Properties

Label 2898.2.a.v.1.1
Level $2898$
Weight $2$
Character 2898.1
Self dual yes
Analytic conductor $23.141$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 2898 = 2 \cdot 3^{2} \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2898.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(23.1406465058\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{10})^+\)
Defining polynomial: \(x^{2} - x - 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 966)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-0.618034\) of defining polynomial
Character \(\chi\) \(=\) 2898.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{4} -2.00000 q^{5} +1.00000 q^{7} -1.00000 q^{8} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{4} -2.00000 q^{5} +1.00000 q^{7} -1.00000 q^{8} +2.00000 q^{10} -4.47214 q^{13} -1.00000 q^{14} +1.00000 q^{16} -4.47214 q^{17} +2.47214 q^{19} -2.00000 q^{20} +1.00000 q^{23} -1.00000 q^{25} +4.47214 q^{26} +1.00000 q^{28} +2.00000 q^{29} -2.47214 q^{31} -1.00000 q^{32} +4.47214 q^{34} -2.00000 q^{35} +6.94427 q^{37} -2.47214 q^{38} +2.00000 q^{40} +6.00000 q^{41} -4.94427 q^{43} -1.00000 q^{46} +2.47214 q^{47} +1.00000 q^{49} +1.00000 q^{50} -4.47214 q^{52} +10.9443 q^{53} -1.00000 q^{56} -2.00000 q^{58} -4.00000 q^{59} -6.00000 q^{61} +2.47214 q^{62} +1.00000 q^{64} +8.94427 q^{65} +4.94427 q^{67} -4.47214 q^{68} +2.00000 q^{70} -4.94427 q^{71} +14.9443 q^{73} -6.94427 q^{74} +2.47214 q^{76} -4.94427 q^{79} -2.00000 q^{80} -6.00000 q^{82} -2.47214 q^{83} +8.94427 q^{85} +4.94427 q^{86} -9.41641 q^{89} -4.47214 q^{91} +1.00000 q^{92} -2.47214 q^{94} -4.94427 q^{95} -0.472136 q^{97} -1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} + 2 q^{4} - 4 q^{5} + 2 q^{7} - 2 q^{8} + O(q^{10}) \) \( 2 q - 2 q^{2} + 2 q^{4} - 4 q^{5} + 2 q^{7} - 2 q^{8} + 4 q^{10} - 2 q^{14} + 2 q^{16} - 4 q^{19} - 4 q^{20} + 2 q^{23} - 2 q^{25} + 2 q^{28} + 4 q^{29} + 4 q^{31} - 2 q^{32} - 4 q^{35} - 4 q^{37} + 4 q^{38} + 4 q^{40} + 12 q^{41} + 8 q^{43} - 2 q^{46} - 4 q^{47} + 2 q^{49} + 2 q^{50} + 4 q^{53} - 2 q^{56} - 4 q^{58} - 8 q^{59} - 12 q^{61} - 4 q^{62} + 2 q^{64} - 8 q^{67} + 4 q^{70} + 8 q^{71} + 12 q^{73} + 4 q^{74} - 4 q^{76} + 8 q^{79} - 4 q^{80} - 12 q^{82} + 4 q^{83} - 8 q^{86} + 8 q^{89} + 2 q^{92} + 4 q^{94} + 8 q^{95} + 8 q^{97} - 2 q^{98} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) 0 0
\(4\) 1.00000 0.500000
\(5\) −2.00000 −0.894427 −0.447214 0.894427i \(-0.647584\pi\)
−0.447214 + 0.894427i \(0.647584\pi\)
\(6\) 0 0
\(7\) 1.00000 0.377964
\(8\) −1.00000 −0.353553
\(9\) 0 0
\(10\) 2.00000 0.632456
\(11\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) 0 0
\(13\) −4.47214 −1.24035 −0.620174 0.784465i \(-0.712938\pi\)
−0.620174 + 0.784465i \(0.712938\pi\)
\(14\) −1.00000 −0.267261
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −4.47214 −1.08465 −0.542326 0.840168i \(-0.682456\pi\)
−0.542326 + 0.840168i \(0.682456\pi\)
\(18\) 0 0
\(19\) 2.47214 0.567147 0.283573 0.958951i \(-0.408480\pi\)
0.283573 + 0.958951i \(0.408480\pi\)
\(20\) −2.00000 −0.447214
\(21\) 0 0
\(22\) 0 0
\(23\) 1.00000 0.208514
\(24\) 0 0
\(25\) −1.00000 −0.200000
\(26\) 4.47214 0.877058
\(27\) 0 0
\(28\) 1.00000 0.188982
\(29\) 2.00000 0.371391 0.185695 0.982607i \(-0.440546\pi\)
0.185695 + 0.982607i \(0.440546\pi\)
\(30\) 0 0
\(31\) −2.47214 −0.444009 −0.222004 0.975046i \(-0.571260\pi\)
−0.222004 + 0.975046i \(0.571260\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0 0
\(34\) 4.47214 0.766965
\(35\) −2.00000 −0.338062
\(36\) 0 0
\(37\) 6.94427 1.14163 0.570816 0.821078i \(-0.306627\pi\)
0.570816 + 0.821078i \(0.306627\pi\)
\(38\) −2.47214 −0.401033
\(39\) 0 0
\(40\) 2.00000 0.316228
\(41\) 6.00000 0.937043 0.468521 0.883452i \(-0.344787\pi\)
0.468521 + 0.883452i \(0.344787\pi\)
\(42\) 0 0
\(43\) −4.94427 −0.753994 −0.376997 0.926214i \(-0.623043\pi\)
−0.376997 + 0.926214i \(0.623043\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) −1.00000 −0.147442
\(47\) 2.47214 0.360598 0.180299 0.983612i \(-0.442293\pi\)
0.180299 + 0.983612i \(0.442293\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 1.00000 0.141421
\(51\) 0 0
\(52\) −4.47214 −0.620174
\(53\) 10.9443 1.50331 0.751656 0.659556i \(-0.229256\pi\)
0.751656 + 0.659556i \(0.229256\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −1.00000 −0.133631
\(57\) 0 0
\(58\) −2.00000 −0.262613
\(59\) −4.00000 −0.520756 −0.260378 0.965507i \(-0.583847\pi\)
−0.260378 + 0.965507i \(0.583847\pi\)
