Properties

Label 3360.2.a.bh.1.2
Level $3360$
Weight $2$
Character 3360.1
Self dual yes
Analytic conductor $26.830$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 3360 = 2^{5} \cdot 3 \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3360.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(26.8297350792\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{10})^+\)
Defining polynomial: \( x^{2} - x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.61803\) of defining polynomial
Character \(\chi\) \(=\) 3360.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{3} +1.00000 q^{5} +1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{3} +1.00000 q^{5} +1.00000 q^{7} +1.00000 q^{9} +4.47214 q^{13} +1.00000 q^{15} +4.47214 q^{17} +1.00000 q^{21} -2.47214 q^{23} +1.00000 q^{25} +1.00000 q^{27} -0.472136 q^{29} +2.47214 q^{31} +1.00000 q^{35} -0.472136 q^{37} +4.47214 q^{39} -6.94427 q^{41} +1.52786 q^{43} +1.00000 q^{45} +6.47214 q^{47} +1.00000 q^{49} +4.47214 q^{51} +2.00000 q^{53} -4.00000 q^{59} +3.52786 q^{61} +1.00000 q^{63} +4.47214 q^{65} -6.47214 q^{67} -2.47214 q^{69} -11.4164 q^{71} -0.472136 q^{73} +1.00000 q^{75} -4.94427 q^{79} +1.00000 q^{81} +0.944272 q^{83} +4.47214 q^{85} -0.472136 q^{87} +10.9443 q^{89} +4.47214 q^{91} +2.47214 q^{93} +4.47214 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{3} + 2 q^{5} + 2 q^{7} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{3} + 2 q^{5} + 2 q^{7} + 2 q^{9} + 2 q^{15} + 2 q^{21} + 4 q^{23} + 2 q^{25} + 2 q^{27} + 8 q^{29} - 4 q^{31} + 2 q^{35} + 8 q^{37} + 4 q^{41} + 12 q^{43} + 2 q^{45} + 4 q^{47} + 2 q^{49} + 4 q^{53} - 8 q^{59} + 16 q^{61} + 2 q^{63} - 4 q^{67} + 4 q^{69} + 4 q^{71} + 8 q^{73} + 2 q^{75} + 8 q^{79} + 2 q^{81} - 16 q^{83} + 8 q^{87} + 4 q^{89} - 4 q^{93}+O(q^{100}) \) Copy content Toggle raw display

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
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) 1.00000 0.377964
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) 0 0
\(13\) 4.47214 1.24035 0.620174 0.784465i \(-0.287062\pi\)
0.620174 + 0.784465i \(0.287062\pi\)
\(14\) 0 0
\(15\) 1.00000 0.258199
\(16\) 0 0
\(17\) 4.47214 1.08465 0.542326 0.840168i \(-0.317544\pi\)
0.542326 + 0.840168i \(0.317544\pi\)
\(18\) 0 0
\(19\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(20\) 0 0
\(21\) 1.00000 0.218218
\(22\) 0 0
\(23\) −2.47214 −0.515476 −0.257738 0.966215i \(-0.582977\pi\)
−0.257738 + 0.966215i \(0.582977\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) −0.472136 −0.0876734 −0.0438367 0.999039i \(-0.513958\pi\)
−0.0438367 + 0.999039i \(0.513958\pi\)
\(30\) 0 0
\(31\) 2.47214 0.444009 0.222004 0.975046i \(-0.428740\pi\)
0.222004 + 0.975046i \(0.428740\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 1.00000 0.169031
\(36\) 0 0
\(37\) −0.472136 −0.0776187 −0.0388093 0.999247i \(-0.512356\pi\)
−0.0388093 + 0.999247i \(0.512356\pi\)
\(38\) 0 0
\(39\) 4.47214 0.716115
\(40\) 0 0
\(41\) −6.94427 −1.08451 −0.542257 0.840213i \(-0.682430\pi\)
−0.542257 + 0.840213i \(0.682430\pi\)
\(42\) 0 0
\(43\) 1.52786 0.232997 0.116499 0.993191i \(-0.462833\pi\)
0.116499 + 0.993191i \(0.462833\pi\)
\(44\) 0 0
\(45\) 1.00000 0.149071
\(46\) 0 0
\(47\) 6.47214 0.944058 0.472029 0.881583i \(-0.343522\pi\)
0.472029 + 0.881583i \(0.343522\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 4.47214 0.626224
\(52\) 0 0
\(53\) 2.00000 0.274721 0.137361 0.990521i \(-0.456138\pi\)
0.137361 + 0.990521i \(0.456138\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −4.00000 −0.520756 −0.260378 0.965507i \(-0.583847\pi\)
−0.260378 + 0.965507i \(0.583847\pi\)
\(60\) 0 0
\(61\) 3.52786 0.451697 0.225848 0.974162i \(-0.427485\pi\)
0.225848 + 0.974162i \(0.427485\pi\)
\(62\) 0 0
\(63\) 1.00000 0.125988
\(64\) 0 0
\(65\) 4.47214 0.554700
\(66\) 0 0
\(67\) −6.47214 −0.790697 −0.395349 0.918531i \(-0.629376\pi\)
−0.395349 + 0.918531i \(0.629376\pi\)
\(68\) 0 0
\(69\) −2.47214 −0.297610
\(70\) 0 0
\(71\) −11.4164 −1.35488 −0.677439 0.735579i \(-0.736910\pi\)
