Properties

Label 7623.2.a.cp.1.2
Level 7623
Weight 2
Character 7623.1
Self dual yes
Analytic conductor 60.870
Analytic rank 1
Dimension 6
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 7623 = 3^{2} \cdot 7 \cdot 11^{2} \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 7623.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(60.8699614608\)
Analytic rank: \(1\)
Dimension: \(6\)
Coefficient field: 6.6.7674048.1
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 847)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-1.10939\)
Character \(\chi\) = 7623.1

$q$-expansion

\(f(q)\) \(=\) \(q-2.10939 q^{2} +2.44952 q^{4} -0.492391 q^{5} +1.00000 q^{7} -0.948212 q^{8} +O(q^{10})\) \(q-2.10939 q^{2} +2.44952 q^{4} -0.492391 q^{5} +1.00000 q^{7} -0.948212 q^{8} +1.03864 q^{10} +5.30029 q^{13} -2.10939 q^{14} -2.89889 q^{16} -3.03721 q^{17} -4.66622 q^{19} -1.20612 q^{20} +5.63835 q^{23} -4.75755 q^{25} -11.1804 q^{26} +2.44952 q^{28} -6.92295 q^{29} -1.26565 q^{31} +8.01131 q^{32} +6.40665 q^{34} -0.492391 q^{35} +10.8759 q^{37} +9.84288 q^{38} +0.466891 q^{40} +1.44322 q^{41} +2.88224 q^{43} -11.8935 q^{46} +8.75522 q^{47} +1.00000 q^{49} +10.0355 q^{50} +12.9832 q^{52} -6.63835 q^{53} -0.948212 q^{56} +14.6032 q^{58} +8.35733 q^{59} -13.8953 q^{61} +2.66975 q^{62} -11.1012 q^{64} -2.60982 q^{65} -9.70431 q^{67} -7.43970 q^{68} +1.03864 q^{70} -5.94751 q^{71} -3.77421 q^{73} -22.9414 q^{74} -11.4300 q^{76} -8.80383 q^{79} +1.42739 q^{80} -3.04431 q^{82} -11.0898 q^{83} +1.49549 q^{85} -6.07976 q^{86} -3.10324 q^{89} +5.30029 q^{91} +13.8113 q^{92} -18.4682 q^{94} +2.29761 q^{95} -6.31676 q^{97} -2.10939 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q - 4q^{2} + 4q^{4} + 4q^{5} + 6q^{7} - 12q^{8} + O(q^{10}) \) \( 6q - 4q^{2} + 4q^{4} + 4q^{5} + 6q^{7} - 12q^{8} - 8q^{10} + 4q^{13} - 4q^{14} + 8q^{16} - 22q^{17} + 6q^{19} - 2q^{20} - 2q^{23} + 4q^{25} - 6q^{26} + 4q^{28} - 12q^{29} - 2q^{31} - 8q^{32} + 24q^{34} + 4q^{35} + 14q^{37} + 22q^{38} + 18q^{40} - 26q^{41} - 4q^{43} + 12q^{46} + 16q^{47} + 6q^{49} + 4q^{50} + 12q^{52} - 4q^{53} - 12q^{56} - 2q^{58} + 4q^{59} - 8q^{61} - 20q^{62} + 26q^{64} - 24q^{65} + 6q^{67} - 12q^{68} - 8q^{70} - 22q^{71} + 14q^{73} - 44q^{74} - 30q^{76} - 28q^{79} + 4q^{80} - 4q^{82} - 22q^{83} - 24q^{85} + 30q^{86} + 4q^{91} - 10q^{92} - 38q^{94} + 24q^{95} - 4q^{97} - 4q^{98} + O(q^{100}) \)

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.10939 −1.49156 −0.745781 0.666191i \(-0.767924\pi\)
−0.745781 + 0.666191i \(0.767924\pi\)
\(3\) 0 0
\(4\) 2.44952 1.22476
\(5\) −0.492391 −0.220204 −0.110102 0.993920i \(-0.535118\pi\)
−0.110102 + 0.993920i \(0.535118\pi\)
\(6\) 0 0
\(7\) 1.00000 0.377964
\(8\) −0.948212 −0.335243
\(9\) 0 0
\(10\) 1.03864 0.328448
\(11\) 0 0
\(12\) 0 0
\(13\) 5.30029 1.47004 0.735018 0.678047i \(-0.237174\pi\)
0.735018 + 0.678047i \(0.237174\pi\)
\(14\) −2.10939 −0.563758
\(15\) 0 0
\(16\) −2.89889 −0.724723
\(17\) −3.03721 −0.736631 −0.368315 0.929701i \(-0.620065\pi\)
−0.368315 + 0.929701i \(0.620065\pi\)
\(18\) 0 0
\(19\) −4.66622 −1.07050 −0.535252 0.844692i \(-0.679784\pi\)
−0.535252 + 0.844692i \(0.679784\pi\)
\(20\) −1.20612 −0.269697
\(21\) 0 0
\(22\) 0 0
\(23\) 5.63835 1.17568 0.587839 0.808978i \(-0.299979\pi\)
0.587839 + 0.808978i \(0.299979\pi\)
\(24\) 0 0
\(25\) −4.75755 −0.951510
\(26\) −11.1804 −2.19265
\(27\) 0 0
\(28\) 2.44952 0.462916
\(29\) −6.92295 −1.28556 −0.642780 0.766051i \(-0.722219\pi\)
−0.642780 + 0.766051i \(0.722219\pi\)
\(30\) 0 0
\(31\) −1.26565 −0.227317 −0.113659 0.993520i \(-0.536257\pi\)
−0.113659 + 0.993520i \(0.536257\pi\)
\(32\) 8.01131 1.41621
\(33\) 0 0
\(34\) 6.40665 1.09873
\(35\) −0.492391 −0.0832293
\(36\) 0 0
\(37\) 10.8759 1.78798 0.893990 0.448087i \(-0.147895\pi\)
0.893990 + 0.448087i \(0.147895\pi\)
\(38\) 9.84288 1.59673
\(39\) 0 0
\(40\) 0.466891 0.0738220
\(41\) 1.44322 0.225393 0.112696 0.993629i \(-0.464051\pi\)
0.112696 + 0.993629i \(0.464051\pi\)
\(42\) 0 0
\(43\) 2.88224 0.439537 0.219769 0.975552i \(-0.429470\pi\)
0.219769 + 0.975552i \(0.429470\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) −11.8935 −1.75360
\(47\) 8.75522 1.27708 0.638540 0.769589i \(-0.279539\pi\)
0.638540 + 0.769589i \(0.279539\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 10.0355 1.41924
\(51\) 0 0
\(52\) 12.9832 1.80044
\(53\) −6.63835 −0.911848 −0.455924 0.890019i \(-0.650691\pi\)
−0.455924 + 0.890019i \(0.650691\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −0.948212 −0.126710
\(57\) 0 0
\(58\) 14.6032 1.91749
