Properties

Label 3675.2.a.bj.1.1
Level $3675$
Weight $2$
Character 3675.1
Self dual yes
Analytic conductor $29.345$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 3675 = 3 \cdot 5^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3675.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(29.3450227428\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.148.1
Defining polynomial: \( x^{3} - x^{2} - 3x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 105)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(0.311108\) of defining polynomial
Character \(\chi\) \(=\) 3675.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.90321 q^{2} +1.00000 q^{3} +1.62222 q^{4} -1.90321 q^{6} +0.719004 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.90321 q^{2} +1.00000 q^{3} +1.62222 q^{4} -1.90321 q^{6} +0.719004 q^{8} +1.00000 q^{9} +2.00000 q^{11} +1.62222 q^{12} +6.42864 q^{13} -4.61285 q^{16} -4.42864 q^{17} -1.90321 q^{18} +2.42864 q^{19} -3.80642 q^{22} +1.37778 q^{23} +0.719004 q^{24} -12.2351 q^{26} +1.00000 q^{27} +0.755569 q^{29} -5.18421 q^{31} +7.34122 q^{32} +2.00000 q^{33} +8.42864 q^{34} +1.62222 q^{36} +7.61285 q^{37} -4.62222 q^{38} +6.42864 q^{39} +8.23506 q^{41} +10.1017 q^{43} +3.24443 q^{44} -2.62222 q^{46} -2.75557 q^{47} -4.61285 q^{48} -4.42864 q^{51} +10.4286 q^{52} +9.18421 q^{53} -1.90321 q^{54} +2.42864 q^{57} -1.43801 q^{58} -14.1017 q^{59} -6.85728 q^{61} +9.86665 q^{62} -4.74620 q^{64} -3.80642 q^{66} +2.75557 q^{67} -7.18421 q^{68} +1.37778 q^{69} +2.00000 q^{71} +0.719004 q^{72} +1.57136 q^{73} -14.4889 q^{74} +3.93978 q^{76} -12.2351 q^{78} -4.85728 q^{79} +1.00000 q^{81} -15.6731 q^{82} -11.6128 q^{83} -19.2257 q^{86} +0.755569 q^{87} +1.43801 q^{88} -4.62222 q^{89} +2.23506 q^{92} -5.18421 q^{93} +5.24443 q^{94} +7.34122 q^{96} +11.9398 q^{97} +2.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + q^{2} + 3 q^{3} + 5 q^{4} + q^{6} + 9 q^{8} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + q^{2} + 3 q^{3} + 5 q^{4} + q^{6} + 9 q^{8} + 3 q^{9} + 6 q^{11} + 5 q^{12} + 6 q^{13} + 13 q^{16} + q^{18} - 6 q^{19} + 2 q^{22} + 4 q^{23} + 9 q^{24} - 10 q^{26} + 3 q^{27} + 2 q^{29} - 2 q^{31} + 29 q^{32} + 6 q^{33} + 12 q^{34} + 5 q^{36} - 4 q^{37} - 14 q^{38} + 6 q^{39} - 2 q^{41} + 4 q^{43} + 10 q^{44} - 8 q^{46} - 8 q^{47} + 13 q^{48} + 18 q^{52} + 14 q^{53} + q^{54} - 6 q^{57} - 18 q^{58} - 16 q^{59} + 6 q^{61} + 30 q^{62} + 13 q^{64} + 2 q^{66} + 8 q^{67} - 8 q^{68} + 4 q^{69} + 6 q^{71} + 9 q^{72} + 18 q^{73} - 44 q^{74} - 2 q^{76} - 10 q^{78} + 12 q^{79} + 3 q^{81} - 34 q^{82} - 8 q^{83} - 4 q^{86} + 2 q^{87} + 18 q^{88} - 14 q^{89} - 20 q^{92} - 2 q^{93} + 16 q^{94} + 29 q^{96} + 22 q^{97} + 6 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.90321 −1.34577 −0.672887 0.739745i \(-0.734946\pi\)
−0.672887 + 0.739745i \(0.734946\pi\)
\(3\) 1.00000 0.577350
\(4\) 1.62222 0.811108
\(5\) 0 0
\(6\) −1.90321 −0.776983
\(7\) 0 0
\(8\) 0.719004 0.254206
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 2.00000 0.603023 0.301511 0.953463i \(-0.402509\pi\)
0.301511 + 0.953463i \(0.402509\pi\)
\(12\) 1.62222 0.468293
\(13\) 6.42864 1.78298 0.891492 0.453037i \(-0.149659\pi\)
0.891492 + 0.453037i \(0.149659\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) −4.61285 −1.15321
\(17\) −4.42864 −1.07410 −0.537051 0.843550i \(-0.680462\pi\)
−0.537051 + 0.843550i \(0.680462\pi\)
\(18\) −1.90321 −0.448591
\(19\) 2.42864 0.557168 0.278584 0.960412i \(-0.410135\pi\)
0.278584 + 0.960412i \(0.410135\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) −3.80642 −0.811532
\(23\) 1.37778 0.287288 0.143644 0.989629i \(-0.454118\pi\)
0.143644 + 0.989629i \(0.454118\pi\)
\(24\) 0.719004 0.146766
\(25\) 0 0
\(26\) −12.2351 −2.39949
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) 0.755569 0.140306 0.0701528 0.997536i \(-0.477651\pi\)
0.0701528 + 0.997536i \(0.477651\pi\)
\(30\) 0 0
\(31\) −5.18421 −0.931111 −0.465556 0.885019i \(-0.654145\pi\)
−0.465556 + 0.885019i \(0.654145\pi\)
\(32\) 7.34122 1.29776
\(33\) 2.00000 0.348155
\(34\) 8.42864 1.44550
\(35\) 0 0
\(36\) 1.62222 0.270369
\(37\) 7.61285 1.25154 0.625772 0.780006i \(-0.284784\pi\)
0.625772 + 0.780006i \(0.284784\pi\)
\(38\) −4.62222 −0.749822
\(39\) 6.42864 1.02941
\(40\) 0 0
\(41\) 8.23506 1.28610 0.643050 0.765824i \(-0.277669\pi\)
0.643050 + 0.765824i \(0.277669\pi\)
\(42\) 0 0
\(43\) 10.1017 1.54050 0.770248 0.637744i \(-0.220132\pi\)
0.770248 + 0.637744i \(0.220132\pi\)
\(44\) 3.24443 0.489116
\(45\) 0 0
\(46\) −2.62222 −0.386625
\(47\) −2.75557 −0.401941 −0.200971 0.979597i \(-0.564410\pi\)
−0.200971 + 0.979597i \(0.564410\pi\)
\(48\) −4.61285 −0.665807
\(49\) 0 0
\(50\) 0 0
\(51\) −4.42864 −0.620134
\(52\) 10.4286 1.44619
\(53\) 9.18421 1.26155 0.630774 0.775967i \(-0.282737\pi\)
0.630774 + 0.775967i \(0.282737\pi\)