\(60\) 0 0
\(61\) −6.00000 −0.768221 −0.384111 0.923287i \(-0.625492\pi\)
−0.384111 + 0.923287i \(0.625492\pi\)
\(62\) 2.47214 0.313962
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 8.94427 1.10940
\(66\) 0 0
\(67\) 4.94427 0.604039 0.302019 0.953302i \(-0.402339\pi\)
0.302019 + 0.953302i \(0.402339\pi\)
\(68\) −4.47214 −0.542326
\(69\) 0 0
\(70\) 2.00000 0.239046
\(71\) −4.94427 −0.586777 −0.293389 0.955993i \(-0.594783\pi\)
−0.293389 + 0.955993i \(0.594783\pi\)
\(72\) 0 0
\(73\) 14.9443 1.74909 0.874547 0.484940i \(-0.161159\pi\)
0.874547 + 0.484940i \(0.161159\pi\)
\(74\) −6.94427 −0.807255
\(75\) 0 0
\(76\) 2.47214 0.283573
\(77\) 0 0
\(78\) 0 0
\(79\) −4.94427 −0.556274 −0.278137 0.960541i \(-0.589717\pi\)
−0.278137 + 0.960541i \(0.589717\pi\)
\(80\) −2.00000 −0.223607
\(81\) 0 0
\(82\) −6.00000 −0.662589
\(83\) −2.47214 −0.271352 −0.135676 0.990753i \(-0.543321\pi\)
−0.135676 + 0.990753i \(0.543321\pi\)
\(84\) 0 0
\(85\) 8.94427 0.970143
\(86\) 4.94427 0.533155
\(87\) 0 0
\(88\) 0 0
\(89\) −9.41641 −0.998137 −0.499069 0.866562i \(-0.666324\pi\)
−0.499069 + 0.866562i \(0.666324\pi\)
\(90\) 0 0
\(91\) −4.47214 −0.468807
\(92\) 1.00000 0.104257
\(93\) 0 0
\(94\) −2.47214 −0.254981
\(95\) −4.94427 −0.507272
\(96\) 0 0
\(97\) −0.472136 −0.0479381 −0.0239691 0.999713i \(-0.507630\pi\)
−0.0239691 + 0.999713i \(0.507630\pi\)
\(98\) −1.00000 −0.101015
\(99\) 0 0
\(100\) −1.00000 −0.100000
\(101\) 9.41641 0.936968 0.468484 0.883472i \(-0.344800\pi\)
0.468484 + 0.883472i \(0.344800\pi\)
\(102\) 0 0
\(103\) −4.94427 −0.487174 −0.243587 0.969879i \(-0.578324\pi\)
−0.243587 + 0.969879i \(0.578324\pi\)
\(104\) 4.47214 0.438529
\(105\) 0 0
\(106\) −10.9443 −1.06300
\(107\) 8.00000 0.773389 0.386695 0.922208i \(-0.373617\pi\)
0.386695 + 0.922208i \(0.373617\pi\)
\(108\) 0 0
\(109\) −2.94427 −0.282010 −0.141005 0.990009i \(-0.545033\pi\)
−0.141005 + 0.990009i \(0.545033\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 1.00000 0.0944911
\(113\) 18.9443 1.78213 0.891064 0.453878i \(-0.149960\pi\)
0.891064 + 0.453878i \(0.149960\pi\)
\(114\) 0 0
\(115\) −2.00000 −0.186501
\(116\) 2.00000 0.185695
\(117\) 0 0
\(118\) 4.00000 0.368230
\(119\) −4.47214 −0.409960
\(120\) 0 0
\(121\) −11.0000 −1.00000
\(122\) 6.00000 0.543214
\(123\) 0 0
\(124\) −2.47214 −0.222004
\(125\) 12.0000 1.07331
\(126\) 0 0
\(127\) 20.9443 1.85850 0.929252 0.369447i \(-0.120453\pi\)
0.929252 + 0.369447i \(0.120453\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 0 0
\(130\) −8.94427 −0.784465
\(131\) −12.0000 −1.04844 −0.524222 0.851581i \(-0.675644\pi\)
−0.524222 + 0.851581i \(0.675644\pi\)
\(132\) 0 0
\(133\) 2.47214 0.214361
\(134\) −4.94427 −0.427120
\(135\) 0 0
\(136\) 4.47214 0.383482
\(137\) 14.0000 1.19610 0.598050 0.801459i \(-0.295942\pi\)
0.598050 + 0.801459i \(0.295942\pi\)
\(138\) 0 0
\(139\) 8.94427 0.758643 0.379322 0.925265i \(-0.376157\pi\)
0.379322 + 0.925265i \(0.376157\pi\)
\(140\) −2.00000 −0.169031
\(141\) 0 0
\(142\) 4.94427 0.414914
\(143\) 0 0
\(144\) 0 0
\(145\) −4.00000 −0.332182
\(146\) −14.9443 −1.23680
\(147\) 0 0
\(148\) 6.94427 0.570816
\(149\) 1.05573 0.0864886 0.0432443 0.999065i \(-0.486231\pi\)
0.0432443 + 0.999065i \(0.486231\pi\)
\(150\) 0 0
\(151\) 4.94427 0.402359 0.201180 0.979554i \(-0.435523\pi\)
0.201180 + 0.979554i \(0.435523\pi\)
\(152\) −2.47214 −0.200517
\(153\) 0 0
\(154\) 0 0
\(155\) 4.94427 0.397133
\(156\) 0 0
\(157\) −2.94427 −0.234978 −0.117489 0.993074i \(-0.537485\pi\)
−0.117489 + 0.993074i \(0.537485\pi\)
\(158\) 4.94427 0.393345
\(159\) 0 0
\(160\) 2.00000 0.158114
\(161\) 1.00000 0.0788110
\(162\) 0 0
\(163\) −8.94427 −0.700569 −0.350285 0.936643i \(-0.613915\pi\)
−0.350285 + 0.936643i \(0.613915\pi\)
\(164\) 6.00000 0.468521
\(165\) 0 0
\(166\) 2.47214 0.191875
\(167\) 15.4164 1.19296 0.596479 0.802629i \(-0.296566\pi\)
0.596479 + 0.802629i \(0.296566\pi\)
\(168\) 0 0
\(169\) 7.00000 0.538462
\(170\) −8.94427 −0.685994
\(171\) 0 0
\(172\) −4.94427 −0.376997
\(173\) −0.472136 −0.0358958 −0.0179479 0.999839i \(-0.505713\pi\)
−0.0179479 + 0.999839i \(0.505713\pi\)
\(174\) 0 0
\(175\) −1.00000 −0.0755929
\(176\) 0 0
\(177\) 0 0
\(178\) 9.41641 0.705790
\(179\) 20.0000 1.49487 0.747435 0.664335i \(-0.231285\pi\)
0.747435 + 0.664335i \(0.231285\pi\)
\(180\) 0 0
\(181\) 2.00000 0.148659 0.0743294 0.997234i \(-0.476318\pi\)
0.0743294 + 0.997234i \(0.476318\pi\)
\(182\) 4.47214 0.331497
\(183\) 0 0
\(184\) −1.00000 −0.0737210
