Properties

Label 43.2.a.b.1.1
Level $43$
Weight $2$
Character 43.1
Self dual yes
Analytic conductor $0.343$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 43 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 43.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(0.343356728692\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{8})^+\)
Defining polynomial: \(x^{2} - 2\)
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 \(\chi\) \(=\) 43.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.41421 q^{2} +1.41421 q^{3} +3.41421 q^{5} -2.00000 q^{6} -3.41421 q^{7} +2.82843 q^{8} -1.00000 q^{9} +O(q^{10})\) \(q-1.41421 q^{2} +1.41421 q^{3} +3.41421 q^{5} -2.00000 q^{6} -3.41421 q^{7} +2.82843 q^{8} -1.00000 q^{9} -4.82843 q^{10} -3.82843 q^{11} -1.82843 q^{13} +4.82843 q^{14} +4.82843 q^{15} -4.00000 q^{16} +2.17157 q^{17} +1.41421 q^{18} +0.828427 q^{19} -4.82843 q^{21} +5.41421 q^{22} +6.65685 q^{23} +4.00000 q^{24} +6.65685 q^{25} +2.58579 q^{26} -5.65685 q^{27} -4.24264 q^{29} -6.82843 q^{30} -3.00000 q^{31} -5.41421 q^{33} -3.07107 q^{34} -11.6569 q^{35} +8.48528 q^{37} -1.17157 q^{38} -2.58579 q^{39} +9.65685 q^{40} +1.82843 q^{41} +6.82843 q^{42} +1.00000 q^{43} -3.41421 q^{45} -9.41421 q^{46} +6.00000 q^{47} -5.65685 q^{48} +4.65685 q^{49} -9.41421 q^{50} +3.07107 q^{51} +13.8284 q^{53} +8.00000 q^{54} -13.0711 q^{55} -9.65685 q^{56} +1.17157 q^{57} +6.00000 q^{58} -4.82843 q^{59} -0.242641 q^{61} +4.24264 q^{62} +3.41421 q^{63} +8.00000 q^{64} -6.24264 q^{65} +7.65685 q^{66} -7.48528 q^{67} +9.41421 q^{69} +16.4853 q^{70} -3.17157 q^{71} -2.82843 q^{72} -16.2426 q^{73} -12.0000 q^{74} +9.41421 q^{75} +13.0711 q^{77} +3.65685 q^{78} +4.82843 q^{79} -13.6569 q^{80} -5.00000 q^{81} -2.58579 q^{82} +3.34315 q^{83} +7.41421 q^{85} -1.41421 q^{86} -6.00000 q^{87} -10.8284 q^{88} -1.75736 q^{89} +4.82843 q^{90} +6.24264 q^{91} -4.24264 q^{93} -8.48528 q^{94} +2.82843 q^{95} +1.82843 q^{97} -6.58579 q^{98} +3.82843 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 4q^{5} - 4q^{6} - 4q^{7} - 2q^{9} + O(q^{10}) \) \( 2q + 4q^{5} - 4q^{6} - 4q^{7} - 2q^{9} - 4q^{10} - 2q^{11} + 2q^{13} + 4q^{14} + 4q^{15} - 8q^{16} + 10q^{17} - 4q^{19} - 4q^{21} + 8q^{22} + 2q^{23} + 8q^{24} + 2q^{25} + 8q^{26} - 8q^{30} - 6q^{31} - 8q^{33} + 8q^{34} - 12q^{35} - 8q^{38} - 8q^{39} + 8q^{40} - 2q^{41} + 8q^{42} + 2q^{43} - 4q^{45} - 16q^{46} + 12q^{47} - 2q^{49} - 16q^{50} - 8q^{51} + 22q^{53} + 16q^{54} - 12q^{55} - 8q^{56} + 8q^{57} + 12q^{58} - 4q^{59} + 8q^{61} + 4q^{63} + 16q^{64} - 4q^{65} + 4q^{66} + 2q^{67} + 16q^{69} + 16q^{70} - 12q^{71} - 24q^{73} - 24q^{74} + 16q^{75} + 12q^{77} - 4q^{78} + 4q^{79} - 16q^{80} - 10q^{81} - 8q^{82} + 18q^{83} + 12q^{85} - 12q^{87} - 16q^{88} - 12q^{89} + 4q^{90} + 4q^{91} - 2q^{97} - 16q^{98} + 2q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.41421 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(3\) 1.41421 0.816497 0.408248 0.912871i \(-0.366140\pi\)
0.408248 + 0.912871i \(0.366140\pi\)
\(4\) 0 0
\(5\) 3.41421 1.52688 0.763441 0.645877i \(-0.223508\pi\)
0.763441 + 0.645877i \(0.223508\pi\)
\(6\) −2.00000 −0.816497
\(7\) −3.41421 −1.29045 −0.645226 0.763992i \(-0.723237\pi\)
−0.645226 + 0.763992i \(0.723237\pi\)
\(8\) 2.82843 1.00000
\(9\) −1.00000 −0.333333
\(10\) −4.82843 −1.52688
\(11\) −3.82843 −1.15431 −0.577157 0.816633i \(-0.695838\pi\)
−0.577157 + 0.816633i \(0.695838\pi\)
\(12\) 0 0
\(13\) −1.82843 −0.507114 −0.253557 0.967320i \(-0.581601\pi\)
−0.253557 + 0.967320i \(0.581601\pi\)
\(14\) 4.82843 1.29045
\(15\) 4.82843 1.24669
\(16\) −4.00000 −1.00000
\(17\) 2.17157 0.526684 0.263342 0.964703i \(-0.415175\pi\)
0.263342 + 0.964703i \(0.415175\pi\)
\(18\) 1.41421 0.333333
\(19\) 0.828427 0.190054 0.0950271 0.995475i \(-0.469706\pi\)
0.0950271 + 0.995475i \(0.469706\pi\)
\(20\) 0 0
\(21\) −4.82843 −1.05365
\(22\) 5.41421 1.15431
\(23\) 6.65685 1.38805 0.694025 0.719951i \(-0.255836\pi\)
0.694025 + 0.719951i \(0.255836\pi\)
\(24\) 4.00000 0.816497
\(25\) 6.65685 1.33137
\(26\) 2.58579 0.507114
\(27\) −5.65685 −1.08866
\(28\) 0 0
\(29\) −4.24264 −0.787839 −0.393919 0.919145i \(-0.628881\pi\)
−0.393919 + 0.919145i \(0.628881\pi\)
\(30\) −6.82843 −1.24669
\(31\) −3.00000 −0.538816 −0.269408 0.963026i \(-0.586828\pi\)
−0.269408 + 0.963026i \(0.586828\pi\)
\(32\) 0 0
\(33\) −5.41421 −0.942494
\(34\) −3.07107 −0.526684
\(35\) −11.6569 −1.97037
\(36\) 0 0
\(37\) 8.48528 1.39497 0.697486 0.716599i \(-0.254302\pi\)
0.697486 + 0.716599i \(0.254302\pi\)
\(38\) −1.17157 −0.190054
\(39\) −2.58579 −0.414057
\(40\) 9.65685 1.52688
\(41\) 1.82843 0.285552 0.142776 0.989755i \(-0.454397\pi\)
0.142776 + 0.989755i \(0.454397\pi\)
\(42\) 6.82843 1.05365
\(43\) 1.00000 0.152499
\(44\) 0 0
\(45\) −3.41421 −0.508961
\(46\) −9.41421 −1.38805
\(47\) 6.00000 0.875190 0.437595 0.899172i \(-0.355830\pi\)
0.437595 + 0.899172i \(0.355830\pi\)
\(48\) −5.65685 −0.816497
\(49\) 4.65685 0.665265
\(50\) −9.41421 −1.33137
\(51\) 3.07107 0.430036
\(52\) 0 0
