Properties

Label 1225.2.b.f.99.1
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 99.1
Root \(-2.56155i\) of defining polynomial
Character \(\chi\) \(=\) 1225.99
Dual form 1225.2.b.f.99.4

$q$-expansion

\(f(q)\) \(=\) \(q-2.56155i q^{2} +1.56155i q^{3} -4.56155 q^{4} +4.00000 q^{6} +6.56155i q^{8} +0.561553 q^{9} +O(q^{10})\) \(q-2.56155i q^{2} +1.56155i q^{3} -4.56155 q^{4} +4.00000 q^{6} +6.56155i q^{8} +0.561553 q^{9} -1.56155 q^{11} -7.12311i q^{12} +0.438447i q^{13} +7.68466 q^{16} +0.438447i q^{17} -1.43845i q^{18} -7.12311 q^{19} +4.00000i q^{22} -3.12311i q^{23} -10.2462 q^{24} +1.12311 q^{26} +5.56155i q^{27} -6.68466 q^{29} -6.56155i q^{32} -2.43845i q^{33} +1.12311 q^{34} -2.56155 q^{36} +6.00000i q^{37} +18.2462i q^{38} -0.684658 q^{39} -5.12311 q^{41} -0.876894i q^{43} +7.12311 q^{44} -8.00000 q^{46} +8.68466i q^{47} +12.0000i q^{48} -0.684658 q^{51} -2.00000i q^{52} +5.12311i q^{53} +14.2462 q^{54} -11.1231i q^{57} +17.1231i q^{58} -4.00000 q^{59} -15.3693 q^{61} -1.43845 q^{64} -6.24621 q^{66} +10.2462i q^{67} -2.00000i q^{68} +4.87689 q^{69} +8.00000 q^{71} +3.68466i q^{72} -12.2462i q^{73} +15.3693 q^{74} +32.4924 q^{76} +1.75379i q^{78} +2.43845 q^{79} -7.00000 q^{81} +13.1231i q^{82} +4.00000i q^{83} -2.24621 q^{86} -10.4384i q^{87} -10.2462i q^{88} -1.12311 q^{89} +14.2462i q^{92} +22.2462 q^{94} +10.2462 q^{96} -5.80776i q^{97} -0.876894 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 10q^{4} + 16q^{6} - 6q^{9} + O(q^{10}) \) \( 4q - 10q^{4} + 16q^{6} - 6q^{9} + 2q^{11} + 6q^{16} - 12q^{19} - 8q^{24} - 12q^{26} - 2q^{29} - 12q^{34} - 2q^{36} + 22q^{39} - 4q^{41} + 12q^{44} - 32q^{46} + 22q^{51} + 24q^{54} - 16q^{59} - 12q^{61} - 14q^{64} + 8q^{66} + 36q^{69} + 32q^{71} + 12q^{74} + 64q^{76} + 18q^{79} - 28q^{81} + 24q^{86} + 12q^{89} + 56q^{94} + 8q^{96} - 20q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 2.56155i − 1.81129i −0.424035 0.905646i \(-0.639387\pi\)
0.424035 0.905646i \(-0.360613\pi\)
\(3\) 1.56155i 0.901563i 0.892634 + 0.450781i \(0.148855\pi\)
−0.892634 + 0.450781i \(0.851145\pi\)
\(4\) −4.56155 −2.28078
\(5\) 0 0
\(6\) 4.00000 1.63299
\(7\) 0 0
\(8\) 6.56155i 2.31986i
\(9\) 0.561553 0.187184
\(10\) 0 0
\(11\) −1.56155 −0.470826 −0.235413 0.971895i \(-0.575644\pi\)
−0.235413 + 0.971895i \(0.575644\pi\)
\(12\) − 7.12311i − 2.05626i
\(13\) 0.438447i 0.121603i 0.998150 + 0.0608017i \(0.0193657\pi\)
−0.998150 + 0.0608017i \(0.980634\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 7.68466 1.92116
\(17\) 0.438447i 0.106339i 0.998586 + 0.0531695i \(0.0169324\pi\)
−0.998586 + 0.0531695i \(0.983068\pi\)
\(18\) − 1.43845i − 0.339045i
\(19\) −7.12311 −1.63415 −0.817076 0.576530i \(-0.804407\pi\)
−0.817076 + 0.576530i \(0.804407\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 4.00000i 0.852803i
\(23\) − 3.12311i − 0.651213i −0.945505 0.325606i \(-0.894432\pi\)
0.945505 0.325606i \(-0.105568\pi\)
\(24\) −10.2462 −2.09150
\(25\) 0 0
\(26\) 1.12311 0.220259
\(27\) 5.56155i 1.07032i
\(28\) 0 0
\(29\) −6.68466 −1.24131 −0.620655 0.784084i \(-0.713133\pi\)
−0.620655 + 0.784084i \(0.713133\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) − 6.56155i − 1.15993i
\(33\) − 2.43845i − 0.424479i
\(34\) 1.12311 0.192611
\(35\) 0 0
\(36\) −2.56155 −0.426925
\(37\) 6.00000i 0.986394i 0.869918 + 0.493197i \(0.164172\pi\)
−0.869918 + 0.493197i \(0.835828\pi\)
\(38\) 18.2462i 2.95993i
\(39\) −0.684658 −0.109633
\(40\) 0 0
\(41\) −5.12311 −0.800095 −0.400047 0.916494i \(-0.631006\pi\)
−0.400047 + 0.916494i \(0.631006\pi\)
\(42\) 0 0
\(43\) − 0.876894i − 0.133725i −0.997762 0.0668626i \(-0.978701\pi\)
0.997762 0.0668626i \(-0.0212989\pi\)
\(44\) 7.12311 1.07385
\(45\) 0 0
\(46\) −8.00000 −1.17954
\(47\) 8.68466i 1.26679i 0.773830 + 0.633394i \(0.218339\pi\)
−0.773830 + 0.633394i \(0.781661\pi\)
\(48\) 12.0000i 1.73205i
\(49\) 0 0
\(50\) 0 0
\(51\) −0.684658 −0.0958714
\(52\) − 2.00000i − 0.277350i
\(53\) 5.12311i 0.703713i 0.936054 + 0.351856i \(0.114449\pi\)
−0.936054 + 0.351856i \(0.885551\pi\)
\(54\) 14.2462 1.93866
\(55\) 0 0
\(56\) 0 0
\(57\) − 11.1231i − 1.47329i
\(58\) 17.1231i 2.24837i
\(59\) −4.00000 −0.520756 −0.260378 0.965507i \(-0.583847\pi\)
−0.260378 + 0.965507i \(0.583847\pi\)
\(60\) 0 0
\(61\) −15.3693 −1.96784 −0.983920 0.178611i \(-0.942839\pi\)
−0.983920 + 0.178611i \(0.942839\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) −1.43845 −0.179806
\(65\) 0 0
\(66\) −6.24621 −0.768855
\(67\) 10.2462i 1.25177i 0.779914 + 0.625887i \(0.215263\pi\)
−0.779914 + 0.625887i \(0.784737\pi\)
\(68\) − 2.00000i − 0.242536i
\(69\) 4.87689 0.587109
\(70\) 0 0
\(71\) 8.00000 0.949425 0.474713 0.880141i \(-0.342552\pi\)
0.474713 + 0.880141i \(0.342552\pi\)
\(72\) 3.68466i 0.434241i
\(73\) − 12.2462i − 1.43331i −0.697428 0.716655i \(-0.745672\pi\)
