Properties

Label 7623.2.a.bz.1.2
Level 7623
Weight 2
Character 7623.1
Self dual yes
Analytic conductor 60.870
Analytic rank 1
Dimension 3
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: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.568.1
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 847)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-0.363328\)
Character \(\chi\) = 7623.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.36333 q^{2} -0.141336 q^{4} -3.14134 q^{5} +1.00000 q^{7} +2.91934 q^{8} +O(q^{10})\) \(q-1.36333 q^{2} -0.141336 q^{4} -3.14134 q^{5} +1.00000 q^{7} +2.91934 q^{8} +4.28267 q^{10} +4.77801 q^{13} -1.36333 q^{14} -3.69735 q^{16} -4.77801 q^{17} +7.00933 q^{19} +0.443984 q^{20} -5.14134 q^{23} +4.86799 q^{25} -6.51399 q^{26} -0.141336 q^{28} +7.00933 q^{29} -3.63667 q^{31} -0.797984 q^{32} +6.51399 q^{34} -3.14134 q^{35} -9.86799 q^{37} -9.55602 q^{38} -9.17064 q^{40} -3.22199 q^{41} -4.28267 q^{43} +7.00933 q^{46} +0.778008 q^{47} +1.00000 q^{49} -6.63667 q^{50} -0.675305 q^{52} -2.28267 q^{53} +2.91934 q^{56} -9.55602 q^{58} +0.363328 q^{59} -3.22199 q^{61} +4.95798 q^{62} +8.48262 q^{64} -15.0093 q^{65} -6.59465 q^{67} +0.675305 q^{68} +4.28267 q^{70} +15.1600 q^{71} +3.22199 q^{73} +13.4533 q^{74} -0.990671 q^{76} +3.71733 q^{79} +11.6146 q^{80} +4.39263 q^{82} +1.55602 q^{83} +15.0093 q^{85} +5.83869 q^{86} +5.58532 q^{89} +4.77801 q^{91} +0.726656 q^{92} -1.06068 q^{94} -22.0187 q^{95} +6.15066 q^{97} -1.36333 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q - 2q^{2} + 8q^{4} - q^{5} + 3q^{7} - 6q^{8} + O(q^{10}) \) \( 3q - 2q^{2} + 8q^{4} - q^{5} + 3q^{7} - 6q^{8} - 4q^{10} + 8q^{13} - 2q^{14} + 10q^{16} - 8q^{17} + 14q^{20} - 7q^{23} + 2q^{25} + 12q^{26} + 8q^{28} - 13q^{31} - 34q^{32} - 12q^{34} - q^{35} - 17q^{37} - 16q^{38} - 36q^{40} - 16q^{41} + 4q^{43} - 4q^{47} + 3q^{49} - 22q^{50} + 10q^{53} - 6q^{56} - 16q^{58} - q^{59} - 16q^{61} - 4q^{62} + 34q^{64} - 24q^{65} - 3q^{67} - 4q^{70} - 5q^{71} + 16q^{73} + 32q^{74} - 24q^{76} + 28q^{79} + 56q^{80} + 28q^{82} - 8q^{83} + 24q^{85} - 12q^{86} + 21q^{89} + 8q^{91} - 2q^{92} + 20q^{94} - 24q^{95} - 11q^{97} - 2q^{98} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.36333 −0.964019 −0.482009 0.876166i \(-0.660093\pi\)
−0.482009 + 0.876166i \(0.660093\pi\)
\(3\) 0 0
\(4\) −0.141336 −0.0706681
\(5\) −3.14134 −1.40485 −0.702424 0.711759i \(-0.747899\pi\)
−0.702424 + 0.711759i \(0.747899\pi\)
\(6\) 0 0
\(7\) 1.00000 0.377964
\(8\) 2.91934 1.03214
\(9\) 0 0
\(10\) 4.28267 1.35430
\(11\) 0 0
\(12\) 0 0
\(13\) 4.77801 1.32518 0.662590 0.748982i \(-0.269457\pi\)
0.662590 + 0.748982i \(0.269457\pi\)
\(14\) −1.36333 −0.364365
\(15\) 0 0
\(16\) −3.69735 −0.924338
\(17\) −4.77801 −1.15884 −0.579419 0.815030i \(-0.696720\pi\)
−0.579419 + 0.815030i \(0.696720\pi\)
\(18\) 0 0
\(19\) 7.00933 1.60805 0.804025 0.594595i \(-0.202688\pi\)
0.804025 + 0.594595i \(0.202688\pi\)
\(20\) 0.443984 0.0992779
\(21\) 0 0
\(22\) 0 0
\(23\) −5.14134 −1.07204 −0.536021 0.844204i \(-0.680073\pi\)
−0.536021 + 0.844204i \(0.680073\pi\)
\(24\) 0 0
\(25\) 4.86799 0.973599
\(26\) −6.51399 −1.27750
\(27\) 0 0
\(28\) −0.141336 −0.0267100
\(29\) 7.00933 1.30160 0.650800 0.759249i \(-0.274434\pi\)
0.650800 + 0.759249i \(0.274434\pi\)
\(30\) 0 0
\(31\) −3.63667 −0.653166 −0.326583 0.945169i \(-0.605897\pi\)
−0.326583 + 0.945169i \(0.605897\pi\)
\(32\) −0.797984 −0.141065
\(33\) 0 0
\(34\) 6.51399 1.11714
\(35\) −3.14134 −0.530983
\(36\) 0 0
\(37\) −9.86799 −1.62229 −0.811144 0.584846i \(-0.801155\pi\)
−0.811144 + 0.584846i \(0.801155\pi\)
\(38\) −9.55602 −1.55019
\(39\) 0 0
\(40\) −9.17064 −1.45001
\(41\) −3.22199 −0.503191 −0.251595 0.967833i \(-0.580955\pi\)
−0.251595 + 0.967833i \(0.580955\pi\)
\(42\) 0 0
\(43\) −4.28267 −0.653101 −0.326551 0.945180i \(-0.605886\pi\)
−0.326551 + 0.945180i \(0.605886\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 7.00933 1.03347
\(47\) 0.778008 0.113484 0.0567421 0.998389i \(-0.481929\pi\)
0.0567421 + 0.998389i \(0.481929\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) −6.63667 −0.938567
\(51\) 0 0
\(52\) −0.675305 −0.0936480
\(53\) −2.28267 −0.313549 −0.156775 0.987634i \(-0.550110\pi\)
−0.156775 + 0.987634i \(0.550110\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 2.91934 0.390114
\(57\) 0 0
\(58\) −9.55602 −1.25477
\(59\) 0.363328 0.0473013 0.0236507 0.999720i \(-0.492471\pi\)
0.0236507 + 0.999720i \(0.492471\pi\)
\(60\) 0 0
\(61\) −3.22199 −0.412534 −0.206267 0.978496i \(-0.566132\pi\)
