Properties

Label 7623.2.a.ch.1.3
Level 7623
Weight 2
Character 7623.1
Self dual yes
Analytic conductor 60.870
Analytic rank 0
Dimension 4
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 7623 = 3^{2} \cdot 7 \cdot 11^{2} \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 7623.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(60.8699614608\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.2525.1
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 77)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-0.777484\)
Character \(\chi\) = 7623.1

$q$-expansion

\(f(q)\) \(=\) \(q+0.777484 q^{2} -1.39552 q^{4} +0.222516 q^{5} -1.00000 q^{7} -2.63996 q^{8} +O(q^{10})\) \(q+0.777484 q^{2} -1.39552 q^{4} +0.222516 q^{5} -1.00000 q^{7} -2.63996 q^{8} +0.173002 q^{10} +6.52954 q^{13} -0.777484 q^{14} +0.738508 q^{16} +4.33461 q^{17} +2.91501 q^{19} -0.310525 q^{20} +3.89796 q^{23} -4.95049 q^{25} +5.07662 q^{26} +1.39552 q^{28} +3.77399 q^{29} -6.88958 q^{31} +5.85410 q^{32} +3.37009 q^{34} -0.222516 q^{35} -5.65351 q^{37} +2.26637 q^{38} -0.587433 q^{40} -1.33811 q^{41} -4.70820 q^{43} +3.03060 q^{46} -6.04554 q^{47} +1.00000 q^{49} -3.84893 q^{50} -9.11210 q^{52} +1.71792 q^{53} +2.63996 q^{56} +2.93422 q^{58} +9.53304 q^{59} -9.62943 q^{61} -5.35654 q^{62} +3.07446 q^{64} +1.45293 q^{65} +1.27155 q^{67} -6.04903 q^{68} -0.173002 q^{70} +9.30919 q^{71} +5.58911 q^{73} -4.39552 q^{74} -4.06794 q^{76} -4.52954 q^{79} +0.164330 q^{80} -1.04036 q^{82} -11.2838 q^{83} +0.964520 q^{85} -3.66055 q^{86} -7.92157 q^{89} -6.52954 q^{91} -5.43967 q^{92} -4.70031 q^{94} +0.648635 q^{95} -9.05391 q^{97} +0.777484 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 2q^{2} + 4q^{4} + 6q^{5} - 4q^{7} - 9q^{8} + O(q^{10}) \) \( 4q - 2q^{2} + 4q^{4} + 6q^{5} - 4q^{7} - 9q^{8} - 14q^{10} + 2q^{14} - 4q^{16} + 3q^{17} + 3q^{19} + 17q^{20} + 8q^{23} + 12q^{26} - 4q^{28} - 3q^{29} - 3q^{31} + 10q^{32} - 12q^{34} - 6q^{35} - 7q^{37} + 20q^{38} - 13q^{40} - 4q^{41} + 8q^{43} - 3q^{46} + 14q^{47} + 4q^{49} - 33q^{50} - 17q^{52} + 9q^{53} + 9q^{56} + 3q^{58} + 25q^{59} - 19q^{61} - 10q^{62} + 3q^{64} - 12q^{65} - 15q^{67} + q^{68} + 14q^{70} + 7q^{71} - 11q^{73} - 8q^{74} - 26q^{76} + 8q^{79} + 4q^{80} + 3q^{82} + q^{83} + 15q^{85} - 4q^{86} + 17q^{89} + 17q^{92} - 20q^{94} - 17q^{95} - 15q^{97} - 2q^{98} + O(q^{100}) \)

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.777484 0.549764 0.274882 0.961478i \(-0.411361\pi\)
0.274882 + 0.961478i \(0.411361\pi\)
\(3\) 0 0
\(4\) −1.39552 −0.697759
\(5\) 0.222516 0.0995121 0.0497560 0.998761i \(-0.484156\pi\)
0.0497560 + 0.998761i \(0.484156\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) −2.63996 −0.933368
\(9\) 0 0
\(10\) 0.173002 0.0547082
\(11\) 0 0
\(12\) 0 0
\(13\) 6.52954 1.81097 0.905485 0.424379i \(-0.139507\pi\)
0.905485 + 0.424379i \(0.139507\pi\)
\(14\) −0.777484 −0.207791
\(15\) 0 0
\(16\) 0.738508 0.184627
\(17\) 4.33461 1.05130 0.525649 0.850701i \(-0.323822\pi\)
0.525649 + 0.850701i \(0.323822\pi\)
\(18\) 0 0
\(19\) 2.91501 0.668748 0.334374 0.942440i \(-0.391475\pi\)
0.334374 + 0.942440i \(0.391475\pi\)
\(20\) −0.310525 −0.0694355
\(21\) 0 0
\(22\) 0 0
\(23\) 3.89796 0.812780 0.406390 0.913700i \(-0.366787\pi\)
0.406390 + 0.913700i \(0.366787\pi\)
\(24\) 0 0
\(25\) −4.95049 −0.990097
\(26\) 5.07662 0.995607
\(27\) 0 0
\(28\) 1.39552 0.263728
\(29\) 3.77399 0.700812 0.350406 0.936598i \(-0.386044\pi\)
0.350406 + 0.936598i \(0.386044\pi\)
\(30\) 0 0
\(31\) −6.88958 −1.23741 −0.618703 0.785625i \(-0.712341\pi\)
−0.618703 + 0.785625i \(0.712341\pi\)
\(32\) 5.85410 1.03487
\(33\) 0 0
\(34\) 3.37009 0.577966
\(35\) −0.222516 −0.0376120
\(36\) 0 0
\(37\) −5.65351 −0.929432 −0.464716 0.885460i \(-0.653844\pi\)
−0.464716 + 0.885460i \(0.653844\pi\)
\(38\) 2.26637 0.367654
\(39\) 0 0
\(40\) −0.587433 −0.0928813
\(41\) −1.33811 −0.208978 −0.104489 0.994526i \(-0.533321\pi\)
−0.104489 + 0.994526i \(0.533321\pi\)
\(42\) 0 0
\(43\) −4.70820 −0.717994 −0.358997 0.933339i \(-0.616881\pi\)
−0.358997 + 0.933339i \(0.616881\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 3.03060 0.446838
\(47\) −6.04554 −0.881832 −0.440916 0.897548i \(-0.645346\pi\)
−0.440916 + 0.897548i \(0.645346\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) −3.84893 −0.544320
\(51\) 0 0
\(52\) −9.11210 −1.26362
\(53\) 1.71792 0.235974 0.117987 0.993015i \(-0.462356\pi\)
0.117987 + 0.993015i \(0.462356\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 2.63996 0.352780
\(57\) 0 0
\(58\) 2.93422 0.385281
\(59\) 9.53304 1.24110 0.620548 0.784168i \(-0.286910\pi\)
0.620548 + 0.784168i \(0.286910\pi\)
\(60\) 0 0
