Properties

Label 825.2.a.k.1.3
Level $825$
Weight $2$
Character 825.1
Self dual yes
Analytic conductor $6.588$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 825 = 3 \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 825.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(6.58765816676\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.148.1
Defining polynomial: \(x^{3} - x^{2} - 3 x + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 165)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(2.17009\) of defining polynomial
Character \(\chi\) \(=\) 825.1

$q$-expansion

\(f(q)\) \(=\) \(q+2.70928 q^{2} -1.00000 q^{3} +5.34017 q^{4} -2.70928 q^{6} -1.07838 q^{7} +9.04945 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+2.70928 q^{2} -1.00000 q^{3} +5.34017 q^{4} -2.70928 q^{6} -1.07838 q^{7} +9.04945 q^{8} +1.00000 q^{9} +1.00000 q^{11} -5.34017 q^{12} +4.34017 q^{13} -2.92162 q^{14} +13.8371 q^{16} -7.75872 q^{17} +2.70928 q^{18} +5.26180 q^{19} +1.07838 q^{21} +2.70928 q^{22} +2.15676 q^{23} -9.04945 q^{24} +11.7587 q^{26} -1.00000 q^{27} -5.75872 q^{28} +1.41855 q^{29} -4.68035 q^{31} +19.3896 q^{32} -1.00000 q^{33} -21.0205 q^{34} +5.34017 q^{36} +2.00000 q^{37} +14.2557 q^{38} -4.34017 q^{39} -9.41855 q^{41} +2.92162 q^{42} -7.60197 q^{43} +5.34017 q^{44} +5.84324 q^{46} -4.68035 q^{47} -13.8371 q^{48} -5.83710 q^{49} +7.75872 q^{51} +23.1773 q^{52} -0.156755 q^{53} -2.70928 q^{54} -9.75872 q^{56} -5.26180 q^{57} +3.84324 q^{58} +6.15676 q^{59} -4.15676 q^{61} -12.6803 q^{62} -1.07838 q^{63} +24.8576 q^{64} -2.70928 q^{66} +8.68035 q^{67} -41.4329 q^{68} -2.15676 q^{69} -4.68035 q^{71} +9.04945 q^{72} +10.4969 q^{73} +5.41855 q^{74} +28.0989 q^{76} -1.07838 q^{77} -11.7587 q^{78} -8.09890 q^{79} +1.00000 q^{81} -25.5174 q^{82} +11.0205 q^{83} +5.75872 q^{84} -20.5958 q^{86} -1.41855 q^{87} +9.04945 q^{88} -12.8371 q^{89} -4.68035 q^{91} +11.5174 q^{92} +4.68035 q^{93} -12.6803 q^{94} -19.3896 q^{96} -14.6803 q^{97} -15.8143 q^{98} +1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q + q^{2} - 3q^{3} + 5q^{4} - q^{6} + 9q^{8} + 3q^{9} + O(q^{10}) \) \( 3q + q^{2} - 3q^{3} + 5q^{4} - q^{6} + 9q^{8} + 3q^{9} + 3q^{11} - 5q^{12} + 2q^{13} - 12q^{14} + 13q^{16} + 2q^{17} + q^{18} + 8q^{19} + q^{22} - 9q^{24} + 10q^{26} - 3q^{27} + 8q^{28} - 10q^{29} + 8q^{31} + 29q^{32} - 3q^{33} - 30q^{34} + 5q^{36} + 6q^{37} - 2q^{39} - 14q^{41} + 12q^{42} - 4q^{43} + 5q^{44} + 24q^{46} + 8q^{47} - 13q^{48} + 11q^{49} - 2q^{51} + 30q^{52} + 6q^{53} - q^{54} - 4q^{56} - 8q^{57} + 18q^{58} + 12q^{59} - 6q^{61} - 16q^{62} + 13q^{64} - q^{66} + 4q^{67} - 42q^{68} + 8q^{71} + 9q^{72} + 14q^{73} + 2q^{74} + 48q^{76} - 10q^{78} + 12q^{79} + 3q^{81} - 26q^{82} - 8q^{84} - 8q^{86} + 10q^{87} + 9q^{88} - 10q^{89} + 8q^{91} - 16q^{92} - 8q^{93} - 16q^{94} - 29q^{96} - 22q^{97} - 39q^{98} + 3q^{99} + O(q^{100}) \)

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.70928 1.91575 0.957873 0.287190i \(-0.0927213\pi\)
0.957873 + 0.287190i \(0.0927213\pi\)
\(3\) −1.00000 −0.577350
\(4\) 5.34017 2.67009
\(5\) 0 0
\(6\) −2.70928 −1.10606
\(7\) −1.07838 −0.407588 −0.203794 0.979014i \(-0.565327\pi\)
−0.203794 + 0.979014i \(0.565327\pi\)
\(8\) 9.04945 3.19946
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 1.00000 0.301511
\(12\) −5.34017 −1.54158
\(13\) 4.34017 1.20375 0.601874 0.798591i \(-0.294421\pi\)
0.601874 + 0.798591i \(0.294421\pi\)
\(14\) −2.92162 −0.780836
\(15\) 0 0
\(16\) 13.8371 3.45928
\(17\) −7.75872 −1.88177 −0.940883 0.338730i \(-0.890003\pi\)
−0.940883 + 0.338730i \(0.890003\pi\)
\(18\) 2.70928 0.638582
\(19\) 5.26180 1.20714 0.603569 0.797311i \(-0.293745\pi\)
0.603569 + 0.797311i \(0.293745\pi\)
\(20\) 0 0
\(21\) 1.07838 0.235321
\(22\) 2.70928 0.577619
\(23\) 2.15676 0.449715 0.224857 0.974392i \(-0.427808\pi\)
0.224857 + 0.974392i \(0.427808\pi\)
\(24\) −9.04945 −1.84721
\(25\) 0 0
\(26\) 11.7587 2.30608
\(27\) −1.00000 −0.192450
\(28\) −5.75872 −1.08830
\(29\) 1.41855 0.263418 0.131709 0.991288i \(-0.457954\pi\)
0.131709 + 0.991288i \(0.457954\pi\)
\(30\) 0 0
\(31\) −4.68035 −0.840615 −0.420307 0.907382i \(-0.638078\pi\)
−0.420307 + 0.907382i \(0.638078\pi\)
\(32\) 19.3896 3.42763
\(33\) −1.00000 −0.174078
\(34\) −21.0205 −3.60499
\(35\) 0 0
\(36\) 5.34017 0.890029
\(37\) 2.00000 0.328798 0.164399 0.986394i \(-0.447432\pi\)
0.164399 + 0.986394i \(0.447432\pi\)
\(38\) 14.2557 2.31257
\(39\) −4.34017 −0.694984
\(40\) 0 0
\(41\) −9.41855 −1.47093 −0.735465 0.677562i \(-0.763036\pi\)
−0.735465 + 0.677562i \(0.763036\pi\)
\(42\) 2.92162 0.450816
\(43\) −7.60197 −1.15929 −0.579645 0.814869i \(-0.696809\pi\)
−0.579645 + 0.814869i \(0.696809\pi\)
\(44\) 5.34017 0.805061
\(45\) 0 0
\(46\) 5.84324 0.861539
\(47\) −4.68035 −0.682699 −0.341349 0.939937i \(-0.610884\pi\)
−0.341349 + 0.939937i \(0.610884\pi\)
\(48\) −13.8371 −1.99721
\(49\) −5.83710 −0.833872
\(50\) 0 0
