Properties

Label 4020.2.a.h.1.5
Level $4020$
Weight $2$
Character 4020.1
Self dual yes
Analytic conductor $32.100$
Analytic rank $0$
Dimension $7$
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] = [4020,2,Mod(1,4020)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4020, 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("4020.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4020 = 2^{2} \cdot 3 \cdot 5 \cdot 67 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4020.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: \(32.0998616126\)
Analytic rank: \(0\)
Dimension: \(7\)
Coefficient field: \(\mathbb{Q}[x]/(x^{7} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{7} - 3x^{6} - 28x^{5} + 90x^{4} + 143x^{3} - 418x^{2} - 256x + 160 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.5
Root \(3.15736\) of defining polynomial
Character \(\chi\) \(=\) 4020.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{3} -1.00000 q^{5} +3.15736 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{3} -1.00000 q^{5} +3.15736 q^{7} +1.00000 q^{9} -4.84615 q^{11} +6.88967 q^{13} +1.00000 q^{15} +5.52837 q^{17} +5.73402 q^{19} -3.15736 q^{21} +4.53661 q^{23} +1.00000 q^{25} -1.00000 q^{27} -3.37924 q^{29} -6.57845 q^{31} +4.84615 q^{33} -3.15736 q^{35} -0.221881 q^{37} -6.88967 q^{39} -3.95590 q^{41} -1.21536 q^{43} -1.00000 q^{45} -2.99348 q^{47} +2.96894 q^{49} -5.52837 q^{51} +7.21887 q^{53} +4.84615 q^{55} -5.73402 q^{57} -12.3745 q^{59} +11.2189 q^{61} +3.15736 q^{63} -6.88967 q^{65} -1.00000 q^{67} -4.53661 q^{69} -6.85861 q^{71} +3.91478 q^{73} -1.00000 q^{75} -15.3010 q^{77} +7.83791 q^{79} +1.00000 q^{81} +12.7490 q^{83} -5.52837 q^{85} +3.37924 q^{87} +2.96242 q^{89} +21.7532 q^{91} +6.57845 q^{93} -5.73402 q^{95} +5.48656 q^{97} -4.84615 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 7 q - 7 q^{3} - 7 q^{5} + 3 q^{7} + 7 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 7 q - 7 q^{3} - 7 q^{5} + 3 q^{7} + 7 q^{9} - 5 q^{11} + 5 q^{13} + 7 q^{15} + 3 q^{17} + 2 q^{19} - 3 q^{21} - 3 q^{23} + 7 q^{25} - 7 q^{27} - 8 q^{29} + 7 q^{31} + 5 q^{33} - 3 q^{35} - 5 q^{37} - 5 q^{39} + 7 q^{41} + 3 q^{43} - 7 q^{45} - 6 q^{47} + 16 q^{49} - 3 q^{51} - 9 q^{53} + 5 q^{55} - 2 q^{57} - 22 q^{59} + 19 q^{61} + 3 q^{63} - 5 q^{65} - 7 q^{67} + 3 q^{69} + 23 q^{73} - 7 q^{75} + 9 q^{77} + 25 q^{79} + 7 q^{81} - 20 q^{83} - 3 q^{85} + 8 q^{87} + q^{89} + 23 q^{91} - 7 q^{93} - 2 q^{95} + 3 q^{97} - 5 q^{99}+O(q^{100}) \) 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
\(3\) −1.00000 −0.577350
\(4\) 0 0
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) 3.15736 1.19337 0.596686 0.802475i \(-0.296484\pi\)
0.596686 + 0.802475i \(0.296484\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −4.84615 −1.46117 −0.730584 0.682823i \(-0.760752\pi\)
−0.730584 + 0.682823i \(0.760752\pi\)
\(12\) 0 0
\(13\) 6.88967 1.91085 0.955425 0.295233i \(-0.0953973\pi\)
0.955425 + 0.295233i \(0.0953973\pi\)
\(14\) 0 0
\(15\) 1.00000 0.258199
\(16\) 0 0
\(17\) 5.52837 1.34083 0.670414 0.741988i \(-0.266117\pi\)
0.670414 + 0.741988i \(0.266117\pi\)
\(18\) 0 0
\(19\) 5.73402 1.31547 0.657737 0.753247i \(-0.271514\pi\)
0.657737 + 0.753247i \(0.271514\pi\)
\(20\) 0 0
\(21\) −3.15736 −0.688993
\(22\) 0 0
\(23\) 4.53661 0.945948 0.472974 0.881076i \(-0.343180\pi\)
0.472974 + 0.881076i \(0.343180\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) −3.37924 −0.627510 −0.313755 0.949504i \(-0.601587\pi\)
−0.313755 + 0.949504i \(0.601587\pi\)
\(30\) 0 0
\(31\) −6.57845 −1.18152 −0.590762 0.806846i \(-0.701173\pi\)
−0.590762 + 0.806846i \(0.701173\pi\)
\(32\) 0 0
\(33\) 4.84615 0.843606
\(34\) 0 0
\(35\) −3.15736 −0.533692
\(36\) 0 0
\(37\) −0.221881 −0.0364771 −0.0182385 0.999834i \(-0.505806\pi\)
−0.0182385 + 0.999834i \(0.505806\pi\)
\(38\) 0 0
\(39\) −6.88967 −1.10323
\(40\) 0 0
\(41\) −3.95590 −0.617808 −0.308904 0.951093i \(-0.599962\pi\)
−0.308904 + 0.951093i \(0.599962\pi\)
\(42\) 0 0
\(43\) −1.21536 −0.185341 −0.0926704 0.995697i \(-0.529540\pi\)
−0.0926704 + 0.995697i \(0.529540\pi\)
\(44\) 0 0
\(45\) −1.00000 −0.149071
\(46\) 0 0
\(47\) −2.99348 −0.436644 −0.218322 0.975877i \(-0.570058\pi\)
−0.218322 + 0.975877i \(0.570058\pi\)
