Properties

Label 6044.2.a.a.1.14
Level $6044$
Weight $2$
Character 6044.1
Self dual yes
Analytic conductor $48.262$
Analytic rank $1$
Dimension $63$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [6044,2,Mod(1,6044)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6044, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("6044.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6044 = 2^{2} \cdot 1511 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6044.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(48.2615829817\)
Analytic rank: \(1\)
Dimension: \(63\)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.14
Character \(\chi\) \(=\) 6044.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.98161 q^{3} -3.75516 q^{5} -4.03693 q^{7} +0.926797 q^{9} +O(q^{10})\) \(q-1.98161 q^{3} -3.75516 q^{5} -4.03693 q^{7} +0.926797 q^{9} +1.98101 q^{11} -1.54356 q^{13} +7.44128 q^{15} -3.21178 q^{17} -4.50606 q^{19} +7.99964 q^{21} +3.07756 q^{23} +9.10121 q^{25} +4.10829 q^{27} -7.71564 q^{29} +6.10333 q^{31} -3.92561 q^{33} +15.1593 q^{35} -8.55476 q^{37} +3.05874 q^{39} +0.512844 q^{41} +0.00742461 q^{43} -3.48027 q^{45} +7.94533 q^{47} +9.29680 q^{49} +6.36451 q^{51} +8.76403 q^{53} -7.43902 q^{55} +8.92928 q^{57} +10.0472 q^{59} +10.8260 q^{61} -3.74142 q^{63} +5.79630 q^{65} +0.290722 q^{67} -6.09854 q^{69} -10.2769 q^{71} +7.17032 q^{73} -18.0351 q^{75} -7.99721 q^{77} -10.0803 q^{79} -10.9214 q^{81} -13.1250 q^{83} +12.0607 q^{85} +15.2894 q^{87} +11.9426 q^{89} +6.23124 q^{91} -12.0945 q^{93} +16.9210 q^{95} +17.1591 q^{97} +1.83600 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 63 q - 7 q^{3} - 7 q^{5} - 22 q^{7} + 62 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 63 q - 7 q^{3} - 7 q^{5} - 22 q^{7} + 62 q^{9} - 21 q^{11} - 19 q^{13} - 30 q^{15} - 5 q^{17} - 59 q^{19} - 30 q^{21} - 24 q^{23} + 60 q^{25} - 34 q^{27} - 28 q^{29} - 48 q^{31} - q^{33} - 44 q^{35} - 29 q^{37} - 75 q^{39} - 3 q^{41} - 88 q^{43} - 21 q^{45} - 21 q^{47} + 63 q^{49} - 85 q^{51} - 24 q^{53} - 85 q^{55} - 35 q^{59} - 78 q^{61} - 74 q^{63} - 13 q^{65} - 68 q^{67} - 43 q^{69} - 59 q^{71} - q^{73} - 45 q^{75} - 33 q^{77} - 140 q^{79} + 51 q^{81} - 27 q^{83} - 84 q^{85} - 61 q^{87} - 2 q^{89} - 92 q^{91} - 51 q^{93} - 51 q^{95} - 10 q^{97} - 115 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −1.98161 −1.14409 −0.572043 0.820224i \(-0.693849\pi\)
−0.572043 + 0.820224i \(0.693849\pi\)
\(4\) 0 0
\(5\) −3.75516 −1.67936 −0.839679 0.543084i \(-0.817257\pi\)
−0.839679 + 0.543084i \(0.817257\pi\)
\(6\) 0 0
\(7\) −4.03693 −1.52582 −0.762908 0.646507i \(-0.776229\pi\)
−0.762908 + 0.646507i \(0.776229\pi\)
\(8\) 0 0
\(9\) 0.926797 0.308932
\(10\) 0 0
\(11\) 1.98101 0.597298 0.298649 0.954363i \(-0.403464\pi\)
0.298649 + 0.954363i \(0.403464\pi\)
\(12\) 0 0
\(13\) −1.54356 −0.428106 −0.214053 0.976822i \(-0.568667\pi\)
−0.214053 + 0.976822i \(0.568667\pi\)
\(14\) 0 0
\(15\) 7.44128 1.92133
\(16\) 0 0
\(17\) −3.21178 −0.778971 −0.389486 0.921033i \(-0.627347\pi\)
−0.389486 + 0.921033i \(0.627347\pi\)
\(18\) 0 0
\(19\) −4.50606 −1.03376 −0.516881 0.856057i \(-0.672907\pi\)
−0.516881 + 0.856057i \(0.672907\pi\)
\(20\) 0 0
\(21\) 7.99964 1.74566
\(22\) 0 0
\(23\) 3.07756 0.641716 0.320858 0.947127i \(-0.396029\pi\)
0.320858 + 0.947127i \(0.396029\pi\)
\(24\) 0 0
\(25\) 9.10121 1.82024
\(26\) 0 0
\(27\) 4.10829 0.790641
\(28\) 0 0
\(29\) −7.71564 −1.43276 −0.716379 0.697711i \(-0.754202\pi\)
−0.716379 + 0.697711i \(0.754202\pi\)
\(30\) 0 0
\(31\) 6.10333 1.09619 0.548095 0.836416i \(-0.315353\pi\)
0.548095 + 0.836416i \(0.315353\pi\)
\(32\) 0 0
\(33\) −3.92561 −0.683360
\(34\) 0 0
\(35\) 15.1593 2.56239
\(36\) 0 0
\(37\) −8.55476 −1.40639 −0.703197 0.710995i \(-0.748245\pi\)
−0.703197 + 0.710995i \(0.748245\pi\)
\(38\) 0 0
\(39\) 3.05874 0.489790
\(40\) 0 0
\(41\) 0.512844 0.0800928 0.0400464 0.999198i \(-0.487249\pi\)
0.0400464 + 0.999198i \(0.487249\pi\)
\(42\) 0 0
\(43\) 0.00742461 0.00113224 0.000566121 1.00000i \(-0.499820\pi\)
0.000566121 1.00000i \(0.499820\pi\)
\(44\) 0 0
\(45\) −3.48027 −0.518808
\(46\) 0 0
\(47\) 7.94533 1.15895 0.579473 0.814992i \(-0.303259\pi\)
0.579473 + 0.814992i \(0.303259\pi\)
\(48\) 0 0
\(49\) 9.29680 1.32811
