Properties

Label 7623.2.a.bm.1.1
Level 7623
Weight 2
Character 7623.1
Self dual yes
Analytic conductor 60.870
Analytic rank 1
Dimension 2
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 7623 = 3^{2} \cdot 7 \cdot 11^{2} \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 7623.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(60.8699614608\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{10})^+\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 231)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-0.618034\)
Character \(\chi\) = 7623.1

$q$-expansion

\(f(q)\) \(=\) \(q-0.618034 q^{2} -1.61803 q^{4} -1.00000 q^{5} -1.00000 q^{7} +2.23607 q^{8} +O(q^{10})\) \(q-0.618034 q^{2} -1.61803 q^{4} -1.00000 q^{5} -1.00000 q^{7} +2.23607 q^{8} +0.618034 q^{10} -3.47214 q^{13} +0.618034 q^{14} +1.85410 q^{16} +5.23607 q^{17} +6.70820 q^{19} +1.61803 q^{20} -5.70820 q^{23} -4.00000 q^{25} +2.14590 q^{26} +1.61803 q^{28} +5.00000 q^{29} -5.23607 q^{31} -5.61803 q^{32} -3.23607 q^{34} +1.00000 q^{35} -7.00000 q^{37} -4.14590 q^{38} -2.23607 q^{40} -2.47214 q^{41} -5.70820 q^{43} +3.52786 q^{46} -0.236068 q^{47} +1.00000 q^{49} +2.47214 q^{50} +5.61803 q^{52} +12.1803 q^{53} -2.23607 q^{56} -3.09017 q^{58} +11.1803 q^{59} -2.00000 q^{61} +3.23607 q^{62} -0.236068 q^{64} +3.47214 q^{65} -9.76393 q^{67} -8.47214 q^{68} -0.618034 q^{70} +2.47214 q^{71} -4.52786 q^{73} +4.32624 q^{74} -10.8541 q^{76} +14.4721 q^{79} -1.85410 q^{80} +1.52786 q^{82} +6.76393 q^{83} -5.23607 q^{85} +3.52786 q^{86} +4.47214 q^{89} +3.47214 q^{91} +9.23607 q^{92} +0.145898 q^{94} -6.70820 q^{95} +9.70820 q^{97} -0.618034 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + q^{2} - q^{4} - 2q^{5} - 2q^{7} + O(q^{10}) \) \( 2q + q^{2} - q^{4} - 2q^{5} - 2q^{7} - q^{10} + 2q^{13} - q^{14} - 3q^{16} + 6q^{17} + q^{20} + 2q^{23} - 8q^{25} + 11q^{26} + q^{28} + 10q^{29} - 6q^{31} - 9q^{32} - 2q^{34} + 2q^{35} - 14q^{37} - 15q^{38} + 4q^{41} + 2q^{43} + 16q^{46} + 4q^{47} + 2q^{49} - 4q^{50} + 9q^{52} + 2q^{53} + 5q^{58} - 4q^{61} + 2q^{62} + 4q^{64} - 2q^{65} - 24q^{67} - 8q^{68} + q^{70} - 4q^{71} - 18q^{73} - 7q^{74} - 15q^{76} + 20q^{79} + 3q^{80} + 12q^{82} + 18q^{83} - 6q^{85} + 16q^{86} - 2q^{91} + 14q^{92} + 7q^{94} + 6q^{97} + q^{98} + O(q^{100}) \)

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.618034 −0.437016 −0.218508 0.975835i \(-0.570119\pi\)
−0.218508 + 0.975835i \(0.570119\pi\)
\(3\) 0 0
\(4\) −1.61803 −0.809017
\(5\) −1.00000 −0.447214 −0.223607 0.974679i \(-0.571783\pi\)
−0.223607 + 0.974679i \(0.571783\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) 2.23607 0.790569
\(9\) 0 0
\(10\) 0.618034 0.195440
\(11\) 0 0
\(12\) 0 0
\(13\) −3.47214 −0.962997 −0.481499 0.876447i \(-0.659907\pi\)
−0.481499 + 0.876447i \(0.659907\pi\)
\(14\) 0.618034 0.165177
\(15\) 0 0
\(16\) 1.85410 0.463525
\(17\) 5.23607 1.26993 0.634967 0.772540i \(-0.281014\pi\)
0.634967 + 0.772540i \(0.281014\pi\)
\(18\) 0 0
\(19\) 6.70820 1.53897 0.769484 0.638666i \(-0.220514\pi\)
0.769484 + 0.638666i \(0.220514\pi\)
\(20\) 1.61803 0.361803
\(21\) 0 0
\(22\) 0 0
\(23\) −5.70820 −1.19024 −0.595121 0.803636i \(-0.702896\pi\)
−0.595121 + 0.803636i \(0.702896\pi\)
\(24\) 0 0
\(25\) −4.00000 −0.800000
\(26\) 2.14590 0.420845
\(27\) 0 0
\(28\) 1.61803 0.305780
\(29\) 5.00000 0.928477 0.464238 0.885710i \(-0.346328\pi\)
0.464238 + 0.885710i \(0.346328\pi\)
\(30\) 0 0
\(31\) −5.23607 −0.940426 −0.470213 0.882553i \(-0.655823\pi\)
−0.470213 + 0.882553i \(0.655823\pi\)
\(32\) −5.61803 −0.993137
\(33\) 0 0
\(34\) −3.23607 −0.554981
\(35\) 1.00000 0.169031
\(36\) 0 0
\(37\) −7.00000 −1.15079 −0.575396 0.817875i \(-0.695152\pi\)
−0.575396 + 0.817875i \(0.695152\pi\)
\(38\) −4.14590 −0.672553
\(39\) 0 0
\(40\) −2.23607 −0.353553
\(41\) −2.47214 −0.386083 −0.193041 0.981191i \(-0.561835\pi\)
−0.193041 + 0.981191i \(0.561835\pi\)
\(42\) 0 0
\(43\) −5.70820 −0.870493 −0.435246 0.900311i \(-0.643339\pi\)
−0.435246 + 0.900311i \(0.643339\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 3.52786 0.520155
\(47\) −0.236068 −0.0344341 −0.0172170 0.999852i \(-0.505481\pi\)
−0.0172170 + 0.999852i \(0.505481\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 2.47214 0.349613
\(51\) 0 0
\(52\) 5.61803 0.779081
\(53\) 12.1803 1.67310 0.836549 0.547892i \(-0.184569\pi\)
0.836549 + 0.547892i \(0.184569\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −2.23607 −0.298807
\(57\) 0 0
\(58\) −3.09017 −0.405759
\(59\) 11.1803 1.45556 0.727778 0.685813i \(-0.240553\pi\)
