Properties

Label 2475.2.c.r.199.2
Level $2475$
Weight $2$
Character 2475.199
Analytic conductor $19.763$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 2475 = 3^{2} \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2475.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(19.7629745003\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.350464.1
Defining polynomial: \(x^{6} - 2 x^{5} + 2 x^{4} + 2 x^{3} + 4 x^{2} - 4 x + 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{5} \)
Twist minimal: no (minimal twist has level 165)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 199.2
Root \(1.45161 + 1.45161i\) of defining polynomial
Character \(\chi\) \(=\) 2475.199
Dual form 2475.2.c.r.199.5

$q$-expansion

\(f(q)\) \(=\) \(q-1.90321i q^{2} -1.62222 q^{4} -4.42864i q^{7} -0.719004i q^{8} +O(q^{10})\) \(q-1.90321i q^{2} -1.62222 q^{4} -4.42864i q^{7} -0.719004i q^{8} -1.00000 q^{11} +0.622216i q^{13} -8.42864 q^{14} -4.61285 q^{16} +5.18421i q^{17} -7.05086 q^{19} +1.90321i q^{22} +8.85728i q^{23} +1.18421 q^{26} +7.18421i q^{28} -7.80642 q^{29} +2.75557 q^{31} +7.34122i q^{32} +9.86665 q^{34} -2.00000i q^{37} +13.4193i q^{38} +0.193576 q^{41} -5.67307i q^{43} +1.62222 q^{44} +16.8573 q^{46} +2.75557i q^{47} -12.6128 q^{49} -1.00937i q^{52} -10.8573i q^{53} -3.18421 q^{56} +14.8573i q^{58} -4.85728 q^{59} +6.85728 q^{61} -5.24443i q^{62} +4.74620 q^{64} -1.24443i q^{67} -8.40990i q^{68} -2.75557 q^{71} -4.23506i q^{73} -3.80642 q^{74} +11.4380 q^{76} +4.42864i q^{77} -8.56199 q^{79} -0.368416i q^{82} +0.133353i q^{83} -10.7971 q^{86} +0.719004i q^{88} +5.61285 q^{89} +2.75557 q^{91} -14.3684i q^{92} +5.24443 q^{94} +7.24443i q^{97} +24.0049i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 10 q^{4} + O(q^{10}) \) \( 6 q - 10 q^{4} - 6 q^{11} - 24 q^{14} + 26 q^{16} - 16 q^{19} - 20 q^{26} - 20 q^{29} + 16 q^{31} + 60 q^{34} + 28 q^{41} + 10 q^{44} + 48 q^{46} - 22 q^{49} + 8 q^{56} + 24 q^{59} - 12 q^{61} - 26 q^{64} - 16 q^{71} + 4 q^{74} + 96 q^{76} - 24 q^{79} + 16 q^{86} - 20 q^{89} + 16 q^{91} + 32 q^{94} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2475\mathbb{Z}\right)^\times\).

\(n\) \(551\) \(2026\) \(2377\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 1.90321i − 1.34577i −0.739745 0.672887i \(-0.765054\pi\)
0.739745 0.672887i \(-0.234946\pi\)
\(3\) 0 0
\(4\) −1.62222 −0.811108
\(5\) 0 0
\(6\) 0 0
\(7\) − 4.42864i − 1.67387i −0.547304 0.836934i \(-0.684346\pi\)
0.547304 0.836934i \(-0.315654\pi\)
\(8\) − 0.719004i − 0.254206i
\(9\) 0 0
\(10\) 0 0
\(11\) −1.00000 −0.301511
\(12\) 0 0
\(13\) 0.622216i 0.172572i 0.996270 + 0.0862858i \(0.0274998\pi\)
−0.996270 + 0.0862858i \(0.972500\pi\)
\(14\) −8.42864 −2.25265
\(15\) 0 0
\(16\) −4.61285 −1.15321
\(17\) 5.18421i 1.25736i 0.777666 + 0.628678i \(0.216403\pi\)
−0.777666 + 0.628678i \(0.783597\pi\)
\(18\) 0 0
\(19\) −7.05086 −1.61758 −0.808789 0.588100i \(-0.799876\pi\)
−0.808789 + 0.588100i \(0.799876\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 1.90321i 0.405766i
\(23\) 8.85728i 1.84687i 0.383754 + 0.923435i \(0.374631\pi\)
−0.383754 + 0.923435i \(0.625369\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 1.18421 0.232242
\(27\) 0 0
\(28\) 7.18421i 1.35769i
\(29\) −7.80642 −1.44962 −0.724808 0.688951i \(-0.758072\pi\)
−0.724808 + 0.688951i \(0.758072\pi\)
\(30\) 0 0
\(31\) 2.75557 0.494915 0.247457 0.968899i \(-0.420405\pi\)
0.247457 + 0.968899i \(0.420405\pi\)
\(32\) 7.34122i 1.29776i
\(33\) 0 0
\(34\) 9.86665 1.69212
\(35\) 0 0
\(36\) 0 0
\(37\) − 2.00000i − 0.328798i −0.986394 0.164399i \(-0.947432\pi\)
0.986394 0.164399i \(-0.0525685\pi\)
\(38\) 13.4193i 2.17689i
\(39\) 0 0
\(40\) 0 0
\(41\) 0.193576 0.0302315 0.0151158 0.999886i \(-0.495188\pi\)
0.0151158 + 0.999886i \(0.495188\pi\)
\(42\) 0 0
\(43\) − 5.67307i − 0.865135i −0.901602 0.432568i \(-0.857608\pi\)
0.901602 0.432568i \(-0.142392\pi\)
\(44\) 1.62222 0.244558
\(45\) 0 0
\(46\) 16.8573 2.48547
\(47\) 2.75557i 0.401941i 0.979597 + 0.200971i \(0.0644095\pi\)
−0.979597 + 0.200971i \(0.935590\pi\)
\(48\) 0 0
\(49\) −12.6128 −1.80184
\(50\) 0 0
\(51\) 0 0
\(52\) − 1.00937i − 0.139974i
\(53\) − 10.8573i − 1.49136i −0.666303 0.745681i \(-0.732124\pi\)
0.666303 0.745681i \(-0.267876\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −3.18421 −0.425508
\(57\) 0 0
\(58\) 14.8573i 1.95086i
\(59\) −4.85728 −0.632364 −0.316182 0.948699i \(-0.602401\pi\)
−0.316182 + 0.948699i \(0.602401\pi\)
\(60\) 0 0
\(61\) 6.85728 0.877985 0.438992 0.898491i \(-0.355336\pi\)
0.438992 + 0.898491i \(0.355336\pi\)