\(61\) −9.62943 −1.23292 −0.616461 0.787386i \(-0.711434\pi\)
−0.616461 + 0.787386i \(0.711434\pi\)
\(62\) −5.35654 −0.680281
\(63\) 0 0
\(64\) 3.07446 0.384307
\(65\) 1.45293 0.180213
\(66\) 0 0
\(67\) 1.27155 0.155344 0.0776722 0.996979i \(-0.475251\pi\)
0.0776722 + 0.996979i \(0.475251\pi\)
\(68\) −6.04903 −0.733553
\(69\) 0 0
\(70\) −0.173002 −0.0206778
\(71\) 9.30919 1.10480 0.552399 0.833580i \(-0.313713\pi\)
0.552399 + 0.833580i \(0.313713\pi\)
\(72\) 0 0
\(73\) 5.58911 0.654156 0.327078 0.944997i \(-0.393936\pi\)
0.327078 + 0.944997i \(0.393936\pi\)
\(74\) −4.39552 −0.510969
\(75\) 0 0
\(76\) −4.06794 −0.466625
\(77\) 0 0
\(78\) 0 0
\(79\) −4.52954 −0.509614 −0.254807 0.966992i \(-0.582012\pi\)
−0.254807 + 0.966992i \(0.582012\pi\)
\(80\) 0.164330 0.0183726
\(81\) 0 0
\(82\) −1.04036 −0.114888
\(83\) −11.2838 −1.23855 −0.619277 0.785173i \(-0.712574\pi\)
−0.619277 + 0.785173i \(0.712574\pi\)
\(84\) 0 0
\(85\) 0.964520 0.104617
\(86\) −3.66055 −0.394728
\(87\) 0 0
\(88\) 0 0
\(89\) −7.92157 −0.839684 −0.419842 0.907597i \(-0.637915\pi\)
−0.419842 + 0.907597i \(0.637915\pi\)
\(90\) 0 0
\(91\) −6.52954 −0.684482
\(92\) −5.43967 −0.567125
\(93\) 0 0
\(94\) −4.70031 −0.484800
\(95\) 0.648635 0.0665485
\(96\) 0 0
\(97\) −9.05391 −0.919285 −0.459643 0.888104i \(-0.652023\pi\)
−0.459643 + 0.888104i \(0.652023\pi\)
\(98\) 0.777484 0.0785378
\(99\) 0 0
\(100\) 6.90849 0.690849
\(101\) 19.2102 1.91148 0.955741 0.294208i \(-0.0950559\pi\)
0.955741 + 0.294208i \(0.0950559\pi\)
\(102\) 0 0
\(103\) 16.2383 1.60001 0.800004 0.599995i \(-0.204831\pi\)
0.800004 + 0.599995i \(0.204831\pi\)
\(104\) −17.2377 −1.69030
\(105\) 0 0
\(106\) 1.33565 0.129730
\(107\) 5.39336 0.521396 0.260698 0.965420i \(-0.416047\pi\)
0.260698 + 0.965420i \(0.416047\pi\)
\(108\) 0 0
\(109\) 5.39901 0.517132 0.258566 0.965994i \(-0.416750\pi\)
0.258566 + 0.965994i \(0.416750\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) −0.738508 −0.0697824
\(113\) 16.4962 1.55183 0.775917 0.630835i \(-0.217287\pi\)
0.775917 + 0.630835i \(0.217287\pi\)
\(114\) 0 0
\(115\) 0.867357 0.0808815
\(116\) −5.26667 −0.488998
\(117\) 0 0
\(118\) 7.41179 0.682310
\(119\) −4.33461 −0.397353
\(120\) 0 0
\(121\) 0 0
\(122\) −7.48673 −0.677816
\(123\) 0 0
\(124\) 9.61454 0.863411
\(125\) −2.21414 −0.198039
\(126\) 0 0
\(127\) 1.99728 0.177230 0.0886150 0.996066i \(-0.471756\pi\)
0.0886150 + 0.996066i \(0.471756\pi\)
\(128\) −9.31786 −0.823590
\(129\) 0 0
\(130\) 1.12963 0.0990749
\(131\) −1.37009 −0.119706 −0.0598528 0.998207i \(-0.519063\pi\)
−0.0598528 + 0.998207i \(0.519063\pi\)
\(132\) 0 0
\(133\) −2.91501 −0.252763
\(134\) 0.988609 0.0854028
\(135\) 0 0
\(136\) −11.4432 −0.981248
\(137\) −2.49924 −0.213525 −0.106762 0.994285i \(-0.534048\pi\)
−0.106762 + 0.994285i \(0.534048\pi\)
\(138\) 0 0
\(139\) −4.76260 −0.403958 −0.201979 0.979390i \(-0.564737\pi\)
−0.201979 + 0.979390i \(0.564737\pi\)
\(140\) 0.310525 0.0262441
\(141\) 0 0
\(142\) 7.23775 0.607378
\(143\) 0 0
\(144\) 0 0
\(145\) 0.839772 0.0697392
\(146\) 4.34545 0.359632
\(147\) 0 0
\(148\) 7.88958 0.648520
\(149\) 7.22985 0.592293 0.296146 0.955143i \(-0.404298\pi\)
0.296146 + 0.955143i \(0.404298\pi\)
\(150\) 0 0
\(151\) 8.13968 0.662398 0.331199 0.943561i \(-0.392547\pi\)
0.331199 + 0.943561i \(0.392547\pi\)
\(152\) −7.69551 −0.624188
\(153\) 0 0
\(154\) 0 0
\(155\) −1.53304 −0.123137
\(156\) 0 0
\(157\) 20.1514 1.60826 0.804129 0.594455i \(-0.202632\pi\)
0.804129 + 0.594455i \(0.202632\pi\)
\(158\) −3.52165 −0.280167
\(159\) 0 0
\(160\) 1.30263 0.102982
\(161\) −3.89796 −0.307202
\(162\) 0 0
\(163\) 7.43449 0.582315 0.291157 0.956675i \(-0.405960\pi\)
0.291157 + 0.956675i \(0.405960\pi\)
\(164\) 1.86736 0.145816
\(165\) 0 0
\(166\) −8.77295 −0.680913
\(167\) 2.21112 0.171102 0.0855510 0.996334i \(-0.472735\pi\)
0.0855510 + 0.996334i \(0.472735\pi\)
\(168\) 0 0
\(169\) 29.6349 2.27961
\(170\) 0.749899 0.0575146
\(171\) 0 0
\(172\) 6.57038 0.500987
\(173\) 10.3622 0.787823 0.393912 0.919148i \(-0.371122\pi\)
0.393912 + 0.919148i \(0.371122\pi\)
\(174\) 0 0
\(175\) 4.95049 0.374222
\(176\) 0 0
\(177\) 0 0
\(178\) −6.15889 −0.461629
\(179\) 23.4094 1.74970 0.874851 0.484392i \(-0.160959\pi\)
0.874851 + 0.484392i \(0.160959\pi\)
\(180\) 0 0
\(181\) −13.7161 −1.01951 −0.509755 0.860320i \(-0.670264\pi\)
−0.509755 + 0.860320i \(0.670264\pi\)
\(182\) −5.07662 −0.376304
\(183\) 0 0
\(184\) −10.2905 −0.758623
\(185\) −1.25800 −0.0924897
\(186\) 0 0
\(187\) 0 0
\(188\) 8.43666 0.615306
\(189\) 0 0
\(190\) 0.504303 0.0365860
\(191\) 12.0249 0.870094 0.435047 0.900408i \(-0.356732\pi\)
0.435047 + 0.900408i \(0.356732\pi\)
\(192\) 0 0
