Properties

Label 5070.2.a.bg.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: \(\Q(\zeta_{12})^+\)
Defining polynomial: \(x^{2} - 3\)
Coefficient ring: \(\Z[a_1, \ldots, a_{23}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 390)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-1.73205\) of defining polynomial
Character \(\chi\) \(=\) 5070.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{6} +2.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{6} +2.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{10} +6.46410 q^{11} -1.00000 q^{12} +2.00000 q^{14} -1.00000 q^{15} +1.00000 q^{16} +4.00000 q^{17} +1.00000 q^{18} +7.46410 q^{19} +1.00000 q^{20} -2.00000 q^{21} +6.46410 q^{22} -3.73205 q^{23} -1.00000 q^{24} +1.00000 q^{25} -1.00000 q^{27} +2.00000 q^{28} +0.267949 q^{29} -1.00000 q^{30} -1.73205 q^{31} +1.00000 q^{32} -6.46410 q^{33} +4.00000 q^{34} +2.00000 q^{35} +1.00000 q^{36} +9.19615 q^{37} +7.46410 q^{38} +1.00000 q^{40} -2.00000 q^{41} -2.00000 q^{42} -11.9282 q^{43} +6.46410 q^{44} +1.00000 q^{45} -3.73205 q^{46} +3.53590 q^{47} -1.00000 q^{48} -3.00000 q^{49} +1.00000 q^{50} -4.00000 q^{51} +0.928203 q^{53} -1.00000 q^{54} +6.46410 q^{55} +2.00000 q^{56} -7.46410 q^{57} +0.267949 q^{58} -8.46410 q^{59} -1.00000 q^{60} -10.3923 q^{61} -1.73205 q^{62} +2.00000 q^{63} +1.00000 q^{64} -6.46410 q^{66} +11.4641 q^{67} +4.00000 q^{68} +3.73205 q^{69} +2.00000 q^{70} -12.3923 q^{71} +1.00000 q^{72} -2.00000 q^{73} +9.19615 q^{74} -1.00000 q^{75} +7.46410 q^{76} +12.9282 q^{77} -13.9282 q^{79} +1.00000 q^{80} +1.00000 q^{81} -2.00000 q^{82} +8.92820 q^{83} -2.00000 q^{84} +4.00000 q^{85} -11.9282 q^{86} -0.267949 q^{87} +6.46410 q^{88} -0.535898 q^{89} +1.00000 q^{90} -3.73205 q^{92} +1.73205 q^{93} +3.53590 q^{94} +7.46410 q^{95} -1.00000 q^{96} +0.535898 q^{97} -3.00000 q^{98} +6.46410 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{2} - 2q^{3} + 2q^{4} + 2q^{5} - 2q^{6} + 4q^{7} + 2q^{8} + 2q^{9} + O(q^{10}) \) \( 2q + 2q^{2} - 2q^{3} + 2q^{4} + 2q^{5} - 2q^{6} + 4q^{7} + 2q^{8} + 2q^{9} + 2q^{10} + 6q^{11} - 2q^{12} + 4q^{14} - 2q^{15} + 2q^{16} + 8q^{17} + 2q^{18} + 8q^{19} + 2q^{20} - 4q^{21} + 6q^{22} - 4q^{23} - 2q^{24} + 2q^{25} - 2q^{27} + 4q^{28} + 4q^{29} - 2q^{30} + 2q^{32} - 6q^{33} + 8q^{34} + 4q^{35} + 2q^{36} + 8q^{37} + 8q^{38} + 2q^{40} - 4q^{41} - 4q^{42} - 10q^{43} + 6q^{44} + 2q^{45} - 4q^{46} + 14q^{47} - 2q^{48} - 6q^{49} + 2q^{50} - 8q^{51} - 12q^{53} - 2q^{54} + 6q^{55} + 4q^{56} - 8q^{57} + 4q^{58} - 10q^{59} - 2q^{60} + 4q^{63} + 2q^{64} - 6q^{66} + 16q^{67} + 8q^{68} + 4q^{69} + 4q^{70} - 4q^{71} + 2q^{72} - 4q^{73} + 8q^{74} - 2q^{75} + 8q^{76} + 12q^{77} - 14q^{79} + 2q^{80} + 2q^{81} - 4q^{82} + 4q^{83} - 4q^{84} + 8q^{85} - 10q^{86} - 4q^{87} + 6q^{88} - 8q^{89} + 2q^{90} - 4q^{92} + 14q^{94} + 8q^{95} - 2q^{96} + 8q^{97} - 6q^{98} + 6q^{99} + O(q^{100}) \)

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
\(5\) 1.00000 0.447214
\(6\) −1.00000 −0.408248
\(7\) 2.00000 0.755929 0.377964 0.925820i \(-0.376624\pi\)
0.377964 + 0.925820i \(0.376624\pi\)
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) 1.00000 0.316228
\(11\) 6.46410 1.94900 0.974500 0.224388i \(-0.0720382\pi\)
0.974500 + 0.224388i \(0.0720382\pi\)
\(12\) −1.00000 −0.288675
\(13\) 0 0
\(14\) 2.00000 0.534522
\(15\) −1.00000 −0.258199
\(16\) 1.00000 0.250000
\(17\) 4.00000 0.970143 0.485071 0.874475i \(-0.338794\pi\)
0.485071 + 0.874475i \(0.338794\pi\)
\(18\) 1.00000 0.235702
\(19\) 7.46410 1.71238 0.856191 0.516659i \(-0.172825\pi\)
0.856191 + 0.516659i \(0.172825\pi\)
\(20\) 1.00000 0.223607
\(21\) −2.00000 −0.436436
\(22\) 6.46410 1.37815
\(23\) −3.73205 −0.778186 −0.389093 0.921198i \(-0.627212\pi\)
−0.389093 + 0.921198i \(0.627212\pi\)
\(24\) −1.00000 −0.204124
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 2.00000 0.377964
\(29\) 0.267949 0.0497569 0.0248785 0.999690i \(-0.492080\pi\)
0.0248785 + 0.999690i \(0.492080\pi\)
\(30\) −1.00000 −0.182574
\(31\) −1.73205 −0.311086 −0.155543 0.987829i \(-0.549713\pi\)
−0.155543 + 0.987829i \(0.549713\pi\)
\(32\) 1.00000 0.176777
\(33\) −6.46410 −1.12526
\(34\) 4.00000 0.685994
\(35\) 2.00000 0.338062
\(36\) 1.00000 0.166667
\(37\) 9.19615 1.51184 0.755919 0.654665i \(-0.227190\pi\)
0.755919 + 0.654665i \(0.227190\pi\)
\(38\) 7.46410 1.21084
\(39\) 0 0
\(40\) 1.00000 0.158114
\(41\) −2.00000 −0.312348 −0.156174 0.987730i \(-0.549916\pi\)
−0.156174 + 0.987730i \(0.549916\pi\)
\(42\) −2.00000 −0.308607
\(43\) −11.9282 −1.81903 −0.909517 0.415667i \(-0.863548\pi\)
−0.909517 + 0.415667i \(0.863548\pi\)
\(44\) 6.46410 0.974500
\(45\) 1.00000 0.149071
\(46\) −3.73205 −0.550261
\(47\) 3.53590 0.515764 0.257882 0.966176i \(-0.416975\pi\)
0.257882 + 0.966176i \(0.416975\pi\)
\(48\) −1.00000 −0.144338
\(49\) −3.00000 −0.428571
\(50\) 1.00000 0.141421
\(51\) −4.00000 −0.560112
\(52\) 0 0
\(53\) 0.928203 0.127499 0.0637493 0.997966i \(-0.479694\pi\)
0.0637493 + 0.997966i \(0.479694\pi\)
\(54\) −1.00000 −0.136083
\(55\) 6.46410 0.871619
\(56\) 2.00000 0.267261
\(57\) −7.46410 −0.988644
\(58\) 0.267949 0.0351835
\(59\) −8.46410 −1.10193 −0.550966 0.834528i \(-0.685741\pi\)
