Properties

Label 7728.2.a.bd.1.1
Level $7728$
Weight $2$
Character 7728.1
Self dual yes
Analytic conductor $61.708$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 7728 = 2^{4} \cdot 3 \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7728.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(61.7083906820\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{10})^+\)
Defining polynomial: \(x^{2} - x - 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 966)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(1.61803\) of defining polynomial
Character \(\chi\) \(=\) 7728.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{3} +2.00000 q^{5} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{3} +2.00000 q^{5} -1.00000 q^{7} +1.00000 q^{9} -4.47214 q^{13} -2.00000 q^{15} +4.47214 q^{17} -2.47214 q^{19} +1.00000 q^{21} +1.00000 q^{23} -1.00000 q^{25} -1.00000 q^{27} -2.00000 q^{29} +2.47214 q^{31} -2.00000 q^{35} +6.94427 q^{37} +4.47214 q^{39} -6.00000 q^{41} +4.94427 q^{43} +2.00000 q^{45} +2.47214 q^{47} +1.00000 q^{49} -4.47214 q^{51} -10.9443 q^{53} +2.47214 q^{57} -4.00000 q^{59} -6.00000 q^{61} -1.00000 q^{63} -8.94427 q^{65} -4.94427 q^{67} -1.00000 q^{69} -4.94427 q^{71} +14.9443 q^{73} +1.00000 q^{75} +4.94427 q^{79} +1.00000 q^{81} -2.47214 q^{83} +8.94427 q^{85} +2.00000 q^{87} +9.41641 q^{89} +4.47214 q^{91} -2.47214 q^{93} -4.94427 q^{95} -0.472136 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{3} + 4 q^{5} - 2 q^{7} + 2 q^{9} + O(q^{10}) \) \( 2 q - 2 q^{3} + 4 q^{5} - 2 q^{7} + 2 q^{9} - 4 q^{15} + 4 q^{19} + 2 q^{21} + 2 q^{23} - 2 q^{25} - 2 q^{27} - 4 q^{29} - 4 q^{31} - 4 q^{35} - 4 q^{37} - 12 q^{41} - 8 q^{43} + 4 q^{45} - 4 q^{47} + 2 q^{49} - 4 q^{53} - 4 q^{57} - 8 q^{59} - 12 q^{61} - 2 q^{63} + 8 q^{67} - 2 q^{69} + 8 q^{71} + 12 q^{73} + 2 q^{75} - 8 q^{79} + 2 q^{81} + 4 q^{83} + 4 q^{87} - 8 q^{89} + 4 q^{93} + 8 q^{95} + 8 q^{97} + O(q^{100}) \)

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
\(5\) 2.00000 0.894427 0.447214 0.894427i \(-0.352416\pi\)
0.447214 + 0.894427i \(0.352416\pi\)
\(6\) 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 \(-0.712938\pi\)
−0.620174 + 0.784465i \(0.712938\pi\)
\(14\) 0 0
\(15\) −2.00000 −0.516398
\(16\) 0 0
\(17\) 4.47214 1.08465 0.542326 0.840168i \(-0.317544\pi\)
0.542326 + 0.840168i \(0.317544\pi\)
\(18\) 0 0
\(19\) −2.47214 −0.567147 −0.283573 0.958951i \(-0.591520\pi\)
−0.283573 + 0.958951i \(0.591520\pi\)
\(20\) 0 0
\(21\) 1.00000 0.218218
\(22\) 0 0
\(23\) 1.00000 0.208514
\(24\) 0 0
\(25\) −1.00000 −0.200000
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) −2.00000 −0.371391 −0.185695 0.982607i \(-0.559454\pi\)
−0.185695 + 0.982607i \(0.559454\pi\)
\(30\) 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
\(35\) −2.00000 −0.338062
\(36\) 0 0
\(37\) 6.94427 1.14163 0.570816 0.821078i \(-0.306627\pi\)
0.570816 + 0.821078i \(0.306627\pi\)
\(38\) 0 0
\(39\) 4.47214 0.716115
\(40\) 0 0
\(41\) −6.00000 −0.937043 −0.468521 0.883452i \(-0.655213\pi\)
−0.468521 + 0.883452i \(0.655213\pi\)
\(42\) 0 0
\(43\) 4.94427 0.753994 0.376997 0.926214i \(-0.376957\pi\)
0.376997 + 0.926214i \(0.376957\pi\)
\(44\) 0 0
\(45\) 2.00000 0.298142
\(46\) 0 0
\(47\) 2.47214 0.360598 0.180299 0.983612i \(-0.442293\pi\)
0.180299 + 0.983612i \(0.442293\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) −4.47214 −0.626224
\(52\) 0 0
\(53\) −10.9443 −1.50331 −0.751656 0.659556i \(-0.770744\pi\)
−0.751656 + 0.659556i \(0.770744\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 2.47214 0.327442
\(58\) 0 0
\(59\) −4.00000 −0.520756 −0.260378 0.965507i \(-0.583847\pi\)
−0.260378 + 0.965507i \(0.583847\pi\)
\(60\) 0 0
\(61\) −6.00000 −0.768221 −0.384111 0.923287i \(-0.625492\pi\)
−0.384111 + 0.923287i \(0.625492\pi\)
\(62\) 0 0
\(63\) −1.00000 −0.125988
\(64\) 0 0
\(65\) −8.94427 −1.10940
\(66\) 0 0
\(67\) −4.94427 −0.604039 −0.302019 0.953302i \(-0.597661\pi\)
−0.302019 + 0.953302i \(0.597661\pi\)
\(68\) 0 0
\(69\) −1.00000 −0.120386
\(70\) 0 0
\(71\) −4.94427 −0.586777 −0.293389 0.955993i \(-0.594783\pi\)
