Properties

Label 6930.2.a.ca.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: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 770)
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

\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{7} +1.00000 q^{8} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{7} +1.00000 q^{8} -1.00000 q^{10} -1.00000 q^{11} -1.46410 q^{13} +1.00000 q^{14} +1.00000 q^{16} -3.46410 q^{17} -2.73205 q^{19} -1.00000 q^{20} -1.00000 q^{22} -1.26795 q^{23} +1.00000 q^{25} -1.46410 q^{26} +1.00000 q^{28} +4.73205 q^{29} +8.92820 q^{31} +1.00000 q^{32} -3.46410 q^{34} -1.00000 q^{35} +3.26795 q^{37} -2.73205 q^{38} -1.00000 q^{40} +4.73205 q^{41} -4.92820 q^{43} -1.00000 q^{44} -1.26795 q^{46} +1.00000 q^{49} +1.00000 q^{50} -1.46410 q^{52} +4.73205 q^{53} +1.00000 q^{55} +1.00000 q^{56} +4.73205 q^{58} +13.8564 q^{59} +2.00000 q^{61} +8.92820 q^{62} +1.00000 q^{64} +1.46410 q^{65} -10.9282 q^{67} -3.46410 q^{68} -1.00000 q^{70} -9.46410 q^{71} +11.4641 q^{73} +3.26795 q^{74} -2.73205 q^{76} -1.00000 q^{77} +6.73205 q^{79} -1.00000 q^{80} +4.73205 q^{82} +4.39230 q^{83} +3.46410 q^{85} -4.92820 q^{86} -1.00000 q^{88} +15.4641 q^{89} -1.46410 q^{91} -1.26795 q^{92} +2.73205 q^{95} +5.80385 q^{97} +1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} + 2 q^{4} - 2 q^{5} + 2 q^{7} + 2 q^{8} + O(q^{10}) \) \( 2 q + 2 q^{2} + 2 q^{4} - 2 q^{5} + 2 q^{7} + 2 q^{8} - 2 q^{10} - 2 q^{11} + 4 q^{13} + 2 q^{14} + 2 q^{16} - 2 q^{19} - 2 q^{20} - 2 q^{22} - 6 q^{23} + 2 q^{25} + 4 q^{26} + 2 q^{28} + 6 q^{29} + 4 q^{31} + 2 q^{32} - 2 q^{35} + 10 q^{37} - 2 q^{38} - 2 q^{40} + 6 q^{41} + 4 q^{43} - 2 q^{44} - 6 q^{46} + 2 q^{49} + 2 q^{50} + 4 q^{52} + 6 q^{53} + 2 q^{55} + 2 q^{56} + 6 q^{58} + 4 q^{61} + 4 q^{62} + 2 q^{64} - 4 q^{65} - 8 q^{67} - 2 q^{70} - 12 q^{71} + 16 q^{73} + 10 q^{74} - 2 q^{76} - 2 q^{77} + 10 q^{79} - 2 q^{80} + 6 q^{82} - 12 q^{83} + 4 q^{86} - 2 q^{88} + 24 q^{89} + 4 q^{91} - 6 q^{92} + 2 q^{95} + 22 q^{97} + 2 q^{98} + O(q^{100}) \)

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
\(13\) −1.46410 −0.406069 −0.203034 0.979172i \(-0.565080\pi\)
−0.203034 + 0.979172i \(0.565080\pi\)
\(14\) 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
\(19\) −2.73205 −0.626775 −0.313388 0.949625i \(-0.601464\pi\)
−0.313388 + 0.949625i \(0.601464\pi\)
\(20\) −1.00000 −0.223607
\(21\) 0 0
\(22\) −1.00000 −0.213201
\(23\) −1.26795 −0.264386 −0.132193 0.991224i \(-0.542202\pi\)
−0.132193 + 0.991224i \(0.542202\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) −1.46410 −0.287134
\(27\) 0 0
\(28\) 1.00000 0.188982
\(29\) 4.73205 0.878720 0.439360 0.898311i \(-0.355205\pi\)
0.439360 + 0.898311i \(0.355205\pi\)
\(30\) 0 0
\(31\) 8.92820 1.60355 0.801776 0.597624i \(-0.203889\pi\)
0.801776 + 0.597624i \(0.203889\pi\)
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) −3.46410 −0.594089
\(35\) −1.00000 −0.169031
\(36\) 0 0
\(37\) 3.26795 0.537248 0.268624 0.963245i \(-0.413431\pi\)
0.268624 + 0.963245i \(0.413431\pi\)
\(38\) −2.73205 −0.443197
\(39\) 0 0
\(40\) −1.00000 −0.158114
\(41\) 4.73205 0.739022 0.369511 0.929226i \(-0.379525\pi\)
0.369511 + 0.929226i \(0.379525\pi\)
\(42\) 0 0
\(43\) −4.92820 −0.751544 −0.375772 0.926712i \(-0.622622\pi\)
−0.375772 + 0.926712i \(0.622622\pi\)
\(44\) −1.00000 −0.150756
\(45\) 0 0
\(46\) −1.26795 −0.186949
\(47\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 1.00000 0.141421
\(51\) 0 0
\(52\) −1.46410 −0.203034
\(53\) 4.73205 0.649997 0.324999 0.945715i \(-0.394636\pi\)
0.324999 + 0.945715i \(0.394636\pi\)
\(54\) 0 0
\(55\) 1.00000 0.134840
\(56\) 1.00000 0.133631
\(57\) 0 0
\(58\) 4.73205 0.621349
\(59\) 13.8564 1.80395 0.901975 0.431788i \(-0.142117\pi\)
0.901975 + 0.431788i \(0.142117\pi\)
\(60\) 0 0
\(61\) 2.00000 0.256074 0.128037 0.991769i \(-0.459132\pi\)
0.128037 + 0.991769i \(0.459132\pi\)
\(62\) 8.92820 1.13388
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 1.46410 0.181599
\(66\) 0 0
\(67\) −10.9282 −1.33509 −0.667546 0.744568i \(-0.732655\pi\)
−0.667546 + 0.744568i \(0.732655\pi\)
\(68\) −3.46410 −0.420084
\(69\) 0 0
\(70\) −1.00000 −0.119523
\(71\) −9.46410 −1.12318 −0.561591 0.827415i \(-0.689811\pi\)
−0.561591 + 0.827415i \(0.689811\pi\)
\(72\) 0 0
\(73\) 11.4641 1.34177 0.670886 0.741561i \(-0.265914\pi\)
0.670886 + 0.741561i \(0.265914\pi\)
\(74\) 3.26795 0.379891
\(75\) 0 0
\(76\) −2.73205 −0.313388
\(77\) −1.00000 −0.113961
