Properties

Label 7728.2.a.bv.1.3
Level $7728$
Weight $2$
Character 7728.1
Self dual yes
Analytic conductor $61.708$
Analytic rank $1$
Dimension $3$
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: \(3\)
Coefficient field: 3.3.1509.1
Defining polynomial: \(x^{3} - x^{2} - 7 x + 4\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1932)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-2.47735\) of defining polynomial
Character \(\chi\) \(=\) 7728.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{3} +2.47735 q^{5} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{3} +2.47735 q^{5} -1.00000 q^{7} +1.00000 q^{9} -5.95470 q^{11} -0.137275 q^{13} +2.47735 q^{15} +4.61463 q^{17} -1.00000 q^{19} -1.00000 q^{21} -1.00000 q^{23} +1.13727 q^{25} +1.00000 q^{27} -1.65992 q^{29} +2.61463 q^{31} -5.95470 q^{33} -2.47735 q^{35} -11.5693 q^{37} -0.137275 q^{39} -3.68016 q^{41} -11.0920 q^{43} +2.47735 q^{45} -5.34008 q^{47} +1.00000 q^{49} +4.61463 q^{51} -5.13727 q^{53} -14.7519 q^{55} -1.00000 q^{57} -5.13727 q^{59} +4.52265 q^{61} -1.00000 q^{63} -0.340078 q^{65} -2.47735 q^{67} -1.00000 q^{69} +11.3665 q^{71} -0.340078 q^{73} +1.13727 q^{75} +5.95470 q^{77} +12.5240 q^{79} +1.00000 q^{81} -1.65992 q^{83} +11.4321 q^{85} -1.65992 q^{87} +3.86273 q^{89} +0.137275 q^{91} +2.61463 q^{93} -2.47735 q^{95} -2.88918 q^{97} -5.95470 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 3 q^{3} - q^{5} - 3 q^{7} + 3 q^{9} + O(q^{10}) \) \( 3 q + 3 q^{3} - q^{5} - 3 q^{7} + 3 q^{9} - q^{11} + 3 q^{13} - q^{15} + 2 q^{17} - 3 q^{19} - 3 q^{21} - 3 q^{23} + 3 q^{27} - 10 q^{29} - 4 q^{31} - q^{33} + q^{35} - 6 q^{37} + 3 q^{39} - q^{41} - 13 q^{43} - q^{45} - 11 q^{47} + 3 q^{49} + 2 q^{51} - 12 q^{53} - 29 q^{55} - 3 q^{57} - 12 q^{59} + 22 q^{61} - 3 q^{63} + 4 q^{65} + q^{67} - 3 q^{69} + 7 q^{71} + 4 q^{73} + q^{77} - 8 q^{79} + 3 q^{81} - 10 q^{83} + 9 q^{85} - 10 q^{87} + 15 q^{89} - 3 q^{91} - 4 q^{93} + q^{95} + 10 q^{97} - q^{99} + O(q^{100}) \)

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
\(5\) 2.47735 1.10791 0.553953 0.832548i \(-0.313119\pi\)
0.553953 + 0.832548i \(0.313119\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −5.95470 −1.79541 −0.897706 0.440596i \(-0.854767\pi\)
−0.897706 + 0.440596i \(0.854767\pi\)
\(12\) 0 0
\(13\) −0.137275 −0.0380731 −0.0190366 0.999819i \(-0.506060\pi\)
−0.0190366 + 0.999819i \(0.506060\pi\)
\(14\) 0 0
\(15\) 2.47735 0.639650
\(16\) 0 0
\(17\) 4.61463 1.11921 0.559606 0.828759i \(-0.310953\pi\)
0.559606 + 0.828759i \(0.310953\pi\)
\(18\) 0 0
\(19\) −1.00000 −0.229416 −0.114708 0.993399i \(-0.536593\pi\)
−0.114708 + 0.993399i \(0.536593\pi\)
\(20\) 0 0
\(21\) −1.00000 −0.218218
\(22\) 0 0
\(23\) −1.00000 −0.208514
\(24\) 0 0
\(25\) 1.13727 0.227455
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) −1.65992 −0.308240 −0.154120 0.988052i \(-0.549254\pi\)
−0.154120 + 0.988052i \(0.549254\pi\)
\(30\) 0 0
\(31\) 2.61463 0.469601 0.234800 0.972044i \(-0.424556\pi\)
0.234800 + 0.972044i \(0.424556\pi\)
\(32\) 0 0
\(33\) −5.95470 −1.03658
\(34\) 0 0
\(35\) −2.47735 −0.418749
\(36\) 0 0
\(37\) −11.5693 −1.90199 −0.950993 0.309212i \(-0.899935\pi\)
−0.950993 + 0.309212i \(0.899935\pi\)
\(38\) 0 0
\(39\) −0.137275 −0.0219815
\(40\) 0 0
\(41\) −3.68016 −0.574744 −0.287372 0.957819i \(-0.592782\pi\)
−0.287372 + 0.957819i \(0.592782\pi\)
\(42\) 0 0
\(43\) −11.0920 −1.69151 −0.845755 0.533571i \(-0.820850\pi\)
−0.845755 + 0.533571i \(0.820850\pi\)
\(44\) 0 0
\(45\) 2.47735 0.369302
\(46\) 0 0
\(47\) −5.34008 −0.778930 −0.389465 0.921041i \(-0.627340\pi\)
−0.389465 + 0.921041i \(0.627340\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 4.61463 0.646177
\(52\) 0 0
\(53\) −5.13727 −0.705659 −0.352829 0.935688i \(-0.614780\pi\)
−0.352829 + 0.935688i \(0.614780\pi\)
\(54\) 0 0
\(55\) −14.7519 −1.98915
\(56\) 0 0
\(57\) −1.00000 −0.132453
\(58\) 0 0
\(59\) −5.13727 −0.668816 −0.334408 0.942428i \(-0.608536\pi\)
−0.334408 + 0.942428i \(0.608536\pi\)
\(60\) 0 0
\(61\) 4.52265 0.579066 0.289533 0.957168i \(-0.406500\pi\)
0.289533 + 0.957168i \(0.406500\pi\)
\(62\) 0 0
\(63\) −1.00000 −0.125988
\(64\) 0 0
\(65\) −0.340078 −0.0421814
\(66\) 0 0