−0.293389 + 0.955993i \(0.594783\pi\)
\(72\) 0 0
\(73\) 14.9443 1.74909 0.874547 0.484940i \(-0.161159\pi\)
0.874547 + 0.484940i \(0.161159\pi\)
\(74\) 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 \(-0.410283\pi\)
0.278137 + 0.960541i \(0.410283\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −2.47214 −0.271352 −0.135676 0.990753i \(-0.543321\pi\)
−0.135676 + 0.990753i \(0.543321\pi\)
\(84\) 0 0
\(85\) 8.94427 0.970143
\(86\) 0 0
\(87\) 2.00000 0.214423
\(88\) 0 0
\(89\) 9.41641 0.998137 0.499069 0.866562i \(-0.333676\pi\)
0.499069 + 0.866562i \(0.333676\pi\)
\(90\) 0 0
\(91\) 4.47214 0.468807
\(92\) 0 0
\(93\) −2.47214 −0.256349
\(94\) 0 0
\(95\) −4.94427 −0.507272
\(96\) 0 0
\(97\) −0.472136 −0.0479381 −0.0239691 0.999713i \(-0.507630\pi\)
−0.0239691 + 0.999713i \(0.507630\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −9.41641 −0.936968 −0.468484 0.883472i \(-0.655200\pi\)
−0.468484 + 0.883472i \(0.655200\pi\)
\(102\) 0 0
\(103\) 4.94427 0.487174 0.243587 0.969879i \(-0.421676\pi\)
0.243587 + 0.969879i \(0.421676\pi\)
\(104\) 0 0
\(105\) 2.00000 0.195180
\(106\) 0 0
\(107\) 8.00000 0.773389 0.386695 0.922208i \(-0.373617\pi\)
0.386695 + 0.922208i \(0.373617\pi\)
\(108\) 0 0
\(109\) −2.94427 −0.282010 −0.141005 0.990009i \(-0.545033\pi\)
−0.141005 + 0.990009i \(0.545033\pi\)
\(110\) 0 0
\(111\) −6.94427 −0.659121
\(112\) 0 0
\(113\) −18.9443 −1.78213 −0.891064 0.453878i \(-0.850040\pi\)
−0.891064 + 0.453878i \(0.850040\pi\)
\(114\) 0 0
\(115\) 2.00000 0.186501
\(116\) 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
\(123\) 6.00000 0.541002
\(124\) 0 0
\(125\) −12.0000 −1.07331
\(126\) 0 0
\(127\) −20.9443 −1.85850 −0.929252 0.369447i \(-0.879547\pi\)
−0.929252 + 0.369447i \(0.879547\pi\)
\(128\) 0 0
\(129\) −4.94427 −0.435319
\(130\) 0 0
\(131\) −12.0000 −1.04844 −0.524222 0.851581i \(-0.675644\pi\)
−0.524222 + 0.851581i \(0.675644\pi\)
\(132\) 0 0
\(133\) 2.47214 0.214361
\(134\) 0 0
\(135\) −2.00000 −0.172133
\(136\) 0 0
\(137\) −14.0000 −1.19610 −0.598050 0.801459i \(-0.704058\pi\)
−0.598050 + 0.801459i \(0.704058\pi\)
\(138\) 0 0
\(139\) −8.94427 −0.758643 −0.379322 0.925265i \(-0.623843\pi\)
−0.379322 + 0.925265i \(0.623843\pi\)
\(140\) 0 0
\(141\) −2.47214 −0.208191
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) −4.00000 −0.332182
\(146\) 0 0
\(147\) −1.00000 −0.0824786
\(148\) 0 0
\(149\) −1.05573 −0.0864886 −0.0432443 0.999065i \(-0.513769\pi\)
−0.0432443 + 0.999065i \(0.513769\pi\)
\(150\) 0 0
\(151\) −4.94427 −0.402359 −0.201180 0.979554i \(-0.564477\pi\)
−0.201180 + 0.979554i \(0.564477\pi\)
\(152\) 0 0
\(153\) 4.47214 0.361551
\(154\) 0 0
\(155\) 4.94427 0.397133
\(156\) 0 0
\(157\) −2.94427 −0.234978 −0.117489 0.993074i \(-0.537485\pi\)
−0.117489 + 0.993074i \(0.537485\pi\)
\(158\) 0 0
\(159\) 10.9443 0.867937
\(160\) 0 0
\(161\) −1.00000 −0.0788110
\(162\) 0 0
\(163\) 8.94427 0.700569 0.350285 0.936643i \(-0.386085\pi\)
0.350285 + 0.936643i \(0.386085\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 15.4164 1.19296 0.596479 0.802629i \(-0.296566\pi\)
0.596479 + 0.802629i \(0.296566\pi\)
\(168\) 0 0
\(169\) 7.00000 0.538462
\(170\) 0 0
\(171\) −2.47214 −0.189049
\(172\) 0 0
\(173\) 0.472136 0.0358958 0.0179479 0.999839i \(-0.494287\pi\)
0.0179479 + 0.999839i \(0.494287\pi\)
\(174\) 0 0
\(175\) 1.00000 0.0755929
\(176\) 0 0
\(177\) 4.00000 0.300658
\(178\) 0 0
\(179\) 20.0000 1.49487 0.747435 0.664335i \(-0.231285\pi\)
0.747435 + 0.664335i \(0.231285\pi\)
\(180\) 0 0
\(181\) 2.00000 0.148659 0.0743294 0.997234i \(-0.476318\pi\)
0.0743294 + 0.997234i \(0.476318\pi\)
\(182\) 0 0
\(183\) 6.00000 0.443533
\(184\) 0 0
\(185\) 13.8885 1.02111
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 1.00000 0.0727393
\(190\) 0 0
\(191\) 3.05573 0.221105 0.110552 0.993870i \(-0.464738\pi\)
0.110552 + 0.993870i \(0.464738\pi\)
\(192\) 0 0
\(193\) 11.8885 0.855756 0.427878 0.903836i \(-0.359261\pi\)
0.427878 + 0.903836i \(0.359261\pi\)
\(194\) 0 0
\(195\) 8.94427 0.640513
\(196\) 0 0
\(197\) −14.9443 −1.06474 −0.532368 0.846513i \(-0.678698\pi\)