\(62\) − 5.24443i − 0.666043i
\(63\) 0 0
\(64\) 4.74620 0.593275
\(65\) 0 0
\(66\) 0 0
\(67\) − 1.24443i − 0.152031i −0.997107 0.0760157i \(-0.975780\pi\)
0.997107 0.0760157i \(-0.0242199\pi\)
\(68\) − 8.40990i − 1.01985i
\(69\) 0 0
\(70\) 0 0
\(71\) −2.75557 −0.327026 −0.163513 0.986541i \(-0.552283\pi\)
−0.163513 + 0.986541i \(0.552283\pi\)
\(72\) 0 0
\(73\) − 4.23506i − 0.495677i −0.968801 0.247838i \(-0.920280\pi\)
0.968801 0.247838i \(-0.0797202\pi\)
\(74\) −3.80642 −0.442488
\(75\) 0 0
\(76\) 11.4380 1.31203
\(77\) 4.42864i 0.504690i
\(78\) 0 0
\(79\) −8.56199 −0.963299 −0.481650 0.876364i \(-0.659962\pi\)
−0.481650 + 0.876364i \(0.659962\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) − 0.368416i − 0.0406848i
\(83\) 0.133353i 0.0146374i 0.999973 + 0.00731870i \(0.00232964\pi\)
−0.999973 + 0.00731870i \(0.997670\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −10.7971 −1.16428
\(87\) 0 0
\(88\) 0.719004i 0.0766461i
\(89\) 5.61285 0.594961 0.297480 0.954728i \(-0.403854\pi\)
0.297480 + 0.954728i \(0.403854\pi\)
\(90\) 0 0
\(91\) 2.75557 0.288862
\(92\) − 14.3684i − 1.49801i
\(93\) 0 0
\(94\) 5.24443 0.540922
\(95\) 0 0
\(96\) 0 0
\(97\) 7.24443i 0.735561i 0.929913 + 0.367780i \(0.119882\pi\)
−0.929913 + 0.367780i \(0.880118\pi\)
\(98\) 24.0049i 2.42486i
\(99\) 0 0
\(100\) 0 0
\(101\) −4.66370 −0.464056 −0.232028 0.972709i \(-0.574536\pi\)
−0.232028 + 0.972709i \(0.574536\pi\)
\(102\) 0 0
\(103\) 11.6128i 1.14425i 0.820167 + 0.572124i \(0.193880\pi\)
−0.820167 + 0.572124i \(0.806120\pi\)
\(104\) 0.447375 0.0438688
\(105\) 0 0
\(106\) −20.6637 −2.00704
\(107\) − 2.62222i − 0.253499i −0.991935 0.126750i \(-0.959546\pi\)
0.991935 0.126750i \(-0.0404545\pi\)
\(108\) 0 0
\(109\) 19.7146 1.88831 0.944156 0.329499i \(-0.106880\pi\)
0.944156 + 0.329499i \(0.106880\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 20.4286i 1.93032i
\(113\) − 6.00000i − 0.564433i −0.959351 0.282216i \(-0.908930\pi\)
0.959351 0.282216i \(-0.0910696\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 12.6637 1.17580
\(117\) 0 0
\(118\) 9.24443i 0.851019i
\(119\) 22.9590 2.10465
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) − 13.0509i − 1.18157i
\(123\) 0 0
\(124\) −4.47013 −0.401429
\(125\) 0 0
\(126\) 0 0
\(127\) − 15.1842i − 1.34738i −0.739014 0.673690i \(-0.764708\pi\)
0.739014 0.673690i \(-0.235292\pi\)
\(128\) 5.64941i 0.499342i
\(129\) 0 0
\(130\) 0 0
\(131\) −1.24443 −0.108726 −0.0543632 0.998521i \(-0.517313\pi\)
−0.0543632 + 0.998521i \(0.517313\pi\)
\(132\) 0 0
\(133\) 31.2257i 2.70761i
\(134\) −2.36842 −0.204600
\(135\) 0 0
\(136\) 3.72746 0.319627
\(137\) 0.488863i 0.0417663i 0.999782 + 0.0208832i \(0.00664780\pi\)
−0.999782 + 0.0208832i \(0.993352\pi\)
\(138\) 0 0
\(139\) −17.8064 −1.51032 −0.755161 0.655540i \(-0.772441\pi\)
−0.755161 + 0.655540i \(0.772441\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 5.24443i 0.440103i
\(143\) − 0.622216i − 0.0520323i
\(144\) 0 0
\(145\) 0 0
\(146\) −8.06022 −0.667069
\(147\) 0 0
\(148\) 3.24443i 0.266691i
\(149\) −1.43801 −0.117806 −0.0589031 0.998264i \(-0.518760\pi\)
−0.0589031 + 0.998264i \(0.518760\pi\)
\(150\) 0 0
\(151\) −12.1748 −0.990774 −0.495387 0.868672i \(-0.664974\pi\)
−0.495387 + 0.868672i \(0.664974\pi\)
\(152\) 5.06959i 0.411198i
\(153\) 0 0
\(154\) 8.42864 0.679199
\(155\) 0 0
\(156\) 0 0
\(157\) 18.4701i 1.47408i 0.675851 + 0.737038i \(0.263776\pi\)
−0.675851 + 0.737038i \(0.736224\pi\)
\(158\) 16.2953i 1.29638i
\(159\) 0 0
\(160\) 0 0
\(161\) 39.2257 3.09142
\(162\) 0 0
\(163\) 10.1017i 0.791227i 0.918417 + 0.395614i \(0.129468\pi\)
−0.918417 + 0.395614i \(0.870532\pi\)
\(164\) −0.314022 −0.0245210
\(165\) 0 0
\(166\) 0.253799 0.0196986
\(167\) 16.3368i 1.26418i 0.774896 + 0.632089i \(0.217802\pi\)
−0.774896 + 0.632089i \(0.782198\pi\)
\(168\) 0 0
\(169\) 12.6128 0.970219
\(170\) 0 0
\(171\) 0 0
\(172\) 9.20294i 0.701718i
\(173\) − 9.18421i − 0.698262i −0.937074 0.349131i \(-0.886477\pi\)
0.937074 0.349131i \(-0.113523\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 4.61285 0.347706
\(177\) 0 0
\(178\) − 10.6824i − 0.800683i
\(179\) −25.3274 −1.89306 −0.946530 0.322617i \(-0.895437\pi\)
−0.946530 + 0.322617i \(0.895437\pi\)
\(180\) 0 0
\(181\) −13.6128 −1.01184 −0.505918 0.862582i \(-0.668846\pi\)
−0.505918 + 0.862582i \(0.668846\pi\)
\(182\) − 5.24443i − 0.388743i
\(183\) 0 0
\(184\) 6.36842 0.469486
\(185\) 0 0
\(186\) 0 0
\(187\) − 5.18421i − 0.379107i
\(188\) − 4.47013i − 0.326017i
\(189\) 0 0
\(190\) 0 0
