Properties

Label 6930.2.a.by.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: \(\Q(\sqrt{33}) \)
Defining polynomial: \(x^{2} - x - 8\)
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 \(-2.37228\) of defining polynomial
Character \(\chi\) \(=\) 6930.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{7} +1.00000 q^{8} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{7} +1.00000 q^{8} -1.00000 q^{10} +1.00000 q^{11} -4.74456 q^{13} -1.00000 q^{14} +1.00000 q^{16} -4.74456 q^{17} -1.00000 q^{20} +1.00000 q^{22} +6.74456 q^{23} +1.00000 q^{25} -4.74456 q^{26} -1.00000 q^{28} -2.00000 q^{29} +1.00000 q^{32} -4.74456 q^{34} +1.00000 q^{35} +8.74456 q^{37} -1.00000 q^{40} -6.00000 q^{41} +4.00000 q^{43} +1.00000 q^{44} +6.74456 q^{46} +4.00000 q^{47} +1.00000 q^{49} +1.00000 q^{50} -4.74456 q^{52} +6.00000 q^{53} -1.00000 q^{55} -1.00000 q^{56} -2.00000 q^{58} -4.00000 q^{59} +6.00000 q^{61} +1.00000 q^{64} +4.74456 q^{65} -1.25544 q^{67} -4.74456 q^{68} +1.00000 q^{70} +6.00000 q^{73} +8.74456 q^{74} -1.00000 q^{77} +12.0000 q^{79} -1.00000 q^{80} -6.00000 q^{82} +4.00000 q^{83} +4.74456 q^{85} +4.00000 q^{86} +1.00000 q^{88} +4.74456 q^{89} +4.74456 q^{91} +6.74456 q^{92} +4.00000 q^{94} -7.48913 q^{97} +1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} + 2 q^{4} - 2 q^{5} - 2 q^{7} + 2 q^{8} + O(q^{10}) \) \( 2 q + 2 q^{2} + 2 q^{4} - 2 q^{5} - 2 q^{7} + 2 q^{8} - 2 q^{10} + 2 q^{11} + 2 q^{13} - 2 q^{14} + 2 q^{16} + 2 q^{17} - 2 q^{20} + 2 q^{22} + 2 q^{23} + 2 q^{25} + 2 q^{26} - 2 q^{28} - 4 q^{29} + 2 q^{32} + 2 q^{34} + 2 q^{35} + 6 q^{37} - 2 q^{40} - 12 q^{41} + 8 q^{43} + 2 q^{44} + 2 q^{46} + 8 q^{47} + 2 q^{49} + 2 q^{50} + 2 q^{52} + 12 q^{53} - 2 q^{55} - 2 q^{56} - 4 q^{58} - 8 q^{59} + 12 q^{61} + 2 q^{64} - 2 q^{65} - 14 q^{67} + 2 q^{68} + 2 q^{70} + 12 q^{73} + 6 q^{74} - 2 q^{77} + 24 q^{79} - 2 q^{80} - 12 q^{82} + 8 q^{83} - 2 q^{85} + 8 q^{86} + 2 q^{88} - 2 q^{89} - 2 q^{91} + 2 q^{92} + 8 q^{94} + 8 q^{97} + 2 q^{98} + O(q^{100}) \)

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
\(13\) −4.74456 −1.31590 −0.657952 0.753059i \(-0.728577\pi\)
−0.657952 + 0.753059i \(0.728577\pi\)
\(14\) −1.00000 −0.267261
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −4.74456 −1.15073 −0.575363 0.817898i \(-0.695139\pi\)
−0.575363 + 0.817898i \(0.695139\pi\)
\(18\) 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
\(23\) 6.74456 1.40634 0.703169 0.711022i \(-0.251768\pi\)
0.703169 + 0.711022i \(0.251768\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) −4.74456 −0.930485
\(27\) 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
\(31\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) −4.74456 −0.813686
\(35\) 1.00000 0.169031
\(36\) 0 0
\(37\) 8.74456 1.43760 0.718799 0.695218i \(-0.244692\pi\)
0.718799 + 0.695218i \(0.244692\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) −1.00000 −0.158114
\(41\) −6.00000 −0.937043 −0.468521 0.883452i \(-0.655213\pi\)
−0.468521 + 0.883452i \(0.655213\pi\)
\(42\) 0 0
\(43\) 4.00000 0.609994 0.304997 0.952353i \(-0.401344\pi\)
0.304997 + 0.952353i \(0.401344\pi\)
\(44\) 1.00000 0.150756
\(45\) 0 0
\(46\) 6.74456 0.994432
\(47\) 4.00000 0.583460 0.291730 0.956501i \(-0.405769\pi\)
0.291730 + 0.956501i \(0.405769\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 1.00000 0.141421
\(51\) 0 0
\(52\) −4.74456 −0.657952
\(53\) 6.00000 0.824163 0.412082 0.911147i \(-0.364802\pi\)
0.412082 + 0.911147i \(0.364802\pi\)
\(54\) 0 0
\(55\) −1.00000 −0.134840
\(56\) −1.00000 −0.133631
\(57\) 0 0
\(58\) −2.00000 −0.262613
\(59\) −4.00000 −0.520756 −0.260378 0.965507i \(-0.583847\pi\)
−0.260378 + 0.965507i \(0.583847\pi\)
\(60\) 0 0
\(61\) 6.00000 0.768221 0.384111 0.923287i \(-0.374508\pi\)
0.384111 + 0.923287i \(0.374508\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 4.74456 0.588491
\(66\) 0 0
\(67\) −1.25544 −0.153376 −0.0766880 0.997055i \(-0.524435\pi\)
−0.0766880 + 0.997055i \(0.524435\pi\)
\(68\) −4.74456 −0.575363
\(69\) 0 0
\(70\) 1.00000 0.119523
\(71\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) 6.00000 0.702247 0.351123 0.936329i \(-0.385800\pi\)
0.351123 + 0.936329i \(0.385800\pi\)
\(74\) 8.74456 1.01653
\(75\) 0 0
\(76\) 0 0
\(77\) −1.00000 −0.113961
\(78\) 0 0
\(79\) 12.0000 1.35011 0.675053 0.737769i \(-0.264121\pi\)
