Properties

Label 825.2.a.e.1.1
Level $825$
Weight $2$
Character 825.1
Self dual yes
Analytic conductor $6.588$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 825 = 3 \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 825.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(6.58765816676\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{12})^+\)
Defining polynomial: \(x^{2} - 3\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 165)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.73205\) of defining polynomial
Character \(\chi\) \(=\) 825.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.73205 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.73205 q^{6} -2.00000 q^{7} +1.73205 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.73205 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.73205 q^{6} -2.00000 q^{7} +1.73205 q^{8} +1.00000 q^{9} -1.00000 q^{11} -1.00000 q^{12} +1.46410 q^{13} +3.46410 q^{14} -5.00000 q^{16} -1.73205 q^{18} -1.46410 q^{19} +2.00000 q^{21} +1.73205 q^{22} +6.92820 q^{23} -1.73205 q^{24} -2.53590 q^{26} -1.00000 q^{27} -2.00000 q^{28} +3.46410 q^{29} +2.92820 q^{31} +5.19615 q^{32} +1.00000 q^{33} +1.00000 q^{36} -8.92820 q^{37} +2.53590 q^{38} -1.46410 q^{39} -3.46410 q^{41} -3.46410 q^{42} -8.92820 q^{43} -1.00000 q^{44} -12.0000 q^{46} -6.92820 q^{47} +5.00000 q^{48} -3.00000 q^{49} +1.46410 q^{52} +12.9282 q^{53} +1.73205 q^{54} -3.46410 q^{56} +1.46410 q^{57} -6.00000 q^{58} +6.92820 q^{59} +2.00000 q^{61} -5.07180 q^{62} -2.00000 q^{63} +1.00000 q^{64} -1.73205 q^{66} -8.00000 q^{67} -6.92820 q^{69} -13.8564 q^{71} +1.73205 q^{72} -12.3923 q^{73} +15.4641 q^{74} -1.46410 q^{76} +2.00000 q^{77} +2.53590 q^{78} -13.4641 q^{79} +1.00000 q^{81} +6.00000 q^{82} -15.4641 q^{83} +2.00000 q^{84} +15.4641 q^{86} -3.46410 q^{87} -1.73205 q^{88} -12.9282 q^{89} -2.92820 q^{91} +6.92820 q^{92} -2.92820 q^{93} +12.0000 q^{94} -5.19615 q^{96} +10.0000 q^{97} +5.19615 q^{98} -1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{3} + 2q^{4} - 4q^{7} + 2q^{9} + O(q^{10}) \) \( 2q - 2q^{3} + 2q^{4} - 4q^{7} + 2q^{9} - 2q^{11} - 2q^{12} - 4q^{13} - 10q^{16} + 4q^{19} + 4q^{21} - 12q^{26} - 2q^{27} - 4q^{28} - 8q^{31} + 2q^{33} + 2q^{36} - 4q^{37} + 12q^{38} + 4q^{39} - 4q^{43} - 2q^{44} - 24q^{46} + 10q^{48} - 6q^{49} - 4q^{52} + 12q^{53} - 4q^{57} - 12q^{58} + 4q^{61} - 24q^{62} - 4q^{63} + 2q^{64} - 16q^{67} - 4q^{73} + 24q^{74} + 4q^{76} + 4q^{77} + 12q^{78} - 20q^{79} + 2q^{81} + 12q^{82} - 24q^{83} + 4q^{84} + 24q^{86} - 12q^{89} + 8q^{91} + 8q^{93} + 24q^{94} + 20q^{97} - 2q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.73205 −1.22474 −0.612372 0.790569i \(-0.709785\pi\)
−0.612372 + 0.790569i \(0.709785\pi\)
\(3\) −1.00000 −0.577350
\(4\) 1.00000 0.500000
\(5\) 0 0
\(6\) 1.73205 0.707107
\(7\) −2.00000 −0.755929 −0.377964 0.925820i \(-0.623376\pi\)
−0.377964 + 0.925820i \(0.623376\pi\)
\(8\) 1.73205 0.612372
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −1.00000 −0.301511
\(12\) −1.00000 −0.288675
\(13\) 1.46410 0.406069 0.203034 0.979172i \(-0.434920\pi\)
0.203034 + 0.979172i \(0.434920\pi\)
\(14\) 3.46410 0.925820
\(15\) 0 0
\(16\) −5.00000 −1.25000
\(17\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(18\) −1.73205 −0.408248
\(19\) −1.46410 −0.335888 −0.167944 0.985797i \(-0.553713\pi\)
−0.167944 + 0.985797i \(0.553713\pi\)
\(20\) 0 0
\(21\) 2.00000 0.436436
\(22\) 1.73205 0.369274
\(23\) 6.92820 1.44463 0.722315 0.691564i \(-0.243078\pi\)
0.722315 + 0.691564i \(0.243078\pi\)
\(24\) −1.73205 −0.353553
\(25\) 0 0
\(26\) −2.53590 −0.497331
\(27\) −1.00000 −0.192450
\(28\) −2.00000 −0.377964
\(29\) 3.46410 0.643268 0.321634 0.946864i \(-0.395768\pi\)
0.321634 + 0.946864i \(0.395768\pi\)
\(30\) 0 0
\(31\) 2.92820 0.525921 0.262960 0.964807i \(-0.415301\pi\)
0.262960 + 0.964807i \(0.415301\pi\)
\(32\) 5.19615 0.918559
\(33\) 1.00000 0.174078
\(34\) 0 0
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) −8.92820 −1.46779 −0.733894 0.679264i \(-0.762299\pi\)
−0.733894 + 0.679264i \(0.762299\pi\)
\(38\) 2.53590 0.411377
\(39\) −1.46410 −0.234444
\(40\) 0 0
\(41\) −3.46410 −0.541002 −0.270501 0.962720i \(-0.587189\pi\)
−0.270501 + 0.962720i \(0.587189\pi\)
\(42\) −3.46410 −0.534522
\(43\) −8.92820 −1.36154 −0.680769 0.732498i \(-0.738354\pi\)
−0.680769 + 0.732498i \(0.738354\pi\)
\(44\) −1.00000 −0.150756
\(45\) 0 0
\(46\) −12.0000 −1.76930
\(47\) −6.92820 −1.01058 −0.505291 0.862949i \(-0.668615\pi\)
−0.505291 + 0.862949i \(0.668615\pi\)
\(48\) 5.00000 0.721688
\(49\) −3.00000 −0.428571
\(50\) 0 0
\(51\) 0 0
\(52\) 1.46410 0.203034
\(53\) 12.9282 1.77583 0.887913 0.460012i \(-0.152155\pi\)
0.887913 + 0.460012i \(0.152155\pi\)
\(54\) 1.73205 0.235702
\(55\) 0 0
\(56\) −3.46410 −0.462910
\(57\) 1.46410 0.193925