−0.532368 + 0.846513i \(0.678698\pi\)
\(198\) 0 0
\(199\) −3.05573 −0.216615 −0.108307 0.994117i \(-0.534543\pi\)
−0.108307 + 0.994117i \(0.534543\pi\)
\(200\) 0 0
\(201\) 4.94427 0.348742
\(202\) 0 0
\(203\) 2.00000 0.140372
\(204\) 0 0
\(205\) −12.0000 −0.838116
\(206\) 0 0
\(207\) 1.00000 0.0695048
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 0.944272 0.0650064 0.0325032 0.999472i \(-0.489652\pi\)
0.0325032 + 0.999472i \(0.489652\pi\)
\(212\) 0 0
\(213\) 4.94427 0.338776
\(214\) 0 0
\(215\) 9.88854 0.674393
\(216\) 0 0
\(217\) −2.47214 −0.167820
\(218\) 0 0
\(219\) −14.9443 −1.00984
\(220\) 0 0
\(221\) −20.0000 −1.34535
\(222\) 0 0
\(223\) −18.4721 −1.23699 −0.618493 0.785790i \(-0.712256\pi\)
−0.618493 + 0.785790i \(0.712256\pi\)
\(224\) 0 0
\(225\) −1.00000 −0.0666667
\(226\) 0 0
\(227\) −5.52786 −0.366897 −0.183449 0.983029i \(-0.558726\pi\)
−0.183449 + 0.983029i \(0.558726\pi\)
\(228\) 0 0
\(229\) −14.0000 −0.925146 −0.462573 0.886581i \(-0.653074\pi\)
−0.462573 + 0.886581i \(0.653074\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −6.00000 −0.393073 −0.196537 0.980497i \(-0.562969\pi\)
−0.196537 + 0.980497i \(0.562969\pi\)
\(234\) 0 0
\(235\) 4.94427 0.322529
\(236\) 0 0
\(237\) −4.94427 −0.321165
\(238\) 0 0
\(239\) −20.9443 −1.35477 −0.677386 0.735628i \(-0.736887\pi\)
−0.677386 + 0.735628i \(0.736887\pi\)
\(240\) 0 0
\(241\) 20.4721 1.31873 0.659363 0.751825i \(-0.270826\pi\)
0.659363 + 0.751825i \(0.270826\pi\)
\(242\) 0 0
\(243\) −1.00000 −0.0641500
\(244\) 0 0
\(245\) 2.00000 0.127775
\(246\) 0 0
\(247\) 11.0557 0.703459
\(248\) 0 0
\(249\) 2.47214 0.156665
\(250\) 0 0
\(251\) 5.52786 0.348916 0.174458 0.984665i \(-0.444183\pi\)
0.174458 + 0.984665i \(0.444183\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) −8.94427 −0.560112
\(256\) 0 0
\(257\) −2.94427 −0.183659 −0.0918293 0.995775i \(-0.529271\pi\)
−0.0918293 + 0.995775i \(0.529271\pi\)
\(258\) 0 0
\(259\) −6.94427 −0.431496
\(260\) 0 0
\(261\) −2.00000 −0.123797
\(262\) 0 0
\(263\) 24.0000 1.47990 0.739952 0.672660i \(-0.234848\pi\)
0.739952 + 0.672660i \(0.234848\pi\)
\(264\) 0 0
\(265\) −21.8885 −1.34460
\(266\) 0 0
\(267\) −9.41641 −0.576275
\(268\) 0 0
\(269\) 16.4721 1.00432 0.502162 0.864774i \(-0.332538\pi\)
0.502162 + 0.864774i \(0.332538\pi\)
\(270\) 0 0
\(271\) −15.4164 −0.936480 −0.468240 0.883601i \(-0.655112\pi\)
−0.468240 + 0.883601i \(0.655112\pi\)
\(272\) 0 0
\(273\) −4.47214 −0.270666
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −11.8885 −0.714313 −0.357157 0.934044i \(-0.616254\pi\)
−0.357157 + 0.934044i \(0.616254\pi\)
\(278\) 0 0
\(279\) 2.47214 0.148003
\(280\) 0 0
\(281\) 10.0000 0.596550 0.298275 0.954480i \(-0.403589\pi\)
0.298275 + 0.954480i \(0.403589\pi\)
\(282\) 0 0
\(283\) −18.4721 −1.09805 −0.549027 0.835804i \(-0.685002\pi\)
−0.549027 + 0.835804i \(0.685002\pi\)
\(284\) 0 0
\(285\) 4.94427 0.292873
\(286\) 0 0
\(287\) 6.00000 0.354169
\(288\) 0 0
\(289\) 3.00000 0.176471
\(290\) 0 0
\(291\) 0.472136 0.0276771
\(292\) 0 0
\(293\) 27.8885 1.62927 0.814633 0.579977i \(-0.196938\pi\)
0.814633 + 0.579977i \(0.196938\pi\)
\(294\) 0 0
\(295\) −8.00000 −0.465778
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −4.47214 −0.258630
\(300\) 0 0
\(301\) −4.94427 −0.284983
\(302\) 0 0
\(303\) 9.41641 0.540958
\(304\) 0 0
\(305\) −12.0000 −0.687118
\(306\) 0 0
\(307\) −24.9443 −1.42364 −0.711822 0.702360i \(-0.752130\pi\)
−0.711822 + 0.702360i \(0.752130\pi\)
\(308\) 0 0
\(309\) −4.94427 −0.281270
\(310\) 0 0
\(311\) 2.47214 0.140182 0.0700910 0.997541i \(-0.477671\pi\)
0.0700910 + 0.997541i \(0.477671\pi\)
\(312\) 0 0
\(313\) −13.4164 −0.758340 −0.379170 0.925327i \(-0.623790\pi\)
−0.379170 + 0.925327i \(0.623790\pi\)
\(314\) 0 0
\(315\) −2.00000 −0.112687
\(316\) 0 0
\(317\) −13.0557 −0.733283 −0.366641 0.930362i \(-0.619492\pi\)
−0.366641 + 0.930362i \(0.619492\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) −8.00000 −0.446516
\(322\) 0 0
\(323\) −11.0557 −0.615157
\(324\) 0 0
\(325\) 4.47214 0.248069
\(326\) 0 0
\(327\) 2.94427 0.162819
\(328\) 0 0
