Properties

Label 7605.2.a.cj.1.3
Level $7605$
Weight $2$
Character 7605.1
Self dual yes
Analytic conductor $60.726$
Analytic rank $1$
Dimension $4$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 7605 = 3^{2} \cdot 5 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7605.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(60.7262307372\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: 4.4.4752.1
Defining polynomial: \(x^{4} - 2 x^{3} - 3 x^{2} + 4 x + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 65)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(1.21969\) of defining polynomial
Character \(\chi\) \(=\) 7605.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.21969 q^{2} -0.512364 q^{4} -1.00000 q^{5} -3.60020 q^{7} -3.06430 q^{8} +O(q^{10})\) \(q+1.21969 q^{2} -0.512364 q^{4} -1.00000 q^{5} -3.60020 q^{7} -3.06430 q^{8} -1.21969 q^{10} +5.37182 q^{11} -4.39111 q^{14} -2.71276 q^{16} +1.13186 q^{17} -2.26795 q^{19} +0.512364 q^{20} +6.55193 q^{22} +3.89287 q^{23} +1.00000 q^{25} +1.84461 q^{28} +0.0247279 q^{29} -5.46410 q^{31} +2.81988 q^{32} +1.38051 q^{34} +3.60020 q^{35} +8.70406 q^{37} -2.76619 q^{38} +3.06430 q^{40} +3.73205 q^{41} -1.13186 q^{43} -2.75232 q^{44} +4.74809 q^{46} +2.58535 q^{47} +5.96141 q^{49} +1.21969 q^{50} +4.43937 q^{53} -5.37182 q^{55} +11.0321 q^{56} +0.0301603 q^{58} -0.171425 q^{59} +3.36023 q^{61} -6.66449 q^{62} +8.86488 q^{64} -6.39980 q^{67} -0.579922 q^{68} +4.39111 q^{70} -10.7973 q^{71} +4.70308 q^{73} +10.6162 q^{74} +1.16202 q^{76} -19.3396 q^{77} -11.9826 q^{79} +2.71276 q^{80} +4.55193 q^{82} -12.1286 q^{83} -1.13186 q^{85} -1.38051 q^{86} -16.4608 q^{88} -16.1540 q^{89} -1.99457 q^{92} +3.15332 q^{94} +2.26795 q^{95} -12.1682 q^{97} +7.27105 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 2q^{2} + 2q^{4} - 4q^{5} - 10q^{7} + 6q^{8} + O(q^{10}) \) \( 4q + 2q^{2} + 2q^{4} - 4q^{5} - 10q^{7} + 6q^{8} - 2q^{10} - 2q^{14} + 2q^{16} + 2q^{17} - 16q^{19} - 2q^{20} + 12q^{22} + 10q^{23} + 4q^{25} - 8q^{28} - 8q^{29} - 8q^{31} + 4q^{32} + 4q^{34} + 10q^{35} + 2q^{37} - 8q^{38} - 6q^{40} + 8q^{41} - 2q^{43} + 12q^{44} - 16q^{46} + 8q^{47} + 12q^{49} + 2q^{50} + 12q^{53} - 12q^{56} - 22q^{58} + 12q^{59} + 28q^{61} - 4q^{62} + 4q^{64} - 30q^{67} - 14q^{68} + 2q^{70} + 4q^{71} + 8q^{73} + 10q^{74} - 20q^{76} - 18q^{77} - 8q^{79} - 2q^{80} + 4q^{82} - 12q^{83} - 2q^{85} - 4q^{86} - 18q^{88} - 12q^{89} - 22q^{92} - 32q^{94} + 16q^{95} - 2q^{97} - 24q^{98} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.21969 0.862449 0.431224 0.902245i \(-0.358082\pi\)
0.431224 + 0.902245i \(0.358082\pi\)
\(3\) 0 0
\(4\) −0.512364 −0.256182
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) −3.60020 −1.36075 −0.680373 0.732866i \(-0.738182\pi\)
−0.680373 + 0.732866i \(0.738182\pi\)
\(8\) −3.06430 −1.08339
\(9\) 0 0
\(10\) −1.21969 −0.385699
\(11\) 5.37182 1.61966 0.809832 0.586662i \(-0.199558\pi\)
0.809832 + 0.586662i \(0.199558\pi\)
\(12\) 0 0
\(13\) 0 0
\(14\) −4.39111 −1.17357
\(15\) 0 0
\(16\) −2.71276 −0.678189
\(17\) 1.13186 0.274515 0.137258 0.990535i \(-0.456171\pi\)
0.137258 + 0.990535i \(0.456171\pi\)
\(18\) 0 0
\(19\) −2.26795 −0.520303 −0.260152 0.965568i \(-0.583773\pi\)
−0.260152 + 0.965568i \(0.583773\pi\)
\(20\) 0.512364 0.114568
\(21\) 0 0
\(22\) 6.55193 1.39688
\(23\) 3.89287 0.811720 0.405860 0.913935i \(-0.366972\pi\)
0.405860 + 0.913935i \(0.366972\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 0 0
\(28\) 1.84461 0.348599
\(29\) 0.0247279 0.00459185 0.00229593 0.999997i \(-0.499269\pi\)
0.00229593 + 0.999997i \(0.499269\pi\)
\(30\) 0 0
\(31\) −5.46410 −0.981382 −0.490691 0.871334i \(-0.663256\pi\)
−0.490691 + 0.871334i \(0.663256\pi\)
\(32\) 2.81988 0.498490
\(33\) 0 0
\(34\) 1.38051 0.236755
\(35\) 3.60020 0.608544
\(36\) 0 0
\(37\) 8.70406 1.43094 0.715470 0.698644i \(-0.246213\pi\)
0.715470 + 0.698644i \(0.246213\pi\)
\(38\) −2.76619 −0.448735
\(39\) 0 0
\(40\) 3.06430 0.484508
\(41\) 3.73205 0.582848 0.291424 0.956594i \(-0.405871\pi\)
0.291424 + 0.956594i \(0.405871\pi\)
\(42\) 0 0
\(43\) −1.13186 −0.172606 −0.0863031 0.996269i \(-0.527505\pi\)
−0.0863031 + 0.996269i \(0.527505\pi\)
\(44\) −2.75232 −0.414929
\(45\) 0 0
\(46\) 4.74809 0.700067
\(47\) 2.58535 0.377113 0.188556 0.982062i \(-0.439619\pi\)
0.188556 + 0.982062i \(0.439619\pi\)
\(48\) 0 0
\(49\) 5.96141 0.851630
\(50\) 1.21969 0.172490
\(51\) 0 0
\(52\) 0 0
\(53\) 4.43937 0.609795 0.304897 0.952385i \(-0.401378\pi\)
0.304897 + 0.952385i \(0.401378\pi\)
\(54\) 0 0
\(55\) −5.37182 −0.724336
\(56\) 11.0321 1.47422
\(57\) 0 0
\(58\) 0.0301603 0.00396024
\(59\) −0.171425 −0.0223176 −0.0111588 0.999938i \(-0.503552\pi\)