\(53\) 13.8284 1.89948 0.949740 0.313039i \(-0.101347\pi\)
0.949740 + 0.313039i \(0.101347\pi\)
\(54\) 8.00000 1.08866
\(55\) −13.0711 −1.76250
\(56\) −9.65685 −1.29045
\(57\) 1.17157 0.155179
\(58\) 6.00000 0.787839
\(59\) −4.82843 −0.628608 −0.314304 0.949322i \(-0.601771\pi\)
−0.314304 + 0.949322i \(0.601771\pi\)
\(60\) 0 0
\(61\) −0.242641 −0.0310670 −0.0155335 0.999879i \(-0.504945\pi\)
−0.0155335 + 0.999879i \(0.504945\pi\)
\(62\) 4.24264 0.538816
\(63\) 3.41421 0.430150
\(64\) 8.00000 1.00000
\(65\) −6.24264 −0.774304
\(66\) 7.65685 0.942494
\(67\) −7.48528 −0.914473 −0.457236 0.889345i \(-0.651161\pi\)
−0.457236 + 0.889345i \(0.651161\pi\)
\(68\) 0 0
\(69\) 9.41421 1.13334
\(70\) 16.4853 1.97037
\(71\) −3.17157 −0.376396 −0.188198 0.982131i \(-0.560265\pi\)
−0.188198 + 0.982131i \(0.560265\pi\)
\(72\) −2.82843 −0.333333
\(73\) −16.2426 −1.90106 −0.950529 0.310637i \(-0.899458\pi\)
−0.950529 + 0.310637i \(0.899458\pi\)
\(74\) −12.0000 −1.39497
\(75\) 9.41421 1.08706
\(76\) 0 0
\(77\) 13.0711 1.48959
\(78\) 3.65685 0.414057
\(79\) 4.82843 0.543240 0.271620 0.962405i \(-0.412441\pi\)
0.271620 + 0.962405i \(0.412441\pi\)
\(80\) −13.6569 −1.52688
\(81\) −5.00000 −0.555556
\(82\) −2.58579 −0.285552
\(83\) 3.34315 0.366958 0.183479 0.983024i \(-0.441264\pi\)
0.183479 + 0.983024i \(0.441264\pi\)
\(84\) 0 0
\(85\) 7.41421 0.804184
\(86\) −1.41421 −0.152499
\(87\) −6.00000 −0.643268
\(88\) −10.8284 −1.15431
\(89\) −1.75736 −0.186280 −0.0931399 0.995653i \(-0.529690\pi\)
−0.0931399 + 0.995653i \(0.529690\pi\)
\(90\) 4.82843 0.508961
\(91\) 6.24264 0.654407
\(92\) 0 0
\(93\) −4.24264 −0.439941
\(94\) −8.48528 −0.875190
\(95\) 2.82843 0.290191
\(96\) 0 0
\(97\) 1.82843 0.185649 0.0928243 0.995683i \(-0.470411\pi\)
0.0928243 + 0.995683i \(0.470411\pi\)
\(98\) −6.58579 −0.665265
\(99\) 3.82843 0.384771
\(100\) 0 0
\(101\) 5.82843 0.579950 0.289975 0.957034i \(-0.406353\pi\)
0.289975 + 0.957034i \(0.406353\pi\)
\(102\) −4.34315 −0.430036
\(103\) 0.514719 0.0507167 0.0253584 0.999678i \(-0.491927\pi\)
0.0253584 + 0.999678i \(0.491927\pi\)
\(104\) −5.17157 −0.507114
\(105\) −16.4853 −1.60880
\(106\) −19.5563 −1.89948
\(107\) −0.343146 −0.0331732 −0.0165866 0.999862i \(-0.505280\pi\)
−0.0165866 + 0.999862i \(0.505280\pi\)
\(108\) 0 0
\(109\) −19.9706 −1.91283 −0.956416 0.292006i \(-0.905677\pi\)
−0.956416 + 0.292006i \(0.905677\pi\)
\(110\) 18.4853 1.76250
\(111\) 12.0000 1.13899
\(112\) 13.6569 1.29045
\(113\) −6.82843 −0.642364 −0.321182 0.947017i \(-0.604080\pi\)
−0.321182 + 0.947017i \(0.604080\pi\)
\(114\) −1.65685 −0.155179
\(115\) 22.7279 2.11939
\(116\) 0 0
\(117\) 1.82843 0.169038
\(118\) 6.82843 0.628608
\(119\) −7.41421 −0.679660
\(120\) 13.6569 1.24669
\(121\) 3.65685 0.332441
\(122\) 0.343146 0.0310670
\(123\) 2.58579 0.233153
\(124\) 0 0
\(125\) 5.65685 0.505964
\(126\) −4.82843 −0.430150
\(127\) 3.82843 0.339718 0.169859 0.985468i \(-0.445669\pi\)
0.169859 + 0.985468i \(0.445669\pi\)
\(128\) −11.3137 −1.00000
\(129\) 1.41421 0.124515
\(130\) 8.82843 0.774304
\(131\) 9.65685 0.843723 0.421862 0.906660i \(-0.361377\pi\)
0.421862 + 0.906660i \(0.361377\pi\)
\(132\) 0 0
\(133\) −2.82843 −0.245256
\(134\) 10.5858 0.914473
\(135\) −19.3137 −1.66226
\(136\) 6.14214 0.526684
\(137\) −14.4853 −1.23756 −0.618781 0.785564i \(-0.712373\pi\)
−0.618781 + 0.785564i \(0.712373\pi\)
\(138\) −13.3137 −1.13334
\(139\) 5.48528 0.465255 0.232628 0.972566i \(-0.425268\pi\)
0.232628 + 0.972566i \(0.425268\pi\)
\(140\) 0 0
\(141\) 8.48528 0.714590
\(142\) 4.48528 0.376396
\(143\) 7.00000 0.585369
\(144\) 4.00000 0.333333
\(145\) −14.4853 −1.20294
\(146\) 22.9706 1.90106
\(147\) 6.58579 0.543187
\(148\) 0 0
\(149\) 8.48528 0.695141 0.347571 0.937654i \(-0.387007\pi\)
0.347571 + 0.937654i \(0.387007\pi\)
\(150\) −13.3137 −1.08706
\(151\) 18.2426 1.48457 0.742283 0.670087i \(-0.233743\pi\)
0.742283 + 0.670087i \(0.233743\pi\)
\(152\) 2.34315 0.190054
\(153\) −2.17157 −0.175561
\(154\) −18.4853 −1.48959
\(155\) −10.2426 −0.822709
\(156\) 0 0
\(157\) −10.0000 −0.798087 −0.399043 0.916932i \(-0.630658\pi\)
−0.399043 + 0.916932i \(0.630658\pi\)
\(158\) −6.82843 −0.543240
\(159\) 19.5563 1.55092
\(160\) 0 0
\(161\) −22.7279 −1.79121
\(162\) 7.07107 0.555556
\(163\) −11.7574 −0.920907 −0.460454 0.887684i \(-0.652313\pi\)
−0.460454 + 0.887684i \(0.652313\pi\)
\(164\) 0 0
\(165\) −18.4853 −1.43908
\(166\) −4.72792 −0.366958
\(167\) −14.3137 −1.10763 −0.553814 0.832640i \(-0.686828\pi\)
−0.553814 + 0.832640i \(0.686828\pi\)
\(168\) −13.6569 −1.05365
\(169\) −9.65685 −0.742835
\(170\) −10.4853 −0.804184
\(171\) −0.828427 −0.0633514
\(172\) 0 0
\(173\) 23.6569 1.79860 0.899299 0.437335i \(-0.144078\pi\)
0.899299 + 0.437335i \(0.144078\pi\)
\(174\) 8.48528 0.643268
\(175\) −22.7279 −1.71807
\(176\) 15.3137 1.15431
\(177\) −6.82843 −0.513256
\(178\) 2.48528 0.186280
\(179\) −4.58579 −0.342758 −0.171379 0.985205i \(-0.554822\pi\)
−0.171379 + 0.985205i \(0.554822\pi\)
\(180\) 0 0
