Properties

Label 1521.2.a.p.1.2
Level $1521$
Weight $2$
Character 1521.1
Self dual yes
Analytic conductor $12.145$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1521 = 3^{2} \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1521.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(12.1452461474\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: \(\Q(\zeta_{14})^+\)
Defining polynomial: \(x^{3} - x^{2} - 2 x + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 507)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(0.445042\) of defining polynomial
Character \(\chi\) \(=\) 1521.1

$q$-expansion

\(f(q)\) \(=\) \(q-0.445042 q^{2} -1.80194 q^{4} +0.246980 q^{5} +1.75302 q^{7} +1.69202 q^{8} +O(q^{10})\) \(q-0.445042 q^{2} -1.80194 q^{4} +0.246980 q^{5} +1.75302 q^{7} +1.69202 q^{8} -0.109916 q^{10} -5.65279 q^{11} -0.780167 q^{14} +2.85086 q^{16} +3.80194 q^{17} +5.58211 q^{19} -0.445042 q^{20} +2.51573 q^{22} -8.34481 q^{23} -4.93900 q^{25} -3.15883 q^{28} +5.93900 q^{29} +5.26875 q^{31} -4.65279 q^{32} -1.69202 q^{34} +0.432960 q^{35} +3.19806 q^{37} -2.48427 q^{38} +0.417895 q^{40} -0.445042 q^{41} +1.71379 q^{43} +10.1860 q^{44} +3.71379 q^{46} +6.73556 q^{47} -3.92692 q^{49} +2.19806 q^{50} +1.06100 q^{53} -1.39612 q^{55} +2.96615 q^{56} -2.64310 q^{58} +13.7017 q^{59} -8.51573 q^{61} -2.34481 q^{62} -3.63102 q^{64} +5.96077 q^{67} -6.85086 q^{68} -0.192685 q^{70} +5.71917 q^{71} +7.35690 q^{73} -1.42327 q^{74} -10.0586 q^{76} -9.90946 q^{77} +4.45473 q^{79} +0.704103 q^{80} +0.198062 q^{82} +10.1860 q^{83} +0.939001 q^{85} -0.762709 q^{86} -9.56465 q^{88} -0.137063 q^{89} +15.0368 q^{92} -2.99761 q^{94} +1.37867 q^{95} -13.6896 q^{97} +1.74764 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q - q^{2} - q^{4} - 4q^{5} + 10q^{7} + O(q^{10}) \) \( 3q - q^{2} - q^{4} - 4q^{5} + 10q^{7} - q^{10} + q^{11} - q^{14} - 5q^{16} + 7q^{17} + 11q^{19} - q^{20} - 5q^{22} - 2q^{23} - 5q^{25} - q^{28} + 8q^{29} + 8q^{31} + 4q^{32} - 18q^{35} + 14q^{37} - 20q^{38} + 7q^{40} - q^{41} - 3q^{43} + 16q^{44} + 3q^{46} + 9q^{47} + 17q^{49} + 11q^{50} + 13q^{53} - 13q^{55} - 7q^{56} - 12q^{58} + 14q^{59} - 13q^{61} + 16q^{62} + 4q^{64} + 5q^{67} - 7q^{68} - 8q^{70} + 6q^{71} + 18q^{73} - 7q^{74} + q^{76} + 15q^{77} - 9q^{79} + 16q^{80} + 5q^{82} + 16q^{83} - 7q^{85} + 15q^{86} - 7q^{88} + 5q^{89} + 17q^{92} + 32q^{94} - 3q^{95} + 5q^{97} + 13q^{98} + O(q^{100}) \)

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.445042 −0.314692 −0.157346 0.987544i \(-0.550294\pi\)
−0.157346 + 0.987544i \(0.550294\pi\)
\(3\) 0 0
\(4\) −1.80194 −0.900969
\(5\) 0.246980 0.110453 0.0552263 0.998474i \(-0.482412\pi\)
0.0552263 + 0.998474i \(0.482412\pi\)
\(6\) 0 0
\(7\) 1.75302 0.662579 0.331290 0.943529i \(-0.392516\pi\)
0.331290 + 0.943529i \(0.392516\pi\)
\(8\) 1.69202 0.598220
\(9\) 0 0
\(10\) −0.109916 −0.0347586
\(11\) −5.65279 −1.70438 −0.852191 0.523232i \(-0.824726\pi\)
−0.852191 + 0.523232i \(0.824726\pi\)
\(12\) 0 0
\(13\) 0 0
\(14\) −0.780167 −0.208509
\(15\) 0 0
\(16\) 2.85086 0.712714
\(17\) 3.80194 0.922105 0.461053 0.887373i \(-0.347472\pi\)
0.461053 + 0.887373i \(0.347472\pi\)
\(18\) 0 0
\(19\) 5.58211 1.28062 0.640311 0.768115i \(-0.278805\pi\)
0.640311 + 0.768115i \(0.278805\pi\)
\(20\) −0.445042 −0.0995144
\(21\) 0 0
\(22\) 2.51573 0.536355
\(23\) −8.34481 −1.74001 −0.870007 0.493039i \(-0.835886\pi\)
−0.870007 + 0.493039i \(0.835886\pi\)
\(24\) 0 0
\(25\) −4.93900 −0.987800
\(26\) 0 0
\(27\) 0 0
\(28\) −3.15883 −0.596963
\(29\) 5.93900 1.10284 0.551422 0.834226i \(-0.314085\pi\)
0.551422 + 0.834226i \(0.314085\pi\)
\(30\) 0 0
\(31\) 5.26875 0.946295 0.473148 0.880983i \(-0.343118\pi\)
0.473148 + 0.880983i \(0.343118\pi\)
\(32\) −4.65279 −0.822505
\(33\) 0 0
\(34\) −1.69202 −0.290179
\(35\) 0.432960 0.0731836
\(36\) 0 0
\(37\) 3.19806 0.525758 0.262879 0.964829i \(-0.415328\pi\)
0.262879 + 0.964829i \(0.415328\pi\)
\(38\) −2.48427 −0.403002
\(39\) 0 0
\(40\) 0.417895 0.0660750
\(41\) −0.445042 −0.0695039 −0.0347519 0.999396i \(-0.511064\pi\)
−0.0347519 + 0.999396i \(0.511064\pi\)
\(42\) 0 0
\(43\) 1.71379 0.261351 0.130675 0.991425i \(-0.458285\pi\)
0.130675 + 0.991425i \(0.458285\pi\)
\(44\) 10.1860 1.53559
\(45\) 0 0
\(46\) 3.71379 0.547569
\(47\) 6.73556 0.982483 0.491241 0.871024i \(-0.336543\pi\)
0.491241 + 0.871024i \(0.336543\pi\)
\(48\) 0 0
\(49\) −3.92692 −0.560988
\(50\) 2.19806 0.310853
\(51\) 0 0
\(52\) 0 0
\(53\) 1.06100 0.145739 0.0728697 0.997341i \(-0.476784\pi\)
0.0728697 + 0.997341i \(0.476784\pi\)
\(54\) 0 0
\(55\) −1.39612 −0.188253
\(56\) 2.96615 0.396368
\(57\) 0 0
