Properties

Label 275.2.b.d.199.3
Level $275$
Weight $2$
Character 275.199
Analytic conductor $2.196$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 275 = 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 275.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(2.19588605559\)
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 \(\chi\) \(=\) 275.199
Dual form 275.2.b.d.199.2

$q$-expansion

\(f(q)\) \(=\) \(q+0.414214i q^{2} +2.82843i q^{3} +1.82843 q^{4} -1.17157 q^{6} +2.00000i q^{7} +1.58579i q^{8} -5.00000 q^{9} +O(q^{10})\) \(q+0.414214i q^{2} +2.82843i q^{3} +1.82843 q^{4} -1.17157 q^{6} +2.00000i q^{7} +1.58579i q^{8} -5.00000 q^{9} +1.00000 q^{11} +5.17157i q^{12} -6.82843i q^{13} -0.828427 q^{14} +3.00000 q^{16} -1.17157i q^{17} -2.07107i q^{18} -5.65685 q^{21} +0.414214i q^{22} +2.82843i q^{23} -4.48528 q^{24} +2.82843 q^{26} -5.65685i q^{27} +3.65685i q^{28} -7.65685 q^{29} +4.41421i q^{32} +2.82843i q^{33} +0.485281 q^{34} -9.14214 q^{36} -3.65685i q^{37} +19.3137 q^{39} +6.00000 q^{41} -2.34315i q^{42} -6.00000i q^{43} +1.82843 q^{44} -1.17157 q^{46} +2.82843i q^{47} +8.48528i q^{48} +3.00000 q^{49} +3.31371 q^{51} -12.4853i q^{52} +0.343146i q^{53} +2.34315 q^{54} -3.17157 q^{56} -3.17157i q^{58} +9.65685 q^{59} +13.3137 q^{61} -10.0000i q^{63} +4.17157 q^{64} -1.17157 q^{66} +4.48528i q^{67} -2.14214i q^{68} -8.00000 q^{69} -11.3137 q^{71} -7.92893i q^{72} -6.82843i q^{73} +1.51472 q^{74} +2.00000i q^{77} +8.00000i q^{78} -4.00000 q^{79} +1.00000 q^{81} +2.48528i q^{82} -6.00000i q^{83} -10.3431 q^{84} +2.48528 q^{86} -21.6569i q^{87} +1.58579i q^{88} -9.31371 q^{89} +13.6569 q^{91} +5.17157i q^{92} -1.17157 q^{94} -12.4853 q^{96} +7.65685i q^{97} +1.24264i q^{98} -5.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{4} - 16 q^{6} - 20 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{4} - 16 q^{6} - 20 q^{9} + 4 q^{11} + 8 q^{14} + 12 q^{16} + 16 q^{24} - 8 q^{29} - 32 q^{34} + 20 q^{36} + 32 q^{39} + 24 q^{41} - 4 q^{44} - 16 q^{46} + 12 q^{49} - 32 q^{51} + 32 q^{54} - 24 q^{56} + 16 q^{59} + 8 q^{61} + 28 q^{64} - 16 q^{66} - 32 q^{69} + 40 q^{74} - 16 q^{79} + 4 q^{81} - 64 q^{84} - 24 q^{86} + 8 q^{89} + 32 q^{91} - 16 q^{94} - 16 q^{96} - 20 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/275\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(177\)
\(\chi(n)\) \(1\) \(-1\)

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 \(0.953216\pi\)
\(3\) 2.82843i 1.63299i 0.577350 + 0.816497i \(0.304087\pi\)
−0.577350 + 0.816497i \(0.695913\pi\)
\(4\) 1.82843 0.914214
\(5\) 0 0
\(6\) −1.17157 −0.478293
\(7\) 2.00000i 0.755929i 0.925820 + 0.377964i \(0.123376\pi\)
−0.925820 + 0.377964i \(0.876624\pi\)
\(8\) 1.58579i 0.560660i
\(9\) −5.00000 −1.66667
\(10\) 0 0
\(11\) 1.00000 0.301511
\(12\) 5.17157i 1.49290i
\(13\) − 6.82843i − 1.89386i −0.321433 0.946932i \(-0.604164\pi\)
0.321433 0.946932i \(-0.395836\pi\)
\(14\) −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 \(-0.0453771\pi\)
\(18\) − 2.07107i − 0.488155i
\(19\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(20\) 0 0
\(21\) −5.65685 −1.23443
\(22\) 0.414214i 0.0883106i
\(23\) 2.82843i 0.589768i 0.955533 + 0.294884i \(0.0952810\pi\)
−0.955533 + 0.294884i \(0.904719\pi\)
\(24\) −4.48528 −0.915554
\(25\) 0 0
\(26\) 2.82843 0.554700
\(27\) − 5.65685i − 1.08866i
\(28\) 3.65685i 0.691080i
\(29\) −7.65685 −1.42184 −0.710921 0.703272i \(-0.751722\pi\)
−0.710921 + 0.703272i \(0.751722\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) 4.41421i 0.780330i
\(33\) 2.82843i 0.492366i
\(34\) 0.485281 0.0832251
\(35\) 0 0
\(36\) −9.14214 −1.52369
\(37\) − 3.65685i − 0.601183i −0.953753 0.300592i \(-0.902816\pi\)
0.953753 0.300592i \(-0.0971841\pi\)
\(38\) 0 0
\(39\) 19.3137 3.09267
\(40\) 0 0
\(41\) 6.00000 0.937043 0.468521 0.883452i \(-0.344787\pi\)
0.468521 + 0.883452i \(0.344787\pi\)
\(42\) − 2.34315i − 0.361555i
\(43\) − 6.00000i − 0.914991i −0.889212 0.457496i \(-0.848747\pi\)
0.889212 0.457496i \(-0.151253\pi\)
\(44\) 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 \(0.933863\pi\)
\(48\) 8.48528i 1.22474i
\(49\) 3.00000 0.428571
\(50\) 0 0
\(51\) 3.31371 0.464012
\(52\) − 12.4853i − 1.73140i
\(53\) 0.343146i 0.0471347i 0.999722 + 0.0235673i \(0.00750241\pi\)
−0.999722 + 0.0235673i \(0.992498\pi\)
\(54\) 2.34315 0.318862
\(55\) 0 0
\(56\) −3.17157 −0.423819
\(57\) 0 0