0.697428 0.716655i \(-0.254328\pi\)
\(74\) 15.3693 1.78665
\(75\) 0 0
\(76\) 32.4924 3.72714
\(77\) 0 0
\(78\) 1.75379i 0.198577i
\(79\) 2.43845 0.274347 0.137173 0.990547i \(-0.456198\pi\)
0.137173 + 0.990547i \(0.456198\pi\)
\(80\) 0 0
\(81\) −7.00000 −0.777778
\(82\) 13.1231i 1.44920i
\(83\) 4.00000i 0.439057i 0.975606 + 0.219529i \(0.0704519\pi\)
−0.975606 + 0.219529i \(0.929548\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −2.24621 −0.242215
\(87\) − 10.4384i − 1.11912i
\(88\) − 10.2462i − 1.09225i
\(89\) −1.12311 −0.119049 −0.0595245 0.998227i \(-0.518958\pi\)
−0.0595245 + 0.998227i \(0.518958\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 14.2462i 1.48527i
\(93\) 0 0
\(94\) 22.2462 2.29452
\(95\) 0 0
\(96\) 10.2462 1.04575
\(97\) − 5.80776i − 0.589689i −0.955545 0.294845i \(-0.904732\pi\)
0.955545 0.294845i \(-0.0952679\pi\)
\(98\) 0 0
\(99\) −0.876894 −0.0881312
\(100\) 0 0
\(101\) 16.2462 1.61656 0.808279 0.588799i \(-0.200399\pi\)
0.808279 + 0.588799i \(0.200399\pi\)
\(102\) 1.75379i 0.173651i
\(103\) 5.56155i 0.547996i 0.961730 + 0.273998i \(0.0883462\pi\)
−0.961730 + 0.273998i \(0.911654\pi\)
\(104\) −2.87689 −0.282103
\(105\) 0 0
\(106\) 13.1231 1.27463
\(107\) 13.3693i 1.29246i 0.763142 + 0.646230i \(0.223655\pi\)
−0.763142 + 0.646230i \(0.776345\pi\)
\(108\) − 25.3693i − 2.44116i
\(109\) −5.31534 −0.509117 −0.254559 0.967057i \(-0.581930\pi\)
−0.254559 + 0.967057i \(0.581930\pi\)
\(110\) 0 0
\(111\) −9.36932 −0.889296
\(112\) 0 0
\(113\) 14.0000i 1.31701i 0.752577 + 0.658505i \(0.228811\pi\)
−0.752577 + 0.658505i \(0.771189\pi\)
\(114\) −28.4924 −2.66856
\(115\) 0 0
\(116\) 30.4924 2.83115
\(117\) 0.246211i 0.0227622i
\(118\) 10.2462i 0.943240i
\(119\) 0 0
\(120\) 0 0
\(121\) −8.56155 −0.778323
\(122\) 39.3693i 3.56433i
\(123\) − 8.00000i − 0.721336i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) − 6.24621i − 0.554262i −0.960832 0.277131i \(-0.910616\pi\)
0.960832 0.277131i \(-0.0893835\pi\)
\(128\) − 9.43845i − 0.834249i
\(129\) 1.36932 0.120562
\(130\) 0 0
\(131\) 0.876894 0.0766146 0.0383073 0.999266i \(-0.487803\pi\)
0.0383073 + 0.999266i \(0.487803\pi\)
\(132\) 11.1231i 0.968142i
\(133\) 0 0
\(134\) 26.2462 2.26733
\(135\) 0 0
\(136\) −2.87689 −0.246692
\(137\) − 17.1231i − 1.46293i −0.681881 0.731463i \(-0.738838\pi\)
0.681881 0.731463i \(-0.261162\pi\)
\(138\) − 12.4924i − 1.06343i
\(139\) −15.1231 −1.28273 −0.641363 0.767238i \(-0.721631\pi\)
−0.641363 + 0.767238i \(0.721631\pi\)
\(140\) 0 0
\(141\) −13.5616 −1.14209
\(142\) − 20.4924i − 1.71969i
\(143\) − 0.684658i − 0.0572540i
\(144\) 4.31534 0.359612
\(145\) 0 0
\(146\) −31.3693 −2.59614
\(147\) 0 0
\(148\) − 27.3693i − 2.24974i
\(149\) −12.2462 −1.00325 −0.501624 0.865086i \(-0.667264\pi\)
−0.501624 + 0.865086i \(0.667264\pi\)
\(150\) 0 0
\(151\) −6.93087 −0.564026 −0.282013 0.959411i \(-0.591002\pi\)
−0.282013 + 0.959411i \(0.591002\pi\)
\(152\) − 46.7386i − 3.79100i
\(153\) 0.246211i 0.0199050i
\(154\) 0 0
\(155\) 0 0
\(156\) 3.12311 0.250049
\(157\) − 20.2462i − 1.61582i −0.589303 0.807912i \(-0.700598\pi\)
0.589303 0.807912i \(-0.299402\pi\)
\(158\) − 6.24621i − 0.496922i
\(159\) −8.00000 −0.634441
\(160\) 0 0
\(161\) 0 0
\(162\) 17.9309i 1.40878i
\(163\) 7.12311i 0.557925i 0.960302 + 0.278962i \(0.0899905\pi\)
−0.960302 + 0.278962i \(0.910010\pi\)
\(164\) 23.3693 1.82484
\(165\) 0 0
\(166\) 10.2462 0.795260
\(167\) 6.93087i 0.536327i 0.963373 + 0.268163i \(0.0864167\pi\)
−0.963373 + 0.268163i \(0.913583\pi\)
\(168\) 0 0
\(169\) 12.8078 0.985213
\(170\) 0 0
\(171\) −4.00000 −0.305888
\(172\) 4.00000i 0.304997i
\(173\) − 4.43845i − 0.337449i −0.985663 0.168724i \(-0.946035\pi\)
0.985663 0.168724i \(-0.0539648\pi\)
\(174\) −26.7386 −2.02705
\(175\) 0 0
\(176\) −12.0000 −0.904534
\(177\) − 6.24621i − 0.469494i
\(178\) 2.87689i 0.215632i
\(179\) −20.0000 −1.49487 −0.747435 0.664335i \(-0.768715\pi\)
−0.747435 + 0.664335i \(0.768715\pi\)
\(180\) 0 0
\(181\) 17.6155 1.30935 0.654676 0.755910i \(-0.272805\pi\)
0.654676 + 0.755910i \(0.272805\pi\)
\(182\) 0 0
\(183\) − 24.0000i − 1.77413i
\(184\) 20.4924 1.51072
\(185\) 0 0
\(186\) 0 0
\(187\) − 0.684658i − 0.0500672i
\(188\) − 39.6155i − 2.88926i
\(189\) 0 0
\(190\) 0 0
\(191\) −13.5616 −0.981280 −0.490640 0.871363i \(-0.663237\pi\)
−0.490640 + 0.871363i \(0.663237\pi\)
\(192\) − 2.24621i − 0.162106i
\(193\) − 19.3693i − 1.39423i −0.716957 0.697117i \(-0.754466\pi\)
0.716957 0.697117i \(-0.245534\pi\)
\(194\) −14.8769 −1.06810
\(195\) 0 0
\(196\) 0 0
\(197\) 1.12311i 0.0800180i 0.999199 + 0.0400090i \(0.0127387\pi\)
−0.999199 + 0.0400090i \(0.987261\pi\)
\(198\) 2.24621i 0.159631i
\(199\) −1.75379 −0.124323 −0.0621614 0.998066i \(-0.519799\pi\)
−0.0621614 + 0.998066i \(0.519799\pi\)
\(200\) 0 0
\(201\) −16.0000 −1.12855
