Properties

Label 4020.2.a.g.1.5
Level $4020$
Weight $2$
Character 4020.1
Self dual yes
Analytic conductor $32.100$
Analytic rank $0$
Dimension $6$
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: \(6\)
Coefficient field: 6.6.195727752.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 15x^{4} - 25x^{3} - 3x^{2} + 9x + 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.5
Root \(-2.33297\) 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)
 
\(f(q)\) \(=\) \(q+1.00000 q^{3} -1.00000 q^{5} +3.15496 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{3} -1.00000 q^{5} +3.15496 q^{7} +1.00000 q^{9} +2.98752 q^{11} +0.285913 q^{13} -1.00000 q^{15} +3.60830 q^{17} -0.827332 q^{19} +3.15496 q^{21} +8.54034 q^{23} +1.00000 q^{25} +1.00000 q^{27} -7.01509 q^{29} -5.11929 q^{31} +2.98752 q^{33} -3.15496 q^{35} +7.86013 q^{37} +0.285913 q^{39} -0.299088 q^{41} -0.325033 q^{43} -1.00000 q^{45} +6.94740 q^{47} +2.95375 q^{49} +3.60830 q^{51} -8.00050 q^{53} -2.98752 q^{55} -0.827332 q^{57} -7.59664 q^{59} -7.05522 q^{61} +3.15496 q^{63} -0.285913 q^{65} +1.00000 q^{67} +8.54034 q^{69} +14.9890 q^{71} -5.66984 q^{73} +1.00000 q^{75} +9.42551 q^{77} -1.40186 q^{79} +1.00000 q^{81} +4.16743 q^{83} -3.60830 q^{85} -7.01509 q^{87} +4.37544 q^{89} +0.902043 q^{91} -5.11929 q^{93} +0.827332 q^{95} +3.52023 q^{97} +2.98752 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 6 q^{3} - 6 q^{5} + 3 q^{7} + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 6 q^{3} - 6 q^{5} + 3 q^{7} + 6 q^{9} + 3 q^{11} + 3 q^{13} - 6 q^{15} + 6 q^{17} - 3 q^{19} + 3 q^{21} + 6 q^{23} + 6 q^{25} + 6 q^{27} + 21 q^{29} - 3 q^{31} + 3 q^{33} - 3 q^{35} + 3 q^{39} + 9 q^{41} - 3 q^{43} - 6 q^{45} + 21 q^{47} + 3 q^{49} + 6 q^{51} + 15 q^{53} - 3 q^{55} - 3 q^{57} + 15 q^{59} + 15 q^{61} + 3 q^{63} - 3 q^{65} + 6 q^{67} + 6 q^{69} + 18 q^{71} - 6 q^{73} + 6 q^{75} + 33 q^{77} + 9 q^{79} + 6 q^{81} + 24 q^{83} - 6 q^{85} + 21 q^{87} + 24 q^{89} - 21 q^{91} - 3 q^{93} + 3 q^{95} + 9 q^{97} + 3 q^{99}+O(q^{100}) \) 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
\(7\) 3.15496 1.19246 0.596231 0.802813i \(-0.296664\pi\)
0.596231 + 0.802813i \(0.296664\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 2.98752 0.900772 0.450386 0.892834i \(-0.351286\pi\)
0.450386 + 0.892834i \(0.351286\pi\)
\(12\) 0 0
\(13\) 0.285913 0.0792980 0.0396490 0.999214i \(-0.487376\pi\)
0.0396490 + 0.999214i \(0.487376\pi\)
\(14\) 0 0
\(15\) −1.00000 −0.258199
\(16\) 0 0
\(17\) 3.60830 0.875142 0.437571 0.899184i \(-0.355839\pi\)
0.437571 + 0.899184i \(0.355839\pi\)
\(18\) 0 0
\(19\) −0.827332 −0.189803 −0.0949014 0.995487i \(-0.530254\pi\)
−0.0949014 + 0.995487i \(0.530254\pi\)
\(20\) 0 0
\(21\) 3.15496 0.688468
\(22\) 0 0
\(23\) 8.54034 1.78078 0.890392 0.455195i \(-0.150431\pi\)
0.890392 + 0.455195i \(0.150431\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) −7.01509 −1.30267 −0.651335 0.758790i \(-0.725791\pi\)
−0.651335 + 0.758790i \(0.725791\pi\)
\(30\) 0 0
\(31\) −5.11929 −0.919452 −0.459726 0.888061i \(-0.652052\pi\)
−0.459726 + 0.888061i \(0.652052\pi\)
\(32\) 0 0
\(33\) 2.98752 0.520061
\(34\) 0 0
\(35\) −3.15496 −0.533285
\(36\) 0 0
\(37\) 7.86013 1.29220 0.646099 0.763254i \(-0.276399\pi\)
0.646099 + 0.763254i \(0.276399\pi\)
\(38\) 0 0
\(39\) 0.285913 0.0457827
\(40\) 0 0
\(41\) −0.299088 −0.0467097 −0.0233549 0.999727i \(-0.507435\pi\)
−0.0233549 + 0.999727i \(0.507435\pi\)
\(42\) 0 0
\(43\) −0.325033 −0.0495671 −0.0247835 0.999693i \(-0.507890\pi\)
−0.0247835 + 0.999693i \(0.507890\pi\)
\(44\) 0 0
\(45\) −1.00000 −0.149071
\(46\) 0 0
\(47\) 6.94740 1.01338 0.506691 0.862128i \(-0.330869\pi\)
0.506691 + 0.862128i \(0.330869\pi\)
\(48\) 0 0
\(49\) 2.95375 0.421964
\(50\) 0 0
\(51\) 3.60830 0.505263
\(52\) 0 0
\(53\) −8.00050 −1.09895 −0.549477 0.835509i \(-0.685173\pi\)
−0.549477 + 0.835509i \(0.685173\pi\)
\(54\) 0 0
\(55\) −2.98752 −0.402838
\(56\) 0 0
\(57\) −0.827332 −0.109583
\(58\) 0 0
\(59\) −7.59664 −0.988998 −0.494499 0.869178i \(-0.664648\pi\)
−0.494499 + 0.869178i \(0.664648\pi\)
\(60\) 0 0
\(61\) −7.05522 −0.903328 −0.451664 0.892188i \(-0.649169\pi\)
−0.451664 + 0.892188i \(0.649169\pi\)