\(58\) − 3.17157i − 0.416448i
\(59\) 9.65685 1.25722 0.628608 0.777723i \(-0.283625\pi\)
0.628608 + 0.777723i \(0.283625\pi\)
\(60\) 0 0
\(61\) 13.3137 1.70465 0.852323 0.523016i \(-0.175193\pi\)
0.852323 + 0.523016i \(0.175193\pi\)
\(62\) 0 0
\(63\) − 10.0000i − 1.25988i
\(64\) 4.17157 0.521447
\(65\) 0 0
\(66\) −1.17157 −0.144211
\(67\) 4.48528i 0.547964i 0.961735 + 0.273982i \(0.0883409\pi\)
−0.961735 + 0.273982i \(0.911659\pi\)
\(68\) − 2.14214i − 0.259772i
\(69\) −8.00000 −0.963087
\(70\) 0 0
\(71\) −11.3137 −1.34269 −0.671345 0.741145i \(-0.734283\pi\)
−0.671345 + 0.741145i \(0.734283\pi\)
\(72\) − 7.92893i − 0.934434i
\(73\) − 6.82843i − 0.799207i −0.916688 0.399603i \(-0.869148\pi\)
0.916688 0.399603i \(-0.130852\pi\)
\(74\) 1.51472 0.176082
\(75\) 0 0
\(76\) 0 0
\(77\) 2.00000i 0.227921i
\(78\) 8.00000i 0.905822i
\(79\) −4.00000 −0.450035 −0.225018 0.974355i \(-0.572244\pi\)
−0.225018 + 0.974355i \(0.572244\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 2.48528i 0.274453i
\(83\) − 6.00000i − 0.658586i −0.944228 0.329293i \(-0.893190\pi\)
0.944228 0.329293i \(-0.106810\pi\)
\(84\) −10.3431 −1.12853
\(85\) 0 0
\(86\) 2.48528 0.267995
\(87\) − 21.6569i − 2.32186i
\(88\) 1.58579i 0.169045i
\(89\) −9.31371 −0.987251 −0.493626 0.869675i \(-0.664329\pi\)
−0.493626 + 0.869675i \(0.664329\pi\)
\(90\) 0 0
\(91\) 13.6569 1.43163
\(92\) 5.17157i 0.539174i
\(93\) 0 0
\(94\) −1.17157 −0.120839
\(95\) 0 0
\(96\) −12.4853 −1.27427
\(97\) 7.65685i 0.777436i 0.921357 + 0.388718i \(0.127082\pi\)
−0.921357 + 0.388718i \(0.872918\pi\)
\(98\) 1.24264i 0.125526i
\(99\) −5.00000 −0.502519
\(100\) 0 0
\(101\) −13.3137 −1.32476 −0.662382 0.749166i \(-0.730454\pi\)
−0.662382 + 0.749166i \(0.730454\pi\)
\(102\) 1.37258i 0.135906i
\(103\) 1.17157i 0.115439i 0.998333 + 0.0577193i \(0.0183828\pi\)
−0.998333 + 0.0577193i \(0.981617\pi\)
\(104\) 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 \(0.943438\pi\)
\(108\) − 10.3431i − 0.995270i
\(109\) −3.65685 −0.350263 −0.175132 0.984545i \(-0.556035\pi\)
−0.175132 + 0.984545i \(0.556035\pi\)
\(110\) 0 0
\(111\) 10.3431 0.981728
\(112\) 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
\(117\) 34.1421i 3.15644i
\(118\) 4.00000i 0.368230i
\(119\) 2.34315 0.214796
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 5.51472i 0.499279i
\(123\) 16.9706i 1.53018i
\(124\) 0 0
\(125\) 0 0
\(126\) 4.14214 0.369011
\(127\) − 15.6569i − 1.38932i −0.719338 0.694661i \(-0.755555\pi\)
0.719338 0.694661i \(-0.244445\pi\)
\(128\) 10.5563i 0.933058i
\(129\) 16.9706 1.49417
\(130\) 0 0
\(131\) 11.3137 0.988483 0.494242 0.869325i \(-0.335446\pi\)
0.494242 + 0.869325i \(0.335446\pi\)
\(132\) 5.17157i 0.450128i
\(133\) 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 \(-0.438269\pi\)
\(138\) − 3.31371i − 0.282082i
\(139\) 4.00000 0.339276 0.169638 0.985506i \(-0.445740\pi\)
0.169638 + 0.985506i \(0.445740\pi\)
\(140\) 0 0
\(141\) −8.00000 −0.673722
\(142\) − 4.68629i − 0.393265i
\(143\) − 6.82843i − 0.571022i
\(144\) −15.0000 −1.25000
\(145\) 0 0
\(146\) 2.82843 0.234082
\(147\) 8.48528i 0.699854i
\(148\) − 6.68629i − 0.549610i
\(149\) −11.6569 −0.954967 −0.477483 0.878641i \(-0.658451\pi\)
−0.477483 + 0.878641i \(0.658451\pi\)
\(150\) 0 0
\(151\) −12.0000 −0.976546 −0.488273 0.872691i \(-0.662373\pi\)
−0.488273 + 0.872691i \(0.662373\pi\)
\(152\) 0 0
\(153\) 5.85786i 0.473580i
\(154\) −0.828427 −0.0667566
\(155\) 0 0
\(156\) 35.3137 2.82736
\(157\) 14.0000i 1.11732i 0.829396 + 0.558661i \(0.188685\pi\)
−0.829396 + 0.558661i \(0.811315\pi\)
\(158\) − 1.65685i − 0.131812i
\(159\) −0.970563 −0.0769706
\(160\) 0 0
\(161\) −5.65685 −0.445823
\(162\) 0.414214i 0.0325437i
\(163\) − 0.485281i − 0.0380102i −0.999819 0.0190051i \(-0.993950\pi\)
0.999819 0.0190051i \(-0.00604987\pi\)
\(164\) 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 \(-0.139537\pi\)
\(168\) − 8.97056i − 0.692094i
\(169\) −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 \(0.924986\pi\)
\(174\) 8.97056 0.680057
\(175\) 0 0
\(176\) 3.00000 0.226134
\(177\) 27.3137i 2.05302i
\(178\) − 3.85786i − 0.289159i
\(179\) 1.65685 0.123839 0.0619196 0.998081i \(-0.480278\pi\)
0.0619196 + 0.998081i \(0.480278\pi\)
\(180\) 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
\(183\) 37.6569i 2.78367i
\(184\) −4.48528 −0.330659
\(185\) 0 0
\(186\) 0 0
\(187\) − 1.17157i − 0.0856739i
