Properties

Label 1815.2.a.j.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 \(1.61803\) of defining polynomial
Character \(\chi\) \(=\) 1815.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.61803 q^{2} -1.00000 q^{3} +0.618034 q^{4} -1.00000 q^{5} -1.61803 q^{6} -1.23607 q^{7} -2.23607 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.61803 q^{2} -1.00000 q^{3} +0.618034 q^{4} -1.00000 q^{5} -1.61803 q^{6} -1.23607 q^{7} -2.23607 q^{8} +1.00000 q^{9} -1.61803 q^{10} -0.618034 q^{12} +0.763932 q^{13} -2.00000 q^{14} +1.00000 q^{15} -4.85410 q^{16} +3.47214 q^{17} +1.61803 q^{18} -2.47214 q^{19} -0.618034 q^{20} +1.23607 q^{21} +8.70820 q^{23} +2.23607 q^{24} +1.00000 q^{25} +1.23607 q^{26} -1.00000 q^{27} -0.763932 q^{28} +3.23607 q^{29} +1.61803 q^{30} +6.70820 q^{31} -3.38197 q^{32} +5.61803 q^{34} +1.23607 q^{35} +0.618034 q^{36} +5.23607 q^{37} -4.00000 q^{38} -0.763932 q^{39} +2.23607 q^{40} +12.4721 q^{41} +2.00000 q^{42} -0.763932 q^{43} -1.00000 q^{45} +14.0902 q^{46} -4.70820 q^{47} +4.85410 q^{48} -5.47214 q^{49} +1.61803 q^{50} -3.47214 q^{51} +0.472136 q^{52} -7.94427 q^{53} -1.61803 q^{54} +2.76393 q^{56} +2.47214 q^{57} +5.23607 q^{58} -1.70820 q^{59} +0.618034 q^{60} +7.47214 q^{61} +10.8541 q^{62} -1.23607 q^{63} +4.23607 q^{64} -0.763932 q^{65} +6.76393 q^{67} +2.14590 q^{68} -8.70820 q^{69} +2.00000 q^{70} -5.52786 q^{71} -2.23607 q^{72} -1.52786 q^{73} +8.47214 q^{74} -1.00000 q^{75} -1.52786 q^{76} -1.23607 q^{78} -0.708204 q^{79} +4.85410 q^{80} +1.00000 q^{81} +20.1803 q^{82} -4.00000 q^{83} +0.763932 q^{84} -3.47214 q^{85} -1.23607 q^{86} -3.23607 q^{87} +9.23607 q^{89} -1.61803 q^{90} -0.944272 q^{91} +5.38197 q^{92} -6.70820 q^{93} -7.61803 q^{94} +2.47214 q^{95} +3.38197 q^{96} +17.2361 q^{97} -8.85410 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + q^{2} - 2q^{3} - q^{4} - 2q^{5} - q^{6} + 2q^{7} + 2q^{9} + O(q^{10}) \) \( 2q + q^{2} - 2q^{3} - q^{4} - 2q^{5} - q^{6} + 2q^{7} + 2q^{9} - q^{10} + q^{12} + 6q^{13} - 4q^{14} + 2q^{15} - 3q^{16} - 2q^{17} + q^{18} + 4q^{19} + q^{20} - 2q^{21} + 4q^{23} + 2q^{25} - 2q^{26} - 2q^{27} - 6q^{28} + 2q^{29} + q^{30} - 9q^{32} + 9q^{34} - 2q^{35} - q^{36} + 6q^{37} - 8q^{38} - 6q^{39} + 16q^{41} + 4q^{42} - 6q^{43} - 2q^{45} + 17q^{46} + 4q^{47} + 3q^{48} - 2q^{49} + q^{50} + 2q^{51} - 8q^{52} + 2q^{53} - q^{54} + 10q^{56} - 4q^{57} + 6q^{58} + 10q^{59} - q^{60} + 6q^{61} + 15q^{62} + 2q^{63} + 4q^{64} - 6q^{65} + 18q^{67} + 11q^{68} - 4q^{69} + 4q^{70} - 20q^{71} - 12q^{73} + 8q^{74} - 2q^{75} - 12q^{76} + 2q^{78} + 12q^{79} + 3q^{80} + 2q^{81} + 18q^{82} - 8q^{83} + 6q^{84} + 2q^{85} + 2q^{86} - 2q^{87} + 14q^{89} - q^{90} + 16q^{91} + 13q^{92} - 13q^{94} - 4q^{95} + 9q^{96} + 30q^{97} - 11q^{98} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.61803 1.14412 0.572061 0.820211i \(-0.306144\pi\)
0.572061 + 0.820211i \(0.306144\pi\)
\(3\) −1.00000 −0.577350
\(4\) 0.618034 0.309017
\(5\) −1.00000 −0.447214
\(6\) −1.61803 −0.660560
\(7\) −1.23607 −0.467190 −0.233595 0.972334i \(-0.575049\pi\)
−0.233595 + 0.972334i \(0.575049\pi\)
\(8\) −2.23607 −0.790569
\(9\) 1.00000 0.333333
\(10\) −1.61803 −0.511667
\(11\) 0 0
\(12\) −0.618034 −0.178411
\(13\) 0.763932 0.211877 0.105938 0.994373i \(-0.466215\pi\)
0.105938 + 0.994373i \(0.466215\pi\)
\(14\) −2.00000 −0.534522
\(15\) 1.00000 0.258199
\(16\) −4.85410 −1.21353
\(17\) 3.47214 0.842117 0.421058 0.907034i \(-0.361659\pi\)
0.421058 + 0.907034i \(0.361659\pi\)
\(18\) 1.61803 0.381374
\(19\) −2.47214 −0.567147 −0.283573 0.958951i \(-0.591520\pi\)
−0.283573 + 0.958951i \(0.591520\pi\)
\(20\) −0.618034 −0.138197
\(21\) 1.23607 0.269732
\(22\) 0 0
\(23\) 8.70820 1.81579 0.907893 0.419202i \(-0.137690\pi\)
0.907893 + 0.419202i \(0.137690\pi\)
\(24\) 2.23607 0.456435
\(25\) 1.00000 0.200000
\(26\) 1.23607 0.242413
\(27\) −1.00000 −0.192450
\(28\) −0.763932 −0.144370
\(29\) 3.23607 0.600923 0.300461 0.953794i \(-0.402859\pi\)
0.300461 + 0.953794i \(0.402859\pi\)
\(30\) 1.61803 0.295411
\(31\) 6.70820 1.20483 0.602414 0.798183i \(-0.294205\pi\)
0.602414 + 0.798183i \(0.294205\pi\)
\(32\) −3.38197 −0.597853
\(33\) 0 0
\(34\) 5.61803 0.963485
\(35\) 1.23607 0.208934
\(36\) 0.618034 0.103006
\(37\) 5.23607 0.860804 0.430402 0.902637i \(-0.358372\pi\)
0.430402 + 0.902637i \(0.358372\pi\)
\(38\) −4.00000 −0.648886
\(39\) −0.763932 −0.122327
\(40\) 2.23607 0.353553
\(41\) 12.4721 1.94782 0.973910 0.226934i \(-0.0728701\pi\)
0.973910 + 0.226934i \(0.0728701\pi\)
\(42\) 2.00000 0.308607
\(43\) −0.763932 −0.116499 −0.0582493 0.998302i \(-0.518552\pi\)
−0.0582493 + 0.998302i \(0.518552\pi\)
\(44\) 0 0
\(45\) −1.00000 −0.149071
\(46\) 14.0902 2.07748
\(47\) −4.70820 −0.686762 −0.343381 0.939196i \(-0.611572\pi\)
−0.343381 + 0.939196i \(0.611572\pi\)
\(48\) 4.85410 0.700629
\(49\) −5.47214 −0.781734
\(50\) 1.61803 0.228825
\(51\) −3.47214 −0.486196
\(52\) 0.472136 0.0654735
\(53\) −7.94427 −1.09123 −0.545615 0.838036i \(-0.683704\pi\)
−0.545615 + 0.838036i \(0.683704\pi\)
\(54\) −1.61803 −0.220187
\(55\) 0 0
