Properties

Label 2475.2.c.f.199.2
Level $2475$
Weight $2$
Character 2475.199
Analytic conductor $19.763$
Analytic rank $0$
Dimension $2$
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: \(2\)
Coefficient field: \(\Q(i)\)
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 55)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 199.2
Root \(1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 2475.199
Dual form 2475.2.c.f.199.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000i q^{2} +1.00000 q^{4} +3.00000i q^{8} +O(q^{10})\) \(q+1.00000i q^{2} +1.00000 q^{4} +3.00000i q^{8} +1.00000 q^{11} +2.00000i q^{13} -1.00000 q^{16} +6.00000i q^{17} +4.00000 q^{19} +1.00000i q^{22} -4.00000i q^{23} -2.00000 q^{26} +6.00000 q^{29} -8.00000 q^{31} +5.00000i q^{32} -6.00000 q^{34} +2.00000i q^{37} +4.00000i q^{38} -2.00000 q^{41} +4.00000i q^{43} +1.00000 q^{44} +4.00000 q^{46} -12.0000i q^{47} +7.00000 q^{49} +2.00000i q^{52} +2.00000i q^{53} +6.00000i q^{58} +4.00000 q^{59} -10.0000 q^{61} -8.00000i q^{62} -7.00000 q^{64} +16.0000i q^{67} +6.00000i q^{68} -8.00000 q^{71} +14.0000i q^{73} -2.00000 q^{74} +4.00000 q^{76} -8.00000 q^{79} -2.00000i q^{82} +4.00000i q^{83} -4.00000 q^{86} +3.00000i q^{88} +10.0000 q^{89} -4.00000i q^{92} +12.0000 q^{94} -10.0000i q^{97} +7.00000i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{4} + 2 q^{11} - 2 q^{16} + 8 q^{19} - 4 q^{26} + 12 q^{29} - 16 q^{31} - 12 q^{34} - 4 q^{41} + 2 q^{44} + 8 q^{46} + 14 q^{49} + 8 q^{59} - 20 q^{61} - 14 q^{64} - 16 q^{71} - 4 q^{74} + 8 q^{76} - 16 q^{79} - 8 q^{86} + 20 q^{89} + 24 q^{94}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000i 0.707107i 0.935414 + 0.353553i \(0.115027\pi\)
−0.935414 + 0.353553i \(0.884973\pi\)
\(3\) 0 0
\(4\) 1.00000 0.500000
\(5\) 0 0
\(6\) 0 0
\(7\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(8\) 3.00000i 1.06066i
\(9\) 0 0
\(10\) 0 0
\(11\) 1.00000 0.301511
\(12\) 0 0
\(13\) 2.00000i 0.554700i 0.960769 + 0.277350i \(0.0894562\pi\)
−0.960769 + 0.277350i \(0.910544\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) −1.00000 −0.250000
\(17\) 6.00000i 1.45521i 0.685994 + 0.727607i \(0.259367\pi\)
−0.685994 + 0.727607i \(0.740633\pi\)
\(18\) 0 0
\(19\) 4.00000 0.917663 0.458831 0.888523i \(-0.348268\pi\)
0.458831 + 0.888523i \(0.348268\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 1.00000i 0.213201i
\(23\) − 4.00000i − 0.834058i −0.908893 0.417029i \(-0.863071\pi\)
0.908893 0.417029i \(-0.136929\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) −2.00000 −0.392232
\(27\) 0 0
\(28\) 0 0
\(29\) 6.00000 1.11417 0.557086 0.830455i \(-0.311919\pi\)
0.557086 + 0.830455i \(0.311919\pi\)
\(30\) 0 0
\(31\) −8.00000 −1.43684 −0.718421 0.695608i \(-0.755135\pi\)
−0.718421 + 0.695608i \(0.755135\pi\)
\(32\) 5.00000i 0.883883i
\(33\) 0 0
\(34\) −6.00000 −1.02899
\(35\) 0 0
\(36\) 0 0
\(37\) 2.00000i 0.328798i 0.986394 + 0.164399i \(0.0525685\pi\)
−0.986394 + 0.164399i \(0.947432\pi\)
\(38\) 4.00000i 0.648886i
\(39\) 0 0
\(40\) 0 0
\(41\) −2.00000 −0.312348 −0.156174 0.987730i \(-0.549916\pi\)
−0.156174 + 0.987730i \(0.549916\pi\)
\(42\) 0 0
\(43\) 4.00000i 0.609994i 0.952353 + 0.304997i \(0.0986555\pi\)
−0.952353 + 0.304997i \(0.901344\pi\)
\(44\) 1.00000 0.150756
\(45\) 0 0
\(46\) 4.00000 0.589768
\(47\) − 12.0000i − 1.75038i −0.483779 0.875190i \(-0.660736\pi\)
0.483779 0.875190i \(-0.339264\pi\)
\(48\) 0 0
\(49\) 7.00000 1.00000
\(50\) 0 0
\(51\) 0 0
\(52\) 2.00000i 0.277350i
\(53\) 2.00000i 0.274721i 0.990521 + 0.137361i \(0.0438619\pi\)
−0.990521 + 0.137361i \(0.956138\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 6.00000i 0.787839i
\(59\) 4.00000 0.520756 0.260378 0.965507i \(-0.416153\pi\)
0.260378 + 0.965507i \(0.416153\pi\)
\(60\) 0 0
\(61\) −10.0000 −1.28037 −0.640184 0.768221i \(-0.721142\pi\)
−0.640184 + 0.768221i \(0.721142\pi\)
\(62\) − 8.00000i − 1.01600i
\(63\) 0 0
\(64\) −7.00000 −0.875000
\(65\) 0 0
\(66\) 0 0
\(67\) 16.0000i 1.95471i 0.211604 + 0.977356i \(0.432131\pi\)
−0.211604 + 0.977356i \(0.567869\pi\)
\(68\) 6.00000i 0.727607i
\(69\) 0 0
\(70\) 0 0
\(71\) −8.00000 −0.949425 −0.474713 0.880141i \(-0.657448\pi\)
−0.474713 + 0.880141i \(0.657448\pi\)
\(72\) 0 0
\(73\) 14.0000i 1.63858i 0.573382 + 0.819288i \(0.305631\pi\)
−0.573382 + 0.819288i \(0.694369\pi\)
\(74\) −2.00000 −0.232495
\(75\) 0 0
\(76\) 4.00000 0.458831
\(77\) 0 0
\(78\) 0 0
