Properties

Label 6930.2.a.bt.1.1
Level $6930$
Weight $2$
Character 6930.1
Self dual yes
Analytic conductor $55.336$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 6930 = 2 \cdot 3^{2} \cdot 5 \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6930.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(55.3363286007\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{12})^+\)
Defining polynomial: \(x^{2} - 3\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 2310)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.73205\) of defining polynomial
Character \(\chi\) \(=\) 6930.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{7} -1.00000 q^{8} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{7} -1.00000 q^{8} -1.00000 q^{10} -1.00000 q^{11} -3.46410 q^{13} -1.00000 q^{14} +1.00000 q^{16} -3.46410 q^{17} +1.00000 q^{20} +1.00000 q^{22} -5.46410 q^{23} +1.00000 q^{25} +3.46410 q^{26} +1.00000 q^{28} -2.00000 q^{29} -6.92820 q^{31} -1.00000 q^{32} +3.46410 q^{34} +1.00000 q^{35} +0.535898 q^{37} -1.00000 q^{40} -4.92820 q^{41} +10.9282 q^{43} -1.00000 q^{44} +5.46410 q^{46} +6.92820 q^{47} +1.00000 q^{49} -1.00000 q^{50} -3.46410 q^{52} +0.928203 q^{53} -1.00000 q^{55} -1.00000 q^{56} +2.00000 q^{58} -10.9282 q^{59} +8.92820 q^{61} +6.92820 q^{62} +1.00000 q^{64} -3.46410 q^{65} +2.53590 q^{67} -3.46410 q^{68} -1.00000 q^{70} +4.00000 q^{71} +12.9282 q^{73} -0.535898 q^{74} -1.00000 q^{77} +8.00000 q^{79} +1.00000 q^{80} +4.92820 q^{82} +6.92820 q^{83} -3.46410 q^{85} -10.9282 q^{86} +1.00000 q^{88} +3.46410 q^{89} -3.46410 q^{91} -5.46410 q^{92} -6.92820 q^{94} +15.8564 q^{97} -1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} + 2 q^{4} + 2 q^{5} + 2 q^{7} - 2 q^{8} + O(q^{10}) \) \( 2 q - 2 q^{2} + 2 q^{4} + 2 q^{5} + 2 q^{7} - 2 q^{8} - 2 q^{10} - 2 q^{11} - 2 q^{14} + 2 q^{16} + 2 q^{20} + 2 q^{22} - 4 q^{23} + 2 q^{25} + 2 q^{28} - 4 q^{29} - 2 q^{32} + 2 q^{35} + 8 q^{37} - 2 q^{40} + 4 q^{41} + 8 q^{43} - 2 q^{44} + 4 q^{46} + 2 q^{49} - 2 q^{50} - 12 q^{53} - 2 q^{55} - 2 q^{56} + 4 q^{58} - 8 q^{59} + 4 q^{61} + 2 q^{64} + 12 q^{67} - 2 q^{70} + 8 q^{71} + 12 q^{73} - 8 q^{74} - 2 q^{77} + 16 q^{79} + 2 q^{80} - 4 q^{82} - 8 q^{86} + 2 q^{88} - 4 q^{92} + 4 q^{97} - 2 q^{98} + O(q^{100}) \)

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) 0 0
\(4\) 1.00000 0.500000
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) 1.00000 0.377964
\(8\) −1.00000 −0.353553
\(9\) 0 0
\(10\) −1.00000 −0.316228
\(11\) −1.00000 −0.301511
\(12\) 0 0
\(13\) −3.46410 −0.960769 −0.480384 0.877058i \(-0.659503\pi\)
−0.480384 + 0.877058i \(0.659503\pi\)
\(14\) −1.00000 −0.267261
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −3.46410 −0.840168 −0.420084 0.907485i \(-0.637999\pi\)
−0.420084 + 0.907485i \(0.637999\pi\)
\(18\) 0 0
\(19\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(20\) 1.00000 0.223607
\(21\) 0 0
\(22\) 1.00000 0.213201
\(23\) −5.46410 −1.13934 −0.569672 0.821872i \(-0.692930\pi\)
−0.569672 + 0.821872i \(0.692930\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 3.46410 0.679366
\(27\) 0 0
\(28\) 1.00000 0.188982
\(29\) −2.00000 −0.371391 −0.185695 0.982607i \(-0.559454\pi\)
−0.185695 + 0.982607i \(0.559454\pi\)
\(30\) 0 0
\(31\) −6.92820 −1.24434 −0.622171 0.782881i \(-0.713749\pi\)
−0.622171 + 0.782881i \(0.713749\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0 0
\(34\) 3.46410 0.594089
\(35\) 1.00000 0.169031
\(36\) 0 0
\(37\) 0.535898 0.0881012 0.0440506 0.999029i \(-0.485974\pi\)
0.0440506 + 0.999029i \(0.485974\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) −1.00000 −0.158114
\(41\) −4.92820 −0.769656 −0.384828 0.922988i \(-0.625739\pi\)
−0.384828 + 0.922988i \(0.625739\pi\)
\(42\) 0 0
\(43\) 10.9282 1.66654 0.833268 0.552870i \(-0.186467\pi\)
0.833268 + 0.552870i \(0.186467\pi\)
\(44\) −1.00000 −0.150756
\(45\) 0 0
\(46\) 5.46410 0.805638
\(47\) 6.92820 1.01058 0.505291 0.862949i \(-0.331385\pi\)
0.505291 + 0.862949i \(0.331385\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) −1.00000 −0.141421
\(51\) 0 0
\(52\) −3.46410 −0.480384
\(53\) 0.928203 0.127499 0.0637493 0.997966i \(-0.479694\pi\)
0.0637493 + 0.997966i \(0.479694\pi\)
\(54\) 0 0
\(55\) −1.00000 −0.134840
\(56\) −1.00000 −0.133631
\(57\) 0 0
\(58\) 2.00000 0.262613
\(59\) −10.9282 −1.42273 −0.711365 0.702822i \(-0.751923\pi\)
−0.711365 + 0.702822i \(0.751923\pi\)
\(60\) 0 0
\(61\) 8.92820 1.14314 0.571570 0.820554i \(-0.306335\pi\)