\(56\) 2.76393 0.369346
\(57\) 2.47214 0.327442
\(58\) 5.23607 0.687529
\(59\) −1.70820 −0.222389 −0.111195 0.993799i \(-0.535468\pi\)
−0.111195 + 0.993799i \(0.535468\pi\)
\(60\) 0.618034 0.0797878
\(61\) 7.47214 0.956709 0.478354 0.878167i \(-0.341233\pi\)
0.478354 + 0.878167i \(0.341233\pi\)
\(62\) 10.8541 1.37847
\(63\) −1.23607 −0.155730
\(64\) 4.23607 0.529508
\(65\) −0.763932 −0.0947541
\(66\) 0 0
\(67\) 6.76393 0.826346 0.413173 0.910653i \(-0.364421\pi\)
0.413173 + 0.910653i \(0.364421\pi\)
\(68\) 2.14590 0.260228
\(69\) −8.70820 −1.04834
\(70\) 2.00000 0.239046
\(71\) −5.52786 −0.656037 −0.328018 0.944671i \(-0.606381\pi\)
−0.328018 + 0.944671i \(0.606381\pi\)
\(72\) −2.23607 −0.263523
\(73\) −1.52786 −0.178823 −0.0894115 0.995995i \(-0.528499\pi\)
−0.0894115 + 0.995995i \(0.528499\pi\)
\(74\) 8.47214 0.984866
\(75\) −1.00000 −0.115470
\(76\) −1.52786 −0.175258
\(77\) 0 0
\(78\) −1.23607 −0.139957
\(79\) −0.708204 −0.0796792 −0.0398396 0.999206i \(-0.512685\pi\)
−0.0398396 + 0.999206i \(0.512685\pi\)
\(80\) 4.85410 0.542705
\(81\) 1.00000 0.111111
\(82\) 20.1803 2.22855
\(83\) −4.00000 −0.439057 −0.219529 0.975606i \(-0.570452\pi\)
−0.219529 + 0.975606i \(0.570452\pi\)
\(84\) 0.763932 0.0833518
\(85\) −3.47214 −0.376606
\(86\) −1.23607 −0.133289
\(87\) −3.23607 −0.346943
\(88\) 0 0
\(89\) 9.23607 0.979021 0.489511 0.871997i \(-0.337175\pi\)
0.489511 + 0.871997i \(0.337175\pi\)
\(90\) −1.61803 −0.170556
\(91\) −0.944272 −0.0989866
\(92\) 5.38197 0.561109
\(93\) −6.70820 −0.695608
\(94\) −7.61803 −0.785740
\(95\) 2.47214 0.253636
\(96\) 3.38197 0.345170
\(97\) 17.2361 1.75006 0.875029 0.484071i \(-0.160842\pi\)
0.875029 + 0.484071i \(0.160842\pi\)
\(98\) −8.85410 −0.894399
\(99\) 0 0
\(100\) 0.618034 0.0618034
\(101\) −14.4721 −1.44003 −0.720016 0.693958i \(-0.755865\pi\)
−0.720016 + 0.693958i \(0.755865\pi\)
\(102\) −5.61803 −0.556268
\(103\) 16.9443 1.66957 0.834784 0.550577i \(-0.185592\pi\)
0.834784 + 0.550577i \(0.185592\pi\)
\(104\) −1.70820 −0.167503
\(105\) −1.23607 −0.120628
\(106\) −12.8541 −1.24850
\(107\) 9.76393 0.943915 0.471957 0.881621i \(-0.343548\pi\)
0.471957 + 0.881621i \(0.343548\pi\)
\(108\) −0.618034 −0.0594703
\(109\) −2.94427 −0.282010 −0.141005 0.990009i \(-0.545033\pi\)
−0.141005 + 0.990009i \(0.545033\pi\)
\(110\) 0 0
\(111\) −5.23607 −0.496986
\(112\) 6.00000 0.566947
\(113\) 14.4164 1.35618 0.678091 0.734978i \(-0.262808\pi\)
0.678091 + 0.734978i \(0.262808\pi\)
\(114\) 4.00000 0.374634
\(115\) −8.70820 −0.812044
\(116\) 2.00000 0.185695
\(117\) 0.763932 0.0706255
\(118\) −2.76393 −0.254441
\(119\) −4.29180 −0.393428
\(120\) −2.23607 −0.204124
\(121\) 0 0
\(122\) 12.0902 1.09459
\(123\) −12.4721 −1.12457
\(124\) 4.14590 0.372313
\(125\) −1.00000 −0.0894427
\(126\) −2.00000 −0.178174
\(127\) −9.70820 −0.861464 −0.430732 0.902480i \(-0.641745\pi\)
−0.430732 + 0.902480i \(0.641745\pi\)
\(128\) 13.6180 1.20368
\(129\) 0.763932 0.0672605
\(130\) −1.23607 −0.108410
\(131\) 8.47214 0.740214 0.370107 0.928989i \(-0.379321\pi\)
0.370107 + 0.928989i \(0.379321\pi\)
\(132\) 0 0
\(133\) 3.05573 0.264965
\(134\) 10.9443 0.945441
\(135\) 1.00000 0.0860663
\(136\) −7.76393 −0.665752
\(137\) 8.52786 0.728585 0.364292 0.931285i \(-0.381311\pi\)
0.364292 + 0.931285i \(0.381311\pi\)
\(138\) −14.0902 −1.19943
\(139\) −20.7082 −1.75645 −0.878223 0.478251i \(-0.841271\pi\)
−0.878223 + 0.478251i \(0.841271\pi\)
\(140\) 0.763932 0.0645640
\(141\) 4.70820 0.396502
\(142\) −8.94427 −0.750587
\(143\) 0 0
\(144\) −4.85410 −0.404508
\(145\) −3.23607 −0.268741
\(146\) −2.47214 −0.204595
\(147\) 5.47214 0.451334
\(148\) 3.23607 0.266003
\(149\) −17.7082 −1.45071 −0.725356 0.688374i \(-0.758325\pi\)
−0.725356 + 0.688374i \(0.758325\pi\)
\(150\) −1.61803 −0.132112
\(151\) 20.7082 1.68521 0.842605 0.538532i \(-0.181021\pi\)
0.842605 + 0.538532i \(0.181021\pi\)
\(152\) 5.52786 0.448369
\(153\) 3.47214 0.280706
\(154\) 0 0
\(155\) −6.70820 −0.538816
\(156\) −0.472136 −0.0378011
\(157\) 15.7082 1.25365 0.626826 0.779160i \(-0.284354\pi\)
0.626826 + 0.779160i \(0.284354\pi\)
\(158\) −1.14590 −0.0911628
\(159\) 7.94427 0.630022
\(160\) 3.38197 0.267368
\(161\) −10.7639 −0.848317
\(162\) 1.61803 0.127125
\(163\) −16.1803 −1.26734 −0.633671 0.773603i \(-0.718453\pi\)
−0.633671 + 0.773603i \(0.718453\pi\)
\(164\) 7.70820 0.601910
\(165\) 0 0
\(166\) −6.47214 −0.502335
\(167\) −19.1803 −1.48422 −0.742110 0.670279i \(-0.766175\pi\)
−0.742110 + 0.670279i \(0.766175\pi\)
\(168\) −2.76393 −0.213242
\(169\) −12.4164 −0.955108
\(170\) −5.61803 −0.430884
\(171\) −2.47214 −0.189049
\(172\) −0.472136 −0.0360000
\(173\) −14.9443 −1.13619 −0.568096 0.822962i \(-0.692320\pi\)
−0.568096 + 0.822962i \(0.692320\pi\)
\(174\) −5.23607 −0.396945
\(175\) −1.23607 −0.0934380
\(176\) 0 0
\(177\) 1.70820 0.128396
\(178\) 14.9443 1.12012
\(179\) 2.18034 0.162966 0.0814831 0.996675i \(-0.474034\pi\)
0.0814831 + 0.996675i \(0.474034\pi\)
\(180\) −0.618034 −0.0460655
\(181\) 5.41641 0.402598 0.201299 0.979530i \(-0.435484\pi\)
0.201299 + 0.979530i \(0.435484\pi\)