−0.550966 + 0.834528i \(0.685741\pi\)
\(60\) −1.00000 −0.129099
\(61\) −10.3923 −1.33060 −0.665299 0.746577i \(-0.731696\pi\)
−0.665299 + 0.746577i \(0.731696\pi\)
\(62\) −1.73205 −0.219971
\(63\) 2.00000 0.251976
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) −6.46410 −0.795676
\(67\) 11.4641 1.40056 0.700281 0.713867i \(-0.253058\pi\)
0.700281 + 0.713867i \(0.253058\pi\)
\(68\) 4.00000 0.485071
\(69\) 3.73205 0.449286
\(70\) 2.00000 0.239046
\(71\) −12.3923 −1.47070 −0.735348 0.677690i \(-0.762981\pi\)
−0.735348 + 0.677690i \(0.762981\pi\)
\(72\) 1.00000 0.117851
\(73\) −2.00000 −0.234082 −0.117041 0.993127i \(-0.537341\pi\)
−0.117041 + 0.993127i \(0.537341\pi\)
\(74\) 9.19615 1.06903
\(75\) −1.00000 −0.115470
\(76\) 7.46410 0.856191
\(77\) 12.9282 1.47331
\(78\) 0 0
\(79\) −13.9282 −1.56705 −0.783523 0.621363i \(-0.786579\pi\)
−0.783523 + 0.621363i \(0.786579\pi\)
\(80\) 1.00000 0.111803
\(81\) 1.00000 0.111111
\(82\) −2.00000 −0.220863
\(83\) 8.92820 0.979998 0.489999 0.871723i \(-0.336997\pi\)
0.489999 + 0.871723i \(0.336997\pi\)
\(84\) −2.00000 −0.218218
\(85\) 4.00000 0.433861
\(86\) −11.9282 −1.28625
\(87\) −0.267949 −0.0287272
\(88\) 6.46410 0.689076
\(89\) −0.535898 −0.0568051 −0.0284026 0.999597i \(-0.509042\pi\)
−0.0284026 + 0.999597i \(0.509042\pi\)
\(90\) 1.00000 0.105409
\(91\) 0 0
\(92\) −3.73205 −0.389093
\(93\) 1.73205 0.179605
\(94\) 3.53590 0.364700
\(95\) 7.46410 0.765801
\(96\) −1.00000 −0.102062
\(97\) 0.535898 0.0544122 0.0272061 0.999630i \(-0.491339\pi\)
0.0272061 + 0.999630i \(0.491339\pi\)
\(98\) −3.00000 −0.303046
\(99\) 6.46410 0.649667
\(100\) 1.00000 0.100000
\(101\) −2.92820 −0.291367 −0.145684 0.989331i \(-0.546538\pi\)
−0.145684 + 0.989331i \(0.546538\pi\)
\(102\) −4.00000 −0.396059
\(103\) 11.8564 1.16825 0.584123 0.811665i \(-0.301438\pi\)
0.584123 + 0.811665i \(0.301438\pi\)
\(104\) 0 0
\(105\) −2.00000 −0.195180
\(106\) 0.928203 0.0901551
\(107\) −7.85641 −0.759507 −0.379754 0.925088i \(-0.623991\pi\)
−0.379754 + 0.925088i \(0.623991\pi\)
\(108\) −1.00000 −0.0962250
\(109\) −15.8564 −1.51877 −0.759384 0.650643i \(-0.774500\pi\)
−0.759384 + 0.650643i \(0.774500\pi\)
\(110\) 6.46410 0.616328
\(111\) −9.19615 −0.872860
\(112\) 2.00000 0.188982
\(113\) 0.803848 0.0756196 0.0378098 0.999285i \(-0.487962\pi\)
0.0378098 + 0.999285i \(0.487962\pi\)
\(114\) −7.46410 −0.699077
\(115\) −3.73205 −0.348016
\(116\) 0.267949 0.0248785
\(117\) 0 0
\(118\) −8.46410 −0.779184
\(119\) 8.00000 0.733359
\(120\) −1.00000 −0.0912871
\(121\) 30.7846 2.79860
\(122\) −10.3923 −0.940875
\(123\) 2.00000 0.180334
\(124\) −1.73205 −0.155543
\(125\) 1.00000 0.0894427
\(126\) 2.00000 0.178174
\(127\) 4.92820 0.437307 0.218654 0.975803i \(-0.429834\pi\)
0.218654 + 0.975803i \(0.429834\pi\)
\(128\) 1.00000 0.0883883
\(129\) 11.9282 1.05022
\(130\) 0 0
\(131\) −18.6603 −1.63035 −0.815177 0.579212i \(-0.803360\pi\)
−0.815177 + 0.579212i \(0.803360\pi\)
\(132\) −6.46410 −0.562628
\(133\) 14.9282 1.29444
\(134\) 11.4641 0.990348
\(135\) −1.00000 −0.0860663
\(136\) 4.00000 0.342997
\(137\) 2.46410 0.210522 0.105261 0.994445i \(-0.466432\pi\)
0.105261 + 0.994445i \(0.466432\pi\)
\(138\) 3.73205 0.317693
\(139\) −12.9282 −1.09656 −0.548278 0.836296i \(-0.684716\pi\)
−0.548278 + 0.836296i \(0.684716\pi\)
\(140\) 2.00000 0.169031
\(141\) −3.53590 −0.297776
\(142\) −12.3923 −1.03994
\(143\) 0 0
\(144\) 1.00000 0.0833333
\(145\) 0.267949 0.0222520
\(146\) −2.00000 −0.165521
\(147\) 3.00000 0.247436
\(148\) 9.19615 0.755919
\(149\) −13.5359 −1.10890 −0.554452 0.832216i \(-0.687072\pi\)
−0.554452 + 0.832216i \(0.687072\pi\)
\(150\) −1.00000 −0.0816497
\(151\) 10.3923 0.845714 0.422857 0.906196i \(-0.361027\pi\)
0.422857 + 0.906196i \(0.361027\pi\)
\(152\) 7.46410 0.605419
\(153\) 4.00000 0.323381
\(154\) 12.9282 1.04178
\(155\) −1.73205 −0.139122
\(156\) 0 0
\(157\) −5.00000 −0.399043 −0.199522 0.979893i \(-0.563939\pi\)
−0.199522 + 0.979893i \(0.563939\pi\)
\(158\) −13.9282 −1.10807
\(159\) −0.928203 −0.0736113
\(160\) 1.00000 0.0790569
\(161\) −7.46410 −0.588254
\(162\) 1.00000 0.0785674
\(163\) −15.0526 −1.17901 −0.589504 0.807766i \(-0.700677\pi\)
−0.589504 + 0.807766i \(0.700677\pi\)
\(164\) −2.00000 −0.156174
\(165\) −6.46410 −0.503230
\(166\) 8.92820 0.692963
\(167\) 16.3205 1.26292 0.631459 0.775409i \(-0.282456\pi\)
0.631459 + 0.775409i \(0.282456\pi\)
\(168\) −2.00000 −0.154303
\(169\) 0 0
\(170\) 4.00000 0.306786
\(171\) 7.46410 0.570794
\(172\) −11.9282 −0.909517
\(173\) −10.9282 −0.830856 −0.415428 0.909626i \(-0.636368\pi\)
−0.415428 + 0.909626i \(0.636368\pi\)
\(174\) −0.267949 −0.0203132
\(175\) 2.00000 0.151186
\(176\) 6.46410 0.487250
\(177\) 8.46410 0.636201
\(178\) −0.535898 −0.0401673
\(179\) 19.7321 1.47484 0.737421 0.675433i \(-0.236043\pi\)
0.737421 + 0.675433i \(0.236043\pi\)
\(180\) 1.00000 0.0745356
\(181\) 2.92820 0.217652 0.108826 0.994061i \(-0.465291\pi\)
0.108826 + 0.994061i \(0.465291\pi\)
\(182\) 0 0
\(183\) 10.3923 0.768221
\(184\) −3.73205 −0.275130
\(185\) 9.19615 0.676115
\(186\) 1.73205 0.127000
\(187\) 25.8564 1.89081
\(188\) 3.53590 0.257882
