Properties

Label 462.2.a.h.1.1
Level $462$
Weight $2$
Character 462.1
Self dual yes
Analytic conductor $3.689$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 462 = 2 \cdot 3 \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 462.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(3.68908857338\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{12})^+\)
Defining polynomial: \(x^{2} - 3\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.73205\) of defining polynomial
Character \(\chi\) \(=\) 462.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -3.46410 q^{5} -1.00000 q^{6} +1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -3.46410 q^{5} -1.00000 q^{6} +1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +3.46410 q^{10} +1.00000 q^{11} +1.00000 q^{12} +2.00000 q^{13} -1.00000 q^{14} -3.46410 q^{15} +1.00000 q^{16} +3.46410 q^{17} -1.00000 q^{18} +5.46410 q^{19} -3.46410 q^{20} +1.00000 q^{21} -1.00000 q^{22} +6.92820 q^{23} -1.00000 q^{24} +7.00000 q^{25} -2.00000 q^{26} +1.00000 q^{27} +1.00000 q^{28} -6.00000 q^{29} +3.46410 q^{30} +5.46410 q^{31} -1.00000 q^{32} +1.00000 q^{33} -3.46410 q^{34} -3.46410 q^{35} +1.00000 q^{36} -4.92820 q^{37} -5.46410 q^{38} +2.00000 q^{39} +3.46410 q^{40} +3.46410 q^{41} -1.00000 q^{42} -10.9282 q^{43} +1.00000 q^{44} -3.46410 q^{45} -6.92820 q^{46} +9.46410 q^{47} +1.00000 q^{48} +1.00000 q^{49} -7.00000 q^{50} +3.46410 q^{51} +2.00000 q^{52} -0.928203 q^{53} -1.00000 q^{54} -3.46410 q^{55} -1.00000 q^{56} +5.46410 q^{57} +6.00000 q^{58} +6.92820 q^{59} -3.46410 q^{60} +2.00000 q^{61} -5.46410 q^{62} +1.00000 q^{63} +1.00000 q^{64} -6.92820 q^{65} -1.00000 q^{66} +14.9282 q^{67} +3.46410 q^{68} +6.92820 q^{69} +3.46410 q^{70} -12.0000 q^{71} -1.00000 q^{72} -0.535898 q^{73} +4.92820 q^{74} +7.00000 q^{75} +5.46410 q^{76} +1.00000 q^{77} -2.00000 q^{78} -10.9282 q^{79} -3.46410 q^{80} +1.00000 q^{81} -3.46410 q^{82} +4.39230 q^{83} +1.00000 q^{84} -12.0000 q^{85} +10.9282 q^{86} -6.00000 q^{87} -1.00000 q^{88} -0.928203 q^{89} +3.46410 q^{90} +2.00000 q^{91} +6.92820 q^{92} +5.46410 q^{93} -9.46410 q^{94} -18.9282 q^{95} -1.00000 q^{96} +2.00000 q^{97} -1.00000 q^{98} +1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{2} + 2q^{3} + 2q^{4} - 2q^{6} + 2q^{7} - 2q^{8} + 2q^{9} + O(q^{10}) \) \( 2q - 2q^{2} + 2q^{3} + 2q^{4} - 2q^{6} + 2q^{7} - 2q^{8} + 2q^{9} + 2q^{11} + 2q^{12} + 4q^{13} - 2q^{14} + 2q^{16} - 2q^{18} + 4q^{19} + 2q^{21} - 2q^{22} - 2q^{24} + 14q^{25} - 4q^{26} + 2q^{27} + 2q^{28} - 12q^{29} + 4q^{31} - 2q^{32} + 2q^{33} + 2q^{36} + 4q^{37} - 4q^{38} + 4q^{39} - 2q^{42} - 8q^{43} + 2q^{44} + 12q^{47} + 2q^{48} + 2q^{49} - 14q^{50} + 4q^{52} + 12q^{53} - 2q^{54} - 2q^{56} + 4q^{57} + 12q^{58} + 4q^{61} - 4q^{62} + 2q^{63} + 2q^{64} - 2q^{66} + 16q^{67} - 24q^{71} - 2q^{72} - 8q^{73} - 4q^{74} + 14q^{75} + 4q^{76} + 2q^{77} - 4q^{78} - 8q^{79} + 2q^{81} - 12q^{83} + 2q^{84} - 24q^{85} + 8q^{86} - 12q^{87} - 2q^{88} + 12q^{89} + 4q^{91} + 4q^{93} - 12q^{94} - 24q^{95} - 2q^{96} + 4q^{97} - 2q^{98} + 2q^{99} + O(q^{100}) \)

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
\(5\) −3.46410 −1.54919 −0.774597 0.632456i \(-0.782047\pi\)
−0.774597 + 0.632456i \(0.782047\pi\)
\(6\) −1.00000 −0.408248
\(7\) 1.00000 0.377964
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) 3.46410 1.09545
\(11\) 1.00000 0.301511
\(12\) 1.00000 0.288675
\(13\) 2.00000 0.554700 0.277350 0.960769i \(-0.410544\pi\)
0.277350 + 0.960769i \(0.410544\pi\)
\(14\) −1.00000 −0.267261
\(15\) −3.46410 −0.894427
\(16\) 1.00000 0.250000
\(17\) 3.46410 0.840168 0.420084 0.907485i \(-0.362001\pi\)
0.420084 + 0.907485i \(0.362001\pi\)
\(18\) −1.00000 −0.235702
\(19\) 5.46410 1.25355 0.626775 0.779200i \(-0.284374\pi\)
0.626775 + 0.779200i \(0.284374\pi\)
\(20\) −3.46410 −0.774597
\(21\) 1.00000 0.218218
\(22\) −1.00000 −0.213201
\(23\) 6.92820 1.44463 0.722315 0.691564i \(-0.243078\pi\)
0.722315 + 0.691564i \(0.243078\pi\)
\(24\) −1.00000 −0.204124
\(25\) 7.00000 1.40000
\(26\) −2.00000 −0.392232
\(27\) 1.00000 0.192450
\(28\) 1.00000 0.188982
\(29\) −6.00000 −1.11417 −0.557086 0.830455i \(-0.688081\pi\)
−0.557086 + 0.830455i \(0.688081\pi\)
\(30\) 3.46410 0.632456
\(31\) 5.46410 0.981382 0.490691 0.871334i \(-0.336744\pi\)
0.490691 + 0.871334i \(0.336744\pi\)
\(32\) −1.00000 −0.176777
\(33\) 1.00000 0.174078
\(34\) −3.46410 −0.594089
\(35\) −3.46410 −0.585540
\(36\) 1.00000 0.166667
\(37\) −4.92820 −0.810192 −0.405096 0.914274i \(-0.632762\pi\)
−0.405096 + 0.914274i \(0.632762\pi\)
\(38\) −5.46410 −0.886394
\(39\) 2.00000 0.320256
\(40\) 3.46410 0.547723
\(41\) 3.46410 0.541002 0.270501 0.962720i \(-0.412811\pi\)
0.270501 + 0.962720i \(0.412811\pi\)
\(42\) −1.00000 −0.154303
\(43\) −10.9282 −1.66654 −0.833268 0.552870i \(-0.813533\pi\)
−0.833268 + 0.552870i \(0.813533\pi\)
\(44\) 1.00000 0.150756
\(45\) −3.46410 −0.516398
\(46\) −6.92820 −1.02151
\(47\) 9.46410 1.38048 0.690241 0.723580i \(-0.257505\pi\)
0.690241 + 0.723580i \(0.257505\pi\)
\(48\) 1.00000 0.144338
\(49\) 1.00000 0.142857
\(50\) −7.00000 −0.989949
\(51\) 3.46410 0.485071
\(52\) 2.00000 0.277350
\(53\) −0.928203 −0.127499 −0.0637493 0.997966i \(-0.520306\pi\)
−0.0637493 + 0.997966i \(0.520306\pi\)