\(182\) −1.52786 −0.113253
\(183\) −7.47214 −0.552356
\(184\) −19.4721 −1.43550
\(185\) −5.23607 −0.384963
\(186\) −10.8541 −0.795861
\(187\) 0 0
\(188\) −2.90983 −0.212221
\(189\) 1.23607 0.0899107
\(190\) 4.00000 0.290191
\(191\) −21.5967 −1.56269 −0.781343 0.624102i \(-0.785465\pi\)
−0.781343 + 0.624102i \(0.785465\pi\)
\(192\) −4.23607 −0.305712
\(193\) −6.18034 −0.444871 −0.222435 0.974947i \(-0.571401\pi\)
−0.222435 + 0.974947i \(0.571401\pi\)
\(194\) 27.8885 2.00228
\(195\) 0.763932 0.0547063
\(196\) −3.38197 −0.241569
\(197\) 25.4164 1.81084 0.905422 0.424513i \(-0.139555\pi\)
0.905422 + 0.424513i \(0.139555\pi\)
\(198\) 0 0
\(199\) 11.7639 0.833923 0.416962 0.908924i \(-0.363095\pi\)
0.416962 + 0.908924i \(0.363095\pi\)
\(200\) −2.23607 −0.158114
\(201\) −6.76393 −0.477091
\(202\) −23.4164 −1.64757
\(203\) −4.00000 −0.280745
\(204\) −2.14590 −0.150243
\(205\) −12.4721 −0.871092
\(206\) 27.4164 1.91019
\(207\) 8.70820 0.605262
\(208\) −3.70820 −0.257118
\(209\) 0 0
\(210\) −2.00000 −0.138013
\(211\) 6.23607 0.429309 0.214654 0.976690i \(-0.431138\pi\)
0.214654 + 0.976690i \(0.431138\pi\)
\(212\) −4.90983 −0.337209
\(213\) 5.52786 0.378763
\(214\) 15.7984 1.07995
\(215\) 0.763932 0.0520997
\(216\) 2.23607 0.152145
\(217\) −8.29180 −0.562884
\(218\) −4.76393 −0.322654
\(219\) 1.52786 0.103243
\(220\) 0 0
\(221\) 2.65248 0.178425
\(222\) −8.47214 −0.568613
\(223\) 26.9443 1.80432 0.902161 0.431400i \(-0.141980\pi\)
0.902161 + 0.431400i \(0.141980\pi\)
\(224\) 4.18034 0.279311
\(225\) 1.00000 0.0666667
\(226\) 23.3262 1.55164
\(227\) 3.29180 0.218484 0.109242 0.994015i \(-0.465158\pi\)
0.109242 + 0.994015i \(0.465158\pi\)
\(228\) 1.52786 0.101185
\(229\) 7.00000 0.462573 0.231287 0.972886i \(-0.425707\pi\)
0.231287 + 0.972886i \(0.425707\pi\)
\(230\) −14.0902 −0.929078
\(231\) 0 0
\(232\) −7.23607 −0.475071
\(233\) −24.8885 −1.63050 −0.815251 0.579107i \(-0.803401\pi\)
−0.815251 + 0.579107i \(0.803401\pi\)
\(234\) 1.23607 0.0808043
\(235\) 4.70820 0.307129
\(236\) −1.05573 −0.0687220
\(237\) 0.708204 0.0460028
\(238\) −6.94427 −0.450130
\(239\) 6.65248 0.430313 0.215156 0.976580i \(-0.430974\pi\)
0.215156 + 0.976580i \(0.430974\pi\)
\(240\) −4.85410 −0.313331
\(241\) −0.0557281 −0.00358976 −0.00179488 0.999998i \(-0.500571\pi\)
−0.00179488 + 0.999998i \(0.500571\pi\)
\(242\) 0 0
\(243\) −1.00000 −0.0641500
\(244\) 4.61803 0.295639
\(245\) 5.47214 0.349602
\(246\) −20.1803 −1.28665
\(247\) −1.88854 −0.120165
\(248\) −15.0000 −0.952501
\(249\) 4.00000 0.253490
\(250\) −1.61803 −0.102333
\(251\) −10.2918 −0.649612 −0.324806 0.945781i \(-0.605299\pi\)
−0.324806 + 0.945781i \(0.605299\pi\)
\(252\) −0.763932 −0.0481232
\(253\) 0 0
\(254\) −15.7082 −0.985620
\(255\) 3.47214 0.217434
\(256\) 13.5623 0.847644
\(257\) −6.88854 −0.429696 −0.214848 0.976648i \(-0.568926\pi\)
−0.214848 + 0.976648i \(0.568926\pi\)
\(258\) 1.23607 0.0769542
\(259\) −6.47214 −0.402159
\(260\) −0.472136 −0.0292806
\(261\) 3.23607 0.200308
\(262\) 13.7082 0.846896
\(263\) −20.1246 −1.24094 −0.620468 0.784231i \(-0.713057\pi\)
−0.620468 + 0.784231i \(0.713057\pi\)
\(264\) 0 0
\(265\) 7.94427 0.488013
\(266\) 4.94427 0.303153
\(267\) −9.23607 −0.565238
\(268\) 4.18034 0.255355
\(269\) 24.9443 1.52088 0.760440 0.649409i \(-0.224984\pi\)
0.760440 + 0.649409i \(0.224984\pi\)
\(270\) 1.61803 0.0984704
\(271\) −19.7639 −1.20057 −0.600287 0.799785i \(-0.704947\pi\)
−0.600287 + 0.799785i \(0.704947\pi\)
\(272\) −16.8541 −1.02193
\(273\) 0.944272 0.0571499
\(274\) 13.7984 0.833590
\(275\) 0 0
\(276\) −5.38197 −0.323956
\(277\) 12.9443 0.777746 0.388873 0.921291i \(-0.372865\pi\)
0.388873 + 0.921291i \(0.372865\pi\)
\(278\) −33.5066 −2.00959
\(279\) 6.70820 0.401610
\(280\) −2.76393 −0.165177
\(281\) 10.7639 0.642122 0.321061 0.947058i \(-0.395960\pi\)
0.321061 + 0.947058i \(0.395960\pi\)
\(282\) 7.61803 0.453647
\(283\) −23.8885 −1.42003 −0.710013 0.704188i \(-0.751311\pi\)
−0.710013 + 0.704188i \(0.751311\pi\)
\(284\) −3.41641 −0.202727
\(285\) −2.47214 −0.146437
\(286\) 0 0
\(287\) −15.4164 −0.910002
\(288\) −3.38197 −0.199284
\(289\) −4.94427 −0.290840
\(290\) −5.23607 −0.307472
\(291\) −17.2361 −1.01040
\(292\) −0.944272 −0.0552593
\(293\) 12.5279 0.731886 0.365943 0.930637i \(-0.380747\pi\)
0.365943 + 0.930637i \(0.380747\pi\)
\(294\) 8.85410 0.516382
\(295\) 1.70820 0.0994555
\(296\) −11.7082 −0.680526
\(297\) 0 0
\(298\) −28.6525 −1.65979
\(299\) 6.65248 0.384723
\(300\) −0.618034 −0.0356822
\(301\) 0.944272 0.0544269
\(302\) 33.5066 1.92809
\(303\) 14.4721 0.831402
\(304\) 12.0000 0.688247
\(305\) −7.47214 −0.427853
\(306\) 5.61803 0.321162
\(307\) 23.8885 1.36339 0.681696 0.731636i \(-0.261243\pi\)
0.681696 + 0.731636i \(0.261243\pi\)
\(308\) 0 0
\(309\) −16.9443 −0.963926
\(310\) −10.8541 −0.616472
\(311\) 1.23607 0.0700910 0.0350455 0.999386i \(-0.488842\pi\)
0.0350455 + 0.999386i \(0.488842\pi\)
\(312\) 1.70820 0.0967080
\(313\) 27.2361 1.53947 0.769737 0.638361i \(-0.220387\pi\)
0.769737 + 0.638361i \(0.220387\pi\)
\(314\) 25.4164 1.43433
