Properties

Label 2925.2.c.p.2224.4
Level $2925$
Weight $2$
Character 2925.2224
Analytic conductor $23.356$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 2925 = 3^{2} \cdot 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2925.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(23.3562425912\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{17})\)
Defining polynomial: \( x^{4} + 9x^{2} + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 585)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 2224.4
Root \(2.56155i\) of defining polynomial
Character \(\chi\) \(=\) 2925.2224
Dual form 2925.2.c.p.2224.1

$q$-expansion

\(f(q)\) \(=\) \(q+2.56155i q^{2} -4.56155 q^{4} -0.438447i q^{7} -6.56155i q^{8} +O(q^{10})\) \(q+2.56155i q^{2} -4.56155 q^{4} -0.438447i q^{7} -6.56155i q^{8} -1.56155 q^{11} +1.00000i q^{13} +1.12311 q^{14} +7.68466 q^{16} +1.56155i q^{17} +5.12311 q^{19} -4.00000i q^{22} -2.43845i q^{23} -2.56155 q^{26} +2.00000i q^{28} +7.12311 q^{29} +6.00000 q^{31} +6.56155i q^{32} -4.00000 q^{34} -10.6847i q^{37} +13.1231i q^{38} +3.56155 q^{41} -3.12311i q^{43} +7.12311 q^{44} +6.24621 q^{46} +11.1231i q^{47} +6.80776 q^{49} -4.56155i q^{52} +4.68466i q^{53} -2.87689 q^{56} +18.2462i q^{58} -12.0000 q^{59} -6.68466 q^{61} +15.3693i q^{62} -1.43845 q^{64} -11.3693i q^{67} -7.12311i q^{68} +10.4384 q^{71} +6.00000i q^{73} +27.3693 q^{74} -23.3693 q^{76} +0.684658i q^{77} -4.68466 q^{79} +9.12311i q^{82} +16.4924i q^{83} +8.00000 q^{86} +10.2462i q^{88} -10.6847 q^{89} +0.438447 q^{91} +11.1231i q^{92} -28.4924 q^{94} +16.9309i q^{97} +17.4384i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 10 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 10 q^{4} + 2 q^{11} - 12 q^{14} + 6 q^{16} + 4 q^{19} - 2 q^{26} + 12 q^{29} + 24 q^{31} - 16 q^{34} + 6 q^{41} + 12 q^{44} - 8 q^{46} - 14 q^{49} - 28 q^{56} - 48 q^{59} - 2 q^{61} - 14 q^{64} + 50 q^{71} + 60 q^{74} - 44 q^{76} + 6 q^{79} + 32 q^{86} - 18 q^{89} + 10 q^{91} - 48 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2925\mathbb{Z}\right)^\times\).

\(n\) \(326\) \(352\) \(2251\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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 \(0.360613\pi\)
−0.424035 + 0.905646i \(0.639387\pi\)
\(3\) 0 0
\(4\) −4.56155 −2.28078
\(5\) 0 0
\(6\) 0 0
\(7\) − 0.438447i − 0.165717i −0.996561 0.0828587i \(-0.973595\pi\)
0.996561 0.0828587i \(-0.0264050\pi\)
\(8\) − 6.56155i − 2.31986i
\(9\) 0 0
\(10\) 0 0
\(11\) −1.56155 −0.470826 −0.235413 0.971895i \(-0.575644\pi\)
−0.235413 + 0.971895i \(0.575644\pi\)
\(12\) 0 0
\(13\) 1.00000i 0.277350i
\(14\) 1.12311 0.300163
\(15\) 0 0
\(16\) 7.68466 1.92116
\(17\) 1.56155i 0.378732i 0.981907 + 0.189366i \(0.0606433\pi\)
−0.981907 + 0.189366i \(0.939357\pi\)
\(18\) 0 0
\(19\) 5.12311 1.17532 0.587661 0.809108i \(-0.300049\pi\)
0.587661 + 0.809108i \(0.300049\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) − 4.00000i − 0.852803i
\(23\) − 2.43845i − 0.508451i −0.967145 0.254226i \(-0.918179\pi\)
0.967145 0.254226i \(-0.0818206\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) −2.56155 −0.502362
\(27\) 0 0
\(28\) 2.00000i 0.377964i
\(29\) 7.12311 1.32273 0.661364 0.750065i \(-0.269978\pi\)
0.661364 + 0.750065i \(0.269978\pi\)
\(30\) 0 0
\(31\) 6.00000 1.07763 0.538816 0.842424i \(-0.318872\pi\)
0.538816 + 0.842424i \(0.318872\pi\)
\(32\) 6.56155i 1.15993i
\(33\) 0 0
\(34\) −4.00000 −0.685994
\(35\) 0 0
\(36\) 0 0
\(37\) − 10.6847i − 1.75655i −0.478159 0.878274i \(-0.658696\pi\)
0.478159 0.878274i \(-0.341304\pi\)
\(38\) 13.1231i 2.12885i
\(39\) 0 0
\(40\) 0 0
\(41\) 3.56155 0.556221 0.278111 0.960549i \(-0.410292\pi\)
0.278111 + 0.960549i \(0.410292\pi\)
\(42\) 0 0
\(43\) − 3.12311i − 0.476269i −0.971232 0.238135i \(-0.923464\pi\)
0.971232 0.238135i \(-0.0765359\pi\)
\(44\) 7.12311 1.07385
\(45\) 0 0
\(46\) 6.24621 0.920954
\(47\) 11.1231i 1.62247i 0.584719 + 0.811236i \(0.301205\pi\)
−0.584719 + 0.811236i \(0.698795\pi\)
\(48\) 0 0
\(49\) 6.80776 0.972538
\(50\) 0 0
\(51\) 0 0
\(52\) − 4.56155i − 0.632574i
\(53\) 4.68466i 0.643487i 0.946827 + 0.321744i \(0.104269\pi\)
−0.946827 + 0.321744i \(0.895731\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −2.87689 −0.384441
\(57\) 0 0
\(58\) 18.2462i 2.39584i
\(59\) −12.0000 −1.56227 −0.781133 0.624364i \(-0.785358\pi\)
−0.781133 + 0.624364i \(0.785358\pi\)
\(60\) 0 0
\(61\) −6.68466 −0.855883 −0.427941 0.903806i \(-0.640761\pi\)
−0.427941 + 0.903806i \(0.640761\pi\)
