\pi\)
\(594\) 0 0
\(595\) 21.2111 0.869570
\(596\) 15.2111i 0.623071i
\(597\) −22.4222 −0.917680
\(598\) 5.02776i 0.205600i
\(599\) −24.0000 −0.980613 −0.490307 0.871550i \(-0.663115\pi\)
−0.490307 + 0.871550i \(0.663115\pi\)
\(600\) 1.00000i 0.0408248i
\(601\) 1.63331 0.0666240 0.0333120 0.999445i \(-0.489394\pi\)
0.0333120 + 0.999445i \(0.489394\pi\)
\(602\) 36.8444 1.50167
\(603\) 3.21110i 0.130766i
\(604\) 6.00000i 0.244137i
\(605\) − 11.0000i − 0.447214i
\(606\) − 7.39445i − 0.300379i
\(607\) 17.2111 0.698577 0.349289 0.937015i \(-0.386423\pi\)
0.349289 + 0.937015i \(0.386423\pi\)
\(608\) 4.60555 0.186780
\(609\) − 21.2111i − 0.859517i
\(610\) 11.2111 0.453924
\(611\) 33.2111i 1.34358i
\(612\) −4.60555 −0.186168
\(613\) 33.2111i 1.34138i 0.741736 + 0.670692i \(0.234003\pi\)
−0.741736 + 0.670692i \(0.765997\pi\)
\(614\) 8.78890 0.354691
\(615\) 3.21110 0.129484
\(616\) 0 0
\(617\) − 12.4222i − 0.500099i −0.968233 0.250050i \(-0.919553\pi\)
0.968233 0.250050i \(-0.0804469\pi\)
\(618\) − 4.00000i − 0.160904i
\(619\) − 25.8167i − 1.03766i −0.854878 0.518829i \(-0.826368\pi\)
0.854878 0.518829i \(-0.173632\pi\)
\(620\) 6.00000 0.240966
\(621\) 1.39445 0.0559573
\(622\) − 12.0000i − 0.481156i
\(623\) 70.0555 2.80671
\(624\) −3.60555 −0.144338
\(625\) 1.00000 0.0400000
\(626\) − 3.57779i − 0.142997i
\(627\) 0 0
\(628\) −20.4222 −0.814935
\(629\) 42.4222i 1.69148i
\(630\) − 4.60555i − 0.183490i
\(631\) − 3.21110i − 0.127832i −0.997955 0.0639160i \(-0.979641\pi\)
0.997955 0.0639160i \(-0.0203590\pi\)
\(632\) − 14.4222i − 0.573685i
\(633\) 17.2111 0.684080
\(634\) 18.0000 0.714871
\(635\) − 1.21110i − 0.0480611i
\(636\) 6.00000 0.237915
\(637\) −51.2389 −2.03016
\(638\) 0 0
\(639\) − 9.21110i − 0.364386i
\(640\) 1.00000 0.0395285
\(641\) 0.422205 0.0166761 0.00833805 0.999965i \(-0.497346\pi\)
0.00833805 + 0.999965i \(0.497346\pi\)
\(642\) 0 0
\(643\) − 9.63331i − 0.379901i −0.981794 0.189950i \(-0.939167\pi\)
0.981794 0.189950i \(-0.0608327\pi\)
\(644\) 6.42221i 0.253070i
\(645\) 8.00000i 0.315000i
\(646\) 21.2111 0.834540
\(647\) −34.6056 −1.36048 −0.680242 0.732987i \(-0.738125\pi\)
−0.680242 + 0.732987i \(0.738125\pi\)
\(648\) 1.00000i 0.0392837i
\(649\) 0 0
\(650\) 3.60555i 0.141421i
\(651\) −27.6333 −1.08303
\(652\) 24.4222i 0.956447i
\(653\) 39.2111 1.53445 0.767225 0.641379i \(-0.221637\pi\)
0.767225 + 0.641379i \(0.221637\pi\)
\(654\) 1.39445 0.0545273
\(655\) − 22.6056i − 0.883272i
\(656\) − 3.21110i − 0.125372i
\(657\) 1.39445i 0.0544026i
\(658\) 42.4222i 1.65379i
\(659\) −26.2389 −1.02212 −0.511060 0.859545i \(-0.670747\pi\)
−0.511060 + 0.859545i \(0.670747\pi\)
\(660\) 0 0
