Properties

Label 3850.2.c.d.1849.1
Level $3850$
Weight $2$
Character 3850.1849
Analytic conductor $30.742$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 3850 = 2 \cdot 5^{2} \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3850.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(30.7424047782\)
Analytic rank: \(1\)
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 154)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1849.1
Root \(-1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 3850.1849
Dual form 3850.2.c.d.1849.2

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000i q^{2} -2.00000i q^{3} -1.00000 q^{4} -2.00000 q^{6} -1.00000i q^{7} +1.00000i q^{8} -1.00000 q^{9} +O(q^{10})\) \(q-1.00000i q^{2} -2.00000i q^{3} -1.00000 q^{4} -2.00000 q^{6} -1.00000i q^{7} +1.00000i q^{8} -1.00000 q^{9} +1.00000 q^{11} +2.00000i q^{12} +4.00000i q^{13} -1.00000 q^{14} +1.00000 q^{16} +1.00000i q^{18} -4.00000 q^{19} -2.00000 q^{21} -1.00000i q^{22} -4.00000i q^{23} +2.00000 q^{24} +4.00000 q^{26} -4.00000i q^{27} +1.00000i q^{28} -2.00000 q^{29} -10.0000 q^{31} -1.00000i q^{32} -2.00000i q^{33} +1.00000 q^{36} -6.00000i q^{37} +4.00000i q^{38} +8.00000 q^{39} +2.00000i q^{42} +4.00000i q^{43} -1.00000 q^{44} -4.00000 q^{46} +10.0000i q^{47} -2.00000i q^{48} -1.00000 q^{49} -4.00000i q^{52} +14.0000i q^{53} -4.00000 q^{54} +1.00000 q^{56} +8.00000i q^{57} +2.00000i q^{58} -10.0000 q^{59} -8.00000 q^{61} +10.0000i q^{62} +1.00000i q^{63} -1.00000 q^{64} -2.00000 q^{66} +8.00000i q^{67} -8.00000 q^{69} -4.00000 q^{71} -1.00000i q^{72} -4.00000i q^{73} -6.00000 q^{74} +4.00000 q^{76} -1.00000i q^{77} -8.00000i q^{78} -16.0000 q^{79} -11.0000 q^{81} -4.00000i q^{83} +2.00000 q^{84} +4.00000 q^{86} +4.00000i q^{87} +1.00000i q^{88} -10.0000 q^{89} +4.00000 q^{91} +4.00000i q^{92} +20.0000i q^{93} +10.0000 q^{94} -2.00000 q^{96} +6.00000i q^{97} +1.00000i q^{98} -1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{4} - 4 q^{6} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{4} - 4 q^{6} - 2 q^{9} + 2 q^{11} - 2 q^{14} + 2 q^{16} - 8 q^{19} - 4 q^{21} + 4 q^{24} + 8 q^{26} - 4 q^{29} - 20 q^{31} + 2 q^{36} + 16 q^{39} - 2 q^{44} - 8 q^{46} - 2 q^{49} - 8 q^{54} + 2 q^{56} - 20 q^{59} - 16 q^{61} - 2 q^{64} - 4 q^{66} - 16 q^{69} - 8 q^{71} - 12 q^{74} + 8 q^{76} - 32 q^{79} - 22 q^{81} + 4 q^{84} + 8 q^{86} - 20 q^{89} + 8 q^{91} + 20 q^{94} - 4 q^{96} - 2 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3850\mathbb{Z}\right)^\times\).

\(n\) \(1751\) \(2201\) \(2927\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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
\(3\) − 2.00000i − 1.15470i −0.816497 0.577350i \(-0.804087\pi\)
0.816497 0.577350i \(-0.195913\pi\)
\(4\) −1.00000 −0.500000
\(5\) 0 0
\(6\) −2.00000 −0.816497
\(7\) − 1.00000i − 0.377964i
\(8\) 1.00000i 0.353553i
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) 1.00000 0.301511
\(12\) 2.00000i 0.577350i
\(13\) 4.00000i 1.10940i 0.832050 + 0.554700i \(0.187167\pi\)
−0.832050 + 0.554700i \(0.812833\pi\)
\(14\) −1.00000 −0.267261
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(18\) 1.00000i 0.235702i
\(19\) −4.00000 −0.917663 −0.458831 0.888523i \(-0.651732\pi\)
−0.458831 + 0.888523i \(0.651732\pi\)
\(20\) 0 0
\(21\) −2.00000 −0.436436
\(22\) − 1.00000i − 0.213201i
\(23\) − 4.00000i − 0.834058i −0.908893 0.417029i \(-0.863071\pi\)
0.908893 0.417029i \(-0.136929\pi\)
\(24\) 2.00000 0.408248
\(25\) 0 0
\(26\) 4.00000 0.784465
\(27\) − 4.00000i − 0.769800i
\(28\) 1.00000i 0.188982i
\(29\) −2.00000 −0.371391 −0.185695 0.982607i \(-0.559454\pi\)
−0.185695 + 0.982607i \(0.559454\pi\)
\(30\) 0 0
\(31\) −10.0000 −1.79605 −0.898027 0.439941i \(-0.854999\pi\)
−0.898027 + 0.439941i \(0.854999\pi\)
\(32\) − 1.00000i − 0.176777i
\(33\) − 2.00000i − 0.348155i
\(34\) 0 0
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) − 6.00000i − 0.986394i −0.869918 0.493197i \(-0.835828\pi\)
0.869918 0.493197i \(-0.164172\pi\)
\(38\) 4.00000i 0.648886i
\(39\) 8.00000 1.28103
\(40\) 0 0
\(41\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(42\) 2.00000i 0.308607i
\(43\) 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
\(47\) 10.0000i 1.45865i 0.684167 + 0.729325i \(0.260166\pi\)
−0.684167 + 0.729325i \(0.739834\pi\)
\(48\) − 2.00000i − 0.288675i
\(49\) −1.00000 −0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) − 4.00000i − 0.554700i
\(53\) 14.0000i 1.92305i 0.274721 + 0.961524i \(0.411414\pi\)
−0.274721 + 0.961524i \(0.588586\pi\)
\(54\) −4.00000 −0.544331
\(55\) 0 0
\(56\) 1.00000 0.133631
\(57\) 8.00000i 1.05963i
\(58\) 2.00000i 0.262613i
\(59\) −10.0000 −1.30189 −0.650945 0.759125i \(-0.725627\pi\)
−0.650945 + 0.759125i \(0.725627\pi\)
\(60\) 0 0
\(61\) −8.00000 −1.02430 −0.512148 0.858898i \(-0.671150\pi\)
−0.512148 + 0.858898i \(0.671150\pi\)
\(62\) 10.0000i 1.27000i