\(202\) − 41.6155i − 2.92806i
\(203\) 0 0
\(204\) 3.12311 0.218661
\(205\) 0 0
\(206\) 14.2462 0.992581
\(207\) − 1.75379i − 0.121897i
\(208\) 3.36932i 0.233620i
\(209\) 11.1231 0.769401
\(210\) 0 0
\(211\) 14.0540 0.967516 0.483758 0.875202i \(-0.339272\pi\)
0.483758 + 0.875202i \(0.339272\pi\)
\(212\) − 23.3693i − 1.60501i
\(213\) 12.4924i 0.855967i
\(214\) 34.2462 2.34102
\(215\) 0 0
\(216\) −36.4924 −2.48299
\(217\) 0 0
\(218\) 13.6155i 0.922160i
\(219\) 19.1231 1.29222
\(220\) 0 0
\(221\) −0.192236 −0.0129312
\(222\) 24.0000i 1.61077i
\(223\) − 2.43845i − 0.163291i −0.996661 0.0816453i \(-0.973983\pi\)
0.996661 0.0816453i \(-0.0260175\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 35.8617 2.38549
\(227\) − 11.3153i − 0.751026i −0.926817 0.375513i \(-0.877467\pi\)
0.926817 0.375513i \(-0.122533\pi\)
\(228\) 50.7386i 3.36025i
\(229\) 10.8769 0.718765 0.359383 0.933190i \(-0.382987\pi\)
0.359383 + 0.933190i \(0.382987\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) − 43.8617i − 2.87966i
\(233\) − 5.12311i − 0.335626i −0.985819 0.167813i \(-0.946330\pi\)
0.985819 0.167813i \(-0.0536704\pi\)
\(234\) 0.630683 0.0412290
\(235\) 0 0
\(236\) 18.2462 1.18773
\(237\) 3.80776i 0.247341i
\(238\) 0 0
\(239\) −19.8078 −1.28126 −0.640629 0.767851i \(-0.721326\pi\)
−0.640629 + 0.767851i \(0.721326\pi\)
\(240\) 0 0
\(241\) 4.24621 0.273523 0.136761 0.990604i \(-0.456331\pi\)
0.136761 + 0.990604i \(0.456331\pi\)
\(242\) 21.9309i 1.40977i
\(243\) 5.75379i 0.369106i
\(244\) 70.1080 4.48820
\(245\) 0 0
\(246\) −20.4924 −1.30655
\(247\) − 3.12311i − 0.198718i
\(248\) 0 0
\(249\) −6.24621 −0.395838
\(250\) 0 0
\(251\) 8.87689 0.560305 0.280152 0.959956i \(-0.409615\pi\)
0.280152 + 0.959956i \(0.409615\pi\)
\(252\) 0 0
\(253\) 4.87689i 0.306608i
\(254\) −16.0000 −1.00393
\(255\) 0 0
\(256\) −27.0540 −1.69087
\(257\) 10.4924i 0.654499i 0.944938 + 0.327250i \(0.106122\pi\)
−0.944938 + 0.327250i \(0.893878\pi\)
\(258\) − 3.50758i − 0.218372i
\(259\) 0 0
\(260\) 0 0
\(261\) −3.75379 −0.232354
\(262\) − 2.24621i − 0.138771i
\(263\) 12.8769i 0.794023i 0.917814 + 0.397012i \(0.129953\pi\)
−0.917814 + 0.397012i \(0.870047\pi\)
\(264\) 16.0000 0.984732
\(265\) 0 0
\(266\) 0 0
\(267\) − 1.75379i − 0.107330i
\(268\) − 46.7386i − 2.85502i
\(269\) −20.7386 −1.26446 −0.632228 0.774782i \(-0.717860\pi\)
−0.632228 + 0.774782i \(0.717860\pi\)
\(270\) 0 0
\(271\) 16.0000 0.971931 0.485965 0.873978i \(-0.338468\pi\)
0.485965 + 0.873978i \(0.338468\pi\)
\(272\) 3.36932i 0.204295i
\(273\) 0 0
\(274\) −43.8617 −2.64978
\(275\) 0 0
\(276\) −22.2462 −1.33906
\(277\) − 0.246211i − 0.0147934i −0.999973 0.00739670i \(-0.997646\pi\)
0.999973 0.00739670i \(-0.00235446\pi\)
\(278\) 38.7386i 2.32339i
\(279\) 0 0
\(280\) 0 0
\(281\) 12.4384 0.742016 0.371008 0.928630i \(-0.379012\pi\)
0.371008 + 0.928630i \(0.379012\pi\)
\(282\) 34.7386i 2.06866i
\(283\) − 11.3153i − 0.672627i −0.941750 0.336314i \(-0.890820\pi\)
0.941750 0.336314i \(-0.109180\pi\)
\(284\) −36.4924 −2.16543
\(285\) 0 0
\(286\) −1.75379 −0.103704
\(287\) 0 0
\(288\) − 3.68466i − 0.217121i
\(289\) 16.8078 0.988692
\(290\) 0 0
\(291\) 9.06913 0.531642
\(292\) 55.8617i 3.26906i
\(293\) − 2.68466i − 0.156839i −0.996920 0.0784197i \(-0.975013\pi\)
0.996920 0.0784197i \(-0.0249874\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) −39.3693 −2.28830
\(297\) − 8.68466i − 0.503935i
\(298\) 31.3693i 1.81718i
\(299\) 1.36932 0.0791896
\(300\) 0 0
\(301\) 0 0
\(302\) 17.7538i 1.02162i
\(303\) 25.3693i 1.45743i
\(304\) −54.7386 −3.13948
\(305\) 0 0
\(306\) 0.630683 0.0360538
\(307\) 19.3153i 1.10238i 0.834378 + 0.551192i \(0.185827\pi\)
−0.834378 + 0.551192i \(0.814173\pi\)
\(308\) 0 0
\(309\) −8.68466 −0.494053
\(310\) 0 0
\(311\) −31.6155 −1.79275 −0.896376 0.443294i \(-0.853810\pi\)
−0.896376 + 0.443294i \(0.853810\pi\)
\(312\) − 4.49242i − 0.254333i
\(313\) − 22.3002i − 1.26048i −0.776400 0.630241i \(-0.782956\pi\)
0.776400 0.630241i \(-0.217044\pi\)
\(314\) −51.8617 −2.92673
\(315\) 0 0
\(316\) −11.1231 −0.625724
\(317\) 10.4924i 0.589313i 0.955603 + 0.294657i \(0.0952053\pi\)
−0.955603 + 0.294657i \(0.904795\pi\)
\(318\) 20.4924i 1.14916i
\(319\) 10.4384 0.584441
\(320\) 0 0
\(321\) −20.8769 −1.16523
\(322\) 0 0
\(323\) − 3.12311i − 0.173774i
\(324\) 31.9309 1.77394
\(325\) 0 0
\(326\) 18.2462 1.01056
\(327\) − 8.30019i − 0.459001i
\(328\) − 33.6155i − 1.85611i
\(329\) 0 0
\(330\) 0 0
\(331\) 12.0000 0.659580 0.329790 0.944054i \(-0.393022\pi\)
0.329790 + 0.944054i \(0.393022\pi\)
\(332\) − 18.2462i − 1.00139i
\(333\) 3.36932i 0.184637i
\(334\) 17.7538 0.971444
\(335\) 0 0
\(336\) 0 0
\(337\) − 1.50758i − 0.0821230i −0.999157 0.0410615i \(-0.986926\pi\)
0.999157 0.0410615i \(-0.0130740\pi\)
\(338\) − 32.8078i − 1.78451i
\(339\) −21.8617 −1.18737
\(340\) 0 0
\(341\) 0 0
