Properties

Label 3040.2.a.m.1.3
Level $3040$
Weight $2$
Character 3040.1
Self dual yes
Analytic conductor $24.275$
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] = [3040,2,Mod(1,3040)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3040, 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("3040.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3040 = 2^{5} \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3040.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: \(24.2745222145\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.148.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 3x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(0.311108\) of defining polynomial
Character \(\chi\) \(=\) 3040.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+2.21432 q^{3} +1.00000 q^{5} -1.52543 q^{7} +1.90321 q^{9} +O(q^{10})\) \(q+2.21432 q^{3} +1.00000 q^{5} -1.52543 q^{7} +1.90321 q^{9} -2.90321 q^{11} -1.31111 q^{13} +2.21432 q^{15} -7.80642 q^{17} -1.00000 q^{19} -3.37778 q^{21} -4.28100 q^{23} +1.00000 q^{25} -2.42864 q^{27} -9.18421 q^{29} +7.80642 q^{31} -6.42864 q^{33} -1.52543 q^{35} +8.16839 q^{37} -2.90321 q^{39} -11.0509 q^{41} -1.09679 q^{43} +1.90321 q^{45} -4.28100 q^{47} -4.67307 q^{49} -17.2859 q^{51} +5.54617 q^{53} -2.90321 q^{55} -2.21432 q^{57} +1.86665 q^{59} +0.709636 q^{61} -2.90321 q^{63} -1.31111 q^{65} -1.45875 q^{67} -9.47949 q^{69} -2.29529 q^{71} +14.0415 q^{73} +2.21432 q^{75} +4.42864 q^{77} +6.13335 q^{79} -11.0874 q^{81} +11.7605 q^{83} -7.80642 q^{85} -20.3368 q^{87} +3.67307 q^{89} +2.00000 q^{91} +17.2859 q^{93} -1.00000 q^{95} -11.2192 q^{97} -5.52543 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 3 q^{5} + 2 q^{7} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 3 q^{5} + 2 q^{7} - q^{9} - 2 q^{11} - 4 q^{13} - 10 q^{17} - 3 q^{19} - 10 q^{21} - 6 q^{23} + 3 q^{25} + 6 q^{27} - 14 q^{29} + 10 q^{31} - 6 q^{33} + 2 q^{35} - 2 q^{37} - 2 q^{39} - 20 q^{41} - 10 q^{43} - q^{45} - 6 q^{47} - q^{49} - 12 q^{51} - 10 q^{53} - 2 q^{55} + 6 q^{59} - 18 q^{61} - 2 q^{63} - 4 q^{65} + 2 q^{67} - 2 q^{69} + 6 q^{71} + 2 q^{73} + 18 q^{79} - 13 q^{81} + 2 q^{83} - 10 q^{85} - 8 q^{87} - 2 q^{89} + 6 q^{91} + 12 q^{93} - 3 q^{95} + 6 q^{97} - 10 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 2.21432 1.27844 0.639219 0.769025i \(-0.279258\pi\)
0.639219 + 0.769025i \(0.279258\pi\)
\(4\) 0 0
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) −1.52543 −0.576557 −0.288279 0.957547i \(-0.593083\pi\)
−0.288279 + 0.957547i \(0.593083\pi\)
