Properties

Label 6930.2.a.bv.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

\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{7} -1.00000 q^{8} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{7} -1.00000 q^{8} -1.00000 q^{10} -1.00000 q^{11} -1.46410 q^{13} -1.00000 q^{14} +1.00000 q^{16} +3.46410 q^{17} +6.73205 q^{19} +1.00000 q^{20} +1.00000 q^{22} +8.19615 q^{23} +1.00000 q^{25} +1.46410 q^{26} +1.00000 q^{28} +4.73205 q^{29} +2.00000 q^{31} -1.00000 q^{32} -3.46410 q^{34} +1.00000 q^{35} +0.732051 q^{37} -6.73205 q^{38} -1.00000 q^{40} +2.19615 q^{41} +2.00000 q^{43} -1.00000 q^{44} -8.19615 q^{46} -6.92820 q^{47} +1.00000 q^{49} -1.00000 q^{50} -1.46410 q^{52} +7.26795 q^{53} -1.00000 q^{55} -1.00000 q^{56} -4.73205 q^{58} -6.92820 q^{59} -4.92820 q^{61} -2.00000 q^{62} +1.00000 q^{64} -1.46410 q^{65} -4.00000 q^{67} +3.46410 q^{68} -1.00000 q^{70} -9.46410 q^{71} -14.3923 q^{73} -0.732051 q^{74} +6.73205 q^{76} -1.00000 q^{77} -12.1962 q^{79} +1.00000 q^{80} -2.19615 q^{82} +16.3923 q^{83} +3.46410 q^{85} -2.00000 q^{86} +1.00000 q^{88} -3.46410 q^{89} -1.46410 q^{91} +8.19615 q^{92} +6.92820 q^{94} +6.73205 q^{95} +14.5885 q^{97} -1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} + 2 q^{4} + 2 q^{5} + 2 q^{7} - 2 q^{8} + O(q^{10}) \) \( 2 q - 2 q^{2} + 2 q^{4} + 2 q^{5} + 2 q^{7} - 2 q^{8} - 2 q^{10} - 2 q^{11} + 4 q^{13} - 2 q^{14} + 2 q^{16} + 10 q^{19} + 2 q^{20} + 2 q^{22} + 6 q^{23} + 2 q^{25} - 4 q^{26} + 2 q^{28} + 6 q^{29} + 4 q^{31} - 2 q^{32} + 2 q^{35} - 2 q^{37} - 10 q^{38} - 2 q^{40} - 6 q^{41} + 4 q^{43} - 2 q^{44} - 6 q^{46} + 2 q^{49} - 2 q^{50} + 4 q^{52} + 18 q^{53} - 2 q^{55} - 2 q^{56} - 6 q^{58} + 4 q^{61} - 4 q^{62} + 2 q^{64} + 4 q^{65} - 8 q^{67} - 2 q^{70} - 12 q^{71} - 8 q^{73} + 2 q^{74} + 10 q^{76} - 2 q^{77} - 14 q^{79} + 2 q^{80} + 6 q^{82} + 12 q^{83} - 4 q^{86} + 2 q^{88} + 4 q^{91} + 6 q^{92} + 10 q^{95} - 2 q^{97} - 2 q^{98} + O(q^{100}) \)

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 \(-0.362001\pi\)
0.420084 + 0.907485i \(0.362001\pi\)
\(18\) 0 0
\(19\) 6.73205 1.54444 0.772219 0.635356i \(-0.219147\pi\)
0.772219 + 0.635356i \(0.219147\pi\)
\(20\) 1.00000 0.223607
\(21\) 0 0
\(22\) 1.00000 0.213201
\(23\) 8.19615 1.70902 0.854508 0.519438i \(-0.173859\pi\)
0.854508 + 0.519438i \(0.173859\pi\)
\(24\) 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
\(31\) 2.00000 0.359211 0.179605 0.983739i \(-0.442518\pi\)
0.179605 + 0.983739i \(0.442518\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0 0
\(34\) −3.46410 −0.594089
\(35\) 1.00000 0.169031
\(36\) 0 0
\(37\) 0.732051 0.120348 0.0601742 0.998188i \(-0.480834\pi\)
0.0601742 + 0.998188i \(0.480834\pi\)
\(38\) −6.73205 −1.09208
\(39\) 0 0
\(40\) −1.00000 −0.158114
\(41\) 2.19615 0.342981 0.171491 0.985186i \(-0.445142\pi\)
0.171491 + 0.985186i \(0.445142\pi\)
\(42\) 0 0
\(43\) 2.00000 0.304997 0.152499 0.988304i \(-0.451268\pi\)
0.152499 + 0.988304i \(0.451268\pi\)
\(44\) −1.00000 −0.150756
\(45\) 0 0
\(46\) −8.19615 −1.20846
\(47\) −6.92820 −1.01058 −0.505291 0.862949i \(-0.668615\pi\)
−0.505291 + 0.862949i \(0.668615\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) −1.00000 −0.141421
\(51\) 0 0
\(52\) −1.46410 −0.203034
\(53\) 7.26795 0.998330 0.499165 0.866507i \(-0.333640\pi\)
0.499165 + 0.866507i \(0.333640\pi\)
\(54\) 0 0
\(55\) −1.00000 −0.134840
\(56\) −1.00000 −0.133631
\(57\) 0 0
\(58\) −4.73205 −0.621349
\(59\) −6.92820 −0.901975 −0.450988 0.892530i \(-0.648928\pi\)
−0.450988 + 0.892530i \(0.648928\pi\)
\(60\) 0 0
\(61\) −4.92820 −0.630992 −0.315496 0.948927i \(-0.602171\pi\)
−0.315496 + 0.948927i \(0.602171\pi\)
\(62\) −2.00000 −0.254000
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) −1.46410 −0.181599
\(66\) 0 0
\(67\) −4.00000 −0.488678 −0.244339 0.969690i \(-0.578571\pi\)
−0.244339 + 0.969690i \(0.578571\pi\)
\(68\) 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
\(73\) −14.3923 −1.68449 −0.842246 0.539093i \(-0.818767\pi\)
−0.842246 + 0.539093i \(0.818767\pi\)
\(74\) −0.732051 −0.0850992
\(75\) 0 0
\(76\) 6.73205 0.772219
\(77\) −1.00000 −0.113961
\(78\) 0 0
\(79\) −12.1962 −1.37217 −0.686087 0.727519i \(-0.740673\pi\)
−0.686087 + 0.727519i \(0.740673\pi\)
\(80\) 1.00000 0.111803
\(81\) 0 0
\(82\) −2.19615 −0.242524
\(83\) 16.3923 1.79929 0.899645 0.436623i \(-0.143826\pi\)
0.899645 + 0.436623i \(0.143826\pi\)
\(84\) 0 0