\(315\) 1.23607 0.0696445
\(316\) −0.437694 −0.0246222
\(317\) 13.9443 0.783188 0.391594 0.920138i \(-0.371924\pi\)
0.391594 + 0.920138i \(0.371924\pi\)
\(318\) 12.8541 0.720822
\(319\) 0 0
\(320\) −4.23607 −0.236803
\(321\) −9.76393 −0.544970
\(322\) −17.4164 −0.970578
\(323\) −8.58359 −0.477604
\(324\) 0.618034 0.0343352
\(325\) 0.763932 0.0423753
\(326\) −26.1803 −1.44999
\(327\) 2.94427 0.162819
\(328\) −27.8885 −1.53989
\(329\) 5.81966 0.320848
\(330\) 0 0
\(331\) 5.18034 0.284737 0.142369 0.989814i \(-0.454528\pi\)
0.142369 + 0.989814i \(0.454528\pi\)
\(332\) −2.47214 −0.135676
\(333\) 5.23607 0.286935
\(334\) −31.0344 −1.69813
\(335\) −6.76393 −0.369553
\(336\) −6.00000 −0.327327
\(337\) 27.8885 1.51919 0.759593 0.650399i \(-0.225398\pi\)
0.759593 + 0.650399i \(0.225398\pi\)
\(338\) −20.0902 −1.09276
\(339\) −14.4164 −0.782992
\(340\) −2.14590 −0.116378
\(341\) 0 0
\(342\) −4.00000 −0.216295
\(343\) 15.4164 0.832408
\(344\) 1.70820 0.0921002
\(345\) 8.70820 0.468834
\(346\) −24.1803 −1.29994
\(347\) −9.76393 −0.524155 −0.262078 0.965047i \(-0.584408\pi\)
−0.262078 + 0.965047i \(0.584408\pi\)
\(348\) −2.00000 −0.107211
\(349\) −15.0000 −0.802932 −0.401466 0.915874i \(-0.631499\pi\)
−0.401466 + 0.915874i \(0.631499\pi\)
\(350\) −2.00000 −0.106904
\(351\) −0.763932 −0.0407757
\(352\) 0 0
\(353\) −7.00000 −0.372572 −0.186286 0.982496i \(-0.559645\pi\)
−0.186286 + 0.982496i \(0.559645\pi\)
\(354\) 2.76393 0.146901
\(355\) 5.52786 0.293389
\(356\) 5.70820 0.302534
\(357\) 4.29180 0.227146
\(358\) 3.52786 0.186453
\(359\) −19.4164 −1.02476 −0.512379 0.858759i \(-0.671236\pi\)
−0.512379 + 0.858759i \(0.671236\pi\)
\(360\) 2.23607 0.117851
\(361\) −12.8885 −0.678344
\(362\) 8.76393 0.460622
\(363\) 0 0
\(364\) −0.583592 −0.0305885
\(365\) 1.52786 0.0799721
\(366\) −12.0902 −0.631963
\(367\) 9.23607 0.482119 0.241059 0.970510i \(-0.422505\pi\)
0.241059 + 0.970510i \(0.422505\pi\)
\(368\) −42.2705 −2.20350
\(369\) 12.4721 0.649273
\(370\) −8.47214 −0.440445
\(371\) 9.81966 0.509811
\(372\) −4.14590 −0.214955
\(373\) −5.88854 −0.304897 −0.152449 0.988311i \(-0.548716\pi\)
−0.152449 + 0.988311i \(0.548716\pi\)
\(374\) 0 0
\(375\) 1.00000 0.0516398
\(376\) 10.5279 0.542933
\(377\) 2.47214 0.127321
\(378\) 2.00000 0.102869
\(379\) −18.5967 −0.955251 −0.477625 0.878564i \(-0.658502\pi\)
−0.477625 + 0.878564i \(0.658502\pi\)
\(380\) 1.52786 0.0783778
\(381\) 9.70820 0.497366
\(382\) −34.9443 −1.78790
\(383\) 16.0000 0.817562 0.408781 0.912633i \(-0.365954\pi\)
0.408781 + 0.912633i \(0.365954\pi\)
\(384\) −13.6180 −0.694942
\(385\) 0 0
\(386\) −10.0000 −0.508987
\(387\) −0.763932 −0.0388328
\(388\) 10.6525 0.540798
\(389\) 27.5967 1.39921 0.699605 0.714529i \(-0.253359\pi\)
0.699605 + 0.714529i \(0.253359\pi\)
\(390\) 1.23607 0.0625907
\(391\) 30.2361 1.52910
\(392\) 12.2361 0.618015
\(393\) −8.47214 −0.427363
\(394\) 41.1246 2.07183
\(395\) 0.708204 0.0356336
\(396\) 0 0
\(397\) 9.41641 0.472596 0.236298 0.971681i \(-0.424066\pi\)
0.236298 + 0.971681i \(0.424066\pi\)
\(398\) 19.0344 0.954110
\(399\) −3.05573 −0.152978
\(400\) −4.85410 −0.242705
\(401\) 3.05573 0.152596 0.0762979 0.997085i \(-0.475690\pi\)
0.0762979 + 0.997085i \(0.475690\pi\)
\(402\) −10.9443 −0.545851
\(403\) 5.12461 0.255275
\(404\) −8.94427 −0.444994
\(405\) −1.00000 −0.0496904
\(406\) −6.47214 −0.321207
\(407\) 0 0
\(408\) 7.76393 0.384372
\(409\) −34.7771 −1.71962 −0.859808 0.510617i \(-0.829417\pi\)
−0.859808 + 0.510617i \(0.829417\pi\)
\(410\) −20.1803 −0.996636
\(411\) −8.52786 −0.420649
\(412\) 10.4721 0.515925
\(413\) 2.11146 0.103898
\(414\) 14.0902 0.692494
\(415\) 4.00000 0.196352
\(416\) −2.58359 −0.126671
\(417\) 20.7082 1.01409
\(418\) 0 0
\(419\) −14.0000 −0.683945 −0.341972 0.939710i \(-0.611095\pi\)
−0.341972 + 0.939710i \(0.611095\pi\)
\(420\) −0.763932 −0.0372761
\(421\) −23.8328 −1.16154 −0.580770 0.814068i \(-0.697249\pi\)
−0.580770 + 0.814068i \(0.697249\pi\)
\(422\) 10.0902 0.491182
\(423\) −4.70820 −0.228921
\(424\) 17.7639 0.862693
\(425\) 3.47214 0.168423
\(426\) 8.94427 0.433351
\(427\) −9.23607 −0.446965
\(428\) 6.03444 0.291686
\(429\) 0 0
\(430\) 1.23607 0.0596085
\(431\) −28.1803 −1.35740 −0.678700 0.734416i \(-0.737456\pi\)
−0.678700 + 0.734416i \(0.737456\pi\)
\(432\) 4.85410 0.233543
\(433\) 20.7639 0.997851 0.498925 0.866645i \(-0.333728\pi\)
0.498925 + 0.866645i \(0.333728\pi\)
\(434\) −13.4164 −0.644008
\(435\) 3.23607 0.155158
\(436\) −1.81966 −0.0871459
\(437\) −21.5279 −1.02982
\(438\) 2.47214 0.118123
\(439\) 32.1246 1.53322 0.766612 0.642111i \(-0.221941\pi\)
0.766612 + 0.642111i \(0.221941\pi\)
\(440\) 0 0
\(441\) −5.47214 −0.260578
\(442\) 4.29180 0.204140
\(443\) −19.4164 −0.922501 −0.461251 0.887270i \(-0.652599\pi\)
−0.461251 + 0.887270i \(0.652599\pi\)
\(444\) −3.23607 −0.153577
\(445\) −9.23607 −0.437832
\(446\) 43.5967 2.06437
\(447\) 17.7082 0.837569
\(448\) −5.23607 −0.247381
\(449\) −10.0000 −0.471929 −0.235965 0.971762i \(-0.575825\pi\)
−0.235965 + 0.971762i \(0.575825\pi\)
\(450\) 1.61803 0.0762749
\(451\) 0 0
