Properties

Label 3450.2.a.bl.1.1
Level $3450$
Weight $2$
Character 3450.1
Self dual yes
Analytic conductor $27.548$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 3450 = 2 \cdot 3 \cdot 5^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3450.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(27.5483886973\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{6}) \)
Defining polynomial: \(x^{2} - 6\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-2.44949\) of defining polynomial
Character \(\chi\) \(=\) 3450.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} -3.44949 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} -3.44949 q^{7} +1.00000 q^{8} +1.00000 q^{9} +2.00000 q^{11} +1.00000 q^{12} +6.89898 q^{13} -3.44949 q^{14} +1.00000 q^{16} -0.550510 q^{17} +1.00000 q^{18} -2.89898 q^{19} -3.44949 q^{21} +2.00000 q^{22} +1.00000 q^{23} +1.00000 q^{24} +6.89898 q^{26} +1.00000 q^{27} -3.44949 q^{28} +5.00000 q^{29} +2.00000 q^{31} +1.00000 q^{32} +2.00000 q^{33} -0.550510 q^{34} +1.00000 q^{36} -6.34847 q^{37} -2.89898 q^{38} +6.89898 q^{39} +4.89898 q^{41} -3.44949 q^{42} +1.10102 q^{43} +2.00000 q^{44} +1.00000 q^{46} +5.89898 q^{47} +1.00000 q^{48} +4.89898 q^{49} -0.550510 q^{51} +6.89898 q^{52} -8.89898 q^{53} +1.00000 q^{54} -3.44949 q^{56} -2.89898 q^{57} +5.00000 q^{58} +10.0000 q^{59} +4.89898 q^{61} +2.00000 q^{62} -3.44949 q^{63} +1.00000 q^{64} +2.00000 q^{66} -2.00000 q^{67} -0.550510 q^{68} +1.00000 q^{69} -8.79796 q^{71} +1.00000 q^{72} +11.8990 q^{73} -6.34847 q^{74} -2.89898 q^{76} -6.89898 q^{77} +6.89898 q^{78} +10.0000 q^{79} +1.00000 q^{81} +4.89898 q^{82} -7.44949 q^{83} -3.44949 q^{84} +1.10102 q^{86} +5.00000 q^{87} +2.00000 q^{88} +4.34847 q^{89} -23.7980 q^{91} +1.00000 q^{92} +2.00000 q^{93} +5.89898 q^{94} +1.00000 q^{96} +16.6969 q^{97} +4.89898 q^{98} +2.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{2} + 2q^{3} + 2q^{4} + 2q^{6} - 2q^{7} + 2q^{8} + 2q^{9} + O(q^{10}) \) \( 2q + 2q^{2} + 2q^{3} + 2q^{4} + 2q^{6} - 2q^{7} + 2q^{8} + 2q^{9} + 4q^{11} + 2q^{12} + 4q^{13} - 2q^{14} + 2q^{16} - 6q^{17} + 2q^{18} + 4q^{19} - 2q^{21} + 4q^{22} + 2q^{23} + 2q^{24} + 4q^{26} + 2q^{27} - 2q^{28} + 10q^{29} + 4q^{31} + 2q^{32} + 4q^{33} - 6q^{34} + 2q^{36} + 2q^{37} + 4q^{38} + 4q^{39} - 2q^{42} + 12q^{43} + 4q^{44} + 2q^{46} + 2q^{47} + 2q^{48} - 6q^{51} + 4q^{52} - 8q^{53} + 2q^{54} - 2q^{56} + 4q^{57} + 10q^{58} + 20q^{59} + 4q^{62} - 2q^{63} + 2q^{64} + 4q^{66} - 4q^{67} - 6q^{68} + 2q^{69} + 2q^{71} + 2q^{72} + 14q^{73} + 2q^{74} + 4q^{76} - 4q^{77} + 4q^{78} + 20q^{79} + 2q^{81} - 10q^{83} - 2q^{84} + 12q^{86} + 10q^{87} + 4q^{88} - 6q^{89} - 28q^{91} + 2q^{92} + 4q^{93} + 2q^{94} + 2q^{96} + 4q^{97} + 4q^{99} + O(q^{100}) \)

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
\(3\) 1.00000 0.577350
\(4\) 1.00000 0.500000
\(5\) 0 0
\(6\) 1.00000 0.408248
\(7\) −3.44949 −1.30378 −0.651892 0.758312i \(-0.726025\pi\)
−0.651892 + 0.758312i \(0.726025\pi\)
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 2.00000 0.603023 0.301511 0.953463i \(-0.402509\pi\)
0.301511 + 0.953463i \(0.402509\pi\)
\(12\) 1.00000 0.288675
\(13\) 6.89898 1.91343 0.956716 0.291022i \(-0.0939953\pi\)
0.956716 + 0.291022i \(0.0939953\pi\)
\(14\) −3.44949 −0.921915
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −0.550510 −0.133518 −0.0667592 0.997769i \(-0.521266\pi\)
−0.0667592 + 0.997769i \(0.521266\pi\)
\(18\) 1.00000 0.235702
\(19\) −2.89898 −0.665072 −0.332536 0.943091i \(-0.607904\pi\)
−0.332536 + 0.943091i \(0.607904\pi\)
\(20\) 0 0
\(21\) −3.44949 −0.752740
\(22\) 2.00000 0.426401
\(23\) 1.00000 0.208514
\(24\) 1.00000 0.204124
\(25\) 0 0
\(26\) 6.89898 1.35300
\(27\) 1.00000 0.192450
\(28\) −3.44949 −0.651892
\(29\) 5.00000 0.928477 0.464238 0.885710i \(-0.346328\pi\)
0.464238 + 0.885710i \(0.346328\pi\)
\(30\) 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
\(33\) 2.00000 0.348155
\(34\) −0.550510 −0.0944117
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) −6.34847 −1.04368 −0.521841 0.853043i \(-0.674755\pi\)
−0.521841 + 0.853043i \(0.674755\pi\)
\(38\) −2.89898 −0.470277
\(39\) 6.89898 1.10472
\(40\) 0 0
\(41\) 4.89898 0.765092 0.382546 0.923936i \(-0.375047\pi\)
0.382546 + 0.923936i \(0.375047\pi\)
\(42\) −3.44949 −0.532268
\(43\) 1.10102 0.167904 0.0839520 0.996470i \(-0.473246\pi\)
0.0839520 + 0.996470i \(0.473246\pi\)
\(44\) 2.00000 0.301511
\(45\) 0 0
\(46\) 1.00000 0.147442
\(47\) 5.89898 0.860455 0.430227 0.902721i \(-0.358433\pi\)
0.430227 + 0.902721i \(0.358433\pi\)
\(48\) 1.00000 0.144338
\(49\) 4.89898 0.699854
\(50\) 0 0
\(51\) −0.550510 −0.0770869
\(52\) 6.89898 0.956716
\(53\) −8.89898 −1.22237 −0.611184 0.791488i \(-0.709307\pi\)
−0.611184 + 0.791488i \(0.709307\pi\)
\(54\) 1.00000 0.136083
\(55\) 0 0
\(56\) −3.44949 −0.460957