\(191\) −6.10171 −0.441504 −0.220752 0.975330i \(-0.570851\pi\)
−0.220752 + 0.975330i \(0.570851\pi\)
\(192\) 0 0
\(193\) − 18.3368i − 1.31991i −0.751305 0.659955i \(-0.770575\pi\)
0.751305 0.659955i \(-0.229425\pi\)
\(194\) 13.7877 0.989898
\(195\) 0 0
\(196\) 20.4608 1.46148
\(197\) 6.69535i 0.477024i 0.971140 + 0.238512i \(0.0766596\pi\)
−0.971140 + 0.238512i \(0.923340\pi\)
\(198\) 0 0
\(199\) −14.1017 −0.999644 −0.499822 0.866128i \(-0.666601\pi\)
−0.499822 + 0.866128i \(0.666601\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 8.87601i 0.624514i
\(203\) 34.5718i 2.42647i
\(204\) 0 0
\(205\) 0 0
\(206\) 22.1017 1.53990
\(207\) 0 0
\(208\) − 2.87019i − 0.199012i
\(209\) 7.05086 0.487718
\(210\) 0 0
\(211\) −10.6637 −0.734120 −0.367060 0.930197i \(-0.619636\pi\)
−0.367060 + 0.930197i \(0.619636\pi\)
\(212\) 17.6128i 1.20966i
\(213\) 0 0
\(214\) −4.99063 −0.341153
\(215\) 0 0
\(216\) 0 0
\(217\) − 12.2034i − 0.828422i
\(218\) − 37.5210i − 2.54124i
\(219\) 0 0
\(220\) 0 0
\(221\) −3.22570 −0.216984
\(222\) 0 0
\(223\) − 8.85728i − 0.593127i −0.955013 0.296564i \(-0.904159\pi\)
0.955013 0.296564i \(-0.0958407\pi\)
\(224\) 32.5116 2.17227
\(225\) 0 0
\(226\) −11.4193 −0.759599
\(227\) − 13.3778i − 0.887915i −0.896048 0.443957i \(-0.853574\pi\)
0.896048 0.443957i \(-0.146426\pi\)
\(228\) 0 0
\(229\) −11.5111 −0.760677 −0.380339 0.924847i \(-0.624193\pi\)
−0.380339 + 0.924847i \(0.624193\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 5.61285i 0.368502i
\(233\) − 4.32693i − 0.283467i −0.989905 0.141733i \(-0.954732\pi\)
0.989905 0.141733i \(-0.0452675\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 7.87955 0.512915
\(237\) 0 0
\(238\) − 43.6958i − 2.83238i
\(239\) −3.34614 −0.216444 −0.108222 0.994127i \(-0.534516\pi\)
−0.108222 + 0.994127i \(0.534516\pi\)
\(240\) 0 0
\(241\) −1.34614 −0.0867126 −0.0433563 0.999060i \(-0.513805\pi\)
−0.0433563 + 0.999060i \(0.513805\pi\)
\(242\) − 1.90321i − 0.122343i
\(243\) 0 0
\(244\) −11.1240 −0.712140
\(245\) 0 0
\(246\) 0 0
\(247\) − 4.38715i − 0.279148i
\(248\) − 1.98126i − 0.125810i
\(249\) 0 0
\(250\) 0 0
\(251\) −22.7556 −1.43632 −0.718159 0.695879i \(-0.755015\pi\)
−0.718159 + 0.695879i \(0.755015\pi\)
\(252\) 0 0
\(253\) − 8.85728i − 0.556852i
\(254\) −28.8988 −1.81327
\(255\) 0 0
\(256\) 20.2444 1.26528
\(257\) 6.85728i 0.427745i 0.976862 + 0.213873i \(0.0686078\pi\)
−0.976862 + 0.213873i \(0.931392\pi\)
\(258\) 0 0
\(259\) −8.85728 −0.550365
\(260\) 0 0
\(261\) 0 0
\(262\) 2.36842i 0.146321i
\(263\) − 29.5812i − 1.82406i −0.410129 0.912028i \(-0.634516\pi\)
0.410129 0.912028i \(-0.365484\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 59.4291 3.64383
\(267\) 0 0
\(268\) 2.01874i 0.123314i
\(269\) 8.48886 0.517575 0.258788 0.965934i \(-0.416677\pi\)
0.258788 + 0.965934i \(0.416677\pi\)
\(270\) 0 0
\(271\) 14.6637 0.890757 0.445378 0.895343i \(-0.353069\pi\)
0.445378 + 0.895343i \(0.353069\pi\)
\(272\) − 23.9140i − 1.45000i
\(273\) 0 0
\(274\) 0.930409 0.0562081
\(275\) 0 0
\(276\) 0 0
\(277\) 14.6035i 0.877438i 0.898624 + 0.438719i \(0.144568\pi\)
−0.898624 + 0.438719i \(0.855432\pi\)
\(278\) 33.8894i 2.03255i
\(279\) 0 0
\(280\) 0 0
\(281\) 0.193576 0.0115478 0.00577389 0.999983i \(-0.498162\pi\)
0.00577389 + 0.999983i \(0.498162\pi\)
\(282\) 0 0
\(283\) − 27.1842i − 1.61593i −0.589228 0.807967i \(-0.700568\pi\)
0.589228 0.807967i \(-0.299432\pi\)
\(284\) 4.47013 0.265253
\(285\) 0 0
\(286\) −1.18421 −0.0700237
\(287\) − 0.857279i − 0.0506036i
\(288\) 0 0
\(289\) −9.87601 −0.580942
\(290\) 0 0
\(291\) 0 0
\(292\) 6.87019i 0.402047i
\(293\) − 2.81579i − 0.164500i −0.996612 0.0822502i \(-0.973789\pi\)
0.996612 0.0822502i \(-0.0262106\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) −1.43801 −0.0835825
\(297\) 0 0
\(298\) 2.73683i 0.158540i
\(299\) −5.51114 −0.318717
\(300\) 0 0
\(301\) −25.1240 −1.44812
\(302\) 23.1713i 1.33336i
\(303\) 0 0
\(304\) 32.5245 1.86541
\(305\) 0 0
\(306\) 0 0
\(307\) − 24.4286i − 1.39422i −0.716966 0.697108i \(-0.754470\pi\)
0.716966 0.697108i \(-0.245530\pi\)
\(308\) − 7.18421i − 0.409358i
\(309\) 0 0
\(310\) 0 0
\(311\) −19.8796 −1.12727 −0.563633 0.826025i \(-0.690597\pi\)
−0.563633 + 0.826025i \(0.690597\pi\)
\(312\) 0 0
\(313\) 15.7146i 0.888239i 0.895967 + 0.444120i \(0.146483\pi\)
−0.895967 + 0.444120i \(0.853517\pi\)
\(314\) 35.1526 1.98377
\(315\) 0 0
\(316\) 13.8894 0.781340
\(317\) − 16.4889i − 0.926107i −0.886330 0.463053i \(-0.846754\pi\)
0.886330 0.463053i \(-0.153246\pi\)
\(318\) 0 0
\(319\) 7.80642 0.437076
\(320\) 0 0
\(321\) 0 0
\(322\) − 74.6548i − 4.16035i