\(185\) −13.8885 −1.02111
\(186\) 0 0
\(187\) 0 0
\(188\) 2.47214 0.180299
\(189\) 0 0
\(190\) 4.94427 0.358695
\(191\) 3.05573 0.221105 0.110552 0.993870i \(-0.464738\pi\)
0.110552 + 0.993870i \(0.464738\pi\)
\(192\) 0 0
\(193\) 11.8885 0.855756 0.427878 0.903836i \(-0.359261\pi\)
0.427878 + 0.903836i \(0.359261\pi\)
\(194\) 0.472136 0.0338974
\(195\) 0 0
\(196\) 1.00000 0.0714286
\(197\) 14.9443 1.06474 0.532368 0.846513i \(-0.321302\pi\)
0.532368 + 0.846513i \(0.321302\pi\)
\(198\) 0 0
\(199\) 3.05573 0.216615 0.108307 0.994117i \(-0.465457\pi\)
0.108307 + 0.994117i \(0.465457\pi\)
\(200\) 1.00000 0.0707107
\(201\) 0 0
\(202\) −9.41641 −0.662536
\(203\) 2.00000 0.140372
\(204\) 0 0
\(205\) −12.0000 −0.838116
\(206\) 4.94427 0.344484
\(207\) 0 0
\(208\) −4.47214 −0.310087
\(209\) 0 0
\(210\) 0 0
\(211\) −0.944272 −0.0650064 −0.0325032 0.999472i \(-0.510348\pi\)
−0.0325032 + 0.999472i \(0.510348\pi\)
\(212\) 10.9443 0.751656
\(213\) 0 0
\(214\) −8.00000 −0.546869
\(215\) 9.88854 0.674393
\(216\) 0 0
\(217\) −2.47214 −0.167820
\(218\) 2.94427 0.199411
\(219\) 0 0
\(220\) 0 0
\(221\) 20.0000 1.34535
\(222\) 0 0
\(223\) 18.4721 1.23699 0.618493 0.785790i \(-0.287744\pi\)
0.618493 + 0.785790i \(0.287744\pi\)
\(224\) −1.00000 −0.0668153
\(225\) 0 0
\(226\) −18.9443 −1.26015
\(227\) −5.52786 −0.366897 −0.183449 0.983029i \(-0.558726\pi\)
−0.183449 + 0.983029i \(0.558726\pi\)
\(228\) 0 0
\(229\) −14.0000 −0.925146 −0.462573 0.886581i \(-0.653074\pi\)
−0.462573 + 0.886581i \(0.653074\pi\)
\(230\) 2.00000 0.131876
\(231\) 0 0
\(232\) −2.00000 −0.131306
\(233\) 6.00000 0.393073 0.196537 0.980497i \(-0.437031\pi\)
0.196537 + 0.980497i \(0.437031\pi\)
\(234\) 0 0
\(235\) −4.94427 −0.322529
\(236\) −4.00000 −0.260378
\(237\) 0 0
\(238\) 4.47214 0.289886
\(239\) −20.9443 −1.35477 −0.677386 0.735628i \(-0.736887\pi\)
−0.677386 + 0.735628i \(0.736887\pi\)
\(240\) 0 0
\(241\) 20.4721 1.31873 0.659363 0.751825i \(-0.270826\pi\)
0.659363 + 0.751825i \(0.270826\pi\)
\(242\) 11.0000 0.707107
\(243\) 0 0
\(244\) −6.00000 −0.384111
\(245\) −2.00000 −0.127775
\(246\) 0 0
\(247\) −11.0557 −0.703459
\(248\) 2.47214 0.156981
\(249\) 0 0
\(250\) −12.0000 −0.758947
\(251\) 5.52786 0.348916 0.174458 0.984665i \(-0.444183\pi\)
0.174458 + 0.984665i \(0.444183\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) −20.9443 −1.31416
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 2.94427 0.183659 0.0918293 0.995775i \(-0.470729\pi\)
0.0918293 + 0.995775i \(0.470729\pi\)
\(258\) 0 0
\(259\) 6.94427 0.431496
\(260\) 8.94427 0.554700
\(261\) 0 0
\(262\) 12.0000 0.741362
\(263\) 24.0000 1.47990 0.739952 0.672660i \(-0.234848\pi\)
0.739952 + 0.672660i \(0.234848\pi\)
\(264\) 0 0
\(265\) −21.8885 −1.34460
\(266\) −2.47214 −0.151576
\(267\) 0 0
\(268\) 4.94427 0.302019
\(269\) −16.4721 −1.00432 −0.502162 0.864774i \(-0.667462\pi\)
−0.502162 + 0.864774i \(0.667462\pi\)
\(270\) 0 0
\(271\) 15.4164 0.936480 0.468240 0.883601i \(-0.344888\pi\)
0.468240 + 0.883601i \(0.344888\pi\)
\(272\) −4.47214 −0.271163
\(273\) 0 0
\(274\) −14.0000 −0.845771
\(275\) 0 0
\(276\) 0 0
\(277\) −11.8885 −0.714313 −0.357157 0.934044i \(-0.616254\pi\)
−0.357157 + 0.934044i \(0.616254\pi\)
\(278\) −8.94427 −0.536442
\(279\) 0 0
\(280\) 2.00000 0.119523
\(281\) −10.0000 −0.596550 −0.298275 0.954480i \(-0.596411\pi\)
−0.298275 + 0.954480i \(0.596411\pi\)
\(282\) 0 0
\(283\) 18.4721 1.09805 0.549027 0.835804i \(-0.314998\pi\)
0.549027 + 0.835804i \(0.314998\pi\)
\(284\) −4.94427 −0.293389
\(285\) 0 0
\(286\) 0 0
\(287\) 6.00000 0.354169
\(288\) 0 0
\(289\) 3.00000 0.176471
\(290\) 4.00000 0.234888
\(291\) 0 0
\(292\) 14.9443 0.874547
\(293\) −27.8885 −1.62927 −0.814633 0.579977i \(-0.803062\pi\)
−0.814633 + 0.579977i \(0.803062\pi\)
\(294\) 0 0
\(295\) 8.00000 0.465778
\(296\) −6.94427 −0.403628
\(297\) 0 0
\(298\) −1.05573 −0.0611567
\(299\) −4.47214 −0.258630
\(300\) 0 0
\(301\) −4.94427 −0.284983
\(302\) −4.94427 −0.284511
\(303\) 0 0
\(304\) 2.47214 0.141787
\(305\) 12.0000 0.687118
\(306\) 0 0
\(307\) 24.9443 1.42364 0.711822 0.702360i \(-0.247870\pi\)
0.711822 + 0.702360i \(0.247870\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) −4.94427 −0.280816
\(311\) 2.47214 0.140182 0.0700910 0.997541i \(-0.477671\pi\)
0.0700910 + 0.997541i \(0.477671\pi\)
\(312\) 0 0
\(313\) −13.4164 −0.758340 −0.379170 0.925327i \(-0.623790\pi\)
−0.379170 + 0.925327i \(0.623790\pi\)
\(314\) 2.94427 0.166155
\(315\) 0 0
\(316\) −4.94427 −0.278137
