Properties

Label 7623.2.a.cl.1.1
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: 4.4.725.1
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 231)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(2.09529\)
Character \(\chi\) = 7623.1

$q$-expansion

\(f(q)\) \(=\) \(q-2.39026 q^{2} +3.71333 q^{4} +2.58993 q^{5} +1.00000 q^{7} -4.09529 q^{8} +O(q^{10})\) \(q-2.39026 q^{2} +3.71333 q^{4} +2.58993 q^{5} +1.00000 q^{7} -4.09529 q^{8} -6.19059 q^{10} -6.78051 q^{13} -2.39026 q^{14} +2.36215 q^{16} -3.14511 q^{17} +1.04548 q^{19} +9.61724 q^{20} +6.52707 q^{23} +1.70772 q^{25} +16.2072 q^{26} +3.71333 q^{28} -0.607298 q^{29} -8.83673 q^{31} +2.54445 q^{32} +7.51762 q^{34} +2.58993 q^{35} +8.94299 q^{37} -2.49897 q^{38} -10.6065 q^{40} +8.69747 q^{41} -4.48159 q^{43} -15.6014 q^{46} -3.26290 q^{47} +1.00000 q^{49} -4.08188 q^{50} -25.1783 q^{52} -1.89958 q^{53} -4.09529 q^{56} +1.45160 q^{58} +0.174006 q^{59} -8.13437 q^{61} +21.1221 q^{62} -10.8062 q^{64} -17.5610 q^{65} -7.33649 q^{67} -11.6788 q^{68} -6.19059 q^{70} -2.70693 q^{71} -0.589926 q^{73} -21.3761 q^{74} +3.88221 q^{76} +9.80862 q^{79} +6.11779 q^{80} -20.7892 q^{82} +8.74577 q^{83} -8.14560 q^{85} +10.7122 q^{86} +11.6788 q^{89} -6.78051 q^{91} +24.2372 q^{92} +7.79916 q^{94} +2.70772 q^{95} +11.8528 q^{97} -2.39026 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + q^{2} + 3q^{4} + 4q^{5} + 4q^{7} - 9q^{8} + O(q^{10}) \) \( 4q + q^{2} + 3q^{4} + 4q^{5} + 4q^{7} - 9q^{8} - 10q^{10} - 6q^{13} + q^{14} - 3q^{16} + 8q^{17} + 10q^{19} + 10q^{23} + 12q^{25} + 20q^{26} + 3q^{28} - 18q^{31} - 2q^{32} + 18q^{34} + 4q^{35} - 2q^{37} + 8q^{38} - 6q^{40} + 10q^{41} + 4q^{43} - 11q^{46} - 4q^{47} + 4q^{49} - 9q^{50} - 20q^{52} - 9q^{56} + 14q^{58} + 16q^{59} - 14q^{61} - 11q^{64} - 28q^{65} - 28q^{67} - 16q^{68} - 10q^{70} + 18q^{71} + 4q^{73} - 41q^{74} + 4q^{76} + 20q^{79} + 36q^{80} - 24q^{82} + 6q^{83} + 20q^{85} + 20q^{86} + 16q^{89} - 6q^{91} + 22q^{92} + 16q^{94} + 16q^{95} + 32q^{97} + q^{98} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.39026 −1.69017 −0.845083 0.534634i \(-0.820449\pi\)
−0.845083 + 0.534634i \(0.820449\pi\)
\(3\) 0 0
\(4\) 3.71333 1.85666
\(5\) 2.58993 1.15825 0.579125 0.815239i \(-0.303394\pi\)
0.579125 + 0.815239i \(0.303394\pi\)
\(6\) 0 0
\(7\) 1.00000 0.377964
\(8\) −4.09529 −1.44791
\(9\) 0 0
\(10\) −6.19059 −1.95764
\(11\) 0 0
\(12\) 0 0
\(13\) −6.78051 −1.88058 −0.940288 0.340380i \(-0.889444\pi\)
−0.940288 + 0.340380i \(0.889444\pi\)
\(14\) −2.39026 −0.638823
\(15\) 0 0
\(16\) 2.36215 0.590537
\(17\) −3.14511 −0.762801 −0.381400 0.924410i \(-0.624558\pi\)
−0.381400 + 0.924410i \(0.624558\pi\)
\(18\) 0 0
\(19\) 1.04548 0.239850 0.119925 0.992783i \(-0.461735\pi\)
0.119925 + 0.992783i \(0.461735\pi\)
\(20\) 9.61724 2.15048
\(21\) 0 0
\(22\) 0 0
\(23\) 6.52707 1.36099 0.680495 0.732753i \(-0.261765\pi\)
0.680495 + 0.732753i \(0.261765\pi\)
\(24\) 0 0
\(25\) 1.70772 0.341543
\(26\) 16.2072 3.17849
\(27\) 0 0
\(28\) 3.71333 0.701753
\(29\) −0.607298 −0.112772 −0.0563862 0.998409i \(-0.517958\pi\)
−0.0563862 + 0.998409i \(0.517958\pi\)
\(30\) 0 0
\(31\) −8.83673 −1.58712 −0.793562 0.608490i \(-0.791776\pi\)
−0.793562 + 0.608490i \(0.791776\pi\)
\(32\) 2.54445 0.449799
\(33\) 0 0
\(34\) 7.51762 1.28926
\(35\) 2.58993 0.437777
\(36\) 0 0
\(37\) 8.94299 1.47022 0.735110 0.677948i \(-0.237131\pi\)
0.735110 + 0.677948i \(0.237131\pi\)
\(38\) −2.49897 −0.405386
\(39\) 0 0
\(40\) −10.6065 −1.67704
\(41\) 8.69747 1.35832 0.679158 0.733992i \(-0.262345\pi\)
0.679158 + 0.733992i \(0.262345\pi\)
\(42\) 0 0
\(43\) −4.48159 −0.683437 −0.341718 0.939802i \(-0.611009\pi\)
−0.341718 + 0.939802i \(0.611009\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) −15.6014 −2.30030
\(47\) −3.26290 −0.475943 −0.237971 0.971272i \(-0.576482\pi\)
−0.237971 + 0.971272i \(0.576482\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) −4.08188 −0.577265
\(51\) 0 0
\(52\) −25.1783 −3.49160
\(53\) −1.89958 −0.260928 −0.130464 0.991453i \(-0.541647\pi\)
−0.130464 + 0.991453i \(0.541647\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −4.09529 −0.547257
\(57\) 0 0
\(58\) 1.45160 0.190604
\(59\) 0.174006 0.0226537 0.0113268 0.999936i \(-0.496394\pi\)
0.0113268 + 0.999936i \(0.496394\pi\)
\(60\) 0 0
\(61\) −8.13437 −1.04150 −0.520750 0.853709i \(-0.674348\pi\)
−0.520750 + 0.853709i \(0.674348\pi\)
\(62\) 21.1221 2.68250
\(63\) 0 0
\(64\) −10.8062 −1.35077