\(323\) − 36.5531i − 2.03387i
\(324\) 0 0
\(325\) 0 0
\(326\) 19.2257 1.06481
\(327\) 0 0
\(328\) − 0.139182i − 0.00768504i
\(329\) 12.2034 0.672796
\(330\) 0 0
\(331\) 15.3461 0.843500 0.421750 0.906712i \(-0.361416\pi\)
0.421750 + 0.906712i \(0.361416\pi\)
\(332\) − 0.216327i − 0.0118725i
\(333\) 0 0
\(334\) 31.0923 1.70130
\(335\) 0 0
\(336\) 0 0
\(337\) 28.2351i 1.53806i 0.639212 + 0.769031i \(0.279261\pi\)
−0.639212 + 0.769031i \(0.720739\pi\)
\(338\) − 24.0049i − 1.30570i
\(339\) 0 0
\(340\) 0 0
\(341\) −2.75557 −0.149222
\(342\) 0 0
\(343\) 24.8573i 1.34217i
\(344\) −4.07896 −0.219923
\(345\) 0 0
\(346\) −17.4795 −0.939703
\(347\) − 2.62222i − 0.140768i −0.997520 0.0703840i \(-0.977578\pi\)
0.997520 0.0703840i \(-0.0224224\pi\)
\(348\) 0 0
\(349\) −5.14272 −0.275284 −0.137642 0.990482i \(-0.543952\pi\)
−0.137642 + 0.990482i \(0.543952\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) − 7.34122i − 0.391288i
\(353\) − 9.34614i − 0.497445i −0.968575 0.248722i \(-0.919989\pi\)
0.968575 0.248722i \(-0.0800107\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) −9.10525 −0.482577
\(357\) 0 0
\(358\) 48.2034i 2.54763i
\(359\) 10.7556 0.567657 0.283829 0.958875i \(-0.408395\pi\)
0.283829 + 0.958875i \(0.408395\pi\)
\(360\) 0 0
\(361\) 30.7146 1.61656
\(362\) 25.9081i 1.36170i
\(363\) 0 0
\(364\) −4.47013 −0.234298
\(365\) 0 0
\(366\) 0 0
\(367\) 33.7975i 1.76422i 0.471046 + 0.882108i \(0.343876\pi\)
−0.471046 + 0.882108i \(0.656124\pi\)
\(368\) − 40.8573i − 2.12983i
\(369\) 0 0
\(370\) 0 0
\(371\) −48.0830 −2.49634
\(372\) 0 0
\(373\) − 33.9496i − 1.75784i −0.476965 0.878922i \(-0.658263\pi\)
0.476965 0.878922i \(-0.341737\pi\)
\(374\) −9.86665 −0.510192
\(375\) 0 0
\(376\) 1.98126 0.102176
\(377\) − 4.85728i − 0.250163i
\(378\) 0 0
\(379\) 20.0000 1.02733 0.513665 0.857991i \(-0.328287\pi\)
0.513665 + 0.857991i \(0.328287\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 11.6128i 0.594165i
\(383\) − 14.6351i − 0.747820i −0.927465 0.373910i \(-0.878017\pi\)
0.927465 0.373910i \(-0.121983\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −34.8988 −1.77630
\(387\) 0 0
\(388\) − 11.7520i − 0.596619i
\(389\) −5.61285 −0.284583 −0.142291 0.989825i \(-0.545447\pi\)
−0.142291 + 0.989825i \(0.545447\pi\)
\(390\) 0 0
\(391\) −45.9180 −2.32217
\(392\) 9.06868i 0.458038i
\(393\) 0 0
\(394\) 12.7427 0.641966
\(395\) 0 0
\(396\) 0 0
\(397\) − 12.7556i − 0.640184i −0.947387 0.320092i \(-0.896286\pi\)
0.947387 0.320092i \(-0.103714\pi\)
\(398\) 26.8385i 1.34529i
\(399\) 0 0
\(400\) 0 0
\(401\) −2.00000 −0.0998752 −0.0499376 0.998752i \(-0.515902\pi\)
−0.0499376 + 0.998752i \(0.515902\pi\)
\(402\) 0 0
\(403\) 1.71456i 0.0854082i
\(404\) 7.56553 0.376399
\(405\) 0 0
\(406\) 65.7975 3.26548
\(407\) 2.00000i 0.0991363i
\(408\) 0 0
\(409\) 7.12399 0.352258 0.176129 0.984367i \(-0.443642\pi\)
0.176129 + 0.984367i \(0.443642\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) − 18.8385i − 0.928108i
\(413\) 21.5111i 1.05849i
\(414\) 0 0
\(415\) 0 0
\(416\) −4.56782 −0.223956
\(417\) 0 0
\(418\) − 13.4193i − 0.656358i
\(419\) 15.6128 0.762738 0.381369 0.924423i \(-0.375453\pi\)
0.381369 + 0.924423i \(0.375453\pi\)
\(420\) 0 0
\(421\) 7.89829 0.384939 0.192470 0.981303i \(-0.438350\pi\)
0.192470 + 0.981303i \(0.438350\pi\)
\(422\) 20.2953i 0.987959i
\(423\) 0 0
\(424\) −7.80642 −0.379113
\(425\) 0 0
\(426\) 0 0
\(427\) − 30.3684i − 1.46963i
\(428\) 4.25380i 0.205615i
\(429\) 0 0
\(430\) 0 0
\(431\) −34.3051 −1.65242 −0.826210 0.563362i \(-0.809508\pi\)
−0.826210 + 0.563362i \(0.809508\pi\)
\(432\) 0 0
\(433\) − 14.4701i − 0.695390i −0.937608 0.347695i \(-0.886964\pi\)
0.937608 0.347695i \(-0.113036\pi\)
\(434\) −23.2257 −1.11487
\(435\) 0 0
\(436\) −31.9813 −1.53162
\(437\) − 62.4514i − 2.98746i
\(438\) 0 0
\(439\) 19.3176 0.921977 0.460988 0.887406i \(-0.347495\pi\)
0.460988 + 0.887406i \(0.347495\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 6.13918i 0.292011i
\(443\) − 13.1240i − 0.623539i −0.950158 0.311770i \(-0.899078\pi\)
0.950158 0.311770i \(-0.100922\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) −16.8573 −0.798215
\(447\) 0 0
\(448\) − 21.0192i − 0.993064i
\(449\) −32.3051 −1.52457 −0.762287 0.647240i \(-0.775923\pi\)
−0.762287 + 0.647240i \(0.775923\pi\)
\(450\) 0 0
\(451\) −0.193576 −0.00911514
\(452\) 9.73329i 0.457816i
\(453\) 0 0
\(454\) −25.4608 −1.19493
\(455\) 0 0
\(456\) 0 0
\(457\) − 23.4608i − 1.09745i −0.836004 0.548724i \(-0.815114\pi\)
0.836004 0.548724i \(-0.184886\pi\)
\(458\) 21.9081i 1.02370i
