Properties

Label 1850.2.b.i.149.2
Level $1850$
Weight $2$
Character 1850.149
Analytic conductor $14.772$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1850 = 2 \cdot 5^{2} \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1850.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(14.7723243739\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{13})\)
Defining polynomial: \( x^{4} + 7x^{2} + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 74)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 149.2
Root \(1.30278i\) of defining polynomial
Character \(\chi\) \(=\) 1850.149
Dual form 1850.2.b.i.149.3

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000i q^{2} +0.302776i q^{3} -1.00000 q^{4} +0.302776 q^{6} +4.60555i q^{7} +1.00000i q^{8} +2.90833 q^{9} +O(q^{10})\) \(q-1.00000i q^{2} +0.302776i q^{3} -1.00000 q^{4} +0.302776 q^{6} +4.60555i q^{7} +1.00000i q^{8} +2.90833 q^{9} +1.30278 q^{11} -0.302776i q^{12} +2.30278i q^{13} +4.60555 q^{14} +1.00000 q^{16} -6.00000i q^{17} -2.90833i q^{18} -2.00000 q^{19} -1.39445 q^{21} -1.30278i q^{22} +6.90833i q^{23} -0.302776 q^{24} +2.30278 q^{26} +1.78890i q^{27} -4.60555i q^{28} -6.90833 q^{29} +3.30278 q^{31} -1.00000i q^{32} +0.394449i q^{33} -6.00000 q^{34} -2.90833 q^{36} +1.00000i q^{37} +2.00000i q^{38} -0.697224 q^{39} -0.908327 q^{41} +1.39445i q^{42} +6.60555i q^{43} -1.30278 q^{44} +6.90833 q^{46} -2.60555i q^{47} +0.302776i q^{48} -14.2111 q^{49} +1.81665 q^{51} -2.30278i q^{52} +6.00000i q^{53} +1.78890 q^{54} -4.60555 q^{56} -0.605551i q^{57} +6.90833i q^{58} -3.39445 q^{59} -10.5139 q^{61} -3.30278i q^{62} +13.3944i q^{63} -1.00000 q^{64} +0.394449 q^{66} +14.5139i q^{67} +6.00000i q^{68} -2.09167 q^{69} +6.00000 q^{71} +2.90833i q^{72} +8.69722i q^{73} +1.00000 q^{74} +2.00000 q^{76} +6.00000i q^{77} +0.697224i q^{78} +16.1194 q^{79} +8.18335 q^{81} +0.908327i q^{82} -17.2111i q^{83} +1.39445 q^{84} +6.60555 q^{86} -2.09167i q^{87} +1.30278i q^{88} -5.21110 q^{89} -10.6056 q^{91} -6.90833i q^{92} +1.00000i q^{93} -2.60555 q^{94} +0.302776 q^{96} +12.4222i q^{97} +14.2111i q^{98} +3.78890 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{4} - 6 q^{6} - 10 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{4} - 6 q^{6} - 10 q^{9} - 2 q^{11} + 4 q^{14} + 4 q^{16} - 8 q^{19} - 20 q^{21} + 6 q^{24} + 2 q^{26} - 6 q^{29} + 6 q^{31} - 24 q^{34} + 10 q^{36} - 10 q^{39} + 18 q^{41} + 2 q^{44} + 6 q^{46} - 28 q^{49} - 36 q^{51} + 36 q^{54} - 4 q^{56} - 28 q^{59} - 6 q^{61} - 4 q^{64} + 16 q^{66} - 30 q^{69} + 24 q^{71} + 4 q^{74} + 8 q^{76} + 14 q^{79} + 76 q^{81} + 20 q^{84} + 12 q^{86} + 8 q^{89} - 28 q^{91} + 4 q^{94} - 6 q^{96} + 44 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1850\mathbb{Z}\right)^\times\).

\(n\) \(1001\) \(1777\)
\(\chi(n)\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 1.00000i − 0.707107i
\(3\) 0.302776i 0.174808i 0.996173 + 0.0874038i \(0.0278570\pi\)
−0.996173 + 0.0874038i \(0.972143\pi\)
\(4\) −1.00000 −0.500000
\(5\) 0 0
\(6\) 0.302776 0.123608
\(7\) 4.60555i 1.74073i 0.492403 + 0.870367i \(0.336119\pi\)
−0.492403 + 0.870367i \(0.663881\pi\)
\(8\) 1.00000i 0.353553i
\(9\) 2.90833 0.969442
\(10\) 0 0
\(11\) 1.30278 0.392802 0.196401 0.980524i \(-0.437075\pi\)
0.196401 + 0.980524i \(0.437075\pi\)
\(12\) − 0.302776i − 0.0874038i
\(13\) 2.30278i 0.638675i 0.947641 + 0.319338i \(0.103460\pi\)
−0.947641 + 0.319338i \(0.896540\pi\)
\(14\) 4.60555 1.23089
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) − 6.00000i − 1.45521i −0.685994 0.727607i \(-0.740633\pi\)
0.685994 0.727607i \(-0.259367\pi\)
\(18\) − 2.90833i − 0.685499i
\(19\) −2.00000 −0.458831 −0.229416 0.973329i \(-0.573682\pi\)
−0.229416 + 0.973329i \(0.573682\pi\)
\(20\) 0 0
\(21\) −1.39445 −0.304294
\(22\) − 1.30278i − 0.277753i
\(23\) 6.90833i 1.44049i 0.693722 + 0.720243i \(0.255970\pi\)
−0.693722 + 0.720243i \(0.744030\pi\)
\(24\) −0.302776 −0.0618038
\(25\) 0 0
\(26\) 2.30278 0.451611
\(27\) 1.78890i 0.344273i
\(28\) − 4.60555i − 0.870367i
\(29\) −6.90833 −1.28284 −0.641422 0.767188i \(-0.721655\pi\)
−0.641422 + 0.767188i \(0.721655\pi\)
\(30\) 0 0
\(31\) 3.30278 0.593196 0.296598 0.955002i \(-0.404148\pi\)
0.296598 + 0.955002i \(0.404148\pi\)
\(32\) − 1.00000i − 0.176777i
\(33\) 0.394449i 0.0686647i
\(34\) −6.00000 −1.02899
\(35\) 0 0
\(36\) −2.90833 −0.484721
\(37\) 1.00000i 0.164399i
\(38\) 2.00000i 0.324443i
\(39\) −0.697224 −0.111645
\(40\) 0 0
\(41\) −0.908327 −0.141857 −0.0709284 0.997481i \(-0.522596\pi\)
−0.0709284 + 0.997481i \(0.522596\pi\)
\(42\) 1.39445i 0.215168i
\(43\) 6.60555i 1.00734i 0.863897 + 0.503669i \(0.168017\pi\)
−0.863897 + 0.503669i \(0.831983\pi\)
\(44\) −1.30278 −0.196401
\(45\) 0 0
\(46\) 6.90833 1.01858
\(47\) − 2.60555i − 0.380059i −0.981778 0.190029i \(-0.939142\pi\)
0.981778 0.190029i \(-0.0608583\pi\)
\(48\) 0.302776i 0.0437019i
\(49\) −14.2111 −2.03016
\(50\) 0 0
\(51\) 1.81665 0.254382
\(52\) − 2.30278i − 0.319338i
\(53\) 6.00000i 0.824163i 0.911147 + 0.412082i \(0.135198\pi\)
−0.911147 + 0.412082i \(0.864802\pi\)
