Properties

Label 2640.2.a.be.1.2
Level $2640$
Weight $2$
Character 2640.1
Self dual yes
Analytic conductor $21.081$
Analytic rank $1$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 2640 = 2^{4} \cdot 3 \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2640.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(21.0805061336\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.148.1
Defining polynomial: \( x^{3} - x^{2} - 3x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 165)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(2.17009\) of defining polynomial
Character \(\chi\) \(=\) 2640.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{3} +1.00000 q^{5} -1.07838 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{3} +1.00000 q^{5} -1.07838 q^{7} +1.00000 q^{9} -1.00000 q^{11} -4.34017 q^{13} -1.00000 q^{15} +7.75872 q^{17} -5.26180 q^{19} +1.07838 q^{21} +2.15676 q^{23} +1.00000 q^{25} -1.00000 q^{27} +1.41855 q^{29} +4.68035 q^{31} +1.00000 q^{33} -1.07838 q^{35} -2.00000 q^{37} +4.34017 q^{39} -9.41855 q^{41} -7.60197 q^{43} +1.00000 q^{45} -4.68035 q^{47} -5.83710 q^{49} -7.75872 q^{51} +0.156755 q^{53} -1.00000 q^{55} +5.26180 q^{57} -6.15676 q^{59} -4.15676 q^{61} -1.07838 q^{63} -4.34017 q^{65} +8.68035 q^{67} -2.15676 q^{69} +4.68035 q^{71} -10.4969 q^{73} -1.00000 q^{75} +1.07838 q^{77} +8.09890 q^{79} +1.00000 q^{81} +11.0205 q^{83} +7.75872 q^{85} -1.41855 q^{87} -12.8371 q^{89} +4.68035 q^{91} -4.68035 q^{93} -5.26180 q^{95} +14.6803 q^{97} -1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 3 q^{3} + 3 q^{5} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 3 q^{3} + 3 q^{5} + 3 q^{9} - 3 q^{11} - 2 q^{13} - 3 q^{15} - 2 q^{17} - 8 q^{19} + 3 q^{25} - 3 q^{27} - 10 q^{29} - 8 q^{31} + 3 q^{33} - 6 q^{37} + 2 q^{39} - 14 q^{41} - 4 q^{43} + 3 q^{45} + 8 q^{47} + 11 q^{49} + 2 q^{51} - 6 q^{53} - 3 q^{55} + 8 q^{57} - 12 q^{59} - 6 q^{61} - 2 q^{65} + 4 q^{67} - 8 q^{71} - 14 q^{73} - 3 q^{75} - 12 q^{79} + 3 q^{81} - 2 q^{85} + 10 q^{87} - 10 q^{89} - 8 q^{91} + 8 q^{93} - 8 q^{95} + 22 q^{97} - 3 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −1.00000 −0.577350
\(4\) 0 0
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) −1.07838 −0.407588 −0.203794 0.979014i \(-0.565327\pi\)
−0.203794 + 0.979014i \(0.565327\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −1.00000 −0.301511
\(12\) 0 0
\(13\) −4.34017 −1.20375 −0.601874 0.798591i \(-0.705579\pi\)
−0.601874 + 0.798591i \(0.705579\pi\)
\(14\) 0 0
\(15\) −1.00000 −0.258199
\(16\) 0 0
\(17\) 7.75872 1.88177 0.940883 0.338730i \(-0.109997\pi\)
0.940883 + 0.338730i \(0.109997\pi\)
\(18\) 0 0
\(19\) −5.26180 −1.20714 −0.603569 0.797311i \(-0.706255\pi\)
−0.603569 + 0.797311i \(0.706255\pi\)
\(20\) 0 0
\(21\) 1.07838 0.235321
\(22\) 0 0
\(23\) 2.15676 0.449715 0.224857 0.974392i \(-0.427808\pi\)
0.224857 + 0.974392i \(0.427808\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) 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 \(-0.361922\pi\)
0.420307 + 0.907382i \(0.361922\pi\)
\(32\) 0 0
\(33\) 1.00000 0.174078
\(34\) 0 0
\(35\) −1.07838 −0.182279
\(36\) 0 0
\(37\) −2.00000 −0.328798 −0.164399 0.986394i \(-0.552568\pi\)
−0.164399 + 0.986394i \(0.552568\pi\)
\(38\) 0 0
\(39\) 4.34017 0.694984
\(40\) 0 0
\(41\) −9.41855 −1.47093 −0.735465 0.677562i \(-0.763036\pi\)
−0.735465 + 0.677562i \(0.763036\pi\)
\(42\) 0 0
\(43\) −7.60197 −1.15929 −0.579645 0.814869i \(-0.696809\pi\)
−0.579645 + 0.814869i \(0.696809\pi\)
\(44\) 0 0
\(45\) 1.00000 0.149071
\(46\) 0 0
\(47\) −4.68035 −0.682699 −0.341349 0.939937i \(-0.610884\pi\)
−0.341349 + 0.939937i \(0.610884\pi\)
\(48\) 0 0
\(49\) −5.83710 −0.833872
\(50\) 0 0
\(51\) −7.75872 −1.08644
\(52\) 0 0
\(53\) 0.156755 0.0215320 0.0107660 0.999942i \(-0.496573\pi\)
0.0107660 + 0.999942i \(0.496573\pi\)
\(54\) 0 0
\(55\) −1.00000 −0.134840
\(56\) 0 0
\(57\) 5.26180 0.696942
\(58\) 0 0
\(59\) −6.15676 −0.801541 −0.400771 0.916178i \(-0.631258\pi\)
−0.400771 + 0.916178i \(0.631258\pi\)
\(60\) 0 0
\(61\) −4.15676 −0.532218 −0.266109 0.963943i \(-0.585738\pi\)