\(317\) 13.0557 0.733283 0.366641 0.930362i \(-0.380508\pi\)
0.366641 + 0.930362i \(0.380508\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) −2.00000 −0.111803
\(321\) 0 0
\(322\) −1.00000 −0.0557278
\(323\) −11.0557 −0.615157
\(324\) 0 0
\(325\) 4.47214 0.248069
\(326\) 8.94427 0.495377
\(327\) 0 0
\(328\) −6.00000 −0.331295
\(329\) 2.47214 0.136293
\(330\) 0 0
\(331\) −13.8885 −0.763383 −0.381692 0.924290i \(-0.624658\pi\)
−0.381692 + 0.924290i \(0.624658\pi\)
\(332\) −2.47214 −0.135676
\(333\) 0 0
\(334\) −15.4164 −0.843548
\(335\) −9.88854 −0.540269
\(336\) 0 0
\(337\) −2.94427 −0.160385 −0.0801924 0.996779i \(-0.525553\pi\)
−0.0801924 + 0.996779i \(0.525553\pi\)
\(338\) −7.00000 −0.380750
\(339\) 0 0
\(340\) 8.94427 0.485071
\(341\) 0 0
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) 4.94427 0.266577
\(345\) 0 0
\(346\) 0.472136 0.0253822
\(347\) 0.944272 0.0506912 0.0253456 0.999679i \(-0.491931\pi\)
0.0253456 + 0.999679i \(0.491931\pi\)
\(348\) 0 0
\(349\) 24.4721 1.30996 0.654982 0.755645i \(-0.272676\pi\)
0.654982 + 0.755645i \(0.272676\pi\)
\(350\) 1.00000 0.0534522
\(351\) 0 0
\(352\) 0 0
\(353\) −2.00000 −0.106449 −0.0532246 0.998583i \(-0.516950\pi\)
−0.0532246 + 0.998583i \(0.516950\pi\)
\(354\) 0 0
\(355\) 9.88854 0.524829
\(356\) −9.41641 −0.499069
\(357\) 0 0
\(358\) −20.0000 −1.05703
\(359\) 30.8328 1.62729 0.813647 0.581359i \(-0.197479\pi\)
0.813647 + 0.581359i \(0.197479\pi\)
\(360\) 0 0
\(361\) −12.8885 −0.678344
\(362\) −2.00000 −0.105118
\(363\) 0 0
\(364\) −4.47214 −0.234404
\(365\) −29.8885 −1.56444
\(366\) 0 0
\(367\) 30.8328 1.60946 0.804730 0.593641i \(-0.202310\pi\)
0.804730 + 0.593641i \(0.202310\pi\)
\(368\) 1.00000 0.0521286
\(369\) 0 0
\(370\) 13.8885 0.722031
\(371\) 10.9443 0.568198
\(372\) 0 0
\(373\) 6.94427 0.359561 0.179780 0.983707i \(-0.442461\pi\)
0.179780 + 0.983707i \(0.442461\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) −2.47214 −0.127491
\(377\) −8.94427 −0.460653
\(378\) 0 0
\(379\) 24.0000 1.23280 0.616399 0.787434i \(-0.288591\pi\)
0.616399 + 0.787434i \(0.288591\pi\)
\(380\) −4.94427 −0.253636
\(381\) 0 0
\(382\) −3.05573 −0.156345
\(383\) −16.0000 −0.817562 −0.408781 0.912633i \(-0.634046\pi\)
−0.408781 + 0.912633i \(0.634046\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −11.8885 −0.605111
\(387\) 0 0
\(388\) −0.472136 −0.0239691
\(389\) 18.9443 0.960513 0.480256 0.877128i \(-0.340544\pi\)
0.480256 + 0.877128i \(0.340544\pi\)
\(390\) 0 0
\(391\) −4.47214 −0.226166
\(392\) −1.00000 −0.0505076
\(393\) 0 0
\(394\) −14.9443 −0.752882
\(395\) 9.88854 0.497547
\(396\) 0 0
\(397\) −36.4721 −1.83048 −0.915242 0.402905i \(-0.868001\pi\)
−0.915242 + 0.402905i \(0.868001\pi\)
\(398\) −3.05573 −0.153170
\(399\) 0 0
\(400\) −1.00000 −0.0500000
\(401\) 22.0000 1.09863 0.549314 0.835616i \(-0.314889\pi\)
0.549314 + 0.835616i \(0.314889\pi\)
\(402\) 0 0
\(403\) 11.0557 0.550725
\(404\) 9.41641 0.468484
\(405\) 0 0
\(406\) −2.00000 −0.0992583
\(407\) 0 0
\(408\) 0 0
\(409\) −15.8885 −0.785638 −0.392819 0.919616i \(-0.628500\pi\)
−0.392819 + 0.919616i \(0.628500\pi\)
\(410\) 12.0000 0.592638
\(411\) 0 0
\(412\) −4.94427 −0.243587
\(413\) −4.00000 −0.196827
\(414\) 0 0
\(415\) 4.94427 0.242705
\(416\) 4.47214 0.219265
\(417\) 0 0
\(418\) 0 0
\(419\) 7.41641 0.362315 0.181158 0.983454i \(-0.442016\pi\)
0.181158 + 0.983454i \(0.442016\pi\)
\(420\) 0 0
\(421\) 32.8328 1.60017 0.800087 0.599884i \(-0.204787\pi\)
0.800087 + 0.599884i \(0.204787\pi\)
\(422\) 0.944272 0.0459664
\(423\) 0 0
\(424\) −10.9443 −0.531501
\(425\) 4.47214 0.216930
\(426\) 0 0
\(427\) −6.00000 −0.290360
\(428\) 8.00000 0.386695
\(429\) 0 0
\(430\) −9.88854 −0.476868
\(431\) −17.8885 −0.861661 −0.430830 0.902433i \(-0.641779\pi\)
−0.430830 + 0.902433i \(0.641779\pi\)
\(432\) 0 0
\(433\) 25.4164 1.22143 0.610717 0.791849i \(-0.290881\pi\)
0.610717 + 0.791849i \(0.290881\pi\)
\(434\) 2.47214 0.118666
\(435\) 0 0
\(436\) −2.94427 −0.141005
\(437\) 2.47214 0.118258
\(438\) 0 0
\(439\) 28.3607 1.35358 0.676791 0.736175i \(-0.263370\pi\)
0.676791 + 0.736175i \(0.263370\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) −20.0000 −0.951303
\(443\) −34.8328 −1.65496 −0.827479 0.561497i \(-0.810225\pi\)
−0.827479 + 0.561497i \(0.810225\pi\)
\(444\) 0 0
\(445\) 18.8328 0.892761
\(446\) −18.4721 −0.874681
\(447\) 0 0
\(448\) 1.00000 0.0472456
\(449\) −18.0000 −0.849473 −0.424736 0.905317i \(-0.639633\pi\)