\(452\) 8.90983 0.419083
\(453\) −20.7082 −0.972956
\(454\) 5.32624 0.249973
\(455\) 0.944272 0.0442681
\(456\) −5.52786 −0.258866
\(457\) 10.3607 0.484652 0.242326 0.970195i \(-0.422090\pi\)
0.242326 + 0.970195i \(0.422090\pi\)
\(458\) 11.3262 0.529240
\(459\) −3.47214 −0.162065
\(460\) −5.38197 −0.250935
\(461\) 7.81966 0.364198 0.182099 0.983280i \(-0.441711\pi\)
0.182099 + 0.983280i \(0.441711\pi\)
\(462\) 0 0
\(463\) 6.00000 0.278844 0.139422 0.990233i \(-0.455476\pi\)
0.139422 + 0.990233i \(0.455476\pi\)
\(464\) −15.7082 −0.729235
\(465\) 6.70820 0.311086
\(466\) −40.2705 −1.86550
\(467\) −8.23607 −0.381120 −0.190560 0.981676i \(-0.561030\pi\)
−0.190560 + 0.981676i \(0.561030\pi\)
\(468\) 0.472136 0.0218245
\(469\) −8.36068 −0.386060
\(470\) 7.61803 0.351394
\(471\) −15.7082 −0.723796
\(472\) 3.81966 0.175814
\(473\) 0 0
\(474\) 1.14590 0.0526328
\(475\) −2.47214 −0.113429
\(476\) −2.65248 −0.121576
\(477\) −7.94427 −0.363743
\(478\) 10.7639 0.492331
\(479\) −13.7082 −0.626344 −0.313172 0.949696i \(-0.601392\pi\)
−0.313172 + 0.949696i \(0.601392\pi\)
\(480\) −3.38197 −0.154365
\(481\) 4.00000 0.182384
\(482\) −0.0901699 −0.00410713
\(483\) 10.7639 0.489776
\(484\) 0 0
\(485\) −17.2361 −0.782650
\(486\) −1.61803 −0.0733955
\(487\) 1.70820 0.0774061 0.0387031 0.999251i \(-0.487677\pi\)
0.0387031 + 0.999251i \(0.487677\pi\)
\(488\) −16.7082 −0.756345
\(489\) 16.1803 0.731700
\(490\) 8.85410 0.399988
\(491\) −29.5279 −1.33257 −0.666287 0.745695i \(-0.732117\pi\)
−0.666287 + 0.745695i \(0.732117\pi\)
\(492\) −7.70820 −0.347513
\(493\) 11.2361 0.506047
\(494\) −3.05573 −0.137484
\(495\) 0 0
\(496\) −32.5623 −1.46209
\(497\) 6.83282 0.306494
\(498\) 6.47214 0.290023
\(499\) 16.9443 0.758530 0.379265 0.925288i \(-0.376177\pi\)
0.379265 + 0.925288i \(0.376177\pi\)
\(500\) −0.618034 −0.0276393
\(501\) 19.1803 0.856914
\(502\) −16.6525 −0.743236
\(503\) −0.819660 −0.0365468 −0.0182734 0.999833i \(-0.505817\pi\)
−0.0182734 + 0.999833i \(0.505817\pi\)
\(504\) 2.76393 0.123115
\(505\) 14.4721 0.644002
\(506\) 0 0
\(507\) 12.4164 0.551432
\(508\) −6.00000 −0.266207
\(509\) −33.5967 −1.48915 −0.744575 0.667539i \(-0.767348\pi\)
−0.744575 + 0.667539i \(0.767348\pi\)
\(510\) 5.61803 0.248771
\(511\) 1.88854 0.0835443
\(512\) −5.29180 −0.233867
\(513\) 2.47214 0.109147
\(514\) −11.1459 −0.491624
\(515\) −16.9443 −0.746654
\(516\) 0.472136 0.0207846
\(517\) 0 0
\(518\) −10.4721 −0.460119
\(519\) 14.9443 0.655981
\(520\) 1.70820 0.0749097
\(521\) 40.1803 1.76033 0.880166 0.474665i \(-0.157431\pi\)
0.880166 + 0.474665i \(0.157431\pi\)
\(522\) 5.23607 0.229176
\(523\) 42.5410 1.86019 0.930094 0.367320i \(-0.119725\pi\)
0.930094 + 0.367320i \(0.119725\pi\)
\(524\) 5.23607 0.228739
\(525\) 1.23607 0.0539464
\(526\) −32.5623 −1.41978
\(527\) 23.2918 1.01461
\(528\) 0 0
\(529\) 52.8328 2.29708
\(530\) 12.8541 0.558347
\(531\) −1.70820 −0.0741297
\(532\) 1.88854 0.0818788
\(533\) 9.52786 0.412698
\(534\) −14.9443 −0.646702
\(535\) −9.76393 −0.422132
\(536\) −15.1246 −0.653284
\(537\) −2.18034 −0.0940886
\(538\) 40.3607 1.74007
\(539\) 0 0
\(540\) 0.618034 0.0265959
\(541\) 19.3050 0.829985 0.414992 0.909825i \(-0.363784\pi\)
0.414992 + 0.909825i \(0.363784\pi\)
\(542\) −31.9787 −1.37360
\(543\) −5.41641 −0.232440
\(544\) −11.7426 −0.503462
\(545\) 2.94427 0.126119
\(546\) 1.52786 0.0653865
\(547\) 39.5967 1.69303 0.846517 0.532361i \(-0.178695\pi\)
0.846517 + 0.532361i \(0.178695\pi\)
\(548\) 5.27051 0.225145
\(549\) 7.47214 0.318903
\(550\) 0 0
\(551\) −8.00000 −0.340811
\(552\) 19.4721 0.828789
\(553\) 0.875388 0.0372253
\(554\) 20.9443 0.889837
\(555\) 5.23607 0.222259
\(556\) −12.7984 −0.542772
\(557\) −17.4721 −0.740318 −0.370159 0.928968i \(-0.620697\pi\)
−0.370159 + 0.928968i \(0.620697\pi\)
\(558\) 10.8541 0.459491
\(559\) −0.583592 −0.0246833
\(560\) −6.00000 −0.253546
\(561\) 0 0
\(562\) 17.4164 0.734667
\(563\) 16.9443 0.714116 0.357058 0.934082i \(-0.383780\pi\)
0.357058 + 0.934082i \(0.383780\pi\)
\(564\) 2.90983 0.122526
\(565\) −14.4164 −0.606503
\(566\) −38.6525 −1.62468
\(567\) −1.23607 −0.0519100
\(568\) 12.3607 0.518643
\(569\) −6.29180 −0.263766 −0.131883 0.991265i \(-0.542102\pi\)
−0.131883 + 0.991265i \(0.542102\pi\)
\(570\) −4.00000 −0.167542
\(571\) −40.1246 −1.67916 −0.839581 0.543234i \(-0.817200\pi\)
−0.839581 + 0.543234i \(0.817200\pi\)
\(572\) 0 0
\(573\) 21.5967 0.902217
\(574\) −24.9443 −1.04115
\(575\) 8.70820 0.363157
\(576\) 4.23607 0.176503
\(577\) 20.0000 0.832611 0.416305 0.909225i \(-0.363325\pi\)
0.416305 + 0.909225i \(0.363325\pi\)
\(578\) −8.00000 −0.332756
\(579\) 6.18034 0.256846
\(580\) −2.00000 −0.0830455
\(581\) 4.94427 0.205123
\(582\) −27.8885 −1.15602
\(583\) 0 0
\(584\) 3.41641 0.141372
\(585\) −0.763932 −0.0315847
\(586\) 20.2705 0.837367
\(587\) 19.6525 0.811144 0.405572 0.914063i \(-0.367072\pi\)
0.405572 + 0.914063i \(0.367072\pi\)
\(588\) 3.38197 0.139470
\(589\) −16.5836 −0.683315
\(590\) 2.76393 0.113789
\(591\) −25.4164 −1.04549
\(592\) −25.4164 −1.04461
\(593\) −17.0557 −0.700395 −0.350197 0.936676i \(-0.613885\pi\)