\(50\) 0 0
\(51\) 6.36451 0.891210
\(52\) 0 0
\(53\) 8.76403 1.20383 0.601916 0.798559i \(-0.294404\pi\)
0.601916 + 0.798559i \(0.294404\pi\)
\(54\) 0 0
\(55\) −7.43902 −1.00308
\(56\) 0 0
\(57\) 8.92928 1.18271
\(58\) 0 0
\(59\) 10.0472 1.30803 0.654016 0.756480i \(-0.273083\pi\)
0.654016 + 0.756480i \(0.273083\pi\)
\(60\) 0 0
\(61\) 10.8260 1.38613 0.693064 0.720876i \(-0.256260\pi\)
0.693064 + 0.720876i \(0.256260\pi\)
\(62\) 0 0
\(63\) −3.74142 −0.471374
\(64\) 0 0
\(65\) 5.79630 0.718943
\(66\) 0 0
\(67\) 0.290722 0.0355174 0.0177587 0.999842i \(-0.494347\pi\)
0.0177587 + 0.999842i \(0.494347\pi\)
\(68\) 0 0
\(69\) −6.09854 −0.734178
\(70\) 0 0
\(71\) −10.2769 −1.21964 −0.609821 0.792540i \(-0.708758\pi\)
−0.609821 + 0.792540i \(0.708758\pi\)
\(72\) 0 0
\(73\) 7.17032 0.839222 0.419611 0.907704i \(-0.362166\pi\)
0.419611 + 0.907704i \(0.362166\pi\)
\(74\) 0 0
\(75\) −18.0351 −2.08251
\(76\) 0 0
\(77\) −7.99721 −0.911367
\(78\) 0 0
\(79\) −10.0803 −1.13412 −0.567059 0.823677i \(-0.691919\pi\)
−0.567059 + 0.823677i \(0.691919\pi\)
\(80\) 0 0
\(81\) −10.9214 −1.21349
\(82\) 0 0
\(83\) −13.1250 −1.44065 −0.720326 0.693636i \(-0.756008\pi\)
−0.720326 + 0.693636i \(0.756008\pi\)
\(84\) 0 0
\(85\) 12.0607 1.30817
\(86\) 0 0
\(87\) 15.2894 1.63920
\(88\) 0 0
\(89\) 11.9426 1.26591 0.632957 0.774187i \(-0.281841\pi\)
0.632957 + 0.774187i \(0.281841\pi\)
\(90\) 0 0
\(91\) 6.23124 0.653211
\(92\) 0 0
\(93\) −12.0945 −1.25414
\(94\) 0 0
\(95\) 16.9210 1.73605
\(96\) 0 0
\(97\) 17.1591 1.74224 0.871121 0.491068i \(-0.163393\pi\)
0.871121 + 0.491068i \(0.163393\pi\)
\(98\) 0 0
\(99\) 1.83600 0.184525
\(100\) 0 0
\(101\) −15.0930 −1.50181 −0.750906 0.660409i \(-0.770383\pi\)
−0.750906 + 0.660409i \(0.770383\pi\)
\(102\) 0 0
\(103\) 18.9503 1.86723 0.933613 0.358284i \(-0.116638\pi\)
0.933613 + 0.358284i \(0.116638\pi\)
\(104\) 0 0
\(105\) −30.0399 −2.93159
\(106\) 0 0
\(107\) 0.820531 0.0793237 0.0396619 0.999213i \(-0.487372\pi\)
0.0396619 + 0.999213i \(0.487372\pi\)
\(108\) 0 0
\(109\) 10.4642 1.00229 0.501144 0.865364i \(-0.332913\pi\)
0.501144 + 0.865364i \(0.332913\pi\)
\(110\) 0 0
\(111\) 16.9522 1.60903
\(112\) 0 0
\(113\) 10.3388 0.972595 0.486298 0.873793i \(-0.338347\pi\)
0.486298 + 0.873793i \(0.338347\pi\)
\(114\) 0 0
\(115\) −11.5567 −1.07767
\(116\) 0 0
\(117\) −1.43057 −0.132256
\(118\) 0 0
\(119\) 12.9657 1.18857
\(120\) 0 0
\(121\) −7.07559 −0.643235
\(122\) 0 0
\(123\) −1.01626 −0.0916331
\(124\) 0 0
\(125\) −15.4007 −1.37748
\(126\) 0 0
\(127\) −5.30930 −0.471124 −0.235562 0.971859i \(-0.575693\pi\)
−0.235562 + 0.971859i \(0.575693\pi\)
\(128\) 0 0
\(129\) −0.0147127 −0.00129538
\(130\) 0 0
\(131\) −11.8227 −1.03295 −0.516475 0.856302i \(-0.672756\pi\)
−0.516475 + 0.856302i \(0.672756\pi\)
\(132\) 0 0
\(133\) 18.1907 1.57733
\(134\) 0 0
\(135\) −15.4273 −1.32777
\(136\) 0 0
\(137\) 6.19520 0.529292 0.264646 0.964346i \(-0.414745\pi\)
0.264646 + 0.964346i \(0.414745\pi\)
\(138\) 0 0
\(139\) 22.8819 1.94081 0.970407 0.241475i \(-0.0776313\pi\)
0.970407 + 0.241475i \(0.0776313\pi\)
\(140\) 0 0
\(141\) −15.7446 −1.32593
\(142\) 0 0
\(143\) −3.05781 −0.255707
\(144\) 0 0
\(145\) 28.9734 2.40611
\(146\) 0 0
\(147\) −18.4227 −1.51948
\(148\) 0 0
\(149\) −21.5624 −1.76646 −0.883229 0.468942i \(-0.844635\pi\)
−0.883229 + 0.468942i \(0.844635\pi\)
\(150\) 0 0
\(151\) −23.6841 −1.92738 −0.963691 0.267022i \(-0.913960\pi\)
−0.963691 + 0.267022i \(0.913960\pi\)
\(152\) 0 0
\(153\) −2.97667 −0.240649
\(154\) 0 0
\(155\) −22.9190 −1.84090
\(156\) 0 0
\(157\) 4.11544 0.328448 0.164224 0.986423i \(-0.447488\pi\)
0.164224 + 0.986423i \(0.447488\pi\)
\(158\) 0 0
\(159\) −17.3669 −1.37729
\(160\) 0 0
\(161\) −12.4239 −0.979140
\(162\) 0 0
\(163\) 8.19809 0.642124 0.321062 0.947058i \(-0.395960\pi\)
0.321062 + 0.947058i \(0.395960\pi\)
\(164\) 0 0
\(165\) 14.7413 1.14761
\(166\) 0 0
\(167\) −13.1489 −1.01749 −0.508745 0.860917i \(-0.669890\pi\)
−0.508745 + 0.860917i \(0.669890\pi\)
\(168\) 0 0
\(169\) −10.6174 −0.816725
\(170\) 0 0
\(171\) −4.17621 −0.319362
\(172\) 0 0
\(173\) 17.8651 1.35826 0.679128 0.734020i \(-0.262358\pi\)
0.679128 + 0.734020i \(0.262358\pi\)
\(174\) 0 0
\(175\) −36.7409 −2.77735
\(176\) 0 0
\(177\) −19.9097 −1.49650
