Properties

Label 55.2.a.b.1.1
Level $55$
Weight $2$
Character 55.1
Self dual yes
Analytic conductor $0.439$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 55 = 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 55.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(0.439177211117\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{8})^+\)
Defining polynomial: \( x^{2} - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.41421\) of defining polynomial
Character \(\chi\) \(=\) 55.1

$q$-expansion

\(f(q)\) \(=\) \(q-0.414214 q^{2} +2.82843 q^{3} -1.82843 q^{4} -1.00000 q^{5} -1.17157 q^{6} -2.00000 q^{7} +1.58579 q^{8} +5.00000 q^{9} +O(q^{10})\) \(q-0.414214 q^{2} +2.82843 q^{3} -1.82843 q^{4} -1.00000 q^{5} -1.17157 q^{6} -2.00000 q^{7} +1.58579 q^{8} +5.00000 q^{9} +0.414214 q^{10} +1.00000 q^{11} -5.17157 q^{12} -6.82843 q^{13} +0.828427 q^{14} -2.82843 q^{15} +3.00000 q^{16} +1.17157 q^{17} -2.07107 q^{18} +1.82843 q^{20} -5.65685 q^{21} -0.414214 q^{22} +2.82843 q^{23} +4.48528 q^{24} +1.00000 q^{25} +2.82843 q^{26} +5.65685 q^{27} +3.65685 q^{28} +7.65685 q^{29} +1.17157 q^{30} -4.41421 q^{32} +2.82843 q^{33} -0.485281 q^{34} +2.00000 q^{35} -9.14214 q^{36} +3.65685 q^{37} -19.3137 q^{39} -1.58579 q^{40} +6.00000 q^{41} +2.34315 q^{42} -6.00000 q^{43} -1.82843 q^{44} -5.00000 q^{45} -1.17157 q^{46} -2.82843 q^{47} +8.48528 q^{48} -3.00000 q^{49} -0.414214 q^{50} +3.31371 q^{51} +12.4853 q^{52} +0.343146 q^{53} -2.34315 q^{54} -1.00000 q^{55} -3.17157 q^{56} -3.17157 q^{58} -9.65685 q^{59} +5.17157 q^{60} +13.3137 q^{61} -10.0000 q^{63} -4.17157 q^{64} +6.82843 q^{65} -1.17157 q^{66} -4.48528 q^{67} -2.14214 q^{68} +8.00000 q^{69} -0.828427 q^{70} -11.3137 q^{71} +7.92893 q^{72} -6.82843 q^{73} -1.51472 q^{74} +2.82843 q^{75} -2.00000 q^{77} +8.00000 q^{78} +4.00000 q^{79} -3.00000 q^{80} +1.00000 q^{81} -2.48528 q^{82} -6.00000 q^{83} +10.3431 q^{84} -1.17157 q^{85} +2.48528 q^{86} +21.6569 q^{87} +1.58579 q^{88} +9.31371 q^{89} +2.07107 q^{90} +13.6569 q^{91} -5.17157 q^{92} +1.17157 q^{94} -12.4853 q^{96} -7.65685 q^{97} +1.24264 q^{98} +5.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} + 2 q^{4} - 2 q^{5} - 8 q^{6} - 4 q^{7} + 6 q^{8} + 10 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{2} + 2 q^{4} - 2 q^{5} - 8 q^{6} - 4 q^{7} + 6 q^{8} + 10 q^{9} - 2 q^{10} + 2 q^{11} - 16 q^{12} - 8 q^{13} - 4 q^{14} + 6 q^{16} + 8 q^{17} + 10 q^{18} - 2 q^{20} + 2 q^{22} - 8 q^{24} + 2 q^{25} - 4 q^{28} + 4 q^{29} + 8 q^{30} - 6 q^{32} + 16 q^{34} + 4 q^{35} + 10 q^{36} - 4 q^{37} - 16 q^{39} - 6 q^{40} + 12 q^{41} + 16 q^{42} - 12 q^{43} + 2 q^{44} - 10 q^{45} - 8 q^{46} - 6 q^{49} + 2 q^{50} - 16 q^{51} + 8 q^{52} + 12 q^{53} - 16 q^{54} - 2 q^{55} - 12 q^{56} - 12 q^{58} - 8 q^{59} + 16 q^{60} + 4 q^{61} - 20 q^{63} - 14 q^{64} + 8 q^{65} - 8 q^{66} + 8 q^{67} + 24 q^{68} + 16 q^{69} + 4 q^{70} + 30 q^{72} - 8 q^{73} - 20 q^{74} - 4 q^{77} + 16 q^{78} + 8 q^{79} - 6 q^{80} + 2 q^{81} + 12 q^{82} - 12 q^{83} + 32 q^{84} - 8 q^{85} - 12 q^{86} + 32 q^{87} + 6 q^{88} - 4 q^{89} - 10 q^{90} + 16 q^{91} - 16 q^{92} + 8 q^{94} - 8 q^{96} - 4 q^{97} - 6 q^{98} + 10 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.414214 −0.292893 −0.146447 0.989219i \(-0.546784\pi\)
−0.146447 + 0.989219i \(0.546784\pi\)
\(3\) 2.82843 1.63299 0.816497 0.577350i \(-0.195913\pi\)
0.816497 + 0.577350i \(0.195913\pi\)
\(4\) −1.82843 −0.914214
\(5\) −1.00000 −0.447214
\(6\) −1.17157 −0.478293
\(7\) −2.00000 −0.755929 −0.377964 0.925820i \(-0.623376\pi\)
−0.377964 + 0.925820i \(0.623376\pi\)
\(8\) 1.58579 0.560660
\(9\) 5.00000 1.66667
\(10\) 0.414214 0.130986
\(11\) 1.00000 0.301511
\(12\) −5.17157 −1.49290
\(13\) −6.82843 −1.89386 −0.946932 0.321433i \(-0.895836\pi\)
−0.946932 + 0.321433i \(0.895836\pi\)
\(14\) 0.828427 0.221406
\(15\) −2.82843 −0.730297
\(16\) 3.00000 0.750000
\(17\) 1.17157 0.284148 0.142074 0.989856i \(-0.454623\pi\)
0.142074 + 0.989856i \(0.454623\pi\)
\(18\) −2.07107 −0.488155
\(19\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(20\) 1.82843 0.408849
\(21\) −5.65685 −1.23443
\(22\) −0.414214 −0.0883106
\(23\) 2.82843 0.589768 0.294884 0.955533i \(-0.404719\pi\)
0.294884 + 0.955533i \(0.404719\pi\)
\(24\) 4.48528 0.915554
\(25\) 1.00000 0.200000
\(26\) 2.82843 0.554700
\(27\) 5.65685 1.08866
\(28\) 3.65685 0.691080
\(29\) 7.65685 1.42184 0.710921 0.703272i \(-0.248278\pi\)
0.710921 + 0.703272i \(0.248278\pi\)
\(30\) 1.17157 0.213899
\(31\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) −4.41421 −0.780330
\(33\) 2.82843 0.492366
\(34\) −0.485281 −0.0832251
\(35\) 2.00000 0.338062
\(36\) −9.14214 −1.52369
\(37\) 3.65685 0.601183 0.300592 0.953753i \(-0.402816\pi\)
0.300592 + 0.953753i \(0.402816\pi\)
\(38\) 0 0
\(39\) −19.3137 −3.09267
\(40\) −1.58579 −0.250735
\(41\) 6.00000 0.937043 0.468521 0.883452i \(-0.344787\pi\)
0.468521 + 0.883452i \(0.344787\pi\)
\(42\) 2.34315 0.361555
\(43\) −6.00000 −0.914991 −0.457496 0.889212i \(-0.651253\pi\)
−0.457496 + 0.889212i \(0.651253\pi\)
\(44\) −1.82843 −0.275646
\(45\) −5.00000 −0.745356
\(46\) −1.17157 −0.172739
\(47\) −2.82843 −0.412568 −0.206284 0.978492i \(-0.566137\pi\)
−0.206284 + 0.978492i \(0.566137\pi\)
\(48\) 8.48528 1.22474
\(49\) −3.00000 −0.428571