\(8\) 0 0
\(9\) 1.90321 0.634404
\(10\) 0 0
\(11\) −2.90321 −0.875351 −0.437676 0.899133i \(-0.644198\pi\)
−0.437676 + 0.899133i \(0.644198\pi\)
\(12\) 0 0
\(13\) −1.31111 −0.363636 −0.181818 0.983332i \(-0.558198\pi\)
−0.181818 + 0.983332i \(0.558198\pi\)
\(14\) 0 0
\(15\) 2.21432 0.571735
\(16\) 0 0
\(17\) −7.80642 −1.89334 −0.946668 0.322211i \(-0.895574\pi\)
−0.946668 + 0.322211i \(0.895574\pi\)
\(18\) 0 0
\(19\) −1.00000 −0.229416
\(20\) 0 0
\(21\) −3.37778 −0.737093
\(22\) 0 0
\(23\) −4.28100 −0.892649 −0.446325 0.894871i \(-0.647267\pi\)
−0.446325 + 0.894871i \(0.647267\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) −2.42864 −0.467392
\(28\) 0 0
\(29\) −9.18421 −1.70546 −0.852732 0.522348i \(-0.825056\pi\)
−0.852732 + 0.522348i \(0.825056\pi\)
\(30\) 0 0
\(31\) 7.80642 1.40208 0.701038 0.713124i \(-0.252721\pi\)
0.701038 + 0.713124i \(0.252721\pi\)
\(32\) 0 0
\(33\) −6.42864 −1.11908
\(34\) 0 0
\(35\) −1.52543 −0.257844
\(36\) 0 0
\(37\) 8.16839 1.34287 0.671437 0.741061i \(-0.265677\pi\)
0.671437 + 0.741061i \(0.265677\pi\)
\(38\) 0 0
\(39\) −2.90321 −0.464886
\(40\) 0 0
\(41\) −11.0509 −1.72585 −0.862927 0.505329i \(-0.831371\pi\)
−0.862927 + 0.505329i \(0.831371\pi\)
\(42\) 0 0
\(43\) −1.09679 −0.167259 −0.0836293 0.996497i \(-0.526651\pi\)
−0.0836293 + 0.996497i \(0.526651\pi\)
\(44\) 0 0
\(45\) 1.90321 0.283714
\(46\) 0 0
\(47\) −4.28100 −0.624447 −0.312224 0.950009i \(-0.601074\pi\)
−0.312224 + 0.950009i \(0.601074\pi\)
\(48\) 0 0
\(49\) −4.67307 −0.667582
\(50\) 0 0
\(51\) −17.2859 −2.42051
\(52\) 0 0
\(53\) 5.54617 0.761825 0.380913 0.924611i \(-0.375610\pi\)
0.380913 + 0.924611i \(0.375610\pi\)
\(54\) 0 0
\(55\) −2.90321 −0.391469
\(56\) 0 0
\(57\) −2.21432 −0.293294
\(58\) 0 0
\(59\) 1.86665 0.243017 0.121508 0.992590i \(-0.461227\pi\)
0.121508 + 0.992590i \(0.461227\pi\)
\(60\) 0 0
\(61\) 0.709636 0.0908596 0.0454298 0.998968i \(-0.485534\pi\)
0.0454298 + 0.998968i \(0.485534\pi\)
\(62\) 0 0
\(63\) −2.90321 −0.365770
\(64\) 0 0
\(65\) −1.31111 −0.162623
\(66\) 0 0
\(67\) −1.45875 −0.178215 −0.0891074 0.996022i \(-0.528401\pi\)
−0.0891074 + 0.996022i \(0.528401\pi\)
\(68\) 0 0
\(69\) −9.47949 −1.14120
\(70\) 0 0
\(71\) −2.29529 −0.272400 −0.136200 0.990681i \(-0.543489\pi\)
−0.136200 + 0.990681i \(0.543489\pi\)
\(72\) 0 0
\(73\) 14.0415 1.64343 0.821716 0.569897i \(-0.193017\pi\)
0.821716 + 0.569897i \(0.193017\pi\)
\(74\) 0 0
\(75\) 2.21432 0.255688
\(76\) 0 0
\(77\) 4.42864 0.504690
\(78\) 0 0
\(79\) 6.13335 0.690056 0.345028 0.938592i \(-0.387869\pi\)