−0.0111588 + 0.999938i \(0.503552\pi\)
\(60\) 0 0
\(61\) 3.36023 0.430234 0.215117 0.976588i \(-0.430987\pi\)
0.215117 + 0.976588i \(0.430987\pi\)
\(62\) −6.66449 −0.846391
\(63\) 0 0
\(64\) 8.86488 1.10811
\(65\) 0 0
\(66\) 0 0
\(67\) −6.39980 −0.781861 −0.390930 0.920420i \(-0.627847\pi\)
−0.390930 + 0.920420i \(0.627847\pi\)
\(68\) −0.579922 −0.0703258
\(69\) 0 0
\(70\) 4.39111 0.524838
\(71\) −10.7973 −1.28141 −0.640703 0.767788i \(-0.721357\pi\)
−0.640703 + 0.767788i \(0.721357\pi\)
\(72\) 0 0
\(73\) 4.70308 0.550454 0.275227 0.961379i \(-0.411247\pi\)
0.275227 + 0.961379i \(0.411247\pi\)
\(74\) 10.6162 1.23411
\(75\) 0 0
\(76\) 1.16202 0.133292
\(77\) −19.3396 −2.20395
\(78\) 0 0
\(79\) −11.9826 −1.34815 −0.674075 0.738663i \(-0.735457\pi\)
−0.674075 + 0.738663i \(0.735457\pi\)
\(80\) 2.71276 0.303295
\(81\) 0 0
\(82\) 4.55193 0.502677
\(83\) −12.1286 −1.33129 −0.665643 0.746270i \(-0.731843\pi\)
−0.665643 + 0.746270i \(0.731843\pi\)
\(84\) 0 0
\(85\) −1.13186 −0.122767
\(86\) −1.38051 −0.148864
\(87\) 0 0
\(88\) −16.4608 −1.75473
\(89\) −16.1540 −1.71232 −0.856162 0.516707i \(-0.827158\pi\)
−0.856162 + 0.516707i \(0.827158\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) −1.99457 −0.207948
\(93\) 0 0
\(94\) 3.15332 0.325240
\(95\) 2.26795 0.232687
\(96\) 0 0
\(97\) −12.1682 −1.23549 −0.617745 0.786379i \(-0.711954\pi\)
−0.617745 + 0.786379i \(0.711954\pi\)
\(98\) 7.27105 0.734487
\(99\) 0 0
\(100\) −0.512364 −0.0512364
\(101\) 4.05441 0.403429 0.201714 0.979444i \(-0.435349\pi\)
0.201714 + 0.979444i \(0.435349\pi\)
\(102\) 0 0
\(103\) −17.9035 −1.76408 −0.882041 0.471173i \(-0.843831\pi\)
−0.882041 + 0.471173i \(0.843831\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 5.41465 0.525917
\(107\) 9.13186 0.882810 0.441405 0.897308i \(-0.354480\pi\)
0.441405 + 0.897308i \(0.354480\pi\)
\(108\) 0 0
\(109\) −7.37605 −0.706498 −0.353249 0.935529i \(-0.614923\pi\)
−0.353249 + 0.935529i \(0.614923\pi\)
\(110\) −6.55193 −0.624702
\(111\) 0 0
\(112\) 9.76645 0.922843
\(113\) −7.07588 −0.665643 −0.332821 0.942990i \(-0.608001\pi\)
−0.332821 + 0.942990i \(0.608001\pi\)
\(114\) 0 0
\(115\) −3.89287 −0.363012
\(116\) −0.0126697 −0.00117635
\(117\) 0 0
\(118\) −0.209084 −0.0192478
\(119\) −4.07490 −0.373545
\(120\) 0 0
\(121\) 17.8564 1.62331
\(122\) 4.09843 0.371055
\(123\) 0 0
\(124\) 2.79961 0.251412
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) −11.4361 −1.01479 −0.507395 0.861713i \(-0.669392\pi\)
−0.507395 + 0.861713i \(0.669392\pi\)
\(128\) 5.17262 0.457199
\(129\) 0 0
\(130\) 0 0
\(131\) 10.5680 0.923328 0.461664 0.887055i \(-0.347253\pi\)
0.461664 + 0.887055i \(0.347253\pi\)
\(132\) 0 0
\(133\) 8.16506 0.708001
\(134\) −7.80576 −0.674315
\(135\) 0 0
\(136\) −3.46834 −0.297408
\(137\) 3.78672 0.323522 0.161761 0.986830i \(-0.448283\pi\)
0.161761 + 0.986830i \(0.448283\pi\)
\(138\) 0 0
\(139\) 2.01386 0.170814 0.0854068 0.996346i \(-0.472781\pi\)
0.0854068 + 0.996346i \(0.472781\pi\)
\(140\) −1.84461 −0.155898
\(141\) 0 0
\(142\) −13.1694 −1.10515
\(143\) 0 0
\(144\) 0 0
\(145\) −0.0247279 −0.00205354
\(146\) 5.73629 0.474739
\(147\) 0 0
\(148\) −4.45965 −0.366581
\(149\) −5.51780 −0.452035 −0.226018 0.974123i \(-0.572571\pi\)
−0.226018 + 0.974123i \(0.572571\pi\)
\(150\) 0 0
\(151\) 4.88961 0.397911 0.198956 0.980009i \(-0.436245\pi\)
0.198956 + 0.980009i \(0.436245\pi\)
\(152\) 6.94967 0.563693
\(153\) 0 0
\(154\) −23.5882 −1.90079
\(155\) 5.46410 0.438887
\(156\) 0 0
\(157\) 10.0405 0.801323 0.400661 0.916226i \(-0.368780\pi\)
0.400661 + 0.916226i \(0.368780\pi\)
\(158\) −14.6150 −1.16271
\(159\) 0 0
\(160\) −2.81988 −0.222931
\(161\) −14.0151 −1.10454
\(162\) 0 0
\(163\) −6.78124 −0.531148 −0.265574 0.964090i \(-0.585561\pi\)
−0.265574 + 0.964090i \(0.585561\pi\)
\(164\) −1.91217 −0.149315
\(165\) 0 0
\(166\) −14.7931 −1.14817
\(167\) 10.4898 0.811726 0.405863 0.913934i \(-0.366971\pi\)
0.405863 + 0.913934i \(0.366971\pi\)
\(168\) 0 0
\(169\) 0 0
\(170\) −1.38051 −0.105880
\(171\) 0 0
\(172\) 0.579922 0.0442186
\(173\) 4.45845 0.338970 0.169485 0.985533i \(-0.445790\pi\)
0.169485 + 0.985533i \(0.445790\pi\)
\(174\) 0 0
\(175\) −3.60020 −0.272149
\(176\) −14.5724 −1.09844
\(177\) 0 0
\(178\) −19.7029 −1.47679
\(179\) −18.6313 −1.39257 −0.696284 0.717766i \(-0.745165\pi\)
−0.696284 + 0.717766i \(0.745165\pi\)
\(180\) 0 0
\(181\) 18.0900 1.34462 0.672310 0.740270i \(-0.265302\pi\)
0.672310 + 0.740270i \(0.265302\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) −11.9289 −0.879412
\(185\) −8.70406 −0.639935
\(186\) 0 0
\(187\) 6.08012 0.444622
\(188\) −1.32464 −0.0966095
\(189\) 0 0
\(190\) 2.76619 0.200680
\(191\) −27.3363 −1.97799 −0.988994 0.147958i \(-0.952730\pi\)