\(50\) −0.414214 −0.0585786
\(51\) 3.31371 0.464012
\(52\) 12.4853 1.73140
\(53\) 0.343146 0.0471347 0.0235673 0.999722i \(-0.492498\pi\)
0.0235673 + 0.999722i \(0.492498\pi\)
\(54\) −2.34315 −0.318862
\(55\) −1.00000 −0.134840
\(56\) −3.17157 −0.423819
\(57\) 0 0
\(58\) −3.17157 −0.416448
\(59\) −9.65685 −1.25722 −0.628608 0.777723i \(-0.716375\pi\)
−0.628608 + 0.777723i \(0.716375\pi\)
\(60\) 5.17157 0.667647
\(61\) 13.3137 1.70465 0.852323 0.523016i \(-0.175193\pi\)
0.852323 + 0.523016i \(0.175193\pi\)
\(62\) 0 0
\(63\) −10.0000 −1.25988
\(64\) −4.17157 −0.521447
\(65\) 6.82843 0.846962
\(66\) −1.17157 −0.144211
\(67\) −4.48528 −0.547964 −0.273982 0.961735i \(-0.588341\pi\)
−0.273982 + 0.961735i \(0.588341\pi\)
\(68\) −2.14214 −0.259772
\(69\) 8.00000 0.963087
\(70\) −0.828427 −0.0990160
\(71\) −11.3137 −1.34269 −0.671345 0.741145i \(-0.734283\pi\)
−0.671345 + 0.741145i \(0.734283\pi\)
\(72\) 7.92893 0.934434
\(73\) −6.82843 −0.799207 −0.399603 0.916688i \(-0.630852\pi\)
−0.399603 + 0.916688i \(0.630852\pi\)
\(74\) −1.51472 −0.176082
\(75\) 2.82843 0.326599
\(76\) 0 0
\(77\) −2.00000 −0.227921
\(78\) 8.00000 0.905822
\(79\) 4.00000 0.450035 0.225018 0.974355i \(-0.427756\pi\)
0.225018 + 0.974355i \(0.427756\pi\)
\(80\) −3.00000 −0.335410
\(81\) 1.00000 0.111111
\(82\) −2.48528 −0.274453
\(83\) −6.00000 −0.658586 −0.329293 0.944228i \(-0.606810\pi\)
−0.329293 + 0.944228i \(0.606810\pi\)
\(84\) 10.3431 1.12853
\(85\) −1.17157 −0.127075
\(86\) 2.48528 0.267995
\(87\) 21.6569 2.32186
\(88\) 1.58579 0.169045
\(89\) 9.31371 0.987251 0.493626 0.869675i \(-0.335671\pi\)
0.493626 + 0.869675i \(0.335671\pi\)
\(90\) 2.07107 0.218310
\(91\) 13.6569 1.43163
\(92\) −5.17157 −0.539174
\(93\) 0 0
\(94\) 1.17157 0.120839
\(95\) 0 0
\(96\) −12.4853 −1.27427
\(97\) −7.65685 −0.777436 −0.388718 0.921357i \(-0.627082\pi\)
−0.388718 + 0.921357i \(0.627082\pi\)
\(98\) 1.24264 0.125526
\(99\) 5.00000 0.502519
\(100\) −1.82843 −0.182843
\(101\) −13.3137 −1.32476 −0.662382 0.749166i \(-0.730454\pi\)
−0.662382 + 0.749166i \(0.730454\pi\)
\(102\) −1.37258 −0.135906
\(103\) 1.17157 0.115439 0.0577193 0.998333i \(-0.481617\pi\)
0.0577193 + 0.998333i \(0.481617\pi\)
\(104\) −10.8284 −1.06181
\(105\) 5.65685 0.552052
\(106\) −0.142136 −0.0138054
\(107\) −3.65685 −0.353521 −0.176761 0.984254i \(-0.556562\pi\)
−0.176761 + 0.984254i \(0.556562\pi\)
\(108\) −10.3431 −0.995270
\(109\) 3.65685 0.350263 0.175132 0.984545i \(-0.443965\pi\)
0.175132 + 0.984545i \(0.443965\pi\)
\(110\) 0.414214 0.0394937
\(111\) 10.3431 0.981728
\(112\) −6.00000 −0.566947
\(113\) 8.34315 0.784857 0.392429 0.919782i \(-0.371635\pi\)
0.392429 + 0.919782i \(0.371635\pi\)
\(114\) 0 0
\(115\) −2.82843 −0.263752
\(116\) −14.0000 −1.29987
\(117\) −34.1421 −3.15644
\(118\) 4.00000 0.368230
\(119\) −2.34315 −0.214796
\(120\) −4.48528 −0.409448
\(121\) 1.00000 0.0909091
\(122\) −5.51472 −0.499279
\(123\) 16.9706 1.53018
\(124\) 0 0
\(125\) −1.00000 −0.0894427
\(126\) 4.14214 0.369011
\(127\) 15.6569 1.38932 0.694661 0.719338i \(-0.255555\pi\)
0.694661 + 0.719338i \(0.255555\pi\)
\(128\) 10.5563 0.933058
\(129\) −16.9706 −1.49417
\(130\) −2.82843 −0.248069
\(131\) 11.3137 0.988483 0.494242 0.869325i \(-0.335446\pi\)
0.494242 + 0.869325i \(0.335446\pi\)
\(132\) −5.17157 −0.450128
\(133\) 0 0
\(134\) 1.85786 0.160495
\(135\) −5.65685 −0.486864
\(136\) 1.85786 0.159311
\(137\) 22.9706 1.96251 0.981254 0.192720i \(-0.0617309\pi\)
0.981254 + 0.192720i \(0.0617309\pi\)
\(138\) −3.31371 −0.282082
\(139\) −4.00000 −0.339276 −0.169638 0.985506i \(-0.554260\pi\)
−0.169638 + 0.985506i \(0.554260\pi\)
\(140\) −3.65685 −0.309061
\(141\) −8.00000 −0.673722
\(142\) 4.68629 0.393265
\(143\) −6.82843 −0.571022
\(144\) 15.0000 1.25000
\(145\) −7.65685 −0.635867
\(146\) 2.82843 0.234082
\(147\) −8.48528 −0.699854
\(148\) −6.68629 −0.549610
\(149\) 11.6569 0.954967 0.477483 0.878641i \(-0.341549\pi\)
0.477483 + 0.878641i \(0.341549\pi\)
\(150\) −1.17157 −0.0956585
\(151\) −12.0000 −0.976546 −0.488273 0.872691i \(-0.662373\pi\)
−0.488273 + 0.872691i \(0.662373\pi\)
\(152\) 0 0
\(153\) 5.85786 0.473580
\(154\) 0.828427 0.0667566
\(155\) 0 0
\(156\) 35.3137 2.82736
\(157\) −14.0000 −1.11732 −0.558661 0.829396i \(-0.688685\pi\)
−0.558661 + 0.829396i \(0.688685\pi\)
\(158\) −1.65685 −0.131812
\(159\) 0.970563 0.0769706
\(160\) 4.41421 0.348974
\(161\) −5.65685 −0.445823
\(162\) −0.414214 −0.0325437
\(163\) −0.485281 −0.0380102 −0.0190051 0.999819i \(-0.506050\pi\)
−0.0190051 + 0.999819i \(0.506050\pi\)
\(164\) −10.9706 −0.856657
\(165\) −2.82843 −0.220193
\(166\) 2.48528 0.192895
\(167\) 10.9706 0.848928 0.424464 0.905445i \(-0.360463\pi\)
0.424464 + 0.905445i \(0.360463\pi\)
\(168\) −8.97056 −0.692094
\(169\) 33.6274 2.58672
\(170\) 0.485281 0.0372194
\(171\) 0 0
\(172\) 10.9706 0.836498
\(173\) 6.14214 0.466978 0.233489 0.972359i \(-0.424986\pi\)
0.233489 + 0.972359i \(0.424986\pi\)
\(174\) −8.97056 −0.680057
\(175\) −2.00000 −0.151186
\(176\) 3.00000 0.226134
\(177\) −27.3137 −2.05302
\(178\) −3.85786 −0.289159
\(179\) −1.65685 −0.123839 −0.0619196 0.998081i \(-0.519722\pi\)
−0.0619196 + 0.998081i \(0.519722\pi\)
\(180\) 9.14214 0.681415
\(181\) −1.31371 −0.0976472 −0.0488236 0.998807i \(-0.515547\pi\)
−0.0488236 + 0.998807i \(0.515547\pi\)
