Properties

Label 2352.2.a.bd.1.1
Level $2352$
Weight $2$
Character 2352.1
Self dual yes
Analytic conductor $18.781$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 2352 = 2^{4} \cdot 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2352.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(18.7808145554\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{8})^+\)
Defining polynomial: \(x^{2} - 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1176)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.41421\) of defining polynomial
Character \(\chi\) \(=\) 2352.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{3} -3.41421 q^{5} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{3} -3.41421 q^{5} +1.00000 q^{9} -4.82843 q^{11} -1.41421 q^{13} -3.41421 q^{15} +6.24264 q^{17} +1.17157 q^{19} +0.828427 q^{23} +6.65685 q^{25} +1.00000 q^{27} -8.48528 q^{29} +10.8284 q^{31} -4.82843 q^{33} -9.65685 q^{37} -1.41421 q^{39} +3.41421 q^{41} +8.00000 q^{43} -3.41421 q^{45} +1.17157 q^{47} +6.24264 q^{51} +9.31371 q^{53} +16.4853 q^{55} +1.17157 q^{57} +10.8284 q^{59} +5.89949 q^{61} +4.82843 q^{65} +8.00000 q^{67} +0.828427 q^{69} -4.82843 q^{71} -3.07107 q^{73} +6.65685 q^{75} +13.6569 q^{79} +1.00000 q^{81} -7.31371 q^{83} -21.3137 q^{85} -8.48528 q^{87} -14.7279 q^{89} +10.8284 q^{93} -4.00000 q^{95} +16.2426 q^{97} -4.82843 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{3} - 4q^{5} + 2q^{9} + O(q^{10}) \) \( 2q + 2q^{3} - 4q^{5} + 2q^{9} - 4q^{11} - 4q^{15} + 4q^{17} + 8q^{19} - 4q^{23} + 2q^{25} + 2q^{27} + 16q^{31} - 4q^{33} - 8q^{37} + 4q^{41} + 16q^{43} - 4q^{45} + 8q^{47} + 4q^{51} - 4q^{53} + 16q^{55} + 8q^{57} + 16q^{59} - 8q^{61} + 4q^{65} + 16q^{67} - 4q^{69} - 4q^{71} + 8q^{73} + 2q^{75} + 16q^{79} + 2q^{81} + 8q^{83} - 20q^{85} - 4q^{89} + 16q^{93} - 8q^{95} + 24q^{97} - 4q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 1.00000 0.577350
\(4\) 0 0
\(5\) −3.41421 −1.52688 −0.763441 0.645877i \(-0.776492\pi\)
−0.763441 + 0.645877i \(0.776492\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −4.82843 −1.45583 −0.727913 0.685670i \(-0.759509\pi\)
−0.727913 + 0.685670i \(0.759509\pi\)
\(12\) 0 0
\(13\) −1.41421 −0.392232 −0.196116 0.980581i \(-0.562833\pi\)
−0.196116 + 0.980581i \(0.562833\pi\)
\(14\) 0 0
\(15\) −3.41421 −0.881546
\(16\) 0 0
\(17\) 6.24264 1.51406 0.757031 0.653379i \(-0.226649\pi\)
0.757031 + 0.653379i \(0.226649\pi\)
\(18\) 0 0
\(19\) 1.17157 0.268777 0.134389 0.990929i \(-0.457093\pi\)
0.134389 + 0.990929i \(0.457093\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0.828427 0.172739 0.0863695 0.996263i \(-0.472473\pi\)
0.0863695 + 0.996263i \(0.472473\pi\)
\(24\) 0 0
\(25\) 6.65685 1.33137
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) −8.48528 −1.57568 −0.787839 0.615882i \(-0.788800\pi\)
−0.787839 + 0.615882i \(0.788800\pi\)
\(30\) 0 0
\(31\) 10.8284 1.94484 0.972421 0.233231i \(-0.0749297\pi\)
0.972421 + 0.233231i \(0.0749297\pi\)
\(32\) 0 0
\(33\) −4.82843 −0.840521
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −9.65685 −1.58758 −0.793789 0.608194i \(-0.791894\pi\)
−0.793789 + 0.608194i \(0.791894\pi\)
\(38\) 0 0
\(39\) −1.41421 −0.226455
\(40\) 0 0
\(41\) 3.41421 0.533211 0.266605 0.963806i \(-0.414098\pi\)
0.266605 + 0.963806i \(0.414098\pi\)
\(42\) 0 0
\(43\) 8.00000 1.21999 0.609994 0.792406i \(-0.291172\pi\)
0.609994 + 0.792406i \(0.291172\pi\)
\(44\) 0 0
\(45\) −3.41421 −0.508961
\(46\) 0 0
\(47\) 1.17157 0.170891 0.0854457 0.996343i \(-0.472769\pi\)
0.0854457 + 0.996343i \(0.472769\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) 6.24264 0.874145
\(52\) 0 0
\(53\) 9.31371 1.27934 0.639668 0.768651i \(-0.279072\pi\)
0.639668 + 0.768651i \(0.279072\pi\)
\(54\) 0 0
\(55\) 16.4853 2.22287
\(56\) 0 0
\(57\) 1.17157 0.155179
\(58\) 0 0
\(59\) 10.8284 1.40974 0.704871 0.709336i \(-0.251005\pi\)
0.704871 + 0.709336i \(0.251005\pi\)
\(60\) 0 0
\(61\) 5.89949 0.755353 0.377676 0.925938i \(-0.376723\pi\)
0.377676 + 0.925938i \(0.376723\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 4.82843 0.598893
\(66\) 0 0
\(67\) 8.00000 0.977356 0.488678 0.872464i \(-0.337479\pi\)
0.488678 + 0.872464i \(0.337479\pi\)
\(68\) 0 0
\(69\) 0.828427 0.0997309
\(70\) 0 0
\(71\) −4.82843 −0.573029 −0.286514 0.958076i \(-0.592497\pi\)