\(181\) 7.31371 0.543624 0.271812 0.962350i \(-0.412377\pi\)
0.271812 + 0.962350i \(0.412377\pi\)
\(182\) −8.82843 −0.654407
\(183\) −0.343146 −0.0253661
\(184\) 18.8284 1.38805
\(185\) 28.9706 2.12996
\(186\) 6.00000 0.439941
\(187\) −8.31371 −0.607959
\(188\) 0 0
\(189\) 19.3137 1.40487
\(190\) −4.00000 −0.290191
\(191\) −6.14214 −0.444429 −0.222215 0.974998i \(-0.571329\pi\)
−0.222215 + 0.974998i \(0.571329\pi\)
\(192\) 11.3137 0.816497
\(193\) 15.9706 1.14959 0.574793 0.818299i \(-0.305083\pi\)
0.574793 + 0.818299i \(0.305083\pi\)
\(194\) −2.58579 −0.185649
\(195\) −8.82843 −0.632217
\(196\) 0 0
\(197\) −14.1421 −1.00759 −0.503793 0.863825i \(-0.668062\pi\)
−0.503793 + 0.863825i \(0.668062\pi\)
\(198\) −5.41421 −0.384771
\(199\) 7.65685 0.542780 0.271390 0.962469i \(-0.412517\pi\)
0.271390 + 0.962469i \(0.412517\pi\)
\(200\) 18.8284 1.33137
\(201\) −10.5858 −0.746664
\(202\) −8.24264 −0.579950
\(203\) 14.4853 1.01667
\(204\) 0 0
\(205\) 6.24264 0.436005
\(206\) −0.727922 −0.0507167
\(207\) −6.65685 −0.462683
\(208\) 7.31371 0.507114
\(209\) −3.17157 −0.219382
\(210\) 23.3137 1.60880
\(211\) −0.142136 −0.00978502 −0.00489251 0.999988i \(-0.501557\pi\)
−0.00489251 + 0.999988i \(0.501557\pi\)
\(212\) 0 0
\(213\) −4.48528 −0.307326
\(214\) 0.485281 0.0331732
\(215\) 3.41421 0.232847
\(216\) −16.0000 −1.08866
\(217\) 10.2426 0.695316
\(218\) 28.2426 1.91283
\(219\) −22.9706 −1.55221
\(220\) 0 0
\(221\) −3.97056 −0.267089
\(222\) −16.9706 −1.13899
\(223\) 4.10051 0.274590 0.137295 0.990530i \(-0.456159\pi\)
0.137295 + 0.990530i \(0.456159\pi\)
\(224\) 0 0
\(225\) −6.65685 −0.443790
\(226\) 9.65685 0.642364
\(227\) −6.34315 −0.421009 −0.210505 0.977593i \(-0.567511\pi\)
−0.210505 + 0.977593i \(0.567511\pi\)
\(228\) 0 0
\(229\) −25.9706 −1.71618 −0.858092 0.513497i \(-0.828350\pi\)
−0.858092 + 0.513497i \(0.828350\pi\)
\(230\) −32.1421 −2.11939
\(231\) 18.4853 1.21624
\(232\) −12.0000 −0.787839
\(233\) 4.82843 0.316321 0.158160 0.987413i \(-0.449444\pi\)
0.158160 + 0.987413i \(0.449444\pi\)
\(234\) −2.58579 −0.169038
\(235\) 20.4853 1.33631
\(236\) 0 0
\(237\) 6.82843 0.443554
\(238\) 10.4853 0.679660
\(239\) 14.4853 0.936975 0.468487 0.883470i \(-0.344799\pi\)
0.468487 + 0.883470i \(0.344799\pi\)
\(240\) −19.3137 −1.24669
\(241\) 4.00000 0.257663 0.128831 0.991667i \(-0.458877\pi\)
0.128831 + 0.991667i \(0.458877\pi\)
\(242\) −5.17157 −0.332441
\(243\) 9.89949 0.635053
\(244\) 0 0
\(245\) 15.8995 1.01578
\(246\) −3.65685 −0.233153
\(247\) −1.51472 −0.0963792
\(248\) −8.48528 −0.538816
\(249\) 4.72792 0.299620
\(250\) −8.00000 −0.505964
\(251\) 23.1421 1.46072 0.730359 0.683063i \(-0.239353\pi\)
0.730359 + 0.683063i \(0.239353\pi\)
\(252\) 0 0
\(253\) −25.4853 −1.60225
\(254\) −5.41421 −0.339718
\(255\) 10.4853 0.656614
\(256\) 0 0
\(257\) −1.75736 −0.109621 −0.0548105 0.998497i \(-0.517455\pi\)
−0.0548105 + 0.998497i \(0.517455\pi\)
\(258\) −2.00000 −0.124515
\(259\) −28.9706 −1.80014
\(260\) 0 0
\(261\) 4.24264 0.262613
\(262\) −13.6569 −0.843723
\(263\) 0.142136 0.00876446 0.00438223 0.999990i \(-0.498605\pi\)
0.00438223 + 0.999990i \(0.498605\pi\)
\(264\) −15.3137 −0.942494
\(265\) 47.2132 2.90028
\(266\) 4.00000 0.245256
\(267\) −2.48528 −0.152097
\(268\) 0 0
\(269\) −3.00000 −0.182913 −0.0914566 0.995809i \(-0.529152\pi\)
−0.0914566 + 0.995809i \(0.529152\pi\)
\(270\) 27.3137 1.66226
\(271\) −25.9706 −1.57760 −0.788800 0.614650i \(-0.789297\pi\)
−0.788800 + 0.614650i \(0.789297\pi\)
\(272\) −8.68629 −0.526684
\(273\) 8.82843 0.534321
\(274\) 20.4853 1.23756
\(275\) −25.4853 −1.53682
\(276\) 0 0
\(277\) 7.89949 0.474635 0.237317 0.971432i \(-0.423732\pi\)
0.237317 + 0.971432i \(0.423732\pi\)
\(278\) −7.75736 −0.465255
\(279\) 3.00000 0.179605
\(280\) −32.9706 −1.97037
\(281\) −8.65685 −0.516425 −0.258212 0.966088i \(-0.583133\pi\)
−0.258212 + 0.966088i \(0.583133\pi\)
\(282\) −12.0000 −0.714590
\(283\) 18.3137 1.08864 0.544318 0.838879i \(-0.316788\pi\)
0.544318 + 0.838879i \(0.316788\pi\)
\(284\) 0 0
\(285\) 4.00000 0.236940
\(286\) −9.89949 −0.585369
\(287\) −6.24264 −0.368491
\(288\) 0 0
\(289\) −12.2843 −0.722604
\(290\) 20.4853 1.20294
\(291\) 2.58579 0.151581
\(292\) 0 0
\(293\) 10.3431 0.604253 0.302127 0.953268i \(-0.402303\pi\)
0.302127 + 0.953268i \(0.402303\pi\)
\(294\) −9.31371 −0.543187
\(295\) −16.4853 −0.959810
\(296\) 24.0000 1.39497
\(297\) 21.6569 1.25666
\(298\) −12.0000 −0.695141
\(299\) −12.1716 −0.703900
\(300\) 0 0
\(301\) −3.41421 −0.196792
\(302\) −25.7990 −1.48457
\(303\) 8.24264 0.473527
\(304\) −3.31371 −0.190054
\(305\) −0.828427 −0.0474356
\(306\) 3.07107 0.175561
\(307\) 26.7990 1.52950 0.764750 0.644328i \(-0.222863\pi\)
0.764750 + 0.644328i \(0.222863\pi\)
\(308\) 0 0
\(309\) 0.727922 0.0414100
\(310\) 14.4853 0.822709
\(311\) −13.9706 −0.792198 −0.396099 0.918208i \(-0.629636\pi\)
−0.396099 + 0.918208i \(0.629636\pi\)
\(312\) −7.31371 −0.414057
\(313\) −25.2132 −1.42513 −0.712567 0.701604i \(-0.752468\pi\)
−0.712567 + 0.701604i \(0.752468\pi\)
\(314\) 14.1421 0.798087