\(459\) 0 0
\(460\) 0 0
\(461\) 28.8671 1.34448 0.672238 0.740335i \(-0.265333\pi\)
0.672238 + 0.740335i \(0.265333\pi\)
\(462\) 0 0
\(463\) 19.3461i 0.899091i 0.893257 + 0.449546i \(0.148414\pi\)
−0.893257 + 0.449546i \(0.851586\pi\)
\(464\) 36.0098 1.67172
\(465\) 0 0
\(466\) −8.23506 −0.381482
\(467\) − 3.14272i − 0.145428i −0.997353 0.0727139i \(-0.976834\pi\)
0.997353 0.0727139i \(-0.0231660\pi\)
\(468\) 0 0
\(469\) −5.51114 −0.254481
\(470\) 0 0
\(471\) 0 0
\(472\) 3.49240i 0.160751i
\(473\) 5.67307i 0.260848i
\(474\) 0 0
\(475\) 0 0
\(476\) −37.2444 −1.70710
\(477\) 0 0
\(478\) 6.36842i 0.291285i
\(479\) −24.8573 −1.13576 −0.567879 0.823112i \(-0.692236\pi\)
−0.567879 + 0.823112i \(0.692236\pi\)
\(480\) 0 0
\(481\) 1.24443 0.0567412
\(482\) 2.56199i 0.116696i
\(483\) 0 0
\(484\) −1.62222 −0.0737371
\(485\) 0 0
\(486\) 0 0
\(487\) − 11.5299i − 0.522468i −0.965275 0.261234i \(-0.915871\pi\)
0.965275 0.261234i \(-0.0841295\pi\)
\(488\) − 4.93041i − 0.223189i
\(489\) 0 0
\(490\) 0 0
\(491\) −16.3872 −0.739542 −0.369771 0.929123i \(-0.620564\pi\)
−0.369771 + 0.929123i \(0.620564\pi\)
\(492\) 0 0
\(493\) − 40.4701i − 1.82268i
\(494\) −8.34968 −0.375670
\(495\) 0 0
\(496\) −12.7110 −0.570742
\(497\) 12.2034i 0.547398i
\(498\) 0 0
\(499\) 25.3274 1.13381 0.566905 0.823783i \(-0.308141\pi\)
0.566905 + 0.823783i \(0.308141\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 43.3087i 1.93296i
\(503\) 19.0923i 0.851285i 0.904891 + 0.425643i \(0.139952\pi\)
−0.904891 + 0.425643i \(0.860048\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) −16.8573 −0.749397
\(507\) 0 0
\(508\) 24.6321i 1.09287i
\(509\) −32.4514 −1.43838 −0.719191 0.694812i \(-0.755488\pi\)
−0.719191 + 0.694812i \(0.755488\pi\)
\(510\) 0 0
\(511\) −18.7556 −0.829698
\(512\) − 27.2306i − 1.20343i
\(513\) 0 0
\(514\) 13.0509 0.575649
\(515\) 0 0
\(516\) 0 0
\(517\) − 2.75557i − 0.121190i
\(518\) 16.8573i 0.740666i
\(519\) 0 0
\(520\) 0 0
\(521\) 29.2257 1.28040 0.640200 0.768208i \(-0.278851\pi\)
0.640200 + 0.768208i \(0.278851\pi\)
\(522\) 0 0
\(523\) − 6.71408i − 0.293586i −0.989167 0.146793i \(-0.953105\pi\)
0.989167 0.146793i \(-0.0468952\pi\)
\(524\) 2.01874 0.0881889
\(525\) 0 0
\(526\) −56.2993 −2.45477
\(527\) 14.2854i 0.622284i
\(528\) 0 0
\(529\) −55.4514 −2.41093
\(530\) 0 0
\(531\) 0 0
\(532\) − 50.6548i − 2.19616i
\(533\) 0.120446i 0.00521710i
\(534\) 0 0
\(535\) 0 0
\(536\) −0.894751 −0.0386473
\(537\) 0 0
\(538\) − 16.1561i − 0.696539i
\(539\) 12.6128 0.543274
\(540\) 0 0
\(541\) −18.0000 −0.773880 −0.386940 0.922105i \(-0.626468\pi\)
−0.386940 + 0.922105i \(0.626468\pi\)
\(542\) − 27.9081i − 1.19876i
\(543\) 0 0
\(544\) −38.0584 −1.63174
\(545\) 0 0
\(546\) 0 0
\(547\) 41.3689i 1.76881i 0.466724 + 0.884403i \(0.345434\pi\)
−0.466724 + 0.884403i \(0.654566\pi\)
\(548\) − 0.793040i − 0.0338770i
\(549\) 0 0
\(550\) 0 0
\(551\) 55.0420 2.34487
\(552\) 0 0
\(553\) 37.9180i 1.61244i
\(554\) 27.7935 1.18083
\(555\) 0 0
\(556\) 28.8859 1.22503
\(557\) 20.7971i 0.881200i 0.897704 + 0.440600i \(0.145234\pi\)
−0.897704 + 0.440600i \(0.854766\pi\)
\(558\) 0 0
\(559\) 3.52987 0.149298
\(560\) 0 0
\(561\) 0 0
\(562\) − 0.368416i − 0.0155407i
\(563\) 37.7275i 1.59002i 0.606594 + 0.795012i \(0.292535\pi\)
−0.606594 + 0.795012i \(0.707465\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) −51.7373 −2.17468
\(567\) 0 0
\(568\) 1.98126i 0.0831320i
\(569\) −7.33630 −0.307554 −0.153777 0.988106i \(-0.549144\pi\)
−0.153777 + 0.988106i \(0.549144\pi\)
\(570\) 0 0
\(571\) 36.6450 1.53354 0.766772 0.641919i \(-0.221862\pi\)
0.766772 + 0.641919i \(0.221862\pi\)
\(572\) 1.00937i 0.0422038i
\(573\) 0 0
\(574\) −1.63158 −0.0681010
\(575\) 0 0
\(576\) 0 0
\(577\) 4.22216i 0.175771i 0.996131 + 0.0878853i \(0.0280109\pi\)
−0.996131 + 0.0878853i \(0.971989\pi\)
\(578\) 18.7961i 0.781817i
\(579\) 0 0
\(580\) 0 0
\(581\) 0.590573 0.0245011
\(582\) 0 0
\(583\) 10.8573i 0.449663i
\(584\) −3.04503 −0.126004
\(585\) 0 0
\(586\) −5.35905 −0.221380
\(587\) − 34.3684i − 1.41854i −0.704939 0.709268i \(-0.749026\pi\)
0.704939 0.709268i \(-0.250974\pi\)
\(588\) 0 0
\(589\) −19.4291 −0.800563
\(590\) 0 0
\(591\) 0 0
\(592\) 9.22570i 0.379174i
\(593\) 27.9398i 1.14735i 0.819083 + 0.573675i \(0.194483\pi\)
−0.819083 + 0.573675i \(0.805517\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 2.33276 0.0955535
\(597\) 0 0
\(598\) 10.4889i 0.428921i
\(599\) −31.2257 −1.27585 −0.637924 0.770100i \(-0.720206\pi\)
−0.637924 + 0.770100i \(0.720206\pi\)
\(600\) 0 0
\(601\) −8.75557 −0.357147 −0.178574 0.983927i \(-0.557148\pi\)