\(59\) 8.35733 1.08803 0.544016 0.839075i \(-0.316903\pi\)
0.544016 + 0.839075i \(0.316903\pi\)
\(60\) 0 0
\(61\) −13.8953 −1.77911 −0.889554 0.456829i \(-0.848985\pi\)
−0.889554 + 0.456829i \(0.848985\pi\)
\(62\) 2.66975 0.339058
\(63\) 0 0
\(64\) −11.1012 −1.38765
\(65\) −2.60982 −0.323708
\(66\) 0 0
\(67\) −9.70431 −1.18557 −0.592785 0.805361i \(-0.701972\pi\)
−0.592785 + 0.805361i \(0.701972\pi\)
\(68\) −7.43970 −0.902196
\(69\) 0 0
\(70\) 1.03864 0.124142
\(71\) −5.94751 −0.705839 −0.352920 0.935654i \(-0.614811\pi\)
−0.352920 + 0.935654i \(0.614811\pi\)
\(72\) 0 0
\(73\) −3.77421 −0.441737 −0.220869 0.975304i \(-0.570889\pi\)
−0.220869 + 0.975304i \(0.570889\pi\)
\(74\) −22.9414 −2.66688
\(75\) 0 0
\(76\) −11.4300 −1.31111
\(77\) 0 0
\(78\) 0 0
\(79\) −8.80383 −0.990508 −0.495254 0.868748i \(-0.664925\pi\)
−0.495254 + 0.868748i \(0.664925\pi\)
\(80\) 1.42739 0.159587
\(81\) 0 0
\(82\) −3.04431 −0.336188
\(83\) −11.0898 −1.21726 −0.608632 0.793453i \(-0.708281\pi\)
−0.608632 + 0.793453i \(0.708281\pi\)
\(84\) 0 0
\(85\) 1.49549 0.162209
\(86\) −6.07976 −0.655597
\(87\) 0 0
\(88\) 0 0
\(89\) −3.10324 −0.328943 −0.164472 0.986382i \(-0.552592\pi\)
−0.164472 + 0.986382i \(0.552592\pi\)
\(90\) 0 0
\(91\) 5.30029 0.555622
\(92\) 13.8113 1.43992
\(93\) 0 0
\(94\) −18.4682 −1.90484
\(95\) 2.29761 0.235730
\(96\) 0 0
\(97\) −6.31676 −0.641370 −0.320685 0.947186i \(-0.603913\pi\)
−0.320685 + 0.947186i \(0.603913\pi\)
\(98\) −2.10939 −0.213080
\(99\) 0 0
\(100\) −11.6537 −1.16537
\(101\) −11.7984 −1.17399 −0.586993 0.809592i \(-0.699688\pi\)
−0.586993 + 0.809592i \(0.699688\pi\)
\(102\) 0 0
\(103\) 7.00565 0.690287 0.345144 0.938550i \(-0.387830\pi\)
0.345144 + 0.938550i \(0.387830\pi\)
\(104\) −5.02580 −0.492820
\(105\) 0 0
\(106\) 14.0029 1.36008
\(107\) −11.3547 −1.09770 −0.548850 0.835921i \(-0.684934\pi\)
−0.548850 + 0.835921i \(0.684934\pi\)
\(108\) 0 0
\(109\) 18.9414 1.81426 0.907129 0.420853i \(-0.138269\pi\)
0.907129 + 0.420853i \(0.138269\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) −2.89889 −0.273920
\(113\) 13.4961 1.26961 0.634804 0.772673i \(-0.281081\pi\)
0.634804 + 0.772673i \(0.281081\pi\)
\(114\) 0 0
\(115\) −2.77627 −0.258889
\(116\) −16.9579 −1.57450
\(117\) 0 0
\(118\) −17.6289 −1.62287
\(119\) −3.03721 −0.278420
\(120\) 0 0
\(121\) 0 0
\(122\) 29.3106 2.65365
\(123\) 0 0
\(124\) −3.10023 −0.278409
\(125\) 4.80453 0.429730
\(126\) 0 0
\(127\) −4.66064 −0.413565 −0.206782 0.978387i \(-0.566299\pi\)
−0.206782 + 0.978387i \(0.566299\pi\)
\(128\) 7.39409 0.653551
\(129\) 0 0
\(130\) 5.50512 0.482831
\(131\) −9.03676 −0.789545 −0.394773 0.918779i \(-0.629177\pi\)
−0.394773 + 0.918779i \(0.629177\pi\)
\(132\) 0 0
\(133\) −4.66622 −0.404613
\(134\) 20.4702 1.76835
\(135\) 0 0
\(136\) 2.87991 0.246951
\(137\) 1.63772 0.139920 0.0699600 0.997550i \(-0.477713\pi\)
0.0699600 + 0.997550i \(0.477713\pi\)
\(138\) 0 0
\(139\) −1.53472 −0.130173 −0.0650866 0.997880i \(-0.520732\pi\)
−0.0650866 + 0.997880i \(0.520732\pi\)
\(140\) −1.20612 −0.101936
\(141\) 0 0
\(142\) 12.5456 1.05280
\(143\) 0 0
\(144\) 0 0
\(145\) 3.40880 0.283086
\(146\) 7.96127 0.658879
\(147\) 0 0
\(148\) 26.6406 2.18985
\(149\) −13.4909 −1.10522 −0.552610 0.833440i \(-0.686368\pi\)
−0.552610 + 0.833440i \(0.686368\pi\)
\(150\) 0 0
\(151\) −12.2370 −0.995835 −0.497917 0.867225i \(-0.665902\pi\)
−0.497917 + 0.867225i \(0.665902\pi\)
\(152\) 4.42457 0.358880
\(153\) 0 0
\(154\) 0 0
\(155\) 0.623194 0.0500562
\(156\) 0 0
\(157\) 2.52042 0.201152 0.100576 0.994929i \(-0.467932\pi\)
0.100576 + 0.994929i \(0.467932\pi\)
\(158\) 18.5707 1.47741
\(159\) 0 0
\(160\) −3.94470 −0.311856
\(161\) 5.63835 0.444364
\(162\) 0 0
\(163\) 7.87905 0.617135 0.308567 0.951203i \(-0.400151\pi\)
0.308567 + 0.951203i \(0.400151\pi\)
\(164\) 3.53519 0.276052
\(165\) 0 0
\(166\) 23.3927 1.81562
\(167\) 2.05485 0.159009 0.0795047 0.996834i \(-0.474666\pi\)
0.0795047 + 0.996834i \(0.474666\pi\)
\(168\) 0 0
\(169\) 15.0931 1.16101
\(170\) −3.15458 −0.241945
\(171\) 0 0
\(172\) 7.06010 0.538327
\(173\) 23.2707 1.76923 0.884617 0.466318i \(-0.154420\pi\)
0.884617 + 0.466318i \(0.154420\pi\)
\(174\) 0 0
\(175\) −4.75755 −0.359637
\(176\) 0 0
\(177\) 0 0
\(178\) 6.54595 0.490640
\(179\) −17.6596 −1.31994 −0.659969 0.751293i \(-0.729431\pi\)
−0.659969 + 0.751293i \(0.729431\pi\)
\(180\) 0 0
\(181\) 15.4701 1.14988 0.574941 0.818195i \(-0.305025\pi\)
0.574941 + 0.818195i \(0.305025\pi\)
\(182\) −11.1804 −0.828745
\(183\) 0 0
\(184\) −5.34635 −0.394138
\(185\) −5.35518 −0.393720
\(186\) 0 0
\(187\) 0 0