\(58\) −2.64310 −0.347057
\(59\) 13.7017 1.78381 0.891905 0.452222i \(-0.149369\pi\)
0.891905 + 0.452222i \(0.149369\pi\)
\(60\) 0 0
\(61\) −8.51573 −1.09033 −0.545164 0.838330i \(-0.683533\pi\)
−0.545164 + 0.838330i \(0.683533\pi\)
\(62\) −2.34481 −0.297792
\(63\) 0 0
\(64\) −3.63102 −0.453878
\(65\) 0 0
\(66\) 0 0
\(67\) 5.96077 0.728224 0.364112 0.931355i \(-0.381373\pi\)
0.364112 + 0.931355i \(0.381373\pi\)
\(68\) −6.85086 −0.830788
\(69\) 0 0
\(70\) −0.192685 −0.0230303
\(71\) 5.71917 0.678740 0.339370 0.940653i \(-0.389786\pi\)
0.339370 + 0.940653i \(0.389786\pi\)
\(72\) 0 0
\(73\) 7.35690 0.861060 0.430530 0.902576i \(-0.358327\pi\)
0.430530 + 0.902576i \(0.358327\pi\)
\(74\) −1.42327 −0.165452
\(75\) 0 0
\(76\) −10.0586 −1.15380
\(77\) −9.90946 −1.12929
\(78\) 0 0
\(79\) 4.45473 0.501196 0.250598 0.968091i \(-0.419373\pi\)
0.250598 + 0.968091i \(0.419373\pi\)
\(80\) 0.704103 0.0787211
\(81\) 0 0
\(82\) 0.198062 0.0218723
\(83\) 10.1860 1.11806 0.559028 0.829149i \(-0.311174\pi\)
0.559028 + 0.829149i \(0.311174\pi\)
\(84\) 0 0
\(85\) 0.939001 0.101849
\(86\) −0.762709 −0.0822450
\(87\) 0 0
\(88\) −9.56465 −1.01959
\(89\) −0.137063 −0.0145287 −0.00726434 0.999974i \(-0.502312\pi\)
−0.00726434 + 0.999974i \(0.502312\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 15.0368 1.56770
\(93\) 0 0
\(94\) −2.99761 −0.309180
\(95\) 1.37867 0.141448
\(96\) 0 0
\(97\) −13.6896 −1.38997 −0.694986 0.719024i \(-0.744589\pi\)
−0.694986 + 0.719024i \(0.744589\pi\)
\(98\) 1.74764 0.176539
\(99\) 0 0
\(100\) 8.89977 0.889977
\(101\) 5.41119 0.538434 0.269217 0.963080i \(-0.413235\pi\)
0.269217 + 0.963080i \(0.413235\pi\)
\(102\) 0 0
\(103\) 13.7560 1.35542 0.677710 0.735330i \(-0.262973\pi\)
0.677710 + 0.735330i \(0.262973\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) −0.472189 −0.0458630
\(107\) 12.8170 1.23907 0.619533 0.784970i \(-0.287322\pi\)
0.619533 + 0.784970i \(0.287322\pi\)
\(108\) 0 0
\(109\) 12.1468 1.16345 0.581724 0.813386i \(-0.302378\pi\)
0.581724 + 0.813386i \(0.302378\pi\)
\(110\) 0.621334 0.0592419
\(111\) 0 0
\(112\) 4.99761 0.472229
\(113\) 1.63773 0.154064 0.0770322 0.997029i \(-0.475456\pi\)
0.0770322 + 0.997029i \(0.475456\pi\)
\(114\) 0 0
\(115\) −2.06100 −0.192189
\(116\) −10.7017 −0.993629
\(117\) 0 0
\(118\) −6.09783 −0.561351
\(119\) 6.66487 0.610968
\(120\) 0 0
\(121\) 20.9541 1.90492
\(122\) 3.78986 0.343117
\(123\) 0 0
\(124\) −9.49396 −0.852583
\(125\) −2.45473 −0.219558
\(126\) 0 0
\(127\) 10.7995 0.958305 0.479152 0.877732i \(-0.340944\pi\)
0.479152 + 0.877732i \(0.340944\pi\)
\(128\) 10.9215 0.965337
\(129\) 0 0
\(130\) 0 0
\(131\) −0.907542 −0.0792923 −0.0396462 0.999214i \(-0.512623\pi\)
−0.0396462 + 0.999214i \(0.512623\pi\)
\(132\) 0 0
\(133\) 9.78554 0.848514
\(134\) −2.65279 −0.229166
\(135\) 0 0
\(136\) 6.43296 0.551622
\(137\) 9.54825 0.815762 0.407881 0.913035i \(-0.366268\pi\)
0.407881 + 0.913035i \(0.366268\pi\)
\(138\) 0 0
\(139\) −4.09246 −0.347118 −0.173559 0.984823i \(-0.555527\pi\)
−0.173559 + 0.984823i \(0.555527\pi\)
\(140\) −0.780167 −0.0659362
\(141\) 0 0
\(142\) −2.54527 −0.213594
\(143\) 0 0
\(144\) 0 0
\(145\) 1.46681 0.121812
\(146\) −3.27413 −0.270969
\(147\) 0 0
\(148\) −5.76271 −0.473692
\(149\) 15.3884 1.26066 0.630332 0.776326i \(-0.282919\pi\)
0.630332 + 0.776326i \(0.282919\pi\)
\(150\) 0 0
\(151\) −3.67456 −0.299032 −0.149516 0.988759i \(-0.547772\pi\)
−0.149516 + 0.988759i \(0.547772\pi\)
\(152\) 9.44504 0.766094
\(153\) 0 0
\(154\) 4.41013 0.355378
\(155\) 1.30127 0.104521
\(156\) 0 0
\(157\) −4.87800 −0.389307 −0.194653 0.980872i \(-0.562358\pi\)
−0.194653 + 0.980872i \(0.562358\pi\)
\(158\) −1.98254 −0.157723
\(159\) 0 0
\(160\) −1.14914 −0.0908479
\(161\) −14.6286 −1.15290
\(162\) 0 0
\(163\) 8.63102 0.676034 0.338017 0.941140i \(-0.390244\pi\)
0.338017 + 0.941140i \(0.390244\pi\)
\(164\) 0.801938 0.0626208
\(165\) 0 0
\(166\) −4.53319 −0.351844
\(167\) −9.46980 −0.732795 −0.366397 0.930458i \(-0.619409\pi\)
−0.366397 + 0.930458i \(0.619409\pi\)
\(168\) 0 0
\(169\) 0 0
\(170\) −0.417895 −0.0320511
\(171\) 0 0
\(172\) −3.08815 −0.235469
\(173\) −4.77479 −0.363021 −0.181510 0.983389i \(-0.558099\pi\)
−0.181510 + 0.983389i \(0.558099\pi\)
\(174\) 0 0
\(175\) −8.65817 −0.654496
\(176\) −16.1153 −1.21474
\(177\) 0 0
\(178\) 0.0609989 0.00457206
\(179\) −3.43535 −0.256770 −0.128385 0.991724i \(-0.540979\pi\)
−0.128385 + 0.991724i \(0.540979\pi\)
\(180\) 0 0
\(181\) −13.4862 −1.00242 −0.501210 0.865326i \(-0.667112\pi\)
−0.501210 + 0.865326i \(0.667112\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) −14.1196 −1.04091
\(185\) 0.789856 0.0580714
