Properties

Label 2475.2.c.q.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: 6.0.5161984.1
Defining polynomial: \( x^{6} - 4x^{3} + 25x^{2} - 20x + 8 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 825)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 199.2
Root \(1.32001 - 1.32001i\) of defining polynomial
Character \(\chi\) \(=\) 2475.199
Dual form 2475.2.c.q.199.5

$q$-expansion

\(f(q)\) \(=\) \(q-2.12489i q^{2} -2.51514 q^{4} +3.64002i q^{7} +1.09461i q^{8} +O(q^{10})\) \(q-2.12489i q^{2} -2.51514 q^{4} +3.64002i q^{7} +1.09461i q^{8} -1.00000 q^{11} -1.51514i q^{13} +7.73463 q^{14} -2.70436 q^{16} -1.15516i q^{17} -2.60975 q^{19} +2.12489i q^{22} -5.73463i q^{23} -3.21949 q^{26} -9.15516i q^{28} +6.24977 q^{29} +5.51514 q^{31} +7.93567i q^{32} -2.45459 q^{34} +0.454586i q^{37} +5.54541i q^{38} -4.12489 q^{41} -11.7044i q^{43} +2.51514 q^{44} -12.1854 q^{46} +3.48486i q^{47} -6.24977 q^{49} +3.81078i q^{52} -12.5601i q^{53} -3.98440 q^{56} -13.2800i q^{58} -7.73463 q^{59} -12.0147 q^{61} -11.7190i q^{62} +11.4537 q^{64} -14.2645i q^{67} +2.90539i q^{68} -8.51514 q^{71} +9.21949i q^{73} +0.965943 q^{74} +6.56387 q^{76} -3.64002i q^{77} -5.09461 q^{79} +8.76491i q^{82} -14.7493i q^{83} -24.8704 q^{86} -1.09461i q^{88} -10.4995 q^{89} +5.51514 q^{91} +14.4234i q^{92} +7.40493 q^{94} -6.77959i q^{97} +13.2800i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 16 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 16 q^{4} - 6 q^{11} + 12 q^{14} + 20 q^{16} + 2 q^{19} + 16 q^{26} + 4 q^{29} + 34 q^{31} - 12 q^{34} - 8 q^{41} + 16 q^{44} - 60 q^{46} - 4 q^{49} + 44 q^{56} - 12 q^{59} - 6 q^{61} - 68 q^{64} - 52 q^{71} + 28 q^{74} - 48 q^{76} - 12 q^{79} - 56 q^{86} + 4 q^{89} + 34 q^{91} - 4 q^{94}+O(q^{100}) \) Copy content Toggle raw display

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 2.12489i − 1.50252i −0.660006 0.751260i \(-0.729446\pi\)
0.660006 0.751260i \(-0.270554\pi\)
\(3\) 0 0
\(4\) −2.51514 −1.25757
\(5\) 0 0
\(6\) 0 0
\(7\) 3.64002i 1.37580i 0.725806 + 0.687900i \(0.241467\pi\)
−0.725806 + 0.687900i \(0.758533\pi\)
\(8\) 1.09461i 0.387003i
\(9\) 0 0
\(10\) 0 0
\(11\) −1.00000 −0.301511
\(12\) 0 0
\(13\) − 1.51514i − 0.420224i −0.977677 0.210112i \(-0.932617\pi\)
0.977677 0.210112i \(-0.0673828\pi\)
\(14\) 7.73463 2.06717
\(15\) 0 0
\(16\) −2.70436 −0.676089
\(17\) − 1.15516i − 0.280168i −0.990140 0.140084i \(-0.955263\pi\)
0.990140 0.140084i \(-0.0447372\pi\)
\(18\) 0 0
\(19\) −2.60975 −0.598717 −0.299359 0.954141i \(-0.596773\pi\)
−0.299359 + 0.954141i \(0.596773\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 2.12489i 0.453027i
\(23\) − 5.73463i − 1.19575i −0.801588 0.597877i \(-0.796011\pi\)
0.801588 0.597877i \(-0.203989\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) −3.21949 −0.631395
\(27\) 0 0
\(28\) − 9.15516i − 1.73016i
\(29\) 6.24977 1.16055 0.580277 0.814419i \(-0.302944\pi\)
0.580277 + 0.814419i \(0.302944\pi\)
\(30\) 0 0
\(31\) 5.51514 0.990548 0.495274 0.868737i \(-0.335068\pi\)
0.495274 + 0.868737i \(0.335068\pi\)
\(32\) 7.93567i 1.40284i
\(33\) 0 0
\(34\) −2.45459 −0.420958
\(35\) 0 0
\(36\) 0 0
\(37\) 0.454586i 0.0747335i 0.999302 + 0.0373667i \(0.0118970\pi\)
−0.999302 + 0.0373667i \(0.988103\pi\)
\(38\) 5.54541i 0.899585i
\(39\) 0 0
\(40\) 0 0
\(41\) −4.12489 −0.644199 −0.322099 0.946706i \(-0.604389\pi\)
−0.322099 + 0.946706i \(0.604389\pi\)
\(42\) 0 0
\(43\) − 11.7044i − 1.78490i −0.451149 0.892449i \(-0.648986\pi\)
0.451149 0.892449i \(-0.351014\pi\)
\(44\) 2.51514 0.379171
\(45\) 0 0
\(46\) −12.1854 −1.79664
\(47\) 3.48486i 0.508319i 0.967162 + 0.254160i \(0.0817989\pi\)
−0.967162 + 0.254160i \(0.918201\pi\)
\(48\) 0 0
\(49\) −6.24977 −0.892824
\(50\) 0 0
\(51\) 0 0
\(52\) 3.81078i 0.528460i
\(53\) − 12.5601i − 1.72526i −0.505834 0.862631i \(-0.668815\pi\)
0.505834 0.862631i \(-0.331185\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −3.98440 −0.532438
\(57\) 0 0
\(58\) − 13.2800i − 1.74376i
\(59\) −7.73463 −1.00696 −0.503482 0.864006i \(-0.667948\pi\)
−0.503482 + 0.864006i \(0.667948\pi\)
\(60\) 0 0
\(61\) −12.0147 −1.53832 −0.769161 0.639055i \(-0.779326\pi\)
−0.769161 + 0.639055i \(0.779326\pi\)
\(62\) − 11.7190i − 1.48832i
\(63\) 0 0
\(64\) 11.4537 1.43171
\(65\) 0 0
\(66\) 0 0
\(67\) − 14.2645i − 1.74268i −0.490680 0.871340i \(-0.663251\pi\)
0.490680 0.871340i \(-0.336749\pi\)
\(68\) 2.90539i 0.352330i
\(69\) 0 0
\(70\) 0 0
\(71\) −8.51514 −1.01056 −0.505280 0.862955i \(-0.668611\pi\)