\(67\) −2.47735 −0.302657 −0.151328 0.988484i \(-0.548355\pi\)
−0.151328 + 0.988484i \(0.548355\pi\)
\(68\) 0 0
\(69\) −1.00000 −0.120386
\(70\) 0 0
\(71\) 11.3665 1.34896 0.674479 0.738294i \(-0.264368\pi\)
0.674479 + 0.738294i \(0.264368\pi\)
\(72\) 0 0
\(73\) −0.340078 −0.0398031 −0.0199015 0.999802i \(-0.506335\pi\)
−0.0199015 + 0.999802i \(0.506335\pi\)
\(74\) 0 0
\(75\) 1.13727 0.131321
\(76\) 0 0
\(77\) 5.95470 0.678602
\(78\) 0 0
\(79\) 12.5240 1.40906 0.704532 0.709672i \(-0.251157\pi\)
0.704532 + 0.709672i \(0.251157\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −1.65992 −0.182200 −0.0911001 0.995842i \(-0.529038\pi\)
−0.0911001 + 0.995842i \(0.529038\pi\)
\(84\) 0 0
\(85\) 11.4321 1.23998
\(86\) 0 0
\(87\) −1.65992 −0.177962
\(88\) 0 0
\(89\) 3.86273 0.409448 0.204724 0.978820i \(-0.434370\pi\)
0.204724 + 0.978820i \(0.434370\pi\)
\(90\) 0 0
\(91\) 0.137275 0.0143903
\(92\) 0 0
\(93\) 2.61463 0.271124
\(94\) 0 0
\(95\) −2.47735 −0.254171
\(96\) 0 0
\(97\) −2.88918 −0.293351 −0.146676 0.989185i \(-0.546857\pi\)
−0.146676 + 0.989185i \(0.546857\pi\)
\(98\) 0 0
\(99\) −5.95470 −0.598470
\(100\) 0 0
\(101\) −19.3212 −1.92253 −0.961267 0.275618i \(-0.911118\pi\)
−0.961267 + 0.275618i \(0.911118\pi\)
\(102\) 0 0
\(103\) −6.38537 −0.629169 −0.314585 0.949229i \(-0.601865\pi\)
−0.314585 + 0.949229i \(0.601865\pi\)
\(104\) 0 0
\(105\) −2.47735 −0.241765
\(106\) 0 0
\(107\) −0.908021 −0.0877817 −0.0438908 0.999036i \(-0.513975\pi\)
−0.0438908 + 0.999036i \(0.513975\pi\)
\(108\) 0 0
\(109\) 6.75190 0.646715 0.323357 0.946277i \(-0.395188\pi\)
0.323357 + 0.946277i \(0.395188\pi\)
\(110\) 0 0
\(111\) −11.5693 −1.09811
\(112\) 0 0
\(113\) 18.6613 1.75551 0.877754 0.479111i \(-0.159041\pi\)
0.877754 + 0.479111i \(0.159041\pi\)
\(114\) 0 0
\(115\) −2.47735 −0.231014
\(116\) 0 0
\(117\) −0.137275 −0.0126910
\(118\) 0 0
\(119\) −4.61463 −0.423022
\(120\) 0 0
\(121\) 24.4585 2.22350
\(122\) 0 0
\(123\) −3.68016 −0.331828
\(124\) 0 0
\(125\) −9.56933 −0.855907
\(126\) 0 0
\(127\) 4.15751 0.368919 0.184460 0.982840i \(-0.440947\pi\)
0.184460 + 0.982840i \(0.440947\pi\)
\(128\) 0 0
\(129\) −11.0920 −0.976594
\(130\) 0 0
\(131\) −14.2293 −1.24322 −0.621608 0.783329i \(-0.713520\pi\)
−0.621608 + 0.783329i \(0.713520\pi\)
\(132\) 0 0
\(133\) 1.00000 0.0867110
\(134\) 0 0
\(135\) 2.47735 0.213217
\(136\) 0 0
\(137\) −7.27455 −0.621507 −0.310753 0.950491i \(-0.600581\pi\)
−0.310753 + 0.950491i \(0.600581\pi\)
\(138\) 0 0
\(139\) −15.4321 −1.30893 −0.654465 0.756092i \(-0.727106\pi\)
−0.654465 + 0.756092i \(0.727106\pi\)
\(140\) 0 0
\(141\) −5.34008 −0.449716
\(142\) 0 0
\(143\) 0.817430 0.0683569
\(144\) 0 0
\(145\) −4.11221 −0.341501
\(146\) 0 0
\(147\) 1.00000 0.0824786
\(148\) 0 0
\(149\) −15.2495 −1.24929 −0.624643 0.780910i \(-0.714756\pi\)
−0.624643 + 0.780910i \(0.714756\pi\)
\(150\) 0 0
\(151\) −16.5693 −1.34839 −0.674197 0.738552i \(-0.735510\pi\)
−0.674197 + 0.738552i \(0.735510\pi\)
\(152\) 0 0
\(153\) 4.61463 0.373070
\(154\) 0 0
\(155\) 6.47735 0.520273
\(156\) 0 0
\(157\) −0.659922 −0.0526675 −0.0263338 0.999653i \(-0.508383\pi\)
−0.0263338 + 0.999653i \(0.508383\pi\)
\(158\) 0 0
\(159\) −5.13727 −0.407412
\(160\) 0 0
\(161\) 1.00000 0.0788110
\(162\) 0 0
\(163\) −23.3212 −1.82666 −0.913330 0.407220i \(-0.866498\pi\)
−0.913330 + 0.407220i \(0.866498\pi\)
\(164\) 0 0
\(165\) −14.7519 −1.14843
\(166\) 0 0
\(167\) −9.00000 −0.696441 −0.348220 0.937413i \(-0.613214\pi\)
−0.348220 + 0.937413i \(0.613214\pi\)
\(168\) 0 0
\(169\) −12.9812 −0.998550
\(170\) 0 0
\(171\) −1.00000 −0.0764719
\(172\) 0 0
\(173\) 15.1637 1.15288 0.576438 0.817141i \(-0.304442\pi\)
0.576438 + 0.817141i \(0.304442\pi\)
\(174\) 0 0
\(175\) −1.13727 −0.0859699
\(176\) 0 0
\(177\) −5.13727 −0.386141
\(178\) 0 0
\(179\) 16.3868 1.22480 0.612402 0.790546i \(-0.290203\pi\)
0.612402 + 0.790546i \(0.290203\pi\)
\(180\) 0 0
\(181\) 11.0202 0.819127 0.409564 0.912282i \(-0.365681\pi\)
0.409564 + 0.912282i \(0.365681\pi\)
\(182\) 0 0
\(183\) 4.52265 0.334324
\(184\) 0 0
\(185\) −28.6613 −2.10722
\(186\) 0 0
\(187\) −27.4787 −2.00944
\(188\) 0 0
\(189\) −1.00000 −0.0727393
\(190\) 0 0
\(191\) −8.02023 −0.580324 −0.290162 0.956978i \(-0.593709\pi\)