−0.266109 + 0.963943i \(0.585738\pi\)
\(62\) 0 0
\(63\) −1.07838 −0.135863
\(64\) 0 0
\(65\) −4.34017 −0.538332
\(66\) 0 0
\(67\) 8.68035 1.06047 0.530237 0.847850i \(-0.322103\pi\)
0.530237 + 0.847850i \(0.322103\pi\)
\(68\) 0 0
\(69\) −2.15676 −0.259643
\(70\) 0 0
\(71\) 4.68035 0.555455 0.277727 0.960660i \(-0.410419\pi\)
0.277727 + 0.960660i \(0.410419\pi\)
\(72\) 0 0
\(73\) −10.4969 −1.22857 −0.614286 0.789083i \(-0.710556\pi\)
−0.614286 + 0.789083i \(0.710556\pi\)
\(74\) 0 0
\(75\) −1.00000 −0.115470
\(76\) 0 0
\(77\) 1.07838 0.122893
\(78\) 0 0
\(79\) 8.09890 0.911197 0.455599 0.890185i \(-0.349425\pi\)
0.455599 + 0.890185i \(0.349425\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 11.0205 1.20966 0.604830 0.796355i \(-0.293241\pi\)
0.604830 + 0.796355i \(0.293241\pi\)
\(84\) 0 0
\(85\) 7.75872 0.841552
\(86\) 0 0
\(87\) −1.41855 −0.152085
\(88\) 0 0
\(89\) −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
\(92\) 0 0
\(93\) −4.68035 −0.485329
\(94\) 0 0
\(95\) −5.26180 −0.539849
\(96\) 0 0
\(97\) 14.6803 1.49056 0.745282 0.666750i \(-0.232315\pi\)
0.745282 + 0.666750i \(0.232315\pi\)
\(98\) 0 0
\(99\) −1.00000 −0.100504
\(100\) 0 0
\(101\) −15.5753 −1.54980 −0.774900 0.632083i \(-0.782200\pi\)
−0.774900 + 0.632083i \(0.782200\pi\)
\(102\) 0 0
\(103\) −6.83710 −0.673680 −0.336840 0.941562i \(-0.609358\pi\)
−0.336840 + 0.941562i \(0.609358\pi\)
\(104\) 0 0
\(105\) 1.07838 0.105239
\(106\) 0 0
\(107\) −6.34017 −0.612928 −0.306464 0.951882i \(-0.599146\pi\)
−0.306464 + 0.951882i \(0.599146\pi\)
\(108\) 0 0
\(109\) 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
\(112\) 0 0
\(113\) −6.00000 −0.564433 −0.282216 0.959351i \(-0.591070\pi\)
−0.282216 + 0.959351i \(0.591070\pi\)
\(114\) 0 0
\(115\) 2.15676 0.201118
\(116\) 0 0
\(117\) −4.34017 −0.401249
\(118\) 0 0
\(119\) −8.36683 −0.766987
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 0 0
\(123\) 9.41855 0.849242
\(124\) 0 0
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) 2.24128 0.198881 0.0994406 0.995044i \(-0.468295\pi\)
0.0994406 + 0.995044i \(0.468295\pi\)
\(128\) 0 0
\(129\) 7.60197 0.669316
\(130\) 0 0
\(131\) −8.68035 −0.758405 −0.379203 0.925314i \(-0.623802\pi\)
−0.379203 + 0.925314i \(0.623802\pi\)
\(132\) 0 0
\(133\) 5.67420 0.492016
\(134\) 0 0
\(135\) −1.00000 −0.0860663
\(136\) 0 0
\(137\) −15.3607 −1.31235 −0.656176 0.754608i \(-0.727827\pi\)
−0.656176 + 0.754608i \(0.727827\pi\)
\(138\) 0 0
\(139\) −8.58145 −0.727869 −0.363935 0.931425i \(-0.618567\pi\)
−0.363935 + 0.931425i \(0.618567\pi\)
\(140\) 0 0
\(141\) 4.68035 0.394156
\(142\) 0 0
\(143\) 4.34017 0.362943
\(144\) 0 0
\(145\) 1.41855 0.117804
\(146\) 0 0
\(147\) 5.83710 0.481436
\(148\) 0 0
\(149\) −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 \(-0.883042\pi\)
−0.933253 + 0.359221i \(0.883042\pi\)
\(152\) 0 0
\(153\) 7.75872 0.627256
\(154\) 0 0
\(155\) 4.68035 0.375934
\(156\) 0 0
\(157\) −10.9939 −0.877405 −0.438703 0.898632i \(-0.644562\pi\)
−0.438703 + 0.898632i \(0.644562\pi\)
\(158\) 0 0
\(159\) −0.156755 −0.0124315
\(160\) 0 0
\(161\) −2.32580 −0.183298
\(162\) 0 0
\(163\) 6.52359 0.510967 0.255484 0.966813i \(-0.417765\pi\)
0.255484 + 0.966813i \(0.417765\pi\)
\(164\) 0 0
\(165\) 1.00000 0.0778499
\(166\) 0 0
\(167\) −1.97334 −0.152701 −0.0763507 0.997081i \(-0.524327\pi\)
−0.0763507 + 0.997081i \(0.524327\pi\)
\(168\) 0 0
\(169\) 5.83710 0.449008
\(170\) 0 0
\(171\) −5.26180 −0.402380
\(172\) 0 0
\(173\) 3.75872 0.285770 0.142885 0.989739i \(-0.454362\pi\)
0.142885 + 0.989739i \(0.454362\pi\)
\(174\) 0 0
\(175\) −1.07838 −0.0815177
\(176\) 0 0
\(177\) 6.15676 0.462770
\(178\) 0 0
\(179\) −15.1506 −1.13241 −0.566205 0.824264i \(-0.691589\pi\)
−0.566205 + 0.824264i \(0.691589\pi\)
\(180\) 0 0
\(181\) 4.83710 0.359539 0.179769 0.983709i \(-0.442465\pi\)
0.179769 + 0.983709i \(0.442465\pi\)
\(182\) 0 0
\(183\) 4.15676 0.307276
\(184\) 0 0
\(185\) −2.00000 −0.147043
\(186\) 0 0
\(187\) −7.75872 −0.567374
\(188\) 0 0
\(189\) 1.07838 0.0784404
\(190\) 0 0
\(191\) −2.52359 −0.182601 −0.0913003 0.995823i \(-0.529102\pi\)
−0.0913003 + 0.995823i \(0.529102\pi\)