\(54\) 1.78890 0.243438
\(55\) 0 0
\(56\) −4.60555 −0.615443
\(57\) − 0.605551i − 0.0802072i
\(58\) 6.90833i 0.907108i
\(59\) −3.39445 −0.441920 −0.220960 0.975283i \(-0.570919\pi\)
−0.220960 + 0.975283i \(0.570919\pi\)
\(60\) 0 0
\(61\) −10.5139 −1.34616 −0.673082 0.739568i \(-0.735030\pi\)
−0.673082 + 0.739568i \(0.735030\pi\)
\(62\) − 3.30278i − 0.419453i
\(63\) 13.3944i 1.68754i
\(64\) −1.00000 −0.125000
\(65\) 0 0
\(66\) 0.394449 0.0485533
\(67\) 14.5139i 1.77315i 0.462583 + 0.886576i \(0.346923\pi\)
−0.462583 + 0.886576i \(0.653077\pi\)
\(68\) 6.00000i 0.727607i
\(69\) −2.09167 −0.251808
\(70\) 0 0
\(71\) 6.00000 0.712069 0.356034 0.934473i \(-0.384129\pi\)
0.356034 + 0.934473i \(0.384129\pi\)
\(72\) 2.90833i 0.342750i
\(73\) 8.69722i 1.01793i 0.860786 + 0.508967i \(0.169972\pi\)
−0.860786 + 0.508967i \(0.830028\pi\)
\(74\) 1.00000 0.116248
\(75\) 0 0
\(76\) 2.00000 0.229416
\(77\) 6.00000i 0.683763i
\(78\) 0.697224i 0.0789451i
\(79\) 16.1194 1.81358 0.906789 0.421585i \(-0.138526\pi\)
0.906789 + 0.421585i \(0.138526\pi\)
\(80\) 0 0
\(81\) 8.18335 0.909261
\(82\) 0.908327i 0.100308i
\(83\) − 17.2111i − 1.88916i −0.328276 0.944582i \(-0.606467\pi\)
0.328276 0.944582i \(-0.393533\pi\)
\(84\) 1.39445 0.152147
\(85\) 0 0
\(86\) 6.60555 0.712295
\(87\) − 2.09167i − 0.224251i
\(88\) 1.30278i 0.138876i
\(89\) −5.21110 −0.552376 −0.276188 0.961104i \(-0.589071\pi\)
−0.276188 + 0.961104i \(0.589071\pi\)
\(90\) 0 0
\(91\) −10.6056 −1.11176
\(92\) − 6.90833i − 0.720243i
\(93\) 1.00000i 0.103695i
\(94\) −2.60555 −0.268742
\(95\) 0 0
\(96\) 0.302776 0.0309019
\(97\) 12.4222i 1.26128i 0.776074 + 0.630642i \(0.217208\pi\)
−0.776074 + 0.630642i \(0.782792\pi\)
\(98\) 14.2111i 1.43554i
\(99\) 3.78890 0.380799
\(100\) 0 0
\(101\) 16.4222 1.63407 0.817035 0.576588i \(-0.195616\pi\)
0.817035 + 0.576588i \(0.195616\pi\)
\(102\) − 1.81665i − 0.179876i
\(103\) − 3.30278i − 0.325432i −0.986673 0.162716i \(-0.947975\pi\)
0.986673 0.162716i \(-0.0520255\pi\)
\(104\) −2.30278 −0.225806
\(105\) 0 0
\(106\) 6.00000 0.582772
\(107\) 4.30278i 0.415965i 0.978133 + 0.207983i \(0.0666897\pi\)
−0.978133 + 0.207983i \(0.933310\pi\)
\(108\) − 1.78890i − 0.172137i
\(109\) −2.00000 −0.191565 −0.0957826 0.995402i \(-0.530535\pi\)
−0.0957826 + 0.995402i \(0.530535\pi\)
\(110\) 0 0
\(111\) −0.302776 −0.0287382
\(112\) 4.60555i 0.435184i
\(113\) − 11.2111i − 1.05465i −0.849663 0.527326i \(-0.823195\pi\)
0.849663 0.527326i \(-0.176805\pi\)
\(114\) −0.605551 −0.0567151
\(115\) 0 0
\(116\) 6.90833 0.641422
\(117\) 6.69722i 0.619159i
\(118\) 3.39445i 0.312484i
\(119\) 27.6333 2.53314
\(120\) 0 0
\(121\) −9.30278 −0.845707
\(122\) 10.5139i 0.951882i
\(123\) − 0.275019i − 0.0247977i
\(124\) −3.30278 −0.296598
\(125\) 0 0
\(126\) 13.3944 1.19327
\(127\) − 4.78890i − 0.424946i −0.977167 0.212473i \(-0.931848\pi\)
0.977167 0.212473i \(-0.0681517\pi\)
\(128\) 1.00000i 0.0883883i
\(129\) −2.00000 −0.176090
\(130\) 0 0
\(131\) 3.39445 0.296574 0.148287 0.988944i \(-0.452624\pi\)
0.148287 + 0.988944i \(0.452624\pi\)
\(132\) − 0.394449i − 0.0343324i
\(133\) − 9.21110i − 0.798704i
\(134\) 14.5139 1.25381
\(135\) 0 0
\(136\) 6.00000 0.514496
\(137\) − 9.90833i − 0.846525i −0.906007 0.423263i \(-0.860885\pi\)
0.906007 0.423263i \(-0.139115\pi\)
\(138\) 2.09167i 0.178055i
\(139\) −8.90833 −0.755594 −0.377797 0.925888i \(-0.623318\pi\)
−0.377797 + 0.925888i \(0.623318\pi\)
\(140\) 0 0
\(141\) 0.788897 0.0664372
\(142\) − 6.00000i − 0.503509i
\(143\) 3.00000i 0.250873i
\(144\) 2.90833 0.242361
\(145\) 0 0
\(146\) 8.69722 0.719787
\(147\) − 4.30278i − 0.354887i
\(148\) − 1.00000i − 0.0821995i
\(149\) 1.81665 0.148826 0.0744130 0.997228i \(-0.476292\pi\)
0.0744130 + 0.997228i \(0.476292\pi\)
\(150\) 0 0
\(151\) −13.3944 −1.09002 −0.545012 0.838428i \(-0.683475\pi\)
−0.545012 + 0.838428i \(0.683475\pi\)
\(152\) − 2.00000i − 0.162221i
\(153\) − 17.4500i − 1.41075i
\(154\) 6.00000 0.483494
\(155\) 0 0
\(156\) 0.697224 0.0558226
\(157\) 7.21110i 0.575509i 0.957704 + 0.287754i \(0.0929087\pi\)
−0.957704 + 0.287754i \(0.907091\pi\)
\(158\) − 16.1194i − 1.28239i
\(159\) −1.81665 −0.144070
\(160\) 0 0
\(161\) −31.8167 −2.50750
\(162\) − 8.18335i − 0.642944i
\(163\) 20.4222i 1.59959i 0.600273 + 0.799795i \(0.295059\pi\)
−0.600273 + 0.799795i \(0.704941\pi\)
\(164\) 0.908327 0.0709284
\(165\) 0 0
\(166\) −17.2111 −1.33584
\(167\) 12.5139i 0.968353i 0.874970 + 0.484176i \(0.160881\pi\)
−0.874970 + 0.484176i \(0.839119\pi\)
\(168\) − 1.39445i − 0.107584i
\(169\) 7.69722 0.592094
\(170\) 0 0
\(171\) −5.81665 −0.444811
\(172\) − 6.60555i − 0.503669i
\(173\) 23.2111i 1.76471i 0.470587 + 0.882354i \(0.344042\pi\)
−0.470587 + 0.882354i \(0.655958\pi\)
\(174\) −2.09167 −0.158569
\(175\) 0 0
\(176\) 1.30278 0.0982004
\(177\) − 1.02776i − 0.0772509i
\(178\) 5.21110i 0.390589i
\(179\) −7.81665 −0.584244 −0.292122 0.956381i \(-0.594361\pi\)
−0.292122 + 0.956381i \(0.594361\pi\)
\(180\) 0 0
\(181\) 20.0000 1.48659 0.743294 0.668965i \(-0.233262\pi\)
0.743294 + 0.668965i \(0.233262\pi\)