\(79\) −8.00000 −0.900070 −0.450035 0.893011i \(-0.648589\pi\)
−0.450035 + 0.893011i \(0.648589\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) − 2.00000i − 0.220863i
\(83\) 4.00000i 0.439057i 0.975606 + 0.219529i \(0.0704519\pi\)
−0.975606 + 0.219529i \(0.929548\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −4.00000 −0.431331
\(87\) 0 0
\(88\) 3.00000i 0.319801i
\(89\) 10.0000 1.06000 0.529999 0.847998i \(-0.322192\pi\)
0.529999 + 0.847998i \(0.322192\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) − 4.00000i − 0.417029i
\(93\) 0 0
\(94\) 12.0000 1.23771
\(95\) 0 0
\(96\) 0 0
\(97\) − 10.0000i − 1.01535i −0.861550 0.507673i \(-0.830506\pi\)
0.861550 0.507673i \(-0.169494\pi\)
\(98\) 7.00000i 0.707107i
\(99\) 0 0
\(100\) 0 0
\(101\) 10.0000 0.995037 0.497519 0.867453i \(-0.334245\pi\)
0.497519 + 0.867453i \(0.334245\pi\)
\(102\) 0 0
\(103\) − 4.00000i − 0.394132i −0.980390 0.197066i \(-0.936859\pi\)
0.980390 0.197066i \(-0.0631413\pi\)
\(104\) −6.00000 −0.588348
\(105\) 0 0
\(106\) −2.00000 −0.194257
\(107\) 12.0000i 1.16008i 0.814587 + 0.580042i \(0.196964\pi\)
−0.814587 + 0.580042i \(0.803036\pi\)
\(108\) 0 0
\(109\) 18.0000 1.72409 0.862044 0.506834i \(-0.169184\pi\)
0.862044 + 0.506834i \(0.169184\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 6.00000i 0.564433i 0.959351 + 0.282216i \(0.0910696\pi\)
−0.959351 + 0.282216i \(0.908930\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 6.00000 0.557086
\(117\) 0 0
\(118\) 4.00000i 0.368230i
\(119\) 0 0
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) − 10.0000i − 0.905357i
\(123\) 0 0
\(124\) −8.00000 −0.718421
\(125\) 0 0
\(126\) 0 0
\(127\) − 16.0000i − 1.41977i −0.704317 0.709885i \(-0.748747\pi\)
0.704317 0.709885i \(-0.251253\pi\)
\(128\) 3.00000i 0.265165i
\(129\) 0 0
\(130\) 0 0
\(131\) 12.0000 1.04844 0.524222 0.851581i \(-0.324356\pi\)
0.524222 + 0.851581i \(0.324356\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) −16.0000 −1.38219
\(135\) 0 0
\(136\) −18.0000 −1.54349
\(137\) 18.0000i 1.53784i 0.639343 + 0.768922i \(0.279207\pi\)
−0.639343 + 0.768922i \(0.720793\pi\)
\(138\) 0 0
\(139\) −12.0000 −1.01783 −0.508913 0.860818i \(-0.669953\pi\)
−0.508913 + 0.860818i \(0.669953\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) − 8.00000i − 0.671345i
\(143\) 2.00000i 0.167248i
\(144\) 0 0
\(145\) 0 0
\(146\) −14.0000 −1.15865
\(147\) 0 0
\(148\) 2.00000i 0.164399i
\(149\) −10.0000 −0.819232 −0.409616 0.912258i \(-0.634337\pi\)
−0.409616 + 0.912258i \(0.634337\pi\)
\(150\) 0 0
\(151\) 8.00000 0.651031 0.325515 0.945537i \(-0.394462\pi\)
0.325515 + 0.945537i \(0.394462\pi\)
\(152\) 12.0000i 0.973329i
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 2.00000i 0.159617i 0.996810 + 0.0798087i \(0.0254309\pi\)
−0.996810 + 0.0798087i \(0.974569\pi\)
\(158\) − 8.00000i − 0.636446i
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 16.0000i 1.25322i 0.779334 + 0.626608i \(0.215557\pi\)
−0.779334 + 0.626608i \(0.784443\pi\)
\(164\) −2.00000 −0.156174
\(165\) 0 0
\(166\) −4.00000 −0.310460
\(167\) − 8.00000i − 0.619059i −0.950890 0.309529i \(-0.899829\pi\)
0.950890 0.309529i \(-0.100171\pi\)
\(168\) 0 0
\(169\) 9.00000 0.692308
\(170\) 0 0
\(171\) 0 0
\(172\) 4.00000i 0.304997i
\(173\) 6.00000i 0.456172i 0.973641 + 0.228086i \(0.0732467\pi\)
−0.973641 + 0.228086i \(0.926753\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −1.00000 −0.0753778
\(177\) 0 0
\(178\) 10.0000i 0.749532i
\(179\) 4.00000 0.298974 0.149487 0.988764i \(-0.452238\pi\)
0.149487 + 0.988764i \(0.452238\pi\)
\(180\) 0 0
\(181\) −10.0000 −0.743294 −0.371647 0.928374i \(-0.621207\pi\)
−0.371647 + 0.928374i \(0.621207\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 12.0000 0.884652
\(185\) 0 0
\(186\) 0 0
\(187\) 6.00000i 0.438763i
\(188\) − 12.0000i − 0.875190i
\(189\) 0 0
\(190\) 0 0
\(191\) −8.00000 −0.578860 −0.289430 0.957199i \(-0.593466\pi\)
−0.289430 + 0.957199i \(0.593466\pi\)
\(192\) 0 0
\(193\) − 26.0000i − 1.87152i −0.352636 0.935760i \(-0.614715\pi\)
0.352636 0.935760i \(-0.385285\pi\)
\(194\) 10.0000 0.717958
\(195\) 0 0
\(196\) 7.00000 0.500000
\(197\) 2.00000i 0.142494i 0.997459 + 0.0712470i \(0.0226979\pi\)
−0.997459 + 0.0712470i \(0.977302\pi\)
\(198\) 0 0
\(199\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 10.0000i 0.703598i
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 4.00000 0.278693
\(207\) 0 0
\(208\) − 2.00000i − 0.138675i