\(329\) −2.47214 −0.136293
\(330\) 0 0
\(331\) 13.8885 0.763383 0.381692 0.924290i \(-0.375342\pi\)
0.381692 + 0.924290i \(0.375342\pi\)
\(332\) 0 0
\(333\) 6.94427 0.380544
\(334\) 0 0
\(335\) −9.88854 −0.540269
\(336\) 0 0
\(337\) −2.94427 −0.160385 −0.0801924 0.996779i \(-0.525553\pi\)
−0.0801924 + 0.996779i \(0.525553\pi\)
\(338\) 0 0
\(339\) 18.9443 1.02891
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) 0 0
\(345\) −2.00000 −0.107676
\(346\) 0 0
\(347\) 0.944272 0.0506912 0.0253456 0.999679i \(-0.491931\pi\)
0.0253456 + 0.999679i \(0.491931\pi\)
\(348\) 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
\(353\) 2.00000 0.106449 0.0532246 0.998583i \(-0.483050\pi\)
0.0532246 + 0.998583i \(0.483050\pi\)
\(354\) 0 0
\(355\) −9.88854 −0.524829
\(356\) 0 0
\(357\) 4.47214 0.236691
\(358\) 0 0
\(359\) 30.8328 1.62729 0.813647 0.581359i \(-0.197479\pi\)
0.813647 + 0.581359i \(0.197479\pi\)
\(360\) 0 0
\(361\) −12.8885 −0.678344
\(362\) 0 0
\(363\) 11.0000 0.577350
\(364\) 0 0
\(365\) 29.8885 1.56444
\(366\) 0 0
\(367\) −30.8328 −1.60946 −0.804730 0.593641i \(-0.797690\pi\)
−0.804730 + 0.593641i \(0.797690\pi\)
\(368\) 0 0
\(369\) −6.00000 −0.312348
\(370\) 0 0
\(371\) 10.9443 0.568198
\(372\) 0 0
\(373\) 6.94427 0.359561 0.179780 0.983707i \(-0.442461\pi\)
0.179780 + 0.983707i \(0.442461\pi\)
\(374\) 0 0
\(375\) 12.0000 0.619677
\(376\) 0 0
\(377\) 8.94427 0.460653
\(378\) 0 0
\(379\) −24.0000 −1.23280 −0.616399 0.787434i \(-0.711409\pi\)
−0.616399 + 0.787434i \(0.711409\pi\)
\(380\) 0 0
\(381\) 20.9443 1.07301
\(382\) 0 0
\(383\) −16.0000 −0.817562 −0.408781 0.912633i \(-0.634046\pi\)
−0.408781 + 0.912633i \(0.634046\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 4.94427 0.251331
\(388\) 0 0
\(389\) −18.9443 −0.960513 −0.480256 0.877128i \(-0.659456\pi\)
−0.480256 + 0.877128i \(0.659456\pi\)
\(390\) 0 0
\(391\) 4.47214 0.226166
\(392\) 0 0
\(393\) 12.0000 0.605320
\(394\) 0 0
\(395\) 9.88854 0.497547
\(396\) 0 0
\(397\) −36.4721 −1.83048 −0.915242 0.402905i \(-0.868001\pi\)
−0.915242 + 0.402905i \(0.868001\pi\)
\(398\) 0 0
\(399\) −2.47214 −0.123762
\(400\) 0 0
\(401\) −22.0000 −1.09863 −0.549314 0.835616i \(-0.685111\pi\)
−0.549314 + 0.835616i \(0.685111\pi\)
\(402\) 0 0
\(403\) −11.0557 −0.550725
\(404\) 0 0
\(405\) 2.00000 0.0993808
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) −15.8885 −0.785638 −0.392819 0.919616i \(-0.628500\pi\)
−0.392819 + 0.919616i \(0.628500\pi\)
\(410\) 0 0
\(411\) 14.0000 0.690569
\(412\) 0 0
\(413\) 4.00000 0.196827
\(414\) 0 0
\(415\) −4.94427 −0.242705
\(416\) 0 0
\(417\) 8.94427 0.438003
\(418\) 0 0
\(419\) 7.41641 0.362315 0.181158 0.983454i \(-0.442016\pi\)
0.181158 + 0.983454i \(0.442016\pi\)
\(420\) 0 0
\(421\) 32.8328 1.60017 0.800087 0.599884i \(-0.204787\pi\)
0.800087 + 0.599884i \(0.204787\pi\)
\(422\) 0 0
\(423\) 2.47214 0.120199
\(424\) 0 0
\(425\) −4.47214 −0.216930
\(426\) 0 0
\(427\) 6.00000 0.290360
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −17.8885 −0.861661 −0.430830 0.902433i \(-0.641779\pi\)
−0.430830 + 0.902433i \(0.641779\pi\)
\(432\) 0 0
\(433\) 25.4164 1.22143 0.610717 0.791849i \(-0.290881\pi\)
0.610717 + 0.791849i \(0.290881\pi\)
\(434\) 0 0
\(435\) 4.00000 0.191785
\(436\) 0 0
\(437\) −2.47214 −0.118258
\(438\) 0 0
\(439\) −28.3607 −1.35358 −0.676791 0.736175i \(-0.736630\pi\)
−0.676791 + 0.736175i \(0.736630\pi\)
\(440\) 0 0
\(441\) 1.00000 0.0476190
\(442\) 0 0
\(443\) −34.8328 −1.65496 −0.827479 0.561497i \(-0.810225\pi\)
−0.827479 + 0.561497i \(0.810225\pi\)
\(444\) 0 0
\(445\) 18.8328 0.892761
\(446\) 0 0
\(447\) 1.05573 0.0499342
\(448\) 0 0
\(449\) 18.0000 0.849473 0.424736 0.905317i \(-0.360367\pi\)
0.424736 + 0.905317i \(0.360367\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 4.94427 0.232302
\(454\) 0 0
\(455\) 8.94427 0.419314
\(456\) 0 0
\(457\) 2.00000 0.0935561 0.0467780 0.998905i \(-0.485105\pi\)
0.0467780 + 0.998905i \(0.485105\pi\)
\(458\) 0 0
\(459\) −4.47214 −0.208741
\(460\) 0 0
\(461\) 6.58359 0.306628 0.153314 0.988177i \(-0.451005\pi\)
0.153314 + 0.988177i \(0.451005\pi\)
