Properties

Label 1840.2.a.q.1.2
Level $1840$
Weight $2$
Character 1840.1
Self dual yes
Analytic conductor $14.692$
Analytic rank $1$
Dimension $3$
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] = [1840,2,Mod(1,1840)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1840, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 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("1840.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1840 = 2^{4} \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1840.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: \(14.6924739719\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.229.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - 4x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 920)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-1.86081\) of defining polynomial
Character \(\chi\) \(=\) 1840.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.46260 q^{3} +1.00000 q^{5} -1.53740 q^{7} -0.860806 q^{9} +O(q^{10})\) \(q-1.46260 q^{3} +1.00000 q^{5} -1.53740 q^{7} -0.860806 q^{9} +0.860806 q^{11} +0.139194 q^{13} -1.46260 q^{15} +5.50761 q^{17} -5.25901 q^{19} +2.24860 q^{21} +1.00000 q^{23} +1.00000 q^{25} +5.64681 q^{27} +9.76663 q^{29} -6.78600 q^{31} -1.25901 q^{33} -1.53740 q^{35} -12.0900 q^{37} -0.203585 q^{39} -9.98062 q^{41} -11.4432 q^{43} -0.860806 q^{45} +2.32340 q^{47} -4.63640 q^{49} -8.05543 q^{51} +0.149606 q^{53} +0.860806 q^{55} +7.69182 q^{57} +11.0152 q^{59} +4.43281 q^{61} +1.32340 q^{63} +0.139194 q^{65} +10.4972 q^{67} -1.46260 q^{69} -7.31299 q^{71} +7.11982 q^{73} -1.46260 q^{75} -1.32340 q^{77} -6.79641 q^{79} -5.67660 q^{81} -10.6468 q^{83} +5.50761 q^{85} -14.2847 q^{87} -17.2936 q^{89} -0.213997 q^{91} +9.92520 q^{93} -5.25901 q^{95} +1.93561 q^{97} -0.740987 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 2 q^{3} + 3 q^{5} - 7 q^{7} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 2 q^{3} + 3 q^{5} - 7 q^{7} + 3 q^{9} - 3 q^{11} + 6 q^{13} - 2 q^{15} - 5 q^{17} - 7 q^{19} - 6 q^{21} + 3 q^{23} + 3 q^{25} + q^{27} - q^{29} - 10 q^{31} + 5 q^{33} - 7 q^{35} + 2 q^{37} - 7 q^{39} - 10 q^{41} - 12 q^{43} + 3 q^{45} - q^{47} + 6 q^{49} - 9 q^{51} + 10 q^{53} - 3 q^{55} - 12 q^{57} - 10 q^{59} - 13 q^{61} - 4 q^{63} + 6 q^{65} + 6 q^{67} - 2 q^{69} - 10 q^{71} + 7 q^{73} - 2 q^{75} + 4 q^{77} - 14 q^{79} - 25 q^{81} - 16 q^{83} - 5 q^{85} + 5 q^{87} - 20 q^{89} - 11 q^{91} + 25 q^{93} - 7 q^{95} + 5 q^{97} - 11 q^{99}+O(q^{100}) \) 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
\(3\) −1.46260 −0.844432 −0.422216 0.906495i \(-0.638748\pi\)
−0.422216 + 0.906495i \(0.638748\pi\)
\(4\) 0 0
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) −1.53740 −0.581083 −0.290542 0.956862i \(-0.593835\pi\)
−0.290542 + 0.956862i \(0.593835\pi\)
\(8\) 0 0
\(9\) −0.860806 −0.286935
\(10\) 0 0
\(11\) 0.860806 0.259543 0.129771 0.991544i \(-0.458576\pi\)
0.129771 + 0.991544i \(0.458576\pi\)
\(12\) 0 0
\(13\) 0.139194 0.0386055 0.0193028 0.999814i \(-0.493855\pi\)
0.0193028 + 0.999814i \(0.493855\pi\)
\(14\) 0 0
\(15\) −1.46260 −0.377641
\(16\) 0 0
\(17\) 5.50761 1.33579 0.667896 0.744254i \(-0.267195\pi\)
0.667896 + 0.744254i \(0.267195\pi\)
\(18\) 0 0
\(19\) −5.25901 −1.20650 −0.603250 0.797552i \(-0.706128\pi\)
−0.603250 + 0.797552i \(0.706128\pi\)
\(20\) 0 0
\(21\) 2.24860 0.490685
\(22\) 0 0
\(23\) 1.00000 0.208514
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 5.64681 1.08673
\(28\) 0 0
\(29\) 9.76663 1.81362 0.906809 0.421543i \(-0.138511\pi\)
0.906809 + 0.421543i \(0.138511\pi\)
\(30\) 0 0
\(31\) −6.78600 −1.21880 −0.609401 0.792862i \(-0.708590\pi\)
−0.609401 + 0.792862i \(0.708590\pi\)
\(32\) 0 0
\(33\) −1.25901 −0.219166
\(34\) 0 0
\(35\) −1.53740 −0.259868
\(36\) 0 0
\(37\) −12.0900 −1.98759 −0.993795 0.111232i \(-0.964520\pi\)
−0.993795 + 0.111232i \(0.964520\pi\)
\(38\) 0 0
\(39\) −0.203585 −0.0325997
\(40\) 0 0
\(41\) −9.98062 −1.55871 −0.779356 0.626582i \(-0.784454\pi\)
−0.779356 + 0.626582i \(0.784454\pi\)
\(42\) 0 0
\(43\) −11.4432 −1.74508 −0.872538 0.488547i \(-0.837527\pi\)
−0.872538 + 0.488547i \(0.837527\pi\)
\(44\) 0 0
\(45\) −0.860806 −0.128321
\(46\) 0 0
\(47\) 2.32340 0.338903 0.169452 0.985538i \(-0.445800\pi\)
0.169452 + 0.985538i \(0.445800\pi\)
\(48\) 0 0
\(49\) −4.63640 −0.662342