−0.206267 + 0.978496i \(0.566132\pi\)
\(62\) 4.95798 0.629664
\(63\) 0 0
\(64\) 8.48262 1.06033
\(65\) −15.0093 −1.86168
\(66\) 0 0
\(67\) −6.59465 −0.805665 −0.402832 0.915274i \(-0.631974\pi\)
−0.402832 + 0.915274i \(0.631974\pi\)
\(68\) 0.675305 0.0818928
\(69\) 0 0
\(70\) 4.28267 0.511877
\(71\) 15.1600 1.79916 0.899580 0.436756i \(-0.143873\pi\)
0.899580 + 0.436756i \(0.143873\pi\)
\(72\) 0 0
\(73\) 3.22199 0.377106 0.188553 0.982063i \(-0.439620\pi\)
0.188553 + 0.982063i \(0.439620\pi\)
\(74\) 13.4533 1.56392
\(75\) 0 0
\(76\) −0.990671 −0.113638
\(77\) 0 0
\(78\) 0 0
\(79\) 3.71733 0.418232 0.209116 0.977891i \(-0.432941\pi\)
0.209116 + 0.977891i \(0.432941\pi\)
\(80\) 11.6146 1.29855
\(81\) 0 0
\(82\) 4.39263 0.485085
\(83\) 1.55602 0.170795 0.0853975 0.996347i \(-0.472784\pi\)
0.0853975 + 0.996347i \(0.472784\pi\)
\(84\) 0 0
\(85\) 15.0093 1.62799
\(86\) 5.83869 0.629602
\(87\) 0 0
\(88\) 0 0
\(89\) 5.58532 0.592043 0.296021 0.955181i \(-0.404340\pi\)
0.296021 + 0.955181i \(0.404340\pi\)
\(90\) 0 0
\(91\) 4.77801 0.500871
\(92\) 0.726656 0.0757592
\(93\) 0 0
\(94\) −1.06068 −0.109401
\(95\) −22.0187 −2.25907
\(96\) 0 0
\(97\) 6.15066 0.624505 0.312253 0.949999i \(-0.398916\pi\)
0.312253 + 0.949999i \(0.398916\pi\)
\(98\) −1.36333 −0.137717
\(99\) 0 0
\(100\) −0.688023 −0.0688023
\(101\) −10.7967 −1.07431 −0.537154 0.843484i \(-0.680501\pi\)
−0.537154 + 0.843484i \(0.680501\pi\)
\(102\) 0 0
\(103\) −13.2406 −1.30464 −0.652320 0.757944i \(-0.726204\pi\)
−0.652320 + 0.757944i \(0.726204\pi\)
\(104\) 13.9486 1.36778
\(105\) 0 0
\(106\) 3.11203 0.302267
\(107\) 14.0187 1.35523 0.677617 0.735415i \(-0.263013\pi\)
0.677617 + 0.735415i \(0.263013\pi\)
\(108\) 0 0
\(109\) 15.5747 1.49178 0.745892 0.666067i \(-0.232024\pi\)
0.745892 + 0.666067i \(0.232024\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) −3.69735 −0.349367
\(113\) −4.13201 −0.388707 −0.194353 0.980932i \(-0.562261\pi\)
−0.194353 + 0.980932i \(0.562261\pi\)
\(114\) 0 0
\(115\) 16.1507 1.50606
\(116\) −0.990671 −0.0919815
\(117\) 0 0
\(118\) −0.495336 −0.0455993
\(119\) −4.77801 −0.437999
\(120\) 0 0
\(121\) 0 0
\(122\) 4.39263 0.397690
\(123\) 0 0
\(124\) 0.513993 0.0461579
\(125\) 0.414680 0.0370901
\(126\) 0 0
\(127\) −22.0187 −1.95384 −0.976920 0.213606i \(-0.931479\pi\)
−0.976920 + 0.213606i \(0.931479\pi\)
\(128\) −9.96862 −0.881110
\(129\) 0 0
\(130\) 20.4626 1.79469
\(131\) 15.0093 1.31137 0.655686 0.755034i \(-0.272380\pi\)
0.655686 + 0.755034i \(0.272380\pi\)
\(132\) 0 0
\(133\) 7.00933 0.607786
\(134\) 8.99067 0.776676
\(135\) 0 0
\(136\) −13.9486 −1.19609
\(137\) 10.6974 0.913936 0.456968 0.889483i \(-0.348935\pi\)
0.456968 + 0.889483i \(0.348935\pi\)
\(138\) 0 0
\(139\) −4.10270 −0.347987 −0.173993 0.984747i \(-0.555667\pi\)
−0.173993 + 0.984747i \(0.555667\pi\)
\(140\) 0.443984 0.0375235
\(141\) 0 0
\(142\) −20.6680 −1.73442
\(143\) 0 0
\(144\) 0 0
\(145\) −22.0187 −1.82855
\(146\) −4.39263 −0.363537
\(147\) 0 0
\(148\) 1.39470 0.114644
\(149\) 14.0187 1.14845 0.574227 0.818696i \(-0.305303\pi\)
0.574227 + 0.818696i \(0.305303\pi\)
\(150\) 0 0
\(151\) 6.82936 0.555765 0.277883 0.960615i \(-0.410367\pi\)
0.277883 + 0.960615i \(0.410367\pi\)
\(152\) 20.4626 1.65974
\(153\) 0 0
\(154\) 0 0
\(155\) 11.4240 0.917598
\(156\) 0 0
\(157\) −23.4427 −1.87093 −0.935464 0.353421i \(-0.885018\pi\)
−0.935464 + 0.353421i \(0.885018\pi\)
\(158\) −5.06794 −0.403183
\(159\) 0 0
\(160\) 2.50674 0.198175
\(161\) −5.14134 −0.405194
\(162\) 0 0
\(163\) −0.990671 −0.0775954 −0.0387977 0.999247i \(-0.512353\pi\)
−0.0387977 + 0.999247i \(0.512353\pi\)
\(164\) 0.455384 0.0355595
\(165\) 0 0
\(166\) −2.12136 −0.164649
\(167\) 0.565344 0.0437477 0.0218738 0.999761i \(-0.493037\pi\)
0.0218738 + 0.999761i \(0.493037\pi\)
\(168\) 0 0
\(169\) 9.82936 0.756105
\(170\) −20.4626 −1.56941
\(171\) 0 0
\(172\) 0.605296 0.0461534
\(173\) −17.2406 −1.31078 −0.655391 0.755290i \(-0.727496\pi\)
−0.655391 + 0.755290i \(0.727496\pi\)
\(174\) 0 0
\(175\) 4.86799 0.367986
\(176\) 0 0
\(177\) 0 0
\(178\) −7.61462 −0.570740
\(179\) −11.3213 −0.846194 −0.423097 0.906084i \(-0.639057\pi\)
−0.423097 + 0.906084i \(0.639057\pi\)
\(180\) 0 0
\(181\) 14.8773 1.10582 0.552911 0.833240i \(-0.313517\pi\)
0.552911 + 0.833240i \(0.313517\pi\)
\(182\) −6.51399 −0.482849
\(183\) 0 0
\(184\) −15.0093 −1.10650
\(185\) 30.9987 2.27907
\(186\) 0 0
\(187\) 0 0
\(188\) −0.109961 −0.00801970
\(189\) 0 0
\(190\) 30.0187 2.17778
\(191\) −3.84934 −0.278528 −0.139264 0.990255i \(-0.544474\pi\)
−0.139264 + 0.990255i \(0.544474\pi\)
