Properties

Label 1850.2.a.u.1.2
Level $1850$
Weight $2$
Character 1850.1
Self dual yes
Analytic conductor $14.772$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1850 = 2 \cdot 5^{2} \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1850.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(14.7723243739\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{13}) \)
Defining polynomial: \(x^{2} - x - 3\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 74)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-1.30278\) of defining polynomial
Character \(\chi\) \(=\) 1850.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} +0.302776 q^{3} +1.00000 q^{4} +0.302776 q^{6} -4.60555 q^{7} +1.00000 q^{8} -2.90833 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} +0.302776 q^{3} +1.00000 q^{4} +0.302776 q^{6} -4.60555 q^{7} +1.00000 q^{8} -2.90833 q^{9} +1.30278 q^{11} +0.302776 q^{12} +2.30278 q^{13} -4.60555 q^{14} +1.00000 q^{16} +6.00000 q^{17} -2.90833 q^{18} +2.00000 q^{19} -1.39445 q^{21} +1.30278 q^{22} +6.90833 q^{23} +0.302776 q^{24} +2.30278 q^{26} -1.78890 q^{27} -4.60555 q^{28} +6.90833 q^{29} +3.30278 q^{31} +1.00000 q^{32} +0.394449 q^{33} +6.00000 q^{34} -2.90833 q^{36} -1.00000 q^{37} +2.00000 q^{38} +0.697224 q^{39} -0.908327 q^{41} -1.39445 q^{42} +6.60555 q^{43} +1.30278 q^{44} +6.90833 q^{46} +2.60555 q^{47} +0.302776 q^{48} +14.2111 q^{49} +1.81665 q^{51} +2.30278 q^{52} +6.00000 q^{53} -1.78890 q^{54} -4.60555 q^{56} +0.605551 q^{57} +6.90833 q^{58} +3.39445 q^{59} -10.5139 q^{61} +3.30278 q^{62} +13.3944 q^{63} +1.00000 q^{64} +0.394449 q^{66} -14.5139 q^{67} +6.00000 q^{68} +2.09167 q^{69} +6.00000 q^{71} -2.90833 q^{72} +8.69722 q^{73} -1.00000 q^{74} +2.00000 q^{76} -6.00000 q^{77} +0.697224 q^{78} -16.1194 q^{79} +8.18335 q^{81} -0.908327 q^{82} -17.2111 q^{83} -1.39445 q^{84} +6.60555 q^{86} +2.09167 q^{87} +1.30278 q^{88} +5.21110 q^{89} -10.6056 q^{91} +6.90833 q^{92} +1.00000 q^{93} +2.60555 q^{94} +0.302776 q^{96} -12.4222 q^{97} +14.2111 q^{98} -3.78890 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} - 3 q^{3} + 2 q^{4} - 3 q^{6} - 2 q^{7} + 2 q^{8} + 5 q^{9} + O(q^{10}) \) \( 2 q + 2 q^{2} - 3 q^{3} + 2 q^{4} - 3 q^{6} - 2 q^{7} + 2 q^{8} + 5 q^{9} - q^{11} - 3 q^{12} + q^{13} - 2 q^{14} + 2 q^{16} + 12 q^{17} + 5 q^{18} + 4 q^{19} - 10 q^{21} - q^{22} + 3 q^{23} - 3 q^{24} + q^{26} - 18 q^{27} - 2 q^{28} + 3 q^{29} + 3 q^{31} + 2 q^{32} + 8 q^{33} + 12 q^{34} + 5 q^{36} - 2 q^{37} + 4 q^{38} + 5 q^{39} + 9 q^{41} - 10 q^{42} + 6 q^{43} - q^{44} + 3 q^{46} - 2 q^{47} - 3 q^{48} + 14 q^{49} - 18 q^{51} + q^{52} + 12 q^{53} - 18 q^{54} - 2 q^{56} - 6 q^{57} + 3 q^{58} + 14 q^{59} - 3 q^{61} + 3 q^{62} + 34 q^{63} + 2 q^{64} + 8 q^{66} - 11 q^{67} + 12 q^{68} + 15 q^{69} + 12 q^{71} + 5 q^{72} + 21 q^{73} - 2 q^{74} + 4 q^{76} - 12 q^{77} + 5 q^{78} - 7 q^{79} + 38 q^{81} + 9 q^{82} - 20 q^{83} - 10 q^{84} + 6 q^{86} + 15 q^{87} - q^{88} - 4 q^{89} - 14 q^{91} + 3 q^{92} + 2 q^{93} - 2 q^{94} - 3 q^{96} + 4 q^{97} + 14 q^{98} - 22 q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) 0.302776 0.174808 0.0874038 0.996173i \(-0.472143\pi\)
0.0874038 + 0.996173i \(0.472143\pi\)
\(4\) 1.00000 0.500000
\(5\) 0 0
\(6\) 0.302776 0.123608
\(7\) −4.60555 −1.74073 −0.870367 0.492403i \(-0.836119\pi\)
−0.870367 + 0.492403i \(0.836119\pi\)
\(8\) 1.00000 0.353553
\(9\) −2.90833 −0.969442
\(10\) 0 0
\(11\) 1.30278 0.392802 0.196401 0.980524i \(-0.437075\pi\)
0.196401 + 0.980524i \(0.437075\pi\)
\(12\) 0.302776 0.0874038
\(13\) 2.30278 0.638675 0.319338 0.947641i \(-0.396540\pi\)
0.319338 + 0.947641i \(0.396540\pi\)
\(14\) −4.60555 −1.23089
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 6.00000 1.45521 0.727607 0.685994i \(-0.240633\pi\)
0.727607 + 0.685994i \(0.240633\pi\)
\(18\) −2.90833 −0.685499
\(19\) 2.00000 0.458831 0.229416 0.973329i \(-0.426318\pi\)
0.229416 + 0.973329i \(0.426318\pi\)
\(20\) 0 0
\(21\) −1.39445 −0.304294
\(22\) 1.30278 0.277753
\(23\) 6.90833 1.44049 0.720243 0.693722i \(-0.244030\pi\)
0.720243 + 0.693722i \(0.244030\pi\)
\(24\) 0.302776 0.0618038
\(25\) 0 0
\(26\) 2.30278 0.451611
\(27\) −1.78890 −0.344273
\(28\) −4.60555 −0.870367
\(29\) 6.90833 1.28284 0.641422 0.767188i \(-0.278345\pi\)
0.641422 + 0.767188i \(0.278345\pi\)
\(30\) 0 0
\(31\) 3.30278 0.593196 0.296598 0.955002i \(-0.404148\pi\)
0.296598 + 0.955002i \(0.404148\pi\)
\(32\) 1.00000 0.176777
\(33\) 0.394449 0.0686647
\(34\) 6.00000 1.02899
\(35\) 0 0
\(36\) −2.90833 −0.484721
\(37\) −1.00000 −0.164399
\(38\) 2.00000 0.324443
\(39\) 0.697224 0.111645
\(40\) 0 0
\(41\) −0.908327 −0.141857 −0.0709284 0.997481i \(-0.522596\pi\)
−0.0709284 + 0.997481i \(0.522596\pi\)
\(42\) −1.39445 −0.215168
\(43\) 6.60555 1.00734 0.503669 0.863897i \(-0.331983\pi\)
0.503669 + 0.863897i \(0.331983\pi\)
\(44\) 1.30278 0.196401
\(45\) 0 0
\(46\) 6.90833 1.01858