\(78\) 0 0
\(79\) 6.73205 0.757415 0.378707 0.925516i \(-0.376369\pi\)
0.378707 + 0.925516i \(0.376369\pi\)
\(80\) −1.00000 −0.111803
\(81\) 0 0
\(82\) 4.73205 0.522568
\(83\) 4.39230 0.482118 0.241059 0.970510i \(-0.422505\pi\)
0.241059 + 0.970510i \(0.422505\pi\)
\(84\) 0 0
\(85\) 3.46410 0.375735
\(86\) −4.92820 −0.531422
\(87\) 0 0
\(88\) −1.00000 −0.106600
\(89\) 15.4641 1.63919 0.819596 0.572942i \(-0.194198\pi\)
0.819596 + 0.572942i \(0.194198\pi\)
\(90\) 0 0
\(91\) −1.46410 −0.153480
\(92\) −1.26795 −0.132193
\(93\) 0 0
\(94\) 0 0
\(95\) 2.73205 0.280302
\(96\) 0 0
\(97\) 5.80385 0.589291 0.294646 0.955607i \(-0.404798\pi\)
0.294646 + 0.955607i \(0.404798\pi\)
\(98\) 1.00000 0.101015
\(99\) 0 0
\(100\) 1.00000 0.100000
\(101\) 6.00000 0.597022 0.298511 0.954406i \(-0.403510\pi\)
0.298511 + 0.954406i \(0.403510\pi\)
\(102\) 0 0
\(103\) 10.5359 1.03813 0.519066 0.854734i \(-0.326280\pi\)
0.519066 + 0.854734i \(0.326280\pi\)
\(104\) −1.46410 −0.143567
\(105\) 0 0
\(106\) 4.73205 0.459617
\(107\) −0.928203 −0.0897328 −0.0448664 0.998993i \(-0.514286\pi\)
−0.0448664 + 0.998993i \(0.514286\pi\)
\(108\) 0 0
\(109\) −1.80385 −0.172777 −0.0863886 0.996262i \(-0.527533\pi\)
−0.0863886 + 0.996262i \(0.527533\pi\)
\(110\) 1.00000 0.0953463
\(111\) 0 0
\(112\) 1.00000 0.0944911
\(113\) −12.9282 −1.21618 −0.608092 0.793867i \(-0.708065\pi\)
−0.608092 + 0.793867i \(0.708065\pi\)
\(114\) 0 0
\(115\) 1.26795 0.118237
\(116\) 4.73205 0.439360
\(117\) 0 0
\(118\) 13.8564 1.27559
\(119\) −3.46410 −0.317554
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 2.00000 0.181071
\(123\) 0 0
\(124\) 8.92820 0.801776
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) 9.85641 0.874615 0.437307 0.899312i \(-0.355932\pi\)
0.437307 + 0.899312i \(0.355932\pi\)
\(128\) 1.00000 0.0883883
\(129\) 0 0
\(130\) 1.46410 0.128410
\(131\) 17.6603 1.54298 0.771492 0.636239i \(-0.219511\pi\)
0.771492 + 0.636239i \(0.219511\pi\)
\(132\) 0 0
\(133\) −2.73205 −0.236899
\(134\) −10.9282 −0.944053
\(135\) 0 0
\(136\) −3.46410 −0.297044
\(137\) −7.85641 −0.671218 −0.335609 0.942001i \(-0.608942\pi\)
−0.335609 + 0.942001i \(0.608942\pi\)
\(138\) 0 0
\(139\) 4.19615 0.355913 0.177957 0.984038i \(-0.443051\pi\)
0.177957 + 0.984038i \(0.443051\pi\)
\(140\) −1.00000 −0.0845154
\(141\) 0 0
\(142\) −9.46410 −0.794210
\(143\) 1.46410 0.122434
\(144\) 0 0
\(145\) −4.73205 −0.392975
\(146\) 11.4641 0.948776
\(147\) 0 0
\(148\) 3.26795 0.268624
\(149\) 7.26795 0.595414 0.297707 0.954657i \(-0.403778\pi\)
0.297707 + 0.954657i \(0.403778\pi\)
\(150\) 0 0
\(151\) −7.80385 −0.635068 −0.317534 0.948247i \(-0.602855\pi\)
−0.317534 + 0.948247i \(0.602855\pi\)
\(152\) −2.73205 −0.221599
\(153\) 0 0
\(154\) −1.00000 −0.0805823
\(155\) −8.92820 −0.717131
\(156\) 0 0
\(157\) 6.39230 0.510161 0.255081 0.966920i \(-0.417898\pi\)
0.255081 + 0.966920i \(0.417898\pi\)
\(158\) 6.73205 0.535573
\(159\) 0 0
\(160\) −1.00000 −0.0790569
\(161\) −1.26795 −0.0999284
\(162\) 0 0
\(163\) 8.00000 0.626608 0.313304 0.949653i \(-0.398564\pi\)
0.313304 + 0.949653i \(0.398564\pi\)
\(164\) 4.73205 0.369511
\(165\) 0 0
\(166\) 4.39230 0.340909
\(167\) 13.8564 1.07224 0.536120 0.844141i \(-0.319889\pi\)
0.536120 + 0.844141i \(0.319889\pi\)
\(168\) 0 0
\(169\) −10.8564 −0.835108
\(170\) 3.46410 0.265684
\(171\) 0 0
\(172\) −4.92820 −0.375772
\(173\) −0.928203 −0.0705700 −0.0352850 0.999377i \(-0.511234\pi\)
−0.0352850 + 0.999377i \(0.511234\pi\)
\(174\) 0 0
\(175\) 1.00000 0.0755929
\(176\) −1.00000 −0.0753778
\(177\) 0 0
\(178\) 15.4641 1.15908
\(179\) 19.8564 1.48414 0.742069 0.670324i \(-0.233845\pi\)
0.742069 + 0.670324i \(0.233845\pi\)
\(180\) 0 0
\(181\) −11.8564 −0.881280 −0.440640 0.897684i \(-0.645248\pi\)
−0.440640 + 0.897684i \(0.645248\pi\)
\(182\) −1.46410 −0.108526
\(183\) 0 0
\(184\) −1.26795 −0.0934745
\(185\) −3.26795 −0.240264
\(186\) 0 0
\(187\) 3.46410 0.253320
\(188\) 0 0
\(189\) 0 0
\(190\) 2.73205 0.198204
\(191\) −6.92820 −0.501307 −0.250654 0.968077i \(-0.580646\pi\)
−0.250654 + 0.968077i \(0.580646\pi\)
\(192\) 0 0
\(193\) −25.4641 −1.83295 −0.916473 0.400096i \(-0.868977\pi\)
−0.916473 + 0.400096i \(0.868977\pi\)
\(194\) 5.80385 0.416692
\(195\) 0 0
\(196\) 1.00000 0.0714286
\(197\) 22.3923 1.59539 0.797693 0.603064i \(-0.206054\pi\)
0.797693 + 0.603064i \(0.206054\pi\)
\(198\) 0 0
\(199\) −24.7846 −1.75693 −0.878467 0.477803i \(-0.841433\pi\)
−0.878467 + 0.477803i \(0.841433\pi\)
\(200\) 1.00000 0.0707107
