Properties

Label 2475.2.c.l.199.3
Level $2475$
Weight $2$
Character 2475.199
Analytic conductor $19.763$
Analytic rank $0$
Dimension $4$
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: \(4\)
Coefficient field: \(\Q(\zeta_{8})\)
Defining polynomial: \( x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: no (minimal twist has level 55)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 199.3
Root \(-0.707107 + 0.707107i\) of defining polynomial
Character \(\chi\) \(=\) 2475.199
Dual form 2475.2.c.l.199.2

$q$-expansion

\(f(q)\) \(=\) \(q+0.414214i q^{2} +1.82843 q^{4} -2.00000i q^{7} +1.58579i q^{8} +O(q^{10})\) \(q+0.414214i q^{2} +1.82843 q^{4} -2.00000i q^{7} +1.58579i q^{8} -1.00000 q^{11} +6.82843i q^{13} +0.828427 q^{14} +3.00000 q^{16} -1.17157i q^{17} -0.414214i q^{22} +2.82843i q^{23} -2.82843 q^{26} -3.65685i q^{28} +7.65685 q^{29} +4.41421i q^{32} +0.485281 q^{34} +3.65685i q^{37} -6.00000 q^{41} +6.00000i q^{43} -1.82843 q^{44} -1.17157 q^{46} +2.82843i q^{47} +3.00000 q^{49} +12.4853i q^{52} +0.343146i q^{53} +3.17157 q^{56} +3.17157i q^{58} -9.65685 q^{59} +13.3137 q^{61} +4.17157 q^{64} -4.48528i q^{67} -2.14214i q^{68} +11.3137 q^{71} +6.82843i q^{73} -1.51472 q^{74} +2.00000i q^{77} -4.00000 q^{79} -2.48528i q^{82} -6.00000i q^{83} -2.48528 q^{86} -1.58579i q^{88} +9.31371 q^{89} +13.6569 q^{91} +5.17157i q^{92} -1.17157 q^{94} -7.65685i q^{97} +1.24264i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{4} - 4 q^{11} - 8 q^{14} + 12 q^{16} + 8 q^{29} - 32 q^{34} - 24 q^{41} + 4 q^{44} - 16 q^{46} + 12 q^{49} + 24 q^{56} - 16 q^{59} + 8 q^{61} + 28 q^{64} - 40 q^{74} - 16 q^{79} + 24 q^{86} - 8 q^{89} + 32 q^{91} - 16 q^{94}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.414214i 0.292893i 0.989219 + 0.146447i \(0.0467837\pi\)
−0.989219 + 0.146447i \(0.953216\pi\)
\(3\) 0 0
\(4\) 1.82843 0.914214
\(5\) 0 0
\(6\) 0 0
\(7\) − 2.00000i − 0.755929i −0.925820 0.377964i \(-0.876624\pi\)
0.925820 0.377964i \(-0.123376\pi\)
\(8\) 1.58579i 0.560660i
\(9\) 0 0
\(10\) 0 0
\(11\) −1.00000 −0.301511
\(12\) 0 0
\(13\) 6.82843i 1.89386i 0.321433 + 0.946932i \(0.395836\pi\)
−0.321433 + 0.946932i \(0.604164\pi\)
\(14\) 0.828427 0.221406
\(15\) 0 0
\(16\) 3.00000 0.750000
\(17\) − 1.17157i − 0.284148i −0.989856 0.142074i \(-0.954623\pi\)
0.989856 0.142074i \(-0.0453771\pi\)
\(18\) 0 0
\(19\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) − 0.414214i − 0.0883106i
\(23\) 2.82843i 0.589768i 0.955533 + 0.294884i \(0.0952810\pi\)
−0.955533 + 0.294884i \(0.904719\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) −2.82843 −0.554700
\(27\) 0 0
\(28\) − 3.65685i − 0.691080i
\(29\) 7.65685 1.42184 0.710921 0.703272i \(-0.248278\pi\)
0.710921 + 0.703272i \(0.248278\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) 4.41421i 0.780330i
\(33\) 0 0
\(34\) 0.485281 0.0832251
\(35\) 0 0
\(36\) 0 0
\(37\) 3.65685i 0.601183i 0.953753 + 0.300592i \(0.0971841\pi\)
−0.953753 + 0.300592i \(0.902816\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −6.00000 −0.937043 −0.468521 0.883452i \(-0.655213\pi\)
−0.468521 + 0.883452i \(0.655213\pi\)
\(42\) 0 0
\(43\) 6.00000i 0.914991i 0.889212 + 0.457496i \(0.151253\pi\)
−0.889212 + 0.457496i \(0.848747\pi\)
\(44\) −1.82843 −0.275646
\(45\) 0 0
\(46\) −1.17157 −0.172739
\(47\) 2.82843i 0.412568i 0.978492 + 0.206284i \(0.0661372\pi\)
−0.978492 + 0.206284i \(0.933863\pi\)
\(48\) 0 0
\(49\) 3.00000 0.428571
\(50\) 0 0
\(51\) 0 0
\(52\) 12.4853i 1.73140i
\(53\) 0.343146i 0.0471347i 0.999722 + 0.0235673i \(0.00750241\pi\)
−0.999722 + 0.0235673i \(0.992498\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 3.17157 0.423819
\(57\) 0 0
\(58\) 3.17157i 0.416448i
\(59\) −9.65685 −1.25722 −0.628608 0.777723i \(-0.716375\pi\)
−0.628608 + 0.777723i \(0.716375\pi\)
\(60\) 0 0
\(61\) 13.3137 1.70465 0.852323 0.523016i \(-0.175193\pi\)
0.852323 + 0.523016i \(0.175193\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 4.17157 0.521447
\(65\) 0 0
\(66\) 0 0
\(67\) − 4.48528i − 0.547964i −0.961735 0.273982i \(-0.911659\pi\)
0.961735 0.273982i \(-0.0883409\pi\)
\(68\) − 2.14214i − 0.259772i
\(69\) 0 0
\(70\) 0 0
\(71\) 11.3137 1.34269 0.671345 0.741145i \(-0.265717\pi\)
0.671345 + 0.741145i \(0.265717\pi\)
\(72\) 0 0
\(73\) 6.82843i 0.799207i 0.916688 + 0.399603i \(0.130852\pi\)
−0.916688 + 0.399603i \(0.869148\pi\)
\(74\) −1.51472 −0.176082
\(75\) 0 0
\(76\) 0 0
\(77\) 2.00000i 0.227921i
\(78\) 0 0
\(79\) −4.00000 −0.450035 −0.225018 0.974355i \(-0.572244\pi\)