\(54\) −1.90321 −0.258994
\(55\) 0 0
\(56\) 0 0
\(57\) 2.42864 0.321681
\(58\) −1.43801 −0.188820
\(59\) −14.1017 −1.83589 −0.917943 0.396712i \(-0.870151\pi\)
−0.917943 + 0.396712i \(0.870151\pi\)
\(60\) 0 0
\(61\) −6.85728 −0.877985 −0.438992 0.898491i \(-0.644664\pi\)
−0.438992 + 0.898491i \(0.644664\pi\)
\(62\) 9.86665 1.25307
\(63\) 0 0
\(64\) −4.74620 −0.593275
\(65\) 0 0
\(66\) −3.80642 −0.468538
\(67\) 2.75557 0.336646 0.168323 0.985732i \(-0.446165\pi\)
0.168323 + 0.985732i \(0.446165\pi\)
\(68\) −7.18421 −0.871213
\(69\) 1.37778 0.165866
\(70\) 0 0
\(71\) 2.00000 0.237356 0.118678 0.992933i \(-0.462134\pi\)
0.118678 + 0.992933i \(0.462134\pi\)
\(72\) 0.719004 0.0847354
\(73\) 1.57136 0.183914 0.0919569 0.995763i \(-0.470688\pi\)
0.0919569 + 0.995763i \(0.470688\pi\)
\(74\) −14.4889 −1.68430
\(75\) 0 0
\(76\) 3.93978 0.451923
\(77\) 0 0
\(78\) −12.2351 −1.38535
\(79\) −4.85728 −0.546487 −0.273243 0.961945i \(-0.588096\pi\)
−0.273243 + 0.961945i \(0.588096\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) −15.6731 −1.73080
\(83\) −11.6128 −1.27468 −0.637338 0.770585i \(-0.719964\pi\)
−0.637338 + 0.770585i \(0.719964\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −19.2257 −2.07316
\(87\) 0.755569 0.0810055
\(88\) 1.43801 0.153292
\(89\) −4.62222 −0.489954 −0.244977 0.969529i \(-0.578780\pi\)
−0.244977 + 0.969529i \(0.578780\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 2.23506 0.233021
\(93\) −5.18421 −0.537577
\(94\) 5.24443 0.540922
\(95\) 0 0
\(96\) 7.34122 0.749260
\(97\) 11.9398 1.21230 0.606150 0.795350i \(-0.292713\pi\)
0.606150 + 0.795350i \(0.292713\pi\)
\(98\) 0 0
\(99\) 2.00000 0.201008
\(100\) 0 0
\(101\) −1.47949 −0.147215 −0.0736076 0.997287i \(-0.523451\pi\)
−0.0736076 + 0.997287i \(0.523451\pi\)
\(102\) 8.42864 0.834560
\(103\) 8.85728 0.872734 0.436367 0.899769i \(-0.356265\pi\)
0.436367 + 0.899769i \(0.356265\pi\)
\(104\) 4.62222 0.453246
\(105\) 0 0
\(106\) −17.4795 −1.69776
\(107\) 1.76494 0.170623 0.0853114 0.996354i \(-0.472811\pi\)
0.0853114 + 0.996354i \(0.472811\pi\)
\(108\) 1.62222 0.156098
\(109\) 5.61285 0.537613 0.268807 0.963194i \(-0.413371\pi\)
0.268807 + 0.963194i \(0.413371\pi\)
\(110\) 0 0
\(111\) 7.61285 0.722580
\(112\) 0 0
\(113\) −11.2859 −1.06169 −0.530845 0.847469i \(-0.678125\pi\)
−0.530845 + 0.847469i \(0.678125\pi\)
\(114\) −4.62222 −0.432910
\(115\) 0 0
\(116\) 1.22570 0.113803
\(117\) 6.42864 0.594328
\(118\) 26.8385 2.47069
\(119\) 0 0
\(120\) 0 0
\(121\) −7.00000 −0.636364
\(122\) 13.0509 1.18157
\(123\) 8.23506 0.742531
\(124\) −8.40990 −0.755232
\(125\) 0 0
\(126\) 0 0
\(127\) −12.8573 −1.14090 −0.570450 0.821333i \(-0.693231\pi\)
−0.570450 + 0.821333i \(0.693231\pi\)
\(128\) −5.64941 −0.499342
\(129\) 10.1017 0.889406
\(130\) 0 0
\(131\) 2.10171 0.183627 0.0918136 0.995776i \(-0.470734\pi\)
0.0918136 + 0.995776i \(0.470734\pi\)
\(132\) 3.24443 0.282391
\(133\) 0 0
\(134\) −5.24443 −0.453050
\(135\) 0 0
\(136\) −3.18421 −0.273044
\(137\) 15.9398 1.36183 0.680914 0.732364i \(-0.261583\pi\)
0.680914 + 0.732364i \(0.261583\pi\)
\(138\) −2.62222 −0.223218
\(139\) −11.6731 −0.990097 −0.495048 0.868865i \(-0.664850\pi\)
−0.495048 + 0.868865i \(0.664850\pi\)
\(140\) 0 0
\(141\) −2.75557 −0.232061
\(142\) −3.80642 −0.319428
\(143\) 12.8573 1.07518
\(144\) −4.61285 −0.384404
\(145\) 0 0
\(146\) −2.99063 −0.247506
\(147\) 0 0
\(148\) 12.3497 1.01514
\(149\) −21.2257 −1.73888 −0.869438 0.494041i \(-0.835519\pi\)
−0.869438 + 0.494041i \(0.835519\pi\)
\(150\) 0 0
\(151\) 16.8573 1.37183 0.685913 0.727684i \(-0.259403\pi\)
0.685913 + 0.727684i \(0.259403\pi\)
\(152\) 1.74620 0.141636
\(153\) −4.42864 −0.358034
\(154\) 0 0
\(155\) 0 0
\(156\) 10.4286 0.834959
\(157\) 10.4286 0.832296 0.416148 0.909297i \(-0.363380\pi\)
0.416148 + 0.909297i \(0.363380\pi\)
\(158\) 9.24443 0.735447
\(159\) 9.18421 0.728355
\(160\) 0 0
\(161\) 0 0
\(162\) −1.90321 −0.149530
\(163\) 20.8573 1.63367 0.816834 0.576873i \(-0.195727\pi\)
0.816834 + 0.576873i \(0.195727\pi\)
\(164\) 13.3590 1.04317
\(165\) 0 0
\(166\) 22.1017 1.71543
\(167\) −15.3461 −1.18752 −0.593760 0.804642i \(-0.702357\pi\)
−0.593760 + 0.804642i \(0.702357\pi\)
\(168\) 0 0
\(169\) 28.3274 2.17903
\(170\) 0 0
\(171\) 2.42864 0.185723
\(172\) 16.3872 1.24951
\(173\) −2.06022 −0.156636 −0.0783179 0.996928i \(-0.524955\pi\)
−0.0783179 + 0.996928i \(0.524955\pi\)
\(174\) −1.43801 −0.109015
\(175\) 0 0
\(176\) −9.22570 −0.695413
\(177\) −14.1017 −1.05995
\(178\) 8.79706 0.659367
\(179\) 10.0000 0.747435 0.373718 0.927543i \(-0.378083\pi\)
0.373718 + 0.927543i \(0.378083\pi\)
\(180\) 0 0
\(181\) 12.1017 0.899513 0.449757 0.893151i \(-0.351511\pi\)
0.449757 + 0.893151i \(0.351511\pi\)
\(182\) 0 0
\(183\) −6.85728 −0.506905
\(184\) 0.990632 0.0730304
\(185\) 0 0
\(186\) 9.86665 0.723458
\(187\) −8.85728 −0.647708
\(188\) −4.47013 −0.326017