\(50\) 0 0
\(51\) −8.05543 −1.12799
\(52\) 0 0
\(53\) 0.149606 0.0205500 0.0102750 0.999947i \(-0.496729\pi\)
0.0102750 + 0.999947i \(0.496729\pi\)
\(54\) 0 0
\(55\) 0.860806 0.116071
\(56\) 0 0
\(57\) 7.69182 1.01881
\(58\) 0 0
\(59\) 11.0152 1.43406 0.717030 0.697042i \(-0.245501\pi\)
0.717030 + 0.697042i \(0.245501\pi\)
\(60\) 0 0
\(61\) 4.43281 0.567563 0.283782 0.958889i \(-0.408411\pi\)
0.283782 + 0.958889i \(0.408411\pi\)
\(62\) 0 0
\(63\) 1.32340 0.166733
\(64\) 0 0
\(65\) 0.139194 0.0172649
\(66\) 0 0
\(67\) 10.4972 1.28244 0.641219 0.767358i \(-0.278429\pi\)
0.641219 + 0.767358i \(0.278429\pi\)
\(68\) 0 0
\(69\) −1.46260 −0.176076
\(70\) 0 0
\(71\) −7.31299 −0.867892 −0.433946 0.900939i \(-0.642879\pi\)
−0.433946 + 0.900939i \(0.642879\pi\)
\(72\) 0 0
\(73\) 7.11982 0.833312 0.416656 0.909064i \(-0.363202\pi\)
0.416656 + 0.909064i \(0.363202\pi\)
\(74\) 0 0
\(75\) −1.46260 −0.168886
\(76\) 0 0
\(77\) −1.32340 −0.150816
\(78\) 0 0
\(79\) −6.79641 −0.764656 −0.382328 0.924027i \(-0.624878\pi\)
−0.382328 + 0.924027i \(0.624878\pi\)
\(80\) 0 0
\(81\) −5.67660 −0.630733
\(82\) 0 0
\(83\) −10.6468 −1.16864 −0.584320 0.811524i \(-0.698639\pi\)
−0.584320 + 0.811524i \(0.698639\pi\)
\(84\) 0 0
\(85\) 5.50761 0.597385
\(86\) 0 0
\(87\) −14.2847 −1.53148
\(88\) 0 0
\(89\) −17.2936 −1.83312 −0.916560 0.399897i \(-0.869046\pi\)
−0.916560 + 0.399897i \(0.869046\pi\)
\(90\) 0 0
\(91\) −0.213997 −0.0224330
\(92\) 0 0
\(93\) 9.92520 1.02919
\(94\) 0 0
\(95\) −5.25901 −0.539563
\(96\) 0 0
\(97\) 1.93561 0.196531 0.0982657 0.995160i \(-0.468671\pi\)
0.0982657 + 0.995160i \(0.468671\pi\)
\(98\) 0 0
\(99\) −0.740987 −0.0744720
\(100\) 0 0
\(101\) −15.4224 −1.53459 −0.767293 0.641297i \(-0.778397\pi\)
−0.767293 + 0.641297i \(0.778397\pi\)
\(102\) 0 0
\(103\) −7.91478 −0.779867 −0.389933 0.920843i \(-0.627502\pi\)
−0.389933 + 0.920843i \(0.627502\pi\)
\(104\) 0 0
\(105\) 2.24860 0.219441
\(106\) 0 0
\(107\) 4.27839 0.413607 0.206804 0.978382i \(-0.433694\pi\)
0.206804 + 0.978382i \(0.433694\pi\)
\(108\) 0 0
\(109\) −17.7770 −1.70273 −0.851366 0.524572i \(-0.824225\pi\)
−0.851366 + 0.524572i \(0.824225\pi\)
\(110\) 0 0
\(111\) 17.6829 1.67838
\(112\) 0 0
\(113\) −3.20359 −0.301368 −0.150684 0.988582i \(-0.548148\pi\)
−0.150684 + 0.988582i \(0.548148\pi\)
\(114\) 0 0
\(115\) 1.00000 0.0932505
\(116\) 0 0
\(117\) −0.119819 −0.0110773
\(118\) 0 0
\(119\) −8.46742 −0.776207
\(120\) 0 0
\(121\) −10.2590 −0.932638
\(122\) 0 0
\(123\) 14.5976 1.31623
\(124\) 0 0
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) 15.2099 1.34966 0.674828 0.737975i \(-0.264218\pi\)
0.674828 + 0.737975i \(0.264218\pi\)
\(128\) 0 0
\(129\) 16.7368 1.47360
\(130\) 0 0
\(131\) −12.6918 −1.10889 −0.554445 0.832220i \(-0.687069\pi\)
−0.554445 + 0.832220i \(0.687069\pi\)
\(132\) 0 0
\(133\) 8.08522 0.701077
\(134\) 0 0
\(135\) 5.64681 0.486000
\(136\) 0 0
\(137\) −14.0346 −1.19906 −0.599529 0.800353i \(-0.704645\pi\)
−0.599529 + 0.800353i \(0.704645\pi\)
\(138\) 0 0
\(139\) −14.2638 −1.20984 −0.604921 0.796285i \(-0.706795\pi\)
−0.604921 + 0.796285i \(0.706795\pi\)
\(140\) 0 0
\(141\) −3.39821 −0.286181
\(142\) 0 0
\(143\) 0.119819 0.0100198
\(144\) 0 0
\(145\) 9.76663 0.811074
\(146\) 0 0
\(147\) 6.78119 0.559303
\(148\) 0 0
\(149\) −4.46260 −0.365590 −0.182795 0.983151i \(-0.558515\pi\)
−0.182795 + 0.983151i \(0.558515\pi\)
\(150\) 0 0
\(151\) −10.7562 −0.875328 −0.437664 0.899139i \(-0.644194\pi\)
−0.437664 + 0.899139i \(0.644194\pi\)
\(152\) 0 0
\(153\) −4.74099 −0.383286
\(154\) 0 0
\(155\) −6.78600 −0.545065
\(156\) 0 0
\(157\) 14.5180 1.15866 0.579332 0.815091i \(-0.303313\pi\)
0.579332 + 0.815091i \(0.303313\pi\)
\(158\) 0 0
\(159\) −0.218814 −0.0173531
\(160\) 0 0
\(161\) −1.53740 −0.121164
\(162\) 0 0
\(163\) 1.30403 0.102139 0.0510697 0.998695i \(-0.483737\pi\)
0.0510697 + 0.998695i \(0.483737\pi\)
\(164\) 0 0
\(165\) −1.25901 −0.0980141
\(166\) 0 0
\(167\) −3.44322 −0.266445 −0.133222 0.991086i \(-0.542532\pi\)
−0.133222 + 0.991086i \(0.542532\pi\)
\(168\) 0 0
\(169\) −12.9806 −0.998510
\(170\) 0 0
\(171\) 4.52699 0.346188
\(172\) 0 0
\(173\) 9.35801 0.711476 0.355738 0.934586i \(-0.384230\pi\)
0.355738 + 0.934586i \(0.384230\pi\)
\(174\) 0 0
\(175\) −1.53740 −0.116217
\(176\) 0 0
\(177\) −16.1109 −1.21097
\(178\) 0 0