\(51\) 7.75872 1.08644
\(52\) 23.1773 3.21411
\(53\) −0.156755 −0.0215320 −0.0107660 0.999942i \(-0.503427\pi\)
−0.0107660 + 0.999942i \(0.503427\pi\)
\(54\) −2.70928 −0.368686
\(55\) 0 0
\(56\) −9.75872 −1.30406
\(57\) −5.26180 −0.696942
\(58\) 3.84324 0.504643
\(59\) 6.15676 0.801541 0.400771 0.916178i \(-0.368742\pi\)
0.400771 + 0.916178i \(0.368742\pi\)
\(60\) 0 0
\(61\) −4.15676 −0.532218 −0.266109 0.963943i \(-0.585738\pi\)
−0.266109 + 0.963943i \(0.585738\pi\)
\(62\) −12.6803 −1.61041
\(63\) −1.07838 −0.135863
\(64\) 24.8576 3.10720
\(65\) 0 0
\(66\) −2.70928 −0.333489
\(67\) 8.68035 1.06047 0.530237 0.847850i \(-0.322103\pi\)
0.530237 + 0.847850i \(0.322103\pi\)
\(68\) −41.4329 −5.02448
\(69\) −2.15676 −0.259643
\(70\) 0 0
\(71\) −4.68035 −0.555455 −0.277727 0.960660i \(-0.589581\pi\)
−0.277727 + 0.960660i \(0.589581\pi\)
\(72\) 9.04945 1.06649
\(73\) 10.4969 1.22857 0.614286 0.789083i \(-0.289444\pi\)
0.614286 + 0.789083i \(0.289444\pi\)
\(74\) 5.41855 0.629894
\(75\) 0 0
\(76\) 28.0989 3.22316
\(77\) −1.07838 −0.122893
\(78\) −11.7587 −1.33141
\(79\) −8.09890 −0.911197 −0.455599 0.890185i \(-0.650575\pi\)
−0.455599 + 0.890185i \(0.650575\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) −25.5174 −2.81793
\(83\) 11.0205 1.20966 0.604830 0.796355i \(-0.293241\pi\)
0.604830 + 0.796355i \(0.293241\pi\)
\(84\) 5.75872 0.628328
\(85\) 0 0
\(86\) −20.5958 −2.22090
\(87\) −1.41855 −0.152085
\(88\) 9.04945 0.964674
\(89\) −12.8371 −1.36073 −0.680365 0.732873i \(-0.738179\pi\)
−0.680365 + 0.732873i \(0.738179\pi\)
\(90\) 0 0
\(91\) −4.68035 −0.490634
\(92\) 11.5174 1.20078
\(93\) 4.68035 0.485329
\(94\) −12.6803 −1.30788
\(95\) 0 0
\(96\) −19.3896 −1.97894
\(97\) −14.6803 −1.49056 −0.745282 0.666750i \(-0.767685\pi\)
−0.745282 + 0.666750i \(0.767685\pi\)
\(98\) −15.8143 −1.59749
\(99\) 1.00000 0.100504
\(100\) 0 0
\(101\) −15.5753 −1.54980 −0.774900 0.632083i \(-0.782200\pi\)
−0.774900 + 0.632083i \(0.782200\pi\)
\(102\) 21.0205 2.08134
\(103\) −6.83710 −0.673680 −0.336840 0.941562i \(-0.609358\pi\)
−0.336840 + 0.941562i \(0.609358\pi\)
\(104\) 39.2762 3.85135
\(105\) 0 0
\(106\) −0.424694 −0.0412499
\(107\) −6.34017 −0.612928 −0.306464 0.951882i \(-0.599146\pi\)
−0.306464 + 0.951882i \(0.599146\pi\)
\(108\) −5.34017 −0.513858
\(109\) 2.31351 0.221594 0.110797 0.993843i \(-0.464660\pi\)
0.110797 + 0.993843i \(0.464660\pi\)
\(110\) 0 0
\(111\) −2.00000 −0.189832
\(112\) −14.9216 −1.40996
\(113\) 6.00000 0.564433 0.282216 0.959351i \(-0.408930\pi\)
0.282216 + 0.959351i \(0.408930\pi\)
\(114\) −14.2557 −1.33516
\(115\) 0 0
\(116\) 7.57531 0.703350
\(117\) 4.34017 0.401249
\(118\) 16.6803 1.53555
\(119\) 8.36683 0.766987
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) −11.2618 −1.01960
\(123\) 9.41855 0.849242
\(124\) −24.9939 −2.24451
\(125\) 0 0
\(126\) −2.92162 −0.260279
\(127\) 2.24128 0.198881 0.0994406 0.995044i \(-0.468295\pi\)
0.0994406 + 0.995044i \(0.468295\pi\)
\(128\) 28.5669 2.52498
\(129\) 7.60197 0.669316
\(130\) 0 0
\(131\) 8.68035 0.758405 0.379203 0.925314i \(-0.376198\pi\)
0.379203 + 0.925314i \(0.376198\pi\)
\(132\) −5.34017 −0.464802
\(133\) −5.67420 −0.492016
\(134\) 23.5174 2.03160
\(135\) 0 0
\(136\) −70.2122 −6.02064
\(137\) 15.3607 1.31235 0.656176 0.754608i \(-0.272173\pi\)
0.656176 + 0.754608i \(0.272173\pi\)
\(138\) −5.84324 −0.497410
\(139\) 8.58145 0.727869 0.363935 0.931425i \(-0.381433\pi\)
0.363935 + 0.931425i \(0.381433\pi\)
\(140\) 0 0
\(141\) 4.68035 0.394156
\(142\) −12.6803 −1.06411
\(143\) 4.34017 0.362943
\(144\) 13.8371 1.15309
\(145\) 0 0
\(146\) 28.4391 2.35363
\(147\) 5.83710 0.481436
\(148\) 10.6803 0.877919
\(149\) −18.0989 −1.48272 −0.741360 0.671108i \(-0.765819\pi\)
−0.741360 + 0.671108i \(0.765819\pi\)
\(150\) 0 0
\(151\) 22.9360 1.86651 0.933253 0.359221i \(-0.116958\pi\)
0.933253 + 0.359221i \(0.116958\pi\)
\(152\) 47.6163 3.86220
\(153\) −7.75872 −0.627256
\(154\) −2.92162 −0.235431
\(155\) 0 0
\(156\) −23.1773 −1.85567
\(157\) 10.9939 0.877405 0.438703 0.898632i \(-0.355438\pi\)
0.438703 + 0.898632i \(0.355438\pi\)
\(158\) −21.9421 −1.74562
\(159\) 0.156755 0.0124315
\(160\) 0 0
\(161\) −2.32580 −0.183298
\(162\) 2.70928 0.212861
\(163\) 6.52359 0.510967 0.255484 0.966813i \(-0.417765\pi\)
0.255484 + 0.966813i \(0.417765\pi\)
\(164\) −50.2967 −3.92751
\(165\) 0 0
\(166\) 29.8576 2.31740
\(167\) −1.97334 −0.152701 −0.0763507 0.997081i \(-0.524327\pi\)
−0.0763507 + 0.997081i \(0.524327\pi\)
\(168\) 9.75872 0.752902
\(169\) 5.83710 0.449008
\(170\) 0 0
\(171\) 5.26180 0.402380
\(172\) −40.5958 −3.09540
\(173\) −3.75872 −0.285770 −0.142885 0.989739i \(-0.545638\pi\)
−0.142885 + 0.989739i \(0.545638\pi\)
\(174\) −3.84324 −0.291356
\(175\) 0 0
\(176\) 13.8371 1.04301
\(177\) −6.15676 −0.462770
\(178\) −34.7792 −2.60681
\(179\) 15.1506 1.13241 0.566205 0.824264i \(-0.308411\pi\)
0.566205 + 0.824264i \(0.308411\pi\)
\(180\) 0 0
