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Properties

Label 690.2.a.d.1.1
Level $690$
Weight $2$
Character 690.1
Self dual yes
Analytic conductor $5.510$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

Related objects

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Newspace parameters

Level: \( N \) \(=\) \( 690 = 2 \cdot 3 \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 690.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(5.50967773947\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 690.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{6} +4.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{6} +4.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} +2.00000 q^{11} -1.00000 q^{12} +4.00000 q^{13} -4.00000 q^{14} -1.00000 q^{15} +1.00000 q^{16} -6.00000 q^{17} -1.00000 q^{18} -4.00000 q^{19} +1.00000 q^{20} -4.00000 q^{21} -2.00000 q^{22} -1.00000 q^{23} +1.00000 q^{24} +1.00000 q^{25} -4.00000 q^{26} -1.00000 q^{27} +4.00000 q^{28} +8.00000 q^{29} +1.00000 q^{30} +8.00000 q^{31} -1.00000 q^{32} -2.00000 q^{33} +6.00000 q^{34} +4.00000 q^{35} +1.00000 q^{36} -10.0000 q^{37} +4.00000 q^{38} -4.00000 q^{39} -1.00000 q^{40} +6.00000 q^{41} +4.00000 q^{42} +6.00000 q^{43} +2.00000 q^{44} +1.00000 q^{45} +1.00000 q^{46} -4.00000 q^{47} -1.00000 q^{48} +9.00000 q^{49} -1.00000 q^{50} +6.00000 q^{51} +4.00000 q^{52} -14.0000 q^{53} +1.00000 q^{54} +2.00000 q^{55} -4.00000 q^{56} +4.00000 q^{57} -8.00000 q^{58} +4.00000 q^{59} -1.00000 q^{60} +6.00000 q^{61} -8.00000 q^{62} +4.00000 q^{63} +1.00000 q^{64} +4.00000 q^{65} +2.00000 q^{66} +14.0000 q^{67} -6.00000 q^{68} +1.00000 q^{69} -4.00000 q^{70} +10.0000 q^{71} -1.00000 q^{72} +14.0000 q^{73} +10.0000 q^{74} -1.00000 q^{75} -4.00000 q^{76} +8.00000 q^{77} +4.00000 q^{78} -8.00000 q^{79} +1.00000 q^{80} +1.00000 q^{81} -6.00000 q^{82} -4.00000 q^{83} -4.00000 q^{84} -6.00000 q^{85} -6.00000 q^{86} -8.00000 q^{87} -2.00000 q^{88} -1.00000 q^{90} +16.0000 q^{91} -1.00000 q^{92} -8.00000 q^{93} +4.00000 q^{94} -4.00000 q^{95} +1.00000 q^{96} -8.00000 q^{97} -9.00000 q^{98} +2.00000 q^{99} +O(q^{100})\)

Display 100 coefficients 10 coefficients

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
\(5\) 1.00000 0.447214
\(6\) 1.00000 0.408248
\(7\) 4.00000 1.51186 0.755929 − 0.654654i \(-0.227186\pi\)
0.755929 + 0.654654i \(0.227186\pi\)
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) −1.00000 −0.316228
\(11\) 2.00000 0.603023 0.301511 − 0.953463i \(-0.402509\pi\)
0.301511 + 0.953463i \(0.402509\pi\)
\(12\) −1.00000 −0.288675
\(13\) 4.00000 1.10940 0.554700 − 0.832050i \(-0.312833\pi\)
0.554700 + 0.832050i \(0.312833\pi\)
\(14\) −4.00000 −1.06904
\(15\) −1.00000 −0.258199
\(16\) 1.00000 0.250000
\(17\) −6.00000 −1.45521 −0.727607 − 0.685994i \(-0.759367\pi\)
−0.727607 + 0.685994i \(0.759367\pi\)
\(18\) −1.00000 −0.235702
\(19\) −4.00000 −0.917663 −0.458831 − 0.888523i \(-0.651732\pi\)
−0.458831 + 0.888523i \(0.651732\pi\)
\(20\) 1.00000 0.223607
\(21\) −4.00000 −0.872872
\(22\) −2.00000 −0.426401
\(23\) −1.00000 −0.208514
\(24\) 1.00000 0.204124
\(25\) 1.00000 0.200000
\(26\) −4.00000 −0.784465
\(27\) −1.00000 −0.192450
\(28\) 4.00000 0.755929
\(29\) 8.00000 1.48556 0.742781 − 0.669534i \(-0.233506\pi\)
0.742781 + 0.669534i \(0.233506\pi\)
\(30\) 1.00000 0.182574
\(31\) 8.00000 1.43684 0.718421 − 0.695608i \(-0.244865\pi\)
0.718421 + 0.695608i \(0.244865\pi\)
\(32\) −1.00000 −0.176777
\(33\) −2.00000 −0.348155
\(34\) 6.00000 1.02899
\(35\) 4.00000 0.676123
\(36\) 1.00000 0.166667
\(37\) −10.0000 −1.64399 −0.821995 − 0.569495i \(-0.807139\pi\)
−0.821995 + 0.569495i \(0.807139\pi\)
\(38\) 4.00000 0.648886
\(39\) −4.00000 −0.640513
\(40\) −1.00000 −0.158114
\(41\) 6.00000 0.937043 0.468521 − 0.883452i \(-0.344787\pi\)
0.468521 + 0.883452i \(0.344787\pi\)
\(42\) 4.00000 0.617213
\(43\) 6.00000 0.914991 0.457496 − 0.889212i \(-0.348747\pi\)
0.457496 + 0.889212i \(0.348747\pi\)
\(44\) 2.00000 0.301511
\(45\) 1.00000 0.149071
\(46\) 1.00000 0.147442
\(47\) −4.00000 −0.583460 −0.291730 − 0.956501i \(-0.594231\pi\)
−0.291730 + 0.956501i \(0.594231\pi\)
\(48\) −1.00000 −0.144338
\(49\) 9.00000 1.28571
\(50\) −1.00000 −0.141421
\(51\) 6.00000 0.840168
\(52\) 4.00000 0.554700
\(53\) −14.0000 −1.92305 −0.961524 − 0.274721i \(-0.911414\pi\)
−0.961524 + 0.274721i \(0.911414\pi\)
\(54\) 1.00000 0.136083
\(55\) 2.00000 0.269680
\(56\) −4.00000 −0.534522
\(57\) 4.00000 0.529813
\(58\) −8.00000 −1.05045
\(59\) 4.00000 0.520756 0.260378 − 0.965507i \(-0.416153\pi\)
0.260378 + 0.965507i \(0.416153\pi\)
\(60\) −1.00000 −0.129099
\(61\) 6.00000 0.768221 0.384111 − 0.923287i \(-0.374508\pi\)
0.384111 + 0.923287i \(0.374508\pi\)
\(62\) −8.00000 −1.01600
\(63\) 4.00000 0.503953
\(64\) 1.00000 0.125000
\(65\) 4.00000 0.496139
\(66\) 2.00000 0.246183
\(67\) 14.0000 1.71037 0.855186 − 0.518321i \(-0.173443\pi\)
0.855186 + 0.518321i \(0.173443\pi\)
\(68\) −6.00000 −0.727607
\(69\) 1.00000 0.120386
\(70\) −4.00000 −0.478091
\(71\) 10.0000 1.18678 0.593391 − 0.804914i \(-0.297789\pi\)
