Properties

Label 1225.2.b.f.99.2
Level $1225$
Weight $2$
Character 1225.99
Analytic conductor $9.782$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1225 = 5^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1225.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(9.78167424761\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{17})\)
Defining polynomial: \(x^{4} + 9 x^{2} + 16\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 35)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 99.2
Root \(-1.56155i\) of defining polynomial
Character \(\chi\) \(=\) 1225.99
Dual form 1225.2.b.f.99.3

$q$-expansion

\(f(q)\) \(=\) \(q-1.56155i q^{2} +2.56155i q^{3} -0.438447 q^{4} +4.00000 q^{6} -2.43845i q^{8} -3.56155 q^{9} +O(q^{10})\) \(q-1.56155i q^{2} +2.56155i q^{3} -0.438447 q^{4} +4.00000 q^{6} -2.43845i q^{8} -3.56155 q^{9} +2.56155 q^{11} -1.12311i q^{12} -4.56155i q^{13} -4.68466 q^{16} -4.56155i q^{17} +5.56155i q^{18} +1.12311 q^{19} -4.00000i q^{22} -5.12311i q^{23} +6.24621 q^{24} -7.12311 q^{26} -1.43845i q^{27} +5.68466 q^{29} +2.43845i q^{32} +6.56155i q^{33} -7.12311 q^{34} +1.56155 q^{36} -6.00000i q^{37} -1.75379i q^{38} +11.6847 q^{39} +3.12311 q^{41} +9.12311i q^{43} -1.12311 q^{44} -8.00000 q^{46} +3.68466i q^{47} -12.0000i q^{48} +11.6847 q^{51} +2.00000i q^{52} +3.12311i q^{53} -2.24621 q^{54} +2.87689i q^{57} -8.87689i q^{58} -4.00000 q^{59} +9.36932 q^{61} -5.56155 q^{64} +10.2462 q^{66} +6.24621i q^{67} +2.00000i q^{68} +13.1231 q^{69} +8.00000 q^{71} +8.68466i q^{72} -4.24621i q^{73} -9.36932 q^{74} -0.492423 q^{76} -18.2462i q^{78} +6.56155 q^{79} -7.00000 q^{81} -4.87689i q^{82} -4.00000i q^{83} +14.2462 q^{86} +14.5616i q^{87} -6.24621i q^{88} +7.12311 q^{89} +2.24621i q^{92} +5.75379 q^{94} -6.24621 q^{96} -14.8078i q^{97} -9.12311 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 10q^{4} + 16q^{6} - 6q^{9} + O(q^{10}) \) \( 4q - 10q^{4} + 16q^{6} - 6q^{9} + 2q^{11} + 6q^{16} - 12q^{19} - 8q^{24} - 12q^{26} - 2q^{29} - 12q^{34} - 2q^{36} + 22q^{39} - 4q^{41} + 12q^{44} - 32q^{46} + 22q^{51} + 24q^{54} - 16q^{59} - 12q^{61} - 14q^{64} + 8q^{66} + 36q^{69} + 32q^{71} + 12q^{74} + 64q^{76} + 18q^{79} - 28q^{81} + 24q^{86} + 12q^{89} + 56q^{94} + 8q^{96} - 20q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1225\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(1177\)
\(\chi(n)\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 1.56155i − 1.10418i −0.833783 0.552092i \(-0.813830\pi\)
0.833783 0.552092i \(-0.186170\pi\)
\(3\) 2.56155i 1.47891i 0.673204 + 0.739457i \(0.264917\pi\)
−0.673204 + 0.739457i \(0.735083\pi\)
\(4\) −0.438447 −0.219224
\(5\) 0 0
\(6\) 4.00000 1.63299
\(7\) 0 0
\(8\) − 2.43845i − 0.862121i
\(9\) −3.56155 −1.18718
\(10\) 0 0
\(11\) 2.56155 0.772337 0.386169 0.922428i \(-0.373798\pi\)
0.386169 + 0.922428i \(0.373798\pi\)
\(12\) − 1.12311i − 0.324213i
\(13\) − 4.56155i − 1.26515i −0.774500 0.632574i \(-0.781999\pi\)
0.774500 0.632574i \(-0.218001\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) −4.68466 −1.17116
\(17\) − 4.56155i − 1.10634i −0.833069 0.553170i \(-0.813418\pi\)
0.833069 0.553170i \(-0.186582\pi\)
\(18\) 5.56155i 1.31087i
\(19\) 1.12311 0.257658 0.128829 0.991667i \(-0.458878\pi\)
0.128829 + 0.991667i \(0.458878\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) − 4.00000i − 0.852803i
\(23\) − 5.12311i − 1.06824i −0.845408 0.534121i \(-0.820643\pi\)
0.845408 0.534121i \(-0.179357\pi\)
\(24\) 6.24621 1.27500
\(25\) 0 0
\(26\) −7.12311 −1.39696
\(27\) − 1.43845i − 0.276829i
\(28\) 0 0
\(29\) 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 \(0.5\pi\)
\(32\) 2.43845i 0.431061i
\(33\) 6.56155i 1.14222i
\(34\) −7.12311 −1.22160
\(35\) 0 0
\(36\) 1.56155 0.260259
\(37\) − 6.00000i − 0.986394i −0.869918 0.493197i \(-0.835828\pi\)
0.869918 0.493197i \(-0.164172\pi\)
\(38\) − 1.75379i − 0.284502i
\(39\) 11.6847 1.87104
\(40\) 0 0
\(41\) 3.12311 0.487747 0.243874 0.969807i \(-0.421582\pi\)
0.243874 + 0.969807i \(0.421582\pi\)
\(42\) 0 0
\(43\) 9.12311i 1.39126i 0.718400 + 0.695630i \(0.244875\pi\)
−0.718400 + 0.695630i \(0.755125\pi\)
\(44\) −1.12311 −0.169315
\(45\) 0 0
\(46\) −8.00000 −1.17954
\(47\) 3.68466i 0.537463i 0.963215 + 0.268731i \(0.0866044\pi\)
−0.963215 + 0.268731i \(0.913396\pi\)
\(48\) − 12.0000i − 1.73205i
\(49\) 0 0
\(50\) 0 0
\(51\) 11.6847 1.63618
\(52\) 2.00000i 0.277350i
\(53\) 3.12311i 0.428992i 0.976725 + 0.214496i \(0.0688108\pi\)
−0.976725 + 0.214496i \(0.931189\pi\)
\(54\) −2.24621 −0.305671
\(55\) 0 0
\(56\) 0 0
\(57\) 2.87689i 0.381054i
\(58\) − 8.87689i − 1.16559i
\(59\) −4.00000 −0.520756 −0.260378 0.965507i \(-0.583847\pi\)
−0.260378 + 0.965507i \(0.583847\pi\)
\(60\) 0 0
\(61\) 9.36932 1.19962 0.599809 0.800143i \(-0.295243\pi\)
