Properties

Label 7098.2.a.bg.1.2
Level $7098$
Weight $2$
Character 7098.1
Self dual yes
Analytic conductor $56.678$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 7098 = 2 \cdot 3 \cdot 7 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7098.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(56.6778153547\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{17}) \)
Defining polynomial: \(x^{2} - x - 4\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 546)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-1.56155\) of defining polynomial
Character \(\chi\) \(=\) 7098.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +0.561553 q^{5} +1.00000 q^{6} -1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +0.561553 q^{5} +1.00000 q^{6} -1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} -0.561553 q^{10} +1.43845 q^{11} -1.00000 q^{12} +1.00000 q^{14} -0.561553 q^{15} +1.00000 q^{16} -5.68466 q^{17} -1.00000 q^{18} -2.56155 q^{19} +0.561553 q^{20} +1.00000 q^{21} -1.43845 q^{22} -5.68466 q^{23} +1.00000 q^{24} -4.68466 q^{25} -1.00000 q^{27} -1.00000 q^{28} -2.56155 q^{29} +0.561553 q^{30} -10.2462 q^{31} -1.00000 q^{32} -1.43845 q^{33} +5.68466 q^{34} -0.561553 q^{35} +1.00000 q^{36} -1.68466 q^{37} +2.56155 q^{38} -0.561553 q^{40} -4.00000 q^{41} -1.00000 q^{42} +10.5616 q^{43} +1.43845 q^{44} +0.561553 q^{45} +5.68466 q^{46} +6.24621 q^{47} -1.00000 q^{48} +1.00000 q^{49} +4.68466 q^{50} +5.68466 q^{51} +13.1231 q^{53} +1.00000 q^{54} +0.807764 q^{55} +1.00000 q^{56} +2.56155 q^{57} +2.56155 q^{58} -12.2462 q^{59} -0.561553 q^{60} +2.56155 q^{61} +10.2462 q^{62} -1.00000 q^{63} +1.00000 q^{64} +1.43845 q^{66} -7.12311 q^{67} -5.68466 q^{68} +5.68466 q^{69} +0.561553 q^{70} +15.3693 q^{71} -1.00000 q^{72} +7.43845 q^{73} +1.68466 q^{74} +4.68466 q^{75} -2.56155 q^{76} -1.43845 q^{77} +16.0000 q^{79} +0.561553 q^{80} +1.00000 q^{81} +4.00000 q^{82} +2.00000 q^{83} +1.00000 q^{84} -3.19224 q^{85} -10.5616 q^{86} +2.56155 q^{87} -1.43845 q^{88} -8.00000 q^{89} -0.561553 q^{90} -5.68466 q^{92} +10.2462 q^{93} -6.24621 q^{94} -1.43845 q^{95} +1.00000 q^{96} -10.0000 q^{97} -1.00000 q^{98} +1.43845 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{2} - 2q^{3} + 2q^{4} - 3q^{5} + 2q^{6} - 2q^{7} - 2q^{8} + 2q^{9} + O(q^{10}) \) \( 2q - 2q^{2} - 2q^{3} + 2q^{4} - 3q^{5} + 2q^{6} - 2q^{7} - 2q^{8} + 2q^{9} + 3q^{10} + 7q^{11} - 2q^{12} + 2q^{14} + 3q^{15} + 2q^{16} + q^{17} - 2q^{18} - q^{19} - 3q^{20} + 2q^{21} - 7q^{22} + q^{23} + 2q^{24} + 3q^{25} - 2q^{27} - 2q^{28} - q^{29} - 3q^{30} - 4q^{31} - 2q^{32} - 7q^{33} - q^{34} + 3q^{35} + 2q^{36} + 9q^{37} + q^{38} + 3q^{40} - 8q^{41} - 2q^{42} + 17q^{43} + 7q^{44} - 3q^{45} - q^{46} - 4q^{47} - 2q^{48} + 2q^{49} - 3q^{50} - q^{51} + 18q^{53} + 2q^{54} - 19q^{55} + 2q^{56} + q^{57} + q^{58} - 8q^{59} + 3q^{60} + q^{61} + 4q^{62} - 2q^{63} + 2q^{64} + 7q^{66} - 6q^{67} + q^{68} - q^{69} - 3q^{70} + 6q^{71} - 2q^{72} + 19q^{73} - 9q^{74} - 3q^{75} - q^{76} - 7q^{77} + 32q^{79} - 3q^{80} + 2q^{81} + 8q^{82} + 4q^{83} + 2q^{84} - 27q^{85} - 17q^{86} + q^{87} - 7q^{88} - 16q^{89} + 3q^{90} + q^{92} + 4q^{93} + 4q^{94} - 7q^{95} + 2q^{96} - 20q^{97} - 2q^{98} + 7q^{99} + O(q^{100}) \)

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
\(3\) −1.00000 −0.577350
\(4\) 1.00000 0.500000
\(5\) 0.561553 0.251134 0.125567 0.992085i \(-0.459925\pi\)
0.125567 + 0.992085i \(0.459925\pi\)
\(6\) 1.00000 0.408248
\(7\) −1.00000 −0.377964
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) −0.561553 −0.177579
\(11\) 1.43845 0.433708 0.216854 0.976204i \(-0.430420\pi\)
0.216854 + 0.976204i \(0.430420\pi\)
\(12\) −1.00000 −0.288675
\(13\) 0 0
\(14\) 1.00000 0.267261
\(15\) −0.561553 −0.144992
\(16\) 1.00000 0.250000
\(17\) −5.68466 −1.37873 −0.689366 0.724413i \(-0.742111\pi\)
−0.689366 + 0.724413i \(0.742111\pi\)
\(18\) −1.00000 −0.235702
\(19\) −2.56155 −0.587661 −0.293830 0.955858i \(-0.594930\pi\)
−0.293830 + 0.955858i \(0.594930\pi\)
\(20\) 0.561553 0.125567
\(21\) 1.00000 0.218218
\(22\) −1.43845 −0.306678
\(23\) −5.68466 −1.18533 −0.592667 0.805448i \(-0.701925\pi\)
−0.592667 + 0.805448i \(0.701925\pi\)
\(24\) 1.00000 0.204124
\(25\) −4.68466 −0.936932
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) −1.00000 −0.188982
\(29\) −2.56155 −0.475668 −0.237834 0.971306i \(-0.576437\pi\)
−0.237834 + 0.971306i \(0.576437\pi\)
\(30\) 0.561553 0.102525
\(31\) −10.2462 −1.84027 −0.920137 0.391597i \(-0.871923\pi\)
−0.920137 + 0.391597i \(0.871923\pi\)
\(32\) −1.00000 −0.176777
\(33\) −1.43845 −0.250402
\(34\) 5.68466 0.974911
\(35\) −0.561553 −0.0949197
\(36\) 1.00000 0.166667
\(37\) −1.68466 −0.276956 −0.138478 0.990366i \(-0.544221\pi\)
−0.138478 + 0.990366i \(0.544221\pi\)
\(38\) 2.56155 0.415539
\(39\) 0 0
\(40\) −0.561553 −0.0887893
\(41\) −4.00000 −0.624695 −0.312348 0.949968i \(-0.601115\pi\)
−0.312348 + 0.949968i \(0.601115\pi\)
\(42\) −1.00000 −0.154303
\(43\) 10.5616 1.61062 0.805311 0.592853i \(-0.201998\pi\)
0.805311 + 0.592853i \(0.201998\pi\)
\(44\) 1.43845 0.216854
\(45\) 0.561553 0.0837114
\(46\) 5.68466 0.838157
\(47\) 6.24621 0.911104 0.455552 0.890209i \(-0.349442\pi\)