0.675053 + 0.737769i \(0.264121\pi\)
\(80\) −1.00000 −0.111803
\(81\) 0 0
\(82\) −6.00000 −0.662589
\(83\) 4.00000 0.439057 0.219529 0.975606i \(-0.429548\pi\)
0.219529 + 0.975606i \(0.429548\pi\)
\(84\) 0 0
\(85\) 4.74456 0.514620
\(86\) 4.00000 0.431331
\(87\) 0 0
\(88\) 1.00000 0.106600
\(89\) 4.74456 0.502923 0.251461 0.967867i \(-0.419089\pi\)
0.251461 + 0.967867i \(0.419089\pi\)
\(90\) 0 0
\(91\) 4.74456 0.497365
\(92\) 6.74456 0.703169
\(93\) 0 0
\(94\) 4.00000 0.412568
\(95\) 0 0
\(96\) 0 0
\(97\) −7.48913 −0.760405 −0.380203 0.924903i \(-0.624146\pi\)
−0.380203 + 0.924903i \(0.624146\pi\)
\(98\) 1.00000 0.101015
\(99\) 0 0
\(100\) 1.00000 0.100000
\(101\) −6.00000 −0.597022 −0.298511 0.954406i \(-0.596490\pi\)
−0.298511 + 0.954406i \(0.596490\pi\)
\(102\) 0 0
\(103\) 9.48913 0.934991 0.467496 0.883995i \(-0.345156\pi\)
0.467496 + 0.883995i \(0.345156\pi\)
\(104\) −4.74456 −0.465243
\(105\) 0 0
\(106\) 6.00000 0.582772
\(107\) 4.00000 0.386695 0.193347 0.981130i \(-0.438066\pi\)
0.193347 + 0.981130i \(0.438066\pi\)
\(108\) 0 0
\(109\) 3.25544 0.311814 0.155907 0.987772i \(-0.450170\pi\)
0.155907 + 0.987772i \(0.450170\pi\)
\(110\) −1.00000 −0.0953463
\(111\) 0 0
\(112\) −1.00000 −0.0944911
\(113\) −0.744563 −0.0700426 −0.0350213 0.999387i \(-0.511150\pi\)
−0.0350213 + 0.999387i \(0.511150\pi\)
\(114\) 0 0
\(115\) −6.74456 −0.628934
\(116\) −2.00000 −0.185695
\(117\) 0 0
\(118\) −4.00000 −0.368230
\(119\) 4.74456 0.434933
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 6.00000 0.543214
\(123\) 0 0
\(124\) 0 0
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) 8.00000 0.709885 0.354943 0.934888i \(-0.384500\pi\)
0.354943 + 0.934888i \(0.384500\pi\)
\(128\) 1.00000 0.0883883
\(129\) 0 0
\(130\) 4.74456 0.416126
\(131\) 13.4891 1.17855 0.589275 0.807932i \(-0.299413\pi\)
0.589275 + 0.807932i \(0.299413\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) −1.25544 −0.108453
\(135\) 0 0
\(136\) −4.74456 −0.406843
\(137\) 18.2337 1.55781 0.778905 0.627142i \(-0.215776\pi\)
0.778905 + 0.627142i \(0.215776\pi\)
\(138\) 0 0
\(139\) 21.4891 1.82268 0.911342 0.411650i \(-0.135047\pi\)
0.911342 + 0.411650i \(0.135047\pi\)
\(140\) 1.00000 0.0845154
\(141\) 0 0
\(142\) 0 0
\(143\) −4.74456 −0.396760
\(144\) 0 0
\(145\) 2.00000 0.166091
\(146\) 6.00000 0.496564
\(147\) 0 0
\(148\) 8.74456 0.718799
\(149\) −15.4891 −1.26892 −0.634459 0.772956i \(-0.718777\pi\)
−0.634459 + 0.772956i \(0.718777\pi\)
\(150\) 0 0
\(151\) −4.00000 −0.325515 −0.162758 0.986666i \(-0.552039\pi\)
−0.162758 + 0.986666i \(0.552039\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) −1.00000 −0.0805823
\(155\) 0 0
\(156\) 0 0
\(157\) 6.00000 0.478852 0.239426 0.970915i \(-0.423041\pi\)
0.239426 + 0.970915i \(0.423041\pi\)
\(158\) 12.0000 0.954669
\(159\) 0 0
\(160\) −1.00000 −0.0790569
\(161\) −6.74456 −0.531546
\(162\) 0 0
\(163\) 14.7446 1.15488 0.577442 0.816432i \(-0.304051\pi\)
0.577442 + 0.816432i \(0.304051\pi\)
\(164\) −6.00000 −0.468521
\(165\) 0 0
\(166\) 4.00000 0.310460
\(167\) −20.2337 −1.56573 −0.782865 0.622192i \(-0.786242\pi\)
−0.782865 + 0.622192i \(0.786242\pi\)
\(168\) 0 0
\(169\) 9.51087 0.731606
\(170\) 4.74456 0.363891
\(171\) 0 0
\(172\) 4.00000 0.304997
\(173\) −23.4891 −1.78585 −0.892923 0.450210i \(-0.851349\pi\)
−0.892923 + 0.450210i \(0.851349\pi\)
\(174\) 0 0
\(175\) −1.00000 −0.0755929
\(176\) 1.00000 0.0753778
\(177\) 0 0
\(178\) 4.74456 0.355620
\(179\) 4.00000 0.298974 0.149487 0.988764i \(-0.452238\pi\)
0.149487 + 0.988764i \(0.452238\pi\)
\(180\) 0 0
\(181\) 12.7446 0.947296 0.473648 0.880714i \(-0.342937\pi\)
0.473648 + 0.880714i \(0.342937\pi\)
\(182\) 4.74456 0.351690
\(183\) 0 0
\(184\) 6.74456 0.497216
\(185\) −8.74456 −0.642913
\(186\) 0 0
\(187\) −4.74456 −0.346957
\(188\) 4.00000 0.291730
\(189\) 0 0
\(190\) 0 0
\(191\) 5.48913 0.397179 0.198590 0.980083i \(-0.436364\pi\)
0.198590 + 0.980083i \(0.436364\pi\)
\(192\) 0 0
\(193\) −20.9783 −1.51005 −0.755024 0.655697i \(-0.772375\pi\)
−0.755024 + 0.655697i \(0.772375\pi\)
\(194\) −7.48913 −0.537688
\(195\) 0 0
\(196\) 1.00000 0.0714286
\(197\) −23.4891 −1.67353 −0.836765 0.547561i \(-0.815556\pi\)
−0.836765 + 0.547561i \(0.815556\pi\)
\(198\) 0 0
\(199\) 26.9783 1.91244 0.956219 0.292653i \(-0.0945380\pi\)
0.956219 + 0.292653i \(0.0945380\pi\)