\(58\) −6.00000 −0.787839
\(59\) 6.92820 0.901975 0.450988 0.892530i \(-0.351072\pi\)
0.450988 + 0.892530i \(0.351072\pi\)
\(60\) 0 0
\(61\) 2.00000 0.256074 0.128037 0.991769i \(-0.459132\pi\)
0.128037 + 0.991769i \(0.459132\pi\)
\(62\) −5.07180 −0.644119
\(63\) −2.00000 −0.251976
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) −1.73205 −0.213201
\(67\) −8.00000 −0.977356 −0.488678 0.872464i \(-0.662521\pi\)
−0.488678 + 0.872464i \(0.662521\pi\)
\(68\) 0 0
\(69\) −6.92820 −0.834058
\(70\) 0 0
\(71\) −13.8564 −1.64445 −0.822226 0.569160i \(-0.807268\pi\)
−0.822226 + 0.569160i \(0.807268\pi\)
\(72\) 1.73205 0.204124
\(73\) −12.3923 −1.45041 −0.725205 0.688533i \(-0.758255\pi\)
−0.725205 + 0.688533i \(0.758255\pi\)
\(74\) 15.4641 1.79767
\(75\) 0 0
\(76\) −1.46410 −0.167944
\(77\) 2.00000 0.227921
\(78\) 2.53590 0.287134
\(79\) −13.4641 −1.51483 −0.757415 0.652934i \(-0.773538\pi\)
−0.757415 + 0.652934i \(0.773538\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 6.00000 0.662589
\(83\) −15.4641 −1.69741 −0.848703 0.528870i \(-0.822616\pi\)
−0.848703 + 0.528870i \(0.822616\pi\)
\(84\) 2.00000 0.218218
\(85\) 0 0
\(86\) 15.4641 1.66754
\(87\) −3.46410 −0.371391
\(88\) −1.73205 −0.184637
\(89\) −12.9282 −1.37039 −0.685193 0.728361i \(-0.740282\pi\)
−0.685193 + 0.728361i \(0.740282\pi\)
\(90\) 0 0
\(91\) −2.92820 −0.306959
\(92\) 6.92820 0.722315
\(93\) −2.92820 −0.303641
\(94\) 12.0000 1.23771
\(95\) 0 0
\(96\) −5.19615 −0.530330
\(97\) 10.0000 1.01535 0.507673 0.861550i \(-0.330506\pi\)
0.507673 + 0.861550i \(0.330506\pi\)
\(98\) 5.19615 0.524891
\(99\) −1.00000 −0.100504
\(100\) 0 0
\(101\) 10.3923 1.03407 0.517036 0.855963i \(-0.327035\pi\)
0.517036 + 0.855963i \(0.327035\pi\)
\(102\) 0 0
\(103\) −8.00000 −0.788263 −0.394132 0.919054i \(-0.628955\pi\)
−0.394132 + 0.919054i \(0.628955\pi\)
\(104\) 2.53590 0.248665
\(105\) 0 0
\(106\) −22.3923 −2.17493
\(107\) −15.4641 −1.49497 −0.747486 0.664278i \(-0.768739\pi\)
−0.747486 + 0.664278i \(0.768739\pi\)
\(108\) −1.00000 −0.0962250
\(109\) −10.0000 −0.957826 −0.478913 0.877862i \(-0.658969\pi\)
−0.478913 + 0.877862i \(0.658969\pi\)
\(110\) 0 0
\(111\) 8.92820 0.847428
\(112\) 10.0000 0.944911
\(113\) −0.928203 −0.0873180 −0.0436590 0.999046i \(-0.513902\pi\)
−0.0436590 + 0.999046i \(0.513902\pi\)
\(114\) −2.53590 −0.237509
\(115\) 0 0
\(116\) 3.46410 0.321634
\(117\) 1.46410 0.135356
\(118\) −12.0000 −1.10469
\(119\) 0 0
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) −3.46410 −0.313625
\(123\) 3.46410 0.312348
\(124\) 2.92820 0.262960
\(125\) 0 0
\(126\) 3.46410 0.308607
\(127\) 4.92820 0.437307 0.218654 0.975803i \(-0.429834\pi\)
0.218654 + 0.975803i \(0.429834\pi\)
\(128\) −12.1244 −1.07165
\(129\) 8.92820 0.786084
\(130\) 0 0
\(131\) 5.07180 0.443125 0.221562 0.975146i \(-0.428884\pi\)
0.221562 + 0.975146i \(0.428884\pi\)
\(132\) 1.00000 0.0870388
\(133\) 2.92820 0.253907
\(134\) 13.8564 1.19701
\(135\) 0 0
\(136\) 0 0
\(137\) 18.0000 1.53784 0.768922 0.639343i \(-0.220793\pi\)
0.768922 + 0.639343i \(0.220793\pi\)
\(138\) 12.0000 1.02151
\(139\) −8.39230 −0.711826 −0.355913 0.934519i \(-0.615830\pi\)
−0.355913 + 0.934519i \(0.615830\pi\)
\(140\) 0 0
\(141\) 6.92820 0.583460
\(142\) 24.0000 2.01404
\(143\) −1.46410 −0.122434
\(144\) −5.00000 −0.416667
\(145\) 0 0
\(146\) 21.4641 1.77638
\(147\) 3.00000 0.247436
\(148\) −8.92820 −0.733894
\(149\) −8.53590 −0.699288 −0.349644 0.936883i \(-0.613697\pi\)
−0.349644 + 0.936883i \(0.613697\pi\)
\(150\) 0 0
\(151\) 0.392305 0.0319253 0.0159627 0.999873i \(-0.494919\pi\)
0.0159627 + 0.999873i \(0.494919\pi\)
\(152\) −2.53590 −0.205689
\(153\) 0 0
\(154\) −3.46410 −0.279145
\(155\) 0 0
\(156\) −1.46410 −0.117222
\(157\) 16.9282 1.35102 0.675509 0.737352i \(-0.263924\pi\)
0.675509 + 0.737352i \(0.263924\pi\)
\(158\) 23.3205 1.85528
\(159\) −12.9282 −1.02527
\(160\) 0 0
\(161\) −13.8564 −1.09204
\(162\) −1.73205 −0.136083
\(163\) 17.8564 1.39862 0.699311 0.714818i \(-0.253490\pi\)
0.699311 + 0.714818i \(0.253490\pi\)
\(164\) −3.46410 −0.270501
\(165\) 0 0
\(166\) 26.7846 2.07889
\(167\) −10.3923 −0.804181 −0.402090 0.915600i \(-0.631716\pi\)
−0.402090 + 0.915600i \(0.631716\pi\)
\(168\) 3.46410 0.267261
\(169\) −10.8564 −0.835108
\(170\) 0 0
\(171\) −1.46410 −0.111963
\(172\) −8.92820 −0.680769
\(173\) 12.0000 0.912343 0.456172 0.889892i \(-0.349220\pi\)
0.456172 + 0.889892i \(0.349220\pi\)
\(174\) 6.00000 0.454859
\(175\) 0 0
\(176\) 5.00000 0.376889
\(177\) −6.92820 −0.520756
\(178\) 22.3923 1.67837
\(179\) −6.92820 −0.517838 −0.258919 0.965899i \(-0.583366\pi\)
−0.258919 + 0.965899i \(0.583366\pi\)
\(180\) 0 0
\(181\) −11.8564 −0.881280 −0.440640 0.897684i \(-0.645248\pi\)
−0.440640 + 0.897684i \(0.645248\pi\)
\(182\) 5.07180 0.375947
\(183\) −2.00000 −0.147844
\(184\) 12.0000 0.884652
\(185\) 0 0