−0.350197 + 0.936676i \(0.613885\pi\)
\(594\) 0 0
\(595\) 4.29180 0.175946
\(596\) −10.9443 −0.448295
\(597\) −11.7639 −0.481466
\(598\) 10.7639 0.440170
\(599\) 45.0132 1.83919 0.919594 0.392870i \(-0.128518\pi\)
0.919594 + 0.392870i \(0.128518\pi\)
\(600\) 2.23607 0.0912871
\(601\) 19.5279 0.796558 0.398279 0.917264i \(-0.369608\pi\)
0.398279 + 0.917264i \(0.369608\pi\)
\(602\) 1.52786 0.0622711
\(603\) 6.76393 0.275449
\(604\) 12.7984 0.520758
\(605\) 0 0
\(606\) 23.4164 0.951227
\(607\) 19.7082 0.799931 0.399966 0.916530i \(-0.369022\pi\)
0.399966 + 0.916530i \(0.369022\pi\)
\(608\) 8.36068 0.339070
\(609\) 4.00000 0.162088
\(610\) −12.0902 −0.489517
\(611\) −3.59675 −0.145509
\(612\) 2.14590 0.0867428
\(613\) 38.8328 1.56844 0.784221 0.620481i \(-0.213063\pi\)
0.784221 + 0.620481i \(0.213063\pi\)
\(614\) 38.6525 1.55989
\(615\) 12.4721 0.502925
\(616\) 0 0
\(617\) −29.7771 −1.19878 −0.599390 0.800457i \(-0.704590\pi\)
−0.599390 + 0.800457i \(0.704590\pi\)
\(618\) −27.4164 −1.10285
\(619\) −18.4721 −0.742458 −0.371229 0.928541i \(-0.621063\pi\)
−0.371229 + 0.928541i \(0.621063\pi\)
\(620\) −4.14590 −0.166503
\(621\) −8.70820 −0.349448
\(622\) 2.00000 0.0801927
\(623\) −11.4164 −0.457389
\(624\) 3.70820 0.148447
\(625\) 1.00000 0.0400000
\(626\) 44.0689 1.76135
\(627\) 0 0
\(628\) 9.70820 0.387400
\(629\) 18.1803 0.724898
\(630\) 2.00000 0.0796819
\(631\) −13.1803 −0.524701 −0.262351 0.964973i \(-0.584498\pi\)
−0.262351 + 0.964973i \(0.584498\pi\)
\(632\) 1.58359 0.0629919
\(633\) −6.23607 −0.247861
\(634\) 22.5623 0.896064
\(635\) 9.70820 0.385258
\(636\) 4.90983 0.194687
\(637\) −4.18034 −0.165631
\(638\) 0 0
\(639\) −5.52786 −0.218679
\(640\) −13.6180 −0.538300
\(641\) 31.1246 1.22935 0.614674 0.788781i \(-0.289288\pi\)
0.614674 + 0.788781i \(0.289288\pi\)
\(642\) −15.7984 −0.623512
\(643\) 24.8328 0.979311 0.489655 0.871916i \(-0.337123\pi\)
0.489655 + 0.871916i \(0.337123\pi\)
\(644\) −6.65248 −0.262144
\(645\) −0.763932 −0.0300798
\(646\) −13.8885 −0.546437
\(647\) −11.7639 −0.462488 −0.231244 0.972896i \(-0.574280\pi\)
−0.231244 + 0.972896i \(0.574280\pi\)
\(648\) −2.23607 −0.0878410
\(649\) 0 0
\(650\) 1.23607 0.0484826
\(651\) 8.29180 0.324981
\(652\) −10.0000 −0.391630
\(653\) −36.8328 −1.44138 −0.720690 0.693258i \(-0.756175\pi\)
−0.720690 + 0.693258i \(0.756175\pi\)
\(654\) 4.76393 0.186284
\(655\) −8.47214 −0.331034
\(656\) −60.5410 −2.36373
\(657\) −1.52786 −0.0596077
\(658\) 9.41641 0.367090
\(659\) −13.5967 −0.529654 −0.264827 0.964296i \(-0.585315\pi\)
−0.264827 + 0.964296i \(0.585315\pi\)
\(660\) 0 0
\(661\) −25.7771 −1.00261 −0.501306 0.865270i \(-0.667147\pi\)
−0.501306 + 0.865270i \(0.667147\pi\)
\(662\) 8.38197 0.325774
\(663\) −2.65248 −0.103014
\(664\) 8.94427 0.347105
\(665\) −3.05573 −0.118496
\(666\) 8.47214 0.328289
\(667\) 28.1803 1.09115
\(668\) −11.8541 −0.458649
\(669\) −26.9443 −1.04173
\(670\) −10.9443 −0.422814
\(671\) 0 0
\(672\) −4.18034 −0.161260
\(673\) 21.5967 0.832493 0.416247 0.909252i \(-0.363345\pi\)
0.416247 + 0.909252i \(0.363345\pi\)
\(674\) 45.1246 1.73814
\(675\) −1.00000 −0.0384900
\(676\) −7.67376 −0.295145
\(677\) −21.4164 −0.823099 −0.411550 0.911387i \(-0.635012\pi\)
−0.411550 + 0.911387i \(0.635012\pi\)
\(678\) −23.3262 −0.895839
\(679\) −21.3050 −0.817609
\(680\) 7.76393 0.297733
\(681\) −3.29180 −0.126142
\(682\) 0 0
\(683\) −24.0000 −0.918334 −0.459167 0.888350i \(-0.651852\pi\)
−0.459167 + 0.888350i \(0.651852\pi\)
\(684\) −1.52786 −0.0584193
\(685\) −8.52786 −0.325833
\(686\) 24.9443 0.952377
\(687\) −7.00000 −0.267067
\(688\) 3.70820 0.141374
\(689\) −6.06888 −0.231206
\(690\) 14.0902 0.536404
\(691\) 29.5410 1.12379 0.561897 0.827207i \(-0.310072\pi\)
0.561897 + 0.827207i \(0.310072\pi\)
\(692\) −9.23607 −0.351103
\(693\) 0 0
\(694\) −15.7984 −0.599698
\(695\) 20.7082 0.785507
\(696\) 7.23607 0.274282
\(697\) 43.3050 1.64029
\(698\) −24.2705 −0.918652
\(699\) 24.8885 0.941371
\(700\) −0.763932 −0.0288739
\(701\) 13.8197 0.521961 0.260981 0.965344i \(-0.415954\pi\)
0.260981 + 0.965344i \(0.415954\pi\)
\(702\) −1.23607 −0.0466524
\(703\) −12.9443 −0.488202
\(704\) 0 0
\(705\) −4.70820 −0.177321
\(706\) −11.3262 −0.426269
\(707\) 17.8885 0.672768
\(708\) 1.05573 0.0396767
\(709\) −0.0557281 −0.00209291 −0.00104646 0.999999i \(-0.500333\pi\)
−0.00104646 + 0.999999i \(0.500333\pi\)
\(710\) 8.94427 0.335673
\(711\) −0.708204 −0.0265597
\(712\) −20.6525 −0.773984
\(713\) 58.4164 2.18771
\(714\) 6.94427 0.259883
\(715\) 0 0
\(716\) 1.34752 0.0503593
\(717\) −6.65248 −0.248441
\(718\) −31.4164 −1.17245
\(719\) −18.3607 −0.684738 −0.342369 0.939566i \(-0.611229\pi\)
−0.342369 + 0.939566i \(0.611229\pi\)
\(720\) 4.85410 0.180902
\(721\) −20.9443 −0.780005
\(722\) −20.8541 −0.776109
\(723\) 0.0557281 0.00207255
\(724\) 3.34752 0.124410
\(725\) 3.23607 0.120185
\(726\) 0 0
\(727\) −8.83282 −0.327591 −0.163796 0.986494i \(-0.552374\pi\)
−0.163796 + 0.986494i \(0.552374\pi\)
\(728\) 2.11146 0.0782558
\(729\) 1.00000 0.0370370
\(730\) 2.47214 0.0914979