\(188\) 5.17157i 0.377176i
\(189\) 11.3137 0.822951
\(190\) 0 0
\(191\) −19.3137 −1.39749 −0.698745 0.715370i \(-0.746258\pi\)
−0.698745 + 0.715370i \(0.746258\pi\)
\(192\) 11.7990i 0.851519i
\(193\) − 6.82843i − 0.491521i −0.969331 0.245760i \(-0.920962\pi\)
0.969331 0.245760i \(-0.0790377\pi\)
\(194\) −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 \(0.941021\pi\)
\(198\) − 2.07107i − 0.147184i
\(199\) −21.6569 −1.53521 −0.767607 0.640921i \(-0.778553\pi\)
−0.767607 + 0.640921i \(0.778553\pi\)
\(200\) 0 0
\(201\) −12.6863 −0.894822
\(202\) − 5.51472i − 0.388014i
\(203\) − 15.3137i − 1.07481i
\(204\) 6.05887 0.424206
\(205\) 0 0
\(206\) −0.485281 −0.0338112
\(207\) − 14.1421i − 0.982946i
\(208\) − 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
\(213\) − 32.0000i − 2.19260i
\(214\) −1.51472 −0.103544
\(215\) 0 0
\(216\) 8.97056 0.610369
\(217\) 0 0
\(218\) − 1.51472i − 0.102590i
\(219\) 19.3137 1.30510
\(220\) 0 0
\(221\) −8.00000 −0.538138
\(222\) 4.28427i 0.287541i
\(223\) − 5.17157i − 0.346314i −0.984894 0.173157i \(-0.944603\pi\)
0.984894 0.173157i \(-0.0553968\pi\)
\(224\) −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
\(231\) −5.65685 −0.372194
\(232\) − 12.1421i − 0.797170i
\(233\) 22.1421i 1.45058i 0.688444 + 0.725290i \(0.258294\pi\)
−0.688444 + 0.725290i \(0.741706\pi\)
\(234\) −14.1421 −0.924500
\(235\) 0 0
\(236\) 17.6569 1.14936
\(237\) − 11.3137i − 0.734904i
\(238\) 0.970563i 0.0629122i
\(239\) 0.686292 0.0443925 0.0221963 0.999754i \(-0.492934\pi\)
0.0221963 + 0.999754i \(0.492934\pi\)
\(240\) 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
\(243\) − 14.1421i − 0.907218i
\(244\) 24.3431 1.55841
\(245\) 0 0
\(246\) −7.02944 −0.448181
\(247\) 0 0
\(248\) 0 0
\(249\) 16.9706 1.07547
\(250\) 0 0
\(251\) 12.0000 0.757433 0.378717 0.925513i \(-0.376365\pi\)
0.378717 + 0.925513i \(0.376365\pi\)
\(252\) − 18.2843i − 1.15180i
\(253\) 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 \(-0.136304\pi\)
\(258\) 7.02944i 0.437634i
\(259\) 7.31371 0.454452
\(260\) 0 0
\(261\) 38.2843 2.36974
\(262\) 4.68629i 0.289520i
\(263\) 22.9706i 1.41643i 0.705999 + 0.708213i \(0.250498\pi\)
−0.705999 + 0.708213i \(0.749502\pi\)
\(264\) −4.48528 −0.276050
\(265\) 0 0
\(266\) 0 0
\(267\) − 26.3431i − 1.61217i
\(268\) 8.20101i 0.500956i
\(269\) 5.31371 0.323983 0.161991 0.986792i \(-0.448208\pi\)
0.161991 + 0.986792i \(0.448208\pi\)
\(270\) 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
\(273\) 38.6274i 2.33784i
\(274\) 9.51472 0.574805
\(275\) 0 0
\(276\) −14.6274 −0.880467
\(277\) − 1.17157i − 0.0703930i −0.999380 0.0351965i \(-0.988794\pi\)
0.999380 0.0351965i \(-0.0112057\pi\)
\(278\) 1.65685i 0.0993715i
\(279\) 0 0
\(280\) 0 0
\(281\) −5.31371 −0.316989 −0.158495 0.987360i \(-0.550664\pi\)
−0.158495 + 0.987360i \(0.550664\pi\)
\(282\) − 3.31371i − 0.197328i
\(283\) − 12.6274i − 0.750622i −0.926899 0.375311i \(-0.877536\pi\)
0.926899 0.375311i \(-0.122464\pi\)
\(284\) −20.6863 −1.22751
\(285\) 0 0
\(286\) 2.82843 0.167248
\(287\) 12.0000i 0.708338i
\(288\) − 22.0711i − 1.30055i
\(289\) 15.6274 0.919260
\(290\) 0 0
\(291\) −21.6569 −1.26955
\(292\) − 12.4853i − 0.730646i
\(293\) − 14.8284i − 0.866286i −0.901325 0.433143i \(-0.857405\pi\)
0.901325 0.433143i \(-0.142595\pi\)
\(294\) −3.51472 −0.204983
\(295\) 0 0
\(296\) 5.79899 0.337059
\(297\) − 5.65685i − 0.328244i
\(298\) − 4.82843i − 0.279703i
\(299\) 19.3137 1.11694
\(300\) 0 0
\(301\) 12.0000 0.691669
\(302\) − 4.97056i − 0.286024i
\(303\) − 37.6569i − 2.16333i
\(304\) 0 0
\(305\) 0 0
\(306\) −2.42641 −0.138708
\(307\) 27.6569i 1.57846i 0.614098 + 0.789230i \(0.289520\pi\)
−0.614098 + 0.789230i \(0.710480\pi\)
\(308\) 3.65685i 0.208369i
\(309\) −3.31371 −0.188510
\(310\) 0 0
\(311\) 27.3137 1.54882 0.774409 0.632685i \(-0.218047\pi\)
0.774409 + 0.632685i \(0.218047\pi\)
\(312\) 30.6274i 1.73394i
\(313\) 21.3137i 1.20472i 0.798224 + 0.602361i \(0.205773\pi\)
−0.798224 + 0.602361i \(0.794227\pi\)
\(314\) −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 \(-0.204256\pi\)
\(318\) − 0.402020i − 0.0225442i
\(319\) −7.65685 −0.428702
\(320\) 0 0
\(321\) −10.3431 −0.577298
\(322\) − 2.34315i − 0.130578i
\(323\) 0 0
\(324\) 1.82843 0.101579
\(325\) 0 0
\(326\) 0.201010 0.0111329
\(327\) − 10.3431i − 0.571977i
\(328\) 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
\(333\) 18.2843i 1.00197i
\(334\) 4.54416 0.248645