\(181\) 4.83710 0.359539 0.179769 0.983709i \(-0.442465\pi\)
0.179769 + 0.983709i \(0.442465\pi\)
\(182\) −12.6803 −0.939930
\(183\) 4.15676 0.307276
\(184\) 19.5174 1.43885
\(185\) 0 0
\(186\) 12.6803 0.929768
\(187\) −7.75872 −0.567374
\(188\) −24.9939 −1.82286
\(189\) 1.07838 0.0784404
\(190\) 0 0
\(191\) 2.52359 0.182601 0.0913003 0.995823i \(-0.470898\pi\)
0.0913003 + 0.995823i \(0.470898\pi\)
\(192\) −24.8576 −1.79394
\(193\) −0.0266620 −0.00191917 −0.000959586 1.00000i \(-0.500305\pi\)
−0.000959586 1.00000i \(0.500305\pi\)
\(194\) −39.7731 −2.85554
\(195\) 0 0
\(196\) −31.1711 −2.22651
\(197\) −21.1194 −1.50470 −0.752348 0.658766i \(-0.771079\pi\)
−0.752348 + 0.658766i \(0.771079\pi\)
\(198\) 2.70928 0.192540
\(199\) 10.5236 0.745998 0.372999 0.927832i \(-0.378330\pi\)
0.372999 + 0.927832i \(0.378330\pi\)
\(200\) 0 0
\(201\) −8.68035 −0.612264
\(202\) −42.1978 −2.96903
\(203\) −1.52973 −0.107366
\(204\) 41.4329 2.90089
\(205\) 0 0
\(206\) −18.5236 −1.29060
\(207\) 2.15676 0.149905
\(208\) 60.0554 4.16409
\(209\) 5.26180 0.363966
\(210\) 0 0
\(211\) 9.57531 0.659191 0.329596 0.944122i \(-0.393088\pi\)
0.329596 + 0.944122i \(0.393088\pi\)
\(212\) −0.837101 −0.0574924
\(213\) 4.68035 0.320692
\(214\) −17.1773 −1.17421
\(215\) 0 0
\(216\) −9.04945 −0.615737
\(217\) 5.04718 0.342625
\(218\) 6.26794 0.424518
\(219\) −10.4969 −0.709317
\(220\) 0 0
\(221\) −33.6742 −2.26517
\(222\) −5.41855 −0.363669
\(223\) 2.15676 0.144427 0.0722135 0.997389i \(-0.476994\pi\)
0.0722135 + 0.997389i \(0.476994\pi\)
\(224\) −20.9093 −1.39706
\(225\) 0 0
\(226\) 16.2557 1.08131
\(227\) −9.65983 −0.641145 −0.320573 0.947224i \(-0.603875\pi\)
−0.320573 + 0.947224i \(0.603875\pi\)
\(228\) −28.0989 −1.86089
\(229\) −3.36069 −0.222081 −0.111040 0.993816i \(-0.535418\pi\)
−0.111040 + 0.993816i \(0.535418\pi\)
\(230\) 0 0
\(231\) 1.07838 0.0709520
\(232\) 12.8371 0.842797
\(233\) 2.39803 0.157100 0.0785501 0.996910i \(-0.474971\pi\)
0.0785501 + 0.996910i \(0.474971\pi\)
\(234\) 11.7587 0.768692
\(235\) 0 0
\(236\) 32.8781 2.14018
\(237\) 8.09890 0.526080
\(238\) 22.6681 1.46935
\(239\) −7.20394 −0.465984 −0.232992 0.972479i \(-0.574852\pi\)
−0.232992 + 0.972479i \(0.574852\pi\)
\(240\) 0 0
\(241\) −5.20394 −0.335215 −0.167608 0.985854i \(-0.553604\pi\)
−0.167608 + 0.985854i \(0.553604\pi\)
\(242\) 2.70928 0.174159
\(243\) −1.00000 −0.0641500
\(244\) −22.1978 −1.42107
\(245\) 0 0
\(246\) 25.5174 1.62693
\(247\) 22.8371 1.45309
\(248\) −42.3545 −2.68952
\(249\) −11.0205 −0.698397
\(250\) 0 0
\(251\) 15.3197 0.966968 0.483484 0.875353i \(-0.339371\pi\)
0.483484 + 0.875353i \(0.339371\pi\)
\(252\) −5.75872 −0.362765
\(253\) 2.15676 0.135594
\(254\) 6.07223 0.381006
\(255\) 0 0
\(256\) 27.6803 1.73002
\(257\) −4.15676 −0.259291 −0.129646 0.991560i \(-0.541384\pi\)
−0.129646 + 0.991560i \(0.541384\pi\)
\(258\) 20.5958 1.28224
\(259\) −2.15676 −0.134014
\(260\) 0 0
\(261\) 1.41855 0.0878061
\(262\) 23.5174 1.45291
\(263\) 18.7070 1.15352 0.576762 0.816912i \(-0.304316\pi\)
0.576762 + 0.816912i \(0.304316\pi\)
\(264\) −9.04945 −0.556955
\(265\) 0 0
\(266\) −15.3730 −0.942578
\(267\) 12.8371 0.785618
\(268\) 46.3545 2.83155
\(269\) 23.3607 1.42433 0.712163 0.702014i \(-0.247716\pi\)
0.712163 + 0.702014i \(0.247716\pi\)
\(270\) 0 0
\(271\) −5.57531 −0.338676 −0.169338 0.985558i \(-0.554163\pi\)
−0.169338 + 0.985558i \(0.554163\pi\)
\(272\) −107.358 −6.50955
\(273\) 4.68035 0.283267
\(274\) 41.6163 2.51414
\(275\) 0 0
\(276\) −11.5174 −0.693269
\(277\) 26.0144 1.56305 0.781526 0.623872i \(-0.214442\pi\)
0.781526 + 0.623872i \(0.214442\pi\)
\(278\) 23.2495 1.39441
\(279\) −4.68035 −0.280205
\(280\) 0 0
\(281\) −9.41855 −0.561864 −0.280932 0.959728i \(-0.590643\pi\)
−0.280932 + 0.959728i \(0.590643\pi\)
\(282\) 12.6803 0.755104
\(283\) −14.2413 −0.846556 −0.423278 0.906000i \(-0.639121\pi\)
−0.423278 + 0.906000i \(0.639121\pi\)
\(284\) −24.9939 −1.48311
\(285\) 0 0
\(286\) 11.7587 0.695308
\(287\) 10.1568 0.599534
\(288\) 19.3896 1.14254
\(289\) 43.1978 2.54105
\(290\) 0 0
\(291\) 14.6803 0.860577
\(292\) 56.0554 3.28039
\(293\) 15.7587 0.920634 0.460317 0.887754i \(-0.347736\pi\)
0.460317 + 0.887754i \(0.347736\pi\)
\(294\) 15.8143 0.922310
\(295\) 0 0
\(296\) 18.0989 1.05198
\(297\) −1.00000 −0.0580259
\(298\) −49.0349 −2.84052
\(299\) 9.36069 0.541343
\(300\) 0 0
\(301\) 8.19779 0.472513
\(302\) 62.1399 3.57575
\(303\) 15.5753 0.894778
\(304\) 72.8080 4.17582
\(305\) 0 0
\(306\) −21.0205 −1.20166
\(307\) 18.9216 1.07991 0.539957 0.841693i \(-0.318440\pi\)
0.539957 + 0.841693i \(0.318440\pi\)
\(308\) −5.75872 −0.328134
\(309\) 6.83710 0.388949
\(310\) 0 0
\(311\) −20.8781 −1.18389 −0.591945 0.805978i \(-0.701640\pi\)
−0.591945 + 0.805978i \(0.701640\pi\)
\(312\) −39.2762 −2.22358
\(313\) −6.31351 −0.356861 −0.178430 0.983953i \(-0.557102\pi\)
−0.178430 + 0.983953i \(0.557102\pi\)
\(314\) 29.7854 1.68089
\(315\) 0 0