\(209\) 4.00000 0.276686
\(210\) 0 0
\(211\) 4.00000 0.275371 0.137686 0.990476i \(-0.456034\pi\)
0.137686 + 0.990476i \(0.456034\pi\)
\(212\) 2.00000i 0.137361i
\(213\) 0 0
\(214\) −12.0000 −0.820303
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) 18.0000i 1.21911i
\(219\) 0 0
\(220\) 0 0
\(221\) −12.0000 −0.807207
\(222\) 0 0
\(223\) − 4.00000i − 0.267860i −0.990991 0.133930i \(-0.957240\pi\)
0.990991 0.133930i \(-0.0427597\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) −6.00000 −0.399114
\(227\) − 20.0000i − 1.32745i −0.747978 0.663723i \(-0.768975\pi\)
0.747978 0.663723i \(-0.231025\pi\)
\(228\) 0 0
\(229\) 10.0000 0.660819 0.330409 0.943838i \(-0.392813\pi\)
0.330409 + 0.943838i \(0.392813\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 18.0000i 1.18176i
\(233\) − 6.00000i − 0.393073i −0.980497 0.196537i \(-0.937031\pi\)
0.980497 0.196537i \(-0.0629694\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 4.00000 0.260378
\(237\) 0 0
\(238\) 0 0
\(239\) 8.00000 0.517477 0.258738 0.965947i \(-0.416693\pi\)
0.258738 + 0.965947i \(0.416693\pi\)
\(240\) 0 0
\(241\) 10.0000 0.644157 0.322078 0.946713i \(-0.395619\pi\)
0.322078 + 0.946713i \(0.395619\pi\)
\(242\) 1.00000i 0.0642824i
\(243\) 0 0
\(244\) −10.0000 −0.640184
\(245\) 0 0
\(246\) 0 0
\(247\) 8.00000i 0.509028i
\(248\) − 24.0000i − 1.52400i
\(249\) 0 0
\(250\) 0 0
\(251\) −12.0000 −0.757433 −0.378717 0.925513i \(-0.623635\pi\)
−0.378717 + 0.925513i \(0.623635\pi\)
\(252\) 0 0
\(253\) − 4.00000i − 0.251478i
\(254\) 16.0000 1.00393
\(255\) 0 0
\(256\) −17.0000 −1.06250
\(257\) 18.0000i 1.12281i 0.827541 + 0.561405i \(0.189739\pi\)
−0.827541 + 0.561405i \(0.810261\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) 12.0000i 0.741362i
\(263\) − 24.0000i − 1.47990i −0.672660 0.739952i \(-0.734848\pi\)
0.672660 0.739952i \(-0.265152\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 16.0000i 0.977356i
\(269\) −18.0000 −1.09748 −0.548740 0.835993i \(-0.684892\pi\)
−0.548740 + 0.835993i \(0.684892\pi\)
\(270\) 0 0
\(271\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(272\) − 6.00000i − 0.363803i
\(273\) 0 0
\(274\) −18.0000 −1.08742
\(275\) 0 0
\(276\) 0 0
\(277\) − 10.0000i − 0.600842i −0.953807 0.300421i \(-0.902873\pi\)
0.953807 0.300421i \(-0.0971271\pi\)
\(278\) − 12.0000i − 0.719712i
\(279\) 0 0
\(280\) 0 0
\(281\) −18.0000 −1.07379 −0.536895 0.843649i \(-0.680403\pi\)
−0.536895 + 0.843649i \(0.680403\pi\)
\(282\) 0 0
\(283\) 4.00000i 0.237775i 0.992908 + 0.118888i \(0.0379328\pi\)
−0.992908 + 0.118888i \(0.962067\pi\)
\(284\) −8.00000 −0.474713
\(285\) 0 0
\(286\) −2.00000 −0.118262
\(287\) 0 0
\(288\) 0 0
\(289\) −19.0000 −1.11765
\(290\) 0 0
\(291\) 0 0
\(292\) 14.0000i 0.819288i
\(293\) − 10.0000i − 0.584206i −0.956387 0.292103i \(-0.905645\pi\)
0.956387 0.292103i \(-0.0943550\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) −6.00000 −0.348743
\(297\) 0 0
\(298\) − 10.0000i − 0.579284i
\(299\) 8.00000 0.462652
\(300\) 0 0
\(301\) 0 0
\(302\) 8.00000i 0.460348i
\(303\) 0 0
\(304\) −4.00000 −0.229416
\(305\) 0 0
\(306\) 0 0
\(307\) − 20.0000i − 1.14146i −0.821138 0.570730i \(-0.806660\pi\)
0.821138 0.570730i \(-0.193340\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 24.0000 1.36092 0.680458 0.732787i \(-0.261781\pi\)
0.680458 + 0.732787i \(0.261781\pi\)
\(312\) 0 0
\(313\) − 22.0000i − 1.24351i −0.783210 0.621757i \(-0.786419\pi\)
0.783210 0.621757i \(-0.213581\pi\)
\(314\) −2.00000 −0.112867
\(315\) 0 0
\(316\) −8.00000 −0.450035
\(317\) − 18.0000i − 1.01098i −0.862832 0.505490i \(-0.831312\pi\)
0.862832 0.505490i \(-0.168688\pi\)
\(318\) 0 0
\(319\) 6.00000 0.335936
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 24.0000i 1.33540i
\(324\) 0 0
\(325\) 0 0
\(326\) −16.0000 −0.886158
\(327\) 0 0
\(328\) − 6.00000i − 0.331295i
\(329\) 0 0
\(330\) 0 0
\(331\) 4.00000 0.219860 0.109930 0.993939i \(-0.464937\pi\)
0.109930 + 0.993939i \(0.464937\pi\)
\(332\) 4.00000i 0.219529i
\(333\) 0 0
\(334\) 8.00000 0.437741
\(335\) 0 0
\(336\) 0 0
\(337\) − 6.00000i − 0.326841i −0.986557 0.163420i \(-0.947747\pi\)
0.986557 0.163420i \(-0.0522527\pi\)
\(338\) 9.00000i 0.489535i
\(339\) 0 0
\(340\) 0 0
\(341\) −8.00000 −0.433224
\(342\) 0 0
\(343\) 0 0
\(344\) −12.0000 −0.646997
\(345\) 0 0
\(346\) −6.00000 −0.322562
\(347\) − 4.00000i − 0.214731i −0.994220 0.107366i \(-0.965758\pi\)
0.994220 0.107366i \(-0.0342415\pi\)