\(62\) 15.3693i 1.95191i
\(63\) 0 0
\(64\) −1.43845 −0.179806
\(65\) 0 0
\(66\) 0 0
\(67\) − 11.3693i − 1.38898i −0.719501 0.694492i \(-0.755629\pi\)
0.719501 0.694492i \(-0.244371\pi\)
\(68\) − 7.12311i − 0.863803i
\(69\) 0 0
\(70\) 0 0
\(71\) 10.4384 1.23882 0.619408 0.785069i \(-0.287373\pi\)
0.619408 + 0.785069i \(0.287373\pi\)
\(72\) 0 0
\(73\) 6.00000i 0.702247i 0.936329 + 0.351123i \(0.114200\pi\)
−0.936329 + 0.351123i \(0.885800\pi\)
\(74\) 27.3693 3.18162
\(75\) 0 0
\(76\) −23.3693 −2.68064
\(77\) 0.684658i 0.0780241i
\(78\) 0 0
\(79\) −4.68466 −0.527065 −0.263533 0.964650i \(-0.584888\pi\)
−0.263533 + 0.964650i \(0.584888\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 9.12311i 1.00748i
\(83\) 16.4924i 1.81028i 0.425115 + 0.905139i \(0.360234\pi\)
−0.425115 + 0.905139i \(0.639766\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 8.00000 0.862662
\(87\) 0 0
\(88\) 10.2462i 1.09225i
\(89\) −10.6847 −1.13257 −0.566286 0.824209i \(-0.691620\pi\)
−0.566286 + 0.824209i \(0.691620\pi\)
\(90\) 0 0
\(91\) 0.438447 0.0459618
\(92\) 11.1231i 1.15966i
\(93\) 0 0
\(94\) −28.4924 −2.93877
\(95\) 0 0
\(96\) 0 0
\(97\) 16.9309i 1.71907i 0.511077 + 0.859535i \(0.329247\pi\)
−0.511077 + 0.859535i \(0.670753\pi\)
\(98\) 17.4384i 1.76155i
\(99\) 0 0
\(100\) 0 0
\(101\) 10.2462 1.01954 0.509768 0.860312i \(-0.329731\pi\)
0.509768 + 0.860312i \(0.329731\pi\)
\(102\) 0 0
\(103\) 15.1231i 1.49012i 0.666995 + 0.745062i \(0.267580\pi\)
−0.666995 + 0.745062i \(0.732420\pi\)
\(104\) 6.56155 0.643413
\(105\) 0 0
\(106\) −12.0000 −1.16554
\(107\) 10.9309i 1.05673i 0.849018 + 0.528364i \(0.177194\pi\)
−0.849018 + 0.528364i \(0.822806\pi\)
\(108\) 0 0
\(109\) 2.00000 0.191565 0.0957826 0.995402i \(-0.469465\pi\)
0.0957826 + 0.995402i \(0.469465\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) − 3.36932i − 0.318371i
\(113\) 4.87689i 0.458780i 0.973335 + 0.229390i \(0.0736731\pi\)
−0.973335 + 0.229390i \(0.926327\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) −32.4924 −3.01685
\(117\) 0 0
\(118\) − 30.7386i − 2.82972i
\(119\) 0.684658 0.0627625
\(120\) 0 0
\(121\) −8.56155 −0.778323
\(122\) − 17.1231i − 1.55025i
\(123\) 0 0
\(124\) −27.3693 −2.45784
\(125\) 0 0
\(126\) 0 0
\(127\) 1.75379i 0.155624i 0.996968 + 0.0778118i \(0.0247933\pi\)
−0.996968 + 0.0778118i \(0.975207\pi\)
\(128\) 9.43845i 0.834249i
\(129\) 0 0
\(130\) 0 0
\(131\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(132\) 0 0
\(133\) − 2.24621i − 0.194771i
\(134\) 29.1231 2.51585
\(135\) 0 0
\(136\) 10.2462 0.878605
\(137\) 1.12311i 0.0959534i 0.998848 + 0.0479767i \(0.0152773\pi\)
−0.998848 + 0.0479767i \(0.984723\pi\)
\(138\) 0 0
\(139\) −3.31534 −0.281204 −0.140602 0.990066i \(-0.544904\pi\)
−0.140602 + 0.990066i \(0.544904\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 26.7386i 2.24386i
\(143\) − 1.56155i − 0.130584i
\(144\) 0 0
\(145\) 0 0
\(146\) −15.3693 −1.27197
\(147\) 0 0
\(148\) 48.7386i 4.00629i
\(149\) 17.8078 1.45887 0.729434 0.684051i \(-0.239783\pi\)
0.729434 + 0.684051i \(0.239783\pi\)
\(150\) 0 0
\(151\) 11.3693 0.925222 0.462611 0.886561i \(-0.346913\pi\)
0.462611 + 0.886561i \(0.346913\pi\)
\(152\) − 33.6155i − 2.72658i
\(153\) 0 0
\(154\) −1.75379 −0.141324
\(155\) 0 0
\(156\) 0 0
\(157\) 3.36932i 0.268901i 0.990920 + 0.134450i \(0.0429269\pi\)
−0.990920 + 0.134450i \(0.957073\pi\)
\(158\) − 12.0000i − 0.954669i
\(159\) 0 0
\(160\) 0 0
\(161\) −1.06913 −0.0842593
\(162\) 0 0
\(163\) − 16.0540i − 1.25744i −0.777630 0.628722i \(-0.783578\pi\)
0.777630 0.628722i \(-0.216422\pi\)
\(164\) −16.2462 −1.26862
\(165\) 0 0
\(166\) −42.2462 −3.27894
\(167\) 4.87689i 0.377385i 0.982036 + 0.188693i \(0.0604250\pi\)
−0.982036 + 0.188693i \(0.939575\pi\)
\(168\) 0 0
\(169\) −1.00000 −0.0769231
\(170\) 0 0
\(171\) 0 0
\(172\) 14.2462i 1.08626i
\(173\) − 12.8769i − 0.979012i −0.872000 0.489506i \(-0.837177\pi\)
0.872000 0.489506i \(-0.162823\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −12.0000 −0.904534
\(177\) 0 0
\(178\) − 27.3693i − 2.05142i
\(179\) 4.87689 0.364516 0.182258 0.983251i \(-0.441659\pi\)
0.182258 + 0.983251i \(0.441659\pi\)
\(180\) 0 0
\(181\) 13.3153 0.989722 0.494861 0.868972i \(-0.335219\pi\)
0.494861 + 0.868972i \(0.335219\pi\)
\(182\) 1.12311i 0.0832501i
\(183\) 0 0
\(184\) −16.0000 −1.17954
\(185\) 0 0
\(186\) 0 0
\(187\) − 2.43845i − 0.178317i
\(188\) − 50.7386i − 3.70050i
\(189\) 0 0
\(190\) 0 0
\(191\) 19.6155 1.41933 0.709665 0.704539i \(-0.248846\pi\)
0.709665 + 0.704539i \(0.248846\pi\)