\(47\) 2.60555 0.380059 0.190029 0.981778i \(-0.439142\pi\)
0.190029 + 0.981778i \(0.439142\pi\)
\(48\) 0.302776 0.0437019
\(49\) 14.2111 2.03016
\(50\) 0 0
\(51\) 1.81665 0.254382
\(52\) 2.30278 0.319338
\(53\) 6.00000 0.824163 0.412082 0.911147i \(-0.364802\pi\)
0.412082 + 0.911147i \(0.364802\pi\)
\(54\) −1.78890 −0.243438
\(55\) 0 0
\(56\) −4.60555 −0.615443
\(57\) 0.605551 0.0802072
\(58\) 6.90833 0.907108
\(59\) 3.39445 0.441920 0.220960 0.975283i \(-0.429081\pi\)
0.220960 + 0.975283i \(0.429081\pi\)
\(60\) 0 0
\(61\) −10.5139 −1.34616 −0.673082 0.739568i \(-0.735030\pi\)
−0.673082 + 0.739568i \(0.735030\pi\)
\(62\) 3.30278 0.419453
\(63\) 13.3944 1.68754
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 0.394449 0.0485533
\(67\) −14.5139 −1.77315 −0.886576 0.462583i \(-0.846923\pi\)
−0.886576 + 0.462583i \(0.846923\pi\)
\(68\) 6.00000 0.727607
\(69\) 2.09167 0.251808
\(70\) 0 0
\(71\) 6.00000 0.712069 0.356034 0.934473i \(-0.384129\pi\)
0.356034 + 0.934473i \(0.384129\pi\)
\(72\) −2.90833 −0.342750
\(73\) 8.69722 1.01793 0.508967 0.860786i \(-0.330028\pi\)
0.508967 + 0.860786i \(0.330028\pi\)
\(74\) −1.00000 −0.116248
\(75\) 0 0
\(76\) 2.00000 0.229416
\(77\) −6.00000 −0.683763
\(78\) 0.697224 0.0789451
\(79\) −16.1194 −1.81358 −0.906789 0.421585i \(-0.861474\pi\)
−0.906789 + 0.421585i \(0.861474\pi\)
\(80\) 0 0
\(81\) 8.18335 0.909261
\(82\) −0.908327 −0.100308
\(83\) −17.2111 −1.88916 −0.944582 0.328276i \(-0.893533\pi\)
−0.944582 + 0.328276i \(0.893533\pi\)
\(84\) −1.39445 −0.152147
\(85\) 0 0
\(86\) 6.60555 0.712295
\(87\) 2.09167 0.224251
\(88\) 1.30278 0.138876
\(89\) 5.21110 0.552376 0.276188 0.961104i \(-0.410929\pi\)
0.276188 + 0.961104i \(0.410929\pi\)
\(90\) 0 0
\(91\) −10.6056 −1.11176
\(92\) 6.90833 0.720243
\(93\) 1.00000 0.103695
\(94\) 2.60555 0.268742
\(95\) 0 0
\(96\) 0.302776 0.0309019
\(97\) −12.4222 −1.26128 −0.630642 0.776074i \(-0.717208\pi\)
−0.630642 + 0.776074i \(0.717208\pi\)
\(98\) 14.2111 1.43554
\(99\) −3.78890 −0.380799
\(100\) 0 0
\(101\) 16.4222 1.63407 0.817035 0.576588i \(-0.195616\pi\)
0.817035 + 0.576588i \(0.195616\pi\)
\(102\) 1.81665 0.179876
\(103\) −3.30278 −0.325432 −0.162716 0.986673i \(-0.552025\pi\)
−0.162716 + 0.986673i \(0.552025\pi\)
\(104\) 2.30278 0.225806
\(105\) 0 0
\(106\) 6.00000 0.582772
\(107\) −4.30278 −0.415965 −0.207983 0.978133i \(-0.566690\pi\)
−0.207983 + 0.978133i \(0.566690\pi\)
\(108\) −1.78890 −0.172137
\(109\) 2.00000 0.191565 0.0957826 0.995402i \(-0.469465\pi\)
0.0957826 + 0.995402i \(0.469465\pi\)
\(110\) 0 0
\(111\) −0.302776 −0.0287382
\(112\) −4.60555 −0.435184
\(113\) −11.2111 −1.05465 −0.527326 0.849663i \(-0.676805\pi\)
−0.527326 + 0.849663i \(0.676805\pi\)
\(114\) 0.605551 0.0567151
\(115\) 0 0
\(116\) 6.90833 0.641422
\(117\) −6.69722 −0.619159
\(118\) 3.39445 0.312484
\(119\) −27.6333 −2.53314
\(120\) 0 0
\(121\) −9.30278 −0.845707
\(122\) −10.5139 −0.951882
\(123\) −0.275019 −0.0247977
\(124\) 3.30278 0.296598
\(125\) 0 0
\(126\) 13.3944 1.19327
\(127\) 4.78890 0.424946 0.212473 0.977167i \(-0.431848\pi\)
0.212473 + 0.977167i \(0.431848\pi\)
\(128\) 1.00000 0.0883883
\(129\) 2.00000 0.176090
\(130\) 0 0
\(131\) 3.39445 0.296574 0.148287 0.988944i \(-0.452624\pi\)
0.148287 + 0.988944i \(0.452624\pi\)
\(132\) 0.394449 0.0343324
\(133\) −9.21110 −0.798704
\(134\) −14.5139 −1.25381
\(135\) 0 0
\(136\) 6.00000 0.514496
\(137\) 9.90833 0.846525 0.423263 0.906007i \(-0.360885\pi\)
0.423263 + 0.906007i \(0.360885\pi\)
\(138\) 2.09167 0.178055
\(139\) 8.90833 0.755594 0.377797 0.925888i \(-0.376682\pi\)
0.377797 + 0.925888i \(0.376682\pi\)
\(140\) 0 0
\(141\) 0.788897 0.0664372
\(142\) 6.00000 0.503509
\(143\) 3.00000 0.250873
\(144\) −2.90833 −0.242361
\(145\) 0 0
\(146\) 8.69722 0.719787
\(147\) 4.30278 0.354887
\(148\) −1.00000 −0.0821995
\(149\) −1.81665 −0.148826 −0.0744130 0.997228i \(-0.523708\pi\)
−0.0744130 + 0.997228i \(0.523708\pi\)
\(150\) 0 0
\(151\) −13.3944 −1.09002 −0.545012 0.838428i \(-0.683475\pi\)
−0.545012 + 0.838428i \(0.683475\pi\)
\(152\) 2.00000 0.162221
\(153\) −17.4500 −1.41075
\(154\) −6.00000 −0.483494
\(155\) 0 0
\(156\) 0.697224 0.0558226
\(157\) −7.21110 −0.575509 −0.287754 0.957704i \(-0.592909\pi\)
−0.287754 + 0.957704i \(0.592909\pi\)
\(158\) −16.1194 −1.28239
\(159\) 1.81665 0.144070
\(160\) 0 0
\(161\) −31.8167 −2.50750
\(162\) 8.18335 0.642944
\(163\) 20.4222 1.59959 0.799795 0.600273i \(-0.204941\pi\)
0.799795 + 0.600273i \(0.204941\pi\)
\(164\) −0.908327 −0.0709284
\(165\) 0 0
\(166\) −17.2111 −1.33584
\(167\) −12.5139 −0.968353 −0.484176 0.874970i \(-0.660881\pi\)
−0.484176 + 0.874970i \(0.660881\pi\)
\(168\) −1.39445 −0.107584
\(169\) −7.69722 −0.592094
\(170\) 0 0
\(171\) −5.81665 −0.444811
\(172\) 6.60555 0.503669
\(173\) 23.2111 1.76471 0.882354 0.470587i \(-0.155958\pi\)
0.882354 + 0.470587i \(0.155958\pi\)
\(174\) 2.09167 0.158569
\(175\) 0 0
\(176\) 1.30278 0.0982004
\(177\) 1.02776 0.0772509