\(462\) 0 0
\(463\) −6.11146 −0.284023 −0.142012 0.989865i \(-0.545357\pi\)
−0.142012 + 0.989865i \(0.545357\pi\)
\(464\) 0 0
\(465\) −4.94427 −0.229285
\(466\) 0 0
\(467\) −33.3050 −1.54117 −0.770585 0.637338i \(-0.780036\pi\)
−0.770585 + 0.637338i \(0.780036\pi\)
\(468\) 0 0
\(469\) 4.94427 0.228305
\(470\) 0 0
\(471\) 2.94427 0.135665
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 2.47214 0.113429
\(476\) 0 0
\(477\) −10.9443 −0.501104
\(478\) 0 0
\(479\) 4.94427 0.225910 0.112955 0.993600i \(-0.463968\pi\)
0.112955 + 0.993600i \(0.463968\pi\)
\(480\) 0 0
\(481\) −31.0557 −1.41602
\(482\) 0 0
\(483\) 1.00000 0.0455016
\(484\) 0 0
\(485\) −0.944272 −0.0428772
\(486\) 0 0
\(487\) −24.0000 −1.08754 −0.543772 0.839233i \(-0.683004\pi\)
−0.543772 + 0.839233i \(0.683004\pi\)
\(488\) 0 0
\(489\) −8.94427 −0.404474
\(490\) 0 0
\(491\) −40.9443 −1.84779 −0.923895 0.382647i \(-0.875013\pi\)
−0.923895 + 0.382647i \(0.875013\pi\)
\(492\) 0 0
\(493\) −8.94427 −0.402830
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 4.94427 0.221781
\(498\) 0 0
\(499\) −32.9443 −1.47479 −0.737394 0.675463i \(-0.763944\pi\)
−0.737394 + 0.675463i \(0.763944\pi\)
\(500\) 0 0
\(501\) −15.4164 −0.688754
\(502\) 0 0
\(503\) 9.88854 0.440908 0.220454 0.975397i \(-0.429246\pi\)
0.220454 + 0.975397i \(0.429246\pi\)
\(504\) 0 0
\(505\) −18.8328 −0.838049
\(506\) 0 0
\(507\) −7.00000 −0.310881
\(508\) 0 0
\(509\) 6.58359 0.291813 0.145906 0.989298i \(-0.453390\pi\)
0.145906 + 0.989298i \(0.453390\pi\)
\(510\) 0 0
\(511\) −14.9443 −0.661096
\(512\) 0 0
\(513\) 2.47214 0.109147
\(514\) 0 0
\(515\) 9.88854 0.435741
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) −0.472136 −0.0207245
\(520\) 0 0
\(521\) 33.4164 1.46400 0.732000 0.681305i \(-0.238587\pi\)
0.732000 + 0.681305i \(0.238587\pi\)
\(522\) 0 0
\(523\) 25.3050 1.10651 0.553254 0.833013i \(-0.313386\pi\)
0.553254 + 0.833013i \(0.313386\pi\)
\(524\) 0 0
\(525\) −1.00000 −0.0436436
\(526\) 0 0
\(527\) 11.0557 0.481595
\(528\) 0 0
\(529\) 1.00000 0.0434783
\(530\) 0 0
\(531\) −4.00000 −0.173585
\(532\) 0 0
\(533\) 26.8328 1.16226
\(534\) 0 0
\(535\) 16.0000 0.691740
\(536\) 0 0
\(537\) −20.0000 −0.863064
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −10.0000 −0.429934 −0.214967 0.976621i \(-0.568964\pi\)
−0.214967 + 0.976621i \(0.568964\pi\)
\(542\) 0 0
\(543\) −2.00000 −0.0858282
\(544\) 0 0
\(545\) −5.88854 −0.252238
\(546\) 0 0
\(547\) −20.0000 −0.855138 −0.427569 0.903983i \(-0.640630\pi\)
−0.427569 + 0.903983i \(0.640630\pi\)
\(548\) 0 0
\(549\) −6.00000 −0.256074
\(550\) 0 0
\(551\) 4.94427 0.210633
\(552\) 0 0
\(553\) −4.94427 −0.210252
\(554\) 0 0
\(555\) −13.8885 −0.589536
\(556\) 0 0
\(557\) −17.0557 −0.722674 −0.361337 0.932435i \(-0.617680\pi\)
−0.361337 + 0.932435i \(0.617680\pi\)
\(558\) 0 0
\(559\) −22.1115 −0.935215
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −26.4721 −1.11567 −0.557834 0.829953i \(-0.688367\pi\)
−0.557834 + 0.829953i \(0.688367\pi\)
\(564\) 0 0
\(565\) −37.8885 −1.59398
\(566\) 0 0
\(567\) −1.00000 −0.0419961
\(568\) 0 0
\(569\) −7.88854 −0.330705 −0.165352 0.986235i \(-0.552876\pi\)
−0.165352 + 0.986235i \(0.552876\pi\)
\(570\) 0 0
\(571\) 12.9443 0.541701 0.270850 0.962621i \(-0.412695\pi\)
0.270850 + 0.962621i \(0.412695\pi\)
\(572\) 0 0
\(573\) −3.05573 −0.127655
\(574\) 0 0
\(575\) −1.00000 −0.0417029
\(576\) 0 0
\(577\) −2.94427 −0.122572 −0.0612858 0.998120i \(-0.519520\pi\)
−0.0612858 + 0.998120i \(0.519520\pi\)
\(578\) 0 0
\(579\) −11.8885 −0.494071
\(580\) 0 0
\(581\) 2.47214 0.102561
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) −8.94427 −0.369800
\(586\) 0 0
\(587\) 0.944272 0.0389743 0.0194871 0.999810i \(-0.493797\pi\)
0.0194871 + 0.999810i \(0.493797\pi\)
\(588\) 0 0
\(589\) −6.11146 −0.251818
\(590\) 0 0
\(591\) 14.9443 0.614725
\(592\) 0 0
\(593\) −28.8328 −1.18402 −0.592011 0.805930i \(-0.701666\pi\)
−0.592011 + 0.805930i \(0.701666\pi\)
\(594\) 0 0
\(595\) −8.94427 −0.366679
\(596\) 0 0
\(597\) 3.05573 0.125063
\(598\) 0 0
\(599\) −33.8885 −1.38465 −0.692324 0.721587i \(-0.743413\pi\)
