Properties

Label 5550.2.a.bf.1.1
Level $5550$
Weight $2$
Character 5550.1
Self dual yes
Analytic conductor $44.317$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 5550 = 2 \cdot 3 \cdot 5^{2} \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5550.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(44.3169731218\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1110)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 5550.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} -3.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} -3.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} -5.00000 q^{11} +1.00000 q^{12} +2.00000 q^{13} -3.00000 q^{14} +1.00000 q^{16} +7.00000 q^{17} +1.00000 q^{18} -2.00000 q^{19} -3.00000 q^{21} -5.00000 q^{22} -4.00000 q^{23} +1.00000 q^{24} +2.00000 q^{26} +1.00000 q^{27} -3.00000 q^{28} -5.00000 q^{29} -7.00000 q^{31} +1.00000 q^{32} -5.00000 q^{33} +7.00000 q^{34} +1.00000 q^{36} -1.00000 q^{37} -2.00000 q^{38} +2.00000 q^{39} +1.00000 q^{41} -3.00000 q^{42} -9.00000 q^{43} -5.00000 q^{44} -4.00000 q^{46} +1.00000 q^{48} +2.00000 q^{49} +7.00000 q^{51} +2.00000 q^{52} -3.00000 q^{53} +1.00000 q^{54} -3.00000 q^{56} -2.00000 q^{57} -5.00000 q^{58} -14.0000 q^{59} -7.00000 q^{61} -7.00000 q^{62} -3.00000 q^{63} +1.00000 q^{64} -5.00000 q^{66} +4.00000 q^{67} +7.00000 q^{68} -4.00000 q^{69} +8.00000 q^{71} +1.00000 q^{72} +12.0000 q^{73} -1.00000 q^{74} -2.00000 q^{76} +15.0000 q^{77} +2.00000 q^{78} +4.00000 q^{79} +1.00000 q^{81} +1.00000 q^{82} +10.0000 q^{83} -3.00000 q^{84} -9.00000 q^{86} -5.00000 q^{87} -5.00000 q^{88} -14.0000 q^{89} -6.00000 q^{91} -4.00000 q^{92} -7.00000 q^{93} +1.00000 q^{96} -1.00000 q^{97} +2.00000 q^{98} -5.00000 q^{99} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) 1.00000 0.577350
\(4\) 1.00000 0.500000
\(5\) 0 0
\(6\) 1.00000 0.408248
\(7\) −3.00000 −1.13389 −0.566947 0.823754i \(-0.691875\pi\)
−0.566947 + 0.823754i \(0.691875\pi\)
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −5.00000 −1.50756 −0.753778 0.657129i \(-0.771771\pi\)
−0.753778 + 0.657129i \(0.771771\pi\)
\(12\) 1.00000 0.288675
\(13\) 2.00000 0.554700 0.277350 0.960769i \(-0.410544\pi\)
0.277350 + 0.960769i \(0.410544\pi\)
\(14\) −3.00000 −0.801784
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 7.00000 1.69775 0.848875 0.528594i \(-0.177281\pi\)
0.848875 + 0.528594i \(0.177281\pi\)
\(18\) 1.00000 0.235702
\(19\) −2.00000 −0.458831 −0.229416 0.973329i \(-0.573682\pi\)
−0.229416 + 0.973329i \(0.573682\pi\)
\(20\) 0 0
\(21\) −3.00000 −0.654654
\(22\) −5.00000 −1.06600
\(23\) −4.00000 −0.834058 −0.417029 0.908893i \(-0.636929\pi\)
−0.417029 + 0.908893i \(0.636929\pi\)
\(24\) 1.00000 0.204124
\(25\) 0 0
\(26\) 2.00000 0.392232
\(27\) 1.00000 0.192450
\(28\) −3.00000 −0.566947
\(29\) −5.00000 −0.928477 −0.464238 0.885710i \(-0.653672\pi\)
−0.464238 + 0.885710i \(0.653672\pi\)
\(30\) 0 0
\(31\) −7.00000 −1.25724 −0.628619 0.777714i \(-0.716379\pi\)
−0.628619 + 0.777714i \(0.716379\pi\)
\(32\) 1.00000 0.176777
\(33\) −5.00000 −0.870388
\(34\) 7.00000 1.20049
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) −1.00000 −0.164399
\(38\) −2.00000 −0.324443
\(39\) 2.00000 0.320256
\(40\) 0 0
\(41\) 1.00000 0.156174 0.0780869 0.996947i \(-0.475119\pi\)
0.0780869 + 0.996947i \(0.475119\pi\)
\(42\) −3.00000 −0.462910
\(43\) −9.00000 −1.37249 −0.686244 0.727372i \(-0.740742\pi\)
−0.686244 + 0.727372i \(0.740742\pi\)
\(44\) −5.00000 −0.753778
\(45\) 0 0
\(46\) −4.00000 −0.589768
\(47\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 1.00000 0.144338
\(49\) 2.00000 0.285714
\(50\) 0 0
\(51\) 7.00000 0.980196
\(52\) 2.00000 0.277350
\(53\) −3.00000 −0.412082 −0.206041 0.978543i \(-0.566058\pi\)
−0.206041 + 0.978543i \(0.566058\pi\)
\(54\) 1.00000 0.136083
\(55\) 0 0
\(56\) −3.00000 −0.400892
\(57\) −2.00000 −0.264906
\(58\) −5.00000 −0.656532
\(59\) −14.0000 −1.82264 −0.911322 0.411693i \(-0.864937\pi\)
−0.911322 + 0.411693i \(0.864937\pi\)
\(60\) 0 0
\(61\) −7.00000 −0.896258 −0.448129 0.893969i \(-0.647910\pi\)
−0.448129 + 0.893969i \(0.647910\pi\)
\(62\) −7.00000 −0.889001
\(63\) −3.00000 −0.377964
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) −5.00000 −0.615457
\(67\) 4.00000 0.488678 0.244339 0.969690i \(-0.421429\pi\)
0.244339 + 0.969690i \(0.421429\pi\)
\(68\) 7.00000 0.848875
\(69\) −4.00000 −0.481543
\(70\) 0 0
\(71\) 8.00000 0.949425 0.474713 0.880141i \(-0.342552\pi\)
0.474713 + 0.880141i \(0.342552\pi\)
\(72\) 1.00000 0.117851
\(73\) 12.0000 1.40449 0.702247 0.711934i \(-0.252180\pi\)
0.702247 + 0.711934i \(0.252180\pi\)