\(661\) − 50.2389i − 1.95407i −0.213090 0.977033i \(-0.568353\pi\)
0.213090 0.977033i \(-0.431647\pi\)
\(662\) 16.6056 0.645393
\(663\) −16.6056 −0.644906
\(664\) 2.78890 0.108230
\(665\) 21.2111i 0.822531i
\(666\) 9.21110 0.356923
\(667\) −6.42221 −0.248669
\(668\) 9.21110i 0.356388i
\(669\) 1.81665i 0.0702359i
\(670\) − 3.21110i − 0.124056i
\(671\) 0 0
\(672\) −4.60555 −0.177663
\(673\) −37.6333 −1.45066 −0.725329 0.688403i \(-0.758312\pi\)
−0.725329 + 0.688403i \(0.758312\pi\)
\(674\) − 13.6333i − 0.525135i
\(675\) 1.00000 0.0384900
\(676\) −13.0000 −0.500000
\(677\) −28.0555 −1.07826 −0.539130 0.842222i \(-0.681247\pi\)
−0.539130 + 0.842222i \(0.681247\pi\)
\(678\) − 13.8167i − 0.530625i
\(679\) 6.42221 0.246462
\(680\) 4.60555 0.176615
\(681\) 24.0000i 0.919682i
\(682\) 0 0
\(683\) − 9.21110i − 0.352453i −0.984350 0.176227i \(-0.943611\pi\)
0.984350 0.176227i \(-0.0563891\pi\)
\(684\) − 4.60555i − 0.176098i
\(685\) −3.21110 −0.122690
\(686\) −33.2111 −1.26801
\(687\) − 19.8167i − 0.756053i
\(688\) 8.00000 0.304997
\(689\) 21.6333 0.824163
\(690\) −1.39445 −0.0530858
\(691\) − 20.2389i − 0.769922i −0.922933 0.384961i \(-0.874215\pi\)
0.922933 0.384961i \(-0.125785\pi\)
\(692\) 12.4222 0.472221
\(693\) 0 0
\(694\) 27.6333i 1.04895i
\(695\) − 17.2111i − 0.652854i
\(696\) − 4.60555i − 0.174573i
\(697\) − 14.7889i − 0.560169i
\(698\) 7.81665 0.295865
\(699\) 1.81665 0.0687122
\(700\) 4.60555i 0.174073i
\(701\) 47.0278 1.77621 0.888107 0.459637i \(-0.152020\pi\)
0.888107 + 0.459637i \(0.152020\pi\)
\(702\) 3.60555i 0.136083i
\(703\) −42.4222 −1.59998
\(704\) 0 0
\(705\) −9.21110 −0.346910
\(706\) 8.78890 0.330775
\(707\) − 34.0555i − 1.28079i
\(708\) 9.21110i 0.346174i
\(709\) 1.39445i 0.0523696i 0.999657 + 0.0261848i \(0.00833584\pi\)
−0.999657 + 0.0261848i \(0.991664\pi\)
\(710\) 9.21110i 0.345687i
\(711\) −14.4222 −0.540875
\(712\) 15.2111 0.570060
\(713\) 8.36669i 0.313335i
\(714\) −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 \(-0.912904\pi\)
−0.962799 + 0.270220i \(0.912904\pi\)
\(720\) − 1.00000i − 0.0372678i
\(721\) − 18.4222i − 0.686079i
\(722\) 2.21110i 0.0822887i
\(723\) − 6.42221i − 0.238844i
\(724\) −8.42221 −0.313009
\(725\) −4.60555 −0.171046
\(726\) 11.0000i 0.408248i
\(727\) −14.4222 −0.534890 −0.267445 0.963573i \(-0.586179\pi\)
−0.267445 + 0.963573i \(0.586179\pi\)
\(728\) −16.6056 −0.615443
\(729\) 1.00000 0.0370370
\(730\) − 1.39445i − 0.0516109i
\(731\) 36.8444 1.36274
\(732\) −11.2111 −0.414374
\(733\) 34.0555i 1.25787i 0.777458 + 0.628935i \(0.216509\pi\)
−0.777458 + 0.628935i \(0.783491\pi\)
\(734\) 19.6333i 0.724679i
\(735\) − 14.2111i − 0.524184i