\(63\) 1.00000i 0.125988i
\(64\) −1.00000 −0.125000
\(65\) 0 0
\(66\) −2.00000 −0.246183
\(67\) 8.00000i 0.977356i 0.872464 + 0.488678i \(0.162521\pi\)
−0.872464 + 0.488678i \(0.837479\pi\)
\(68\) 0 0
\(69\) −8.00000 −0.963087
\(70\) 0 0
\(71\) −4.00000 −0.474713 −0.237356 0.971423i \(-0.576281\pi\)
−0.237356 + 0.971423i \(0.576281\pi\)
\(72\) − 1.00000i − 0.117851i
\(73\) − 4.00000i − 0.468165i −0.972217 0.234082i \(-0.924791\pi\)
0.972217 0.234082i \(-0.0752085\pi\)
\(74\) −6.00000 −0.697486
\(75\) 0 0
\(76\) 4.00000 0.458831
\(77\) − 1.00000i − 0.113961i
\(78\) − 8.00000i − 0.905822i
\(79\) −16.0000 −1.80014 −0.900070 0.435745i \(-0.856485\pi\)
−0.900070 + 0.435745i \(0.856485\pi\)
\(80\) 0 0
\(81\) −11.0000 −1.22222
\(82\) 0 0
\(83\) − 4.00000i − 0.439057i −0.975606 0.219529i \(-0.929548\pi\)
0.975606 0.219529i \(-0.0704519\pi\)
\(84\) 2.00000 0.218218
\(85\) 0 0
\(86\) 4.00000 0.431331
\(87\) 4.00000i 0.428845i
\(88\) 1.00000i 0.106600i
\(89\) −10.0000 −1.06000 −0.529999 0.847998i \(-0.677808\pi\)
−0.529999 + 0.847998i \(0.677808\pi\)
\(90\) 0 0
\(91\) 4.00000 0.419314
\(92\) 4.00000i 0.417029i
\(93\) 20.0000i 2.07390i
\(94\) 10.0000 1.03142
\(95\) 0 0
\(96\) −2.00000 −0.204124
\(97\) 6.00000i 0.609208i 0.952479 + 0.304604i \(0.0985241\pi\)
−0.952479 + 0.304604i \(0.901476\pi\)
\(98\) 1.00000i 0.101015i
\(99\) −1.00000 −0.100504
\(100\) 0 0
\(101\) 12.0000 1.19404 0.597022 0.802225i \(-0.296350\pi\)
0.597022 + 0.802225i \(0.296350\pi\)
\(102\) 0 0
\(103\) − 2.00000i − 0.197066i −0.995134 0.0985329i \(-0.968585\pi\)
0.995134 0.0985329i \(-0.0314150\pi\)
\(104\) −4.00000 −0.392232
\(105\) 0 0
\(106\) 14.0000 1.35980
\(107\) − 12.0000i − 1.16008i −0.814587 0.580042i \(-0.803036\pi\)
0.814587 0.580042i \(-0.196964\pi\)
\(108\) 4.00000i 0.384900i
\(109\) 14.0000 1.34096 0.670478 0.741929i \(-0.266089\pi\)
0.670478 + 0.741929i \(0.266089\pi\)
\(110\) 0 0
\(111\) −12.0000 −1.13899
\(112\) − 1.00000i − 0.0944911i
\(113\) 14.0000i 1.31701i 0.752577 + 0.658505i \(0.228811\pi\)
−0.752577 + 0.658505i \(0.771189\pi\)
\(114\) 8.00000 0.749269
\(115\) 0 0
\(116\) 2.00000 0.185695
\(117\) − 4.00000i − 0.369800i
\(118\) 10.0000i 0.920575i
\(119\) 0 0
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 8.00000i 0.724286i
\(123\) 0 0
\(124\) 10.0000 0.898027
\(125\) 0 0
\(126\) 1.00000 0.0890871
\(127\) − 16.0000i − 1.41977i −0.704317 0.709885i \(-0.748747\pi\)
0.704317 0.709885i \(-0.251253\pi\)
\(128\) 1.00000i 0.0883883i
\(129\) 8.00000 0.704361
\(130\) 0 0
\(131\) 8.00000 0.698963 0.349482 0.936943i \(-0.386358\pi\)
0.349482 + 0.936943i \(0.386358\pi\)
\(132\) 2.00000i 0.174078i
\(133\) 4.00000i 0.346844i
\(134\) 8.00000 0.691095
\(135\) 0 0
\(136\) 0 0
\(137\) 6.00000i 0.512615i 0.966595 + 0.256307i \(0.0825059\pi\)
−0.966595 + 0.256307i \(0.917494\pi\)
\(138\) 8.00000i 0.681005i
\(139\) −20.0000 −1.69638 −0.848189 0.529694i \(-0.822307\pi\)
−0.848189 + 0.529694i \(0.822307\pi\)
\(140\) 0 0
\(141\) 20.0000 1.68430
\(142\) 4.00000i 0.335673i
\(143\) 4.00000i 0.334497i
\(144\) −1.00000 −0.0833333
\(145\) 0 0
\(146\) −4.00000 −0.331042
\(147\) 2.00000i 0.164957i
\(148\) 6.00000i 0.493197i
\(149\) −22.0000 −1.80231 −0.901155 0.433497i \(-0.857280\pi\)
−0.901155 + 0.433497i \(0.857280\pi\)
\(150\) 0 0
\(151\) 16.0000 1.30206 0.651031 0.759051i \(-0.274337\pi\)
0.651031 + 0.759051i \(0.274337\pi\)
\(152\) − 4.00000i − 0.324443i
\(153\) 0 0
\(154\) −1.00000 −0.0805823
\(155\) 0 0
\(156\) −8.00000 −0.640513
\(157\) 10.0000i 0.798087i 0.916932 + 0.399043i \(0.130658\pi\)
−0.916932 + 0.399043i \(0.869342\pi\)
\(158\) 16.0000i 1.27289i
\(159\) 28.0000 2.22054
\(160\) 0 0
\(161\) −4.00000 −0.315244
\(162\) 11.0000i 0.864242i
\(163\) − 24.0000i − 1.87983i −0.341415 0.939913i \(-0.610906\pi\)
0.341415 0.939913i \(-0.389094\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) −4.00000 −0.310460
\(167\) − 8.00000i − 0.619059i −0.950890 0.309529i \(-0.899829\pi\)
0.950890 0.309529i \(-0.100171\pi\)
\(168\) − 2.00000i − 0.154303i
\(169\) −3.00000 −0.230769
\(170\) 0 0
\(171\) 4.00000 0.305888
\(172\) − 4.00000i − 0.304997i
\(173\) − 4.00000i − 0.304114i −0.988372 0.152057i \(-0.951410\pi\)
0.988372 0.152057i \(-0.0485898\pi\)
\(174\) 4.00000 0.303239
\(175\) 0 0
\(176\) 1.00000 0.0753778
\(177\) 20.0000i 1.50329i
\(178\) 10.0000i 0.749532i
\(179\) −12.0000 −0.896922 −0.448461 0.893802i \(-0.648028\pi\)
−0.448461 + 0.893802i \(0.648028\pi\)
\(180\) 0 0
\(181\) 14.0000 1.04061 0.520306 0.853980i \(-0.325818\pi\)
0.520306 + 0.853980i \(0.325818\pi\)
\(182\) − 4.00000i − 0.296500i
\(183\) 16.0000i 1.18275i
\(184\) 4.00000 0.294884
\(185\) 0 0
\(186\) 20.0000 1.46647
\(187\) 0 0
\(188\) − 10.0000i − 0.729325i
\(189\) −4.00000 −0.290957
\(190\) 0 0
\(191\) 8.00000 0.578860 0.289430 0.957199i \(-0.406534\pi\)
0.289430 + 0.957199i \(0.406534\pi\)
\(192\) 2.00000i 0.144338i