\(192\) 0 0
\(193\) 0.0266620 0.00191917 0.000959586 1.00000i \(-0.499695\pi\)
0.000959586 1.00000i \(0.499695\pi\)
\(194\) 0 0
\(195\) 4.34017 0.310806
\(196\) 0 0
\(197\) 21.1194 1.50470 0.752348 0.658766i \(-0.228921\pi\)
0.752348 + 0.658766i \(0.228921\pi\)
\(198\) 0 0
\(199\) −10.5236 −0.745998 −0.372999 0.927832i \(-0.621670\pi\)
−0.372999 + 0.927832i \(0.621670\pi\)
\(200\) 0 0
\(201\) −8.68035 −0.612264
\(202\) 0 0
\(203\) −1.52973 −0.107366
\(204\) 0 0
\(205\) −9.41855 −0.657820
\(206\) 0 0
\(207\) 2.15676 0.149905
\(208\) 0 0
\(209\) 5.26180 0.363966
\(210\) 0 0
\(211\) −9.57531 −0.659191 −0.329596 0.944122i \(-0.606912\pi\)
−0.329596 + 0.944122i \(0.606912\pi\)
\(212\) 0 0
\(213\) −4.68035 −0.320692
\(214\) 0 0
\(215\) −7.60197 −0.518450
\(216\) 0 0
\(217\) −5.04718 −0.342625
\(218\) 0 0
\(219\) 10.4969 0.709317
\(220\) 0 0
\(221\) −33.6742 −2.26517
\(222\) 0 0
\(223\) 2.15676 0.144427 0.0722135 0.997389i \(-0.476994\pi\)
0.0722135 + 0.997389i \(0.476994\pi\)
\(224\) 0 0
\(225\) 1.00000 0.0666667
\(226\) 0 0
\(227\) −9.65983 −0.641145 −0.320573 0.947224i \(-0.603875\pi\)
−0.320573 + 0.947224i \(0.603875\pi\)
\(228\) 0 0
\(229\) −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
\(232\) 0 0
\(233\) −2.39803 −0.157100 −0.0785501 0.996910i \(-0.525029\pi\)
−0.0785501 + 0.996910i \(0.525029\pi\)
\(234\) 0 0
\(235\) −4.68035 −0.305312
\(236\) 0 0
\(237\) −8.09890 −0.526080
\(238\) 0 0
\(239\) 7.20394 0.465984 0.232992 0.972479i \(-0.425148\pi\)
0.232992 + 0.972479i \(0.425148\pi\)
\(240\) 0 0
\(241\) −5.20394 −0.335215 −0.167608 0.985854i \(-0.553604\pi\)
−0.167608 + 0.985854i \(0.553604\pi\)
\(242\) 0 0
\(243\) −1.00000 −0.0641500
\(244\) 0 0
\(245\) −5.83710 −0.372919
\(246\) 0 0
\(247\) 22.8371 1.45309
\(248\) 0 0
\(249\) −11.0205 −0.698397
\(250\) 0 0
\(251\) −15.3197 −0.966968 −0.483484 0.875353i \(-0.660629\pi\)
−0.483484 + 0.875353i \(0.660629\pi\)
\(252\) 0 0
\(253\) −2.15676 −0.135594
\(254\) 0 0
\(255\) −7.75872 −0.485870
\(256\) 0 0
\(257\) 4.15676 0.259291 0.129646 0.991560i \(-0.458616\pi\)
0.129646 + 0.991560i \(0.458616\pi\)
\(258\) 0 0
\(259\) 2.15676 0.134014
\(260\) 0 0
\(261\) 1.41855 0.0878061
\(262\) 0 0
\(263\) 18.7070 1.15352 0.576762 0.816912i \(-0.304316\pi\)
0.576762 + 0.816912i \(0.304316\pi\)
\(264\) 0 0
\(265\) 0.156755 0.00962941
\(266\) 0 0
\(267\) 12.8371 0.785618
\(268\) 0 0
\(269\) 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 \(-0.445837\pi\)
0.169338 + 0.985558i \(0.445837\pi\)
\(272\) 0 0
\(273\) −4.68035 −0.283267
\(274\) 0 0
\(275\) −1.00000 −0.0603023
\(276\) 0 0
\(277\) −26.0144 −1.56305 −0.781526 0.623872i \(-0.785558\pi\)
−0.781526 + 0.623872i \(0.785558\pi\)
\(278\) 0 0
\(279\) 4.68035 0.280205
\(280\) 0 0
\(281\) −9.41855 −0.561864 −0.280932 0.959728i \(-0.590643\pi\)
−0.280932 + 0.959728i \(0.590643\pi\)
\(282\) 0 0
\(283\) −14.2413 −0.846556 −0.423278 0.906000i \(-0.639121\pi\)
−0.423278 + 0.906000i \(0.639121\pi\)
\(284\) 0 0
\(285\) 5.26180 0.311682
\(286\) 0 0
\(287\) 10.1568 0.599534
\(288\) 0 0
\(289\) 43.1978 2.54105
\(290\) 0 0
\(291\) −14.6803 −0.860577
\(292\) 0 0
\(293\) −15.7587 −0.920634 −0.460317 0.887754i \(-0.652264\pi\)
−0.460317 + 0.887754i \(0.652264\pi\)
\(294\) 0 0
\(295\) −6.15676 −0.358460
\(296\) 0 0
\(297\) 1.00000 0.0580259
\(298\) 0 0
\(299\) −9.36069 −0.541343
\(300\) 0 0
\(301\) 8.19779 0.472513
\(302\) 0 0
\(303\) 15.5753 0.894778
\(304\) 0 0
\(305\) −4.15676 −0.238015
\(306\) 0 0
\(307\) 18.9216 1.07991 0.539957 0.841693i \(-0.318440\pi\)
0.539957 + 0.841693i \(0.318440\pi\)
\(308\) 0 0
\(309\) 6.83710 0.388949
\(310\) 0 0
\(311\) 20.8781 1.18389 0.591945 0.805978i \(-0.298360\pi\)
0.591945 + 0.805978i \(0.298360\pi\)
\(312\) 0 0
\(313\) 6.31351 0.356861 0.178430 0.983953i \(-0.442898\pi\)
0.178430 + 0.983953i \(0.442898\pi\)
\(314\) 0 0
\(315\) −1.07838 −0.0607597
\(316\) 0 0
\(317\) 31.3607 1.76139 0.880696 0.473682i \(-0.157075\pi\)
0.880696 + 0.473682i \(0.157075\pi\)
\(318\) 0 0
\(319\) −1.41855 −0.0794236
\(320\) 0 0
\(321\) 6.34017 0.353874
\(322\) 0 0
\(323\) −40.8248 −2.27155
\(324\) 0 0
\(325\) −4.34017 −0.240749