\(57\) −2.89898 −0.383979
\(58\) 5.00000 0.656532
\(59\) 10.0000 1.30189 0.650945 0.759125i \(-0.274373\pi\)
0.650945 + 0.759125i \(0.274373\pi\)
\(60\) 0 0
\(61\) 4.89898 0.627250 0.313625 0.949547i \(-0.398457\pi\)
0.313625 + 0.949547i \(0.398457\pi\)
\(62\) 2.00000 0.254000
\(63\) −3.44949 −0.434595
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 2.00000 0.246183
\(67\) −2.00000 −0.244339 −0.122169 0.992509i \(-0.538985\pi\)
−0.122169 + 0.992509i \(0.538985\pi\)
\(68\) −0.550510 −0.0667592
\(69\) 1.00000 0.120386
\(70\) 0 0
\(71\) −8.79796 −1.04413 −0.522063 0.852907i \(-0.674837\pi\)
−0.522063 + 0.852907i \(0.674837\pi\)
\(72\) 1.00000 0.117851
\(73\) 11.8990 1.39267 0.696335 0.717717i \(-0.254813\pi\)
0.696335 + 0.717717i \(0.254813\pi\)
\(74\) −6.34847 −0.737995
\(75\) 0 0
\(76\) −2.89898 −0.332536
\(77\) −6.89898 −0.786212
\(78\) 6.89898 0.781156
\(79\) 10.0000 1.12509 0.562544 0.826767i \(-0.309823\pi\)
0.562544 + 0.826767i \(0.309823\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 4.89898 0.541002
\(83\) −7.44949 −0.817688 −0.408844 0.912604i \(-0.634068\pi\)
−0.408844 + 0.912604i \(0.634068\pi\)
\(84\) −3.44949 −0.376370
\(85\) 0 0
\(86\) 1.10102 0.118726
\(87\) 5.00000 0.536056
\(88\) 2.00000 0.213201
\(89\) 4.34847 0.460937 0.230468 0.973080i \(-0.425974\pi\)
0.230468 + 0.973080i \(0.425974\pi\)
\(90\) 0 0
\(91\) −23.7980 −2.49470
\(92\) 1.00000 0.104257
\(93\) 2.00000 0.207390
\(94\) 5.89898 0.608433
\(95\) 0 0
\(96\) 1.00000 0.102062
\(97\) 16.6969 1.69532 0.847659 0.530542i \(-0.178012\pi\)
0.847659 + 0.530542i \(0.178012\pi\)
\(98\) 4.89898 0.494872
\(99\) 2.00000 0.201008
\(100\) 0 0
\(101\) 15.6969 1.56190 0.780952 0.624591i \(-0.214734\pi\)
0.780952 + 0.624591i \(0.214734\pi\)
\(102\) −0.550510 −0.0545086
\(103\) 11.2474 1.10824 0.554122 0.832435i \(-0.313054\pi\)
0.554122 + 0.832435i \(0.313054\pi\)
\(104\) 6.89898 0.676501
\(105\) 0 0
\(106\) −8.89898 −0.864345
\(107\) −2.00000 −0.193347 −0.0966736 0.995316i \(-0.530820\pi\)
−0.0966736 + 0.995316i \(0.530820\pi\)
\(108\) 1.00000 0.0962250
\(109\) −1.44949 −0.138836 −0.0694180 0.997588i \(-0.522114\pi\)
−0.0694180 + 0.997588i \(0.522114\pi\)
\(110\) 0 0
\(111\) −6.34847 −0.602570
\(112\) −3.44949 −0.325946
\(113\) −0.348469 −0.0327812 −0.0163906 0.999866i \(-0.505218\pi\)
−0.0163906 + 0.999866i \(0.505218\pi\)
\(114\) −2.89898 −0.271514
\(115\) 0 0
\(116\) 5.00000 0.464238
\(117\) 6.89898 0.637811
\(118\) 10.0000 0.920575
\(119\) 1.89898 0.174079
\(120\) 0 0
\(121\) −7.00000 −0.636364
\(122\) 4.89898 0.443533
\(123\) 4.89898 0.441726
\(124\) 2.00000 0.179605
\(125\) 0 0
\(126\) −3.44949 −0.307305
\(127\) −17.7980 −1.57931 −0.789657 0.613549i \(-0.789741\pi\)
−0.789657 + 0.613549i \(0.789741\pi\)
\(128\) 1.00000 0.0883883
\(129\) 1.10102 0.0969395
\(130\) 0 0
\(131\) −0.898979 −0.0785442 −0.0392721 0.999229i \(-0.512504\pi\)
−0.0392721 + 0.999229i \(0.512504\pi\)
\(132\) 2.00000 0.174078
\(133\) 10.0000 0.867110
\(134\) −2.00000 −0.172774
\(135\) 0 0
\(136\) −0.550510 −0.0472059
\(137\) −7.65153 −0.653714 −0.326857 0.945074i \(-0.605990\pi\)
−0.326857 + 0.945074i \(0.605990\pi\)
\(138\) 1.00000 0.0851257
\(139\) −13.6969 −1.16176 −0.580880 0.813990i \(-0.697291\pi\)
−0.580880 + 0.813990i \(0.697291\pi\)
\(140\) 0 0
\(141\) 5.89898 0.496784
\(142\) −8.79796 −0.738308
\(143\) 13.7980 1.15384
\(144\) 1.00000 0.0833333
\(145\) 0 0
\(146\) 11.8990 0.984767
\(147\) 4.89898 0.404061
\(148\) −6.34847 −0.521841
\(149\) 20.0000 1.63846 0.819232 0.573462i \(-0.194400\pi\)
0.819232 + 0.573462i \(0.194400\pi\)
\(150\) 0 0
\(151\) 10.6969 0.870505 0.435252 0.900309i \(-0.356659\pi\)
0.435252 + 0.900309i \(0.356659\pi\)
\(152\) −2.89898 −0.235138
\(153\) −0.550510 −0.0445061
\(154\) −6.89898 −0.555936
\(155\) 0 0
\(156\) 6.89898 0.552360
\(157\) 0.898979 0.0717464 0.0358732 0.999356i \(-0.488579\pi\)
0.0358732 + 0.999356i \(0.488579\pi\)
\(158\) 10.0000 0.795557
\(159\) −8.89898 −0.705735
\(160\) 0 0
\(161\) −3.44949 −0.271858
\(162\) 1.00000 0.0785674
\(163\) −1.79796 −0.140827 −0.0704135 0.997518i \(-0.522432\pi\)
−0.0704135 + 0.997518i \(0.522432\pi\)
\(164\) 4.89898 0.382546
\(165\) 0 0
\(166\) −7.44949 −0.578193
\(167\) −7.00000 −0.541676 −0.270838 0.962625i \(-0.587301\pi\)
−0.270838 + 0.962625i \(0.587301\pi\)
\(168\) −3.44949 −0.266134
\(169\) 34.5959 2.66122
\(170\) 0 0
\(171\) −2.89898 −0.221691
\(172\) 1.10102 0.0839520
\(173\) −0.202041 −0.0153609 −0.00768045 0.999971i \(-0.502445\pi\)
−0.00768045 + 0.999971i \(0.502445\pi\)
\(174\) 5.00000 0.379049
\(175\) 0 0
\(176\) 2.00000 0.150756
\(177\) 10.0000 0.751646
\(178\) 4.34847 0.325932
\(179\) 12.8990 0.964115 0.482057 0.876140i \(-0.339890\pi\)
0.482057 + 0.876140i \(0.339890\pi\)
\(180\) 0 0
\(181\) −3.65153 −0.271416 −0.135708 0.990749i \(-0.543331\pi\)
−0.135708 + 0.990749i \(0.543331\pi\)
