Properties

Label 5520.2.a.bi.1.1
Level $5520$
Weight $2$
Character 5520.1
Self dual yes
Analytic conductor $44.077$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 5520 = 2^{4} \cdot 3 \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5520.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(44.0774219157\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{6}) \)
Defining polynomial: \(x^{2} - 6\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 345)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(2.44949\) of defining polynomial
Character \(\chi\) \(=\) 5520.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{3} -1.00000 q^{5} +1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{3} -1.00000 q^{5} +1.00000 q^{7} +1.00000 q^{9} -2.44949 q^{11} +4.44949 q^{13} +1.00000 q^{15} -5.44949 q^{17} -4.44949 q^{19} -1.00000 q^{21} +1.00000 q^{23} +1.00000 q^{25} -1.00000 q^{27} +10.3485 q^{29} -0.101021 q^{31} +2.44949 q^{33} -1.00000 q^{35} +3.89898 q^{37} -4.44949 q^{39} +5.44949 q^{41} -2.00000 q^{43} -1.00000 q^{45} -8.44949 q^{47} -6.00000 q^{49} +5.44949 q^{51} +0.550510 q^{53} +2.44949 q^{55} +4.44949 q^{57} -10.3485 q^{59} +0.651531 q^{61} +1.00000 q^{63} -4.44949 q^{65} +7.00000 q^{67} -1.00000 q^{69} +4.34847 q^{71} -5.34847 q^{73} -1.00000 q^{75} -2.44949 q^{77} +4.00000 q^{79} +1.00000 q^{81} -15.2474 q^{83} +5.44949 q^{85} -10.3485 q^{87} -16.8990 q^{89} +4.44949 q^{91} +0.101021 q^{93} +4.44949 q^{95} +3.10102 q^{97} -2.44949 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{3} - 2q^{5} + 2q^{7} + 2q^{9} + O(q^{10}) \) \( 2q - 2q^{3} - 2q^{5} + 2q^{7} + 2q^{9} + 4q^{13} + 2q^{15} - 6q^{17} - 4q^{19} - 2q^{21} + 2q^{23} + 2q^{25} - 2q^{27} + 6q^{29} - 10q^{31} - 2q^{35} - 2q^{37} - 4q^{39} + 6q^{41} - 4q^{43} - 2q^{45} - 12q^{47} - 12q^{49} + 6q^{51} + 6q^{53} + 4q^{57} - 6q^{59} + 16q^{61} + 2q^{63} - 4q^{65} + 14q^{67} - 2q^{69} - 6q^{71} + 4q^{73} - 2q^{75} + 8q^{79} + 2q^{81} - 6q^{83} + 6q^{85} - 6q^{87} - 24q^{89} + 4q^{91} + 10q^{93} + 4q^{95} + 16q^{97} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −1.00000 −0.577350
\(4\) 0 0
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) 1.00000 0.377964 0.188982 0.981981i \(-0.439481\pi\)
0.188982 + 0.981981i \(0.439481\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −2.44949 −0.738549 −0.369274 0.929320i \(-0.620394\pi\)
−0.369274 + 0.929320i \(0.620394\pi\)
\(12\) 0 0
\(13\) 4.44949 1.23407 0.617033 0.786937i \(-0.288334\pi\)
0.617033 + 0.786937i \(0.288334\pi\)
\(14\) 0 0
\(15\) 1.00000 0.258199
\(16\) 0 0
\(17\) −5.44949 −1.32170 −0.660848 0.750520i \(-0.729803\pi\)
−0.660848 + 0.750520i \(0.729803\pi\)
\(18\) 0 0
\(19\) −4.44949 −1.02078 −0.510391 0.859942i \(-0.670499\pi\)
−0.510391 + 0.859942i \(0.670499\pi\)
\(20\) 0 0
\(21\) −1.00000 −0.218218
\(22\) 0 0
\(23\) 1.00000 0.208514
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) 10.3485 1.92166 0.960831 0.277134i \(-0.0893846\pi\)
0.960831 + 0.277134i \(0.0893846\pi\)
\(30\) 0 0
\(31\) −0.101021 −0.0181438 −0.00907191 0.999959i \(-0.502888\pi\)
−0.00907191 + 0.999959i \(0.502888\pi\)
\(32\) 0 0
\(33\) 2.44949 0.426401
\(34\) 0 0
\(35\) −1.00000 −0.169031
\(36\) 0 0
\(37\) 3.89898 0.640988 0.320494 0.947250i \(-0.396151\pi\)
0.320494 + 0.947250i \(0.396151\pi\)
\(38\) 0 0
\(39\) −4.44949 −0.712489
\(40\) 0 0
\(41\) 5.44949 0.851067 0.425534 0.904943i \(-0.360086\pi\)
0.425534 + 0.904943i \(0.360086\pi\)
\(42\) 0 0
\(43\) −2.00000 −0.304997 −0.152499 0.988304i \(-0.548732\pi\)
−0.152499 + 0.988304i \(0.548732\pi\)
\(44\) 0 0
\(45\) −1.00000 −0.149071
\(46\) 0 0
\(47\) −8.44949 −1.23248 −0.616242 0.787557i \(-0.711346\pi\)
−0.616242 + 0.787557i \(0.711346\pi\)
\(48\) 0 0
\(49\) −6.00000 −0.857143
\(50\) 0 0
\(51\) 5.44949 0.763081
\(52\) 0 0
\(53\) 0.550510 0.0756184 0.0378092 0.999285i \(-0.487962\pi\)
0.0378092 + 0.999285i \(0.487962\pi\)
\(54\) 0 0
\(55\) 2.44949 0.330289
\(56\) 0 0
\(57\) 4.44949 0.589349
\(58\) 0 0
\(59\) −10.3485 −1.34726 −0.673628 0.739071i \(-0.735265\pi\)
−0.673628 + 0.739071i \(0.735265\pi\)
\(60\) 0 0
\(61\) 0.651531 0.0834200 0.0417100 0.999130i \(-0.486719\pi\)
0.0417100 + 0.999130i \(0.486719\pi\)
\(62\) 0 0
\(63\) 1.00000 0.125988
\(64\) 0 0