0.599809 + 0.800143i \(0.295243\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) −5.56155 −0.695194
\(65\) 0 0
\(66\) 10.2462 1.26122
\(67\) 6.24621i 0.763096i 0.924349 + 0.381548i \(0.124609\pi\)
−0.924349 + 0.381548i \(0.875391\pi\)
\(68\) 2.00000i 0.242536i
\(69\) 13.1231 1.57984
\(70\) 0 0
\(71\) 8.00000 0.949425 0.474713 0.880141i \(-0.342552\pi\)
0.474713 + 0.880141i \(0.342552\pi\)
\(72\) 8.68466i 1.02350i
\(73\) − 4.24621i − 0.496981i −0.968634 0.248491i \(-0.920065\pi\)
0.968634 0.248491i \(-0.0799345\pi\)
\(74\) −9.36932 −1.08916
\(75\) 0 0
\(76\) −0.492423 −0.0564847
\(77\) 0 0
\(78\) − 18.2462i − 2.06598i
\(79\) 6.56155 0.738232 0.369116 0.929383i \(-0.379660\pi\)
0.369116 + 0.929383i \(0.379660\pi\)
\(80\) 0 0
\(81\) −7.00000 −0.777778
\(82\) − 4.87689i − 0.538563i
\(83\) − 4.00000i − 0.439057i −0.975606 0.219529i \(-0.929548\pi\)
0.975606 0.219529i \(-0.0704519\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 14.2462 1.53621
\(87\) 14.5616i 1.56116i
\(88\) − 6.24621i − 0.665848i
\(89\) 7.12311 0.755048 0.377524 0.926000i \(-0.376776\pi\)
0.377524 + 0.926000i \(0.376776\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 2.24621i 0.234184i
\(93\) 0 0
\(94\) 5.75379 0.593458
\(95\) 0 0
\(96\) −6.24621 −0.637501
\(97\) − 14.8078i − 1.50350i −0.659448 0.751750i \(-0.729210\pi\)
0.659448 0.751750i \(-0.270790\pi\)
\(98\) 0 0
\(99\) −9.12311 −0.916907
\(100\) 0 0
\(101\) −0.246211 −0.0244989 −0.0122495 0.999925i \(-0.503899\pi\)
−0.0122495 + 0.999925i \(0.503899\pi\)
\(102\) − 18.2462i − 1.80664i
\(103\) − 1.43845i − 0.141734i −0.997486 0.0708672i \(-0.977423\pi\)
0.997486 0.0708672i \(-0.0225767\pi\)
\(104\) −11.1231 −1.09071
\(105\) 0 0
\(106\) 4.87689 0.473686
\(107\) 11.3693i 1.09911i 0.835456 + 0.549557i \(0.185203\pi\)
−0.835456 + 0.549557i \(0.814797\pi\)
\(108\) 0.630683i 0.0606875i
\(109\) −17.6847 −1.69388 −0.846942 0.531686i \(-0.821559\pi\)
−0.846942 + 0.531686i \(0.821559\pi\)
\(110\) 0 0
\(111\) 15.3693 1.45879
\(112\) 0 0
\(113\) − 14.0000i − 1.31701i −0.752577 0.658505i \(-0.771189\pi\)
0.752577 0.658505i \(-0.228811\pi\)
\(114\) 4.49242 0.420754
\(115\) 0 0
\(116\) −2.49242 −0.231416
\(117\) 16.2462i 1.50196i
\(118\) 6.24621i 0.575010i
\(119\) 0 0
\(120\) 0 0
\(121\) −4.43845 −0.403495
\(122\) − 14.6307i − 1.32460i
\(123\) 8.00000i 0.721336i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) − 10.2462i − 0.909204i −0.890695 0.454602i \(-0.849781\pi\)
0.890695 0.454602i \(-0.150219\pi\)
\(128\) 13.5616i 1.19868i
\(129\) −23.3693 −2.05755
\(130\) 0 0
\(131\) 9.12311 0.797089 0.398545 0.917149i \(-0.369515\pi\)
0.398545 + 0.917149i \(0.369515\pi\)
\(132\) − 2.87689i − 0.250402i
\(133\) 0 0
\(134\) 9.75379 0.842599
\(135\) 0 0
\(136\) −11.1231 −0.953798
\(137\) 8.87689i 0.758404i 0.925314 + 0.379202i \(0.123801\pi\)
−0.925314 + 0.379202i \(0.876199\pi\)
\(138\) − 20.4924i − 1.74443i
\(139\) −6.87689 −0.583291 −0.291645 0.956527i \(-0.594203\pi\)
−0.291645 + 0.956527i \(0.594203\pi\)
\(140\) 0 0
\(141\) −9.43845 −0.794861
\(142\) − 12.4924i − 1.04834i
\(143\) − 11.6847i − 0.977120i
\(144\) 16.6847 1.39039
\(145\) 0 0
\(146\) −6.63068 −0.548759
\(147\) 0 0
\(148\) 2.63068i 0.216241i
\(149\) 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 \(0.149052\pi\)
\(152\) − 2.73863i − 0.222133i
\(153\) 16.2462i 1.31343i
\(154\) 0 0
\(155\) 0 0
\(156\) −5.12311 −0.410177
\(157\) 3.75379i 0.299585i 0.988717 + 0.149792i \(0.0478606\pi\)
−0.988717 + 0.149792i \(0.952139\pi\)
\(158\) − 10.2462i − 0.815145i
\(159\) −8.00000 −0.634441
\(160\) 0 0
\(161\) 0 0
\(162\) 10.9309i 0.858810i
\(163\) 1.12311i 0.0879684i 0.999032 + 0.0439842i \(0.0140051\pi\)
−0.999032 + 0.0439842i \(0.985995\pi\)
\(164\) −1.36932 −0.106926
\(165\) 0 0
\(166\) −6.24621 −0.484800
\(167\) 21.9309i 1.69706i 0.529146 + 0.848531i \(0.322512\pi\)
−0.529146 + 0.848531i \(0.677488\pi\)
\(168\) 0 0
\(169\) −7.80776 −0.600597
\(170\) 0 0
\(171\) −4.00000 −0.305888
\(172\) − 4.00000i − 0.304997i
\(173\) 8.56155i 0.650923i 0.945555 + 0.325461i \(0.105520\pi\)
−0.945555 + 0.325461i \(0.894480\pi\)
\(174\) 22.7386 1.72381
\(175\) 0 0
\(176\) −12.0000 −0.904534
\(177\) − 10.2462i − 0.770152i
\(178\) − 11.1231i − 0.833712i
\(179\) −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 \(-0.840899\pi\)
−0.877664 + 0.479276i \(0.840899\pi\)
\(182\) 0 0
\(183\) 24.0000i 1.77413i
\(184\) −12.4924 −0.920954
\(185\) 0 0
\(186\) 0 0
\(187\) − 11.6847i − 0.854467i
\(188\) − 1.61553i − 0.117824i
\(189\) 0 0
\(190\) 0 0
\(191\) −9.43845 −0.682942 −0.341471 0.939892i \(-0.610925\pi\)
−0.341471 + 0.939892i \(0.610925\pi\)
\(192\) − 14.2462i − 1.02813i
\(193\) − 5.36932i − 0.386492i −0.981150 0.193246i \(-0.938098\pi\)