\(85\) 3.46410 0.375735
\(86\) −2.00000 −0.215666
\(87\) 0 0
\(88\) 1.00000 0.106600
\(89\) −3.46410 −0.367194 −0.183597 0.983002i \(-0.558774\pi\)
−0.183597 + 0.983002i \(0.558774\pi\)
\(90\) 0 0
\(91\) −1.46410 −0.153480
\(92\) 8.19615 0.854508
\(93\) 0 0
\(94\) 6.92820 0.714590
\(95\) 6.73205 0.690694
\(96\) 0 0
\(97\) 14.5885 1.48123 0.740617 0.671928i \(-0.234533\pi\)
0.740617 + 0.671928i \(0.234533\pi\)
\(98\) −1.00000 −0.101015
\(99\) 0 0
\(100\) 1.00000 0.100000
\(101\) 7.85641 0.781742 0.390871 0.920446i \(-0.372174\pi\)
0.390871 + 0.920446i \(0.372174\pi\)
\(102\) 0 0
\(103\) 12.3923 1.22105 0.610525 0.791997i \(-0.290958\pi\)
0.610525 + 0.791997i \(0.290958\pi\)
\(104\) 1.46410 0.143567
\(105\) 0 0
\(106\) −7.26795 −0.705926
\(107\) −7.85641 −0.759507 −0.379754 0.925088i \(-0.623991\pi\)
−0.379754 + 0.925088i \(0.623991\pi\)
\(108\) 0 0
\(109\) −15.6603 −1.49998 −0.749990 0.661449i \(-0.769942\pi\)
−0.749990 + 0.661449i \(0.769942\pi\)
\(110\) 1.00000 0.0953463
\(111\) 0 0
\(112\) 1.00000 0.0944911
\(113\) −7.85641 −0.739069 −0.369534 0.929217i \(-0.620483\pi\)
−0.369534 + 0.929217i \(0.620483\pi\)
\(114\) 0 0
\(115\) 8.19615 0.764295
\(116\) 4.73205 0.439360
\(117\) 0 0
\(118\) 6.92820 0.637793
\(119\) 3.46410 0.317554
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 4.92820 0.446179
\(123\) 0 0
\(124\) 2.00000 0.179605
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) 1.07180 0.0951066 0.0475533 0.998869i \(-0.484858\pi\)
0.0475533 + 0.998869i \(0.484858\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 0 0
\(130\) 1.46410 0.128410
\(131\) −5.66025 −0.494539 −0.247269 0.968947i \(-0.579533\pi\)
−0.247269 + 0.968947i \(0.579533\pi\)
\(132\) 0 0
\(133\) 6.73205 0.583743
\(134\) 4.00000 0.345547
\(135\) 0 0
\(136\) −3.46410 −0.297044
\(137\) 0.928203 0.0793018 0.0396509 0.999214i \(-0.487375\pi\)
0.0396509 + 0.999214i \(0.487375\pi\)
\(138\) 0 0
\(139\) 13.6603 1.15865 0.579324 0.815097i \(-0.303317\pi\)
0.579324 + 0.815097i \(0.303317\pi\)
\(140\) 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
\(146\) 14.3923 1.19112
\(147\) 0 0
\(148\) 0.732051 0.0601742
\(149\) 7.26795 0.595414 0.297707 0.954657i \(-0.403778\pi\)
0.297707 + 0.954657i \(0.403778\pi\)
\(150\) 0 0
\(151\) 11.1244 0.905287 0.452644 0.891692i \(-0.350481\pi\)
0.452644 + 0.891692i \(0.350481\pi\)
\(152\) −6.73205 −0.546041
\(153\) 0 0
\(154\) 1.00000 0.0805823
\(155\) 2.00000 0.160644
\(156\) 0 0
\(157\) 4.53590 0.362004 0.181002 0.983483i \(-0.442066\pi\)
0.181002 + 0.983483i \(0.442066\pi\)
\(158\) 12.1962 0.970274
\(159\) 0 0
\(160\) −1.00000 −0.0790569
\(161\) 8.19615 0.645947
\(162\) 0 0
\(163\) −4.00000 −0.313304 −0.156652 0.987654i \(-0.550070\pi\)
−0.156652 + 0.987654i \(0.550070\pi\)
\(164\) 2.19615 0.171491
\(165\) 0 0
\(166\) −16.3923 −1.27229
\(167\) −13.8564 −1.07224 −0.536120 0.844141i \(-0.680111\pi\)
−0.536120 + 0.844141i \(0.680111\pi\)
\(168\) 0 0
\(169\) −10.8564 −0.835108
\(170\) −3.46410 −0.265684
\(171\) 0 0
\(172\) 2.00000 0.152499
\(173\) −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
\(178\) 3.46410 0.259645
\(179\) 6.00000 0.448461 0.224231 0.974536i \(-0.428013\pi\)
0.224231 + 0.974536i \(0.428013\pi\)
\(180\) 0 0
\(181\) 14.0000 1.04061 0.520306 0.853980i \(-0.325818\pi\)
0.520306 + 0.853980i \(0.325818\pi\)
\(182\) 1.46410 0.108526
\(183\) 0 0
\(184\) −8.19615 −0.604228
\(185\) 0.732051 0.0538214
\(186\) 0 0
\(187\) −3.46410 −0.253320
\(188\) −6.92820 −0.505291
\(189\) 0 0
\(190\) −6.73205 −0.488394
\(191\) 12.0000 0.868290 0.434145 0.900843i \(-0.357051\pi\)
0.434145 + 0.900843i \(0.357051\pi\)
\(192\) 0 0
\(193\) 12.3923 0.892018 0.446009 0.895029i \(-0.352845\pi\)
0.446009 + 0.895029i \(0.352845\pi\)
\(194\) −14.5885 −1.04739
\(195\) 0 0
\(196\) 1.00000 0.0714286
\(197\) 24.2487 1.72765 0.863825 0.503793i \(-0.168062\pi\)
0.863825 + 0.503793i \(0.168062\pi\)
\(198\) 0 0
\(199\) 2.92820 0.207575 0.103787 0.994600i \(-0.466904\pi\)
0.103787 + 0.994600i \(0.466904\pi\)
\(200\) −1.00000 −0.0707107
\(201\) 0 0
\(202\) −7.85641 −0.552775
\(203\) 4.73205 0.332125
\(204\) 0 0
\(205\) 2.19615 0.153386
\(206\) −12.3923 −0.863413
\(207\) 0 0
\(208\) −1.46410 −0.101517
\(209\) −6.73205 −0.465666
\(210\) 0 0
\(211\) 26.9282 1.85381 0.926907 0.375291i \(-0.122457\pi\)
0.926907 + 0.375291i \(0.122457\pi\)
\(212\) 7.26795 0.499165
\(213\) 0 0
\(214\) 7.85641 0.537053