−0.988994 + 0.147958i \(0.952730\pi\)
\(192\) 0 0
\(193\) −21.7674 −1.56685 −0.783425 0.621486i \(-0.786529\pi\)
−0.783425 + 0.621486i \(0.786529\pi\)
\(194\) −14.8413 −1.06555
\(195\) 0 0
\(196\) −3.05441 −0.218172
\(197\) −1.69672 −0.120886 −0.0604432 0.998172i \(-0.519251\pi\)
−0.0604432 + 0.998172i \(0.519251\pi\)
\(198\) 0 0
\(199\) 25.3255 1.79527 0.897637 0.440735i \(-0.145282\pi\)
0.897637 + 0.440735i \(0.145282\pi\)
\(200\) −3.06430 −0.216679
\(201\) 0 0
\(202\) 4.94511 0.347937
\(203\) −0.0890252 −0.00624834
\(204\) 0 0
\(205\) −3.73205 −0.260658
\(206\) −21.8366 −1.52143
\(207\) 0 0
\(208\) 0 0
\(209\) −12.1830 −0.842716
\(210\) 0 0
\(211\) −0.335507 −0.0230973 −0.0115486 0.999933i \(-0.503676\pi\)
−0.0115486 + 0.999933i \(0.503676\pi\)
\(212\) −2.27458 −0.156218
\(213\) 0 0
\(214\) 11.1380 0.761378
\(215\) 1.13186 0.0771919
\(216\) 0 0
\(217\) 19.6718 1.33541
\(218\) −8.99648 −0.609318
\(219\) 0 0
\(220\) 2.75232 0.185562
\(221\) 0 0
\(222\) 0 0
\(223\) 12.2968 0.823452 0.411726 0.911308i \(-0.364926\pi\)
0.411726 + 0.911308i \(0.364926\pi\)
\(224\) −10.1521 −0.678318
\(225\) 0 0
\(226\) −8.63036 −0.574083
\(227\) −7.63227 −0.506571 −0.253286 0.967392i \(-0.581511\pi\)
−0.253286 + 0.967392i \(0.581511\pi\)
\(228\) 0 0
\(229\) 14.4008 0.951631 0.475815 0.879545i \(-0.342153\pi\)
0.475815 + 0.879545i \(0.342153\pi\)
\(230\) −4.74809 −0.313080
\(231\) 0 0
\(232\) −0.0757736 −0.00497478
\(233\) −9.49617 −0.622115 −0.311057 0.950391i \(-0.600683\pi\)
−0.311057 + 0.950391i \(0.600683\pi\)
\(234\) 0 0
\(235\) −2.58535 −0.168650
\(236\) 0.0878318 0.00571736
\(237\) 0 0
\(238\) −4.97010 −0.322164
\(239\) −19.9143 −1.28815 −0.644076 0.764962i \(-0.722758\pi\)
−0.644076 + 0.764962i \(0.722758\pi\)
\(240\) 0 0
\(241\) −23.2664 −1.49872 −0.749360 0.662163i \(-0.769639\pi\)
−0.749360 + 0.662163i \(0.769639\pi\)
\(242\) 21.7792 1.40002
\(243\) 0 0
\(244\) −1.72166 −0.110218
\(245\) −5.96141 −0.380860
\(246\) 0 0
\(247\) 0 0
\(248\) 16.7436 1.06322
\(249\) 0 0
\(250\) −1.21969 −0.0771398
\(251\) −11.8402 −0.747344 −0.373672 0.927561i \(-0.621901\pi\)
−0.373672 + 0.927561i \(0.621901\pi\)
\(252\) 0 0
\(253\) 20.9118 1.31471
\(254\) −13.9485 −0.875205
\(255\) 0 0
\(256\) −11.4208 −0.713800
\(257\) 5.55002 0.346201 0.173100 0.984904i \(-0.444621\pi\)
0.173100 + 0.984904i \(0.444621\pi\)
\(258\) 0 0
\(259\) −31.3363 −1.94714
\(260\) 0 0
\(261\) 0 0
\(262\) 12.8896 0.796323
\(263\) −6.85967 −0.422985 −0.211493 0.977380i \(-0.567832\pi\)
−0.211493 + 0.977380i \(0.567832\pi\)
\(264\) 0 0
\(265\) −4.43937 −0.272709
\(266\) 9.95882 0.610614
\(267\) 0 0
\(268\) 3.27903 0.200299
\(269\) 1.42199 0.0867001 0.0433501 0.999060i \(-0.486197\pi\)
0.0433501 + 0.999060i \(0.486197\pi\)
\(270\) 0 0
\(271\) 9.96947 0.605602 0.302801 0.953054i \(-0.402078\pi\)
0.302801 + 0.953054i \(0.402078\pi\)
\(272\) −3.07045 −0.186173
\(273\) 0 0
\(274\) 4.61862 0.279021
\(275\) 5.37182 0.323933
\(276\) 0 0
\(277\) −17.5237 −1.05290 −0.526449 0.850206i \(-0.676477\pi\)
−0.526449 + 0.850206i \(0.676477\pi\)
\(278\) 2.45628 0.147318
\(279\) 0 0
\(280\) −11.0321 −0.659292
\(281\) 10.7352 0.640406 0.320203 0.947349i \(-0.396249\pi\)
0.320203 + 0.947349i \(0.396249\pi\)
\(282\) 0 0
\(283\) 1.31838 0.0783698 0.0391849 0.999232i \(-0.487524\pi\)
0.0391849 + 0.999232i \(0.487524\pi\)
\(284\) 5.53216 0.328273
\(285\) 0 0
\(286\) 0 0
\(287\) −13.4361 −0.793109
\(288\) 0 0
\(289\) −15.7189 −0.924641
\(290\) −0.0301603 −0.00177107
\(291\) 0 0
\(292\) −2.40969 −0.141016
\(293\) 18.7427 1.09496 0.547479 0.836820i \(-0.315588\pi\)
0.547479 + 0.836820i \(0.315588\pi\)
\(294\) 0 0
\(295\) 0.171425 0.00998072
\(296\) −26.6718 −1.55027
\(297\) 0 0
\(298\) −6.72998 −0.389857
\(299\) 0 0
\(300\) 0 0
\(301\) 4.07490 0.234873
\(302\) 5.96380 0.343178
\(303\) 0 0
\(304\) 6.15239 0.352864
\(305\) −3.36023 −0.192406
\(306\) 0 0
\(307\) −14.3043 −0.816387 −0.408194 0.912895i \(-0.633841\pi\)
−0.408194 + 0.912895i \(0.633841\pi\)
\(308\) 9.90891 0.564612
\(309\) 0 0
\(310\) 6.66449 0.378518
\(311\) −2.76102 −0.156563 −0.0782815 0.996931i \(-0.524943\pi\)
−0.0782815 + 0.996931i \(0.524943\pi\)
\(312\) 0 0
\(313\) −16.3858 −0.926179 −0.463090 0.886311i \(-0.653259\pi\)
−0.463090 + 0.886311i \(0.653259\pi\)
\(314\) 12.2463 0.691100
\(315\) 0 0
\(316\) 6.13946 0.345372
\(317\) 1.78575 0.100297 0.0501487 0.998742i \(-0.484030\pi\)
0.0501487 + 0.998742i \(0.484030\pi\)
\(318\) 0 0
\(319\) 0.132834 0.00743725
\(320\) −8.86488 −0.495562
\(321\) 0 0
\(322\) −17.0940 −0.952613
\(323\) −2.56699 −0.142831
\(324\) 0 0
\(325\) 0 0
\(326\) −8.27099 −0.458088
\(327\) 0 0
\(328\) −11.4361 −0.631454
\(329\) −9.30778 −0.513155