\(193\) 15.8839 1.14335 0.571675 0.820480i \(-0.306294\pi\)
0.571675 + 0.820480i \(0.306294\pi\)
\(194\) −7.03927 −0.505390
\(195\) 0 0
\(196\) −1.39552 −0.0996799
\(197\) −12.3035 −0.876590 −0.438295 0.898831i \(-0.644418\pi\)
−0.438295 + 0.898831i \(0.644418\pi\)
\(198\) 0 0
\(199\) 15.2615 1.08186 0.540929 0.841068i \(-0.318073\pi\)
0.540929 + 0.841068i \(0.318073\pi\)
\(200\) 13.0691 0.924125
\(201\) 0 0
\(202\) 14.9356 1.05087
\(203\) −3.77399 −0.264882
\(204\) 0 0
\(205\) −0.297751 −0.0207958
\(206\) 12.6250 0.879627
\(207\) 0 0
\(208\) 4.82212 0.334354
\(209\) 0 0
\(210\) 0 0
\(211\) 8.93344 0.615003 0.307502 0.951548i \(-0.400507\pi\)
0.307502 + 0.951548i \(0.400507\pi\)
\(212\) −2.39738 −0.164653
\(213\) 0 0
\(214\) 4.19325 0.286645
\(215\) −1.04765 −0.0714491
\(216\) 0 0
\(217\) 6.88958 0.467695
\(218\) 4.19765 0.284301
\(219\) 0 0
\(220\) 0 0
\(221\) 28.3031 1.90387
\(222\) 0 0
\(223\) 0.715244 0.0478963 0.0239481 0.999713i \(-0.492376\pi\)
0.0239481 + 0.999713i \(0.492376\pi\)
\(224\) −5.85410 −0.391144
\(225\) 0 0
\(226\) 12.8256 0.853143
\(227\) 5.32162 0.353208 0.176604 0.984282i \(-0.443489\pi\)
0.176604 + 0.984282i \(0.443489\pi\)
\(228\) 0 0
\(229\) −6.60832 −0.436690 −0.218345 0.975872i \(-0.570066\pi\)
−0.218345 + 0.975872i \(0.570066\pi\)
\(230\) 0.674356 0.0444657
\(231\) 0 0
\(232\) −9.96318 −0.654115
\(233\) 9.55713 0.626108 0.313054 0.949735i \(-0.398648\pi\)
0.313054 + 0.949735i \(0.398648\pi\)
\(234\) 0 0
\(235\) −1.34523 −0.0877529
\(236\) −13.3035 −0.865986
\(237\) 0 0
\(238\) −3.37009 −0.218451
\(239\) −26.8466 −1.73656 −0.868282 0.496071i \(-0.834776\pi\)
−0.868282 + 0.496071i \(0.834776\pi\)
\(240\) 0 0
\(241\) 18.8663 1.21529 0.607643 0.794210i \(-0.292115\pi\)
0.607643 + 0.794210i \(0.292115\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 13.4380 0.860282
\(245\) 0.222516 0.0142160
\(246\) 0 0
\(247\) 19.0337 1.21108
\(248\) 18.1882 1.15495
\(249\) 0 0
\(250\) −1.72146 −0.108875
\(251\) −29.3732 −1.85402 −0.927009 0.375040i \(-0.877629\pi\)
−0.927009 + 0.375040i \(0.877629\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 1.55285 0.0974348
\(255\) 0 0
\(256\) −13.3934 −0.837088
\(257\) 16.9164 1.05522 0.527608 0.849488i \(-0.323089\pi\)
0.527608 + 0.849488i \(0.323089\pi\)
\(258\) 0 0
\(259\) 5.65351 0.351292
\(260\) −2.02759 −0.125746
\(261\) 0 0
\(262\) −1.06523 −0.0658099
\(263\) 8.18034 0.504421 0.252211 0.967672i \(-0.418842\pi\)
0.252211 + 0.967672i \(0.418842\pi\)
\(264\) 0 0
\(265\) 0.382263 0.0234822
\(266\) −2.26637 −0.138960
\(267\) 0 0
\(268\) −1.77447 −0.108393
\(269\) 6.24716 0.380896 0.190448 0.981697i \(-0.439006\pi\)
0.190448 + 0.981697i \(0.439006\pi\)
\(270\) 0 0
\(271\) 7.86688 0.477879 0.238939 0.971034i \(-0.423200\pi\)
0.238939 + 0.971034i \(0.423200\pi\)
\(272\) 3.20115 0.194098
\(273\) 0 0
\(274\) −1.94312 −0.117388
\(275\) 0 0
\(276\) 0 0
\(277\) 12.0476 0.723873 0.361937 0.932203i \(-0.382116\pi\)
0.361937 + 0.932203i \(0.382116\pi\)
\(278\) −3.70284 −0.222082
\(279\) 0 0
\(280\) 0.587433 0.0351058
\(281\) −21.1549 −1.26200 −0.630998 0.775784i \(-0.717354\pi\)
−0.630998 + 0.775784i \(0.717354\pi\)
\(282\) 0 0
\(283\) −25.0034 −1.48630 −0.743148 0.669127i \(-0.766668\pi\)
−0.743148 + 0.669127i \(0.766668\pi\)
\(284\) −12.9911 −0.770883
\(285\) 0 0
\(286\) 0 0
\(287\) 1.33811 0.0789861
\(288\) 0 0
\(289\) 1.78888 0.105228
\(290\) 0.652909 0.0383402
\(291\) 0 0
\(292\) −7.79971 −0.456443
\(293\) 0.455087 0.0265865 0.0132932 0.999912i \(-0.495769\pi\)
0.0132932 + 0.999912i \(0.495769\pi\)
\(294\) 0 0
\(295\) 2.12125 0.123504
\(296\) 14.9251 0.867502
\(297\) 0 0
\(298\) 5.62110 0.325621
\(299\) 25.4519 1.47192
\(300\) 0 0
\(301\) 4.70820 0.271376
\(302\) 6.32848 0.364163
\(303\) 0 0
\(304\) 2.15275 0.123469
\(305\) −2.14270 −0.122691
\(306\) 0 0
\(307\) −8.03578 −0.458626 −0.229313 0.973353i \(-0.573648\pi\)
−0.229313 + 0.973353i \(0.573648\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) −1.19191 −0.0676962
\(311\) −0.136965 −0.00776656 −0.00388328 0.999992i \(-0.501236\pi\)
−0.00388328 + 0.999992i \(0.501236\pi\)
\(312\) 0 0
\(313\) −15.2899 −0.864238 −0.432119 0.901817i \(-0.642234\pi\)
−0.432119 + 0.901817i \(0.642234\pi\)
\(314\) 15.6674 0.884163
\(315\) 0 0
\(316\) 6.32106 0.355587
\(317\) 32.9925 1.85304 0.926522 0.376239i \(-0.122783\pi\)
0.926522 + 0.376239i \(0.122783\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0.684115 0.0382432
\(321\) 0 0
\(322\) −3.03060 −0.168889
\(323\) 12.6354 0.703054
\(324\) 0 0
\(325\) −32.3244 −1.79304
\(326\) 5.78020 0.320136
\(327\) 0 0
\(328\) 3.53256 0.195053
\(329\) 6.04554 0.333301
\(330\) 0 0
\(331\) 29.5335 1.62331 0.811653 0.584140i \(-0.198568\pi\)
