Properties

Label 3234.2.a.o.1.1
Level $3234$
Weight $2$
Character 3234.1
Self dual yes
Analytic conductor $25.824$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 3234 = 2 \cdot 3 \cdot 7^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3234.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(25.8236200137\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 462)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 3234.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +3.00000 q^{5} -1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +3.00000 q^{5} -1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} -3.00000 q^{10} -1.00000 q^{11} +1.00000 q^{12} +6.00000 q^{13} +3.00000 q^{15} +1.00000 q^{16} -5.00000 q^{17} -1.00000 q^{18} +6.00000 q^{19} +3.00000 q^{20} +1.00000 q^{22} +5.00000 q^{23} -1.00000 q^{24} +4.00000 q^{25} -6.00000 q^{26} +1.00000 q^{27} -6.00000 q^{29} -3.00000 q^{30} +4.00000 q^{31} -1.00000 q^{32} -1.00000 q^{33} +5.00000 q^{34} +1.00000 q^{36} -2.00000 q^{37} -6.00000 q^{38} +6.00000 q^{39} -3.00000 q^{40} +5.00000 q^{41} -10.0000 q^{43} -1.00000 q^{44} +3.00000 q^{45} -5.00000 q^{46} +9.00000 q^{47} +1.00000 q^{48} -4.00000 q^{50} -5.00000 q^{51} +6.00000 q^{52} +2.00000 q^{53} -1.00000 q^{54} -3.00000 q^{55} +6.00000 q^{57} +6.00000 q^{58} -12.0000 q^{59} +3.00000 q^{60} -5.00000 q^{61} -4.00000 q^{62} +1.00000 q^{64} +18.0000 q^{65} +1.00000 q^{66} +5.00000 q^{67} -5.00000 q^{68} +5.00000 q^{69} +4.00000 q^{71} -1.00000 q^{72} +12.0000 q^{73} +2.00000 q^{74} +4.00000 q^{75} +6.00000 q^{76} -6.00000 q^{78} -1.00000 q^{79} +3.00000 q^{80} +1.00000 q^{81} -5.00000 q^{82} +1.00000 q^{83} -15.0000 q^{85} +10.0000 q^{86} -6.00000 q^{87} +1.00000 q^{88} +6.00000 q^{89} -3.00000 q^{90} +5.00000 q^{92} +4.00000 q^{93} -9.00000 q^{94} +18.0000 q^{95} -1.00000 q^{96} +9.00000 q^{97} -1.00000 q^{99} +O(q^{100})\)

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) 1.00000 0.577350
\(4\) 1.00000 0.500000
\(5\) 3.00000 1.34164 0.670820 0.741620i \(-0.265942\pi\)
0.670820 + 0.741620i \(0.265942\pi\)
\(6\) −1.00000 −0.408248
\(7\) 0 0
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) −3.00000 −0.948683
\(11\) −1.00000 −0.301511
\(12\) 1.00000 0.288675
\(13\) 6.00000 1.66410 0.832050 0.554700i \(-0.187167\pi\)
0.832050 + 0.554700i \(0.187167\pi\)
\(14\) 0 0
\(15\) 3.00000 0.774597
\(16\) 1.00000 0.250000
\(17\) −5.00000 −1.21268 −0.606339 0.795206i \(-0.707363\pi\)
−0.606339 + 0.795206i \(0.707363\pi\)
\(18\) −1.00000 −0.235702
\(19\) 6.00000 1.37649 0.688247 0.725476i \(-0.258380\pi\)
0.688247 + 0.725476i \(0.258380\pi\)
\(20\) 3.00000 0.670820
\(21\) 0 0
\(22\) 1.00000 0.213201
\(23\) 5.00000 1.04257 0.521286 0.853382i \(-0.325452\pi\)
0.521286 + 0.853382i \(0.325452\pi\)
\(24\) −1.00000 −0.204124
\(25\) 4.00000 0.800000
\(26\) −6.00000 −1.17670
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) −6.00000 −1.11417 −0.557086 0.830455i \(-0.688081\pi\)
−0.557086 + 0.830455i \(0.688081\pi\)
\(30\) −3.00000 −0.547723
\(31\) 4.00000 0.718421 0.359211 0.933257i \(-0.383046\pi\)
0.359211 + 0.933257i \(0.383046\pi\)
\(32\) −1.00000 −0.176777
\(33\) −1.00000 −0.174078
\(34\) 5.00000 0.857493
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) −2.00000 −0.328798 −0.164399 0.986394i \(-0.552568\pi\)
−0.164399 + 0.986394i \(0.552568\pi\)
\(38\) −6.00000 −0.973329
\(39\) 6.00000 0.960769
\(40\) −3.00000 −0.474342
\(41\) 5.00000 0.780869 0.390434 0.920631i \(-0.372325\pi\)
0.390434 + 0.920631i \(0.372325\pi\)
\(42\) 0 0
\(43\) −10.0000 −1.52499 −0.762493 0.646997i \(-0.776025\pi\)
−0.762493 + 0.646997i \(0.776025\pi\)
\(44\) −1.00000 −0.150756
\(45\) 3.00000 0.447214
\(46\) −5.00000 −0.737210
\(47\) 9.00000 1.31278 0.656392 0.754420i \(-0.272082\pi\)
0.656392 + 0.754420i \(0.272082\pi\)
\(48\) 1.00000 0.144338
\(49\) 0 0
\(50\) −4.00000 −0.565685
\(51\) −5.00000 −0.700140
\(52\) 6.00000 0.832050
\(53\) 2.00000 0.274721 0.137361 0.990521i \(-0.456138\pi\)
0.137361 + 0.990521i \(0.456138\pi\)
\(54\) −1.00000 −0.136083
\(55\) −3.00000 −0.404520
\(56\) 0 0
\(57\) 6.00000 0.794719
\(58\) 6.00000 0.787839
\(59\) −12.0000 −1.56227 −0.781133 0.624364i \(-0.785358\pi\)
−0.781133 + 0.624364i \(0.785358\pi\)
\(60\) 3.00000 0.387298
\(61\) −5.00000 −0.640184 −0.320092 0.947386i \(-0.603714\pi\)
−0.320092 + 0.947386i \(0.603714\pi\)
\(62\) −4.00000 −0.508001
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 18.0000 2.23263
\(66\) 1.00000 0.123091
\(67\) 5.00000 0.610847 0.305424 0.952217i \(-0.401202\pi\)
0.305424 + 0.952217i \(0.401202\pi\)
\(68\) −5.00000 −0.606339
\(69\) 5.00000 0.601929
\(70\) 0 0
\(71\) 4.00000 0.474713 0.237356 0.971423i \(-0.423719\pi\)
0.237356 + 0.971423i \(0.423719\pi\)
\(72\) −1.00000 −0.117851
\(73\) 12.0000 1.40449 0.702247 0.711934i \(-0.252180\pi\)