−0.225018 + 0.974355i \(0.572244\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) − 2.48528i − 0.274453i
\(83\) − 6.00000i − 0.658586i −0.944228 0.329293i \(-0.893190\pi\)
0.944228 0.329293i \(-0.106810\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −2.48528 −0.267995
\(87\) 0 0
\(88\) − 1.58579i − 0.169045i
\(89\) 9.31371 0.987251 0.493626 0.869675i \(-0.335671\pi\)
0.493626 + 0.869675i \(0.335671\pi\)
\(90\) 0 0
\(91\) 13.6569 1.43163
\(92\) 5.17157i 0.539174i
\(93\) 0 0
\(94\) −1.17157 −0.120839
\(95\) 0 0
\(96\) 0 0
\(97\) − 7.65685i − 0.777436i −0.921357 0.388718i \(-0.872918\pi\)
0.921357 0.388718i \(-0.127082\pi\)
\(98\) 1.24264i 0.125526i
\(99\) 0 0
\(100\) 0 0
\(101\) 13.3137 1.32476 0.662382 0.749166i \(-0.269546\pi\)
0.662382 + 0.749166i \(0.269546\pi\)
\(102\) 0 0
\(103\) − 1.17157i − 0.115439i −0.998333 0.0577193i \(-0.981617\pi\)
0.998333 0.0577193i \(-0.0183828\pi\)
\(104\) −10.8284 −1.06181
\(105\) 0 0
\(106\) −0.142136 −0.0138054
\(107\) 3.65685i 0.353521i 0.984254 + 0.176761i \(0.0565619\pi\)
−0.984254 + 0.176761i \(0.943438\pi\)
\(108\) 0 0
\(109\) −3.65685 −0.350263 −0.175132 0.984545i \(-0.556035\pi\)
−0.175132 + 0.984545i \(0.556035\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) − 6.00000i − 0.566947i
\(113\) 8.34315i 0.784857i 0.919782 + 0.392429i \(0.128365\pi\)
−0.919782 + 0.392429i \(0.871635\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 14.0000 1.29987
\(117\) 0 0
\(118\) − 4.00000i − 0.368230i
\(119\) −2.34315 −0.214796
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 5.51472i 0.499279i
\(123\) 0 0
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 15.6569i 1.38932i 0.719338 + 0.694661i \(0.244445\pi\)
−0.719338 + 0.694661i \(0.755555\pi\)
\(128\) 10.5563i 0.933058i
\(129\) 0 0
\(130\) 0 0
\(131\) −11.3137 −0.988483 −0.494242 0.869325i \(-0.664554\pi\)
−0.494242 + 0.869325i \(0.664554\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 1.85786 0.160495
\(135\) 0 0
\(136\) 1.85786 0.159311
\(137\) − 22.9706i − 1.96251i −0.192720 0.981254i \(-0.561731\pi\)
0.192720 0.981254i \(-0.438269\pi\)
\(138\) 0 0
\(139\) 4.00000 0.339276 0.169638 0.985506i \(-0.445740\pi\)
0.169638 + 0.985506i \(0.445740\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 4.68629i 0.393265i
\(143\) − 6.82843i − 0.571022i
\(144\) 0 0
\(145\) 0 0
\(146\) −2.82843 −0.234082
\(147\) 0 0
\(148\) 6.68629i 0.549610i
\(149\) 11.6569 0.954967 0.477483 0.878641i \(-0.341549\pi\)
0.477483 + 0.878641i \(0.341549\pi\)
\(150\) 0 0
\(151\) −12.0000 −0.976546 −0.488273 0.872691i \(-0.662373\pi\)
−0.488273 + 0.872691i \(0.662373\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) −0.828427 −0.0667566
\(155\) 0 0
\(156\) 0 0
\(157\) − 14.0000i − 1.11732i −0.829396 0.558661i \(-0.811315\pi\)
0.829396 0.558661i \(-0.188685\pi\)
\(158\) − 1.65685i − 0.131812i
\(159\) 0 0
\(160\) 0 0
\(161\) 5.65685 0.445823
\(162\) 0 0
\(163\) 0.485281i 0.0380102i 0.999819 + 0.0190051i \(0.00604987\pi\)
−0.999819 + 0.0190051i \(0.993950\pi\)
\(164\) −10.9706 −0.856657
\(165\) 0 0
\(166\) 2.48528 0.192895
\(167\) − 10.9706i − 0.848928i −0.905445 0.424464i \(-0.860463\pi\)
0.905445 0.424464i \(-0.139537\pi\)
\(168\) 0 0
\(169\) −33.6274 −2.58672
\(170\) 0 0
\(171\) 0 0
\(172\) 10.9706i 0.836498i
\(173\) 6.14214i 0.466978i 0.972359 + 0.233489i \(0.0750143\pi\)
−0.972359 + 0.233489i \(0.924986\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −3.00000 −0.226134
\(177\) 0 0
\(178\) 3.85786i 0.289159i
\(179\) −1.65685 −0.123839 −0.0619196 0.998081i \(-0.519722\pi\)
−0.0619196 + 0.998081i \(0.519722\pi\)
\(180\) 0 0
\(181\) −1.31371 −0.0976472 −0.0488236 0.998807i \(-0.515547\pi\)
−0.0488236 + 0.998807i \(0.515547\pi\)
\(182\) 5.65685i 0.419314i
\(183\) 0 0
\(184\) −4.48528 −0.330659
\(185\) 0 0
\(186\) 0 0
\(187\) 1.17157i 0.0856739i
\(188\) 5.17157i 0.377176i
\(189\) 0 0
\(190\) 0 0
\(191\) 19.3137 1.39749 0.698745 0.715370i \(-0.253742\pi\)
0.698745 + 0.715370i \(0.253742\pi\)
\(192\) 0 0
\(193\) 6.82843i 0.491521i 0.969331 + 0.245760i \(0.0790377\pi\)
−0.969331 + 0.245760i \(0.920962\pi\)
\(194\) 3.17157 0.227706
\(195\) 0 0
\(196\) 5.48528 0.391806
\(197\) 5.17157i 0.368459i 0.982883 + 0.184230i \(0.0589790\pi\)
−0.982883 + 0.184230i \(0.941021\pi\)
\(198\) 0 0
\(199\) −21.6569 −1.53521 −0.767607 0.640921i \(-0.778553\pi\)
−0.767607 + 0.640921i \(0.778553\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 5.51472i 0.388014i
\(203\) − 15.3137i − 1.07481i
\(204\) 0 0
\(205\) 0 0
\(206\) 0.485281 0.0338112
\(207\) 0 0