−0.290162 + 0.956978i \(0.593709\pi\)
\(192\) 0 0
\(193\) −3.88918 −0.279949 −0.139975 0.990155i \(-0.544702\pi\)
−0.139975 + 0.990155i \(0.544702\pi\)
\(194\) 0 0
\(195\) −0.340078 −0.0243535
\(196\) 0 0
\(197\) 1.09198 0.0778003 0.0389002 0.999243i \(-0.487615\pi\)
0.0389002 + 0.999243i \(0.487615\pi\)
\(198\) 0 0
\(199\) 25.2306 1.78855 0.894276 0.447515i \(-0.147691\pi\)
0.894276 + 0.447515i \(0.147691\pi\)
\(200\) 0 0
\(201\) −2.47735 −0.174739
\(202\) 0 0
\(203\) 1.65992 0.116504
\(204\) 0 0
\(205\) −9.11704 −0.636762
\(206\) 0 0
\(207\) −1.00000 −0.0695048
\(208\) 0 0
\(209\) 5.95470 0.411896
\(210\) 0 0
\(211\) 17.9547 1.23605 0.618026 0.786157i \(-0.287932\pi\)
0.618026 + 0.786157i \(0.287932\pi\)
\(212\) 0 0
\(213\) 11.3665 0.778822
\(214\) 0 0
\(215\) −27.4787 −1.87403
\(216\) 0 0
\(217\) −2.61463 −0.177492
\(218\) 0 0
\(219\) −0.340078 −0.0229803
\(220\) 0 0
\(221\) −0.633471 −0.0426119
\(222\) 0 0
\(223\) −28.0216 −1.87647 −0.938233 0.346003i \(-0.887539\pi\)
−0.938233 + 0.346003i \(0.887539\pi\)
\(224\) 0 0
\(225\) 1.13727 0.0758183
\(226\) 0 0
\(227\) 4.47735 0.297172 0.148586 0.988899i \(-0.452528\pi\)
0.148586 + 0.988899i \(0.452528\pi\)
\(228\) 0 0
\(229\) −4.49759 −0.297209 −0.148604 0.988897i \(-0.547478\pi\)
−0.148604 + 0.988897i \(0.547478\pi\)
\(230\) 0 0
\(231\) 5.95470 0.391791
\(232\) 0 0
\(233\) 11.7066 0.766925 0.383463 0.923556i \(-0.374731\pi\)
0.383463 + 0.923556i \(0.374731\pi\)
\(234\) 0 0
\(235\) −13.2293 −0.862981
\(236\) 0 0
\(237\) 12.5240 0.813524
\(238\) 0 0
\(239\) −2.34630 −0.151769 −0.0758846 0.997117i \(-0.524178\pi\)
−0.0758846 + 0.997117i \(0.524178\pi\)
\(240\) 0 0
\(241\) 1.27455 0.0821009 0.0410505 0.999157i \(-0.486930\pi\)
0.0410505 + 0.999157i \(0.486930\pi\)
\(242\) 0 0
\(243\) 1.00000 0.0641500
\(244\) 0 0
\(245\) 2.47735 0.158272
\(246\) 0 0
\(247\) 0.137275 0.00873458
\(248\) 0 0
\(249\) −1.65992 −0.105193
\(250\) 0 0
\(251\) 16.1840 1.02152 0.510761 0.859723i \(-0.329364\pi\)
0.510761 + 0.859723i \(0.329364\pi\)
\(252\) 0 0
\(253\) 5.95470 0.374369
\(254\) 0 0
\(255\) 11.4321 0.715903
\(256\) 0 0
\(257\) −25.1184 −1.56684 −0.783422 0.621490i \(-0.786528\pi\)
−0.783422 + 0.621490i \(0.786528\pi\)
\(258\) 0 0
\(259\) 11.5693 0.718883
\(260\) 0 0
\(261\) −1.65992 −0.102747
\(262\) 0 0
\(263\) 18.1387 1.11848 0.559239 0.829007i \(-0.311093\pi\)
0.559239 + 0.829007i \(0.311093\pi\)
\(264\) 0 0
\(265\) −12.7268 −0.781804
\(266\) 0 0
\(267\) 3.86273 0.236395
\(268\) 0 0
\(269\) 30.1401 1.83767 0.918836 0.394640i \(-0.129131\pi\)
0.918836 + 0.394640i \(0.129131\pi\)
\(270\) 0 0
\(271\) 0.209021 0.0126971 0.00634856 0.999980i \(-0.497979\pi\)
0.00634856 + 0.999980i \(0.497979\pi\)
\(272\) 0 0
\(273\) 0.137275 0.00830824
\(274\) 0 0
\(275\) −6.77213 −0.408375
\(276\) 0 0
\(277\) −10.7972 −0.648741 −0.324370 0.945930i \(-0.605152\pi\)
−0.324370 + 0.945930i \(0.605152\pi\)
\(278\) 0 0
\(279\) 2.61463 0.156534
\(280\) 0 0
\(281\) −26.2745 −1.56741 −0.783704 0.621134i \(-0.786672\pi\)
−0.783704 + 0.621134i \(0.786672\pi\)
\(282\) 0 0
\(283\) 11.7066 0.695886 0.347943 0.937516i \(-0.386880\pi\)
0.347943 + 0.937516i \(0.386880\pi\)
\(284\) 0 0
\(285\) −2.47735 −0.146746
\(286\) 0 0
\(287\) 3.68016 0.217233
\(288\) 0 0
\(289\) 4.29478 0.252634
\(290\) 0 0
\(291\) −2.88918 −0.169367
\(292\) 0 0
\(293\) −22.4585 −1.31204 −0.656020 0.754743i \(-0.727761\pi\)
−0.656020 + 0.754743i \(0.727761\pi\)
\(294\) 0 0
\(295\) −12.7268 −0.740985
\(296\) 0 0
\(297\) −5.95470 −0.345527
\(298\) 0 0
\(299\) 0.137275 0.00793880
\(300\) 0 0
\(301\) 11.0920 0.639331
\(302\) 0 0
\(303\) −19.3212 −1.10998
\(304\) 0 0
\(305\) 11.2042 0.641550
\(306\) 0 0
\(307\) −11.9345 −0.681136 −0.340568 0.940220i \(-0.610619\pi\)
−0.340568 + 0.940220i \(0.610619\pi\)
\(308\) 0 0
\(309\) −6.38537 −0.363251
\(310\) 0 0
\(311\) 7.75190 0.439570 0.219785 0.975548i \(-0.429464\pi\)
0.219785 + 0.975548i \(0.429464\pi\)
\(312\) 0 0
\(313\) 6.93447 0.391960 0.195980 0.980608i \(-0.437211\pi\)
0.195980 + 0.980608i \(0.437211\pi\)
\(314\) 0 0
\(315\) −2.47735 −0.139583
\(316\) 0 0
\(317\) −26.3868 −1.48203 −0.741014 0.671489i \(-0.765655\pi\)
−0.741014 + 0.671489i \(0.765655\pi\)
\(318\) 0 0