0.981150 0.193246i \(-0.0619015\pi\)
\(194\) −23.1231 −1.66014
\(195\) 0 0
\(196\) 0 0
\(197\) 7.12311i 0.507500i 0.967270 + 0.253750i \(0.0816641\pi\)
−0.967270 + 0.253750i \(0.918336\pi\)
\(198\) 14.2462i 1.01243i
\(199\) −18.2462 −1.29344 −0.646720 0.762728i \(-0.723860\pi\)
−0.646720 + 0.762728i \(0.723860\pi\)
\(200\) 0 0
\(201\) −16.0000 −1.12855
\(202\) 0.384472i 0.0270513i
\(203\) 0 0
\(204\) −5.12311 −0.358689
\(205\) 0 0
\(206\) −2.24621 −0.156501
\(207\) 18.2462i 1.26820i
\(208\) 21.3693i 1.48170i
\(209\) 2.87689 0.198999
\(210\) 0 0
\(211\) −23.0540 −1.58710 −0.793551 0.608504i \(-0.791770\pi\)
−0.793551 + 0.608504i \(0.791770\pi\)
\(212\) − 1.36932i − 0.0940451i
\(213\) 20.4924i 1.40412i
\(214\) 17.7538 1.21362
\(215\) 0 0
\(216\) −3.50758 −0.238660
\(217\) 0 0
\(218\) 27.6155i 1.87036i
\(219\) 10.8769 0.734992
\(220\) 0 0
\(221\) −20.8078 −1.39968
\(222\) − 24.0000i − 1.61077i
\(223\) 6.56155i 0.439394i 0.975568 + 0.219697i \(0.0705069\pi\)
−0.975568 + 0.219697i \(0.929493\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) −21.8617 −1.45422
\(227\) 23.6847i 1.57201i 0.618223 + 0.786003i \(0.287853\pi\)
−0.618223 + 0.786003i \(0.712147\pi\)
\(228\) − 1.26137i − 0.0835360i
\(229\) 19.1231 1.26369 0.631845 0.775095i \(-0.282298\pi\)
0.631845 + 0.775095i \(0.282298\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) − 13.8617i − 0.910068i
\(233\) − 3.12311i − 0.204601i −0.994754 0.102301i \(-0.967380\pi\)
0.994754 0.102301i \(-0.0326204\pi\)
\(234\) 25.3693 1.65844
\(235\) 0 0
\(236\) 1.75379 0.114162
\(237\) 16.8078i 1.09178i
\(238\) 0 0
\(239\) 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 \(-0.629056\pi\)
−0.394424 + 0.918929i \(0.629056\pi\)
\(242\) 6.93087i 0.445533i
\(243\) − 22.2462i − 1.42710i
\(244\) −4.10795 −0.262985
\(245\) 0 0
\(246\) 12.4924 0.796488
\(247\) − 5.12311i − 0.325975i
\(248\) 0 0
\(249\) 10.2462 0.649327
\(250\) 0 0
\(251\) 17.1231 1.08080 0.540400 0.841408i \(-0.318273\pi\)
0.540400 + 0.841408i \(0.318273\pi\)
\(252\) 0 0
\(253\) − 13.1231i − 0.825043i
\(254\) −16.0000 −1.00393
\(255\) 0 0
\(256\) 10.0540 0.628373
\(257\) 22.4924i 1.40304i 0.712650 + 0.701519i \(0.247495\pi\)
−0.712650 + 0.701519i \(0.752505\pi\)
\(258\) 36.4924i 2.27192i
\(259\) 0 0
\(260\) 0 0
\(261\) −20.2462 −1.25321
\(262\) − 14.2462i − 0.880134i
\(263\) − 21.1231i − 1.30251i −0.758860 0.651253i \(-0.774244\pi\)
0.758860 0.651253i \(-0.225756\pi\)
\(264\) 16.0000 0.984732
\(265\) 0 0
\(266\) 0 0
\(267\) 18.2462i 1.11665i
\(268\) − 2.73863i − 0.167289i
\(269\) 28.7386 1.75223 0.876113 0.482106i \(-0.160128\pi\)
0.876113 + 0.482106i \(0.160128\pi\)
\(270\) 0 0
\(271\) 16.0000 0.971931 0.485965 0.873978i \(-0.338468\pi\)
0.485965 + 0.873978i \(0.338468\pi\)
\(272\) 21.3693i 1.29571i
\(273\) 0 0
\(274\) 13.8617 0.837418
\(275\) 0 0
\(276\) −5.75379 −0.346337
\(277\) − 16.2462i − 0.976140i −0.872804 0.488070i \(-0.837701\pi\)
0.872804 0.488070i \(-0.162299\pi\)
\(278\) 10.7386i 0.644060i
\(279\) 0 0
\(280\) 0 0
\(281\) 16.5616 0.987979 0.493990 0.869468i \(-0.335538\pi\)
0.493990 + 0.869468i \(0.335538\pi\)
\(282\) 14.7386i 0.877673i
\(283\) 23.6847i 1.40791i 0.710246 + 0.703953i \(0.248584\pi\)
−0.710246 + 0.703953i \(0.751416\pi\)
\(284\) −3.50758 −0.208136
\(285\) 0 0
\(286\) −18.2462 −1.07892
\(287\) 0 0
\(288\) − 8.68466i − 0.511748i
\(289\) −3.80776 −0.223986
\(290\) 0 0
\(291\) 37.9309 2.22355
\(292\) 1.86174i 0.108950i
\(293\) − 9.68466i − 0.565784i −0.959152 0.282892i \(-0.908706\pi\)
0.959152 0.282892i \(-0.0912938\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) −14.6307 −0.850391
\(297\) − 3.68466i − 0.213806i
\(298\) − 6.63068i − 0.384105i
\(299\) −23.3693 −1.35148
\(300\) 0 0
\(301\) 0 0
\(302\) − 34.2462i − 1.97065i
\(303\) − 0.630683i − 0.0362318i
\(304\) −5.26137 −0.301760
\(305\) 0 0
\(306\) 25.3693 1.45027
\(307\) − 31.6847i − 1.80834i −0.427174 0.904169i \(-0.640491\pi\)
0.427174 0.904169i \(-0.359509\pi\)
\(308\) 0 0
\(309\) 3.68466 0.209613
\(310\) 0 0
\(311\) 9.61553 0.545247 0.272623 0.962121i \(-0.412109\pi\)
0.272623 + 0.962121i \(0.412109\pi\)
\(312\) − 28.4924i − 1.61307i
\(313\) − 31.3002i − 1.76919i −0.466359 0.884596i \(-0.654434\pi\)
0.466359 0.884596i \(-0.345566\pi\)
\(314\) 5.86174 0.330797
\(315\) 0 0
\(316\) −2.87689 −0.161838
\(317\) 22.4924i 1.26330i 0.775254 + 0.631650i \(0.217622\pi\)
−0.775254 + 0.631650i \(0.782378\pi\)
\(318\) 12.4924i 0.700540i
\(319\) 14.5616 0.815290
\(320\) 0 0
\(321\) −29.1231 −1.62549
\(322\) 0 0
\(323\) − 5.12311i − 0.285057i
\(324\) 3.06913 0.170507
\(325\) 0 0
\(326\) 1.75379 0.0971334
\(327\) − 45.3002i − 2.50511i
\(328\) − 7.61553i − 0.420497i
\(329\) 0 0
\(330\) 0 0
\(331\) 12.0000 0.659580 0.329790 0.944054i \(-0.393022\pi\)