\(330\) 0 0
\(331\) −7.22440 −0.397089 −0.198545 0.980092i \(-0.563621\pi\)
−0.198545 + 0.980092i \(0.563621\pi\)
\(332\) 6.21425 0.341052
\(333\) 0 0
\(334\) 12.7943 0.700072
\(335\) 6.39980 0.349659
\(336\) 0 0
\(337\) −4.36219 −0.237624 −0.118812 0.992917i \(-0.537909\pi\)
−0.118812 + 0.992917i \(0.537909\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0.579922 0.0314507
\(341\) −29.3521 −1.58951
\(342\) 0 0
\(343\) 3.73913 0.201894
\(344\) 3.46834 0.187000
\(345\) 0 0
\(346\) 5.43792 0.292344
\(347\) 26.7072 1.43372 0.716858 0.697219i \(-0.245580\pi\)
0.716858 + 0.697219i \(0.245580\pi\)
\(348\) 0 0
\(349\) −23.5711 −1.26173 −0.630865 0.775892i \(-0.717300\pi\)
−0.630865 + 0.775892i \(0.717300\pi\)
\(350\) −4.39111 −0.234715
\(351\) 0 0
\(352\) 15.1479 0.807385
\(353\) 5.73727 0.305364 0.152682 0.988275i \(-0.451209\pi\)
0.152682 + 0.988275i \(0.451209\pi\)
\(354\) 0 0
\(355\) 10.7973 0.573063
\(356\) 8.27675 0.438667
\(357\) 0 0
\(358\) −22.7243 −1.20102
\(359\) −24.7583 −1.30669 −0.653347 0.757059i \(-0.726636\pi\)
−0.653347 + 0.757059i \(0.726636\pi\)
\(360\) 0 0
\(361\) −13.8564 −0.729285
\(362\) 22.0641 1.15967
\(363\) 0 0
\(364\) 0 0
\(365\) −4.70308 −0.246171
\(366\) 0 0
\(367\) 26.0535 1.35998 0.679992 0.733220i \(-0.261983\pi\)
0.679992 + 0.733220i \(0.261983\pi\)
\(368\) −10.5604 −0.550499
\(369\) 0 0
\(370\) −10.6162 −0.551912
\(371\) −15.9826 −0.829776
\(372\) 0 0
\(373\) 13.2045 0.683702 0.341851 0.939754i \(-0.388946\pi\)
0.341851 + 0.939754i \(0.388946\pi\)
\(374\) 7.41584 0.383464
\(375\) 0 0
\(376\) −7.92229 −0.408561
\(377\) 0 0
\(378\) 0 0
\(379\) −25.9977 −1.33541 −0.667707 0.744425i \(-0.732724\pi\)
−0.667707 + 0.744425i \(0.732724\pi\)
\(380\) −1.16202 −0.0596101
\(381\) 0 0
\(382\) −33.3418 −1.70591
\(383\) −9.60020 −0.490547 −0.245274 0.969454i \(-0.578878\pi\)
−0.245274 + 0.969454i \(0.578878\pi\)
\(384\) 0 0
\(385\) 19.3396 0.985637
\(386\) −26.5494 −1.35133
\(387\) 0 0
\(388\) 6.23453 0.316510
\(389\) −5.63129 −0.285518 −0.142759 0.989758i \(-0.545597\pi\)
−0.142759 + 0.989758i \(0.545597\pi\)
\(390\) 0 0
\(391\) 4.40617 0.222829
\(392\) −18.2675 −0.922650
\(393\) 0 0
\(394\) −2.06947 −0.104258
\(395\) 11.9826 0.602911
\(396\) 0 0
\(397\) 16.7658 0.841452 0.420726 0.907188i \(-0.361775\pi\)
0.420726 + 0.907188i \(0.361775\pi\)
\(398\) 30.8891 1.54833
\(399\) 0 0
\(400\) −2.71276 −0.135638
\(401\) −13.8780 −0.693036 −0.346518 0.938043i \(-0.612636\pi\)
−0.346518 + 0.938043i \(0.612636\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) −2.07733 −0.103351
\(405\) 0 0
\(406\) −0.108583 −0.00538888
\(407\) 46.7566 2.31764
\(408\) 0 0
\(409\) −29.4251 −1.45498 −0.727489 0.686120i \(-0.759313\pi\)
−0.727489 + 0.686120i \(0.759313\pi\)
\(410\) −4.55193 −0.224804
\(411\) 0 0
\(412\) 9.17310 0.451926
\(413\) 0.617162 0.0303686
\(414\) 0 0
\(415\) 12.1286 0.595369
\(416\) 0 0
\(417\) 0 0
\(418\) −14.8595 −0.726800
\(419\) −6.96793 −0.340406 −0.170203 0.985409i \(-0.554442\pi\)
−0.170203 + 0.985409i \(0.554442\pi\)
\(420\) 0 0
\(421\) −7.12125 −0.347069 −0.173534 0.984828i \(-0.555519\pi\)
−0.173534 + 0.984828i \(0.555519\pi\)
\(422\) −0.409213 −0.0199202
\(423\) 0 0
\(424\) −13.6036 −0.660647
\(425\) 1.13186 0.0549030
\(426\) 0 0
\(427\) −12.0975 −0.585439
\(428\) −4.67883 −0.226160
\(429\) 0 0
\(430\) 1.38051 0.0665740
\(431\) 30.2144 1.45537 0.727687 0.685909i \(-0.240595\pi\)
0.727687 + 0.685909i \(0.240595\pi\)
\(432\) 0 0
\(433\) 1.20013 0.0576745 0.0288373 0.999584i \(-0.490820\pi\)
0.0288373 + 0.999584i \(0.490820\pi\)
\(434\) 23.9935 1.15172
\(435\) 0 0
\(436\) 3.77922 0.180992
\(437\) −8.82884 −0.422341
\(438\) 0 0
\(439\) −16.5541 −0.790084 −0.395042 0.918663i \(-0.629270\pi\)
−0.395042 + 0.918663i \(0.629270\pi\)
\(440\) 16.4608 0.784740
\(441\) 0 0
\(442\) 0 0
\(443\) −4.55949 −0.216628 −0.108314 0.994117i \(-0.534545\pi\)
−0.108314 + 0.994117i \(0.534545\pi\)
\(444\) 0 0
\(445\) 16.1540 0.765775
\(446\) 14.9982 0.710185
\(447\) 0 0
\(448\) −31.9153 −1.50786
\(449\) 13.8522 0.653724 0.326862 0.945072i \(-0.394009\pi\)
0.326862 + 0.945072i \(0.394009\pi\)
\(450\) 0 0
\(451\) 20.0479 0.944018
\(452\) 3.62542 0.170526
\(453\) 0 0
\(454\) −9.30897 −0.436892
\(455\) 0 0
\(456\) 0 0
\(457\) −40.1146 −1.87648 −0.938240 0.345984i \(-0.887545\pi\)
−0.938240 + 0.345984i \(0.887545\pi\)
\(458\) 17.5644 0.820733
\(459\) 0 0
\(460\) 1.99457 0.0929972
\(461\) −7.53900 −0.351126 −0.175563 0.984468i \(-0.556175\pi\)
−0.175563 + 0.984468i \(0.556175\pi\)
\(462\) 0 0
\(463\) −23.3031 −1.08299 −0.541494 0.840705i \(-0.682141\pi\)
−0.541494 + 0.840705i \(0.682141\pi\)
\(464\) −0.0670807 −0.00311414
\(465\) 0 0
\(466\) −11.5824 −0.536542