\(182\) 10.6056i 0.786136i
\(183\) − 3.18335i − 0.235320i
\(184\) −6.90833 −0.509289
\(185\) 0 0
\(186\) 1.00000 0.0733236
\(187\) − 7.81665i − 0.571610i
\(188\) 2.60555i 0.190029i
\(189\) −8.23886 −0.599289
\(190\) 0 0
\(191\) 12.5139 0.905472 0.452736 0.891644i \(-0.350448\pi\)
0.452736 + 0.891644i \(0.350448\pi\)
\(192\) − 0.302776i − 0.0218509i
\(193\) 4.00000i 0.287926i 0.989583 + 0.143963i \(0.0459847\pi\)
−0.989583 + 0.143963i \(0.954015\pi\)
\(194\) 12.4222 0.891862
\(195\) 0 0
\(196\) 14.2111 1.01508
\(197\) − 6.00000i − 0.427482i −0.976890 0.213741i \(-0.931435\pi\)
0.976890 0.213741i \(-0.0685649\pi\)
\(198\) − 3.78890i − 0.269265i
\(199\) 2.42221 0.171706 0.0858528 0.996308i \(-0.472639\pi\)
0.0858528 + 0.996308i \(0.472639\pi\)
\(200\) 0 0
\(201\) −4.39445 −0.309961
\(202\) − 16.4222i − 1.15546i
\(203\) − 31.8167i − 2.23309i
\(204\) −1.81665 −0.127191
\(205\) 0 0
\(206\) −3.30278 −0.230115
\(207\) 20.0917i 1.39647i
\(208\) 2.30278i 0.159669i
\(209\) −2.60555 −0.180230
\(210\) 0 0
\(211\) 6.69722 0.461056 0.230528 0.973066i \(-0.425955\pi\)
0.230528 + 0.973066i \(0.425955\pi\)
\(212\) − 6.00000i − 0.412082i
\(213\) 1.81665i 0.124475i
\(214\) 4.30278 0.294132
\(215\) 0 0
\(216\) −1.78890 −0.121719
\(217\) 15.2111i 1.03260i
\(218\) 2.00000i 0.135457i
\(219\) −2.63331 −0.177942
\(220\) 0 0
\(221\) 13.8167 0.929409
\(222\) 0.302776i 0.0203210i
\(223\) − 15.8167i − 1.05916i −0.848260 0.529581i \(-0.822349\pi\)
0.848260 0.529581i \(-0.177651\pi\)
\(224\) 4.60555 0.307721
\(225\) 0 0
\(226\) −11.2111 −0.745751
\(227\) − 7.81665i − 0.518810i −0.965769 0.259405i \(-0.916474\pi\)
0.965769 0.259405i \(-0.0835264\pi\)
\(228\) 0.605551i 0.0401036i
\(229\) −17.3944 −1.14946 −0.574729 0.818344i \(-0.694892\pi\)
−0.574729 + 0.818344i \(0.694892\pi\)
\(230\) 0 0
\(231\) −1.81665 −0.119527
\(232\) − 6.90833i − 0.453554i
\(233\) 9.51388i 0.623275i 0.950201 + 0.311637i \(0.100877\pi\)
−0.950201 + 0.311637i \(0.899123\pi\)
\(234\) 6.69722 0.437811
\(235\) 0 0
\(236\) 3.39445 0.220960
\(237\) 4.88057i 0.317027i
\(238\) − 27.6333i − 1.79120i
\(239\) −0.513878 −0.0332400 −0.0166200 0.999862i \(-0.505291\pi\)
−0.0166200 + 0.999862i \(0.505291\pi\)
\(240\) 0 0
\(241\) 8.00000 0.515325 0.257663 0.966235i \(-0.417048\pi\)
0.257663 + 0.966235i \(0.417048\pi\)
\(242\) 9.30278i 0.598005i
\(243\) 7.84441i 0.503219i
\(244\) 10.5139 0.673082
\(245\) 0 0
\(246\) −0.275019 −0.0175346
\(247\) − 4.60555i − 0.293044i
\(248\) 3.30278i 0.209726i
\(249\) 5.21110 0.330240
\(250\) 0 0
\(251\) −6.78890 −0.428511 −0.214256 0.976778i \(-0.568733\pi\)
−0.214256 + 0.976778i \(0.568733\pi\)
\(252\) − 13.3944i − 0.843771i
\(253\) 9.00000i 0.565825i
\(254\) −4.78890 −0.300482
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) − 11.2111i − 0.699329i −0.936875 0.349665i \(-0.886296\pi\)
0.936875 0.349665i \(-0.113704\pi\)
\(258\) 2.00000i 0.124515i
\(259\) −4.60555 −0.286175
\(260\) 0 0
\(261\) −20.0917 −1.24364
\(262\) − 3.39445i − 0.209710i
\(263\) 7.81665i 0.481996i 0.970526 + 0.240998i \(0.0774746\pi\)
−0.970526 + 0.240998i \(0.922525\pi\)
\(264\) −0.394449 −0.0242766
\(265\) 0 0
\(266\) −9.21110 −0.564769
\(267\) − 1.57779i − 0.0965595i
\(268\) − 14.5139i − 0.886576i
\(269\) 6.78890 0.413926 0.206963 0.978349i \(-0.433642\pi\)
0.206963 + 0.978349i \(0.433642\pi\)
\(270\) 0 0
\(271\) 6.42221 0.390121 0.195061 0.980791i \(-0.437510\pi\)
0.195061 + 0.980791i \(0.437510\pi\)
\(272\) − 6.00000i − 0.363803i
\(273\) − 3.21110i − 0.194345i
\(274\) −9.90833 −0.598584
\(275\) 0 0
\(276\) 2.09167 0.125904
\(277\) − 25.1194i − 1.50928i −0.656139 0.754640i \(-0.727811\pi\)
0.656139 0.754640i \(-0.272189\pi\)
\(278\) 8.90833i 0.534286i
\(279\) 9.60555 0.575069
\(280\) 0 0
\(281\) −12.0000 −0.715860 −0.357930 0.933748i \(-0.616517\pi\)
−0.357930 + 0.933748i \(0.616517\pi\)
\(282\) − 0.788897i − 0.0469782i
\(283\) − 17.3944i − 1.03399i −0.855988 0.516996i \(-0.827050\pi\)
0.855988 0.516996i \(-0.172950\pi\)
\(284\) −6.00000 −0.356034
\(285\) 0 0
\(286\) 3.00000 0.177394
\(287\) − 4.18335i − 0.246935i
\(288\) − 2.90833i − 0.171375i
\(289\) −19.0000 −1.11765
\(290\) 0 0
\(291\) −3.76114 −0.220482
\(292\) − 8.69722i − 0.508967i
\(293\) 25.0278i 1.46214i 0.682304 + 0.731069i \(0.260978\pi\)
−0.682304 + 0.731069i \(0.739022\pi\)
\(294\) −4.30278 −0.250943
\(295\) 0 0
\(296\) −1.00000 −0.0581238
\(297\) 2.33053i 0.135231i
\(298\) − 1.81665i − 0.105236i
\(299\) −15.9083 −0.920002
\(300\) 0 0
\(301\) −30.4222 −1.75351
\(302\) 13.3944i 0.770764i
\(303\) 4.97224i 0.285648i
\(304\) −2.00000 −0.114708
\(305\) 0 0
\(306\) −17.4500 −0.997548
\(307\) 7.09167i 0.404743i 0.979309 + 0.202372i \(0.0648649\pi\)
−0.979309 + 0.202372i \(0.935135\pi\)
\(308\) − 6.00000i − 0.341882i
\(309\) 1.00000 0.0568880
\(310\) 0 0
\(311\) 5.09167 0.288722 0.144361 0.989525i \(-0.453887\pi\)
0.144361 + 0.989525i \(0.453887\pi\)
\(312\) − 0.697224i − 0.0394726i
\(313\) − 27.0278i − 1.52770i −0.645394 0.763850i \(-0.723307\pi\)
0.645394 0.763850i \(-0.276693\pi\)
\(314\) 7.21110 0.406946
\(315\) 0 0
\(316\) −16.1194 −0.906789