\(182\) −5.65685 −0.419314
\(183\) 37.6569 2.78367
\(184\) 4.48528 0.330659
\(185\) −3.65685 −0.268857
\(186\) 0 0
\(187\) 1.17157 0.0856739
\(188\) 5.17157 0.377176
\(189\) −11.3137 −0.822951
\(190\) 0 0
\(191\) −19.3137 −1.39749 −0.698745 0.715370i \(-0.746258\pi\)
−0.698745 + 0.715370i \(0.746258\pi\)
\(192\) −11.7990 −0.851519
\(193\) −6.82843 −0.491521 −0.245760 0.969331i \(-0.579038\pi\)
−0.245760 + 0.969331i \(0.579038\pi\)
\(194\) 3.17157 0.227706
\(195\) 19.3137 1.38308
\(196\) 5.48528 0.391806
\(197\) −5.17157 −0.368459 −0.184230 0.982883i \(-0.558979\pi\)
−0.184230 + 0.982883i \(0.558979\pi\)
\(198\) −2.07107 −0.147184
\(199\) 21.6569 1.53521 0.767607 0.640921i \(-0.221447\pi\)
0.767607 + 0.640921i \(0.221447\pi\)
\(200\) 1.58579 0.112132
\(201\) −12.6863 −0.894822
\(202\) 5.51472 0.388014
\(203\) −15.3137 −1.07481
\(204\) −6.05887 −0.424206
\(205\) −6.00000 −0.419058
\(206\) −0.485281 −0.0338112
\(207\) 14.1421 0.982946
\(208\) −20.4853 −1.42040
\(209\) 0 0
\(210\) −2.34315 −0.161692
\(211\) −16.0000 −1.10149 −0.550743 0.834675i \(-0.685655\pi\)
−0.550743 + 0.834675i \(0.685655\pi\)
\(212\) −0.627417 −0.0430912
\(213\) −32.0000 −2.19260
\(214\) 1.51472 0.103544
\(215\) 6.00000 0.409197
\(216\) 8.97056 0.610369
\(217\) 0 0
\(218\) −1.51472 −0.102590
\(219\) −19.3137 −1.30510
\(220\) 1.82843 0.123273
\(221\) −8.00000 −0.538138
\(222\) −4.28427 −0.287541
\(223\) −5.17157 −0.346314 −0.173157 0.984894i \(-0.555397\pi\)
−0.173157 + 0.984894i \(0.555397\pi\)
\(224\) 8.82843 0.589874
\(225\) 5.00000 0.333333
\(226\) −3.45584 −0.229879
\(227\) 2.68629 0.178295 0.0891477 0.996018i \(-0.471586\pi\)
0.0891477 + 0.996018i \(0.471586\pi\)
\(228\) 0 0
\(229\) −21.3137 −1.40845 −0.704225 0.709977i \(-0.748705\pi\)
−0.704225 + 0.709977i \(0.748705\pi\)
\(230\) 1.17157 0.0772512
\(231\) −5.65685 −0.372194
\(232\) 12.1421 0.797170
\(233\) 22.1421 1.45058 0.725290 0.688444i \(-0.241706\pi\)
0.725290 + 0.688444i \(0.241706\pi\)
\(234\) 14.1421 0.924500
\(235\) 2.82843 0.184506
\(236\) 17.6569 1.14936
\(237\) 11.3137 0.734904
\(238\) 0.970563 0.0629122
\(239\) −0.686292 −0.0443925 −0.0221963 0.999754i \(-0.507066\pi\)
−0.0221963 + 0.999754i \(0.507066\pi\)
\(240\) −8.48528 −0.547723
\(241\) 6.00000 0.386494 0.193247 0.981150i \(-0.438098\pi\)
0.193247 + 0.981150i \(0.438098\pi\)
\(242\) −0.414214 −0.0266267
\(243\) −14.1421 −0.907218
\(244\) −24.3431 −1.55841
\(245\) 3.00000 0.191663
\(246\) −7.02944 −0.448181
\(247\) 0 0
\(248\) 0 0
\(249\) −16.9706 −1.07547
\(250\) 0.414214 0.0261972
\(251\) 12.0000 0.757433 0.378717 0.925513i \(-0.376365\pi\)
0.378717 + 0.925513i \(0.376365\pi\)
\(252\) 18.2843 1.15180
\(253\) 2.82843 0.177822
\(254\) −6.48528 −0.406923
\(255\) −3.31371 −0.207512
\(256\) 3.97056 0.248160
\(257\) 13.3137 0.830486 0.415243 0.909710i \(-0.363696\pi\)
0.415243 + 0.909710i \(0.363696\pi\)
\(258\) 7.02944 0.437634
\(259\) −7.31371 −0.454452
\(260\) −12.4853 −0.774304
\(261\) 38.2843 2.36974
\(262\) −4.68629 −0.289520
\(263\) 22.9706 1.41643 0.708213 0.705999i \(-0.249502\pi\)
0.708213 + 0.705999i \(0.249502\pi\)
\(264\) 4.48528 0.276050
\(265\) −0.343146 −0.0210793
\(266\) 0 0
\(267\) 26.3431 1.61217
\(268\) 8.20101 0.500956
\(269\) −5.31371 −0.323983 −0.161991 0.986792i \(-0.551792\pi\)
−0.161991 + 0.986792i \(0.551792\pi\)
\(270\) 2.34315 0.142599
\(271\) −15.3137 −0.930242 −0.465121 0.885247i \(-0.653989\pi\)
−0.465121 + 0.885247i \(0.653989\pi\)
\(272\) 3.51472 0.213111
\(273\) 38.6274 2.33784
\(274\) −9.51472 −0.574805
\(275\) 1.00000 0.0603023
\(276\) −14.6274 −0.880467
\(277\) 1.17157 0.0703930 0.0351965 0.999380i \(-0.488794\pi\)
0.0351965 + 0.999380i \(0.488794\pi\)
\(278\) 1.65685 0.0993715
\(279\) 0 0
\(280\) 3.17157 0.189538
\(281\) −5.31371 −0.316989 −0.158495 0.987360i \(-0.550664\pi\)
−0.158495 + 0.987360i \(0.550664\pi\)
\(282\) 3.31371 0.197328
\(283\) −12.6274 −0.750622 −0.375311 0.926899i \(-0.622464\pi\)
−0.375311 + 0.926899i \(0.622464\pi\)
\(284\) 20.6863 1.22751
\(285\) 0 0
\(286\) 2.82843 0.167248
\(287\) −12.0000 −0.708338
\(288\) −22.0711 −1.30055
\(289\) −15.6274 −0.919260
\(290\) 3.17157 0.186241
\(291\) −21.6569 −1.26955
\(292\) 12.4853 0.730646
\(293\) −14.8284 −0.866286 −0.433143 0.901325i \(-0.642595\pi\)
−0.433143 + 0.901325i \(0.642595\pi\)
\(294\) 3.51472 0.204983
\(295\) 9.65685 0.562244
\(296\) 5.79899 0.337059
\(297\) 5.65685 0.328244
\(298\) −4.82843 −0.279703
\(299\) −19.3137 −1.11694
\(300\) −5.17157 −0.298581
\(301\) 12.0000 0.691669
\(302\) 4.97056 0.286024
\(303\) −37.6569 −2.16333
\(304\) 0 0
\(305\) −13.3137 −0.762341
\(306\) −2.42641 −0.138708
\(307\) −27.6569 −1.57846 −0.789230 0.614098i \(-0.789520\pi\)
−0.789230 + 0.614098i \(0.789520\pi\)
\(308\) 3.65685 0.208369
\(309\) 3.31371 0.188510
\(310\) 0 0
\(311\) 27.3137 1.54882 0.774409 0.632685i \(-0.218047\pi\)
0.774409 + 0.632685i \(0.218047\pi\)
\(312\) −30.6274 −1.73394
\(313\) 21.3137 1.20472 0.602361 0.798224i \(-0.294227\pi\)
0.602361 + 0.798224i \(0.294227\pi\)
\(314\) 5.79899 0.327256
\(315\) 10.0000 0.563436
\(316\) −7.31371 −0.411428
\(317\) 21.3137 1.19710 0.598549 0.801087i \(-0.295744\pi\)
0.598549 + 0.801087i \(0.295744\pi\)
\(318\) −0.402020 −0.0225442
\(319\) 7.65685 0.428702
\(320\) 4.17157 0.233198