0.345028 + 0.938592i \(0.387869\pi\)
\(80\) 0 0
\(81\) −11.0874 −1.23194
\(82\) 0 0
\(83\) 11.7605 1.29088 0.645441 0.763810i \(-0.276674\pi\)
0.645441 + 0.763810i \(0.276674\pi\)
\(84\) 0 0
\(85\) −7.80642 −0.846726
\(86\) 0 0
\(87\) −20.3368 −2.18033
\(88\) 0 0
\(89\) 3.67307 0.389345 0.194672 0.980868i \(-0.437636\pi\)
0.194672 + 0.980868i \(0.437636\pi\)
\(90\) 0 0
\(91\) 2.00000 0.209657
\(92\) 0 0
\(93\) 17.2859 1.79247
\(94\) 0 0
\(95\) −1.00000 −0.102598
\(96\) 0 0
\(97\) −11.2192 −1.13914 −0.569571 0.821942i \(-0.692891\pi\)
−0.569571 + 0.821942i \(0.692891\pi\)
\(98\) 0 0
\(99\) −5.52543 −0.555326
\(100\) 0 0
\(101\) −0.963435 −0.0958654 −0.0479327 0.998851i \(-0.515263\pi\)
−0.0479327 + 0.998851i \(0.515263\pi\)
\(102\) 0 0
\(103\) 17.8874 1.76250 0.881248 0.472653i \(-0.156704\pi\)
0.881248 + 0.472653i \(0.156704\pi\)
\(104\) 0 0
\(105\) −3.37778 −0.329638
\(106\) 0 0
\(107\) −13.5002 −1.30512 −0.652559 0.757738i \(-0.726304\pi\)
−0.652559 + 0.757738i \(0.726304\pi\)
\(108\) 0 0
\(109\) 10.5303 1.00862 0.504312 0.863521i \(-0.331746\pi\)
0.504312 + 0.863521i \(0.331746\pi\)
\(110\) 0 0
\(111\) 18.0874 1.71678
\(112\) 0 0
\(113\) −18.5970 −1.74946 −0.874731 0.484610i \(-0.838962\pi\)
−0.874731 + 0.484610i \(0.838962\pi\)
\(114\) 0 0
\(115\) −4.28100 −0.399205
\(116\) 0 0
\(117\) −2.49532 −0.230692
\(118\) 0 0
\(119\) 11.9081 1.09162
\(120\) 0 0
\(121\) −2.57136 −0.233760
\(122\) 0 0
\(123\) −24.4701 −2.20640
\(124\) 0 0
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) −15.2050 −1.34922 −0.674611 0.738174i \(-0.735688\pi\)
−0.674611 + 0.738174i \(0.735688\pi\)
\(128\) 0 0
\(129\) −2.42864 −0.213830
\(130\) 0 0
\(131\) 11.0923 0.969142 0.484571 0.874752i \(-0.338976\pi\)
0.484571 + 0.874752i \(0.338976\pi\)
\(132\) 0 0
\(133\) 1.52543 0.132271
\(134\) 0 0
\(135\) −2.42864 −0.209024
\(136\) 0 0
\(137\) 12.1017 1.03392 0.516959 0.856010i \(-0.327064\pi\)
0.516959 + 0.856010i \(0.327064\pi\)
\(138\) 0 0
\(139\) −10.3412 −0.877131 −0.438565 0.898699i \(-0.644513\pi\)
−0.438565 + 0.898699i \(0.644513\pi\)
\(140\) 0 0
\(141\) −9.47949 −0.798317
\(142\) 0 0
\(143\) 3.80642 0.318309
\(144\) 0 0
\(145\) −9.18421 −0.762707
\(146\) 0 0
\(147\) −10.3477 −0.853462
\(148\) 0 0
\(149\) −5.65878 −0.463585 −0.231793 0.972765i \(-0.574459\pi\)
−0.231793 + 0.972765i \(0.574459\pi\)
\(150\) 0 0
\(151\) 11.4795 0.934188 0.467094 0.884208i \(-0.345301\pi\)
0.467094 + 0.884208i \(0.345301\pi\)
\(152\) 0 0
\(153\) −14.8573 −1.20114
\(154\) 0 0
\(155\) 7.80642 0.627027
\(156\) 0 0
\(157\) −0.488863 −0.0390155 −0.0195077 0.999810i \(-0.506210\pi\)
