Properties

Label 315.2.a.e.1.1
Level $315$
Weight $2$
Character 315.1
Self dual yes
Analytic conductor $2.515$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 315 = 3^{2} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 315.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(2.51528766367\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{17}) \)
Defining polynomial: \(x^{2} - x - 4\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 35)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.56155\) of defining polynomial
Character \(\chi\) \(=\) 315.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.56155 q^{2} +0.438447 q^{4} -1.00000 q^{5} -1.00000 q^{7} +2.43845 q^{8} +O(q^{10})\) \(q-1.56155 q^{2} +0.438447 q^{4} -1.00000 q^{5} -1.00000 q^{7} +2.43845 q^{8} +1.56155 q^{10} -2.56155 q^{11} +4.56155 q^{13} +1.56155 q^{14} -4.68466 q^{16} +4.56155 q^{17} +1.12311 q^{19} -0.438447 q^{20} +4.00000 q^{22} +5.12311 q^{23} +1.00000 q^{25} -7.12311 q^{26} -0.438447 q^{28} +5.68466 q^{29} +2.43845 q^{32} -7.12311 q^{34} +1.00000 q^{35} +6.00000 q^{37} -1.75379 q^{38} -2.43845 q^{40} +3.12311 q^{41} +9.12311 q^{43} -1.12311 q^{44} -8.00000 q^{46} -3.68466 q^{47} +1.00000 q^{49} -1.56155 q^{50} +2.00000 q^{52} -3.12311 q^{53} +2.56155 q^{55} -2.43845 q^{56} -8.87689 q^{58} +4.00000 q^{59} -9.36932 q^{61} +5.56155 q^{64} -4.56155 q^{65} -6.24621 q^{67} +2.00000 q^{68} -1.56155 q^{70} -8.00000 q^{71} +4.24621 q^{73} -9.36932 q^{74} +0.492423 q^{76} +2.56155 q^{77} -6.56155 q^{79} +4.68466 q^{80} -4.87689 q^{82} -4.00000 q^{83} -4.56155 q^{85} -14.2462 q^{86} -6.24621 q^{88} -7.12311 q^{89} -4.56155 q^{91} +2.24621 q^{92} +5.75379 q^{94} -1.12311 q^{95} -14.8078 q^{97} -1.56155 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + q^{2} + 5q^{4} - 2q^{5} - 2q^{7} + 9q^{8} + O(q^{10}) \) \( 2q + q^{2} + 5q^{4} - 2q^{5} - 2q^{7} + 9q^{8} - q^{10} - q^{11} + 5q^{13} - q^{14} + 3q^{16} + 5q^{17} - 6q^{19} - 5q^{20} + 8q^{22} + 2q^{23} + 2q^{25} - 6q^{26} - 5q^{28} - q^{29} + 9q^{32} - 6q^{34} + 2q^{35} + 12q^{37} - 20q^{38} - 9q^{40} - 2q^{41} + 10q^{43} + 6q^{44} - 16q^{46} + 5q^{47} + 2q^{49} + q^{50} + 4q^{52} + 2q^{53} + q^{55} - 9q^{56} - 26q^{58} + 8q^{59} + 6q^{61} + 7q^{64} - 5q^{65} + 4q^{67} + 4q^{68} + q^{70} - 16q^{71} - 8q^{73} + 6q^{74} - 32q^{76} + q^{77} - 9q^{79} - 3q^{80} - 18q^{82} - 8q^{83} - 5q^{85} - 12q^{86} + 4q^{88} - 6q^{89} - 5q^{91} - 12q^{92} + 28q^{94} + 6q^{95} - 9q^{97} + q^{98} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.56155 −1.10418 −0.552092 0.833783i \(-0.686170\pi\)
−0.552092 + 0.833783i \(0.686170\pi\)
\(3\) 0 0
\(4\) 0.438447 0.219224
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) 2.43845 0.862121
\(9\) 0 0
\(10\) 1.56155 0.493806
\(11\) −2.56155 −0.772337 −0.386169 0.922428i \(-0.626202\pi\)
−0.386169 + 0.922428i \(0.626202\pi\)
\(12\) 0 0
\(13\) 4.56155 1.26515 0.632574 0.774500i \(-0.281999\pi\)
0.632574 + 0.774500i \(0.281999\pi\)
\(14\) 1.56155 0.417343
\(15\) 0 0
\(16\) −4.68466 −1.17116
\(17\) 4.56155 1.10634 0.553170 0.833069i \(-0.313418\pi\)
0.553170 + 0.833069i \(0.313418\pi\)
\(18\) 0 0
\(19\) 1.12311 0.257658 0.128829 0.991667i \(-0.458878\pi\)
0.128829 + 0.991667i \(0.458878\pi\)
\(20\) −0.438447 −0.0980398
\(21\) 0 0
\(22\) 4.00000 0.852803
\(23\) 5.12311 1.06824 0.534121 0.845408i \(-0.320643\pi\)
0.534121 + 0.845408i \(0.320643\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) −7.12311 −1.39696
\(27\) 0 0
\(28\) −0.438447 −0.0828587
\(29\) 5.68466 1.05561 0.527807 0.849364i \(-0.323014\pi\)
0.527807 + 0.849364i \(0.323014\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) 2.43845 0.431061
\(33\) 0 0
\(34\) −7.12311 −1.22160
\(35\) 1.00000 0.169031
\(36\) 0 0
\(37\) 6.00000 0.986394 0.493197 0.869918i \(-0.335828\pi\)
0.493197 + 0.869918i \(0.335828\pi\)
\(38\) −1.75379 −0.284502
\(39\) 0 0
\(40\) −2.43845 −0.385552
\(41\) 3.12311 0.487747 0.243874 0.969807i \(-0.421582\pi\)
0.243874 + 0.969807i \(0.421582\pi\)
\(42\) 0 0
\(43\) 9.12311 1.39126 0.695630 0.718400i \(-0.255125\pi\)
0.695630 + 0.718400i \(0.255125\pi\)
\(44\) −1.12311 −0.169315
\(45\) 0 0
\(46\) −8.00000 −1.17954
\(47\) −3.68466 −0.537463 −0.268731 0.963215i \(-0.586604\pi\)
−0.268731 + 0.963215i \(0.586604\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) −1.56155 −0.220837
\(51\) 0 0
\(52\) 2.00000 0.277350
\(53\) −3.12311 −0.428992 −0.214496 0.976725i \(-0.568811\pi\)
−0.214496 + 0.976725i \(0.568811\pi\)
\(54\) 0 0
\(55\) 2.56155 0.345400
\(56\) −2.43845 −0.325851
\(57\) 0 0
\(58\) −8.87689 −1.16559
\(59\) 4.00000 0.520756 0.260378 0.965507i \(-0.416153\pi\)
0.260378 + 0.965507i \(0.416153\pi\)
\(60\) 0 0