\(179\) 18.4134 1.37628 0.688142 0.725576i \(-0.258426\pi\)
0.688142 + 0.725576i \(0.258426\pi\)
\(180\) 0 0
\(181\) −1.94457 −0.144539 −0.0722694 0.997385i \(-0.523024\pi\)
−0.0722694 + 0.997385i \(0.523024\pi\)
\(182\) 0 0
\(183\) −6.48342 −0.479268
\(184\) 0 0
\(185\) −12.0900 −0.888877
\(186\) 0 0
\(187\) 4.74099 0.346695
\(188\) 0 0
\(189\) −8.68141 −0.631480
\(190\) 0 0
\(191\) 0.775591 0.0561198 0.0280599 0.999606i \(-0.491067\pi\)
0.0280599 + 0.999606i \(0.491067\pi\)
\(192\) 0 0
\(193\) −11.7666 −0.846980 −0.423490 0.905901i \(-0.639195\pi\)
−0.423490 + 0.905901i \(0.639195\pi\)
\(194\) 0 0
\(195\) −0.203585 −0.0145790
\(196\) 0 0
\(197\) 9.68141 0.689772 0.344886 0.938645i \(-0.387918\pi\)
0.344886 + 0.938645i \(0.387918\pi\)
\(198\) 0 0
\(199\) 3.29362 0.233478 0.116739 0.993163i \(-0.462756\pi\)
0.116739 + 0.993163i \(0.462756\pi\)
\(200\) 0 0
\(201\) −15.3532 −1.08293
\(202\) 0 0
\(203\) −15.0152 −1.05386
\(204\) 0 0
\(205\) −9.98062 −0.697077
\(206\) 0 0
\(207\) −0.860806 −0.0598301
\(208\) 0 0
\(209\) −4.52699 −0.313138
\(210\) 0 0
\(211\) 16.2396 1.11798 0.558991 0.829173i \(-0.311188\pi\)
0.558991 + 0.829173i \(0.311188\pi\)
\(212\) 0 0
\(213\) 10.6960 0.732876
\(214\) 0 0
\(215\) −11.4432 −0.780421
\(216\) 0 0
\(217\) 10.4328 0.708225
\(218\) 0 0
\(219\) −10.4134 −0.703675
\(220\) 0 0
\(221\) 0.766628 0.0515690
\(222\) 0 0
\(223\) −21.2936 −1.42593 −0.712963 0.701202i \(-0.752647\pi\)
−0.712963 + 0.701202i \(0.752647\pi\)
\(224\) 0 0
\(225\) −0.860806 −0.0573871
\(226\) 0 0
\(227\) 29.6620 1.96874 0.984369 0.176117i \(-0.0563536\pi\)
0.984369 + 0.176117i \(0.0563536\pi\)
\(228\) 0 0
\(229\) 19.6829 1.30068 0.650340 0.759643i \(-0.274626\pi\)
0.650340 + 0.759643i \(0.274626\pi\)
\(230\) 0 0
\(231\) 1.93561 0.127354
\(232\) 0 0
\(233\) −5.52699 −0.362085 −0.181043 0.983475i \(-0.557947\pi\)
−0.181043 + 0.983475i \(0.557947\pi\)
\(234\) 0 0
\(235\) 2.32340 0.151562
\(236\) 0 0
\(237\) 9.94043 0.645700
\(238\) 0 0
\(239\) 17.3982 1.12540 0.562698 0.826662i \(-0.309763\pi\)
0.562698 + 0.826662i \(0.309763\pi\)
\(240\) 0 0
\(241\) −16.0900 −1.03645 −0.518225 0.855244i \(-0.673407\pi\)
−0.518225 + 0.855244i \(0.673407\pi\)
\(242\) 0 0
\(243\) −8.63785 −0.554118
\(244\) 0 0
\(245\) −4.63640 −0.296209
\(246\) 0 0
\(247\) −0.732024 −0.0465776
\(248\) 0 0
\(249\) 15.5720 0.986836
\(250\) 0 0
\(251\) 29.0498 1.83361 0.916805 0.399336i \(-0.130759\pi\)
0.916805 + 0.399336i \(0.130759\pi\)
\(252\) 0 0
\(253\) 0.860806 0.0541184
\(254\) 0 0
\(255\) −8.05543 −0.504450
\(256\) 0 0
\(257\) −6.56304 −0.409391 −0.204696 0.978826i \(-0.565620\pi\)
−0.204696 + 0.978826i \(0.565620\pi\)
\(258\) 0 0
\(259\) 18.5872 1.15495
\(260\) 0 0
\(261\) −8.40717 −0.520391
\(262\) 0 0
\(263\) 0.561593 0.0346293 0.0173147 0.999850i \(-0.494488\pi\)
0.0173147 + 0.999850i \(0.494488\pi\)
\(264\) 0 0
\(265\) 0.149606 0.00919024
\(266\) 0 0
\(267\) 25.2936 1.54794
\(268\) 0 0
\(269\) −7.26943 −0.443225 −0.221612 0.975135i \(-0.571132\pi\)
−0.221612 + 0.975135i \(0.571132\pi\)
\(270\) 0 0
\(271\) −0.184210 −0.0111900 −0.00559498 0.999984i \(-0.501781\pi\)
−0.00559498 + 0.999984i \(0.501781\pi\)
\(272\) 0 0
\(273\) 0.312992 0.0189431
\(274\) 0 0
\(275\) 0.860806 0.0519085
\(276\) 0 0
\(277\) 25.8954 1.55590 0.777952 0.628323i \(-0.216259\pi\)
0.777952 + 0.628323i \(0.216259\pi\)
\(278\) 0 0
\(279\) 5.84143 0.349717
\(280\) 0 0
\(281\) −26.7756 −1.59730 −0.798649 0.601797i \(-0.794452\pi\)
−0.798649 + 0.601797i \(0.794452\pi\)
\(282\) 0 0
\(283\) −5.70079 −0.338877 −0.169438 0.985541i \(-0.554195\pi\)
−0.169438 + 0.985541i \(0.554195\pi\)
\(284\) 0 0
\(285\) 7.69182 0.455624
\(286\) 0 0
\(287\) 15.3442 0.905741
\(288\) 0 0
\(289\) 13.3338 0.784342
\(290\) 0 0
\(291\) −2.83102 −0.165957
\(292\) 0 0
\(293\) 1.05398 0.0615741 0.0307871 0.999526i \(-0.490199\pi\)
0.0307871 + 0.999526i \(0.490199\pi\)
\(294\) 0 0
\(295\) 11.0152 0.641331
\(296\) 0 0
\(297\) 4.86081 0.282053
\(298\) 0 0
\(299\) 0.139194 0.00804981
\(300\) 0 0
\(301\) 17.5928 1.01403
\(302\) 0 0
\(303\) 22.5568 1.29585
\(304\) 0 0
\(305\) 4.43281 0.253822
\(306\) 0 0
\(307\) −17.8760 −1.02024 −0.510120 0.860103i \(-0.670399\pi\)
−0.510120 + 0.860103i \(0.670399\pi\)
\(308\) 0 0
\(309\) 11.5762 0.658544
\(310\) 0 0
\(311\) −16.9315 −0.960095 −0.480048 0.877242i \(-0.659381\pi\)
−0.480048 + 0.877242i \(0.659381\pi\)