\(74\) −1.00000 −0.116248
\(75\) 0 0
\(76\) −2.00000 −0.229416
\(77\) 15.0000 1.70941
\(78\) 2.00000 0.226455
\(79\) 4.00000 0.450035 0.225018 0.974355i \(-0.427756\pi\)
0.225018 + 0.974355i \(0.427756\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 1.00000 0.110432
\(83\) 10.0000 1.09764 0.548821 0.835940i \(-0.315077\pi\)
0.548821 + 0.835940i \(0.315077\pi\)
\(84\) −3.00000 −0.327327
\(85\) 0 0
\(86\) −9.00000 −0.970495
\(87\) −5.00000 −0.536056
\(88\) −5.00000 −0.533002
\(89\) −14.0000 −1.48400 −0.741999 0.670402i \(-0.766122\pi\)
−0.741999 + 0.670402i \(0.766122\pi\)
\(90\) 0 0
\(91\) −6.00000 −0.628971
\(92\) −4.00000 −0.417029
\(93\) −7.00000 −0.725866
\(94\) 0 0
\(95\) 0 0
\(96\) 1.00000 0.102062
\(97\) −1.00000 −0.101535 −0.0507673 0.998711i \(-0.516167\pi\)
−0.0507673 + 0.998711i \(0.516167\pi\)
\(98\) 2.00000 0.202031
\(99\) −5.00000 −0.502519
\(100\) 0 0
\(101\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(102\) 7.00000 0.693103
\(103\) −8.00000 −0.788263 −0.394132 0.919054i \(-0.628955\pi\)
−0.394132 + 0.919054i \(0.628955\pi\)
\(104\) 2.00000 0.196116
\(105\) 0 0
\(106\) −3.00000 −0.291386
\(107\) −14.0000 −1.35343 −0.676716 0.736245i \(-0.736597\pi\)
−0.676716 + 0.736245i \(0.736597\pi\)
\(108\) 1.00000 0.0962250
\(109\) −7.00000 −0.670478 −0.335239 0.942133i \(-0.608817\pi\)
−0.335239 + 0.942133i \(0.608817\pi\)
\(110\) 0 0
\(111\) −1.00000 −0.0949158
\(112\) −3.00000 −0.283473
\(113\) 5.00000 0.470360 0.235180 0.971952i \(-0.424432\pi\)
0.235180 + 0.971952i \(0.424432\pi\)
\(114\) −2.00000 −0.187317
\(115\) 0 0
\(116\) −5.00000 −0.464238
\(117\) 2.00000 0.184900
\(118\) −14.0000 −1.28880
\(119\) −21.0000 −1.92507
\(120\) 0 0
\(121\) 14.0000 1.27273
\(122\) −7.00000 −0.633750
\(123\) 1.00000 0.0901670
\(124\) −7.00000 −0.628619
\(125\) 0 0
\(126\) −3.00000 −0.267261
\(127\) −16.0000 −1.41977 −0.709885 0.704317i \(-0.751253\pi\)
−0.709885 + 0.704317i \(0.751253\pi\)
\(128\) 1.00000 0.0883883
\(129\) −9.00000 −0.792406
\(130\) 0 0
\(131\) −6.00000 −0.524222 −0.262111 0.965038i \(-0.584419\pi\)
−0.262111 + 0.965038i \(0.584419\pi\)
\(132\) −5.00000 −0.435194
\(133\) 6.00000 0.520266
\(134\) 4.00000 0.345547
\(135\) 0 0
\(136\) 7.00000 0.600245
\(137\) 22.0000 1.87959 0.939793 0.341743i \(-0.111017\pi\)
0.939793 + 0.341743i \(0.111017\pi\)
\(138\) −4.00000 −0.340503
\(139\) −3.00000 −0.254457 −0.127228 0.991873i \(-0.540608\pi\)
−0.127228 + 0.991873i \(0.540608\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 8.00000 0.671345
\(143\) −10.0000 −0.836242
\(144\) 1.00000 0.0833333
\(145\) 0 0
\(146\) 12.0000 0.993127
\(147\) 2.00000 0.164957
\(148\) −1.00000 −0.0821995
\(149\) −6.00000 −0.491539 −0.245770 0.969328i \(-0.579041\pi\)
−0.245770 + 0.969328i \(0.579041\pi\)
\(150\) 0 0
\(151\) 10.0000 0.813788 0.406894 0.913475i \(-0.366612\pi\)
0.406894 + 0.913475i \(0.366612\pi\)
\(152\) −2.00000 −0.162221
\(153\) 7.00000 0.565916
\(154\) 15.0000 1.20873
\(155\) 0 0
\(156\) 2.00000 0.160128
\(157\) 11.0000 0.877896 0.438948 0.898513i \(-0.355351\pi\)
0.438948 + 0.898513i \(0.355351\pi\)
\(158\) 4.00000 0.318223
\(159\) −3.00000 −0.237915
\(160\) 0 0
\(161\) 12.0000 0.945732
\(162\) 1.00000 0.0785674
\(163\) 15.0000 1.17489 0.587445 0.809264i \(-0.300134\pi\)
0.587445 + 0.809264i \(0.300134\pi\)
\(164\) 1.00000 0.0780869
\(165\) 0 0
\(166\) 10.0000 0.776151
\(167\) −20.0000 −1.54765 −0.773823 0.633402i \(-0.781658\pi\)
−0.773823 + 0.633402i \(0.781658\pi\)
\(168\) −3.00000 −0.231455
\(169\) −9.00000 −0.692308
\(170\) 0 0
\(171\) −2.00000 −0.152944
\(172\) −9.00000 −0.686244
\(173\) −19.0000 −1.44454 −0.722272 0.691609i \(-0.756902\pi\)
−0.722272 + 0.691609i \(0.756902\pi\)
\(174\) −5.00000 −0.379049
\(175\) 0 0
\(176\) −5.00000 −0.376889
\(177\) −14.0000 −1.05230
\(178\) −14.0000 −1.04934
\(179\) −24.0000 −1.79384 −0.896922 0.442189i \(-0.854202\pi\)
−0.896922 + 0.442189i \(0.854202\pi\)
\(180\) 0 0
\(181\) 14.0000 1.04061 0.520306 0.853980i \(-0.325818\pi\)
0.520306 + 0.853980i \(0.325818\pi\)
\(182\) −6.00000 −0.444750
\(183\) −7.00000 −0.517455
\(184\) −4.00000 −0.294884
\(185\) 0 0
\(186\) −7.00000 −0.513265
\(187\) −35.0000 −2.55945
\(188\) 0 0
\(189\) −3.00000 −0.218218
\(190\) 0 0
\(191\) −7.00000 −0.506502 −0.253251 0.967401i \(-0.581500\pi\)
−0.253251 + 0.967401i \(0.581500\pi\)
\(192\) 1.00000 0.0721688
\(193\) −14.0000 −1.00774 −0.503871 0.863779i \(-0.668091\pi\)
−0.503871 + 0.863779i \(0.668091\pi\)
\(194\) −1.00000 −0.0717958
\(195\) 0 0
\(196\) 2.00000 0.142857
\(197\) −10.0000 −0.712470 −0.356235 0.934396i \(-0.615940\pi\)
−0.356235 + 0.934396i \(0.615940\pi\)