\(61\) −9.36932 −1.19962 −0.599809 0.800143i \(-0.704757\pi\)
−0.599809 + 0.800143i \(0.704757\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 5.56155 0.695194
\(65\) −4.56155 −0.565791
\(66\) 0 0
\(67\) −6.24621 −0.763096 −0.381548 0.924349i \(-0.624609\pi\)
−0.381548 + 0.924349i \(0.624609\pi\)
\(68\) 2.00000 0.242536
\(69\) 0 0
\(70\) −1.56155 −0.186641
\(71\) −8.00000 −0.949425 −0.474713 0.880141i \(-0.657448\pi\)
−0.474713 + 0.880141i \(0.657448\pi\)
\(72\) 0 0
\(73\) 4.24621 0.496981 0.248491 0.968634i \(-0.420065\pi\)
0.248491 + 0.968634i \(0.420065\pi\)
\(74\) −9.36932 −1.08916
\(75\) 0 0
\(76\) 0.492423 0.0564847
\(77\) 2.56155 0.291916
\(78\) 0 0
\(79\) −6.56155 −0.738232 −0.369116 0.929383i \(-0.620340\pi\)
−0.369116 + 0.929383i \(0.620340\pi\)
\(80\) 4.68466 0.523761
\(81\) 0 0
\(82\) −4.87689 −0.538563
\(83\) −4.00000 −0.439057 −0.219529 0.975606i \(-0.570452\pi\)
−0.219529 + 0.975606i \(0.570452\pi\)
\(84\) 0 0
\(85\) −4.56155 −0.494770
\(86\) −14.2462 −1.53621
\(87\) 0 0
\(88\) −6.24621 −0.665848
\(89\) −7.12311 −0.755048 −0.377524 0.926000i \(-0.623224\pi\)
−0.377524 + 0.926000i \(0.623224\pi\)
\(90\) 0 0
\(91\) −4.56155 −0.478181
\(92\) 2.24621 0.234184
\(93\) 0 0
\(94\) 5.75379 0.593458
\(95\) −1.12311 −0.115228
\(96\) 0 0
\(97\) −14.8078 −1.50350 −0.751750 0.659448i \(-0.770790\pi\)
−0.751750 + 0.659448i \(0.770790\pi\)
\(98\) −1.56155 −0.157741
\(99\) 0 0
\(100\) 0.438447 0.0438447
\(101\) −0.246211 −0.0244989 −0.0122495 0.999925i \(-0.503899\pi\)
−0.0122495 + 0.999925i \(0.503899\pi\)
\(102\) 0 0
\(103\) 1.43845 0.141734 0.0708672 0.997486i \(-0.477423\pi\)
0.0708672 + 0.997486i \(0.477423\pi\)
\(104\) 11.1231 1.09071
\(105\) 0 0
\(106\) 4.87689 0.473686
\(107\) 11.3693 1.09911 0.549557 0.835456i \(-0.314797\pi\)
0.549557 + 0.835456i \(0.314797\pi\)
\(108\) 0 0
\(109\) 17.6847 1.69388 0.846942 0.531686i \(-0.178441\pi\)
0.846942 + 0.531686i \(0.178441\pi\)
\(110\) −4.00000 −0.381385
\(111\) 0 0
\(112\) 4.68466 0.442659
\(113\) 14.0000 1.31701 0.658505 0.752577i \(-0.271189\pi\)
0.658505 + 0.752577i \(0.271189\pi\)
\(114\) 0 0
\(115\) −5.12311 −0.477732
\(116\) 2.49242 0.231416
\(117\) 0 0
\(118\) −6.24621 −0.575010
\(119\) −4.56155 −0.418157
\(120\) 0 0
\(121\) −4.43845 −0.403495
\(122\) 14.6307 1.32460
\(123\) 0 0
\(124\) 0 0
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) 10.2462 0.909204 0.454602 0.890695i \(-0.349781\pi\)
0.454602 + 0.890695i \(0.349781\pi\)
\(128\) −13.5616 −1.19868
\(129\) 0 0
\(130\) 7.12311 0.624738
\(131\) 9.12311 0.797089 0.398545 0.917149i \(-0.369515\pi\)
0.398545 + 0.917149i \(0.369515\pi\)
\(132\) 0 0
\(133\) −1.12311 −0.0973856
\(134\) 9.75379 0.842599
\(135\) 0 0
\(136\) 11.1231 0.953798
\(137\) 8.87689 0.758404 0.379202 0.925314i \(-0.376199\pi\)
0.379202 + 0.925314i \(0.376199\pi\)
\(138\) 0 0
\(139\) −6.87689 −0.583291 −0.291645 0.956527i \(-0.594203\pi\)
−0.291645 + 0.956527i \(0.594203\pi\)
\(140\) 0.438447 0.0370556
\(141\) 0 0
\(142\) 12.4924 1.04834
\(143\) −11.6847 −0.977120
\(144\) 0 0
\(145\) −5.68466 −0.472085
\(146\) −6.63068 −0.548759
\(147\) 0 0
\(148\) 2.63068 0.216241
\(149\) 4.24621 0.347863 0.173932 0.984758i \(-0.444353\pi\)
0.173932 + 0.984758i \(0.444353\pi\)
\(150\) 0 0
\(151\) 21.9309 1.78471 0.892354 0.451335i \(-0.149052\pi\)
0.892354 + 0.451335i \(0.149052\pi\)
\(152\) 2.73863 0.222133
\(153\) 0 0
\(154\) −4.00000 −0.322329
\(155\) 0 0
\(156\) 0 0
\(157\) 3.75379 0.299585 0.149792 0.988717i \(-0.452139\pi\)
0.149792 + 0.988717i \(0.452139\pi\)
\(158\) 10.2462 0.815145
\(159\) 0 0
\(160\) −2.43845 −0.192776
\(161\) −5.12311 −0.403757
\(162\) 0 0
\(163\) 1.12311 0.0879684 0.0439842 0.999032i \(-0.485995\pi\)
0.0439842 + 0.999032i \(0.485995\pi\)
\(164\) 1.36932 0.106926
\(165\) 0 0
\(166\) 6.24621 0.484800
\(167\) −21.9309 −1.69706 −0.848531 0.529146i \(-0.822512\pi\)
−0.848531 + 0.529146i \(0.822512\pi\)
\(168\) 0 0
\(169\) 7.80776 0.600597
\(170\) 7.12311 0.546317
\(171\) 0 0
\(172\) 4.00000 0.304997
\(173\) 8.56155 0.650923 0.325461 0.945555i \(-0.394480\pi\)
0.325461 + 0.945555i \(0.394480\pi\)
\(174\) 0 0
\(175\) −1.00000 −0.0755929
\(176\) 12.0000 0.904534
\(177\) 0 0
\(178\) 11.1231 0.833712
\(179\) −20.0000 −1.49487 −0.747435 0.664335i \(-0.768715\pi\)
−0.747435 + 0.664335i \(0.768715\pi\)
\(180\) 0 0
\(181\) 23.6155 1.75533 0.877664 0.479276i \(-0.159101\pi\)
0.877664 + 0.479276i \(0.159101\pi\)
\(182\) 7.12311 0.528000
\(183\) 0 0
\(184\) 12.4924 0.920954
\(185\) −6.00000 −0.441129
\(186\) 0 0
\(187\) −11.6847 −0.854467
\(188\) −1.61553 −0.117824
\(189\) 0 0
\(190\) 1.75379 0.127233
\(191\) 9.43845 0.682942 0.341471 0.939892i \(-0.389075\pi\)
0.341471 + 0.939892i \(0.389075\pi\)
\(192\) 0 0