\(65\) −4.44949 −0.551891
\(66\) 0 0
\(67\) 7.00000 0.855186 0.427593 0.903971i \(-0.359362\pi\)
0.427593 + 0.903971i \(0.359362\pi\)
\(68\) 0 0
\(69\) −1.00000 −0.120386
\(70\) 0 0
\(71\) 4.34847 0.516068 0.258034 0.966136i \(-0.416925\pi\)
0.258034 + 0.966136i \(0.416925\pi\)
\(72\) 0 0
\(73\) −5.34847 −0.625991 −0.312995 0.949755i \(-0.601332\pi\)
−0.312995 + 0.949755i \(0.601332\pi\)
\(74\) 0 0
\(75\) −1.00000 −0.115470
\(76\) 0 0
\(77\) −2.44949 −0.279145
\(78\) 0 0
\(79\) 4.00000 0.450035 0.225018 0.974355i \(-0.427756\pi\)
0.225018 + 0.974355i \(0.427756\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −15.2474 −1.67362 −0.836812 0.547490i \(-0.815584\pi\)
−0.836812 + 0.547490i \(0.815584\pi\)
\(84\) 0 0
\(85\) 5.44949 0.591080
\(86\) 0 0
\(87\) −10.3485 −1.10947
\(88\) 0 0
\(89\) −16.8990 −1.79129 −0.895644 0.444771i \(-0.853285\pi\)
−0.895644 + 0.444771i \(0.853285\pi\)
\(90\) 0 0
\(91\) 4.44949 0.466433
\(92\) 0 0
\(93\) 0.101021 0.0104753
\(94\) 0 0
\(95\) 4.44949 0.456508
\(96\) 0 0
\(97\) 3.10102 0.314861 0.157430 0.987530i \(-0.449679\pi\)
0.157430 + 0.987530i \(0.449679\pi\)
\(98\) 0 0
\(99\) −2.44949 −0.246183
\(100\) 0 0
\(101\) −12.5505 −1.24882 −0.624411 0.781096i \(-0.714661\pi\)
−0.624411 + 0.781096i \(0.714661\pi\)
\(102\) 0 0
\(103\) 12.6969 1.25107 0.625533 0.780198i \(-0.284881\pi\)
0.625533 + 0.780198i \(0.284881\pi\)
\(104\) 0 0
\(105\) 1.00000 0.0975900
\(106\) 0 0
\(107\) 5.44949 0.526822 0.263411 0.964684i \(-0.415152\pi\)
0.263411 + 0.964684i \(0.415152\pi\)
\(108\) 0 0
\(109\) 11.5505 1.10634 0.553169 0.833069i \(-0.313418\pi\)
0.553169 + 0.833069i \(0.313418\pi\)
\(110\) 0 0
\(111\) −3.89898 −0.370075
\(112\) 0 0
\(113\) −1.65153 −0.155363 −0.0776815 0.996978i \(-0.524752\pi\)
−0.0776815 + 0.996978i \(0.524752\pi\)
\(114\) 0 0
\(115\) −1.00000 −0.0932505
\(116\) 0 0
\(117\) 4.44949 0.411355
\(118\) 0 0
\(119\) −5.44949 −0.499554
\(120\) 0 0
\(121\) −5.00000 −0.454545
\(122\) 0 0
\(123\) −5.44949 −0.491364
\(124\) 0 0
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) 1.55051 0.137586 0.0687928 0.997631i \(-0.478085\pi\)
0.0687928 + 0.997631i \(0.478085\pi\)
\(128\) 0 0
\(129\) 2.00000 0.176090
\(130\) 0 0
\(131\) 19.5959 1.71210 0.856052 0.516890i \(-0.172910\pi\)
0.856052 + 0.516890i \(0.172910\pi\)
\(132\) 0 0
\(133\) −4.44949 −0.385820
\(134\) 0 0
\(135\) 1.00000 0.0860663
\(136\) 0 0
\(137\) −16.8990 −1.44378 −0.721889 0.692009i \(-0.756726\pi\)
−0.721889 + 0.692009i \(0.756726\pi\)
\(138\) 0 0
\(139\) 4.79796 0.406958 0.203479 0.979079i \(-0.434775\pi\)
0.203479 + 0.979079i \(0.434775\pi\)
\(140\) 0 0
\(141\) 8.44949 0.711575
\(142\) 0 0
\(143\) −10.8990 −0.911418
\(144\) 0 0
\(145\) −10.3485 −0.859394
\(146\) 0 0
\(147\) 6.00000 0.494872
\(148\) 0 0
\(149\) −1.34847 −0.110471 −0.0552355 0.998473i \(-0.517591\pi\)
−0.0552355 + 0.998473i \(0.517591\pi\)
\(150\) 0 0
\(151\) −14.0000 −1.13930 −0.569652 0.821886i \(-0.692922\pi\)
−0.569652 + 0.821886i \(0.692922\pi\)
\(152\) 0 0
\(153\) −5.44949 −0.440565
\(154\) 0 0
\(155\) 0.101021 0.00811416
\(156\) 0 0
\(157\) 18.5959 1.48412 0.742058 0.670336i \(-0.233850\pi\)
0.742058 + 0.670336i \(0.233850\pi\)
\(158\) 0 0
\(159\) −0.550510 −0.0436583
\(160\) 0 0
\(161\) 1.00000 0.0788110
\(162\) 0 0
\(163\) 10.0000 0.783260 0.391630 0.920123i \(-0.371911\pi\)
0.391630 + 0.920123i \(0.371911\pi\)
\(164\) 0 0
\(165\) −2.44949 −0.190693
\(166\) 0 0
\(167\) −13.3485 −1.03294 −0.516468 0.856307i \(-0.672753\pi\)
−0.516468 + 0.856307i \(0.672753\pi\)
\(168\) 0 0
\(169\) 6.79796 0.522920
\(170\) 0 0
\(171\) −4.44949 −0.340261
\(172\) 0 0
\(173\) 19.5959 1.48985 0.744925 0.667148i \(-0.232485\pi\)
0.744925 + 0.667148i \(0.232485\pi\)
\(174\) 0 0
\(175\) 1.00000 0.0755929
\(176\) 0 0
\(177\) 10.3485 0.777839
\(178\) 0 0
\(179\) 6.00000 0.448461 0.224231 0.974536i \(-0.428013\pi\)
0.224231 + 0.974536i \(0.428013\pi\)
\(180\) 0 0
\(181\) −8.89898 −0.661456 −0.330728 0.943726i \(-0.607294\pi\)
−0.330728 + 0.943726i \(0.607294\pi\)
\(182\) 0 0
\(183\) −0.651531 −0.0481625
\(184\) 0 0
\(185\) −3.89898 −0.286659
\(186\) 0 0
\(187\) 13.3485 0.976137
\(188\) 0 0
\(189\) −1.00000 −0.0727393
\(190\) 0 0
\(191\) 18.2474 1.32034 0.660170 0.751117i \(-0.270484\pi\)