\(215\) 2.00000 0.136399
\(216\) 0 0
\(217\) 2.00000 0.135769
\(218\) 15.6603 1.06065
\(219\) 0 0
\(220\) −1.00000 −0.0674200
\(221\) −5.07180 −0.341166
\(222\) 0 0
\(223\) −25.4641 −1.70520 −0.852601 0.522562i \(-0.824976\pi\)
−0.852601 + 0.522562i \(0.824976\pi\)
\(224\) −1.00000 −0.0668153
\(225\) 0 0
\(226\) 7.85641 0.522600
\(227\) −6.92820 −0.459841 −0.229920 0.973209i \(-0.573847\pi\)
−0.229920 + 0.973209i \(0.573847\pi\)
\(228\) 0 0
\(229\) 24.3923 1.61189 0.805944 0.591991i \(-0.201658\pi\)
0.805944 + 0.591991i \(0.201658\pi\)
\(230\) −8.19615 −0.540438
\(231\) 0 0
\(232\) −4.73205 −0.310674
\(233\) 7.85641 0.514690 0.257345 0.966320i \(-0.417152\pi\)
0.257345 + 0.966320i \(0.417152\pi\)
\(234\) 0 0
\(235\) −6.92820 −0.451946
\(236\) −6.92820 −0.450988
\(237\) 0 0
\(238\) −3.46410 −0.224544
\(239\) −1.26795 −0.0820168 −0.0410084 0.999159i \(-0.513057\pi\)
−0.0410084 + 0.999159i \(0.513057\pi\)
\(240\) 0 0
\(241\) 3.26795 0.210507 0.105254 0.994445i \(-0.466435\pi\)
0.105254 + 0.994445i \(0.466435\pi\)
\(242\) −1.00000 −0.0642824
\(243\) 0 0
\(244\) −4.92820 −0.315496
\(245\) 1.00000 0.0638877
\(246\) 0 0
\(247\) −9.85641 −0.627148
\(248\) −2.00000 −0.127000
\(249\) 0 0
\(250\) −1.00000 −0.0632456
\(251\) −12.0000 −0.757433 −0.378717 0.925513i \(-0.623635\pi\)
−0.378717 + 0.925513i \(0.623635\pi\)
\(252\) 0 0
\(253\) −8.19615 −0.515288
\(254\) −1.07180 −0.0672505
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 23.6603 1.47589 0.737943 0.674863i \(-0.235797\pi\)
0.737943 + 0.674863i \(0.235797\pi\)
\(258\) 0 0
\(259\) 0.732051 0.0454874
\(260\) −1.46410 −0.0907997
\(261\) 0 0
\(262\) 5.66025 0.349692
\(263\) 24.0000 1.47990 0.739952 0.672660i \(-0.234848\pi\)
0.739952 + 0.672660i \(0.234848\pi\)
\(264\) 0 0
\(265\) 7.26795 0.446467
\(266\) −6.73205 −0.412769
\(267\) 0 0
\(268\) −4.00000 −0.244339
\(269\) −28.3923 −1.73111 −0.865555 0.500814i \(-0.833034\pi\)
−0.865555 + 0.500814i \(0.833034\pi\)
\(270\) 0 0
\(271\) 0.392305 0.0238308 0.0119154 0.999929i \(-0.496207\pi\)
0.0119154 + 0.999929i \(0.496207\pi\)
\(272\) 3.46410 0.210042
\(273\) 0 0
\(274\) −0.928203 −0.0560748
\(275\) −1.00000 −0.0603023
\(276\) 0 0
\(277\) 7.07180 0.424903 0.212452 0.977172i \(-0.431855\pi\)
0.212452 + 0.977172i \(0.431855\pi\)
\(278\) −13.6603 −0.819288
\(279\) 0 0
\(280\) −1.00000 −0.0597614
\(281\) −22.3923 −1.33581 −0.667906 0.744245i \(-0.732809\pi\)
−0.667906 + 0.744245i \(0.732809\pi\)
\(282\) 0 0
\(283\) −31.7128 −1.88513 −0.942566 0.334021i \(-0.891594\pi\)
−0.942566 + 0.334021i \(0.891594\pi\)
\(284\) −9.46410 −0.561591
\(285\) 0 0
\(286\) −1.46410 −0.0865741
\(287\) 2.19615 0.129635
\(288\) 0 0
\(289\) −5.00000 −0.294118
\(290\) −4.73205 −0.277876
\(291\) 0 0
\(292\) −14.3923 −0.842246
\(293\) −21.4641 −1.25395 −0.626973 0.779041i \(-0.715706\pi\)
−0.626973 + 0.779041i \(0.715706\pi\)
\(294\) 0 0
\(295\) −6.92820 −0.403376
\(296\) −0.732051 −0.0425496
\(297\) 0 0
\(298\) −7.26795 −0.421021
\(299\) −12.0000 −0.693978
\(300\) 0 0
\(301\) 2.00000 0.115278
\(302\) −11.1244 −0.640135
\(303\) 0 0
\(304\) 6.73205 0.386110
\(305\) −4.92820 −0.282188
\(306\) 0 0
\(307\) 24.3923 1.39214 0.696071 0.717973i \(-0.254930\pi\)
0.696071 + 0.717973i \(0.254930\pi\)
\(308\) −1.00000 −0.0569803
\(309\) 0 0
\(310\) −2.00000 −0.113592
\(311\) −24.9282 −1.41355 −0.706774 0.707439i \(-0.749850\pi\)
−0.706774 + 0.707439i \(0.749850\pi\)
\(312\) 0 0
\(313\) 22.1962 1.25460 0.627300 0.778777i \(-0.284160\pi\)
0.627300 + 0.778777i \(0.284160\pi\)
\(314\) −4.53590 −0.255976
\(315\) 0 0
\(316\) −12.1962 −0.686087
\(317\) −30.5885 −1.71802 −0.859009 0.511960i \(-0.828920\pi\)
−0.859009 + 0.511960i \(0.828920\pi\)
\(318\) 0 0
\(319\) −4.73205 −0.264944
\(320\) 1.00000 0.0559017
\(321\) 0 0
\(322\) −8.19615 −0.456754
\(323\) 23.3205 1.29759
\(324\) 0 0
\(325\) −1.46410 −0.0812137
\(326\) 4.00000 0.221540
\(327\) 0 0
\(328\) −2.19615 −0.121262
\(329\) −6.92820 −0.381964
\(330\) 0 0
\(331\) −18.7846 −1.03250 −0.516248 0.856439i \(-0.672672\pi\)
−0.516248 + 0.856439i \(0.672672\pi\)
\(332\) 16.3923 0.899645
\(333\) 0 0
\(334\) 13.8564 0.758189
\(335\) −4.00000 −0.218543
\(336\) 0 0
\(337\) 22.7846 1.24116 0.620578 0.784144i \(-0.286898\pi\)
0.620578 + 0.784144i \(0.286898\pi\)
\(338\) 10.8564 0.590511
\(339\) 0 0
\(340\) 3.46410 0.187867
\(341\) −2.00000 −0.108306
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) −2.00000 −0.107833