\(188\) 21.4461 1.56412
\(189\) 0 0
\(190\) −4.84655 −0.351605
\(191\) −15.9385 −1.15327 −0.576635 0.817002i \(-0.695634\pi\)
−0.576635 + 0.817002i \(0.695634\pi\)
\(192\) 0 0
\(193\) 8.45386 0.608522 0.304261 0.952589i \(-0.401590\pi\)
0.304261 + 0.952589i \(0.401590\pi\)
\(194\) 13.3245 0.956643
\(195\) 0 0
\(196\) 2.44952 0.174966
\(197\) 14.3384 1.02157 0.510785 0.859708i \(-0.329355\pi\)
0.510785 + 0.859708i \(0.329355\pi\)
\(198\) 0 0
\(199\) −22.1343 −1.56906 −0.784528 0.620094i \(-0.787095\pi\)
−0.784528 + 0.620094i \(0.787095\pi\)
\(200\) 4.51117 0.318988
\(201\) 0 0
\(202\) 24.8874 1.75107
\(203\) −6.92295 −0.485896
\(204\) 0 0
\(205\) −0.710628 −0.0496324
\(206\) −14.7776 −1.02961
\(207\) 0 0
\(208\) −15.3650 −1.06537
\(209\) 0 0
\(210\) 0 0
\(211\) 2.18302 0.150286 0.0751428 0.997173i \(-0.476059\pi\)
0.0751428 + 0.997173i \(0.476059\pi\)
\(212\) −16.2608 −1.11679
\(213\) 0 0
\(214\) 23.9515 1.63729
\(215\) −1.41919 −0.0967878
\(216\) 0 0
\(217\) −1.26565 −0.0859178
\(218\) −39.9548 −2.70608
\(219\) 0 0
\(220\) 0 0
\(221\) −16.0981 −1.08287
\(222\) 0 0
\(223\) 27.2603 1.82549 0.912744 0.408533i \(-0.133960\pi\)
0.912744 + 0.408533i \(0.133960\pi\)
\(224\) 8.01131 0.535278
\(225\) 0 0
\(226\) −28.4685 −1.89370
\(227\) 13.5892 0.901945 0.450973 0.892538i \(-0.351077\pi\)
0.450973 + 0.892538i \(0.351077\pi\)
\(228\) 0 0
\(229\) −8.30141 −0.548573 −0.274286 0.961648i \(-0.588442\pi\)
−0.274286 + 0.961648i \(0.588442\pi\)
\(230\) 5.85624 0.386149
\(231\) 0 0
\(232\) 6.56443 0.430976
\(233\) −12.9476 −0.848229 −0.424114 0.905609i \(-0.639415\pi\)
−0.424114 + 0.905609i \(0.639415\pi\)
\(234\) 0 0
\(235\) −4.31099 −0.281218
\(236\) 20.4715 1.33258
\(237\) 0 0
\(238\) 6.40665 0.415281
\(239\) −1.89342 −0.122475 −0.0612375 0.998123i \(-0.519505\pi\)
−0.0612375 + 0.998123i \(0.519505\pi\)
\(240\) 0 0
\(241\) −11.6983 −0.753557 −0.376778 0.926303i \(-0.622968\pi\)
−0.376778 + 0.926303i \(0.622968\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) −34.0368 −2.17898
\(245\) −0.492391 −0.0314577
\(246\) 0 0
\(247\) −24.7323 −1.57368
\(248\) 1.20010 0.0762066
\(249\) 0 0
\(250\) −10.1346 −0.640970
\(251\) 13.1860 0.832291 0.416146 0.909298i \(-0.363381\pi\)
0.416146 + 0.909298i \(0.363381\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 9.83110 0.616858
\(255\) 0 0
\(256\) 6.60537 0.412836
\(257\) 15.3445 0.957165 0.478582 0.878043i \(-0.341151\pi\)
0.478582 + 0.878043i \(0.341151\pi\)
\(258\) 0 0
\(259\) 10.8759 0.675793
\(260\) −6.39280 −0.396465
\(261\) 0 0
\(262\) 19.0620 1.17766
\(263\) −10.4197 −0.642507 −0.321254 0.946993i \(-0.604104\pi\)
−0.321254 + 0.946993i \(0.604104\pi\)
\(264\) 0 0
\(265\) 3.26867 0.200792
\(266\) 9.84288 0.603506
\(267\) 0 0
\(268\) −23.7709 −1.45204
\(269\) −16.2712 −0.992074 −0.496037 0.868301i \(-0.665212\pi\)
−0.496037 + 0.868301i \(0.665212\pi\)
\(270\) 0 0
\(271\) −5.83922 −0.354707 −0.177354 0.984147i \(-0.556754\pi\)
−0.177354 + 0.984147i \(0.556754\pi\)
\(272\) 8.80454 0.533853
\(273\) 0 0
\(274\) −3.45459 −0.208700
\(275\) 0 0
\(276\) 0 0
\(277\) 15.9255 0.956868 0.478434 0.878123i \(-0.341205\pi\)
0.478434 + 0.878123i \(0.341205\pi\)
\(278\) 3.23732 0.194162
\(279\) 0 0
\(280\) 0.466891 0.0279021
\(281\) −10.2004 −0.608504 −0.304252 0.952592i \(-0.598406\pi\)
−0.304252 + 0.952592i \(0.598406\pi\)
\(282\) 0 0
\(283\) −16.1634 −0.960812 −0.480406 0.877046i \(-0.659511\pi\)
−0.480406 + 0.877046i \(0.659511\pi\)
\(284\) −14.5685 −0.864483
\(285\) 0 0
\(286\) 0 0
\(287\) 1.44322 0.0851905
\(288\) 0 0
\(289\) −7.77538 −0.457375
\(290\) −7.19049 −0.422240
\(291\) 0 0
\(292\) −9.24499 −0.541022
\(293\) −27.0517 −1.58038 −0.790188 0.612864i \(-0.790017\pi\)
−0.790188 + 0.612864i \(0.790017\pi\)
\(294\) 0 0
\(295\) −4.11508 −0.239589
\(296\) −10.3126 −0.599408
\(297\) 0 0
\(298\) 28.4576 1.64851
\(299\) 29.8849 1.72829
\(300\) 0 0
\(301\) 2.88224 0.166129
\(302\) 25.8126 1.48535
\(303\) 0 0
\(304\) 13.5269 0.775820
\(305\) 6.84192 0.391767
\(306\) 0 0
\(307\) 29.7251 1.69650 0.848250 0.529596i \(-0.177657\pi\)
0.848250 + 0.529596i \(0.177657\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) −1.31456 −0.0746619
\(311\) −22.4029 −1.27035 −0.635176 0.772367i \(-0.719072\pi\)
−0.635176 + 0.772367i \(0.719072\pi\)
\(312\) 0 0
\(313\) 9.94833 0.562313 0.281157 0.959662i \(-0.409282\pi\)
0.281157 + 0.959662i \(0.409282\pi\)
\(314\) −5.31655 −0.300030
\(315\) 0 0
\(316\) −21.5652 −1.21313
\(317\) 11.1420 0.625796 0.312898 0.949787i \(-0.398700\pi\)
0.312898 + 0.949787i \(0.398700\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 5.46613 0.305566