\(335\) 0 0
\(336\) −16.9706 −0.925820
\(337\) 3.51472i 0.191459i 0.995407 + 0.0957295i \(0.0305184\pi\)
−0.995407 + 0.0957295i \(0.969482\pi\)
\(338\) − 13.9289i − 0.757634i
\(339\) −23.5980 −1.28167
\(340\) 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 \(0.788525\pi\)
\(348\) − 39.5980i − 2.12267i
\(349\) 6.97056 0.373126 0.186563 0.982443i \(-0.440265\pi\)
0.186563 + 0.982443i \(0.440265\pi\)
\(350\) 0 0
\(351\) −38.6274 −2.06178
\(352\) 4.41421i 0.235278i
\(353\) − 1.31371i − 0.0699216i −0.999389 0.0349608i \(-0.988869\pi\)
0.999389 0.0349608i \(-0.0111306\pi\)
\(354\) −11.3137 −0.601317
\(355\) 0 0
\(356\) −17.0294 −0.902558
\(357\) 6.62742i 0.350760i
\(358\) 0.686292i 0.0362716i
\(359\) −23.3137 −1.23045 −0.615225 0.788351i \(-0.710935\pi\)
−0.615225 + 0.788351i \(0.710935\pi\)
\(360\) 0 0
\(361\) −19.0000 −1.00000
\(362\) − 0.544156i − 0.0286002i
\(363\) 2.82843i 0.148454i
\(364\) 24.9706 1.30881
\(365\) 0 0
\(366\) −15.5980 −0.815319
\(367\) − 8.48528i − 0.442928i −0.975169 0.221464i \(-0.928916\pi\)
0.975169 0.221464i \(-0.0710835\pi\)
\(368\) 8.48528i 0.442326i
\(369\) −30.0000 −1.56174
\(370\) 0 0
\(371\) −0.686292 −0.0356305
\(372\) 0 0
\(373\) − 3.79899i − 0.196704i −0.995152 0.0983521i \(-0.968643\pi\)
0.995152 0.0983521i \(-0.0313571\pi\)
\(374\) 0.485281 0.0250933
\(375\) 0 0
\(376\) −4.48528 −0.231311
\(377\) 52.2843i 2.69278i
\(378\) 4.68629i 0.241037i
\(379\) −22.3431 −1.14769 −0.573845 0.818964i \(-0.694549\pi\)
−0.573845 + 0.818964i \(0.694549\pi\)
\(380\) 0 0
\(381\) 44.2843 2.26875
\(382\) − 8.00000i − 0.409316i
\(383\) − 34.1421i − 1.74458i −0.488987 0.872291i \(-0.662634\pi\)
0.488987 0.872291i \(-0.337366\pi\)
\(384\) −29.8579 −1.52368
\(385\) 0 0
\(386\) 2.82843 0.143963
\(387\) 30.0000i 1.52499i
\(388\) 14.0000i 0.710742i
\(389\) 24.6274 1.24866 0.624330 0.781161i \(-0.285372\pi\)
0.624330 + 0.781161i \(0.285372\pi\)
\(390\) 0 0
\(391\) 3.31371 0.167581
\(392\) 4.75736i 0.240283i
\(393\) 32.0000i 1.61419i
\(394\) −2.14214 −0.107919
\(395\) 0 0
\(396\) −9.14214 −0.459410
\(397\) − 13.3137i − 0.668196i −0.942538 0.334098i \(-0.891568\pi\)
0.942538 0.334098i \(-0.108432\pi\)
\(398\) − 8.97056i − 0.449654i
\(399\) 0 0
\(400\) 0 0
\(401\) 17.3137 0.864605 0.432303 0.901729i \(-0.357701\pi\)
0.432303 + 0.901729i \(0.357701\pi\)
\(402\) − 5.25483i − 0.262087i
\(403\) 0 0
\(404\) −24.3431 −1.21112
\(405\) 0 0
\(406\) 6.34315 0.314805
\(407\) − 3.65685i − 0.181264i
\(408\) 5.25483i 0.260153i
\(409\) −34.9706 −1.72918 −0.864592 0.502475i \(-0.832423\pi\)
−0.864592 + 0.502475i \(0.832423\pi\)
\(410\) 0 0
\(411\) 64.9706 3.20476
\(412\) 2.14214i 0.105535i
\(413\) 19.3137i 0.950365i
\(414\) 5.85786 0.287898
\(415\) 0 0
\(416\) 30.1421 1.47784
\(417\) 11.3137i 0.554035i
\(418\) 0 0
\(419\) 14.3431 0.700709 0.350354 0.936617i \(-0.386061\pi\)
0.350354 + 0.936617i \(0.386061\pi\)
\(420\) 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
\(423\) − 14.1421i − 0.687614i
\(424\) −0.544156 −0.0264265
\(425\) 0 0
\(426\) 13.2548 0.642199
\(427\) 26.6274i 1.28859i
\(428\) 6.68629i 0.323194i
\(429\) 19.3137 0.932475
\(430\) 0 0
\(431\) 11.3137 0.544962 0.272481 0.962161i \(-0.412156\pi\)
0.272481 + 0.962161i \(0.412156\pi\)
\(432\) − 16.9706i − 0.816497i
\(433\) 3.65685i 0.175737i 0.996132 + 0.0878686i \(0.0280056\pi\)
−0.996132 + 0.0878686i \(0.971994\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −6.68629 −0.320215
\(437\) 0 0
\(438\) 8.00000i 0.382255i
\(439\) 16.0000 0.763638 0.381819 0.924237i \(-0.375298\pi\)
0.381819 + 0.924237i \(0.375298\pi\)
\(440\) 0 0
\(441\) −15.0000 −0.714286
\(442\) − 3.31371i − 0.157617i
\(443\) − 21.1716i − 1.00589i −0.864318 0.502946i \(-0.832249\pi\)
0.864318 0.502946i \(-0.167751\pi\)
\(444\) 18.9117 0.897509
\(445\) 0 0
\(446\) 2.14214 0.101433
\(447\) − 32.9706i − 1.55945i
\(448\) 8.34315i 0.394177i
\(449\) 16.6274 0.784696 0.392348 0.919817i \(-0.371663\pi\)
0.392348 + 0.919817i \(0.371663\pi\)
\(450\) 0 0
\(451\) 6.00000 0.282529
\(452\) 15.2548i 0.717527i
\(453\) − 33.9411i − 1.59469i
\(454\) 1.11270 0.0522215
\(455\) 0 0
\(456\) 0 0
\(457\) 16.4853i 0.771149i 0.922677 + 0.385574i \(0.125997\pi\)
−0.922677 + 0.385574i \(0.874003\pi\)
\(458\) 8.82843i 0.412525i
\(459\) −6.62742 −0.309341
\(460\) 0 0
\(461\) −32.6274 −1.51961 −0.759805 0.650151i \(-0.774706\pi\)
−0.759805 + 0.650151i \(0.774706\pi\)
