Properties

Label 3640.2.a.l.1.1
Level $3640$
Weight $2$
Character 3640.1
Self dual yes
Analytic conductor $29.066$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 3640 = 2^{3} \cdot 5 \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3640.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(29.0655463357\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{13}) \)
Defining polynomial: \( x^{2} - x - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(2.30278\) of defining polynomial
Character \(\chi\) \(=\) 3640.1

$q$-expansion

\(f(q)\) \(=\) \(q-2.30278 q^{3} +1.00000 q^{5} +1.00000 q^{7} +2.30278 q^{9} +O(q^{10})\) \(q-2.30278 q^{3} +1.00000 q^{5} +1.00000 q^{7} +2.30278 q^{9} +2.60555 q^{11} -1.00000 q^{13} -2.30278 q^{15} +2.69722 q^{17} -2.69722 q^{19} -2.30278 q^{21} -8.00000 q^{23} +1.00000 q^{25} +1.60555 q^{27} -8.90833 q^{29} -4.30278 q^{31} -6.00000 q^{33} +1.00000 q^{35} +6.90833 q^{37} +2.30278 q^{39} -0.697224 q^{41} -2.00000 q^{43} +2.30278 q^{45} +2.60555 q^{47} +1.00000 q^{49} -6.21110 q^{51} +4.60555 q^{53} +2.60555 q^{55} +6.21110 q^{57} -14.1194 q^{59} +13.2111 q^{61} +2.30278 q^{63} -1.00000 q^{65} -1.90833 q^{67} +18.4222 q^{69} +4.60555 q^{71} -1.39445 q^{73} -2.30278 q^{75} +2.60555 q^{77} -14.5139 q^{79} -10.6056 q^{81} +9.81665 q^{83} +2.69722 q^{85} +20.5139 q^{87} -3.69722 q^{89} -1.00000 q^{91} +9.90833 q^{93} -2.69722 q^{95} +1.21110 q^{97} +6.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{3} + 2 q^{5} + 2 q^{7} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - q^{3} + 2 q^{5} + 2 q^{7} + q^{9} - 2 q^{11} - 2 q^{13} - q^{15} + 9 q^{17} - 9 q^{19} - q^{21} - 16 q^{23} + 2 q^{25} - 4 q^{27} - 7 q^{29} - 5 q^{31} - 12 q^{33} + 2 q^{35} + 3 q^{37} + q^{39} - 5 q^{41} - 4 q^{43} + q^{45} - 2 q^{47} + 2 q^{49} + 2 q^{51} + 2 q^{53} - 2 q^{55} - 2 q^{57} - 3 q^{59} + 12 q^{61} + q^{63} - 2 q^{65} + 7 q^{67} + 8 q^{69} + 2 q^{71} - 10 q^{73} - q^{75} - 2 q^{77} - 11 q^{79} - 14 q^{81} - 2 q^{83} + 9 q^{85} + 23 q^{87} - 11 q^{89} - 2 q^{91} + 9 q^{93} - 9 q^{95} - 12 q^{97} + 12 q^{99}+O(q^{100}) \) 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
\(3\) −2.30278 −1.32951 −0.664754 0.747062i \(-0.731464\pi\)
−0.664754 + 0.747062i \(0.731464\pi\)
\(4\) 0 0
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) 1.00000 0.377964
\(8\) 0 0
\(9\) 2.30278 0.767592
\(10\) 0 0
\(11\) 2.60555 0.785603 0.392802 0.919623i \(-0.371506\pi\)
0.392802 + 0.919623i \(0.371506\pi\)
\(12\) 0 0
\(13\) −1.00000 −0.277350
\(14\) 0 0
\(15\) −2.30278 −0.594574
\(16\) 0 0
\(17\) 2.69722 0.654173 0.327086 0.944994i \(-0.393933\pi\)
0.327086 + 0.944994i \(0.393933\pi\)
\(18\) 0 0
\(19\) −2.69722 −0.618786 −0.309393 0.950934i \(-0.600126\pi\)
−0.309393 + 0.950934i \(0.600126\pi\)
\(20\) 0 0
\(21\) −2.30278 −0.502507
\(22\) 0 0
\(23\) −8.00000 −1.66812 −0.834058 0.551677i \(-0.813988\pi\)
−0.834058 + 0.551677i \(0.813988\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 1.60555 0.308988
\(28\) 0 0
\(29\) −8.90833 −1.65423 −0.827117 0.562029i \(-0.810021\pi\)
−0.827117 + 0.562029i \(0.810021\pi\)
\(30\) 0 0
\(31\) −4.30278 −0.772801 −0.386401 0.922331i \(-0.626282\pi\)
−0.386401 + 0.922331i \(0.626282\pi\)
\(32\) 0 0
\(33\) −6.00000 −1.04447
\(34\) 0 0
\(35\) 1.00000 0.169031
\(36\) 0 0
\(37\) 6.90833 1.13572 0.567861 0.823124i \(-0.307771\pi\)
0.567861 + 0.823124i \(0.307771\pi\)
\(38\) 0 0
\(39\) 2.30278 0.368739
\(40\) 0 0
\(41\) −0.697224 −0.108888 −0.0544441 0.998517i \(-0.517339\pi\)
−0.0544441 + 0.998517i \(0.517339\pi\)
\(42\) 0 0
\(43\) −2.00000 −0.304997 −0.152499 0.988304i \(-0.548732\pi\)
−0.152499 + 0.988304i \(0.548732\pi\)
\(44\) 0 0
\(45\) 2.30278 0.343278
\(46\) 0 0
\(47\) 2.60555 0.380059 0.190029 0.981778i \(-0.439142\pi\)
0.190029 + 0.981778i \(0.439142\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) −6.21110 −0.869728
\(52\) 0 0
\(53\) 4.60555 0.632621 0.316311 0.948656i \(-0.397556\pi\)
0.316311 + 0.948656i \(0.397556\pi\)
\(54\) 0 0
\(55\) 2.60555 0.351332
\(56\) 0 0
\(57\) 6.21110 0.822681
\(58\) 0 0
\(59\) −14.1194 −1.83819 −0.919097 0.394032i \(-0.871080\pi\)
−0.919097 + 0.394032i \(0.871080\pi\)
\(60\) 0 0