−0.286514 + 0.958076i \(0.592497\pi\)
\(72\) 0 0
\(73\) −3.07107 −0.359441 −0.179721 0.983718i \(-0.557519\pi\)
−0.179721 + 0.983718i \(0.557519\pi\)
\(74\) 0 0
\(75\) 6.65685 0.768667
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 13.6569 1.53652 0.768258 0.640140i \(-0.221124\pi\)
0.768258 + 0.640140i \(0.221124\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −7.31371 −0.802784 −0.401392 0.915906i \(-0.631473\pi\)
−0.401392 + 0.915906i \(0.631473\pi\)
\(84\) 0 0
\(85\) −21.3137 −2.31180
\(86\) 0 0
\(87\) −8.48528 −0.909718
\(88\) 0 0
\(89\) −14.7279 −1.56116 −0.780578 0.625058i \(-0.785075\pi\)
−0.780578 + 0.625058i \(0.785075\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 10.8284 1.12286
\(94\) 0 0
\(95\) −4.00000 −0.410391
\(96\) 0 0
\(97\) 16.2426 1.64919 0.824595 0.565723i \(-0.191403\pi\)
0.824595 + 0.565723i \(0.191403\pi\)
\(98\) 0 0
\(99\) −4.82843 −0.485275
\(100\) 0 0
\(101\) −0.585786 −0.0582879 −0.0291440 0.999575i \(-0.509278\pi\)
−0.0291440 + 0.999575i \(0.509278\pi\)
\(102\) 0 0
\(103\) 5.17157 0.509570 0.254785 0.966998i \(-0.417995\pi\)
0.254785 + 0.966998i \(0.417995\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 2.48528 0.240261 0.120131 0.992758i \(-0.461669\pi\)
0.120131 + 0.992758i \(0.461669\pi\)
\(108\) 0 0
\(109\) 11.3137 1.08366 0.541828 0.840489i \(-0.317732\pi\)
0.541828 + 0.840489i \(0.317732\pi\)
\(110\) 0 0
\(111\) −9.65685 −0.916588
\(112\) 0 0
\(113\) 6.00000 0.564433 0.282216 0.959351i \(-0.408930\pi\)
0.282216 + 0.959351i \(0.408930\pi\)
\(114\) 0 0
\(115\) −2.82843 −0.263752
\(116\) 0 0
\(117\) −1.41421 −0.130744
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 12.3137 1.11943
\(122\) 0 0
\(123\) 3.41421 0.307849
\(124\) 0 0
\(125\) −5.65685 −0.505964
\(126\) 0 0
\(127\) −7.31371 −0.648987 −0.324493 0.945888i \(-0.605194\pi\)
−0.324493 + 0.945888i \(0.605194\pi\)
\(128\) 0 0
\(129\) 8.00000 0.704361
\(130\) 0 0
\(131\) 15.3137 1.33796 0.668982 0.743278i \(-0.266730\pi\)
0.668982 + 0.743278i \(0.266730\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) −3.41421 −0.293849
\(136\) 0 0
\(137\) −12.4853 −1.06669 −0.533345 0.845898i \(-0.679065\pi\)
−0.533345 + 0.845898i \(0.679065\pi\)
\(138\) 0 0
\(139\) 9.65685 0.819084 0.409542 0.912291i \(-0.365689\pi\)
0.409542 + 0.912291i \(0.365689\pi\)
\(140\) 0 0
\(141\) 1.17157 0.0986642
\(142\) 0 0
\(143\) 6.82843 0.571022
\(144\) 0 0
\(145\) 28.9706 2.40587
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −10.0000 −0.819232 −0.409616 0.912258i \(-0.634337\pi\)
−0.409616 + 0.912258i \(0.634337\pi\)
\(150\) 0 0
\(151\) 1.65685 0.134833 0.0674164 0.997725i \(-0.478524\pi\)
0.0674164 + 0.997725i \(0.478524\pi\)
\(152\) 0 0
\(153\) 6.24264 0.504688
\(154\) 0 0
\(155\) −36.9706 −2.96955
\(156\) 0 0
\(157\) −5.89949 −0.470831 −0.235415 0.971895i \(-0.575645\pi\)
−0.235415 + 0.971895i \(0.575645\pi\)
\(158\) 0 0
\(159\) 9.31371 0.738625
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 2.34315 0.183529 0.0917647 0.995781i \(-0.470749\pi\)
0.0917647 + 0.995781i \(0.470749\pi\)
\(164\) 0 0
\(165\) 16.4853 1.28338
\(166\) 0 0
\(167\) 6.82843 0.528400 0.264200 0.964468i \(-0.414892\pi\)
0.264200 + 0.964468i \(0.414892\pi\)
\(168\) 0 0
\(169\) −11.0000 −0.846154
\(170\) 0 0
\(171\) 1.17157 0.0895924
\(172\) 0 0
\(173\) 0.585786 0.0445365 0.0222683 0.999752i \(-0.492911\pi\)
0.0222683 + 0.999752i \(0.492911\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 10.8284 0.813914
\(178\) 0 0
\(179\) 21.7990 1.62933 0.814667 0.579930i \(-0.196920\pi\)
0.814667 + 0.579930i \(0.196920\pi\)
\(180\) 0 0
\(181\) −9.89949 −0.735824 −0.367912 0.929861i \(-0.619927\pi\)
−0.367912 + 0.929861i \(0.619927\pi\)
\(182\) 0 0
\(183\) 5.89949 0.436103
\(184\) 0 0
\(185\) 32.9706 2.42404
\(186\) 0 0
\(187\) −30.1421 −2.20421
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −20.8284 −1.50709 −0.753546 0.657395i \(-0.771658\pi\)
−0.753546 + 0.657395i \(0.771658\pi\)
\(192\) 0 0
\(193\) −20.6274 −1.48479 −0.742397 0.669960i \(-0.766311\pi\)
−0.742397 + 0.669960i \(0.766311\pi\)
\(194\) 0 0
\(195\) 4.82843 0.345771
\(196\) 0 0
\(197\) 2.00000 0.142494 0.0712470 0.997459i \(-0.477302\pi\)
0.0712470 + 0.997459i \(0.477302\pi\)
\(198\) 0 0