\(178\) 5.21110 0.390589
\(179\) 7.81665 0.584244 0.292122 0.956381i \(-0.405639\pi\)
0.292122 + 0.956381i \(0.405639\pi\)
\(180\) 0 0
\(181\) 20.0000 1.48659 0.743294 0.668965i \(-0.233262\pi\)
0.743294 + 0.668965i \(0.233262\pi\)
\(182\) −10.6056 −0.786136
\(183\) −3.18335 −0.235320
\(184\) 6.90833 0.509289
\(185\) 0 0
\(186\) 1.00000 0.0733236
\(187\) 7.81665 0.571610
\(188\) 2.60555 0.190029
\(189\) 8.23886 0.599289
\(190\) 0 0
\(191\) 12.5139 0.905472 0.452736 0.891644i \(-0.350448\pi\)
0.452736 + 0.891644i \(0.350448\pi\)
\(192\) 0.302776 0.0218509
\(193\) 4.00000 0.287926 0.143963 0.989583i \(-0.454015\pi\)
0.143963 + 0.989583i \(0.454015\pi\)
\(194\) −12.4222 −0.891862
\(195\) 0 0
\(196\) 14.2111 1.01508
\(197\) 6.00000 0.427482 0.213741 0.976890i \(-0.431435\pi\)
0.213741 + 0.976890i \(0.431435\pi\)
\(198\) −3.78890 −0.269265
\(199\) −2.42221 −0.171706 −0.0858528 0.996308i \(-0.527361\pi\)
−0.0858528 + 0.996308i \(0.527361\pi\)
\(200\) 0 0
\(201\) −4.39445 −0.309961
\(202\) 16.4222 1.15546
\(203\) −31.8167 −2.23309
\(204\) 1.81665 0.127191
\(205\) 0 0
\(206\) −3.30278 −0.230115
\(207\) −20.0917 −1.39647
\(208\) 2.30278 0.159669
\(209\) 2.60555 0.180230
\(210\) 0 0
\(211\) 6.69722 0.461056 0.230528 0.973066i \(-0.425955\pi\)
0.230528 + 0.973066i \(0.425955\pi\)
\(212\) 6.00000 0.412082
\(213\) 1.81665 0.124475
\(214\) −4.30278 −0.294132
\(215\) 0 0
\(216\) −1.78890 −0.121719
\(217\) −15.2111 −1.03260
\(218\) 2.00000 0.135457
\(219\) 2.63331 0.177942
\(220\) 0 0
\(221\) 13.8167 0.929409
\(222\) −0.302776 −0.0203210
\(223\) −15.8167 −1.05916 −0.529581 0.848260i \(-0.677651\pi\)
−0.529581 + 0.848260i \(0.677651\pi\)
\(224\) −4.60555 −0.307721
\(225\) 0 0
\(226\) −11.2111 −0.745751
\(227\) 7.81665 0.518810 0.259405 0.965769i \(-0.416474\pi\)
0.259405 + 0.965769i \(0.416474\pi\)
\(228\) 0.605551 0.0401036
\(229\) 17.3944 1.14946 0.574729 0.818344i \(-0.305108\pi\)
0.574729 + 0.818344i \(0.305108\pi\)
\(230\) 0 0
\(231\) −1.81665 −0.119527
\(232\) 6.90833 0.453554
\(233\) 9.51388 0.623275 0.311637 0.950201i \(-0.399123\pi\)
0.311637 + 0.950201i \(0.399123\pi\)
\(234\) −6.69722 −0.437811
\(235\) 0 0
\(236\) 3.39445 0.220960
\(237\) −4.88057 −0.317027
\(238\) −27.6333 −1.79120
\(239\) 0.513878 0.0332400 0.0166200 0.999862i \(-0.494709\pi\)
0.0166200 + 0.999862i \(0.494709\pi\)
\(240\) 0 0
\(241\) 8.00000 0.515325 0.257663 0.966235i \(-0.417048\pi\)
0.257663 + 0.966235i \(0.417048\pi\)
\(242\) −9.30278 −0.598005
\(243\) 7.84441 0.503219
\(244\) −10.5139 −0.673082
\(245\) 0 0
\(246\) −0.275019 −0.0175346
\(247\) 4.60555 0.293044
\(248\) 3.30278 0.209726
\(249\) −5.21110 −0.330240
\(250\) 0 0
\(251\) −6.78890 −0.428511 −0.214256 0.976778i \(-0.568733\pi\)
−0.214256 + 0.976778i \(0.568733\pi\)
\(252\) 13.3944 0.843771
\(253\) 9.00000 0.565825
\(254\) 4.78890 0.300482
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 11.2111 0.699329 0.349665 0.936875i \(-0.386296\pi\)
0.349665 + 0.936875i \(0.386296\pi\)
\(258\) 2.00000 0.124515
\(259\) 4.60555 0.286175
\(260\) 0 0
\(261\) −20.0917 −1.24364
\(262\) 3.39445 0.209710
\(263\) 7.81665 0.481996 0.240998 0.970526i \(-0.422525\pi\)
0.240998 + 0.970526i \(0.422525\pi\)
\(264\) 0.394449 0.0242766
\(265\) 0 0
\(266\) −9.21110 −0.564769
\(267\) 1.57779 0.0965595
\(268\) −14.5139 −0.886576
\(269\) −6.78890 −0.413926 −0.206963 0.978349i \(-0.566358\pi\)
−0.206963 + 0.978349i \(0.566358\pi\)
\(270\) 0 0
\(271\) 6.42221 0.390121 0.195061 0.980791i \(-0.437510\pi\)
0.195061 + 0.980791i \(0.437510\pi\)
\(272\) 6.00000 0.363803
\(273\) −3.21110 −0.194345
\(274\) 9.90833 0.598584
\(275\) 0 0
\(276\) 2.09167 0.125904
\(277\) 25.1194 1.50928 0.754640 0.656139i \(-0.227811\pi\)
0.754640 + 0.656139i \(0.227811\pi\)
\(278\) 8.90833 0.534286
\(279\) −9.60555 −0.575069
\(280\) 0 0
\(281\) −12.0000 −0.715860 −0.357930 0.933748i \(-0.616517\pi\)
−0.357930 + 0.933748i \(0.616517\pi\)
\(282\) 0.788897 0.0469782
\(283\) −17.3944 −1.03399 −0.516996 0.855988i \(-0.672950\pi\)
−0.516996 + 0.855988i \(0.672950\pi\)
\(284\) 6.00000 0.356034
\(285\) 0 0
\(286\) 3.00000 0.177394
\(287\) 4.18335 0.246935
\(288\) −2.90833 −0.171375
\(289\) 19.0000 1.11765
\(290\) 0 0
\(291\) −3.76114 −0.220482
\(292\) 8.69722 0.508967
\(293\) 25.0278 1.46214 0.731069 0.682304i \(-0.239022\pi\)
0.731069 + 0.682304i \(0.239022\pi\)
\(294\) 4.30278 0.250943
\(295\) 0 0
\(296\) −1.00000 −0.0581238
\(297\) −2.33053 −0.135231
\(298\) −1.81665 −0.105236
\(299\) 15.9083 0.920002
\(300\) 0 0
\(301\) −30.4222 −1.75351
\(302\) −13.3944 −0.770764
\(303\) 4.97224 0.285648
\(304\) 2.00000 0.114708
\(305\) 0 0
\(306\) −17.4500 −0.997548
\(307\) −7.09167 −0.404743 −0.202372 0.979309i \(-0.564865\pi\)
−0.202372 + 0.979309i \(0.564865\pi\)
\(308\) −6.00000 −0.341882
\(309\) −1.00000 −0.0568880
\(310\) 0 0
\(311\) 5.09167 0.288722 0.144361 0.989525i \(-0.453887\pi\)
0.144361 + 0.989525i \(0.453887\pi\)