\(186\) 0 0
\(187\) −21.4916 −1.57162
\(188\) −12.1371 −0.885186
\(189\) 0 0
\(190\) −0.613564 −0.0445126
\(191\) 1.30127 0.0941569 0.0470784 0.998891i \(-0.485009\pi\)
0.0470784 + 0.998891i \(0.485009\pi\)
\(192\) 0 0
\(193\) −9.19567 −0.661919 −0.330959 0.943645i \(-0.607372\pi\)
−0.330959 + 0.943645i \(0.607372\pi\)
\(194\) 6.09246 0.437413
\(195\) 0 0
\(196\) 7.07606 0.505433
\(197\) −4.11231 −0.292990 −0.146495 0.989211i \(-0.546799\pi\)
−0.146495 + 0.989211i \(0.546799\pi\)
\(198\) 0 0
\(199\) −24.7724 −1.75607 −0.878034 0.478598i \(-0.841145\pi\)
−0.878034 + 0.478598i \(0.841145\pi\)
\(200\) −8.35690 −0.590922
\(201\) 0 0
\(202\) −2.40821 −0.169441
\(203\) 10.4112 0.730722
\(204\) 0 0
\(205\) −0.109916 −0.00767688
\(206\) −6.12200 −0.426540
\(207\) 0 0
\(208\) 0 0
\(209\) −31.5545 −2.18267
\(210\) 0 0
\(211\) −5.93900 −0.408858 −0.204429 0.978881i \(-0.565534\pi\)
−0.204429 + 0.978881i \(0.565534\pi\)
\(212\) −1.91185 −0.131307
\(213\) 0 0
\(214\) −5.70410 −0.389924
\(215\) 0.423272 0.0288669
\(216\) 0 0
\(217\) 9.23623 0.626996
\(218\) −5.40581 −0.366128
\(219\) 0 0
\(220\) 2.51573 0.169610
\(221\) 0 0
\(222\) 0 0
\(223\) 14.2010 0.950972 0.475486 0.879723i \(-0.342272\pi\)
0.475486 + 0.879723i \(0.342272\pi\)
\(224\) −8.15644 −0.544975
\(225\) 0 0
\(226\) −0.728857 −0.0484829
\(227\) −16.0073 −1.06244 −0.531221 0.847233i \(-0.678267\pi\)
−0.531221 + 0.847233i \(0.678267\pi\)
\(228\) 0 0
\(229\) 1.84117 0.121668 0.0608339 0.998148i \(-0.480624\pi\)
0.0608339 + 0.998148i \(0.480624\pi\)
\(230\) 0.917231 0.0604804
\(231\) 0 0
\(232\) 10.0489 0.659744
\(233\) 23.4252 1.53464 0.767318 0.641267i \(-0.221591\pi\)
0.767318 + 0.641267i \(0.221591\pi\)
\(234\) 0 0
\(235\) 1.66355 0.108518
\(236\) −24.6896 −1.60716
\(237\) 0 0
\(238\) −2.96615 −0.192267
\(239\) −14.6015 −0.944491 −0.472246 0.881467i \(-0.656556\pi\)
−0.472246 + 0.881467i \(0.656556\pi\)
\(240\) 0 0
\(241\) 8.63102 0.555973 0.277987 0.960585i \(-0.410333\pi\)
0.277987 + 0.960585i \(0.410333\pi\)
\(242\) −9.32544 −0.599462
\(243\) 0 0
\(244\) 15.3448 0.982351
\(245\) −0.969869 −0.0619627
\(246\) 0 0
\(247\) 0 0
\(248\) 8.91484 0.566093
\(249\) 0 0
\(250\) 1.09246 0.0690931
\(251\) 3.80194 0.239976 0.119988 0.992775i \(-0.461714\pi\)
0.119988 + 0.992775i \(0.461714\pi\)
\(252\) 0 0
\(253\) 47.1715 2.96565
\(254\) −4.80625 −0.301571
\(255\) 0 0
\(256\) 2.40150 0.150094
\(257\) 20.5961 1.28475 0.642375 0.766391i \(-0.277949\pi\)
0.642375 + 0.766391i \(0.277949\pi\)
\(258\) 0 0
\(259\) 5.60627 0.348357
\(260\) 0 0
\(261\) 0 0
\(262\) 0.403894 0.0249527
\(263\) −0.332733 −0.0205172 −0.0102586 0.999947i \(-0.503265\pi\)
−0.0102586 + 0.999947i \(0.503265\pi\)
\(264\) 0 0
\(265\) 0.262045 0.0160973
\(266\) −4.35498 −0.267021
\(267\) 0 0
\(268\) −10.7409 −0.656107
\(269\) −27.3032 −1.66471 −0.832353 0.554247i \(-0.813006\pi\)
−0.832353 + 0.554247i \(0.813006\pi\)
\(270\) 0 0
\(271\) 27.9855 1.70000 0.850000 0.526783i \(-0.176602\pi\)
0.850000 + 0.526783i \(0.176602\pi\)
\(272\) 10.8388 0.657197
\(273\) 0 0
\(274\) −4.24937 −0.256714
\(275\) 27.9191 1.68359
\(276\) 0 0
\(277\) −2.10321 −0.126370 −0.0631849 0.998002i \(-0.520126\pi\)
−0.0631849 + 0.998002i \(0.520126\pi\)
\(278\) 1.82132 0.109235
\(279\) 0 0
\(280\) 0.732578 0.0437799
\(281\) −27.2349 −1.62470 −0.812349 0.583172i \(-0.801811\pi\)
−0.812349 + 0.583172i \(0.801811\pi\)
\(282\) 0 0
\(283\) 5.28382 0.314090 0.157045 0.987591i \(-0.449803\pi\)
0.157045 + 0.987591i \(0.449803\pi\)
\(284\) −10.3056 −0.611524
\(285\) 0 0
\(286\) 0 0
\(287\) −0.780167 −0.0460518
\(288\) 0 0
\(289\) −2.54527 −0.149722
\(290\) −0.652793 −0.0383333
\(291\) 0 0
\(292\) −13.2567 −0.775788
\(293\) −32.6625 −1.90816 −0.954081 0.299548i \(-0.903164\pi\)
−0.954081 + 0.299548i \(0.903164\pi\)
\(294\) 0 0
\(295\) 3.38404 0.197027
\(296\) 5.41119 0.314519
\(297\) 0 0
\(298\) −6.84846 −0.396721
\(299\) 0 0
\(300\) 0 0
\(301\) 3.00431 0.173166
\(302\) 1.63533 0.0941029
\(303\) 0 0
\(304\) 15.9138 0.912717
\(305\) −2.10321 −0.120430
\(306\) 0 0
\(307\) −20.7614 −1.18491 −0.592457 0.805602i \(-0.701842\pi\)
−0.592457 + 0.805602i \(0.701842\pi\)
\(308\) 17.8562 1.01745
\(309\) 0 0
\(310\) −0.579121 −0.0328919
\(311\) 11.3013 0.640836 0.320418 0.947276i \(-0.396177\pi\)
0.320418 + 0.947276i \(0.396177\pi\)
\(312\) 0 0
\(313\) −4.27173 −0.241453 −0.120726 0.992686i \(-0.538522\pi\)
−0.120726 + 0.992686i \(0.538522\pi\)
\(314\) 2.17092 0.122512
\(315\) 0 0
\(316\) −8.02715 −0.451562
\(317\) −15.4776 −0.869307 −0.434653 0.900598i \(-0.643129\pi\)
−0.434653 + 0.900598i \(0.643129\pi\)
\(318\) 0 0
\(319\) −33.5719 −1.87967