\(348\) 0 0
\(349\) 10.0000 0.535288 0.267644 0.963518i \(-0.413755\pi\)
0.267644 + 0.963518i \(0.413755\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 5.00000i 0.266501i
\(353\) − 18.0000i − 0.958043i −0.877803 0.479022i \(-0.840992\pi\)
0.877803 0.479022i \(-0.159008\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 10.0000 0.529999
\(357\) 0 0
\(358\) 4.00000i 0.211407i
\(359\) −32.0000 −1.68890 −0.844448 0.535638i \(-0.820071\pi\)
−0.844448 + 0.535638i \(0.820071\pi\)
\(360\) 0 0
\(361\) −3.00000 −0.157895
\(362\) − 10.0000i − 0.525588i
\(363\) 0 0
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) − 4.00000i − 0.208798i −0.994535 0.104399i \(-0.966708\pi\)
0.994535 0.104399i \(-0.0332919\pi\)
\(368\) 4.00000i 0.208514i
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 18.0000i 0.932005i 0.884783 + 0.466002i \(0.154306\pi\)
−0.884783 + 0.466002i \(0.845694\pi\)
\(374\) −6.00000 −0.310253
\(375\) 0 0
\(376\) 36.0000 1.85656
\(377\) 12.0000i 0.618031i
\(378\) 0 0
\(379\) −20.0000 −1.02733 −0.513665 0.857991i \(-0.671713\pi\)
−0.513665 + 0.857991i \(0.671713\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) − 8.00000i − 0.409316i
\(383\) 12.0000i 0.613171i 0.951843 + 0.306586i \(0.0991866\pi\)
−0.951843 + 0.306586i \(0.900813\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 26.0000 1.32337
\(387\) 0 0
\(388\) − 10.0000i − 0.507673i
\(389\) 6.00000 0.304212 0.152106 0.988364i \(-0.451394\pi\)
0.152106 + 0.988364i \(0.451394\pi\)
\(390\) 0 0
\(391\) 24.0000 1.21373
\(392\) 21.0000i 1.06066i
\(393\) 0 0
\(394\) −2.00000 −0.100759
\(395\) 0 0
\(396\) 0 0
\(397\) − 30.0000i − 1.50566i −0.658217 0.752828i \(-0.728689\pi\)
0.658217 0.752828i \(-0.271311\pi\)
\(398\) 0 0
\(399\) 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
\(403\) − 16.0000i − 0.797017i
\(404\) 10.0000 0.497519
\(405\) 0 0
\(406\) 0 0
\(407\) 2.00000i 0.0991363i
\(408\) 0 0
\(409\) 6.00000 0.296681 0.148340 0.988936i \(-0.452607\pi\)
0.148340 + 0.988936i \(0.452607\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) − 4.00000i − 0.197066i
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) −10.0000 −0.490290
\(417\) 0 0
\(418\) 4.00000i 0.195646i
\(419\) −28.0000 −1.36789 −0.683945 0.729534i \(-0.739737\pi\)
−0.683945 + 0.729534i \(0.739737\pi\)
\(420\) 0 0
\(421\) 6.00000 0.292422 0.146211 0.989253i \(-0.453292\pi\)
0.146211 + 0.989253i \(0.453292\pi\)
\(422\) 4.00000i 0.194717i
\(423\) 0 0
\(424\) −6.00000 −0.291386
\(425\) 0 0
\(426\) 0 0
\(427\) 0 0
\(428\) 12.0000i 0.580042i
\(429\) 0 0
\(430\) 0 0
\(431\) 24.0000 1.15604 0.578020 0.816023i \(-0.303826\pi\)
0.578020 + 0.816023i \(0.303826\pi\)
\(432\) 0 0
\(433\) − 22.0000i − 1.05725i −0.848855 0.528626i \(-0.822707\pi\)
0.848855 0.528626i \(-0.177293\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 18.0000 0.862044
\(437\) − 16.0000i − 0.765384i
\(438\) 0 0
\(439\) 8.00000 0.381819 0.190910 0.981608i \(-0.438856\pi\)
0.190910 + 0.981608i \(0.438856\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) − 12.0000i − 0.570782i
\(443\) − 8.00000i − 0.380091i −0.981775 0.190046i \(-0.939136\pi\)
0.981775 0.190046i \(-0.0608636\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 4.00000 0.189405
\(447\) 0 0
\(448\) 0 0
\(449\) 2.00000 0.0943858 0.0471929 0.998886i \(-0.484972\pi\)
0.0471929 + 0.998886i \(0.484972\pi\)
\(450\) 0 0
\(451\) −2.00000 −0.0941763
\(452\) 6.00000i 0.282216i
\(453\) 0 0
\(454\) 20.0000 0.938647
\(455\) 0 0
\(456\) 0 0
\(457\) 26.0000i 1.21623i 0.793849 + 0.608114i \(0.208074\pi\)
−0.793849 + 0.608114i \(0.791926\pi\)
\(458\) 10.0000i 0.467269i
\(459\) 0 0
\(460\) 0 0
\(461\) 34.0000 1.58354 0.791769 0.610821i \(-0.209160\pi\)
0.791769 + 0.610821i \(0.209160\pi\)
\(462\) 0 0
\(463\) − 36.0000i − 1.67306i −0.547920 0.836531i \(-0.684580\pi\)
0.547920 0.836531i \(-0.315420\pi\)
\(464\) −6.00000 −0.278543
\(465\) 0 0
\(466\) 6.00000 0.277945
\(467\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 12.0000i 0.552345i
\(473\) 4.00000i 0.183920i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 0 0
\(478\) 8.00000i 0.365911i
\(479\) 24.0000 1.09659 0.548294 0.836286i \(-0.315277\pi\)
0.548294 + 0.836286i \(0.315277\pi\)
\(480\) 0 0
\(481\) −4.00000 −0.182384
\(482\) 10.0000i 0.455488i
\(483\) 0 0
\(484\) 1.00000 0.0454545
\(485\) 0 0
\(486\) 0 0
\(487\) − 28.0000i − 1.26880i −0.773004 0.634401i \(-0.781247\pi\)