\(200\) 1.00000 0.0707107
\(201\) 0 0
\(202\) −6.00000 −0.422159
\(203\) 2.00000 0.140372
\(204\) 0 0
\(205\) 6.00000 0.419058
\(206\) 9.48913 0.661139
\(207\) 0 0
\(208\) −4.74456 −0.328976
\(209\) 0 0
\(210\) 0 0
\(211\) 13.2554 0.912542 0.456271 0.889841i \(-0.349185\pi\)
0.456271 + 0.889841i \(0.349185\pi\)
\(212\) 6.00000 0.412082
\(213\) 0 0
\(214\) 4.00000 0.273434
\(215\) −4.00000 −0.272798
\(216\) 0 0
\(217\) 0 0
\(218\) 3.25544 0.220486
\(219\) 0 0
\(220\) −1.00000 −0.0674200
\(221\) 22.5109 1.51425
\(222\) 0 0
\(223\) −1.48913 −0.0997192 −0.0498596 0.998756i \(-0.515877\pi\)
−0.0498596 + 0.998756i \(0.515877\pi\)
\(224\) −1.00000 −0.0668153
\(225\) 0 0
\(226\) −0.744563 −0.0495276
\(227\) −4.00000 −0.265489 −0.132745 0.991150i \(-0.542379\pi\)
−0.132745 + 0.991150i \(0.542379\pi\)
\(228\) 0 0
\(229\) 15.2554 1.00811 0.504054 0.863672i \(-0.331841\pi\)
0.504054 + 0.863672i \(0.331841\pi\)
\(230\) −6.74456 −0.444723
\(231\) 0 0
\(232\) −2.00000 −0.131306
\(233\) −22.0000 −1.44127 −0.720634 0.693316i \(-0.756149\pi\)
−0.720634 + 0.693316i \(0.756149\pi\)
\(234\) 0 0
\(235\) −4.00000 −0.260931
\(236\) −4.00000 −0.260378
\(237\) 0 0
\(238\) 4.74456 0.307544
\(239\) 16.2337 1.05007 0.525035 0.851081i \(-0.324052\pi\)
0.525035 + 0.851081i \(0.324052\pi\)
\(240\) 0 0
\(241\) 14.0000 0.901819 0.450910 0.892570i \(-0.351100\pi\)
0.450910 + 0.892570i \(0.351100\pi\)
\(242\) 1.00000 0.0642824
\(243\) 0 0
\(244\) 6.00000 0.384111
\(245\) −1.00000 −0.0638877
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 0 0
\(250\) −1.00000 −0.0632456
\(251\) 4.00000 0.252478 0.126239 0.992000i \(-0.459709\pi\)
0.126239 + 0.992000i \(0.459709\pi\)
\(252\) 0 0
\(253\) 6.74456 0.424027
\(254\) 8.00000 0.501965
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −0.510875 −0.0318675 −0.0159337 0.999873i \(-0.505072\pi\)
−0.0159337 + 0.999873i \(0.505072\pi\)
\(258\) 0 0
\(259\) −8.74456 −0.543361
\(260\) 4.74456 0.294245
\(261\) 0 0
\(262\) 13.4891 0.833361
\(263\) 2.51087 0.154827 0.0774136 0.996999i \(-0.475334\pi\)
0.0774136 + 0.996999i \(0.475334\pi\)
\(264\) 0 0
\(265\) −6.00000 −0.368577
\(266\) 0 0
\(267\) 0 0
\(268\) −1.25544 −0.0766880
\(269\) −0.510875 −0.0311486 −0.0155743 0.999879i \(-0.504958\pi\)
−0.0155743 + 0.999879i \(0.504958\pi\)
\(270\) 0 0
\(271\) 8.00000 0.485965 0.242983 0.970031i \(-0.421874\pi\)
0.242983 + 0.970031i \(0.421874\pi\)
\(272\) −4.74456 −0.287681
\(273\) 0 0
\(274\) 18.2337 1.10154
\(275\) 1.00000 0.0603023
\(276\) 0 0
\(277\) −3.48913 −0.209641 −0.104821 0.994491i \(-0.533427\pi\)
−0.104821 + 0.994491i \(0.533427\pi\)
\(278\) 21.4891 1.28883
\(279\) 0 0
\(280\) 1.00000 0.0597614
\(281\) −14.2337 −0.849111 −0.424555 0.905402i \(-0.639570\pi\)
−0.424555 + 0.905402i \(0.639570\pi\)
\(282\) 0 0
\(283\) −24.2337 −1.44054 −0.720272 0.693692i \(-0.755983\pi\)
−0.720272 + 0.693692i \(0.755983\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) −4.74456 −0.280552
\(287\) 6.00000 0.354169
\(288\) 0 0
\(289\) 5.51087 0.324169
\(290\) 2.00000 0.117444
\(291\) 0 0
\(292\) 6.00000 0.351123
\(293\) 6.00000 0.350524 0.175262 0.984522i \(-0.443923\pi\)
0.175262 + 0.984522i \(0.443923\pi\)
\(294\) 0 0
\(295\) 4.00000 0.232889
\(296\) 8.74456 0.508267
\(297\) 0 0
\(298\) −15.4891 −0.897261
\(299\) −32.0000 −1.85061
\(300\) 0 0
\(301\) −4.00000 −0.230556
\(302\) −4.00000 −0.230174
\(303\) 0 0
\(304\) 0 0
\(305\) −6.00000 −0.343559
\(306\) 0 0
\(307\) −16.2337 −0.926506 −0.463253 0.886226i \(-0.653318\pi\)
−0.463253 + 0.886226i \(0.653318\pi\)
\(308\) −1.00000 −0.0569803
\(309\) 0 0
\(310\) 0 0
\(311\) 6.74456 0.382449 0.191225 0.981546i \(-0.438754\pi\)
0.191225 + 0.981546i \(0.438754\pi\)
\(312\) 0 0
\(313\) 6.00000 0.339140 0.169570 0.985518i \(-0.445762\pi\)
0.169570 + 0.985518i \(0.445762\pi\)
\(314\) 6.00000 0.338600
\(315\) 0 0
\(316\) 12.0000 0.675053
\(317\) −28.9783 −1.62758 −0.813790 0.581159i \(-0.802600\pi\)
−0.813790 + 0.581159i \(0.802600\pi\)
\(318\) 0 0
\(319\) −2.00000 −0.111979
\(320\) −1.00000 −0.0559017
\(321\) 0 0
\(322\) −6.74456 −0.375860
\(323\) 0 0
\(324\) 0 0
\(325\) −4.74456 −0.263181
\(326\) 14.7446 0.816626
\(327\) 0 0
\(328\) −6.00000 −0.331295
\(329\) −4.00000 −0.220527
\(330\) 0 0
\(331\) −20.0000 −1.09930 −0.549650 0.835395i \(-0.685239\pi\)
−0.549650 + 0.835395i \(0.685239\pi\)