0.811653 + 0.584140i \(0.198568\pi\)
\(332\) 15.7467 0.864212
\(333\) 0 0
\(334\) 1.71911 0.0940658
\(335\) 0.282939 0.0154586
\(336\) 0 0
\(337\) 5.63025 0.306699 0.153350 0.988172i \(-0.450994\pi\)
0.153350 + 0.988172i \(0.450994\pi\)
\(338\) 23.0407 1.25325
\(339\) 0 0
\(340\) −1.34600 −0.0729974
\(341\) 0 0
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) 12.4295 0.670153
\(345\) 0 0
\(346\) 8.05645 0.433117
\(347\) −8.73061 −0.468684 −0.234342 0.972154i \(-0.575294\pi\)
−0.234342 + 0.972154i \(0.575294\pi\)
\(348\) 0 0
\(349\) 18.4670 0.988514 0.494257 0.869316i \(-0.335440\pi\)
0.494257 + 0.869316i \(0.335440\pi\)
\(350\) 3.84893 0.205734
\(351\) 0 0
\(352\) 0 0
\(353\) 23.4857 1.25002 0.625009 0.780618i \(-0.285095\pi\)
0.625009 + 0.780618i \(0.285095\pi\)
\(354\) 0 0
\(355\) 2.07144 0.109941
\(356\) 11.0547 0.585897
\(357\) 0 0
\(358\) 18.2005 0.961924
\(359\) −14.8812 −0.785400 −0.392700 0.919667i \(-0.628459\pi\)
−0.392700 + 0.919667i \(0.628459\pi\)
\(360\) 0 0
\(361\) −10.5027 −0.552776
\(362\) −10.6641 −0.560490
\(363\) 0 0
\(364\) 9.11210 0.477604
\(365\) 1.24367 0.0650964
\(366\) 0 0
\(367\) −34.2259 −1.78658 −0.893289 0.449482i \(-0.851609\pi\)
−0.893289 + 0.449482i \(0.851609\pi\)
\(368\) 2.87867 0.150061
\(369\) 0 0
\(370\) −0.978072 −0.0508475
\(371\) −1.71792 −0.0891897
\(372\) 0 0
\(373\) 3.01739 0.156235 0.0781173 0.996944i \(-0.475109\pi\)
0.0781173 + 0.996944i \(0.475109\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 15.9600 0.823073
\(377\) 24.6424 1.26915
\(378\) 0 0
\(379\) −6.12431 −0.314585 −0.157292 0.987552i \(-0.550277\pi\)
−0.157292 + 0.987552i \(0.550277\pi\)
\(380\) −0.905182 −0.0464348
\(381\) 0 0
\(382\) 9.34920 0.478347
\(383\) −4.39098 −0.224369 −0.112184 0.993687i \(-0.535785\pi\)
−0.112184 + 0.993687i \(0.535785\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 12.3495 0.628573
\(387\) 0 0
\(388\) 12.6349 0.641440
\(389\) −10.4960 −0.532169 −0.266084 0.963950i \(-0.585730\pi\)
−0.266084 + 0.963950i \(0.585730\pi\)
\(390\) 0 0
\(391\) 16.8961 0.854475
\(392\) −2.63996 −0.133338
\(393\) 0 0
\(394\) −9.56580 −0.481918
\(395\) −1.00789 −0.0507127
\(396\) 0 0
\(397\) 11.3888 0.571589 0.285794 0.958291i \(-0.407743\pi\)
0.285794 + 0.958291i \(0.407743\pi\)
\(398\) 11.8656 0.594767
\(399\) 0 0
\(400\) −3.65597 −0.182799
\(401\) 4.72496 0.235953 0.117977 0.993016i \(-0.462359\pi\)
0.117977 + 0.993016i \(0.462359\pi\)
\(402\) 0 0
\(403\) −44.9858 −2.24090
\(404\) −26.8081 −1.33375
\(405\) 0 0
\(406\) −2.93422 −0.145623
\(407\) 0 0
\(408\) 0 0
\(409\) 2.46544 0.121908 0.0609541 0.998141i \(-0.480586\pi\)
0.0609541 + 0.998141i \(0.480586\pi\)
\(410\) −0.231496 −0.0114328
\(411\) 0 0
\(412\) −22.6609 −1.11642
\(413\) −9.53304 −0.469090
\(414\) 0 0
\(415\) −2.51082 −0.123251
\(416\) 38.2246 1.87412
\(417\) 0 0
\(418\) 0 0
\(419\) 14.3399 0.700548 0.350274 0.936647i \(-0.386088\pi\)
0.350274 + 0.936647i \(0.386088\pi\)
\(420\) 0 0
\(421\) −17.3157 −0.843918 −0.421959 0.906615i \(-0.638657\pi\)
−0.421959 + 0.906615i \(0.638657\pi\)
\(422\) 6.94561 0.338107
\(423\) 0 0
\(424\) −4.53523 −0.220250
\(425\) −21.4584 −1.04089
\(426\) 0 0
\(427\) 9.62943 0.466001
\(428\) −7.52653 −0.363808
\(429\) 0 0
\(430\) −0.814531 −0.0392802
\(431\) −27.8923 −1.34352 −0.671762 0.740767i \(-0.734462\pi\)
−0.671762 + 0.740767i \(0.734462\pi\)
\(432\) 0 0
\(433\) −29.5470 −1.41994 −0.709969 0.704232i \(-0.751291\pi\)
−0.709969 + 0.704232i \(0.751291\pi\)
\(434\) 5.35654 0.257122
\(435\) 0 0
\(436\) −7.53442 −0.360833
\(437\) 11.3626 0.543546
\(438\) 0 0
\(439\) −33.6655 −1.60677 −0.803384 0.595461i \(-0.796970\pi\)
−0.803384 + 0.595461i \(0.796970\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 22.0052 1.04668
\(443\) −10.3267 −0.490637 −0.245319 0.969443i \(-0.578893\pi\)
−0.245319 + 0.969443i \(0.578893\pi\)
\(444\) 0 0
\(445\) −1.76267 −0.0835587
\(446\) 0.556091 0.0263317
\(447\) 0 0
\(448\) −3.07446 −0.145254
\(449\) 2.75785 0.130151 0.0650756 0.997880i \(-0.479271\pi\)
0.0650756 + 0.997880i \(0.479271\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) −23.0208 −1.08281
\(453\) 0 0
\(454\) 4.13747 0.194181
\(455\) −1.45293 −0.0681142
\(456\) 0 0
\(457\) −40.3305 −1.88658 −0.943291 0.331968i \(-0.892287\pi\)
−0.943291 + 0.331968i \(0.892287\pi\)
\(458\) −5.13787 −0.240077
\(459\) 0 0
\(460\) −1.21041 −0.0564358
\(461\) 34.2251 1.59402 0.797011 0.603965i \(-0.206413\pi\)
0.797011 + 0.603965i \(0.206413\pi\)
\(462\) 0 0
\(463\) 0.707349 0.0328733 0.0164367 0.999865i \(-0.494768\pi\)
0.0164367 + 0.999865i \(0.494768\pi\)
\(464\) 2.78712 0.129389
\(465\) 0 0
\(466\) 7.43052 0.344212
\(467\) 28.5924 1.32310 0.661550 0.749901i \(-0.269899\pi\)