\(192\) 0 0
\(193\) 2.54669 0.183315 0.0916573 0.995791i \(-0.470784\pi\)
0.0916573 + 0.995791i \(0.470784\pi\)
\(194\) −8.38538 −0.602035
\(195\) 0 0
\(196\) −0.141336 −0.0100954
\(197\) −10.5467 −0.751420 −0.375710 0.926737i \(-0.622601\pi\)
−0.375710 + 0.926737i \(0.622601\pi\)
\(198\) 0 0
\(199\) 11.6846 0.828302 0.414151 0.910208i \(-0.364079\pi\)
0.414151 + 0.910208i \(0.364079\pi\)
\(200\) 14.2113 1.00489
\(201\) 0 0
\(202\) 14.7194 1.03565
\(203\) 7.00933 0.491958
\(204\) 0 0
\(205\) 10.1214 0.706906
\(206\) 18.0514 1.25770
\(207\) 0 0
\(208\) −17.6660 −1.22492
\(209\) 0 0
\(210\) 0 0
\(211\) −11.1120 −0.764984 −0.382492 0.923959i \(-0.624934\pi\)
−0.382492 + 0.923959i \(0.624934\pi\)
\(212\) 0.322624 0.0221579
\(213\) 0 0
\(214\) −19.1120 −1.30647
\(215\) 13.4533 0.917508
\(216\) 0 0
\(217\) −3.63667 −0.246873
\(218\) −21.2334 −1.43811
\(219\) 0 0
\(220\) 0 0
\(221\) −22.8294 −1.53567
\(222\) 0 0
\(223\) 28.0407 1.87774 0.938872 0.344266i \(-0.111872\pi\)
0.938872 + 0.344266i \(0.111872\pi\)
\(224\) −0.797984 −0.0533175
\(225\) 0 0
\(226\) 5.63328 0.374720
\(227\) −23.0093 −1.52718 −0.763591 0.645700i \(-0.776566\pi\)
−0.763591 + 0.645700i \(0.776566\pi\)
\(228\) 0 0
\(229\) 13.4240 0.887083 0.443542 0.896254i \(-0.353722\pi\)
0.443542 + 0.896254i \(0.353722\pi\)
\(230\) −22.0187 −1.45187
\(231\) 0 0
\(232\) 20.4626 1.34344
\(233\) −3.53736 −0.231740 −0.115870 0.993264i \(-0.536966\pi\)
−0.115870 + 0.993264i \(0.536966\pi\)
\(234\) 0 0
\(235\) −2.44398 −0.159428
\(236\) −0.0513514 −0.00334269
\(237\) 0 0
\(238\) 6.51399 0.422239
\(239\) 22.0187 1.42427 0.712134 0.702043i \(-0.247729\pi\)
0.712134 + 0.702043i \(0.247729\pi\)
\(240\) 0 0
\(241\) 0.315366 0.0203145 0.0101573 0.999948i \(-0.496767\pi\)
0.0101573 + 0.999948i \(0.496767\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0.455384 0.0291530
\(245\) −3.14134 −0.200693
\(246\) 0 0
\(247\) 33.4906 2.13096
\(248\) −10.6167 −0.674161
\(249\) 0 0
\(250\) −0.565344 −0.0357555
\(251\) −18.6460 −1.17693 −0.588463 0.808524i \(-0.700267\pi\)
−0.588463 + 0.808524i \(0.700267\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 30.0187 1.88354
\(255\) 0 0
\(256\) −3.37473 −0.210920
\(257\) 8.54669 0.533128 0.266564 0.963817i \(-0.414112\pi\)
0.266564 + 0.963817i \(0.414112\pi\)
\(258\) 0 0
\(259\) −9.86799 −0.613167
\(260\) 2.12136 0.131561
\(261\) 0 0
\(262\) −20.4626 −1.26419
\(263\) 16.0000 0.986602 0.493301 0.869859i \(-0.335790\pi\)
0.493301 + 0.869859i \(0.335790\pi\)
\(264\) 0 0
\(265\) 7.17064 0.440489
\(266\) −9.55602 −0.585917
\(267\) 0 0
\(268\) 0.932062 0.0569348
\(269\) −7.55602 −0.460698 −0.230349 0.973108i \(-0.573987\pi\)
−0.230349 + 0.973108i \(0.573987\pi\)
\(270\) 0 0
\(271\) −14.0187 −0.851573 −0.425786 0.904824i \(-0.640002\pi\)
−0.425786 + 0.904824i \(0.640002\pi\)
\(272\) 17.6660 1.07116
\(273\) 0 0
\(274\) −14.5840 −0.881052
\(275\) 0 0
\(276\) 0 0
\(277\) 6.01866 0.361626 0.180813 0.983517i \(-0.442127\pi\)
0.180813 + 0.983517i \(0.442127\pi\)
\(278\) 5.59333 0.335466
\(279\) 0 0
\(280\) −9.17064 −0.548051
\(281\) −20.6680 −1.23295 −0.616476 0.787374i \(-0.711440\pi\)
−0.616476 + 0.787374i \(0.711440\pi\)
\(282\) 0 0
\(283\) −9.91595 −0.589442 −0.294721 0.955583i \(-0.595227\pi\)
−0.294721 + 0.955583i \(0.595227\pi\)
\(284\) −2.14265 −0.127143
\(285\) 0 0
\(286\) 0 0
\(287\) −3.22199 −0.190188
\(288\) 0 0
\(289\) 5.82936 0.342903
\(290\) 30.0187 1.76276
\(291\) 0 0
\(292\) −0.455384 −0.0266493
\(293\) −26.7967 −1.56548 −0.782739 0.622350i \(-0.786178\pi\)
−0.782739 + 0.622350i \(0.786178\pi\)
\(294\) 0 0
\(295\) −1.14134 −0.0664512
\(296\) −28.8081 −1.67443
\(297\) 0 0
\(298\) −19.1120 −1.10713
\(299\) −24.5653 −1.42065
\(300\) 0 0
\(301\) −4.28267 −0.246849
\(302\) −9.31066 −0.535768
\(303\) 0 0
\(304\) −25.9160 −1.48638
\(305\) 10.1214 0.579547
\(306\) 0 0
\(307\) −11.8973 −0.679015 −0.339507 0.940603i \(-0.610260\pi\)
−0.339507 + 0.940603i \(0.610260\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) −15.5747 −0.884582
\(311\) 21.2406 1.20445 0.602223 0.798328i \(-0.294282\pi\)
0.602223 + 0.798328i \(0.294282\pi\)
\(312\) 0 0
\(313\) −12.6974 −0.717697 −0.358848 0.933396i \(-0.616830\pi\)
−0.358848 + 0.933396i \(0.616830\pi\)
\(314\) 31.9600 1.80361
\(315\) 0 0
\(316\) −0.525393 −0.0295556
\(317\) 11.8680 0.666573 0.333286 0.942826i \(-0.391842\pi\)
0.333286 + 0.942826i \(0.391842\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) −26.6468 −1.48960
\(321\) 0 0
\(322\) 7.00933 0.390615
\(323\) −33.4906 −1.86347
\(324\) 0 0