\(61\) 13.2111 1.69151 0.845754 0.533573i \(-0.179151\pi\)
0.845754 + 0.533573i \(0.179151\pi\)
\(62\) 0 0
\(63\) 2.30278 0.290122
\(64\) 0 0
\(65\) −1.00000 −0.124035
\(66\) 0 0
\(67\) −1.90833 −0.233139 −0.116570 0.993183i \(-0.537190\pi\)
−0.116570 + 0.993183i \(0.537190\pi\)
\(68\) 0 0
\(69\) 18.4222 2.21777
\(70\) 0 0
\(71\) 4.60555 0.546578 0.273289 0.961932i \(-0.411888\pi\)
0.273289 + 0.961932i \(0.411888\pi\)
\(72\) 0 0
\(73\) −1.39445 −0.163208 −0.0816039 0.996665i \(-0.526004\pi\)
−0.0816039 + 0.996665i \(0.526004\pi\)
\(74\) 0 0
\(75\) −2.30278 −0.265902
\(76\) 0 0
\(77\) 2.60555 0.296930
\(78\) 0 0
\(79\) −14.5139 −1.63294 −0.816469 0.577389i \(-0.804072\pi\)
−0.816469 + 0.577389i \(0.804072\pi\)
\(80\) 0 0
\(81\) −10.6056 −1.17839
\(82\) 0 0
\(83\) 9.81665 1.07752 0.538759 0.842460i \(-0.318893\pi\)
0.538759 + 0.842460i \(0.318893\pi\)
\(84\) 0 0
\(85\) 2.69722 0.292555
\(86\) 0 0
\(87\) 20.5139 2.19932
\(88\) 0 0
\(89\) −3.69722 −0.391905 −0.195952 0.980613i \(-0.562780\pi\)
−0.195952 + 0.980613i \(0.562780\pi\)
\(90\) 0 0
\(91\) −1.00000 −0.104828
\(92\) 0 0
\(93\) 9.90833 1.02745
\(94\) 0 0
\(95\) −2.69722 −0.276729
\(96\) 0 0
\(97\) 1.21110 0.122969 0.0614844 0.998108i \(-0.480417\pi\)
0.0614844 + 0.998108i \(0.480417\pi\)
\(98\) 0 0
\(99\) 6.00000 0.603023
\(100\) 0 0
\(101\) 6.00000 0.597022 0.298511 0.954406i \(-0.403510\pi\)
0.298511 + 0.954406i \(0.403510\pi\)
\(102\) 0 0
\(103\) 1.90833 0.188033 0.0940165 0.995571i \(-0.470029\pi\)
0.0940165 + 0.995571i \(0.470029\pi\)
\(104\) 0 0
\(105\) −2.30278 −0.224728
\(106\) 0 0
\(107\) −6.00000 −0.580042 −0.290021 0.957020i \(-0.593662\pi\)
−0.290021 + 0.957020i \(0.593662\pi\)
\(108\) 0 0
\(109\) −4.00000 −0.383131 −0.191565 0.981480i \(-0.561356\pi\)
−0.191565 + 0.981480i \(0.561356\pi\)
\(110\) 0 0
\(111\) −15.9083 −1.50995
\(112\) 0 0
\(113\) −13.8167 −1.29976 −0.649881 0.760036i \(-0.725181\pi\)
−0.649881 + 0.760036i \(0.725181\pi\)
\(114\) 0 0
\(115\) −8.00000 −0.746004
\(116\) 0 0
\(117\) −2.30278 −0.212892
\(118\) 0 0
\(119\) 2.69722 0.247254
\(120\) 0 0
\(121\) −4.21110 −0.382828
\(122\) 0 0
\(123\) 1.60555 0.144768
\(124\) 0 0
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) −1.39445 −0.123737 −0.0618687 0.998084i \(-0.519706\pi\)
−0.0618687 + 0.998084i \(0.519706\pi\)
\(128\) 0 0
\(129\) 4.60555 0.405496
\(130\) 0 0
\(131\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(132\) 0 0
\(133\) −2.69722 −0.233879
\(134\) 0 0
\(135\) 1.60555 0.138184
\(136\) 0 0
\(137\) 6.30278 0.538482 0.269241 0.963073i \(-0.413227\pi\)
0.269241 + 0.963073i \(0.413227\pi\)
\(138\) 0 0
\(139\) −12.4222 −1.05364 −0.526819 0.849978i \(-0.676615\pi\)
−0.526819 + 0.849978i \(0.676615\pi\)
\(140\) 0 0
\(141\) −6.00000 −0.505291
\(142\) 0 0
\(143\) −2.60555 −0.217887
\(144\) 0 0
\(145\) −8.90833 −0.739796
\(146\) 0 0
\(147\) −2.30278 −0.189930
\(148\) 0 0
\(149\) 8.00000 0.655386 0.327693 0.944784i \(-0.393729\pi\)
0.327693 + 0.944784i \(0.393729\pi\)
\(150\) 0 0
\(151\) −10.0000 −0.813788 −0.406894 0.913475i \(-0.633388\pi\)
−0.406894 + 0.913475i \(0.633388\pi\)
\(152\) 0 0
\(153\) 6.21110 0.502138
\(154\) 0 0
\(155\) −4.30278 −0.345607
\(156\) 0 0
\(157\) −7.51388 −0.599673 −0.299836 0.953991i \(-0.596932\pi\)
−0.299836 + 0.953991i \(0.596932\pi\)
\(158\) 0 0
\(159\) −10.6056 −0.841075
\(160\) 0 0
\(161\) −8.00000 −0.630488
\(162\) 0 0
\(163\) −0.119429 −0.00935444 −0.00467722 0.999989i \(-0.501489\pi\)
−0.00467722 + 0.999989i \(0.501489\pi\)
\(164\) 0 0
\(165\) −6.00000 −0.467099
\(166\) 0 0
\(167\) −8.78890 −0.680105 −0.340053 0.940406i \(-0.610445\pi\)
−0.340053 + 0.940406i \(0.610445\pi\)
\(168\) 0 0
\(169\) 1.00000 0.0769231
\(170\) 0 0
\(171\) −6.21110 −0.474975
\(172\) 0 0
\(173\) 22.3028 1.69565 0.847824 0.530277i \(-0.177912\pi\)
0.847824 + 0.530277i \(0.177912\pi\)
\(174\) 0 0
\(175\) 1.00000 0.0755929
\(176\) 0 0
\(177\) 32.5139 2.44389
\(178\) 0 0
\(179\) −19.9083 −1.48802 −0.744009 0.668169i \(-0.767078\pi\)
−0.744009 + 0.668169i \(0.767078\pi\)
\(180\) 0 0
\(181\) −11.8167 −0.878325 −0.439162 0.898408i \(-0.644725\pi\)
−0.439162 + 0.898408i \(0.644725\pi\)
\(182\) 0 0
\(183\) −30.4222 −2.24887
\(184\) 0 0
\(185\) 6.90833 0.507910