0.571570 + 0.820554i \(0.306335\pi\)
\(62\) 6.92820 0.879883
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) −3.46410 −0.429669
\(66\) 0 0
\(67\) 2.53590 0.309809 0.154905 0.987929i \(-0.450493\pi\)
0.154905 + 0.987929i \(0.450493\pi\)
\(68\) −3.46410 −0.420084
\(69\) 0 0
\(70\) −1.00000 −0.119523
\(71\) 4.00000 0.474713 0.237356 0.971423i \(-0.423719\pi\)
0.237356 + 0.971423i \(0.423719\pi\)
\(72\) 0 0
\(73\) 12.9282 1.51313 0.756566 0.653917i \(-0.226876\pi\)
0.756566 + 0.653917i \(0.226876\pi\)
\(74\) −0.535898 −0.0622969
\(75\) 0 0
\(76\) 0 0
\(77\) −1.00000 −0.113961
\(78\) 0 0
\(79\) 8.00000 0.900070 0.450035 0.893011i \(-0.351411\pi\)
0.450035 + 0.893011i \(0.351411\pi\)
\(80\) 1.00000 0.111803
\(81\) 0 0
\(82\) 4.92820 0.544229
\(83\) 6.92820 0.760469 0.380235 0.924890i \(-0.375843\pi\)
0.380235 + 0.924890i \(0.375843\pi\)
\(84\) 0 0
\(85\) −3.46410 −0.375735
\(86\) −10.9282 −1.17842
\(87\) 0 0
\(88\) 1.00000 0.106600
\(89\) 3.46410 0.367194 0.183597 0.983002i \(-0.441226\pi\)
0.183597 + 0.983002i \(0.441226\pi\)
\(90\) 0 0
\(91\) −3.46410 −0.363137
\(92\) −5.46410 −0.569672
\(93\) 0 0
\(94\) −6.92820 −0.714590
\(95\) 0 0
\(96\) 0 0
\(97\) 15.8564 1.60997 0.804987 0.593292i \(-0.202172\pi\)
0.804987 + 0.593292i \(0.202172\pi\)
\(98\) −1.00000 −0.101015
\(99\) 0 0
\(100\) 1.00000 0.100000
\(101\) 15.8564 1.57777 0.788886 0.614540i \(-0.210658\pi\)
0.788886 + 0.614540i \(0.210658\pi\)
\(102\) 0 0
\(103\) 16.0000 1.57653 0.788263 0.615338i \(-0.210980\pi\)
0.788263 + 0.615338i \(0.210980\pi\)
\(104\) 3.46410 0.339683
\(105\) 0 0
\(106\) −0.928203 −0.0901551
\(107\) −6.92820 −0.669775 −0.334887 0.942258i \(-0.608698\pi\)
−0.334887 + 0.942258i \(0.608698\pi\)
\(108\) 0 0
\(109\) −6.39230 −0.612272 −0.306136 0.951988i \(-0.599036\pi\)
−0.306136 + 0.951988i \(0.599036\pi\)
\(110\) 1.00000 0.0953463
\(111\) 0 0
\(112\) 1.00000 0.0944911
\(113\) −4.53590 −0.426701 −0.213351 0.976976i \(-0.568438\pi\)
−0.213351 + 0.976976i \(0.568438\pi\)
\(114\) 0 0
\(115\) −5.46410 −0.509530
\(116\) −2.00000 −0.185695
\(117\) 0 0
\(118\) 10.9282 1.00602
\(119\) −3.46410 −0.317554
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) −8.92820 −0.808322
\(123\) 0 0
\(124\) −6.92820 −0.622171
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) −2.92820 −0.259836 −0.129918 0.991525i \(-0.541471\pi\)
−0.129918 + 0.991525i \(0.541471\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 0 0
\(130\) 3.46410 0.303822
\(131\) 14.9282 1.30428 0.652142 0.758097i \(-0.273871\pi\)
0.652142 + 0.758097i \(0.273871\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) −2.53590 −0.219068
\(135\) 0 0
\(136\) 3.46410 0.297044
\(137\) 11.4641 0.979444 0.489722 0.871879i \(-0.337098\pi\)
0.489722 + 0.871879i \(0.337098\pi\)
\(138\) 0 0
\(139\) 10.9282 0.926918 0.463459 0.886118i \(-0.346608\pi\)
0.463459 + 0.886118i \(0.346608\pi\)
\(140\) 1.00000 0.0845154
\(141\) 0 0
\(142\) −4.00000 −0.335673
\(143\) 3.46410 0.289683
\(144\) 0 0
\(145\) −2.00000 −0.166091
\(146\) −12.9282 −1.06995
\(147\) 0 0
\(148\) 0.535898 0.0440506
\(149\) 0.928203 0.0760414 0.0380207 0.999277i \(-0.487895\pi\)
0.0380207 + 0.999277i \(0.487895\pi\)
\(150\) 0 0
\(151\) 8.00000 0.651031 0.325515 0.945537i \(-0.394462\pi\)
0.325515 + 0.945537i \(0.394462\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 1.00000 0.0805823
\(155\) −6.92820 −0.556487
\(156\) 0 0
\(157\) 22.7846 1.81841 0.909205 0.416349i \(-0.136691\pi\)
0.909205 + 0.416349i \(0.136691\pi\)
\(158\) −8.00000 −0.636446
\(159\) 0 0
\(160\) −1.00000 −0.0790569
\(161\) −5.46410 −0.430632
\(162\) 0 0
\(163\) 2.53590 0.198627 0.0993134 0.995056i \(-0.468335\pi\)
0.0993134 + 0.995056i \(0.468335\pi\)
\(164\) −4.92820 −0.384828
\(165\) 0 0
\(166\) −6.92820 −0.537733
\(167\) 2.53590 0.196234 0.0981169 0.995175i \(-0.468718\pi\)
0.0981169 + 0.995175i \(0.468718\pi\)
\(168\) 0 0
\(169\) −1.00000 −0.0769231
\(170\) 3.46410 0.265684
\(171\) 0 0
\(172\) 10.9282 0.833268
\(173\) −4.92820 −0.374684 −0.187342 0.982295i \(-0.559987\pi\)
−0.187342 + 0.982295i \(0.559987\pi\)
\(174\) 0 0
\(175\) 1.00000 0.0755929
\(176\) −1.00000 −0.0753778
\(177\) 0 0
\(178\) −3.46410 −0.259645
\(179\) −20.7846 −1.55351 −0.776757 0.629800i \(-0.783137\pi\)
−0.776757 + 0.629800i \(0.783137\pi\)
\(180\) 0 0
\(181\) −23.4641 −1.74407 −0.872036 0.489441i \(-0.837201\pi\)
−0.872036 + 0.489441i \(0.837201\pi\)
\(182\) 3.46410 0.256776
\(183\) 0 0
\(184\) 5.46410 0.402819
\(185\) 0.535898 0.0394000