\(48\) 0 0
\(49\) 2.96894 0.424135
\(50\) 0 0
\(51\) −5.52837 −0.774127
\(52\) 0 0
\(53\) 7.21887 0.991588 0.495794 0.868440i \(-0.334877\pi\)
0.495794 + 0.868440i \(0.334877\pi\)
\(54\) 0 0
\(55\) 4.84615 0.653454
\(56\) 0 0
\(57\) −5.73402 −0.759490
\(58\) 0 0
\(59\) −12.3745 −1.61103 −0.805513 0.592579i \(-0.798110\pi\)
−0.805513 + 0.592579i \(0.798110\pi\)
\(60\) 0 0
\(61\) 11.2189 1.43643 0.718215 0.695822i \(-0.244960\pi\)
0.718215 + 0.695822i \(0.244960\pi\)
\(62\) 0 0
\(63\) 3.15736 0.397790
\(64\) 0 0
\(65\) −6.88967 −0.854558
\(66\) 0 0
\(67\) −1.00000 −0.122169
\(68\) 0 0
\(69\) −4.53661 −0.546143
\(70\) 0 0
\(71\) −6.85861 −0.813968 −0.406984 0.913435i \(-0.633419\pi\)
−0.406984 + 0.913435i \(0.633419\pi\)
\(72\) 0 0
\(73\) 3.91478 0.458190 0.229095 0.973404i \(-0.426423\pi\)
0.229095 + 0.973404i \(0.426423\pi\)
\(74\) 0 0
\(75\) −1.00000 −0.115470
\(76\) 0 0
\(77\) −15.3010 −1.74372
\(78\) 0 0
\(79\) 7.83791 0.881834 0.440917 0.897548i \(-0.354653\pi\)
0.440917 + 0.897548i \(0.354653\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 12.7490 1.39938 0.699690 0.714447i \(-0.253321\pi\)
0.699690 + 0.714447i \(0.253321\pi\)
\(84\) 0 0
\(85\) −5.52837 −0.599636
\(86\) 0 0
\(87\) 3.37924 0.362293
\(88\) 0 0
\(89\) 2.96242 0.314016 0.157008 0.987597i \(-0.449815\pi\)
0.157008 + 0.987597i \(0.449815\pi\)
\(90\) 0 0
\(91\) 21.7532 2.28035
\(92\) 0 0
\(93\) 6.57845 0.682154
\(94\) 0 0
\(95\) −5.73402 −0.588298
\(96\) 0 0
\(97\) 5.48656 0.557076 0.278538 0.960425i \(-0.410150\pi\)
0.278538 + 0.960425i \(0.410150\pi\)
\(98\) 0 0
\(99\) −4.84615 −0.487056
\(100\) 0 0
\(101\) −3.57769 −0.355994 −0.177997 0.984031i \(-0.556962\pi\)
−0.177997 + 0.984031i \(0.556962\pi\)
\(102\) 0 0
\(103\) 0.784639 0.0773128 0.0386564 0.999253i \(-0.487692\pi\)
0.0386564 + 0.999253i \(0.487692\pi\)
\(104\) 0 0
\(105\) 3.15736 0.308127
\(106\) 0 0
\(107\) 5.08465 0.491551 0.245776 0.969327i \(-0.420957\pi\)
0.245776 + 0.969327i \(0.420957\pi\)
\(108\) 0 0
\(109\) −1.79230 −0.171671 −0.0858356 0.996309i \(-0.527356\pi\)
−0.0858356 + 0.996309i \(0.527356\pi\)
\(110\) 0 0
\(111\) 0.221881 0.0210600
\(112\) 0 0
\(113\) −5.45562 −0.513222 −0.256611 0.966515i \(-0.582606\pi\)
−0.256611 + 0.966515i \(0.582606\pi\)
\(114\) 0 0
\(115\) −4.53661 −0.423041
\(116\) 0 0
\(117\) 6.88967 0.636950
\(118\) 0 0
\(119\) 17.4551 1.60010
\(120\) 0 0
\(121\) 12.4851 1.13501
\(122\) 0 0
\(123\) 3.95590 0.356692
\(124\) 0 0
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) 13.1240 1.16457 0.582284 0.812985i \(-0.302159\pi\)
0.582284 + 0.812985i \(0.302159\pi\)
\(128\) 0 0
\(129\) 1.21536 0.107007
\(130\) 0 0
\(131\) 4.58250 0.400375 0.200187 0.979758i \(-0.435845\pi\)
0.200187 + 0.979758i \(0.435845\pi\)
\(132\) 0 0
\(133\) 18.1044 1.56985
\(134\) 0 0
\(135\) 1.00000 0.0860663
\(136\) 0 0
\(137\) 11.8684 1.01399 0.506993 0.861950i \(-0.330757\pi\)
0.506993 + 0.861950i \(0.330757\pi\)
\(138\) 0 0
\(139\) −10.5254 −0.892750 −0.446375 0.894846i \(-0.647285\pi\)
−0.446375 + 0.894846i \(0.647285\pi\)
\(140\) 0 0
\(141\) 2.99348 0.252096
\(142\) 0 0
\(143\) −33.3883 −2.79207
\(144\) 0 0
\(145\) 3.37924 0.280631
\(146\) 0 0
\(147\) −2.96894 −0.244874
\(148\) 0 0
\(149\) 3.73059 0.305622 0.152811 0.988255i \(-0.451167\pi\)
0.152811 + 0.988255i \(0.451167\pi\)
\(150\) 0 0
\(151\) 2.21837 0.180529 0.0902643 0.995918i \(-0.471229\pi\)
0.0902643 + 0.995918i \(0.471229\pi\)
\(152\) 0 0
\(153\) 5.52837 0.446942
\(154\) 0 0
\(155\) 6.57845 0.528394
\(156\) 0 0
\(157\) −22.0531 −1.76003 −0.880016 0.474945i \(-0.842468\pi\)
−0.880016 + 0.474945i \(0.842468\pi\)
\(158\) 0 0
\(159\) −7.21887 −0.572494
\(160\) 0 0
\(161\) 14.3237 1.12887
\(162\) 0 0
\(163\) −12.5802 −0.985355 −0.492677 0.870212i \(-0.663982\pi\)
−0.492677 + 0.870212i \(0.663982\pi\)
\(164\) 0 0
\(165\) −4.84615 −0.377272
\(166\) 0 0
\(167\) 11.2328 0.869218 0.434609 0.900619i \(-0.356887\pi\)
0.434609 + 0.900619i \(0.356887\pi\)
\(168\) 0 0
\(169\) 34.4675 2.65135
\(170\) 0 0
\(171\) 5.73402 0.438492
\(172\) 0 0
\(173\) −5.79138 −0.440310 −0.220155 0.975465i \(-0.570656\pi\)
−0.220155 + 0.975465i \(0.570656\pi\)
\(174\) 0 0
\(175\) 3.15736 0.238674
\(176\) 0 0
\(177\) 12.3745 0.930126
\(178\) 0 0