\(201\) 0 0
\(202\) 6.00000 0.422159
\(203\) 4.73205 0.332125
\(204\) 0 0
\(205\) −4.73205 −0.330501
\(206\) 10.5359 0.734071
\(207\) 0 0
\(208\) −1.46410 −0.101517
\(209\) 2.73205 0.188980
\(210\) 0 0
\(211\) 8.00000 0.550743 0.275371 0.961338i \(-0.411199\pi\)
0.275371 + 0.961338i \(0.411199\pi\)
\(212\) 4.73205 0.324999
\(213\) 0 0
\(214\) −0.928203 −0.0634507
\(215\) 4.92820 0.336101
\(216\) 0 0
\(217\) 8.92820 0.606086
\(218\) −1.80385 −0.122172
\(219\) 0 0
\(220\) 1.00000 0.0674200
\(221\) 5.07180 0.341166
\(222\) 0 0
\(223\) −8.39230 −0.561990 −0.280995 0.959709i \(-0.590664\pi\)
−0.280995 + 0.959709i \(0.590664\pi\)
\(224\) 1.00000 0.0668153
\(225\) 0 0
\(226\) −12.9282 −0.859971
\(227\) −13.8564 −0.919682 −0.459841 0.888001i \(-0.652094\pi\)
−0.459841 + 0.888001i \(0.652094\pi\)
\(228\) 0 0
\(229\) −13.4641 −0.889733 −0.444866 0.895597i \(-0.646749\pi\)
−0.444866 + 0.895597i \(0.646749\pi\)
\(230\) 1.26795 0.0836061
\(231\) 0 0
\(232\) 4.73205 0.310674
\(233\) −7.85641 −0.514690 −0.257345 0.966320i \(-0.582848\pi\)
−0.257345 + 0.966320i \(0.582848\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 13.8564 0.901975
\(237\) 0 0
\(238\) −3.46410 −0.224544
\(239\) 3.80385 0.246050 0.123025 0.992404i \(-0.460740\pi\)
0.123025 + 0.992404i \(0.460740\pi\)
\(240\) 0 0
\(241\) −18.1962 −1.17212 −0.586059 0.810269i \(-0.699321\pi\)
−0.586059 + 0.810269i \(0.699321\pi\)
\(242\) 1.00000 0.0642824
\(243\) 0 0
\(244\) 2.00000 0.128037
\(245\) −1.00000 −0.0638877
\(246\) 0 0
\(247\) 4.00000 0.254514
\(248\) 8.92820 0.566941
\(249\) 0 0
\(250\) −1.00000 −0.0632456
\(251\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(252\) 0 0
\(253\) 1.26795 0.0797153
\(254\) 9.85641 0.618446
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −22.9808 −1.43350 −0.716750 0.697330i \(-0.754371\pi\)
−0.716750 + 0.697330i \(0.754371\pi\)
\(258\) 0 0
\(259\) 3.26795 0.203060
\(260\) 1.46410 0.0907997
\(261\) 0 0
\(262\) 17.6603 1.09105
\(263\) −18.9282 −1.16716 −0.583582 0.812055i \(-0.698349\pi\)
−0.583582 + 0.812055i \(0.698349\pi\)
\(264\) 0 0
\(265\) −4.73205 −0.290688
\(266\) −2.73205 −0.167513
\(267\) 0 0
\(268\) −10.9282 −0.667546
\(269\) −4.39230 −0.267804 −0.133902 0.990995i \(-0.542751\pi\)
−0.133902 + 0.990995i \(0.542751\pi\)
\(270\) 0 0
\(271\) 24.3923 1.48173 0.740863 0.671656i \(-0.234417\pi\)
0.740863 + 0.671656i \(0.234417\pi\)
\(272\) −3.46410 −0.210042
\(273\) 0 0
\(274\) −7.85641 −0.474623
\(275\) −1.00000 −0.0603023
\(276\) 0 0
\(277\) 27.8564 1.67373 0.836865 0.547410i \(-0.184386\pi\)
0.836865 + 0.547410i \(0.184386\pi\)
\(278\) 4.19615 0.251668
\(279\) 0 0
\(280\) −1.00000 −0.0597614
\(281\) 10.3923 0.619953 0.309976 0.950744i \(-0.399679\pi\)
0.309976 + 0.950744i \(0.399679\pi\)
\(282\) 0 0
\(283\) −10.9282 −0.649614 −0.324807 0.945780i \(-0.605299\pi\)
−0.324807 + 0.945780i \(0.605299\pi\)
\(284\) −9.46410 −0.561591
\(285\) 0 0
\(286\) 1.46410 0.0865741
\(287\) 4.73205 0.279324
\(288\) 0 0
\(289\) −5.00000 −0.294118
\(290\) −4.73205 −0.277876
\(291\) 0 0
\(292\) 11.4641 0.670886
\(293\) 2.53590 0.148149 0.0740744 0.997253i \(-0.476400\pi\)
0.0740744 + 0.997253i \(0.476400\pi\)
\(294\) 0 0
\(295\) −13.8564 −0.806751
\(296\) 3.26795 0.189946
\(297\) 0 0
\(298\) 7.26795 0.421021
\(299\) 1.85641 0.107359
\(300\) 0 0
\(301\) −4.92820 −0.284057
\(302\) −7.80385 −0.449061
\(303\) 0 0
\(304\) −2.73205 −0.156694
\(305\) −2.00000 −0.114520
\(306\) 0 0
\(307\) 31.3205 1.78756 0.893778 0.448510i \(-0.148045\pi\)
0.893778 + 0.448510i \(0.148045\pi\)
\(308\) −1.00000 −0.0569803
\(309\) 0 0
\(310\) −8.92820 −0.507088
\(311\) −7.85641 −0.445496 −0.222748 0.974876i \(-0.571503\pi\)
−0.222748 + 0.974876i \(0.571503\pi\)
\(312\) 0 0
\(313\) 27.2679 1.54128 0.770638 0.637273i \(-0.219938\pi\)
0.770638 + 0.637273i \(0.219938\pi\)
\(314\) 6.39230 0.360739
\(315\) 0 0
\(316\) 6.73205 0.378707
\(317\) 32.4449 1.82229 0.911143 0.412091i \(-0.135202\pi\)
0.911143 + 0.412091i \(0.135202\pi\)
\(318\) 0 0
\(319\) −4.73205 −0.264944
\(320\) −1.00000 −0.0559017
\(321\) 0 0
\(322\) −1.26795 −0.0706600
\(323\) 9.46410 0.526597
\(324\) 0 0
\(325\) −1.46410 −0.0812137
\(326\) 8.00000 0.443079
\(327\) 0 0
\(328\) 4.73205 0.261284
\(329\) 0 0
\(330\) 0 0
\(331\) −4.92820 −0.270879 −0.135439 0.990786i \(-0.543245\pi\)
−0.135439 + 0.990786i \(0.543245\pi\)
\(332\) 4.39230 0.241059
\(333\) 0 0
\(334\) 13.8564 0.758189
\(335\) 10.9282 0.597072
\(336\) 0 0