\(193\) 6.00000i 0.431889i 0.976406 + 0.215945i \(0.0692831\pi\)
−0.976406 + 0.215945i \(0.930717\pi\)
\(194\) 6.00000 0.430775
\(195\) 0 0
\(196\) 1.00000 0.0714286
\(197\) − 18.0000i − 1.28245i −0.767354 0.641223i \(-0.778427\pi\)
0.767354 0.641223i \(-0.221573\pi\)
\(198\) 1.00000i 0.0710669i
\(199\) 14.0000 0.992434 0.496217 0.868199i \(-0.334722\pi\)
0.496217 + 0.868199i \(0.334722\pi\)
\(200\) 0 0
\(201\) 16.0000 1.12855
\(202\) − 12.0000i − 0.844317i
\(203\) 2.00000i 0.140372i
\(204\) 0 0
\(205\) 0 0
\(206\) −2.00000 −0.139347
\(207\) 4.00000i 0.278019i
\(208\) 4.00000i 0.277350i
\(209\) −4.00000 −0.276686
\(210\) 0 0
\(211\) −4.00000 −0.275371 −0.137686 0.990476i \(-0.543966\pi\)
−0.137686 + 0.990476i \(0.543966\pi\)
\(212\) − 14.0000i − 0.961524i
\(213\) 8.00000i 0.548151i
\(214\) −12.0000 −0.820303
\(215\) 0 0
\(216\) 4.00000 0.272166
\(217\) 10.0000i 0.678844i
\(218\) − 14.0000i − 0.948200i
\(219\) −8.00000 −0.540590
\(220\) 0 0
\(221\) 0 0
\(222\) 12.0000i 0.805387i
\(223\) 14.0000i 0.937509i 0.883328 + 0.468755i \(0.155297\pi\)
−0.883328 + 0.468755i \(0.844703\pi\)
\(224\) −1.00000 −0.0668153
\(225\) 0 0
\(226\) 14.0000 0.931266
\(227\) 8.00000i 0.530979i 0.964114 + 0.265489i \(0.0855335\pi\)
−0.964114 + 0.265489i \(0.914466\pi\)
\(228\) − 8.00000i − 0.529813i
\(229\) 10.0000 0.660819 0.330409 0.943838i \(-0.392813\pi\)
0.330409 + 0.943838i \(0.392813\pi\)
\(230\) 0 0
\(231\) −2.00000 −0.131590
\(232\) − 2.00000i − 0.131306i
\(233\) − 6.00000i − 0.393073i −0.980497 0.196537i \(-0.937031\pi\)
0.980497 0.196537i \(-0.0629694\pi\)
\(234\) −4.00000 −0.261488
\(235\) 0 0
\(236\) 10.0000 0.650945
\(237\) 32.0000i 2.07862i
\(238\) 0 0
\(239\) 8.00000 0.517477 0.258738 0.965947i \(-0.416693\pi\)
0.258738 + 0.965947i \(0.416693\pi\)
\(240\) 0 0
\(241\) 8.00000 0.515325 0.257663 0.966235i \(-0.417048\pi\)
0.257663 + 0.966235i \(0.417048\pi\)
\(242\) − 1.00000i − 0.0642824i
\(243\) 10.0000i 0.641500i
\(244\) 8.00000 0.512148
\(245\) 0 0
\(246\) 0 0
\(247\) − 16.0000i − 1.01806i
\(248\) − 10.0000i − 0.635001i
\(249\) −8.00000 −0.506979
\(250\) 0 0
\(251\) −26.0000 −1.64111 −0.820553 0.571571i \(-0.806334\pi\)
−0.820553 + 0.571571i \(0.806334\pi\)
\(252\) − 1.00000i − 0.0629941i
\(253\) − 4.00000i − 0.251478i
\(254\) −16.0000 −1.00393
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 2.00000i 0.124757i 0.998053 + 0.0623783i \(0.0198685\pi\)
−0.998053 + 0.0623783i \(0.980131\pi\)
\(258\) − 8.00000i − 0.498058i
\(259\) −6.00000 −0.372822
\(260\) 0 0
\(261\) 2.00000 0.123797
\(262\) − 8.00000i − 0.494242i
\(263\) 24.0000i 1.47990i 0.672660 + 0.739952i \(0.265152\pi\)
−0.672660 + 0.739952i \(0.734848\pi\)
\(264\) 2.00000 0.123091
\(265\) 0 0
\(266\) 4.00000 0.245256
\(267\) 20.0000i 1.22398i
\(268\) − 8.00000i − 0.488678i
\(269\) −14.0000 −0.853595 −0.426798 0.904347i \(-0.640358\pi\)
−0.426798 + 0.904347i \(0.640358\pi\)
\(270\) 0 0
\(271\) −28.0000 −1.70088 −0.850439 0.526073i \(-0.823664\pi\)
−0.850439 + 0.526073i \(0.823664\pi\)
\(272\) 0 0
\(273\) − 8.00000i − 0.484182i
\(274\) 6.00000 0.362473
\(275\) 0 0
\(276\) 8.00000 0.481543
\(277\) 6.00000i 0.360505i 0.983620 + 0.180253i \(0.0576915\pi\)
−0.983620 + 0.180253i \(0.942309\pi\)
\(278\) 20.0000i 1.19952i
\(279\) 10.0000 0.598684
\(280\) 0 0
\(281\) 30.0000 1.78965 0.894825 0.446417i \(-0.147300\pi\)
0.894825 + 0.446417i \(0.147300\pi\)
\(282\) − 20.0000i − 1.19098i
\(283\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(284\) 4.00000 0.237356
\(285\) 0 0
\(286\) 4.00000 0.236525
\(287\) 0 0
\(288\) 1.00000i 0.0589256i
\(289\) 17.0000 1.00000
\(290\) 0 0
\(291\) 12.0000 0.703452
\(292\) 4.00000i 0.234082i
\(293\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(294\) 2.00000 0.116642
\(295\) 0 0
\(296\) 6.00000 0.348743
\(297\) − 4.00000i − 0.232104i
\(298\) 22.0000i 1.27443i
\(299\) 16.0000 0.925304
\(300\) 0 0
\(301\) 4.00000 0.230556
\(302\) − 16.0000i − 0.920697i
\(303\) − 24.0000i − 1.37876i
\(304\) −4.00000 −0.229416
\(305\) 0 0
\(306\) 0 0
\(307\) 16.0000i 0.913168i 0.889680 + 0.456584i \(0.150927\pi\)
−0.889680 + 0.456584i \(0.849073\pi\)
\(308\) 1.00000i 0.0569803i
\(309\) −4.00000 −0.227552
\(310\) 0 0
\(311\) −6.00000 −0.340229 −0.170114 0.985424i \(-0.554414\pi\)
−0.170114 + 0.985424i \(0.554414\pi\)
\(312\) 8.00000i 0.452911i
\(313\) 6.00000i 0.339140i 0.985518 + 0.169570i \(0.0542379\pi\)
−0.985518 + 0.169570i \(0.945762\pi\)
\(314\) 10.0000 0.564333
\(315\) 0 0
\(316\) 16.0000 0.900070
\(317\) − 18.0000i − 1.01098i −0.862832 0.505490i \(-0.831312\pi\)
0.862832 0.505490i \(-0.168688\pi\)
\(318\) − 28.0000i − 1.57016i
\(319\) −2.00000 −0.111979
\(320\) 0 0
\(321\) −24.0000 −1.33955
\(322\) 4.00000i 0.222911i
\(323\) 0 0
\(324\) 11.0000 0.611111
\(325\) 0 0
\(326\) −24.0000 −1.32924
\(327\) − 28.0000i − 1.54840i
\(328\) 0 0
\(329\) 10.0000 0.551318
\(330\) 0 0
\(331\) −20.0000 −1.09930 −0.549650 0.835395i \(-0.685239\pi\)