\(179\) −0.461711 −0.0345099 −0.0172549 0.999851i \(-0.505493\pi\)
−0.0172549 + 0.999851i \(0.505493\pi\)
\(180\) 0 0
\(181\) −23.9076 −1.77704 −0.888520 0.458838i \(-0.848266\pi\)
−0.888520 + 0.458838i \(0.848266\pi\)
\(182\) 0 0
\(183\) −11.2189 −0.829323
\(184\) 0 0
\(185\) 0.221881 0.0163130
\(186\) 0 0
\(187\) −26.7913 −1.95917
\(188\) 0 0
\(189\) −3.15736 −0.229664
\(190\) 0 0
\(191\) 11.9649 0.865749 0.432875 0.901454i \(-0.357499\pi\)
0.432875 + 0.901454i \(0.357499\pi\)
\(192\) 0 0
\(193\) 3.67742 0.264707 0.132353 0.991203i \(-0.457747\pi\)
0.132353 + 0.991203i \(0.457747\pi\)
\(194\) 0 0
\(195\) 6.88967 0.493379
\(196\) 0 0
\(197\) 17.9063 1.27577 0.637886 0.770131i \(-0.279809\pi\)
0.637886 + 0.770131i \(0.279809\pi\)
\(198\) 0 0
\(199\) −12.8328 −0.909697 −0.454848 0.890569i \(-0.650307\pi\)
−0.454848 + 0.890569i \(0.650307\pi\)
\(200\) 0 0
\(201\) 1.00000 0.0705346
\(202\) 0 0
\(203\) −10.6695 −0.748852
\(204\) 0 0
\(205\) 3.95590 0.276292
\(206\) 0 0
\(207\) 4.53661 0.315316
\(208\) 0 0
\(209\) −27.7879 −1.92213
\(210\) 0 0
\(211\) 5.45959 0.375854 0.187927 0.982183i \(-0.439823\pi\)
0.187927 + 0.982183i \(0.439823\pi\)
\(212\) 0 0
\(213\) 6.85861 0.469944
\(214\) 0 0
\(215\) 1.21536 0.0828869
\(216\) 0 0
\(217\) −20.7706 −1.41000
\(218\) 0 0
\(219\) −3.91478 −0.264536
\(220\) 0 0
\(221\) 38.0887 2.56212
\(222\) 0 0
\(223\) 25.2071 1.68799 0.843994 0.536352i \(-0.180198\pi\)
0.843994 + 0.536352i \(0.180198\pi\)
\(224\) 0 0
\(225\) 1.00000 0.0666667
\(226\) 0 0
\(227\) −9.46282 −0.628069 −0.314035 0.949412i \(-0.601681\pi\)
−0.314035 + 0.949412i \(0.601681\pi\)
\(228\) 0 0
\(229\) 1.16836 0.0772077 0.0386038 0.999255i \(-0.487709\pi\)
0.0386038 + 0.999255i \(0.487709\pi\)
\(230\) 0 0
\(231\) 15.3010 1.00673
\(232\) 0 0
\(233\) 7.95443 0.521112 0.260556 0.965459i \(-0.416094\pi\)
0.260556 + 0.965459i \(0.416094\pi\)
\(234\) 0 0
\(235\) 2.99348 0.195273
\(236\) 0 0
\(237\) −7.83791 −0.509127
\(238\) 0 0
\(239\) 21.6435 1.40000 0.700001 0.714142i \(-0.253183\pi\)
0.700001 + 0.714142i \(0.253183\pi\)
\(240\) 0 0
\(241\) 25.3117 1.63047 0.815235 0.579130i \(-0.196607\pi\)
0.815235 + 0.579130i \(0.196607\pi\)
\(242\) 0 0
\(243\) −1.00000 −0.0641500
\(244\) 0 0
\(245\) −2.96894 −0.189679
\(246\) 0 0
\(247\) 39.5055 2.51368
\(248\) 0 0
\(249\) −12.7490 −0.807932
\(250\) 0 0
\(251\) 15.4411 0.974634 0.487317 0.873225i \(-0.337976\pi\)
0.487317 + 0.873225i \(0.337976\pi\)
\(252\) 0 0
\(253\) −21.9851 −1.38219
\(254\) 0 0
\(255\) 5.52837 0.346200
\(256\) 0 0
\(257\) 9.95691 0.621095 0.310547 0.950558i \(-0.399488\pi\)
0.310547 + 0.950558i \(0.399488\pi\)
\(258\) 0 0
\(259\) −0.700560 −0.0435307
\(260\) 0 0
\(261\) −3.37924 −0.209170
\(262\) 0 0
\(263\) −13.4957 −0.832179 −0.416089 0.909324i \(-0.636600\pi\)
−0.416089 + 0.909324i \(0.636600\pi\)
\(264\) 0 0
\(265\) −7.21887 −0.443452
\(266\) 0 0
\(267\) −2.96242 −0.181297
\(268\) 0 0
\(269\) −22.4578 −1.36928 −0.684639 0.728882i \(-0.740040\pi\)
−0.684639 + 0.728882i \(0.740040\pi\)
\(270\) 0 0
\(271\) −16.8705 −1.02481 −0.512406 0.858743i \(-0.671246\pi\)
−0.512406 + 0.858743i \(0.671246\pi\)
\(272\) 0 0
\(273\) −21.7532 −1.31656
\(274\) 0 0
\(275\) −4.84615 −0.292234
\(276\) 0 0
\(277\) 28.6173 1.71945 0.859723 0.510761i \(-0.170636\pi\)
0.859723 + 0.510761i \(0.170636\pi\)
\(278\) 0 0
\(279\) −6.57845 −0.393842
\(280\) 0 0
\(281\) 11.2629 0.671888 0.335944 0.941882i \(-0.390945\pi\)
0.335944 + 0.941882i \(0.390945\pi\)
\(282\) 0 0
\(283\) −30.1828 −1.79418 −0.897090 0.441847i \(-0.854323\pi\)
−0.897090 + 0.441847i \(0.854323\pi\)
\(284\) 0 0
\(285\) 5.73402 0.339654
\(286\) 0 0
\(287\) −12.4902 −0.737274
\(288\) 0 0
\(289\) 13.5629 0.797818
\(290\) 0 0
\(291\) −5.48656 −0.321628
\(292\) 0 0
\(293\) 10.1216 0.591313 0.295657 0.955294i \(-0.404462\pi\)
0.295657 + 0.955294i \(0.404462\pi\)
\(294\) 0 0
\(295\) 12.3745 0.720472
\(296\) 0 0
\(297\) 4.84615 0.281202
\(298\) 0 0
\(299\) 31.2557 1.80757
\(300\) 0 0
\(301\) −3.83734 −0.221180
\(302\) 0 0
\(303\) 3.57769 0.205533
\(304\) 0 0
\(305\) −11.2189 −0.642391
\(306\) 0 0
\(307\) 6.50547 0.371287 0.185643 0.982617i \(-0.440563\pi\)
0.185643 + 0.982617i \(0.440563\pi\)
\(308\) 0 0
\(309\) −0.784639 −0.0446365
\(310\) 0 0