\(312\) 0.697224 0.0394726
\(313\) −27.0278 −1.52770 −0.763850 0.645394i \(-0.776693\pi\)
−0.763850 + 0.645394i \(0.776693\pi\)
\(314\) −7.21110 −0.406946
\(315\) 0 0
\(316\) −16.1194 −0.906789
\(317\) 5.21110 0.292685 0.146342 0.989234i \(-0.453250\pi\)
0.146342 + 0.989234i \(0.453250\pi\)
\(318\) 1.81665 0.101873
\(319\) 9.00000 0.503903
\(320\) 0 0
\(321\) −1.30278 −0.0727138
\(322\) −31.8167 −1.77307
\(323\) 12.0000 0.667698
\(324\) 8.18335 0.454630
\(325\) 0 0
\(326\) 20.4222 1.13108
\(327\) 0.605551 0.0334871
\(328\) −0.908327 −0.0501540
\(329\) −12.0000 −0.661581
\(330\) 0 0
\(331\) 1.21110 0.0665682 0.0332841 0.999446i \(-0.489403\pi\)
0.0332841 + 0.999446i \(0.489403\pi\)
\(332\) −17.2111 −0.944582
\(333\) 2.90833 0.159375
\(334\) −12.5139 −0.684729
\(335\) 0 0
\(336\) −1.39445 −0.0760734
\(337\) 19.1194 1.04150 0.520751 0.853709i \(-0.325652\pi\)
0.520751 + 0.853709i \(0.325652\pi\)
\(338\) −7.69722 −0.418674
\(339\) −3.39445 −0.184361
\(340\) 0 0
\(341\) 4.30278 0.233008
\(342\) −5.81665 −0.314529
\(343\) −33.2111 −1.79323
\(344\) 6.60555 0.356147
\(345\) 0 0
\(346\) 23.2111 1.24784
\(347\) −31.8167 −1.70801 −0.854004 0.520267i \(-0.825832\pi\)
−0.854004 + 0.520267i \(0.825832\pi\)
\(348\) 2.09167 0.112125
\(349\) −22.2389 −1.19042 −0.595209 0.803571i \(-0.702931\pi\)
−0.595209 + 0.803571i \(0.702931\pi\)
\(350\) 0 0
\(351\) −4.11943 −0.219879
\(352\) 1.30278 0.0694382
\(353\) −31.8167 −1.69343 −0.846715 0.532047i \(-0.821423\pi\)
−0.846715 + 0.532047i \(0.821423\pi\)
\(354\) 1.02776 0.0546246
\(355\) 0 0
\(356\) 5.21110 0.276188
\(357\) −8.36669 −0.442812
\(358\) 7.81665 0.413123
\(359\) −11.2111 −0.591699 −0.295850 0.955235i \(-0.595603\pi\)
−0.295850 + 0.955235i \(0.595603\pi\)
\(360\) 0 0
\(361\) −15.0000 −0.789474
\(362\) 20.0000 1.05118
\(363\) −2.81665 −0.147836
\(364\) −10.6056 −0.555882
\(365\) 0 0
\(366\) −3.18335 −0.166396
\(367\) 17.8167 0.930022 0.465011 0.885305i \(-0.346050\pi\)
0.465011 + 0.885305i \(0.346050\pi\)
\(368\) 6.90833 0.360121
\(369\) 2.64171 0.137522
\(370\) 0 0
\(371\) −27.6333 −1.43465
\(372\) 1.00000 0.0518476
\(373\) −3.81665 −0.197619 −0.0988094 0.995106i \(-0.531503\pi\)
−0.0988094 + 0.995106i \(0.531503\pi\)
\(374\) 7.81665 0.404190
\(375\) 0 0
\(376\) 2.60555 0.134371
\(377\) 15.9083 0.819321
\(378\) 8.23886 0.423761
\(379\) −15.3305 −0.787477 −0.393738 0.919223i \(-0.628818\pi\)
−0.393738 + 0.919223i \(0.628818\pi\)
\(380\) 0 0
\(381\) 1.44996 0.0742838
\(382\) 12.5139 0.640266
\(383\) −20.8444 −1.06510 −0.532550 0.846399i \(-0.678766\pi\)
−0.532550 + 0.846399i \(0.678766\pi\)
\(384\) 0.302776 0.0154510
\(385\) 0 0
\(386\) 4.00000 0.203595
\(387\) −19.2111 −0.976555
\(388\) −12.4222 −0.630642
\(389\) −11.8806 −0.602369 −0.301184 0.953566i \(-0.597382\pi\)
−0.301184 + 0.953566i \(0.597382\pi\)
\(390\) 0 0
\(391\) 41.4500 2.09621
\(392\) 14.2111 0.717769
\(393\) 1.02776 0.0518435
\(394\) 6.00000 0.302276
\(395\) 0 0
\(396\) −3.78890 −0.190399
\(397\) −27.8167 −1.39608 −0.698039 0.716060i \(-0.745944\pi\)
−0.698039 + 0.716060i \(0.745944\pi\)
\(398\) −2.42221 −0.121414
\(399\) −2.78890 −0.139620
\(400\) 0 0
\(401\) 13.8167 0.689971 0.344985 0.938608i \(-0.387884\pi\)
0.344985 + 0.938608i \(0.387884\pi\)
\(402\) −4.39445 −0.219175
\(403\) 7.60555 0.378859
\(404\) 16.4222 0.817035
\(405\) 0 0
\(406\) −31.8167 −1.57903
\(407\) −1.30278 −0.0645762
\(408\) 1.81665 0.0899378
\(409\) −5.02776 −0.248607 −0.124303 0.992244i \(-0.539670\pi\)
−0.124303 + 0.992244i \(0.539670\pi\)
\(410\) 0 0
\(411\) 3.00000 0.147979
\(412\) −3.30278 −0.162716
\(413\) −15.6333 −0.769265
\(414\) −20.0917 −0.987452
\(415\) 0 0
\(416\) 2.30278 0.112903
\(417\) 2.69722 0.132084
\(418\) 2.60555 0.127442
\(419\) −25.1472 −1.22852 −0.614260 0.789104i \(-0.710545\pi\)
−0.614260 + 0.789104i \(0.710545\pi\)
\(420\) 0 0
\(421\) 28.7250 1.39997 0.699985 0.714158i \(-0.253190\pi\)
0.699985 + 0.714158i \(0.253190\pi\)
\(422\) 6.69722 0.326016
\(423\) −7.57779 −0.368445
\(424\) 6.00000 0.291386
\(425\) 0 0
\(426\) 1.81665 0.0880172
\(427\) 48.4222 2.34331
\(428\) −4.30278 −0.207983
\(429\) 0.908327 0.0438544
\(430\) 0 0
\(431\) −5.21110 −0.251010 −0.125505 0.992093i \(-0.540055\pi\)
−0.125505 + 0.992093i \(0.540055\pi\)
\(432\) −1.78890 −0.0860684
\(433\) 11.9361 0.573612 0.286806 0.957989i \(-0.407407\pi\)
0.286806 + 0.957989i \(0.407407\pi\)
\(434\) −15.2111 −0.730156
\(435\) 0 0
\(436\) 2.00000 0.0957826
\(437\) 13.8167 0.660940
\(438\) 2.63331 0.125824
\(439\) −9.33053 −0.445322 −0.222661 0.974896i \(-0.571474\pi\)
−0.222661 + 0.974896i \(0.571474\pi\)
\(440\) 0 0
\(441\) −41.3305 −1.96812
\(442\) 13.8167 0.657191
\(443\) −0.275019 −0.0130666 −0.00653328 0.999979i \(-0.502080\pi\)
−0.00653328 + 0.999979i \(0.502080\pi\)
\(444\) −0.302776 −0.0143691
\(445\) 0 0
\(446\) −15.8167 −0.748940
\(447\) −0.550039 −0.0260159
\(448\) −4.60555 −0.217592
\(449\) −0.788897 −0.0372304 −0.0186152 0.999827i \(-0.505926\pi\)