0.455552 + 0.890209i \(0.349442\pi\)
\(48\) −1.00000 −0.144338
\(49\) 1.00000 0.142857
\(50\) 4.68466 0.662511
\(51\) 5.68466 0.796011
\(52\) 0 0
\(53\) 13.1231 1.80260 0.901299 0.433198i \(-0.142615\pi\)
0.901299 + 0.433198i \(0.142615\pi\)
\(54\) 1.00000 0.136083
\(55\) 0.807764 0.108919
\(56\) 1.00000 0.133631
\(57\) 2.56155 0.339286
\(58\) 2.56155 0.336348
\(59\) −12.2462 −1.59432 −0.797160 0.603768i \(-0.793666\pi\)
−0.797160 + 0.603768i \(0.793666\pi\)
\(60\) −0.561553 −0.0724962
\(61\) 2.56155 0.327973 0.163987 0.986463i \(-0.447565\pi\)
0.163987 + 0.986463i \(0.447565\pi\)
\(62\) 10.2462 1.30127
\(63\) −1.00000 −0.125988
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 1.43845 0.177061
\(67\) −7.12311 −0.870226 −0.435113 0.900376i \(-0.643292\pi\)
−0.435113 + 0.900376i \(0.643292\pi\)
\(68\) −5.68466 −0.689366
\(69\) 5.68466 0.684352
\(70\) 0.561553 0.0671184
\(71\) 15.3693 1.82400 0.912001 0.410188i \(-0.134537\pi\)
0.912001 + 0.410188i \(0.134537\pi\)
\(72\) −1.00000 −0.117851
\(73\) 7.43845 0.870604 0.435302 0.900284i \(-0.356642\pi\)
0.435302 + 0.900284i \(0.356642\pi\)
\(74\) 1.68466 0.195838
\(75\) 4.68466 0.540938
\(76\) −2.56155 −0.293830
\(77\) −1.43845 −0.163926
\(78\) 0 0
\(79\) 16.0000 1.80014 0.900070 0.435745i \(-0.143515\pi\)
0.900070 + 0.435745i \(0.143515\pi\)
\(80\) 0.561553 0.0627835
\(81\) 1.00000 0.111111
\(82\) 4.00000 0.441726
\(83\) 2.00000 0.219529 0.109764 0.993958i \(-0.464990\pi\)
0.109764 + 0.993958i \(0.464990\pi\)
\(84\) 1.00000 0.109109
\(85\) −3.19224 −0.346247
\(86\) −10.5616 −1.13888
\(87\) 2.56155 0.274627
\(88\) −1.43845 −0.153339
\(89\) −8.00000 −0.847998 −0.423999 0.905663i \(-0.639374\pi\)
−0.423999 + 0.905663i \(0.639374\pi\)
\(90\) −0.561553 −0.0591929
\(91\) 0 0
\(92\) −5.68466 −0.592667
\(93\) 10.2462 1.06248
\(94\) −6.24621 −0.644247
\(95\) −1.43845 −0.147582
\(96\) 1.00000 0.102062
\(97\) −10.0000 −1.01535 −0.507673 0.861550i \(-0.669494\pi\)
−0.507673 + 0.861550i \(0.669494\pi\)
\(98\) −1.00000 −0.101015
\(99\) 1.43845 0.144569
\(100\) −4.68466 −0.468466
\(101\) −6.00000 −0.597022 −0.298511 0.954406i \(-0.596490\pi\)
−0.298511 + 0.954406i \(0.596490\pi\)
\(102\) −5.68466 −0.562865
\(103\) 18.8078 1.85318 0.926592 0.376068i \(-0.122724\pi\)
0.926592 + 0.376068i \(0.122724\pi\)
\(104\) 0 0
\(105\) 0.561553 0.0548019
\(106\) −13.1231 −1.27463
\(107\) −13.1231 −1.26866 −0.634329 0.773063i \(-0.718724\pi\)
−0.634329 + 0.773063i \(0.718724\pi\)
\(108\) −1.00000 −0.0962250
\(109\) 15.9309 1.52590 0.762950 0.646457i \(-0.223750\pi\)
0.762950 + 0.646457i \(0.223750\pi\)
\(110\) −0.807764 −0.0770173
\(111\) 1.68466 0.159901
\(112\) −1.00000 −0.0944911
\(113\) 3.75379 0.353127 0.176563 0.984289i \(-0.443502\pi\)
0.176563 + 0.984289i \(0.443502\pi\)
\(114\) −2.56155 −0.239911
\(115\) −3.19224 −0.297678
\(116\) −2.56155 −0.237834
\(117\) 0 0
\(118\) 12.2462 1.12736
\(119\) 5.68466 0.521112
\(120\) 0.561553 0.0512625
\(121\) −8.93087 −0.811897
\(122\) −2.56155 −0.231912
\(123\) 4.00000 0.360668
\(124\) −10.2462 −0.920137
\(125\) −5.43845 −0.486430
\(126\) 1.00000 0.0890871
\(127\) 6.24621 0.554262 0.277131 0.960832i \(-0.410616\pi\)
0.277131 + 0.960832i \(0.410616\pi\)
\(128\) −1.00000 −0.0883883
\(129\) −10.5616 −0.929893
\(130\) 0 0
\(131\) 2.56155 0.223804 0.111902 0.993719i \(-0.464306\pi\)
0.111902 + 0.993719i \(0.464306\pi\)
\(132\) −1.43845 −0.125201
\(133\) 2.56155 0.222115
\(134\) 7.12311 0.615343
\(135\) −0.561553 −0.0483308
\(136\) 5.68466 0.487455
\(137\) −5.68466 −0.485673 −0.242837 0.970067i \(-0.578078\pi\)
−0.242837 + 0.970067i \(0.578078\pi\)
\(138\) −5.68466 −0.483910
\(139\) −6.24621 −0.529797 −0.264898 0.964276i \(-0.585338\pi\)
−0.264898 + 0.964276i \(0.585338\pi\)
\(140\) −0.561553 −0.0474599
\(141\) −6.24621 −0.526026
\(142\) −15.3693 −1.28976
\(143\) 0 0
\(144\) 1.00000 0.0833333
\(145\) −1.43845 −0.119457
\(146\) −7.43845 −0.615610
\(147\) −1.00000 −0.0824786
\(148\) −1.68466 −0.138478
\(149\) −10.0000 −0.819232 −0.409616 0.912258i \(-0.634337\pi\)
−0.409616 + 0.912258i \(0.634337\pi\)
\(150\) −4.68466 −0.382501
\(151\) 4.31534 0.351178 0.175589 0.984464i \(-0.443817\pi\)
0.175589 + 0.984464i \(0.443817\pi\)
\(152\) 2.56155 0.207769
\(153\) −5.68466 −0.459577
\(154\) 1.43845 0.115913
\(155\) −5.75379 −0.462155
\(156\) 0 0
\(157\) 8.80776 0.702936 0.351468 0.936200i \(-0.385683\pi\)
0.351468 + 0.936200i \(0.385683\pi\)
\(158\) −16.0000 −1.27289
\(159\) −13.1231 −1.04073
\(160\) −0.561553 −0.0443946
\(161\) 5.68466 0.448014
\(162\) −1.00000 −0.0785674
\(163\) 0.876894 0.0686837 0.0343418 0.999410i \(-0.489067\pi\)
0.0343418 + 0.999410i \(0.489067\pi\)
\(164\) −4.00000 −0.312348
\(165\) −0.807764 −0.0628843
\(166\) −2.00000 −0.155230
\(167\) 8.80776 0.681565 0.340783 0.940142i \(-0.389308\pi\)
0.340783 + 0.940142i \(0.389308\pi\)
\(168\) −1.00000 −0.0771517
\(169\) 0 0
\(170\) 3.19224 0.244833
\(171\) −2.56155 −0.195887
\(172\) 10.5616 0.805311
\(173\) 20.2462 1.53929 0.769645 0.638471i \(-0.220433\pi\)
0.769645 + 0.638471i \(0.220433\pi\)
\(174\) −2.56155 −0.194191
\(175\) 4.68466 0.354127
\(176\) 1.43845 0.108427
\(177\) 12.2462 0.920482
\(178\) 8.00000 0.599625