−0.677439 + 0.735579i \(0.736910\pi\)
\(72\) 0 0
\(73\) −0.472136 −0.0552593 −0.0276297 0.999618i \(-0.508796\pi\)
−0.0276297 + 0.999618i \(0.508796\pi\)
\(74\) 0 0
\(75\) 1.00000 0.115470
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −4.94427 −0.556274 −0.278137 0.960541i \(-0.589717\pi\)
−0.278137 + 0.960541i \(0.589717\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 0.944272 0.103647 0.0518237 0.998656i \(-0.483497\pi\)
0.0518237 + 0.998656i \(0.483497\pi\)
\(84\) 0 0
\(85\) 4.47214 0.485071
\(86\) 0 0
\(87\) −0.472136 −0.0506183
\(88\) 0 0
\(89\) 10.9443 1.16009 0.580045 0.814584i \(-0.303035\pi\)
0.580045 + 0.814584i \(0.303035\pi\)
\(90\) 0 0
\(91\) 4.47214 0.468807
\(92\) 0 0
\(93\) 2.47214 0.256349
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 4.47214 0.454077 0.227038 0.973886i \(-0.427096\pi\)
0.227038 + 0.973886i \(0.427096\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 10.9443 1.08900 0.544498 0.838762i \(-0.316720\pi\)
0.544498 + 0.838762i \(0.316720\pi\)
\(102\) 0 0
\(103\) 17.8885 1.76261 0.881305 0.472547i \(-0.156665\pi\)
0.881305 + 0.472547i \(0.156665\pi\)
\(104\) 0 0
\(105\) 1.00000 0.0975900
\(106\) 0 0
\(107\) 2.47214 0.238990 0.119495 0.992835i \(-0.461872\pi\)
0.119495 + 0.992835i \(0.461872\pi\)
\(108\) 0 0
\(109\) 14.0000 1.34096 0.670478 0.741929i \(-0.266089\pi\)
0.670478 + 0.741929i \(0.266089\pi\)
\(110\) 0 0
\(111\) −0.472136 −0.0448132
\(112\) 0 0
\(113\) −2.00000 −0.188144 −0.0940721 0.995565i \(-0.529988\pi\)
−0.0940721 + 0.995565i \(0.529988\pi\)
\(114\) 0 0
\(115\) −2.47214 −0.230528
\(116\) 0 0
\(117\) 4.47214 0.413449
\(118\) 0 0
\(119\) 4.47214 0.409960
\(120\) 0 0
\(121\) −11.0000 −1.00000
\(122\) 0 0
\(123\) −6.94427 −0.626144
\(124\) 0 0
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(128\) 0 0
\(129\) 1.52786 0.134521
\(130\) 0 0
\(131\) −16.9443 −1.48043 −0.740214 0.672371i \(-0.765276\pi\)
−0.740214 + 0.672371i \(0.765276\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 1.00000 0.0860663
\(136\) 0 0
\(137\) −6.94427 −0.593289 −0.296645 0.954988i \(-0.595868\pi\)
−0.296645 + 0.954988i \(0.595868\pi\)
\(138\) 0 0
\(139\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(140\) 0 0
\(141\) 6.47214 0.545052
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) −0.472136 −0.0392088
\(146\) 0 0
\(147\) 1.00000 0.0824786
\(148\) 0 0
\(149\) 12.4721 1.02176 0.510879 0.859653i \(-0.329320\pi\)
0.510879 + 0.859653i \(0.329320\pi\)
\(150\) 0 0
\(151\) −3.05573 −0.248672 −0.124336 0.992240i \(-0.539680\pi\)
−0.124336 + 0.992240i \(0.539680\pi\)
\(152\) 0 0
\(153\) 4.47214 0.361551
\(154\) 0 0
\(155\) 2.47214 0.198567
\(156\) 0 0
\(157\) 7.52786 0.600789 0.300394 0.953815i \(-0.402882\pi\)
0.300394 + 0.953815i \(0.402882\pi\)
\(158\) 0 0
\(159\) 2.00000 0.158610
\(160\) 0 0
\(161\) −2.47214 −0.194832
\(162\) 0 0
\(163\) 6.47214 0.506937 0.253468 0.967344i \(-0.418429\pi\)
0.253468 + 0.967344i \(0.418429\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −1.52786 −0.118230 −0.0591148 0.998251i \(-0.518828\pi\)
−0.0591148 + 0.998251i \(0.518828\pi\)
\(168\) 0 0
\(169\) 7.00000 0.538462
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 6.00000 0.456172 0.228086 0.973641i \(-0.426753\pi\)
0.228086 + 0.973641i \(0.426753\pi\)
\(174\) 0 0
\(175\) 1.00000 0.0755929
\(176\) 0 0
\(177\) −4.00000 −0.300658
\(178\) 0 0
\(179\) −16.0000 −1.19590 −0.597948 0.801535i \(-0.704017\pi\)
−0.597948 + 0.801535i \(0.704017\pi\)
\(180\) 0 0
\(181\) 13.4164 0.997234 0.498617 0.866822i \(-0.333841\pi\)
0.498617 + 0.866822i \(0.333841\pi\)
\(182\) 0 0
\(183\) 3.52786 0.260787
\(184\) 0 0
\(185\) −0.472136 −0.0347121
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 1.00000 0.0727393
\(190\) 0 0
\(191\) 3.41641 0.247203 0.123601 0.992332i \(-0.460556\pi\)
0.123601 + 0.992332i \(0.460556\pi\)
\(192\) 0 0
\(193\) −1.05573 −0.0759930 −0.0379965 0.999278i \(-0.512098\pi\)
−0.0379965 + 0.999278i \(0.512098\pi\)
\(194\) 0 0
\(195\) 4.47214 0.320256
\(196\) 0 0
\(197\) −2.94427 −0.209771 −0.104885 0.994484i \(-0.533448\pi\)