\(192\) 0 0
\(193\) − 19.5616i − 1.40807i −0.710165 0.704036i \(-0.751379\pi\)
0.710165 0.704036i \(-0.248621\pi\)
\(194\) −43.3693 −3.11374
\(195\) 0 0
\(196\) −31.0540 −2.21814
\(197\) − 3.36932i − 0.240054i −0.992771 0.120027i \(-0.961702\pi\)
0.992771 0.120027i \(-0.0382981\pi\)
\(198\) 0 0
\(199\) 8.00000 0.567105 0.283552 0.958957i \(-0.408487\pi\)
0.283552 + 0.958957i \(0.408487\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 26.2462i 1.84668i
\(203\) − 3.12311i − 0.219199i
\(204\) 0 0
\(205\) 0 0
\(206\) −38.7386 −2.69905
\(207\) 0 0
\(208\) 7.68466i 0.532835i
\(209\) −8.00000 −0.553372
\(210\) 0 0
\(211\) 6.24621 0.430007 0.215003 0.976613i \(-0.431024\pi\)
0.215003 + 0.976613i \(0.431024\pi\)
\(212\) − 21.3693i − 1.46765i
\(213\) 0 0
\(214\) −28.0000 −1.91404
\(215\) 0 0
\(216\) 0 0
\(217\) − 2.63068i − 0.178582i
\(218\) 5.12311i 0.346980i
\(219\) 0 0
\(220\) 0 0
\(221\) −1.56155 −0.105041
\(222\) 0 0
\(223\) 15.3693i 1.02921i 0.857429 + 0.514603i \(0.172061\pi\)
−0.857429 + 0.514603i \(0.827939\pi\)
\(224\) 2.87689 0.192221
\(225\) 0 0
\(226\) −12.4924 −0.830984
\(227\) 5.75379i 0.381892i 0.981601 + 0.190946i \(0.0611556\pi\)
−0.981601 + 0.190946i \(0.938844\pi\)
\(228\) 0 0
\(229\) −17.1231 −1.13153 −0.565763 0.824568i \(-0.691418\pi\)
−0.565763 + 0.824568i \(0.691418\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) − 46.7386i − 3.06854i
\(233\) 27.8078i 1.82175i 0.412685 + 0.910874i \(0.364591\pi\)
−0.412685 + 0.910874i \(0.635409\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 54.7386 3.56318
\(237\) 0 0
\(238\) 1.75379i 0.113681i
\(239\) −22.9309 −1.48327 −0.741637 0.670801i \(-0.765950\pi\)
−0.741637 + 0.670801i \(0.765950\pi\)
\(240\) 0 0
\(241\) 24.7386 1.59356 0.796778 0.604272i \(-0.206536\pi\)
0.796778 + 0.604272i \(0.206536\pi\)
\(242\) − 21.9309i − 1.40977i
\(243\) 0 0
\(244\) 30.4924 1.95208
\(245\) 0 0
\(246\) 0 0
\(247\) 5.12311i 0.325975i
\(248\) − 39.3693i − 2.49995i
\(249\) 0 0
\(250\) 0 0
\(251\) 26.2462 1.65665 0.828323 0.560251i \(-0.189295\pi\)
0.828323 + 0.560251i \(0.189295\pi\)
\(252\) 0 0
\(253\) 3.80776i 0.239392i
\(254\) −4.49242 −0.281880
\(255\) 0 0
\(256\) −27.0540 −1.69087
\(257\) − 12.8769i − 0.803239i −0.915807 0.401619i \(-0.868448\pi\)
0.915807 0.401619i \(-0.131552\pi\)
\(258\) 0 0
\(259\) −4.68466 −0.291091
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 2.24621i 0.138507i 0.997599 + 0.0692537i \(0.0220618\pi\)
−0.997599 + 0.0692537i \(0.977938\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 5.75379 0.352787
\(267\) 0 0
\(268\) 51.8617i 3.16796i
\(269\) 0.876894 0.0534652 0.0267326 0.999643i \(-0.491490\pi\)
0.0267326 + 0.999643i \(0.491490\pi\)
\(270\) 0 0
\(271\) −19.3693 −1.17660 −0.588301 0.808642i \(-0.700203\pi\)
−0.588301 + 0.808642i \(0.700203\pi\)
\(272\) 12.0000i 0.727607i
\(273\) 0 0
\(274\) −2.87689 −0.173800
\(275\) 0 0
\(276\) 0 0
\(277\) − 12.2462i − 0.735804i −0.929865 0.367902i \(-0.880076\pi\)
0.929865 0.367902i \(-0.119924\pi\)
\(278\) − 8.49242i − 0.509342i
\(279\) 0 0
\(280\) 0 0
\(281\) 4.24621 0.253308 0.126654 0.991947i \(-0.459576\pi\)
0.126654 + 0.991947i \(0.459576\pi\)
\(282\) 0 0
\(283\) 4.00000i 0.237775i 0.992908 + 0.118888i \(0.0379328\pi\)
−0.992908 + 0.118888i \(0.962067\pi\)
\(284\) −47.6155 −2.82546
\(285\) 0 0
\(286\) 4.00000 0.236525
\(287\) − 1.56155i − 0.0921755i
\(288\) 0 0
\(289\) 14.5616 0.856562
\(290\) 0 0
\(291\) 0 0
\(292\) − 27.3693i − 1.60167i
\(293\) 20.2462i 1.18280i 0.806380 + 0.591398i \(0.201424\pi\)
−0.806380 + 0.591398i \(0.798576\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) −70.1080 −4.07494
\(297\) 0 0
\(298\) 45.6155i 2.64244i
\(299\) 2.43845 0.141019
\(300\) 0 0
\(301\) −1.36932 −0.0789261
\(302\) 29.1231i 1.67585i
\(303\) 0 0
\(304\) 39.3693 2.25799
\(305\) 0 0
\(306\) 0 0
\(307\) 7.56155i 0.431561i 0.976442 + 0.215780i \(0.0692295\pi\)
−0.976442 + 0.215780i \(0.930770\pi\)
\(308\) − 3.12311i − 0.177955i
\(309\) 0 0
\(310\) 0 0
\(311\) 2.63068 0.149172 0.0745862 0.997215i \(-0.476236\pi\)
0.0745862 + 0.997215i \(0.476236\pi\)
\(312\) 0 0
\(313\) − 29.1231i − 1.64614i −0.567943 0.823068i \(-0.692261\pi\)
0.567943 0.823068i \(-0.307739\pi\)
\(314\) −8.63068 −0.487058
\(315\) 0 0
\(316\) 21.3693 1.20212
\(317\) 1.50758i 0.0846740i 0.999103 + 0.0423370i \(0.0134803\pi\)
−0.999103 + 0.0423370i \(0.986520\pi\)
\(318\) 0 0
\(319\) −11.1231 −0.622774
\(320\) 0 0
\(321\) 0 0
\(322\) − 2.73863i − 0.152618i
\(323\) 8.00000i 0.445132i
\(324\) 0 0
\(325\) 0 0