\(736\) 1.39445i 0.0514001i
\(737\) 0 0
\(738\) −3.21110 −0.118202
\(739\) − 20.2389i − 0.744498i −0.928133 0.372249i \(-0.878587\pi\)
0.928133 0.372249i \(-0.121413\pi\)
\(740\) −9.21110 −0.338607
\(741\) − 16.6056i − 0.610020i
\(742\) 27.6333 1.01445
\(743\) − 36.8444i − 1.35169i −0.737044 0.675845i \(-0.763779\pi\)
0.737044 0.675845i \(-0.236221\pi\)
\(744\) −6.00000 −0.219971
\(745\) −15.2111 −0.557292
\(746\) 20.4222i 0.747710i
\(747\) − 2.78890i − 0.102040i
\(748\) 0 0
\(749\) 0 0
\(750\) −1.00000 −0.0365148
\(751\) −10.4222 −0.380312 −0.190156 0.981754i \(-0.560899\pi\)
−0.190156 + 0.981754i \(0.560899\pi\)
\(752\) 9.21110i 0.335894i
\(753\) 13.3944 0.488121
\(754\) − 16.6056i − 0.604739i
\(755\) −6.00000 −0.218362
\(756\) 4.60555i 0.167502i
\(757\) 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
\(760\) 4.60555i 0.167061i
\(761\) − 33.6333i − 1.21921i −0.792707 0.609603i \(-0.791329\pi\)
0.792707 0.609603i \(-0.208671\pi\)
\(762\) 1.21110i 0.0438736i
\(763\) 6.42221 0.232499
\(764\) 12.0000 0.434145
\(765\) − 4.60555i − 0.166514i
\(766\) 27.6333 0.998432
\(767\) 33.2111i 1.19918i
\(768\) −1.00000 −0.0360844
\(769\) 12.8444i 0.463181i 0.972813 + 0.231591i \(0.0743930\pi\)
−0.972813 + 0.231591i \(0.925607\pi\)
\(770\) 0 0
\(771\) 28.6056 1.03020
\(772\) 7.81665i 0.281328i
\(773\) − 30.0000i − 1.07903i −0.841978 0.539513i \(-0.818609\pi\)
0.841978 0.539513i \(-0.181391\pi\)
\(774\) − 8.00000i − 0.287554i
\(775\) 6.00000i 0.215526i
\(776\) 1.39445 0.0500578
\(777\) 42.4222 1.52189
\(778\) − 4.60555i − 0.165117i
\(779\) 14.7889 0.529867
\(780\) − 3.60555i − 0.129099i
\(781\) 0 0
\(782\) 6.42221i 0.229658i
\(783\) −4.60555 −0.164589
\(784\) −14.2111 −0.507539
\(785\) − 20.4222i − 0.728900i
\(786\) 22.6056i 0.806313i
\(787\) − 49.2666i − 1.75617i −0.478509 0.878083i \(-0.658823\pi\)
0.478509 0.878083i \(-0.341177\pi\)
\(788\) − 6.00000i − 0.213741i
\(789\) 7.81665 0.278280
\(790\) 14.4222 0.513119
\(791\) − 63.6333i − 2.26254i
\(792\) 0 0
\(793\) −40.4222 −1.43543
\(794\) 3.63331 0.128941
\(795\) 6.00000i 0.212798i
\(796\) −22.4222 −0.794734
\(797\) −6.00000 −0.212531 −0.106265 0.994338i \(-0.533889\pi\)
−0.106265 + 0.994338i \(0.533889\pi\)
\(798\) − 21.2111i − 0.750865i
\(799\) 42.4222i 1.50079i
\(800\) 1.00000i 0.0353553i
\(801\) − 15.2111i − 0.537458i
\(802\) −8.78890 −0.310347
\(803\) 0 0
\(804\) 3.21110i 0.113247i
\(805\) −6.42221 −0.226353
\(806\) −21.6333 −0.762001
\(807\) −25.8167 −0.908789
\(808\) − 7.39445i − 0.260136i
\(809\) −6.84441 −0.240637 −0.120318 0.992735i \(-0.538392\pi\)
−0.120318 + 0.992735i \(0.538392\pi\)
\(810\) −1.00000 −0.0351364