\(193\) −5.36932 −0.386492 −0.193246 0.981150i \(-0.561902\pi\)
−0.193246 + 0.981150i \(0.561902\pi\)
\(194\) 23.1231 1.66014
\(195\) 0 0
\(196\) 0.438447 0.0313177
\(197\) 7.12311 0.507500 0.253750 0.967270i \(-0.418336\pi\)
0.253750 + 0.967270i \(0.418336\pi\)
\(198\) 0 0
\(199\) −18.2462 −1.29344 −0.646720 0.762728i \(-0.723860\pi\)
−0.646720 + 0.762728i \(0.723860\pi\)
\(200\) 2.43845 0.172424
\(201\) 0 0
\(202\) 0.384472 0.0270513
\(203\) −5.68466 −0.398985
\(204\) 0 0
\(205\) −3.12311 −0.218127
\(206\) −2.24621 −0.156501
\(207\) 0 0
\(208\) −21.3693 −1.48170
\(209\) −2.87689 −0.198999
\(210\) 0 0
\(211\) −23.0540 −1.58710 −0.793551 0.608504i \(-0.791770\pi\)
−0.793551 + 0.608504i \(0.791770\pi\)
\(212\) −1.36932 −0.0940451
\(213\) 0 0
\(214\) −17.7538 −1.21362
\(215\) −9.12311 −0.622191
\(216\) 0 0
\(217\) 0 0
\(218\) −27.6155 −1.87036
\(219\) 0 0
\(220\) 1.12311 0.0757198
\(221\) 20.8078 1.39968
\(222\) 0 0
\(223\) −6.56155 −0.439394 −0.219697 0.975568i \(-0.570507\pi\)
−0.219697 + 0.975568i \(0.570507\pi\)
\(224\) −2.43845 −0.162926
\(225\) 0 0
\(226\) −21.8617 −1.45422
\(227\) −23.6847 −1.57201 −0.786003 0.618223i \(-0.787853\pi\)
−0.786003 + 0.618223i \(0.787853\pi\)
\(228\) 0 0
\(229\) 19.1231 1.26369 0.631845 0.775095i \(-0.282298\pi\)
0.631845 + 0.775095i \(0.282298\pi\)
\(230\) 8.00000 0.527504
\(231\) 0 0
\(232\) 13.8617 0.910068
\(233\) 3.12311 0.204601 0.102301 0.994754i \(-0.467380\pi\)
0.102301 + 0.994754i \(0.467380\pi\)
\(234\) 0 0
\(235\) 3.68466 0.240361
\(236\) 1.75379 0.114162
\(237\) 0 0
\(238\) 7.12311 0.461722
\(239\) 0.807764 0.0522499 0.0261250 0.999659i \(-0.491683\pi\)
0.0261250 + 0.999659i \(0.491683\pi\)
\(240\) 0 0
\(241\) 12.2462 0.788848 0.394424 0.918929i \(-0.370944\pi\)
0.394424 + 0.918929i \(0.370944\pi\)
\(242\) 6.93087 0.445533
\(243\) 0 0
\(244\) −4.10795 −0.262985
\(245\) −1.00000 −0.0638877
\(246\) 0 0
\(247\) 5.12311 0.325975
\(248\) 0 0
\(249\) 0 0
\(250\) 1.56155 0.0987613
\(251\) 17.1231 1.08080 0.540400 0.841408i \(-0.318273\pi\)
0.540400 + 0.841408i \(0.318273\pi\)
\(252\) 0 0
\(253\) −13.1231 −0.825043
\(254\) −16.0000 −1.00393
\(255\) 0 0
\(256\) 10.0540 0.628373
\(257\) −22.4924 −1.40304 −0.701519 0.712650i \(-0.747495\pi\)
−0.701519 + 0.712650i \(0.747495\pi\)
\(258\) 0 0
\(259\) −6.00000 −0.372822
\(260\) −2.00000 −0.124035
\(261\) 0 0
\(262\) −14.2462 −0.880134
\(263\) 21.1231 1.30251 0.651253 0.758860i \(-0.274244\pi\)
0.651253 + 0.758860i \(0.274244\pi\)
\(264\) 0 0
\(265\) 3.12311 0.191851
\(266\) 1.75379 0.107532
\(267\) 0 0
\(268\) −2.73863 −0.167289
\(269\) −28.7386 −1.75223 −0.876113 0.482106i \(-0.839872\pi\)
−0.876113 + 0.482106i \(0.839872\pi\)
\(270\) 0 0
\(271\) −16.0000 −0.971931 −0.485965 0.873978i \(-0.661532\pi\)
−0.485965 + 0.873978i \(0.661532\pi\)
\(272\) −21.3693 −1.29571
\(273\) 0 0
\(274\) −13.8617 −0.837418
\(275\) −2.56155 −0.154467
\(276\) 0 0
\(277\) 16.2462 0.976140 0.488070 0.872804i \(-0.337701\pi\)
0.488070 + 0.872804i \(0.337701\pi\)
\(278\) 10.7386 0.644060
\(279\) 0 0
\(280\) 2.43845 0.145725
\(281\) −16.5616 −0.987979 −0.493990 0.869468i \(-0.664462\pi\)
−0.493990 + 0.869468i \(0.664462\pi\)
\(282\) 0 0
\(283\) −23.6847 −1.40791 −0.703953 0.710246i \(-0.748584\pi\)
−0.703953 + 0.710246i \(0.748584\pi\)
\(284\) −3.50758 −0.208136
\(285\) 0 0
\(286\) 18.2462 1.07892
\(287\) −3.12311 −0.184351
\(288\) 0 0
\(289\) 3.80776 0.223986
\(290\) 8.87689 0.521269
\(291\) 0 0
\(292\) 1.86174 0.108950
\(293\) −9.68466 −0.565784 −0.282892 0.959152i \(-0.591294\pi\)
−0.282892 + 0.959152i \(0.591294\pi\)
\(294\) 0 0
\(295\) −4.00000 −0.232889
\(296\) 14.6307 0.850391
\(297\) 0 0
\(298\) −6.63068 −0.384105
\(299\) 23.3693 1.35148
\(300\) 0 0
\(301\) −9.12311 −0.525847
\(302\) −34.2462 −1.97065
\(303\) 0 0
\(304\) −5.26137 −0.301760
\(305\) 9.36932 0.536486
\(306\) 0 0
\(307\) −31.6847 −1.80834 −0.904169 0.427174i \(-0.859509\pi\)
−0.904169 + 0.427174i \(0.859509\pi\)
\(308\) 1.12311 0.0639949
\(309\) 0 0
\(310\) 0 0
\(311\) 9.61553 0.545247 0.272623 0.962121i \(-0.412109\pi\)
0.272623 + 0.962121i \(0.412109\pi\)
\(312\) 0 0
\(313\) 31.3002 1.76919 0.884596 0.466359i \(-0.154434\pi\)
0.884596 + 0.466359i \(0.154434\pi\)
\(314\) −5.86174 −0.330797
\(315\) 0 0
\(316\) −2.87689 −0.161838
\(317\) 22.4924 1.26330 0.631650 0.775254i \(-0.282378\pi\)
0.631650 + 0.775254i \(0.282378\pi\)
\(318\) 0 0
\(319\) −14.5616 −0.815290
\(320\) −5.56155 −0.310900
\(321\) 0 0
\(322\) 8.00000 0.445823
\(323\) 5.12311 0.285057
\(324\) 0 0
\(325\) 4.56155 0.253029
\(326\) −1.75379 −0.0971334
\(327\) 0 0
\(328\) 7.61553 0.420497
\(329\) 3.68466 0.203142
\(330\) 0 0
\(331\) 12.0000 0.659580 0.329790 0.944054i \(-0.393022\pi\)
0.329790 + 0.944054i \(0.393022\pi\)