\(186\) 0 0
\(187\) 3.46410 0.253320
\(188\) 6.92820 0.505291
\(189\) 0 0
\(190\) 0 0
\(191\) −1.07180 −0.0775525 −0.0387762 0.999248i \(-0.512346\pi\)
−0.0387762 + 0.999248i \(0.512346\pi\)
\(192\) 0 0
\(193\) 24.9282 1.79437 0.897186 0.441654i \(-0.145608\pi\)
0.897186 + 0.441654i \(0.145608\pi\)
\(194\) −15.8564 −1.13842
\(195\) 0 0
\(196\) 1.00000 0.0714286
\(197\) 18.0000 1.28245 0.641223 0.767354i \(-0.278427\pi\)
0.641223 + 0.767354i \(0.278427\pi\)
\(198\) 0 0
\(199\) 6.92820 0.491127 0.245564 0.969380i \(-0.421027\pi\)
0.245564 + 0.969380i \(0.421027\pi\)
\(200\) −1.00000 −0.0707107
\(201\) 0 0
\(202\) −15.8564 −1.11565
\(203\) −2.00000 −0.140372
\(204\) 0 0
\(205\) −4.92820 −0.344201
\(206\) −16.0000 −1.11477
\(207\) 0 0
\(208\) −3.46410 −0.240192
\(209\) 0 0
\(210\) 0 0
\(211\) 11.3205 0.779336 0.389668 0.920955i \(-0.372590\pi\)
0.389668 + 0.920955i \(0.372590\pi\)
\(212\) 0.928203 0.0637493
\(213\) 0 0
\(214\) 6.92820 0.473602
\(215\) 10.9282 0.745297
\(216\) 0 0
\(217\) −6.92820 −0.470317
\(218\) 6.39230 0.432942
\(219\) 0 0
\(220\) −1.00000 −0.0674200
\(221\) 12.0000 0.807207
\(222\) 0 0
\(223\) −16.7846 −1.12398 −0.561990 0.827144i \(-0.689964\pi\)
−0.561990 + 0.827144i \(0.689964\pi\)
\(224\) −1.00000 −0.0668153
\(225\) 0 0
\(226\) 4.53590 0.301723
\(227\) 6.92820 0.459841 0.229920 0.973209i \(-0.426153\pi\)
0.229920 + 0.973209i \(0.426153\pi\)
\(228\) 0 0
\(229\) −18.3923 −1.21540 −0.607699 0.794168i \(-0.707907\pi\)
−0.607699 + 0.794168i \(0.707907\pi\)
\(230\) 5.46410 0.360292
\(231\) 0 0
\(232\) 2.00000 0.131306
\(233\) −2.00000 −0.131024 −0.0655122 0.997852i \(-0.520868\pi\)
−0.0655122 + 0.997852i \(0.520868\pi\)
\(234\) 0 0
\(235\) 6.92820 0.451946
\(236\) −10.9282 −0.711365
\(237\) 0 0
\(238\) 3.46410 0.224544
\(239\) 16.3923 1.06033 0.530165 0.847894i \(-0.322130\pi\)
0.530165 + 0.847894i \(0.322130\pi\)
\(240\) 0 0
\(241\) −20.9282 −1.34810 −0.674052 0.738684i \(-0.735448\pi\)
−0.674052 + 0.738684i \(0.735448\pi\)
\(242\) −1.00000 −0.0642824
\(243\) 0 0
\(244\) 8.92820 0.571570
\(245\) 1.00000 0.0638877
\(246\) 0 0
\(247\) 0 0
\(248\) 6.92820 0.439941
\(249\) 0 0
\(250\) −1.00000 −0.0632456
\(251\) 18.9282 1.19474 0.597369 0.801967i \(-0.296213\pi\)
0.597369 + 0.801967i \(0.296213\pi\)
\(252\) 0 0
\(253\) 5.46410 0.343525
\(254\) 2.92820 0.183732
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 3.07180 0.191613 0.0958067 0.995400i \(-0.469457\pi\)
0.0958067 + 0.995400i \(0.469457\pi\)
\(258\) 0 0
\(259\) 0.535898 0.0332991
\(260\) −3.46410 −0.214834
\(261\) 0 0
\(262\) −14.9282 −0.922267
\(263\) −24.7846 −1.52828 −0.764142 0.645048i \(-0.776837\pi\)
−0.764142 + 0.645048i \(0.776837\pi\)
\(264\) 0 0
\(265\) 0.928203 0.0570191
\(266\) 0 0
\(267\) 0 0
\(268\) 2.53590 0.154905
\(269\) 3.07180 0.187291 0.0936454 0.995606i \(-0.470148\pi\)
0.0936454 + 0.995606i \(0.470148\pi\)
\(270\) 0 0
\(271\) 21.8564 1.32768 0.663841 0.747874i \(-0.268925\pi\)
0.663841 + 0.747874i \(0.268925\pi\)
\(272\) −3.46410 −0.210042
\(273\) 0 0
\(274\) −11.4641 −0.692572
\(275\) −1.00000 −0.0603023
\(276\) 0 0
\(277\) −10.0000 −0.600842 −0.300421 0.953807i \(-0.597127\pi\)
−0.300421 + 0.953807i \(0.597127\pi\)
\(278\) −10.9282 −0.655430
\(279\) 0 0
\(280\) −1.00000 −0.0597614
\(281\) −6.39230 −0.381333 −0.190666 0.981655i \(-0.561065\pi\)
−0.190666 + 0.981655i \(0.561065\pi\)
\(282\) 0 0
\(283\) −5.46410 −0.324807 −0.162404 0.986724i \(-0.551925\pi\)
−0.162404 + 0.986724i \(0.551925\pi\)
\(284\) 4.00000 0.237356
\(285\) 0 0
\(286\) −3.46410 −0.204837
\(287\) −4.92820 −0.290903
\(288\) 0 0
\(289\) −5.00000 −0.294118
\(290\) 2.00000 0.117444
\(291\) 0 0
\(292\) 12.9282 0.756566
\(293\) 14.0000 0.817889 0.408944 0.912559i \(-0.365897\pi\)
0.408944 + 0.912559i \(0.365897\pi\)
\(294\) 0 0
\(295\) −10.9282 −0.636265
\(296\) −0.535898 −0.0311485
\(297\) 0 0
\(298\) −0.928203 −0.0537694
\(299\) 18.9282 1.09465
\(300\) 0 0
\(301\) 10.9282 0.629891
\(302\) −8.00000 −0.460348
\(303\) 0 0
\(304\) 0 0
\(305\) 8.92820 0.511227
\(306\) 0 0
\(307\) −27.3205 −1.55926 −0.779632 0.626238i \(-0.784594\pi\)
−0.779632 + 0.626238i \(0.784594\pi\)
\(308\) −1.00000 −0.0569803
\(309\) 0 0
\(310\) 6.92820 0.393496
\(311\) −13.4641 −0.763479 −0.381740 0.924270i \(-0.624675\pi\)
−0.381740 + 0.924270i \(0.624675\pi\)
\(312\) 0 0
\(313\) −33.7128 −1.90556 −0.952780 0.303660i \(-0.901791\pi\)
−0.952780 + 0.303660i \(0.901791\pi\)
\(314\) −22.7846 −1.28581
\(315\) 0 0
\(316\) 8.00000 0.450035
