Properties

Label 1815.2.a.f.1.2
Level $1815$
Weight $2$
Character 1815.1
Self dual yes
Analytic conductor $14.493$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1815 = 3 \cdot 5 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1815.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(14.4928479669\)
Analytic rank: \(0\)
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: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-0.618034\) of defining polynomial
Character \(\chi\) \(=\) 1815.1

$q$-expansion

\(f(q)\) \(=\) \(q+0.618034 q^{2} -1.00000 q^{3} -1.61803 q^{4} -1.00000 q^{5} -0.618034 q^{6} -3.23607 q^{7} -2.23607 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+0.618034 q^{2} -1.00000 q^{3} -1.61803 q^{4} -1.00000 q^{5} -0.618034 q^{6} -3.23607 q^{7} -2.23607 q^{8} +1.00000 q^{9} -0.618034 q^{10} +1.61803 q^{12} -5.23607 q^{13} -2.00000 q^{14} +1.00000 q^{15} +1.85410 q^{16} +5.47214 q^{17} +0.618034 q^{18} -6.47214 q^{19} +1.61803 q^{20} +3.23607 q^{21} -4.70820 q^{23} +2.23607 q^{24} +1.00000 q^{25} -3.23607 q^{26} -1.00000 q^{27} +5.23607 q^{28} +1.23607 q^{29} +0.618034 q^{30} -6.70820 q^{31} +5.61803 q^{32} +3.38197 q^{34} +3.23607 q^{35} -1.61803 q^{36} +0.763932 q^{37} -4.00000 q^{38} +5.23607 q^{39} +2.23607 q^{40} -3.52786 q^{41} +2.00000 q^{42} +5.23607 q^{43} -1.00000 q^{45} -2.90983 q^{46} +8.70820 q^{47} -1.85410 q^{48} +3.47214 q^{49} +0.618034 q^{50} -5.47214 q^{51} +8.47214 q^{52} +9.94427 q^{53} -0.618034 q^{54} +7.23607 q^{56} +6.47214 q^{57} +0.763932 q^{58} +11.7082 q^{59} -1.61803 q^{60} +1.47214 q^{61} -4.14590 q^{62} -3.23607 q^{63} -0.236068 q^{64} +5.23607 q^{65} +11.2361 q^{67} -8.85410 q^{68} +4.70820 q^{69} +2.00000 q^{70} -14.4721 q^{71} -2.23607 q^{72} +10.4721 q^{73} +0.472136 q^{74} -1.00000 q^{75} +10.4721 q^{76} +3.23607 q^{78} -12.7082 q^{79} -1.85410 q^{80} +1.00000 q^{81} -2.18034 q^{82} +4.00000 q^{83} -5.23607 q^{84} -5.47214 q^{85} +3.23607 q^{86} -1.23607 q^{87} +4.76393 q^{89} -0.618034 q^{90} +16.9443 q^{91} +7.61803 q^{92} +6.70820 q^{93} +5.38197 q^{94} +6.47214 q^{95} -5.61803 q^{96} +12.7639 q^{97} +2.14590 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - q^{2} - 2q^{3} - q^{4} - 2q^{5} + q^{6} - 2q^{7} + 2q^{9} + O(q^{10}) \) \( 2q - q^{2} - 2q^{3} - q^{4} - 2q^{5} + q^{6} - 2q^{7} + 2q^{9} + q^{10} + q^{12} - 6q^{13} - 4q^{14} + 2q^{15} - 3q^{16} + 2q^{17} - q^{18} - 4q^{19} + q^{20} + 2q^{21} + 4q^{23} + 2q^{25} - 2q^{26} - 2q^{27} + 6q^{28} - 2q^{29} - q^{30} + 9q^{32} + 9q^{34} + 2q^{35} - q^{36} + 6q^{37} - 8q^{38} + 6q^{39} - 16q^{41} + 4q^{42} + 6q^{43} - 2q^{45} - 17q^{46} + 4q^{47} + 3q^{48} - 2q^{49} - q^{50} - 2q^{51} + 8q^{52} + 2q^{53} + q^{54} + 10q^{56} + 4q^{57} + 6q^{58} + 10q^{59} - q^{60} - 6q^{61} - 15q^{62} - 2q^{63} + 4q^{64} + 6q^{65} + 18q^{67} - 11q^{68} - 4q^{69} + 4q^{70} - 20q^{71} + 12q^{73} - 8q^{74} - 2q^{75} + 12q^{76} + 2q^{78} - 12q^{79} + 3q^{80} + 2q^{81} + 18q^{82} + 8q^{83} - 6q^{84} - 2q^{85} + 2q^{86} + 2q^{87} + 14q^{89} + q^{90} + 16q^{91} + 13q^{92} + 13q^{94} + 4q^{95} - 9q^{96} + 30q^{97} + 11q^{98} + O(q^{100}) \)

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
\(6\) −0.618034 −0.252311
\(7\) −3.23607 −1.22312 −0.611559 0.791199i \(-0.709457\pi\)
−0.611559 + 0.791199i \(0.709457\pi\)
\(8\) −2.23607 −0.790569
\(9\) 1.00000 0.333333
\(10\) −0.618034 −0.195440
\(11\) 0 0
\(12\) 1.61803 0.467086
\(13\) −5.23607 −1.45222 −0.726112 0.687576i \(-0.758675\pi\)
−0.726112 + 0.687576i \(0.758675\pi\)
\(14\) −2.00000 −0.534522
\(15\) 1.00000 0.258199
\(16\) 1.85410 0.463525
\(17\) 5.47214 1.32719 0.663594 0.748093i \(-0.269030\pi\)
0.663594 + 0.748093i \(0.269030\pi\)
\(18\) 0.618034 0.145672
\(19\) −6.47214 −1.48481 −0.742405 0.669951i \(-0.766315\pi\)
−0.742405 + 0.669951i \(0.766315\pi\)
\(20\) 1.61803 0.361803
\(21\) 3.23607 0.706168
\(22\) 0 0
\(23\) −4.70820 −0.981728 −0.490864 0.871236i \(-0.663319\pi\)
−0.490864 + 0.871236i \(0.663319\pi\)
\(24\) 2.23607 0.456435
\(25\) 1.00000 0.200000
\(26\) −3.23607 −0.634645
\(27\) −1.00000 −0.192450
\(28\) 5.23607 0.989524
\(29\) 1.23607 0.229532 0.114766 0.993393i \(-0.463388\pi\)
0.114766 + 0.993393i \(0.463388\pi\)
\(30\) 0.618034 0.112837
\(31\) −6.70820 −1.20483 −0.602414 0.798183i \(-0.705795\pi\)
−0.602414 + 0.798183i \(0.705795\pi\)
\(32\) 5.61803 0.993137
\(33\) 0 0
\(34\) 3.38197 0.580002
\(35\) 3.23607 0.546995
\(36\) −1.61803 −0.269672
\(37\) 0.763932 0.125590 0.0627948 0.998026i \(-0.479999\pi\)
0.0627948 + 0.998026i \(0.479999\pi\)
\(38\) −4.00000 −0.648886
\(39\) 5.23607 0.838442
\(40\) 2.23607 0.353553
\(41\) −3.52786 −0.550960 −0.275480 0.961307i \(-0.588837\pi\)
−0.275480 + 0.961307i \(0.588837\pi\)
\(42\) 2.00000 0.308607
\(43\) 5.23607 0.798493 0.399246 0.916844i \(-0.369272\pi\)
0.399246 + 0.916844i \(0.369272\pi\)
\(44\) 0 0
\(45\) −1.00000 −0.149071
\(46\) −2.90983 −0.429031
\(47\) 8.70820 1.27022 0.635111 0.772421i \(-0.280954\pi\)
0.635111 + 0.772421i \(0.280954\pi\)
\(48\) −1.85410 −0.267617
\(49\) 3.47214 0.496019
\(50\) 0.618034 0.0874032