−0.0186152 + 0.999827i \(0.505926\pi\)
\(450\) 0 0
\(451\) −1.18335 −0.0557216
\(452\) −11.2111 −0.527326
\(453\) −4.05551 −0.190545
\(454\) 7.81665 0.366854
\(455\) 0 0
\(456\) 0.605551 0.0283575
\(457\) −4.60555 −0.215439 −0.107719 0.994181i \(-0.534355\pi\)
−0.107719 + 0.994181i \(0.534355\pi\)
\(458\) 17.3944 0.812789
\(459\) −10.7334 −0.500991
\(460\) 0 0
\(461\) −16.4222 −0.764858 −0.382429 0.923985i \(-0.624912\pi\)
−0.382429 + 0.923985i \(0.624912\pi\)
\(462\) −1.81665 −0.0845184
\(463\) −30.3028 −1.40829 −0.704145 0.710056i \(-0.748669\pi\)
−0.704145 + 0.710056i \(0.748669\pi\)
\(464\) 6.90833 0.320711
\(465\) 0 0
\(466\) 9.51388 0.440722
\(467\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(468\) −6.69722 −0.309579
\(469\) 66.8444 3.08659
\(470\) 0 0
\(471\) −2.18335 −0.100603
\(472\) 3.39445 0.156242
\(473\) 8.60555 0.395684
\(474\) −4.88057 −0.224172
\(475\) 0 0
\(476\) −27.6333 −1.26657
\(477\) −17.4500 −0.798979
\(478\) 0.513878 0.0235042
\(479\) 12.1194 0.553751 0.276875 0.960906i \(-0.410701\pi\)
0.276875 + 0.960906i \(0.410701\pi\)
\(480\) 0 0
\(481\) −2.30278 −0.104998
\(482\) 8.00000 0.364390
\(483\) −9.63331 −0.438331
\(484\) −9.30278 −0.422853
\(485\) 0 0
\(486\) 7.84441 0.355830
\(487\) 22.7889 1.03266 0.516332 0.856389i \(-0.327297\pi\)
0.516332 + 0.856389i \(0.327297\pi\)
\(488\) −10.5139 −0.475941
\(489\) 6.18335 0.279621
\(490\) 0 0
\(491\) −14.7250 −0.664529 −0.332265 0.943186i \(-0.607813\pi\)
−0.332265 + 0.943186i \(0.607813\pi\)
\(492\) −0.275019 −0.0123988
\(493\) 41.4500 1.86681
\(494\) 4.60555 0.207214
\(495\) 0 0
\(496\) 3.30278 0.148299
\(497\) −27.6333 −1.23952
\(498\) −5.21110 −0.233515
\(499\) 8.23886 0.368822 0.184411 0.982849i \(-0.440962\pi\)
0.184411 + 0.982849i \(0.440962\pi\)
\(500\) 0 0
\(501\) −3.78890 −0.169275
\(502\) −6.78890 −0.303003
\(503\) 24.5139 1.09302 0.546510 0.837453i \(-0.315956\pi\)
0.546510 + 0.837453i \(0.315956\pi\)
\(504\) 13.3944 0.596636
\(505\) 0 0
\(506\) 9.00000 0.400099
\(507\) −2.33053 −0.103503
\(508\) 4.78890 0.212473
\(509\) −25.8167 −1.14430 −0.572152 0.820148i \(-0.693891\pi\)
−0.572152 + 0.820148i \(0.693891\pi\)
\(510\) 0 0
\(511\) −40.0555 −1.77195
\(512\) 1.00000 0.0441942
\(513\) −3.57779 −0.157964
\(514\) 11.2111 0.494501
\(515\) 0 0
\(516\) 2.00000 0.0880451
\(517\) 3.39445 0.149288
\(518\) 4.60555 0.202356
\(519\) 7.02776 0.308484
\(520\) 0 0
\(521\) −9.63331 −0.422043 −0.211021 0.977481i \(-0.567679\pi\)
−0.211021 + 0.977481i \(0.567679\pi\)
\(522\) −20.0917 −0.879389
\(523\) −32.2389 −1.40971 −0.704853 0.709353i \(-0.748987\pi\)
−0.704853 + 0.709353i \(0.748987\pi\)
\(524\) 3.39445 0.148287
\(525\) 0 0
\(526\) 7.81665 0.340822
\(527\) 19.8167 0.863227
\(528\) 0.394449 0.0171662
\(529\) 24.7250 1.07500
\(530\) 0 0
\(531\) −9.87217 −0.428416
\(532\) −9.21110 −0.399352
\(533\) −2.09167 −0.0906004
\(534\) 1.57779 0.0682779
\(535\) 0 0
\(536\) −14.5139 −0.626904
\(537\) 2.36669 0.102130
\(538\) −6.78890 −0.292690
\(539\) 18.5139 0.797449
\(540\) 0 0
\(541\) −20.9361 −0.900113 −0.450056 0.893000i \(-0.648596\pi\)
−0.450056 + 0.893000i \(0.648596\pi\)
\(542\) 6.42221 0.275857
\(543\) 6.05551 0.259867
\(544\) 6.00000 0.257248
\(545\) 0 0
\(546\) −3.21110 −0.137423
\(547\) 13.3944 0.572705 0.286353 0.958124i \(-0.407557\pi\)
0.286353 + 0.958124i \(0.407557\pi\)
\(548\) 9.90833 0.423263
\(549\) 30.5778 1.30503
\(550\) 0 0
\(551\) 13.8167 0.588609
\(552\) 2.09167 0.0890275
\(553\) 74.2389 3.15696
\(554\) 25.1194 1.06722
\(555\) 0 0
\(556\) 8.90833 0.377797
\(557\) 6.51388 0.276002 0.138001 0.990432i \(-0.455932\pi\)
0.138001 + 0.990432i \(0.455932\pi\)
\(558\) −9.60555 −0.406635
\(559\) 15.2111 0.643361
\(560\) 0 0
\(561\) 2.36669 0.0999218
\(562\) −12.0000 −0.506189
\(563\) −44.0555 −1.85672 −0.928359 0.371684i \(-0.878780\pi\)
−0.928359 + 0.371684i \(0.878780\pi\)
\(564\) 0.788897 0.0332186
\(565\) 0 0
\(566\) −17.3944 −0.731143
\(567\) −37.6888 −1.58278
\(568\) 6.00000 0.251754
\(569\) −10.4222 −0.436922 −0.218461 0.975846i \(-0.570104\pi\)
−0.218461 + 0.975846i \(0.570104\pi\)
\(570\) 0 0
\(571\) −20.3028 −0.849645 −0.424822 0.905277i \(-0.639663\pi\)
−0.424822 + 0.905277i \(0.639663\pi\)
\(572\) 3.00000 0.125436
\(573\) 3.78890 0.158283
\(574\) 4.18335 0.174609
\(575\) 0 0
\(576\) −2.90833 −0.121180
\(577\) 28.2389 1.17560 0.587800 0.809007i \(-0.299994\pi\)
0.587800 + 0.809007i \(0.299994\pi\)
\(578\) 19.0000 0.790296
\(579\) 1.21110 0.0503317
\(580\) 0 0
\(581\) 79.2666 3.28853
\(582\) −3.76114 −0.155904
\(583\) 7.81665 0.323733
\(584\) 8.69722 0.359894
\(585\) 0 0
\(586\) 25.0278 1.03389
\(587\) 2.36669 0.0976838 0.0488419 0.998807i \(-0.484447\pi\)
0.0488419 + 0.998807i \(0.484447\pi\)
\(588\) 4.30278 0.177443
\(589\) 6.60555 0.272177
\(590\) 0 0
\(591\) 1.81665 0.0747272
\(592\) −1.00000 −0.0410997
\(593\) −36.5139 −1.49945 −0.749723 0.661752i \(-0.769813\pi\)
−0.749723 + 0.661752i \(0.769813\pi\)