−0.549650 + 0.835395i \(0.685239\pi\)
\(332\) 4.00000i 0.219529i
\(333\) 6.00000i 0.328798i
\(334\) −8.00000 −0.437741
\(335\) 0 0
\(336\) −2.00000 −0.109109
\(337\) − 34.0000i − 1.85210i −0.377403 0.926049i \(-0.623183\pi\)
0.377403 0.926049i \(-0.376817\pi\)
\(338\) 3.00000i 0.163178i
\(339\) 28.0000 1.52075
\(340\) 0 0
\(341\) −10.0000 −0.541530
\(342\) − 4.00000i − 0.216295i
\(343\) 1.00000i 0.0539949i
\(344\) −4.00000 −0.215666
\(345\) 0 0
\(346\) −4.00000 −0.215041
\(347\) − 12.0000i − 0.644194i −0.946707 0.322097i \(-0.895612\pi\)
0.946707 0.322097i \(-0.104388\pi\)
\(348\) − 4.00000i − 0.214423i
\(349\) −32.0000 −1.71292 −0.856460 0.516213i \(-0.827341\pi\)
−0.856460 + 0.516213i \(0.827341\pi\)
\(350\) 0 0
\(351\) 16.0000 0.854017
\(352\) − 1.00000i − 0.0533002i
\(353\) − 2.00000i − 0.106449i −0.998583 0.0532246i \(-0.983050\pi\)
0.998583 0.0532246i \(-0.0169499\pi\)
\(354\) 20.0000 1.06299
\(355\) 0 0
\(356\) 10.0000 0.529999
\(357\) 0 0
\(358\) 12.0000i 0.634220i
\(359\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(360\) 0 0
\(361\) −3.00000 −0.157895
\(362\) − 14.0000i − 0.735824i
\(363\) − 2.00000i − 0.104973i
\(364\) −4.00000 −0.209657
\(365\) 0 0
\(366\) 16.0000 0.836333
\(367\) 18.0000i 0.939592i 0.882775 + 0.469796i \(0.155673\pi\)
−0.882775 + 0.469796i \(0.844327\pi\)
\(368\) − 4.00000i − 0.208514i
\(369\) 0 0
\(370\) 0 0
\(371\) 14.0000 0.726844
\(372\) − 20.0000i − 1.03695i
\(373\) 34.0000i 1.76045i 0.474554 + 0.880227i \(0.342610\pi\)
−0.474554 + 0.880227i \(0.657390\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) −10.0000 −0.515711
\(377\) − 8.00000i − 0.412021i
\(378\) 4.00000i 0.205738i
\(379\) −8.00000 −0.410932 −0.205466 0.978664i \(-0.565871\pi\)
−0.205466 + 0.978664i \(0.565871\pi\)
\(380\) 0 0
\(381\) −32.0000 −1.63941
\(382\) − 8.00000i − 0.409316i
\(383\) − 14.0000i − 0.715367i −0.933843 0.357683i \(-0.883567\pi\)
0.933843 0.357683i \(-0.116433\pi\)
\(384\) 2.00000 0.102062
\(385\) 0 0
\(386\) 6.00000 0.305392
\(387\) − 4.00000i − 0.203331i
\(388\) − 6.00000i − 0.304604i
\(389\) 18.0000 0.912636 0.456318 0.889817i \(-0.349168\pi\)
0.456318 + 0.889817i \(0.349168\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) − 1.00000i − 0.0505076i
\(393\) − 16.0000i − 0.807093i
\(394\) −18.0000 −0.906827
\(395\) 0 0
\(396\) 1.00000 0.0502519
\(397\) 18.0000i 0.903394i 0.892171 + 0.451697i \(0.149181\pi\)
−0.892171 + 0.451697i \(0.850819\pi\)
\(398\) − 14.0000i − 0.701757i
\(399\) 8.00000 0.400501
\(400\) 0 0
\(401\) 10.0000 0.499376 0.249688 0.968326i \(-0.419672\pi\)
0.249688 + 0.968326i \(0.419672\pi\)
\(402\) − 16.0000i − 0.798007i
\(403\) − 40.0000i − 1.99254i
\(404\) −12.0000 −0.597022
\(405\) 0 0
\(406\) 2.00000 0.0992583
\(407\) − 6.00000i − 0.297409i
\(408\) 0 0
\(409\) 4.00000 0.197787 0.0988936 0.995098i \(-0.468470\pi\)
0.0988936 + 0.995098i \(0.468470\pi\)
\(410\) 0 0
\(411\) 12.0000 0.591916
\(412\) 2.00000i 0.0985329i
\(413\) 10.0000i 0.492068i
\(414\) 4.00000 0.196589
\(415\) 0 0
\(416\) 4.00000 0.196116
\(417\) 40.0000i 1.95881i
\(418\) 4.00000i 0.195646i
\(419\) 30.0000 1.46560 0.732798 0.680446i \(-0.238214\pi\)
0.732798 + 0.680446i \(0.238214\pi\)
\(420\) 0 0
\(421\) 10.0000 0.487370 0.243685 0.969854i \(-0.421644\pi\)
0.243685 + 0.969854i \(0.421644\pi\)
\(422\) 4.00000i 0.194717i
\(423\) − 10.0000i − 0.486217i
\(424\) −14.0000 −0.679900
\(425\) 0 0
\(426\) 8.00000 0.387601
\(427\) 8.00000i 0.387147i
\(428\) 12.0000i 0.580042i
\(429\) 8.00000 0.386244
\(430\) 0 0
\(431\) −16.0000 −0.770693 −0.385346 0.922772i \(-0.625918\pi\)
−0.385346 + 0.922772i \(0.625918\pi\)
\(432\) − 4.00000i − 0.192450i
\(433\) 10.0000i 0.480569i 0.970702 + 0.240285i \(0.0772408\pi\)
−0.970702 + 0.240285i \(0.922759\pi\)
\(434\) 10.0000 0.480015
\(435\) 0 0
\(436\) −14.0000 −0.670478
\(437\) 16.0000i 0.765384i
\(438\) 8.00000i 0.382255i
\(439\) 28.0000 1.33637 0.668184 0.743996i \(-0.267072\pi\)
0.668184 + 0.743996i \(0.267072\pi\)
\(440\) 0 0
\(441\) 1.00000 0.0476190
\(442\) 0 0
\(443\) − 4.00000i − 0.190046i −0.995475 0.0950229i \(-0.969708\pi\)
0.995475 0.0950229i \(-0.0302924\pi\)
\(444\) 12.0000 0.569495
\(445\) 0 0
\(446\) 14.0000 0.662919
\(447\) 44.0000i 2.08113i
\(448\) 1.00000i 0.0472456i
\(449\) −6.00000 −0.283158 −0.141579 0.989927i \(-0.545218\pi\)
−0.141579 + 0.989927i \(0.545218\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) − 14.0000i − 0.658505i
\(453\) − 32.0000i − 1.50349i
\(454\) 8.00000 0.375459
\(455\) 0 0
\(456\) −8.00000 −0.374634
\(457\) − 38.0000i − 1.77757i −0.458329 0.888783i \(-0.651552\pi\)
0.458329 0.888783i \(-0.348448\pi\)
\(458\) − 10.0000i − 0.467269i
\(459\) 0 0
\(460\) 0 0
\(461\) 32.0000 1.49039 0.745194 0.666847i \(-0.232357\pi\)
0.745194 + 0.666847i \(0.232357\pi\)
\(462\) 2.00000i 0.0930484i
\(463\) − 12.0000i − 0.557687i −0.960337 0.278844i \(-0.910049\pi\)
0.960337 0.278844i \(-0.0899511\pi\)