\(189\) −2.00000 −0.145479
\(190\) 7.46410 0.541503
\(191\) 21.4641 1.55309 0.776544 0.630063i \(-0.216971\pi\)
0.776544 + 0.630063i \(0.216971\pi\)
\(192\) −1.00000 −0.0721688
\(193\) 11.3205 0.814868 0.407434 0.913235i \(-0.366424\pi\)
0.407434 + 0.913235i \(0.366424\pi\)
\(194\) 0.535898 0.0384753
\(195\) 0 0
\(196\) −3.00000 −0.214286
\(197\) −4.39230 −0.312939 −0.156469 0.987683i \(-0.550011\pi\)
−0.156469 + 0.987683i \(0.550011\pi\)
\(198\) 6.46410 0.459384
\(199\) 5.07180 0.359530 0.179765 0.983710i \(-0.442466\pi\)
0.179765 + 0.983710i \(0.442466\pi\)
\(200\) 1.00000 0.0707107
\(201\) −11.4641 −0.808615
\(202\) −2.92820 −0.206028
\(203\) 0.535898 0.0376127
\(204\) −4.00000 −0.280056
\(205\) −2.00000 −0.139686
\(206\) 11.8564 0.826075
\(207\) −3.73205 −0.259395
\(208\) 0 0
\(209\) 48.2487 3.33743
\(210\) −2.00000 −0.138013
\(211\) −11.3205 −0.779336 −0.389668 0.920955i \(-0.627410\pi\)
−0.389668 + 0.920955i \(0.627410\pi\)
\(212\) 0.928203 0.0637493
\(213\) 12.3923 0.849107
\(214\) −7.85641 −0.537053
\(215\) −11.9282 −0.813497
\(216\) −1.00000 −0.0680414
\(217\) −3.46410 −0.235159
\(218\) −15.8564 −1.07393
\(219\) 2.00000 0.135147
\(220\) 6.46410 0.435810
\(221\) 0 0
\(222\) −9.19615 −0.617205
\(223\) 20.5359 1.37519 0.687593 0.726097i \(-0.258667\pi\)
0.687593 + 0.726097i \(0.258667\pi\)
\(224\) 2.00000 0.133631
\(225\) 1.00000 0.0666667
\(226\) 0.803848 0.0534711
\(227\) −16.3923 −1.08800 −0.543998 0.839087i \(-0.683090\pi\)
−0.543998 + 0.839087i \(0.683090\pi\)
\(228\) −7.46410 −0.494322
\(229\) 7.85641 0.519166 0.259583 0.965721i \(-0.416415\pi\)
0.259583 + 0.965721i \(0.416415\pi\)
\(230\) −3.73205 −0.246084
\(231\) −12.9282 −0.850613
\(232\) 0.267949 0.0175917
\(233\) −6.12436 −0.401220 −0.200610 0.979671i \(-0.564292\pi\)
−0.200610 + 0.979671i \(0.564292\pi\)
\(234\) 0 0
\(235\) 3.53590 0.230657
\(236\) −8.46410 −0.550966
\(237\) 13.9282 0.904734
\(238\) 8.00000 0.518563
\(239\) −16.3923 −1.06033 −0.530165 0.847894i \(-0.677870\pi\)
−0.530165 + 0.847894i \(0.677870\pi\)
\(240\) −1.00000 −0.0645497
\(241\) −17.7321 −1.14222 −0.571111 0.820873i \(-0.693487\pi\)
−0.571111 + 0.820873i \(0.693487\pi\)
\(242\) 30.7846 1.97891
\(243\) −1.00000 −0.0641500
\(244\) −10.3923 −0.665299
\(245\) −3.00000 −0.191663
\(246\) 2.00000 0.127515
\(247\) 0 0
\(248\) −1.73205 −0.109985
\(249\) −8.92820 −0.565802
\(250\) 1.00000 0.0632456
\(251\) −15.7321 −0.992998 −0.496499 0.868037i \(-0.665381\pi\)
−0.496499 + 0.868037i \(0.665381\pi\)
\(252\) 2.00000 0.125988
\(253\) −24.1244 −1.51669
\(254\) 4.92820 0.309223
\(255\) −4.00000 −0.250490
\(256\) 1.00000 0.0625000
\(257\) 5.33975 0.333084 0.166542 0.986034i \(-0.446740\pi\)
0.166542 + 0.986034i \(0.446740\pi\)
\(258\) 11.9282 0.742617
\(259\) 18.3923 1.14284
\(260\) 0 0
\(261\) 0.267949 0.0165856
\(262\) −18.6603 −1.15283
\(263\) 6.12436 0.377644 0.188822 0.982011i \(-0.439533\pi\)
0.188822 + 0.982011i \(0.439533\pi\)
\(264\) −6.46410 −0.397838
\(265\) 0.928203 0.0570191
\(266\) 14.9282 0.915307
\(267\) 0.535898 0.0327964
\(268\) 11.4641 0.700281
\(269\) 12.0000 0.731653 0.365826 0.930683i \(-0.380786\pi\)
0.365826 + 0.930683i \(0.380786\pi\)
\(270\) −1.00000 −0.0608581
\(271\) 1.19615 0.0726611 0.0363305 0.999340i \(-0.488433\pi\)
0.0363305 + 0.999340i \(0.488433\pi\)
\(272\) 4.00000 0.242536
\(273\) 0 0
\(274\) 2.46410 0.148862
\(275\) 6.46410 0.389800
\(276\) 3.73205 0.224643
\(277\) 3.92820 0.236023 0.118011 0.993012i \(-0.462348\pi\)
0.118011 + 0.993012i \(0.462348\pi\)
\(278\) −12.9282 −0.775382
\(279\) −1.73205 −0.103695
\(280\) 2.00000 0.119523
\(281\) 8.92820 0.532612 0.266306 0.963889i \(-0.414197\pi\)
0.266306 + 0.963889i \(0.414197\pi\)
\(282\) −3.53590 −0.210560
\(283\) 9.92820 0.590170 0.295085 0.955471i \(-0.404652\pi\)
0.295085 + 0.955471i \(0.404652\pi\)
\(284\) −12.3923 −0.735348
\(285\) −7.46410 −0.442135
\(286\) 0 0
\(287\) −4.00000 −0.236113
\(288\) 1.00000 0.0589256
\(289\) −1.00000 −0.0588235
\(290\) 0.267949 0.0157345
\(291\) −0.535898 −0.0314149
\(292\) −2.00000 −0.117041
\(293\) 31.8564 1.86107 0.930536 0.366202i \(-0.119342\pi\)
0.930536 + 0.366202i \(0.119342\pi\)
\(294\) 3.00000 0.174964
\(295\) −8.46410 −0.492799
\(296\) 9.19615 0.534516
\(297\) −6.46410 −0.375085
\(298\) −13.5359 −0.784114
\(299\) 0 0
\(300\) −1.00000 −0.0577350
\(301\) −23.8564 −1.37506
\(302\) 10.3923 0.598010
\(303\) 2.92820 0.168221
\(304\) 7.46410 0.428096
\(305\) −10.3923 −0.595062
\(306\) 4.00000 0.228665
\(307\) −19.4641 −1.11087 −0.555437 0.831558i \(-0.687449\pi\)
−0.555437 + 0.831558i \(0.687449\pi\)
\(308\) 12.9282 0.736653
\(309\) −11.8564 −0.674487
\(310\) −1.73205 −0.0983739
\(311\) 28.3923 1.60998 0.804990 0.593288i \(-0.202171\pi\)
0.804990 + 0.593288i \(0.202171\pi\)
\(312\) 0 0
\(313\) −28.0000 −1.58265 −0.791327 0.611393i \(-0.790609\pi\)
−0.791327 + 0.611393i \(0.790609\pi\)
\(314\) −5.00000 −0.282166
\(315\) 2.00000 0.112687
\(316\) −13.9282 −0.783523
\(317\) 14.5359 0.816417 0.408209 0.912889i \(-0.366154\pi\)
0.408209 + 0.912889i \(0.366154\pi\)
\(318\) −0.928203 −0.0520511
\(319\) 1.73205 0.0969762
\(320\) 1.00000 0.0559017
\(321\) 7.85641 0.438502
\(322\) −7.46410 −0.415958