−0.505280 + 0.862955i \(0.668611\pi\)
\(72\) 0 0
\(73\) 9.21949i 1.07906i 0.841966 + 0.539530i \(0.181398\pi\)
−0.841966 + 0.539530i \(0.818602\pi\)
\(74\) 0.965943 0.112289
\(75\) 0 0
\(76\) 6.56387 0.752928
\(77\) − 3.64002i − 0.414819i
\(78\) 0 0
\(79\) −5.09461 −0.573188 −0.286594 0.958052i \(-0.592523\pi\)
−0.286594 + 0.958052i \(0.592523\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 8.76491i 0.967922i
\(83\) − 14.7493i − 1.61895i −0.587156 0.809474i \(-0.699753\pi\)
0.587156 0.809474i \(-0.300247\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −24.8704 −2.68185
\(87\) 0 0
\(88\) − 1.09461i − 0.116686i
\(89\) −10.4995 −1.11295 −0.556475 0.830865i \(-0.687846\pi\)
−0.556475 + 0.830865i \(0.687846\pi\)
\(90\) 0 0
\(91\) 5.51514 0.578144
\(92\) 14.4234i 1.50374i
\(93\) 0 0
\(94\) 7.40493 0.763760
\(95\) 0 0
\(96\) 0 0
\(97\) − 6.77959i − 0.688363i −0.938903 0.344181i \(-0.888156\pi\)
0.938903 0.344181i \(-0.111844\pi\)
\(98\) 13.2800i 1.34149i
\(99\) 0 0
\(100\) 0 0
\(101\) −7.40493 −0.736818 −0.368409 0.929664i \(-0.620097\pi\)
−0.368409 + 0.929664i \(0.620097\pi\)
\(102\) 0 0
\(103\) 16.4995i 1.62575i 0.582439 + 0.812874i \(0.302098\pi\)
−0.582439 + 0.812874i \(0.697902\pi\)
\(104\) 1.65848 0.162628
\(105\) 0 0
\(106\) −26.6888 −2.59224
\(107\) − 3.93945i − 0.380841i −0.981703 0.190420i \(-0.939015\pi\)
0.981703 0.190420i \(-0.0609851\pi\)
\(108\) 0 0
\(109\) −6.73463 −0.645061 −0.322530 0.946559i \(-0.604533\pi\)
−0.322530 + 0.946559i \(0.604533\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) − 9.84392i − 0.930163i
\(113\) − 6.00000i − 0.564433i −0.959351 0.282216i \(-0.908930\pi\)
0.959351 0.282216i \(-0.0910696\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) −15.7190 −1.45948
\(117\) 0 0
\(118\) 16.4352i 1.51298i
\(119\) 4.20482 0.385455
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 25.5298i 2.31136i
\(123\) 0 0
\(124\) −13.8713 −1.24568
\(125\) 0 0
\(126\) 0 0
\(127\) 8.06433i 0.715594i 0.933799 + 0.357797i \(0.116472\pi\)
−0.933799 + 0.357797i \(0.883528\pi\)
\(128\) − 8.46640i − 0.748331i
\(129\) 0 0
\(130\) 0 0
\(131\) 12.8099 1.11920 0.559602 0.828762i \(-0.310954\pi\)
0.559602 + 0.828762i \(0.310954\pi\)
\(132\) 0 0
\(133\) − 9.49954i − 0.823715i
\(134\) −30.3103 −2.61841
\(135\) 0 0
\(136\) 1.26445 0.108426
\(137\) − 22.8099i − 1.94878i −0.224868 0.974389i \(-0.572195\pi\)
0.224868 0.974389i \(-0.427805\pi\)
\(138\) 0 0
\(139\) −7.59037 −0.643807 −0.321903 0.946773i \(-0.604323\pi\)
−0.321903 + 0.946773i \(0.604323\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 18.0937i 1.51839i
\(143\) 1.51514i 0.126702i
\(144\) 0 0
\(145\) 0 0
\(146\) 19.5904 1.62131
\(147\) 0 0
\(148\) − 1.14335i − 0.0939825i
\(149\) −1.81456 −0.148655 −0.0743274 0.997234i \(-0.523681\pi\)
−0.0743274 + 0.997234i \(0.523681\pi\)
\(150\) 0 0
\(151\) 24.3250 1.97954 0.989770 0.142670i \(-0.0455687\pi\)
0.989770 + 0.142670i \(0.0455687\pi\)
\(152\) − 2.85665i − 0.231705i
\(153\) 0 0
\(154\) −7.73463 −0.623274
\(155\) 0 0
\(156\) 0 0
\(157\) 9.76491i 0.779325i 0.920958 + 0.389662i \(0.127408\pi\)
−0.920958 + 0.389662i \(0.872592\pi\)
\(158\) 10.8255i 0.861227i
\(159\) 0 0
\(160\) 0 0
\(161\) 20.8742 1.64512
\(162\) 0 0
\(163\) 6.98440i 0.547061i 0.961863 + 0.273530i \(0.0881914\pi\)
−0.961863 + 0.273530i \(0.911809\pi\)
\(164\) 10.3747 0.810125
\(165\) 0 0
\(166\) −31.3406 −2.43250
\(167\) 6.31032i 0.488307i 0.969737 + 0.244154i \(0.0785102\pi\)
−0.969737 + 0.244154i \(0.921490\pi\)
\(168\) 0 0
\(169\) 10.7044 0.823412
\(170\) 0 0
\(171\) 0 0
\(172\) 29.4381i 2.24463i
\(173\) − 12.8448i − 0.976575i −0.872683 0.488287i \(-0.837622\pi\)
0.872683 0.488287i \(-0.162378\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 2.70436 0.203849
\(177\) 0 0
\(178\) 22.3103i 1.67223i
\(179\) 13.4849 1.00791 0.503953 0.863731i \(-0.331878\pi\)
0.503953 + 0.863731i \(0.331878\pi\)
\(180\) 0 0
\(181\) −23.0899 −1.71626 −0.858130 0.513433i \(-0.828374\pi\)
−0.858130 + 0.513433i \(0.828374\pi\)
\(182\) − 11.7190i − 0.868673i
\(183\) 0 0
\(184\) 6.27718 0.462760
\(185\) 0 0
\(186\) 0 0
\(187\) 1.15516i 0.0844738i
\(188\) − 8.76491i − 0.639247i
\(189\) 0 0
\(190\) 0 0
\(191\) 7.98440 0.577731 0.288866 0.957370i \(-0.406722\pi\)
0.288866 + 0.957370i \(0.406722\pi\)
\(192\) 0 0
\(193\) − 11.7649i − 0.846857i −0.905930 0.423428i \(-0.860827\pi\)
0.905930 0.423428i \(-0.139173\pi\)
\(194\) −14.4058 −1.03428
\(195\) 0 0