0.660170 + 0.751117i \(0.270484\pi\)
\(192\) 0 0
\(193\) −6.69694 −0.482056 −0.241028 0.970518i \(-0.577485\pi\)
−0.241028 + 0.970518i \(0.577485\pi\)
\(194\) 0 0
\(195\) 4.44949 0.318635
\(196\) 0 0
\(197\) −22.8990 −1.63148 −0.815742 0.578415i \(-0.803671\pi\)
−0.815742 + 0.578415i \(0.803671\pi\)
\(198\) 0 0
\(199\) 2.89898 0.205503 0.102752 0.994707i \(-0.467235\pi\)
0.102752 + 0.994707i \(0.467235\pi\)
\(200\) 0 0
\(201\) −7.00000 −0.493742
\(202\) 0 0
\(203\) 10.3485 0.726320
\(204\) 0 0
\(205\) −5.44949 −0.380609
\(206\) 0 0
\(207\) 1.00000 0.0695048
\(208\) 0 0
\(209\) 10.8990 0.753898
\(210\) 0 0
\(211\) −11.0000 −0.757271 −0.378636 0.925546i \(-0.623607\pi\)
−0.378636 + 0.925546i \(0.623607\pi\)
\(212\) 0 0
\(213\) −4.34847 −0.297952
\(214\) 0 0
\(215\) 2.00000 0.136399
\(216\) 0 0
\(217\) −0.101021 −0.00685772
\(218\) 0 0
\(219\) 5.34847 0.361416
\(220\) 0 0
\(221\) −24.2474 −1.63106
\(222\) 0 0
\(223\) −21.5959 −1.44617 −0.723085 0.690759i \(-0.757276\pi\)
−0.723085 + 0.690759i \(0.757276\pi\)
\(224\) 0 0
\(225\) 1.00000 0.0666667
\(226\) 0 0
\(227\) −21.7980 −1.44678 −0.723391 0.690439i \(-0.757417\pi\)
−0.723391 + 0.690439i \(0.757417\pi\)
\(228\) 0 0
\(229\) −28.4949 −1.88300 −0.941498 0.337019i \(-0.890581\pi\)
−0.941498 + 0.337019i \(0.890581\pi\)
\(230\) 0 0
\(231\) 2.44949 0.161165
\(232\) 0 0
\(233\) 21.7980 1.42803 0.714016 0.700129i \(-0.246874\pi\)
0.714016 + 0.700129i \(0.246874\pi\)
\(234\) 0 0
\(235\) 8.44949 0.551184
\(236\) 0 0
\(237\) −4.00000 −0.259828
\(238\) 0 0
\(239\) −6.55051 −0.423717 −0.211859 0.977300i \(-0.567952\pi\)
−0.211859 + 0.977300i \(0.567952\pi\)
\(240\) 0 0
\(241\) −29.3485 −1.89050 −0.945251 0.326346i \(-0.894183\pi\)
−0.945251 + 0.326346i \(0.894183\pi\)
\(242\) 0 0
\(243\) −1.00000 −0.0641500
\(244\) 0 0
\(245\) 6.00000 0.383326
\(246\) 0 0
\(247\) −19.7980 −1.25971
\(248\) 0 0
\(249\) 15.2474 0.966268
\(250\) 0 0
\(251\) −18.0000 −1.13615 −0.568075 0.822977i \(-0.692312\pi\)
−0.568075 + 0.822977i \(0.692312\pi\)
\(252\) 0 0
\(253\) −2.44949 −0.153998
\(254\) 0 0
\(255\) −5.44949 −0.341260
\(256\) 0 0
\(257\) −25.3485 −1.58119 −0.790597 0.612337i \(-0.790230\pi\)
−0.790597 + 0.612337i \(0.790230\pi\)
\(258\) 0 0
\(259\) 3.89898 0.242271
\(260\) 0 0
\(261\) 10.3485 0.640554
\(262\) 0 0
\(263\) −27.2474 −1.68015 −0.840075 0.542471i \(-0.817489\pi\)
−0.840075 + 0.542471i \(0.817489\pi\)
\(264\) 0 0
\(265\) −0.550510 −0.0338176
\(266\) 0 0
\(267\) 16.8990 1.03420
\(268\) 0 0
\(269\) −5.44949 −0.332261 −0.166131 0.986104i \(-0.553127\pi\)
−0.166131 + 0.986104i \(0.553127\pi\)
\(270\) 0 0
\(271\) −31.6969 −1.92545 −0.962726 0.270479i \(-0.912818\pi\)
−0.962726 + 0.270479i \(0.912818\pi\)
\(272\) 0 0
\(273\) −4.44949 −0.269295
\(274\) 0 0
\(275\) −2.44949 −0.147710
\(276\) 0 0
\(277\) −10.0000 −0.600842 −0.300421 0.953807i \(-0.597127\pi\)
−0.300421 + 0.953807i \(0.597127\pi\)
\(278\) 0 0
\(279\) −0.101021 −0.00604794
\(280\) 0 0
\(281\) −19.3485 −1.15423 −0.577116 0.816662i \(-0.695822\pi\)
−0.577116 + 0.816662i \(0.695822\pi\)
\(282\) 0 0
\(283\) 2.10102 0.124893 0.0624464 0.998048i \(-0.480110\pi\)
0.0624464 + 0.998048i \(0.480110\pi\)
\(284\) 0 0
\(285\) −4.44949 −0.263565
\(286\) 0 0
\(287\) 5.44949 0.321673
\(288\) 0 0
\(289\) 12.6969 0.746879
\(290\) 0 0
\(291\) −3.10102 −0.181785
\(292\) 0 0
\(293\) −21.2474 −1.24129 −0.620645 0.784092i \(-0.713129\pi\)
−0.620645 + 0.784092i \(0.713129\pi\)
\(294\) 0 0
\(295\) 10.3485 0.602511
\(296\) 0 0
\(297\) 2.44949 0.142134
\(298\) 0 0
\(299\) 4.44949 0.257321
\(300\) 0 0
\(301\) −2.00000 −0.115278
\(302\) 0 0
\(303\) 12.5505 0.721008
\(304\) 0 0
\(305\) −0.651531 −0.0373065
\(306\) 0 0
\(307\) 2.65153 0.151331 0.0756654 0.997133i \(-0.475892\pi\)
0.0756654 + 0.997133i \(0.475892\pi\)
\(308\) 0 0
\(309\) −12.6969 −0.722304
\(310\) 0 0
\(311\) −12.0000 −0.680458 −0.340229 0.940343i \(-0.610505\pi\)
−0.340229 + 0.940343i \(0.610505\pi\)
\(312\) 0 0
\(313\) 19.6969 1.11334 0.556668 0.830735i \(-0.312079\pi\)
0.556668 + 0.830735i \(0.312079\pi\)
\(314\) 0 0
\(315\) −1.00000 −0.0563436
\(316\) 0 0
\(317\) 6.24745 0.350892 0.175446 0.984489i \(-0.443863\pi\)
0.175446 + 0.984489i \(0.443863\pi\)
\(318\) 0 0
\(319\) −25.3485 −1.41924
\(320\) 0 0
