Properties

Label 2541.2.a.t.1.2
Level $2541$
Weight $2$
Character 2541.1
Self dual yes
Analytic conductor $20.290$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 2541 = 3 \cdot 7 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2541.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(20.2899871536\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{10})^+\)
Defining polynomial: \(x^{2} - x - 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 231)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-0.618034\) of defining polynomial
Character \(\chi\) \(=\) 2541.1

$q$-expansion

\(f(q)\) \(=\) \(q+0.618034 q^{2} +1.00000 q^{3} -1.61803 q^{4} +1.00000 q^{5} +0.618034 q^{6} -1.00000 q^{7} -2.23607 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+0.618034 q^{2} +1.00000 q^{3} -1.61803 q^{4} +1.00000 q^{5} +0.618034 q^{6} -1.00000 q^{7} -2.23607 q^{8} +1.00000 q^{9} +0.618034 q^{10} -1.61803 q^{12} -3.47214 q^{13} -0.618034 q^{14} +1.00000 q^{15} +1.85410 q^{16} -5.23607 q^{17} +0.618034 q^{18} +6.70820 q^{19} -1.61803 q^{20} -1.00000 q^{21} +5.70820 q^{23} -2.23607 q^{24} -4.00000 q^{25} -2.14590 q^{26} +1.00000 q^{27} +1.61803 q^{28} -5.00000 q^{29} +0.618034 q^{30} -5.23607 q^{31} +5.61803 q^{32} -3.23607 q^{34} -1.00000 q^{35} -1.61803 q^{36} -7.00000 q^{37} +4.14590 q^{38} -3.47214 q^{39} -2.23607 q^{40} +2.47214 q^{41} -0.618034 q^{42} -5.70820 q^{43} +1.00000 q^{45} +3.52786 q^{46} +0.236068 q^{47} +1.85410 q^{48} +1.00000 q^{49} -2.47214 q^{50} -5.23607 q^{51} +5.61803 q^{52} -12.1803 q^{53} +0.618034 q^{54} +2.23607 q^{56} +6.70820 q^{57} -3.09017 q^{58} -11.1803 q^{59} -1.61803 q^{60} -2.00000 q^{61} -3.23607 q^{62} -1.00000 q^{63} -0.236068 q^{64} -3.47214 q^{65} -9.76393 q^{67} +8.47214 q^{68} +5.70820 q^{69} -0.618034 q^{70} -2.47214 q^{71} -2.23607 q^{72} -4.52786 q^{73} -4.32624 q^{74} -4.00000 q^{75} -10.8541 q^{76} -2.14590 q^{78} +14.4721 q^{79} +1.85410 q^{80} +1.00000 q^{81} +1.52786 q^{82} -6.76393 q^{83} +1.61803 q^{84} -5.23607 q^{85} -3.52786 q^{86} -5.00000 q^{87} -4.47214 q^{89} +0.618034 q^{90} +3.47214 q^{91} -9.23607 q^{92} -5.23607 q^{93} +0.145898 q^{94} +6.70820 q^{95} +5.61803 q^{96} +9.70820 q^{97} +0.618034 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{2} + 2 q^{3} - q^{4} + 2 q^{5} - q^{6} - 2 q^{7} + 2 q^{9} + O(q^{10}) \) \( 2 q - q^{2} + 2 q^{3} - q^{4} + 2 q^{5} - q^{6} - 2 q^{7} + 2 q^{9} - q^{10} - q^{12} + 2 q^{13} + q^{14} + 2 q^{15} - 3 q^{16} - 6 q^{17} - q^{18} - q^{20} - 2 q^{21} - 2 q^{23} - 8 q^{25} - 11 q^{26} + 2 q^{27} + q^{28} - 10 q^{29} - q^{30} - 6 q^{31} + 9 q^{32} - 2 q^{34} - 2 q^{35} - q^{36} - 14 q^{37} + 15 q^{38} + 2 q^{39} - 4 q^{41} + q^{42} + 2 q^{43} + 2 q^{45} + 16 q^{46} - 4 q^{47} - 3 q^{48} + 2 q^{49} + 4 q^{50} - 6 q^{51} + 9 q^{52} - 2 q^{53} - q^{54} + 5 q^{58} - q^{60} - 4 q^{61} - 2 q^{62} - 2 q^{63} + 4 q^{64} + 2 q^{65} - 24 q^{67} + 8 q^{68} - 2 q^{69} + q^{70} + 4 q^{71} - 18 q^{73} + 7 q^{74} - 8 q^{75} - 15 q^{76} - 11 q^{78} + 20 q^{79} - 3 q^{80} + 2 q^{81} + 12 q^{82} - 18 q^{83} + q^{84} - 6 q^{85} - 16 q^{86} - 10 q^{87} - q^{90} - 2 q^{91} - 14 q^{92} - 6 q^{93} + 7 q^{94} + 9 q^{96} + 6 q^{97} - q^{98} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.618034 0.437016 0.218508 0.975835i \(-0.429881\pi\)
0.218508 + 0.975835i \(0.429881\pi\)
\(3\) 1.00000 0.577350
\(4\) −1.61803 −0.809017
\(5\) 1.00000 0.447214 0.223607 0.974679i \(-0.428217\pi\)
0.223607 + 0.974679i \(0.428217\pi\)
\(6\) 0.618034 0.252311
\(7\) −1.00000 −0.377964
\(8\) −2.23607 −0.790569
\(9\) 1.00000 0.333333
\(10\) 0.618034 0.195440
\(11\) 0 0
\(12\) −1.61803 −0.467086
\(13\) −3.47214 −0.962997 −0.481499 0.876447i \(-0.659907\pi\)
−0.481499 + 0.876447i \(0.659907\pi\)
\(14\) −0.618034 −0.165177
\(15\) 1.00000 0.258199
\(16\) 1.85410 0.463525
\(17\) −5.23607 −1.26993 −0.634967 0.772540i \(-0.718986\pi\)
−0.634967 + 0.772540i \(0.718986\pi\)
\(18\) 0.618034 0.145672
\(19\) 6.70820 1.53897 0.769484 0.638666i \(-0.220514\pi\)
0.769484 + 0.638666i \(0.220514\pi\)
\(20\) −1.61803 −0.361803
\(21\) −1.00000 −0.218218
\(22\) 0 0
\(23\) 5.70820 1.19024 0.595121 0.803636i \(-0.297104\pi\)
0.595121 + 0.803636i \(0.297104\pi\)
\(24\) −2.23607 −0.456435
\(25\) −4.00000 −0.800000
\(26\) −2.14590 −0.420845
\(27\) 1.00000 0.192450
\(28\) 1.61803 0.305780
\(29\) −5.00000 −0.928477 −0.464238 0.885710i \(-0.653672\pi\)
−0.464238 + 0.885710i \(0.653672\pi\)
\(30\) 0.618034 0.112837
\(31\) −5.23607 −0.940426 −0.470213 0.882553i \(-0.655823\pi\)
−0.470213 + 0.882553i \(0.655823\pi\)
\(32\) 5.61803 0.993137
\(33\) 0 0
\(34\) −3.23607 −0.554981
\(35\) −1.00000 −0.169031
\(36\) −1.61803 −0.269672
\(37\) −7.00000 −1.15079 −0.575396 0.817875i \(-0.695152\pi\)
−0.575396 + 0.817875i \(0.695152\pi\)
\(38\) 4.14590 0.672553
\(39\) −3.47214 −0.555987
\(40\) −2.23607 −0.353553
\(41\) 2.47214 0.386083 0.193041 0.981191i \(-0.438165\pi\)
0.193041 + 0.981191i \(0.438165\pi\)
\(42\) −0.618034 −0.0953647
\(43\) −5.70820 −0.870493 −0.435246 0.900311i \(-0.643339\pi\)
−0.435246 + 0.900311i \(0.643339\pi\)
\(44\) 0 0
\(45\) 1.00000 0.149071
\(46\) 3.52786 0.520155