0.329790 + 0.944054i \(0.393022\pi\)
\(332\) 1.75379i 0.0962517i
\(333\) 21.3693i 1.17103i
\(334\) 34.2462 1.87387
\(335\) 0 0
\(336\) 0 0
\(337\) 34.4924i 1.87892i 0.342656 + 0.939461i \(0.388674\pi\)
−0.342656 + 0.939461i \(0.611326\pi\)
\(338\) 12.1922i 0.663170i
\(339\) 35.8617 1.94774
\(340\) 0 0
\(341\) 0 0
\(342\) 6.24621i 0.337756i
\(343\) 0 0
\(344\) 22.2462 1.19944
\(345\) 0 0
\(346\) 13.3693 0.718739
\(347\) 1.12311i 0.0602915i 0.999546 + 0.0301457i \(0.00959714\pi\)
−0.999546 + 0.0301457i \(0.990403\pi\)
\(348\) − 6.38447i − 0.342244i
\(349\) −22.4924 −1.20399 −0.601996 0.798499i \(-0.705628\pi\)
−0.601996 + 0.798499i \(0.705628\pi\)
\(350\) 0 0
\(351\) −6.56155 −0.350230
\(352\) 6.24621i 0.332924i
\(353\) 14.8078i 0.788138i 0.919081 + 0.394069i \(0.128933\pi\)
−0.919081 + 0.394069i \(0.871067\pi\)
\(354\) −16.0000 −0.850390
\(355\) 0 0
\(356\) −3.12311 −0.165524
\(357\) 0 0
\(358\) 31.2311i 1.65061i
\(359\) −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
\(362\) 36.8769i 1.93821i
\(363\) − 11.3693i − 0.596734i
\(364\) 0 0
\(365\) 0 0
\(366\) 37.4773 1.95897
\(367\) 3.68466i 0.192338i 0.995365 + 0.0961688i \(0.0306589\pi\)
−0.995365 + 0.0961688i \(0.969341\pi\)
\(368\) 24.0000i 1.25109i
\(369\) −11.1231 −0.579046
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 29.3693i 1.52069i 0.649522 + 0.760343i \(0.274969\pi\)
−0.649522 + 0.760343i \(0.725031\pi\)
\(374\) −18.2462 −0.943489
\(375\) 0 0
\(376\) 8.98485 0.463358
\(377\) − 25.9309i − 1.33551i
\(378\) 0 0
\(379\) −16.4924 −0.847159 −0.423579 0.905859i \(-0.639227\pi\)
−0.423579 + 0.905859i \(0.639227\pi\)
\(380\) 0 0
\(381\) 26.2462 1.34463
\(382\) 14.7386i 0.754094i
\(383\) 10.2462i 0.523557i 0.965128 + 0.261778i \(0.0843090\pi\)
−0.965128 + 0.261778i \(0.915691\pi\)
\(384\) −34.7386 −1.77275
\(385\) 0 0
\(386\) −8.38447 −0.426758
\(387\) − 32.4924i − 1.65168i
\(388\) 6.49242i 0.329603i
\(389\) −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
\(392\) 0 0
\(393\) 23.3693i 1.17883i
\(394\) 11.1231 0.560374
\(395\) 0 0
\(396\) 4.00000 0.201008
\(397\) 23.4384i 1.17634i 0.808737 + 0.588171i \(0.200152\pi\)
−0.808737 + 0.588171i \(0.799848\pi\)
\(398\) 28.4924i 1.42820i
\(399\) 0 0
\(400\) 0 0
\(401\) 27.4384 1.37021 0.685105 0.728444i \(-0.259756\pi\)
0.685105 + 0.728444i \(0.259756\pi\)
\(402\) 24.9848i 1.24613i
\(403\) 0 0
\(404\) 0.107951 0.00537074
\(405\) 0 0
\(406\) 0 0
\(407\) − 15.3693i − 0.761829i
\(408\) − 28.4924i − 1.41059i
\(409\) −26.4924 −1.30997 −0.654983 0.755644i \(-0.727324\pi\)
−0.654983 + 0.755644i \(0.727324\pi\)
\(410\) 0 0
\(411\) −22.7386 −1.12161
\(412\) 0.630683i 0.0310715i
\(413\) 0 0
\(414\) 28.4924 1.40033
\(415\) 0 0
\(416\) 11.1231 0.545355
\(417\) − 17.6155i − 0.862636i
\(418\) − 4.49242i − 0.219732i
\(419\) 9.75379 0.476504 0.238252 0.971203i \(-0.423426\pi\)
0.238252 + 0.971203i \(0.423426\pi\)
\(420\) 0 0
\(421\) 9.68466 0.472001 0.236001 0.971753i \(-0.424163\pi\)
0.236001 + 0.971753i \(0.424163\pi\)
\(422\) 36.0000i 1.75245i
\(423\) − 13.1231i − 0.638067i
\(424\) 7.61553 0.369843
\(425\) 0 0
\(426\) 32.0000 1.55041
\(427\) 0 0
\(428\) − 4.98485i − 0.240952i
\(429\) 29.9309 1.44508
\(430\) 0 0
\(431\) 0.807764 0.0389086 0.0194543 0.999811i \(-0.493807\pi\)
0.0194543 + 0.999811i \(0.493807\pi\)
\(432\) 6.73863i 0.324213i
\(433\) 8.24621i 0.396288i 0.980173 + 0.198144i \(0.0634913\pi\)
−0.980173 + 0.198144i \(0.936509\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 7.75379 0.371339
\(437\) − 5.75379i − 0.275241i
\(438\) − 16.9848i − 0.811567i
\(439\) −15.3693 −0.733537 −0.366769 0.930312i \(-0.619536\pi\)
−0.366769 + 0.930312i \(0.619536\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 32.4924i 1.54551i
\(443\) − 27.3693i − 1.30036i −0.759782 0.650178i \(-0.774694\pi\)
0.759782 0.650178i \(-0.225306\pi\)
\(444\) −6.73863 −0.319801
\(445\) 0 0
\(446\) 10.2462 0.485172
\(447\) 10.8769i 0.514459i
\(448\) 0 0
\(449\) −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
\(452\) 6.13826i 0.288719i
\(453\) 56.1771i 2.63943i
\(454\) 36.9848 1.73578
\(455\) 0 0
\(456\) 7.01515 0.328515
\(457\) 8.87689i 0.415244i 0.978209 + 0.207622i \(0.0665723\pi\)
−0.978209 + 0.207622i \(0.933428\pi\)
\(458\) − 29.8617i − 1.39535i
\(459\) −6.56155 −0.306267
\(460\) 0 0
\(461\) 4.87689 0.227140 0.113570 0.993530i \(-0.463771\pi\)
0.113570 + 0.993530i \(0.463771\pi\)
\(462\) 0 0
\(463\) − 20.4924i − 0.952364i −0.879347 0.476182i \(-0.842020\pi\)
0.879347 0.476182i \(-0.157980\pi\)
\(464\) −26.6307 −1.23630
\(465\) 0 0
\(466\) −4.87689 −0.225918
\(467\) 26.5616i 1.22912i 0.788869 + 0.614561i \(0.210667\pi\)
−0.788869 + 0.614561i \(0.789333\pi\)
\(468\) − 7.12311i − 0.329266i
\(469\) 0 0
\(470\) 0 0
\(471\) −9.61553 −0.443060
\(472\) 9.75379i 0.448955i