\(321\) −5.44949 −0.304161
\(322\) 0 0
\(323\) 24.2474 1.34916
\(324\) 0 0
\(325\) 4.44949 0.246813
\(326\) 0 0
\(327\) −11.5505 −0.638745
\(328\) 0 0
\(329\) −8.44949 −0.465835
\(330\) 0 0
\(331\) −24.5959 −1.35191 −0.675957 0.736941i \(-0.736269\pi\)
−0.675957 + 0.736941i \(0.736269\pi\)
\(332\) 0 0
\(333\) 3.89898 0.213663
\(334\) 0 0
\(335\) −7.00000 −0.382451
\(336\) 0 0
\(337\) −19.7980 −1.07846 −0.539232 0.842157i \(-0.681285\pi\)
−0.539232 + 0.842157i \(0.681285\pi\)
\(338\) 0 0
\(339\) 1.65153 0.0896988
\(340\) 0 0
\(341\) 0.247449 0.0134001
\(342\) 0 0
\(343\) −13.0000 −0.701934
\(344\) 0 0
\(345\) 1.00000 0.0538382
\(346\) 0 0
\(347\) 28.8990 1.55138 0.775689 0.631115i \(-0.217402\pi\)
0.775689 + 0.631115i \(0.217402\pi\)
\(348\) 0 0
\(349\) 23.4949 1.25765 0.628827 0.777546i \(-0.283536\pi\)
0.628827 + 0.777546i \(0.283536\pi\)
\(350\) 0 0
\(351\) −4.44949 −0.237496
\(352\) 0 0
\(353\) −15.5505 −0.827670 −0.413835 0.910352i \(-0.635811\pi\)
−0.413835 + 0.910352i \(0.635811\pi\)
\(354\) 0 0
\(355\) −4.34847 −0.230793
\(356\) 0 0
\(357\) 5.44949 0.288418
\(358\) 0 0
\(359\) 3.55051 0.187389 0.0936944 0.995601i \(-0.470132\pi\)
0.0936944 + 0.995601i \(0.470132\pi\)
\(360\) 0 0
\(361\) 0.797959 0.0419978
\(362\) 0 0
\(363\) 5.00000 0.262432
\(364\) 0 0
\(365\) 5.34847 0.279952
\(366\) 0 0
\(367\) −13.6969 −0.714974 −0.357487 0.933918i \(-0.616366\pi\)
−0.357487 + 0.933918i \(0.616366\pi\)
\(368\) 0 0
\(369\) 5.44949 0.283689
\(370\) 0 0
\(371\) 0.550510 0.0285811
\(372\) 0 0
\(373\) 27.5959 1.42886 0.714431 0.699706i \(-0.246686\pi\)
0.714431 + 0.699706i \(0.246686\pi\)
\(374\) 0 0
\(375\) 1.00000 0.0516398
\(376\) 0 0
\(377\) 46.0454 2.37146
\(378\) 0 0
\(379\) −25.3939 −1.30440 −0.652198 0.758049i \(-0.726153\pi\)
−0.652198 + 0.758049i \(0.726153\pi\)
\(380\) 0 0
\(381\) −1.55051 −0.0794350
\(382\) 0 0
\(383\) −1.65153 −0.0843893 −0.0421946 0.999109i \(-0.513435\pi\)
−0.0421946 + 0.999109i \(0.513435\pi\)
\(384\) 0 0
\(385\) 2.44949 0.124838
\(386\) 0 0
\(387\) −2.00000 −0.101666
\(388\) 0 0
\(389\) −3.30306 −0.167472 −0.0837359 0.996488i \(-0.526685\pi\)
−0.0837359 + 0.996488i \(0.526685\pi\)
\(390\) 0 0
\(391\) −5.44949 −0.275593
\(392\) 0 0
\(393\) −19.5959 −0.988483
\(394\) 0 0
\(395\) −4.00000 −0.201262
\(396\) 0 0
\(397\) −13.7980 −0.692500 −0.346250 0.938142i \(-0.612545\pi\)
−0.346250 + 0.938142i \(0.612545\pi\)
\(398\) 0 0
\(399\) 4.44949 0.222753
\(400\) 0 0
\(401\) −13.1010 −0.654234 −0.327117 0.944984i \(-0.606077\pi\)
−0.327117 + 0.944984i \(0.606077\pi\)
\(402\) 0 0
\(403\) −0.449490 −0.0223907
\(404\) 0 0
\(405\) −1.00000 −0.0496904
\(406\) 0 0
\(407\) −9.55051 −0.473401
\(408\) 0 0
\(409\) 21.8990 1.08283 0.541417 0.840754i \(-0.317888\pi\)
0.541417 + 0.840754i \(0.317888\pi\)
\(410\) 0 0
\(411\) 16.8990 0.833565
\(412\) 0 0
\(413\) −10.3485 −0.509215
\(414\) 0 0
\(415\) 15.2474 0.748468
\(416\) 0 0
\(417\) −4.79796 −0.234957
\(418\) 0 0
\(419\) −33.5505 −1.63905 −0.819525 0.573044i \(-0.805763\pi\)
−0.819525 + 0.573044i \(0.805763\pi\)
\(420\) 0 0
\(421\) −22.2474 −1.08427 −0.542137 0.840290i \(-0.682385\pi\)
−0.542137 + 0.840290i \(0.682385\pi\)
\(422\) 0 0
\(423\) −8.44949 −0.410828
\(424\) 0 0
\(425\) −5.44949 −0.264339
\(426\) 0 0
\(427\) 0.651531 0.0315298
\(428\) 0 0
\(429\) 10.8990 0.526208
\(430\) 0 0
\(431\) 24.4949 1.17988 0.589939 0.807448i \(-0.299152\pi\)
0.589939 + 0.807448i \(0.299152\pi\)
\(432\) 0 0
\(433\) 2.79796 0.134461 0.0672307 0.997737i \(-0.478584\pi\)
0.0672307 + 0.997737i \(0.478584\pi\)
\(434\) 0 0
\(435\) 10.3485 0.496171
\(436\) 0 0
\(437\) −4.44949 −0.212848
\(438\) 0 0
\(439\) 36.6969 1.75145 0.875725 0.482811i \(-0.160384\pi\)
0.875725 + 0.482811i \(0.160384\pi\)
\(440\) 0 0
\(441\) −6.00000 −0.285714
\(442\) 0 0
\(443\) 20.4495 0.971585 0.485792 0.874074i \(-0.338531\pi\)
0.485792 + 0.874074i \(0.338531\pi\)
\(444\) 0 0
\(445\) 16.8990 0.801088
\(446\) 0 0
\(447\) 1.34847 0.0637804
\(448\) 0 0
\(449\) −21.2474 −1.00273 −0.501365 0.865236i \(-0.667168\pi\)
−0.501365 + 0.865236i \(0.667168\pi\)
\(450\) 0 0
\(451\) −13.3485 −0.628555
\(452\) 0 0
\(453\) 14.0000 0.657777
\(454\) 0 0
\(455\) −4.44949 −0.208595
\(456\) 0 0
\(457\) 0.101021 0.00472554 0.00236277 0.999997i \(-0.499248\pi\)