−0.104885 + 0.994484i \(0.533448\pi\)
\(198\) 0 0
\(199\) −15.4164 −1.09284 −0.546420 0.837511i \(-0.684010\pi\)
−0.546420 + 0.837511i \(0.684010\pi\)
\(200\) 0 0
\(201\) −6.47214 −0.456509
\(202\) 0 0
\(203\) −0.472136 −0.0331374
\(204\) 0 0
\(205\) −6.94427 −0.485009
\(206\) 0 0
\(207\) −2.47214 −0.171825
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 8.00000 0.550743 0.275371 0.961338i \(-0.411199\pi\)
0.275371 + 0.961338i \(0.411199\pi\)
\(212\) 0 0
\(213\) −11.4164 −0.782239
\(214\) 0 0
\(215\) 1.52786 0.104199
\(216\) 0 0
\(217\) 2.47214 0.167820
\(218\) 0 0
\(219\) −0.472136 −0.0319040
\(220\) 0 0
\(221\) 20.0000 1.34535
\(222\) 0 0
\(223\) −9.88854 −0.662186 −0.331093 0.943598i \(-0.607417\pi\)
−0.331093 + 0.943598i \(0.607417\pi\)
\(224\) 0 0
\(225\) 1.00000 0.0666667
\(226\) 0 0
\(227\) −16.9443 −1.12463 −0.562315 0.826923i \(-0.690089\pi\)
−0.562315 + 0.826923i \(0.690089\pi\)
\(228\) 0 0
\(229\) 11.5279 0.761783 0.380891 0.924620i \(-0.375617\pi\)
0.380891 + 0.924620i \(0.375617\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −14.9443 −0.979032 −0.489516 0.871994i \(-0.662826\pi\)
−0.489516 + 0.871994i \(0.662826\pi\)
\(234\) 0 0
\(235\) 6.47214 0.422196
\(236\) 0 0
\(237\) −4.94427 −0.321165
\(238\) 0 0
\(239\) 24.3607 1.57576 0.787881 0.615828i \(-0.211178\pi\)
0.787881 + 0.615828i \(0.211178\pi\)
\(240\) 0 0
\(241\) −6.00000 −0.386494 −0.193247 0.981150i \(-0.561902\pi\)
−0.193247 + 0.981150i \(0.561902\pi\)
\(242\) 0 0
\(243\) 1.00000 0.0641500
\(244\) 0 0
\(245\) 1.00000 0.0638877
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 0.944272 0.0598408
\(250\) 0 0
\(251\) −24.9443 −1.57447 −0.787234 0.616654i \(-0.788488\pi\)
−0.787234 + 0.616654i \(0.788488\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 4.47214 0.280056
\(256\) 0 0
\(257\) −0.472136 −0.0294510 −0.0147255 0.999892i \(-0.504687\pi\)
−0.0147255 + 0.999892i \(0.504687\pi\)
\(258\) 0 0
\(259\) −0.472136 −0.0293371
\(260\) 0 0
\(261\) −0.472136 −0.0292245
\(262\) 0 0
\(263\) −18.4721 −1.13904 −0.569520 0.821977i \(-0.692871\pi\)
−0.569520 + 0.821977i \(0.692871\pi\)
\(264\) 0 0
\(265\) 2.00000 0.122859
\(266\) 0 0
\(267\) 10.9443 0.669779
\(268\) 0 0
\(269\) 22.0000 1.34136 0.670682 0.741745i \(-0.266002\pi\)
0.670682 + 0.741745i \(0.266002\pi\)
\(270\) 0 0
\(271\) −20.3607 −1.23682 −0.618412 0.785854i \(-0.712224\pi\)
−0.618412 + 0.785854i \(0.712224\pi\)
\(272\) 0 0
\(273\) 4.47214 0.270666
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 4.47214 0.268705 0.134352 0.990934i \(-0.457105\pi\)
0.134352 + 0.990934i \(0.457105\pi\)
\(278\) 0 0
\(279\) 2.47214 0.148003
\(280\) 0 0
\(281\) 3.88854 0.231971 0.115986 0.993251i \(-0.462997\pi\)
0.115986 + 0.993251i \(0.462997\pi\)
\(282\) 0 0
\(283\) 29.8885 1.77669 0.888345 0.459177i \(-0.151856\pi\)
0.888345 + 0.459177i \(0.151856\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −6.94427 −0.409907
\(288\) 0 0
\(289\) 3.00000 0.176471
\(290\) 0 0
\(291\) 4.47214 0.262161
\(292\) 0 0
\(293\) −22.9443 −1.34042 −0.670209 0.742172i \(-0.733796\pi\)
−0.670209 + 0.742172i \(0.733796\pi\)
\(294\) 0 0
\(295\) −4.00000 −0.232889
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −11.0557 −0.639369
\(300\) 0 0
\(301\) 1.52786 0.0880646
\(302\) 0 0
\(303\) 10.9443 0.628732
\(304\) 0 0
\(305\) 3.52786 0.202005
\(306\) 0 0
\(307\) −7.05573 −0.402692 −0.201346 0.979520i \(-0.564532\pi\)
−0.201346 + 0.979520i \(0.564532\pi\)
\(308\) 0 0
\(309\) 17.8885 1.01764
\(310\) 0 0
\(311\) 1.88854 0.107089 0.0535447 0.998565i \(-0.482948\pi\)
0.0535447 + 0.998565i \(0.482948\pi\)
\(312\) 0 0
\(313\) −8.47214 −0.478873 −0.239437 0.970912i \(-0.576963\pi\)
−0.239437 + 0.970912i \(0.576963\pi\)
\(314\) 0 0
\(315\) 1.00000 0.0563436
\(316\) 0 0
\(317\) 19.8885 1.11705 0.558526 0.829487i \(-0.311367\pi\)
0.558526 + 0.829487i \(0.311367\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 2.47214 0.137981
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) 4.47214 0.248069
\(326\) 0 0
\(327\) 14.0000 0.774202