\(317\) − 5.21110i − 0.292685i −0.989234 0.146342i \(-0.953250\pi\)
0.989234 0.146342i \(-0.0467501\pi\)
\(318\) 1.81665i 0.101873i
\(319\) −9.00000 −0.503903
\(320\) 0 0
\(321\) −1.30278 −0.0727138
\(322\) 31.8167i 1.77307i
\(323\) 12.0000i 0.667698i
\(324\) −8.18335 −0.454630
\(325\) 0 0
\(326\) 20.4222 1.13108
\(327\) − 0.605551i − 0.0334871i
\(328\) − 0.908327i − 0.0501540i
\(329\) 12.0000 0.661581
\(330\) 0 0
\(331\) 1.21110 0.0665682 0.0332841 0.999446i \(-0.489403\pi\)
0.0332841 + 0.999446i \(0.489403\pi\)
\(332\) 17.2111i 0.944582i
\(333\) 2.90833i 0.159375i
\(334\) 12.5139 0.684729
\(335\) 0 0
\(336\) −1.39445 −0.0760734
\(337\) − 19.1194i − 1.04150i −0.853709 0.520751i \(-0.825652\pi\)
0.853709 0.520751i \(-0.174348\pi\)
\(338\) − 7.69722i − 0.418674i
\(339\) 3.39445 0.184361
\(340\) 0 0
\(341\) 4.30278 0.233008
\(342\) 5.81665i 0.314529i
\(343\) − 33.2111i − 1.79323i
\(344\) −6.60555 −0.356147
\(345\) 0 0
\(346\) 23.2111 1.24784
\(347\) 31.8167i 1.70801i 0.520267 + 0.854004i \(0.325832\pi\)
−0.520267 + 0.854004i \(0.674168\pi\)
\(348\) 2.09167i 0.112125i
\(349\) 22.2389 1.19042 0.595209 0.803571i \(-0.297069\pi\)
0.595209 + 0.803571i \(0.297069\pi\)
\(350\) 0 0
\(351\) −4.11943 −0.219879
\(352\) − 1.30278i − 0.0694382i
\(353\) − 31.8167i − 1.69343i −0.532047 0.846715i \(-0.678577\pi\)
0.532047 0.846715i \(-0.321423\pi\)
\(354\) −1.02776 −0.0546246
\(355\) 0 0
\(356\) 5.21110 0.276188
\(357\) 8.36669i 0.442812i
\(358\) 7.81665i 0.413123i
\(359\) 11.2111 0.591699 0.295850 0.955235i \(-0.404397\pi\)
0.295850 + 0.955235i \(0.404397\pi\)
\(360\) 0 0
\(361\) −15.0000 −0.789474
\(362\) − 20.0000i − 1.05118i
\(363\) − 2.81665i − 0.147836i
\(364\) 10.6056 0.555882
\(365\) 0 0
\(366\) −3.18335 −0.166396
\(367\) − 17.8167i − 0.930022i −0.885305 0.465011i \(-0.846050\pi\)
0.885305 0.465011i \(-0.153950\pi\)
\(368\) 6.90833i 0.360121i
\(369\) −2.64171 −0.137522
\(370\) 0 0
\(371\) −27.6333 −1.43465
\(372\) − 1.00000i − 0.0518476i
\(373\) − 3.81665i − 0.197619i −0.995106 0.0988094i \(-0.968497\pi\)
0.995106 0.0988094i \(-0.0315034\pi\)
\(374\) −7.81665 −0.404190
\(375\) 0 0
\(376\) 2.60555 0.134371
\(377\) − 15.9083i − 0.819321i
\(378\) 8.23886i 0.423761i
\(379\) 15.3305 0.787477 0.393738 0.919223i \(-0.371182\pi\)
0.393738 + 0.919223i \(0.371182\pi\)
\(380\) 0 0
\(381\) 1.44996 0.0742838
\(382\) − 12.5139i − 0.640266i
\(383\) − 20.8444i − 1.06510i −0.846399 0.532550i \(-0.821234\pi\)
0.846399 0.532550i \(-0.178766\pi\)
\(384\) −0.302776 −0.0154510
\(385\) 0 0
\(386\) 4.00000 0.203595
\(387\) 19.2111i 0.976555i
\(388\) − 12.4222i − 0.630642i
\(389\) 11.8806 0.602369 0.301184 0.953566i \(-0.402618\pi\)
0.301184 + 0.953566i \(0.402618\pi\)
\(390\) 0 0
\(391\) 41.4500 2.09621
\(392\) − 14.2111i − 0.717769i
\(393\) 1.02776i 0.0518435i
\(394\) −6.00000 −0.302276
\(395\) 0 0
\(396\) −3.78890 −0.190399
\(397\) 27.8167i 1.39608i 0.716060 + 0.698039i \(0.245944\pi\)
−0.716060 + 0.698039i \(0.754056\pi\)
\(398\) − 2.42221i − 0.121414i
\(399\) 2.78890 0.139620
\(400\) 0 0
\(401\) 13.8167 0.689971 0.344985 0.938608i \(-0.387884\pi\)
0.344985 + 0.938608i \(0.387884\pi\)
\(402\) 4.39445i 0.219175i
\(403\) 7.60555i 0.378859i
\(404\) −16.4222 −0.817035
\(405\) 0 0
\(406\) −31.8167 −1.57903
\(407\) 1.30278i 0.0645762i
\(408\) 1.81665i 0.0899378i
\(409\) 5.02776 0.248607 0.124303 0.992244i \(-0.460330\pi\)
0.124303 + 0.992244i \(0.460330\pi\)
\(410\) 0 0
\(411\) 3.00000 0.147979
\(412\) 3.30278i 0.162716i
\(413\) − 15.6333i − 0.769265i
\(414\) 20.0917 0.987452
\(415\) 0 0
\(416\) 2.30278 0.112903
\(417\) − 2.69722i − 0.132084i
\(418\) 2.60555i 0.127442i
\(419\) 25.1472 1.22852 0.614260 0.789104i \(-0.289455\pi\)
0.614260 + 0.789104i \(0.289455\pi\)
\(420\) 0 0
\(421\) 28.7250 1.39997 0.699985 0.714158i \(-0.253190\pi\)
0.699985 + 0.714158i \(0.253190\pi\)
\(422\) − 6.69722i − 0.326016i
\(423\) − 7.57779i − 0.368445i
\(424\) −6.00000 −0.291386
\(425\) 0 0
\(426\) 1.81665 0.0880172
\(427\) − 48.4222i − 2.34331i
\(428\) − 4.30278i − 0.207983i
\(429\) −0.908327 −0.0438544
\(430\) 0 0
\(431\) −5.21110 −0.251010 −0.125505 0.992093i \(-0.540055\pi\)
−0.125505 + 0.992093i \(0.540055\pi\)
\(432\) 1.78890i 0.0860684i
\(433\) 11.9361i 0.573612i 0.957989 + 0.286806i \(0.0925934\pi\)
−0.957989 + 0.286806i \(0.907407\pi\)
\(434\) 15.2111 0.730156
\(435\) 0 0
\(436\) 2.00000 0.0957826
\(437\) − 13.8167i − 0.660940i
\(438\) 2.63331i 0.125824i
\(439\) 9.33053 0.445322 0.222661 0.974896i \(-0.428526\pi\)
0.222661 + 0.974896i \(0.428526\pi\)
\(440\) 0 0
\(441\) −41.3305 −1.96812
\(442\) − 13.8167i − 0.657191i
\(443\) − 0.275019i − 0.0130666i −0.999979 0.00653328i \(-0.997920\pi\)
0.999979 0.00653328i \(-0.00207962\pi\)
\(444\) 0.302776 0.0143691
\(445\) 0 0
\(446\) −15.8167 −0.748940
\(447\) 0.550039i 0.0260159i
\(448\) − 4.60555i − 0.217592i
\(449\) 0.788897 0.0372304 0.0186152 0.999827i \(-0.494074\pi\)
0.0186152 + 0.999827i \(0.494074\pi\)
\(450\) 0 0
\(451\) −1.18335 −0.0557216
\(452\) 11.2111i 0.527326i
\(453\) − 4.05551i − 0.190545i
\(454\) −7.81665 −0.366854