0.593391 + 0.804914i \(0.297789\pi\)
\(72\) −1.00000 −0.117851
\(73\) 14.0000 1.63858 0.819288 − 0.573382i \(-0.194369\pi\)
0.819288 + 0.573382i \(0.194369\pi\)
\(74\) 10.0000 1.16248
\(75\) −1.00000 −0.115470
\(76\) −4.00000 −0.458831
\(77\) 8.00000 0.911685
\(78\) 4.00000 0.452911
\(79\) −8.00000 −0.900070 −0.450035 − 0.893011i \(-0.648589\pi\)
−0.450035 + 0.893011i \(0.648589\pi\)
\(80\) 1.00000 0.111803
\(81\) 1.00000 0.111111
\(82\) −6.00000 −0.662589
\(83\) −4.00000 −0.439057 −0.219529 − 0.975606i \(-0.570452\pi\)
−0.219529 + 0.975606i \(0.570452\pi\)
\(84\) −4.00000 −0.436436
\(85\) −6.00000 −0.650791
\(86\) −6.00000 −0.646997
\(87\) −8.00000 −0.857690
\(88\) −2.00000 −0.213201
\(89\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(90\) −1.00000 −0.105409
\(91\) 16.0000 1.67726
\(92\) −1.00000 −0.104257
\(93\) −8.00000 −0.829561
\(94\) 4.00000 0.412568
\(95\) −4.00000 −0.410391
\(96\) 1.00000 0.102062
\(97\) −8.00000 −0.812277 −0.406138 − 0.913812i \(-0.633125\pi\)
−0.406138 + 0.913812i \(0.633125\pi\)
\(98\) −9.00000 −0.909137
\(99\) 2.00000 0.201008
\(100\) 1.00000 0.100000
\(101\) −8.00000 −0.796030 −0.398015 − 0.917379i \(-0.630301\pi\)
−0.398015 + 0.917379i \(0.630301\pi\)
\(102\) −6.00000 −0.594089
\(103\) 4.00000 0.394132 0.197066 − 0.980390i \(-0.436859\pi\)
0.197066 + 0.980390i \(0.436859\pi\)
\(104\) −4.00000 −0.392232
\(105\) −4.00000 −0.390360
\(106\) 14.0000 1.35980
\(107\) 8.00000 0.773389 0.386695 − 0.922208i \(-0.373617\pi\)
0.386695 + 0.922208i \(0.373617\pi\)
\(108\) −1.00000 −0.0962250
\(109\) −14.0000 −1.34096 −0.670478 − 0.741929i \(-0.733911\pi\)
−0.670478 + 0.741929i \(0.733911\pi\)
\(110\) −2.00000 −0.190693
\(111\) 10.0000 0.949158
\(112\) 4.00000 0.377964
\(113\) −10.0000 −0.940721 −0.470360 − 0.882474i \(-0.655876\pi\)
−0.470360 + 0.882474i \(0.655876\pi\)
\(114\) −4.00000 −0.374634
\(115\) −1.00000 −0.0932505
\(116\) 8.00000 0.742781
\(117\) 4.00000 0.369800
\(118\) −4.00000 −0.368230
\(119\) −24.0000 −2.20008
\(120\) 1.00000 0.0912871
\(121\) −7.00000 −0.636364
\(122\) −6.00000 −0.543214
\(123\) −6.00000 −0.541002
\(124\) 8.00000 0.718421
\(125\) 1.00000 0.0894427
\(126\) −4.00000 −0.356348
\(127\) 2.00000 0.177471 0.0887357 − 0.996055i \(-0.471717\pi\)
0.0887357 + 0.996055i \(0.471717\pi\)
\(128\) −1.00000 −0.0883883
\(129\) −6.00000 −0.528271
\(130\) −4.00000 −0.350823
\(131\) 16.0000 1.39793 0.698963 − 0.715158i \(-0.253645\pi\)
0.698963 + 0.715158i \(0.253645\pi\)
\(132\) −2.00000 −0.174078
\(133\) −16.0000 −1.38738
\(134\) −14.0000 −1.20942
\(135\) −1.00000 −0.0860663
\(136\) 6.00000 0.514496
\(137\) 6.00000 0.512615 0.256307 − 0.966595i \(-0.417494\pi\)
0.256307 + 0.966595i \(0.417494\pi\)
\(138\) −1.00000 −0.0851257
\(139\) −20.0000 −1.69638 −0.848189 − 0.529694i \(-0.822307\pi\)
−0.848189 + 0.529694i \(0.822307\pi\)
\(140\) 4.00000 0.338062
\(141\) 4.00000 0.336861
\(142\) −10.0000 −0.839181
\(143\) 8.00000 0.668994
\(144\) 1.00000 0.0833333
\(145\) 8.00000 0.664364
\(146\) −14.0000 −1.15865
\(147\) −9.00000 −0.742307
\(148\) −10.0000 −0.821995
\(149\) −10.0000 −0.819232 −0.409616 − 0.912258i \(-0.634337\pi\)
−0.409616 + 0.912258i \(0.634337\pi\)
\(150\) 1.00000 0.0816497
\(151\) 20.0000 1.62758 0.813788 − 0.581161i \(-0.197401\pi\)
0.813788 + 0.581161i \(0.197401\pi\)
\(152\) 4.00000 0.324443
\(153\) −6.00000 −0.485071
\(154\) −8.00000 −0.644658
\(155\) 8.00000 0.642575
\(156\) −4.00000 −0.320256
\(157\) −2.00000 −0.159617 −0.0798087 − 0.996810i \(-0.525431\pi\)
−0.0798087 + 0.996810i \(0.525431\pi\)
\(158\) 8.00000 0.636446
\(159\) 14.0000 1.11027
\(160\) −1.00000 −0.0790569
\(161\) −4.00000 −0.315244
\(162\) −1.00000 −0.0785674
\(163\) −8.00000 −0.626608 −0.313304 − 0.949653i \(-0.601436\pi\)
−0.313304 + 0.949653i \(0.601436\pi\)
\(164\) 6.00000 0.468521
\(165\) −2.00000 −0.155700
\(166\) 4.00000 0.310460
\(167\) −8.00000 −0.619059 −0.309529 − 0.950890i \(-0.600171\pi\)
−0.309529 + 0.950890i \(0.600171\pi\)
\(168\) 4.00000 0.308607
\(169\) 3.00000 0.230769
\(170\) 6.00000 0.460179
\(171\) −4.00000 −0.305888
\(172\) 6.00000 0.457496
\(173\) 10.0000 0.760286 0.380143 − 0.924928i \(-0.375875\pi\)
0.380143 + 0.924928i \(0.375875\pi\)
\(174\) 8.00000 0.606478
\(175\) 4.00000 0.302372
\(176\) 2.00000 0.150756
\(177\) −4.00000 −0.300658
\(178\) 0 0
\(179\) −4.00000 −0.298974 −0.149487 − 0.988764i \(-0.547762\pi\)
−0.149487 + 0.988764i \(0.547762\pi\)
\(180\) 1.00000 0.0745356
\(181\) −2.00000 −0.148659 −0.0743294 − 0.997234i \(-0.523682\pi\)
−0.0743294 + 0.997234i \(0.523682\pi\)
\(182\) −16.0000 −1.18600
\(183\) −6.00000 −0.443533
\(184\) 1.00000 0.0737210
\(185\) −10.0000 −0.735215
\(186\) 8.00000 0.586588
\(187\) −12.0000 −0.877527
\(188\) −4.00000 −0.291730
\(189\) −4.00000 −0.290957
\(190\) 4.00000 0.290191
\(191\) −16.0000 −1.15772 −0.578860 − 0.815427i \(-0.696502\pi\)
−0.578860 + 0.815427i \(0.696502\pi\)
\(192\) −1.00000 −0.0721688
\(193\) −22.0000 −1.58359 −0.791797 − 0.610784i \(-0.790854\pi\)
−0.791797 + 0.610784i \(0.790854\pi\)
\(194\) 8.00000 0.574367
\(195\) −4.00000 −0.286446
\(196\) 9.00000 0.642857