0.727778 + 0.685813i \(0.240553\pi\)
\(60\) 0 0
\(61\) −2.00000 −0.256074 −0.128037 0.991769i \(-0.540868\pi\)
−0.128037 + 0.991769i \(0.540868\pi\)
\(62\) 3.23607 0.410981
\(63\) 0 0
\(64\) −0.236068 −0.0295085
\(65\) 3.47214 0.430665
\(66\) 0 0
\(67\) −9.76393 −1.19285 −0.596427 0.802667i \(-0.703414\pi\)
−0.596427 + 0.802667i \(0.703414\pi\)
\(68\) −8.47214 −1.02740
\(69\) 0 0
\(70\) −0.618034 −0.0738692
\(71\) 2.47214 0.293389 0.146694 0.989182i \(-0.453137\pi\)
0.146694 + 0.989182i \(0.453137\pi\)
\(72\) 0 0
\(73\) −4.52786 −0.529946 −0.264973 0.964256i \(-0.585363\pi\)
−0.264973 + 0.964256i \(0.585363\pi\)
\(74\) 4.32624 0.502915
\(75\) 0 0
\(76\) −10.8541 −1.24505
\(77\) 0 0
\(78\) 0 0
\(79\) 14.4721 1.62824 0.814121 0.580695i \(-0.197219\pi\)
0.814121 + 0.580695i \(0.197219\pi\)
\(80\) −1.85410 −0.207295
\(81\) 0 0
\(82\) 1.52786 0.168724
\(83\) 6.76393 0.742438 0.371219 0.928545i \(-0.378940\pi\)
0.371219 + 0.928545i \(0.378940\pi\)
\(84\) 0 0
\(85\) −5.23607 −0.567931
\(86\) 3.52786 0.380419
\(87\) 0 0
\(88\) 0 0
\(89\) 4.47214 0.474045 0.237023 0.971504i \(-0.423828\pi\)
0.237023 + 0.971504i \(0.423828\pi\)
\(90\) 0 0
\(91\) 3.47214 0.363979
\(92\) 9.23607 0.962927
\(93\) 0 0
\(94\) 0.145898 0.0150482
\(95\) −6.70820 −0.688247
\(96\) 0 0
\(97\) 9.70820 0.985719 0.492859 0.870109i \(-0.335952\pi\)
0.492859 + 0.870109i \(0.335952\pi\)
\(98\) −0.618034 −0.0624309
\(99\) 0 0
\(100\) 6.47214 0.647214
\(101\) 18.1803 1.80901 0.904506 0.426461i \(-0.140240\pi\)
0.904506 + 0.426461i \(0.140240\pi\)
\(102\) 0 0
\(103\) 17.4164 1.71609 0.858045 0.513575i \(-0.171679\pi\)
0.858045 + 0.513575i \(0.171679\pi\)
\(104\) −7.76393 −0.761316
\(105\) 0 0
\(106\) −7.52786 −0.731171
\(107\) −4.23607 −0.409516 −0.204758 0.978813i \(-0.565641\pi\)
−0.204758 + 0.978813i \(0.565641\pi\)
\(108\) 0 0
\(109\) −2.76393 −0.264737 −0.132368 0.991201i \(-0.542258\pi\)
−0.132368 + 0.991201i \(0.542258\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) −1.85410 −0.175196
\(113\) 0.472136 0.0444148 0.0222074 0.999753i \(-0.492931\pi\)
0.0222074 + 0.999753i \(0.492931\pi\)
\(114\) 0 0
\(115\) 5.70820 0.532293
\(116\) −8.09017 −0.751153
\(117\) 0 0
\(118\) −6.90983 −0.636101
\(119\) −5.23607 −0.479990
\(120\) 0 0
\(121\) 0 0
\(122\) 1.23607 0.111908
\(123\) 0 0
\(124\) 8.47214 0.760820
\(125\) 9.00000 0.804984
\(126\) 0 0
\(127\) 12.6525 1.12273 0.561363 0.827570i \(-0.310277\pi\)
0.561363 + 0.827570i \(0.310277\pi\)
\(128\) 11.3820 1.00603
\(129\) 0 0
\(130\) −2.14590 −0.188208
\(131\) 0.944272 0.0825014 0.0412507 0.999149i \(-0.486866\pi\)
0.0412507 + 0.999149i \(0.486866\pi\)
\(132\) 0 0
\(133\) −6.70820 −0.581675
\(134\) 6.03444 0.521296
\(135\) 0 0
\(136\) 11.7082 1.00397
\(137\) −19.7082 −1.68379 −0.841893 0.539645i \(-0.818559\pi\)
−0.841893 + 0.539645i \(0.818559\pi\)
\(138\) 0 0
\(139\) −14.4721 −1.22751 −0.613755 0.789496i \(-0.710342\pi\)
−0.613755 + 0.789496i \(0.710342\pi\)
\(140\) −1.61803 −0.136749
\(141\) 0 0
\(142\) −1.52786 −0.128216
\(143\) 0 0
\(144\) 0 0
\(145\) −5.00000 −0.415227
\(146\) 2.79837 0.231595
\(147\) 0 0
\(148\) 11.3262 0.931011
\(149\) 5.00000 0.409616 0.204808 0.978802i \(-0.434343\pi\)
0.204808 + 0.978802i \(0.434343\pi\)
\(150\) 0 0
\(151\) 14.1803 1.15398 0.576990 0.816751i \(-0.304227\pi\)
0.576990 + 0.816751i \(0.304227\pi\)
\(152\) 15.0000 1.21666
\(153\) 0 0
\(154\) 0 0
\(155\) 5.23607 0.420571
\(156\) 0 0
\(157\) −15.4164 −1.23036 −0.615182 0.788385i \(-0.710917\pi\)
−0.615182 + 0.788385i \(0.710917\pi\)
\(158\) −8.94427 −0.711568
\(159\) 0 0
\(160\) 5.61803 0.444145
\(161\) 5.70820 0.449869
\(162\) 0 0
\(163\) −22.7082 −1.77864 −0.889322 0.457282i \(-0.848823\pi\)
−0.889322 + 0.457282i \(0.848823\pi\)
\(164\) 4.00000 0.312348
\(165\) 0 0
\(166\) −4.18034 −0.324457
\(167\) −22.6525 −1.75290 −0.876451 0.481492i \(-0.840095\pi\)
−0.876451 + 0.481492i \(0.840095\pi\)
\(168\) 0 0
\(169\) −0.944272 −0.0726363
\(170\) 3.23607 0.248195
\(171\) 0 0
\(172\) 9.23607 0.704244
\(173\) −1.52786 −0.116161 −0.0580807 0.998312i \(-0.518498\pi\)
−0.0580807 + 0.998312i \(0.518498\pi\)
\(174\) 0 0
\(175\) 4.00000 0.302372
\(176\) 0 0
\(177\) 0 0
\(178\) −2.76393 −0.207165
\(179\) −8.94427 −0.668526 −0.334263 0.942480i \(-0.608487\pi\)
−0.334263 + 0.942480i \(0.608487\pi\)
\(180\) 0 0
\(181\) −0.763932 −0.0567826 −0.0283913 0.999597i \(-0.509038\pi\)
−0.0283913 + 0.999597i \(0.509038\pi\)
\(182\) −2.14590 −0.159065
\(183\) 0 0
\(184\) −12.7639 −0.940970
\(185\) 7.00000 0.514650