\(325\) 23.2593 1.29019
\(326\) 1.35061 0.0748034
\(327\) 0 0
\(328\) −9.40610 −0.519365
\(329\) 0.778008 0.0428930
\(330\) 0 0
\(331\) −11.8867 −0.653349 −0.326675 0.945137i \(-0.605928\pi\)
−0.326675 + 0.945137i \(0.605928\pi\)
\(332\) −0.219921 −0.0120697
\(333\) 0 0
\(334\) −0.770750 −0.0421736
\(335\) 20.7160 1.13184
\(336\) 0 0
\(337\) −0.990671 −0.0539653 −0.0269827 0.999636i \(-0.508590\pi\)
−0.0269827 + 0.999636i \(0.508590\pi\)
\(338\) −13.4006 −0.728899
\(339\) 0 0
\(340\) −2.12136 −0.115047
\(341\) 0 0
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) −12.5026 −0.674095
\(345\) 0 0
\(346\) 23.5047 1.26362
\(347\) −15.1893 −0.815404 −0.407702 0.913115i \(-0.633670\pi\)
−0.407702 + 0.913115i \(0.633670\pi\)
\(348\) 0 0
\(349\) 29.7033 1.58998 0.794990 0.606622i \(-0.207476\pi\)
0.794990 + 0.606622i \(0.207476\pi\)
\(350\) −6.63667 −0.354745
\(351\) 0 0
\(352\) 0 0
\(353\) 6.71601 0.357457 0.178729 0.983898i \(-0.442802\pi\)
0.178729 + 0.983898i \(0.442802\pi\)
\(354\) 0 0
\(355\) −47.6226 −2.52755
\(356\) −0.789407 −0.0418385
\(357\) 0 0
\(358\) 15.4347 0.815747
\(359\) 17.7360 0.936069 0.468035 0.883710i \(-0.344962\pi\)
0.468035 + 0.883710i \(0.344962\pi\)
\(360\) 0 0
\(361\) 30.1307 1.58583
\(362\) −20.2827 −1.06603
\(363\) 0 0
\(364\) −0.675305 −0.0353956
\(365\) −10.1214 −0.529776
\(366\) 0 0
\(367\) −8.09931 −0.422781 −0.211390 0.977402i \(-0.567799\pi\)
−0.211390 + 0.977402i \(0.567799\pi\)
\(368\) 19.0093 0.990930
\(369\) 0 0
\(370\) −42.2614 −2.19706
\(371\) −2.28267 −0.118510
\(372\) 0 0
\(373\) −0.565344 −0.0292724 −0.0146362 0.999893i \(-0.504659\pi\)
−0.0146362 + 0.999893i \(0.504659\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 2.27127 0.117132
\(377\) 33.4906 1.72486
\(378\) 0 0
\(379\) −22.5360 −1.15760 −0.578799 0.815470i \(-0.696479\pi\)
−0.578799 + 0.815470i \(0.696479\pi\)
\(380\) 3.11203 0.159644
\(381\) 0 0
\(382\) 5.24791 0.268506
\(383\) −21.3913 −1.09305 −0.546523 0.837444i \(-0.684049\pi\)
−0.546523 + 0.837444i \(0.684049\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −3.47197 −0.176719
\(387\) 0 0
\(388\) −0.869311 −0.0441326
\(389\) −10.6787 −0.541431 −0.270716 0.962659i \(-0.587260\pi\)
−0.270716 + 0.962659i \(0.587260\pi\)
\(390\) 0 0
\(391\) 24.5653 1.24232
\(392\) 2.91934 0.147449
\(393\) 0 0
\(394\) 14.3786 0.724383
\(395\) −11.6774 −0.587553
\(396\) 0 0
\(397\) −23.9160 −1.20031 −0.600154 0.799885i \(-0.704894\pi\)
−0.600154 + 0.799885i \(0.704894\pi\)
\(398\) −15.9300 −0.798498
\(399\) 0 0
\(400\) −17.9987 −0.899934
\(401\) −11.6587 −0.582209 −0.291104 0.956691i \(-0.594023\pi\)
−0.291104 + 0.956691i \(0.594023\pi\)
\(402\) 0 0
\(403\) −17.3760 −0.865563
\(404\) 1.52596 0.0759193
\(405\) 0 0
\(406\) −9.55602 −0.474257
\(407\) 0 0
\(408\) 0 0
\(409\) −32.8153 −1.62261 −0.811307 0.584621i \(-0.801243\pi\)
−0.811307 + 0.584621i \(0.801243\pi\)
\(410\) −13.7987 −0.681471
\(411\) 0 0
\(412\) 1.87138 0.0921964
\(413\) 0.363328 0.0178782
\(414\) 0 0
\(415\) −4.88797 −0.239941
\(416\) −3.81277 −0.186937
\(417\) 0 0
\(418\) 0 0
\(419\) −10.7967 −0.527452 −0.263726 0.964598i \(-0.584951\pi\)
−0.263726 + 0.964598i \(0.584951\pi\)
\(420\) 0 0
\(421\) −1.15198 −0.0561442 −0.0280721 0.999606i \(-0.508937\pi\)
−0.0280721 + 0.999606i \(0.508937\pi\)
\(422\) 15.1493 0.737459
\(423\) 0 0
\(424\) −6.66391 −0.323628
\(425\) −23.2593 −1.12824
\(426\) 0 0
\(427\) −3.22199 −0.155923
\(428\) −1.98134 −0.0957718
\(429\) 0 0
\(430\) −18.3413 −0.884495
\(431\) −31.9600 −1.53946 −0.769731 0.638369i \(-0.779609\pi\)
−0.769731 + 0.638369i \(0.779609\pi\)
\(432\) 0 0
\(433\) −4.43334 −0.213053 −0.106526 0.994310i \(-0.533973\pi\)
−0.106526 + 0.994310i \(0.533973\pi\)
\(434\) 4.95798 0.237991
\(435\) 0 0
\(436\) −2.20126 −0.105421
\(437\) −36.0373 −1.72390
\(438\) 0 0
\(439\) 27.4720 1.31117 0.655583 0.755123i \(-0.272423\pi\)
0.655583 + 0.755123i \(0.272423\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 31.1239 1.48041
\(443\) 7.79073 0.370149 0.185074 0.982725i \(-0.440747\pi\)
0.185074 + 0.982725i \(0.440747\pi\)
\(444\) 0 0
\(445\) −17.5454 −0.831730
\(446\) −38.2287 −1.81018
\(447\) 0 0
\(448\) 8.48262 0.400766
\(449\) −20.5106 −0.967955 −0.483978 0.875080i \(-0.660808\pi\)
−0.483978 + 0.875080i \(0.660808\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0.584002 0.0274691
\(453\) 0 0
\(454\) 31.3693 1.47223
\(455\) −15.0093 −0.703648
\(456\) 0 0
\(457\) 8.56534 0.400670 0.200335 0.979727i \(-0.435797\pi\)
0.200335 + 0.979727i \(0.435797\pi\)
\(458\) −18.3013 −0.855165