\(197\) −22.0000 −1.56744 −0.783718 − 0.621117i \(-0.786679\pi\)
−0.783718 + 0.621117i \(0.786679\pi\)
\(198\) −2.00000 −0.142134
\(199\) 16.0000 1.13421 0.567105 − 0.823646i \(-0.308063\pi\)
0.567105 + 0.823646i \(0.308063\pi\)
\(200\) −1.00000 −0.0707107
\(201\) −14.0000 −0.987484
\(202\) 8.00000 0.562878
\(203\) 32.0000 2.24596
\(204\) 6.00000 0.420084
\(205\) 6.00000 0.419058
\(206\) −4.00000 −0.278693
\(207\) −1.00000 −0.0695048
\(208\) 4.00000 0.277350
\(209\) −8.00000 −0.553372
\(210\) 4.00000 0.276026
\(211\) −20.0000 −1.37686 −0.688428 − 0.725304i \(-0.741699\pi\)
−0.688428 + 0.725304i \(0.741699\pi\)
\(212\) −14.0000 −0.961524
\(213\) −10.0000 −0.685189
\(214\) −8.00000 −0.546869
\(215\) 6.00000 0.409197
\(216\) 1.00000 0.0680414
\(217\) 32.0000 2.17230
\(218\) 14.0000 0.948200
\(219\) −14.0000 −0.946032
\(220\) 2.00000 0.134840
\(221\) −24.0000 −1.61441
\(222\) −10.0000 −0.671156
\(223\) 14.0000 0.937509 0.468755 − 0.883328i \(-0.344703\pi\)
0.468755 + 0.883328i \(0.344703\pi\)
\(224\) −4.00000 −0.267261
\(225\) 1.00000 0.0666667
\(226\) 10.0000 0.665190
\(227\) 8.00000 0.530979 0.265489 − 0.964114i \(-0.414466\pi\)
0.265489 + 0.964114i \(0.414466\pi\)
\(228\) 4.00000 0.264906
\(229\) −22.0000 −1.45380 −0.726900 − 0.686743i \(-0.759040\pi\)
−0.726900 + 0.686743i \(0.759040\pi\)
\(230\) 1.00000 0.0659380
\(231\) −8.00000 −0.526361
\(232\) −8.00000 −0.525226
\(233\) −6.00000 −0.393073 −0.196537 − 0.980497i \(-0.562969\pi\)
−0.196537 + 0.980497i \(0.562969\pi\)
\(234\) −4.00000 −0.261488
\(235\) −4.00000 −0.260931
\(236\) 4.00000 0.260378
\(237\) 8.00000 0.519656
\(238\) 24.0000 1.55569
\(239\) 26.0000 1.68180 0.840900 − 0.541190i \(-0.182026\pi\)
0.840900 + 0.541190i \(0.182026\pi\)
\(240\) −1.00000 −0.0645497
\(241\) 10.0000 0.644157 0.322078 − 0.946713i \(-0.395619\pi\)
0.322078 + 0.946713i \(0.395619\pi\)
\(242\) 7.00000 0.449977
\(243\) −1.00000 −0.0641500
\(244\) 6.00000 0.384111
\(245\) 9.00000 0.574989
\(246\) 6.00000 0.382546
\(247\) −16.0000 −1.01806
\(248\) −8.00000 −0.508001
\(249\) 4.00000 0.253490
\(250\) −1.00000 −0.0632456
\(251\) −2.00000 −0.126239 −0.0631194 − 0.998006i \(-0.520105\pi\)
−0.0631194 + 0.998006i \(0.520105\pi\)
\(252\) 4.00000 0.251976
\(253\) −2.00000 −0.125739
\(254\) −2.00000 −0.125491
\(255\) 6.00000 0.375735
\(256\) 1.00000 0.0625000
\(257\) 18.0000 1.12281 0.561405 − 0.827541i \(-0.310261\pi\)
0.561405 + 0.827541i \(0.310261\pi\)
\(258\) 6.00000 0.373544
\(259\) −40.0000 −2.48548
\(260\) 4.00000 0.248069
\(261\) 8.00000 0.495188
\(262\) −16.0000 −0.988483
\(263\) 16.0000 0.986602 0.493301 − 0.869859i \(-0.335790\pi\)
0.493301 + 0.869859i \(0.335790\pi\)
\(264\) 2.00000 0.123091
\(265\) −14.0000 −0.860013
\(266\) 16.0000 0.981023
\(267\) 0 0
\(268\) 14.0000 0.855186
\(269\) 12.0000 0.731653 0.365826 − 0.930683i \(-0.380786\pi\)
0.365826 + 0.930683i \(0.380786\pi\)
\(270\) 1.00000 0.0608581
\(271\) −8.00000 −0.485965 −0.242983 − 0.970031i \(-0.578126\pi\)
−0.242983 + 0.970031i \(0.578126\pi\)
\(272\) −6.00000 −0.363803
\(273\) −16.0000 −0.968364
\(274\) −6.00000 −0.362473
\(275\) 2.00000 0.120605
\(276\) 1.00000 0.0601929
\(277\) −12.0000 −0.721010 −0.360505 − 0.932757i \(-0.617396\pi\)
−0.360505 + 0.932757i \(0.617396\pi\)
\(278\) 20.0000 1.19952
\(279\) 8.00000 0.478947
\(280\) −4.00000 −0.239046
\(281\) 8.00000 0.477240 0.238620 − 0.971113i \(-0.423305\pi\)
0.238620 + 0.971113i \(0.423305\pi\)
\(282\) −4.00000 −0.238197
\(283\) −22.0000 −1.30776 −0.653882 − 0.756596i \(-0.726861\pi\)
−0.653882 + 0.756596i \(0.726861\pi\)
\(284\) 10.0000 0.593391
\(285\) 4.00000 0.236940
\(286\) −8.00000 −0.473050
\(287\) 24.0000 1.41668
\(288\) −1.00000 −0.0589256
\(289\) 19.0000 1.11765
\(290\) −8.00000 −0.469776
\(291\) 8.00000 0.468968
\(292\) 14.0000 0.819288
\(293\) 26.0000 1.51894 0.759468 − 0.650545i \(-0.225459\pi\)
0.759468 + 0.650545i \(0.225459\pi\)
\(294\) 9.00000 0.524891
\(295\) 4.00000 0.232889
\(296\) 10.0000 0.581238
\(297\) −2.00000 −0.116052
\(298\) 10.0000 0.579284
\(299\) −4.00000 −0.231326
\(300\) −1.00000 −0.0577350
\(301\) 24.0000 1.38334
\(302\) −20.0000 −1.15087
\(303\) 8.00000 0.459588
\(304\) −4.00000 −0.229416
\(305\) 6.00000 0.343559
\(306\) 6.00000 0.342997
\(307\) 4.00000 0.228292 0.114146 − 0.993464i \(-0.463587\pi\)
0.114146 + 0.993464i \(0.463587\pi\)
\(308\) 8.00000 0.455842
\(309\) −4.00000 −0.227552
\(310\) −8.00000 −0.454369
\(311\) −14.0000 −0.793867 −0.396934 − 0.917847i \(-0.629926\pi\)
−0.396934 + 0.917847i \(0.629926\pi\)
\(312\) 4.00000 0.226455
\(313\) −4.00000 −0.226093 −0.113047 − 0.993590i \(-0.536061\pi\)
−0.113047 + 0.993590i \(0.536061\pi\)
\(314\) 2.00000 0.112867
\(315\) 4.00000 0.225374
\(316\) −8.00000 −0.450035
\(317\) −18.0000 −1.01098 −0.505490 − 0.862832i \(-0.668688\pi\)
−0.505490 + 0.862832i \(0.668688\pi\)
\(318\) −14.0000 −0.785081
\(319\) 16.0000 0.895828
\(320\) 1.00000 0.0559017
\(321\) −8.00000 −0.446516
\(322\) 4.00000 0.222911
\(323\) 24.0000 1.33540
\(324\) 1.00000 0.0555556
\(325\) 4.00000 0.221880
\(326\) 8.00000 0.443079
\(327\) 14.0000 0.774202
\(328\) −6.00000 −0.331295