\(54\) −1.00000 −0.136083
\(55\) −3.46410 −0.467099
\(56\) −1.00000 −0.133631
\(57\) 5.46410 0.723738
\(58\) 6.00000 0.787839
\(59\) 6.92820 0.901975 0.450988 0.892530i \(-0.351072\pi\)
0.450988 + 0.892530i \(0.351072\pi\)
\(60\) −3.46410 −0.447214
\(61\) 2.00000 0.256074 0.128037 0.991769i \(-0.459132\pi\)
0.128037 + 0.991769i \(0.459132\pi\)
\(62\) −5.46410 −0.693942
\(63\) 1.00000 0.125988
\(64\) 1.00000 0.125000
\(65\) −6.92820 −0.859338
\(66\) −1.00000 −0.123091
\(67\) 14.9282 1.82377 0.911885 0.410445i \(-0.134627\pi\)
0.911885 + 0.410445i \(0.134627\pi\)
\(68\) 3.46410 0.420084
\(69\) 6.92820 0.834058
\(70\) 3.46410 0.414039
\(71\) −12.0000 −1.42414 −0.712069 0.702109i \(-0.752242\pi\)
−0.712069 + 0.702109i \(0.752242\pi\)
\(72\) −1.00000 −0.117851
\(73\) −0.535898 −0.0627222 −0.0313611 0.999508i \(-0.509984\pi\)
−0.0313611 + 0.999508i \(0.509984\pi\)
\(74\) 4.92820 0.572892
\(75\) 7.00000 0.808290
\(76\) 5.46410 0.626775
\(77\) 1.00000 0.113961
\(78\) −2.00000 −0.226455
\(79\) −10.9282 −1.22952 −0.614759 0.788715i \(-0.710747\pi\)
−0.614759 + 0.788715i \(0.710747\pi\)
\(80\) −3.46410 −0.387298
\(81\) 1.00000 0.111111
\(82\) −3.46410 −0.382546
\(83\) 4.39230 0.482118 0.241059 0.970510i \(-0.422505\pi\)
0.241059 + 0.970510i \(0.422505\pi\)
\(84\) 1.00000 0.109109
\(85\) −12.0000 −1.30158
\(86\) 10.9282 1.17842
\(87\) −6.00000 −0.643268
\(88\) −1.00000 −0.106600
\(89\) −0.928203 −0.0983893 −0.0491947 0.998789i \(-0.515665\pi\)
−0.0491947 + 0.998789i \(0.515665\pi\)
\(90\) 3.46410 0.365148
\(91\) 2.00000 0.209657
\(92\) 6.92820 0.722315
\(93\) 5.46410 0.566601
\(94\) −9.46410 −0.976148
\(95\) −18.9282 −1.94199
\(96\) −1.00000 −0.102062
\(97\) 2.00000 0.203069 0.101535 0.994832i \(-0.467625\pi\)
0.101535 + 0.994832i \(0.467625\pi\)
\(98\) −1.00000 −0.101015
\(99\) 1.00000 0.100504
\(100\) 7.00000 0.700000
\(101\) −19.8564 −1.97579 −0.987893 0.155136i \(-0.950418\pi\)
−0.987893 + 0.155136i \(0.950418\pi\)
\(102\) −3.46410 −0.342997
\(103\) −13.4641 −1.32666 −0.663329 0.748328i \(-0.730857\pi\)
−0.663329 + 0.748328i \(0.730857\pi\)
\(104\) −2.00000 −0.196116
\(105\) −3.46410 −0.338062
\(106\) 0.928203 0.0901551
\(107\) −6.92820 −0.669775 −0.334887 0.942258i \(-0.608698\pi\)
−0.334887 + 0.942258i \(0.608698\pi\)
\(108\) 1.00000 0.0962250
\(109\) 15.8564 1.51877 0.759384 0.650643i \(-0.225500\pi\)
0.759384 + 0.650643i \(0.225500\pi\)
\(110\) 3.46410 0.330289
\(111\) −4.92820 −0.467764
\(112\) 1.00000 0.0944911
\(113\) 7.85641 0.739069 0.369534 0.929217i \(-0.379517\pi\)
0.369534 + 0.929217i \(0.379517\pi\)
\(114\) −5.46410 −0.511760
\(115\) −24.0000 −2.23801
\(116\) −6.00000 −0.557086
\(117\) 2.00000 0.184900
\(118\) −6.92820 −0.637793
\(119\) 3.46410 0.317554
\(120\) 3.46410 0.316228
\(121\) 1.00000 0.0909091
\(122\) −2.00000 −0.181071
\(123\) 3.46410 0.312348
\(124\) 5.46410 0.490691
\(125\) −6.92820 −0.619677
\(126\) −1.00000 −0.0890871
\(127\) −10.9282 −0.969721 −0.484861 0.874591i \(-0.661130\pi\)
−0.484861 + 0.874591i \(0.661130\pi\)
\(128\) −1.00000 −0.0883883
\(129\) −10.9282 −0.962175
\(130\) 6.92820 0.607644
\(131\) −4.39230 −0.383757 −0.191879 0.981419i \(-0.561458\pi\)
−0.191879 + 0.981419i \(0.561458\pi\)
\(132\) 1.00000 0.0870388
\(133\) 5.46410 0.473798
\(134\) −14.9282 −1.28960
\(135\) −3.46410 −0.298142
\(136\) −3.46410 −0.297044
\(137\) 7.85641 0.671218 0.335609 0.942001i \(-0.391058\pi\)
0.335609 + 0.942001i \(0.391058\pi\)
\(138\) −6.92820 −0.589768
\(139\) −13.4641 −1.14201 −0.571005 0.820947i \(-0.693446\pi\)
−0.571005 + 0.820947i \(0.693446\pi\)
\(140\) −3.46410 −0.292770
\(141\) 9.46410 0.797021
\(142\) 12.0000 1.00702
\(143\) 2.00000 0.167248
\(144\) 1.00000 0.0833333
\(145\) 20.7846 1.72607
\(146\) 0.535898 0.0443513
\(147\) 1.00000 0.0824786
\(148\) −4.92820 −0.405096
\(149\) −6.00000 −0.491539 −0.245770 0.969328i \(-0.579041\pi\)
−0.245770 + 0.969328i \(0.579041\pi\)
\(150\) −7.00000 −0.571548
\(151\) −16.0000 −1.30206 −0.651031 0.759051i \(-0.725663\pi\)
−0.651031 + 0.759051i \(0.725663\pi\)
\(152\) −5.46410 −0.443197
\(153\) 3.46410 0.280056
\(154\) −1.00000 −0.0805823
\(155\) −18.9282 −1.52035
\(156\) 2.00000 0.160128
\(157\) −2.39230 −0.190927 −0.0954634 0.995433i \(-0.530433\pi\)
−0.0954634 + 0.995433i \(0.530433\pi\)
\(158\) 10.9282 0.869401
\(159\) −0.928203 −0.0736113
\(160\) 3.46410 0.273861
\(161\) 6.92820 0.546019
\(162\) −1.00000 −0.0785674
\(163\) −4.00000 −0.313304 −0.156652 0.987654i \(-0.550070\pi\)
−0.156652 + 0.987654i \(0.550070\pi\)
\(164\) 3.46410 0.270501
\(165\) −3.46410 −0.269680
\(166\) −4.39230 −0.340909
\(167\) 18.9282 1.46471 0.732354 0.680924i \(-0.238422\pi\)
0.732354 + 0.680924i \(0.238422\pi\)
\(168\) −1.00000 −0.0771517
\(169\) −9.00000 −0.692308
\(170\) 12.0000 0.920358
\(171\) 5.46410 0.417850
\(172\) −10.9282 −0.833268
\(173\) −0.928203 −0.0705700 −0.0352850 0.999377i \(-0.511234\pi\)
−0.0352850 + 0.999377i \(0.511234\pi\)
\(174\) 6.00000 0.454859
\(175\) 7.00000 0.529150
\(176\) 1.00000 0.0753778
\(177\) 6.92820 0.520756
\(178\) 0.928203 0.0695718
\(179\) −20.7846 −1.55351 −0.776757 0.629800i \(-0.783137\pi\)
−0.776757 + 0.629800i \(0.783137\pi\)
\(180\) −3.46410 −0.258199
\(181\) 6.39230 0.475136 0.237568 0.971371i \(-0.423650\pi\)