\(321\) −10.3431 −0.577298
\(322\) 2.34315 0.130578
\(323\) 0 0
\(324\) −1.82843 −0.101579
\(325\) −6.82843 −0.378773
\(326\) 0.201010 0.0111329
\(327\) 10.3431 0.571977
\(328\) 9.51472 0.525362
\(329\) 5.65685 0.311872
\(330\) 1.17157 0.0644930
\(331\) 15.3137 0.841718 0.420859 0.907126i \(-0.361729\pi\)
0.420859 + 0.907126i \(0.361729\pi\)
\(332\) 10.9706 0.602088
\(333\) 18.2843 1.00197
\(334\) −4.54416 −0.248645
\(335\) 4.48528 0.245057
\(336\) −16.9706 −0.925820
\(337\) −3.51472 −0.191459 −0.0957295 0.995407i \(-0.530518\pi\)
−0.0957295 + 0.995407i \(0.530518\pi\)
\(338\) −13.9289 −0.757634
\(339\) 23.5980 1.28167
\(340\) 2.14214 0.116174
\(341\) 0 0
\(342\) 0 0
\(343\) 20.0000 1.07990
\(344\) −9.51472 −0.512999
\(345\) −8.00000 −0.430706
\(346\) −2.54416 −0.136775
\(347\) −22.9706 −1.23312 −0.616562 0.787306i \(-0.711475\pi\)
−0.616562 + 0.787306i \(0.711475\pi\)
\(348\) −39.5980 −2.12267
\(349\) −6.97056 −0.373126 −0.186563 0.982443i \(-0.559735\pi\)
−0.186563 + 0.982443i \(0.559735\pi\)
\(350\) 0.828427 0.0442813
\(351\) −38.6274 −2.06178
\(352\) −4.41421 −0.235278
\(353\) −1.31371 −0.0699216 −0.0349608 0.999389i \(-0.511131\pi\)
−0.0349608 + 0.999389i \(0.511131\pi\)
\(354\) 11.3137 0.601317
\(355\) 11.3137 0.600469
\(356\) −17.0294 −0.902558
\(357\) −6.62742 −0.350760
\(358\) 0.686292 0.0362716
\(359\) 23.3137 1.23045 0.615225 0.788351i \(-0.289065\pi\)
0.615225 + 0.788351i \(0.289065\pi\)
\(360\) −7.92893 −0.417891
\(361\) −19.0000 −1.00000
\(362\) 0.544156 0.0286002
\(363\) 2.82843 0.148454
\(364\) −24.9706 −1.30881
\(365\) 6.82843 0.357416
\(366\) −15.5980 −0.815319
\(367\) 8.48528 0.442928 0.221464 0.975169i \(-0.428916\pi\)
0.221464 + 0.975169i \(0.428916\pi\)
\(368\) 8.48528 0.442326
\(369\) 30.0000 1.56174
\(370\) 1.51472 0.0787465
\(371\) −0.686292 −0.0356305
\(372\) 0 0
\(373\) −3.79899 −0.196704 −0.0983521 0.995152i \(-0.531357\pi\)
−0.0983521 + 0.995152i \(0.531357\pi\)
\(374\) −0.485281 −0.0250933
\(375\) −2.82843 −0.146059
\(376\) −4.48528 −0.231311
\(377\) −52.2843 −2.69278
\(378\) 4.68629 0.241037
\(379\) 22.3431 1.14769 0.573845 0.818964i \(-0.305451\pi\)
0.573845 + 0.818964i \(0.305451\pi\)
\(380\) 0 0
\(381\) 44.2843 2.26875
\(382\) 8.00000 0.409316
\(383\) −34.1421 −1.74458 −0.872291 0.488987i \(-0.837366\pi\)
−0.872291 + 0.488987i \(0.837366\pi\)
\(384\) 29.8579 1.52368
\(385\) 2.00000 0.101929
\(386\) 2.82843 0.143963
\(387\) −30.0000 −1.52499
\(388\) 14.0000 0.710742
\(389\) −24.6274 −1.24866 −0.624330 0.781161i \(-0.714628\pi\)
−0.624330 + 0.781161i \(0.714628\pi\)
\(390\) −8.00000 −0.405096
\(391\) 3.31371 0.167581
\(392\) −4.75736 −0.240283
\(393\) 32.0000 1.61419
\(394\) 2.14214 0.107919
\(395\) −4.00000 −0.201262
\(396\) −9.14214 −0.459410
\(397\) 13.3137 0.668196 0.334098 0.942538i \(-0.391568\pi\)
0.334098 + 0.942538i \(0.391568\pi\)
\(398\) −8.97056 −0.449654
\(399\) 0 0
\(400\) 3.00000 0.150000
\(401\) 17.3137 0.864605 0.432303 0.901729i \(-0.357701\pi\)
0.432303 + 0.901729i \(0.357701\pi\)
\(402\) 5.25483 0.262087
\(403\) 0 0
\(404\) 24.3431 1.21112
\(405\) −1.00000 −0.0496904
\(406\) 6.34315 0.314805
\(407\) 3.65685 0.181264
\(408\) 5.25483 0.260153
\(409\) 34.9706 1.72918 0.864592 0.502475i \(-0.167577\pi\)
0.864592 + 0.502475i \(0.167577\pi\)
\(410\) 2.48528 0.122739
\(411\) 64.9706 3.20476
\(412\) −2.14214 −0.105535
\(413\) 19.3137 0.950365
\(414\) −5.85786 −0.287898
\(415\) 6.00000 0.294528
\(416\) 30.1421 1.47784
\(417\) −11.3137 −0.554035
\(418\) 0 0
\(419\) −14.3431 −0.700709 −0.350354 0.936617i \(-0.613939\pi\)
−0.350354 + 0.936617i \(0.613939\pi\)
\(420\) −10.3431 −0.504694
\(421\) −6.00000 −0.292422 −0.146211 0.989253i \(-0.546708\pi\)
−0.146211 + 0.989253i \(0.546708\pi\)
\(422\) 6.62742 0.322618
\(423\) −14.1421 −0.687614
\(424\) 0.544156 0.0264265
\(425\) 1.17157 0.0568296
\(426\) 13.2548 0.642199
\(427\) −26.6274 −1.28859
\(428\) 6.68629 0.323194
\(429\) −19.3137 −0.932475
\(430\) −2.48528 −0.119851
\(431\) 11.3137 0.544962 0.272481 0.962161i \(-0.412156\pi\)
0.272481 + 0.962161i \(0.412156\pi\)
\(432\) 16.9706 0.816497
\(433\) 3.65685 0.175737 0.0878686 0.996132i \(-0.471994\pi\)
0.0878686 + 0.996132i \(0.471994\pi\)
\(434\) 0 0
\(435\) −21.6569 −1.03837
\(436\) −6.68629 −0.320215
\(437\) 0 0
\(438\) 8.00000 0.382255
\(439\) −16.0000 −0.763638 −0.381819 0.924237i \(-0.624702\pi\)
−0.381819 + 0.924237i \(0.624702\pi\)
\(440\) −1.58579 −0.0755994
\(441\) −15.0000 −0.714286
\(442\) 3.31371 0.157617
\(443\) −21.1716 −1.00589 −0.502946 0.864318i \(-0.667751\pi\)
−0.502946 + 0.864318i \(0.667751\pi\)
\(444\) −18.9117 −0.897509
\(445\) −9.31371 −0.441512
\(446\) 2.14214 0.101433
\(447\) 32.9706 1.55945
\(448\) 8.34315 0.394177
\(449\) −16.6274 −0.784696 −0.392348 0.919817i \(-0.628337\pi\)
−0.392348 + 0.919817i \(0.628337\pi\)
\(450\) −2.07107 −0.0976311
\(451\) 6.00000 0.282529
\(452\) −15.2548 −0.717527
\(453\) −33.9411 −1.59469
\(454\) −1.11270 −0.0522215
\(455\) −13.6569 −0.640243
\(456\) 0 0
\(457\) −16.4853 −0.771149 −0.385574 0.922677i \(-0.625997\pi\)
−0.385574 + 0.922677i \(0.625997\pi\)
\(458\) 8.82843 0.412525
\(459\) 6.62742 0.309341
\(460\) 5.17157 0.241126
\(461\) −32.6274 −1.51961 −0.759805 0.650151i \(-0.774706\pi\)
−0.759805 + 0.650151i \(0.774706\pi\)
\(462\) 2.34315 0.109013