\(329\) −16.0000 −0.882109
\(330\) 2.00000 0.110096
\(331\) 12.0000 0.659580 0.329790 − 0.944054i \(-0.393022\pi\)
0.329790 + 0.944054i \(0.393022\pi\)
\(332\) −4.00000 −0.219529
\(333\) −10.0000 −0.547997
\(334\) 8.00000 0.437741
\(335\) 14.0000 0.764902
\(336\) −4.00000 −0.218218
\(337\) −20.0000 −1.08947 −0.544735 − 0.838608i \(-0.683370\pi\)
−0.544735 + 0.838608i \(0.683370\pi\)
\(338\) −3.00000 −0.163178
\(339\) 10.0000 0.543125
\(340\) −6.00000 −0.325396
\(341\) 16.0000 0.866449
\(342\) 4.00000 0.216295
\(343\) 8.00000 0.431959
\(344\) −6.00000 −0.323498
\(345\) 1.00000 0.0538382
\(346\) −10.0000 −0.537603
\(347\) 12.0000 0.644194 0.322097 − 0.946707i \(-0.395612\pi\)
0.322097 + 0.946707i \(0.395612\pi\)
\(348\) −8.00000 −0.428845
\(349\) −30.0000 −1.60586 −0.802932 − 0.596071i \(-0.796728\pi\)
−0.802932 + 0.596071i \(0.796728\pi\)
\(350\) −4.00000 −0.213809
\(351\) −4.00000 −0.213504
\(352\) −2.00000 −0.106600
\(353\) 14.0000 0.745145 0.372572 − 0.928003i \(-0.378476\pi\)
0.372572 + 0.928003i \(0.378476\pi\)
\(354\) 4.00000 0.212598
\(355\) 10.0000 0.530745
\(356\) 0 0
\(357\) 24.0000 1.27021
\(358\) 4.00000 0.211407
\(359\) −16.0000 −0.844448 −0.422224 − 0.906492i \(-0.638750\pi\)
−0.422224 + 0.906492i \(0.638750\pi\)
\(360\) −1.00000 −0.0527046
\(361\) −3.00000 −0.157895
\(362\) 2.00000 0.105118
\(363\) 7.00000 0.367405
\(364\) 16.0000 0.838628
\(365\) 14.0000 0.732793
\(366\) 6.00000 0.313625
\(367\) −12.0000 −0.626395 −0.313197 − 0.949688i \(-0.601400\pi\)
−0.313197 + 0.949688i \(0.601400\pi\)
\(368\) −1.00000 −0.0521286
\(369\) 6.00000 0.312348
\(370\) 10.0000 0.519875
\(371\) −56.0000 −2.90738
\(372\) −8.00000 −0.414781
\(373\) −18.0000 −0.932005 −0.466002 − 0.884783i \(-0.654306\pi\)
−0.466002 + 0.884783i \(0.654306\pi\)
\(374\) 12.0000 0.620505
\(375\) −1.00000 −0.0516398
\(376\) 4.00000 0.206284
\(377\) 32.0000 1.64808
\(378\) 4.00000 0.205738
\(379\) −16.0000 −0.821865 −0.410932 − 0.911666i \(-0.634797\pi\)
−0.410932 + 0.911666i \(0.634797\pi\)
\(380\) −4.00000 −0.205196
\(381\) −2.00000 −0.102463
\(382\) 16.0000 0.818631
\(383\) −24.0000 −1.22634 −0.613171 − 0.789950i \(-0.710106\pi\)
−0.613171 + 0.789950i \(0.710106\pi\)
\(384\) 1.00000 0.0510310
\(385\) 8.00000 0.407718
\(386\) 22.0000 1.11977
\(387\) 6.00000 0.304997
\(388\) −8.00000 −0.406138
\(389\) −14.0000 −0.709828 −0.354914 − 0.934899i \(-0.615490\pi\)
−0.354914 + 0.934899i \(0.615490\pi\)
\(390\) 4.00000 0.202548
\(391\) 6.00000 0.303433
\(392\) −9.00000 −0.454569
\(393\) −16.0000 −0.807093
\(394\) 22.0000 1.10834
\(395\) −8.00000 −0.402524
\(396\) 2.00000 0.100504
\(397\) −28.0000 −1.40528 −0.702640 − 0.711546i \(-0.747995\pi\)
−0.702640 + 0.711546i \(0.747995\pi\)
\(398\) −16.0000 −0.802008
\(399\) 16.0000 0.801002
\(400\) 1.00000 0.0500000
\(401\) 20.0000 0.998752 0.499376 − 0.866385i \(-0.333563\pi\)
0.499376 + 0.866385i \(0.333563\pi\)
\(402\) 14.0000 0.698257
\(403\) 32.0000 1.59403
\(404\) −8.00000 −0.398015
\(405\) 1.00000 0.0496904
\(406\) −32.0000 −1.58813
\(407\) −20.0000 −0.991363
\(408\) −6.00000 −0.297044
\(409\) −10.0000 −0.494468 −0.247234 − 0.968956i \(-0.579522\pi\)
−0.247234 + 0.968956i \(0.579522\pi\)
\(410\) −6.00000 −0.296319
\(411\) −6.00000 −0.295958
\(412\) 4.00000 0.197066
\(413\) 16.0000 0.787309
\(414\) 1.00000 0.0491473
\(415\) −4.00000 −0.196352
\(416\) −4.00000 −0.196116
\(417\) 20.0000 0.979404
\(418\) 8.00000 0.391293
\(419\) −30.0000 −1.46560 −0.732798 − 0.680446i \(-0.761786\pi\)
−0.732798 + 0.680446i \(0.761786\pi\)
\(420\) −4.00000 −0.195180
\(421\) 6.00000 0.292422 0.146211 − 0.989253i \(-0.453292\pi\)
0.146211 + 0.989253i \(0.453292\pi\)
\(422\) 20.0000 0.973585
\(423\) −4.00000 −0.194487
\(424\) 14.0000 0.679900
\(425\) −6.00000 −0.291043
\(426\) 10.0000 0.484502
\(427\) 24.0000 1.16144
\(428\) 8.00000 0.386695
\(429\) −8.00000 −0.386244
\(430\) −6.00000 −0.289346
\(431\) 8.00000 0.385346 0.192673 − 0.981263i \(-0.438284\pi\)
0.192673 + 0.981263i \(0.438284\pi\)
\(432\) −1.00000 −0.0481125
\(433\) 32.0000 1.53782 0.768911 − 0.639356i \(-0.220799\pi\)
0.768911 + 0.639356i \(0.220799\pi\)
\(434\) −32.0000 −1.53605
\(435\) −8.00000 −0.383571
\(436\) −14.0000 −0.670478
\(437\) 4.00000 0.191346
\(438\) 14.0000 0.668946
\(439\) −20.0000 −0.954548 −0.477274 − 0.878755i \(-0.658375\pi\)
−0.477274 + 0.878755i \(0.658375\pi\)
\(440\) −2.00000 −0.0953463
\(441\) 9.00000 0.428571
\(442\) 24.0000 1.14156
\(443\) −28.0000 −1.33032 −0.665160 − 0.746701i \(-0.731637\pi\)
−0.665160 + 0.746701i \(0.731637\pi\)
\(444\) 10.0000 0.474579
\(445\) 0 0
\(446\) −14.0000 −0.662919
\(447\) 10.0000 0.472984
\(448\) 4.00000 0.188982
\(449\) 30.0000 1.41579 0.707894 − 0.706319i \(-0.249646\pi\)
0.707894 + 0.706319i \(0.249646\pi\)
\(450\) −1.00000 −0.0471405
\(451\) 12.0000 0.565058
\(452\) −10.0000 −0.470360
\(453\) −20.0000 −0.939682
\(454\) −8.00000 −0.375459
\(455\) 16.0000 0.750092
\(456\) −4.00000 −0.187317
\(457\) 16.0000 0.748448 0.374224 − 0.927338i \(-0.377909\pi\)
0.374224 + 0.927338i \(0.377909\pi\)
\(458\) 22.0000 1.02799
\(459\) 6.00000 0.280056
\(460\) −1.00000 −0.0466252