\(186\) 0 0
\(187\) 0 0
\(188\) 0.381966 0.0278577
\(189\) 0 0
\(190\) 4.14590 0.300775
\(191\) 10.7639 0.778851 0.389425 0.921058i \(-0.372674\pi\)
0.389425 + 0.921058i \(0.372674\pi\)
\(192\) 0 0
\(193\) −14.6525 −1.05471 −0.527354 0.849646i \(-0.676816\pi\)
−0.527354 + 0.849646i \(0.676816\pi\)
\(194\) −6.00000 −0.430775
\(195\) 0 0
\(196\) −1.61803 −0.115574
\(197\) −16.4721 −1.17359 −0.586796 0.809735i \(-0.699611\pi\)
−0.586796 + 0.809735i \(0.699611\pi\)
\(198\) 0 0
\(199\) −3.81966 −0.270769 −0.135384 0.990793i \(-0.543227\pi\)
−0.135384 + 0.990793i \(0.543227\pi\)
\(200\) −8.94427 −0.632456
\(201\) 0 0
\(202\) −11.2361 −0.790567
\(203\) −5.00000 −0.350931
\(204\) 0 0
\(205\) 2.47214 0.172661
\(206\) −10.7639 −0.749959
\(207\) 0 0
\(208\) −6.43769 −0.446374
\(209\) 0 0
\(210\) 0 0
\(211\) −5.41641 −0.372881 −0.186440 0.982466i \(-0.559695\pi\)
−0.186440 + 0.982466i \(0.559695\pi\)
\(212\) −19.7082 −1.35357
\(213\) 0 0
\(214\) 2.61803 0.178965
\(215\) 5.70820 0.389296
\(216\) 0 0
\(217\) 5.23607 0.355447
\(218\) 1.70820 0.115694
\(219\) 0 0
\(220\) 0 0
\(221\) −18.1803 −1.22294
\(222\) 0 0
\(223\) −6.00000 −0.401790 −0.200895 0.979613i \(-0.564385\pi\)
−0.200895 + 0.979613i \(0.564385\pi\)
\(224\) 5.61803 0.375371
\(225\) 0 0
\(226\) −0.291796 −0.0194100
\(227\) −2.00000 −0.132745 −0.0663723 0.997795i \(-0.521143\pi\)
−0.0663723 + 0.997795i \(0.521143\pi\)
\(228\) 0 0
\(229\) 7.23607 0.478173 0.239086 0.970998i \(-0.423152\pi\)
0.239086 + 0.970998i \(0.423152\pi\)
\(230\) −3.52786 −0.232620
\(231\) 0 0
\(232\) 11.1803 0.734025
\(233\) −14.9443 −0.979032 −0.489516 0.871994i \(-0.662826\pi\)
−0.489516 + 0.871994i \(0.662826\pi\)
\(234\) 0 0
\(235\) 0.236068 0.0153994
\(236\) −18.0902 −1.17757
\(237\) 0 0
\(238\) 3.23607 0.209763
\(239\) 10.1246 0.654907 0.327453 0.944867i \(-0.393810\pi\)
0.327453 + 0.944867i \(0.393810\pi\)
\(240\) 0 0
\(241\) −25.9443 −1.67122 −0.835609 0.549325i \(-0.814885\pi\)
−0.835609 + 0.549325i \(0.814885\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 3.23607 0.207168
\(245\) −1.00000 −0.0638877
\(246\) 0 0
\(247\) −23.2918 −1.48202
\(248\) −11.7082 −0.743472
\(249\) 0 0
\(250\) −5.56231 −0.351791
\(251\) −12.1246 −0.765299 −0.382649 0.923894i \(-0.624988\pi\)
−0.382649 + 0.923894i \(0.624988\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) −7.81966 −0.490649
\(255\) 0 0
\(256\) −6.56231 −0.410144
\(257\) 7.00000 0.436648 0.218324 0.975876i \(-0.429941\pi\)
0.218324 + 0.975876i \(0.429941\pi\)
\(258\) 0 0
\(259\) 7.00000 0.434959
\(260\) −5.61803 −0.348416
\(261\) 0 0
\(262\) −0.583592 −0.0360544
\(263\) −26.1246 −1.61091 −0.805456 0.592655i \(-0.798080\pi\)
−0.805456 + 0.592655i \(0.798080\pi\)
\(264\) 0 0
\(265\) −12.1803 −0.748232
\(266\) 4.14590 0.254201
\(267\) 0 0
\(268\) 15.7984 0.965039
\(269\) −1.05573 −0.0643689 −0.0321844 0.999482i \(-0.510246\pi\)
−0.0321844 + 0.999482i \(0.510246\pi\)
\(270\) 0 0
\(271\) −5.29180 −0.321454 −0.160727 0.986999i \(-0.551384\pi\)
−0.160727 + 0.986999i \(0.551384\pi\)
\(272\) 9.70820 0.588646
\(273\) 0 0
\(274\) 12.1803 0.735841
\(275\) 0 0
\(276\) 0 0
\(277\) 6.47214 0.388873 0.194436 0.980915i \(-0.437712\pi\)
0.194436 + 0.980915i \(0.437712\pi\)
\(278\) 8.94427 0.536442
\(279\) 0 0
\(280\) 2.23607 0.133631
\(281\) 11.4721 0.684370 0.342185 0.939633i \(-0.388833\pi\)
0.342185 + 0.939633i \(0.388833\pi\)
\(282\) 0 0
\(283\) 13.7639 0.818181 0.409090 0.912494i \(-0.365846\pi\)
0.409090 + 0.912494i \(0.365846\pi\)
\(284\) −4.00000 −0.237356
\(285\) 0 0
\(286\) 0 0
\(287\) 2.47214 0.145926
\(288\) 0 0
\(289\) 10.4164 0.612730
\(290\) 3.09017 0.181461
\(291\) 0 0
\(292\) 7.32624 0.428736
\(293\) −16.0000 −0.934730 −0.467365 0.884064i \(-0.654797\pi\)
−0.467365 + 0.884064i \(0.654797\pi\)
\(294\) 0 0
\(295\) −11.1803 −0.650945
\(296\) −15.6525 −0.909782
\(297\) 0 0
\(298\) −3.09017 −0.179009
\(299\) 19.8197 1.14620
\(300\) 0 0
\(301\) 5.70820 0.329015
\(302\) −8.76393 −0.504308
\(303\) 0 0
\(304\) 12.4377 0.713351
\(305\) 2.00000 0.114520
\(306\) 0 0
\(307\) 20.9443 1.19535 0.597676 0.801737i \(-0.296091\pi\)
0.597676 + 0.801737i \(0.296091\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) −3.23607 −0.183796
\(311\) −9.88854 −0.560728 −0.280364 0.959894i \(-0.590455\pi\)
−0.280364 + 0.959894i \(0.590455\pi\)
\(312\) 0 0
\(313\) 24.6525 1.39344 0.696720 0.717343i \(-0.254642\pi\)
0.696720 + 0.717343i \(0.254642\pi\)
\(314\) 9.52786 0.537688
\(315\) 0 0
\(316\) −23.4164 −1.31728