\(342\) 10.2462i 0.554052i
\(343\) 0 0
\(344\) 5.75379 0.310223
\(345\) 0 0
\(346\) −11.3693 −0.611218
\(347\) 7.12311i 0.382388i 0.981552 + 0.191194i \(0.0612360\pi\)
−0.981552 + 0.191194i \(0.938764\pi\)
\(348\) 47.6155i 2.55246i
\(349\) 10.4924 0.561646 0.280823 0.959760i \(-0.409393\pi\)
0.280823 + 0.959760i \(0.409393\pi\)
\(350\) 0 0
\(351\) −2.43845 −0.130155
\(352\) 10.2462i 0.546125i
\(353\) 5.80776i 0.309116i 0.987984 + 0.154558i \(0.0493954\pi\)
−0.987984 + 0.154558i \(0.950605\pi\)
\(354\) −16.0000 −0.850390
\(355\) 0 0
\(356\) 5.12311 0.271524
\(357\) 0 0
\(358\) 51.2311i 2.70765i
\(359\) −8.00000 −0.422224 −0.211112 0.977462i \(-0.567708\pi\)
−0.211112 + 0.977462i \(0.567708\pi\)
\(360\) 0 0
\(361\) 31.7386 1.67045
\(362\) − 45.1231i − 2.37162i
\(363\) − 13.3693i − 0.701707i
\(364\) 0 0
\(365\) 0 0
\(366\) −61.4773 −3.21347
\(367\) 8.68466i 0.453335i 0.973972 + 0.226668i \(0.0727831\pi\)
−0.973972 + 0.226668i \(0.927217\pi\)
\(368\) − 24.0000i − 1.25109i
\(369\) −2.87689 −0.149765
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) − 4.63068i − 0.239768i −0.992788 0.119884i \(-0.961748\pi\)
0.992788 0.119884i \(-0.0382522\pi\)
\(374\) −1.75379 −0.0906863
\(375\) 0 0
\(376\) −56.9848 −2.93877
\(377\) − 2.93087i − 0.150947i
\(378\) 0 0
\(379\) 16.4924 0.847159 0.423579 0.905859i \(-0.360773\pi\)
0.423579 + 0.905859i \(0.360773\pi\)
\(380\) 0 0
\(381\) 9.75379 0.499702
\(382\) 34.7386i 1.77738i
\(383\) 6.24621i 0.319166i 0.987184 + 0.159583i \(0.0510150\pi\)
−0.987184 + 0.159583i \(0.948985\pi\)
\(384\) 14.7386 0.752128
\(385\) 0 0
\(386\) −49.6155 −2.52536
\(387\) − 0.492423i − 0.0250312i
\(388\) 26.4924i 1.34495i
\(389\) 24.9309 1.26405 0.632023 0.774950i \(-0.282225\pi\)
0.632023 + 0.774950i \(0.282225\pi\)
\(390\) 0 0
\(391\) 1.36932 0.0692493
\(392\) 0 0
\(393\) 1.36932i 0.0690729i
\(394\) 2.87689 0.144936
\(395\) 0 0
\(396\) 4.00000 0.201008
\(397\) − 27.5616i − 1.38327i −0.722245 0.691637i \(-0.756890\pi\)
0.722245 0.691637i \(-0.243110\pi\)
\(398\) 4.49242i 0.225185i
\(399\) 0 0
\(400\) 0 0
\(401\) 31.5616 1.57611 0.788054 0.615606i \(-0.211089\pi\)
0.788054 + 0.615606i \(0.211089\pi\)
\(402\) 40.9848i 2.04414i
\(403\) 0 0
\(404\) −74.1080 −3.68701
\(405\) 0 0
\(406\) 0 0
\(407\) − 9.36932i − 0.464420i
\(408\) − 4.49242i − 0.222408i
\(409\) 6.49242 0.321030 0.160515 0.987033i \(-0.448685\pi\)
0.160515 + 0.987033i \(0.448685\pi\)
\(410\) 0 0
\(411\) 26.7386 1.31892
\(412\) − 25.3693i − 1.24986i
\(413\) 0 0
\(414\) −4.49242 −0.220791
\(415\) 0 0
\(416\) 2.87689 0.141051
\(417\) − 23.6155i − 1.15646i
\(418\) − 28.4924i − 1.39361i
\(419\) 26.2462 1.28221 0.641106 0.767453i \(-0.278476\pi\)
0.641106 + 0.767453i \(0.278476\pi\)
\(420\) 0 0
\(421\) −2.68466 −0.130842 −0.0654211 0.997858i \(-0.520839\pi\)
−0.0654211 + 0.997858i \(0.520839\pi\)
\(422\) − 36.0000i − 1.75245i
\(423\) 4.87689i 0.237123i
\(424\) −33.6155 −1.63251
\(425\) 0 0
\(426\) 32.0000 1.55041
\(427\) 0 0
\(428\) − 60.9848i − 2.94781i
\(429\) 1.06913 0.0516181
\(430\) 0 0
\(431\) −19.8078 −0.954106 −0.477053 0.878874i \(-0.658295\pi\)
−0.477053 + 0.878874i \(0.658295\pi\)
\(432\) 42.7386i 2.05626i
\(433\) 8.24621i 0.396288i 0.980173 + 0.198144i \(0.0634913\pi\)
−0.980173 + 0.198144i \(0.936509\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 24.2462 1.16118
\(437\) 22.2462i 1.06418i
\(438\) − 48.9848i − 2.34059i
\(439\) 9.36932 0.447173 0.223587 0.974684i \(-0.428223\pi\)
0.223587 + 0.974684i \(0.428223\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0.492423i 0.0234221i
\(443\) 2.63068i 0.124988i 0.998045 + 0.0624938i \(0.0199054\pi\)
−0.998045 + 0.0624938i \(0.980095\pi\)
\(444\) 42.7386 2.02829
\(445\) 0 0
\(446\) −6.24621 −0.295767
\(447\) − 19.1231i − 0.904492i
\(448\) 0 0
\(449\) 1.80776 0.0853137 0.0426568 0.999090i \(-0.486418\pi\)
0.0426568 + 0.999090i \(0.486418\pi\)
\(450\) 0 0
\(451\) 8.00000 0.376705
\(452\) − 63.8617i − 3.00380i
\(453\) − 10.8229i − 0.508505i
\(454\) −28.9848 −1.36033
\(455\) 0 0
\(456\) 72.9848 3.41783
\(457\) − 17.1231i − 0.800985i −0.916300 0.400493i \(-0.868839\pi\)
0.916300 0.400493i \(-0.131161\pi\)
\(458\) − 27.8617i − 1.30189i
\(459\) −2.43845 −0.113817
\(460\) 0 0
\(461\) 13.1231 0.611204 0.305602 0.952159i \(-0.401142\pi\)
0.305602 + 0.952159i \(0.401142\pi\)
\(462\) 0 0
\(463\) − 12.4924i − 0.580572i −0.956940 0.290286i \(-0.906250\pi\)
0.956940 0.290286i \(-0.0937505\pi\)
\(464\) −51.3693 −2.38476
\(465\) 0 0
\(466\) −13.1231 −0.607916
\(467\) − 22.4384i − 1.03833i −0.854675 0.519164i \(-0.826243\pi\)
0.854675 0.519164i \(-0.173757\pi\)
\(468\) − 1.12311i − 0.0519156i
\(469\) 0 0
\(470\) 0 0
\(471\) 31.6155 1.45677
\(472\) − 26.2462i − 1.20808i
\(473\) 1.36932i 0.0629613i
\(474\) 9.75379 0.448006
\(475\) 0 0
\(476\) 0 0
\(477\) 2.87689i 0.131724i