\(326\) 0 0
\(327\) −2.31351 −0.127937
\(328\) 0 0
\(329\) 5.04718 0.278260
\(330\) 0 0
\(331\) −19.2039 −1.05554 −0.527772 0.849386i \(-0.676972\pi\)
−0.527772 + 0.849386i \(0.676972\pi\)
\(332\) 0 0
\(333\) −2.00000 −0.109599
\(334\) 0 0
\(335\) 8.68035 0.474258
\(336\) 0 0
\(337\) 13.5031 0.735559 0.367780 0.929913i \(-0.380118\pi\)
0.367780 + 0.929913i \(0.380118\pi\)
\(338\) 0 0
\(339\) 6.00000 0.325875
\(340\) 0 0
\(341\) −4.68035 −0.253455
\(342\) 0 0
\(343\) 13.8432 0.747465
\(344\) 0 0
\(345\) −2.15676 −0.116116
\(346\) 0 0
\(347\) −6.34017 −0.340358 −0.170179 0.985413i \(-0.554435\pi\)
−0.170179 + 0.985413i \(0.554435\pi\)
\(348\) 0 0
\(349\) 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
\(352\) 0 0
\(353\) −13.2039 −0.702775 −0.351387 0.936230i \(-0.614290\pi\)
−0.351387 + 0.936230i \(0.614290\pi\)
\(354\) 0 0
\(355\) 4.68035 0.248407
\(356\) 0 0
\(357\) 8.36683 0.442820
\(358\) 0 0
\(359\) −3.31965 −0.175205 −0.0876023 0.996156i \(-0.527920\pi\)
−0.0876023 + 0.996156i \(0.527920\pi\)
\(360\) 0 0
\(361\) 8.68649 0.457184
\(362\) 0 0
\(363\) −1.00000 −0.0524864
\(364\) 0 0
\(365\) −10.4969 −0.549434
\(366\) 0 0
\(367\) 36.1445 1.88673 0.943363 0.331762i \(-0.107643\pi\)
0.943363 + 0.331762i \(0.107643\pi\)
\(368\) 0 0
\(369\) −9.41855 −0.490310
\(370\) 0 0
\(371\) −0.169042 −0.00877620
\(372\) 0 0
\(373\) −2.81044 −0.145519 −0.0727595 0.997350i \(-0.523181\pi\)
−0.0727595 + 0.997350i \(0.523181\pi\)
\(374\) 0 0
\(375\) −1.00000 −0.0516398
\(376\) 0 0
\(377\) −6.15676 −0.317089
\(378\) 0 0
\(379\) 20.0000 1.02733 0.513665 0.857991i \(-0.328287\pi\)
0.513665 + 0.857991i \(0.328287\pi\)
\(380\) 0 0
\(381\) −2.24128 −0.114824
\(382\) 0 0
\(383\) −33.5585 −1.71476 −0.857379 0.514685i \(-0.827909\pi\)
−0.857379 + 0.514685i \(0.827909\pi\)
\(384\) 0 0
\(385\) 1.07838 0.0549592
\(386\) 0 0
\(387\) −7.60197 −0.386430
\(388\) 0 0
\(389\) 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
\(392\) 0 0
\(393\) 8.68035 0.437866
\(394\) 0 0
\(395\) 8.09890 0.407500
\(396\) 0 0
\(397\) −5.31965 −0.266986 −0.133493 0.991050i \(-0.542619\pi\)
−0.133493 + 0.991050i \(0.542619\pi\)
\(398\) 0 0
\(399\) −5.67420 −0.284065
\(400\) 0 0
\(401\) 2.00000 0.0998752 0.0499376 0.998752i \(-0.484098\pi\)
0.0499376 + 0.998752i \(0.484098\pi\)
\(402\) 0 0
\(403\) −20.3135 −1.01189
\(404\) 0 0
\(405\) 1.00000 0.0496904
\(406\) 0 0
\(407\) 2.00000 0.0991363
\(408\) 0 0
\(409\) 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
\(412\) 0 0
\(413\) 6.63931 0.326699
\(414\) 0 0
\(415\) 11.0205 0.540976
\(416\) 0 0
\(417\) 8.58145 0.420235
\(418\) 0 0
\(419\) 2.83710 0.138601 0.0693007 0.997596i \(-0.477923\pi\)
0.0693007 + 0.997596i \(0.477923\pi\)
\(420\) 0 0
\(421\) 11.4764 0.559326 0.279663 0.960098i \(-0.409777\pi\)
0.279663 + 0.960098i \(0.409777\pi\)
\(422\) 0 0
\(423\) −4.68035 −0.227566
\(424\) 0 0
\(425\) 7.75872 0.376353
\(426\) 0 0
\(427\) 4.48255 0.216926
\(428\) 0 0
\(429\) −4.34017 −0.209546
\(430\) 0 0
\(431\) −23.5708 −1.13536 −0.567682 0.823248i \(-0.692160\pi\)
−0.567682 + 0.823248i \(0.692160\pi\)
\(432\) 0 0
\(433\) −14.9939 −0.720559 −0.360279 0.932844i \(-0.617319\pi\)
−0.360279 + 0.932844i \(0.617319\pi\)
\(434\) 0 0
\(435\) −1.41855 −0.0680143
\(436\) 0 0
\(437\) −11.3484 −0.542868
\(438\) 0 0
\(439\) −4.77924 −0.228101 −0.114050 0.993475i \(-0.536383\pi\)
−0.114050 + 0.993475i \(0.536383\pi\)
\(440\) 0 0
\(441\) −5.83710 −0.277957
\(442\) 0 0
\(443\) −20.1978 −0.959626 −0.479813 0.877371i \(-0.659296\pi\)
−0.479813 + 0.877371i \(0.659296\pi\)
\(444\) 0 0
\(445\) −12.8371 −0.608537
\(446\) 0 0
\(447\) 18.0989 0.856048
\(448\) 0 0
\(449\) −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
\(452\) 0 0
\(453\) 22.9360 1.07763
\(454\) 0 0
\(455\) 4.68035 0.219418
\(456\) 0 0
\(457\) 28.1711 1.31779 0.658895 0.752235i \(-0.271024\pi\)
0.658895 + 0.752235i \(0.271024\pi\)
\(458\) 0 0
\(459\) −7.75872 −0.362146
\(460\) 0 0
\(461\) −1.47187 −0.0685520 −0.0342760 0.999412i \(-0.510913\pi\)
−0.0342760 + 0.999412i \(0.510913\pi\)
\(462\) 0 0
\(463\) 23.2039 1.07838 0.539189 0.842185i \(-0.318731\pi\)