0.237568 + 0.971371i \(0.423650\pi\)
\(182\) −2.00000 −0.148250
\(183\) 2.00000 0.147844
\(184\) −6.92820 −0.510754
\(185\) 17.0718 1.25514
\(186\) −5.46410 −0.400647
\(187\) 3.46410 0.253320
\(188\) 9.46410 0.690241
\(189\) 1.00000 0.0727393
\(190\) 18.9282 1.37320
\(191\) 20.7846 1.50392 0.751961 0.659208i \(-0.229108\pi\)
0.751961 + 0.659208i \(0.229108\pi\)
\(192\) 1.00000 0.0721688
\(193\) 26.0000 1.87152 0.935760 0.352636i \(-0.114715\pi\)
0.935760 + 0.352636i \(0.114715\pi\)
\(194\) −2.00000 −0.143592
\(195\) −6.92820 −0.496139
\(196\) 1.00000 0.0714286
\(197\) 18.0000 1.28245 0.641223 0.767354i \(-0.278427\pi\)
0.641223 + 0.767354i \(0.278427\pi\)
\(198\) −1.00000 −0.0710669
\(199\) 0.392305 0.0278098 0.0139049 0.999903i \(-0.495574\pi\)
0.0139049 + 0.999903i \(0.495574\pi\)
\(200\) −7.00000 −0.494975
\(201\) 14.9282 1.05295
\(202\) 19.8564 1.39709
\(203\) −6.00000 −0.421117
\(204\) 3.46410 0.242536
\(205\) −12.0000 −0.838116
\(206\) 13.4641 0.938088
\(207\) 6.92820 0.481543
\(208\) 2.00000 0.138675
\(209\) 5.46410 0.377960
\(210\) 3.46410 0.239046
\(211\) 16.7846 1.15550 0.577750 0.816214i \(-0.303931\pi\)
0.577750 + 0.816214i \(0.303931\pi\)
\(212\) −0.928203 −0.0637493
\(213\) −12.0000 −0.822226
\(214\) 6.92820 0.473602
\(215\) 37.8564 2.58179
\(216\) −1.00000 −0.0680414
\(217\) 5.46410 0.370927
\(218\) −15.8564 −1.07393
\(219\) −0.535898 −0.0362127
\(220\) −3.46410 −0.233550
\(221\) 6.92820 0.466041
\(222\) 4.92820 0.330759
\(223\) −8.39230 −0.561990 −0.280995 0.959709i \(-0.590664\pi\)
−0.280995 + 0.959709i \(0.590664\pi\)
\(224\) −1.00000 −0.0668153
\(225\) 7.00000 0.466667
\(226\) −7.85641 −0.522600
\(227\) −4.39230 −0.291528 −0.145764 0.989319i \(-0.546564\pi\)
−0.145764 + 0.989319i \(0.546564\pi\)
\(228\) 5.46410 0.361869
\(229\) −21.3205 −1.40890 −0.704449 0.709754i \(-0.748806\pi\)
−0.704449 + 0.709754i \(0.748806\pi\)
\(230\) 24.0000 1.58251
\(231\) 1.00000 0.0657952
\(232\) 6.00000 0.393919
\(233\) −7.85641 −0.514690 −0.257345 0.966320i \(-0.582848\pi\)
−0.257345 + 0.966320i \(0.582848\pi\)
\(234\) −2.00000 −0.130744
\(235\) −32.7846 −2.13863
\(236\) 6.92820 0.450988
\(237\) −10.9282 −0.709863
\(238\) −3.46410 −0.224544
\(239\) −24.0000 −1.55243 −0.776215 0.630468i \(-0.782863\pi\)
−0.776215 + 0.630468i \(0.782863\pi\)
\(240\) −3.46410 −0.223607
\(241\) 9.60770 0.618886 0.309443 0.950918i \(-0.399857\pi\)
0.309443 + 0.950918i \(0.399857\pi\)
\(242\) −1.00000 −0.0642824
\(243\) 1.00000 0.0641500
\(244\) 2.00000 0.128037
\(245\) −3.46410 −0.221313
\(246\) −3.46410 −0.220863
\(247\) 10.9282 0.695345
\(248\) −5.46410 −0.346971
\(249\) 4.39230 0.278351
\(250\) 6.92820 0.438178
\(251\) 30.9282 1.95217 0.976085 0.217387i \(-0.0697534\pi\)
0.976085 + 0.217387i \(0.0697534\pi\)
\(252\) 1.00000 0.0629941
\(253\) 6.92820 0.435572
\(254\) 10.9282 0.685696
\(255\) −12.0000 −0.751469
\(256\) 1.00000 0.0625000
\(257\) 12.9282 0.806439 0.403220 0.915103i \(-0.367891\pi\)
0.403220 + 0.915103i \(0.367891\pi\)
\(258\) 10.9282 0.680360
\(259\) −4.92820 −0.306224
\(260\) −6.92820 −0.429669
\(261\) −6.00000 −0.371391
\(262\) 4.39230 0.271357
\(263\) −5.07180 −0.312740 −0.156370 0.987699i \(-0.549979\pi\)
−0.156370 + 0.987699i \(0.549979\pi\)
\(264\) −1.00000 −0.0615457
\(265\) 3.21539 0.197520
\(266\) −5.46410 −0.335026
\(267\) −0.928203 −0.0568051
\(268\) 14.9282 0.911885
\(269\) 24.2487 1.47847 0.739235 0.673448i \(-0.235187\pi\)
0.739235 + 0.673448i \(0.235187\pi\)
\(270\) 3.46410 0.210819
\(271\) −24.7846 −1.50556 −0.752779 0.658273i \(-0.771287\pi\)
−0.752779 + 0.658273i \(0.771287\pi\)
\(272\) 3.46410 0.210042
\(273\) 2.00000 0.121046
\(274\) −7.85641 −0.474623
\(275\) 7.00000 0.422116
\(276\) 6.92820 0.417029
\(277\) −11.8564 −0.712382 −0.356191 0.934413i \(-0.615925\pi\)
−0.356191 + 0.934413i \(0.615925\pi\)
\(278\) 13.4641 0.807523
\(279\) 5.46410 0.327127
\(280\) 3.46410 0.207020
\(281\) −7.85641 −0.468674 −0.234337 0.972155i \(-0.575292\pi\)
−0.234337 + 0.972155i \(0.575292\pi\)
\(282\) −9.46410 −0.563579
\(283\) −13.4641 −0.800358 −0.400179 0.916437i \(-0.631052\pi\)
−0.400179 + 0.916437i \(0.631052\pi\)
\(284\) −12.0000 −0.712069
\(285\) −18.9282 −1.12121
\(286\) −2.00000 −0.118262
\(287\) 3.46410 0.204479
\(288\) −1.00000 −0.0589256
\(289\) −5.00000 −0.294118
\(290\) −20.7846 −1.22051
\(291\) 2.00000 0.117242
\(292\) −0.535898 −0.0313611
\(293\) −24.9282 −1.45632 −0.728161 0.685407i \(-0.759625\pi\)
−0.728161 + 0.685407i \(0.759625\pi\)
\(294\) −1.00000 −0.0583212
\(295\) −24.0000 −1.39733
\(296\) 4.92820 0.286446
\(297\) 1.00000 0.0580259
\(298\) 6.00000 0.347571
\(299\) 13.8564 0.801337
\(300\) 7.00000 0.404145
\(301\) −10.9282 −0.629891
\(302\) 16.0000 0.920697
\(303\) −19.8564 −1.14072
\(304\) 5.46410 0.313388
\(305\) −6.92820 −0.396708
\(306\) −3.46410 −0.198030
\(307\) 10.5359 0.601315 0.300658 0.953732i \(-0.402794\pi\)
0.300658 + 0.953732i \(0.402794\pi\)
\(308\) 1.00000 0.0569803
\(309\) −13.4641 −0.765946
\(310\) 18.9282 1.07505
\(311\) 28.3923 1.60998 0.804990 0.593288i \(-0.202171\pi\)
0.804990 + 0.593288i \(0.202171\pi\)
\(312\) −2.00000 −0.113228
\(313\) −16.9282 −0.956839 −0.478419 0.878132i \(-0.658790\pi\)
−0.478419 + 0.878132i \(0.658790\pi\)