\(455\) 0 0
\(456\) 0.605551 0.0283575
\(457\) 4.60555i 0.215439i 0.994181 + 0.107719i \(0.0343548\pi\)
−0.994181 + 0.107719i \(0.965645\pi\)
\(458\) 17.3944i 0.812789i
\(459\) 10.7334 0.500991
\(460\) 0 0
\(461\) −16.4222 −0.764858 −0.382429 0.923985i \(-0.624912\pi\)
−0.382429 + 0.923985i \(0.624912\pi\)
\(462\) 1.81665i 0.0845184i
\(463\) − 30.3028i − 1.40829i −0.710056 0.704145i \(-0.751331\pi\)
0.710056 0.704145i \(-0.248669\pi\)
\(464\) −6.90833 −0.320711
\(465\) 0 0
\(466\) 9.51388 0.440722
\(467\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(468\) − 6.69722i − 0.309579i
\(469\) −66.8444 −3.08659
\(470\) 0 0
\(471\) −2.18335 −0.100603
\(472\) − 3.39445i − 0.156242i
\(473\) 8.60555i 0.395684i
\(474\) 4.88057 0.224172
\(475\) 0 0
\(476\) −27.6333 −1.26657
\(477\) 17.4500i 0.798979i
\(478\) 0.513878i 0.0235042i
\(479\) −12.1194 −0.553751 −0.276875 0.960906i \(-0.589299\pi\)
−0.276875 + 0.960906i \(0.589299\pi\)
\(480\) 0 0
\(481\) −2.30278 −0.104998
\(482\) − 8.00000i − 0.364390i
\(483\) − 9.63331i − 0.438331i
\(484\) 9.30278 0.422853
\(485\) 0 0
\(486\) 7.84441 0.355830
\(487\) − 22.7889i − 1.03266i −0.856389 0.516332i \(-0.827297\pi\)
0.856389 0.516332i \(-0.172703\pi\)
\(488\) − 10.5139i − 0.475941i
\(489\) −6.18335 −0.279621
\(490\) 0 0
\(491\) −14.7250 −0.664529 −0.332265 0.943186i \(-0.607813\pi\)
−0.332265 + 0.943186i \(0.607813\pi\)
\(492\) 0.275019i 0.0123988i
\(493\) 41.4500i 1.86681i
\(494\) −4.60555 −0.207214
\(495\) 0 0
\(496\) 3.30278 0.148299
\(497\) 27.6333i 1.23952i
\(498\) − 5.21110i − 0.233515i
\(499\) −8.23886 −0.368822 −0.184411 0.982849i \(-0.559038\pi\)
−0.184411 + 0.982849i \(0.559038\pi\)
\(500\) 0 0
\(501\) −3.78890 −0.169275
\(502\) 6.78890i 0.303003i
\(503\) 24.5139i 1.09302i 0.837453 + 0.546510i \(0.184044\pi\)
−0.837453 + 0.546510i \(0.815956\pi\)
\(504\) −13.3944 −0.596636
\(505\) 0 0
\(506\) 9.00000 0.400099
\(507\) 2.33053i 0.103503i
\(508\) 4.78890i 0.212473i
\(509\) 25.8167 1.14430 0.572152 0.820148i \(-0.306109\pi\)
0.572152 + 0.820148i \(0.306109\pi\)
\(510\) 0 0
\(511\) −40.0555 −1.77195
\(512\) − 1.00000i − 0.0441942i
\(513\) − 3.57779i − 0.157964i
\(514\) −11.2111 −0.494501
\(515\) 0 0
\(516\) 2.00000 0.0880451
\(517\) − 3.39445i − 0.149288i
\(518\) 4.60555i 0.202356i
\(519\) −7.02776 −0.308484
\(520\) 0 0
\(521\) −9.63331 −0.422043 −0.211021 0.977481i \(-0.567679\pi\)
−0.211021 + 0.977481i \(0.567679\pi\)
\(522\) 20.0917i 0.879389i
\(523\) − 32.2389i − 1.40971i −0.709353 0.704853i \(-0.751013\pi\)
0.709353 0.704853i \(-0.248987\pi\)
\(524\) −3.39445 −0.148287
\(525\) 0 0
\(526\) 7.81665 0.340822
\(527\) − 19.8167i − 0.863227i
\(528\) 0.394449i 0.0171662i
\(529\) −24.7250 −1.07500
\(530\) 0 0
\(531\) −9.87217 −0.428416
\(532\) 9.21110i 0.399352i
\(533\) − 2.09167i − 0.0906004i
\(534\) −1.57779 −0.0682779
\(535\) 0 0
\(536\) −14.5139 −0.626904
\(537\) − 2.36669i − 0.102130i
\(538\) − 6.78890i − 0.292690i
\(539\) −18.5139 −0.797449
\(540\) 0 0
\(541\) −20.9361 −0.900113 −0.450056 0.893000i \(-0.648596\pi\)
−0.450056 + 0.893000i \(0.648596\pi\)
\(542\) − 6.42221i − 0.275857i
\(543\) 6.05551i 0.259867i
\(544\) −6.00000 −0.257248
\(545\) 0 0
\(546\) −3.21110 −0.137423
\(547\) − 13.3944i − 0.572705i −0.958124 0.286353i \(-0.907557\pi\)
0.958124 0.286353i \(-0.0924429\pi\)
\(548\) 9.90833i 0.423263i
\(549\) −30.5778 −1.30503
\(550\) 0 0
\(551\) 13.8167 0.588609
\(552\) − 2.09167i − 0.0890275i
\(553\) 74.2389i 3.15696i
\(554\) −25.1194 −1.06722
\(555\) 0 0
\(556\) 8.90833 0.377797
\(557\) − 6.51388i − 0.276002i −0.990432 0.138001i \(-0.955932\pi\)
0.990432 0.138001i \(-0.0440677\pi\)
\(558\) − 9.60555i − 0.406635i
\(559\) −15.2111 −0.643361
\(560\) 0 0
\(561\) 2.36669 0.0999218
\(562\) 12.0000i 0.506189i
\(563\) − 44.0555i − 1.85672i −0.371684 0.928359i \(-0.621220\pi\)
0.371684 0.928359i \(-0.378780\pi\)
\(564\) −0.788897 −0.0332186
\(565\) 0 0
\(566\) −17.3944 −0.731143
\(567\) 37.6888i 1.58278i
\(568\) 6.00000i 0.251754i
\(569\) 10.4222 0.436922 0.218461 0.975846i \(-0.429896\pi\)
0.218461 + 0.975846i \(0.429896\pi\)
\(570\) 0 0
\(571\) −20.3028 −0.849645 −0.424822 0.905277i \(-0.639663\pi\)
−0.424822 + 0.905277i \(0.639663\pi\)
\(572\) − 3.00000i − 0.125436i
\(573\) 3.78890i 0.158283i
\(574\) −4.18335 −0.174609
\(575\) 0 0
\(576\) −2.90833 −0.121180
\(577\) − 28.2389i − 1.17560i −0.809007 0.587800i \(-0.799994\pi\)
0.809007 0.587800i \(-0.200006\pi\)
\(578\) 19.0000i 0.790296i
\(579\) −1.21110 −0.0503317
\(580\) 0 0
\(581\) 79.2666 3.28853
\(582\) 3.76114i 0.155904i
\(583\) 7.81665i 0.323733i
\(584\) −8.69722 −0.359894
\(585\) 0 0
\(586\) 25.0278 1.03389
\(587\) − 2.36669i − 0.0976838i −0.998807 0.0488419i \(-0.984447\pi\)
0.998807 0.0488419i \(-0.0155531\pi\)
\(588\) 4.30278i 0.177443i
\(589\) −6.60555 −0.272177
\(590\) 0 0
\(591\) 1.81665 0.0747272
\(592\) 1.00000i 0.0410997i
\(593\) − 36.5139i − 1.49945i −0.661752 0.749723i \(-0.730187\pi\)
0.661752 0.749723i \(-0.269813\pi\)
\(594\) 2.33053 0.0956229
\(595\) 0 0
\(596\) −1.81665 −0.0744130
\(597\) 0.733385i 0.0300154i