\(315\) 11.6569 0.656789
\(316\) 0 0
\(317\) 25.8284 1.45067 0.725334 0.688397i \(-0.241685\pi\)
0.725334 + 0.688397i \(0.241685\pi\)
\(318\) −27.6569 −1.55092
\(319\) 16.2426 0.909413
\(320\) 27.3137 1.52688
\(321\) −0.485281 −0.0270858
\(322\) 32.1421 1.79121
\(323\) 1.79899 0.100098
\(324\) 0 0
\(325\) −12.1716 −0.675157
\(326\) 16.6274 0.920907
\(327\) −28.2426 −1.56182
\(328\) 5.17157 0.285552
\(329\) −20.4853 −1.12939
\(330\) 26.1421 1.43908
\(331\) 21.4142 1.17703 0.588516 0.808486i \(-0.299712\pi\)
0.588516 + 0.808486i \(0.299712\pi\)
\(332\) 0 0
\(333\) −8.48528 −0.464991
\(334\) 20.2426 1.10763
\(335\) −25.5563 −1.39629
\(336\) 19.3137 1.05365
\(337\) −5.00000 −0.272367 −0.136184 0.990684i \(-0.543484\pi\)
−0.136184 + 0.990684i \(0.543484\pi\)
\(338\) 13.6569 0.742835
\(339\) −9.65685 −0.524488
\(340\) 0 0
\(341\) 11.4853 0.621963
\(342\) 1.17157 0.0633514
\(343\) 8.00000 0.431959
\(344\) 2.82843 0.152499
\(345\) 32.1421 1.73047
\(346\) −33.4558 −1.79860
\(347\) −4.58579 −0.246178 −0.123089 0.992396i \(-0.539280\pi\)
−0.123089 + 0.992396i \(0.539280\pi\)
\(348\) 0 0
\(349\) −20.2426 −1.08356 −0.541782 0.840519i \(-0.682250\pi\)
−0.541782 + 0.840519i \(0.682250\pi\)
\(350\) 32.1421 1.71807
\(351\) 10.3431 0.552076
\(352\) 0 0
\(353\) 5.82843 0.310216 0.155108 0.987898i \(-0.450427\pi\)
0.155108 + 0.987898i \(0.450427\pi\)
\(354\) 9.65685 0.513256
\(355\) −10.8284 −0.574713
\(356\) 0 0
\(357\) −10.4853 −0.554940
\(358\) 6.48528 0.342758
\(359\) −21.3431 −1.12645 −0.563224 0.826304i \(-0.690439\pi\)
−0.563224 + 0.826304i \(0.690439\pi\)
\(360\) −9.65685 −0.508961
\(361\) −18.3137 −0.963879
\(362\) −10.3431 −0.543624
\(363\) 5.17157 0.271437
\(364\) 0 0
\(365\) −55.4558 −2.90269
\(366\) 0.485281 0.0253661
\(367\) 9.79899 0.511503 0.255752 0.966743i \(-0.417677\pi\)
0.255752 + 0.966743i \(0.417677\pi\)
\(368\) −26.6274 −1.38805
\(369\) −1.82843 −0.0951841
\(370\) −40.9706 −2.12996
\(371\) −47.2132 −2.45119
\(372\) 0 0
\(373\) 8.48528 0.439351 0.219676 0.975573i \(-0.429500\pi\)
0.219676 + 0.975573i \(0.429500\pi\)
\(374\) 11.7574 0.607959
\(375\) 8.00000 0.413118
\(376\) 16.9706 0.875190
\(377\) 7.75736 0.399524
\(378\) −27.3137 −1.40487
\(379\) 30.3137 1.55711 0.778555 0.627576i \(-0.215953\pi\)
0.778555 + 0.627576i \(0.215953\pi\)
\(380\) 0 0
\(381\) 5.41421 0.277379
\(382\) 8.68629 0.444429
\(383\) −20.4853 −1.04675 −0.523374 0.852103i \(-0.675327\pi\)
−0.523374 + 0.852103i \(0.675327\pi\)
\(384\) −16.0000 −0.816497
\(385\) 44.6274 2.27442
\(386\) −22.5858 −1.14959
\(387\) −1.00000 −0.0508329
\(388\) 0 0
\(389\) −16.6274 −0.843044 −0.421522 0.906818i \(-0.638504\pi\)
−0.421522 + 0.906818i \(0.638504\pi\)
\(390\) 12.4853 0.632217
\(391\) 14.4558 0.731063
\(392\) 13.1716 0.665265
\(393\) 13.6569 0.688897
\(394\) 20.0000 1.00759
\(395\) 16.4853 0.829465
\(396\) 0 0
\(397\) 29.4558 1.47835 0.739173 0.673515i \(-0.235216\pi\)
0.739173 + 0.673515i \(0.235216\pi\)
\(398\) −10.8284 −0.542780
\(399\) −4.00000 −0.200250
\(400\) −26.6274 −1.33137
\(401\) −12.5147 −0.624955 −0.312478 0.949925i \(-0.601159\pi\)
−0.312478 + 0.949925i \(0.601159\pi\)
\(402\) 14.9706 0.746664
\(403\) 5.48528 0.273241
\(404\) 0 0
\(405\) −17.0711 −0.848268
\(406\) −20.4853 −1.01667
\(407\) −32.4853 −1.61024
\(408\) 8.68629 0.430036
\(409\) −16.8701 −0.834171 −0.417085 0.908867i \(-0.636948\pi\)
−0.417085 + 0.908867i \(0.636948\pi\)
\(410\) −8.82843 −0.436005
\(411\) −20.4853 −1.01046
\(412\) 0 0
\(413\) 16.4853 0.811188
\(414\) 9.41421 0.462683
\(415\) 11.4142 0.560302
\(416\) 0 0
\(417\) 7.75736 0.379880
\(418\) 4.48528 0.219382
\(419\) −23.8995 −1.16757 −0.583783 0.811909i \(-0.698428\pi\)
−0.583783 + 0.811909i \(0.698428\pi\)
\(420\) 0 0
\(421\) −15.6569 −0.763068 −0.381534 0.924355i \(-0.624604\pi\)
−0.381534 + 0.924355i \(0.624604\pi\)
\(422\) 0.201010 0.00978502
\(423\) −6.00000 −0.291730
\(424\) 39.1127 1.89948
\(425\) 14.4558 0.701211
\(426\) 6.34315 0.307326
\(427\) 0.828427 0.0400904
\(428\) 0 0
\(429\) 9.89949 0.477952
\(430\) −4.82843 −0.232847
\(431\) 23.2843 1.12156 0.560782 0.827964i \(-0.310501\pi\)
0.560782 + 0.827964i \(0.310501\pi\)
\(432\) 22.6274 1.08866
\(433\) 21.7574 1.04559 0.522796 0.852458i \(-0.324889\pi\)
0.522796 + 0.852458i \(0.324889\pi\)
\(434\) −14.4853 −0.695316
\(435\) −20.4853 −0.982194
\(436\) 0 0
\(437\) 5.51472 0.263805
\(438\) 32.4853 1.55221
\(439\) 11.4853 0.548163 0.274081 0.961707i \(-0.411626\pi\)
0.274081 + 0.961707i \(0.411626\pi\)
\(440\) −36.9706 −1.76250
\(441\) −4.65685 −0.221755
\(442\) 5.61522 0.267089
\(443\) 7.85786 0.373338 0.186669 0.982423i \(-0.440231\pi\)
0.186669 + 0.982423i \(0.440231\pi\)
\(444\) 0 0
\(445\) −6.00000 −0.284427
\(446\) −5.79899 −0.274590
\(447\) 12.0000 0.567581
\(448\) −27.3137 −1.29045
\(449\) 33.2132 1.56743 0.783714 0.621122i \(-0.213323\pi\)
0.783714 + 0.621122i \(0.213323\pi\)
\(450\) 9.41421 0.443790
\(451\) −7.00000 −0.329617
\(452\) 0 0
\(453\) 25.7990 1.21214
\(454\) 8.97056 0.421009
\(455\) 21.3137 0.999202
