Properties

Label 7800.2.a.bu.1.4
Level $7800$
Weight $2$
Character 7800.1
Self dual yes
Analytic conductor $62.283$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 7800 = 2^{3} \cdot 3 \cdot 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7800.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(62.2833135766\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.461844.1
Defining polynomial: \(x^{4} - x^{3} - 14 x^{2} - 12 x + 6\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(4.57275\) of defining polynomial
Character \(\chi\) \(=\) 7800.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{3} +3.57275 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{3} +3.57275 q^{7} +1.00000 q^{9} +5.26846 q^{11} -1.00000 q^{13} -7.06883 q^{17} +5.76455 q^{19} -3.57275 q^{21} -2.30429 q^{23} -1.00000 q^{27} -8.14550 q^{29} +5.87704 q^{31} -5.26846 q^{33} -3.38095 q^{37} +1.00000 q^{39} +4.77238 q^{41} +7.44979 q^{43} +9.03301 q^{47} +5.76455 q^{49} +7.06883 q^{51} +5.45762 q^{53} -5.76455 q^{57} -8.56492 q^{59} +8.07667 q^{61} +3.57275 q^{63} +8.95370 q^{67} +2.30429 q^{69} +9.76455 q^{71} +12.4420 q^{73} +18.8229 q^{77} +1.38095 q^{79} +1.00000 q^{81} -9.57275 q^{83} +8.14550 q^{87} -4.53693 q^{89} -3.57275 q^{91} -5.87704 q^{93} -7.14550 q^{97} +5.26846 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 4q^{3} - 3q^{7} + 4q^{9} + O(q^{10}) \) \( 4q - 4q^{3} - 3q^{7} + 4q^{9} + 5q^{11} - 4q^{13} - 7q^{17} + 3q^{19} + 3q^{21} - 8q^{23} - 4q^{27} + 2q^{29} + 5q^{31} - 5q^{33} + q^{37} + 4q^{39} + 7q^{41} - 6q^{43} + 3q^{49} + 7q^{51} - 6q^{53} - 3q^{57} - 9q^{59} + 19q^{61} - 3q^{63} + 4q^{67} + 8q^{69} + 19q^{71} + 6q^{73} + 17q^{77} - 9q^{79} + 4q^{81} - 21q^{83} - 2q^{87} + 14q^{89} + 3q^{91} - 5q^{93} + 6q^{97} + 5q^{99} + O(q^{100}) \)

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
\(5\) 0 0
\(6\) 0 0
\(7\) 3.57275 1.35037 0.675186 0.737647i \(-0.264063\pi\)
0.675186 + 0.737647i \(0.264063\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 5.26846 1.58850 0.794251 0.607590i \(-0.207864\pi\)
0.794251 + 0.607590i \(0.207864\pi\)
\(12\) 0 0
\(13\) −1.00000 −0.277350
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −7.06883 −1.71444 −0.857222 0.514947i \(-0.827812\pi\)
−0.857222 + 0.514947i \(0.827812\pi\)
\(18\) 0 0
\(19\) 5.76455 1.32248 0.661239 0.750175i \(-0.270031\pi\)
0.661239 + 0.750175i \(0.270031\pi\)
\(20\) 0 0
\(21\) −3.57275 −0.779638
\(22\) 0 0
\(23\) −2.30429 −0.480477 −0.240238 0.970714i \(-0.577226\pi\)
−0.240238 + 0.970714i \(0.577226\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) −8.14550 −1.51258 −0.756291 0.654236i \(-0.772990\pi\)
−0.756291 + 0.654236i \(0.772990\pi\)
\(30\) 0 0
\(31\) 5.87704 1.05555 0.527774 0.849385i \(-0.323027\pi\)
0.527774 + 0.849385i \(0.323027\pi\)
\(32\) 0 0
\(33\) −5.26846 −0.917122
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −3.38095 −0.555825 −0.277913 0.960606i \(-0.589643\pi\)
−0.277913 + 0.960606i \(0.589643\pi\)
\(38\) 0 0
\(39\) 1.00000 0.160128
\(40\) 0 0
\(41\) 4.77238 0.745321 0.372660 0.927968i \(-0.378446\pi\)
0.372660 + 0.927968i \(0.378446\pi\)
\(42\) 0 0
\(43\) 7.44979 1.13608 0.568041 0.823000i \(-0.307701\pi\)
0.568041 + 0.823000i \(0.307701\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 9.03301 1.31760 0.658800 0.752318i \(-0.271064\pi\)
0.658800 + 0.752318i \(0.271064\pi\)
\(48\) 0 0
\(49\) 5.76455 0.823507
\(50\) 0 0
\(51\) 7.06883 0.989835
\(52\) 0 0
\(53\) 5.45762 0.749662 0.374831 0.927093i \(-0.377701\pi\)
0.374831 + 0.927093i \(0.377701\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −5.76455 −0.763533
\(58\) 0 0
\(59\) −8.56492 −1.11506 −0.557529 0.830158i \(-0.688250\pi\)
−0.557529 + 0.830158i \(0.688250\pi\)
\(60\) 0 0
\(61\) 8.07667 1.03411 0.517056 0.855952i \(-0.327028\pi\)
0.517056 + 0.855952i \(0.327028\pi\)
\(62\) 0 0
\(63\) 3.57275 0.450124
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 8.95370 1.09387 0.546935 0.837175i \(-0.315795\pi\)
0.546935 + 0.837175i \(0.315795\pi\)
\(68\) 0 0
\(69\) 2.30429 0.277404
\(70\) 0 0