\(65\) −17.5610 −2.17818
\(66\) 0 0
\(67\) −7.33649 −0.896294 −0.448147 0.893960i \(-0.647916\pi\)
−0.448147 + 0.893960i \(0.647916\pi\)
\(68\) −11.6788 −1.41626
\(69\) 0 0
\(70\) −6.19059 −0.739917
\(71\) −2.70693 −0.321253 −0.160626 0.987015i \(-0.551351\pi\)
−0.160626 + 0.987015i \(0.551351\pi\)
\(72\) 0 0
\(73\) −0.589926 −0.0690456 −0.0345228 0.999404i \(-0.510991\pi\)
−0.0345228 + 0.999404i \(0.510991\pi\)
\(74\) −21.3761 −2.48492
\(75\) 0 0
\(76\) 3.88221 0.445320
\(77\) 0 0
\(78\) 0 0
\(79\) 9.80862 1.10356 0.551778 0.833991i \(-0.313950\pi\)
0.551778 + 0.833991i \(0.313950\pi\)
\(80\) 6.11779 0.683990
\(81\) 0 0
\(82\) −20.7892 −2.29578
\(83\) 8.74577 0.959973 0.479986 0.877276i \(-0.340642\pi\)
0.479986 + 0.877276i \(0.340642\pi\)
\(84\) 0 0
\(85\) −8.14560 −0.883514
\(86\) 10.7122 1.15512
\(87\) 0 0
\(88\) 0 0
\(89\) 11.6788 1.23795 0.618976 0.785410i \(-0.287548\pi\)
0.618976 + 0.785410i \(0.287548\pi\)
\(90\) 0 0
\(91\) −6.78051 −0.710791
\(92\) 24.2372 2.52690
\(93\) 0 0
\(94\) 7.79916 0.804422
\(95\) 2.70772 0.277806
\(96\) 0 0
\(97\) 11.8528 1.20347 0.601736 0.798695i \(-0.294476\pi\)
0.601736 + 0.798695i \(0.294476\pi\)
\(98\) −2.39026 −0.241452
\(99\) 0 0
\(100\) 6.34131 0.634131
\(101\) 3.71845 0.370000 0.185000 0.982739i \(-0.440772\pi\)
0.185000 + 0.982739i \(0.440772\pi\)
\(102\) 0 0
\(103\) 7.32703 0.721954 0.360977 0.932575i \(-0.382443\pi\)
0.360977 + 0.932575i \(0.382443\pi\)
\(104\) 27.7682 2.72290
\(105\) 0 0
\(106\) 4.54049 0.441011
\(107\) 13.3804 1.29353 0.646765 0.762689i \(-0.276121\pi\)
0.646765 + 0.762689i \(0.276121\pi\)
\(108\) 0 0
\(109\) −14.1228 −1.35272 −0.676362 0.736570i \(-0.736444\pi\)
−0.676362 + 0.736570i \(0.736444\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 2.36215 0.223202
\(113\) 2.66351 0.250562 0.125281 0.992121i \(-0.460017\pi\)
0.125281 + 0.992121i \(0.460017\pi\)
\(114\) 0 0
\(115\) 16.9046 1.57637
\(116\) −2.25510 −0.209380
\(117\) 0 0
\(118\) −0.415920 −0.0382885
\(119\) −3.14511 −0.288312
\(120\) 0 0
\(121\) 0 0
\(122\) 19.4432 1.76031
\(123\) 0 0
\(124\) −32.8137 −2.94676
\(125\) −8.52677 −0.762658
\(126\) 0 0
\(127\) 10.3633 0.919596 0.459798 0.888024i \(-0.347922\pi\)
0.459798 + 0.888024i \(0.347922\pi\)
\(128\) 20.7406 1.83323
\(129\) 0 0
\(130\) 41.9754 3.68148
\(131\) −11.6441 −1.01735 −0.508674 0.860959i \(-0.669864\pi\)
−0.508674 + 0.860959i \(0.669864\pi\)
\(132\) 0 0
\(133\) 1.04548 0.0906546
\(134\) 17.5361 1.51489
\(135\) 0 0
\(136\) 12.8801 1.10446
\(137\) 15.4895 1.32336 0.661679 0.749787i \(-0.269844\pi\)
0.661679 + 0.749787i \(0.269844\pi\)
\(138\) 0 0
\(139\) 5.81808 0.493483 0.246742 0.969081i \(-0.420640\pi\)
0.246742 + 0.969081i \(0.420640\pi\)
\(140\) 9.61724 0.812805
\(141\) 0 0
\(142\) 6.47025 0.542971
\(143\) 0 0
\(144\) 0 0
\(145\) −1.57286 −0.130619
\(146\) 1.41007 0.116699
\(147\) 0 0
\(148\) 33.2083 2.72970
\(149\) −14.4143 −1.18087 −0.590434 0.807086i \(-0.701044\pi\)
−0.590434 + 0.807086i \(0.701044\pi\)
\(150\) 0 0
\(151\) 22.2171 1.80800 0.904002 0.427529i \(-0.140616\pi\)
0.904002 + 0.427529i \(0.140616\pi\)
\(152\) −4.28155 −0.347279
\(153\) 0 0
\(154\) 0 0
\(155\) −22.8865 −1.83829
\(156\) 0 0
\(157\) −9.79125 −0.781427 −0.390713 0.920512i \(-0.627772\pi\)
−0.390713 + 0.920512i \(0.627772\pi\)
\(158\) −23.4451 −1.86519
\(159\) 0 0
\(160\) 6.58993 0.520979
\(161\) 6.52707 0.514405
\(162\) 0 0
\(163\) 11.3638 0.890082 0.445041 0.895510i \(-0.353189\pi\)
0.445041 + 0.895510i \(0.353189\pi\)
\(164\) 32.2965 2.52194
\(165\) 0 0
\(166\) −20.9046 −1.62251
\(167\) −5.08889 −0.393790 −0.196895 0.980425i \(-0.563086\pi\)
−0.196895 + 0.980425i \(0.563086\pi\)
\(168\) 0 0
\(169\) 32.9754 2.53657
\(170\) 19.4701 1.49329
\(171\) 0 0
\(172\) −16.6416 −1.26891
\(173\) 7.77467 0.591097 0.295549 0.955328i \(-0.404498\pi\)
0.295549 + 0.955328i \(0.404498\pi\)
\(174\) 0 0
\(175\) 1.70772 0.129091
\(176\) 0 0
\(177\) 0 0
\(178\) −27.9154 −2.09235
\(179\) 6.14590 0.459366 0.229683 0.973265i \(-0.426231\pi\)
0.229683 + 0.973265i \(0.426231\pi\)
\(180\) 0 0
\(181\) −9.25801 −0.688142 −0.344071 0.938944i \(-0.611806\pi\)
−0.344071 + 0.938944i \(0.611806\pi\)
\(182\) 16.2072 1.20136
\(183\) 0 0
\(184\) −26.7303 −1.97058
\(185\) 23.1617 1.70288
\(186\) 0 0
\(187\) 0 0
\(188\) −12.1162 −0.883665
\(189\) 0 0
\(190\) −6.47214 −0.469538
\(191\) 14.6374 1.05913 0.529564 0.848270i \(-0.322356\pi\)
0.529564 + 0.848270i \(0.322356\pi\)
\(192\) 0 0