\(467\) 22.6297 1.04718 0.523589 0.851971i \(-0.324593\pi\)
0.523589 + 0.851971i \(0.324593\pi\)
\(468\) 0 0
\(469\) 23.0405 1.06391
\(470\) −3.15332 −0.145452
\(471\) 0 0
\(472\) 0.525296 0.0241787
\(473\) −6.08012 −0.279564
\(474\) 0 0
\(475\) −2.26795 −0.104061
\(476\) 2.08783 0.0956956
\(477\) 0 0
\(478\) −24.2893 −1.11096
\(479\) 20.6448 0.943286 0.471643 0.881790i \(-0.343661\pi\)
0.471643 + 0.881790i \(0.343661\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) −28.3777 −1.29257
\(483\) 0 0
\(484\) −9.14898 −0.415863
\(485\) 12.1682 0.552528
\(486\) 0 0
\(487\) −3.03605 −0.137576 −0.0687882 0.997631i \(-0.521913\pi\)
−0.0687882 + 0.997631i \(0.521913\pi\)
\(488\) −10.2968 −0.466112
\(489\) 0 0
\(490\) −7.27105 −0.328473
\(491\) −10.6680 −0.481441 −0.240720 0.970595i \(-0.577384\pi\)
−0.240720 + 0.970595i \(0.577384\pi\)
\(492\) 0 0
\(493\) 0.0279884 0.00126053
\(494\) 0 0
\(495\) 0 0
\(496\) 14.8228 0.665562
\(497\) 38.8725 1.74367
\(498\) 0 0
\(499\) −33.9143 −1.51821 −0.759107 0.650966i \(-0.774364\pi\)
−0.759107 + 0.650966i \(0.774364\pi\)
\(500\) 0.512364 0.0229136
\(501\) 0 0
\(502\) −14.4413 −0.644546
\(503\) 12.6276 0.563037 0.281518 0.959556i \(-0.409162\pi\)
0.281518 + 0.959556i \(0.409162\pi\)
\(504\) 0 0
\(505\) −4.05441 −0.180419
\(506\) 25.5058 1.13387
\(507\) 0 0
\(508\) 5.85945 0.259971
\(509\) −24.1526 −1.07055 −0.535273 0.844679i \(-0.679791\pi\)
−0.535273 + 0.844679i \(0.679791\pi\)
\(510\) 0 0
\(511\) −16.9320 −0.749029
\(512\) −24.2750 −1.07281
\(513\) 0 0
\(514\) 6.76929 0.298581
\(515\) 17.9035 0.788921
\(516\) 0 0
\(517\) 13.8880 0.610796
\(518\) −38.2205 −1.67931
\(519\) 0 0
\(520\) 0 0
\(521\) 24.7521 1.08441 0.542205 0.840246i \(-0.317590\pi\)
0.542205 + 0.840246i \(0.317590\pi\)
\(522\) 0 0
\(523\) 37.0326 1.61932 0.809662 0.586897i \(-0.199650\pi\)
0.809662 + 0.586897i \(0.199650\pi\)
\(524\) −5.41465 −0.236540
\(525\) 0 0
\(526\) −8.36665 −0.364803
\(527\) −6.18457 −0.269404
\(528\) 0 0
\(529\) −7.84554 −0.341111
\(530\) −5.41465 −0.235197
\(531\) 0 0
\(532\) −4.18348 −0.181377
\(533\) 0 0
\(534\) 0 0
\(535\) −9.13186 −0.394805
\(536\) 19.6109 0.847062
\(537\) 0 0
\(538\) 1.73438 0.0747744
\(539\) 32.0236 1.37935
\(540\) 0 0
\(541\) 8.38144 0.360346 0.180173 0.983635i \(-0.442334\pi\)
0.180173 + 0.983635i \(0.442334\pi\)
\(542\) 12.1596 0.522301
\(543\) 0 0
\(544\) 3.19170 0.136843
\(545\) 7.37605 0.315955
\(546\) 0 0
\(547\) −22.7842 −0.974181 −0.487091 0.873351i \(-0.661942\pi\)
−0.487091 + 0.873351i \(0.661942\pi\)
\(548\) −1.94018 −0.0828804
\(549\) 0 0
\(550\) 6.55193 0.279375
\(551\) −0.0560816 −0.00238915
\(552\) 0 0
\(553\) 43.1398 1.83449
\(554\) −21.3735 −0.908071
\(555\) 0 0
\(556\) −1.03183 −0.0437594
\(557\) −28.1527 −1.19287 −0.596435 0.802662i \(-0.703417\pi\)
−0.596435 + 0.802662i \(0.703417\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) −9.76645 −0.412708
\(561\) 0 0
\(562\) 13.0935 0.552317
\(563\) 18.1303 0.764101 0.382050 0.924142i \(-0.375218\pi\)
0.382050 + 0.924142i \(0.375218\pi\)
\(564\) 0 0
\(565\) 7.07588 0.297684
\(566\) 1.60801 0.0675899
\(567\) 0 0
\(568\) 33.0862 1.38827
\(569\) −40.5985 −1.70198 −0.850988 0.525185i \(-0.823996\pi\)
−0.850988 + 0.525185i \(0.823996\pi\)
\(570\) 0 0
\(571\) 24.7159 1.03433 0.517164 0.855886i \(-0.326988\pi\)
0.517164 + 0.855886i \(0.326988\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) −16.3879 −0.684016
\(575\) 3.89287 0.162344
\(576\) 0 0
\(577\) −23.0691 −0.960379 −0.480189 0.877165i \(-0.659432\pi\)
−0.480189 + 0.877165i \(0.659432\pi\)
\(578\) −19.1721 −0.797456
\(579\) 0 0
\(580\) 0.0126697 0.000526079 0
\(581\) 43.6653 1.81154
\(582\) 0 0
\(583\) 23.8475 0.987662
\(584\) −14.4116 −0.596358
\(585\) 0 0
\(586\) 22.8602 0.944345
\(587\) −20.3523 −0.840030 −0.420015 0.907517i \(-0.637975\pi\)
−0.420015 + 0.907517i \(0.637975\pi\)
\(588\) 0 0
\(589\) 12.3923 0.510616
\(590\) 0.209084 0.00860786
\(591\) 0 0
\(592\) −23.6120 −0.970447
\(593\) 10.3834 0.426395 0.213198 0.977009i \(-0.431612\pi\)
0.213198 + 0.977009i \(0.431612\pi\)
\(594\) 0 0
\(595\) 4.07490 0.167055
\(596\) 2.82712 0.115803
\(597\) 0 0
\(598\) 0 0
\(599\) 31.5965 1.29100 0.645499 0.763761i \(-0.276649\pi\)
0.645499 + 0.763761i \(0.276649\pi\)
\(600\) 0 0
\(601\) −43.8845 −1.79009 −0.895044 0.445979i \(-0.852856\pi\)
−0.895044 + 0.445979i \(0.852856\pi\)
\(602\) 4.97010 0.202566
\(603\) 0 0
\(604\) −2.50526 −0.101938
\(605\) −17.8564 −0.725966
\(606\) 0 0
\(607\) 2.17540 0.0882968 0.0441484 0.999025i \(-0.485943\pi\)
0.0441484 + 0.999025i \(0.485943\pi\)
\(608\) −6.39535 −0.259366
\(609\) 0 0
\(610\) −4.09843 −0.165941
\(611\) 0 0
\(612\) 0 0
\(613\) 14.7620 0.596231 0.298116 0.954530i \(-0.403642\pi\)