\(473\) 23.3693i 1.07452i
\(474\) 26.2462 1.20553
\(475\) 0 0
\(476\) 0 0
\(477\) − 11.1231i − 0.509292i
\(478\) − 1.26137i − 0.0576935i
\(479\) 13.1231 0.599610 0.299805 0.954001i \(-0.403078\pi\)
0.299805 + 0.954001i \(0.403078\pi\)
\(480\) 0 0
\(481\) −27.3693 −1.24793
\(482\) 19.1231i 0.871034i
\(483\) 0 0
\(484\) 1.94602 0.0884557
\(485\) 0 0
\(486\) −34.7386 −1.57578
\(487\) − 5.12311i − 0.232150i −0.993240 0.116075i \(-0.962969\pi\)
0.993240 0.116075i \(-0.0370313\pi\)
\(488\) − 22.8466i − 1.03422i
\(489\) −2.87689 −0.130098
\(490\) 0 0
\(491\) 4.17708 0.188509 0.0942545 0.995548i \(-0.469953\pi\)
0.0942545 + 0.995548i \(0.469953\pi\)
\(492\) − 3.50758i − 0.158134i
\(493\) − 25.9309i − 1.16787i
\(494\) −8.00000 −0.359937
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) − 16.0000i − 0.716977i
\(499\) 4.17708 0.186992 0.0934959 0.995620i \(-0.470196\pi\)
0.0934959 + 0.995620i \(0.470196\pi\)
\(500\) 0 0
\(501\) −56.1771 −2.50981
\(502\) − 26.7386i − 1.19340i
\(503\) − 10.0691i − 0.448960i −0.974479 0.224480i \(-0.927932\pi\)
0.974479 0.224480i \(-0.0720684\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) −20.4924 −0.910999
\(507\) − 20.0000i − 0.888231i
\(508\) 4.49242i 0.199319i
\(509\) −28.2462 −1.25199 −0.625996 0.779827i \(-0.715307\pi\)
−0.625996 + 0.779827i \(0.715307\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 11.4233i 0.504843i
\(513\) − 1.61553i − 0.0713273i
\(514\) 35.1231 1.54921
\(515\) 0 0
\(516\) 10.2462 0.451064
\(517\) 9.43845i 0.415102i
\(518\) 0 0
\(519\) −21.9309 −0.962658
\(520\) 0 0
\(521\) −10.0000 −0.438108 −0.219054 0.975713i \(-0.570297\pi\)
−0.219054 + 0.975713i \(0.570297\pi\)
\(522\) 31.6155i 1.38377i
\(523\) − 7.50758i − 0.328283i −0.986437 0.164142i \(-0.947515\pi\)
0.986437 0.164142i \(-0.0524854\pi\)
\(524\) −4.00000 −0.174741
\(525\) 0 0
\(526\) −32.9848 −1.43821
\(527\) 0 0
\(528\) − 30.7386i − 1.33773i
\(529\) −3.24621 −0.141140
\(530\) 0 0
\(531\) 14.2462 0.618233
\(532\) 0 0
\(533\) − 14.2462i − 0.617072i
\(534\) 28.4924 1.23299
\(535\) 0 0
\(536\) 15.2311 0.657881
\(537\) − 51.2311i − 2.21078i
\(538\) − 44.8769i − 1.93478i
\(539\) 0 0
\(540\) 0 0
\(541\) −17.1922 −0.739152 −0.369576 0.929201i \(-0.620497\pi\)
−0.369576 + 0.929201i \(0.620497\pi\)
\(542\) − 24.9848i − 1.07319i
\(543\) − 60.4924i − 2.59598i
\(544\) 11.1231 0.476899
\(545\) 0 0
\(546\) 0 0
\(547\) − 14.2462i − 0.609124i −0.952493 0.304562i \(-0.901490\pi\)
0.952493 0.304562i \(-0.0985101\pi\)
\(548\) − 3.89205i − 0.166260i
\(549\) −33.3693 −1.42417
\(550\) 0 0
\(551\) 6.38447 0.271988
\(552\) − 32.0000i − 1.36201i
\(553\) 0 0
\(554\) −25.3693 −1.07784
\(555\) 0 0
\(556\) 3.01515 0.127871
\(557\) 4.87689i 0.206641i 0.994648 + 0.103320i \(0.0329467\pi\)
−0.994648 + 0.103320i \(0.967053\pi\)
\(558\) 0 0
\(559\) 41.6155 1.76015
\(560\) 0 0
\(561\) 29.9309 1.26368
\(562\) − 25.8617i − 1.09091i
\(563\) 28.0000i 1.18006i 0.807382 + 0.590030i \(0.200884\pi\)
−0.807382 + 0.590030i \(0.799116\pi\)
\(564\) 4.13826 0.174252
\(565\) 0 0
\(566\) 36.9848 1.55459
\(567\) 0 0
\(568\) − 19.5076i − 0.818520i
\(569\) −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 \(0.449788\pi\)
\(572\) 5.12311i 0.214208i
\(573\) − 24.1771i − 1.01001i
\(574\) 0 0
\(575\) 0 0
\(576\) 19.8078 0.825324
\(577\) 13.0540i 0.543444i 0.962376 + 0.271722i \(0.0875931\pi\)
−0.962376 + 0.271722i \(0.912407\pi\)
\(578\) 5.94602i 0.247322i
\(579\) 13.7538 0.571588
\(580\) 0 0
\(581\) 0 0
\(582\) − 59.2311i − 2.45521i
\(583\) 8.00000i 0.331326i
\(584\) −10.3542 −0.428458
\(585\) 0 0
\(586\) −15.1231 −0.624730
\(587\) − 9.75379i − 0.402582i −0.979531 0.201291i \(-0.935486\pi\)
0.979531 0.201291i \(-0.0645137\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) −18.2462 −0.750549
\(592\) 28.1080i 1.15523i
\(593\) 23.4384i 0.962502i 0.876583 + 0.481251i \(0.159817\pi\)
−0.876583 + 0.481251i \(0.840183\pi\)
\(594\) −5.75379 −0.236081
\(595\) 0 0
\(596\) −1.86174 −0.0762598
\(597\) − 46.7386i − 1.91288i
\(598\) 36.4924i 1.49229i
\(599\) −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 \(-0.318302\pi\)
0.540324 + 0.841457i \(0.318302\pi\)
\(602\) 0 0
\(603\) − 22.2462i − 0.905936i
\(604\) −9.61553 −0.391250
\(605\) 0 0
\(606\) −0.984845 −0.0400066
\(607\) − 4.94602i − 0.200753i −0.994950 0.100376i \(-0.967995\pi\)
0.994950 0.100376i \(-0.0320047\pi\)
\(608\) 2.73863i 0.111066i
\(609\) 0 0
\(610\) 0 0
\(611\) 16.8078 0.679969
\(612\) − 7.12311i − 0.287934i
\(613\) − 8.73863i − 0.352950i −0.984305 0.176475i \(-0.943531\pi\)
0.984305 0.176475i \(-0.0564695\pi\)
\(614\) −49.4773 −1.99674
\(615\) 0 0
\(616\) 0 0
\(617\) − 15.7538i − 0.634224i −0.948388 0.317112i \(-0.897287\pi\)
0.948388 0.317112i \(-0.102713\pi\)