\(464\) −2.00000 −0.0928477
\(465\) 0 0
\(466\) −6.00000 −0.277945
\(467\) 14.0000i 0.647843i 0.946084 + 0.323921i \(0.105001\pi\)
−0.946084 + 0.323921i \(0.894999\pi\)
\(468\) 4.00000i 0.184900i
\(469\) 8.00000 0.369406
\(470\) 0 0
\(471\) 20.0000 0.921551
\(472\) − 10.0000i − 0.460287i
\(473\) 4.00000i 0.183920i
\(474\) 32.0000 1.46981
\(475\) 0 0
\(476\) 0 0
\(477\) − 14.0000i − 0.641016i
\(478\) − 8.00000i − 0.365911i
\(479\) −12.0000 −0.548294 −0.274147 0.961688i \(-0.588395\pi\)
−0.274147 + 0.961688i \(0.588395\pi\)
\(480\) 0 0
\(481\) 24.0000 1.09431
\(482\) − 8.00000i − 0.364390i
\(483\) 8.00000i 0.364013i
\(484\) −1.00000 −0.0454545
\(485\) 0 0
\(486\) 10.0000 0.453609
\(487\) 12.0000i 0.543772i 0.962329 + 0.271886i \(0.0876473\pi\)
−0.962329 + 0.271886i \(0.912353\pi\)
\(488\) − 8.00000i − 0.362143i
\(489\) −48.0000 −2.17064
\(490\) 0 0
\(491\) −28.0000 −1.26362 −0.631811 0.775122i \(-0.717688\pi\)
−0.631811 + 0.775122i \(0.717688\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) −16.0000 −0.719874
\(495\) 0 0
\(496\) −10.0000 −0.449013
\(497\) 4.00000i 0.179425i
\(498\) 8.00000i 0.358489i
\(499\) 16.0000 0.716258 0.358129 0.933672i \(-0.383415\pi\)
0.358129 + 0.933672i \(0.383415\pi\)
\(500\) 0 0
\(501\) −16.0000 −0.714827
\(502\) 26.0000i 1.16044i
\(503\) − 12.0000i − 0.535054i −0.963550 0.267527i \(-0.913794\pi\)
0.963550 0.267527i \(-0.0862064\pi\)
\(504\) −1.00000 −0.0445435
\(505\) 0 0
\(506\) −4.00000 −0.177822
\(507\) 6.00000i 0.266469i
\(508\) 16.0000i 0.709885i
\(509\) −38.0000 −1.68432 −0.842160 0.539227i \(-0.818716\pi\)
−0.842160 + 0.539227i \(0.818716\pi\)
\(510\) 0 0
\(511\) −4.00000 −0.176950
\(512\) − 1.00000i − 0.0441942i
\(513\) 16.0000i 0.706417i
\(514\) 2.00000 0.0882162
\(515\) 0 0
\(516\) −8.00000 −0.352180
\(517\) 10.0000i 0.439799i
\(518\) 6.00000i 0.263625i
\(519\) −8.00000 −0.351161
\(520\) 0 0
\(521\) −42.0000 −1.84005 −0.920027 0.391856i \(-0.871833\pi\)
−0.920027 + 0.391856i \(0.871833\pi\)
\(522\) − 2.00000i − 0.0875376i
\(523\) − 16.0000i − 0.699631i −0.936819 0.349816i \(-0.886244\pi\)
0.936819 0.349816i \(-0.113756\pi\)
\(524\) −8.00000 −0.349482
\(525\) 0 0
\(526\) 24.0000 1.04645
\(527\) 0 0
\(528\) − 2.00000i − 0.0870388i
\(529\) 7.00000 0.304348
\(530\) 0 0
\(531\) 10.0000 0.433963
\(532\) − 4.00000i − 0.173422i
\(533\) 0 0
\(534\) 20.0000 0.865485
\(535\) 0 0
\(536\) −8.00000 −0.345547
\(537\) 24.0000i 1.03568i
\(538\) 14.0000i 0.603583i
\(539\) −1.00000 −0.0430730
\(540\) 0 0
\(541\) −2.00000 −0.0859867 −0.0429934 0.999075i \(-0.513689\pi\)
−0.0429934 + 0.999075i \(0.513689\pi\)
\(542\) 28.0000i 1.20270i
\(543\) − 28.0000i − 1.20160i
\(544\) 0 0
\(545\) 0 0
\(546\) −8.00000 −0.342368
\(547\) 4.00000i 0.171028i 0.996337 + 0.0855138i \(0.0272532\pi\)
−0.996337 + 0.0855138i \(0.972747\pi\)
\(548\) − 6.00000i − 0.256307i
\(549\) 8.00000 0.341432
\(550\) 0 0
\(551\) 8.00000 0.340811
\(552\) − 8.00000i − 0.340503i
\(553\) 16.0000i 0.680389i
\(554\) 6.00000 0.254916
\(555\) 0 0
\(556\) 20.0000 0.848189
\(557\) 30.0000i 1.27114i 0.772043 + 0.635570i \(0.219235\pi\)
−0.772043 + 0.635570i \(0.780765\pi\)
\(558\) − 10.0000i − 0.423334i
\(559\) −16.0000 −0.676728
\(560\) 0 0
\(561\) 0 0
\(562\) − 30.0000i − 1.26547i
\(563\) − 12.0000i − 0.505740i −0.967500 0.252870i \(-0.918626\pi\)
0.967500 0.252870i \(-0.0813744\pi\)
\(564\) −20.0000 −0.842152
\(565\) 0 0
\(566\) 0 0
\(567\) 11.0000i 0.461957i
\(568\) − 4.00000i − 0.167836i
\(569\) −14.0000 −0.586911 −0.293455 0.955973i \(-0.594805\pi\)
−0.293455 + 0.955973i \(0.594805\pi\)
\(570\) 0 0
\(571\) −28.0000 −1.17176 −0.585882 0.810397i \(-0.699252\pi\)
−0.585882 + 0.810397i \(0.699252\pi\)
\(572\) − 4.00000i − 0.167248i
\(573\) − 16.0000i − 0.668410i
\(574\) 0 0
\(575\) 0 0
\(576\) 1.00000 0.0416667
\(577\) − 42.0000i − 1.74848i −0.485491 0.874241i \(-0.661359\pi\)
0.485491 0.874241i \(-0.338641\pi\)
\(578\) − 17.0000i − 0.707107i
\(579\) 12.0000 0.498703
\(580\) 0 0
\(581\) −4.00000 −0.165948
\(582\) − 12.0000i − 0.497416i
\(583\) 14.0000i 0.579821i
\(584\) 4.00000 0.165521
\(585\) 0 0
\(586\) 0 0
\(587\) − 42.0000i − 1.73353i −0.498721 0.866763i \(-0.666197\pi\)
0.498721 0.866763i \(-0.333803\pi\)
\(588\) − 2.00000i − 0.0824786i
\(589\) 40.0000 1.64817
\(590\) 0 0
\(591\) −36.0000 −1.48084
\(592\) − 6.00000i − 0.246598i
\(593\) − 12.0000i − 0.492781i −0.969171 0.246390i \(-0.920755\pi\)
0.969171 0.246390i \(-0.0792446\pi\)
\(594\) −4.00000 −0.164122
\(595\) 0 0
\(596\) 22.0000 0.901155
\(597\) − 28.0000i − 1.14596i
\(598\) − 16.0000i − 0.654289i
\(599\) 12.0000 0.490307 0.245153 0.969484i \(-0.421162\pi\)
0.245153 + 0.969484i \(0.421162\pi\)
\(600\) 0 0
\(601\) −24.0000 −0.978980 −0.489490 0.872009i \(-0.662817\pi\)
−0.489490 + 0.872009i \(0.662817\pi\)
\(602\) − 4.00000i − 0.163028i
\(603\) − 8.00000i − 0.325785i
\(604\) −16.0000 −0.651031
\(605\) 0 0