\(314\) 2.39230 0.135006
\(315\) −3.46410 −0.195180
\(316\) −10.9282 −0.614759
\(317\) 12.9282 0.726120 0.363060 0.931766i \(-0.381732\pi\)
0.363060 + 0.931766i \(0.381732\pi\)
\(318\) 0.928203 0.0520511
\(319\) −6.00000 −0.335936
\(320\) −3.46410 −0.193649
\(321\) −6.92820 −0.386695
\(322\) −6.92820 −0.386094
\(323\) 18.9282 1.05319
\(324\) 1.00000 0.0555556
\(325\) 14.0000 0.776580
\(326\) 4.00000 0.221540
\(327\) 15.8564 0.876861
\(328\) −3.46410 −0.191273
\(329\) 9.46410 0.521773
\(330\) 3.46410 0.190693
\(331\) −9.07180 −0.498631 −0.249316 0.968422i \(-0.580206\pi\)
−0.249316 + 0.968422i \(0.580206\pi\)
\(332\) 4.39230 0.241059
\(333\) −4.92820 −0.270064
\(334\) −18.9282 −1.03571
\(335\) −51.7128 −2.82537
\(336\) 1.00000 0.0545545
\(337\) 20.9282 1.14003 0.570016 0.821634i \(-0.306937\pi\)
0.570016 + 0.821634i \(0.306937\pi\)
\(338\) 9.00000 0.489535
\(339\) 7.85641 0.426701
\(340\) −12.0000 −0.650791
\(341\) 5.46410 0.295898
\(342\) −5.46410 −0.295465
\(343\) 1.00000 0.0539949
\(344\) 10.9282 0.589209
\(345\) −24.0000 −1.29212
\(346\) 0.928203 0.0499005
\(347\) −20.7846 −1.11578 −0.557888 0.829916i \(-0.688388\pi\)
−0.557888 + 0.829916i \(0.688388\pi\)
\(348\) −6.00000 −0.321634
\(349\) 10.7846 0.577287 0.288643 0.957437i \(-0.406796\pi\)
0.288643 + 0.957437i \(0.406796\pi\)
\(350\) −7.00000 −0.374166
\(351\) 2.00000 0.106752
\(352\) −1.00000 −0.0533002
\(353\) 12.9282 0.688099 0.344049 0.938952i \(-0.388201\pi\)
0.344049 + 0.938952i \(0.388201\pi\)
\(354\) −6.92820 −0.368230
\(355\) 41.5692 2.20627
\(356\) −0.928203 −0.0491947
\(357\) 3.46410 0.183340
\(358\) 20.7846 1.09850
\(359\) −5.07180 −0.267679 −0.133840 0.991003i \(-0.542731\pi\)
−0.133840 + 0.991003i \(0.542731\pi\)
\(360\) 3.46410 0.182574
\(361\) 10.8564 0.571390
\(362\) −6.39230 −0.335972
\(363\) 1.00000 0.0524864
\(364\) 2.00000 0.104828
\(365\) 1.85641 0.0971688
\(366\) −2.00000 −0.104542
\(367\) 19.3205 1.00852 0.504261 0.863551i \(-0.331765\pi\)
0.504261 + 0.863551i \(0.331765\pi\)
\(368\) 6.92820 0.361158
\(369\) 3.46410 0.180334
\(370\) −17.0718 −0.887520
\(371\) −0.928203 −0.0481899
\(372\) 5.46410 0.283300
\(373\) 29.7128 1.53847 0.769236 0.638965i \(-0.220637\pi\)
0.769236 + 0.638965i \(0.220637\pi\)
\(374\) −3.46410 −0.179124
\(375\) −6.92820 −0.357771
\(376\) −9.46410 −0.488074
\(377\) −12.0000 −0.618031
\(378\) −1.00000 −0.0514344
\(379\) −17.8564 −0.917222 −0.458611 0.888637i \(-0.651653\pi\)
−0.458611 + 0.888637i \(0.651653\pi\)
\(380\) −18.9282 −0.970996
\(381\) −10.9282 −0.559869
\(382\) −20.7846 −1.06343
\(383\) 14.5359 0.742750 0.371375 0.928483i \(-0.378886\pi\)
0.371375 + 0.928483i \(0.378886\pi\)
\(384\) −1.00000 −0.0510310
\(385\) −3.46410 −0.176547
\(386\) −26.0000 −1.32337
\(387\) −10.9282 −0.555512
\(388\) 2.00000 0.101535
\(389\) −30.0000 −1.52106 −0.760530 0.649303i \(-0.775061\pi\)
−0.760530 + 0.649303i \(0.775061\pi\)
\(390\) 6.92820 0.350823
\(391\) 24.0000 1.21373
\(392\) −1.00000 −0.0505076
\(393\) −4.39230 −0.221562
\(394\) −18.0000 −0.906827
\(395\) 37.8564 1.90476
\(396\) 1.00000 0.0502519
\(397\) −2.39230 −0.120066 −0.0600332 0.998196i \(-0.519121\pi\)
−0.0600332 + 0.998196i \(0.519121\pi\)
\(398\) −0.392305 −0.0196645
\(399\) 5.46410 0.273547
\(400\) 7.00000 0.350000
\(401\) 4.14359 0.206921 0.103461 0.994634i \(-0.467008\pi\)
0.103461 + 0.994634i \(0.467008\pi\)
\(402\) −14.9282 −0.744551
\(403\) 10.9282 0.544373
\(404\) −19.8564 −0.987893
\(405\) −3.46410 −0.172133
\(406\) 6.00000 0.297775
\(407\) −4.92820 −0.244282
\(408\) −3.46410 −0.171499
\(409\) −9.32051 −0.460869 −0.230435 0.973088i \(-0.574015\pi\)
−0.230435 + 0.973088i \(0.574015\pi\)
\(410\) 12.0000 0.592638
\(411\) 7.85641 0.387528
\(412\) −13.4641 −0.663329
\(413\) 6.92820 0.340915
\(414\) −6.92820 −0.340503
\(415\) −15.2154 −0.746894
\(416\) −2.00000 −0.0980581
\(417\) −13.4641 −0.659340
\(418\) −5.46410 −0.267258
\(419\) −36.0000 −1.75872 −0.879358 0.476162i \(-0.842028\pi\)
−0.879358 + 0.476162i \(0.842028\pi\)
\(420\) −3.46410 −0.169031
\(421\) −37.7128 −1.83801 −0.919005 0.394246i \(-0.871006\pi\)
−0.919005 + 0.394246i \(0.871006\pi\)
\(422\) −16.7846 −0.817062
\(423\) 9.46410 0.460160
\(424\) 0.928203 0.0450775
\(425\) 24.2487 1.17624
\(426\) 12.0000 0.581402
\(427\) 2.00000 0.0967868
\(428\) −6.92820 −0.334887
\(429\) 2.00000 0.0965609
\(430\) −37.8564 −1.82560
\(431\) −18.9282 −0.911739 −0.455870 0.890047i \(-0.650672\pi\)
−0.455870 + 0.890047i \(0.650672\pi\)
\(432\) 1.00000 0.0481125
\(433\) 2.00000 0.0961139 0.0480569 0.998845i \(-0.484697\pi\)
0.0480569 + 0.998845i \(0.484697\pi\)
\(434\) −5.46410 −0.262285
\(435\) 20.7846 0.996546
\(436\) 15.8564 0.759384
\(437\) 37.8564 1.81092
\(438\) 0.535898 0.0256062
\(439\) −24.7846 −1.18290 −0.591452 0.806340i \(-0.701445\pi\)
−0.591452 + 0.806340i \(0.701445\pi\)
\(440\) 3.46410 0.165145
\(441\) 1.00000 0.0476190
\(442\) −6.92820 −0.329541
\(443\) −20.7846 −0.987507 −0.493753 0.869602i \(-0.664375\pi\)
−0.493753 + 0.869602i \(0.664375\pi\)
\(444\) −4.92820 −0.233882
\(445\) 3.21539 0.152424
\(446\) 8.39230 0.397387
\(447\) −6.00000 −0.283790
\(448\) 1.00000 0.0472456
\(449\) −19.8564 −0.937082 −0.468541 0.883442i \(-0.655220\pi\)
−0.468541 + 0.883442i \(0.655220\pi\)