\(186\) 5.07180 0.371882
\(187\) 0 0
\(188\) −6.92820 −0.505291
\(189\) 2.00000 0.145479
\(190\) 0 0
\(191\) −5.07180 −0.366982 −0.183491 0.983021i \(-0.558740\pi\)
−0.183491 + 0.983021i \(0.558740\pi\)
\(192\) −1.00000 −0.0721688
\(193\) −3.60770 −0.259688 −0.129844 0.991534i \(-0.541448\pi\)
−0.129844 + 0.991534i \(0.541448\pi\)
\(194\) −17.3205 −1.24354
\(195\) 0 0
\(196\) −3.00000 −0.214286
\(197\) 12.0000 0.854965 0.427482 0.904024i \(-0.359401\pi\)
0.427482 + 0.904024i \(0.359401\pi\)
\(198\) 1.73205 0.123091
\(199\) 16.7846 1.18983 0.594915 0.803789i \(-0.297186\pi\)
0.594915 + 0.803789i \(0.297186\pi\)
\(200\) 0 0
\(201\) 8.00000 0.564276
\(202\) −18.0000 −1.26648
\(203\) −6.92820 −0.486265
\(204\) 0 0
\(205\) 0 0
\(206\) 13.8564 0.965422
\(207\) 6.92820 0.481543
\(208\) −7.32051 −0.507586
\(209\) 1.46410 0.101274
\(210\) 0 0
\(211\) 12.3923 0.853121 0.426561 0.904459i \(-0.359725\pi\)
0.426561 + 0.904459i \(0.359725\pi\)
\(212\) 12.9282 0.887913
\(213\) 13.8564 0.949425
\(214\) 26.7846 1.83096
\(215\) 0 0
\(216\) −1.73205 −0.117851
\(217\) −5.85641 −0.397559
\(218\) 17.3205 1.17309
\(219\) 12.3923 0.837394
\(220\) 0 0
\(221\) 0 0
\(222\) −15.4641 −1.03788
\(223\) 17.8564 1.19575 0.597877 0.801588i \(-0.296011\pi\)
0.597877 + 0.801588i \(0.296011\pi\)
\(224\) −10.3923 −0.694365
\(225\) 0 0
\(226\) 1.60770 0.106942
\(227\) −8.53590 −0.566547 −0.283274 0.959039i \(-0.591420\pi\)
−0.283274 + 0.959039i \(0.591420\pi\)
\(228\) 1.46410 0.0969625
\(229\) 3.85641 0.254839 0.127419 0.991849i \(-0.459331\pi\)
0.127419 + 0.991849i \(0.459331\pi\)
\(230\) 0 0
\(231\) −2.00000 −0.131590
\(232\) 6.00000 0.393919
\(233\) −12.0000 −0.786146 −0.393073 0.919507i \(-0.628588\pi\)
−0.393073 + 0.919507i \(0.628588\pi\)
\(234\) −2.53590 −0.165777
\(235\) 0 0
\(236\) 6.92820 0.450988
\(237\) 13.4641 0.874587
\(238\) 0 0
\(239\) −12.0000 −0.776215 −0.388108 0.921614i \(-0.626871\pi\)
−0.388108 + 0.921614i \(0.626871\pi\)
\(240\) 0 0
\(241\) 27.8564 1.79439 0.897194 0.441636i \(-0.145602\pi\)
0.897194 + 0.441636i \(0.145602\pi\)
\(242\) −1.73205 −0.111340
\(243\) −1.00000 −0.0641500
\(244\) 2.00000 0.128037
\(245\) 0 0
\(246\) −6.00000 −0.382546
\(247\) −2.14359 −0.136394
\(248\) 5.07180 0.322059
\(249\) 15.4641 0.979998
\(250\) 0 0
\(251\) 25.8564 1.63204 0.816021 0.578022i \(-0.196175\pi\)
0.816021 + 0.578022i \(0.196175\pi\)
\(252\) −2.00000 −0.125988
\(253\) −6.92820 −0.435572
\(254\) −8.53590 −0.535590
\(255\) 0 0
\(256\) 19.0000 1.18750
\(257\) −7.85641 −0.490069 −0.245035 0.969514i \(-0.578799\pi\)
−0.245035 + 0.969514i \(0.578799\pi\)
\(258\) −15.4641 −0.962753
\(259\) 17.8564 1.10954
\(260\) 0 0
\(261\) 3.46410 0.214423
\(262\) −8.78461 −0.542715
\(263\) −27.4641 −1.69351 −0.846755 0.531984i \(-0.821447\pi\)
−0.846755 + 0.531984i \(0.821447\pi\)
\(264\) 1.73205 0.106600
\(265\) 0 0
\(266\) −5.07180 −0.310972
\(267\) 12.9282 0.791193
\(268\) −8.00000 −0.488678
\(269\) −7.85641 −0.479014 −0.239507 0.970895i \(-0.576986\pi\)
−0.239507 + 0.970895i \(0.576986\pi\)
\(270\) 0 0
\(271\) −32.3923 −1.96769 −0.983846 0.179016i \(-0.942709\pi\)
−0.983846 + 0.179016i \(0.942709\pi\)
\(272\) 0 0
\(273\) 2.92820 0.177223
\(274\) −31.1769 −1.88347
\(275\) 0 0
\(276\) −6.92820 −0.417029
\(277\) −22.5359 −1.35405 −0.677025 0.735960i \(-0.736731\pi\)
−0.677025 + 0.735960i \(0.736731\pi\)
\(278\) 14.5359 0.871805
\(279\) 2.92820 0.175307
\(280\) 0 0
\(281\) −3.46410 −0.206651 −0.103325 0.994648i \(-0.532948\pi\)
−0.103325 + 0.994648i \(0.532948\pi\)
\(282\) −12.0000 −0.714590
\(283\) −8.92820 −0.530727 −0.265363 0.964148i \(-0.585492\pi\)
−0.265363 + 0.964148i \(0.585492\pi\)
\(284\) −13.8564 −0.822226
\(285\) 0 0
\(286\) 2.53590 0.149951
\(287\) 6.92820 0.408959
\(288\) 5.19615 0.306186
\(289\) −17.0000 −1.00000
\(290\) 0 0
\(291\) −10.0000 −0.586210
\(292\) −12.3923 −0.725205
\(293\) −13.8564 −0.809500 −0.404750 0.914427i \(-0.632641\pi\)
−0.404750 + 0.914427i \(0.632641\pi\)
\(294\) −5.19615 −0.303046
\(295\) 0 0
\(296\) −15.4641 −0.898833
\(297\) 1.00000 0.0580259
\(298\) 14.7846 0.856449
\(299\) 10.1436 0.586619
\(300\) 0 0
\(301\) 17.8564 1.02923
\(302\) −0.679492 −0.0391004
\(303\) −10.3923 −0.597022
\(304\) 7.32051 0.419860
\(305\) 0 0
\(306\) 0 0
\(307\) −14.0000 −0.799022 −0.399511 0.916728i \(-0.630820\pi\)
−0.399511 + 0.916728i \(0.630820\pi\)
\(308\) 2.00000 0.113961
\(309\) 8.00000 0.455104
\(310\) 0 0
\(311\) 18.9282 1.07332 0.536660 0.843799i \(-0.319686\pi\)
0.536660 + 0.843799i \(0.319686\pi\)
\(312\) −2.53590 −0.143567
\(313\) −7.07180 −0.399722 −0.199861 0.979824i \(-0.564049\pi\)
−0.199861 + 0.979824i \(0.564049\pi\)
\(314\) −29.3205 −1.65465
\(315\) 0 0
\(316\) −13.4641 −0.757415
\(317\) 11.0718 0.621854 0.310927 0.950434i \(-0.399361\pi\)
0.310927 + 0.950434i \(0.399361\pi\)
\(318\) 22.3923 1.25570