\(179\) −2.87689 −0.215029 −0.107515 0.994204i \(-0.534289\pi\)
−0.107515 + 0.994204i \(0.534289\pi\)
\(180\) 0.561553 0.0418557
\(181\) −11.3693 −0.845075 −0.422537 0.906346i \(-0.638860\pi\)
−0.422537 + 0.906346i \(0.638860\pi\)
\(182\) 0 0
\(183\) −2.56155 −0.189355
\(184\) 5.68466 0.419079
\(185\) −0.946025 −0.0695531
\(186\) −10.2462 −0.751289
\(187\) −8.17708 −0.597967
\(188\) 6.24621 0.455552
\(189\) 1.00000 0.0727393
\(190\) 1.43845 0.104356
\(191\) 13.0540 0.944553 0.472276 0.881451i \(-0.343432\pi\)
0.472276 + 0.881451i \(0.343432\pi\)
\(192\) −1.00000 −0.0721688
\(193\) −12.0000 −0.863779 −0.431889 0.901927i \(-0.642153\pi\)
−0.431889 + 0.901927i \(0.642153\pi\)
\(194\) 10.0000 0.717958
\(195\) 0 0
\(196\) 1.00000 0.0714286
\(197\) 6.00000 0.427482 0.213741 0.976890i \(-0.431435\pi\)
0.213741 + 0.976890i \(0.431435\pi\)
\(198\) −1.43845 −0.102226
\(199\) 23.9309 1.69641 0.848207 0.529665i \(-0.177682\pi\)
0.848207 + 0.529665i \(0.177682\pi\)
\(200\) 4.68466 0.331255
\(201\) 7.12311 0.502425
\(202\) 6.00000 0.422159
\(203\) 2.56155 0.179786
\(204\) 5.68466 0.398006
\(205\) −2.24621 −0.156882
\(206\) −18.8078 −1.31040
\(207\) −5.68466 −0.395111
\(208\) 0 0
\(209\) −3.68466 −0.254873
\(210\) −0.561553 −0.0387508
\(211\) −12.8078 −0.881723 −0.440861 0.897575i \(-0.645327\pi\)
−0.440861 + 0.897575i \(0.645327\pi\)
\(212\) 13.1231 0.901299
\(213\) −15.3693 −1.05309
\(214\) 13.1231 0.897077
\(215\) 5.93087 0.404482
\(216\) 1.00000 0.0680414
\(217\) 10.2462 0.695558
\(218\) −15.9309 −1.07897
\(219\) −7.43845 −0.502644
\(220\) 0.807764 0.0544594
\(221\) 0 0
\(222\) −1.68466 −0.113067
\(223\) 18.2462 1.22186 0.610928 0.791686i \(-0.290796\pi\)
0.610928 + 0.791686i \(0.290796\pi\)
\(224\) 1.00000 0.0668153
\(225\) −4.68466 −0.312311
\(226\) −3.75379 −0.249698
\(227\) −23.6155 −1.56742 −0.783709 0.621128i \(-0.786675\pi\)
−0.783709 + 0.621128i \(0.786675\pi\)
\(228\) 2.56155 0.169643
\(229\) 2.00000 0.132164 0.0660819 0.997814i \(-0.478950\pi\)
0.0660819 + 0.997814i \(0.478950\pi\)
\(230\) 3.19224 0.210490
\(231\) 1.43845 0.0946429
\(232\) 2.56155 0.168174
\(233\) −2.00000 −0.131024 −0.0655122 0.997852i \(-0.520868\pi\)
−0.0655122 + 0.997852i \(0.520868\pi\)
\(234\) 0 0
\(235\) 3.50758 0.228809
\(236\) −12.2462 −0.797160
\(237\) −16.0000 −1.03931
\(238\) −5.68466 −0.368482
\(239\) 2.24621 0.145295 0.0726477 0.997358i \(-0.476855\pi\)
0.0726477 + 0.997358i \(0.476855\pi\)
\(240\) −0.561553 −0.0362481
\(241\) 6.00000 0.386494 0.193247 0.981150i \(-0.438098\pi\)
0.193247 + 0.981150i \(0.438098\pi\)
\(242\) 8.93087 0.574098
\(243\) −1.00000 −0.0641500
\(244\) 2.56155 0.163987
\(245\) 0.561553 0.0358763
\(246\) −4.00000 −0.255031
\(247\) 0 0
\(248\) 10.2462 0.650635
\(249\) −2.00000 −0.126745
\(250\) 5.43845 0.343958
\(251\) 15.0540 0.950198 0.475099 0.879932i \(-0.342412\pi\)
0.475099 + 0.879932i \(0.342412\pi\)
\(252\) −1.00000 −0.0629941
\(253\) −8.17708 −0.514089
\(254\) −6.24621 −0.391922
\(255\) 3.19224 0.199906
\(256\) 1.00000 0.0625000
\(257\) 26.0000 1.62184 0.810918 0.585160i \(-0.198968\pi\)
0.810918 + 0.585160i \(0.198968\pi\)
\(258\) 10.5616 0.657534
\(259\) 1.68466 0.104680
\(260\) 0 0
\(261\) −2.56155 −0.158556
\(262\) −2.56155 −0.158253
\(263\) 1.36932 0.0844357 0.0422178 0.999108i \(-0.486558\pi\)
0.0422178 + 0.999108i \(0.486558\pi\)
\(264\) 1.43845 0.0885303
\(265\) 7.36932 0.452694
\(266\) −2.56155 −0.157059
\(267\) 8.00000 0.489592
\(268\) −7.12311 −0.435113
\(269\) 6.63068 0.404280 0.202140 0.979357i \(-0.435210\pi\)
0.202140 + 0.979357i \(0.435210\pi\)
\(270\) 0.561553 0.0341750
\(271\) −8.00000 −0.485965 −0.242983 0.970031i \(-0.578126\pi\)
−0.242983 + 0.970031i \(0.578126\pi\)
\(272\) −5.68466 −0.344683
\(273\) 0 0
\(274\) 5.68466 0.343423
\(275\) −6.73863 −0.406355
\(276\) 5.68466 0.342176
\(277\) 19.1231 1.14900 0.574498 0.818506i \(-0.305197\pi\)
0.574498 + 0.818506i \(0.305197\pi\)
\(278\) 6.24621 0.374623
\(279\) −10.2462 −0.613425
\(280\) 0.561553 0.0335592
\(281\) 6.00000 0.357930 0.178965 0.983855i \(-0.442725\pi\)
0.178965 + 0.983855i \(0.442725\pi\)
\(282\) 6.24621 0.371956
\(283\) 13.1231 0.780088 0.390044 0.920796i \(-0.372460\pi\)
0.390044 + 0.920796i \(0.372460\pi\)
\(284\) 15.3693 0.912001
\(285\) 1.43845 0.0852063
\(286\) 0 0
\(287\) 4.00000 0.236113
\(288\) −1.00000 −0.0589256
\(289\) 15.3153 0.900902
\(290\) 1.43845 0.0844685
\(291\) 10.0000 0.586210
\(292\) 7.43845 0.435302
\(293\) −24.2462 −1.41648 −0.708239 0.705972i \(-0.750510\pi\)
−0.708239 + 0.705972i \(0.750510\pi\)
\(294\) 1.00000 0.0583212
\(295\) −6.87689 −0.400388
\(296\) 1.68466 0.0979188
\(297\) −1.43845 −0.0834672
\(298\) 10.0000 0.579284
\(299\) 0 0
\(300\) 4.68466 0.270469
\(301\) −10.5616 −0.608758
\(302\) −4.31534 −0.248320
\(303\) 6.00000 0.344691
\(304\) −2.56155 −0.146915
\(305\) 1.43845 0.0823652
\(306\) 5.68466 0.324970
\(307\) 9.75379 0.556678 0.278339 0.960483i \(-0.410216\pi\)
0.278339 + 0.960483i \(0.410216\pi\)
\(308\) −1.43845 −0.0819631
\(309\) −18.8078 −1.06994
\(310\) 5.75379 0.326793
\(311\) −32.4924 −1.84248 −0.921238 0.388999i \(-0.872821\pi\)
−0.921238 + 0.388999i \(0.872821\pi\)
\(312\) 0 0