\(317\) −24.1803 −1.35810 −0.679052 0.734091i \(-0.737609\pi\)
−0.679052 + 0.734091i \(0.737609\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0.236068 0.0131966
\(321\) 0 0
\(322\) −3.52786 −0.196600
\(323\) 35.1246 1.95439
\(324\) 0 0
\(325\) 13.8885 0.770398
\(326\) 14.0344 0.777296
\(327\) 0 0
\(328\) −5.52786 −0.305225
\(329\) 0.236068 0.0130148
\(330\) 0 0
\(331\) −11.4164 −0.627503 −0.313751 0.949505i \(-0.601586\pi\)
−0.313751 + 0.949505i \(0.601586\pi\)
\(332\) −10.9443 −0.600645
\(333\) 0 0
\(334\) 14.0000 0.766046
\(335\) 9.76393 0.533461
\(336\) 0 0
\(337\) 8.18034 0.445612 0.222806 0.974863i \(-0.428478\pi\)
0.222806 + 0.974863i \(0.428478\pi\)
\(338\) 0.583592 0.0317432
\(339\) 0 0
\(340\) 8.47214 0.459466
\(341\) 0 0
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) −12.7639 −0.688185
\(345\) 0 0
\(346\) 0.944272 0.0507644
\(347\) 28.0000 1.50312 0.751559 0.659665i \(-0.229302\pi\)
0.751559 + 0.659665i \(0.229302\pi\)
\(348\) 0 0
\(349\) −1.58359 −0.0847677 −0.0423839 0.999101i \(-0.513495\pi\)
−0.0423839 + 0.999101i \(0.513495\pi\)
\(350\) −2.47214 −0.132141
\(351\) 0 0
\(352\) 0 0
\(353\) −24.5279 −1.30549 −0.652743 0.757579i \(-0.726382\pi\)
−0.652743 + 0.757579i \(0.726382\pi\)
\(354\) 0 0
\(355\) −2.47214 −0.131207
\(356\) −7.23607 −0.383511
\(357\) 0 0
\(358\) 5.52786 0.292157
\(359\) 23.4164 1.23587 0.617935 0.786229i \(-0.287969\pi\)
0.617935 + 0.786229i \(0.287969\pi\)
\(360\) 0 0
\(361\) 26.0000 1.36842
\(362\) 0.472136 0.0248149
\(363\) 0 0
\(364\) −5.61803 −0.294465
\(365\) 4.52786 0.236999
\(366\) 0 0
\(367\) −19.8885 −1.03817 −0.519087 0.854722i \(-0.673728\pi\)
−0.519087 + 0.854722i \(0.673728\pi\)
\(368\) −10.5836 −0.551708
\(369\) 0 0
\(370\) −4.32624 −0.224910
\(371\) −12.1803 −0.632372
\(372\) 0 0
\(373\) −4.65248 −0.240896 −0.120448 0.992720i \(-0.538433\pi\)
−0.120448 + 0.992720i \(0.538433\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) −0.527864 −0.0272225
\(377\) −17.3607 −0.894120
\(378\) 0 0
\(379\) −31.1803 −1.60163 −0.800813 0.598914i \(-0.795599\pi\)
−0.800813 + 0.598914i \(0.795599\pi\)
\(380\) 10.8541 0.556804
\(381\) 0 0
\(382\) −6.65248 −0.340370
\(383\) −32.9443 −1.68337 −0.841687 0.539966i \(-0.818437\pi\)
−0.841687 + 0.539966i \(0.818437\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 9.05573 0.460924
\(387\) 0 0
\(388\) −15.7082 −0.797463
\(389\) −11.0557 −0.560548 −0.280274 0.959920i \(-0.590425\pi\)
−0.280274 + 0.959920i \(0.590425\pi\)
\(390\) 0 0
\(391\) −29.8885 −1.51153
\(392\) 2.23607 0.112938
\(393\) 0 0
\(394\) 10.1803 0.512878
\(395\) −14.4721 −0.728172
\(396\) 0 0
\(397\) 23.1246 1.16059 0.580295 0.814406i \(-0.302937\pi\)
0.580295 + 0.814406i \(0.302937\pi\)
\(398\) 2.36068 0.118330
\(399\) 0 0
\(400\) −7.41641 −0.370820
\(401\) 29.7082 1.48356 0.741778 0.670645i \(-0.233983\pi\)
0.741778 + 0.670645i \(0.233983\pi\)
\(402\) 0 0
\(403\) 18.1803 0.905627
\(404\) −29.4164 −1.46352
\(405\) 0 0
\(406\) 3.09017 0.153363
\(407\) 0 0
\(408\) 0 0
\(409\) −21.0557 −1.04114 −0.520569 0.853819i \(-0.674280\pi\)
−0.520569 + 0.853819i \(0.674280\pi\)
\(410\) −1.52786 −0.0754558
\(411\) 0 0
\(412\) −28.1803 −1.38835
\(413\) −11.1803 −0.550149
\(414\) 0 0
\(415\) −6.76393 −0.332028
\(416\) 19.5066 0.956389
\(417\) 0 0
\(418\) 0 0
\(419\) 1.18034 0.0576634 0.0288317 0.999584i \(-0.490821\pi\)
0.0288317 + 0.999584i \(0.490821\pi\)
\(420\) 0 0
\(421\) −13.0000 −0.633581 −0.316791 0.948495i \(-0.602605\pi\)
−0.316791 + 0.948495i \(0.602605\pi\)
\(422\) 3.34752 0.162955
\(423\) 0 0
\(424\) 27.2361 1.32270
\(425\) −20.9443 −1.01595
\(426\) 0 0
\(427\) 2.00000 0.0967868
\(428\) 6.85410 0.331306
\(429\) 0 0
\(430\) −3.52786 −0.170129
\(431\) 8.70820 0.419459 0.209730 0.977759i \(-0.432742\pi\)
0.209730 + 0.977759i \(0.432742\pi\)
\(432\) 0 0
\(433\) −10.4721 −0.503259 −0.251629 0.967824i \(-0.580966\pi\)
−0.251629 + 0.967824i \(0.580966\pi\)
\(434\) −3.23607 −0.155336
\(435\) 0 0
\(436\) 4.47214 0.214176
\(437\) −38.2918 −1.83175
\(438\) 0 0
\(439\) −11.1803 −0.533609 −0.266804 0.963751i \(-0.585968\pi\)
−0.266804 + 0.963751i \(0.585968\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 11.2361 0.534445
\(443\) −18.4721 −0.877638 −0.438819 0.898576i \(-0.644603\pi\)
−0.438819 + 0.898576i \(0.644603\pi\)
\(444\) 0 0
\(445\) −4.47214 −0.212000
\(446\) 3.70820 0.175589
\(447\) 0 0
\(448\) 0.236068 0.0111532
\(449\) −20.0000 −0.943858 −0.471929 0.881636i \(-0.656442\pi\)
−0.471929 + 0.881636i \(0.656442\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) −0.763932 −0.0359323