0.298116 + 0.954530i \(0.403642\pi\)
\(614\) −17.4467 −0.704092
\(615\) 0 0
\(616\) 59.2622 2.38774
\(617\) −20.2972 −0.817134 −0.408567 0.912728i \(-0.633971\pi\)
−0.408567 + 0.912728i \(0.633971\pi\)
\(618\) 0 0
\(619\) −9.94207 −0.399605 −0.199803 0.979836i \(-0.564030\pi\)
−0.199803 + 0.979836i \(0.564030\pi\)
\(620\) −2.79961 −0.112435
\(621\) 0 0
\(622\) −3.36758 −0.135028
\(623\) 58.1577 2.33004
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) −19.9855 −0.798782
\(627\) 0 0
\(628\) −5.14441 −0.205284
\(629\) 9.85174 0.392815
\(630\) 0 0
\(631\) −0.973420 −0.0387512 −0.0193756 0.999812i \(-0.506168\pi\)
−0.0193756 + 0.999812i \(0.506168\pi\)
\(632\) 36.7183 1.46058
\(633\) 0 0
\(634\) 2.17805 0.0865014
\(635\) 11.4361 0.453828
\(636\) 0 0
\(637\) 0 0
\(638\) 0.162015 0.00641425
\(639\) 0 0
\(640\) −5.17262 −0.204466
\(641\) 12.6209 0.498497 0.249249 0.968440i \(-0.419816\pi\)
0.249249 + 0.968440i \(0.419816\pi\)
\(642\) 0 0
\(643\) −9.96043 −0.392801 −0.196401 0.980524i \(-0.562925\pi\)
−0.196401 + 0.980524i \(0.562925\pi\)
\(644\) 7.18083 0.282964
\(645\) 0 0
\(646\) −3.13092 −0.123185
\(647\) 36.2763 1.42617 0.713084 0.701079i \(-0.247298\pi\)
0.713084 + 0.701079i \(0.247298\pi\)
\(648\) 0 0
\(649\) −0.920861 −0.0361470
\(650\) 0 0
\(651\) 0 0
\(652\) 3.47447 0.136071
\(653\) −13.7554 −0.538289 −0.269145 0.963100i \(-0.586741\pi\)
−0.269145 + 0.963100i \(0.586741\pi\)
\(654\) 0 0
\(655\) −10.5680 −0.412925
\(656\) −10.1241 −0.395281
\(657\) 0 0
\(658\) −11.3526 −0.442570
\(659\) 2.58183 0.100574 0.0502869 0.998735i \(-0.483986\pi\)
0.0502869 + 0.998735i \(0.483986\pi\)
\(660\) 0 0
\(661\) −24.8765 −0.967582 −0.483791 0.875183i \(-0.660741\pi\)
−0.483791 + 0.875183i \(0.660741\pi\)
\(662\) −8.81151 −0.342469
\(663\) 0 0
\(664\) 37.1656 1.44231
\(665\) −8.16506 −0.316627
\(666\) 0 0
\(667\) 0.0962625 0.00372730
\(668\) −5.37460 −0.207949
\(669\) 0 0
\(670\) 7.80576 0.301563
\(671\) 18.0506 0.696834
\(672\) 0 0
\(673\) 43.3222 1.66995 0.834974 0.550289i \(-0.185483\pi\)
0.834974 + 0.550289i \(0.185483\pi\)
\(674\) −5.32051 −0.204938
\(675\) 0 0
\(676\) 0 0
\(677\) 41.3625 1.58969 0.794845 0.606813i \(-0.207552\pi\)
0.794845 + 0.606813i \(0.207552\pi\)
\(678\) 0 0
\(679\) 43.8078 1.68119
\(680\) 3.46834 0.133005
\(681\) 0 0
\(682\) −35.8004 −1.37087
\(683\) 2.62688 0.100515 0.0502574 0.998736i \(-0.483996\pi\)
0.0502574 + 0.998736i \(0.483996\pi\)
\(684\) 0 0
\(685\) −3.78672 −0.144683
\(686\) 4.56057 0.174123
\(687\) 0 0
\(688\) 3.07045 0.117060
\(689\) 0 0
\(690\) 0 0
\(691\) 15.2753 0.581099 0.290550 0.956860i \(-0.406162\pi\)
0.290550 + 0.956860i \(0.406162\pi\)
\(692\) −2.28435 −0.0868380
\(693\) 0 0
\(694\) 32.5744 1.23651
\(695\) −2.01386 −0.0763902
\(696\) 0 0
\(697\) 4.22414 0.160001
\(698\) −28.7493 −1.08818
\(699\) 0 0
\(700\) 1.84461 0.0697197
\(701\) 48.1947 1.82029 0.910144 0.414292i \(-0.135971\pi\)
0.910144 + 0.414292i \(0.135971\pi\)
\(702\) 0 0
\(703\) −19.7404 −0.744522
\(704\) 47.6205 1.79477
\(705\) 0 0
\(706\) 6.99767 0.263361
\(707\) −14.5967 −0.548964
\(708\) 0 0
\(709\) 38.8699 1.45979 0.729896 0.683559i \(-0.239569\pi\)
0.729896 + 0.683559i \(0.239569\pi\)
\(710\) 13.1694 0.494237
\(711\) 0 0
\(712\) 49.5008 1.85512
\(713\) −21.2711 −0.796607
\(714\) 0 0
\(715\) 0 0
\(716\) 9.54600 0.356751
\(717\) 0 0
\(718\) −30.1974 −1.12696
\(719\) −6.61660 −0.246758 −0.123379 0.992360i \(-0.539373\pi\)
−0.123379 + 0.992360i \(0.539373\pi\)
\(720\) 0 0
\(721\) 64.4560 2.40047
\(722\) −16.9005 −0.628971
\(723\) 0 0
\(724\) −9.26867 −0.344467
\(725\) 0.0247279 0.000918370 0
\(726\) 0 0
\(727\) −18.3735 −0.681435 −0.340717 0.940166i \(-0.610670\pi\)
−0.340717 + 0.940166i \(0.610670\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) −5.73629 −0.212310
\(731\) −1.28110 −0.0473830
\(732\) 0 0
\(733\) −0.791131 −0.0292211 −0.0146105 0.999893i \(-0.504651\pi\)
−0.0146105 + 0.999893i \(0.504651\pi\)
\(734\) 31.7772 1.17292
\(735\) 0 0
\(736\) 10.9774 0.404634
\(737\) −34.3786 −1.26635
\(738\) 0 0
\(739\) −31.1853 −1.14717 −0.573585 0.819146i \(-0.694448\pi\)
−0.573585 + 0.819146i \(0.694448\pi\)
\(740\) 4.45965 0.163940
\(741\) 0 0
\(742\) −19.4938 −0.715639
\(743\) 5.56304 0.204088 0.102044 0.994780i \(-0.467462\pi\)
0.102044 + 0.994780i \(0.467462\pi\)
\(744\) 0 0
\(745\) 5.51780 0.202156
\(746\) 16.1053 0.589658
\(747\) 0 0
\(748\) −3.11523 −0.113904
\(749\) −32.8765 −1.20128
\(750\) 0 0
\(751\) −35.2097 −1.28482 −0.642410 0.766361i \(-0.722065\pi\)
−0.642410 + 0.766361i \(0.722065\pi\)
\(752\) −7.01343 −0.255754
\(753\) 0 0
\(754\) 0 0
\(755\) −4.88961 −0.177951
\(756\) 0 0
\(757\) 50.0446 1.81890 0.909451 0.415810i \(-0.136502\pi\)