\(323\) 29.8564 1.66125
\(324\) 1.00000 0.0555556
\(325\) 0 0
\(326\) −15.0526 −0.833684
\(327\) 15.8564 0.876861
\(328\) −2.00000 −0.110432
\(329\) 7.07180 0.389881
\(330\) −6.46410 −0.355837
\(331\) −16.7846 −0.922566 −0.461283 0.887253i \(-0.652611\pi\)
−0.461283 + 0.887253i \(0.652611\pi\)
\(332\) 8.92820 0.489999
\(333\) 9.19615 0.503946
\(334\) 16.3205 0.893018
\(335\) 11.4641 0.626351
\(336\) −2.00000 −0.109109
\(337\) −9.32051 −0.507720 −0.253860 0.967241i \(-0.581700\pi\)
−0.253860 + 0.967241i \(0.581700\pi\)
\(338\) 0 0
\(339\) −0.803848 −0.0436590
\(340\) 4.00000 0.216930
\(341\) −11.1962 −0.606306
\(342\) 7.46410 0.403612
\(343\) −20.0000 −1.07990
\(344\) −11.9282 −0.643126
\(345\) 3.73205 0.200927
\(346\) −10.9282 −0.587504
\(347\) 1.60770 0.0863056 0.0431528 0.999068i \(-0.486260\pi\)
0.0431528 + 0.999068i \(0.486260\pi\)
\(348\) −0.267949 −0.0143636
\(349\) 21.4641 1.14895 0.574474 0.818523i \(-0.305207\pi\)
0.574474 + 0.818523i \(0.305207\pi\)
\(350\) 2.00000 0.106904
\(351\) 0 0
\(352\) 6.46410 0.344538
\(353\) −2.00000 −0.106449 −0.0532246 0.998583i \(-0.516950\pi\)
−0.0532246 + 0.998583i \(0.516950\pi\)
\(354\) 8.46410 0.449862
\(355\) −12.3923 −0.657715
\(356\) −0.535898 −0.0284026
\(357\) −8.00000 −0.423405
\(358\) 19.7321 1.04287
\(359\) −5.07180 −0.267679 −0.133840 0.991003i \(-0.542731\pi\)
−0.133840 + 0.991003i \(0.542731\pi\)
\(360\) 1.00000 0.0527046
\(361\) 36.7128 1.93225
\(362\) 2.92820 0.153903
\(363\) −30.7846 −1.61577
\(364\) 0 0
\(365\) −2.00000 −0.104685
\(366\) 10.3923 0.543214
\(367\) 15.6077 0.814715 0.407358 0.913269i \(-0.366450\pi\)
0.407358 + 0.913269i \(0.366450\pi\)
\(368\) −3.73205 −0.194547
\(369\) −2.00000 −0.104116
\(370\) 9.19615 0.478085
\(371\) 1.85641 0.0963798
\(372\) 1.73205 0.0898027
\(373\) −15.7846 −0.817296 −0.408648 0.912692i \(-0.634000\pi\)
−0.408648 + 0.912692i \(0.634000\pi\)
\(374\) 25.8564 1.33700
\(375\) −1.00000 −0.0516398
\(376\) 3.53590 0.182350
\(377\) 0 0
\(378\) −2.00000 −0.102869
\(379\) −27.8564 −1.43089 −0.715444 0.698670i \(-0.753775\pi\)
−0.715444 + 0.698670i \(0.753775\pi\)
\(380\) 7.46410 0.382900
\(381\) −4.92820 −0.252479
\(382\) 21.4641 1.09820
\(383\) 25.3923 1.29749 0.648743 0.761007i \(-0.275295\pi\)
0.648743 + 0.761007i \(0.275295\pi\)
\(384\) −1.00000 −0.0510310
\(385\) 12.9282 0.658882
\(386\) 11.3205 0.576199
\(387\) −11.9282 −0.606345
\(388\) 0.535898 0.0272061
\(389\) 23.7321 1.20326 0.601631 0.798774i \(-0.294518\pi\)
0.601631 + 0.798774i \(0.294518\pi\)
\(390\) 0 0
\(391\) −14.9282 −0.754952
\(392\) −3.00000 −0.151523
\(393\) 18.6603 0.941285
\(394\) −4.39230 −0.221281
\(395\) −13.9282 −0.700804
\(396\) 6.46410 0.324833
\(397\) −12.1244 −0.608504 −0.304252 0.952592i \(-0.598406\pi\)
−0.304252 + 0.952592i \(0.598406\pi\)
\(398\) 5.07180 0.254226
\(399\) −14.9282 −0.747345
\(400\) 1.00000 0.0500000
\(401\) −32.0000 −1.59800 −0.799002 0.601329i \(-0.794638\pi\)
−0.799002 + 0.601329i \(0.794638\pi\)
\(402\) −11.4641 −0.571777
\(403\) 0 0
\(404\) −2.92820 −0.145684
\(405\) 1.00000 0.0496904
\(406\) 0.535898 0.0265962
\(407\) 59.4449 2.94657
\(408\) −4.00000 −0.198030
\(409\) 4.00000 0.197787 0.0988936 0.995098i \(-0.468470\pi\)
0.0988936 + 0.995098i \(0.468470\pi\)
\(410\) −2.00000 −0.0987730
\(411\) −2.46410 −0.121545
\(412\) 11.8564 0.584123
\(413\) −16.9282 −0.832982
\(414\) −3.73205 −0.183420
\(415\) 8.92820 0.438268
\(416\) 0 0
\(417\) 12.9282 0.633097
\(418\) 48.2487 2.35992
\(419\) 22.3923 1.09394 0.546968 0.837154i \(-0.315782\pi\)
0.546968 + 0.837154i \(0.315782\pi\)
\(420\) −2.00000 −0.0975900
\(421\) 4.39230 0.214068 0.107034 0.994255i \(-0.465865\pi\)
0.107034 + 0.994255i \(0.465865\pi\)
\(422\) −11.3205 −0.551074
\(423\) 3.53590 0.171921
\(424\) 0.928203 0.0450775
\(425\) 4.00000 0.194029
\(426\) 12.3923 0.600409
\(427\) −20.7846 −1.00584
\(428\) −7.85641 −0.379754
\(429\) 0 0
\(430\) −11.9282 −0.575229
\(431\) −28.3923 −1.36761 −0.683805 0.729665i \(-0.739676\pi\)
−0.683805 + 0.729665i \(0.739676\pi\)
\(432\) −1.00000 −0.0481125
\(433\) −19.3205 −0.928484 −0.464242 0.885708i \(-0.653673\pi\)
−0.464242 + 0.885708i \(0.653673\pi\)
\(434\) −3.46410 −0.166282
\(435\) −0.267949 −0.0128472
\(436\) −15.8564 −0.759384
\(437\) −27.8564 −1.33255
\(438\) 2.00000 0.0955637
\(439\) 17.8564 0.852240 0.426120 0.904667i \(-0.359880\pi\)
0.426120 + 0.904667i \(0.359880\pi\)
\(440\) 6.46410 0.308164
\(441\) −3.00000 −0.142857
\(442\) 0 0
\(443\) 16.3923 0.778822 0.389411 0.921064i \(-0.372679\pi\)
0.389411 + 0.921064i \(0.372679\pi\)
\(444\) −9.19615 −0.436430
\(445\) −0.535898 −0.0254040
\(446\) 20.5359 0.972403
\(447\) 13.5359 0.640226
\(448\) 2.00000 0.0944911
\(449\) 15.7128 0.741533 0.370767 0.928726i \(-0.379095\pi\)
0.370767 + 0.928726i \(0.379095\pi\)
\(450\) 1.00000 0.0471405
\(451\) −12.9282 −0.608765
\(452\) 0.803848 0.0378098
\(453\) −10.3923 −0.488273
\(454\) −16.3923 −0.769329
\(455\) 0 0
\(456\) −7.46410 −0.349539
\(457\) −24.5359 −1.14774 −0.573870 0.818946i \(-0.694559\pi\)
−0.573870 + 0.818946i \(0.694559\pi\)
\(458\) 7.85641 0.367106
\(459\) −4.00000 −0.186704
\(460\) −3.73205 −0.174008