\(198\) −5.00000 −0.355335
\(199\) 4.00000 0.283552 0.141776 0.989899i \(-0.454719\pi\)
0.141776 + 0.989899i \(0.454719\pi\)
\(200\) 0 0
\(201\) 4.00000 0.282138
\(202\) 0 0
\(203\) 15.0000 1.05279
\(204\) 7.00000 0.490098
\(205\) 0 0
\(206\) −8.00000 −0.557386
\(207\) −4.00000 −0.278019
\(208\) 2.00000 0.138675
\(209\) 10.0000 0.691714
\(210\) 0 0
\(211\) −13.0000 −0.894957 −0.447478 0.894295i \(-0.647678\pi\)
−0.447478 + 0.894295i \(0.647678\pi\)
\(212\) −3.00000 −0.206041
\(213\) 8.00000 0.548151
\(214\) −14.0000 −0.957020
\(215\) 0 0
\(216\) 1.00000 0.0680414
\(217\) 21.0000 1.42557
\(218\) −7.00000 −0.474100
\(219\) 12.0000 0.810885
\(220\) 0 0
\(221\) 14.0000 0.941742
\(222\) −1.00000 −0.0671156
\(223\) −9.00000 −0.602685 −0.301342 0.953516i \(-0.597435\pi\)
−0.301342 + 0.953516i \(0.597435\pi\)
\(224\) −3.00000 −0.200446
\(225\) 0 0
\(226\) 5.00000 0.332595
\(227\) 7.00000 0.464606 0.232303 0.972643i \(-0.425374\pi\)
0.232303 + 0.972643i \(0.425374\pi\)
\(228\) −2.00000 −0.132453
\(229\) −28.0000 −1.85029 −0.925146 0.379611i \(-0.876058\pi\)
−0.925146 + 0.379611i \(0.876058\pi\)
\(230\) 0 0
\(231\) 15.0000 0.986928
\(232\) −5.00000 −0.328266
\(233\) 20.0000 1.31024 0.655122 0.755523i \(-0.272617\pi\)
0.655122 + 0.755523i \(0.272617\pi\)
\(234\) 2.00000 0.130744
\(235\) 0 0
\(236\) −14.0000 −0.911322
\(237\) 4.00000 0.259828
\(238\) −21.0000 −1.36123
\(239\) 17.0000 1.09964 0.549819 0.835284i \(-0.314697\pi\)
0.549819 + 0.835284i \(0.314697\pi\)
\(240\) 0 0
\(241\) −2.00000 −0.128831 −0.0644157 0.997923i \(-0.520518\pi\)
−0.0644157 + 0.997923i \(0.520518\pi\)
\(242\) 14.0000 0.899954
\(243\) 1.00000 0.0641500
\(244\) −7.00000 −0.448129
\(245\) 0 0
\(246\) 1.00000 0.0637577
\(247\) −4.00000 −0.254514
\(248\) −7.00000 −0.444500
\(249\) 10.0000 0.633724
\(250\) 0 0
\(251\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(252\) −3.00000 −0.188982
\(253\) 20.0000 1.25739
\(254\) −16.0000 −1.00393
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −14.0000 −0.873296 −0.436648 0.899632i \(-0.643834\pi\)
−0.436648 + 0.899632i \(0.643834\pi\)
\(258\) −9.00000 −0.560316
\(259\) 3.00000 0.186411
\(260\) 0 0
\(261\) −5.00000 −0.309492
\(262\) −6.00000 −0.370681
\(263\) −11.0000 −0.678289 −0.339145 0.940734i \(-0.610138\pi\)
−0.339145 + 0.940734i \(0.610138\pi\)
\(264\) −5.00000 −0.307729
\(265\) 0 0
\(266\) 6.00000 0.367884
\(267\) −14.0000 −0.856786
\(268\) 4.00000 0.244339
\(269\) 16.0000 0.975537 0.487769 0.872973i \(-0.337811\pi\)
0.487769 + 0.872973i \(0.337811\pi\)
\(270\) 0 0
\(271\) 20.0000 1.21491 0.607457 0.794353i \(-0.292190\pi\)
0.607457 + 0.794353i \(0.292190\pi\)
\(272\) 7.00000 0.424437
\(273\) −6.00000 −0.363137
\(274\) 22.0000 1.32907
\(275\) 0 0
\(276\) −4.00000 −0.240772
\(277\) −16.0000 −0.961347 −0.480673 0.876900i \(-0.659608\pi\)
−0.480673 + 0.876900i \(0.659608\pi\)
\(278\) −3.00000 −0.179928
\(279\) −7.00000 −0.419079
\(280\) 0 0
\(281\) 10.0000 0.596550 0.298275 0.954480i \(-0.403589\pi\)
0.298275 + 0.954480i \(0.403589\pi\)
\(282\) 0 0
\(283\) 12.0000 0.713326 0.356663 0.934233i \(-0.383914\pi\)
0.356663 + 0.934233i \(0.383914\pi\)
\(284\) 8.00000 0.474713
\(285\) 0 0
\(286\) −10.0000 −0.591312
\(287\) −3.00000 −0.177084
\(288\) 1.00000 0.0589256
\(289\) 32.0000 1.88235
\(290\) 0 0
\(291\) −1.00000 −0.0586210
\(292\) 12.0000 0.702247
\(293\) −9.00000 −0.525786 −0.262893 0.964825i \(-0.584677\pi\)
−0.262893 + 0.964825i \(0.584677\pi\)
\(294\) 2.00000 0.116642
\(295\) 0 0
\(296\) −1.00000 −0.0581238
\(297\) −5.00000 −0.290129
\(298\) −6.00000 −0.347571
\(299\) −8.00000 −0.462652
\(300\) 0 0
\(301\) 27.0000 1.55625
\(302\) 10.0000 0.575435
\(303\) 0 0
\(304\) −2.00000 −0.114708
\(305\) 0 0
\(306\) 7.00000 0.400163
\(307\) −6.00000 −0.342438 −0.171219 0.985233i \(-0.554771\pi\)
−0.171219 + 0.985233i \(0.554771\pi\)
\(308\) 15.0000 0.854704
\(309\) −8.00000 −0.455104
\(310\) 0 0
\(311\) 21.0000 1.19080 0.595400 0.803429i \(-0.296993\pi\)
0.595400 + 0.803429i \(0.296993\pi\)
\(312\) 2.00000 0.113228
\(313\) 10.0000 0.565233 0.282617 0.959233i \(-0.408798\pi\)
0.282617 + 0.959233i \(0.408798\pi\)
\(314\) 11.0000 0.620766
\(315\) 0 0
\(316\) 4.00000 0.225018
\(317\) 21.0000 1.17948 0.589739 0.807594i \(-0.299231\pi\)
0.589739 + 0.807594i \(0.299231\pi\)
\(318\) −3.00000 −0.168232
\(319\) 25.0000 1.39973
\(320\) 0 0
\(321\) −14.0000 −0.781404
\(322\) 12.0000 0.668734
\(323\) −14.0000 −0.778981
\(324\) 1.00000 0.0555556
\(325\) 0 0
\(326\) 15.0000 0.830773
\(327\) −7.00000 −0.387101
\(328\) 1.00000 0.0552158
\(329\) 0 0
\(330\) 0 0
\(331\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(332\) 10.0000 0.548821