\(461\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(462\) 8.00000 0.372194
\(463\) −10.0000 −0.464739 −0.232370 − 0.972628i \(-0.574648\pi\)
−0.232370 + 0.972628i \(0.574648\pi\)
\(464\) 8.00000 0.371391
\(465\) −8.00000 −0.370991
\(466\) 6.00000 0.277945
\(467\) −36.0000 −1.66588 −0.832941 − 0.553362i \(-0.813345\pi\)
−0.832941 + 0.553362i \(0.813345\pi\)
\(468\) 4.00000 0.184900
\(469\) 56.0000 2.58584
\(470\) 4.00000 0.184506
\(471\) 2.00000 0.0921551
\(472\) −4.00000 −0.184115
\(473\) 12.0000 0.551761
\(474\) −8.00000 −0.367452
\(475\) −4.00000 −0.183533
\(476\) −24.0000 −1.10004
\(477\) −14.0000 −0.641016
\(478\) −26.0000 −1.18921
\(479\) 24.0000 1.09659 0.548294 − 0.836286i \(-0.315277\pi\)
0.548294 + 0.836286i \(0.315277\pi\)
\(480\) 1.00000 0.0456435
\(481\) −40.0000 −1.82384
\(482\) −10.0000 −0.455488
\(483\) 4.00000 0.182006
\(484\) −7.00000 −0.318182
\(485\) −8.00000 −0.363261
\(486\) 1.00000 0.0453609
\(487\) −10.0000 −0.453143 −0.226572 − 0.973995i \(-0.572752\pi\)
−0.226572 + 0.973995i \(0.572752\pi\)
\(488\) −6.00000 −0.271607
\(489\) 8.00000 0.361773
\(490\) −9.00000 −0.406579
\(491\) −24.0000 −1.08310 −0.541552 − 0.840667i \(-0.682163\pi\)
−0.541552 + 0.840667i \(0.682163\pi\)
\(492\) −6.00000 −0.270501
\(493\) −48.0000 −2.16181
\(494\) 16.0000 0.719874
\(495\) 2.00000 0.0898933
\(496\) 8.00000 0.359211
\(497\) 40.0000 1.79425
\(498\) −4.00000 −0.179244
\(499\) 20.0000 0.895323 0.447661 − 0.894203i \(-0.352257\pi\)
0.447661 + 0.894203i \(0.352257\pi\)
\(500\) 1.00000 0.0447214
\(501\) 8.00000 0.357414
\(502\) 2.00000 0.0892644
\(503\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(504\) −4.00000 −0.178174
\(505\) −8.00000 −0.355995
\(506\) 2.00000 0.0889108
\(507\) −3.00000 −0.133235
\(508\) 2.00000 0.0887357
\(509\) 24.0000 1.06378 0.531891 − 0.846813i \(-0.321482\pi\)
0.531891 + 0.846813i \(0.321482\pi\)
\(510\) −6.00000 −0.265684
\(511\) 56.0000 2.47729
\(512\) −1.00000 −0.0441942
\(513\) 4.00000 0.176604
\(514\) −18.0000 −0.793946
\(515\) 4.00000 0.176261
\(516\) −6.00000 −0.264135
\(517\) −8.00000 −0.351840
\(518\) 40.0000 1.75750
\(519\) −10.0000 −0.438951
\(520\) −4.00000 −0.175412
\(521\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(522\) −8.00000 −0.350150
\(523\) −14.0000 −0.612177 −0.306089 − 0.952003i \(-0.599020\pi\)
−0.306089 + 0.952003i \(0.599020\pi\)
\(524\) 16.0000 0.698963
\(525\) −4.00000 −0.174574
\(526\) −16.0000 −0.697633
\(527\) −48.0000 −2.09091
\(528\) −2.00000 −0.0870388
\(529\) 1.00000 0.0434783
\(530\) 14.0000 0.608121
\(531\) 4.00000 0.173585
\(532\) −16.0000 −0.693688
\(533\) 24.0000 1.03956
\(534\) 0 0
\(535\) 8.00000 0.345870
\(536\) −14.0000 −0.604708
\(537\) 4.00000 0.172613
\(538\) −12.0000 −0.517357
\(539\) 18.0000 0.775315
\(540\) −1.00000 −0.0430331
\(541\) 10.0000 0.429934 0.214967 − 0.976621i \(-0.431036\pi\)
0.214967 + 0.976621i \(0.431036\pi\)
\(542\) 8.00000 0.343629
\(543\) 2.00000 0.0858282
\(544\) 6.00000 0.257248
\(545\) −14.0000 −0.599694
\(546\) 16.0000 0.684737
\(547\) 28.0000 1.19719 0.598597 − 0.801050i \(-0.295725\pi\)
0.598597 + 0.801050i \(0.295725\pi\)
\(548\) 6.00000 0.256307
\(549\) 6.00000 0.256074
\(550\) −2.00000 −0.0852803
\(551\) −32.0000 −1.36325
\(552\) −1.00000 −0.0425628
\(553\) −32.0000 −1.36078
\(554\) 12.0000 0.509831
\(555\) 10.0000 0.424476
\(556\) −20.0000 −0.848189
\(557\) −2.00000 −0.0847427 −0.0423714 − 0.999102i \(-0.513491\pi\)
−0.0423714 + 0.999102i \(0.513491\pi\)
\(558\) −8.00000 −0.338667
\(559\) 24.0000 1.01509
\(560\) 4.00000 0.169031
\(561\) 12.0000 0.506640
\(562\) −8.00000 −0.337460
\(563\) −12.0000 −0.505740 −0.252870 − 0.967500i \(-0.581374\pi\)
−0.252870 + 0.967500i \(0.581374\pi\)
\(564\) 4.00000 0.168430
\(565\) −10.0000 −0.420703
\(566\) 22.0000 0.924729
\(567\) 4.00000 0.167984
\(568\) −10.0000 −0.419591
\(569\) −44.0000 −1.84458 −0.922288 − 0.386503i \(-0.873683\pi\)
−0.922288 + 0.386503i \(0.873683\pi\)
\(570\) −4.00000 −0.167542
\(571\) −24.0000 −1.00437 −0.502184 − 0.864761i \(-0.667470\pi\)
−0.502184 + 0.864761i \(0.667470\pi\)
\(572\) 8.00000 0.334497
\(573\) 16.0000 0.668410
\(574\) −24.0000 −1.00174
\(575\) −1.00000 −0.0417029
\(576\) 1.00000 0.0416667
\(577\) −34.0000 −1.41544 −0.707719 − 0.706494i \(-0.750276\pi\)
−0.707719 + 0.706494i \(0.750276\pi\)
\(578\) −19.0000 −0.790296
\(579\) 22.0000 0.914289
\(580\) 8.00000 0.332182
\(581\) −16.0000 −0.663792
\(582\) −8.00000 −0.331611
\(583\) −28.0000 −1.15964
\(584\) −14.0000 −0.579324
\(585\) 4.00000 0.165380
\(586\) −26.0000 −1.07405
\(587\) 12.0000 0.495293 0.247647 − 0.968850i \(-0.420343\pi\)
0.247647 + 0.968850i \(0.420343\pi\)
\(588\) −9.00000 −0.371154
\(589\) −32.0000 −1.31854
\(590\) −4.00000 −0.164677
\(591\) 22.0000 0.904959
\(592\) −10.0000 −0.410997
\(593\) 18.0000 0.739171 0.369586 − 0.929197i \(-0.379500\pi\)
0.369586 + 0.929197i \(0.379500\pi\)
\(594\) 2.00000 0.0820610
\(595\) −24.0000 −0.983904
\(596\) −10.0000 −0.409616
\(597\) −16.0000 −0.654836
\(598\) 4.00000 0.163572
\(599\) 14.0000 0.572024 0.286012 − 0.958226i \(-0.407670\pi\)
0.286012 + 0.958226i \(0.407670\pi\)