\(193\) 16.4713 1.18563 0.592817 0.805337i \(-0.298016\pi\)
0.592817 + 0.805337i \(0.298016\pi\)
\(194\) −28.3313 −2.03407
\(195\) 0 0
\(196\) 3.71333 0.265238
\(197\) −6.81654 −0.485658 −0.242829 0.970069i \(-0.578075\pi\)
−0.242829 + 0.970069i \(0.578075\pi\)
\(198\) 0 0
\(199\) 15.5561 1.10275 0.551373 0.834259i \(-0.314104\pi\)
0.551373 + 0.834259i \(0.314104\pi\)
\(200\) −6.99360 −0.494522
\(201\) 0 0
\(202\) −8.88806 −0.625361
\(203\) −0.607298 −0.0426239
\(204\) 0 0
\(205\) 22.5258 1.57327
\(206\) −17.5135 −1.22022
\(207\) 0 0
\(208\) −16.0166 −1.11055
\(209\) 0 0
\(210\) 0 0
\(211\) −1.56464 −0.107714 −0.0538571 0.998549i \(-0.517152\pi\)
−0.0538571 + 0.998549i \(0.517152\pi\)
\(212\) −7.05377 −0.484455
\(213\) 0 0
\(214\) −31.9826 −2.18628
\(215\) −11.6070 −0.791591
\(216\) 0 0
\(217\) −8.83673 −0.599876
\(218\) 33.7572 2.28633
\(219\) 0 0
\(220\) 0 0
\(221\) 21.3254 1.43450
\(222\) 0 0
\(223\) 4.13926 0.277186 0.138593 0.990349i \(-0.455742\pi\)
0.138593 + 0.990349i \(0.455742\pi\)
\(224\) 2.54445 0.170008
\(225\) 0 0
\(226\) −6.36648 −0.423492
\(227\) −5.57761 −0.370199 −0.185099 0.982720i \(-0.559261\pi\)
−0.185099 + 0.982720i \(0.559261\pi\)
\(228\) 0 0
\(229\) −0.788428 −0.0521008 −0.0260504 0.999661i \(-0.508293\pi\)
−0.0260504 + 0.999661i \(0.508293\pi\)
\(230\) −40.4064 −2.66432
\(231\) 0 0
\(232\) 2.48706 0.163284
\(233\) 27.7066 1.81512 0.907561 0.419921i \(-0.137942\pi\)
0.907561 + 0.419921i \(0.137942\pi\)
\(234\) 0 0
\(235\) −8.45066 −0.551260
\(236\) 0.646142 0.0420603
\(237\) 0 0
\(238\) 7.51762 0.487295
\(239\) −8.58122 −0.555073 −0.277537 0.960715i \(-0.589518\pi\)
−0.277537 + 0.960715i \(0.589518\pi\)
\(240\) 0 0
\(241\) 19.6392 1.26507 0.632535 0.774531i \(-0.282014\pi\)
0.632535 + 0.774531i \(0.282014\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) −30.2056 −1.93371
\(245\) 2.58993 0.165464
\(246\) 0 0
\(247\) −7.08889 −0.451055
\(248\) 36.1890 2.29800
\(249\) 0 0
\(250\) 20.3812 1.28902
\(251\) 29.8538 1.88436 0.942178 0.335114i \(-0.108775\pi\)
0.942178 + 0.335114i \(0.108775\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) −24.7710 −1.55427
\(255\) 0 0
\(256\) −27.9631 −1.74769
\(257\) −2.83673 −0.176950 −0.0884752 0.996078i \(-0.528199\pi\)
−0.0884752 + 0.996078i \(0.528199\pi\)
\(258\) 0 0
\(259\) 8.94299 0.555691
\(260\) −65.2099 −4.04414
\(261\) 0 0
\(262\) 27.8323 1.71949
\(263\) 10.2373 0.631262 0.315631 0.948882i \(-0.397784\pi\)
0.315631 + 0.948882i \(0.397784\pi\)
\(264\) 0 0
\(265\) −4.91978 −0.302219
\(266\) −2.49897 −0.153221
\(267\) 0 0
\(268\) −27.2428 −1.66412
\(269\) −10.0483 −0.612656 −0.306328 0.951926i \(-0.599100\pi\)
−0.306328 + 0.951926i \(0.599100\pi\)
\(270\) 0 0
\(271\) 2.50432 0.152127 0.0760634 0.997103i \(-0.475765\pi\)
0.0760634 + 0.997103i \(0.475765\pi\)
\(272\) −7.42921 −0.450462
\(273\) 0 0
\(274\) −37.0239 −2.23670
\(275\) 0 0
\(276\) 0 0
\(277\) 14.9108 0.895904 0.447952 0.894058i \(-0.352154\pi\)
0.447952 + 0.894058i \(0.352154\pi\)
\(278\) −13.9067 −0.834069
\(279\) 0 0
\(280\) −10.6065 −0.633860
\(281\) 2.04676 0.122099 0.0610497 0.998135i \(-0.480555\pi\)
0.0610497 + 0.998135i \(0.480555\pi\)
\(282\) 0 0
\(283\) −17.0723 −1.01484 −0.507422 0.861698i \(-0.669401\pi\)
−0.507422 + 0.861698i \(0.669401\pi\)
\(284\) −10.0517 −0.596459
\(285\) 0 0
\(286\) 0 0
\(287\) 8.69747 0.513395
\(288\) 0 0
\(289\) −7.10830 −0.418135
\(290\) 3.75953 0.220767
\(291\) 0 0
\(292\) −2.19059 −0.128194
\(293\) −26.7355 −1.56191 −0.780953 0.624590i \(-0.785266\pi\)
−0.780953 + 0.624590i \(0.785266\pi\)
\(294\) 0 0
\(295\) 0.450663 0.0262386
\(296\) −36.6242 −2.12874
\(297\) 0 0
\(298\) 34.4540 1.99587
\(299\) −44.2569 −2.55944
\(300\) 0 0
\(301\) −4.48159 −0.258315
\(302\) −53.1046 −3.05583
\(303\) 0 0
\(304\) 2.46958 0.141640
\(305\) −21.0674 −1.20632
\(306\) 0 0
\(307\) 16.3329 0.932166 0.466083 0.884741i \(-0.345665\pi\)
0.466083 + 0.884741i \(0.345665\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 54.7046 3.10701
\(311\) −26.0663 −1.47809 −0.739043 0.673658i \(-0.764722\pi\)
−0.739043 + 0.673658i \(0.764722\pi\)
\(312\) 0 0
\(313\) −21.4866 −1.21450 −0.607249 0.794512i \(-0.707727\pi\)
−0.607249 + 0.794512i \(0.707727\pi\)
\(314\) 23.4036 1.32074
\(315\) 0 0
\(316\) 36.4226 2.04893
\(317\) 24.8208 1.39408 0.697038 0.717035i \(-0.254501\pi\)
0.697038 + 0.717035i \(0.254501\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) −27.9872 −1.56453
\(321\) 0 0
\(322\) −15.6014 −0.869431