0.773004 0.634401i \(-0.218753\pi\)
\(488\) − 30.0000i − 1.35804i
\(489\) 0 0
\(490\) 0 0
\(491\) 28.0000 1.26362 0.631811 0.775122i \(-0.282312\pi\)
0.631811 + 0.775122i \(0.282312\pi\)
\(492\) 0 0
\(493\) 36.0000i 1.62136i
\(494\) −8.00000 −0.359937
\(495\) 0 0
\(496\) 8.00000 0.359211
\(497\) 0 0
\(498\) 0 0
\(499\) 36.0000 1.61158 0.805791 0.592200i \(-0.201741\pi\)
0.805791 + 0.592200i \(0.201741\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) − 12.0000i − 0.535586i
\(503\) 16.0000i 0.713405i 0.934218 + 0.356702i \(0.116099\pi\)
−0.934218 + 0.356702i \(0.883901\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 4.00000 0.177822
\(507\) 0 0
\(508\) − 16.0000i − 0.709885i
\(509\) 14.0000 0.620539 0.310270 0.950649i \(-0.399581\pi\)
0.310270 + 0.950649i \(0.399581\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) − 11.0000i − 0.486136i
\(513\) 0 0
\(514\) −18.0000 −0.793946
\(515\) 0 0
\(516\) 0 0
\(517\) − 12.0000i − 0.527759i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 6.00000 0.262865 0.131432 0.991325i \(-0.458042\pi\)
0.131432 + 0.991325i \(0.458042\pi\)
\(522\) 0 0
\(523\) − 20.0000i − 0.874539i −0.899331 0.437269i \(-0.855946\pi\)
0.899331 0.437269i \(-0.144054\pi\)
\(524\) 12.0000 0.524222
\(525\) 0 0
\(526\) 24.0000 1.04645
\(527\) − 48.0000i − 2.09091i
\(528\) 0 0
\(529\) 7.00000 0.304348
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) − 4.00000i − 0.173259i
\(534\) 0 0
\(535\) 0 0
\(536\) −48.0000 −2.07328
\(537\) 0 0
\(538\) − 18.0000i − 0.776035i
\(539\) 7.00000 0.301511
\(540\) 0 0
\(541\) −34.0000 −1.46177 −0.730887 0.682498i \(-0.760893\pi\)
−0.730887 + 0.682498i \(0.760893\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) −30.0000 −1.28624
\(545\) 0 0
\(546\) 0 0
\(547\) − 12.0000i − 0.513083i −0.966533 0.256541i \(-0.917417\pi\)
0.966533 0.256541i \(-0.0825830\pi\)
\(548\) 18.0000i 0.768922i
\(549\) 0 0
\(550\) 0 0
\(551\) 24.0000 1.02243
\(552\) 0 0
\(553\) 0 0
\(554\) 10.0000 0.424859
\(555\) 0 0
\(556\) −12.0000 −0.508913
\(557\) 10.0000i 0.423714i 0.977301 + 0.211857i \(0.0679510\pi\)
−0.977301 + 0.211857i \(0.932049\pi\)
\(558\) 0 0
\(559\) −8.00000 −0.338364
\(560\) 0 0
\(561\) 0 0
\(562\) − 18.0000i − 0.759284i
\(563\) − 36.0000i − 1.51722i −0.651546 0.758610i \(-0.725879\pi\)
0.651546 0.758610i \(-0.274121\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) −4.00000 −0.168133
\(567\) 0 0
\(568\) − 24.0000i − 1.00702i
\(569\) 26.0000 1.08998 0.544988 0.838444i \(-0.316534\pi\)
0.544988 + 0.838444i \(0.316534\pi\)
\(570\) 0 0
\(571\) 36.0000 1.50655 0.753277 0.657704i \(-0.228472\pi\)
0.753277 + 0.657704i \(0.228472\pi\)
\(572\) 2.00000i 0.0836242i
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 22.0000i 0.915872i 0.888985 + 0.457936i \(0.151411\pi\)
−0.888985 + 0.457936i \(0.848589\pi\)
\(578\) − 19.0000i − 0.790296i
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 2.00000i 0.0828315i
\(584\) −42.0000 −1.73797
\(585\) 0 0
\(586\) 10.0000 0.413096
\(587\) − 24.0000i − 0.990586i −0.868726 0.495293i \(-0.835061\pi\)
0.868726 0.495293i \(-0.164939\pi\)
\(588\) 0 0
\(589\) −32.0000 −1.31854
\(590\) 0 0
\(591\) 0 0
\(592\) − 2.00000i − 0.0821995i
\(593\) − 22.0000i − 0.903432i −0.892162 0.451716i \(-0.850812\pi\)
0.892162 0.451716i \(-0.149188\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −10.0000 −0.409616
\(597\) 0 0
\(598\) 8.00000i 0.327144i
\(599\) 24.0000 0.980613 0.490307 0.871550i \(-0.336885\pi\)
0.490307 + 0.871550i \(0.336885\pi\)
\(600\) 0 0
\(601\) 2.00000 0.0815817 0.0407909 0.999168i \(-0.487012\pi\)
0.0407909 + 0.999168i \(0.487012\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 8.00000 0.325515
\(605\) 0 0
\(606\) 0 0
\(607\) 32.0000i 1.29884i 0.760430 + 0.649420i \(0.224988\pi\)
−0.760430 + 0.649420i \(0.775012\pi\)
\(608\) 20.0000i 0.811107i
\(609\) 0 0
\(610\) 0 0
\(611\) 24.0000 0.970936
\(612\) 0 0
\(613\) 34.0000i 1.37325i 0.727013 + 0.686624i \(0.240908\pi\)
−0.727013 + 0.686624i \(0.759092\pi\)
\(614\) 20.0000 0.807134
\(615\) 0 0
\(616\) 0 0
\(617\) 2.00000i 0.0805170i 0.999189 + 0.0402585i \(0.0128181\pi\)
−0.999189 + 0.0402585i \(0.987182\pi\)
\(618\) 0 0
\(619\) 20.0000 0.803868 0.401934 0.915669i \(-0.368338\pi\)
0.401934 + 0.915669i \(0.368338\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 24.0000i 0.962312i
\(623\) 0 0
\(624\) 0 0
\(625\) 0 0
\(626\) 22.0000 0.879297
\(627\) 0 0
\(628\) 2.00000i 0.0798087i