0.539189 + 0.842185i \(0.318731\pi\)
\(464\) 0 0
\(465\) −4.68035 −0.217046
\(466\) 0 0
\(467\) −14.1568 −0.655097 −0.327548 0.944834i \(-0.606222\pi\)
−0.327548 + 0.944834i \(0.606222\pi\)
\(468\) 0 0
\(469\) −9.36069 −0.432237
\(470\) 0 0
\(471\) 10.9939 0.506570
\(472\) 0 0
\(473\) 7.60197 0.349539
\(474\) 0 0
\(475\) −5.26180 −0.241428
\(476\) 0 0
\(477\) 0.156755 0.00717734
\(478\) 0 0
\(479\) 13.8432 0.632514 0.316257 0.948674i \(-0.397574\pi\)
0.316257 + 0.948674i \(0.397574\pi\)
\(480\) 0 0
\(481\) 8.68035 0.395790
\(482\) 0 0
\(483\) 2.32580 0.105827
\(484\) 0 0
\(485\) 14.6803 0.666600
\(486\) 0 0
\(487\) 40.9939 1.85761 0.928804 0.370570i \(-0.120838\pi\)
0.928804 + 0.370570i \(0.120838\pi\)
\(488\) 0 0
\(489\) −6.52359 −0.295007
\(490\) 0 0
\(491\) −34.8371 −1.57218 −0.786088 0.618114i \(-0.787897\pi\)
−0.786088 + 0.618114i \(0.787897\pi\)
\(492\) 0 0
\(493\) 11.0061 0.495692
\(494\) 0 0
\(495\) −1.00000 −0.0449467
\(496\) 0 0
\(497\) −5.04718 −0.226397
\(498\) 0 0
\(499\) −15.1506 −0.678235 −0.339117 0.940744i \(-0.610128\pi\)
−0.339117 + 0.940744i \(0.610128\pi\)
\(500\) 0 0
\(501\) 1.97334 0.0881622
\(502\) 0 0
\(503\) 6.65368 0.296673 0.148337 0.988937i \(-0.452608\pi\)
0.148337 + 0.988937i \(0.452608\pi\)
\(504\) 0 0
\(505\) −15.5753 −0.693092
\(506\) 0 0
\(507\) −5.83710 −0.259235
\(508\) 0 0
\(509\) 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
\(512\) 0 0
\(513\) 5.26180 0.232314
\(514\) 0 0
\(515\) −6.83710 −0.301279
\(516\) 0 0
\(517\) 4.68035 0.205841
\(518\) 0 0
\(519\) −3.75872 −0.164990
\(520\) 0 0
\(521\) 7.67420 0.336213 0.168106 0.985769i \(-0.446235\pi\)
0.168106 + 0.985769i \(0.446235\pi\)
\(522\) 0 0
\(523\) −23.2351 −1.01600 −0.508001 0.861357i \(-0.669615\pi\)
−0.508001 + 0.861357i \(0.669615\pi\)
\(524\) 0 0
\(525\) 1.07838 0.0470643
\(526\) 0 0
\(527\) 36.3135 1.58184
\(528\) 0 0
\(529\) −18.3484 −0.797757
\(530\) 0 0
\(531\) −6.15676 −0.267180
\(532\) 0 0
\(533\) 40.8781 1.77063
\(534\) 0 0
\(535\) −6.34017 −0.274110
\(536\) 0 0
\(537\) 15.1506 0.653797
\(538\) 0 0
\(539\) 5.83710 0.251422
\(540\) 0 0
\(541\) −18.0000 −0.773880 −0.386940 0.922105i \(-0.626468\pi\)
−0.386940 + 0.922105i \(0.626468\pi\)
\(542\) 0 0
\(543\) −4.83710 −0.207580
\(544\) 0 0
\(545\) 2.31351 0.0990999
\(546\) 0 0
\(547\) 23.0661 0.986235 0.493117 0.869963i \(-0.335857\pi\)
0.493117 + 0.869963i \(0.335857\pi\)
\(548\) 0 0
\(549\) −4.15676 −0.177406
\(550\) 0 0
\(551\) −7.46412 −0.317982
\(552\) 0 0
\(553\) −8.73367 −0.371393
\(554\) 0 0
\(555\) 2.00000 0.0848953
\(556\) 0 0
\(557\) 10.5958 0.448960 0.224480 0.974479i \(-0.427932\pi\)
0.224480 + 0.974479i \(0.427932\pi\)
\(558\) 0 0
\(559\) 32.9939 1.39549
\(560\) 0 0
\(561\) 7.75872 0.327574
\(562\) 0 0
\(563\) 36.2122 1.52616 0.763080 0.646303i \(-0.223686\pi\)
0.763080 + 0.646303i \(0.223686\pi\)
\(564\) 0 0
\(565\) −6.00000 −0.252422
\(566\) 0 0
\(567\) −1.07838 −0.0452876
\(568\) 0 0
\(569\) −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 \(-0.301324\pi\)
0.584414 + 0.811456i \(0.301324\pi\)
\(572\) 0 0
\(573\) 2.52359 0.105425
\(574\) 0 0
\(575\) 2.15676 0.0899429
\(576\) 0 0
\(577\) 41.4017 1.72358 0.861788 0.507268i \(-0.169345\pi\)
0.861788 + 0.507268i \(0.169345\pi\)
\(578\) 0 0
\(579\) −0.0266620 −0.00110803
\(580\) 0 0
\(581\) −11.8843 −0.493043
\(582\) 0 0
\(583\) −0.156755 −0.00649215
\(584\) 0 0
\(585\) −4.34017 −0.179444
\(586\) 0 0
\(587\) −8.48255 −0.350112 −0.175056 0.984558i \(-0.556011\pi\)
−0.175056 + 0.984558i \(0.556011\pi\)
\(588\) 0 0
\(589\) −24.6270 −1.01474
\(590\) 0 0
\(591\) −21.1194 −0.868737
\(592\) 0 0
\(593\) 7.56093 0.310490 0.155245 0.987876i \(-0.450383\pi\)
0.155245 + 0.987876i \(0.450383\pi\)
\(594\) 0 0
\(595\) −8.36683 −0.343007
\(596\) 0 0
\(597\) 10.5236 0.430702
\(598\) 0 0
\(599\) −5.67420 −0.231842 −0.115921 0.993258i \(-0.536982\pi\)
−0.115921 + 0.993258i \(0.536982\pi\)
\(600\) 0 0
\(601\) −1.31965 −0.0538298 −0.0269149 0.999638i \(-0.508568\pi\)
−0.0269149 + 0.999638i \(0.508568\pi\)
\(602\) 0 0
\(603\) 8.68035 0.353491
\(604\) 0 0
\(605\) 1.00000 0.0406558
\(606\) 0 0