\(606\) −24.0000 −0.974933
\(607\) − 24.0000i − 0.974130i −0.873366 0.487065i \(-0.838067\pi\)
0.873366 0.487065i \(-0.161933\pi\)
\(608\) 4.00000i 0.162221i
\(609\) 4.00000 0.162088
\(610\) 0 0
\(611\) −40.0000 −1.61823
\(612\) 0 0
\(613\) − 2.00000i − 0.0807792i −0.999184 0.0403896i \(-0.987140\pi\)
0.999184 0.0403896i \(-0.0128599\pi\)
\(614\) 16.0000 0.645707
\(615\) 0 0
\(616\) 1.00000 0.0402911
\(617\) 38.0000i 1.52982i 0.644136 + 0.764911i \(0.277217\pi\)
−0.644136 + 0.764911i \(0.722783\pi\)
\(618\) 4.00000i 0.160904i
\(619\) 2.00000 0.0803868 0.0401934 0.999192i \(-0.487203\pi\)
0.0401934 + 0.999192i \(0.487203\pi\)
\(620\) 0 0
\(621\) −16.0000 −0.642058
\(622\) 6.00000i 0.240578i
\(623\) 10.0000i 0.400642i
\(624\) 8.00000 0.320256
\(625\) 0 0
\(626\) 6.00000 0.239808
\(627\) 8.00000i 0.319489i
\(628\) − 10.0000i − 0.399043i
\(629\) 0 0
\(630\) 0 0
\(631\) 8.00000 0.318475 0.159237 0.987240i \(-0.449096\pi\)
0.159237 + 0.987240i \(0.449096\pi\)
\(632\) − 16.0000i − 0.636446i
\(633\) 8.00000i 0.317971i
\(634\) −18.0000 −0.714871
\(635\) 0 0
\(636\) −28.0000 −1.11027
\(637\) − 4.00000i − 0.158486i
\(638\) 2.00000i 0.0791808i
\(639\) 4.00000 0.158238
\(640\) 0 0
\(641\) −18.0000 −0.710957 −0.355479 0.934684i \(-0.615682\pi\)
−0.355479 + 0.934684i \(0.615682\pi\)
\(642\) 24.0000i 0.947204i
\(643\) − 22.0000i − 0.867595i −0.901010 0.433798i \(-0.857173\pi\)
0.901010 0.433798i \(-0.142827\pi\)
\(644\) 4.00000 0.157622
\(645\) 0 0
\(646\) 0 0
\(647\) 6.00000i 0.235884i 0.993020 + 0.117942i \(0.0376297\pi\)
−0.993020 + 0.117942i \(0.962370\pi\)
\(648\) − 11.0000i − 0.432121i
\(649\) −10.0000 −0.392534
\(650\) 0 0
\(651\) 20.0000 0.783862
\(652\) 24.0000i 0.939913i
\(653\) − 46.0000i − 1.80012i −0.435767 0.900060i \(-0.643523\pi\)
0.435767 0.900060i \(-0.356477\pi\)
\(654\) −28.0000 −1.09489
\(655\) 0 0
\(656\) 0 0
\(657\) 4.00000i 0.156055i
\(658\) − 10.0000i − 0.389841i
\(659\) −20.0000 −0.779089 −0.389545 0.921008i \(-0.627368\pi\)
−0.389545 + 0.921008i \(0.627368\pi\)
\(660\) 0 0
\(661\) −38.0000 −1.47803 −0.739014 0.673690i \(-0.764708\pi\)
−0.739014 + 0.673690i \(0.764708\pi\)
\(662\) 20.0000i 0.777322i
\(663\) 0 0
\(664\) 4.00000 0.155230
\(665\) 0 0
\(666\) 6.00000 0.232495
\(667\) 8.00000i 0.309761i
\(668\) 8.00000i 0.309529i
\(669\) 28.0000 1.08254
\(670\) 0 0
\(671\) −8.00000 −0.308837
\(672\) 2.00000i 0.0771517i
\(673\) 10.0000i 0.385472i 0.981251 + 0.192736i \(0.0617360\pi\)
−0.981251 + 0.192736i \(0.938264\pi\)
\(674\) −34.0000 −1.30963
\(675\) 0 0
\(676\) 3.00000 0.115385
\(677\) 12.0000i 0.461197i 0.973049 + 0.230599i \(0.0740685\pi\)
−0.973049 + 0.230599i \(0.925932\pi\)
\(678\) − 28.0000i − 1.07533i
\(679\) 6.00000 0.230259
\(680\) 0 0
\(681\) 16.0000 0.613121
\(682\) 10.0000i 0.382920i
\(683\) − 12.0000i − 0.459167i −0.973289 0.229584i \(-0.926264\pi\)
0.973289 0.229584i \(-0.0737364\pi\)
\(684\) −4.00000 −0.152944
\(685\) 0 0
\(686\) 1.00000 0.0381802
\(687\) − 20.0000i − 0.763048i
\(688\) 4.00000i 0.152499i
\(689\) −56.0000 −2.13343
\(690\) 0 0
\(691\) 42.0000 1.59776 0.798878 0.601494i \(-0.205427\pi\)
0.798878 + 0.601494i \(0.205427\pi\)
\(692\) 4.00000i 0.152057i
\(693\) 1.00000i 0.0379869i
\(694\) −12.0000 −0.455514
\(695\) 0 0
\(696\) −4.00000 −0.151620
\(697\) 0 0
\(698\) 32.0000i 1.21122i
\(699\) −12.0000 −0.453882
\(700\) 0 0
\(701\) −6.00000 −0.226617 −0.113308 0.993560i \(-0.536145\pi\)
−0.113308 + 0.993560i \(0.536145\pi\)
\(702\) − 16.0000i − 0.603881i
\(703\) 24.0000i 0.905177i
\(704\) −1.00000 −0.0376889
\(705\) 0 0
\(706\) −2.00000 −0.0752710
\(707\) − 12.0000i − 0.451306i
\(708\) − 20.0000i − 0.751646i
\(709\) 10.0000 0.375558 0.187779 0.982211i \(-0.439871\pi\)
0.187779 + 0.982211i \(0.439871\pi\)
\(710\) 0 0
\(711\) 16.0000 0.600047
\(712\) − 10.0000i − 0.374766i
\(713\) 40.0000i 1.49801i
\(714\) 0 0
\(715\) 0 0
\(716\) 12.0000 0.448461
\(717\) − 16.0000i − 0.597531i
\(718\) 0 0
\(719\) −6.00000 −0.223762 −0.111881 0.993722i \(-0.535688\pi\)
−0.111881 + 0.993722i \(0.535688\pi\)
\(720\) 0 0
\(721\) −2.00000 −0.0744839
\(722\) 3.00000i 0.111648i
\(723\) − 16.0000i − 0.595046i
\(724\) −14.0000 −0.520306
\(725\) 0 0
\(726\) −2.00000 −0.0742270
\(727\) 46.0000i 1.70605i 0.521874 + 0.853023i \(0.325233\pi\)
−0.521874 + 0.853023i \(0.674767\pi\)
\(728\) 4.00000i 0.148250i
\(729\) −13.0000 −0.481481
\(730\) 0 0
\(731\) 0 0
\(732\) − 16.0000i − 0.591377i
\(733\) 8.00000i 0.295487i 0.989026 + 0.147743i \(0.0472010\pi\)
−0.989026 + 0.147743i \(0.952799\pi\)
\(734\) 18.0000 0.664392
\(735\) 0 0
\(736\) −4.00000 −0.147442
\(737\) 8.00000i 0.294684i
\(738\) 0 0
\(739\) 52.0000 1.91285 0.956425 0.291977i \(-0.0943129\pi\)
0.956425 + 0.291977i \(0.0943129\pi\)
\(740\) 0 0
\(741\) −32.0000 −1.17555
\(742\) − 14.0000i − 0.513956i
\(743\) − 16.0000i − 0.586983i −0.955962 0.293492i \(-0.905183\pi\)
0.955962 0.293492i \(-0.0948173\pi\)
\(744\) −20.0000 −0.733236