\(71\) 9.76455 1.15884 0.579419 0.815030i \(-0.303279\pi\)
0.579419 + 0.815030i \(0.303279\pi\)
\(72\) 0 0
\(73\) 12.4420 1.45622 0.728110 0.685460i \(-0.240399\pi\)
0.728110 + 0.685460i \(0.240399\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 18.8229 2.14507
\(78\) 0 0
\(79\) 1.38095 0.155369 0.0776847 0.996978i \(-0.475247\pi\)
0.0776847 + 0.996978i \(0.475247\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −9.57275 −1.05075 −0.525373 0.850872i \(-0.676074\pi\)
−0.525373 + 0.850872i \(0.676074\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 8.14550 0.873289
\(88\) 0 0
\(89\) −4.53693 −0.480913 −0.240457 0.970660i \(-0.577297\pi\)
−0.240457 + 0.970660i \(0.577297\pi\)
\(90\) 0 0
\(91\) −3.57275 −0.374526
\(92\) 0 0
\(93\) −5.87704 −0.609420
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −7.14550 −0.725516 −0.362758 0.931883i \(-0.618165\pi\)
−0.362758 + 0.931883i \(0.618165\pi\)
\(98\) 0 0
\(99\) 5.26846 0.529501
\(100\) 0 0
\(101\) −0.380954 −0.0379064 −0.0189532 0.999820i \(-0.506033\pi\)
−0.0189532 + 0.999820i \(0.506033\pi\)
\(102\) 0 0
\(103\) −14.0584 −1.38521 −0.692606 0.721316i \(-0.743537\pi\)
−0.692606 + 0.721316i \(0.743537\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −5.68524 −0.549613 −0.274806 0.961500i \(-0.588614\pi\)
−0.274806 + 0.961500i \(0.588614\pi\)
\(108\) 0 0
\(109\) −1.61121 −0.154326 −0.0771631 0.997018i \(-0.524586\pi\)
−0.0771631 + 0.997018i \(0.524586\pi\)
\(110\) 0 0
\(111\) 3.38095 0.320906
\(112\) 0 0
\(113\) 0.383593 0.0360854 0.0180427 0.999837i \(-0.494257\pi\)
0.0180427 + 0.999837i \(0.494257\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) −1.00000 −0.0924500
\(118\) 0 0
\(119\) −25.2552 −2.31514
\(120\) 0 0
\(121\) 16.7567 1.52334
\(122\) 0 0
\(123\) −4.77238 −0.430311
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) −14.5265 −1.28901 −0.644507 0.764598i \(-0.722937\pi\)
−0.644507 + 0.764598i \(0.722937\pi\)
\(128\) 0 0
\(129\) −7.44979 −0.655917
\(130\) 0 0
\(131\) 2.84403 0.248484 0.124242 0.992252i \(-0.460350\pi\)
0.124242 + 0.992252i \(0.460350\pi\)
\(132\) 0 0
\(133\) 20.5953 1.78584
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −2.46809 −0.210864 −0.105432 0.994427i \(-0.533622\pi\)
−0.105432 + 0.994427i \(0.533622\pi\)
\(138\) 0 0
\(139\) 6.13767 0.520590 0.260295 0.965529i \(-0.416180\pi\)
0.260295 + 0.965529i \(0.416180\pi\)
\(140\) 0 0
\(141\) −9.03301 −0.760717
\(142\) 0 0
\(143\) −5.26846 −0.440571
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) −5.76455 −0.475452
\(148\) 0 0
\(149\) −3.14550 −0.257690 −0.128845 0.991665i \(-0.541127\pi\)
−0.128845 + 0.991665i \(0.541127\pi\)
\(150\) 0 0
\(151\) −1.95634 −0.159205 −0.0796025 0.996827i \(-0.525365\pi\)
−0.0796025 + 0.996827i \(0.525365\pi\)
\(152\) 0 0
\(153\) −7.06883 −0.571481
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 5.23545 0.417835 0.208917 0.977933i \(-0.433006\pi\)
0.208917 + 0.977933i \(0.433006\pi\)
\(158\) 0 0
\(159\) −5.45762 −0.432818
\(160\) 0 0
\(161\) −8.23264 −0.648823
\(162\) 0 0
\(163\) −15.2248 −1.19250 −0.596249 0.802799i \(-0.703343\pi\)
−0.596249 + 0.802799i \(0.703343\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 21.6720 1.67703 0.838513 0.544881i \(-0.183425\pi\)
0.838513 + 0.544881i \(0.183425\pi\)
\(168\) 0 0
\(169\) 1.00000 0.0769231
\(170\) 0 0
\(171\) 5.76455 0.440826
\(172\) 0 0
\(173\) 16.7617 1.27437 0.637186 0.770710i \(-0.280098\pi\)
0.637186 + 0.770710i \(0.280098\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 8.56492 0.643779
\(178\) 0 0
\(179\) 23.1322 1.72898 0.864492 0.502647i \(-0.167641\pi\)
0.864492 + 0.502647i \(0.167641\pi\)
\(180\) 0 0
\(181\) −3.91005 −0.290632 −0.145316 0.989385i \(-0.546420\pi\)
−0.145316 + 0.989385i \(0.546420\pi\)
\(182\) 0 0
\(183\) −8.07667 −0.597044
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) −37.2419 −2.72340
\(188\) 0 0
\(189\) −3.57275 −0.259879
\(190\) 0 0
\(191\) −4.29100 −0.310486 −0.155243 0.987876i \(-0.549616\pi\)
−0.155243 + 0.987876i \(0.549616\pi\)
\(192\) 0 0
\(193\) −26.5796 −1.91324 −0.956622 0.291334i \(-0.905901\pi\)