\(731\) −2.65248 −0.0981054
\(732\) −4.61803 −0.170687
\(733\) 12.1115 0.447347 0.223673 0.974664i \(-0.428195\pi\)
0.223673 + 0.974664i \(0.428195\pi\)
\(734\) 14.9443 0.551603
\(735\) −5.47214 −0.201843
\(736\) −29.4508 −1.08557
\(737\) 0 0
\(738\) 20.1803 0.742849
\(739\) −29.6525 −1.09078 −0.545392 0.838181i \(-0.683619\pi\)
−0.545392 + 0.838181i \(0.683619\pi\)
\(740\) −3.23607 −0.118960
\(741\) 1.88854 0.0693774
\(742\) 15.8885 0.583287
\(743\) −2.23607 −0.0820334 −0.0410167 0.999158i \(-0.513060\pi\)
−0.0410167 + 0.999158i \(0.513060\pi\)
\(744\) 15.0000 0.549927
\(745\) 17.7082 0.648778
\(746\) −9.52786 −0.348840
\(747\) −4.00000 −0.146352
\(748\) 0 0
\(749\) −12.0689 −0.440987
\(750\) 1.61803 0.0590822
\(751\) 42.0132 1.53308 0.766541 0.642195i \(-0.221976\pi\)
0.766541 + 0.642195i \(0.221976\pi\)
\(752\) 22.8541 0.833403
\(753\) 10.2918 0.375054
\(754\) 4.00000 0.145671
\(755\) −20.7082 −0.753649
\(756\) 0.763932 0.0277839
\(757\) 1.05573 0.0383711 0.0191855 0.999816i \(-0.493893\pi\)
0.0191855 + 0.999816i \(0.493893\pi\)
\(758\) −30.0902 −1.09292
\(759\) 0 0
\(760\) −5.52786 −0.200517
\(761\) −13.3475 −0.483847 −0.241924 0.970295i \(-0.577778\pi\)
−0.241924 + 0.970295i \(0.577778\pi\)
\(762\) 15.7082 0.569048
\(763\) 3.63932 0.131752
\(764\) −13.3475 −0.482896
\(765\) −3.47214 −0.125535
\(766\) 25.8885 0.935391
\(767\) −1.30495 −0.0471191
\(768\) −13.5623 −0.489388
\(769\) −33.4721 −1.20704 −0.603518 0.797349i \(-0.706235\pi\)
−0.603518 + 0.797349i \(0.706235\pi\)
\(770\) 0 0
\(771\) 6.88854 0.248085
\(772\) −3.81966 −0.137473
\(773\) 9.94427 0.357671 0.178835 0.983879i \(-0.442767\pi\)
0.178835 + 0.983879i \(0.442767\pi\)
\(774\) −1.23607 −0.0444295
\(775\) 6.70820 0.240966
\(776\) −38.5410 −1.38354
\(777\) 6.47214 0.232187
\(778\) 44.6525 1.60087
\(779\) −30.8328 −1.10470
\(780\) 0.472136 0.0169052
\(781\) 0 0
\(782\) 48.9230 1.74948
\(783\) −3.23607 −0.115648
\(784\) 26.5623 0.948654
\(785\) −15.7082 −0.560650
\(786\) −13.7082 −0.488955
\(787\) −44.9443 −1.60209 −0.801045 0.598604i \(-0.795722\pi\)
−0.801045 + 0.598604i \(0.795722\pi\)
\(788\) 15.7082 0.559582
\(789\) 20.1246 0.716455
\(790\) 1.14590 0.0407692
\(791\) −17.8197 −0.633594
\(792\) 0 0
\(793\) 5.70820 0.202704
\(794\) 15.2361 0.540708
\(795\) −7.94427 −0.281754
\(796\) 7.27051 0.257696
\(797\) 30.0000 1.06265 0.531327 0.847167i \(-0.321693\pi\)
0.531327 + 0.847167i \(0.321693\pi\)
\(798\) −4.94427 −0.175025
\(799\) −16.3475 −0.578334
\(800\) −3.38197 −0.119571
\(801\) 9.23607 0.326340
\(802\) 4.94427 0.174588
\(803\) 0 0
\(804\) −4.18034 −0.147429
\(805\) 10.7639 0.379379
\(806\) 8.29180 0.292066
\(807\) −24.9443 −0.878080
\(808\) 32.3607 1.13844
\(809\) −8.00000 −0.281265 −0.140633 0.990062i \(-0.544914\pi\)
−0.140633 + 0.990062i \(0.544914\pi\)
\(810\) −1.61803 −0.0568519
\(811\) 48.1246 1.68988 0.844942 0.534858i \(-0.179635\pi\)
0.844942 + 0.534858i \(0.179635\pi\)
\(812\) −2.47214 −0.0867550
\(813\) 19.7639 0.693151
\(814\) 0 0
\(815\) 16.1803 0.566773
\(816\) 16.8541 0.590012
\(817\) 1.88854 0.0660718
\(818\) −56.2705 −1.96745
\(819\) −0.944272 −0.0329955
\(820\) −7.70820 −0.269182
\(821\) 37.7771 1.31843 0.659215 0.751955i \(-0.270889\pi\)
0.659215 + 0.751955i \(0.270889\pi\)
\(822\) −13.7984 −0.481274
\(823\) −42.2492 −1.47272 −0.736358 0.676592i \(-0.763456\pi\)
−0.736358 + 0.676592i \(0.763456\pi\)
\(824\) −37.8885 −1.31991
\(825\) 0 0
\(826\) 3.41641 0.118872
\(827\) −8.94427 −0.311023 −0.155511 0.987834i \(-0.549703\pi\)
−0.155511 + 0.987834i \(0.549703\pi\)
\(828\) 5.38197 0.187036
\(829\) −11.0000 −0.382046 −0.191023 0.981586i \(-0.561180\pi\)
−0.191023 + 0.981586i \(0.561180\pi\)
\(830\) 6.47214 0.224651
\(831\) −12.9443 −0.449032
\(832\) 3.23607 0.112190
\(833\) −19.0000 −0.658311
\(834\) 33.5066 1.16024
\(835\) 19.1803 0.663763
\(836\) 0 0
\(837\) −6.70820 −0.231869
\(838\) −22.6525 −0.782517
\(839\) −16.3607 −0.564833 −0.282417 0.959292i \(-0.591136\pi\)
−0.282417 + 0.959292i \(0.591136\pi\)
\(840\) 2.76393 0.0953647
\(841\) −18.5279 −0.638892
\(842\) −38.5623 −1.32894
\(843\) −10.7639 −0.370730
\(844\) 3.85410 0.132664
\(845\) 12.4164 0.427137
\(846\) −7.61803 −0.261913
\(847\) 0 0
\(848\) 38.5623 1.32424
\(849\) 23.8885 0.819853
\(850\) 5.61803 0.192697
\(851\) 45.5967 1.56304
\(852\) 3.41641 0.117044
\(853\) 3.88854 0.133141 0.0665706 0.997782i \(-0.478794\pi\)
0.0665706 + 0.997782i \(0.478794\pi\)
\(854\) −14.9443 −0.511382
\(855\) 2.47214 0.0845453
\(856\) −21.8328 −0.746230
\(857\) 1.00000 0.0341593 0.0170797 0.999854i \(-0.494563\pi\)
0.0170797 + 0.999854i \(0.494563\pi\)
\(858\) 0 0
\(859\) 26.8328 0.915524 0.457762 0.889075i \(-0.348651\pi\)
0.457762 + 0.889075i \(0.348651\pi\)
\(860\) 0.472136 0.0160997
\(861\) 15.4164 0.525390
\(862\) −45.5967 −1.55303
\(863\) −2.11146 −0.0718748 −0.0359374 0.999354i \(-0.511442\pi\)
−0.0359374 + 0.999354i \(0.511442\pi\)
\(864\) 3.38197 0.115057
\(865\) 14.9443 0.508120
\(866\) 33.5967 1.14166
\(867\) 4.94427 0.167916
\(868\) −5.12461 −0.173941
\(869\) 0 0
\(870\) 5.23607 0.177519