\(316\) −43.2495 −2.43297
\(317\) −31.3607 −1.76139 −0.880696 0.473682i \(-0.842925\pi\)
−0.880696 + 0.473682i \(0.842925\pi\)
\(318\) 0.424694 0.0238156
\(319\) 1.41855 0.0794236
\(320\) 0 0
\(321\) 6.34017 0.353874
\(322\) −6.30122 −0.351154
\(323\) −40.8248 −2.27155
\(324\) 5.34017 0.296676
\(325\) 0 0
\(326\) 17.6742 0.978884
\(327\) −2.31351 −0.127937
\(328\) −85.2327 −4.70619
\(329\) 5.04718 0.278260
\(330\) 0 0
\(331\) 19.2039 1.05554 0.527772 0.849386i \(-0.323028\pi\)
0.527772 + 0.849386i \(0.323028\pi\)
\(332\) 58.8515 3.22989
\(333\) 2.00000 0.109599
\(334\) −5.34632 −0.292537
\(335\) 0 0
\(336\) 14.9216 0.814041
\(337\) −13.5031 −0.735559 −0.367780 0.929913i \(-0.619882\pi\)
−0.367780 + 0.929913i \(0.619882\pi\)
\(338\) 15.8143 0.860185
\(339\) −6.00000 −0.325875
\(340\) 0 0
\(341\) −4.68035 −0.253455
\(342\) 14.2557 0.770857
\(343\) 13.8432 0.747465
\(344\) −68.7936 −3.70910
\(345\) 0 0
\(346\) −10.1834 −0.547464
\(347\) −6.34017 −0.340358 −0.170179 0.985413i \(-0.554435\pi\)
−0.170179 + 0.985413i \(0.554435\pi\)
\(348\) −7.57531 −0.406079
\(349\) 16.1568 0.864851 0.432426 0.901670i \(-0.357658\pi\)
0.432426 + 0.901670i \(0.357658\pi\)
\(350\) 0 0
\(351\) −4.34017 −0.231661
\(352\) 19.3896 1.03347
\(353\) 13.2039 0.702775 0.351387 0.936230i \(-0.385710\pi\)
0.351387 + 0.936230i \(0.385710\pi\)
\(354\) −16.6803 −0.886550
\(355\) 0 0
\(356\) −68.5523 −3.63327
\(357\) −8.36683 −0.442820
\(358\) 41.0472 2.16941
\(359\) 3.31965 0.175205 0.0876023 0.996156i \(-0.472080\pi\)
0.0876023 + 0.996156i \(0.472080\pi\)
\(360\) 0 0
\(361\) 8.68649 0.457184
\(362\) 13.1050 0.688786
\(363\) −1.00000 −0.0524864
\(364\) −24.9939 −1.31003
\(365\) 0 0
\(366\) 11.2618 0.588663
\(367\) 36.1445 1.88673 0.943363 0.331762i \(-0.107643\pi\)
0.943363 + 0.331762i \(0.107643\pi\)
\(368\) 29.8432 1.55569
\(369\) −9.41855 −0.490310
\(370\) 0 0
\(371\) 0.169042 0.00877620
\(372\) 24.9939 1.29587
\(373\) 2.81044 0.145519 0.0727595 0.997350i \(-0.476819\pi\)
0.0727595 + 0.997350i \(0.476819\pi\)
\(374\) −21.0205 −1.08695
\(375\) 0 0
\(376\) −42.3545 −2.18427
\(377\) 6.15676 0.317089
\(378\) 2.92162 0.150272
\(379\) −20.0000 −1.02733 −0.513665 0.857991i \(-0.671713\pi\)
−0.513665 + 0.857991i \(0.671713\pi\)
\(380\) 0 0
\(381\) −2.24128 −0.114824
\(382\) 6.83710 0.349817
\(383\) −33.5585 −1.71476 −0.857379 0.514685i \(-0.827909\pi\)
−0.857379 + 0.514685i \(0.827909\pi\)
\(384\) −28.5669 −1.45780
\(385\) 0 0
\(386\) −0.0722347 −0.00367665
\(387\) −7.60197 −0.386430
\(388\) −78.3956 −3.97993
\(389\) 12.8371 0.650867 0.325433 0.945565i \(-0.394490\pi\)
0.325433 + 0.945565i \(0.394490\pi\)
\(390\) 0 0
\(391\) −16.7337 −0.846258
\(392\) −52.8225 −2.66794
\(393\) −8.68035 −0.437866
\(394\) −57.2183 −2.88262
\(395\) 0 0
\(396\) 5.34017 0.268354
\(397\) 5.31965 0.266986 0.133493 0.991050i \(-0.457381\pi\)
0.133493 + 0.991050i \(0.457381\pi\)
\(398\) 28.5113 1.42914
\(399\) 5.67420 0.284065
\(400\) 0 0
\(401\) 2.00000 0.0998752 0.0499376 0.998752i \(-0.484098\pi\)
0.0499376 + 0.998752i \(0.484098\pi\)
\(402\) −23.5174 −1.17294
\(403\) −20.3135 −1.01189
\(404\) −83.1748 −4.13810
\(405\) 0 0
\(406\) −4.14447 −0.205687
\(407\) 2.00000 0.0991363
\(408\) 70.2122 3.47602
\(409\) 26.1978 1.29540 0.647699 0.761897i \(-0.275732\pi\)
0.647699 + 0.761897i \(0.275732\pi\)
\(410\) 0 0
\(411\) −15.3607 −0.757687
\(412\) −36.5113 −1.79878
\(413\) −6.63931 −0.326699
\(414\) 5.84324 0.287180
\(415\) 0 0
\(416\) 84.1543 4.12600
\(417\) −8.58145 −0.420235
\(418\) 14.2557 0.697267
\(419\) −2.83710 −0.138601 −0.0693007 0.997596i \(-0.522077\pi\)
−0.0693007 + 0.997596i \(0.522077\pi\)
\(420\) 0 0
\(421\) 11.4764 0.559326 0.279663 0.960098i \(-0.409777\pi\)
0.279663 + 0.960098i \(0.409777\pi\)
\(422\) 25.9421 1.26284
\(423\) −4.68035 −0.227566
\(424\) −1.41855 −0.0688909
\(425\) 0 0
\(426\) 12.6803 0.614365
\(427\) 4.48255 0.216926
\(428\) −33.8576 −1.63657
\(429\) −4.34017 −0.209546
\(430\) 0 0
\(431\) 23.5708 1.13536 0.567682 0.823248i \(-0.307840\pi\)
0.567682 + 0.823248i \(0.307840\pi\)
\(432\) −13.8371 −0.665738
\(433\) 14.9939 0.720559 0.360279 0.932844i \(-0.382681\pi\)
0.360279 + 0.932844i \(0.382681\pi\)
\(434\) 13.6742 0.656383
\(435\) 0 0
\(436\) 12.3545 0.591676
\(437\) 11.3484 0.542868
\(438\) −28.4391 −1.35887
\(439\) 4.77924 0.228101 0.114050 0.993475i \(-0.463617\pi\)
0.114050 + 0.993475i \(0.463617\pi\)
\(440\) 0 0
\(441\) −5.83710 −0.277957
\(442\) −91.2327 −4.33950
\(443\) −20.1978 −0.959626 −0.479813 0.877371i \(-0.659296\pi\)
−0.479813 + 0.877371i \(0.659296\pi\)
\(444\) −10.6803 −0.506867
\(445\) 0 0
\(446\) 5.84324 0.276686
\(447\) 18.0989 0.856048
\(448\) −26.8059 −1.26646
\(449\) −21.5708 −1.01799 −0.508994 0.860770i \(-0.669982\pi\)
−0.508994 + 0.860770i \(0.669982\pi\)
\(450\) 0 0
\(451\) −9.41855 −0.443502
\(452\) 32.0410 1.50708
\(453\) −22.9360 −1.07763
\(454\) −26.1711 −1.22827
\(455\) 0 0
\(456\) −47.6163 −2.22984