\(189\) 0 0
\(190\) 0 0
\(191\) 0.488863 0.0353729 0.0176864 0.999844i \(-0.494370\pi\)
0.0176864 + 0.999844i \(0.494370\pi\)
\(192\) −4.74620 −0.342528
\(193\) −22.9590 −1.65262 −0.826312 0.563212i \(-0.809565\pi\)
−0.826312 + 0.563212i \(0.809565\pi\)
\(194\) −22.7239 −1.63148
\(195\) 0 0
\(196\) 0 0
\(197\) −1.18421 −0.0843713 −0.0421857 0.999110i \(-0.513432\pi\)
−0.0421857 + 0.999110i \(0.513432\pi\)
\(198\) −3.80642 −0.270511
\(199\) −8.79706 −0.623607 −0.311803 0.950147i \(-0.600933\pi\)
−0.311803 + 0.950147i \(0.600933\pi\)
\(200\) 0 0
\(201\) 2.75557 0.194363
\(202\) 2.81579 0.198118
\(203\) 0 0
\(204\) −7.18421 −0.502995
\(205\) 0 0
\(206\) −16.8573 −1.17450
\(207\) 1.37778 0.0957626
\(208\) −29.6543 −2.05616
\(209\) 4.85728 0.335985
\(210\) 0 0
\(211\) 23.2257 1.59892 0.799461 0.600717i \(-0.205118\pi\)
0.799461 + 0.600717i \(0.205118\pi\)
\(212\) 14.8988 1.02325
\(213\) 2.00000 0.137038
\(214\) −3.35905 −0.229620
\(215\) 0 0
\(216\) 0.719004 0.0489220
\(217\) 0 0
\(218\) −10.6824 −0.723506
\(219\) 1.57136 0.106183
\(220\) 0 0
\(221\) −28.4701 −1.91511
\(222\) −14.4889 −0.972429
\(223\) 15.2257 1.01959 0.509794 0.860297i \(-0.329722\pi\)
0.509794 + 0.860297i \(0.329722\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 21.4795 1.42879
\(227\) 14.3684 0.953665 0.476833 0.878994i \(-0.341785\pi\)
0.476833 + 0.878994i \(0.341785\pi\)
\(228\) 3.93978 0.260918
\(229\) 5.61285 0.370907 0.185454 0.982653i \(-0.440625\pi\)
0.185454 + 0.982653i \(0.440625\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0.543257 0.0356666
\(233\) 23.2859 1.52551 0.762756 0.646687i \(-0.223846\pi\)
0.762756 + 0.646687i \(0.223846\pi\)
\(234\) −12.2351 −0.799831
\(235\) 0 0
\(236\) −22.8760 −1.48910
\(237\) −4.85728 −0.315514
\(238\) 0 0
\(239\) −8.48886 −0.549099 −0.274549 0.961573i \(-0.588529\pi\)
−0.274549 + 0.961573i \(0.588529\pi\)
\(240\) 0 0
\(241\) 7.24443 0.466655 0.233327 0.972398i \(-0.425039\pi\)
0.233327 + 0.972398i \(0.425039\pi\)
\(242\) 13.3225 0.856402
\(243\) 1.00000 0.0641500
\(244\) −11.1240 −0.712140
\(245\) 0 0
\(246\) −15.6731 −0.999278
\(247\) 15.6128 0.993422
\(248\) −3.72746 −0.236694
\(249\) −11.6128 −0.735934
\(250\) 0 0
\(251\) −27.6128 −1.74291 −0.871454 0.490478i \(-0.836822\pi\)
−0.871454 + 0.490478i \(0.836822\pi\)
\(252\) 0 0
\(253\) 2.75557 0.173241
\(254\) 24.4701 1.53539
\(255\) 0 0
\(256\) 20.2444 1.26528
\(257\) 0.428639 0.0267378 0.0133689 0.999911i \(-0.495744\pi\)
0.0133689 + 0.999911i \(0.495744\pi\)
\(258\) −19.2257 −1.19694
\(259\) 0 0
\(260\) 0 0
\(261\) 0.755569 0.0467685
\(262\) −4.00000 −0.247121
\(263\) −9.37778 −0.578259 −0.289129 0.957290i \(-0.593366\pi\)
−0.289129 + 0.957290i \(0.593366\pi\)
\(264\) 1.43801 0.0885032
\(265\) 0 0
\(266\) 0 0
\(267\) −4.62222 −0.282875
\(268\) 4.47013 0.273056
\(269\) −1.74620 −0.106468 −0.0532339 0.998582i \(-0.516953\pi\)
−0.0532339 + 0.998582i \(0.516953\pi\)
\(270\) 0 0
\(271\) 2.69535 0.163731 0.0818653 0.996643i \(-0.473912\pi\)
0.0818653 + 0.996643i \(0.473912\pi\)
\(272\) 20.4286 1.23867
\(273\) 0 0
\(274\) −30.3368 −1.83271
\(275\) 0 0
\(276\) 2.23506 0.134535
\(277\) −5.12399 −0.307870 −0.153935 0.988081i \(-0.549195\pi\)
−0.153935 + 0.988081i \(0.549195\pi\)
\(278\) 22.2163 1.33245
\(279\) −5.18421 −0.310370
\(280\) 0 0
\(281\) 23.9813 1.43060 0.715301 0.698816i \(-0.246290\pi\)
0.715301 + 0.698816i \(0.246290\pi\)
\(282\) 5.24443 0.312301
\(283\) −2.36842 −0.140788 −0.0703939 0.997519i \(-0.522426\pi\)
−0.0703939 + 0.997519i \(0.522426\pi\)
\(284\) 3.24443 0.192522
\(285\) 0 0
\(286\) −24.4701 −1.44695
\(287\) 0 0
\(288\) 7.34122 0.432585
\(289\) 2.61285 0.153697
\(290\) 0 0
\(291\) 11.9398 0.699922
\(292\) 2.54909 0.149174
\(293\) −8.42864 −0.492406 −0.246203 0.969218i \(-0.579183\pi\)
−0.246203 + 0.969218i \(0.579183\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 5.47367 0.318150
\(297\) 2.00000 0.116052
\(298\) 40.3970 2.34014
\(299\) 8.85728 0.512230
\(300\) 0 0
\(301\) 0 0
\(302\) −32.0830 −1.84617
\(303\) −1.47949 −0.0849947
\(304\) −11.2029 −0.642533
\(305\) 0 0
\(306\) 8.42864 0.481833
\(307\) 22.5718 1.28824 0.644121 0.764923i \(-0.277223\pi\)
0.644121 + 0.764923i \(0.277223\pi\)
\(308\) 0 0
\(309\) 8.85728 0.503873
\(310\) 0 0
\(311\) 24.0830 1.36562 0.682810 0.730596i \(-0.260758\pi\)
0.682810 + 0.730596i \(0.260758\pi\)
\(312\) 4.62222 0.261681
\(313\) −9.65433 −0.545695 −0.272848 0.962057i \(-0.587965\pi\)
−0.272848 + 0.962057i \(0.587965\pi\)
\(314\) −19.8479 −1.12008
\(315\) 0 0
\(316\) −7.87955 −0.443260
\(317\) −6.04149 −0.339324 −0.169662 0.985502i \(-0.554268\pi\)
−0.169662 + 0.985502i \(0.554268\pi\)
\(318\) −17.4795 −0.980201
\(319\) 1.51114 0.0846075
\(320\) 0 0
\(321\) 1.76494 0.0985092
\(322\) 0 0
\(323\) −10.7556 −0.598456
\(324\) 1.62222 0.0901231
\(325\) 0 0
\(326\) −39.6958 −2.19855
\(327\) 5.61285 0.310391