\(345\) 0 0
\(346\) 0.928203 0.0499005
\(347\) 23.0718 1.23856 0.619279 0.785171i \(-0.287425\pi\)
0.619279 + 0.785171i \(0.287425\pi\)
\(348\) 0 0
\(349\) −5.60770 −0.300173 −0.150087 0.988673i \(-0.547955\pi\)
−0.150087 + 0.988673i \(0.547955\pi\)
\(350\) −1.00000 −0.0534522
\(351\) 0 0
\(352\) 1.00000 0.0533002
\(353\) −14.1962 −0.755585 −0.377792 0.925890i \(-0.623317\pi\)
−0.377792 + 0.925890i \(0.623317\pi\)
\(354\) 0 0
\(355\) −9.46410 −0.502302
\(356\) −3.46410 −0.183597
\(357\) 0 0
\(358\) −6.00000 −0.317110
\(359\) 34.0526 1.79723 0.898613 0.438743i \(-0.144576\pi\)
0.898613 + 0.438743i \(0.144576\pi\)
\(360\) 0 0
\(361\) 26.3205 1.38529
\(362\) −14.0000 −0.735824
\(363\) 0 0
\(364\) −1.46410 −0.0767398
\(365\) −14.3923 −0.753328
\(366\) 0 0
\(367\) 12.3923 0.646873 0.323437 0.946250i \(-0.395162\pi\)
0.323437 + 0.946250i \(0.395162\pi\)
\(368\) 8.19615 0.427254
\(369\) 0 0
\(370\) −0.732051 −0.0380575
\(371\) 7.26795 0.377333
\(372\) 0 0
\(373\) 18.3923 0.952317 0.476159 0.879359i \(-0.342029\pi\)
0.476159 + 0.879359i \(0.342029\pi\)
\(374\) 3.46410 0.179124
\(375\) 0 0
\(376\) 6.92820 0.357295
\(377\) −6.92820 −0.356821
\(378\) 0 0
\(379\) 6.14359 0.315575 0.157788 0.987473i \(-0.449564\pi\)
0.157788 + 0.987473i \(0.449564\pi\)
\(380\) 6.73205 0.345347
\(381\) 0 0
\(382\) −12.0000 −0.613973
\(383\) 33.4641 1.70994 0.854968 0.518681i \(-0.173577\pi\)
0.854968 + 0.518681i \(0.173577\pi\)
\(384\) 0 0
\(385\) −1.00000 −0.0509647
\(386\) −12.3923 −0.630752
\(387\) 0 0
\(388\) 14.5885 0.740617
\(389\) 1.60770 0.0815134 0.0407567 0.999169i \(-0.487023\pi\)
0.0407567 + 0.999169i \(0.487023\pi\)
\(390\) 0 0
\(391\) 28.3923 1.43586
\(392\) −1.00000 −0.0505076
\(393\) 0 0
\(394\) −24.2487 −1.22163
\(395\) −12.1962 −0.613655
\(396\) 0 0
\(397\) 30.3923 1.52535 0.762673 0.646784i \(-0.223887\pi\)
0.762673 + 0.646784i \(0.223887\pi\)
\(398\) −2.92820 −0.146778
\(399\) 0 0
\(400\) 1.00000 0.0500000
\(401\) −2.53590 −0.126637 −0.0633184 0.997993i \(-0.520168\pi\)
−0.0633184 + 0.997993i \(0.520168\pi\)
\(402\) 0 0
\(403\) −2.92820 −0.145864
\(404\) 7.85641 0.390871
\(405\) 0 0
\(406\) −4.73205 −0.234848
\(407\) −0.732051 −0.0362864
\(408\) 0 0
\(409\) 10.8756 0.537766 0.268883 0.963173i \(-0.413346\pi\)
0.268883 + 0.963173i \(0.413346\pi\)
\(410\) −2.19615 −0.108460
\(411\) 0 0
\(412\) 12.3923 0.610525
\(413\) −6.92820 −0.340915
\(414\) 0 0
\(415\) 16.3923 0.804667
\(416\) 1.46410 0.0717835
\(417\) 0 0
\(418\) 6.73205 0.329275
\(419\) −30.9282 −1.51094 −0.755471 0.655182i \(-0.772592\pi\)
−0.755471 + 0.655182i \(0.772592\pi\)
\(420\) 0 0
\(421\) −35.8564 −1.74753 −0.873767 0.486344i \(-0.838330\pi\)
−0.873767 + 0.486344i \(0.838330\pi\)
\(422\) −26.9282 −1.31084
\(423\) 0 0
\(424\) −7.26795 −0.352963
\(425\) 3.46410 0.168034
\(426\) 0 0
\(427\) −4.92820 −0.238492
\(428\) −7.85641 −0.379754
\(429\) 0 0
\(430\) −2.00000 −0.0964486
\(431\) −3.12436 −0.150495 −0.0752475 0.997165i \(-0.523975\pi\)
−0.0752475 + 0.997165i \(0.523975\pi\)
\(432\) 0 0
\(433\) −18.1962 −0.874451 −0.437226 0.899352i \(-0.644039\pi\)
−0.437226 + 0.899352i \(0.644039\pi\)
\(434\) −2.00000 −0.0960031
\(435\) 0 0
\(436\) −15.6603 −0.749990
\(437\) 55.1769 2.63947
\(438\) 0 0
\(439\) 26.2487 1.25278 0.626391 0.779509i \(-0.284531\pi\)
0.626391 + 0.779509i \(0.284531\pi\)
\(440\) 1.00000 0.0476731
\(441\) 0 0
\(442\) 5.07180 0.241241
\(443\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(444\) 0 0
\(445\) −3.46410 −0.164214
\(446\) 25.4641 1.20576
\(447\) 0 0
\(448\) 1.00000 0.0472456
\(449\) −40.3923 −1.90623 −0.953115 0.302607i \(-0.902143\pi\)
−0.953115 + 0.302607i \(0.902143\pi\)
\(450\) 0 0
\(451\) −2.19615 −0.103413
\(452\) −7.85641 −0.369534
\(453\) 0 0
\(454\) 6.92820 0.325157
\(455\) −1.46410 −0.0686381
\(456\) 0 0
\(457\) 2.00000 0.0935561 0.0467780 0.998905i \(-0.485105\pi\)
0.0467780 + 0.998905i \(0.485105\pi\)
\(458\) −24.3923 −1.13978
\(459\) 0 0
\(460\) 8.19615 0.382148
\(461\) −22.3923 −1.04291 −0.521457 0.853278i \(-0.674611\pi\)
−0.521457 + 0.853278i \(0.674611\pi\)
\(462\) 0 0
\(463\) 27.5167 1.27881 0.639404 0.768871i \(-0.279181\pi\)
0.639404 + 0.768871i \(0.279181\pi\)
\(464\) 4.73205 0.219680
\(465\) 0 0
\(466\) −7.85641 −0.363941
\(467\) 34.0526 1.57576 0.787882 0.615826i \(-0.211178\pi\)
0.787882 + 0.615826i \(0.211178\pi\)
\(468\) 0 0
\(469\) −4.00000 −0.184703
\(470\) 6.92820 0.319574