\(313\) 15.7538 0.890457 0.445228 0.895417i \(-0.353122\pi\)
0.445228 + 0.895417i \(0.353122\pi\)
\(314\) −8.80776 −0.497051
\(315\) −0.561553 −0.0316399
\(316\) 16.0000 0.900070
\(317\) −21.3693 −1.20022 −0.600110 0.799917i \(-0.704877\pi\)
−0.600110 + 0.799917i \(0.704877\pi\)
\(318\) 13.1231 0.735907
\(319\) −3.68466 −0.206301
\(320\) 0.561553 0.0313918
\(321\) 13.1231 0.732460
\(322\) −5.68466 −0.316794
\(323\) 14.5616 0.810226
\(324\) 1.00000 0.0555556
\(325\) 0 0
\(326\) −0.876894 −0.0485667
\(327\) −15.9309 −0.880979
\(328\) 4.00000 0.220863
\(329\) −6.24621 −0.344365
\(330\) 0.807764 0.0444659
\(331\) −7.12311 −0.391521 −0.195761 0.980652i \(-0.562718\pi\)
−0.195761 + 0.980652i \(0.562718\pi\)
\(332\) 2.00000 0.109764
\(333\) −1.68466 −0.0923187
\(334\) −8.80776 −0.481939
\(335\) −4.00000 −0.218543
\(336\) 1.00000 0.0545545
\(337\) 17.0540 0.928989 0.464495 0.885576i \(-0.346236\pi\)
0.464495 + 0.885576i \(0.346236\pi\)
\(338\) 0 0
\(339\) −3.75379 −0.203878
\(340\) −3.19224 −0.173123
\(341\) −14.7386 −0.798142
\(342\) 2.56155 0.138513
\(343\) −1.00000 −0.0539949
\(344\) −10.5616 −0.569441
\(345\) 3.19224 0.171864
\(346\) −20.2462 −1.08844
\(347\) 8.49242 0.455897 0.227949 0.973673i \(-0.426798\pi\)
0.227949 + 0.973673i \(0.426798\pi\)
\(348\) 2.56155 0.137314
\(349\) −21.3693 −1.14387 −0.571937 0.820298i \(-0.693808\pi\)
−0.571937 + 0.820298i \(0.693808\pi\)
\(350\) −4.68466 −0.250406
\(351\) 0 0
\(352\) −1.43845 −0.0766695
\(353\) 1.75379 0.0933448 0.0466724 0.998910i \(-0.485138\pi\)
0.0466724 + 0.998910i \(0.485138\pi\)
\(354\) −12.2462 −0.650879
\(355\) 8.63068 0.458069
\(356\) −8.00000 −0.423999
\(357\) −5.68466 −0.300864
\(358\) 2.87689 0.152049
\(359\) −17.6155 −0.929712 −0.464856 0.885386i \(-0.653894\pi\)
−0.464856 + 0.885386i \(0.653894\pi\)
\(360\) −0.561553 −0.0295964
\(361\) −12.4384 −0.654655
\(362\) 11.3693 0.597558
\(363\) 8.93087 0.468749
\(364\) 0 0
\(365\) 4.17708 0.218638
\(366\) 2.56155 0.133895
\(367\) 25.3693 1.32427 0.662134 0.749386i \(-0.269651\pi\)
0.662134 + 0.749386i \(0.269651\pi\)
\(368\) −5.68466 −0.296333
\(369\) −4.00000 −0.208232
\(370\) 0.946025 0.0491815
\(371\) −13.1231 −0.681318
\(372\) 10.2462 0.531241
\(373\) −6.00000 −0.310668 −0.155334 0.987862i \(-0.549645\pi\)
−0.155334 + 0.987862i \(0.549645\pi\)
\(374\) 8.17708 0.422827
\(375\) 5.43845 0.280840
\(376\) −6.24621 −0.322124
\(377\) 0 0
\(378\) −1.00000 −0.0514344
\(379\) 27.1231 1.39322 0.696610 0.717450i \(-0.254691\pi\)
0.696610 + 0.717450i \(0.254691\pi\)
\(380\) −1.43845 −0.0737908
\(381\) −6.24621 −0.320003
\(382\) −13.0540 −0.667899
\(383\) −13.4384 −0.686673 −0.343336 0.939213i \(-0.611557\pi\)
−0.343336 + 0.939213i \(0.611557\pi\)
\(384\) 1.00000 0.0510310
\(385\) −0.807764 −0.0411675
\(386\) 12.0000 0.610784
\(387\) 10.5616 0.536874
\(388\) −10.0000 −0.507673
\(389\) −18.8769 −0.957097 −0.478548 0.878061i \(-0.658837\pi\)
−0.478548 + 0.878061i \(0.658837\pi\)
\(390\) 0 0
\(391\) 32.3153 1.63426
\(392\) −1.00000 −0.0505076
\(393\) −2.56155 −0.129213
\(394\) −6.00000 −0.302276
\(395\) 8.98485 0.452077
\(396\) 1.43845 0.0722847
\(397\) −9.36932 −0.470233 −0.235116 0.971967i \(-0.575547\pi\)
−0.235116 + 0.971967i \(0.575547\pi\)
\(398\) −23.9309 −1.19955
\(399\) −2.56155 −0.128238
\(400\) −4.68466 −0.234233
\(401\) 34.9848 1.74706 0.873530 0.486771i \(-0.161825\pi\)
0.873530 + 0.486771i \(0.161825\pi\)
\(402\) −7.12311 −0.355268
\(403\) 0 0
\(404\) −6.00000 −0.298511
\(405\) 0.561553 0.0279038
\(406\) −2.56155 −0.127128
\(407\) −2.42329 −0.120118
\(408\) −5.68466 −0.281433
\(409\) 6.80776 0.336622 0.168311 0.985734i \(-0.446169\pi\)
0.168311 + 0.985734i \(0.446169\pi\)
\(410\) 2.24621 0.110932
\(411\) 5.68466 0.280404
\(412\) 18.8078 0.926592
\(413\) 12.2462 0.602597
\(414\) 5.68466 0.279386
\(415\) 1.12311 0.0551311
\(416\) 0 0
\(417\) 6.24621 0.305878
\(418\) 3.68466 0.180223
\(419\) −8.31534 −0.406231 −0.203116 0.979155i \(-0.565107\pi\)
−0.203116 + 0.979155i \(0.565107\pi\)
\(420\) 0.561553 0.0274010
\(421\) 10.0000 0.487370 0.243685 0.969854i \(-0.421644\pi\)
0.243685 + 0.969854i \(0.421644\pi\)
\(422\) 12.8078 0.623472
\(423\) 6.24621 0.303701
\(424\) −13.1231 −0.637314
\(425\) 26.6307 1.29178
\(426\) 15.3693 0.744646
\(427\) −2.56155 −0.123962
\(428\) −13.1231 −0.634329
\(429\) 0 0
\(430\) −5.93087 −0.286012
\(431\) 27.8617 1.34205 0.671026 0.741433i \(-0.265854\pi\)
0.671026 + 0.741433i \(0.265854\pi\)
\(432\) −1.00000 −0.0481125
\(433\) 1.36932 0.0658052 0.0329026 0.999459i \(-0.489525\pi\)
0.0329026 + 0.999459i \(0.489525\pi\)
\(434\) −10.2462 −0.491834
\(435\) 1.43845 0.0689683
\(436\) 15.9309 0.762950
\(437\) 14.5616 0.696574
\(438\) 7.43845 0.355423
\(439\) −19.9309 −0.951249 −0.475624 0.879649i \(-0.657778\pi\)
−0.475624 + 0.879649i \(0.657778\pi\)
\(440\) −0.807764 −0.0385086
\(441\) 1.00000 0.0476190
\(442\) 0 0
\(443\) 31.3693 1.49040 0.745201 0.666840i \(-0.232354\pi\)
0.745201 + 0.666840i \(0.232354\pi\)
\(444\) 1.68466 0.0799504
\(445\) −4.49242 −0.212961
\(446\) −18.2462 −0.863983
\(447\) 10.0000 0.472984
\(448\) −1.00000 −0.0472456
\(449\) −21.6847 −1.02336 −0.511681 0.859175i \(-0.670977\pi\)
−0.511681 + 0.859175i \(0.670977\pi\)