−0.178574 + 0.983927i \(0.557148\pi\)
\(602\) 47.8163i 1.94885i
\(603\) 0 0
\(604\) 19.7502 0.803625
\(605\) 0 0
\(606\) 0 0
\(607\) − 15.1842i − 0.616308i −0.951336 0.308154i \(-0.900289\pi\)
0.951336 0.308154i \(-0.0997112\pi\)
\(608\) − 51.7619i − 2.09922i
\(609\) 0 0
\(610\) 0 0
\(611\) −1.71456 −0.0693636
\(612\) 0 0
\(613\) − 42.7239i − 1.72560i −0.505543 0.862802i \(-0.668708\pi\)
0.505543 0.862802i \(-0.331292\pi\)
\(614\) −46.4929 −1.87630
\(615\) 0 0
\(616\) 3.18421 0.128295
\(617\) 3.51114i 0.141353i 0.997499 + 0.0706765i \(0.0225158\pi\)
−0.997499 + 0.0706765i \(0.977484\pi\)
\(618\) 0 0
\(619\) −17.5941 −0.707167 −0.353584 0.935403i \(-0.615037\pi\)
−0.353584 + 0.935403i \(0.615037\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 37.8350i 1.51705i
\(623\) − 24.8573i − 0.995886i
\(624\) 0 0
\(625\) 0 0
\(626\) 29.9081 1.19537
\(627\) 0 0
\(628\) − 29.9625i − 1.19564i
\(629\) 10.3684 0.413416
\(630\) 0 0
\(631\) 15.8163 0.629636 0.314818 0.949152i \(-0.398057\pi\)
0.314818 + 0.949152i \(0.398057\pi\)
\(632\) 6.15610i 0.244877i
\(633\) 0 0
\(634\) −31.3818 −1.24633
\(635\) 0 0
\(636\) 0 0
\(637\) − 7.84791i − 0.310946i
\(638\) − 14.8573i − 0.588205i
\(639\) 0 0
\(640\) 0 0
\(641\) −25.8163 −1.01968 −0.509841 0.860269i \(-0.670296\pi\)
−0.509841 + 0.860269i \(0.670296\pi\)
\(642\) 0 0
\(643\) − 18.1017i − 0.713862i −0.934131 0.356931i \(-0.883823\pi\)
0.934131 0.356931i \(-0.116177\pi\)
\(644\) −63.6325 −2.50747
\(645\) 0 0
\(646\) −69.5683 −2.73713
\(647\) 47.0420i 1.84941i 0.380684 + 0.924705i \(0.375689\pi\)
−0.380684 + 0.924705i \(0.624311\pi\)
\(648\) 0 0
\(649\) 4.85728 0.190665
\(650\) 0 0
\(651\) 0 0
\(652\) − 16.3872i − 0.641770i
\(653\) 30.0830i 1.17724i 0.808411 + 0.588619i \(0.200328\pi\)
−0.808411 + 0.588619i \(0.799672\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) −0.892937 −0.0348633
\(657\) 0 0
\(658\) − 23.2257i − 0.905432i
\(659\) −10.2854 −0.400664 −0.200332 0.979728i \(-0.564202\pi\)
−0.200332 + 0.979728i \(0.564202\pi\)
\(660\) 0 0
\(661\) −27.7146 −1.07797 −0.538986 0.842315i \(-0.681192\pi\)
−0.538986 + 0.842315i \(0.681192\pi\)
\(662\) − 29.2070i − 1.13516i
\(663\) 0 0
\(664\) 0.0958814 0.00372092
\(665\) 0 0
\(666\) 0 0
\(667\) − 69.1437i − 2.67725i
\(668\) − 26.5018i − 1.02538i
\(669\) 0 0
\(670\) 0 0
\(671\) −6.85728 −0.264722
\(672\) 0 0
\(673\) 9.86665i 0.380331i 0.981752 + 0.190166i \(0.0609025\pi\)
−0.981752 + 0.190166i \(0.939097\pi\)
\(674\) 53.7373 2.06988
\(675\) 0 0
\(676\) −20.4608 −0.786952
\(677\) 5.65433i 0.217314i 0.994079 + 0.108657i \(0.0346550\pi\)
−0.994079 + 0.108657i \(0.965345\pi\)
\(678\) 0 0
\(679\) 32.0830 1.23123
\(680\) 0 0
\(681\) 0 0
\(682\) 5.24443i 0.200820i
\(683\) 34.1847i 1.30804i 0.756477 + 0.654020i \(0.226919\pi\)
−0.756477 + 0.654020i \(0.773081\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 47.3087 1.80625
\(687\) 0 0
\(688\) 26.1690i 0.997684i
\(689\) 6.75557 0.257367
\(690\) 0 0
\(691\) −19.2257 −0.731380 −0.365690 0.930737i \(-0.619167\pi\)
−0.365690 + 0.930737i \(0.619167\pi\)
\(692\) 14.8988i 0.566366i
\(693\) 0 0
\(694\) −4.99063 −0.189442
\(695\) 0 0
\(696\) 0 0
\(697\) 1.00354i 0.0380118i
\(698\) 9.78769i 0.370469i
\(699\) 0 0
\(700\) 0 0
\(701\) 29.9081 1.12961 0.564807 0.825223i \(-0.308950\pi\)
0.564807 + 0.825223i \(0.308950\pi\)
\(702\) 0 0
\(703\) 14.1017i 0.531856i
\(704\) −4.74620 −0.178879
\(705\) 0 0
\(706\) −17.7877 −0.669448
\(707\) 20.6539i 0.776768i
\(708\) 0 0
\(709\) 15.3274 0.575633 0.287816 0.957686i \(-0.407071\pi\)
0.287816 + 0.957686i \(0.407071\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) − 4.03566i − 0.151243i
\(713\) 24.4068i 0.914043i
\(714\) 0 0
\(715\) 0 0
\(716\) 41.0865 1.53548
\(717\) 0 0
\(718\) − 20.4701i − 0.763938i
\(719\) 23.8163 0.888197 0.444098 0.895978i \(-0.353524\pi\)
0.444098 + 0.895978i \(0.353524\pi\)
\(720\) 0 0
\(721\) 51.4291 1.91532
\(722\) − 58.4563i − 2.17552i
\(723\) 0 0
\(724\) 22.0830 0.820707
\(725\) 0 0
\(726\) 0 0
\(727\) − 32.9403i − 1.22169i −0.791752 0.610843i \(-0.790831\pi\)
0.791752 0.610843i \(-0.209169\pi\)
\(728\) − 1.98126i − 0.0734305i
\(729\) 0 0
\(730\) 0 0
\(731\) 29.4104 1.08778
\(732\) 0 0
\(733\) 29.8666i 1.10315i 0.834125 + 0.551575i \(0.185973\pi\)
−0.834125 + 0.551575i \(0.814027\pi\)
\(734\) 64.3239 2.37424
\(735\) 0 0
\(736\) −65.0232 −2.39679
\(737\) 1.24443i 0.0458392i
\(738\) 0 0
\(739\) 5.06959 0.186488 0.0932440 0.995643i \(-0.470276\pi\)
0.0932440 + 0.995643i \(0.470276\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 91.5121i 3.35951i