\(453\) 0 0
\(454\) 1.23607 0.0580115
\(455\) −3.47214 −0.162776
\(456\) 0 0
\(457\) −15.2361 −0.712713 −0.356357 0.934350i \(-0.615981\pi\)
−0.356357 + 0.934350i \(0.615981\pi\)
\(458\) −4.47214 −0.208969
\(459\) 0 0
\(460\) −9.23607 −0.430634
\(461\) −28.0000 −1.30409 −0.652045 0.758180i \(-0.726089\pi\)
−0.652045 + 0.758180i \(0.726089\pi\)
\(462\) 0 0
\(463\) −4.81966 −0.223989 −0.111994 0.993709i \(-0.535724\pi\)
−0.111994 + 0.993709i \(0.535724\pi\)
\(464\) 9.27051 0.430373
\(465\) 0 0
\(466\) 9.23607 0.427853
\(467\) −29.1803 −1.35031 −0.675153 0.737678i \(-0.735922\pi\)
−0.675153 + 0.737678i \(0.735922\pi\)
\(468\) 0 0
\(469\) 9.76393 0.450856
\(470\) −0.145898 −0.00672977
\(471\) 0 0
\(472\) 25.0000 1.15072
\(473\) 0 0
\(474\) 0 0
\(475\) −26.8328 −1.23117
\(476\) 8.47214 0.388320
\(477\) 0 0
\(478\) −6.25735 −0.286205
\(479\) 11.7082 0.534961 0.267481 0.963563i \(-0.413809\pi\)
0.267481 + 0.963563i \(0.413809\pi\)
\(480\) 0 0
\(481\) 24.3050 1.10821
\(482\) 16.0344 0.730349
\(483\) 0 0
\(484\) 0 0
\(485\) −9.70820 −0.440827
\(486\) 0 0
\(487\) 16.9443 0.767818 0.383909 0.923371i \(-0.374578\pi\)
0.383909 + 0.923371i \(0.374578\pi\)
\(488\) −4.47214 −0.202444
\(489\) 0 0
\(490\) 0.618034 0.0279199
\(491\) −18.1246 −0.817952 −0.408976 0.912545i \(-0.634114\pi\)
−0.408976 + 0.912545i \(0.634114\pi\)
\(492\) 0 0
\(493\) 26.1803 1.17910
\(494\) 14.3951 0.647667
\(495\) 0 0
\(496\) −9.70820 −0.435911
\(497\) −2.47214 −0.110890
\(498\) 0 0
\(499\) −2.23607 −0.100100 −0.0500501 0.998747i \(-0.515938\pi\)
−0.0500501 + 0.998747i \(0.515938\pi\)
\(500\) −14.5623 −0.651246
\(501\) 0 0
\(502\) 7.49342 0.334448
\(503\) 2.29180 0.102186 0.0510931 0.998694i \(-0.483729\pi\)
0.0510931 + 0.998694i \(0.483729\pi\)
\(504\) 0 0
\(505\) −18.1803 −0.809015
\(506\) 0 0
\(507\) 0 0
\(508\) −20.4721 −0.908304
\(509\) 30.0000 1.32973 0.664863 0.746965i \(-0.268490\pi\)
0.664863 + 0.746965i \(0.268490\pi\)
\(510\) 0 0
\(511\) 4.52786 0.200301
\(512\) −18.7082 −0.826794
\(513\) 0 0
\(514\) −4.32624 −0.190822
\(515\) −17.4164 −0.767459
\(516\) 0 0
\(517\) 0 0
\(518\) −4.32624 −0.190084
\(519\) 0 0
\(520\) 7.76393 0.340471
\(521\) −38.3050 −1.67817 −0.839085 0.544000i \(-0.816909\pi\)
−0.839085 + 0.544000i \(0.816909\pi\)
\(522\) 0 0
\(523\) 21.6525 0.946797 0.473398 0.880848i \(-0.343027\pi\)
0.473398 + 0.880848i \(0.343027\pi\)
\(524\) −1.52786 −0.0667451
\(525\) 0 0
\(526\) 16.1459 0.703995
\(527\) −27.4164 −1.19428
\(528\) 0 0
\(529\) 9.58359 0.416678
\(530\) 7.52786 0.326990
\(531\) 0 0
\(532\) 10.8541 0.470585
\(533\) 8.58359 0.371797
\(534\) 0 0
\(535\) 4.23607 0.183141
\(536\) −21.8328 −0.943034
\(537\) 0 0
\(538\) 0.652476 0.0281302
\(539\) 0 0
\(540\) 0 0
\(541\) 36.9443 1.58836 0.794179 0.607684i \(-0.207901\pi\)
0.794179 + 0.607684i \(0.207901\pi\)
\(542\) 3.27051 0.140480
\(543\) 0 0
\(544\) −29.4164 −1.26122
\(545\) 2.76393 0.118394
\(546\) 0 0
\(547\) −14.8328 −0.634205 −0.317103 0.948391i \(-0.602710\pi\)
−0.317103 + 0.948391i \(0.602710\pi\)
\(548\) 31.8885 1.36221
\(549\) 0 0
\(550\) 0 0
\(551\) 33.5410 1.42890
\(552\) 0 0
\(553\) −14.4721 −0.615418
\(554\) −4.00000 −0.169944
\(555\) 0 0
\(556\) 23.4164 0.993077
\(557\) −12.5279 −0.530823 −0.265411 0.964135i \(-0.585508\pi\)
−0.265411 + 0.964135i \(0.585508\pi\)
\(558\) 0 0
\(559\) 19.8197 0.838282
\(560\) 1.85410 0.0783501
\(561\) 0 0
\(562\) −7.09017 −0.299081
\(563\) 34.6525 1.46043 0.730214 0.683219i \(-0.239420\pi\)
0.730214 + 0.683219i \(0.239420\pi\)
\(564\) 0 0
\(565\) −0.472136 −0.0198629
\(566\) −8.50658 −0.357558
\(567\) 0 0
\(568\) 5.52786 0.231944
\(569\) 30.0000 1.25767 0.628833 0.777541i \(-0.283533\pi\)
0.628833 + 0.777541i \(0.283533\pi\)
\(570\) 0 0
\(571\) −17.5279 −0.733518 −0.366759 0.930316i \(-0.619533\pi\)
−0.366759 + 0.930316i \(0.619533\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) −1.52786 −0.0637718
\(575\) 22.8328 0.952194
\(576\) 0 0
\(577\) 7.34752 0.305881 0.152941 0.988235i \(-0.451126\pi\)
0.152941 + 0.988235i \(0.451126\pi\)
\(578\) −6.43769 −0.267773
\(579\) 0 0
\(580\) 8.09017 0.335926
\(581\) −6.76393 −0.280615
\(582\) 0 0
\(583\) 0 0
\(584\) −10.1246 −0.418959
\(585\) 0 0
\(586\) 9.88854 0.408492
\(587\) −46.0132 −1.89917 −0.949583 0.313515i \(-0.898493\pi\)
−0.949583 + 0.313515i \(0.898493\pi\)
\(588\) 0 0
\(589\) −35.1246 −1.44728
\(590\) 6.90983 0.284473
\(591\) 0 0
\(592\) −12.9787 −0.533422
\(593\) −22.8328 −0.937631 −0.468816 0.883296i \(-0.655319\pi\)
−0.468816 + 0.883296i \(0.655319\pi\)