\(607\) 2.24128 0.0909706 0.0454853 0.998965i \(-0.485517\pi\)
0.0454853 + 0.998965i \(0.485517\pi\)
\(608\) 0 0
\(609\) 1.52973 0.0619879
\(610\) 0 0
\(611\) 20.3135 0.821797
\(612\) 0 0
\(613\) 42.8638 1.73125 0.865626 0.500692i \(-0.166921\pi\)
0.865626 + 0.500692i \(0.166921\pi\)
\(614\) 0 0
\(615\) 9.41855 0.379793
\(616\) 0 0
\(617\) 11.3607 0.457364 0.228682 0.973501i \(-0.426558\pi\)
0.228682 + 0.973501i \(0.426558\pi\)
\(618\) 0 0
\(619\) 45.1917 1.81641 0.908203 0.418530i \(-0.137455\pi\)
0.908203 + 0.418530i \(0.137455\pi\)
\(620\) 0 0
\(621\) −2.15676 −0.0865476
\(622\) 0 0
\(623\) 13.8432 0.554618
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) −5.26180 −0.210136
\(628\) 0 0
\(629\) −15.5174 −0.618721
\(630\) 0 0
\(631\) 9.78992 0.389731 0.194865 0.980830i \(-0.437573\pi\)
0.194865 + 0.980830i \(0.437573\pi\)
\(632\) 0 0
\(633\) 9.57531 0.380584
\(634\) 0 0
\(635\) 2.24128 0.0889423
\(636\) 0 0
\(637\) 25.3340 1.00377
\(638\) 0 0
\(639\) 4.68035 0.185152
\(640\) 0 0
\(641\) 0.210079 0.00829764 0.00414882 0.999991i \(-0.498679\pi\)
0.00414882 + 0.999991i \(0.498679\pi\)
\(642\) 0 0
\(643\) −14.5236 −0.572754 −0.286377 0.958117i \(-0.592451\pi\)
−0.286377 + 0.958117i \(0.592451\pi\)
\(644\) 0 0
\(645\) 7.60197 0.299327
\(646\) 0 0
\(647\) −15.4641 −0.607957 −0.303979 0.952679i \(-0.598315\pi\)
−0.303979 + 0.952679i \(0.598315\pi\)
\(648\) 0 0
\(649\) 6.15676 0.241674
\(650\) 0 0
\(651\) 5.04718 0.197815
\(652\) 0 0
\(653\) −17.8310 −0.697779 −0.348890 0.937164i \(-0.613441\pi\)
−0.348890 + 0.937164i \(0.613441\pi\)
\(654\) 0 0
\(655\) −8.68035 −0.339169
\(656\) 0 0
\(657\) −10.4969 −0.409524
\(658\) 0 0
\(659\) 32.3135 1.25876 0.629378 0.777099i \(-0.283310\pi\)
0.629378 + 0.777099i \(0.283310\pi\)
\(660\) 0 0
\(661\) −5.68649 −0.221179 −0.110589 0.993866i \(-0.535274\pi\)
−0.110589 + 0.993866i \(0.535274\pi\)
\(662\) 0 0
\(663\) 33.6742 1.30780
\(664\) 0 0
\(665\) 5.67420 0.220036
\(666\) 0 0
\(667\) 3.05947 0.118463
\(668\) 0 0
\(669\) −2.15676 −0.0833850
\(670\) 0 0
\(671\) 4.15676 0.160470
\(672\) 0 0
\(673\) −21.0205 −0.810281 −0.405141 0.914254i \(-0.632777\pi\)
−0.405141 + 0.914254i \(0.632777\pi\)
\(674\) 0 0
\(675\) −1.00000 −0.0384900
\(676\) 0 0
\(677\) 36.7526 1.41252 0.706258 0.707954i \(-0.250382\pi\)
0.706258 + 0.707954i \(0.250382\pi\)
\(678\) 0 0
\(679\) −15.8310 −0.607536
\(680\) 0 0
\(681\) 9.65983 0.370165
\(682\) 0 0
\(683\) 17.3074 0.662248 0.331124 0.943587i \(-0.392572\pi\)
0.331124 + 0.943587i \(0.392572\pi\)
\(684\) 0 0
\(685\) −15.3607 −0.586902
\(686\) 0 0
\(687\) 3.36069 0.128218
\(688\) 0 0
\(689\) −0.680346 −0.0259191
\(690\) 0 0
\(691\) −17.6742 −0.672358 −0.336179 0.941798i \(-0.609135\pi\)
−0.336179 + 0.941798i \(0.609135\pi\)
\(692\) 0 0
\(693\) 1.07838 0.0409642
\(694\) 0 0
\(695\) −8.58145 −0.325513
\(696\) 0 0
\(697\) −73.0759 −2.76795
\(698\) 0 0
\(699\) 2.39803 0.0907019
\(700\) 0 0
\(701\) −17.1050 −0.646048 −0.323024 0.946391i \(-0.604700\pi\)
−0.323024 + 0.946391i \(0.604700\pi\)
\(702\) 0 0
\(703\) 10.5236 0.396905
\(704\) 0 0
\(705\) 4.68035 0.176272
\(706\) 0 0
\(707\) 16.7961 0.631681
\(708\) 0 0
\(709\) 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
\(712\) 0 0
\(713\) 10.0944 0.378037
\(714\) 0 0
\(715\) 4.34017 0.162313
\(716\) 0 0
\(717\) −7.20394 −0.269036
\(718\) 0 0
\(719\) 1.78992 0.0667528 0.0333764 0.999443i \(-0.489374\pi\)
0.0333764 + 0.999443i \(0.489374\pi\)
\(720\) 0 0
\(721\) 7.37298 0.274584
\(722\) 0 0
\(723\) 5.20394 0.193536
\(724\) 0 0
\(725\) 1.41855 0.0526837
\(726\) 0 0
\(727\) −25.9877 −0.963831 −0.481915 0.876218i \(-0.660059\pi\)
−0.481915 + 0.876218i \(0.660059\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −58.9816 −2.18151
\(732\) 0 0
\(733\) −41.0205 −1.51513 −0.757564 0.652761i \(-0.773610\pi\)
−0.757564 + 0.652761i \(0.773610\pi\)
\(734\) 0 0
\(735\) 5.83710 0.215305
\(736\) 0 0
\(737\) −8.68035 −0.319745
\(738\) 0 0
\(739\) 47.6163 1.75160 0.875798 0.482678i \(-0.160336\pi\)
0.875798 + 0.482678i \(0.160336\pi\)
\(740\) 0 0
\(741\) −22.8371 −0.838942
\(742\) 0 0
\(743\) −0.550252 −0.0201868 −0.0100934 0.999949i \(-0.503213\pi\)