\(47\) 0.236068 0.0344341 0.0172170 0.999852i \(-0.494519\pi\)
0.0172170 + 0.999852i \(0.494519\pi\)
\(48\) 1.85410 0.267617
\(49\) 1.00000 0.142857
\(50\) −2.47214 −0.349613
\(51\) −5.23607 −0.733196
\(52\) 5.61803 0.779081
\(53\) −12.1803 −1.67310 −0.836549 0.547892i \(-0.815431\pi\)
−0.836549 + 0.547892i \(0.815431\pi\)
\(54\) 0.618034 0.0841038
\(55\) 0 0
\(56\) 2.23607 0.298807
\(57\) 6.70820 0.888523
\(58\) −3.09017 −0.405759
\(59\) −11.1803 −1.45556 −0.727778 0.685813i \(-0.759447\pi\)
−0.727778 + 0.685813i \(0.759447\pi\)
\(60\) −1.61803 −0.208887
\(61\) −2.00000 −0.256074 −0.128037 0.991769i \(-0.540868\pi\)
−0.128037 + 0.991769i \(0.540868\pi\)
\(62\) −3.23607 −0.410981
\(63\) −1.00000 −0.125988
\(64\) −0.236068 −0.0295085
\(65\) −3.47214 −0.430665
\(66\) 0 0
\(67\) −9.76393 −1.19285 −0.596427 0.802667i \(-0.703414\pi\)
−0.596427 + 0.802667i \(0.703414\pi\)
\(68\) 8.47214 1.02740
\(69\) 5.70820 0.687187
\(70\) −0.618034 −0.0738692
\(71\) −2.47214 −0.293389 −0.146694 0.989182i \(-0.546863\pi\)
−0.146694 + 0.989182i \(0.546863\pi\)
\(72\) −2.23607 −0.263523
\(73\) −4.52786 −0.529946 −0.264973 0.964256i \(-0.585363\pi\)
−0.264973 + 0.964256i \(0.585363\pi\)
\(74\) −4.32624 −0.502915
\(75\) −4.00000 −0.461880
\(76\) −10.8541 −1.24505
\(77\) 0 0
\(78\) −2.14590 −0.242975
\(79\) 14.4721 1.62824 0.814121 0.580695i \(-0.197219\pi\)
0.814121 + 0.580695i \(0.197219\pi\)
\(80\) 1.85410 0.207295
\(81\) 1.00000 0.111111
\(82\) 1.52786 0.168724
\(83\) −6.76393 −0.742438 −0.371219 0.928545i \(-0.621060\pi\)
−0.371219 + 0.928545i \(0.621060\pi\)
\(84\) 1.61803 0.176542
\(85\) −5.23607 −0.567931
\(86\) −3.52786 −0.380419
\(87\) −5.00000 −0.536056
\(88\) 0 0
\(89\) −4.47214 −0.474045 −0.237023 0.971504i \(-0.576172\pi\)
−0.237023 + 0.971504i \(0.576172\pi\)
\(90\) 0.618034 0.0651465
\(91\) 3.47214 0.363979
\(92\) −9.23607 −0.962927
\(93\) −5.23607 −0.542955
\(94\) 0.145898 0.0150482
\(95\) 6.70820 0.688247
\(96\) 5.61803 0.573388
\(97\) 9.70820 0.985719 0.492859 0.870109i \(-0.335952\pi\)
0.492859 + 0.870109i \(0.335952\pi\)
\(98\) 0.618034 0.0624309
\(99\) 0 0
\(100\) 6.47214 0.647214
\(101\) −18.1803 −1.80901 −0.904506 0.426461i \(-0.859760\pi\)
−0.904506 + 0.426461i \(0.859760\pi\)
\(102\) −3.23607 −0.320418
\(103\) 17.4164 1.71609 0.858045 0.513575i \(-0.171679\pi\)
0.858045 + 0.513575i \(0.171679\pi\)
\(104\) 7.76393 0.761316
\(105\) −1.00000 −0.0975900
\(106\) −7.52786 −0.731171
\(107\) 4.23607 0.409516 0.204758 0.978813i \(-0.434359\pi\)
0.204758 + 0.978813i \(0.434359\pi\)
\(108\) −1.61803 −0.155695
\(109\) −2.76393 −0.264737 −0.132368 0.991201i \(-0.542258\pi\)
−0.132368 + 0.991201i \(0.542258\pi\)
\(110\) 0 0
\(111\) −7.00000 −0.664411
\(112\) −1.85410 −0.175196
\(113\) −0.472136 −0.0444148 −0.0222074 0.999753i \(-0.507069\pi\)
−0.0222074 + 0.999753i \(0.507069\pi\)
\(114\) 4.14590 0.388299
\(115\) 5.70820 0.532293
\(116\) 8.09017 0.751153
\(117\) −3.47214 −0.320999
\(118\) −6.90983 −0.636101
\(119\) 5.23607 0.479990
\(120\) −2.23607 −0.204124
\(121\) 0 0
\(122\) −1.23607 −0.111908
\(123\) 2.47214 0.222905
\(124\) 8.47214 0.760820
\(125\) −9.00000 −0.804984
\(126\) −0.618034 −0.0550588
\(127\) 12.6525 1.12273 0.561363 0.827570i \(-0.310277\pi\)
0.561363 + 0.827570i \(0.310277\pi\)
\(128\) −11.3820 −1.00603
\(129\) −5.70820 −0.502579
\(130\) −2.14590 −0.188208
\(131\) −0.944272 −0.0825014 −0.0412507 0.999149i \(-0.513134\pi\)
−0.0412507 + 0.999149i \(0.513134\pi\)
\(132\) 0 0
\(133\) −6.70820 −0.581675
\(134\) −6.03444 −0.521296
\(135\) 1.00000 0.0860663
\(136\) 11.7082 1.00397
\(137\) 19.7082 1.68379 0.841893 0.539645i \(-0.181441\pi\)
0.841893 + 0.539645i \(0.181441\pi\)
\(138\) 3.52786 0.300312
\(139\) −14.4721 −1.22751 −0.613755 0.789496i \(-0.710342\pi\)
−0.613755 + 0.789496i \(0.710342\pi\)
\(140\) 1.61803 0.136749
\(141\) 0.236068 0.0198805
\(142\) −1.52786 −0.128216
\(143\) 0 0
\(144\) 1.85410 0.154508
\(145\) −5.00000 −0.415227
\(146\) −2.79837 −0.231595
\(147\) 1.00000 0.0824786
\(148\) 11.3262 0.931011
\(149\) −5.00000 −0.409616 −0.204808 0.978802i \(-0.565657\pi\)
−0.204808 + 0.978802i \(0.565657\pi\)
\(150\) −2.47214 −0.201849
\(151\) 14.1803 1.15398 0.576990 0.816751i \(-0.304227\pi\)
0.576990 + 0.816751i \(0.304227\pi\)
\(152\) −15.0000 −1.21666
\(153\) −5.23607 −0.423311
\(154\) 0 0
\(155\) −5.23607 −0.420571
\(156\) 5.61803 0.449803
\(157\) −15.4164 −1.23036 −0.615182 0.788385i \(-0.710917\pi\)
−0.615182 + 0.788385i \(0.710917\pi\)
\(158\) 8.94427 0.711568
\(159\) −12.1803 −0.965964
\(160\) 5.61803 0.444145
\(161\) −5.70820 −0.449869
\(162\) 0.618034 0.0485573
\(163\) −22.7082 −1.77864 −0.889322 0.457282i \(-0.848823\pi\)
−0.889322 + 0.457282i \(0.848823\pi\)
\(164\) −4.00000 −0.312348
\(165\) 0 0
\(166\) −4.18034 −0.324457
\(167\) 22.6525 1.75290 0.876451 0.481492i \(-0.159905\pi\)
0.876451 + 0.481492i \(0.159905\pi\)
\(168\) 2.23607 0.172516
\(169\) −0.944272 −0.0726363
\(170\) −3.23607 −0.248195
\(171\) 6.70820 0.512989
\(172\) 9.23607 0.704244
\(173\) 1.52786 0.116161 0.0580807 0.998312i \(-0.481502\pi\)
0.0580807 + 0.998312i \(0.481502\pi\)
\(174\) −3.09017 −0.234265