−0.424736 + 0.905317i \(0.639633\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 18.9443 0.891064
\(453\) 0 0
\(454\) 5.52786 0.259436
\(455\) 8.94427 0.419314
\(456\) 0 0
\(457\) 2.00000 0.0935561 0.0467780 0.998905i \(-0.485105\pi\)
0.0467780 + 0.998905i \(0.485105\pi\)
\(458\) 14.0000 0.654177
\(459\) 0 0
\(460\) −2.00000 −0.0932505
\(461\) −6.58359 −0.306628 −0.153314 0.988177i \(-0.548995\pi\)
−0.153314 + 0.988177i \(0.548995\pi\)
\(462\) 0 0
\(463\) 6.11146 0.284023 0.142012 0.989865i \(-0.454643\pi\)
0.142012 + 0.989865i \(0.454643\pi\)
\(464\) 2.00000 0.0928477
\(465\) 0 0
\(466\) −6.00000 −0.277945
\(467\) −33.3050 −1.54117 −0.770585 0.637338i \(-0.780036\pi\)
−0.770585 + 0.637338i \(0.780036\pi\)
\(468\) 0 0
\(469\) 4.94427 0.228305
\(470\) 4.94427 0.228062
\(471\) 0 0
\(472\) 4.00000 0.184115
\(473\) 0 0
\(474\) 0 0
\(475\) −2.47214 −0.113429
\(476\) −4.47214 −0.204980
\(477\) 0 0
\(478\) 20.9443 0.957969
\(479\) 4.94427 0.225910 0.112955 0.993600i \(-0.463968\pi\)
0.112955 + 0.993600i \(0.463968\pi\)
\(480\) 0 0
\(481\) −31.0557 −1.41602
\(482\) −20.4721 −0.932480
\(483\) 0 0
\(484\) −11.0000 −0.500000
\(485\) 0.944272 0.0428772
\(486\) 0 0
\(487\) 24.0000 1.08754 0.543772 0.839233i \(-0.316996\pi\)
0.543772 + 0.839233i \(0.316996\pi\)
\(488\) 6.00000 0.271607
\(489\) 0 0
\(490\) 2.00000 0.0903508
\(491\) −40.9443 −1.84779 −0.923895 0.382647i \(-0.875013\pi\)
−0.923895 + 0.382647i \(0.875013\pi\)
\(492\) 0 0
\(493\) −8.94427 −0.402830
\(494\) 11.0557 0.497421
\(495\) 0 0
\(496\) −2.47214 −0.111002
\(497\) −4.94427 −0.221781
\(498\) 0 0
\(499\) 32.9443 1.47479 0.737394 0.675463i \(-0.236056\pi\)
0.737394 + 0.675463i \(0.236056\pi\)
\(500\) 12.0000 0.536656
\(501\) 0 0
\(502\) −5.52786 −0.246721
\(503\) 9.88854 0.440908 0.220454 0.975397i \(-0.429246\pi\)
0.220454 + 0.975397i \(0.429246\pi\)
\(504\) 0 0
\(505\) −18.8328 −0.838049
\(506\) 0 0
\(507\) 0 0
\(508\) 20.9443 0.929252
\(509\) −6.58359 −0.291813 −0.145906 0.989298i \(-0.546610\pi\)
−0.145906 + 0.989298i \(0.546610\pi\)
\(510\) 0 0
\(511\) 14.9443 0.661096
\(512\) −1.00000 −0.0441942
\(513\) 0 0
\(514\) −2.94427 −0.129866
\(515\) 9.88854 0.435741
\(516\) 0 0
\(517\) 0 0
\(518\) −6.94427 −0.305114
\(519\) 0 0
\(520\) −8.94427 −0.392232
\(521\) −33.4164 −1.46400 −0.732000 0.681305i \(-0.761413\pi\)
−0.732000 + 0.681305i \(0.761413\pi\)
\(522\) 0 0
\(523\) −25.3050 −1.10651 −0.553254 0.833013i \(-0.686614\pi\)
−0.553254 + 0.833013i \(0.686614\pi\)
\(524\) −12.0000 −0.524222
\(525\) 0 0
\(526\) −24.0000 −1.04645
\(527\) 11.0557 0.481595
\(528\) 0 0
\(529\) 1.00000 0.0434783
\(530\) 21.8885 0.950778
\(531\) 0 0
\(532\) 2.47214 0.107181
\(533\) −26.8328 −1.16226
\(534\) 0 0
\(535\) −16.0000 −0.691740
\(536\) −4.94427 −0.213560
\(537\) 0 0
\(538\) 16.4721 0.710164
\(539\) 0 0
\(540\) 0 0
\(541\) −10.0000 −0.429934 −0.214967 0.976621i \(-0.568964\pi\)
−0.214967 + 0.976621i \(0.568964\pi\)
\(542\) −15.4164 −0.662191
\(543\) 0 0
\(544\) 4.47214 0.191741
\(545\) 5.88854 0.252238
\(546\) 0 0
\(547\) 20.0000 0.855138 0.427569 0.903983i \(-0.359370\pi\)
0.427569 + 0.903983i \(0.359370\pi\)
\(548\) 14.0000 0.598050
\(549\) 0 0
\(550\) 0 0
\(551\) 4.94427 0.210633
\(552\) 0 0
\(553\) −4.94427 −0.210252
\(554\) 11.8885 0.505096
\(555\) 0 0
\(556\) 8.94427 0.379322
\(557\) 17.0557 0.722674 0.361337 0.932435i \(-0.382320\pi\)
0.361337 + 0.932435i \(0.382320\pi\)
\(558\) 0 0
\(559\) 22.1115 0.935215
\(560\) −2.00000 −0.0845154
\(561\) 0 0
\(562\) 10.0000 0.421825
\(563\) −26.4721 −1.11567 −0.557834 0.829953i \(-0.688367\pi\)
−0.557834 + 0.829953i \(0.688367\pi\)
\(564\) 0 0
\(565\) −37.8885 −1.59398
\(566\) −18.4721 −0.776442
\(567\) 0 0
\(568\) 4.94427 0.207457
\(569\) 7.88854 0.330705 0.165352 0.986235i \(-0.447124\pi\)
0.165352 + 0.986235i \(0.447124\pi\)
\(570\) 0 0
\(571\) −12.9443 −0.541701 −0.270850 0.962621i \(-0.587305\pi\)
−0.270850 + 0.962621i \(0.587305\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) −6.00000 −0.250435
\(575\) −1.00000 −0.0417029
\(576\) 0 0
\(577\) −2.94427 −0.122572 −0.0612858 0.998120i \(-0.519520\pi\)
−0.0612858 + 0.998120i \(0.519520\pi\)
\(578\) −3.00000 −0.124784
\(579\) 0 0
\(580\) −4.00000 −0.166091
\(581\) −2.47214 −0.102561
\(582\) 0 0
\(583\) 0 0
\(584\) −14.9443 −0.618398
\(585\) 0 0
\(586\) 27.8885 1.15207
\(587\) 0.944272 0.0389743 0.0194871 0.999810i \(-0.493797\pi\)
0.0194871 + 0.999810i \(0.493797\pi\)
\(588\) 0 0
\(589\) −6.11146 −0.251818
\(590\) −8.00000 −0.329355
\(591\) 0 0
\(592\) 6.94427 0.285408