\(312\) 0 0
\(313\) −19.9959 −1.13023 −0.565116 0.825011i \(-0.691169\pi\)
−0.565116 + 0.825011i \(0.691169\pi\)
\(314\) 0 0
\(315\) 1.32340 0.0745654
\(316\) 0 0
\(317\) 3.29776 0.185221 0.0926104 0.995702i \(-0.470479\pi\)
0.0926104 + 0.995702i \(0.470479\pi\)
\(318\) 0 0
\(319\) 8.40717 0.470711
\(320\) 0 0
\(321\) −6.25756 −0.349263
\(322\) 0 0
\(323\) −28.9646 −1.61163
\(324\) 0 0
\(325\) 0.139194 0.00772110
\(326\) 0 0
\(327\) 26.0007 1.43784
\(328\) 0 0
\(329\) −3.57201 −0.196931
\(330\) 0 0
\(331\) 12.8802 0.707959 0.353979 0.935253i \(-0.384828\pi\)
0.353979 + 0.935253i \(0.384828\pi\)
\(332\) 0 0
\(333\) 10.4072 0.570309
\(334\) 0 0
\(335\) 10.4972 0.573523
\(336\) 0 0
\(337\) −9.22026 −0.502260 −0.251130 0.967953i \(-0.580802\pi\)
−0.251130 + 0.967953i \(0.580802\pi\)
\(338\) 0 0
\(339\) 4.68556 0.254485
\(340\) 0 0
\(341\) −5.84143 −0.316331
\(342\) 0 0
\(343\) 17.8898 0.965959
\(344\) 0 0
\(345\) −1.46260 −0.0787437
\(346\) 0 0
\(347\) −31.2984 −1.68019 −0.840094 0.542441i \(-0.817500\pi\)
−0.840094 + 0.542441i \(0.817500\pi\)
\(348\) 0 0
\(349\) 9.89541 0.529689 0.264845 0.964291i \(-0.414679\pi\)
0.264845 + 0.964291i \(0.414679\pi\)
\(350\) 0 0
\(351\) 0.786003 0.0419537
\(352\) 0 0
\(353\) 19.4287 1.03408 0.517042 0.855960i \(-0.327033\pi\)
0.517042 + 0.855960i \(0.327033\pi\)
\(354\) 0 0
\(355\) −7.31299 −0.388133
\(356\) 0 0
\(357\) 12.3844 0.655453
\(358\) 0 0
\(359\) 13.1440 0.693714 0.346857 0.937918i \(-0.387249\pi\)
0.346857 + 0.937918i \(0.387249\pi\)
\(360\) 0 0
\(361\) 8.65722 0.455643
\(362\) 0 0
\(363\) 15.0048 0.787549
\(364\) 0 0
\(365\) 7.11982 0.372668
\(366\) 0 0
\(367\) 15.7424 0.821748 0.410874 0.911692i \(-0.365224\pi\)
0.410874 + 0.911692i \(0.365224\pi\)
\(368\) 0 0
\(369\) 8.59138 0.447249
\(370\) 0 0
\(371\) −0.230005 −0.0119413
\(372\) 0 0
\(373\) 27.6620 1.43229 0.716143 0.697954i \(-0.245906\pi\)
0.716143 + 0.697954i \(0.245906\pi\)
\(374\) 0 0
\(375\) −1.46260 −0.0755283
\(376\) 0 0
\(377\) 1.35946 0.0700156
\(378\) 0 0
\(379\) −28.2140 −1.44926 −0.724628 0.689140i \(-0.757988\pi\)
−0.724628 + 0.689140i \(0.757988\pi\)
\(380\) 0 0
\(381\) −22.2459 −1.13969
\(382\) 0 0
\(383\) 21.5928 1.10334 0.551671 0.834062i \(-0.313990\pi\)
0.551671 + 0.834062i \(0.313990\pi\)
\(384\) 0 0
\(385\) −1.32340 −0.0674469
\(386\) 0 0
\(387\) 9.85039 0.500724
\(388\) 0 0
\(389\) 17.2501 0.874612 0.437306 0.899313i \(-0.355933\pi\)
0.437306 + 0.899313i \(0.355933\pi\)
\(390\) 0 0
\(391\) 5.50761 0.278532
\(392\) 0 0
\(393\) 18.5630 0.936382
\(394\) 0 0
\(395\) −6.79641 −0.341965
\(396\) 0 0
\(397\) 34.8850 1.75083 0.875414 0.483374i \(-0.160589\pi\)
0.875414 + 0.483374i \(0.160589\pi\)
\(398\) 0 0
\(399\) −11.8254 −0.592012
\(400\) 0 0
\(401\) 4.21881 0.210678 0.105339 0.994436i \(-0.466407\pi\)
0.105339 + 0.994436i \(0.466407\pi\)
\(402\) 0 0
\(403\) −0.944572 −0.0470525
\(404\) 0 0
\(405\) −5.67660 −0.282072
\(406\) 0 0
\(407\) −10.4072 −0.515864
\(408\) 0 0
\(409\) 38.8165 1.91935 0.959675 0.281111i \(-0.0907030\pi\)
0.959675 + 0.281111i \(0.0907030\pi\)
\(410\) 0 0
\(411\) 20.5270 1.01252
\(412\) 0 0
\(413\) −16.9348 −0.833309
\(414\) 0 0
\(415\) −10.6468 −0.522631
\(416\) 0 0
\(417\) 20.8623 1.02163
\(418\) 0 0
\(419\) −22.0305 −1.07626 −0.538129 0.842862i \(-0.680869\pi\)
−0.538129 + 0.842862i \(0.680869\pi\)
\(420\) 0 0
\(421\) −17.2203 −0.839264 −0.419632 0.907694i \(-0.637841\pi\)
−0.419632 + 0.907694i \(0.637841\pi\)
\(422\) 0 0
\(423\) −2.00000 −0.0972433
\(424\) 0 0
\(425\) 5.50761 0.267159
\(426\) 0 0
\(427\) −6.81501 −0.329801
\(428\) 0 0
\(429\) −0.175247 −0.00846102
\(430\) 0 0
\(431\) 11.8116 0.568947 0.284473 0.958684i \(-0.408181\pi\)
0.284473 + 0.958684i \(0.408181\pi\)
\(432\) 0 0
\(433\) 19.8642 0.954611 0.477306 0.878737i \(-0.341613\pi\)
0.477306 + 0.878737i \(0.341613\pi\)
\(434\) 0 0
\(435\) −14.2847 −0.684897
\(436\) 0 0
\(437\) −5.25901 −0.251573
\(438\) 0 0
\(439\) 5.04647 0.240855 0.120427 0.992722i \(-0.461574\pi\)
0.120427 + 0.992722i \(0.461574\pi\)
\(440\) 0 0
\(441\) 3.99104 0.190049
\(442\) 0 0
\(443\) 14.3193 0.680328 0.340164 0.940366i \(-0.389517\pi\)
0.340164 + 0.940366i \(0.389517\pi\)
\(444\) 0 0
\(445\) −17.2936 −0.819796
\(446\) 0 0
\(447\) 6.52699 0.308716
\(448\) 0 0
\(449\) −13.3699 −0.630963 −0.315482 0.948932i \(-0.602166\pi\)