\(600\) 1.00000 0.0408248
\(601\) −26.0000 −1.06056 −0.530281 − 0.847822i \(-0.677914\pi\)
−0.530281 + 0.847822i \(0.677914\pi\)
\(602\) −24.0000 −0.978167
\(603\) 14.0000 0.570124
\(604\) 20.0000 0.813788
\(605\) −7.00000 −0.284590
\(606\) −8.00000 −0.324978
\(607\) 30.0000 1.21766 0.608831 − 0.793300i \(-0.291639\pi\)
0.608831 + 0.793300i \(0.291639\pi\)
\(608\) 4.00000 0.162221
\(609\) −32.0000 −1.29671
\(610\) −6.00000 −0.242933
\(611\) −16.0000 −0.647291
\(612\) −6.00000 −0.242536
\(613\) −14.0000 −0.565455 −0.282727 − 0.959200i \(-0.591239\pi\)
−0.282727 + 0.959200i \(0.591239\pi\)
\(614\) −4.00000 −0.161427
\(615\) −6.00000 −0.241943
\(616\) −8.00000 −0.322329
\(617\) 18.0000 0.724653 0.362326 − 0.932051i \(-0.381983\pi\)
0.362326 + 0.932051i \(0.381983\pi\)
\(618\) 4.00000 0.160904
\(619\) −20.0000 −0.803868 −0.401934 − 0.915669i \(-0.631662\pi\)
−0.401934 + 0.915669i \(0.631662\pi\)
\(620\) 8.00000 0.321288
\(621\) 1.00000 0.0401286
\(622\) 14.0000 0.561349
\(623\) 0 0
\(624\) −4.00000 −0.160128
\(625\) 1.00000 0.0400000
\(626\) 4.00000 0.159872
\(627\) 8.00000 0.319489
\(628\) −2.00000 −0.0798087
\(629\) 60.0000 2.39236
\(630\) −4.00000 −0.159364
\(631\) −8.00000 −0.318475 −0.159237 − 0.987240i \(-0.550904\pi\)
−0.159237 + 0.987240i \(0.550904\pi\)
\(632\) 8.00000 0.318223
\(633\) 20.0000 0.794929
\(634\) 18.0000 0.714871
\(635\) 2.00000 0.0793676
\(636\) 14.0000 0.555136
\(637\) 36.0000 1.42637
\(638\) −16.0000 −0.633446
\(639\) 10.0000 0.395594
\(640\) −1.00000 −0.0395285
\(641\) −24.0000 −0.947943 −0.473972 − 0.880540i \(-0.657180\pi\)
−0.473972 + 0.880540i \(0.657180\pi\)
\(642\) 8.00000 0.315735
\(643\) 34.0000 1.34083 0.670415 − 0.741987i \(-0.266116\pi\)
0.670415 + 0.741987i \(0.266116\pi\)
\(644\) −4.00000 −0.157622
\(645\) −6.00000 −0.236250
\(646\) −24.0000 −0.944267
\(647\) −28.0000 −1.10079 −0.550397 − 0.834903i \(-0.685524\pi\)
−0.550397 + 0.834903i \(0.685524\pi\)
\(648\) −1.00000 −0.0392837
\(649\) 8.00000 0.314027
\(650\) −4.00000 −0.156893
\(651\) −32.0000 −1.25418
\(652\) −8.00000 −0.313304
\(653\) −42.0000 −1.64359 −0.821794 − 0.569785i \(-0.807026\pi\)
−0.821794 + 0.569785i \(0.807026\pi\)
\(654\) −14.0000 −0.547443
\(655\) 16.0000 0.625172
\(656\) 6.00000 0.234261
\(657\) 14.0000 0.546192
\(658\) 16.0000 0.623745
\(659\) −6.00000 −0.233727 −0.116863 − 0.993148i \(-0.537284\pi\)
−0.116863 + 0.993148i \(0.537284\pi\)
\(660\) −2.00000 −0.0778499
\(661\) −22.0000 −0.855701 −0.427850 − 0.903850i \(-0.640729\pi\)
−0.427850 + 0.903850i \(0.640729\pi\)
\(662\) −12.0000 −0.466393
\(663\) 24.0000 0.932083
\(664\) 4.00000 0.155230
\(665\) −16.0000 −0.620453
\(666\) 10.0000 0.387492
\(667\) −8.00000 −0.309761
\(668\) −8.00000 −0.309529
\(669\) −14.0000 −0.541271
\(670\) −14.0000 −0.540867
\(671\) 12.0000 0.463255
\(672\) 4.00000 0.154303
\(673\) −34.0000 −1.31060 −0.655302 − 0.755367i \(-0.727459\pi\)
−0.655302 + 0.755367i \(0.727459\pi\)
\(674\) 20.0000 0.770371
\(675\) −1.00000 −0.0384900
\(676\) 3.00000 0.115385
\(677\) −26.0000 −0.999261 −0.499631 − 0.866239i \(-0.666531\pi\)
−0.499631 + 0.866239i \(0.666531\pi\)
\(678\) −10.0000 −0.384048
\(679\) −32.0000 −1.22805
\(680\) 6.00000 0.230089
\(681\) −8.00000 −0.306561
\(682\) −16.0000 −0.612672
\(683\) 12.0000 0.459167 0.229584 − 0.973289i \(-0.426264\pi\)
0.229584 + 0.973289i \(0.426264\pi\)
\(684\) −4.00000 −0.152944
\(685\) 6.00000 0.229248
\(686\) −8.00000 −0.305441
\(687\) 22.0000 0.839352
\(688\) 6.00000 0.228748
\(689\) −56.0000 −2.13343
\(690\) −1.00000 −0.0380693
\(691\) −4.00000 −0.152167 −0.0760836 − 0.997101i \(-0.524242\pi\)
−0.0760836 + 0.997101i \(0.524242\pi\)
\(692\) 10.0000 0.380143
\(693\) 8.00000 0.303895
\(694\) −12.0000 −0.455514
\(695\) −20.0000 −0.758643
\(696\) 8.00000 0.303239
\(697\) −36.0000 −1.36360
\(698\) 30.0000 1.13552
\(699\) 6.00000 0.226941
\(700\) 4.00000 0.151186
\(701\) −6.00000 −0.226617 −0.113308 − 0.993560i \(-0.536145\pi\)
−0.113308 + 0.993560i \(0.536145\pi\)
\(702\) 4.00000 0.150970
\(703\) 40.0000 1.50863
\(704\) 2.00000 0.0753778
\(705\) 4.00000 0.150649
\(706\) −14.0000 −0.526897
\(707\) −32.0000 −1.20348
\(708\) −4.00000 −0.150329
\(709\) 34.0000 1.27690 0.638448 − 0.769665i \(-0.279577\pi\)
0.638448 + 0.769665i \(0.279577\pi\)
\(710\) −10.0000 −0.375293
\(711\) −8.00000 −0.300023
\(712\) 0 0
\(713\) −8.00000 −0.299602
\(714\) −24.0000 −0.898177
\(715\) 8.00000 0.299183
\(716\) −4.00000 −0.149487
\(717\) −26.0000 −0.970988
\(718\) 16.0000 0.597115
\(719\) 46.0000 1.71551 0.857755 − 0.514058i \(-0.171858\pi\)
0.857755 + 0.514058i \(0.171858\pi\)
\(720\) 1.00000 0.0372678
\(721\) 16.0000 0.595871
\(722\) 3.00000 0.111648
\(723\) −10.0000 −0.371904
\(724\) −2.00000 −0.0743294
\(725\) 8.00000 0.297113
\(726\) −7.00000 −0.259794
\(727\) 32.0000 1.18681 0.593407 − 0.804902i \(-0.297782\pi\)
0.593407 + 0.804902i \(0.297782\pi\)
\(728\) −16.0000 −0.592999
\(729\) 1.00000 0.0370370
\(730\) −14.0000 −0.518163
\(731\) −36.0000 −1.33151
\(732\) −6.00000 −0.221766
\(733\) 14.0000 0.517102 0.258551 − 0.965998i \(-0.416755\pi\)