\(593\) 28.8328 1.18402 0.592011 0.805930i \(-0.298334\pi\)
0.592011 + 0.805930i \(0.298334\pi\)
\(594\) 0 0
\(595\) 8.94427 0.366679
\(596\) 1.05573 0.0432443
\(597\) 0 0
\(598\) 4.47214 0.182879
\(599\) −33.8885 −1.38465 −0.692324 0.721587i \(-0.743413\pi\)
−0.692324 + 0.721587i \(0.743413\pi\)
\(600\) 0 0
\(601\) 40.8328 1.66561 0.832803 0.553570i \(-0.186735\pi\)
0.832803 + 0.553570i \(0.186735\pi\)
\(602\) 4.94427 0.201513
\(603\) 0 0
\(604\) 4.94427 0.201180
\(605\) 22.0000 0.894427
\(606\) 0 0
\(607\) −28.3607 −1.15112 −0.575562 0.817758i \(-0.695217\pi\)
−0.575562 + 0.817758i \(0.695217\pi\)
\(608\) −2.47214 −0.100258
\(609\) 0 0
\(610\) −12.0000 −0.485866
\(611\) −11.0557 −0.447267
\(612\) 0 0
\(613\) 40.8328 1.64922 0.824611 0.565700i \(-0.191394\pi\)
0.824611 + 0.565700i \(0.191394\pi\)
\(614\) −24.9443 −1.00667
\(615\) 0 0
\(616\) 0 0
\(617\) −26.0000 −1.04672 −0.523360 0.852111i \(-0.675322\pi\)
−0.523360 + 0.852111i \(0.675322\pi\)
\(618\) 0 0
\(619\) −34.4721 −1.38555 −0.692776 0.721153i \(-0.743613\pi\)
−0.692776 + 0.721153i \(0.743613\pi\)
\(620\) 4.94427 0.198567
\(621\) 0 0
\(622\) −2.47214 −0.0991236
\(623\) −9.41641 −0.377260
\(624\) 0 0
\(625\) −19.0000 −0.760000
\(626\) 13.4164 0.536228
\(627\) 0 0
\(628\) −2.94427 −0.117489
\(629\) −31.0557 −1.23827
\(630\) 0 0
\(631\) −11.0557 −0.440122 −0.220061 0.975486i \(-0.570626\pi\)
−0.220061 + 0.975486i \(0.570626\pi\)
\(632\) 4.94427 0.196673
\(633\) 0 0
\(634\) −13.0557 −0.518509
\(635\) −41.8885 −1.66230
\(636\) 0 0
\(637\) −4.47214 −0.177192
\(638\) 0 0
\(639\) 0 0
\(640\) 2.00000 0.0790569
\(641\) −10.0000 −0.394976 −0.197488 0.980305i \(-0.563278\pi\)
−0.197488 + 0.980305i \(0.563278\pi\)
\(642\) 0 0
\(643\) −10.4721 −0.412981 −0.206490 0.978449i \(-0.566204\pi\)
−0.206490 + 0.978449i \(0.566204\pi\)
\(644\) 1.00000 0.0394055
\(645\) 0 0
\(646\) 11.0557 0.434982
\(647\) −12.3607 −0.485948 −0.242974 0.970033i \(-0.578123\pi\)
−0.242974 + 0.970033i \(0.578123\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) −4.47214 −0.175412
\(651\) 0 0
\(652\) −8.94427 −0.350285
\(653\) −6.00000 −0.234798 −0.117399 0.993085i \(-0.537456\pi\)
−0.117399 + 0.993085i \(0.537456\pi\)
\(654\) 0 0
\(655\) 24.0000 0.937758
\(656\) 6.00000 0.234261
\(657\) 0 0
\(658\) −2.47214 −0.0963739
\(659\) −46.8328 −1.82435 −0.912174 0.409804i \(-0.865597\pi\)
−0.912174 + 0.409804i \(0.865597\pi\)
\(660\) 0 0
\(661\) −30.0000 −1.16686 −0.583432 0.812162i \(-0.698291\pi\)
−0.583432 + 0.812162i \(0.698291\pi\)
\(662\) 13.8885 0.539794
\(663\) 0 0
\(664\) 2.47214 0.0959375
\(665\) −4.94427 −0.191731
\(666\) 0 0
\(667\) 2.00000 0.0774403
\(668\) 15.4164 0.596479
\(669\) 0 0
\(670\) 9.88854 0.382028
\(671\) 0 0
\(672\) 0 0
\(673\) −30.0000 −1.15642 −0.578208 0.815890i \(-0.696248\pi\)
−0.578208 + 0.815890i \(0.696248\pi\)
\(674\) 2.94427 0.113409
\(675\) 0 0
\(676\) 7.00000 0.269231
\(677\) 4.11146 0.158016 0.0790080 0.996874i \(-0.474825\pi\)
0.0790080 + 0.996874i \(0.474825\pi\)
\(678\) 0 0
\(679\) −0.472136 −0.0181189
\(680\) −8.94427 −0.342997
\(681\) 0 0
\(682\) 0 0
\(683\) −16.9443 −0.648355 −0.324177 0.945996i \(-0.605087\pi\)
−0.324177 + 0.945996i \(0.605087\pi\)
\(684\) 0 0
\(685\) −28.0000 −1.06983
\(686\) −1.00000 −0.0381802
\(687\) 0 0
\(688\) −4.94427 −0.188499
\(689\) −48.9443 −1.86463
\(690\) 0 0
\(691\) −15.0557 −0.572747 −0.286373 0.958118i \(-0.592450\pi\)
−0.286373 + 0.958118i \(0.592450\pi\)
\(692\) −0.472136 −0.0179479
\(693\) 0 0
\(694\) −0.944272 −0.0358441
\(695\) −17.8885 −0.678551
\(696\) 0 0
\(697\) −26.8328 −1.01637
\(698\) −24.4721 −0.926284
\(699\) 0 0
\(700\) −1.00000 −0.0377964
\(701\) 20.8328 0.786845 0.393422 0.919358i \(-0.371291\pi\)
0.393422 + 0.919358i \(0.371291\pi\)
\(702\) 0 0
\(703\) 17.1672 0.647473
\(704\) 0 0
\(705\) 0 0
\(706\) 2.00000 0.0752710
\(707\) 9.41641 0.354140
\(708\) 0 0
\(709\) 6.94427 0.260798 0.130399 0.991462i \(-0.458374\pi\)
0.130399 + 0.991462i \(0.458374\pi\)
\(710\) −9.88854 −0.371110
\(711\) 0 0
\(712\) 9.41641 0.352895
\(713\) −2.47214 −0.0925822
\(714\) 0 0
\(715\) 0 0
\(716\) 20.0000 0.747435
\(717\) 0 0
\(718\) −30.8328 −1.15067
\(719\) 5.52786 0.206155 0.103077 0.994673i \(-0.467131\pi\)
0.103077 + 0.994673i \(0.467131\pi\)
\(720\) 0 0
\(721\) −4.94427 −0.184134
\(722\) 12.8885 0.479662
\(723\) 0 0
\(724\) 2.00000 0.0743294
\(725\) −2.00000 −0.0742781
\(726\) 0 0
\(727\) 3.05573 0.113331 0.0566653 0.998393i \(-0.481953\pi\)
0.0566653 + 0.998393i \(0.481953\pi\)