−0.692324 + 0.721587i \(0.743413\pi\)
\(600\) 0 0
\(601\) 40.8328 1.66561 0.832803 0.553570i \(-0.186735\pi\)
0.832803 + 0.553570i \(0.186735\pi\)
\(602\) 0 0
\(603\) −4.94427 −0.201346
\(604\) 0 0
\(605\) −22.0000 −0.894427
\(606\) 0 0
\(607\) 28.3607 1.15112 0.575562 0.817758i \(-0.304783\pi\)
0.575562 + 0.817758i \(0.304783\pi\)
\(608\) 0 0
\(609\) −2.00000 −0.0810441
\(610\) 0 0
\(611\) −11.0557 −0.447267
\(612\) 0 0
\(613\) 40.8328 1.64922 0.824611 0.565700i \(-0.191394\pi\)
0.824611 + 0.565700i \(0.191394\pi\)
\(614\) 0 0
\(615\) 12.0000 0.483887
\(616\) 0 0
\(617\) 26.0000 1.04672 0.523360 0.852111i \(-0.324678\pi\)
0.523360 + 0.852111i \(0.324678\pi\)
\(618\) 0 0
\(619\) 34.4721 1.38555 0.692776 0.721153i \(-0.256387\pi\)
0.692776 + 0.721153i \(0.256387\pi\)
\(620\) 0 0
\(621\) −1.00000 −0.0401286
\(622\) 0 0
\(623\) −9.41641 −0.377260
\(624\) 0 0
\(625\) −19.0000 −0.760000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 31.0557 1.23827
\(630\) 0 0
\(631\) 11.0557 0.440122 0.220061 0.975486i \(-0.429374\pi\)
0.220061 + 0.975486i \(0.429374\pi\)
\(632\) 0 0
\(633\) −0.944272 −0.0375314
\(634\) 0 0
\(635\) −41.8885 −1.66230
\(636\) 0 0
\(637\) −4.47214 −0.177192
\(638\) 0 0
\(639\) −4.94427 −0.195592
\(640\) 0 0
\(641\) 10.0000 0.394976 0.197488 0.980305i \(-0.436722\pi\)
0.197488 + 0.980305i \(0.436722\pi\)
\(642\) 0 0
\(643\) 10.4721 0.412981 0.206490 0.978449i \(-0.433796\pi\)
0.206490 + 0.978449i \(0.433796\pi\)
\(644\) 0 0
\(645\) −9.88854 −0.389361
\(646\) 0 0
\(647\) −12.3607 −0.485948 −0.242974 0.970033i \(-0.578123\pi\)
−0.242974 + 0.970033i \(0.578123\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 2.47214 0.0968906
\(652\) 0 0
\(653\) 6.00000 0.234798 0.117399 0.993085i \(-0.462544\pi\)
0.117399 + 0.993085i \(0.462544\pi\)
\(654\) 0 0
\(655\) −24.0000 −0.937758
\(656\) 0 0
\(657\) 14.9443 0.583032
\(658\) 0 0
\(659\) −46.8328 −1.82435 −0.912174 0.409804i \(-0.865597\pi\)
−0.912174 + 0.409804i \(0.865597\pi\)
\(660\) 0 0
\(661\) −30.0000 −1.16686 −0.583432 0.812162i \(-0.698291\pi\)
−0.583432 + 0.812162i \(0.698291\pi\)
\(662\) 0 0
\(663\) 20.0000 0.776736
\(664\) 0 0
\(665\) 4.94427 0.191731
\(666\) 0 0
\(667\) −2.00000 −0.0774403
\(668\) 0 0
\(669\) 18.4721 0.714174
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) −30.0000 −1.15642 −0.578208 0.815890i \(-0.696248\pi\)
−0.578208 + 0.815890i \(0.696248\pi\)
\(674\) 0 0
\(675\) 1.00000 0.0384900
\(676\) 0 0
\(677\) −4.11146 −0.158016 −0.0790080 0.996874i \(-0.525175\pi\)
−0.0790080 + 0.996874i \(0.525175\pi\)
\(678\) 0 0
\(679\) 0.472136 0.0181189
\(680\) 0 0
\(681\) 5.52786 0.211828
\(682\) 0 0
\(683\) −16.9443 −0.648355 −0.324177 0.945996i \(-0.605087\pi\)
−0.324177 + 0.945996i \(0.605087\pi\)
\(684\) 0 0
\(685\) −28.0000 −1.06983
\(686\) 0 0
\(687\) 14.0000 0.534133
\(688\) 0 0
\(689\) 48.9443 1.86463
\(690\) 0 0
\(691\) 15.0557 0.572747 0.286373 0.958118i \(-0.407550\pi\)
0.286373 + 0.958118i \(0.407550\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −17.8885 −0.678551
\(696\) 0 0
\(697\) −26.8328 −1.01637
\(698\) 0 0
\(699\) 6.00000 0.226941
\(700\) 0 0
\(701\) −20.8328 −0.786845 −0.393422 0.919358i \(-0.628709\pi\)
−0.393422 + 0.919358i \(0.628709\pi\)
\(702\) 0 0
\(703\) −17.1672 −0.647473
\(704\) 0 0
\(705\) −4.94427 −0.186212
\(706\) 0 0
\(707\) 9.41641 0.354140
\(708\) 0 0
\(709\) 6.94427 0.260798 0.130399 0.991462i \(-0.458374\pi\)
0.130399 + 0.991462i \(0.458374\pi\)
\(710\) 0 0
\(711\) 4.94427 0.185425
\(712\) 0 0
\(713\) 2.47214 0.0925822
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 20.9443 0.782178
\(718\) 0 0
\(719\) 5.52786 0.206155 0.103077 0.994673i \(-0.467131\pi\)
0.103077 + 0.994673i \(0.467131\pi\)
\(720\) 0 0
\(721\) −4.94427 −0.184134
\(722\) 0 0
\(723\) −20.4721 −0.761367
\(724\) 0 0
\(725\) 2.00000 0.0742781
\(726\) 0 0
\(727\) −3.05573 −0.113331 −0.0566653 0.998393i \(-0.518047\pi\)
−0.0566653 + 0.998393i \(0.518047\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 22.1115 0.817822
\(732\) 0 0
\(733\) −50.9443 −1.88167 −0.940835 0.338866i \(-0.889957\pi\)