\(461\) 0.464102 0.0216154 0.0108077 0.999942i \(-0.496560\pi\)
0.0108077 + 0.999942i \(0.496560\pi\)
\(462\) −12.9282 −0.601474
\(463\) −7.07180 −0.328654 −0.164327 0.986406i \(-0.552545\pi\)
−0.164327 + 0.986406i \(0.552545\pi\)
\(464\) 0.267949 0.0124392
\(465\) 1.73205 0.0803219
\(466\) −6.12436 −0.283705
\(467\) 15.8564 0.733747 0.366873 0.930271i \(-0.380428\pi\)
0.366873 + 0.930271i \(0.380428\pi\)
\(468\) 0 0
\(469\) 22.9282 1.05873
\(470\) 3.53590 0.163099
\(471\) 5.00000 0.230388
\(472\) −8.46410 −0.389592
\(473\) −77.1051 −3.54530
\(474\) 13.9282 0.639744
\(475\) 7.46410 0.342476
\(476\) 8.00000 0.366679
\(477\) 0.928203 0.0424995
\(478\) −16.3923 −0.749767
\(479\) 5.46410 0.249661 0.124831 0.992178i \(-0.460161\pi\)
0.124831 + 0.992178i \(0.460161\pi\)
\(480\) −1.00000 −0.0456435
\(481\) 0 0
\(482\) −17.7321 −0.807673
\(483\) 7.46410 0.339628
\(484\) 30.7846 1.39930
\(485\) 0.535898 0.0243339
\(486\) −1.00000 −0.0453609
\(487\) 23.1769 1.05025 0.525123 0.851026i \(-0.324019\pi\)
0.525123 + 0.851026i \(0.324019\pi\)
\(488\) −10.3923 −0.470438
\(489\) 15.0526 0.680700
\(490\) −3.00000 −0.135526
\(491\) 17.3205 0.781664 0.390832 0.920462i \(-0.372187\pi\)
0.390832 + 0.920462i \(0.372187\pi\)
\(492\) 2.00000 0.0901670
\(493\) 1.07180 0.0482713
\(494\) 0 0
\(495\) 6.46410 0.290540
\(496\) −1.73205 −0.0777714
\(497\) −24.7846 −1.11174
\(498\) −8.92820 −0.400082
\(499\) 6.53590 0.292587 0.146293 0.989241i \(-0.453266\pi\)
0.146293 + 0.989241i \(0.453266\pi\)
\(500\) 1.00000 0.0447214
\(501\) −16.3205 −0.729147
\(502\) −15.7321 −0.702156
\(503\) −31.1769 −1.39011 −0.695055 0.718957i \(-0.744620\pi\)
−0.695055 + 0.718957i \(0.744620\pi\)
\(504\) 2.00000 0.0890871
\(505\) −2.92820 −0.130303
\(506\) −24.1244 −1.07246
\(507\) 0 0
\(508\) 4.92820 0.218654
\(509\) −1.39230 −0.0617128 −0.0308564 0.999524i \(-0.509823\pi\)
−0.0308564 + 0.999524i \(0.509823\pi\)
\(510\) −4.00000 −0.177123
\(511\) −4.00000 −0.176950
\(512\) 1.00000 0.0441942
\(513\) −7.46410 −0.329548
\(514\) 5.33975 0.235526
\(515\) 11.8564 0.522456
\(516\) 11.9282 0.525110
\(517\) 22.8564 1.00522
\(518\) 18.3923 0.808111
\(519\) 10.9282 0.479695
\(520\) 0 0
\(521\) −17.3205 −0.758825 −0.379413 0.925228i \(-0.623874\pi\)
−0.379413 + 0.925228i \(0.623874\pi\)
\(522\) 0.267949 0.0117278
\(523\) 11.7846 0.515305 0.257653 0.966238i \(-0.417051\pi\)
0.257653 + 0.966238i \(0.417051\pi\)
\(524\) −18.6603 −0.815177
\(525\) −2.00000 −0.0872872
\(526\) 6.12436 0.267035
\(527\) −6.92820 −0.301797
\(528\) −6.46410 −0.281314
\(529\) −9.07180 −0.394426
\(530\) 0.928203 0.0403186
\(531\) −8.46410 −0.367311
\(532\) 14.9282 0.647220
\(533\) 0 0
\(534\) 0.535898 0.0231906
\(535\) −7.85641 −0.339662
\(536\) 11.4641 0.495174
\(537\) −19.7321 −0.851501
\(538\) 12.0000 0.517357
\(539\) −19.3923 −0.835286
\(540\) −1.00000 −0.0430331
\(541\) −26.9282 −1.15773 −0.578867 0.815422i \(-0.696505\pi\)
−0.578867 + 0.815422i \(0.696505\pi\)
\(542\) 1.19615 0.0513791
\(543\) −2.92820 −0.125661
\(544\) 4.00000 0.171499
\(545\) −15.8564 −0.679214
\(546\) 0 0
\(547\) 22.9282 0.980339 0.490170 0.871627i \(-0.336935\pi\)
0.490170 + 0.871627i \(0.336935\pi\)
\(548\) 2.46410 0.105261
\(549\) −10.3923 −0.443533
\(550\) 6.46410 0.275630
\(551\) 2.00000 0.0852029
\(552\) 3.73205 0.158847
\(553\) −27.8564 −1.18457
\(554\) 3.92820 0.166893
\(555\) −9.19615 −0.390355
\(556\) −12.9282 −0.548278
\(557\) 17.7128 0.750516 0.375258 0.926920i \(-0.377554\pi\)
0.375258 + 0.926920i \(0.377554\pi\)
\(558\) −1.73205 −0.0733236
\(559\) 0 0
\(560\) 2.00000 0.0845154
\(561\) −25.8564 −1.09166
\(562\) 8.92820 0.376614
\(563\) 4.67949 0.197217 0.0986085 0.995126i \(-0.468561\pi\)
0.0986085 + 0.995126i \(0.468561\pi\)
\(564\) −3.53590 −0.148888
\(565\) 0.803848 0.0338181
\(566\) 9.92820 0.417314
\(567\) 2.00000 0.0839921
\(568\) −12.3923 −0.519970
\(569\) −29.3205 −1.22918 −0.614590 0.788847i \(-0.710678\pi\)
−0.614590 + 0.788847i \(0.710678\pi\)
\(570\) −7.46410 −0.312637
\(571\) 17.1769 0.718832 0.359416 0.933178i \(-0.382976\pi\)
0.359416 + 0.933178i \(0.382976\pi\)
\(572\) 0 0
\(573\) −21.4641 −0.896676
\(574\) −4.00000 −0.166957
\(575\) −3.73205 −0.155637
\(576\) 1.00000 0.0416667
\(577\) −10.0000 −0.416305 −0.208153 0.978096i \(-0.566745\pi\)
−0.208153 + 0.978096i \(0.566745\pi\)
\(578\) −1.00000 −0.0415945
\(579\) −11.3205 −0.470464
\(580\) 0.267949 0.0111260
\(581\) 17.8564 0.740809
\(582\) −0.535898 −0.0222137
\(583\) 6.00000 0.248495
\(584\) −2.00000 −0.0827606
\(585\) 0 0
\(586\) 31.8564 1.31598
\(587\) 2.39230 0.0987410 0.0493705 0.998781i \(-0.484278\pi\)
0.0493705 + 0.998781i \(0.484278\pi\)
\(588\) 3.00000 0.123718
\(589\) −12.9282 −0.532697
\(590\) −8.46410 −0.348462
\(591\) 4.39230 0.180675
\(592\) 9.19615 0.377960
\(593\) −45.1051 −1.85225 −0.926123 0.377223i \(-0.876879\pi\)
−0.926123 + 0.377223i \(0.876879\pi\)
\(594\) −6.46410 −0.265225
\(595\) 8.00000 0.327968
\(596\) −13.5359 −0.554452
\(597\) −5.07180 −0.207575
\(598\) 0 0
\(599\) −10.3923 −0.424618 −0.212309 0.977203i \(-0.568098\pi\)
−0.212309 + 0.977203i \(0.568098\pi\)
\(600\) −1.00000 −0.0408248
\(601\) −19.7846 −0.807031 −0.403516 0.914973i \(-0.632212\pi\)