\(811\) − 32.2389i − 1.13206i −0.824385 0.566030i \(-0.808479\pi\)
0.824385 0.566030i \(-0.191521\pi\)
\(812\) − 21.2111i − 0.744364i
\(813\) − 0.422205i − 0.0148074i
\(814\) 0 0
\(815\) −24.4222 −0.855473
\(816\) −4.60555 −0.161227
\(817\) 36.8444i 1.28902i
\(818\) 14.7889 0.517082
\(819\) 16.6056i 0.580245i
\(820\) 3.21110 0.112137
\(821\) 3.21110i 0.112068i 0.998429 + 0.0560341i \(0.0178456\pi\)
−0.998429 + 0.0560341i \(0.982154\pi\)
\(822\) 3.21110 0.112000
\(823\) −4.00000 −0.139431 −0.0697156 0.997567i \(-0.522209\pi\)
−0.0697156 + 0.997567i \(0.522209\pi\)
\(824\) − 4.00000i − 0.139347i
\(825\) 0 0
\(826\) 42.4222i 1.47606i
\(827\) 27.6333i 0.960904i 0.877021 + 0.480452i \(0.159527\pi\)
−0.877021 + 0.480452i \(0.840473\pi\)
\(828\) 1.39445 0.0484604
\(829\) 46.8444 1.62697 0.813487 0.581583i \(-0.197567\pi\)
0.813487 + 0.581583i \(0.197567\pi\)
\(830\) 2.78890i 0.0968040i
\(831\) 16.4222 0.569680
\(832\) −3.60555 −0.125000
\(833\) −65.4500 −2.26771
\(834\) 17.2111i 0.595972i
\(835\) −9.21110 −0.318763
\(836\) 0 0
\(837\) 6.00000i 0.207390i
\(838\) − 4.18335i − 0.144511i
\(839\) 18.4222i 0.636005i 0.948090 + 0.318003i \(0.103012\pi\)
−0.948090 + 0.318003i \(0.896988\pi\)
\(840\) − 4.60555i − 0.158907i
\(841\) −7.78890 −0.268583
\(842\) 19.8167 0.682927
\(843\) 27.2111i 0.937200i
\(844\) 17.2111 0.592431
\(845\) − 13.0000i − 0.447214i
\(846\) 9.21110 0.316684
\(847\) 50.6611i 1.74073i
\(848\) 6.00000 0.206041
\(849\) −10.4222 −0.357689
\(850\) 4.60555i 0.157969i
\(851\) − 12.8444i − 0.440301i
\(852\) − 9.21110i − 0.315567i
\(853\) 14.7889i 0.506362i 0.967419 + 0.253181i \(0.0814769\pi\)
−0.967419 + 0.253181i \(0.918523\pi\)
\(854\) −51.6333 −1.76686
\(855\) 4.60555 0.157507
\(856\) 0 0
\(857\) −23.0278 −0.786613 −0.393307 0.919407i \(-0.628669\pi\)
−0.393307 + 0.919407i \(0.628669\pi\)
\(858\) 0 0
\(859\) 25.2111 0.860192 0.430096 0.902783i \(-0.358480\pi\)
0.430096 + 0.902783i \(0.358480\pi\)
\(860\) 8.00000i 0.272798i
\(861\) −14.7889 −0.504004
\(862\) −12.0000 −0.408722
\(863\) − 51.6333i − 1.75762i −0.477173 0.878809i \(-0.658339\pi\)
0.477173 0.878809i \(-0.341661\pi\)
\(864\) 1.00000i 0.0340207i
\(865\) 12.4222i 0.422368i
\(866\) − 19.2111i − 0.652820i
\(867\) −4.21110 −0.143017
\(868\) −27.6333 −0.937936
\(869\) 0 0
\(870\) 4.60555 0.156143
\(871\) 11.5778i 0.392299i
\(872\) 1.39445 0.0472220
\(873\) − 1.39445i − 0.0471949i
\(874\) −6.42221 −0.217234
\(875\) −4.60555 −0.155696
\(876\) 1.39445i 0.0471141i
\(877\) 24.8444i 0.838936i 0.907770 + 0.419468i \(0.137783\pi\)
−0.907770 + 0.419468i \(0.862217\pi\)
\(878\) − 8.00000i − 0.269987i
\(879\) 18.0000i 0.607125i
\(880\) 0 0