\(51\) −5.47214 −0.766252
\(52\) 8.47214 1.17487
\(53\) 9.94427 1.36595 0.682975 0.730441i \(-0.260686\pi\)
0.682975 + 0.730441i \(0.260686\pi\)
\(54\) −0.618034 −0.0841038
\(55\) 0 0
\(56\) 7.23607 0.966960
\(57\) 6.47214 0.857255
\(58\) 0.763932 0.100309
\(59\) 11.7082 1.52428 0.762139 0.647413i \(-0.224149\pi\)
0.762139 + 0.647413i \(0.224149\pi\)
\(60\) −1.61803 −0.208887
\(61\) 1.47214 0.188488 0.0942438 0.995549i \(-0.469957\pi\)
0.0942438 + 0.995549i \(0.469957\pi\)
\(62\) −4.14590 −0.526530
\(63\) −3.23607 −0.407706
\(64\) −0.236068 −0.0295085
\(65\) 5.23607 0.649454
\(66\) 0 0
\(67\) 11.2361 1.37270 0.686352 0.727269i \(-0.259211\pi\)
0.686352 + 0.727269i \(0.259211\pi\)
\(68\) −8.85410 −1.07372
\(69\) 4.70820 0.566801
\(70\) 2.00000 0.239046
\(71\) −14.4721 −1.71753 −0.858763 0.512373i \(-0.828767\pi\)
−0.858763 + 0.512373i \(0.828767\pi\)
\(72\) −2.23607 −0.263523
\(73\) 10.4721 1.22567 0.612835 0.790211i \(-0.290029\pi\)
0.612835 + 0.790211i \(0.290029\pi\)
\(74\) 0.472136 0.0548847
\(75\) −1.00000 −0.115470
\(76\) 10.4721 1.20124
\(77\) 0 0
\(78\) 3.23607 0.366413
\(79\) −12.7082 −1.42978 −0.714892 0.699235i \(-0.753524\pi\)
−0.714892 + 0.699235i \(0.753524\pi\)
\(80\) −1.85410 −0.207295
\(81\) 1.00000 0.111111
\(82\) −2.18034 −0.240778
\(83\) 4.00000 0.439057 0.219529 0.975606i \(-0.429548\pi\)
0.219529 + 0.975606i \(0.429548\pi\)
\(84\) −5.23607 −0.571302
\(85\) −5.47214 −0.593536
\(86\) 3.23607 0.348954
\(87\) −1.23607 −0.132520
\(88\) 0 0
\(89\) 4.76393 0.504976 0.252488 0.967600i \(-0.418751\pi\)
0.252488 + 0.967600i \(0.418751\pi\)
\(90\) −0.618034 −0.0651465
\(91\) 16.9443 1.77624
\(92\) 7.61803 0.794235
\(93\) 6.70820 0.695608
\(94\) 5.38197 0.555107
\(95\) 6.47214 0.664027
\(96\) −5.61803 −0.573388
\(97\) 12.7639 1.29598 0.647990 0.761648i \(-0.275610\pi\)
0.647990 + 0.761648i \(0.275610\pi\)
\(98\) 2.14590 0.216768
\(99\) 0 0
\(100\) −1.61803 −0.161803
\(101\) 5.52786 0.550043 0.275022 0.961438i \(-0.411315\pi\)
0.275022 + 0.961438i \(0.411315\pi\)
\(102\) −3.38197 −0.334865
\(103\) −0.944272 −0.0930419 −0.0465209 0.998917i \(-0.514813\pi\)
−0.0465209 + 0.998917i \(0.514813\pi\)
\(104\) 11.7082 1.14808
\(105\) −3.23607 −0.315808
\(106\) 6.14590 0.596942
\(107\) −14.2361 −1.37625 −0.688126 0.725591i \(-0.741567\pi\)
−0.688126 + 0.725591i \(0.741567\pi\)
\(108\) 1.61803 0.155695
\(109\) −14.9443 −1.43140 −0.715701 0.698407i \(-0.753893\pi\)
−0.715701 + 0.698407i \(0.753893\pi\)
\(110\) 0 0
\(111\) −0.763932 −0.0725092
\(112\) −6.00000 −0.566947
\(113\) −12.4164 −1.16804 −0.584019 0.811740i \(-0.698521\pi\)
−0.584019 + 0.811740i \(0.698521\pi\)
\(114\) 4.00000 0.374634
\(115\) 4.70820 0.439042
\(116\) −2.00000 −0.185695
\(117\) −5.23607 −0.484075
\(118\) 7.23607 0.666134
\(119\) −17.7082 −1.62331
\(120\) −2.23607 −0.204124
\(121\) 0 0
\(122\) 0.909830 0.0823721
\(123\) 3.52786 0.318097
\(124\) 10.8541 0.974727
\(125\) −1.00000 −0.0894427
\(126\) −2.00000 −0.178174
\(127\) −3.70820 −0.329050 −0.164525 0.986373i \(-0.552609\pi\)
−0.164525 + 0.986373i \(0.552609\pi\)
\(128\) −11.3820 −1.00603
\(129\) −5.23607 −0.461010
\(130\) 3.23607 0.283822
\(131\) 0.472136 0.0412507 0.0206254 0.999787i \(-0.493434\pi\)
0.0206254 + 0.999787i \(0.493434\pi\)
\(132\) 0 0
\(133\) 20.9443 1.81610
\(134\) 6.94427 0.599894
\(135\) 1.00000 0.0860663
\(136\) −12.2361 −1.04923
\(137\) 17.4721 1.49275 0.746373 0.665528i \(-0.231794\pi\)
0.746373 + 0.665528i \(0.231794\pi\)
\(138\) 2.90983 0.247701
\(139\) 7.29180 0.618482 0.309241 0.950984i \(-0.399925\pi\)
0.309241 + 0.950984i \(0.399925\pi\)
\(140\) −5.23607 −0.442529
\(141\) −8.70820 −0.733363
\(142\) −8.94427 −0.750587
\(143\) 0 0
\(144\) 1.85410 0.154508
\(145\) −1.23607 −0.102650
\(146\) 6.47214 0.535638
\(147\) −3.47214 −0.286377
\(148\) −1.23607 −0.101604
\(149\) 4.29180 0.351598 0.175799 0.984426i \(-0.443749\pi\)
0.175799 + 0.984426i \(0.443749\pi\)
\(150\) −0.618034 −0.0504623
\(151\) −7.29180 −0.593398 −0.296699 0.954971i \(-0.595886\pi\)
−0.296699 + 0.954971i \(0.595886\pi\)
\(152\) 14.4721 1.17385
\(153\) 5.47214 0.442396
\(154\) 0 0
\(155\) 6.70820 0.538816
\(156\) −8.47214 −0.678314
\(157\) 2.29180 0.182905 0.0914526 0.995809i \(-0.470849\pi\)
0.0914526 + 0.995809i \(0.470849\pi\)
\(158\) −7.85410 −0.624839
\(159\) −9.94427 −0.788632
\(160\) −5.61803 −0.444145
\(161\) 15.2361 1.20077
\(162\) 0.618034 0.0485573
\(163\) 6.18034 0.484082 0.242041 0.970266i \(-0.422183\pi\)
0.242041 + 0.970266i \(0.422183\pi\)
\(164\) 5.70820 0.445736
\(165\) 0 0
\(166\) 2.47214 0.191875
\(167\) −3.18034 −0.246102 −0.123051 0.992400i \(-0.539268\pi\)
−0.123051 + 0.992400i \(0.539268\pi\)
\(168\) −7.23607 −0.558275
\(169\) 14.4164 1.10895
\(170\) −3.38197 −0.259385
\(171\) −6.47214 −0.494937
\(172\) −8.47214 −0.645994
\(173\) −2.94427 −0.223849 −0.111924 0.993717i \(-0.535701\pi\)
−0.111924 + 0.993717i \(0.535701\pi\)
\(174\) −0.763932 −0.0579135
\(175\) −3.23607 −0.244624
\(176\) 0 0
\(177\) −11.7082 −0.880042
\(178\) 2.94427 0.220683
\(179\) −20.1803 −1.50835 −0.754175 0.656674i \(-0.771963\pi\)