\(186\) 0 0
\(187\) 7.02776 0.513920
\(188\) 0 0
\(189\) 1.60555 0.116787
\(190\) 0 0
\(191\) 1.51388 0.109540 0.0547702 0.998499i \(-0.482557\pi\)
0.0547702 + 0.998499i \(0.482557\pi\)
\(192\) 0 0
\(193\) 0.0916731 0.00659877 0.00329939 0.999995i \(-0.498950\pi\)
0.00329939 + 0.999995i \(0.498950\pi\)
\(194\) 0 0
\(195\) 2.30278 0.164905
\(196\) 0 0
\(197\) −12.1194 −0.863474 −0.431737 0.902000i \(-0.642099\pi\)
−0.431737 + 0.902000i \(0.642099\pi\)
\(198\) 0 0
\(199\) −13.2111 −0.936510 −0.468255 0.883593i \(-0.655117\pi\)
−0.468255 + 0.883593i \(0.655117\pi\)
\(200\) 0 0
\(201\) 4.39445 0.309961
\(202\) 0 0
\(203\) −8.90833 −0.625242
\(204\) 0 0
\(205\) −0.697224 −0.0486963
\(206\) 0 0
\(207\) −18.4222 −1.28043
\(208\) 0 0
\(209\) −7.02776 −0.486120
\(210\) 0 0
\(211\) −6.51388 −0.448434 −0.224217 0.974539i \(-0.571982\pi\)
−0.224217 + 0.974539i \(0.571982\pi\)
\(212\) 0 0
\(213\) −10.6056 −0.726680
\(214\) 0 0
\(215\) −2.00000 −0.136399
\(216\) 0 0
\(217\) −4.30278 −0.292091
\(218\) 0 0
\(219\) 3.21110 0.216986
\(220\) 0 0
\(221\) −2.69722 −0.181435
\(222\) 0 0
\(223\) −1.21110 −0.0811014 −0.0405507 0.999177i \(-0.512911\pi\)
−0.0405507 + 0.999177i \(0.512911\pi\)
\(224\) 0 0
\(225\) 2.30278 0.153518
\(226\) 0 0
\(227\) 1.81665 0.120576 0.0602878 0.998181i \(-0.480798\pi\)
0.0602878 + 0.998181i \(0.480798\pi\)
\(228\) 0 0
\(229\) 3.30278 0.218254 0.109127 0.994028i \(-0.465195\pi\)
0.109127 + 0.994028i \(0.465195\pi\)
\(230\) 0 0
\(231\) −6.00000 −0.394771
\(232\) 0 0
\(233\) 19.8167 1.29823 0.649116 0.760689i \(-0.275139\pi\)
0.649116 + 0.760689i \(0.275139\pi\)
\(234\) 0 0
\(235\) 2.60555 0.169967
\(236\) 0 0
\(237\) 33.4222 2.17101
\(238\) 0 0
\(239\) −21.6333 −1.39934 −0.699671 0.714465i \(-0.746670\pi\)
−0.699671 + 0.714465i \(0.746670\pi\)
\(240\) 0 0
\(241\) −6.09167 −0.392399 −0.196200 0.980564i \(-0.562860\pi\)
−0.196200 + 0.980564i \(0.562860\pi\)
\(242\) 0 0
\(243\) 19.6056 1.25770
\(244\) 0 0
\(245\) 1.00000 0.0638877
\(246\) 0 0
\(247\) 2.69722 0.171620
\(248\) 0 0
\(249\) −22.6056 −1.43257
\(250\) 0 0
\(251\) 15.6333 0.986766 0.493383 0.869812i \(-0.335760\pi\)
0.493383 + 0.869812i \(0.335760\pi\)
\(252\) 0 0
\(253\) −20.8444 −1.31048
\(254\) 0 0
\(255\) −6.21110 −0.388954
\(256\) 0 0
\(257\) −21.6333 −1.34945 −0.674724 0.738070i \(-0.735737\pi\)
−0.674724 + 0.738070i \(0.735737\pi\)
\(258\) 0 0
\(259\) 6.90833 0.429263
\(260\) 0 0
\(261\) −20.5139 −1.26978
\(262\) 0 0
\(263\) −27.6333 −1.70394 −0.851971 0.523588i \(-0.824593\pi\)
−0.851971 + 0.523588i \(0.824593\pi\)
\(264\) 0 0
\(265\) 4.60555 0.282917
\(266\) 0 0
\(267\) 8.51388 0.521041
\(268\) 0 0
\(269\) 3.81665 0.232705 0.116353 0.993208i \(-0.462880\pi\)
0.116353 + 0.993208i \(0.462880\pi\)
\(270\) 0 0
\(271\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(272\) 0 0
\(273\) 2.30278 0.139370
\(274\) 0 0
\(275\) 2.60555 0.157121
\(276\) 0 0
\(277\) −13.2111 −0.793778 −0.396889 0.917867i \(-0.629910\pi\)
−0.396889 + 0.917867i \(0.629910\pi\)
\(278\) 0 0
\(279\) −9.90833 −0.593196
\(280\) 0 0
\(281\) −32.0000 −1.90896 −0.954480 0.298275i \(-0.903589\pi\)
−0.954480 + 0.298275i \(0.903589\pi\)
\(282\) 0 0
\(283\) −25.9083 −1.54009 −0.770045 0.637989i \(-0.779766\pi\)
−0.770045 + 0.637989i \(0.779766\pi\)
\(284\) 0 0
\(285\) 6.21110 0.367914
\(286\) 0 0
\(287\) −0.697224 −0.0411559
\(288\) 0 0
\(289\) −9.72498 −0.572058
\(290\) 0 0
\(291\) −2.78890 −0.163488
\(292\) 0 0
\(293\) −26.4222 −1.54360 −0.771801 0.635864i \(-0.780644\pi\)
−0.771801 + 0.635864i \(0.780644\pi\)
\(294\) 0 0
\(295\) −14.1194 −0.822065
\(296\) 0 0
\(297\) 4.18335 0.242742
\(298\) 0 0
\(299\) 8.00000 0.462652
\(300\) 0 0
\(301\) −2.00000 −0.115278
\(302\) 0 0
\(303\) −13.8167 −0.793746
\(304\) 0 0
\(305\) 13.2111 0.756466
\(306\) 0 0
\(307\) 14.6056 0.833583 0.416791 0.909002i \(-0.363155\pi\)
0.416791 + 0.909002i \(0.363155\pi\)
\(308\) 0 0
\(309\) −4.39445 −0.249991
\(310\) 0 0
\(311\) 13.0278 0.738736 0.369368 0.929283i \(-0.379574\pi\)
0.369368 + 0.929283i \(0.379574\pi\)
\(312\) 0 0
\(313\) 9.33053 0.527393 0.263696 0.964606i \(-0.415058\pi\)
0.263696 + 0.964606i \(0.415058\pi\)
\(314\) 0 0
\(315\) 2.30278 0.129747
\(316\) 0 0