\(320\) −0.896789 −0.0501320
\(321\) 0 0
\(322\) 6.51035 0.362808
\(323\) 21.2228 1.18087
\(324\) 0 0
\(325\) 0 0
\(326\) −3.84117 −0.212743
\(327\) 0 0
\(328\) −0.753020 −0.0415786
\(329\) 11.8076 0.650973
\(330\) 0 0
\(331\) 6.06829 0.333544 0.166772 0.985996i \(-0.446666\pi\)
0.166772 + 0.985996i \(0.446666\pi\)
\(332\) −18.3545 −1.00733
\(333\) 0 0
\(334\) 4.21446 0.230605
\(335\) 1.47219 0.0804343
\(336\) 0 0
\(337\) 12.1239 0.660432 0.330216 0.943905i \(-0.392878\pi\)
0.330216 + 0.943905i \(0.392878\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) −1.69202 −0.0917627
\(341\) −29.7832 −1.61285
\(342\) 0 0
\(343\) −19.1551 −1.03428
\(344\) 2.89977 0.156345
\(345\) 0 0
\(346\) 2.12498 0.114240
\(347\) −23.1497 −1.24274 −0.621371 0.783516i \(-0.713424\pi\)
−0.621371 + 0.783516i \(0.713424\pi\)
\(348\) 0 0
\(349\) −22.1957 −1.18811 −0.594053 0.804426i \(-0.702473\pi\)
−0.594053 + 0.804426i \(0.702473\pi\)
\(350\) 3.85325 0.205965
\(351\) 0 0
\(352\) 26.3013 1.40186
\(353\) 5.07069 0.269885 0.134943 0.990853i \(-0.456915\pi\)
0.134943 + 0.990853i \(0.456915\pi\)
\(354\) 0 0
\(355\) 1.41252 0.0749687
\(356\) 0.246980 0.0130899
\(357\) 0 0
\(358\) 1.52888 0.0808036
\(359\) 16.6746 0.880050 0.440025 0.897986i \(-0.354970\pi\)
0.440025 + 0.897986i \(0.354970\pi\)
\(360\) 0 0
\(361\) 12.1599 0.639995
\(362\) 6.00192 0.315454
\(363\) 0 0
\(364\) 0 0
\(365\) 1.81700 0.0951063
\(366\) 0 0
\(367\) −1.17928 −0.0615577 −0.0307789 0.999526i \(-0.509799\pi\)
−0.0307789 + 0.999526i \(0.509799\pi\)
\(368\) −23.7899 −1.24013
\(369\) 0 0
\(370\) −0.351519 −0.0182746
\(371\) 1.85995 0.0965639
\(372\) 0 0
\(373\) −30.0925 −1.55813 −0.779064 0.626944i \(-0.784305\pi\)
−0.779064 + 0.626944i \(0.784305\pi\)
\(374\) 9.56465 0.494576
\(375\) 0 0
\(376\) 11.3967 0.587741
\(377\) 0 0
\(378\) 0 0
\(379\) −19.1631 −0.984345 −0.492172 0.870498i \(-0.663797\pi\)
−0.492172 + 0.870498i \(0.663797\pi\)
\(380\) −2.48427 −0.127440
\(381\) 0 0
\(382\) −0.579121 −0.0296304
\(383\) 15.3884 0.786308 0.393154 0.919473i \(-0.371384\pi\)
0.393154 + 0.919473i \(0.371384\pi\)
\(384\) 0 0
\(385\) −2.44743 −0.124733
\(386\) 4.09246 0.208301
\(387\) 0 0
\(388\) 24.6679 1.25232
\(389\) 24.0315 1.21844 0.609222 0.793000i \(-0.291482\pi\)
0.609222 + 0.793000i \(0.291482\pi\)
\(390\) 0 0
\(391\) −31.7265 −1.60448
\(392\) −6.64443 −0.335594
\(393\) 0 0
\(394\) 1.83015 0.0922016
\(395\) 1.10023 0.0553585
\(396\) 0 0
\(397\) 29.6015 1.48566 0.742828 0.669482i \(-0.233484\pi\)
0.742828 + 0.669482i \(0.233484\pi\)
\(398\) 11.0248 0.552621
\(399\) 0 0
\(400\) −14.0804 −0.704019
\(401\) 21.1032 1.05384 0.526922 0.849914i \(-0.323346\pi\)
0.526922 + 0.849914i \(0.323346\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) −9.75063 −0.485112
\(405\) 0 0
\(406\) −4.63342 −0.229953
\(407\) −18.0780 −0.896092
\(408\) 0 0
\(409\) −36.0224 −1.78119 −0.890596 0.454796i \(-0.849712\pi\)
−0.890596 + 0.454796i \(0.849712\pi\)
\(410\) 0.0489173 0.00241586
\(411\) 0 0
\(412\) −24.7875 −1.22119
\(413\) 24.0194 1.18192
\(414\) 0 0
\(415\) 2.51573 0.123492
\(416\) 0 0
\(417\) 0 0
\(418\) 14.0431 0.686869
\(419\) −5.96854 −0.291582 −0.145791 0.989315i \(-0.546573\pi\)
−0.145791 + 0.989315i \(0.546573\pi\)
\(420\) 0 0
\(421\) −2.09544 −0.102126 −0.0510628 0.998695i \(-0.516261\pi\)
−0.0510628 + 0.998695i \(0.516261\pi\)
\(422\) 2.64310 0.128664
\(423\) 0 0
\(424\) 1.79523 0.0871842
\(425\) −18.7778 −0.910856
\(426\) 0 0
\(427\) −14.9282 −0.722429
\(428\) −23.0954 −1.11636
\(429\) 0 0
\(430\) −0.188374 −0.00908418
\(431\) 2.88577 0.139003 0.0695014 0.997582i \(-0.477859\pi\)
0.0695014 + 0.997582i \(0.477859\pi\)
\(432\) 0 0
\(433\) −12.6485 −0.607847 −0.303924 0.952696i \(-0.598297\pi\)
−0.303924 + 0.952696i \(0.598297\pi\)
\(434\) −4.11051 −0.197311
\(435\) 0 0
\(436\) −21.8877 −1.04823
\(437\) −46.5816 −2.22830
\(438\) 0 0
\(439\) −10.9269 −0.521513 −0.260757 0.965405i \(-0.583972\pi\)
−0.260757 + 0.965405i \(0.583972\pi\)
\(440\) −2.36227 −0.112617
\(441\) 0 0
\(442\) 0 0
\(443\) 19.2403 0.914133 0.457067 0.889433i \(-0.348900\pi\)
0.457067 + 0.889433i \(0.348900\pi\)
\(444\) 0 0
\(445\) −0.0338518 −0.00160473
\(446\) −6.32006 −0.299264
\(447\) 0 0
\(448\) −6.36526 −0.300730
\(449\) 28.8200 1.36010 0.680050 0.733166i \(-0.261958\pi\)
0.680050 + 0.733166i \(0.261958\pi\)
\(450\) 0 0
\(451\) 2.51573 0.118461
\(452\) −2.95108 −0.138807
\(453\) 0 0
\(454\) 7.12392 0.334342
\(455\) 0 0
\(456\) 0 0
\(457\) −18.0707 −0.845311 −0.422656 0.906290i \(-0.638902\pi\)
−0.422656 + 0.906290i \(0.638902\pi\)
\(458\) −0.819396 −0.0382879
\(459\) 0 0
\(460\) 3.71379 0.173156