\(326\) 41.1231 2.27760
\(327\) 0 0
\(328\) − 23.3693i − 1.29035i
\(329\) 4.87689 0.268872
\(330\) 0 0
\(331\) 29.1231 1.60075 0.800375 0.599499i \(-0.204634\pi\)
0.800375 + 0.599499i \(0.204634\pi\)
\(332\) − 75.2311i − 4.12884i
\(333\) 0 0
\(334\) −12.4924 −0.683555
\(335\) 0 0
\(336\) 0 0
\(337\) − 30.4924i − 1.66103i −0.556998 0.830514i \(-0.688047\pi\)
0.556998 0.830514i \(-0.311953\pi\)
\(338\) − 2.56155i − 0.139330i
\(339\) 0 0
\(340\) 0 0
\(341\) −9.36932 −0.507377
\(342\) 0 0
\(343\) − 6.05398i − 0.326884i
\(344\) −20.4924 −1.10488
\(345\) 0 0
\(346\) 32.9848 1.77328
\(347\) − 26.0540i − 1.39865i −0.714804 0.699325i \(-0.753484\pi\)
0.714804 0.699325i \(-0.246516\pi\)
\(348\) 0 0
\(349\) 23.3693 1.25093 0.625465 0.780252i \(-0.284909\pi\)
0.625465 + 0.780252i \(0.284909\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) − 10.2462i − 0.546125i
\(353\) 22.4924i 1.19715i 0.801066 + 0.598575i \(0.204266\pi\)
−0.801066 + 0.598575i \(0.795734\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 48.7386 2.58314
\(357\) 0 0
\(358\) 12.4924i 0.660245i
\(359\) 14.2462 0.751886 0.375943 0.926643i \(-0.377319\pi\)
0.375943 + 0.926643i \(0.377319\pi\)
\(360\) 0 0
\(361\) 7.24621 0.381380
\(362\) 34.1080i 1.79267i
\(363\) 0 0
\(364\) −2.00000 −0.104828
\(365\) 0 0
\(366\) 0 0
\(367\) − 1.75379i − 0.0915470i −0.998952 0.0457735i \(-0.985425\pi\)
0.998952 0.0457735i \(-0.0145753\pi\)
\(368\) − 18.7386i − 0.976819i
\(369\) 0 0
\(370\) 0 0
\(371\) 2.05398 0.106637
\(372\) 0 0
\(373\) − 12.2462i − 0.634085i −0.948411 0.317042i \(-0.897310\pi\)
0.948411 0.317042i \(-0.102690\pi\)
\(374\) 6.24621 0.322984
\(375\) 0 0
\(376\) 72.9848 3.76391
\(377\) 7.12311i 0.366859i
\(378\) 0 0
\(379\) 30.4924 1.56629 0.783145 0.621839i \(-0.213614\pi\)
0.783145 + 0.621839i \(0.213614\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 50.2462i 2.57082i
\(383\) − 3.50758i − 0.179229i −0.995977 0.0896144i \(-0.971437\pi\)
0.995977 0.0896144i \(-0.0285635\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 50.1080 2.55043
\(387\) 0 0
\(388\) − 77.2311i − 3.92081i
\(389\) −37.8617 −1.91967 −0.959833 0.280571i \(-0.909476\pi\)
−0.959833 + 0.280571i \(0.909476\pi\)
\(390\) 0 0
\(391\) 3.80776 0.192567
\(392\) − 44.6695i − 2.25615i
\(393\) 0 0
\(394\) 8.63068 0.434808
\(395\) 0 0
\(396\) 0 0
\(397\) − 4.43845i − 0.222759i −0.993778 0.111380i \(-0.964473\pi\)
0.993778 0.111380i \(-0.0355270\pi\)
\(398\) 20.4924i 1.02719i
\(399\) 0 0
\(400\) 0 0
\(401\) 3.75379 0.187455 0.0937276 0.995598i \(-0.470122\pi\)
0.0937276 + 0.995598i \(0.470122\pi\)
\(402\) 0 0
\(403\) 6.00000i 0.298881i
\(404\) −46.7386 −2.32533
\(405\) 0 0
\(406\) 8.00000 0.397033
\(407\) 16.6847i 0.827028i
\(408\) 0 0
\(409\) 6.87689 0.340041 0.170020 0.985441i \(-0.445617\pi\)
0.170020 + 0.985441i \(0.445617\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) − 68.9848i − 3.39864i
\(413\) 5.26137i 0.258895i
\(414\) 0 0
\(415\) 0 0
\(416\) −6.56155 −0.321707
\(417\) 0 0
\(418\) − 20.4924i − 1.00232i
\(419\) −7.61553 −0.372043 −0.186021 0.982546i \(-0.559559\pi\)
−0.186021 + 0.982546i \(0.559559\pi\)
\(420\) 0 0
\(421\) 39.3693 1.91874 0.959372 0.282146i \(-0.0910462\pi\)
0.959372 + 0.282146i \(0.0910462\pi\)
\(422\) 16.0000i 0.778868i
\(423\) 0 0
\(424\) 30.7386 1.49280
\(425\) 0 0
\(426\) 0 0
\(427\) 2.93087i 0.141835i
\(428\) − 49.8617i − 2.41016i
\(429\) 0 0
\(430\) 0 0
\(431\) −3.50758 −0.168954 −0.0844770 0.996425i \(-0.526922\pi\)
−0.0844770 + 0.996425i \(0.526922\pi\)
\(432\) 0 0
\(433\) 9.61553i 0.462093i 0.972943 + 0.231046i \(0.0742149\pi\)
−0.972943 + 0.231046i \(0.925785\pi\)
\(434\) 6.73863 0.323465
\(435\) 0 0
\(436\) −9.12311 −0.436918
\(437\) − 12.4924i − 0.597594i
\(438\) 0 0
\(439\) −22.0540 −1.05258 −0.526289 0.850306i \(-0.676417\pi\)
−0.526289 + 0.850306i \(0.676417\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) − 4.00000i − 0.190261i
\(443\) 7.80776i 0.370958i 0.982648 + 0.185479i \(0.0593837\pi\)
−0.982648 + 0.185479i \(0.940616\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) −39.3693 −1.86419
\(447\) 0 0
\(448\) 0.630683i 0.0297970i
\(449\) 35.1771 1.66011 0.830055 0.557682i \(-0.188309\pi\)
0.830055 + 0.557682i \(0.188309\pi\)
\(450\) 0 0
\(451\) −5.56155 −0.261883
\(452\) − 22.2462i − 1.04637i
\(453\) 0 0
\(454\) −14.7386 −0.691718
\(455\) 0 0
\(456\) 0 0
\(457\) − 13.3153i − 0.622865i −0.950268 0.311433i \(-0.899191\pi\)
0.950268 0.311433i \(-0.100809\pi\)
\(458\) − 43.8617i − 2.04952i
\(459\) 0 0
\(460\) 0 0