\(881\) 39.2111 1.32106 0.660528 0.750802i \(-0.270333\pi\)
0.660528 + 0.750802i \(0.270333\pi\)
\(882\) 14.2111i 0.478513i
\(883\) 9.57779 0.322318 0.161159 0.986928i \(-0.448477\pi\)
0.161159 + 0.986928i \(0.448477\pi\)
\(884\) −16.6056 −0.558505
\(885\) −9.21110 −0.309628
\(886\) − 15.6333i − 0.525211i
\(887\) 6.97224 0.234105 0.117053 0.993126i \(-0.462655\pi\)
0.117053 + 0.993126i \(0.462655\pi\)
\(888\) 9.21110 0.309104
\(889\) 5.57779i 0.187073i
\(890\) 15.2111i 0.509877i
\(891\) 0 0
\(892\) 1.81665i 0.0608261i
\(893\) −42.4222 −1.41960
\(894\) 15.2111 0.508735
\(895\) 19.8167i 0.662398i
\(896\) −4.60555 −0.153861
\(897\) 5.02776 0.167872
\(898\) −33.6333 −1.12236
\(899\) − 27.6333i − 0.921622i
\(900\) 1.00000 0.0333333
\(901\) 27.6333 0.920599
\(902\) 0 0
\(903\) − 36.8444i − 1.22611i
\(904\) − 13.8167i − 0.459535i
\(905\) − 8.42221i − 0.279964i
\(906\) 6.00000 0.199337
\(907\) 21.5778 0.716479 0.358239 0.933630i \(-0.383377\pi\)
0.358239 + 0.933630i \(0.383377\pi\)
\(908\) 24.0000i 0.796468i
\(909\) −7.39445 −0.245258
\(910\) − 16.6056i − 0.550469i
\(911\) −27.6333 −0.915532 −0.457766 0.889073i \(-0.651350\pi\)
−0.457766 + 0.889073i \(0.651350\pi\)
\(912\) − 4.60555i − 0.152505i
\(913\) 0 0
\(914\) −38.2389 −1.26483
\(915\) − 11.2111i − 0.370628i
\(916\) − 19.8167i − 0.654761i
\(917\) 104.111i 3.43805i
\(918\) 4.60555i 0.152006i
\(919\) 16.0000 0.527791 0.263896 0.964551i \(-0.414993\pi\)
0.263896 + 0.964551i \(0.414993\pi\)
\(920\) −1.39445 −0.0459736
\(921\) − 8.78890i − 0.289604i
\(922\) 33.6333 1.10765
\(923\) − 33.2111i − 1.09316i
\(924\) 0 0
\(925\) − 9.21110i − 0.302859i
\(926\) 31.3944 1.03169
\(927\) −4.00000 −0.131377
\(928\) − 4.60555i − 0.151185i
\(929\) − 39.2111i − 1.28647i −0.765667 0.643237i \(-0.777591\pi\)
0.765667 0.643237i \(-0.222409\pi\)
\(930\) − 6.00000i − 0.196748i
\(931\) − 65.4500i − 2.14504i
\(932\) 1.81665 0.0595065
\(933\) −12.0000 −0.392862
\(934\) 30.4222i 0.995445i
\(935\) 0 0
\(936\) 3.60555i 0.117851i
\(937\) −10.3667 −0.338665 −0.169333 0.985559i \(-0.554161\pi\)
−0.169333 + 0.985559i \(0.554161\pi\)
\(938\) 14.7889i 0.482875i
\(939\) −3.57779 −0.116757
\(940\) −9.21110 −0.300433
\(941\) 54.0000i 1.76035i 0.474650 + 0.880175i \(0.342575\pi\)
−0.474650 + 0.880175i \(0.657425\pi\)
\(942\) 20.4222i 0.665391i
\(943\) 4.47772i 0.145815i
\(944\) 9.21110i 0.299796i
\(945\) −4.60555 −0.149819
\(946\) 0 0
\(947\) 15.6333i 0.508014i 0.967202 + 0.254007i \(0.0817487\pi\)
−0.967202 + 0.254007i \(0.918251\pi\)
\(948\) −14.4222 −0.468411
\(949\) 5.02776i 0.163208i
\(950\) −4.60555 −0.149424
\(951\) − 18.0000i − 0.583690i
\(952\) −21.2111 −0.687456