0.702247 + 0.711934i \(0.252180\pi\)
\(74\) 2.00000 0.232495
\(75\) 4.00000 0.461880
\(76\) 6.00000 0.688247
\(77\) 0 0
\(78\) −6.00000 −0.679366
\(79\) −1.00000 −0.112509 −0.0562544 0.998416i \(-0.517916\pi\)
−0.0562544 + 0.998416i \(0.517916\pi\)
\(80\) 3.00000 0.335410
\(81\) 1.00000 0.111111
\(82\) −5.00000 −0.552158
\(83\) 1.00000 0.109764 0.0548821 0.998493i \(-0.482522\pi\)
0.0548821 + 0.998493i \(0.482522\pi\)
\(84\) 0 0
\(85\) −15.0000 −1.62698
\(86\) 10.0000 1.07833
\(87\) −6.00000 −0.643268
\(88\) 1.00000 0.106600
\(89\) 6.00000 0.635999 0.317999 0.948091i \(-0.396989\pi\)
0.317999 + 0.948091i \(0.396989\pi\)
\(90\) −3.00000 −0.316228
\(91\) 0 0
\(92\) 5.00000 0.521286
\(93\) 4.00000 0.414781
\(94\) −9.00000 −0.928279
\(95\) 18.0000 1.84676
\(96\) −1.00000 −0.102062
\(97\) 9.00000 0.913812 0.456906 0.889515i \(-0.348958\pi\)
0.456906 + 0.889515i \(0.348958\pi\)
\(98\) 0 0
\(99\) −1.00000 −0.100504
\(100\) 4.00000 0.400000
\(101\) −8.00000 −0.796030 −0.398015 0.917379i \(-0.630301\pi\)
−0.398015 + 0.917379i \(0.630301\pi\)
\(102\) 5.00000 0.495074
\(103\) −2.00000 −0.197066 −0.0985329 0.995134i \(-0.531415\pi\)
−0.0985329 + 0.995134i \(0.531415\pi\)
\(104\) −6.00000 −0.588348
\(105\) 0 0
\(106\) −2.00000 −0.194257
\(107\) −13.0000 −1.25676 −0.628379 0.777908i \(-0.716281\pi\)
−0.628379 + 0.777908i \(0.716281\pi\)
\(108\) 1.00000 0.0962250
\(109\) −11.0000 −1.05361 −0.526804 0.849987i \(-0.676610\pi\)
−0.526804 + 0.849987i \(0.676610\pi\)
\(110\) 3.00000 0.286039
\(111\) −2.00000 −0.189832
\(112\) 0 0
\(113\) 10.0000 0.940721 0.470360 0.882474i \(-0.344124\pi\)
0.470360 + 0.882474i \(0.344124\pi\)
\(114\) −6.00000 −0.561951
\(115\) 15.0000 1.39876
\(116\) −6.00000 −0.557086
\(117\) 6.00000 0.554700
\(118\) 12.0000 1.10469
\(119\) 0 0
\(120\) −3.00000 −0.273861
\(121\) 1.00000 0.0909091
\(122\) 5.00000 0.452679
\(123\) 5.00000 0.450835
\(124\) 4.00000 0.359211
\(125\) −3.00000 −0.268328
\(126\) 0 0
\(127\) 5.00000 0.443678 0.221839 0.975083i \(-0.428794\pi\)
0.221839 + 0.975083i \(0.428794\pi\)
\(128\) −1.00000 −0.0883883
\(129\) −10.0000 −0.880451
\(130\) −18.0000 −1.57870
\(131\) −4.00000 −0.349482 −0.174741 0.984614i \(-0.555909\pi\)
−0.174741 + 0.984614i \(0.555909\pi\)
\(132\) −1.00000 −0.0870388
\(133\) 0 0
\(134\) −5.00000 −0.431934
\(135\) 3.00000 0.258199
\(136\) 5.00000 0.428746
\(137\) −2.00000 −0.170872 −0.0854358 0.996344i \(-0.527228\pi\)
−0.0854358 + 0.996344i \(0.527228\pi\)
\(138\) −5.00000 −0.425628
\(139\) 22.0000 1.86602 0.933008 0.359856i \(-0.117174\pi\)
0.933008 + 0.359856i \(0.117174\pi\)
\(140\) 0 0
\(141\) 9.00000 0.757937
\(142\) −4.00000 −0.335673
\(143\) −6.00000 −0.501745
\(144\) 1.00000 0.0833333
\(145\) −18.0000 −1.49482
\(146\) −12.0000 −0.993127
\(147\) 0 0
\(148\) −2.00000 −0.164399
\(149\) 18.0000 1.47462 0.737309 0.675556i \(-0.236096\pi\)
0.737309 + 0.675556i \(0.236096\pi\)
\(150\) −4.00000 −0.326599
\(151\) 9.00000 0.732410 0.366205 0.930534i \(-0.380657\pi\)
0.366205 + 0.930534i \(0.380657\pi\)
\(152\) −6.00000 −0.486664
\(153\) −5.00000 −0.404226
\(154\) 0 0
\(155\) 12.0000 0.963863
\(156\) 6.00000 0.480384
\(157\) −24.0000 −1.91541 −0.957704 0.287754i \(-0.907091\pi\)
−0.957704 + 0.287754i \(0.907091\pi\)
\(158\) 1.00000 0.0795557
\(159\) 2.00000 0.158610
\(160\) −3.00000 −0.237171
\(161\) 0 0
\(162\) −1.00000 −0.0785674
\(163\) −1.00000 −0.0783260 −0.0391630 0.999233i \(-0.512469\pi\)
−0.0391630 + 0.999233i \(0.512469\pi\)
\(164\) 5.00000 0.390434
\(165\) −3.00000 −0.233550
\(166\) −1.00000 −0.0776151
\(167\) −24.0000 −1.85718 −0.928588 0.371113i \(-0.878976\pi\)
−0.928588 + 0.371113i \(0.878976\pi\)
\(168\) 0 0
\(169\) 23.0000 1.76923
\(170\) 15.0000 1.15045
\(171\) 6.00000 0.458831
\(172\) −10.0000 −0.762493
\(173\) −12.0000 −0.912343 −0.456172 0.889892i \(-0.650780\pi\)
−0.456172 + 0.889892i \(0.650780\pi\)
\(174\) 6.00000 0.454859
\(175\) 0 0
\(176\) −1.00000 −0.0753778
\(177\) −12.0000 −0.901975
\(178\) −6.00000 −0.449719
\(179\) −8.00000 −0.597948 −0.298974 0.954261i \(-0.596644\pi\)
−0.298974 + 0.954261i \(0.596644\pi\)
\(180\) 3.00000 0.223607
\(181\) −22.0000 −1.63525 −0.817624 0.575753i \(-0.804709\pi\)
−0.817624 + 0.575753i \(0.804709\pi\)
\(182\) 0 0
\(183\) −5.00000 −0.369611
\(184\) −5.00000 −0.368605
\(185\) −6.00000 −0.441129
\(186\) −4.00000 −0.293294
\(187\) 5.00000 0.365636
\(188\) 9.00000 0.656392
\(189\) 0 0
\(190\) −18.0000 −1.30586
\(191\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(192\) 1.00000 0.0721688
\(193\) 6.00000 0.431889 0.215945 0.976406i \(-0.430717\pi\)
0.215945 + 0.976406i \(0.430717\pi\)
\(194\) −9.00000 −0.646162
\(195\) 18.0000 1.28901
\(196\) 0 0
\(197\) −24.0000 −1.70993 −0.854965 0.518686i \(-0.826421\pi\)
−0.854965 + 0.518686i \(0.826421\pi\)
\(198\) 1.00000 0.0710669
\(199\) 10.0000 0.708881 0.354441 0.935079i \(-0.384671\pi\)