\(199\) −5.65685 −0.401004 −0.200502 0.979693i \(-0.564257\pi\)
−0.200502 + 0.979693i \(0.564257\pi\)
\(200\) 0 0
\(201\) 8.00000 0.564276
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) −11.6569 −0.814150
\(206\) 0 0
\(207\) 0.828427 0.0575797
\(208\) 0 0
\(209\) −5.65685 −0.391293
\(210\) 0 0
\(211\) 25.6569 1.76629 0.883145 0.469099i \(-0.155421\pi\)
0.883145 + 0.469099i \(0.155421\pi\)
\(212\) 0 0
\(213\) −4.82843 −0.330838
\(214\) 0 0
\(215\) −27.3137 −1.86278
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) −3.07107 −0.207524
\(220\) 0 0
\(221\) −8.82843 −0.593864
\(222\) 0 0
\(223\) −2.34315 −0.156909 −0.0784543 0.996918i \(-0.524998\pi\)
−0.0784543 + 0.996918i \(0.524998\pi\)
\(224\) 0 0
\(225\) 6.65685 0.443790
\(226\) 0 0
\(227\) 19.7990 1.31411 0.657053 0.753845i \(-0.271803\pi\)
0.657053 + 0.753845i \(0.271803\pi\)
\(228\) 0 0
\(229\) −0.928932 −0.0613856 −0.0306928 0.999529i \(-0.509771\pi\)
−0.0306928 + 0.999529i \(0.509771\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −11.5147 −0.754354 −0.377177 0.926141i \(-0.623105\pi\)
−0.377177 + 0.926141i \(0.623105\pi\)
\(234\) 0 0
\(235\) −4.00000 −0.260931
\(236\) 0 0
\(237\) 13.6569 0.887108
\(238\) 0 0
\(239\) 8.82843 0.571063 0.285532 0.958369i \(-0.407830\pi\)
0.285532 + 0.958369i \(0.407830\pi\)
\(240\) 0 0
\(241\) 2.10051 0.135305 0.0676527 0.997709i \(-0.478449\pi\)
0.0676527 + 0.997709i \(0.478449\pi\)
\(242\) 0 0
\(243\) 1.00000 0.0641500
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −1.65685 −0.105423
\(248\) 0 0
\(249\) −7.31371 −0.463487
\(250\) 0 0
\(251\) −8.48528 −0.535586 −0.267793 0.963476i \(-0.586294\pi\)
−0.267793 + 0.963476i \(0.586294\pi\)
\(252\) 0 0
\(253\) −4.00000 −0.251478
\(254\) 0 0
\(255\) −21.3137 −1.33472
\(256\) 0 0
\(257\) 21.7574 1.35719 0.678593 0.734514i \(-0.262590\pi\)
0.678593 + 0.734514i \(0.262590\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) −8.48528 −0.525226
\(262\) 0 0
\(263\) −19.1716 −1.18217 −0.591085 0.806609i \(-0.701300\pi\)
−0.591085 + 0.806609i \(0.701300\pi\)
\(264\) 0 0
\(265\) −31.7990 −1.95340
\(266\) 0 0
\(267\) −14.7279 −0.901334
\(268\) 0 0
\(269\) 18.0416 1.10002 0.550009 0.835159i \(-0.314624\pi\)
0.550009 + 0.835159i \(0.314624\pi\)
\(270\) 0 0
\(271\) −18.8284 −1.14375 −0.571873 0.820342i \(-0.693783\pi\)
−0.571873 + 0.820342i \(0.693783\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −32.1421 −1.93824
\(276\) 0 0
\(277\) −6.00000 −0.360505 −0.180253 0.983620i \(-0.557691\pi\)
−0.180253 + 0.983620i \(0.557691\pi\)
\(278\) 0 0
\(279\) 10.8284 0.648281
\(280\) 0 0
\(281\) 4.48528 0.267569 0.133785 0.991010i \(-0.457287\pi\)
0.133785 + 0.991010i \(0.457287\pi\)
\(282\) 0 0
\(283\) −9.17157 −0.545193 −0.272597 0.962128i \(-0.587882\pi\)
−0.272597 + 0.962128i \(0.587882\pi\)
\(284\) 0 0
\(285\) −4.00000 −0.236940
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 21.9706 1.29239
\(290\) 0 0
\(291\) 16.2426 0.952160
\(292\) 0 0
\(293\) 13.0711 0.763620 0.381810 0.924241i \(-0.375301\pi\)
0.381810 + 0.924241i \(0.375301\pi\)
\(294\) 0 0
\(295\) −36.9706 −2.15251
\(296\) 0 0
\(297\) −4.82843 −0.280174
\(298\) 0 0
\(299\) −1.17157 −0.0677538
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) −0.585786 −0.0336526
\(304\) 0 0
\(305\) −20.1421 −1.15334
\(306\) 0 0
\(307\) −28.4853 −1.62574 −0.812870 0.582445i \(-0.802096\pi\)
−0.812870 + 0.582445i \(0.802096\pi\)
\(308\) 0 0
\(309\) 5.17157 0.294201
\(310\) 0 0
\(311\) 2.14214 0.121469 0.0607347 0.998154i \(-0.480656\pi\)
0.0607347 + 0.998154i \(0.480656\pi\)
\(312\) 0 0
\(313\) 14.5858 0.824437 0.412219 0.911085i \(-0.364754\pi\)
0.412219 + 0.911085i \(0.364754\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −1.31371 −0.0737852 −0.0368926 0.999319i \(-0.511746\pi\)
−0.0368926 + 0.999319i \(0.511746\pi\)
\(318\) 0 0
\(319\) 40.9706 2.29391
\(320\) 0 0
\(321\) 2.48528 0.138715
\(322\) 0 0
\(323\) 7.31371 0.406946
\(324\) 0 0
\(325\) −9.41421 −0.522207
\(326\) 0 0
\(327\) 11.3137 0.625650
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 31.3137 1.72116 0.860579 0.509318i \(-0.170102\pi\)
0.860579 + 0.509318i \(0.170102\pi\)
\(332\) 0 0
\(333\) −9.65685 −0.529192
\(334\) 0 0
\(335\) −27.3137 −1.49231
\(336\) 0 0
\(337\) 16.9706 0.924445 0.462223 0.886764i \(-0.347052\pi\)