\(196\) 15.7190 1.12279
\(197\) 3.81456i 0.271776i 0.990724 + 0.135888i \(0.0433888\pi\)
−0.990724 + 0.135888i \(0.956611\pi\)
\(198\) 0 0
\(199\) 12.0752 0.855990 0.427995 0.903781i \(-0.359220\pi\)
0.427995 + 0.903781i \(0.359220\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 15.7346i 1.10708i
\(203\) 22.7493i 1.59669i
\(204\) 0 0
\(205\) 0 0
\(206\) 35.0596 2.44272
\(207\) 0 0
\(208\) 4.09747i 0.284109i
\(209\) 2.60975 0.180520
\(210\) 0 0
\(211\) 10.2645 0.706634 0.353317 0.935504i \(-0.385054\pi\)
0.353317 + 0.935504i \(0.385054\pi\)
\(212\) 31.5904i 2.16964i
\(213\) 0 0
\(214\) −8.37088 −0.572221
\(215\) 0 0
\(216\) 0 0
\(217\) 20.0752i 1.36280i
\(218\) 14.3103i 0.969217i
\(219\) 0 0
\(220\) 0 0
\(221\) −1.75023 −0.117733
\(222\) 0 0
\(223\) − 12.9239i − 0.865445i −0.901527 0.432723i \(-0.857553\pi\)
0.901527 0.432723i \(-0.142447\pi\)
\(224\) −28.8860 −1.93003
\(225\) 0 0
\(226\) −12.7493 −0.848072
\(227\) − 22.8099i − 1.51394i −0.653447 0.756972i \(-0.726678\pi\)
0.653447 0.756972i \(-0.273322\pi\)
\(228\) 0 0
\(229\) −14.7796 −0.976663 −0.488331 0.872658i \(-0.662394\pi\)
−0.488331 + 0.872658i \(0.662394\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 6.84106i 0.449137i
\(233\) 4.96594i 0.325330i 0.986681 + 0.162665i \(0.0520089\pi\)
−0.986681 + 0.162665i \(0.947991\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 19.4537 1.26633
\(237\) 0 0
\(238\) − 8.93475i − 0.579154i
\(239\) −14.9991 −0.970210 −0.485105 0.874456i \(-0.661219\pi\)
−0.485105 + 0.874456i \(0.661219\pi\)
\(240\) 0 0
\(241\) 5.04496 0.324974 0.162487 0.986711i \(-0.448048\pi\)
0.162487 + 0.986711i \(0.448048\pi\)
\(242\) − 2.12489i − 0.136593i
\(243\) 0 0
\(244\) 30.2186 1.93455
\(245\) 0 0
\(246\) 0 0
\(247\) 3.95413i 0.251595i
\(248\) 6.03692i 0.383345i
\(249\) 0 0
\(250\) 0 0
\(251\) 3.03028 0.191269 0.0956347 0.995417i \(-0.469512\pi\)
0.0956347 + 0.995417i \(0.469512\pi\)
\(252\) 0 0
\(253\) 5.73463i 0.360533i
\(254\) 17.1358 1.07519
\(255\) 0 0
\(256\) 4.91721 0.307325
\(257\) 13.6509i 0.851521i 0.904836 + 0.425761i \(0.139993\pi\)
−0.904836 + 0.425761i \(0.860007\pi\)
\(258\) 0 0
\(259\) −1.65470 −0.102818
\(260\) 0 0
\(261\) 0 0
\(262\) − 27.2195i − 1.68163i
\(263\) 12.5601i 0.774489i 0.921977 + 0.387244i \(0.126573\pi\)
−0.921977 + 0.387244i \(0.873427\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) −20.1854 −1.23765
\(267\) 0 0
\(268\) 35.8771i 2.19154i
\(269\) −24.6888 −1.50530 −0.752650 0.658421i \(-0.771225\pi\)
−0.752650 + 0.658421i \(0.771225\pi\)
\(270\) 0 0
\(271\) 7.56479 0.459528 0.229764 0.973246i \(-0.426205\pi\)
0.229764 + 0.973246i \(0.426205\pi\)
\(272\) 3.12397i 0.189418i
\(273\) 0 0
\(274\) −48.4683 −2.92808
\(275\) 0 0
\(276\) 0 0
\(277\) 1.92477i 0.115648i 0.998327 + 0.0578241i \(0.0184162\pi\)
−0.998327 + 0.0578241i \(0.981584\pi\)
\(278\) 16.1287i 0.967333i
\(279\) 0 0
\(280\) 0 0
\(281\) 1.87511 0.111860 0.0559300 0.998435i \(-0.482188\pi\)
0.0559300 + 0.998435i \(0.482188\pi\)
\(282\) 0 0
\(283\) − 30.1396i − 1.79161i −0.444446 0.895806i \(-0.646599\pi\)
0.444446 0.895806i \(-0.353401\pi\)
\(284\) 21.4167 1.27085
\(285\) 0 0
\(286\) 3.21949 0.190373
\(287\) − 15.0147i − 0.886289i
\(288\) 0 0
\(289\) 15.6656 0.921506
\(290\) 0 0
\(291\) 0 0
\(292\) − 23.1883i − 1.35699i
\(293\) − 29.1552i − 1.70326i −0.524141 0.851631i \(-0.675614\pi\)
0.524141 0.851631i \(-0.324386\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) −0.497594 −0.0289221
\(297\) 0 0
\(298\) 3.85574i 0.223357i
\(299\) −8.68876 −0.502484
\(300\) 0 0
\(301\) 42.6041 2.45566
\(302\) − 51.6878i − 2.97430i
\(303\) 0 0
\(304\) 7.05769 0.404786
\(305\) 0 0
\(306\) 0 0
\(307\) 27.8548i 1.58976i 0.606768 + 0.794879i \(0.292466\pi\)
−0.606768 + 0.794879i \(0.707534\pi\)
\(308\) 9.15516i 0.521664i
\(309\) 0 0
\(310\) 0 0
\(311\) −23.9083 −1.35571 −0.677856 0.735194i \(-0.737091\pi\)
−0.677856 + 0.735194i \(0.737091\pi\)
\(312\) 0 0
\(313\) − 28.3094i − 1.60014i −0.599905 0.800071i \(-0.704795\pi\)
0.599905 0.800071i \(-0.295205\pi\)
\(314\) 20.7493 1.17095
\(315\) 0 0
\(316\) 12.8136 0.720824
\(317\) 8.80986i 0.494811i 0.968912 + 0.247406i \(0.0795780\pi\)
−0.968912 + 0.247406i \(0.920422\pi\)
\(318\) 0 0
\(319\) −6.24977 −0.349920
\(320\) 0 0
\(321\) 0 0
\(322\) − 44.3553i − 2.47182i
\(323\) 3.01468i 0.167741i
\(324\) 0 0
\(325\) 0 0
\(326\) 14.8411 0.821970
\(327\) 0 0
\(328\) − 4.51514i − 0.249307i
\(329\) −12.6850 −0.699346
\(330\) 0 0