\(317\) 12.4222 0.697701 0.348850 0.937178i \(-0.386572\pi\)
0.348850 + 0.937178i \(0.386572\pi\)
\(318\) 0 0
\(319\) −23.2111 −1.29957
\(320\) 0 0
\(321\) 13.8167 0.771170
\(322\) 0 0
\(323\) −7.27502 −0.404793
\(324\) 0 0
\(325\) −1.00000 −0.0554700
\(326\) 0 0
\(327\) 9.21110 0.509375
\(328\) 0 0
\(329\) 2.60555 0.143649
\(330\) 0 0
\(331\) 18.0000 0.989369 0.494685 0.869072i \(-0.335284\pi\)
0.494685 + 0.869072i \(0.335284\pi\)
\(332\) 0 0
\(333\) 15.9083 0.871771
\(334\) 0 0
\(335\) −1.90833 −0.104263
\(336\) 0 0
\(337\) 4.42221 0.240893 0.120446 0.992720i \(-0.461567\pi\)
0.120446 + 0.992720i \(0.461567\pi\)
\(338\) 0 0
\(339\) 31.8167 1.72804
\(340\) 0 0
\(341\) −11.2111 −0.607115
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) 0 0
\(345\) 18.4222 0.991818
\(346\) 0 0
\(347\) −8.18335 −0.439305 −0.219653 0.975578i \(-0.570492\pi\)
−0.219653 + 0.975578i \(0.570492\pi\)
\(348\) 0 0
\(349\) 23.9361 1.28127 0.640635 0.767846i \(-0.278671\pi\)
0.640635 + 0.767846i \(0.278671\pi\)
\(350\) 0 0
\(351\) −1.60555 −0.0856980
\(352\) 0 0
\(353\) −25.6333 −1.36432 −0.682162 0.731201i \(-0.738960\pi\)
−0.682162 + 0.731201i \(0.738960\pi\)
\(354\) 0 0
\(355\) 4.60555 0.244437
\(356\) 0 0
\(357\) −6.21110 −0.328726
\(358\) 0 0
\(359\) −11.0278 −0.582023 −0.291011 0.956720i \(-0.593992\pi\)
−0.291011 + 0.956720i \(0.593992\pi\)
\(360\) 0 0
\(361\) −11.7250 −0.617104
\(362\) 0 0
\(363\) 9.69722 0.508972
\(364\) 0 0
\(365\) −1.39445 −0.0729888
\(366\) 0 0
\(367\) 1.57779 0.0823602 0.0411801 0.999152i \(-0.486888\pi\)
0.0411801 + 0.999152i \(0.486888\pi\)
\(368\) 0 0
\(369\) −1.60555 −0.0835817
\(370\) 0 0
\(371\) 4.60555 0.239108
\(372\) 0 0
\(373\) −14.0000 −0.724893 −0.362446 0.932005i \(-0.618058\pi\)
−0.362446 + 0.932005i \(0.618058\pi\)
\(374\) 0 0
\(375\) −2.30278 −0.118915
\(376\) 0 0
\(377\) 8.90833 0.458802
\(378\) 0 0
\(379\) 13.6333 0.700296 0.350148 0.936694i \(-0.386131\pi\)
0.350148 + 0.936694i \(0.386131\pi\)
\(380\) 0 0
\(381\) 3.21110 0.164510
\(382\) 0 0
\(383\) 26.2389 1.34074 0.670372 0.742026i \(-0.266135\pi\)
0.670372 + 0.742026i \(0.266135\pi\)
\(384\) 0 0
\(385\) 2.60555 0.132791
\(386\) 0 0
\(387\) −4.60555 −0.234113
\(388\) 0 0
\(389\) −8.51388 −0.431671 −0.215835 0.976430i \(-0.569247\pi\)
−0.215835 + 0.976430i \(0.569247\pi\)
\(390\) 0 0
\(391\) −21.5778 −1.09124
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −14.5139 −0.730272
\(396\) 0 0
\(397\) 8.18335 0.410710 0.205355 0.978688i \(-0.434165\pi\)
0.205355 + 0.978688i \(0.434165\pi\)
\(398\) 0 0
\(399\) 6.21110 0.310944
\(400\) 0 0
\(401\) −24.0000 −1.19850 −0.599251 0.800561i \(-0.704535\pi\)
−0.599251 + 0.800561i \(0.704535\pi\)
\(402\) 0 0
\(403\) 4.30278 0.214337
\(404\) 0 0
\(405\) −10.6056 −0.526994
\(406\) 0 0
\(407\) 18.0000 0.892227
\(408\) 0 0
\(409\) −2.00000 −0.0988936 −0.0494468 0.998777i \(-0.515746\pi\)
−0.0494468 + 0.998777i \(0.515746\pi\)
\(410\) 0 0
\(411\) −14.5139 −0.715917
\(412\) 0 0
\(413\) −14.1194 −0.694772
\(414\) 0 0
\(415\) 9.81665 0.481881
\(416\) 0 0
\(417\) 28.6056 1.40082
\(418\) 0 0
\(419\) −15.6333 −0.763737 −0.381869 0.924217i \(-0.624719\pi\)
−0.381869 + 0.924217i \(0.624719\pi\)
\(420\) 0 0
\(421\) −12.4222 −0.605421 −0.302711 0.953083i \(-0.597892\pi\)
−0.302711 + 0.953083i \(0.597892\pi\)
\(422\) 0 0
\(423\) 6.00000 0.291730
\(424\) 0 0
\(425\) 2.69722 0.130835
\(426\) 0 0
\(427\) 13.2111 0.639330
\(428\) 0 0
\(429\) 6.00000 0.289683
\(430\) 0 0
\(431\) −34.0000 −1.63772 −0.818861 0.573992i \(-0.805394\pi\)
−0.818861 + 0.573992i \(0.805394\pi\)
\(432\) 0 0
\(433\) −24.9361 −1.19835 −0.599176 0.800617i \(-0.704505\pi\)
−0.599176 + 0.800617i \(0.704505\pi\)
\(434\) 0 0
\(435\) 20.5139 0.983565
\(436\) 0 0
\(437\) 21.5778 1.03221
\(438\) 0 0
\(439\) 19.2111 0.916896 0.458448 0.888721i \(-0.348406\pi\)
0.458448 + 0.888721i \(0.348406\pi\)
\(440\) 0 0
\(441\) 2.30278 0.109656
\(442\) 0 0
\(443\) 16.6056 0.788954 0.394477 0.918906i \(-0.370926\pi\)
0.394477 + 0.918906i \(0.370926\pi\)
\(444\) 0 0
\(445\) −3.69722 −0.175265
\(446\) 0 0
\(447\) −18.4222 −0.871340
\(448\) 0 0
\(449\) −35.0278 −1.65306 −0.826531 0.562891i \(-0.809689\pi\)
−0.826531 + 0.562891i \(0.809689\pi\)