\(462\) − 2.34315i − 0.109013i
\(463\) 22.1421i 1.02903i 0.857481 + 0.514516i \(0.172028\pi\)
−0.857481 + 0.514516i \(0.827972\pi\)
\(464\) −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 \(0.931936\pi\)
\(468\) 62.4264i 2.88566i
\(469\) −8.97056 −0.414222
\(470\) 0 0
\(471\) −39.5980 −1.82458
\(472\) 15.3137i 0.704871i
\(473\) − 6.00000i − 0.275880i
\(474\) 4.68629 0.215248
\(475\) 0 0
\(476\) 4.28427 0.196369
\(477\) − 1.71573i − 0.0785578i
\(478\) 0.284271i 0.0130023i
\(479\) 36.0000 1.64488 0.822441 0.568850i \(-0.192612\pi\)
0.822441 + 0.568850i \(0.192612\pi\)
\(480\) 0 0
\(481\) −24.9706 −1.13856
\(482\) 2.48528i 0.113201i
\(483\) − 16.0000i − 0.728025i
\(484\) 1.82843 0.0831103
\(485\) 0 0
\(486\) 5.85786 0.265718
\(487\) 7.51472i 0.340524i 0.985399 + 0.170262i \(0.0544615\pi\)
−0.985399 + 0.170262i \(0.945539\pi\)
\(488\) 21.1127i 0.955727i
\(489\) 1.37258 0.0620703
\(490\) 0 0
\(491\) −23.3137 −1.05213 −0.526066 0.850443i \(-0.676334\pi\)
−0.526066 + 0.850443i \(0.676334\pi\)
\(492\) 31.0294i 1.39892i
\(493\) 8.97056i 0.404014i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) − 22.6274i − 1.01498i
\(498\) 7.02944i 0.314997i
\(499\) 1.65685 0.0741710 0.0370855 0.999312i \(-0.488193\pi\)
0.0370855 + 0.999312i \(0.488193\pi\)
\(500\) 0 0
\(501\) 31.0294 1.38629
\(502\) 4.97056i 0.221847i
\(503\) − 28.6274i − 1.27643i −0.769857 0.638217i \(-0.779672\pi\)
0.769857 0.638217i \(-0.220328\pi\)
\(504\) 15.8579 0.706365
\(505\) 0 0
\(506\) −1.17157 −0.0520828
\(507\) − 95.1127i − 4.22410i
\(508\) − 28.6274i − 1.27014i
\(509\) −9.31371 −0.412823 −0.206411 0.978465i \(-0.566179\pi\)
−0.206411 + 0.978465i \(0.566179\pi\)
\(510\) 0 0
\(511\) 13.6569 0.604144
\(512\) 22.7574i 1.00574i
\(513\) 0 0
\(514\) 5.51472 0.243244
\(515\) 0 0
\(516\) 31.0294 1.36599
\(517\) 2.82843i 0.124394i
\(518\) 3.02944i 0.133106i
\(519\) −17.3726 −0.762572
\(520\) 0 0
\(521\) 2.68629 0.117689 0.0588443 0.998267i \(-0.481258\pi\)
0.0588443 + 0.998267i \(0.481258\pi\)
\(522\) 15.8579i 0.694080i
\(523\) 37.5980i 1.64404i 0.569455 + 0.822022i \(0.307154\pi\)
−0.569455 + 0.822022i \(0.692846\pi\)
\(524\) 20.6863 0.903685
\(525\) 0 0
\(526\) −9.51472 −0.414861
\(527\) 0 0
\(528\) 8.48528i 0.369274i
\(529\) 15.0000 0.652174
\(530\) 0 0
\(531\) −48.2843 −2.09536
\(532\) 0 0
\(533\) − 40.9706i − 1.77463i
\(534\) 10.9117 0.472195
\(535\) 0 0
\(536\) −7.11270 −0.307222
\(537\) 4.68629i 0.202228i
\(538\) 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
\(543\) − 3.71573i − 0.159457i
\(544\) 5.17157 0.221729
\(545\) 0 0
\(546\) −16.0000 −0.684737
\(547\) 34.0000i 1.45374i 0.686778 + 0.726868i \(0.259025\pi\)
−0.686778 + 0.726868i \(0.740975\pi\)
\(548\) − 42.0000i − 1.79415i
\(549\) −66.5685 −2.84108
\(550\) 0 0
\(551\) 0 0
\(552\) − 12.6863i − 0.539964i
\(553\) − 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
\(561\) 3.31371 0.139905
\(562\) − 2.20101i − 0.0928440i
\(563\) 11.6569i 0.491278i 0.969361 + 0.245639i \(0.0789977\pi\)
−0.969361 + 0.245639i \(0.921002\pi\)
\(564\) −14.6274 −0.615925
\(565\) 0 0
\(566\) 5.23045 0.219852
\(567\) 2.00000i 0.0839921i
\(568\) − 17.9411i − 0.752793i
\(569\) −20.3431 −0.852829 −0.426415 0.904528i \(-0.640224\pi\)
−0.426415 + 0.904528i \(0.640224\pi\)
\(570\) 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
\(573\) − 54.6274i − 2.28209i
\(574\) −4.97056 −0.207467
\(575\) 0 0
\(576\) −20.8579 −0.869078
\(577\) − 6.97056i − 0.290188i −0.989418 0.145094i \(-0.953651\pi\)
0.989418 0.145094i \(-0.0463485\pi\)
\(578\) 6.47309i 0.269245i
\(579\) 19.3137 0.802650
\(580\) 0 0
\(581\) 12.0000 0.497844
\(582\) − 8.97056i − 0.371842i
\(583\) 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 \(-0.181387\pi\)
\(588\) 15.5147i 0.639816i
\(589\) 0 0
\(590\) 0 0
\(591\) −14.6274 −0.601692
\(592\) − 10.9706i − 0.450887i
\(593\) 20.4853i 0.841230i 0.907239 + 0.420615i \(0.138186\pi\)
−0.907239 + 0.420615i \(0.861814\pi\)
\(594\) 2.34315 0.0961404
\(595\) 0 0
\(596\) −21.3137 −0.873044
\(597\) − 61.2548i − 2.50699i
\(598\) 8.00000i 0.327144i
\(599\) −5.65685 −0.231133 −0.115566 0.993300i \(-0.536868\pi\)
−0.115566 + 0.993300i \(0.536868\pi\)
\(600\) 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
\(603\) − 22.4264i − 0.913274i
\(604\) −21.9411 −0.892772
\(605\) 0 0
\(606\) 15.5980 0.633625
\(607\) 18.2843i 0.742136i 0.928606 + 0.371068i \(0.121008\pi\)