\(178\) 0 0
\(179\) 0.0452376 0.00338122 0.00169061 0.999999i \(-0.499462\pi\)
0.00169061 + 0.999999i \(0.499462\pi\)
\(180\) 0 0
\(181\) −17.4560 −1.29750 −0.648749 0.761003i \(-0.724707\pi\)
−0.648749 + 0.761003i \(0.724707\pi\)
\(182\) 0 0
\(183\) −21.4530 −1.58585
\(184\) 0 0
\(185\) 32.1245 2.36184
\(186\) 0 0
\(187\) −6.36258 −0.465278
\(188\) 0 0
\(189\) −16.5849 −1.20637
\(190\) 0 0
\(191\) −6.77631 −0.490316 −0.245158 0.969483i \(-0.578840\pi\)
−0.245158 + 0.969483i \(0.578840\pi\)
\(192\) 0 0
\(193\) 20.1719 1.45200 0.726002 0.687693i \(-0.241376\pi\)
0.726002 + 0.687693i \(0.241376\pi\)
\(194\) 0 0
\(195\) −11.4860 −0.822532
\(196\) 0 0
\(197\) 14.0795 1.00312 0.501561 0.865122i \(-0.332759\pi\)
0.501561 + 0.865122i \(0.332759\pi\)
\(198\) 0 0
\(199\) −26.8299 −1.90192 −0.950959 0.309317i \(-0.899900\pi\)
−0.950959 + 0.309317i \(0.899900\pi\)
\(200\) 0 0
\(201\) −0.576099 −0.0406349
\(202\) 0 0
\(203\) 31.1475 2.18613
\(204\) 0 0
\(205\) −1.92581 −0.134504
\(206\) 0 0
\(207\) 2.85227 0.198247
\(208\) 0 0
\(209\) −8.92657 −0.617464
\(210\) 0 0
\(211\) −24.2087 −1.66659 −0.833296 0.552827i \(-0.813549\pi\)
−0.833296 + 0.552827i \(0.813549\pi\)
\(212\) 0 0
\(213\) 20.3648 1.39537
\(214\) 0 0
\(215\) −0.0278806 −0.00190144
\(216\) 0 0
\(217\) −24.6387 −1.67259
\(218\) 0 0
\(219\) −14.2088 −0.960142
\(220\) 0 0
\(221\) 4.95757 0.333482
\(222\) 0 0
\(223\) −20.2940 −1.35899 −0.679495 0.733680i \(-0.737801\pi\)
−0.679495 + 0.733680i \(0.737801\pi\)
\(224\) 0 0
\(225\) 8.43497 0.562332
\(226\) 0 0
\(227\) −4.65300 −0.308831 −0.154415 0.988006i \(-0.549349\pi\)
−0.154415 + 0.988006i \(0.549349\pi\)
\(228\) 0 0
\(229\) 16.4665 1.08814 0.544069 0.839041i \(-0.316883\pi\)
0.544069 + 0.839041i \(0.316883\pi\)
\(230\) 0 0
\(231\) 15.8474 1.04268
\(232\) 0 0
\(233\) 25.9380 1.69925 0.849627 0.527383i \(-0.176827\pi\)
0.849627 + 0.527383i \(0.176827\pi\)
\(234\) 0 0
\(235\) −29.8360 −1.94628
\(236\) 0 0
\(237\) 19.9752 1.29753
\(238\) 0 0
\(239\) 26.1687 1.69271 0.846356 0.532617i \(-0.178791\pi\)
0.846356 + 0.532617i \(0.178791\pi\)
\(240\) 0 0
\(241\) −15.5894 −1.00420 −0.502100 0.864810i \(-0.667439\pi\)
−0.502100 + 0.864810i \(0.667439\pi\)
\(242\) 0 0
\(243\) 9.31722 0.597700
\(244\) 0 0
\(245\) −34.9110 −2.23038
\(246\) 0 0
\(247\) 6.95537 0.442559
\(248\) 0 0
\(249\) 26.0086 1.64823
\(250\) 0 0
\(251\) −2.73428 −0.172586 −0.0862930 0.996270i \(-0.527502\pi\)
−0.0862930 + 0.996270i \(0.527502\pi\)
\(252\) 0 0
\(253\) 6.09669 0.383295
\(254\) 0 0
\(255\) −23.8997 −1.49666
\(256\) 0 0
\(257\) 0.326482 0.0203654 0.0101827 0.999948i \(-0.496759\pi\)
0.0101827 + 0.999948i \(0.496759\pi\)
\(258\) 0 0
\(259\) 34.5350 2.14590
\(260\) 0 0
\(261\) −7.15083 −0.442626
\(262\) 0 0
\(263\) 10.7711 0.664173 0.332087 0.943249i \(-0.392247\pi\)
0.332087 + 0.943249i \(0.392247\pi\)
\(264\) 0 0
\(265\) −32.9103 −2.02166
\(266\) 0 0
\(267\) −23.6656 −1.44831
\(268\) 0 0
\(269\) −20.7801 −1.26699 −0.633493 0.773748i \(-0.718380\pi\)
−0.633493 + 0.773748i \(0.718380\pi\)
\(270\) 0 0
\(271\) −7.97548 −0.484476 −0.242238 0.970217i \(-0.577881\pi\)
−0.242238 + 0.970217i \(0.577881\pi\)
\(272\) 0 0
\(273\) −12.3479 −0.747329
\(274\) 0 0
\(275\) 18.0296 1.08723
\(276\) 0 0
\(277\) 28.5965 1.71820 0.859099 0.511810i \(-0.171025\pi\)
0.859099 + 0.511810i \(0.171025\pi\)
\(278\) 0 0
\(279\) 5.65655 0.338649
\(280\) 0 0
\(281\) −1.04543 −0.0623651 −0.0311826 0.999514i \(-0.509927\pi\)
−0.0311826 + 0.999514i \(0.509927\pi\)
\(282\) 0 0
\(283\) −0.591169 −0.0351413 −0.0175707 0.999846i \(-0.505593\pi\)
−0.0175707 + 0.999846i \(0.505593\pi\)
\(284\) 0 0
\(285\) −33.5308 −1.98620
\(286\) 0 0
\(287\) −2.07032 −0.122207
\(288\) 0 0
\(289\) −6.68447 −0.393204
\(290\) 0 0
\(291\) −34.0027 −1.99328
\(292\) 0 0
\(293\) −0.956681 −0.0558899 −0.0279449 0.999609i \(-0.508896\pi\)
−0.0279449 + 0.999609i \(0.508896\pi\)
\(294\) 0 0
\(295\) −37.7288 −2.19665
\(296\) 0 0
\(297\) 8.13857 0.472248
\(298\) 0 0
\(299\) −4.75039 −0.274722
\(300\) 0 0
\(301\) −0.0299726 −0.00172759
\(302\) 0 0
\(303\) 29.9086 1.71820
\(304\) 0 0
\(305\) −40.6534 −2.32781
\(306\) 0 0
\(307\) −15.7592 −0.899423 −0.449711 0.893174i \(-0.648473\pi\)
−0.449711 + 0.893174i \(0.648473\pi\)