0.354441 + 0.935079i \(0.384671\pi\)
\(200\) −4.00000 −0.282843
\(201\) 5.00000 0.352673
\(202\) 8.00000 0.562878
\(203\) 0 0
\(204\) −5.00000 −0.350070
\(205\) 15.0000 1.04765
\(206\) 2.00000 0.139347
\(207\) 5.00000 0.347524
\(208\) 6.00000 0.416025
\(209\) −6.00000 −0.415029
\(210\) 0 0
\(211\) 14.0000 0.963800 0.481900 0.876226i \(-0.339947\pi\)
0.481900 + 0.876226i \(0.339947\pi\)
\(212\) 2.00000 0.137361
\(213\) 4.00000 0.274075
\(214\) 13.0000 0.888662
\(215\) −30.0000 −2.04598
\(216\) −1.00000 −0.0680414
\(217\) 0 0
\(218\) 11.0000 0.745014
\(219\) 12.0000 0.810885
\(220\) −3.00000 −0.202260
\(221\) −30.0000 −2.01802
\(222\) 2.00000 0.134231
\(223\) 16.0000 1.07144 0.535720 0.844396i \(-0.320040\pi\)
0.535720 + 0.844396i \(0.320040\pi\)
\(224\) 0 0
\(225\) 4.00000 0.266667
\(226\) −10.0000 −0.665190
\(227\) 3.00000 0.199117 0.0995585 0.995032i \(-0.468257\pi\)
0.0995585 + 0.995032i \(0.468257\pi\)
\(228\) 6.00000 0.397360
\(229\) 26.0000 1.71813 0.859064 0.511868i \(-0.171046\pi\)
0.859064 + 0.511868i \(0.171046\pi\)
\(230\) −15.0000 −0.989071
\(231\) 0 0
\(232\) 6.00000 0.393919
\(233\) −29.0000 −1.89985 −0.949927 0.312473i \(-0.898843\pi\)
−0.949927 + 0.312473i \(0.898843\pi\)
\(234\) −6.00000 −0.392232
\(235\) 27.0000 1.76129
\(236\) −12.0000 −0.781133
\(237\) −1.00000 −0.0649570
\(238\) 0 0
\(239\) 26.0000 1.68180 0.840900 0.541190i \(-0.182026\pi\)
0.840900 + 0.541190i \(0.182026\pi\)
\(240\) 3.00000 0.193649
\(241\) −26.0000 −1.67481 −0.837404 0.546585i \(-0.815928\pi\)
−0.837404 + 0.546585i \(0.815928\pi\)
\(242\) −1.00000 −0.0642824
\(243\) 1.00000 0.0641500
\(244\) −5.00000 −0.320092
\(245\) 0 0
\(246\) −5.00000 −0.318788
\(247\) 36.0000 2.29063
\(248\) −4.00000 −0.254000
\(249\) 1.00000 0.0633724
\(250\) 3.00000 0.189737
\(251\) 12.0000 0.757433 0.378717 0.925513i \(-0.376365\pi\)
0.378717 + 0.925513i \(0.376365\pi\)
\(252\) 0 0
\(253\) −5.00000 −0.314347
\(254\) −5.00000 −0.313728
\(255\) −15.0000 −0.939336
\(256\) 1.00000 0.0625000
\(257\) −14.0000 −0.873296 −0.436648 0.899632i \(-0.643834\pi\)
−0.436648 + 0.899632i \(0.643834\pi\)
\(258\) 10.0000 0.622573
\(259\) 0 0
\(260\) 18.0000 1.11631
\(261\) −6.00000 −0.371391
\(262\) 4.00000 0.247121
\(263\) 24.0000 1.47990 0.739952 0.672660i \(-0.234848\pi\)
0.739952 + 0.672660i \(0.234848\pi\)
\(264\) 1.00000 0.0615457
\(265\) 6.00000 0.368577
\(266\) 0 0
\(267\) 6.00000 0.367194
\(268\) 5.00000 0.305424
\(269\) −9.00000 −0.548740 −0.274370 0.961624i \(-0.588469\pi\)
−0.274370 + 0.961624i \(0.588469\pi\)
\(270\) −3.00000 −0.182574
\(271\) −24.0000 −1.45790 −0.728948 0.684569i \(-0.759990\pi\)
−0.728948 + 0.684569i \(0.759990\pi\)
\(272\) −5.00000 −0.303170
\(273\) 0 0
\(274\) 2.00000 0.120824
\(275\) −4.00000 −0.241209
\(276\) 5.00000 0.300965
\(277\) 18.0000 1.08152 0.540758 0.841178i \(-0.318138\pi\)
0.540758 + 0.841178i \(0.318138\pi\)
\(278\) −22.0000 −1.31947
\(279\) 4.00000 0.239474
\(280\) 0 0
\(281\) 11.0000 0.656205 0.328102 0.944642i \(-0.393591\pi\)
0.328102 + 0.944642i \(0.393591\pi\)
\(282\) −9.00000 −0.535942
\(283\) −6.00000 −0.356663 −0.178331 0.983970i \(-0.557070\pi\)
−0.178331 + 0.983970i \(0.557070\pi\)
\(284\) 4.00000 0.237356
\(285\) 18.0000 1.06623
\(286\) 6.00000 0.354787
\(287\) 0 0
\(288\) −1.00000 −0.0589256
\(289\) 8.00000 0.470588
\(290\) 18.0000 1.05700
\(291\) 9.00000 0.527589
\(292\) 12.0000 0.702247
\(293\) 8.00000 0.467365 0.233682 0.972313i \(-0.424922\pi\)
0.233682 + 0.972313i \(0.424922\pi\)
\(294\) 0 0
\(295\) −36.0000 −2.09600
\(296\) 2.00000 0.116248
\(297\) −1.00000 −0.0580259
\(298\) −18.0000 −1.04271
\(299\) 30.0000 1.73494
\(300\) 4.00000 0.230940
\(301\) 0 0
\(302\) −9.00000 −0.517892
\(303\) −8.00000 −0.459588
\(304\) 6.00000 0.344124
\(305\) −15.0000 −0.858898
\(306\) 5.00000 0.285831
\(307\) 18.0000 1.02731 0.513657 0.857996i \(-0.328290\pi\)
0.513657 + 0.857996i \(0.328290\pi\)
\(308\) 0 0
\(309\) −2.00000 −0.113776
\(310\) −12.0000 −0.681554
\(311\) −21.0000 −1.19080 −0.595400 0.803429i \(-0.703007\pi\)
−0.595400 + 0.803429i \(0.703007\pi\)
\(312\) −6.00000 −0.339683
\(313\) −6.00000 −0.339140 −0.169570 0.985518i \(-0.554238\pi\)
−0.169570 + 0.985518i \(0.554238\pi\)
\(314\) 24.0000 1.35440
\(315\) 0 0
\(316\) −1.00000 −0.0562544
\(317\) 9.00000 0.505490 0.252745 0.967533i \(-0.418667\pi\)
0.252745 + 0.967533i \(0.418667\pi\)
\(318\) −2.00000 −0.112154
\(319\) 6.00000 0.335936
\(320\) 3.00000 0.167705
\(321\) −13.0000 −0.725589
\(322\) 0 0
\(323\) −30.0000 −1.66924
\(324\) 1.00000 0.0555556
\(325\) 24.0000 1.33128
\(326\) 1.00000 0.0553849
\(327\) −11.0000 −0.608301
\(328\) −5.00000 −0.276079
\(329\) 0 0
\(330\) 3.00000 0.165145
\(331\) 35.0000 1.92377 0.961887 0.273447i \(-0.0881639\pi\)
0.961887 + 0.273447i \(0.0881639\pi\)
\(332\) 1.00000 0.0548821
\(333\) −2.00000 −0.109599