\(456\) 3.31371 0.155179
\(457\) 0.727922 0.0340508 0.0170254 0.999855i \(-0.494580\pi\)
0.0170254 + 0.999855i \(0.494580\pi\)
\(458\) 36.7279 1.71618
\(459\) −12.2843 −0.573381
\(460\) 0 0
\(461\) 42.6274 1.98536 0.992678 0.120788i \(-0.0385420\pi\)
0.992678 + 0.120788i \(0.0385420\pi\)
\(462\) −26.1421 −1.21624
\(463\) −30.7279 −1.42805 −0.714024 0.700121i \(-0.753129\pi\)
−0.714024 + 0.700121i \(0.753129\pi\)
\(464\) 16.9706 0.787839
\(465\) −14.4853 −0.671739
\(466\) −6.82843 −0.316321
\(467\) 23.6569 1.09471 0.547354 0.836901i \(-0.315635\pi\)
0.547354 + 0.836901i \(0.315635\pi\)
\(468\) 0 0
\(469\) 25.5563 1.18008
\(470\) −28.9706 −1.33631
\(471\) −14.1421 −0.651635
\(472\) −13.6569 −0.628608
\(473\) −3.82843 −0.176031
\(474\) −9.65685 −0.443554
\(475\) 5.51472 0.253033
\(476\) 0 0
\(477\) −13.8284 −0.633160
\(478\) −20.4853 −0.936975
\(479\) −6.65685 −0.304159 −0.152080 0.988368i \(-0.548597\pi\)
−0.152080 + 0.988368i \(0.548597\pi\)
\(480\) 0 0
\(481\) −15.5147 −0.707410
\(482\) −5.65685 −0.257663
\(483\) −32.1421 −1.46252
\(484\) 0 0
\(485\) 6.24264 0.283464
\(486\) −14.0000 −0.635053
\(487\) 22.3431 1.01246 0.506232 0.862397i \(-0.331038\pi\)
0.506232 + 0.862397i \(0.331038\pi\)
\(488\) −0.686292 −0.0310670
\(489\) −16.6274 −0.751918
\(490\) −22.4853 −1.01578
\(491\) 13.4142 0.605375 0.302687 0.953090i \(-0.402116\pi\)
0.302687 + 0.953090i \(0.402116\pi\)
\(492\) 0 0
\(493\) −9.21320 −0.414942
\(494\) 2.14214 0.0963792
\(495\) 13.0711 0.587501
\(496\) 12.0000 0.538816
\(497\) 10.8284 0.485721
\(498\) −6.68629 −0.299620
\(499\) −25.7574 −1.15306 −0.576529 0.817077i \(-0.695593\pi\)
−0.576529 + 0.817077i \(0.695593\pi\)
\(500\) 0 0
\(501\) −20.2426 −0.904374
\(502\) −32.7279 −1.46072
\(503\) −30.7696 −1.37195 −0.685973 0.727627i \(-0.740623\pi\)
−0.685973 + 0.727627i \(0.740623\pi\)
\(504\) 9.65685 0.430150
\(505\) 19.8995 0.885516
\(506\) 36.0416 1.60225
\(507\) −13.6569 −0.606522
\(508\) 0 0
\(509\) 0.514719 0.0228145 0.0114073 0.999935i \(-0.496369\pi\)
0.0114073 + 0.999935i \(0.496369\pi\)
\(510\) −14.8284 −0.656614
\(511\) 55.4558 2.45322
\(512\) 22.6274 1.00000
\(513\) −4.68629 −0.206905
\(514\) 2.48528 0.109621
\(515\) 1.75736 0.0774385
\(516\) 0 0
\(517\) −22.9706 −1.01024
\(518\) 40.9706 1.80014
\(519\) 33.4558 1.46855
\(520\) −17.6569 −0.774304
\(521\) 16.9289 0.741670 0.370835 0.928699i \(-0.379072\pi\)
0.370835 + 0.928699i \(0.379072\pi\)
\(522\) −6.00000 −0.262613
\(523\) −39.2132 −1.71467 −0.857337 0.514756i \(-0.827883\pi\)
−0.857337 + 0.514756i \(0.827883\pi\)
\(524\) 0 0
\(525\) −32.1421 −1.40280
\(526\) −0.201010 −0.00876446
\(527\) −6.51472 −0.283786
\(528\) 21.6569 0.942494
\(529\) 21.3137 0.926683
\(530\) −66.7696 −2.90028
\(531\) 4.82843 0.209536
\(532\) 0 0
\(533\) −3.34315 −0.144808
\(534\) 3.51472 0.152097
\(535\) −1.17157 −0.0506515
\(536\) −21.1716 −0.914473
\(537\) −6.48528 −0.279861
\(538\) 4.24264 0.182913
\(539\) −17.8284 −0.767925
\(540\) 0 0
\(541\) 8.79899 0.378298 0.189149 0.981948i \(-0.439427\pi\)
0.189149 + 0.981948i \(0.439427\pi\)
\(542\) 36.7279 1.57760
\(543\) 10.3431 0.443867
\(544\) 0 0
\(545\) −68.1838 −2.92067
\(546\) −12.4853 −0.534321
\(547\) 9.00000 0.384812 0.192406 0.981315i \(-0.438371\pi\)
0.192406 + 0.981315i \(0.438371\pi\)
\(548\) 0 0
\(549\) 0.242641 0.0103557
\(550\) 36.0416 1.53682
\(551\) −3.51472 −0.149732
\(552\) 26.6274 1.13334
\(553\) −16.4853 −0.701025
\(554\) −11.1716 −0.474635
\(555\) 40.9706 1.73910
\(556\) 0 0
\(557\) −9.00000 −0.381342 −0.190671 0.981654i \(-0.561066\pi\)
−0.190671 + 0.981654i \(0.561066\pi\)
\(558\) −4.24264 −0.179605
\(559\) −1.82843 −0.0773342
\(560\) 46.6274 1.97037
\(561\) −11.7574 −0.496396
\(562\) 12.2426 0.516425
\(563\) 39.6274 1.67010 0.835048 0.550177i \(-0.185440\pi\)
0.835048 + 0.550177i \(0.185440\pi\)
\(564\) 0 0
\(565\) −23.3137 −0.980815
\(566\) −25.8995 −1.08864
\(567\) 17.0711 0.716917
\(568\) −8.97056 −0.376396
\(569\) 13.3431 0.559374 0.279687 0.960091i \(-0.409769\pi\)
0.279687 + 0.960091i \(0.409769\pi\)
\(570\) −5.65685 −0.236940
\(571\) −3.07107 −0.128520 −0.0642601 0.997933i \(-0.520469\pi\)
−0.0642601 + 0.997933i \(0.520469\pi\)
\(572\) 0 0
\(573\) −8.68629 −0.362875
\(574\) 8.82843 0.368491
\(575\) 44.3137 1.84801
\(576\) −8.00000 −0.333333
\(577\) 33.0711 1.37677 0.688383 0.725347i \(-0.258321\pi\)
0.688383 + 0.725347i \(0.258321\pi\)
\(578\) 17.3726 0.722604
\(579\) 22.5858 0.938633
\(580\) 0 0
\(581\) −11.4142 −0.473541
\(582\) −3.65685 −0.151581
\(583\) −52.9411 −2.19260
\(584\) −45.9411 −1.90106
\(585\) 6.24264 0.258101
\(586\) −14.6274 −0.604253
\(587\) 33.7990 1.39503 0.697517 0.716568i \(-0.254288\pi\)
0.697517 + 0.716568i \(0.254288\pi\)
\(588\) 0 0
\(589\) −2.48528 −0.102404
\(590\) 23.3137 0.959810
\(591\) −20.0000 −0.822690
\(592\) −33.9411 −1.39497
\(593\) −9.07107 −0.372504 −0.186252 0.982502i \(-0.559634\pi\)
−0.186252 + 0.982502i \(0.559634\pi\)
\(594\) −30.6274 −1.25666
\(595\) −25.3137 −1.03776
\(596\) 0 0
\(597\) 10.8284 0.443178
\(598\) 17.2132 0.703900