\(208\) 20.4853i 1.42040i
\(209\) 0 0
\(210\) 0 0
\(211\) −16.0000 −1.10149 −0.550743 0.834675i \(-0.685655\pi\)
−0.550743 + 0.834675i \(0.685655\pi\)
\(212\) 0.627417i 0.0430912i
\(213\) 0 0
\(214\) −1.51472 −0.103544
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) − 1.51472i − 0.102590i
\(219\) 0 0
\(220\) 0 0
\(221\) 8.00000 0.538138
\(222\) 0 0
\(223\) 5.17157i 0.346314i 0.984894 + 0.173157i \(0.0553968\pi\)
−0.984894 + 0.173157i \(0.944603\pi\)
\(224\) 8.82843 0.589874
\(225\) 0 0
\(226\) −3.45584 −0.229879
\(227\) − 2.68629i − 0.178295i −0.996018 0.0891477i \(-0.971586\pi\)
0.996018 0.0891477i \(-0.0284143\pi\)
\(228\) 0 0
\(229\) 21.3137 1.40845 0.704225 0.709977i \(-0.251295\pi\)
0.704225 + 0.709977i \(0.251295\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 12.1421i 0.797170i
\(233\) 22.1421i 1.45058i 0.688444 + 0.725290i \(0.258294\pi\)
−0.688444 + 0.725290i \(0.741706\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) −17.6569 −1.14936
\(237\) 0 0
\(238\) − 0.970563i − 0.0629122i
\(239\) −0.686292 −0.0443925 −0.0221963 0.999754i \(-0.507066\pi\)
−0.0221963 + 0.999754i \(0.507066\pi\)
\(240\) 0 0
\(241\) 6.00000 0.386494 0.193247 0.981150i \(-0.438098\pi\)
0.193247 + 0.981150i \(0.438098\pi\)
\(242\) 0.414214i 0.0266267i
\(243\) 0 0
\(244\) 24.3431 1.55841
\(245\) 0 0
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 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
\(253\) − 2.82843i − 0.177822i
\(254\) −6.48528 −0.406923
\(255\) 0 0
\(256\) 3.97056 0.248160
\(257\) − 13.3137i − 0.830486i −0.909710 0.415243i \(-0.863696\pi\)
0.909710 0.415243i \(-0.136304\pi\)
\(258\) 0 0
\(259\) 7.31371 0.454452
\(260\) 0 0
\(261\) 0 0
\(262\) − 4.68629i − 0.289520i
\(263\) 22.9706i 1.41643i 0.705999 + 0.708213i \(0.250498\pi\)
−0.705999 + 0.708213i \(0.749502\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) − 8.20101i − 0.500956i
\(269\) −5.31371 −0.323983 −0.161991 0.986792i \(-0.551792\pi\)
−0.161991 + 0.986792i \(0.551792\pi\)
\(270\) 0 0
\(271\) −15.3137 −0.930242 −0.465121 0.885247i \(-0.653989\pi\)
−0.465121 + 0.885247i \(0.653989\pi\)
\(272\) − 3.51472i − 0.213111i
\(273\) 0 0
\(274\) 9.51472 0.574805
\(275\) 0 0
\(276\) 0 0
\(277\) 1.17157i 0.0703930i 0.999380 + 0.0351965i \(0.0112057\pi\)
−0.999380 + 0.0351965i \(0.988794\pi\)
\(278\) 1.65685i 0.0993715i
\(279\) 0 0
\(280\) 0 0
\(281\) 5.31371 0.316989 0.158495 0.987360i \(-0.449336\pi\)
0.158495 + 0.987360i \(0.449336\pi\)
\(282\) 0 0
\(283\) 12.6274i 0.750622i 0.926899 + 0.375311i \(0.122464\pi\)
−0.926899 + 0.375311i \(0.877536\pi\)
\(284\) 20.6863 1.22751
\(285\) 0 0
\(286\) 2.82843 0.167248
\(287\) 12.0000i 0.708338i
\(288\) 0 0
\(289\) 15.6274 0.919260
\(290\) 0 0
\(291\) 0 0
\(292\) 12.4853i 0.730646i
\(293\) − 14.8284i − 0.866286i −0.901325 0.433143i \(-0.857405\pi\)
0.901325 0.433143i \(-0.142595\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) −5.79899 −0.337059
\(297\) 0 0
\(298\) 4.82843i 0.279703i
\(299\) −19.3137 −1.11694
\(300\) 0 0
\(301\) 12.0000 0.691669
\(302\) − 4.97056i − 0.286024i
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) − 27.6569i − 1.57846i −0.614098 0.789230i \(-0.710480\pi\)
0.614098 0.789230i \(-0.289520\pi\)
\(308\) 3.65685i 0.208369i
\(309\) 0 0
\(310\) 0 0
\(311\) −27.3137 −1.54882 −0.774409 0.632685i \(-0.781953\pi\)
−0.774409 + 0.632685i \(0.781953\pi\)
\(312\) 0 0
\(313\) − 21.3137i − 1.20472i −0.798224 0.602361i \(-0.794227\pi\)
0.798224 0.602361i \(-0.205773\pi\)
\(314\) 5.79899 0.327256
\(315\) 0 0
\(316\) −7.31371 −0.411428
\(317\) − 21.3137i − 1.19710i −0.801087 0.598549i \(-0.795744\pi\)
0.801087 0.598549i \(-0.204256\pi\)
\(318\) 0 0
\(319\) −7.65685 −0.428702
\(320\) 0 0
\(321\) 0 0
\(322\) 2.34315i 0.130578i
\(323\) 0 0
\(324\) 0 0
\(325\) 0 0
\(326\) −0.201010 −0.0111329
\(327\) 0 0
\(328\) − 9.51472i − 0.525362i
\(329\) 5.65685 0.311872
\(330\) 0 0
\(331\) 15.3137 0.841718 0.420859 0.907126i \(-0.361729\pi\)
0.420859 + 0.907126i \(0.361729\pi\)
\(332\) − 10.9706i − 0.602088i
\(333\) 0 0
\(334\) 4.54416 0.248645
\(335\) 0 0
\(336\) 0 0
\(337\) − 3.51472i − 0.191459i −0.995407 0.0957295i \(-0.969482\pi\)
0.995407 0.0957295i \(-0.0305184\pi\)
\(338\) − 13.9289i − 0.757634i
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) − 20.0000i − 1.07990i
\(344\) −9.51472 −0.512999
\(345\) 0 0
\(346\) −2.54416 −0.136775
\(347\) 22.9706i 1.23312i 0.787306 + 0.616562i \(0.211475\pi\)
−0.787306 + 0.616562i \(0.788525\pi\)
\(348\) 0 0
\(349\) 6.97056 0.373126 0.186563 0.982443i \(-0.440265\pi\)