\(331\) 32.2498 1.77261 0.886304 0.463104i \(-0.153264\pi\)
0.886304 + 0.463104i \(0.153264\pi\)
\(332\) 37.0966i 2.03594i
\(333\) 0 0
\(334\) 13.4087 0.733692
\(335\) 0 0
\(336\) 0 0
\(337\) 28.9844i 1.57888i 0.613827 + 0.789441i \(0.289629\pi\)
−0.613827 + 0.789441i \(0.710371\pi\)
\(338\) − 22.7455i − 1.23719i
\(339\) 0 0
\(340\) 0 0
\(341\) −5.51514 −0.298661
\(342\) 0 0
\(343\) 2.73085i 0.147452i
\(344\) 12.8117 0.690760
\(345\) 0 0
\(346\) −27.2938 −1.46732
\(347\) 35.7190i 1.91750i 0.284253 + 0.958749i \(0.408254\pi\)
−0.284253 + 0.958749i \(0.591746\pi\)
\(348\) 0 0
\(349\) 23.2800 1.24615 0.623076 0.782161i \(-0.285883\pi\)
0.623076 + 0.782161i \(0.285883\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) − 7.93567i − 0.422972i
\(353\) − 9.75023i − 0.518952i −0.965750 0.259476i \(-0.916450\pi\)
0.965750 0.259476i \(-0.0835499\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 26.4078 1.39961
\(357\) 0 0
\(358\) − 28.6538i − 1.51440i
\(359\) 33.9007 1.78921 0.894605 0.446858i \(-0.147457\pi\)
0.894605 + 0.446858i \(0.147457\pi\)
\(360\) 0 0
\(361\) −12.1892 −0.641538
\(362\) 49.0634i 2.57872i
\(363\) 0 0
\(364\) −13.8713 −0.727055
\(365\) 0 0
\(366\) 0 0
\(367\) 1.88601i 0.0984491i 0.998788 + 0.0492245i \(0.0156750\pi\)
−0.998788 + 0.0492245i \(0.984325\pi\)
\(368\) 15.5085i 0.808436i
\(369\) 0 0
\(370\) 0 0
\(371\) 45.7190 2.37361
\(372\) 0 0
\(373\) 16.3250i 0.845277i 0.906298 + 0.422638i \(0.138896\pi\)
−0.906298 + 0.422638i \(0.861104\pi\)
\(374\) 2.45459 0.126924
\(375\) 0 0
\(376\) −3.81456 −0.196721
\(377\) − 9.46927i − 0.487692i
\(378\) 0 0
\(379\) 26.0440 1.33779 0.668896 0.743356i \(-0.266767\pi\)
0.668896 + 0.743356i \(0.266767\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) − 16.9659i − 0.868053i
\(383\) 12.4702i 0.637197i 0.947890 + 0.318598i \(0.103212\pi\)
−0.947890 + 0.318598i \(0.896788\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −24.9991 −1.27242
\(387\) 0 0
\(388\) 17.0516i 0.865664i
\(389\) −18.0899 −0.917195 −0.458597 0.888644i \(-0.651648\pi\)
−0.458597 + 0.888644i \(0.651648\pi\)
\(390\) 0 0
\(391\) −6.62443 −0.335012
\(392\) − 6.84106i − 0.345526i
\(393\) 0 0
\(394\) 8.10551 0.408350
\(395\) 0 0
\(396\) 0 0
\(397\) − 15.2342i − 0.764581i −0.924042 0.382291i \(-0.875135\pi\)
0.924042 0.382291i \(-0.124865\pi\)
\(398\) − 25.6585i − 1.28614i
\(399\) 0 0
\(400\) 0 0
\(401\) −2.74931 −0.137294 −0.0686471 0.997641i \(-0.521868\pi\)
−0.0686471 + 0.997641i \(0.521868\pi\)
\(402\) 0 0
\(403\) − 8.35620i − 0.416252i
\(404\) 18.6244 0.926600
\(405\) 0 0
\(406\) 48.3397 2.39906
\(407\) − 0.454586i − 0.0225330i
\(408\) 0 0
\(409\) 3.98532 0.197061 0.0985307 0.995134i \(-0.468586\pi\)
0.0985307 + 0.995134i \(0.468586\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) − 41.4986i − 2.04449i
\(413\) − 28.1542i − 1.38538i
\(414\) 0 0
\(415\) 0 0
\(416\) 12.0236 0.589507
\(417\) 0 0
\(418\) − 5.54541i − 0.271235i
\(419\) −5.13578 −0.250899 −0.125450 0.992100i \(-0.540037\pi\)
−0.125450 + 0.992100i \(0.540037\pi\)
\(420\) 0 0
\(421\) −8.94657 −0.436029 −0.218014 0.975946i \(-0.569958\pi\)
−0.218014 + 0.975946i \(0.569958\pi\)
\(422\) − 21.8108i − 1.06173i
\(423\) 0 0
\(424\) 13.7484 0.667681
\(425\) 0 0
\(426\) 0 0
\(427\) − 43.7337i − 2.11642i
\(428\) 9.90826i 0.478934i
\(429\) 0 0
\(430\) 0 0
\(431\) −22.7493 −1.09580 −0.547898 0.836545i \(-0.684572\pi\)
−0.547898 + 0.836545i \(0.684572\pi\)
\(432\) 0 0
\(433\) − 7.58325i − 0.364428i −0.983259 0.182214i \(-0.941674\pi\)
0.983259 0.182214i \(-0.0583263\pi\)
\(434\) 42.6576 2.04763
\(435\) 0 0
\(436\) 16.9385 0.811209
\(437\) 14.9659i 0.715918i
\(438\) 0 0
\(439\) −17.3903 −0.829991 −0.414996 0.909823i \(-0.636217\pi\)
−0.414996 + 0.909823i \(0.636217\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 3.71904i 0.176897i
\(443\) 10.6438i 0.505702i 0.967505 + 0.252851i \(0.0813683\pi\)
−0.967505 + 0.252851i \(0.918632\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) −27.4617 −1.30035
\(447\) 0 0
\(448\) 41.6916i 1.96974i
\(449\) −26.9310 −1.27095 −0.635476 0.772121i \(-0.719196\pi\)
−0.635476 + 0.772121i \(0.719196\pi\)
\(450\) 0 0
\(451\) 4.12489 0.194233
\(452\) 15.0908i 0.709813i
\(453\) 0 0
\(454\) −48.4683 −2.27473
\(455\) 0 0
\(456\) 0 0
\(457\) − 15.7796i − 0.738138i −0.929402 0.369069i \(-0.879677\pi\)
0.929402 0.369069i \(-0.120323\pi\)
\(458\) 31.4049i 1.46746i
\(459\) 0 0
\(460\) 0 0
\(461\) −8.18922 −0.381410 −0.190705 0.981647i \(-0.561077\pi\)