\(328\) 5.92104 0.326935
\(329\) 0 0
\(330\) 0 0
\(331\) 13.5111 0.742639 0.371320 0.928505i \(-0.378905\pi\)
0.371320 + 0.928505i \(0.378905\pi\)
\(332\) −18.8385 −1.03390
\(333\) 7.61285 0.417181
\(334\) 29.2070 1.59813
\(335\) 0 0
\(336\) 0 0
\(337\) 10.4889 0.571365 0.285682 0.958324i \(-0.407780\pi\)
0.285682 + 0.958324i \(0.407780\pi\)
\(338\) −53.9131 −2.93248
\(339\) −11.2859 −0.612967
\(340\) 0 0
\(341\) −10.3684 −0.561481
\(342\) −4.62222 −0.249941
\(343\) 0 0
\(344\) 7.26317 0.391604
\(345\) 0 0
\(346\) 3.92104 0.210796
\(347\) −16.7239 −0.897787 −0.448894 0.893585i \(-0.648182\pi\)
−0.448894 + 0.893585i \(0.648182\pi\)
\(348\) 1.22570 0.0657042
\(349\) 16.3684 0.876181 0.438091 0.898931i \(-0.355655\pi\)
0.438091 + 0.898931i \(0.355655\pi\)
\(350\) 0 0
\(351\) 6.42864 0.343135
\(352\) 14.6824 0.782577
\(353\) −0.549086 −0.0292249 −0.0146124 0.999893i \(-0.504651\pi\)
−0.0146124 + 0.999893i \(0.504651\pi\)
\(354\) 26.8385 1.42645
\(355\) 0 0
\(356\) −7.49823 −0.397405
\(357\) 0 0
\(358\) −19.0321 −1.00588
\(359\) −0.285442 −0.0150651 −0.00753253 0.999972i \(-0.502398\pi\)
−0.00753253 + 0.999972i \(0.502398\pi\)
\(360\) 0 0
\(361\) −13.1017 −0.689564
\(362\) −23.0321 −1.21054
\(363\) −7.00000 −0.367405
\(364\) 0 0
\(365\) 0 0
\(366\) 13.0509 0.682179
\(367\) −1.71456 −0.0894992 −0.0447496 0.998998i \(-0.514249\pi\)
−0.0447496 + 0.998998i \(0.514249\pi\)
\(368\) −6.35551 −0.331304
\(369\) 8.23506 0.428700
\(370\) 0 0
\(371\) 0 0
\(372\) −8.40990 −0.436033
\(373\) 16.0000 0.828449 0.414224 0.910175i \(-0.364053\pi\)
0.414224 + 0.910175i \(0.364053\pi\)
\(374\) 16.8573 0.871669
\(375\) 0 0
\(376\) −1.98126 −0.102176
\(377\) 4.85728 0.250163
\(378\) 0 0
\(379\) −4.85728 −0.249502 −0.124751 0.992188i \(-0.539813\pi\)
−0.124751 + 0.992188i \(0.539813\pi\)
\(380\) 0 0
\(381\) −12.8573 −0.658698
\(382\) −0.930409 −0.0476039
\(383\) 8.38715 0.428563 0.214282 0.976772i \(-0.431259\pi\)
0.214282 + 0.976772i \(0.431259\pi\)
\(384\) −5.64941 −0.288295
\(385\) 0 0
\(386\) 43.6958 2.22406
\(387\) 10.1017 0.513499
\(388\) 19.3689 0.983307
\(389\) 8.95899 0.454239 0.227119 0.973867i \(-0.427069\pi\)
0.227119 + 0.973867i \(0.427069\pi\)
\(390\) 0 0
\(391\) −6.10171 −0.308577
\(392\) 0 0
\(393\) 2.10171 0.106017
\(394\) 2.25380 0.113545
\(395\) 0 0
\(396\) 3.24443 0.163039
\(397\) 2.54909 0.127935 0.0639675 0.997952i \(-0.479625\pi\)
0.0639675 + 0.997952i \(0.479625\pi\)
\(398\) 16.7427 0.839234
\(399\) 0 0
\(400\) 0 0
\(401\) 0.958989 0.0478896 0.0239448 0.999713i \(-0.492377\pi\)
0.0239448 + 0.999713i \(0.492377\pi\)
\(402\) −5.24443 −0.261568
\(403\) −33.3274 −1.66016
\(404\) −2.40006 −0.119407
\(405\) 0 0
\(406\) 0 0
\(407\) 15.2257 0.754710
\(408\) −3.18421 −0.157642
\(409\) 31.9813 1.58137 0.790686 0.612222i \(-0.209724\pi\)
0.790686 + 0.612222i \(0.209724\pi\)
\(410\) 0 0
\(411\) 15.9398 0.786251
\(412\) 14.3684 0.707881
\(413\) 0 0
\(414\) −2.62222 −0.128875
\(415\) 0 0
\(416\) 47.1941 2.31388
\(417\) −11.6731 −0.571633
\(418\) −9.24443 −0.452160
\(419\) 0.470127 0.0229672 0.0114836 0.999934i \(-0.496345\pi\)
0.0114836 + 0.999934i \(0.496345\pi\)
\(420\) 0 0
\(421\) −33.6128 −1.63819 −0.819095 0.573658i \(-0.805524\pi\)
−0.819095 + 0.573658i \(0.805524\pi\)
\(422\) −44.2034 −2.15179
\(423\) −2.75557 −0.133980
\(424\) 6.60348 0.320693
\(425\) 0 0
\(426\) −3.80642 −0.184422
\(427\) 0 0
\(428\) 2.86311 0.138394
\(429\) 12.8573 0.620755
\(430\) 0 0
\(431\) 11.7146 0.564270 0.282135 0.959375i \(-0.408957\pi\)
0.282135 + 0.959375i \(0.408957\pi\)
\(432\) −4.61285 −0.221936
\(433\) 0.0602231 0.00289414 0.00144707 0.999999i \(-0.499539\pi\)
0.00144707 + 0.999999i \(0.499539\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 9.10525 0.436062
\(437\) 3.34614 0.160068
\(438\) −2.99063 −0.142898
\(439\) 22.4286 1.07046 0.535230 0.844706i \(-0.320225\pi\)
0.535230 + 0.844706i \(0.320225\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 54.1847 2.57730
\(443\) −23.9496 −1.13788 −0.568940 0.822379i \(-0.692647\pi\)
−0.568940 + 0.822379i \(0.692647\pi\)
\(444\) 12.3497 0.586090
\(445\) 0 0
\(446\) −28.9777 −1.37214
\(447\) −21.2257 −1.00394
\(448\) 0 0
\(449\) 29.4291 1.38885 0.694423 0.719567i \(-0.255660\pi\)
0.694423 + 0.719567i \(0.255660\pi\)
\(450\) 0 0
\(451\) 16.4701 0.775548
\(452\) −18.3082 −0.861145
\(453\) 16.8573 0.792024
\(454\) −27.3461 −1.28342
\(455\) 0 0
\(456\) 1.74620 0.0817733
\(457\) −3.14272 −0.147010 −0.0735051 0.997295i \(-0.523419\pi\)
−0.0735051 + 0.997295i \(0.523419\pi\)
\(458\) −10.6824 −0.499158
\(459\) −4.42864 −0.206711
\(460\) 0 0
\(461\) 3.37778 0.157319 0.0786596 0.996902i \(-0.474936\pi\)
0.0786596 + 0.996902i \(0.474936\pi\)
\(462\) 0 0
\(463\) 20.8573 0.969320 0.484660 0.874703i \(-0.338943\pi\)
0.484660 + 0.874703i \(0.338943\pi\)
\(464\) −3.48532 −0.161802
\(465\) 0 0
\(466\) −44.3180 −2.05299