\(319\) −3.46410 −0.193952
\(320\) 0 0
\(321\) 15.4641 0.863122
\(322\) 24.0000 1.33747
\(323\) 0 0
\(324\) 1.00000 0.0555556
\(325\) 0 0
\(326\) −30.9282 −1.71295
\(327\) 10.0000 0.553001
\(328\) −6.00000 −0.331295
\(329\) 13.8564 0.763928
\(330\) 0 0
\(331\) −17.8564 −0.981477 −0.490738 0.871307i \(-0.663273\pi\)
−0.490738 + 0.871307i \(0.663273\pi\)
\(332\) −15.4641 −0.848703
\(333\) −8.92820 −0.489263
\(334\) 18.0000 0.984916
\(335\) 0 0
\(336\) −10.0000 −0.545545
\(337\) 29.1769 1.58937 0.794684 0.607023i \(-0.207637\pi\)
0.794684 + 0.607023i \(0.207637\pi\)
\(338\) 18.8038 1.02279
\(339\) 0.928203 0.0504131
\(340\) 0 0
\(341\) −2.92820 −0.158571
\(342\) 2.53590 0.137126
\(343\) 20.0000 1.07990
\(344\) −15.4641 −0.833768
\(345\) 0 0
\(346\) −20.7846 −1.11739
\(347\) −1.60770 −0.0863056 −0.0431528 0.999068i \(-0.513740\pi\)
−0.0431528 + 0.999068i \(0.513740\pi\)
\(348\) −3.46410 −0.185695
\(349\) −35.8564 −1.91935 −0.959675 0.281113i \(-0.909296\pi\)
−0.959675 + 0.281113i \(0.909296\pi\)
\(350\) 0 0
\(351\) −1.46410 −0.0781480
\(352\) −5.19615 −0.276956
\(353\) 0.928203 0.0494033 0.0247016 0.999695i \(-0.492136\pi\)
0.0247016 + 0.999695i \(0.492136\pi\)
\(354\) 12.0000 0.637793
\(355\) 0 0
\(356\) −12.9282 −0.685193
\(357\) 0 0
\(358\) 12.0000 0.634220
\(359\) 20.7846 1.09697 0.548485 0.836160i \(-0.315205\pi\)
0.548485 + 0.836160i \(0.315205\pi\)
\(360\) 0 0
\(361\) −16.8564 −0.887179
\(362\) 20.5359 1.07934
\(363\) −1.00000 −0.0524864
\(364\) −2.92820 −0.153480
\(365\) 0 0
\(366\) 3.46410 0.181071
\(367\) −20.0000 −1.04399 −0.521996 0.852948i \(-0.674812\pi\)
−0.521996 + 0.852948i \(0.674812\pi\)
\(368\) −34.6410 −1.80579
\(369\) −3.46410 −0.180334
\(370\) 0 0
\(371\) −25.8564 −1.34240
\(372\) −2.92820 −0.151820
\(373\) −0.392305 −0.0203128 −0.0101564 0.999948i \(-0.503233\pi\)
−0.0101564 + 0.999948i \(0.503233\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) −12.0000 −0.618853
\(377\) 5.07180 0.261211
\(378\) −3.46410 −0.178174
\(379\) 9.85641 0.506290 0.253145 0.967428i \(-0.418535\pi\)
0.253145 + 0.967428i \(0.418535\pi\)
\(380\) 0 0
\(381\) −4.92820 −0.252479
\(382\) 8.78461 0.449460
\(383\) −13.8564 −0.708029 −0.354015 0.935240i \(-0.615184\pi\)
−0.354015 + 0.935240i \(0.615184\pi\)
\(384\) 12.1244 0.618718
\(385\) 0 0
\(386\) 6.24871 0.318051
\(387\) −8.92820 −0.453846
\(388\) 10.0000 0.507673
\(389\) 24.9282 1.26391 0.631955 0.775005i \(-0.282253\pi\)
0.631955 + 0.775005i \(0.282253\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) −5.19615 −0.262445
\(393\) −5.07180 −0.255838
\(394\) −20.7846 −1.04711
\(395\) 0 0
\(396\) −1.00000 −0.0502519
\(397\) −2.00000 −0.100377 −0.0501886 0.998740i \(-0.515982\pi\)
−0.0501886 + 0.998740i \(0.515982\pi\)
\(398\) −29.0718 −1.45724
\(399\) −2.92820 −0.146594
\(400\) 0 0
\(401\) 19.8564 0.991582 0.495791 0.868442i \(-0.334878\pi\)
0.495791 + 0.868442i \(0.334878\pi\)
\(402\) −13.8564 −0.691095
\(403\) 4.28719 0.213560
\(404\) 10.3923 0.517036
\(405\) 0 0
\(406\) 12.0000 0.595550
\(407\) 8.92820 0.442555
\(408\) 0 0
\(409\) 34.7846 1.71999 0.859994 0.510304i \(-0.170467\pi\)
0.859994 + 0.510304i \(0.170467\pi\)
\(410\) 0 0
\(411\) −18.0000 −0.887875
\(412\) −8.00000 −0.394132
\(413\) −13.8564 −0.681829
\(414\) −12.0000 −0.589768
\(415\) 0 0
\(416\) 7.60770 0.372998
\(417\) 8.39230 0.410973
\(418\) −2.53590 −0.124035
\(419\) 17.0718 0.834012 0.417006 0.908904i \(-0.363079\pi\)
0.417006 + 0.908904i \(0.363079\pi\)
\(420\) 0 0
\(421\) 2.00000 0.0974740 0.0487370 0.998812i \(-0.484480\pi\)
0.0487370 + 0.998812i \(0.484480\pi\)
\(422\) −21.4641 −1.04486
\(423\) −6.92820 −0.336861
\(424\) 22.3923 1.08747
\(425\) 0 0
\(426\) −24.0000 −1.16280
\(427\) −4.00000 −0.193574
\(428\) −15.4641 −0.747486
\(429\) 1.46410 0.0706875
\(430\) 0 0
\(431\) −32.7846 −1.57918 −0.789590 0.613635i \(-0.789706\pi\)
−0.789590 + 0.613635i \(0.789706\pi\)
\(432\) 5.00000 0.240563
\(433\) −27.8564 −1.33869 −0.669347 0.742950i \(-0.733426\pi\)
−0.669347 + 0.742950i \(0.733426\pi\)
\(434\) 10.1436 0.486908
\(435\) 0 0
\(436\) −10.0000 −0.478913
\(437\) −10.1436 −0.485234
\(438\) −21.4641 −1.02559
\(439\) −29.1769 −1.39254 −0.696269 0.717781i \(-0.745158\pi\)
−0.696269 + 0.717781i \(0.745158\pi\)
\(440\) 0 0
\(441\) −3.00000 −0.142857
\(442\) 0 0
\(443\) −12.0000 −0.570137 −0.285069 0.958507i \(-0.592016\pi\)
−0.285069 + 0.958507i \(0.592016\pi\)
\(444\) 8.92820 0.423714
\(445\) 0 0
\(446\) −30.9282 −1.46449
\(447\) 8.53590 0.403734
\(448\) −2.00000 −0.0944911
\(449\) 14.7846 0.697729 0.348864 0.937173i \(-0.386567\pi\)
0.348864 + 0.937173i \(0.386567\pi\)
\(450\) 0 0
\(451\) 3.46410 0.163118
\(452\) −0.928203 −0.0436590
\(453\) −0.392305 −0.0184321
\(454\) 14.7846 0.693876
\(455\) 0 0
\(456\) 2.53590 0.118754
\(457\) 8.39230 0.392575 0.196288 0.980546i \(-0.437111\pi\)