\(62\) 0 0
\(63\) 3.15496 0.397487
\(64\) 0 0
\(65\) −0.285913 −0.0354631
\(66\) 0 0
\(67\) 1.00000 0.122169
\(68\) 0 0
\(69\) 8.54034 1.02814
\(70\) 0 0
\(71\) 14.9890 1.77887 0.889436 0.457061i \(-0.151098\pi\)
0.889436 + 0.457061i \(0.151098\pi\)
\(72\) 0 0
\(73\) −5.66984 −0.663604 −0.331802 0.943349i \(-0.607657\pi\)
−0.331802 + 0.943349i \(0.607657\pi\)
\(74\) 0 0
\(75\) 1.00000 0.115470
\(76\) 0 0
\(77\) 9.42551 1.07414
\(78\) 0 0
\(79\) −1.40186 −0.157722 −0.0788609 0.996886i \(-0.525128\pi\)
−0.0788609 + 0.996886i \(0.525128\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 4.16743 0.457435 0.228718 0.973493i \(-0.426547\pi\)
0.228718 + 0.973493i \(0.426547\pi\)
\(84\) 0 0
\(85\) −3.60830 −0.391375
\(86\) 0 0
\(87\) −7.01509 −0.752097
\(88\) 0 0
\(89\) 4.37544 0.463796 0.231898 0.972740i \(-0.425506\pi\)
0.231898 + 0.972740i \(0.425506\pi\)
\(90\) 0 0
\(91\) 0.902043 0.0945598
\(92\) 0 0
\(93\) −5.11929 −0.530846
\(94\) 0 0
\(95\) 0.827332 0.0848824
\(96\) 0 0
\(97\) 3.52023 0.357425 0.178713 0.983901i \(-0.442807\pi\)
0.178713 + 0.983901i \(0.442807\pi\)
\(98\) 0 0
\(99\) 2.98752 0.300257
\(100\) 0 0
\(101\) −8.93919 −0.889482 −0.444741 0.895659i \(-0.646704\pi\)
−0.444741 + 0.895659i \(0.646704\pi\)
\(102\) 0 0
\(103\) 11.5575 1.13880 0.569398 0.822062i \(-0.307176\pi\)
0.569398 + 0.822062i \(0.307176\pi\)
\(104\) 0 0
\(105\) −3.15496 −0.307892
\(106\) 0 0
\(107\) 0.378522 0.0365931 0.0182966 0.999833i \(-0.494176\pi\)
0.0182966 + 0.999833i \(0.494176\pi\)
\(108\) 0 0
\(109\) 4.16387 0.398827 0.199413 0.979915i \(-0.436096\pi\)
0.199413 + 0.979915i \(0.436096\pi\)
\(110\) 0 0
\(111\) 7.86013 0.746051
\(112\) 0 0
\(113\) −3.08719 −0.290418 −0.145209 0.989401i \(-0.546385\pi\)
−0.145209 + 0.989401i \(0.546385\pi\)
\(114\) 0 0
\(115\) −8.54034 −0.796391
\(116\) 0 0
\(117\) 0.285913 0.0264327
\(118\) 0 0
\(119\) 11.3840 1.04357
\(120\) 0 0
\(121\) −2.07470 −0.188609
\(122\) 0 0
\(123\) −0.299088 −0.0269679
\(124\) 0 0
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) −5.16100 −0.457965 −0.228983 0.973431i \(-0.573540\pi\)
−0.228983 + 0.973431i \(0.573540\pi\)
\(128\) 0 0
\(129\) −0.325033 −0.0286176
\(130\) 0 0
\(131\) 20.2863 1.77243 0.886213 0.463278i \(-0.153327\pi\)
0.886213 + 0.463278i \(0.153327\pi\)
\(132\) 0 0
\(133\) −2.61020 −0.226333
\(134\) 0 0
\(135\) −1.00000 −0.0860663
\(136\) 0 0
\(137\) 3.78286 0.323192 0.161596 0.986857i \(-0.448336\pi\)
0.161596 + 0.986857i \(0.448336\pi\)
\(138\) 0 0
\(139\) −2.25651 −0.191395 −0.0956975 0.995410i \(-0.530508\pi\)
−0.0956975 + 0.995410i \(0.530508\pi\)
\(140\) 0 0
\(141\) 6.94740 0.585076
\(142\) 0 0
\(143\) 0.854172 0.0714294
\(144\) 0 0
\(145\) 7.01509 0.582572
\(146\) 0 0
\(147\) 2.95375 0.243621
\(148\) 0 0
\(149\) 10.5973 0.868163 0.434082 0.900874i \(-0.357073\pi\)
0.434082 + 0.900874i \(0.357073\pi\)
\(150\) 0 0
\(151\) −8.20194 −0.667465 −0.333732 0.942668i \(-0.608308\pi\)
−0.333732 + 0.942668i \(0.608308\pi\)
\(152\) 0 0
\(153\) 3.60830 0.291714
\(154\) 0 0
\(155\) 5.11929 0.411191
\(156\) 0 0
\(157\) −20.3258 −1.62218 −0.811088 0.584924i \(-0.801124\pi\)
−0.811088 + 0.584924i \(0.801124\pi\)
\(158\) 0 0
\(159\) −8.00050 −0.634481
\(160\) 0 0
\(161\) 26.9444 2.12352
\(162\) 0 0
\(163\) −10.7073 −0.838660 −0.419330 0.907834i \(-0.637735\pi\)
−0.419330 + 0.907834i \(0.637735\pi\)
\(164\) 0 0
\(165\) −2.98752 −0.232578
\(166\) 0 0
\(167\) 23.8494 1.84552 0.922760 0.385376i \(-0.125928\pi\)
0.922760 + 0.385376i \(0.125928\pi\)
\(168\) 0 0
\(169\) −12.9183 −0.993712
\(170\) 0 0
\(171\) −0.827332 −0.0632676
\(172\) 0 0
\(173\) 18.7835 1.42808 0.714042 0.700103i \(-0.246863\pi\)
0.714042 + 0.700103i \(0.246863\pi\)
\(174\) 0 0
\(175\) 3.15496 0.238492
\(176\) 0 0
\(177\) −7.59664 −0.570998
\(178\) 0 0
\(179\) −2.89439 −0.216337 −0.108168 0.994133i \(-0.534499\pi\)
−0.108168 + 0.994133i \(0.534499\pi\)
\(180\) 0 0
\(181\) 7.61165 0.565769 0.282885 0.959154i \(-0.408709\pi\)
0.282885 + 0.959154i \(0.408709\pi\)
\(182\) 0 0
\(183\) −7.05522 −0.521537
\(184\) 0 0
\(185\) −7.86013 −0.577889
\(186\) 0 0
\(187\) 10.7799 0.788304
\(188\) 0 0
\(189\) 3.15496 0.229489
\(190\) 0 0
\(191\) 3.14590 0.227629 0.113815 0.993502i \(-0.463693\pi\)