−0.0195077 + 0.999810i \(0.506210\pi\)
\(158\) 0 0
\(159\) 12.2810 0.973946
\(160\) 0 0
\(161\) 6.53035 0.514664
\(162\) 0 0
\(163\) 21.1798 1.65893 0.829464 0.558561i \(-0.188646\pi\)
0.829464 + 0.558561i \(0.188646\pi\)
\(164\) 0 0
\(165\) −6.42864 −0.500469
\(166\) 0 0
\(167\) 16.4494 1.27289 0.636446 0.771321i \(-0.280404\pi\)
0.636446 + 0.771321i \(0.280404\pi\)
\(168\) 0 0
\(169\) −11.2810 −0.867769
\(170\) 0 0
\(171\) −1.90321 −0.145542
\(172\) 0 0
\(173\) −19.8415 −1.50852 −0.754259 0.656577i \(-0.772004\pi\)
−0.754259 + 0.656577i \(0.772004\pi\)
\(174\) 0 0
\(175\) −1.52543 −0.115311
\(176\) 0 0
\(177\) 4.13335 0.310682
\(178\) 0 0
\(179\) 20.7052 1.54758 0.773789 0.633443i \(-0.218359\pi\)
0.773789 + 0.633443i \(0.218359\pi\)
\(180\) 0 0
\(181\) −11.5111 −0.855616 −0.427808 0.903870i \(-0.640714\pi\)
−0.427808 + 0.903870i \(0.640714\pi\)
\(182\) 0 0
\(183\) 1.57136 0.116158
\(184\) 0 0
\(185\) 8.16839 0.600552
\(186\) 0 0
\(187\) 22.6637 1.65733
\(188\) 0 0
\(189\) 3.70471 0.269478
\(190\) 0 0
\(191\) −10.0731 −0.728866 −0.364433 0.931230i \(-0.618737\pi\)
−0.364433 + 0.931230i \(0.618737\pi\)
\(192\) 0 0
\(193\) −12.7304 −0.916353 −0.458176 0.888861i \(-0.651497\pi\)
−0.458176 + 0.888861i \(0.651497\pi\)
\(194\) 0 0
\(195\) −2.90321 −0.207903
\(196\) 0 0
\(197\) 12.2351 0.871712 0.435856 0.900016i \(-0.356446\pi\)
0.435856 + 0.900016i \(0.356446\pi\)
\(198\) 0 0
\(199\) 5.24443 0.371768 0.185884 0.982572i \(-0.440485\pi\)
0.185884 + 0.982572i \(0.440485\pi\)
\(200\) 0 0
\(201\) −3.23014 −0.227837
\(202\) 0 0
\(203\) 14.0098 0.983298
\(204\) 0 0
\(205\) −11.0509 −0.771825
\(206\) 0 0
\(207\) −8.14764 −0.566300
\(208\) 0 0
\(209\) 2.90321 0.200819
\(210\) 0 0
\(211\) −16.7239 −1.15132 −0.575661 0.817688i \(-0.695255\pi\)
−0.575661 + 0.817688i \(0.695255\pi\)
\(212\) 0 0
\(213\) −5.08250 −0.348247
\(214\) 0 0
\(215\) −1.09679 −0.0748003
\(216\) 0 0
\(217\) −11.9081 −0.808377
\(218\) 0 0
\(219\) 31.0923 2.10103
\(220\) 0 0
\(221\) 10.2351 0.688485
\(222\) 0 0
\(223\) 27.5002 1.84155 0.920776 0.390092i \(-0.127557\pi\)
0.920776 + 0.390092i \(0.127557\pi\)
\(224\) 0 0
\(225\) 1.90321 0.126881
\(226\) 0 0
\(227\) −19.3067 −1.28143 −0.640714 0.767780i \(-0.721362\pi\)
−0.640714 + 0.767780i \(0.721362\pi\)
\(228\) 0 0
\(229\) −19.2400 −1.27141 −0.635707 0.771930i \(-0.719291\pi\)
−0.635707 + 0.771930i \(0.719291\pi\)
\(230\) 0 0
\(231\) 9.80642 0.645215
\(232\) 0 0
\(233\) −14.9906 −0.982069 −0.491034 0.871140i \(-0.663381\pi\)
−0.491034 + 0.871140i \(0.663381\pi\)
\(234\) 0 0