\(328\) 0 0
\(329\) 6.47214 0.356820
\(330\) 0 0
\(331\) 32.0000 1.75888 0.879440 0.476011i \(-0.157918\pi\)
0.879440 + 0.476011i \(0.157918\pi\)
\(332\) 0 0
\(333\) −0.472136 −0.0258729
\(334\) 0 0
\(335\) −6.47214 −0.353611
\(336\) 0 0
\(337\) −10.9443 −0.596172 −0.298086 0.954539i \(-0.596348\pi\)
−0.298086 + 0.954539i \(0.596348\pi\)
\(338\) 0 0
\(339\) −2.00000 −0.108625
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) 0 0
\(345\) −2.47214 −0.133095
\(346\) 0 0
\(347\) −18.4721 −0.991636 −0.495818 0.868426i \(-0.665132\pi\)
−0.495818 + 0.868426i \(0.665132\pi\)
\(348\) 0 0
\(349\) 24.4721 1.30996 0.654982 0.755645i \(-0.272676\pi\)
0.654982 + 0.755645i \(0.272676\pi\)
\(350\) 0 0
\(351\) 4.47214 0.238705
\(352\) 0 0
\(353\) 9.41641 0.501185 0.250592 0.968093i \(-0.419375\pi\)
0.250592 + 0.968093i \(0.419375\pi\)
\(354\) 0 0
\(355\) −11.4164 −0.605920
\(356\) 0 0
\(357\) 4.47214 0.236691
\(358\) 0 0
\(359\) −11.4164 −0.602535 −0.301267 0.953540i \(-0.597410\pi\)
−0.301267 + 0.953540i \(0.597410\pi\)
\(360\) 0 0
\(361\) −19.0000 −1.00000
\(362\) 0 0
\(363\) −11.0000 −0.577350
\(364\) 0 0
\(365\) −0.472136 −0.0247127
\(366\) 0 0
\(367\) −3.05573 −0.159508 −0.0797539 0.996815i \(-0.525413\pi\)
−0.0797539 + 0.996815i \(0.525413\pi\)
\(368\) 0 0
\(369\) −6.94427 −0.361504
\(370\) 0 0
\(371\) 2.00000 0.103835
\(372\) 0 0
\(373\) −29.4164 −1.52312 −0.761562 0.648092i \(-0.775567\pi\)
−0.761562 + 0.648092i \(0.775567\pi\)
\(374\) 0 0
\(375\) 1.00000 0.0516398
\(376\) 0 0
\(377\) −2.11146 −0.108746
\(378\) 0 0
\(379\) 36.9443 1.89770 0.948850 0.315728i \(-0.102249\pi\)
0.948850 + 0.315728i \(0.102249\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −3.41641 −0.174570 −0.0872851 0.996183i \(-0.527819\pi\)
−0.0872851 + 0.996183i \(0.527819\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 1.52786 0.0776657
\(388\) 0 0
\(389\) 15.5279 0.787294 0.393647 0.919262i \(-0.371213\pi\)
0.393647 + 0.919262i \(0.371213\pi\)
\(390\) 0 0
\(391\) −11.0557 −0.559112
\(392\) 0 0
\(393\) −16.9443 −0.854725
\(394\) 0 0
\(395\) −4.94427 −0.248773
\(396\) 0 0
\(397\) 25.4164 1.27561 0.637806 0.770197i \(-0.279842\pi\)
0.637806 + 0.770197i \(0.279842\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −9.05573 −0.452221 −0.226111 0.974102i \(-0.572601\pi\)
−0.226111 + 0.974102i \(0.572601\pi\)
\(402\) 0 0
\(403\) 11.0557 0.550725
\(404\) 0 0
\(405\) 1.00000 0.0496904
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 22.9443 1.13452 0.567261 0.823538i \(-0.308003\pi\)
0.567261 + 0.823538i \(0.308003\pi\)
\(410\) 0 0
\(411\) −6.94427 −0.342536
\(412\) 0 0
\(413\) −4.00000 −0.196827
\(414\) 0 0
\(415\) 0.944272 0.0463525
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −32.9443 −1.60943 −0.804717 0.593659i \(-0.797683\pi\)
−0.804717 + 0.593659i \(0.797683\pi\)
\(420\) 0 0
\(421\) −11.8885 −0.579412 −0.289706 0.957116i \(-0.593558\pi\)
−0.289706 + 0.957116i \(0.593558\pi\)
\(422\) 0 0
\(423\) 6.47214 0.314686
\(424\) 0 0
\(425\) 4.47214 0.216930
\(426\) 0 0
\(427\) 3.52786 0.170725
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −25.5279 −1.22963 −0.614817 0.788670i \(-0.710770\pi\)
−0.614817 + 0.788670i \(0.710770\pi\)
\(432\) 0 0
\(433\) −26.3607 −1.26681 −0.633407 0.773819i \(-0.718344\pi\)
−0.633407 + 0.773819i \(0.718344\pi\)
\(434\) 0 0
\(435\) −0.472136 −0.0226372
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) 21.5279 1.02747 0.513734 0.857949i \(-0.328262\pi\)
0.513734 + 0.857949i \(0.328262\pi\)
\(440\) 0 0
\(441\) 1.00000 0.0476190
\(442\) 0 0
\(443\) −10.4721 −0.497546 −0.248773 0.968562i \(-0.580027\pi\)
−0.248773 + 0.968562i \(0.580027\pi\)
\(444\) 0 0
\(445\) 10.9443 0.518808
\(446\) 0 0
\(447\) 12.4721 0.589912
\(448\) 0 0
\(449\) −20.8328 −0.983161 −0.491581 0.870832i \(-0.663581\pi\)
−0.491581 + 0.870832i \(0.663581\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) −3.05573 −0.143571
\(454\) 0 0
\(455\) 4.47214 0.209657
\(456\) 0 0
\(457\) −17.0557 −0.797833 −0.398917 0.916987i \(-0.630614\pi\)
−0.398917 + 0.916987i \(0.630614\pi\)