0.113815 + 0.993502i \(0.463693\pi\)
\(192\) 0 0
\(193\) −9.62036 −0.692489 −0.346244 0.938144i \(-0.612543\pi\)
−0.346244 + 0.938144i \(0.612543\pi\)
\(194\) 0 0
\(195\) −0.285913 −0.0204747
\(196\) 0 0
\(197\) −20.2285 −1.44122 −0.720612 0.693339i \(-0.756139\pi\)
−0.720612 + 0.693339i \(0.756139\pi\)
\(198\) 0 0
\(199\) 3.64854 0.258638 0.129319 0.991603i \(-0.458721\pi\)
0.129319 + 0.991603i \(0.458721\pi\)
\(200\) 0 0
\(201\) 1.00000 0.0705346
\(202\) 0 0
\(203\) −22.1323 −1.55338
\(204\) 0 0
\(205\) 0.299088 0.0208892
\(206\) 0 0
\(207\) 8.54034 0.593594
\(208\) 0 0
\(209\) −2.47167 −0.170969
\(210\) 0 0
\(211\) −7.82327 −0.538576 −0.269288 0.963060i \(-0.586788\pi\)
−0.269288 + 0.963060i \(0.586788\pi\)
\(212\) 0 0
\(213\) 14.9890 1.02703
\(214\) 0 0
\(215\) 0.325033 0.0221671
\(216\) 0 0
\(217\) −16.1511 −1.09641
\(218\) 0 0
\(219\) −5.66984 −0.383132
\(220\) 0 0
\(221\) 1.03166 0.0693970
\(222\) 0 0
\(223\) 2.55347 0.170993 0.0854966 0.996338i \(-0.472752\pi\)
0.0854966 + 0.996338i \(0.472752\pi\)
\(224\) 0 0
\(225\) 1.00000 0.0666667
\(226\) 0 0
\(227\) 20.2142 1.34167 0.670833 0.741608i \(-0.265937\pi\)
0.670833 + 0.741608i \(0.265937\pi\)
\(228\) 0 0
\(229\) 3.86312 0.255282 0.127641 0.991820i \(-0.459259\pi\)
0.127641 + 0.991820i \(0.459259\pi\)
\(230\) 0 0
\(231\) 9.42551 0.620153
\(232\) 0 0
\(233\) 2.98051 0.195260 0.0976298 0.995223i \(-0.468874\pi\)
0.0976298 + 0.995223i \(0.468874\pi\)
\(234\) 0 0
\(235\) −6.94740 −0.453198
\(236\) 0 0
\(237\) −1.40186 −0.0910607
\(238\) 0 0
\(239\) 7.31022 0.472859 0.236429 0.971649i \(-0.424023\pi\)
0.236429 + 0.971649i \(0.424023\pi\)
\(240\) 0 0
\(241\) 25.4246 1.63774 0.818871 0.573978i \(-0.194600\pi\)
0.818871 + 0.573978i \(0.194600\pi\)
\(242\) 0 0
\(243\) 1.00000 0.0641500
\(244\) 0 0
\(245\) −2.95375 −0.188708
\(246\) 0 0
\(247\) −0.236545 −0.0150510
\(248\) 0 0
\(249\) 4.16743 0.264100
\(250\) 0 0
\(251\) −29.6780 −1.87326 −0.936630 0.350321i \(-0.886073\pi\)
−0.936630 + 0.350321i \(0.886073\pi\)
\(252\) 0 0
\(253\) 25.5145 1.60408
\(254\) 0 0
\(255\) −3.60830 −0.225961
\(256\) 0 0
\(257\) 17.6045 1.09814 0.549068 0.835778i \(-0.314983\pi\)
0.549068 + 0.835778i \(0.314983\pi\)
\(258\) 0 0
\(259\) 24.7984 1.54090
\(260\) 0 0
\(261\) −7.01509 −0.434223
\(262\) 0 0
\(263\) 7.47086 0.460673 0.230337 0.973111i \(-0.426017\pi\)
0.230337 + 0.973111i \(0.426017\pi\)
\(264\) 0 0
\(265\) 8.00050 0.491467
\(266\) 0 0
\(267\) 4.37544 0.267773
\(268\) 0 0
\(269\) 1.84115 0.112257 0.0561285 0.998424i \(-0.482124\pi\)
0.0561285 + 0.998424i \(0.482124\pi\)
\(270\) 0 0
\(271\) 13.6863 0.831386 0.415693 0.909505i \(-0.363539\pi\)
0.415693 + 0.909505i \(0.363539\pi\)
\(272\) 0 0
\(273\) 0.902043 0.0545941
\(274\) 0 0
\(275\) 2.98752 0.180154
\(276\) 0 0
\(277\) 7.01952 0.421762 0.210881 0.977512i \(-0.432367\pi\)
0.210881 + 0.977512i \(0.432367\pi\)
\(278\) 0 0
\(279\) −5.11929 −0.306484
\(280\) 0 0
\(281\) 5.76047 0.343641 0.171820 0.985128i \(-0.445035\pi\)
0.171820 + 0.985128i \(0.445035\pi\)
\(282\) 0 0
\(283\) 28.2822 1.68120 0.840601 0.541655i \(-0.182202\pi\)
0.840601 + 0.541655i \(0.182202\pi\)
\(284\) 0 0
\(285\) 0.827332 0.0490069
\(286\) 0 0
\(287\) −0.943611 −0.0556996
\(288\) 0 0
\(289\) −3.98015 −0.234127
\(290\) 0 0
\(291\) 3.52023 0.206360
\(292\) 0 0
\(293\) 9.17258 0.535868 0.267934 0.963437i \(-0.413659\pi\)
0.267934 + 0.963437i \(0.413659\pi\)
\(294\) 0 0
\(295\) 7.59664 0.442293
\(296\) 0 0
\(297\) 2.98752 0.173354
\(298\) 0 0
\(299\) 2.44179 0.141213
\(300\) 0 0
\(301\) −1.02547 −0.0591068
\(302\) 0 0
\(303\) −8.93919 −0.513543
\(304\) 0 0
\(305\) 7.05522 0.403981
\(306\) 0 0
\(307\) −5.81781 −0.332040 −0.166020 0.986122i \(-0.553092\pi\)
−0.166020 + 0.986122i \(0.553092\pi\)
\(308\) 0 0
\(309\) 11.5575 0.657484
\(310\) 0 0
\(311\) −0.795437 −0.0451051 −0.0225525 0.999746i \(-0.507179\pi\)
−0.0225525 + 0.999746i \(0.507179\pi\)
\(312\) 0 0
\(313\) −25.8364 −1.46036 −0.730181 0.683254i \(-0.760564\pi\)
−0.730181 + 0.683254i \(0.760564\pi\)
\(314\) 0 0
\(315\) −3.15496 −0.177762
\(316\) 0 0
\(317\) 20.0724 1.12738 0.563688 0.825987i \(-0.309382\pi\)
0.563688 + 0.825987i \(0.309382\pi\)