\(459\) 0 0
\(460\) −2.28267 −0.106430
\(461\) 9.66598 0.450189 0.225095 0.974337i \(-0.427731\pi\)
0.225095 + 0.974337i \(0.427731\pi\)
\(462\) 0 0
\(463\) 11.4240 0.530919 0.265459 0.964122i \(-0.414476\pi\)
0.265459 + 0.964122i \(0.414476\pi\)
\(464\) −25.9160 −1.20312
\(465\) 0 0
\(466\) 4.82258 0.223402
\(467\) −18.2207 −0.843152 −0.421576 0.906793i \(-0.638523\pi\)
−0.421576 + 0.906793i \(0.638523\pi\)
\(468\) 0 0
\(469\) −6.59465 −0.304513
\(470\) 3.33195 0.153692
\(471\) 0 0
\(472\) 1.06068 0.0488218
\(473\) 0 0
\(474\) 0 0
\(475\) 34.1214 1.56560
\(476\) 0.675305 0.0309526
\(477\) 0 0
\(478\) −30.0187 −1.37302
\(479\) 3.47197 0.158638 0.0793192 0.996849i \(-0.474725\pi\)
0.0793192 + 0.996849i \(0.474725\pi\)
\(480\) 0 0
\(481\) −47.1493 −2.14983
\(482\) −0.429948 −0.0195836
\(483\) 0 0
\(484\) 0 0
\(485\) −19.3213 −0.877335
\(486\) 0 0
\(487\) −27.4613 −1.24439 −0.622196 0.782862i \(-0.713759\pi\)
−0.622196 + 0.782862i \(0.713759\pi\)
\(488\) −9.40610 −0.425794
\(489\) 0 0
\(490\) 4.28267 0.193471
\(491\) −6.26401 −0.282691 −0.141346 0.989960i \(-0.545143\pi\)
−0.141346 + 0.989960i \(0.545143\pi\)
\(492\) 0 0
\(493\) −33.4906 −1.50834
\(494\) −45.6587 −2.05428
\(495\) 0 0
\(496\) 13.4461 0.603746
\(497\) 15.1600 0.680019
\(498\) 0 0
\(499\) −20.6680 −0.925229 −0.462614 0.886560i \(-0.653089\pi\)
−0.462614 + 0.886560i \(0.653089\pi\)
\(500\) −0.0586092 −0.00262108
\(501\) 0 0
\(502\) 25.4206 1.13458
\(503\) 22.0187 0.981763 0.490882 0.871226i \(-0.336675\pi\)
0.490882 + 0.871226i \(0.336675\pi\)
\(504\) 0 0
\(505\) 33.9160 1.50924
\(506\) 0 0
\(507\) 0 0
\(508\) 3.11203 0.138074
\(509\) −21.4240 −0.949602 −0.474801 0.880093i \(-0.657480\pi\)
−0.474801 + 0.880093i \(0.657480\pi\)
\(510\) 0 0
\(511\) 3.22199 0.142533
\(512\) 24.5381 1.08444
\(513\) 0 0
\(514\) −11.6519 −0.513945
\(515\) 41.5933 1.83282
\(516\) 0 0
\(517\) 0 0
\(518\) 13.4533 0.591105
\(519\) 0 0
\(520\) −43.8174 −1.92152
\(521\) −13.9453 −0.610953 −0.305476 0.952200i \(-0.598816\pi\)
−0.305476 + 0.952200i \(0.598816\pi\)
\(522\) 0 0
\(523\) 18.4813 0.808131 0.404065 0.914730i \(-0.367597\pi\)
0.404065 + 0.914730i \(0.367597\pi\)
\(524\) −2.12136 −0.0926721
\(525\) 0 0
\(526\) −21.8133 −0.951103
\(527\) 17.3760 0.756912
\(528\) 0 0
\(529\) 3.43334 0.149276
\(530\) −9.77594 −0.424640
\(531\) 0 0
\(532\) −0.990671 −0.0429510
\(533\) −15.3947 −0.666819
\(534\) 0 0
\(535\) −44.0373 −1.90390
\(536\) −19.2520 −0.831562
\(537\) 0 0
\(538\) 10.3013 0.444122
\(539\) 0 0
\(540\) 0 0
\(541\) −37.4533 −1.61024 −0.805122 0.593109i \(-0.797900\pi\)
−0.805122 + 0.593109i \(0.797900\pi\)
\(542\) 19.1120 0.820932
\(543\) 0 0
\(544\) 3.81277 0.163471
\(545\) −48.9253 −2.09573
\(546\) 0 0
\(547\) 17.1307 0.732455 0.366228 0.930525i \(-0.380649\pi\)
0.366228 + 0.930525i \(0.380649\pi\)
\(548\) −1.51192 −0.0645861
\(549\) 0 0
\(550\) 0 0
\(551\) 49.1307 2.09304
\(552\) 0 0
\(553\) 3.71733 0.158077
\(554\) −8.20541 −0.348614
\(555\) 0 0
\(556\) 0.579860 0.0245915
\(557\) 24.5653 1.04087 0.520434 0.853902i \(-0.325770\pi\)
0.520434 + 0.853902i \(0.325770\pi\)
\(558\) 0 0
\(559\) −20.4626 −0.865478
\(560\) 11.6146 0.490807
\(561\) 0 0
\(562\) 28.1773 1.18859
\(563\) −15.0093 −0.632568 −0.316284 0.948665i \(-0.602435\pi\)
−0.316284 + 0.948665i \(0.602435\pi\)
\(564\) 0 0
\(565\) 12.9800 0.546074
\(566\) 13.5187 0.568233
\(567\) 0 0
\(568\) 44.2572 1.85699
\(569\) 12.1027 0.507372 0.253686 0.967287i \(-0.418357\pi\)
0.253686 + 0.967287i \(0.418357\pi\)
\(570\) 0 0
\(571\) −38.9439 −1.62975 −0.814877 0.579634i \(-0.803195\pi\)
−0.814877 + 0.579634i \(0.803195\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 4.39263 0.183345
\(575\) −25.0280 −1.04374
\(576\) 0 0
\(577\) 38.1507 1.58823 0.794116 0.607766i \(-0.207934\pi\)
0.794116 + 0.607766i \(0.207934\pi\)
\(578\) −7.94733 −0.330565
\(579\) 0 0
\(580\) 3.11203 0.129220
\(581\) 1.55602 0.0645544
\(582\) 0 0
\(583\) 0 0
\(584\) 9.40610 0.389227
\(585\) 0 0
\(586\) 36.5327 1.50915
\(587\) 34.7967 1.43621 0.718106 0.695934i \(-0.245009\pi\)
0.718106 + 0.695934i \(0.245009\pi\)
\(588\) 0 0
\(589\) −25.4906 −1.05032
\(590\) 1.55602 0.0640602
\(591\) 0 0
\(592\) 36.4854 1.49954
\(593\) −18.5913 −0.763452 −0.381726 0.924276i \(-0.624670\pi\)
−0.381726 + 0.924276i \(0.624670\pi\)
\(594\) 0 0
\(595\) 15.0093 0.615322
\(596\) −1.98134 −0.0811590
\(597\) 0 0
\(598\) 33.4906 1.36953
\(599\) −24.6680 −1.00791 −0.503955 0.863730i \(-0.668122\pi\)
−0.503955 + 0.863730i \(0.668122\pi\)
\(600\) 0 0
\(601\) 30.1286 1.22897 0.614486 0.788928i \(-0.289363\pi\)