\(311\) 27.5371 1.56149 0.780743 0.624852i \(-0.214841\pi\)
0.780743 + 0.624852i \(0.214841\pi\)
\(312\) 0 0
\(313\) 11.4107 0.644971 0.322485 0.946574i \(-0.395482\pi\)
0.322485 + 0.946574i \(0.395482\pi\)
\(314\) 0 0
\(315\) −3.15736 −0.177897
\(316\) 0 0
\(317\) −20.7200 −1.16375 −0.581875 0.813278i \(-0.697681\pi\)
−0.581875 + 0.813278i \(0.697681\pi\)
\(318\) 0 0
\(319\) 16.3763 0.916897
\(320\) 0 0
\(321\) −5.08465 −0.283797
\(322\) 0 0
\(323\) 31.6998 1.76382
\(324\) 0 0
\(325\) 6.88967 0.382170
\(326\) 0 0
\(327\) 1.79230 0.0991144
\(328\) 0 0
\(329\) −9.45150 −0.521078
\(330\) 0 0
\(331\) 13.3874 0.735837 0.367919 0.929858i \(-0.380071\pi\)
0.367919 + 0.929858i \(0.380071\pi\)
\(332\) 0 0
\(333\) −0.221881 −0.0121590
\(334\) 0 0
\(335\) 1.00000 0.0546358
\(336\) 0 0
\(337\) 20.1569 1.09802 0.549009 0.835816i \(-0.315005\pi\)
0.549009 + 0.835816i \(0.315005\pi\)
\(338\) 0 0
\(339\) 5.45562 0.296309
\(340\) 0 0
\(341\) 31.8801 1.72641
\(342\) 0 0
\(343\) −12.7275 −0.687221
\(344\) 0 0
\(345\) 4.53661 0.244243
\(346\) 0 0
\(347\) 6.95526 0.373378 0.186689 0.982419i \(-0.440224\pi\)
0.186689 + 0.982419i \(0.440224\pi\)
\(348\) 0 0
\(349\) 0.149016 0.00797664 0.00398832 0.999992i \(-0.498730\pi\)
0.00398832 + 0.999992i \(0.498730\pi\)
\(350\) 0 0
\(351\) −6.88967 −0.367743
\(352\) 0 0
\(353\) 29.8624 1.58941 0.794707 0.606993i \(-0.207624\pi\)
0.794707 + 0.606993i \(0.207624\pi\)
\(354\) 0 0
\(355\) 6.85861 0.364017
\(356\) 0 0
\(357\) −17.4551 −0.923821
\(358\) 0 0
\(359\) 10.7189 0.565722 0.282861 0.959161i \(-0.408716\pi\)
0.282861 + 0.959161i \(0.408716\pi\)
\(360\) 0 0
\(361\) 13.8790 0.730474
\(362\) 0 0
\(363\) −12.4851 −0.655299
\(364\) 0 0
\(365\) −3.91478 −0.204909
\(366\) 0 0
\(367\) −0.0871262 −0.00454795 −0.00227398 0.999997i \(-0.500724\pi\)
−0.00227398 + 0.999997i \(0.500724\pi\)
\(368\) 0 0
\(369\) −3.95590 −0.205936
\(370\) 0 0
\(371\) 22.7926 1.18333
\(372\) 0 0
\(373\) −12.2747 −0.635561 −0.317780 0.948164i \(-0.602937\pi\)
−0.317780 + 0.948164i \(0.602937\pi\)
\(374\) 0 0
\(375\) 1.00000 0.0516398
\(376\) 0 0
\(377\) −23.2819 −1.19908
\(378\) 0 0
\(379\) 18.4419 0.947299 0.473649 0.880714i \(-0.342936\pi\)
0.473649 + 0.880714i \(0.342936\pi\)
\(380\) 0 0
\(381\) −13.1240 −0.672364
\(382\) 0 0
\(383\) −30.0990 −1.53799 −0.768993 0.639257i \(-0.779242\pi\)
−0.768993 + 0.639257i \(0.779242\pi\)
\(384\) 0 0
\(385\) 15.3010 0.779813
\(386\) 0 0
\(387\) −1.21536 −0.0617803
\(388\) 0 0
\(389\) 4.74599 0.240631 0.120316 0.992736i \(-0.461609\pi\)
0.120316 + 0.992736i \(0.461609\pi\)
\(390\) 0 0
\(391\) 25.0801 1.26835
\(392\) 0 0
\(393\) −4.58250 −0.231157
\(394\) 0 0
\(395\) −7.83791 −0.394368
\(396\) 0 0
\(397\) −15.2317 −0.764455 −0.382227 0.924068i \(-0.624843\pi\)
−0.382227 + 0.924068i \(0.624843\pi\)
\(398\) 0 0
\(399\) −18.1044 −0.906353
\(400\) 0 0
\(401\) 30.5296 1.52458 0.762288 0.647239i \(-0.224076\pi\)
0.762288 + 0.647239i \(0.224076\pi\)
\(402\) 0 0
\(403\) −45.3234 −2.25772
\(404\) 0 0
\(405\) −1.00000 −0.0496904
\(406\) 0 0
\(407\) 1.07527 0.0532991
\(408\) 0 0
\(409\) 22.2610 1.10074 0.550369 0.834922i \(-0.314487\pi\)
0.550369 + 0.834922i \(0.314487\pi\)
\(410\) 0 0
\(411\) −11.8684 −0.585425
\(412\) 0 0
\(413\) −39.0709 −1.92255
\(414\) 0 0
\(415\) −12.7490 −0.625822
\(416\) 0 0
\(417\) 10.5254 0.515429
\(418\) 0 0
\(419\) −36.2422 −1.77055 −0.885274 0.465069i \(-0.846029\pi\)
−0.885274 + 0.465069i \(0.846029\pi\)
\(420\) 0 0
\(421\) −1.00730 −0.0490929 −0.0245464 0.999699i \(-0.507814\pi\)
−0.0245464 + 0.999699i \(0.507814\pi\)
\(422\) 0 0
\(423\) −2.99348 −0.145548
\(424\) 0 0
\(425\) 5.52837 0.268165
\(426\) 0 0
\(427\) 35.4220 1.71419
\(428\) 0 0
\(429\) 33.3883 1.61200
\(430\) 0 0
\(431\) −28.9295 −1.39348 −0.696742 0.717321i \(-0.745368\pi\)
−0.696742 + 0.717321i \(0.745368\pi\)
\(432\) 0 0
\(433\) 20.0203 0.962113 0.481056 0.876690i \(-0.340253\pi\)
0.481056 + 0.876690i \(0.340253\pi\)
\(434\) 0 0
\(435\) −3.37924 −0.162022
\(436\) 0 0
\(437\) 26.0130 1.24437
\(438\) 0 0
\(439\) 14.6347 0.698475 0.349238 0.937034i \(-0.386441\pi\)
0.349238 + 0.937034i \(0.386441\pi\)
\(440\) 0 0
\(441\) 2.96894 0.141378
\(442\) 0 0
\(443\) 13.0508 0.620060 0.310030 0.950727i \(-0.399661\pi\)
0.310030 + 0.950727i \(0.399661\pi\)