\(953\) −20.2389 −0.655601 −0.327800 0.944747i \(-0.606307\pi\)
−0.327800 + 0.944747i \(0.606307\pi\)
\(954\) − 6.00000i − 0.194257i
\(955\) 12.0000i 0.388311i
\(956\) 0 0
\(957\) 0 0
\(958\) −5.57779 −0.180210
\(959\) 14.7889 0.477558
\(960\) − 1.00000i − 0.0322749i
\(961\) −5.00000 −0.161290
\(962\) 33.2111 1.07077
\(963\) 0 0
\(964\) − 6.42221i − 0.206845i
\(965\) −7.81665 −0.251627
\(966\) 6.42221 0.206631
\(967\) 8.23886i 0.264944i 0.991187 + 0.132472i \(0.0422914\pi\)
−0.991187 + 0.132472i \(0.957709\pi\)
\(968\) 11.0000i 0.353553i
\(969\) − 21.2111i − 0.681399i
\(970\) 1.39445i 0.0447731i
\(971\) −53.0278 −1.70174 −0.850871 0.525375i \(-0.823925\pi\)
−0.850871 + 0.525375i \(0.823925\pi\)
\(972\) 1.00000 0.0320750
\(973\) 79.2666i 2.54117i
\(974\) 0.972244 0.0311527
\(975\) 3.60555 0.115470
\(976\) −11.2111 −0.358859
\(977\) − 18.8444i − 0.602886i −0.953484 0.301443i \(-0.902532\pi\)
0.953484 0.301443i \(-0.0974683\pi\)
\(978\) 24.4222 0.780936
\(979\) 0 0
\(980\) − 14.2111i − 0.453957i
\(981\) − 1.39445i − 0.0445213i
\(982\) − 7.81665i − 0.249439i
\(983\) 42.4222i 1.35306i 0.736416 + 0.676529i \(0.236517\pi\)
−0.736416 + 0.676529i \(0.763483\pi\)
\(984\) −3.21110 −0.102366
\(985\) 6.00000 0.191176
\(986\) − 21.2111i − 0.675499i
\(987\) 42.4222 1.35031
\(988\) − 16.6056i − 0.528293i
\(989\) −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
\(993\) − 16.6056i − 0.526961i
\(994\) − 42.4222i − 1.34555i
\(995\) − 22.4222i − 0.710832i
\(996\) − 2.78890i − 0.0883696i
\(997\) −16.4222 −0.520096 −0.260048 0.965596i \(-0.583738\pi\)
−0.260048 + 0.965596i \(0.583738\pi\)
\(998\) −23.0278 −0.728931
\(999\) − 9.21110i − 0.291426i
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 390.2.b.c.181.2 4
3.2 odd 2 1170.2.b.d.181.4 4
4.3 odd 2 3120.2.g.q.961.1 4
5.2 odd 4 1950.2.f.n.649.3 4
5.3 odd 4 1950.2.f.m.649.2 4
5.4 even 2 1950.2.b.k.1351.3 4
13.5 odd 4 5070.2.a.z.1.1 2
13.8 odd 4 5070.2.a.bf.1.2 2
13.12 even 2 inner 390.2.b.c.181.3 yes 4
39.38 odd 2 1170.2.b.d.181.1 4
52.51 odd 2 3120.2.g.q.961.4 4
65.12 odd 4 1950.2.f.m.649.4 4
65.38 odd 4 1950.2.f.n.649.1 4
65.64 even 2 1950.2.b.k.1351.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
390.2.b.c.181.2 4 1.1 even 1 trivial
390.2.b.c.181.3 yes 4 13.12 even 2 inner
1170.2.b.d.181.1 4 39.38 odd 2
1170.2.b.d.181.4 4 3.2 odd 2
1950.2.b.k.1351.2 4 65.64 even 2
1950.2.b.k.1351.3 4 5.4 even 2
1950.2.f.m.649.2 4 5.3 odd 4
1950.2.f.m.649.4 4 65.12 odd 4
1950.2.f.n.649.1 4 65.38 odd 4
1950.2.f.n.649.3 4 5.2 odd 4
3120.2.g.q.961.1 4 4.3 odd 2
3120.2.g.q.961.4 4 52.51 odd 2
5070.2.a.z.1.1 2 13.5 odd 4
5070.2.a.bf.1.2 2 13.8 odd 4