\(332\) −1.75379 −0.0962517
\(333\) 0 0
\(334\) 34.2462 1.87387
\(335\) 6.24621 0.341267
\(336\) 0 0
\(337\) −34.4924 −1.87892 −0.939461 0.342656i \(-0.888674\pi\)
−0.939461 + 0.342656i \(0.888674\pi\)
\(338\) −12.1922 −0.663170
\(339\) 0 0
\(340\) −2.00000 −0.108465
\(341\) 0 0
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) 22.2462 1.19944
\(345\) 0 0
\(346\) −13.3693 −0.718739
\(347\) 1.12311 0.0602915 0.0301457 0.999546i \(-0.490403\pi\)
0.0301457 + 0.999546i \(0.490403\pi\)
\(348\) 0 0
\(349\) −22.4924 −1.20399 −0.601996 0.798499i \(-0.705628\pi\)
−0.601996 + 0.798499i \(0.705628\pi\)
\(350\) 1.56155 0.0834685
\(351\) 0 0
\(352\) −6.24621 −0.332924
\(353\) 14.8078 0.788138 0.394069 0.919081i \(-0.371067\pi\)
0.394069 + 0.919081i \(0.371067\pi\)
\(354\) 0 0
\(355\) 8.00000 0.424596
\(356\) −3.12311 −0.165524
\(357\) 0 0
\(358\) 31.2311 1.65061
\(359\) −8.00000 −0.422224 −0.211112 0.977462i \(-0.567708\pi\)
−0.211112 + 0.977462i \(0.567708\pi\)
\(360\) 0 0
\(361\) −17.7386 −0.933612
\(362\) −36.8769 −1.93821
\(363\) 0 0
\(364\) −2.00000 −0.104828
\(365\) −4.24621 −0.222257
\(366\) 0 0
\(367\) 3.68466 0.192338 0.0961688 0.995365i \(-0.469341\pi\)
0.0961688 + 0.995365i \(0.469341\pi\)
\(368\) −24.0000 −1.25109
\(369\) 0 0
\(370\) 9.36932 0.487088
\(371\) 3.12311 0.162144
\(372\) 0 0
\(373\) 29.3693 1.52069 0.760343 0.649522i \(-0.225031\pi\)
0.760343 + 0.649522i \(0.225031\pi\)
\(374\) 18.2462 0.943489
\(375\) 0 0
\(376\) −8.98485 −0.463358
\(377\) 25.9309 1.33551
\(378\) 0 0
\(379\) 16.4924 0.847159 0.423579 0.905859i \(-0.360773\pi\)
0.423579 + 0.905859i \(0.360773\pi\)
\(380\) −0.492423 −0.0252607
\(381\) 0 0
\(382\) −14.7386 −0.754094
\(383\) 10.2462 0.523557 0.261778 0.965128i \(-0.415691\pi\)
0.261778 + 0.965128i \(0.415691\pi\)
\(384\) 0 0
\(385\) −2.56155 −0.130549
\(386\) 8.38447 0.426758
\(387\) 0 0
\(388\) −6.49242 −0.329603
\(389\) −3.93087 −0.199303 −0.0996515 0.995022i \(-0.531773\pi\)
−0.0996515 + 0.995022i \(0.531773\pi\)
\(390\) 0 0
\(391\) 23.3693 1.18184
\(392\) 2.43845 0.123160
\(393\) 0 0
\(394\) −11.1231 −0.560374
\(395\) 6.56155 0.330148
\(396\) 0 0
\(397\) 23.4384 1.17634 0.588171 0.808737i \(-0.299848\pi\)
0.588171 + 0.808737i \(0.299848\pi\)
\(398\) 28.4924 1.42820
\(399\) 0 0
\(400\) −4.68466 −0.234233
\(401\) −27.4384 −1.37021 −0.685105 0.728444i \(-0.740244\pi\)
−0.685105 + 0.728444i \(0.740244\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) −0.107951 −0.00537074
\(405\) 0 0
\(406\) 8.87689 0.440553
\(407\) −15.3693 −0.761829
\(408\) 0 0
\(409\) −26.4924 −1.30997 −0.654983 0.755644i \(-0.727324\pi\)
−0.654983 + 0.755644i \(0.727324\pi\)
\(410\) 4.87689 0.240853
\(411\) 0 0
\(412\) 0.630683 0.0310715
\(413\) −4.00000 −0.196827
\(414\) 0 0
\(415\) 4.00000 0.196352
\(416\) 11.1231 0.545355
\(417\) 0 0
\(418\) 4.49242 0.219732
\(419\) −9.75379 −0.476504 −0.238252 0.971203i \(-0.576574\pi\)
−0.238252 + 0.971203i \(0.576574\pi\)
\(420\) 0 0
\(421\) 9.68466 0.472001 0.236001 0.971753i \(-0.424163\pi\)
0.236001 + 0.971753i \(0.424163\pi\)
\(422\) 36.0000 1.75245
\(423\) 0 0
\(424\) −7.61553 −0.369843
\(425\) 4.56155 0.221268
\(426\) 0 0
\(427\) 9.36932 0.453413
\(428\) 4.98485 0.240952
\(429\) 0 0
\(430\) 14.2462 0.687013
\(431\) −0.807764 −0.0389086 −0.0194543 0.999811i \(-0.506193\pi\)
−0.0194543 + 0.999811i \(0.506193\pi\)
\(432\) 0 0
\(433\) −8.24621 −0.396288 −0.198144 0.980173i \(-0.563491\pi\)
−0.198144 + 0.980173i \(0.563491\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 7.75379 0.371339
\(437\) 5.75379 0.275241
\(438\) 0 0
\(439\) −15.3693 −0.733537 −0.366769 0.930312i \(-0.619536\pi\)
−0.366769 + 0.930312i \(0.619536\pi\)
\(440\) 6.24621 0.297776
\(441\) 0 0
\(442\) −32.4924 −1.54551
\(443\) 27.3693 1.30036 0.650178 0.759782i \(-0.274694\pi\)
0.650178 + 0.759782i \(0.274694\pi\)
\(444\) 0 0
\(445\) 7.12311 0.337668
\(446\) 10.2462 0.485172
\(447\) 0 0
\(448\) −5.56155 −0.262759
\(449\) −18.8078 −0.887593 −0.443797 0.896128i \(-0.646369\pi\)
−0.443797 + 0.896128i \(0.646369\pi\)
\(450\) 0 0
\(451\) −8.00000 −0.376705
\(452\) 6.13826 0.288719
\(453\) 0 0
\(454\) 36.9848 1.73578
\(455\) 4.56155 0.213849
\(456\) 0 0
\(457\) −8.87689 −0.415244 −0.207622 0.978209i \(-0.566572\pi\)
−0.207622 + 0.978209i \(0.566572\pi\)
\(458\) −29.8617 −1.39535
\(459\) 0 0
\(460\) −2.24621 −0.104730
\(461\) 4.87689 0.227140 0.113570 0.993530i \(-0.463771\pi\)
0.113570 + 0.993530i \(0.463771\pi\)
\(462\) 0 0
\(463\) −20.4924 −0.952364 −0.476182 0.879347i \(-0.657980\pi\)
−0.476182 + 0.879347i \(0.657980\pi\)
\(464\) −26.6307 −1.23630
\(465\) 0 0
\(466\) −4.87689 −0.225918
\(467\) −26.5616 −1.22912 −0.614561 0.788869i \(-0.710667\pi\)