\(317\) 16.9282 0.950783 0.475391 0.879774i \(-0.342306\pi\)
0.475391 + 0.879774i \(0.342306\pi\)
\(318\) 0 0
\(319\) 2.00000 0.111979
\(320\) 1.00000 0.0559017
\(321\) 0 0
\(322\) 5.46410 0.304502
\(323\) 0 0
\(324\) 0 0
\(325\) −3.46410 −0.192154
\(326\) −2.53590 −0.140450
\(327\) 0 0
\(328\) 4.92820 0.272115
\(329\) 6.92820 0.381964
\(330\) 0 0
\(331\) −4.00000 −0.219860 −0.109930 0.993939i \(-0.535063\pi\)
−0.109930 + 0.993939i \(0.535063\pi\)
\(332\) 6.92820 0.380235
\(333\) 0 0
\(334\) −2.53590 −0.138758
\(335\) 2.53590 0.138551
\(336\) 0 0
\(337\) −18.7846 −1.02326 −0.511631 0.859205i \(-0.670959\pi\)
−0.511631 + 0.859205i \(0.670959\pi\)
\(338\) 1.00000 0.0543928
\(339\) 0 0
\(340\) −3.46410 −0.187867
\(341\) 6.92820 0.375183
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) −10.9282 −0.589209
\(345\) 0 0
\(346\) 4.92820 0.264942
\(347\) 14.9282 0.801388 0.400694 0.916212i \(-0.368769\pi\)
0.400694 + 0.916212i \(0.368769\pi\)
\(348\) 0 0
\(349\) −15.0718 −0.806775 −0.403387 0.915029i \(-0.632167\pi\)
−0.403387 + 0.915029i \(0.632167\pi\)
\(350\) −1.00000 −0.0534522
\(351\) 0 0
\(352\) 1.00000 0.0533002
\(353\) −7.07180 −0.376394 −0.188197 0.982131i \(-0.560264\pi\)
−0.188197 + 0.982131i \(0.560264\pi\)
\(354\) 0 0
\(355\) 4.00000 0.212298
\(356\) 3.46410 0.183597
\(357\) 0 0
\(358\) 20.7846 1.09850
\(359\) −2.53590 −0.133840 −0.0669198 0.997758i \(-0.521317\pi\)
−0.0669198 + 0.997758i \(0.521317\pi\)
\(360\) 0 0
\(361\) −19.0000 −1.00000
\(362\) 23.4641 1.23325
\(363\) 0 0
\(364\) −3.46410 −0.181568
\(365\) 12.9282 0.676693
\(366\) 0 0
\(367\) 16.0000 0.835193 0.417597 0.908633i \(-0.362873\pi\)
0.417597 + 0.908633i \(0.362873\pi\)
\(368\) −5.46410 −0.284836
\(369\) 0 0
\(370\) −0.535898 −0.0278600
\(371\) 0.928203 0.0481899
\(372\) 0 0
\(373\) 8.92820 0.462285 0.231142 0.972920i \(-0.425754\pi\)
0.231142 + 0.972920i \(0.425754\pi\)
\(374\) −3.46410 −0.179124
\(375\) 0 0
\(376\) −6.92820 −0.357295
\(377\) 6.92820 0.356821
\(378\) 0 0
\(379\) −28.0000 −1.43826 −0.719132 0.694874i \(-0.755460\pi\)
−0.719132 + 0.694874i \(0.755460\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 1.07180 0.0548379
\(383\) −28.7846 −1.47082 −0.735412 0.677620i \(-0.763012\pi\)
−0.735412 + 0.677620i \(0.763012\pi\)
\(384\) 0 0
\(385\) −1.00000 −0.0509647
\(386\) −24.9282 −1.26881
\(387\) 0 0
\(388\) 15.8564 0.804987
\(389\) 33.7128 1.70931 0.854654 0.519198i \(-0.173769\pi\)
0.854654 + 0.519198i \(0.173769\pi\)
\(390\) 0 0
\(391\) 18.9282 0.957240
\(392\) −1.00000 −0.0505076
\(393\) 0 0
\(394\) −18.0000 −0.906827
\(395\) 8.00000 0.402524
\(396\) 0 0
\(397\) −10.0000 −0.501886 −0.250943 0.968002i \(-0.580741\pi\)
−0.250943 + 0.968002i \(0.580741\pi\)
\(398\) −6.92820 −0.347279
\(399\) 0 0
\(400\) 1.00000 0.0500000
\(401\) −29.7128 −1.48379 −0.741894 0.670518i \(-0.766072\pi\)
−0.741894 + 0.670518i \(0.766072\pi\)
\(402\) 0 0
\(403\) 24.0000 1.19553
\(404\) 15.8564 0.788886
\(405\) 0 0
\(406\) 2.00000 0.0992583
\(407\) −0.535898 −0.0265635
\(408\) 0 0
\(409\) 0.928203 0.0458967 0.0229483 0.999737i \(-0.492695\pi\)
0.0229483 + 0.999737i \(0.492695\pi\)
\(410\) 4.92820 0.243387
\(411\) 0 0
\(412\) 16.0000 0.788263
\(413\) −10.9282 −0.537742
\(414\) 0 0
\(415\) 6.92820 0.340092
\(416\) 3.46410 0.169842
\(417\) 0 0
\(418\) 0 0
\(419\) 13.8564 0.676930 0.338465 0.940979i \(-0.390092\pi\)
0.338465 + 0.940979i \(0.390092\pi\)
\(420\) 0 0
\(421\) −26.0000 −1.26716 −0.633581 0.773676i \(-0.718416\pi\)
−0.633581 + 0.773676i \(0.718416\pi\)
\(422\) −11.3205 −0.551074
\(423\) 0 0
\(424\) −0.928203 −0.0450775
\(425\) −3.46410 −0.168034
\(426\) 0 0
\(427\) 8.92820 0.432066
\(428\) −6.92820 −0.334887
\(429\) 0 0
\(430\) −10.9282 −0.527005
\(431\) −11.3205 −0.545290 −0.272645 0.962115i \(-0.587898\pi\)
−0.272645 + 0.962115i \(0.587898\pi\)
\(432\) 0 0
\(433\) 37.7128 1.81236 0.906181 0.422890i \(-0.138984\pi\)
0.906181 + 0.422890i \(0.138984\pi\)
\(434\) 6.92820 0.332564
\(435\) 0 0
\(436\) −6.39230 −0.306136
\(437\) 0 0
\(438\) 0 0
\(439\) 24.7846 1.18290 0.591452 0.806340i \(-0.298555\pi\)
0.591452 + 0.806340i \(0.298555\pi\)
\(440\) 1.00000 0.0476731
\(441\) 0 0
\(442\) −12.0000 −0.570782
\(443\) −13.0718 −0.621060 −0.310530 0.950564i \(-0.600506\pi\)
−0.310530 + 0.950564i \(0.600506\pi\)
\(444\) 0 0
\(445\) 3.46410 0.164214
\(446\) 16.7846 0.794774
\(447\) 0 0
\(448\) 1.00000 0.0472456
\(449\) 22.0000 1.03824 0.519122 0.854700i \(-0.326259\pi\)
0.519122 + 0.854700i \(0.326259\pi\)
\(450\) 0 0