0.258551 + 0.965998i \(0.416755\pi\)
\(734\) 12.0000 0.442928
\(735\) −9.00000 −0.331970
\(736\) 1.00000 0.0368605
\(737\) 28.0000 1.03139
\(738\) −6.00000 −0.220863
\(739\) −4.00000 −0.147142 −0.0735712 − 0.997290i \(-0.523440\pi\)
−0.0735712 + 0.997290i \(0.523440\pi\)
\(740\) −10.0000 −0.367607
\(741\) 16.0000 0.587775
\(742\) 56.0000 2.05582
\(743\) 32.0000 1.17397 0.586983 − 0.809599i \(-0.300316\pi\)
0.586983 + 0.809599i \(0.300316\pi\)
\(744\) 8.00000 0.293294
\(745\) −10.0000 −0.366372
\(746\) 18.0000 0.659027
\(747\) −4.00000 −0.146352
\(748\) −12.0000 −0.438763
\(749\) 32.0000 1.16925
\(750\) 1.00000 0.0365148
\(751\) −16.0000 −0.583848 −0.291924 − 0.956441i \(-0.594295\pi\)
−0.291924 + 0.956441i \(0.594295\pi\)
\(752\) −4.00000 −0.145865
\(753\) 2.00000 0.0728841
\(754\) −32.0000 −1.16537
\(755\) 20.0000 0.727875
\(756\) −4.00000 −0.145479
\(757\) 34.0000 1.23575 0.617876 − 0.786276i \(-0.287994\pi\)
0.617876 + 0.786276i \(0.287994\pi\)
\(758\) 16.0000 0.581146
\(759\) 2.00000 0.0725954
\(760\) 4.00000 0.145095
\(761\) −6.00000 −0.217500 −0.108750 − 0.994069i \(-0.534685\pi\)
−0.108750 + 0.994069i \(0.534685\pi\)
\(762\) 2.00000 0.0724524
\(763\) −56.0000 −2.02734
\(764\) −16.0000 −0.578860
\(765\) −6.00000 −0.216930
\(766\) 24.0000 0.867155
\(767\) 16.0000 0.577727
\(768\) −1.00000 −0.0360844
\(769\) 14.0000 0.504853 0.252426 − 0.967616i \(-0.418771\pi\)
0.252426 + 0.967616i \(0.418771\pi\)
\(770\) −8.00000 −0.288300
\(771\) −18.0000 −0.648254
\(772\) −22.0000 −0.791797
\(773\) 42.0000 1.51064 0.755318 − 0.655359i \(-0.227483\pi\)
0.755318 + 0.655359i \(0.227483\pi\)
\(774\) −6.00000 −0.215666
\(775\) 8.00000 0.287368
\(776\) 8.00000 0.287183
\(777\) 40.0000 1.43499
\(778\) 14.0000 0.501924
\(779\) −24.0000 −0.859889
\(780\) −4.00000 −0.143223
\(781\) 20.0000 0.715656
\(782\) −6.00000 −0.214560
\(783\) −8.00000 −0.285897
\(784\) 9.00000 0.321429
\(785\) −2.00000 −0.0713831
\(786\) 16.0000 0.570701
\(787\) 50.0000 1.78231 0.891154 − 0.453701i \(-0.149897\pi\)
0.891154 + 0.453701i \(0.149897\pi\)
\(788\) −22.0000 −0.783718
\(789\) −16.0000 −0.569615
\(790\) 8.00000 0.284627
\(791\) −40.0000 −1.42224
\(792\) −2.00000 −0.0710669
\(793\) 24.0000 0.852265
\(794\) 28.0000 0.993683
\(795\) 14.0000 0.496529
\(796\) 16.0000 0.567105
\(797\) 54.0000 1.91278 0.956389 − 0.292096i \(-0.0943526\pi\)
0.956389 + 0.292096i \(0.0943526\pi\)
\(798\) −16.0000 −0.566394
\(799\) 24.0000 0.849059
\(800\) −1.00000 −0.0353553
\(801\) 0 0
\(802\) −20.0000 −0.706225
\(803\) 28.0000 0.988099
\(804\) −14.0000 −0.493742
\(805\) −4.00000 −0.140981
\(806\) −32.0000 −1.12715
\(807\) −12.0000 −0.422420
\(808\) 8.00000 0.281439
\(809\) −6.00000 −0.210949 −0.105474 − 0.994422i \(-0.533636\pi\)
−0.105474 + 0.994422i \(0.533636\pi\)
\(810\) −1.00000 −0.0351364
\(811\) −12.0000 −0.421377 −0.210688 − 0.977553i \(-0.567571\pi\)
−0.210688 + 0.977553i \(0.567571\pi\)
\(812\) 32.0000 1.12298
\(813\) 8.00000 0.280572
\(814\) 20.0000 0.701000
\(815\) −8.00000 −0.280228
\(816\) 6.00000 0.210042
\(817\) −24.0000 −0.839654
\(818\) 10.0000 0.349642
\(819\) 16.0000 0.559085
\(820\) 6.00000 0.209529
\(821\) 12.0000 0.418803 0.209401 − 0.977830i \(-0.432848\pi\)
0.209401 + 0.977830i \(0.432848\pi\)
\(822\) 6.00000 0.209274
\(823\) 42.0000 1.46403 0.732014 − 0.681290i \(-0.238581\pi\)
0.732014 + 0.681290i \(0.238581\pi\)
\(824\) −4.00000 −0.139347
\(825\) −2.00000 −0.0696311
\(826\) −16.0000 −0.556711
\(827\) −48.0000 −1.66912 −0.834562 − 0.550914i \(-0.814279\pi\)
−0.834562 + 0.550914i \(0.814279\pi\)
\(828\) −1.00000 −0.0347524
\(829\) −38.0000 −1.31979 −0.659897 − 0.751356i \(-0.729400\pi\)
−0.659897 + 0.751356i \(0.729400\pi\)
\(830\) 4.00000 0.138842
\(831\) 12.0000 0.416275
\(832\) 4.00000 0.138675
\(833\) −54.0000 −1.87099
\(834\) −20.0000 −0.692543
\(835\) −8.00000 −0.276851
\(836\) −8.00000 −0.276686
\(837\) −8.00000 −0.276520
\(838\) 30.0000 1.03633
\(839\) −12.0000 −0.414286 −0.207143 − 0.978311i \(-0.566417\pi\)
−0.207143 + 0.978311i \(0.566417\pi\)
\(840\) 4.00000 0.138013
\(841\) 35.0000 1.20690
\(842\) −6.00000 −0.206774
\(843\) −8.00000 −0.275535
\(844\) −20.0000 −0.688428
\(845\) 3.00000 0.103203
\(846\) 4.00000 0.137523
\(847\) −28.0000 −0.962091
\(848\) −14.0000 −0.480762
\(849\) 22.0000 0.755038
\(850\) 6.00000 0.205798
\(851\) 10.0000 0.342796
\(852\) −10.0000 −0.342594
\(853\) −4.00000 −0.136957 −0.0684787 − 0.997653i \(-0.521815\pi\)
−0.0684787 + 0.997653i \(0.521815\pi\)
\(854\) −24.0000 −0.821263
\(855\) −4.00000 −0.136797
\(856\) −8.00000 −0.273434
\(857\) 6.00000 0.204956 0.102478 − 0.994735i \(-0.467323\pi\)
0.102478 + 0.994735i \(0.467323\pi\)
\(858\) 8.00000 0.273115
\(859\) −4.00000 −0.136478 −0.0682391 − 0.997669i \(-0.521738\pi\)
−0.0682391 + 0.997669i \(0.521738\pi\)
\(860\) 6.00000 0.204598
\(861\) −24.0000 −0.817918
\(862\) −8.00000 −0.272481
\(863\) −36.0000 −1.22545 −0.612727 − 0.790295i \(-0.709928\pi\)
−0.612727 + 0.790295i \(0.709928\pi\)
\(864\) 1.00000 0.0340207
\(865\) 10.0000 0.340010
\(866\) −32.0000 −1.08740
\(867\) −19.0000 −0.645274
\(868\) 32.0000 1.08615
\(869\) −16.0000 −0.542763