0.186563 + 0.982443i \(0.440265\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) − 4.41421i − 0.235278i
\(353\) − 1.31371i − 0.0699216i −0.999389 0.0349608i \(-0.988869\pi\)
0.999389 0.0349608i \(-0.0111306\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 17.0294 0.902558
\(357\) 0 0
\(358\) − 0.686292i − 0.0362716i
\(359\) 23.3137 1.23045 0.615225 0.788351i \(-0.289065\pi\)
0.615225 + 0.788351i \(0.289065\pi\)
\(360\) 0 0
\(361\) −19.0000 −1.00000
\(362\) − 0.544156i − 0.0286002i
\(363\) 0 0
\(364\) 24.9706 1.30881
\(365\) 0 0
\(366\) 0 0
\(367\) 8.48528i 0.442928i 0.975169 + 0.221464i \(0.0710835\pi\)
−0.975169 + 0.221464i \(0.928916\pi\)
\(368\) 8.48528i 0.442326i
\(369\) 0 0
\(370\) 0 0
\(371\) 0.686292 0.0356305
\(372\) 0 0
\(373\) 3.79899i 0.196704i 0.995152 + 0.0983521i \(0.0313571\pi\)
−0.995152 + 0.0983521i \(0.968643\pi\)
\(374\) −0.485281 −0.0250933
\(375\) 0 0
\(376\) −4.48528 −0.231311
\(377\) 52.2843i 2.69278i
\(378\) 0 0
\(379\) −22.3431 −1.14769 −0.573845 0.818964i \(-0.694549\pi\)
−0.573845 + 0.818964i \(0.694549\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 8.00000i 0.409316i
\(383\) − 34.1421i − 1.74458i −0.488987 0.872291i \(-0.662634\pi\)
0.488987 0.872291i \(-0.337366\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −2.82843 −0.143963
\(387\) 0 0
\(388\) − 14.0000i − 0.710742i
\(389\) −24.6274 −1.24866 −0.624330 0.781161i \(-0.714628\pi\)
−0.624330 + 0.781161i \(0.714628\pi\)
\(390\) 0 0
\(391\) 3.31371 0.167581
\(392\) 4.75736i 0.240283i
\(393\) 0 0
\(394\) −2.14214 −0.107919
\(395\) 0 0
\(396\) 0 0
\(397\) 13.3137i 0.668196i 0.942538 + 0.334098i \(0.108432\pi\)
−0.942538 + 0.334098i \(0.891568\pi\)
\(398\) − 8.97056i − 0.449654i
\(399\) 0 0
\(400\) 0 0
\(401\) −17.3137 −0.864605 −0.432303 0.901729i \(-0.642299\pi\)
−0.432303 + 0.901729i \(0.642299\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 24.3431 1.21112
\(405\) 0 0
\(406\) 6.34315 0.314805
\(407\) − 3.65685i − 0.181264i
\(408\) 0 0
\(409\) −34.9706 −1.72918 −0.864592 0.502475i \(-0.832423\pi\)
−0.864592 + 0.502475i \(0.832423\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) − 2.14214i − 0.105535i
\(413\) 19.3137i 0.950365i
\(414\) 0 0
\(415\) 0 0
\(416\) −30.1421 −1.47784
\(417\) 0 0
\(418\) 0 0
\(419\) −14.3431 −0.700709 −0.350354 0.936617i \(-0.613939\pi\)
−0.350354 + 0.936617i \(0.613939\pi\)
\(420\) 0 0
\(421\) −6.00000 −0.292422 −0.146211 0.989253i \(-0.546708\pi\)
−0.146211 + 0.989253i \(0.546708\pi\)
\(422\) − 6.62742i − 0.322618i
\(423\) 0 0
\(424\) −0.544156 −0.0264265
\(425\) 0 0
\(426\) 0 0
\(427\) − 26.6274i − 1.28859i
\(428\) 6.68629i 0.323194i
\(429\) 0 0
\(430\) 0 0
\(431\) −11.3137 −0.544962 −0.272481 0.962161i \(-0.587844\pi\)
−0.272481 + 0.962161i \(0.587844\pi\)
\(432\) 0 0
\(433\) − 3.65685i − 0.175737i −0.996132 0.0878686i \(-0.971994\pi\)
0.996132 0.0878686i \(-0.0280056\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −6.68629 −0.320215
\(437\) 0 0
\(438\) 0 0
\(439\) 16.0000 0.763638 0.381819 0.924237i \(-0.375298\pi\)
0.381819 + 0.924237i \(0.375298\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 3.31371i 0.157617i
\(443\) − 21.1716i − 1.00589i −0.864318 0.502946i \(-0.832249\pi\)
0.864318 0.502946i \(-0.167751\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) −2.14214 −0.101433
\(447\) 0 0
\(448\) − 8.34315i − 0.394177i
\(449\) −16.6274 −0.784696 −0.392348 0.919817i \(-0.628337\pi\)
−0.392348 + 0.919817i \(0.628337\pi\)
\(450\) 0 0
\(451\) 6.00000 0.282529
\(452\) 15.2548i 0.717527i
\(453\) 0 0
\(454\) 1.11270 0.0522215
\(455\) 0 0
\(456\) 0 0
\(457\) − 16.4853i − 0.771149i −0.922677 0.385574i \(-0.874003\pi\)
0.922677 0.385574i \(-0.125997\pi\)
\(458\) 8.82843i 0.412525i
\(459\) 0 0
\(460\) 0 0
\(461\) 32.6274 1.51961 0.759805 0.650151i \(-0.225294\pi\)
0.759805 + 0.650151i \(0.225294\pi\)
\(462\) 0 0
\(463\) − 22.1421i − 1.02903i −0.857481 0.514516i \(-0.827972\pi\)
0.857481 0.514516i \(-0.172028\pi\)
\(464\) 22.9706 1.06638
\(465\) 0 0
\(466\) −9.17157 −0.424865
\(467\) 9.17157i 0.424410i 0.977225 + 0.212205i \(0.0680644\pi\)
−0.977225 + 0.212205i \(0.931936\pi\)
\(468\) 0 0
\(469\) −8.97056 −0.414222
\(470\) 0 0
\(471\) 0 0
\(472\) − 15.3137i − 0.704871i
\(473\) − 6.00000i − 0.275880i
\(474\) 0 0
\(475\) 0 0
\(476\) −4.28427 −0.196369
\(477\) 0 0
\(478\) − 0.284271i − 0.0130023i
\(479\) −36.0000 −1.64488 −0.822441 0.568850i \(-0.807388\pi\)
−0.822441 + 0.568850i \(0.807388\pi\)
\(480\) 0 0
\(481\) −24.9706 −1.13856
\(482\) 2.48528i 0.113201i