\(457\) −28.1711 −1.31779 −0.658895 0.752235i \(-0.728976\pi\)
−0.658895 + 0.752235i \(0.728976\pi\)
\(458\) −9.10504 −0.425451
\(459\) 7.75872 0.362146
\(460\) 0 0
\(461\) −1.47187 −0.0685520 −0.0342760 0.999412i \(-0.510913\pi\)
−0.0342760 + 0.999412i \(0.510913\pi\)
\(462\) 2.92162 0.135926
\(463\) 23.2039 1.07838 0.539189 0.842185i \(-0.318731\pi\)
0.539189 + 0.842185i \(0.318731\pi\)
\(464\) 19.6286 0.911236
\(465\) 0 0
\(466\) 6.49693 0.300964
\(467\) −14.1568 −0.655097 −0.327548 0.944834i \(-0.606222\pi\)
−0.327548 + 0.944834i \(0.606222\pi\)
\(468\) 23.1773 1.07137
\(469\) −9.36069 −0.432237
\(470\) 0 0
\(471\) −10.9939 −0.506570
\(472\) 55.7152 2.56450
\(473\) −7.60197 −0.349539
\(474\) 21.9421 1.00784
\(475\) 0 0
\(476\) 44.6803 2.04792
\(477\) −0.156755 −0.00717734
\(478\) −19.5174 −0.892707
\(479\) −13.8432 −0.632514 −0.316257 0.948674i \(-0.602426\pi\)
−0.316257 + 0.948674i \(0.602426\pi\)
\(480\) 0 0
\(481\) 8.68035 0.395790
\(482\) −14.0989 −0.642187
\(483\) 2.32580 0.105827
\(484\) 5.34017 0.242735
\(485\) 0 0
\(486\) −2.70928 −0.122895
\(487\) 40.9939 1.85761 0.928804 0.370570i \(-0.120838\pi\)
0.928804 + 0.370570i \(0.120838\pi\)
\(488\) −37.6163 −1.70281
\(489\) −6.52359 −0.295007
\(490\) 0 0
\(491\) 34.8371 1.57218 0.786088 0.618114i \(-0.212103\pi\)
0.786088 + 0.618114i \(0.212103\pi\)
\(492\) 50.2967 2.26755
\(493\) −11.0061 −0.495692
\(494\) 61.8720 2.78375
\(495\) 0 0
\(496\) −64.7624 −2.90792
\(497\) 5.04718 0.226397
\(498\) −29.8576 −1.33795
\(499\) 15.1506 0.678235 0.339117 0.940744i \(-0.389872\pi\)
0.339117 + 0.940744i \(0.389872\pi\)
\(500\) 0 0
\(501\) 1.97334 0.0881622
\(502\) 41.5052 1.85247
\(503\) 6.65368 0.296673 0.148337 0.988937i \(-0.452608\pi\)
0.148337 + 0.988937i \(0.452608\pi\)
\(504\) −9.75872 −0.434688
\(505\) 0 0
\(506\) 5.84324 0.259764
\(507\) −5.83710 −0.259235
\(508\) 11.9688 0.531030
\(509\) 41.3484 1.83274 0.916368 0.400337i \(-0.131107\pi\)
0.916368 + 0.400337i \(0.131107\pi\)
\(510\) 0 0
\(511\) −11.3197 −0.500752
\(512\) 17.8599 0.789303
\(513\) −5.26180 −0.232314
\(514\) −11.2618 −0.496736
\(515\) 0 0
\(516\) 40.5958 1.78713
\(517\) −4.68035 −0.205841
\(518\) −5.84324 −0.256737
\(519\) 3.75872 0.164990
\(520\) 0 0
\(521\) 7.67420 0.336213 0.168106 0.985769i \(-0.446235\pi\)
0.168106 + 0.985769i \(0.446235\pi\)
\(522\) 3.84324 0.168214
\(523\) −23.2351 −1.01600 −0.508001 0.861357i \(-0.669615\pi\)
−0.508001 + 0.861357i \(0.669615\pi\)
\(524\) 46.3545 2.02501
\(525\) 0 0
\(526\) 50.6824 2.20986
\(527\) 36.3135 1.58184
\(528\) −13.8371 −0.602183
\(529\) −18.3484 −0.797757
\(530\) 0 0
\(531\) 6.15676 0.267180
\(532\) −30.3012 −1.31372
\(533\) −40.8781 −1.77063
\(534\) 34.7792 1.50505
\(535\) 0 0
\(536\) 78.5523 3.39294
\(537\) −15.1506 −0.653797
\(538\) 63.2905 2.72865
\(539\) −5.83710 −0.251422
\(540\) 0 0
\(541\) −18.0000 −0.773880 −0.386940 0.922105i \(-0.626468\pi\)
−0.386940 + 0.922105i \(0.626468\pi\)
\(542\) −15.1050 −0.648817
\(543\) −4.83710 −0.207580
\(544\) −150.439 −6.45001
\(545\) 0 0
\(546\) 12.6803 0.542669
\(547\) 23.0661 0.986235 0.493117 0.869963i \(-0.335857\pi\)
0.493117 + 0.869963i \(0.335857\pi\)
\(548\) 82.0288 3.50409
\(549\) −4.15676 −0.177406
\(550\) 0 0
\(551\) 7.46412 0.317982
\(552\) −19.5174 −0.830718
\(553\) 8.73367 0.371393
\(554\) 70.4801 2.99441
\(555\) 0 0
\(556\) 45.8264 1.94347
\(557\) −10.5958 −0.448960 −0.224480 0.974479i \(-0.572068\pi\)
−0.224480 + 0.974479i \(0.572068\pi\)
\(558\) −12.6803 −0.536802
\(559\) −32.9939 −1.39549
\(560\) 0 0
\(561\) 7.75872 0.327574
\(562\) −25.5174 −1.07639
\(563\) 36.2122 1.52616 0.763080 0.646303i \(-0.223686\pi\)
0.763080 + 0.646303i \(0.223686\pi\)
\(564\) 24.9939 1.05243
\(565\) 0 0
\(566\) −38.5835 −1.62179
\(567\) −1.07838 −0.0452876
\(568\) −42.3545 −1.77716
\(569\) −27.5753 −1.15602 −0.578008 0.816031i \(-0.696170\pi\)
−0.578008 + 0.816031i \(0.696170\pi\)
\(570\) 0 0
\(571\) −27.9299 −1.16883 −0.584414 0.811456i \(-0.698676\pi\)
−0.584414 + 0.811456i \(0.698676\pi\)
\(572\) 23.1773 0.969091
\(573\) −2.52359 −0.105425
\(574\) 27.5174 1.14856
\(575\) 0 0
\(576\) 24.8576 1.03573
\(577\) −41.4017 −1.72358 −0.861788 0.507268i \(-0.830655\pi\)
−0.861788 + 0.507268i \(0.830655\pi\)
\(578\) 117.035 4.86800
\(579\) 0.0266620 0.00110803
\(580\) 0 0
\(581\) −11.8843 −0.493043
\(582\) 39.7731 1.64865
\(583\) −0.156755 −0.00649215
\(584\) 94.9914 3.93077
\(585\) 0 0
\(586\) 42.6947 1.76370
\(587\) −8.48255 −0.350112 −0.175056 0.984558i \(-0.556011\pi\)
−0.175056 + 0.984558i \(0.556011\pi\)
\(588\) 31.1711 1.28548
\(589\) −24.6270 −1.01474
\(590\) 0 0
\(591\) 21.1194 0.868737
\(592\) 27.6742 1.13740
\(593\) −7.56093 −0.310490 −0.155245 0.987876i \(-0.549617\pi\)
−0.155245 + 0.987876i \(0.549617\pi\)
\(594\) −2.70928 −0.111163
\(595\) 0 0
\(596\) −96.6512 −3.95899
\(597\) −10.5236 −0.430702
\(598\) 25.3607 1.03708
\(599\) 5.67420 0.231842 0.115921 0.993258i \(-0.463018\pi\)