\(594\) 0 0
\(595\) 5.23607 0.214658
\(596\) −8.09017 −0.331386
\(597\) 0 0
\(598\) −12.2492 −0.500908
\(599\) 25.5279 1.04304 0.521520 0.853239i \(-0.325365\pi\)
0.521520 + 0.853239i \(0.325365\pi\)
\(600\) 0 0
\(601\) −27.0000 −1.10135 −0.550676 0.834719i \(-0.685630\pi\)
−0.550676 + 0.834719i \(0.685630\pi\)
\(602\) −3.52786 −0.143785
\(603\) 0 0
\(604\) −22.9443 −0.933589
\(605\) 0 0
\(606\) 0 0
\(607\) −6.81966 −0.276801 −0.138401 0.990376i \(-0.544196\pi\)
−0.138401 + 0.990376i \(0.544196\pi\)
\(608\) −37.6869 −1.52841
\(609\) 0 0
\(610\) −1.23607 −0.0500469
\(611\) 0.819660 0.0331599
\(612\) 0 0
\(613\) −13.5967 −0.549167 −0.274584 0.961563i \(-0.588540\pi\)
−0.274584 + 0.961563i \(0.588540\pi\)
\(614\) −12.9443 −0.522388
\(615\) 0 0
\(616\) 0 0
\(617\) −32.4721 −1.30728 −0.653639 0.756806i \(-0.726759\pi\)
−0.653639 + 0.756806i \(0.726759\pi\)
\(618\) 0 0
\(619\) −44.0689 −1.77128 −0.885639 0.464374i \(-0.846279\pi\)
−0.885639 + 0.464374i \(0.846279\pi\)
\(620\) −8.47214 −0.340249
\(621\) 0 0
\(622\) 6.11146 0.245047
\(623\) −4.47214 −0.179172
\(624\) 0 0
\(625\) 11.0000 0.440000
\(626\) −15.2361 −0.608956
\(627\) 0 0
\(628\) 24.9443 0.995385
\(629\) −36.6525 −1.46143
\(630\) 0 0
\(631\) 44.3607 1.76597 0.882985 0.469400i \(-0.155530\pi\)
0.882985 + 0.469400i \(0.155530\pi\)
\(632\) 32.3607 1.28724
\(633\) 0 0
\(634\) 14.9443 0.593513
\(635\) −12.6525 −0.502098
\(636\) 0 0
\(637\) −3.47214 −0.137571
\(638\) 0 0
\(639\) 0 0
\(640\) −11.3820 −0.449912
\(641\) 46.5410 1.83826 0.919130 0.393955i \(-0.128893\pi\)
0.919130 + 0.393955i \(0.128893\pi\)
\(642\) 0 0
\(643\) −47.9574 −1.89126 −0.945628 0.325250i \(-0.894552\pi\)
−0.945628 + 0.325250i \(0.894552\pi\)
\(644\) −9.23607 −0.363952
\(645\) 0 0
\(646\) −21.7082 −0.854098
\(647\) −12.3475 −0.485431 −0.242716 0.970097i \(-0.578038\pi\)
−0.242716 + 0.970097i \(0.578038\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) −8.58359 −0.336676
\(651\) 0 0
\(652\) 36.7426 1.43895
\(653\) 44.9443 1.75881 0.879403 0.476079i \(-0.157942\pi\)
0.879403 + 0.476079i \(0.157942\pi\)
\(654\) 0 0
\(655\) −0.944272 −0.0368958
\(656\) −4.58359 −0.178959
\(657\) 0 0
\(658\) −0.145898 −0.00568770
\(659\) 23.5410 0.917028 0.458514 0.888687i \(-0.348382\pi\)
0.458514 + 0.888687i \(0.348382\pi\)
\(660\) 0 0
\(661\) 40.5410 1.57686 0.788431 0.615123i \(-0.210894\pi\)
0.788431 + 0.615123i \(0.210894\pi\)
\(662\) 7.05573 0.274229
\(663\) 0 0
\(664\) 15.1246 0.586949
\(665\) 6.70820 0.260133
\(666\) 0 0
\(667\) −28.5410 −1.10511
\(668\) 36.6525 1.41813
\(669\) 0 0
\(670\) −6.03444 −0.233131
\(671\) 0 0
\(672\) 0 0
\(673\) −3.59675 −0.138644 −0.0693222 0.997594i \(-0.522084\pi\)
−0.0693222 + 0.997594i \(0.522084\pi\)
\(674\) −5.05573 −0.194739
\(675\) 0 0
\(676\) 1.52786 0.0587640
\(677\) 19.3050 0.741950 0.370975 0.928643i \(-0.379024\pi\)
0.370975 + 0.928643i \(0.379024\pi\)
\(678\) 0 0
\(679\) −9.70820 −0.372567
\(680\) −11.7082 −0.448989
\(681\) 0 0
\(682\) 0 0
\(683\) −37.0132 −1.41627 −0.708135 0.706078i \(-0.750463\pi\)
−0.708135 + 0.706078i \(0.750463\pi\)
\(684\) 0 0
\(685\) 19.7082 0.753012
\(686\) 0.618034 0.0235966
\(687\) 0 0
\(688\) −10.5836 −0.403496
\(689\) −42.2918 −1.61119
\(690\) 0 0
\(691\) 2.00000 0.0760836 0.0380418 0.999276i \(-0.487888\pi\)
0.0380418 + 0.999276i \(0.487888\pi\)
\(692\) 2.47214 0.0939765
\(693\) 0 0
\(694\) −17.3050 −0.656887
\(695\) 14.4721 0.548959
\(696\) 0 0
\(697\) −12.9443 −0.490299
\(698\) 0.978714 0.0370449
\(699\) 0 0
\(700\) −6.47214 −0.244624
\(701\) 4.11146 0.155288 0.0776438 0.996981i \(-0.475260\pi\)
0.0776438 + 0.996981i \(0.475260\pi\)
\(702\) 0 0
\(703\) −46.9574 −1.77103
\(704\) 0 0
\(705\) 0 0
\(706\) 15.1591 0.570519
\(707\) −18.1803 −0.683742
\(708\) 0 0
\(709\) −49.7214 −1.86732 −0.933662 0.358154i \(-0.883406\pi\)
−0.933662 + 0.358154i \(0.883406\pi\)
\(710\) 1.52786 0.0573397
\(711\) 0 0
\(712\) 10.0000 0.374766
\(713\) 29.8885 1.11933
\(714\) 0 0
\(715\) 0 0
\(716\) 14.4721 0.540849
\(717\) 0 0
\(718\) −14.4721 −0.540095
\(719\) −7.76393 −0.289546 −0.144773 0.989465i \(-0.546245\pi\)
−0.144773 + 0.989465i \(0.546245\pi\)
\(720\) 0 0
\(721\) −17.4164 −0.648621
\(722\) −16.0689 −0.598022
\(723\) 0 0
\(724\) 1.23607 0.0459381
\(725\) −20.0000 −0.742781
\(726\) 0 0
\(727\) 24.1803 0.896799 0.448400 0.893833i \(-0.351994\pi\)
0.448400 + 0.893833i \(0.351994\pi\)
\(728\) 7.76393 0.287750
\(729\) 0 0
\(730\) −2.79837 −0.103572
\(731\) −29.8885 −1.10547
\(732\) 0 0
\(733\) 8.11146 0.299603 0.149802 0.988716i \(-0.452136\pi\)