\(463\) 22.1421 1.02903 0.514516 0.857481i \(-0.327972\pi\)
0.514516 + 0.857481i \(0.327972\pi\)
\(464\) 22.9706 1.06638
\(465\) 0 0
\(466\) −9.17157 −0.424865
\(467\) −9.17157 −0.424410 −0.212205 0.977225i \(-0.568064\pi\)
−0.212205 + 0.977225i \(0.568064\pi\)
\(468\) 62.4264 2.88566
\(469\) 8.97056 0.414222
\(470\) −1.17157 −0.0540406
\(471\) −39.5980 −1.82458
\(472\) −15.3137 −0.704871
\(473\) −6.00000 −0.275880
\(474\) −4.68629 −0.215248
\(475\) 0 0
\(476\) 4.28427 0.196369
\(477\) 1.71573 0.0785578
\(478\) 0.284271 0.0130023
\(479\) −36.0000 −1.64488 −0.822441 0.568850i \(-0.807388\pi\)
−0.822441 + 0.568850i \(0.807388\pi\)
\(480\) 12.4853 0.569873
\(481\) −24.9706 −1.13856
\(482\) −2.48528 −0.113201
\(483\) −16.0000 −0.728025
\(484\) −1.82843 −0.0831103
\(485\) 7.65685 0.347680
\(486\) 5.85786 0.265718
\(487\) −7.51472 −0.340524 −0.170262 0.985399i \(-0.554461\pi\)
−0.170262 + 0.985399i \(0.554461\pi\)
\(488\) 21.1127 0.955727
\(489\) −1.37258 −0.0620703
\(490\) −1.24264 −0.0561368
\(491\) −23.3137 −1.05213 −0.526066 0.850443i \(-0.676334\pi\)
−0.526066 + 0.850443i \(0.676334\pi\)
\(492\) −31.0294 −1.39892
\(493\) 8.97056 0.404014
\(494\) 0 0
\(495\) −5.00000 −0.224733
\(496\) 0 0
\(497\) 22.6274 1.01498
\(498\) 7.02944 0.314997
\(499\) −1.65685 −0.0741710 −0.0370855 0.999312i \(-0.511807\pi\)
−0.0370855 + 0.999312i \(0.511807\pi\)
\(500\) 1.82843 0.0817697
\(501\) 31.0294 1.38629
\(502\) −4.97056 −0.221847
\(503\) −28.6274 −1.27643 −0.638217 0.769857i \(-0.720328\pi\)
−0.638217 + 0.769857i \(0.720328\pi\)
\(504\) −15.8579 −0.706365
\(505\) 13.3137 0.592452
\(506\) −1.17157 −0.0520828
\(507\) 95.1127 4.22410
\(508\) −28.6274 −1.27014
\(509\) 9.31371 0.412823 0.206411 0.978465i \(-0.433821\pi\)
0.206411 + 0.978465i \(0.433821\pi\)
\(510\) 1.37258 0.0607790
\(511\) 13.6569 0.604144
\(512\) −22.7574 −1.00574
\(513\) 0 0
\(514\) −5.51472 −0.243244
\(515\) −1.17157 −0.0516257
\(516\) 31.0294 1.36599
\(517\) −2.82843 −0.124394
\(518\) 3.02944 0.133106
\(519\) 17.3726 0.762572
\(520\) 10.8284 0.474858
\(521\) 2.68629 0.117689 0.0588443 0.998267i \(-0.481258\pi\)
0.0588443 + 0.998267i \(0.481258\pi\)
\(522\) −15.8579 −0.694080
\(523\) 37.5980 1.64404 0.822022 0.569455i \(-0.192846\pi\)
0.822022 + 0.569455i \(0.192846\pi\)
\(524\) −20.6863 −0.903685
\(525\) −5.65685 −0.246885
\(526\) −9.51472 −0.414861
\(527\) 0 0
\(528\) 8.48528 0.369274
\(529\) −15.0000 −0.652174
\(530\) 0.142136 0.00617398
\(531\) −48.2843 −2.09536
\(532\) 0 0
\(533\) −40.9706 −1.77463
\(534\) −10.9117 −0.472195
\(535\) 3.65685 0.158100
\(536\) −7.11270 −0.307222
\(537\) −4.68629 −0.202228
\(538\) 2.20101 0.0948923
\(539\) −3.00000 −0.129219
\(540\) 10.3431 0.445098
\(541\) 6.00000 0.257960 0.128980 0.991647i \(-0.458830\pi\)
0.128980 + 0.991647i \(0.458830\pi\)
\(542\) 6.34315 0.272461
\(543\) −3.71573 −0.159457
\(544\) −5.17157 −0.221729
\(545\) −3.65685 −0.156642
\(546\) −16.0000 −0.684737
\(547\) −34.0000 −1.45374 −0.726868 0.686778i \(-0.759025\pi\)
−0.726868 + 0.686778i \(0.759025\pi\)
\(548\) −42.0000 −1.79415
\(549\) 66.5685 2.84108
\(550\) −0.414214 −0.0176621
\(551\) 0 0
\(552\) 12.6863 0.539964
\(553\) −8.00000 −0.340195
\(554\) −0.485281 −0.0206176
\(555\) −10.3431 −0.439042
\(556\) 7.31371 0.310170
\(557\) 38.1421 1.61613 0.808067 0.589090i \(-0.200514\pi\)
0.808067 + 0.589090i \(0.200514\pi\)
\(558\) 0 0
\(559\) 40.9706 1.73287
\(560\) 6.00000 0.253546
\(561\) 3.31371 0.139905
\(562\) 2.20101 0.0928440
\(563\) 11.6569 0.491278 0.245639 0.969361i \(-0.421002\pi\)
0.245639 + 0.969361i \(0.421002\pi\)
\(564\) 14.6274 0.615925
\(565\) −8.34315 −0.350999
\(566\) 5.23045 0.219852
\(567\) −2.00000 −0.0839921
\(568\) −17.9411 −0.752793
\(569\) 20.3431 0.852829 0.426415 0.904528i \(-0.359776\pi\)
0.426415 + 0.904528i \(0.359776\pi\)
\(570\) 0 0
\(571\) 45.9411 1.92258 0.961288 0.275545i \(-0.0888584\pi\)
0.961288 + 0.275545i \(0.0888584\pi\)
\(572\) 12.4853 0.522036
\(573\) −54.6274 −2.28209
\(574\) 4.97056 0.207467
\(575\) 2.82843 0.117954
\(576\) −20.8579 −0.869078
\(577\) 6.97056 0.290188 0.145094 0.989418i \(-0.453651\pi\)
0.145094 + 0.989418i \(0.453651\pi\)
\(578\) 6.47309 0.269245
\(579\) −19.3137 −0.802650
\(580\) 14.0000 0.581318
\(581\) 12.0000 0.497844
\(582\) 8.97056 0.371842
\(583\) 0.343146 0.0142116
\(584\) −10.8284 −0.448084
\(585\) 34.1421 1.41160
\(586\) 6.14214 0.253729
\(587\) 26.1421 1.07900 0.539501 0.841985i \(-0.318613\pi\)
0.539501 + 0.841985i \(0.318613\pi\)
\(588\) 15.5147 0.639816
\(589\) 0 0
\(590\) −4.00000 −0.164677
\(591\) −14.6274 −0.601692
\(592\) 10.9706 0.450887
\(593\) 20.4853 0.841230 0.420615 0.907239i \(-0.361814\pi\)
0.420615 + 0.907239i \(0.361814\pi\)
\(594\) −2.34315 −0.0961404
\(595\) 2.34315 0.0960596
\(596\) −21.3137 −0.873044
\(597\) 61.2548 2.50699
\(598\) 8.00000 0.327144
\(599\) 5.65685 0.231133 0.115566 0.993300i \(-0.463132\pi\)
0.115566 + 0.993300i \(0.463132\pi\)
\(600\) 4.48528 0.183111
\(601\) −43.9411 −1.79240 −0.896198 0.443654i \(-0.853682\pi\)
−0.896198 + 0.443654i \(0.853682\pi\)
\(602\) −4.97056 −0.202585
\(603\) −22.4264 −0.913274
\(604\) 21.9411 0.892772
\(605\) −1.00000 −0.0406558
\(606\) 15.5980 0.633625
\(607\) −18.2843 −0.742136 −0.371068 0.928606i \(-0.621008\pi\)
−0.371068 + 0.928606i \(0.621008\pi\)