\(743\) 22.4385i 0.823188i 0.911367 + 0.411594i \(0.135028\pi\)
−0.911367 + 0.411594i \(0.864972\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) −64.6133 −2.36566
\(747\) 0 0
\(748\) 8.40990i 0.307497i
\(749\) −11.6128 −0.424324
\(750\) 0 0
\(751\) −6.63512 −0.242119 −0.121060 0.992645i \(-0.538629\pi\)
−0.121060 + 0.992645i \(0.538629\pi\)
\(752\) − 12.7110i − 0.463523i
\(753\) 0 0
\(754\) −9.24443 −0.336662
\(755\) 0 0
\(756\) 0 0
\(757\) 8.75557i 0.318227i 0.987260 + 0.159113i \(0.0508635\pi\)
−0.987260 + 0.159113i \(0.949136\pi\)
\(758\) − 38.0642i − 1.38256i
\(759\) 0 0
\(760\) 0 0
\(761\) −3.15257 −0.114280 −0.0571402 0.998366i \(-0.518198\pi\)
−0.0571402 + 0.998366i \(0.518198\pi\)
\(762\) 0 0
\(763\) − 87.3087i − 3.16079i
\(764\) 9.89829 0.358108
\(765\) 0 0
\(766\) −27.8537 −1.00640
\(767\) − 3.02227i − 0.109128i
\(768\) 0 0
\(769\) 28.9590 1.04429 0.522144 0.852857i \(-0.325132\pi\)
0.522144 + 0.852857i \(0.325132\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 29.7462i 1.07059i
\(773\) 29.1427i 1.04819i 0.851660 + 0.524095i \(0.175596\pi\)
−0.851660 + 0.524095i \(0.824404\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 5.20877 0.186984
\(777\) 0 0
\(778\) 10.6824i 0.382984i
\(779\) −1.36488 −0.0489018
\(780\) 0 0
\(781\) 2.75557 0.0986020
\(782\) 87.3916i 3.12512i
\(783\) 0 0
\(784\) 58.1811 2.07790
\(785\) 0 0
\(786\) 0 0
\(787\) − 11.2672i − 0.401632i −0.979629 0.200816i \(-0.935641\pi\)
0.979629 0.200816i \(-0.0643593\pi\)
\(788\) − 10.8613i − 0.386918i
\(789\) 0 0
\(790\) 0 0
\(791\) −26.5718 −0.944786
\(792\) 0 0
\(793\) 4.26671i 0.151515i
\(794\) −24.2766 −0.861543
\(795\) 0 0
\(796\) 22.8760 0.810819
\(797\) − 41.9625i − 1.48639i −0.669075 0.743195i \(-0.733310\pi\)
0.669075 0.743195i \(-0.266690\pi\)
\(798\) 0 0
\(799\) −14.2854 −0.505383
\(800\) 0 0
\(801\) 0 0
\(802\) 3.80642i 0.134409i
\(803\) 4.23506i 0.149452i
\(804\) 0 0
\(805\) 0 0
\(806\) 3.26317 0.114940
\(807\) 0 0
\(808\) 3.35322i 0.117966i
\(809\) −27.8064 −0.977622 −0.488811 0.872390i \(-0.662569\pi\)
−0.488811 + 0.872390i \(0.662569\pi\)
\(810\) 0 0
\(811\) 6.78415 0.238224 0.119112 0.992881i \(-0.461995\pi\)
0.119112 + 0.992881i \(0.461995\pi\)
\(812\) − 56.0830i − 1.96813i
\(813\) 0 0
\(814\) 3.80642 0.133415
\(815\) 0 0
\(816\) 0 0
\(817\) 40.0000i 1.39942i
\(818\) − 13.5585i − 0.474060i
\(819\) 0 0
\(820\) 0 0
\(821\) −3.62269 −0.126433 −0.0632164 0.998000i \(-0.520136\pi\)
−0.0632164 + 0.998000i \(0.520136\pi\)
\(822\) 0 0
\(823\) − 42.0642i − 1.46627i −0.680085 0.733134i \(-0.738057\pi\)
0.680085 0.733134i \(-0.261943\pi\)
\(824\) 8.34968 0.290875
\(825\) 0 0
\(826\) 40.9403 1.42449
\(827\) − 30.8256i − 1.07191i −0.844246 0.535956i \(-0.819951\pi\)
0.844246 0.535956i \(-0.180049\pi\)
\(828\) 0 0
\(829\) −7.12399 −0.247426 −0.123713 0.992318i \(-0.539480\pi\)
−0.123713 + 0.992318i \(0.539480\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 2.95316i 0.102382i
\(833\) − 65.3876i − 2.26555i
\(834\) 0 0
\(835\) 0 0
\(836\) −11.4380 −0.395592
\(837\) 0 0
\(838\) − 29.7146i − 1.02647i
\(839\) 3.34614 0.115522 0.0577608 0.998330i \(-0.481604\pi\)
0.0577608 + 0.998330i \(0.481604\pi\)
\(840\) 0 0
\(841\) 31.9403 1.10139
\(842\) − 15.0321i − 0.518041i
\(843\) 0 0
\(844\) 17.2988 0.595450
\(845\) 0 0
\(846\) 0 0
\(847\) − 4.42864i − 0.152170i
\(848\) 50.0830i 1.71986i
\(849\) 0 0
\(850\) 0 0
\(851\) 17.7146 0.607247
\(852\) 0 0
\(853\) 26.4197i 0.904595i 0.891867 + 0.452297i \(0.149395\pi\)
−0.891867 + 0.452297i \(0.850605\pi\)
\(854\) −57.7975 −1.97779
\(855\) 0 0
\(856\) −1.88538 −0.0644411
\(857\) − 38.7783i − 1.32464i −0.749220 0.662321i \(-0.769571\pi\)
0.749220 0.662321i \(-0.230429\pi\)
\(858\) 0 0
\(859\) 27.3087 0.931760 0.465880 0.884848i \(-0.345738\pi\)
0.465880 + 0.884848i \(0.345738\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 65.2899i 2.22378i
\(863\) − 49.5308i − 1.68605i −0.537875 0.843024i \(-0.680773\pi\)
0.537875 0.843024i \(-0.319227\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) −27.5397 −0.935838
\(867\) 0 0
\(868\) 19.7966i 0.671940i
\(869\) 8.56199 0.290446
\(870\) 0 0
\(871\) 0.774305 0.0262363
\(872\) − 14.1748i − 0.480021i
\(873\) 0 0
\(874\) −118.858 −4.02044
\(875\) 0 0
\(876\) 0 0
\(877\) − 4.50177i − 0.152014i −0.997107 0.0760070i \(-0.975783\pi\)
0.997107 0.0760070i \(-0.0242171\pi\)
\(878\) − 36.7654i − 1.24077i
\(879\) 0 0
\(880\) 0 0
\(881\) 15.1240 0.509540 0.254770 0.967002i \(-0.418000\pi\)
0.254770 + 0.967002i \(0.418000\pi\)
\(882\) 0 0