\(332\) 4.00000 0.219529
\(333\) 0 0
\(334\) −20.2337 −1.10714
\(335\) 1.25544 0.0685919
\(336\) 0 0
\(337\) −7.48913 −0.407959 −0.203979 0.978975i \(-0.565388\pi\)
−0.203979 + 0.978975i \(0.565388\pi\)
\(338\) 9.51087 0.517323
\(339\) 0 0
\(340\) 4.74456 0.257310
\(341\) 0 0
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) 4.00000 0.215666
\(345\) 0 0
\(346\) −23.4891 −1.26278
\(347\) 20.0000 1.07366 0.536828 0.843692i \(-0.319622\pi\)
0.536828 + 0.843692i \(0.319622\pi\)
\(348\) 0 0
\(349\) 24.9783 1.33706 0.668528 0.743687i \(-0.266925\pi\)
0.668528 + 0.743687i \(0.266925\pi\)
\(350\) −1.00000 −0.0534522
\(351\) 0 0
\(352\) 1.00000 0.0533002
\(353\) 31.4891 1.67600 0.837999 0.545673i \(-0.183726\pi\)
0.837999 + 0.545673i \(0.183726\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 4.74456 0.251461
\(357\) 0 0
\(358\) 4.00000 0.211407
\(359\) 18.7446 0.989300 0.494650 0.869092i \(-0.335296\pi\)
0.494650 + 0.869092i \(0.335296\pi\)
\(360\) 0 0
\(361\) −19.0000 −1.00000
\(362\) 12.7446 0.669839
\(363\) 0 0
\(364\) 4.74456 0.248683
\(365\) −6.00000 −0.314054
\(366\) 0 0
\(367\) 1.48913 0.0777317 0.0388659 0.999244i \(-0.487625\pi\)
0.0388659 + 0.999244i \(0.487625\pi\)
\(368\) 6.74456 0.351585
\(369\) 0 0
\(370\) −8.74456 −0.454608
\(371\) −6.00000 −0.311504
\(372\) 0 0
\(373\) 20.9783 1.08621 0.543106 0.839664i \(-0.317248\pi\)
0.543106 + 0.839664i \(0.317248\pi\)
\(374\) −4.74456 −0.245335
\(375\) 0 0
\(376\) 4.00000 0.206284
\(377\) 9.48913 0.488715
\(378\) 0 0
\(379\) 30.9783 1.59125 0.795623 0.605792i \(-0.207144\pi\)
0.795623 + 0.605792i \(0.207144\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 5.48913 0.280848
\(383\) −20.0000 −1.02195 −0.510976 0.859595i \(-0.670716\pi\)
−0.510976 + 0.859595i \(0.670716\pi\)
\(384\) 0 0
\(385\) 1.00000 0.0509647
\(386\) −20.9783 −1.06776
\(387\) 0 0
\(388\) −7.48913 −0.380203
\(389\) 4.51087 0.228710 0.114355 0.993440i \(-0.463520\pi\)
0.114355 + 0.993440i \(0.463520\pi\)
\(390\) 0 0
\(391\) −32.0000 −1.61831
\(392\) 1.00000 0.0505076
\(393\) 0 0
\(394\) −23.4891 −1.18337
\(395\) −12.0000 −0.603786
\(396\) 0 0
\(397\) 30.4674 1.52911 0.764557 0.644556i \(-0.222958\pi\)
0.764557 + 0.644556i \(0.222958\pi\)
\(398\) 26.9783 1.35230
\(399\) 0 0
\(400\) 1.00000 0.0500000
\(401\) −12.9783 −0.648103 −0.324051 0.946039i \(-0.605045\pi\)
−0.324051 + 0.946039i \(0.605045\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) −6.00000 −0.298511
\(405\) 0 0
\(406\) 2.00000 0.0992583
\(407\) 8.74456 0.433452
\(408\) 0 0
\(409\) 0.510875 0.0252611 0.0126306 0.999920i \(-0.495979\pi\)
0.0126306 + 0.999920i \(0.495979\pi\)
\(410\) 6.00000 0.296319
\(411\) 0 0
\(412\) 9.48913 0.467496
\(413\) 4.00000 0.196827
\(414\) 0 0
\(415\) −4.00000 −0.196352
\(416\) −4.74456 −0.232621
\(417\) 0 0
\(418\) 0 0
\(419\) 25.4891 1.24523 0.622613 0.782530i \(-0.286071\pi\)
0.622613 + 0.782530i \(0.286071\pi\)
\(420\) 0 0
\(421\) 16.9783 0.827469 0.413735 0.910398i \(-0.364224\pi\)
0.413735 + 0.910398i \(0.364224\pi\)
\(422\) 13.2554 0.645265
\(423\) 0 0
\(424\) 6.00000 0.291386
\(425\) −4.74456 −0.230145
\(426\) 0 0
\(427\) −6.00000 −0.290360
\(428\) 4.00000 0.193347
\(429\) 0 0
\(430\) −4.00000 −0.192897
\(431\) 13.2554 0.638492 0.319246 0.947672i \(-0.396570\pi\)
0.319246 + 0.947672i \(0.396570\pi\)
\(432\) 0 0
\(433\) −4.51087 −0.216779 −0.108389 0.994109i \(-0.534569\pi\)
−0.108389 + 0.994109i \(0.534569\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 3.25544 0.155907
\(437\) 0 0
\(438\) 0 0
\(439\) 16.0000 0.763638 0.381819 0.924237i \(-0.375298\pi\)
0.381819 + 0.924237i \(0.375298\pi\)
\(440\) −1.00000 −0.0476731
\(441\) 0 0
\(442\) 22.5109 1.07073
\(443\) 10.9783 0.521592 0.260796 0.965394i \(-0.416015\pi\)
0.260796 + 0.965394i \(0.416015\pi\)
\(444\) 0 0
\(445\) −4.74456 −0.224914
\(446\) −1.48913 −0.0705121
\(447\) 0 0
\(448\) −1.00000 −0.0472456
\(449\) 14.0000 0.660701 0.330350 0.943858i \(-0.392833\pi\)
0.330350 + 0.943858i \(0.392833\pi\)
\(450\) 0 0
\(451\) −6.00000 −0.282529
\(452\) −0.744563 −0.0350213
\(453\) 0 0
\(454\) −4.00000 −0.187729
\(455\) −4.74456 −0.222429
\(456\) 0 0
\(457\) 16.9783 0.794209 0.397105 0.917773i \(-0.370015\pi\)
0.397105 + 0.917773i \(0.370015\pi\)
\(458\) 15.2554 0.712840
\(459\) 0 0
\(460\) −6.74456 −0.314467
\(461\) 23.4891 1.09400 0.546999 0.837133i \(-0.315770\pi\)