\(318\) 0 0
\(319\) −20.9578 −1.17341
\(320\) 0 0
\(321\) 0.378522 0.0211271
\(322\) 0 0
\(323\) −2.98526 −0.166104
\(324\) 0 0
\(325\) 0.285913 0.0158596
\(326\) 0 0
\(327\) 4.16387 0.230263
\(328\) 0 0
\(329\) 21.9187 1.20842
\(330\) 0 0
\(331\) −24.6748 −1.35625 −0.678126 0.734946i \(-0.737207\pi\)
−0.678126 + 0.734946i \(0.737207\pi\)
\(332\) 0 0
\(333\) 7.86013 0.430733
\(334\) 0 0
\(335\) −1.00000 −0.0546358
\(336\) 0 0
\(337\) 8.95874 0.488014 0.244007 0.969774i \(-0.421538\pi\)
0.244007 + 0.969774i \(0.421538\pi\)
\(338\) 0 0
\(339\) −3.08719 −0.167673
\(340\) 0 0
\(341\) −15.2940 −0.828217
\(342\) 0 0
\(343\) −12.7657 −0.689285
\(344\) 0 0
\(345\) −8.54034 −0.459796
\(346\) 0 0
\(347\) −19.7693 −1.06127 −0.530635 0.847601i \(-0.678046\pi\)
−0.530635 + 0.847601i \(0.678046\pi\)
\(348\) 0 0
\(349\) −10.1902 −0.545470 −0.272735 0.962089i \(-0.587928\pi\)
−0.272735 + 0.962089i \(0.587928\pi\)
\(350\) 0 0
\(351\) 0.285913 0.0152609
\(352\) 0 0
\(353\) −20.2854 −1.07968 −0.539842 0.841767i \(-0.681516\pi\)
−0.539842 + 0.841767i \(0.681516\pi\)
\(354\) 0 0
\(355\) −14.9890 −0.795535
\(356\) 0 0
\(357\) 11.3840 0.602507
\(358\) 0 0
\(359\) −23.6854 −1.25007 −0.625034 0.780598i \(-0.714915\pi\)
−0.625034 + 0.780598i \(0.714915\pi\)
\(360\) 0 0
\(361\) −18.3155 −0.963975
\(362\) 0 0
\(363\) −2.07470 −0.108894
\(364\) 0 0
\(365\) 5.66984 0.296773
\(366\) 0 0
\(367\) 9.98082 0.520994 0.260497 0.965475i \(-0.416114\pi\)
0.260497 + 0.965475i \(0.416114\pi\)
\(368\) 0 0
\(369\) −0.299088 −0.0155699
\(370\) 0 0
\(371\) −25.2412 −1.31046
\(372\) 0 0
\(373\) −21.8677 −1.13227 −0.566133 0.824314i \(-0.691561\pi\)
−0.566133 + 0.824314i \(0.691561\pi\)
\(374\) 0 0
\(375\) −1.00000 −0.0516398
\(376\) 0 0
\(377\) −2.00571 −0.103299
\(378\) 0 0
\(379\) 12.2895 0.631270 0.315635 0.948881i \(-0.397783\pi\)
0.315635 + 0.948881i \(0.397783\pi\)
\(380\) 0 0
\(381\) −5.16100 −0.264406
\(382\) 0 0
\(383\) −7.13827 −0.364749 −0.182374 0.983229i \(-0.558378\pi\)
−0.182374 + 0.983229i \(0.558378\pi\)
\(384\) 0 0
\(385\) −9.42551 −0.480368
\(386\) 0 0
\(387\) −0.325033 −0.0165224
\(388\) 0 0
\(389\) −9.63393 −0.488459 −0.244230 0.969717i \(-0.578535\pi\)
−0.244230 + 0.969717i \(0.578535\pi\)
\(390\) 0 0
\(391\) 30.8161 1.55844
\(392\) 0 0
\(393\) 20.2863 1.02331
\(394\) 0 0
\(395\) 1.40186 0.0705353
\(396\) 0 0
\(397\) 30.3923 1.52534 0.762672 0.646786i \(-0.223887\pi\)
0.762672 + 0.646786i \(0.223887\pi\)
\(398\) 0 0
\(399\) −2.61020 −0.130673
\(400\) 0 0
\(401\) −28.1718 −1.40683 −0.703415 0.710779i \(-0.748343\pi\)
−0.703415 + 0.710779i \(0.748343\pi\)
\(402\) 0 0
\(403\) −1.46367 −0.0729107
\(404\) 0 0
\(405\) −1.00000 −0.0496904
\(406\) 0 0
\(407\) 23.4823 1.16398
\(408\) 0 0
\(409\) 3.98115 0.196855 0.0984275 0.995144i \(-0.468619\pi\)
0.0984275 + 0.995144i \(0.468619\pi\)
\(410\) 0 0
\(411\) 3.78286 0.186595
\(412\) 0 0
\(413\) −23.9671 −1.17934
\(414\) 0 0
\(415\) −4.16743 −0.204571
\(416\) 0 0
\(417\) −2.25651 −0.110502
\(418\) 0 0
\(419\) 11.5070 0.562152 0.281076 0.959686i \(-0.409309\pi\)
0.281076 + 0.959686i \(0.409309\pi\)
\(420\) 0 0
\(421\) −6.00861 −0.292842 −0.146421 0.989222i \(-0.546775\pi\)
−0.146421 + 0.989222i \(0.546775\pi\)
\(422\) 0 0
\(423\) 6.94740 0.337794
\(424\) 0 0
\(425\) 3.60830 0.175028
\(426\) 0 0
\(427\) −22.2589 −1.07718
\(428\) 0 0
\(429\) 0.854172 0.0412398
\(430\) 0 0
\(431\) 0.782291 0.0376816 0.0188408 0.999822i \(-0.494002\pi\)
0.0188408 + 0.999822i \(0.494002\pi\)
\(432\) 0 0
\(433\) −28.2474 −1.35748 −0.678742 0.734376i \(-0.737475\pi\)
−0.678742 + 0.734376i \(0.737475\pi\)
\(434\) 0 0
\(435\) 7.01509 0.336348
\(436\) 0 0
\(437\) −7.06569 −0.337998
\(438\) 0 0
\(439\) −13.5072 −0.644664 −0.322332 0.946627i \(-0.604467\pi\)
−0.322332 + 0.946627i \(0.604467\pi\)
\(440\) 0 0
\(441\) 2.95375 0.140655
\(442\) 0 0
\(443\) −37.7621 −1.79413 −0.897066 0.441897i \(-0.854306\pi\)
−0.897066 + 0.441897i \(0.854306\pi\)
\(444\) 0 0
\(445\) −4.37544 −0.207416
\(446\) 0 0
\(447\) 10.5973 0.501234
\(448\) 0 0
\(449\) 40.5523 1.91378 0.956890 0.290452i \(-0.0938056\pi\)
0.956890 + 0.290452i \(0.0938056\pi\)
\(450\) 0 0
\(451\) −0.893533 −0.0420748
\(452\) 0 0