\(333\) −1.00000 −0.0547997
\(334\) −20.0000 −1.09435
\(335\) 0 0
\(336\) −3.00000 −0.163663
\(337\) 22.0000 1.19842 0.599208 0.800593i \(-0.295482\pi\)
0.599208 + 0.800593i \(0.295482\pi\)
\(338\) −9.00000 −0.489535
\(339\) 5.00000 0.271563
\(340\) 0 0
\(341\) 35.0000 1.89536
\(342\) −2.00000 −0.108148
\(343\) 15.0000 0.809924
\(344\) −9.00000 −0.485247
\(345\) 0 0
\(346\) −19.0000 −1.02145
\(347\) 12.0000 0.644194 0.322097 0.946707i \(-0.395612\pi\)
0.322097 + 0.946707i \(0.395612\pi\)
\(348\) −5.00000 −0.268028
\(349\) −16.0000 −0.856460 −0.428230 0.903670i \(-0.640863\pi\)
−0.428230 + 0.903670i \(0.640863\pi\)
\(350\) 0 0
\(351\) 2.00000 0.106752
\(352\) −5.00000 −0.266501
\(353\) −17.0000 −0.904819 −0.452409 0.891810i \(-0.649435\pi\)
−0.452409 + 0.891810i \(0.649435\pi\)
\(354\) −14.0000 −0.744092
\(355\) 0 0
\(356\) −14.0000 −0.741999
\(357\) −21.0000 −1.11144
\(358\) −24.0000 −1.26844
\(359\) 34.0000 1.79445 0.897226 0.441572i \(-0.145579\pi\)
0.897226 + 0.441572i \(0.145579\pi\)
\(360\) 0 0
\(361\) −15.0000 −0.789474
\(362\) 14.0000 0.735824
\(363\) 14.0000 0.734809
\(364\) −6.00000 −0.314485
\(365\) 0 0
\(366\) −7.00000 −0.365896
\(367\) 13.0000 0.678594 0.339297 0.940679i \(-0.389811\pi\)
0.339297 + 0.940679i \(0.389811\pi\)
\(368\) −4.00000 −0.208514
\(369\) 1.00000 0.0520579
\(370\) 0 0
\(371\) 9.00000 0.467257
\(372\) −7.00000 −0.362933
\(373\) −22.0000 −1.13912 −0.569558 0.821951i \(-0.692886\pi\)
−0.569558 + 0.821951i \(0.692886\pi\)
\(374\) −35.0000 −1.80981
\(375\) 0 0
\(376\) 0 0
\(377\) −10.0000 −0.515026
\(378\) −3.00000 −0.154303
\(379\) 16.0000 0.821865 0.410932 0.911666i \(-0.365203\pi\)
0.410932 + 0.911666i \(0.365203\pi\)
\(380\) 0 0
\(381\) −16.0000 −0.819705
\(382\) −7.00000 −0.358151
\(383\) −4.00000 −0.204390 −0.102195 0.994764i \(-0.532587\pi\)
−0.102195 + 0.994764i \(0.532587\pi\)
\(384\) 1.00000 0.0510310
\(385\) 0 0
\(386\) −14.0000 −0.712581
\(387\) −9.00000 −0.457496
\(388\) −1.00000 −0.0507673
\(389\) −31.0000 −1.57176 −0.785881 0.618378i \(-0.787790\pi\)
−0.785881 + 0.618378i \(0.787790\pi\)
\(390\) 0 0
\(391\) −28.0000 −1.41602
\(392\) 2.00000 0.101015
\(393\) −6.00000 −0.302660
\(394\) −10.0000 −0.503793
\(395\) 0 0
\(396\) −5.00000 −0.251259
\(397\) 34.0000 1.70641 0.853206 0.521575i \(-0.174655\pi\)
0.853206 + 0.521575i \(0.174655\pi\)
\(398\) 4.00000 0.200502
\(399\) 6.00000 0.300376
\(400\) 0 0
\(401\) 22.0000 1.09863 0.549314 0.835616i \(-0.314889\pi\)
0.549314 + 0.835616i \(0.314889\pi\)
\(402\) 4.00000 0.199502
\(403\) −14.0000 −0.697390
\(404\) 0 0
\(405\) 0 0
\(406\) 15.0000 0.744438
\(407\) 5.00000 0.247841
\(408\) 7.00000 0.346552
\(409\) 34.0000 1.68119 0.840596 0.541663i \(-0.182205\pi\)
0.840596 + 0.541663i \(0.182205\pi\)
\(410\) 0 0
\(411\) 22.0000 1.08518
\(412\) −8.00000 −0.394132
\(413\) 42.0000 2.06668
\(414\) −4.00000 −0.196589
\(415\) 0 0
\(416\) 2.00000 0.0980581
\(417\) −3.00000 −0.146911
\(418\) 10.0000 0.489116
\(419\) 20.0000 0.977064 0.488532 0.872546i \(-0.337533\pi\)
0.488532 + 0.872546i \(0.337533\pi\)
\(420\) 0 0
\(421\) 10.0000 0.487370 0.243685 0.969854i \(-0.421644\pi\)
0.243685 + 0.969854i \(0.421644\pi\)
\(422\) −13.0000 −0.632830
\(423\) 0 0
\(424\) −3.00000 −0.145693
\(425\) 0 0
\(426\) 8.00000 0.387601
\(427\) 21.0000 1.01626
\(428\) −14.0000 −0.676716
\(429\) −10.0000 −0.482805
\(430\) 0 0
\(431\) 21.0000 1.01153 0.505767 0.862670i \(-0.331209\pi\)
0.505767 + 0.862670i \(0.331209\pi\)
\(432\) 1.00000 0.0481125
\(433\) 16.0000 0.768911 0.384455 0.923144i \(-0.374389\pi\)
0.384455 + 0.923144i \(0.374389\pi\)
\(434\) 21.0000 1.00803
\(435\) 0 0
\(436\) −7.00000 −0.335239
\(437\) 8.00000 0.382692
\(438\) 12.0000 0.573382
\(439\) −33.0000 −1.57500 −0.787502 0.616312i \(-0.788626\pi\)
−0.787502 + 0.616312i \(0.788626\pi\)
\(440\) 0 0
\(441\) 2.00000 0.0952381
\(442\) 14.0000 0.665912
\(443\) −42.0000 −1.99548 −0.997740 0.0671913i \(-0.978596\pi\)
−0.997740 + 0.0671913i \(0.978596\pi\)
\(444\) −1.00000 −0.0474579
\(445\) 0 0
\(446\) −9.00000 −0.426162
\(447\) −6.00000 −0.283790
\(448\) −3.00000 −0.141737
\(449\) −12.0000 −0.566315 −0.283158 0.959073i \(-0.591382\pi\)
−0.283158 + 0.959073i \(0.591382\pi\)
\(450\) 0 0
\(451\) −5.00000 −0.235441
\(452\) 5.00000 0.235180
\(453\) 10.0000 0.469841
\(454\) 7.00000 0.328526
\(455\) 0 0
\(456\) −2.00000 −0.0936586
\(457\) 39.0000 1.82434 0.912172 0.409809i \(-0.134405\pi\)
0.912172 + 0.409809i \(0.134405\pi\)
\(458\) −28.0000 −1.30835
\(459\) 7.00000 0.326732
\(460\) 0 0
\(461\) −15.0000 −0.698620 −0.349310 0.937007i \(-0.613584\pi\)
−0.349310 + 0.937007i \(0.613584\pi\)