\(871\) 5.16718 0.175083
\(872\) 6.58359 0.222949
\(873\) 17.2361 0.583353
\(874\) −34.8328 −1.17824
\(875\) 1.23607 0.0417867
\(876\) 0.944272 0.0319040
\(877\) −51.3050 −1.73245 −0.866223 0.499658i \(-0.833459\pi\)
−0.866223 + 0.499658i \(0.833459\pi\)
\(878\) 51.9787 1.75420
\(879\) −12.5279 −0.422554
\(880\) 0 0
\(881\) −14.8328 −0.499730 −0.249865 0.968281i \(-0.580386\pi\)
−0.249865 + 0.968281i \(0.580386\pi\)
\(882\) −8.85410 −0.298133
\(883\) −12.1803 −0.409901 −0.204951 0.978772i \(-0.565703\pi\)
−0.204951 + 0.978772i \(0.565703\pi\)
\(884\) 1.63932 0.0551363
\(885\) −1.70820 −0.0574206
\(886\) −31.4164 −1.05545
\(887\) −0.944272 −0.0317055 −0.0158528 0.999874i \(-0.505046\pi\)
−0.0158528 + 0.999874i \(0.505046\pi\)
\(888\) 11.7082 0.392902
\(889\) 12.0000 0.402467
\(890\) −14.9443 −0.500933
\(891\) 0 0
\(892\) 16.6525 0.557566
\(893\) 11.6393 0.389495
\(894\) 28.6525 0.958282
\(895\) −2.18034 −0.0728807
\(896\) −16.8328 −0.562345
\(897\) −6.65248 −0.222120
\(898\) −16.1803 −0.539945
\(899\) 21.7082 0.724009
\(900\) 0.618034 0.0206011
\(901\) −27.5836 −0.918943
\(902\) 0 0
\(903\) −0.944272 −0.0314234
\(904\) −32.2361 −1.07216
\(905\) −5.41641 −0.180047
\(906\) −33.5066 −1.11318
\(907\) −42.0000 −1.39459 −0.697294 0.716786i \(-0.745613\pi\)
−0.697294 + 0.716786i \(0.745613\pi\)
\(908\) 2.03444 0.0675153
\(909\) −14.4721 −0.480010
\(910\) 1.52786 0.0506482
\(911\) −45.9574 −1.52264 −0.761319 0.648378i \(-0.775448\pi\)
−0.761319 + 0.648378i \(0.775448\pi\)
\(912\) −12.0000 −0.397360
\(913\) 0 0
\(914\) 16.7639 0.554502
\(915\) 7.47214 0.247021
\(916\) 4.32624 0.142943
\(917\) −10.4721 −0.345820
\(918\) −5.61803 −0.185423
\(919\) −23.4164 −0.772436 −0.386218 0.922408i \(-0.626219\pi\)
−0.386218 + 0.922408i \(0.626219\pi\)
\(920\) 19.4721 0.641977
\(921\) −23.8885 −0.787154
\(922\) 12.6525 0.416687
\(923\) −4.22291 −0.138999
\(924\) 0 0
\(925\) 5.23607 0.172161
\(926\) 9.70820 0.319031
\(927\) 16.9443 0.556523
\(928\) −10.9443 −0.359263
\(929\) 33.0557 1.08452 0.542262 0.840210i \(-0.317568\pi\)
0.542262 + 0.840210i \(0.317568\pi\)
\(930\) 10.8541 0.355920
\(931\) 13.5279 0.443358
\(932\) −15.3820 −0.503853
\(933\) −1.23607 −0.0404670
\(934\) −13.3262 −0.436048
\(935\) 0 0
\(936\) −1.70820 −0.0558344
\(937\) −7.12461 −0.232751 −0.116375 0.993205i \(-0.537128\pi\)
−0.116375 + 0.993205i \(0.537128\pi\)
\(938\) −13.5279 −0.441700
\(939\) −27.2361 −0.888815
\(940\) 2.90983 0.0949082
\(941\) −40.7639 −1.32887 −0.664433 0.747348i \(-0.731327\pi\)
−0.664433 + 0.747348i \(0.731327\pi\)
\(942\) −25.4164 −0.828111
\(943\) 108.610 3.53683
\(944\) 8.29180 0.269875
\(945\) −1.23607 −0.0402093
\(946\) 0 0
\(947\) −13.7639 −0.447268 −0.223634 0.974673i \(-0.571792\pi\)
−0.223634 + 0.974673i \(0.571792\pi\)
\(948\) 0.437694 0.0142156
\(949\) −1.16718 −0.0378884
\(950\) −4.00000 −0.129777
\(951\) −13.9443 −0.452174
\(952\) 9.59675 0.311032
\(953\) −3.88854 −0.125962 −0.0629811 0.998015i \(-0.520061\pi\)
−0.0629811 + 0.998015i \(0.520061\pi\)
\(954\) −12.8541 −0.416167
\(955\) 21.5967 0.698854
\(956\) 4.11146 0.132974
\(957\) 0 0
\(958\) −22.1803 −0.716614
\(959\) −10.5410 −0.340387
\(960\) 4.23607 0.136719
\(961\) 14.0000 0.451613
\(962\) 6.47214 0.208670
\(963\) 9.76393 0.314638
\(964\) −0.0344419 −0.00110930
\(965\) 6.18034 0.198952
\(966\) 17.4164 0.560364
\(967\) −32.0000 −1.02905 −0.514525 0.857475i \(-0.672032\pi\)
−0.514525 + 0.857475i \(0.672032\pi\)
\(968\) 0 0
\(969\) 8.58359 0.275745
\(970\) −27.8885 −0.895447
\(971\) 24.7639 0.794712 0.397356 0.917664i \(-0.369928\pi\)
0.397356 + 0.917664i \(0.369928\pi\)
\(972\) −0.618034 −0.0198234
\(973\) 25.5967 0.820594
\(974\) 2.76393 0.0885621
\(975\) −0.763932 −0.0244654
\(976\) −36.2705 −1.16099
\(977\) 1.94427 0.0622028 0.0311014 0.999516i \(-0.490099\pi\)
0.0311014 + 0.999516i \(0.490099\pi\)
\(978\) 26.1803 0.837155
\(979\) 0 0
\(980\) 3.38197 0.108033
\(981\) −2.94427 −0.0940034
\(982\) −47.7771 −1.52463
\(983\) −32.8197 −1.04678 −0.523392 0.852092i \(-0.675334\pi\)
−0.523392 + 0.852092i \(0.675334\pi\)
\(984\) 27.8885 0.889054
\(985\) −25.4164 −0.809834
\(986\) 18.1803 0.578980
\(987\) −5.81966 −0.185242
\(988\) −1.16718 −0.0371331
\(989\) −6.65248 −0.211536
\(990\) 0 0
\(991\) 1.76393 0.0560331 0.0280166 0.999607i \(-0.491081\pi\)
0.0280166 + 0.999607i \(0.491081\pi\)
\(992\) −22.6869 −0.720310
\(993\) −5.18034 −0.164393
\(994\) 11.0557 0.350666
\(995\) −11.7639 −0.372942
\(996\) 2.47214 0.0783326
\(997\) 43.2361 1.36930 0.684650 0.728872i \(-0.259955\pi\)
0.684650 + 0.728872i \(0.259955\pi\)
\(998\) 27.4164 0.867851
\(999\) −5.23607 −0.165662
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.j.1.2 yes 2
3.2 odd 2 5445.2.a.o.1.1 2
5.4 even 2 9075.2.a.bd.1.1 2
11.10 odd 2 1815.2.a.f.1.1 2
33.32 even 2 5445.2.a.x.1.2 2
55.54 odd 2 9075.2.a.bx.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1815.2.a.f.1.1 2 11.10 odd 2
1815.2.a.j.1.2 yes 2 1.1 even 1 trivial
5445.2.a.o.1.1 2 3.2 odd 2
5445.2.a.x.1.2 2 33.32 even 2
9075.2.a.bd.1.1 2 5.4 even 2
9075.2.a.bx.1.2 2 55.54 odd 2