\(444\) 0 0
\(445\) −2.96242 −0.140432
\(446\) 0 0
\(447\) −3.73059 −0.176451
\(448\) 0 0
\(449\) 4.14737 0.195726 0.0978632 0.995200i \(-0.468799\pi\)
0.0978632 + 0.995200i \(0.468799\pi\)
\(450\) 0 0
\(451\) 19.1709 0.902722
\(452\) 0 0
\(453\) −2.21837 −0.104228
\(454\) 0 0
\(455\) −21.7532 −1.01981
\(456\) 0 0
\(457\) −31.8504 −1.48990 −0.744950 0.667121i \(-0.767526\pi\)
−0.744950 + 0.667121i \(0.767526\pi\)
\(458\) 0 0
\(459\) −5.52837 −0.258042
\(460\) 0 0
\(461\) −29.5357 −1.37561 −0.687807 0.725894i \(-0.741426\pi\)
−0.687807 + 0.725894i \(0.741426\pi\)
\(462\) 0 0
\(463\) −41.2328 −1.91625 −0.958126 0.286346i \(-0.907559\pi\)
−0.958126 + 0.286346i \(0.907559\pi\)
\(464\) 0 0
\(465\) −6.57845 −0.305068
\(466\) 0 0
\(467\) 29.0990 1.34654 0.673272 0.739395i \(-0.264889\pi\)
0.673272 + 0.739395i \(0.264889\pi\)
\(468\) 0 0
\(469\) −3.15736 −0.145793
\(470\) 0 0
\(471\) 22.0531 1.01615
\(472\) 0 0
\(473\) 5.88982 0.270814
\(474\) 0 0
\(475\) 5.73402 0.263095
\(476\) 0 0
\(477\) 7.21887 0.330529
\(478\) 0 0
\(479\) −11.0991 −0.507129 −0.253564 0.967319i \(-0.581603\pi\)
−0.253564 + 0.967319i \(0.581603\pi\)
\(480\) 0 0
\(481\) −1.52869 −0.0697022
\(482\) 0 0
\(483\) −14.3237 −0.651752
\(484\) 0 0
\(485\) −5.48656 −0.249132
\(486\) 0 0
\(487\) 9.44172 0.427845 0.213923 0.976851i \(-0.431376\pi\)
0.213923 + 0.976851i \(0.431376\pi\)
\(488\) 0 0
\(489\) 12.5802 0.568895
\(490\) 0 0
\(491\) −33.9696 −1.53302 −0.766512 0.642230i \(-0.778010\pi\)
−0.766512 + 0.642230i \(0.778010\pi\)
\(492\) 0 0
\(493\) −18.6817 −0.841383
\(494\) 0 0
\(495\) 4.84615 0.217818
\(496\) 0 0
\(497\) −21.6551 −0.971365
\(498\) 0 0
\(499\) −32.2275 −1.44270 −0.721351 0.692570i \(-0.756478\pi\)
−0.721351 + 0.692570i \(0.756478\pi\)
\(500\) 0 0
\(501\) −11.2328 −0.501843
\(502\) 0 0
\(503\) −23.6427 −1.05417 −0.527087 0.849811i \(-0.676716\pi\)
−0.527087 + 0.849811i \(0.676716\pi\)
\(504\) 0 0
\(505\) 3.57769 0.159205
\(506\) 0 0
\(507\) −34.4675 −1.53076
\(508\) 0 0
\(509\) −19.9633 −0.884859 −0.442429 0.896803i \(-0.645883\pi\)
−0.442429 + 0.896803i \(0.645883\pi\)
\(510\) 0 0
\(511\) 12.3604 0.546791
\(512\) 0 0
\(513\) −5.73402 −0.253163
\(514\) 0 0
\(515\) −0.784639 −0.0345753
\(516\) 0 0
\(517\) 14.5068 0.638010
\(518\) 0 0
\(519\) 5.79138 0.254213
\(520\) 0 0
\(521\) 29.5165 1.29314 0.646571 0.762854i \(-0.276202\pi\)
0.646571 + 0.762854i \(0.276202\pi\)
\(522\) 0 0
\(523\) 23.6977 1.03623 0.518113 0.855312i \(-0.326635\pi\)
0.518113 + 0.855312i \(0.326635\pi\)
\(524\) 0 0
\(525\) −3.15736 −0.137799
\(526\) 0 0
\(527\) −36.3681 −1.58422
\(528\) 0 0
\(529\) −2.41919 −0.105182
\(530\) 0 0
\(531\) −12.3745 −0.537008
\(532\) 0 0
\(533\) −27.2549 −1.18054
\(534\) 0 0
\(535\) −5.08465 −0.219828
\(536\) 0 0
\(537\) 0.461711 0.0199243
\(538\) 0 0
\(539\) −14.3879 −0.619732
\(540\) 0 0
\(541\) 21.3066 0.916041 0.458020 0.888942i \(-0.348559\pi\)
0.458020 + 0.888942i \(0.348559\pi\)
\(542\) 0 0
\(543\) 23.9076 1.02597
\(544\) 0 0
\(545\) 1.79230 0.0767737
\(546\) 0 0
\(547\) −23.5676 −1.00768 −0.503839 0.863798i \(-0.668079\pi\)
−0.503839 + 0.863798i \(0.668079\pi\)
\(548\) 0 0
\(549\) 11.2189 0.478810
\(550\) 0 0
\(551\) −19.3767 −0.825474
\(552\) 0 0
\(553\) 24.7471 1.05236
\(554\) 0 0
\(555\) −0.221881 −0.00941834
\(556\) 0 0
\(557\) −13.4238 −0.568783 −0.284392 0.958708i \(-0.591792\pi\)
−0.284392 + 0.958708i \(0.591792\pi\)
\(558\) 0 0
\(559\) −8.37344 −0.354159
\(560\) 0 0
\(561\) 26.7913 1.13113
\(562\) 0 0
\(563\) −16.5293 −0.696628 −0.348314 0.937378i \(-0.613246\pi\)
−0.348314 + 0.937378i \(0.613246\pi\)
\(564\) 0 0
\(565\) 5.45562 0.229520
\(566\) 0 0
\(567\) 3.15736 0.132597
\(568\) 0 0
\(569\) 8.97352 0.376190 0.188095 0.982151i \(-0.439769\pi\)
0.188095 + 0.982151i \(0.439769\pi\)
\(570\) 0 0
\(571\) −31.2167 −1.30638 −0.653189 0.757194i \(-0.726569\pi\)
−0.653189 + 0.757194i \(0.726569\pi\)
\(572\) 0 0
\(573\) −11.9649 −0.499841
\(574\) 0 0
\(575\) 4.53661 0.189190
\(576\) 0 0
\(577\) −19.5281 −0.812965 −0.406482 0.913659i \(-0.633245\pi\)
−0.406482 + 0.913659i \(0.633245\pi\)
\(578\) 0 0
\(579\) −3.67742 −0.152828
\(580\) 0 0
\(581\) 40.2531 1.66998
\(582\) 0 0
\(583\) −34.9837 −1.44888
\(584\) 0 0
\(585\) −6.88967 −0.284853
\(586\) 0 0