\(319\) 9.88435 0.553417
\(320\) 0 0
\(321\) −0.908021 −0.0506808
\(322\) 0 0
\(323\) −4.61463 −0.256765
\(324\) 0 0
\(325\) −0.156119 −0.00865992
\(326\) 0 0
\(327\) 6.75190 0.373381
\(328\) 0 0
\(329\) 5.34008 0.294408
\(330\) 0 0
\(331\) −7.61463 −0.418538 −0.209269 0.977858i \(-0.567108\pi\)
−0.209269 + 0.977858i \(0.567108\pi\)
\(332\) 0 0
\(333\) −11.5693 −0.633995
\(334\) 0 0
\(335\) −6.13727 −0.335315
\(336\) 0 0
\(337\) 17.0920 0.931059 0.465530 0.885032i \(-0.345864\pi\)
0.465530 + 0.885032i \(0.345864\pi\)
\(338\) 0 0
\(339\) 18.6613 1.01354
\(340\) 0 0
\(341\) −15.5693 −0.843127
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) 0 0
\(345\) −2.47735 −0.133376
\(346\) 0 0
\(347\) 35.1136 1.88500 0.942498 0.334211i \(-0.108470\pi\)
0.942498 + 0.334211i \(0.108470\pi\)
\(348\) 0 0
\(349\) 24.1651 1.29353 0.646764 0.762690i \(-0.276122\pi\)
0.646764 + 0.762690i \(0.276122\pi\)
\(350\) 0 0
\(351\) −0.137275 −0.00732718
\(352\) 0 0
\(353\) −0.156119 −0.00830938 −0.00415469 0.999991i \(-0.501322\pi\)
−0.00415469 + 0.999991i \(0.501322\pi\)
\(354\) 0 0
\(355\) 28.1589 1.49452
\(356\) 0 0
\(357\) −4.61463 −0.244232
\(358\) 0 0
\(359\) 12.1122 0.639258 0.319629 0.947543i \(-0.396442\pi\)
0.319629 + 0.947543i \(0.396442\pi\)
\(360\) 0 0
\(361\) −18.0000 −0.947368
\(362\) 0 0
\(363\) 24.4585 1.28374
\(364\) 0 0
\(365\) −0.842492 −0.0440981
\(366\) 0 0
\(367\) 21.8830 1.14228 0.571141 0.820852i \(-0.306501\pi\)
0.571141 + 0.820852i \(0.306501\pi\)
\(368\) 0 0
\(369\) −3.68016 −0.191581
\(370\) 0 0
\(371\) 5.13727 0.266714
\(372\) 0 0
\(373\) 6.88918 0.356708 0.178354 0.983966i \(-0.442923\pi\)
0.178354 + 0.983966i \(0.442923\pi\)
\(374\) 0 0
\(375\) −9.56933 −0.494158
\(376\) 0 0
\(377\) 0.227865 0.0117357
\(378\) 0 0
\(379\) 29.1387 1.49675 0.748376 0.663274i \(-0.230834\pi\)
0.748376 + 0.663274i \(0.230834\pi\)
\(380\) 0 0
\(381\) 4.15751 0.212996
\(382\) 0 0
\(383\) −16.4307 −0.839568 −0.419784 0.907624i \(-0.637894\pi\)
−0.419784 + 0.907624i \(0.637894\pi\)
\(384\) 0 0
\(385\) 14.7519 0.751827
\(386\) 0 0
\(387\) −11.0920 −0.563837
\(388\) 0 0
\(389\) −21.3854 −1.08428 −0.542141 0.840288i \(-0.682386\pi\)
−0.542141 + 0.840288i \(0.682386\pi\)
\(390\) 0 0
\(391\) −4.61463 −0.233372
\(392\) 0 0
\(393\) −14.2293 −0.717771
\(394\) 0 0
\(395\) 31.0265 1.56111
\(396\) 0 0
\(397\) −25.4132 −1.27545 −0.637726 0.770263i \(-0.720125\pi\)
−0.637726 + 0.770263i \(0.720125\pi\)
\(398\) 0 0
\(399\) 1.00000 0.0500626
\(400\) 0 0
\(401\) 7.00000 0.349563 0.174782 0.984607i \(-0.444078\pi\)
0.174782 + 0.984607i \(0.444078\pi\)
\(402\) 0 0
\(403\) −0.358922 −0.0178792
\(404\) 0 0
\(405\) 2.47735 0.123101
\(406\) 0 0
\(407\) 68.8920 3.41485
\(408\) 0 0
\(409\) −6.06553 −0.299921 −0.149961 0.988692i \(-0.547915\pi\)
−0.149961 + 0.988692i \(0.547915\pi\)
\(410\) 0 0
\(411\) −7.27455 −0.358827
\(412\) 0 0
\(413\) 5.13727 0.252789
\(414\) 0 0
\(415\) −4.11221 −0.201861
\(416\) 0 0
\(417\) −15.4321 −0.755711
\(418\) 0 0
\(419\) 19.3415 0.944892 0.472446 0.881359i \(-0.343371\pi\)
0.472446 + 0.881359i \(0.343371\pi\)
\(420\) 0 0
\(421\) −2.54288 −0.123932 −0.0619662 0.998078i \(-0.519737\pi\)
−0.0619662 + 0.998078i \(0.519737\pi\)
\(422\) 0 0
\(423\) −5.34008 −0.259643
\(424\) 0 0
\(425\) 5.24810 0.254570
\(426\) 0 0
\(427\) −4.52265 −0.218866
\(428\) 0 0
\(429\) 0.817430 0.0394659
\(430\) 0 0
\(431\) −2.79720 −0.134736 −0.0673681 0.997728i \(-0.521460\pi\)
−0.0673681 + 0.997728i \(0.521460\pi\)
\(432\) 0 0
\(433\) −11.7080 −0.562650 −0.281325 0.959612i \(-0.590774\pi\)
−0.281325 + 0.959612i \(0.590774\pi\)
\(434\) 0 0
\(435\) −4.11221 −0.197165
\(436\) 0 0
\(437\) 1.00000 0.0478365
\(438\) 0 0
\(439\) −17.9749 −0.857897 −0.428948 0.903329i \(-0.641116\pi\)
−0.428948 + 0.903329i \(0.641116\pi\)
\(440\) 0 0
\(441\) 1.00000 0.0476190
\(442\) 0 0
\(443\) 7.81882 0.371483 0.185742 0.982599i \(-0.440531\pi\)
0.185742 + 0.982599i \(0.440531\pi\)
\(444\) 0 0
\(445\) 9.56933 0.453630
\(446\) 0 0
\(447\) −15.2495 −0.721276
\(448\) 0 0
\(449\) −25.0265 −1.18107 −0.590536 0.807012i \(-0.701083\pi\)
−0.590536 + 0.807012i \(0.701083\pi\)
\(450\) 0 0
\(451\) 21.9142 1.03190
\(452\) 0 0
\(453\) −16.5693 −0.778495
\(454\) 0 0