\(471\) 0 0
\(472\) 6.92820 0.318896
\(473\) −2.00000 −0.0919601
\(474\) 0 0
\(475\) 6.73205 0.308888
\(476\) 3.46410 0.158777
\(477\) 0 0
\(478\) 1.26795 0.0579946
\(479\) 32.7846 1.49797 0.748984 0.662589i \(-0.230542\pi\)
0.748984 + 0.662589i \(0.230542\pi\)
\(480\) 0 0
\(481\) −1.07180 −0.0488697
\(482\) −3.26795 −0.148851
\(483\) 0 0
\(484\) 1.00000 0.0454545
\(485\) 14.5885 0.662428
\(486\) 0 0
\(487\) 4.19615 0.190146 0.0950729 0.995470i \(-0.469692\pi\)
0.0950729 + 0.995470i \(0.469692\pi\)
\(488\) 4.92820 0.223089
\(489\) 0 0
\(490\) −1.00000 −0.0451754
\(491\) 27.7128 1.25066 0.625331 0.780360i \(-0.284964\pi\)
0.625331 + 0.780360i \(0.284964\pi\)
\(492\) 0 0
\(493\) 16.3923 0.738272
\(494\) 9.85641 0.443461
\(495\) 0 0
\(496\) 2.00000 0.0898027
\(497\) −9.46410 −0.424523
\(498\) 0 0
\(499\) 12.1436 0.543622 0.271811 0.962351i \(-0.412377\pi\)
0.271811 + 0.962351i \(0.412377\pi\)
\(500\) 1.00000 0.0447214
\(501\) 0 0
\(502\) 12.0000 0.535586
\(503\) −8.78461 −0.391686 −0.195843 0.980635i \(-0.562744\pi\)
−0.195843 + 0.980635i \(0.562744\pi\)
\(504\) 0 0
\(505\) 7.85641 0.349605
\(506\) 8.19615 0.364363
\(507\) 0 0
\(508\) 1.07180 0.0475533
\(509\) −11.0718 −0.490749 −0.245374 0.969428i \(-0.578911\pi\)
−0.245374 + 0.969428i \(0.578911\pi\)
\(510\) 0 0
\(511\) −14.3923 −0.636678
\(512\) −1.00000 −0.0441942
\(513\) 0 0
\(514\) −23.6603 −1.04361
\(515\) 12.3923 0.546070
\(516\) 0 0
\(517\) 6.92820 0.304702
\(518\) −0.732051 −0.0321645
\(519\) 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
\(523\) −29.1769 −1.27582 −0.637909 0.770112i \(-0.720200\pi\)
−0.637909 + 0.770112i \(0.720200\pi\)
\(524\) −5.66025 −0.247269
\(525\) 0 0
\(526\) −24.0000 −1.04645
\(527\) 6.92820 0.301797
\(528\) 0 0
\(529\) 44.1769 1.92074
\(530\) −7.26795 −0.315700
\(531\) 0 0
\(532\) 6.73205 0.291871
\(533\) −3.21539 −0.139274
\(534\) 0 0
\(535\) −7.85641 −0.339662
\(536\) 4.00000 0.172774
\(537\) 0 0
\(538\) 28.3923 1.22408
\(539\) −1.00000 −0.0430730
\(540\) 0 0
\(541\) −20.7321 −0.891340 −0.445670 0.895197i \(-0.647035\pi\)
−0.445670 + 0.895197i \(0.647035\pi\)
\(542\) −0.392305 −0.0168509
\(543\) 0 0
\(544\) −3.46410 −0.148522
\(545\) −15.6603 −0.670812
\(546\) 0 0
\(547\) 28.7846 1.23074 0.615371 0.788238i \(-0.289006\pi\)
0.615371 + 0.788238i \(0.289006\pi\)
\(548\) 0.928203 0.0396509
\(549\) 0 0
\(550\) 1.00000 0.0426401
\(551\) 31.8564 1.35713
\(552\) 0 0
\(553\) −12.1962 −0.518633
\(554\) −7.07180 −0.300452
\(555\) 0 0
\(556\) 13.6603 0.579324
\(557\) 25.6077 1.08503 0.542516 0.840045i \(-0.317472\pi\)
0.542516 + 0.840045i \(0.317472\pi\)
\(558\) 0 0
\(559\) −2.92820 −0.123850
\(560\) 1.00000 0.0422577
\(561\) 0 0
\(562\) 22.3923 0.944562
\(563\) −5.07180 −0.213751 −0.106875 0.994272i \(-0.534085\pi\)
−0.106875 + 0.994272i \(0.534085\pi\)
\(564\) 0 0
\(565\) −7.85641 −0.330522
\(566\) 31.7128 1.33299
\(567\) 0 0
\(568\) 9.46410 0.397105
\(569\) 6.00000 0.251533 0.125767 0.992060i \(-0.459861\pi\)
0.125767 + 0.992060i \(0.459861\pi\)
\(570\) 0 0
\(571\) 3.60770 0.150977 0.0754887 0.997147i \(-0.475948\pi\)
0.0754887 + 0.997147i \(0.475948\pi\)
\(572\) 1.46410 0.0612172
\(573\) 0 0
\(574\) −2.19615 −0.0916656
\(575\) 8.19615 0.341803
\(576\) 0 0
\(577\) −34.5885 −1.43994 −0.719968 0.694007i \(-0.755844\pi\)
−0.719968 + 0.694007i \(0.755844\pi\)
\(578\) 5.00000 0.207973
\(579\) 0 0
\(580\) 4.73205 0.196488
\(581\) 16.3923 0.680067
\(582\) 0 0
\(583\) −7.26795 −0.301008
\(584\) 14.3923 0.595558
\(585\) 0 0
\(586\) 21.4641 0.886674
\(587\) 11.4115 0.471005 0.235502 0.971874i \(-0.424326\pi\)
0.235502 + 0.971874i \(0.424326\pi\)
\(588\) 0 0
\(589\) 13.4641 0.554779
\(590\) 6.92820 0.285230
\(591\) 0 0
\(592\) 0.732051 0.0300871
\(593\) −0.248711 −0.0102133 −0.00510667 0.999987i \(-0.501626\pi\)
−0.00510667 + 0.999987i \(0.501626\pi\)
\(594\) 0 0
\(595\) 3.46410 0.142014
\(596\) 7.26795 0.297707
\(597\) 0 0
\(598\) 12.0000 0.490716
\(599\) −37.1769 −1.51901 −0.759504 0.650503i \(-0.774558\pi\)
−0.759504 + 0.650503i \(0.774558\pi\)
\(600\) 0 0
\(601\) −5.51666 −0.225029 −0.112515 0.993650i \(-0.535891\pi\)
−0.112515 + 0.993650i \(0.535891\pi\)
\(602\) −2.00000 −0.0815139
\(603\) 0 0
\(604\) 11.1244 0.452644
\(605\) 1.00000 0.0406558
\(606\) 0 0
\(607\) 20.9282 0.849450 0.424725 0.905323i \(-0.360371\pi\)
0.424725 + 0.905323i \(0.360371\pi\)
\(608\) −6.73205 −0.273021
\(609\) 0 0