\(308\) 0 0
\(309\) −37.5521 −2.13627
\(310\) 0 0
\(311\) 20.4808 1.16136 0.580679 0.814133i \(-0.302787\pi\)
0.580679 + 0.814133i \(0.302787\pi\)
\(312\) 0 0
\(313\) −2.57876 −0.145760 −0.0728802 0.997341i \(-0.523219\pi\)
−0.0728802 + 0.997341i \(0.523219\pi\)
\(314\) 0 0
\(315\) 14.0496 0.791606
\(316\) 0 0
\(317\) −15.6486 −0.878911 −0.439455 0.898264i \(-0.644829\pi\)
−0.439455 + 0.898264i \(0.644829\pi\)
\(318\) 0 0
\(319\) −15.2848 −0.855784
\(320\) 0 0
\(321\) −1.62598 −0.0907532
\(322\) 0 0
\(323\) 14.4725 0.805270
\(324\) 0 0
\(325\) −14.0482 −0.779256
\(326\) 0 0
\(327\) −20.7360 −1.14670
\(328\) 0 0
\(329\) −32.0747 −1.76834
\(330\) 0 0
\(331\) 5.07476 0.278934 0.139467 0.990227i \(-0.455461\pi\)
0.139467 + 0.990227i \(0.455461\pi\)
\(332\) 0 0
\(333\) −7.92853 −0.434481
\(334\) 0 0
\(335\) −1.09171 −0.0596463
\(336\) 0 0
\(337\) −34.2593 −1.86622 −0.933111 0.359590i \(-0.882917\pi\)
−0.933111 + 0.359590i \(0.882917\pi\)
\(338\) 0 0
\(339\) −20.4876 −1.11273
\(340\) 0 0
\(341\) 12.0908 0.654752
\(342\) 0 0
\(343\) −9.27203 −0.500643
\(344\) 0 0
\(345\) 22.9010 1.23295
\(346\) 0 0
\(347\) −27.4058 −1.47122 −0.735610 0.677405i \(-0.763104\pi\)
−0.735610 + 0.677405i \(0.763104\pi\)
\(348\) 0 0
\(349\) −31.1293 −1.66632 −0.833158 0.553035i \(-0.813469\pi\)
−0.833158 + 0.553035i \(0.813469\pi\)
\(350\) 0 0
\(351\) −6.34138 −0.338478
\(352\) 0 0
\(353\) 0.376805 0.0200553 0.0100277 0.999950i \(-0.496808\pi\)
0.0100277 + 0.999950i \(0.496808\pi\)
\(354\) 0 0
\(355\) 38.5913 2.04821
\(356\) 0 0
\(357\) −25.6931 −1.35982
\(358\) 0 0
\(359\) 8.54113 0.450784 0.225392 0.974268i \(-0.427634\pi\)
0.225392 + 0.974268i \(0.427634\pi\)
\(360\) 0 0
\(361\) 1.30459 0.0686626
\(362\) 0 0
\(363\) 14.0211 0.735916
\(364\) 0 0
\(365\) −26.9257 −1.40935
\(366\) 0 0
\(367\) 31.5264 1.64566 0.822832 0.568285i \(-0.192393\pi\)
0.822832 + 0.568285i \(0.192393\pi\)
\(368\) 0 0
\(369\) 0.475303 0.0247433
\(370\) 0 0
\(371\) −35.3798 −1.83683
\(372\) 0 0
\(373\) −12.8971 −0.667786 −0.333893 0.942611i \(-0.608362\pi\)
−0.333893 + 0.942611i \(0.608362\pi\)
\(374\) 0 0
\(375\) 30.5182 1.57595
\(376\) 0 0
\(377\) 11.9095 0.613372
\(378\) 0 0
\(379\) 14.3680 0.738034 0.369017 0.929423i \(-0.379694\pi\)
0.369017 + 0.929423i \(0.379694\pi\)
\(380\) 0 0
\(381\) 10.5210 0.539007
\(382\) 0 0
\(383\) −6.19620 −0.316611 −0.158306 0.987390i \(-0.550603\pi\)
−0.158306 + 0.987390i \(0.550603\pi\)
\(384\) 0 0
\(385\) 30.0308 1.53051
\(386\) 0 0
\(387\) 0.00688111 0.000349786 0
\(388\) 0 0
\(389\) 10.2489 0.519642 0.259821 0.965657i \(-0.416337\pi\)
0.259821 + 0.965657i \(0.416337\pi\)
\(390\) 0 0
\(391\) −9.88445 −0.499878
\(392\) 0 0
\(393\) 23.4280 1.18178
\(394\) 0 0
\(395\) 37.8530 1.90459
\(396\) 0 0
\(397\) 19.9947 1.00351 0.501754 0.865011i \(-0.332688\pi\)
0.501754 + 0.865011i \(0.332688\pi\)
\(398\) 0 0
\(399\) −36.0469 −1.80460
\(400\) 0 0
\(401\) 7.93525 0.396267 0.198134 0.980175i \(-0.436512\pi\)
0.198134 + 0.980175i \(0.436512\pi\)
\(402\) 0 0
\(403\) −9.42085 −0.469286
\(404\) 0 0
\(405\) 41.0117 2.03789
\(406\) 0 0
\(407\) −16.9471 −0.840036
\(408\) 0 0
\(409\) 22.8734 1.13102 0.565508 0.824743i \(-0.308680\pi\)
0.565508 + 0.824743i \(0.308680\pi\)
\(410\) 0 0
\(411\) −12.2765 −0.605555
\(412\) 0 0
\(413\) −40.5598 −1.99582
\(414\) 0 0
\(415\) 49.2863 2.41937
\(416\) 0 0
\(417\) −45.3430 −2.22046
\(418\) 0 0
\(419\) 40.1560 1.96175 0.980875 0.194637i \(-0.0623527\pi\)
0.980875 + 0.194637i \(0.0623527\pi\)
\(420\) 0 0
\(421\) 36.8384 1.79540 0.897698 0.440612i \(-0.145239\pi\)
0.897698 + 0.440612i \(0.145239\pi\)
\(422\) 0 0
\(423\) 7.36371 0.358036
\(424\) 0 0
\(425\) −29.2311 −1.41792
\(426\) 0 0
\(427\) −43.7038 −2.11498
\(428\) 0 0
\(429\) 6.05940 0.292551
\(430\) 0 0
\(431\) 20.7592 0.999937 0.499969 0.866044i \(-0.333345\pi\)
0.499969 + 0.866044i \(0.333345\pi\)
\(432\) 0 0
\(433\) −0.900824 −0.0432908 −0.0216454 0.999766i \(-0.506890\pi\)
−0.0216454 + 0.999766i \(0.506890\pi\)
\(434\) 0 0
\(435\) −57.4142 −2.75280
\(436\) 0 0
\(437\) −13.8677 −0.663381
\(438\) 0 0
\(439\) −10.4142 −0.497041 −0.248521 0.968627i \(-0.579944\pi\)
−0.248521 + 0.968627i \(0.579944\pi\)
\(440\) 0 0
\(441\) 8.61625 0.410298
\(442\) 0 0
\(443\) 4.20294 0.199688 0.0998438 0.995003i \(-0.468166\pi\)