\(450\) −7.00000 −0.329983
\(451\) 3.46410 0.163118
\(452\) 7.85641 0.369534
\(453\) −16.0000 −0.751746
\(454\) 4.39230 0.206141
\(455\) −6.92820 −0.324799
\(456\) −5.46410 −0.255880
\(457\) 15.8564 0.741731 0.370866 0.928687i \(-0.379061\pi\)
0.370866 + 0.928687i \(0.379061\pi\)
\(458\) 21.3205 0.996242
\(459\) 3.46410 0.161690
\(460\) −24.0000 −1.11901
\(461\) −6.00000 −0.279448 −0.139724 0.990190i \(-0.544622\pi\)
−0.139724 + 0.990190i \(0.544622\pi\)
\(462\) −1.00000 −0.0465242
\(463\) 8.00000 0.371792 0.185896 0.982569i \(-0.440481\pi\)
0.185896 + 0.982569i \(0.440481\pi\)
\(464\) −6.00000 −0.278543
\(465\) −18.9282 −0.877774
\(466\) 7.85641 0.363941
\(467\) −15.7128 −0.727102 −0.363551 0.931574i \(-0.618436\pi\)
−0.363551 + 0.931574i \(0.618436\pi\)
\(468\) 2.00000 0.0924500
\(469\) 14.9282 0.689320
\(470\) 32.7846 1.51224
\(471\) −2.39230 −0.110232
\(472\) −6.92820 −0.318896
\(473\) −10.9282 −0.502479
\(474\) 10.9282 0.501949
\(475\) 38.2487 1.75497
\(476\) 3.46410 0.158777
\(477\) −0.928203 −0.0424995
\(478\) 24.0000 1.09773
\(479\) −13.8564 −0.633115 −0.316558 0.948573i \(-0.602527\pi\)
−0.316558 + 0.948573i \(0.602527\pi\)
\(480\) 3.46410 0.158114
\(481\) −9.85641 −0.449413
\(482\) −9.60770 −0.437619
\(483\) 6.92820 0.315244
\(484\) 1.00000 0.0454545
\(485\) −6.92820 −0.314594
\(486\) −1.00000 −0.0453609
\(487\) 40.7846 1.84813 0.924064 0.382239i \(-0.124847\pi\)
0.924064 + 0.382239i \(0.124847\pi\)
\(488\) −2.00000 −0.0905357
\(489\) −4.00000 −0.180886
\(490\) 3.46410 0.156492
\(491\) −1.85641 −0.0837785 −0.0418892 0.999122i \(-0.513338\pi\)
−0.0418892 + 0.999122i \(0.513338\pi\)
\(492\) 3.46410 0.156174
\(493\) −20.7846 −0.936092
\(494\) −10.9282 −0.491683
\(495\) −3.46410 −0.155700
\(496\) 5.46410 0.245345
\(497\) −12.0000 −0.538274
\(498\) −4.39230 −0.196824
\(499\) −9.07180 −0.406109 −0.203055 0.979167i \(-0.565087\pi\)
−0.203055 + 0.979167i \(0.565087\pi\)
\(500\) −6.92820 −0.309839
\(501\) 18.9282 0.845650
\(502\) −30.9282 −1.38039
\(503\) 18.9282 0.843967 0.421983 0.906604i \(-0.361334\pi\)
0.421983 + 0.906604i \(0.361334\pi\)
\(504\) −1.00000 −0.0445435
\(505\) 68.7846 3.06087
\(506\) −6.92820 −0.307996
\(507\) −9.00000 −0.399704
\(508\) −10.9282 −0.484861
\(509\) −41.3205 −1.83150 −0.915750 0.401749i \(-0.868402\pi\)
−0.915750 + 0.401749i \(0.868402\pi\)
\(510\) 12.0000 0.531369
\(511\) −0.535898 −0.0237067
\(512\) −1.00000 −0.0441942
\(513\) 5.46410 0.241246
\(514\) −12.9282 −0.570239
\(515\) 46.6410 2.05525
\(516\) −10.9282 −0.481087
\(517\) 9.46410 0.416231
\(518\) 4.92820 0.216533
\(519\) −0.928203 −0.0407436
\(520\) 6.92820 0.303822
\(521\) 36.9282 1.61785 0.808927 0.587909i \(-0.200049\pi\)
0.808927 + 0.587909i \(0.200049\pi\)
\(522\) 6.00000 0.262613
\(523\) 24.3923 1.06660 0.533301 0.845926i \(-0.320952\pi\)
0.533301 + 0.845926i \(0.320952\pi\)
\(524\) −4.39230 −0.191879
\(525\) 7.00000 0.305505
\(526\) 5.07180 0.221141
\(527\) 18.9282 0.824525
\(528\) 1.00000 0.0435194
\(529\) 25.0000 1.08696
\(530\) −3.21539 −0.139668
\(531\) 6.92820 0.300658
\(532\) 5.46410 0.236899
\(533\) 6.92820 0.300094
\(534\) 0.928203 0.0401673
\(535\) 24.0000 1.03761
\(536\) −14.9282 −0.644800
\(537\) −20.7846 −0.896922
\(538\) −24.2487 −1.04544
\(539\) 1.00000 0.0430730
\(540\) −3.46410 −0.149071
\(541\) −25.7128 −1.10548 −0.552740 0.833354i \(-0.686418\pi\)
−0.552740 + 0.833354i \(0.686418\pi\)
\(542\) 24.7846 1.06459
\(543\) 6.39230 0.274320
\(544\) −3.46410 −0.148522
\(545\) −54.9282 −2.35287
\(546\) −2.00000 −0.0855921
\(547\) −16.0000 −0.684111 −0.342055 0.939680i \(-0.611123\pi\)
−0.342055 + 0.939680i \(0.611123\pi\)
\(548\) 7.85641 0.335609
\(549\) 2.00000 0.0853579
\(550\) −7.00000 −0.298481
\(551\) −32.7846 −1.39667
\(552\) −6.92820 −0.294884
\(553\) −10.9282 −0.464714
\(554\) 11.8564 0.503730
\(555\) 17.0718 0.724657
\(556\) −13.4641 −0.571005
\(557\) 21.7128 0.920001 0.460001 0.887919i \(-0.347849\pi\)
0.460001 + 0.887919i \(0.347849\pi\)
\(558\) −5.46410 −0.231314
\(559\) −21.8564 −0.924427
\(560\) −3.46410 −0.146385
\(561\) 3.46410 0.146254
\(562\) 7.85641 0.331403
\(563\) −18.2487 −0.769091 −0.384546 0.923106i \(-0.625642\pi\)
−0.384546 + 0.923106i \(0.625642\pi\)
\(564\) 9.46410 0.398511
\(565\) −27.2154 −1.14496
\(566\) 13.4641 0.565938
\(567\) 1.00000 0.0419961
\(568\) 12.0000 0.503509
\(569\) −31.8564 −1.33549 −0.667745 0.744390i \(-0.732740\pi\)
−0.667745 + 0.744390i \(0.732740\pi\)
\(570\) 18.9282 0.792815
\(571\) 8.00000 0.334790 0.167395 0.985890i \(-0.446465\pi\)
0.167395 + 0.985890i \(0.446465\pi\)
\(572\) 2.00000 0.0836242
\(573\) 20.7846 0.868290
\(574\) −3.46410 −0.144589
\(575\) 48.4974 2.02248
\(576\) 1.00000 0.0416667
\(577\) −30.7846 −1.28158 −0.640790 0.767716i \(-0.721393\pi\)
−0.640790 + 0.767716i \(0.721393\pi\)
\(578\) 5.00000 0.207973
\(579\) 26.0000 1.08052
\(580\) 20.7846 0.863034
\(581\) 4.39230 0.182224
\(582\) −2.00000 −0.0829027
\(583\) −0.928203 −0.0384422
\(584\) 0.535898 0.0221756
\(585\) −6.92820 −0.286446
\(586\) 24.9282 1.02977
\(587\) −20.7846 −0.857873 −0.428936 0.903335i \(-0.641112\pi\)
−0.428936 + 0.903335i \(0.641112\pi\)
\(588\) 1.00000 0.0412393
\(589\) 29.8564 1.23021
\(590\) 24.0000 0.988064
\(591\) 18.0000 0.740421
\(592\) −4.92820 −0.202548