\(455\) 0.340078 0.0159431
\(456\) 0 0
\(457\) 26.3868 1.23432 0.617160 0.786837i \(-0.288283\pi\)
0.617160 + 0.786837i \(0.288283\pi\)
\(458\) 0 0
\(459\) 4.61463 0.215392
\(460\) 0 0
\(461\) −19.2481 −0.896473 −0.448237 0.893915i \(-0.647948\pi\)
−0.448237 + 0.893915i \(0.647948\pi\)
\(462\) 0 0
\(463\) −9.04047 −0.420146 −0.210073 0.977686i \(-0.567370\pi\)
−0.210073 + 0.977686i \(0.567370\pi\)
\(464\) 0 0
\(465\) 6.47735 0.300380
\(466\) 0 0
\(467\) −35.2571 −1.63150 −0.815752 0.578402i \(-0.803677\pi\)
−0.815752 + 0.578402i \(0.803677\pi\)
\(468\) 0 0
\(469\) 2.47735 0.114394
\(470\) 0 0
\(471\) −0.659922 −0.0304076
\(472\) 0 0
\(473\) 66.0495 3.03696
\(474\) 0 0
\(475\) −1.13727 −0.0521817
\(476\) 0 0
\(477\) −5.13727 −0.235220
\(478\) 0 0
\(479\) −35.5721 −1.62533 −0.812666 0.582730i \(-0.801984\pi\)
−0.812666 + 0.582730i \(0.801984\pi\)
\(480\) 0 0
\(481\) 1.58818 0.0724146
\(482\) 0 0
\(483\) 1.00000 0.0455016
\(484\) 0 0
\(485\) −7.15751 −0.325006
\(486\) 0 0
\(487\) −0.614627 −0.0278514 −0.0139257 0.999903i \(-0.504433\pi\)
−0.0139257 + 0.999903i \(0.504433\pi\)
\(488\) 0 0
\(489\) −23.3212 −1.05462
\(490\) 0 0
\(491\) 12.2683 0.553662 0.276831 0.960919i \(-0.410716\pi\)
0.276831 + 0.960919i \(0.410716\pi\)
\(492\) 0 0
\(493\) −7.65992 −0.344986
\(494\) 0 0
\(495\) −14.7519 −0.663049
\(496\) 0 0
\(497\) −11.3665 −0.509858
\(498\) 0 0
\(499\) 24.1122 1.07941 0.539705 0.841854i \(-0.318536\pi\)
0.539705 + 0.841854i \(0.318536\pi\)
\(500\) 0 0
\(501\) −9.00000 −0.402090
\(502\) 0 0
\(503\) −22.7519 −1.01446 −0.507229 0.861812i \(-0.669330\pi\)
−0.507229 + 0.861812i \(0.669330\pi\)
\(504\) 0 0
\(505\) −47.8655 −2.12999
\(506\) 0 0
\(507\) −12.9812 −0.576513
\(508\) 0 0
\(509\) −15.7581 −0.698466 −0.349233 0.937036i \(-0.613558\pi\)
−0.349233 + 0.937036i \(0.613558\pi\)
\(510\) 0 0
\(511\) 0.340078 0.0150442
\(512\) 0 0
\(513\) −1.00000 −0.0441511
\(514\) 0 0
\(515\) −15.8188 −0.697060
\(516\) 0 0
\(517\) 31.7986 1.39850
\(518\) 0 0
\(519\) 15.1637 0.665614
\(520\) 0 0
\(521\) 9.45090 0.414052 0.207026 0.978335i \(-0.433622\pi\)
0.207026 + 0.978335i \(0.433622\pi\)
\(522\) 0 0
\(523\) −4.93447 −0.215769 −0.107885 0.994163i \(-0.534408\pi\)
−0.107885 + 0.994163i \(0.534408\pi\)
\(524\) 0 0
\(525\) −1.13727 −0.0496347
\(526\) 0 0
\(527\) 12.0655 0.525583
\(528\) 0 0
\(529\) 1.00000 0.0434783
\(530\) 0 0
\(531\) −5.13727 −0.222939
\(532\) 0 0
\(533\) 0.505192 0.0218823
\(534\) 0 0
\(535\) −2.24949 −0.0972538
\(536\) 0 0
\(537\) 16.3868 0.707141
\(538\) 0 0
\(539\) −5.95470 −0.256487
\(540\) 0 0
\(541\) 43.8391 1.88479 0.942394 0.334505i \(-0.108569\pi\)
0.942394 + 0.334505i \(0.108569\pi\)
\(542\) 0 0
\(543\) 11.0202 0.472923
\(544\) 0 0
\(545\) 16.7268 0.716499
\(546\) 0 0
\(547\) −9.61602 −0.411151 −0.205576 0.978641i \(-0.565907\pi\)
−0.205576 + 0.978641i \(0.565907\pi\)
\(548\) 0 0
\(549\) 4.52265 0.193022
\(550\) 0 0
\(551\) 1.65992 0.0707151
\(552\) 0 0
\(553\) −12.5240 −0.532576
\(554\) 0 0
\(555\) −28.6613 −1.21660
\(556\) 0 0
\(557\) −1.90663 −0.0807866 −0.0403933 0.999184i \(-0.512861\pi\)
−0.0403933 + 0.999184i \(0.512861\pi\)
\(558\) 0 0
\(559\) 1.52265 0.0644011
\(560\) 0 0
\(561\) −27.4787 −1.16015
\(562\) 0 0
\(563\) −15.3665 −0.647622 −0.323811 0.946122i \(-0.604964\pi\)
−0.323811 + 0.946122i \(0.604964\pi\)
\(564\) 0 0
\(565\) 46.2306 1.94494
\(566\) 0 0
\(567\) −1.00000 −0.0419961
\(568\) 0 0
\(569\) −18.1637 −0.761463 −0.380731 0.924686i \(-0.624328\pi\)
−0.380731 + 0.924686i \(0.624328\pi\)
\(570\) 0 0
\(571\) 16.4334 0.687718 0.343859 0.939021i \(-0.388266\pi\)
0.343859 + 0.939021i \(0.388266\pi\)
\(572\) 0 0
\(573\) −8.02023 −0.335050
\(574\) 0 0
\(575\) −1.13727 −0.0474276
\(576\) 0 0
\(577\) 26.3679 1.09771 0.548855 0.835917i \(-0.315064\pi\)
0.548855 + 0.835917i \(0.315064\pi\)
\(578\) 0 0
\(579\) −3.88918 −0.161629
\(580\) 0 0
\(581\) 1.65992 0.0688652
\(582\) 0 0
\(583\) 30.5910 1.26695
\(584\) 0 0
\(585\) −0.340078 −0.0140605
\(586\) 0 0
\(587\) 3.11221 0.128455 0.0642274 0.997935i \(-0.479542\pi\)
0.0642274 + 0.997935i \(0.479542\pi\)
\(588\) 0 0
\(589\) −2.61463 −0.107734
\(590\) 0 0
\(591\) 1.09198 0.0449180
\(592\) 0 0
\(593\) 10.2745 0.421925 0.210963 0.977494i \(-0.432340\pi\)