0.661550 + 0.749901i \(0.269899\pi\)
\(468\) 0 0
\(469\) −1.27155 −0.0587146
\(470\) −1.04589 −0.0482434
\(471\) 0 0
\(472\) −25.1669 −1.15840
\(473\) 0 0
\(474\) 0 0
\(475\) −14.4307 −0.662126
\(476\) 6.04903 0.277257
\(477\) 0 0
\(478\) −20.8728 −0.954701
\(479\) 5.09716 0.232895 0.116448 0.993197i \(-0.462849\pi\)
0.116448 + 0.993197i \(0.462849\pi\)
\(480\) 0 0
\(481\) −36.9149 −1.68317
\(482\) 14.6683 0.668121
\(483\) 0 0
\(484\) 0 0
\(485\) −2.01464 −0.0914800
\(486\) 0 0
\(487\) 29.1883 1.32265 0.661324 0.750101i \(-0.269995\pi\)
0.661324 + 0.750101i \(0.269995\pi\)
\(488\) 25.4213 1.15077
\(489\) 0 0
\(490\) 0.173002 0.00781546
\(491\) −30.1439 −1.36038 −0.680189 0.733037i \(-0.738102\pi\)
−0.680189 + 0.733037i \(0.738102\pi\)
\(492\) 0 0
\(493\) 16.3588 0.736762
\(494\) 14.7984 0.665810
\(495\) 0 0
\(496\) −5.08801 −0.228458
\(497\) −9.30919 −0.417574
\(498\) 0 0
\(499\) 5.95104 0.266405 0.133203 0.991089i \(-0.457474\pi\)
0.133203 + 0.991089i \(0.457474\pi\)
\(500\) 3.08987 0.138183
\(501\) 0 0
\(502\) −22.8372 −1.01927
\(503\) 4.97855 0.221983 0.110991 0.993821i \(-0.464597\pi\)
0.110991 + 0.993821i \(0.464597\pi\)
\(504\) 0 0
\(505\) 4.27456 0.190216
\(506\) 0 0
\(507\) 0 0
\(508\) −2.78724 −0.123664
\(509\) 21.3285 0.945370 0.472685 0.881231i \(-0.343285\pi\)
0.472685 + 0.881231i \(0.343285\pi\)
\(510\) 0 0
\(511\) −5.58911 −0.247248
\(512\) 8.22256 0.363389
\(513\) 0 0
\(514\) 13.1522 0.580120
\(515\) 3.61328 0.159220
\(516\) 0 0
\(517\) 0 0
\(518\) 4.39552 0.193128
\(519\) 0 0
\(520\) −3.83567 −0.168205
\(521\) 22.1383 0.969899 0.484949 0.874542i \(-0.338838\pi\)
0.484949 + 0.874542i \(0.338838\pi\)
\(522\) 0 0
\(523\) 4.91146 0.214763 0.107382 0.994218i \(-0.465753\pi\)
0.107382 + 0.994218i \(0.465753\pi\)
\(524\) 1.91199 0.0835257
\(525\) 0 0
\(526\) 6.36009 0.277313
\(527\) −29.8637 −1.30088
\(528\) 0 0
\(529\) −7.80592 −0.339388
\(530\) 0.297204 0.0129097
\(531\) 0 0
\(532\) 4.06794 0.176368
\(533\) −8.73725 −0.378452
\(534\) 0 0
\(535\) 1.20011 0.0518851
\(536\) −3.35684 −0.144993
\(537\) 0 0
\(538\) 4.85707 0.209403
\(539\) 0 0
\(540\) 0 0
\(541\) 19.3845 0.833404 0.416702 0.909043i \(-0.363186\pi\)
0.416702 + 0.909043i \(0.363186\pi\)
\(542\) 6.11637 0.262721
\(543\) 0 0
\(544\) 25.3753 1.08796
\(545\) 1.20137 0.0514609
\(546\) 0 0
\(547\) 14.4501 0.617840 0.308920 0.951088i \(-0.400032\pi\)
0.308920 + 0.951088i \(0.400032\pi\)
\(548\) 3.48774 0.148989
\(549\) 0 0
\(550\) 0 0
\(551\) 11.0012 0.468667
\(552\) 0 0
\(553\) 4.52954 0.192616
\(554\) 9.36686 0.397960
\(555\) 0 0
\(556\) 6.64629 0.281865
\(557\) −18.4708 −0.782634 −0.391317 0.920256i \(-0.627980\pi\)
−0.391317 + 0.920256i \(0.627980\pi\)
\(558\) 0 0
\(559\) −30.7424 −1.30027
\(560\) −0.164330 −0.00694419
\(561\) 0 0
\(562\) −16.4476 −0.693801
\(563\) −31.2838 −1.31846 −0.659228 0.751943i \(-0.729117\pi\)
−0.659228 + 0.751943i \(0.729117\pi\)
\(564\) 0 0
\(565\) 3.67067 0.154426
\(566\) −19.4397 −0.817112
\(567\) 0 0
\(568\) −24.5759 −1.03118
\(569\) −1.73605 −0.0727790 −0.0363895 0.999338i \(-0.511586\pi\)
−0.0363895 + 0.999338i \(0.511586\pi\)
\(570\) 0 0
\(571\) 12.5309 0.524403 0.262201 0.965013i \(-0.415551\pi\)
0.262201 + 0.965013i \(0.415551\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 1.04036 0.0434238
\(575\) −19.2968 −0.804732
\(576\) 0 0
\(577\) 20.1896 0.840505 0.420252 0.907407i \(-0.361942\pi\)
0.420252 + 0.907407i \(0.361942\pi\)
\(578\) 1.39082 0.0578506
\(579\) 0 0
\(580\) −1.17192 −0.0486612
\(581\) 11.2838 0.468129
\(582\) 0 0
\(583\) 0 0
\(584\) −14.7550 −0.610568
\(585\) 0 0
\(586\) 0.353823 0.0146163
\(587\) 0.00837574 0.000345704 0 0.000172852 1.00000i \(-0.499945\pi\)
0.000172852 1.00000i \(0.499945\pi\)
\(588\) 0 0
\(589\) −20.0832 −0.827513
\(590\) 1.64924 0.0678981
\(591\) 0 0
\(592\) −4.17516 −0.171598
\(593\) 0.439298 0.0180398 0.00901989 0.999959i \(-0.497129\pi\)
0.00901989 + 0.999959i \(0.497129\pi\)
\(594\) 0 0
\(595\) −0.964520 −0.0395415
\(596\) −10.0894 −0.413278
\(597\) 0 0
\(598\) 19.7884 0.809210
\(599\) −45.1664 −1.84545 −0.922724 0.385462i \(-0.874042\pi\)
−0.922724 + 0.385462i \(0.874042\pi\)
\(600\) 0 0
\(601\) 11.1437 0.454559 0.227280 0.973830i \(-0.427017\pi\)
0.227280 + 0.973830i \(0.427017\pi\)
\(602\) 3.66055 0.149193
\(603\) 0 0
\(604\) −11.3591 −0.462194
\(605\) 0 0
\(606\) 0 0
\(607\) −35.6998 −1.44901 −0.724504 0.689270i \(-0.757931\pi\)
−0.724504 + 0.689270i \(0.757931\pi\)
\(608\) 17.0647 0.692067
\(609\) 0 0
\(610\) −1.66591 −0.0674509
\(611\) −39.4746 −1.59697
\(612\) 0 0
\(613\) 17.1980 0.694620 0.347310 0.937750i \(-0.387095\pi\)
0.347310 + 0.937750i \(0.387095\pi\)