\(458\) 0 0
\(459\) 4.47214 0.208741
\(460\) 0 0
\(461\) −24.8328 −1.15658 −0.578290 0.815831i \(-0.696280\pi\)
−0.578290 + 0.815831i \(0.696280\pi\)
\(462\) 0 0
\(463\) 20.9443 0.973363 0.486681 0.873580i \(-0.338207\pi\)
0.486681 + 0.873580i \(0.338207\pi\)
\(464\) 0 0
\(465\) 2.47214 0.114643
\(466\) 0 0
\(467\) 5.88854 0.272489 0.136245 0.990675i \(-0.456497\pi\)
0.136245 + 0.990675i \(0.456497\pi\)
\(468\) 0 0
\(469\) −6.47214 −0.298855
\(470\) 0 0
\(471\) 7.52786 0.346866
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 2.00000 0.0915737
\(478\) 0 0
\(479\) −25.8885 −1.18288 −0.591439 0.806350i \(-0.701440\pi\)
−0.591439 + 0.806350i \(0.701440\pi\)
\(480\) 0 0
\(481\) −2.11146 −0.0962741
\(482\) 0 0
\(483\) −2.47214 −0.112486
\(484\) 0 0
\(485\) 4.47214 0.203069
\(486\) 0 0
\(487\) −36.9443 −1.67410 −0.837052 0.547123i \(-0.815723\pi\)
−0.837052 + 0.547123i \(0.815723\pi\)
\(488\) 0 0
\(489\) 6.47214 0.292680
\(490\) 0 0
\(491\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(492\) 0 0
\(493\) −2.11146 −0.0950952
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −11.4164 −0.512096
\(498\) 0 0
\(499\) 28.9443 1.29572 0.647862 0.761758i \(-0.275663\pi\)
0.647862 + 0.761758i \(0.275663\pi\)
\(500\) 0 0
\(501\) −1.52786 −0.0682599
\(502\) 0 0
\(503\) −30.4721 −1.35869 −0.679343 0.733821i \(-0.737735\pi\)
−0.679343 + 0.733821i \(0.737735\pi\)
\(504\) 0 0
\(505\) 10.9443 0.487014
\(506\) 0 0
\(507\) 7.00000 0.310881
\(508\) 0 0
\(509\) 22.0000 0.975133 0.487566 0.873086i \(-0.337885\pi\)
0.487566 + 0.873086i \(0.337885\pi\)
\(510\) 0 0
\(511\) −0.472136 −0.0208861
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 17.8885 0.788263
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 6.00000 0.263371
\(520\) 0 0
\(521\) −34.0000 −1.48957 −0.744784 0.667306i \(-0.767447\pi\)
−0.744784 + 0.667306i \(0.767447\pi\)
\(522\) 0 0
\(523\) 7.05573 0.308525 0.154263 0.988030i \(-0.450700\pi\)
0.154263 + 0.988030i \(0.450700\pi\)
\(524\) 0 0
\(525\) 1.00000 0.0436436
\(526\) 0 0
\(527\) 11.0557 0.481595
\(528\) 0 0
\(529\) −16.8885 −0.734285
\(530\) 0 0
\(531\) −4.00000 −0.173585
\(532\) 0 0
\(533\) −31.0557 −1.34517
\(534\) 0 0
\(535\) 2.47214 0.106880
\(536\) 0 0
\(537\) −16.0000 −0.690451
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 2.94427 0.126584 0.0632921 0.997995i \(-0.479840\pi\)
0.0632921 + 0.997995i \(0.479840\pi\)
\(542\) 0 0
\(543\) 13.4164 0.575753
\(544\) 0 0
\(545\) 14.0000 0.599694
\(546\) 0 0
\(547\) 11.4164 0.488130 0.244065 0.969759i \(-0.421519\pi\)
0.244065 + 0.969759i \(0.421519\pi\)
\(548\) 0 0
\(549\) 3.52786 0.150566
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) −4.94427 −0.210252
\(554\) 0 0
\(555\) −0.472136 −0.0200411
\(556\) 0 0
\(557\) 10.0000 0.423714 0.211857 0.977301i \(-0.432049\pi\)
0.211857 + 0.977301i \(0.432049\pi\)
\(558\) 0 0
\(559\) 6.83282 0.288997
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 12.0000 0.505740 0.252870 0.967500i \(-0.418626\pi\)
0.252870 + 0.967500i \(0.418626\pi\)
\(564\) 0 0
\(565\) −2.00000 −0.0841406
\(566\) 0 0
\(567\) 1.00000 0.0419961
\(568\) 0 0
\(569\) −20.8328 −0.873357 −0.436679 0.899618i \(-0.643845\pi\)
−0.436679 + 0.899618i \(0.643845\pi\)
\(570\) 0 0
\(571\) 12.9443 0.541701 0.270850 0.962621i \(-0.412695\pi\)
0.270850 + 0.962621i \(0.412695\pi\)
\(572\) 0 0
\(573\) 3.41641 0.142722
\(574\) 0 0
\(575\) −2.47214 −0.103095
\(576\) 0 0
\(577\) −2.36068 −0.0982764 −0.0491382 0.998792i \(-0.515647\pi\)
−0.0491382 + 0.998792i \(0.515647\pi\)
\(578\) 0 0
\(579\) −1.05573 −0.0438746
\(580\) 0 0
\(581\) 0.944272 0.0391750
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 4.47214 0.184900
\(586\) 0 0
\(587\) −29.8885 −1.23363 −0.616816 0.787107i \(-0.711578\pi\)
−0.616816 + 0.787107i \(0.711578\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) −2.94427 −0.121111
\(592\) 0 0
\(593\) −3.52786 −0.144872 −0.0724360 0.997373i \(-0.523077\pi\)
−0.0724360 + 0.997373i \(0.523077\pi\)
\(594\) 0 0
\(595\) 4.47214 0.183340
\(596\) 0 0
\(597\) −15.4164 −0.630952