−0.315482 + 0.948932i \(0.602166\pi\)
\(450\) 0 0
\(451\) −8.59138 −0.404552
\(452\) 0 0
\(453\) 15.7320 0.739155
\(454\) 0 0
\(455\) −0.213997 −0.0100323
\(456\) 0 0
\(457\) 2.38924 0.111764 0.0558821 0.998437i \(-0.482203\pi\)
0.0558821 + 0.998437i \(0.482203\pi\)
\(458\) 0 0
\(459\) 31.1004 1.45164
\(460\) 0 0
\(461\) 21.2278 0.988676 0.494338 0.869270i \(-0.335410\pi\)
0.494338 + 0.869270i \(0.335410\pi\)
\(462\) 0 0
\(463\) −35.1261 −1.63245 −0.816224 0.577736i \(-0.803936\pi\)
−0.816224 + 0.577736i \(0.803936\pi\)
\(464\) 0 0
\(465\) 9.92520 0.460270
\(466\) 0 0
\(467\) −16.7756 −0.776282 −0.388141 0.921600i \(-0.626883\pi\)
−0.388141 + 0.921600i \(0.626883\pi\)
\(468\) 0 0
\(469\) −16.1384 −0.745203
\(470\) 0 0
\(471\) −21.2340 −0.978413
\(472\) 0 0
\(473\) −9.85039 −0.452922
\(474\) 0 0
\(475\) −5.25901 −0.241300
\(476\) 0 0
\(477\) −0.128782 −0.00589652
\(478\) 0 0
\(479\) 5.61076 0.256362 0.128181 0.991751i \(-0.459086\pi\)
0.128181 + 0.991751i \(0.459086\pi\)
\(480\) 0 0
\(481\) −1.68286 −0.0767319
\(482\) 0 0
\(483\) 2.24860 0.102315
\(484\) 0 0
\(485\) 1.93561 0.0878915
\(486\) 0 0
\(487\) −10.2638 −0.465099 −0.232549 0.972585i \(-0.574707\pi\)
−0.232549 + 0.972585i \(0.574707\pi\)
\(488\) 0 0
\(489\) −1.90727 −0.0862498
\(490\) 0 0
\(491\) −9.28735 −0.419132 −0.209566 0.977794i \(-0.567205\pi\)
−0.209566 + 0.977794i \(0.567205\pi\)
\(492\) 0 0
\(493\) 53.7908 2.42262
\(494\) 0 0
\(495\) −0.740987 −0.0333049
\(496\) 0 0
\(497\) 11.2430 0.504318
\(498\) 0 0
\(499\) −13.3082 −0.595756 −0.297878 0.954604i \(-0.596279\pi\)
−0.297878 + 0.954604i \(0.596279\pi\)
\(500\) 0 0
\(501\) 5.03605 0.224994
\(502\) 0 0
\(503\) 37.7383 1.68267 0.841334 0.540516i \(-0.181771\pi\)
0.841334 + 0.540516i \(0.181771\pi\)
\(504\) 0 0
\(505\) −15.4224 −0.686288
\(506\) 0 0
\(507\) 18.9854 0.843173
\(508\) 0 0
\(509\) −1.11982 −0.0496351 −0.0248176 0.999692i \(-0.507900\pi\)
−0.0248176 + 0.999692i \(0.507900\pi\)
\(510\) 0 0
\(511\) −10.9460 −0.484223
\(512\) 0 0
\(513\) −29.6966 −1.31114
\(514\) 0 0
\(515\) −7.91478 −0.348767
\(516\) 0 0
\(517\) 2.00000 0.0879599
\(518\) 0 0
\(519\) −13.6870 −0.600793
\(520\) 0 0
\(521\) −13.9100 −0.609407 −0.304703 0.952447i \(-0.598557\pi\)
−0.304703 + 0.952447i \(0.598557\pi\)
\(522\) 0 0
\(523\) −8.19799 −0.358473 −0.179237 0.983806i \(-0.557363\pi\)
−0.179237 + 0.983806i \(0.557363\pi\)
\(524\) 0 0
\(525\) 2.24860 0.0981370
\(526\) 0 0
\(527\) −37.3747 −1.62807
\(528\) 0 0
\(529\) 1.00000 0.0434783
\(530\) 0 0
\(531\) −9.48197 −0.411483
\(532\) 0 0
\(533\) −1.38924 −0.0601749
\(534\) 0 0
\(535\) 4.27839 0.184971
\(536\) 0 0
\(537\) −26.9315 −1.16218
\(538\) 0 0
\(539\) −3.99104 −0.171906
\(540\) 0 0
\(541\) 23.5478 1.01240 0.506200 0.862416i \(-0.331050\pi\)
0.506200 + 0.862416i \(0.331050\pi\)
\(542\) 0 0
\(543\) 2.84413 0.122053
\(544\) 0 0
\(545\) −17.7770 −0.761485
\(546\) 0 0
\(547\) −1.77077 −0.0757128 −0.0378564 0.999283i \(-0.512053\pi\)
−0.0378564 + 0.999283i \(0.512053\pi\)
\(548\) 0 0
\(549\) −3.81579 −0.162854
\(550\) 0 0
\(551\) −51.3628 −2.18813
\(552\) 0 0
\(553\) 10.4488 0.444329
\(554\) 0 0
\(555\) 17.6829 0.750596
\(556\) 0 0
\(557\) −22.5872 −0.957052 −0.478526 0.878073i \(-0.658829\pi\)
−0.478526 + 0.878073i \(0.658829\pi\)
\(558\) 0 0
\(559\) −1.59283 −0.0673695
\(560\) 0 0
\(561\) −6.93416 −0.292760
\(562\) 0 0
\(563\) −21.8504 −0.920884 −0.460442 0.887690i \(-0.652309\pi\)
−0.460442 + 0.887690i \(0.652309\pi\)
\(564\) 0 0
\(565\) −3.20359 −0.134776
\(566\) 0 0
\(567\) 8.72721 0.366508
\(568\) 0 0
\(569\) 11.4737 0.481002 0.240501 0.970649i \(-0.422688\pi\)
0.240501 + 0.970649i \(0.422688\pi\)
\(570\) 0 0
\(571\) −27.7085 −1.15956 −0.579782 0.814771i \(-0.696862\pi\)
−0.579782 + 0.814771i \(0.696862\pi\)
\(572\) 0 0
\(573\) −1.13438 −0.0473893
\(574\) 0 0
\(575\) 1.00000 0.0417029
\(576\) 0 0
\(577\) 20.3442 0.846941 0.423471 0.905910i \(-0.360812\pi\)
0.423471 + 0.905910i \(0.360812\pi\)
\(578\) 0 0
\(579\) 17.2099 0.715217
\(580\) 0 0
\(581\) 16.3684 0.679076
\(582\) 0 0
\(583\) 0.128782 0.00533360
\(584\) 0 0
\(585\) −0.119819 −0.00495391
\(586\) 0 0
\(587\) −12.1690 −0.502268 −0.251134 0.967952i \(-0.580803\pi\)
−0.251134 + 0.967952i \(0.580803\pi\)
\(588\) 0 0
\(589\) 35.6877 1.47049
\(590\) 0 0
\(591\) −14.1600 −0.582465
\(592\) 0 0
\(593\) −40.5180 −1.66388 −0.831938 0.554869i \(-0.812769\pi\)