0.210963 + 0.977494i \(0.432340\pi\)
\(594\) 0 0
\(595\) −11.4321 −0.468669
\(596\) 0 0
\(597\) 25.2306 1.03262
\(598\) 0 0
\(599\) −17.9032 −0.731505 −0.365752 0.930712i \(-0.619188\pi\)
−0.365752 + 0.930712i \(0.619188\pi\)
\(600\) 0 0
\(601\) −1.06692 −0.0435205 −0.0217602 0.999763i \(-0.506927\pi\)
−0.0217602 + 0.999763i \(0.506927\pi\)
\(602\) 0 0
\(603\) −2.47735 −0.100886
\(604\) 0 0
\(605\) 60.5923 2.46343
\(606\) 0 0
\(607\) −3.77213 −0.153106 −0.0765531 0.997066i \(-0.524391\pi\)
−0.0765531 + 0.997066i \(0.524391\pi\)
\(608\) 0 0
\(609\) 1.65992 0.0672634
\(610\) 0 0
\(611\) 0.733057 0.0296563
\(612\) 0 0
\(613\) −3.97494 −0.160546 −0.0802731 0.996773i \(-0.525579\pi\)
−0.0802731 + 0.996773i \(0.525579\pi\)
\(614\) 0 0
\(615\) −9.11704 −0.367635
\(616\) 0 0
\(617\) 25.2355 1.01594 0.507971 0.861374i \(-0.330396\pi\)
0.507971 + 0.861374i \(0.330396\pi\)
\(618\) 0 0
\(619\) −2.23064 −0.0896571 −0.0448286 0.998995i \(-0.514274\pi\)
−0.0448286 + 0.998995i \(0.514274\pi\)
\(620\) 0 0
\(621\) −1.00000 −0.0401286
\(622\) 0 0
\(623\) −3.86273 −0.154757
\(624\) 0 0
\(625\) −29.3930 −1.17572
\(626\) 0 0
\(627\) 5.95470 0.237808
\(628\) 0 0
\(629\) −53.3882 −2.12872
\(630\) 0 0
\(631\) 6.47113 0.257612 0.128806 0.991670i \(-0.458886\pi\)
0.128806 + 0.991670i \(0.458886\pi\)
\(632\) 0 0
\(633\) 17.9547 0.713635
\(634\) 0 0
\(635\) 10.2996 0.408728
\(636\) 0 0
\(637\) −0.137275 −0.00543902
\(638\) 0 0
\(639\) 11.3665 0.449653
\(640\) 0 0
\(641\) −22.7471 −0.898455 −0.449228 0.893417i \(-0.648301\pi\)
−0.449228 + 0.893417i \(0.648301\pi\)
\(642\) 0 0
\(643\) 19.5707 0.771794 0.385897 0.922542i \(-0.373892\pi\)
0.385897 + 0.922542i \(0.373892\pi\)
\(644\) 0 0
\(645\) −27.4787 −1.08197
\(646\) 0 0
\(647\) −23.9812 −0.942797 −0.471398 0.881920i \(-0.656251\pi\)
−0.471398 + 0.881920i \(0.656251\pi\)
\(648\) 0 0
\(649\) 30.5910 1.20080
\(650\) 0 0
\(651\) −2.61463 −0.102475
\(652\) 0 0
\(653\) −15.1198 −0.591684 −0.295842 0.955237i \(-0.595600\pi\)
−0.295842 + 0.955237i \(0.595600\pi\)
\(654\) 0 0
\(655\) −35.2509 −1.37737
\(656\) 0 0
\(657\) −0.340078 −0.0132677
\(658\) 0 0
\(659\) 17.5693 0.684404 0.342202 0.939626i \(-0.388827\pi\)
0.342202 + 0.939626i \(0.388827\pi\)
\(660\) 0 0
\(661\) −43.7331 −1.70102 −0.850509 0.525960i \(-0.823706\pi\)
−0.850509 + 0.525960i \(0.823706\pi\)
\(662\) 0 0
\(663\) −0.633471 −0.0246020
\(664\) 0 0
\(665\) 2.47735 0.0960676
\(666\) 0 0
\(667\) 1.65992 0.0642724
\(668\) 0 0
\(669\) −28.0216 −1.08338
\(670\) 0 0
\(671\) −26.9310 −1.03966
\(672\) 0 0
\(673\) 7.09059 0.273322 0.136661 0.990618i \(-0.456363\pi\)
0.136661 + 0.990618i \(0.456363\pi\)
\(674\) 0 0
\(675\) 1.13727 0.0437737
\(676\) 0 0
\(677\) −6.67394 −0.256500 −0.128250 0.991742i \(-0.540936\pi\)
−0.128250 + 0.991742i \(0.540936\pi\)
\(678\) 0 0
\(679\) 2.88918 0.110876
\(680\) 0 0
\(681\) 4.47735 0.171573
\(682\) 0 0
\(683\) 39.5721 1.51418 0.757092 0.653308i \(-0.226619\pi\)
0.757092 + 0.653308i \(0.226619\pi\)
\(684\) 0 0
\(685\) −18.0216 −0.688571
\(686\) 0 0
\(687\) −4.49759 −0.171594
\(688\) 0 0
\(689\) 0.705218 0.0268667
\(690\) 0 0
\(691\) −24.4146 −0.928775 −0.464388 0.885632i \(-0.653726\pi\)
−0.464388 + 0.885632i \(0.653726\pi\)
\(692\) 0 0
\(693\) 5.95470 0.226201
\(694\) 0 0
\(695\) −38.2306 −1.45017
\(696\) 0 0
\(697\) −16.9825 −0.643260
\(698\) 0 0
\(699\) 11.7066 0.442785
\(700\) 0 0
\(701\) −32.3693 −1.22257 −0.611286 0.791410i \(-0.709347\pi\)
−0.611286 + 0.791410i \(0.709347\pi\)
\(702\) 0 0
\(703\) 11.5693 0.436346
\(704\) 0 0
\(705\) −13.2293 −0.498243
\(706\) 0 0
\(707\) 19.3212 0.726650
\(708\) 0 0
\(709\) −34.5457 −1.29739 −0.648695 0.761049i \(-0.724685\pi\)
−0.648695 + 0.761049i \(0.724685\pi\)
\(710\) 0 0
\(711\) 12.5240 0.469688
\(712\) 0 0
\(713\) −2.61463 −0.0979186
\(714\) 0 0
\(715\) 2.02506 0.0757330
\(716\) 0 0
\(717\) −2.34630 −0.0876240
\(718\) 0 0
\(719\) −25.0202 −0.933097 −0.466549 0.884496i \(-0.654503\pi\)
−0.466549 + 0.884496i \(0.654503\pi\)
\(720\) 0 0
\(721\) 6.38537 0.237804
\(722\) 0 0
\(723\) 1.27455 0.0474010
\(724\) 0 0
\(725\) −1.88779 −0.0701107
\(726\) 0 0
\(727\) 33.5491 1.24427 0.622134 0.782911i \(-0.286266\pi\)