\(614\) −6.24769 −0.252136
\(615\) 0 0
\(616\) 0 0
\(617\) 16.8852 0.679774 0.339887 0.940466i \(-0.389611\pi\)
0.339887 + 0.940466i \(0.389611\pi\)
\(618\) 0 0
\(619\) 31.0692 1.24878 0.624389 0.781114i \(-0.285348\pi\)
0.624389 + 0.781114i \(0.285348\pi\)
\(620\) 2.13939 0.0859198
\(621\) 0 0
\(622\) −0.106488 −0.00426978
\(623\) 7.92157 0.317371
\(624\) 0 0
\(625\) 24.2598 0.970390
\(626\) −11.8877 −0.475127
\(627\) 0 0
\(628\) −28.1217 −1.12218
\(629\) −24.5058 −0.977110
\(630\) 0 0
\(631\) 7.25364 0.288763 0.144382 0.989522i \(-0.453881\pi\)
0.144382 + 0.989522i \(0.453881\pi\)
\(632\) 11.9578 0.475657
\(633\) 0 0
\(634\) 25.6512 1.01874
\(635\) 0.444427 0.0176365
\(636\) 0 0
\(637\) 6.52954 0.258710
\(638\) 0 0
\(639\) 0 0
\(640\) −2.07337 −0.0819572
\(641\) 20.7499 0.819572 0.409786 0.912182i \(-0.365603\pi\)
0.409786 + 0.912182i \(0.365603\pi\)
\(642\) 0 0
\(643\) 7.11127 0.280441 0.140221 0.990120i \(-0.455219\pi\)
0.140221 + 0.990120i \(0.455219\pi\)
\(644\) 5.43967 0.214353
\(645\) 0 0
\(646\) 9.82385 0.386514
\(647\) −27.5789 −1.08424 −0.542119 0.840302i \(-0.682378\pi\)
−0.542119 + 0.840302i \(0.682378\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) −25.1317 −0.985748
\(651\) 0 0
\(652\) −10.3750 −0.406315
\(653\) −16.1541 −0.632160 −0.316080 0.948732i \(-0.602367\pi\)
−0.316080 + 0.948732i \(0.602367\pi\)
\(654\) 0 0
\(655\) −0.304867 −0.0119122
\(656\) −0.988205 −0.0385829
\(657\) 0 0
\(658\) 4.70031 0.183237
\(659\) 13.2085 0.514531 0.257266 0.966341i \(-0.417178\pi\)
0.257266 + 0.966341i \(0.417178\pi\)
\(660\) 0 0
\(661\) −4.90660 −0.190845 −0.0954223 0.995437i \(-0.530420\pi\)
−0.0954223 + 0.995437i \(0.530420\pi\)
\(662\) 22.9618 0.892436
\(663\) 0 0
\(664\) 29.7887 1.15603
\(665\) −0.648635 −0.0251530
\(666\) 0 0
\(667\) 14.7108 0.569606
\(668\) −3.08566 −0.119388
\(669\) 0 0
\(670\) 0.219981 0.00849861
\(671\) 0 0
\(672\) 0 0
\(673\) −29.9872 −1.15592 −0.577960 0.816065i \(-0.696151\pi\)
−0.577960 + 0.816065i \(0.696151\pi\)
\(674\) 4.37743 0.168612
\(675\) 0 0
\(676\) −41.3561 −1.59062
\(677\) −12.5889 −0.483832 −0.241916 0.970297i \(-0.577776\pi\)
−0.241916 + 0.970297i \(0.577776\pi\)
\(678\) 0 0
\(679\) 9.05391 0.347457
\(680\) −2.54630 −0.0976460
\(681\) 0 0
\(682\) 0 0
\(683\) 28.5342 1.09183 0.545916 0.837840i \(-0.316182\pi\)
0.545916 + 0.837840i \(0.316182\pi\)
\(684\) 0 0
\(685\) −0.556120 −0.0212483
\(686\) −0.777484 −0.0296845
\(687\) 0 0
\(688\) −3.47704 −0.132561
\(689\) 11.2172 0.427341
\(690\) 0 0
\(691\) 25.3742 0.965281 0.482641 0.875818i \(-0.339678\pi\)
0.482641 + 0.875818i \(0.339678\pi\)
\(692\) −14.4606 −0.549711
\(693\) 0 0
\(694\) −6.78791 −0.257666
\(695\) −1.05975 −0.0401987
\(696\) 0 0
\(697\) −5.80019 −0.219698
\(698\) 14.3578 0.543450
\(699\) 0 0
\(700\) −6.90849 −0.261117
\(701\) 45.7021 1.72615 0.863073 0.505079i \(-0.168537\pi\)
0.863073 + 0.505079i \(0.168537\pi\)
\(702\) 0 0
\(703\) −16.4800 −0.621556
\(704\) 0 0
\(705\) 0 0
\(706\) 18.2598 0.687215
\(707\) −19.2102 −0.722473
\(708\) 0 0
\(709\) 38.5304 1.44704 0.723520 0.690304i \(-0.242523\pi\)
0.723520 + 0.690304i \(0.242523\pi\)
\(710\) 1.61051 0.0604415
\(711\) 0 0
\(712\) 20.9126 0.783734
\(713\) −26.8553 −1.00574
\(714\) 0 0
\(715\) 0 0
\(716\) −32.6683 −1.22087
\(717\) 0 0
\(718\) −11.5699 −0.431785
\(719\) −5.70213 −0.212653 −0.106327 0.994331i \(-0.533909\pi\)
−0.106327 + 0.994331i \(0.533909\pi\)
\(720\) 0 0
\(721\) −16.2383 −0.604746
\(722\) −8.16571 −0.303896
\(723\) 0 0
\(724\) 19.1411 0.711372
\(725\) −18.6831 −0.693872
\(726\) 0 0
\(727\) 11.8221 0.438458 0.219229 0.975673i \(-0.429646\pi\)
0.219229 + 0.975673i \(0.429646\pi\)
\(728\) 17.2377 0.638873
\(729\) 0 0
\(730\) 0.966931 0.0357877
\(731\) −20.4082 −0.754826
\(732\) 0 0
\(733\) 8.81193 0.325476 0.162738 0.986669i \(-0.447967\pi\)
0.162738 + 0.986669i \(0.447967\pi\)
\(734\) −26.6101 −0.982197
\(735\) 0 0
\(736\) 22.8190 0.841121
\(737\) 0 0
\(738\) 0 0
\(739\) 0.480809 0.0176868 0.00884342 0.999961i \(-0.497185\pi\)
0.00884342 + 0.999961i \(0.497185\pi\)
\(740\) 1.75556 0.0645355
\(741\) 0 0
\(742\) −1.33565 −0.0490333
\(743\) −33.2443 −1.21961 −0.609807 0.792550i \(-0.708753\pi\)
−0.609807 + 0.792550i \(0.708753\pi\)
\(744\) 0 0
\(745\) 1.60876 0.0589403
\(746\) 2.34598 0.0858923
\(747\) 0 0
\(748\) 0 0
\(749\) −5.39336 −0.197069
\(750\) 0 0
\(751\) 28.0066 1.02198 0.510988 0.859588i \(-0.329280\pi\)
0.510988 + 0.859588i \(0.329280\pi\)
\(752\) −4.46467 −0.162810
\(753\) 0 0
\(754\) 19.1591 0.697733
\(755\) 1.81121 0.0659166
\(756\) 0 0
\(757\) −22.2468 −0.808575 −0.404288 0.914632i \(-0.632481\pi\)
−0.404288 + 0.914632i \(0.632481\pi\)
\(758\) −4.76156 −0.172948
\(759\) 0 0