\(461\) 8.05398 0.375111 0.187556 0.982254i \(-0.439944\pi\)
0.187556 + 0.982254i \(0.439944\pi\)
\(462\) 0 0
\(463\) 28.9309i 1.34453i 0.740310 + 0.672266i \(0.234679\pi\)
−0.740310 + 0.672266i \(0.765321\pi\)
\(464\) 54.7386 2.54118
\(465\) 0 0
\(466\) −71.2311 −3.29971
\(467\) 6.93087i 0.320722i 0.987058 + 0.160361i \(0.0512659\pi\)
−0.987058 + 0.160361i \(0.948734\pi\)
\(468\) 0 0
\(469\) −4.98485 −0.230179
\(470\) 0 0
\(471\) 0 0
\(472\) 78.7386i 3.62424i
\(473\) 4.87689i 0.224240i
\(474\) 0 0
\(475\) 0 0
\(476\) −3.12311 −0.143147
\(477\) 0 0
\(478\) − 58.7386i − 2.68664i
\(479\) −26.0540 −1.19044 −0.595218 0.803564i \(-0.702934\pi\)
−0.595218 + 0.803564i \(0.702934\pi\)
\(480\) 0 0
\(481\) 10.6847 0.487178
\(482\) 63.3693i 2.88639i
\(483\) 0 0
\(484\) 39.0540 1.77518
\(485\) 0 0
\(486\) 0 0
\(487\) − 32.0540i − 1.45250i −0.687428 0.726252i \(-0.741260\pi\)
0.687428 0.726252i \(-0.258740\pi\)
\(488\) 43.8617i 1.98553i
\(489\) 0 0
\(490\) 0 0
\(491\) −27.6155 −1.24627 −0.623136 0.782114i \(-0.714142\pi\)
−0.623136 + 0.782114i \(0.714142\pi\)
\(492\) 0 0
\(493\) 11.1231i 0.500959i
\(494\) −13.1231 −0.590436
\(495\) 0 0
\(496\) 46.1080 2.07031
\(497\) − 4.57671i − 0.205293i
\(498\) 0 0
\(499\) −34.9848 −1.56614 −0.783068 0.621936i \(-0.786347\pi\)
−0.783068 + 0.621936i \(0.786347\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 67.2311i 3.00067i
\(503\) − 13.7538i − 0.613251i −0.951830 0.306626i \(-0.900800\pi\)
0.951830 0.306626i \(-0.0991999\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) −9.75379 −0.433609
\(507\) 0 0
\(508\) − 8.00000i − 0.354943i
\(509\) 23.5616 1.04435 0.522174 0.852839i \(-0.325121\pi\)
0.522174 + 0.852839i \(0.325121\pi\)
\(510\) 0 0
\(511\) 2.63068 0.116375
\(512\) − 50.4233i − 2.22842i
\(513\) 0 0
\(514\) 32.9848 1.45490
\(515\) 0 0
\(516\) 0 0
\(517\) − 17.3693i − 0.763902i
\(518\) − 12.0000i − 0.527250i
\(519\) 0 0
\(520\) 0 0
\(521\) −25.8617 −1.13302 −0.566512 0.824054i \(-0.691707\pi\)
−0.566512 + 0.824054i \(0.691707\pi\)
\(522\) 0 0
\(523\) 30.7386i 1.34411i 0.740503 + 0.672053i \(0.234587\pi\)
−0.740503 + 0.672053i \(0.765413\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) −5.75379 −0.250877
\(527\) 9.36932i 0.408134i
\(528\) 0 0
\(529\) 17.0540 0.741477
\(530\) 0 0
\(531\) 0 0
\(532\) 10.2462i 0.444230i
\(533\) 3.56155i 0.154268i
\(534\) 0 0
\(535\) 0 0
\(536\) −74.6004 −3.22225
\(537\) 0 0
\(538\) 2.24621i 0.0968410i
\(539\) −10.6307 −0.457896
\(540\) 0 0
\(541\) 18.8769 0.811581 0.405791 0.913966i \(-0.366996\pi\)
0.405791 + 0.913966i \(0.366996\pi\)
\(542\) − 49.6155i − 2.13117i
\(543\) 0 0
\(544\) −10.2462 −0.439303
\(545\) 0 0
\(546\) 0 0
\(547\) − 5.36932i − 0.229575i −0.993390 0.114788i \(-0.963381\pi\)
0.993390 0.114788i \(-0.0366188\pi\)
\(548\) − 5.12311i − 0.218848i
\(549\) 0 0
\(550\) 0 0
\(551\) 36.4924 1.55463
\(552\) 0 0
\(553\) 2.05398i 0.0873439i
\(554\) 31.3693 1.33275
\(555\) 0 0
\(556\) 15.1231 0.641363
\(557\) 6.49242i 0.275093i 0.990495 + 0.137546i \(0.0439216\pi\)
−0.990495 + 0.137546i \(0.956078\pi\)
\(558\) 0 0
\(559\) 3.12311 0.132093
\(560\) 0 0
\(561\) 0 0
\(562\) 10.8769i 0.458814i
\(563\) 19.3153i 0.814045i 0.913418 + 0.407022i \(0.133433\pi\)
−0.913418 + 0.407022i \(0.866567\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) −10.2462 −0.430680
\(567\) 0 0
\(568\) − 68.4924i − 2.87388i
\(569\) 32.8769 1.37827 0.689136 0.724632i \(-0.257990\pi\)
0.689136 + 0.724632i \(0.257990\pi\)
\(570\) 0 0
\(571\) −22.0540 −0.922930 −0.461465 0.887158i \(-0.652676\pi\)
−0.461465 + 0.887158i \(0.652676\pi\)
\(572\) 7.12311i 0.297832i
\(573\) 0 0
\(574\) 4.00000 0.166957
\(575\) 0 0
\(576\) 0 0
\(577\) 24.4384i 1.01739i 0.860948 + 0.508693i \(0.169871\pi\)
−0.860948 + 0.508693i \(0.830129\pi\)
\(578\) 37.3002i 1.55148i
\(579\) 0 0
\(580\) 0 0
\(581\) 7.23106 0.299995
\(582\) 0 0
\(583\) − 7.31534i − 0.302970i
\(584\) 39.3693 1.62911
\(585\) 0 0
\(586\) −51.8617 −2.14239
\(587\) − 32.4924i − 1.34111i −0.741862 0.670553i \(-0.766057\pi\)
0.741862 0.670553i \(-0.233943\pi\)
\(588\) 0 0
\(589\) 30.7386 1.26656
\(590\) 0 0
\(591\) 0 0
\(592\) − 82.1080i − 3.37462i
\(593\) − 24.2462i − 0.995673i −0.867271 0.497836i \(-0.834128\pi\)
0.867271 0.497836i \(-0.165872\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −81.2311 −3.32735
\(597\) 0 0
\(598\) 6.24621i 0.255427i
\(599\) 9.36932 0.382820 0.191410 0.981510i \(-0.438694\pi\)
0.191410 + 0.981510i \(0.438694\pi\)
\(600\) 0 0
\(601\) −28.5464 −1.16443 −0.582216 0.813034i \(-0.697814\pi\)