\(453\) −8.20194 −0.385361
\(454\) 0 0
\(455\) −0.902043 −0.0422884
\(456\) 0 0
\(457\) 1.83602 0.0858855 0.0429427 0.999078i \(-0.486327\pi\)
0.0429427 + 0.999078i \(0.486327\pi\)
\(458\) 0 0
\(459\) 3.60830 0.168421
\(460\) 0 0
\(461\) 11.1909 0.521213 0.260606 0.965445i \(-0.416078\pi\)
0.260606 + 0.965445i \(0.416078\pi\)
\(462\) 0 0
\(463\) −4.88392 −0.226975 −0.113488 0.993539i \(-0.536202\pi\)
−0.113488 + 0.993539i \(0.536202\pi\)
\(464\) 0 0
\(465\) 5.11929 0.237401
\(466\) 0 0
\(467\) −5.42945 −0.251245 −0.125622 0.992078i \(-0.540093\pi\)
−0.125622 + 0.992078i \(0.540093\pi\)
\(468\) 0 0
\(469\) 3.15496 0.145682
\(470\) 0 0
\(471\) −20.3258 −0.936564
\(472\) 0 0
\(473\) −0.971044 −0.0446487
\(474\) 0 0
\(475\) −0.827332 −0.0379606
\(476\) 0 0
\(477\) −8.00050 −0.366318
\(478\) 0 0
\(479\) −29.0597 −1.32777 −0.663886 0.747834i \(-0.731094\pi\)
−0.663886 + 0.747834i \(0.731094\pi\)
\(480\) 0 0
\(481\) 2.24731 0.102469
\(482\) 0 0
\(483\) 26.9444 1.22601
\(484\) 0 0
\(485\) −3.52023 −0.159845
\(486\) 0 0
\(487\) −17.6239 −0.798614 −0.399307 0.916817i \(-0.630749\pi\)
−0.399307 + 0.916817i \(0.630749\pi\)
\(488\) 0 0
\(489\) −10.7073 −0.484200
\(490\) 0 0
\(491\) 13.2557 0.598219 0.299110 0.954219i \(-0.403310\pi\)
0.299110 + 0.954219i \(0.403310\pi\)
\(492\) 0 0
\(493\) −25.3126 −1.14002
\(494\) 0 0
\(495\) −2.98752 −0.134279
\(496\) 0 0
\(497\) 47.2898 2.12124
\(498\) 0 0
\(499\) 8.82162 0.394910 0.197455 0.980312i \(-0.436732\pi\)
0.197455 + 0.980312i \(0.436732\pi\)
\(500\) 0 0
\(501\) 23.8494 1.06551
\(502\) 0 0
\(503\) −26.7640 −1.19335 −0.596674 0.802484i \(-0.703511\pi\)
−0.596674 + 0.802484i \(0.703511\pi\)
\(504\) 0 0
\(505\) 8.93919 0.397789
\(506\) 0 0
\(507\) −12.9183 −0.573720
\(508\) 0 0
\(509\) −13.2099 −0.585520 −0.292760 0.956186i \(-0.594574\pi\)
−0.292760 + 0.956186i \(0.594574\pi\)
\(510\) 0 0
\(511\) −17.8881 −0.791323
\(512\) 0 0
\(513\) −0.827332 −0.0365276
\(514\) 0 0
\(515\) −11.5575 −0.509285
\(516\) 0 0
\(517\) 20.7555 0.912826
\(518\) 0 0
\(519\) 18.7835 0.824505
\(520\) 0 0
\(521\) −2.07464 −0.0908915 −0.0454458 0.998967i \(-0.514471\pi\)
−0.0454458 + 0.998967i \(0.514471\pi\)
\(522\) 0 0
\(523\) −20.0122 −0.875072 −0.437536 0.899201i \(-0.644149\pi\)
−0.437536 + 0.899201i \(0.644149\pi\)
\(524\) 0 0
\(525\) 3.15496 0.137694
\(526\) 0 0
\(527\) −18.4720 −0.804651
\(528\) 0 0
\(529\) 49.9374 2.17119
\(530\) 0 0
\(531\) −7.59664 −0.329666
\(532\) 0 0
\(533\) −0.0855132 −0.00370399
\(534\) 0 0
\(535\) −0.378522 −0.0163649
\(536\) 0 0
\(537\) −2.89439 −0.124902
\(538\) 0 0
\(539\) 8.82440 0.380094
\(540\) 0 0
\(541\) 21.3776 0.919097 0.459548 0.888153i \(-0.348011\pi\)
0.459548 + 0.888153i \(0.348011\pi\)
\(542\) 0 0
\(543\) 7.61165 0.326647
\(544\) 0 0
\(545\) −4.16387 −0.178361
\(546\) 0 0
\(547\) −9.86325 −0.421722 −0.210861 0.977516i \(-0.567627\pi\)
−0.210861 + 0.977516i \(0.567627\pi\)
\(548\) 0 0
\(549\) −7.05522 −0.301109
\(550\) 0 0
\(551\) 5.80381 0.247250
\(552\) 0 0
\(553\) −4.42281 −0.188077
\(554\) 0 0
\(555\) −7.86013 −0.333644
\(556\) 0 0
\(557\) 21.0273 0.890955 0.445478 0.895293i \(-0.353034\pi\)
0.445478 + 0.895293i \(0.353034\pi\)
\(558\) 0 0
\(559\) −0.0929312 −0.00393057
\(560\) 0 0
\(561\) 10.7799 0.455127
\(562\) 0 0
\(563\) 16.9495 0.714337 0.357169 0.934040i \(-0.383742\pi\)
0.357169 + 0.934040i \(0.383742\pi\)
\(564\) 0 0
\(565\) 3.08719 0.129879
\(566\) 0 0
\(567\) 3.15496 0.132496
\(568\) 0 0
\(569\) −8.64662 −0.362485 −0.181243 0.983438i \(-0.558012\pi\)
−0.181243 + 0.983438i \(0.558012\pi\)
\(570\) 0 0
\(571\) −35.7902 −1.49777 −0.748887 0.662698i \(-0.769411\pi\)
−0.748887 + 0.662698i \(0.769411\pi\)
\(572\) 0 0
\(573\) 3.14590 0.131422
\(574\) 0 0
\(575\) 8.54034 0.356157
\(576\) 0 0
\(577\) −4.73298 −0.197037 −0.0985183 0.995135i \(-0.531410\pi\)
−0.0985183 + 0.995135i \(0.531410\pi\)
\(578\) 0 0
\(579\) −9.62036 −0.399808
\(580\) 0 0
\(581\) 13.1481 0.545474
\(582\) 0 0
\(583\) −23.9017 −0.989907
\(584\) 0 0
\(585\) −0.285913 −0.0118210
\(586\) 0 0
\(587\) 38.7594 1.59977 0.799886 0.600151i \(-0.204893\pi\)
0.799886 + 0.600151i \(0.204893\pi\)
\(588\) 0 0
\(589\) 4.23535 0.174515
\(590\) 0 0
\(591\) −20.2285 −0.832091