\(618\) − 5.75379i − 0.231451i
\(619\) −42.1080 −1.69246 −0.846231 0.532817i \(-0.821134\pi\)
−0.846231 + 0.532817i \(0.821134\pi\)
\(620\) 0 0
\(621\) −7.36932 −0.295720
\(622\) − 15.0152i − 0.602053i
\(623\) 0 0
\(624\) −54.7386 −2.19130
\(625\) 0 0
\(626\) −48.8769 −1.95351
\(627\) 7.36932i 0.294302i
\(628\) − 1.64584i − 0.0656761i
\(629\) −27.3693 −1.09129
\(630\) 0 0
\(631\) 8.80776 0.350632 0.175316 0.984512i \(-0.443905\pi\)
0.175316 + 0.984512i \(0.443905\pi\)
\(632\) − 16.0000i − 0.636446i
\(633\) − 59.0540i − 2.34718i
\(634\) 35.1231 1.39492
\(635\) 0 0
\(636\) 3.50758 0.139084
\(637\) 0 0
\(638\) − 22.7386i − 0.900231i
\(639\) −28.4924 −1.12714
\(640\) 0 0
\(641\) 2.00000 0.0789953 0.0394976 0.999220i \(-0.487424\pi\)
0.0394976 + 0.999220i \(0.487424\pi\)
\(642\) 45.4773i 1.79484i
\(643\) 2.56155i 0.101018i 0.998724 + 0.0505089i \(0.0160843\pi\)
−0.998724 + 0.0505089i \(0.983916\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) −8.00000 −0.314756
\(647\) 3.50758i 0.137897i 0.997620 + 0.0689486i \(0.0219644\pi\)
−0.997620 + 0.0689486i \(0.978036\pi\)
\(648\) 17.0691i 0.670539i
\(649\) −10.2462 −0.402199
\(650\) 0 0
\(651\) 0 0
\(652\) − 0.492423i − 0.0192848i
\(653\) 49.2311i 1.92656i 0.268499 + 0.963280i \(0.413473\pi\)
−0.268499 + 0.963280i \(0.586527\pi\)
\(654\) −70.7386 −2.76610
\(655\) 0 0
\(656\) −14.6307 −0.571232
\(657\) 15.1231i 0.590009i
\(658\) 0 0
\(659\) 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 \(-0.519345\pi\)
−0.0607374 + 0.998154i \(0.519345\pi\)
\(662\) − 18.7386i − 0.728298i
\(663\) − 53.3002i − 2.07001i
\(664\) −9.75379 −0.378520
\(665\) 0 0
\(666\) 33.3693 1.29303
\(667\) − 29.1231i − 1.12765i
\(668\) − 9.61553i − 0.372036i
\(669\) −16.8078 −0.649826
\(670\) 0 0
\(671\) 24.0000 0.926510
\(672\) 0 0
\(673\) − 25.8617i − 0.996897i −0.866919 0.498448i \(-0.833903\pi\)
0.866919 0.498448i \(-0.166097\pi\)
\(674\) 53.8617 2.07468
\(675\) 0 0
\(676\) 3.42329 0.131665
\(677\) − 23.9309i − 0.919738i −0.887987 0.459869i \(-0.847896\pi\)
0.887987 0.459869i \(-0.152104\pi\)
\(678\) − 56.0000i − 2.15067i
\(679\) 0 0
\(680\) 0 0
\(681\) −60.6695 −2.32486
\(682\) 0 0
\(683\) 42.7386i 1.63535i 0.575681 + 0.817674i \(0.304737\pi\)
−0.575681 + 0.817674i \(0.695263\pi\)
\(684\) 1.75379 0.0670578
\(685\) 0 0
\(686\) 0 0
\(687\) 48.9848i 1.86889i
\(688\) − 42.7386i − 1.62940i
\(689\) 14.2462 0.542737
\(690\) 0 0
\(691\) −8.49242 −0.323067 −0.161533 0.986867i \(-0.551644\pi\)
−0.161533 + 0.986867i \(0.551644\pi\)
\(692\) − 3.75379i − 0.142698i
\(693\) 0 0
\(694\) 1.75379 0.0665729
\(695\) 0 0
\(696\) 35.5076 1.34591
\(697\) − 14.2462i − 0.539614i
\(698\) 35.1231i 1.32943i
\(699\) 8.00000 0.302588
\(700\) 0 0
\(701\) 0.0691303 0.00261102 0.00130551 0.999999i \(-0.499584\pi\)
0.00130551 + 0.999999i \(0.499584\pi\)
\(702\) 10.2462i 0.386718i
\(703\) − 6.73863i − 0.254152i
\(704\) −14.2462 −0.536924
\(705\) 0 0
\(706\) 23.1231 0.870250
\(707\) 0 0
\(708\) 4.49242i 0.168836i
\(709\) 18.1771 0.682655 0.341327 0.939945i \(-0.389124\pi\)
0.341327 + 0.939945i \(0.389124\pi\)
\(710\) 0 0
\(711\) −23.3693 −0.876418
\(712\) − 17.3693i − 0.650943i
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 8.76894 0.327711
\(717\) 2.06913i 0.0772731i
\(718\) 12.4924i 0.466213i
\(719\) 49.6155 1.85035 0.925173 0.379544i \(-0.123919\pi\)
0.925173 + 0.379544i \(0.123919\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 27.6998i 1.03088i
\(723\) − 31.3693i − 1.16664i
\(724\) 10.3542 0.384809
\(725\) 0 0
\(726\) −17.7538 −0.658905
\(727\) 19.5076i 0.723496i 0.932276 + 0.361748i \(0.117820\pi\)
−0.932276 + 0.361748i \(0.882180\pi\)
\(728\) 0 0
\(729\) 35.9848 1.33277
\(730\) 0 0
\(731\) 41.6155 1.53921
\(732\) − 10.5227i − 0.388931i
\(733\) 5.68466i 0.209968i 0.994474 + 0.104984i \(0.0334791\pi\)
−0.994474 + 0.104984i \(0.966521\pi\)
\(734\) 5.75379 0.212376
\(735\) 0 0
\(736\) 12.4924 0.460477
\(737\) 16.0000i 0.589368i
\(738\) 17.3693i 0.639373i
\(739\) −6.06913 −0.223257 −0.111628 0.993750i \(-0.535607\pi\)
−0.111628 + 0.993750i \(0.535607\pi\)
\(740\) 0 0
\(741\) 13.1231 0.482089
\(742\) 0 0
\(743\) 32.9848i 1.21010i 0.796189 + 0.605048i \(0.206846\pi\)
−0.796189 + 0.605048i \(0.793154\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 45.8617 1.67912
\(747\) 14.2462i 0.521242i
\(748\) 5.12311i 0.187319i
\(749\) 0 0
\(750\) 0 0
\(751\) 45.9309 1.67604 0.838021 0.545639i \(-0.183713\pi\)
0.838021 + 0.545639i \(0.183713\pi\)
\(752\) − 17.2614i − 0.629457i
\(753\) 43.8617i 1.59841i
\(754\) −40.4924 −1.47465
\(755\) 0 0
\(756\) 0 0
\(757\) − 14.6307i − 0.531761i −0.964006 0.265881i \(-0.914337\pi\)
0.964006 0.265881i \(-0.0856627\pi\)
\(758\) 25.7538i 0.935420i
\(759\) 33.6155 1.22017
\(760\) 0 0
\(761\) −31.7538 −1.15107 −0.575537 0.817776i \(-0.695207\pi\)