\(235\) −4.28100 −0.279261
\(236\) 0 0
\(237\) 13.5812 0.882194
\(238\) 0 0
\(239\) 11.7748 0.761647 0.380824 0.924648i \(-0.375641\pi\)
0.380824 + 0.924648i \(0.375641\pi\)
\(240\) 0 0
\(241\) −10.4889 −0.675647 −0.337824 0.941209i \(-0.609691\pi\)
−0.337824 + 0.941209i \(0.609691\pi\)
\(242\) 0 0
\(243\) −17.2652 −1.10756
\(244\) 0 0
\(245\) −4.67307 −0.298552
\(246\) 0 0
\(247\) 1.31111 0.0834238
\(248\) 0 0
\(249\) 26.0415 1.65031
\(250\) 0 0
\(251\) 11.0509 0.697524 0.348762 0.937211i \(-0.386602\pi\)
0.348762 + 0.937211i \(0.386602\pi\)
\(252\) 0 0
\(253\) 12.4286 0.781382
\(254\) 0 0
\(255\) −17.2859 −1.08249
\(256\) 0 0
\(257\) 17.6795 1.10282 0.551409 0.834235i \(-0.314090\pi\)
0.551409 + 0.834235i \(0.314090\pi\)
\(258\) 0 0
\(259\) −12.4603 −0.774244
\(260\) 0 0
\(261\) −17.4795 −1.08195
\(262\) 0 0
\(263\) 21.1383 1.30344 0.651721 0.758459i \(-0.274047\pi\)
0.651721 + 0.758459i \(0.274047\pi\)
\(264\) 0 0
\(265\) 5.54617 0.340699
\(266\) 0 0
\(267\) 8.13335 0.497753
\(268\) 0 0
\(269\) −17.5812 −1.07194 −0.535972 0.844235i \(-0.680055\pi\)
−0.535972 + 0.844235i \(0.680055\pi\)
\(270\) 0 0
\(271\) −21.0049 −1.27596 −0.637979 0.770054i \(-0.720229\pi\)
−0.637979 + 0.770054i \(0.720229\pi\)
\(272\) 0 0
\(273\) 4.42864 0.268033
\(274\) 0 0
\(275\) −2.90321 −0.175070
\(276\) 0 0
\(277\) −6.85728 −0.412014 −0.206007 0.978551i \(-0.566047\pi\)
−0.206007 + 0.978551i \(0.566047\pi\)
\(278\) 0 0
\(279\) 14.8573 0.889482
\(280\) 0 0
\(281\) 17.0321 1.01605 0.508026 0.861342i \(-0.330376\pi\)
0.508026 + 0.861342i \(0.330376\pi\)
\(282\) 0 0
\(283\) −27.7605 −1.65019 −0.825095 0.564994i \(-0.808878\pi\)
−0.825095 + 0.564994i \(0.808878\pi\)
\(284\) 0 0
\(285\) −2.21432 −0.131165
\(286\) 0 0
\(287\) 16.8573 0.995054
\(288\) 0 0
\(289\) 43.9403 2.58472
\(290\) 0 0
\(291\) −24.8430 −1.45632
\(292\) 0 0
\(293\) −32.3432 −1.88951 −0.944756 0.327775i \(-0.893701\pi\)
−0.944756 + 0.327775i \(0.893701\pi\)
\(294\) 0 0
\(295\) 1.86665 0.108680
\(296\) 0 0
\(297\) 7.05086 0.409132
\(298\) 0 0
\(299\) 5.61285 0.324599
\(300\) 0 0
\(301\) 1.67307 0.0964342
\(302\) 0 0
\(303\) −2.13335 −0.122558
\(304\) 0 0
\(305\) 0.709636 0.0406336
\(306\) 0 0
\(307\) −26.1541 −1.49269 −0.746347 0.665558i \(-0.768194\pi\)
−0.746347 + 0.665558i \(0.768194\pi\)
\(308\) 0 0
\(309\) 39.6084 2.25324
\(310\) 0 0
\(311\) −14.1289 −0.801177 −0.400588 0.916258i \(-0.631194\pi\)
−0.400588 + 0.916258i \(0.631194\pi\)
\(312\) 0 0
\(313\) −6.99063 −0.395134 −0.197567 0.980289i \(-0.563304\pi\)
−0.197567 + 0.980289i \(0.563304\pi\)
\(314\) 0 0