\(745\) 0 0
\(746\) 34.0000 1.24483
\(747\) 4.00000i 0.146352i
\(748\) 0 0
\(749\) −12.0000 −0.438470
\(750\) 0 0
\(751\) 20.0000 0.729810 0.364905 0.931045i \(-0.381101\pi\)
0.364905 + 0.931045i \(0.381101\pi\)
\(752\) 10.0000i 0.364662i
\(753\) 52.0000i 1.89499i
\(754\) −8.00000 −0.291343
\(755\) 0 0
\(756\) 4.00000 0.145479
\(757\) 30.0000i 1.09037i 0.838316 + 0.545184i \(0.183540\pi\)
−0.838316 + 0.545184i \(0.816460\pi\)
\(758\) 8.00000i 0.290573i
\(759\) −8.00000 −0.290382
\(760\) 0 0
\(761\) −12.0000 −0.435000 −0.217500 0.976060i \(-0.569790\pi\)
−0.217500 + 0.976060i \(0.569790\pi\)
\(762\) 32.0000i 1.15924i
\(763\) − 14.0000i − 0.506834i
\(764\) −8.00000 −0.289430
\(765\) 0 0
\(766\) −14.0000 −0.505841
\(767\) − 40.0000i − 1.44432i
\(768\) − 2.00000i − 0.0721688i
\(769\) −4.00000 −0.144244 −0.0721218 0.997396i \(-0.522977\pi\)
−0.0721218 + 0.997396i \(0.522977\pi\)
\(770\) 0 0
\(771\) 4.00000 0.144056
\(772\) − 6.00000i − 0.215945i
\(773\) 34.0000i 1.22290i 0.791285 + 0.611448i \(0.209412\pi\)
−0.791285 + 0.611448i \(0.790588\pi\)
\(774\) −4.00000 −0.143777
\(775\) 0 0
\(776\) −6.00000 −0.215387
\(777\) 12.0000i 0.430498i
\(778\) − 18.0000i − 0.645331i
\(779\) 0 0
\(780\) 0 0
\(781\) −4.00000 −0.143131
\(782\) 0 0
\(783\) 8.00000i 0.285897i
\(784\) −1.00000 −0.0357143
\(785\) 0 0
\(786\) −16.0000 −0.570701
\(787\) 20.0000i 0.712923i 0.934310 + 0.356462i \(0.116017\pi\)
−0.934310 + 0.356462i \(0.883983\pi\)
\(788\) 18.0000i 0.641223i
\(789\) 48.0000 1.70885
\(790\) 0 0
\(791\) 14.0000 0.497783
\(792\) − 1.00000i − 0.0355335i
\(793\) − 32.0000i − 1.13635i
\(794\) 18.0000 0.638796
\(795\) 0 0
\(796\) −14.0000 −0.496217
\(797\) 26.0000i 0.920967i 0.887668 + 0.460484i \(0.152324\pi\)
−0.887668 + 0.460484i \(0.847676\pi\)
\(798\) − 8.00000i − 0.283197i
\(799\) 0 0
\(800\) 0 0
\(801\) 10.0000 0.353333
\(802\) − 10.0000i − 0.353112i
\(803\) − 4.00000i − 0.141157i
\(804\) −16.0000 −0.564276
\(805\) 0 0
\(806\) −40.0000 −1.40894
\(807\) 28.0000i 0.985647i
\(808\) 12.0000i 0.422159i
\(809\) −26.0000 −0.914111 −0.457056 0.889438i \(-0.651096\pi\)
−0.457056 + 0.889438i \(0.651096\pi\)
\(810\) 0 0
\(811\) −40.0000 −1.40459 −0.702295 0.711886i \(-0.747841\pi\)
−0.702295 + 0.711886i \(0.747841\pi\)
\(812\) − 2.00000i − 0.0701862i
\(813\) 56.0000i 1.96401i
\(814\) −6.00000 −0.210300
\(815\) 0 0
\(816\) 0 0
\(817\) − 16.0000i − 0.559769i
\(818\) − 4.00000i − 0.139857i
\(819\) −4.00000 −0.139771
\(820\) 0 0
\(821\) −50.0000 −1.74501 −0.872506 0.488603i \(-0.837507\pi\)
−0.872506 + 0.488603i \(0.837507\pi\)
\(822\) − 12.0000i − 0.418548i
\(823\) − 8.00000i − 0.278862i −0.990232 0.139431i \(-0.955473\pi\)
0.990232 0.139431i \(-0.0445274\pi\)
\(824\) 2.00000 0.0696733
\(825\) 0 0
\(826\) 10.0000 0.347945
\(827\) − 20.0000i − 0.695468i −0.937593 0.347734i \(-0.886951\pi\)
0.937593 0.347734i \(-0.113049\pi\)
\(828\) − 4.00000i − 0.139010i
\(829\) 14.0000 0.486240 0.243120 0.969996i \(-0.421829\pi\)
0.243120 + 0.969996i \(0.421829\pi\)
\(830\) 0 0
\(831\) 12.0000 0.416275
\(832\) − 4.00000i − 0.138675i
\(833\) 0 0
\(834\) 40.0000 1.38509
\(835\) 0 0
\(836\) 4.00000 0.138343
\(837\) 40.0000i 1.38260i
\(838\) − 30.0000i − 1.03633i
\(839\) −30.0000 −1.03572 −0.517858 0.855467i \(-0.673270\pi\)
−0.517858 + 0.855467i \(0.673270\pi\)
\(840\) 0 0
\(841\) −25.0000 −0.862069
\(842\) − 10.0000i − 0.344623i
\(843\) − 60.0000i − 2.06651i
\(844\) 4.00000 0.137686
\(845\) 0 0
\(846\) −10.0000 −0.343807
\(847\) − 1.00000i − 0.0343604i
\(848\) 14.0000i 0.480762i
\(849\) 0 0
\(850\) 0 0
\(851\) −24.0000 −0.822709
\(852\) − 8.00000i − 0.274075i
\(853\) − 4.00000i − 0.136957i −0.997653 0.0684787i \(-0.978185\pi\)
0.997653 0.0684787i \(-0.0218145\pi\)
\(854\) 8.00000 0.273754
\(855\) 0 0
\(856\) 12.0000 0.410152
\(857\) 12.0000i 0.409912i 0.978771 + 0.204956i \(0.0657052\pi\)
−0.978771 + 0.204956i \(0.934295\pi\)
\(858\) − 8.00000i − 0.273115i
\(859\) −14.0000 −0.477674 −0.238837 0.971060i \(-0.576766\pi\)
−0.238837 + 0.971060i \(0.576766\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 16.0000i 0.544962i
\(863\) − 40.0000i − 1.36162i −0.732462 0.680808i \(-0.761629\pi\)
0.732462 0.680808i \(-0.238371\pi\)
\(864\) −4.00000 −0.136083
\(865\) 0 0
\(866\) 10.0000 0.339814
\(867\) − 34.0000i − 1.15470i
\(868\) − 10.0000i − 0.339422i
\(869\) −16.0000 −0.542763
\(870\) 0 0
\(871\) −32.0000 −1.08428
\(872\) 14.0000i 0.474100i
\(873\) − 6.00000i − 0.203069i
\(874\) 16.0000 0.541208
\(875\) 0 0
\(876\) 8.00000 0.270295
\(877\) − 22.0000i − 0.742887i −0.928456 0.371444i \(-0.878863\pi\)
0.928456 0.371444i \(-0.121137\pi\)
\(878\) − 28.0000i − 0.944954i
\(879\) 0 0
\(880\) 0 0
\(881\) −42.0000 −1.41502 −0.707508 0.706705i \(-0.750181\pi\)
−0.707508 + 0.706705i \(0.750181\pi\)
\(882\) − 1.00000i − 0.0336718i
\(883\) 4.00000i 0.134611i 0.997732 + 0.0673054i \(0.0214402\pi\)