\(323\) −3.28815 −0.182957
\(324\) 0 0
\(325\) −11.5792 −0.642298
\(326\) −27.1624 −1.50439
\(327\) 0 0
\(328\) −35.6187 −1.96671
\(329\) −3.26290 −0.179889
\(330\) 0 0
\(331\) −31.6351 −1.73882 −0.869411 0.494089i \(-0.835502\pi\)
−0.869411 + 0.494089i \(0.835502\pi\)
\(332\) 32.4759 1.78235
\(333\) 0 0
\(334\) 12.1638 0.665571
\(335\) −19.0010 −1.03813
\(336\) 0 0
\(337\) 32.8075 1.78714 0.893569 0.448925i \(-0.148193\pi\)
0.893569 + 0.448925i \(0.148193\pi\)
\(338\) −78.8196 −4.28722
\(339\) 0 0
\(340\) −30.2473 −1.64039
\(341\) 0 0
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) 18.3534 0.989551
\(345\) 0 0
\(346\) −18.5835 −0.999053
\(347\) 14.0886 0.756315 0.378158 0.925741i \(-0.376558\pi\)
0.378158 + 0.925741i \(0.376558\pi\)
\(348\) 0 0
\(349\) 22.2403 1.19050 0.595249 0.803541i \(-0.297053\pi\)
0.595249 + 0.803541i \(0.297053\pi\)
\(350\) −4.08188 −0.218186
\(351\) 0 0
\(352\) 0 0
\(353\) −15.2345 −0.810850 −0.405425 0.914128i \(-0.632876\pi\)
−0.405425 + 0.914128i \(0.632876\pi\)
\(354\) 0 0
\(355\) −7.01074 −0.372091
\(356\) 43.3673 2.29846
\(357\) 0 0
\(358\) −14.6903 −0.776405
\(359\) 15.4483 0.815330 0.407665 0.913132i \(-0.366343\pi\)
0.407665 + 0.913132i \(0.366343\pi\)
\(360\) 0 0
\(361\) −17.9070 −0.942472
\(362\) 22.1290 1.16308
\(363\) 0 0
\(364\) −25.1783 −1.31970
\(365\) −1.52786 −0.0799721
\(366\) 0 0
\(367\) 28.3390 1.47928 0.739641 0.673001i \(-0.234995\pi\)
0.739641 + 0.673001i \(0.234995\pi\)
\(368\) 15.4179 0.803715
\(369\) 0 0
\(370\) −55.3624 −2.87815
\(371\) −1.89958 −0.0986214
\(372\) 0 0
\(373\) 1.01940 0.0527828 0.0263914 0.999652i \(-0.491598\pi\)
0.0263914 + 0.999652i \(0.491598\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 13.3625 0.689120
\(377\) 4.11779 0.212077
\(378\) 0 0
\(379\) −8.90750 −0.457547 −0.228774 0.973480i \(-0.573472\pi\)
−0.228774 + 0.973480i \(0.573472\pi\)
\(380\) 10.0546 0.515792
\(381\) 0 0
\(382\) −34.9872 −1.79010
\(383\) −9.80783 −0.501157 −0.250578 0.968096i \(-0.580621\pi\)
−0.250578 + 0.968096i \(0.580621\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −39.3707 −2.00392
\(387\) 0 0
\(388\) 44.0134 2.23444
\(389\) −24.1420 −1.22405 −0.612024 0.790840i \(-0.709644\pi\)
−0.612024 + 0.790840i \(0.709644\pi\)
\(390\) 0 0
\(391\) −20.5284 −1.03816
\(392\) −4.09529 −0.206844
\(393\) 0 0
\(394\) 16.2933 0.820843
\(395\) 25.4036 1.27819
\(396\) 0 0
\(397\) 20.6730 1.03755 0.518773 0.854912i \(-0.326389\pi\)
0.518773 + 0.854912i \(0.326389\pi\)
\(398\) −37.1832 −1.86382
\(399\) 0 0
\(400\) 4.03388 0.201694
\(401\) 25.2166 1.25926 0.629629 0.776896i \(-0.283207\pi\)
0.629629 + 0.776896i \(0.283207\pi\)
\(402\) 0 0
\(403\) 59.9176 2.98471
\(404\) 13.8078 0.686965
\(405\) 0 0
\(406\) 1.45160 0.0720416
\(407\) 0 0
\(408\) 0 0
\(409\) −10.1558 −0.502174 −0.251087 0.967964i \(-0.580788\pi\)
−0.251087 + 0.967964i \(0.580788\pi\)
\(410\) −53.8424 −2.65909
\(411\) 0 0
\(412\) 27.2077 1.34043
\(413\) 0.174006 0.00856229
\(414\) 0 0
\(415\) 22.6509 1.11189
\(416\) −17.2526 −0.845881
\(417\) 0 0
\(418\) 0 0
\(419\) 28.3684 1.38589 0.692943 0.720993i \(-0.256314\pi\)
0.692943 + 0.720993i \(0.256314\pi\)
\(420\) 0 0
\(421\) −27.6839 −1.34923 −0.674615 0.738170i \(-0.735690\pi\)
−0.674615 + 0.738170i \(0.735690\pi\)
\(422\) 3.73989 0.182055
\(423\) 0 0
\(424\) 7.77935 0.377798
\(425\) −5.37095 −0.260529
\(426\) 0 0
\(427\) −8.13437 −0.393650
\(428\) 49.6858 2.40165
\(429\) 0 0
\(430\) 27.7437 1.33792
\(431\) 7.31248 0.352230 0.176115 0.984370i \(-0.443647\pi\)
0.176115 + 0.984370i \(0.443647\pi\)
\(432\) 0 0
\(433\) −10.5321 −0.506142 −0.253071 0.967448i \(-0.581441\pi\)
−0.253071 + 0.967448i \(0.581441\pi\)
\(434\) 21.1221 1.01389
\(435\) 0 0
\(436\) −52.4428 −2.51155
\(437\) 6.82393 0.326433
\(438\) 0 0
\(439\) 0.574098 0.0274002 0.0137001 0.999906i \(-0.495639\pi\)
0.0137001 + 0.999906i \(0.495639\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) −50.9733 −2.42455
\(443\) 18.2117 0.865266 0.432633 0.901570i \(-0.357585\pi\)
0.432633 + 0.901570i \(0.357585\pi\)
\(444\) 0 0
\(445\) 30.2473 1.43386
\(446\) −9.89390 −0.468490
\(447\) 0 0
\(448\) −10.8062 −0.510544
\(449\) −4.43378 −0.209243 −0.104622 0.994512i \(-0.533363\pi\)
−0.104622 + 0.994512i \(0.533363\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 9.89050 0.465210
\(453\) 0 0
\(454\) 13.3319 0.625698
\(455\) −17.5610 −0.823274
\(456\) 0 0
\(457\) 20.2931 0.949270 0.474635 0.880183i \(-0.342580\pi\)