−0.928606 + 0.371068i \(0.878992\pi\)
\(608\) 0 0
\(609\) 43.3137 1.75516
\(610\) 0 0
\(611\) 19.3137 0.781349
\(612\) 10.7107i 0.432954i
\(613\) 25.4558i 1.02815i 0.857745 + 0.514076i \(0.171865\pi\)
−0.857745 + 0.514076i \(0.828135\pi\)
\(614\) −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 \(-0.0753923\pi\)
\(618\) − 1.37258i − 0.0552134i
\(619\) 25.6569 1.03124 0.515618 0.856819i \(-0.327562\pi\)
0.515618 + 0.856819i \(0.327562\pi\)
\(620\) 0 0
\(621\) 16.0000 0.642058
\(622\) 11.3137i 0.453638i
\(623\) − 18.6274i − 0.746292i
\(624\) 57.9411 2.31950
\(625\) 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
\(633\) − 45.2548i − 1.79872i
\(634\) 8.82843 0.350622
\(635\) 0 0
\(636\) −1.77460 −0.0703676
\(637\) − 20.4853i − 0.811656i
\(638\) − 3.17157i − 0.125564i
\(639\) 56.5685 2.23782
\(640\) 0 0
\(641\) 30.0000 1.18493 0.592464 0.805597i \(-0.298155\pi\)
0.592464 + 0.805597i \(0.298155\pi\)
\(642\) − 4.28427i − 0.169087i
\(643\) 49.4558i 1.95035i 0.221440 + 0.975174i \(0.428924\pi\)
−0.221440 + 0.975174i \(0.571076\pi\)
\(644\) −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 \(0.757518\pi\)
\(648\) 1.58579i 0.0622956i
\(649\) 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 \(0.997863\pi\)
\(654\) 4.28427 0.167528
\(655\) 0 0
\(656\) 18.0000 0.702782
\(657\) 34.1421i 1.33201i
\(658\) − 2.34315i − 0.0913453i
\(659\) 21.9411 0.854705 0.427352 0.904085i \(-0.359446\pi\)
0.427352 + 0.904085i \(0.359446\pi\)
\(660\) 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
\(663\) − 22.6274i − 0.878776i
\(664\) 9.51472 0.369243
\(665\) 0 0
\(666\) −7.57359 −0.293471
\(667\) − 21.6569i − 0.838557i
\(668\) − 20.0589i − 0.776101i
\(669\) 14.6274 0.565529
\(670\) 0 0
\(671\) 13.3137 0.513970
\(672\) − 24.9706i − 0.963260i
\(673\) 4.48528i 0.172895i 0.996256 + 0.0864474i \(0.0275515\pi\)
−0.996256 + 0.0864474i \(0.972449\pi\)
\(674\) −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 \(-0.107042\pi\)
\(678\) − 9.77460i − 0.375391i
\(679\) −15.3137 −0.587686
\(680\) 0 0
\(681\) 7.59798 0.291155
\(682\) 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
\(687\) 60.2843i 2.29999i
\(688\) − 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
\(693\) − 10.0000i − 0.379869i
\(694\) −9.51472 −0.361174
\(695\) 0 0
\(696\) 34.3431 1.30177
\(697\) − 7.02944i − 0.266259i
\(698\) 2.88730i 0.109286i
\(699\) −62.6274 −2.36879
\(700\) 0 0
\(701\) 32.6274 1.23232 0.616160 0.787621i \(-0.288687\pi\)
0.616160 + 0.787621i \(0.288687\pi\)
\(702\) − 16.0000i − 0.603881i
\(703\) 0 0
\(704\) 4.17157 0.157222
\(705\) 0 0
\(706\) 0.544156 0.0204796
\(707\) − 26.6274i − 1.00143i
\(708\) 49.9411i 1.87690i
\(709\) 20.6274 0.774679 0.387339 0.921937i \(-0.373394\pi\)
0.387339 + 0.921937i \(0.373394\pi\)
\(710\) 0 0
\(711\) 20.0000 0.750059
\(712\) − 14.7696i − 0.553512i
\(713\) 0 0
\(714\) −2.74517 −0.102735
\(715\) 0 0
\(716\) 3.02944 0.113215
\(717\) 1.94113i 0.0724927i
\(718\) − 9.65685i − 0.360391i
\(719\) −29.6569 −1.10601 −0.553007 0.833177i \(-0.686520\pi\)
−0.553007 + 0.833177i \(0.686520\pi\)
\(720\) 0 0
\(721\) −2.34315 −0.0872633
\(722\) − 7.87006i − 0.292893i
\(723\) 16.9706i 0.631142i
\(724\) −2.40202 −0.0892704
\(725\) 0 0
\(726\) −1.17157 −0.0434811
\(727\) 36.4853i 1.35316i 0.736367 + 0.676582i \(0.236540\pi\)
−0.736367 + 0.676582i \(0.763460\pi\)
\(728\) 21.6569i 0.802656i
\(729\) 43.0000 1.59259
\(730\) 0 0
\(731\) −7.02944 −0.259993
\(732\) 68.8528i 2.54487i
\(733\) 33.4558i 1.23572i 0.786288 + 0.617860i \(0.212000\pi\)
−0.786288 + 0.617860i \(0.788000\pi\)
\(734\) 3.51472 0.129731
\(735\) 0 0
\(736\) −12.4853 −0.460214
\(737\) 4.48528i 0.165217i
\(738\) − 12.4264i − 0.457422i
\(739\) 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
\(747\) 30.0000i 1.09764i
\(748\) − 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
\(753\) 33.9411i 1.23688i
\(754\) −21.6569 −0.788696
\(755\) 0 0
\(756\) 20.6863 0.752353
\(757\) 9.31371i 0.338512i 0.985572 + 0.169256i \(0.0541365\pi\)
−0.985572 + 0.169256i \(0.945863\pi\)
\(758\) − 9.25483i − 0.336151i
\(759\) −8.00000 −0.290382
\(760\) 0 0
\(761\) −30.0000 −1.08750 −0.543750 0.839248i \(-0.682996\pi\)
−0.543750 + 0.839248i \(0.682996\pi\)
\(762\) 18.3431i 0.664502i
\(763\) − 7.31371i − 0.264774i
\(764\) −35.3137 −1.27761
\(765\) 0 0
\(766\) 14.1421 0.510976
\(767\) − 65.9411i − 2.38100i
\(768\) 11.2304i 0.405244i