\(462\) 15.0000 0.697863
\(463\) 6.00000 0.278844 0.139422 0.990233i \(-0.455476\pi\)
0.139422 + 0.990233i \(0.455476\pi\)
\(464\) −5.00000 −0.232119
\(465\) 0 0
\(466\) 20.0000 0.926482
\(467\) −3.00000 −0.138823 −0.0694117 0.997588i \(-0.522112\pi\)
−0.0694117 + 0.997588i \(0.522112\pi\)
\(468\) 2.00000 0.0924500
\(469\) −12.0000 −0.554109
\(470\) 0 0
\(471\) 11.0000 0.506853
\(472\) −14.0000 −0.644402
\(473\) 45.0000 2.06910
\(474\) 4.00000 0.183726
\(475\) 0 0
\(476\) −21.0000 −0.962533
\(477\) −3.00000 −0.137361
\(478\) 17.0000 0.777562
\(479\) −24.0000 −1.09659 −0.548294 0.836286i \(-0.684723\pi\)
−0.548294 + 0.836286i \(0.684723\pi\)
\(480\) 0 0
\(481\) −2.00000 −0.0911922
\(482\) −2.00000 −0.0910975
\(483\) 12.0000 0.546019
\(484\) 14.0000 0.636364
\(485\) 0 0
\(486\) 1.00000 0.0453609
\(487\) 30.0000 1.35943 0.679715 0.733476i \(-0.262104\pi\)
0.679715 + 0.733476i \(0.262104\pi\)
\(488\) −7.00000 −0.316875
\(489\) 15.0000 0.678323
\(490\) 0 0
\(491\) 28.0000 1.26362 0.631811 0.775122i \(-0.282312\pi\)
0.631811 + 0.775122i \(0.282312\pi\)
\(492\) 1.00000 0.0450835
\(493\) −35.0000 −1.57632
\(494\) −4.00000 −0.179969
\(495\) 0 0
\(496\) −7.00000 −0.314309
\(497\) −24.0000 −1.07655
\(498\) 10.0000 0.448111
\(499\) 4.00000 0.179065 0.0895323 0.995984i \(-0.471463\pi\)
0.0895323 + 0.995984i \(0.471463\pi\)
\(500\) 0 0
\(501\) −20.0000 −0.893534
\(502\) 0 0
\(503\) −24.0000 −1.07011 −0.535054 0.844818i \(-0.679709\pi\)
−0.535054 + 0.844818i \(0.679709\pi\)
\(504\) −3.00000 −0.133631
\(505\) 0 0
\(506\) 20.0000 0.889108
\(507\) −9.00000 −0.399704
\(508\) −16.0000 −0.709885
\(509\) −20.0000 −0.886484 −0.443242 0.896402i \(-0.646172\pi\)
−0.443242 + 0.896402i \(0.646172\pi\)
\(510\) 0 0
\(511\) −36.0000 −1.59255
\(512\) 1.00000 0.0441942
\(513\) −2.00000 −0.0883022
\(514\) −14.0000 −0.617514
\(515\) 0 0
\(516\) −9.00000 −0.396203
\(517\) 0 0
\(518\) 3.00000 0.131812
\(519\) −19.0000 −0.834007
\(520\) 0 0
\(521\) 3.00000 0.131432 0.0657162 0.997838i \(-0.479067\pi\)
0.0657162 + 0.997838i \(0.479067\pi\)
\(522\) −5.00000 −0.218844
\(523\) −16.0000 −0.699631 −0.349816 0.936819i \(-0.613756\pi\)
−0.349816 + 0.936819i \(0.613756\pi\)
\(524\) −6.00000 −0.262111
\(525\) 0 0
\(526\) −11.0000 −0.479623
\(527\) −49.0000 −2.13447
\(528\) −5.00000 −0.217597
\(529\) −7.00000 −0.304348
\(530\) 0 0
\(531\) −14.0000 −0.607548
\(532\) 6.00000 0.260133
\(533\) 2.00000 0.0866296
\(534\) −14.0000 −0.605839
\(535\) 0 0
\(536\) 4.00000 0.172774
\(537\) −24.0000 −1.03568
\(538\) 16.0000 0.689809
\(539\) −10.0000 −0.430730
\(540\) 0 0
\(541\) 42.0000 1.80572 0.902861 0.429934i \(-0.141463\pi\)
0.902861 + 0.429934i \(0.141463\pi\)
\(542\) 20.0000 0.859074
\(543\) 14.0000 0.600798
\(544\) 7.00000 0.300123
\(545\) 0 0
\(546\) −6.00000 −0.256776
\(547\) 7.00000 0.299298 0.149649 0.988739i \(-0.452186\pi\)
0.149649 + 0.988739i \(0.452186\pi\)
\(548\) 22.0000 0.939793
\(549\) −7.00000 −0.298753
\(550\) 0 0
\(551\) 10.0000 0.426014
\(552\) −4.00000 −0.170251
\(553\) −12.0000 −0.510292
\(554\) −16.0000 −0.679775
\(555\) 0 0
\(556\) −3.00000 −0.127228
\(557\) 18.0000 0.762684 0.381342 0.924434i \(-0.375462\pi\)
0.381342 + 0.924434i \(0.375462\pi\)
\(558\) −7.00000 −0.296334
\(559\) −18.0000 −0.761319
\(560\) 0 0
\(561\) −35.0000 −1.47770
\(562\) 10.0000 0.421825
\(563\) 3.00000 0.126435 0.0632175 0.998000i \(-0.479864\pi\)
0.0632175 + 0.998000i \(0.479864\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 12.0000 0.504398
\(567\) −3.00000 −0.125988
\(568\) 8.00000 0.335673
\(569\) −10.0000 −0.419222 −0.209611 0.977785i \(-0.567220\pi\)
−0.209611 + 0.977785i \(0.567220\pi\)
\(570\) 0 0
\(571\) 27.0000 1.12991 0.564957 0.825120i \(-0.308893\pi\)
0.564957 + 0.825120i \(0.308893\pi\)
\(572\) −10.0000 −0.418121
\(573\) −7.00000 −0.292429
\(574\) −3.00000 −0.125218
\(575\) 0 0
\(576\) 1.00000 0.0416667
\(577\) −10.0000 −0.416305 −0.208153 0.978096i \(-0.566745\pi\)
−0.208153 + 0.978096i \(0.566745\pi\)
\(578\) 32.0000 1.33102
\(579\) −14.0000 −0.581820
\(580\) 0 0
\(581\) −30.0000 −1.24461
\(582\) −1.00000 −0.0414513
\(583\) 15.0000 0.621237
\(584\) 12.0000 0.496564
\(585\) 0 0
\(586\) −9.00000 −0.371787
\(587\) 21.0000 0.866763 0.433381 0.901211i \(-0.357320\pi\)
0.433381 + 0.901211i \(0.357320\pi\)
\(588\) 2.00000 0.0824786
\(589\) 14.0000 0.576860
\(590\) 0 0
\(591\) −10.0000 −0.411345
\(592\) −1.00000 −0.0410997
\(593\) −46.0000 −1.88899 −0.944497 0.328521i \(-0.893450\pi\)
−0.944497 + 0.328521i \(0.893450\pi\)
\(594\) −5.00000 −0.205152
\(595\) 0 0
\(596\) −6.00000 −0.245770
\(597\) 4.00000 0.163709
\(598\) −8.00000 −0.327144