\(315\) −2.90321 −0.163577
\(316\) 0 0
\(317\) −8.62867 −0.484634 −0.242317 0.970197i \(-0.577907\pi\)
−0.242317 + 0.970197i \(0.577907\pi\)
\(318\) 0 0
\(319\) 26.6637 1.49288
\(320\) 0 0
\(321\) −29.8938 −1.66851
\(322\) 0 0
\(323\) 7.80642 0.434361
\(324\) 0 0
\(325\) −1.31111 −0.0727272
\(326\) 0 0
\(327\) 23.3176 1.28946
\(328\) 0 0
\(329\) 6.53035 0.360030
\(330\) 0 0
\(331\) 12.4286 0.683140 0.341570 0.939856i \(-0.389041\pi\)
0.341570 + 0.939856i \(0.389041\pi\)
\(332\) 0 0
\(333\) 15.5462 0.851925
\(334\) 0 0
\(335\) −1.45875 −0.0797001
\(336\) 0 0
\(337\) −5.05731 −0.275489 −0.137745 0.990468i \(-0.543985\pi\)
−0.137745 + 0.990468i \(0.543985\pi\)
\(338\) 0 0
\(339\) −41.1798 −2.23658
\(340\) 0 0
\(341\) −22.6637 −1.22731
\(342\) 0 0
\(343\) 17.8064 0.961457
\(344\) 0 0
\(345\) −9.47949 −0.510359
\(346\) 0 0
\(347\) 3.59856 0.193181 0.0965903 0.995324i \(-0.469206\pi\)
0.0965903 + 0.995324i \(0.469206\pi\)
\(348\) 0 0
\(349\) 0.917502 0.0491128 0.0245564 0.999698i \(-0.492183\pi\)
0.0245564 + 0.999698i \(0.492183\pi\)
\(350\) 0 0
\(351\) 3.18421 0.169960
\(352\) 0 0
\(353\) 20.9304 1.11401 0.557007 0.830508i \(-0.311950\pi\)
0.557007 + 0.830508i \(0.311950\pi\)
\(354\) 0 0
\(355\) −2.29529 −0.121821
\(356\) 0 0
\(357\) 26.3684 1.39556
\(358\) 0 0
\(359\) 4.41435 0.232980 0.116490 0.993192i \(-0.462836\pi\)
0.116490 + 0.993192i \(0.462836\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 0 0
\(363\) −5.69381 −0.298848
\(364\) 0 0
\(365\) 14.0415 0.734965
\(366\) 0 0
\(367\) 17.8336 0.930907 0.465454 0.885072i \(-0.345891\pi\)
0.465454 + 0.885072i \(0.345891\pi\)
\(368\) 0 0
\(369\) −21.0321 −1.09489
\(370\) 0 0
\(371\) −8.46028 −0.439236
\(372\) 0 0
\(373\) 3.73975 0.193637 0.0968184 0.995302i \(-0.469133\pi\)
0.0968184 + 0.995302i \(0.469133\pi\)
\(374\) 0 0
\(375\) 2.21432 0.114347
\(376\) 0 0
\(377\) 12.0415 0.620168
\(378\) 0 0
\(379\) −29.3274 −1.50645 −0.753224 0.657764i \(-0.771502\pi\)
−0.753224 + 0.657764i \(0.771502\pi\)
\(380\) 0 0
\(381\) −33.6686 −1.72490
\(382\) 0 0
\(383\) −26.7447 −1.36659 −0.683294 0.730143i \(-0.739453\pi\)
−0.683294 + 0.730143i \(0.739453\pi\)
\(384\) 0 0
\(385\) 4.42864 0.225704
\(386\) 0 0
\(387\) −2.08742 −0.106110
\(388\) 0 0
\(389\) −17.3876 −0.881588 −0.440794 0.897608i \(-0.645303\pi\)
−0.440794 + 0.897608i \(0.645303\pi\)
\(390\) 0 0
\(391\) 33.4193 1.69009
\(392\) 0 0
\(393\) 24.5620 1.23899
\(394\) 0 0
\(395\) 6.13335 0.308602
\(396\) 0 0
\(397\) 0.152089 0.00763313 0.00381656 0.999993i \(-0.498785\pi\)
0.00381656 + 0.999993i \(0.498785\pi\)