−0.190705 + 0.981647i \(0.561077\pi\)
\(462\) 0 0
\(463\) − 16.0899i − 0.747762i −0.927477 0.373881i \(-0.878027\pi\)
0.927477 0.373881i \(-0.121973\pi\)
\(464\) −16.9016 −0.784638
\(465\) 0 0
\(466\) 10.5521 0.488815
\(467\) 29.4693i 1.36367i 0.731504 + 0.681837i \(0.238819\pi\)
−0.731504 + 0.681837i \(0.761181\pi\)
\(468\) 0 0
\(469\) 51.9229 2.39758
\(470\) 0 0
\(471\) 0 0
\(472\) − 8.46640i − 0.389698i
\(473\) 11.7044i 0.538167i
\(474\) 0 0
\(475\) 0 0
\(476\) −10.5757 −0.484736
\(477\) 0 0
\(478\) 31.8713i 1.45776i
\(479\) 32.2186 1.47210 0.736052 0.676925i \(-0.236688\pi\)
0.736052 + 0.676925i \(0.236688\pi\)
\(480\) 0 0
\(481\) 0.688760 0.0314048
\(482\) − 10.7200i − 0.488280i
\(483\) 0 0
\(484\) −2.51514 −0.114324
\(485\) 0 0
\(486\) 0 0
\(487\) − 35.8936i − 1.62649i −0.581919 0.813247i \(-0.697698\pi\)
0.581919 0.813247i \(-0.302302\pi\)
\(488\) − 13.1514i − 0.595335i
\(489\) 0 0
\(490\) 0 0
\(491\) 7.15894 0.323079 0.161539 0.986866i \(-0.448354\pi\)
0.161539 + 0.986866i \(0.448354\pi\)
\(492\) 0 0
\(493\) − 7.21949i − 0.325150i
\(494\) 8.40207 0.378027
\(495\) 0 0
\(496\) −14.9149 −0.669699
\(497\) − 30.9953i − 1.39033i
\(498\) 0 0
\(499\) −27.0743 −1.21201 −0.606006 0.795460i \(-0.707229\pi\)
−0.606006 + 0.795460i \(0.707229\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) − 6.43899i − 0.287386i
\(503\) 26.9991i 1.20383i 0.798560 + 0.601915i \(0.205595\pi\)
−0.798560 + 0.601915i \(0.794405\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 12.1854 0.541709
\(507\) 0 0
\(508\) − 20.2829i − 0.899909i
\(509\) 15.5904 0.691031 0.345515 0.938413i \(-0.387704\pi\)
0.345515 + 0.938413i \(0.387704\pi\)
\(510\) 0 0
\(511\) −33.5592 −1.48457
\(512\) − 27.3813i − 1.21009i
\(513\) 0 0
\(514\) 29.0066 1.27943
\(515\) 0 0
\(516\) 0 0
\(517\) − 3.48486i − 0.153264i
\(518\) 3.51605i 0.154487i
\(519\) 0 0
\(520\) 0 0
\(521\) −11.1589 −0.488882 −0.244441 0.969664i \(-0.578604\pi\)
−0.244441 + 0.969664i \(0.578604\pi\)
\(522\) 0 0
\(523\) 10.5786i 0.462568i 0.972886 + 0.231284i \(0.0742926\pi\)
−0.972886 + 0.231284i \(0.925707\pi\)
\(524\) −32.2186 −1.40748
\(525\) 0 0
\(526\) 26.6888 1.16369
\(527\) − 6.37088i − 0.277520i
\(528\) 0 0
\(529\) −9.88601 −0.429827
\(530\) 0 0
\(531\) 0 0
\(532\) 23.8927i 1.03588i
\(533\) 6.24977i 0.270708i
\(534\) 0 0
\(535\) 0 0
\(536\) 15.6140 0.674422
\(537\) 0 0
\(538\) 52.4608i 2.26175i
\(539\) 6.24977 0.269197
\(540\) 0 0
\(541\) −11.2947 −0.485598 −0.242799 0.970077i \(-0.578066\pi\)
−0.242799 + 0.970077i \(0.578066\pi\)
\(542\) − 16.0743i − 0.690451i
\(543\) 0 0
\(544\) 9.16698 0.393031
\(545\) 0 0
\(546\) 0 0
\(547\) 6.09369i 0.260547i 0.991478 + 0.130274i \(0.0415856\pi\)
−0.991478 + 0.130274i \(0.958414\pi\)
\(548\) 57.3700i 2.45072i
\(549\) 0 0
\(550\) 0 0
\(551\) −16.3103 −0.694843
\(552\) 0 0
\(553\) − 18.5445i − 0.788592i
\(554\) 4.08991 0.173764
\(555\) 0 0
\(556\) 19.0908 0.809631
\(557\) 5.90069i 0.250020i 0.992155 + 0.125010i \(0.0398964\pi\)
−0.992155 + 0.125010i \(0.960104\pi\)
\(558\) 0 0
\(559\) −17.7337 −0.750056
\(560\) 0 0
\(561\) 0 0
\(562\) − 3.98440i − 0.168072i
\(563\) 3.03028i 0.127711i 0.997959 + 0.0638555i \(0.0203397\pi\)
−0.997959 + 0.0638555i \(0.979660\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) −64.0431 −2.69193
\(567\) 0 0
\(568\) − 9.32075i − 0.391090i
\(569\) 13.4049 0.561964 0.280982 0.959713i \(-0.409340\pi\)
0.280982 + 0.959713i \(0.409340\pi\)
\(570\) 0 0
\(571\) 26.8851 1.12511 0.562553 0.826761i \(-0.309819\pi\)
0.562553 + 0.826761i \(0.309819\pi\)
\(572\) − 3.81078i − 0.159337i
\(573\) 0 0
\(574\) −31.9045 −1.33167
\(575\) 0 0
\(576\) 0 0
\(577\) 2.03028i 0.0845215i 0.999107 + 0.0422607i \(0.0134560\pi\)
−0.999107 + 0.0422607i \(0.986544\pi\)
\(578\) − 33.2876i − 1.38458i
\(579\) 0 0
\(580\) 0 0
\(581\) 53.6878 2.22735
\(582\) 0 0
\(583\) 12.5601i 0.520186i
\(584\) −10.0917 −0.417599
\(585\) 0 0
\(586\) −61.9514 −2.55919
\(587\) − 21.8245i − 0.900795i −0.892828 0.450398i \(-0.851282\pi\)
0.892828 0.450398i \(-0.148718\pi\)
\(588\) 0 0
\(589\) −14.3931 −0.593058
\(590\) 0 0
\(591\) 0 0
\(592\) − 1.22936i − 0.0505265i
\(593\) 8.06811i 0.331318i 0.986183 + 0.165659i \(0.0529751\pi\)
−0.986183 + 0.165659i \(0.947025\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 4.56387 0.186944
\(597\) 0 0
\(598\) 18.4626i 0.754993i
\(599\) 7.61353 0.311080 0.155540 0.987830i \(-0.450288\pi\)
0.155540 + 0.987830i \(0.450288\pi\)
\(600\) 0 0
\(601\) 3.57569 0.145855 0.0729277 0.997337i \(-0.476766\pi\)