\(593\) −20.5359 −0.843308 −0.421654 0.906757i \(-0.638550\pi\)
−0.421654 + 0.906757i \(0.638550\pi\)
\(594\) −1.00000 −0.0410305
\(595\) −12.0000 −0.491952
\(596\) −6.00000 −0.245770
\(597\) 0.392305 0.0160560
\(598\) −13.8564 −0.566631
\(599\) 1.85641 0.0758507 0.0379254 0.999281i \(-0.487925\pi\)
0.0379254 + 0.999281i \(0.487925\pi\)
\(600\) −7.00000 −0.285774
\(601\) 18.3923 0.750238 0.375119 0.926977i \(-0.377602\pi\)
0.375119 + 0.926977i \(0.377602\pi\)
\(602\) 10.9282 0.445400
\(603\) 14.9282 0.607923
\(604\) −16.0000 −0.651031
\(605\) −3.46410 −0.140836
\(606\) 19.8564 0.806611
\(607\) −19.7128 −0.800118 −0.400059 0.916489i \(-0.631010\pi\)
−0.400059 + 0.916489i \(0.631010\pi\)
\(608\) −5.46410 −0.221599
\(609\) −6.00000 −0.243132
\(610\) 6.92820 0.280515
\(611\) 18.9282 0.765753
\(612\) 3.46410 0.140028
\(613\) −22.0000 −0.888572 −0.444286 0.895885i \(-0.646543\pi\)
−0.444286 + 0.895885i \(0.646543\pi\)
\(614\) −10.5359 −0.425194
\(615\) −12.0000 −0.483887
\(616\) −1.00000 −0.0402911
\(617\) −19.8564 −0.799389 −0.399694 0.916648i \(-0.630884\pi\)
−0.399694 + 0.916648i \(0.630884\pi\)
\(618\) 13.4641 0.541606
\(619\) −31.7128 −1.27465 −0.637323 0.770597i \(-0.719958\pi\)
−0.637323 + 0.770597i \(0.719958\pi\)
\(620\) −18.9282 −0.760175
\(621\) 6.92820 0.278019
\(622\) −28.3923 −1.13843
\(623\) −0.928203 −0.0371877
\(624\) 2.00000 0.0800641
\(625\) −11.0000 −0.440000
\(626\) 16.9282 0.676587
\(627\) 5.46410 0.218215
\(628\) −2.39230 −0.0954634
\(629\) −17.0718 −0.680697
\(630\) 3.46410 0.138013
\(631\) −0.784610 −0.0312348 −0.0156174 0.999878i \(-0.504971\pi\)
−0.0156174 + 0.999878i \(0.504971\pi\)
\(632\) 10.9282 0.434701
\(633\) 16.7846 0.667128
\(634\) −12.9282 −0.513445
\(635\) 37.8564 1.50229
\(636\) −0.928203 −0.0368057
\(637\) 2.00000 0.0792429
\(638\) 6.00000 0.237542
\(639\) −12.0000 −0.474713
\(640\) 3.46410 0.136931
\(641\) 31.8564 1.25825 0.629126 0.777303i \(-0.283413\pi\)
0.629126 + 0.777303i \(0.283413\pi\)
\(642\) 6.92820 0.273434
\(643\) 14.9282 0.588711 0.294355 0.955696i \(-0.404895\pi\)
0.294355 + 0.955696i \(0.404895\pi\)
\(644\) 6.92820 0.273009
\(645\) 37.8564 1.49059
\(646\) −18.9282 −0.744720
\(647\) 0.679492 0.0267136 0.0133568 0.999911i \(-0.495748\pi\)
0.0133568 + 0.999911i \(0.495748\pi\)
\(648\) −1.00000 −0.0392837
\(649\) 6.92820 0.271956
\(650\) −14.0000 −0.549125
\(651\) 5.46410 0.214155
\(652\) −4.00000 −0.156652
\(653\) −33.7128 −1.31928 −0.659642 0.751580i \(-0.729292\pi\)
−0.659642 + 0.751580i \(0.729292\pi\)
\(654\) −15.8564 −0.620035
\(655\) 15.2154 0.594514
\(656\) 3.46410 0.135250
\(657\) −0.535898 −0.0209074
\(658\) −9.46410 −0.368949
\(659\) −17.0718 −0.665023 −0.332511 0.943099i \(-0.607896\pi\)
−0.332511 + 0.943099i \(0.607896\pi\)
\(660\) −3.46410 −0.134840
\(661\) 44.2487 1.72108 0.860538 0.509387i \(-0.170128\pi\)
0.860538 + 0.509387i \(0.170128\pi\)
\(662\) 9.07180 0.352585
\(663\) 6.92820 0.269069
\(664\) −4.39230 −0.170454
\(665\) −18.9282 −0.734004
\(666\) 4.92820 0.190964
\(667\) −41.5692 −1.60957
\(668\) 18.9282 0.732354
\(669\) −8.39230 −0.324465
\(670\) 51.7128 1.99784
\(671\) 2.00000 0.0772091
\(672\) −1.00000 −0.0385758
\(673\) 20.9282 0.806723 0.403361 0.915041i \(-0.367842\pi\)
0.403361 + 0.915041i \(0.367842\pi\)
\(674\) −20.9282 −0.806124
\(675\) 7.00000 0.269430
\(676\) −9.00000 −0.346154
\(677\) 2.78461 0.107021 0.0535106 0.998567i \(-0.482959\pi\)
0.0535106 + 0.998567i \(0.482959\pi\)
\(678\) −7.85641 −0.301723
\(679\) 2.00000 0.0767530
\(680\) 12.0000 0.460179
\(681\) −4.39230 −0.168313
\(682\) −5.46410 −0.209231
\(683\) 3.21539 0.123033 0.0615167 0.998106i \(-0.480406\pi\)
0.0615167 + 0.998106i \(0.480406\pi\)
\(684\) 5.46410 0.208925
\(685\) −27.2154 −1.03985
\(686\) −1.00000 −0.0381802
\(687\) −21.3205 −0.813428
\(688\) −10.9282 −0.416634
\(689\) −1.85641 −0.0707235
\(690\) 24.0000 0.913664
\(691\) −33.0718 −1.25811 −0.629055 0.777361i \(-0.716558\pi\)
−0.629055 + 0.777361i \(0.716558\pi\)
\(692\) −0.928203 −0.0352850
\(693\) 1.00000 0.0379869
\(694\) 20.7846 0.788973
\(695\) 46.6410 1.76919
\(696\) 6.00000 0.227429
\(697\) 12.0000 0.454532
\(698\) −10.7846 −0.408203
\(699\) −7.85641 −0.297157
\(700\) 7.00000 0.264575
\(701\) 45.7128 1.72655 0.863275 0.504735i \(-0.168410\pi\)
0.863275 + 0.504735i \(0.168410\pi\)
\(702\) −2.00000 −0.0754851
\(703\) −26.9282 −1.01562
\(704\) 1.00000 0.0376889
\(705\) −32.7846 −1.23474
\(706\) −12.9282 −0.486559
\(707\) −19.8564 −0.746777
\(708\) 6.92820 0.260378
\(709\) 46.7846 1.75703 0.878516 0.477712i \(-0.158534\pi\)
0.878516 + 0.477712i \(0.158534\pi\)
\(710\) −41.5692 −1.56007
\(711\) −10.9282 −0.409840
\(712\) 0.928203 0.0347859
\(713\) 37.8564 1.41773
\(714\) −3.46410 −0.129641
\(715\) −6.92820 −0.259100
\(716\) −20.7846 −0.776757
\(717\) −24.0000 −0.896296
\(718\) 5.07180 0.189278
\(719\) −37.1769 −1.38646 −0.693232 0.720714i \(-0.743814\pi\)
−0.693232 + 0.720714i \(0.743814\pi\)
\(720\) −3.46410 −0.129099
\(721\) −13.4641 −0.501429
\(722\) −10.8564 −0.404034
\(723\) 9.60770 0.357314
\(724\) 6.39230 0.237568
\(725\) −42.0000 −1.55984
\(726\) −1.00000 −0.0371135
\(727\) 33.1769 1.23046 0.615232 0.788346i \(-0.289062\pi\)
0.615232 + 0.788346i \(0.289062\pi\)