0.462223 + 0.886764i \(0.347052\pi\)
\(338\) 0 0
\(339\) 6.00000 0.325875
\(340\) 0 0
\(341\) −52.2843 −2.83135
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) −2.82843 −0.152277
\(346\) 0 0
\(347\) −24.1421 −1.29602 −0.648009 0.761633i \(-0.724398\pi\)
−0.648009 + 0.761633i \(0.724398\pi\)
\(348\) 0 0
\(349\) 6.38478 0.341769 0.170885 0.985291i \(-0.445337\pi\)
0.170885 + 0.985291i \(0.445337\pi\)
\(350\) 0 0
\(351\) −1.41421 −0.0754851
\(352\) 0 0
\(353\) −3.89949 −0.207549 −0.103775 0.994601i \(-0.533092\pi\)
−0.103775 + 0.994601i \(0.533092\pi\)
\(354\) 0 0
\(355\) 16.4853 0.874948
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −15.4558 −0.815728 −0.407864 0.913043i \(-0.633726\pi\)
−0.407864 + 0.913043i \(0.633726\pi\)
\(360\) 0 0
\(361\) −17.6274 −0.927759
\(362\) 0 0
\(363\) 12.3137 0.646302
\(364\) 0 0
\(365\) 10.4853 0.548825
\(366\) 0 0
\(367\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(368\) 0 0
\(369\) 3.41421 0.177737
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 37.3137 1.93203 0.966015 0.258485i \(-0.0832232\pi\)
0.966015 + 0.258485i \(0.0832232\pi\)
\(374\) 0 0
\(375\) −5.65685 −0.292119
\(376\) 0 0
\(377\) 12.0000 0.618031
\(378\) 0 0
\(379\) 23.3137 1.19754 0.598772 0.800919i \(-0.295655\pi\)
0.598772 + 0.800919i \(0.295655\pi\)
\(380\) 0 0
\(381\) −7.31371 −0.374693
\(382\) 0 0
\(383\) −8.97056 −0.458374 −0.229187 0.973382i \(-0.573607\pi\)
−0.229187 + 0.973382i \(0.573607\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 8.00000 0.406663
\(388\) 0 0
\(389\) 14.1421 0.717035 0.358517 0.933523i \(-0.383282\pi\)
0.358517 + 0.933523i \(0.383282\pi\)
\(390\) 0 0
\(391\) 5.17157 0.261538
\(392\) 0 0
\(393\) 15.3137 0.772474
\(394\) 0 0
\(395\) −46.6274 −2.34608
\(396\) 0 0
\(397\) −32.7279 −1.64257 −0.821284 0.570520i \(-0.806742\pi\)
−0.821284 + 0.570520i \(0.806742\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 4.48528 0.223984 0.111992 0.993709i \(-0.464277\pi\)
0.111992 + 0.993709i \(0.464277\pi\)
\(402\) 0 0
\(403\) −15.3137 −0.762830
\(404\) 0 0
\(405\) −3.41421 −0.169654
\(406\) 0 0
\(407\) 46.6274 2.31124
\(408\) 0 0
\(409\) −7.75736 −0.383577 −0.191788 0.981436i \(-0.561429\pi\)
−0.191788 + 0.981436i \(0.561429\pi\)
\(410\) 0 0
\(411\) −12.4853 −0.615854
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 24.9706 1.22576
\(416\) 0 0
\(417\) 9.65685 0.472898
\(418\) 0 0
\(419\) 5.17157 0.252648 0.126324 0.991989i \(-0.459682\pi\)
0.126324 + 0.991989i \(0.459682\pi\)
\(420\) 0 0
\(421\) 29.3137 1.42866 0.714331 0.699808i \(-0.246731\pi\)
0.714331 + 0.699808i \(0.246731\pi\)
\(422\) 0 0
\(423\) 1.17157 0.0569638
\(424\) 0 0
\(425\) 41.5563 2.01578
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 6.82843 0.329680
\(430\) 0 0
\(431\) 13.5147 0.650981 0.325491 0.945545i \(-0.394471\pi\)
0.325491 + 0.945545i \(0.394471\pi\)
\(432\) 0 0
\(433\) 10.3848 0.499061 0.249530 0.968367i \(-0.419724\pi\)
0.249530 + 0.968367i \(0.419724\pi\)
\(434\) 0 0
\(435\) 28.9706 1.38903
\(436\) 0 0
\(437\) 0.970563 0.0464283
\(438\) 0 0
\(439\) 19.3137 0.921793 0.460897 0.887454i \(-0.347528\pi\)
0.460897 + 0.887454i \(0.347528\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −21.5147 −1.02220 −0.511098 0.859523i \(-0.670761\pi\)
−0.511098 + 0.859523i \(0.670761\pi\)
\(444\) 0 0
\(445\) 50.2843 2.38370
\(446\) 0 0
\(447\) −10.0000 −0.472984
\(448\) 0 0
\(449\) −10.0000 −0.471929 −0.235965 0.971762i \(-0.575825\pi\)
−0.235965 + 0.971762i \(0.575825\pi\)
\(450\) 0 0
\(451\) −16.4853 −0.776262
\(452\) 0 0
\(453\) 1.65685 0.0778458
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −24.6274 −1.15202 −0.576011 0.817442i \(-0.695391\pi\)
−0.576011 + 0.817442i \(0.695391\pi\)
\(458\) 0 0
\(459\) 6.24264 0.291382
\(460\) 0 0
\(461\) −9.75736 −0.454446 −0.227223 0.973843i \(-0.572965\pi\)
−0.227223 + 0.973843i \(0.572965\pi\)
\(462\) 0 0
\(463\) 12.9706 0.602793 0.301397 0.953499i \(-0.402547\pi\)
0.301397 + 0.953499i \(0.402547\pi\)
\(464\) 0 0
\(465\) −36.9706 −1.71447
\(466\) 0 0
\(467\) −5.17157 −0.239312 −0.119656 0.992815i \(-0.538179\pi\)
−0.119656 + 0.992815i \(0.538179\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) −5.89949 −0.271834
\(472\) 0 0
\(473\) −38.6274 −1.77609
\(474\) 0 0
\(475\) 7.79899 0.357842