\(450\) 4.68466 0.220837
\(451\) −5.75379 −0.270935
\(452\) 3.75379 0.176563
\(453\) −4.31534 −0.202752
\(454\) 23.6155 1.10833
\(455\) 0 0
\(456\) −2.56155 −0.119956
\(457\) 16.0000 0.748448 0.374224 0.927338i \(-0.377909\pi\)
0.374224 + 0.927338i \(0.377909\pi\)
\(458\) −2.00000 −0.0934539
\(459\) 5.68466 0.265337
\(460\) −3.19224 −0.148839
\(461\) −8.06913 −0.375817 −0.187908 0.982187i \(-0.560171\pi\)
−0.187908 + 0.982187i \(0.560171\pi\)
\(462\) −1.43845 −0.0669226
\(463\) 39.5464 1.83788 0.918938 0.394401i \(-0.129048\pi\)
0.918938 + 0.394401i \(0.129048\pi\)
\(464\) −2.56155 −0.118917
\(465\) 5.75379 0.266826
\(466\) 2.00000 0.0926482
\(467\) 2.56155 0.118535 0.0592673 0.998242i \(-0.481124\pi\)
0.0592673 + 0.998242i \(0.481124\pi\)
\(468\) 0 0
\(469\) 7.12311 0.328914
\(470\) −3.50758 −0.161792
\(471\) −8.80776 −0.405840
\(472\) 12.2462 0.563678
\(473\) 15.1922 0.698540
\(474\) 16.0000 0.734904
\(475\) 12.0000 0.550598
\(476\) 5.68466 0.260556
\(477\) 13.1231 0.600866
\(478\) −2.24621 −0.102739
\(479\) −41.3002 −1.88705 −0.943527 0.331296i \(-0.892514\pi\)
−0.943527 + 0.331296i \(0.892514\pi\)
\(480\) 0.561553 0.0256313
\(481\) 0 0
\(482\) −6.00000 −0.273293
\(483\) −5.68466 −0.258661
\(484\) −8.93087 −0.405949
\(485\) −5.61553 −0.254988
\(486\) 1.00000 0.0453609
\(487\) 30.2462 1.37059 0.685293 0.728267i \(-0.259674\pi\)
0.685293 + 0.728267i \(0.259674\pi\)
\(488\) −2.56155 −0.115956
\(489\) −0.876894 −0.0396545
\(490\) −0.561553 −0.0253684
\(491\) −6.24621 −0.281888 −0.140944 0.990018i \(-0.545014\pi\)
−0.140944 + 0.990018i \(0.545014\pi\)
\(492\) 4.00000 0.180334
\(493\) 14.5616 0.655819
\(494\) 0 0
\(495\) 0.807764 0.0363063
\(496\) −10.2462 −0.460068
\(497\) −15.3693 −0.689408
\(498\) 2.00000 0.0896221
\(499\) 34.4924 1.54409 0.772046 0.635566i \(-0.219233\pi\)
0.772046 + 0.635566i \(0.219233\pi\)
\(500\) −5.43845 −0.243215
\(501\) −8.80776 −0.393502
\(502\) −15.0540 −0.671892
\(503\) −5.61553 −0.250384 −0.125192 0.992133i \(-0.539955\pi\)
−0.125192 + 0.992133i \(0.539955\pi\)
\(504\) 1.00000 0.0445435
\(505\) −3.36932 −0.149933
\(506\) 8.17708 0.363516
\(507\) 0 0
\(508\) 6.24621 0.277131
\(509\) 9.68466 0.429265 0.214632 0.976695i \(-0.431145\pi\)
0.214632 + 0.976695i \(0.431145\pi\)
\(510\) −3.19224 −0.141355
\(511\) −7.43845 −0.329058
\(512\) −1.00000 −0.0441942
\(513\) 2.56155 0.113095
\(514\) −26.0000 −1.14681
\(515\) 10.5616 0.465398
\(516\) −10.5616 −0.464946
\(517\) 8.98485 0.395153
\(518\) −1.68466 −0.0740196
\(519\) −20.2462 −0.888710
\(520\) 0 0
\(521\) −29.0540 −1.27288 −0.636439 0.771327i \(-0.719593\pi\)
−0.636439 + 0.771327i \(0.719593\pi\)
\(522\) 2.56155 0.112116
\(523\) 24.4924 1.07098 0.535489 0.844542i \(-0.320127\pi\)
0.535489 + 0.844542i \(0.320127\pi\)
\(524\) 2.56155 0.111902
\(525\) −4.68466 −0.204455
\(526\) −1.36932 −0.0597051
\(527\) 58.2462 2.53724
\(528\) −1.43845 −0.0626004
\(529\) 9.31534 0.405015
\(530\) −7.36932 −0.320103
\(531\) −12.2462 −0.531440
\(532\) 2.56155 0.111057
\(533\) 0 0
\(534\) −8.00000 −0.346194
\(535\) −7.36932 −0.318603
\(536\) 7.12311 0.307671
\(537\) 2.87689 0.124147
\(538\) −6.63068 −0.285869
\(539\) 1.43845 0.0619583
\(540\) −0.561553 −0.0241654
\(541\) −26.8078 −1.15256 −0.576278 0.817254i \(-0.695495\pi\)
−0.576278 + 0.817254i \(0.695495\pi\)
\(542\) 8.00000 0.343629
\(543\) 11.3693 0.487904
\(544\) 5.68466 0.243728
\(545\) 8.94602 0.383206
\(546\) 0 0
\(547\) −36.9848 −1.58136 −0.790679 0.612231i \(-0.790272\pi\)
−0.790679 + 0.612231i \(0.790272\pi\)
\(548\) −5.68466 −0.242837
\(549\) 2.56155 0.109324
\(550\) 6.73863 0.287336
\(551\) 6.56155 0.279532
\(552\) −5.68466 −0.241955
\(553\) −16.0000 −0.680389
\(554\) −19.1231 −0.812463
\(555\) 0.946025 0.0401565
\(556\) −6.24621 −0.264898
\(557\) −0.246211 −0.0104323 −0.00521615 0.999986i \(-0.501660\pi\)
−0.00521615 + 0.999986i \(0.501660\pi\)
\(558\) 10.2462 0.433757
\(559\) 0 0
\(560\) −0.561553 −0.0237299
\(561\) 8.17708 0.345237
\(562\) −6.00000 −0.253095
\(563\) −0.946025 −0.0398702 −0.0199351 0.999801i \(-0.506346\pi\)
−0.0199351 + 0.999801i \(0.506346\pi\)
\(564\) −6.24621 −0.263013
\(565\) 2.10795 0.0886821
\(566\) −13.1231 −0.551605
\(567\) −1.00000 −0.0419961
\(568\) −15.3693 −0.644882
\(569\) 18.0000 0.754599 0.377300 0.926091i \(-0.376853\pi\)
0.377300 + 0.926091i \(0.376853\pi\)
\(570\) −1.43845 −0.0602499
\(571\) 1.75379 0.0733938 0.0366969 0.999326i \(-0.488316\pi\)
0.0366969 + 0.999326i \(0.488316\pi\)
\(572\) 0 0
\(573\) −13.0540 −0.545338
\(574\) −4.00000 −0.166957
\(575\) 26.6307 1.11058
\(576\) 1.00000 0.0416667
\(577\) −24.2462 −1.00938 −0.504691 0.863300i \(-0.668394\pi\)
−0.504691 + 0.863300i \(0.668394\pi\)
\(578\) −15.3153 −0.637034
\(579\) 12.0000 0.498703
\(580\) −1.43845 −0.0597283
\(581\) −2.00000 −0.0829740
\(582\) −10.0000 −0.414513
\(583\) 18.8769 0.781801
\(584\) −7.43845 −0.307805
\(585\) 0 0
\(586\) 24.2462 1.00160
\(587\) 28.8769 1.19188 0.595938 0.803030i \(-0.296780\pi\)
0.595938 + 0.803030i \(0.296780\pi\)
\(588\) −1.00000 −0.0412393
\(589\) 26.2462 1.08146
\(590\) 6.87689 0.283117
\(591\) −6.00000 −0.246807
\(592\) −1.68466 −0.0692390
\(593\) −28.0000 −1.14982 −0.574911 0.818216i \(-0.694963\pi\)