\(451\) 4.92820 0.232060
\(452\) −4.53590 −0.213351
\(453\) 0 0
\(454\) −6.92820 −0.325157
\(455\) −3.46410 −0.162400
\(456\) 0 0
\(457\) 32.9282 1.54032 0.770158 0.637853i \(-0.220177\pi\)
0.770158 + 0.637853i \(0.220177\pi\)
\(458\) 18.3923 0.859416
\(459\) 0 0
\(460\) −5.46410 −0.254765
\(461\) 18.0000 0.838344 0.419172 0.907907i \(-0.362320\pi\)
0.419172 + 0.907907i \(0.362320\pi\)
\(462\) 0 0
\(463\) −20.7846 −0.965943 −0.482971 0.875636i \(-0.660442\pi\)
−0.482971 + 0.875636i \(0.660442\pi\)
\(464\) −2.00000 −0.0928477
\(465\) 0 0
\(466\) 2.00000 0.0926482
\(467\) 17.0718 0.789989 0.394994 0.918684i \(-0.370747\pi\)
0.394994 + 0.918684i \(0.370747\pi\)
\(468\) 0 0
\(469\) 2.53590 0.117097
\(470\) −6.92820 −0.319574
\(471\) 0 0
\(472\) 10.9282 0.503011
\(473\) −10.9282 −0.502479
\(474\) 0 0
\(475\) 0 0
\(476\) −3.46410 −0.158777
\(477\) 0 0
\(478\) −16.3923 −0.749767
\(479\) 13.8564 0.633115 0.316558 0.948573i \(-0.397473\pi\)
0.316558 + 0.948573i \(0.397473\pi\)
\(480\) 0 0
\(481\) −1.85641 −0.0846448
\(482\) 20.9282 0.953254
\(483\) 0 0
\(484\) 1.00000 0.0454545
\(485\) 15.8564 0.720002
\(486\) 0 0
\(487\) 14.9282 0.676461 0.338231 0.941063i \(-0.390172\pi\)
0.338231 + 0.941063i \(0.390172\pi\)
\(488\) −8.92820 −0.404161
\(489\) 0 0
\(490\) −1.00000 −0.0451754
\(491\) 25.8564 1.16688 0.583442 0.812155i \(-0.301706\pi\)
0.583442 + 0.812155i \(0.301706\pi\)
\(492\) 0 0
\(493\) 6.92820 0.312031
\(494\) 0 0
\(495\) 0 0
\(496\) −6.92820 −0.311086
\(497\) 4.00000 0.179425
\(498\) 0 0
\(499\) 30.9282 1.38454 0.692268 0.721640i \(-0.256611\pi\)
0.692268 + 0.721640i \(0.256611\pi\)
\(500\) 1.00000 0.0447214
\(501\) 0 0
\(502\) −18.9282 −0.844807
\(503\) 8.39230 0.374194 0.187097 0.982341i \(-0.440092\pi\)
0.187097 + 0.982341i \(0.440092\pi\)
\(504\) 0 0
\(505\) 15.8564 0.705601
\(506\) −5.46410 −0.242909
\(507\) 0 0
\(508\) −2.92820 −0.129918
\(509\) 19.0718 0.845343 0.422671 0.906283i \(-0.361092\pi\)
0.422671 + 0.906283i \(0.361092\pi\)
\(510\) 0 0
\(511\) 12.9282 0.571910
\(512\) −1.00000 −0.0441942
\(513\) 0 0
\(514\) −3.07180 −0.135491
\(515\) 16.0000 0.705044
\(516\) 0 0
\(517\) −6.92820 −0.304702
\(518\) −0.535898 −0.0235460
\(519\) 0 0
\(520\) 3.46410 0.151911
\(521\) −20.5359 −0.899694 −0.449847 0.893106i \(-0.648521\pi\)
−0.449847 + 0.893106i \(0.648521\pi\)
\(522\) 0 0
\(523\) 23.6077 1.03229 0.516146 0.856500i \(-0.327366\pi\)
0.516146 + 0.856500i \(0.327366\pi\)
\(524\) 14.9282 0.652142
\(525\) 0 0
\(526\) 24.7846 1.08066
\(527\) 24.0000 1.04546
\(528\) 0 0
\(529\) 6.85641 0.298105
\(530\) −0.928203 −0.0403186
\(531\) 0 0
\(532\) 0 0
\(533\) 17.0718 0.739462
\(534\) 0 0
\(535\) −6.92820 −0.299532
\(536\) −2.53590 −0.109534
\(537\) 0 0
\(538\) −3.07180 −0.132435
\(539\) −1.00000 −0.0430730
\(540\) 0 0
\(541\) −25.3205 −1.08861 −0.544307 0.838886i \(-0.683207\pi\)
−0.544307 + 0.838886i \(0.683207\pi\)
\(542\) −21.8564 −0.938813
\(543\) 0 0
\(544\) 3.46410 0.148522
\(545\) −6.39230 −0.273816
\(546\) 0 0
\(547\) −8.00000 −0.342055 −0.171028 0.985266i \(-0.554709\pi\)
−0.171028 + 0.985266i \(0.554709\pi\)
\(548\) 11.4641 0.489722
\(549\) 0 0
\(550\) 1.00000 0.0426401
\(551\) 0 0
\(552\) 0 0
\(553\) 8.00000 0.340195
\(554\) 10.0000 0.424859
\(555\) 0 0
\(556\) 10.9282 0.463459
\(557\) 4.14359 0.175570 0.0877848 0.996139i \(-0.472021\pi\)
0.0877848 + 0.996139i \(0.472021\pi\)
\(558\) 0 0
\(559\) −37.8564 −1.60116
\(560\) 1.00000 0.0422577
\(561\) 0 0
\(562\) 6.39230 0.269643
\(563\) −23.7128 −0.999376 −0.499688 0.866205i \(-0.666552\pi\)
−0.499688 + 0.866205i \(0.666552\pi\)
\(564\) 0 0
\(565\) −4.53590 −0.190827
\(566\) 5.46410 0.229673
\(567\) 0 0
\(568\) −4.00000 −0.167836
\(569\) −23.1769 −0.971627 −0.485813 0.874063i \(-0.661477\pi\)
−0.485813 + 0.874063i \(0.661477\pi\)
\(570\) 0 0
\(571\) −11.3205 −0.473749 −0.236874 0.971540i \(-0.576123\pi\)
−0.236874 + 0.971540i \(0.576123\pi\)
\(572\) 3.46410 0.144841
\(573\) 0 0
\(574\) 4.92820 0.205699
\(575\) −5.46410 −0.227869
\(576\) 0 0
\(577\) −9.71281 −0.404350 −0.202175 0.979349i \(-0.564801\pi\)
−0.202175 + 0.979349i \(0.564801\pi\)
\(578\) 5.00000 0.207973
\(579\) 0 0
\(580\) −2.00000 −0.0830455
\(581\) 6.92820 0.287430
\(582\) 0 0
\(583\) −0.928203 −0.0384422
\(584\) −12.9282 −0.534973
\(585\) 0 0
\(586\) −14.0000 −0.578335
\(587\) −20.7846 −0.857873 −0.428936 0.903335i \(-0.641112\pi\)
−0.428936 + 0.903335i \(0.641112\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 10.9282 0.449907
\(591\) 0 0