\(610\) 4.92820 0.199537
\(611\) 10.1436 0.410366
\(612\) 0 0
\(613\) −35.1769 −1.42078 −0.710391 0.703807i \(-0.751482\pi\)
−0.710391 + 0.703807i \(0.751482\pi\)
\(614\) −24.3923 −0.984393
\(615\) 0 0
\(616\) 1.00000 0.0402911
\(617\) −17.3205 −0.697297 −0.348649 0.937253i \(-0.613359\pi\)
−0.348649 + 0.937253i \(0.613359\pi\)
\(618\) 0 0
\(619\) 28.7846 1.15695 0.578476 0.815700i \(-0.303648\pi\)
0.578476 + 0.815700i \(0.303648\pi\)
\(620\) 2.00000 0.0803219
\(621\) 0 0
\(622\) 24.9282 0.999530
\(623\) −3.46410 −0.138786
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) −22.1962 −0.887137
\(627\) 0 0
\(628\) 4.53590 0.181002
\(629\) 2.53590 0.101113
\(630\) 0 0
\(631\) −46.9282 −1.86818 −0.934091 0.357035i \(-0.883788\pi\)
−0.934091 + 0.357035i \(0.883788\pi\)
\(632\) 12.1962 0.485137
\(633\) 0 0
\(634\) 30.5885 1.21482
\(635\) 1.07180 0.0425330
\(636\) 0 0
\(637\) −1.46410 −0.0580098
\(638\) 4.73205 0.187344
\(639\) 0 0
\(640\) −1.00000 −0.0395285
\(641\) 35.5692 1.40490 0.702450 0.711733i \(-0.252090\pi\)
0.702450 + 0.711733i \(0.252090\pi\)
\(642\) 0 0
\(643\) 33.2679 1.31196 0.655980 0.754778i \(-0.272256\pi\)
0.655980 + 0.754778i \(0.272256\pi\)
\(644\) 8.19615 0.322974
\(645\) 0 0
\(646\) −23.3205 −0.917533
\(647\) 26.5359 1.04323 0.521617 0.853180i \(-0.325329\pi\)
0.521617 + 0.853180i \(0.325329\pi\)
\(648\) 0 0
\(649\) 6.92820 0.271956
\(650\) 1.46410 0.0574268
\(651\) 0 0
\(652\) −4.00000 −0.156652
\(653\) −11.6603 −0.456301 −0.228151 0.973626i \(-0.573268\pi\)
−0.228151 + 0.973626i \(0.573268\pi\)
\(654\) 0 0
\(655\) −5.66025 −0.221164
\(656\) 2.19615 0.0857453
\(657\) 0 0
\(658\) 6.92820 0.270089
\(659\) −30.2487 −1.17832 −0.589161 0.808015i \(-0.700542\pi\)
−0.589161 + 0.808015i \(0.700542\pi\)
\(660\) 0 0
\(661\) −46.0000 −1.78919 −0.894596 0.446875i \(-0.852537\pi\)
−0.894596 + 0.446875i \(0.852537\pi\)
\(662\) 18.7846 0.730085
\(663\) 0 0
\(664\) −16.3923 −0.636145
\(665\) 6.73205 0.261058
\(666\) 0 0
\(667\) 38.7846 1.50175
\(668\) −13.8564 −0.536120
\(669\) 0 0
\(670\) 4.00000 0.154533
\(671\) 4.92820 0.190251
\(672\) 0 0
\(673\) 2.00000 0.0770943 0.0385472 0.999257i \(-0.487727\pi\)
0.0385472 + 0.999257i \(0.487727\pi\)
\(674\) −22.7846 −0.877630
\(675\) 0 0
\(676\) −10.8564 −0.417554
\(677\) 4.14359 0.159251 0.0796256 0.996825i \(-0.474628\pi\)
0.0796256 + 0.996825i \(0.474628\pi\)
\(678\) 0 0
\(679\) 14.5885 0.559854
\(680\) −3.46410 −0.132842
\(681\) 0 0
\(682\) 2.00000 0.0765840
\(683\) −6.24871 −0.239100 −0.119550 0.992828i \(-0.538145\pi\)
−0.119550 + 0.992828i \(0.538145\pi\)
\(684\) 0 0
\(685\) 0.928203 0.0354648
\(686\) −1.00000 −0.0381802
\(687\) 0 0
\(688\) 2.00000 0.0762493
\(689\) −10.6410 −0.405390
\(690\) 0 0
\(691\) −13.4641 −0.512199 −0.256099 0.966650i \(-0.582437\pi\)
−0.256099 + 0.966650i \(0.582437\pi\)
\(692\) −0.928203 −0.0352850
\(693\) 0 0
\(694\) −23.0718 −0.875793
\(695\) 13.6603 0.518163
\(696\) 0 0
\(697\) 7.60770 0.288162
\(698\) 5.60770 0.212254
\(699\) 0 0
\(700\) 1.00000 0.0377964
\(701\) −41.9090 −1.58288 −0.791440 0.611247i \(-0.790668\pi\)
−0.791440 + 0.611247i \(0.790668\pi\)
\(702\) 0 0
\(703\) 4.92820 0.185871
\(704\) −1.00000 −0.0376889
\(705\) 0 0
\(706\) 14.1962 0.534279
\(707\) 7.85641 0.295471
\(708\) 0 0
\(709\) 25.3205 0.950932 0.475466 0.879734i \(-0.342280\pi\)
0.475466 + 0.879734i \(0.342280\pi\)
\(710\) 9.46410 0.355181
\(711\) 0 0
\(712\) 3.46410 0.129823
\(713\) 16.3923 0.613897
\(714\) 0 0
\(715\) 1.46410 0.0547543
\(716\) 6.00000 0.224231
\(717\) 0 0
\(718\) −34.0526 −1.27083
\(719\) 27.7128 1.03351 0.516757 0.856132i \(-0.327139\pi\)
0.516757 + 0.856132i \(0.327139\pi\)
\(720\) 0 0
\(721\) 12.3923 0.461514
\(722\) −26.3205 −0.979548
\(723\) 0 0
\(724\) 14.0000 0.520306
\(725\) 4.73205 0.175744
\(726\) 0 0
\(727\) −4.00000 −0.148352 −0.0741759 0.997245i \(-0.523633\pi\)
−0.0741759 + 0.997245i \(0.523633\pi\)
\(728\) 1.46410 0.0542632
\(729\) 0 0
\(730\) 14.3923 0.532683
\(731\) 6.92820 0.256249
\(732\) 0 0
\(733\) 26.0000 0.960332 0.480166 0.877178i \(-0.340576\pi\)
0.480166 + 0.877178i \(0.340576\pi\)
\(734\) −12.3923 −0.457408
\(735\) 0 0
\(736\) −8.19615 −0.302114
\(737\) 4.00000 0.147342
\(738\) 0 0
\(739\) 11.7128 0.430863 0.215431 0.976519i \(-0.430884\pi\)
0.215431 + 0.976519i \(0.430884\pi\)
\(740\) 0.732051 0.0269107
\(741\) 0 0
\(742\) −7.26795 −0.266815
\(743\) 32.7846 1.20275 0.601375 0.798967i \(-0.294620\pi\)