0.196288 + 0.980546i \(0.437111\pi\)
\(458\) −6.67949 −0.312112
\(459\) 0 0
\(460\) 0 0
\(461\) −12.2487 −0.570479 −0.285240 0.958456i \(-0.592073\pi\)
−0.285240 + 0.958456i \(0.592073\pi\)
\(462\) 3.46410 0.161165
\(463\) 28.0000 1.30127 0.650635 0.759390i \(-0.274503\pi\)
0.650635 + 0.759390i \(0.274503\pi\)
\(464\) −17.3205 −0.804084
\(465\) 0 0
\(466\) 20.7846 0.962828
\(467\) −18.9282 −0.875893 −0.437946 0.899001i \(-0.644294\pi\)
−0.437946 + 0.899001i \(0.644294\pi\)
\(468\) 1.46410 0.0676781
\(469\) 16.0000 0.738811
\(470\) 0 0
\(471\) −16.9282 −0.780010
\(472\) 12.0000 0.552345
\(473\) 8.92820 0.410519
\(474\) −23.3205 −1.07115
\(475\) 0 0
\(476\) 0 0
\(477\) 12.9282 0.591942
\(478\) 20.7846 0.950666
\(479\) 12.0000 0.548294 0.274147 0.961688i \(-0.411605\pi\)
0.274147 + 0.961688i \(0.411605\pi\)
\(480\) 0 0
\(481\) −13.0718 −0.596023
\(482\) −48.2487 −2.19767
\(483\) 13.8564 0.630488
\(484\) 1.00000 0.0454545
\(485\) 0 0
\(486\) 1.73205 0.0785674
\(487\) −23.7128 −1.07453 −0.537265 0.843413i \(-0.680543\pi\)
−0.537265 + 0.843413i \(0.680543\pi\)
\(488\) 3.46410 0.156813
\(489\) −17.8564 −0.807495
\(490\) 0 0
\(491\) −17.0718 −0.770439 −0.385220 0.922825i \(-0.625874\pi\)
−0.385220 + 0.922825i \(0.625874\pi\)
\(492\) 3.46410 0.156174
\(493\) 0 0
\(494\) 3.71281 0.167047
\(495\) 0 0
\(496\) −14.6410 −0.657401
\(497\) 27.7128 1.24309
\(498\) −26.7846 −1.20025
\(499\) −12.7846 −0.572318 −0.286159 0.958182i \(-0.592379\pi\)
−0.286159 + 0.958182i \(0.592379\pi\)
\(500\) 0 0
\(501\) 10.3923 0.464294
\(502\) −44.7846 −1.99883
\(503\) 31.1769 1.39011 0.695055 0.718957i \(-0.255380\pi\)
0.695055 + 0.718957i \(0.255380\pi\)
\(504\) −3.46410 −0.154303
\(505\) 0 0
\(506\) 12.0000 0.533465
\(507\) 10.8564 0.482150
\(508\) 4.92820 0.218654
\(509\) −7.85641 −0.348229 −0.174115 0.984725i \(-0.555706\pi\)
−0.174115 + 0.984725i \(0.555706\pi\)
\(510\) 0 0
\(511\) 24.7846 1.09641
\(512\) −8.66025 −0.382733
\(513\) 1.46410 0.0646417
\(514\) 13.6077 0.600210
\(515\) 0 0
\(516\) 8.92820 0.393042
\(517\) 6.92820 0.304702
\(518\) −30.9282 −1.35891
\(519\) −12.0000 −0.526742
\(520\) 0 0
\(521\) 6.00000 0.262865 0.131432 0.991325i \(-0.458042\pi\)
0.131432 + 0.991325i \(0.458042\pi\)
\(522\) −6.00000 −0.262613
\(523\) 22.0000 0.961993 0.480996 0.876723i \(-0.340275\pi\)
0.480996 + 0.876723i \(0.340275\pi\)
\(524\) 5.07180 0.221562
\(525\) 0 0
\(526\) 47.5692 2.07412
\(527\) 0 0
\(528\) −5.00000 −0.217597
\(529\) 25.0000 1.08696
\(530\) 0 0
\(531\) 6.92820 0.300658
\(532\) 2.92820 0.126954
\(533\) −5.07180 −0.219684
\(534\) −22.3923 −0.969010
\(535\) 0 0
\(536\) −13.8564 −0.598506
\(537\) 6.92820 0.298974
\(538\) 13.6077 0.586669
\(539\) 3.00000 0.129219
\(540\) 0 0
\(541\) 0.143594 0.00617357 0.00308678 0.999995i \(-0.499017\pi\)
0.00308678 + 0.999995i \(0.499017\pi\)
\(542\) 56.1051 2.40992
\(543\) 11.8564 0.508807
\(544\) 0 0
\(545\) 0 0
\(546\) −5.07180 −0.217053
\(547\) −2.00000 −0.0855138 −0.0427569 0.999086i \(-0.513614\pi\)
−0.0427569 + 0.999086i \(0.513614\pi\)
\(548\) 18.0000 0.768922
\(549\) 2.00000 0.0853579
\(550\) 0 0
\(551\) −5.07180 −0.216066
\(552\) −12.0000 −0.510754
\(553\) 26.9282 1.14510
\(554\) 39.0333 1.65837
\(555\) 0 0
\(556\) −8.39230 −0.355913
\(557\) −44.7846 −1.89758 −0.948792 0.315900i \(-0.897694\pi\)
−0.948792 + 0.315900i \(0.897694\pi\)
\(558\) −5.07180 −0.214706
\(559\) −13.0718 −0.552878
\(560\) 0 0
\(561\) 0 0
\(562\) 6.00000 0.253095
\(563\) 10.3923 0.437983 0.218992 0.975727i \(-0.429723\pi\)
0.218992 + 0.975727i \(0.429723\pi\)
\(564\) 6.92820 0.291730
\(565\) 0 0
\(566\) 15.4641 0.650005
\(567\) −2.00000 −0.0839921
\(568\) −24.0000 −1.00702
\(569\) −29.3205 −1.22918 −0.614590 0.788847i \(-0.710678\pi\)
−0.614590 + 0.788847i \(0.710678\pi\)
\(570\) 0 0
\(571\) 24.3923 1.02079 0.510393 0.859941i \(-0.329500\pi\)
0.510393 + 0.859941i \(0.329500\pi\)
\(572\) −1.46410 −0.0612172
\(573\) 5.07180 0.211877
\(574\) −12.0000 −0.500870
\(575\) 0 0
\(576\) 1.00000 0.0416667
\(577\) −22.7846 −0.948536 −0.474268 0.880381i \(-0.657287\pi\)
−0.474268 + 0.880381i \(0.657287\pi\)
\(578\) 29.4449 1.22474
\(579\) 3.60770 0.149931
\(580\) 0 0
\(581\) 30.9282 1.28312
\(582\) 17.3205 0.717958
\(583\) −12.9282 −0.535431
\(584\) −21.4641 −0.888191
\(585\) 0 0
\(586\) 24.0000 0.991431
\(587\) 5.07180 0.209335 0.104668 0.994507i \(-0.466622\pi\)
0.104668 + 0.994507i \(0.466622\pi\)
\(588\) 3.00000 0.123718
\(589\) −4.28719 −0.176650
\(590\) 0 0
\(591\) −12.0000 −0.493614
\(592\) 44.6410 1.83473
\(593\) 32.7846 1.34630 0.673151 0.739505i \(-0.264940\pi\)
0.673151 + 0.739505i \(0.264940\pi\)
\(594\) −1.73205 −0.0710669
\(595\) 0 0
\(596\) −8.53590 −0.349644
\(597\) −16.7846 −0.686948
\(598\) −17.5692 −0.718459
\(599\) −10.1436 −0.414456 −0.207228 0.978293i \(-0.566444\pi\)