\(598\) 0 0
\(599\) −6.47214 −0.264444 −0.132222 0.991220i \(-0.542211\pi\)
−0.132222 + 0.991220i \(0.542211\pi\)
\(600\) 0 0
\(601\) −26.9443 −1.09908 −0.549540 0.835467i \(-0.685197\pi\)
−0.549540 + 0.835467i \(0.685197\pi\)
\(602\) 0 0
\(603\) −6.47214 −0.263566
\(604\) 0 0
\(605\) −11.0000 −0.447214
\(606\) 0 0
\(607\) 25.8885 1.05078 0.525392 0.850860i \(-0.323919\pi\)
0.525392 + 0.850860i \(0.323919\pi\)
\(608\) 0 0
\(609\) −0.472136 −0.0191319
\(610\) 0 0
\(611\) 28.9443 1.17096
\(612\) 0 0
\(613\) −32.4721 −1.31154 −0.655769 0.754962i \(-0.727655\pi\)
−0.655769 + 0.754962i \(0.727655\pi\)
\(614\) 0 0
\(615\) −6.94427 −0.280020
\(616\) 0 0
\(617\) −21.0557 −0.847672 −0.423836 0.905739i \(-0.639317\pi\)
−0.423836 + 0.905739i \(0.639317\pi\)
\(618\) 0 0
\(619\) 27.0557 1.08746 0.543731 0.839260i \(-0.317011\pi\)
0.543731 + 0.839260i \(0.317011\pi\)
\(620\) 0 0
\(621\) −2.47214 −0.0992034
\(622\) 0 0
\(623\) 10.9443 0.438473
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −2.11146 −0.0841893
\(630\) 0 0
\(631\) −1.88854 −0.0751817 −0.0375909 0.999293i \(-0.511968\pi\)
−0.0375909 + 0.999293i \(0.511968\pi\)
\(632\) 0 0
\(633\) 8.00000 0.317971
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 4.47214 0.177192
\(638\) 0 0
\(639\) −11.4164 −0.451626
\(640\) 0 0
\(641\) −10.9443 −0.432273 −0.216136 0.976363i \(-0.569346\pi\)
−0.216136 + 0.976363i \(0.569346\pi\)
\(642\) 0 0
\(643\) −12.0000 −0.473234 −0.236617 0.971603i \(-0.576039\pi\)
−0.236617 + 0.971603i \(0.576039\pi\)
\(644\) 0 0
\(645\) 1.52786 0.0601596
\(646\) 0 0
\(647\) −4.58359 −0.180200 −0.0900998 0.995933i \(-0.528719\pi\)
−0.0900998 + 0.995933i \(0.528719\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 2.47214 0.0968906
\(652\) 0 0
\(653\) −4.11146 −0.160894 −0.0804469 0.996759i \(-0.525635\pi\)
−0.0804469 + 0.996759i \(0.525635\pi\)
\(654\) 0 0
\(655\) −16.9443 −0.662067
\(656\) 0 0
\(657\) −0.472136 −0.0184198
\(658\) 0 0
\(659\) −30.8328 −1.20108 −0.600538 0.799596i \(-0.705047\pi\)
−0.600538 + 0.799596i \(0.705047\pi\)
\(660\) 0 0
\(661\) −46.3607 −1.80322 −0.901611 0.432548i \(-0.857614\pi\)
−0.901611 + 0.432548i \(0.857614\pi\)
\(662\) 0 0
\(663\) 20.0000 0.776736
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 1.16718 0.0451936
\(668\) 0 0
\(669\) −9.88854 −0.382313
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 45.7771 1.76458 0.882289 0.470709i \(-0.156002\pi\)
0.882289 + 0.470709i \(0.156002\pi\)
\(674\) 0 0
\(675\) 1.00000 0.0384900
\(676\) 0 0
\(677\) −11.8885 −0.456914 −0.228457 0.973554i \(-0.573368\pi\)
−0.228457 + 0.973554i \(0.573368\pi\)
\(678\) 0 0
\(679\) 4.47214 0.171625
\(680\) 0 0
\(681\) −16.9443 −0.649306
\(682\) 0 0
\(683\) 12.3607 0.472968 0.236484 0.971635i \(-0.424005\pi\)
0.236484 + 0.971635i \(0.424005\pi\)
\(684\) 0 0
\(685\) −6.94427 −0.265327
\(686\) 0 0
\(687\) 11.5279 0.439815
\(688\) 0 0
\(689\) 8.94427 0.340750
\(690\) 0 0
\(691\) 25.8885 0.984847 0.492423 0.870356i \(-0.336111\pi\)
0.492423 + 0.870356i \(0.336111\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) −31.0557 −1.17632
\(698\) 0 0
\(699\) −14.9443 −0.565244
\(700\) 0 0
\(701\) 9.41641 0.355653 0.177826 0.984062i \(-0.443093\pi\)
0.177826 + 0.984062i \(0.443093\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) 6.47214 0.243755
\(706\) 0 0
\(707\) 10.9443 0.411602
\(708\) 0 0
\(709\) 12.8328 0.481947 0.240973 0.970532i \(-0.422533\pi\)
0.240973 + 0.970532i \(0.422533\pi\)
\(710\) 0 0
\(711\) −4.94427 −0.185425
\(712\) 0 0
\(713\) −6.11146 −0.228876
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 24.3607 0.909766
\(718\) 0 0
\(719\) 16.0000 0.596699 0.298350 0.954457i \(-0.403564\pi\)
0.298350 + 0.954457i \(0.403564\pi\)
\(720\) 0 0
\(721\) 17.8885 0.666204
\(722\) 0 0
\(723\) −6.00000 −0.223142
\(724\) 0 0
\(725\) −0.472136 −0.0175347
\(726\) 0 0
\(727\) −28.9443 −1.07348 −0.536742 0.843747i \(-0.680345\pi\)
−0.536742 + 0.843747i \(0.680345\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 6.83282 0.252721
\(732\) 0 0