\(608\) 0 0
\(609\) −43.3137 −1.75516
\(610\) 5.51472 0.223284
\(611\) 19.3137 0.781349
\(612\) −10.7107 −0.432954
\(613\) 25.4558 1.02815 0.514076 0.857745i \(-0.328135\pi\)
0.514076 + 0.857745i \(0.328135\pi\)
\(614\) 11.4558 0.462320
\(615\) −16.9706 −0.684319
\(616\) −3.17157 −0.127786
\(617\) 11.6569 0.469287 0.234644 0.972081i \(-0.424608\pi\)
0.234644 + 0.972081i \(0.424608\pi\)
\(618\) −1.37258 −0.0552134
\(619\) −25.6569 −1.03124 −0.515618 0.856819i \(-0.672438\pi\)
−0.515618 + 0.856819i \(0.672438\pi\)
\(620\) 0 0
\(621\) 16.0000 0.642058
\(622\) −11.3137 −0.453638
\(623\) −18.6274 −0.746292
\(624\) −57.9411 −2.31950
\(625\) 1.00000 0.0400000
\(626\) −8.82843 −0.352855
\(627\) 0 0
\(628\) 25.5980 1.02147
\(629\) 4.28427 0.170825
\(630\) −4.14214 −0.165027
\(631\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(632\) 6.34315 0.252317
\(633\) −45.2548 −1.79872
\(634\) −8.82843 −0.350622
\(635\) −15.6569 −0.621323
\(636\) −1.77460 −0.0703676
\(637\) 20.4853 0.811656
\(638\) −3.17157 −0.125564
\(639\) −56.5685 −2.23782
\(640\) −10.5563 −0.417276
\(641\) 30.0000 1.18493 0.592464 0.805597i \(-0.298155\pi\)
0.592464 + 0.805597i \(0.298155\pi\)
\(642\) 4.28427 0.169087
\(643\) 49.4558 1.95035 0.975174 0.221440i \(-0.0710756\pi\)
0.975174 + 0.221440i \(0.0710756\pi\)
\(644\) 10.3431 0.407577
\(645\) 16.9706 0.668215
\(646\) 0 0
\(647\) −35.1127 −1.38042 −0.690211 0.723608i \(-0.742482\pi\)
−0.690211 + 0.723608i \(0.742482\pi\)
\(648\) 1.58579 0.0622956
\(649\) −9.65685 −0.379065
\(650\) 2.82843 0.110940
\(651\) 0 0
\(652\) 0.887302 0.0347494
\(653\) 0.343146 0.0134283 0.00671417 0.999977i \(-0.497863\pi\)
0.00671417 + 0.999977i \(0.497863\pi\)
\(654\) −4.28427 −0.167528
\(655\) −11.3137 −0.442063
\(656\) 18.0000 0.702782
\(657\) −34.1421 −1.33201
\(658\) −2.34315 −0.0913453
\(659\) −21.9411 −0.854705 −0.427352 0.904085i \(-0.640554\pi\)
−0.427352 + 0.904085i \(0.640554\pi\)
\(660\) 5.17157 0.201303
\(661\) −0.627417 −0.0244037 −0.0122018 0.999926i \(-0.503884\pi\)
−0.0122018 + 0.999926i \(0.503884\pi\)
\(662\) −6.34315 −0.246533
\(663\) −22.6274 −0.878776
\(664\) −9.51472 −0.369243
\(665\) 0 0
\(666\) −7.57359 −0.293471
\(667\) 21.6569 0.838557
\(668\) −20.0589 −0.776101
\(669\) −14.6274 −0.565529
\(670\) −1.85786 −0.0717756
\(671\) 13.3137 0.513970
\(672\) 24.9706 0.963260
\(673\) 4.48528 0.172895 0.0864474 0.996256i \(-0.472449\pi\)
0.0864474 + 0.996256i \(0.472449\pi\)
\(674\) 1.45584 0.0560770
\(675\) 5.65685 0.217732
\(676\) −61.4853 −2.36482
\(677\) 17.1716 0.659957 0.329979 0.943988i \(-0.392958\pi\)
0.329979 + 0.943988i \(0.392958\pi\)
\(678\) −9.77460 −0.375391
\(679\) 15.3137 0.587686
\(680\) −1.85786 −0.0712458
\(681\) 7.59798 0.291155
\(682\) 0 0
\(683\) 31.7990 1.21675 0.608377 0.793648i \(-0.291821\pi\)
0.608377 + 0.793648i \(0.291821\pi\)
\(684\) 0 0
\(685\) −22.9706 −0.877660
\(686\) −8.28427 −0.316295
\(687\) −60.2843 −2.29999
\(688\) −18.0000 −0.686244
\(689\) −2.34315 −0.0892667
\(690\) 3.31371 0.126151
\(691\) −16.6863 −0.634776 −0.317388 0.948296i \(-0.602806\pi\)
−0.317388 + 0.948296i \(0.602806\pi\)
\(692\) −11.2304 −0.426918
\(693\) −10.0000 −0.379869
\(694\) 9.51472 0.361174
\(695\) 4.00000 0.151729
\(696\) 34.3431 1.30177
\(697\) 7.02944 0.266259
\(698\) 2.88730 0.109286
\(699\) 62.6274 2.36879
\(700\) 3.65685 0.138216
\(701\) 32.6274 1.23232 0.616160 0.787621i \(-0.288687\pi\)
0.616160 + 0.787621i \(0.288687\pi\)
\(702\) 16.0000 0.603881
\(703\) 0 0
\(704\) −4.17157 −0.157222
\(705\) 8.00000 0.301297
\(706\) 0.544156 0.0204796
\(707\) 26.6274 1.00143
\(708\) 49.9411 1.87690
\(709\) −20.6274 −0.774679 −0.387339 0.921937i \(-0.626606\pi\)
−0.387339 + 0.921937i \(0.626606\pi\)
\(710\) −4.68629 −0.175873
\(711\) 20.0000 0.750059
\(712\) 14.7696 0.553512
\(713\) 0 0
\(714\) 2.74517 0.102735
\(715\) 6.82843 0.255369
\(716\) 3.02944 0.113215
\(717\) −1.94113 −0.0724927
\(718\) −9.65685 −0.360391
\(719\) 29.6569 1.10601 0.553007 0.833177i \(-0.313480\pi\)
0.553007 + 0.833177i \(0.313480\pi\)
\(720\) −15.0000 −0.559017
\(721\) −2.34315 −0.0872633
\(722\) 7.87006 0.292893
\(723\) 16.9706 0.631142
\(724\) 2.40202 0.0892704
\(725\) 7.65685 0.284368
\(726\) −1.17157 −0.0434811
\(727\) −36.4853 −1.35316 −0.676582 0.736367i \(-0.736540\pi\)
−0.676582 + 0.736367i \(0.736540\pi\)
\(728\) 21.6569 0.802656
\(729\) −43.0000 −1.59259
\(730\) −2.82843 −0.104685
\(731\) −7.02944 −0.259993
\(732\) −68.8528 −2.54487
\(733\) 33.4558 1.23572 0.617860 0.786288i \(-0.288000\pi\)
0.617860 + 0.786288i \(0.288000\pi\)
\(734\) −3.51472 −0.129731
\(735\) 8.48528 0.312984
\(736\) −12.4853 −0.460214
\(737\) −4.48528 −0.165217
\(738\) −12.4264 −0.457422
\(739\) −37.9411 −1.39569 −0.697843 0.716250i \(-0.745857\pi\)
−0.697843 + 0.716250i \(0.745857\pi\)
\(740\) 6.68629 0.245793
\(741\) 0 0
\(742\) 0.284271 0.0104359
\(743\) 29.5980 1.08584 0.542922 0.839783i \(-0.317318\pi\)
0.542922 + 0.839783i \(0.317318\pi\)
\(744\) 0 0
\(745\) −11.6569 −0.427074
\(746\) 1.57359 0.0576133
\(747\) −30.0000 −1.09764
\(748\) −2.14214 −0.0783242
\(749\) 7.31371 0.267237
\(750\) 1.17157 0.0427798
\(751\) −16.0000 −0.583848 −0.291924 0.956441i \(-0.594295\pi\)
−0.291924 + 0.956441i \(0.594295\pi\)
\(752\) −8.48528 −0.309426
\(753\) 33.9411 1.23688
\(754\) 21.6569 0.788696