\(175\) 4.00000 0.302372
\(176\) 0 0
\(177\) −11.1803 −0.840366
\(178\) −2.76393 −0.207165
\(179\) 8.94427 0.668526 0.334263 0.942480i \(-0.391513\pi\)
0.334263 + 0.942480i \(0.391513\pi\)
\(180\) −1.61803 −0.120601
\(181\) −0.763932 −0.0567826 −0.0283913 0.999597i \(-0.509038\pi\)
−0.0283913 + 0.999597i \(0.509038\pi\)
\(182\) 2.14590 0.159065
\(183\) −2.00000 −0.147844
\(184\) −12.7639 −0.940970
\(185\) −7.00000 −0.514650
\(186\) −3.23607 −0.237280
\(187\) 0 0
\(188\) −0.381966 −0.0278577
\(189\) −1.00000 −0.0727393
\(190\) 4.14590 0.300775
\(191\) −10.7639 −0.778851 −0.389425 0.921058i \(-0.627326\pi\)
−0.389425 + 0.921058i \(0.627326\pi\)
\(192\) −0.236068 −0.0170367
\(193\) −14.6525 −1.05471 −0.527354 0.849646i \(-0.676816\pi\)
−0.527354 + 0.849646i \(0.676816\pi\)
\(194\) 6.00000 0.430775
\(195\) −3.47214 −0.248645
\(196\) −1.61803 −0.115574
\(197\) 16.4721 1.17359 0.586796 0.809735i \(-0.300389\pi\)
0.586796 + 0.809735i \(0.300389\pi\)
\(198\) 0 0
\(199\) −3.81966 −0.270769 −0.135384 0.990793i \(-0.543227\pi\)
−0.135384 + 0.990793i \(0.543227\pi\)
\(200\) 8.94427 0.632456
\(201\) −9.76393 −0.688695
\(202\) −11.2361 −0.790567
\(203\) 5.00000 0.350931
\(204\) 8.47214 0.593168
\(205\) 2.47214 0.172661
\(206\) 10.7639 0.749959
\(207\) 5.70820 0.396748
\(208\) −6.43769 −0.446374
\(209\) 0 0
\(210\) −0.618034 −0.0426484
\(211\) −5.41641 −0.372881 −0.186440 0.982466i \(-0.559695\pi\)
−0.186440 + 0.982466i \(0.559695\pi\)
\(212\) 19.7082 1.35357
\(213\) −2.47214 −0.169388
\(214\) 2.61803 0.178965
\(215\) −5.70820 −0.389296
\(216\) −2.23607 −0.152145
\(217\) 5.23607 0.355447
\(218\) −1.70820 −0.115694
\(219\) −4.52786 −0.305965
\(220\) 0 0
\(221\) 18.1803 1.22294
\(222\) −4.32624 −0.290358
\(223\) −6.00000 −0.401790 −0.200895 0.979613i \(-0.564385\pi\)
−0.200895 + 0.979613i \(0.564385\pi\)
\(224\) −5.61803 −0.375371
\(225\) −4.00000 −0.266667
\(226\) −0.291796 −0.0194100
\(227\) 2.00000 0.132745 0.0663723 0.997795i \(-0.478857\pi\)
0.0663723 + 0.997795i \(0.478857\pi\)
\(228\) −10.8541 −0.718830
\(229\) 7.23607 0.478173 0.239086 0.970998i \(-0.423152\pi\)
0.239086 + 0.970998i \(0.423152\pi\)
\(230\) 3.52786 0.232620
\(231\) 0 0
\(232\) 11.1803 0.734025
\(233\) 14.9443 0.979032 0.489516 0.871994i \(-0.337174\pi\)
0.489516 + 0.871994i \(0.337174\pi\)
\(234\) −2.14590 −0.140282
\(235\) 0.236068 0.0153994
\(236\) 18.0902 1.17757
\(237\) 14.4721 0.940066
\(238\) 3.23607 0.209763
\(239\) −10.1246 −0.654907 −0.327453 0.944867i \(-0.606190\pi\)
−0.327453 + 0.944867i \(0.606190\pi\)
\(240\) 1.85410 0.119682
\(241\) −25.9443 −1.67122 −0.835609 0.549325i \(-0.814885\pi\)
−0.835609 + 0.549325i \(0.814885\pi\)
\(242\) 0 0
\(243\) 1.00000 0.0641500
\(244\) 3.23607 0.207168
\(245\) 1.00000 0.0638877
\(246\) 1.52786 0.0974131
\(247\) −23.2918 −1.48202
\(248\) 11.7082 0.743472
\(249\) −6.76393 −0.428647
\(250\) −5.56231 −0.351791
\(251\) 12.1246 0.765299 0.382649 0.923894i \(-0.375012\pi\)
0.382649 + 0.923894i \(0.375012\pi\)
\(252\) 1.61803 0.101927
\(253\) 0 0
\(254\) 7.81966 0.490649
\(255\) −5.23607 −0.327895
\(256\) −6.56231 −0.410144
\(257\) −7.00000 −0.436648 −0.218324 0.975876i \(-0.570059\pi\)
−0.218324 + 0.975876i \(0.570059\pi\)
\(258\) −3.52786 −0.219635
\(259\) 7.00000 0.434959
\(260\) 5.61803 0.348416
\(261\) −5.00000 −0.309492
\(262\) −0.583592 −0.0360544
\(263\) 26.1246 1.61091 0.805456 0.592655i \(-0.201920\pi\)
0.805456 + 0.592655i \(0.201920\pi\)
\(264\) 0 0
\(265\) −12.1803 −0.748232
\(266\) −4.14590 −0.254201
\(267\) −4.47214 −0.273690
\(268\) 15.7984 0.965039
\(269\) 1.05573 0.0643689 0.0321844 0.999482i \(-0.489754\pi\)
0.0321844 + 0.999482i \(0.489754\pi\)
\(270\) 0.618034 0.0376124
\(271\) −5.29180 −0.321454 −0.160727 0.986999i \(-0.551384\pi\)
−0.160727 + 0.986999i \(0.551384\pi\)
\(272\) −9.70820 −0.588646
\(273\) 3.47214 0.210143
\(274\) 12.1803 0.735841
\(275\) 0 0
\(276\) −9.23607 −0.555946
\(277\) 6.47214 0.388873 0.194436 0.980915i \(-0.437712\pi\)
0.194436 + 0.980915i \(0.437712\pi\)
\(278\) −8.94427 −0.536442
\(279\) −5.23607 −0.313475
\(280\) 2.23607 0.133631
\(281\) −11.4721 −0.684370 −0.342185 0.939633i \(-0.611167\pi\)
−0.342185 + 0.939633i \(0.611167\pi\)
\(282\) 0.145898 0.00868810
\(283\) 13.7639 0.818181 0.409090 0.912494i \(-0.365846\pi\)
0.409090 + 0.912494i \(0.365846\pi\)
\(284\) 4.00000 0.237356
\(285\) 6.70820 0.397360
\(286\) 0 0
\(287\) −2.47214 −0.145926
\(288\) 5.61803 0.331046
\(289\) 10.4164 0.612730
\(290\) −3.09017 −0.181461
\(291\) 9.70820 0.569105
\(292\) 7.32624 0.428736
\(293\) 16.0000 0.934730 0.467365 0.884064i \(-0.345203\pi\)
0.467365 + 0.884064i \(0.345203\pi\)
\(294\) 0.618034 0.0360445
\(295\) −11.1803 −0.650945
\(296\) 15.6525 0.909782
\(297\) 0 0
\(298\) −3.09017 −0.179009
\(299\) −19.8197 −1.14620
\(300\) 6.47214 0.373669
\(301\) 5.70820 0.329015
\(302\) 8.76393 0.504308
\(303\) −18.1803 −1.04443
\(304\) 12.4377 0.713351
\(305\) −2.00000 −0.114520
\(306\) −3.23607 −0.184994
\(307\) 20.9443 1.19535 0.597676 0.801737i \(-0.296091\pi\)
0.597676 + 0.801737i \(0.296091\pi\)
\(308\) 0 0
\(309\) 17.4164 0.990785
\(310\) −3.23607 −0.183796
\(311\) 9.88854 0.560728 0.280364 0.959894i \(-0.409545\pi\)