\(483\) 0 0
\(484\) 1.82843 0.0831103
\(485\) 0 0
\(486\) 0 0
\(487\) − 7.51472i − 0.340524i −0.985399 0.170262i \(-0.945539\pi\)
0.985399 0.170262i \(-0.0544615\pi\)
\(488\) 21.1127i 0.955727i
\(489\) 0 0
\(490\) 0 0
\(491\) 23.3137 1.05213 0.526066 0.850443i \(-0.323666\pi\)
0.526066 + 0.850443i \(0.323666\pi\)
\(492\) 0 0
\(493\) − 8.97056i − 0.404014i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) − 22.6274i − 1.01498i
\(498\) 0 0
\(499\) 1.65685 0.0741710 0.0370855 0.999312i \(-0.488193\pi\)
0.0370855 + 0.999312i \(0.488193\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) − 4.97056i − 0.221847i
\(503\) − 28.6274i − 1.27643i −0.769857 0.638217i \(-0.779672\pi\)
0.769857 0.638217i \(-0.220328\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 1.17157 0.0520828
\(507\) 0 0
\(508\) 28.6274i 1.27014i
\(509\) 9.31371 0.412823 0.206411 0.978465i \(-0.433821\pi\)
0.206411 + 0.978465i \(0.433821\pi\)
\(510\) 0 0
\(511\) 13.6569 0.604144
\(512\) 22.7574i 1.00574i
\(513\) 0 0
\(514\) 5.51472 0.243244
\(515\) 0 0
\(516\) 0 0
\(517\) − 2.82843i − 0.124394i
\(518\) 3.02944i 0.133106i
\(519\) 0 0
\(520\) 0 0
\(521\) −2.68629 −0.117689 −0.0588443 0.998267i \(-0.518742\pi\)
−0.0588443 + 0.998267i \(0.518742\pi\)
\(522\) 0 0
\(523\) − 37.5980i − 1.64404i −0.569455 0.822022i \(-0.692846\pi\)
0.569455 0.822022i \(-0.307154\pi\)
\(524\) −20.6863 −0.903685
\(525\) 0 0
\(526\) −9.51472 −0.414861
\(527\) 0 0
\(528\) 0 0
\(529\) 15.0000 0.652174
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) − 40.9706i − 1.77463i
\(534\) 0 0
\(535\) 0 0
\(536\) 7.11270 0.307222
\(537\) 0 0
\(538\) − 2.20101i − 0.0948923i
\(539\) −3.00000 −0.129219
\(540\) 0 0
\(541\) 6.00000 0.257960 0.128980 0.991647i \(-0.458830\pi\)
0.128980 + 0.991647i \(0.458830\pi\)
\(542\) − 6.34315i − 0.272461i
\(543\) 0 0
\(544\) 5.17157 0.221729
\(545\) 0 0
\(546\) 0 0
\(547\) − 34.0000i − 1.45374i −0.686778 0.726868i \(-0.740975\pi\)
0.686778 0.726868i \(-0.259025\pi\)
\(548\) − 42.0000i − 1.79415i
\(549\) 0 0
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 8.00000i 0.340195i
\(554\) −0.485281 −0.0206176
\(555\) 0 0
\(556\) 7.31371 0.310170
\(557\) − 38.1421i − 1.61613i −0.589090 0.808067i \(-0.700514\pi\)
0.589090 0.808067i \(-0.299486\pi\)
\(558\) 0 0
\(559\) −40.9706 −1.73287
\(560\) 0 0
\(561\) 0 0
\(562\) 2.20101i 0.0928440i
\(563\) 11.6569i 0.491278i 0.969361 + 0.245639i \(0.0789977\pi\)
−0.969361 + 0.245639i \(0.921002\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) −5.23045 −0.219852
\(567\) 0 0
\(568\) 17.9411i 0.752793i
\(569\) 20.3431 0.852829 0.426415 0.904528i \(-0.359776\pi\)
0.426415 + 0.904528i \(0.359776\pi\)
\(570\) 0 0
\(571\) 45.9411 1.92258 0.961288 0.275545i \(-0.0888584\pi\)
0.961288 + 0.275545i \(0.0888584\pi\)
\(572\) − 12.4853i − 0.522036i
\(573\) 0 0
\(574\) −4.97056 −0.207467
\(575\) 0 0
\(576\) 0 0
\(577\) 6.97056i 0.290188i 0.989418 + 0.145094i \(0.0463485\pi\)
−0.989418 + 0.145094i \(0.953651\pi\)
\(578\) 6.47309i 0.269245i
\(579\) 0 0
\(580\) 0 0
\(581\) −12.0000 −0.497844
\(582\) 0 0
\(583\) − 0.343146i − 0.0142116i
\(584\) −10.8284 −0.448084
\(585\) 0 0
\(586\) 6.14214 0.253729
\(587\) − 26.1421i − 1.07900i −0.841985 0.539501i \(-0.818613\pi\)
0.841985 0.539501i \(-0.181387\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) 10.9706i 0.450887i
\(593\) 20.4853i 0.841230i 0.907239 + 0.420615i \(0.138186\pi\)
−0.907239 + 0.420615i \(0.861814\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 21.3137 0.873044
\(597\) 0 0
\(598\) − 8.00000i − 0.327144i
\(599\) 5.65685 0.231133 0.115566 0.993300i \(-0.463132\pi\)
0.115566 + 0.993300i \(0.463132\pi\)
\(600\) 0 0
\(601\) −43.9411 −1.79240 −0.896198 0.443654i \(-0.853682\pi\)
−0.896198 + 0.443654i \(0.853682\pi\)
\(602\) 4.97056i 0.202585i
\(603\) 0 0
\(604\) −21.9411 −0.892772
\(605\) 0 0
\(606\) 0 0
\(607\) − 18.2843i − 0.742136i −0.928606 0.371068i \(-0.878992\pi\)
0.928606 0.371068i \(-0.121008\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −19.3137 −0.781349
\(612\) 0 0
\(613\) − 25.4558i − 1.02815i −0.857745 0.514076i \(-0.828135\pi\)
0.857745 0.514076i \(-0.171865\pi\)
\(614\) 11.4558 0.462320
\(615\) 0 0
\(616\) −3.17157 −0.127786
\(617\) − 11.6569i − 0.469287i −0.972081 0.234644i \(-0.924608\pi\)
0.972081 0.234644i \(-0.0753923\pi\)
\(618\) 0 0
\(619\) 25.6569 1.03124 0.515618 0.856819i \(-0.327562\pi\)
0.515618 + 0.856819i \(0.327562\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) − 11.3137i − 0.453638i
\(623\) − 18.6274i − 0.746292i