\(598\) 15.9083i 0.650540i
\(599\) 35.2111 1.43869 0.719343 0.694655i \(-0.244443\pi\)
0.719343 + 0.694655i \(0.244443\pi\)
\(600\) 0 0
\(601\) −20.6972 −0.844257 −0.422129 0.906536i \(-0.638717\pi\)
−0.422129 + 0.906536i \(0.638717\pi\)
\(602\) 30.4222i 1.23992i
\(603\) 42.2111i 1.71897i
\(604\) 13.3944 0.545012
\(605\) 0 0
\(606\) 4.97224 0.201984
\(607\) − 31.5139i − 1.27911i −0.768746 0.639554i \(-0.779119\pi\)
0.768746 0.639554i \(-0.220881\pi\)
\(608\) 2.00000i 0.0811107i
\(609\) 9.63331 0.390361
\(610\) 0 0
\(611\) 6.00000 0.242734
\(612\) 17.4500i 0.705373i
\(613\) 8.18335i 0.330522i 0.986250 + 0.165261i \(0.0528467\pi\)
−0.986250 + 0.165261i \(0.947153\pi\)
\(614\) 7.09167 0.286197
\(615\) 0 0
\(616\) −6.00000 −0.241747
\(617\) 47.5694i 1.91507i 0.288314 + 0.957536i \(0.406905\pi\)
−0.288314 + 0.957536i \(0.593095\pi\)
\(618\) − 1.00000i − 0.0402259i
\(619\) 2.69722 0.108411 0.0542053 0.998530i \(-0.482737\pi\)
0.0542053 + 0.998530i \(0.482737\pi\)
\(620\) 0 0
\(621\) −12.3583 −0.495921
\(622\) − 5.09167i − 0.204157i
\(623\) − 24.0000i − 0.961540i
\(624\) −0.697224 −0.0279113
\(625\) 0 0
\(626\) −27.0278 −1.08025
\(627\) − 0.788897i − 0.0315055i
\(628\) − 7.21110i − 0.287754i
\(629\) 6.00000 0.239236
\(630\) 0 0
\(631\) 18.3028 0.728622 0.364311 0.931277i \(-0.381305\pi\)
0.364311 + 0.931277i \(0.381305\pi\)
\(632\) 16.1194i 0.641196i
\(633\) 2.02776i 0.0805961i
\(634\) −5.21110 −0.206959
\(635\) 0 0
\(636\) 1.81665 0.0720350
\(637\) − 32.7250i − 1.29661i
\(638\) 9.00000i 0.356313i
\(639\) 17.4500 0.690310
\(640\) 0 0
\(641\) −2.48612 −0.0981959 −0.0490980 0.998794i \(-0.515635\pi\)
−0.0490980 + 0.998794i \(0.515635\pi\)
\(642\) 1.30278i 0.0514165i
\(643\) 29.8167i 1.17585i 0.808914 + 0.587927i \(0.200056\pi\)
−0.808914 + 0.587927i \(0.799944\pi\)
\(644\) 31.8167 1.25375
\(645\) 0 0
\(646\) 12.0000 0.472134
\(647\) − 25.9361i − 1.01965i −0.860277 0.509826i \(-0.829710\pi\)
0.860277 0.509826i \(-0.170290\pi\)
\(648\) 8.18335i 0.321472i
\(649\) −4.42221 −0.173587
\(650\) 0 0
\(651\) −4.60555 −0.180506
\(652\) − 20.4222i − 0.799795i
\(653\) − 6.90833i − 0.270344i −0.990822 0.135172i \(-0.956841\pi\)
0.990822 0.135172i \(-0.0431587\pi\)
\(654\) −0.605551 −0.0236789
\(655\) 0 0
\(656\) −0.908327 −0.0354642
\(657\) 25.2944i 0.986827i
\(658\) − 12.0000i − 0.467809i
\(659\) 42.1194 1.64074 0.820370 0.571833i \(-0.193767\pi\)
0.820370 + 0.571833i \(0.193767\pi\)
\(660\) 0 0
\(661\) −12.4861 −0.485654 −0.242827 0.970070i \(-0.578075\pi\)
−0.242827 + 0.970070i \(0.578075\pi\)
\(662\) − 1.21110i − 0.0470708i
\(663\) 4.18335i 0.162468i
\(664\) 17.2111 0.667920
\(665\) 0 0
\(666\) 2.90833 0.112695
\(667\) − 47.7250i − 1.84792i
\(668\) − 12.5139i − 0.484176i
\(669\) 4.78890 0.185149
\(670\) 0 0
\(671\) −13.6972 −0.528775
\(672\) 1.39445i 0.0537920i
\(673\) − 24.3028i − 0.936803i −0.883516 0.468402i \(-0.844830\pi\)
0.883516 0.468402i \(-0.155170\pi\)
\(674\) −19.1194 −0.736453
\(675\) 0 0
\(676\) −7.69722 −0.296047
\(677\) − 36.2389i − 1.39277i −0.717667 0.696386i \(-0.754790\pi\)
0.717667 0.696386i \(-0.245210\pi\)
\(678\) − 3.39445i − 0.130363i
\(679\) −57.2111 −2.19556
\(680\) 0 0
\(681\) 2.36669 0.0906918
\(682\) − 4.30278i − 0.164762i
\(683\) − 12.0000i − 0.459167i −0.973289 0.229584i \(-0.926264\pi\)
0.973289 0.229584i \(-0.0737364\pi\)
\(684\) 5.81665 0.222405
\(685\) 0 0
\(686\) −33.2111 −1.26801
\(687\) − 5.26662i − 0.200934i
\(688\) 6.60555i 0.251834i
\(689\) −13.8167 −0.526373
\(690\) 0 0
\(691\) 8.00000 0.304334 0.152167 0.988355i \(-0.451375\pi\)
0.152167 + 0.988355i \(0.451375\pi\)
\(692\) − 23.2111i − 0.882354i
\(693\) 17.4500i 0.662869i
\(694\) 31.8167 1.20774
\(695\) 0 0
\(696\) 2.09167 0.0792847
\(697\) 5.44996i 0.206432i
\(698\) − 22.2389i − 0.841753i
\(699\) −2.88057 −0.108953
\(700\) 0 0
\(701\) 14.8806 0.562031 0.281016 0.959703i \(-0.409329\pi\)
0.281016 + 0.959703i \(0.409329\pi\)
\(702\) 4.11943i 0.155478i
\(703\) − 2.00000i − 0.0754314i
\(704\) −1.30278 −0.0491002
\(705\) 0 0
\(706\) −31.8167 −1.19744
\(707\) 75.6333i 2.84448i
\(708\) 1.02776i 0.0386254i
\(709\) 1.66947 0.0626982 0.0313491 0.999508i \(-0.490020\pi\)
0.0313491 + 0.999508i \(0.490020\pi\)
\(710\) 0 0
\(711\) 46.8806 1.75816
\(712\) − 5.21110i − 0.195294i
\(713\) 22.8167i 0.854490i
\(714\) 8.36669 0.313116
\(715\) 0 0
\(716\) 7.81665 0.292122
\(717\) − 0.155590i − 0.00581061i
\(718\) − 11.2111i − 0.418395i
\(719\) 8.36669 0.312025 0.156012 0.987755i \(-0.450136\pi\)
0.156012 + 0.987755i \(0.450136\pi\)
\(720\) 0 0
\(721\) 15.2111 0.566491
\(722\) 15.0000i 0.558242i
\(723\) 2.42221i 0.0900828i
\(724\) −20.0000 −0.743294
\(725\) 0 0
\(726\) −2.81665 −0.104536
\(727\) 29.9083i 1.10924i 0.832104 + 0.554619i \(0.187136\pi\)
−0.832104 + 0.554619i \(0.812864\pi\)
\(728\) − 10.6056i − 0.393068i
\(729\) 22.1749 0.821294
\(730\) 0 0
\(731\) 39.6333 1.46589
\(732\) 3.18335i 0.117660i
\(733\) − 29.6333i − 1.09453i −0.836959 0.547266i \(-0.815669\pi\)
0.836959 0.547266i \(-0.184331\pi\)
\(734\) −17.8167 −0.657625
\(735\) 0 0
\(736\) 6.90833 0.254644
\(737\) 18.9083i 0.696497i