\(461\) 7.56763 0.352460 0.176230 0.984349i \(-0.443610\pi\)
0.176230 + 0.984349i \(0.443610\pi\)
\(462\) 0 0
\(463\) 35.3551 1.64309 0.821545 0.570143i \(-0.193112\pi\)
0.821545 + 0.570143i \(0.193112\pi\)
\(464\) 16.9312 0.786013
\(465\) 0 0
\(466\) −10.4252 −0.482938
\(467\) −13.0000 −0.601568 −0.300784 0.953692i \(-0.597248\pi\)
−0.300784 + 0.953692i \(0.597248\pi\)
\(468\) 0 0
\(469\) 10.4494 0.482506
\(470\) −0.740348 −0.0341497
\(471\) 0 0
\(472\) 23.1836 1.06711
\(473\) −9.68771 −0.445441
\(474\) 0 0
\(475\) −27.5700 −1.26500
\(476\) −12.0097 −0.550463
\(477\) 0 0
\(478\) 6.49827 0.297224
\(479\) 25.5265 1.16633 0.583167 0.812352i \(-0.301813\pi\)
0.583167 + 0.812352i \(0.301813\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) −3.84117 −0.174960
\(483\) 0 0
\(484\) −37.7579 −1.71627
\(485\) −3.38106 −0.153526
\(486\) 0 0
\(487\) 16.0073 0.725360 0.362680 0.931914i \(-0.381862\pi\)
0.362680 + 0.931914i \(0.381862\pi\)
\(488\) −14.4088 −0.652256
\(489\) 0 0
\(490\) 0.431632 0.0194992
\(491\) −20.7385 −0.935917 −0.467959 0.883750i \(-0.655010\pi\)
−0.467959 + 0.883750i \(0.655010\pi\)
\(492\) 0 0
\(493\) 22.5797 1.01694
\(494\) 0 0
\(495\) 0 0
\(496\) 15.0204 0.674438
\(497\) 10.0258 0.449719
\(498\) 0 0
\(499\) −8.06770 −0.361160 −0.180580 0.983560i \(-0.557797\pi\)
−0.180580 + 0.983560i \(0.557797\pi\)
\(500\) 4.42327 0.197815
\(501\) 0 0
\(502\) −1.69202 −0.0755186
\(503\) 30.2422 1.34843 0.674216 0.738534i \(-0.264482\pi\)
0.674216 + 0.738534i \(0.264482\pi\)
\(504\) 0 0
\(505\) 1.33645 0.0594714
\(506\) −20.9933 −0.933266
\(507\) 0 0
\(508\) −19.4601 −0.863403
\(509\) −16.0495 −0.711382 −0.355691 0.934604i \(-0.615754\pi\)
−0.355691 + 0.934604i \(0.615754\pi\)
\(510\) 0 0
\(511\) 12.8968 0.570520
\(512\) −22.9119 −1.01257
\(513\) 0 0
\(514\) −9.16613 −0.404301
\(515\) 3.39745 0.149710
\(516\) 0 0
\(517\) −38.0747 −1.67452
\(518\) −2.49502 −0.109625
\(519\) 0 0
\(520\) 0 0
\(521\) 2.69309 0.117986 0.0589931 0.998258i \(-0.481211\pi\)
0.0589931 + 0.998258i \(0.481211\pi\)
\(522\) 0 0
\(523\) 35.3957 1.54774 0.773872 0.633342i \(-0.218317\pi\)
0.773872 + 0.633342i \(0.218317\pi\)
\(524\) 1.63533 0.0714399
\(525\) 0 0
\(526\) 0.148080 0.00645659
\(527\) 20.0315 0.872584
\(528\) 0 0
\(529\) 46.6359 2.02765
\(530\) −0.116621 −0.00506569
\(531\) 0 0
\(532\) −17.6329 −0.764485
\(533\) 0 0
\(534\) 0 0
\(535\) 3.16554 0.136858
\(536\) 10.0858 0.435638
\(537\) 0 0
\(538\) 12.1511 0.523870
\(539\) 22.1981 0.956138
\(540\) 0 0
\(541\) −34.7338 −1.49332 −0.746660 0.665205i \(-0.768344\pi\)
−0.746660 + 0.665205i \(0.768344\pi\)
\(542\) −12.4547 −0.534976
\(543\) 0 0
\(544\) −17.6896 −0.758437
\(545\) 3.00000 0.128506
\(546\) 0 0
\(547\) −26.1183 −1.11674 −0.558368 0.829593i \(-0.688572\pi\)
−0.558368 + 0.829593i \(0.688572\pi\)
\(548\) −17.2054 −0.734976
\(549\) 0 0
\(550\) −12.4252 −0.529812
\(551\) 33.1521 1.41233
\(552\) 0 0
\(553\) 7.80923 0.332082
\(554\) 0.936017 0.0397676
\(555\) 0 0
\(556\) 7.37435 0.312742
\(557\) −24.7748 −1.04974 −0.524871 0.851182i \(-0.675886\pi\)
−0.524871 + 0.851182i \(0.675886\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 1.23431 0.0521590
\(561\) 0 0
\(562\) 12.1207 0.511280
\(563\) −5.26098 −0.221724 −0.110862 0.993836i \(-0.535361\pi\)
−0.110862 + 0.993836i \(0.535361\pi\)
\(564\) 0 0
\(565\) 0.404485 0.0170168
\(566\) −2.35152 −0.0988417
\(567\) 0 0
\(568\) 9.67696 0.406036
\(569\) 33.7458 1.41470 0.707350 0.706864i \(-0.249891\pi\)
0.707350 + 0.706864i \(0.249891\pi\)
\(570\) 0 0
\(571\) 23.0887 0.966234 0.483117 0.875556i \(-0.339505\pi\)
0.483117 + 0.875556i \(0.339505\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0.347207 0.0144921
\(575\) 41.2150 1.71879
\(576\) 0 0
\(577\) 3.57002 0.148622 0.0743110 0.997235i \(-0.476324\pi\)
0.0743110 + 0.997235i \(0.476324\pi\)
\(578\) 1.13275 0.0471162
\(579\) 0 0
\(580\) −2.64310 −0.109749
\(581\) 17.8562 0.740801
\(582\) 0 0
\(583\) −5.99761 −0.248396
\(584\) 12.4480 0.515103
\(585\) 0 0
\(586\) 14.5362 0.600484
\(587\) 11.4625 0.473108 0.236554 0.971618i \(-0.423982\pi\)
0.236554 + 0.971618i \(0.423982\pi\)
\(588\) 0 0
\(589\) 29.4107 1.21185
\(590\) −1.50604 −0.0620027
\(591\) 0 0
\(592\) 9.11721 0.374715
\(593\) 21.8538 0.897430 0.448715 0.893675i \(-0.351882\pi\)
0.448715 + 0.893675i \(0.351882\pi\)
\(594\) 0 0
\(595\) 1.64609 0.0674830
\(596\) −27.7289 −1.13582
\(597\) 0 0
\(598\) 0 0
\(599\) −27.0573 −1.10553 −0.552765 0.833337i \(-0.686427\pi\)
−0.552765 + 0.833337i \(0.686427\pi\)
\(600\) 0 0
\(601\) 10.8780 0.443723 0.221861 0.975078i \(-0.428787\pi\)
0.221861 + 0.975078i \(0.428787\pi\)