\(624\) 0 0
\(625\) 0 0
\(626\) 8.82843 0.352855
\(627\) 0 0
\(628\) − 25.5980i − 1.02147i
\(629\) 4.28427 0.170825
\(630\) 0 0
\(631\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(632\) − 6.34315i − 0.252317i
\(633\) 0 0
\(634\) 8.82843 0.350622
\(635\) 0 0
\(636\) 0 0
\(637\) 20.4853i 0.811656i
\(638\) − 3.17157i − 0.125564i
\(639\) 0 0
\(640\) 0 0
\(641\) −30.0000 −1.18493 −0.592464 0.805597i \(-0.701845\pi\)
−0.592464 + 0.805597i \(0.701845\pi\)
\(642\) 0 0
\(643\) − 49.4558i − 1.95035i −0.221440 0.975174i \(-0.571076\pi\)
0.221440 0.975174i \(-0.428924\pi\)
\(644\) 10.3431 0.407577
\(645\) 0 0
\(646\) 0 0
\(647\) 35.1127i 1.38042i 0.723608 + 0.690211i \(0.242482\pi\)
−0.723608 + 0.690211i \(0.757518\pi\)
\(648\) 0 0
\(649\) 9.65685 0.379065
\(650\) 0 0
\(651\) 0 0
\(652\) 0.887302i 0.0347494i
\(653\) 0.343146i 0.0134283i 0.999977 + 0.00671417i \(0.00213720\pi\)
−0.999977 + 0.00671417i \(0.997863\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) −18.0000 −0.702782
\(657\) 0 0
\(658\) 2.34315i 0.0913453i
\(659\) −21.9411 −0.854705 −0.427352 0.904085i \(-0.640554\pi\)
−0.427352 + 0.904085i \(0.640554\pi\)
\(660\) 0 0
\(661\) −0.627417 −0.0244037 −0.0122018 0.999926i \(-0.503884\pi\)
−0.0122018 + 0.999926i \(0.503884\pi\)
\(662\) 6.34315i 0.246533i
\(663\) 0 0
\(664\) 9.51472 0.369243
\(665\) 0 0
\(666\) 0 0
\(667\) 21.6569i 0.838557i
\(668\) − 20.0589i − 0.776101i
\(669\) 0 0
\(670\) 0 0
\(671\) −13.3137 −0.513970
\(672\) 0 0
\(673\) − 4.48528i − 0.172895i −0.996256 0.0864474i \(-0.972449\pi\)
0.996256 0.0864474i \(-0.0275515\pi\)
\(674\) 1.45584 0.0560770
\(675\) 0 0
\(676\) −61.4853 −2.36482
\(677\) − 17.1716i − 0.659957i −0.943988 0.329979i \(-0.892958\pi\)
0.943988 0.329979i \(-0.107042\pi\)
\(678\) 0 0
\(679\) −15.3137 −0.587686
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 31.7990i 1.21675i 0.793648 + 0.608377i \(0.208179\pi\)
−0.793648 + 0.608377i \(0.791821\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 8.28427 0.316295
\(687\) 0 0
\(688\) 18.0000i 0.686244i
\(689\) −2.34315 −0.0892667
\(690\) 0 0
\(691\) −16.6863 −0.634776 −0.317388 0.948296i \(-0.602806\pi\)
−0.317388 + 0.948296i \(0.602806\pi\)
\(692\) 11.2304i 0.426918i
\(693\) 0 0
\(694\) −9.51472 −0.361174
\(695\) 0 0
\(696\) 0 0
\(697\) 7.02944i 0.266259i
\(698\) 2.88730i 0.109286i
\(699\) 0 0
\(700\) 0 0
\(701\) −32.6274 −1.23232 −0.616160 0.787621i \(-0.711313\pi\)
−0.616160 + 0.787621i \(0.711313\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) −4.17157 −0.157222
\(705\) 0 0
\(706\) 0.544156 0.0204796
\(707\) − 26.6274i − 1.00143i
\(708\) 0 0
\(709\) 20.6274 0.774679 0.387339 0.921937i \(-0.373394\pi\)
0.387339 + 0.921937i \(0.373394\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 14.7696i 0.553512i
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) −3.02944 −0.113215
\(717\) 0 0
\(718\) 9.65685i 0.360391i
\(719\) 29.6569 1.10601 0.553007 0.833177i \(-0.313480\pi\)
0.553007 + 0.833177i \(0.313480\pi\)
\(720\) 0 0
\(721\) −2.34315 −0.0872633
\(722\) − 7.87006i − 0.292893i
\(723\) 0 0
\(724\) −2.40202 −0.0892704
\(725\) 0 0
\(726\) 0 0
\(727\) − 36.4853i − 1.35316i −0.736367 0.676582i \(-0.763460\pi\)
0.736367 0.676582i \(-0.236540\pi\)
\(728\) 21.6569i 0.802656i
\(729\) 0 0
\(730\) 0 0
\(731\) 7.02944 0.259993
\(732\) 0 0
\(733\) − 33.4558i − 1.23572i −0.786288 0.617860i \(-0.788000\pi\)
0.786288 0.617860i \(-0.212000\pi\)
\(734\) −3.51472 −0.129731
\(735\) 0 0
\(736\) −12.4853 −0.460214
\(737\) 4.48528i 0.165217i
\(738\) 0 0
\(739\) 37.9411 1.39569 0.697843 0.716250i \(-0.254143\pi\)
0.697843 + 0.716250i \(0.254143\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0.284271i 0.0104359i
\(743\) 29.5980i 1.08584i 0.839783 + 0.542922i \(0.182682\pi\)
−0.839783 + 0.542922i \(0.817318\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) −1.57359 −0.0576133
\(747\) 0 0
\(748\) 2.14214i 0.0783242i
\(749\) 7.31371 0.267237
\(750\) 0 0
\(751\) −16.0000 −0.583848 −0.291924 0.956441i \(-0.594295\pi\)
−0.291924 + 0.956441i \(0.594295\pi\)
\(752\) 8.48528i 0.309426i
\(753\) 0 0
\(754\) −21.6569 −0.788696
\(755\) 0 0
\(756\) 0 0
\(757\) − 9.31371i − 0.338512i −0.985572 0.169256i \(-0.945863\pi\)
0.985572 0.169256i \(-0.0541365\pi\)
\(758\) − 9.25483i − 0.336151i
\(759\) 0 0
\(760\) 0 0
\(761\) 30.0000 1.08750 0.543750 0.839248i \(-0.317004\pi\)
0.543750 + 0.839248i \(0.317004\pi\)
\(762\) 0 0
\(763\) 7.31371i 0.264774i
\(764\) 35.3137 1.27761
\(765\) 0 0
\(766\) 14.1421 0.510976
\(767\) − 65.9411i − 2.38100i