\(476\) 0 0
\(477\) 9.31371 0.426445
\(478\) 0 0
\(479\) 19.1127 0.873281 0.436641 0.899636i \(-0.356168\pi\)
0.436641 + 0.899636i \(0.356168\pi\)
\(480\) 0 0
\(481\) 13.6569 0.622699
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −55.4558 −2.51812
\(486\) 0 0
\(487\) −12.9706 −0.587752 −0.293876 0.955844i \(-0.594945\pi\)
−0.293876 + 0.955844i \(0.594945\pi\)
\(488\) 0 0
\(489\) 2.34315 0.105961
\(490\) 0 0
\(491\) 4.14214 0.186932 0.0934660 0.995622i \(-0.470205\pi\)
0.0934660 + 0.995622i \(0.470205\pi\)
\(492\) 0 0
\(493\) −52.9706 −2.38567
\(494\) 0 0
\(495\) 16.4853 0.740958
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) −28.2843 −1.26618 −0.633089 0.774079i \(-0.718213\pi\)
−0.633089 + 0.774079i \(0.718213\pi\)
\(500\) 0 0
\(501\) 6.82843 0.305072
\(502\) 0 0
\(503\) 30.6274 1.36561 0.682805 0.730601i \(-0.260760\pi\)
0.682805 + 0.730601i \(0.260760\pi\)
\(504\) 0 0
\(505\) 2.00000 0.0889988
\(506\) 0 0
\(507\) −11.0000 −0.488527
\(508\) 0 0
\(509\) −22.9289 −1.01631 −0.508154 0.861267i \(-0.669672\pi\)
−0.508154 + 0.861267i \(0.669672\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 1.17157 0.0517262
\(514\) 0 0
\(515\) −17.6569 −0.778054
\(516\) 0 0
\(517\) −5.65685 −0.248788
\(518\) 0 0
\(519\) 0.585786 0.0257132
\(520\) 0 0
\(521\) −5.27208 −0.230974 −0.115487 0.993309i \(-0.536843\pi\)
−0.115487 + 0.993309i \(0.536843\pi\)
\(522\) 0 0
\(523\) −1.65685 −0.0724492 −0.0362246 0.999344i \(-0.511533\pi\)
−0.0362246 + 0.999344i \(0.511533\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 67.5980 2.94461
\(528\) 0 0
\(529\) −22.3137 −0.970161
\(530\) 0 0
\(531\) 10.8284 0.469914
\(532\) 0 0
\(533\) −4.82843 −0.209142
\(534\) 0 0
\(535\) −8.48528 −0.366851
\(536\) 0 0
\(537\) 21.7990 0.940696
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −8.62742 −0.370922 −0.185461 0.982652i \(-0.559378\pi\)
−0.185461 + 0.982652i \(0.559378\pi\)
\(542\) 0 0
\(543\) −9.89949 −0.424828
\(544\) 0 0
\(545\) −38.6274 −1.65462
\(546\) 0 0
\(547\) 4.97056 0.212526 0.106263 0.994338i \(-0.466111\pi\)
0.106263 + 0.994338i \(0.466111\pi\)
\(548\) 0 0
\(549\) 5.89949 0.251784
\(550\) 0 0
\(551\) −9.94113 −0.423506
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 32.9706 1.39952
\(556\) 0 0
\(557\) 43.9411 1.86185 0.930923 0.365216i \(-0.119005\pi\)
0.930923 + 0.365216i \(0.119005\pi\)
\(558\) 0 0
\(559\) −11.3137 −0.478519
\(560\) 0 0
\(561\) −30.1421 −1.27260
\(562\) 0 0
\(563\) 26.8284 1.13068 0.565342 0.824857i \(-0.308744\pi\)
0.565342 + 0.824857i \(0.308744\pi\)
\(564\) 0 0
\(565\) −20.4853 −0.861822
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 6.82843 0.286263 0.143131 0.989704i \(-0.454283\pi\)
0.143131 + 0.989704i \(0.454283\pi\)
\(570\) 0 0
\(571\) −40.2843 −1.68584 −0.842922 0.538036i \(-0.819167\pi\)
−0.842922 + 0.538036i \(0.819167\pi\)
\(572\) 0 0
\(573\) −20.8284 −0.870120
\(574\) 0 0
\(575\) 5.51472 0.229980
\(576\) 0 0
\(577\) 9.41421 0.391919 0.195959 0.980612i \(-0.437218\pi\)
0.195959 + 0.980612i \(0.437218\pi\)
\(578\) 0 0
\(579\) −20.6274 −0.857246
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) −44.9706 −1.86249
\(584\) 0 0
\(585\) 4.82843 0.199631
\(586\) 0 0
\(587\) −26.8284 −1.10733 −0.553664 0.832740i \(-0.686771\pi\)
−0.553664 + 0.832740i \(0.686771\pi\)
\(588\) 0 0
\(589\) 12.6863 0.522730
\(590\) 0 0
\(591\) 2.00000 0.0822690
\(592\) 0 0
\(593\) −29.0711 −1.19381 −0.596903 0.802314i \(-0.703602\pi\)
−0.596903 + 0.802314i \(0.703602\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −5.65685 −0.231520
\(598\) 0 0
\(599\) −16.1421 −0.659550 −0.329775 0.944060i \(-0.606973\pi\)
−0.329775 + 0.944060i \(0.606973\pi\)
\(600\) 0 0
\(601\) 12.2426 0.499388 0.249694 0.968325i \(-0.419670\pi\)
0.249694 + 0.968325i \(0.419670\pi\)
\(602\) 0 0
\(603\) 8.00000 0.325785
\(604\) 0 0
\(605\) −42.0416 −1.70924
\(606\) 0 0
\(607\) 31.5980 1.28252 0.641261 0.767323i \(-0.278412\pi\)
0.641261 + 0.767323i \(0.278412\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −1.65685 −0.0670291
\(612\) 0 0
\(613\) −2.34315 −0.0946388 −0.0473194 0.998880i \(-0.515068\pi\)
−0.0473194 + 0.998880i \(0.515068\pi\)
\(614\) 0 0
\(615\) −11.6569 −0.470050
\(616\) 0 0
\(617\) 21.4558 0.863780 0.431890 0.901926i \(-0.357847\pi\)