\(733\) −29.4164 −1.08652 −0.543260 0.839565i \(-0.682810\pi\)
−0.543260 + 0.839565i \(0.682810\pi\)
\(734\) 0 0
\(735\) 1.00000 0.0368856
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 16.0000 0.588570 0.294285 0.955718i \(-0.404919\pi\)
0.294285 + 0.955718i \(0.404919\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 7.41641 0.272082 0.136041 0.990703i \(-0.456562\pi\)
0.136041 + 0.990703i \(0.456562\pi\)
\(744\) 0 0
\(745\) 12.4721 0.456944
\(746\) 0 0
\(747\) 0.944272 0.0345491
\(748\) 0 0
\(749\) 2.47214 0.0903299
\(750\) 0 0
\(751\) 11.0557 0.403429 0.201715 0.979444i \(-0.435349\pi\)
0.201715 + 0.979444i \(0.435349\pi\)
\(752\) 0 0
\(753\) −24.9443 −0.909020
\(754\) 0 0
\(755\) −3.05573 −0.111209
\(756\) 0 0
\(757\) −11.5279 −0.418987 −0.209494 0.977810i \(-0.567182\pi\)
−0.209494 + 0.977810i \(0.567182\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −35.8885 −1.30096 −0.650479 0.759524i \(-0.725432\pi\)
−0.650479 + 0.759524i \(0.725432\pi\)
\(762\) 0 0
\(763\) 14.0000 0.506834
\(764\) 0 0
\(765\) 4.47214 0.161690
\(766\) 0 0
\(767\) −17.8885 −0.645918
\(768\) 0 0
\(769\) 18.0000 0.649097 0.324548 0.945869i \(-0.394788\pi\)
0.324548 + 0.945869i \(0.394788\pi\)
\(770\) 0 0
\(771\) −0.472136 −0.0170036
\(772\) 0 0
\(773\) −38.9443 −1.40073 −0.700364 0.713786i \(-0.746979\pi\)
−0.700364 + 0.713786i \(0.746979\pi\)
\(774\) 0 0
\(775\) 2.47214 0.0888017
\(776\) 0 0
\(777\) −0.472136 −0.0169378
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) −0.472136 −0.0168728
\(784\) 0 0
\(785\) 7.52786 0.268681
\(786\) 0 0
\(787\) −47.7771 −1.70307 −0.851535 0.524298i \(-0.824328\pi\)
−0.851535 + 0.524298i \(0.824328\pi\)
\(788\) 0 0
\(789\) −18.4721 −0.657625
\(790\) 0 0
\(791\) −2.00000 −0.0711118
\(792\) 0 0
\(793\) 15.7771 0.560261
\(794\) 0 0
\(795\) 2.00000 0.0709327
\(796\) 0 0
\(797\) −42.0000 −1.48772 −0.743858 0.668338i \(-0.767006\pi\)
−0.743858 + 0.668338i \(0.767006\pi\)
\(798\) 0 0
\(799\) 28.9443 1.02397
\(800\) 0 0
\(801\) 10.9443 0.386697
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) −2.47214 −0.0871313
\(806\) 0 0
\(807\) 22.0000 0.774437
\(808\) 0 0
\(809\) 40.8328 1.43561 0.717803 0.696247i \(-0.245148\pi\)
0.717803 + 0.696247i \(0.245148\pi\)
\(810\) 0 0
\(811\) −43.7771 −1.53722 −0.768611 0.639717i \(-0.779052\pi\)
−0.768611 + 0.639717i \(0.779052\pi\)
\(812\) 0 0
\(813\) −20.3607 −0.714080
\(814\) 0 0
\(815\) 6.47214 0.226709
\(816\) 0 0
\(817\) 0 0
\(818\) 0 0
\(819\) 4.47214 0.156269
\(820\) 0 0
\(821\) −31.3050 −1.09255 −0.546275 0.837606i \(-0.683955\pi\)
−0.546275 + 0.837606i \(0.683955\pi\)
\(822\) 0 0
\(823\) −20.9443 −0.730071 −0.365036 0.930994i \(-0.618943\pi\)
−0.365036 + 0.930994i \(0.618943\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 15.4164 0.536081 0.268041 0.963408i \(-0.413624\pi\)
0.268041 + 0.963408i \(0.413624\pi\)
\(828\) 0 0
\(829\) 6.58359 0.228658 0.114329 0.993443i \(-0.463528\pi\)
0.114329 + 0.993443i \(0.463528\pi\)
\(830\) 0 0
\(831\) 4.47214 0.155137
\(832\) 0 0
\(833\) 4.47214 0.154950
\(834\) 0 0
\(835\) −1.52786 −0.0528739
\(836\) 0 0
\(837\) 2.47214 0.0854495
\(838\) 0 0
\(839\) −14.1115 −0.487182 −0.243591 0.969878i \(-0.578325\pi\)
−0.243591 + 0.969878i \(0.578325\pi\)
\(840\) 0 0
\(841\) −28.7771 −0.992313
\(842\) 0 0
\(843\) 3.88854 0.133929
\(844\) 0 0
\(845\) 7.00000 0.240807
\(846\) 0 0
\(847\) −11.0000 −0.377964
\(848\) 0 0
\(849\) 29.8885 1.02577
\(850\) 0 0
\(851\) 1.16718 0.0400106
\(852\) 0 0
\(853\) −40.4721 −1.38574 −0.692870 0.721063i \(-0.743654\pi\)
−0.692870 + 0.721063i \(0.743654\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −26.3607 −0.900464 −0.450232 0.892912i \(-0.648659\pi\)
−0.450232 + 0.892912i \(0.648659\pi\)
\(858\) 0 0
\(859\) 12.9443 0.441653 0.220826 0.975313i \(-0.429125\pi\)
0.220826 + 0.975313i \(0.429125\pi\)
\(860\) 0 0
\(861\) −6.94427 −0.236660
\(862\) 0 0
\(863\) −3.63932 −0.123884 −0.0619420 0.998080i \(-0.519729\pi\)
−0.0619420 + 0.998080i \(0.519729\pi\)
\(864\) 0 0
\(865\) 6.00000 0.204006
\(866\) 0 0
\(867\) 3.00000 0.101885