\(587\) −37.3779 −1.54275 −0.771375 0.636381i \(-0.780431\pi\)
−0.771375 + 0.636381i \(0.780431\pi\)
\(588\) 0 0
\(589\) −37.7210 −1.55427
\(590\) 0 0
\(591\) −17.9063 −0.736568
\(592\) 0 0
\(593\) −18.7885 −0.771550 −0.385775 0.922593i \(-0.626066\pi\)
−0.385775 + 0.922593i \(0.626066\pi\)
\(594\) 0 0
\(595\) −17.4551 −0.715589
\(596\) 0 0
\(597\) 12.8328 0.525214
\(598\) 0 0
\(599\) −19.7557 −0.807197 −0.403599 0.914936i \(-0.632241\pi\)
−0.403599 + 0.914936i \(0.632241\pi\)
\(600\) 0 0
\(601\) 41.2216 1.68146 0.840732 0.541452i \(-0.182125\pi\)
0.840732 + 0.541452i \(0.182125\pi\)
\(602\) 0 0
\(603\) −1.00000 −0.0407231
\(604\) 0 0
\(605\) −12.4851 −0.507593
\(606\) 0 0
\(607\) 46.8922 1.90330 0.951648 0.307190i \(-0.0993886\pi\)
0.951648 + 0.307190i \(0.0993886\pi\)
\(608\) 0 0
\(609\) 10.6695 0.432350
\(610\) 0 0
\(611\) −20.6241 −0.834361
\(612\) 0 0
\(613\) −3.19718 −0.129133 −0.0645664 0.997913i \(-0.520566\pi\)
−0.0645664 + 0.997913i \(0.520566\pi\)
\(614\) 0 0
\(615\) −3.95590 −0.159517
\(616\) 0 0
\(617\) −45.8138 −1.84440 −0.922198 0.386718i \(-0.873609\pi\)
−0.922198 + 0.386718i \(0.873609\pi\)
\(618\) 0 0
\(619\) 44.0330 1.76984 0.884918 0.465747i \(-0.154214\pi\)
0.884918 + 0.465747i \(0.154214\pi\)
\(620\) 0 0
\(621\) −4.53661 −0.182048
\(622\) 0 0
\(623\) 9.35345 0.374738
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 27.7879 1.10974
\(628\) 0 0
\(629\) −1.22664 −0.0489095
\(630\) 0 0
\(631\) −37.4041 −1.48903 −0.744516 0.667604i \(-0.767320\pi\)
−0.744516 + 0.667604i \(0.767320\pi\)
\(632\) 0 0
\(633\) −5.45959 −0.216999
\(634\) 0 0
\(635\) −13.1240 −0.520811
\(636\) 0 0
\(637\) 20.4550 0.810458
\(638\) 0 0
\(639\) −6.85861 −0.271323
\(640\) 0 0
\(641\) 10.9524 0.432593 0.216297 0.976328i \(-0.430602\pi\)
0.216297 + 0.976328i \(0.430602\pi\)
\(642\) 0 0
\(643\) −40.5462 −1.59899 −0.799493 0.600675i \(-0.794899\pi\)
−0.799493 + 0.600675i \(0.794899\pi\)
\(644\) 0 0
\(645\) −1.21536 −0.0478548
\(646\) 0 0
\(647\) −11.6696 −0.458781 −0.229390 0.973334i \(-0.573673\pi\)
−0.229390 + 0.973334i \(0.573673\pi\)
\(648\) 0 0
\(649\) 59.9687 2.35398
\(650\) 0 0
\(651\) 20.7706 0.814063
\(652\) 0 0
\(653\) 12.3282 0.482441 0.241221 0.970470i \(-0.422452\pi\)
0.241221 + 0.970470i \(0.422452\pi\)
\(654\) 0 0
\(655\) −4.58250 −0.179053
\(656\) 0 0
\(657\) 3.91478 0.152730
\(658\) 0 0
\(659\) −9.47250 −0.368996 −0.184498 0.982833i \(-0.559066\pi\)
−0.184498 + 0.982833i \(0.559066\pi\)
\(660\) 0 0
\(661\) −19.8017 −0.770198 −0.385099 0.922875i \(-0.625833\pi\)
−0.385099 + 0.922875i \(0.625833\pi\)
\(662\) 0 0
\(663\) −38.0887 −1.47924
\(664\) 0 0
\(665\) −18.1044 −0.702058
\(666\) 0 0
\(667\) −15.3303 −0.593592
\(668\) 0 0
\(669\) −25.2071 −0.974561
\(670\) 0 0
\(671\) −54.3683 −2.09886
\(672\) 0 0
\(673\) 9.90899 0.381964 0.190982 0.981594i \(-0.438833\pi\)
0.190982 + 0.981594i \(0.438833\pi\)
\(674\) 0 0
\(675\) −1.00000 −0.0384900
\(676\) 0 0
\(677\) 44.3704 1.70529 0.852647 0.522487i \(-0.174996\pi\)
0.852647 + 0.522487i \(0.174996\pi\)
\(678\) 0 0
\(679\) 17.3231 0.664799
\(680\) 0 0
\(681\) 9.46282 0.362616
\(682\) 0 0
\(683\) −6.09371 −0.233169 −0.116585 0.993181i \(-0.537195\pi\)
−0.116585 + 0.993181i \(0.537195\pi\)
\(684\) 0 0
\(685\) −11.8684 −0.453468
\(686\) 0 0
\(687\) −1.16836 −0.0445759
\(688\) 0 0
\(689\) 49.7356 1.89478
\(690\) 0 0
\(691\) −2.18330 −0.0830566 −0.0415283 0.999137i \(-0.513223\pi\)
−0.0415283 + 0.999137i \(0.513223\pi\)
\(692\) 0 0
\(693\) −15.3010 −0.581239
\(694\) 0 0
\(695\) 10.5254 0.399250
\(696\) 0 0
\(697\) −21.8697 −0.828374
\(698\) 0 0
\(699\) −7.95443 −0.300864
\(700\) 0 0
\(701\) 24.7026 0.933006 0.466503 0.884520i \(-0.345514\pi\)
0.466503 + 0.884520i \(0.345514\pi\)
\(702\) 0 0
\(703\) −1.27227 −0.0479847
\(704\) 0 0
\(705\) −2.99348 −0.112741
\(706\) 0 0
\(707\) −11.2961 −0.424833
\(708\) 0 0
\(709\) −14.0093 −0.526130 −0.263065 0.964778i \(-0.584733\pi\)
−0.263065 + 0.964778i \(0.584733\pi\)
\(710\) 0 0
\(711\) 7.83791 0.293945
\(712\) 0 0
\(713\) −29.8439 −1.11766
\(714\) 0 0
\(715\) 33.3883 1.24865
\(716\) 0 0
\(717\) −21.6435 −0.808291
\(718\) 0 0
\(719\) −25.3733 −0.946263 −0.473131 0.880992i \(-0.656876\pi\)
−0.473131 + 0.880992i \(0.656876\pi\)
\(720\) 0 0