−0.831938 + 0.554869i \(0.812769\pi\)
\(594\) 0 0
\(595\) −8.46742 −0.347130
\(596\) 0 0
\(597\) −4.81724 −0.197156
\(598\) 0 0
\(599\) 28.8913 1.18047 0.590233 0.807233i \(-0.299036\pi\)
0.590233 + 0.807233i \(0.299036\pi\)
\(600\) 0 0
\(601\) 5.34278 0.217937 0.108968 0.994045i \(-0.465245\pi\)
0.108968 + 0.994045i \(0.465245\pi\)
\(602\) 0 0
\(603\) −9.03605 −0.367977
\(604\) 0 0
\(605\) −10.2590 −0.417088
\(606\) 0 0
\(607\) 40.4376 1.64131 0.820656 0.571422i \(-0.193608\pi\)
0.820656 + 0.571422i \(0.193608\pi\)
\(608\) 0 0
\(609\) 21.9612 0.889915
\(610\) 0 0
\(611\) 0.323404 0.0130835
\(612\) 0 0
\(613\) −5.31154 −0.214531 −0.107266 0.994230i \(-0.534210\pi\)
−0.107266 + 0.994230i \(0.534210\pi\)
\(614\) 0 0
\(615\) 14.5976 0.588634
\(616\) 0 0
\(617\) 9.71680 0.391183 0.195592 0.980685i \(-0.437337\pi\)
0.195592 + 0.980685i \(0.437337\pi\)
\(618\) 0 0
\(619\) 23.7562 0.954843 0.477421 0.878674i \(-0.341572\pi\)
0.477421 + 0.878674i \(0.341572\pi\)
\(620\) 0 0
\(621\) 5.64681 0.226599
\(622\) 0 0
\(623\) 26.5872 1.06520
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 6.62117 0.264424
\(628\) 0 0
\(629\) −66.5872 −2.65501
\(630\) 0 0
\(631\) −1.24234 −0.0494566 −0.0247283 0.999694i \(-0.507872\pi\)
−0.0247283 + 0.999694i \(0.507872\pi\)
\(632\) 0 0
\(633\) −23.7521 −0.944060
\(634\) 0 0
\(635\) 15.2099 0.603585
\(636\) 0 0
\(637\) −0.645359 −0.0255701
\(638\) 0 0
\(639\) 6.29507 0.249029
\(640\) 0 0
\(641\) −34.8864 −1.37793 −0.688966 0.724794i \(-0.741935\pi\)
−0.688966 + 0.724794i \(0.741935\pi\)
\(642\) 0 0
\(643\) −32.1592 −1.26824 −0.634118 0.773236i \(-0.718637\pi\)
−0.634118 + 0.773236i \(0.718637\pi\)
\(644\) 0 0
\(645\) 16.7368 0.659012
\(646\) 0 0
\(647\) −23.9550 −0.941768 −0.470884 0.882195i \(-0.656065\pi\)
−0.470884 + 0.882195i \(0.656065\pi\)
\(648\) 0 0
\(649\) 9.48197 0.372200
\(650\) 0 0
\(651\) −15.2590 −0.598048
\(652\) 0 0
\(653\) 4.31926 0.169026 0.0845128 0.996422i \(-0.473067\pi\)
0.0845128 + 0.996422i \(0.473067\pi\)
\(654\) 0 0
\(655\) −12.6918 −0.495911
\(656\) 0 0
\(657\) −6.12878 −0.239107
\(658\) 0 0
\(659\) 6.14961 0.239555 0.119777 0.992801i \(-0.461782\pi\)
0.119777 + 0.992801i \(0.461782\pi\)
\(660\) 0 0
\(661\) −15.7473 −0.612497 −0.306249 0.951952i \(-0.599074\pi\)
−0.306249 + 0.951952i \(0.599074\pi\)
\(662\) 0 0
\(663\) −1.12127 −0.0435465
\(664\) 0 0
\(665\) 8.08522 0.313531
\(666\) 0 0
\(667\) 9.76663 0.378165
\(668\) 0 0
\(669\) 31.1440 1.20410
\(670\) 0 0
\(671\) 3.81579 0.147307
\(672\) 0 0
\(673\) 43.5783 1.67982 0.839909 0.542727i \(-0.182608\pi\)
0.839909 + 0.542727i \(0.182608\pi\)
\(674\) 0 0
\(675\) 5.64681 0.217346
\(676\) 0 0
\(677\) 30.1801 1.15991 0.579957 0.814647i \(-0.303069\pi\)
0.579957 + 0.814647i \(0.303069\pi\)
\(678\) 0 0
\(679\) −2.97581 −0.114201
\(680\) 0 0
\(681\) −43.3836 −1.66247
\(682\) 0 0
\(683\) −29.2832 −1.12049 −0.560245 0.828327i \(-0.689293\pi\)
−0.560245 + 0.828327i \(0.689293\pi\)
\(684\) 0 0
\(685\) −14.0346 −0.536235
\(686\) 0 0
\(687\) −28.7881 −1.09834
\(688\) 0 0
\(689\) 0.0208243 0.000793344 0
\(690\) 0 0
\(691\) 14.9557 0.568940 0.284470 0.958685i \(-0.408182\pi\)
0.284470 + 0.958685i \(0.408182\pi\)
\(692\) 0 0
\(693\) 1.13919 0.0432744
\(694\) 0 0
\(695\) −14.2638 −0.541058
\(696\) 0 0
\(697\) −54.9694 −2.08212
\(698\) 0 0
\(699\) 8.08377 0.305756
\(700\) 0 0
\(701\) 24.9627 0.942828 0.471414 0.881912i \(-0.343744\pi\)
0.471414 + 0.881912i \(0.343744\pi\)
\(702\) 0 0
\(703\) 63.5816 2.39803
\(704\) 0 0
\(705\) −3.39821 −0.127984
\(706\) 0 0
\(707\) 23.7104 0.891722
\(708\) 0 0
\(709\) −38.9717 −1.46361 −0.731806 0.681513i \(-0.761322\pi\)
−0.731806 + 0.681513i \(0.761322\pi\)
\(710\) 0 0
\(711\) 5.85039 0.219407
\(712\) 0 0
\(713\) −6.78600 −0.254138
\(714\) 0 0
\(715\) 0.119819 0.00448098
\(716\) 0 0
\(717\) −25.4466 −0.950320
\(718\) 0 0
\(719\) 25.2978 0.943447 0.471724 0.881746i \(-0.343632\pi\)
0.471724 + 0.881746i \(0.343632\pi\)
\(720\) 0 0
\(721\) 12.1682 0.453168
\(722\) 0 0
\(723\) 23.5333 0.875211
\(724\) 0 0
\(725\) 9.76663 0.362723
\(726\) 0 0
\(727\) −12.7417 −0.472562 −0.236281 0.971685i \(-0.575929\pi\)
−0.236281 + 0.971685i \(0.575929\pi\)
\(728\) 0 0
\(729\) 29.6635 1.09865
\(730\) 0 0
\(731\) −63.0249 −2.33106
\(732\) 0 0
\(733\) −9.83247 −0.363170 −0.181585 0.983375i \(-0.558123\pi\)