−0.940835 + 0.338866i \(0.889957\pi\)
\(734\) 0 0
\(735\) −2.00000 −0.0737711
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 8.94427 0.329020 0.164510 0.986375i \(-0.447396\pi\)
0.164510 + 0.986375i \(0.447396\pi\)
\(740\) 0 0
\(741\) −11.0557 −0.406142
\(742\) 0 0
\(743\) 22.8328 0.837655 0.418827 0.908066i \(-0.362441\pi\)
0.418827 + 0.908066i \(0.362441\pi\)
\(744\) 0 0
\(745\) −2.11146 −0.0773578
\(746\) 0 0
\(747\) −2.47214 −0.0904507
\(748\) 0 0
\(749\) −8.00000 −0.292314
\(750\) 0 0
\(751\) −17.8885 −0.652762 −0.326381 0.945238i \(-0.605829\pi\)
−0.326381 + 0.945238i \(0.605829\pi\)
\(752\) 0 0
\(753\) −5.52786 −0.201447
\(754\) 0 0
\(755\) −9.88854 −0.359881
\(756\) 0 0
\(757\) −42.9443 −1.56084 −0.780418 0.625258i \(-0.784994\pi\)
−0.780418 + 0.625258i \(0.784994\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 21.0557 0.763270 0.381635 0.924313i \(-0.375361\pi\)
0.381635 + 0.924313i \(0.375361\pi\)
\(762\) 0 0
\(763\) 2.94427 0.106590
\(764\) 0 0
\(765\) 8.94427 0.323381
\(766\) 0 0
\(767\) 17.8885 0.645918
\(768\) 0 0
\(769\) 30.3607 1.09483 0.547417 0.836860i \(-0.315611\pi\)
0.547417 + 0.836860i \(0.315611\pi\)
\(770\) 0 0
\(771\) 2.94427 0.106035
\(772\) 0 0
\(773\) 5.05573 0.181842 0.0909210 0.995858i \(-0.471019\pi\)
0.0909210 + 0.995858i \(0.471019\pi\)
\(774\) 0 0
\(775\) −2.47214 −0.0888017
\(776\) 0 0
\(777\) 6.94427 0.249124
\(778\) 0 0
\(779\) 14.8328 0.531441
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 2.00000 0.0714742
\(784\) 0 0
\(785\) −5.88854 −0.210171
\(786\) 0 0
\(787\) 0.583592 0.0208028 0.0104014 0.999946i \(-0.496689\pi\)
0.0104014 + 0.999946i \(0.496689\pi\)
\(788\) 0 0
\(789\) −24.0000 −0.854423
\(790\) 0 0
\(791\) 18.9443 0.673581
\(792\) 0 0
\(793\) 26.8328 0.952861
\(794\) 0 0
\(795\) 21.8885 0.776307
\(796\) 0 0
\(797\) −28.8328 −1.02131 −0.510655 0.859785i \(-0.670597\pi\)
−0.510655 + 0.859785i \(0.670597\pi\)
\(798\) 0 0
\(799\) 11.0557 0.391124
\(800\) 0 0
\(801\) 9.41641 0.332712
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) −2.00000 −0.0704907
\(806\) 0 0
\(807\) −16.4721 −0.579847
\(808\) 0 0
\(809\) 35.8885 1.26177 0.630887 0.775875i \(-0.282691\pi\)
0.630887 + 0.775875i \(0.282691\pi\)
\(810\) 0 0
\(811\) 37.8885 1.33045 0.665223 0.746644i \(-0.268336\pi\)
0.665223 + 0.746644i \(0.268336\pi\)
\(812\) 0 0
\(813\) 15.4164 0.540677
\(814\) 0 0
\(815\) 17.8885 0.626608
\(816\) 0 0
\(817\) −12.2229 −0.427626
\(818\) 0 0
\(819\) 4.47214 0.156269
\(820\) 0 0
\(821\) −16.8328 −0.587469 −0.293735 0.955887i \(-0.594898\pi\)
−0.293735 + 0.955887i \(0.594898\pi\)
\(822\) 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
\(827\) −33.8885 −1.17842 −0.589210 0.807980i \(-0.700561\pi\)
−0.589210 + 0.807980i \(0.700561\pi\)
\(828\) 0 0
\(829\) 13.4164 0.465971 0.232986 0.972480i \(-0.425151\pi\)
0.232986 + 0.972480i \(0.425151\pi\)
\(830\) 0 0
\(831\) 11.8885 0.412409
\(832\) 0 0
\(833\) 4.47214 0.154950
\(834\) 0 0
\(835\) 30.8328 1.06701
\(836\) 0 0
\(837\) −2.47214 −0.0854495
\(838\) 0 0
\(839\) 24.0000 0.828572 0.414286 0.910147i \(-0.364031\pi\)
0.414286 + 0.910147i \(0.364031\pi\)
\(840\) 0 0
\(841\) −25.0000 −0.862069
\(842\) 0 0
\(843\) −10.0000 −0.344418
\(844\) 0 0
\(845\) 14.0000 0.481615
\(846\) 0 0
\(847\) 11.0000 0.377964
\(848\) 0 0
\(849\) 18.4721 0.633962
\(850\) 0 0
\(851\) 6.94427 0.238047
\(852\) 0 0
\(853\) −6.36068 −0.217786 −0.108893 0.994054i \(-0.534731\pi\)
−0.108893 + 0.994054i \(0.534731\pi\)
\(854\) 0 0
\(855\) −4.94427 −0.169091
\(856\) 0 0
\(857\) −47.8885 −1.63584 −0.817921 0.575331i \(-0.804873\pi\)
−0.817921 + 0.575331i \(0.804873\pi\)
\(858\) 0 0
\(859\) −32.9443 −1.12404 −0.562022 0.827122i \(-0.689976\pi\)
−0.562022 + 0.827122i \(0.689976\pi\)
\(860\) 0 0
\(861\) −6.00000 −0.204479
\(862\) 0 0
\(863\) −20.9443 −0.712951 −0.356476 0.934305i \(-0.616022\pi\)
−0.356476 + 0.934305i \(0.616022\pi\)
\(864\) 0 0
\(865\) 0.944272 0.0321062
\(866\) 0 0
\(867\) −3.00000 −0.101885
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 22.1115 0.749218
\(872\) 0 0