0.614486 + 0.788928i \(0.289363\pi\)
\(602\) 5.83869 0.237967
\(603\) 0 0
\(604\) −0.965235 −0.0392749
\(605\) 0 0
\(606\) 0 0
\(607\) 13.6587 0.554390 0.277195 0.960814i \(-0.410595\pi\)
0.277195 + 0.960814i \(0.410595\pi\)
\(608\) −5.59333 −0.226840
\(609\) 0 0
\(610\) −13.7987 −0.558694
\(611\) 3.71733 0.150387
\(612\) 0 0
\(613\) 45.0280 1.81866 0.909332 0.416072i \(-0.136594\pi\)
0.909332 + 0.416072i \(0.136594\pi\)
\(614\) 16.2199 0.654583
\(615\) 0 0
\(616\) 0 0
\(617\) 8.26401 0.332697 0.166348 0.986067i \(-0.446802\pi\)
0.166348 + 0.986067i \(0.446802\pi\)
\(618\) 0 0
\(619\) 43.0500 1.73033 0.865163 0.501490i \(-0.167215\pi\)
0.865163 + 0.501490i \(0.167215\pi\)
\(620\) −1.61462 −0.0648449
\(621\) 0 0
\(622\) −28.9580 −1.16111
\(623\) 5.58532 0.223771
\(624\) 0 0
\(625\) −25.6426 −1.02570
\(626\) 17.3107 0.691873
\(627\) 0 0
\(628\) 3.31330 0.132215
\(629\) 47.1493 1.87997
\(630\) 0 0
\(631\) 1.03863 0.0413473 0.0206737 0.999786i \(-0.493419\pi\)
0.0206737 + 0.999786i \(0.493419\pi\)
\(632\) 10.8522 0.431676
\(633\) 0 0
\(634\) −16.1800 −0.642589
\(635\) 69.1680 2.74485
\(636\) 0 0
\(637\) 4.77801 0.189312
\(638\) 0 0
\(639\) 0 0
\(640\) 31.3148 1.23783
\(641\) −33.6413 −1.32875 −0.664376 0.747399i \(-0.731302\pi\)
−0.664376 + 0.747399i \(0.731302\pi\)
\(642\) 0 0
\(643\) −21.0314 −0.829396 −0.414698 0.909959i \(-0.636113\pi\)
−0.414698 + 0.909959i \(0.636113\pi\)
\(644\) 0.726656 0.0286343
\(645\) 0 0
\(646\) 45.6587 1.79642
\(647\) −25.2300 −0.991894 −0.495947 0.868353i \(-0.665179\pi\)
−0.495947 + 0.868353i \(0.665179\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) −31.7101 −1.24377
\(651\) 0 0
\(652\) 0.140018 0.00548351
\(653\) −20.6387 −0.807656 −0.403828 0.914835i \(-0.632321\pi\)
−0.403828 + 0.914835i \(0.632321\pi\)
\(654\) 0 0
\(655\) −47.1493 −1.84228
\(656\) 11.9128 0.465118
\(657\) 0 0
\(658\) −1.06068 −0.0413496
\(659\) −18.5067 −0.720920 −0.360460 0.932775i \(-0.617380\pi\)
−0.360460 + 0.932775i \(0.617380\pi\)
\(660\) 0 0
\(661\) 16.8001 0.653446 0.326723 0.945120i \(-0.394056\pi\)
0.326723 + 0.945120i \(0.394056\pi\)
\(662\) 16.2054 0.629841
\(663\) 0 0
\(664\) 4.54255 0.176285
\(665\) −22.0187 −0.853847
\(666\) 0 0
\(667\) −36.0373 −1.39537
\(668\) −0.0799036 −0.00309156
\(669\) 0 0
\(670\) −28.2427 −1.09111
\(671\) 0 0
\(672\) 0 0
\(673\) −17.5560 −0.676735 −0.338367 0.941014i \(-0.609875\pi\)
−0.338367 + 0.941014i \(0.609875\pi\)
\(674\) 1.35061 0.0520236
\(675\) 0 0
\(676\) −1.38924 −0.0534324
\(677\) −3.64732 −0.140178 −0.0700889 0.997541i \(-0.522328\pi\)
−0.0700889 + 0.997541i \(0.522328\pi\)
\(678\) 0 0
\(679\) 6.15066 0.236041
\(680\) 43.8174 1.68032
\(681\) 0 0
\(682\) 0 0
\(683\) −27.4720 −1.05119 −0.525593 0.850736i \(-0.676156\pi\)
−0.525593 + 0.850736i \(0.676156\pi\)
\(684\) 0 0
\(685\) −33.6040 −1.28394
\(686\) −1.36333 −0.0520521
\(687\) 0 0
\(688\) 15.8345 0.603686
\(689\) −10.9066 −0.415509
\(690\) 0 0
\(691\) −5.95666 −0.226602 −0.113301 0.993561i \(-0.536142\pi\)
−0.113301 + 0.993561i \(0.536142\pi\)
\(692\) 2.43673 0.0926304
\(693\) 0 0
\(694\) 20.7080 0.786065
\(695\) 12.8880 0.488869
\(696\) 0 0
\(697\) 15.3947 0.583116
\(698\) −40.4953 −1.53277
\(699\) 0 0
\(700\) −0.688023 −0.0260048
\(701\) 7.36927 0.278333 0.139167 0.990269i \(-0.455558\pi\)
0.139167 + 0.990269i \(0.455558\pi\)
\(702\) 0 0
\(703\) −69.1680 −2.60872
\(704\) 0 0
\(705\) 0 0
\(706\) −9.15613 −0.344595
\(707\) −10.7967 −0.406050
\(708\) 0 0
\(709\) −34.7160 −1.30379 −0.651894 0.758310i \(-0.726025\pi\)
−0.651894 + 0.758310i \(0.726025\pi\)
\(710\) 64.9253 2.43660
\(711\) 0 0
\(712\) 16.3055 0.611073
\(713\) 18.6974 0.700221
\(714\) 0 0
\(715\) 0 0
\(716\) 1.60011 0.0597989
\(717\) 0 0
\(718\) −24.1800 −0.902388
\(719\) 4.06200 0.151487 0.0757435 0.997127i \(-0.475867\pi\)
0.0757435 + 0.997127i \(0.475867\pi\)
\(720\) 0 0
\(721\) −13.2406 −0.493108
\(722\) −41.0780 −1.52877
\(723\) 0 0
\(724\) −2.10270 −0.0781463
\(725\) 34.1214 1.26724
\(726\) 0 0
\(727\) −30.5433 −1.13279 −0.566394 0.824135i \(-0.691662\pi\)
−0.566394 + 0.824135i \(0.691662\pi\)
\(728\) 13.9486 0.516971
\(729\) 0 0
\(730\) 13.7987 0.510714
\(731\) 20.4626 0.756838
\(732\) 0 0
\(733\) 30.8340 1.13888 0.569440 0.822033i \(-0.307160\pi\)
0.569440 + 0.822033i \(0.307160\pi\)
\(734\) 11.0420 0.407568
\(735\) 0 0
\(736\) 4.10270 0.151228
\(737\) 0 0
\(738\) 0 0
\(739\) −22.2241 −0.817525 −0.408763 0.912641i \(-0.634040\pi\)
−0.408763 + 0.912641i \(0.634040\pi\)
\(740\) −4.38123 −0.161057
\(741\) 0 0
\(742\) 3.11203 0.114246