−0.956622 + 0.291334i \(0.905901\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 2.09497 0.149261 0.0746304 0.997211i \(-0.476222\pi\)
0.0746304 + 0.997211i \(0.476222\pi\)
\(198\) 0 0
\(199\) −3.93117 −0.278673 −0.139336 0.990245i \(-0.544497\pi\)
−0.139336 + 0.990245i \(0.544497\pi\)
\(200\) 0 0
\(201\) −8.95370 −0.631546
\(202\) 0 0
\(203\) −29.1018 −2.04255
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) −2.30429 −0.160159
\(208\) 0 0
\(209\) 30.3703 2.10076
\(210\) 0 0
\(211\) 6.30429 0.434005 0.217002 0.976171i \(-0.430372\pi\)
0.217002 + 0.976171i \(0.430372\pi\)
\(212\) 0 0
\(213\) −9.76455 −0.669056
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 20.9972 1.42538
\(218\) 0 0
\(219\) −12.4420 −0.840749
\(220\) 0 0
\(221\) 7.06883 0.475501
\(222\) 0 0
\(223\) 20.7696 1.39083 0.695417 0.718607i \(-0.255220\pi\)
0.695417 + 0.718607i \(0.255220\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −26.9352 −1.78775 −0.893877 0.448313i \(-0.852025\pi\)
−0.893877 + 0.448313i \(0.852025\pi\)
\(228\) 0 0
\(229\) 25.7513 1.70169 0.850846 0.525416i \(-0.176090\pi\)
0.850846 + 0.525416i \(0.176090\pi\)
\(230\) 0 0
\(231\) −18.8229 −1.23846
\(232\) 0 0
\(233\) 2.53693 0.166200 0.0830998 0.996541i \(-0.473518\pi\)
0.0830998 + 0.996541i \(0.473518\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) −1.38095 −0.0897026
\(238\) 0 0
\(239\) 20.8559 1.34906 0.674529 0.738249i \(-0.264347\pi\)
0.674529 + 0.738249i \(0.264347\pi\)
\(240\) 0 0
\(241\) 5.69571 0.366893 0.183447 0.983030i \(-0.441275\pi\)
0.183447 + 0.983030i \(0.441275\pi\)
\(242\) 0 0
\(243\) −1.00000 −0.0641500
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −5.76455 −0.366789
\(248\) 0 0
\(249\) 9.57275 0.606648
\(250\) 0 0
\(251\) 14.7724 0.932424 0.466212 0.884673i \(-0.345618\pi\)
0.466212 + 0.884673i \(0.345618\pi\)
\(252\) 0 0
\(253\) −12.1401 −0.763238
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −19.5108 −1.21705 −0.608525 0.793535i \(-0.708238\pi\)
−0.608525 + 0.793535i \(0.708238\pi\)
\(258\) 0 0
\(259\) −12.0793 −0.750572
\(260\) 0 0
\(261\) −8.14550 −0.504194
\(262\) 0 0
\(263\) −5.31212 −0.327559 −0.163780 0.986497i \(-0.552369\pi\)
−0.163780 + 0.986497i \(0.552369\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 4.53693 0.277655
\(268\) 0 0
\(269\) 2.23809 0.136459 0.0682294 0.997670i \(-0.478265\pi\)
0.0682294 + 0.997670i \(0.478265\pi\)
\(270\) 0 0
\(271\) 24.4933 1.48786 0.743930 0.668257i \(-0.232960\pi\)
0.743930 + 0.668257i \(0.232960\pi\)
\(272\) 0 0
\(273\) 3.57275 0.216233
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 12.2250 0.734528 0.367264 0.930117i \(-0.380295\pi\)
0.367264 + 0.930117i \(0.380295\pi\)
\(278\) 0 0
\(279\) 5.87704 0.351849
\(280\) 0 0
\(281\) −10.2222 −0.609803 −0.304902 0.952384i \(-0.598624\pi\)
−0.304902 + 0.952384i \(0.598624\pi\)
\(282\) 0 0
\(283\) 25.0372 1.48831 0.744155 0.668007i \(-0.232852\pi\)
0.744155 + 0.668007i \(0.232852\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 17.0505 1.00646
\(288\) 0 0
\(289\) 32.9684 1.93932
\(290\) 0 0
\(291\) 7.14550 0.418877
\(292\) 0 0
\(293\) −27.2115 −1.58971 −0.794857 0.606797i \(-0.792454\pi\)
−0.794857 + 0.606797i \(0.792454\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) −5.26846 −0.305707
\(298\) 0 0
\(299\) 2.30429 0.133260
\(300\) 0 0
\(301\) 26.6162 1.53413
\(302\) 0 0
\(303\) 0.380954 0.0218852
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 22.9100 1.30755 0.653773 0.756691i \(-0.273185\pi\)
0.653773 + 0.756691i \(0.273185\pi\)
\(308\) 0 0
\(309\) 14.0584 0.799752
\(310\) 0 0
\(311\) 3.40709 0.193199 0.0965993 0.995323i \(-0.469203\pi\)
0.0965993 + 0.995323i \(0.469203\pi\)
\(312\) 0 0
\(313\) −30.5315 −1.72574 −0.862871 0.505425i \(-0.831336\pi\)
−0.862871 + 0.505425i \(0.831336\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 29.2091 1.64055 0.820274 0.571970i \(-0.193821\pi\)
0.820274 + 0.571970i \(0.193821\pi\)
\(318\) 0 0
\(319\) −42.9143 −2.40274
\(320\) 0 0
\(321\) 5.68524 0.317319
\(322\) 0 0