\(467\) 14.3684 0.664891 0.332446 0.943122i \(-0.392126\pi\)
0.332446 + 0.943122i \(0.392126\pi\)
\(468\) 10.4286 0.482064
\(469\) 0 0
\(470\) 0 0
\(471\) 10.4286 0.480526
\(472\) −10.1392 −0.466694
\(473\) 20.2034 0.928954
\(474\) 9.24443 0.424611
\(475\) 0 0
\(476\) 0 0
\(477\) 9.18421 0.420516
\(478\) 16.1561 0.738963
\(479\) −6.36842 −0.290980 −0.145490 0.989360i \(-0.546476\pi\)
−0.145490 + 0.989360i \(0.546476\pi\)
\(480\) 0 0
\(481\) 48.9403 2.23148
\(482\) −13.7877 −0.628012
\(483\) 0 0
\(484\) −11.3555 −0.516160
\(485\) 0 0
\(486\) −1.90321 −0.0863314
\(487\) 17.3274 0.785180 0.392590 0.919714i \(-0.371579\pi\)
0.392590 + 0.919714i \(0.371579\pi\)
\(488\) −4.93041 −0.223189
\(489\) 20.8573 0.943199
\(490\) 0 0
\(491\) 2.00000 0.0902587 0.0451294 0.998981i \(-0.485630\pi\)
0.0451294 + 0.998981i \(0.485630\pi\)
\(492\) 13.3590 0.602272
\(493\) −3.34614 −0.150703
\(494\) −29.7146 −1.33692
\(495\) 0 0
\(496\) 23.9140 1.07377
\(497\) 0 0
\(498\) 22.1017 0.990401
\(499\) 23.3461 1.04512 0.522558 0.852603i \(-0.324978\pi\)
0.522558 + 0.852603i \(0.324978\pi\)
\(500\) 0 0
\(501\) −15.3461 −0.685615
\(502\) 52.5531 2.34556
\(503\) −0.387152 −0.0172623 −0.00863113 0.999963i \(-0.502747\pi\)
−0.00863113 + 0.999963i \(0.502747\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) −5.24443 −0.233143
\(507\) 28.3274 1.25806
\(508\) −20.8573 −0.925392
\(509\) 29.9496 1.32749 0.663747 0.747957i \(-0.268965\pi\)
0.663747 + 0.747957i \(0.268965\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) −27.2306 −1.20343
\(513\) 2.42864 0.107227
\(514\) −0.815792 −0.0359830
\(515\) 0 0
\(516\) 16.3872 0.721404
\(517\) −5.51114 −0.242380
\(518\) 0 0
\(519\) −2.06022 −0.0904338
\(520\) 0 0
\(521\) 18.5205 0.811398 0.405699 0.914007i \(-0.367028\pi\)
0.405699 + 0.914007i \(0.367028\pi\)
\(522\) −1.43801 −0.0629399
\(523\) 4.00000 0.174908 0.0874539 0.996169i \(-0.472127\pi\)
0.0874539 + 0.996169i \(0.472127\pi\)
\(524\) 3.40943 0.148942
\(525\) 0 0
\(526\) 17.8479 0.778206
\(527\) 22.9590 1.00011
\(528\) −9.22570 −0.401497
\(529\) −21.1017 −0.917466
\(530\) 0 0
\(531\) −14.1017 −0.611962
\(532\) 0 0
\(533\) 52.9403 2.29310
\(534\) 8.79706 0.380686
\(535\) 0 0
\(536\) 1.98126 0.0855776
\(537\) 10.0000 0.431532
\(538\) 3.32339 0.143282
\(539\) 0 0
\(540\) 0 0
\(541\) 14.5906 0.627298 0.313649 0.949539i \(-0.398449\pi\)
0.313649 + 0.949539i \(0.398449\pi\)
\(542\) −5.12981 −0.220344
\(543\) 12.1017 0.519334
\(544\) −32.5116 −1.39392
\(545\) 0 0
\(546\) 0 0
\(547\) −18.7556 −0.801930 −0.400965 0.916093i \(-0.631325\pi\)
−0.400965 + 0.916093i \(0.631325\pi\)
\(548\) 25.8578 1.10459
\(549\) −6.85728 −0.292662
\(550\) 0 0
\(551\) 1.83500 0.0781738
\(552\) 0.990632 0.0421641
\(553\) 0 0
\(554\) 9.75203 0.414324
\(555\) 0 0
\(556\) −18.9362 −0.803075
\(557\) 31.8765 1.35065 0.675325 0.737520i \(-0.264003\pi\)
0.675325 + 0.737520i \(0.264003\pi\)
\(558\) 9.86665 0.417688
\(559\) 64.9403 2.74668
\(560\) 0 0
\(561\) −8.85728 −0.373955
\(562\) −45.6414 −1.92527
\(563\) 2.01874 0.0850796 0.0425398 0.999095i \(-0.486455\pi\)
0.0425398 + 0.999095i \(0.486455\pi\)
\(564\) −4.47013 −0.188226
\(565\) 0 0
\(566\) 4.50760 0.189468
\(567\) 0 0
\(568\) 1.43801 0.0603375
\(569\) −28.9590 −1.21402 −0.607012 0.794693i \(-0.707632\pi\)
−0.607012 + 0.794693i \(0.707632\pi\)
\(570\) 0 0
\(571\) 8.97773 0.375706 0.187853 0.982197i \(-0.439847\pi\)
0.187853 + 0.982197i \(0.439847\pi\)
\(572\) 20.8573 0.872087
\(573\) 0.488863 0.0204225
\(574\) 0 0
\(575\) 0 0
\(576\) −4.74620 −0.197758
\(577\) −28.6766 −1.19382 −0.596911 0.802307i \(-0.703606\pi\)
−0.596911 + 0.802307i \(0.703606\pi\)
\(578\) −4.97280 −0.206841
\(579\) −22.9590 −0.954143
\(580\) 0 0
\(581\) 0 0
\(582\) −22.7239 −0.941937
\(583\) 18.3684 0.760742
\(584\) 1.12981 0.0467520
\(585\) 0 0
\(586\) 16.0415 0.662668
\(587\) −45.2070 −1.86589 −0.932945 0.360018i \(-0.882771\pi\)
−0.932945 + 0.360018i \(0.882771\pi\)
\(588\) 0 0
\(589\) −12.5906 −0.518786
\(590\) 0 0
\(591\) −1.18421 −0.0487118
\(592\) −35.1169 −1.44330
\(593\) 18.2636 0.749998 0.374999 0.927025i \(-0.377643\pi\)
0.374999 + 0.927025i \(0.377643\pi\)
\(594\) −3.80642 −0.156179
\(595\) 0 0
\(596\) −34.4327 −1.41042
\(597\) −8.79706 −0.360040
\(598\) −16.8573 −0.689345
\(599\) −22.7368 −0.929002 −0.464501 0.885573i \(-0.653766\pi\)
−0.464501 + 0.885573i \(0.653766\pi\)
\(600\) 0 0
\(601\) −0.488863 −0.0199411 −0.00997056 0.999950i \(-0.503174\pi\)
−0.00997056 + 0.999950i \(0.503174\pi\)
\(602\) 0 0
\(603\) 2.75557 0.112215
\(604\) 27.3461 1.11270
\(605\) 0 0
\(606\) 2.81579 0.114384
\(607\) −20.2034 −0.820032 −0.410016 0.912078i \(-0.634477\pi\)
−0.410016 + 0.912078i \(0.634477\pi\)
\(608\) 17.8292 0.723069
\(609\) 0 0
\(610\) 0 0
\(611\) −17.7146 −0.716654
\(612\) −7.18421 −0.290404
\(613\) −10.3684 −0.418776 −0.209388 0.977833i \(-0.567147\pi\)