\(337\) −30.7846 −1.67694 −0.838472 0.544944i \(-0.816551\pi\)
−0.838472 + 0.544944i \(0.816551\pi\)
\(338\) −10.8564 −0.590511
\(339\) 0 0
\(340\) 3.46410 0.187867
\(341\) −8.92820 −0.483489
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) −4.92820 −0.265711
\(345\) 0 0
\(346\) −0.928203 −0.0499005
\(347\) −7.85641 −0.421754 −0.210877 0.977513i \(-0.567632\pi\)
−0.210877 + 0.977513i \(0.567632\pi\)
\(348\) 0 0
\(349\) 30.3923 1.62686 0.813431 0.581661i \(-0.197597\pi\)
0.813431 + 0.581661i \(0.197597\pi\)
\(350\) 1.00000 0.0534522
\(351\) 0 0
\(352\) −1.00000 −0.0533002
\(353\) 32.4449 1.72687 0.863433 0.504464i \(-0.168310\pi\)
0.863433 + 0.504464i \(0.168310\pi\)
\(354\) 0 0
\(355\) 9.46410 0.502302
\(356\) 15.4641 0.819596
\(357\) 0 0
\(358\) 19.8564 1.04944
\(359\) 1.26795 0.0669198 0.0334599 0.999440i \(-0.489347\pi\)
0.0334599 + 0.999440i \(0.489347\pi\)
\(360\) 0 0
\(361\) −11.5359 −0.607153
\(362\) −11.8564 −0.623159
\(363\) 0 0
\(364\) −1.46410 −0.0767398
\(365\) −11.4641 −0.600059
\(366\) 0 0
\(367\) 29.4641 1.53801 0.769007 0.639241i \(-0.220751\pi\)
0.769007 + 0.639241i \(0.220751\pi\)
\(368\) −1.26795 −0.0660964
\(369\) 0 0
\(370\) −3.26795 −0.169893
\(371\) 4.73205 0.245676
\(372\) 0 0
\(373\) 30.3923 1.57365 0.786827 0.617174i \(-0.211722\pi\)
0.786827 + 0.617174i \(0.211722\pi\)
\(374\) 3.46410 0.179124
\(375\) 0 0
\(376\) 0 0
\(377\) −6.92820 −0.356821
\(378\) 0 0
\(379\) 23.7128 1.21805 0.609023 0.793153i \(-0.291562\pi\)
0.609023 + 0.793153i \(0.291562\pi\)
\(380\) 2.73205 0.140151
\(381\) 0 0
\(382\) −6.92820 −0.354478
\(383\) −16.3923 −0.837608 −0.418804 0.908077i \(-0.637551\pi\)
−0.418804 + 0.908077i \(0.637551\pi\)
\(384\) 0 0
\(385\) 1.00000 0.0509647
\(386\) −25.4641 −1.29609
\(387\) 0 0
\(388\) 5.80385 0.294646
\(389\) 24.2487 1.22946 0.614729 0.788738i \(-0.289265\pi\)
0.614729 + 0.788738i \(0.289265\pi\)
\(390\) 0 0
\(391\) 4.39230 0.222128
\(392\) 1.00000 0.0505076
\(393\) 0 0
\(394\) 22.3923 1.12811
\(395\) −6.73205 −0.338726
\(396\) 0 0
\(397\) −38.3923 −1.92685 −0.963427 0.267970i \(-0.913647\pi\)
−0.963427 + 0.267970i \(0.913647\pi\)
\(398\) −24.7846 −1.24234
\(399\) 0 0
\(400\) 1.00000 0.0500000
\(401\) 25.1769 1.25728 0.628638 0.777698i \(-0.283613\pi\)
0.628638 + 0.777698i \(0.283613\pi\)
\(402\) 0 0
\(403\) −13.0718 −0.651153
\(404\) 6.00000 0.298511
\(405\) 0 0
\(406\) 4.73205 0.234848
\(407\) −3.26795 −0.161986
\(408\) 0 0
\(409\) 22.1962 1.09753 0.548765 0.835977i \(-0.315098\pi\)
0.548765 + 0.835977i \(0.315098\pi\)
\(410\) −4.73205 −0.233699
\(411\) 0 0
\(412\) 10.5359 0.519066
\(413\) 13.8564 0.681829
\(414\) 0 0
\(415\) −4.39230 −0.215610
\(416\) −1.46410 −0.0717835
\(417\) 0 0
\(418\) 2.73205 0.133629
\(419\) −32.7846 −1.60163 −0.800816 0.598910i \(-0.795601\pi\)
−0.800816 + 0.598910i \(0.795601\pi\)
\(420\) 0 0
\(421\) 10.7846 0.525610 0.262805 0.964849i \(-0.415352\pi\)
0.262805 + 0.964849i \(0.415352\pi\)
\(422\) 8.00000 0.389434
\(423\) 0 0
\(424\) 4.73205 0.229809
\(425\) −3.46410 −0.168034
\(426\) 0 0
\(427\) 2.00000 0.0967868
\(428\) −0.928203 −0.0448664
\(429\) 0 0
\(430\) 4.92820 0.237659
\(431\) −18.3397 −0.883394 −0.441697 0.897164i \(-0.645623\pi\)
−0.441697 + 0.897164i \(0.645623\pi\)
\(432\) 0 0
\(433\) 0.732051 0.0351801 0.0175901 0.999845i \(-0.494401\pi\)
0.0175901 + 0.999845i \(0.494401\pi\)
\(434\) 8.92820 0.428567
\(435\) 0 0
\(436\) −1.80385 −0.0863886
\(437\) 3.46410 0.165710
\(438\) 0 0
\(439\) 36.3923 1.73691 0.868455 0.495768i \(-0.165113\pi\)
0.868455 + 0.495768i \(0.165113\pi\)
\(440\) 1.00000 0.0476731
\(441\) 0 0
\(442\) 5.07180 0.241241
\(443\) −20.7846 −0.987507 −0.493753 0.869602i \(-0.664375\pi\)
−0.493753 + 0.869602i \(0.664375\pi\)
\(444\) 0 0
\(445\) −15.4641 −0.733069
\(446\) −8.39230 −0.397387
\(447\) 0 0
\(448\) 1.00000 0.0472456
\(449\) 35.3205 1.66688 0.833439 0.552612i \(-0.186369\pi\)
0.833439 + 0.552612i \(0.186369\pi\)
\(450\) 0 0
\(451\) −4.73205 −0.222824
\(452\) −12.9282 −0.608092
\(453\) 0 0
\(454\) −13.8564 −0.650313
\(455\) 1.46410 0.0686381
\(456\) 0 0
\(457\) −34.0000 −1.59045 −0.795226 0.606313i \(-0.792648\pi\)
−0.795226 + 0.606313i \(0.792648\pi\)
\(458\) −13.4641 −0.629136
\(459\) 0 0
\(460\) 1.26795 0.0591184
\(461\) −29.3205 −1.36559 −0.682796 0.730609i \(-0.739236\pi\)
−0.682796 + 0.730609i \(0.739236\pi\)
\(462\) 0 0
\(463\) 12.9808 0.603267 0.301634 0.953424i \(-0.402468\pi\)
0.301634 + 0.953424i \(0.402468\pi\)
\(464\) 4.73205 0.219680
\(465\) 0 0