−0.181585 + 0.983375i \(0.558123\pi\)
\(734\) 0 0
\(735\) 6.78119 0.250128
\(736\) 0 0
\(737\) 9.03605 0.332847
\(738\) 0 0
\(739\) −22.6918 −0.834732 −0.417366 0.908738i \(-0.637047\pi\)
−0.417366 + 0.908738i \(0.637047\pi\)
\(740\) 0 0
\(741\) 1.07066 0.0393316
\(742\) 0 0
\(743\) −3.08858 −0.113309 −0.0566546 0.998394i \(-0.518043\pi\)
−0.0566546 + 0.998394i \(0.518043\pi\)
\(744\) 0 0
\(745\) −4.46260 −0.163497
\(746\) 0 0
\(747\) 9.16484 0.335324
\(748\) 0 0
\(749\) −6.57760 −0.240340
\(750\) 0 0
\(751\) 10.7577 0.392553 0.196276 0.980549i \(-0.437115\pi\)
0.196276 + 0.980549i \(0.437115\pi\)
\(752\) 0 0
\(753\) −42.4882 −1.54836
\(754\) 0 0
\(755\) −10.7562 −0.391459
\(756\) 0 0
\(757\) −19.5333 −0.709948 −0.354974 0.934876i \(-0.615510\pi\)
−0.354974 + 0.934876i \(0.615510\pi\)
\(758\) 0 0
\(759\) −1.25901 −0.0456993
\(760\) 0 0
\(761\) −15.6933 −0.568881 −0.284440 0.958694i \(-0.591808\pi\)
−0.284440 + 0.958694i \(0.591808\pi\)
\(762\) 0 0
\(763\) 27.3304 0.989429
\(764\) 0 0
\(765\) −4.74099 −0.171411
\(766\) 0 0
\(767\) 1.53326 0.0553626
\(768\) 0 0
\(769\) 11.4128 0.411555 0.205777 0.978599i \(-0.434028\pi\)
0.205777 + 0.978599i \(0.434028\pi\)
\(770\) 0 0
\(771\) 9.59910 0.345703
\(772\) 0 0
\(773\) −22.5872 −0.812406 −0.406203 0.913783i \(-0.633147\pi\)
−0.406203 + 0.913783i \(0.633147\pi\)
\(774\) 0 0
\(775\) −6.78600 −0.243760
\(776\) 0 0
\(777\) −27.1857 −0.975280
\(778\) 0 0
\(779\) 52.4882 1.88059
\(780\) 0 0
\(781\) −6.29507 −0.225255
\(782\) 0 0
\(783\) 55.1503 1.97091
\(784\) 0 0
\(785\) 14.5180 0.518171
\(786\) 0 0
\(787\) 4.42799 0.157841 0.0789205 0.996881i \(-0.474853\pi\)
0.0789205 + 0.996881i \(0.474853\pi\)
\(788\) 0 0
\(789\) −0.821385 −0.0292421
\(790\) 0 0
\(791\) 4.92520 0.175120
\(792\) 0 0
\(793\) 0.617021 0.0219111
\(794\) 0 0
\(795\) −0.218814 −0.00776053
\(796\) 0 0
\(797\) 21.5124 0.762009 0.381005 0.924573i \(-0.375578\pi\)
0.381005 + 0.924573i \(0.375578\pi\)
\(798\) 0 0
\(799\) 12.7964 0.452705
\(800\) 0 0
\(801\) 14.8864 0.525987
\(802\) 0 0
\(803\) 6.12878 0.216280
\(804\) 0 0
\(805\) −1.53740 −0.0541863
\(806\) 0 0
\(807\) 10.6323 0.374273
\(808\) 0 0
\(809\) 7.37883 0.259426 0.129713 0.991552i \(-0.458594\pi\)
0.129713 + 0.991552i \(0.458594\pi\)
\(810\) 0 0
\(811\) −29.0423 −1.01981 −0.509907 0.860230i \(-0.670320\pi\)
−0.509907 + 0.860230i \(0.670320\pi\)
\(812\) 0 0
\(813\) 0.269425 0.00944916
\(814\) 0 0
\(815\) 1.30403 0.0456782
\(816\) 0 0
\(817\) 60.1801 2.10543
\(818\) 0 0
\(819\) 0.184210 0.00643682
\(820\) 0 0
\(821\) 5.70079 0.198959 0.0994794 0.995040i \(-0.468282\pi\)
0.0994794 + 0.995040i \(0.468282\pi\)
\(822\) 0 0
\(823\) 16.0034 0.557842 0.278921 0.960314i \(-0.410023\pi\)
0.278921 + 0.960314i \(0.410023\pi\)
\(824\) 0 0
\(825\) −1.25901 −0.0438332
\(826\) 0 0
\(827\) −39.7937 −1.38376 −0.691882 0.722011i \(-0.743218\pi\)
−0.691882 + 0.722011i \(0.743218\pi\)
\(828\) 0 0
\(829\) −23.3353 −0.810467 −0.405234 0.914213i \(-0.632810\pi\)
−0.405234 + 0.914213i \(0.632810\pi\)
\(830\) 0 0
\(831\) −37.8746 −1.31385
\(832\) 0 0
\(833\) −25.5355 −0.884752
\(834\) 0 0
\(835\) −3.44322 −0.119158
\(836\) 0 0
\(837\) −38.3193 −1.32451
\(838\) 0 0
\(839\) −37.1469 −1.28245 −0.641227 0.767351i \(-0.721574\pi\)
−0.641227 + 0.767351i \(0.721574\pi\)
\(840\) 0 0
\(841\) 66.3870 2.28921
\(842\) 0 0
\(843\) 39.1619 1.34881
\(844\) 0 0
\(845\) −12.9806 −0.446547
\(846\) 0 0
\(847\) 15.7722 0.541940
\(848\) 0 0
\(849\) 8.33796 0.286158
\(850\) 0 0
\(851\) −12.0900 −0.414441
\(852\) 0 0
\(853\) −53.7658 −1.84091 −0.920454 0.390851i \(-0.872181\pi\)
−0.920454 + 0.390851i \(0.872181\pi\)
\(854\) 0 0
\(855\) 4.52699 0.154820
\(856\) 0 0
\(857\) −23.9675 −0.818715 −0.409357 0.912374i \(-0.634247\pi\)
−0.409357 + 0.912374i \(0.634247\pi\)
\(858\) 0 0
\(859\) −3.70705 −0.126483 −0.0632415 0.997998i \(-0.520144\pi\)
−0.0632415 + 0.997998i \(0.520144\pi\)
\(860\) 0 0
\(861\) −22.4424 −0.764836
\(862\) 0 0
\(863\) −49.6296 −1.68941 −0.844705 0.535232i \(-0.820224\pi\)
−0.844705 + 0.535232i \(0.820224\pi\)
\(864\) 0 0
\(865\) 9.35801 0.318182
\(866\) 0 0
\(867\) −19.5020 −0.662323
\(868\) 0 0
\(869\) −5.85039 −0.198461
\(870\) 0 0
\(871\) 1.46115 0.0495091
\(872\) 0 0
\(873\) −1.66618 −0.0563918
\(874\) 0 0
\(875\) −1.53740 −0.0519737
\(876\) 0 0