−0.575537 + 0.817776i \(0.695207\pi\)
\(762\) − 40.9848i − 1.48472i
\(763\) 0 0
\(764\) 4.13826 0.149717
\(765\) 0 0
\(766\) 16.0000 0.578103
\(767\) 18.2462i 0.658833i
\(768\) 25.7538i 0.929310i
\(769\) −9.50758 −0.342852 −0.171426 0.985197i \(-0.554837\pi\)
−0.171426 + 0.985197i \(0.554837\pi\)
\(770\) 0 0
\(771\) −57.6155 −2.07497
\(772\) 2.35416i 0.0847281i
\(773\) − 8.06913i − 0.290226i −0.989415 0.145113i \(-0.953645\pi\)
0.989415 0.145113i \(-0.0463546\pi\)
\(774\) −50.7386 −1.82376
\(775\) 0 0
\(776\) −36.1080 −1.29620
\(777\) 0 0
\(778\) 6.13826i 0.220067i
\(779\) 3.50758 0.125672
\(780\) 0 0
\(781\) 20.4924 0.733277
\(782\) 36.4924i 1.30497i
\(783\) − 8.17708i − 0.292225i
\(784\) 0 0
\(785\) 0 0
\(786\) 36.4924 1.30164
\(787\) − 3.82292i − 0.136272i −0.997676 0.0681362i \(-0.978295\pi\)
0.997676 0.0681362i \(-0.0217052\pi\)
\(788\) − 3.12311i − 0.111256i
\(789\) 54.1080 1.92629
\(790\) 0 0
\(791\) 0 0
\(792\) 22.2462i 0.790485i
\(793\) − 42.7386i − 1.51769i
\(794\) 36.6004 1.29890
\(795\) 0 0
\(796\) 8.00000 0.283552
\(797\) − 13.0540i − 0.462396i −0.972907 0.231198i \(-0.925736\pi\)
0.972907 0.231198i \(-0.0742644\pi\)
\(798\) 0 0
\(799\) 16.8078 0.594616
\(800\) 0 0
\(801\) −25.3693 −0.896381
\(802\) − 42.8466i − 1.51297i
\(803\) − 10.8769i − 0.383837i
\(804\) 7.01515 0.247405
\(805\) 0 0
\(806\) 0 0
\(807\) 73.6155i 2.59139i
\(808\) 0.600373i 0.0211211i
\(809\) 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 \(-0.376092\pi\)
0.379512 + 0.925187i \(0.376092\pi\)
\(812\) 0 0
\(813\) 40.9848i 1.43740i
\(814\) −24.0000 −0.841200
\(815\) 0 0
\(816\) −54.7386 −1.91624
\(817\) 10.2462i 0.358470i
\(818\) 41.3693i 1.44644i
\(819\) 0 0
\(820\) 0 0
\(821\) 40.4233 1.41078 0.705391 0.708818i \(-0.250771\pi\)
0.705391 + 0.708818i \(0.250771\pi\)
\(822\) 35.5076i 1.23847i
\(823\) − 3.50758i − 0.122266i −0.998130 0.0611332i \(-0.980529\pi\)
0.998130 0.0611332i \(-0.0194715\pi\)
\(824\) −3.50758 −0.122192
\(825\) 0 0
\(826\) 0 0
\(827\) − 19.3693i − 0.673537i −0.941587 0.336769i \(-0.890666\pi\)
0.941587 0.336769i \(-0.109334\pi\)
\(828\) − 8.00000i − 0.278019i
\(829\) 43.1231 1.49773 0.748864 0.662724i \(-0.230600\pi\)
0.748864 + 0.662724i \(0.230600\pi\)
\(830\) 0 0
\(831\) 41.6155 1.44363
\(832\) 25.3693i 0.879523i
\(833\) 0 0
\(834\) −27.5076 −0.952510
\(835\) 0 0
\(836\) −1.26137 −0.0436253
\(837\) 0 0
\(838\) − 15.2311i − 0.526148i
\(839\) −37.1231 −1.28163 −0.640816 0.767695i \(-0.721404\pi\)
−0.640816 + 0.767695i \(0.721404\pi\)
\(840\) 0 0
\(841\) 3.31534 0.114322
\(842\) − 15.1231i − 0.521177i
\(843\) 42.4233i 1.46114i
\(844\) 10.1080 0.347930
\(845\) 0 0
\(846\) −20.4924 −0.704544
\(847\) 0 0
\(848\) − 14.6307i − 0.502420i
\(849\) −60.6695 −2.08217
\(850\) 0 0
\(851\) −30.7386 −1.05371
\(852\) − 8.98485i − 0.307816i
\(853\) 56.7386i 1.94269i 0.237666 + 0.971347i \(0.423618\pi\)
−0.237666 + 0.971347i \(0.576382\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 27.7235 0.947569
\(857\) − 32.2462i − 1.10151i −0.834667 0.550755i \(-0.814340\pi\)
0.834667 0.550755i \(-0.185660\pi\)
\(858\) − 46.7386i − 1.59563i
\(859\) 16.4924 0.562714 0.281357 0.959603i \(-0.409215\pi\)
0.281357 + 0.959603i \(0.409215\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) − 1.26137i − 0.0429623i
\(863\) − 42.2462i − 1.43808i −0.694970 0.719039i \(-0.744582\pi\)
0.694970 0.719039i \(-0.255418\pi\)
\(864\) 3.50758 0.119330
\(865\) 0 0
\(866\) 12.8769 0.437575
\(867\) − 9.75379i − 0.331256i
\(868\) 0 0
\(869\) 16.8078 0.570164
\(870\) 0 0
\(871\) 28.4924 0.965429
\(872\) 43.1231i 1.46033i
\(873\) 52.7386i 1.78493i
\(874\) −8.98485 −0.303917
\(875\) 0 0
\(876\) −4.76894 −0.161128
\(877\) 23.7538i 0.802108i 0.916054 + 0.401054i \(0.131356\pi\)
−0.916054 + 0.401054i \(0.868644\pi\)
\(878\) 24.0000i 0.809961i
\(879\) 24.8078 0.836745
\(880\) 0 0
\(881\) −45.8617 −1.54512 −0.772561 0.634941i \(-0.781024\pi\)
−0.772561 + 0.634941i \(0.781024\pi\)
\(882\) 0 0
\(883\) 24.4924i 0.824236i 0.911131 + 0.412118i \(0.135211\pi\)
−0.911131 + 0.412118i \(0.864789\pi\)
\(884\) 9.12311 0.306843
\(885\) 0 0
\(886\) −42.7386 −1.43583
\(887\) − 12.4924i − 0.419454i −0.977760 0.209727i \(-0.932742\pi\)
0.977760 0.209727i \(-0.0672576\pi\)
\(888\) − 37.4773i − 1.25765i
\(889\) 0 0
\(890\) 0 0
\(891\) −17.9309 −0.600707
\(892\) − 2.87689i − 0.0963255i
\(893\) 4.13826i 0.138482i
\(894\) 16.9848 0.568058
\(895\) 0 0
\(896\) 0 0
\(897\) − 59.8617i − 1.99873i
\(898\) 29.3693i 0.980067i
\(899\) 0 0
\(900\) 0 0
\(901\) 14.2462 0.474610
\(902\) − 12.4924i − 0.415952i
\(903\) 0 0
\(904\) −34.1383 −1.13542
\(905\) 0 0
\(906\) 87.7235 2.91442
\(907\) − 50.1080i − 1.66381i −0.554920 0.831904i \(-0.687251\pi\)
0.554920 0.831904i \(-0.312749\pi\)