\(592\) 0.535898 0.0220253
\(593\) 35.1769 1.44454 0.722271 0.691610i \(-0.243098\pi\)
0.722271 + 0.691610i \(0.243098\pi\)
\(594\) 0 0
\(595\) −3.46410 −0.142014
\(596\) 0.928203 0.0380207
\(597\) 0 0
\(598\) −18.9282 −0.774032
\(599\) 44.7846 1.82985 0.914925 0.403624i \(-0.132250\pi\)
0.914925 + 0.403624i \(0.132250\pi\)
\(600\) 0 0
\(601\) −36.9282 −1.50633 −0.753166 0.657830i \(-0.771475\pi\)
−0.753166 + 0.657830i \(0.771475\pi\)
\(602\) −10.9282 −0.445400
\(603\) 0 0
\(604\) 8.00000 0.325515
\(605\) 1.00000 0.0406558
\(606\) 0 0
\(607\) 16.0000 0.649420 0.324710 0.945814i \(-0.394733\pi\)
0.324710 + 0.945814i \(0.394733\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) −8.92820 −0.361492
\(611\) −24.0000 −0.970936
\(612\) 0 0
\(613\) −2.00000 −0.0807792 −0.0403896 0.999184i \(-0.512860\pi\)
−0.0403896 + 0.999184i \(0.512860\pi\)
\(614\) 27.3205 1.10257
\(615\) 0 0
\(616\) 1.00000 0.0402911
\(617\) −18.3923 −0.740446 −0.370223 0.928943i \(-0.620719\pi\)
−0.370223 + 0.928943i \(0.620719\pi\)
\(618\) 0 0
\(619\) −19.3205 −0.776557 −0.388278 0.921542i \(-0.626930\pi\)
−0.388278 + 0.921542i \(0.626930\pi\)
\(620\) −6.92820 −0.278243
\(621\) 0 0
\(622\) 13.4641 0.539861
\(623\) 3.46410 0.138786
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 33.7128 1.34743
\(627\) 0 0
\(628\) 22.7846 0.909205
\(629\) −1.85641 −0.0740198
\(630\) 0 0
\(631\) −24.7846 −0.986660 −0.493330 0.869842i \(-0.664220\pi\)
−0.493330 + 0.869842i \(0.664220\pi\)
\(632\) −8.00000 −0.318223
\(633\) 0 0
\(634\) −16.9282 −0.672305
\(635\) −2.92820 −0.116202
\(636\) 0 0
\(637\) −3.46410 −0.137253
\(638\) −2.00000 −0.0791808
\(639\) 0 0
\(640\) −1.00000 −0.0395285
\(641\) 6.00000 0.236986 0.118493 0.992955i \(-0.462194\pi\)
0.118493 + 0.992955i \(0.462194\pi\)
\(642\) 0 0
\(643\) 31.7128 1.25063 0.625316 0.780372i \(-0.284970\pi\)
0.625316 + 0.780372i \(0.284970\pi\)
\(644\) −5.46410 −0.215316
\(645\) 0 0
\(646\) 0 0
\(647\) 9.07180 0.356649 0.178325 0.983972i \(-0.442932\pi\)
0.178325 + 0.983972i \(0.442932\pi\)
\(648\) 0 0
\(649\) 10.9282 0.428969
\(650\) 3.46410 0.135873
\(651\) 0 0
\(652\) 2.53590 0.0993134
\(653\) −4.14359 −0.162151 −0.0810757 0.996708i \(-0.525836\pi\)
−0.0810757 + 0.996708i \(0.525836\pi\)
\(654\) 0 0
\(655\) 14.9282 0.583293
\(656\) −4.92820 −0.192414
\(657\) 0 0
\(658\) −6.92820 −0.270089
\(659\) −15.7128 −0.612084 −0.306042 0.952018i \(-0.599005\pi\)
−0.306042 + 0.952018i \(0.599005\pi\)
\(660\) 0 0
\(661\) 22.3923 0.870960 0.435480 0.900198i \(-0.356579\pi\)
0.435480 + 0.900198i \(0.356579\pi\)
\(662\) 4.00000 0.155464
\(663\) 0 0
\(664\) −6.92820 −0.268866
\(665\) 0 0
\(666\) 0 0
\(667\) 10.9282 0.423142
\(668\) 2.53590 0.0981169
\(669\) 0 0
\(670\) −2.53590 −0.0979703
\(671\) −8.92820 −0.344669
\(672\) 0 0
\(673\) −28.9282 −1.11510 −0.557550 0.830143i \(-0.688259\pi\)
−0.557550 + 0.830143i \(0.688259\pi\)
\(674\) 18.7846 0.723556
\(675\) 0 0
\(676\) −1.00000 −0.0384615
\(677\) 27.0718 1.04045 0.520227 0.854028i \(-0.325847\pi\)
0.520227 + 0.854028i \(0.325847\pi\)
\(678\) 0 0
\(679\) 15.8564 0.608513
\(680\) 3.46410 0.132842
\(681\) 0 0
\(682\) −6.92820 −0.265295
\(683\) −13.0718 −0.500178 −0.250089 0.968223i \(-0.580460\pi\)
−0.250089 + 0.968223i \(0.580460\pi\)
\(684\) 0 0
\(685\) 11.4641 0.438021
\(686\) −1.00000 −0.0381802
\(687\) 0 0
\(688\) 10.9282 0.416634
\(689\) −3.21539 −0.122497
\(690\) 0 0
\(691\) 24.3923 0.927927 0.463964 0.885854i \(-0.346427\pi\)
0.463964 + 0.885854i \(0.346427\pi\)
\(692\) −4.92820 −0.187342
\(693\) 0 0
\(694\) −14.9282 −0.566667
\(695\) 10.9282 0.414530
\(696\) 0 0
\(697\) 17.0718 0.646640
\(698\) 15.0718 0.570476
\(699\) 0 0
\(700\) 1.00000 0.0377964
\(701\) −2.00000 −0.0755390 −0.0377695 0.999286i \(-0.512025\pi\)
−0.0377695 + 0.999286i \(0.512025\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) −1.00000 −0.0376889
\(705\) 0 0
\(706\) 7.07180 0.266151
\(707\) 15.8564 0.596342
\(708\) 0 0
\(709\) 19.8564 0.745723 0.372861 0.927887i \(-0.378377\pi\)
0.372861 + 0.927887i \(0.378377\pi\)
\(710\) −4.00000 −0.150117
\(711\) 0 0
\(712\) −3.46410 −0.129823
\(713\) 37.8564 1.41773
\(714\) 0 0
\(715\) 3.46410 0.129550
\(716\) −20.7846 −0.776757
\(717\) 0 0
\(718\) 2.53590 0.0946389
\(719\) −16.3923 −0.611330 −0.305665 0.952139i \(-0.598879\pi\)
−0.305665 + 0.952139i \(0.598879\pi\)
\(720\) 0 0
\(721\) 16.0000 0.595871
\(722\) 19.0000 0.707107
\(723\) 0 0
\(724\) −23.4641 −0.872036
\(725\) −2.00000 −0.0742781
\(726\) 0 0