\(323\) −40.7486 −2.26731
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 1.61121 0.0891002
\(328\) 0 0
\(329\) 32.2727 1.77925
\(330\) 0 0
\(331\) 0.166619 0.00915822 0.00457911 0.999990i \(-0.498542\pi\)
0.00457911 + 0.999990i \(0.498542\pi\)
\(332\) 0 0
\(333\) −3.38095 −0.185275
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 24.6929 1.34511 0.672554 0.740048i \(-0.265197\pi\)
0.672554 + 0.740048i \(0.265197\pi\)
\(338\) 0 0
\(339\) −0.383593 −0.0208339
\(340\) 0 0
\(341\) 30.9630 1.67674
\(342\) 0 0
\(343\) −4.41397 −0.238332
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −9.61360 −0.516085 −0.258042 0.966134i \(-0.583077\pi\)
−0.258042 + 0.966134i \(0.583077\pi\)
\(348\) 0 0
\(349\) −18.1960 −0.974011 −0.487006 0.873399i \(-0.661911\pi\)
−0.487006 + 0.873399i \(0.661911\pi\)
\(350\) 0 0
\(351\) 1.00000 0.0533761
\(352\) 0 0
\(353\) 28.2805 1.50522 0.752610 0.658466i \(-0.228794\pi\)
0.752610 + 0.658466i \(0.228794\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 25.2552 1.33665
\(358\) 0 0
\(359\) 0.742009 0.0391617 0.0195809 0.999808i \(-0.493767\pi\)
0.0195809 + 0.999808i \(0.493767\pi\)
\(360\) 0 0
\(361\) 14.2300 0.748948
\(362\) 0 0
\(363\) −16.7567 −0.879499
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) −3.30693 −0.172620 −0.0863100 0.996268i \(-0.527508\pi\)
−0.0863100 + 0.996268i \(0.527508\pi\)
\(368\) 0 0
\(369\) 4.77238 0.248440
\(370\) 0 0
\(371\) 19.4987 1.01232
\(372\) 0 0
\(373\) −12.0477 −0.623807 −0.311904 0.950114i \(-0.600967\pi\)
−0.311904 + 0.950114i \(0.600967\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 8.14550 0.419515
\(378\) 0 0
\(379\) −11.6521 −0.598526 −0.299263 0.954171i \(-0.596741\pi\)
−0.299263 + 0.954171i \(0.596741\pi\)
\(380\) 0 0
\(381\) 14.5265 0.744213
\(382\) 0 0
\(383\) −25.9022 −1.32354 −0.661771 0.749706i \(-0.730195\pi\)
−0.661771 + 0.749706i \(0.730195\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 7.44979 0.378694
\(388\) 0 0
\(389\) 8.93900 0.453225 0.226613 0.973985i \(-0.427235\pi\)
0.226613 + 0.973985i \(0.427235\pi\)
\(390\) 0 0
\(391\) 16.2886 0.823751
\(392\) 0 0
\(393\) −2.84403 −0.143462
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 10.1561 0.509722 0.254861 0.966978i \(-0.417970\pi\)
0.254861 + 0.966978i \(0.417970\pi\)
\(398\) 0 0
\(399\) −20.5953 −1.03105
\(400\) 0 0
\(401\) −36.3703 −1.81625 −0.908123 0.418703i \(-0.862485\pi\)
−0.908123 + 0.418703i \(0.862485\pi\)
\(402\) 0 0
\(403\) −5.87704 −0.292756
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −17.8124 −0.882930
\(408\) 0 0
\(409\) −24.6613 −1.21942 −0.609712 0.792623i \(-0.708715\pi\)
−0.609712 + 0.792623i \(0.708715\pi\)
\(410\) 0 0
\(411\) 2.46809 0.121742
\(412\) 0 0
\(413\) −30.6003 −1.50574
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) −6.13767 −0.300563
\(418\) 0 0
\(419\) 26.4391 1.29164 0.645818 0.763491i \(-0.276516\pi\)
0.645818 + 0.763491i \(0.276516\pi\)
\(420\) 0 0
\(421\) 14.7619 0.719451 0.359726 0.933058i \(-0.382870\pi\)
0.359726 + 0.933058i \(0.382870\pi\)
\(422\) 0 0
\(423\) 9.03301 0.439200
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 28.8559 1.39644
\(428\) 0 0
\(429\) 5.26846 0.254364
\(430\) 0 0
\(431\) 8.21979 0.395933 0.197967 0.980209i \(-0.436566\pi\)
0.197967 + 0.980209i \(0.436566\pi\)
\(432\) 0 0
\(433\) 22.7591 1.09373 0.546866 0.837220i \(-0.315821\pi\)
0.546866 + 0.837220i \(0.315821\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −13.2832 −0.635420
\(438\) 0 0
\(439\) −29.4470 −1.40543 −0.702714 0.711473i \(-0.748029\pi\)
−0.702714 + 0.711473i \(0.748029\pi\)
\(440\) 0 0
\(441\) 5.76455 0.274502
\(442\) 0 0
\(443\) −8.02614 −0.381333 −0.190667 0.981655i \(-0.561065\pi\)
−0.190667 + 0.981655i \(0.561065\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 3.14550 0.148777
\(448\) 0 0
\(449\) −4.05317 −0.191281 −0.0956404 0.995416i \(-0.530490\pi\)
−0.0956404 + 0.995416i \(0.530490\pi\)
\(450\) 0 0
\(451\) 25.1431 1.18394
\(452\) 0 0
\(453\) 1.95634 0.0919170
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 31.5875 1.47760 0.738799 0.673925i \(-0.235393\pi\)