\(743\) −31.1493 −1.14276 −0.571379 0.820686i \(-0.693591\pi\)
−0.571379 + 0.820686i \(0.693591\pi\)
\(744\) 0 0
\(745\) −44.0373 −1.61340
\(746\) 0.770750 0.0282192
\(747\) 0 0
\(748\) 0 0
\(749\) 14.0187 0.512231
\(750\) 0 0
\(751\) 11.9894 0.437498 0.218749 0.975781i \(-0.429802\pi\)
0.218749 + 0.975781i \(0.429802\pi\)
\(752\) −2.87657 −0.104898
\(753\) 0 0
\(754\) −45.6587 −1.66279
\(755\) −21.4533 −0.780766
\(756\) 0 0
\(757\) −11.4533 −0.416278 −0.208139 0.978099i \(-0.566741\pi\)
−0.208139 + 0.978099i \(0.566741\pi\)
\(758\) 30.7240 1.11595
\(759\) 0 0
\(760\) −64.2800 −2.33168
\(761\) 7.89004 0.286014 0.143007 0.989722i \(-0.454323\pi\)
0.143007 + 0.989722i \(0.454323\pi\)
\(762\) 0 0
\(763\) 15.5747 0.563841
\(764\) 0.544050 0.0196830
\(765\) 0 0
\(766\) 29.1634 1.05372
\(767\) 1.73599 0.0626828
\(768\) 0 0
\(769\) 48.8153 1.76033 0.880163 0.474672i \(-0.157433\pi\)
0.880163 + 0.474672i \(0.157433\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −0.359939 −0.0129545
\(773\) 3.19608 0.114955 0.0574774 0.998347i \(-0.481694\pi\)
0.0574774 + 0.998347i \(0.481694\pi\)
\(774\) 0 0
\(775\) −17.7033 −0.635921
\(776\) 17.9559 0.644579
\(777\) 0 0
\(778\) 14.5586 0.521950
\(779\) −22.5840 −0.809156
\(780\) 0 0
\(781\) 0 0
\(782\) −33.4906 −1.19762
\(783\) 0 0
\(784\) −3.69735 −0.132048
\(785\) 73.6413 2.62837
\(786\) 0 0
\(787\) 42.4813 1.51429 0.757147 0.653244i \(-0.226592\pi\)
0.757147 + 0.653244i \(0.226592\pi\)
\(788\) 1.49063 0.0531014
\(789\) 0 0
\(790\) 15.9201 0.566412
\(791\) −4.13201 −0.146917
\(792\) 0 0
\(793\) −15.3947 −0.546682
\(794\) 32.6053 1.15712
\(795\) 0 0
\(796\) −1.65146 −0.0585345
\(797\) 28.8587 1.02223 0.511113 0.859513i \(-0.329233\pi\)
0.511113 + 0.859513i \(0.329233\pi\)
\(798\) 0 0
\(799\) −3.71733 −0.131510
\(800\) −3.88458 −0.137341
\(801\) 0 0
\(802\) 15.8947 0.561260
\(803\) 0 0
\(804\) 0 0
\(805\) 16.1507 0.569236
\(806\) 23.6893 0.834418
\(807\) 0 0
\(808\) −31.5192 −1.10884
\(809\) −29.4533 −1.03552 −0.517762 0.855525i \(-0.673235\pi\)
−0.517762 + 0.855525i \(0.673235\pi\)
\(810\) 0 0
\(811\) 2.48130 0.0871302 0.0435651 0.999051i \(-0.486128\pi\)
0.0435651 + 0.999051i \(0.486128\pi\)
\(812\) −0.990671 −0.0347657
\(813\) 0 0
\(814\) 0 0
\(815\) 3.11203 0.109010
\(816\) 0 0
\(817\) −30.0187 −1.05022
\(818\) 44.7381 1.56423
\(819\) 0 0
\(820\) −1.43051 −0.0499557
\(821\) −14.0187 −0.489255 −0.244627 0.969617i \(-0.578666\pi\)
−0.244627 + 0.969617i \(0.578666\pi\)
\(822\) 0 0
\(823\) −42.7933 −1.49168 −0.745840 0.666125i \(-0.767952\pi\)
−0.745840 + 0.666125i \(0.767952\pi\)
\(824\) −38.6540 −1.34658
\(825\) 0 0
\(826\) −0.495336 −0.0172349
\(827\) 9.17064 0.318894 0.159447 0.987206i \(-0.449029\pi\)
0.159447 + 0.987206i \(0.449029\pi\)
\(828\) 0 0
\(829\) 29.1973 1.01406 0.507032 0.861927i \(-0.330743\pi\)
0.507032 + 0.861927i \(0.330743\pi\)
\(830\) 6.66391 0.231308
\(831\) 0 0
\(832\) 40.5300 1.40513
\(833\) −4.77801 −0.165548
\(834\) 0 0
\(835\) −1.77594 −0.0614588
\(836\) 0 0
\(837\) 0 0
\(838\) 14.7194 0.508473
\(839\) −19.7326 −0.681245 −0.340622 0.940200i \(-0.610638\pi\)
−0.340622 + 0.940200i \(0.610638\pi\)
\(840\) 0 0
\(841\) 20.1307 0.694162
\(842\) 1.57053 0.0541241
\(843\) 0 0
\(844\) 1.57053 0.0540599
\(845\) −30.8773 −1.06221
\(846\) 0 0
\(847\) 0 0
\(848\) 8.43984 0.289825
\(849\) 0 0
\(850\) 31.7101 1.08765
\(851\) 50.7347 1.73916
\(852\) 0 0
\(853\) −28.3527 −0.970777 −0.485389 0.874298i \(-0.661322\pi\)
−0.485389 + 0.874298i \(0.661322\pi\)
\(854\) 4.39263 0.150313
\(855\) 0 0
\(856\) 40.9253 1.39880
\(857\) −13.2033 −0.451017 −0.225509 0.974241i \(-0.572404\pi\)
−0.225509 + 0.974241i \(0.572404\pi\)
\(858\) 0 0
\(859\) −20.8260 −0.710573 −0.355286 0.934757i \(-0.615617\pi\)
−0.355286 + 0.934757i \(0.615617\pi\)
\(860\) −1.90144 −0.0648385
\(861\) 0 0
\(862\) 43.5720 1.48407
\(863\) −34.9439 −1.18951 −0.594753 0.803909i \(-0.702750\pi\)
−0.594753 + 0.803909i \(0.702750\pi\)
\(864\) 0 0
\(865\) 54.1587 1.84145
\(866\) 6.04409 0.205387
\(867\) 0 0
\(868\) 0.513993 0.0174461
\(869\) 0 0
\(870\) 0 0
\(871\) −31.5093 −1.06765
\(872\) 45.4678 1.53973
\(873\) 0 0
\(874\) 49.1307 1.66187
\(875\) 0.414680 0.0140187
\(876\) 0 0
\(877\) −26.4813 −0.894210 −0.447105 0.894481i \(-0.647545\pi\)
−0.447105 + 0.894481i \(0.647545\pi\)
\(878\) −37.4533 −1.26399
\(879\) 0 0
\(880\) 0 0
\(881\) −17.2627 −0.581595 −0.290798 0.956785i \(-0.593921\pi\)
−0.290798 + 0.956785i \(0.593921\pi\)
\(882\) 0 0
\(883\) −44.6027 −1.50100 −0.750499 0.660871i \(-0.770187\pi\)
−0.750499 + 0.660871i \(0.770187\pi\)