\(602\) −1.33704 −0.0544939
\(603\) 0 0
\(604\) 6.62133 0.269418
\(605\) 5.17523 0.210403
\(606\) 0 0
\(607\) −29.6359 −1.20289 −0.601443 0.798916i \(-0.705407\pi\)
−0.601443 + 0.798916i \(0.705407\pi\)
\(608\) −25.9724 −1.05332
\(609\) 0 0
\(610\) 0.936017 0.0378982
\(611\) 0 0
\(612\) 0 0
\(613\) 10.2343 0.413360 0.206680 0.978409i \(-0.433734\pi\)
0.206680 + 0.978409i \(0.433734\pi\)
\(614\) 9.23968 0.372883
\(615\) 0 0
\(616\) −16.7670 −0.675563
\(617\) −26.2828 −1.05810 −0.529052 0.848590i \(-0.677452\pi\)
−0.529052 + 0.848590i \(0.677452\pi\)
\(618\) 0 0
\(619\) 29.0834 1.16896 0.584479 0.811408i \(-0.301299\pi\)
0.584479 + 0.811408i \(0.301299\pi\)
\(620\) −2.34481 −0.0941700
\(621\) 0 0
\(622\) −5.02954 −0.201666
\(623\) −0.240275 −0.00962641
\(624\) 0 0
\(625\) 24.0887 0.963549
\(626\) 1.90110 0.0759833
\(627\) 0 0
\(628\) 8.78986 0.350753
\(629\) 12.1588 0.484804
\(630\) 0 0
\(631\) 25.4480 1.01307 0.506535 0.862219i \(-0.330926\pi\)
0.506535 + 0.862219i \(0.330926\pi\)
\(632\) 7.53750 0.299826
\(633\) 0 0
\(634\) 6.88816 0.273564
\(635\) 2.66727 0.105847
\(636\) 0 0
\(637\) 0 0
\(638\) 14.9409 0.591517
\(639\) 0 0
\(640\) 2.69740 0.106624
\(641\) 26.7409 1.05620 0.528102 0.849181i \(-0.322904\pi\)
0.528102 + 0.849181i \(0.322904\pi\)
\(642\) 0 0
\(643\) −32.9614 −1.29987 −0.649935 0.759990i \(-0.725204\pi\)
−0.649935 + 0.759990i \(0.725204\pi\)
\(644\) 26.3599 1.03872
\(645\) 0 0
\(646\) −9.44504 −0.371610
\(647\) −34.4946 −1.35612 −0.678060 0.735006i \(-0.737179\pi\)
−0.678060 + 0.735006i \(0.737179\pi\)
\(648\) 0 0
\(649\) −77.4529 −3.04029
\(650\) 0 0
\(651\) 0 0
\(652\) −15.5526 −0.609085
\(653\) −36.1517 −1.41472 −0.707362 0.706852i \(-0.750115\pi\)
−0.707362 + 0.706852i \(0.750115\pi\)
\(654\) 0 0
\(655\) −0.224144 −0.00875805
\(656\) −1.26875 −0.0495364
\(657\) 0 0
\(658\) −5.25487 −0.204856
\(659\) 6.81700 0.265553 0.132776 0.991146i \(-0.457611\pi\)
0.132776 + 0.991146i \(0.457611\pi\)
\(660\) 0 0
\(661\) −10.8944 −0.423743 −0.211871 0.977298i \(-0.567956\pi\)
−0.211871 + 0.977298i \(0.567956\pi\)
\(662\) −2.70065 −0.104964
\(663\) 0 0
\(664\) 17.2349 0.668844
\(665\) 2.41683 0.0937206
\(666\) 0 0
\(667\) −49.5599 −1.91897
\(668\) 17.0640 0.660225
\(669\) 0 0
\(670\) −0.655186 −0.0253120
\(671\) 48.1377 1.85833
\(672\) 0 0
\(673\) 20.7385 0.799412 0.399706 0.916643i \(-0.369112\pi\)
0.399706 + 0.916643i \(0.369112\pi\)
\(674\) −5.39565 −0.207833
\(675\) 0 0
\(676\) 0 0
\(677\) 25.5786 0.983067 0.491534 0.870859i \(-0.336436\pi\)
0.491534 + 0.870859i \(0.336436\pi\)
\(678\) 0 0
\(679\) −23.9982 −0.920966
\(680\) 1.58881 0.0609281
\(681\) 0 0
\(682\) 13.2547 0.507551
\(683\) −21.6310 −0.827688 −0.413844 0.910348i \(-0.635814\pi\)
−0.413844 + 0.910348i \(0.635814\pi\)
\(684\) 0 0
\(685\) 2.35822 0.0901031
\(686\) 8.52483 0.325479
\(687\) 0 0
\(688\) 4.88577 0.186268
\(689\) 0 0
\(690\) 0 0
\(691\) 2.62996 0.100048 0.0500242 0.998748i \(-0.484070\pi\)
0.0500242 + 0.998748i \(0.484070\pi\)
\(692\) 8.60388 0.327070
\(693\) 0 0
\(694\) 10.3026 0.391081
\(695\) −1.01075 −0.0383401
\(696\) 0 0
\(697\) −1.69202 −0.0640899
\(698\) 9.87800 0.373888
\(699\) 0 0
\(700\) 15.6015 0.589681
\(701\) −40.0925 −1.51427 −0.757136 0.653258i \(-0.773402\pi\)
−0.757136 + 0.653258i \(0.773402\pi\)
\(702\) 0 0
\(703\) 17.8519 0.673298
\(704\) 20.5254 0.773581
\(705\) 0 0
\(706\) −2.25667 −0.0849308
\(707\) 9.48593 0.356755
\(708\) 0 0
\(709\) 23.2097 0.871657 0.435829 0.900030i \(-0.356455\pi\)
0.435829 + 0.900030i \(0.356455\pi\)
\(710\) −0.628630 −0.0235921
\(711\) 0 0
\(712\) −0.231914 −0.00869135
\(713\) −43.9667 −1.64657
\(714\) 0 0
\(715\) 0 0
\(716\) 6.19029 0.231342
\(717\) 0 0
\(718\) −7.42088 −0.276945
\(719\) 26.0146 0.970181 0.485090 0.874464i \(-0.338787\pi\)
0.485090 + 0.874464i \(0.338787\pi\)
\(720\) 0 0
\(721\) 24.1146 0.898073
\(722\) −5.41166 −0.201401
\(723\) 0 0
\(724\) 24.3013 0.903150
\(725\) −29.3327 −1.08939
\(726\) 0 0
\(727\) −16.5472 −0.613701 −0.306851 0.951758i \(-0.599275\pi\)
−0.306851 + 0.951758i \(0.599275\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) −0.808643 −0.0299292
\(731\) 6.51573 0.240993
\(732\) 0 0
\(733\) 18.8750 0.697165 0.348582 0.937278i \(-0.386663\pi\)
0.348582 + 0.937278i \(0.386663\pi\)
\(734\) 0.524827 0.0193717
\(735\) 0 0
\(736\) 38.8267 1.43117
\(737\) −33.6950 −1.24117
\(738\) 0 0
\(739\) 47.3239 1.74084 0.870419 0.492312i \(-0.163848\pi\)
0.870419 + 0.492312i \(0.163848\pi\)
\(740\) −1.42327 −0.0523205
\(741\) 0 0
\(742\) −0.827757 −0.0303879
\(743\) 8.88769 0.326058 0.163029 0.986621i \(-0.447874\pi\)
0.163029 + 0.986621i \(0.447874\pi\)
\(744\) 0 0