\(592\) 0 0
\(593\) 0.562880 0.0231147 0.0115574 0.999933i \(-0.496321\pi\)
0.0115574 + 0.999933i \(0.496321\pi\)
\(594\) 0 0
\(595\) −11.3840 −0.466700
\(596\) 0 0
\(597\) 3.64854 0.149325
\(598\) 0 0
\(599\) 26.8815 1.09835 0.549174 0.835708i \(-0.314943\pi\)
0.549174 + 0.835708i \(0.314943\pi\)
\(600\) 0 0
\(601\) 35.1263 1.43283 0.716416 0.697673i \(-0.245781\pi\)
0.716416 + 0.697673i \(0.245781\pi\)
\(602\) 0 0
\(603\) 1.00000 0.0407231
\(604\) 0 0
\(605\) 2.07470 0.0843486
\(606\) 0 0
\(607\) 16.9876 0.689506 0.344753 0.938693i \(-0.387963\pi\)
0.344753 + 0.938693i \(0.387963\pi\)
\(608\) 0 0
\(609\) −22.1323 −0.896846
\(610\) 0 0
\(611\) 1.98635 0.0803591
\(612\) 0 0
\(613\) 4.17805 0.168750 0.0843750 0.996434i \(-0.473111\pi\)
0.0843750 + 0.996434i \(0.473111\pi\)
\(614\) 0 0
\(615\) 0.299088 0.0120604
\(616\) 0 0
\(617\) −42.7969 −1.72294 −0.861469 0.507810i \(-0.830455\pi\)
−0.861469 + 0.507810i \(0.830455\pi\)
\(618\) 0 0
\(619\) −34.4662 −1.38531 −0.692656 0.721268i \(-0.743560\pi\)
−0.692656 + 0.721268i \(0.743560\pi\)
\(620\) 0 0
\(621\) 8.54034 0.342712
\(622\) 0 0
\(623\) 13.8043 0.553059
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) −2.47167 −0.0987091
\(628\) 0 0
\(629\) 28.3617 1.13086
\(630\) 0 0
\(631\) −31.5909 −1.25762 −0.628808 0.777561i \(-0.716457\pi\)
−0.628808 + 0.777561i \(0.716457\pi\)
\(632\) 0 0
\(633\) −7.82327 −0.310947
\(634\) 0 0
\(635\) 5.16100 0.204808
\(636\) 0 0
\(637\) 0.844516 0.0334609
\(638\) 0 0
\(639\) 14.9890 0.592957
\(640\) 0 0
\(641\) 41.9149 1.65554 0.827770 0.561068i \(-0.189609\pi\)
0.827770 + 0.561068i \(0.189609\pi\)
\(642\) 0 0
\(643\) −47.2476 −1.86326 −0.931631 0.363405i \(-0.881614\pi\)
−0.931631 + 0.363405i \(0.881614\pi\)
\(644\) 0 0
\(645\) 0.325033 0.0127982
\(646\) 0 0
\(647\) 4.69644 0.184636 0.0923181 0.995730i \(-0.470572\pi\)
0.0923181 + 0.995730i \(0.470572\pi\)
\(648\) 0 0
\(649\) −22.6951 −0.890862
\(650\) 0 0
\(651\) −16.1511 −0.633013
\(652\) 0 0
\(653\) 11.0808 0.433625 0.216813 0.976213i \(-0.430434\pi\)
0.216813 + 0.976213i \(0.430434\pi\)
\(654\) 0 0
\(655\) −20.2863 −0.792653
\(656\) 0 0
\(657\) −5.66984 −0.221201
\(658\) 0 0
\(659\) 13.6262 0.530801 0.265401 0.964138i \(-0.414496\pi\)
0.265401 + 0.964138i \(0.414496\pi\)
\(660\) 0 0
\(661\) 37.1582 1.44529 0.722643 0.691221i \(-0.242927\pi\)
0.722643 + 0.691221i \(0.242927\pi\)
\(662\) 0 0
\(663\) 1.03166 0.0400664
\(664\) 0 0
\(665\) 2.61020 0.101219
\(666\) 0 0
\(667\) −59.9112 −2.31977
\(668\) 0 0
\(669\) 2.55347 0.0987230
\(670\) 0 0
\(671\) −21.0776 −0.813693
\(672\) 0 0
\(673\) −34.8829 −1.34464 −0.672319 0.740262i \(-0.734702\pi\)
−0.672319 + 0.740262i \(0.734702\pi\)
\(674\) 0 0
\(675\) 1.00000 0.0384900
\(676\) 0 0
\(677\) −4.33989 −0.166796 −0.0833978 0.996516i \(-0.526577\pi\)
−0.0833978 + 0.996516i \(0.526577\pi\)
\(678\) 0 0
\(679\) 11.1062 0.426216
\(680\) 0 0
\(681\) 20.2142 0.774611
\(682\) 0 0
\(683\) −40.8172 −1.56183 −0.780913 0.624640i \(-0.785246\pi\)
−0.780913 + 0.624640i \(0.785246\pi\)
\(684\) 0 0
\(685\) −3.78286 −0.144536
\(686\) 0 0
\(687\) 3.86312 0.147387
\(688\) 0 0
\(689\) −2.28745 −0.0871448
\(690\) 0 0
\(691\) 15.7801 0.600303 0.300151 0.953892i \(-0.402963\pi\)
0.300151 + 0.953892i \(0.402963\pi\)
\(692\) 0 0
\(693\) 9.42551 0.358045
\(694\) 0 0
\(695\) 2.25651 0.0855944
\(696\) 0 0
\(697\) −1.07920 −0.0408777
\(698\) 0 0
\(699\) 2.98051 0.112733
\(700\) 0 0
\(701\) −26.0045 −0.982177 −0.491088 0.871110i \(-0.663401\pi\)
−0.491088 + 0.871110i \(0.663401\pi\)
\(702\) 0 0
\(703\) −6.50294 −0.245263
\(704\) 0 0
\(705\) −6.94740 −0.261654
\(706\) 0 0
\(707\) −28.2027 −1.06067
\(708\) 0 0
\(709\) −6.80831 −0.255691 −0.127846 0.991794i \(-0.540806\pi\)
−0.127846 + 0.991794i \(0.540806\pi\)
\(710\) 0 0
\(711\) −1.40186 −0.0525739
\(712\) 0 0
\(713\) −43.7205 −1.63734
\(714\) 0 0
\(715\) −0.854172 −0.0319442
\(716\) 0 0
\(717\) 7.31022 0.273005
\(718\) 0 0
\(719\) −31.8383 −1.18737 −0.593684 0.804698i \(-0.702327\pi\)
−0.593684 + 0.804698i \(0.702327\pi\)
\(720\) 0 0
\(721\) 36.4635 1.35797
\(722\) 0 0
\(723\) 25.4246 0.945551
\(724\) 0 0
\(725\) −7.01509 −0.260534
\(726\) 0 0
\(727\) 26.7120 0.990694 0.495347 0.868695i \(-0.335041\pi\)