−0.754175 + 0.656674i \(0.771963\pi\)
\(180\) 1.61803 0.120601
\(181\) −21.4164 −1.59187 −0.795935 0.605383i \(-0.793020\pi\)
−0.795935 + 0.605383i \(0.793020\pi\)
\(182\) 10.4721 0.776246
\(183\) −1.47214 −0.108823
\(184\) 10.5279 0.776124
\(185\) −0.763932 −0.0561654
\(186\) 4.14590 0.303992
\(187\) 0 0
\(188\) −14.0902 −1.02763
\(189\) 3.23607 0.235389
\(190\) 4.00000 0.290191
\(191\) 27.5967 1.99683 0.998415 0.0562752i \(-0.0179224\pi\)
0.998415 + 0.0562752i \(0.0179224\pi\)
\(192\) 0.236068 0.0170367
\(193\) −16.1803 −1.16469 −0.582343 0.812943i \(-0.697864\pi\)
−0.582343 + 0.812943i \(0.697864\pi\)
\(194\) 7.88854 0.566364
\(195\) −5.23607 −0.374963
\(196\) −5.61803 −0.401288
\(197\) 1.41641 0.100915 0.0504574 0.998726i \(-0.483932\pi\)
0.0504574 + 0.998726i \(0.483932\pi\)
\(198\) 0 0
\(199\) 16.2361 1.15094 0.575472 0.817821i \(-0.304818\pi\)
0.575472 + 0.817821i \(0.304818\pi\)
\(200\) −2.23607 −0.158114
\(201\) −11.2361 −0.792531
\(202\) 3.41641 0.240378
\(203\) −4.00000 −0.280745
\(204\) 8.85410 0.619911
\(205\) 3.52786 0.246397
\(206\) −0.583592 −0.0406608
\(207\) −4.70820 −0.327243
\(208\) −9.70820 −0.673143
\(209\) 0 0
\(210\) −2.00000 −0.138013
\(211\) −1.76393 −0.121434 −0.0607170 0.998155i \(-0.519339\pi\)
−0.0607170 + 0.998155i \(0.519339\pi\)
\(212\) −16.0902 −1.10508
\(213\) 14.4721 0.991614
\(214\) −8.79837 −0.601444
\(215\) −5.23607 −0.357097
\(216\) 2.23607 0.152145
\(217\) 21.7082 1.47365
\(218\) −9.23607 −0.625545
\(219\) −10.4721 −0.707641
\(220\) 0 0
\(221\) −28.6525 −1.92737
\(222\) −0.472136 −0.0316877
\(223\) 9.05573 0.606416 0.303208 0.952924i \(-0.401942\pi\)
0.303208 + 0.952924i \(0.401942\pi\)
\(224\) −18.1803 −1.21473
\(225\) 1.00000 0.0666667
\(226\) −7.67376 −0.510451
\(227\) −16.7082 −1.10896 −0.554481 0.832196i \(-0.687083\pi\)
−0.554481 + 0.832196i \(0.687083\pi\)
\(228\) −10.4721 −0.693534
\(229\) 7.00000 0.462573 0.231287 0.972886i \(-0.425707\pi\)
0.231287 + 0.972886i \(0.425707\pi\)
\(230\) 2.90983 0.191869
\(231\) 0 0
\(232\) −2.76393 −0.181461
\(233\) −10.8885 −0.713332 −0.356666 0.934232i \(-0.616087\pi\)
−0.356666 + 0.934232i \(0.616087\pi\)
\(234\) −3.23607 −0.211548
\(235\) −8.70820 −0.568061
\(236\) −18.9443 −1.23317
\(237\) 12.7082 0.825487
\(238\) −10.9443 −0.709412
\(239\) 24.6525 1.59464 0.797318 0.603559i \(-0.206251\pi\)
0.797318 + 0.603559i \(0.206251\pi\)
\(240\) 1.85410 0.119682
\(241\) 17.9443 1.15589 0.577946 0.816075i \(-0.303854\pi\)
0.577946 + 0.816075i \(0.303854\pi\)
\(242\) 0 0
\(243\) −1.00000 −0.0641500
\(244\) −2.38197 −0.152490
\(245\) −3.47214 −0.221827
\(246\) 2.18034 0.139013
\(247\) 33.8885 2.15628
\(248\) 15.0000 0.952501
\(249\) −4.00000 −0.253490
\(250\) −0.618034 −0.0390879
\(251\) −23.7082 −1.49645 −0.748224 0.663446i \(-0.769093\pi\)
−0.748224 + 0.663446i \(0.769093\pi\)
\(252\) 5.23607 0.329841
\(253\) 0 0
\(254\) −2.29180 −0.143800
\(255\) 5.47214 0.342678
\(256\) −6.56231 −0.410144
\(257\) 28.8885 1.80202 0.901009 0.433801i \(-0.142828\pi\)
0.901009 + 0.433801i \(0.142828\pi\)
\(258\) −3.23607 −0.201469
\(259\) −2.47214 −0.153611
\(260\) −8.47214 −0.525420
\(261\) 1.23607 0.0765107
\(262\) 0.291796 0.0180272
\(263\) −20.1246 −1.24094 −0.620468 0.784231i \(-0.713057\pi\)
−0.620468 + 0.784231i \(0.713057\pi\)
\(264\) 0 0
\(265\) −9.94427 −0.610872
\(266\) 12.9443 0.793664
\(267\) −4.76393 −0.291548
\(268\) −18.1803 −1.11054
\(269\) 7.05573 0.430195 0.215098 0.976593i \(-0.430993\pi\)
0.215098 + 0.976593i \(0.430993\pi\)
\(270\) 0.618034 0.0376124
\(271\) 24.2361 1.47224 0.736118 0.676853i \(-0.236657\pi\)
0.736118 + 0.676853i \(0.236657\pi\)
\(272\) 10.1459 0.615185
\(273\) −16.9443 −1.02551
\(274\) 10.7984 0.652354
\(275\) 0 0
\(276\) −7.61803 −0.458552
\(277\) 4.94427 0.297073 0.148536 0.988907i \(-0.452544\pi\)
0.148536 + 0.988907i \(0.452544\pi\)
\(278\) 4.50658 0.270287
\(279\) −6.70820 −0.401610
\(280\) −7.23607 −0.432438
\(281\) −15.2361 −0.908908 −0.454454 0.890770i \(-0.650166\pi\)
−0.454454 + 0.890770i \(0.650166\pi\)
\(282\) −5.38197 −0.320491
\(283\) −11.8885 −0.706701 −0.353350 0.935491i \(-0.614958\pi\)
−0.353350 + 0.935491i \(0.614958\pi\)
\(284\) 23.4164 1.38951
\(285\) −6.47214 −0.383376
\(286\) 0 0
\(287\) 11.4164 0.673889
\(288\) 5.61803 0.331046
\(289\) 12.9443 0.761428
\(290\) −0.763932 −0.0448596
\(291\) −12.7639 −0.748235
\(292\) −16.9443 −0.991589
\(293\) −21.4721 −1.25442 −0.627208 0.778852i \(-0.715802\pi\)
−0.627208 + 0.778852i \(0.715802\pi\)
\(294\) −2.14590 −0.125151
\(295\) −11.7082 −0.681678
\(296\) −1.70820 −0.0992873
\(297\) 0 0
\(298\) 2.65248 0.153654
\(299\) 24.6525 1.42569
\(300\) 1.61803 0.0934172
\(301\) −16.9443 −0.976652
\(302\) −4.50658 −0.259324
\(303\) −5.52786 −0.317567
\(304\) −12.0000 −0.688247
\(305\) −1.47214 −0.0842943
\(306\) 3.38197 0.193334
\(307\) 11.8885 0.678515 0.339258 0.940694i \(-0.389824\pi\)
0.339258 + 0.940694i \(0.389824\pi\)
\(308\) 0 0
\(309\) 0.944272 0.0537178
\(310\) 4.14590 0.235471
\(311\) −3.23607 −0.183501 −0.0917503 0.995782i \(-0.529246\pi\)
−0.0917503 + 0.995782i \(0.529246\pi\)
\(312\) −11.7082 −0.662847