0.149802 + 0.988716i \(0.452136\pi\)
\(734\) 12.2918 0.453698
\(735\) 0 0
\(736\) 32.0689 1.18207
\(737\) 0 0
\(738\) 0 0
\(739\) −34.0689 −1.25324 −0.626622 0.779323i \(-0.715563\pi\)
−0.626622 + 0.779323i \(0.715563\pi\)
\(740\) −11.3262 −0.416361
\(741\) 0 0
\(742\) 7.52786 0.276357
\(743\) 2.81966 0.103443 0.0517216 0.998662i \(-0.483529\pi\)
0.0517216 + 0.998662i \(0.483529\pi\)
\(744\) 0 0
\(745\) −5.00000 −0.183186
\(746\) 2.87539 0.105275
\(747\) 0 0
\(748\) 0 0
\(749\) 4.23607 0.154783
\(750\) 0 0
\(751\) 39.7639 1.45101 0.725503 0.688219i \(-0.241607\pi\)
0.725503 + 0.688219i \(0.241607\pi\)
\(752\) −0.437694 −0.0159611
\(753\) 0 0
\(754\) 10.7295 0.390745
\(755\) −14.1803 −0.516075
\(756\) 0 0
\(757\) −51.7214 −1.87984 −0.939922 0.341388i \(-0.889103\pi\)
−0.939922 + 0.341388i \(0.889103\pi\)
\(758\) 19.2705 0.699936
\(759\) 0 0
\(760\) −15.0000 −0.544107
\(761\) 27.7771 1.00692 0.503459 0.864019i \(-0.332060\pi\)
0.503459 + 0.864019i \(0.332060\pi\)
\(762\) 0 0
\(763\) 2.76393 0.100061
\(764\) −17.4164 −0.630104
\(765\) 0 0
\(766\) 20.3607 0.735661
\(767\) −38.8197 −1.40170
\(768\) 0 0
\(769\) −13.9443 −0.502843 −0.251422 0.967878i \(-0.580898\pi\)
−0.251422 + 0.967878i \(0.580898\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 23.7082 0.853277
\(773\) 5.47214 0.196819 0.0984095 0.995146i \(-0.468624\pi\)
0.0984095 + 0.995146i \(0.468624\pi\)
\(774\) 0 0
\(775\) 20.9443 0.752340
\(776\) 21.7082 0.779279
\(777\) 0 0
\(778\) 6.83282 0.244968
\(779\) −16.5836 −0.594169
\(780\) 0 0
\(781\) 0 0
\(782\) 18.4721 0.660562
\(783\) 0 0
\(784\) 1.85410 0.0662179
\(785\) 15.4164 0.550235
\(786\) 0 0
\(787\) −3.65248 −0.130197 −0.0650984 0.997879i \(-0.520736\pi\)
−0.0650984 + 0.997879i \(0.520736\pi\)
\(788\) 26.6525 0.949455
\(789\) 0 0
\(790\) 8.94427 0.318223
\(791\) −0.472136 −0.0167872
\(792\) 0 0
\(793\) 6.94427 0.246598
\(794\) −14.2918 −0.507197
\(795\) 0 0
\(796\) 6.18034 0.219056
\(797\) 2.52786 0.0895415 0.0447708 0.998997i \(-0.485744\pi\)
0.0447708 + 0.998997i \(0.485744\pi\)
\(798\) 0 0
\(799\) −1.23607 −0.0437289
\(800\) 22.4721 0.794510
\(801\) 0 0
\(802\) −18.3607 −0.648338
\(803\) 0 0
\(804\) 0 0
\(805\) −5.70820 −0.201188
\(806\) −11.2361 −0.395774
\(807\) 0 0
\(808\) 40.6525 1.43015
\(809\) −56.3050 −1.97958 −0.989788 0.142545i \(-0.954471\pi\)
−0.989788 + 0.142545i \(0.954471\pi\)
\(810\) 0 0
\(811\) −15.2918 −0.536968 −0.268484 0.963284i \(-0.586523\pi\)
−0.268484 + 0.963284i \(0.586523\pi\)
\(812\) 8.09017 0.283909
\(813\) 0 0
\(814\) 0 0
\(815\) 22.7082 0.795434
\(816\) 0 0
\(817\) −38.2918 −1.33966
\(818\) 13.0132 0.454994
\(819\) 0 0
\(820\) −4.00000 −0.139686
\(821\) −7.47214 −0.260779 −0.130390 0.991463i \(-0.541623\pi\)
−0.130390 + 0.991463i \(0.541623\pi\)
\(822\) 0 0
\(823\) −17.1803 −0.598869 −0.299435 0.954117i \(-0.596798\pi\)
−0.299435 + 0.954117i \(0.596798\pi\)
\(824\) 38.9443 1.35669
\(825\) 0 0
\(826\) 6.90983 0.240424
\(827\) 12.3475 0.429365 0.214683 0.976684i \(-0.431128\pi\)
0.214683 + 0.976684i \(0.431128\pi\)
\(828\) 0 0
\(829\) 35.7771 1.24259 0.621295 0.783577i \(-0.286607\pi\)
0.621295 + 0.783577i \(0.286607\pi\)
\(830\) 4.18034 0.145102
\(831\) 0 0
\(832\) 0.819660 0.0284166
\(833\) 5.23607 0.181419
\(834\) 0 0
\(835\) 22.6525 0.783921
\(836\) 0 0
\(837\) 0 0
\(838\) −0.729490 −0.0251998
\(839\) 23.5410 0.812726 0.406363 0.913712i \(-0.366797\pi\)
0.406363 + 0.913712i \(0.366797\pi\)
\(840\) 0 0
\(841\) −4.00000 −0.137931
\(842\) 8.03444 0.276885
\(843\) 0 0
\(844\) 8.76393 0.301667
\(845\) 0.944272 0.0324839
\(846\) 0 0
\(847\) 0 0
\(848\) 22.5836 0.775524
\(849\) 0 0
\(850\) 12.9443 0.443985
\(851\) 39.9574 1.36972
\(852\) 0 0
\(853\) 29.4164 1.00720 0.503599 0.863937i \(-0.332009\pi\)
0.503599 + 0.863937i \(0.332009\pi\)
\(854\) −1.23607 −0.0422974
\(855\) 0 0
\(856\) −9.47214 −0.323751
\(857\) 0.111456 0.00380727 0.00190364 0.999998i \(-0.499394\pi\)
0.00190364 + 0.999998i \(0.499394\pi\)
\(858\) 0 0
\(859\) −40.0000 −1.36478 −0.682391 0.730987i \(-0.739060\pi\)
−0.682391 + 0.730987i \(0.739060\pi\)
\(860\) −9.23607 −0.314947
\(861\) 0 0
\(862\) −5.38197 −0.183310
\(863\) 43.2361 1.47177 0.735886 0.677105i \(-0.236766\pi\)
0.735886 + 0.677105i \(0.236766\pi\)
\(864\) 0 0
\(865\) 1.52786 0.0519489
\(866\) 6.47214 0.219932
\(867\) 0 0
\(868\) −8.47214 −0.287563
\(869\) 0 0
\(870\) 0 0
\(871\) 33.9017 1.14872
\(872\) −6.18034 −0.209293
\(873\) 0 0
\(874\) 23.6656 0.800502
\(875\) −9.00000 −0.304256
\(876\) 0 0