\(728\) 4.47214 0.165748
\(729\) 0 0
\(730\) 29.8885 1.10622
\(731\) 22.1115 0.817822
\(732\) 0 0
\(733\) −50.9443 −1.88167 −0.940835 0.338866i \(-0.889957\pi\)
−0.940835 + 0.338866i \(0.889957\pi\)
\(734\) −30.8328 −1.13806
\(735\) 0 0
\(736\) −1.00000 −0.0368605
\(737\) 0 0
\(738\) 0 0
\(739\) −8.94427 −0.329020 −0.164510 0.986375i \(-0.552604\pi\)
−0.164510 + 0.986375i \(0.552604\pi\)
\(740\) −13.8885 −0.510553
\(741\) 0 0
\(742\) −10.9443 −0.401777
\(743\) 22.8328 0.837655 0.418827 0.908066i \(-0.362441\pi\)
0.418827 + 0.908066i \(0.362441\pi\)
\(744\) 0 0
\(745\) −2.11146 −0.0773578
\(746\) −6.94427 −0.254248
\(747\) 0 0
\(748\) 0 0
\(749\) 8.00000 0.292314
\(750\) 0 0
\(751\) 17.8885 0.652762 0.326381 0.945238i \(-0.394171\pi\)
0.326381 + 0.945238i \(0.394171\pi\)
\(752\) 2.47214 0.0901495
\(753\) 0 0
\(754\) 8.94427 0.325731
\(755\) −9.88854 −0.359881
\(756\) 0 0
\(757\) −42.9443 −1.56084 −0.780418 0.625258i \(-0.784994\pi\)
−0.780418 + 0.625258i \(0.784994\pi\)
\(758\) −24.0000 −0.871719
\(759\) 0 0
\(760\) 4.94427 0.179348
\(761\) −21.0557 −0.763270 −0.381635 0.924313i \(-0.624639\pi\)
−0.381635 + 0.924313i \(0.624639\pi\)
\(762\) 0 0
\(763\) −2.94427 −0.106590
\(764\) 3.05573 0.110552
\(765\) 0 0
\(766\) 16.0000 0.578103
\(767\) 17.8885 0.645918
\(768\) 0 0
\(769\) 30.3607 1.09483 0.547417 0.836860i \(-0.315611\pi\)
0.547417 + 0.836860i \(0.315611\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 11.8885 0.427878
\(773\) −5.05573 −0.181842 −0.0909210 0.995858i \(-0.528981\pi\)
−0.0909210 + 0.995858i \(0.528981\pi\)
\(774\) 0 0
\(775\) 2.47214 0.0888017
\(776\) 0.472136 0.0169487
\(777\) 0 0
\(778\) −18.9443 −0.679185
\(779\) 14.8328 0.531441
\(780\) 0 0
\(781\) 0 0
\(782\) 4.47214 0.159923
\(783\) 0 0
\(784\) 1.00000 0.0357143
\(785\) 5.88854 0.210171
\(786\) 0 0
\(787\) −0.583592 −0.0208028 −0.0104014 0.999946i \(-0.503311\pi\)
−0.0104014 + 0.999946i \(0.503311\pi\)
\(788\) 14.9443 0.532368
\(789\) 0 0
\(790\) −9.88854 −0.351819
\(791\) 18.9443 0.673581
\(792\) 0 0
\(793\) 26.8328 0.952861
\(794\) 36.4721 1.29435
\(795\) 0 0
\(796\) 3.05573 0.108307
\(797\) 28.8328 1.02131 0.510655 0.859785i \(-0.329403\pi\)
0.510655 + 0.859785i \(0.329403\pi\)
\(798\) 0 0
\(799\) −11.0557 −0.391124
\(800\) 1.00000 0.0353553
\(801\) 0 0
\(802\) −22.0000 −0.776847
\(803\) 0 0
\(804\) 0 0
\(805\) −2.00000 −0.0704907
\(806\) −11.0557 −0.389421
\(807\) 0 0
\(808\) −9.41641 −0.331268
\(809\) −35.8885 −1.26177 −0.630887 0.775875i \(-0.717309\pi\)
−0.630887 + 0.775875i \(0.717309\pi\)
\(810\) 0 0
\(811\) −37.8885 −1.33045 −0.665223 0.746644i \(-0.731664\pi\)
−0.665223 + 0.746644i \(0.731664\pi\)
\(812\) 2.00000 0.0701862
\(813\) 0 0
\(814\) 0 0
\(815\) 17.8885 0.626608
\(816\) 0 0
\(817\) −12.2229 −0.427626
\(818\) 15.8885 0.555530
\(819\) 0 0
\(820\) −12.0000 −0.419058
\(821\) 16.8328 0.587469 0.293735 0.955887i \(-0.405102\pi\)
0.293735 + 0.955887i \(0.405102\pi\)
\(822\) 0 0
\(823\) 20.9443 0.730071 0.365036 0.930994i \(-0.381057\pi\)
0.365036 + 0.930994i \(0.381057\pi\)
\(824\) 4.94427 0.172242
\(825\) 0 0
\(826\) 4.00000 0.139178
\(827\) −33.8885 −1.17842 −0.589210 0.807980i \(-0.700561\pi\)
−0.589210 + 0.807980i \(0.700561\pi\)
\(828\) 0 0
\(829\) 13.4164 0.465971 0.232986 0.972480i \(-0.425151\pi\)
0.232986 + 0.972480i \(0.425151\pi\)
\(830\) −4.94427 −0.171618
\(831\) 0 0
\(832\) −4.47214 −0.155043
\(833\) −4.47214 −0.154950
\(834\) 0 0
\(835\) −30.8328 −1.06701
\(836\) 0 0
\(837\) 0 0
\(838\) −7.41641 −0.256196
\(839\) 24.0000 0.828572 0.414286 0.910147i \(-0.364031\pi\)
0.414286 + 0.910147i \(0.364031\pi\)
\(840\) 0 0
\(841\) −25.0000 −0.862069
\(842\) −32.8328 −1.13149
\(843\) 0 0
\(844\) −0.944272 −0.0325032
\(845\) −14.0000 −0.481615
\(846\) 0 0
\(847\) −11.0000 −0.377964
\(848\) 10.9443 0.375828
\(849\) 0 0
\(850\) −4.47214 −0.153393
\(851\) 6.94427 0.238047
\(852\) 0 0
\(853\) −6.36068 −0.217786 −0.108893 0.994054i \(-0.534731\pi\)
−0.108893 + 0.994054i \(0.534731\pi\)
\(854\) 6.00000 0.205316
\(855\) 0 0
\(856\) −8.00000 −0.273434
\(857\) 47.8885 1.63584 0.817921 0.575331i \(-0.195127\pi\)
0.817921 + 0.575331i \(0.195127\pi\)
\(858\) 0 0
\(859\) 32.9443 1.12404 0.562022 0.827122i \(-0.310024\pi\)
0.562022 + 0.827122i \(0.310024\pi\)
\(860\) 9.88854 0.337197
\(861\) 0 0
\(862\) 17.8885 0.609286
\(863\) −20.9443 −0.712951 −0.356476 0.934305i \(-0.616022\pi\)
−0.356476 + 0.934305i \(0.616022\pi\)
\(864\) 0 0
\(865\) 0.944272 0.0321062
\(866\) −25.4164 −0.863685