0.115921 + 0.993258i \(0.463018\pi\)
\(600\) 0 0
\(601\) −1.31965 −0.0538298 −0.0269149 0.999638i \(-0.508568\pi\)
−0.0269149 + 0.999638i \(0.508568\pi\)
\(602\) 22.2101 0.905215
\(603\) 8.68035 0.353491
\(604\) 122.482 4.98373
\(605\) 0 0
\(606\) 42.1978 1.71417
\(607\) 2.24128 0.0909706 0.0454853 0.998965i \(-0.485517\pi\)
0.0454853 + 0.998965i \(0.485517\pi\)
\(608\) 102.024 4.13763
\(609\) 1.52973 0.0619879
\(610\) 0 0
\(611\) −20.3135 −0.821797
\(612\) −41.4329 −1.67483
\(613\) −42.8638 −1.73125 −0.865626 0.500692i \(-0.833079\pi\)
−0.865626 + 0.500692i \(0.833079\pi\)
\(614\) 51.2639 2.06884
\(615\) 0 0
\(616\) −9.75872 −0.393190
\(617\) −11.3607 −0.457364 −0.228682 0.973501i \(-0.573442\pi\)
−0.228682 + 0.973501i \(0.573442\pi\)
\(618\) 18.5236 0.745128
\(619\) −45.1917 −1.81641 −0.908203 0.418530i \(-0.862545\pi\)
−0.908203 + 0.418530i \(0.862545\pi\)
\(620\) 0 0
\(621\) −2.15676 −0.0865476
\(622\) −56.5646 −2.26803
\(623\) 13.8432 0.554618
\(624\) −60.0554 −2.40414
\(625\) 0 0
\(626\) −17.1050 −0.683655
\(627\) −5.26180 −0.210136
\(628\) 58.7091 2.34275
\(629\) −15.5174 −0.618721
\(630\) 0 0
\(631\) −9.78992 −0.389731 −0.194865 0.980830i \(-0.562427\pi\)
−0.194865 + 0.980830i \(0.562427\pi\)
\(632\) −73.2905 −2.91534
\(633\) −9.57531 −0.380584
\(634\) −84.9647 −3.37438
\(635\) 0 0
\(636\) 0.837101 0.0331932
\(637\) −25.3340 −1.00377
\(638\) 3.84324 0.152156
\(639\) −4.68035 −0.185152
\(640\) 0 0
\(641\) 0.210079 0.00829764 0.00414882 0.999991i \(-0.498679\pi\)
0.00414882 + 0.999991i \(0.498679\pi\)
\(642\) 17.1773 0.677933
\(643\) −14.5236 −0.572754 −0.286377 0.958117i \(-0.592451\pi\)
−0.286377 + 0.958117i \(0.592451\pi\)
\(644\) −12.4202 −0.489423
\(645\) 0 0
\(646\) −110.606 −4.35172
\(647\) −15.4641 −0.607957 −0.303979 0.952679i \(-0.598315\pi\)
−0.303979 + 0.952679i \(0.598315\pi\)
\(648\) 9.04945 0.355496
\(649\) 6.15676 0.241674
\(650\) 0 0
\(651\) −5.04718 −0.197815
\(652\) 34.8371 1.36433
\(653\) 17.8310 0.697779 0.348890 0.937164i \(-0.386559\pi\)
0.348890 + 0.937164i \(0.386559\pi\)
\(654\) −6.26794 −0.245096
\(655\) 0 0
\(656\) −130.325 −5.08835
\(657\) 10.4969 0.409524
\(658\) 13.6742 0.533076
\(659\) −32.3135 −1.25876 −0.629378 0.777099i \(-0.716690\pi\)
−0.629378 + 0.777099i \(0.716690\pi\)
\(660\) 0 0
\(661\) −5.68649 −0.221179 −0.110589 0.993866i \(-0.535274\pi\)
−0.110589 + 0.993866i \(0.535274\pi\)
\(662\) 52.0288 2.02215
\(663\) 33.6742 1.30780
\(664\) 99.7296 3.87026
\(665\) 0 0
\(666\) 5.41855 0.209965
\(667\) 3.05947 0.118463
\(668\) −10.5380 −0.407726
\(669\) −2.15676 −0.0833850
\(670\) 0 0
\(671\) −4.15676 −0.160470
\(672\) 20.9093 0.806595
\(673\) 21.0205 0.810281 0.405141 0.914254i \(-0.367223\pi\)
0.405141 + 0.914254i \(0.367223\pi\)
\(674\) −36.5835 −1.40915
\(675\) 0 0
\(676\) 31.1711 1.19889
\(677\) −36.7526 −1.41252 −0.706258 0.707954i \(-0.749618\pi\)
−0.706258 + 0.707954i \(0.749618\pi\)
\(678\) −16.2557 −0.624295
\(679\) 15.8310 0.607536
\(680\) 0 0
\(681\) 9.65983 0.370165
\(682\) −12.6803 −0.485556
\(683\) 17.3074 0.662248 0.331124 0.943587i \(-0.392572\pi\)
0.331124 + 0.943587i \(0.392572\pi\)
\(684\) 28.0989 1.07439
\(685\) 0 0
\(686\) 37.5052 1.43195
\(687\) 3.36069 0.128218
\(688\) −105.189 −4.01030
\(689\) −0.680346 −0.0259191
\(690\) 0 0
\(691\) 17.6742 0.672358 0.336179 0.941798i \(-0.390865\pi\)
0.336179 + 0.941798i \(0.390865\pi\)
\(692\) −20.0722 −0.763032
\(693\) −1.07838 −0.0409642
\(694\) −17.1773 −0.652040
\(695\) 0 0
\(696\) −12.8371 −0.486589
\(697\) 73.0759 2.76795
\(698\) 43.7731 1.65684
\(699\) −2.39803 −0.0907019
\(700\) 0 0
\(701\) −17.1050 −0.646048 −0.323024 0.946391i \(-0.604700\pi\)
−0.323024 + 0.946391i \(0.604700\pi\)
\(702\) −11.7587 −0.443804
\(703\) 10.5236 0.396905
\(704\) 24.8576 0.936857
\(705\) 0 0
\(706\) 35.7731 1.34634
\(707\) 16.7961 0.631681
\(708\) −32.8781 −1.23564
\(709\) 25.1506 0.944551 0.472276 0.881451i \(-0.343433\pi\)
0.472276 + 0.881451i \(0.343433\pi\)
\(710\) 0 0
\(711\) −8.09890 −0.303732
\(712\) −116.169 −4.35361
\(713\) −10.0944 −0.378037
\(714\) −22.6681 −0.848331
\(715\) 0 0
\(716\) 80.9069 3.02363
\(717\) 7.20394 0.269036
\(718\) 8.99386 0.335648
\(719\) −1.78992 −0.0667528 −0.0333764 0.999443i \(-0.510626\pi\)
−0.0333764 + 0.999443i \(0.510626\pi\)
\(720\) 0 0
\(721\) 7.37298 0.274584
\(722\) 23.5341 0.875848
\(723\) 5.20394 0.193536
\(724\) 25.8310 0.960000
\(725\) 0 0
\(726\) −2.70928 −0.100551
\(727\) −25.9877 −0.963831 −0.481915 0.876218i \(-0.660059\pi\)
−0.481915 + 0.876218i \(0.660059\pi\)
\(728\) −42.3545 −1.56976
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 58.9816 2.18151
\(732\) 22.1978 0.820454
\(733\) 41.0205 1.51513 0.757564 0.652761i \(-0.226390\pi\)
0.757564 + 0.652761i \(0.226390\pi\)
\(734\) 97.9253 3.61449
\(735\) 0 0
\(736\) 41.8187 1.54146
\(737\) 8.68035 0.319745
\(738\) −25.5174 −0.939310
\(739\) −47.6163 −1.75160 −0.875798 0.482678i \(-0.839664\pi\)