−0.582216 + 0.813034i \(0.697814\pi\)
\(602\) − 3.50758i − 0.142958i
\(603\) 0 0
\(604\) −51.8617 −2.11022
\(605\) 0 0
\(606\) 0 0
\(607\) − 24.0000i − 0.974130i −0.873366 0.487065i \(-0.838067\pi\)
0.873366 0.487065i \(-0.161933\pi\)
\(608\) 33.6155i 1.36329i
\(609\) 0 0
\(610\) 0 0
\(611\) −11.1231 −0.449993
\(612\) 0 0
\(613\) 32.5464i 1.31454i 0.753657 + 0.657268i \(0.228288\pi\)
−0.753657 + 0.657268i \(0.771712\pi\)
\(614\) −19.3693 −0.781682
\(615\) 0 0
\(616\) 4.49242 0.181005
\(617\) 0.738634i 0.0297363i 0.999889 + 0.0148681i \(0.00473285\pi\)
−0.999889 + 0.0148681i \(0.995267\pi\)
\(618\) 0 0
\(619\) 8.63068 0.346896 0.173448 0.984843i \(-0.444509\pi\)
0.173448 + 0.984843i \(0.444509\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 6.73863i 0.270195i
\(623\) 4.68466i 0.187687i
\(624\) 0 0
\(625\) 0 0
\(626\) 74.6004 2.98163
\(627\) 0 0
\(628\) − 15.3693i − 0.613303i
\(629\) 16.6847 0.665261
\(630\) 0 0
\(631\) −32.7386 −1.30330 −0.651652 0.758518i \(-0.725924\pi\)
−0.651652 + 0.758518i \(0.725924\pi\)
\(632\) 30.7386i 1.22272i
\(633\) 0 0
\(634\) −3.86174 −0.153369
\(635\) 0 0
\(636\) 0 0
\(637\) 6.80776i 0.269733i
\(638\) − 28.4924i − 1.12803i
\(639\) 0 0
\(640\) 0 0
\(641\) −32.9848 −1.30282 −0.651412 0.758725i \(-0.725823\pi\)
−0.651412 + 0.758725i \(0.725823\pi\)
\(642\) 0 0
\(643\) 10.6847i 0.421362i 0.977555 + 0.210681i \(0.0675681\pi\)
−0.977555 + 0.210681i \(0.932432\pi\)
\(644\) 4.87689 0.192177
\(645\) 0 0
\(646\) −20.4924 −0.806264
\(647\) 31.8078i 1.25049i 0.780428 + 0.625246i \(0.215001\pi\)
−0.780428 + 0.625246i \(0.784999\pi\)
\(648\) 0 0
\(649\) 18.7386 0.735556
\(650\) 0 0
\(651\) 0 0
\(652\) 73.2311i 2.86795i
\(653\) 33.3693i 1.30584i 0.757426 + 0.652921i \(0.226457\pi\)
−0.757426 + 0.652921i \(0.773543\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 27.3693 1.06859
\(657\) 0 0
\(658\) 12.4924i 0.487005i
\(659\) −0.876894 −0.0341590 −0.0170795 0.999854i \(-0.505437\pi\)
−0.0170795 + 0.999854i \(0.505437\pi\)
\(660\) 0 0
\(661\) 6.49242 0.252526 0.126263 0.991997i \(-0.459702\pi\)
0.126263 + 0.991997i \(0.459702\pi\)
\(662\) 74.6004i 2.89943i
\(663\) 0 0
\(664\) 108.216 4.19959
\(665\) 0 0
\(666\) 0 0
\(667\) − 17.3693i − 0.672543i
\(668\) − 22.2462i − 0.860732i
\(669\) 0 0
\(670\) 0 0
\(671\) 10.4384 0.402972
\(672\) 0 0
\(673\) − 16.7386i − 0.645227i −0.946531 0.322613i \(-0.895439\pi\)
0.946531 0.322613i \(-0.104561\pi\)
\(674\) 78.1080 3.00861
\(675\) 0 0
\(676\) 4.56155 0.175444
\(677\) − 14.4384i − 0.554915i −0.960738 0.277457i \(-0.910508\pi\)
0.960738 0.277457i \(-0.0894917\pi\)
\(678\) 0 0
\(679\) 7.42329 0.284880
\(680\) 0 0
\(681\) 0 0
\(682\) − 24.0000i − 0.919007i
\(683\) − 32.4924i − 1.24329i −0.783300 0.621644i \(-0.786465\pi\)
0.783300 0.621644i \(-0.213535\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 15.5076 0.592082
\(687\) 0 0
\(688\) − 24.0000i − 0.914991i
\(689\) −4.68466 −0.178471
\(690\) 0 0
\(691\) −21.6155 −0.822293 −0.411147 0.911569i \(-0.634872\pi\)
−0.411147 + 0.911569i \(0.634872\pi\)
\(692\) 58.7386i 2.23291i
\(693\) 0 0
\(694\) 66.7386 2.53336
\(695\) 0 0
\(696\) 0 0
\(697\) 5.56155i 0.210659i
\(698\) 59.8617i 2.26580i
\(699\) 0 0
\(700\) 0 0
\(701\) −48.9848 −1.85013 −0.925066 0.379806i \(-0.875991\pi\)
−0.925066 + 0.379806i \(0.875991\pi\)
\(702\) 0 0
\(703\) − 54.7386i − 2.06451i
\(704\) 2.24621 0.0846573
\(705\) 0 0
\(706\) −57.6155 −2.16839
\(707\) − 4.49242i − 0.168955i
\(708\) 0 0
\(709\) −9.12311 −0.342625 −0.171313 0.985217i \(-0.554801\pi\)
−0.171313 + 0.985217i \(0.554801\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 70.1080i 2.62741i
\(713\) − 14.6307i − 0.547923i
\(714\) 0 0
\(715\) 0 0
\(716\) −22.2462 −0.831380
\(717\) 0 0
\(718\) 36.4924i 1.36189i
\(719\) −36.0000 −1.34257 −0.671287 0.741198i \(-0.734258\pi\)
−0.671287 + 0.741198i \(0.734258\pi\)
\(720\) 0 0
\(721\) 6.63068 0.246940
\(722\) 18.5616i 0.690789i
\(723\) 0 0
\(724\) −60.7386 −2.25733
\(725\) 0 0
\(726\) 0 0
\(727\) 12.8769i 0.477578i 0.971072 + 0.238789i \(0.0767504\pi\)
−0.971072 + 0.238789i \(0.923250\pi\)
\(728\) − 2.87689i − 0.106625i
\(729\) 0 0
\(730\) 0 0
\(731\) 4.87689 0.180378
\(732\) 0 0
\(733\) 16.4384i 0.607168i 0.952805 + 0.303584i \(0.0981833\pi\)
−0.952805 + 0.303584i \(0.901817\pi\)
\(734\) 4.49242 0.165818
\(735\) 0 0
\(736\) 16.0000 0.589768
\(737\) 17.7538i 0.653969i
\(738\) 0 0
\(739\) −48.7386 −1.79288 −0.896440 0.443166i \(-0.853855\pi\)
−0.896440 + 0.443166i \(0.853855\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 5.26137i 0.193151i