\(478\) 50.7386i 2.32073i
\(479\) 4.87689 0.222831 0.111415 0.993774i \(-0.464462\pi\)
0.111415 + 0.993774i \(0.464462\pi\)
\(480\) 0 0
\(481\) −2.63068 −0.119949
\(482\) − 10.8769i − 0.495429i
\(483\) 0 0
\(484\) 39.0540 1.77518
\(485\) 0 0
\(486\) 14.7386 0.668558
\(487\) − 3.12311i − 0.141521i −0.997493 0.0707607i \(-0.977457\pi\)
0.997493 0.0707607i \(-0.0225427\pi\)
\(488\) − 100.847i − 4.56511i
\(489\) −11.1231 −0.503004
\(490\) 0 0
\(491\) −41.1771 −1.85830 −0.929148 0.369708i \(-0.879458\pi\)
−0.929148 + 0.369708i \(0.879458\pi\)
\(492\) 36.4924i 1.64521i
\(493\) − 2.93087i − 0.132000i
\(494\) −8.00000 −0.359937
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 16.0000i 0.716977i
\(499\) −41.1771 −1.84334 −0.921670 0.387976i \(-0.873174\pi\)
−0.921670 + 0.387976i \(0.873174\pi\)
\(500\) 0 0
\(501\) −10.8229 −0.483532
\(502\) − 22.7386i − 1.01487i
\(503\) 38.9309i 1.73584i 0.496703 + 0.867921i \(0.334544\pi\)
−0.496703 + 0.867921i \(0.665456\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 12.4924 0.555356
\(507\) 20.0000i 0.888231i
\(508\) 28.4924i 1.26415i
\(509\) −11.7538 −0.520978 −0.260489 0.965477i \(-0.583884\pi\)
−0.260489 + 0.965477i \(0.583884\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 50.4233i 2.22842i
\(513\) − 39.6155i − 1.74907i
\(514\) 26.8769 1.18549
\(515\) 0 0
\(516\) −6.24621 −0.274974
\(517\) − 13.5616i − 0.596436i
\(518\) 0 0
\(519\) 6.93087 0.304231
\(520\) 0 0
\(521\) −10.0000 −0.438108 −0.219054 0.975713i \(-0.570297\pi\)
−0.219054 + 0.975713i \(0.570297\pi\)
\(522\) 9.61553i 0.420860i
\(523\) 40.4924i 1.77061i 0.465011 + 0.885305i \(0.346050\pi\)
−0.465011 + 0.885305i \(0.653950\pi\)
\(524\) −4.00000 −0.174741
\(525\) 0 0
\(526\) 32.9848 1.43821
\(527\) 0 0
\(528\) − 18.7386i − 0.815494i
\(529\) 13.2462 0.575922
\(530\) 0 0
\(531\) −2.24621 −0.0974773
\(532\) 0 0
\(533\) − 2.24621i − 0.0972942i
\(534\) −4.49242 −0.194406
\(535\) 0 0
\(536\) −67.2311 −2.90394
\(537\) − 31.2311i − 1.34772i
\(538\) 53.1231i 2.29030i
\(539\) 0 0
\(540\) 0 0
\(541\) −37.8078 −1.62548 −0.812741 0.582625i \(-0.802026\pi\)
−0.812741 + 0.582625i \(0.802026\pi\)
\(542\) − 40.9848i − 1.76045i
\(543\) 27.5076i 1.18046i
\(544\) 2.87689 0.123346
\(545\) 0 0
\(546\) 0 0
\(547\) − 2.24621i − 0.0960411i −0.998846 0.0480205i \(-0.984709\pi\)
0.998846 0.0480205i \(-0.0152913\pi\)
\(548\) 78.1080i 3.33661i
\(549\) −8.63068 −0.368349
\(550\) 0 0
\(551\) 47.6155 2.02849
\(552\) 32.0000i 1.36201i
\(553\) 0 0
\(554\) −0.630683 −0.0267952
\(555\) 0 0
\(556\) 68.9848 2.92561
\(557\) − 13.1231i − 0.556044i −0.960575 0.278022i \(-0.910321\pi\)
0.960575 0.278022i \(-0.0896788\pi\)
\(558\) 0 0
\(559\) 0.384472 0.0162614
\(560\) 0 0
\(561\) 1.06913 0.0451387
\(562\) − 31.8617i − 1.34401i
\(563\) − 28.0000i − 1.18006i −0.807382 0.590030i \(-0.799116\pi\)
0.807382 0.590030i \(-0.200884\pi\)
\(564\) 61.8617 2.60485
\(565\) 0 0
\(566\) −28.9848 −1.21832
\(567\) 0 0
\(568\) 52.4924i 2.20253i
\(569\) 30.9848 1.29895 0.649476 0.760382i \(-0.274988\pi\)
0.649476 + 0.760382i \(0.274988\pi\)
\(570\) 0 0
\(571\) 40.4924 1.69456 0.847278 0.531150i \(-0.178240\pi\)
0.847278 + 0.531150i \(0.178240\pi\)
\(572\) 3.12311i 0.130584i
\(573\) − 21.1771i − 0.884685i
\(574\) 0 0
\(575\) 0 0
\(576\) −0.807764 −0.0336568
\(577\) 24.0540i 1.00138i 0.865627 + 0.500690i \(0.166920\pi\)
−0.865627 + 0.500690i \(0.833080\pi\)
\(578\) − 43.0540i − 1.79081i
\(579\) 30.2462 1.25699
\(580\) 0 0
\(581\) 0 0
\(582\) − 23.2311i − 0.962958i
\(583\) − 8.00000i − 0.331326i
\(584\) 80.3542 3.32508
\(585\) 0 0
\(586\) −6.87689 −0.284082
\(587\) 26.2462i 1.08330i 0.840605 + 0.541649i \(0.182200\pi\)
−0.840605 + 0.541649i \(0.817800\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) −1.75379 −0.0721412
\(592\) 46.1080i 1.89503i
\(593\) − 27.5616i − 1.13182i −0.824468 0.565909i \(-0.808525\pi\)
0.824468 0.565909i \(-0.191475\pi\)
\(594\) −22.2462 −0.912773
\(595\) 0 0
\(596\) 55.8617 2.28819
\(597\) − 2.73863i − 0.112085i
\(598\) − 3.50758i − 0.143436i
\(599\) 11.8078 0.482452 0.241226 0.970469i \(-0.422450\pi\)
0.241226 + 0.970469i \(0.422450\pi\)
\(600\) 0 0
\(601\) −6.49242 −0.264831 −0.132416 0.991194i \(-0.542273\pi\)
−0.132416 + 0.991194i \(0.542273\pi\)
\(602\) 0 0
\(603\) 5.75379i 0.234312i
\(604\) 31.6155 1.28642
\(605\) 0 0
\(606\) 64.9848 2.63983
\(607\) 42.0540i 1.70692i 0.521160 + 0.853459i \(0.325500\pi\)
−0.521160 + 0.853459i \(0.674500\pi\)
\(608\) 46.7386i 1.89550i
\(609\) 0 0
\(610\) 0 0
\(611\) −3.80776 −0.154046
\(612\) − 1.12311i − 0.0453989i
\(613\) − 40.7386i − 1.64542i −0.568463 0.822709i \(-0.692462\pi\)
0.568463 0.822709i \(-0.307538\pi\)
\(614\) 49.4773 1.99674
\(615\) 0 0
\(616\) 0 0
\(617\) 32.2462i 1.29818i 0.760710 + 0.649092i \(0.224851\pi\)
−0.760710 + 0.649092i \(0.775149\pi\)
\(618\) 22.2462i 0.894874i