0.601375 + 0.798967i \(0.294620\pi\)
\(744\) 0 0
\(745\) 7.26795 0.266277
\(746\) −18.3923 −0.673390
\(747\) 0 0
\(748\) −3.46410 −0.126660
\(749\) −7.85641 −0.287067
\(750\) 0 0
\(751\) −11.6077 −0.423571 −0.211785 0.977316i \(-0.567928\pi\)
−0.211785 + 0.977316i \(0.567928\pi\)
\(752\) −6.92820 −0.252646
\(753\) 0 0
\(754\) 6.92820 0.252310
\(755\) 11.1244 0.404857
\(756\) 0 0
\(757\) −43.3731 −1.57642 −0.788210 0.615406i \(-0.788992\pi\)
−0.788210 + 0.615406i \(0.788992\pi\)
\(758\) −6.14359 −0.223145
\(759\) 0 0
\(760\) −6.73205 −0.244197
\(761\) 12.3397 0.447315 0.223658 0.974668i \(-0.428200\pi\)
0.223658 + 0.974668i \(0.428200\pi\)
\(762\) 0 0
\(763\) −15.6603 −0.566939
\(764\) 12.0000 0.434145
\(765\) 0 0
\(766\) −33.4641 −1.20911
\(767\) 10.1436 0.366264
\(768\) 0 0
\(769\) −18.1962 −0.656170 −0.328085 0.944648i \(-0.606403\pi\)
−0.328085 + 0.944648i \(0.606403\pi\)
\(770\) 1.00000 0.0360375
\(771\) 0 0
\(772\) 12.3923 0.446009
\(773\) 0.928203 0.0333851 0.0166926 0.999861i \(-0.494686\pi\)
0.0166926 + 0.999861i \(0.494686\pi\)
\(774\) 0 0
\(775\) 2.00000 0.0718421
\(776\) −14.5885 −0.523695
\(777\) 0 0
\(778\) −1.60770 −0.0576387
\(779\) 14.7846 0.529714
\(780\) 0 0
\(781\) 9.46410 0.338652
\(782\) −28.3923 −1.01531
\(783\) 0 0
\(784\) 1.00000 0.0357143
\(785\) 4.53590 0.161893
\(786\) 0 0
\(787\) 18.1436 0.646749 0.323375 0.946271i \(-0.395183\pi\)
0.323375 + 0.946271i \(0.395183\pi\)
\(788\) 24.2487 0.863825
\(789\) 0 0
\(790\) 12.1962 0.433920
\(791\) −7.85641 −0.279342
\(792\) 0 0
\(793\) 7.21539 0.256226
\(794\) −30.3923 −1.07858
\(795\) 0 0
\(796\) 2.92820 0.103787
\(797\) 25.6077 0.907071 0.453536 0.891238i \(-0.350163\pi\)
0.453536 + 0.891238i \(0.350163\pi\)
\(798\) 0 0
\(799\) −24.0000 −0.849059
\(800\) −1.00000 −0.0353553
\(801\) 0 0
\(802\) 2.53590 0.0895457
\(803\) 14.3923 0.507893
\(804\) 0 0
\(805\) 8.19615 0.288876
\(806\) 2.92820 0.103142
\(807\) 0 0
\(808\) −7.85641 −0.276387
\(809\) 31.1769 1.09612 0.548061 0.836438i \(-0.315366\pi\)
0.548061 + 0.836438i \(0.315366\pi\)
\(810\) 0 0
\(811\) −43.1244 −1.51430 −0.757150 0.653241i \(-0.773409\pi\)
−0.757150 + 0.653241i \(0.773409\pi\)
\(812\) 4.73205 0.166062
\(813\) 0 0
\(814\) 0.732051 0.0256584
\(815\) −4.00000 −0.140114
\(816\) 0 0
\(817\) 13.4641 0.471049
\(818\) −10.8756 −0.380258
\(819\) 0 0
\(820\) 2.19615 0.0766930
\(821\) −17.9090 −0.625027 −0.312514 0.949913i \(-0.601171\pi\)
−0.312514 + 0.949913i \(0.601171\pi\)
\(822\) 0 0
\(823\) −0.875644 −0.0305230 −0.0152615 0.999884i \(-0.504858\pi\)
−0.0152615 + 0.999884i \(0.504858\pi\)
\(824\) −12.3923 −0.431706
\(825\) 0 0
\(826\) 6.92820 0.241063
\(827\) 25.8564 0.899115 0.449558 0.893251i \(-0.351582\pi\)
0.449558 + 0.893251i \(0.351582\pi\)
\(828\) 0 0
\(829\) 2.24871 0.0781010 0.0390505 0.999237i \(-0.487567\pi\)
0.0390505 + 0.999237i \(0.487567\pi\)
\(830\) −16.3923 −0.568985
\(831\) 0 0
\(832\) −1.46410 −0.0507586
\(833\) 3.46410 0.120024
\(834\) 0 0
\(835\) −13.8564 −0.479521
\(836\) −6.73205 −0.232833
\(837\) 0 0
\(838\) 30.9282 1.06840
\(839\) 31.8564 1.09981 0.549903 0.835229i \(-0.314665\pi\)
0.549903 + 0.835229i \(0.314665\pi\)
\(840\) 0 0
\(841\) −6.60770 −0.227852
\(842\) 35.8564 1.23569
\(843\) 0 0
\(844\) 26.9282 0.926907
\(845\) −10.8564 −0.373472
\(846\) 0 0
\(847\) 1.00000 0.0343604
\(848\) 7.26795 0.249582
\(849\) 0 0
\(850\) −3.46410 −0.118818
\(851\) 6.00000 0.205677
\(852\) 0 0
\(853\) −54.7846 −1.87579 −0.937895 0.346920i \(-0.887227\pi\)
−0.937895 + 0.346920i \(0.887227\pi\)
\(854\) 4.92820 0.168640
\(855\) 0 0
\(856\) 7.85641 0.268526
\(857\) 27.4641 0.938156 0.469078 0.883157i \(-0.344586\pi\)
0.469078 + 0.883157i \(0.344586\pi\)
\(858\) 0 0
\(859\) −12.7846 −0.436205 −0.218103 0.975926i \(-0.569987\pi\)
−0.218103 + 0.975926i \(0.569987\pi\)
\(860\) 2.00000 0.0681994
\(861\) 0 0
\(862\) 3.12436 0.106416
\(863\) 7.51666 0.255870 0.127935 0.991783i \(-0.459165\pi\)
0.127935 + 0.991783i \(0.459165\pi\)
\(864\) 0 0
\(865\) −0.928203 −0.0315599
\(866\) 18.1962 0.618330
\(867\) 0 0
\(868\) 2.00000 0.0678844
\(869\) 12.1962 0.413726
\(870\) 0 0
\(871\) 5.85641 0.198437
\(872\) 15.6603 0.530323
\(873\) 0 0
\(874\) −55.1769 −1.86639
\(875\) 1.00000 0.0338062
\(876\) 0 0
\(877\) −21.3205 −0.719942 −0.359971 0.932963i \(-0.617213\pi\)
−0.359971 + 0.932963i \(0.617213\pi\)
\(878\) −26.2487 −0.885851
\(879\) 0 0
\(880\) −1.00000 −0.0337100