\(594\) −2.33053 −0.0956229
\(595\) 0 0
\(596\) −1.81665 −0.0744130
\(597\) −0.733385 −0.0300154
\(598\) 15.9083 0.650540
\(599\) −35.2111 −1.43869 −0.719343 0.694655i \(-0.755557\pi\)
−0.719343 + 0.694655i \(0.755557\pi\)
\(600\) 0 0
\(601\) −20.6972 −0.844257 −0.422129 0.906536i \(-0.638717\pi\)
−0.422129 + 0.906536i \(0.638717\pi\)
\(602\) −30.4222 −1.23992
\(603\) 42.2111 1.71897
\(604\) −13.3944 −0.545012
\(605\) 0 0
\(606\) 4.97224 0.201984
\(607\) 31.5139 1.27911 0.639554 0.768746i \(-0.279119\pi\)
0.639554 + 0.768746i \(0.279119\pi\)
\(608\) 2.00000 0.0811107
\(609\) −9.63331 −0.390361
\(610\) 0 0
\(611\) 6.00000 0.242734
\(612\) −17.4500 −0.705373
\(613\) 8.18335 0.330522 0.165261 0.986250i \(-0.447153\pi\)
0.165261 + 0.986250i \(0.447153\pi\)
\(614\) −7.09167 −0.286197
\(615\) 0 0
\(616\) −6.00000 −0.241747
\(617\) −47.5694 −1.91507 −0.957536 0.288314i \(-0.906905\pi\)
−0.957536 + 0.288314i \(0.906905\pi\)
\(618\) −1.00000 −0.0402259
\(619\) −2.69722 −0.108411 −0.0542053 0.998530i \(-0.517263\pi\)
−0.0542053 + 0.998530i \(0.517263\pi\)
\(620\) 0 0
\(621\) −12.3583 −0.495921
\(622\) 5.09167 0.204157
\(623\) −24.0000 −0.961540
\(624\) 0.697224 0.0279113
\(625\) 0 0
\(626\) −27.0278 −1.08025
\(627\) 0.788897 0.0315055
\(628\) −7.21110 −0.287754
\(629\) −6.00000 −0.239236
\(630\) 0 0
\(631\) 18.3028 0.728622 0.364311 0.931277i \(-0.381305\pi\)
0.364311 + 0.931277i \(0.381305\pi\)
\(632\) −16.1194 −0.641196
\(633\) 2.02776 0.0805961
\(634\) 5.21110 0.206959
\(635\) 0 0
\(636\) 1.81665 0.0720350
\(637\) 32.7250 1.29661
\(638\) 9.00000 0.356313
\(639\) −17.4500 −0.690310
\(640\) 0 0
\(641\) −2.48612 −0.0981959 −0.0490980 0.998794i \(-0.515635\pi\)
−0.0490980 + 0.998794i \(0.515635\pi\)
\(642\) −1.30278 −0.0514165
\(643\) 29.8167 1.17585 0.587927 0.808914i \(-0.299944\pi\)
0.587927 + 0.808914i \(0.299944\pi\)
\(644\) −31.8167 −1.25375
\(645\) 0 0
\(646\) 12.0000 0.472134
\(647\) 25.9361 1.01965 0.509826 0.860277i \(-0.329710\pi\)
0.509826 + 0.860277i \(0.329710\pi\)
\(648\) 8.18335 0.321472
\(649\) 4.42221 0.173587
\(650\) 0 0
\(651\) −4.60555 −0.180506
\(652\) 20.4222 0.799795
\(653\) −6.90833 −0.270344 −0.135172 0.990822i \(-0.543159\pi\)
−0.135172 + 0.990822i \(0.543159\pi\)
\(654\) 0.605551 0.0236789
\(655\) 0 0
\(656\) −0.908327 −0.0354642
\(657\) −25.2944 −0.986827
\(658\) −12.0000 −0.467809
\(659\) −42.1194 −1.64074 −0.820370 0.571833i \(-0.806233\pi\)
−0.820370 + 0.571833i \(0.806233\pi\)
\(660\) 0 0
\(661\) −12.4861 −0.485654 −0.242827 0.970070i \(-0.578075\pi\)
−0.242827 + 0.970070i \(0.578075\pi\)
\(662\) 1.21110 0.0470708
\(663\) 4.18335 0.162468
\(664\) −17.2111 −0.667920
\(665\) 0 0
\(666\) 2.90833 0.112695
\(667\) 47.7250 1.84792
\(668\) −12.5139 −0.484176
\(669\) −4.78890 −0.185149
\(670\) 0 0
\(671\) −13.6972 −0.528775
\(672\) −1.39445 −0.0537920
\(673\) −24.3028 −0.936803 −0.468402 0.883516i \(-0.655170\pi\)
−0.468402 + 0.883516i \(0.655170\pi\)
\(674\) 19.1194 0.736453
\(675\) 0 0
\(676\) −7.69722 −0.296047
\(677\) 36.2389 1.39277 0.696386 0.717667i \(-0.254790\pi\)
0.696386 + 0.717667i \(0.254790\pi\)
\(678\) −3.39445 −0.130363
\(679\) 57.2111 2.19556
\(680\) 0 0
\(681\) 2.36669 0.0906918
\(682\) 4.30278 0.164762
\(683\) −12.0000 −0.459167 −0.229584 0.973289i \(-0.573736\pi\)
−0.229584 + 0.973289i \(0.573736\pi\)
\(684\) −5.81665 −0.222405
\(685\) 0 0
\(686\) −33.2111 −1.26801
\(687\) 5.26662 0.200934
\(688\) 6.60555 0.251834
\(689\) 13.8167 0.526373
\(690\) 0 0
\(691\) 8.00000 0.304334 0.152167 0.988355i \(-0.451375\pi\)
0.152167 + 0.988355i \(0.451375\pi\)
\(692\) 23.2111 0.882354
\(693\) 17.4500 0.662869
\(694\) −31.8167 −1.20774
\(695\) 0 0
\(696\) 2.09167 0.0792847
\(697\) −5.44996 −0.206432
\(698\) −22.2389 −0.841753
\(699\) 2.88057 0.108953
\(700\) 0 0
\(701\) 14.8806 0.562031 0.281016 0.959703i \(-0.409329\pi\)
0.281016 + 0.959703i \(0.409329\pi\)
\(702\) −4.11943 −0.155478
\(703\) −2.00000 −0.0754314
\(704\) 1.30278 0.0491002
\(705\) 0 0
\(706\) −31.8167 −1.19744
\(707\) −75.6333 −2.84448
\(708\) 1.02776 0.0386254
\(709\) −1.66947 −0.0626982 −0.0313491 0.999508i \(-0.509980\pi\)
−0.0313491 + 0.999508i \(0.509980\pi\)
\(710\) 0 0
\(711\) 46.8806 1.75816
\(712\) 5.21110 0.195294
\(713\) 22.8167 0.854490
\(714\) −8.36669 −0.313116
\(715\) 0 0
\(716\) 7.81665 0.292122
\(717\) 0.155590 0.00581061
\(718\) −11.2111 −0.418395
\(719\) −8.36669 −0.312025 −0.156012 0.987755i \(-0.549864\pi\)
−0.156012 + 0.987755i \(0.549864\pi\)
\(720\) 0 0
\(721\) 15.2111 0.566491
\(722\) −15.0000 −0.558242
\(723\) 2.42221 0.0900828
\(724\) 20.0000 0.743294
\(725\) 0 0
\(726\) −2.81665 −0.104536
\(727\) −29.9083 −1.10924 −0.554619 0.832104i \(-0.687136\pi\)
−0.554619 + 0.832104i \(0.687136\pi\)
\(728\) −10.6056 −0.393068
\(729\) −22.1749 −0.821294
\(730\) 0 0
\(731\) 39.6333 1.46589
\(732\) −3.18335 −0.117660
\(733\) −29.6333 −1.09453 −0.547266 0.836959i \(-0.684331\pi\)
−0.547266 + 0.836959i \(0.684331\pi\)