\(619\) 32.1080 1.29053 0.645264 0.763960i \(-0.276747\pi\)
0.645264 + 0.763960i \(0.276747\pi\)
\(620\) 0 0
\(621\) 17.3693 0.697007
\(622\) 80.9848i 3.24720i
\(623\) 0 0
\(624\) −5.26137 −0.210623
\(625\) 0 0
\(626\) −57.1231 −2.28310
\(627\) 17.3693i 0.693664i
\(628\) 92.3542i 3.68533i
\(629\) −2.63068 −0.104892
\(630\) 0 0
\(631\) −11.8078 −0.470060 −0.235030 0.971988i \(-0.575519\pi\)
−0.235030 + 0.971988i \(0.575519\pi\)
\(632\) 16.0000i 0.636446i
\(633\) 21.9460i 0.872276i
\(634\) 26.8769 1.06742
\(635\) 0 0
\(636\) 36.4924 1.44702
\(637\) 0 0
\(638\) − 26.7386i − 1.05859i
\(639\) 4.49242 0.177717
\(640\) 0 0
\(641\) 2.00000 0.0789953 0.0394976 0.999220i \(-0.487424\pi\)
0.0394976 + 0.999220i \(0.487424\pi\)
\(642\) 53.4773i 2.11058i
\(643\) 1.56155i 0.0615816i 0.999526 + 0.0307908i \(0.00980257\pi\)
−0.999526 + 0.0307908i \(0.990197\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) −8.00000 −0.314756
\(647\) − 36.4924i − 1.43467i −0.696731 0.717333i \(-0.745363\pi\)
0.696731 0.717333i \(-0.254637\pi\)
\(648\) − 45.9309i − 1.80433i
\(649\) 6.24621 0.245185
\(650\) 0 0
\(651\) 0 0
\(652\) − 32.4924i − 1.27250i
\(653\) 33.2311i 1.30043i 0.759750 + 0.650216i \(0.225322\pi\)
−0.759750 + 0.650216i \(0.774678\pi\)
\(654\) −21.2614 −0.831385
\(655\) 0 0
\(656\) −39.3693 −1.53711
\(657\) − 6.87689i − 0.268293i
\(658\) 0 0
\(659\) −9.17708 −0.357488 −0.178744 0.983896i \(-0.557203\pi\)
−0.178744 + 0.983896i \(0.557203\pi\)
\(660\) 0 0
\(661\) 5.12311 0.199266 0.0996329 0.995024i \(-0.468233\pi\)
0.0996329 + 0.995024i \(0.468233\pi\)
\(662\) − 30.7386i − 1.19469i
\(663\) − 0.300187i − 0.0116583i
\(664\) −26.2462 −1.01855
\(665\) 0 0
\(666\) 8.63068 0.334432
\(667\) 20.8769i 0.808357i
\(668\) − 31.6155i − 1.22324i
\(669\) 3.80776 0.147217
\(670\) 0 0
\(671\) 24.0000 0.926510
\(672\) 0 0
\(673\) − 31.8617i − 1.22818i −0.789236 0.614090i \(-0.789523\pi\)
0.789236 0.614090i \(-0.210477\pi\)
\(674\) −3.86174 −0.148749
\(675\) 0 0
\(676\) −58.4233 −2.24705
\(677\) − 4.93087i − 0.189509i −0.995501 0.0947544i \(-0.969793\pi\)
0.995501 0.0947544i \(-0.0302066\pi\)
\(678\) 56.0000i 2.15067i
\(679\) 0 0
\(680\) 0 0
\(681\) 17.6695 0.677097
\(682\) 0 0
\(683\) 6.73863i 0.257847i 0.991655 + 0.128923i \(0.0411521\pi\)
−0.991655 + 0.128923i \(0.958848\pi\)
\(684\) 18.2462 0.697661
\(685\) 0 0
\(686\) 0 0
\(687\) 16.9848i 0.648012i
\(688\) − 6.73863i − 0.256908i
\(689\) −2.24621 −0.0855738
\(690\) 0 0
\(691\) 24.4924 0.931736 0.465868 0.884854i \(-0.345742\pi\)
0.465868 + 0.884854i \(0.345742\pi\)
\(692\) 20.2462i 0.769645i
\(693\) 0 0
\(694\) 18.2462 0.692617
\(695\) 0 0
\(696\) 68.4924 2.59620
\(697\) − 2.24621i − 0.0850813i
\(698\) − 26.8769i − 1.01731i
\(699\) 8.00000 0.302588
\(700\) 0 0
\(701\) 28.9309 1.09270 0.546352 0.837556i \(-0.316016\pi\)
0.546352 + 0.837556i \(0.316016\pi\)
\(702\) 6.24621i 0.235748i
\(703\) − 42.7386i − 1.61192i
\(704\) 2.24621 0.0846573
\(705\) 0 0
\(706\) 14.8769 0.559899
\(707\) 0 0
\(708\) 28.4924i 1.07081i
\(709\) −27.1771 −1.02066 −0.510328 0.859980i \(-0.670476\pi\)
−0.510328 + 0.859980i \(0.670476\pi\)
\(710\) 0 0
\(711\) 1.36932 0.0513534
\(712\) − 7.36932i − 0.276177i
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 91.2311 3.40946
\(717\) − 30.9309i − 1.15513i
\(718\) 20.4924i 0.764770i
\(719\) 8.38447 0.312688 0.156344 0.987703i \(-0.450029\pi\)
0.156344 + 0.987703i \(0.450029\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) − 81.3002i − 3.02568i
\(723\) 6.63068i 0.246598i
\(724\) −80.3542 −2.98634
\(725\) 0 0
\(726\) −34.2462 −1.27100
\(727\) − 52.4924i − 1.94684i −0.229035 0.973418i \(-0.573557\pi\)
0.229035 0.973418i \(-0.426443\pi\)
\(728\) 0 0
\(729\) −29.9848 −1.11055
\(730\) 0 0
\(731\) 0.384472 0.0142202
\(732\) 109.477i 4.04640i
\(733\) 6.68466i 0.246903i 0.992351 + 0.123452i \(0.0393964\pi\)
−0.992351 + 0.123452i \(0.960604\pi\)
\(734\) 22.2462 0.821123
\(735\) 0 0
\(736\) −20.4924 −0.755361
\(737\) − 16.0000i − 0.589368i
\(738\) 7.36932i 0.271268i
\(739\) −34.9309 −1.28495 −0.642476 0.766305i \(-0.722093\pi\)
−0.642476 + 0.766305i \(0.722093\pi\)
\(740\) 0 0
\(741\) 4.87689 0.179157
\(742\) 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
\(746\) −11.8617 −0.434289
\(747\) 2.24621i 0.0821846i
\(748\) 3.12311i 0.114192i
\(749\) 0 0
\(750\) 0 0
\(751\) 17.0691 0.622861 0.311431 0.950269i \(-0.399192\pi\)
0.311431 + 0.950269i \(0.399192\pi\)
\(752\) 66.7386i 2.43371i
\(753\) 13.8617i 0.505150i
\(754\) −7.50758 −0.273410
\(755\) 0 0
\(756\) 0 0
\(757\) 39.3693i 1.43090i 0.698663 + 0.715451i \(0.253779\pi\)
−0.698663 + 0.715451i \(0.746221\pi\)
\(758\) − 42.2462i − 1.53445i
\(759\) −7.61553 −0.276426
\(760\) 0 0
\(761\) −48.2462 −1.74892 −0.874462 0.485094i \(-0.838785\pi\)
−0.874462 + 0.485094i \(0.838785\pi\)