−0.614561 + 0.788869i \(0.710667\pi\)
\(468\) 0 0
\(469\) 6.24621 0.288423
\(470\) −5.75379 −0.265402
\(471\) 0 0
\(472\) 9.75379 0.448955
\(473\) −23.3693 −1.07452
\(474\) 0 0
\(475\) 1.12311 0.0515316
\(476\) −2.00000 −0.0916698
\(477\) 0 0
\(478\) −1.26137 −0.0576935
\(479\) −13.1231 −0.599610 −0.299805 0.954001i \(-0.596922\pi\)
−0.299805 + 0.954001i \(0.596922\pi\)
\(480\) 0 0
\(481\) 27.3693 1.24793
\(482\) −19.1231 −0.871034
\(483\) 0 0
\(484\) −1.94602 −0.0884557
\(485\) 14.8078 0.672386
\(486\) 0 0
\(487\) 5.12311 0.232150 0.116075 0.993240i \(-0.462969\pi\)
0.116075 + 0.993240i \(0.462969\pi\)
\(488\) −22.8466 −1.03422
\(489\) 0 0
\(490\) 1.56155 0.0705438
\(491\) −4.17708 −0.188509 −0.0942545 0.995548i \(-0.530047\pi\)
−0.0942545 + 0.995548i \(0.530047\pi\)
\(492\) 0 0
\(493\) 25.9309 1.16787
\(494\) −8.00000 −0.359937
\(495\) 0 0
\(496\) 0 0
\(497\) 8.00000 0.358849
\(498\) 0 0
\(499\) −4.17708 −0.186992 −0.0934959 0.995620i \(-0.529804\pi\)
−0.0934959 + 0.995620i \(0.529804\pi\)
\(500\) −0.438447 −0.0196080
\(501\) 0 0
\(502\) −26.7386 −1.19340
\(503\) −10.0691 −0.448960 −0.224480 0.974479i \(-0.572068\pi\)
−0.224480 + 0.974479i \(0.572068\pi\)
\(504\) 0 0
\(505\) 0.246211 0.0109563
\(506\) 20.4924 0.910999
\(507\) 0 0
\(508\) 4.49242 0.199319
\(509\) 28.2462 1.25199 0.625996 0.779827i \(-0.284693\pi\)
0.625996 + 0.779827i \(0.284693\pi\)
\(510\) 0 0
\(511\) −4.24621 −0.187841
\(512\) 11.4233 0.504843
\(513\) 0 0
\(514\) 35.1231 1.54921
\(515\) −1.43845 −0.0633856
\(516\) 0 0
\(517\) 9.43845 0.415102
\(518\) 9.36932 0.411664
\(519\) 0 0
\(520\) −11.1231 −0.487780
\(521\) −10.0000 −0.438108 −0.219054 0.975713i \(-0.570297\pi\)
−0.219054 + 0.975713i \(0.570297\pi\)
\(522\) 0 0
\(523\) 7.50758 0.328283 0.164142 0.986437i \(-0.447515\pi\)
0.164142 + 0.986437i \(0.447515\pi\)
\(524\) 4.00000 0.174741
\(525\) 0 0
\(526\) −32.9848 −1.43821
\(527\) 0 0
\(528\) 0 0
\(529\) 3.24621 0.141140
\(530\) −4.87689 −0.211839
\(531\) 0 0
\(532\) −0.492423 −0.0213492
\(533\) 14.2462 0.617072
\(534\) 0 0
\(535\) −11.3693 −0.491538
\(536\) −15.2311 −0.657881
\(537\) 0 0
\(538\) 44.8769 1.93478
\(539\) −2.56155 −0.110334
\(540\) 0 0
\(541\) −17.1922 −0.739152 −0.369576 0.929201i \(-0.620497\pi\)
−0.369576 + 0.929201i \(0.620497\pi\)
\(542\) 24.9848 1.07319
\(543\) 0 0
\(544\) 11.1231 0.476899
\(545\) −17.6847 −0.757528
\(546\) 0 0
\(547\) 14.2462 0.609124 0.304562 0.952493i \(-0.401490\pi\)
0.304562 + 0.952493i \(0.401490\pi\)
\(548\) 3.89205 0.166260
\(549\) 0 0
\(550\) 4.00000 0.170561
\(551\) 6.38447 0.271988
\(552\) 0 0
\(553\) 6.56155 0.279026
\(554\) −25.3693 −1.07784
\(555\) 0 0
\(556\) −3.01515 −0.127871
\(557\) 4.87689 0.206641 0.103320 0.994648i \(-0.467053\pi\)
0.103320 + 0.994648i \(0.467053\pi\)
\(558\) 0 0
\(559\) 41.6155 1.76015
\(560\) −4.68466 −0.197963
\(561\) 0 0
\(562\) 25.8617 1.09091
\(563\) 28.0000 1.18006 0.590030 0.807382i \(-0.299116\pi\)
0.590030 + 0.807382i \(0.299116\pi\)
\(564\) 0 0
\(565\) −14.0000 −0.588984
\(566\) 36.9848 1.55459
\(567\) 0 0
\(568\) −19.5076 −0.818520
\(569\) −34.9848 −1.46664 −0.733320 0.679883i \(-0.762031\pi\)
−0.733320 + 0.679883i \(0.762031\pi\)
\(570\) 0 0
\(571\) 7.50758 0.314182 0.157091 0.987584i \(-0.449788\pi\)
0.157091 + 0.987584i \(0.449788\pi\)
\(572\) −5.12311 −0.214208
\(573\) 0 0
\(574\) 4.87689 0.203558
\(575\) 5.12311 0.213648
\(576\) 0 0
\(577\) 13.0540 0.543444 0.271722 0.962376i \(-0.412407\pi\)
0.271722 + 0.962376i \(0.412407\pi\)
\(578\) −5.94602 −0.247322
\(579\) 0 0
\(580\) −2.49242 −0.103492
\(581\) 4.00000 0.165948
\(582\) 0 0
\(583\) 8.00000 0.331326
\(584\) 10.3542 0.428458
\(585\) 0 0
\(586\) 15.1231 0.624730
\(587\) 9.75379 0.402582 0.201291 0.979531i \(-0.435486\pi\)
0.201291 + 0.979531i \(0.435486\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 6.24621 0.257152
\(591\) 0 0
\(592\) −28.1080 −1.15523
\(593\) 23.4384 0.962502 0.481251 0.876583i \(-0.340183\pi\)
0.481251 + 0.876583i \(0.340183\pi\)
\(594\) 0 0
\(595\) 4.56155 0.187005
\(596\) 1.86174 0.0762598
\(597\) 0 0
\(598\) −36.4924 −1.49229
\(599\) −8.80776 −0.359875 −0.179938 0.983678i \(-0.557590\pi\)
−0.179938 + 0.983678i \(0.557590\pi\)
\(600\) 0 0
\(601\) −26.4924 −1.08065 −0.540324 0.841457i \(-0.681698\pi\)
−0.540324 + 0.841457i \(0.681698\pi\)
\(602\) 14.2462 0.580632
\(603\) 0 0
\(604\) 9.61553 0.391250
\(605\) 4.43845 0.180449
\(606\) 0 0
\(607\) −4.94602 −0.200753 −0.100376 0.994950i \(-0.532005\pi\)
−0.100376 + 0.994950i \(0.532005\pi\)
\(608\) 2.73863 0.111066
\(609\) 0 0
\(610\) −14.6307 −0.592379
\(611\) −16.8078 −0.679969
\(612\) 0 0
\(613\) −8.73863 −0.352950 −0.176475 0.984305i \(-0.556469\pi\)
−0.176475 + 0.984305i \(0.556469\pi\)
\(614\) 49.4773 1.99674