\(321\) 0 0
\(322\) −11.8935 −0.662797
\(323\) 14.1723 0.788567
\(324\) 0 0
\(325\) −25.2164 −1.39875
\(326\) −16.6200 −0.920495
\(327\) 0 0
\(328\) −1.36848 −0.0755615
\(329\) 8.75522 0.482691
\(330\) 0 0
\(331\) 14.5950 0.802214 0.401107 0.916031i \(-0.368626\pi\)
0.401107 + 0.916031i \(0.368626\pi\)
\(332\) −27.1647 −1.49086
\(333\) 0 0
\(334\) −4.33449 −0.237172
\(335\) 4.77832 0.261067
\(336\) 0 0
\(337\) 12.6059 0.686688 0.343344 0.939210i \(-0.388440\pi\)
0.343344 + 0.939210i \(0.388440\pi\)
\(338\) −31.8372 −1.73172
\(339\) 0 0
\(340\) 3.66324 0.198667
\(341\) 0 0
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) −2.73297 −0.147352
\(345\) 0 0
\(346\) −49.0868 −2.63893
\(347\) −0.410734 −0.0220494 −0.0110247 0.999939i \(-0.503509\pi\)
−0.0110247 + 0.999939i \(0.503509\pi\)
\(348\) 0 0
\(349\) 13.4025 0.717422 0.358711 0.933449i \(-0.383216\pi\)
0.358711 + 0.933449i \(0.383216\pi\)
\(350\) 10.0355 0.536421
\(351\) 0 0
\(352\) 0 0
\(353\) −12.3419 −0.656892 −0.328446 0.944523i \(-0.606525\pi\)
−0.328446 + 0.944523i \(0.606525\pi\)
\(354\) 0 0
\(355\) 2.92850 0.155429
\(356\) −7.60146 −0.402877
\(357\) 0 0
\(358\) 37.2509 1.96877
\(359\) 25.0097 1.31996 0.659981 0.751283i \(-0.270565\pi\)
0.659981 + 0.751283i \(0.270565\pi\)
\(360\) 0 0
\(361\) 2.77364 0.145981
\(362\) −32.6324 −1.71512
\(363\) 0 0
\(364\) 12.9832 0.680503
\(365\) 1.85839 0.0972724
\(366\) 0 0
\(367\) −2.06915 −0.108009 −0.0540043 0.998541i \(-0.517198\pi\)
−0.0540043 + 0.998541i \(0.517198\pi\)
\(368\) −16.3450 −0.852041
\(369\) 0 0
\(370\) 11.2961 0.587259
\(371\) −6.63835 −0.344646
\(372\) 0 0
\(373\) −14.7623 −0.764365 −0.382183 0.924087i \(-0.624828\pi\)
−0.382183 + 0.924087i \(0.624828\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) −8.30180 −0.428133
\(377\) −36.6937 −1.88982
\(378\) 0 0
\(379\) 27.7508 1.42546 0.712730 0.701438i \(-0.247458\pi\)
0.712730 + 0.701438i \(0.247458\pi\)
\(380\) 5.62803 0.288712
\(381\) 0 0
\(382\) 33.6205 1.72017
\(383\) 18.0334 0.921464 0.460732 0.887539i \(-0.347587\pi\)
0.460732 + 0.887539i \(0.347587\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −17.8325 −0.907649
\(387\) 0 0
\(388\) −15.4730 −0.785524
\(389\) −13.5412 −0.686564 −0.343282 0.939232i \(-0.611539\pi\)
−0.343282 + 0.939232i \(0.611539\pi\)
\(390\) 0 0
\(391\) −17.1248 −0.866040
\(392\) −0.948212 −0.0478919
\(393\) 0 0
\(394\) −30.2453 −1.52374
\(395\) 4.33493 0.218114
\(396\) 0 0
\(397\) −24.6525 −1.23727 −0.618637 0.785677i \(-0.712315\pi\)
−0.618637 + 0.785677i \(0.712315\pi\)
\(398\) 46.6897 2.34035
\(399\) 0 0
\(400\) 13.7916 0.689581
\(401\) −18.7389 −0.935778 −0.467889 0.883787i \(-0.654985\pi\)
−0.467889 + 0.883787i \(0.654985\pi\)
\(402\) 0 0
\(403\) −6.70831 −0.334165
\(404\) −28.9004 −1.43785
\(405\) 0 0
\(406\) 14.6032 0.724745
\(407\) 0 0
\(408\) 0 0
\(409\) 15.7098 0.776801 0.388401 0.921491i \(-0.373028\pi\)
0.388401 + 0.921491i \(0.373028\pi\)
\(410\) 1.49899 0.0740299
\(411\) 0 0
\(412\) 17.1605 0.845436
\(413\) 8.35733 0.411238
\(414\) 0 0
\(415\) 5.46052 0.268046
\(416\) 42.4623 2.08189
\(417\) 0 0
\(418\) 0 0
\(419\) −22.6034 −1.10425 −0.552125 0.833761i \(-0.686183\pi\)
−0.552125 + 0.833761i \(0.686183\pi\)
\(420\) 0 0
\(421\) −23.3311 −1.13709 −0.568544 0.822653i \(-0.692493\pi\)
−0.568544 + 0.822653i \(0.692493\pi\)
\(422\) −4.60485 −0.224160
\(423\) 0 0
\(424\) 6.29456 0.305691
\(425\) 14.4497 0.700912
\(426\) 0 0
\(427\) −13.8953 −0.672440
\(428\) −27.8136 −1.34442
\(429\) 0 0
\(430\) 2.99362 0.144365
\(431\) 9.53898 0.459476 0.229738 0.973252i \(-0.426213\pi\)
0.229738 + 0.973252i \(0.426213\pi\)
\(432\) 0 0
\(433\) −23.0105 −1.10581 −0.552907 0.833243i \(-0.686482\pi\)
−0.552907 + 0.833243i \(0.686482\pi\)
\(434\) 2.66975 0.128152
\(435\) 0 0
\(436\) 46.3973 2.22203
\(437\) −26.3098 −1.25857
\(438\) 0 0
\(439\) −27.6434 −1.31935 −0.659673 0.751553i \(-0.729305\pi\)
−0.659673 + 0.751553i \(0.729305\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 33.9571 1.61518
\(443\) −14.7713 −0.701807 −0.350904 0.936412i \(-0.614125\pi\)
−0.350904 + 0.936412i \(0.614125\pi\)
\(444\) 0 0
\(445\) 1.52801 0.0724346
\(446\) −57.5026 −2.72283
\(447\) 0 0
\(448\) −11.1012 −0.524482
\(449\) −30.2669 −1.42838 −0.714192 0.699950i \(-0.753205\pi\)
−0.714192 + 0.699950i \(0.753205\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 33.0590 1.55496
\(453\) 0 0
\(454\) −28.6648 −1.34531
\(455\) −2.60982 −0.122350
\(456\) 0 0
\(457\) −20.2565 −0.947558 −0.473779 0.880644i \(-0.657111\pi\)
−0.473779 + 0.880644i \(0.657111\pi\)