−0.997732 + 0.0673054i \(0.978560\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) −4.00000 −0.134383
\(887\) − 28.0000i − 0.940148i −0.882627 0.470074i \(-0.844227\pi\)
0.882627 0.470074i \(-0.155773\pi\)
\(888\) − 12.0000i − 0.402694i
\(889\) −16.0000 −0.536623
\(890\) 0 0
\(891\) −11.0000 −0.368514
\(892\) − 14.0000i − 0.468755i
\(893\) − 40.0000i − 1.33855i
\(894\) 44.0000 1.47158
\(895\) 0 0
\(896\) 1.00000 0.0334077
\(897\) − 32.0000i − 1.06845i
\(898\) 6.00000i 0.200223i
\(899\) 20.0000 0.667037
\(900\) 0 0
\(901\) 0 0
\(902\) 0 0
\(903\) − 8.00000i − 0.266223i
\(904\) −14.0000 −0.465633
\(905\) 0 0
\(906\) −32.0000 −1.06313
\(907\) 48.0000i 1.59381i 0.604102 + 0.796907i \(0.293532\pi\)
−0.604102 + 0.796907i \(0.706468\pi\)
\(908\) − 8.00000i − 0.265489i
\(909\) −12.0000 −0.398015
\(910\) 0 0
\(911\) −4.00000 −0.132526 −0.0662630 0.997802i \(-0.521108\pi\)
−0.0662630 + 0.997802i \(0.521108\pi\)
\(912\) 8.00000i 0.264906i
\(913\) − 4.00000i − 0.132381i
\(914\) −38.0000 −1.25693
\(915\) 0 0
\(916\) −10.0000 −0.330409
\(917\) − 8.00000i − 0.264183i
\(918\) 0 0
\(919\) 56.0000 1.84727 0.923635 0.383274i \(-0.125203\pi\)
0.923635 + 0.383274i \(0.125203\pi\)
\(920\) 0 0
\(921\) 32.0000 1.05444
\(922\) − 32.0000i − 1.05386i
\(923\) − 16.0000i − 0.526646i
\(924\) 2.00000 0.0657952
\(925\) 0 0
\(926\) −12.0000 −0.394344
\(927\) 2.00000i 0.0656886i
\(928\) 2.00000i 0.0656532i
\(929\) 10.0000 0.328089 0.164045 0.986453i \(-0.447546\pi\)
0.164045 + 0.986453i \(0.447546\pi\)
\(930\) 0 0
\(931\) 4.00000 0.131095
\(932\) 6.00000i 0.196537i
\(933\) 12.0000i 0.392862i
\(934\) 14.0000 0.458094
\(935\) 0 0
\(936\) 4.00000 0.130744
\(937\) − 52.0000i − 1.69877i −0.527777 0.849383i \(-0.676974\pi\)
0.527777 0.849383i \(-0.323026\pi\)
\(938\) − 8.00000i − 0.261209i
\(939\) 12.0000 0.391605
\(940\) 0 0
\(941\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(942\) − 20.0000i − 0.651635i
\(943\) 0 0
\(944\) −10.0000 −0.325472
\(945\) 0 0
\(946\) 4.00000 0.130051
\(947\) 12.0000i 0.389948i 0.980808 + 0.194974i \(0.0624622\pi\)
−0.980808 + 0.194974i \(0.937538\pi\)
\(948\) − 32.0000i − 1.03931i
\(949\) 16.0000 0.519382
\(950\) 0 0
\(951\) −36.0000 −1.16738
\(952\) 0 0
\(953\) − 34.0000i − 1.10137i −0.834714 0.550684i \(-0.814367\pi\)
0.834714 0.550684i \(-0.185633\pi\)
\(954\) −14.0000 −0.453267
\(955\) 0 0
\(956\) −8.00000 −0.258738
\(957\) 4.00000i 0.129302i
\(958\) 12.0000i 0.387702i
\(959\) 6.00000 0.193750
\(960\) 0 0
\(961\) 69.0000 2.22581
\(962\) − 24.0000i − 0.773791i
\(963\) 12.0000i 0.386695i
\(964\) −8.00000 −0.257663
\(965\) 0 0
\(966\) 8.00000 0.257396
\(967\) − 24.0000i − 0.771788i −0.922543 0.385894i \(-0.873893\pi\)
0.922543 0.385894i \(-0.126107\pi\)
\(968\) 1.00000i 0.0321412i
\(969\) 0 0
\(970\) 0 0
\(971\) 30.0000 0.962746 0.481373 0.876516i \(-0.340138\pi\)
0.481373 + 0.876516i \(0.340138\pi\)
\(972\) − 10.0000i − 0.320750i
\(973\) 20.0000i 0.641171i
\(974\) 12.0000 0.384505
\(975\) 0 0
\(976\) −8.00000 −0.256074
\(977\) − 22.0000i − 0.703842i −0.936030 0.351921i \(-0.885529\pi\)
0.936030 0.351921i \(-0.114471\pi\)
\(978\) 48.0000i 1.53487i
\(979\) −10.0000 −0.319601
\(980\) 0 0
\(981\) −14.0000 −0.446986
\(982\) 28.0000i 0.893516i
\(983\) − 26.0000i − 0.829271i −0.909988 0.414636i \(-0.863909\pi\)
0.909988 0.414636i \(-0.136091\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) − 20.0000i − 0.636607i
\(988\) 16.0000i 0.509028i
\(989\) 16.0000 0.508770
\(990\) 0 0
\(991\) −4.00000 −0.127064 −0.0635321 0.997980i \(-0.520237\pi\)
−0.0635321 + 0.997980i \(0.520237\pi\)
\(992\) 10.0000i 0.317500i
\(993\) 40.0000i 1.26936i
\(994\) 4.00000 0.126872
\(995\) 0 0
\(996\) 8.00000 0.253490
\(997\) 36.0000i 1.14013i 0.821599 + 0.570066i \(0.193082\pi\)
−0.821599 + 0.570066i \(0.806918\pi\)
\(998\) − 16.0000i − 0.506471i
\(999\) −24.0000 −0.759326
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 3850.2.c.d.1849.1 2
5.2 odd 4 3850.2.a.o.1.1 1
5.3 odd 4 154.2.a.b.1.1 1
5.4 even 2 inner 3850.2.c.d.1849.2 2
15.8 even 4 1386.2.a.f.1.1 1
20.3 even 4 1232.2.a.c.1.1 1
35.3 even 12 1078.2.e.l.177.1 2
35.13 even 4 1078.2.a.b.1.1 1
35.18 odd 12 1078.2.e.h.177.1 2
35.23 odd 12 1078.2.e.h.67.1 2
35.33 even 12 1078.2.e.l.67.1 2
40.3 even 4 4928.2.a.bf.1.1 1
40.13 odd 4 4928.2.a.d.1.1 1
55.43 even 4 1694.2.a.i.1.1 1
105.83 odd 4 9702.2.a.bz.1.1 1
140.83 odd 4 8624.2.a.z.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
154.2.a.b.1.1 1 5.3 odd 4
1078.2.a.b.1.1 1 35.13 even 4
1078.2.e.h.67.1 2 35.23 odd 12
1078.2.e.h.177.1 2 35.18 odd 12
1078.2.e.l.67.1 2 35.33 even 12
1078.2.e.l.177.1 2 35.3 even 12
1232.2.a.c.1.1 1 20.3 even 4
1386.2.a.f.1.1 1 15.8 even 4
1694.2.a.i.1.1 1 55.43 even 4
3850.2.a.o.1.1 1 5.2 odd 4
3850.2.c.d.1849.1 2 1.1 even 1 trivial
3850.2.c.d.1849.2 2 5.4 even 2 inner
4928.2.a.d.1.1 1 40.13 odd 4
4928.2.a.bf.1.1 1 40.3 even 4
8624.2.a.z.1.1 1 140.83 odd 4
9702.2.a.bz.1.1 1 105.83 odd 4