\(615\) 0 0
\(616\) 6.24621 0.251667
\(617\) −15.7538 −0.634224 −0.317112 0.948388i \(-0.602713\pi\)
−0.317112 + 0.948388i \(0.602713\pi\)
\(618\) 0 0
\(619\) −42.1080 −1.69246 −0.846231 0.532817i \(-0.821134\pi\)
−0.846231 + 0.532817i \(0.821134\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) −15.0152 −0.602053
\(623\) 7.12311 0.285381
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) −48.8769 −1.95351
\(627\) 0 0
\(628\) 1.64584 0.0656761
\(629\) 27.3693 1.09129
\(630\) 0 0
\(631\) 8.80776 0.350632 0.175316 0.984512i \(-0.443905\pi\)
0.175316 + 0.984512i \(0.443905\pi\)
\(632\) −16.0000 −0.636446
\(633\) 0 0
\(634\) −35.1231 −1.39492
\(635\) −10.2462 −0.406608
\(636\) 0 0
\(637\) 4.56155 0.180735
\(638\) 22.7386 0.900231
\(639\) 0 0
\(640\) 13.5616 0.536067
\(641\) −2.00000 −0.0789953 −0.0394976 0.999220i \(-0.512576\pi\)
−0.0394976 + 0.999220i \(0.512576\pi\)
\(642\) 0 0
\(643\) −2.56155 −0.101018 −0.0505089 0.998724i \(-0.516084\pi\)
−0.0505089 + 0.998724i \(0.516084\pi\)
\(644\) −2.24621 −0.0885131
\(645\) 0 0
\(646\) −8.00000 −0.314756
\(647\) −3.50758 −0.137897 −0.0689486 0.997620i \(-0.521964\pi\)
−0.0689486 + 0.997620i \(0.521964\pi\)
\(648\) 0 0
\(649\) −10.2462 −0.402199
\(650\) −7.12311 −0.279391
\(651\) 0 0
\(652\) 0.492423 0.0192848
\(653\) −49.2311 −1.92656 −0.963280 0.268499i \(-0.913473\pi\)
−0.963280 + 0.268499i \(0.913473\pi\)
\(654\) 0 0
\(655\) −9.12311 −0.356469
\(656\) −14.6307 −0.571232
\(657\) 0 0
\(658\) −5.75379 −0.224306
\(659\) 36.1771 1.40926 0.704629 0.709575i \(-0.251113\pi\)
0.704629 + 0.709575i \(0.251113\pi\)
\(660\) 0 0
\(661\) 3.12311 0.121475 0.0607374 0.998154i \(-0.480655\pi\)
0.0607374 + 0.998154i \(0.480655\pi\)
\(662\) −18.7386 −0.728298
\(663\) 0 0
\(664\) −9.75379 −0.378520
\(665\) 1.12311 0.0435522
\(666\) 0 0
\(667\) 29.1231 1.12765
\(668\) −9.61553 −0.372036
\(669\) 0 0
\(670\) −9.75379 −0.376822
\(671\) 24.0000 0.926510
\(672\) 0 0
\(673\) −25.8617 −0.996897 −0.498448 0.866919i \(-0.666097\pi\)
−0.498448 + 0.866919i \(0.666097\pi\)
\(674\) 53.8617 2.07468
\(675\) 0 0
\(676\) 3.42329 0.131665
\(677\) 23.9309 0.919738 0.459869 0.887987i \(-0.347896\pi\)
0.459869 + 0.887987i \(0.347896\pi\)
\(678\) 0 0
\(679\) 14.8078 0.568270
\(680\) −11.1231 −0.426552
\(681\) 0 0
\(682\) 0 0
\(683\) −42.7386 −1.63535 −0.817674 0.575681i \(-0.804737\pi\)
−0.817674 + 0.575681i \(0.804737\pi\)
\(684\) 0 0
\(685\) −8.87689 −0.339169
\(686\) 1.56155 0.0596204
\(687\) 0 0
\(688\) −42.7386 −1.62940
\(689\) −14.2462 −0.542737
\(690\) 0 0
\(691\) 8.49242 0.323067 0.161533 0.986867i \(-0.448356\pi\)
0.161533 + 0.986867i \(0.448356\pi\)
\(692\) 3.75379 0.142698
\(693\) 0 0
\(694\) −1.75379 −0.0665729
\(695\) 6.87689 0.260855
\(696\) 0 0
\(697\) 14.2462 0.539614
\(698\) 35.1231 1.32943
\(699\) 0 0
\(700\) −0.438447 −0.0165717
\(701\) −0.0691303 −0.00261102 −0.00130551 0.999999i \(-0.500416\pi\)
−0.00130551 + 0.999999i \(0.500416\pi\)
\(702\) 0 0
\(703\) 6.73863 0.254152
\(704\) −14.2462 −0.536924
\(705\) 0 0
\(706\) −23.1231 −0.870250
\(707\) 0.246211 0.00925973
\(708\) 0 0
\(709\) −18.1771 −0.682655 −0.341327 0.939945i \(-0.610876\pi\)
−0.341327 + 0.939945i \(0.610876\pi\)
\(710\) −12.4924 −0.468832
\(711\) 0 0
\(712\) −17.3693 −0.650943
\(713\) 0 0
\(714\) 0 0
\(715\) 11.6847 0.436981
\(716\) −8.76894 −0.327711
\(717\) 0 0
\(718\) 12.4924 0.466213
\(719\) −49.6155 −1.85035 −0.925173 0.379544i \(-0.876081\pi\)
−0.925173 + 0.379544i \(0.876081\pi\)
\(720\) 0 0
\(721\) −1.43845 −0.0535706
\(722\) 27.6998 1.03088
\(723\) 0 0
\(724\) 10.3542 0.384809
\(725\) 5.68466 0.211123
\(726\) 0 0
\(727\) 19.5076 0.723496 0.361748 0.932276i \(-0.382180\pi\)
0.361748 + 0.932276i \(0.382180\pi\)
\(728\) −11.1231 −0.412250
\(729\) 0 0
\(730\) 6.63068 0.245413
\(731\) 41.6155 1.53921
\(732\) 0 0
\(733\) −5.68466 −0.209968 −0.104984 0.994474i \(-0.533479\pi\)
−0.104984 + 0.994474i \(0.533479\pi\)
\(734\) −5.75379 −0.212376
\(735\) 0 0
\(736\) 12.4924 0.460477
\(737\) 16.0000 0.589368
\(738\) 0 0
\(739\) 6.06913 0.223257 0.111628 0.993750i \(-0.464393\pi\)
0.111628 + 0.993750i \(0.464393\pi\)
\(740\) −2.63068 −0.0967058
\(741\) 0 0
\(742\) −4.87689 −0.179036
\(743\) −32.9848 −1.21010 −0.605048 0.796189i \(-0.706846\pi\)
−0.605048 + 0.796189i \(0.706846\pi\)
\(744\) 0 0
\(745\) −4.24621 −0.155569
\(746\) −45.8617 −1.67912
\(747\) 0 0
\(748\) −5.12311 −0.187319
\(749\) −11.3693 −0.415426
\(750\) 0 0
\(751\) 45.9309 1.67604 0.838021 0.545639i \(-0.183713\pi\)
0.838021 + 0.545639i \(0.183713\pi\)
\(752\) 17.2614 0.629457
\(753\) 0 0
\(754\) −40.4924 −1.47465
\(755\) −21.9309 −0.798146
\(756\) 0 0
\(757\) 14.6307 0.531761 0.265881 0.964006i \(-0.414337\pi\)