\(629\) −12.0000 −0.478471
\(630\) 0 0
\(631\) 40.0000 1.59237 0.796187 0.605050i \(-0.206847\pi\)
0.796187 + 0.605050i \(0.206847\pi\)
\(632\) − 24.0000i − 0.954669i
\(633\) 0 0
\(634\) 18.0000 0.714871
\(635\) 0 0
\(636\) 0 0
\(637\) 14.0000i 0.554700i
\(638\) 6.00000i 0.237542i
\(639\) 0 0
\(640\) 0 0
\(641\) −34.0000 −1.34292 −0.671460 0.741041i \(-0.734332\pi\)
−0.671460 + 0.741041i \(0.734332\pi\)
\(642\) 0 0
\(643\) − 16.0000i − 0.630978i −0.948929 0.315489i \(-0.897831\pi\)
0.948929 0.315489i \(-0.102169\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) −24.0000 −0.944267
\(647\) 20.0000i 0.786281i 0.919478 + 0.393141i \(0.128611\pi\)
−0.919478 + 0.393141i \(0.871389\pi\)
\(648\) 0 0
\(649\) 4.00000 0.157014
\(650\) 0 0
\(651\) 0 0
\(652\) 16.0000i 0.626608i
\(653\) 10.0000i 0.391330i 0.980671 + 0.195665i \(0.0626866\pi\)
−0.980671 + 0.195665i \(0.937313\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 2.00000 0.0780869
\(657\) 0 0
\(658\) 0 0
\(659\) 36.0000 1.40236 0.701180 0.712984i \(-0.252657\pi\)
0.701180 + 0.712984i \(0.252657\pi\)
\(660\) 0 0
\(661\) −10.0000 −0.388955 −0.194477 0.980907i \(-0.562301\pi\)
−0.194477 + 0.980907i \(0.562301\pi\)
\(662\) 4.00000i 0.155464i
\(663\) 0 0
\(664\) −12.0000 −0.465690
\(665\) 0 0
\(666\) 0 0
\(667\) − 24.0000i − 0.929284i
\(668\) − 8.00000i − 0.309529i
\(669\) 0 0
\(670\) 0 0
\(671\) −10.0000 −0.386046
\(672\) 0 0
\(673\) − 26.0000i − 1.00223i −0.865382 0.501113i \(-0.832924\pi\)
0.865382 0.501113i \(-0.167076\pi\)
\(674\) 6.00000 0.231111
\(675\) 0 0
\(676\) 9.00000 0.346154
\(677\) − 38.0000i − 1.46046i −0.683202 0.730229i \(-0.739413\pi\)
0.683202 0.730229i \(-0.260587\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) − 8.00000i − 0.306336i
\(683\) 16.0000i 0.612223i 0.951996 + 0.306111i \(0.0990280\pi\)
−0.951996 + 0.306111i \(0.900972\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 0 0
\(688\) − 4.00000i − 0.152499i
\(689\) −4.00000 −0.152388
\(690\) 0 0
\(691\) −28.0000 −1.06517 −0.532585 0.846376i \(-0.678779\pi\)
−0.532585 + 0.846376i \(0.678779\pi\)
\(692\) 6.00000i 0.228086i
\(693\) 0 0
\(694\) 4.00000 0.151838
\(695\) 0 0
\(696\) 0 0
\(697\) − 12.0000i − 0.454532i
\(698\) 10.0000i 0.378506i
\(699\) 0 0
\(700\) 0 0
\(701\) −22.0000 −0.830929 −0.415464 0.909610i \(-0.636381\pi\)
−0.415464 + 0.909610i \(0.636381\pi\)
\(702\) 0 0
\(703\) 8.00000i 0.301726i
\(704\) −7.00000 −0.263822
\(705\) 0 0
\(706\) 18.0000 0.677439
\(707\) 0 0
\(708\) 0 0
\(709\) 10.0000 0.375558 0.187779 0.982211i \(-0.439871\pi\)
0.187779 + 0.982211i \(0.439871\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 30.0000i 1.12430i
\(713\) 32.0000i 1.19841i
\(714\) 0 0
\(715\) 0 0
\(716\) 4.00000 0.149487
\(717\) 0 0
\(718\) − 32.0000i − 1.19423i
\(719\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) − 3.00000i − 0.111648i
\(723\) 0 0
\(724\) −10.0000 −0.371647
\(725\) 0 0
\(726\) 0 0
\(727\) − 52.0000i − 1.92857i −0.264861 0.964287i \(-0.585326\pi\)
0.264861 0.964287i \(-0.414674\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −24.0000 −0.887672
\(732\) 0 0
\(733\) 42.0000i 1.55131i 0.631160 + 0.775653i \(0.282579\pi\)
−0.631160 + 0.775653i \(0.717421\pi\)
\(734\) 4.00000 0.147643
\(735\) 0 0
\(736\) 20.0000 0.737210
\(737\) 16.0000i 0.589368i
\(738\) 0 0
\(739\) −4.00000 −0.147142 −0.0735712 0.997290i \(-0.523440\pi\)
−0.0735712 + 0.997290i \(0.523440\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) − 40.0000i − 1.46746i −0.679442 0.733729i \(-0.737778\pi\)
0.679442 0.733729i \(-0.262222\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) −18.0000 −0.659027
\(747\) 0 0
\(748\) 6.00000i 0.219382i
\(749\) 0 0
\(750\) 0 0
\(751\) 16.0000 0.583848 0.291924 0.956441i \(-0.405705\pi\)
0.291924 + 0.956441i \(0.405705\pi\)
\(752\) 12.0000i 0.437595i
\(753\) 0 0
\(754\) −12.0000 −0.437014
\(755\) 0 0
\(756\) 0 0
\(757\) − 6.00000i − 0.218074i −0.994038 0.109037i \(-0.965223\pi\)
0.994038 0.109037i \(-0.0347767\pi\)
\(758\) − 20.0000i − 0.726433i
\(759\) 0 0
\(760\) 0 0
\(761\) −10.0000 −0.362500 −0.181250 0.983437i \(-0.558014\pi\)
−0.181250 + 0.983437i \(0.558014\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) −8.00000 −0.289430
\(765\) 0 0
\(766\) −12.0000 −0.433578
\(767\) 8.00000i 0.288863i
\(768\) 0 0
\(769\) 22.0000 0.793340 0.396670 0.917961i \(-0.370166\pi\)
0.396670 + 0.917961i \(0.370166\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) − 26.0000i − 0.935760i