\(599\) −24.0000 −0.980613 −0.490307 0.871550i \(-0.663115\pi\)
−0.490307 + 0.871550i \(0.663115\pi\)
\(600\) 0 0
\(601\) −7.00000 −0.285536 −0.142768 0.989756i \(-0.545600\pi\)
−0.142768 + 0.989756i \(0.545600\pi\)
\(602\) 27.0000 1.10044
\(603\) 4.00000 0.162893
\(604\) 10.0000 0.406894
\(605\) 0 0
\(606\) 0 0
\(607\) 14.0000 0.568242 0.284121 0.958788i \(-0.408298\pi\)
0.284121 + 0.958788i \(0.408298\pi\)
\(608\) −2.00000 −0.0811107
\(609\) 15.0000 0.607831
\(610\) 0 0
\(611\) 0 0
\(612\) 7.00000 0.282958
\(613\) −43.0000 −1.73675 −0.868377 0.495905i \(-0.834836\pi\)
−0.868377 + 0.495905i \(0.834836\pi\)
\(614\) −6.00000 −0.242140
\(615\) 0 0
\(616\) 15.0000 0.604367
\(617\) −6.00000 −0.241551 −0.120775 0.992680i \(-0.538538\pi\)
−0.120775 + 0.992680i \(0.538538\pi\)
\(618\) −8.00000 −0.321807
\(619\) 5.00000 0.200967 0.100483 0.994939i \(-0.467961\pi\)
0.100483 + 0.994939i \(0.467961\pi\)
\(620\) 0 0
\(621\) −4.00000 −0.160514
\(622\) 21.0000 0.842023
\(623\) 42.0000 1.68269
\(624\) 2.00000 0.0800641
\(625\) 0 0
\(626\) 10.0000 0.399680
\(627\) 10.0000 0.399362
\(628\) 11.0000 0.438948
\(629\) −7.00000 −0.279108
\(630\) 0 0
\(631\) −7.00000 −0.278666 −0.139333 0.990246i \(-0.544496\pi\)
−0.139333 + 0.990246i \(0.544496\pi\)
\(632\) 4.00000 0.159111
\(633\) −13.0000 −0.516704
\(634\) 21.0000 0.834017
\(635\) 0 0
\(636\) −3.00000 −0.118958
\(637\) 4.00000 0.158486
\(638\) 25.0000 0.989759
\(639\) 8.00000 0.316475
\(640\) 0 0
\(641\) 41.0000 1.61940 0.809701 0.586842i \(-0.199629\pi\)
0.809701 + 0.586842i \(0.199629\pi\)
\(642\) −14.0000 −0.552536
\(643\) 43.0000 1.69575 0.847877 0.530193i \(-0.177880\pi\)
0.847877 + 0.530193i \(0.177880\pi\)
\(644\) 12.0000 0.472866
\(645\) 0 0
\(646\) −14.0000 −0.550823
\(647\) −40.0000 −1.57256 −0.786281 0.617869i \(-0.787996\pi\)
−0.786281 + 0.617869i \(0.787996\pi\)
\(648\) 1.00000 0.0392837
\(649\) 70.0000 2.74774
\(650\) 0 0
\(651\) 21.0000 0.823055
\(652\) 15.0000 0.587445
\(653\) −36.0000 −1.40879 −0.704394 0.709809i \(-0.748781\pi\)
−0.704394 + 0.709809i \(0.748781\pi\)
\(654\) −7.00000 −0.273722
\(655\) 0 0
\(656\) 1.00000 0.0390434
\(657\) 12.0000 0.468165
\(658\) 0 0
\(659\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(660\) 0 0
\(661\) 31.0000 1.20576 0.602880 0.797832i \(-0.294020\pi\)
0.602880 + 0.797832i \(0.294020\pi\)
\(662\) 0 0
\(663\) 14.0000 0.543715
\(664\) 10.0000 0.388075
\(665\) 0 0
\(666\) −1.00000 −0.0387492
\(667\) 20.0000 0.774403
\(668\) −20.0000 −0.773823
\(669\) −9.00000 −0.347960
\(670\) 0 0
\(671\) 35.0000 1.35116
\(672\) −3.00000 −0.115728
\(673\) 8.00000 0.308377 0.154189 0.988041i \(-0.450724\pi\)
0.154189 + 0.988041i \(0.450724\pi\)
\(674\) 22.0000 0.847408
\(675\) 0 0
\(676\) −9.00000 −0.346154
\(677\) 42.0000 1.61419 0.807096 0.590421i \(-0.201038\pi\)
0.807096 + 0.590421i \(0.201038\pi\)
\(678\) 5.00000 0.192024
\(679\) 3.00000 0.115129
\(680\) 0 0
\(681\) 7.00000 0.268241
\(682\) 35.0000 1.34022
\(683\) 47.0000 1.79841 0.899203 0.437533i \(-0.144148\pi\)
0.899203 + 0.437533i \(0.144148\pi\)
\(684\) −2.00000 −0.0764719
\(685\) 0 0
\(686\) 15.0000 0.572703
\(687\) −28.0000 −1.06827
\(688\) −9.00000 −0.343122
\(689\) −6.00000 −0.228582
\(690\) 0 0
\(691\) −41.0000 −1.55971 −0.779857 0.625958i \(-0.784708\pi\)
−0.779857 + 0.625958i \(0.784708\pi\)
\(692\) −19.0000 −0.722272
\(693\) 15.0000 0.569803
\(694\) 12.0000 0.455514
\(695\) 0 0
\(696\) −5.00000 −0.189525
\(697\) 7.00000 0.265144
\(698\) −16.0000 −0.605609
\(699\) 20.0000 0.756469
\(700\) 0 0
\(701\) −34.0000 −1.28416 −0.642081 0.766637i \(-0.721929\pi\)
−0.642081 + 0.766637i \(0.721929\pi\)
\(702\) 2.00000 0.0754851
\(703\) 2.00000 0.0754314
\(704\) −5.00000 −0.188445
\(705\) 0 0
\(706\) −17.0000 −0.639803
\(707\) 0 0
\(708\) −14.0000 −0.526152
\(709\) −27.0000 −1.01401 −0.507003 0.861944i \(-0.669247\pi\)
−0.507003 + 0.861944i \(0.669247\pi\)
\(710\) 0 0
\(711\) 4.00000 0.150012
\(712\) −14.0000 −0.524672
\(713\) 28.0000 1.04861
\(714\) −21.0000 −0.785905
\(715\) 0 0
\(716\) −24.0000 −0.896922
\(717\) 17.0000 0.634877
\(718\) 34.0000 1.26887
\(719\) 34.0000 1.26799 0.633993 0.773339i \(-0.281415\pi\)
0.633993 + 0.773339i \(0.281415\pi\)
\(720\) 0 0
\(721\) 24.0000 0.893807
\(722\) −15.0000 −0.558242
\(723\) −2.00000 −0.0743808
\(724\) 14.0000 0.520306
\(725\) 0 0
\(726\) 14.0000 0.519589
\(727\) 18.0000 0.667583 0.333792 0.942647i \(-0.391672\pi\)
0.333792 + 0.942647i \(0.391672\pi\)
\(728\) −6.00000 −0.222375
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −63.0000 −2.33014
\(732\) −7.00000 −0.258727
\(733\) 13.0000 0.480166 0.240083 0.970752i \(-0.422825\pi\)
0.240083 + 0.970752i \(0.422825\pi\)