\(182\) −23.7980 −1.76402
\(183\) 4.89898 0.362143
\(184\) 1.00000 0.0737210
\(185\) 0 0
\(186\) 2.00000 0.146647
\(187\) −1.10102 −0.0805146
\(188\) 5.89898 0.430227
\(189\) −3.44949 −0.250913
\(190\) 0 0
\(191\) 2.00000 0.144715 0.0723575 0.997379i \(-0.476948\pi\)
0.0723575 + 0.997379i \(0.476948\pi\)
\(192\) 1.00000 0.0721688
\(193\) −19.6969 −1.41782 −0.708908 0.705301i \(-0.750812\pi\)
−0.708908 + 0.705301i \(0.750812\pi\)
\(194\) 16.6969 1.19877
\(195\) 0 0
\(196\) 4.89898 0.349927
\(197\) 5.89898 0.420285 0.210142 0.977671i \(-0.432607\pi\)
0.210142 + 0.977671i \(0.432607\pi\)
\(198\) 2.00000 0.142134
\(199\) −10.1464 −0.719261 −0.359631 0.933095i \(-0.617097\pi\)
−0.359631 + 0.933095i \(0.617097\pi\)
\(200\) 0 0
\(201\) −2.00000 −0.141069
\(202\) 15.6969 1.10443
\(203\) −17.2474 −1.21053
\(204\) −0.550510 −0.0385434
\(205\) 0 0
\(206\) 11.2474 0.783647
\(207\) 1.00000 0.0695048
\(208\) 6.89898 0.478358
\(209\) −5.79796 −0.401053
\(210\) 0 0
\(211\) −8.79796 −0.605676 −0.302838 0.953042i \(-0.597934\pi\)
−0.302838 + 0.953042i \(0.597934\pi\)
\(212\) −8.89898 −0.611184
\(213\) −8.79796 −0.602826
\(214\) −2.00000 −0.136717
\(215\) 0 0
\(216\) 1.00000 0.0680414
\(217\) −6.89898 −0.468333
\(218\) −1.44949 −0.0981718
\(219\) 11.8990 0.804059
\(220\) 0 0
\(221\) −3.79796 −0.255478
\(222\) −6.34847 −0.426081
\(223\) −4.69694 −0.314530 −0.157265 0.987556i \(-0.550268\pi\)
−0.157265 + 0.987556i \(0.550268\pi\)
\(224\) −3.44949 −0.230479
\(225\) 0 0
\(226\) −0.348469 −0.0231798
\(227\) −29.2474 −1.94122 −0.970611 0.240655i \(-0.922638\pi\)
−0.970611 + 0.240655i \(0.922638\pi\)
\(228\) −2.89898 −0.191990
\(229\) 12.8990 0.852389 0.426194 0.904632i \(-0.359854\pi\)
0.426194 + 0.904632i \(0.359854\pi\)
\(230\) 0 0
\(231\) −6.89898 −0.453920
\(232\) 5.00000 0.328266
\(233\) 2.69694 0.176682 0.0883412 0.996090i \(-0.471843\pi\)
0.0883412 + 0.996090i \(0.471843\pi\)
\(234\) 6.89898 0.451000
\(235\) 0 0
\(236\) 10.0000 0.650945
\(237\) 10.0000 0.649570
\(238\) 1.89898 0.123093
\(239\) −17.8990 −1.15779 −0.578894 0.815403i \(-0.696516\pi\)
−0.578894 + 0.815403i \(0.696516\pi\)
\(240\) 0 0
\(241\) −8.00000 −0.515325 −0.257663 0.966235i \(-0.582952\pi\)
−0.257663 + 0.966235i \(0.582952\pi\)
\(242\) −7.00000 −0.449977
\(243\) 1.00000 0.0641500
\(244\) 4.89898 0.313625
\(245\) 0 0
\(246\) 4.89898 0.312348
\(247\) −20.0000 −1.27257
\(248\) 2.00000 0.127000
\(249\) −7.44949 −0.472092
\(250\) 0 0
\(251\) 19.2474 1.21489 0.607444 0.794362i \(-0.292195\pi\)
0.607444 + 0.794362i \(0.292195\pi\)
\(252\) −3.44949 −0.217297
\(253\) 2.00000 0.125739
\(254\) −17.7980 −1.11674
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −27.7980 −1.73399 −0.866995 0.498318i \(-0.833951\pi\)
−0.866995 + 0.498318i \(0.833951\pi\)
\(258\) 1.10102 0.0685465
\(259\) 21.8990 1.36074
\(260\) 0 0
\(261\) 5.00000 0.309492
\(262\) −0.898979 −0.0555391
\(263\) −0.202041 −0.0124584 −0.00622919 0.999981i \(-0.501983\pi\)
−0.00622919 + 0.999981i \(0.501983\pi\)
\(264\) 2.00000 0.123091
\(265\) 0 0
\(266\) 10.0000 0.613139
\(267\) 4.34847 0.266122
\(268\) −2.00000 −0.122169
\(269\) −15.7980 −0.963219 −0.481609 0.876386i \(-0.659948\pi\)
−0.481609 + 0.876386i \(0.659948\pi\)
\(270\) 0 0
\(271\) −8.00000 −0.485965 −0.242983 0.970031i \(-0.578126\pi\)
−0.242983 + 0.970031i \(0.578126\pi\)
\(272\) −0.550510 −0.0333796
\(273\) −23.7980 −1.44032
\(274\) −7.65153 −0.462246
\(275\) 0 0
\(276\) 1.00000 0.0601929
\(277\) −13.5959 −0.816900 −0.408450 0.912781i \(-0.633930\pi\)
−0.408450 + 0.912781i \(0.633930\pi\)
\(278\) −13.6969 −0.821488
\(279\) 2.00000 0.119737
\(280\) 0 0
\(281\) −25.2474 −1.50614 −0.753068 0.657942i \(-0.771427\pi\)
−0.753068 + 0.657942i \(0.771427\pi\)
\(282\) 5.89898 0.351279
\(283\) −27.5959 −1.64041 −0.820204 0.572072i \(-0.806140\pi\)
−0.820204 + 0.572072i \(0.806140\pi\)
\(284\) −8.79796 −0.522063
\(285\) 0 0
\(286\) 13.7980 0.815890
\(287\) −16.8990 −0.997515
\(288\) 1.00000 0.0589256
\(289\) −16.6969 −0.982173
\(290\) 0 0
\(291\) 16.6969 0.978792
\(292\) 11.8990 0.696335
\(293\) −11.7980 −0.689244 −0.344622 0.938742i \(-0.611993\pi\)
−0.344622 + 0.938742i \(0.611993\pi\)
\(294\) 4.89898 0.285714
\(295\) 0 0
\(296\) −6.34847 −0.368997
\(297\) 2.00000 0.116052
\(298\) 20.0000 1.15857
\(299\) 6.89898 0.398978
\(300\) 0 0
\(301\) −3.79796 −0.218911
\(302\) 10.6969 0.615540
\(303\) 15.6969 0.901766
\(304\) −2.89898 −0.166268
\(305\) 0 0
\(306\) −0.550510 −0.0314706
\(307\) −29.8990 −1.70642 −0.853212 0.521564i \(-0.825349\pi\)
−0.853212 + 0.521564i \(0.825349\pi\)
\(308\) −6.89898 −0.393106
\(309\) 11.2474 0.639845
\(310\) 0 0
\(311\) −24.5959 −1.39471 −0.697353 0.716728i \(-0.745639\pi\)
−0.697353 + 0.716728i \(0.745639\pi\)
\(312\) 6.89898 0.390578
\(313\) −3.10102 −0.175280 −0.0876400 0.996152i \(-0.527933\pi\)
−0.0876400 + 0.996152i \(0.527933\pi\)