\(743\) − 18.7386i − 0.687454i −0.939070 0.343727i \(-0.888311\pi\)
0.939070 0.343727i \(-0.111689\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 31.3693 1.14851
\(747\) 0 0
\(748\) 11.1231i 0.406701i
\(749\) 4.79261 0.175118
\(750\) 0 0
\(751\) 14.0540 0.512837 0.256418 0.966566i \(-0.417458\pi\)
0.256418 + 0.966566i \(0.417458\pi\)
\(752\) 85.4773i 3.11704i
\(753\) 0 0
\(754\) −18.2462 −0.664488
\(755\) 0 0
\(756\) 0 0
\(757\) − 14.4924i − 0.526736i −0.964695 0.263368i \(-0.915167\pi\)
0.964695 0.263368i \(-0.0848334\pi\)
\(758\) 78.1080i 2.83701i
\(759\) 0 0
\(760\) 0 0
\(761\) 45.2311 1.63962 0.819812 0.572632i \(-0.194078\pi\)
0.819812 + 0.572632i \(0.194078\pi\)
\(762\) 0 0
\(763\) − 0.876894i − 0.0317457i
\(764\) −89.4773 −3.23717
\(765\) 0 0
\(766\) 8.98485 0.324636
\(767\) − 12.0000i − 0.433295i
\(768\) 0 0
\(769\) 30.0000 1.08183 0.540914 0.841078i \(-0.318079\pi\)
0.540914 + 0.841078i \(0.318079\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 89.2311i 3.21150i
\(773\) 9.12311i 0.328135i 0.986449 + 0.164068i \(0.0524615\pi\)
−0.986449 + 0.164068i \(0.947538\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 111.093 3.98800
\(777\) 0 0
\(778\) − 96.9848i − 3.47708i
\(779\) 18.2462 0.653738
\(780\) 0 0
\(781\) −16.3002 −0.583267
\(782\) 9.75379i 0.348795i
\(783\) 0 0
\(784\) 52.3153 1.86841
\(785\) 0 0
\(786\) 0 0
\(787\) 11.3693i 0.405272i 0.979254 + 0.202636i \(0.0649509\pi\)
−0.979254 + 0.202636i \(0.935049\pi\)
\(788\) 15.3693i 0.547509i
\(789\) 0 0
\(790\) 0 0
\(791\) 2.13826 0.0760278
\(792\) 0 0
\(793\) − 6.68466i − 0.237379i
\(794\) 11.3693 0.403482
\(795\) 0 0
\(796\) −36.4924 −1.29344
\(797\) 4.68466i 0.165939i 0.996552 + 0.0829696i \(0.0264404\pi\)
−0.996552 + 0.0829696i \(0.973560\pi\)
\(798\) 0 0
\(799\) −17.3693 −0.614482
\(800\) 0 0
\(801\) 0 0
\(802\) 9.61553i 0.339536i
\(803\) − 9.36932i − 0.330636i
\(804\) 0 0
\(805\) 0 0
\(806\) −15.3693 −0.541361
\(807\) 0 0
\(808\) − 67.2311i − 2.36518i
\(809\) −7.50758 −0.263952 −0.131976 0.991253i \(-0.542132\pi\)
−0.131976 + 0.991253i \(0.542132\pi\)
\(810\) 0 0
\(811\) 4.24621 0.149105 0.0745523 0.997217i \(-0.476247\pi\)
0.0745523 + 0.997217i \(0.476247\pi\)
\(812\) 14.2462i 0.499944i
\(813\) 0 0
\(814\) −42.7386 −1.49799
\(815\) 0 0
\(816\) 0 0
\(817\) − 16.0000i − 0.559769i
\(818\) 17.6155i 0.615912i
\(819\) 0 0
\(820\) 0 0
\(821\) 48.5464 1.69428 0.847140 0.531369i \(-0.178322\pi\)
0.847140 + 0.531369i \(0.178322\pi\)
\(822\) 0 0
\(823\) 29.7538i 1.03715i 0.855032 + 0.518576i \(0.173538\pi\)
−0.855032 + 0.518576i \(0.826462\pi\)
\(824\) 99.2311 3.45688
\(825\) 0 0
\(826\) −13.4773 −0.468934
\(827\) 7.12311i 0.247695i 0.992301 + 0.123847i \(0.0395233\pi\)
−0.992301 + 0.123847i \(0.960477\pi\)
\(828\) 0 0
\(829\) −0.738634 −0.0256538 −0.0128269 0.999918i \(-0.504083\pi\)
−0.0128269 + 0.999918i \(0.504083\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) − 1.43845i − 0.0498692i
\(833\) 10.6307i 0.368331i
\(834\) 0 0
\(835\) 0 0
\(836\) 36.4924 1.26212
\(837\) 0 0
\(838\) − 19.5076i − 0.673878i
\(839\) 44.7926 1.54641 0.773206 0.634155i \(-0.218652\pi\)
0.773206 + 0.634155i \(0.218652\pi\)
\(840\) 0 0
\(841\) 21.7386 0.749608
\(842\) 100.847i 3.47540i
\(843\) 0 0
\(844\) −28.4924 −0.980750
\(845\) 0 0
\(846\) 0 0
\(847\) 3.75379i 0.128982i
\(848\) 36.0000i 1.23625i
\(849\) 0 0
\(850\) 0 0
\(851\) −26.0540 −0.893119
\(852\) 0 0
\(853\) − 41.4233i − 1.41831i −0.705054 0.709153i \(-0.749077\pi\)
0.705054 0.709153i \(-0.250923\pi\)
\(854\) −7.50758 −0.256904
\(855\) 0 0
\(856\) 71.7235 2.45146
\(857\) 2.43845i 0.0832958i 0.999132 + 0.0416479i \(0.0132608\pi\)
−0.999132 + 0.0416479i \(0.986739\pi\)
\(858\) 0 0
\(859\) −3.80776 −0.129919 −0.0649596 0.997888i \(-0.520692\pi\)
−0.0649596 + 0.997888i \(0.520692\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) − 8.98485i − 0.306025i
\(863\) 9.36932i 0.318935i 0.987203 + 0.159468i \(0.0509778\pi\)
−0.987203 + 0.159468i \(0.949022\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) −24.6307 −0.836985
\(867\) 0 0
\(868\) 12.0000i 0.407307i
\(869\) 7.31534 0.248156
\(870\) 0 0
\(871\) 11.3693 0.385235
\(872\) − 13.1231i − 0.444404i
\(873\) 0 0
\(874\) 32.0000 1.08242
\(875\) 0 0
\(876\) 0 0
\(877\) 46.9848i 1.58657i 0.608853 + 0.793283i \(0.291630\pi\)
−0.608853 + 0.793283i \(0.708370\pi\)
\(878\) − 56.4924i − 1.90653i
\(879\) 0 0
\(880\) 0 0
\(881\) −21.3693 −0.719951 −0.359975 0.932962i \(-0.617215\pi\)