\(599\) 36.6569 1.49776 0.748879 0.662706i \(-0.230592\pi\)
0.748879 + 0.662706i \(0.230592\pi\)
\(600\) 26.6274 1.08706
\(601\) −24.9706 −1.01857 −0.509285 0.860598i \(-0.670090\pi\)
−0.509285 + 0.860598i \(0.670090\pi\)
\(602\) 4.82843 0.196792
\(603\) 7.48528 0.304824
\(604\) 0 0
\(605\) 12.4853 0.507599
\(606\) −11.6569 −0.473527
\(607\) −32.9706 −1.33823 −0.669117 0.743157i \(-0.733327\pi\)
−0.669117 + 0.743157i \(0.733327\pi\)
\(608\) 0 0
\(609\) 20.4853 0.830105
\(610\) 1.17157 0.0474356
\(611\) −10.9706 −0.443821
\(612\) 0 0
\(613\) 6.82843 0.275798 0.137899 0.990446i \(-0.455965\pi\)
0.137899 + 0.990446i \(0.455965\pi\)
\(614\) −37.8995 −1.52950
\(615\) 8.82843 0.355997
\(616\) 36.9706 1.48959
\(617\) 19.9706 0.803985 0.401992 0.915643i \(-0.368318\pi\)
0.401992 + 0.915643i \(0.368318\pi\)
\(618\) −1.02944 −0.0414100
\(619\) 28.9706 1.16443 0.582213 0.813037i \(-0.302187\pi\)
0.582213 + 0.813037i \(0.302187\pi\)
\(620\) 0 0
\(621\) −37.6569 −1.51112
\(622\) 19.7574 0.792198
\(623\) 6.00000 0.240385
\(624\) 10.3431 0.414057
\(625\) −13.9706 −0.558823
\(626\) 35.6569 1.42513
\(627\) −4.48528 −0.179125
\(628\) 0 0
\(629\) 18.4264 0.734709
\(630\) −16.4853 −0.656789
\(631\) 46.7696 1.86187 0.930933 0.365189i \(-0.118996\pi\)
0.930933 + 0.365189i \(0.118996\pi\)
\(632\) 13.6569 0.543240
\(633\) −0.201010 −0.00798944
\(634\) −36.5269 −1.45067
\(635\) 13.0711 0.518710
\(636\) 0 0
\(637\) −8.51472 −0.337365
\(638\) −22.9706 −0.909413
\(639\) 3.17157 0.125465
\(640\) −38.6274 −1.52688
\(641\) −29.5563 −1.16741 −0.583703 0.811967i \(-0.698397\pi\)
−0.583703 + 0.811967i \(0.698397\pi\)
\(642\) 0.686292 0.0270858
\(643\) −26.4853 −1.04448 −0.522239 0.852799i \(-0.674903\pi\)
−0.522239 + 0.852799i \(0.674903\pi\)
\(644\) 0 0
\(645\) 4.82843 0.190119
\(646\) −2.54416 −0.100098
\(647\) −1.17157 −0.0460593 −0.0230296 0.999735i \(-0.507331\pi\)
−0.0230296 + 0.999735i \(0.507331\pi\)
\(648\) −14.1421 −0.555556
\(649\) 18.4853 0.725611
\(650\) 17.2132 0.675157
\(651\) 14.4853 0.567723
\(652\) 0 0
\(653\) 15.1716 0.593710 0.296855 0.954923i \(-0.404062\pi\)
0.296855 + 0.954923i \(0.404062\pi\)
\(654\) 39.9411 1.56182
\(655\) 32.9706 1.28827
\(656\) −7.31371 −0.285552
\(657\) 16.2426 0.633686
\(658\) 28.9706 1.12939
\(659\) 16.3137 0.635492 0.317746 0.948176i \(-0.397074\pi\)
0.317746 + 0.948176i \(0.397074\pi\)
\(660\) 0 0
\(661\) −17.0000 −0.661223 −0.330612 0.943767i \(-0.607255\pi\)
−0.330612 + 0.943767i \(0.607255\pi\)
\(662\) −30.2843 −1.17703
\(663\) −5.61522 −0.218077
\(664\) 9.45584 0.366958
\(665\) −9.65685 −0.374477
\(666\) 12.0000 0.464991
\(667\) −28.2426 −1.09356
\(668\) 0 0
\(669\) 5.79899 0.224202
\(670\) 36.1421 1.39629
\(671\) 0.928932 0.0358610
\(672\) 0 0
\(673\) −16.7279 −0.644814 −0.322407 0.946601i \(-0.604492\pi\)
−0.322407 + 0.946601i \(0.604492\pi\)
\(674\) 7.07107 0.272367
\(675\) −37.6569 −1.44941
\(676\) 0 0
\(677\) 11.1716 0.429358 0.214679 0.976685i \(-0.431129\pi\)
0.214679 + 0.976685i \(0.431129\pi\)
\(678\) 13.6569 0.524488
\(679\) −6.24264 −0.239571
\(680\) 20.9706 0.804184
\(681\) −8.97056 −0.343753
\(682\) −16.2426 −0.621963
\(683\) −46.4558 −1.77758 −0.888792 0.458311i \(-0.848454\pi\)
−0.888792 + 0.458311i \(0.848454\pi\)
\(684\) 0 0
\(685\) −49.4558 −1.88961
\(686\) −11.3137 −0.431959
\(687\) −36.7279 −1.40126
\(688\) −4.00000 −0.152499
\(689\) −25.2843 −0.963254
\(690\) −45.4558 −1.73047
\(691\) −26.7279 −1.01678 −0.508389 0.861128i \(-0.669759\pi\)
−0.508389 + 0.861128i \(0.669759\pi\)
\(692\) 0 0
\(693\) −13.0711 −0.496529
\(694\) 6.48528 0.246178
\(695\) 18.7279 0.710391
\(696\) −16.9706 −0.643268
\(697\) 3.97056 0.150396
\(698\) 28.6274 1.08356
\(699\) 6.82843 0.258275
\(700\) 0 0
\(701\) −29.6569 −1.12012 −0.560062 0.828451i \(-0.689223\pi\)
−0.560062 + 0.828451i \(0.689223\pi\)
\(702\) −14.6274 −0.552076
\(703\) 7.02944 0.265120
\(704\) −30.6274 −1.15431
\(705\) 28.9706 1.09109
\(706\) −8.24264 −0.310216
\(707\) −19.8995 −0.748398
\(708\) 0 0
\(709\) −38.1127 −1.43135 −0.715676 0.698432i \(-0.753881\pi\)
−0.715676 + 0.698432i \(0.753881\pi\)
\(710\) 15.3137 0.574713
\(711\) −4.82843 −0.181080
\(712\) −4.97056 −0.186280
\(713\) −19.9706 −0.747903
\(714\) 14.8284 0.554940
\(715\) 23.8995 0.893790
\(716\) 0 0
\(717\) 20.4853 0.765037
\(718\) 30.1838 1.12645
\(719\) −22.6274 −0.843860 −0.421930 0.906628i \(-0.638647\pi\)
−0.421930 + 0.906628i \(0.638647\pi\)
\(720\) 13.6569 0.508961
\(721\) −1.75736 −0.0654475
\(722\) 25.8995 0.963879
\(723\) 5.65685 0.210381
\(724\) 0 0
\(725\) −28.2426 −1.04891
\(726\) −7.31371 −0.271437
\(727\) −24.9706 −0.926107 −0.463053 0.886330i \(-0.653246\pi\)
−0.463053 + 0.886330i \(0.653246\pi\)
\(728\) 17.6569 0.654407
\(729\) 29.0000 1.07407
\(730\) 78.4264 2.90269
\(731\) 2.17157 0.0803185
\(732\) 0 0
\(733\) 16.9706 0.626822 0.313411 0.949618i \(-0.398528\pi\)
0.313411 + 0.949618i \(0.398528\pi\)
\(734\) −13.8579 −0.511503
\(735\) 22.4853 0.829382
\(736\) 0 0
\(737\) 28.6569 1.05559
\(738\) 2.58579 0.0951841