0.280364 + 0.959894i \(0.409545\pi\)
\(312\) 7.76393 0.439546
\(313\) 24.6525 1.39344 0.696720 0.717343i \(-0.254642\pi\)
0.696720 + 0.717343i \(0.254642\pi\)
\(314\) −9.52786 −0.537688
\(315\) −1.00000 −0.0563436
\(316\) −23.4164 −1.31728
\(317\) 24.1803 1.35810 0.679052 0.734091i \(-0.262391\pi\)
0.679052 + 0.734091i \(0.262391\pi\)
\(318\) −7.52786 −0.422142
\(319\) 0 0
\(320\) −0.236068 −0.0131966
\(321\) 4.23607 0.236434
\(322\) −3.52786 −0.196600
\(323\) −35.1246 −1.95439
\(324\) −1.61803 −0.0898908
\(325\) 13.8885 0.770398
\(326\) −14.0344 −0.777296
\(327\) −2.76393 −0.152846
\(328\) −5.52786 −0.305225
\(329\) −0.236068 −0.0130148
\(330\) 0 0
\(331\) −11.4164 −0.627503 −0.313751 0.949505i \(-0.601586\pi\)
−0.313751 + 0.949505i \(0.601586\pi\)
\(332\) 10.9443 0.600645
\(333\) −7.00000 −0.383598
\(334\) 14.0000 0.766046
\(335\) −9.76393 −0.533461
\(336\) −1.85410 −0.101150
\(337\) 8.18034 0.445612 0.222806 0.974863i \(-0.428478\pi\)
0.222806 + 0.974863i \(0.428478\pi\)
\(338\) −0.583592 −0.0317432
\(339\) −0.472136 −0.0256429
\(340\) 8.47214 0.459466
\(341\) 0 0
\(342\) 4.14590 0.224184
\(343\) −1.00000 −0.0539949
\(344\) 12.7639 0.688185
\(345\) 5.70820 0.307319
\(346\) 0.944272 0.0507644
\(347\) −28.0000 −1.50312 −0.751559 0.659665i \(-0.770698\pi\)
−0.751559 + 0.659665i \(0.770698\pi\)
\(348\) 8.09017 0.433679
\(349\) −1.58359 −0.0847677 −0.0423839 0.999101i \(-0.513495\pi\)
−0.0423839 + 0.999101i \(0.513495\pi\)
\(350\) 2.47214 0.132141
\(351\) −3.47214 −0.185329
\(352\) 0 0
\(353\) 24.5279 1.30549 0.652743 0.757579i \(-0.273618\pi\)
0.652743 + 0.757579i \(0.273618\pi\)
\(354\) −6.90983 −0.367253
\(355\) −2.47214 −0.131207
\(356\) 7.23607 0.383511
\(357\) 5.23607 0.277122
\(358\) 5.52786 0.292157
\(359\) −23.4164 −1.23587 −0.617935 0.786229i \(-0.712031\pi\)
−0.617935 + 0.786229i \(0.712031\pi\)
\(360\) −2.23607 −0.117851
\(361\) 26.0000 1.36842
\(362\) −0.472136 −0.0248149
\(363\) 0 0
\(364\) −5.61803 −0.294465
\(365\) −4.52786 −0.236999
\(366\) −1.23607 −0.0646103
\(367\) −19.8885 −1.03817 −0.519087 0.854722i \(-0.673728\pi\)
−0.519087 + 0.854722i \(0.673728\pi\)
\(368\) 10.5836 0.551708
\(369\) 2.47214 0.128694
\(370\) −4.32624 −0.224910
\(371\) 12.1803 0.632372
\(372\) 8.47214 0.439260
\(373\) −4.65248 −0.240896 −0.120448 0.992720i \(-0.538433\pi\)
−0.120448 + 0.992720i \(0.538433\pi\)
\(374\) 0 0
\(375\) −9.00000 −0.464758
\(376\) −0.527864 −0.0272225
\(377\) 17.3607 0.894120
\(378\) −0.618034 −0.0317882
\(379\) −31.1803 −1.60163 −0.800813 0.598914i \(-0.795599\pi\)
−0.800813 + 0.598914i \(0.795599\pi\)
\(380\) −10.8541 −0.556804
\(381\) 12.6525 0.648206
\(382\) −6.65248 −0.340370
\(383\) 32.9443 1.68337 0.841687 0.539966i \(-0.181563\pi\)
0.841687 + 0.539966i \(0.181563\pi\)
\(384\) −11.3820 −0.580834
\(385\) 0 0
\(386\) −9.05573 −0.460924
\(387\) −5.70820 −0.290164
\(388\) −15.7082 −0.797463
\(389\) 11.0557 0.560548 0.280274 0.959920i \(-0.409575\pi\)
0.280274 + 0.959920i \(0.409575\pi\)
\(390\) −2.14590 −0.108662
\(391\) −29.8885 −1.51153
\(392\) −2.23607 −0.112938
\(393\) −0.944272 −0.0476322
\(394\) 10.1803 0.512878
\(395\) 14.4721 0.728172
\(396\) 0 0
\(397\) 23.1246 1.16059 0.580295 0.814406i \(-0.302937\pi\)
0.580295 + 0.814406i \(0.302937\pi\)
\(398\) −2.36068 −0.118330
\(399\) −6.70820 −0.335830
\(400\) −7.41641 −0.370820
\(401\) −29.7082 −1.48356 −0.741778 0.670645i \(-0.766017\pi\)
−0.741778 + 0.670645i \(0.766017\pi\)
\(402\) −6.03444 −0.300971
\(403\) 18.1803 0.905627
\(404\) 29.4164 1.46352
\(405\) 1.00000 0.0496904
\(406\) 3.09017 0.153363
\(407\) 0 0
\(408\) 11.7082 0.579642
\(409\) −21.0557 −1.04114 −0.520569 0.853819i \(-0.674280\pi\)
−0.520569 + 0.853819i \(0.674280\pi\)
\(410\) 1.52786 0.0754558
\(411\) 19.7082 0.972134
\(412\) −28.1803 −1.38835
\(413\) 11.1803 0.550149
\(414\) 3.52786 0.173385
\(415\) −6.76393 −0.332028
\(416\) −19.5066 −0.956389
\(417\) −14.4721 −0.708704
\(418\) 0 0
\(419\) −1.18034 −0.0576634 −0.0288317 0.999584i \(-0.509179\pi\)
−0.0288317 + 0.999584i \(0.509179\pi\)
\(420\) 1.61803 0.0789520
\(421\) −13.0000 −0.633581 −0.316791 0.948495i \(-0.602605\pi\)
−0.316791 + 0.948495i \(0.602605\pi\)
\(422\) −3.34752 −0.162955
\(423\) 0.236068 0.0114780
\(424\) 27.2361 1.32270
\(425\) 20.9443 1.01595
\(426\) −1.52786 −0.0740253
\(427\) 2.00000 0.0967868
\(428\) −6.85410 −0.331306
\(429\) 0 0
\(430\) −3.52786 −0.170129
\(431\) −8.70820 −0.419459 −0.209730 0.977759i \(-0.567258\pi\)
−0.209730 + 0.977759i \(0.567258\pi\)
\(432\) 1.85410 0.0892055
\(433\) −10.4721 −0.503259 −0.251629 0.967824i \(-0.580966\pi\)
−0.251629 + 0.967824i \(0.580966\pi\)
\(434\) 3.23607 0.155336
\(435\) −5.00000 −0.239732
\(436\) 4.47214 0.214176
\(437\) 38.2918 1.83175
\(438\) −2.79837 −0.133711
\(439\) −11.1803 −0.533609 −0.266804 0.963751i \(-0.585968\pi\)
−0.266804 + 0.963751i \(0.585968\pi\)
\(440\) 0 0
\(441\) 1.00000 0.0476190
\(442\) 11.2361 0.534445
\(443\) 18.4721 0.877638 0.438819 0.898576i \(-0.355397\pi\)
0.438819 + 0.898576i \(0.355397\pi\)
\(444\) 11.3262 0.537519
\(445\) −4.47214 −0.212000
\(446\) −3.70820 −0.175589
\(447\) −5.00000 −0.236492