0.495347 + 0.868695i \(0.335041\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −1.17282 −0.0433782
\(732\) 0 0
\(733\) 7.50114 0.277061 0.138530 0.990358i \(-0.455762\pi\)
0.138530 + 0.990358i \(0.455762\pi\)
\(734\) 0 0
\(735\) −2.95375 −0.108951
\(736\) 0 0
\(737\) 2.98752 0.110047
\(738\) 0 0
\(739\) −46.5034 −1.71066 −0.855328 0.518087i \(-0.826644\pi\)
−0.855328 + 0.518087i \(0.826644\pi\)
\(740\) 0 0
\(741\) −0.236545 −0.00868969
\(742\) 0 0
\(743\) −20.8678 −0.765567 −0.382783 0.923838i \(-0.625034\pi\)
−0.382783 + 0.923838i \(0.625034\pi\)
\(744\) 0 0
\(745\) −10.5973 −0.388254
\(746\) 0 0
\(747\) 4.16743 0.152478
\(748\) 0 0
\(749\) 1.19422 0.0436359
\(750\) 0 0
\(751\) 19.7525 0.720779 0.360390 0.932802i \(-0.382644\pi\)
0.360390 + 0.932802i \(0.382644\pi\)
\(752\) 0 0
\(753\) −29.6780 −1.08153
\(754\) 0 0
\(755\) 8.20194 0.298499
\(756\) 0 0
\(757\) −22.5601 −0.819961 −0.409981 0.912094i \(-0.634465\pi\)
−0.409981 + 0.912094i \(0.634465\pi\)
\(758\) 0 0
\(759\) 25.5145 0.926116
\(760\) 0 0
\(761\) 4.24842 0.154005 0.0770026 0.997031i \(-0.475465\pi\)
0.0770026 + 0.997031i \(0.475465\pi\)
\(762\) 0 0
\(763\) 13.1368 0.475585
\(764\) 0 0
\(765\) −3.60830 −0.130458
\(766\) 0 0
\(767\) −2.17198 −0.0784255
\(768\) 0 0
\(769\) 20.1223 0.725627 0.362814 0.931862i \(-0.381816\pi\)
0.362814 + 0.931862i \(0.381816\pi\)
\(770\) 0 0
\(771\) 17.6045 0.634009
\(772\) 0 0
\(773\) −5.02404 −0.180702 −0.0903511 0.995910i \(-0.528799\pi\)
−0.0903511 + 0.995910i \(0.528799\pi\)
\(774\) 0 0
\(775\) −5.11929 −0.183890
\(776\) 0 0
\(777\) 24.7984 0.889637
\(778\) 0 0
\(779\) 0.247445 0.00886564
\(780\) 0 0
\(781\) 44.7801 1.60236
\(782\) 0 0
\(783\) −7.01509 −0.250699
\(784\) 0 0
\(785\) 20.3258 0.725459
\(786\) 0 0
\(787\) 5.10979 0.182144 0.0910721 0.995844i \(-0.470971\pi\)
0.0910721 + 0.995844i \(0.470971\pi\)
\(788\) 0 0
\(789\) 7.47086 0.265970
\(790\) 0 0
\(791\) −9.73994 −0.346313
\(792\) 0 0
\(793\) −2.01718 −0.0716321
\(794\) 0 0
\(795\) 8.00050 0.283749
\(796\) 0 0
\(797\) −35.3566 −1.25239 −0.626197 0.779665i \(-0.715389\pi\)
−0.626197 + 0.779665i \(0.715389\pi\)
\(798\) 0 0
\(799\) 25.0683 0.886853
\(800\) 0 0
\(801\) 4.37544 0.154599
\(802\) 0 0
\(803\) −16.9388 −0.597756
\(804\) 0 0
\(805\) −26.9444 −0.949665
\(806\) 0 0
\(807\) 1.84115 0.0648116
\(808\) 0 0
\(809\) 8.85483 0.311319 0.155660 0.987811i \(-0.450250\pi\)
0.155660 + 0.987811i \(0.450250\pi\)
\(810\) 0 0
\(811\) −25.8145 −0.906471 −0.453236 0.891391i \(-0.649730\pi\)
−0.453236 + 0.891391i \(0.649730\pi\)
\(812\) 0 0
\(813\) 13.6863 0.480001
\(814\) 0 0
\(815\) 10.7073 0.375060
\(816\) 0 0
\(817\) 0.268910 0.00940798
\(818\) 0 0
\(819\) 0.902043 0.0315199
\(820\) 0 0
\(821\) 19.2283 0.671073 0.335536 0.942027i \(-0.391082\pi\)
0.335536 + 0.942027i \(0.391082\pi\)
\(822\) 0 0
\(823\) 45.1086 1.57239 0.786193 0.617981i \(-0.212049\pi\)
0.786193 + 0.617981i \(0.212049\pi\)
\(824\) 0 0
\(825\) 2.98752 0.104012
\(826\) 0 0
\(827\) 10.6280 0.369573 0.184787 0.982779i \(-0.440841\pi\)
0.184787 + 0.982779i \(0.440841\pi\)
\(828\) 0 0
\(829\) 1.25278 0.0435110 0.0217555 0.999763i \(-0.493074\pi\)
0.0217555 + 0.999763i \(0.493074\pi\)
\(830\) 0 0
\(831\) 7.01952 0.243504
\(832\) 0 0
\(833\) 10.6580 0.369279
\(834\) 0 0
\(835\) −23.8494 −0.825341
\(836\) 0 0
\(837\) −5.11929 −0.176949
\(838\) 0 0
\(839\) 4.38362 0.151339 0.0756696 0.997133i \(-0.475891\pi\)
0.0756696 + 0.997133i \(0.475891\pi\)
\(840\) 0 0
\(841\) 20.2115 0.696948
\(842\) 0 0
\(843\) 5.76047 0.198401
\(844\) 0 0
\(845\) 12.9183 0.444401
\(846\) 0 0
\(847\) −6.54559 −0.224909
\(848\) 0 0
\(849\) 28.2822 0.970642
\(850\) 0 0
\(851\) 67.1282 2.30113
\(852\) 0 0
\(853\) −17.7363 −0.607280 −0.303640 0.952787i \(-0.598202\pi\)
−0.303640 + 0.952787i \(0.598202\pi\)
\(854\) 0 0
\(855\) 0.827332 0.0282941
\(856\) 0 0
\(857\) 9.03497 0.308629 0.154314 0.988022i \(-0.450683\pi\)
0.154314 + 0.988022i \(0.450683\pi\)
\(858\) 0 0
\(859\) −37.9596 −1.29516 −0.647582 0.761996i \(-0.724220\pi\)
−0.647582 + 0.761996i \(0.724220\pi\)
\(860\) 0 0
\(861\) −0.943611 −0.0321582
\(862\) 0 0
\(863\) −27.6735 −0.942017 −0.471008 0.882129i \(-0.656110\pi\)