0.546999 + 0.837133i \(0.315770\pi\)
\(462\) 0 0
\(463\) −37.4891 −1.74227 −0.871134 0.491046i \(-0.836615\pi\)
−0.871134 + 0.491046i \(0.836615\pi\)
\(464\) −2.00000 −0.0928477
\(465\) 0 0
\(466\) −22.0000 −1.01913
\(467\) −4.00000 −0.185098 −0.0925490 0.995708i \(-0.529501\pi\)
−0.0925490 + 0.995708i \(0.529501\pi\)
\(468\) 0 0
\(469\) 1.25544 0.0579707
\(470\) −4.00000 −0.184506
\(471\) 0 0
\(472\) −4.00000 −0.184115
\(473\) 4.00000 0.183920
\(474\) 0 0
\(475\) 0 0
\(476\) 4.74456 0.217467
\(477\) 0 0
\(478\) 16.2337 0.742512
\(479\) 8.00000 0.365529 0.182765 0.983157i \(-0.441495\pi\)
0.182765 + 0.983157i \(0.441495\pi\)
\(480\) 0 0
\(481\) −41.4891 −1.89174
\(482\) 14.0000 0.637683
\(483\) 0 0
\(484\) 1.00000 0.0454545
\(485\) 7.48913 0.340064
\(486\) 0 0
\(487\) −10.5109 −0.476293 −0.238147 0.971229i \(-0.576540\pi\)
−0.238147 + 0.971229i \(0.576540\pi\)
\(488\) 6.00000 0.271607
\(489\) 0 0
\(490\) −1.00000 −0.0451754
\(491\) 22.9783 1.03699 0.518497 0.855079i \(-0.326492\pi\)
0.518497 + 0.855079i \(0.326492\pi\)
\(492\) 0 0
\(493\) 9.48913 0.427369
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) −25.4891 −1.14105 −0.570525 0.821280i \(-0.693260\pi\)
−0.570525 + 0.821280i \(0.693260\pi\)
\(500\) −1.00000 −0.0447214
\(501\) 0 0
\(502\) 4.00000 0.178529
\(503\) −17.2554 −0.769382 −0.384691 0.923045i \(-0.625692\pi\)
−0.384691 + 0.923045i \(0.625692\pi\)
\(504\) 0 0
\(505\) 6.00000 0.266996
\(506\) 6.74456 0.299832
\(507\) 0 0
\(508\) 8.00000 0.354943
\(509\) −16.5109 −0.731832 −0.365916 0.930648i \(-0.619244\pi\)
−0.365916 + 0.930648i \(0.619244\pi\)
\(510\) 0 0
\(511\) −6.00000 −0.265424
\(512\) 1.00000 0.0441942
\(513\) 0 0
\(514\) −0.510875 −0.0225337
\(515\) −9.48913 −0.418141
\(516\) 0 0
\(517\) 4.00000 0.175920
\(518\) −8.74456 −0.384214
\(519\) 0 0
\(520\) 4.74456 0.208063
\(521\) 31.7228 1.38980 0.694901 0.719106i \(-0.255448\pi\)
0.694901 + 0.719106i \(0.255448\pi\)
\(522\) 0 0
\(523\) 5.25544 0.229804 0.114902 0.993377i \(-0.463345\pi\)
0.114902 + 0.993377i \(0.463345\pi\)
\(524\) 13.4891 0.589275
\(525\) 0 0
\(526\) 2.51087 0.109479
\(527\) 0 0
\(528\) 0 0
\(529\) 22.4891 0.977788
\(530\) −6.00000 −0.260623
\(531\) 0 0
\(532\) 0 0
\(533\) 28.4674 1.23306
\(534\) 0 0
\(535\) −4.00000 −0.172935
\(536\) −1.25544 −0.0542266
\(537\) 0 0
\(538\) −0.510875 −0.0220254
\(539\) 1.00000 0.0430730
\(540\) 0 0
\(541\) −18.2337 −0.783927 −0.391964 0.919981i \(-0.628204\pi\)
−0.391964 + 0.919981i \(0.628204\pi\)
\(542\) 8.00000 0.343629
\(543\) 0 0
\(544\) −4.74456 −0.203421
\(545\) −3.25544 −0.139448
\(546\) 0 0
\(547\) −36.4674 −1.55923 −0.779616 0.626258i \(-0.784586\pi\)
−0.779616 + 0.626258i \(0.784586\pi\)
\(548\) 18.2337 0.778905
\(549\) 0 0
\(550\) 1.00000 0.0426401
\(551\) 0 0
\(552\) 0 0
\(553\) −12.0000 −0.510292
\(554\) −3.48913 −0.148239
\(555\) 0 0
\(556\) 21.4891 0.911342
\(557\) 11.4891 0.486810 0.243405 0.969925i \(-0.421736\pi\)
0.243405 + 0.969925i \(0.421736\pi\)
\(558\) 0 0
\(559\) −18.9783 −0.802694
\(560\) 1.00000 0.0422577
\(561\) 0 0
\(562\) −14.2337 −0.600412
\(563\) 17.4891 0.737079 0.368539 0.929612i \(-0.379858\pi\)
0.368539 + 0.929612i \(0.379858\pi\)
\(564\) 0 0
\(565\) 0.744563 0.0313240
\(566\) −24.2337 −1.01862
\(567\) 0 0
\(568\) 0 0
\(569\) 15.2554 0.639541 0.319771 0.947495i \(-0.396394\pi\)
0.319771 + 0.947495i \(0.396394\pi\)
\(570\) 0 0
\(571\) −29.2554 −1.22430 −0.612151 0.790741i \(-0.709696\pi\)
−0.612151 + 0.790741i \(0.709696\pi\)
\(572\) −4.74456 −0.198380
\(573\) 0 0
\(574\) 6.00000 0.250435
\(575\) 6.74456 0.281268
\(576\) 0 0
\(577\) −10.0000 −0.416305 −0.208153 0.978096i \(-0.566745\pi\)
−0.208153 + 0.978096i \(0.566745\pi\)
\(578\) 5.51087 0.229222
\(579\) 0 0
\(580\) 2.00000 0.0830455
\(581\) −4.00000 −0.165948
\(582\) 0 0
\(583\) 6.00000 0.248495
\(584\) 6.00000 0.248282
\(585\) 0 0
\(586\) 6.00000 0.247858
\(587\) −28.0000 −1.15568 −0.577842 0.816149i \(-0.696105\pi\)
−0.577842 + 0.816149i \(0.696105\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 4.00000 0.164677
\(591\) 0 0
\(592\) 8.74456 0.359399
\(593\) −34.2337 −1.40581 −0.702905 0.711284i \(-0.748114\pi\)
−0.702905 + 0.711284i \(0.748114\pi\)
\(594\) 0 0
\(595\) −4.74456 −0.194508
\(596\) −15.4891 −0.634459
\(597\) 0 0
\(598\) −32.0000 −1.30858
\(599\) −24.4674 −0.999710 −0.499855 0.866109i \(-0.666613\pi\)