\(448\) 0.236068 0.0111532
\(449\) 20.0000 0.943858 0.471929 0.881636i \(-0.343558\pi\)
0.471929 + 0.881636i \(0.343558\pi\)
\(450\) −2.47214 −0.116538
\(451\) 0 0
\(452\) 0.763932 0.0359323
\(453\) 14.1803 0.666250
\(454\) 1.23607 0.0580115
\(455\) 3.47214 0.162776
\(456\) −15.0000 −0.702439
\(457\) −15.2361 −0.712713 −0.356357 0.934350i \(-0.615981\pi\)
−0.356357 + 0.934350i \(0.615981\pi\)
\(458\) 4.47214 0.208969
\(459\) −5.23607 −0.244399
\(460\) −9.23607 −0.430634
\(461\) 28.0000 1.30409 0.652045 0.758180i \(-0.273911\pi\)
0.652045 + 0.758180i \(0.273911\pi\)
\(462\) 0 0
\(463\) −4.81966 −0.223989 −0.111994 0.993709i \(-0.535724\pi\)
−0.111994 + 0.993709i \(0.535724\pi\)
\(464\) −9.27051 −0.430373
\(465\) −5.23607 −0.242817
\(466\) 9.23607 0.427853
\(467\) 29.1803 1.35031 0.675153 0.737678i \(-0.264078\pi\)
0.675153 + 0.737678i \(0.264078\pi\)
\(468\) 5.61803 0.259694
\(469\) 9.76393 0.450856
\(470\) 0.145898 0.00672977
\(471\) −15.4164 −0.710351
\(472\) 25.0000 1.15072
\(473\) 0 0
\(474\) 8.94427 0.410824
\(475\) −26.8328 −1.23117
\(476\) −8.47214 −0.388320
\(477\) −12.1803 −0.557699
\(478\) −6.25735 −0.286205
\(479\) −11.7082 −0.534961 −0.267481 0.963563i \(-0.586191\pi\)
−0.267481 + 0.963563i \(0.586191\pi\)
\(480\) 5.61803 0.256427
\(481\) 24.3050 1.10821
\(482\) −16.0344 −0.730349
\(483\) −5.70820 −0.259732
\(484\) 0 0
\(485\) 9.70820 0.440827
\(486\) 0.618034 0.0280346
\(487\) 16.9443 0.767818 0.383909 0.923371i \(-0.374578\pi\)
0.383909 + 0.923371i \(0.374578\pi\)
\(488\) 4.47214 0.202444
\(489\) −22.7082 −1.02690
\(490\) 0.618034 0.0279199
\(491\) 18.1246 0.817952 0.408976 0.912545i \(-0.365886\pi\)
0.408976 + 0.912545i \(0.365886\pi\)
\(492\) −4.00000 −0.180334
\(493\) 26.1803 1.17910
\(494\) −14.3951 −0.647667
\(495\) 0 0
\(496\) −9.70820 −0.435911
\(497\) 2.47214 0.110890
\(498\) −4.18034 −0.187326
\(499\) −2.23607 −0.100100 −0.0500501 0.998747i \(-0.515938\pi\)
−0.0500501 + 0.998747i \(0.515938\pi\)
\(500\) 14.5623 0.651246
\(501\) 22.6525 1.01204
\(502\) 7.49342 0.334448
\(503\) −2.29180 −0.102186 −0.0510931 0.998694i \(-0.516271\pi\)
−0.0510931 + 0.998694i \(0.516271\pi\)
\(504\) 2.23607 0.0996024
\(505\) −18.1803 −0.809015
\(506\) 0 0
\(507\) −0.944272 −0.0419366
\(508\) −20.4721 −0.908304
\(509\) −30.0000 −1.32973 −0.664863 0.746965i \(-0.731510\pi\)
−0.664863 + 0.746965i \(0.731510\pi\)
\(510\) −3.23607 −0.143295
\(511\) 4.52786 0.200301
\(512\) 18.7082 0.826794
\(513\) 6.70820 0.296174
\(514\) −4.32624 −0.190822
\(515\) 17.4164 0.767459
\(516\) 9.23607 0.406595
\(517\) 0 0
\(518\) 4.32624 0.190084
\(519\) 1.52786 0.0670658
\(520\) 7.76393 0.340471
\(521\) 38.3050 1.67817 0.839085 0.544000i \(-0.183091\pi\)
0.839085 + 0.544000i \(0.183091\pi\)
\(522\) −3.09017 −0.135253
\(523\) 21.6525 0.946797 0.473398 0.880848i \(-0.343027\pi\)
0.473398 + 0.880848i \(0.343027\pi\)
\(524\) 1.52786 0.0667451
\(525\) 4.00000 0.174574
\(526\) 16.1459 0.703995
\(527\) 27.4164 1.19428
\(528\) 0 0
\(529\) 9.58359 0.416678
\(530\) −7.52786 −0.326990
\(531\) −11.1803 −0.485185
\(532\) 10.8541 0.470585
\(533\) −8.58359 −0.371797
\(534\) −2.76393 −0.119607
\(535\) 4.23607 0.183141
\(536\) 21.8328 0.943034
\(537\) 8.94427 0.385974
\(538\) 0.652476 0.0281302
\(539\) 0 0
\(540\) −1.61803 −0.0696291
\(541\) 36.9443 1.58836 0.794179 0.607684i \(-0.207901\pi\)
0.794179 + 0.607684i \(0.207901\pi\)
\(542\) −3.27051 −0.140480
\(543\) −0.763932 −0.0327835
\(544\) −29.4164 −1.26122
\(545\) −2.76393 −0.118394
\(546\) 2.14590 0.0918360
\(547\) −14.8328 −0.634205 −0.317103 0.948391i \(-0.602710\pi\)
−0.317103 + 0.948391i \(0.602710\pi\)
\(548\) −31.8885 −1.36221
\(549\) −2.00000 −0.0853579
\(550\) 0 0
\(551\) −33.5410 −1.42890
\(552\) −12.7639 −0.543269
\(553\) −14.4721 −0.615418
\(554\) 4.00000 0.169944
\(555\) −7.00000 −0.297133
\(556\) 23.4164 0.993077
\(557\) 12.5279 0.530823 0.265411 0.964135i \(-0.414492\pi\)
0.265411 + 0.964135i \(0.414492\pi\)
\(558\) −3.23607 −0.136994
\(559\) 19.8197 0.838282
\(560\) −1.85410 −0.0783501
\(561\) 0 0
\(562\) −7.09017 −0.299081
\(563\) −34.6525 −1.46043 −0.730214 0.683219i \(-0.760580\pi\)
−0.730214 + 0.683219i \(0.760580\pi\)
\(564\) −0.381966 −0.0160837
\(565\) −0.472136 −0.0198629
\(566\) 8.50658 0.357558
\(567\) −1.00000 −0.0419961
\(568\) 5.52786 0.231944
\(569\) −30.0000 −1.25767 −0.628833 0.777541i \(-0.716467\pi\)
−0.628833 + 0.777541i \(0.716467\pi\)
\(570\) 4.14590 0.173653
\(571\) −17.5279 −0.733518 −0.366759 0.930316i \(-0.619533\pi\)
−0.366759 + 0.930316i \(0.619533\pi\)
\(572\) 0 0
\(573\) −10.7639 −0.449670
\(574\) −1.52786 −0.0637718
\(575\) −22.8328 −0.952194
\(576\) −0.236068 −0.00983617
\(577\) 7.34752 0.305881 0.152941 0.988235i \(-0.451126\pi\)
0.152941 + 0.988235i \(0.451126\pi\)
\(578\) 6.43769 0.267773
\(579\) −14.6525 −0.608936
\(580\) 8.09017 0.335926
\(581\) 6.76393 0.280615
\(582\) 6.00000 0.248708
\(583\) 0 0
\(584\) 10.1246 0.418959
\(585\) −3.47214 −0.143555
\(586\) 9.88854 0.408492
\(587\) 46.0132 1.89917 0.949583 0.313515i \(-0.101507\pi\)
0.949583 + 0.313515i \(0.101507\pi\)
\(588\) −1.61803 −0.0667266
\(589\) −35.1246 −1.44728
\(590\) −6.90983 −0.284473