0.00236277 + 0.999997i \(0.499248\pi\)
\(458\) 0 0
\(459\) 5.44949 0.254360
\(460\) 0 0
\(461\) −18.4949 −0.861393 −0.430697 0.902497i \(-0.641732\pi\)
−0.430697 + 0.902497i \(0.641732\pi\)
\(462\) 0 0
\(463\) 24.9444 1.15926 0.579632 0.814878i \(-0.303196\pi\)
0.579632 + 0.814878i \(0.303196\pi\)
\(464\) 0 0
\(465\) −0.101021 −0.00468471
\(466\) 0 0
\(467\) 34.8434 1.61236 0.806179 0.591671i \(-0.201532\pi\)
0.806179 + 0.591671i \(0.201532\pi\)
\(468\) 0 0
\(469\) 7.00000 0.323230
\(470\) 0 0
\(471\) −18.5959 −0.856855
\(472\) 0 0
\(473\) 4.89898 0.225255
\(474\) 0 0
\(475\) −4.44949 −0.204157
\(476\) 0 0
\(477\) 0.550510 0.0252061
\(478\) 0 0
\(479\) −32.9444 −1.50527 −0.752634 0.658439i \(-0.771217\pi\)
−0.752634 + 0.658439i \(0.771217\pi\)
\(480\) 0 0
\(481\) 17.3485 0.791022
\(482\) 0 0
\(483\) −1.00000 −0.0455016
\(484\) 0 0
\(485\) −3.10102 −0.140810
\(486\) 0 0
\(487\) 18.9444 0.858452 0.429226 0.903197i \(-0.358786\pi\)
0.429226 + 0.903197i \(0.358786\pi\)
\(488\) 0 0
\(489\) −10.0000 −0.452216
\(490\) 0 0
\(491\) −12.5505 −0.566397 −0.283198 0.959061i \(-0.591395\pi\)
−0.283198 + 0.959061i \(0.591395\pi\)
\(492\) 0 0
\(493\) −56.3939 −2.53985
\(494\) 0 0
\(495\) 2.44949 0.110096
\(496\) 0 0
\(497\) 4.34847 0.195056
\(498\) 0 0
\(499\) 1.00000 0.0447661 0.0223831 0.999749i \(-0.492875\pi\)
0.0223831 + 0.999749i \(0.492875\pi\)
\(500\) 0 0
\(501\) 13.3485 0.596366
\(502\) 0 0
\(503\) −7.04541 −0.314139 −0.157070 0.987588i \(-0.550205\pi\)
−0.157070 + 0.987588i \(0.550205\pi\)
\(504\) 0 0
\(505\) 12.5505 0.558490
\(506\) 0 0
\(507\) −6.79796 −0.301908
\(508\) 0 0
\(509\) 28.8990 1.28092 0.640462 0.767990i \(-0.278743\pi\)
0.640462 + 0.767990i \(0.278743\pi\)
\(510\) 0 0
\(511\) −5.34847 −0.236602
\(512\) 0 0
\(513\) 4.44949 0.196450
\(514\) 0 0
\(515\) −12.6969 −0.559494
\(516\) 0 0
\(517\) 20.6969 0.910250
\(518\) 0 0
\(519\) −19.5959 −0.860165
\(520\) 0 0
\(521\) −43.8434 −1.92081 −0.960406 0.278603i \(-0.910129\pi\)
−0.960406 + 0.278603i \(0.910129\pi\)
\(522\) 0 0
\(523\) 33.3939 1.46021 0.730106 0.683334i \(-0.239471\pi\)
0.730106 + 0.683334i \(0.239471\pi\)
\(524\) 0 0
\(525\) −1.00000 −0.0436436
\(526\) 0 0
\(527\) 0.550510 0.0239806
\(528\) 0 0
\(529\) 1.00000 0.0434783
\(530\) 0 0
\(531\) −10.3485 −0.449085
\(532\) 0 0
\(533\) 24.2474 1.05027
\(534\) 0 0
\(535\) −5.44949 −0.235602
\(536\) 0 0
\(537\) −6.00000 −0.258919
\(538\) 0 0
\(539\) 14.6969 0.633042
\(540\) 0 0
\(541\) 12.8990 0.554570 0.277285 0.960788i \(-0.410565\pi\)
0.277285 + 0.960788i \(0.410565\pi\)
\(542\) 0 0
\(543\) 8.89898 0.381892
\(544\) 0 0
\(545\) −11.5505 −0.494769
\(546\) 0 0
\(547\) 28.0000 1.19719 0.598597 0.801050i \(-0.295725\pi\)
0.598597 + 0.801050i \(0.295725\pi\)
\(548\) 0 0
\(549\) 0.651531 0.0278067
\(550\) 0 0
\(551\) −46.0454 −1.96160
\(552\) 0 0
\(553\) 4.00000 0.170097
\(554\) 0 0
\(555\) 3.89898 0.165502
\(556\) 0 0
\(557\) −19.0454 −0.806980 −0.403490 0.914984i \(-0.632203\pi\)
−0.403490 + 0.914984i \(0.632203\pi\)
\(558\) 0 0
\(559\) −8.89898 −0.376387
\(560\) 0 0
\(561\) −13.3485 −0.563573
\(562\) 0 0
\(563\) −5.44949 −0.229669 −0.114834 0.993385i \(-0.536634\pi\)
−0.114834 + 0.993385i \(0.536634\pi\)
\(564\) 0 0
\(565\) 1.65153 0.0694804
\(566\) 0 0
\(567\) 1.00000 0.0419961
\(568\) 0 0
\(569\) −28.2929 −1.18610 −0.593049 0.805166i \(-0.702076\pi\)
−0.593049 + 0.805166i \(0.702076\pi\)
\(570\) 0 0
\(571\) −13.1464 −0.550161 −0.275080 0.961421i \(-0.588704\pi\)
−0.275080 + 0.961421i \(0.588704\pi\)
\(572\) 0 0
\(573\) −18.2474 −0.762298
\(574\) 0 0
\(575\) 1.00000 0.0417029
\(576\) 0 0
\(577\) −28.0000 −1.16566 −0.582828 0.812596i \(-0.698054\pi\)
−0.582828 + 0.812596i \(0.698054\pi\)
\(578\) 0 0
\(579\) 6.69694 0.278315
\(580\) 0 0
\(581\) −15.2474 −0.632571
\(582\) 0 0
\(583\) −1.34847 −0.0558479
\(584\) 0 0
\(585\) −4.44949 −0.183964
\(586\) 0 0
\(587\) 6.00000 0.247647 0.123823 0.992304i \(-0.460484\pi\)
0.123823 + 0.992304i \(0.460484\pi\)
\(588\) 0 0
\(589\) 0.449490 0.0185209
\(590\) 0 0
\(591\) 22.8990 0.941938
\(592\) 0 0
\(593\) 27.5505 1.13136 0.565682 0.824624i \(-0.308613\pi\)
0.565682 + 0.824624i \(0.308613\pi\)
\(594\) 0 0
\(595\) 5.44949 0.223407
\(596\) 0 0
\(597\) −2.89898 −0.118647
\(598\) 0 0