\(313\) 22.7639 1.28669 0.643347 0.765575i \(-0.277545\pi\)
0.643347 + 0.765575i \(0.277545\pi\)
\(314\) 1.41641 0.0799325
\(315\) 3.23607 0.182332
\(316\) 20.5623 1.15672
\(317\) −3.94427 −0.221532 −0.110766 0.993846i \(-0.535330\pi\)
−0.110766 + 0.993846i \(0.535330\pi\)
\(318\) −6.14590 −0.344645
\(319\) 0 0
\(320\) 0.236068 0.0131966
\(321\) 14.2361 0.794580
\(322\) 9.41641 0.524756
\(323\) −35.4164 −1.97062
\(324\) −1.61803 −0.0898908
\(325\) −5.23607 −0.290445
\(326\) 3.81966 0.211551
\(327\) 14.9443 0.826420
\(328\) 7.88854 0.435572
\(329\) −28.1803 −1.55363
\(330\) 0 0
\(331\) −17.1803 −0.944317 −0.472158 0.881514i \(-0.656525\pi\)
−0.472158 + 0.881514i \(0.656525\pi\)
\(332\) −6.47214 −0.355205
\(333\) 0.763932 0.0418632
\(334\) −1.96556 −0.107551
\(335\) −11.2361 −0.613892
\(336\) 6.00000 0.327327
\(337\) 7.88854 0.429716 0.214858 0.976645i \(-0.431071\pi\)
0.214858 + 0.976645i \(0.431071\pi\)
\(338\) 8.90983 0.484631
\(339\) 12.4164 0.674367
\(340\) 8.85410 0.480181
\(341\) 0 0
\(342\) −4.00000 −0.216295
\(343\) 11.4164 0.616428
\(344\) −11.7082 −0.631264
\(345\) −4.70820 −0.253481
\(346\) −1.81966 −0.0978255
\(347\) 14.2361 0.764232 0.382116 0.924114i \(-0.375195\pi\)
0.382116 + 0.924114i \(0.375195\pi\)
\(348\) 2.00000 0.107211
\(349\) 15.0000 0.802932 0.401466 0.915874i \(-0.368501\pi\)
0.401466 + 0.915874i \(0.368501\pi\)
\(350\) −2.00000 −0.106904
\(351\) 5.23607 0.279481
\(352\) 0 0
\(353\) −7.00000 −0.372572 −0.186286 0.982496i \(-0.559645\pi\)
−0.186286 + 0.982496i \(0.559645\pi\)
\(354\) −7.23607 −0.384593
\(355\) 14.4721 0.768101
\(356\) −7.70820 −0.408534
\(357\) 17.7082 0.937218
\(358\) −12.4721 −0.659173
\(359\) −7.41641 −0.391423 −0.195712 0.980662i \(-0.562702\pi\)
−0.195712 + 0.980662i \(0.562702\pi\)
\(360\) 2.23607 0.117851
\(361\) 22.8885 1.20466
\(362\) −13.2361 −0.695672
\(363\) 0 0
\(364\) −27.4164 −1.43701
\(365\) −10.4721 −0.548137
\(366\) −0.909830 −0.0475576
\(367\) 4.76393 0.248675 0.124338 0.992240i \(-0.460319\pi\)
0.124338 + 0.992240i \(0.460319\pi\)
\(368\) −8.72949 −0.455056
\(369\) −3.52786 −0.183653
\(370\) −0.472136 −0.0245452
\(371\) −32.1803 −1.67072
\(372\) −10.8541 −0.562759
\(373\) −29.8885 −1.54757 −0.773785 0.633448i \(-0.781639\pi\)
−0.773785 + 0.633448i \(0.781639\pi\)
\(374\) 0 0
\(375\) 1.00000 0.0516398
\(376\) −19.4721 −1.00420
\(377\) −6.47214 −0.333332
\(378\) 2.00000 0.102869
\(379\) 30.5967 1.57165 0.785825 0.618449i \(-0.212239\pi\)
0.785825 + 0.618449i \(0.212239\pi\)
\(380\) −10.4721 −0.537209
\(381\) 3.70820 0.189977
\(382\) 17.0557 0.872647
\(383\) 16.0000 0.817562 0.408781 0.912633i \(-0.365954\pi\)
0.408781 + 0.912633i \(0.365954\pi\)
\(384\) 11.3820 0.580834
\(385\) 0 0
\(386\) −10.0000 −0.508987
\(387\) 5.23607 0.266164
\(388\) −20.6525 −1.04847
\(389\) −21.5967 −1.09500 −0.547499 0.836806i \(-0.684420\pi\)
−0.547499 + 0.836806i \(0.684420\pi\)
\(390\) −3.23607 −0.163865
\(391\) −25.7639 −1.30294
\(392\) −7.76393 −0.392138
\(393\) −0.472136 −0.0238161
\(394\) 0.875388 0.0441014
\(395\) 12.7082 0.639419
\(396\) 0 0
\(397\) −17.4164 −0.874104 −0.437052 0.899436i \(-0.643978\pi\)
−0.437052 + 0.899436i \(0.643978\pi\)
\(398\) 10.0344 0.502981
\(399\) −20.9443 −1.04853
\(400\) 1.85410 0.0927051
\(401\) 20.9443 1.04591 0.522954 0.852361i \(-0.324830\pi\)
0.522954 + 0.852361i \(0.324830\pi\)
\(402\) −6.94427 −0.346349
\(403\) 35.1246 1.74968
\(404\) −8.94427 −0.444994
\(405\) −1.00000 −0.0496904
\(406\) −2.47214 −0.122690
\(407\) 0 0
\(408\) 12.2361 0.605776
\(409\) −36.7771 −1.81851 −0.909255 0.416240i \(-0.863348\pi\)
−0.909255 + 0.416240i \(0.863348\pi\)
\(410\) 2.18034 0.107679
\(411\) −17.4721 −0.861837
\(412\) 1.52786 0.0752725
\(413\) −37.8885 −1.86437
\(414\) −2.90983 −0.143010
\(415\) −4.00000 −0.196352
\(416\) −29.4164 −1.44226
\(417\) −7.29180 −0.357081
\(418\) 0 0
\(419\) −14.0000 −0.683945 −0.341972 0.939710i \(-0.611095\pi\)
−0.341972 + 0.939710i \(0.611095\pi\)
\(420\) 5.23607 0.255494
\(421\) 29.8328 1.45396 0.726981 0.686657i \(-0.240923\pi\)
0.726981 + 0.686657i \(0.240923\pi\)
\(422\) −1.09017 −0.0530686
\(423\) 8.70820 0.423407
\(424\) −22.2361 −1.07988
\(425\) 5.47214 0.265438
\(426\) 8.94427 0.433351
\(427\) −4.76393 −0.230543
\(428\) 23.0344 1.11341
\(429\) 0 0
\(430\) −3.23607 −0.156057
\(431\) 5.81966 0.280323 0.140162 0.990129i \(-0.455238\pi\)
0.140162 + 0.990129i \(0.455238\pi\)
\(432\) −1.85410 −0.0892055
\(433\) 25.2361 1.21277 0.606384 0.795172i \(-0.292619\pi\)
0.606384 + 0.795172i \(0.292619\pi\)
\(434\) 13.4164 0.644008
\(435\) 1.23607 0.0592649
\(436\) 24.1803 1.15803
\(437\) 30.4721 1.45768
\(438\) −6.47214 −0.309251
\(439\) 8.12461 0.387767 0.193883 0.981025i \(-0.437892\pi\)
0.193883 + 0.981025i \(0.437892\pi\)
\(440\) 0 0
\(441\) 3.47214 0.165340
\(442\) −17.7082 −0.842293
\(443\) 7.41641 0.352364 0.176182 0.984358i \(-0.443625\pi\)
0.176182 + 0.984358i \(0.443625\pi\)
\(444\) 1.23607 0.0586612
\(445\) −4.76393 −0.225832
\(446\) 5.59675 0.265014
\(447\) −4.29180 −0.202995
\(448\) 0.763932 0.0360924
\(449\) −10.0000 −0.471929 −0.235965 0.971762i \(-0.575825\pi\)