0.909451 + 0.415810i \(0.136502\pi\)
\(758\) −31.7091 −1.15173
\(759\) 0 0
\(760\) −6.94967 −0.252091
\(761\) 44.8209 1.62476 0.812379 0.583130i \(-0.198172\pi\)
0.812379 + 0.583130i \(0.198172\pi\)
\(762\) 0 0
\(763\) 26.5552 0.961364
\(764\) 14.0061 0.506725
\(765\) 0 0
\(766\) −11.7092 −0.423072
\(767\) 0 0
\(768\) 0 0
\(769\) 39.3633 1.41948 0.709739 0.704465i \(-0.248813\pi\)
0.709739 + 0.704465i \(0.248813\pi\)
\(770\) 23.5882 0.850061
\(771\) 0 0
\(772\) 11.1528 0.401399
\(773\) 48.7805 1.75451 0.877256 0.480022i \(-0.159371\pi\)
0.877256 + 0.480022i \(0.159371\pi\)
\(774\) 0 0
\(775\) −5.46410 −0.196276
\(776\) 37.2869 1.33852
\(777\) 0 0
\(778\) −6.86841 −0.246244
\(779\) −8.46410 −0.303258
\(780\) 0 0
\(781\) −58.0013 −2.07545
\(782\) 5.37415 0.192179
\(783\) 0 0
\(784\) −16.1718 −0.577566
\(785\) −10.0405 −0.358363
\(786\) 0 0
\(787\) −39.8608 −1.42088 −0.710442 0.703756i \(-0.751505\pi\)
−0.710442 + 0.703756i \(0.751505\pi\)
\(788\) 0.869338 0.0309689
\(789\) 0 0
\(790\) 14.6150 0.519980
\(791\) 25.4745 0.905771
\(792\) 0 0
\(793\) 0 0
\(794\) 20.4490 0.725709
\(795\) 0 0
\(796\) −12.9759 −0.459917
\(797\) −26.2118 −0.928470 −0.464235 0.885712i \(-0.653671\pi\)
−0.464235 + 0.885712i \(0.653671\pi\)
\(798\) 0 0
\(799\) 2.92625 0.103523
\(800\) 2.81988 0.0996979
\(801\) 0 0
\(802\) −16.9269 −0.597708
\(803\) 25.2641 0.891551
\(804\) 0 0
\(805\) 14.0151 0.493968
\(806\) 0 0
\(807\) 0 0
\(808\) −12.4239 −0.437072
\(809\) −22.2136 −0.780990 −0.390495 0.920605i \(-0.627696\pi\)
−0.390495 + 0.920605i \(0.627696\pi\)
\(810\) 0 0
\(811\) −19.0950 −0.670515 −0.335257 0.942127i \(-0.608823\pi\)
−0.335257 + 0.942127i \(0.608823\pi\)
\(812\) 0.0456133 0.00160071
\(813\) 0 0
\(814\) 57.0284 1.99885
\(815\) 6.78124 0.237537
\(816\) 0 0
\(817\) 2.56699 0.0898076
\(818\) −35.8894 −1.25484
\(819\) 0 0
\(820\) 1.91217 0.0667758
\(821\) −34.1584 −1.19214 −0.596068 0.802934i \(-0.703271\pi\)
−0.596068 + 0.802934i \(0.703271\pi\)
\(822\) 0 0
\(823\) 4.07940 0.142199 0.0710995 0.997469i \(-0.477349\pi\)
0.0710995 + 0.997469i \(0.477349\pi\)
\(824\) 54.8616 1.91119
\(825\) 0 0
\(826\) 0.752744 0.0261913
\(827\) −54.8780 −1.90830 −0.954148 0.299337i \(-0.903235\pi\)
−0.954148 + 0.299337i \(0.903235\pi\)
\(828\) 0 0
\(829\) 8.14950 0.283044 0.141522 0.989935i \(-0.454800\pi\)
0.141522 + 0.989935i \(0.454800\pi\)
\(830\) 14.7931 0.513476
\(831\) 0 0
\(832\) 0 0
\(833\) 6.74745 0.233785
\(834\) 0 0
\(835\) −10.4898 −0.363015
\(836\) 6.24213 0.215889
\(837\) 0 0
\(838\) −8.49869 −0.293582
\(839\) −21.8865 −0.755606 −0.377803 0.925886i \(-0.623320\pi\)
−0.377803 + 0.925886i \(0.623320\pi\)
\(840\) 0 0
\(841\) −28.9994 −0.999979
\(842\) −8.68570 −0.299329
\(843\) 0 0
\(844\) 0.171902 0.00591710
\(845\) 0 0
\(846\) 0 0
\(847\) −64.2866 −2.20891
\(848\) −12.0429 −0.413556
\(849\) 0 0
\(850\) 1.38051 0.0473511
\(851\) 33.8838 1.16152
\(852\) 0 0
\(853\) 19.2240 0.658217 0.329108 0.944292i \(-0.393252\pi\)
0.329108 + 0.944292i \(0.393252\pi\)
\(854\) −14.7552 −0.504911
\(855\) 0 0
\(856\) −27.9827 −0.956430
\(857\) 27.8197 0.950302 0.475151 0.879904i \(-0.342393\pi\)
0.475151 + 0.879904i \(0.342393\pi\)
\(858\) 0 0
\(859\) 45.7355 1.56048 0.780238 0.625482i \(-0.215098\pi\)
0.780238 + 0.625482i \(0.215098\pi\)
\(860\) −0.579922 −0.0197752
\(861\) 0 0
\(862\) 36.8521 1.25519
\(863\) 54.8186 1.86605 0.933024 0.359814i \(-0.117160\pi\)
0.933024 + 0.359814i \(0.117160\pi\)
\(864\) 0 0
\(865\) −4.45845 −0.151592
\(866\) 1.46378 0.0497413
\(867\) 0 0
\(868\) −10.0791 −0.342108
\(869\) −64.3684 −2.18355
\(870\) 0 0
\(871\) 0 0
\(872\) 22.6024 0.765415
\(873\) 0 0
\(874\) −10.7684 −0.364247
\(875\) 3.60020 0.121709
\(876\) 0 0
\(877\) 27.3794 0.924537 0.462269 0.886740i \(-0.347036\pi\)
0.462269 + 0.886740i \(0.347036\pi\)
\(878\) −20.1908 −0.681407
\(879\) 0 0
\(880\) 14.5724 0.491236
\(881\) 34.4426 1.16040 0.580200 0.814474i \(-0.302974\pi\)
0.580200 + 0.814474i \(0.302974\pi\)
\(882\) 0 0
\(883\) −17.3592 −0.584183 −0.292092 0.956390i \(-0.594351\pi\)
−0.292092 + 0.956390i \(0.594351\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) −5.56115 −0.186831
\(887\) 31.1427 1.04567 0.522835 0.852434i \(-0.324874\pi\)
0.522835 + 0.852434i \(0.324874\pi\)
\(888\) 0 0
\(889\) 41.1722 1.38087
\(890\) 19.7029 0.660442
\(891\) 0 0
\(892\) −6.30042 −0.210954
\(893\) −5.86345 −0.196213
\(894\) 0 0
\(895\) 18.6313 0.622775
\(896\) −18.6224 −0.622132
\(897\) 0 0
\(898\) 16.8953 0.563804
\(899\) −0.135116 −0.00450636
\(900\) 0 0
\(901\) 5.02473 0.167398
\(902\) 24.4521 0.814167
\(903\) 0 0
\(904\) 21.6826 0.721152
\(905\) −18.0900 −0.601332
\(906\) 0 0
\(907\) 17.6057 0.584587 0.292294 0.956329i \(-0.405582\pi\)