−0.207228 + 0.978293i \(0.566444\pi\)
\(600\) 0 0
\(601\) 36.6410 1.49462 0.747309 0.664477i \(-0.231345\pi\)
0.747309 + 0.664477i \(0.231345\pi\)
\(602\) −30.9282 −1.26054
\(603\) −8.00000 −0.325785
\(604\) 0.392305 0.0159627
\(605\) 0 0
\(606\) 18.0000 0.731200
\(607\) −22.7846 −0.924799 −0.462399 0.886672i \(-0.653011\pi\)
−0.462399 + 0.886672i \(0.653011\pi\)
\(608\) −7.60770 −0.308533
\(609\) 6.92820 0.280745
\(610\) 0 0
\(611\) −10.1436 −0.410366
\(612\) 0 0
\(613\) −0.392305 −0.0158450 −0.00792252 0.999969i \(-0.502522\pi\)
−0.00792252 + 0.999969i \(0.502522\pi\)
\(614\) 24.2487 0.978598
\(615\) 0 0
\(616\) 3.46410 0.139573
\(617\) 23.0718 0.928836 0.464418 0.885616i \(-0.346264\pi\)
0.464418 + 0.885616i \(0.346264\pi\)
\(618\) −13.8564 −0.557386
\(619\) 20.0000 0.803868 0.401934 0.915669i \(-0.368338\pi\)
0.401934 + 0.915669i \(0.368338\pi\)
\(620\) 0 0
\(621\) −6.92820 −0.278019
\(622\) −32.7846 −1.31454
\(623\) 25.8564 1.03592
\(624\) 7.32051 0.293055
\(625\) 0 0
\(626\) 12.2487 0.489557
\(627\) −1.46410 −0.0584706
\(628\) 16.9282 0.675509
\(629\) 0 0
\(630\) 0 0
\(631\) −34.9282 −1.39047 −0.695235 0.718783i \(-0.744700\pi\)
−0.695235 + 0.718783i \(0.744700\pi\)
\(632\) −23.3205 −0.927640
\(633\) −12.3923 −0.492550
\(634\) −19.1769 −0.761613
\(635\) 0 0
\(636\) −12.9282 −0.512637
\(637\) −4.39230 −0.174029
\(638\) 6.00000 0.237542
\(639\) −13.8564 −0.548151
\(640\) 0 0
\(641\) 0.928203 0.0366618 0.0183309 0.999832i \(-0.494165\pi\)
0.0183309 + 0.999832i \(0.494165\pi\)
\(642\) −26.7846 −1.05710
\(643\) 45.5692 1.79707 0.898537 0.438897i \(-0.144631\pi\)
0.898537 + 0.438897i \(0.144631\pi\)
\(644\) −13.8564 −0.546019
\(645\) 0 0
\(646\) 0 0
\(647\) 27.7128 1.08950 0.544752 0.838597i \(-0.316624\pi\)
0.544752 + 0.838597i \(0.316624\pi\)
\(648\) 1.73205 0.0680414
\(649\) −6.92820 −0.271956
\(650\) 0 0
\(651\) 5.85641 0.229531
\(652\) 17.8564 0.699311
\(653\) 7.85641 0.307445 0.153722 0.988114i \(-0.450874\pi\)
0.153722 + 0.988114i \(0.450874\pi\)
\(654\) −17.3205 −0.677285
\(655\) 0 0
\(656\) 17.3205 0.676252
\(657\) −12.3923 −0.483470
\(658\) −24.0000 −0.935617
\(659\) 39.7128 1.54699 0.773496 0.633801i \(-0.218506\pi\)
0.773496 + 0.633801i \(0.218506\pi\)
\(660\) 0 0
\(661\) 14.0000 0.544537 0.272268 0.962221i \(-0.412226\pi\)
0.272268 + 0.962221i \(0.412226\pi\)
\(662\) 30.9282 1.20206
\(663\) 0 0
\(664\) −26.7846 −1.03944
\(665\) 0 0
\(666\) 15.4641 0.599222
\(667\) 24.0000 0.929284
\(668\) −10.3923 −0.402090
\(669\) −17.8564 −0.690369
\(670\) 0 0
\(671\) −2.00000 −0.0772091
\(672\) 10.3923 0.400892
\(673\) −31.3205 −1.20732 −0.603658 0.797243i \(-0.706291\pi\)
−0.603658 + 0.797243i \(0.706291\pi\)
\(674\) −50.5359 −1.94657
\(675\) 0 0
\(676\) −10.8564 −0.417554
\(677\) 32.7846 1.26001 0.630007 0.776589i \(-0.283052\pi\)
0.630007 + 0.776589i \(0.283052\pi\)
\(678\) −1.60770 −0.0617432
\(679\) −20.0000 −0.767530
\(680\) 0 0
\(681\) 8.53590 0.327096
\(682\) 5.07180 0.194209
\(683\) −8.78461 −0.336134 −0.168067 0.985776i \(-0.553752\pi\)
−0.168067 + 0.985776i \(0.553752\pi\)
\(684\) −1.46410 −0.0559813
\(685\) 0 0
\(686\) −34.6410 −1.32260
\(687\) −3.85641 −0.147131
\(688\) 44.6410 1.70192
\(689\) 18.9282 0.721107
\(690\) 0 0
\(691\) −7.71281 −0.293409 −0.146705 0.989180i \(-0.546867\pi\)
−0.146705 + 0.989180i \(0.546867\pi\)
\(692\) 12.0000 0.456172
\(693\) 2.00000 0.0759737
\(694\) 2.78461 0.105702
\(695\) 0 0
\(696\) −6.00000 −0.227429
\(697\) 0 0
\(698\) 62.1051 2.35071
\(699\) 12.0000 0.453882
\(700\) 0 0
\(701\) 32.5359 1.22886 0.614432 0.788970i \(-0.289385\pi\)
0.614432 + 0.788970i \(0.289385\pi\)
\(702\) 2.53590 0.0957113
\(703\) 13.0718 0.493012
\(704\) −1.00000 −0.0376889
\(705\) 0 0
\(706\) −1.60770 −0.0605064
\(707\) −20.7846 −0.781686
\(708\) −6.92820 −0.260378
\(709\) 15.8564 0.595500 0.297750 0.954644i \(-0.403764\pi\)
0.297750 + 0.954644i \(0.403764\pi\)
\(710\) 0 0
\(711\) −13.4641 −0.504943
\(712\) −22.3923 −0.839187
\(713\) 20.2872 0.759761
\(714\) 0 0
\(715\) 0 0
\(716\) −6.92820 −0.258919
\(717\) 12.0000 0.448148
\(718\) −36.0000 −1.34351
\(719\) −18.9282 −0.705903 −0.352951 0.935642i \(-0.614822\pi\)
−0.352951 + 0.935642i \(0.614822\pi\)
\(720\) 0 0
\(721\) 16.0000 0.595871
\(722\) 29.1962 1.08657
\(723\) −27.8564 −1.03599
\(724\) −11.8564 −0.440640
\(725\) 0 0
\(726\) 1.73205 0.0642824
\(727\) −8.00000 −0.296704 −0.148352 0.988935i \(-0.547397\pi\)
−0.148352 + 0.988935i \(0.547397\pi\)
\(728\) −5.07180 −0.187973
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 0 0
\(732\) −2.00000 −0.0739221
\(733\) 49.9615 1.84537 0.922686 0.385553i \(-0.125989\pi\)
0.922686 + 0.385553i \(0.125989\pi\)
\(734\) 34.6410 1.27862
\(735\) 0 0
\(736\) 36.0000 1.32698
\(737\) 8.00000 0.294684
\(738\) 6.00000 0.220863
\(739\) 10.5359 0.387569 0.193785 0.981044i \(-0.437924\pi\)