\(739\) −39.4558 −1.45141 −0.725703 0.688008i \(-0.758485\pi\)
−0.725703 + 0.688008i \(0.758485\pi\)
\(740\) 0 0
\(741\) −2.14214 −0.0786933
\(742\) 66.7696 2.45119
\(743\) 15.1127 0.554431 0.277216 0.960808i \(-0.410588\pi\)
0.277216 + 0.960808i \(0.410588\pi\)
\(744\) −12.0000 −0.439941
\(745\) 28.9706 1.06140
\(746\) −12.0000 −0.439351
\(747\) −3.34315 −0.122319
\(748\) 0 0
\(749\) 1.17157 0.0428083
\(750\) −11.3137 −0.413118
\(751\) −11.7574 −0.429032 −0.214516 0.976720i \(-0.568817\pi\)
−0.214516 + 0.976720i \(0.568817\pi\)
\(752\) −24.0000 −0.875190
\(753\) 32.7279 1.19267
\(754\) −10.9706 −0.399524
\(755\) 62.2843 2.26676
\(756\) 0 0
\(757\) 3.51472 0.127745 0.0638723 0.997958i \(-0.479655\pi\)
0.0638723 + 0.997958i \(0.479655\pi\)
\(758\) −42.8701 −1.55711
\(759\) −36.0416 −1.30823
\(760\) 8.00000 0.290191
\(761\) 19.1127 0.692835 0.346417 0.938080i \(-0.387398\pi\)
0.346417 + 0.938080i \(0.387398\pi\)
\(762\) −7.65685 −0.277379
\(763\) 68.1838 2.46842
\(764\) 0 0
\(765\) −7.41421 −0.268061
\(766\) 28.9706 1.04675
\(767\) 8.82843 0.318776
\(768\) 0 0
\(769\) 44.7696 1.61443 0.807216 0.590257i \(-0.200973\pi\)
0.807216 + 0.590257i \(0.200973\pi\)
\(770\) −63.1127 −2.27442
\(771\) −2.48528 −0.0895052
\(772\) 0 0
\(773\) 8.10051 0.291355 0.145677 0.989332i \(-0.453464\pi\)
0.145677 + 0.989332i \(0.453464\pi\)
\(774\) 1.41421 0.0508329
\(775\) −19.9706 −0.717364
\(776\) 5.17157 0.185649
\(777\) −40.9706 −1.46981
\(778\) 23.5147 0.843044
\(779\) 1.51472 0.0542704
\(780\) 0 0
\(781\) 12.1421 0.434480
\(782\) −20.4437 −0.731063
\(783\) 24.0000 0.857690
\(784\) −18.6274 −0.665265
\(785\) −34.1421 −1.21859
\(786\) −19.3137 −0.688897
\(787\) 9.79899 0.349296 0.174648 0.984631i \(-0.444121\pi\)
0.174648 + 0.984631i \(0.444121\pi\)
\(788\) 0 0
\(789\) 0.201010 0.00715615
\(790\) −23.3137 −0.829465
\(791\) 23.3137 0.828940
\(792\) 10.8284 0.384771
\(793\) 0.443651 0.0157545
\(794\) −41.6569 −1.47835
\(795\) 66.7696 2.36807
\(796\) 0 0
\(797\) 35.3137 1.25088 0.625438 0.780274i \(-0.284920\pi\)
0.625438 + 0.780274i \(0.284920\pi\)
\(798\) 5.65685 0.200250
\(799\) 13.0294 0.460948
\(800\) 0 0
\(801\) 1.75736 0.0620932
\(802\) 17.6985 0.624955
\(803\) 62.1838 2.19442
\(804\) 0 0
\(805\) −77.5980 −2.73497
\(806\) −7.75736 −0.273241
\(807\) −4.24264 −0.149348
\(808\) 16.4853 0.579950
\(809\) −5.65685 −0.198884 −0.0994422 0.995043i \(-0.531706\pi\)
−0.0994422 + 0.995043i \(0.531706\pi\)
\(810\) 24.1421 0.848268
\(811\) 23.2721 0.817193 0.408597 0.912715i \(-0.366018\pi\)
0.408597 + 0.912715i \(0.366018\pi\)
\(812\) 0 0
\(813\) −36.7279 −1.28810
\(814\) 45.9411 1.61024
\(815\) −40.1421 −1.40612
\(816\) −12.2843 −0.430036
\(817\) 0.828427 0.0289830
\(818\) 23.8579 0.834171
\(819\) −6.24264 −0.218136
\(820\) 0 0
\(821\) −52.1127 −1.81875 −0.909373 0.415982i \(-0.863438\pi\)
−0.909373 + 0.415982i \(0.863438\pi\)
\(822\) 28.9706 1.01046
\(823\) −54.6569 −1.90522 −0.952609 0.304197i \(-0.901612\pi\)
−0.952609 + 0.304197i \(0.901612\pi\)
\(824\) 1.45584 0.0507167
\(825\) −36.0416 −1.25481
\(826\) −23.3137 −0.811188
\(827\) −1.65685 −0.0576145 −0.0288072 0.999585i \(-0.509171\pi\)
−0.0288072 + 0.999585i \(0.509171\pi\)
\(828\) 0 0
\(829\) 23.7990 0.826573 0.413287 0.910601i \(-0.364381\pi\)
0.413287 + 0.910601i \(0.364381\pi\)
\(830\) −16.1421 −0.560302
\(831\) 11.1716 0.387538
\(832\) −14.6274 −0.507114
\(833\) 10.1127 0.350384
\(834\) −10.9706 −0.379880
\(835\) −48.8701 −1.69122
\(836\) 0 0
\(837\) 16.9706 0.586588
\(838\) 33.7990 1.16757
\(839\) −48.8701 −1.68718 −0.843591 0.536986i \(-0.819563\pi\)
−0.843591 + 0.536986i \(0.819563\pi\)
\(840\) −46.6274 −1.60880
\(841\) −11.0000 −0.379310
\(842\) 22.1421 0.763068
\(843\) −12.2426 −0.421659
\(844\) 0 0
\(845\) −32.9706 −1.13422
\(846\) 8.48528 0.291730
\(847\) −12.4853 −0.428999
\(848\) −55.3137 −1.89948
\(849\) 25.8995 0.888868
\(850\) −20.4437 −0.701211
\(851\) 56.4853 1.93629
\(852\) 0 0
\(853\) 46.5980 1.59548 0.797742 0.602999i \(-0.206028\pi\)
0.797742 + 0.602999i \(0.206028\pi\)
\(854\) −1.17157 −0.0400904
\(855\) −2.82843 −0.0967302
\(856\) −0.970563 −0.0331732
\(857\) 7.65685 0.261553 0.130777 0.991412i \(-0.458253\pi\)
0.130777 + 0.991412i \(0.458253\pi\)
\(858\) −14.0000 −0.477952
\(859\) 16.9706 0.579028 0.289514 0.957174i \(-0.406506\pi\)
0.289514 + 0.957174i \(0.406506\pi\)
\(860\) 0 0
\(861\) −8.82843 −0.300872
\(862\) −32.9289 −1.12156
\(863\) −47.2548 −1.60857 −0.804287 0.594242i \(-0.797452\pi\)
−0.804287 + 0.594242i \(0.797452\pi\)
\(864\) 0 0
\(865\) 80.7696 2.74625
\(866\) −30.7696 −1.04559
\(867\) −17.3726 −0.590004
\(868\) 0 0
\(869\) −18.4853 −0.627070
\(870\) 28.9706 0.982194
\(871\) 13.6863 0.463742
\(872\) −56.4853 −1.91283
\(873\) −1.82843 −0.0618829
\(874\) −7.79899 −0.263805
\(875\) −19.3137 −0.652923
\(876\) 0 0
\(877\) 36.7990 1.24261 0.621307 0.783567i \(-0.286602\pi\)
0.621307 + 0.783567i \(0.286602\pi\)
\(878\) −16.2426 −0.548163
\(879\) 14.6274 0.493371
\(880\) 52.2843 1.76250