−0.403516 + 0.914973i \(0.632212\pi\)
\(602\) −23.8564 −0.972315
\(603\) 11.4641 0.466854
\(604\) 10.3923 0.422857
\(605\) 30.7846 1.25157
\(606\) 2.92820 0.118950
\(607\) −19.1769 −0.778367 −0.389183 0.921160i \(-0.627243\pi\)
−0.389183 + 0.921160i \(0.627243\pi\)
\(608\) 7.46410 0.302709
\(609\) −0.535898 −0.0217157
\(610\) −10.3923 −0.420772
\(611\) 0 0
\(612\) 4.00000 0.161690
\(613\) 39.0526 1.57732 0.788659 0.614831i \(-0.210776\pi\)
0.788659 + 0.614831i \(0.210776\pi\)
\(614\) −19.4641 −0.785507
\(615\) 2.00000 0.0806478
\(616\) 12.9282 0.520892
\(617\) −22.4641 −0.904371 −0.452185 0.891924i \(-0.649355\pi\)
−0.452185 + 0.891924i \(0.649355\pi\)
\(618\) −11.8564 −0.476935
\(619\) 24.2487 0.974638 0.487319 0.873224i \(-0.337975\pi\)
0.487319 + 0.873224i \(0.337975\pi\)
\(620\) −1.73205 −0.0695608
\(621\) 3.73205 0.149762
\(622\) 28.3923 1.13843
\(623\) −1.07180 −0.0429406
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) −28.0000 −1.11911
\(627\) −48.2487 −1.92687
\(628\) −5.00000 −0.199522
\(629\) 36.7846 1.46670
\(630\) 2.00000 0.0796819
\(631\) −31.4641 −1.25257 −0.626283 0.779596i \(-0.715425\pi\)
−0.626283 + 0.779596i \(0.715425\pi\)
\(632\) −13.9282 −0.554034
\(633\) 11.3205 0.449950
\(634\) 14.5359 0.577294
\(635\) 4.92820 0.195570
\(636\) −0.928203 −0.0368057
\(637\) 0 0
\(638\) 1.73205 0.0685725
\(639\) −12.3923 −0.490232
\(640\) 1.00000 0.0395285
\(641\) −0.143594 −0.00567160 −0.00283580 0.999996i \(-0.500903\pi\)
−0.00283580 + 0.999996i \(0.500903\pi\)
\(642\) 7.85641 0.310068
\(643\) −20.5359 −0.809857 −0.404928 0.914348i \(-0.632704\pi\)
−0.404928 + 0.914348i \(0.632704\pi\)
\(644\) −7.46410 −0.294127
\(645\) 11.9282 0.469673
\(646\) 29.8564 1.17468
\(647\) 13.3205 0.523683 0.261842 0.965111i \(-0.415670\pi\)
0.261842 + 0.965111i \(0.415670\pi\)
\(648\) 1.00000 0.0392837
\(649\) −54.7128 −2.14767
\(650\) 0 0
\(651\) 3.46410 0.135769
\(652\) −15.0526 −0.589504
\(653\) 4.24871 0.166265 0.0831325 0.996539i \(-0.473508\pi\)
0.0831325 + 0.996539i \(0.473508\pi\)
\(654\) 15.8564 0.620035
\(655\) −18.6603 −0.729116
\(656\) −2.00000 −0.0780869
\(657\) −2.00000 −0.0780274
\(658\) 7.07180 0.275687
\(659\) −0.267949 −0.0104378 −0.00521891 0.999986i \(-0.501661\pi\)
−0.00521891 + 0.999986i \(0.501661\pi\)
\(660\) −6.46410 −0.251615
\(661\) −8.67949 −0.337593 −0.168797 0.985651i \(-0.553988\pi\)
−0.168797 + 0.985651i \(0.553988\pi\)
\(662\) −16.7846 −0.652352
\(663\) 0 0
\(664\) 8.92820 0.346481
\(665\) 14.9282 0.578891
\(666\) 9.19615 0.356344
\(667\) −1.00000 −0.0387202
\(668\) 16.3205 0.631459
\(669\) −20.5359 −0.793964
\(670\) 11.4641 0.442897
\(671\) −67.1769 −2.59334
\(672\) −2.00000 −0.0771517
\(673\) −32.0000 −1.23351 −0.616755 0.787155i \(-0.711553\pi\)
−0.616755 + 0.787155i \(0.711553\pi\)
\(674\) −9.32051 −0.359013
\(675\) −1.00000 −0.0384900
\(676\) 0 0
\(677\) 32.3923 1.24494 0.622469 0.782645i \(-0.286130\pi\)
0.622469 + 0.782645i \(0.286130\pi\)
\(678\) −0.803848 −0.0308716
\(679\) 1.07180 0.0411318
\(680\) 4.00000 0.153393
\(681\) 16.3923 0.628154
\(682\) −11.1962 −0.428723
\(683\) 40.7846 1.56058 0.780290 0.625418i \(-0.215072\pi\)
0.780290 + 0.625418i \(0.215072\pi\)
\(684\) 7.46410 0.285397
\(685\) 2.46410 0.0941485
\(686\) −20.0000 −0.763604
\(687\) −7.85641 −0.299741
\(688\) −11.9282 −0.454758
\(689\) 0 0
\(690\) 3.73205 0.142077
\(691\) 27.1769 1.03386 0.516929 0.856028i \(-0.327075\pi\)
0.516929 + 0.856028i \(0.327075\pi\)
\(692\) −10.9282 −0.415428
\(693\) 12.9282 0.491102
\(694\) 1.60770 0.0610273
\(695\) −12.9282 −0.490395
\(696\) −0.267949 −0.0101566
\(697\) −8.00000 −0.303022
\(698\) 21.4641 0.812428
\(699\) 6.12436 0.231644
\(700\) 2.00000 0.0755929
\(701\) 0.267949 0.0101203 0.00506015 0.999987i \(-0.498389\pi\)
0.00506015 + 0.999987i \(0.498389\pi\)
\(702\) 0 0
\(703\) 68.6410 2.58884
\(704\) 6.46410 0.243625
\(705\) −3.53590 −0.133170
\(706\) −2.00000 −0.0752710
\(707\) −5.85641 −0.220253
\(708\) 8.46410 0.318100
\(709\) 22.9282 0.861087 0.430543 0.902570i \(-0.358322\pi\)
0.430543 + 0.902570i \(0.358322\pi\)
\(710\) −12.3923 −0.465075
\(711\) −13.9282 −0.522348
\(712\) −0.535898 −0.0200836
\(713\) 6.46410 0.242083
\(714\) −8.00000 −0.299392
\(715\) 0 0
\(716\) 19.7321 0.737421
\(717\) 16.3923 0.612182
\(718\) −5.07180 −0.189278
\(719\) 34.6410 1.29189 0.645946 0.763383i \(-0.276463\pi\)
0.645946 + 0.763383i \(0.276463\pi\)
\(720\) 1.00000 0.0372678
\(721\) 23.7128 0.883111
\(722\) 36.7128 1.36631
\(723\) 17.7321 0.659462
\(724\) 2.92820 0.108826
\(725\) 0.267949 0.00995138
\(726\) −30.7846 −1.14252
\(727\) 31.7128 1.17616 0.588082 0.808802i \(-0.299883\pi\)
0.588082 + 0.808802i \(0.299883\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) −2.00000 −0.0740233
\(731\) −47.7128 −1.76472
\(732\) 10.3923 0.384111
\(733\) −50.9282 −1.88108 −0.940538 0.339688i \(-0.889678\pi\)
−0.940538 + 0.339688i \(0.889678\pi\)
\(734\) 15.6077 0.576091
\(735\) 3.00000 0.110657
\(736\) −3.73205 −0.137565
\(737\) 74.1051 2.72970
\(738\) −2.00000 −0.0736210
\(739\) −19.3205 −0.710716 −0.355358 0.934730i \(-0.615641\pi\)
−0.355358 + 0.934730i \(0.615641\pi\)
\(740\) 9.19615 0.338057
\(741\) 0 0