−0.235965 + 0.971762i \(0.575825\pi\)
\(450\) 0.618034 0.0291344
\(451\) 0 0
\(452\) 20.0902 0.944962
\(453\) 7.29180 0.342598
\(454\) −10.3262 −0.484634
\(455\) −16.9443 −0.794360
\(456\) −14.4721 −0.677720
\(457\) 34.3607 1.60732 0.803662 0.595085i \(-0.202882\pi\)
0.803662 + 0.595085i \(0.202882\pi\)
\(458\) 4.32624 0.202152
\(459\) −5.47214 −0.255417
\(460\) −7.61803 −0.355193
\(461\) −30.1803 −1.40564 −0.702819 0.711368i \(-0.748076\pi\)
−0.702819 + 0.711368i \(0.748076\pi\)
\(462\) 0 0
\(463\) 6.00000 0.278844 0.139422 0.990233i \(-0.455476\pi\)
0.139422 + 0.990233i \(0.455476\pi\)
\(464\) 2.29180 0.106394
\(465\) −6.70820 −0.311086
\(466\) −6.72949 −0.311738
\(467\) −3.76393 −0.174174 −0.0870870 0.996201i \(-0.527756\pi\)
−0.0870870 + 0.996201i \(0.527756\pi\)
\(468\) 8.47214 0.391625
\(469\) −36.3607 −1.67898
\(470\) −5.38197 −0.248252
\(471\) −2.29180 −0.105600
\(472\) −26.1803 −1.20505
\(473\) 0 0
\(474\) 7.85410 0.360751
\(475\) −6.47214 −0.296962
\(476\) 28.6525 1.31328
\(477\) 9.94427 0.455317
\(478\) 15.2361 0.696882
\(479\) 0.291796 0.0133325 0.00666625 0.999978i \(-0.497878\pi\)
0.00666625 + 0.999978i \(0.497878\pi\)
\(480\) 5.61803 0.256427
\(481\) −4.00000 −0.182384
\(482\) 11.0902 0.505143
\(483\) −15.2361 −0.693265
\(484\) 0 0
\(485\) −12.7639 −0.579580
\(486\) −0.618034 −0.0280346
\(487\) −11.7082 −0.530549 −0.265275 0.964173i \(-0.585463\pi\)
−0.265275 + 0.964173i \(0.585463\pi\)
\(488\) −3.29180 −0.149013
\(489\) −6.18034 −0.279485
\(490\) −2.14590 −0.0969418
\(491\) 38.4721 1.73622 0.868112 0.496369i \(-0.165334\pi\)
0.868112 + 0.496369i \(0.165334\pi\)
\(492\) −5.70820 −0.257346
\(493\) 6.76393 0.304632
\(494\) 20.9443 0.942327
\(495\) 0 0
\(496\) −12.4377 −0.558469
\(497\) 46.8328 2.10074
\(498\) −2.47214 −0.110779
\(499\) −0.944272 −0.0422714 −0.0211357 0.999777i \(-0.506728\pi\)
−0.0211357 + 0.999777i \(0.506728\pi\)
\(500\) 1.61803 0.0723607
\(501\) 3.18034 0.142087
\(502\) −14.6525 −0.653972
\(503\) 23.1803 1.03356 0.516780 0.856118i \(-0.327130\pi\)
0.516780 + 0.856118i \(0.327130\pi\)
\(504\) 7.23607 0.322320
\(505\) −5.52786 −0.245987
\(506\) 0 0
\(507\) −14.4164 −0.640255
\(508\) 6.00000 0.266207
\(509\) 15.5967 0.691314 0.345657 0.938361i \(-0.387656\pi\)
0.345657 + 0.938361i \(0.387656\pi\)
\(510\) 3.38197 0.149756
\(511\) −33.8885 −1.49914
\(512\) 18.7082 0.826794
\(513\) 6.47214 0.285752
\(514\) 17.8541 0.787511
\(515\) 0.944272 0.0416096
\(516\) 8.47214 0.372965
\(517\) 0 0
\(518\) −1.52786 −0.0671305
\(519\) 2.94427 0.129239
\(520\) −11.7082 −0.513439
\(521\) 17.8197 0.780693 0.390347 0.920668i \(-0.372355\pi\)
0.390347 + 0.920668i \(0.372355\pi\)
\(522\) 0.763932 0.0334364
\(523\) 24.5410 1.07310 0.536552 0.843867i \(-0.319727\pi\)
0.536552 + 0.843867i \(0.319727\pi\)
\(524\) −0.763932 −0.0333725
\(525\) 3.23607 0.141234
\(526\) −12.4377 −0.542309
\(527\) −36.7082 −1.59903
\(528\) 0 0
\(529\) −0.832816 −0.0362094
\(530\) −6.14590 −0.266961
\(531\) 11.7082 0.508093
\(532\) −33.8885 −1.46925
\(533\) 18.4721 0.800117
\(534\) −2.94427 −0.127411
\(535\) 14.2361 0.615479
\(536\) −25.1246 −1.08522
\(537\) 20.1803 0.870846
\(538\) 4.36068 0.188002
\(539\) 0 0
\(540\) −1.61803 −0.0696291
\(541\) 43.3050 1.86183 0.930913 0.365242i \(-0.119014\pi\)
0.930913 + 0.365242i \(0.119014\pi\)
\(542\) 14.9787 0.643391
\(543\) 21.4164 0.919066
\(544\) 30.7426 1.31808
\(545\) 14.9443 0.640142
\(546\) −10.4721 −0.448166
\(547\) 9.59675 0.410327 0.205164 0.978728i \(-0.434227\pi\)
0.205164 + 0.978728i \(0.434227\pi\)
\(548\) −28.2705 −1.20766
\(549\) 1.47214 0.0628292
\(550\) 0 0
\(551\) −8.00000 −0.340811
\(552\) −10.5279 −0.448096
\(553\) 41.1246 1.74880
\(554\) 3.05573 0.129825
\(555\) 0.763932 0.0324271
\(556\) −11.7984 −0.500363
\(557\) 8.52786 0.361337 0.180669 0.983544i \(-0.442174\pi\)
0.180669 + 0.983544i \(0.442174\pi\)
\(558\) −4.14590 −0.175510
\(559\) −27.4164 −1.15959
\(560\) 6.00000 0.253546
\(561\) 0 0
\(562\) −9.41641 −0.397207
\(563\) 0.944272 0.0397963 0.0198982 0.999802i \(-0.493666\pi\)
0.0198982 + 0.999802i \(0.493666\pi\)
\(564\) 14.0902 0.593303
\(565\) 12.4164 0.522362
\(566\) −7.34752 −0.308839
\(567\) −3.23607 −0.135902
\(568\) 32.3607 1.35782
\(569\) 19.7082 0.826211 0.413105 0.910683i \(-0.364444\pi\)
0.413105 + 0.910683i \(0.364444\pi\)
\(570\) −4.00000 −0.167542
\(571\) −0.124612 −0.00521484 −0.00260742 0.999997i \(-0.500830\pi\)
−0.00260742 + 0.999997i \(0.500830\pi\)
\(572\) 0 0
\(573\) −27.5967 −1.15287
\(574\) 7.05573 0.294500
\(575\) −4.70820 −0.196346
\(576\) −0.236068 −0.00983617
\(577\) 20.0000 0.832611 0.416305 0.909225i \(-0.363325\pi\)
0.416305 + 0.909225i \(0.363325\pi\)
\(578\) 8.00000 0.332756
\(579\) 16.1803 0.672432
\(580\) 2.00000 0.0830455
\(581\) −12.9443 −0.537019
\(582\) −7.88854 −0.326991
\(583\) 0 0
\(584\) −23.4164 −0.968978
\(585\) 5.23607 0.216485
\(586\) −13.2705 −0.548200
\(587\) −11.6525 −0.480949 −0.240475 0.970655i \(-0.577303\pi\)
−0.240475 + 0.970655i \(0.577303\pi\)
\(588\) 5.61803 0.231684
\(589\) 43.4164 1.78894
\(590\) −7.23607 −0.297904