\(883\) 30.2480i 1.01793i 0.860789 + 0.508963i \(0.169971\pi\)
−0.860789 + 0.508963i \(0.830029\pi\)
\(884\) 5.23277 0.175997
\(885\) 0 0
\(886\) −24.9777 −0.839143
\(887\) − 57.1941i − 1.92039i −0.279333 0.960194i \(-0.590113\pi\)
0.279333 0.960194i \(-0.409887\pi\)
\(888\) 0 0
\(889\) −67.2454 −2.25534
\(890\) 0 0
\(891\) 0 0
\(892\) 14.3684i 0.481090i
\(893\) − 19.4291i − 0.650171i
\(894\) 0 0
\(895\) 0 0
\(896\) 25.0192 0.835833
\(897\) 0 0
\(898\) 61.4835i 2.05173i
\(899\) −21.5111 −0.717437
\(900\) 0 0
\(901\) 56.2864 1.87517
\(902\) 0.368416i 0.0122669i
\(903\) 0 0
\(904\) −4.31402 −0.143482
\(905\) 0 0
\(906\) 0 0
\(907\) − 53.2641i − 1.76861i −0.466913 0.884303i \(-0.654634\pi\)
0.466913 0.884303i \(-0.345366\pi\)
\(908\) 21.7017i 0.720195i
\(909\) 0 0
\(910\) 0 0
\(911\) 0.590573 0.0195665 0.00978327 0.999952i \(-0.496886\pi\)
0.00978327 + 0.999952i \(0.496886\pi\)
\(912\) 0 0
\(913\) − 0.133353i − 0.00441334i
\(914\) −44.6508 −1.47692
\(915\) 0 0
\(916\) 18.6735 0.616991
\(917\) 5.51114i 0.181994i
\(918\) 0 0
\(919\) −55.8707 −1.84300 −0.921502 0.388375i \(-0.873037\pi\)
−0.921502 + 0.388375i \(0.873037\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) − 54.9403i − 1.80936i
\(923\) − 1.71456i − 0.0564354i
\(924\) 0 0
\(925\) 0 0
\(926\) 36.8198 1.20997
\(927\) 0 0
\(928\) − 57.3087i − 1.88125i
\(929\) 15.3274 0.502876 0.251438 0.967873i \(-0.419097\pi\)
0.251438 + 0.967873i \(0.419097\pi\)
\(930\) 0 0
\(931\) 88.9314 2.91461
\(932\) 7.01921i 0.229922i
\(933\) 0 0
\(934\) −5.98126 −0.195713
\(935\) 0 0
\(936\) 0 0
\(937\) − 27.8479i − 0.909752i −0.890555 0.454876i \(-0.849684\pi\)
0.890555 0.454876i \(-0.150316\pi\)
\(938\) 10.4889i 0.342474i
\(939\) 0 0
\(940\) 0 0
\(941\) −10.4157 −0.339543 −0.169772 0.985483i \(-0.554303\pi\)
−0.169772 + 0.985483i \(0.554303\pi\)
\(942\) 0 0
\(943\) 1.71456i 0.0558337i
\(944\) 22.4059 0.729250
\(945\) 0 0
\(946\) 10.7971 0.351043
\(947\) − 8.47013i − 0.275242i −0.990485 0.137621i \(-0.956054\pi\)
0.990485 0.137621i \(-0.0439456\pi\)
\(948\) 0 0
\(949\) 2.63512 0.0855397
\(950\) 0 0
\(951\) 0 0
\(952\) − 16.5076i − 0.535014i
\(953\) − 8.71408i − 0.282277i −0.989990 0.141138i \(-0.954924\pi\)
0.989990 0.141138i \(-0.0450763\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 5.42816 0.175559
\(957\) 0 0
\(958\) 47.3087i 1.52847i
\(959\) 2.16500 0.0699114
\(960\) 0 0
\(961\) −23.4068 −0.755059
\(962\) − 2.36842i − 0.0763608i
\(963\) 0 0
\(964\) 2.18373 0.0703333
\(965\) 0 0
\(966\) 0 0
\(967\) 44.2449i 1.42282i 0.702777 + 0.711410i \(0.251943\pi\)
−0.702777 + 0.711410i \(0.748057\pi\)
\(968\) − 0.719004i − 0.0231097i
\(969\) 0 0
\(970\) 0 0
\(971\) 57.1437 1.83383 0.916914 0.399085i \(-0.130672\pi\)
0.916914 + 0.399085i \(0.130672\pi\)
\(972\) 0 0
\(973\) 78.8582i 2.52808i
\(974\) −21.9438 −0.703124
\(975\) 0 0
\(976\) −31.6316 −1.01250
\(977\) 16.2480i 0.519819i 0.965633 + 0.259909i \(0.0836927\pi\)
−0.965633 + 0.259909i \(0.916307\pi\)
\(978\) 0 0
\(979\) −5.61285 −0.179387
\(980\) 0 0
\(981\) 0 0
\(982\) 31.1882i 0.995256i
\(983\) 1.12399i 0.0358496i 0.999839 + 0.0179248i \(0.00570594\pi\)
−0.999839 + 0.0179248i \(0.994294\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) −77.0232 −2.45292
\(987\) 0 0
\(988\) 7.11691i 0.226419i
\(989\) 50.2480 1.59779
\(990\) 0 0
\(991\) −53.6513 −1.70429 −0.852144 0.523307i \(-0.824698\pi\)
−0.852144 + 0.523307i \(0.824698\pi\)
\(992\) 20.2292i 0.642279i
\(993\) 0 0
\(994\) 23.2257 0.736674
\(995\) 0 0
\(996\) 0 0
\(997\) − 35.7275i − 1.13150i −0.824577 0.565750i \(-0.808587\pi\)
0.824577 0.565750i \(-0.191413\pi\)
\(998\) − 48.2034i − 1.52585i
\(999\) 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 2475.2.c.r.199.2 6
3.2 odd 2 825.2.c.g.199.5 6
5.2 odd 4 2475.2.a.bb.1.3 3
5.3 odd 4 495.2.a.e.1.1 3
5.4 even 2 inner 2475.2.c.r.199.5 6
15.2 even 4 825.2.a.k.1.1 3
15.8 even 4 165.2.a.c.1.3 3
15.14 odd 2 825.2.c.g.199.2 6
20.3 even 4 7920.2.a.cj.1.3 3
55.43 even 4 5445.2.a.z.1.3 3
60.23 odd 4 2640.2.a.be.1.3 3
105.83 odd 4 8085.2.a.bk.1.3 3
165.32 odd 4 9075.2.a.cf.1.3 3
165.98 odd 4 1815.2.a.m.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
165.2.a.c.1.3 3 15.8 even 4
495.2.a.e.1.1 3 5.3 odd 4
825.2.a.k.1.1 3 15.2 even 4
825.2.c.g.199.2 6 15.14 odd 2
825.2.c.g.199.5 6 3.2 odd 2
1815.2.a.m.1.1 3 165.98 odd 4
2475.2.a.bb.1.3 3 5.2 odd 4
2475.2.c.r.199.2 6 1.1 even 1 trivial
2475.2.c.r.199.5 6 5.4 even 2 inner
2640.2.a.be.1.3 3 60.23 odd 4
5445.2.a.z.1.3 3 55.43 even 4
7920.2.a.cj.1.3 3 20.3 even 4
8085.2.a.bk.1.3 3 105.83 odd 4
9075.2.a.cf.1.3 3 165.32 odd 4