\(773\) − 14.0000i − 0.503545i −0.967786 0.251773i \(-0.918987\pi\)
0.967786 0.251773i \(-0.0810135\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 30.0000 1.07694
\(777\) 0 0
\(778\) 6.00000i 0.215110i
\(779\) −8.00000 −0.286630
\(780\) 0 0
\(781\) −8.00000 −0.286263
\(782\) 24.0000i 0.858238i
\(783\) 0 0
\(784\) −7.00000 −0.250000
\(785\) 0 0
\(786\) 0 0
\(787\) − 52.0000i − 1.85360i −0.375555 0.926800i \(-0.622548\pi\)
0.375555 0.926800i \(-0.377452\pi\)
\(788\) 2.00000i 0.0712470i
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) − 20.0000i − 0.710221i
\(794\) 30.0000 1.06466
\(795\) 0 0
\(796\) 0 0
\(797\) − 2.00000i − 0.0708436i −0.999372 0.0354218i \(-0.988723\pi\)
0.999372 0.0354218i \(-0.0112775\pi\)
\(798\) 0 0
\(799\) 72.0000 2.54718
\(800\) 0 0
\(801\) 0 0
\(802\) − 2.00000i − 0.0706225i
\(803\) 14.0000i 0.494049i
\(804\) 0 0
\(805\) 0 0
\(806\) 16.0000 0.563576
\(807\) 0 0
\(808\) 30.0000i 1.05540i
\(809\) 18.0000 0.632846 0.316423 0.948618i \(-0.397518\pi\)
0.316423 + 0.948618i \(0.397518\pi\)
\(810\) 0 0
\(811\) −12.0000 −0.421377 −0.210688 0.977553i \(-0.567571\pi\)
−0.210688 + 0.977553i \(0.567571\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) −2.00000 −0.0701000
\(815\) 0 0
\(816\) 0 0
\(817\) 16.0000i 0.559769i
\(818\) 6.00000i 0.209785i
\(819\) 0 0
\(820\) 0 0
\(821\) 18.0000 0.628204 0.314102 0.949389i \(-0.398297\pi\)
0.314102 + 0.949389i \(0.398297\pi\)
\(822\) 0 0
\(823\) − 36.0000i − 1.25488i −0.778664 0.627441i \(-0.784103\pi\)
0.778664 0.627441i \(-0.215897\pi\)
\(824\) 12.0000 0.418040
\(825\) 0 0
\(826\) 0 0
\(827\) 12.0000i 0.417281i 0.977992 + 0.208640i \(0.0669038\pi\)
−0.977992 + 0.208640i \(0.933096\pi\)
\(828\) 0 0
\(829\) 2.00000 0.0694629 0.0347314 0.999397i \(-0.488942\pi\)
0.0347314 + 0.999397i \(0.488942\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) − 14.0000i − 0.485363i
\(833\) 42.0000i 1.45521i
\(834\) 0 0
\(835\) 0 0
\(836\) 4.00000 0.138343
\(837\) 0 0
\(838\) − 28.0000i − 0.967244i
\(839\) −32.0000 −1.10476 −0.552381 0.833592i \(-0.686281\pi\)
−0.552381 + 0.833592i \(0.686281\pi\)
\(840\) 0 0
\(841\) 7.00000 0.241379
\(842\) 6.00000i 0.206774i
\(843\) 0 0
\(844\) 4.00000 0.137686
\(845\) 0 0
\(846\) 0 0
\(847\) 0 0
\(848\) − 2.00000i − 0.0686803i
\(849\) 0 0
\(850\) 0 0
\(851\) 8.00000 0.274236
\(852\) 0 0
\(853\) 2.00000i 0.0684787i 0.999414 + 0.0342393i \(0.0109009\pi\)
−0.999414 + 0.0342393i \(0.989099\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) −36.0000 −1.23045
\(857\) − 18.0000i − 0.614868i −0.951569 0.307434i \(-0.900530\pi\)
0.951569 0.307434i \(-0.0994704\pi\)
\(858\) 0 0
\(859\) −28.0000 −0.955348 −0.477674 0.878537i \(-0.658520\pi\)
−0.477674 + 0.878537i \(0.658520\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 24.0000i 0.817443i
\(863\) − 4.00000i − 0.136162i −0.997680 0.0680808i \(-0.978312\pi\)
0.997680 0.0680808i \(-0.0216876\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 22.0000 0.747590
\(867\) 0 0
\(868\) 0 0
\(869\) −8.00000 −0.271381
\(870\) 0 0
\(871\) −32.0000 −1.08428
\(872\) 54.0000i 1.82867i
\(873\) 0 0
\(874\) 16.0000 0.541208
\(875\) 0 0
\(876\) 0 0
\(877\) 22.0000i 0.742887i 0.928456 + 0.371444i \(0.121137\pi\)
−0.928456 + 0.371444i \(0.878863\pi\)
\(878\) 8.00000i 0.269987i
\(879\) 0 0
\(880\) 0 0
\(881\) −18.0000 −0.606435 −0.303218 0.952921i \(-0.598061\pi\)
−0.303218 + 0.952921i \(0.598061\pi\)
\(882\) 0 0
\(883\) − 16.0000i − 0.538443i −0.963078 0.269221i \(-0.913234\pi\)
0.963078 0.269221i \(-0.0867663\pi\)
\(884\) −12.0000 −0.403604
\(885\) 0 0
\(886\) 8.00000 0.268765
\(887\) 56.0000i 1.88030i 0.340766 + 0.940148i \(0.389313\pi\)
−0.340766 + 0.940148i \(0.610687\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 0 0
\(892\) − 4.00000i − 0.133930i
\(893\) − 48.0000i − 1.60626i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) 2.00000i 0.0667409i
\(899\) −48.0000 −1.60089
\(900\) 0 0
\(901\) −12.0000 −0.399778
\(902\) − 2.00000i − 0.0665927i
\(903\) 0 0
\(904\) −18.0000 −0.598671
\(905\) 0 0
\(906\) 0 0
\(907\) 40.0000i 1.32818i 0.747653 + 0.664089i \(0.231180\pi\)
−0.747653 + 0.664089i \(0.768820\pi\)
\(908\) − 20.0000i − 0.663723i
\(909\) 0 0
\(910\) 0 0
\(911\) −8.00000 −0.265052 −0.132526 0.991180i \(-0.542309\pi\)
−0.132526 + 0.991180i \(0.542309\pi\)
\(912\) 0 0
\(913\) 4.00000i 0.132381i
\(914\) −26.0000 −0.860004