\(466\) −7.85641 −0.363941
\(467\) −17.6603 −0.817219 −0.408610 0.912709i \(-0.633986\pi\)
−0.408610 + 0.912709i \(0.633986\pi\)
\(468\) 0 0
\(469\) −10.9282 −0.504618
\(470\) 0 0
\(471\) 0 0
\(472\) 13.8564 0.637793
\(473\) 4.92820 0.226599
\(474\) 0 0
\(475\) −2.73205 −0.125355
\(476\) −3.46410 −0.158777
\(477\) 0 0
\(478\) 3.80385 0.173984
\(479\) −13.8564 −0.633115 −0.316558 0.948573i \(-0.602527\pi\)
−0.316558 + 0.948573i \(0.602527\pi\)
\(480\) 0 0
\(481\) −4.78461 −0.218159
\(482\) −18.1962 −0.828812
\(483\) 0 0
\(484\) 1.00000 0.0454545
\(485\) −5.80385 −0.263539
\(486\) 0 0
\(487\) 4.87564 0.220937 0.110468 0.993880i \(-0.464765\pi\)
0.110468 + 0.993880i \(0.464765\pi\)
\(488\) 2.00000 0.0905357
\(489\) 0 0
\(490\) −1.00000 −0.0451754
\(491\) −42.9282 −1.93732 −0.968661 0.248385i \(-0.920100\pi\)
−0.968661 + 0.248385i \(0.920100\pi\)
\(492\) 0 0
\(493\) −16.3923 −0.738272
\(494\) 4.00000 0.179969
\(495\) 0 0
\(496\) 8.92820 0.400888
\(497\) −9.46410 −0.424523
\(498\) 0 0
\(499\) 2.00000 0.0895323 0.0447661 0.998997i \(-0.485746\pi\)
0.0447661 + 0.998997i \(0.485746\pi\)
\(500\) −1.00000 −0.0447214
\(501\) 0 0
\(502\) 0 0
\(503\) 18.9282 0.843967 0.421983 0.906604i \(-0.361334\pi\)
0.421983 + 0.906604i \(0.361334\pi\)
\(504\) 0 0
\(505\) −6.00000 −0.266996
\(506\) 1.26795 0.0563672
\(507\) 0 0
\(508\) 9.85641 0.437307
\(509\) −2.78461 −0.123426 −0.0617128 0.998094i \(-0.519656\pi\)
−0.0617128 + 0.998094i \(0.519656\pi\)
\(510\) 0 0
\(511\) 11.4641 0.507142
\(512\) 1.00000 0.0441942
\(513\) 0 0
\(514\) −22.9808 −1.01364
\(515\) −10.5359 −0.464267
\(516\) 0 0
\(517\) 0 0
\(518\) 3.26795 0.143585
\(519\) 0 0
\(520\) 1.46410 0.0642051
\(521\) −10.3923 −0.455295 −0.227648 0.973744i \(-0.573103\pi\)
−0.227648 + 0.973744i \(0.573103\pi\)
\(522\) 0 0
\(523\) 19.3205 0.844827 0.422413 0.906403i \(-0.361183\pi\)
0.422413 + 0.906403i \(0.361183\pi\)
\(524\) 17.6603 0.771492
\(525\) 0 0
\(526\) −18.9282 −0.825309
\(527\) −30.9282 −1.34725
\(528\) 0 0
\(529\) −21.3923 −0.930100
\(530\) −4.73205 −0.205547
\(531\) 0 0
\(532\) −2.73205 −0.118449
\(533\) −6.92820 −0.300094
\(534\) 0 0
\(535\) 0.928203 0.0401297
\(536\) −10.9282 −0.472026
\(537\) 0 0
\(538\) −4.39230 −0.189366
\(539\) −1.00000 −0.0430730
\(540\) 0 0
\(541\) 17.1244 0.736234 0.368117 0.929780i \(-0.380003\pi\)
0.368117 + 0.929780i \(0.380003\pi\)
\(542\) 24.3923 1.04774
\(543\) 0 0
\(544\) −3.46410 −0.148522
\(545\) 1.80385 0.0772683
\(546\) 0 0
\(547\) 4.78461 0.204575 0.102288 0.994755i \(-0.467384\pi\)
0.102288 + 0.994755i \(0.467384\pi\)
\(548\) −7.85641 −0.335609
\(549\) 0 0
\(550\) −1.00000 −0.0426401
\(551\) −12.9282 −0.550760
\(552\) 0 0
\(553\) 6.73205 0.286276
\(554\) 27.8564 1.18351
\(555\) 0 0
\(556\) 4.19615 0.177957
\(557\) 12.2487 0.518995 0.259497 0.965744i \(-0.416443\pi\)
0.259497 + 0.965744i \(0.416443\pi\)
\(558\) 0 0
\(559\) 7.21539 0.305178
\(560\) −1.00000 −0.0422577
\(561\) 0 0
\(562\) 10.3923 0.438373
\(563\) 20.7846 0.875967 0.437983 0.898983i \(-0.355693\pi\)
0.437983 + 0.898983i \(0.355693\pi\)
\(564\) 0 0
\(565\) 12.9282 0.543894
\(566\) −10.9282 −0.459347
\(567\) 0 0
\(568\) −9.46410 −0.397105
\(569\) 30.0000 1.25767 0.628833 0.777541i \(-0.283533\pi\)
0.628833 + 0.777541i \(0.283533\pi\)
\(570\) 0 0
\(571\) −20.3923 −0.853391 −0.426696 0.904395i \(-0.640322\pi\)
−0.426696 + 0.904395i \(0.640322\pi\)
\(572\) 1.46410 0.0612172
\(573\) 0 0
\(574\) 4.73205 0.197512
\(575\) −1.26795 −0.0528771
\(576\) 0 0
\(577\) 8.33975 0.347188 0.173594 0.984817i \(-0.444462\pi\)
0.173594 + 0.984817i \(0.444462\pi\)
\(578\) −5.00000 −0.207973
\(579\) 0 0
\(580\) −4.73205 −0.196488
\(581\) 4.39230 0.182224
\(582\) 0 0
\(583\) −4.73205 −0.195982
\(584\) 11.4641 0.474388
\(585\) 0 0
\(586\) 2.53590 0.104757
\(587\) 28.9808 1.19616 0.598082 0.801435i \(-0.295930\pi\)
0.598082 + 0.801435i \(0.295930\pi\)
\(588\) 0 0
\(589\) −24.3923 −1.00507
\(590\) −13.8564 −0.570459
\(591\) 0 0
\(592\) 3.26795 0.134312
\(593\) −32.5359 −1.33609 −0.668045 0.744121i \(-0.732868\pi\)
−0.668045 + 0.744121i \(0.732868\pi\)
\(594\) 0 0
\(595\) 3.46410 0.142014
\(596\) 7.26795 0.297707
\(597\) 0 0
\(598\) 1.85641 0.0759141
\(599\) 32.1051 1.31178 0.655890 0.754857i \(-0.272294\pi\)
0.655890 + 0.754857i \(0.272294\pi\)
\(600\) 0 0
\(601\) 28.4449 1.16029 0.580145 0.814513i \(-0.302996\pi\)
0.580145 + 0.814513i \(0.302996\pi\)
\(602\) −4.92820 −0.200859
\(603\) 0 0
\(604\) −7.80385 −0.317534
\(605\) −1.00000 −0.0406558