0.292294 + 0.956329i \(0.405582\pi\)
\(908\) 3.91050 0.129774
\(909\) 0 0
\(910\) 0 0
\(911\) −50.0232 −1.65734 −0.828671 0.559737i \(-0.810902\pi\)
−0.828671 + 0.559737i \(0.810902\pi\)
\(912\) 0 0
\(913\) −65.1526 −2.15624
\(914\) −48.9272 −1.61837
\(915\) 0 0
\(916\) −7.37844 −0.243791
\(917\) −38.0468 −1.25641
\(918\) 0 0
\(919\) −7.61556 −0.251214 −0.125607 0.992080i \(-0.540088\pi\)
−0.125607 + 0.992080i \(0.540088\pi\)
\(920\) 11.9289 0.393285
\(921\) 0 0
\(922\) −9.19522 −0.302828
\(923\) 0 0
\(924\) 0 0
\(925\) 8.70406 0.286188
\(926\) −28.4225 −0.934022
\(927\) 0 0
\(928\) 0.0697297 0.00228899
\(929\) −14.1239 −0.463391 −0.231695 0.972788i \(-0.574427\pi\)
−0.231695 + 0.972788i \(0.574427\pi\)
\(930\) 0 0
\(931\) −13.5202 −0.443106
\(932\) 4.86550 0.159375
\(933\) 0 0
\(934\) 27.6012 0.903138
\(935\) −6.08012 −0.198841
\(936\) 0 0
\(937\) −23.9317 −0.781815 −0.390908 0.920430i \(-0.627839\pi\)
−0.390908 + 0.920430i \(0.627839\pi\)
\(938\) 28.1023 0.917571
\(939\) 0 0
\(940\) 1.32464 0.0432051
\(941\) −25.3591 −0.826683 −0.413342 0.910576i \(-0.635638\pi\)
−0.413342 + 0.910576i \(0.635638\pi\)
\(942\) 0 0
\(943\) 14.5284 0.473110
\(944\) 0.465033 0.0151355
\(945\) 0 0
\(946\) −7.41584 −0.241110
\(947\) 41.4223 1.34604 0.673021 0.739623i \(-0.264996\pi\)
0.673021 + 0.739623i \(0.264996\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) −2.76619 −0.0897470
\(951\) 0 0
\(952\) 12.4867 0.404696
\(953\) −24.3026 −0.787237 −0.393619 0.919274i \(-0.628777\pi\)
−0.393619 + 0.919274i \(0.628777\pi\)
\(954\) 0 0
\(955\) 27.3363 0.884583
\(956\) 10.2034 0.330001
\(957\) 0 0
\(958\) 25.1802 0.813536
\(959\) −13.6329 −0.440231
\(960\) 0 0
\(961\) −1.14359 −0.0368901
\(962\) 0 0
\(963\) 0 0
\(964\) 11.9209 0.383945
\(965\) 21.7674 0.700717
\(966\) 0 0
\(967\) −23.6784 −0.761445 −0.380722 0.924689i \(-0.624325\pi\)
−0.380722 + 0.924689i \(0.624325\pi\)
\(968\) −54.7173 −1.75868
\(969\) 0 0
\(970\) 14.8413 0.476527
\(971\) 16.9722 0.544663 0.272332 0.962203i \(-0.412205\pi\)
0.272332 + 0.962203i \(0.412205\pi\)
\(972\) 0 0
\(973\) −7.25030 −0.232434
\(974\) −3.70303 −0.118653
\(975\) 0 0
\(976\) −9.11550 −0.291780
\(977\) 24.9994 0.799802 0.399901 0.916558i \(-0.369044\pi\)
0.399901 + 0.916558i \(0.369044\pi\)
\(978\) 0 0
\(979\) −86.7765 −2.77339
\(980\) 3.05441 0.0975696
\(981\) 0 0
\(982\) −13.0116 −0.415218
\(983\) −27.3418 −0.872068 −0.436034 0.899930i \(-0.643617\pi\)
−0.436034 + 0.899930i \(0.643617\pi\)
\(984\) 0 0
\(985\) 1.69672 0.0540620
\(986\) 0.0341370 0.00108715
\(987\) 0 0
\(988\) 0 0
\(989\) −4.40617 −0.140108
\(990\) 0 0
\(991\) 16.0760 0.510672 0.255336 0.966852i \(-0.417814\pi\)
0.255336 + 0.966852i \(0.417814\pi\)
\(992\) −15.4081 −0.489208
\(993\) 0 0
\(994\) 47.4123 1.50383
\(995\) −25.3255 −0.802871
\(996\) 0 0
\(997\) −34.5612 −1.09456 −0.547282 0.836948i \(-0.684337\pi\)
−0.547282 + 0.836948i \(0.684337\pi\)
\(998\) −41.3649 −1.30938
\(999\) 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 7605.2.a.cj.1.3 4
3.2 odd 2 845.2.a.l.1.2 4
13.6 odd 12 585.2.bu.c.361.2 8
13.11 odd 12 585.2.bu.c.316.2 8
13.12 even 2 7605.2.a.cf.1.2 4
15.14 odd 2 4225.2.a.bl.1.3 4
39.2 even 12 845.2.m.g.316.2 8
39.5 even 4 845.2.c.g.506.6 8
39.8 even 4 845.2.c.g.506.3 8
39.11 even 12 65.2.m.a.56.3 yes 8
39.17 odd 6 845.2.e.m.146.2 8
39.20 even 12 845.2.m.g.361.2 8
39.23 odd 6 845.2.e.m.191.2 8
39.29 odd 6 845.2.e.n.191.3 8
39.32 even 12 65.2.m.a.36.3 8
39.35 odd 6 845.2.e.n.146.3 8
39.38 odd 2 845.2.a.m.1.3 4
156.11 odd 12 1040.2.da.b.641.4 8
156.71 odd 12 1040.2.da.b.881.4 8
195.32 odd 12 325.2.m.c.49.3 8
195.89 even 12 325.2.n.d.251.2 8
195.128 odd 12 325.2.m.c.199.3 8
195.149 even 12 325.2.n.d.101.2 8
195.167 odd 12 325.2.m.b.199.2 8
195.188 odd 12 325.2.m.b.49.2 8
195.194 odd 2 4225.2.a.bi.1.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
65.2.m.a.36.3 8 39.32 even 12
65.2.m.a.56.3 yes 8 39.11 even 12
325.2.m.b.49.2 8 195.188 odd 12
325.2.m.b.199.2 8 195.167 odd 12
325.2.m.c.49.3 8 195.32 odd 12
325.2.m.c.199.3 8 195.128 odd 12
325.2.n.d.101.2 8 195.149 even 12
325.2.n.d.251.2 8 195.89 even 12
585.2.bu.c.316.2 8 13.11 odd 12
585.2.bu.c.361.2 8 13.6 odd 12
845.2.a.l.1.2 4 3.2 odd 2
845.2.a.m.1.3 4 39.38 odd 2
845.2.c.g.506.3 8 39.8 even 4
845.2.c.g.506.6 8 39.5 even 4
845.2.e.m.146.2 8 39.17 odd 6
845.2.e.m.191.2 8 39.23 odd 6
845.2.e.n.146.3 8 39.35 odd 6
845.2.e.n.191.3 8 39.29 odd 6
845.2.m.g.316.2 8 39.2 even 12
845.2.m.g.361.2 8 39.20 even 12
1040.2.da.b.641.4 8 156.11 odd 12
1040.2.da.b.881.4 8 156.71 odd 12
4225.2.a.bi.1.2 4 195.194 odd 2
4225.2.a.bl.1.3 4 15.14 odd 2
7605.2.a.cf.1.2 4 13.12 even 2
7605.2.a.cj.1.3 4 1.1 even 1 trivial