\(591\) −1.41641 −0.0582632
\(592\) 1.41641 0.0582140
\(593\) 34.9443 1.43499 0.717495 0.696564i \(-0.245289\pi\)
0.717495 + 0.696564i \(0.245289\pi\)
\(594\) 0 0
\(595\) 17.7082 0.725966
\(596\) −6.94427 −0.284448
\(597\) −16.2361 −0.664498
\(598\) 15.2361 0.623049
\(599\) −31.0132 −1.26716 −0.633582 0.773676i \(-0.718416\pi\)
−0.633582 + 0.773676i \(0.718416\pi\)
\(600\) 2.23607 0.0912871
\(601\) −28.4721 −1.16140 −0.580701 0.814117i \(-0.697222\pi\)
−0.580701 + 0.814117i \(0.697222\pi\)
\(602\) −10.4721 −0.426812
\(603\) 11.2361 0.457568
\(604\) 11.7984 0.480069
\(605\) 0 0
\(606\) −3.41641 −0.138782
\(607\) −6.29180 −0.255376 −0.127688 0.991814i \(-0.540756\pi\)
−0.127688 + 0.991814i \(0.540756\pi\)
\(608\) −36.3607 −1.47462
\(609\) 4.00000 0.162088
\(610\) −0.909830 −0.0368379
\(611\) −45.5967 −1.84465
\(612\) −8.85410 −0.357906
\(613\) 14.8328 0.599092 0.299546 0.954082i \(-0.403165\pi\)
0.299546 + 0.954082i \(0.403165\pi\)
\(614\) 7.34752 0.296522
\(615\) −3.52786 −0.142257
\(616\) 0 0
\(617\) 41.7771 1.68188 0.840941 0.541127i \(-0.182002\pi\)
0.840941 + 0.541127i \(0.182002\pi\)
\(618\) 0.583592 0.0234755
\(619\) −9.52786 −0.382957 −0.191479 0.981497i \(-0.561328\pi\)
−0.191479 + 0.981497i \(0.561328\pi\)
\(620\) −10.8541 −0.435911
\(621\) 4.70820 0.188934
\(622\) −2.00000 −0.0801927
\(623\) −15.4164 −0.617645
\(624\) 9.70820 0.388639
\(625\) 1.00000 0.0400000
\(626\) 14.0689 0.562306
\(627\) 0 0
\(628\) −3.70820 −0.147973
\(629\) 4.18034 0.166681
\(630\) 2.00000 0.0796819
\(631\) 9.18034 0.365464 0.182732 0.983163i \(-0.441506\pi\)
0.182732 + 0.983163i \(0.441506\pi\)
\(632\) 28.4164 1.13034
\(633\) 1.76393 0.0701100
\(634\) −2.43769 −0.0968132
\(635\) 3.70820 0.147156
\(636\) 16.0902 0.638017
\(637\) −18.1803 −0.720331
\(638\) 0 0
\(639\) −14.4721 −0.572509
\(640\) 11.3820 0.449912
\(641\) −9.12461 −0.360400 −0.180200 0.983630i \(-0.557675\pi\)
−0.180200 + 0.983630i \(0.557675\pi\)
\(642\) 8.79837 0.347244
\(643\) −28.8328 −1.13706 −0.568528 0.822664i \(-0.692487\pi\)
−0.568528 + 0.822664i \(0.692487\pi\)
\(644\) −24.6525 −0.971444
\(645\) 5.23607 0.206170
\(646\) −21.8885 −0.861193
\(647\) −16.2361 −0.638306 −0.319153 0.947703i \(-0.603398\pi\)
−0.319153 + 0.947703i \(0.603398\pi\)
\(648\) −2.23607 −0.0878410
\(649\) 0 0
\(650\) −3.23607 −0.126929
\(651\) −21.7082 −0.850812
\(652\) −10.0000 −0.391630
\(653\) 16.8328 0.658719 0.329359 0.944205i \(-0.393167\pi\)
0.329359 + 0.944205i \(0.393167\pi\)
\(654\) 9.23607 0.361159
\(655\) −0.472136 −0.0184479
\(656\) −6.54102 −0.255384
\(657\) 10.4721 0.408557
\(658\) −17.4164 −0.678962
\(659\) −35.5967 −1.38665 −0.693326 0.720624i \(-0.743855\pi\)
−0.693326 + 0.720624i \(0.743855\pi\)
\(660\) 0 0
\(661\) 45.7771 1.78052 0.890261 0.455450i \(-0.150522\pi\)
0.890261 + 0.455450i \(0.150522\pi\)
\(662\) −10.6180 −0.412682
\(663\) 28.6525 1.11277
\(664\) −8.94427 −0.347105
\(665\) −20.9443 −0.812184
\(666\) 0.472136 0.0182949
\(667\) −5.81966 −0.225338
\(668\) 5.14590 0.199101
\(669\) −9.05573 −0.350115
\(670\) −6.94427 −0.268281
\(671\) 0 0
\(672\) 18.1803 0.701322
\(673\) 27.5967 1.06378 0.531888 0.846815i \(-0.321483\pi\)
0.531888 + 0.846815i \(0.321483\pi\)
\(674\) 4.87539 0.187793
\(675\) −1.00000 −0.0384900
\(676\) −23.3262 −0.897163
\(677\) −5.41641 −0.208169 −0.104085 0.994568i \(-0.533191\pi\)
−0.104085 + 0.994568i \(0.533191\pi\)
\(678\) 7.67376 0.294709
\(679\) −41.3050 −1.58514
\(680\) 12.2361 0.469232
\(681\) 16.7082 0.640260
\(682\) 0 0
\(683\) −24.0000 −0.918334 −0.459167 0.888350i \(-0.651852\pi\)
−0.459167 + 0.888350i \(0.651852\pi\)
\(684\) 10.4721 0.400412
\(685\) −17.4721 −0.667576
\(686\) 7.05573 0.269389
\(687\) −7.00000 −0.267067
\(688\) 9.70820 0.370122
\(689\) −52.0689 −1.98367
\(690\) −2.90983 −0.110775
\(691\) −37.5410 −1.42813 −0.714064 0.700081i \(-0.753147\pi\)
−0.714064 + 0.700081i \(0.753147\pi\)
\(692\) 4.76393 0.181098
\(693\) 0 0
\(694\) 8.79837 0.333982
\(695\) −7.29180 −0.276594
\(696\) 2.76393 0.104767
\(697\) −19.3050 −0.731227
\(698\) 9.27051 0.350894
\(699\) 10.8885 0.411843
\(700\) 5.23607 0.197905
\(701\) −36.1803 −1.36651 −0.683256 0.730179i \(-0.739437\pi\)
−0.683256 + 0.730179i \(0.739437\pi\)
\(702\) 3.23607 0.122138
\(703\) −4.94427 −0.186477
\(704\) 0 0
\(705\) 8.70820 0.327970
\(706\) −4.32624 −0.162820
\(707\) −17.8885 −0.672768
\(708\) 18.9443 0.711969
\(709\) −17.9443 −0.673911 −0.336956 0.941521i \(-0.609397\pi\)
−0.336956 + 0.941521i \(0.609397\pi\)
\(710\) 8.94427 0.335673
\(711\) −12.7082 −0.476595
\(712\) −10.6525 −0.399218
\(713\) 31.5836 1.18281
\(714\) 10.9443 0.409579
\(715\) 0 0
\(716\) 32.6525 1.22028
\(717\) −24.6525 −0.920664
\(718\) −4.58359 −0.171058
\(719\) 26.3607 0.983087 0.491544 0.870853i \(-0.336433\pi\)
0.491544 + 0.870853i \(0.336433\pi\)
\(720\) −1.85410 −0.0690983
\(721\) 3.05573 0.113801
\(722\) 14.1459 0.526456
\(723\) −17.9443 −0.667355
\(724\) 34.6525 1.28785
\(725\) 1.23607 0.0459064
\(726\) 0 0
\(727\) 44.8328 1.66276 0.831379 0.555706i \(-0.187552\pi\)
0.831379 + 0.555706i \(0.187552\pi\)
\(728\) −37.8885 −1.40424