−0.499855 + 0.866109i \(0.666613\pi\)
\(600\) 0 0
\(601\) 38.4674 1.56912 0.784558 0.620055i \(-0.212890\pi\)
0.784558 + 0.620055i \(0.212890\pi\)
\(602\) −4.00000 −0.163028
\(603\) 0 0
\(604\) −4.00000 −0.162758
\(605\) −1.00000 −0.0406558
\(606\) 0 0
\(607\) 10.9783 0.445593 0.222797 0.974865i \(-0.428481\pi\)
0.222797 + 0.974865i \(0.428481\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) −6.00000 −0.242933
\(611\) −18.9783 −0.767778
\(612\) 0 0
\(613\) −8.51087 −0.343751 −0.171875 0.985119i \(-0.554983\pi\)
−0.171875 + 0.985119i \(0.554983\pi\)
\(614\) −16.2337 −0.655138
\(615\) 0 0
\(616\) −1.00000 −0.0402911
\(617\) 42.2337 1.70026 0.850132 0.526569i \(-0.176522\pi\)
0.850132 + 0.526569i \(0.176522\pi\)
\(618\) 0 0
\(619\) 32.2337 1.29558 0.647791 0.761818i \(-0.275693\pi\)
0.647791 + 0.761818i \(0.275693\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 6.74456 0.270432
\(623\) −4.74456 −0.190087
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 6.00000 0.239808
\(627\) 0 0
\(628\) 6.00000 0.239426
\(629\) −41.4891 −1.65428
\(630\) 0 0
\(631\) −18.9783 −0.755512 −0.377756 0.925905i \(-0.623304\pi\)
−0.377756 + 0.925905i \(0.623304\pi\)
\(632\) 12.0000 0.477334
\(633\) 0 0
\(634\) −28.9783 −1.15087
\(635\) −8.00000 −0.317470
\(636\) 0 0
\(637\) −4.74456 −0.187986
\(638\) −2.00000 −0.0791808
\(639\) 0 0
\(640\) −1.00000 −0.0395285
\(641\) 3.02175 0.119352 0.0596760 0.998218i \(-0.480993\pi\)
0.0596760 + 0.998218i \(0.480993\pi\)
\(642\) 0 0
\(643\) −4.00000 −0.157745 −0.0788723 0.996885i \(-0.525132\pi\)
−0.0788723 + 0.996885i \(0.525132\pi\)
\(644\) −6.74456 −0.265773
\(645\) 0 0
\(646\) 0 0
\(647\) −20.0000 −0.786281 −0.393141 0.919478i \(-0.628611\pi\)
−0.393141 + 0.919478i \(0.628611\pi\)
\(648\) 0 0
\(649\) −4.00000 −0.157014
\(650\) −4.74456 −0.186097
\(651\) 0 0
\(652\) 14.7446 0.577442
\(653\) 11.4891 0.449604 0.224802 0.974404i \(-0.427826\pi\)
0.224802 + 0.974404i \(0.427826\pi\)
\(654\) 0 0
\(655\) −13.4891 −0.527064
\(656\) −6.00000 −0.234261
\(657\) 0 0
\(658\) −4.00000 −0.155936
\(659\) −6.97825 −0.271834 −0.135917 0.990720i \(-0.543398\pi\)
−0.135917 + 0.990720i \(0.543398\pi\)
\(660\) 0 0
\(661\) −46.2337 −1.79828 −0.899141 0.437659i \(-0.855808\pi\)
−0.899141 + 0.437659i \(0.855808\pi\)
\(662\) −20.0000 −0.777322
\(663\) 0 0
\(664\) 4.00000 0.155230
\(665\) 0 0
\(666\) 0 0
\(667\) −13.4891 −0.522301
\(668\) −20.2337 −0.782865
\(669\) 0 0
\(670\) 1.25544 0.0485018
\(671\) 6.00000 0.231627
\(672\) 0 0
\(673\) 8.51087 0.328070 0.164035 0.986455i \(-0.447549\pi\)
0.164035 + 0.986455i \(0.447549\pi\)
\(674\) −7.48913 −0.288470
\(675\) 0 0
\(676\) 9.51087 0.365803
\(677\) 46.4674 1.78589 0.892943 0.450169i \(-0.148636\pi\)
0.892943 + 0.450169i \(0.148636\pi\)
\(678\) 0 0
\(679\) 7.48913 0.287406
\(680\) 4.74456 0.181946
\(681\) 0 0
\(682\) 0 0
\(683\) −10.9783 −0.420071 −0.210036 0.977694i \(-0.567358\pi\)
−0.210036 + 0.977694i \(0.567358\pi\)
\(684\) 0 0
\(685\) −18.2337 −0.696673
\(686\) −1.00000 −0.0381802
\(687\) 0 0
\(688\) 4.00000 0.152499
\(689\) −28.4674 −1.08452
\(690\) 0 0
\(691\) 16.2337 0.617559 0.308779 0.951134i \(-0.400080\pi\)
0.308779 + 0.951134i \(0.400080\pi\)
\(692\) −23.4891 −0.892923
\(693\) 0 0
\(694\) 20.0000 0.759190
\(695\) −21.4891 −0.815129
\(696\) 0 0
\(697\) 28.4674 1.07828
\(698\) 24.9783 0.945441
\(699\) 0 0
\(700\) −1.00000 −0.0377964
\(701\) −26.0000 −0.982006 −0.491003 0.871158i \(-0.663370\pi\)
−0.491003 + 0.871158i \(0.663370\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 1.00000 0.0376889
\(705\) 0 0
\(706\) 31.4891 1.18511
\(707\) 6.00000 0.225653
\(708\) 0 0
\(709\) −28.9783 −1.08830 −0.544151 0.838988i \(-0.683148\pi\)
−0.544151 + 0.838988i \(0.683148\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 4.74456 0.177810
\(713\) 0 0
\(714\) 0 0
\(715\) 4.74456 0.177437
\(716\) 4.00000 0.149487
\(717\) 0 0
\(718\) 18.7446 0.699541
\(719\) −33.7228 −1.25765 −0.628824 0.777547i \(-0.716464\pi\)
−0.628824 + 0.777547i \(0.716464\pi\)
\(720\) 0 0
\(721\) −9.48913 −0.353393
\(722\) −19.0000 −0.707107
\(723\) 0 0
\(724\) 12.7446 0.473648
\(725\) −2.00000 −0.0742781
\(726\) 0 0
\(727\) −33.4891 −1.24204 −0.621021 0.783794i \(-0.713282\pi\)
−0.621021 + 0.783794i \(0.713282\pi\)
\(728\) 4.74456 0.175845
\(729\) 0 0