\(591\) 16.4721 0.677573
\(592\) −12.9787 −0.533422
\(593\) 22.8328 0.937631 0.468816 0.883296i \(-0.344681\pi\)
0.468816 + 0.883296i \(0.344681\pi\)
\(594\) 0 0
\(595\) 5.23607 0.214658
\(596\) 8.09017 0.331386
\(597\) −3.81966 −0.156328
\(598\) −12.2492 −0.500908
\(599\) −25.5279 −1.04304 −0.521520 0.853239i \(-0.674635\pi\)
−0.521520 + 0.853239i \(0.674635\pi\)
\(600\) 8.94427 0.365148
\(601\) −27.0000 −1.10135 −0.550676 0.834719i \(-0.685630\pi\)
−0.550676 + 0.834719i \(0.685630\pi\)
\(602\) 3.52786 0.143785
\(603\) −9.76393 −0.397618
\(604\) −22.9443 −0.933589
\(605\) 0 0
\(606\) −11.2361 −0.456434
\(607\) −6.81966 −0.276801 −0.138401 0.990376i \(-0.544196\pi\)
−0.138401 + 0.990376i \(0.544196\pi\)
\(608\) 37.6869 1.52841
\(609\) 5.00000 0.202610
\(610\) −1.23607 −0.0500469
\(611\) −0.819660 −0.0331599
\(612\) 8.47214 0.342466
\(613\) −13.5967 −0.549167 −0.274584 0.961563i \(-0.588540\pi\)
−0.274584 + 0.961563i \(0.588540\pi\)
\(614\) 12.9443 0.522388
\(615\) 2.47214 0.0996861
\(616\) 0 0
\(617\) 32.4721 1.30728 0.653639 0.756806i \(-0.273241\pi\)
0.653639 + 0.756806i \(0.273241\pi\)
\(618\) 10.7639 0.432989
\(619\) −44.0689 −1.77128 −0.885639 0.464374i \(-0.846279\pi\)
−0.885639 + 0.464374i \(0.846279\pi\)
\(620\) 8.47214 0.340249
\(621\) 5.70820 0.229062
\(622\) 6.11146 0.245047
\(623\) 4.47214 0.179172
\(624\) −6.43769 −0.257714
\(625\) 11.0000 0.440000
\(626\) 15.2361 0.608956
\(627\) 0 0
\(628\) 24.9443 0.995385
\(629\) 36.6525 1.46143
\(630\) −0.618034 −0.0246231
\(631\) 44.3607 1.76597 0.882985 0.469400i \(-0.155530\pi\)
0.882985 + 0.469400i \(0.155530\pi\)
\(632\) −32.3607 −1.28724
\(633\) −5.41641 −0.215283
\(634\) 14.9443 0.593513
\(635\) 12.6525 0.502098
\(636\) 19.7082 0.781481
\(637\) −3.47214 −0.137571
\(638\) 0 0
\(639\) −2.47214 −0.0977962
\(640\) −11.3820 −0.449912
\(641\) −46.5410 −1.83826 −0.919130 0.393955i \(-0.871107\pi\)
−0.919130 + 0.393955i \(0.871107\pi\)
\(642\) 2.61803 0.103326
\(643\) −47.9574 −1.89126 −0.945628 0.325250i \(-0.894552\pi\)
−0.945628 + 0.325250i \(0.894552\pi\)
\(644\) 9.23607 0.363952
\(645\) −5.70820 −0.224760
\(646\) −21.7082 −0.854098
\(647\) 12.3475 0.485431 0.242716 0.970097i \(-0.421962\pi\)
0.242716 + 0.970097i \(0.421962\pi\)
\(648\) −2.23607 −0.0878410
\(649\) 0 0
\(650\) 8.58359 0.336676
\(651\) 5.23607 0.205218
\(652\) 36.7426 1.43895
\(653\) −44.9443 −1.75881 −0.879403 0.476079i \(-0.842058\pi\)
−0.879403 + 0.476079i \(0.842058\pi\)
\(654\) −1.70820 −0.0667961
\(655\) −0.944272 −0.0368958
\(656\) 4.58359 0.178959
\(657\) −4.52786 −0.176649
\(658\) −0.145898 −0.00568770
\(659\) −23.5410 −0.917028 −0.458514 0.888687i \(-0.651618\pi\)
−0.458514 + 0.888687i \(0.651618\pi\)
\(660\) 0 0
\(661\) 40.5410 1.57686 0.788431 0.615123i \(-0.210894\pi\)
0.788431 + 0.615123i \(0.210894\pi\)
\(662\) −7.05573 −0.274229
\(663\) 18.1803 0.706066
\(664\) 15.1246 0.586949
\(665\) −6.70820 −0.260133
\(666\) −4.32624 −0.167638
\(667\) −28.5410 −1.10511
\(668\) −36.6525 −1.41813
\(669\) −6.00000 −0.231973
\(670\) −6.03444 −0.233131
\(671\) 0 0
\(672\) −5.61803 −0.216720
\(673\) −3.59675 −0.138644 −0.0693222 0.997594i \(-0.522084\pi\)
−0.0693222 + 0.997594i \(0.522084\pi\)
\(674\) 5.05573 0.194739
\(675\) −4.00000 −0.153960
\(676\) 1.52786 0.0587640
\(677\) −19.3050 −0.741950 −0.370975 0.928643i \(-0.620976\pi\)
−0.370975 + 0.928643i \(0.620976\pi\)
\(678\) −0.291796 −0.0112064
\(679\) −9.70820 −0.372567
\(680\) 11.7082 0.448989
\(681\) 2.00000 0.0766402
\(682\) 0 0
\(683\) 37.0132 1.41627 0.708135 0.706078i \(-0.249537\pi\)
0.708135 + 0.706078i \(0.249537\pi\)
\(684\) −10.8541 −0.415017
\(685\) 19.7082 0.753012
\(686\) −0.618034 −0.0235966
\(687\) 7.23607 0.276073
\(688\) −10.5836 −0.403496
\(689\) 42.2918 1.61119
\(690\) 3.52786 0.134303
\(691\) 2.00000 0.0760836 0.0380418 0.999276i \(-0.487888\pi\)
0.0380418 + 0.999276i \(0.487888\pi\)
\(692\) −2.47214 −0.0939765
\(693\) 0 0
\(694\) −17.3050 −0.656887
\(695\) −14.4721 −0.548959
\(696\) 11.1803 0.423790
\(697\) −12.9443 −0.490299
\(698\) −0.978714 −0.0370449
\(699\) 14.9443 0.565244
\(700\) −6.47214 −0.244624
\(701\) −4.11146 −0.155288 −0.0776438 0.996981i \(-0.524740\pi\)
−0.0776438 + 0.996981i \(0.524740\pi\)
\(702\) −2.14590 −0.0809917
\(703\) −46.9574 −1.77103
\(704\) 0 0
\(705\) 0.236068 0.00889083
\(706\) 15.1591 0.570519
\(707\) 18.1803 0.683742
\(708\) 18.0902 0.679870
\(709\) −49.7214 −1.86732 −0.933662 0.358154i \(-0.883406\pi\)
−0.933662 + 0.358154i \(0.883406\pi\)
\(710\) −1.52786 −0.0573397
\(711\) 14.4721 0.542748
\(712\) 10.0000 0.374766
\(713\) −29.8885 −1.11933
\(714\) 3.23607 0.121107
\(715\) 0 0
\(716\) −14.4721 −0.540849
\(717\) −10.1246 −0.378111
\(718\) −14.4721 −0.540095
\(719\) 7.76393 0.289546 0.144773 0.989465i \(-0.453755\pi\)
0.144773 + 0.989465i \(0.453755\pi\)
\(720\) 1.85410 0.0690983
\(721\) −17.4164 −0.648621
\(722\) 16.0689 0.598022
\(723\) −25.9443 −0.964878
\(724\) 1.23607 0.0459381
\(725\) 20.0000 0.742781
\(726\) 0 0
\(727\) 24.1803 0.896799 0.448400 0.893833i \(-0.351994\pi\)
0.448400 + 0.893833i \(0.351994\pi\)
\(728\) −7.76393 −0.287750
\(729\) 1.00000 0.0370370