\(742\) 1.85641 0.0681508
\(743\) 40.4641 1.48448 0.742242 0.670132i \(-0.233763\pi\)
0.742242 + 0.670132i \(0.233763\pi\)
\(744\) 1.73205 0.0635001
\(745\) −13.5359 −0.495917
\(746\) −15.7846 −0.577916
\(747\) 8.92820 0.326666
\(748\) 25.8564 0.945404
\(749\) −15.7128 −0.574134
\(750\) −1.00000 −0.0365148
\(751\) −14.0718 −0.513487 −0.256744 0.966480i \(-0.582650\pi\)
−0.256744 + 0.966480i \(0.582650\pi\)
\(752\) 3.53590 0.128941
\(753\) 15.7321 0.573308
\(754\) 0 0
\(755\) 10.3923 0.378215
\(756\) −2.00000 −0.0727393
\(757\) −18.0000 −0.654221 −0.327111 0.944986i \(-0.606075\pi\)
−0.327111 + 0.944986i \(0.606075\pi\)
\(758\) −27.8564 −1.01179
\(759\) 24.1244 0.875659
\(760\) 7.46410 0.270751
\(761\) −5.07180 −0.183852 −0.0919262 0.995766i \(-0.529302\pi\)
−0.0919262 + 0.995766i \(0.529302\pi\)
\(762\) −4.92820 −0.178530
\(763\) −31.7128 −1.14808
\(764\) 21.4641 0.776544
\(765\) 4.00000 0.144620
\(766\) 25.3923 0.917461
\(767\) 0 0
\(768\) −1.00000 −0.0360844
\(769\) 11.5885 0.417890 0.208945 0.977927i \(-0.432997\pi\)
0.208945 + 0.977927i \(0.432997\pi\)
\(770\) 12.9282 0.465900
\(771\) −5.33975 −0.192306
\(772\) 11.3205 0.407434
\(773\) 27.7128 0.996761 0.498380 0.866959i \(-0.333928\pi\)
0.498380 + 0.866959i \(0.333928\pi\)
\(774\) −11.9282 −0.428750
\(775\) −1.73205 −0.0622171
\(776\) 0.535898 0.0192376
\(777\) −18.3923 −0.659820
\(778\) 23.7321 0.850835
\(779\) −14.9282 −0.534858
\(780\) 0 0
\(781\) −80.1051 −2.86639
\(782\) −14.9282 −0.533831
\(783\) −0.267949 −0.00957572
\(784\) −3.00000 −0.107143
\(785\) −5.00000 −0.178458
\(786\) 18.6603 0.665589
\(787\) −1.73205 −0.0617409 −0.0308705 0.999523i \(-0.509828\pi\)
−0.0308705 + 0.999523i \(0.509828\pi\)
\(788\) −4.39230 −0.156469
\(789\) −6.12436 −0.218033
\(790\) −13.9282 −0.495543
\(791\) 1.60770 0.0571631
\(792\) 6.46410 0.229692
\(793\) 0 0
\(794\) −12.1244 −0.430277
\(795\) −0.928203 −0.0329200
\(796\) 5.07180 0.179765
\(797\) 10.1436 0.359305 0.179652 0.983730i \(-0.442503\pi\)
0.179652 + 0.983730i \(0.442503\pi\)
\(798\) −14.9282 −0.528453
\(799\) 14.1436 0.500364
\(800\) 1.00000 0.0353553
\(801\) −0.535898 −0.0189350
\(802\) −32.0000 −1.12996
\(803\) −12.9282 −0.456226
\(804\) −11.4641 −0.404308
\(805\) −7.46410 −0.263075
\(806\) 0 0
\(807\) −12.0000 −0.422420
\(808\) −2.92820 −0.103014
\(809\) −42.7846 −1.50423 −0.752113 0.659034i \(-0.770965\pi\)
−0.752113 + 0.659034i \(0.770965\pi\)
\(810\) 1.00000 0.0351364
\(811\) 40.7846 1.43214 0.716071 0.698028i \(-0.245939\pi\)
0.716071 + 0.698028i \(0.245939\pi\)
\(812\) 0.535898 0.0188063
\(813\) −1.19615 −0.0419509
\(814\) 59.4449 2.08354
\(815\) −15.0526 −0.527268
\(816\) −4.00000 −0.140028
\(817\) −89.0333 −3.11488
\(818\) 4.00000 0.139857
\(819\) 0 0
\(820\) −2.00000 −0.0698430
\(821\) −14.6077 −0.509812 −0.254906 0.966966i \(-0.582045\pi\)
−0.254906 + 0.966966i \(0.582045\pi\)
\(822\) −2.46410 −0.0859454
\(823\) −39.1769 −1.36562 −0.682811 0.730595i \(-0.739243\pi\)
−0.682811 + 0.730595i \(0.739243\pi\)
\(824\) 11.8564 0.413037
\(825\) −6.46410 −0.225051
\(826\) −16.9282 −0.589008
\(827\) −17.3205 −0.602293 −0.301147 0.953578i \(-0.597369\pi\)
−0.301147 + 0.953578i \(0.597369\pi\)
\(828\) −3.73205 −0.129698
\(829\) −16.5359 −0.574315 −0.287158 0.957883i \(-0.592710\pi\)
−0.287158 + 0.957883i \(0.592710\pi\)
\(830\) 8.92820 0.309902
\(831\) −3.92820 −0.136268
\(832\) 0 0
\(833\) −12.0000 −0.415775
\(834\) 12.9282 0.447667
\(835\) 16.3205 0.564794
\(836\) 48.2487 1.66872
\(837\) 1.73205 0.0598684
\(838\) 22.3923 0.773529
\(839\) 43.5692 1.50418 0.752088 0.659062i \(-0.229047\pi\)
0.752088 + 0.659062i \(0.229047\pi\)
\(840\) −2.00000 −0.0690066
\(841\) −28.9282 −0.997524
\(842\) 4.39230 0.151369
\(843\) −8.92820 −0.307504
\(844\) −11.3205 −0.389668
\(845\) 0 0
\(846\) 3.53590 0.121567
\(847\) 61.5692 2.11554
\(848\) 0.928203 0.0318746
\(849\) −9.92820 −0.340735
\(850\) 4.00000 0.137199
\(851\) −34.3205 −1.17649
\(852\) 12.3923 0.424553
\(853\) −43.8372 −1.50096 −0.750478 0.660895i \(-0.770177\pi\)
−0.750478 + 0.660895i \(0.770177\pi\)
\(854\) −20.7846 −0.711235
\(855\) 7.46410 0.255267
\(856\) −7.85641 −0.268526
\(857\) −24.5167 −0.837473 −0.418737 0.908108i \(-0.637527\pi\)
−0.418737 + 0.908108i \(0.637527\pi\)
\(858\) 0 0
\(859\) −35.1769 −1.20022 −0.600110 0.799917i \(-0.704877\pi\)
−0.600110 + 0.799917i \(0.704877\pi\)
\(860\) −11.9282 −0.406748
\(861\) 4.00000 0.136320
\(862\) −28.3923 −0.967046
\(863\) 35.5359 1.20966 0.604828 0.796356i \(-0.293242\pi\)
0.604828 + 0.796356i \(0.293242\pi\)
\(864\) −1.00000 −0.0340207
\(865\) −10.9282 −0.371570
\(866\) −19.3205 −0.656538
\(867\) 1.00000 0.0339618
\(868\) −3.46410 −0.117579
\(869\) −90.0333 −3.05417
\(870\) −0.267949 −0.00908433
\(871\) 0 0
\(872\) −15.8564 −0.536966
\(873\) 0.535898 0.0181374
\(874\) −27.8564 −0.942257
\(875\) 2.00000 0.0676123
\(876\) 2.00000 0.0675737
\(877\) −3.87564 −0.130871 −0.0654356 0.997857i \(-0.520844\pi\)
−0.0654356 + 0.997857i \(0.520844\pi\)
\(878\) 17.8564 0.602625
\(879\) −31.8564 −1.07449
\(880\) 6.46410 0.217905
\(881\) 14.0000 0.471672 0.235836 0.971793i \(-0.424217\pi\)
0.235836 + 0.971793i \(0.424217\pi\)