0.738799 + 0.673925i \(0.235393\pi\)
\(458\) 0 0
\(459\) 7.06883 0.329945
\(460\) 0 0
\(461\) −6.47091 −0.301380 −0.150690 0.988581i \(-0.548150\pi\)
−0.150690 + 0.988581i \(0.548150\pi\)
\(462\) 0 0
\(463\) 12.6389 0.587382 0.293691 0.955900i \(-0.405116\pi\)
0.293691 + 0.955900i \(0.405116\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 27.9815 1.29483 0.647415 0.762138i \(-0.275850\pi\)
0.647415 + 0.762138i \(0.275850\pi\)
\(468\) 0 0
\(469\) 31.9894 1.47713
\(470\) 0 0
\(471\) −5.23545 −0.241237
\(472\) 0 0
\(473\) 39.2489 1.80467
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 5.45762 0.249887
\(478\) 0 0
\(479\) −26.1652 −1.19552 −0.597760 0.801675i \(-0.703942\pi\)
−0.597760 + 0.801675i \(0.703942\pi\)
\(480\) 0 0
\(481\) 3.38095 0.154158
\(482\) 0 0
\(483\) 8.23264 0.374598
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 16.9509 0.768118 0.384059 0.923308i \(-0.374526\pi\)
0.384059 + 0.923308i \(0.374526\pi\)
\(488\) 0 0
\(489\) 15.2248 0.688490
\(490\) 0 0
\(491\) 37.6458 1.69893 0.849466 0.527643i \(-0.176924\pi\)
0.849466 + 0.527643i \(0.176924\pi\)
\(492\) 0 0
\(493\) 57.5792 2.59324
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 34.8863 1.56486
\(498\) 0 0
\(499\) 28.2445 1.26440 0.632200 0.774806i \(-0.282152\pi\)
0.632200 + 0.774806i \(0.282152\pi\)
\(500\) 0 0
\(501\) −21.6720 −0.968232
\(502\) 0 0
\(503\) 20.4420 0.911462 0.455731 0.890118i \(-0.349378\pi\)
0.455731 + 0.890118i \(0.349378\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) −1.00000 −0.0444116
\(508\) 0 0
\(509\) −37.6691 −1.66966 −0.834828 0.550511i \(-0.814433\pi\)
−0.834828 + 0.550511i \(0.814433\pi\)
\(510\) 0 0
\(511\) 44.4520 1.96644
\(512\) 0 0
\(513\) −5.76455 −0.254511
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 47.5901 2.09301
\(518\) 0 0
\(519\) −16.7617 −0.735759
\(520\) 0 0
\(521\) −3.84666 −0.168525 −0.0842627 0.996444i \(-0.526853\pi\)
−0.0842627 + 0.996444i \(0.526853\pi\)
\(522\) 0 0
\(523\) 39.1399 1.71147 0.855734 0.517417i \(-0.173106\pi\)
0.855734 + 0.517417i \(0.173106\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −41.5438 −1.80968
\(528\) 0 0
\(529\) −17.6903 −0.769142
\(530\) 0 0
\(531\) −8.56492 −0.371686
\(532\) 0 0
\(533\) −4.77238 −0.206715
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) −23.1322 −0.998229
\(538\) 0 0
\(539\) 30.3703 1.30814
\(540\) 0 0
\(541\) 12.9129 0.555167 0.277584 0.960701i \(-0.410466\pi\)
0.277584 + 0.960701i \(0.410466\pi\)
\(542\) 0 0
\(543\) 3.91005 0.167796
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) −30.5027 −1.30420 −0.652101 0.758132i \(-0.726112\pi\)
−0.652101 + 0.758132i \(0.726112\pi\)
\(548\) 0 0
\(549\) 8.07667 0.344704
\(550\) 0 0
\(551\) −46.9551 −2.00036
\(552\) 0 0
\(553\) 4.93380 0.209807
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −36.7330 −1.55643 −0.778213 0.628001i \(-0.783873\pi\)
−0.778213 + 0.628001i \(0.783873\pi\)
\(558\) 0 0
\(559\) −7.44979 −0.315092
\(560\) 0 0
\(561\) 37.2419 1.57235
\(562\) 0 0
\(563\) −9.47354 −0.399262 −0.199631 0.979871i \(-0.563974\pi\)
−0.199631 + 0.979871i \(0.563974\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 3.57275 0.150041
\(568\) 0 0
\(569\) 20.7086 0.868148 0.434074 0.900877i \(-0.357076\pi\)
0.434074 + 0.900877i \(0.357076\pi\)
\(570\) 0 0
\(571\) −5.36247 −0.224413 −0.112206 0.993685i \(-0.535792\pi\)
−0.112206 + 0.993685i \(0.535792\pi\)
\(572\) 0 0
\(573\) 4.29100 0.179259
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 2.87017 0.119487 0.0597433 0.998214i \(-0.480972\pi\)
0.0597433 + 0.998214i \(0.480972\pi\)
\(578\) 0 0
\(579\) 26.5796 1.10461
\(580\) 0 0
\(581\) −34.2011 −1.41890
\(582\) 0 0
\(583\) 28.7533 1.19084
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −5.08856 −0.210027 −0.105014 0.994471i \(-0.533489\pi\)
−0.105014 + 0.994471i \(0.533489\pi\)
\(588\) 0 0
\(589\) 33.8785 1.39594
\(590\) 0 0
\(591\) −2.09497 −0.0861757
\(592\) 0 0
\(593\) −23.4393 −0.962537 −0.481269 0.876573i \(-0.659824\pi\)