−0.471008 + 0.882129i \(0.656110\pi\)
\(864\) 0 0
\(865\) −18.7835 −0.638659
\(866\) 0 0
\(867\) −3.98015 −0.135173
\(868\) 0 0
\(869\) −4.18809 −0.142071
\(870\) 0 0
\(871\) 0.285913 0.00968779
\(872\) 0 0
\(873\) 3.52023 0.119142
\(874\) 0 0
\(875\) −3.15496 −0.106657
\(876\) 0 0
\(877\) −24.1259 −0.814673 −0.407337 0.913278i \(-0.633542\pi\)
−0.407337 + 0.913278i \(0.633542\pi\)
\(878\) 0 0
\(879\) 9.17258 0.309383
\(880\) 0 0
\(881\) 0.512416 0.0172637 0.00863187 0.999963i \(-0.497252\pi\)
0.00863187 + 0.999963i \(0.497252\pi\)
\(882\) 0 0
\(883\) −31.3694 −1.05567 −0.527833 0.849348i \(-0.676995\pi\)
−0.527833 + 0.849348i \(0.676995\pi\)
\(884\) 0 0
\(885\) 7.59664 0.255358
\(886\) 0 0
\(887\) 28.1787 0.946148 0.473074 0.881023i \(-0.343144\pi\)
0.473074 + 0.881023i \(0.343144\pi\)
\(888\) 0 0
\(889\) −16.2827 −0.546106
\(890\) 0 0
\(891\) 2.98752 0.100086
\(892\) 0 0
\(893\) −5.74780 −0.192343
\(894\) 0 0
\(895\) 2.89439 0.0967487
\(896\) 0 0
\(897\) 2.44179 0.0815291
\(898\) 0 0
\(899\) 35.9123 1.19774
\(900\) 0 0
\(901\) −28.8682 −0.961740
\(902\) 0 0
\(903\) −1.02547 −0.0341254
\(904\) 0 0
\(905\) −7.61165 −0.253020
\(906\) 0 0
\(907\) 3.52920 0.117185 0.0585926 0.998282i \(-0.481339\pi\)
0.0585926 + 0.998282i \(0.481339\pi\)
\(908\) 0 0
\(909\) −8.93919 −0.296494
\(910\) 0 0
\(911\) −0.394566 −0.0130726 −0.00653628 0.999979i \(-0.502081\pi\)
−0.00653628 + 0.999979i \(0.502081\pi\)
\(912\) 0 0
\(913\) 12.4503 0.412045
\(914\) 0 0
\(915\) 7.05522 0.233238
\(916\) 0 0
\(917\) 64.0025 2.11355
\(918\) 0 0
\(919\) −32.3198 −1.06613 −0.533066 0.846074i \(-0.678960\pi\)
−0.533066 + 0.846074i \(0.678960\pi\)
\(920\) 0 0
\(921\) −5.81781 −0.191703
\(922\) 0 0
\(923\) 4.28556 0.141061
\(924\) 0 0
\(925\) 7.86013 0.258440
\(926\) 0 0
\(927\) 11.5575 0.379599
\(928\) 0 0
\(929\) −25.0133 −0.820661 −0.410330 0.911937i \(-0.634587\pi\)
−0.410330 + 0.911937i \(0.634587\pi\)
\(930\) 0 0
\(931\) −2.44373 −0.0800901
\(932\) 0 0
\(933\) −0.795437 −0.0260414
\(934\) 0 0
\(935\) −10.7799 −0.352540
\(936\) 0 0
\(937\) −13.2156 −0.431733 −0.215867 0.976423i \(-0.569258\pi\)
−0.215867 + 0.976423i \(0.569258\pi\)
\(938\) 0 0
\(939\) −25.8364 −0.843140
\(940\) 0 0
\(941\) −29.6504 −0.966574 −0.483287 0.875462i \(-0.660557\pi\)
−0.483287 + 0.875462i \(0.660557\pi\)
\(942\) 0 0
\(943\) −2.55432 −0.0831799
\(944\) 0 0
\(945\) −3.15496 −0.102631
\(946\) 0 0
\(947\) 19.7923 0.643162 0.321581 0.946882i \(-0.395786\pi\)
0.321581 + 0.946882i \(0.395786\pi\)
\(948\) 0 0
\(949\) −1.62108 −0.0526225
\(950\) 0 0
\(951\) 20.0724 0.650891
\(952\) 0 0
\(953\) −43.2637 −1.40145 −0.700724 0.713432i \(-0.747140\pi\)
−0.700724 + 0.713432i \(0.747140\pi\)
\(954\) 0 0
\(955\) −3.14590 −0.101799
\(956\) 0 0
\(957\) −20.9578 −0.677468
\(958\) 0 0
\(959\) 11.9348 0.385394
\(960\) 0 0
\(961\) −4.79286 −0.154608
\(962\) 0 0
\(963\) 0.378522 0.0121977
\(964\) 0 0
\(965\) 9.62036 0.309690
\(966\) 0 0
\(967\) −19.1463 −0.615703 −0.307852 0.951434i \(-0.599610\pi\)
−0.307852 + 0.951434i \(0.599610\pi\)
\(968\) 0 0
\(969\) −2.98526 −0.0959004
\(970\) 0 0
\(971\) −47.8352 −1.53510 −0.767552 0.640987i \(-0.778525\pi\)
−0.767552 + 0.640987i \(0.778525\pi\)
\(972\) 0 0
\(973\) −7.11920 −0.228231
\(974\) 0 0
\(975\) 0.285913 0.00915654
\(976\) 0 0
\(977\) −37.2324 −1.19117 −0.595585 0.803292i \(-0.703080\pi\)
−0.595585 + 0.803292i \(0.703080\pi\)
\(978\) 0 0
\(979\) 13.0717 0.417774
\(980\) 0 0
\(981\) 4.16387 0.132942
\(982\) 0 0
\(983\) −60.4931 −1.92943 −0.964715 0.263297i \(-0.915190\pi\)
−0.964715 + 0.263297i \(0.915190\pi\)
\(984\) 0 0
\(985\) 20.2285 0.644535
\(986\) 0 0
\(987\) 21.9187 0.697681
\(988\) 0 0
\(989\) −2.77589 −0.0882683
\(990\) 0 0
\(991\) −24.9987 −0.794111 −0.397055 0.917795i \(-0.629968\pi\)
−0.397055 + 0.917795i \(0.629968\pi\)
\(992\) 0 0
\(993\) −24.6748 −0.783032
\(994\) 0 0
\(995\) −3.64854 −0.115667
\(996\) 0 0
\(997\) −42.9031 −1.35875 −0.679377 0.733789i \(-0.737750\pi\)
−0.679377 + 0.733789i \(0.737750\pi\)
\(998\) 0 0
\(999\) 7.86013 0.248684
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.g.1.5 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4020.2.a.g.1.5 6 1.1 even 1 trivial