\(334\) 24.0000 1.31322
\(335\) 15.0000 0.819538
\(336\) 0 0
\(337\) −2.00000 −0.108947 −0.0544735 0.998515i \(-0.517348\pi\)
−0.0544735 + 0.998515i \(0.517348\pi\)
\(338\) −23.0000 −1.25104
\(339\) 10.0000 0.543125
\(340\) −15.0000 −0.813489
\(341\) −4.00000 −0.216612
\(342\) −6.00000 −0.324443
\(343\) 0 0
\(344\) 10.0000 0.539164
\(345\) 15.0000 0.807573
\(346\) 12.0000 0.645124
\(347\) 7.00000 0.375780 0.187890 0.982190i \(-0.439835\pi\)
0.187890 + 0.982190i \(0.439835\pi\)
\(348\) −6.00000 −0.321634
\(349\) −19.0000 −1.01705 −0.508523 0.861048i \(-0.669808\pi\)
−0.508523 + 0.861048i \(0.669808\pi\)
\(350\) 0 0
\(351\) 6.00000 0.320256
\(352\) 1.00000 0.0533002
\(353\) 24.0000 1.27739 0.638696 0.769460i \(-0.279474\pi\)
0.638696 + 0.769460i \(0.279474\pi\)
\(354\) 12.0000 0.637793
\(355\) 12.0000 0.636894
\(356\) 6.00000 0.317999
\(357\) 0 0
\(358\) 8.00000 0.422813
\(359\) −10.0000 −0.527780 −0.263890 0.964553i \(-0.585006\pi\)
−0.263890 + 0.964553i \(0.585006\pi\)
\(360\) −3.00000 −0.158114
\(361\) 17.0000 0.894737
\(362\) 22.0000 1.15629
\(363\) 1.00000 0.0524864
\(364\) 0 0
\(365\) 36.0000 1.88433
\(366\) 5.00000 0.261354
\(367\) −14.0000 −0.730794 −0.365397 0.930852i \(-0.619067\pi\)
−0.365397 + 0.930852i \(0.619067\pi\)
\(368\) 5.00000 0.260643
\(369\) 5.00000 0.260290
\(370\) 6.00000 0.311925
\(371\) 0 0
\(372\) 4.00000 0.207390
\(373\) −7.00000 −0.362446 −0.181223 0.983442i \(-0.558006\pi\)
−0.181223 + 0.983442i \(0.558006\pi\)
\(374\) −5.00000 −0.258544
\(375\) −3.00000 −0.154919
\(376\) −9.00000 −0.464140
\(377\) −36.0000 −1.85409
\(378\) 0 0
\(379\) 13.0000 0.667765 0.333883 0.942615i \(-0.391641\pi\)
0.333883 + 0.942615i \(0.391641\pi\)
\(380\) 18.0000 0.923381
\(381\) 5.00000 0.256158
\(382\) 0 0
\(383\) 8.00000 0.408781 0.204390 0.978889i \(-0.434479\pi\)
0.204390 + 0.978889i \(0.434479\pi\)
\(384\) −1.00000 −0.0510310
\(385\) 0 0
\(386\) −6.00000 −0.305392
\(387\) −10.0000 −0.508329
\(388\) 9.00000 0.456906
\(389\) −21.0000 −1.06474 −0.532371 0.846511i \(-0.678699\pi\)
−0.532371 + 0.846511i \(0.678699\pi\)
\(390\) −18.0000 −0.911465
\(391\) −25.0000 −1.26430
\(392\) 0 0
\(393\) −4.00000 −0.201773
\(394\) 24.0000 1.20910
\(395\) −3.00000 −0.150946
\(396\) −1.00000 −0.0502519
\(397\) 26.0000 1.30490 0.652451 0.757831i \(-0.273741\pi\)
0.652451 + 0.757831i \(0.273741\pi\)
\(398\) −10.0000 −0.501255
\(399\) 0 0
\(400\) 4.00000 0.200000
\(401\) −6.00000 −0.299626 −0.149813 0.988714i \(-0.547867\pi\)
−0.149813 + 0.988714i \(0.547867\pi\)
\(402\) −5.00000 −0.249377
\(403\) 24.0000 1.19553
\(404\) −8.00000 −0.398015
\(405\) 3.00000 0.149071
\(406\) 0 0
\(407\) 2.00000 0.0991363
\(408\) 5.00000 0.247537
\(409\) 10.0000 0.494468 0.247234 0.968956i \(-0.420478\pi\)
0.247234 + 0.968956i \(0.420478\pi\)
\(410\) −15.0000 −0.740797
\(411\) −2.00000 −0.0986527
\(412\) −2.00000 −0.0985329
\(413\) 0 0
\(414\) −5.00000 −0.245737
\(415\) 3.00000 0.147264
\(416\) −6.00000 −0.294174
\(417\) 22.0000 1.07734
\(418\) 6.00000 0.293470
\(419\) −20.0000 −0.977064 −0.488532 0.872546i \(-0.662467\pi\)
−0.488532 + 0.872546i \(0.662467\pi\)
\(420\) 0 0
\(421\) 8.00000 0.389896 0.194948 0.980814i \(-0.437546\pi\)
0.194948 + 0.980814i \(0.437546\pi\)
\(422\) −14.0000 −0.681509
\(423\) 9.00000 0.437595
\(424\) −2.00000 −0.0971286
\(425\) −20.0000 −0.970143
\(426\) −4.00000 −0.193801
\(427\) 0 0
\(428\) −13.0000 −0.628379
\(429\) −6.00000 −0.289683
\(430\) 30.0000 1.44673
\(431\) 22.0000 1.05970 0.529851 0.848091i \(-0.322248\pi\)
0.529851 + 0.848091i \(0.322248\pi\)
\(432\) 1.00000 0.0481125
\(433\) 11.0000 0.528626 0.264313 0.964437i \(-0.414855\pi\)
0.264313 + 0.964437i \(0.414855\pi\)
\(434\) 0 0
\(435\) −18.0000 −0.863034
\(436\) −11.0000 −0.526804
\(437\) 30.0000 1.43509
\(438\) −12.0000 −0.573382
\(439\) −5.00000 −0.238637 −0.119318 0.992856i \(-0.538071\pi\)
−0.119318 + 0.992856i \(0.538071\pi\)
\(440\) 3.00000 0.143019
\(441\) 0 0
\(442\) 30.0000 1.42695
\(443\) 2.00000 0.0950229 0.0475114 0.998871i \(-0.484871\pi\)
0.0475114 + 0.998871i \(0.484871\pi\)
\(444\) −2.00000 −0.0949158
\(445\) 18.0000 0.853282
\(446\) −16.0000 −0.757622
\(447\) 18.0000 0.851371
\(448\) 0 0
\(449\) 2.00000 0.0943858 0.0471929 0.998886i \(-0.484972\pi\)
0.0471929 + 0.998886i \(0.484972\pi\)
\(450\) −4.00000 −0.188562
\(451\) −5.00000 −0.235441
\(452\) 10.0000 0.470360
\(453\) 9.00000 0.422857
\(454\) −3.00000 −0.140797
\(455\) 0 0
\(456\) −6.00000 −0.280976
\(457\) −26.0000 −1.21623 −0.608114 0.793849i \(-0.708074\pi\)
−0.608114 + 0.793849i \(0.708074\pi\)
\(458\) −26.0000 −1.21490
\(459\) −5.00000 −0.233380
\(460\) 15.0000 0.699379
\(461\) −10.0000 −0.465746 −0.232873 0.972507i \(-0.574813\pi\)
−0.232873 + 0.972507i \(0.574813\pi\)
\(462\) 0 0
\(463\) 28.0000 1.30127 0.650635 0.759390i \(-0.274503\pi\)
0.650635 + 0.759390i \(0.274503\pi\)