\(450\) 0 0
\(451\) −1.81665 −0.0855429
\(452\) 0 0
\(453\) 23.0278 1.08194
\(454\) 0 0
\(455\) −1.00000 −0.0468807
\(456\) 0 0
\(457\) 27.5139 1.28704 0.643522 0.765427i \(-0.277472\pi\)
0.643522 + 0.765427i \(0.277472\pi\)
\(458\) 0 0
\(459\) 4.33053 0.202132
\(460\) 0 0
\(461\) 24.9083 1.16010 0.580048 0.814582i \(-0.303034\pi\)
0.580048 + 0.814582i \(0.303034\pi\)
\(462\) 0 0
\(463\) 5.72498 0.266062 0.133031 0.991112i \(-0.457529\pi\)
0.133031 + 0.991112i \(0.457529\pi\)
\(464\) 0 0
\(465\) 9.90833 0.459488
\(466\) 0 0
\(467\) −19.4861 −0.901710 −0.450855 0.892597i \(-0.648881\pi\)
−0.450855 + 0.892597i \(0.648881\pi\)
\(468\) 0 0
\(469\) −1.90833 −0.0881183
\(470\) 0 0
\(471\) 17.3028 0.797270
\(472\) 0 0
\(473\) −5.21110 −0.239607
\(474\) 0 0
\(475\) −2.69722 −0.123757
\(476\) 0 0
\(477\) 10.6056 0.485595
\(478\) 0 0
\(479\) 40.9083 1.86915 0.934575 0.355767i \(-0.115780\pi\)
0.934575 + 0.355767i \(0.115780\pi\)
\(480\) 0 0
\(481\) −6.90833 −0.314993
\(482\) 0 0
\(483\) 18.4222 0.838239
\(484\) 0 0
\(485\) 1.21110 0.0549933
\(486\) 0 0
\(487\) 30.3028 1.37315 0.686575 0.727059i \(-0.259113\pi\)
0.686575 + 0.727059i \(0.259113\pi\)
\(488\) 0 0
\(489\) 0.275019 0.0124368
\(490\) 0 0
\(491\) −30.4222 −1.37293 −0.686467 0.727161i \(-0.740840\pi\)
−0.686467 + 0.727161i \(0.740840\pi\)
\(492\) 0 0
\(493\) −24.0278 −1.08216
\(494\) 0 0
\(495\) 6.00000 0.269680
\(496\) 0 0
\(497\) 4.60555 0.206587
\(498\) 0 0
\(499\) 26.0000 1.16392 0.581960 0.813217i \(-0.302286\pi\)
0.581960 + 0.813217i \(0.302286\pi\)
\(500\) 0 0
\(501\) 20.2389 0.904206
\(502\) 0 0
\(503\) −27.6333 −1.23211 −0.616054 0.787704i \(-0.711270\pi\)
−0.616054 + 0.787704i \(0.711270\pi\)
\(504\) 0 0
\(505\) 6.00000 0.266996
\(506\) 0 0
\(507\) −2.30278 −0.102270
\(508\) 0 0
\(509\) −24.3305 −1.07843 −0.539216 0.842168i \(-0.681279\pi\)
−0.539216 + 0.842168i \(0.681279\pi\)
\(510\) 0 0
\(511\) −1.39445 −0.0616868
\(512\) 0 0
\(513\) −4.33053 −0.191198
\(514\) 0 0
\(515\) 1.90833 0.0840909
\(516\) 0 0
\(517\) 6.78890 0.298575
\(518\) 0 0
\(519\) −51.3583 −2.25438
\(520\) 0 0
\(521\) −22.2389 −0.974302 −0.487151 0.873318i \(-0.661964\pi\)
−0.487151 + 0.873318i \(0.661964\pi\)
\(522\) 0 0
\(523\) 34.4222 1.50518 0.752589 0.658491i \(-0.228805\pi\)
0.752589 + 0.658491i \(0.228805\pi\)
\(524\) 0 0
\(525\) −2.30278 −0.100501
\(526\) 0 0
\(527\) −11.6056 −0.505546
\(528\) 0 0
\(529\) 41.0000 1.78261
\(530\) 0 0
\(531\) −32.5139 −1.41098
\(532\) 0 0
\(533\) 0.697224 0.0302001
\(534\) 0 0
\(535\) −6.00000 −0.259403
\(536\) 0 0
\(537\) 45.8444 1.97833
\(538\) 0 0
\(539\) 2.60555 0.112229
\(540\) 0 0
\(541\) 38.0555 1.63613 0.818067 0.575123i \(-0.195046\pi\)
0.818067 + 0.575123i \(0.195046\pi\)
\(542\) 0 0
\(543\) 27.2111 1.16774
\(544\) 0 0
\(545\) −4.00000 −0.171341
\(546\) 0 0
\(547\) −37.6333 −1.60908 −0.804542 0.593896i \(-0.797589\pi\)
−0.804542 + 0.593896i \(0.797589\pi\)
\(548\) 0 0
\(549\) 30.4222 1.29839
\(550\) 0 0
\(551\) 24.0278 1.02362
\(552\) 0 0
\(553\) −14.5139 −0.617193
\(554\) 0 0
\(555\) −15.9083 −0.675271
\(556\) 0 0
\(557\) 4.09167 0.173370 0.0866849 0.996236i \(-0.472373\pi\)
0.0866849 + 0.996236i \(0.472373\pi\)
\(558\) 0 0
\(559\) 2.00000 0.0845910
\(560\) 0 0
\(561\) −16.1833 −0.683261
\(562\) 0 0
\(563\) 41.5416 1.75077 0.875386 0.483425i \(-0.160608\pi\)
0.875386 + 0.483425i \(0.160608\pi\)
\(564\) 0 0
\(565\) −13.8167 −0.581271
\(566\) 0 0
\(567\) −10.6056 −0.445391
\(568\) 0 0
\(569\) 45.7527 1.91805 0.959027 0.283314i \(-0.0914338\pi\)
0.959027 + 0.283314i \(0.0914338\pi\)
\(570\) 0 0
\(571\) −10.7250 −0.448826 −0.224413 0.974494i \(-0.572047\pi\)
−0.224413 + 0.974494i \(0.572047\pi\)
\(572\) 0 0
\(573\) −3.48612 −0.145635
\(574\) 0 0
\(575\) −8.00000 −0.333623
\(576\) 0 0
\(577\) −14.6056 −0.608037 −0.304019 0.952666i \(-0.598328\pi\)
−0.304019 + 0.952666i \(0.598328\pi\)
\(578\) 0 0
\(579\) −0.211103 −0.00877312
\(580\) 0 0
\(581\) 9.81665 0.407263
\(582\) 0 0
\(583\) 12.0000 0.496989
\(584\) 0 0
\(585\) −2.30278 −0.0952081
\(586\) 0 0
\(587\) 1.81665 0.0749813 0.0374907 0.999297i \(-0.488064\pi\)
0.0374907 + 0.999297i \(0.488064\pi\)
\(588\) 0 0
\(589\) 11.6056 0.478198
\(590\) 0 0