\(755\) 12.0000 0.436725
\(756\) 20.6863 0.752353
\(757\) −9.31371 −0.338512 −0.169256 0.985572i \(-0.554137\pi\)
−0.169256 + 0.985572i \(0.554137\pi\)
\(758\) −9.25483 −0.336151
\(759\) 8.00000 0.290382
\(760\) 0 0
\(761\) −30.0000 −1.08750 −0.543750 0.839248i \(-0.682996\pi\)
−0.543750 + 0.839248i \(0.682996\pi\)
\(762\) −18.3431 −0.664502
\(763\) −7.31371 −0.264774
\(764\) 35.3137 1.27761
\(765\) −5.85786 −0.211792
\(766\) 14.1421 0.510976
\(767\) 65.9411 2.38100
\(768\) 11.2304 0.405244
\(769\) 14.9706 0.539852 0.269926 0.962881i \(-0.413001\pi\)
0.269926 + 0.962881i \(0.413001\pi\)
\(770\) −0.828427 −0.0298544
\(771\) 37.6569 1.35618
\(772\) 12.4853 0.449355
\(773\) −30.2843 −1.08925 −0.544625 0.838680i \(-0.683328\pi\)
−0.544625 + 0.838680i \(0.683328\pi\)
\(774\) 12.4264 0.446658
\(775\) 0 0
\(776\) −12.1421 −0.435877
\(777\) −20.6863 −0.742117
\(778\) 10.2010 0.365724
\(779\) 0 0
\(780\) −35.3137 −1.26443
\(781\) −11.3137 −0.404836
\(782\) −1.37258 −0.0490835
\(783\) 43.3137 1.54791
\(784\) −9.00000 −0.321429
\(785\) 14.0000 0.499681
\(786\) −13.2548 −0.472784
\(787\) 18.9706 0.676228 0.338114 0.941105i \(-0.390211\pi\)
0.338114 + 0.941105i \(0.390211\pi\)
\(788\) 9.45584 0.336850
\(789\) 64.9706 2.31301
\(790\) 1.65685 0.0589482
\(791\) −16.6863 −0.593296
\(792\) 7.92893 0.281742
\(793\) −90.9117 −3.22837
\(794\) −5.51472 −0.195710
\(795\) −0.970563 −0.0344223
\(796\) −39.5980 −1.40351
\(797\) −12.6274 −0.447286 −0.223643 0.974671i \(-0.571795\pi\)
−0.223643 + 0.974671i \(0.571795\pi\)
\(798\) 0 0
\(799\) −3.31371 −0.117231
\(800\) −4.41421 −0.156066
\(801\) 46.5685 1.64542
\(802\) −7.17157 −0.253237
\(803\) −6.82843 −0.240970
\(804\) 23.1960 0.818058
\(805\) 5.65685 0.199378
\(806\) 0 0
\(807\) −15.0294 −0.529061
\(808\) −21.1127 −0.742742
\(809\) 22.9706 0.807602 0.403801 0.914847i \(-0.367689\pi\)
0.403801 + 0.914847i \(0.367689\pi\)
\(810\) 0.414214 0.0145540
\(811\) −13.9411 −0.489539 −0.244770 0.969581i \(-0.578712\pi\)
−0.244770 + 0.969581i \(0.578712\pi\)
\(812\) 28.0000 0.982607
\(813\) −43.3137 −1.51908
\(814\) −1.51472 −0.0530909
\(815\) 0.485281 0.0169987
\(816\) 9.94113 0.348009
\(817\) 0 0
\(818\) −14.4853 −0.506466
\(819\) 68.2843 2.38605
\(820\) 10.9706 0.383109
\(821\) −18.6863 −0.652156 −0.326078 0.945343i \(-0.605727\pi\)
−0.326078 + 0.945343i \(0.605727\pi\)
\(822\) −26.9117 −0.938653
\(823\) 36.4853 1.27180 0.635898 0.771773i \(-0.280630\pi\)
0.635898 + 0.771773i \(0.280630\pi\)
\(824\) 1.85786 0.0647218
\(825\) 2.82843 0.0984732
\(826\) −8.00000 −0.278356
\(827\) −34.2843 −1.19218 −0.596090 0.802917i \(-0.703280\pi\)
−0.596090 + 0.802917i \(0.703280\pi\)
\(828\) −25.8579 −0.898623
\(829\) 18.0000 0.625166 0.312583 0.949890i \(-0.398806\pi\)
0.312583 + 0.949890i \(0.398806\pi\)
\(830\) −2.48528 −0.0862654
\(831\) 3.31371 0.114951
\(832\) 28.4853 0.987549
\(833\) −3.51472 −0.121778
\(834\) 4.68629 0.162273
\(835\) −10.9706 −0.379652
\(836\) 0 0
\(837\) 0 0
\(838\) 5.94113 0.205233
\(839\) 37.6569 1.30006 0.650029 0.759909i \(-0.274757\pi\)
0.650029 + 0.759909i \(0.274757\pi\)
\(840\) 8.97056 0.309514
\(841\) 29.6274 1.02164
\(842\) 2.48528 0.0856485
\(843\) −15.0294 −0.517641
\(844\) 29.2548 1.00699
\(845\) −33.6274 −1.15682
\(846\) 5.85786 0.201398
\(847\) −2.00000 −0.0687208
\(848\) 1.02944 0.0353510
\(849\) −35.7157 −1.22576
\(850\) −0.485281 −0.0166450
\(851\) 10.3431 0.354558
\(852\) 58.5097 2.00451
\(853\) −32.4853 −1.11227 −0.556137 0.831090i \(-0.687717\pi\)
−0.556137 + 0.831090i \(0.687717\pi\)
\(854\) 11.0294 0.377420
\(855\) 0 0
\(856\) −5.79899 −0.198205
\(857\) −48.7696 −1.66594 −0.832968 0.553321i \(-0.813360\pi\)
−0.832968 + 0.553321i \(0.813360\pi\)
\(858\) 8.00000 0.273115
\(859\) −32.2843 −1.10153 −0.550763 0.834662i \(-0.685663\pi\)
−0.550763 + 0.834662i \(0.685663\pi\)
\(860\) −10.9706 −0.374093
\(861\) −33.9411 −1.15671
\(862\) −4.68629 −0.159616
\(863\) −14.8284 −0.504766 −0.252383 0.967627i \(-0.581214\pi\)
−0.252383 + 0.967627i \(0.581214\pi\)
\(864\) −24.9706 −0.849516
\(865\) −6.14214 −0.208839
\(866\) −1.51472 −0.0514722
\(867\) −44.2010 −1.50115
\(868\) 0 0
\(869\) 4.00000 0.135691
\(870\) 8.97056 0.304131
\(871\) 30.6274 1.03777
\(872\) 5.79899 0.196379
\(873\) −38.2843 −1.29573
\(874\) 0 0
\(875\) 2.00000 0.0676123
\(876\) 35.3137 1.19314
\(877\) 1.45584 0.0491604 0.0245802 0.999698i \(-0.492175\pi\)
0.0245802 + 0.999698i \(0.492175\pi\)
\(878\) 6.62742 0.223664
\(879\) −41.9411 −1.41464
\(880\) −3.00000 −0.101130
\(881\) −52.6274 −1.77306 −0.886531 0.462668i \(-0.846892\pi\)
−0.886531 + 0.462668i \(0.846892\pi\)
\(882\) 6.21320 0.209209
\(883\) 42.8284 1.44129 0.720646 0.693304i \(-0.243845\pi\)
0.720646 + 0.693304i \(0.243845\pi\)
\(884\) 14.6274 0.491973
\(885\) 27.3137 0.918140
\(886\) 8.76955 0.294619
\(887\) −18.2843 −0.613926 −0.306963 0.951721i \(-0.599313\pi\)
−0.306963 + 0.951721i \(0.599313\pi\)
\(888\) 16.4020 0.550416
\(889\) −31.3137 −1.05023
\(890\) 3.85786 0.129316
\(891\) 1.00000 0.0335013
\(892\) 9.45584 0.316605
\(893\) 0 0
\(894\) −13.6569 −0.456754
\(895\) 1.65685 0.0553825
\(896\) −21.1127 −0.705326
\(897\) −54.6274 −1.82396
\(898\) 6.88730 0.229832
\(899\) 0 0
\(900\) −9.14214 −0.304738
\(901\) 0.402020 0.0133932
\(902\) −2.48528 −0.0827508