\(314\) 0.898979 0.0507323
\(315\) 0 0
\(316\) 10.0000 0.562544
\(317\) −28.5959 −1.60611 −0.803053 0.595907i \(-0.796793\pi\)
−0.803053 + 0.595907i \(0.796793\pi\)
\(318\) −8.89898 −0.499030
\(319\) 10.0000 0.559893
\(320\) 0 0
\(321\) −2.00000 −0.111629
\(322\) −3.44949 −0.192233
\(323\) 1.59592 0.0887992
\(324\) 1.00000 0.0555556
\(325\) 0 0
\(326\) −1.79796 −0.0995797
\(327\) −1.44949 −0.0801570
\(328\) 4.89898 0.270501
\(329\) −20.3485 −1.12185
\(330\) 0 0
\(331\) 12.7980 0.703439 0.351720 0.936105i \(-0.385597\pi\)
0.351720 + 0.936105i \(0.385597\pi\)
\(332\) −7.44949 −0.408844
\(333\) −6.34847 −0.347894
\(334\) −7.00000 −0.383023
\(335\) 0 0
\(336\) −3.44949 −0.188185
\(337\) 32.4949 1.77011 0.885055 0.465487i \(-0.154121\pi\)
0.885055 + 0.465487i \(0.154121\pi\)
\(338\) 34.5959 1.88177
\(339\) −0.348469 −0.0189263
\(340\) 0 0
\(341\) 4.00000 0.216612
\(342\) −2.89898 −0.156759
\(343\) 7.24745 0.391325
\(344\) 1.10102 0.0593630
\(345\) 0 0
\(346\) −0.202041 −0.0108618
\(347\) −3.30306 −0.177318 −0.0886588 0.996062i \(-0.528258\pi\)
−0.0886588 + 0.996062i \(0.528258\pi\)
\(348\) 5.00000 0.268028
\(349\) 22.8990 1.22575 0.612877 0.790178i \(-0.290012\pi\)
0.612877 + 0.790178i \(0.290012\pi\)
\(350\) 0 0
\(351\) 6.89898 0.368240
\(352\) 2.00000 0.106600
\(353\) −26.0000 −1.38384 −0.691920 0.721974i \(-0.743235\pi\)
−0.691920 + 0.721974i \(0.743235\pi\)
\(354\) 10.0000 0.531494
\(355\) 0 0
\(356\) 4.34847 0.230468
\(357\) 1.89898 0.100505
\(358\) 12.8990 0.681732
\(359\) −25.7980 −1.36156 −0.680782 0.732486i \(-0.738360\pi\)
−0.680782 + 0.732486i \(0.738360\pi\)
\(360\) 0 0
\(361\) −10.5959 −0.557680
\(362\) −3.65153 −0.191920
\(363\) −7.00000 −0.367405
\(364\) −23.7980 −1.24735
\(365\) 0 0
\(366\) 4.89898 0.256074
\(367\) −27.7980 −1.45104 −0.725521 0.688200i \(-0.758401\pi\)
−0.725521 + 0.688200i \(0.758401\pi\)
\(368\) 1.00000 0.0521286
\(369\) 4.89898 0.255031
\(370\) 0 0
\(371\) 30.6969 1.59371
\(372\) 2.00000 0.103695
\(373\) 21.2474 1.10015 0.550076 0.835115i \(-0.314599\pi\)
0.550076 + 0.835115i \(0.314599\pi\)
\(374\) −1.10102 −0.0569324
\(375\) 0 0
\(376\) 5.89898 0.304217
\(377\) 34.4949 1.77658
\(378\) −3.44949 −0.177423
\(379\) 30.2929 1.55604 0.778020 0.628240i \(-0.216224\pi\)
0.778020 + 0.628240i \(0.216224\pi\)
\(380\) 0 0
\(381\) −17.7980 −0.911817
\(382\) 2.00000 0.102329
\(383\) 9.79796 0.500652 0.250326 0.968162i \(-0.419462\pi\)
0.250326 + 0.968162i \(0.419462\pi\)
\(384\) 1.00000 0.0510310
\(385\) 0 0
\(386\) −19.6969 −1.00255
\(387\) 1.10102 0.0559680
\(388\) 16.6969 0.847659
\(389\) −4.20204 −0.213052 −0.106526 0.994310i \(-0.533973\pi\)
−0.106526 + 0.994310i \(0.533973\pi\)
\(390\) 0 0
\(391\) −0.550510 −0.0278405
\(392\) 4.89898 0.247436
\(393\) −0.898979 −0.0453475
\(394\) 5.89898 0.297186
\(395\) 0 0
\(396\) 2.00000 0.100504
\(397\) 28.0000 1.40528 0.702640 0.711546i \(-0.252005\pi\)
0.702640 + 0.711546i \(0.252005\pi\)
\(398\) −10.1464 −0.508594
\(399\) 10.0000 0.500626
\(400\) 0 0
\(401\) 10.6969 0.534180 0.267090 0.963672i \(-0.413938\pi\)
0.267090 + 0.963672i \(0.413938\pi\)
\(402\) −2.00000 −0.0997509
\(403\) 13.7980 0.687325
\(404\) 15.6969 0.780952
\(405\) 0 0
\(406\) −17.2474 −0.855977
\(407\) −12.6969 −0.629364
\(408\) −0.550510 −0.0272543
\(409\) −3.69694 −0.182802 −0.0914009 0.995814i \(-0.529134\pi\)
−0.0914009 + 0.995814i \(0.529134\pi\)
\(410\) 0 0
\(411\) −7.65153 −0.377422
\(412\) 11.2474 0.554122
\(413\) −34.4949 −1.69738
\(414\) 1.00000 0.0491473
\(415\) 0 0
\(416\) 6.89898 0.338250
\(417\) −13.6969 −0.670742
\(418\) −5.79796 −0.283587
\(419\) −20.1464 −0.984217 −0.492109 0.870534i \(-0.663774\pi\)
−0.492109 + 0.870534i \(0.663774\pi\)
\(420\) 0 0
\(421\) 36.4949 1.77865 0.889326 0.457273i \(-0.151174\pi\)
0.889326 + 0.457273i \(0.151174\pi\)
\(422\) −8.79796 −0.428278
\(423\) 5.89898 0.286818
\(424\) −8.89898 −0.432173
\(425\) 0 0
\(426\) −8.79796 −0.426263
\(427\) −16.8990 −0.817799
\(428\) −2.00000 −0.0966736
\(429\) 13.7980 0.666172
\(430\) 0 0
\(431\) 40.6969 1.96030 0.980151 0.198251i \(-0.0635261\pi\)
0.980151 + 0.198251i \(0.0635261\pi\)
\(432\) 1.00000 0.0481125
\(433\) 12.6969 0.610176 0.305088 0.952324i \(-0.401314\pi\)
0.305088 + 0.952324i \(0.401314\pi\)
\(434\) −6.89898 −0.331162
\(435\) 0 0
\(436\) −1.44949 −0.0694180
\(437\) −2.89898 −0.138677
\(438\) 11.8990 0.568555
\(439\) 18.6969 0.892356 0.446178 0.894944i \(-0.352785\pi\)
0.446178 + 0.894944i \(0.352785\pi\)
\(440\) 0 0
\(441\) 4.89898 0.233285
\(442\) −3.79796 −0.180650
\(443\) −34.6969 −1.64850 −0.824251 0.566225i \(-0.808403\pi\)
−0.824251 + 0.566225i \(0.808403\pi\)
\(444\) −6.34847 −0.301285
\(445\) 0 0
\(446\) −4.69694 −0.222406
\(447\) 20.0000 0.945968
\(448\) −3.44949 −0.162973
\(449\) −15.7980 −0.745552 −0.372776 0.927921i \(-0.621594\pi\)
−0.372776 + 0.927921i \(0.621594\pi\)