−0.0100934 + 0.999949i \(0.503213\pi\)
\(744\) 0 0
\(745\) −18.0989 −0.663092
\(746\) 0 0
\(747\) 11.0205 0.403220
\(748\) 0 0
\(749\) 6.83710 0.249822
\(750\) 0 0
\(751\) −41.5585 −1.51649 −0.758245 0.651969i \(-0.773943\pi\)
−0.758245 + 0.651969i \(0.773943\pi\)
\(752\) 0 0
\(753\) 15.3197 0.558279
\(754\) 0 0
\(755\) −22.9360 −0.834726
\(756\) 0 0
\(757\) 1.31965 0.0479636 0.0239818 0.999712i \(-0.492366\pi\)
0.0239818 + 0.999712i \(0.492366\pi\)
\(758\) 0 0
\(759\) 2.15676 0.0782853
\(760\) 0 0
\(761\) −2.21461 −0.0802797 −0.0401399 0.999194i \(-0.512780\pi\)
−0.0401399 + 0.999194i \(0.512780\pi\)
\(762\) 0 0
\(763\) −2.49484 −0.0903192
\(764\) 0 0
\(765\) 7.75872 0.280517
\(766\) 0 0
\(767\) 26.7214 0.964853
\(768\) 0 0
\(769\) −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 0
\(773\) 40.1568 1.44434 0.722169 0.691717i \(-0.243145\pi\)
0.722169 + 0.691717i \(0.243145\pi\)
\(774\) 0 0
\(775\) 4.68035 0.168123
\(776\) 0 0
\(777\) −2.15676 −0.0773732
\(778\) 0 0
\(779\) 49.5585 1.77562
\(780\) 0 0
\(781\) −4.68035 −0.167476
\(782\) 0 0
\(783\) −1.41855 −0.0506949
\(784\) 0 0
\(785\) −10.9939 −0.392388
\(786\) 0 0
\(787\) −49.5897 −1.76768 −0.883841 0.467788i \(-0.845051\pi\)
−0.883841 + 0.467788i \(0.845051\pi\)
\(788\) 0 0
\(789\) −18.7070 −0.665987
\(790\) 0 0
\(791\) 6.47027 0.230056
\(792\) 0 0
\(793\) 18.0410 0.640656
\(794\) 0 0
\(795\) −0.156755 −0.00555954
\(796\) 0 0
\(797\) −46.7091 −1.65452 −0.827261 0.561818i \(-0.810102\pi\)
−0.827261 + 0.561818i \(0.810102\pi\)
\(798\) 0 0
\(799\) −36.3135 −1.28468
\(800\) 0 0
\(801\) −12.8371 −0.453577
\(802\) 0 0
\(803\) 10.4969 0.370429
\(804\) 0 0
\(805\) −2.32580 −0.0819736
\(806\) 0 0
\(807\) −23.3607 −0.822335
\(808\) 0 0
\(809\) −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 \(-0.659133\pi\)
−0.479366 + 0.877615i \(0.659133\pi\)
\(812\) 0 0
\(813\) −5.57531 −0.195535
\(814\) 0 0
\(815\) 6.52359 0.228511
\(816\) 0 0
\(817\) 40.0000 1.39942
\(818\) 0 0
\(819\) 4.68035 0.163545
\(820\) 0 0
\(821\) −31.2085 −1.08918 −0.544592 0.838701i \(-0.683315\pi\)
−0.544592 + 0.838701i \(0.683315\pi\)
\(822\) 0 0
\(823\) 50.1855 1.74936 0.874678 0.484704i \(-0.161073\pi\)
0.874678 + 0.484704i \(0.161073\pi\)
\(824\) 0 0
\(825\) 1.00000 0.0348155
\(826\) 0 0
\(827\) −27.3874 −0.952352 −0.476176 0.879350i \(-0.657977\pi\)
−0.476176 + 0.879350i \(0.657977\pi\)
\(828\) 0 0
\(829\) −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
\(832\) 0 0
\(833\) −45.2885 −1.56915
\(834\) 0 0
\(835\) −1.97334 −0.0682902
\(836\) 0 0
\(837\) −4.68035 −0.161776
\(838\) 0 0
\(839\) −7.20394 −0.248708 −0.124354 0.992238i \(-0.539686\pi\)
−0.124354 + 0.992238i \(0.539686\pi\)
\(840\) 0 0
\(841\) −26.9877 −0.930611
\(842\) 0 0
\(843\) 9.41855 0.324392
\(844\) 0 0
\(845\) 5.83710 0.200802
\(846\) 0 0
\(847\) −1.07838 −0.0370535
\(848\) 0 0
\(849\) 14.2413 0.488759
\(850\) 0 0
\(851\) −4.31351 −0.147865
\(852\) 0 0
\(853\) 39.8043 1.36287 0.681437 0.731877i \(-0.261356\pi\)
0.681437 + 0.731877i \(0.261356\pi\)
\(854\) 0 0
\(855\) −5.26180 −0.179950
\(856\) 0 0
\(857\) −36.9504 −1.26220 −0.631100 0.775701i \(-0.717396\pi\)
−0.631100 + 0.775701i \(0.717396\pi\)
\(858\) 0 0
\(859\) −57.5052 −1.96205 −0.981025 0.193879i \(-0.937893\pi\)
−0.981025 + 0.193879i \(0.937893\pi\)
\(860\) 0 0
\(861\) −10.1568 −0.346141
\(862\) 0 0
\(863\) 1.89657 0.0645599 0.0322800 0.999479i \(-0.489723\pi\)
0.0322800 + 0.999479i \(0.489723\pi\)
\(864\) 0 0
\(865\) 3.75872 0.127800
\(866\) 0 0
\(867\) −43.1978 −1.46707
\(868\) 0 0
\(869\) −8.09890 −0.274736
\(870\) 0 0
\(871\) −37.6742 −1.27654
\(872\) 0 0
\(873\) 14.6803 0.496854
\(874\) 0 0
\(875\) −1.07838 −0.0364558
\(876\) 0 0
\(877\) 32.5380 1.09873 0.549365 0.835583i \(-0.314870\pi\)
0.549365 + 0.835583i \(0.314870\pi\)
\(878\) 0 0
\(879\) 15.7587 0.531529
\(880\) 0 0
\(881\) 18.1978 0.613099 0.306550 0.951855i \(-0.400825\pi\)
0.306550 + 0.951855i \(0.400825\pi\)
\(882\) 0 0
\(883\) −36.3956 −1.22481 −0.612405 0.790545i \(-0.709798\pi\)
−0.612405 + 0.790545i \(0.709798\pi\)
\(884\) 0 0
\(885\) 6.15676 0.206957
\(886\) 0 0