\(903\) 33.9411 1.12949
\(904\) 13.2304 0.440038
\(905\) 1.31371 0.0436691
\(906\) 14.0589 0.467075
\(907\) −44.4853 −1.47711 −0.738555 0.674193i \(-0.764491\pi\)
−0.738555 + 0.674193i \(0.764491\pi\)
\(908\) −4.91169 −0.163000
\(909\) −66.5685 −2.20794
\(910\) 5.65685 0.187523
\(911\) 57.9411 1.91968 0.959838 0.280556i \(-0.0905189\pi\)
0.959838 + 0.280556i \(0.0905189\pi\)
\(912\) 0 0
\(913\) −6.00000 −0.198571
\(914\) 6.82843 0.225864
\(915\) −37.6569 −1.24490
\(916\) 38.9706 1.28762
\(917\) −22.6274 −0.747223
\(918\) −2.74517 −0.0906040
\(919\) −32.0000 −1.05558 −0.527791 0.849374i \(-0.676980\pi\)
−0.527791 + 0.849374i \(0.676980\pi\)
\(920\) −4.48528 −0.147875
\(921\) −78.2254 −2.57761
\(922\) 13.5147 0.445084
\(923\) 77.2548 2.54287
\(924\) 10.3431 0.340265
\(925\) 3.65685 0.120237
\(926\) −9.17157 −0.301397
\(927\) 5.85786 0.192398
\(928\) −33.7990 −1.10951
\(929\) −17.3137 −0.568044 −0.284022 0.958818i \(-0.591669\pi\)
−0.284022 + 0.958818i \(0.591669\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) −40.4853 −1.32614
\(933\) 77.2548 2.52921
\(934\) 3.79899 0.124307
\(935\) −1.17157 −0.0383145
\(936\) −54.1421 −1.76969
\(937\) 49.4558 1.61565 0.807826 0.589421i \(-0.200644\pi\)
0.807826 + 0.589421i \(0.200644\pi\)
\(938\) −3.71573 −0.121323
\(939\) 60.2843 1.96730
\(940\) −5.17157 −0.168678
\(941\) −29.3137 −0.955600 −0.477800 0.878469i \(-0.658566\pi\)
−0.477800 + 0.878469i \(0.658566\pi\)
\(942\) 16.4020 0.534407
\(943\) 16.9706 0.552638
\(944\) −28.9706 −0.942912
\(945\) 11.3137 0.368035
\(946\) 2.48528 0.0808035
\(947\) 46.8284 1.52172 0.760860 0.648916i \(-0.224778\pi\)
0.760860 + 0.648916i \(0.224778\pi\)
\(948\) −20.6863 −0.671860
\(949\) 46.6274 1.51359
\(950\) 0 0
\(951\) 60.2843 1.95485
\(952\) −3.71573 −0.120427
\(953\) 58.8284 1.90564 0.952820 0.303536i \(-0.0981674\pi\)
0.952820 + 0.303536i \(0.0981674\pi\)
\(954\) −0.710678 −0.0230091
\(955\) 19.3137 0.624977
\(956\) 1.25483 0.0405842
\(957\) 21.6569 0.700067
\(958\) 14.9117 0.481775
\(959\) −45.9411 −1.48352
\(960\) 11.7990 0.380811
\(961\) −31.0000 −1.00000
\(962\) 10.3431 0.333476
\(963\) −18.2843 −0.589202
\(964\) −10.9706 −0.353338
\(965\) 6.82843 0.219815
\(966\) 6.62742 0.213234
\(967\) 18.9706 0.610052 0.305026 0.952344i \(-0.401335\pi\)
0.305026 + 0.952344i \(0.401335\pi\)
\(968\) 1.58579 0.0509691
\(969\) 0 0
\(970\) −3.17157 −0.101833
\(971\) −31.3137 −1.00490 −0.502452 0.864605i \(-0.667569\pi\)
−0.502452 + 0.864605i \(0.667569\pi\)
\(972\) 25.8579 0.829391
\(973\) 8.00000 0.256468
\(974\) 3.11270 0.0997373
\(975\) −19.3137 −0.618534
\(976\) 39.9411 1.27848
\(977\) 43.6569 1.39671 0.698353 0.715753i \(-0.253916\pi\)
0.698353 + 0.715753i \(0.253916\pi\)
\(978\) 0.568542 0.0181800
\(979\) 9.31371 0.297667
\(980\) −5.48528 −0.175221
\(981\) 18.2843 0.583772
\(982\) 9.65685 0.308163
\(983\) −50.1421 −1.59929 −0.799643 0.600476i \(-0.794978\pi\)
−0.799643 + 0.600476i \(0.794978\pi\)
\(984\) 26.9117 0.857913
\(985\) 5.17157 0.164780
\(986\) −3.71573 −0.118333
\(987\) 16.0000 0.509286
\(988\) 0 0
\(989\) −16.9706 −0.539633
\(990\) 2.07107 0.0658229
\(991\) 9.94113 0.315790 0.157895 0.987456i \(-0.449529\pi\)
0.157895 + 0.987456i \(0.449529\pi\)
\(992\) 0 0
\(993\) 43.3137 1.37452
\(994\) −9.37258 −0.297280
\(995\) −21.6569 −0.686568
\(996\) 31.0294 0.983205
\(997\) 9.45584 0.299470 0.149735 0.988726i \(-0.452158\pi\)
0.149735 + 0.988726i \(0.452158\pi\)
\(998\) 0.686292 0.0217242
\(999\) 20.6863 0.654485
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 55.2.a.b.1.1 2
3.2 odd 2 495.2.a.b.1.2 2
4.3 odd 2 880.2.a.m.1.1 2
5.2 odd 4 275.2.b.d.199.2 4
5.3 odd 4 275.2.b.d.199.3 4
5.4 even 2 275.2.a.c.1.2 2
7.6 odd 2 2695.2.a.f.1.1 2
8.3 odd 2 3520.2.a.bo.1.2 2
8.5 even 2 3520.2.a.bn.1.1 2
11.2 odd 10 605.2.g.l.81.1 8
11.3 even 5 605.2.g.f.251.1 8
11.4 even 5 605.2.g.f.511.1 8
11.5 even 5 605.2.g.f.366.2 8
11.6 odd 10 605.2.g.l.366.1 8
11.7 odd 10 605.2.g.l.511.2 8
11.8 odd 10 605.2.g.l.251.2 8
11.9 even 5 605.2.g.f.81.2 8
11.10 odd 2 605.2.a.d.1.2 2
12.11 even 2 7920.2.a.ch.1.1 2
13.12 even 2 9295.2.a.g.1.2 2
15.2 even 4 2475.2.c.l.199.3 4
15.8 even 4 2475.2.c.l.199.2 4
15.14 odd 2 2475.2.a.x.1.1 2
20.3 even 4 4400.2.b.q.4049.1 4
20.7 even 4 4400.2.b.q.4049.4 4
20.19 odd 2 4400.2.a.bn.1.2 2
33.32 even 2 5445.2.a.y.1.1 2
44.43 even 2 9680.2.a.bn.1.1 2
55.54 odd 2 3025.2.a.o.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
55.2.a.b.1.1 2 1.1 even 1 trivial
275.2.a.c.1.2 2 5.4 even 2
275.2.b.d.199.2 4 5.2 odd 4
275.2.b.d.199.3 4 5.3 odd 4
495.2.a.b.1.2 2 3.2 odd 2
605.2.a.d.1.2 2 11.10 odd 2
605.2.g.f.81.2 8 11.9 even 5
605.2.g.f.251.1 8 11.3 even 5
605.2.g.f.366.2 8 11.5 even 5
605.2.g.f.511.1 8 11.4 even 5
605.2.g.l.81.1 8 11.2 odd 10
605.2.g.l.251.2 8 11.8 odd 10
605.2.g.l.366.1 8 11.6 odd 10
605.2.g.l.511.2 8 11.7 odd 10
880.2.a.m.1.1 2 4.3 odd 2
2475.2.a.x.1.1 2 15.14 odd 2
2475.2.c.l.199.2 4 15.8 even 4
2475.2.c.l.199.3 4 15.2 even 4
2695.2.a.f.1.1 2 7.6 odd 2
3025.2.a.o.1.1 2 55.54 odd 2
3520.2.a.bn.1.1 2 8.5 even 2
3520.2.a.bo.1.2 2 8.3 odd 2
4400.2.a.bn.1.2 2 20.19 odd 2
4400.2.b.q.4049.1 4 20.3 even 4
4400.2.b.q.4049.4 4 20.7 even 4
5445.2.a.y.1.1 2 33.32 even 2
7920.2.a.ch.1.1 2 12.11 even 2
9295.2.a.g.1.2 2 13.12 even 2
9680.2.a.bn.1.1 2 44.43 even 2