\(915\) 0 0
\(916\) 10.0000 0.330409
\(917\) 0 0
\(918\) 0 0
\(919\) −24.0000 −0.791687 −0.395843 0.918318i \(-0.629548\pi\)
−0.395843 + 0.918318i \(0.629548\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 34.0000i 1.11973i
\(923\) − 16.0000i − 0.526646i
\(924\) 0 0
\(925\) 0 0
\(926\) 36.0000 1.18303
\(927\) 0 0
\(928\) 30.0000i 0.984798i
\(929\) −14.0000 −0.459325 −0.229663 0.973270i \(-0.573762\pi\)
−0.229663 + 0.973270i \(0.573762\pi\)
\(930\) 0 0
\(931\) 28.0000 0.917663
\(932\) − 6.00000i − 0.196537i
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) − 6.00000i − 0.196011i −0.995186 0.0980057i \(-0.968754\pi\)
0.995186 0.0980057i \(-0.0312463\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 2.00000 0.0651981 0.0325991 0.999469i \(-0.489622\pi\)
0.0325991 + 0.999469i \(0.489622\pi\)
\(942\) 0 0
\(943\) 8.00000i 0.260516i
\(944\) −4.00000 −0.130189
\(945\) 0 0
\(946\) −4.00000 −0.130051
\(947\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(948\) 0 0
\(949\) −28.0000 −0.908918
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 42.0000i 1.36051i 0.732974 + 0.680257i \(0.238132\pi\)
−0.732974 + 0.680257i \(0.761868\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 8.00000 0.258738
\(957\) 0 0
\(958\) 24.0000i 0.775405i
\(959\) 0 0
\(960\) 0 0
\(961\) 33.0000 1.06452
\(962\) − 4.00000i − 0.128965i
\(963\) 0 0
\(964\) 10.0000 0.322078
\(965\) 0 0
\(966\) 0 0
\(967\) − 8.00000i − 0.257263i −0.991692 0.128631i \(-0.958942\pi\)
0.991692 0.128631i \(-0.0410584\pi\)
\(968\) 3.00000i 0.0964237i
\(969\) 0 0
\(970\) 0 0
\(971\) −36.0000 −1.15529 −0.577647 0.816286i \(-0.696029\pi\)
−0.577647 + 0.816286i \(0.696029\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 28.0000 0.897178
\(975\) 0 0
\(976\) 10.0000 0.320092
\(977\) 10.0000i 0.319928i 0.987123 + 0.159964i \(0.0511379\pi\)
−0.987123 + 0.159964i \(0.948862\pi\)
\(978\) 0 0
\(979\) 10.0000 0.319601
\(980\) 0 0
\(981\) 0 0
\(982\) 28.0000i 0.893516i
\(983\) 4.00000i 0.127580i 0.997963 + 0.0637901i \(0.0203188\pi\)
−0.997963 + 0.0637901i \(0.979681\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) −36.0000 −1.14647
\(987\) 0 0
\(988\) 8.00000i 0.254514i
\(989\) 16.0000 0.508770
\(990\) 0 0
\(991\) 8.00000 0.254128 0.127064 0.991894i \(-0.459445\pi\)
0.127064 + 0.991894i \(0.459445\pi\)
\(992\) − 40.0000i − 1.27000i
\(993\) 0 0
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 46.0000i 1.45683i 0.685134 + 0.728417i \(0.259744\pi\)
−0.685134 + 0.728417i \(0.740256\pi\)
\(998\) 36.0000i 1.13956i
\(999\) 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.f.199.2 2
3.2 odd 2 275.2.b.b.199.1 2
5.2 odd 4 495.2.a.a.1.1 1
5.3 odd 4 2475.2.a.i.1.1 1
5.4 even 2 inner 2475.2.c.f.199.1 2
12.11 even 2 4400.2.b.n.4049.2 2
15.2 even 4 55.2.a.a.1.1 1
15.8 even 4 275.2.a.a.1.1 1
15.14 odd 2 275.2.b.b.199.2 2
20.7 even 4 7920.2.a.i.1.1 1
55.32 even 4 5445.2.a.i.1.1 1
60.23 odd 4 4400.2.a.p.1.1 1
60.47 odd 4 880.2.a.h.1.1 1
60.59 even 2 4400.2.b.n.4049.1 2
105.62 odd 4 2695.2.a.c.1.1 1
120.77 even 4 3520.2.a.p.1.1 1
120.107 odd 4 3520.2.a.n.1.1 1
165.2 odd 20 605.2.g.c.81.1 4
165.17 odd 20 605.2.g.c.366.1 4
165.32 odd 4 605.2.a.b.1.1 1
165.47 even 20 605.2.g.a.251.1 4
165.62 odd 20 605.2.g.c.511.1 4
165.92 even 20 605.2.g.a.511.1 4
165.98 odd 4 3025.2.a.f.1.1 1
165.107 odd 20 605.2.g.c.251.1 4
165.137 even 20 605.2.g.a.366.1 4
165.152 even 20 605.2.g.a.81.1 4
195.77 even 4 9295.2.a.b.1.1 1
660.527 even 4 9680.2.a.r.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
55.2.a.a.1.1 1 15.2 even 4
275.2.a.a.1.1 1 15.8 even 4
275.2.b.b.199.1 2 3.2 odd 2
275.2.b.b.199.2 2 15.14 odd 2
495.2.a.a.1.1 1 5.2 odd 4
605.2.a.b.1.1 1 165.32 odd 4
605.2.g.a.81.1 4 165.152 even 20
605.2.g.a.251.1 4 165.47 even 20
605.2.g.a.366.1 4 165.137 even 20
605.2.g.a.511.1 4 165.92 even 20
605.2.g.c.81.1 4 165.2 odd 20
605.2.g.c.251.1 4 165.107 odd 20
605.2.g.c.366.1 4 165.17 odd 20
605.2.g.c.511.1 4 165.62 odd 20
880.2.a.h.1.1 1 60.47 odd 4
2475.2.a.i.1.1 1 5.3 odd 4
2475.2.c.f.199.1 2 5.4 even 2 inner
2475.2.c.f.199.2 2 1.1 even 1 trivial
2695.2.a.c.1.1 1 105.62 odd 4
3025.2.a.f.1.1 1 165.98 odd 4
3520.2.a.n.1.1 1 120.107 odd 4
3520.2.a.p.1.1 1 120.77 even 4
4400.2.a.p.1.1 1 60.23 odd 4
4400.2.b.n.4049.1 2 60.59 even 2
4400.2.b.n.4049.2 2 12.11 even 2
5445.2.a.i.1.1 1 55.32 even 4
7920.2.a.i.1.1 1 20.7 even 4
9295.2.a.b.1.1 1 195.77 even 4
9680.2.a.r.1.1 1 660.527 even 4