\(877\) −13.1004 −0.442371 −0.221185 0.975232i \(-0.570993\pi\)
−0.221185 + 0.975232i \(0.570993\pi\)
\(878\) 0 0
\(879\) −1.54155 −0.0519951
\(880\) 0 0
\(881\) 3.88085 0.130749 0.0653746 0.997861i \(-0.479176\pi\)
0.0653746 + 0.997861i \(0.479176\pi\)
\(882\) 0 0
\(883\) 10.8131 0.363890 0.181945 0.983309i \(-0.441761\pi\)
0.181945 + 0.983309i \(0.441761\pi\)
\(884\) 0 0
\(885\) −16.1109 −0.541561
\(886\) 0 0
\(887\) 8.58387 0.288218 0.144109 0.989562i \(-0.453968\pi\)
0.144109 + 0.989562i \(0.453968\pi\)
\(888\) 0 0
\(889\) −23.3836 −0.784262
\(890\) 0 0
\(891\) −4.88645 −0.163702
\(892\) 0 0
\(893\) −12.2188 −0.408887
\(894\) 0 0
\(895\) 18.4134 0.615493
\(896\) 0 0
\(897\) −0.203585 −0.00679751
\(898\) 0 0
\(899\) −66.2764 −2.21044
\(900\) 0 0
\(901\) 0.823974 0.0274505
\(902\) 0 0
\(903\) −25.7312 −0.856282
\(904\) 0 0
\(905\) −1.94457 −0.0646398
\(906\) 0 0
\(907\) 21.2728 0.706351 0.353176 0.935557i \(-0.385102\pi\)
0.353176 + 0.935557i \(0.385102\pi\)
\(908\) 0 0
\(909\) 13.2757 0.440327
\(910\) 0 0
\(911\) 27.6441 0.915890 0.457945 0.888980i \(-0.348586\pi\)
0.457945 + 0.888980i \(0.348586\pi\)
\(912\) 0 0
\(913\) −9.16484 −0.303312
\(914\) 0 0
\(915\) −6.48342 −0.214335
\(916\) 0 0
\(917\) 19.5124 0.644357
\(918\) 0 0
\(919\) 0.994404 0.0328024 0.0164012 0.999865i \(-0.494779\pi\)
0.0164012 + 0.999865i \(0.494779\pi\)
\(920\) 0 0
\(921\) 26.1455 0.861522
\(922\) 0 0
\(923\) −1.01793 −0.0335054
\(924\) 0 0
\(925\) −12.0900 −0.397518
\(926\) 0 0
\(927\) 6.81309 0.223771
\(928\) 0 0
\(929\) −18.3747 −0.602854 −0.301427 0.953489i \(-0.597463\pi\)
−0.301427 + 0.953489i \(0.597463\pi\)
\(930\) 0 0
\(931\) 24.3829 0.799116
\(932\) 0 0
\(933\) 24.7639 0.810735
\(934\) 0 0
\(935\) 4.74099 0.155047
\(936\) 0 0
\(937\) −24.0942 −0.787122 −0.393561 0.919298i \(-0.628757\pi\)
−0.393561 + 0.919298i \(0.628757\pi\)
\(938\) 0 0
\(939\) 29.2459 0.954404
\(940\) 0 0
\(941\) 51.9244 1.69269 0.846344 0.532637i \(-0.178799\pi\)
0.846344 + 0.532637i \(0.178799\pi\)
\(942\) 0 0
\(943\) −9.98062 −0.325014
\(944\) 0 0
\(945\) −8.68141 −0.282406
\(946\) 0 0
\(947\) 4.72643 0.153588 0.0767941 0.997047i \(-0.475532\pi\)
0.0767941 + 0.997047i \(0.475532\pi\)
\(948\) 0 0
\(949\) 0.991037 0.0321704
\(950\) 0 0
\(951\) −4.82330 −0.156406
\(952\) 0 0
\(953\) 3.57682 0.115865 0.0579323 0.998321i \(-0.481549\pi\)
0.0579323 + 0.998321i \(0.481549\pi\)
\(954\) 0 0
\(955\) 0.775591 0.0250975
\(956\) 0 0
\(957\) −12.2963 −0.397483
\(958\) 0 0
\(959\) 21.5768 0.696752
\(960\) 0 0
\(961\) 15.0498 0.485478
\(962\) 0 0
\(963\) −3.68286 −0.118679
\(964\) 0 0
\(965\) −11.7666 −0.378781
\(966\) 0 0
\(967\) −33.9646 −1.09223 −0.546114 0.837711i \(-0.683894\pi\)
−0.546114 + 0.837711i \(0.683894\pi\)
\(968\) 0 0
\(969\) 42.3636 1.36092
\(970\) 0 0
\(971\) 56.0561 1.79893 0.899463 0.436997i \(-0.143958\pi\)
0.899463 + 0.436997i \(0.143958\pi\)
\(972\) 0 0
\(973\) 21.9292 0.703019
\(974\) 0 0
\(975\) −0.203585 −0.00651994
\(976\) 0 0
\(977\) −53.1219 −1.69952 −0.849761 0.527169i \(-0.823254\pi\)
−0.849761 + 0.527169i \(0.823254\pi\)
\(978\) 0 0
\(979\) −14.8864 −0.475773
\(980\) 0 0
\(981\) 15.3026 0.488574
\(982\) 0 0
\(983\) −28.7833 −0.918045 −0.459022 0.888425i \(-0.651800\pi\)
−0.459022 + 0.888425i \(0.651800\pi\)
\(984\) 0 0
\(985\) 9.68141 0.308475
\(986\) 0 0
\(987\) 5.22441 0.166295
\(988\) 0 0
\(989\) −11.4432 −0.363873
\(990\) 0 0
\(991\) −48.1641 −1.52998 −0.764991 0.644041i \(-0.777257\pi\)
−0.764991 + 0.644041i \(0.777257\pi\)
\(992\) 0 0
\(993\) −18.8385 −0.597823
\(994\) 0 0
\(995\) 3.29362 0.104415
\(996\) 0 0
\(997\) 43.5153 1.37814 0.689072 0.724693i \(-0.258018\pi\)
0.689072 + 0.724693i \(0.258018\pi\)
\(998\) 0 0
\(999\) −68.2701 −2.15997
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 1840.2.a.q.1.2 3
4.3 odd 2 920.2.a.i.1.2 3
5.4 even 2 9200.2.a.ci.1.2 3
8.3 odd 2 7360.2.a.bw.1.2 3
8.5 even 2 7360.2.a.cf.1.2 3
12.11 even 2 8280.2.a.bl.1.2 3
20.3 even 4 4600.2.e.q.4049.4 6
20.7 even 4 4600.2.e.q.4049.3 6
20.19 odd 2 4600.2.a.v.1.2 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
920.2.a.i.1.2 3 4.3 odd 2
1840.2.a.q.1.2 3 1.1 even 1 trivial
4600.2.a.v.1.2 3 20.19 odd 2
4600.2.e.q.4049.3 6 20.7 even 4
4600.2.e.q.4049.4 6 20.3 even 4
7360.2.a.bw.1.2 3 8.3 odd 2
7360.2.a.cf.1.2 3 8.5 even 2
8280.2.a.bl.1.2 3 12.11 even 2
9200.2.a.ci.1.2 3 5.4 even 2