−0.481269 + 0.876573i \(0.659824\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 3.93117 0.160892
\(598\) 0 0
\(599\) 26.9229 1.10004 0.550020 0.835151i \(-0.314620\pi\)
0.550020 + 0.835151i \(0.314620\pi\)
\(600\) 0 0
\(601\) −28.8124 −1.17528 −0.587642 0.809121i \(-0.699944\pi\)
−0.587642 + 0.809121i \(0.699944\pi\)
\(602\) 0 0
\(603\) 8.95370 0.364623
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 24.1349 0.979604 0.489802 0.871834i \(-0.337069\pi\)
0.489802 + 0.871834i \(0.337069\pi\)
\(608\) 0 0
\(609\) 29.1018 1.17927
\(610\) 0 0
\(611\) −9.03301 −0.365437
\(612\) 0 0
\(613\) 43.0739 1.73974 0.869868 0.493284i \(-0.164204\pi\)
0.869868 + 0.493284i \(0.164204\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 32.7225 1.31736 0.658679 0.752424i \(-0.271116\pi\)
0.658679 + 0.752424i \(0.271116\pi\)
\(618\) 0 0
\(619\) −14.5236 −0.583755 −0.291877 0.956456i \(-0.594280\pi\)
−0.291877 + 0.956456i \(0.594280\pi\)
\(620\) 0 0
\(621\) 2.30429 0.0924678
\(622\) 0 0
\(623\) −16.2093 −0.649412
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) −30.3703 −1.21287
\(628\) 0 0
\(629\) 23.8994 0.952932
\(630\) 0 0
\(631\) −31.8994 −1.26990 −0.634948 0.772555i \(-0.718978\pi\)
−0.634948 + 0.772555i \(0.718978\pi\)
\(632\) 0 0
\(633\) −6.30429 −0.250573
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) −5.76455 −0.228400
\(638\) 0 0
\(639\) 9.76455 0.386280
\(640\) 0 0
\(641\) 15.5132 0.612733 0.306367 0.951914i \(-0.400887\pi\)
0.306367 + 0.951914i \(0.400887\pi\)
\(642\) 0 0
\(643\) 16.5474 0.652566 0.326283 0.945272i \(-0.394204\pi\)
0.326283 + 0.945272i \(0.394204\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −35.4574 −1.39398 −0.696988 0.717083i \(-0.745477\pi\)
−0.696988 + 0.717083i \(0.745477\pi\)
\(648\) 0 0
\(649\) −45.1240 −1.77127
\(650\) 0 0
\(651\) −20.9972 −0.822945
\(652\) 0 0
\(653\) −32.2804 −1.26323 −0.631614 0.775283i \(-0.717607\pi\)
−0.631614 + 0.775283i \(0.717607\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 12.4420 0.485407
\(658\) 0 0
\(659\) −40.5768 −1.58065 −0.790324 0.612689i \(-0.790088\pi\)
−0.790324 + 0.612689i \(0.790088\pi\)
\(660\) 0 0
\(661\) −29.2167 −1.13640 −0.568199 0.822891i \(-0.692360\pi\)
−0.568199 + 0.822891i \(0.692360\pi\)
\(662\) 0 0
\(663\) −7.06883 −0.274531
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 18.7696 0.726761
\(668\) 0 0
\(669\) −20.7696 −0.802998
\(670\) 0 0
\(671\) 42.5516 1.64269
\(672\) 0 0
\(673\) −46.1165 −1.77766 −0.888831 0.458235i \(-0.848482\pi\)
−0.888831 + 0.458235i \(0.848482\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 26.6086 1.02265 0.511325 0.859387i \(-0.329155\pi\)
0.511325 + 0.859387i \(0.329155\pi\)
\(678\) 0 0
\(679\) −25.5291 −0.979717
\(680\) 0 0
\(681\) 26.9352 1.03216
\(682\) 0 0
\(683\) −25.4778 −0.974880 −0.487440 0.873156i \(-0.662069\pi\)
−0.487440 + 0.873156i \(0.662069\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) −25.7513 −0.982472
\(688\) 0 0
\(689\) −5.45762 −0.207919
\(690\) 0 0
\(691\) 0.374084 0.0142308 0.00711540 0.999975i \(-0.497735\pi\)
0.00711540 + 0.999975i \(0.497735\pi\)
\(692\) 0 0
\(693\) 18.8229 0.715023
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) −33.7352 −1.27781
\(698\) 0 0
\(699\) −2.53693 −0.0959554
\(700\) 0 0
\(701\) 41.1083 1.55264 0.776319 0.630340i \(-0.217084\pi\)
0.776319 + 0.630340i \(0.217084\pi\)
\(702\) 0 0
\(703\) −19.4897 −0.735067
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −1.36105 −0.0511877
\(708\) 0 0
\(709\) −31.1479 −1.16978 −0.584892 0.811111i \(-0.698863\pi\)
−0.584892 + 0.811111i \(0.698863\pi\)
\(710\) 0 0
\(711\) 1.38095 0.0517898
\(712\) 0 0
\(713\) −13.5424 −0.507166
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) −20.8559 −0.778879
\(718\) 0 0
\(719\) 30.9736 1.15512 0.577560 0.816348i \(-0.304005\pi\)
0.577560 + 0.816348i \(0.304005\pi\)
\(720\) 0 0
\(721\) −50.2270 −1.87055
\(722\) 0 0
\(723\) −5.69571 −0.211826
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) −21.6084 −0.801411 −0.400706 0.916207i \(-0.631235\pi\)
−0.400706 + 0.916207i \(0.631235\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −52.6613 −1.94775