0.0998438 + 0.995003i \(0.468166\pi\)
\(444\) 0 0
\(445\) −44.8464 −2.12592
\(446\) 0 0
\(447\) 42.7283 2.02098
\(448\) 0 0
\(449\) 33.7299 1.59181 0.795906 0.605420i \(-0.206995\pi\)
0.795906 + 0.605420i \(0.206995\pi\)
\(450\) 0 0
\(451\) 1.01595 0.0478393
\(452\) 0 0
\(453\) 46.9327 2.20509
\(454\) 0 0
\(455\) −23.3993 −1.09697
\(456\) 0 0
\(457\) −6.79665 −0.317934 −0.158967 0.987284i \(-0.550816\pi\)
−0.158967 + 0.987284i \(0.550816\pi\)
\(458\) 0 0
\(459\) −13.1949 −0.615886
\(460\) 0 0
\(461\) 1.69822 0.0790939 0.0395470 0.999218i \(-0.487409\pi\)
0.0395470 + 0.999218i \(0.487409\pi\)
\(462\) 0 0
\(463\) 12.7065 0.590519 0.295259 0.955417i \(-0.404594\pi\)
0.295259 + 0.955417i \(0.404594\pi\)
\(464\) 0 0
\(465\) 45.4166 2.10614
\(466\) 0 0
\(467\) −18.5427 −0.858056 −0.429028 0.903291i \(-0.641144\pi\)
−0.429028 + 0.903291i \(0.641144\pi\)
\(468\) 0 0
\(469\) −1.17362 −0.0541930
\(470\) 0 0
\(471\) −8.15521 −0.375772
\(472\) 0 0
\(473\) 0.0147082 0.000676286 0
\(474\) 0 0
\(475\) −41.0106 −1.88170
\(476\) 0 0
\(477\) 8.12248 0.371903
\(478\) 0 0
\(479\) 12.7299 0.581643 0.290822 0.956777i \(-0.406071\pi\)
0.290822 + 0.956777i \(0.406071\pi\)
\(480\) 0 0
\(481\) 13.2048 0.602085
\(482\) 0 0
\(483\) 24.6194 1.12022
\(484\) 0 0
\(485\) −64.4351 −2.92585
\(486\) 0 0
\(487\) −26.7859 −1.21378 −0.606891 0.794785i \(-0.707584\pi\)
−0.606891 + 0.794785i \(0.707584\pi\)
\(488\) 0 0
\(489\) −16.2455 −0.734645
\(490\) 0 0
\(491\) −29.8966 −1.34921 −0.674607 0.738177i \(-0.735687\pi\)
−0.674607 + 0.738177i \(0.735687\pi\)
\(492\) 0 0
\(493\) 24.7809 1.11608
\(494\) 0 0
\(495\) −6.89446 −0.309883
\(496\) 0 0
\(497\) 41.4870 1.86095
\(498\) 0 0
\(499\) −17.1329 −0.766973 −0.383487 0.923546i \(-0.625277\pi\)
−0.383487 + 0.923546i \(0.625277\pi\)
\(500\) 0 0
\(501\) 26.0560 1.16410
\(502\) 0 0
\(503\) 32.3119 1.44072 0.720358 0.693603i \(-0.243978\pi\)
0.720358 + 0.693603i \(0.243978\pi\)
\(504\) 0 0
\(505\) 56.6767 2.52208
\(506\) 0 0
\(507\) 21.0397 0.934404
\(508\) 0 0
\(509\) 21.6430 0.959308 0.479654 0.877458i \(-0.340762\pi\)
0.479654 + 0.877458i \(0.340762\pi\)
\(510\) 0 0
\(511\) −28.9461 −1.28050
\(512\) 0 0
\(513\) −18.5122 −0.817334
\(514\) 0 0
\(515\) −71.1612 −3.13574
\(516\) 0 0
\(517\) 15.7398 0.692236
\(518\) 0 0
\(519\) −35.4017 −1.55396
\(520\) 0 0
\(521\) 11.7723 0.515754 0.257877 0.966178i \(-0.416977\pi\)
0.257877 + 0.966178i \(0.416977\pi\)
\(522\) 0 0
\(523\) −30.2208 −1.32146 −0.660732 0.750622i \(-0.729754\pi\)
−0.660732 + 0.750622i \(0.729754\pi\)
\(524\) 0 0
\(525\) 72.8064 3.17753
\(526\) 0 0
\(527\) −19.6026 −0.853901
\(528\) 0 0
\(529\) −13.5286 −0.588201
\(530\) 0 0
\(531\) 9.31171 0.404094
\(532\) 0 0
\(533\) −0.791605 −0.0342882
\(534\) 0 0
\(535\) −3.08122 −0.133213
\(536\) 0 0
\(537\) −0.0896434 −0.00386840
\(538\) 0 0
\(539\) 18.4171 0.793280
\(540\) 0 0
\(541\) −5.78626 −0.248771 −0.124385 0.992234i \(-0.539696\pi\)
−0.124385 + 0.992234i \(0.539696\pi\)
\(542\) 0 0
\(543\) 34.5911 1.48445
\(544\) 0 0
\(545\) −39.2947 −1.68320
\(546\) 0 0
\(547\) 38.9157 1.66392 0.831958 0.554839i \(-0.187220\pi\)
0.831958 + 0.554839i \(0.187220\pi\)
\(548\) 0 0
\(549\) 10.0335 0.428220
\(550\) 0 0
\(551\) 34.7671 1.48113
\(552\) 0 0
\(553\) 40.6933 1.73046
\(554\) 0 0
\(555\) −63.6583 −2.70214
\(556\) 0 0
\(557\) −8.80599 −0.373122 −0.186561 0.982443i \(-0.559734\pi\)
−0.186561 + 0.982443i \(0.559734\pi\)
\(558\) 0 0
\(559\) −0.0114603 −0.000484720 0
\(560\) 0 0
\(561\) 12.6082 0.532318
\(562\) 0 0
\(563\) 20.5774 0.867236 0.433618 0.901097i \(-0.357237\pi\)
0.433618 + 0.901097i \(0.357237\pi\)
\(564\) 0 0
\(565\) −38.8239 −1.63333
\(566\) 0 0
\(567\) 44.0891 1.85157
\(568\) 0 0
\(569\) 11.7000 0.490489 0.245244 0.969461i \(-0.421132\pi\)
0.245244 + 0.969461i \(0.421132\pi\)
\(570\) 0 0
\(571\) −36.8079 −1.54036 −0.770181 0.637825i \(-0.779834\pi\)
−0.770181 + 0.637825i \(0.779834\pi\)
\(572\) 0 0
\(573\) 13.4280 0.560964
\(574\) 0 0
\(575\) 28.0095 1.16808
\(576\) 0 0
\(577\) 3.86103 0.160737 0.0803683 0.996765i \(-0.474390\pi\)
0.0803683 + 0.996765i \(0.474390\pi\)
\(578\) 0 0
\(579\) −39.9729 −1.66122
\(580\) 0 0
\(581\) 52.9845 2.19817
\(582\) 0 0
\(583\) 17.3617 0.719047
\(584\) 0 0