\(599\) 2.69694 0.110194 0.0550970 0.998481i \(-0.482453\pi\)
0.0550970 + 0.998481i \(0.482453\pi\)
\(600\) 0 0
\(601\) 5.00000 0.203954 0.101977 0.994787i \(-0.467483\pi\)
0.101977 + 0.994787i \(0.467483\pi\)
\(602\) 0 0
\(603\) 7.00000 0.285062
\(604\) 0 0
\(605\) 5.00000 0.203279
\(606\) 0 0
\(607\) 16.2474 0.659464 0.329732 0.944075i \(-0.393042\pi\)
0.329732 + 0.944075i \(0.393042\pi\)
\(608\) 0 0
\(609\) −10.3485 −0.419341
\(610\) 0 0
\(611\) −37.5959 −1.52097
\(612\) 0 0
\(613\) 14.0000 0.565455 0.282727 0.959200i \(-0.408761\pi\)
0.282727 + 0.959200i \(0.408761\pi\)
\(614\) 0 0
\(615\) 5.44949 0.219745
\(616\) 0 0
\(617\) 12.5505 0.505265 0.252632 0.967562i \(-0.418704\pi\)
0.252632 + 0.967562i \(0.418704\pi\)
\(618\) 0 0
\(619\) 4.00000 0.160774 0.0803868 0.996764i \(-0.474384\pi\)
0.0803868 + 0.996764i \(0.474384\pi\)
\(620\) 0 0
\(621\) −1.00000 −0.0401286
\(622\) 0 0
\(623\) −16.8990 −0.677043
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) −10.8990 −0.435263
\(628\) 0 0
\(629\) −21.2474 −0.847191
\(630\) 0 0
\(631\) −29.5505 −1.17639 −0.588194 0.808720i \(-0.700161\pi\)
−0.588194 + 0.808720i \(0.700161\pi\)
\(632\) 0 0
\(633\) 11.0000 0.437211
\(634\) 0 0
\(635\) −1.55051 −0.0615301
\(636\) 0 0
\(637\) −26.6969 −1.05777
\(638\) 0 0
\(639\) 4.34847 0.172023
\(640\) 0 0
\(641\) 26.9444 1.06424 0.532120 0.846669i \(-0.321396\pi\)
0.532120 + 0.846669i \(0.321396\pi\)
\(642\) 0 0
\(643\) 9.69694 0.382410 0.191205 0.981550i \(-0.438760\pi\)
0.191205 + 0.981550i \(0.438760\pi\)
\(644\) 0 0
\(645\) −2.00000 −0.0787499
\(646\) 0 0
\(647\) −1.95459 −0.0768430 −0.0384215 0.999262i \(-0.512233\pi\)
−0.0384215 + 0.999262i \(0.512233\pi\)
\(648\) 0 0
\(649\) 25.3485 0.995014
\(650\) 0 0
\(651\) 0.101021 0.00395931
\(652\) 0 0
\(653\) 43.8434 1.71572 0.857862 0.513881i \(-0.171793\pi\)
0.857862 + 0.513881i \(0.171793\pi\)
\(654\) 0 0
\(655\) −19.5959 −0.765676
\(656\) 0 0
\(657\) −5.34847 −0.208664
\(658\) 0 0
\(659\) 5.14643 0.200476 0.100238 0.994963i \(-0.468040\pi\)
0.100238 + 0.994963i \(0.468040\pi\)
\(660\) 0 0
\(661\) 46.0908 1.79272 0.896362 0.443322i \(-0.146200\pi\)
0.896362 + 0.443322i \(0.146200\pi\)
\(662\) 0 0
\(663\) 24.2474 0.941693
\(664\) 0 0
\(665\) 4.44949 0.172544
\(666\) 0 0
\(667\) 10.3485 0.400694
\(668\) 0 0
\(669\) 21.5959 0.834946
\(670\) 0 0
\(671\) −1.59592 −0.0616097
\(672\) 0 0
\(673\) 22.4495 0.865364 0.432682 0.901547i \(-0.357567\pi\)
0.432682 + 0.901547i \(0.357567\pi\)
\(674\) 0 0
\(675\) −1.00000 −0.0384900
\(676\) 0 0
\(677\) −12.5505 −0.482355 −0.241178 0.970481i \(-0.577534\pi\)
−0.241178 + 0.970481i \(0.577534\pi\)
\(678\) 0 0
\(679\) 3.10102 0.119006
\(680\) 0 0
\(681\) 21.7980 0.835300
\(682\) 0 0
\(683\) −18.2474 −0.698219 −0.349110 0.937082i \(-0.613516\pi\)
−0.349110 + 0.937082i \(0.613516\pi\)
\(684\) 0 0
\(685\) 16.8990 0.645677
\(686\) 0 0
\(687\) 28.4949 1.08715
\(688\) 0 0
\(689\) 2.44949 0.0933181
\(690\) 0 0
\(691\) −16.2020 −0.616355 −0.308177 0.951329i \(-0.599719\pi\)
−0.308177 + 0.951329i \(0.599719\pi\)
\(692\) 0 0
\(693\) −2.44949 −0.0930484
\(694\) 0 0
\(695\) −4.79796 −0.181997
\(696\) 0 0
\(697\) −29.6969 −1.12485
\(698\) 0 0
\(699\) −21.7980 −0.824475
\(700\) 0 0
\(701\) −52.0454 −1.96573 −0.982864 0.184332i \(-0.940988\pi\)
−0.982864 + 0.184332i \(0.940988\pi\)
\(702\) 0 0
\(703\) −17.3485 −0.654310
\(704\) 0 0
\(705\) −8.44949 −0.318226
\(706\) 0 0
\(707\) −12.5505 −0.472011
\(708\) 0 0
\(709\) −28.7423 −1.07944 −0.539721 0.841844i \(-0.681470\pi\)
−0.539721 + 0.841844i \(0.681470\pi\)
\(710\) 0 0
\(711\) 4.00000 0.150012
\(712\) 0 0
\(713\) −0.101021 −0.00378325
\(714\) 0 0
\(715\) 10.8990 0.407599
\(716\) 0 0
\(717\) 6.55051 0.244633
\(718\) 0 0
\(719\) 11.9444 0.445450 0.222725 0.974881i \(-0.428505\pi\)
0.222725 + 0.974881i \(0.428505\pi\)
\(720\) 0 0
\(721\) 12.6969 0.472859
\(722\) 0 0
\(723\) 29.3485 1.09148
\(724\) 0 0
\(725\) 10.3485 0.384332
\(726\) 0 0
\(727\) 16.3031 0.604647 0.302324 0.953205i \(-0.402238\pi\)
0.302324 + 0.953205i \(0.402238\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 10.8990 0.403113
\(732\) 0 0
\(733\) −8.59592 −0.317497 −0.158749 0.987319i \(-0.550746\pi\)
−0.158749 + 0.987319i \(0.550746\pi\)
\(734\) 0 0