\(745\) 3.80061 0.139244
\(746\) 13.3924 0.490331
\(747\) 0 0
\(748\) 38.7265 1.41598
\(749\) 22.4685 0.820980
\(750\) 0 0
\(751\) 0.710808 0.0259377 0.0129689 0.999916i \(-0.495872\pi\)
0.0129689 + 0.999916i \(0.495872\pi\)
\(752\) 19.2021 0.700229
\(753\) 0 0
\(754\) 0 0
\(755\) −0.907542 −0.0330288
\(756\) 0 0
\(757\) 9.78554 0.355662 0.177831 0.984061i \(-0.443092\pi\)
0.177831 + 0.984061i \(0.443092\pi\)
\(758\) 8.52840 0.309766
\(759\) 0 0
\(760\) 2.33273 0.0846171
\(761\) 18.8810 0.684435 0.342218 0.939621i \(-0.388822\pi\)
0.342218 + 0.939621i \(0.388822\pi\)
\(762\) 0 0
\(763\) 21.2935 0.770877
\(764\) −2.34481 −0.0848324
\(765\) 0 0
\(766\) −6.84846 −0.247445
\(767\) 0 0
\(768\) 0 0
\(769\) −12.4349 −0.448413 −0.224207 0.974542i \(-0.571979\pi\)
−0.224207 + 0.974542i \(0.571979\pi\)
\(770\) 1.08921 0.0392524
\(771\) 0 0
\(772\) 16.5700 0.596368
\(773\) −45.6746 −1.64280 −0.821400 0.570353i \(-0.806807\pi\)
−0.821400 + 0.570353i \(0.806807\pi\)
\(774\) 0 0
\(775\) −26.0224 −0.934751
\(776\) −23.1631 −0.831508
\(777\) 0 0
\(778\) −10.6950 −0.383435
\(779\) −2.48427 −0.0890082
\(780\) 0 0
\(781\) −32.3293 −1.15683
\(782\) 14.1196 0.504916
\(783\) 0 0
\(784\) −11.1951 −0.399824
\(785\) −1.20477 −0.0430000
\(786\) 0 0
\(787\) −4.51871 −0.161075 −0.0805374 0.996752i \(-0.525664\pi\)
−0.0805374 + 0.996752i \(0.525664\pi\)
\(788\) 7.41013 0.263975
\(789\) 0 0
\(790\) −0.489647 −0.0174209
\(791\) 2.87097 0.102080
\(792\) 0 0
\(793\) 0 0
\(794\) −13.1739 −0.467524
\(795\) 0 0
\(796\) 44.6383 1.58216
\(797\) 28.7391 1.01799 0.508996 0.860769i \(-0.330017\pi\)
0.508996 + 0.860769i \(0.330017\pi\)
\(798\) 0 0
\(799\) 25.6082 0.905953
\(800\) 22.9801 0.812471
\(801\) 0 0
\(802\) −9.39181 −0.331636
\(803\) −41.5870 −1.46757
\(804\) 0 0
\(805\) −3.61297 −0.127341
\(806\) 0 0
\(807\) 0 0
\(808\) 9.15585 0.322102
\(809\) −5.42891 −0.190870 −0.0954352 0.995436i \(-0.530424\pi\)
−0.0954352 + 0.995436i \(0.530424\pi\)
\(810\) 0 0
\(811\) 0.629104 0.0220908 0.0110454 0.999939i \(-0.496484\pi\)
0.0110454 + 0.999939i \(0.496484\pi\)
\(812\) −18.7603 −0.658358
\(813\) 0 0
\(814\) 8.04546 0.281993
\(815\) 2.13169 0.0746697
\(816\) 0 0
\(817\) 9.56657 0.334692
\(818\) 16.0315 0.560527
\(819\) 0 0
\(820\) 0.198062 0.00691663
\(821\) −36.2640 −1.26562 −0.632811 0.774307i \(-0.718099\pi\)
−0.632811 + 0.774307i \(0.718099\pi\)
\(822\) 0 0
\(823\) 41.7396 1.45495 0.727476 0.686133i \(-0.240693\pi\)
0.727476 + 0.686133i \(0.240693\pi\)
\(824\) 23.2755 0.810839
\(825\) 0 0
\(826\) −10.6896 −0.371940
\(827\) −38.1997 −1.32833 −0.664167 0.747584i \(-0.731214\pi\)
−0.664167 + 0.747584i \(0.731214\pi\)
\(828\) 0 0
\(829\) 15.9788 0.554967 0.277484 0.960730i \(-0.410500\pi\)
0.277484 + 0.960730i \(0.410500\pi\)
\(830\) −1.11960 −0.0388621
\(831\) 0 0
\(832\) 0 0
\(833\) −14.9299 −0.517290
\(834\) 0 0
\(835\) −2.33885 −0.0809391
\(836\) 56.8592 1.96652
\(837\) 0 0
\(838\) 2.65625 0.0917587
\(839\) 4.63879 0.160149 0.0800744 0.996789i \(-0.474484\pi\)
0.0800744 + 0.996789i \(0.474484\pi\)
\(840\) 0 0
\(841\) 6.27173 0.216267
\(842\) 0.932559 0.0321381
\(843\) 0 0
\(844\) 10.7017 0.368368
\(845\) 0 0
\(846\) 0 0
\(847\) 36.7329 1.26216
\(848\) 3.02475 0.103870
\(849\) 0 0
\(850\) 8.35690 0.286639
\(851\) −26.6872 −0.914827
\(852\) 0 0
\(853\) 18.3884 0.629605 0.314803 0.949157i \(-0.398062\pi\)
0.314803 + 0.949157i \(0.398062\pi\)
\(854\) 6.64370 0.227343
\(855\) 0 0
\(856\) 21.6866 0.741234
\(857\) −28.1849 −0.962778 −0.481389 0.876507i \(-0.659868\pi\)
−0.481389 + 0.876507i \(0.659868\pi\)
\(858\) 0 0
\(859\) 33.3957 1.13944 0.569722 0.821837i \(-0.307051\pi\)
0.569722 + 0.821837i \(0.307051\pi\)
\(860\) −0.762709 −0.0260082
\(861\) 0 0
\(862\) −1.28429 −0.0437431
\(863\) 43.0640 1.46592 0.732958 0.680274i \(-0.238139\pi\)
0.732958 + 0.680274i \(0.238139\pi\)
\(864\) 0 0
\(865\) −1.17928 −0.0400966
\(866\) 5.62910 0.191285
\(867\) 0 0
\(868\) −16.6431 −0.564904
\(869\) −25.1817 −0.854230
\(870\) 0 0
\(871\) 0 0
\(872\) 20.5526 0.695998
\(873\) 0 0
\(874\) 20.7308 0.701229
\(875\) −4.30319 −0.145474
\(876\) 0 0
\(877\) −30.1702 −1.01877 −0.509387 0.860537i \(-0.670128\pi\)
−0.509387 + 0.860537i \(0.670128\pi\)
\(878\) 4.86294 0.164116
\(879\) 0 0
\(880\) −3.98015 −0.134171
\(881\) 3.56273 0.120031 0.0600157 0.998197i \(-0.480885\pi\)
0.0600157 + 0.998197i \(0.480885\pi\)
\(882\) 0 0
\(883\) −10.2088 −0.343554 −0.171777 0.985136i \(-0.554951\pi\)
−0.171777 + 0.985136i \(0.554951\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) −8.56273 −0.287670
\(887\) 9.33645 0.313487 0.156744 0.987639i \(-0.449900\pi\)
0.156744 + 0.987639i \(0.449900\pi\)