0.265881 + 0.964006i \(0.414337\pi\)
\(758\) −25.7538 −0.935420
\(759\) 0 0
\(760\) −2.73863 −0.0993407
\(761\) −31.7538 −1.15107 −0.575537 0.817776i \(-0.695207\pi\)
−0.575537 + 0.817776i \(0.695207\pi\)
\(762\) 0 0
\(763\) −17.6847 −0.640228
\(764\) 4.13826 0.149717
\(765\) 0 0
\(766\) −16.0000 −0.578103
\(767\) 18.2462 0.658833
\(768\) 0 0
\(769\) −9.50758 −0.342852 −0.171426 0.985197i \(-0.554837\pi\)
−0.171426 + 0.985197i \(0.554837\pi\)
\(770\) 4.00000 0.144150
\(771\) 0 0
\(772\) −2.35416 −0.0847281
\(773\) −8.06913 −0.290226 −0.145113 0.989415i \(-0.546355\pi\)
−0.145113 + 0.989415i \(0.546355\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) −36.1080 −1.29620
\(777\) 0 0
\(778\) 6.13826 0.220067
\(779\) 3.50758 0.125672
\(780\) 0 0
\(781\) 20.4924 0.733277
\(782\) −36.4924 −1.30497
\(783\) 0 0
\(784\) −4.68466 −0.167309
\(785\) −3.75379 −0.133978
\(786\) 0 0
\(787\) −3.82292 −0.136272 −0.0681362 0.997676i \(-0.521705\pi\)
−0.0681362 + 0.997676i \(0.521705\pi\)
\(788\) 3.12311 0.111256
\(789\) 0 0
\(790\) −10.2462 −0.364544
\(791\) −14.0000 −0.497783
\(792\) 0 0
\(793\) −42.7386 −1.51769
\(794\) −36.6004 −1.29890
\(795\) 0 0
\(796\) −8.00000 −0.283552
\(797\) 13.0540 0.462396 0.231198 0.972907i \(-0.425736\pi\)
0.231198 + 0.972907i \(0.425736\pi\)
\(798\) 0 0
\(799\) −16.8078 −0.594616
\(800\) 2.43845 0.0862121
\(801\) 0 0
\(802\) 42.8466 1.51297
\(803\) −10.8769 −0.383837
\(804\) 0 0
\(805\) 5.12311 0.180566
\(806\) 0 0
\(807\) 0 0
\(808\) −0.600373 −0.0211211
\(809\) 53.5464 1.88259 0.941296 0.337584i \(-0.109610\pi\)
0.941296 + 0.337584i \(0.109610\pi\)
\(810\) 0 0
\(811\) −21.6155 −0.759024 −0.379512 0.925187i \(-0.623908\pi\)
−0.379512 + 0.925187i \(0.623908\pi\)
\(812\) −2.49242 −0.0874669
\(813\) 0 0
\(814\) 24.0000 0.841200
\(815\) −1.12311 −0.0393407
\(816\) 0 0
\(817\) 10.2462 0.358470
\(818\) 41.3693 1.44644
\(819\) 0 0
\(820\) −1.36932 −0.0478186
\(821\) −40.4233 −1.41078 −0.705391 0.708818i \(-0.749229\pi\)
−0.705391 + 0.708818i \(0.749229\pi\)
\(822\) 0 0
\(823\) −3.50758 −0.122266 −0.0611332 0.998130i \(-0.519471\pi\)
−0.0611332 + 0.998130i \(0.519471\pi\)
\(824\) 3.50758 0.122192
\(825\) 0 0
\(826\) 6.24621 0.217333
\(827\) −19.3693 −0.673537 −0.336769 0.941587i \(-0.609334\pi\)
−0.336769 + 0.941587i \(0.609334\pi\)
\(828\) 0 0
\(829\) 43.1231 1.49773 0.748864 0.662724i \(-0.230600\pi\)
0.748864 + 0.662724i \(0.230600\pi\)
\(830\) −6.24621 −0.216809
\(831\) 0 0
\(832\) 25.3693 0.879523
\(833\) 4.56155 0.158048
\(834\) 0 0
\(835\) 21.9309 0.758949
\(836\) −1.26137 −0.0436253
\(837\) 0 0
\(838\) 15.2311 0.526148
\(839\) 37.1231 1.28163 0.640816 0.767695i \(-0.278596\pi\)
0.640816 + 0.767695i \(0.278596\pi\)
\(840\) 0 0
\(841\) 3.31534 0.114322
\(842\) −15.1231 −0.521177
\(843\) 0 0
\(844\) −10.1080 −0.347930
\(845\) −7.80776 −0.268595
\(846\) 0 0
\(847\) 4.43845 0.152507
\(848\) 14.6307 0.502420
\(849\) 0 0
\(850\) −7.12311 −0.244321
\(851\) 30.7386 1.05371
\(852\) 0 0
\(853\) −56.7386 −1.94269 −0.971347 0.237666i \(-0.923618\pi\)
−0.971347 + 0.237666i \(0.923618\pi\)
\(854\) −14.6307 −0.500652
\(855\) 0 0
\(856\) 27.7235 0.947569
\(857\) 32.2462 1.10151 0.550755 0.834667i \(-0.314340\pi\)
0.550755 + 0.834667i \(0.314340\pi\)
\(858\) 0 0
\(859\) 16.4924 0.562714 0.281357 0.959603i \(-0.409215\pi\)
0.281357 + 0.959603i \(0.409215\pi\)
\(860\) −4.00000 −0.136399
\(861\) 0 0
\(862\) 1.26137 0.0429623
\(863\) 42.2462 1.43808 0.719039 0.694970i \(-0.244582\pi\)
0.719039 + 0.694970i \(0.244582\pi\)
\(864\) 0 0
\(865\) −8.56155 −0.291102
\(866\) 12.8769 0.437575
\(867\) 0 0
\(868\) 0 0
\(869\) 16.8078 0.570164
\(870\) 0 0
\(871\) −28.4924 −0.965429
\(872\) 43.1231 1.46033
\(873\) 0 0
\(874\) −8.98485 −0.303917
\(875\) 1.00000 0.0338062
\(876\) 0 0
\(877\) −23.7538 −0.802108 −0.401054 0.916054i \(-0.631356\pi\)
−0.401054 + 0.916054i \(0.631356\pi\)
\(878\) 24.0000 0.809961
\(879\) 0 0
\(880\) −12.0000 −0.404520
\(881\) −45.8617 −1.54512 −0.772561 0.634941i \(-0.781024\pi\)
−0.772561 + 0.634941i \(0.781024\pi\)
\(882\) 0 0
\(883\) 24.4924 0.824236 0.412118 0.911131i \(-0.364789\pi\)
0.412118 + 0.911131i \(0.364789\pi\)
\(884\) 9.12311 0.306843
\(885\) 0 0
\(886\) −42.7386 −1.43583
\(887\) 12.4924 0.419454 0.209727 0.977760i \(-0.432742\pi\)
0.209727 + 0.977760i \(0.432742\pi\)
\(888\) 0 0
\(889\) −10.2462 −0.343647
\(890\) −11.1231 −0.372847
\(891\) 0 0
\(892\) −2.87689 −0.0963255
\(893\) −4.13826 −0.138482
\(894\) 0 0
\(895\) 20.0000 0.668526
\(896\) 13.5616 0.453060
\(897\) 0 0
\(898\) 29.3693 0.980067
\(899\) 0 0
\(900\) 0 0
\(901\) −14.2462 −0.474610
\(902\) 12.4924 0.415952
\(903\) 0 0
\(904\) 34.1383 1.13542
\(905\) −23.6155 −0.785007
\(906\) 0 0