0.474635 + 0.880183i \(0.342580\pi\)
\(458\) 1.88454 0.0880590
\(459\) 0 0
\(460\) 62.7725 2.92678
\(461\) −4.85723 −0.226224 −0.113112 0.993582i \(-0.536082\pi\)
−0.113112 + 0.993582i \(0.536082\pi\)
\(462\) 0 0
\(463\) −19.5168 −0.907024 −0.453512 0.891250i \(-0.649829\pi\)
−0.453512 + 0.891250i \(0.649829\pi\)
\(464\) −1.43453 −0.0665963
\(465\) 0 0
\(466\) −66.2259 −3.06786
\(467\) 2.74624 0.127081 0.0635403 0.997979i \(-0.479761\pi\)
0.0635403 + 0.997979i \(0.479761\pi\)
\(468\) 0 0
\(469\) −7.33649 −0.338767
\(470\) 20.1993 0.931722
\(471\) 0 0
\(472\) −0.712607 −0.0328004
\(473\) 0 0
\(474\) 0 0
\(475\) 1.78538 0.0819190
\(476\) −11.6788 −0.535298
\(477\) 0 0
\(478\) 20.5113 0.938166
\(479\) −18.0252 −0.823595 −0.411797 0.911275i \(-0.635099\pi\)
−0.411797 + 0.911275i \(0.635099\pi\)
\(480\) 0 0
\(481\) −60.6381 −2.76486
\(482\) −46.9427 −2.13818
\(483\) 0 0
\(484\) 0 0
\(485\) 30.6979 1.39392
\(486\) 0 0
\(487\) −12.3834 −0.561146 −0.280573 0.959833i \(-0.590524\pi\)
−0.280573 + 0.959833i \(0.590524\pi\)
\(488\) 33.3126 1.50799
\(489\) 0 0
\(490\) −6.19059 −0.279662
\(491\) 16.6168 0.749904 0.374952 0.927044i \(-0.377659\pi\)
0.374952 + 0.927044i \(0.377659\pi\)
\(492\) 0 0
\(493\) 1.91002 0.0860228
\(494\) 16.9443 0.762359
\(495\) 0 0
\(496\) −20.8737 −0.937255
\(497\) −2.70693 −0.121422
\(498\) 0 0
\(499\) −20.3850 −0.912557 −0.456279 0.889837i \(-0.650818\pi\)
−0.456279 + 0.889837i \(0.650818\pi\)
\(500\) −31.6627 −1.41600
\(501\) 0 0
\(502\) −71.3583 −3.18487
\(503\) 6.68164 0.297920 0.148960 0.988843i \(-0.452407\pi\)
0.148960 + 0.988843i \(0.452407\pi\)
\(504\) 0 0
\(505\) 9.63051 0.428552
\(506\) 0 0
\(507\) 0 0
\(508\) 38.4824 1.70738
\(509\) 18.3114 0.811639 0.405819 0.913953i \(-0.366986\pi\)
0.405819 + 0.913953i \(0.366986\pi\)
\(510\) 0 0
\(511\) −0.589926 −0.0260968
\(512\) 25.3577 1.12066
\(513\) 0 0
\(514\) 6.78051 0.299076
\(515\) 18.9765 0.836203
\(516\) 0 0
\(517\) 0 0
\(518\) −21.3761 −0.939210
\(519\) 0 0
\(520\) 71.9176 3.15379
\(521\) 0.587369 0.0257331 0.0128666 0.999917i \(-0.495904\pi\)
0.0128666 + 0.999917i \(0.495904\pi\)
\(522\) 0 0
\(523\) −28.6098 −1.25102 −0.625510 0.780216i \(-0.715109\pi\)
−0.625510 + 0.780216i \(0.715109\pi\)
\(524\) −43.2383 −1.88887
\(525\) 0 0
\(526\) −24.4699 −1.06694
\(527\) 27.7925 1.21066
\(528\) 0 0
\(529\) 19.6027 0.852291
\(530\) 11.7595 0.510801
\(531\) 0 0
\(532\) 3.88221 0.168315
\(533\) −58.9733 −2.55442
\(534\) 0 0
\(535\) 34.6542 1.49823
\(536\) 30.0451 1.29775
\(537\) 0 0
\(538\) 24.0180 1.03549
\(539\) 0 0
\(540\) 0 0
\(541\) −6.17906 −0.265659 −0.132829 0.991139i \(-0.542406\pi\)
−0.132829 + 0.991139i \(0.542406\pi\)
\(542\) −5.98597 −0.257120
\(543\) 0 0
\(544\) −8.00256 −0.343107
\(545\) −36.5771 −1.56679
\(546\) 0 0
\(547\) 6.05464 0.258878 0.129439 0.991587i \(-0.458682\pi\)
0.129439 + 0.991587i \(0.458682\pi\)
\(548\) 57.5176 2.45703
\(549\) 0 0
\(550\) 0 0
\(551\) −0.634918 −0.0270484
\(552\) 0 0
\(553\) 9.80862 0.417105
\(554\) −35.6407 −1.51423
\(555\) 0 0
\(556\) 21.6044 0.916232
\(557\) −5.98799 −0.253719 −0.126860 0.991921i \(-0.540490\pi\)
−0.126860 + 0.991921i \(0.540490\pi\)
\(558\) 0 0
\(559\) 30.3875 1.28525
\(560\) 6.11779 0.258524
\(561\) 0 0
\(562\) −4.89228 −0.206368
\(563\) 36.1739 1.52455 0.762273 0.647255i \(-0.224083\pi\)
0.762273 + 0.647255i \(0.224083\pi\)
\(564\) 0 0
\(565\) 6.89830 0.290214
\(566\) 40.8072 1.71525
\(567\) 0 0
\(568\) 11.0857 0.465144
\(569\) 33.4701 1.40314 0.701569 0.712601i \(-0.252483\pi\)
0.701569 + 0.712601i \(0.252483\pi\)
\(570\) 0 0
\(571\) 4.96371 0.207725 0.103862 0.994592i \(-0.466880\pi\)
0.103862 + 0.994592i \(0.466880\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) −20.7892 −0.867724
\(575\) 11.1464 0.464836
\(576\) 0 0
\(577\) 12.1249 0.504765 0.252383 0.967628i \(-0.418786\pi\)
0.252383 + 0.967628i \(0.418786\pi\)
\(578\) 16.9907 0.706718
\(579\) 0 0
\(580\) −5.84053 −0.242515
\(581\) 8.74577 0.362836
\(582\) 0 0
\(583\) 0 0
\(584\) 2.41592 0.0999715
\(585\) 0 0
\(586\) 63.9048 2.63988
\(587\) 4.79565 0.197938 0.0989689 0.995091i \(-0.468446\pi\)
0.0989689 + 0.995091i \(0.468446\pi\)
\(588\) 0 0
\(589\) −9.23862 −0.380671
\(590\) −1.07720 −0.0443477
\(591\) 0 0
\(592\) 21.1247 0.868219
\(593\) 15.3387 0.629886 0.314943 0.949111i \(-0.398015\pi\)
0.314943 + 0.949111i \(0.398015\pi\)
\(594\) 0 0
\(595\) −8.14560 −0.333937
\(596\) −53.5252 −2.19248
\(597\) 0 0
\(598\) 105.785 4.32589