\(762\) − 24.9848i − 0.905105i
\(763\) 0 0
\(764\) 61.8617 2.23808
\(765\) 0 0
\(766\) 16.0000 0.578103
\(767\) − 1.75379i − 0.0633256i
\(768\) − 42.2462i − 1.52443i
\(769\) −42.4924 −1.53232 −0.766158 0.642652i \(-0.777834\pi\)
−0.766158 + 0.642652i \(0.777834\pi\)
\(770\) 0 0
\(771\) −16.3845 −0.590072
\(772\) 88.3542i 3.17994i
\(773\) 36.9309i 1.32831i 0.747594 + 0.664156i \(0.231209\pi\)
−0.747594 + 0.664156i \(0.768791\pi\)
\(774\) −1.26137 −0.0453389
\(775\) 0 0
\(776\) 38.1080 1.36800
\(777\) 0 0
\(778\) − 63.8617i − 2.28955i
\(779\) 36.4924 1.30748
\(780\) 0 0
\(781\) −12.4924 −0.447014
\(782\) − 3.50758i − 0.125431i
\(783\) − 37.1771i − 1.32860i
\(784\) 0 0
\(785\) 0 0
\(786\) 3.50758 0.125111
\(787\) 49.1771i 1.75297i 0.481426 + 0.876487i \(0.340119\pi\)
−0.481426 + 0.876487i \(0.659881\pi\)
\(788\) − 5.12311i − 0.182503i
\(789\) −20.1080 −0.715862
\(790\) 0 0
\(791\) 0 0
\(792\) − 5.75379i − 0.204452i
\(793\) − 6.73863i − 0.239296i
\(794\) −70.6004 −2.50551
\(795\) 0 0
\(796\) 8.00000 0.283552
\(797\) − 24.0540i − 0.852036i −0.904715 0.426018i \(-0.859916\pi\)
0.904715 0.426018i \(-0.140084\pi\)
\(798\) 0 0
\(799\) −3.80776 −0.134709
\(800\) 0 0
\(801\) −0.630683 −0.0222841
\(802\) − 80.8466i − 2.85479i
\(803\) 19.1231i 0.674840i
\(804\) 72.9848 2.57398
\(805\) 0 0
\(806\) 0 0
\(807\) − 32.3845i − 1.13999i
\(808\) 106.600i 3.75019i
\(809\) −16.5464 −0.581740 −0.290870 0.956763i \(-0.593945\pi\)
−0.290870 + 0.956763i \(0.593945\pi\)
\(810\) 0 0
\(811\) −19.6155 −0.688794 −0.344397 0.938824i \(-0.611917\pi\)
−0.344397 + 0.938824i \(0.611917\pi\)
\(812\) 0 0
\(813\) 24.9848i 0.876257i
\(814\) −24.0000 −0.841200
\(815\) 0 0
\(816\) −5.26137 −0.184185
\(817\) 6.24621i 0.218527i
\(818\) − 16.6307i − 0.581478i
\(819\) 0 0
\(820\) 0 0
\(821\) −21.4233 −0.747678 −0.373839 0.927494i \(-0.621959\pi\)
−0.373839 + 0.927494i \(0.621959\pi\)
\(822\) − 68.4924i − 2.38895i
\(823\) 36.4924i 1.27205i 0.771670 + 0.636023i \(0.219422\pi\)
−0.771670 + 0.636023i \(0.780578\pi\)
\(824\) −36.4924 −1.27127
\(825\) 0 0
\(826\) 0 0
\(827\) − 5.36932i − 0.186709i −0.995633 0.0933547i \(-0.970241\pi\)
0.995633 0.0933547i \(-0.0297591\pi\)
\(828\) 8.00000i 0.278019i
\(829\) 34.8769 1.21132 0.605662 0.795722i \(-0.292908\pi\)
0.605662 + 0.795722i \(0.292908\pi\)
\(830\) 0 0
\(831\) 0.384472 0.0133372
\(832\) − 0.630683i − 0.0218650i
\(833\) 0 0
\(834\) −60.4924 −2.09468
\(835\) 0 0
\(836\) −50.7386 −1.75483
\(837\) 0 0
\(838\) − 67.2311i − 2.32246i
\(839\) −28.8769 −0.996941 −0.498471 0.866907i \(-0.666105\pi\)
−0.498471 + 0.866907i \(0.666105\pi\)
\(840\) 0 0
\(841\) 15.6847 0.540850
\(842\) 6.87689i 0.236993i
\(843\) 19.4233i 0.668974i
\(844\) −64.1080 −2.20669
\(845\) 0 0
\(846\) 12.4924 0.429498
\(847\) 0 0
\(848\) 39.3693i 1.35195i
\(849\) 17.6695 0.606416
\(850\) 0 0
\(851\) 18.7386 0.642352
\(852\) − 56.9848i − 1.95227i
\(853\) − 7.26137i − 0.248624i −0.992243 0.124312i \(-0.960328\pi\)
0.992243 0.124312i \(-0.0396724\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) −87.7235 −2.99833
\(857\) 15.7538i 0.538139i 0.963121 + 0.269070i \(0.0867162\pi\)
−0.963121 + 0.269070i \(0.913284\pi\)
\(858\) − 2.73863i − 0.0934954i
\(859\) −16.4924 −0.562714 −0.281357 0.959603i \(-0.590785\pi\)
−0.281357 + 0.959603i \(0.590785\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 50.7386i 1.72816i
\(863\) 25.7538i 0.876669i 0.898812 + 0.438335i \(0.144431\pi\)
−0.898812 + 0.438335i \(0.855569\pi\)
\(864\) 36.4924 1.24150
\(865\) 0 0
\(866\) 21.1231 0.717792
\(867\) 26.2462i 0.891368i
\(868\) 0 0
\(869\) −3.80776 −0.129170
\(870\) 0 0
\(871\) −4.49242 −0.152220
\(872\) − 34.8769i − 1.18108i
\(873\) − 3.26137i − 0.110381i
\(874\) 56.9848 1.92754
\(875\) 0 0
\(876\) −87.2311 −2.94726
\(877\) − 40.2462i − 1.35902i −0.733667 0.679509i \(-0.762193\pi\)
0.733667 0.679509i \(-0.237807\pi\)
\(878\) − 24.0000i − 0.809961i
\(879\) 4.19224 0.141401
\(880\) 0 0
\(881\) 11.8617 0.399632 0.199816 0.979833i \(-0.435966\pi\)
0.199816 + 0.979833i \(0.435966\pi\)
\(882\) 0 0
\(883\) 8.49242i 0.285793i 0.989738 + 0.142896i \(0.0456416\pi\)
−0.989738 + 0.142896i \(0.954358\pi\)
\(884\) 0.876894 0.0294931
\(885\) 0 0
\(886\) 6.73863 0.226389
\(887\) − 20.4924i − 0.688068i −0.938957 0.344034i \(-0.888206\pi\)
0.938957 0.344034i \(-0.111794\pi\)
\(888\) − 61.4773i − 2.06304i
\(889\) 0 0
\(890\) 0 0
\(891\) 10.9309 0.366198
\(892\) 11.1231i 0.372429i
\(893\) − 61.8617i − 2.07012i
\(894\) −48.9848 −1.63830
\(895\) 0 0
\(896\) 0 0
\(897\) 2.13826i 0.0713944i
\(898\) − 4.63068i − 0.154528i
\(899\) 0 0
\(900\) 0 0
\(901\) −2.24621 −0.0748321
\(902\) − 20.4924i − 0.682323i
\(903\) 0 0
\(904\) −91.8617 −3.05528
\(905\) 0 0
\(906\) −27.7235 −0.921051
\(907\) − 24.1080i − 0.800491i −0.916408 0.400246i \(-0.868925\pi\)
0.916408 0.400246i \(-0.131075\pi\)