\(760\) −1.71237 −0.0621142
\(761\) 48.2254 1.74817 0.874085 0.485774i \(-0.161462\pi\)
0.874085 + 0.485774i \(0.161462\pi\)
\(762\) 0 0
\(763\) −5.39901 −0.195457
\(764\) −16.7810 −0.607116
\(765\) 0 0
\(766\) −3.41392 −0.123350
\(767\) 62.2464 2.24759
\(768\) 0 0
\(769\) 43.6883 1.57544 0.787721 0.616032i \(-0.211261\pi\)
0.787721 + 0.616032i \(0.211261\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −22.1663 −0.797783
\(773\) 0.585569 0.0210615 0.0105307 0.999945i \(-0.496648\pi\)
0.0105307 + 0.999945i \(0.496648\pi\)
\(774\) 0 0
\(775\) 34.1068 1.22515
\(776\) 23.9020 0.858031
\(777\) 0 0
\(778\) −8.16048 −0.292567
\(779\) −3.90060 −0.139753
\(780\) 0 0
\(781\) 0 0
\(782\) 13.1365 0.469760
\(783\) 0 0
\(784\) 0.738508 0.0263753
\(785\) 4.48401 0.160041
\(786\) 0 0
\(787\) 30.7166 1.09493 0.547464 0.836829i \(-0.315593\pi\)
0.547464 + 0.836829i \(0.315593\pi\)
\(788\) 17.1698 0.611649
\(789\) 0 0
\(790\) −0.783622 −0.0278800
\(791\) −16.4962 −0.586538
\(792\) 0 0
\(793\) −62.8758 −2.23278
\(794\) 8.85463 0.314239
\(795\) 0 0
\(796\) −21.2977 −0.754877
\(797\) 10.8333 0.383735 0.191867 0.981421i \(-0.438546\pi\)
0.191867 + 0.981421i \(0.438546\pi\)
\(798\) 0 0
\(799\) −26.2051 −0.927068
\(800\) −28.9807 −1.02462
\(801\) 0 0
\(802\) 3.67358 0.129719
\(803\) 0 0
\(804\) 0 0
\(805\) −0.867357 −0.0305703
\(806\) −34.9758 −1.23197
\(807\) 0 0
\(808\) −50.7141 −1.78412
\(809\) 38.5901 1.35675 0.678377 0.734714i \(-0.262684\pi\)
0.678377 + 0.734714i \(0.262684\pi\)
\(810\) 0 0
\(811\) −50.8935 −1.78711 −0.893556 0.448951i \(-0.851798\pi\)
−0.893556 + 0.448951i \(0.851798\pi\)
\(812\) 5.26667 0.184824
\(813\) 0 0
\(814\) 0 0
\(815\) 1.65429 0.0579473
\(816\) 0 0
\(817\) −13.7244 −0.480158
\(818\) 1.91684 0.0670208
\(819\) 0 0
\(820\) 0.415516 0.0145105
\(821\) 40.3094 1.40681 0.703404 0.710790i \(-0.251662\pi\)
0.703404 + 0.710790i \(0.251662\pi\)
\(822\) 0 0
\(823\) 25.6817 0.895209 0.447605 0.894232i \(-0.352277\pi\)
0.447605 + 0.894232i \(0.352277\pi\)
\(824\) −42.8685 −1.49340
\(825\) 0 0
\(826\) −7.41179 −0.257889
\(827\) 31.9557 1.11121 0.555604 0.831447i \(-0.312487\pi\)
0.555604 + 0.831447i \(0.312487\pi\)
\(828\) 0 0
\(829\) 48.4002 1.68101 0.840504 0.541805i \(-0.182259\pi\)
0.840504 + 0.541805i \(0.182259\pi\)
\(830\) −1.95212 −0.0677591
\(831\) 0 0
\(832\) 20.0748 0.695969
\(833\) 4.33461 0.150185
\(834\) 0 0
\(835\) 0.492010 0.0170267
\(836\) 0 0
\(837\) 0 0
\(838\) 11.1490 0.385137
\(839\) 37.7165 1.30212 0.651059 0.759027i \(-0.274325\pi\)
0.651059 + 0.759027i \(0.274325\pi\)
\(840\) 0 0
\(841\) −14.7570 −0.508863
\(842\) −13.4627 −0.463956
\(843\) 0 0
\(844\) −12.4668 −0.429124
\(845\) 6.59424 0.226849
\(846\) 0 0
\(847\) 0 0
\(848\) 1.26869 0.0435671
\(849\) 0 0
\(850\) −16.6836 −0.572243
\(851\) −22.0372 −0.755424
\(852\) 0 0
\(853\) 9.28864 0.318037 0.159019 0.987276i \(-0.449167\pi\)
0.159019 + 0.987276i \(0.449167\pi\)
\(854\) 7.48673 0.256191
\(855\) 0 0
\(856\) −14.2383 −0.486654
\(857\) 29.7644 1.01673 0.508365 0.861141i \(-0.330250\pi\)
0.508365 + 0.861141i \(0.330250\pi\)
\(858\) 0 0
\(859\) 33.2611 1.13485 0.567427 0.823424i \(-0.307939\pi\)
0.567427 + 0.823424i \(0.307939\pi\)
\(860\) 1.46201 0.0498543
\(861\) 0 0
\(862\) −21.6858 −0.738622
\(863\) 18.5677 0.632051 0.316025 0.948751i \(-0.397652\pi\)
0.316025 + 0.948751i \(0.397652\pi\)
\(864\) 0 0
\(865\) 2.30575 0.0783979
\(866\) −22.9723 −0.780632
\(867\) 0 0
\(868\) −9.61454 −0.326339
\(869\) 0 0
\(870\) 0 0
\(871\) 8.30263 0.281324
\(872\) −14.2532 −0.482674
\(873\) 0 0
\(874\) 8.83422 0.298822
\(875\) 2.21414 0.0748516
\(876\) 0 0
\(877\) −22.4921 −0.759505 −0.379752 0.925088i \(-0.623991\pi\)
−0.379752 + 0.925088i \(0.623991\pi\)
\(878\) −26.1744 −0.883344
\(879\) 0 0
\(880\) 0 0
\(881\) −7.06565 −0.238048 −0.119024 0.992891i \(-0.537977\pi\)
−0.119024 + 0.992891i \(0.537977\pi\)
\(882\) 0 0
\(883\) −19.9382 −0.670974 −0.335487 0.942045i \(-0.608901\pi\)
−0.335487 + 0.942045i \(0.608901\pi\)
\(884\) −39.4974 −1.32844
\(885\) 0 0
\(886\) −8.02886 −0.269735
\(887\) −6.95481 −0.233520 −0.116760 0.993160i \(-0.537251\pi\)
−0.116760 + 0.993160i \(0.537251\pi\)
\(888\) 0 0
\(889\) −1.99728 −0.0669867
\(890\) −1.37045 −0.0459376
\(891\) 0 0
\(892\) −0.998136 −0.0334201
\(893\) −17.6228 −0.589724
\(894\) 0 0
\(895\) 5.20896 0.174116
\(896\) 9.31786 0.311288
\(897\) 0 0
\(898\) 2.14419 0.0715525
\(899\) −26.0012 −0.867189
\(900\) 0 0
\(901\) 7.44650 0.248079
\(902\) 0 0
\(903\) 0 0
\(904\) −43.5494 −1.44843
\(905\) −3.05205 −0.101454
\(906\) 0 0
\(907\) −23.4401 −0.778317 −0.389159 0.921171i \(-0.627234\pi\)
−0.389159 + 0.921171i \(0.627234\pi\)
\(908\) −7.42642 −0.246454