\(884\) 3.22661 0.108523
\(885\) 0 0
\(886\) −10.6213 −0.356830
\(887\) −30.7894 −1.03381 −0.516904 0.856044i \(-0.672915\pi\)
−0.516904 + 0.856044i \(0.672915\pi\)
\(888\) 0 0
\(889\) −22.0187 −0.738482
\(890\) 23.9201 0.801803
\(891\) 0 0
\(892\) −3.96316 −0.132697
\(893\) 5.45331 0.182488
\(894\) 0 0
\(895\) 35.5640 1.18877
\(896\) −9.96862 −0.333028
\(897\) 0 0
\(898\) 27.9627 0.933127
\(899\) −25.4906 −0.850160
\(900\) 0 0
\(901\) 10.9066 0.363352
\(902\) 0 0
\(903\) 0 0
\(904\) −12.0628 −0.401201
\(905\) −46.7347 −1.55351
\(906\) 0 0
\(907\) 23.7801 0.789605 0.394802 0.918766i \(-0.370813\pi\)
0.394802 + 0.918766i \(0.370813\pi\)
\(908\) 3.25205 0.107923
\(909\) 0 0
\(910\) 20.4626 0.678330
\(911\) −49.0280 −1.62437 −0.812185 0.583400i \(-0.801722\pi\)
−0.812185 + 0.583400i \(0.801722\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) −11.6774 −0.386253
\(915\) 0 0
\(916\) −1.89730 −0.0626885
\(917\) 15.0093 0.495652
\(918\) 0 0
\(919\) −40.0000 −1.31948 −0.659739 0.751495i \(-0.729333\pi\)
−0.659739 + 0.751495i \(0.729333\pi\)
\(920\) 47.1493 1.55447
\(921\) 0 0
\(922\) −13.1779 −0.433991
\(923\) 72.4346 2.38421
\(924\) 0 0
\(925\) −48.0373 −1.57946
\(926\) −15.5747 −0.511816
\(927\) 0 0
\(928\) −5.59333 −0.183610
\(929\) 56.5840 1.85646 0.928230 0.372006i \(-0.121330\pi\)
0.928230 + 0.372006i \(0.121330\pi\)
\(930\) 0 0
\(931\) 7.00933 0.229721
\(932\) 0.499956 0.0163766
\(933\) 0 0
\(934\) 24.8408 0.812814
\(935\) 0 0
\(936\) 0 0
\(937\) 16.8153 0.549333 0.274666 0.961540i \(-0.411433\pi\)
0.274666 + 0.961540i \(0.411433\pi\)
\(938\) 8.99067 0.293556
\(939\) 0 0
\(940\) 0.345423 0.0112665
\(941\) 7.25931 0.236647 0.118323 0.992975i \(-0.462248\pi\)
0.118323 + 0.992975i \(0.462248\pi\)
\(942\) 0 0
\(943\) 16.5653 0.539442
\(944\) −1.34335 −0.0437224
\(945\) 0 0
\(946\) 0 0
\(947\) 25.6040 0.832017 0.416009 0.909361i \(-0.363429\pi\)
0.416009 + 0.909361i \(0.363429\pi\)
\(948\) 0 0
\(949\) 15.3947 0.499733
\(950\) −46.5186 −1.50926
\(951\) 0 0
\(952\) −13.9486 −0.452078
\(953\) 7.14935 0.231590 0.115795 0.993273i \(-0.463058\pi\)
0.115795 + 0.993273i \(0.463058\pi\)
\(954\) 0 0
\(955\) 12.0921 0.391290
\(956\) −3.11203 −0.100650
\(957\) 0 0
\(958\) −4.73344 −0.152930
\(959\) 10.6974 0.345436
\(960\) 0 0
\(961\) −17.7746 −0.573375
\(962\) 64.2800 2.07247
\(963\) 0 0
\(964\) −0.0445726 −0.00143559
\(965\) −8.00000 −0.257529
\(966\) 0 0
\(967\) −15.9600 −0.513241 −0.256620 0.966512i \(-0.582609\pi\)
−0.256620 + 0.966512i \(0.582609\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 26.3413 0.845768
\(971\) 30.4779 0.978083 0.489041 0.872261i \(-0.337347\pi\)
0.489041 + 0.872261i \(0.337347\pi\)
\(972\) 0 0
\(973\) −4.10270 −0.131527
\(974\) 37.4388 1.19962
\(975\) 0 0
\(976\) 11.9128 0.381321
\(977\) −4.13201 −0.132195 −0.0660973 0.997813i \(-0.521055\pi\)
−0.0660973 + 0.997813i \(0.521055\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0.443984 0.0141826
\(981\) 0 0
\(982\) 8.53991 0.272519
\(983\) 25.0246 0.798161 0.399080 0.916916i \(-0.369329\pi\)
0.399080 + 0.916916i \(0.369329\pi\)
\(984\) 0 0
\(985\) 33.1307 1.05563
\(986\) 45.6587 1.45407
\(987\) 0 0
\(988\) −4.73344 −0.150591
\(989\) 22.0187 0.700153
\(990\) 0 0
\(991\) −27.5747 −0.875938 −0.437969 0.898990i \(-0.644302\pi\)
−0.437969 + 0.898990i \(0.644302\pi\)
\(992\) 2.90201 0.0921388
\(993\) 0 0
\(994\) −20.6680 −0.655551
\(995\) −36.7054 −1.16364
\(996\) 0 0
\(997\) −23.6846 −0.750100 −0.375050 0.927005i \(-0.622374\pi\)
−0.375050 + 0.927005i \(0.622374\pi\)
\(998\) 28.1773 0.891938
\(999\) 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.bz.1.2 3
3.2 odd 2 847.2.a.j.1.2 yes 3
11.10 odd 2 7623.2.a.ce.1.2 3
21.20 even 2 5929.2.a.y.1.2 3
33.2 even 10 847.2.f.u.323.2 12
33.5 odd 10 847.2.f.t.729.2 12
33.8 even 10 847.2.f.u.372.2 12
33.14 odd 10 847.2.f.t.372.2 12
33.17 even 10 847.2.f.u.729.2 12
33.20 odd 10 847.2.f.t.323.2 12
33.26 odd 10 847.2.f.t.148.2 12
33.29 even 10 847.2.f.u.148.2 12
33.32 even 2 847.2.a.i.1.2 3
231.230 odd 2 5929.2.a.t.1.2 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
847.2.a.i.1.2 3 33.32 even 2
847.2.a.j.1.2 yes 3 3.2 odd 2
847.2.f.t.148.2 12 33.26 odd 10
847.2.f.t.323.2 12 33.20 odd 10
847.2.f.t.372.2 12 33.14 odd 10
847.2.f.t.729.2 12 33.5 odd 10
847.2.f.u.148.2 12 33.29 even 10
847.2.f.u.323.2 12 33.2 even 10
847.2.f.u.372.2 12 33.8 even 10
847.2.f.u.729.2 12 33.17 even 10
5929.2.a.t.1.2 3 231.230 odd 2
5929.2.a.y.1.2 3 21.20 even 2
7623.2.a.bz.1.2 3 1.1 even 1 trivial
7623.2.a.ce.1.2 3 11.10 odd 2