\(873\) −0.472136 −0.0159794
\(874\) 0 0
\(875\) 12.0000 0.405674
\(876\) 0 0
\(877\) −34.7214 −1.17246 −0.586229 0.810146i \(-0.699388\pi\)
−0.586229 + 0.810146i \(0.699388\pi\)
\(878\) 0 0
\(879\) −27.8885 −0.940657
\(880\) 0 0
\(881\) 22.3607 0.753350 0.376675 0.926345i \(-0.377067\pi\)
0.376675 + 0.926345i \(0.377067\pi\)
\(882\) 0 0
\(883\) −2.83282 −0.0953318 −0.0476659 0.998863i \(-0.515178\pi\)
−0.0476659 + 0.998863i \(0.515178\pi\)
\(884\) 0 0
\(885\) 8.00000 0.268917
\(886\) 0 0
\(887\) 5.52786 0.185608 0.0928038 0.995684i \(-0.470417\pi\)
0.0928038 + 0.995684i \(0.470417\pi\)
\(888\) 0 0
\(889\) 20.9443 0.702448
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −6.11146 −0.204512
\(894\) 0 0
\(895\) 40.0000 1.33705
\(896\) 0 0
\(897\) 4.47214 0.149320
\(898\) 0 0
\(899\) −4.94427 −0.164901
\(900\) 0 0
\(901\) −48.9443 −1.63057
\(902\) 0 0
\(903\) 4.94427 0.164535
\(904\) 0 0
\(905\) 4.00000 0.132964
\(906\) 0 0
\(907\) 41.8885 1.39089 0.695443 0.718581i \(-0.255208\pi\)
0.695443 + 0.718581i \(0.255208\pi\)
\(908\) 0 0
\(909\) −9.41641 −0.312323
\(910\) 0 0
\(911\) 48.7214 1.61421 0.807105 0.590407i \(-0.201033\pi\)
0.807105 + 0.590407i \(0.201033\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) 12.0000 0.396708
\(916\) 0 0
\(917\) 12.0000 0.396275
\(918\) 0 0
\(919\) −35.0557 −1.15638 −0.578191 0.815902i \(-0.696241\pi\)
−0.578191 + 0.815902i \(0.696241\pi\)
\(920\) 0 0
\(921\) 24.9443 0.821942
\(922\) 0 0
\(923\) 22.1115 0.727807
\(924\) 0 0
\(925\) −6.94427 −0.228326
\(926\) 0 0
\(927\) 4.94427 0.162391
\(928\) 0 0
\(929\) 18.0000 0.590561 0.295280 0.955411i \(-0.404587\pi\)
0.295280 + 0.955411i \(0.404587\pi\)
\(930\) 0 0
\(931\) −2.47214 −0.0810210
\(932\) 0 0
\(933\) −2.47214 −0.0809341
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 40.2492 1.31488 0.657442 0.753505i \(-0.271638\pi\)
0.657442 + 0.753505i \(0.271638\pi\)
\(938\) 0 0
\(939\) 13.4164 0.437828
\(940\) 0 0
\(941\) −6.00000 −0.195594 −0.0977972 0.995206i \(-0.531180\pi\)
−0.0977972 + 0.995206i \(0.531180\pi\)
\(942\) 0 0
\(943\) −6.00000 −0.195387
\(944\) 0 0
\(945\) 2.00000 0.0650600
\(946\) 0 0
\(947\) −0.944272 −0.0306847 −0.0153424 0.999882i \(-0.504884\pi\)
−0.0153424 + 0.999882i \(0.504884\pi\)
\(948\) 0 0
\(949\) −66.8328 −2.16949
\(950\) 0 0
\(951\) 13.0557 0.423361
\(952\) 0 0
\(953\) −9.05573 −0.293344 −0.146672 0.989185i \(-0.546856\pi\)
−0.146672 + 0.989185i \(0.546856\pi\)
\(954\) 0 0
\(955\) 6.11146 0.197762
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 14.0000 0.452084
\(960\) 0 0
\(961\) −24.8885 −0.802856
\(962\) 0 0
\(963\) 8.00000 0.257796
\(964\) 0 0
\(965\) 23.7771 0.765412
\(966\) 0 0
\(967\) 16.0000 0.514525 0.257263 0.966342i \(-0.417179\pi\)
0.257263 + 0.966342i \(0.417179\pi\)
\(968\) 0 0
\(969\) 11.0557 0.355161
\(970\) 0 0
\(971\) 23.4164 0.751468 0.375734 0.926727i \(-0.377391\pi\)
0.375734 + 0.926727i \(0.377391\pi\)
\(972\) 0 0
\(973\) 8.94427 0.286740
\(974\) 0 0
\(975\) −4.47214 −0.143223
\(976\) 0 0
\(977\) −6.00000 −0.191957 −0.0959785 0.995383i \(-0.530598\pi\)
−0.0959785 + 0.995383i \(0.530598\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) −2.94427 −0.0940034
\(982\) 0 0
\(983\) 22.1115 0.705246 0.352623 0.935765i \(-0.385290\pi\)
0.352623 + 0.935765i \(0.385290\pi\)
\(984\) 0 0
\(985\) −29.8885 −0.952328
\(986\) 0 0
\(987\) 2.47214 0.0786890
\(988\) 0 0
\(989\) 4.94427 0.157219
\(990\) 0 0
\(991\) −60.9443 −1.93596 −0.967979 0.251030i \(-0.919231\pi\)
−0.967979 + 0.251030i \(0.919231\pi\)
\(992\) 0 0
\(993\) −13.8885 −0.440740
\(994\) 0 0
\(995\) −6.11146 −0.193746
\(996\) 0 0
\(997\) −46.3607 −1.46826 −0.734129 0.679010i \(-0.762409\pi\)
−0.734129 + 0.679010i \(0.762409\pi\)
\(998\) 0 0
\(999\) −6.94427 −0.219707
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 7728.2.a.bd.1.1 2
4.3 odd 2 966.2.a.p.1.1 2
12.11 even 2 2898.2.a.v.1.1 2
28.27 even 2 6762.2.a.bz.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
966.2.a.p.1.1 2 4.3 odd 2
2898.2.a.v.1.1 2 12.11 even 2
6762.2.a.bz.1.2 2 28.27 even 2
7728.2.a.bd.1.1 2 1.1 even 1 trivial