\(888\) 0 0
\(889\) 18.9318 0.634953
\(890\) 0.0150655 0.000504996 0
\(891\) 0 0
\(892\) −25.5894 −0.856797
\(893\) 37.5986 1.25819
\(894\) 0 0
\(895\) −0.848462 −0.0283610
\(896\) 19.1457 0.639613
\(897\) 0 0
\(898\) −12.8261 −0.428013
\(899\) 31.2911 1.04362
\(900\) 0 0
\(901\) 4.03385 0.134387
\(902\) −1.11960 −0.0372788
\(903\) 0 0
\(904\) 2.77107 0.0921644
\(905\) −3.33081 −0.110720
\(906\) 0 0
\(907\) −41.0804 −1.36405 −0.682026 0.731328i \(-0.738901\pi\)
−0.682026 + 0.731328i \(0.738901\pi\)
\(908\) 28.8442 0.957227
\(909\) 0 0
\(910\) 0 0
\(911\) 18.9705 0.628519 0.314260 0.949337i \(-0.398244\pi\)
0.314260 + 0.949337i \(0.398244\pi\)
\(912\) 0 0
\(913\) −57.5792 −1.90559
\(914\) 8.04221 0.266013
\(915\) 0 0
\(916\) −3.31767 −0.109619
\(917\) −1.59094 −0.0525375
\(918\) 0 0
\(919\) 29.0019 0.956685 0.478343 0.878173i \(-0.341238\pi\)
0.478343 + 0.878173i \(0.341238\pi\)
\(920\) −3.48725 −0.114971
\(921\) 0 0
\(922\) −3.36791 −0.110916
\(923\) 0 0
\(924\) 0 0
\(925\) −15.7952 −0.519344
\(926\) −15.7345 −0.517068
\(927\) 0 0
\(928\) −27.6329 −0.907096
\(929\) 50.7211 1.66410 0.832052 0.554697i \(-0.187166\pi\)
0.832052 + 0.554697i \(0.187166\pi\)
\(930\) 0 0
\(931\) −21.9205 −0.718415
\(932\) −42.2107 −1.38266
\(933\) 0 0
\(934\) 5.78554 0.189309
\(935\) −5.30798 −0.173589
\(936\) 0 0
\(937\) 51.3051 1.67606 0.838032 0.545620i \(-0.183706\pi\)
0.838032 + 0.545620i \(0.183706\pi\)
\(938\) −4.65040 −0.151841
\(939\) 0 0
\(940\) −2.99761 −0.0977712
\(941\) 34.7036 1.13131 0.565653 0.824643i \(-0.308624\pi\)
0.565653 + 0.824643i \(0.308624\pi\)
\(942\) 0 0
\(943\) 3.71379 0.120938
\(944\) 39.0616 1.27135
\(945\) 0 0
\(946\) 4.31144 0.140177
\(947\) 13.0127 0.422855 0.211428 0.977394i \(-0.432189\pi\)
0.211428 + 0.977394i \(0.432189\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 12.2698 0.398085
\(951\) 0 0
\(952\) 11.2771 0.365493
\(953\) −26.0151 −0.842711 −0.421355 0.906896i \(-0.638445\pi\)
−0.421355 + 0.906896i \(0.638445\pi\)
\(954\) 0 0
\(955\) 0.321388 0.0103999
\(956\) 26.3110 0.850957
\(957\) 0 0
\(958\) −11.3604 −0.367036
\(959\) 16.7383 0.540507
\(960\) 0 0
\(961\) −3.24027 −0.104525
\(962\) 0 0
\(963\) 0 0
\(964\) −15.5526 −0.500914
\(965\) −2.27114 −0.0731107
\(966\) 0 0
\(967\) 36.6644 1.17905 0.589524 0.807751i \(-0.299315\pi\)
0.589524 + 0.807751i \(0.299315\pi\)
\(968\) 35.4547 1.13956
\(969\) 0 0
\(970\) 1.50471 0.0483134
\(971\) 37.8465 1.21455 0.607277 0.794490i \(-0.292262\pi\)
0.607277 + 0.794490i \(0.292262\pi\)
\(972\) 0 0
\(973\) −7.17416 −0.229993
\(974\) −7.12392 −0.228265
\(975\) 0 0
\(976\) −24.2771 −0.777091
\(977\) −28.8998 −0.924586 −0.462293 0.886727i \(-0.652973\pi\)
−0.462293 + 0.886727i \(0.652973\pi\)
\(978\) 0 0
\(979\) 0.774791 0.0247624
\(980\) 1.74764 0.0558264
\(981\) 0 0
\(982\) 9.22952 0.294526
\(983\) 19.3991 0.618735 0.309368 0.950942i \(-0.399883\pi\)
0.309368 + 0.950942i \(0.399883\pi\)
\(984\) 0 0
\(985\) −1.01566 −0.0323615
\(986\) −10.0489 −0.320023
\(987\) 0 0
\(988\) 0 0
\(989\) −14.3013 −0.454754
\(990\) 0 0
\(991\) −5.18300 −0.164643 −0.0823217 0.996606i \(-0.526233\pi\)
−0.0823217 + 0.996606i \(0.526233\pi\)
\(992\) −24.5144 −0.778333
\(993\) 0 0
\(994\) −4.46191 −0.141523
\(995\) −6.11828 −0.193962
\(996\) 0 0
\(997\) −49.3642 −1.56338 −0.781690 0.623667i \(-0.785642\pi\)
−0.781690 + 0.623667i \(0.785642\pi\)
\(998\) 3.59047 0.113654
\(999\) 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 1521.2.a.p.1.2 3
3.2 odd 2 507.2.a.k.1.2 yes 3
12.11 even 2 8112.2.a.cf.1.1 3
13.5 odd 4 1521.2.b.m.1351.4 6
13.8 odd 4 1521.2.b.m.1351.3 6
13.12 even 2 1521.2.a.q.1.2 3
39.2 even 12 507.2.j.h.316.4 12
39.5 even 4 507.2.b.g.337.3 6
39.8 even 4 507.2.b.g.337.4 6
39.11 even 12 507.2.j.h.316.3 12
39.17 odd 6 507.2.e.k.484.2 6
39.20 even 12 507.2.j.h.361.4 12
39.23 odd 6 507.2.e.k.22.2 6
39.29 odd 6 507.2.e.j.22.2 6
39.32 even 12 507.2.j.h.361.3 12
39.35 odd 6 507.2.e.j.484.2 6
39.38 odd 2 507.2.a.j.1.2 3
156.155 even 2 8112.2.a.by.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
507.2.a.j.1.2 3 39.38 odd 2
507.2.a.k.1.2 yes 3 3.2 odd 2
507.2.b.g.337.3 6 39.5 even 4
507.2.b.g.337.4 6 39.8 even 4
507.2.e.j.22.2 6 39.29 odd 6
507.2.e.j.484.2 6 39.35 odd 6
507.2.e.k.22.2 6 39.23 odd 6
507.2.e.k.484.2 6 39.17 odd 6
507.2.j.h.316.3 12 39.11 even 12
507.2.j.h.316.4 12 39.2 even 12
507.2.j.h.361.3 12 39.32 even 12
507.2.j.h.361.4 12 39.20 even 12
1521.2.a.p.1.2 3 1.1 even 1 trivial
1521.2.a.q.1.2 3 13.12 even 2
1521.2.b.m.1351.3 6 13.8 odd 4
1521.2.b.m.1351.4 6 13.5 odd 4
8112.2.a.by.1.3 3 156.155 even 2
8112.2.a.cf.1.1 3 12.11 even 2