0.431890 + 0.901926i \(0.357847\pi\)
\(618\) 0 0
\(619\) −40.2843 −1.61916 −0.809581 0.587008i \(-0.800306\pi\)
−0.809581 + 0.587008i \(0.800306\pi\)
\(620\) 0 0
\(621\) 0.828427 0.0332436
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −13.9706 −0.558823
\(626\) 0 0
\(627\) −5.65685 −0.225913
\(628\) 0 0
\(629\) −60.2843 −2.40369
\(630\) 0 0
\(631\) 8.28427 0.329792 0.164896 0.986311i \(-0.447271\pi\)
0.164896 + 0.986311i \(0.447271\pi\)
\(632\) 0 0
\(633\) 25.6569 1.01977
\(634\) 0 0
\(635\) 24.9706 0.990927
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) −4.82843 −0.191010
\(640\) 0 0
\(641\) 25.1716 0.994217 0.497109 0.867688i \(-0.334395\pi\)
0.497109 + 0.867688i \(0.334395\pi\)
\(642\) 0 0
\(643\) 19.5147 0.769585 0.384793 0.923003i \(-0.374273\pi\)
0.384793 + 0.923003i \(0.374273\pi\)
\(644\) 0 0
\(645\) −27.3137 −1.07548
\(646\) 0 0
\(647\) 14.8284 0.582966 0.291483 0.956576i \(-0.405851\pi\)
0.291483 + 0.956576i \(0.405851\pi\)
\(648\) 0 0
\(649\) −52.2843 −2.05234
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 2.82843 0.110685 0.0553425 0.998467i \(-0.482375\pi\)
0.0553425 + 0.998467i \(0.482375\pi\)
\(654\) 0 0
\(655\) −52.2843 −2.04292
\(656\) 0 0
\(657\) −3.07107 −0.119814
\(658\) 0 0
\(659\) −25.5147 −0.993912 −0.496956 0.867776i \(-0.665549\pi\)
−0.496956 + 0.867776i \(0.665549\pi\)
\(660\) 0 0
\(661\) 23.3553 0.908417 0.454209 0.890895i \(-0.349922\pi\)
0.454209 + 0.890895i \(0.349922\pi\)
\(662\) 0 0
\(663\) −8.82843 −0.342868
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −7.02944 −0.272181
\(668\) 0 0
\(669\) −2.34315 −0.0905912
\(670\) 0 0
\(671\) −28.4853 −1.09966
\(672\) 0 0
\(673\) 15.3137 0.590300 0.295150 0.955451i \(-0.404630\pi\)
0.295150 + 0.955451i \(0.404630\pi\)
\(674\) 0 0
\(675\) 6.65685 0.256222
\(676\) 0 0
\(677\) 1.55635 0.0598154 0.0299077 0.999553i \(-0.490479\pi\)
0.0299077 + 0.999553i \(0.490479\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 19.7990 0.758699
\(682\) 0 0
\(683\) −12.1421 −0.464606 −0.232303 0.972643i \(-0.574626\pi\)
−0.232303 + 0.972643i \(0.574626\pi\)
\(684\) 0 0
\(685\) 42.6274 1.62871
\(686\) 0 0
\(687\) −0.928932 −0.0354410
\(688\) 0 0
\(689\) −13.1716 −0.501797
\(690\) 0 0
\(691\) 28.0000 1.06517 0.532585 0.846376i \(-0.321221\pi\)
0.532585 + 0.846376i \(0.321221\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −32.9706 −1.25064
\(696\) 0 0
\(697\) 21.3137 0.807314
\(698\) 0 0
\(699\) −11.5147 −0.435527
\(700\) 0 0
\(701\) −10.8284 −0.408984 −0.204492 0.978868i \(-0.565554\pi\)
−0.204492 + 0.978868i \(0.565554\pi\)
\(702\) 0 0
\(703\) −11.3137 −0.426705
\(704\) 0 0
\(705\) −4.00000 −0.150649
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −8.00000 −0.300446 −0.150223 0.988652i \(-0.547999\pi\)
−0.150223 + 0.988652i \(0.547999\pi\)
\(710\) 0 0
\(711\) 13.6569 0.512172
\(712\) 0 0
\(713\) 8.97056 0.335950
\(714\) 0 0
\(715\) −23.3137 −0.871883
\(716\) 0 0
\(717\) 8.82843 0.329704
\(718\) 0 0
\(719\) −27.3137 −1.01863 −0.509315 0.860580i \(-0.670101\pi\)
−0.509315 + 0.860580i \(0.670101\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 2.10051 0.0781186
\(724\) 0 0
\(725\) −56.4853 −2.09781
\(726\) 0 0
\(727\) −25.4558 −0.944105 −0.472052 0.881570i \(-0.656487\pi\)
−0.472052 + 0.881570i \(0.656487\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 49.9411 1.84714
\(732\) 0 0
\(733\) −20.2426 −0.747679 −0.373839 0.927493i \(-0.621959\pi\)
−0.373839 + 0.927493i \(0.621959\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −38.6274 −1.42286
\(738\) 0 0
\(739\) 36.2843 1.33474 0.667369 0.744727i \(-0.267420\pi\)
0.667369 + 0.744727i \(0.267420\pi\)
\(740\) 0 0
\(741\) −1.65685 −0.0608661
\(742\) 0 0
\(743\) 16.8284 0.617375 0.308688 0.951163i \(-0.400110\pi\)
0.308688 + 0.951163i \(0.400110\pi\)
\(744\) 0 0
\(745\) 34.1421 1.25087
\(746\) 0 0
\(747\) −7.31371 −0.267595
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) −18.3431 −0.669351 −0.334675 0.942333i \(-0.608627\pi\)
−0.334675 + 0.942333i \(0.608627\pi\)
\(752\) 0 0
\(753\) −8.48528 −0.309221
\(754\) 0 0
\(755\) −5.65685 −0.205874
\(756\) 0 0
\(757\) 11.3137 0.411204 0.205602 0.978636i \(-0.434085\pi\)
0.205602 + 0.978636i \(0.434085\pi\)
\(758\) 0 0
\(759\) −4.00000 −0.145191
\(760\) 0 0