\(721\) 2.47739 0.0922628
\(722\) 0 0
\(723\) −25.3117 −0.941353
\(724\) 0 0
\(725\) −3.37924 −0.125502
\(726\) 0 0
\(727\) 39.7313 1.47355 0.736776 0.676137i \(-0.236347\pi\)
0.736776 + 0.676137i \(0.236347\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −6.71897 −0.248510
\(732\) 0 0
\(733\) 2.65713 0.0981432 0.0490716 0.998795i \(-0.484374\pi\)
0.0490716 + 0.998795i \(0.484374\pi\)
\(734\) 0 0
\(735\) 2.96894 0.109511
\(736\) 0 0
\(737\) 4.84615 0.178510
\(738\) 0 0
\(739\) −28.5553 −1.05042 −0.525212 0.850971i \(-0.676014\pi\)
−0.525212 + 0.850971i \(0.676014\pi\)
\(740\) 0 0
\(741\) −39.5055 −1.45127
\(742\) 0 0
\(743\) 34.3117 1.25877 0.629387 0.777092i \(-0.283306\pi\)
0.629387 + 0.777092i \(0.283306\pi\)
\(744\) 0 0
\(745\) −3.73059 −0.136678
\(746\) 0 0
\(747\) 12.7490 0.466460
\(748\) 0 0
\(749\) 16.0541 0.586603
\(750\) 0 0
\(751\) −48.6656 −1.77583 −0.887917 0.460003i \(-0.847848\pi\)
−0.887917 + 0.460003i \(0.847848\pi\)
\(752\) 0 0
\(753\) −15.4411 −0.562705
\(754\) 0 0
\(755\) −2.21837 −0.0807348
\(756\) 0 0
\(757\) 37.7595 1.37239 0.686197 0.727416i \(-0.259279\pi\)
0.686197 + 0.727416i \(0.259279\pi\)
\(758\) 0 0
\(759\) 21.9851 0.798007
\(760\) 0 0
\(761\) 52.6281 1.90777 0.953883 0.300178i \(-0.0970460\pi\)
0.953883 + 0.300178i \(0.0970460\pi\)
\(762\) 0 0
\(763\) −5.65894 −0.204867
\(764\) 0 0
\(765\) −5.52837 −0.199879
\(766\) 0 0
\(767\) −85.2563 −3.07843
\(768\) 0 0
\(769\) −23.4558 −0.845838 −0.422919 0.906168i \(-0.638995\pi\)
−0.422919 + 0.906168i \(0.638995\pi\)
\(770\) 0 0
\(771\) −9.95691 −0.358589
\(772\) 0 0
\(773\) 37.6368 1.35370 0.676851 0.736120i \(-0.263344\pi\)
0.676851 + 0.736120i \(0.263344\pi\)
\(774\) 0 0
\(775\) −6.57845 −0.236305
\(776\) 0 0
\(777\) 0.700560 0.0251325
\(778\) 0 0
\(779\) −22.6832 −0.812711
\(780\) 0 0
\(781\) 33.2378 1.18934
\(782\) 0 0
\(783\) 3.37924 0.120764
\(784\) 0 0
\(785\) 22.0531 0.787110
\(786\) 0 0
\(787\) 20.6751 0.736989 0.368494 0.929630i \(-0.379873\pi\)
0.368494 + 0.929630i \(0.379873\pi\)
\(788\) 0 0
\(789\) 13.4957 0.480459
\(790\) 0 0
\(791\) −17.2254 −0.612464
\(792\) 0 0
\(793\) 77.2943 2.74480
\(794\) 0 0
\(795\) 7.21887 0.256027
\(796\) 0 0
\(797\) 43.3374 1.53509 0.767546 0.640994i \(-0.221478\pi\)
0.767546 + 0.640994i \(0.221478\pi\)
\(798\) 0 0
\(799\) −16.5491 −0.585464
\(800\) 0 0
\(801\) 2.96242 0.104672
\(802\) 0 0
\(803\) −18.9716 −0.669493
\(804\) 0 0
\(805\) −14.3237 −0.504845
\(806\) 0 0
\(807\) 22.4578 0.790553
\(808\) 0 0
\(809\) 25.2355 0.887234 0.443617 0.896217i \(-0.353695\pi\)
0.443617 + 0.896217i \(0.353695\pi\)
\(810\) 0 0
\(811\) −24.7789 −0.870104 −0.435052 0.900405i \(-0.643270\pi\)
−0.435052 + 0.900405i \(0.643270\pi\)
\(812\) 0 0
\(813\) 16.8705 0.591676
\(814\) 0 0
\(815\) 12.5802 0.440664
\(816\) 0 0
\(817\) −6.96891 −0.243811
\(818\) 0 0
\(819\) 21.7532 0.760118
\(820\) 0 0
\(821\) 40.2166 1.40357 0.701785 0.712389i \(-0.252387\pi\)
0.701785 + 0.712389i \(0.252387\pi\)
\(822\) 0 0
\(823\) −19.9065 −0.693897 −0.346949 0.937884i \(-0.612782\pi\)
−0.346949 + 0.937884i \(0.612782\pi\)
\(824\) 0 0
\(825\) 4.84615 0.168721
\(826\) 0 0
\(827\) −35.8464 −1.24650 −0.623250 0.782023i \(-0.714188\pi\)
−0.623250 + 0.782023i \(0.714188\pi\)
\(828\) 0 0
\(829\) −22.2844 −0.773970 −0.386985 0.922086i \(-0.626484\pi\)
−0.386985 + 0.922086i \(0.626484\pi\)
\(830\) 0 0
\(831\) −28.6173 −0.992723
\(832\) 0 0
\(833\) 16.4134 0.568691
\(834\) 0 0
\(835\) −11.2328 −0.388726
\(836\) 0 0
\(837\) 6.57845 0.227385
\(838\) 0 0
\(839\) −8.05570 −0.278114 −0.139057 0.990284i \(-0.544407\pi\)
−0.139057 + 0.990284i \(0.544407\pi\)
\(840\) 0 0
\(841\) −17.5807 −0.606231
\(842\) 0 0
\(843\) −11.2629 −0.387914
\(844\) 0 0
\(845\) −34.4675 −1.18572
\(846\) 0 0
\(847\) 39.4201 1.35449
\(848\) 0 0
\(849\) 30.1828 1.03587
\(850\) 0 0
\(851\) −1.00659 −0.0345054
\(852\) 0 0
\(853\) −27.6279 −0.945960 −0.472980 0.881073i \(-0.656822\pi\)
−0.472980 + 0.881073i \(0.656822\pi\)
\(854\) 0 0
\(855\) −5.73402 −0.196099
\(856\) 0 0
\(857\) 16.8455 0.575430 0.287715 0.957716i \(-0.407104\pi\)
0.287715 + 0.957716i \(0.407104\pi\)
\(858\) 0 0
\(859\) 49.6982 1.69568 0.847840 0.530253i \(-0.177903\pi\)
0.847840 + 0.530253i \(0.177903\pi\)
\(860\) 0 0
\(861\) 12.4902 0.425666