\(729\) 1.00000 0.0370370
\(730\) −6.47214 −0.239544
\(731\) 28.6525 1.05975
\(732\) 2.38197 0.0880400
\(733\) −47.8885 −1.76880 −0.884402 0.466726i \(-0.845433\pi\)
−0.884402 + 0.466726i \(0.845433\pi\)
\(734\) 2.94427 0.108675
\(735\) 3.47214 0.128072
\(736\) −26.4508 −0.974991
\(737\) 0 0
\(738\) −2.18034 −0.0802594
\(739\) −1.65248 −0.0607873 −0.0303937 0.999538i \(-0.509676\pi\)
−0.0303937 + 0.999538i \(0.509676\pi\)
\(740\) 1.23607 0.0454388
\(741\) −33.8885 −1.24493
\(742\) −19.8885 −0.730131
\(743\) −2.23607 −0.0820334 −0.0410167 0.999158i \(-0.513060\pi\)
−0.0410167 + 0.999158i \(0.513060\pi\)
\(744\) −15.0000 −0.549927
\(745\) −4.29180 −0.157239
\(746\) −18.4721 −0.676313
\(747\) 4.00000 0.146352
\(748\) 0 0
\(749\) 46.0689 1.68332
\(750\) 0.618034 0.0225674
\(751\) −34.0132 −1.24116 −0.620579 0.784144i \(-0.713102\pi\)
−0.620579 + 0.784144i \(0.713102\pi\)
\(752\) 16.1459 0.588780
\(753\) 23.7082 0.863975
\(754\) −4.00000 −0.145671
\(755\) 7.29180 0.265376
\(756\) −5.23607 −0.190434
\(757\) 18.9443 0.688541 0.344271 0.938870i \(-0.388126\pi\)
0.344271 + 0.938870i \(0.388126\pi\)
\(758\) 18.9098 0.686836
\(759\) 0 0
\(760\) −14.4721 −0.524960
\(761\) 44.6525 1.61865 0.809325 0.587360i \(-0.199833\pi\)
0.809325 + 0.587360i \(0.199833\pi\)
\(762\) 2.29180 0.0830230
\(763\) 48.3607 1.75077
\(764\) −44.6525 −1.61547
\(765\) −5.47214 −0.197845
\(766\) 9.88854 0.357288
\(767\) −61.3050 −2.21359
\(768\) 6.56231 0.236797
\(769\) 24.5279 0.884497 0.442249 0.896892i \(-0.354181\pi\)
0.442249 + 0.896892i \(0.354181\pi\)
\(770\) 0 0
\(771\) −28.8885 −1.04040
\(772\) 26.1803 0.942251
\(773\) −7.94427 −0.285736 −0.142868 0.989742i \(-0.545632\pi\)
−0.142868 + 0.989742i \(0.545632\pi\)
\(774\) 3.23607 0.116318
\(775\) −6.70820 −0.240966
\(776\) −28.5410 −1.02456
\(777\) 2.47214 0.0886874
\(778\) −13.3475 −0.478532
\(779\) 22.8328 0.818071
\(780\) 8.47214 0.303351
\(781\) 0 0
\(782\) −15.9230 −0.569405
\(783\) −1.23607 −0.0441735
\(784\) 6.43769 0.229918
\(785\) −2.29180 −0.0817977
\(786\) −0.291796 −0.0104080
\(787\) 27.0557 0.964433 0.482216 0.876052i \(-0.339832\pi\)
0.482216 + 0.876052i \(0.339832\pi\)
\(788\) −2.29180 −0.0816419
\(789\) 20.1246 0.716455
\(790\) 7.85410 0.279436
\(791\) 40.1803 1.42865
\(792\) 0 0
\(793\) −7.70820 −0.273726
\(794\) −10.7639 −0.381998
\(795\) 9.94427 0.352687
\(796\) −26.2705 −0.931134
\(797\) 30.0000 1.06265 0.531327 0.847167i \(-0.321693\pi\)
0.531327 + 0.847167i \(0.321693\pi\)
\(798\) −12.9443 −0.458222
\(799\) 47.6525 1.68582
\(800\) 5.61803 0.198627
\(801\) 4.76393 0.168325
\(802\) 12.9443 0.457078
\(803\) 0 0
\(804\) 18.1803 0.641171
\(805\) −15.2361 −0.537001
\(806\) 21.7082 0.764639
\(807\) −7.05573 −0.248373
\(808\) −12.3607 −0.434847
\(809\) 8.00000 0.281265 0.140633 0.990062i \(-0.455086\pi\)
0.140633 + 0.990062i \(0.455086\pi\)
\(810\) −0.618034 −0.0217155
\(811\) −7.87539 −0.276542 −0.138271 0.990394i \(-0.544155\pi\)
−0.138271 + 0.990394i \(0.544155\pi\)
\(812\) 6.47214 0.227127
\(813\) −24.2361 −0.849996
\(814\) 0 0
\(815\) −6.18034 −0.216488
\(816\) −10.1459 −0.355177
\(817\) −33.8885 −1.18561
\(818\) −22.7295 −0.794718
\(819\) 16.9443 0.592081
\(820\) −5.70820 −0.199339
\(821\) 33.7771 1.17883 0.589414 0.807831i \(-0.299359\pi\)
0.589414 + 0.807831i \(0.299359\pi\)
\(822\) −10.7984 −0.376637
\(823\) 38.2492 1.33328 0.666642 0.745378i \(-0.267731\pi\)
0.666642 + 0.745378i \(0.267731\pi\)
\(824\) 2.11146 0.0735561
\(825\) 0 0
\(826\) −23.4164 −0.814761
\(827\) −8.94427 −0.311023 −0.155511 0.987834i \(-0.549703\pi\)
−0.155511 + 0.987834i \(0.549703\pi\)
\(828\) 7.61803 0.264745
\(829\) −11.0000 −0.382046 −0.191023 0.981586i \(-0.561180\pi\)
−0.191023 + 0.981586i \(0.561180\pi\)
\(830\) −2.47214 −0.0858091
\(831\) −4.94427 −0.171515
\(832\) 1.23607 0.0428529
\(833\) 19.0000 0.658311
\(834\) −4.50658 −0.156050
\(835\) 3.18034 0.110060
\(836\) 0 0
\(837\) 6.70820 0.231869
\(838\) −8.65248 −0.298895
\(839\) 28.3607 0.979119 0.489560 0.871970i \(-0.337158\pi\)
0.489560 + 0.871970i \(0.337158\pi\)
\(840\) 7.23607 0.249668
\(841\) −27.4721 −0.947315
\(842\) 18.4377 0.635405
\(843\) 15.2361 0.524758
\(844\) 2.85410 0.0982422
\(845\) −14.4164 −0.495940
\(846\) 5.38197 0.185036
\(847\) 0 0
\(848\) 18.4377 0.633153
\(849\) 11.8885 0.408014
\(850\) 3.38197 0.116000
\(851\) −3.59675 −0.123295
\(852\) −23.4164 −0.802233
\(853\) 31.8885 1.09184 0.545921 0.837836i \(-0.316180\pi\)
0.545921 + 0.837836i \(0.316180\pi\)
\(854\) −2.94427 −0.100751
\(855\) 6.47214 0.221342
\(856\) 31.8328 1.08802
\(857\) −1.00000 −0.0341593 −0.0170797 0.999854i \(-0.505437\pi\)
−0.0170797 + 0.999854i \(0.505437\pi\)
\(858\) 0 0
\(859\) −26.8328 −0.915524 −0.457762 0.889075i \(-0.651349\pi\)
−0.457762 + 0.889075i \(0.651349\pi\)
\(860\) 8.47214 0.288897
\(861\) −11.4164 −0.389070
\(862\) 3.59675 0.122506
\(863\) −37.8885 −1.28974 −0.644871 0.764292i \(-0.723089\pi\)
−0.644871 + 0.764292i \(0.723089\pi\)
\(864\) −5.61803 −0.191129
\(865\) 2.94427 0.100108
\(866\) 15.5967 0.529999
\(867\) −12.9443 −0.439611
\(868\) −35.1246 −1.19221