\(450\) 0 0
\(451\) 9.79796 0.461368
\(452\) −0.348469 −0.0163906
\(453\) 10.6969 0.502586
\(454\) −29.2474 −1.37265
\(455\) 0 0
\(456\) −2.89898 −0.135757
\(457\) 39.5959 1.85222 0.926109 0.377255i \(-0.123132\pi\)
0.926109 + 0.377255i \(0.123132\pi\)
\(458\) 12.8990 0.602730
\(459\) −0.550510 −0.0256956
\(460\) 0 0
\(461\) −27.4949 −1.28057 −0.640283 0.768140i \(-0.721183\pi\)
−0.640283 + 0.768140i \(0.721183\pi\)
\(462\) −6.89898 −0.320970
\(463\) 4.00000 0.185896 0.0929479 0.995671i \(-0.470371\pi\)
0.0929479 + 0.995671i \(0.470371\pi\)
\(464\) 5.00000 0.232119
\(465\) 0 0
\(466\) 2.69694 0.124933
\(467\) 23.9444 1.10801 0.554007 0.832512i \(-0.313098\pi\)
0.554007 + 0.832512i \(0.313098\pi\)
\(468\) 6.89898 0.318905
\(469\) 6.89898 0.318565
\(470\) 0 0
\(471\) 0.898979 0.0414228
\(472\) 10.0000 0.460287
\(473\) 2.20204 0.101250
\(474\) 10.0000 0.459315
\(475\) 0 0
\(476\) 1.89898 0.0870396
\(477\) −8.89898 −0.407456
\(478\) −17.8990 −0.818680
\(479\) −30.0000 −1.37073 −0.685367 0.728197i \(-0.740358\pi\)
−0.685367 + 0.728197i \(0.740358\pi\)
\(480\) 0 0
\(481\) −43.7980 −1.99702
\(482\) −8.00000 −0.364390
\(483\) −3.44949 −0.156957
\(484\) −7.00000 −0.318182
\(485\) 0 0
\(486\) 1.00000 0.0453609
\(487\) −24.8990 −1.12828 −0.564140 0.825679i \(-0.690792\pi\)
−0.564140 + 0.825679i \(0.690792\pi\)
\(488\) 4.89898 0.221766
\(489\) −1.79796 −0.0813065
\(490\) 0 0
\(491\) 34.8990 1.57497 0.787484 0.616335i \(-0.211383\pi\)
0.787484 + 0.616335i \(0.211383\pi\)
\(492\) 4.89898 0.220863
\(493\) −2.75255 −0.123969
\(494\) −20.0000 −0.899843
\(495\) 0 0
\(496\) 2.00000 0.0898027
\(497\) 30.3485 1.36131
\(498\) −7.44949 −0.333820
\(499\) 12.1010 0.541716 0.270858 0.962619i \(-0.412693\pi\)
0.270858 + 0.962619i \(0.412693\pi\)
\(500\) 0 0
\(501\) −7.00000 −0.312737
\(502\) 19.2474 0.859056
\(503\) −14.6969 −0.655304 −0.327652 0.944798i \(-0.606257\pi\)
−0.327652 + 0.944798i \(0.606257\pi\)
\(504\) −3.44949 −0.153652
\(505\) 0 0
\(506\) 2.00000 0.0889108
\(507\) 34.5959 1.53646
\(508\) −17.7980 −0.789657
\(509\) −15.0000 −0.664863 −0.332432 0.943127i \(-0.607869\pi\)
−0.332432 + 0.943127i \(0.607869\pi\)
\(510\) 0 0
\(511\) −41.0454 −1.81574
\(512\) 1.00000 0.0441942
\(513\) −2.89898 −0.127993
\(514\) −27.7980 −1.22612
\(515\) 0 0
\(516\) 1.10102 0.0484697
\(517\) 11.7980 0.518874
\(518\) 21.8990 0.962186
\(519\) −0.202041 −0.00886862
\(520\) 0 0
\(521\) −8.14643 −0.356901 −0.178451 0.983949i \(-0.557109\pi\)
−0.178451 + 0.983949i \(0.557109\pi\)
\(522\) 5.00000 0.218844
\(523\) −16.0000 −0.699631 −0.349816 0.936819i \(-0.613756\pi\)
−0.349816 + 0.936819i \(0.613756\pi\)
\(524\) −0.898979 −0.0392721
\(525\) 0 0
\(526\) −0.202041 −0.00880941
\(527\) −1.10102 −0.0479612
\(528\) 2.00000 0.0870388
\(529\) 1.00000 0.0434783
\(530\) 0 0
\(531\) 10.0000 0.433963
\(532\) 10.0000 0.433555
\(533\) 33.7980 1.46395
\(534\) 4.34847 0.188177
\(535\) 0 0
\(536\) −2.00000 −0.0863868
\(537\) 12.8990 0.556632
\(538\) −15.7980 −0.681098
\(539\) 9.79796 0.422028
\(540\) 0 0
\(541\) 23.5959 1.01447 0.507234 0.861808i \(-0.330668\pi\)
0.507234 + 0.861808i \(0.330668\pi\)
\(542\) −8.00000 −0.343629
\(543\) −3.65153 −0.156702
\(544\) −0.550510 −0.0236029
\(545\) 0 0
\(546\) −23.7980 −1.01846
\(547\) 8.79796 0.376174 0.188087 0.982152i \(-0.439771\pi\)
0.188087 + 0.982152i \(0.439771\pi\)
\(548\) −7.65153 −0.326857
\(549\) 4.89898 0.209083
\(550\) 0 0
\(551\) −14.4949 −0.617503
\(552\) 1.00000 0.0425628
\(553\) −34.4949 −1.46687
\(554\) −13.5959 −0.577635
\(555\) 0 0
\(556\) −13.6969 −0.580880
\(557\) −0.696938 −0.0295302 −0.0147651 0.999891i \(-0.504700\pi\)
−0.0147651 + 0.999891i \(0.504700\pi\)
\(558\) 2.00000 0.0846668
\(559\) 7.59592 0.321273
\(560\) 0 0
\(561\) −1.10102 −0.0464851
\(562\) −25.2474 −1.06500
\(563\) −6.00000 −0.252870 −0.126435 0.991975i \(-0.540353\pi\)
−0.126435 + 0.991975i \(0.540353\pi\)
\(564\) 5.89898 0.248392
\(565\) 0 0
\(566\) −27.5959 −1.15994
\(567\) −3.44949 −0.144865
\(568\) −8.79796 −0.369154
\(569\) 4.49490 0.188436 0.0942180 0.995552i \(-0.469965\pi\)
0.0942180 + 0.995552i \(0.469965\pi\)
\(570\) 0 0
\(571\) 3.30306 0.138229 0.0691144 0.997609i \(-0.477983\pi\)
0.0691144 + 0.997609i \(0.477983\pi\)
\(572\) 13.7980 0.576922
\(573\) 2.00000 0.0835512
\(574\) −16.8990 −0.705350
\(575\) 0 0
\(576\) 1.00000 0.0416667
\(577\) −2.00000 −0.0832611 −0.0416305 0.999133i \(-0.513255\pi\)
−0.0416305 + 0.999133i \(0.513255\pi\)
\(578\) −16.6969 −0.694501
\(579\) −19.6969 −0.818577
\(580\) 0 0
\(581\) 25.6969 1.06609
\(582\) 16.6969 0.692110
\(583\) −17.7980 −0.737116
\(584\) 11.8990 0.492383
\(585\) 0 0
\(586\) −11.7980 −0.487369
\(587\) −4.89898 −0.202203 −0.101101 0.994876i \(-0.532237\pi\)
−0.101101 + 0.994876i \(0.532237\pi\)
\(588\) 4.89898 0.202031
\(589\) −5.79796 −0.238901
\(590\) 0 0
\(591\) 5.89898 0.242652