\(591\) 27.9083 1.14800
\(592\) 0 0
\(593\) −22.1833 −0.910961 −0.455480 0.890246i \(-0.650532\pi\)
−0.455480 + 0.890246i \(0.650532\pi\)
\(594\) 0 0
\(595\) 2.69722 0.110575
\(596\) 0 0
\(597\) 30.4222 1.24510
\(598\) 0 0
\(599\) 18.4222 0.752711 0.376355 0.926475i \(-0.377177\pi\)
0.376355 + 0.926475i \(0.377177\pi\)
\(600\) 0 0
\(601\) 37.2111 1.51787 0.758936 0.651165i \(-0.225719\pi\)
0.758936 + 0.651165i \(0.225719\pi\)
\(602\) 0 0
\(603\) −4.39445 −0.178956
\(604\) 0 0
\(605\) −4.21110 −0.171206
\(606\) 0 0
\(607\) −11.5139 −0.467334 −0.233667 0.972317i \(-0.575073\pi\)
−0.233667 + 0.972317i \(0.575073\pi\)
\(608\) 0 0
\(609\) 20.5139 0.831264
\(610\) 0 0
\(611\) −2.60555 −0.105409
\(612\) 0 0
\(613\) −18.8444 −0.761119 −0.380559 0.924757i \(-0.624269\pi\)
−0.380559 + 0.924757i \(0.624269\pi\)
\(614\) 0 0
\(615\) 1.60555 0.0647421
\(616\) 0 0
\(617\) 3.90833 0.157343 0.0786717 0.996901i \(-0.474932\pi\)
0.0786717 + 0.996901i \(0.474932\pi\)
\(618\) 0 0
\(619\) 10.7250 0.431073 0.215537 0.976496i \(-0.430850\pi\)
0.215537 + 0.976496i \(0.430850\pi\)
\(620\) 0 0
\(621\) −12.8444 −0.515428
\(622\) 0 0
\(623\) −3.69722 −0.148126
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 16.1833 0.646301
\(628\) 0 0
\(629\) 18.6333 0.742959
\(630\) 0 0
\(631\) 5.81665 0.231557 0.115779 0.993275i \(-0.463064\pi\)
0.115779 + 0.993275i \(0.463064\pi\)
\(632\) 0 0
\(633\) 15.0000 0.596196
\(634\) 0 0
\(635\) −1.39445 −0.0553370
\(636\) 0 0
\(637\) −1.00000 −0.0396214
\(638\) 0 0
\(639\) 10.6056 0.419549
\(640\) 0 0
\(641\) −38.7250 −1.52954 −0.764772 0.644301i \(-0.777149\pi\)
−0.764772 + 0.644301i \(0.777149\pi\)
\(642\) 0 0
\(643\) 42.8444 1.68962 0.844809 0.535068i \(-0.179714\pi\)
0.844809 + 0.535068i \(0.179714\pi\)
\(644\) 0 0
\(645\) 4.60555 0.181343
\(646\) 0 0
\(647\) 0.486122 0.0191114 0.00955571 0.999954i \(-0.496958\pi\)
0.00955571 + 0.999954i \(0.496958\pi\)
\(648\) 0 0
\(649\) −36.7889 −1.44409
\(650\) 0 0
\(651\) 9.90833 0.388338
\(652\) 0 0
\(653\) 2.42221 0.0947882 0.0473941 0.998876i \(-0.484908\pi\)
0.0473941 + 0.998876i \(0.484908\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) −3.21110 −0.125277
\(658\) 0 0
\(659\) −30.1472 −1.17437 −0.587184 0.809454i \(-0.699763\pi\)
−0.587184 + 0.809454i \(0.699763\pi\)
\(660\) 0 0
\(661\) 15.4861 0.602340 0.301170 0.953570i \(-0.402623\pi\)
0.301170 + 0.953570i \(0.402623\pi\)
\(662\) 0 0
\(663\) 6.21110 0.241219
\(664\) 0 0
\(665\) −2.69722 −0.104594
\(666\) 0 0
\(667\) 71.2666 2.75945
\(668\) 0 0
\(669\) 2.78890 0.107825
\(670\) 0 0
\(671\) 34.4222 1.32885
\(672\) 0 0
\(673\) 7.81665 0.301310 0.150655 0.988586i \(-0.451862\pi\)
0.150655 + 0.988586i \(0.451862\pi\)
\(674\) 0 0
\(675\) 1.60555 0.0617977
\(676\) 0 0
\(677\) −8.42221 −0.323692 −0.161846 0.986816i \(-0.551745\pi\)
−0.161846 + 0.986816i \(0.551745\pi\)
\(678\) 0 0
\(679\) 1.21110 0.0464779
\(680\) 0 0
\(681\) −4.18335 −0.160306
\(682\) 0 0
\(683\) 2.93608 0.112346 0.0561731 0.998421i \(-0.482110\pi\)
0.0561731 + 0.998421i \(0.482110\pi\)
\(684\) 0 0
\(685\) 6.30278 0.240817
\(686\) 0 0
\(687\) −7.60555 −0.290170
\(688\) 0 0
\(689\) −4.60555 −0.175458
\(690\) 0 0
\(691\) −3.09167 −0.117613 −0.0588064 0.998269i \(-0.518729\pi\)
−0.0588064 + 0.998269i \(0.518729\pi\)
\(692\) 0 0
\(693\) 6.00000 0.227921
\(694\) 0 0
\(695\) −12.4222 −0.471201
\(696\) 0 0
\(697\) −1.88057 −0.0712317
\(698\) 0 0
\(699\) −45.6333 −1.72601
\(700\) 0 0
\(701\) −35.1194 −1.32644 −0.663221 0.748423i \(-0.730811\pi\)
−0.663221 + 0.748423i \(0.730811\pi\)
\(702\) 0 0
\(703\) −18.6333 −0.702769
\(704\) 0 0
\(705\) −6.00000 −0.225973
\(706\) 0 0
\(707\) 6.00000 0.225653
\(708\) 0 0
\(709\) 29.2111 1.09705 0.548523 0.836135i \(-0.315190\pi\)
0.548523 + 0.836135i \(0.315190\pi\)
\(710\) 0 0
\(711\) −33.4222 −1.25343
\(712\) 0 0
\(713\) 34.4222 1.28912
\(714\) 0 0
\(715\) −2.60555 −0.0974421
\(716\) 0 0
\(717\) 49.8167 1.86044
\(718\) 0 0
\(719\) −2.42221 −0.0903330 −0.0451665 0.998979i \(-0.514382\pi\)
−0.0451665 + 0.998979i \(0.514382\pi\)
\(720\) 0 0
\(721\) 1.90833 0.0710698
\(722\) 0 0
\(723\) 14.0278 0.521698
\(724\) 0 0
\(725\) −8.90833 −0.330847
\(726\) 0 0
\(727\) −25.6972 −0.953057 −0.476529 0.879159i \(-0.658105\pi\)