\(585\) 5.37200 0.222105
\(586\) 0 0
\(587\) 16.6309 0.686429 0.343215 0.939257i \(-0.388484\pi\)
0.343215 + 0.939257i \(0.388484\pi\)
\(588\) 0 0
\(589\) −27.5020 −1.13320
\(590\) 0 0
\(591\) −27.9001 −1.14766
\(592\) 0 0
\(593\) 30.2687 1.24299 0.621493 0.783420i \(-0.286527\pi\)
0.621493 + 0.783420i \(0.286527\pi\)
\(594\) 0 0
\(595\) −48.6884 −1.99603
\(596\) 0 0
\(597\) 53.1664 2.17596
\(598\) 0 0
\(599\) 22.8264 0.932662 0.466331 0.884610i \(-0.345576\pi\)
0.466331 + 0.884610i \(0.345576\pi\)
\(600\) 0 0
\(601\) 43.2656 1.76484 0.882421 0.470461i \(-0.155913\pi\)
0.882421 + 0.470461i \(0.155913\pi\)
\(602\) 0 0
\(603\) 0.269440 0.0109725
\(604\) 0 0
\(605\) 26.5699 1.08022
\(606\) 0 0
\(607\) 45.5044 1.84697 0.923484 0.383638i \(-0.125329\pi\)
0.923484 + 0.383638i \(0.125329\pi\)
\(608\) 0 0
\(609\) −61.7223 −2.50112
\(610\) 0 0
\(611\) −12.2641 −0.496151
\(612\) 0 0
\(613\) −19.8696 −0.802525 −0.401263 0.915963i \(-0.631428\pi\)
−0.401263 + 0.915963i \(0.631428\pi\)
\(614\) 0 0
\(615\) 3.81622 0.153885
\(616\) 0 0
\(617\) −22.6405 −0.911472 −0.455736 0.890115i \(-0.650624\pi\)
−0.455736 + 0.890115i \(0.650624\pi\)
\(618\) 0 0
\(619\) 9.04376 0.363499 0.181750 0.983345i \(-0.441824\pi\)
0.181750 + 0.983345i \(0.441824\pi\)
\(620\) 0 0
\(621\) 12.6435 0.507366
\(622\) 0 0
\(623\) −48.2115 −1.93155
\(624\) 0 0
\(625\) 12.3259 0.493037
\(626\) 0 0
\(627\) 17.6890 0.706431
\(628\) 0 0
\(629\) 27.4760 1.09554
\(630\) 0 0
\(631\) −28.2681 −1.12534 −0.562668 0.826683i \(-0.690225\pi\)
−0.562668 + 0.826683i \(0.690225\pi\)
\(632\) 0 0
\(633\) 47.9722 1.90673
\(634\) 0 0
\(635\) 19.9373 0.791186
\(636\) 0 0
\(637\) −14.3502 −0.568574
\(638\) 0 0
\(639\) −9.52458 −0.376787
\(640\) 0 0
\(641\) 2.94615 0.116366 0.0581830 0.998306i \(-0.481469\pi\)
0.0581830 + 0.998306i \(0.481469\pi\)
\(642\) 0 0
\(643\) −26.6487 −1.05092 −0.525461 0.850818i \(-0.676107\pi\)
−0.525461 + 0.850818i \(0.676107\pi\)
\(644\) 0 0
\(645\) 0.0552486 0.00217541
\(646\) 0 0
\(647\) −22.0182 −0.865624 −0.432812 0.901484i \(-0.642479\pi\)
−0.432812 + 0.901484i \(0.642479\pi\)
\(648\) 0 0
\(649\) 19.9036 0.781285
\(650\) 0 0
\(651\) 48.8245 1.91358
\(652\) 0 0
\(653\) −18.7833 −0.735049 −0.367524 0.930014i \(-0.619795\pi\)
−0.367524 + 0.930014i \(0.619795\pi\)
\(654\) 0 0
\(655\) 44.3960 1.73469
\(656\) 0 0
\(657\) 6.64543 0.259263
\(658\) 0 0
\(659\) 10.5192 0.409770 0.204885 0.978786i \(-0.434318\pi\)
0.204885 + 0.978786i \(0.434318\pi\)
\(660\) 0 0
\(661\) −32.9363 −1.28107 −0.640536 0.767928i \(-0.721288\pi\)
−0.640536 + 0.767928i \(0.721288\pi\)
\(662\) 0 0
\(663\) −9.82399 −0.381532
\(664\) 0 0
\(665\) −68.3088 −2.64890
\(666\) 0 0
\(667\) −23.7453 −0.919423
\(668\) 0 0
\(669\) 40.2150 1.55480
\(670\) 0 0
\(671\) 21.4465 0.827932
\(672\) 0 0
\(673\) 22.8990 0.882692 0.441346 0.897337i \(-0.354501\pi\)
0.441346 + 0.897337i \(0.354501\pi\)
\(674\) 0 0
\(675\) 37.3904 1.43916
\(676\) 0 0
\(677\) 39.1741 1.50558 0.752791 0.658259i \(-0.228707\pi\)
0.752791 + 0.658259i \(0.228707\pi\)
\(678\) 0 0
\(679\) −69.2701 −2.65834
\(680\) 0 0
\(681\) 9.22046 0.353329
\(682\) 0 0
\(683\) −33.0617 −1.26507 −0.632536 0.774531i \(-0.717986\pi\)
−0.632536 + 0.774531i \(0.717986\pi\)
\(684\) 0 0
\(685\) −23.2640 −0.888870
\(686\) 0 0
\(687\) −32.6303 −1.24492
\(688\) 0 0
\(689\) −13.5278 −0.515368
\(690\) 0 0
\(691\) 12.6344 0.480634 0.240317 0.970695i \(-0.422749\pi\)
0.240317 + 0.970695i \(0.422749\pi\)
\(692\) 0 0
\(693\) −7.41179 −0.281551
\(694\) 0 0
\(695\) −85.9250 −3.25932
\(696\) 0 0
\(697\) −1.64714 −0.0623900
\(698\) 0 0
\(699\) −51.3991 −1.94409
\(700\) 0 0
\(701\) −42.1698 −1.59273 −0.796365 0.604816i \(-0.793246\pi\)
−0.796365 + 0.604816i \(0.793246\pi\)
\(702\) 0 0
\(703\) 38.5483 1.45388
\(704\) 0 0
\(705\) 59.1234 2.22672
\(706\) 0 0
\(707\) 60.9295 2.29149
\(708\) 0 0
\(709\) 17.3281 0.650770 0.325385 0.945582i \(-0.394506\pi\)
0.325385 + 0.945582i \(0.394506\pi\)
\(710\) 0 0
\(711\) −9.34236 −0.350366
\(712\) 0 0
\(713\) 18.7834 0.703443
\(714\) 0 0
\(715\) 11.4826 0.429423
\(716\) 0 0
\(717\) −51.8563 −1.93661
\(718\) 0 0
\(719\) 21.0531 0.785146 0.392573 0.919721i \(-0.371585\pi\)
0.392573 + 0.919721i \(0.371585\pi\)
\(720\) 0 0
\(721\) −76.5009 −2.84904