\(769\) −14.9706 −0.539852 −0.269926 0.962881i \(-0.586999\pi\)
−0.269926 + 0.962881i \(0.586999\pi\)
\(770\) 0 0
\(771\) 37.6569 1.35618
\(772\) − 12.4853i − 0.449355i
\(773\) − 30.2843i − 1.08925i −0.838680 0.544625i \(-0.816672\pi\)
0.838680 0.544625i \(-0.183328\pi\)
\(774\) −12.4264 −0.446658
\(775\) 0 0
\(776\) −12.1421 −0.435877
\(777\) 20.6863i 0.742117i
\(778\) 10.2010i 0.365724i
\(779\) 0 0
\(780\) 0 0
\(781\) −11.3137 −0.404836
\(782\) 1.37258i 0.0490835i
\(783\) 43.3137i 1.54791i
\(784\) 9.00000 0.321429
\(785\) 0 0
\(786\) −13.2548 −0.472784
\(787\) − 18.9706i − 0.676228i −0.941105 0.338114i \(-0.890211\pi\)
0.941105 0.338114i \(-0.109789\pi\)
\(788\) 9.45584i 0.336850i
\(789\) −64.9706 −2.31301
\(790\) 0 0
\(791\) −16.6863 −0.593296
\(792\) − 7.92893i − 0.281742i
\(793\) − 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
\(801\) 46.5685 1.64542
\(802\) 7.17157i 0.253237i
\(803\) − 6.82843i − 0.240970i
\(804\) −23.1960 −0.818058
\(805\) 0 0
\(806\) 0 0
\(807\) 15.0294i 0.529061i
\(808\) − 21.1127i − 0.742742i
\(809\) −22.9706 −0.807602 −0.403801 0.914847i \(-0.632311\pi\)
−0.403801 + 0.914847i \(0.632311\pi\)
\(810\) 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
\(813\) − 43.3137i − 1.51908i
\(814\) 1.51472 0.0530909
\(815\) 0 0
\(816\) 9.94113 0.348009
\(817\) 0 0
\(818\) − 14.4853i − 0.506466i
\(819\) −68.2843 −2.38605
\(820\) 0 0
\(821\) −18.6863 −0.652156 −0.326078 0.945343i \(-0.605727\pi\)
−0.326078 + 0.945343i \(0.605727\pi\)
\(822\) 26.9117i 0.938653i
\(823\) 36.4853i 1.27180i 0.771773 + 0.635898i \(0.219370\pi\)
−0.771773 + 0.635898i \(0.780630\pi\)
\(824\) −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 \(0.796720\pi\)
\(828\) − 25.8579i − 0.898623i
\(829\) −18.0000 −0.625166 −0.312583 0.949890i \(-0.601194\pi\)
−0.312583 + 0.949890i \(0.601194\pi\)
\(830\) 0 0
\(831\) 3.31371 0.114951
\(832\) − 28.4853i − 0.987549i
\(833\) − 3.51472i − 0.121778i
\(834\) −4.68629 −0.162273
\(835\) 0 0
\(836\) 0 0
\(837\) 0 0
\(838\) 5.94113i 0.205233i
\(839\) −37.6569 −1.30006 −0.650029 0.759909i \(-0.725243\pi\)
−0.650029 + 0.759909i \(0.725243\pi\)
\(840\) 0 0
\(841\) 29.6274 1.02164
\(842\) − 2.48528i − 0.0856485i
\(843\) − 15.0294i − 0.517641i
\(844\) −29.2548 −1.00699
\(845\) 0 0
\(846\) 5.85786 0.201398
\(847\) 2.00000i 0.0687208i
\(848\) 1.02944i 0.0353510i
\(849\) 35.7157 1.22576
\(850\) 0 0
\(851\) 10.3431 0.354558
\(852\) − 58.5097i − 2.00451i
\(853\) − 32.4853i − 1.11227i −0.831090 0.556137i \(-0.812283\pi\)
0.831090 0.556137i \(-0.187717\pi\)
\(854\) −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 \(0.686640\pi\)
\(858\) 8.00000i 0.273115i
\(859\) 32.2843 1.10153 0.550763 0.834662i \(-0.314337\pi\)
0.550763 + 0.834662i \(0.314337\pi\)
\(860\) 0 0
\(861\) −33.9411 −1.15671
\(862\) 4.68629i 0.159616i
\(863\) − 14.8284i − 0.504766i −0.967627 0.252383i \(-0.918786\pi\)
0.967627 0.252383i \(-0.0812142\pi\)
\(864\) 24.9706 0.849516
\(865\) 0 0
\(866\) −1.51472 −0.0514722
\(867\) 44.2010i 1.50115i
\(868\) 0 0
\(869\) −4.00000 −0.135691
\(870\) 0 0
\(871\) 30.6274 1.03777
\(872\) − 5.79899i − 0.196379i
\(873\) − 38.2843i − 1.29573i
\(874\) 0 0
\(875\) 0 0
\(876\) 35.3137 1.19314
\(877\) − 1.45584i − 0.0491604i −0.999698 0.0245802i \(-0.992175\pi\)
0.999698 0.0245802i \(-0.00782490\pi\)
\(878\) 6.62742i 0.223664i
\(879\) 41.9411 1.41464
\(880\) 0 0
\(881\) −52.6274 −1.77306 −0.886531 0.462668i \(-0.846892\pi\)
−0.886531 + 0.462668i \(0.846892\pi\)
\(882\) − 6.21320i − 0.209209i
\(883\) 42.8284i 1.44129i 0.693304 + 0.720646i \(0.256155\pi\)
−0.693304 + 0.720646i \(0.743845\pi\)
\(884\) −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 \(0.900687\pi\)
\(888\) 16.4020i 0.550416i
\(889\) 31.3137 1.05023
\(890\) 0 0
\(891\) 1.00000 0.0335013
\(892\) − 9.45584i − 0.316605i
\(893\) 0 0
\(894\) 13.6569 0.456754
\(895\) 0 0
\(896\) −21.1127 −0.705326
\(897\) 54.6274i 1.82396i
\(898\) 6.88730i 0.229832i
\(899\) 0 0
\(900\) 0 0
\(901\) 0.402020 0.0133932
\(902\) 2.48528i 0.0827508i
\(903\) 33.9411i 1.12949i
\(904\) −13.2304 −0.440038
\(905\) 0 0
\(906\) 14.0589 0.467075
\(907\) 44.4853i 1.47711i 0.674193 + 0.738555i \(0.264491\pi\)
−0.674193 + 0.738555i \(0.735509\pi\)
\(908\) − 4.91169i − 0.163000i
\(909\) 66.5685 2.20794
\(910\) 0 0
\(911\) 57.9411 1.91968 0.959838 0.280556i \(-0.0905189\pi\)
0.959838 + 0.280556i \(0.0905189\pi\)
\(912\) 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