\(727\) 26.9282 0.998712 0.499356 0.866397i \(-0.333570\pi\)
0.499356 + 0.866397i \(0.333570\pi\)
\(728\) 3.46410 0.128388
\(729\) 0 0
\(730\) −12.9282 −0.478494
\(731\) −37.8564 −1.40017
\(732\) 0 0
\(733\) 24.2487 0.895647 0.447823 0.894122i \(-0.352199\pi\)
0.447823 + 0.894122i \(0.352199\pi\)
\(734\) −16.0000 −0.590571
\(735\) 0 0
\(736\) 5.46410 0.201409
\(737\) −2.53590 −0.0934110
\(738\) 0 0
\(739\) −19.3205 −0.710716 −0.355358 0.934730i \(-0.615641\pi\)
−0.355358 + 0.934730i \(0.615641\pi\)
\(740\) 0.535898 0.0197000
\(741\) 0 0
\(742\) −0.928203 −0.0340754
\(743\) 13.8564 0.508342 0.254171 0.967159i \(-0.418197\pi\)
0.254171 + 0.967159i \(0.418197\pi\)
\(744\) 0 0
\(745\) 0.928203 0.0340067
\(746\) −8.92820 −0.326885
\(747\) 0 0
\(748\) 3.46410 0.126660
\(749\) −6.92820 −0.253151
\(750\) 0 0
\(751\) 40.7846 1.48825 0.744126 0.668040i \(-0.232866\pi\)
0.744126 + 0.668040i \(0.232866\pi\)
\(752\) 6.92820 0.252646
\(753\) 0 0
\(754\) −6.92820 −0.252310
\(755\) 8.00000 0.291150
\(756\) 0 0
\(757\) 26.1051 0.948807 0.474403 0.880308i \(-0.342664\pi\)
0.474403 + 0.880308i \(0.342664\pi\)
\(758\) 28.0000 1.01701
\(759\) 0 0
\(760\) 0 0
\(761\) −2.78461 −0.100942 −0.0504710 0.998726i \(-0.516072\pi\)
−0.0504710 + 0.998726i \(0.516072\pi\)
\(762\) 0 0
\(763\) −6.39230 −0.231417
\(764\) −1.07180 −0.0387762
\(765\) 0 0
\(766\) 28.7846 1.04003
\(767\) 37.8564 1.36692
\(768\) 0 0
\(769\) −2.78461 −0.100416 −0.0502078 0.998739i \(-0.515988\pi\)
−0.0502078 + 0.998739i \(0.515988\pi\)
\(770\) 1.00000 0.0360375
\(771\) 0 0
\(772\) 24.9282 0.897186
\(773\) −26.0000 −0.935155 −0.467578 0.883952i \(-0.654873\pi\)
−0.467578 + 0.883952i \(0.654873\pi\)
\(774\) 0 0
\(775\) −6.92820 −0.248868
\(776\) −15.8564 −0.569212
\(777\) 0 0
\(778\) −33.7128 −1.20866
\(779\) 0 0
\(780\) 0 0
\(781\) −4.00000 −0.143131
\(782\) −18.9282 −0.676871
\(783\) 0 0
\(784\) 1.00000 0.0357143
\(785\) 22.7846 0.813218
\(786\) 0 0
\(787\) 34.5359 1.23107 0.615536 0.788109i \(-0.288940\pi\)
0.615536 + 0.788109i \(0.288940\pi\)
\(788\) 18.0000 0.641223
\(789\) 0 0
\(790\) −8.00000 −0.284627
\(791\) −4.53590 −0.161278
\(792\) 0 0
\(793\) −30.9282 −1.09829
\(794\) 10.0000 0.354887
\(795\) 0 0
\(796\) 6.92820 0.245564
\(797\) −4.14359 −0.146774 −0.0733868 0.997304i \(-0.523381\pi\)
−0.0733868 + 0.997304i \(0.523381\pi\)
\(798\) 0 0
\(799\) −24.0000 −0.849059
\(800\) −1.00000 −0.0353553
\(801\) 0 0
\(802\) 29.7128 1.04920
\(803\) −12.9282 −0.456226
\(804\) 0 0
\(805\) −5.46410 −0.192584
\(806\) −24.0000 −0.845364
\(807\) 0 0
\(808\) −15.8564 −0.557826
\(809\) 4.53590 0.159474 0.0797368 0.996816i \(-0.474592\pi\)
0.0797368 + 0.996816i \(0.474592\pi\)
\(810\) 0 0
\(811\) −13.8564 −0.486564 −0.243282 0.969956i \(-0.578224\pi\)
−0.243282 + 0.969956i \(0.578224\pi\)
\(812\) −2.00000 −0.0701862
\(813\) 0 0
\(814\) 0.535898 0.0187832
\(815\) 2.53590 0.0888286
\(816\) 0 0
\(817\) 0 0
\(818\) −0.928203 −0.0324539
\(819\) 0 0
\(820\) −4.92820 −0.172100
\(821\) −39.8564 −1.39100 −0.695499 0.718527i \(-0.744817\pi\)
−0.695499 + 0.718527i \(0.744817\pi\)
\(822\) 0 0
\(823\) 22.1436 0.771877 0.385939 0.922524i \(-0.373878\pi\)
0.385939 + 0.922524i \(0.373878\pi\)
\(824\) −16.0000 −0.557386
\(825\) 0 0
\(826\) 10.9282 0.380241
\(827\) 20.7846 0.722752 0.361376 0.932420i \(-0.382307\pi\)
0.361376 + 0.932420i \(0.382307\pi\)
\(828\) 0 0
\(829\) −11.1769 −0.388190 −0.194095 0.980983i \(-0.562177\pi\)
−0.194095 + 0.980983i \(0.562177\pi\)
\(830\) −6.92820 −0.240481
\(831\) 0 0
\(832\) −3.46410 −0.120096
\(833\) −3.46410 −0.120024
\(834\) 0 0
\(835\) 2.53590 0.0877584
\(836\) 0 0
\(837\) 0 0
\(838\) −13.8564 −0.478662
\(839\) −37.4641 −1.29340 −0.646702 0.762743i \(-0.723852\pi\)
−0.646702 + 0.762743i \(0.723852\pi\)
\(840\) 0 0
\(841\) −25.0000 −0.862069
\(842\) 26.0000 0.896019
\(843\) 0 0
\(844\) 11.3205 0.389668
\(845\) −1.00000 −0.0344010
\(846\) 0 0
\(847\) 1.00000 0.0343604
\(848\) 0.928203 0.0318746
\(849\) 0 0
\(850\) 3.46410 0.118818
\(851\) −2.92820 −0.100378
\(852\) 0 0
\(853\) −47.1769 −1.61531 −0.807653 0.589658i \(-0.799263\pi\)
−0.807653 + 0.589658i \(0.799263\pi\)
\(854\) −8.92820 −0.305517
\(855\) 0 0
\(856\) 6.92820 0.236801
\(857\) 7.46410 0.254969 0.127484 0.991841i \(-0.459310\pi\)
0.127484 + 0.991841i \(0.459310\pi\)
\(858\) 0 0
\(859\) −0.392305 −0.0133853 −0.00669263 0.999978i \(-0.502130\pi\)
−0.00669263 + 0.999978i \(0.502130\pi\)
\(860\) 10.9282 0.372649
\(861\) 0 0
\(862\) 11.3205 0.385578
\(863\) 6.24871 0.212709 0.106354 0.994328i \(-0.466082\pi\)