\(606\) 0 0
\(607\) 34.7846 1.41186 0.705932 0.708280i \(-0.250528\pi\)
0.705932 + 0.708280i \(0.250528\pi\)
\(608\) −2.73205 −0.110799
\(609\) 0 0
\(610\) −2.00000 −0.0809776
\(611\) 0 0
\(612\) 0 0
\(613\) 27.1769 1.09767 0.548833 0.835932i \(-0.315072\pi\)
0.548833 + 0.835932i \(0.315072\pi\)
\(614\) 31.3205 1.26399
\(615\) 0 0
\(616\) −1.00000 −0.0402911
\(617\) −8.53590 −0.343642 −0.171821 0.985128i \(-0.554965\pi\)
−0.171821 + 0.985128i \(0.554965\pi\)
\(618\) 0 0
\(619\) −10.9282 −0.439242 −0.219621 0.975585i \(-0.570482\pi\)
−0.219621 + 0.975585i \(0.570482\pi\)
\(620\) −8.92820 −0.358565
\(621\) 0 0
\(622\) −7.85641 −0.315013
\(623\) 15.4641 0.619556
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 27.2679 1.08985
\(627\) 0 0
\(628\) 6.39230 0.255081
\(629\) −11.3205 −0.451378
\(630\) 0 0
\(631\) −12.7846 −0.508947 −0.254474 0.967080i \(-0.581902\pi\)
−0.254474 + 0.967080i \(0.581902\pi\)
\(632\) 6.73205 0.267787
\(633\) 0 0
\(634\) 32.4449 1.28855
\(635\) −9.85641 −0.391140
\(636\) 0 0
\(637\) −1.46410 −0.0580098
\(638\) −4.73205 −0.187344
\(639\) 0 0
\(640\) −1.00000 −0.0395285
\(641\) −19.8564 −0.784281 −0.392140 0.919905i \(-0.628265\pi\)
−0.392140 + 0.919905i \(0.628265\pi\)
\(642\) 0 0
\(643\) −14.7321 −0.580975 −0.290488 0.956879i \(-0.593818\pi\)
−0.290488 + 0.956879i \(0.593818\pi\)
\(644\) −1.26795 −0.0499642
\(645\) 0 0
\(646\) 9.46410 0.372360
\(647\) −37.1769 −1.46158 −0.730788 0.682605i \(-0.760847\pi\)
−0.730788 + 0.682605i \(0.760847\pi\)
\(648\) 0 0
\(649\) −13.8564 −0.543912
\(650\) −1.46410 −0.0574268
\(651\) 0 0
\(652\) 8.00000 0.313304
\(653\) 9.80385 0.383654 0.191827 0.981429i \(-0.438559\pi\)
0.191827 + 0.981429i \(0.438559\pi\)
\(654\) 0 0
\(655\) −17.6603 −0.690043
\(656\) 4.73205 0.184756
\(657\) 0 0
\(658\) 0 0
\(659\) −6.24871 −0.243415 −0.121708 0.992566i \(-0.538837\pi\)
−0.121708 + 0.992566i \(0.538837\pi\)
\(660\) 0 0
\(661\) 3.85641 0.149997 0.0749984 0.997184i \(-0.476105\pi\)
0.0749984 + 0.997184i \(0.476105\pi\)
\(662\) −4.92820 −0.191540
\(663\) 0 0
\(664\) 4.39230 0.170454
\(665\) 2.73205 0.105944
\(666\) 0 0
\(667\) −6.00000 −0.232321
\(668\) 13.8564 0.536120
\(669\) 0 0
\(670\) 10.9282 0.422193
\(671\) −2.00000 −0.0772091
\(672\) 0 0
\(673\) −20.1436 −0.776478 −0.388239 0.921559i \(-0.626917\pi\)
−0.388239 + 0.921559i \(0.626917\pi\)
\(674\) −30.7846 −1.18578
\(675\) 0 0
\(676\) −10.8564 −0.417554
\(677\) −19.8564 −0.763144 −0.381572 0.924339i \(-0.624617\pi\)
−0.381572 + 0.924339i \(0.624617\pi\)
\(678\) 0 0
\(679\) 5.80385 0.222731
\(680\) 3.46410 0.132842
\(681\) 0 0
\(682\) −8.92820 −0.341879
\(683\) −42.2487 −1.61660 −0.808301 0.588769i \(-0.799613\pi\)
−0.808301 + 0.588769i \(0.799613\pi\)
\(684\) 0 0
\(685\) 7.85641 0.300178
\(686\) 1.00000 0.0381802
\(687\) 0 0
\(688\) −4.92820 −0.187886
\(689\) −6.92820 −0.263944
\(690\) 0 0
\(691\) 31.3205 1.19149 0.595744 0.803174i \(-0.296857\pi\)
0.595744 + 0.803174i \(0.296857\pi\)
\(692\) −0.928203 −0.0352850
\(693\) 0 0
\(694\) −7.85641 −0.298225
\(695\) −4.19615 −0.159169
\(696\) 0 0
\(697\) −16.3923 −0.620903
\(698\) 30.3923 1.15037
\(699\) 0 0
\(700\) 1.00000 0.0377964
\(701\) −14.1962 −0.536181 −0.268091 0.963394i \(-0.586393\pi\)
−0.268091 + 0.963394i \(0.586393\pi\)
\(702\) 0 0
\(703\) −8.92820 −0.336734
\(704\) −1.00000 −0.0376889
\(705\) 0 0
\(706\) 32.4449 1.22108
\(707\) 6.00000 0.225653
\(708\) 0 0
\(709\) −2.39230 −0.0898449 −0.0449224 0.998990i \(-0.514304\pi\)
−0.0449224 + 0.998990i \(0.514304\pi\)
\(710\) 9.46410 0.355181
\(711\) 0 0
\(712\) 15.4641 0.579542
\(713\) −11.3205 −0.423956
\(714\) 0 0
\(715\) −1.46410 −0.0547543
\(716\) 19.8564 0.742069
\(717\) 0 0
\(718\) 1.26795 0.0473194
\(719\) −24.0000 −0.895049 −0.447524 0.894272i \(-0.647694\pi\)
−0.447524 + 0.894272i \(0.647694\pi\)
\(720\) 0 0
\(721\) 10.5359 0.392377
\(722\) −11.5359 −0.429322
\(723\) 0 0
\(724\) −11.8564 −0.440640
\(725\) 4.73205 0.175744
\(726\) 0 0
\(727\) −38.6410 −1.43312 −0.716558 0.697528i \(-0.754283\pi\)
−0.716558 + 0.697528i \(0.754283\pi\)
\(728\) −1.46410 −0.0542632
\(729\) 0 0
\(730\) −11.4641 −0.424305
\(731\) 17.0718 0.631423
\(732\) 0 0
\(733\) 2.00000 0.0738717 0.0369358 0.999318i \(-0.488240\pi\)
0.0369358 + 0.999318i \(0.488240\pi\)
\(734\) 29.4641 1.08754
\(735\) 0 0
\(736\) −1.26795 −0.0467372
\(737\) 10.9282 0.402546
\(738\) 0 0
\(739\) 18.1436 0.667423 0.333711 0.942675i \(-0.391699\pi\)
0.333711 + 0.942675i \(0.391699\pi\)