0.193785 + 0.981044i \(0.437924\pi\)
\(740\) 0 0
\(741\) 2.14359 0.0787469
\(742\) 44.7846 1.64409
\(743\) −46.3923 −1.70197 −0.850984 0.525191i \(-0.823994\pi\)
−0.850984 + 0.525191i \(0.823994\pi\)
\(744\) −5.07180 −0.185941
\(745\) 0 0
\(746\) 0.679492 0.0248780
\(747\) −15.4641 −0.565802
\(748\) 0 0
\(749\) 30.9282 1.13009
\(750\) 0 0
\(751\) 13.0718 0.476997 0.238498 0.971143i \(-0.423345\pi\)
0.238498 + 0.971143i \(0.423345\pi\)
\(752\) 34.6410 1.26323
\(753\) −25.8564 −0.942260
\(754\) −8.78461 −0.319917
\(755\) 0 0
\(756\) 2.00000 0.0727393
\(757\) 6.78461 0.246591 0.123295 0.992370i \(-0.460654\pi\)
0.123295 + 0.992370i \(0.460654\pi\)
\(758\) −17.0718 −0.620076
\(759\) 6.92820 0.251478
\(760\) 0 0
\(761\) −39.4641 −1.43057 −0.715286 0.698832i \(-0.753704\pi\)
−0.715286 + 0.698832i \(0.753704\pi\)
\(762\) 8.53590 0.309223
\(763\) 20.0000 0.724049
\(764\) −5.07180 −0.183491
\(765\) 0 0
\(766\) 24.0000 0.867155
\(767\) 10.1436 0.366264
\(768\) −19.0000 −0.685603
\(769\) −46.4974 −1.67674 −0.838370 0.545102i \(-0.816491\pi\)
−0.838370 + 0.545102i \(0.816491\pi\)
\(770\) 0 0
\(771\) 7.85641 0.282942
\(772\) −3.60770 −0.129844
\(773\) 31.8564 1.14580 0.572898 0.819627i \(-0.305819\pi\)
0.572898 + 0.819627i \(0.305819\pi\)
\(774\) 15.4641 0.555846
\(775\) 0 0
\(776\) 17.3205 0.621770
\(777\) −17.8564 −0.640595
\(778\) −43.1769 −1.54797
\(779\) 5.07180 0.181716
\(780\) 0 0
\(781\) 13.8564 0.495821
\(782\) 0 0
\(783\) −3.46410 −0.123797
\(784\) 15.0000 0.535714
\(785\) 0 0
\(786\) 8.78461 0.313337
\(787\) 18.7846 0.669599 0.334800 0.942289i \(-0.391331\pi\)
0.334800 + 0.942289i \(0.391331\pi\)
\(788\) 12.0000 0.427482
\(789\) 27.4641 0.977748
\(790\) 0 0
\(791\) 1.85641 0.0660062
\(792\) −1.73205 −0.0615457
\(793\) 2.92820 0.103984
\(794\) 3.46410 0.122936
\(795\) 0 0
\(796\) 16.7846 0.594915
\(797\) −16.6410 −0.589455 −0.294728 0.955581i \(-0.595229\pi\)
−0.294728 + 0.955581i \(0.595229\pi\)
\(798\) 5.07180 0.179540
\(799\) 0 0
\(800\) 0 0
\(801\) −12.9282 −0.456796
\(802\) −34.3923 −1.21443
\(803\) 12.3923 0.437315
\(804\) 8.00000 0.282138
\(805\) 0 0
\(806\) −7.42563 −0.261557
\(807\) 7.85641 0.276559
\(808\) 18.0000 0.633238
\(809\) −8.53590 −0.300106 −0.150053 0.988678i \(-0.547944\pi\)
−0.150053 + 0.988678i \(0.547944\pi\)
\(810\) 0 0
\(811\) −8.39230 −0.294694 −0.147347 0.989085i \(-0.547073\pi\)
−0.147347 + 0.989085i \(0.547073\pi\)
\(812\) −6.92820 −0.243132
\(813\) 32.3923 1.13605
\(814\) −15.4641 −0.542016
\(815\) 0 0
\(816\) 0 0
\(817\) 13.0718 0.457324
\(818\) −60.2487 −2.10655
\(819\) −2.92820 −0.102320
\(820\) 0 0
\(821\) 27.4641 0.958504 0.479252 0.877677i \(-0.340908\pi\)
0.479252 + 0.877677i \(0.340908\pi\)
\(822\) 31.1769 1.08742
\(823\) −49.5692 −1.72787 −0.863937 0.503600i \(-0.832009\pi\)
−0.863937 + 0.503600i \(0.832009\pi\)
\(824\) −13.8564 −0.482711
\(825\) 0 0
\(826\) 24.0000 0.835067
\(827\) −1.60770 −0.0559050 −0.0279525 0.999609i \(-0.508899\pi\)
−0.0279525 + 0.999609i \(0.508899\pi\)
\(828\) 6.92820 0.240772
\(829\) −25.7128 −0.893043 −0.446521 0.894773i \(-0.647337\pi\)
−0.446521 + 0.894773i \(0.647337\pi\)
\(830\) 0 0
\(831\) 22.5359 0.781762
\(832\) 1.46410 0.0507586
\(833\) 0 0
\(834\) −14.5359 −0.503337
\(835\) 0 0
\(836\) 1.46410 0.0506370
\(837\) −2.92820 −0.101214
\(838\) −29.5692 −1.02145
\(839\) 15.2154 0.525294 0.262647 0.964892i \(-0.415405\pi\)
0.262647 + 0.964892i \(0.415405\pi\)
\(840\) 0 0
\(841\) −17.0000 −0.586207
\(842\) −3.46410 −0.119381
\(843\) 3.46410 0.119310
\(844\) 12.3923 0.426561
\(845\) 0 0
\(846\) 12.0000 0.412568
\(847\) −2.00000 −0.0687208
\(848\) −64.6410 −2.21978
\(849\) 8.92820 0.306415
\(850\) 0 0
\(851\) −61.8564 −2.12041
\(852\) 13.8564 0.474713
\(853\) −24.3923 −0.835177 −0.417588 0.908636i \(-0.637125\pi\)
−0.417588 + 0.908636i \(0.637125\pi\)
\(854\) 6.92820 0.237078
\(855\) 0 0
\(856\) −26.7846 −0.915479
\(857\) 10.1436 0.346499 0.173249 0.984878i \(-0.444573\pi\)
0.173249 + 0.984878i \(0.444573\pi\)
\(858\) −2.53590 −0.0865741
\(859\) 47.7128 1.62794 0.813970 0.580907i \(-0.197302\pi\)
0.813970 + 0.580907i \(0.197302\pi\)
\(860\) 0 0
\(861\) −6.92820 −0.236113
\(862\) 56.7846 1.93409
\(863\) 10.1436 0.345292 0.172646 0.984984i \(-0.444768\pi\)
0.172646 + 0.984984i \(0.444768\pi\)
\(864\) −5.19615 −0.176777
\(865\) 0 0
\(866\) 48.2487 1.63956
\(867\) 17.0000 0.577350
\(868\) −5.85641 −0.198779
\(869\) 13.4641 0.456738
\(870\) 0 0
\(871\) −11.7128 −0.396874
\(872\) −17.3205 −0.586546
\(873\) 10.0000 0.338449
\(874\) 17.5692 0.594288
\(875\) 0 0
\(876\) 12.3923 0.418697
\(877\) −14.2487 −0.481145 −0.240572 0.970631i \(-0.577335\pi\)
−0.240572 + 0.970631i \(0.577335\pi\)
\(878\) 50.5359 1.70550
\(879\) 13.8564 0.467365
\(880\) 0 0
\(881\) 12.9282 0.435562 0.217781 0.975998i \(-0.430118\pi\)
0.217781 + 0.975998i \(0.430118\pi\)