\(735\) −6.00000 −0.221313
\(736\) 0 0
\(737\) −17.1464 −0.631597
\(738\) 0 0
\(739\) 9.20204 0.338503 0.169251 0.985573i \(-0.445865\pi\)
0.169251 + 0.985573i \(0.445865\pi\)
\(740\) 0 0
\(741\) 19.7980 0.727296
\(742\) 0 0
\(743\) 21.7980 0.799690 0.399845 0.916583i \(-0.369064\pi\)
0.399845 + 0.916583i \(0.369064\pi\)
\(744\) 0 0
\(745\) 1.34847 0.0494041
\(746\) 0 0
\(747\) −15.2474 −0.557875
\(748\) 0 0
\(749\) 5.44949 0.199120
\(750\) 0 0
\(751\) −30.6515 −1.11849 −0.559245 0.829002i \(-0.688909\pi\)
−0.559245 + 0.829002i \(0.688909\pi\)
\(752\) 0 0
\(753\) 18.0000 0.655956
\(754\) 0 0
\(755\) 14.0000 0.509512
\(756\) 0 0
\(757\) −3.69694 −0.134368 −0.0671838 0.997741i \(-0.521401\pi\)
−0.0671838 + 0.997741i \(0.521401\pi\)
\(758\) 0 0
\(759\) 2.44949 0.0889108
\(760\) 0 0
\(761\) 14.1464 0.512808 0.256404 0.966570i \(-0.417462\pi\)
0.256404 + 0.966570i \(0.417462\pi\)
\(762\) 0 0
\(763\) 11.5505 0.418157
\(764\) 0 0
\(765\) 5.44949 0.197027
\(766\) 0 0
\(767\) −46.0454 −1.66260
\(768\) 0 0
\(769\) 10.4495 0.376818 0.188409 0.982091i \(-0.439667\pi\)
0.188409 + 0.982091i \(0.439667\pi\)
\(770\) 0 0
\(771\) 25.3485 0.912903
\(772\) 0 0
\(773\) −16.2929 −0.586013 −0.293007 0.956110i \(-0.594656\pi\)
−0.293007 + 0.956110i \(0.594656\pi\)
\(774\) 0 0
\(775\) −0.101021 −0.00362876
\(776\) 0 0
\(777\) −3.89898 −0.139875
\(778\) 0 0
\(779\) −24.2474 −0.868755
\(780\) 0 0
\(781\) −10.6515 −0.381142
\(782\) 0 0
\(783\) −10.3485 −0.369824
\(784\) 0 0
\(785\) −18.5959 −0.663717
\(786\) 0 0
\(787\) 21.6969 0.773412 0.386706 0.922203i \(-0.373613\pi\)
0.386706 + 0.922203i \(0.373613\pi\)
\(788\) 0 0
\(789\) 27.2474 0.970035
\(790\) 0 0
\(791\) −1.65153 −0.0587217
\(792\) 0 0
\(793\) 2.89898 0.102946
\(794\) 0 0
\(795\) 0.550510 0.0195246
\(796\) 0 0
\(797\) 9.24745 0.327561 0.163781 0.986497i \(-0.447631\pi\)
0.163781 + 0.986497i \(0.447631\pi\)
\(798\) 0 0
\(799\) 46.0454 1.62897
\(800\) 0 0
\(801\) −16.8990 −0.597096
\(802\) 0 0
\(803\) 13.1010 0.462325
\(804\) 0 0
\(805\) −1.00000 −0.0352454
\(806\) 0 0
\(807\) 5.44949 0.191831
\(808\) 0 0
\(809\) 6.55051 0.230304 0.115152 0.993348i \(-0.463265\pi\)
0.115152 + 0.993348i \(0.463265\pi\)
\(810\) 0 0
\(811\) 42.3939 1.48865 0.744325 0.667817i \(-0.232771\pi\)
0.744325 + 0.667817i \(0.232771\pi\)
\(812\) 0 0
\(813\) 31.6969 1.11166
\(814\) 0 0
\(815\) −10.0000 −0.350285
\(816\) 0 0
\(817\) 8.89898 0.311336
\(818\) 0 0
\(819\) 4.44949 0.155478
\(820\) 0 0
\(821\) 21.3031 0.743482 0.371741 0.928336i \(-0.378761\pi\)
0.371741 + 0.928336i \(0.378761\pi\)
\(822\) 0 0
\(823\) −3.59592 −0.125346 −0.0626729 0.998034i \(-0.519962\pi\)
−0.0626729 + 0.998034i \(0.519962\pi\)
\(824\) 0 0
\(825\) 2.44949 0.0852803
\(826\) 0 0
\(827\) −32.1464 −1.11784 −0.558920 0.829221i \(-0.688784\pi\)
−0.558920 + 0.829221i \(0.688784\pi\)
\(828\) 0 0
\(829\) −27.6969 −0.961954 −0.480977 0.876733i \(-0.659718\pi\)
−0.480977 + 0.876733i \(0.659718\pi\)
\(830\) 0 0
\(831\) 10.0000 0.346896
\(832\) 0 0
\(833\) 32.6969 1.13288
\(834\) 0 0
\(835\) 13.3485 0.461943
\(836\) 0 0
\(837\) 0.101021 0.00349178
\(838\) 0 0
\(839\) 45.1918 1.56020 0.780098 0.625658i \(-0.215169\pi\)
0.780098 + 0.625658i \(0.215169\pi\)
\(840\) 0 0
\(841\) 78.0908 2.69279
\(842\) 0 0
\(843\) 19.3485 0.666397
\(844\) 0 0
\(845\) −6.79796 −0.233857
\(846\) 0 0
\(847\) −5.00000 −0.171802
\(848\) 0 0
\(849\) −2.10102 −0.0721068
\(850\) 0 0
\(851\) 3.89898 0.133655
\(852\) 0 0
\(853\) 46.0908 1.57812 0.789060 0.614316i \(-0.210568\pi\)
0.789060 + 0.614316i \(0.210568\pi\)
\(854\) 0 0
\(855\) 4.44949 0.152169
\(856\) 0 0
\(857\) −13.5959 −0.464428 −0.232214 0.972665i \(-0.574597\pi\)
−0.232214 + 0.972665i \(0.574597\pi\)
\(858\) 0 0
\(859\) −38.1918 −1.30309 −0.651544 0.758611i \(-0.725879\pi\)
−0.651544 + 0.758611i \(0.725879\pi\)
\(860\) 0 0
\(861\) −5.44949 −0.185718
\(862\) 0 0
\(863\) −6.49490 −0.221089 −0.110544 0.993871i \(-0.535259\pi\)
−0.110544 + 0.993871i \(0.535259\pi\)
\(864\) 0 0
\(865\) −19.5959 −0.666281
\(866\) 0 0
\(867\) −12.6969 −0.431211
\(868\) 0 0
\(869\) −9.79796 −0.332373
\(870\) 0 0
\(871\) 31.1464 1.05536
\(872\) 0 0
\(873\) 3.10102 0.104954
\(874\) 0 0
\(875\) −1.00000 −0.0338062
\(876\) 0 0
\(877\) −10.4949 −0.354388 −0.177194 0.984176i \(-0.556702\pi\)