−0.574911 + 0.818216i \(0.694963\pi\)
\(594\) 1.43845 0.0590202
\(595\) 3.19224 0.130869
\(596\) −10.0000 −0.409616
\(597\) −23.9309 −0.979425
\(598\) 0 0
\(599\) −23.3002 −0.952020 −0.476010 0.879440i \(-0.657917\pi\)
−0.476010 + 0.879440i \(0.657917\pi\)
\(600\) −4.68466 −0.191250
\(601\) 25.8617 1.05492 0.527461 0.849579i \(-0.323144\pi\)
0.527461 + 0.849579i \(0.323144\pi\)
\(602\) 10.5616 0.430457
\(603\) −7.12311 −0.290075
\(604\) 4.31534 0.175589
\(605\) −5.01515 −0.203895
\(606\) −6.00000 −0.243733
\(607\) 41.5464 1.68632 0.843158 0.537666i \(-0.180694\pi\)
0.843158 + 0.537666i \(0.180694\pi\)
\(608\) 2.56155 0.103885
\(609\) −2.56155 −0.103799
\(610\) −1.43845 −0.0582410
\(611\) 0 0
\(612\) −5.68466 −0.229789
\(613\) −18.8078 −0.759638 −0.379819 0.925061i \(-0.624014\pi\)
−0.379819 + 0.925061i \(0.624014\pi\)
\(614\) −9.75379 −0.393631
\(615\) 2.24621 0.0905760
\(616\) 1.43845 0.0579567
\(617\) −26.8078 −1.07924 −0.539620 0.841909i \(-0.681432\pi\)
−0.539620 + 0.841909i \(0.681432\pi\)
\(618\) 18.8078 0.756559
\(619\) −2.06913 −0.0831654 −0.0415827 0.999135i \(-0.513240\pi\)
−0.0415827 + 0.999135i \(0.513240\pi\)
\(620\) −5.75379 −0.231078
\(621\) 5.68466 0.228117
\(622\) 32.4924 1.30283
\(623\) 8.00000 0.320513
\(624\) 0 0
\(625\) 20.3693 0.814773
\(626\) −15.7538 −0.629648
\(627\) 3.68466 0.147151
\(628\) 8.80776 0.351468
\(629\) 9.57671 0.381848
\(630\) 0.561553 0.0223728
\(631\) 0.315342 0.0125535 0.00627677 0.999980i \(-0.498002\pi\)
0.00627677 + 0.999980i \(0.498002\pi\)
\(632\) −16.0000 −0.636446
\(633\) 12.8078 0.509063
\(634\) 21.3693 0.848684
\(635\) 3.50758 0.139194
\(636\) −13.1231 −0.520365
\(637\) 0 0
\(638\) 3.68466 0.145877
\(639\) 15.3693 0.608001
\(640\) −0.561553 −0.0221973
\(641\) 44.1080 1.74216 0.871080 0.491142i \(-0.163420\pi\)
0.871080 + 0.491142i \(0.163420\pi\)
\(642\) −13.1231 −0.517928
\(643\) 20.8078 0.820578 0.410289 0.911956i \(-0.365428\pi\)
0.410289 + 0.911956i \(0.365428\pi\)
\(644\) 5.68466 0.224007
\(645\) −5.93087 −0.233528
\(646\) −14.5616 −0.572917
\(647\) −35.3693 −1.39051 −0.695256 0.718763i \(-0.744709\pi\)
−0.695256 + 0.718763i \(0.744709\pi\)
\(648\) −1.00000 −0.0392837
\(649\) −17.6155 −0.691470
\(650\) 0 0
\(651\) −10.2462 −0.401581
\(652\) 0.876894 0.0343418
\(653\) −25.4384 −0.995483 −0.497742 0.867325i \(-0.665837\pi\)
−0.497742 + 0.867325i \(0.665837\pi\)
\(654\) 15.9309 0.622946
\(655\) 1.43845 0.0562048
\(656\) −4.00000 −0.156174
\(657\) 7.43845 0.290201
\(658\) 6.24621 0.243503
\(659\) 21.1231 0.822839 0.411420 0.911446i \(-0.365033\pi\)
0.411420 + 0.911446i \(0.365033\pi\)
\(660\) −0.807764 −0.0314422
\(661\) 4.73863 0.184311 0.0921557 0.995745i \(-0.470624\pi\)
0.0921557 + 0.995745i \(0.470624\pi\)
\(662\) 7.12311 0.276847
\(663\) 0 0
\(664\) −2.00000 −0.0776151
\(665\) 1.43845 0.0557806
\(666\) 1.68466 0.0652792
\(667\) 14.5616 0.563826
\(668\) 8.80776 0.340783
\(669\) −18.2462 −0.705439
\(670\) 4.00000 0.154533
\(671\) 3.68466 0.142245
\(672\) −1.00000 −0.0385758
\(673\) −17.1922 −0.662712 −0.331356 0.943506i \(-0.607506\pi\)
−0.331356 + 0.943506i \(0.607506\pi\)
\(674\) −17.0540 −0.656895
\(675\) 4.68466 0.180313
\(676\) 0 0
\(677\) 6.63068 0.254838 0.127419 0.991849i \(-0.459331\pi\)
0.127419 + 0.991849i \(0.459331\pi\)
\(678\) 3.75379 0.144163
\(679\) 10.0000 0.383765
\(680\) 3.19224 0.122417
\(681\) 23.6155 0.904949
\(682\) 14.7386 0.564371
\(683\) −11.6847 −0.447101 −0.223551 0.974692i \(-0.571765\pi\)
−0.223551 + 0.974692i \(0.571765\pi\)
\(684\) −2.56155 −0.0979434
\(685\) −3.19224 −0.121969
\(686\) 1.00000 0.0381802
\(687\) −2.00000 −0.0763048
\(688\) 10.5616 0.402655
\(689\) 0 0
\(690\) −3.19224 −0.121526
\(691\) −3.50758 −0.133435 −0.0667173 0.997772i \(-0.521253\pi\)
−0.0667173 + 0.997772i \(0.521253\pi\)
\(692\) 20.2462 0.769645
\(693\) −1.43845 −0.0546421
\(694\) −8.49242 −0.322368
\(695\) −3.50758 −0.133050
\(696\) −2.56155 −0.0970954
\(697\) 22.7386 0.861287
\(698\) 21.3693 0.808841
\(699\) 2.00000 0.0756469
\(700\) 4.68466 0.177063
\(701\) −37.6155 −1.42072 −0.710359 0.703839i \(-0.751468\pi\)
−0.710359 + 0.703839i \(0.751468\pi\)
\(702\) 0 0
\(703\) 4.31534 0.162756
\(704\) 1.43845 0.0542135
\(705\) −3.50758 −0.132103
\(706\) −1.75379 −0.0660047
\(707\) 6.00000 0.225653
\(708\) 12.2462 0.460241
\(709\) 48.2462 1.81192 0.905962 0.423358i \(-0.139149\pi\)
0.905962 + 0.423358i \(0.139149\pi\)
\(710\) −8.63068 −0.323904
\(711\) 16.0000 0.600047
\(712\) 8.00000 0.299813
\(713\) 58.2462 2.18134
\(714\) 5.68466 0.212743
\(715\) 0 0
\(716\) −2.87689 −0.107515
\(717\) −2.24621 −0.0838863
\(718\) 17.6155 0.657406
\(719\) −30.2462 −1.12799 −0.563997 0.825777i \(-0.690737\pi\)
−0.563997 + 0.825777i \(0.690737\pi\)
\(720\) 0.561553 0.0209278
\(721\) −18.8078 −0.700438
\(722\) 12.4384 0.462911
\(723\) −6.00000 −0.223142
\(724\) −11.3693 −0.422537
\(725\) 12.0000 0.445669
\(726\) −8.93087 −0.331456
\(727\) 20.0691 0.744323 0.372161 0.928168i \(-0.378617\pi\)
0.372161 + 0.928168i \(0.378617\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) −4.17708 −0.154601
\(731\) −60.0388 −2.22062
\(732\) −2.56155 −0.0946777
\(733\) 31.1231 1.14956 0.574779 0.818309i \(-0.305088\pi\)