\(738\) 2.64171i 0.0972427i
\(739\) 42.3305 1.55715 0.778577 0.627549i \(-0.215942\pi\)
0.778577 + 0.627549i \(0.215942\pi\)
\(740\) 0 0
\(741\) 1.39445 0.0512264
\(742\) 27.6333i 1.01445i
\(743\) − 35.4500i − 1.30053i −0.759706 0.650266i \(-0.774657\pi\)
0.759706 0.650266i \(-0.225343\pi\)
\(744\) −1.00000 −0.0366618
\(745\) 0 0
\(746\) −3.81665 −0.139738
\(747\) − 50.0555i − 1.83144i
\(748\) 7.81665i 0.285805i
\(749\) −19.8167 −0.724085
\(750\) 0 0
\(751\) 14.0000 0.510867 0.255434 0.966827i \(-0.417782\pi\)
0.255434 + 0.966827i \(0.417782\pi\)
\(752\) − 2.60555i − 0.0950147i
\(753\) − 2.05551i − 0.0749070i
\(754\) −15.9083 −0.579347
\(755\) 0 0
\(756\) 8.23886 0.299644
\(757\) 9.30278i 0.338115i 0.985606 + 0.169058i \(0.0540724\pi\)
−0.985606 + 0.169058i \(0.945928\pi\)
\(758\) − 15.3305i − 0.556830i
\(759\) −2.72498 −0.0989105
\(760\) 0 0
\(761\) 42.1194 1.52683 0.763414 0.645909i \(-0.223522\pi\)
0.763414 + 0.645909i \(0.223522\pi\)
\(762\) − 1.44996i − 0.0525266i
\(763\) − 9.21110i − 0.333464i
\(764\) −12.5139 −0.452736
\(765\) 0 0
\(766\) −20.8444 −0.753139
\(767\) − 7.81665i − 0.282243i
\(768\) 0.302776i 0.0109255i
\(769\) 22.0000 0.793340 0.396670 0.917961i \(-0.370166\pi\)
0.396670 + 0.917961i \(0.370166\pi\)
\(770\) 0 0
\(771\) 3.39445 0.122248
\(772\) − 4.00000i − 0.143963i
\(773\) 50.0555i 1.80037i 0.435506 + 0.900186i \(0.356569\pi\)
−0.435506 + 0.900186i \(0.643431\pi\)
\(774\) 19.2111 0.690529
\(775\) 0 0
\(776\) −12.4222 −0.445931
\(777\) − 1.39445i − 0.0500256i
\(778\) − 11.8806i − 0.425939i
\(779\) 1.81665 0.0650884
\(780\) 0 0
\(781\) 7.81665 0.279702
\(782\) − 41.4500i − 1.48225i
\(783\) − 12.3583i − 0.441649i
\(784\) −14.2111 −0.507539
\(785\) 0 0
\(786\) 1.02776 0.0366589
\(787\) 25.2111i 0.898679i 0.893361 + 0.449339i \(0.148341\pi\)
−0.893361 + 0.449339i \(0.851659\pi\)
\(788\) 6.00000i 0.213741i
\(789\) −2.36669 −0.0842565
\(790\) 0 0
\(791\) 51.6333 1.83587
\(792\) 3.78890i 0.134633i
\(793\) − 24.2111i − 0.859761i
\(794\) 27.8167 0.987176
\(795\) 0 0
\(796\) −2.42221 −0.0858528
\(797\) − 17.3305i − 0.613879i −0.951729 0.306939i \(-0.900695\pi\)
0.951729 0.306939i \(-0.0993049\pi\)
\(798\) − 2.78890i − 0.0987259i
\(799\) −15.6333 −0.553067
\(800\) 0 0
\(801\) −15.1556 −0.535496
\(802\) − 13.8167i − 0.487883i
\(803\) 11.3305i 0.399846i
\(804\) 4.39445 0.154980
\(805\) 0 0
\(806\) 7.60555 0.267894
\(807\) 2.05551i 0.0723575i
\(808\) 16.4222i 0.577731i
\(809\) −29.4500 −1.03541 −0.517703 0.855561i \(-0.673213\pi\)
−0.517703 + 0.855561i \(0.673213\pi\)
\(810\) 0 0
\(811\) 54.1472 1.90136 0.950682 0.310166i \(-0.100385\pi\)
0.950682 + 0.310166i \(0.100385\pi\)
\(812\) 31.8167i 1.11655i
\(813\) 1.94449i 0.0681961i
\(814\) 1.30278 0.0456623
\(815\) 0 0
\(816\) 1.81665 0.0635956
\(817\) − 13.2111i − 0.462198i
\(818\) − 5.02776i − 0.175791i
\(819\) −30.8444 −1.07779
\(820\) 0 0
\(821\) 11.2111 0.391270 0.195635 0.980677i \(-0.437323\pi\)
0.195635 + 0.980677i \(0.437323\pi\)
\(822\) − 3.00000i − 0.104637i
\(823\) 12.8444i 0.447728i 0.974620 + 0.223864i \(0.0718671\pi\)
−0.974620 + 0.223864i \(0.928133\pi\)
\(824\) 3.30278 0.115058
\(825\) 0 0
\(826\) −15.6333 −0.543952
\(827\) 27.3944i 0.952598i 0.879283 + 0.476299i \(0.158022\pi\)
−0.879283 + 0.476299i \(0.841978\pi\)
\(828\) − 20.0917i − 0.698234i
\(829\) −4.72498 −0.164105 −0.0820527 0.996628i \(-0.526148\pi\)
−0.0820527 + 0.996628i \(0.526148\pi\)
\(830\) 0 0
\(831\) 7.60555 0.263834
\(832\) − 2.30278i − 0.0798344i
\(833\) 85.2666i 2.95431i
\(834\) −2.69722 −0.0933972
\(835\) 0 0
\(836\) 2.60555 0.0901149
\(837\) 5.90833i 0.204222i
\(838\) − 25.1472i − 0.868695i
\(839\) 49.0278 1.69263 0.846313 0.532686i \(-0.178817\pi\)
0.846313 + 0.532686i \(0.178817\pi\)
\(840\) 0 0
\(841\) 18.7250 0.645689
\(842\) − 28.7250i − 0.989928i
\(843\) − 3.63331i − 0.125138i
\(844\) −6.69722 −0.230528
\(845\) 0 0
\(846\) −7.57779 −0.260530
\(847\) − 42.8444i − 1.47215i
\(848\) 6.00000i 0.206041i
\(849\) 5.26662 0.180750
\(850\) 0 0
\(851\) −6.90833 −0.236814
\(852\) − 1.81665i − 0.0622375i
\(853\) 11.5416i 0.395178i 0.980285 + 0.197589i \(0.0633111\pi\)
−0.980285 + 0.197589i \(0.936689\pi\)
\(854\) −48.4222 −1.65697
\(855\) 0 0
\(856\) −4.30278 −0.147066
\(857\) − 14.8444i − 0.507075i −0.967326 0.253538i \(-0.918406\pi\)
0.967326 0.253538i \(-0.0815942\pi\)
\(858\) 0.908327i 0.0310098i
\(859\) 24.0555 0.820764 0.410382 0.911914i \(-0.365395\pi\)
0.410382 + 0.911914i \(0.365395\pi\)
\(860\) 0 0
\(861\) 1.26662 0.0431661
\(862\) 5.21110i 0.177491i
\(863\) − 12.0000i − 0.408485i −0.978920 0.204242i \(-0.934527\pi\)
0.978920 0.204242i \(-0.0654731\pi\)
\(864\) 1.78890 0.0608595
\(865\) 0 0
\(866\) 11.9361 0.405605
\(867\) − 5.75274i − 0.195373i
\(868\) − 15.2111i − 0.516298i
\(869\) 21.0000 0.712376
\(870\) 0 0
\(871\) −33.4222 −1.13247
\(872\) − 2.00000i − 0.0677285i
\(873\) 36.1278i 1.22274i
\(874\) −13.8167 −0.467355
\(875\) 0 0
\(876\) 2.63331 0.0889712
\(877\) 7.21110i 0.243502i 0.992561 + 0.121751i \(0.0388509\pi\)
−0.992561 + 0.121751i \(0.961149\pi\)