0.622134 + 0.782911i \(0.286266\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −51.1853 −1.89316
\(732\) 0 0
\(733\) −4.73306 −0.174819 −0.0874097 0.996172i \(-0.527859\pi\)
−0.0874097 + 0.996172i \(0.527859\pi\)
\(734\) 0 0
\(735\) 2.47735 0.0913785
\(736\) 0 0
\(737\) 14.7519 0.543393
\(738\) 0 0
\(739\) −29.0202 −1.06753 −0.533763 0.845634i \(-0.679223\pi\)
−0.533763 + 0.845634i \(0.679223\pi\)
\(740\) 0 0
\(741\) 0.137275 0.00504291
\(742\) 0 0
\(743\) 11.2279 0.411910 0.205955 0.978561i \(-0.433970\pi\)
0.205955 + 0.978561i \(0.433970\pi\)
\(744\) 0 0
\(745\) −37.7784 −1.38409
\(746\) 0 0
\(747\) −1.65992 −0.0607334
\(748\) 0 0
\(749\) 0.908021 0.0331784
\(750\) 0 0
\(751\) −19.3010 −0.704304 −0.352152 0.935943i \(-0.614550\pi\)
−0.352152 + 0.935943i \(0.614550\pi\)
\(752\) 0 0
\(753\) 16.1840 0.589776
\(754\) 0 0
\(755\) −41.0481 −1.49389
\(756\) 0 0
\(757\) −7.65992 −0.278405 −0.139202 0.990264i \(-0.544454\pi\)
−0.139202 + 0.990264i \(0.544454\pi\)
\(758\) 0 0
\(759\) 5.95470 0.216142
\(760\) 0 0
\(761\) 22.9345 0.831374 0.415687 0.909508i \(-0.363541\pi\)
0.415687 + 0.909508i \(0.363541\pi\)
\(762\) 0 0
\(763\) −6.75190 −0.244435
\(764\) 0 0
\(765\) 11.4321 0.413327
\(766\) 0 0
\(767\) 0.705218 0.0254639
\(768\) 0 0
\(769\) 54.6425 1.97046 0.985229 0.171243i \(-0.0547782\pi\)
0.985229 + 0.171243i \(0.0547782\pi\)
\(770\) 0 0
\(771\) −25.1184 −0.904618
\(772\) 0 0
\(773\) −8.02784 −0.288741 −0.144371 0.989524i \(-0.546116\pi\)
−0.144371 + 0.989524i \(0.546116\pi\)
\(774\) 0 0
\(775\) 2.97355 0.106813
\(776\) 0 0
\(777\) 11.5693 0.415047
\(778\) 0 0
\(779\) 3.68016 0.131855
\(780\) 0 0
\(781\) −67.6843 −2.42194
\(782\) 0 0
\(783\) −1.65992 −0.0593208
\(784\) 0 0
\(785\) −1.63486 −0.0583507
\(786\) 0 0
\(787\) 43.3617 1.54568 0.772839 0.634602i \(-0.218836\pi\)
0.772839 + 0.634602i \(0.218836\pi\)
\(788\) 0 0
\(789\) 18.1387 0.645754
\(790\) 0 0
\(791\) −18.6613 −0.663520
\(792\) 0 0
\(793\) −0.620845 −0.0220468
\(794\) 0 0
\(795\) −12.7268 −0.451374
\(796\) 0 0
\(797\) 44.8641 1.58917 0.794584 0.607154i \(-0.207689\pi\)
0.794584 + 0.607154i \(0.207689\pi\)
\(798\) 0 0
\(799\) −24.6425 −0.871788
\(800\) 0 0
\(801\) 3.86273 0.136483
\(802\) 0 0
\(803\) 2.02506 0.0714629
\(804\) 0 0
\(805\) 2.47735 0.0873152
\(806\) 0 0
\(807\) 30.1401 1.06098
\(808\) 0 0
\(809\) −5.40422 −0.190002 −0.0950011 0.995477i \(-0.530285\pi\)
−0.0950011 + 0.995477i \(0.530285\pi\)
\(810\) 0 0
\(811\) 32.7485 1.14995 0.574977 0.818170i \(-0.305011\pi\)
0.574977 + 0.818170i \(0.305011\pi\)
\(812\) 0 0
\(813\) 0.209021 0.00733068
\(814\) 0 0
\(815\) −57.7749 −2.02377
\(816\) 0 0
\(817\) 11.0920 0.388059
\(818\) 0 0
\(819\) 0.137275 0.00479676
\(820\) 0 0
\(821\) 14.9142 0.520511 0.260255 0.965540i \(-0.416193\pi\)
0.260255 + 0.965540i \(0.416193\pi\)
\(822\) 0 0
\(823\) 4.00139 0.139480 0.0697398 0.997565i \(-0.477783\pi\)
0.0697398 + 0.997565i \(0.477783\pi\)
\(824\) 0 0
\(825\) −6.77213 −0.235775
\(826\) 0 0
\(827\) 6.91563 0.240480 0.120240 0.992745i \(-0.461634\pi\)
0.120240 + 0.992745i \(0.461634\pi\)
\(828\) 0 0
\(829\) 15.7128 0.545729 0.272864 0.962052i \(-0.412029\pi\)
0.272864 + 0.962052i \(0.412029\pi\)
\(830\) 0 0
\(831\) −10.7972 −0.374551
\(832\) 0 0
\(833\) 4.61463 0.159887
\(834\) 0 0
\(835\) −22.2962 −0.771591
\(836\) 0 0
\(837\) 2.61463 0.0903747
\(838\) 0 0
\(839\) −37.9966 −1.31179 −0.655893 0.754853i \(-0.727708\pi\)
−0.655893 + 0.754853i \(0.727708\pi\)
\(840\) 0 0
\(841\) −26.2447 −0.904988
\(842\) 0 0
\(843\) −26.2745 −0.904944
\(844\) 0 0
\(845\) −32.1589 −1.10630
\(846\) 0 0
\(847\) −24.4585 −0.840404
\(848\) 0 0
\(849\) 11.7066 0.401770
\(850\) 0 0
\(851\) 11.5693 0.396592
\(852\) 0 0
\(853\) 21.7659 0.745251 0.372625 0.927982i \(-0.378458\pi\)
0.372625 + 0.927982i \(0.378458\pi\)
\(854\) 0 0
\(855\) −2.47735 −0.0847237
\(856\) 0 0
\(857\) −54.9777 −1.87800 −0.939001 0.343913i \(-0.888247\pi\)
−0.939001 + 0.343913i \(0.888247\pi\)
\(858\) 0 0
\(859\) 54.4084 1.85639 0.928195 0.372094i \(-0.121360\pi\)
0.928195 + 0.372094i \(0.121360\pi\)
\(860\) 0 0
\(861\) 3.68016 0.125419
\(862\) 0 0
\(863\) −7.81604 −0.266061 −0.133031 0.991112i \(-0.542471\pi\)
−0.133031 + 0.991112i \(0.542471\pi\)
\(864\) 0 0