\(592\) −6.34847 −0.260920
\(593\) −4.40408 −0.180854 −0.0904270 0.995903i \(-0.528823\pi\)
−0.0904270 + 0.995903i \(0.528823\pi\)
\(594\) 2.00000 0.0820610
\(595\) 0 0
\(596\) 20.0000 0.819232
\(597\) −10.1464 −0.415266
\(598\) 6.89898 0.282120
\(599\) 37.3939 1.52787 0.763936 0.645292i \(-0.223264\pi\)
0.763936 + 0.645292i \(0.223264\pi\)
\(600\) 0 0
\(601\) 37.0000 1.50926 0.754631 0.656150i \(-0.227816\pi\)
0.754631 + 0.656150i \(0.227816\pi\)
\(602\) −3.79796 −0.154793
\(603\) −2.00000 −0.0814463
\(604\) 10.6969 0.435252
\(605\) 0 0
\(606\) 15.6969 0.637645
\(607\) 18.0000 0.730597 0.365299 0.930890i \(-0.380967\pi\)
0.365299 + 0.930890i \(0.380967\pi\)
\(608\) −2.89898 −0.117569
\(609\) −17.2474 −0.698902
\(610\) 0 0
\(611\) 40.6969 1.64642
\(612\) −0.550510 −0.0222531
\(613\) 34.1464 1.37916 0.689581 0.724209i \(-0.257795\pi\)
0.689581 + 0.724209i \(0.257795\pi\)
\(614\) −29.8990 −1.20662
\(615\) 0 0
\(616\) −6.89898 −0.277968
\(617\) −36.4949 −1.46923 −0.734615 0.678485i \(-0.762637\pi\)
−0.734615 + 0.678485i \(0.762637\pi\)
\(618\) 11.2474 0.452439
\(619\) 4.49490 0.180665 0.0903326 0.995912i \(-0.471207\pi\)
0.0903326 + 0.995912i \(0.471207\pi\)
\(620\) 0 0
\(621\) 1.00000 0.0401286
\(622\) −24.5959 −0.986206
\(623\) −15.0000 −0.600962
\(624\) 6.89898 0.276180
\(625\) 0 0
\(626\) −3.10102 −0.123942
\(627\) −5.79796 −0.231548
\(628\) 0.898979 0.0358732
\(629\) 3.49490 0.139351
\(630\) 0 0
\(631\) 10.5505 0.420009 0.210005 0.977700i \(-0.432652\pi\)
0.210005 + 0.977700i \(0.432652\pi\)
\(632\) 10.0000 0.397779
\(633\) −8.79796 −0.349687
\(634\) −28.5959 −1.13569
\(635\) 0 0
\(636\) −8.89898 −0.352867
\(637\) 33.7980 1.33912
\(638\) 10.0000 0.395904
\(639\) −8.79796 −0.348042
\(640\) 0 0
\(641\) −29.4495 −1.16318 −0.581592 0.813480i \(-0.697570\pi\)
−0.581592 + 0.813480i \(0.697570\pi\)
\(642\) −2.00000 −0.0789337
\(643\) −0.202041 −0.00796772 −0.00398386 0.999992i \(-0.501268\pi\)
−0.00398386 + 0.999992i \(0.501268\pi\)
\(644\) −3.44949 −0.135929
\(645\) 0 0
\(646\) 1.59592 0.0627906
\(647\) −18.5959 −0.731081 −0.365540 0.930795i \(-0.619116\pi\)
−0.365540 + 0.930795i \(0.619116\pi\)
\(648\) 1.00000 0.0392837
\(649\) 20.0000 0.785069
\(650\) 0 0
\(651\) −6.89898 −0.270392
\(652\) −1.79796 −0.0704135
\(653\) −29.6969 −1.16213 −0.581066 0.813857i \(-0.697364\pi\)
−0.581066 + 0.813857i \(0.697364\pi\)
\(654\) −1.44949 −0.0566795
\(655\) 0 0
\(656\) 4.89898 0.191273
\(657\) 11.8990 0.464223
\(658\) −20.3485 −0.793266
\(659\) −18.5505 −0.722625 −0.361313 0.932445i \(-0.617671\pi\)
−0.361313 + 0.932445i \(0.617671\pi\)
\(660\) 0 0
\(661\) −16.5505 −0.643740 −0.321870 0.946784i \(-0.604311\pi\)
−0.321870 + 0.946784i \(0.604311\pi\)
\(662\) 12.7980 0.497407
\(663\) −3.79796 −0.147501
\(664\) −7.44949 −0.289096
\(665\) 0 0
\(666\) −6.34847 −0.245998
\(667\) 5.00000 0.193601
\(668\) −7.00000 −0.270838
\(669\) −4.69694 −0.181594
\(670\) 0 0
\(671\) 9.79796 0.378246
\(672\) −3.44949 −0.133067
\(673\) −12.5959 −0.485537 −0.242768 0.970084i \(-0.578056\pi\)
−0.242768 + 0.970084i \(0.578056\pi\)
\(674\) 32.4949 1.25166
\(675\) 0 0
\(676\) 34.5959 1.33061
\(677\) 25.3939 0.975966 0.487983 0.872853i \(-0.337733\pi\)
0.487983 + 0.872853i \(0.337733\pi\)
\(678\) −0.348469 −0.0133829
\(679\) −57.5959 −2.21033
\(680\) 0 0
\(681\) −29.2474 −1.12076
\(682\) 4.00000 0.153168
\(683\) 16.8990 0.646621 0.323311 0.946293i \(-0.395204\pi\)
0.323311 + 0.946293i \(0.395204\pi\)
\(684\) −2.89898 −0.110845
\(685\) 0 0
\(686\) 7.24745 0.276709
\(687\) 12.8990 0.492127
\(688\) 1.10102 0.0419760
\(689\) −61.3939 −2.33892
\(690\) 0 0
\(691\) −28.7980 −1.09553 −0.547763 0.836634i \(-0.684520\pi\)
−0.547763 + 0.836634i \(0.684520\pi\)
\(692\) −0.202041 −0.00768045
\(693\) −6.89898 −0.262071
\(694\) −3.30306 −0.125383
\(695\) 0 0
\(696\) 5.00000 0.189525
\(697\) −2.69694 −0.102154
\(698\) 22.8990 0.866739
\(699\) 2.69694 0.102008
\(700\) 0 0
\(701\) −18.0000 −0.679851 −0.339925 0.940452i \(-0.610402\pi\)
−0.339925 + 0.940452i \(0.610402\pi\)
\(702\) 6.89898 0.260385
\(703\) 18.4041 0.694123
\(704\) 2.00000 0.0753778
\(705\) 0 0
\(706\) −26.0000 −0.978523
\(707\) −54.1464 −2.03639
\(708\) 10.0000 0.375823
\(709\) −25.9444 −0.974362 −0.487181 0.873301i \(-0.661975\pi\)
−0.487181 + 0.873301i \(0.661975\pi\)
\(710\) 0 0
\(711\) 10.0000 0.375029
\(712\) 4.34847 0.162966
\(713\) 2.00000 0.0749006
\(714\) 1.89898 0.0710675
\(715\) 0 0
\(716\) 12.8990 0.482057
\(717\) −17.8990 −0.668450
\(718\) −25.7980 −0.962771
\(719\) 9.20204 0.343178 0.171589 0.985169i \(-0.445110\pi\)
0.171589 + 0.985169i \(0.445110\pi\)
\(720\) 0 0
\(721\) −38.7980 −1.44491
\(722\) −10.5959 −0.394339
\(723\) −8.00000 −0.297523
\(724\) −3.65153 −0.135708
\(725\) 0 0
\(726\) −7.00000 −0.259794
\(727\) −19.3939 −0.719279 −0.359640 0.933091i \(-0.617100\pi\)