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) −28.9443 −0.980739
\(872\) 0 0
\(873\) 4.47214 0.151359
\(874\) 0 0
\(875\) 1.00000 0.0338062
\(876\) 0 0
\(877\) −15.3050 −0.516811 −0.258406 0.966036i \(-0.583197\pi\)
−0.258406 + 0.966036i \(0.583197\pi\)
\(878\) 0 0
\(879\) −22.9443 −0.773891
\(880\) 0 0
\(881\) 20.8328 0.701875 0.350938 0.936399i \(-0.385863\pi\)
0.350938 + 0.936399i \(0.385863\pi\)
\(882\) 0 0
\(883\) −37.3050 −1.25541 −0.627706 0.778451i \(-0.716006\pi\)
−0.627706 + 0.778451i \(0.716006\pi\)
\(884\) 0 0
\(885\) −4.00000 −0.134459
\(886\) 0 0
\(887\) 51.4164 1.72639 0.863197 0.504867i \(-0.168459\pi\)
0.863197 + 0.504867i \(0.168459\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) −16.0000 −0.534821
\(896\) 0 0
\(897\) −11.0557 −0.369140
\(898\) 0 0
\(899\) −1.16718 −0.0389278
\(900\) 0 0
\(901\) 8.94427 0.297977
\(902\) 0 0
\(903\) 1.52786 0.0508441
\(904\) 0 0
\(905\) 13.4164 0.445976
\(906\) 0 0
\(907\) 8.36068 0.277612 0.138806 0.990320i \(-0.455674\pi\)
0.138806 + 0.990320i \(0.455674\pi\)
\(908\) 0 0
\(909\) 10.9443 0.362999
\(910\) 0 0
\(911\) 27.4164 0.908346 0.454173 0.890913i \(-0.349935\pi\)
0.454173 + 0.890913i \(0.349935\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) 3.52786 0.116628
\(916\) 0 0
\(917\) −16.9443 −0.559549
\(918\) 0 0
\(919\) 32.7214 1.07938 0.539689 0.841864i \(-0.318542\pi\)
0.539689 + 0.841864i \(0.318542\pi\)
\(920\) 0 0
\(921\) −7.05573 −0.232494
\(922\) 0 0
\(923\) −51.0557 −1.68052
\(924\) 0 0
\(925\) −0.472136 −0.0155237
\(926\) 0 0
\(927\) 17.8885 0.587537
\(928\) 0 0
\(929\) 39.8885 1.30870 0.654350 0.756192i \(-0.272942\pi\)
0.654350 + 0.756192i \(0.272942\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 1.88854 0.0618281
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −31.3050 −1.02269 −0.511344 0.859376i \(-0.670852\pi\)
−0.511344 + 0.859376i \(0.670852\pi\)
\(938\) 0 0
\(939\) −8.47214 −0.276478
\(940\) 0 0
\(941\) −2.00000 −0.0651981 −0.0325991 0.999469i \(-0.510378\pi\)
−0.0325991 + 0.999469i \(0.510378\pi\)
\(942\) 0 0
\(943\) 17.1672 0.559040
\(944\) 0 0
\(945\) 1.00000 0.0325300
\(946\) 0 0
\(947\) 11.6393 0.378227 0.189114 0.981955i \(-0.439439\pi\)
0.189114 + 0.981955i \(0.439439\pi\)
\(948\) 0 0
\(949\) −2.11146 −0.0685408
\(950\) 0 0
\(951\) 19.8885 0.644930
\(952\) 0 0
\(953\) −29.7771 −0.964574 −0.482287 0.876013i \(-0.660194\pi\)
−0.482287 + 0.876013i \(0.660194\pi\)
\(954\) 0 0
\(955\) 3.41641 0.110552
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −6.94427 −0.224242
\(960\) 0 0
\(961\) −24.8885 −0.802856
\(962\) 0 0
\(963\) 2.47214 0.0796635
\(964\) 0 0
\(965\) −1.05573 −0.0339851
\(966\) 0 0
\(967\) −20.9443 −0.673522 −0.336761 0.941590i \(-0.609332\pi\)
−0.336761 + 0.941590i \(0.609332\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −12.0000 −0.385098 −0.192549 0.981287i \(-0.561675\pi\)
−0.192549 + 0.981287i \(0.561675\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 4.47214 0.143223
\(976\) 0 0
\(977\) 28.8328 0.922443 0.461222 0.887285i \(-0.347411\pi\)
0.461222 + 0.887285i \(0.347411\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) 14.0000 0.446986
\(982\) 0 0
\(983\) −50.2492 −1.60270 −0.801351 0.598195i \(-0.795885\pi\)
−0.801351 + 0.598195i \(0.795885\pi\)
\(984\) 0 0
\(985\) −2.94427 −0.0938123
\(986\) 0 0
\(987\) 6.47214 0.206010
\(988\) 0 0
\(989\) −3.77709 −0.120104
\(990\) 0 0
\(991\) −19.0557 −0.605325 −0.302663 0.953098i \(-0.597876\pi\)
−0.302663 + 0.953098i \(0.597876\pi\)
\(992\) 0 0
\(993\) 32.0000 1.01549
\(994\) 0 0
\(995\) −15.4164 −0.488733
\(996\) 0 0
\(997\) 9.41641 0.298221 0.149110 0.988821i \(-0.452359\pi\)
0.149110 + 0.988821i \(0.452359\pi\)
\(998\) 0 0
\(999\) −0.472136 −0.0149377
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 3360.2.a.bh.1.2 yes 2
4.3 odd 2 3360.2.a.bd.1.2 2
8.3 odd 2 6720.2.a.cv.1.1 2
8.5 even 2 6720.2.a.cp.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3360.2.a.bd.1.2 2 4.3 odd 2
3360.2.a.bh.1.2 yes 2 1.1 even 1 trivial
6720.2.a.cp.1.1 2 8.5 even 2
6720.2.a.cv.1.1 2 8.3 odd 2