\(730\) −2.79837 −0.103572
\(731\) 29.8885 1.10547
\(732\) 3.23607 0.119609
\(733\) 8.11146 0.299603 0.149802 0.988716i \(-0.452136\pi\)
0.149802 + 0.988716i \(0.452136\pi\)
\(734\) −12.2918 −0.453698
\(735\) 1.00000 0.0368856
\(736\) 32.0689 1.18207
\(737\) 0 0
\(738\) 1.52786 0.0562415
\(739\) −34.0689 −1.25324 −0.626622 0.779323i \(-0.715563\pi\)
−0.626622 + 0.779323i \(0.715563\pi\)
\(740\) 11.3262 0.416361
\(741\) −23.2918 −0.855646
\(742\) 7.52786 0.276357
\(743\) −2.81966 −0.103443 −0.0517216 0.998662i \(-0.516471\pi\)
−0.0517216 + 0.998662i \(0.516471\pi\)
\(744\) 11.7082 0.429244
\(745\) −5.00000 −0.183186
\(746\) −2.87539 −0.105275
\(747\) −6.76393 −0.247479
\(748\) 0 0
\(749\) −4.23607 −0.154783
\(750\) −5.56231 −0.203107
\(751\) 39.7639 1.45101 0.725503 0.688219i \(-0.241607\pi\)
0.725503 + 0.688219i \(0.241607\pi\)
\(752\) 0.437694 0.0159611
\(753\) 12.1246 0.441845
\(754\) 10.7295 0.390745
\(755\) 14.1803 0.516075
\(756\) 1.61803 0.0588473
\(757\) −51.7214 −1.87984 −0.939922 0.341388i \(-0.889103\pi\)
−0.939922 + 0.341388i \(0.889103\pi\)
\(758\) −19.2705 −0.699936
\(759\) 0 0
\(760\) −15.0000 −0.544107
\(761\) −27.7771 −1.00692 −0.503459 0.864019i \(-0.667940\pi\)
−0.503459 + 0.864019i \(0.667940\pi\)
\(762\) 7.81966 0.283276
\(763\) 2.76393 0.100061
\(764\) 17.4164 0.630104
\(765\) −5.23607 −0.189310
\(766\) 20.3607 0.735661
\(767\) 38.8197 1.40170
\(768\) −6.56231 −0.236797
\(769\) −13.9443 −0.502843 −0.251422 0.967878i \(-0.580898\pi\)
−0.251422 + 0.967878i \(0.580898\pi\)
\(770\) 0 0
\(771\) −7.00000 −0.252099
\(772\) 23.7082 0.853277
\(773\) −5.47214 −0.196819 −0.0984095 0.995146i \(-0.531376\pi\)
−0.0984095 + 0.995146i \(0.531376\pi\)
\(774\) −3.52786 −0.126806
\(775\) 20.9443 0.752340
\(776\) −21.7082 −0.779279
\(777\) 7.00000 0.251124
\(778\) 6.83282 0.244968
\(779\) 16.5836 0.594169
\(780\) 5.61803 0.201158
\(781\) 0 0
\(782\) −18.4721 −0.660562
\(783\) −5.00000 −0.178685
\(784\) 1.85410 0.0662179
\(785\) −15.4164 −0.550235
\(786\) −0.583592 −0.0208160
\(787\) −3.65248 −0.130197 −0.0650984 0.997879i \(-0.520736\pi\)
−0.0650984 + 0.997879i \(0.520736\pi\)
\(788\) −26.6525 −0.949455
\(789\) 26.1246 0.930061
\(790\) 8.94427 0.318223
\(791\) 0.472136 0.0167872
\(792\) 0 0
\(793\) 6.94427 0.246598
\(794\) 14.2918 0.507197
\(795\) −12.1803 −0.431992
\(796\) 6.18034 0.219056
\(797\) −2.52786 −0.0895415 −0.0447708 0.998997i \(-0.514256\pi\)
−0.0447708 + 0.998997i \(0.514256\pi\)
\(798\) −4.14590 −0.146763
\(799\) −1.23607 −0.0437289
\(800\) −22.4721 −0.794510
\(801\) −4.47214 −0.158015
\(802\) −18.3607 −0.648338
\(803\) 0 0
\(804\) 15.7984 0.557166
\(805\) −5.70820 −0.201188
\(806\) 11.2361 0.395774
\(807\) 1.05573 0.0371634
\(808\) 40.6525 1.43015
\(809\) 56.3050 1.97958 0.989788 0.142545i \(-0.0455285\pi\)
0.989788 + 0.142545i \(0.0455285\pi\)
\(810\) 0.618034 0.0217155
\(811\) −15.2918 −0.536968 −0.268484 0.963284i \(-0.586523\pi\)
−0.268484 + 0.963284i \(0.586523\pi\)
\(812\) −8.09017 −0.283909
\(813\) −5.29180 −0.185591
\(814\) 0 0
\(815\) −22.7082 −0.795434
\(816\) −9.70820 −0.339855
\(817\) −38.2918 −1.33966
\(818\) −13.0132 −0.454994
\(819\) 3.47214 0.121326
\(820\) −4.00000 −0.139686
\(821\) 7.47214 0.260779 0.130390 0.991463i \(-0.458377\pi\)
0.130390 + 0.991463i \(0.458377\pi\)
\(822\) 12.1803 0.424838
\(823\) −17.1803 −0.598869 −0.299435 0.954117i \(-0.596798\pi\)
−0.299435 + 0.954117i \(0.596798\pi\)
\(824\) −38.9443 −1.35669
\(825\) 0 0
\(826\) 6.90983 0.240424
\(827\) −12.3475 −0.429365 −0.214683 0.976684i \(-0.568872\pi\)
−0.214683 + 0.976684i \(0.568872\pi\)
\(828\) −9.23607 −0.320976
\(829\) 35.7771 1.24259 0.621295 0.783577i \(-0.286607\pi\)
0.621295 + 0.783577i \(0.286607\pi\)
\(830\) −4.18034 −0.145102
\(831\) 6.47214 0.224516
\(832\) 0.819660 0.0284166
\(833\) −5.23607 −0.181419
\(834\) −8.94427 −0.309715
\(835\) 22.6525 0.783921
\(836\) 0 0
\(837\) −5.23607 −0.180985
\(838\) −0.729490 −0.0251998
\(839\) −23.5410 −0.812726 −0.406363 0.913712i \(-0.633203\pi\)
−0.406363 + 0.913712i \(0.633203\pi\)
\(840\) 2.23607 0.0771517
\(841\) −4.00000 −0.137931
\(842\) −8.03444 −0.276885
\(843\) −11.4721 −0.395121
\(844\) 8.76393 0.301667
\(845\) −0.944272 −0.0324839
\(846\) 0.145898 0.00501608
\(847\) 0 0
\(848\) −22.5836 −0.775524
\(849\) 13.7639 0.472377
\(850\) 12.9443 0.443985
\(851\) −39.9574 −1.36972
\(852\) 4.00000 0.137038
\(853\) 29.4164 1.00720 0.503599 0.863937i \(-0.332009\pi\)
0.503599 + 0.863937i \(0.332009\pi\)
\(854\) 1.23607 0.0422974
\(855\) 6.70820 0.229416
\(856\) −9.47214 −0.323751
\(857\) −0.111456 −0.00380727 −0.00190364 0.999998i \(-0.500606\pi\)
−0.00190364 + 0.999998i \(0.500606\pi\)
\(858\) 0 0
\(859\) −40.0000 −1.36478 −0.682391 0.730987i \(-0.739060\pi\)
−0.682391 + 0.730987i \(0.739060\pi\)
\(860\) 9.23607 0.314947
\(861\) −2.47214 −0.0842502
\(862\) −5.38197 −0.183310
\(863\) −43.2361 −1.47177 −0.735886 0.677105i \(-0.763234\pi\)
−0.735886 + 0.677105i \(0.763234\pi\)
\(864\) 5.61803 0.191129
\(865\) 1.52786 0.0519489
\(866\) −6.47214 −0.219932
\(867\) 10.4164 0.353760
\(868\) −8.47214 −0.287563
\(869\) 0 0
\(870\) −3.09017 −0.104767
\(871\) 33.9017 1.14872