−0.476529 + 0.879159i \(0.658105\pi\)
\(728\) 0 0
\(729\) −13.3305 −0.493723
\(730\) 0 0
\(731\) −5.39445 −0.199521
\(732\) 0 0
\(733\) 12.0000 0.443230 0.221615 0.975134i \(-0.428867\pi\)
0.221615 + 0.975134i \(0.428867\pi\)
\(734\) 0 0
\(735\) −2.30278 −0.0849392
\(736\) 0 0
\(737\) −4.97224 −0.183155
\(738\) 0 0
\(739\) 46.0555 1.69418 0.847090 0.531450i \(-0.178353\pi\)
0.847090 + 0.531450i \(0.178353\pi\)
\(740\) 0 0
\(741\) −6.21110 −0.228171
\(742\) 0 0
\(743\) 26.0917 0.957211 0.478605 0.878030i \(-0.341142\pi\)
0.478605 + 0.878030i \(0.341142\pi\)
\(744\) 0 0
\(745\) 8.00000 0.293097
\(746\) 0 0
\(747\) 22.6056 0.827094
\(748\) 0 0
\(749\) −6.00000 −0.219235
\(750\) 0 0
\(751\) −5.51388 −0.201204 −0.100602 0.994927i \(-0.532077\pi\)
−0.100602 + 0.994927i \(0.532077\pi\)
\(752\) 0 0
\(753\) −36.0000 −1.31191
\(754\) 0 0
\(755\) −10.0000 −0.363937
\(756\) 0 0
\(757\) 24.1833 0.878959 0.439479 0.898253i \(-0.355163\pi\)
0.439479 + 0.898253i \(0.355163\pi\)
\(758\) 0 0
\(759\) 48.0000 1.74229
\(760\) 0 0
\(761\) −6.48612 −0.235122 −0.117561 0.993066i \(-0.537508\pi\)
−0.117561 + 0.993066i \(0.537508\pi\)
\(762\) 0 0
\(763\) −4.00000 −0.144810
\(764\) 0 0
\(765\) 6.21110 0.224563
\(766\) 0 0
\(767\) 14.1194 0.509823
\(768\) 0 0
\(769\) −44.4222 −1.60191 −0.800953 0.598727i \(-0.795673\pi\)
−0.800953 + 0.598727i \(0.795673\pi\)
\(770\) 0 0
\(771\) 49.8167 1.79410
\(772\) 0 0
\(773\) 37.8167 1.36017 0.680085 0.733133i \(-0.261943\pi\)
0.680085 + 0.733133i \(0.261943\pi\)
\(774\) 0 0
\(775\) −4.30278 −0.154560
\(776\) 0 0
\(777\) −15.9083 −0.570708
\(778\) 0 0
\(779\) 1.88057 0.0673784
\(780\) 0 0
\(781\) 12.0000 0.429394
\(782\) 0 0
\(783\) −14.3028 −0.511140
\(784\) 0 0
\(785\) −7.51388 −0.268182
\(786\) 0 0
\(787\) 14.4222 0.514096 0.257048 0.966399i \(-0.417250\pi\)
0.257048 + 0.966399i \(0.417250\pi\)
\(788\) 0 0
\(789\) 63.6333 2.26541
\(790\) 0 0
\(791\) −13.8167 −0.491264
\(792\) 0 0
\(793\) −13.2111 −0.469140
\(794\) 0 0
\(795\) −10.6056 −0.376140
\(796\) 0 0
\(797\) 10.5416 0.373404 0.186702 0.982417i \(-0.440220\pi\)
0.186702 + 0.982417i \(0.440220\pi\)
\(798\) 0 0
\(799\) 7.02776 0.248624
\(800\) 0 0
\(801\) −8.51388 −0.300823
\(802\) 0 0
\(803\) −3.63331 −0.128217
\(804\) 0 0
\(805\) −8.00000 −0.281963
\(806\) 0 0
\(807\) −8.78890 −0.309384
\(808\) 0 0
\(809\) −3.48612 −0.122566 −0.0612828 0.998120i \(-0.519519\pi\)
−0.0612828 + 0.998120i \(0.519519\pi\)
\(810\) 0 0
\(811\) 11.6333 0.408501 0.204250 0.978919i \(-0.434524\pi\)
0.204250 + 0.978919i \(0.434524\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −0.119429 −0.00418343
\(816\) 0 0
\(817\) 5.39445 0.188728
\(818\) 0 0
\(819\) −2.30278 −0.0804655
\(820\) 0 0
\(821\) −54.4777 −1.90129 −0.950643 0.310288i \(-0.899575\pi\)
−0.950643 + 0.310288i \(0.899575\pi\)
\(822\) 0 0
\(823\) 33.0278 1.15128 0.575638 0.817705i \(-0.304754\pi\)
0.575638 + 0.817705i \(0.304754\pi\)
\(824\) 0 0
\(825\) −6.00000 −0.208893
\(826\) 0 0
\(827\) 39.6333 1.37819 0.689093 0.724673i \(-0.258009\pi\)
0.689093 + 0.724673i \(0.258009\pi\)
\(828\) 0 0
\(829\) −5.21110 −0.180989 −0.0904945 0.995897i \(-0.528845\pi\)
−0.0904945 + 0.995897i \(0.528845\pi\)
\(830\) 0 0
\(831\) 30.4222 1.05533
\(832\) 0 0
\(833\) 2.69722 0.0934533
\(834\) 0 0
\(835\) −8.78890 −0.304152
\(836\) 0 0
\(837\) −6.90833 −0.238787
\(838\) 0 0
\(839\) 20.8444 0.719629 0.359814 0.933024i \(-0.382840\pi\)
0.359814 + 0.933024i \(0.382840\pi\)
\(840\) 0 0
\(841\) 50.3583 1.73649
\(842\) 0 0
\(843\) 73.6888 2.53798
\(844\) 0 0
\(845\) 1.00000 0.0344010
\(846\) 0 0
\(847\) −4.21110 −0.144695
\(848\) 0 0
\(849\) 59.6611 2.04756
\(850\) 0 0
\(851\) −55.2666 −1.89452
\(852\) 0 0
\(853\) 45.8167 1.56873 0.784366 0.620298i \(-0.212988\pi\)
0.784366 + 0.620298i \(0.212988\pi\)
\(854\) 0 0
\(855\) −6.21110 −0.212415
\(856\) 0 0
\(857\) 45.6972 1.56099 0.780494 0.625164i \(-0.214968\pi\)
0.780494 + 0.625164i \(0.214968\pi\)
\(858\) 0 0
\(859\) −10.4222 −0.355601 −0.177801 0.984067i \(-0.556898\pi\)
−0.177801 + 0.984067i \(0.556898\pi\)
\(860\) 0 0
\(861\) 1.60555 0.0547170
\(862\) 0 0
\(863\) 18.2750 0.622089 0.311044 0.950395i \(-0.399321\pi\)