\(881\) −11.0718 −0.373018 −0.186509 0.982453i \(-0.559717\pi\)
−0.186509 + 0.982453i \(0.559717\pi\)
\(882\) 0 0
\(883\) −35.6077 −1.19829 −0.599147 0.800639i \(-0.704494\pi\)
−0.599147 + 0.800639i \(0.704494\pi\)
\(884\) −5.07180 −0.170583
\(885\) 0 0
\(886\) 0 0
\(887\) −13.8564 −0.465253 −0.232626 0.972566i \(-0.574732\pi\)
−0.232626 + 0.972566i \(0.574732\pi\)
\(888\) 0 0
\(889\) 1.07180 0.0359469
\(890\) 3.46410 0.116117
\(891\) 0 0
\(892\) −25.4641 −0.852601
\(893\) −46.6410 −1.56078
\(894\) 0 0
\(895\) 6.00000 0.200558
\(896\) −1.00000 −0.0334077
\(897\) 0 0
\(898\) 40.3923 1.34791
\(899\) 9.46410 0.315645
\(900\) 0 0
\(901\) 25.1769 0.838765
\(902\) 2.19615 0.0731239
\(903\) 0 0
\(904\) 7.85641 0.261300
\(905\) 14.0000 0.465376
\(906\) 0 0
\(907\) −31.0333 −1.03044 −0.515222 0.857057i \(-0.672291\pi\)
−0.515222 + 0.857057i \(0.672291\pi\)
\(908\) −6.92820 −0.229920
\(909\) 0 0
\(910\) 1.46410 0.0485345
\(911\) −9.46410 −0.313560 −0.156780 0.987634i \(-0.550111\pi\)
−0.156780 + 0.987634i \(0.550111\pi\)
\(912\) 0 0
\(913\) −16.3923 −0.542506
\(914\) −2.00000 −0.0661541
\(915\) 0 0
\(916\) 24.3923 0.805944
\(917\) −5.66025 −0.186918
\(918\) 0 0
\(919\) −25.3731 −0.836980 −0.418490 0.908221i \(-0.637441\pi\)
−0.418490 + 0.908221i \(0.637441\pi\)
\(920\) −8.19615 −0.270219
\(921\) 0 0
\(922\) 22.3923 0.737451
\(923\) 13.8564 0.456089
\(924\) 0 0
\(925\) 0.732051 0.0240697
\(926\) −27.5167 −0.904254
\(927\) 0 0
\(928\) −4.73205 −0.155337
\(929\) 25.6077 0.840161 0.420081 0.907487i \(-0.362002\pi\)
0.420081 + 0.907487i \(0.362002\pi\)
\(930\) 0 0
\(931\) 6.73205 0.220634
\(932\) 7.85641 0.257345
\(933\) 0 0
\(934\) −34.0526 −1.11423
\(935\) −3.46410 −0.113288
\(936\) 0 0
\(937\) −1.21539 −0.0397051 −0.0198525 0.999803i \(-0.506320\pi\)
−0.0198525 + 0.999803i \(0.506320\pi\)
\(938\) 4.00000 0.130605
\(939\) 0 0
\(940\) −6.92820 −0.225973
\(941\) 14.7846 0.481965 0.240982 0.970530i \(-0.422530\pi\)
0.240982 + 0.970530i \(0.422530\pi\)
\(942\) 0 0
\(943\) 18.0000 0.586161
\(944\) −6.92820 −0.225494
\(945\) 0 0
\(946\) 2.00000 0.0650256
\(947\) −47.3205 −1.53771 −0.768855 0.639423i \(-0.779173\pi\)
−0.768855 + 0.639423i \(0.779173\pi\)
\(948\) 0 0
\(949\) 21.0718 0.684019
\(950\) −6.73205 −0.218417
\(951\) 0 0
\(952\) −3.46410 −0.112272
\(953\) −26.5359 −0.859582 −0.429791 0.902928i \(-0.641413\pi\)
−0.429791 + 0.902928i \(0.641413\pi\)
\(954\) 0 0
\(955\) 12.0000 0.388311
\(956\) −1.26795 −0.0410084
\(957\) 0 0
\(958\) −32.7846 −1.05922
\(959\) 0.928203 0.0299732
\(960\) 0 0
\(961\) −27.0000 −0.870968
\(962\) 1.07180 0.0345561
\(963\) 0 0
\(964\) 3.26795 0.105254
\(965\) 12.3923 0.398922
\(966\) 0 0
\(967\) 26.9282 0.865953 0.432976 0.901405i \(-0.357463\pi\)
0.432976 + 0.901405i \(0.357463\pi\)
\(968\) −1.00000 −0.0321412
\(969\) 0 0
\(970\) −14.5885 −0.468407
\(971\) −42.9282 −1.37763 −0.688816 0.724936i \(-0.741869\pi\)
−0.688816 + 0.724936i \(0.741869\pi\)
\(972\) 0 0
\(973\) 13.6603 0.437928
\(974\) −4.19615 −0.134453
\(975\) 0 0
\(976\) −4.92820 −0.157748
\(977\) 33.7128 1.07857 0.539284 0.842124i \(-0.318695\pi\)
0.539284 + 0.842124i \(0.318695\pi\)
\(978\) 0 0
\(979\) 3.46410 0.110713
\(980\) 1.00000 0.0319438
\(981\) 0 0
\(982\) −27.7128 −0.884351
\(983\) 37.1769 1.18576 0.592880 0.805291i \(-0.297991\pi\)
0.592880 + 0.805291i \(0.297991\pi\)
\(984\) 0 0
\(985\) 24.2487 0.772628
\(986\) −16.3923 −0.522037
\(987\) 0 0
\(988\) −9.85641 −0.313574
\(989\) 16.3923 0.521245
\(990\) 0 0
\(991\) −4.00000 −0.127064 −0.0635321 0.997980i \(-0.520237\pi\)
−0.0635321 + 0.997980i \(0.520237\pi\)
\(992\) −2.00000 −0.0635001
\(993\) 0 0
\(994\) 9.46410 0.300183
\(995\) 2.92820 0.0928303
\(996\) 0 0
\(997\) 17.7128 0.560970 0.280485 0.959858i \(-0.409505\pi\)
0.280485 + 0.959858i \(0.409505\pi\)
\(998\) −12.1436 −0.384399
\(999\) 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.bv.1.1 2
3.2 odd 2 770.2.a.j.1.2 2
12.11 even 2 6160.2.a.t.1.1 2
15.2 even 4 3850.2.c.x.1849.3 4
15.8 even 4 3850.2.c.x.1849.2 4
15.14 odd 2 3850.2.a.bd.1.1 2
21.20 even 2 5390.2.a.bs.1.1 2
33.32 even 2 8470.2.a.br.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
770.2.a.j.1.2 2 3.2 odd 2
3850.2.a.bd.1.1 2 15.14 odd 2
3850.2.c.x.1849.2 4 15.8 even 4
3850.2.c.x.1849.3 4 15.2 even 4
5390.2.a.bs.1.1 2 21.20 even 2
6160.2.a.t.1.1 2 12.11 even 2
6930.2.a.bv.1.1 2 1.1 even 1 trivial
8470.2.a.br.1.2 2 33.32 even 2