\(881\) 36.2548 1.22146 0.610728 0.791840i \(-0.290877\pi\)
0.610728 + 0.791840i \(0.290877\pi\)
\(882\) 6.58579 0.221755
\(883\) −8.02944 −0.270212 −0.135106 0.990831i \(-0.543138\pi\)
−0.135106 + 0.990831i \(0.543138\pi\)
\(884\) 0 0
\(885\) −23.3137 −0.783682
\(886\) −11.1127 −0.373338
\(887\) 58.9706 1.98004 0.990019 0.140935i \(-0.0450108\pi\)
0.990019 + 0.140935i \(0.0450108\pi\)
\(888\) 33.9411 1.13899
\(889\) −13.0711 −0.438390
\(890\) 8.48528 0.284427
\(891\) 19.1421 0.641286
\(892\) 0 0
\(893\) 4.97056 0.166334
\(894\) −16.9706 −0.567581
\(895\) −15.6569 −0.523351
\(896\) 38.6274 1.29045
\(897\) −17.2132 −0.574732
\(898\) −46.9706 −1.56743
\(899\) 12.7279 0.424500
\(900\) 0 0
\(901\) 30.0294 1.00043
\(902\) 9.89949 0.329617
\(903\) −4.82843 −0.160680
\(904\) −19.3137 −0.642364
\(905\) 24.9706 0.830050
\(906\) −36.4853 −1.21214
\(907\) −19.9706 −0.663112 −0.331556 0.943436i \(-0.607574\pi\)
−0.331556 + 0.943436i \(0.607574\pi\)
\(908\) 0 0
\(909\) −5.82843 −0.193317
\(910\) −30.1421 −0.999202
\(911\) −4.24264 −0.140565 −0.0702825 0.997527i \(-0.522390\pi\)
−0.0702825 + 0.997527i \(0.522390\pi\)
\(912\) −4.68629 −0.155179
\(913\) −12.7990 −0.423585
\(914\) −1.02944 −0.0340508
\(915\) −1.17157 −0.0387310
\(916\) 0 0
\(917\) −32.9706 −1.08878
\(918\) 17.3726 0.573381
\(919\) −12.5147 −0.412822 −0.206411 0.978465i \(-0.566178\pi\)
−0.206411 + 0.978465i \(0.566178\pi\)
\(920\) 64.2843 2.11939
\(921\) 37.8995 1.24883
\(922\) −60.2843 −1.98536
\(923\) 5.79899 0.190876
\(924\) 0 0
\(925\) 56.4853 1.85722
\(926\) 43.4558 1.42805
\(927\) −0.514719 −0.0169056
\(928\) 0 0
\(929\) 39.1716 1.28518 0.642589 0.766211i \(-0.277860\pi\)
0.642589 + 0.766211i \(0.277860\pi\)
\(930\) 20.4853 0.671739
\(931\) 3.85786 0.126436
\(932\) 0 0
\(933\) −19.7574 −0.646827
\(934\) −33.4558 −1.09471
\(935\) −28.3848 −0.928281
\(936\) 5.17157 0.169038
\(937\) −51.0122 −1.66650 −0.833248 0.552900i \(-0.813521\pi\)
−0.833248 + 0.552900i \(0.813521\pi\)
\(938\) −36.1421 −1.18008
\(939\) −35.6569 −1.16362
\(940\) 0 0
\(941\) −31.6274 −1.03102 −0.515512 0.856882i \(-0.672398\pi\)
−0.515512 + 0.856882i \(0.672398\pi\)
\(942\) 20.0000 0.651635
\(943\) 12.1716 0.396361
\(944\) 19.3137 0.628608
\(945\) 65.9411 2.14506
\(946\) 5.41421 0.176031
\(947\) −5.82843 −0.189398 −0.0946992 0.995506i \(-0.530189\pi\)
−0.0946992 + 0.995506i \(0.530189\pi\)
\(948\) 0 0
\(949\) 29.6985 0.964054
\(950\) −7.79899 −0.253033
\(951\) 36.5269 1.18447
\(952\) −20.9706 −0.679660
\(953\) −24.0416 −0.778785 −0.389392 0.921072i \(-0.627315\pi\)
−0.389392 + 0.921072i \(0.627315\pi\)
\(954\) 19.5563 0.633160
\(955\) −20.9706 −0.678591
\(956\) 0 0
\(957\) 22.9706 0.742533
\(958\) 9.41421 0.304159
\(959\) 49.4558 1.59701
\(960\) 38.6274 1.24669
\(961\) −22.0000 −0.709677
\(962\) 21.9411 0.707410
\(963\) 0.343146 0.0110577
\(964\) 0 0
\(965\) 54.5269 1.75528
\(966\) 45.4558 1.46252
\(967\) −40.9411 −1.31658 −0.658289 0.752765i \(-0.728719\pi\)
−0.658289 + 0.752765i \(0.728719\pi\)
\(968\) 10.3431 0.332441
\(969\) 2.54416 0.0817301
\(970\) −8.82843 −0.283464
\(971\) −3.14214 −0.100836 −0.0504180 0.998728i \(-0.516055\pi\)
−0.0504180 + 0.998728i \(0.516055\pi\)
\(972\) 0 0
\(973\) −18.7279 −0.600390
\(974\) −31.5980 −1.01246
\(975\) −17.2132 −0.551264
\(976\) 0.970563 0.0310670
\(977\) −23.3137 −0.745872 −0.372936 0.927857i \(-0.621649\pi\)
−0.372936 + 0.927857i \(0.621649\pi\)
\(978\) 23.5147 0.751918
\(979\) 6.72792 0.215025
\(980\) 0 0
\(981\) 19.9706 0.637611
\(982\) −18.9706 −0.605375
\(983\) −62.5269 −1.99430 −0.997149 0.0754527i \(-0.975960\pi\)
−0.997149 + 0.0754527i \(0.975960\pi\)
\(984\) 7.31371 0.233153
\(985\) −48.2843 −1.53846
\(986\) 13.0294 0.414942
\(987\) −28.9706 −0.922143
\(988\) 0 0
\(989\) 6.65685 0.211676
\(990\) −18.4853 −0.587501
\(991\) 11.5563 0.367100 0.183550 0.983010i \(-0.441241\pi\)
0.183550 + 0.983010i \(0.441241\pi\)
\(992\) 0 0
\(993\) 30.2843 0.961042
\(994\) −15.3137 −0.485721
\(995\) 26.1421 0.828761
\(996\) 0 0
\(997\) 2.92893 0.0927602 0.0463801 0.998924i \(-0.485231\pi\)
0.0463801 + 0.998924i \(0.485231\pi\)
\(998\) 36.4264 1.15306
\(999\) −48.0000 −1.51865
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 43.2.a.b.1.1 2
3.2 odd 2 387.2.a.h.1.2 2
4.3 odd 2 688.2.a.f.1.1 2
5.2 odd 4 1075.2.b.f.474.1 4
5.3 odd 4 1075.2.b.f.474.4 4
5.4 even 2 1075.2.a.i.1.2 2
7.6 odd 2 2107.2.a.b.1.1 2
8.3 odd 2 2752.2.a.m.1.2 2
8.5 even 2 2752.2.a.l.1.1 2
11.10 odd 2 5203.2.a.f.1.2 2
12.11 even 2 6192.2.a.bd.1.1 2
13.12 even 2 7267.2.a.b.1.2 2
15.14 odd 2 9675.2.a.bf.1.1 2
43.42 odd 2 1849.2.a.f.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
43.2.a.b.1.1 2 1.1 even 1 trivial
387.2.a.h.1.2 2 3.2 odd 2
688.2.a.f.1.1 2 4.3 odd 2
1075.2.a.i.1.2 2 5.4 even 2
1075.2.b.f.474.1 4 5.2 odd 4
1075.2.b.f.474.4 4 5.3 odd 4
1849.2.a.f.1.2 2 43.42 odd 2
2107.2.a.b.1.1 2 7.6 odd 2
2752.2.a.l.1.1 2 8.5 even 2
2752.2.a.m.1.2 2 8.3 odd 2
5203.2.a.f.1.2 2 11.10 odd 2
6192.2.a.bd.1.1 2 12.11 even 2
7267.2.a.b.1.2 2 13.12 even 2
9675.2.a.bf.1.1 2 15.14 odd 2