\(740\) −3.26795 −0.120132
\(741\) 0 0
\(742\) 4.73205 0.173719
\(743\) −3.71281 −0.136210 −0.0681049 0.997678i \(-0.521695\pi\)
−0.0681049 + 0.997678i \(0.521695\pi\)
\(744\) 0 0
\(745\) −7.26795 −0.266277
\(746\) 30.3923 1.11274
\(747\) 0 0
\(748\) 3.46410 0.126660
\(749\) −0.928203 −0.0339158
\(750\) 0 0
\(751\) 2.24871 0.0820566 0.0410283 0.999158i \(-0.486937\pi\)
0.0410283 + 0.999158i \(0.486937\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) −6.92820 −0.252310
\(755\) 7.80385 0.284011
\(756\) 0 0
\(757\) 23.3731 0.849509 0.424754 0.905309i \(-0.360360\pi\)
0.424754 + 0.905309i \(0.360360\pi\)
\(758\) 23.7128 0.861288
\(759\) 0 0
\(760\) 2.73205 0.0991019
\(761\) −19.2679 −0.698463 −0.349231 0.937037i \(-0.613557\pi\)
−0.349231 + 0.937037i \(0.613557\pi\)
\(762\) 0 0
\(763\) −1.80385 −0.0653037
\(764\) −6.92820 −0.250654
\(765\) 0 0
\(766\) −16.3923 −0.592278
\(767\) −20.2872 −0.732528
\(768\) 0 0
\(769\) −24.4449 −0.881504 −0.440752 0.897629i \(-0.645288\pi\)
−0.440752 + 0.897629i \(0.645288\pi\)
\(770\) 1.00000 0.0360375
\(771\) 0 0
\(772\) −25.4641 −0.916473
\(773\) −30.0000 −1.07903 −0.539513 0.841978i \(-0.681391\pi\)
−0.539513 + 0.841978i \(0.681391\pi\)
\(774\) 0 0
\(775\) 8.92820 0.320711
\(776\) 5.80385 0.208346
\(777\) 0 0
\(778\) 24.2487 0.869358
\(779\) −12.9282 −0.463201
\(780\) 0 0
\(781\) 9.46410 0.338652
\(782\) 4.39230 0.157069
\(783\) 0 0
\(784\) 1.00000 0.0357143
\(785\) −6.39230 −0.228151
\(786\) 0 0
\(787\) −28.0000 −0.998092 −0.499046 0.866575i \(-0.666316\pi\)
−0.499046 + 0.866575i \(0.666316\pi\)
\(788\) 22.3923 0.797693
\(789\) 0 0
\(790\) −6.73205 −0.239516
\(791\) −12.9282 −0.459674
\(792\) 0 0
\(793\) −2.92820 −0.103984
\(794\) −38.3923 −1.36249
\(795\) 0 0
\(796\) −24.7846 −0.878467
\(797\) −6.67949 −0.236600 −0.118300 0.992978i \(-0.537744\pi\)
−0.118300 + 0.992978i \(0.537744\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 1.00000 0.0353553
\(801\) 0 0
\(802\) 25.1769 0.889028
\(803\) −11.4641 −0.404559
\(804\) 0 0
\(805\) 1.26795 0.0446893
\(806\) −13.0718 −0.460434
\(807\) 0 0
\(808\) 6.00000 0.211079
\(809\) −5.32051 −0.187059 −0.0935296 0.995617i \(-0.529815\pi\)
−0.0935296 + 0.995617i \(0.529815\pi\)
\(810\) 0 0
\(811\) −4.58846 −0.161123 −0.0805613 0.996750i \(-0.525671\pi\)
−0.0805613 + 0.996750i \(0.525671\pi\)
\(812\) 4.73205 0.166062
\(813\) 0 0
\(814\) −3.26795 −0.114542
\(815\) −8.00000 −0.280228
\(816\) 0 0
\(817\) 13.4641 0.471049
\(818\) 22.1962 0.776070
\(819\) 0 0
\(820\) −4.73205 −0.165250
\(821\) −52.0526 −1.81665 −0.908323 0.418269i \(-0.862637\pi\)
−0.908323 + 0.418269i \(0.862637\pi\)
\(822\) 0 0
\(823\) −24.1962 −0.843425 −0.421712 0.906730i \(-0.638571\pi\)
−0.421712 + 0.906730i \(0.638571\pi\)
\(824\) 10.5359 0.367035
\(825\) 0 0
\(826\) 13.8564 0.482126
\(827\) −53.5692 −1.86278 −0.931392 0.364017i \(-0.881405\pi\)
−0.931392 + 0.364017i \(0.881405\pi\)
\(828\) 0 0
\(829\) 40.1051 1.39291 0.696454 0.717601i \(-0.254760\pi\)
0.696454 + 0.717601i \(0.254760\pi\)
\(830\) −4.39230 −0.152459
\(831\) 0 0
\(832\) −1.46410 −0.0507586
\(833\) −3.46410 −0.120024
\(834\) 0 0
\(835\) −13.8564 −0.479521
\(836\) 2.73205 0.0944900
\(837\) 0 0
\(838\) −32.7846 −1.13253
\(839\) 38.7846 1.33899 0.669497 0.742815i \(-0.266510\pi\)
0.669497 + 0.742815i \(0.266510\pi\)
\(840\) 0 0
\(841\) −6.60770 −0.227852
\(842\) 10.7846 0.371662
\(843\) 0 0
\(844\) 8.00000 0.275371
\(845\) 10.8564 0.373472
\(846\) 0 0
\(847\) 1.00000 0.0343604
\(848\) 4.73205 0.162499
\(849\) 0 0
\(850\) −3.46410 −0.118818
\(851\) −4.14359 −0.142041
\(852\) 0 0
\(853\) 20.9282 0.716568 0.358284 0.933613i \(-0.383362\pi\)
0.358284 + 0.933613i \(0.383362\pi\)
\(854\) 2.00000 0.0684386
\(855\) 0 0
\(856\) −0.928203 −0.0317253
\(857\) 19.1769 0.655071 0.327535 0.944839i \(-0.393782\pi\)
0.327535 + 0.944839i \(0.393782\pi\)
\(858\) 0 0
\(859\) 40.7846 1.39155 0.695776 0.718258i \(-0.255060\pi\)
0.695776 + 0.718258i \(0.255060\pi\)
\(860\) 4.92820 0.168050
\(861\) 0 0
\(862\) −18.3397 −0.624654
\(863\) −24.5885 −0.837001 −0.418500 0.908217i \(-0.637444\pi\)
−0.418500 + 0.908217i \(0.637444\pi\)
\(864\) 0 0
\(865\) 0.928203 0.0315599
\(866\) 0.732051 0.0248761
\(867\) 0 0
\(868\) 8.92820 0.303043
\(869\) −6.73205 −0.228369
\(870\) 0 0
\(871\) 16.0000 0.542139
\(872\) −1.80385 −0.0610860
\(873\) 0 0
\(874\) 3.46410 0.117175
\(875\) −1.00000 −0.0338062
\(876\) 0 0
\(877\) −23.1769 −0.782629 −0.391314 0.920257i \(-0.627979\pi\)