0.574779 + 0.818309i \(0.305088\pi\)
\(734\) −25.3693 −0.936399
\(735\) −0.561553 −0.0207132
\(736\) 5.68466 0.209539
\(737\) −10.2462 −0.377424
\(738\) 4.00000 0.147242
\(739\) 23.6155 0.868711 0.434356 0.900741i \(-0.356976\pi\)
0.434356 + 0.900741i \(0.356976\pi\)
\(740\) −0.946025 −0.0347766
\(741\) 0 0
\(742\) 13.1231 0.481764
\(743\) 16.0000 0.586983 0.293492 0.955962i \(-0.405183\pi\)
0.293492 + 0.955962i \(0.405183\pi\)
\(744\) −10.2462 −0.375644
\(745\) −5.61553 −0.205737
\(746\) 6.00000 0.219676
\(747\) 2.00000 0.0731762
\(748\) −8.17708 −0.298984
\(749\) 13.1231 0.479508
\(750\) −5.43845 −0.198584
\(751\) 14.2462 0.519852 0.259926 0.965629i \(-0.416302\pi\)
0.259926 + 0.965629i \(0.416302\pi\)
\(752\) 6.24621 0.227776
\(753\) −15.0540 −0.548597
\(754\) 0 0
\(755\) 2.42329 0.0881926
\(756\) 1.00000 0.0363696
\(757\) 38.0000 1.38113 0.690567 0.723269i \(-0.257361\pi\)
0.690567 + 0.723269i \(0.257361\pi\)
\(758\) −27.1231 −0.985156
\(759\) 8.17708 0.296809
\(760\) 1.43845 0.0521780
\(761\) −30.8769 −1.11929 −0.559643 0.828734i \(-0.689062\pi\)
−0.559643 + 0.828734i \(0.689062\pi\)
\(762\) 6.24621 0.226276
\(763\) −15.9309 −0.576736
\(764\) 13.0540 0.472276
\(765\) −3.19224 −0.115416
\(766\) 13.4384 0.485551
\(767\) 0 0
\(768\) −1.00000 −0.0360844
\(769\) 12.5616 0.452981 0.226491 0.974013i \(-0.427275\pi\)
0.226491 + 0.974013i \(0.427275\pi\)
\(770\) 0.807764 0.0291098
\(771\) −26.0000 −0.936367
\(772\) −12.0000 −0.431889
\(773\) 15.9309 0.572994 0.286497 0.958081i \(-0.407509\pi\)
0.286497 + 0.958081i \(0.407509\pi\)
\(774\) −10.5616 −0.379627
\(775\) 48.0000 1.72421
\(776\) 10.0000 0.358979
\(777\) −1.68466 −0.0604368
\(778\) 18.8769 0.676769
\(779\) 10.2462 0.367109
\(780\) 0 0
\(781\) 22.1080 0.791085
\(782\) −32.3153 −1.15559
\(783\) 2.56155 0.0915424
\(784\) 1.00000 0.0357143
\(785\) 4.94602 0.176531
\(786\) 2.56155 0.0913676
\(787\) 45.9309 1.63726 0.818629 0.574322i \(-0.194734\pi\)
0.818629 + 0.574322i \(0.194734\pi\)
\(788\) 6.00000 0.213741
\(789\) −1.36932 −0.0487490
\(790\) −8.98485 −0.319666
\(791\) −3.75379 −0.133469
\(792\) −1.43845 −0.0511130
\(793\) 0 0
\(794\) 9.36932 0.332505
\(795\) −7.36932 −0.261363
\(796\) 23.9309 0.848207
\(797\) −20.2462 −0.717158 −0.358579 0.933499i \(-0.616739\pi\)
−0.358579 + 0.933499i \(0.616739\pi\)
\(798\) 2.56155 0.0906780
\(799\) −35.5076 −1.25617
\(800\) 4.68466 0.165628
\(801\) −8.00000 −0.282666
\(802\) −34.9848 −1.23536
\(803\) 10.6998 0.377588
\(804\) 7.12311 0.251213
\(805\) 3.19224 0.112512
\(806\) 0 0
\(807\) −6.63068 −0.233411
\(808\) 6.00000 0.211079
\(809\) −30.6307 −1.07692 −0.538459 0.842652i \(-0.680993\pi\)
−0.538459 + 0.842652i \(0.680993\pi\)
\(810\) −0.561553 −0.0197310
\(811\) −35.0540 −1.23091 −0.615456 0.788171i \(-0.711028\pi\)
−0.615456 + 0.788171i \(0.711028\pi\)
\(812\) 2.56155 0.0898929
\(813\) 8.00000 0.280572
\(814\) 2.42329 0.0849363
\(815\) 0.492423 0.0172488
\(816\) 5.68466 0.199003
\(817\) −27.0540 −0.946499
\(818\) −6.80776 −0.238028
\(819\) 0 0
\(820\) −2.24621 −0.0784411
\(821\) 39.1231 1.36541 0.682703 0.730696i \(-0.260804\pi\)
0.682703 + 0.730696i \(0.260804\pi\)
\(822\) −5.68466 −0.198275
\(823\) −18.7386 −0.653188 −0.326594 0.945165i \(-0.605901\pi\)
−0.326594 + 0.945165i \(0.605901\pi\)
\(824\) −18.8078 −0.655200
\(825\) 6.73863 0.234609
\(826\) −12.2462 −0.426100
\(827\) −15.0540 −0.523478 −0.261739 0.965139i \(-0.584296\pi\)
−0.261739 + 0.965139i \(0.584296\pi\)
\(828\) −5.68466 −0.197556
\(829\) −4.94602 −0.171783 −0.0858913 0.996305i \(-0.527374\pi\)
−0.0858913 + 0.996305i \(0.527374\pi\)
\(830\) −1.12311 −0.0389836
\(831\) −19.1231 −0.663373
\(832\) 0 0
\(833\) −5.68466 −0.196962
\(834\) −6.24621 −0.216289
\(835\) 4.94602 0.171164
\(836\) −3.68466 −0.127437
\(837\) 10.2462 0.354161
\(838\) 8.31534 0.287249
\(839\) 42.2462 1.45850 0.729251 0.684247i \(-0.239869\pi\)
0.729251 + 0.684247i \(0.239869\pi\)
\(840\) −0.561553 −0.0193754
\(841\) −22.4384 −0.773740
\(842\) −10.0000 −0.344623
\(843\) −6.00000 −0.206651
\(844\) −12.8078 −0.440861
\(845\) 0 0
\(846\) −6.24621 −0.214749
\(847\) 8.93087 0.306868
\(848\) 13.1231 0.450649
\(849\) −13.1231 −0.450384
\(850\) −26.6307 −0.913425
\(851\) 9.57671 0.328285
\(852\) −15.3693 −0.526544
\(853\) 19.1231 0.654763 0.327381 0.944892i \(-0.393834\pi\)
0.327381 + 0.944892i \(0.393834\pi\)
\(854\) 2.56155 0.0876545
\(855\) −1.43845 −0.0491939
\(856\) 13.1231 0.448539
\(857\) 38.0000 1.29806 0.649028 0.760765i \(-0.275176\pi\)
0.649028 + 0.760765i \(0.275176\pi\)
\(858\) 0 0
\(859\) 41.6155 1.41990 0.709952 0.704250i \(-0.248717\pi\)
0.709952 + 0.704250i \(0.248717\pi\)
\(860\) 5.93087 0.202241
\(861\) −4.00000 −0.136320
\(862\) −27.8617 −0.948975
\(863\) −6.73863 −0.229386 −0.114693 0.993401i \(-0.536588\pi\)
−0.114693 + 0.993401i \(0.536588\pi\)
\(864\) 1.00000 0.0340207
\(865\) 11.3693 0.386568
\(866\) −1.36932 −0.0465313
\(867\) −15.3153 −0.520136
\(868\) 10.2462 0.347779
\(869\) 23.0152 0.780736
\(870\) −1.43845 −0.0487679
\(871\) 0 0
\(872\) −15.9309 −0.539487
\(873\) −10.0000 −0.338449
\(874\) −14.5616 −0.492552
\(875\) 5.43845 0.183853
\(876\) −7.43845 −0.251322