\(730\) −6.00000 −0.222070
\(731\) −18.9783 −0.701936
\(732\) 0 0
\(733\) −12.7446 −0.470731 −0.235366 0.971907i \(-0.575629\pi\)
−0.235366 + 0.971907i \(0.575629\pi\)
\(734\) 1.48913 0.0549646
\(735\) 0 0
\(736\) 6.74456 0.248608
\(737\) −1.25544 −0.0462446
\(738\) 0 0
\(739\) 42.7446 1.57238 0.786192 0.617982i \(-0.212050\pi\)
0.786192 + 0.617982i \(0.212050\pi\)
\(740\) −8.74456 −0.321457
\(741\) 0 0
\(742\) −6.00000 −0.220267
\(743\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(744\) 0 0
\(745\) 15.4891 0.567478
\(746\) 20.9783 0.768068
\(747\) 0 0
\(748\) −4.74456 −0.173478
\(749\) −4.00000 −0.146157
\(750\) 0 0
\(751\) −50.9783 −1.86022 −0.930111 0.367278i \(-0.880290\pi\)
−0.930111 + 0.367278i \(0.880290\pi\)
\(752\) 4.00000 0.145865
\(753\) 0 0
\(754\) 9.48913 0.345574
\(755\) 4.00000 0.145575
\(756\) 0 0
\(757\) 51.7228 1.87990 0.939949 0.341315i \(-0.110872\pi\)
0.939949 + 0.341315i \(0.110872\pi\)
\(758\) 30.9783 1.12518
\(759\) 0 0
\(760\) 0 0
\(761\) 39.9565 1.44842 0.724211 0.689578i \(-0.242204\pi\)
0.724211 + 0.689578i \(0.242204\pi\)
\(762\) 0 0
\(763\) −3.25544 −0.117855
\(764\) 5.48913 0.198590
\(765\) 0 0
\(766\) −20.0000 −0.722629
\(767\) 18.9783 0.685265
\(768\) 0 0
\(769\) 8.51087 0.306910 0.153455 0.988156i \(-0.450960\pi\)
0.153455 + 0.988156i \(0.450960\pi\)
\(770\) 1.00000 0.0360375
\(771\) 0 0
\(772\) −20.9783 −0.755024
\(773\) −54.4674 −1.95906 −0.979528 0.201310i \(-0.935480\pi\)
−0.979528 + 0.201310i \(0.935480\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) −7.48913 −0.268844
\(777\) 0 0
\(778\) 4.51087 0.161723
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) −32.0000 −1.14432
\(783\) 0 0
\(784\) 1.00000 0.0357143
\(785\) −6.00000 −0.214149
\(786\) 0 0
\(787\) −40.2337 −1.43418 −0.717088 0.696983i \(-0.754525\pi\)
−0.717088 + 0.696983i \(0.754525\pi\)
\(788\) −23.4891 −0.836765
\(789\) 0 0
\(790\) −12.0000 −0.426941
\(791\) 0.744563 0.0264736
\(792\) 0 0
\(793\) −28.4674 −1.01091
\(794\) 30.4674 1.08125
\(795\) 0 0
\(796\) 26.9783 0.956219
\(797\) −24.5109 −0.868220 −0.434110 0.900860i \(-0.642937\pi\)
−0.434110 + 0.900860i \(0.642937\pi\)
\(798\) 0 0
\(799\) −18.9783 −0.671402
\(800\) 1.00000 0.0353553
\(801\) 0 0
\(802\) −12.9783 −0.458278
\(803\) 6.00000 0.211735
\(804\) 0 0
\(805\) 6.74456 0.237715
\(806\) 0 0
\(807\) 0 0
\(808\) −6.00000 −0.211079
\(809\) 26.2337 0.922327 0.461164 0.887315i \(-0.347432\pi\)
0.461164 + 0.887315i \(0.347432\pi\)
\(810\) 0 0
\(811\) −16.0000 −0.561836 −0.280918 0.959732i \(-0.590639\pi\)
−0.280918 + 0.959732i \(0.590639\pi\)
\(812\) 2.00000 0.0701862
\(813\) 0 0
\(814\) 8.74456 0.306497
\(815\) −14.7446 −0.516480
\(816\) 0 0
\(817\) 0 0
\(818\) 0.510875 0.0178623
\(819\) 0 0
\(820\) 6.00000 0.209529
\(821\) −44.9783 −1.56975 −0.784876 0.619653i \(-0.787273\pi\)
−0.784876 + 0.619653i \(0.787273\pi\)
\(822\) 0 0
\(823\) 8.00000 0.278862 0.139431 0.990232i \(-0.455473\pi\)
0.139431 + 0.990232i \(0.455473\pi\)
\(824\) 9.48913 0.330569
\(825\) 0 0
\(826\) 4.00000 0.139178
\(827\) −46.9783 −1.63359 −0.816797 0.576925i \(-0.804252\pi\)
−0.816797 + 0.576925i \(0.804252\pi\)
\(828\) 0 0
\(829\) −6.23369 −0.216505 −0.108252 0.994123i \(-0.534525\pi\)
−0.108252 + 0.994123i \(0.534525\pi\)
\(830\) −4.00000 −0.138842
\(831\) 0 0
\(832\) −4.74456 −0.164488
\(833\) −4.74456 −0.164389
\(834\) 0 0
\(835\) 20.2337 0.700216
\(836\) 0 0
\(837\) 0 0
\(838\) 25.4891 0.880507
\(839\) −17.2554 −0.595724 −0.297862 0.954609i \(-0.596274\pi\)
−0.297862 + 0.954609i \(0.596274\pi\)
\(840\) 0 0
\(841\) −25.0000 −0.862069
\(842\) 16.9783 0.585109
\(843\) 0 0
\(844\) 13.2554 0.456271
\(845\) −9.51087 −0.327184
\(846\) 0 0
\(847\) −1.00000 −0.0343604
\(848\) 6.00000 0.206041
\(849\) 0 0
\(850\) −4.74456 −0.162737
\(851\) 58.9783 2.02175
\(852\) 0 0
\(853\) −9.76631 −0.334392 −0.167196 0.985924i \(-0.553471\pi\)
−0.167196 + 0.985924i \(0.553471\pi\)
\(854\) −6.00000 −0.205316
\(855\) 0 0
\(856\) 4.00000 0.136717
\(857\) −15.2554 −0.521116 −0.260558 0.965458i \(-0.583906\pi\)
−0.260558 + 0.965458i \(0.583906\pi\)
\(858\) 0 0
\(859\) 23.7663 0.810896 0.405448 0.914118i \(-0.367116\pi\)
0.405448 + 0.914118i \(0.367116\pi\)
\(860\) −4.00000 −0.136399
\(861\) 0 0
\(862\) 13.2554 0.451482
\(863\) −9.25544 −0.315059 −0.157529 0.987514i \(-0.550353\pi\)
−0.157529 + 0.987514i \(0.550353\pi\)