\(878\) − 9.33053i − 0.314890i
\(879\) −7.57779 −0.255593
\(880\) 0 0
\(881\) 25.5416 0.860520 0.430260 0.902705i \(-0.358422\pi\)
0.430260 + 0.902705i \(0.358422\pi\)
\(882\) 41.3305i 1.39167i
\(883\) 2.42221i 0.0815137i 0.999169 + 0.0407568i \(0.0129769\pi\)
−0.999169 + 0.0407568i \(0.987023\pi\)
\(884\) −13.8167 −0.464704
\(885\) 0 0
\(886\) −0.275019 −0.00923945
\(887\) 28.4222i 0.954324i 0.878815 + 0.477162i \(0.158335\pi\)
−0.878815 + 0.477162i \(0.841665\pi\)
\(888\) − 0.302776i − 0.0101605i
\(889\) 22.0555 0.739718
\(890\) 0 0
\(891\) 10.6611 0.357159
\(892\) 15.8167i 0.529581i
\(893\) 5.21110i 0.174383i
\(894\) 0.550039 0.0183960
\(895\) 0 0
\(896\) −4.60555 −0.153861
\(897\) − 4.81665i − 0.160823i
\(898\) − 0.788897i − 0.0263258i
\(899\) −22.8167 −0.760978
\(900\) 0 0
\(901\) 36.0000 1.19933
\(902\) 1.18335i 0.0394011i
\(903\) − 9.21110i − 0.306526i
\(904\) 11.2111 0.372876
\(905\) 0 0
\(906\) −4.05551 −0.134735
\(907\) 26.0000i 0.863316i 0.902037 + 0.431658i \(0.142071\pi\)
−0.902037 + 0.431658i \(0.857929\pi\)
\(908\) 7.81665i 0.259405i
\(909\) 47.7611 1.58414
\(910\) 0 0
\(911\) −46.4222 −1.53804 −0.769018 0.639227i \(-0.779254\pi\)
−0.769018 + 0.639227i \(0.779254\pi\)
\(912\) − 0.605551i − 0.0200518i
\(913\) − 22.4222i − 0.742067i
\(914\) 4.60555 0.152338
\(915\) 0 0
\(916\) 17.3944 0.574729
\(917\) 15.6333i 0.516257i
\(918\) − 10.7334i − 0.354254i
\(919\) 38.4222 1.26743 0.633716 0.773566i \(-0.281529\pi\)
0.633716 + 0.773566i \(0.281529\pi\)
\(920\) 0 0
\(921\) −2.14719 −0.0707522
\(922\) 16.4222i 0.540837i
\(923\) 13.8167i 0.454781i
\(924\) 1.81665 0.0597635
\(925\) 0 0
\(926\) −30.3028 −0.995811
\(927\) − 9.60555i − 0.315488i
\(928\) 6.90833i 0.226777i
\(929\) 36.5139 1.19798 0.598991 0.800756i \(-0.295569\pi\)
0.598991 + 0.800756i \(0.295569\pi\)
\(930\) 0 0
\(931\) 28.4222 0.931500
\(932\) − 9.51388i − 0.311637i
\(933\) 1.54163i 0.0504708i
\(934\) 0 0
\(935\) 0 0
\(936\) −6.69722 −0.218906
\(937\) − 28.9083i − 0.944394i −0.881493 0.472197i \(-0.843461\pi\)
0.881493 0.472197i \(-0.156539\pi\)
\(938\) 66.8444i 2.18255i
\(939\) 8.18335 0.267053
\(940\) 0 0
\(941\) −7.81665 −0.254816 −0.127408 0.991850i \(-0.540666\pi\)
−0.127408 + 0.991850i \(0.540666\pi\)
\(942\) 2.18335i 0.0711373i
\(943\) − 6.27502i − 0.204343i
\(944\) −3.39445 −0.110480
\(945\) 0 0
\(946\) 8.60555 0.279791
\(947\) 39.6333i 1.28791i 0.765064 + 0.643955i \(0.222707\pi\)
−0.765064 + 0.643955i \(0.777293\pi\)
\(948\) − 4.88057i − 0.158514i
\(949\) −20.0278 −0.650128
\(950\) 0 0
\(951\) 1.57779 0.0511635
\(952\) 27.6333i 0.895601i
\(953\) 18.7527i 0.607461i 0.952758 + 0.303730i \(0.0982322\pi\)
−0.952758 + 0.303730i \(0.901768\pi\)
\(954\) 17.4500 0.564963
\(955\) 0 0
\(956\) 0.513878 0.0166200
\(957\) − 2.72498i − 0.0880861i
\(958\) 12.1194i 0.391561i
\(959\) 45.6333 1.47358
\(960\) 0 0
\(961\) −20.0917 −0.648118
\(962\) 2.30278i 0.0742445i
\(963\) 12.5139i 0.403254i
\(964\) −8.00000 −0.257663
\(965\) 0 0
\(966\) −9.63331 −0.309947
\(967\) 25.7250i 0.827260i 0.910445 + 0.413630i \(0.135739\pi\)
−0.910445 + 0.413630i \(0.864261\pi\)
\(968\) − 9.30278i − 0.299003i
\(969\) −3.63331 −0.116719
\(970\) 0 0
\(971\) 31.5416 1.01222 0.506110 0.862469i \(-0.331083\pi\)
0.506110 + 0.862469i \(0.331083\pi\)
\(972\) − 7.84441i − 0.251610i
\(973\) − 41.0278i − 1.31529i
\(974\) −22.7889 −0.730203
\(975\) 0 0
\(976\) −10.5139 −0.336541
\(977\) 18.0000i 0.575871i 0.957650 + 0.287936i \(0.0929689\pi\)
−0.957650 + 0.287936i \(0.907031\pi\)
\(978\) 6.18335i 0.197722i
\(979\) −6.78890 −0.216974
\(980\) 0 0
\(981\) −5.81665 −0.185711
\(982\) 14.7250i 0.469893i
\(983\) 12.0000i 0.382741i 0.981518 + 0.191370i \(0.0612931\pi\)
−0.981518 + 0.191370i \(0.938707\pi\)
\(984\) 0.275019 0.00876729
\(985\) 0 0
\(986\) 41.4500 1.32004
\(987\) 3.63331i 0.115649i
\(988\) 4.60555i 0.146522i
\(989\) −45.6333 −1.45105
\(990\) 0 0
\(991\) 54.3028 1.72498 0.862492 0.506070i \(-0.168902\pi\)
0.862492 + 0.506070i \(0.168902\pi\)
\(992\) − 3.30278i − 0.104863i
\(993\) 0.366692i 0.0116366i
\(994\) 27.6333 0.876475
\(995\) 0 0
\(996\) −5.21110 −0.165120
\(997\) − 23.5778i − 0.746716i −0.927687 0.373358i \(-0.878206\pi\)
0.927687 0.373358i \(-0.121794\pi\)
\(998\) 8.23886i 0.260797i
\(999\) −1.78890 −0.0565982
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 1850.2.b.i.149.2 4
5.2 odd 4 1850.2.a.u.1.2 2
5.3 odd 4 74.2.a.a.1.1 2
5.4 even 2 inner 1850.2.b.i.149.3 4
15.8 even 4 666.2.a.j.1.1 2
20.3 even 4 592.2.a.f.1.2 2
35.13 even 4 3626.2.a.a.1.2 2
40.3 even 4 2368.2.a.ba.1.1 2
40.13 odd 4 2368.2.a.s.1.2 2
55.43 even 4 8954.2.a.p.1.1 2
60.23 odd 4 5328.2.a.bf.1.1 2
185.73 odd 4 2738.2.a.l.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
74.2.a.a.1.1 2 5.3 odd 4
592.2.a.f.1.2 2 20.3 even 4
666.2.a.j.1.1 2 15.8 even 4
1850.2.a.u.1.2 2 5.2 odd 4
1850.2.b.i.149.2 4 1.1 even 1 trivial
1850.2.b.i.149.3 4 5.4 even 2 inner
2368.2.a.s.1.2 2 40.13 odd 4
2368.2.a.ba.1.1 2 40.3 even 4
2738.2.a.l.1.1 2 185.73 odd 4
3626.2.a.a.1.2 2 35.13 even 4
5328.2.a.bf.1.1 2 60.23 odd 4
8954.2.a.p.1.1 2 55.43 even 4