\(732\) 0 0
\(733\) 26.0688 0.962874 0.481437 0.876481i \(-0.340115\pi\)
0.481437 + 0.876481i \(0.340115\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 47.1723 1.73761
\(738\) 0 0
\(739\) 47.9402 1.76351 0.881755 0.471707i \(-0.156362\pi\)
0.881755 + 0.471707i \(0.156362\pi\)
\(740\) 0 0
\(741\) 5.76455 0.211766
\(742\) 0 0
\(743\) −2.80037 −0.102736 −0.0513678 0.998680i \(-0.516358\pi\)
−0.0513678 + 0.998680i \(0.516358\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) −9.57275 −0.350249
\(748\) 0 0
\(749\) −20.3119 −0.742182
\(750\) 0 0
\(751\) 13.1194 0.478732 0.239366 0.970929i \(-0.423060\pi\)
0.239366 + 0.970929i \(0.423060\pi\)
\(752\) 0 0
\(753\) −14.7724 −0.538335
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −6.91022 −0.251156 −0.125578 0.992084i \(-0.540079\pi\)
−0.125578 + 0.992084i \(0.540079\pi\)
\(758\) 0 0
\(759\) 12.1401 0.440656
\(760\) 0 0
\(761\) −53.7380 −1.94800 −0.974000 0.226548i \(-0.927256\pi\)
−0.974000 + 0.226548i \(0.927256\pi\)
\(762\) 0 0
\(763\) −5.75646 −0.208398
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 8.56492 0.309261
\(768\) 0 0
\(769\) 28.3727 1.02315 0.511573 0.859240i \(-0.329063\pi\)
0.511573 + 0.859240i \(0.329063\pi\)
\(770\) 0 0
\(771\) 19.5108 0.702664
\(772\) 0 0
\(773\) 48.2194 1.73433 0.867165 0.498021i \(-0.165940\pi\)
0.867165 + 0.498021i \(0.165940\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 12.0793 0.433343
\(778\) 0 0
\(779\) 27.5106 0.985670
\(780\) 0 0
\(781\) 51.4442 1.84082
\(782\) 0 0
\(783\) 8.14550 0.291096
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 38.6736 1.37857 0.689283 0.724492i \(-0.257926\pi\)
0.689283 + 0.724492i \(0.257926\pi\)
\(788\) 0 0
\(789\) 5.31212 0.189116
\(790\) 0 0
\(791\) 1.37048 0.0487287
\(792\) 0 0
\(793\) −8.07667 −0.286811
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −2.38334 −0.0844222 −0.0422111 0.999109i \(-0.513440\pi\)
−0.0422111 + 0.999109i \(0.513440\pi\)
\(798\) 0 0
\(799\) −63.8529 −2.25895
\(800\) 0 0
\(801\) −4.53693 −0.160304
\(802\) 0 0
\(803\) 65.5500 2.31321
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) −2.23809 −0.0787846
\(808\) 0 0
\(809\) −41.1242 −1.44585 −0.722925 0.690926i \(-0.757203\pi\)
−0.722925 + 0.690926i \(0.757203\pi\)
\(810\) 0 0
\(811\) 8.49327 0.298239 0.149120 0.988819i \(-0.452356\pi\)
0.149120 + 0.988819i \(0.452356\pi\)
\(812\) 0 0
\(813\) −24.4933 −0.859017
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 42.9447 1.50244
\(818\) 0 0
\(819\) −3.57275 −0.124842
\(820\) 0 0
\(821\) −12.3703 −0.431727 −0.215863 0.976424i \(-0.569257\pi\)
−0.215863 + 0.976424i \(0.569257\pi\)
\(822\) 0 0
\(823\) −18.5848 −0.647826 −0.323913 0.946087i \(-0.604999\pi\)
−0.323913 + 0.946087i \(0.604999\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 11.4375 0.397720 0.198860 0.980028i \(-0.436276\pi\)
0.198860 + 0.980028i \(0.436276\pi\)
\(828\) 0 0
\(829\) 54.3254 1.88680 0.943400 0.331657i \(-0.107608\pi\)
0.943400 + 0.331657i \(0.107608\pi\)
\(830\) 0 0
\(831\) −12.2250 −0.424080
\(832\) 0 0
\(833\) −40.7486 −1.41186
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) −5.87704 −0.203140
\(838\) 0 0
\(839\) 19.5267 0.674137 0.337068 0.941480i \(-0.390565\pi\)
0.337068 + 0.941480i \(0.390565\pi\)
\(840\) 0 0
\(841\) 37.3492 1.28790
\(842\) 0 0
\(843\) 10.2222 0.352070
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 59.8676 2.05707
\(848\) 0 0
\(849\) −25.0372 −0.859276
\(850\) 0 0
\(851\) 7.79069 0.267061
\(852\) 0 0
\(853\) 9.01592 0.308699 0.154350 0.988016i \(-0.450672\pi\)
0.154350 + 0.988016i \(0.450672\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −38.5820 −1.31794 −0.658968 0.752171i \(-0.729007\pi\)
−0.658968 + 0.752171i \(0.729007\pi\)
\(858\) 0 0
\(859\) −35.4232 −1.20862 −0.604312 0.796748i \(-0.706552\pi\)
−0.604312 + 0.796748i \(0.706552\pi\)
\(860\) 0 0
\(861\) −17.0505 −0.581080
\(862\) 0 0
\(863\) 3.11794 0.106136 0.0530680 0.998591i \(-0.483100\pi\)
0.0530680 + 0.998591i \(0.483100\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) −32.9684 −1.11967