\(722\) 0 0
\(723\) 30.8921 1.14889
\(724\) 0 0
\(725\) −70.2216 −2.60797
\(726\) 0 0
\(727\) −46.4416 −1.72242 −0.861211 0.508247i \(-0.830294\pi\)
−0.861211 + 0.508247i \(0.830294\pi\)
\(728\) 0 0
\(729\) 14.3012 0.529673
\(730\) 0 0
\(731\) −0.0238462 −0.000881984 0
\(732\) 0 0
\(733\) −11.1293 −0.411068 −0.205534 0.978650i \(-0.565893\pi\)
−0.205534 + 0.978650i \(0.565893\pi\)
\(734\) 0 0
\(735\) 69.1801 2.55175
\(736\) 0 0
\(737\) 0.575924 0.0212144
\(738\) 0 0
\(739\) 18.8979 0.695171 0.347585 0.937648i \(-0.387002\pi\)
0.347585 + 0.937648i \(0.387002\pi\)
\(740\) 0 0
\(741\) −13.7829 −0.506326
\(742\) 0 0
\(743\) −20.7542 −0.761397 −0.380699 0.924699i \(-0.624317\pi\)
−0.380699 + 0.924699i \(0.624317\pi\)
\(744\) 0 0
\(745\) 80.9701 2.96651
\(746\) 0 0
\(747\) −12.1642 −0.445064
\(748\) 0 0
\(749\) −3.31243 −0.121033
\(750\) 0 0
\(751\) −28.9709 −1.05716 −0.528582 0.848882i \(-0.677276\pi\)
−0.528582 + 0.848882i \(0.677276\pi\)
\(752\) 0 0
\(753\) 5.41829 0.197453
\(754\) 0 0
\(755\) 88.9374 3.23676
\(756\) 0 0
\(757\) −18.0656 −0.656605 −0.328302 0.944573i \(-0.606477\pi\)
−0.328302 + 0.944573i \(0.606477\pi\)
\(758\) 0 0
\(759\) −12.0813 −0.438523
\(760\) 0 0
\(761\) −8.06197 −0.292246 −0.146123 0.989266i \(-0.546680\pi\)
−0.146123 + 0.989266i \(0.546680\pi\)
\(762\) 0 0
\(763\) −42.2432 −1.52931
\(764\) 0 0
\(765\) 11.1779 0.404136
\(766\) 0 0
\(767\) −15.5084 −0.559977
\(768\) 0 0
\(769\) 54.0895 1.95052 0.975259 0.221067i \(-0.0709538\pi\)
0.975259 + 0.221067i \(0.0709538\pi\)
\(770\) 0 0
\(771\) −0.646961 −0.0232997
\(772\) 0 0
\(773\) 5.44412 0.195811 0.0979057 0.995196i \(-0.468786\pi\)
0.0979057 + 0.995196i \(0.468786\pi\)
\(774\) 0 0
\(775\) 55.5477 1.99533
\(776\) 0 0
\(777\) −68.4350 −2.45509
\(778\) 0 0
\(779\) −2.31091 −0.0827969
\(780\) 0 0
\(781\) −20.3586 −0.728489
\(782\) 0 0
\(783\) −31.6981 −1.13280
\(784\) 0 0
\(785\) −15.4541 −0.551581
\(786\) 0 0
\(787\) −25.3518 −0.903695 −0.451848 0.892095i \(-0.649235\pi\)
−0.451848 + 0.892095i \(0.649235\pi\)
\(788\) 0 0
\(789\) −21.3441 −0.759871
\(790\) 0 0
\(791\) −41.7371 −1.48400
\(792\) 0 0
\(793\) −16.7106 −0.593410
\(794\) 0 0
\(795\) 65.2156 2.31296
\(796\) 0 0
\(797\) 4.06807 0.144098 0.0720492 0.997401i \(-0.477046\pi\)
0.0720492 + 0.997401i \(0.477046\pi\)
\(798\) 0 0
\(799\) −25.5186 −0.902785
\(800\) 0 0
\(801\) 11.0684 0.391082
\(802\) 0 0
\(803\) 14.2045 0.501266
\(804\) 0 0
\(805\) 46.6537 1.64433
\(806\) 0 0
\(807\) 41.1782 1.44954
\(808\) 0 0
\(809\) 44.1759 1.55314 0.776571 0.630030i \(-0.216958\pi\)
0.776571 + 0.630030i \(0.216958\pi\)
\(810\) 0 0
\(811\) −19.0741 −0.669781 −0.334890 0.942257i \(-0.608699\pi\)
−0.334890 + 0.942257i \(0.608699\pi\)
\(812\) 0 0
\(813\) 15.8043 0.554282
\(814\) 0 0
\(815\) −30.7851 −1.07836
\(816\) 0 0
\(817\) −0.0334557 −0.00117047
\(818\) 0 0
\(819\) 5.77509 0.201798
\(820\) 0 0
\(821\) 10.6173 0.370547 0.185274 0.982687i \(-0.440683\pi\)
0.185274 + 0.982687i \(0.440683\pi\)
\(822\) 0 0
\(823\) −13.1191 −0.457302 −0.228651 0.973509i \(-0.573431\pi\)
−0.228651 + 0.973509i \(0.573431\pi\)
\(824\) 0 0
\(825\) −35.7277 −1.24388
\(826\) 0 0
\(827\) 23.8082 0.827891 0.413945 0.910302i \(-0.364150\pi\)
0.413945 + 0.910302i \(0.364150\pi\)
\(828\) 0 0
\(829\) 17.6548 0.613177 0.306588 0.951842i \(-0.400813\pi\)
0.306588 + 0.951842i \(0.400813\pi\)
\(830\) 0 0
\(831\) −56.6672 −1.96576
\(832\) 0 0
\(833\) −29.8593 −1.03456
\(834\) 0 0
\(835\) 49.3761 1.70873
\(836\) 0 0
\(837\) 25.0742 0.866693
\(838\) 0 0
\(839\) −29.2252 −1.00897 −0.504484 0.863421i \(-0.668317\pi\)
−0.504484 + 0.863421i \(0.668317\pi\)
\(840\) 0 0
\(841\) 30.5311 1.05280
\(842\) 0 0
\(843\) 2.07164 0.0713511
\(844\) 0 0
\(845\) 39.8701 1.37157
\(846\) 0 0
\(847\) 28.5636 0.981459
\(848\) 0 0
\(849\) 1.17147 0.0402047
\(850\) 0 0
\(851\) −26.3278 −0.902505
\(852\) 0 0
\(853\) −36.4350 −1.24751 −0.623756 0.781619i \(-0.714394\pi\)
−0.623756 + 0.781619i \(0.714394\pi\)
\(854\) 0 0
\(855\) 15.6823 0.536324
\(856\) 0 0
\(857\) −11.8969 −0.406390 −0.203195 0.979138i \(-0.565132\pi\)
−0.203195 + 0.979138i \(0.565132\pi\)
\(858\) 0 0
\(859\) −1.47013 −0.0501601 −0.0250800 0.999685i \(-0.507984\pi\)
−0.0250800 + 0.999685i \(0.507984\pi\)
\(860\) 0 0