\(464\) −6.00000 −0.278543
\(465\) 12.0000 0.556487
\(466\) 29.0000 1.34340
\(467\) 16.0000 0.740392 0.370196 0.928954i \(-0.379291\pi\)
0.370196 + 0.928954i \(0.379291\pi\)
\(468\) 6.00000 0.277350
\(469\) 0 0
\(470\) −27.0000 −1.24542
\(471\) −24.0000 −1.10586
\(472\) 12.0000 0.552345
\(473\) 10.0000 0.459800
\(474\) 1.00000 0.0459315
\(475\) 24.0000 1.10120
\(476\) 0 0
\(477\) 2.00000 0.0915737
\(478\) −26.0000 −1.18921
\(479\) 12.0000 0.548294 0.274147 0.961688i \(-0.411605\pi\)
0.274147 + 0.961688i \(0.411605\pi\)
\(480\) −3.00000 −0.136931
\(481\) −12.0000 −0.547153
\(482\) 26.0000 1.18427
\(483\) 0 0
\(484\) 1.00000 0.0454545
\(485\) 27.0000 1.22601
\(486\) −1.00000 −0.0453609
\(487\) −4.00000 −0.181257 −0.0906287 0.995885i \(-0.528888\pi\)
−0.0906287 + 0.995885i \(0.528888\pi\)
\(488\) 5.00000 0.226339
\(489\) −1.00000 −0.0452216
\(490\) 0 0
\(491\) −3.00000 −0.135388 −0.0676941 0.997706i \(-0.521564\pi\)
−0.0676941 + 0.997706i \(0.521564\pi\)
\(492\) 5.00000 0.225417
\(493\) 30.0000 1.35113
\(494\) −36.0000 −1.61972
\(495\) −3.00000 −0.134840
\(496\) 4.00000 0.179605
\(497\) 0 0
\(498\) −1.00000 −0.0448111
\(499\) 12.0000 0.537194 0.268597 0.963253i \(-0.413440\pi\)
0.268597 + 0.963253i \(0.413440\pi\)
\(500\) −3.00000 −0.134164
\(501\) −24.0000 −1.07224
\(502\) −12.0000 −0.535586
\(503\) −42.0000 −1.87269 −0.936344 0.351085i \(-0.885813\pi\)
−0.936344 + 0.351085i \(0.885813\pi\)
\(504\) 0 0
\(505\) −24.0000 −1.06799
\(506\) 5.00000 0.222277
\(507\) 23.0000 1.02147
\(508\) 5.00000 0.221839
\(509\) 2.00000 0.0886484 0.0443242 0.999017i \(-0.485887\pi\)
0.0443242 + 0.999017i \(0.485887\pi\)
\(510\) 15.0000 0.664211
\(511\) 0 0
\(512\) −1.00000 −0.0441942
\(513\) 6.00000 0.264906
\(514\) 14.0000 0.617514
\(515\) −6.00000 −0.264392
\(516\) −10.0000 −0.440225
\(517\) −9.00000 −0.395820
\(518\) 0 0
\(519\) −12.0000 −0.526742
\(520\) −18.0000 −0.789352
\(521\) 2.00000 0.0876216 0.0438108 0.999040i \(-0.486050\pi\)
0.0438108 + 0.999040i \(0.486050\pi\)
\(522\) 6.00000 0.262613
\(523\) 6.00000 0.262362 0.131181 0.991358i \(-0.458123\pi\)
0.131181 + 0.991358i \(0.458123\pi\)
\(524\) −4.00000 −0.174741
\(525\) 0 0
\(526\) −24.0000 −1.04645
\(527\) −20.0000 −0.871214
\(528\) −1.00000 −0.0435194
\(529\) 2.00000 0.0869565
\(530\) −6.00000 −0.260623
\(531\) −12.0000 −0.520756
\(532\) 0 0
\(533\) 30.0000 1.29944
\(534\) −6.00000 −0.259645
\(535\) −39.0000 −1.68612
\(536\) −5.00000 −0.215967
\(537\) −8.00000 −0.345225
\(538\) 9.00000 0.388018
\(539\) 0 0
\(540\) 3.00000 0.129099
\(541\) −39.0000 −1.67674 −0.838370 0.545101i \(-0.816491\pi\)
−0.838370 + 0.545101i \(0.816491\pi\)
\(542\) 24.0000 1.03089
\(543\) −22.0000 −0.944110
\(544\) 5.00000 0.214373
\(545\) −33.0000 −1.41356
\(546\) 0 0
\(547\) −42.0000 −1.79579 −0.897895 0.440209i \(-0.854904\pi\)
−0.897895 + 0.440209i \(0.854904\pi\)
\(548\) −2.00000 −0.0854358
\(549\) −5.00000 −0.213395
\(550\) 4.00000 0.170561
\(551\) −36.0000 −1.53365
\(552\) −5.00000 −0.212814
\(553\) 0 0
\(554\) −18.0000 −0.764747
\(555\) −6.00000 −0.254686
\(556\) 22.0000 0.933008
\(557\) −32.0000 −1.35588 −0.677942 0.735116i \(-0.737128\pi\)
−0.677942 + 0.735116i \(0.737128\pi\)
\(558\) −4.00000 −0.169334
\(559\) −60.0000 −2.53773
\(560\) 0 0
\(561\) 5.00000 0.211100
\(562\) −11.0000 −0.464007
\(563\) −16.0000 −0.674320 −0.337160 0.941447i \(-0.609466\pi\)
−0.337160 + 0.941447i \(0.609466\pi\)
\(564\) 9.00000 0.378968
\(565\) 30.0000 1.26211
\(566\) 6.00000 0.252199
\(567\) 0 0
\(568\) −4.00000 −0.167836
\(569\) −6.00000 −0.251533 −0.125767 0.992060i \(-0.540139\pi\)
−0.125767 + 0.992060i \(0.540139\pi\)
\(570\) −18.0000 −0.753937
\(571\) −34.0000 −1.42286 −0.711428 0.702759i \(-0.751951\pi\)
−0.711428 + 0.702759i \(0.751951\pi\)
\(572\) −6.00000 −0.250873
\(573\) 0 0
\(574\) 0 0
\(575\) 20.0000 0.834058
\(576\) 1.00000 0.0416667
\(577\) −7.00000 −0.291414 −0.145707 0.989328i \(-0.546546\pi\)
−0.145707 + 0.989328i \(0.546546\pi\)
\(578\) −8.00000 −0.332756
\(579\) 6.00000 0.249351
\(580\) −18.0000 −0.747409
\(581\) 0 0
\(582\) −9.00000 −0.373062
\(583\) −2.00000 −0.0828315
\(584\) −12.0000 −0.496564
\(585\) 18.0000 0.744208
\(586\) −8.00000 −0.330477
\(587\) −24.0000 −0.990586 −0.495293 0.868726i \(-0.664939\pi\)
−0.495293 + 0.868726i \(0.664939\pi\)
\(588\) 0 0
\(589\) 24.0000 0.988903
\(590\) 36.0000 1.48210
\(591\) −24.0000 −0.987228
\(592\) −2.00000 −0.0821995
\(593\) 14.0000 0.574911 0.287456 0.957794i \(-0.407191\pi\)
0.287456 + 0.957794i \(0.407191\pi\)
\(594\) 1.00000 0.0410305
\(595\) 0 0
\(596\) 18.0000 0.737309
\(597\) 10.0000 0.409273
\(598\) −30.0000 −1.22679
\(599\) −3.00000 −0.122577 −0.0612883 0.998120i \(-0.519521\pi\)
−0.0612883 + 0.998120i \(0.519521\pi\)
\(600\) −4.00000 −0.163299
\(601\) −44.0000 −1.79480 −0.897399 0.441221i \(-0.854546\pi\)
−0.897399 + 0.441221i \(0.854546\pi\)