\(458\) 17.5109 0.818231
\(459\) 0 0
\(460\) −6.80054 −0.317077
\(461\) −8.51184 −0.396436 −0.198218 0.980158i \(-0.563515\pi\)
−0.198218 + 0.980158i \(0.563515\pi\)
\(462\) 0 0
\(463\) −0.591469 −0.0274879 −0.0137440 0.999906i \(-0.504375\pi\)
−0.0137440 + 0.999906i \(0.504375\pi\)
\(464\) 20.0689 0.931675
\(465\) 0 0
\(466\) 27.3116 1.26519
\(467\) 41.0347 1.89886 0.949430 0.313979i \(-0.101662\pi\)
0.949430 + 0.313979i \(0.101662\pi\)
\(468\) 0 0
\(469\) −9.70431 −0.448103
\(470\) 9.09356 0.419454
\(471\) 0 0
\(472\) −7.92452 −0.364756
\(473\) 0 0
\(474\) 0 0
\(475\) 22.1998 1.01860
\(476\) −7.43970 −0.340998
\(477\) 0 0
\(478\) 3.99395 0.182679
\(479\) −20.3437 −0.929527 −0.464763 0.885435i \(-0.653861\pi\)
−0.464763 + 0.885435i \(0.653861\pi\)
\(480\) 0 0
\(481\) 57.6452 2.62840
\(482\) 24.6764 1.12398
\(483\) 0 0
\(484\) 0 0
\(485\) 3.11032 0.141232
\(486\) 0 0
\(487\) −28.8165 −1.30580 −0.652900 0.757444i \(-0.726448\pi\)
−0.652900 + 0.757444i \(0.726448\pi\)
\(488\) 13.1757 0.596435
\(489\) 0 0
\(490\) 1.03864 0.0469212
\(491\) −2.68045 −0.120967 −0.0604834 0.998169i \(-0.519264\pi\)
−0.0604834 + 0.998169i \(0.519264\pi\)
\(492\) 0 0
\(493\) 21.0264 0.946983
\(494\) 52.1701 2.34725
\(495\) 0 0
\(496\) 3.66898 0.164742
\(497\) −5.94751 −0.266782
\(498\) 0 0
\(499\) 22.3425 1.00019 0.500095 0.865971i \(-0.333298\pi\)
0.500095 + 0.865971i \(0.333298\pi\)
\(500\) 11.7688 0.526317
\(501\) 0 0
\(502\) −27.8143 −1.24141
\(503\) 4.47599 0.199575 0.0997873 0.995009i \(-0.468184\pi\)
0.0997873 + 0.995009i \(0.468184\pi\)
\(504\) 0 0
\(505\) 5.80943 0.258516
\(506\) 0 0
\(507\) 0 0
\(508\) −11.4163 −0.506518
\(509\) 17.1547 0.760368 0.380184 0.924911i \(-0.375861\pi\)
0.380184 + 0.924911i \(0.375861\pi\)
\(510\) 0 0
\(511\) −3.77421 −0.166961
\(512\) −28.7215 −1.26932
\(513\) 0 0
\(514\) −32.3676 −1.42767
\(515\) −3.44952 −0.152004
\(516\) 0 0
\(517\) 0 0
\(518\) −22.9414 −1.00799
\(519\) 0 0
\(520\) 2.47466 0.108521
\(521\) −1.00957 −0.0442300 −0.0221150 0.999755i \(-0.507040\pi\)
−0.0221150 + 0.999755i \(0.507040\pi\)
\(522\) 0 0
\(523\) −13.6433 −0.596578 −0.298289 0.954476i \(-0.596416\pi\)
−0.298289 + 0.954476i \(0.596416\pi\)
\(524\) −22.1357 −0.967004
\(525\) 0 0
\(526\) 21.9792 0.958340
\(527\) 3.84404 0.167449
\(528\) 0 0
\(529\) 8.79099 0.382217
\(530\) −6.89488 −0.299495
\(531\) 0 0
\(532\) −11.4300 −0.495554
\(533\) 7.64948 0.331336
\(534\) 0 0
\(535\) 5.59095 0.241718
\(536\) 9.20174 0.397455
\(537\) 0 0
\(538\) 34.3223 1.47974
\(539\) 0 0
\(540\) 0 0
\(541\) −2.76335 −0.118806 −0.0594028 0.998234i \(-0.518920\pi\)
−0.0594028 + 0.998234i \(0.518920\pi\)
\(542\) 12.3172 0.529068
\(543\) 0 0
\(544\) −24.3320 −1.04323
\(545\) −9.32658 −0.399507
\(546\) 0 0
\(547\) 15.2417 0.651687 0.325843 0.945424i \(-0.394352\pi\)
0.325843 + 0.945424i \(0.394352\pi\)
\(548\) 4.01163 0.171368
\(549\) 0 0
\(550\) 0 0
\(551\) 32.3041 1.37620
\(552\) 0 0
\(553\) −8.80383 −0.374377
\(554\) −33.5930 −1.42723
\(555\) 0 0
\(556\) −3.75933 −0.159431
\(557\) −3.56730 −0.151151 −0.0755757 0.997140i \(-0.524079\pi\)
−0.0755757 + 0.997140i \(0.524079\pi\)
\(558\) 0 0
\(559\) 15.2767 0.646136
\(560\) 1.42739 0.0603182
\(561\) 0 0
\(562\) 21.5166 0.907621
\(563\) −36.7500 −1.54883 −0.774414 0.632679i \(-0.781955\pi\)
−0.774414 + 0.632679i \(0.781955\pi\)
\(564\) 0 0
\(565\) −6.64537 −0.279573
\(566\) 34.0948 1.43311
\(567\) 0 0
\(568\) 5.63949 0.236628
\(569\) 34.0802 1.42872 0.714359 0.699779i \(-0.246718\pi\)
0.714359 + 0.699779i \(0.246718\pi\)
\(570\) 0 0
\(571\) −5.79312 −0.242434 −0.121217 0.992626i \(-0.538680\pi\)
−0.121217 + 0.992626i \(0.538680\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) −3.04431 −0.127067
\(575\) −26.8247 −1.11867
\(576\) 0 0
\(577\) −31.1231 −1.29567 −0.647837 0.761779i \(-0.724326\pi\)
−0.647837 + 0.761779i \(0.724326\pi\)
\(578\) 16.4013 0.682204
\(579\) 0 0
\(580\) 8.34993 0.346712
\(581\) −11.0898 −0.460082
\(582\) 0 0
\(583\) 0 0
\(584\) 3.57875 0.148090
\(585\) 0 0
\(586\) 57.0625 2.35723
\(587\) 47.6946 1.96857 0.984283 0.176601i \(-0.0565102\pi\)
0.984283 + 0.176601i \(0.0565102\pi\)
\(588\) 0 0
\(589\) 5.90580 0.243344
\(590\) 8.68030 0.357362
\(591\) 0 0
\(592\) −31.5279 −1.29579
\(593\) −44.5859 −1.83092 −0.915462 0.402404i \(-0.868175\pi\)
−0.915462 + 0.402404i \(0.868175\pi\)
\(594\) 0 0
\(595\) 1.49549 0.0613093
\(596\) −33.0463 −1.35363
\(597\) 0 0
\(598\) −63.0389 −2.57785
\(599\) 8.06937 0.329706 0.164853 0.986318i \(-0.447285\pi\)
0.164853 + 0.986318i \(0.447285\pi\)
\(600\) 0 0