\(918\) − 2.74517i − 0.0906040i
\(919\) 32.0000 1.05558 0.527791 0.849374i \(-0.323020\pi\)
0.527791 + 0.849374i \(0.323020\pi\)
\(920\) 0 0
\(921\) −78.2254 −2.57761
\(922\) − 13.5147i − 0.445084i
\(923\) 77.2548i 2.54287i
\(924\) −10.3431 −0.340265
\(925\) 0 0
\(926\) −9.17157 −0.301397
\(927\) − 5.85786i − 0.192398i
\(928\) − 33.7990i − 1.10951i
\(929\) 17.3137 0.568044 0.284022 0.958818i \(-0.408331\pi\)
0.284022 + 0.958818i \(0.408331\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 40.4853i 1.32614i
\(933\) 77.2548i 2.52921i
\(934\) −3.79899 −0.124307
\(935\) 0 0
\(936\) −54.1421 −1.76969
\(937\) − 49.4558i − 1.61565i −0.589421 0.807826i \(-0.700644\pi\)
0.589421 0.807826i \(-0.299356\pi\)
\(938\) − 3.71573i − 0.121323i
\(939\) −60.2843 −1.96730
\(940\) 0 0
\(941\) −29.3137 −0.955600 −0.477800 0.878469i \(-0.658566\pi\)
−0.477800 + 0.878469i \(0.658566\pi\)
\(942\) − 16.4020i − 0.534407i
\(943\) 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 \(-0.275222\pi\)
\(948\) − 20.6863i − 0.671860i
\(949\) −46.6274 −1.51359
\(950\) 0 0
\(951\) 60.2843 1.95485
\(952\) 3.71573i 0.120427i
\(953\) 58.8284i 1.90564i 0.303536 + 0.952820i \(0.401833\pi\)
−0.303536 + 0.952820i \(0.598167\pi\)
\(954\) 0.710678 0.0230091
\(955\) 0 0
\(956\) 1.25483 0.0405842
\(957\) − 21.6569i − 0.700067i
\(958\) 14.9117i 0.481775i
\(959\) 45.9411 1.48352
\(960\) 0 0
\(961\) −31.0000 −1.00000
\(962\) − 10.3431i − 0.333476i
\(963\) − 18.2843i − 0.589202i
\(964\) 10.9706 0.353338
\(965\) 0 0
\(966\) 6.62742 0.213234
\(967\) − 18.9706i − 0.610052i −0.952344 0.305026i \(-0.901335\pi\)
0.952344 0.305026i \(-0.0986652\pi\)
\(968\) 1.58579i 0.0509691i
\(969\) 0 0
\(970\) 0 0
\(971\) −31.3137 −1.00490 −0.502452 0.864605i \(-0.667569\pi\)
−0.502452 + 0.864605i \(0.667569\pi\)
\(972\) − 25.8579i − 0.829391i
\(973\) 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 \(-0.246084\pi\)
\(978\) 0.568542i 0.0181800i
\(979\) −9.31371 −0.297667
\(980\) 0 0
\(981\) 18.2843 0.583772
\(982\) − 9.65685i − 0.308163i
\(983\) − 50.1421i − 1.59929i −0.600476 0.799643i \(-0.705022\pi\)
0.600476 0.799643i \(-0.294978\pi\)
\(984\) −26.9117 −0.857913
\(985\) 0 0
\(986\) −3.71573 −0.118333
\(987\) − 16.0000i − 0.509286i
\(988\) 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
\(993\) 43.3137i 1.37452i
\(994\) 9.37258 0.297280
\(995\) 0 0
\(996\) 31.0294 0.983205
\(997\) − 9.45584i − 0.299470i −0.988726 0.149735i \(-0.952158\pi\)
0.988726 0.149735i \(-0.0478420\pi\)
\(998\) 0.686292i 0.0217242i
\(999\) −20.6863 −0.654485
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 275.2.b.d.199.3 4
3.2 odd 2 2475.2.c.l.199.2 4
4.3 odd 2 4400.2.b.q.4049.1 4
5.2 odd 4 55.2.a.b.1.1 2
5.3 odd 4 275.2.a.c.1.2 2
5.4 even 2 inner 275.2.b.d.199.2 4
15.2 even 4 495.2.a.b.1.2 2
15.8 even 4 2475.2.a.x.1.1 2
15.14 odd 2 2475.2.c.l.199.3 4
20.3 even 4 4400.2.a.bn.1.2 2
20.7 even 4 880.2.a.m.1.1 2
20.19 odd 2 4400.2.b.q.4049.4 4
35.27 even 4 2695.2.a.f.1.1 2
40.27 even 4 3520.2.a.bo.1.2 2
40.37 odd 4 3520.2.a.bn.1.1 2
55.2 even 20 605.2.g.l.81.1 8
55.7 even 20 605.2.g.l.511.2 8
55.17 even 20 605.2.g.l.366.1 8
55.27 odd 20 605.2.g.f.366.2 8
55.32 even 4 605.2.a.d.1.2 2
55.37 odd 20 605.2.g.f.511.1 8
55.42 odd 20 605.2.g.f.81.2 8
55.43 even 4 3025.2.a.o.1.1 2
55.47 odd 20 605.2.g.f.251.1 8
55.52 even 20 605.2.g.l.251.2 8
60.47 odd 4 7920.2.a.ch.1.1 2
65.12 odd 4 9295.2.a.g.1.2 2
165.32 odd 4 5445.2.a.y.1.1 2
220.87 odd 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 5.2 odd 4
275.2.a.c.1.2 2 5.3 odd 4
275.2.b.d.199.2 4 5.4 even 2 inner
275.2.b.d.199.3 4 1.1 even 1 trivial
495.2.a.b.1.2 2 15.2 even 4
605.2.a.d.1.2 2 55.32 even 4
605.2.g.f.81.2 8 55.42 odd 20
605.2.g.f.251.1 8 55.47 odd 20
605.2.g.f.366.2 8 55.27 odd 20
605.2.g.f.511.1 8 55.37 odd 20
605.2.g.l.81.1 8 55.2 even 20
605.2.g.l.251.2 8 55.52 even 20
605.2.g.l.366.1 8 55.17 even 20
605.2.g.l.511.2 8 55.7 even 20
880.2.a.m.1.1 2 20.7 even 4
2475.2.a.x.1.1 2 15.8 even 4
2475.2.c.l.199.2 4 3.2 odd 2
2475.2.c.l.199.3 4 15.14 odd 2
2695.2.a.f.1.1 2 35.27 even 4
3025.2.a.o.1.1 2 55.43 even 4
3520.2.a.bn.1.1 2 40.37 odd 4
3520.2.a.bo.1.2 2 40.27 even 4
4400.2.a.bn.1.2 2 20.3 even 4
4400.2.b.q.4049.1 4 4.3 odd 2
4400.2.b.q.4049.4 4 20.19 odd 2
5445.2.a.y.1.1 2 165.32 odd 4
7920.2.a.ch.1.1 2 60.47 odd 4
9295.2.a.g.1.2 2 65.12 odd 4
9680.2.a.bn.1.1 2 220.87 odd 4