\(761\) 18.2426 0.661295 0.330648 0.943754i \(-0.392733\pi\)
0.330648 + 0.943754i \(0.392733\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) −21.3137 −0.770599
\(766\) 0 0
\(767\) −15.3137 −0.552946
\(768\) 0 0
\(769\) −0.928932 −0.0334982 −0.0167491 0.999860i \(-0.505332\pi\)
−0.0167491 + 0.999860i \(0.505332\pi\)
\(770\) 0 0
\(771\) 21.7574 0.783572
\(772\) 0 0
\(773\) 0.585786 0.0210693 0.0105346 0.999945i \(-0.496647\pi\)
0.0105346 + 0.999945i \(0.496647\pi\)
\(774\) 0 0
\(775\) 72.0833 2.58931
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 4.00000 0.143315
\(780\) 0 0
\(781\) 23.3137 0.834230
\(782\) 0 0
\(783\) −8.48528 −0.303239
\(784\) 0 0
\(785\) 20.1421 0.718904
\(786\) 0 0
\(787\) 34.6274 1.23433 0.617167 0.786832i \(-0.288280\pi\)
0.617167 + 0.786832i \(0.288280\pi\)
\(788\) 0 0
\(789\) −19.1716 −0.682526
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) −8.34315 −0.296274
\(794\) 0 0
\(795\) −31.7990 −1.12779
\(796\) 0 0
\(797\) 0.786797 0.0278698 0.0139349 0.999903i \(-0.495564\pi\)
0.0139349 + 0.999903i \(0.495564\pi\)
\(798\) 0 0
\(799\) 7.31371 0.258740
\(800\) 0 0
\(801\) −14.7279 −0.520386
\(802\) 0 0
\(803\) 14.8284 0.523284
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 18.0416 0.635095
\(808\) 0 0
\(809\) −30.0000 −1.05474 −0.527372 0.849635i \(-0.676823\pi\)
−0.527372 + 0.849635i \(0.676823\pi\)
\(810\) 0 0
\(811\) 12.9706 0.455458 0.227729 0.973725i \(-0.426870\pi\)
0.227729 + 0.973725i \(0.426870\pi\)
\(812\) 0 0
\(813\) −18.8284 −0.660342
\(814\) 0 0
\(815\) −8.00000 −0.280228
\(816\) 0 0
\(817\) 9.37258 0.327905
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 10.0000 0.349002 0.174501 0.984657i \(-0.444169\pi\)
0.174501 + 0.984657i \(0.444169\pi\)
\(822\) 0 0
\(823\) 13.6569 0.476048 0.238024 0.971259i \(-0.423500\pi\)
0.238024 + 0.971259i \(0.423500\pi\)
\(824\) 0 0
\(825\) −32.1421 −1.11905
\(826\) 0 0
\(827\) −31.4558 −1.09383 −0.546913 0.837189i \(-0.684197\pi\)
−0.546913 + 0.837189i \(0.684197\pi\)
\(828\) 0 0
\(829\) −28.7279 −0.997762 −0.498881 0.866671i \(-0.666256\pi\)
−0.498881 + 0.866671i \(0.666256\pi\)
\(830\) 0 0
\(831\) −6.00000 −0.208138
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) −23.3137 −0.806804
\(836\) 0 0
\(837\) 10.8284 0.374285
\(838\) 0 0
\(839\) 54.8284 1.89289 0.946444 0.322869i \(-0.104647\pi\)
0.946444 + 0.322869i \(0.104647\pi\)
\(840\) 0 0
\(841\) 43.0000 1.48276
\(842\) 0 0
\(843\) 4.48528 0.154481
\(844\) 0 0
\(845\) 37.5563 1.29198
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) −9.17157 −0.314768
\(850\) 0 0
\(851\) −8.00000 −0.274236
\(852\) 0 0
\(853\) 12.0416 0.412298 0.206149 0.978521i \(-0.433907\pi\)
0.206149 + 0.978521i \(0.433907\pi\)
\(854\) 0 0
\(855\) −4.00000 −0.136797
\(856\) 0 0
\(857\) −15.6985 −0.536250 −0.268125 0.963384i \(-0.586404\pi\)
−0.268125 + 0.963384i \(0.586404\pi\)
\(858\) 0 0
\(859\) −33.1716 −1.13180 −0.565900 0.824474i \(-0.691471\pi\)
−0.565900 + 0.824474i \(0.691471\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 46.4853 1.58238 0.791189 0.611572i \(-0.209463\pi\)
0.791189 + 0.611572i \(0.209463\pi\)
\(864\) 0 0
\(865\) −2.00000 −0.0680020
\(866\) 0 0
\(867\) 21.9706 0.746159
\(868\) 0 0
\(869\) −65.9411 −2.23690
\(870\) 0 0
\(871\) −11.3137 −0.383350
\(872\) 0 0
\(873\) 16.2426 0.549730
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −36.2843 −1.22523 −0.612616 0.790380i \(-0.709883\pi\)
−0.612616 + 0.790380i \(0.709883\pi\)
\(878\) 0 0
\(879\) 13.0711 0.440876
\(880\) 0 0
\(881\) 10.9289 0.368205 0.184103 0.982907i \(-0.441062\pi\)
0.184103 + 0.982907i \(0.441062\pi\)
\(882\) 0 0
\(883\) −29.6569 −0.998033 −0.499016 0.866593i \(-0.666305\pi\)
−0.499016 + 0.866593i \(0.666305\pi\)
\(884\) 0 0
\(885\) −36.9706 −1.24275
\(886\) 0 0
\(887\) −29.4558 −0.989030 −0.494515 0.869169i \(-0.664654\pi\)
−0.494515 + 0.869169i \(0.664654\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) −4.82843 −0.161758
\(892\) 0 0
\(893\) 1.37258 0.0459317
\(894\) 0 0
\(895\) −74.4264 −2.48780
\(896\) 0 0
\(897\) −1.17157 −0.0391177
\(898\) 0 0
\(899\) −91.8823 −3.06444
\(900\) 0 0
\(901\) 58.1421 1.93700
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 33.7990 1.12352
\(906\) 0 0
\(907\) 40.9706 1.36041 0.680203 0.733024i \(-0.261892\pi\)
0.680203 + 0.733024i \(0.261892\pi\)