\(877\) −42.9848 −1.45150 −0.725748 0.687961i \(-0.758506\pi\)
−0.725748 + 0.687961i \(0.758506\pi\)
\(878\) 19.9309 0.672634
\(879\) 24.2462 0.817804
\(880\) 0.807764 0.0272297
\(881\) 39.3002 1.32406 0.662028 0.749479i \(-0.269696\pi\)
0.662028 + 0.749479i \(0.269696\pi\)
\(882\) −1.00000 −0.0336718
\(883\) 18.5616 0.624646 0.312323 0.949976i \(-0.398893\pi\)
0.312323 + 0.949976i \(0.398893\pi\)
\(884\) 0 0
\(885\) 6.87689 0.231164
\(886\) −31.3693 −1.05387
\(887\) 19.8617 0.666892 0.333446 0.942769i \(-0.391789\pi\)
0.333446 + 0.942769i \(0.391789\pi\)
\(888\) −1.68466 −0.0565334
\(889\) −6.24621 −0.209491
\(890\) 4.49242 0.150586
\(891\) 1.43845 0.0481898
\(892\) 18.2462 0.610928
\(893\) −16.0000 −0.535420
\(894\) −10.0000 −0.334450
\(895\) −1.61553 −0.0540011
\(896\) 1.00000 0.0334077
\(897\) 0 0
\(898\) 21.6847 0.723626
\(899\) 26.2462 0.875360
\(900\) −4.68466 −0.156155
\(901\) −74.6004 −2.48530
\(902\) 5.75379 0.191580
\(903\) 10.5616 0.351466
\(904\) −3.75379 −0.124849
\(905\) −6.38447 −0.212227
\(906\) 4.31534 0.143368
\(907\) −8.49242 −0.281986 −0.140993 0.990011i \(-0.545030\pi\)
−0.140993 + 0.990011i \(0.545030\pi\)
\(908\) −23.6155 −0.783709
\(909\) −6.00000 −0.199007
\(910\) 0 0
\(911\) −8.56155 −0.283657 −0.141828 0.989891i \(-0.545298\pi\)
−0.141828 + 0.989891i \(0.545298\pi\)
\(912\) 2.56155 0.0848215
\(913\) 2.87689 0.0952113
\(914\) −16.0000 −0.529233
\(915\) −1.43845 −0.0475536
\(916\) 2.00000 0.0660819
\(917\) −2.56155 −0.0845899
\(918\) −5.68466 −0.187622
\(919\) 39.8617 1.31492 0.657459 0.753491i \(-0.271631\pi\)
0.657459 + 0.753491i \(0.271631\pi\)
\(920\) 3.19224 0.105245
\(921\) −9.75379 −0.321398
\(922\) 8.06913 0.265743
\(923\) 0 0
\(924\) 1.43845 0.0473214
\(925\) 7.89205 0.259489
\(926\) −39.5464 −1.29958
\(927\) 18.8078 0.617728
\(928\) 2.56155 0.0840871
\(929\) 49.1231 1.61168 0.805838 0.592136i \(-0.201715\pi\)
0.805838 + 0.592136i \(0.201715\pi\)
\(930\) −5.75379 −0.188674
\(931\) −2.56155 −0.0839515
\(932\) −2.00000 −0.0655122
\(933\) 32.4924 1.06375
\(934\) −2.56155 −0.0838166
\(935\) −4.59186 −0.150170
\(936\) 0 0
\(937\) 37.8617 1.23689 0.618445 0.785828i \(-0.287763\pi\)
0.618445 + 0.785828i \(0.287763\pi\)
\(938\) −7.12311 −0.232578
\(939\) −15.7538 −0.514105
\(940\) 3.50758 0.114405
\(941\) −6.49242 −0.211647 −0.105823 0.994385i \(-0.533748\pi\)
−0.105823 + 0.994385i \(0.533748\pi\)
\(942\) 8.80776 0.286972
\(943\) 22.7386 0.740472
\(944\) −12.2462 −0.398580
\(945\) 0.561553 0.0182673
\(946\) −15.1922 −0.493942
\(947\) 7.19224 0.233716 0.116858 0.993149i \(-0.462718\pi\)
0.116858 + 0.993149i \(0.462718\pi\)
\(948\) −16.0000 −0.519656
\(949\) 0 0
\(950\) −12.0000 −0.389331
\(951\) 21.3693 0.692948
\(952\) −5.68466 −0.184241
\(953\) 48.7386 1.57880 0.789400 0.613880i \(-0.210392\pi\)
0.789400 + 0.613880i \(0.210392\pi\)
\(954\) −13.1231 −0.424876
\(955\) 7.33050 0.237209
\(956\) 2.24621 0.0726477
\(957\) 3.68466 0.119108
\(958\) 41.3002 1.33435
\(959\) 5.68466 0.183567
\(960\) −0.561553 −0.0181240
\(961\) 73.9848 2.38661
\(962\) 0 0
\(963\) −13.1231 −0.422886
\(964\) 6.00000 0.193247
\(965\) −6.73863 −0.216924
\(966\) 5.68466 0.182901
\(967\) 25.9309 0.833881 0.416940 0.908934i \(-0.363102\pi\)
0.416940 + 0.908934i \(0.363102\pi\)
\(968\) 8.93087 0.287049
\(969\) −14.5616 −0.467784
\(970\) 5.61553 0.180304
\(971\) 4.00000 0.128366 0.0641831 0.997938i \(-0.479556\pi\)
0.0641831 + 0.997938i \(0.479556\pi\)
\(972\) −1.00000 −0.0320750
\(973\) 6.24621 0.200244
\(974\) −30.2462 −0.969151
\(975\) 0 0
\(976\) 2.56155 0.0819933
\(977\) −39.7926 −1.27308 −0.636539 0.771244i \(-0.719635\pi\)
−0.636539 + 0.771244i \(0.719635\pi\)
\(978\) 0.876894 0.0280400
\(979\) −11.5076 −0.367784
\(980\) 0.561553 0.0179381
\(981\) 15.9309 0.508634
\(982\) 6.24621 0.199325
\(983\) −6.56155 −0.209281 −0.104641 0.994510i \(-0.533369\pi\)
−0.104641 + 0.994510i \(0.533369\pi\)
\(984\) −4.00000 −0.127515
\(985\) 3.36932 0.107355
\(986\) −14.5616 −0.463734
\(987\) 6.24621 0.198819
\(988\) 0 0
\(989\) −60.0388 −1.90912
\(990\) −0.807764 −0.0256724
\(991\) −13.7538 −0.436903 −0.218452 0.975848i \(-0.570101\pi\)
−0.218452 + 0.975848i \(0.570101\pi\)
\(992\) 10.2462 0.325318
\(993\) 7.12311 0.226045
\(994\) 15.3693 0.487485
\(995\) 13.4384 0.426027
\(996\) −2.00000 −0.0633724
\(997\) −59.3693 −1.88025 −0.940123 0.340837i \(-0.889290\pi\)
−0.940123 + 0.340837i \(0.889290\pi\)
\(998\) −34.4924 −1.09184
\(999\) 1.68466 0.0533002
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 7098.2.a.bg.1.2 2
13.5 odd 4 546.2.c.e.337.3 yes 4
13.8 odd 4 546.2.c.e.337.2 4
13.12 even 2 7098.2.a.bv.1.1 2
39.5 even 4 1638.2.c.h.883.2 4
39.8 even 4 1638.2.c.h.883.3 4
52.31 even 4 4368.2.h.n.337.2 4
52.47 even 4 4368.2.h.n.337.3 4
91.34 even 4 3822.2.c.h.883.1 4
91.83 even 4 3822.2.c.h.883.4 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
546.2.c.e.337.2 4 13.8 odd 4
546.2.c.e.337.3 yes 4 13.5 odd 4
1638.2.c.h.883.2 4 39.5 even 4
1638.2.c.h.883.3 4 39.8 even 4
3822.2.c.h.883.1 4 91.34 even 4
3822.2.c.h.883.4 4 91.83 even 4
4368.2.h.n.337.2 4 52.31 even 4
4368.2.h.n.337.3 4 52.47 even 4
7098.2.a.bg.1.2 2 1.1 even 1 trivial
7098.2.a.bv.1.1 2 13.12 even 2