\(602\) 0 0
\(603\) 5.00000 0.203616
\(604\) 9.00000 0.366205
\(605\) 3.00000 0.121967
\(606\) 8.00000 0.324978
\(607\) 33.0000 1.33943 0.669714 0.742619i \(-0.266417\pi\)
0.669714 + 0.742619i \(0.266417\pi\)
\(608\) −6.00000 −0.243332
\(609\) 0 0
\(610\) 15.0000 0.607332
\(611\) 54.0000 2.18461
\(612\) −5.00000 −0.202113
\(613\) −41.0000 −1.65597 −0.827987 0.560747i \(-0.810514\pi\)
−0.827987 + 0.560747i \(0.810514\pi\)
\(614\) −18.0000 −0.726421
\(615\) 15.0000 0.604858
\(616\) 0 0
\(617\) −6.00000 −0.241551 −0.120775 0.992680i \(-0.538538\pi\)
−0.120775 + 0.992680i \(0.538538\pi\)
\(618\) 2.00000 0.0804518
\(619\) −29.0000 −1.16561 −0.582804 0.812613i \(-0.698045\pi\)
−0.582804 + 0.812613i \(0.698045\pi\)
\(620\) 12.0000 0.481932
\(621\) 5.00000 0.200643
\(622\) 21.0000 0.842023
\(623\) 0 0
\(624\) 6.00000 0.240192
\(625\) −29.0000 −1.16000
\(626\) 6.00000 0.239808
\(627\) −6.00000 −0.239617
\(628\) −24.0000 −0.957704
\(629\) 10.0000 0.398726
\(630\) 0 0
\(631\) 28.0000 1.11466 0.557331 0.830290i \(-0.311825\pi\)
0.557331 + 0.830290i \(0.311825\pi\)
\(632\) 1.00000 0.0397779
\(633\) 14.0000 0.556450
\(634\) −9.00000 −0.357436
\(635\) 15.0000 0.595257
\(636\) 2.00000 0.0793052
\(637\) 0 0
\(638\) −6.00000 −0.237542
\(639\) 4.00000 0.158238
\(640\) −3.00000 −0.118585
\(641\) −40.0000 −1.57991 −0.789953 0.613168i \(-0.789895\pi\)
−0.789953 + 0.613168i \(0.789895\pi\)
\(642\) 13.0000 0.513069
\(643\) 36.0000 1.41970 0.709851 0.704352i \(-0.248762\pi\)
0.709851 + 0.704352i \(0.248762\pi\)
\(644\) 0 0
\(645\) −30.0000 −1.18125
\(646\) 30.0000 1.18033
\(647\) 17.0000 0.668339 0.334169 0.942513i \(-0.391544\pi\)
0.334169 + 0.942513i \(0.391544\pi\)
\(648\) −1.00000 −0.0392837
\(649\) 12.0000 0.471041
\(650\) −24.0000 −0.941357
\(651\) 0 0
\(652\) −1.00000 −0.0391630
\(653\) −27.0000 −1.05659 −0.528296 0.849060i \(-0.677169\pi\)
−0.528296 + 0.849060i \(0.677169\pi\)
\(654\) 11.0000 0.430134
\(655\) −12.0000 −0.468879
\(656\) 5.00000 0.195217
\(657\) 12.0000 0.468165
\(658\) 0 0
\(659\) 31.0000 1.20759 0.603794 0.797140i \(-0.293655\pi\)
0.603794 + 0.797140i \(0.293655\pi\)
\(660\) −3.00000 −0.116775
\(661\) 4.00000 0.155582 0.0777910 0.996970i \(-0.475213\pi\)
0.0777910 + 0.996970i \(0.475213\pi\)
\(662\) −35.0000 −1.36031
\(663\) −30.0000 −1.16510
\(664\) −1.00000 −0.0388075
\(665\) 0 0
\(666\) 2.00000 0.0774984
\(667\) −30.0000 −1.16160
\(668\) −24.0000 −0.928588
\(669\) 16.0000 0.618596
\(670\) −15.0000 −0.579501
\(671\) 5.00000 0.193023
\(672\) 0 0
\(673\) −26.0000 −1.00223 −0.501113 0.865382i \(-0.667076\pi\)
−0.501113 + 0.865382i \(0.667076\pi\)
\(674\) 2.00000 0.0770371
\(675\) 4.00000 0.153960
\(676\) 23.0000 0.884615
\(677\) −18.0000 −0.691796 −0.345898 0.938272i \(-0.612426\pi\)
−0.345898 + 0.938272i \(0.612426\pi\)
\(678\) −10.0000 −0.384048
\(679\) 0 0
\(680\) 15.0000 0.575224
\(681\) 3.00000 0.114960
\(682\) 4.00000 0.153168
\(683\) −12.0000 −0.459167 −0.229584 0.973289i \(-0.573736\pi\)
−0.229584 + 0.973289i \(0.573736\pi\)
\(684\) 6.00000 0.229416
\(685\) −6.00000 −0.229248
\(686\) 0 0
\(687\) 26.0000 0.991962
\(688\) −10.0000 −0.381246
\(689\) 12.0000 0.457164
\(690\) −15.0000 −0.571040
\(691\) −27.0000 −1.02713 −0.513564 0.858051i \(-0.671675\pi\)
−0.513564 + 0.858051i \(0.671675\pi\)
\(692\) −12.0000 −0.456172
\(693\) 0 0
\(694\) −7.00000 −0.265716
\(695\) 66.0000 2.50352
\(696\) 6.00000 0.227429
\(697\) −25.0000 −0.946943
\(698\) 19.0000 0.719161
\(699\) −29.0000 −1.09688
\(700\) 0 0
\(701\) −10.0000 −0.377695 −0.188847 0.982006i \(-0.560475\pi\)
−0.188847 + 0.982006i \(0.560475\pi\)
\(702\) −6.00000 −0.226455
\(703\) −12.0000 −0.452589
\(704\) −1.00000 −0.0376889
\(705\) 27.0000 1.01688
\(706\) −24.0000 −0.903252
\(707\) 0 0
\(708\) −12.0000 −0.450988
\(709\) 20.0000 0.751116 0.375558 0.926799i \(-0.377451\pi\)
0.375558 + 0.926799i \(0.377451\pi\)
\(710\) −12.0000 −0.450352
\(711\) −1.00000 −0.0375029
\(712\) −6.00000 −0.224860
\(713\) 20.0000 0.749006
\(714\) 0 0
\(715\) −18.0000 −0.673162
\(716\) −8.00000 −0.298974
\(717\) 26.0000 0.970988
\(718\) 10.0000 0.373197
\(719\) 37.0000 1.37987 0.689934 0.723873i \(-0.257640\pi\)
0.689934 + 0.723873i \(0.257640\pi\)
\(720\) 3.00000 0.111803
\(721\) 0 0
\(722\) −17.0000 −0.632674
\(723\) −26.0000 −0.966950
\(724\) −22.0000 −0.817624
\(725\) −24.0000 −0.891338
\(726\) −1.00000 −0.0371135
\(727\) 22.0000 0.815935 0.407967 0.912996i \(-0.366238\pi\)
0.407967 + 0.912996i \(0.366238\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) −36.0000 −1.33242
\(731\) 50.0000 1.84932
\(732\) −5.00000 −0.184805
\(733\) −15.0000 −0.554038 −0.277019 0.960864i \(-0.589346\pi\)
−0.277019 + 0.960864i \(0.589346\pi\)
\(734\) 14.0000 0.516749
\(735\) 0 0
\(736\) −5.00000 −0.184302
\(737\) −5.00000 −0.184177
\(738\) −5.00000 −0.184053
\(739\) −10.0000 −0.367856 −0.183928 0.982940i \(-0.558881\pi\)