−0.875798 + 0.482678i \(0.839664\pi\)
\(740\) 0 0
\(741\) −22.8371 −0.838942
\(742\) 0.457980 0.0168130
\(743\) −0.550252 −0.0201868 −0.0100934 0.999949i \(-0.503213\pi\)
−0.0100934 + 0.999949i \(0.503213\pi\)
\(744\) 42.3545 1.55279
\(745\) 0 0
\(746\) 7.61425 0.278778
\(747\) 11.0205 0.403220
\(748\) −41.4329 −1.51494
\(749\) 6.83710 0.249822
\(750\) 0 0
\(751\) 41.5585 1.51649 0.758245 0.651969i \(-0.226057\pi\)
0.758245 + 0.651969i \(0.226057\pi\)
\(752\) −64.7624 −2.36164
\(753\) −15.3197 −0.558279
\(754\) 16.6803 0.607462
\(755\) 0 0
\(756\) 5.75872 0.209443
\(757\) −1.31965 −0.0479636 −0.0239818 0.999712i \(-0.507634\pi\)
−0.0239818 + 0.999712i \(0.507634\pi\)
\(758\) −54.1855 −1.96811
\(759\) −2.15676 −0.0782853
\(760\) 0 0
\(761\) −2.21461 −0.0802797 −0.0401399 0.999194i \(-0.512780\pi\)
−0.0401399 + 0.999194i \(0.512780\pi\)
\(762\) −6.07223 −0.219974
\(763\) −2.49484 −0.0903192
\(764\) 13.4764 0.487559
\(765\) 0 0
\(766\) −90.9192 −3.28504
\(767\) 26.7214 0.964853
\(768\) −27.6803 −0.998828
\(769\) −14.3668 −0.518081 −0.259041 0.965866i \(-0.583406\pi\)
−0.259041 + 0.965866i \(0.583406\pi\)
\(770\) 0 0
\(771\) 4.15676 0.149702
\(772\) −0.142380 −0.00512436
\(773\) −40.1568 −1.44434 −0.722169 0.691717i \(-0.756855\pi\)
−0.722169 + 0.691717i \(0.756855\pi\)
\(774\) −20.5958 −0.740302
\(775\) 0 0
\(776\) −132.849 −4.76900
\(777\) 2.15676 0.0773732
\(778\) 34.7792 1.24690
\(779\) −49.5585 −1.77562
\(780\) 0 0
\(781\) −4.68035 −0.167476
\(782\) −45.3361 −1.62122
\(783\) −1.41855 −0.0506949
\(784\) −80.7686 −2.88459
\(785\) 0 0
\(786\) −23.5174 −0.838840
\(787\) −49.5897 −1.76768 −0.883841 0.467788i \(-0.845051\pi\)
−0.883841 + 0.467788i \(0.845051\pi\)
\(788\) −112.781 −4.01767
\(789\) −18.7070 −0.665987
\(790\) 0 0
\(791\) −6.47027 −0.230056
\(792\) 9.04945 0.321558
\(793\) −18.0410 −0.640656
\(794\) 14.4124 0.511477
\(795\) 0 0
\(796\) 56.1978 1.99188
\(797\) 46.7091 1.65452 0.827261 0.561818i \(-0.189898\pi\)
0.827261 + 0.561818i \(0.189898\pi\)
\(798\) 15.3730 0.544198
\(799\) 36.3135 1.28468
\(800\) 0 0
\(801\) −12.8371 −0.453577
\(802\) 5.41855 0.191336
\(803\) 10.4969 0.370429
\(804\) −46.3545 −1.63480
\(805\) 0 0
\(806\) −55.0349 −1.93852
\(807\) −23.3607 −0.822335
\(808\) −140.948 −4.95853
\(809\) −18.5814 −0.653289 −0.326644 0.945147i \(-0.605918\pi\)
−0.326644 + 0.945147i \(0.605918\pi\)
\(810\) 0 0
\(811\) 27.3028 0.958732 0.479366 0.877615i \(-0.340867\pi\)
0.479366 + 0.877615i \(0.340867\pi\)
\(812\) −8.16904 −0.286677
\(813\) 5.57531 0.195535
\(814\) 5.41855 0.189920
\(815\) 0 0
\(816\) 107.358 3.75829
\(817\) −40.0000 −1.39942
\(818\) 70.9770 2.48165
\(819\) −4.68035 −0.163545
\(820\) 0 0
\(821\) −31.2085 −1.08918 −0.544592 0.838701i \(-0.683315\pi\)
−0.544592 + 0.838701i \(0.683315\pi\)
\(822\) −41.6163 −1.45154
\(823\) 50.1855 1.74936 0.874678 0.484704i \(-0.161073\pi\)
0.874678 + 0.484704i \(0.161073\pi\)
\(824\) −61.8720 −2.15541
\(825\) 0 0
\(826\) −17.9877 −0.625873
\(827\) −27.3874 −0.952352 −0.476176 0.879350i \(-0.657977\pi\)
−0.476176 + 0.879350i \(0.657977\pi\)
\(828\) 11.5174 0.400259
\(829\) −26.1978 −0.909887 −0.454943 0.890520i \(-0.650341\pi\)
−0.454943 + 0.890520i \(0.650341\pi\)
\(830\) 0 0
\(831\) −26.0144 −0.902429
\(832\) 107.886 3.74029
\(833\) 45.2885 1.56915
\(834\) −23.2495 −0.805065
\(835\) 0 0
\(836\) 28.0989 0.971821
\(837\) 4.68035 0.161776
\(838\) −7.68649 −0.265525
\(839\) 7.20394 0.248708 0.124354 0.992238i \(-0.460314\pi\)
0.124354 + 0.992238i \(0.460314\pi\)
\(840\) 0 0
\(841\) −26.9877 −0.930611
\(842\) 31.0928 1.07153
\(843\) 9.41855 0.324392
\(844\) 51.1338 1.76010
\(845\) 0 0
\(846\) −12.6803 −0.435959
\(847\) −1.07838 −0.0370535
\(848\) −2.16904 −0.0744852
\(849\) 14.2413 0.488759
\(850\) 0 0
\(851\) 4.31351 0.147865
\(852\) 24.9939 0.856275
\(853\) −39.8043 −1.36287 −0.681437 0.731877i \(-0.738644\pi\)
−0.681437 + 0.731877i \(0.738644\pi\)
\(854\) 12.1445 0.415575
\(855\) 0 0
\(856\) −57.3751 −1.96104
\(857\) 36.9504 1.26220 0.631100 0.775701i \(-0.282604\pi\)
0.631100 + 0.775701i \(0.282604\pi\)
\(858\) −11.7587 −0.401436
\(859\) 57.5052 1.96205 0.981025 0.193879i \(-0.0621070\pi\)
0.981025 + 0.193879i \(0.0621070\pi\)
\(860\) 0 0
\(861\) −10.1568 −0.346141
\(862\) 63.8597 2.17507
\(863\) 1.89657 0.0645599 0.0322800 0.999479i \(-0.489723\pi\)
0.0322800 + 0.999479i \(0.489723\pi\)
\(864\) −19.3896 −0.659648
\(865\) 0 0
\(866\) 40.6225 1.38041
\(867\) −43.1978 −1.46707
\(868\) 26.9528 0.914838
\(869\) −8.09890 −0.274736
\(870\) 0 0
\(871\) 37.6742 1.27654
\(872\) 20.9360 0.708982
\(873\) −14.6803 −0.496854
\(874\) 30.7460 1.04000
\(875\) 0 0
\(876\) −56.0554 −1.89394
\(877\) −32.5380 −1.09873 −0.549365 0.835583i \(-0.685130\pi\)
−0.549365 + 0.835583i \(0.685130\pi\)
\(878\) 12.9483 0.436983
\(879\) −15.7587 −0.531529
\(880\) 0 0
\(881\) 18.1978 0.613099 0.306550 0.951855i \(-0.400825\pi\)