−0.359975 + 0.932962i \(0.617215\pi\)
\(882\) 0 0
\(883\) 56.1080i 1.88818i 0.329684 + 0.944091i \(0.393058\pi\)
−0.329684 + 0.944091i \(0.606942\pi\)
\(884\) 7.12311 0.239576
\(885\) 0 0
\(886\) −20.0000 −0.671913
\(887\) 5.56155i 0.186739i 0.995632 + 0.0933693i \(0.0297637\pi\)
−0.995632 + 0.0933693i \(0.970236\pi\)
\(888\) 0 0
\(889\) 0.768944 0.0257895
\(890\) 0 0
\(891\) 0 0
\(892\) − 70.1080i − 2.34739i
\(893\) 56.9848i 1.90693i
\(894\) 0 0
\(895\) 0 0
\(896\) 4.13826 0.138250
\(897\) 0 0
\(898\) 90.1080i 3.00694i
\(899\) 42.7386 1.42541
\(900\) 0 0
\(901\) −7.31534 −0.243709
\(902\) − 14.2462i − 0.474347i
\(903\) 0 0
\(904\) 32.0000 1.06430
\(905\) 0 0
\(906\) 0 0
\(907\) − 2.63068i − 0.0873504i −0.999046 0.0436752i \(-0.986093\pi\)
0.999046 0.0436752i \(-0.0139067\pi\)
\(908\) − 26.2462i − 0.871011i
\(909\) 0 0
\(910\) 0 0
\(911\) −12.0000 −0.397578 −0.198789 0.980042i \(-0.563701\pi\)
−0.198789 + 0.980042i \(0.563701\pi\)
\(912\) 0 0
\(913\) − 25.7538i − 0.852326i
\(914\) 34.1080 1.12819
\(915\) 0 0
\(916\) 78.1080 2.58076
\(917\) 0 0
\(918\) 0 0
\(919\) −1.94602 −0.0641934 −0.0320967 0.999485i \(-0.510218\pi\)
−0.0320967 + 0.999485i \(0.510218\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 20.6307i 0.679435i
\(923\) 10.4384i 0.343586i
\(924\) 0 0
\(925\) 0 0
\(926\) −74.1080 −2.43534
\(927\) 0 0
\(928\) 46.7386i 1.53427i
\(929\) −39.6695 −1.30151 −0.650757 0.759286i \(-0.725548\pi\)
−0.650757 + 0.759286i \(0.725548\pi\)
\(930\) 0 0
\(931\) 34.8769 1.14304
\(932\) − 126.847i − 4.15500i
\(933\) 0 0
\(934\) −17.7538 −0.580922
\(935\) 0 0
\(936\) 0 0
\(937\) − 27.3693i − 0.894117i −0.894505 0.447058i \(-0.852472\pi\)
0.894505 0.447058i \(-0.147528\pi\)
\(938\) − 12.7689i − 0.416921i
\(939\) 0 0
\(940\) 0 0
\(941\) −41.8078 −1.36289 −0.681447 0.731867i \(-0.738649\pi\)
−0.681447 + 0.731867i \(0.738649\pi\)
\(942\) 0 0
\(943\) − 8.68466i − 0.282811i
\(944\) −92.2159 −3.00137
\(945\) 0 0
\(946\) −12.4924 −0.406164
\(947\) 60.6004i 1.96925i 0.174688 + 0.984624i \(0.444108\pi\)
−0.174688 + 0.984624i \(0.555892\pi\)
\(948\) 0 0
\(949\) −6.00000 −0.194768
\(950\) 0 0
\(951\) 0 0
\(952\) − 4.49242i − 0.145600i
\(953\) − 29.5616i − 0.957593i −0.877926 0.478796i \(-0.841073\pi\)
0.877926 0.478796i \(-0.158927\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 104.600 3.38302
\(957\) 0 0
\(958\) − 66.7386i − 2.15623i
\(959\) 0.492423 0.0159012
\(960\) 0 0
\(961\) 5.00000 0.161290
\(962\) 27.3693i 0.882422i
\(963\) 0 0
\(964\) −112.847 −3.63454
\(965\) 0 0
\(966\) 0 0
\(967\) − 7.36932i − 0.236981i −0.992955 0.118491i \(-0.962194\pi\)
0.992955 0.118491i \(-0.0378056\pi\)
\(968\) 56.1771i 1.80560i
\(969\) 0 0
\(970\) 0 0
\(971\) −24.4924 −0.785999 −0.393000 0.919539i \(-0.628563\pi\)
−0.393000 + 0.919539i \(0.628563\pi\)
\(972\) 0 0
\(973\) 1.45360i 0.0466003i
\(974\) 82.1080 2.63091
\(975\) 0 0
\(976\) −51.3693 −1.64429
\(977\) − 43.4773i − 1.39096i −0.718545 0.695481i \(-0.755192\pi\)
0.718545 0.695481i \(-0.244808\pi\)
\(978\) 0 0
\(979\) 16.6847 0.533244
\(980\) 0 0
\(981\) 0 0
\(982\) − 70.7386i − 2.25736i
\(983\) − 42.7386i − 1.36315i −0.731748 0.681575i \(-0.761295\pi\)
0.731748 0.681575i \(-0.238705\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) −28.4924 −0.907384
\(987\) 0 0
\(988\) − 23.3693i − 0.743477i
\(989\) −7.61553 −0.242160
\(990\) 0 0
\(991\) 10.0540 0.319375 0.159688 0.987168i \(-0.448951\pi\)
0.159688 + 0.987168i \(0.448951\pi\)
\(992\) 39.3693i 1.24998i
\(993\) 0 0
\(994\) 11.7235 0.371846
\(995\) 0 0
\(996\) 0 0
\(997\) 28.2462i 0.894566i 0.894392 + 0.447283i \(0.147608\pi\)
−0.894392 + 0.447283i \(0.852392\pi\)
\(998\) − 89.6155i − 2.83673i
\(999\) 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 2925.2.c.p.2224.4 4
3.2 odd 2 2925.2.c.o.2224.1 4
5.2 odd 4 2925.2.a.x.1.1 2
5.3 odd 4 585.2.a.l.1.2 yes 2
5.4 even 2 inner 2925.2.c.p.2224.1 4
15.2 even 4 2925.2.a.bc.1.2 2
15.8 even 4 585.2.a.j.1.1 2
15.14 odd 2 2925.2.c.o.2224.4 4
20.3 even 4 9360.2.a.cw.1.1 2
60.23 odd 4 9360.2.a.cl.1.1 2
65.38 odd 4 7605.2.a.bd.1.1 2
195.38 even 4 7605.2.a.bi.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
585.2.a.j.1.1 2 15.8 even 4
585.2.a.l.1.2 yes 2 5.3 odd 4
2925.2.a.x.1.1 2 5.2 odd 4
2925.2.a.bc.1.2 2 15.2 even 4
2925.2.c.o.2224.1 4 3.2 odd 2
2925.2.c.o.2224.4 4 15.14 odd 2
2925.2.c.p.2224.1 4 5.4 even 2 inner
2925.2.c.p.2224.4 4 1.1 even 1 trivial
7605.2.a.bd.1.1 2 65.38 odd 4
7605.2.a.bi.1.2 2 195.38 even 4
9360.2.a.cl.1.1 2 60.23 odd 4
9360.2.a.cw.1.1 2 20.3 even 4