\(865\) 37.5659 1.27728
\(866\) 0 0
\(867\) 4.29478 0.145858
\(868\) 0 0
\(869\) −74.5769 −2.52985
\(870\) 0 0
\(871\) 0.340078 0.0115231
\(872\) 0 0
\(873\) −2.88918 −0.0977838
\(874\) 0 0
\(875\) 9.56933 0.323502
\(876\) 0 0
\(877\) 45.2118 1.52669 0.763347 0.645989i \(-0.223555\pi\)
0.763347 + 0.645989i \(0.223555\pi\)
\(878\) 0 0
\(879\) −22.4585 −0.757507
\(880\) 0 0
\(881\) 37.5443 1.26490 0.632449 0.774602i \(-0.282050\pi\)
0.632449 + 0.774602i \(0.282050\pi\)
\(882\) 0 0
\(883\) −23.5910 −0.793899 −0.396949 0.917840i \(-0.629931\pi\)
−0.396949 + 0.917840i \(0.629931\pi\)
\(884\) 0 0
\(885\) −12.7268 −0.427808
\(886\) 0 0
\(887\) −42.4118 −1.42405 −0.712025 0.702154i \(-0.752222\pi\)
−0.712025 + 0.702154i \(0.752222\pi\)
\(888\) 0 0
\(889\) −4.15751 −0.139438
\(890\) 0 0
\(891\) −5.95470 −0.199490
\(892\) 0 0
\(893\) 5.34008 0.178699
\(894\) 0 0
\(895\) 40.5958 1.35697
\(896\) 0 0
\(897\) 0.137275 0.00458347
\(898\) 0 0
\(899\) −4.34008 −0.144750
\(900\) 0 0
\(901\) −23.7066 −0.789782
\(902\) 0 0
\(903\) 11.0920 0.369118
\(904\) 0 0
\(905\) 27.3010 0.907516
\(906\) 0 0
\(907\) 40.0745 1.33065 0.665326 0.746553i \(-0.268292\pi\)
0.665326 + 0.746553i \(0.268292\pi\)
\(908\) 0 0
\(909\) −19.3212 −0.640845
\(910\) 0 0
\(911\) 13.0858 0.433551 0.216775 0.976222i \(-0.430446\pi\)
0.216775 + 0.976222i \(0.430446\pi\)
\(912\) 0 0
\(913\) 9.88435 0.327124
\(914\) 0 0
\(915\) 11.2042 0.370399
\(916\) 0 0
\(917\) 14.2293 0.469891
\(918\) 0 0
\(919\) 49.6599 1.63813 0.819065 0.573701i \(-0.194493\pi\)
0.819065 + 0.573701i \(0.194493\pi\)
\(920\) 0 0
\(921\) −11.9345 −0.393254
\(922\) 0 0
\(923\) −1.56034 −0.0513591
\(924\) 0 0
\(925\) −13.1575 −0.432616
\(926\) 0 0
\(927\) −6.38537 −0.209723
\(928\) 0 0
\(929\) 29.3944 0.964398 0.482199 0.876062i \(-0.339838\pi\)
0.482199 + 0.876062i \(0.339838\pi\)
\(930\) 0 0
\(931\) −1.00000 −0.0327737
\(932\) 0 0
\(933\) 7.75190 0.253786
\(934\) 0 0
\(935\) −68.0745 −2.22627
\(936\) 0 0
\(937\) 44.2697 1.44623 0.723114 0.690728i \(-0.242710\pi\)
0.723114 + 0.690728i \(0.242710\pi\)
\(938\) 0 0
\(939\) 6.93447 0.226298
\(940\) 0 0
\(941\) −15.2697 −0.497779 −0.248889 0.968532i \(-0.580066\pi\)
−0.248889 + 0.968532i \(0.580066\pi\)
\(942\) 0 0
\(943\) 3.68016 0.119842
\(944\) 0 0
\(945\) −2.47735 −0.0805883
\(946\) 0 0
\(947\) 25.2293 0.819841 0.409920 0.912121i \(-0.365557\pi\)
0.409920 + 0.912121i \(0.365557\pi\)
\(948\) 0 0
\(949\) 0.0466840 0.00151543
\(950\) 0 0
\(951\) −26.3868 −0.855649
\(952\) 0 0
\(953\) 27.1527 0.879562 0.439781 0.898105i \(-0.355056\pi\)
0.439781 + 0.898105i \(0.355056\pi\)
\(954\) 0 0
\(955\) −19.8689 −0.642944
\(956\) 0 0
\(957\) 9.88435 0.319516
\(958\) 0 0
\(959\) 7.27455 0.234907
\(960\) 0 0
\(961\) −24.1637 −0.779475
\(962\) 0 0
\(963\) −0.908021 −0.0292606
\(964\) 0 0
\(965\) −9.63486 −0.310157
\(966\) 0 0
\(967\) 58.9575 1.89594 0.947972 0.318353i \(-0.103130\pi\)
0.947972 + 0.318353i \(0.103130\pi\)
\(968\) 0 0
\(969\) −4.61463 −0.148243
\(970\) 0 0
\(971\) −40.2306 −1.29106 −0.645531 0.763734i \(-0.723364\pi\)
−0.645531 + 0.763734i \(0.723364\pi\)
\(972\) 0 0
\(973\) 15.4321 0.494729
\(974\) 0 0
\(975\) −0.156119 −0.00499981
\(976\) 0 0
\(977\) −0.0669171 −0.00214087 −0.00107043 0.999999i \(-0.500341\pi\)
−0.00107043 + 0.999999i \(0.500341\pi\)
\(978\) 0 0
\(979\) −23.0014 −0.735128
\(980\) 0 0
\(981\) 6.75190 0.215572
\(982\) 0 0
\(983\) 38.0278 1.21290 0.606450 0.795122i \(-0.292593\pi\)
0.606450 + 0.795122i \(0.292593\pi\)
\(984\) 0 0
\(985\) 2.70522 0.0861954
\(986\) 0 0
\(987\) 5.34008 0.169977
\(988\) 0 0
\(989\) 11.0920 0.352704
\(990\) 0 0
\(991\) −34.6565 −1.10090 −0.550450 0.834868i \(-0.685544\pi\)
−0.550450 + 0.834868i \(0.685544\pi\)
\(992\) 0 0
\(993\) −7.61463 −0.241643
\(994\) 0 0
\(995\) 62.5052 1.98155
\(996\) 0 0
\(997\) −21.1637 −0.670262 −0.335131 0.942172i \(-0.608781\pi\)
−0.335131 + 0.942172i \(0.608781\pi\)
\(998\) 0 0
\(999\) −11.5693 −0.366037
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.bv.1.3 3
4.3 odd 2 1932.2.a.i.1.3 3
12.11 even 2 5796.2.a.p.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1932.2.a.i.1.3 3 4.3 odd 2
5796.2.a.p.1.1 3 12.11 even 2
7728.2.a.bv.1.3 3 1.1 even 1 trivial