−0.183928 + 0.982940i \(0.558881\pi\)
\(740\) −6.00000 −0.220564
\(741\) 36.0000 1.32249
\(742\) 0 0
\(743\) 22.0000 0.807102 0.403551 0.914957i \(-0.367776\pi\)
0.403551 + 0.914957i \(0.367776\pi\)
\(744\) −4.00000 −0.146647
\(745\) 54.0000 1.97841
\(746\) 7.00000 0.256288
\(747\) 1.00000 0.0365881
\(748\) 5.00000 0.182818
\(749\) 0 0
\(750\) 3.00000 0.109545
\(751\) −2.00000 −0.0729810 −0.0364905 0.999334i \(-0.511618\pi\)
−0.0364905 + 0.999334i \(0.511618\pi\)
\(752\) 9.00000 0.328196
\(753\) 12.0000 0.437304
\(754\) 36.0000 1.31104
\(755\) 27.0000 0.982631
\(756\) 0 0
\(757\) −2.00000 −0.0726912 −0.0363456 0.999339i \(-0.511572\pi\)
−0.0363456 + 0.999339i \(0.511572\pi\)
\(758\) −13.0000 −0.472181
\(759\) −5.00000 −0.181489
\(760\) −18.0000 −0.652929
\(761\) 47.0000 1.70375 0.851874 0.523746i \(-0.175466\pi\)
0.851874 + 0.523746i \(0.175466\pi\)
\(762\) −5.00000 −0.181131
\(763\) 0 0
\(764\) 0 0
\(765\) −15.0000 −0.542326
\(766\) −8.00000 −0.289052
\(767\) −72.0000 −2.59977
\(768\) 1.00000 0.0360844
\(769\) 46.0000 1.65880 0.829401 0.558653i \(-0.188682\pi\)
0.829401 + 0.558653i \(0.188682\pi\)
\(770\) 0 0
\(771\) −14.0000 −0.504198
\(772\) 6.00000 0.215945
\(773\) 11.0000 0.395643 0.197821 0.980238i \(-0.436613\pi\)
0.197821 + 0.980238i \(0.436613\pi\)
\(774\) 10.0000 0.359443
\(775\) 16.0000 0.574737
\(776\) −9.00000 −0.323081
\(777\) 0 0
\(778\) 21.0000 0.752886
\(779\) 30.0000 1.07486
\(780\) 18.0000 0.644503
\(781\) −4.00000 −0.143131
\(782\) 25.0000 0.893998
\(783\) −6.00000 −0.214423
\(784\) 0 0
\(785\) −72.0000 −2.56979
\(786\) 4.00000 0.142675
\(787\) −50.0000 −1.78231 −0.891154 0.453701i \(-0.850103\pi\)
−0.891154 + 0.453701i \(0.850103\pi\)
\(788\) −24.0000 −0.854965
\(789\) 24.0000 0.854423
\(790\) 3.00000 0.106735
\(791\) 0 0
\(792\) 1.00000 0.0355335
\(793\) −30.0000 −1.06533
\(794\) −26.0000 −0.922705
\(795\) 6.00000 0.212798
\(796\) 10.0000 0.354441
\(797\) −53.0000 −1.87736 −0.938678 0.344795i \(-0.887949\pi\)
−0.938678 + 0.344795i \(0.887949\pi\)
\(798\) 0 0
\(799\) −45.0000 −1.59199
\(800\) −4.00000 −0.141421
\(801\) 6.00000 0.212000
\(802\) 6.00000 0.211867
\(803\) −12.0000 −0.423471
\(804\) 5.00000 0.176336
\(805\) 0 0
\(806\) −24.0000 −0.845364
\(807\) −9.00000 −0.316815
\(808\) 8.00000 0.281439
\(809\) 15.0000 0.527372 0.263686 0.964609i \(-0.415062\pi\)
0.263686 + 0.964609i \(0.415062\pi\)
\(810\) −3.00000 −0.105409
\(811\) 42.0000 1.47482 0.737410 0.675446i \(-0.236049\pi\)
0.737410 + 0.675446i \(0.236049\pi\)
\(812\) 0 0
\(813\) −24.0000 −0.841717
\(814\) −2.00000 −0.0701000
\(815\) −3.00000 −0.105085
\(816\) −5.00000 −0.175035
\(817\) −60.0000 −2.09913
\(818\) −10.0000 −0.349642
\(819\) 0 0
\(820\) 15.0000 0.523823
\(821\) −8.00000 −0.279202 −0.139601 0.990208i \(-0.544582\pi\)
−0.139601 + 0.990208i \(0.544582\pi\)
\(822\) 2.00000 0.0697580
\(823\) 10.0000 0.348578 0.174289 0.984695i \(-0.444237\pi\)
0.174289 + 0.984695i \(0.444237\pi\)
\(824\) 2.00000 0.0696733
\(825\) −4.00000 −0.139262
\(826\) 0 0
\(827\) −15.0000 −0.521601 −0.260801 0.965393i \(-0.583986\pi\)
−0.260801 + 0.965393i \(0.583986\pi\)
\(828\) 5.00000 0.173762
\(829\) −54.0000 −1.87550 −0.937749 0.347314i \(-0.887094\pi\)
−0.937749 + 0.347314i \(0.887094\pi\)
\(830\) −3.00000 −0.104132
\(831\) 18.0000 0.624413
\(832\) 6.00000 0.208013
\(833\) 0 0
\(834\) −22.0000 −0.761798
\(835\) −72.0000 −2.49166
\(836\) −6.00000 −0.207514
\(837\) 4.00000 0.138260
\(838\) 20.0000 0.690889
\(839\) −9.00000 −0.310715 −0.155357 0.987858i \(-0.549653\pi\)
−0.155357 + 0.987858i \(0.549653\pi\)
\(840\) 0 0
\(841\) 7.00000 0.241379
\(842\) −8.00000 −0.275698
\(843\) 11.0000 0.378860
\(844\) 14.0000 0.481900
\(845\) 69.0000 2.37367
\(846\) −9.00000 −0.309426
\(847\) 0 0
\(848\) 2.00000 0.0686803
\(849\) −6.00000 −0.205919
\(850\) 20.0000 0.685994
\(851\) −10.0000 −0.342796
\(852\) 4.00000 0.137038
\(853\) 15.0000 0.513590 0.256795 0.966466i \(-0.417333\pi\)
0.256795 + 0.966466i \(0.417333\pi\)
\(854\) 0 0
\(855\) 18.0000 0.615587
\(856\) 13.0000 0.444331
\(857\) 3.00000 0.102478 0.0512390 0.998686i \(-0.483683\pi\)
0.0512390 + 0.998686i \(0.483683\pi\)
\(858\) 6.00000 0.204837
\(859\) −45.0000 −1.53538 −0.767690 0.640821i \(-0.778594\pi\)
−0.767690 + 0.640821i \(0.778594\pi\)
\(860\) −30.0000 −1.02299
\(861\) 0 0
\(862\) −22.0000 −0.749323
\(863\) −39.0000 −1.32758 −0.663788 0.747921i \(-0.731052\pi\)
−0.663788 + 0.747921i \(0.731052\pi\)
\(864\) −1.00000 −0.0340207
\(865\) −36.0000 −1.22404
\(866\) −11.0000 −0.373795
\(867\) 8.00000 0.271694
\(868\) 0 0
\(869\) 1.00000 0.0339227
\(870\) 18.0000 0.610257
\(871\) 30.0000 1.01651
\(872\) 11.0000 0.372507
\(873\) 9.00000 0.304604
\(874\) −30.0000 −1.01477
\(875\) 0 0
\(876\) 12.0000 0.405442
\(877\) −3.00000 −0.101303 −0.0506514 0.998716i \(-0.516130\pi\)