0.311044 + 0.950395i \(0.399321\pi\)
\(864\) 0 0
\(865\) 22.3028 0.758317
\(866\) 0 0
\(867\) 22.3944 0.760555
\(868\) 0 0
\(869\) −37.8167 −1.28284
\(870\) 0 0
\(871\) 1.90833 0.0646612
\(872\) 0 0
\(873\) 2.78890 0.0943899
\(874\) 0 0
\(875\) 1.00000 0.0338062
\(876\) 0 0
\(877\) 31.9083 1.07747 0.538734 0.842476i \(-0.318903\pi\)
0.538734 + 0.842476i \(0.318903\pi\)
\(878\) 0 0
\(879\) 60.8444 2.05223
\(880\) 0 0
\(881\) −35.4500 −1.19434 −0.597170 0.802115i \(-0.703708\pi\)
−0.597170 + 0.802115i \(0.703708\pi\)
\(882\) 0 0
\(883\) 21.3944 0.719981 0.359990 0.932956i \(-0.382780\pi\)
0.359990 + 0.932956i \(0.382780\pi\)
\(884\) 0 0
\(885\) 32.5139 1.09294
\(886\) 0 0
\(887\) −35.1472 −1.18013 −0.590064 0.807357i \(-0.700897\pi\)
−0.590064 + 0.807357i \(0.700897\pi\)
\(888\) 0 0
\(889\) −1.39445 −0.0467683
\(890\) 0 0
\(891\) −27.6333 −0.925751
\(892\) 0 0
\(893\) −7.02776 −0.235175
\(894\) 0 0
\(895\) −19.9083 −0.665462
\(896\) 0 0
\(897\) −18.4222 −0.615100
\(898\) 0 0
\(899\) 38.3305 1.27839
\(900\) 0 0
\(901\) 12.4222 0.413844
\(902\) 0 0
\(903\) 4.60555 0.153263
\(904\) 0 0
\(905\) −11.8167 −0.392799
\(906\) 0 0
\(907\) −4.60555 −0.152925 −0.0764624 0.997072i \(-0.524363\pi\)
−0.0764624 + 0.997072i \(0.524363\pi\)
\(908\) 0 0
\(909\) 13.8167 0.458269
\(910\) 0 0
\(911\) −23.9083 −0.792118 −0.396059 0.918225i \(-0.629622\pi\)
−0.396059 + 0.918225i \(0.629622\pi\)
\(912\) 0 0
\(913\) 25.5778 0.846501
\(914\) 0 0
\(915\) −30.4222 −1.00573
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) −32.5416 −1.07345 −0.536725 0.843757i \(-0.680339\pi\)
−0.536725 + 0.843757i \(0.680339\pi\)
\(920\) 0 0
\(921\) −33.6333 −1.10826
\(922\) 0 0
\(923\) −4.60555 −0.151594
\(924\) 0 0
\(925\) 6.90833 0.227144
\(926\) 0 0
\(927\) 4.39445 0.144333
\(928\) 0 0
\(929\) −33.7527 −1.10739 −0.553696 0.832719i \(-0.686783\pi\)
−0.553696 + 0.832719i \(0.686783\pi\)
\(930\) 0 0
\(931\) −2.69722 −0.0883980
\(932\) 0 0
\(933\) −30.0000 −0.982156
\(934\) 0 0
\(935\) 7.02776 0.229832
\(936\) 0 0
\(937\) −37.3028 −1.21863 −0.609314 0.792929i \(-0.708555\pi\)
−0.609314 + 0.792929i \(0.708555\pi\)
\(938\) 0 0
\(939\) −21.4861 −0.701173
\(940\) 0 0
\(941\) −27.1194 −0.884068 −0.442034 0.896998i \(-0.645743\pi\)
−0.442034 + 0.896998i \(0.645743\pi\)
\(942\) 0 0
\(943\) 5.57779 0.181638
\(944\) 0 0
\(945\) 1.60555 0.0522286
\(946\) 0 0
\(947\) −56.1472 −1.82454 −0.912269 0.409591i \(-0.865671\pi\)
−0.912269 + 0.409591i \(0.865671\pi\)
\(948\) 0 0
\(949\) 1.39445 0.0452657
\(950\) 0 0
\(951\) −28.6056 −0.927599
\(952\) 0 0
\(953\) 11.0278 0.357224 0.178612 0.983920i \(-0.442839\pi\)
0.178612 + 0.983920i \(0.442839\pi\)
\(954\) 0 0
\(955\) 1.51388 0.0489879
\(956\) 0 0
\(957\) 53.4500 1.72779
\(958\) 0 0
\(959\) 6.30278 0.203527
\(960\) 0 0
\(961\) −12.4861 −0.402778
\(962\) 0 0
\(963\) −13.8167 −0.445235
\(964\) 0 0
\(965\) 0.0916731 0.00295106
\(966\) 0 0
\(967\) 12.4861 0.401527 0.200763 0.979640i \(-0.435658\pi\)
0.200763 + 0.979640i \(0.435658\pi\)
\(968\) 0 0
\(969\) 16.7527 0.538175
\(970\) 0 0
\(971\) 49.0278 1.57338 0.786688 0.617351i \(-0.211794\pi\)
0.786688 + 0.617351i \(0.211794\pi\)
\(972\) 0 0
\(973\) −12.4222 −0.398238
\(974\) 0 0
\(975\) 2.30278 0.0737478
\(976\) 0 0
\(977\) −7.90833 −0.253010 −0.126505 0.991966i \(-0.540376\pi\)
−0.126505 + 0.991966i \(0.540376\pi\)
\(978\) 0 0
\(979\) −9.63331 −0.307882
\(980\) 0 0
\(981\) −9.21110 −0.294088
\(982\) 0 0
\(983\) −53.4500 −1.70479 −0.852395 0.522899i \(-0.824850\pi\)
−0.852395 + 0.522899i \(0.824850\pi\)
\(984\) 0 0
\(985\) −12.1194 −0.386157
\(986\) 0 0
\(987\) −6.00000 −0.190982
\(988\) 0 0
\(989\) 16.0000 0.508770
\(990\) 0 0
\(991\) −34.1472 −1.08472 −0.542361 0.840146i \(-0.682469\pi\)
−0.542361 + 0.840146i \(0.682469\pi\)
\(992\) 0 0
\(993\) −41.4500 −1.31537
\(994\) 0 0
\(995\) −13.2111 −0.418820
\(996\) 0 0
\(997\) −6.14719 −0.194683 −0.0973417 0.995251i \(-0.531034\pi\)
−0.0973417 + 0.995251i \(0.531034\pi\)
\(998\) 0 0
\(999\) 11.0917 0.350925
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 3640.2.a.l.1.1 2
4.3 odd 2 7280.2.a.bh.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3640.2.a.l.1.1 2 1.1 even 1 trivial
7280.2.a.bh.1.2 2 4.3 odd 2