\(734\) 13.0000 0.479839
\(735\) 0 0
\(736\) −4.00000 −0.147442
\(737\) −20.0000 −0.736709
\(738\) 1.00000 0.0368105
\(739\) 33.0000 1.21392 0.606962 0.794731i \(-0.292388\pi\)
0.606962 + 0.794731i \(0.292388\pi\)
\(740\) 0 0
\(741\) −4.00000 −0.146944
\(742\) 9.00000 0.330400
\(743\) 13.0000 0.476924 0.238462 0.971152i \(-0.423357\pi\)
0.238462 + 0.971152i \(0.423357\pi\)
\(744\) −7.00000 −0.256632
\(745\) 0 0
\(746\) −22.0000 −0.805477
\(747\) 10.0000 0.365881
\(748\) −35.0000 −1.27973
\(749\) 42.0000 1.53465
\(750\) 0 0
\(751\) 20.0000 0.729810 0.364905 0.931045i \(-0.381101\pi\)
0.364905 + 0.931045i \(0.381101\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) −10.0000 −0.364179
\(755\) 0 0
\(756\) −3.00000 −0.109109
\(757\) −22.0000 −0.799604 −0.399802 0.916602i \(-0.630921\pi\)
−0.399802 + 0.916602i \(0.630921\pi\)
\(758\) 16.0000 0.581146
\(759\) 20.0000 0.725954
\(760\) 0 0
\(761\) −47.0000 −1.70375 −0.851874 0.523746i \(-0.824534\pi\)
−0.851874 + 0.523746i \(0.824534\pi\)
\(762\) −16.0000 −0.579619
\(763\) 21.0000 0.760251
\(764\) −7.00000 −0.253251
\(765\) 0 0
\(766\) −4.00000 −0.144526
\(767\) −28.0000 −1.01102
\(768\) 1.00000 0.0360844
\(769\) 2.00000 0.0721218 0.0360609 0.999350i \(-0.488519\pi\)
0.0360609 + 0.999350i \(0.488519\pi\)
\(770\) 0 0
\(771\) −14.0000 −0.504198
\(772\) −14.0000 −0.503871
\(773\) −33.0000 −1.18693 −0.593464 0.804861i \(-0.702240\pi\)
−0.593464 + 0.804861i \(0.702240\pi\)
\(774\) −9.00000 −0.323498
\(775\) 0 0
\(776\) −1.00000 −0.0358979
\(777\) 3.00000 0.107624
\(778\) −31.0000 −1.11140
\(779\) −2.00000 −0.0716574
\(780\) 0 0
\(781\) −40.0000 −1.43131
\(782\) −28.0000 −1.00128
\(783\) −5.00000 −0.178685
\(784\) 2.00000 0.0714286
\(785\) 0 0
\(786\) −6.00000 −0.214013
\(787\) 4.00000 0.142585 0.0712923 0.997455i \(-0.477288\pi\)
0.0712923 + 0.997455i \(0.477288\pi\)
\(788\) −10.0000 −0.356235
\(789\) −11.0000 −0.391610
\(790\) 0 0
\(791\) −15.0000 −0.533339
\(792\) −5.00000 −0.177667
\(793\) −14.0000 −0.497155
\(794\) 34.0000 1.20661
\(795\) 0 0
\(796\) 4.00000 0.141776
\(797\) −32.0000 −1.13350 −0.566749 0.823890i \(-0.691799\pi\)
−0.566749 + 0.823890i \(0.691799\pi\)
\(798\) 6.00000 0.212398
\(799\) 0 0
\(800\) 0 0
\(801\) −14.0000 −0.494666
\(802\) 22.0000 0.776847
\(803\) −60.0000 −2.11735
\(804\) 4.00000 0.141069
\(805\) 0 0
\(806\) −14.0000 −0.493129
\(807\) 16.0000 0.563227
\(808\) 0 0
\(809\) −4.00000 −0.140633 −0.0703163 0.997525i \(-0.522401\pi\)
−0.0703163 + 0.997525i \(0.522401\pi\)
\(810\) 0 0
\(811\) 20.0000 0.702295 0.351147 0.936320i \(-0.385792\pi\)
0.351147 + 0.936320i \(0.385792\pi\)
\(812\) 15.0000 0.526397
\(813\) 20.0000 0.701431
\(814\) 5.00000 0.175250
\(815\) 0 0
\(816\) 7.00000 0.245049
\(817\) 18.0000 0.629740
\(818\) 34.0000 1.18878
\(819\) −6.00000 −0.209657
\(820\) 0 0
\(821\) −6.00000 −0.209401 −0.104701 0.994504i \(-0.533388\pi\)
−0.104701 + 0.994504i \(0.533388\pi\)
\(822\) 22.0000 0.767338
\(823\) 32.0000 1.11545 0.557725 0.830026i \(-0.311674\pi\)
0.557725 + 0.830026i \(0.311674\pi\)
\(824\) −8.00000 −0.278693
\(825\) 0 0
\(826\) 42.0000 1.46137
\(827\) −21.0000 −0.730242 −0.365121 0.930960i \(-0.618972\pi\)
−0.365121 + 0.930960i \(0.618972\pi\)
\(828\) −4.00000 −0.139010
\(829\) 29.0000 1.00721 0.503606 0.863934i \(-0.332006\pi\)
0.503606 + 0.863934i \(0.332006\pi\)
\(830\) 0 0
\(831\) −16.0000 −0.555034
\(832\) 2.00000 0.0693375
\(833\) 14.0000 0.485071
\(834\) −3.00000 −0.103882
\(835\) 0 0
\(836\) 10.0000 0.345857
\(837\) −7.00000 −0.241955
\(838\) 20.0000 0.690889
\(839\) 18.0000 0.621429 0.310715 0.950503i \(-0.399432\pi\)
0.310715 + 0.950503i \(0.399432\pi\)
\(840\) 0 0
\(841\) −4.00000 −0.137931
\(842\) 10.0000 0.344623
\(843\) 10.0000 0.344418
\(844\) −13.0000 −0.447478
\(845\) 0 0
\(846\) 0 0
\(847\) −42.0000 −1.44314
\(848\) −3.00000 −0.103020
\(849\) 12.0000 0.411839
\(850\) 0 0
\(851\) 4.00000 0.137118
\(852\) 8.00000 0.274075
\(853\) −6.00000 −0.205436 −0.102718 0.994711i \(-0.532754\pi\)
−0.102718 + 0.994711i \(0.532754\pi\)
\(854\) 21.0000 0.718605
\(855\) 0 0
\(856\) −14.0000 −0.478510
\(857\) −39.0000 −1.33221 −0.666107 0.745856i \(-0.732041\pi\)
−0.666107 + 0.745856i \(0.732041\pi\)
\(858\) −10.0000 −0.341394
\(859\) −6.00000 −0.204717 −0.102359 0.994748i \(-0.532639\pi\)
−0.102359 + 0.994748i \(0.532639\pi\)
\(860\) 0 0
\(861\) −3.00000 −0.102240
\(862\) 21.0000 0.715263
\(863\) 51.0000 1.73606 0.868030 0.496512i \(-0.165386\pi\)
0.868030 + 0.496512i \(0.165386\pi\)
\(864\) 1.00000 0.0340207
\(865\) 0 0
\(866\) 16.0000 0.543702
\(867\) 32.0000 1.08678
\(868\) 21.0000 0.712786
\(869\) −20.0000 −0.678454