\(870\) 8.00000 0.271225
\(871\) 56.0000 1.89749
\(872\) 14.0000 0.474100
\(873\) −8.00000 −0.270759
\(874\) −4.00000 −0.135302
\(875\) 4.00000 0.135225
\(876\) −14.0000 −0.473016
\(877\) 20.0000 0.675352 0.337676 − 0.941262i \(-0.390359\pi\)
0.337676 + 0.941262i \(0.390359\pi\)
\(878\) 20.0000 0.674967
\(879\) −26.0000 −0.876958
\(880\) 2.00000 0.0674200
\(881\) −48.0000 −1.61716 −0.808581 − 0.588386i \(-0.799764\pi\)
−0.808581 + 0.588386i \(0.799764\pi\)
\(882\) −9.00000 −0.303046
\(883\) 8.00000 0.269221 0.134611 − 0.990899i \(-0.457022\pi\)
0.134611 + 0.990899i \(0.457022\pi\)
\(884\) −24.0000 −0.807207
\(885\) −4.00000 −0.134459
\(886\) 28.0000 0.940678
\(887\) 12.0000 0.402921 0.201460 − 0.979497i \(-0.435431\pi\)
0.201460 + 0.979497i \(0.435431\pi\)
\(888\) −10.0000 −0.335578
\(889\) 8.00000 0.268311
\(890\) 0 0
\(891\) 2.00000 0.0670025
\(892\) 14.0000 0.468755
\(893\) 16.0000 0.535420
\(894\) −10.0000 −0.334450
\(895\) −4.00000 −0.133705
\(896\) −4.00000 −0.133631
\(897\) 4.00000 0.133556
\(898\) −30.0000 −1.00111
\(899\) 64.0000 2.13452
\(900\) 1.00000 0.0333333
\(901\) 84.0000 2.79845
\(902\) −12.0000 −0.399556
\(903\) −24.0000 −0.798670
\(904\) 10.0000 0.332595
\(905\) −2.00000 −0.0664822
\(906\) 20.0000 0.664455
\(907\) 46.0000 1.52740 0.763702 − 0.645568i \(-0.223379\pi\)
0.763702 + 0.645568i \(0.223379\pi\)
\(908\) 8.00000 0.265489
\(909\) −8.00000 −0.265343
\(910\) −16.0000 −0.530395
\(911\) −36.0000 −1.19273 −0.596367 − 0.802712i \(-0.703390\pi\)
−0.596367 + 0.802712i \(0.703390\pi\)
\(912\) 4.00000 0.132453
\(913\) −8.00000 −0.264761
\(914\) −16.0000 −0.529233
\(915\) −6.00000 −0.198354
\(916\) −22.0000 −0.726900
\(917\) 64.0000 2.11347
\(918\) −6.00000 −0.198030
\(919\) −16.0000 −0.527791 −0.263896 − 0.964551i \(-0.585007\pi\)
−0.263896 + 0.964551i \(0.585007\pi\)
\(920\) 1.00000 0.0329690
\(921\) −4.00000 −0.131804
\(922\) 0 0
\(923\) 40.0000 1.31662
\(924\) −8.00000 −0.263181
\(925\) −10.0000 −0.328798
\(926\) 10.0000 0.328620
\(927\) 4.00000 0.131377
\(928\) −8.00000 −0.262613
\(929\) −18.0000 −0.590561 −0.295280 − 0.955411i \(-0.595413\pi\)
−0.295280 + 0.955411i \(0.595413\pi\)
\(930\) 8.00000 0.262330
\(931\) −36.0000 −1.17985
\(932\) −6.00000 −0.196537
\(933\) 14.0000 0.458339
\(934\) 36.0000 1.17796
\(935\) −12.0000 −0.392442
\(936\) −4.00000 −0.130744
\(937\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(938\) −56.0000 −1.82846
\(939\) 4.00000 0.130535
\(940\) −4.00000 −0.130466
\(941\) 38.0000 1.23876 0.619382 − 0.785090i \(-0.287383\pi\)
0.619382 + 0.785090i \(0.287383\pi\)
\(942\) −2.00000 −0.0651635
\(943\) −6.00000 −0.195387
\(944\) 4.00000 0.130189
\(945\) −4.00000 −0.130120
\(946\) −12.0000 −0.390154
\(947\) −20.0000 −0.649913 −0.324956 − 0.945729i \(-0.605350\pi\)
−0.324956 + 0.945729i \(0.605350\pi\)
\(948\) 8.00000 0.259828
\(949\) 56.0000 1.81784
\(950\) 4.00000 0.129777
\(951\) 18.0000 0.583690
\(952\) 24.0000 0.777844
\(953\) 38.0000 1.23094 0.615470 − 0.788160i \(-0.288966\pi\)
0.615470 + 0.788160i \(0.288966\pi\)
\(954\) 14.0000 0.453267
\(955\) −16.0000 −0.517748
\(956\) 26.0000 0.840900
\(957\) −16.0000 −0.517207
\(958\) −24.0000 −0.775405
\(959\) 24.0000 0.775000
\(960\) −1.00000 −0.0322749
\(961\) 33.0000 1.06452
\(962\) 40.0000 1.28965
\(963\) 8.00000 0.257796
\(964\) 10.0000 0.322078
\(965\) −22.0000 −0.708205
\(966\) −4.00000 −0.128698
\(967\) 22.0000 0.707472 0.353736 − 0.935345i \(-0.384911\pi\)
0.353736 + 0.935345i \(0.384911\pi\)
\(968\) 7.00000 0.224989
\(969\) −24.0000 −0.770991
\(970\) 8.00000 0.256865
\(971\) 6.00000 0.192549 0.0962746 − 0.995355i \(-0.469307\pi\)
0.0962746 + 0.995355i \(0.469307\pi\)
\(972\) −1.00000 −0.0320750
\(973\) −80.0000 −2.56468
\(974\) 10.0000 0.320421
\(975\) −4.00000 −0.128103
\(976\) 6.00000 0.192055
\(977\) 10.0000 0.319928 0.159964 − 0.987123i \(-0.448862\pi\)
0.159964 + 0.987123i \(0.448862\pi\)
\(978\) −8.00000 −0.255812
\(979\) 0 0
\(980\) 9.00000 0.287494
\(981\) −14.0000 −0.446986
\(982\) 24.0000 0.765871
\(983\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(984\) 6.00000 0.191273
\(985\) −22.0000 −0.700978
\(986\) 48.0000 1.52863
\(987\) 16.0000 0.509286
\(988\) −16.0000 −0.509028
\(989\) −6.00000 −0.190789
\(990\) −2.00000 −0.0635642
\(991\) 4.00000 0.127064 0.0635321 − 0.997980i \(-0.479763\pi\)
0.0635321 + 0.997980i \(0.479763\pi\)
\(992\) −8.00000 −0.254000
\(993\) −12.0000 −0.380808
\(994\) −40.0000 −1.26872
\(995\) 16.0000 0.507234
\(996\) 4.00000 0.126745
\(997\) 4.00000 0.126681 0.0633406 − 0.997992i \(-0.479825\pi\)
0.0633406 + 0.997992i \(0.479825\pi\)
\(998\) −20.0000 −0.633089
\(999\) 10.0000 0.316386
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 690.2.a.d.1.1 ✓ 1
3.2 odd 2 2070.2.a.o.1.1 1
4.3 odd 2 5520.2.a.bb.1.1 1
5.2 odd 4 3450.2.d.s.2899.1 2
5.3 odd 4 3450.2.d.s.2899.2 2
5.4 even 2 3450.2.a.u.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
690.2.a.d.1.1 ✓ 1 1.1 even 1 trivial
2070.2.a.o.1.1 1 3.2 odd 2
3450.2.a.u.1.1 1 5.4 even 2
3450.2.d.s.2899.1 2 5.2 odd 4
3450.2.d.s.2899.2 2 5.3 odd 4
5520.2.a.bb.1.1 1 4.3 odd 2