\(601\) −28.9086 −1.17921 −0.589603 0.807693i \(-0.700716\pi\)
−0.589603 + 0.807693i \(0.700716\pi\)
\(602\) −6.07976 −0.247792
\(603\) 0 0
\(604\) −29.9748 −1.21966
\(605\) 0 0
\(606\) 0 0
\(607\) −9.85310 −0.399925 −0.199962 0.979804i \(-0.564082\pi\)
−0.199962 + 0.979804i \(0.564082\pi\)
\(608\) −37.3826 −1.51606
\(609\) 0 0
\(610\) −14.4323 −0.584345
\(611\) 46.4052 1.87735
\(612\) 0 0
\(613\) −35.8329 −1.44728 −0.723638 0.690180i \(-0.757531\pi\)
−0.723638 + 0.690180i \(0.757531\pi\)
\(614\) −62.7017 −2.53044
\(615\) 0 0
\(616\) 0 0
\(617\) 38.4398 1.54753 0.773764 0.633474i \(-0.218372\pi\)
0.773764 + 0.633474i \(0.218372\pi\)
\(618\) 0 0
\(619\) 4.19552 0.168632 0.0843162 0.996439i \(-0.473129\pi\)
0.0843162 + 0.996439i \(0.473129\pi\)
\(620\) 1.52653 0.0613068
\(621\) 0 0
\(622\) 47.2564 1.89481
\(623\) −3.10324 −0.124329
\(624\) 0 0
\(625\) 21.4220 0.856882
\(626\) −20.9849 −0.838725
\(627\) 0 0
\(628\) 6.17382 0.246362
\(629\) −33.0322 −1.31708
\(630\) 0 0
\(631\) −3.51798 −0.140049 −0.0700243 0.997545i \(-0.522308\pi\)
−0.0700243 + 0.997545i \(0.522308\pi\)
\(632\) 8.34790 0.332061
\(633\) 0 0
\(634\) −23.5028 −0.933414
\(635\) 2.29486 0.0910686
\(636\) 0 0
\(637\) 5.30029 0.210005
\(638\) 0 0
\(639\) 0 0
\(640\) −3.64078 −0.143915
\(641\) −16.0952 −0.635723 −0.317862 0.948137i \(-0.602965\pi\)
−0.317862 + 0.948137i \(0.602965\pi\)
\(642\) 0 0
\(643\) −34.8261 −1.37341 −0.686704 0.726938i \(-0.740943\pi\)
−0.686704 + 0.726938i \(0.740943\pi\)
\(644\) 13.8113 0.544239
\(645\) 0 0
\(646\) −29.8948 −1.17620
\(647\) −16.9051 −0.664607 −0.332304 0.943172i \(-0.607826\pi\)
−0.332304 + 0.943172i \(0.607826\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 53.1912 2.08633
\(651\) 0 0
\(652\) 19.2999 0.755842
\(653\) −3.52799 −0.138061 −0.0690304 0.997615i \(-0.521991\pi\)
−0.0690304 + 0.997615i \(0.521991\pi\)
\(654\) 0 0
\(655\) 4.44962 0.173861
\(656\) −4.18374 −0.163347
\(657\) 0 0
\(658\) −18.4682 −0.719964
\(659\) −29.4409 −1.14686 −0.573428 0.819256i \(-0.694387\pi\)
−0.573428 + 0.819256i \(0.694387\pi\)
\(660\) 0 0
\(661\) 15.2989 0.595059 0.297529 0.954713i \(-0.403837\pi\)
0.297529 + 0.954713i \(0.403837\pi\)
\(662\) −30.7865 −1.19655
\(663\) 0 0
\(664\) 10.5155 0.408080
\(665\) 2.29761 0.0890974
\(666\) 0 0
\(667\) −39.0340 −1.51140
\(668\) 5.03341 0.194748
\(669\) 0 0
\(670\) −10.0793 −0.389398
\(671\) 0 0
\(672\) 0 0
\(673\) 12.1652 0.468936 0.234468 0.972124i \(-0.424665\pi\)
0.234468 + 0.972124i \(0.424665\pi\)
\(674\) −26.5908 −1.02424
\(675\) 0 0
\(676\) 36.9709 1.42196
\(677\) 0.313619 0.0120533 0.00602667 0.999982i \(-0.498082\pi\)
0.00602667 + 0.999982i \(0.498082\pi\)
\(678\) 0 0
\(679\) −6.31676 −0.242415
\(680\) −1.41804 −0.0543795
\(681\) 0 0
\(682\) 0 0
\(683\) −38.2419 −1.46328 −0.731642 0.681689i \(-0.761246\pi\)
−0.731642 + 0.681689i \(0.761246\pi\)
\(684\) 0 0
\(685\) −0.806400 −0.0308110
\(686\) −2.10939 −0.0805368
\(687\) 0 0
\(688\) −8.35530 −0.318543
\(689\) −35.1852 −1.34045
\(690\) 0 0
\(691\) −10.2754 −0.390895 −0.195448 0.980714i \(-0.562616\pi\)
−0.195448 + 0.980714i \(0.562616\pi\)
\(692\) 57.0019 2.16689
\(693\) 0 0
\(694\) 0.866398 0.0328880
\(695\) 0.755683 0.0286647
\(696\) 0 0
\(697\) −4.38335 −0.166031
\(698\) −28.2712 −1.07008
\(699\) 0 0
\(700\) −11.6537 −0.440469
\(701\) −18.9188 −0.714552 −0.357276 0.933999i \(-0.616294\pi\)
−0.357276 + 0.933999i \(0.616294\pi\)
\(702\) 0 0
\(703\) −50.7492 −1.91404
\(704\) 0 0
\(705\) 0 0
\(706\) 26.0338 0.979795
\(707\) −11.7984 −0.443725
\(708\) 0 0
\(709\) −14.1498 −0.531409 −0.265704 0.964055i \(-0.585605\pi\)
−0.265704 + 0.964055i \(0.585605\pi\)
\(710\) −6.17734 −0.231832
\(711\) 0 0
\(712\) 2.94253 0.110276
\(713\) −7.13617 −0.267252
\(714\) 0 0
\(715\) 0 0
\(716\) −43.2575 −1.61661
\(717\) 0 0
\(718\) −52.7552 −1.96880
\(719\) 15.9330 0.594201 0.297101 0.954846i \(-0.403980\pi\)
0.297101 + 0.954846i \(0.403980\pi\)
\(720\) 0 0
\(721\) 7.00565 0.260904
\(722\) −5.85068 −0.217740
\(723\) 0 0
\(724\) 37.8943 1.40833
\(725\) 32.9363 1.22322
\(726\) 0 0
\(727\) −4.20455 −0.155938 −0.0779691 0.996956i \(-0.524844\pi\)
−0.0779691 + 0.996956i \(0.524844\pi\)
\(728\) −5.02580 −0.186269
\(729\) 0 0
\(730\) −3.92006 −0.145088
\(731\) −8.75395 −0.323777
\(732\) 0 0
\(733\) 34.0777 1.25869 0.629345 0.777126i \(-0.283323\pi\)
0.629345 + 0.777126i \(0.283323\pi\)
\(734\) 4.36464 0.161102
\(735\) 0 0
\(736\) 45.1706 1.66501
\(737\) 0 0
\(738\) 0 0
\(739\) 26.1306 0.961230 0.480615 0.876932i \(-0.340413\pi\)
0.480615 + 0.876932i \(0.340413\pi\)
\(740\) −13.1176 −0.482213
\(741\) 0 0