\(882\) 5.19615 0.174964
\(883\) 4.00000 0.134611 0.0673054 0.997732i \(-0.478560\pi\)
0.0673054 + 0.997732i \(0.478560\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 20.7846 0.698273
\(887\) 36.2487 1.21711 0.608556 0.793511i \(-0.291749\pi\)
0.608556 + 0.793511i \(0.291749\pi\)
\(888\) 15.4641 0.518941
\(889\) −9.85641 −0.330573
\(890\) 0 0
\(891\) −1.00000 −0.0335013
\(892\) 17.8564 0.597877
\(893\) 10.1436 0.339442
\(894\) −14.7846 −0.494471
\(895\) 0 0
\(896\) 24.2487 0.810093
\(897\) −10.1436 −0.338685
\(898\) −25.6077 −0.854540
\(899\) 10.1436 0.338308
\(900\) 0 0
\(901\) 0 0
\(902\) −6.00000 −0.199778
\(903\) −17.8564 −0.594224
\(904\) −1.60770 −0.0534711
\(905\) 0 0
\(906\) 0.679492 0.0225746
\(907\) −45.8564 −1.52264 −0.761318 0.648378i \(-0.775448\pi\)
−0.761318 + 0.648378i \(0.775448\pi\)
\(908\) −8.53590 −0.283274
\(909\) 10.3923 0.344691
\(910\) 0 0
\(911\) 5.07180 0.168036 0.0840181 0.996464i \(-0.473225\pi\)
0.0840181 + 0.996464i \(0.473225\pi\)
\(912\) −7.32051 −0.242406
\(913\) 15.4641 0.511787
\(914\) −14.5359 −0.480805
\(915\) 0 0
\(916\) 3.85641 0.127419
\(917\) −10.1436 −0.334971
\(918\) 0 0
\(919\) −11.6077 −0.382903 −0.191451 0.981502i \(-0.561319\pi\)
−0.191451 + 0.981502i \(0.561319\pi\)
\(920\) 0 0
\(921\) 14.0000 0.461316
\(922\) 21.2154 0.698692
\(923\) −20.2872 −0.667761
\(924\) −2.00000 −0.0657952
\(925\) 0 0
\(926\) −48.4974 −1.59372
\(927\) −8.00000 −0.262754
\(928\) 18.0000 0.590879
\(929\) −38.7846 −1.27248 −0.636241 0.771490i \(-0.719512\pi\)
−0.636241 + 0.771490i \(0.719512\pi\)
\(930\) 0 0
\(931\) 4.39230 0.143952
\(932\) −12.0000 −0.393073
\(933\) −18.9282 −0.619682
\(934\) 32.7846 1.07275
\(935\) 0 0
\(936\) 2.53590 0.0828884
\(937\) −0.392305 −0.0128160 −0.00640802 0.999979i \(-0.502040\pi\)
−0.00640802 + 0.999979i \(0.502040\pi\)
\(938\) −27.7128 −0.904855
\(939\) 7.07180 0.230779
\(940\) 0 0
\(941\) 20.5359 0.669451 0.334726 0.942316i \(-0.391356\pi\)
0.334726 + 0.942316i \(0.391356\pi\)
\(942\) 29.3205 0.955314
\(943\) −24.0000 −0.781548
\(944\) −34.6410 −1.12747
\(945\) 0 0
\(946\) −15.4641 −0.502781
\(947\) 5.07180 0.164811 0.0824056 0.996599i \(-0.473740\pi\)
0.0824056 + 0.996599i \(0.473740\pi\)
\(948\) 13.4641 0.437294
\(949\) −18.1436 −0.588966
\(950\) 0 0
\(951\) −11.0718 −0.359028
\(952\) 0 0
\(953\) 44.7846 1.45072 0.725358 0.688372i \(-0.241674\pi\)
0.725358 + 0.688372i \(0.241674\pi\)
\(954\) −22.3923 −0.724978
\(955\) 0 0
\(956\) −12.0000 −0.388108
\(957\) 3.46410 0.111979
\(958\) −20.7846 −0.671520
\(959\) −36.0000 −1.16250
\(960\) 0 0
\(961\) −22.4256 −0.723407
\(962\) 22.6410 0.729976
\(963\) −15.4641 −0.498324
\(964\) 27.8564 0.897194
\(965\) 0 0
\(966\) −24.0000 −0.772187
\(967\) 18.7846 0.604072 0.302036 0.953296i \(-0.402334\pi\)
0.302036 + 0.953296i \(0.402334\pi\)
\(968\) 1.73205 0.0556702
\(969\) 0 0
\(970\) 0 0
\(971\) −1.85641 −0.0595749 −0.0297875 0.999556i \(-0.509483\pi\)
−0.0297875 + 0.999556i \(0.509483\pi\)
\(972\) −1.00000 −0.0320750
\(973\) 16.7846 0.538090
\(974\) 41.0718 1.31603
\(975\) 0 0
\(976\) −10.0000 −0.320092
\(977\) 35.5692 1.13796 0.568980 0.822351i \(-0.307338\pi\)
0.568980 + 0.822351i \(0.307338\pi\)
\(978\) 30.9282 0.988975
\(979\) 12.9282 0.413187
\(980\) 0 0
\(981\) −10.0000 −0.319275
\(982\) 29.5692 0.943592
\(983\) −24.0000 −0.765481 −0.382741 0.923856i \(-0.625020\pi\)
−0.382741 + 0.923856i \(0.625020\pi\)
\(984\) 6.00000 0.191273
\(985\) 0 0
\(986\) 0 0
\(987\) −13.8564 −0.441054
\(988\) −2.14359 −0.0681968
\(989\) −61.8564 −1.96692
\(990\) 0 0
\(991\) −48.7846 −1.54969 −0.774847 0.632149i \(-0.782173\pi\)
−0.774847 + 0.632149i \(0.782173\pi\)
\(992\) 15.2154 0.483089
\(993\) 17.8564 0.566656
\(994\) −48.0000 −1.52247
\(995\) 0 0
\(996\) 15.4641 0.489999
\(997\) −48.3923 −1.53260 −0.766300 0.642483i \(-0.777904\pi\)
−0.766300 + 0.642483i \(0.777904\pi\)
\(998\) 22.1436 0.700943
\(999\) 8.92820 0.282476
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 825.2.a.e.1.1 2
3.2 odd 2 2475.2.a.r.1.2 2
5.2 odd 4 825.2.c.c.199.2 4
5.3 odd 4 825.2.c.c.199.3 4
5.4 even 2 165.2.a.b.1.2 2
11.10 odd 2 9075.2.a.bh.1.2 2
15.2 even 4 2475.2.c.n.199.3 4
15.8 even 4 2475.2.c.n.199.2 4
15.14 odd 2 495.2.a.c.1.1 2
20.19 odd 2 2640.2.a.x.1.1 2
35.34 odd 2 8085.2.a.bd.1.2 2
55.54 odd 2 1815.2.a.i.1.1 2
60.59 even 2 7920.2.a.bz.1.1 2
165.164 even 2 5445.2.a.s.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
165.2.a.b.1.2 2 5.4 even 2
495.2.a.c.1.1 2 15.14 odd 2
825.2.a.e.1.1 2 1.1 even 1 trivial
825.2.c.c.199.2 4 5.2 odd 4
825.2.c.c.199.3 4 5.3 odd 4
1815.2.a.i.1.1 2 55.54 odd 2
2475.2.a.r.1.2 2 3.2 odd 2
2475.2.c.n.199.2 4 15.8 even 4
2475.2.c.n.199.3 4 15.2 even 4
2640.2.a.x.1.1 2 20.19 odd 2
5445.2.a.s.1.2 2 165.164 even 2
7920.2.a.bz.1.1 2 60.59 even 2
8085.2.a.bd.1.2 2 35.34 odd 2
9075.2.a.bh.1.2 2 11.10 odd 2