\(868\) 0 0
\(869\) 7.27551 0.246805
\(870\) 0 0
\(871\) −8.95370 −0.303385
\(872\) 0 0
\(873\) −7.14550 −0.241839
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 31.8229 1.07458 0.537292 0.843397i \(-0.319447\pi\)
0.537292 + 0.843397i \(0.319447\pi\)
\(878\) 0 0
\(879\) 27.2115 0.917822
\(880\) 0 0
\(881\) 27.9761 0.942538 0.471269 0.881990i \(-0.343796\pi\)
0.471269 + 0.881990i \(0.343796\pi\)
\(882\) 0 0
\(883\) −45.1399 −1.51908 −0.759539 0.650462i \(-0.774575\pi\)
−0.759539 + 0.650462i \(0.774575\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −8.52927 −0.286385 −0.143192 0.989695i \(-0.545737\pi\)
−0.143192 + 0.989695i \(0.545737\pi\)
\(888\) 0 0
\(889\) −51.8994 −1.74065
\(890\) 0 0
\(891\) 5.26846 0.176500
\(892\) 0 0
\(893\) 52.0712 1.74250
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) −2.30429 −0.0769379
\(898\) 0 0
\(899\) −47.8714 −1.59660
\(900\) 0 0
\(901\) −38.5790 −1.28525
\(902\) 0 0
\(903\) −26.6162 −0.885733
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) −13.8467 −0.459771 −0.229885 0.973218i \(-0.573835\pi\)
−0.229885 + 0.973218i \(0.573835\pi\)
\(908\) 0 0
\(909\) −0.380954 −0.0126355
\(910\) 0 0
\(911\) −41.8068 −1.38512 −0.692561 0.721360i \(-0.743518\pi\)
−0.692561 + 0.721360i \(0.743518\pi\)
\(912\) 0 0
\(913\) −50.4337 −1.66911
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 10.1610 0.335546
\(918\) 0 0
\(919\) −47.1987 −1.55694 −0.778470 0.627682i \(-0.784004\pi\)
−0.778470 + 0.627682i \(0.784004\pi\)
\(920\) 0 0
\(921\) −22.9100 −0.754912
\(922\) 0 0
\(923\) −9.76455 −0.321404
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) −14.0584 −0.461737
\(928\) 0 0
\(929\) 34.0789 1.11809 0.559046 0.829137i \(-0.311168\pi\)
0.559046 + 0.829137i \(0.311168\pi\)
\(930\) 0 0
\(931\) 33.2300 1.08907
\(932\) 0 0
\(933\) −3.40709 −0.111543
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −5.58482 −0.182448 −0.0912240 0.995830i \(-0.529078\pi\)
−0.0912240 + 0.995830i \(0.529078\pi\)
\(938\) 0 0
\(939\) 30.5315 0.996357
\(940\) 0 0
\(941\) 12.1401 0.395754 0.197877 0.980227i \(-0.436595\pi\)
0.197877 + 0.980227i \(0.436595\pi\)
\(942\) 0 0
\(943\) −10.9969 −0.358109
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 33.2472 1.08039 0.540194 0.841541i \(-0.318351\pi\)
0.540194 + 0.841541i \(0.318351\pi\)
\(948\) 0 0
\(949\) −12.4420 −0.403883
\(950\) 0 0
\(951\) −29.2091 −0.947171
\(952\) 0 0
\(953\) −14.8362 −0.480591 −0.240296 0.970700i \(-0.577244\pi\)
−0.240296 + 0.970700i \(0.577244\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 42.9143 1.38722
\(958\) 0 0
\(959\) −8.81789 −0.284744
\(960\) 0 0
\(961\) 3.53957 0.114180
\(962\) 0 0
\(963\) −5.68524 −0.183204
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) −59.4355 −1.91132 −0.955659 0.294475i \(-0.904855\pi\)
−0.955659 + 0.294475i \(0.904855\pi\)
\(968\) 0 0
\(969\) 40.7486 1.30903
\(970\) 0 0
\(971\) 25.8752 0.830374 0.415187 0.909736i \(-0.363716\pi\)
0.415187 + 0.909736i \(0.363716\pi\)
\(972\) 0 0
\(973\) 21.9284 0.702991
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 7.07913 0.226482 0.113241 0.993568i \(-0.463877\pi\)
0.113241 + 0.993568i \(0.463877\pi\)
\(978\) 0 0
\(979\) −23.9026 −0.763932
\(980\) 0 0
\(981\) −1.61121 −0.0514420
\(982\) 0 0
\(983\) −31.8054 −1.01443 −0.507217 0.861818i \(-0.669326\pi\)
−0.507217 + 0.861818i \(0.669326\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) −32.2727 −1.02725
\(988\) 0 0
\(989\) −17.1664 −0.545861
\(990\) 0 0
\(991\) −30.3888 −0.965332 −0.482666 0.875805i \(-0.660331\pi\)
−0.482666 + 0.875805i \(0.660331\pi\)
\(992\) 0 0
\(993\) −0.166619 −0.00528750
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) −30.6639 −0.971137 −0.485569 0.874199i \(-0.661387\pi\)
−0.485569 + 0.874199i \(0.661387\pi\)
\(998\) 0 0
\(999\) 3.38095 0.106969
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 7800.2.a.bu.1.4 4
5.4 even 2 7800.2.a.bx.1.1 yes 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
7800.2.a.bu.1.4 4 1.1 even 1 trivial
7800.2.a.bx.1.1 yes 4 5.4 even 2