\(728\) −2.00000 −0.0741249
\(729\) 1.00000 0.0370370
\(730\) −1.85641 −0.0687087
\(731\) −37.8564 −1.40017
\(732\) 2.00000 0.0739221
\(733\) 12.1436 0.448534 0.224267 0.974528i \(-0.428001\pi\)
0.224267 + 0.974528i \(0.428001\pi\)
\(734\) −19.3205 −0.713133
\(735\) −3.46410 −0.127775
\(736\) −6.92820 −0.255377
\(737\) 14.9282 0.549887
\(738\) −3.46410 −0.127515
\(739\) 8.00000 0.294285 0.147142 0.989115i \(-0.452992\pi\)
0.147142 + 0.989115i \(0.452992\pi\)
\(740\) 17.0718 0.627572
\(741\) 10.9282 0.401458
\(742\) 0.928203 0.0340754
\(743\) −41.5692 −1.52503 −0.762513 0.646972i \(-0.776035\pi\)
−0.762513 + 0.646972i \(0.776035\pi\)
\(744\) −5.46410 −0.200324
\(745\) 20.7846 0.761489
\(746\) −29.7128 −1.08786
\(747\) 4.39230 0.160706
\(748\) 3.46410 0.126660
\(749\) −6.92820 −0.253151
\(750\) 6.92820 0.252982
\(751\) −24.7846 −0.904403 −0.452202 0.891916i \(-0.649361\pi\)
−0.452202 + 0.891916i \(0.649361\pi\)
\(752\) 9.46410 0.345120
\(753\) 30.9282 1.12709
\(754\) 12.0000 0.437014
\(755\) 55.4256 2.01715
\(756\) 1.00000 0.0363696
\(757\) 22.7846 0.828121 0.414060 0.910249i \(-0.364110\pi\)
0.414060 + 0.910249i \(0.364110\pi\)
\(758\) 17.8564 0.648574
\(759\) 6.92820 0.251478
\(760\) 18.9282 0.686598
\(761\) −25.6077 −0.928278 −0.464139 0.885762i \(-0.653636\pi\)
−0.464139 + 0.885762i \(0.653636\pi\)
\(762\) 10.9282 0.395887
\(763\) 15.8564 0.574040
\(764\) 20.7846 0.751961
\(765\) −12.0000 −0.433861
\(766\) −14.5359 −0.525203
\(767\) 13.8564 0.500326
\(768\) 1.00000 0.0360844
\(769\) 28.5359 1.02903 0.514515 0.857481i \(-0.327972\pi\)
0.514515 + 0.857481i \(0.327972\pi\)
\(770\) 3.46410 0.124838
\(771\) 12.9282 0.465598
\(772\) 26.0000 0.935760
\(773\) −32.5359 −1.17023 −0.585117 0.810949i \(-0.698952\pi\)
−0.585117 + 0.810949i \(0.698952\pi\)
\(774\) 10.9282 0.392806
\(775\) 38.2487 1.37393
\(776\) −2.00000 −0.0717958
\(777\) −4.92820 −0.176798
\(778\) 30.0000 1.07555
\(779\) 18.9282 0.678173
\(780\) −6.92820 −0.248069
\(781\) −12.0000 −0.429394
\(782\) −24.0000 −0.858238
\(783\) −6.00000 −0.214423
\(784\) 1.00000 0.0357143
\(785\) 8.28719 0.295782
\(786\) 4.39230 0.156668
\(787\) 33.1769 1.18263 0.591315 0.806441i \(-0.298609\pi\)
0.591315 + 0.806441i \(0.298609\pi\)
\(788\) 18.0000 0.641223
\(789\) −5.07180 −0.180561
\(790\) −37.8564 −1.34687
\(791\) 7.85641 0.279342
\(792\) −1.00000 −0.0355335
\(793\) 4.00000 0.142044
\(794\) 2.39230 0.0848997
\(795\) 3.21539 0.114038
\(796\) 0.392305 0.0139049
\(797\) −32.5359 −1.15248 −0.576240 0.817280i \(-0.695481\pi\)
−0.576240 + 0.817280i \(0.695481\pi\)
\(798\) −5.46410 −0.193427
\(799\) 32.7846 1.15984
\(800\) −7.00000 −0.247487
\(801\) −0.928203 −0.0327964
\(802\) −4.14359 −0.146315
\(803\) −0.535898 −0.0189114
\(804\) 14.9282 0.526477
\(805\) −24.0000 −0.845889
\(806\) −10.9282 −0.384930
\(807\) 24.2487 0.853595
\(808\) 19.8564 0.698546
\(809\) 11.0718 0.389264 0.194632 0.980876i \(-0.437649\pi\)
0.194632 + 0.980876i \(0.437649\pi\)
\(810\) 3.46410 0.121716
\(811\) −17.1769 −0.603163 −0.301582 0.953440i \(-0.597515\pi\)
−0.301582 + 0.953440i \(0.597515\pi\)
\(812\) −6.00000 −0.210559
\(813\) −24.7846 −0.869234
\(814\) 4.92820 0.172733
\(815\) 13.8564 0.485369
\(816\) 3.46410 0.121268
\(817\) −59.7128 −2.08909
\(818\) 9.32051 0.325884
\(819\) 2.00000 0.0698857
\(820\) −12.0000 −0.419058
\(821\) −6.00000 −0.209401 −0.104701 0.994504i \(-0.533388\pi\)
−0.104701 + 0.994504i \(0.533388\pi\)
\(822\) −7.85641 −0.274024
\(823\) 40.7846 1.42166 0.710831 0.703363i \(-0.248319\pi\)
0.710831 + 0.703363i \(0.248319\pi\)
\(824\) 13.4641 0.469044
\(825\) 7.00000 0.243709
\(826\) −6.92820 −0.241063
\(827\) 12.0000 0.417281 0.208640 0.977992i \(-0.433096\pi\)
0.208640 + 0.977992i \(0.433096\pi\)
\(828\) 6.92820 0.240772
\(829\) 44.2487 1.53682 0.768411 0.639957i \(-0.221048\pi\)
0.768411 + 0.639957i \(0.221048\pi\)
\(830\) 15.2154 0.528134
\(831\) −11.8564 −0.411294
\(832\) 2.00000 0.0693375
\(833\) 3.46410 0.120024
\(834\) 13.4641 0.466224
\(835\) −65.5692 −2.26912
\(836\) 5.46410 0.188980
\(837\) 5.46410 0.188867
\(838\) 36.0000 1.24360
\(839\) −33.4641 −1.15531 −0.577655 0.816281i \(-0.696032\pi\)
−0.577655 + 0.816281i \(0.696032\pi\)
\(840\) 3.46410 0.119523
\(841\) 7.00000 0.241379
\(842\) 37.7128 1.29967
\(843\) −7.85641 −0.270589
\(844\) 16.7846 0.577750
\(845\) 31.1769 1.07252
\(846\) −9.46410 −0.325383
\(847\) 1.00000 0.0343604
\(848\) −0.928203 −0.0318746
\(849\) −13.4641 −0.462087
\(850\) −24.2487 −0.831724
\(851\) −34.1436 −1.17043
\(852\) −12.0000 −0.411113
\(853\) −49.7128 −1.70213 −0.851067 0.525057i \(-0.824044\pi\)
−0.851067 + 0.525057i \(0.824044\pi\)
\(854\) −2.00000 −0.0684386
\(855\) −18.9282 −0.647331
\(856\) 6.92820 0.236801
\(857\) −5.32051 −0.181745 −0.0908725 0.995863i \(-0.528966\pi\)
−0.0908725 + 0.995863i \(0.528966\pi\)
\(858\) −2.00000 −0.0682789
\(859\) 52.7846 1.80099 0.900494 0.434869i \(-0.143205\pi\)
0.900494 + 0.434869i \(0.143205\pi\)
\(860\) 37.8564 1.29089
\(861\) 3.46410 0.118056
\(862\) 18.9282 0.644697
\(863\) 44.7846 1.52449 0.762243 0.647291i \(-0.224098\pi\)
0.762243 + 0.647291i \(0.224098\pi\)
\(864\) −1.00000 −0.0340207
\(865\) 3.21539 0.109327
\(866\) −2.00000 −0.0679628
\(867\) −5.00000 −0.169809
\(868\) 5.46410 0.185464