−0.209388 + 0.977833i \(0.567147\pi\)
\(614\) −42.9590 −1.73368
\(615\) 0 0
\(616\) 0 0
\(617\) 39.2859 1.58159 0.790796 0.612080i \(-0.209667\pi\)
0.790796 + 0.612080i \(0.209667\pi\)
\(618\) −16.8573 −0.678099
\(619\) −42.8988 −1.72425 −0.862123 0.506698i \(-0.830866\pi\)
−0.862123 + 0.506698i \(0.830866\pi\)
\(620\) 0 0
\(621\) 1.37778 0.0552886
\(622\) −45.8350 −1.83782
\(623\) 0 0
\(624\) −29.6543 −1.18712
\(625\) 0 0
\(626\) 18.3742 0.734383
\(627\) 4.85728 0.193981
\(628\) 16.9175 0.675082
\(629\) −33.7146 −1.34429
\(630\) 0 0
\(631\) 15.3461 0.610920 0.305460 0.952205i \(-0.401190\pi\)
0.305460 + 0.952205i \(0.401190\pi\)
\(632\) −3.49240 −0.138920
\(633\) 23.2257 0.923139
\(634\) 11.4982 0.456653
\(635\) 0 0
\(636\) 14.8988 0.590775
\(637\) 0 0
\(638\) −2.87601 −0.113863
\(639\) 2.00000 0.0791188
\(640\) 0 0
\(641\) 30.6735 1.21153 0.605766 0.795643i \(-0.292867\pi\)
0.605766 + 0.795643i \(0.292867\pi\)
\(642\) −3.35905 −0.132571
\(643\) −49.0607 −1.93477 −0.967383 0.253320i \(-0.918477\pi\)
−0.967383 + 0.253320i \(0.918477\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 20.4701 0.805386
\(647\) −15.3461 −0.603319 −0.301660 0.953416i \(-0.597541\pi\)
−0.301660 + 0.953416i \(0.597541\pi\)
\(648\) 0.719004 0.0282451
\(649\) −28.2034 −1.10708
\(650\) 0 0
\(651\) 0 0
\(652\) 33.8350 1.32508
\(653\) 19.4697 0.761906 0.380953 0.924594i \(-0.375596\pi\)
0.380953 + 0.924594i \(0.375596\pi\)
\(654\) −10.6824 −0.417716
\(655\) 0 0
\(656\) −37.9871 −1.48315
\(657\) 1.57136 0.0613046
\(658\) 0 0
\(659\) 30.9403 1.20526 0.602631 0.798020i \(-0.294119\pi\)
0.602631 + 0.798020i \(0.294119\pi\)
\(660\) 0 0
\(661\) −47.7975 −1.85911 −0.929554 0.368685i \(-0.879808\pi\)
−0.929554 + 0.368685i \(0.879808\pi\)
\(662\) −25.7146 −0.999425
\(663\) −28.4701 −1.10569
\(664\) −8.34968 −0.324030
\(665\) 0 0
\(666\) −14.4889 −0.561432
\(667\) 1.04101 0.0403081
\(668\) −24.8948 −0.963207
\(669\) 15.2257 0.588659
\(670\) 0 0
\(671\) −13.7146 −0.529445
\(672\) 0 0
\(673\) −27.8163 −1.07224 −0.536119 0.844142i \(-0.680110\pi\)
−0.536119 + 0.844142i \(0.680110\pi\)
\(674\) −19.9625 −0.768928
\(675\) 0 0
\(676\) 45.9532 1.76743
\(677\) −19.0005 −0.730248 −0.365124 0.930959i \(-0.618973\pi\)
−0.365124 + 0.930959i \(0.618973\pi\)
\(678\) 21.4795 0.824915
\(679\) 0 0
\(680\) 0 0
\(681\) 14.3684 0.550599
\(682\) 19.7333 0.755627
\(683\) −4.52051 −0.172972 −0.0864862 0.996253i \(-0.527564\pi\)
−0.0864862 + 0.996253i \(0.527564\pi\)
\(684\) 3.93978 0.150641
\(685\) 0 0
\(686\) 0 0
\(687\) 5.61285 0.214143
\(688\) −46.5977 −1.77652
\(689\) 59.0420 2.24932
\(690\) 0 0
\(691\) 1.18421 0.0450494 0.0225247 0.999746i \(-0.492830\pi\)
0.0225247 + 0.999746i \(0.492830\pi\)
\(692\) −3.34213 −0.127049
\(693\) 0 0
\(694\) 31.8292 1.20822
\(695\) 0 0
\(696\) 0.543257 0.0205921
\(697\) −36.4701 −1.38140
\(698\) −31.1526 −1.17914
\(699\) 23.2859 0.880754
\(700\) 0 0
\(701\) −26.6735 −1.00745 −0.503723 0.863865i \(-0.668037\pi\)
−0.503723 + 0.863865i \(0.668037\pi\)
\(702\) −12.2351 −0.461783
\(703\) 18.4889 0.697321
\(704\) −9.49240 −0.357758
\(705\) 0 0
\(706\) 1.04503 0.0393301
\(707\) 0 0
\(708\) −22.8760 −0.859733
\(709\) −18.2034 −0.683644 −0.341822 0.939765i \(-0.611044\pi\)
−0.341822 + 0.939765i \(0.611044\pi\)
\(710\) 0 0
\(711\) −4.85728 −0.182162
\(712\) −3.32339 −0.124549
\(713\) −7.14272 −0.267497
\(714\) 0 0
\(715\) 0 0
\(716\) 16.2222 0.606250
\(717\) −8.48886 −0.317022
\(718\) 0.543257 0.0202742
\(719\) −4.85728 −0.181146 −0.0905730 0.995890i \(-0.528870\pi\)
−0.0905730 + 0.995890i \(0.528870\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 24.9353 0.927997
\(723\) 7.24443 0.269423
\(724\) 19.6316 0.729602
\(725\) 0 0
\(726\) 13.3225 0.494444
\(727\) 21.0607 0.781098 0.390549 0.920582i \(-0.372285\pi\)
0.390549 + 0.920582i \(0.372285\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −44.7368 −1.65465
\(732\) −11.1240 −0.411154
\(733\) 9.45091 0.349077 0.174539 0.984650i \(-0.444157\pi\)
0.174539 + 0.984650i \(0.444157\pi\)
\(734\) 3.26317 0.120446
\(735\) 0 0
\(736\) 10.1146 0.372830
\(737\) 5.51114 0.203005
\(738\) −15.6731 −0.576934
\(739\) −8.20342 −0.301768 −0.150884 0.988551i \(-0.548212\pi\)
−0.150884 + 0.988551i \(0.548212\pi\)
\(740\) 0 0
\(741\) 15.6128 0.573552
\(742\) 0 0
\(743\) −8.33677 −0.305847 −0.152923 0.988238i \(-0.548869\pi\)
−0.152923 + 0.988238i \(0.548869\pi\)
\(744\) −3.72746 −0.136655
\(745\) 0 0
\(746\) −30.4514 −1.11490
\(747\) −11.6128 −0.424892
\(748\) −14.3684 −0.525361
\(749\) 0 0
\(750\) 0 0
\(751\) −25.9180 −0.945760 −0.472880 0.881127i \(-0.656786\pi\)
−0.472880 + 0.881127i \(0.656786\pi\)
\(752\) 12.7110 0.463523
\(753\) −27.6128 −1.00627
\(754\) −9.24443 −0.336662
\(755\) 0 0
\(756\) 0 0
\(757\) −8.94025 −0.324939 −0.162470 0.986714i \(-0.551946\pi\)
−0.162470 + 0.986714i \(0.551946\pi\)
\(758\) 9.24443 0.335773