\(861\) 4.10257 0.139815
\(862\) 0 0
\(863\) 9.68482 0.329675 0.164838 0.986321i \(-0.447290\pi\)
0.164838 + 0.986321i \(0.447290\pi\)
\(864\) 0 0
\(865\) −67.0861 −2.28100
\(866\) 0 0
\(867\) 13.2460 0.449859
\(868\) 0 0
\(869\) −19.9691 −0.677406
\(870\) 0 0
\(871\) −0.448746 −0.0152052
\(872\) 0 0
\(873\) 15.9030 0.538235
\(874\) 0 0
\(875\) 62.1714 2.10178
\(876\) 0 0
\(877\) 41.0596 1.38648 0.693242 0.720705i \(-0.256182\pi\)
0.693242 + 0.720705i \(0.256182\pi\)
\(878\) 0 0
\(879\) 1.89577 0.0639428
\(880\) 0 0
\(881\) 14.4068 0.485379 0.242689 0.970104i \(-0.421970\pi\)
0.242689 + 0.970104i \(0.421970\pi\)
\(882\) 0 0
\(883\) −19.1639 −0.644916 −0.322458 0.946584i \(-0.604509\pi\)
−0.322458 + 0.946584i \(0.604509\pi\)
\(884\) 0 0
\(885\) 74.7639 2.51316
\(886\) 0 0
\(887\) −12.0994 −0.406258 −0.203129 0.979152i \(-0.565111\pi\)
−0.203129 + 0.979152i \(0.565111\pi\)
\(888\) 0 0
\(889\) 21.4333 0.718849
\(890\) 0 0
\(891\) −21.6355 −0.724817
\(892\) 0 0
\(893\) −35.8021 −1.19807
\(894\) 0 0
\(895\) −0.169874 −0.00567827
\(896\) 0 0
\(897\) 9.41345 0.314306
\(898\) 0 0
\(899\) −47.0911 −1.57058
\(900\) 0 0
\(901\) −28.1481 −0.937751
\(902\) 0 0
\(903\) 0.0593942 0.00197651
\(904\) 0 0
\(905\) 65.5502 2.17896
\(906\) 0 0
\(907\) 27.7405 0.921109 0.460555 0.887631i \(-0.347651\pi\)
0.460555 + 0.887631i \(0.347651\pi\)
\(908\) 0 0
\(909\) −13.9882 −0.463959
\(910\) 0 0
\(911\) −21.4823 −0.711741 −0.355870 0.934535i \(-0.615816\pi\)
−0.355870 + 0.934535i \(0.615816\pi\)
\(912\) 0 0
\(913\) −26.0007 −0.860498
\(914\) 0 0
\(915\) 80.5593 2.66321
\(916\) 0 0
\(917\) 47.7273 1.57609
\(918\) 0 0
\(919\) −38.5645 −1.27213 −0.636063 0.771637i \(-0.719438\pi\)
−0.636063 + 0.771637i \(0.719438\pi\)
\(920\) 0 0
\(921\) 31.2286 1.02902
\(922\) 0 0
\(923\) 15.8630 0.522136
\(924\) 0 0
\(925\) −77.8586 −2.55998
\(926\) 0 0
\(927\) 17.5631 0.576846
\(928\) 0 0
\(929\) −3.84049 −0.126003 −0.0630013 0.998013i \(-0.520067\pi\)
−0.0630013 + 0.998013i \(0.520067\pi\)
\(930\) 0 0
\(931\) −41.8920 −1.37295
\(932\) 0 0
\(933\) −40.5850 −1.32869
\(934\) 0 0
\(935\) 23.8925 0.781368
\(936\) 0 0
\(937\) 6.95905 0.227342 0.113671 0.993518i \(-0.463739\pi\)
0.113671 + 0.993518i \(0.463739\pi\)
\(938\) 0 0
\(939\) 5.11012 0.166762
\(940\) 0 0
\(941\) −32.5955 −1.06258 −0.531292 0.847189i \(-0.678293\pi\)
−0.531292 + 0.847189i \(0.678293\pi\)
\(942\) 0 0
\(943\) 1.57831 0.0513968
\(944\) 0 0
\(945\) 62.2788 2.02593
\(946\) 0 0
\(947\) −38.6901 −1.25726 −0.628630 0.777705i \(-0.716384\pi\)
−0.628630 + 0.777705i \(0.716384\pi\)
\(948\) 0 0
\(949\) −11.0678 −0.359276
\(950\) 0 0
\(951\) 31.0094 1.00555
\(952\) 0 0
\(953\) 14.1166 0.457280 0.228640 0.973511i \(-0.426572\pi\)
0.228640 + 0.973511i \(0.426572\pi\)
\(954\) 0 0
\(955\) 25.4461 0.823417
\(956\) 0 0
\(957\) 30.2886 0.979090
\(958\) 0 0
\(959\) −25.0096 −0.807602
\(960\) 0 0
\(961\) 6.25065 0.201634
\(962\) 0 0
\(963\) 0.760466 0.0245057
\(964\) 0 0
\(965\) −75.7486 −2.43843
\(966\) 0 0
\(967\) 1.35611 0.0436095 0.0218047 0.999762i \(-0.493059\pi\)
0.0218047 + 0.999762i \(0.493059\pi\)
\(968\) 0 0
\(969\) −28.6789 −0.921298
\(970\) 0 0
\(971\) 49.3188 1.58271 0.791357 0.611354i \(-0.209375\pi\)
0.791357 + 0.611354i \(0.209375\pi\)
\(972\) 0 0
\(973\) −92.3725 −2.96132
\(974\) 0 0
\(975\) 27.8382 0.891536
\(976\) 0 0
\(977\) −16.1966 −0.518175 −0.259088 0.965854i \(-0.583422\pi\)
−0.259088 + 0.965854i \(0.583422\pi\)
\(978\) 0 0
\(979\) 23.6585 0.756128
\(980\) 0 0
\(981\) 9.69819 0.309639
\(982\) 0 0
\(983\) 21.6120 0.689316 0.344658 0.938728i \(-0.387995\pi\)
0.344658 + 0.938728i \(0.387995\pi\)
\(984\) 0 0
\(985\) −52.8707 −1.68460
\(986\) 0 0
\(987\) 63.5598 2.02313
\(988\) 0 0
\(989\) 0.0228497 0.000726578 0
\(990\) 0 0
\(991\) −12.2359 −0.388688 −0.194344 0.980933i \(-0.562258\pi\)
−0.194344 + 0.980933i \(0.562258\pi\)
\(992\) 0 0
\(993\) −10.0562 −0.319124
\(994\) 0 0
\(995\) 100.750 3.19400
\(996\) 0 0
\(997\) −41.1225 −1.30236 −0.651182 0.758922i \(-0.725726\pi\)
−0.651182 + 0.758922i \(0.725726\pi\)
\(998\) 0 0
\(999\) −35.1454 −1.11195
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 6044.2.a.a.1.14 63
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
6044.2.a.a.1.14 63 1.1 even 1 trivial