\(869\) −10.9282 −0.370714
\(870\) −20.7846 −0.704664
\(871\) 29.8564 1.01165
\(872\) −15.8564 −0.536966
\(873\) 2.00000 0.0676897
\(874\) −37.8564 −1.28051
\(875\) −6.92820 −0.234216
\(876\) −0.535898 −0.0181063
\(877\) −46.0000 −1.55331 −0.776655 0.629926i \(-0.783085\pi\)
−0.776655 + 0.629926i \(0.783085\pi\)
\(878\) 24.7846 0.836440
\(879\) −24.9282 −0.840807
\(880\) −3.46410 −0.116775
\(881\) −6.00000 −0.202145 −0.101073 0.994879i \(-0.532227\pi\)
−0.101073 + 0.994879i \(0.532227\pi\)
\(882\) −1.00000 −0.0336718
\(883\) −41.8564 −1.40858 −0.704290 0.709912i \(-0.748735\pi\)
−0.704290 + 0.709912i \(0.748735\pi\)
\(884\) 6.92820 0.233021
\(885\) −24.0000 −0.806751
\(886\) 20.7846 0.698273
\(887\) 10.1436 0.340589 0.170294 0.985393i \(-0.445528\pi\)
0.170294 + 0.985393i \(0.445528\pi\)
\(888\) 4.92820 0.165380
\(889\) −10.9282 −0.366520
\(890\) −3.21539 −0.107780
\(891\) 1.00000 0.0335013
\(892\) −8.39230 −0.280995
\(893\) 51.7128 1.73050
\(894\) 6.00000 0.200670
\(895\) 72.0000 2.40669
\(896\) −1.00000 −0.0334077
\(897\) 13.8564 0.462652
\(898\) 19.8564 0.662617
\(899\) −32.7846 −1.09343
\(900\) 7.00000 0.233333
\(901\) −3.21539 −0.107120
\(902\) −3.46410 −0.115342
\(903\) −10.9282 −0.363668
\(904\) −7.85641 −0.261300
\(905\) −22.1436 −0.736078
\(906\) 16.0000 0.531564
\(907\) 14.9282 0.495683 0.247841 0.968801i \(-0.420279\pi\)
0.247841 + 0.968801i \(0.420279\pi\)
\(908\) −4.39230 −0.145764
\(909\) −19.8564 −0.658595
\(910\) 6.92820 0.229668
\(911\) −48.4974 −1.60679 −0.803396 0.595446i \(-0.796976\pi\)
−0.803396 + 0.595446i \(0.796976\pi\)
\(912\) 5.46410 0.180934
\(913\) 4.39230 0.145364
\(914\) −15.8564 −0.524483
\(915\) −6.92820 −0.229039
\(916\) −21.3205 −0.704449
\(917\) −4.39230 −0.145047
\(918\) −3.46410 −0.114332
\(919\) 21.8564 0.720976 0.360488 0.932764i \(-0.382610\pi\)
0.360488 + 0.932764i \(0.382610\pi\)
\(920\) 24.0000 0.791257
\(921\) 10.5359 0.347170
\(922\) 6.00000 0.197599
\(923\) −24.0000 −0.789970
\(924\) 1.00000 0.0328976
\(925\) −34.4974 −1.13427
\(926\) −8.00000 −0.262896
\(927\) −13.4641 −0.442219
\(928\) 6.00000 0.196960
\(929\) 40.6410 1.33339 0.666694 0.745331i \(-0.267709\pi\)
0.666694 + 0.745331i \(0.267709\pi\)
\(930\) 18.9282 0.620680
\(931\) 5.46410 0.179079
\(932\) −7.85641 −0.257345
\(933\) 28.3923 0.929522
\(934\) 15.7128 0.514139
\(935\) −12.0000 −0.392442
\(936\) −2.00000 −0.0653720
\(937\) −14.3923 −0.470176 −0.235088 0.971974i \(-0.575538\pi\)
−0.235088 + 0.971974i \(0.575538\pi\)
\(938\) −14.9282 −0.487423
\(939\) −16.9282 −0.552431
\(940\) −32.7846 −1.06932
\(941\) −11.0718 −0.360930 −0.180465 0.983581i \(-0.557760\pi\)
−0.180465 + 0.983581i \(0.557760\pi\)
\(942\) 2.39230 0.0779455
\(943\) 24.0000 0.781548
\(944\) 6.92820 0.225494
\(945\) −3.46410 −0.112687
\(946\) 10.9282 0.355307
\(947\) 49.8564 1.62012 0.810058 0.586350i \(-0.199436\pi\)
0.810058 + 0.586350i \(0.199436\pi\)
\(948\) −10.9282 −0.354932
\(949\) −1.07180 −0.0347920
\(950\) −38.2487 −1.24095
\(951\) 12.9282 0.419226
\(952\) −3.46410 −0.112272
\(953\) −26.7846 −0.867639 −0.433819 0.901000i \(-0.642834\pi\)
−0.433819 + 0.901000i \(0.642834\pi\)
\(954\) 0.928203 0.0300517
\(955\) −72.0000 −2.32987
\(956\) −24.0000 −0.776215
\(957\) −6.00000 −0.193952
\(958\) 13.8564 0.447680
\(959\) 7.85641 0.253697
\(960\) −3.46410 −0.111803
\(961\) −1.14359 −0.0368901
\(962\) 9.85641 0.317783
\(963\) −6.92820 −0.223258
\(964\) 9.60770 0.309443
\(965\) −90.0666 −2.89935
\(966\) −6.92820 −0.222911
\(967\) 32.0000 1.02905 0.514525 0.857475i \(-0.327968\pi\)
0.514525 + 0.857475i \(0.327968\pi\)
\(968\) −1.00000 −0.0321412
\(969\) 18.9282 0.608061
\(970\) 6.92820 0.222451
\(971\) −15.7128 −0.504248 −0.252124 0.967695i \(-0.581129\pi\)
−0.252124 + 0.967695i \(0.581129\pi\)
\(972\) 1.00000 0.0320750
\(973\) −13.4641 −0.431639
\(974\) −40.7846 −1.30682
\(975\) 14.0000 0.448359
\(976\) 2.00000 0.0640184
\(977\) −47.5692 −1.52187 −0.760937 0.648826i \(-0.775260\pi\)
−0.760937 + 0.648826i \(0.775260\pi\)
\(978\) 4.00000 0.127906
\(979\) −0.928203 −0.0296655
\(980\) −3.46410 −0.110657
\(981\) 15.8564 0.506256
\(982\) 1.85641 0.0592403
\(983\) 23.3205 0.743809 0.371904 0.928271i \(-0.378705\pi\)
0.371904 + 0.928271i \(0.378705\pi\)
\(984\) −3.46410 −0.110432
\(985\) −62.3538 −1.98676
\(986\) 20.7846 0.661917
\(987\) 9.46410 0.301246
\(988\) 10.9282 0.347672
\(989\) −75.7128 −2.40753
\(990\) 3.46410 0.110096
\(991\) −2.14359 −0.0680935 −0.0340467 0.999420i \(-0.510840\pi\)
−0.0340467 + 0.999420i \(0.510840\pi\)
\(992\) −5.46410 −0.173485
\(993\) −9.07180 −0.287885
\(994\) 12.0000 0.380617
\(995\) −1.35898 −0.0430827
\(996\) 4.39230 0.139176
\(997\) 24.6410 0.780389 0.390194 0.920732i \(-0.372408\pi\)
0.390194 + 0.920732i \(0.372408\pi\)
\(998\) 9.07180 0.287163
\(999\) −4.92820 −0.155921
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 462.2.a.h.1.1 2
3.2 odd 2 1386.2.a.p.1.2 2
4.3 odd 2 3696.2.a.bc.1.1 2
7.6 odd 2 3234.2.a.x.1.2 2
11.10 odd 2 5082.2.a.bu.1.1 2
21.20 even 2 9702.2.a.dd.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
462.2.a.h.1.1 2 1.1 even 1 trivial
1386.2.a.p.1.2 2 3.2 odd 2
3234.2.a.x.1.2 2 7.6 odd 2
3696.2.a.bc.1.1 2 4.3 odd 2
5082.2.a.bu.1.1 2 11.10 odd 2
9702.2.a.dd.1.1 2 21.20 even 2