0.0729277 + 0.997337i \(0.476766\pi\)
\(602\) − 90.5289i − 3.68968i
\(603\) 0 0
\(604\) −61.1807 −2.48941
\(605\) 0 0
\(606\) 0 0
\(607\) 17.5298i 0.711513i 0.934579 + 0.355757i \(0.115777\pi\)
−0.934579 + 0.355757i \(0.884223\pi\)
\(608\) − 20.7101i − 0.839905i
\(609\) 0 0
\(610\) 0 0
\(611\) 5.28005 0.213608
\(612\) 0 0
\(613\) − 12.5601i − 0.507297i −0.967296 0.253649i \(-0.918369\pi\)
0.967296 0.253649i \(-0.0816307\pi\)
\(614\) 59.1883 2.38865
\(615\) 0 0
\(616\) 3.98440 0.160536
\(617\) − 15.9612i − 0.642576i −0.946982 0.321288i \(-0.895884\pi\)
0.946982 0.321288i \(-0.104116\pi\)
\(618\) 0 0
\(619\) 9.23417 0.371153 0.185576 0.982630i \(-0.440585\pi\)
0.185576 + 0.982630i \(0.440585\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 50.8023i 2.03699i
\(623\) − 38.2186i − 1.53119i
\(624\) 0 0
\(625\) 0 0
\(626\) −60.1542 −2.40425
\(627\) 0 0
\(628\) − 24.5601i − 0.980054i
\(629\) 0.525120 0.0209379
\(630\) 0 0
\(631\) 29.2342 1.16379 0.581897 0.813262i \(-0.302311\pi\)
0.581897 + 0.813262i \(0.302311\pi\)
\(632\) − 5.57661i − 0.221826i
\(633\) 0 0
\(634\) 18.7200 0.743464
\(635\) 0 0
\(636\) 0 0
\(637\) 9.46927i 0.375186i
\(638\) 13.2800i 0.525762i
\(639\) 0 0
\(640\) 0 0
\(641\) −13.9612 −0.551436 −0.275718 0.961239i \(-0.588916\pi\)
−0.275718 + 0.961239i \(0.588916\pi\)
\(642\) 0 0
\(643\) − 12.6206i − 0.497710i −0.968541 0.248855i \(-0.919946\pi\)
0.968541 0.248855i \(-0.0800542\pi\)
\(644\) −52.5015 −2.06885
\(645\) 0 0
\(646\) 6.40585 0.252035
\(647\) − 29.6429i − 1.16538i −0.812694 0.582691i \(-0.802000\pi\)
0.812694 0.582691i \(-0.198000\pi\)
\(648\) 0 0
\(649\) 7.73463 0.303611
\(650\) 0 0
\(651\) 0 0
\(652\) − 17.5667i − 0.687967i
\(653\) 9.90069i 0.387444i 0.981056 + 0.193722i \(0.0620560\pi\)
−0.981056 + 0.193722i \(0.937944\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 11.1552 0.435536
\(657\) 0 0
\(658\) 26.9541i 1.05078i
\(659\) −5.28005 −0.205681 −0.102841 0.994698i \(-0.532793\pi\)
−0.102841 + 0.994698i \(0.532793\pi\)
\(660\) 0 0
\(661\) −26.8548 −1.04453 −0.522266 0.852783i \(-0.674913\pi\)
−0.522266 + 0.852783i \(0.674913\pi\)
\(662\) − 68.5271i − 2.66338i
\(663\) 0 0
\(664\) 16.1447 0.626537
\(665\) 0 0
\(666\) 0 0
\(667\) − 35.8401i − 1.38774i
\(668\) − 15.8713i − 0.614080i
\(669\) 0 0
\(670\) 0 0
\(671\) 12.0147 0.463822
\(672\) 0 0
\(673\) 3.81834i 0.147186i 0.997288 + 0.0735932i \(0.0234466\pi\)
−0.997288 + 0.0735932i \(0.976553\pi\)
\(674\) 61.5885 2.37230
\(675\) 0 0
\(676\) −26.9229 −1.03550
\(677\) 15.6897i 0.603003i 0.953466 + 0.301502i \(0.0974879\pi\)
−0.953466 + 0.301502i \(0.902512\pi\)
\(678\) 0 0
\(679\) 24.6779 0.947049
\(680\) 0 0
\(681\) 0 0
\(682\) 11.7190i 0.448745i
\(683\) − 15.6353i − 0.598269i −0.954211 0.299135i \(-0.903302\pi\)
0.954211 0.299135i \(-0.0966979\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 5.80275 0.221550
\(687\) 0 0
\(688\) 31.6528i 1.20675i
\(689\) −19.0303 −0.724996
\(690\) 0 0
\(691\) −31.4305 −1.19567 −0.597836 0.801618i \(-0.703973\pi\)
−0.597836 + 0.801618i \(0.703973\pi\)
\(692\) 32.3065i 1.22811i
\(693\) 0 0
\(694\) 75.8989 2.88108
\(695\) 0 0
\(696\) 0 0
\(697\) 4.76491i 0.180484i
\(698\) − 49.4674i − 1.87237i
\(699\) 0 0
\(700\) 0 0
\(701\) −3.24507 −0.122565 −0.0612824 0.998120i \(-0.519519\pi\)
−0.0612824 + 0.998120i \(0.519519\pi\)
\(702\) 0 0
\(703\) − 1.18635i − 0.0447442i
\(704\) −11.4537 −0.431676
\(705\) 0 0
\(706\) −20.7181 −0.779737
\(707\) − 26.9541i − 1.01371i
\(708\) 0 0
\(709\) 26.7190 1.00345 0.501727 0.865026i \(-0.332698\pi\)
0.501727 + 0.865026i \(0.332698\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) − 11.4929i − 0.430714i
\(713\) − 31.6273i − 1.18445i
\(714\) 0 0
\(715\) 0 0
\(716\) −33.9163 −1.26751
\(717\) 0 0
\(718\) − 72.0351i − 2.68833i
\(719\) −6.78807 −0.253152 −0.126576 0.991957i \(-0.540399\pi\)
−0.126576 + 0.991957i \(0.540399\pi\)
\(720\) 0 0
\(721\) −60.0587 −2.23670
\(722\) 25.9007i 0.963924i
\(723\) 0 0
\(724\) 58.0743 2.15831
\(725\) 0 0
\(726\) 0 0
\(727\) − 19.9154i − 0.738620i −0.929306 0.369310i \(-0.879594\pi\)
0.929306 0.369310i \(-0.120406\pi\)
\(728\) 6.03692i 0.223743i
\(729\) 0 0
\(730\) 0 0
\(731\) −13.5204 −0.500071
\(732\) 0 0
\(733\) 31.3388i 1.15752i 0.815497 + 0.578762i \(0.196464\pi\)
−0.815497 + 0.578762i \(0.803536\pi\)
\(734\) 4.00756 0.147922
\(735\) 0 0
\(736\) 45.5081 1.67745
\(737\) 14.2645i 0.525438i
\(738\) 0 0
\(739\) 2.25355 0.0828982 0.0414491 0.999141i \(-0.486803\pi\)