\(768\) 0 0
\(769\) −14.9706 −0.539852 −0.269926 0.962881i \(-0.586999\pi\)
−0.269926 + 0.962881i \(0.586999\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 12.4853i 0.449355i
\(773\) − 30.2843i − 1.08925i −0.838680 0.544625i \(-0.816672\pi\)
0.838680 0.544625i \(-0.183328\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 12.1421 0.435877
\(777\) 0 0
\(778\) − 10.2010i − 0.365724i
\(779\) 0 0
\(780\) 0 0
\(781\) −11.3137 −0.404836
\(782\) 1.37258i 0.0490835i
\(783\) 0 0
\(784\) 9.00000 0.321429
\(785\) 0 0
\(786\) 0 0
\(787\) 18.9706i 0.676228i 0.941105 + 0.338114i \(0.109789\pi\)
−0.941105 + 0.338114i \(0.890211\pi\)
\(788\) 9.45584i 0.336850i
\(789\) 0 0
\(790\) 0 0
\(791\) 16.6863 0.593296
\(792\) 0 0
\(793\) 90.9117i 3.22837i
\(794\) −5.51472 −0.195710
\(795\) 0 0
\(796\) −39.5980 −1.40351
\(797\) 12.6274i 0.447286i 0.974671 + 0.223643i \(0.0717950\pi\)
−0.974671 + 0.223643i \(0.928205\pi\)
\(798\) 0 0
\(799\) 3.31371 0.117231
\(800\) 0 0
\(801\) 0 0
\(802\) − 7.17157i − 0.253237i
\(803\) − 6.82843i − 0.240970i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 21.1127i 0.742742i
\(809\) 22.9706 0.807602 0.403801 0.914847i \(-0.367689\pi\)
0.403801 + 0.914847i \(0.367689\pi\)
\(810\) 0 0
\(811\) −13.9411 −0.489539 −0.244770 0.969581i \(-0.578712\pi\)
−0.244770 + 0.969581i \(0.578712\pi\)
\(812\) − 28.0000i − 0.982607i
\(813\) 0 0
\(814\) 1.51472 0.0530909
\(815\) 0 0
\(816\) 0 0
\(817\) 0 0
\(818\) − 14.4853i − 0.506466i
\(819\) 0 0
\(820\) 0 0
\(821\) 18.6863 0.652156 0.326078 0.945343i \(-0.394273\pi\)
0.326078 + 0.945343i \(0.394273\pi\)
\(822\) 0 0
\(823\) − 36.4853i − 1.27180i −0.771773 0.635898i \(-0.780630\pi\)
0.771773 0.635898i \(-0.219370\pi\)
\(824\) 1.85786 0.0647218
\(825\) 0 0
\(826\) −8.00000 −0.278356
\(827\) 34.2843i 1.19218i 0.802917 + 0.596090i \(0.203280\pi\)
−0.802917 + 0.596090i \(0.796720\pi\)
\(828\) 0 0
\(829\) −18.0000 −0.625166 −0.312583 0.949890i \(-0.601194\pi\)
−0.312583 + 0.949890i \(0.601194\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 28.4853i 0.987549i
\(833\) − 3.51472i − 0.121778i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 0 0
\(838\) − 5.94113i − 0.205233i
\(839\) 37.6569 1.30006 0.650029 0.759909i \(-0.274757\pi\)
0.650029 + 0.759909i \(0.274757\pi\)
\(840\) 0 0
\(841\) 29.6274 1.02164
\(842\) − 2.48528i − 0.0856485i
\(843\) 0 0
\(844\) −29.2548 −1.00699
\(845\) 0 0
\(846\) 0 0
\(847\) − 2.00000i − 0.0687208i
\(848\) 1.02944i 0.0353510i
\(849\) 0 0
\(850\) 0 0
\(851\) −10.3431 −0.354558
\(852\) 0 0
\(853\) 32.4853i 1.11227i 0.831090 + 0.556137i \(0.187717\pi\)
−0.831090 + 0.556137i \(0.812283\pi\)
\(854\) 11.0294 0.377420
\(855\) 0 0
\(856\) −5.79899 −0.198205
\(857\) 48.7696i 1.66594i 0.553321 + 0.832968i \(0.313360\pi\)
−0.553321 + 0.832968i \(0.686640\pi\)
\(858\) 0 0
\(859\) 32.2843 1.10153 0.550763 0.834662i \(-0.314337\pi\)
0.550763 + 0.834662i \(0.314337\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) − 4.68629i − 0.159616i
\(863\) − 14.8284i − 0.504766i −0.967627 0.252383i \(-0.918786\pi\)
0.967627 0.252383i \(-0.0812142\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 1.51472 0.0514722
\(867\) 0 0
\(868\) 0 0
\(869\) 4.00000 0.135691
\(870\) 0 0
\(871\) 30.6274 1.03777
\(872\) − 5.79899i − 0.196379i
\(873\) 0 0
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 1.45584i 0.0491604i 0.999698 + 0.0245802i \(0.00782490\pi\)
−0.999698 + 0.0245802i \(0.992175\pi\)
\(878\) 6.62742i 0.223664i
\(879\) 0 0
\(880\) 0 0
\(881\) 52.6274 1.77306 0.886531 0.462668i \(-0.153108\pi\)
0.886531 + 0.462668i \(0.153108\pi\)
\(882\) 0 0
\(883\) − 42.8284i − 1.44129i −0.693304 0.720646i \(-0.743845\pi\)
0.693304 0.720646i \(-0.256155\pi\)
\(884\) 14.6274 0.491973
\(885\) 0 0
\(886\) 8.76955 0.294619
\(887\) 18.2843i 0.613926i 0.951721 + 0.306963i \(0.0993127\pi\)
−0.951721 + 0.306963i \(0.900687\pi\)
\(888\) 0 0
\(889\) 31.3137 1.05023
\(890\) 0 0
\(891\) 0 0
\(892\) 9.45584i 0.316605i
\(893\) 0 0
\(894\) 0 0
\(895\) 0 0
\(896\) 21.1127 0.705326
\(897\) 0 0
\(898\) − 6.88730i − 0.229832i
\(899\) 0 0
\(900\) 0 0
\(901\) 0.402020 0.0133932
\(902\) 2.48528i 0.0827508i
\(903\) 0 0
\(904\) −13.2304 −0.440038
\(905\) 0 0
\(906\) 0 0
\(907\) − 44.4853i − 1.47711i −0.674193 0.738555i \(-0.735509\pi\)
0.674193 0.738555i \(-0.264491\pi\)
\(908\) − 4.91169i − 0.163000i
\(909\) 0 0
\(910\) 0 0
\(911\) −57.9411 −1.91968 −0.959838 0.280556i \(-0.909481\pi\)
−0.959838 + 0.280556i \(0.909481\pi\)
\(912\) 0 0
\(913\) 6.00000i 0.198571i
\(914\) 6.82843 0.225864