0.106354 + 0.994328i \(0.466082\pi\)
\(864\) 0 0
\(865\) −4.92820 −0.167564
\(866\) −37.7128 −1.28153
\(867\) 0 0
\(868\) −6.92820 −0.235159
\(869\) −8.00000 −0.271381
\(870\) 0 0
\(871\) −8.78461 −0.297655
\(872\) 6.39230 0.216471
\(873\) 0 0
\(874\) 0 0
\(875\) 1.00000 0.0338062
\(876\) 0 0
\(877\) 19.8564 0.670503 0.335252 0.942129i \(-0.391179\pi\)
0.335252 + 0.942129i \(0.391179\pi\)
\(878\) −24.7846 −0.836440
\(879\) 0 0
\(880\) −1.00000 −0.0337100
\(881\) 17.3205 0.583543 0.291771 0.956488i \(-0.405755\pi\)
0.291771 + 0.956488i \(0.405755\pi\)
\(882\) 0 0
\(883\) 28.1051 0.945813 0.472906 0.881113i \(-0.343205\pi\)
0.472906 + 0.881113i \(0.343205\pi\)
\(884\) 12.0000 0.403604
\(885\) 0 0
\(886\) 13.0718 0.439156
\(887\) 51.3205 1.72317 0.861587 0.507610i \(-0.169471\pi\)
0.861587 + 0.507610i \(0.169471\pi\)
\(888\) 0 0
\(889\) −2.92820 −0.0982088
\(890\) −3.46410 −0.116117
\(891\) 0 0
\(892\) −16.7846 −0.561990
\(893\) 0 0
\(894\) 0 0
\(895\) −20.7846 −0.694753
\(896\) −1.00000 −0.0334077
\(897\) 0 0
\(898\) −22.0000 −0.734150
\(899\) 13.8564 0.462137
\(900\) 0 0
\(901\) −3.21539 −0.107120
\(902\) −4.92820 −0.164091
\(903\) 0 0
\(904\) 4.53590 0.150862
\(905\) −23.4641 −0.779973
\(906\) 0 0
\(907\) 41.1769 1.36726 0.683629 0.729830i \(-0.260401\pi\)
0.683629 + 0.729830i \(0.260401\pi\)
\(908\) 6.92820 0.229920
\(909\) 0 0
\(910\) 3.46410 0.114834
\(911\) 4.00000 0.132526 0.0662630 0.997802i \(-0.478892\pi\)
0.0662630 + 0.997802i \(0.478892\pi\)
\(912\) 0 0
\(913\) −6.92820 −0.229290
\(914\) −32.9282 −1.08917
\(915\) 0 0
\(916\) −18.3923 −0.607699
\(917\) 14.9282 0.492973
\(918\) 0 0
\(919\) 58.9282 1.94386 0.971931 0.235266i \(-0.0755961\pi\)
0.971931 + 0.235266i \(0.0755961\pi\)
\(920\) 5.46410 0.180146
\(921\) 0 0
\(922\) −18.0000 −0.592798
\(923\) −13.8564 −0.456089
\(924\) 0 0
\(925\) 0.535898 0.0176202
\(926\) 20.7846 0.683025
\(927\) 0 0
\(928\) 2.00000 0.0656532
\(929\) 46.3923 1.52208 0.761041 0.648704i \(-0.224689\pi\)
0.761041 + 0.648704i \(0.224689\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) −2.00000 −0.0655122
\(933\) 0 0
\(934\) −17.0718 −0.558606
\(935\) 3.46410 0.113288
\(936\) 0 0
\(937\) 23.8564 0.779355 0.389677 0.920951i \(-0.372587\pi\)
0.389677 + 0.920951i \(0.372587\pi\)
\(938\) −2.53590 −0.0828000
\(939\) 0 0
\(940\) 6.92820 0.225973
\(941\) −22.0000 −0.717180 −0.358590 0.933495i \(-0.616742\pi\)
−0.358590 + 0.933495i \(0.616742\pi\)
\(942\) 0 0
\(943\) 26.9282 0.876903
\(944\) −10.9282 −0.355683
\(945\) 0 0
\(946\) 10.9282 0.355307
\(947\) 37.8564 1.23017 0.615084 0.788462i \(-0.289122\pi\)
0.615084 + 0.788462i \(0.289122\pi\)
\(948\) 0 0
\(949\) −44.7846 −1.45377
\(950\) 0 0
\(951\) 0 0
\(952\) 3.46410 0.112272
\(953\) −40.6410 −1.31649 −0.658246 0.752803i \(-0.728701\pi\)
−0.658246 + 0.752803i \(0.728701\pi\)
\(954\) 0 0
\(955\) −1.07180 −0.0346825
\(956\) 16.3923 0.530165
\(957\) 0 0
\(958\) −13.8564 −0.447680
\(959\) 11.4641 0.370195
\(960\) 0 0
\(961\) 17.0000 0.548387
\(962\) 1.85641 0.0598529
\(963\) 0 0
\(964\) −20.9282 −0.674052
\(965\) 24.9282 0.802467
\(966\) 0 0
\(967\) −13.0718 −0.420361 −0.210180 0.977663i \(-0.567405\pi\)
−0.210180 + 0.977663i \(0.567405\pi\)
\(968\) −1.00000 −0.0321412
\(969\) 0 0
\(970\) −15.8564 −0.509119
\(971\) 19.7128 0.632614 0.316307 0.948657i \(-0.397557\pi\)
0.316307 + 0.948657i \(0.397557\pi\)
\(972\) 0 0
\(973\) 10.9282 0.350342
\(974\) −14.9282 −0.478330
\(975\) 0 0
\(976\) 8.92820 0.285785
\(977\) 2.67949 0.0857245 0.0428623 0.999081i \(-0.486352\pi\)
0.0428623 + 0.999081i \(0.486352\pi\)
\(978\) 0 0
\(979\) −3.46410 −0.110713
\(980\) 1.00000 0.0319438
\(981\) 0 0
\(982\) −25.8564 −0.825111
\(983\) −44.7846 −1.42841 −0.714204 0.699938i \(-0.753211\pi\)
−0.714204 + 0.699938i \(0.753211\pi\)
\(984\) 0 0
\(985\) 18.0000 0.573528
\(986\) −6.92820 −0.220639
\(987\) 0 0
\(988\) 0 0
\(989\) −59.7128 −1.89876
\(990\) 0 0
\(991\) 0.784610 0.0249239 0.0124620 0.999922i \(-0.496033\pi\)
0.0124620 + 0.999922i \(0.496033\pi\)
\(992\) 6.92820 0.219971
\(993\) 0 0
\(994\) −4.00000 −0.126872
\(995\) 6.92820 0.219639
\(996\) 0 0
\(997\) 53.3205 1.68868 0.844339 0.535810i \(-0.179994\pi\)
0.844339 + 0.535810i \(0.179994\pi\)
\(998\) −30.9282 −0.979015
\(999\) 0 0
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 6930.2.a.bt.1.1 2
3.2 odd 2 2310.2.a.bb.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2310.2.a.bb.1.1 2 3.2 odd 2
6930.2.a.bt.1.1 2 1.1 even 1 trivial