0.0414491 + 0.999141i \(0.486803\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) − 97.1477i − 3.56640i
\(743\) − 6.74931i − 0.247608i −0.992307 0.123804i \(-0.960491\pi\)
0.992307 0.123804i \(-0.0395095\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 34.6888 1.27005
\(747\) 0 0
\(748\) − 2.90539i − 0.106232i
\(749\) 14.3397 0.523961
\(750\) 0 0
\(751\) 22.4390 0.818810 0.409405 0.912353i \(-0.365736\pi\)
0.409405 + 0.912353i \(0.365736\pi\)
\(752\) − 9.42431i − 0.343669i
\(753\) 0 0
\(754\) −20.1211 −0.732767
\(755\) 0 0
\(756\) 0 0
\(757\) 25.4158i 0.923754i 0.886944 + 0.461877i \(0.152824\pi\)
−0.886944 + 0.461877i \(0.847176\pi\)
\(758\) − 55.3406i − 2.01006i
\(759\) 0 0
\(760\) 0 0
\(761\) 30.7493 1.11466 0.557331 0.830291i \(-0.311826\pi\)
0.557331 + 0.830291i \(0.311826\pi\)
\(762\) 0 0
\(763\) − 24.5142i − 0.887474i
\(764\) −20.0819 −0.726537
\(765\) 0 0
\(766\) 26.4977 0.957401
\(767\) 11.7190i 0.423150i
\(768\) 0 0
\(769\) 16.2956 0.587636 0.293818 0.955861i \(-0.405074\pi\)
0.293818 + 0.955861i \(0.405074\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 29.5904i 1.06498i
\(773\) 48.7787i 1.75445i 0.480082 + 0.877223i \(0.340607\pi\)
−0.480082 + 0.877223i \(0.659393\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 7.42100 0.266398
\(777\) 0 0
\(778\) 38.4390i 1.37810i
\(779\) 10.7649 0.385693
\(780\) 0 0
\(781\) 8.51514 0.304696
\(782\) 14.0761i 0.503362i
\(783\) 0 0
\(784\) 16.9016 0.603629
\(785\) 0 0
\(786\) 0 0
\(787\) − 46.2001i − 1.64686i −0.567421 0.823428i \(-0.692059\pi\)
0.567421 0.823428i \(-0.307941\pi\)
\(788\) − 9.59415i − 0.341777i
\(789\) 0 0
\(790\) 0 0
\(791\) 21.8401 0.776546
\(792\) 0 0
\(793\) 18.2039i 0.646439i
\(794\) −32.3709 −1.14880
\(795\) 0 0
\(796\) −30.3709 −1.07647
\(797\) − 36.3784i − 1.28859i −0.764777 0.644295i \(-0.777151\pi\)
0.764777 0.644295i \(-0.222849\pi\)
\(798\) 0 0
\(799\) 4.02558 0.142415
\(800\) 0 0
\(801\) 0 0
\(802\) 5.84197i 0.206287i
\(803\) − 9.21949i − 0.325349i
\(804\) 0 0
\(805\) 0 0
\(806\) −17.7560 −0.625427
\(807\) 0 0
\(808\) − 8.10551i − 0.285151i
\(809\) 11.3737 0.399879 0.199940 0.979808i \(-0.435925\pi\)
0.199940 + 0.979808i \(0.435925\pi\)
\(810\) 0 0
\(811\) −13.3903 −0.470195 −0.235098 0.971972i \(-0.575541\pi\)
−0.235098 + 0.971972i \(0.575541\pi\)
\(812\) − 57.2177i − 2.00795i
\(813\) 0 0
\(814\) −0.965943 −0.0338563
\(815\) 0 0
\(816\) 0 0
\(817\) 30.5454i 1.06865i
\(818\) − 8.46835i − 0.296089i
\(819\) 0 0
\(820\) 0 0
\(821\) −32.0975 −1.12021 −0.560105 0.828422i \(-0.689239\pi\)
−0.560105 + 0.828422i \(0.689239\pi\)
\(822\) 0 0
\(823\) − 16.7952i − 0.585443i −0.956198 0.292722i \(-0.905439\pi\)
0.956198 0.292722i \(-0.0945609\pi\)
\(824\) −18.0606 −0.629169
\(825\) 0 0
\(826\) −59.8245 −2.08156
\(827\) 45.5904i 1.58533i 0.609656 + 0.792666i \(0.291308\pi\)
−0.609656 + 0.792666i \(0.708692\pi\)
\(828\) 0 0
\(829\) −12.9385 −0.449374 −0.224687 0.974431i \(-0.572136\pi\)
−0.224687 + 0.974431i \(0.572136\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) − 17.3539i − 0.601638i
\(833\) 7.21949i 0.250141i
\(834\) 0 0
\(835\) 0 0
\(836\) −6.56387 −0.227016
\(837\) 0 0
\(838\) 10.9130i 0.376982i
\(839\) −1.59037 −0.0549057 −0.0274528 0.999623i \(-0.508740\pi\)
−0.0274528 + 0.999623i \(0.508740\pi\)
\(840\) 0 0
\(841\) 10.0596 0.346884
\(842\) 19.0104i 0.655143i
\(843\) 0 0
\(844\) −25.8165 −0.888641
\(845\) 0 0
\(846\) 0 0
\(847\) 3.64002i 0.125073i
\(848\) 33.9670i 1.16643i
\(849\) 0 0
\(850\) 0 0
\(851\) 2.60688 0.0893628
\(852\) 0 0
\(853\) − 10.5161i − 0.360063i −0.983661 0.180031i \(-0.942380\pi\)
0.983661 0.180031i \(-0.0576199\pi\)
\(854\) −92.9291 −3.17997
\(855\) 0 0
\(856\) 4.31216 0.147386
\(857\) 57.4637i 1.96292i 0.191665 + 0.981460i \(0.438611\pi\)
−0.191665 + 0.981460i \(0.561389\pi\)
\(858\) 0 0
\(859\) −32.7181 −1.11633 −0.558164 0.829731i \(-0.688494\pi\)
−0.558164 + 0.829731i \(0.688494\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 48.3397i 1.64646i
\(863\) − 43.1807i − 1.46989i −0.678127 0.734945i \(-0.737208\pi\)
0.678127 0.734945i \(-0.262792\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) −16.1135 −0.547560
\(867\) 0 0
\(868\) − 50.4920i − 1.71381i
\(869\) 5.09461 0.172823
\(870\) 0 0
\(871\) −21.6126 −0.732315
\(872\) − 7.37179i − 0.249640i
\(873\) 0 0
\(874\) 31.8009 1.07568
\(875\) 0 0
\(876\) 0 0
\(877\) − 16.0752i − 0.542822i −0.962464 0.271411i \(-0.912510\pi\)
0.962464 0.271411i \(-0.0874903\pi\)