−0.391314 + 0.920257i \(0.627979\pi\)
\(878\) 36.3923 1.22818
\(879\) 0 0
\(880\) 1.00000 0.0337100
\(881\) −24.9282 −0.839853 −0.419926 0.907558i \(-0.637944\pi\)
−0.419926 + 0.907558i \(0.637944\pi\)
\(882\) 0 0
\(883\) −41.1769 −1.38571 −0.692857 0.721075i \(-0.743648\pi\)
−0.692857 + 0.721075i \(0.743648\pi\)
\(884\) 5.07180 0.170583
\(885\) 0 0
\(886\) −20.7846 −0.698273
\(887\) −24.0000 −0.805841 −0.402921 0.915235i \(-0.632005\pi\)
−0.402921 + 0.915235i \(0.632005\pi\)
\(888\) 0 0
\(889\) 9.85641 0.330573
\(890\) −15.4641 −0.518358
\(891\) 0 0
\(892\) −8.39230 −0.280995
\(893\) 0 0
\(894\) 0 0
\(895\) −19.8564 −0.663726
\(896\) 1.00000 0.0334077
\(897\) 0 0
\(898\) 35.3205 1.17866
\(899\) 42.2487 1.40907
\(900\) 0 0
\(901\) −16.3923 −0.546107
\(902\) −4.73205 −0.157560
\(903\) 0 0
\(904\) −12.9282 −0.429986
\(905\) 11.8564 0.394120
\(906\) 0 0
\(907\) −15.3205 −0.508709 −0.254355 0.967111i \(-0.581863\pi\)
−0.254355 + 0.967111i \(0.581863\pi\)
\(908\) −13.8564 −0.459841
\(909\) 0 0
\(910\) 1.46410 0.0485345
\(911\) −14.5359 −0.481596 −0.240798 0.970575i \(-0.577409\pi\)
−0.240798 + 0.970575i \(0.577409\pi\)
\(912\) 0 0
\(913\) −4.39230 −0.145364
\(914\) −34.0000 −1.12462
\(915\) 0 0
\(916\) −13.4641 −0.444866
\(917\) 17.6603 0.583193
\(918\) 0 0
\(919\) 24.9808 0.824039 0.412020 0.911175i \(-0.364824\pi\)
0.412020 + 0.911175i \(0.364824\pi\)
\(920\) 1.26795 0.0418030
\(921\) 0 0
\(922\) −29.3205 −0.965620
\(923\) 13.8564 0.456089
\(924\) 0 0
\(925\) 3.26795 0.107450
\(926\) 12.9808 0.426574
\(927\) 0 0
\(928\) 4.73205 0.155337
\(929\) −26.1051 −0.856481 −0.428241 0.903665i \(-0.640866\pi\)
−0.428241 + 0.903665i \(0.640866\pi\)
\(930\) 0 0
\(931\) −2.73205 −0.0895393
\(932\) −7.85641 −0.257345
\(933\) 0 0
\(934\) −17.6603 −0.577861
\(935\) −3.46410 −0.113288
\(936\) 0 0
\(937\) 26.0000 0.849383 0.424691 0.905338i \(-0.360383\pi\)
0.424691 + 0.905338i \(0.360383\pi\)
\(938\) −10.9282 −0.356818
\(939\) 0 0
\(940\) 0 0
\(941\) 35.5692 1.15952 0.579762 0.814786i \(-0.303146\pi\)
0.579762 + 0.814786i \(0.303146\pi\)
\(942\) 0 0
\(943\) −6.00000 −0.195387
\(944\) 13.8564 0.450988
\(945\) 0 0
\(946\) 4.92820 0.160230
\(947\) 25.1769 0.818140 0.409070 0.912503i \(-0.365853\pi\)
0.409070 + 0.912503i \(0.365853\pi\)
\(948\) 0 0
\(949\) −16.7846 −0.544851
\(950\) −2.73205 −0.0886394
\(951\) 0 0
\(952\) −3.46410 −0.112272
\(953\) 11.3205 0.366707 0.183354 0.983047i \(-0.441305\pi\)
0.183354 + 0.983047i \(0.441305\pi\)
\(954\) 0 0
\(955\) 6.92820 0.224191
\(956\) 3.80385 0.123025
\(957\) 0 0
\(958\) −13.8564 −0.447680
\(959\) −7.85641 −0.253697
\(960\) 0 0
\(961\) 48.7128 1.57138
\(962\) −4.78461 −0.154262
\(963\) 0 0
\(964\) −18.1962 −0.586059
\(965\) 25.4641 0.819718
\(966\) 0 0
\(967\) −26.1436 −0.840721 −0.420361 0.907357i \(-0.638096\pi\)
−0.420361 + 0.907357i \(0.638096\pi\)
\(968\) 1.00000 0.0321412
\(969\) 0 0
\(970\) −5.80385 −0.186350
\(971\) −49.8564 −1.59997 −0.799984 0.600021i \(-0.795159\pi\)
−0.799984 + 0.600021i \(0.795159\pi\)
\(972\) 0 0
\(973\) 4.19615 0.134522
\(974\) 4.87564 0.156226
\(975\) 0 0
\(976\) 2.00000 0.0640184
\(977\) 0.928203 0.0296959 0.0148479 0.999890i \(-0.495274\pi\)
0.0148479 + 0.999890i \(0.495274\pi\)
\(978\) 0 0
\(979\) −15.4641 −0.494235
\(980\) −1.00000 −0.0319438
\(981\) 0 0
\(982\) −42.9282 −1.36989
\(983\) 7.60770 0.242648 0.121324 0.992613i \(-0.461286\pi\)
0.121324 + 0.992613i \(0.461286\pi\)
\(984\) 0 0
\(985\) −22.3923 −0.713478
\(986\) −16.3923 −0.522037
\(987\) 0 0
\(988\) 4.00000 0.127257
\(989\) 6.24871 0.198697
\(990\) 0 0
\(991\) −22.9282 −0.728338 −0.364169 0.931333i \(-0.618647\pi\)
−0.364169 + 0.931333i \(0.618647\pi\)
\(992\) 8.92820 0.283471
\(993\) 0 0
\(994\) −9.46410 −0.300183
\(995\) 24.7846 0.785725
\(996\) 0 0
\(997\) −23.8564 −0.755540 −0.377770 0.925899i \(-0.623309\pi\)
−0.377770 + 0.925899i \(0.623309\pi\)
\(998\) 2.00000 0.0633089
\(999\) 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.ca.1.1 2
3.2 odd 2 770.2.a.h.1.2 2
12.11 even 2 6160.2.a.v.1.1 2
15.2 even 4 3850.2.c.s.1849.1 4
15.8 even 4 3850.2.c.s.1849.4 4
15.14 odd 2 3850.2.a.bm.1.1 2
21.20 even 2 5390.2.a.bk.1.1 2
33.32 even 2 8470.2.a.ce.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
770.2.a.h.1.2 2 3.2 odd 2
3850.2.a.bm.1.1 2 15.14 odd 2
3850.2.c.s.1849.1 4 15.2 even 4
3850.2.c.s.1849.4 4 15.8 even 4
5390.2.a.bk.1.1 2 21.20 even 2
6160.2.a.v.1.1 2 12.11 even 2
6930.2.a.ca.1.1 2 1.1 even 1 trivial
8470.2.a.ce.1.2 2 33.32 even 2