\(877\) −4.58359 −0.154777 −0.0773885 0.997001i \(-0.524658\pi\)
−0.0773885 + 0.997001i \(0.524658\pi\)
\(878\) 6.90983 0.233195
\(879\) 0 0
\(880\) 0 0
\(881\) −54.8885 −1.84924 −0.924621 0.380888i \(-0.875618\pi\)
−0.924621 + 0.380888i \(0.875618\pi\)
\(882\) 0 0
\(883\) −14.8197 −0.498721 −0.249361 0.968411i \(-0.580220\pi\)
−0.249361 + 0.968411i \(0.580220\pi\)
\(884\) 29.4164 0.989381
\(885\) 0 0
\(886\) 11.4164 0.383542
\(887\) 10.7639 0.361417 0.180709 0.983537i \(-0.442161\pi\)
0.180709 + 0.983537i \(0.442161\pi\)
\(888\) 0 0
\(889\) −12.6525 −0.424350
\(890\) 2.76393 0.0926472
\(891\) 0 0
\(892\) 9.70820 0.325055
\(893\) −1.58359 −0.0529929
\(894\) 0 0
\(895\) 8.94427 0.298974
\(896\) −11.3820 −0.380245
\(897\) 0 0
\(898\) 12.3607 0.412481
\(899\) −26.1803 −0.873163
\(900\) 0 0
\(901\) 63.7771 2.12472
\(902\) 0 0
\(903\) 0 0
\(904\) 1.05573 0.0351130
\(905\) 0.763932 0.0253940
\(906\) 0 0
\(907\) 28.0000 0.929725 0.464862 0.885383i \(-0.346104\pi\)
0.464862 + 0.885383i \(0.346104\pi\)
\(908\) 3.23607 0.107393
\(909\) 0 0
\(910\) 2.14590 0.0711358
\(911\) 14.5836 0.483176 0.241588 0.970379i \(-0.422332\pi\)
0.241588 + 0.970379i \(0.422332\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 9.41641 0.311467
\(915\) 0 0
\(916\) −11.7082 −0.386850
\(917\) −0.944272 −0.0311826
\(918\) 0 0
\(919\) −39.5967 −1.30618 −0.653088 0.757282i \(-0.726527\pi\)
−0.653088 + 0.757282i \(0.726527\pi\)
\(920\) 12.7639 0.420814
\(921\) 0 0
\(922\) 17.3050 0.569908
\(923\) −8.58359 −0.282532
\(924\) 0 0
\(925\) 28.0000 0.920634
\(926\) 2.97871 0.0978866
\(927\) 0 0
\(928\) −28.0902 −0.922105
\(929\) −12.8885 −0.422859 −0.211430 0.977393i \(-0.567812\pi\)
−0.211430 + 0.977393i \(0.567812\pi\)
\(930\) 0 0
\(931\) 6.70820 0.219853
\(932\) 24.1803 0.792053
\(933\) 0 0
\(934\) 18.0344 0.590105
\(935\) 0 0
\(936\) 0 0
\(937\) 39.8885 1.30310 0.651551 0.758605i \(-0.274119\pi\)
0.651551 + 0.758605i \(0.274119\pi\)
\(938\) −6.03444 −0.197032
\(939\) 0 0
\(940\) −0.381966 −0.0124584
\(941\) −60.3607 −1.96770 −0.983851 0.178990i \(-0.942717\pi\)
−0.983851 + 0.178990i \(0.942717\pi\)
\(942\) 0 0
\(943\) 14.1115 0.459532
\(944\) 20.7295 0.674687
\(945\) 0 0
\(946\) 0 0
\(947\) 5.41641 0.176010 0.0880048 0.996120i \(-0.471951\pi\)
0.0880048 + 0.996120i \(0.471951\pi\)
\(948\) 0 0
\(949\) 15.7214 0.510337
\(950\) 16.5836 0.538043
\(951\) 0 0
\(952\) −11.7082 −0.379465
\(953\) 44.7771 1.45047 0.725236 0.688500i \(-0.241731\pi\)
0.725236 + 0.688500i \(0.241731\pi\)
\(954\) 0 0
\(955\) −10.7639 −0.348313
\(956\) −16.3820 −0.529831
\(957\) 0 0
\(958\) −7.23607 −0.233787
\(959\) 19.7082 0.636411
\(960\) 0 0
\(961\) −3.58359 −0.115600
\(962\) −15.0213 −0.484306
\(963\) 0 0
\(964\) 41.9787 1.35204
\(965\) 14.6525 0.471680
\(966\) 0 0
\(967\) 61.1935 1.96785 0.983925 0.178582i \(-0.0571509\pi\)
0.983925 + 0.178582i \(0.0571509\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 6.00000 0.192648
\(971\) −32.1246 −1.03093 −0.515464 0.856911i \(-0.672380\pi\)
−0.515464 + 0.856911i \(0.672380\pi\)
\(972\) 0 0
\(973\) 14.4721 0.463955
\(974\) −10.4721 −0.335549
\(975\) 0 0
\(976\) −3.70820 −0.118697
\(977\) −4.18034 −0.133741 −0.0668705 0.997762i \(-0.521301\pi\)
−0.0668705 + 0.997762i \(0.521301\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 1.61803 0.0516862
\(981\) 0 0
\(982\) 11.2016 0.357458
\(983\) −27.4164 −0.874448 −0.437224 0.899353i \(-0.644038\pi\)
−0.437224 + 0.899353i \(0.644038\pi\)
\(984\) 0 0
\(985\) 16.4721 0.524846
\(986\) −16.1803 −0.515287
\(987\) 0 0
\(988\) 37.6869 1.19898
\(989\) 32.5836 1.03610
\(990\) 0 0
\(991\) −29.1803 −0.926944 −0.463472 0.886112i \(-0.653397\pi\)
−0.463472 + 0.886112i \(0.653397\pi\)
\(992\) 29.4164 0.933972
\(993\) 0 0
\(994\) 1.52786 0.0484609
\(995\) 3.81966 0.121091
\(996\) 0 0
\(997\) −26.9443 −0.853334 −0.426667 0.904409i \(-0.640312\pi\)
−0.426667 + 0.904409i \(0.640312\pi\)
\(998\) 1.38197 0.0437454
\(999\) 0 0
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 7623.2.a.bm.1.1 2
3.2 odd 2 2541.2.a.t.1.2 2
11.10 odd 2 693.2.a.f.1.2 2
33.32 even 2 231.2.a.c.1.1 2
77.76 even 2 4851.2.a.w.1.2 2
132.131 odd 2 3696.2.a.be.1.2 2
165.164 even 2 5775.2.a.be.1.2 2
231.230 odd 2 1617.2.a.p.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
231.2.a.c.1.1 2 33.32 even 2
693.2.a.f.1.2 2 11.10 odd 2
1617.2.a.p.1.1 2 231.230 odd 2
2541.2.a.t.1.2 2 3.2 odd 2
3696.2.a.be.1.2 2 132.131 odd 2
4851.2.a.w.1.2 2 77.76 even 2
5775.2.a.be.1.2 2 165.164 even 2
7623.2.a.bm.1.1 2 1.1 even 1 trivial