−0.359640 + 0.933091i \(0.617100\pi\)
\(728\) −23.7980 −0.882011
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −0.606123 −0.0224183
\(732\) 4.89898 0.181071
\(733\) −5.85357 −0.216207 −0.108103 0.994140i \(-0.534478\pi\)
−0.108103 + 0.994140i \(0.534478\pi\)
\(734\) −27.7980 −1.02604
\(735\) 0 0
\(736\) 1.00000 0.0368605
\(737\) −4.00000 −0.147342
\(738\) 4.89898 0.180334
\(739\) −26.5959 −0.978347 −0.489173 0.872187i \(-0.662701\pi\)
−0.489173 + 0.872187i \(0.662701\pi\)
\(740\) 0 0
\(741\) −20.0000 −0.734718
\(742\) 30.6969 1.12692
\(743\) 44.2929 1.62495 0.812474 0.582998i \(-0.198120\pi\)
0.812474 + 0.582998i \(0.198120\pi\)
\(744\) 2.00000 0.0733236
\(745\) 0 0
\(746\) 21.2474 0.777924
\(747\) −7.44949 −0.272563
\(748\) −1.10102 −0.0402573
\(749\) 6.89898 0.252083
\(750\) 0 0
\(751\) −0.752551 −0.0274610 −0.0137305 0.999906i \(-0.504371\pi\)
−0.0137305 + 0.999906i \(0.504371\pi\)
\(752\) 5.89898 0.215114
\(753\) 19.2474 0.701416
\(754\) 34.4949 1.25623
\(755\) 0 0
\(756\) −3.44949 −0.125457
\(757\) 35.2474 1.28109 0.640545 0.767921i \(-0.278708\pi\)
0.640545 + 0.767921i \(0.278708\pi\)
\(758\) 30.2929 1.10029
\(759\) 2.00000 0.0725954
\(760\) 0 0
\(761\) 38.0908 1.38079 0.690395 0.723432i \(-0.257437\pi\)
0.690395 + 0.723432i \(0.257437\pi\)
\(762\) −17.7980 −0.644752
\(763\) 5.00000 0.181012
\(764\) 2.00000 0.0723575
\(765\) 0 0
\(766\) 9.79796 0.354015
\(767\) 68.9898 2.49108
\(768\) 1.00000 0.0360844
\(769\) −20.2929 −0.731779 −0.365890 0.930658i \(-0.619235\pi\)
−0.365890 + 0.930658i \(0.619235\pi\)
\(770\) 0 0
\(771\) −27.7980 −1.00112
\(772\) −19.6969 −0.708908
\(773\) 1.10102 0.0396010 0.0198005 0.999804i \(-0.493697\pi\)
0.0198005 + 0.999804i \(0.493697\pi\)
\(774\) 1.10102 0.0395754
\(775\) 0 0
\(776\) 16.6969 0.599385
\(777\) 21.8990 0.785622
\(778\) −4.20204 −0.150650
\(779\) −14.2020 −0.508841
\(780\) 0 0
\(781\) −17.5959 −0.629631
\(782\) −0.550510 −0.0196862
\(783\) 5.00000 0.178685
\(784\) 4.89898 0.174964
\(785\) 0 0
\(786\) −0.898979 −0.0320655
\(787\) 28.0000 0.998092 0.499046 0.866575i \(-0.333684\pi\)
0.499046 + 0.866575i \(0.333684\pi\)
\(788\) 5.89898 0.210142
\(789\) −0.202041 −0.00719285
\(790\) 0 0
\(791\) 1.20204 0.0427397
\(792\) 2.00000 0.0710669
\(793\) 33.7980 1.20020
\(794\) 28.0000 0.993683
\(795\) 0 0
\(796\) −10.1464 −0.359631
\(797\) 25.1010 0.889124 0.444562 0.895748i \(-0.353359\pi\)
0.444562 + 0.895748i \(0.353359\pi\)
\(798\) 10.0000 0.353996
\(799\) −3.24745 −0.114886
\(800\) 0 0
\(801\) 4.34847 0.153646
\(802\) 10.6969 0.377722
\(803\) 23.7980 0.839812
\(804\) −2.00000 −0.0705346
\(805\) 0 0
\(806\) 13.7980 0.486012
\(807\) −15.7980 −0.556114
\(808\) 15.6969 0.552216
\(809\) −47.3939 −1.66628 −0.833140 0.553062i \(-0.813459\pi\)
−0.833140 + 0.553062i \(0.813459\pi\)
\(810\) 0 0
\(811\) 26.2020 0.920078 0.460039 0.887899i \(-0.347835\pi\)
0.460039 + 0.887899i \(0.347835\pi\)
\(812\) −17.2474 −0.605267
\(813\) −8.00000 −0.280572
\(814\) −12.6969 −0.445027
\(815\) 0 0
\(816\) −0.550510 −0.0192717
\(817\) −3.19184 −0.111668
\(818\) −3.69694 −0.129260
\(819\) −23.7980 −0.831568
\(820\) 0 0
\(821\) −32.2020 −1.12386 −0.561929 0.827185i \(-0.689941\pi\)
−0.561929 + 0.827185i \(0.689941\pi\)
\(822\) −7.65153 −0.266878
\(823\) −32.0908 −1.11862 −0.559308 0.828960i \(-0.688933\pi\)
−0.559308 + 0.828960i \(0.688933\pi\)
\(824\) 11.2474 0.391823
\(825\) 0 0
\(826\) −34.4949 −1.20023
\(827\) −39.2474 −1.36477 −0.682384 0.730994i \(-0.739057\pi\)
−0.682384 + 0.730994i \(0.739057\pi\)
\(828\) 1.00000 0.0347524
\(829\) −42.8990 −1.48994 −0.744972 0.667096i \(-0.767537\pi\)
−0.744972 + 0.667096i \(0.767537\pi\)
\(830\) 0 0
\(831\) −13.5959 −0.471637
\(832\) 6.89898 0.239179
\(833\) −2.69694 −0.0934434
\(834\) −13.6969 −0.474286
\(835\) 0 0
\(836\) −5.79796 −0.200527
\(837\) 2.00000 0.0691301
\(838\) −20.1464 −0.695947
\(839\) −38.6969 −1.33597 −0.667983 0.744176i \(-0.732842\pi\)
−0.667983 + 0.744176i \(0.732842\pi\)
\(840\) 0 0
\(841\) −4.00000 −0.137931
\(842\) 36.4949 1.25770
\(843\) −25.2474 −0.869568
\(844\) −8.79796 −0.302838
\(845\) 0 0
\(846\) 5.89898 0.202811
\(847\) 24.1464 0.829681
\(848\) −8.89898 −0.305592
\(849\) −27.5959 −0.947089
\(850\) 0 0
\(851\) −6.34847 −0.217623
\(852\) −8.79796 −0.301413
\(853\) 52.9898 1.81434 0.907168 0.420769i \(-0.138240\pi\)
0.907168 + 0.420769i \(0.138240\pi\)
\(854\) −16.8990 −0.578271
\(855\) 0 0
\(856\) −2.00000 −0.0683586
\(857\) −6.49490 −0.221861 −0.110931 0.993828i \(-0.535383\pi\)
−0.110931 + 0.993828i \(0.535383\pi\)
\(858\) 13.7980 0.471055
\(859\) 20.0000 0.682391 0.341196 0.939992i \(-0.389168\pi\)
0.341196 + 0.939992i \(0.389168\pi\)
\(860\) 0 0
\(861\) −16.8990 −0.575916
\(862\) 40.6969 1.38614
\(863\) −38.3939 −1.30694 −0.653471 0.756951i \(-0.726688\pi\)
−0.653471 + 0.756951i \(0.726688\pi\)
\(864\) 1.00000 0.0340207