\(882\) −3.00000 −0.101015
\(883\) −29.9282 −1.00716 −0.503582 0.863947i \(-0.667985\pi\)
−0.503582 + 0.863947i \(0.667985\pi\)
\(884\) 0 0
\(885\) 8.46410 0.284518
\(886\) 16.3923 0.550710
\(887\) 14.1244 0.474249 0.237125 0.971479i \(-0.423795\pi\)
0.237125 + 0.971479i \(0.423795\pi\)
\(888\) −9.19615 −0.308603
\(889\) 9.85641 0.330573
\(890\) −0.535898 −0.0179634
\(891\) 6.46410 0.216556
\(892\) 20.5359 0.687593
\(893\) 26.3923 0.883185
\(894\) 13.5359 0.452708
\(895\) 19.7321 0.659570
\(896\) 2.00000 0.0668153
\(897\) 0 0
\(898\) 15.7128 0.524343
\(899\) −0.464102 −0.0154787
\(900\) 1.00000 0.0333333
\(901\) 3.71281 0.123692
\(902\) −12.9282 −0.430462
\(903\) 23.8564 0.793891
\(904\) 0.803848 0.0267356
\(905\) 2.92820 0.0973368
\(906\) −10.3923 −0.345261
\(907\) 20.8564 0.692526 0.346263 0.938138i \(-0.387451\pi\)
0.346263 + 0.938138i \(0.387451\pi\)
\(908\) −16.3923 −0.543998
\(909\) −2.92820 −0.0971224
\(910\) 0 0
\(911\) 24.2487 0.803396 0.401698 0.915772i \(-0.368420\pi\)
0.401698 + 0.915772i \(0.368420\pi\)
\(912\) −7.46410 −0.247161
\(913\) 57.7128 1.91002
\(914\) −24.5359 −0.811575
\(915\) 10.3923 0.343559
\(916\) 7.85641 0.259583
\(917\) −37.3205 −1.23243
\(918\) −4.00000 −0.132020
\(919\) 14.6410 0.482963 0.241481 0.970405i \(-0.422367\pi\)
0.241481 + 0.970405i \(0.422367\pi\)
\(920\) −3.73205 −0.123042
\(921\) 19.4641 0.641364
\(922\) 0.464102 0.0152844
\(923\) 0 0
\(924\) −12.9282 −0.425307
\(925\) 9.19615 0.302368
\(926\) −7.07180 −0.232394
\(927\) 11.8564 0.389415
\(928\) 0.267949 0.00879586
\(929\) −6.14359 −0.201565 −0.100782 0.994908i \(-0.532135\pi\)
−0.100782 + 0.994908i \(0.532135\pi\)
\(930\) 1.73205 0.0567962
\(931\) −22.3923 −0.733878
\(932\) −6.12436 −0.200610
\(933\) −28.3923 −0.929522
\(934\) 15.8564 0.518837
\(935\) 25.8564 0.845595
\(936\) 0 0
\(937\) −24.6410 −0.804987 −0.402493 0.915423i \(-0.631856\pi\)
−0.402493 + 0.915423i \(0.631856\pi\)
\(938\) 22.9282 0.748632
\(939\) 28.0000 0.913745
\(940\) 3.53590 0.115328
\(941\) −14.7846 −0.481965 −0.240982 0.970530i \(-0.577470\pi\)
−0.240982 + 0.970530i \(0.577470\pi\)
\(942\) 5.00000 0.162909
\(943\) 7.46410 0.243065
\(944\) −8.46410 −0.275483
\(945\) −2.00000 −0.0650600
\(946\) −77.1051 −2.50690
\(947\) 9.46410 0.307542 0.153771 0.988107i \(-0.450858\pi\)
0.153771 + 0.988107i \(0.450858\pi\)
\(948\) 13.9282 0.452367
\(949\) 0 0
\(950\) 7.46410 0.242167
\(951\) −14.5359 −0.471359
\(952\) 8.00000 0.259281
\(953\) 12.2679 0.397398 0.198699 0.980061i \(-0.436328\pi\)
0.198699 + 0.980061i \(0.436328\pi\)
\(954\) 0.928203 0.0300517
\(955\) 21.4641 0.694562
\(956\) −16.3923 −0.530165
\(957\) −1.73205 −0.0559893
\(958\) 5.46410 0.176537
\(959\) 4.92820 0.159140
\(960\) −1.00000 −0.0322749
\(961\) −28.0000 −0.903226
\(962\) 0 0
\(963\) −7.85641 −0.253169
\(964\) −17.7321 −0.571111
\(965\) 11.3205 0.364420
\(966\) 7.46410 0.240154
\(967\) −41.4641 −1.33340 −0.666698 0.745328i \(-0.732293\pi\)
−0.666698 + 0.745328i \(0.732293\pi\)
\(968\) 30.7846 0.989455
\(969\) −29.8564 −0.959126
\(970\) 0.535898 0.0172067
\(971\) 4.53590 0.145564 0.0727820 0.997348i \(-0.476812\pi\)
0.0727820 + 0.997348i \(0.476812\pi\)
\(972\) −1.00000 −0.0320750
\(973\) −25.8564 −0.828918
\(974\) 23.1769 0.742636
\(975\) 0 0
\(976\) −10.3923 −0.332650
\(977\) −12.6077 −0.403356 −0.201678 0.979452i \(-0.564639\pi\)
−0.201678 + 0.979452i \(0.564639\pi\)
\(978\) 15.0526 0.481328
\(979\) −3.46410 −0.110713
\(980\) −3.00000 −0.0958315
\(981\) −15.8564 −0.506256
\(982\) 17.3205 0.552720
\(983\) 36.6077 1.16760 0.583802 0.811896i \(-0.301564\pi\)
0.583802 + 0.811896i \(0.301564\pi\)
\(984\) 2.00000 0.0637577
\(985\) −4.39230 −0.139950
\(986\) 1.07180 0.0341330
\(987\) −7.07180 −0.225098
\(988\) 0 0
\(989\) 44.5167 1.41555
\(990\) 6.46410 0.205443
\(991\) −52.8564 −1.67904 −0.839520 0.543329i \(-0.817163\pi\)
−0.839520 + 0.543329i \(0.817163\pi\)
\(992\) −1.73205 −0.0549927
\(993\) 16.7846 0.532643
\(994\) −24.7846 −0.786120
\(995\) 5.07180 0.160787
\(996\) −8.92820 −0.282901
\(997\) 35.5692 1.12649 0.563244 0.826290i \(-0.309553\pi\)
0.563244 + 0.826290i \(0.309553\pi\)
\(998\) 6.53590 0.206890
\(999\) −9.19615 −0.290953
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.bg.1.2 2
13.5 odd 4 5070.2.b.o.1351.2 4
13.6 odd 12 390.2.bb.b.361.1 yes 4
13.8 odd 4 5070.2.b.o.1351.3 4
13.11 odd 12 390.2.bb.b.121.1 4
13.12 even 2 5070.2.a.y.1.1 2
39.11 even 12 1170.2.bs.e.901.2 4
39.32 even 12 1170.2.bs.e.361.2 4
65.19 odd 12 1950.2.bc.b.751.2 4
65.24 odd 12 1950.2.bc.b.901.2 4
65.32 even 12 1950.2.y.c.49.2 4
65.37 even 12 1950.2.y.f.199.1 4
65.58 even 12 1950.2.y.f.49.1 4
65.63 even 12 1950.2.y.c.199.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
390.2.bb.b.121.1 4 13.11 odd 12
390.2.bb.b.361.1 yes 4 13.6 odd 12
1170.2.bs.e.361.2 4 39.32 even 12
1170.2.bs.e.901.2 4 39.11 even 12
1950.2.y.c.49.2 4 65.32 even 12
1950.2.y.c.199.2 4 65.63 even 12
1950.2.y.f.49.1 4 65.58 even 12
1950.2.y.f.199.1 4 65.37 even 12
1950.2.bc.b.751.2 4 65.19 odd 12
1950.2.bc.b.901.2 4 65.24 odd 12
5070.2.a.y.1.1 2 13.12 even 2
5070.2.a.bg.1.2 2 1.1 even 1 trivial
5070.2.b.o.1351.2 4 13.5 odd 4
5070.2.b.o.1351.3 4 13.8 odd 4