\(872\) 6.18034 0.209293
\(873\) 9.70820 0.328573
\(874\) 23.6656 0.800502
\(875\) 9.00000 0.304256
\(876\) 7.32624 0.247531
\(877\) −4.58359 −0.154777 −0.0773885 0.997001i \(-0.524658\pi\)
−0.0773885 + 0.997001i \(0.524658\pi\)
\(878\) −6.90983 −0.233195
\(879\) 16.0000 0.539667
\(880\) 0 0
\(881\) 54.8885 1.84924 0.924621 0.380888i \(-0.124382\pi\)
0.924621 + 0.380888i \(0.124382\pi\)
\(882\) 0.618034 0.0208103
\(883\) −14.8197 −0.498721 −0.249361 0.968411i \(-0.580220\pi\)
−0.249361 + 0.968411i \(0.580220\pi\)
\(884\) −29.4164 −0.989381
\(885\) −11.1803 −0.375823
\(886\) 11.4164 0.383542
\(887\) −10.7639 −0.361417 −0.180709 0.983537i \(-0.557839\pi\)
−0.180709 + 0.983537i \(0.557839\pi\)
\(888\) 15.6525 0.525263
\(889\) −12.6525 −0.424350
\(890\) −2.76393 −0.0926472
\(891\) 0 0
\(892\) 9.70820 0.325055
\(893\) 1.58359 0.0529929
\(894\) −3.09017 −0.103351
\(895\) 8.94427 0.298974
\(896\) 11.3820 0.380245
\(897\) −19.8197 −0.661759
\(898\) 12.3607 0.412481
\(899\) 26.1803 0.873163
\(900\) 6.47214 0.215738
\(901\) 63.7771 2.12472
\(902\) 0 0
\(903\) 5.70820 0.189957
\(904\) 1.05573 0.0351130
\(905\) −0.763932 −0.0253940
\(906\) 8.76393 0.291162
\(907\) 28.0000 0.929725 0.464862 0.885383i \(-0.346104\pi\)
0.464862 + 0.885383i \(0.346104\pi\)
\(908\) −3.23607 −0.107393
\(909\) −18.1803 −0.603004
\(910\) 2.14590 0.0711358
\(911\) −14.5836 −0.483176 −0.241588 0.970379i \(-0.577668\pi\)
−0.241588 + 0.970379i \(0.577668\pi\)
\(912\) 12.4377 0.411853
\(913\) 0 0
\(914\) −9.41641 −0.311467
\(915\) −2.00000 −0.0661180
\(916\) −11.7082 −0.386850
\(917\) 0.944272 0.0311826
\(918\) −3.23607 −0.106806
\(919\) −39.5967 −1.30618 −0.653088 0.757282i \(-0.726527\pi\)
−0.653088 + 0.757282i \(0.726527\pi\)
\(920\) −12.7639 −0.420814
\(921\) 20.9443 0.690137
\(922\) 17.3050 0.569908
\(923\) 8.58359 0.282532
\(924\) 0 0
\(925\) 28.0000 0.920634
\(926\) −2.97871 −0.0978866
\(927\) 17.4164 0.572030
\(928\) −28.0902 −0.922105
\(929\) 12.8885 0.422859 0.211430 0.977393i \(-0.432188\pi\)
0.211430 + 0.977393i \(0.432188\pi\)
\(930\) −3.23607 −0.106115
\(931\) 6.70820 0.219853
\(932\) −24.1803 −0.792053
\(933\) 9.88854 0.323736
\(934\) 18.0344 0.590105
\(935\) 0 0
\(936\) 7.76393 0.253772
\(937\) 39.8885 1.30310 0.651551 0.758605i \(-0.274119\pi\)
0.651551 + 0.758605i \(0.274119\pi\)
\(938\) 6.03444 0.197032
\(939\) 24.6525 0.804503
\(940\) −0.381966 −0.0124584
\(941\) 60.3607 1.96770 0.983851 0.178990i \(-0.0572829\pi\)
0.983851 + 0.178990i \(0.0572829\pi\)
\(942\) −9.52786 −0.310435
\(943\) 14.1115 0.459532
\(944\) −20.7295 −0.674687
\(945\) −1.00000 −0.0325300
\(946\) 0 0
\(947\) −5.41641 −0.176010 −0.0880048 0.996120i \(-0.528049\pi\)
−0.0880048 + 0.996120i \(0.528049\pi\)
\(948\) −23.4164 −0.760530
\(949\) 15.7214 0.510337
\(950\) −16.5836 −0.538043
\(951\) 24.1803 0.784101
\(952\) −11.7082 −0.379465
\(953\) −44.7771 −1.45047 −0.725236 0.688500i \(-0.758269\pi\)
−0.725236 + 0.688500i \(0.758269\pi\)
\(954\) −7.52786 −0.243724
\(955\) −10.7639 −0.348313
\(956\) 16.3820 0.529831
\(957\) 0 0
\(958\) −7.23607 −0.233787
\(959\) −19.7082 −0.636411
\(960\) −0.236068 −0.00761906
\(961\) −3.58359 −0.115600
\(962\) 15.0213 0.484306
\(963\) 4.23607 0.136505
\(964\) 41.9787 1.35204
\(965\) −14.6525 −0.471680
\(966\) −3.52786 −0.113507
\(967\) 61.1935 1.96785 0.983925 0.178582i \(-0.0571509\pi\)
0.983925 + 0.178582i \(0.0571509\pi\)
\(968\) 0 0
\(969\) −35.1246 −1.12837
\(970\) 6.00000 0.192648
\(971\) 32.1246 1.03093 0.515464 0.856911i \(-0.327620\pi\)
0.515464 + 0.856911i \(0.327620\pi\)
\(972\) −1.61803 −0.0518985
\(973\) 14.4721 0.463955
\(974\) 10.4721 0.335549
\(975\) 13.8885 0.444789
\(976\) −3.70820 −0.118697
\(977\) 4.18034 0.133741 0.0668705 0.997762i \(-0.478699\pi\)
0.0668705 + 0.997762i \(0.478699\pi\)
\(978\) −14.0344 −0.448772
\(979\) 0 0
\(980\) −1.61803 −0.0516862
\(981\) −2.76393 −0.0882456
\(982\) 11.2016 0.357458
\(983\) 27.4164 0.874448 0.437224 0.899353i \(-0.355962\pi\)
0.437224 + 0.899353i \(0.355962\pi\)
\(984\) −5.52786 −0.176222
\(985\) 16.4721 0.524846
\(986\) 16.1803 0.515287
\(987\) −0.236068 −0.00751413
\(988\) 37.6869 1.19898
\(989\) −32.5836 −1.03610
\(990\) 0 0
\(991\) −29.1803 −0.926944 −0.463472 0.886112i \(-0.653397\pi\)
−0.463472 + 0.886112i \(0.653397\pi\)
\(992\) −29.4164 −0.933972
\(993\) −11.4164 −0.362289
\(994\) 1.52786 0.0484609
\(995\) −3.81966 −0.121091
\(996\) 10.9443 0.346783
\(997\) −26.9443 −0.853334 −0.426667 0.904409i \(-0.640312\pi\)
−0.426667 + 0.904409i \(0.640312\pi\)
\(998\) −1.38197 −0.0437454
\(999\) −7.00000 −0.221470
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 2541.2.a.t.1.2 2
3.2 odd 2 7623.2.a.bm.1.1 2
11.10 odd 2 231.2.a.c.1.1 2
33.32 even 2 693.2.a.f.1.2 2
44.43 even 2 3696.2.a.be.1.2 2
55.54 odd 2 5775.2.a.be.1.2 2
77.76 even 2 1617.2.a.p.1.1 2
231.230 odd 2 4851.2.a.w.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
231.2.a.c.1.1 2 11.10 odd 2
693.2.a.f.1.2 2 33.32 even 2
1617.2.a.p.1.1 2 77.76 even 2
2541.2.a.t.1.2 2 1.1 even 1 trivial
3696.2.a.be.1.2 2 44.43 even 2
4851.2.a.w.1.2 2 231.230 odd 2
5775.2.a.be.1.2 2 55.54 odd 2
7623.2.a.bm.1.1 2 3.2 odd 2