\(907\) 50.1080 1.66381 0.831904 0.554920i \(-0.187251\pi\)
0.831904 + 0.554920i \(0.187251\pi\)
\(908\) −10.3845 −0.344621
\(909\) 0 0
\(910\) −7.12311 −0.236129
\(911\) −4.49242 −0.148841 −0.0744203 0.997227i \(-0.523711\pi\)
−0.0744203 + 0.997227i \(0.523711\pi\)
\(912\) 0 0
\(913\) 10.2462 0.339100
\(914\) 13.8617 0.458506
\(915\) 0 0
\(916\) 8.38447 0.277031
\(917\) −9.12311 −0.301271
\(918\) 0 0
\(919\) −13.3002 −0.438733 −0.219366 0.975643i \(-0.570399\pi\)
−0.219366 + 0.975643i \(0.570399\pi\)
\(920\) −12.4924 −0.411863
\(921\) 0 0
\(922\) −7.61553 −0.250804
\(923\) −36.4924 −1.20116
\(924\) 0 0
\(925\) 6.00000 0.197279
\(926\) 32.0000 1.05159
\(927\) 0 0
\(928\) 13.8617 0.455034
\(929\) 52.1080 1.70961 0.854803 0.518952i \(-0.173678\pi\)
0.854803 + 0.518952i \(0.173678\pi\)
\(930\) 0 0
\(931\) 1.12311 0.0368083
\(932\) 1.36932 0.0448535
\(933\) 0 0
\(934\) 41.4773 1.35718
\(935\) 11.6847 0.382129
\(936\) 0 0
\(937\) 22.6695 0.740580 0.370290 0.928916i \(-0.379258\pi\)
0.370290 + 0.928916i \(0.379258\pi\)
\(938\) −9.75379 −0.318472
\(939\) 0 0
\(940\) 1.61553 0.0526927
\(941\) 13.8617 0.451880 0.225940 0.974141i \(-0.427455\pi\)
0.225940 + 0.974141i \(0.427455\pi\)
\(942\) 0 0
\(943\) 16.0000 0.521032
\(944\) −18.7386 −0.609891
\(945\) 0 0
\(946\) 36.4924 1.18647
\(947\) −4.00000 −0.129983 −0.0649913 0.997886i \(-0.520702\pi\)
−0.0649913 + 0.997886i \(0.520702\pi\)
\(948\) 0 0
\(949\) 19.3693 0.628755
\(950\) −1.75379 −0.0569004
\(951\) 0 0
\(952\) −11.1231 −0.360502
\(953\) 24.8769 0.805842 0.402921 0.915235i \(-0.367995\pi\)
0.402921 + 0.915235i \(0.367995\pi\)
\(954\) 0 0
\(955\) −9.43845 −0.305421
\(956\) 0.354162 0.0114544
\(957\) 0 0
\(958\) 20.4924 0.662080
\(959\) −8.87689 −0.286650
\(960\) 0 0
\(961\) −31.0000 −1.00000
\(962\) −42.7386 −1.37795
\(963\) 0 0
\(964\) 5.36932 0.172934
\(965\) 5.36932 0.172844
\(966\) 0 0
\(967\) −26.8769 −0.864303 −0.432151 0.901801i \(-0.642245\pi\)
−0.432151 + 0.901801i \(0.642245\pi\)
\(968\) −10.8229 −0.347862
\(969\) 0 0
\(970\) −23.1231 −0.742438
\(971\) 49.4773 1.58780 0.793901 0.608048i \(-0.208047\pi\)
0.793901 + 0.608048i \(0.208047\pi\)
\(972\) 0 0
\(973\) 6.87689 0.220463
\(974\) −8.00000 −0.256337
\(975\) 0 0
\(976\) 43.8920 1.40495
\(977\) 49.2311 1.57504 0.787521 0.616288i \(-0.211364\pi\)
0.787521 + 0.616288i \(0.211364\pi\)
\(978\) 0 0
\(979\) 18.2462 0.583151
\(980\) −0.438447 −0.0140057
\(981\) 0 0
\(982\) 6.52273 0.208149
\(983\) 10.4233 0.332451 0.166226 0.986088i \(-0.446842\pi\)
0.166226 + 0.986088i \(0.446842\pi\)
\(984\) 0 0
\(985\) −7.12311 −0.226961
\(986\) −40.4924 −1.28954
\(987\) 0 0
\(988\) 2.24621 0.0714615
\(989\) 46.7386 1.48620
\(990\) 0 0
\(991\) −20.4924 −0.650963 −0.325482 0.945548i \(-0.605526\pi\)
−0.325482 + 0.945548i \(0.605526\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) −12.4924 −0.396236
\(995\) 18.2462 0.578444
\(996\) 0 0
\(997\) 9.68466 0.306716 0.153358 0.988171i \(-0.450991\pi\)
0.153358 + 0.988171i \(0.450991\pi\)
\(998\) 6.52273 0.206473
\(999\) 0 0
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 315.2.a.e.1.1 2
3.2 odd 2 35.2.a.b.1.2 2
4.3 odd 2 5040.2.a.bt.1.2 2
5.2 odd 4 1575.2.d.e.1324.2 4
5.3 odd 4 1575.2.d.e.1324.3 4
5.4 even 2 1575.2.a.p.1.2 2
7.6 odd 2 2205.2.a.x.1.1 2
12.11 even 2 560.2.a.i.1.2 2
15.2 even 4 175.2.b.b.99.3 4
15.8 even 4 175.2.b.b.99.2 4
15.14 odd 2 175.2.a.f.1.1 2
21.2 odd 6 245.2.e.i.116.1 4
21.5 even 6 245.2.e.h.116.1 4
21.11 odd 6 245.2.e.i.226.1 4
21.17 even 6 245.2.e.h.226.1 4
21.20 even 2 245.2.a.d.1.2 2
24.5 odd 2 2240.2.a.bh.1.2 2
24.11 even 2 2240.2.a.bd.1.1 2
33.32 even 2 4235.2.a.m.1.1 2
39.38 odd 2 5915.2.a.l.1.1 2
60.23 odd 4 2800.2.g.t.449.4 4
60.47 odd 4 2800.2.g.t.449.1 4
60.59 even 2 2800.2.a.bi.1.1 2
84.83 odd 2 3920.2.a.bs.1.1 2
105.62 odd 4 1225.2.b.f.99.3 4
105.83 odd 4 1225.2.b.f.99.2 4
105.104 even 2 1225.2.a.s.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
35.2.a.b.1.2 2 3.2 odd 2
175.2.a.f.1.1 2 15.14 odd 2
175.2.b.b.99.2 4 15.8 even 4
175.2.b.b.99.3 4 15.2 even 4
245.2.a.d.1.2 2 21.20 even 2
245.2.e.h.116.1 4 21.5 even 6
245.2.e.h.226.1 4 21.17 even 6
245.2.e.i.116.1 4 21.2 odd 6
245.2.e.i.226.1 4 21.11 odd 6
315.2.a.e.1.1 2 1.1 even 1 trivial
560.2.a.i.1.2 2 12.11 even 2
1225.2.a.s.1.1 2 105.104 even 2
1225.2.b.f.99.2 4 105.83 odd 4
1225.2.b.f.99.3 4 105.62 odd 4
1575.2.a.p.1.2 2 5.4 even 2
1575.2.d.e.1324.2 4 5.2 odd 4
1575.2.d.e.1324.3 4 5.3 odd 4
2205.2.a.x.1.1 2 7.6 odd 2
2240.2.a.bd.1.1 2 24.11 even 2
2240.2.a.bh.1.2 2 24.5 odd 2
2800.2.a.bi.1.1 2 60.59 even 2
2800.2.g.t.449.1 4 60.47 odd 4
2800.2.g.t.449.4 4 60.23 odd 4
3920.2.a.bs.1.1 2 84.83 odd 2
4235.2.a.m.1.1 2 33.32 even 2
5040.2.a.bt.1.2 2 4.3 odd 2
5915.2.a.l.1.1 2 39.38 odd 2