\(759\) 2.75557 0.100021
\(760\) 0 0
\(761\) 0.825636 0.0299293 0.0149646 0.999888i \(-0.495236\pi\)
0.0149646 + 0.999888i \(0.495236\pi\)
\(762\) 24.4701 0.886459
\(763\) 0 0
\(764\) 0.793040 0.0286912
\(765\) 0 0
\(766\) −15.9625 −0.576750
\(767\) −90.6548 −3.27336
\(768\) 20.2444 0.730508
\(769\) −21.2257 −0.765418 −0.382709 0.923869i \(-0.625009\pi\)
−0.382709 + 0.923869i \(0.625009\pi\)
\(770\) 0 0
\(771\) 0.428639 0.0154371
\(772\) −37.2444 −1.34046
\(773\) 29.4893 1.06066 0.530329 0.847792i \(-0.322068\pi\)
0.530329 + 0.847792i \(0.322068\pi\)
\(774\) −19.2257 −0.691053
\(775\) 0 0
\(776\) 8.58474 0.308174
\(777\) 0 0
\(778\) −17.0509 −0.611303
\(779\) 20.0000 0.716574
\(780\) 0 0
\(781\) 4.00000 0.143131
\(782\) 11.6128 0.415275
\(783\) 0.755569 0.0270018
\(784\) 0 0
\(785\) 0 0
\(786\) −4.00000 −0.142675
\(787\) −34.4514 −1.22806 −0.614030 0.789283i \(-0.710453\pi\)
−0.614030 + 0.789283i \(0.710453\pi\)
\(788\) −1.92104 −0.0684343
\(789\) −9.37778 −0.333858
\(790\) 0 0
\(791\) 0 0
\(792\) 1.43801 0.0510974
\(793\) −44.0830 −1.56543
\(794\) −4.85145 −0.172172
\(795\) 0 0
\(796\) −14.2707 −0.505812
\(797\) 18.9175 0.670092 0.335046 0.942202i \(-0.391248\pi\)
0.335046 + 0.942202i \(0.391248\pi\)
\(798\) 0 0
\(799\) 12.2034 0.431726
\(800\) 0 0
\(801\) −4.62222 −0.163318
\(802\) −1.82516 −0.0644486
\(803\) 3.14272 0.110904
\(804\) 4.47013 0.157649
\(805\) 0 0
\(806\) 63.4291 2.23420
\(807\) −1.74620 −0.0614692
\(808\) −1.06376 −0.0374230
\(809\) 21.2257 0.746256 0.373128 0.927780i \(-0.378285\pi\)
0.373128 + 0.927780i \(0.378285\pi\)
\(810\) 0 0
\(811\) 21.5081 0.755251 0.377625 0.925958i \(-0.376741\pi\)
0.377625 + 0.925958i \(0.376741\pi\)
\(812\) 0 0
\(813\) 2.69535 0.0945299
\(814\) −28.9777 −1.01567
\(815\) 0 0
\(816\) 20.4286 0.715145
\(817\) 24.5334 0.858315
\(818\) −60.8671 −2.12817
\(819\) 0 0
\(820\) 0 0
\(821\) −46.2034 −1.61251 −0.806255 0.591568i \(-0.798509\pi\)
−0.806255 + 0.591568i \(0.798509\pi\)
\(822\) −30.3368 −1.05812
\(823\) 17.8350 0.621689 0.310845 0.950461i \(-0.399388\pi\)
0.310845 + 0.950461i \(0.399388\pi\)
\(824\) 6.36842 0.221854
\(825\) 0 0
\(826\) 0 0
\(827\) −35.2128 −1.22447 −0.612234 0.790676i \(-0.709729\pi\)
−0.612234 + 0.790676i \(0.709729\pi\)
\(828\) 2.23506 0.0776738
\(829\) −14.3872 −0.499686 −0.249843 0.968286i \(-0.580379\pi\)
−0.249843 + 0.968286i \(0.580379\pi\)
\(830\) 0 0
\(831\) −5.12399 −0.177749
\(832\) −30.5116 −1.05780
\(833\) 0 0
\(834\) 22.2163 0.769289
\(835\) 0 0
\(836\) 7.87955 0.272520
\(837\) −5.18421 −0.179192
\(838\) −0.894751 −0.0309087
\(839\) −1.51114 −0.0521703 −0.0260851 0.999660i \(-0.508304\pi\)
−0.0260851 + 0.999660i \(0.508304\pi\)
\(840\) 0 0
\(841\) −28.4291 −0.980314
\(842\) 63.9724 2.20463
\(843\) 23.9813 0.825959
\(844\) 37.6771 1.29690
\(845\) 0 0
\(846\) 5.24443 0.180307
\(847\) 0 0
\(848\) −42.3654 −1.45483
\(849\) −2.36842 −0.0812838
\(850\) 0 0
\(851\) 10.4889 0.359554
\(852\) 3.24443 0.111152
\(853\) −15.4064 −0.527504 −0.263752 0.964591i \(-0.584960\pi\)
−0.263752 + 0.964591i \(0.584960\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 1.26900 0.0433734
\(857\) 19.8578 0.678328 0.339164 0.940727i \(-0.389856\pi\)
0.339164 + 0.940727i \(0.389856\pi\)
\(858\) −24.4701 −0.835396
\(859\) −2.42864 −0.0828641 −0.0414321 0.999141i \(-0.513192\pi\)
−0.0414321 + 0.999141i \(0.513192\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) −22.2953 −0.759380
\(863\) 39.2958 1.33764 0.668822 0.743423i \(-0.266799\pi\)
0.668822 + 0.743423i \(0.266799\pi\)
\(864\) 7.34122 0.249753
\(865\) 0 0
\(866\) −0.114617 −0.00389485
\(867\) 2.61285 0.0887370
\(868\) 0 0
\(869\) −9.71456 −0.329544
\(870\) 0 0
\(871\) 17.7146 0.600235
\(872\) 4.03566 0.136665
\(873\) 11.9398 0.404100
\(874\) −6.36842 −0.215415
\(875\) 0 0
\(876\) 2.54909 0.0861256
\(877\) 56.2864 1.90066 0.950328 0.311249i \(-0.100747\pi\)
0.950328 + 0.311249i \(0.100747\pi\)
\(878\) −42.6865 −1.44060
\(879\) −8.42864 −0.284291
\(880\) 0 0
\(881\) 2.33677 0.0787279 0.0393640 0.999225i \(-0.487467\pi\)
0.0393640 + 0.999225i \(0.487467\pi\)
\(882\) 0 0
\(883\) −33.7146 −1.13459 −0.567293 0.823516i \(-0.692009\pi\)
−0.567293 + 0.823516i \(0.692009\pi\)
\(884\) −46.1847 −1.55336
\(885\) 0 0
\(886\) 45.5812 1.53133
\(887\) 47.8992 1.60830 0.804150 0.594427i \(-0.202621\pi\)
0.804150 + 0.594427i \(0.202621\pi\)
\(888\) 5.47367 0.183684
\(889\) 0 0
\(890\) 0 0
\(891\) 2.00000 0.0670025
\(892\) 24.6994 0.826996
\(893\) −6.69228 −0.223949
\(894\) 40.3970 1.35108
\(895\) 0 0
\(896\) 0 0
\(897\) 8.85728 0.295736
\(898\) −56.0098 −1.86907
\(899\) −3.91703 −0.130640
\(900\) 0 0
\(901\) −40.6735 −1.35503
\(902\) −31.3461 −1.04371
\(903\) 0 0
\(904\) −8.11462 −0.269888
\(905\) 0 0
\(906\) −32.0830 −1.06589
\(907\) −23.7591 −0.788908 −0.394454 0.918916i \(-0.629066\pi\)
−0.394454 + 0.918916i \(0.629066\pi\)