\(599\) 46.1427 1.88534 0.942671 0.333725i \(-0.108306\pi\)
0.942671 + 0.333725i \(0.108306\pi\)
\(600\) 0 0
\(601\) −31.2352 −1.27411 −0.637056 0.770817i \(-0.719848\pi\)
−0.637056 + 0.770817i \(0.719848\pi\)
\(602\) 10.7122 0.436595
\(603\) 0 0
\(604\) 82.4994 3.35685
\(605\) 0 0
\(606\) 0 0
\(607\) 14.7306 0.597898 0.298949 0.954269i \(-0.403364\pi\)
0.298949 + 0.954269i \(0.403364\pi\)
\(608\) 2.66017 0.107884
\(609\) 0 0
\(610\) 50.3565 2.03888
\(611\) 22.1241 0.895046
\(612\) 0 0
\(613\) 27.6123 1.11525 0.557625 0.830093i \(-0.311713\pi\)
0.557625 + 0.830093i \(0.311713\pi\)
\(614\) −39.0398 −1.57552
\(615\) 0 0
\(616\) 0 0
\(617\) −3.29610 −0.132696 −0.0663479 0.997797i \(-0.521135\pi\)
−0.0663479 + 0.997797i \(0.521135\pi\)
\(618\) 0 0
\(619\) −3.60713 −0.144983 −0.0724915 0.997369i \(-0.523095\pi\)
−0.0724915 + 0.997369i \(0.523095\pi\)
\(620\) −84.9850 −3.41308
\(621\) 0 0
\(622\) 62.3052 2.49821
\(623\) 11.6788 0.467902
\(624\) 0 0
\(625\) −30.6223 −1.22489
\(626\) 51.3586 2.05270
\(627\) 0 0
\(628\) −36.3581 −1.45085
\(629\) −28.1267 −1.12148
\(630\) 0 0
\(631\) −43.1721 −1.71865 −0.859327 0.511426i \(-0.829117\pi\)
−0.859327 + 0.511426i \(0.829117\pi\)
\(632\) −40.1692 −1.59784
\(633\) 0 0
\(634\) −59.3281 −2.35622
\(635\) 26.8402 1.06512
\(636\) 0 0
\(637\) −6.78051 −0.268654
\(638\) 0 0
\(639\) 0 0
\(640\) 53.7167 2.12334
\(641\) −24.9711 −0.986298 −0.493149 0.869945i \(-0.664154\pi\)
−0.493149 + 0.869945i \(0.664154\pi\)
\(642\) 0 0
\(643\) −8.14304 −0.321130 −0.160565 0.987025i \(-0.551332\pi\)
−0.160565 + 0.987025i \(0.551332\pi\)
\(644\) 24.2372 0.955078
\(645\) 0 0
\(646\) 7.85952 0.309229
\(647\) −39.8790 −1.56781 −0.783904 0.620883i \(-0.786774\pi\)
−0.783904 + 0.620883i \(0.786774\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 27.6772 1.08559
\(651\) 0 0
\(652\) 42.1975 1.65258
\(653\) −1.63668 −0.0640484 −0.0320242 0.999487i \(-0.510195\pi\)
−0.0320242 + 0.999487i \(0.510195\pi\)
\(654\) 0 0
\(655\) −30.1573 −1.17834
\(656\) 20.5447 0.802136
\(657\) 0 0
\(658\) 7.79916 0.304043
\(659\) −7.63149 −0.297281 −0.148640 0.988891i \(-0.547490\pi\)
−0.148640 + 0.988891i \(0.547490\pi\)
\(660\) 0 0
\(661\) 20.4402 0.795031 0.397516 0.917595i \(-0.369872\pi\)
0.397516 + 0.917595i \(0.369872\pi\)
\(662\) 75.6160 2.93890
\(663\) 0 0
\(664\) −35.8165 −1.38995
\(665\) 2.70772 0.105001
\(666\) 0 0
\(667\) −3.96388 −0.153482
\(668\) −18.8967 −0.731136
\(669\) 0 0
\(670\) 45.4172 1.75462
\(671\) 0 0
\(672\) 0 0
\(673\) 24.0554 0.927269 0.463634 0.886027i \(-0.346545\pi\)
0.463634 + 0.886027i \(0.346545\pi\)
\(674\) −78.4184 −3.02056
\(675\) 0 0
\(676\) 122.448 4.70955
\(677\) −8.04672 −0.309261 −0.154630 0.987972i \(-0.549419\pi\)
−0.154630 + 0.987972i \(0.549419\pi\)
\(678\) 0 0
\(679\) 11.8528 0.454870
\(680\) 33.3586 1.27924
\(681\) 0 0
\(682\) 0 0
\(683\) 19.2894 0.738089 0.369045 0.929412i \(-0.379685\pi\)
0.369045 + 0.929412i \(0.379685\pi\)
\(684\) 0 0
\(685\) 40.1167 1.53278
\(686\) −2.39026 −0.0912604
\(687\) 0 0
\(688\) −10.5862 −0.403595
\(689\) 12.8801 0.490694
\(690\) 0 0
\(691\) −31.8589 −1.21197 −0.605985 0.795476i \(-0.707221\pi\)
−0.605985 + 0.795476i \(0.707221\pi\)
\(692\) 28.8699 1.09747
\(693\) 0 0
\(694\) −33.6753 −1.27830
\(695\) 15.0684 0.571577
\(696\) 0 0
\(697\) −27.3545 −1.03612
\(698\) −53.1601 −2.01214
\(699\) 0 0
\(700\) 6.34131 0.239679
\(701\) −35.0720 −1.32465 −0.662326 0.749216i \(-0.730430\pi\)
−0.662326 + 0.749216i \(0.730430\pi\)
\(702\) 0 0
\(703\) 9.34972 0.352631
\(704\) 0 0
\(705\) 0 0
\(706\) 36.4143 1.37047
\(707\) 3.71845 0.139847
\(708\) 0 0
\(709\) −13.9057 −0.522239 −0.261120 0.965306i \(-0.584092\pi\)
−0.261120 + 0.965306i \(0.584092\pi\)
\(710\) 16.7575 0.628896
\(711\) 0 0
\(712\) −47.8282 −1.79244
\(713\) −57.6780 −2.16006
\(714\) 0 0
\(715\) 0 0
\(716\) 22.8217 0.852888
\(717\) 0 0
\(718\) −36.9254 −1.37804
\(719\) 40.9046 1.52549 0.762743 0.646702i \(-0.223852\pi\)
0.762743 + 0.646702i \(0.223852\pi\)
\(720\) 0 0
\(721\) 7.32703 0.272873
\(722\) 42.8023 1.59294
\(723\) 0 0
\(724\) −34.3780 −1.27765
\(725\) −1.03709 −0.0385166
\(726\) 0 0
\(727\) −28.5963 −1.06058 −0.530288 0.847817i \(-0.677916\pi\)
−0.530288 + 0.847817i \(0.677916\pi\)
\(728\) 27.7682 1.02916
\(729\) 0 0
\(730\) 3.65199 0.135166
\(731\) 14.0951 0.521326
\(732\) 0 0
\(733\) −15.3513 −0.567013 −0.283507 0.958970i \(-0.591498\pi\)
−0.283507 + 0.958970i \(0.591498\pi\)
\(734\) −67.7375 −2.50024
\(735\) 0 0