\(742\) 14.0029 0.514061
\(743\) 23.6263 0.866765 0.433383 0.901210i \(-0.357320\pi\)
0.433383 + 0.901210i \(0.357320\pi\)
\(744\) 0 0
\(745\) 6.64282 0.243374
\(746\) 31.1395 1.14010
\(747\) 0 0
\(748\) 0 0
\(749\) −11.3547 −0.414892
\(750\) 0 0
\(751\) −2.32359 −0.0847889 −0.0423945 0.999101i \(-0.513499\pi\)
−0.0423945 + 0.999101i \(0.513499\pi\)
\(752\) −25.3804 −0.925529
\(753\) 0 0
\(754\) 77.4012 2.81879
\(755\) 6.02540 0.219287
\(756\) 0 0
\(757\) −14.7651 −0.536647 −0.268324 0.963329i \(-0.586470\pi\)
−0.268324 + 0.963329i \(0.586470\pi\)
\(758\) −58.5371 −2.12616
\(759\) 0 0
\(760\) −2.17862 −0.0790268
\(761\) 47.1876 1.71055 0.855274 0.518175i \(-0.173389\pi\)
0.855274 + 0.518175i \(0.173389\pi\)
\(762\) 0 0
\(763\) 18.9414 0.685725
\(764\) −39.0417 −1.41248
\(765\) 0 0
\(766\) −38.0395 −1.37442
\(767\) 44.2963 1.59945
\(768\) 0 0
\(769\) −12.4418 −0.448662 −0.224331 0.974513i \(-0.572020\pi\)
−0.224331 + 0.974513i \(0.572020\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 20.7079 0.745294
\(773\) 35.2698 1.26857 0.634284 0.773101i \(-0.281295\pi\)
0.634284 + 0.773101i \(0.281295\pi\)
\(774\) 0 0
\(775\) 6.02139 0.216295
\(776\) 5.98962 0.215015
\(777\) 0 0
\(778\) 28.5636 1.02405
\(779\) −6.73438 −0.241284
\(780\) 0 0
\(781\) 0 0
\(782\) 36.1229 1.29175
\(783\) 0 0
\(784\) −2.89889 −0.103532
\(785\) −1.24103 −0.0442944
\(786\) 0 0
\(787\) −16.3383 −0.582397 −0.291198 0.956663i \(-0.594054\pi\)
−0.291198 + 0.956663i \(0.594054\pi\)
\(788\) 35.1223 1.25118
\(789\) 0 0
\(790\) −9.14405 −0.325331
\(791\) 13.4961 0.479867
\(792\) 0 0
\(793\) −73.6491 −2.61536
\(794\) 52.0017 1.84547
\(795\) 0 0
\(796\) −54.2183 −1.92172
\(797\) 0.292902 0.0103751 0.00518755 0.999987i \(-0.498349\pi\)
0.00518755 + 0.999987i \(0.498349\pi\)
\(798\) 0 0
\(799\) −26.5914 −0.940736
\(800\) −38.1142 −1.34754
\(801\) 0 0
\(802\) 39.5277 1.39577
\(803\) 0 0
\(804\) 0 0
\(805\) −2.77627 −0.0978508
\(806\) 14.1504 0.498428
\(807\) 0 0
\(808\) 11.1874 0.393571
\(809\) 11.0373 0.388050 0.194025 0.980997i \(-0.437846\pi\)
0.194025 + 0.980997i \(0.437846\pi\)
\(810\) 0 0
\(811\) −5.21763 −0.183216 −0.0916078 0.995795i \(-0.529201\pi\)
−0.0916078 + 0.995795i \(0.529201\pi\)
\(812\) −16.9579 −0.595106
\(813\) 0 0
\(814\) 0 0
\(815\) −3.87957 −0.135896
\(816\) 0 0
\(817\) −13.4492 −0.470527
\(818\) −33.1381 −1.15865
\(819\) 0 0
\(820\) −1.74070 −0.0607878
\(821\) −26.4468 −0.923000 −0.461500 0.887140i \(-0.652689\pi\)
−0.461500 + 0.887140i \(0.652689\pi\)
\(822\) 0 0
\(823\) −49.1895 −1.71464 −0.857319 0.514786i \(-0.827871\pi\)
−0.857319 + 0.514786i \(0.827871\pi\)
\(824\) −6.64284 −0.231414
\(825\) 0 0
\(826\) −17.6289 −0.613387
\(827\) 41.8006 1.45355 0.726774 0.686876i \(-0.241019\pi\)
0.726774 + 0.686876i \(0.241019\pi\)
\(828\) 0 0
\(829\) −24.8988 −0.864771 −0.432385 0.901689i \(-0.642328\pi\)
−0.432385 + 0.901689i \(0.642328\pi\)
\(830\) −11.5184 −0.399808
\(831\) 0 0
\(832\) −58.8395 −2.03989
\(833\) −3.03721 −0.105233
\(834\) 0 0
\(835\) −1.01179 −0.0350145
\(836\) 0 0
\(837\) 0 0
\(838\) 47.6794 1.64706
\(839\) 53.2912 1.83982 0.919908 0.392135i \(-0.128263\pi\)
0.919908 + 0.392135i \(0.128263\pi\)
\(840\) 0 0
\(841\) 18.9273 0.652666
\(842\) 49.2143 1.69604
\(843\) 0 0
\(844\) 5.34736 0.184064
\(845\) −7.43171 −0.255659
\(846\) 0 0
\(847\) 0 0
\(848\) 19.2439 0.660837
\(849\) 0 0
\(850\) −30.4800 −1.04545
\(851\) 61.3219 2.10209
\(852\) 0 0
\(853\) 13.3706 0.457800 0.228900 0.973450i \(-0.426487\pi\)
0.228900 + 0.973450i \(0.426487\pi\)
\(854\) 29.3106 1.00299
\(855\) 0 0
\(856\) 10.7667 0.367997
\(857\) −43.4419 −1.48395 −0.741974 0.670429i \(-0.766110\pi\)
−0.741974 + 0.670429i \(0.766110\pi\)
\(858\) 0 0
\(859\) −7.96898 −0.271898 −0.135949 0.990716i \(-0.543408\pi\)
−0.135949 + 0.990716i \(0.543408\pi\)
\(860\) −3.47633 −0.118542
\(861\) 0 0
\(862\) −20.1214 −0.685338
\(863\) −50.3471 −1.71384 −0.856918 0.515452i \(-0.827624\pi\)
−0.856918 + 0.515452i \(0.827624\pi\)
\(864\) 0 0
\(865\) −11.4583 −0.389593
\(866\) 48.5381 1.64939
\(867\) 0 0
\(868\) −3.10023 −0.105229
\(869\) 0 0
\(870\) 0 0
\(871\) −51.4357 −1.74283
\(872\) −17.9605 −0.608218
\(873\) 0 0
\(874\) 55.4976 1.87723
\(875\) 4.80453 0.162423
\(876\) 0 0
\(877\) −55.1801 −1.86330 −0.931649 0.363359i \(-0.881630\pi\)
−0.931649 + 0.363359i \(0.881630\pi\)
\(878\) 58.3106 1.96789
\(879\) 0 0
\(880\) 0 0
\(881\) 22.6475 0.763014 0.381507 0.924366i \(-0.375405\pi\)
0.381507 + 0.924366i \(0.375405\pi\)
\(882\) 0 0
\(883\) −12.7050 −0.427557 −0.213779 0.976882i \(-0.568577\pi\)
−0.213779 + 0.976882i \(0.568577\pi\)