\(908\) − 10.3845i − 0.344621i
\(909\) 0.876894 0.0290848
\(910\) 0 0
\(911\) 4.49242 0.148841 0.0744203 0.997227i \(-0.476289\pi\)
0.0744203 + 0.997227i \(0.476289\pi\)
\(912\) − 13.4773i − 0.446277i
\(913\) − 10.2462i − 0.339100i
\(914\) 13.8617 0.458506
\(915\) 0 0
\(916\) −8.38447 −0.277031
\(917\) 0 0
\(918\) 10.2462i 0.338175i
\(919\) 13.3002 0.438733 0.219366 0.975643i \(-0.429601\pi\)
0.219366 + 0.975643i \(0.429601\pi\)
\(920\) 0 0
\(921\) 81.1619 2.67438
\(922\) − 7.61553i − 0.250804i
\(923\) − 36.4924i − 1.20116i
\(924\) 0 0
\(925\) 0 0
\(926\) −32.0000 −1.05159
\(927\) 5.12311i 0.168265i
\(928\) 13.8617i 0.455034i
\(929\) −52.1080 −1.70961 −0.854803 0.518952i \(-0.826322\pi\)
−0.854803 + 0.518952i \(0.826322\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 1.36932i 0.0448535i
\(933\) 24.6307i 0.806372i
\(934\) 41.4773 1.35718
\(935\) 0 0
\(936\) 39.6155 1.29487
\(937\) 22.6695i 0.740580i 0.928916 + 0.370290i \(0.120742\pi\)
−0.928916 + 0.370290i \(0.879258\pi\)
\(938\) 0 0
\(939\) 80.1771 2.61648
\(940\) 0 0
\(941\) 13.8617 0.451880 0.225940 0.974141i \(-0.427455\pi\)
0.225940 + 0.974141i \(0.427455\pi\)
\(942\) 15.0152i 0.489220i
\(943\) − 16.0000i − 0.521032i
\(944\) 18.7386 0.609891
\(945\) 0 0
\(946\) 36.4924 1.18647
\(947\) − 4.00000i − 0.129983i −0.997886 0.0649913i \(-0.979298\pi\)
0.997886 0.0649913i \(-0.0207020\pi\)
\(948\) − 7.36932i − 0.239344i
\(949\) −19.3693 −0.628755
\(950\) 0 0
\(951\) −57.6155 −1.86831
\(952\) 0 0
\(953\) − 24.8769i − 0.805842i −0.915235 0.402921i \(-0.867995\pi\)
0.915235 0.402921i \(-0.132005\pi\)
\(954\) −17.3693 −0.562352
\(955\) 0 0
\(956\) −0.354162 −0.0114544
\(957\) 37.3002i 1.20574i
\(958\) − 20.4924i − 0.662080i
\(959\) 0 0
\(960\) 0 0
\(961\) −31.0000 −1.00000
\(962\) 42.7386i 1.37795i
\(963\) − 40.4924i − 1.30485i
\(964\) 5.36932 0.172934
\(965\) 0 0
\(966\) 0 0
\(967\) 26.8769i 0.864303i 0.901801 + 0.432151i \(0.142245\pi\)
−0.901801 + 0.432151i \(0.857755\pi\)
\(968\) 10.8229i 0.347862i
\(969\) 13.1231 0.421575
\(970\) 0 0
\(971\) 49.4773 1.58780 0.793901 0.608048i \(-0.208047\pi\)
0.793901 + 0.608048i \(0.208047\pi\)
\(972\) 9.75379i 0.312853i
\(973\) 0 0
\(974\) −8.00000 −0.256337
\(975\) 0 0
\(976\) −43.8920 −1.40495
\(977\) 49.2311i 1.57504i 0.616288 + 0.787521i \(0.288636\pi\)
−0.616288 + 0.787521i \(0.711364\pi\)
\(978\) 4.49242i 0.143652i
\(979\) 18.2462 0.583151
\(980\) 0 0
\(981\) 62.9848 2.01095
\(982\) − 6.52273i − 0.208149i
\(983\) 10.4233i 0.332451i 0.986088 + 0.166226i \(0.0531580\pi\)
−0.986088 + 0.166226i \(0.946842\pi\)
\(984\) 19.5076 0.621879
\(985\) 0 0
\(986\) −40.4924 −1.28954
\(987\) 0 0
\(988\) 2.24621i 0.0714615i
\(989\) 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
\(993\) 30.7386i 0.975461i
\(994\) 0 0
\(995\) 0 0
\(996\) −4.49242 −0.142348
\(997\) 9.68466i 0.306716i 0.988171 + 0.153358i \(0.0490088\pi\)
−0.988171 + 0.153358i \(0.950991\pi\)
\(998\) − 6.52273i − 0.206473i
\(999\) −8.63068 −0.273063
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 1225.2.b.f.99.2 4
5.2 odd 4 245.2.a.d.1.2 2
5.3 odd 4 1225.2.a.s.1.1 2
5.4 even 2 inner 1225.2.b.f.99.3 4
7.6 odd 2 175.2.b.b.99.2 4
15.2 even 4 2205.2.a.x.1.1 2
20.7 even 4 3920.2.a.bs.1.1 2
21.20 even 2 1575.2.d.e.1324.3 4
28.27 even 2 2800.2.g.t.449.4 4
35.2 odd 12 245.2.e.h.116.1 4
35.12 even 12 245.2.e.i.116.1 4
35.13 even 4 175.2.a.f.1.1 2
35.17 even 12 245.2.e.i.226.1 4
35.27 even 4 35.2.a.b.1.2 2
35.32 odd 12 245.2.e.h.226.1 4
35.34 odd 2 175.2.b.b.99.3 4
105.62 odd 4 315.2.a.e.1.1 2
105.83 odd 4 1575.2.a.p.1.2 2
105.104 even 2 1575.2.d.e.1324.2 4
140.27 odd 4 560.2.a.i.1.2 2
140.83 odd 4 2800.2.a.bi.1.1 2
140.139 even 2 2800.2.g.t.449.1 4
280.27 odd 4 2240.2.a.bd.1.1 2
280.237 even 4 2240.2.a.bh.1.2 2
385.307 odd 4 4235.2.a.m.1.1 2
420.167 even 4 5040.2.a.bt.1.2 2
455.272 even 4 5915.2.a.l.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
35.2.a.b.1.2 2 35.27 even 4
175.2.a.f.1.1 2 35.13 even 4
175.2.b.b.99.2 4 7.6 odd 2
175.2.b.b.99.3 4 35.34 odd 2
245.2.a.d.1.2 2 5.2 odd 4
245.2.e.h.116.1 4 35.2 odd 12
245.2.e.h.226.1 4 35.32 odd 12
245.2.e.i.116.1 4 35.12 even 12
245.2.e.i.226.1 4 35.17 even 12
315.2.a.e.1.1 2 105.62 odd 4
560.2.a.i.1.2 2 140.27 odd 4
1225.2.a.s.1.1 2 5.3 odd 4
1225.2.b.f.99.2 4 1.1 even 1 trivial
1225.2.b.f.99.3 4 5.4 even 2 inner
1575.2.a.p.1.2 2 105.83 odd 4
1575.2.d.e.1324.2 4 105.104 even 2
1575.2.d.e.1324.3 4 21.20 even 2
2205.2.a.x.1.1 2 15.2 even 4
2240.2.a.bd.1.1 2 280.27 odd 4
2240.2.a.bh.1.2 2 280.237 even 4
2800.2.a.bi.1.1 2 140.83 odd 4
2800.2.g.t.449.1 4 140.139 even 2
2800.2.g.t.449.4 4 28.27 even 2
3920.2.a.bs.1.1 2 20.7 even 4
4235.2.a.m.1.1 2 385.307 odd 4
5040.2.a.bt.1.2 2 420.167 even 4
5915.2.a.l.1.1 2 455.272 even 4