\(736\) 16.6078 0.612171
\(737\) 0 0
\(738\) 0 0
\(739\) −30.8571 −1.13510 −0.567549 0.823340i \(-0.692108\pi\)
−0.567549 + 0.823340i \(0.692108\pi\)
\(740\) 86.0070 3.16168
\(741\) 0 0
\(742\) 4.54049 0.166687
\(743\) −42.5061 −1.55940 −0.779699 0.626155i \(-0.784628\pi\)
−0.779699 + 0.626155i \(0.784628\pi\)
\(744\) 0 0
\(745\) −37.3321 −1.36774
\(746\) −2.43664 −0.0892117
\(747\) 0 0
\(748\) 0 0
\(749\) 13.3804 0.488909
\(750\) 0 0
\(751\) 21.0811 0.769262 0.384631 0.923070i \(-0.374329\pi\)
0.384631 + 0.923070i \(0.374329\pi\)
\(752\) −7.70745 −0.281062
\(753\) 0 0
\(754\) −9.84258 −0.358445
\(755\) 57.5407 2.09412
\(756\) 0 0
\(757\) 42.7360 1.55326 0.776632 0.629954i \(-0.216926\pi\)
0.776632 + 0.629954i \(0.216926\pi\)
\(758\) 21.2912 0.773331
\(759\) 0 0
\(760\) −11.0889 −0.402236
\(761\) −13.9961 −0.507358 −0.253679 0.967288i \(-0.581641\pi\)
−0.253679 + 0.967288i \(0.581641\pi\)
\(762\) 0 0
\(763\) −14.1228 −0.511281
\(764\) 54.3536 1.96644
\(765\) 0 0
\(766\) 23.4432 0.847039
\(767\) −1.17985 −0.0426020
\(768\) 0 0
\(769\) −23.8142 −0.858761 −0.429380 0.903124i \(-0.641268\pi\)
−0.429380 + 0.903124i \(0.641268\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 61.1635 2.20132
\(773\) 43.0321 1.54776 0.773878 0.633335i \(-0.218314\pi\)
0.773878 + 0.633335i \(0.218314\pi\)
\(774\) 0 0
\(775\) −15.0906 −0.542071
\(776\) −48.5408 −1.74251
\(777\) 0 0
\(778\) 57.7055 2.06884
\(779\) 9.09303 0.325792
\(780\) 0 0
\(781\) 0 0
\(782\) 49.0680 1.75467
\(783\) 0 0
\(784\) 2.36215 0.0843625
\(785\) −25.3586 −0.905088
\(786\) 0 0
\(787\) −31.3512 −1.11755 −0.558774 0.829320i \(-0.688728\pi\)
−0.558774 + 0.829320i \(0.688728\pi\)
\(788\) −25.3120 −0.901704
\(789\) 0 0
\(790\) −60.7211 −2.16036
\(791\) 2.66351 0.0947037
\(792\) 0 0
\(793\) 55.1552 1.95862
\(794\) −49.4137 −1.75363
\(795\) 0 0
\(796\) 57.7650 2.04743
\(797\) 6.95708 0.246432 0.123216 0.992380i \(-0.460679\pi\)
0.123216 + 0.992380i \(0.460679\pi\)
\(798\) 0 0
\(799\) 10.2622 0.363049
\(800\) 4.34519 0.153626
\(801\) 0 0
\(802\) −60.2742 −2.12836
\(803\) 0 0
\(804\) 0 0
\(805\) 16.9046 0.595810
\(806\) −143.218 −5.04465
\(807\) 0 0
\(808\) −15.2282 −0.535725
\(809\) 49.7873 1.75043 0.875214 0.483736i \(-0.160721\pi\)
0.875214 + 0.483736i \(0.160721\pi\)
\(810\) 0 0
\(811\) 4.75368 0.166924 0.0834622 0.996511i \(-0.473402\pi\)
0.0834622 + 0.996511i \(0.473402\pi\)
\(812\) −2.25510 −0.0791383
\(813\) 0 0
\(814\) 0 0
\(815\) 29.4314 1.03094
\(816\) 0 0
\(817\) −4.68542 −0.163922
\(818\) 24.2751 0.848758
\(819\) 0 0
\(820\) 83.6457 2.92103
\(821\) 17.6949 0.617557 0.308779 0.951134i \(-0.400080\pi\)
0.308779 + 0.951134i \(0.400080\pi\)
\(822\) 0 0
\(823\) −25.6614 −0.894502 −0.447251 0.894409i \(-0.647597\pi\)
−0.447251 + 0.894409i \(0.647597\pi\)
\(824\) −30.0063 −1.04532
\(825\) 0 0
\(826\) −0.415920 −0.0144717
\(827\) 41.2901 1.43580 0.717898 0.696148i \(-0.245104\pi\)
0.717898 + 0.696148i \(0.245104\pi\)
\(828\) 0 0
\(829\) 22.3238 0.775339 0.387670 0.921798i \(-0.373280\pi\)
0.387670 + 0.921798i \(0.373280\pi\)
\(830\) −54.1415 −1.87928
\(831\) 0 0
\(832\) 73.2714 2.54023
\(833\) −3.14511 −0.108972
\(834\) 0 0
\(835\) −13.1799 −0.456108
\(836\) 0 0
\(837\) 0 0
\(838\) −67.8077 −2.34238
\(839\) 3.18377 0.109916 0.0549579 0.998489i \(-0.482498\pi\)
0.0549579 + 0.998489i \(0.482498\pi\)
\(840\) 0 0
\(841\) −28.6312 −0.987282
\(842\) 66.1716 2.28042
\(843\) 0 0
\(844\) −5.81002 −0.199989
\(845\) 85.4038 2.93798
\(846\) 0 0
\(847\) 0 0
\(848\) −4.48710 −0.154087
\(849\) 0 0
\(850\) 12.8380 0.440338
\(851\) 58.3716 2.00095
\(852\) 0 0
\(853\) 7.23751 0.247808 0.123904 0.992294i \(-0.460459\pi\)
0.123904 + 0.992294i \(0.460459\pi\)
\(854\) 19.4432 0.665334
\(855\) 0 0
\(856\) −54.7966 −1.87291
\(857\) 9.16216 0.312973 0.156487 0.987680i \(-0.449983\pi\)
0.156487 + 0.987680i \(0.449983\pi\)
\(858\) 0 0
\(859\) −18.2015 −0.621028 −0.310514 0.950569i \(-0.600501\pi\)
−0.310514 + 0.950569i \(0.600501\pi\)
\(860\) −43.1006 −1.46972
\(861\) 0 0
\(862\) −17.4787 −0.595327
\(863\) −19.2933 −0.656750 −0.328375 0.944547i \(-0.606501\pi\)
−0.328375 + 0.944547i \(0.606501\pi\)
\(864\) 0 0
\(865\) 20.1358 0.684638
\(866\) 25.1745 0.855464
\(867\) 0 0
\(868\) −32.8137 −1.11377
\(869\) 0 0
\(870\) 0 0
\(871\) 49.7451 1.68555
\(872\) 57.8372 1.95861
\(873\) 0 0
\(874\) −16.3109 −0.551726
\(875\) −8.52677 −0.288258
\(876\) 0 0
\(877\) −4.04722 −0.136665 −0.0683325 0.997663i \(-0.521768\pi\)