\(734\) 17.8167 0.657625
\(735\) 0 0
\(736\) 6.90833 0.254644
\(737\) −18.9083 −0.696497
\(738\) 2.64171 0.0972427
\(739\) −42.3305 −1.55715 −0.778577 0.627549i \(-0.784058\pi\)
−0.778577 + 0.627549i \(0.784058\pi\)
\(740\) 0 0
\(741\) 1.39445 0.0512264
\(742\) −27.6333 −1.01445
\(743\) −35.4500 −1.30053 −0.650266 0.759706i \(-0.725343\pi\)
−0.650266 + 0.759706i \(0.725343\pi\)
\(744\) 1.00000 0.0366618
\(745\) 0 0
\(746\) −3.81665 −0.139738
\(747\) 50.0555 1.83144
\(748\) 7.81665 0.285805
\(749\) 19.8167 0.724085
\(750\) 0 0
\(751\) 14.0000 0.510867 0.255434 0.966827i \(-0.417782\pi\)
0.255434 + 0.966827i \(0.417782\pi\)
\(752\) 2.60555 0.0950147
\(753\) −2.05551 −0.0749070
\(754\) 15.9083 0.579347
\(755\) 0 0
\(756\) 8.23886 0.299644
\(757\) −9.30278 −0.338115 −0.169058 0.985606i \(-0.554072\pi\)
−0.169058 + 0.985606i \(0.554072\pi\)
\(758\) −15.3305 −0.556830
\(759\) 2.72498 0.0989105
\(760\) 0 0
\(761\) 42.1194 1.52683 0.763414 0.645909i \(-0.223522\pi\)
0.763414 + 0.645909i \(0.223522\pi\)
\(762\) 1.44996 0.0525266
\(763\) −9.21110 −0.333464
\(764\) 12.5139 0.452736
\(765\) 0 0
\(766\) −20.8444 −0.753139
\(767\) 7.81665 0.282243
\(768\) 0.302776 0.0109255
\(769\) −22.0000 −0.793340 −0.396670 0.917961i \(-0.629834\pi\)
−0.396670 + 0.917961i \(0.629834\pi\)
\(770\) 0 0
\(771\) 3.39445 0.122248
\(772\) 4.00000 0.143963
\(773\) 50.0555 1.80037 0.900186 0.435506i \(-0.143431\pi\)
0.900186 + 0.435506i \(0.143431\pi\)
\(774\) −19.2111 −0.690529
\(775\) 0 0
\(776\) −12.4222 −0.445931
\(777\) 1.39445 0.0500256
\(778\) −11.8806 −0.425939
\(779\) −1.81665 −0.0650884
\(780\) 0 0
\(781\) 7.81665 0.279702
\(782\) 41.4500 1.48225
\(783\) −12.3583 −0.441649
\(784\) 14.2111 0.507539
\(785\) 0 0
\(786\) 1.02776 0.0366589
\(787\) −25.2111 −0.898679 −0.449339 0.893361i \(-0.648341\pi\)
−0.449339 + 0.893361i \(0.648341\pi\)
\(788\) 6.00000 0.213741
\(789\) 2.36669 0.0842565
\(790\) 0 0
\(791\) 51.6333 1.83587
\(792\) −3.78890 −0.134633
\(793\) −24.2111 −0.859761
\(794\) −27.8167 −0.987176
\(795\) 0 0
\(796\) −2.42221 −0.0858528
\(797\) 17.3305 0.613879 0.306939 0.951729i \(-0.400695\pi\)
0.306939 + 0.951729i \(0.400695\pi\)
\(798\) −2.78890 −0.0987259
\(799\) 15.6333 0.553067
\(800\) 0 0
\(801\) −15.1556 −0.535496
\(802\) 13.8167 0.487883
\(803\) 11.3305 0.399846
\(804\) −4.39445 −0.154980
\(805\) 0 0
\(806\) 7.60555 0.267894
\(807\) −2.05551 −0.0723575
\(808\) 16.4222 0.577731
\(809\) 29.4500 1.03541 0.517703 0.855561i \(-0.326787\pi\)
0.517703 + 0.855561i \(0.326787\pi\)
\(810\) 0 0
\(811\) 54.1472 1.90136 0.950682 0.310166i \(-0.100385\pi\)
0.950682 + 0.310166i \(0.100385\pi\)
\(812\) −31.8167 −1.11655
\(813\) 1.94449 0.0681961
\(814\) −1.30278 −0.0456623
\(815\) 0 0
\(816\) 1.81665 0.0635956
\(817\) 13.2111 0.462198
\(818\) −5.02776 −0.175791
\(819\) 30.8444 1.07779
\(820\) 0 0
\(821\) 11.2111 0.391270 0.195635 0.980677i \(-0.437323\pi\)
0.195635 + 0.980677i \(0.437323\pi\)
\(822\) 3.00000 0.104637
\(823\) 12.8444 0.447728 0.223864 0.974620i \(-0.428133\pi\)
0.223864 + 0.974620i \(0.428133\pi\)
\(824\) −3.30278 −0.115058
\(825\) 0 0
\(826\) −15.6333 −0.543952
\(827\) −27.3944 −0.952598 −0.476299 0.879283i \(-0.658022\pi\)
−0.476299 + 0.879283i \(0.658022\pi\)
\(828\) −20.0917 −0.698234
\(829\) 4.72498 0.164105 0.0820527 0.996628i \(-0.473852\pi\)
0.0820527 + 0.996628i \(0.473852\pi\)
\(830\) 0 0
\(831\) 7.60555 0.263834
\(832\) 2.30278 0.0798344
\(833\) 85.2666 2.95431
\(834\) 2.69722 0.0933972
\(835\) 0 0
\(836\) 2.60555 0.0901149
\(837\) −5.90833 −0.204222
\(838\) −25.1472 −0.868695
\(839\) −49.0278 −1.69263 −0.846313 0.532686i \(-0.821183\pi\)
−0.846313 + 0.532686i \(0.821183\pi\)
\(840\) 0 0
\(841\) 18.7250 0.645689
\(842\) 28.7250 0.989928
\(843\) −3.63331 −0.125138
\(844\) 6.69722 0.230528
\(845\) 0 0
\(846\) −7.57779 −0.260530
\(847\) 42.8444 1.47215
\(848\) 6.00000 0.206041
\(849\) −5.26662 −0.180750
\(850\) 0 0
\(851\) −6.90833 −0.236814
\(852\) 1.81665 0.0622375
\(853\) 11.5416 0.395178 0.197589 0.980285i \(-0.436689\pi\)
0.197589 + 0.980285i \(0.436689\pi\)
\(854\) 48.4222 1.65697
\(855\) 0 0
\(856\) −4.30278 −0.147066
\(857\) 14.8444 0.507075 0.253538 0.967326i \(-0.418406\pi\)
0.253538 + 0.967326i \(0.418406\pi\)
\(858\) 0.908327 0.0310098
\(859\) −24.0555 −0.820764 −0.410382 0.911914i \(-0.634605\pi\)
−0.410382 + 0.911914i \(0.634605\pi\)
\(860\) 0 0
\(861\) 1.26662 0.0431661
\(862\) −5.21110 −0.177491
\(863\) −12.0000 −0.408485 −0.204242 0.978920i \(-0.565473\pi\)
−0.204242 + 0.978920i \(0.565473\pi\)
\(864\) −1.78890 −0.0608595
\(865\) 0 0
\(866\) 11.9361 0.405605
\(867\) 5.75274 0.195373
\(868\) −15.2111 −0.516298
\(869\) −21.0000 −0.712376
\(870\) 0 0
\(871\) −33.4222 −1.13247
\(872\) 2.00000 0.0677285
\(873\) 36.1278 1.22274
\(874\) 13.8167 0.467355
\(875\) 0 0
\(876\) 2.63331 0.0889712
\(877\) −7.21110 −0.243502 −0.121751 0.992561i \(-0.538851\pi\)
−0.121751 + 0.992561i \(0.538851\pi\)