\(909\) 0 0
\(910\) −1.12963 −0.0374468
\(911\) −46.8096 −1.55087 −0.775436 0.631426i \(-0.782470\pi\)
−0.775436 + 0.631426i \(0.782470\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) −31.3563 −1.03718
\(915\) 0 0
\(916\) 9.22203 0.304705
\(917\) 1.37009 0.0452445
\(918\) 0 0
\(919\) 10.6273 0.350563 0.175281 0.984518i \(-0.443916\pi\)
0.175281 + 0.984518i \(0.443916\pi\)
\(920\) −2.28979 −0.0754921
\(921\) 0 0
\(922\) 26.6095 0.876336
\(923\) 60.7848 2.00075
\(924\) 0 0
\(925\) 27.9876 0.920228
\(926\) 0.549953 0.0180726
\(927\) 0 0
\(928\) 22.0933 0.725248
\(929\) 36.6710 1.20313 0.601567 0.798822i \(-0.294543\pi\)
0.601567 + 0.798822i \(0.294543\pi\)
\(930\) 0 0
\(931\) 2.91501 0.0955355
\(932\) −13.3371 −0.436873
\(933\) 0 0
\(934\) 22.2302 0.727393
\(935\) 0 0
\(936\) 0 0
\(937\) −40.2174 −1.31384 −0.656922 0.753959i \(-0.728142\pi\)
−0.656922 + 0.753959i \(0.728142\pi\)
\(938\) −0.988609 −0.0322792
\(939\) 0 0
\(940\) 1.87729 0.0612304
\(941\) −24.9097 −0.812032 −0.406016 0.913866i \(-0.633082\pi\)
−0.406016 + 0.913866i \(0.633082\pi\)
\(942\) 0 0
\(943\) −5.21590 −0.169853
\(944\) 7.04022 0.229140
\(945\) 0 0
\(946\) 0 0
\(947\) 32.2061 1.04656 0.523279 0.852161i \(-0.324708\pi\)
0.523279 + 0.852161i \(0.324708\pi\)
\(948\) 0 0
\(949\) 36.4944 1.18466
\(950\) −11.2196 −0.364013
\(951\) 0 0
\(952\) 11.4432 0.370877
\(953\) −45.0611 −1.45967 −0.729837 0.683621i \(-0.760404\pi\)
−0.729837 + 0.683621i \(0.760404\pi\)
\(954\) 0 0
\(955\) 2.67574 0.0865849
\(956\) 37.4650 1.21170
\(957\) 0 0
\(958\) 3.96296 0.128038
\(959\) 2.49924 0.0807047
\(960\) 0 0
\(961\) 16.4663 0.531172
\(962\) −28.7007 −0.925349
\(963\) 0 0
\(964\) −26.3283 −0.847977
\(965\) 3.53442 0.113777
\(966\) 0 0
\(967\) 1.81387 0.0583300 0.0291650 0.999575i \(-0.490715\pi\)
0.0291650 + 0.999575i \(0.490715\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) −1.56635 −0.0502924
\(971\) −23.0539 −0.739835 −0.369918 0.929065i \(-0.620614\pi\)
−0.369918 + 0.929065i \(0.620614\pi\)
\(972\) 0 0
\(973\) 4.76260 0.152682
\(974\) 22.6934 0.727144
\(975\) 0 0
\(976\) −7.11140 −0.227630
\(977\) 7.09449 0.226973 0.113486 0.993540i \(-0.463798\pi\)
0.113486 + 0.993540i \(0.463798\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) −0.310525 −0.00991935
\(981\) 0 0
\(982\) −23.4364 −0.747887
\(983\) −38.3502 −1.22318 −0.611591 0.791174i \(-0.709470\pi\)
−0.611591 + 0.791174i \(0.709470\pi\)
\(984\) 0 0
\(985\) −2.73773 −0.0872313
\(986\) 12.7187 0.405046
\(987\) 0 0
\(988\) −26.5618 −0.845044
\(989\) −18.3524 −0.583572
\(990\) 0 0
\(991\) 20.2722 0.643967 0.321984 0.946745i \(-0.395650\pi\)
0.321984 + 0.946745i \(0.395650\pi\)
\(992\) −40.3323 −1.28055
\(993\) 0 0
\(994\) −7.23775 −0.229567
\(995\) 3.39592 0.107658
\(996\) 0 0
\(997\) 15.4107 0.488062 0.244031 0.969767i \(-0.421530\pi\)
0.244031 + 0.969767i \(0.421530\pi\)
\(998\) 4.62684 0.146460
\(999\) 0 0
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 7623.2.a.ch.1.3 4
3.2 odd 2 847.2.a.l.1.2 4
11.3 even 5 693.2.m.g.64.2 8
11.4 even 5 693.2.m.g.379.2 8
11.10 odd 2 7623.2.a.co.1.2 4
21.20 even 2 5929.2.a.bi.1.2 4
33.2 even 10 847.2.f.s.323.1 8
33.5 odd 10 847.2.f.p.729.2 8
33.8 even 10 847.2.f.q.372.2 8
33.14 odd 10 77.2.f.a.64.1 8
33.17 even 10 847.2.f.s.729.1 8
33.20 odd 10 847.2.f.p.323.2 8
33.26 odd 10 77.2.f.a.71.1 yes 8
33.29 even 10 847.2.f.q.148.2 8
33.32 even 2 847.2.a.k.1.3 4
231.26 even 30 539.2.q.b.214.1 16
231.47 even 30 539.2.q.b.361.2 16
231.59 even 30 539.2.q.b.324.2 16
231.80 even 30 539.2.q.b.471.1 16
231.125 even 10 539.2.f.d.148.1 8
231.146 even 10 539.2.f.d.295.1 8
231.158 odd 30 539.2.q.c.324.2 16
231.179 odd 30 539.2.q.c.471.1 16
231.191 odd 30 539.2.q.c.214.1 16
231.212 odd 30 539.2.q.c.361.2 16
231.230 odd 2 5929.2.a.bb.1.3 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
77.2.f.a.64.1 8 33.14 odd 10
77.2.f.a.71.1 yes 8 33.26 odd 10
539.2.f.d.148.1 8 231.125 even 10
539.2.f.d.295.1 8 231.146 even 10
539.2.q.b.214.1 16 231.26 even 30
539.2.q.b.324.2 16 231.59 even 30
539.2.q.b.361.2 16 231.47 even 30
539.2.q.b.471.1 16 231.80 even 30
539.2.q.c.214.1 16 231.191 odd 30
539.2.q.c.324.2 16 231.158 odd 30
539.2.q.c.361.2 16 231.212 odd 30
539.2.q.c.471.1 16 231.179 odd 30
693.2.m.g.64.2 8 11.3 even 5
693.2.m.g.379.2 8 11.4 even 5
847.2.a.k.1.3 4 33.32 even 2
847.2.a.l.1.2 4 3.2 odd 2
847.2.f.p.323.2 8 33.20 odd 10
847.2.f.p.729.2 8 33.5 odd 10
847.2.f.q.148.2 8 33.29 even 10
847.2.f.q.372.2 8 33.8 even 10
847.2.f.s.323.1 8 33.2 even 10
847.2.f.s.729.1 8 33.17 even 10
5929.2.a.bb.1.3 4 231.230 odd 2
5929.2.a.bi.1.2 4 21.20 even 2
7623.2.a.ch.1.3 4 1.1 even 1 trivial
7623.2.a.co.1.2 4 11.10 odd 2