\(887\) −27.8699 −0.935780 −0.467890 0.883787i \(-0.654986\pi\)
−0.467890 + 0.883787i \(0.654986\pi\)
\(888\) 0 0
\(889\) −2.41694 −0.0810616
\(890\) 0 0
\(891\) −1.00000 −0.0335013
\(892\) 0 0
\(893\) 24.6270 0.824112
\(894\) 0 0
\(895\) −15.1506 −0.506429
\(896\) 0 0
\(897\) 9.36069 0.312544
\(898\) 0 0
\(899\) 6.63931 0.221433
\(900\) 0 0
\(901\) 1.21622 0.0405182
\(902\) 0 0
\(903\) −8.19779 −0.272805
\(904\) 0 0
\(905\) 4.83710 0.160791
\(906\) 0 0
\(907\) 27.9376 0.927653 0.463826 0.885926i \(-0.346476\pi\)
0.463826 + 0.885926i \(0.346476\pi\)
\(908\) 0 0
\(909\) −15.5753 −0.516600
\(910\) 0 0
\(911\) 11.8843 0.393744 0.196872 0.980429i \(-0.436922\pi\)
0.196872 + 0.980429i \(0.436922\pi\)
\(912\) 0 0
\(913\) −11.0205 −0.364726
\(914\) 0 0
\(915\) 4.15676 0.137418
\(916\) 0 0
\(917\) 9.36069 0.309117
\(918\) 0 0
\(919\) 45.6041 1.50434 0.752170 0.658970i \(-0.229007\pi\)
0.752170 + 0.658970i \(0.229007\pi\)
\(920\) 0 0
\(921\) −18.9216 −0.623489
\(922\) 0 0
\(923\) −20.3135 −0.668627
\(924\) 0 0
\(925\) −2.00000 −0.0657596
\(926\) 0 0
\(927\) −6.83710 −0.224560
\(928\) 0 0
\(929\) −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
\(932\) 0 0
\(933\) −20.8781 −0.683520
\(934\) 0 0
\(935\) −7.75872 −0.253737
\(936\) 0 0
\(937\) 5.33403 0.174255 0.0871276 0.996197i \(-0.472231\pi\)
0.0871276 + 0.996197i \(0.472231\pi\)
\(938\) 0 0
\(939\) −6.31351 −0.206034
\(940\) 0 0
\(941\) 56.8203 1.85229 0.926144 0.377170i \(-0.123103\pi\)
0.926144 + 0.377170i \(0.123103\pi\)
\(942\) 0 0
\(943\) −20.3135 −0.661499
\(944\) 0 0
\(945\) 1.07838 0.0350796
\(946\) 0 0
\(947\) 20.9939 0.682209 0.341104 0.940025i \(-0.389199\pi\)
0.341104 + 0.940025i \(0.389199\pi\)
\(948\) 0 0
\(949\) 45.5585 1.47889
\(950\) 0 0
\(951\) −31.3607 −1.01694
\(952\) 0 0
\(953\) −25.2351 −0.817446 −0.408723 0.912658i \(-0.634026\pi\)
−0.408723 + 0.912658i \(0.634026\pi\)
\(954\) 0 0
\(955\) −2.52359 −0.0816615
\(956\) 0 0
\(957\) 1.41855 0.0458552
\(958\) 0 0
\(959\) 16.5646 0.534900
\(960\) 0 0
\(961\) −9.09436 −0.293367
\(962\) 0 0
\(963\) −6.34017 −0.204309
\(964\) 0 0
\(965\) 0.0266620 0.000858280 0
\(966\) 0 0
\(967\) −13.1317 −0.422287 −0.211144 0.977455i \(-0.567719\pi\)
−0.211144 + 0.977455i \(0.567719\pi\)
\(968\) 0 0
\(969\) 40.8248 1.31148
\(970\) 0 0
\(971\) −8.94053 −0.286915 −0.143458 0.989656i \(-0.545822\pi\)
−0.143458 + 0.989656i \(0.545822\pi\)
\(972\) 0 0
\(973\) 9.25404 0.296671
\(974\) 0 0
\(975\) 4.34017 0.138997
\(976\) 0 0
\(977\) 50.3956 1.61230 0.806149 0.591713i \(-0.201548\pi\)
0.806149 + 0.591713i \(0.201548\pi\)
\(978\) 0 0
\(979\) 12.8371 0.410276
\(980\) 0 0
\(981\) 2.31351 0.0738647
\(982\) 0 0
\(983\) 32.1978 1.02695 0.513475 0.858105i \(-0.328358\pi\)
0.513475 + 0.858105i \(0.328358\pi\)
\(984\) 0 0
\(985\) 21.1194 0.672921
\(986\) 0 0
\(987\) −5.04718 −0.160654
\(988\) 0 0
\(989\) −16.3956 −0.521349
\(990\) 0 0
\(991\) 46.7747 1.48585 0.742924 0.669376i \(-0.233438\pi\)
0.742924 + 0.669376i \(0.233438\pi\)
\(992\) 0 0
\(993\) 19.2039 0.609419
\(994\) 0 0
\(995\) −10.5236 −0.333620
\(996\) 0 0
\(997\) 38.2122 1.21019 0.605096 0.796153i \(-0.293135\pi\)
0.605096 + 0.796153i \(0.293135\pi\)
\(998\) 0 0
\(999\) 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 2640.2.a.be.1.2 3
3.2 odd 2 7920.2.a.cj.1.2 3
4.3 odd 2 165.2.a.c.1.1 3
12.11 even 2 495.2.a.e.1.3 3
20.3 even 4 825.2.c.g.199.6 6
20.7 even 4 825.2.c.g.199.1 6
20.19 odd 2 825.2.a.k.1.3 3
28.27 even 2 8085.2.a.bk.1.1 3
44.43 even 2 1815.2.a.m.1.3 3
60.23 odd 4 2475.2.c.r.199.1 6
60.47 odd 4 2475.2.c.r.199.6 6
60.59 even 2 2475.2.a.bb.1.1 3
132.131 odd 2 5445.2.a.z.1.1 3
220.219 even 2 9075.2.a.cf.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
165.2.a.c.1.1 3 4.3 odd 2
495.2.a.e.1.3 3 12.11 even 2
825.2.a.k.1.3 3 20.19 odd 2
825.2.c.g.199.1 6 20.7 even 4
825.2.c.g.199.6 6 20.3 even 4
1815.2.a.m.1.3 3 44.43 even 2
2475.2.a.bb.1.1 3 60.59 even 2
2475.2.c.r.199.1 6 60.23 odd 4
2475.2.c.r.199.6 6 60.47 odd 4
2640.2.a.be.1.2 3 1.1 even 1 trivial
5445.2.a.z.1.1 3 132.131 odd 2
7920.2.a.cj.1.2 3 3.2 odd 2
8085.2.a.bk.1.1 3 28.27 even 2
9075.2.a.cf.1.1 3 220.219 even 2