\(908\) 23.3087 0.773525
\(909\) −1.47949 −0.0490717
\(910\) 0 0
\(911\) 22.9403 0.760045 0.380022 0.924977i \(-0.375916\pi\)
0.380022 + 0.924977i \(0.375916\pi\)
\(912\) −11.2029 −0.370967
\(913\) −23.2257 −0.768658
\(914\) 5.98126 0.197843
\(915\) 0 0
\(916\) 9.10525 0.300846
\(917\) 0 0
\(918\) 8.42864 0.278187
\(919\) 16.9777 0.560043 0.280022 0.959994i \(-0.409658\pi\)
0.280022 + 0.959994i \(0.409658\pi\)
\(920\) 0 0
\(921\) 22.5718 0.743767
\(922\) −6.42864 −0.211716
\(923\) 12.8573 0.423202
\(924\) 0 0
\(925\) 0 0
\(926\) −39.6958 −1.30449
\(927\) 8.85728 0.290911
\(928\) 5.54680 0.182082
\(929\) −39.3403 −1.29071 −0.645357 0.763881i \(-0.723291\pi\)
−0.645357 + 0.763881i \(0.723291\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 37.7748 1.23735
\(933\) 24.0830 0.788441
\(934\) −27.3461 −0.894793
\(935\) 0 0
\(936\) 4.62222 0.151082
\(937\) −17.7748 −0.580677 −0.290338 0.956924i \(-0.593768\pi\)
−0.290338 + 0.956924i \(0.593768\pi\)
\(938\) 0 0
\(939\) −9.65433 −0.315057
\(940\) 0 0
\(941\) −35.5812 −1.15991 −0.579957 0.814647i \(-0.696931\pi\)
−0.579957 + 0.814647i \(0.696931\pi\)
\(942\) −19.8479 −0.646680
\(943\) 11.3461 0.369481
\(944\) 65.0490 2.11717
\(945\) 0 0
\(946\) −38.4514 −1.25016
\(947\) −30.5018 −0.991174 −0.495587 0.868558i \(-0.665047\pi\)
−0.495587 + 0.868558i \(0.665047\pi\)
\(948\) −7.87955 −0.255916
\(949\) 10.1017 0.327915
\(950\) 0 0
\(951\) −6.04149 −0.195909
\(952\) 0 0
\(953\) 51.1655 1.65741 0.828706 0.559684i \(-0.189077\pi\)
0.828706 + 0.559684i \(0.189077\pi\)
\(954\) −17.4795 −0.565920
\(955\) 0 0
\(956\) −13.7708 −0.445378
\(957\) 1.51114 0.0488481
\(958\) 12.1204 0.391594
\(959\) 0 0
\(960\) 0 0
\(961\) −4.12399 −0.133032
\(962\) −93.1437 −3.00307
\(963\) 1.76494 0.0568743
\(964\) 11.7520 0.378507
\(965\) 0 0
\(966\) 0 0
\(967\) −47.8992 −1.54034 −0.770168 0.637841i \(-0.779828\pi\)
−0.770168 + 0.637841i \(0.779828\pi\)
\(968\) −5.03303 −0.161768
\(969\) −10.7556 −0.345519
\(970\) 0 0
\(971\) −40.6735 −1.30528 −0.652638 0.757670i \(-0.726338\pi\)
−0.652638 + 0.757670i \(0.726338\pi\)
\(972\) 1.62222 0.0520326
\(973\) 0 0
\(974\) −32.9777 −1.05667
\(975\) 0 0
\(976\) 31.6316 1.01250
\(977\) 27.4893 0.879462 0.439731 0.898130i \(-0.355074\pi\)
0.439731 + 0.898130i \(0.355074\pi\)
\(978\) −39.6958 −1.26933
\(979\) −9.24443 −0.295453
\(980\) 0 0
\(981\) 5.61285 0.179204
\(982\) −3.80642 −0.121468
\(983\) 24.0000 0.765481 0.382741 0.923856i \(-0.374980\pi\)
0.382741 + 0.923856i \(0.374980\pi\)
\(984\) 5.92104 0.188756
\(985\) 0 0
\(986\) 6.36842 0.202812
\(987\) 0 0
\(988\) 25.3274 0.805772
\(989\) 13.9180 0.442566
\(990\) 0 0
\(991\) −34.6923 −1.10204 −0.551018 0.834493i \(-0.685761\pi\)
−0.551018 + 0.834493i \(0.685761\pi\)
\(992\) −38.0584 −1.20836
\(993\) 13.5111 0.428763
\(994\) 0 0
\(995\) 0 0
\(996\) −18.8385 −0.596922
\(997\) −28.6766 −0.908197 −0.454099 0.890951i \(-0.650039\pi\)
−0.454099 + 0.890951i \(0.650039\pi\)
\(998\) −44.4327 −1.40649
\(999\) 7.61285 0.240860
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 3675.2.a.bj.1.1 3
5.2 odd 4 735.2.d.b.589.2 6
5.3 odd 4 735.2.d.b.589.5 6
5.4 even 2 3675.2.a.bi.1.3 3
7.6 odd 2 525.2.a.k.1.1 3
15.2 even 4 2205.2.d.l.1324.5 6
15.8 even 4 2205.2.d.l.1324.2 6
21.20 even 2 1575.2.a.w.1.3 3
28.27 even 2 8400.2.a.dj.1.1 3
35.2 odd 12 735.2.q.f.214.2 12
35.3 even 12 735.2.q.e.79.2 12
35.12 even 12 735.2.q.e.214.2 12
35.13 even 4 105.2.d.b.64.5 yes 6
35.17 even 12 735.2.q.e.79.5 12
35.18 odd 12 735.2.q.f.79.2 12
35.23 odd 12 735.2.q.f.214.5 12
35.27 even 4 105.2.d.b.64.2 6
35.32 odd 12 735.2.q.f.79.5 12
35.33 even 12 735.2.q.e.214.5 12
35.34 odd 2 525.2.a.j.1.3 3
105.62 odd 4 315.2.d.e.64.5 6
105.83 odd 4 315.2.d.e.64.2 6
105.104 even 2 1575.2.a.x.1.1 3
140.27 odd 4 1680.2.t.k.1009.2 6
140.83 odd 4 1680.2.t.k.1009.5 6
140.139 even 2 8400.2.a.dg.1.3 3
420.83 even 4 5040.2.t.v.1009.3 6
420.167 even 4 5040.2.t.v.1009.4 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
105.2.d.b.64.2 6 35.27 even 4
105.2.d.b.64.5 yes 6 35.13 even 4
315.2.d.e.64.2 6 105.83 odd 4
315.2.d.e.64.5 6 105.62 odd 4
525.2.a.j.1.3 3 35.34 odd 2
525.2.a.k.1.1 3 7.6 odd 2
735.2.d.b.589.2 6 5.2 odd 4
735.2.d.b.589.5 6 5.3 odd 4
735.2.q.e.79.2 12 35.3 even 12
735.2.q.e.79.5 12 35.17 even 12
735.2.q.e.214.2 12 35.12 even 12
735.2.q.e.214.5 12 35.33 even 12
735.2.q.f.79.2 12 35.18 odd 12
735.2.q.f.79.5 12 35.32 odd 12
735.2.q.f.214.2 12 35.2 odd 12
735.2.q.f.214.5 12 35.23 odd 12
1575.2.a.w.1.3 3 21.20 even 2
1575.2.a.x.1.1 3 105.104 even 2
1680.2.t.k.1009.2 6 140.27 odd 4
1680.2.t.k.1009.5 6 140.83 odd 4
2205.2.d.l.1324.2 6 15.8 even 4
2205.2.d.l.1324.5 6 15.2 even 4
3675.2.a.bi.1.3 3 5.4 even 2
3675.2.a.bj.1.1 3 1.1 even 1 trivial
5040.2.t.v.1009.3 6 420.83 even 4
5040.2.t.v.1009.4 6 420.167 even 4
8400.2.a.dg.1.3 3 140.139 even 2
8400.2.a.dj.1.1 3 28.27 even 2