\(908\) 0 0
\(909\) −0.585786 −0.0194293
\(910\) 0 0
\(911\) 37.5147 1.24292 0.621459 0.783447i \(-0.286540\pi\)
0.621459 + 0.783447i \(0.286540\pi\)
\(912\) 0 0
\(913\) 35.3137 1.16871
\(914\) 0 0
\(915\) −20.1421 −0.665878
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) −41.2548 −1.36087 −0.680436 0.732808i \(-0.738209\pi\)
−0.680436 + 0.732808i \(0.738209\pi\)
\(920\) 0 0
\(921\) −28.4853 −0.938622
\(922\) 0 0
\(923\) 6.82843 0.224760
\(924\) 0 0
\(925\) −64.2843 −2.11365
\(926\) 0 0
\(927\) 5.17157 0.169857
\(928\) 0 0
\(929\) −7.12994 −0.233926 −0.116963 0.993136i \(-0.537316\pi\)
−0.116963 + 0.993136i \(0.537316\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 2.14214 0.0701304
\(934\) 0 0
\(935\) 102.912 3.36557
\(936\) 0 0
\(937\) 25.8995 0.846100 0.423050 0.906106i \(-0.360960\pi\)
0.423050 + 0.906106i \(0.360960\pi\)
\(938\) 0 0
\(939\) 14.5858 0.475989
\(940\) 0 0
\(941\) 24.1005 0.785654 0.392827 0.919612i \(-0.371497\pi\)
0.392827 + 0.919612i \(0.371497\pi\)
\(942\) 0 0
\(943\) 2.82843 0.0921063
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 8.14214 0.264584 0.132292 0.991211i \(-0.457766\pi\)
0.132292 + 0.991211i \(0.457766\pi\)
\(948\) 0 0
\(949\) 4.34315 0.140984
\(950\) 0 0
\(951\) −1.31371 −0.0425999
\(952\) 0 0
\(953\) −52.6274 −1.70477 −0.852385 0.522915i \(-0.824844\pi\)
−0.852385 + 0.522915i \(0.824844\pi\)
\(954\) 0 0
\(955\) 71.1127 2.30115
\(956\) 0 0
\(957\) 40.9706 1.32439
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 86.2548 2.78241
\(962\) 0 0
\(963\) 2.48528 0.0800871
\(964\) 0 0
\(965\) 70.4264 2.26711
\(966\) 0 0
\(967\) −10.6274 −0.341755 −0.170877 0.985292i \(-0.554660\pi\)
−0.170877 + 0.985292i \(0.554660\pi\)
\(968\) 0 0
\(969\) 7.31371 0.234950
\(970\) 0 0
\(971\) 33.2548 1.06720 0.533599 0.845737i \(-0.320839\pi\)
0.533599 + 0.845737i \(0.320839\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) −9.41421 −0.301496
\(976\) 0 0
\(977\) 50.1421 1.60419 0.802095 0.597197i \(-0.203719\pi\)
0.802095 + 0.597197i \(0.203719\pi\)
\(978\) 0 0
\(979\) 71.1127 2.27277
\(980\) 0 0
\(981\) 11.3137 0.361219
\(982\) 0 0
\(983\) 30.6274 0.976863 0.488431 0.872602i \(-0.337569\pi\)
0.488431 + 0.872602i \(0.337569\pi\)
\(984\) 0 0
\(985\) −6.82843 −0.217572
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 6.62742 0.210740
\(990\) 0 0
\(991\) −0.686292 −0.0218008 −0.0109004 0.999941i \(-0.503470\pi\)
−0.0109004 + 0.999941i \(0.503470\pi\)
\(992\) 0 0
\(993\) 31.3137 0.993711
\(994\) 0 0
\(995\) 19.3137 0.612286
\(996\) 0 0
\(997\) 37.4142 1.18492 0.592460 0.805600i \(-0.298157\pi\)
0.592460 + 0.805600i \(0.298157\pi\)
\(998\) 0 0
\(999\) −9.65685 −0.305529
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 2352.2.a.bd.1.1 2
3.2 odd 2 7056.2.a.cx.1.2 2
4.3 odd 2 1176.2.a.j.1.1 2
7.2 even 3 2352.2.q.bc.1537.2 4
7.3 odd 6 2352.2.q.be.961.1 4
7.4 even 3 2352.2.q.bc.961.2 4
7.5 odd 6 2352.2.q.be.1537.1 4
7.6 odd 2 2352.2.a.bb.1.2 2
8.3 odd 2 9408.2.a.ee.1.2 2
8.5 even 2 9408.2.a.ds.1.2 2
12.11 even 2 3528.2.a.bl.1.2 2
21.20 even 2 7056.2.a.cg.1.1 2
28.3 even 6 1176.2.q.k.961.1 4
28.11 odd 6 1176.2.q.o.961.2 4
28.19 even 6 1176.2.q.k.361.1 4
28.23 odd 6 1176.2.q.o.361.2 4
28.27 even 2 1176.2.a.o.1.2 yes 2
56.13 odd 2 9408.2.a.du.1.1 2
56.27 even 2 9408.2.a.dg.1.1 2
84.11 even 6 3528.2.s.bd.3313.1 4
84.23 even 6 3528.2.s.bd.361.1 4
84.47 odd 6 3528.2.s.bm.361.2 4
84.59 odd 6 3528.2.s.bm.3313.2 4
84.83 odd 2 3528.2.a.bb.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1176.2.a.j.1.1 2 4.3 odd 2
1176.2.a.o.1.2 yes 2 28.27 even 2
1176.2.q.k.361.1 4 28.19 even 6
1176.2.q.k.961.1 4 28.3 even 6
1176.2.q.o.361.2 4 28.23 odd 6
1176.2.q.o.961.2 4 28.11 odd 6
2352.2.a.bb.1.2 2 7.6 odd 2
2352.2.a.bd.1.1 2 1.1 even 1 trivial
2352.2.q.bc.961.2 4 7.4 even 3
2352.2.q.bc.1537.2 4 7.2 even 3
2352.2.q.be.961.1 4 7.3 odd 6
2352.2.q.be.1537.1 4 7.5 odd 6
3528.2.a.bb.1.1 2 84.83 odd 2
3528.2.a.bl.1.2 2 12.11 even 2
3528.2.s.bd.361.1 4 84.23 even 6
3528.2.s.bd.3313.1 4 84.11 even 6
3528.2.s.bm.361.2 4 84.47 odd 6
3528.2.s.bm.3313.2 4 84.59 odd 6
7056.2.a.cg.1.1 2 21.20 even 2
7056.2.a.cx.1.2 2 3.2 odd 2
9408.2.a.dg.1.1 2 56.27 even 2
9408.2.a.ds.1.2 2 8.5 even 2
9408.2.a.du.1.1 2 56.13 odd 2
9408.2.a.ee.1.2 2 8.3 odd 2