−0.0683325 + 0.997663i \(0.521768\pi\)
\(878\) −1.37224 −0.0463109
\(879\) 0 0
\(880\) 0 0
\(881\) −6.68797 −0.225324 −0.112662 0.993633i \(-0.535938\pi\)
−0.112662 + 0.993633i \(0.535938\pi\)
\(882\) 0 0
\(883\) −1.85027 −0.0622664 −0.0311332 0.999515i \(-0.509912\pi\)
−0.0311332 + 0.999515i \(0.509912\pi\)
\(884\) 79.1884 2.66339
\(885\) 0 0
\(886\) −43.5307 −1.46244
\(887\) 14.1006 0.473452 0.236726 0.971576i \(-0.423926\pi\)
0.236726 + 0.971576i \(0.423926\pi\)
\(888\) 0 0
\(889\) 10.3633 0.347574
\(890\) −72.2987 −2.42346
\(891\) 0 0
\(892\) 15.3704 0.514640
\(893\) −3.41129 −0.114155
\(894\) 0 0
\(895\) 15.9174 0.532061
\(896\) 20.7406 0.692896
\(897\) 0 0
\(898\) 10.5979 0.353656
\(899\) 5.36653 0.178984
\(900\) 0 0
\(901\) 5.97439 0.199036
\(902\) 0 0
\(903\) 0 0
\(904\) −10.9079 −0.362790
\(905\) −23.9776 −0.797041
\(906\) 0 0
\(907\) −2.00946 −0.0667230 −0.0333615 0.999443i \(-0.510621\pi\)
−0.0333615 + 0.999443i \(0.510621\pi\)
\(908\) −20.7115 −0.687335
\(909\) 0 0
\(910\) 41.9754 1.39147
\(911\) −41.5512 −1.37665 −0.688327 0.725400i \(-0.741655\pi\)
−0.688327 + 0.725400i \(0.741655\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) −48.5057 −1.60442
\(915\) 0 0
\(916\) −2.92769 −0.0967336
\(917\) −11.6441 −0.384521
\(918\) 0 0
\(919\) −35.0390 −1.15583 −0.577915 0.816097i \(-0.696133\pi\)
−0.577915 + 0.816097i \(0.696133\pi\)
\(920\) −69.2295 −2.28243
\(921\) 0 0
\(922\) 11.6100 0.382356
\(923\) 18.3543 0.604141
\(924\) 0 0
\(925\) 15.2721 0.502143
\(926\) 46.6502 1.53302
\(927\) 0 0
\(928\) −1.54524 −0.0507249
\(929\) 1.54300 0.0506243 0.0253121 0.999680i \(-0.491942\pi\)
0.0253121 + 0.999680i \(0.491942\pi\)
\(930\) 0 0
\(931\) 1.04548 0.0342642
\(932\) 102.884 3.37007
\(933\) 0 0
\(934\) −6.56421 −0.214788
\(935\) 0 0
\(936\) 0 0
\(937\) 42.4690 1.38740 0.693700 0.720264i \(-0.255979\pi\)
0.693700 + 0.720264i \(0.255979\pi\)
\(938\) 17.5361 0.572574
\(939\) 0 0
\(940\) −31.3801 −1.02351
\(941\) 4.81759 0.157049 0.0785245 0.996912i \(-0.474979\pi\)
0.0785245 + 0.996912i \(0.474979\pi\)
\(942\) 0 0
\(943\) 56.7690 1.84865
\(944\) 0.411029 0.0133778
\(945\) 0 0
\(946\) 0 0
\(947\) −18.0449 −0.586379 −0.293189 0.956054i \(-0.594717\pi\)
−0.293189 + 0.956054i \(0.594717\pi\)
\(948\) 0 0
\(949\) 4.00000 0.129845
\(950\) −4.26752 −0.138457
\(951\) 0 0
\(952\) 12.8801 0.417448
\(953\) −37.5309 −1.21574 −0.607872 0.794035i \(-0.707977\pi\)
−0.607872 + 0.794035i \(0.707977\pi\)
\(954\) 0 0
\(955\) 37.9099 1.22673
\(956\) −31.8649 −1.03058
\(957\) 0 0
\(958\) 43.0850 1.39201
\(959\) 15.4895 0.500182
\(960\) 0 0
\(961\) 47.0878 1.51896
\(962\) 144.941 4.67307
\(963\) 0 0
\(964\) 72.9267 2.34881
\(965\) 42.6596 1.37326
\(966\) 0 0
\(967\) −1.06636 −0.0342919 −0.0171460 0.999853i \(-0.505458\pi\)
−0.0171460 + 0.999853i \(0.505458\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) −73.3759 −2.35596
\(971\) 45.7575 1.46843 0.734214 0.678918i \(-0.237551\pi\)
0.734214 + 0.678918i \(0.237551\pi\)
\(972\) 0 0
\(973\) 5.81808 0.186519
\(974\) 29.5995 0.948430
\(975\) 0 0
\(976\) −19.2146 −0.615044
\(977\) 34.2304 1.09513 0.547564 0.836764i \(-0.315555\pi\)
0.547564 + 0.836764i \(0.315555\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 9.61724 0.307212
\(981\) 0 0
\(982\) −39.7183 −1.26746
\(983\) 29.3390 0.935769 0.467884 0.883790i \(-0.345016\pi\)
0.467884 + 0.883790i \(0.345016\pi\)
\(984\) 0 0
\(985\) −17.6543 −0.562513
\(986\) −4.56543 −0.145393
\(987\) 0 0
\(988\) −26.3234 −0.837458
\(989\) −29.2517 −0.930150
\(990\) 0 0
\(991\) 19.0194 0.604171 0.302086 0.953281i \(-0.402317\pi\)
0.302086 + 0.953281i \(0.402317\pi\)
\(992\) −22.4846 −0.713886
\(993\) 0 0
\(994\) 6.47025 0.205224
\(995\) 40.2892 1.27725
\(996\) 0 0
\(997\) 4.43152 0.140348 0.0701739 0.997535i \(-0.477645\pi\)
0.0701739 + 0.997535i \(0.477645\pi\)
\(998\) 48.7254 1.54237
\(999\) 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.cl.1.1 4
3.2 odd 2 2541.2.a.bm.1.4 4
11.2 odd 10 693.2.m.f.631.1 8
11.6 odd 10 693.2.m.f.190.1 8
11.10 odd 2 7623.2.a.ci.1.4 4
33.2 even 10 231.2.j.f.169.2 8
33.17 even 10 231.2.j.f.190.2 yes 8
33.32 even 2 2541.2.a.bn.1.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
231.2.j.f.169.2 8 33.2 even 10
231.2.j.f.190.2 yes 8 33.17 even 10
693.2.m.f.190.1 8 11.6 odd 10
693.2.m.f.631.1 8 11.2 odd 10
2541.2.a.bm.1.4 4 3.2 odd 2
2541.2.a.bn.1.1 4 33.32 even 2
7623.2.a.ci.1.4 4 11.10 odd 2
7623.2.a.cl.1.1 4 1.1 even 1 trivial