\(865\) 0 0
\(866\) 12.6969 0.431460
\(867\) −16.6969 −0.567058
\(868\) −6.89898 −0.234167
\(869\) 20.0000 0.678454
\(870\) 0 0
\(871\) −13.7980 −0.467526
\(872\) −1.44949 −0.0490859
\(873\) 16.6969 0.565106
\(874\) −2.89898 −0.0980594
\(875\) 0 0
\(876\) 11.8990 0.402029
\(877\) 29.5959 0.999383 0.499692 0.866203i \(-0.333447\pi\)
0.499692 + 0.866203i \(0.333447\pi\)
\(878\) 18.6969 0.630991
\(879\) −11.7980 −0.397935
\(880\) 0 0
\(881\) −40.8990 −1.37792 −0.688961 0.724799i \(-0.741933\pi\)
−0.688961 + 0.724799i \(0.741933\pi\)
\(882\) 4.89898 0.164957
\(883\) −9.40408 −0.316473 −0.158236 0.987401i \(-0.550581\pi\)
−0.158236 + 0.987401i \(0.550581\pi\)
\(884\) −3.79796 −0.127739
\(885\) 0 0
\(886\) −34.6969 −1.16567
\(887\) −19.8990 −0.668142 −0.334071 0.942548i \(-0.608423\pi\)
−0.334071 + 0.942548i \(0.608423\pi\)
\(888\) −6.34847 −0.213041
\(889\) 61.3939 2.05908
\(890\) 0 0
\(891\) 2.00000 0.0670025
\(892\) −4.69694 −0.157265
\(893\) −17.1010 −0.572264
\(894\) 20.0000 0.668900
\(895\) 0 0
\(896\) −3.44949 −0.115239
\(897\) 6.89898 0.230350
\(898\) −15.7980 −0.527185
\(899\) 10.0000 0.333519
\(900\) 0 0
\(901\) 4.89898 0.163209
\(902\) 9.79796 0.326236
\(903\) −3.79796 −0.126388
\(904\) −0.348469 −0.0115899
\(905\) 0 0
\(906\) 10.6969 0.355382
\(907\) −46.4949 −1.54384 −0.771919 0.635721i \(-0.780703\pi\)
−0.771919 + 0.635721i \(0.780703\pi\)
\(908\) −29.2474 −0.970611
\(909\) 15.6969 0.520635
\(910\) 0 0
\(911\) 50.6969 1.67966 0.839832 0.542846i \(-0.182653\pi\)
0.839832 + 0.542846i \(0.182653\pi\)
\(912\) −2.89898 −0.0959948
\(913\) −14.8990 −0.493084
\(914\) 39.5959 1.30972
\(915\) 0 0
\(916\) 12.8990 0.426194
\(917\) 3.10102 0.102405
\(918\) −0.550510 −0.0181695
\(919\) 25.9444 0.855826 0.427913 0.903820i \(-0.359249\pi\)
0.427913 + 0.903820i \(0.359249\pi\)
\(920\) 0 0
\(921\) −29.8990 −0.985205
\(922\) −27.4949 −0.905496
\(923\) −60.6969 −1.99786
\(924\) −6.89898 −0.226960
\(925\) 0 0
\(926\) 4.00000 0.131448
\(927\) 11.2474 0.369415
\(928\) 5.00000 0.164133
\(929\) 15.7980 0.518314 0.259157 0.965835i \(-0.416555\pi\)
0.259157 + 0.965835i \(0.416555\pi\)
\(930\) 0 0
\(931\) −14.2020 −0.465453
\(932\) 2.69694 0.0883412
\(933\) −24.5959 −0.805234
\(934\) 23.9444 0.783484
\(935\) 0 0
\(936\) 6.89898 0.225500
\(937\) 3.79796 0.124074 0.0620370 0.998074i \(-0.480240\pi\)
0.0620370 + 0.998074i \(0.480240\pi\)
\(938\) 6.89898 0.225260
\(939\) −3.10102 −0.101198
\(940\) 0 0
\(941\) 2.00000 0.0651981 0.0325991 0.999469i \(-0.489622\pi\)
0.0325991 + 0.999469i \(0.489622\pi\)
\(942\) 0.898979 0.0292903
\(943\) 4.89898 0.159533
\(944\) 10.0000 0.325472
\(945\) 0 0
\(946\) 2.20204 0.0715945
\(947\) 39.5959 1.28669 0.643347 0.765575i \(-0.277545\pi\)
0.643347 + 0.765575i \(0.277545\pi\)
\(948\) 10.0000 0.324785
\(949\) 82.0908 2.66478
\(950\) 0 0
\(951\) −28.5959 −0.927286
\(952\) 1.89898 0.0615463
\(953\) 32.8434 1.06390 0.531951 0.846775i \(-0.321459\pi\)
0.531951 + 0.846775i \(0.321459\pi\)
\(954\) −8.89898 −0.288115
\(955\) 0 0
\(956\) −17.8990 −0.578894
\(957\) 10.0000 0.323254
\(958\) −30.0000 −0.969256
\(959\) 26.3939 0.852303
\(960\) 0 0
\(961\) −27.0000 −0.870968
\(962\) −43.7980 −1.41210
\(963\) −2.00000 −0.0644491
\(964\) −8.00000 −0.257663
\(965\) 0 0
\(966\) −3.44949 −0.110986
\(967\) 9.59592 0.308584 0.154292 0.988025i \(-0.450690\pi\)
0.154292 + 0.988025i \(0.450690\pi\)
\(968\) −7.00000 −0.224989
\(969\) 1.59592 0.0512683
\(970\) 0 0
\(971\) −8.14643 −0.261431 −0.130716 0.991420i \(-0.541727\pi\)
−0.130716 + 0.991420i \(0.541727\pi\)
\(972\) 1.00000 0.0320750
\(973\) 47.2474 1.51468
\(974\) −24.8990 −0.797815
\(975\) 0 0
\(976\) 4.89898 0.156813
\(977\) −49.2474 −1.57557 −0.787783 0.615953i \(-0.788771\pi\)
−0.787783 + 0.615953i \(0.788771\pi\)
\(978\) −1.79796 −0.0574924
\(979\) 8.69694 0.277955
\(980\) 0 0
\(981\) −1.44949 −0.0462786
\(982\) 34.8990 1.11367
\(983\) 24.0000 0.765481 0.382741 0.923856i \(-0.374980\pi\)
0.382741 + 0.923856i \(0.374980\pi\)
\(984\) 4.89898 0.156174
\(985\) 0 0
\(986\) −2.75255 −0.0876591
\(987\) −20.3485 −0.647699
\(988\) −20.0000 −0.636285
\(989\) 1.10102 0.0350104
\(990\) 0 0
\(991\) 37.7980 1.20069 0.600346 0.799740i \(-0.295030\pi\)
0.600346 + 0.799740i \(0.295030\pi\)
\(992\) 2.00000 0.0635001
\(993\) 12.7980 0.406131
\(994\) 30.3485 0.962595
\(995\) 0 0
\(996\) −7.44949 −0.236046
\(997\) 30.8990 0.978580 0.489290 0.872121i \(-0.337256\pi\)
0.489290 + 0.872121i \(0.337256\pi\)
\(998\) 12.1010 0.383051
\(999\) −6.34847 −0.200857
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 3450.2.a.bl.1.1 yes 2
5.2 odd 4 3450.2.d.y.2899.3 4
5.3 odd 4 3450.2.d.y.2899.2 4
5.4 even 2 3450.2.a.bf.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3450.2.a.bf.1.2 2 5.4 even 2
3450.2.a.bl.1.1 yes 2 1.1 even 1 trivial
3450.2.d.y.2899.2 4 5.3 odd 4
3450.2.d.y.2899.3 4 5.2 odd 4