\(867\) 0 0
\(868\) −2.47214 −0.0839098
\(869\) 0 0
\(870\) 0 0
\(871\) −22.1115 −0.749218
\(872\) 2.94427 0.0997056
\(873\) 0 0
\(874\) −2.47214 −0.0836212
\(875\) 12.0000 0.405674
\(876\) 0 0
\(877\) −34.7214 −1.17246 −0.586229 0.810146i \(-0.699388\pi\)
−0.586229 + 0.810146i \(0.699388\pi\)
\(878\) −28.3607 −0.957127
\(879\) 0 0
\(880\) 0 0
\(881\) −22.3607 −0.753350 −0.376675 0.926345i \(-0.622933\pi\)
−0.376675 + 0.926345i \(0.622933\pi\)
\(882\) 0 0
\(883\) 2.83282 0.0953318 0.0476659 0.998863i \(-0.484822\pi\)
0.0476659 + 0.998863i \(0.484822\pi\)
\(884\) 20.0000 0.672673
\(885\) 0 0
\(886\) 34.8328 1.17023
\(887\) 5.52786 0.185608 0.0928038 0.995684i \(-0.470417\pi\)
0.0928038 + 0.995684i \(0.470417\pi\)
\(888\) 0 0
\(889\) 20.9443 0.702448
\(890\) −18.8328 −0.631277
\(891\) 0 0
\(892\) 18.4721 0.618493
\(893\) 6.11146 0.204512
\(894\) 0 0
\(895\) −40.0000 −1.33705
\(896\) −1.00000 −0.0334077
\(897\) 0 0
\(898\) 18.0000 0.600668
\(899\) −4.94427 −0.164901
\(900\) 0 0
\(901\) −48.9443 −1.63057
\(902\) 0 0
\(903\) 0 0
\(904\) −18.9443 −0.630077
\(905\) −4.00000 −0.132964
\(906\) 0 0
\(907\) −41.8885 −1.39089 −0.695443 0.718581i \(-0.744792\pi\)
−0.695443 + 0.718581i \(0.744792\pi\)
\(908\) −5.52786 −0.183449
\(909\) 0 0
\(910\) −8.94427 −0.296500
\(911\) 48.7214 1.61421 0.807105 0.590407i \(-0.201033\pi\)
0.807105 + 0.590407i \(0.201033\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) −2.00000 −0.0661541
\(915\) 0 0
\(916\) −14.0000 −0.462573
\(917\) −12.0000 −0.396275
\(918\) 0 0
\(919\) 35.0557 1.15638 0.578191 0.815902i \(-0.303759\pi\)
0.578191 + 0.815902i \(0.303759\pi\)
\(920\) 2.00000 0.0659380
\(921\) 0 0
\(922\) 6.58359 0.216819
\(923\) 22.1115 0.727807
\(924\) 0 0
\(925\) −6.94427 −0.228326
\(926\) −6.11146 −0.200835
\(927\) 0 0
\(928\) −2.00000 −0.0656532
\(929\) −18.0000 −0.590561 −0.295280 0.955411i \(-0.595413\pi\)
−0.295280 + 0.955411i \(0.595413\pi\)
\(930\) 0 0
\(931\) 2.47214 0.0810210
\(932\) 6.00000 0.196537
\(933\) 0 0
\(934\) 33.3050 1.08977
\(935\) 0 0
\(936\) 0 0
\(937\) 40.2492 1.31488 0.657442 0.753505i \(-0.271638\pi\)
0.657442 + 0.753505i \(0.271638\pi\)
\(938\) −4.94427 −0.161436
\(939\) 0 0
\(940\) −4.94427 −0.161264
\(941\) 6.00000 0.195594 0.0977972 0.995206i \(-0.468820\pi\)
0.0977972 + 0.995206i \(0.468820\pi\)
\(942\) 0 0
\(943\) 6.00000 0.195387
\(944\) −4.00000 −0.130189
\(945\) 0 0
\(946\) 0 0
\(947\) −0.944272 −0.0306847 −0.0153424 0.999882i \(-0.504884\pi\)
−0.0153424 + 0.999882i \(0.504884\pi\)
\(948\) 0 0
\(949\) −66.8328 −2.16949
\(950\) 2.47214 0.0802067
\(951\) 0 0
\(952\) 4.47214 0.144943
\(953\) 9.05573 0.293344 0.146672 0.989185i \(-0.453144\pi\)
0.146672 + 0.989185i \(0.453144\pi\)
\(954\) 0 0
\(955\) −6.11146 −0.197762
\(956\) −20.9443 −0.677386
\(957\) 0 0
\(958\) −4.94427 −0.159742
\(959\) 14.0000 0.452084
\(960\) 0 0
\(961\) −24.8885 −0.802856
\(962\) 31.0557 1.00128
\(963\) 0 0
\(964\) 20.4721 0.659363
\(965\) −23.7771 −0.765412
\(966\) 0 0
\(967\) −16.0000 −0.514525 −0.257263 0.966342i \(-0.582821\pi\)
−0.257263 + 0.966342i \(0.582821\pi\)
\(968\) 11.0000 0.353553
\(969\) 0 0
\(970\) −0.944272 −0.0303187
\(971\) 23.4164 0.751468 0.375734 0.926727i \(-0.377391\pi\)
0.375734 + 0.926727i \(0.377391\pi\)
\(972\) 0 0
\(973\) 8.94427 0.286740
\(974\) −24.0000 −0.769010
\(975\) 0 0
\(976\) −6.00000 −0.192055
\(977\) 6.00000 0.191957 0.0959785 0.995383i \(-0.469402\pi\)
0.0959785 + 0.995383i \(0.469402\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) −2.00000 −0.0638877
\(981\) 0 0
\(982\) 40.9443 1.30658
\(983\) 22.1115 0.705246 0.352623 0.935765i \(-0.385290\pi\)
0.352623 + 0.935765i \(0.385290\pi\)
\(984\) 0 0
\(985\) −29.8885 −0.952328
\(986\) 8.94427 0.284844
\(987\) 0 0
\(988\) −11.0557 −0.351730
\(989\) −4.94427 −0.157219
\(990\) 0 0
\(991\) 60.9443 1.93596 0.967979 0.251030i \(-0.0807693\pi\)
0.967979 + 0.251030i \(0.0807693\pi\)
\(992\) 2.47214 0.0784904
\(993\) 0 0
\(994\) 4.94427 0.156823
\(995\) −6.11146 −0.193746
\(996\) 0 0
\(997\) −46.3607 −1.46826 −0.734129 0.679010i \(-0.762409\pi\)
−0.734129 + 0.679010i \(0.762409\pi\)
\(998\) −32.9443 −1.04283
\(999\) 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 2898.2.a.v.1.1 2
3.2 odd 2 966.2.a.p.1.1 2
12.11 even 2 7728.2.a.bd.1.1 2
21.20 even 2 6762.2.a.bz.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
966.2.a.p.1.1 2 3.2 odd 2
2898.2.a.v.1.1 2 1.1 even 1 trivial
6762.2.a.bz.1.2 2 21.20 even 2
7728.2.a.bd.1.1 2 12.11 even 2