\(878\) 36.9523i 1.24708i
\(879\) 0 0
\(880\) 0 0
\(881\) −31.2876 −1.05411 −0.527053 0.849832i \(-0.676703\pi\)
−0.527053 + 0.849832i \(0.676703\pi\)
\(882\) 0 0
\(883\) − 24.7640i − 0.833375i −0.909050 0.416687i \(-0.863191\pi\)
0.909050 0.416687i \(-0.136809\pi\)
\(884\) 4.40207 0.148058
\(885\) 0 0
\(886\) 22.6169 0.759828
\(887\) 26.3085i 0.883353i 0.897174 + 0.441676i \(0.145616\pi\)
−0.897174 + 0.441676i \(0.854384\pi\)
\(888\) 0 0
\(889\) −29.3544 −0.984514
\(890\) 0 0
\(891\) 0 0
\(892\) 32.5053i 1.08836i
\(893\) − 9.09461i − 0.304339i
\(894\) 0 0
\(895\) 0 0
\(896\) 30.8179 1.02955
\(897\) 0 0
\(898\) 57.2252i 1.90963i
\(899\) 34.4683 1.14958
\(900\) 0 0
\(901\) −14.5089 −0.483363
\(902\) − 8.76491i − 0.291840i
\(903\) 0 0
\(904\) 6.56766 0.218437
\(905\) 0 0
\(906\) 0 0
\(907\) − 55.9301i − 1.85713i −0.371174 0.928563i \(-0.621045\pi\)
0.371174 0.928563i \(-0.378955\pi\)
\(908\) 57.3700i 1.90389i
\(909\) 0 0
\(910\) 0 0
\(911\) 15.3931 0.509997 0.254998 0.966941i \(-0.417925\pi\)
0.254998 + 0.966941i \(0.417925\pi\)
\(912\) 0 0
\(913\) 14.7493i 0.488131i
\(914\) −33.5298 −1.10907
\(915\) 0 0
\(916\) 37.1727 1.22822
\(917\) 46.6282i 1.53980i
\(918\) 0 0
\(919\) 51.2598 1.69090 0.845452 0.534052i \(-0.179331\pi\)
0.845452 + 0.534052i \(0.179331\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 17.4012i 0.573076i
\(923\) 12.9016i 0.424662i
\(924\) 0 0
\(925\) 0 0
\(926\) −34.1892 −1.12353
\(927\) 0 0
\(928\) 49.5961i 1.62807i
\(929\) 14.8099 0.485896 0.242948 0.970039i \(-0.421886\pi\)
0.242948 + 0.970039i \(0.421886\pi\)
\(930\) 0 0
\(931\) 16.3103 0.534549
\(932\) − 12.4900i − 0.409125i
\(933\) 0 0
\(934\) 62.6188 2.04895
\(935\) 0 0
\(936\) 0 0
\(937\) 27.2654i 0.890721i 0.895351 + 0.445360i \(0.146924\pi\)
−0.895351 + 0.445360i \(0.853076\pi\)
\(938\) − 110.330i − 3.60241i
\(939\) 0 0
\(940\) 0 0
\(941\) 49.5630 1.61571 0.807853 0.589384i \(-0.200629\pi\)
0.807853 + 0.589384i \(0.200629\pi\)
\(942\) 0 0
\(943\) 23.6547i 0.770303i
\(944\) 20.9172 0.680797
\(945\) 0 0
\(946\) 24.8704 0.808607
\(947\) 13.9844i 0.454432i 0.973844 + 0.227216i \(0.0729623\pi\)
−0.973844 + 0.227216i \(0.927038\pi\)
\(948\) 0 0
\(949\) 13.9688 0.453447
\(950\) 0 0
\(951\) 0 0
\(952\) 4.60263i 0.149172i
\(953\) 17.1240i 0.554700i 0.960769 + 0.277350i \(0.0894561\pi\)
−0.960769 + 0.277350i \(0.910544\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 37.7248 1.22011
\(957\) 0 0
\(958\) − 68.4608i − 2.21187i
\(959\) 83.0284 2.68113
\(960\) 0 0
\(961\) −0.583252 −0.0188146
\(962\) − 1.46354i − 0.0471863i
\(963\) 0 0
\(964\) −12.6888 −0.408677
\(965\) 0 0
\(966\) 0 0
\(967\) 1.90826i 0.0613653i 0.999529 + 0.0306827i \(0.00976813\pi\)
−0.999529 + 0.0306827i \(0.990232\pi\)
\(968\) 1.09461i 0.0351821i
\(969\) 0 0
\(970\) 0 0
\(971\) −31.3856 −1.00721 −0.503605 0.863934i \(-0.667993\pi\)
−0.503605 + 0.863934i \(0.667993\pi\)
\(972\) 0 0
\(973\) − 27.6291i − 0.885749i
\(974\) −76.2697 −2.44384
\(975\) 0 0
\(976\) 32.4920 1.04004
\(977\) − 41.4693i − 1.32672i −0.748301 0.663360i \(-0.769130\pi\)
0.748301 0.663360i \(-0.230870\pi\)
\(978\) 0 0
\(979\) 10.4995 0.335567
\(980\) 0 0
\(981\) 0 0
\(982\) − 15.2119i − 0.485432i
\(983\) − 6.23601i − 0.198898i −0.995043 0.0994489i \(-0.968292\pi\)
0.995043 0.0994489i \(-0.0317080\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) −15.3406 −0.488544
\(987\) 0 0
\(988\) − 9.94518i − 0.316398i
\(989\) −67.1202 −2.13430
\(990\) 0 0
\(991\) 27.2048 0.864189 0.432095 0.901828i \(-0.357775\pi\)
0.432095 + 0.901828i \(0.357775\pi\)
\(992\) 43.7663i 1.38958i
\(993\) 0 0
\(994\) −65.8615 −2.08900
\(995\) 0 0
\(996\) 0 0
\(997\) 28.0606i 0.888687i 0.895857 + 0.444343i \(0.146563\pi\)
−0.895857 + 0.444343i \(0.853437\pi\)
\(998\) 57.5298i 1.82107i
\(999\) 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.q.199.2 6
3.2 odd 2 825.2.c.f.199.5 6
5.2 odd 4 2475.2.a.z.1.3 3
5.3 odd 4 2475.2.a.bd.1.1 3
5.4 even 2 inner 2475.2.c.q.199.5 6
15.2 even 4 825.2.a.m.1.1 yes 3
15.8 even 4 825.2.a.i.1.3 3
15.14 odd 2 825.2.c.f.199.2 6
165.32 odd 4 9075.2.a.cd.1.3 3
165.98 odd 4 9075.2.a.cj.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
825.2.a.i.1.3 3 15.8 even 4
825.2.a.m.1.1 yes 3 15.2 even 4
825.2.c.f.199.2 6 15.14 odd 2
825.2.c.f.199.5 6 3.2 odd 2
2475.2.a.z.1.3 3 5.2 odd 4
2475.2.a.bd.1.1 3 5.3 odd 4
2475.2.c.q.199.2 6 1.1 even 1 trivial
2475.2.c.q.199.5 6 5.4 even 2 inner
9075.2.a.cd.1.3 3 165.32 odd 4
9075.2.a.cj.1.1 3 165.98 odd 4