−0.0506514 + 0.998716i \(0.516130\pi\)
\(878\) 5.00000 0.168742
\(879\) 8.00000 0.269833
\(880\) −3.00000 −0.101130
\(881\) −26.0000 −0.875962 −0.437981 0.898984i \(-0.644306\pi\)
−0.437981 + 0.898984i \(0.644306\pi\)
\(882\) 0 0
\(883\) −13.0000 −0.437485 −0.218742 0.975783i \(-0.570195\pi\)
−0.218742 + 0.975783i \(0.570195\pi\)
\(884\) −30.0000 −1.00901
\(885\) −36.0000 −1.21013
\(886\) −2.00000 −0.0671913
\(887\) −24.0000 −0.805841 −0.402921 0.915235i \(-0.632005\pi\)
−0.402921 + 0.915235i \(0.632005\pi\)
\(888\) 2.00000 0.0671156
\(889\) 0 0
\(890\) −18.0000 −0.603361
\(891\) −1.00000 −0.0335013
\(892\) 16.0000 0.535720
\(893\) 54.0000 1.80704
\(894\) −18.0000 −0.602010
\(895\) −24.0000 −0.802232
\(896\) 0 0
\(897\) 30.0000 1.00167
\(898\) −2.00000 −0.0667409
\(899\) −24.0000 −0.800445
\(900\) 4.00000 0.133333
\(901\) −10.0000 −0.333148
\(902\) 5.00000 0.166482
\(903\) 0 0
\(904\) −10.0000 −0.332595
\(905\) −66.0000 −2.19391
\(906\) −9.00000 −0.299005
\(907\) 23.0000 0.763702 0.381851 0.924224i \(-0.375287\pi\)
0.381851 + 0.924224i \(0.375287\pi\)
\(908\) 3.00000 0.0995585
\(909\) −8.00000 −0.265343
\(910\) 0 0
\(911\) 51.0000 1.68971 0.844853 0.534999i \(-0.179688\pi\)
0.844853 + 0.534999i \(0.179688\pi\)
\(912\) 6.00000 0.198680
\(913\) −1.00000 −0.0330952
\(914\) 26.0000 0.860004
\(915\) −15.0000 −0.495885
\(916\) 26.0000 0.859064
\(917\) 0 0
\(918\) 5.00000 0.165025
\(919\) −17.0000 −0.560778 −0.280389 0.959886i \(-0.590464\pi\)
−0.280389 + 0.959886i \(0.590464\pi\)
\(920\) −15.0000 −0.494535
\(921\) 18.0000 0.593120
\(922\) 10.0000 0.329332
\(923\) 24.0000 0.789970
\(924\) 0 0
\(925\) −8.00000 −0.263038
\(926\) −28.0000 −0.920137
\(927\) −2.00000 −0.0656886
\(928\) 6.00000 0.196960
\(929\) −26.0000 −0.853032 −0.426516 0.904480i \(-0.640259\pi\)
−0.426516 + 0.904480i \(0.640259\pi\)
\(930\) −12.0000 −0.393496
\(931\) 0 0
\(932\) −29.0000 −0.949927
\(933\) −21.0000 −0.687509
\(934\) −16.0000 −0.523536
\(935\) 15.0000 0.490552
\(936\) −6.00000 −0.196116
\(937\) 30.0000 0.980057 0.490029 0.871706i \(-0.336986\pi\)
0.490029 + 0.871706i \(0.336986\pi\)
\(938\) 0 0
\(939\) −6.00000 −0.195803
\(940\) 27.0000 0.880643
\(941\) 6.00000 0.195594 0.0977972 0.995206i \(-0.468820\pi\)
0.0977972 + 0.995206i \(0.468820\pi\)
\(942\) 24.0000 0.781962
\(943\) 25.0000 0.814112
\(944\) −12.0000 −0.390567
\(945\) 0 0
\(946\) −10.0000 −0.325128
\(947\) −30.0000 −0.974869 −0.487435 0.873160i \(-0.662067\pi\)
−0.487435 + 0.873160i \(0.662067\pi\)
\(948\) −1.00000 −0.0324785
\(949\) 72.0000 2.33722
\(950\) −24.0000 −0.778663
\(951\) 9.00000 0.291845
\(952\) 0 0
\(953\) −21.0000 −0.680257 −0.340128 0.940379i \(-0.610471\pi\)
−0.340128 + 0.940379i \(0.610471\pi\)
\(954\) −2.00000 −0.0647524
\(955\) 0 0
\(956\) 26.0000 0.840900
\(957\) 6.00000 0.193952
\(958\) −12.0000 −0.387702
\(959\) 0 0
\(960\) 3.00000 0.0968246
\(961\) −15.0000 −0.483871
\(962\) 12.0000 0.386896
\(963\) −13.0000 −0.418919
\(964\) −26.0000 −0.837404
\(965\) 18.0000 0.579441
\(966\) 0 0
\(967\) −21.0000 −0.675314 −0.337657 0.941269i \(-0.609634\pi\)
−0.337657 + 0.941269i \(0.609634\pi\)
\(968\) −1.00000 −0.0321412
\(969\) −30.0000 −0.963739
\(970\) −27.0000 −0.866918
\(971\) −34.0000 −1.09111 −0.545556 0.838074i \(-0.683681\pi\)
−0.545556 + 0.838074i \(0.683681\pi\)
\(972\) 1.00000 0.0320750
\(973\) 0 0
\(974\) 4.00000 0.128168
\(975\) 24.0000 0.768615
\(976\) −5.00000 −0.160046
\(977\) −28.0000 −0.895799 −0.447900 0.894084i \(-0.647828\pi\)
−0.447900 + 0.894084i \(0.647828\pi\)
\(978\) 1.00000 0.0319765
\(979\) −6.00000 −0.191761
\(980\) 0 0
\(981\) −11.0000 −0.351203
\(982\) 3.00000 0.0957338
\(983\) −59.0000 −1.88181 −0.940904 0.338674i \(-0.890022\pi\)
−0.940904 + 0.338674i \(0.890022\pi\)
\(984\) −5.00000 −0.159394
\(985\) −72.0000 −2.29411
\(986\) −30.0000 −0.955395
\(987\) 0 0
\(988\) 36.0000 1.14531
\(989\) −50.0000 −1.58991
\(990\) 3.00000 0.0953463
\(991\) 22.0000 0.698853 0.349427 0.936964i \(-0.386376\pi\)
0.349427 + 0.936964i \(0.386376\pi\)
\(992\) −4.00000 −0.127000
\(993\) 35.0000 1.11069
\(994\) 0 0
\(995\) 30.0000 0.951064
\(996\) 1.00000 0.0316862
\(997\) 14.0000 0.443384 0.221692 0.975117i \(-0.428842\pi\)
0.221692 + 0.975117i \(0.428842\pi\)
\(998\) −12.0000 −0.379853
\(999\) −2.00000 −0.0632772
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 3234.2.a.o.1.1 1
3.2 odd 2 9702.2.a.be.1.1 1
7.2 even 3 462.2.i.a.67.1 2
7.4 even 3 462.2.i.a.331.1 yes 2
7.6 odd 2 3234.2.a.a.1.1 1
21.2 odd 6 1386.2.k.j.991.1 2
21.11 odd 6 1386.2.k.j.793.1 2
21.20 even 2 9702.2.a.ce.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
462.2.i.a.67.1 2 7.2 even 3
462.2.i.a.331.1 yes 2 7.4 even 3
1386.2.k.j.793.1 2 21.11 odd 6
1386.2.k.j.991.1 2 21.2 odd 6
3234.2.a.a.1.1 1 7.6 odd 2
3234.2.a.o.1.1 1 1.1 even 1 trivial
9702.2.a.be.1.1 1 3.2 odd 2
9702.2.a.ce.1.1 1 21.20 even 2