−0.177194 + 0.984176i \(0.556702\pi\)
\(878\) 0 0
\(879\) 21.2474 0.716659
\(880\) 0 0
\(881\) 8.44949 0.284671 0.142335 0.989819i \(-0.454539\pi\)
0.142335 + 0.989819i \(0.454539\pi\)
\(882\) 0 0
\(883\) 37.5505 1.26368 0.631838 0.775101i \(-0.282301\pi\)
0.631838 + 0.775101i \(0.282301\pi\)
\(884\) 0 0
\(885\) −10.3485 −0.347860
\(886\) 0 0
\(887\) 6.49490 0.218077 0.109039 0.994038i \(-0.465223\pi\)
0.109039 + 0.994038i \(0.465223\pi\)
\(888\) 0 0
\(889\) 1.55051 0.0520024
\(890\) 0 0
\(891\) −2.44949 −0.0820610
\(892\) 0 0
\(893\) 37.5959 1.25810
\(894\) 0 0
\(895\) −6.00000 −0.200558
\(896\) 0 0
\(897\) −4.44949 −0.148564
\(898\) 0 0
\(899\) −1.04541 −0.0348663
\(900\) 0 0
\(901\) −3.00000 −0.0999445
\(902\) 0 0
\(903\) 2.00000 0.0665558
\(904\) 0 0
\(905\) 8.89898 0.295812
\(906\) 0 0
\(907\) −38.7980 −1.28827 −0.644133 0.764914i \(-0.722781\pi\)
−0.644133 + 0.764914i \(0.722781\pi\)
\(908\) 0 0
\(909\) −12.5505 −0.416274
\(910\) 0 0
\(911\) 14.2020 0.470535 0.235267 0.971931i \(-0.424403\pi\)
0.235267 + 0.971931i \(0.424403\pi\)
\(912\) 0 0
\(913\) 37.3485 1.23605
\(914\) 0 0
\(915\) 0.651531 0.0215389
\(916\) 0 0
\(917\) 19.5959 0.647114
\(918\) 0 0
\(919\) 45.3939 1.49741 0.748703 0.662906i \(-0.230677\pi\)
0.748703 + 0.662906i \(0.230677\pi\)
\(920\) 0 0
\(921\) −2.65153 −0.0873709
\(922\) 0 0
\(923\) 19.3485 0.636863
\(924\) 0 0
\(925\) 3.89898 0.128198
\(926\) 0 0
\(927\) 12.6969 0.417022
\(928\) 0 0
\(929\) 22.8434 0.749467 0.374733 0.927133i \(-0.377734\pi\)
0.374733 + 0.927133i \(0.377734\pi\)
\(930\) 0 0
\(931\) 26.6969 0.874957
\(932\) 0 0
\(933\) 12.0000 0.392862
\(934\) 0 0
\(935\) −13.3485 −0.436542
\(936\) 0 0
\(937\) −8.89898 −0.290717 −0.145358 0.989379i \(-0.546434\pi\)
−0.145358 + 0.989379i \(0.546434\pi\)
\(938\) 0 0
\(939\) −19.6969 −0.642785
\(940\) 0 0
\(941\) −24.8536 −0.810203 −0.405102 0.914272i \(-0.632764\pi\)
−0.405102 + 0.914272i \(0.632764\pi\)
\(942\) 0 0
\(943\) 5.44949 0.177460
\(944\) 0 0
\(945\) 1.00000 0.0325300
\(946\) 0 0
\(947\) 0.494897 0.0160820 0.00804100 0.999968i \(-0.497440\pi\)
0.00804100 + 0.999968i \(0.497440\pi\)
\(948\) 0 0
\(949\) −23.7980 −0.772514
\(950\) 0 0
\(951\) −6.24745 −0.202587
\(952\) 0 0
\(953\) 10.4041 0.337021 0.168511 0.985700i \(-0.446104\pi\)
0.168511 + 0.985700i \(0.446104\pi\)
\(954\) 0 0
\(955\) −18.2474 −0.590474
\(956\) 0 0
\(957\) 25.3485 0.819400
\(958\) 0 0
\(959\) −16.8990 −0.545697
\(960\) 0 0
\(961\) −30.9898 −0.999671
\(962\) 0 0
\(963\) 5.44949 0.175607
\(964\) 0 0
\(965\) 6.69694 0.215582
\(966\) 0 0
\(967\) −54.0454 −1.73798 −0.868992 0.494827i \(-0.835231\pi\)
−0.868992 + 0.494827i \(0.835231\pi\)
\(968\) 0 0
\(969\) −24.2474 −0.778940
\(970\) 0 0
\(971\) −22.2929 −0.715412 −0.357706 0.933834i \(-0.616441\pi\)
−0.357706 + 0.933834i \(0.616441\pi\)
\(972\) 0 0
\(973\) 4.79796 0.153816
\(974\) 0 0
\(975\) −4.44949 −0.142498
\(976\) 0 0
\(977\) 19.6515 0.628708 0.314354 0.949306i \(-0.398212\pi\)
0.314354 + 0.949306i \(0.398212\pi\)
\(978\) 0 0
\(979\) 41.3939 1.32295
\(980\) 0 0
\(981\) 11.5505 0.368779
\(982\) 0 0
\(983\) 33.7423 1.07621 0.538107 0.842877i \(-0.319140\pi\)
0.538107 + 0.842877i \(0.319140\pi\)
\(984\) 0 0
\(985\) 22.8990 0.729622
\(986\) 0 0
\(987\) 8.44949 0.268950
\(988\) 0 0
\(989\) −2.00000 −0.0635963
\(990\) 0 0
\(991\) −57.7878 −1.83569 −0.917844 0.396941i \(-0.870072\pi\)
−0.917844 + 0.396941i \(0.870072\pi\)
\(992\) 0 0
\(993\) 24.5959 0.780528
\(994\) 0 0
\(995\) −2.89898 −0.0919038
\(996\) 0 0
\(997\) 40.0908 1.26969 0.634844 0.772640i \(-0.281064\pi\)
0.634844 + 0.772640i \(0.281064\pi\)
\(998\) 0 0
\(999\) −3.89898 −0.123358
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 5520.2.a.bi.1.1 2
4.3 odd 2 345.2.a.i.1.1 2
12.11 even 2 1035.2.a.k.1.2 2
20.3 even 4 1725.2.b.m.1174.4 4
20.7 even 4 1725.2.b.m.1174.1 4
20.19 odd 2 1725.2.a.y.1.2 2
60.59 even 2 5175.2.a.bl.1.1 2
92.91 even 2 7935.2.a.t.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
345.2.a.i.1.1 2 4.3 odd 2
1035.2.a.k.1.2 2 12.11 even 2
1725.2.a.y.1.2 2 20.19 odd 2
1725.2.b.m.1174.1 4 20.7 even 4
1725.2.b.m.1174.4 4 20.3 even 4
5175.2.a.bl.1.1 2 60.59 even 2
5520.2.a.bi.1.1 2 1.1 even 1 trivial
7935.2.a.t.1.1 2 92.91 even 2