\(862\) 0 0
\(863\) 31.5659 1.07452 0.537258 0.843418i \(-0.319460\pi\)
0.537258 + 0.843418i \(0.319460\pi\)
\(864\) 0 0
\(865\) 5.79138 0.196913
\(866\) 0 0
\(867\) −13.5629 −0.460620
\(868\) 0 0
\(869\) −37.9837 −1.28851
\(870\) 0 0
\(871\) −6.88967 −0.233448
\(872\) 0 0
\(873\) 5.48656 0.185692
\(874\) 0 0
\(875\) −3.15736 −0.106738
\(876\) 0 0
\(877\) −2.27991 −0.0769871 −0.0384935 0.999259i \(-0.512256\pi\)
−0.0384935 + 0.999259i \(0.512256\pi\)
\(878\) 0 0
\(879\) −10.1216 −0.341395
\(880\) 0 0
\(881\) −19.2139 −0.647331 −0.323666 0.946172i \(-0.604915\pi\)
−0.323666 + 0.946172i \(0.604915\pi\)
\(882\) 0 0
\(883\) 42.8453 1.44186 0.720929 0.693009i \(-0.243715\pi\)
0.720929 + 0.693009i \(0.243715\pi\)
\(884\) 0 0
\(885\) −12.3745 −0.415965
\(886\) 0 0
\(887\) 16.9128 0.567875 0.283938 0.958843i \(-0.408359\pi\)
0.283938 + 0.958843i \(0.408359\pi\)
\(888\) 0 0
\(889\) 41.4373 1.38976
\(890\) 0 0
\(891\) −4.84615 −0.162352
\(892\) 0 0
\(893\) −17.1647 −0.574394
\(894\) 0 0
\(895\) 0.461711 0.0154333
\(896\) 0 0
\(897\) −31.2557 −1.04360
\(898\) 0 0
\(899\) 22.2302 0.741419
\(900\) 0 0
\(901\) 39.9086 1.32955
\(902\) 0 0
\(903\) 3.83734 0.127699
\(904\) 0 0
\(905\) 23.9076 0.794716
\(906\) 0 0
\(907\) 28.7377 0.954221 0.477110 0.878843i \(-0.341684\pi\)
0.477110 + 0.878843i \(0.341684\pi\)
\(908\) 0 0
\(909\) −3.57769 −0.118665
\(910\) 0 0
\(911\) −39.6087 −1.31230 −0.656148 0.754632i \(-0.727815\pi\)
−0.656148 + 0.754632i \(0.727815\pi\)
\(912\) 0 0
\(913\) −61.7833 −2.04473
\(914\) 0 0
\(915\) 11.2189 0.370884
\(916\) 0 0
\(917\) 14.4686 0.477796
\(918\) 0 0
\(919\) −16.0473 −0.529350 −0.264675 0.964338i \(-0.585265\pi\)
−0.264675 + 0.964338i \(0.585265\pi\)
\(920\) 0 0
\(921\) −6.50547 −0.214363
\(922\) 0 0
\(923\) −47.2536 −1.55537
\(924\) 0 0
\(925\) −0.221881 −0.00729542
\(926\) 0 0
\(927\) 0.784639 0.0257709
\(928\) 0 0
\(929\) −17.1604 −0.563016 −0.281508 0.959559i \(-0.590835\pi\)
−0.281508 + 0.959559i \(0.590835\pi\)
\(930\) 0 0
\(931\) 17.0240 0.557938
\(932\) 0 0
\(933\) −27.5371 −0.901524
\(934\) 0 0
\(935\) 26.7913 0.876169
\(936\) 0 0
\(937\) −19.3225 −0.631239 −0.315619 0.948886i \(-0.602212\pi\)
−0.315619 + 0.948886i \(0.602212\pi\)
\(938\) 0 0
\(939\) −11.4107 −0.372374
\(940\) 0 0
\(941\) −5.07888 −0.165567 −0.0827834 0.996568i \(-0.526381\pi\)
−0.0827834 + 0.996568i \(0.526381\pi\)
\(942\) 0 0
\(943\) −17.9464 −0.584415
\(944\) 0 0
\(945\) 3.15736 0.102709
\(946\) 0 0
\(947\) 35.4433 1.15175 0.575876 0.817537i \(-0.304661\pi\)
0.575876 + 0.817537i \(0.304661\pi\)
\(948\) 0 0
\(949\) 26.9715 0.875533
\(950\) 0 0
\(951\) 20.7200 0.671891
\(952\) 0 0
\(953\) −29.4982 −0.955539 −0.477770 0.878485i \(-0.658555\pi\)
−0.477770 + 0.878485i \(0.658555\pi\)
\(954\) 0 0
\(955\) −11.9649 −0.387175
\(956\) 0 0
\(957\) −16.3763 −0.529371
\(958\) 0 0
\(959\) 37.4728 1.21006
\(960\) 0 0
\(961\) 12.2760 0.396001
\(962\) 0 0
\(963\) 5.08465 0.163850
\(964\) 0 0
\(965\) −3.67742 −0.118380
\(966\) 0 0
\(967\) 50.1741 1.61349 0.806745 0.590900i \(-0.201227\pi\)
0.806745 + 0.590900i \(0.201227\pi\)
\(968\) 0 0
\(969\) −31.6998 −1.01834
\(970\) 0 0
\(971\) −29.6964 −0.953002 −0.476501 0.879174i \(-0.658095\pi\)
−0.476501 + 0.879174i \(0.658095\pi\)
\(972\) 0 0
\(973\) −33.2324 −1.06538
\(974\) 0 0
\(975\) −6.88967 −0.220646
\(976\) 0 0
\(977\) −37.5776 −1.20221 −0.601106 0.799169i \(-0.705273\pi\)
−0.601106 + 0.799169i \(0.705273\pi\)
\(978\) 0 0
\(979\) −14.3563 −0.458830
\(980\) 0 0
\(981\) −1.79230 −0.0572237
\(982\) 0 0
\(983\) 32.7906 1.04586 0.522928 0.852377i \(-0.324840\pi\)
0.522928 + 0.852377i \(0.324840\pi\)
\(984\) 0 0
\(985\) −17.9063 −0.570543
\(986\) 0 0
\(987\) 9.45150 0.300845
\(988\) 0 0
\(989\) −5.51362 −0.175323
\(990\) 0 0
\(991\) 41.5492 1.31986 0.659928 0.751329i \(-0.270587\pi\)
0.659928 + 0.751329i \(0.270587\pi\)
\(992\) 0 0
\(993\) −13.3874 −0.424836
\(994\) 0 0
\(995\) 12.8328 0.406829
\(996\) 0 0
\(997\) −32.2867 −1.02253 −0.511265 0.859423i \(-0.670823\pi\)
−0.511265 + 0.859423i \(0.670823\pi\)
\(998\) 0 0
\(999\) 0.221881 0.00702002
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 4020.2.a.h.1.5 7
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4020.2.a.h.1.5 7 1.1 even 1 trivial