\(915\) 0 0
\(916\) 38.9706 1.28762
\(917\) 22.6274i 0.747223i
\(918\) 0 0
\(919\) 32.0000 1.05558 0.527791 0.849374i \(-0.323020\pi\)
0.527791 + 0.849374i \(0.323020\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 13.5147i 0.445084i
\(923\) 77.2548i 2.54287i
\(924\) 0 0
\(925\) 0 0
\(926\) 9.17157 0.301397
\(927\) 0 0
\(928\) 33.7990i 1.10951i
\(929\) −17.3137 −0.568044 −0.284022 0.958818i \(-0.591669\pi\)
−0.284022 + 0.958818i \(0.591669\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 40.4853i 1.32614i
\(933\) 0 0
\(934\) −3.79899 −0.124307
\(935\) 0 0
\(936\) 0 0
\(937\) 49.4558i 1.61565i 0.589421 + 0.807826i \(0.299356\pi\)
−0.589421 + 0.807826i \(0.700644\pi\)
\(938\) − 3.71573i − 0.121323i
\(939\) 0 0
\(940\) 0 0
\(941\) 29.3137 0.955600 0.477800 0.878469i \(-0.341434\pi\)
0.477800 + 0.878469i \(0.341434\pi\)
\(942\) 0 0
\(943\) − 16.9706i − 0.552638i
\(944\) −28.9706 −0.942912
\(945\) 0 0
\(946\) 2.48528 0.0808035
\(947\) − 46.8284i − 1.52172i −0.648916 0.760860i \(-0.724778\pi\)
0.648916 0.760860i \(-0.275222\pi\)
\(948\) 0 0
\(949\) −46.6274 −1.51359
\(950\) 0 0
\(951\) 0 0
\(952\) − 3.71573i − 0.120427i
\(953\) 58.8284i 1.90564i 0.303536 + 0.952820i \(0.401833\pi\)
−0.303536 + 0.952820i \(0.598167\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) −1.25483 −0.0405842
\(957\) 0 0
\(958\) − 14.9117i − 0.481775i
\(959\) −45.9411 −1.48352
\(960\) 0 0
\(961\) −31.0000 −1.00000
\(962\) − 10.3431i − 0.333476i
\(963\) 0 0
\(964\) 10.9706 0.353338
\(965\) 0 0
\(966\) 0 0
\(967\) 18.9706i 0.610052i 0.952344 + 0.305026i \(0.0986652\pi\)
−0.952344 + 0.305026i \(0.901335\pi\)
\(968\) 1.58579i 0.0509691i
\(969\) 0 0
\(970\) 0 0
\(971\) 31.3137 1.00490 0.502452 0.864605i \(-0.332431\pi\)
0.502452 + 0.864605i \(0.332431\pi\)
\(972\) 0 0
\(973\) − 8.00000i − 0.256468i
\(974\) 3.11270 0.0997373
\(975\) 0 0
\(976\) 39.9411 1.27848
\(977\) − 43.6569i − 1.39671i −0.715753 0.698353i \(-0.753916\pi\)
0.715753 0.698353i \(-0.246084\pi\)
\(978\) 0 0
\(979\) −9.31371 −0.297667
\(980\) 0 0
\(981\) 0 0
\(982\) 9.65685i 0.308163i
\(983\) − 50.1421i − 1.59929i −0.600476 0.799643i \(-0.705022\pi\)
0.600476 0.799643i \(-0.294978\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 3.71573 0.118333
\(987\) 0 0
\(988\) 0 0
\(989\) −16.9706 −0.539633
\(990\) 0 0
\(991\) 9.94113 0.315790 0.157895 0.987456i \(-0.449529\pi\)
0.157895 + 0.987456i \(0.449529\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 9.37258 0.297280
\(995\) 0 0
\(996\) 0 0
\(997\) 9.45584i 0.299470i 0.988726 + 0.149735i \(0.0478420\pi\)
−0.988726 + 0.149735i \(0.952158\pi\)
\(998\) 0.686292i 0.0217242i
\(999\) 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.l.199.3 4
3.2 odd 2 275.2.b.d.199.2 4
5.2 odd 4 2475.2.a.x.1.1 2
5.3 odd 4 495.2.a.b.1.2 2
5.4 even 2 inner 2475.2.c.l.199.2 4
12.11 even 2 4400.2.b.q.4049.4 4
15.2 even 4 275.2.a.c.1.2 2
15.8 even 4 55.2.a.b.1.1 2
15.14 odd 2 275.2.b.d.199.3 4
20.3 even 4 7920.2.a.ch.1.1 2
55.43 even 4 5445.2.a.y.1.1 2
60.23 odd 4 880.2.a.m.1.1 2
60.47 odd 4 4400.2.a.bn.1.2 2
60.59 even 2 4400.2.b.q.4049.1 4
105.83 odd 4 2695.2.a.f.1.1 2
120.53 even 4 3520.2.a.bn.1.1 2
120.83 odd 4 3520.2.a.bo.1.2 2
165.8 odd 20 605.2.g.l.251.2 8
165.32 odd 4 3025.2.a.o.1.1 2
165.38 even 20 605.2.g.f.366.2 8
165.53 even 20 605.2.g.f.81.2 8
165.68 odd 20 605.2.g.l.81.1 8
165.83 odd 20 605.2.g.l.366.1 8
165.98 odd 4 605.2.a.d.1.2 2
165.113 even 20 605.2.g.f.251.1 8
165.128 odd 20 605.2.g.l.511.2 8
165.158 even 20 605.2.g.f.511.1 8
195.38 even 4 9295.2.a.g.1.2 2
660.263 even 4 9680.2.a.bn.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
55.2.a.b.1.1 2 15.8 even 4
275.2.a.c.1.2 2 15.2 even 4
275.2.b.d.199.2 4 3.2 odd 2
275.2.b.d.199.3 4 15.14 odd 2
495.2.a.b.1.2 2 5.3 odd 4
605.2.a.d.1.2 2 165.98 odd 4
605.2.g.f.81.2 8 165.53 even 20
605.2.g.f.251.1 8 165.113 even 20
605.2.g.f.366.2 8 165.38 even 20
605.2.g.f.511.1 8 165.158 even 20
605.2.g.l.81.1 8 165.68 odd 20
605.2.g.l.251.2 8 165.8 odd 20
605.2.g.l.366.1 8 165.83 odd 20
605.2.g.l.511.2 8 165.128 odd 20
880.2.a.m.1.1 2 60.23 odd 4
2475.2.a.x.1.1 2 5.2 odd 4
2475.2.c.l.199.2 4 5.4 even 2 inner
2475.2.c.l.199.3 4 1.1 even 1 trivial
2695.2.a.f.1.1 2 105.83 odd 4
3025.2.a.o.1.1 2 165.32 odd 4
3520.2.a.bn.1.1 2 120.53 even 4
3520.2.a.bo.1.2 2 120.83 odd 4
4400.2.a.bn.1.2 2 60.47 odd 4
4400.2.b.q.4049.1 4 60.59 even 2
4400.2.b.q.4049.4 4 12.11 even 2
5445.2.a.y.1.1 2 55.43 even 4
7920.2.a.ch.1.1 2 20.3 even 4
9295.2.a.g.1.2 2 195.38 even 4
9680.2.a.bn.1.1 2 660.263 even 4