\(908\) 51.6155i 1.71292i
\(909\) 9.12311 0.302594
\(910\) 0 0
\(911\) −28.4924 −0.943996 −0.471998 0.881600i \(-0.656467\pi\)
−0.471998 + 0.881600i \(0.656467\pi\)
\(912\) − 85.4773i − 2.83044i
\(913\) − 6.24621i − 0.206719i
\(914\) −43.8617 −1.45082
\(915\) 0 0
\(916\) −49.6155 −1.63934
\(917\) 0 0
\(918\) 6.24621i 0.206156i
\(919\) −40.3002 −1.32938 −0.664690 0.747119i \(-0.731436\pi\)
−0.664690 + 0.747119i \(0.731436\pi\)
\(920\) 0 0
\(921\) −30.1619 −0.993869
\(922\) − 33.6155i − 1.10707i
\(923\) 3.50758i 0.115453i
\(924\) 0 0
\(925\) 0 0
\(926\) −32.0000 −1.05159
\(927\) 3.12311i 0.102576i
\(928\) 43.8617i 1.43983i
\(929\) 22.1080 0.725338 0.362669 0.931918i \(-0.381866\pi\)
0.362669 + 0.931918i \(0.381866\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 23.3693i 0.765487i
\(933\) − 49.3693i − 1.61628i
\(934\) −57.4773 −1.88071
\(935\) 0 0
\(936\) −1.61553 −0.0528052
\(937\) 55.6695i 1.81864i 0.416094 + 0.909322i \(0.363399\pi\)
−0.416094 + 0.909322i \(0.636601\pi\)
\(938\) 0 0
\(939\) 34.8229 1.13640
\(940\) 0 0
\(941\) −43.8617 −1.42985 −0.714926 0.699200i \(-0.753540\pi\)
−0.714926 + 0.699200i \(0.753540\pi\)
\(942\) − 80.9848i − 2.63863i
\(943\) 16.0000i 0.521032i
\(944\) −30.7386 −1.00046
\(945\) 0 0
\(946\) 3.50758 0.114041
\(947\) 4.00000i 0.129983i 0.997886 + 0.0649913i \(0.0207020\pi\)
−0.997886 + 0.0649913i \(0.979298\pi\)
\(948\) − 17.3693i − 0.564129i
\(949\) 5.36932 0.174295
\(950\) 0 0
\(951\) −16.3845 −0.531303
\(952\) 0 0
\(953\) 33.1231i 1.07296i 0.843912 + 0.536481i \(0.180247\pi\)
−0.843912 + 0.536481i \(0.819753\pi\)
\(954\) 7.36932 0.238590
\(955\) 0 0
\(956\) 90.3542 2.92226
\(957\) 16.3002i 0.526910i
\(958\) − 12.4924i − 0.403612i
\(959\) 0 0
\(960\) 0 0
\(961\) −31.0000 −1.00000
\(962\) 6.73863i 0.217262i
\(963\) 7.50758i 0.241928i
\(964\) −19.3693 −0.623844
\(965\) 0 0
\(966\) 0 0
\(967\) − 35.1231i − 1.12948i −0.825268 0.564741i \(-0.808976\pi\)
0.825268 0.564741i \(-0.191024\pi\)
\(968\) − 56.1771i − 1.80560i
\(969\) 4.87689 0.156668
\(970\) 0 0
\(971\) −49.4773 −1.58780 −0.793901 0.608048i \(-0.791953\pi\)
−0.793901 + 0.608048i \(0.791953\pi\)
\(972\) − 26.2462i − 0.841848i
\(973\) 0 0
\(974\) −8.00000 −0.256337
\(975\) 0 0
\(976\) −118.108 −3.78054
\(977\) 33.2311i 1.06316i 0.847009 + 0.531578i \(0.178401\pi\)
−0.847009 + 0.531578i \(0.821599\pi\)
\(978\) 28.4924i 0.911087i
\(979\) 1.75379 0.0560513
\(980\) 0 0
\(981\) −2.98485 −0.0952988
\(982\) 105.477i 3.36591i
\(983\) 51.4233i 1.64015i 0.572257 + 0.820074i \(0.306068\pi\)
−0.572257 + 0.820074i \(0.693932\pi\)
\(984\) 52.4924 1.67340
\(985\) 0 0
\(986\) −7.50758 −0.239090
\(987\) 0 0
\(988\) 14.2462i 0.453232i
\(989\) −2.73863 −0.0870835
\(990\) 0 0
\(991\) 12.4924 0.396835 0.198417 0.980118i \(-0.436420\pi\)
0.198417 + 0.980118i \(0.436420\pi\)
\(992\) 0 0
\(993\) 18.7386i 0.594653i
\(994\) 0 0
\(995\) 0 0
\(996\) 28.4924 0.902817
\(997\) 2.68466i 0.0850240i 0.999096 + 0.0425120i \(0.0135361\pi\)
−0.999096 + 0.0425120i \(0.986464\pi\)
\(998\) 105.477i 3.33882i
\(999\) −33.3693 −1.05576
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.99.1 4
5.2 odd 4 1225.2.a.s.1.2 2
5.3 odd 4 245.2.a.d.1.1 2
5.4 even 2 inner 1225.2.b.f.99.4 4
7.6 odd 2 175.2.b.b.99.1 4
15.8 even 4 2205.2.a.x.1.2 2
20.3 even 4 3920.2.a.bs.1.2 2
21.20 even 2 1575.2.d.e.1324.4 4
28.27 even 2 2800.2.g.t.449.3 4
35.3 even 12 245.2.e.i.226.2 4
35.13 even 4 35.2.a.b.1.1 2
35.18 odd 12 245.2.e.h.226.2 4
35.23 odd 12 245.2.e.h.116.2 4
35.27 even 4 175.2.a.f.1.2 2
35.33 even 12 245.2.e.i.116.2 4
35.34 odd 2 175.2.b.b.99.4 4
105.62 odd 4 1575.2.a.p.1.1 2
105.83 odd 4 315.2.a.e.1.2 2
105.104 even 2 1575.2.d.e.1324.1 4
140.27 odd 4 2800.2.a.bi.1.2 2
140.83 odd 4 560.2.a.i.1.1 2
140.139 even 2 2800.2.g.t.449.2 4
280.13 even 4 2240.2.a.bh.1.1 2
280.83 odd 4 2240.2.a.bd.1.2 2
385.153 odd 4 4235.2.a.m.1.2 2
420.83 even 4 5040.2.a.bt.1.1 2
455.363 even 4 5915.2.a.l.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
35.2.a.b.1.1 2 35.13 even 4
175.2.a.f.1.2 2 35.27 even 4
175.2.b.b.99.1 4 7.6 odd 2
175.2.b.b.99.4 4 35.34 odd 2
245.2.a.d.1.1 2 5.3 odd 4
245.2.e.h.116.2 4 35.23 odd 12
245.2.e.h.226.2 4 35.18 odd 12
245.2.e.i.116.2 4 35.33 even 12
245.2.e.i.226.2 4 35.3 even 12
315.2.a.e.1.2 2 105.83 odd 4
560.2.a.i.1.1 2 140.83 odd 4
1225.2.a.s.1.2 2 5.2 odd 4
1225.2.b.f.99.1 4 1.1 even 1 trivial
1225.2.b.f.99.4 4 5.4 even 2 inner
1575.2.a.p.1.1 2 105.62 odd 4
1575.2.d.e.1324.1 4 105.104 even 2
1575.2.d.e.1324.4 4 21.20 even 2
2205.2.a.x.1.2 2 15.8 even 4
2240.2.a.bd.1.2 2 280.83 odd 4
2240.2.a.bh.1.1 2 280.13 even 4
2800.2.a.bi.1.2 2 140.27 odd 4
2800.2.g.t.449.2 4 140.139 even 2
2800.2.g.t.449.3 4 28.27 even 2
3920.2.a.bs.1.2 2 20.3 even 4
4235.2.a.m.1.2 2 385.153 odd 4
5040.2.a.bt.1.1 2 420.83 even 4
5915.2.a.l.1.2 2 455.363 even 4