\(884\) −39.4326 −1.32626
\(885\) 0 0
\(886\) 31.1585 1.04679
\(887\) −6.70004 −0.224965 −0.112483 0.993654i \(-0.535880\pi\)
−0.112483 + 0.993654i \(0.535880\pi\)
\(888\) 0 0
\(889\) −4.66064 −0.156313
\(890\) −3.22317 −0.108041
\(891\) 0 0
\(892\) 66.7747 2.23578
\(893\) −40.8538 −1.36712
\(894\) 0 0
\(895\) 8.69542 0.290656
\(896\) 7.39409 0.247019
\(897\) 0 0
\(898\) 63.8446 2.13052
\(899\) 8.76203 0.292230
\(900\) 0 0
\(901\) 20.1620 0.671695
\(902\) 0 0
\(903\) 0 0
\(904\) −12.7972 −0.425628
\(905\) −7.61733 −0.253209
\(906\) 0 0
\(907\) −57.7184 −1.91651 −0.958254 0.285918i \(-0.907701\pi\)
−0.958254 + 0.285918i \(0.907701\pi\)
\(908\) 33.2869 1.10467
\(909\) 0 0
\(910\) 5.50512 0.182493
\(911\) 2.84362 0.0942134 0.0471067 0.998890i \(-0.485000\pi\)
0.0471067 + 0.998890i \(0.485000\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 42.7288 1.41334
\(915\) 0 0
\(916\) −20.3345 −0.671870
\(917\) −9.03676 −0.298420
\(918\) 0 0
\(919\) −9.36824 −0.309030 −0.154515 0.987990i \(-0.549381\pi\)
−0.154515 + 0.987990i \(0.549381\pi\)
\(920\) 2.63250 0.0867908
\(921\) 0 0
\(922\) 17.9548 0.591309
\(923\) −31.5235 −1.03761
\(924\) 0 0
\(925\) −51.7424 −1.70128
\(926\) 1.24764 0.0410000
\(927\) 0 0
\(928\) −55.4620 −1.82063
\(929\) 43.8437 1.43847 0.719233 0.694769i \(-0.244494\pi\)
0.719233 + 0.694769i \(0.244494\pi\)
\(930\) 0 0
\(931\) −4.66622 −0.152929
\(932\) −31.7155 −1.03888
\(933\) 0 0
\(934\) −86.5581 −2.83227
\(935\) 0 0
\(936\) 0 0
\(937\) 3.20925 0.104842 0.0524208 0.998625i \(-0.483306\pi\)
0.0524208 + 0.998625i \(0.483306\pi\)
\(938\) 20.4702 0.668374
\(939\) 0 0
\(940\) −10.5599 −0.344425
\(941\) −11.1320 −0.362893 −0.181446 0.983401i \(-0.558078\pi\)
−0.181446 + 0.983401i \(0.558078\pi\)
\(942\) 0 0
\(943\) 8.13737 0.264989
\(944\) −24.2270 −0.788522
\(945\) 0 0
\(946\) 0 0
\(947\) −51.6934 −1.67981 −0.839905 0.542733i \(-0.817390\pi\)
−0.839905 + 0.542733i \(0.817390\pi\)
\(948\) 0 0
\(949\) −20.0044 −0.649370
\(950\) −46.8280 −1.51930
\(951\) 0 0
\(952\) 2.87991 0.0933386
\(953\) −28.0305 −0.907996 −0.453998 0.891003i \(-0.650003\pi\)
−0.453998 + 0.891003i \(0.650003\pi\)
\(954\) 0 0
\(955\) 7.84798 0.253955
\(956\) −4.63796 −0.150002
\(957\) 0 0
\(958\) 42.9127 1.38645
\(959\) 1.63772 0.0528848
\(960\) 0 0
\(961\) −29.3981 −0.948327
\(962\) −121.596 −3.92042
\(963\) 0 0
\(964\) −28.6553 −0.922926
\(965\) −4.16261 −0.133999
\(966\) 0 0
\(967\) −25.5912 −0.822957 −0.411479 0.911419i \(-0.634988\pi\)
−0.411479 + 0.911419i \(0.634988\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) −6.56086 −0.210657
\(971\) −12.5737 −0.403508 −0.201754 0.979436i \(-0.564664\pi\)
−0.201754 + 0.979436i \(0.564664\pi\)
\(972\) 0 0
\(973\) −1.53472 −0.0492009
\(974\) 60.7852 1.94768
\(975\) 0 0
\(976\) 40.2809 1.28936
\(977\) 14.5164 0.464420 0.232210 0.972666i \(-0.425404\pi\)
0.232210 + 0.972666i \(0.425404\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) −1.20612 −0.0385281
\(981\) 0 0
\(982\) 5.65410 0.180430
\(983\) 12.9883 0.414264 0.207132 0.978313i \(-0.433587\pi\)
0.207132 + 0.978313i \(0.433587\pi\)
\(984\) 0 0
\(985\) −7.06012 −0.224954
\(986\) −44.3529 −1.41249
\(987\) 0 0
\(988\) −60.5824 −1.92738
\(989\) 16.2511 0.516754
\(990\) 0 0
\(991\) 7.01006 0.222682 0.111341 0.993782i \(-0.464485\pi\)
0.111341 + 0.993782i \(0.464485\pi\)
\(992\) −10.1395 −0.321930
\(993\) 0 0
\(994\) 12.5456 0.397922
\(995\) 10.8987 0.345512
\(996\) 0 0
\(997\) −31.8789 −1.00961 −0.504807 0.863232i \(-0.668436\pi\)
−0.504807 + 0.863232i \(0.668436\pi\)
\(998\) −47.1291 −1.49185
\(999\) 0 0
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 7623.2.a.cp.1.2 6
3.2 odd 2 847.2.a.n.1.5 yes 6
11.10 odd 2 7623.2.a.cs.1.5 6
21.20 even 2 5929.2.a.bm.1.5 6
33.2 even 10 847.2.f.z.323.5 24
33.5 odd 10 847.2.f.y.729.2 24
33.8 even 10 847.2.f.z.372.2 24
33.14 odd 10 847.2.f.y.372.5 24
33.17 even 10 847.2.f.z.729.5 24
33.20 odd 10 847.2.f.y.323.2 24
33.26 odd 10 847.2.f.y.148.5 24
33.29 even 10 847.2.f.z.148.2 24
33.32 even 2 847.2.a.m.1.2 6
231.230 odd 2 5929.2.a.bj.1.2 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
847.2.a.m.1.2 6 33.32 even 2
847.2.a.n.1.5 yes 6 3.2 odd 2
847.2.f.y.148.5 24 33.26 odd 10
847.2.f.y.323.2 24 33.20 odd 10
847.2.f.y.372.5 24 33.14 odd 10
847.2.f.y.729.2 24 33.5 odd 10
847.2.f.z.148.2 24 33.29 even 10
847.2.f.z.323.5 24 33.2 even 10
847.2.f.z.372.2 24 33.8 even 10
847.2.f.z.729.5 24 33.17 even 10
5929.2.a.bj.1.2 6 231.230 odd 2
5929.2.a.bm.1.5 6 21.20 even 2
7623.2.a.cp.1.2 6 1.1 even 1 trivial
7623.2.a.cs.1.5 6 11.10 odd 2