0.306550 + 0.951855i \(0.400825\pi\)
\(882\) −15.8143 −0.532496
\(883\) −36.3956 −1.22481 −0.612405 0.790545i \(-0.709798\pi\)
−0.612405 + 0.790545i \(0.709798\pi\)
\(884\) −179.826 −6.04821
\(885\) 0 0
\(886\) −54.7214 −1.83840
\(887\) −27.8699 −0.935780 −0.467890 0.883787i \(-0.654986\pi\)
−0.467890 + 0.883787i \(0.654986\pi\)
\(888\) −18.0989 −0.607359
\(889\) −2.41694 −0.0810616
\(890\) 0 0
\(891\) 1.00000 0.0335013
\(892\) 11.5174 0.385633
\(893\) −24.6270 −0.824112
\(894\) 49.0349 1.63997
\(895\) 0 0
\(896\) −30.8059 −1.02915
\(897\) −9.36069 −0.312544
\(898\) −58.4412 −1.95021
\(899\) −6.63931 −0.221433
\(900\) 0 0
\(901\) 1.21622 0.0405182
\(902\) −25.5174 −0.849638
\(903\) −8.19779 −0.272805
\(904\) 54.2967 1.80588
\(905\) 0 0
\(906\) −62.1399 −2.06446
\(907\) 27.9376 0.927653 0.463826 0.885926i \(-0.346476\pi\)
0.463826 + 0.885926i \(0.346476\pi\)
\(908\) −51.5851 −1.71191
\(909\) −15.5753 −0.516600
\(910\) 0 0
\(911\) −11.8843 −0.393744 −0.196872 0.980429i \(-0.563078\pi\)
−0.196872 + 0.980429i \(0.563078\pi\)
\(912\) −72.8080 −2.41091
\(913\) 11.0205 0.364726
\(914\) −76.3234 −2.52455
\(915\) 0 0
\(916\) −17.9467 −0.592975
\(917\) −9.36069 −0.309117
\(918\) 21.0205 0.693781
\(919\) −45.6041 −1.50434 −0.752170 0.658970i \(-0.770993\pi\)
−0.752170 + 0.658970i \(0.770993\pi\)
\(920\) 0 0
\(921\) −18.9216 −0.623489
\(922\) −3.98771 −0.131328
\(923\) −20.3135 −0.668627
\(924\) 5.75872 0.189448
\(925\) 0 0
\(926\) 62.8659 2.06590
\(927\) −6.83710 −0.224560
\(928\) 27.5052 0.902901
\(929\) −25.1506 −0.825165 −0.412582 0.910920i \(-0.635373\pi\)
−0.412582 + 0.910920i \(0.635373\pi\)
\(930\) 0 0
\(931\) −30.7136 −1.00660
\(932\) 12.8059 0.419471
\(933\) 20.8781 0.683520
\(934\) −38.3545 −1.25500
\(935\) 0 0
\(936\) 39.2762 1.28378
\(937\) −5.33403 −0.174255 −0.0871276 0.996197i \(-0.527769\pi\)
−0.0871276 + 0.996197i \(0.527769\pi\)
\(938\) −25.3607 −0.828056
\(939\) 6.31351 0.206034
\(940\) 0 0
\(941\) 56.8203 1.85229 0.926144 0.377170i \(-0.123103\pi\)
0.926144 + 0.377170i \(0.123103\pi\)
\(942\) −29.7854 −0.970460
\(943\) −20.3135 −0.661499
\(944\) 85.1917 2.77275
\(945\) 0 0
\(946\) −20.5958 −0.669628
\(947\) 20.9939 0.682209 0.341104 0.940025i \(-0.389199\pi\)
0.341104 + 0.940025i \(0.389199\pi\)
\(948\) 43.2495 1.40468
\(949\) 45.5585 1.47889
\(950\) 0 0
\(951\) 31.3607 1.01694
\(952\) 75.7152 2.45395
\(953\) 25.2351 0.817446 0.408723 0.912658i \(-0.365974\pi\)
0.408723 + 0.912658i \(0.365974\pi\)
\(954\) −0.424694 −0.0137500
\(955\) 0 0
\(956\) −38.4703 −1.24422
\(957\) −1.41855 −0.0458552
\(958\) −37.5052 −1.21174
\(959\) −16.5646 −0.534900
\(960\) 0 0
\(961\) −9.09436 −0.293367
\(962\) 23.5174 0.758233
\(963\) −6.34017 −0.204309
\(964\) −27.7899 −0.895053
\(965\) 0 0
\(966\) 6.30122 0.202739
\(967\) −13.1317 −0.422287 −0.211144 0.977455i \(-0.567719\pi\)
−0.211144 + 0.977455i \(0.567719\pi\)
\(968\) 9.04945 0.290860
\(969\) 40.8248 1.31148
\(970\) 0 0
\(971\) 8.94053 0.286915 0.143458 0.989656i \(-0.454178\pi\)
0.143458 + 0.989656i \(0.454178\pi\)
\(972\) −5.34017 −0.171286
\(973\) −9.25404 −0.296671
\(974\) 111.064 3.55871
\(975\) 0 0
\(976\) −57.5174 −1.84109
\(977\) −50.3956 −1.61230 −0.806149 0.591713i \(-0.798452\pi\)
−0.806149 + 0.591713i \(0.798452\pi\)
\(978\) −17.6742 −0.565159
\(979\) −12.8371 −0.410276
\(980\) 0 0
\(981\) 2.31351 0.0738647
\(982\) 94.3833 3.01189
\(983\) 32.1978 1.02695 0.513475 0.858105i \(-0.328358\pi\)
0.513475 + 0.858105i \(0.328358\pi\)
\(984\) 85.2327 2.71712
\(985\) 0 0
\(986\) −29.8187 −0.949620
\(987\) −5.04718 −0.160654
\(988\) 121.954 3.87988
\(989\) −16.3956 −0.521349
\(990\) 0 0
\(991\) −46.7747 −1.48585 −0.742924 0.669376i \(-0.766562\pi\)
−0.742924 + 0.669376i \(0.766562\pi\)
\(992\) −90.7501 −2.88132
\(993\) −19.2039 −0.609419
\(994\) 13.6742 0.433719
\(995\) 0 0
\(996\) −58.8515 −1.86478
\(997\) −38.2122 −1.21019 −0.605096 0.796153i \(-0.706865\pi\)
−0.605096 + 0.796153i \(0.706865\pi\)
\(998\) 41.0472 1.29933
\(999\) −2.00000 −0.0632772
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 825.2.a.k.1.3 3
3.2 odd 2 2475.2.a.bb.1.1 3
5.2 odd 4 825.2.c.g.199.6 6
5.3 odd 4 825.2.c.g.199.1 6
5.4 even 2 165.2.a.c.1.1 3
11.10 odd 2 9075.2.a.cf.1.1 3
15.2 even 4 2475.2.c.r.199.1 6
15.8 even 4 2475.2.c.r.199.6 6
15.14 odd 2 495.2.a.e.1.3 3
20.19 odd 2 2640.2.a.be.1.2 3
35.34 odd 2 8085.2.a.bk.1.1 3
55.54 odd 2 1815.2.a.m.1.3 3
60.59 even 2 7920.2.a.cj.1.2 3
165.164 even 2 5445.2.a.z.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
165.2.a.c.1.1 3 5.4 even 2
495.2.a.e.1.3 3 15.14 odd 2
825.2.a.k.1.3 3 1.1 even 1 trivial
825.2.c.g.199.1 6 5.3 odd 4
825.2.c.g.199.6 6 5.2 odd 4
1815.2.a.m.1.3 3 55.54 odd 2
2475.2.a.bb.1.1 3 3.2 odd 2
2475.2.c.r.199.1 6 15.2 even 4
2475.2.c.r.199.6 6 15.8 even 4
2640.2.a.be.1.2 3 20.19 odd 2
5445.2.a.z.1.1 3 165.164 even 2
7920.2.a.cj.1.2 3 60.59 even 2
8085.2.a.bk.1.1 3 35.34 odd 2
9075.2.a.cf.1.1 3 11.10 odd 2