\(870\) 0 0
\(871\) 8.00000 0.271070
\(872\) −7.00000 −0.237050
\(873\) −1.00000 −0.0338449
\(874\) 8.00000 0.270604
\(875\) 0 0
\(876\) 12.0000 0.405442
\(877\) 3.00000 0.101303 0.0506514 0.998716i \(-0.483870\pi\)
0.0506514 + 0.998716i \(0.483870\pi\)
\(878\) −33.0000 −1.11370
\(879\) −9.00000 −0.303562
\(880\) 0 0
\(881\) −15.0000 −0.505363 −0.252681 0.967550i \(-0.581312\pi\)
−0.252681 + 0.967550i \(0.581312\pi\)
\(882\) 2.00000 0.0673435
\(883\) 49.0000 1.64898 0.824491 0.565876i \(-0.191462\pi\)
0.824491 + 0.565876i \(0.191462\pi\)
\(884\) 14.0000 0.470871
\(885\) 0 0
\(886\) −42.0000 −1.41102
\(887\) −15.0000 −0.503651 −0.251825 0.967773i \(-0.581031\pi\)
−0.251825 + 0.967773i \(0.581031\pi\)
\(888\) −1.00000 −0.0335578
\(889\) 48.0000 1.60987
\(890\) 0 0
\(891\) −5.00000 −0.167506
\(892\) −9.00000 −0.301342
\(893\) 0 0
\(894\) −6.00000 −0.200670
\(895\) 0 0
\(896\) −3.00000 −0.100223
\(897\) −8.00000 −0.267112
\(898\) −12.0000 −0.400445
\(899\) 35.0000 1.16732
\(900\) 0 0
\(901\) −21.0000 −0.699611
\(902\) −5.00000 −0.166482
\(903\) 27.0000 0.898504
\(904\) 5.00000 0.166298
\(905\) 0 0
\(906\) 10.0000 0.332228
\(907\) 36.0000 1.19536 0.597680 0.801735i \(-0.296089\pi\)
0.597680 + 0.801735i \(0.296089\pi\)
\(908\) 7.00000 0.232303
\(909\) 0 0
\(910\) 0 0
\(911\) −40.0000 −1.32526 −0.662630 0.748947i \(-0.730560\pi\)
−0.662630 + 0.748947i \(0.730560\pi\)
\(912\) −2.00000 −0.0662266
\(913\) −50.0000 −1.65476
\(914\) 39.0000 1.29001
\(915\) 0 0
\(916\) −28.0000 −0.925146
\(917\) 18.0000 0.594412
\(918\) 7.00000 0.231034
\(919\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(920\) 0 0
\(921\) −6.00000 −0.197707
\(922\) −15.0000 −0.493999
\(923\) 16.0000 0.526646
\(924\) 15.0000 0.493464
\(925\) 0 0
\(926\) 6.00000 0.197172
\(927\) −8.00000 −0.262754
\(928\) −5.00000 −0.164133
\(929\) −41.0000 −1.34517 −0.672583 0.740022i \(-0.734815\pi\)
−0.672583 + 0.740022i \(0.734815\pi\)
\(930\) 0 0
\(931\) −4.00000 −0.131095
\(932\) 20.0000 0.655122
\(933\) 21.0000 0.687509
\(934\) −3.00000 −0.0981630
\(935\) 0 0
\(936\) 2.00000 0.0653720
\(937\) −32.0000 −1.04539 −0.522697 0.852518i \(-0.675074\pi\)
−0.522697 + 0.852518i \(0.675074\pi\)
\(938\) −12.0000 −0.391814
\(939\) 10.0000 0.326338
\(940\) 0 0
\(941\) 6.00000 0.195594 0.0977972 0.995206i \(-0.468820\pi\)
0.0977972 + 0.995206i \(0.468820\pi\)
\(942\) 11.0000 0.358399
\(943\) −4.00000 −0.130258
\(944\) −14.0000 −0.455661
\(945\) 0 0
\(946\) 45.0000 1.46308
\(947\) −49.0000 −1.59229 −0.796143 0.605108i \(-0.793130\pi\)
−0.796143 + 0.605108i \(0.793130\pi\)
\(948\) 4.00000 0.129914
\(949\) 24.0000 0.779073
\(950\) 0 0
\(951\) 21.0000 0.680972
\(952\) −21.0000 −0.680614
\(953\) 4.00000 0.129573 0.0647864 0.997899i \(-0.479363\pi\)
0.0647864 + 0.997899i \(0.479363\pi\)
\(954\) −3.00000 −0.0971286
\(955\) 0 0
\(956\) 17.0000 0.549819
\(957\) 25.0000 0.808135
\(958\) −24.0000 −0.775405
\(959\) −66.0000 −2.13125
\(960\) 0 0
\(961\) 18.0000 0.580645
\(962\) −2.00000 −0.0644826
\(963\) −14.0000 −0.451144
\(964\) −2.00000 −0.0644157
\(965\) 0 0
\(966\) 12.0000 0.386094
\(967\) 4.00000 0.128631 0.0643157 0.997930i \(-0.479514\pi\)
0.0643157 + 0.997930i \(0.479514\pi\)
\(968\) 14.0000 0.449977
\(969\) −14.0000 −0.449745
\(970\) 0 0
\(971\) 53.0000 1.70085 0.850425 0.526096i \(-0.176345\pi\)
0.850425 + 0.526096i \(0.176345\pi\)
\(972\) 1.00000 0.0320750
\(973\) 9.00000 0.288527
\(974\) 30.0000 0.961262
\(975\) 0 0
\(976\) −7.00000 −0.224065
\(977\) −5.00000 −0.159964 −0.0799821 0.996796i \(-0.525486\pi\)
−0.0799821 + 0.996796i \(0.525486\pi\)
\(978\) 15.0000 0.479647
\(979\) 70.0000 2.23721
\(980\) 0 0
\(981\) −7.00000 −0.223493
\(982\) 28.0000 0.893516
\(983\) −45.0000 −1.43528 −0.717639 0.696416i \(-0.754777\pi\)
−0.717639 + 0.696416i \(0.754777\pi\)
\(984\) 1.00000 0.0318788
\(985\) 0 0
\(986\) −35.0000 −1.11463
\(987\) 0 0
\(988\) −4.00000 −0.127257
\(989\) 36.0000 1.14473
\(990\) 0 0
\(991\) 19.0000 0.603555 0.301777 0.953378i \(-0.402420\pi\)
0.301777 + 0.953378i \(0.402420\pi\)
\(992\) −7.00000 −0.222250
\(993\) 0 0
\(994\) −24.0000 −0.761234
\(995\) 0 0
\(996\) 10.0000 0.316862
\(997\) −6.00000 −0.190022 −0.0950110 0.995476i \(-0.530289\pi\)
−0.0950110 + 0.995476i \(0.530289\pi\)
\(998\) 4.00000 0.126618
\(999\) −1.00000 −0.0316386
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 5550.2.a.bf.1.1 1
5.4 even 2 1110.2.a.d.1.1 1
15.14 odd 2 3330.2.a.q.1.1 1
20.19 odd 2 8880.2.a.z.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1110.2.a.d.1.1 1 5.4 even 2
3330.2.a.q.1.1 1 15.14 odd 2
5550.2.a.bf.1.1 1 1.1 even 1 trivial
8880.2.a.z.1.1 1 20.19 odd 2