\(878\) −9.33053 −0.314890
\(879\) 7.57779 0.255593
\(880\) 0 0
\(881\) 25.5416 0.860520 0.430260 0.902705i \(-0.358422\pi\)
0.430260 + 0.902705i \(0.358422\pi\)
\(882\) −41.3305 −1.39167
\(883\) 2.42221 0.0815137 0.0407568 0.999169i \(-0.487023\pi\)
0.0407568 + 0.999169i \(0.487023\pi\)
\(884\) 13.8167 0.464704
\(885\) 0 0
\(886\) −0.275019 −0.00923945
\(887\) −28.4222 −0.954324 −0.477162 0.878815i \(-0.658335\pi\)
−0.477162 + 0.878815i \(0.658335\pi\)
\(888\) −0.302776 −0.0101605
\(889\) −22.0555 −0.739718
\(890\) 0 0
\(891\) 10.6611 0.357159
\(892\) −15.8167 −0.529581
\(893\) 5.21110 0.174383
\(894\) −0.550039 −0.0183960
\(895\) 0 0
\(896\) −4.60555 −0.153861
\(897\) 4.81665 0.160823
\(898\) −0.788897 −0.0263258
\(899\) 22.8167 0.760978
\(900\) 0 0
\(901\) 36.0000 1.19933
\(902\) −1.18335 −0.0394011
\(903\) −9.21110 −0.306526
\(904\) −11.2111 −0.372876
\(905\) 0 0
\(906\) −4.05551 −0.134735
\(907\) −26.0000 −0.863316 −0.431658 0.902037i \(-0.642071\pi\)
−0.431658 + 0.902037i \(0.642071\pi\)
\(908\) 7.81665 0.259405
\(909\) −47.7611 −1.58414
\(910\) 0 0
\(911\) −46.4222 −1.53804 −0.769018 0.639227i \(-0.779254\pi\)
−0.769018 + 0.639227i \(0.779254\pi\)
\(912\) 0.605551 0.0200518
\(913\) −22.4222 −0.742067
\(914\) −4.60555 −0.152338
\(915\) 0 0
\(916\) 17.3944 0.574729
\(917\) −15.6333 −0.516257
\(918\) −10.7334 −0.354254
\(919\) −38.4222 −1.26743 −0.633716 0.773566i \(-0.718471\pi\)
−0.633716 + 0.773566i \(0.718471\pi\)
\(920\) 0 0
\(921\) −2.14719 −0.0707522
\(922\) −16.4222 −0.540837
\(923\) 13.8167 0.454781
\(924\) −1.81665 −0.0597635
\(925\) 0 0
\(926\) −30.3028 −0.995811
\(927\) 9.60555 0.315488
\(928\) 6.90833 0.226777
\(929\) −36.5139 −1.19798 −0.598991 0.800756i \(-0.704431\pi\)
−0.598991 + 0.800756i \(0.704431\pi\)
\(930\) 0 0
\(931\) 28.4222 0.931500
\(932\) 9.51388 0.311637
\(933\) 1.54163 0.0504708
\(934\) 0 0
\(935\) 0 0
\(936\) −6.69722 −0.218906
\(937\) 28.9083 0.944394 0.472197 0.881493i \(-0.343461\pi\)
0.472197 + 0.881493i \(0.343461\pi\)
\(938\) 66.8444 2.18255
\(939\) −8.18335 −0.267053
\(940\) 0 0
\(941\) −7.81665 −0.254816 −0.127408 0.991850i \(-0.540666\pi\)
−0.127408 + 0.991850i \(0.540666\pi\)
\(942\) −2.18335 −0.0711373
\(943\) −6.27502 −0.204343
\(944\) 3.39445 0.110480
\(945\) 0 0
\(946\) 8.60555 0.279791
\(947\) −39.6333 −1.28791 −0.643955 0.765064i \(-0.722707\pi\)
−0.643955 + 0.765064i \(0.722707\pi\)
\(948\) −4.88057 −0.158514
\(949\) 20.0278 0.650128
\(950\) 0 0
\(951\) 1.57779 0.0511635
\(952\) −27.6333 −0.895601
\(953\) 18.7527 0.607461 0.303730 0.952758i \(-0.401768\pi\)
0.303730 + 0.952758i \(0.401768\pi\)
\(954\) −17.4500 −0.564963
\(955\) 0 0
\(956\) 0.513878 0.0166200
\(957\) 2.72498 0.0880861
\(958\) 12.1194 0.391561
\(959\) −45.6333 −1.47358
\(960\) 0 0
\(961\) −20.0917 −0.648118
\(962\) −2.30278 −0.0742445
\(963\) 12.5139 0.403254
\(964\) 8.00000 0.257663
\(965\) 0 0
\(966\) −9.63331 −0.309947
\(967\) −25.7250 −0.827260 −0.413630 0.910445i \(-0.635739\pi\)
−0.413630 + 0.910445i \(0.635739\pi\)
\(968\) −9.30278 −0.299003
\(969\) 3.63331 0.116719
\(970\) 0 0
\(971\) 31.5416 1.01222 0.506110 0.862469i \(-0.331083\pi\)
0.506110 + 0.862469i \(0.331083\pi\)
\(972\) 7.84441 0.251610
\(973\) −41.0278 −1.31529
\(974\) 22.7889 0.730203
\(975\) 0 0
\(976\) −10.5139 −0.336541
\(977\) −18.0000 −0.575871 −0.287936 0.957650i \(-0.592969\pi\)
−0.287936 + 0.957650i \(0.592969\pi\)
\(978\) 6.18335 0.197722
\(979\) 6.78890 0.216974
\(980\) 0 0
\(981\) −5.81665 −0.185711
\(982\) −14.7250 −0.469893
\(983\) 12.0000 0.382741 0.191370 0.981518i \(-0.438707\pi\)
0.191370 + 0.981518i \(0.438707\pi\)
\(984\) −0.275019 −0.00876729
\(985\) 0 0
\(986\) 41.4500 1.32004
\(987\) −3.63331 −0.115649
\(988\) 4.60555 0.146522
\(989\) 45.6333 1.45105
\(990\) 0 0
\(991\) 54.3028 1.72498 0.862492 0.506070i \(-0.168902\pi\)
0.862492 + 0.506070i \(0.168902\pi\)
\(992\) 3.30278 0.104863
\(993\) 0.366692 0.0116366
\(994\) −27.6333 −0.876475
\(995\) 0 0
\(996\) −5.21110 −0.165120
\(997\) 23.5778 0.746716 0.373358 0.927687i \(-0.378206\pi\)
0.373358 + 0.927687i \(0.378206\pi\)
\(998\) 8.23886 0.260797
\(999\) 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.a.u.1.2 2
5.2 odd 4 1850.2.b.i.149.3 4
5.3 odd 4 1850.2.b.i.149.2 4
5.4 even 2 74.2.a.a.1.1 2
15.14 odd 2 666.2.a.j.1.1 2
20.19 odd 2 592.2.a.f.1.2 2
35.34 odd 2 3626.2.a.a.1.2 2
40.19 odd 2 2368.2.a.ba.1.1 2
40.29 even 2 2368.2.a.s.1.2 2
55.54 odd 2 8954.2.a.p.1.1 2
60.59 even 2 5328.2.a.bf.1.1 2
185.184 even 2 2738.2.a.l.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
74.2.a.a.1.1 2 5.4 even 2
592.2.a.f.1.2 2 20.19 odd 2
666.2.a.j.1.1 2 15.14 odd 2
1850.2.a.u.1.2 2 1.1 even 1 trivial
1850.2.b.i.149.2 4 5.3 odd 4
1850.2.b.i.149.3 4 5.2 odd 4
2368.2.a.s.1.2 2 40.29 even 2
2368.2.a.ba.1.1 2 40.19 odd 2
2738.2.a.l.1.1 2 185.184 even 2
3626.2.a.a.1.2 2 35.34 odd 2
5328.2.a.bf.1.1 2 60.59 even 2
8954.2.a.p.1.1 2 55.54 odd 2