\(864\) 0 0
\(865\) 23.4891 0.798654
\(866\) −4.51087 −0.153286
\(867\) 0 0
\(868\) 0 0
\(869\) 12.0000 0.407072
\(870\) 0 0
\(871\) 5.95650 0.201828
\(872\) 3.25544 0.110243
\(873\) 0 0
\(874\) 0 0
\(875\) 1.00000 0.0338062
\(876\) 0 0
\(877\) −27.4891 −0.928242 −0.464121 0.885772i \(-0.653630\pi\)
−0.464121 + 0.885772i \(0.653630\pi\)
\(878\) 16.0000 0.539974
\(879\) 0 0
\(880\) −1.00000 −0.0337100
\(881\) 20.7446 0.698902 0.349451 0.936955i \(-0.386368\pi\)
0.349451 + 0.936955i \(0.386368\pi\)
\(882\) 0 0
\(883\) 44.7011 1.50431 0.752155 0.658986i \(-0.229014\pi\)
0.752155 + 0.658986i \(0.229014\pi\)
\(884\) 22.5109 0.757123
\(885\) 0 0
\(886\) 10.9783 0.368822
\(887\) −33.7228 −1.13230 −0.566151 0.824302i \(-0.691568\pi\)
−0.566151 + 0.824302i \(0.691568\pi\)
\(888\) 0 0
\(889\) −8.00000 −0.268311
\(890\) −4.74456 −0.159038
\(891\) 0 0
\(892\) −1.48913 −0.0498596
\(893\) 0 0
\(894\) 0 0
\(895\) −4.00000 −0.133705
\(896\) −1.00000 −0.0334077
\(897\) 0 0
\(898\) 14.0000 0.467186
\(899\) 0 0
\(900\) 0 0
\(901\) −28.4674 −0.948386
\(902\) −6.00000 −0.199778
\(903\) 0 0
\(904\) −0.744563 −0.0247638
\(905\) −12.7446 −0.423644
\(906\) 0 0
\(907\) 1.25544 0.0416861 0.0208431 0.999783i \(-0.493365\pi\)
0.0208431 + 0.999783i \(0.493365\pi\)
\(908\) −4.00000 −0.132745
\(909\) 0 0
\(910\) −4.74456 −0.157281
\(911\) 10.9783 0.363726 0.181863 0.983324i \(-0.441787\pi\)
0.181863 + 0.983324i \(0.441787\pi\)
\(912\) 0 0
\(913\) 4.00000 0.132381
\(914\) 16.9783 0.561591
\(915\) 0 0
\(916\) 15.2554 0.504054
\(917\) −13.4891 −0.445450
\(918\) 0 0
\(919\) −6.51087 −0.214774 −0.107387 0.994217i \(-0.534248\pi\)
−0.107387 + 0.994217i \(0.534248\pi\)
\(920\) −6.74456 −0.222362
\(921\) 0 0
\(922\) 23.4891 0.773573
\(923\) 0 0
\(924\) 0 0
\(925\) 8.74456 0.287519
\(926\) −37.4891 −1.23197
\(927\) 0 0
\(928\) −2.00000 −0.0656532
\(929\) 15.2554 0.500515 0.250257 0.968179i \(-0.419485\pi\)
0.250257 + 0.968179i \(0.419485\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) −22.0000 −0.720634
\(933\) 0 0
\(934\) −4.00000 −0.130884
\(935\) 4.74456 0.155164
\(936\) 0 0
\(937\) 19.4891 0.636682 0.318341 0.947976i \(-0.396874\pi\)
0.318341 + 0.947976i \(0.396874\pi\)
\(938\) 1.25544 0.0409915
\(939\) 0 0
\(940\) −4.00000 −0.130466
\(941\) 39.9565 1.30254 0.651272 0.758844i \(-0.274236\pi\)
0.651272 + 0.758844i \(0.274236\pi\)
\(942\) 0 0
\(943\) −40.4674 −1.31780
\(944\) −4.00000 −0.130189
\(945\) 0 0
\(946\) 4.00000 0.130051
\(947\) 18.5109 0.601523 0.300761 0.953699i \(-0.402759\pi\)
0.300761 + 0.953699i \(0.402759\pi\)
\(948\) 0 0
\(949\) −28.4674 −0.924090
\(950\) 0 0
\(951\) 0 0
\(952\) 4.74456 0.153772
\(953\) 39.4891 1.27918 0.639589 0.768717i \(-0.279105\pi\)
0.639589 + 0.768717i \(0.279105\pi\)
\(954\) 0 0
\(955\) −5.48913 −0.177624
\(956\) 16.2337 0.525035
\(957\) 0 0
\(958\) 8.00000 0.258468
\(959\) −18.2337 −0.588796
\(960\) 0 0
\(961\) −31.0000 −1.00000
\(962\) −41.4891 −1.33766
\(963\) 0 0
\(964\) 14.0000 0.450910
\(965\) 20.9783 0.675314
\(966\) 0 0
\(967\) 26.9783 0.867562 0.433781 0.901018i \(-0.357179\pi\)
0.433781 + 0.901018i \(0.357179\pi\)
\(968\) 1.00000 0.0321412
\(969\) 0 0
\(970\) 7.48913 0.240461
\(971\) −17.4891 −0.561253 −0.280626 0.959817i \(-0.590542\pi\)
−0.280626 + 0.959817i \(0.590542\pi\)
\(972\) 0 0
\(973\) −21.4891 −0.688910
\(974\) −10.5109 −0.336790
\(975\) 0 0
\(976\) 6.00000 0.192055
\(977\) −3.25544 −0.104151 −0.0520753 0.998643i \(-0.516584\pi\)
−0.0520753 + 0.998643i \(0.516584\pi\)
\(978\) 0 0
\(979\) 4.74456 0.151637
\(980\) −1.00000 −0.0319438
\(981\) 0 0
\(982\) 22.9783 0.733265
\(983\) 9.02175 0.287749 0.143875 0.989596i \(-0.454044\pi\)
0.143875 + 0.989596i \(0.454044\pi\)
\(984\) 0 0
\(985\) 23.4891 0.748426
\(986\) 9.48913 0.302195
\(987\) 0 0
\(988\) 0 0
\(989\) 26.9783 0.857858
\(990\) 0 0
\(991\) −34.9783 −1.11112 −0.555560 0.831476i \(-0.687496\pi\)
−0.555560 + 0.831476i \(0.687496\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −26.9783 −0.855268
\(996\) 0 0
\(997\) 13.7663 0.435983 0.217992 0.975951i \(-0.430049\pi\)
0.217992 + 0.975951i \(0.430049\pi\)
\(998\) −25.4891 −0.806844
\(999\) 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.by.1.1 2
3.2 odd 2 2310.2.a.x.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2310.2.a.x.1.1 2 3.2 odd 2
6930.2.a.by.1.1 2 1.1 even 1 trivial