\(869\) 0 0
\(870\) 0.763932 0.0258997
\(871\) −58.8328 −1.99347
\(872\) 33.4164 1.13162
\(873\) 12.7639 0.431994
\(874\) 18.8328 0.637029
\(875\) 3.23607 0.109399
\(876\) 16.9443 0.572494
\(877\) −11.3050 −0.381741 −0.190871 0.981615i \(-0.561131\pi\)
−0.190871 + 0.981615i \(0.561131\pi\)
\(878\) 5.02129 0.169460
\(879\) 21.4721 0.724237
\(880\) 0 0
\(881\) 38.8328 1.30831 0.654155 0.756360i \(-0.273024\pi\)
0.654155 + 0.756360i \(0.273024\pi\)
\(882\) 2.14590 0.0722561
\(883\) 10.1803 0.342596 0.171298 0.985219i \(-0.445204\pi\)
0.171298 + 0.985219i \(0.445204\pi\)
\(884\) 46.3607 1.55928
\(885\) 11.7082 0.393567
\(886\) 4.58359 0.153989
\(887\) −16.9443 −0.568933 −0.284466 0.958686i \(-0.591816\pi\)
−0.284466 + 0.958686i \(0.591816\pi\)
\(888\) 1.70820 0.0573236
\(889\) 12.0000 0.402467
\(890\) −2.94427 −0.0986922
\(891\) 0 0
\(892\) −14.6525 −0.490601
\(893\) −56.3607 −1.88604
\(894\) −2.65248 −0.0887121
\(895\) 20.1803 0.674554
\(896\) 36.8328 1.23050
\(897\) −24.6525 −0.823122
\(898\) −6.18034 −0.206241
\(899\) −8.29180 −0.276547
\(900\) −1.61803 −0.0539345
\(901\) 54.4164 1.81287
\(902\) 0 0
\(903\) 16.9443 0.563870
\(904\) 27.7639 0.923415
\(905\) 21.4164 0.711905
\(906\) 4.50658 0.149721
\(907\) −42.0000 −1.39459 −0.697294 0.716786i \(-0.745613\pi\)
−0.697294 + 0.716786i \(0.745613\pi\)
\(908\) 27.0344 0.897169
\(909\) 5.52786 0.183348
\(910\) −10.4721 −0.347148
\(911\) 47.9574 1.58890 0.794450 0.607329i \(-0.207759\pi\)
0.794450 + 0.607329i \(0.207759\pi\)
\(912\) 12.0000 0.397360
\(913\) 0 0
\(914\) 21.2361 0.702427
\(915\) 1.47214 0.0486673
\(916\) −11.3262 −0.374229
\(917\) −1.52786 −0.0504545
\(918\) −3.38197 −0.111622
\(919\) −3.41641 −0.112697 −0.0563484 0.998411i \(-0.517946\pi\)
−0.0563484 + 0.998411i \(0.517946\pi\)
\(920\) −10.5279 −0.347093
\(921\) −11.8885 −0.391741
\(922\) −18.6525 −0.614287
\(923\) 75.7771 2.49423
\(924\) 0 0
\(925\) 0.763932 0.0251179
\(926\) 3.70820 0.121859
\(927\) −0.944272 −0.0310140
\(928\) 6.94427 0.227957
\(929\) 50.9443 1.67143 0.835714 0.549165i \(-0.185054\pi\)
0.835714 + 0.549165i \(0.185054\pi\)
\(930\) −4.14590 −0.135949
\(931\) −22.4721 −0.736495
\(932\) 17.6180 0.577098
\(933\) 3.23607 0.105944
\(934\) −2.32624 −0.0761168
\(935\) 0 0
\(936\) 11.7082 0.382695
\(937\) −33.1246 −1.08213 −0.541067 0.840980i \(-0.681979\pi\)
−0.541067 + 0.840980i \(0.681979\pi\)
\(938\) −22.4721 −0.733741
\(939\) −22.7639 −0.742873
\(940\) 14.0902 0.459571
\(941\) 45.2361 1.47465 0.737327 0.675536i \(-0.236088\pi\)
0.737327 + 0.675536i \(0.236088\pi\)
\(942\) −1.41641 −0.0461491
\(943\) 16.6099 0.540893
\(944\) 21.7082 0.706542
\(945\) −3.23607 −0.105269
\(946\) 0 0
\(947\) −18.2361 −0.592593 −0.296296 0.955096i \(-0.595752\pi\)
−0.296296 + 0.955096i \(0.595752\pi\)
\(948\) −20.5623 −0.667833
\(949\) −54.8328 −1.77995
\(950\) −4.00000 −0.129777
\(951\) 3.94427 0.127902
\(952\) 39.5967 1.28334
\(953\) −31.8885 −1.03297 −0.516486 0.856296i \(-0.672760\pi\)
−0.516486 + 0.856296i \(0.672760\pi\)
\(954\) 6.14590 0.198981
\(955\) −27.5967 −0.893010
\(956\) −39.8885 −1.29009
\(957\) 0 0
\(958\) 0.180340 0.00582652
\(959\) −56.5410 −1.82580
\(960\) −0.236068 −0.00761906
\(961\) 14.0000 0.451613
\(962\) −2.47214 −0.0797049
\(963\) −14.2361 −0.458751
\(964\) −29.0344 −0.935136
\(965\) 16.1803 0.520864
\(966\) −9.41641 −0.302968
\(967\) 32.0000 1.02905 0.514525 0.857475i \(-0.327968\pi\)
0.514525 + 0.857475i \(0.327968\pi\)
\(968\) 0 0
\(969\) 35.4164 1.13774
\(970\) −7.88854 −0.253286
\(971\) 29.2361 0.938230 0.469115 0.883137i \(-0.344573\pi\)
0.469115 + 0.883137i \(0.344573\pi\)
\(972\) 1.61803 0.0518985
\(973\) −23.5967 −0.756477
\(974\) −7.23607 −0.231859
\(975\) 5.23607 0.167688
\(976\) 2.72949 0.0873689
\(977\) −15.9443 −0.510102 −0.255051 0.966928i \(-0.582092\pi\)
−0.255051 + 0.966928i \(0.582092\pi\)
\(978\) −3.81966 −0.122139
\(979\) 0 0
\(980\) 5.61803 0.179462
\(981\) −14.9443 −0.477134
\(982\) 23.7771 0.758757
\(983\) −55.1803 −1.75998 −0.879990 0.474993i \(-0.842451\pi\)
−0.879990 + 0.474993i \(0.842451\pi\)
\(984\) −7.88854 −0.251478
\(985\) −1.41641 −0.0451305
\(986\) 4.18034 0.133129
\(987\) 28.1803 0.896990
\(988\) −54.8328 −1.74446
\(989\) −24.6525 −0.783903
\(990\) 0 0
\(991\) 6.23607 0.198095 0.0990476 0.995083i \(-0.468420\pi\)
0.0990476 + 0.995083i \(0.468420\pi\)
\(992\) −37.6869 −1.19656
\(993\) 17.1803 0.545202
\(994\) 28.9443 0.918057
\(995\) −16.2361 −0.514718
\(996\) 6.47214 0.205077
\(997\) −38.7639 −1.22767 −0.613833 0.789436i \(-0.710373\pi\)
−0.613833 + 0.789436i \(0.710373\pi\)
\(998\) −0.583592 −0.0184733
\(999\) −0.763932 −0.0241697
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 1815.2.a.f.1.2 2
3.2 odd 2 5445.2.a.x.1.1 2
5.4 even 2 9075.2.a.bx.1.1 2
11.10 odd 2 1815.2.a.j.1.1 yes 2
33.32 even 2 5445.2.a.o.1.2 2
55.54 odd 2 9075.2.a.bd.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1815.2.a.f.1.2 2 1.1 even 1 trivial
1815.2.a.j.1.1 yes 2 11.10 odd 2
5445.2.a.o.1.2 2 33.32 even 2
5445.2.a.x.1.1 2 3.2 odd 2
9075.2.a.bd.1.2 2 55.54 odd 2
9075.2.a.bx.1.1 2 5.4 even 2