\(398\) 0 0
\(399\) 3.37778 0.169101
\(400\) 0 0
\(401\) −34.3783 −1.71677 −0.858384 0.513007i \(-0.828531\pi\)
−0.858384 + 0.513007i \(0.828531\pi\)
\(402\) 0 0
\(403\) −10.2351 −0.509845
\(404\) 0 0
\(405\) −11.0874 −0.550938
\(406\) 0 0
\(407\) −23.7146 −1.17549
\(408\) 0 0
\(409\) 16.3368 0.807801 0.403901 0.914803i \(-0.367654\pi\)
0.403901 + 0.914803i \(0.367654\pi\)
\(410\) 0 0
\(411\) 26.7971 1.32180
\(412\) 0 0
\(413\) −2.84743 −0.140113
\(414\) 0 0
\(415\) 11.7605 0.577300
\(416\) 0 0
\(417\) −22.8988 −1.12136
\(418\) 0 0
\(419\) −0.161933 −0.00791096 −0.00395548 0.999992i \(-0.501259\pi\)
−0.00395548 + 0.999992i \(0.501259\pi\)
\(420\) 0 0
\(421\) −14.9590 −0.729057 −0.364528 0.931192i \(-0.618770\pi\)
−0.364528 + 0.931192i \(0.618770\pi\)
\(422\) 0 0
\(423\) −8.14764 −0.396152
\(424\) 0 0
\(425\) −7.80642 −0.378667
\(426\) 0 0
\(427\) −1.08250 −0.0523857
\(428\) 0 0
\(429\) 8.42864 0.406939
\(430\) 0 0
\(431\) −10.3269 −0.497431 −0.248715 0.968577i \(-0.580008\pi\)
−0.248715 + 0.968577i \(0.580008\pi\)
\(432\) 0 0
\(433\) 12.7491 0.612683 0.306342 0.951922i \(-0.400895\pi\)
0.306342 + 0.951922i \(0.400895\pi\)
\(434\) 0 0
\(435\) −20.3368 −0.975074
\(436\) 0 0
\(437\) 4.28100 0.204788
\(438\) 0 0
\(439\) 13.5714 0.647726 0.323863 0.946104i \(-0.395018\pi\)
0.323863 + 0.946104i \(0.395018\pi\)
\(440\) 0 0
\(441\) −8.89384 −0.423516
\(442\) 0 0
\(443\) −0.280996 −0.0133505 −0.00667527 0.999978i \(-0.502125\pi\)
−0.00667527 + 0.999978i \(0.502125\pi\)
\(444\) 0 0
\(445\) 3.67307 0.174120
\(446\) 0 0
\(447\) −12.5303 −0.592665
\(448\) 0 0
\(449\) −12.4286 −0.586544 −0.293272 0.956029i \(-0.594744\pi\)
−0.293272 + 0.956029i \(0.594744\pi\)
\(450\) 0 0
\(451\) 32.0830 1.51073
\(452\) 0 0
\(453\) 25.4193 1.19430
\(454\) 0 0
\(455\) 2.00000 0.0937614
\(456\) 0 0
\(457\) 7.24443 0.338880 0.169440 0.985540i \(-0.445804\pi\)
0.169440 + 0.985540i \(0.445804\pi\)
\(458\) 0 0
\(459\) 18.9590 0.884930
\(460\) 0 0
\(461\) 7.32741 0.341271 0.170636 0.985334i \(-0.445418\pi\)
0.170636 + 0.985334i \(0.445418\pi\)
\(462\) 0 0
\(463\) −22.8528 −1.06206 −0.531031 0.847353i \(-0.678195\pi\)
−0.531031 + 0.847353i \(0.678195\pi\)
\(464\) 0 0
\(465\) 17.2859 0.801615
\(466\) 0 0
\(467\) −10.6494 −0.492796 −0.246398 0.969169i \(-0.579247\pi\)
−0.246398 + 0.969169i \(0.579247\pi\)
\(468\) 0 0
\(469\) 2.22522 0.102751
\(470\) 0 0
\(471\) −1.08250 −0.0498789
\(472\) 0 0
\(473\) 3.18421 0.146410
\(474\) 0 0
\(475\) −1.00000 −0.0458831
\(476\) 0 0
\(477\) 10.5555 0.483305
\(478\) 0 0
\(479\) 14.9032 0.680945 0.340473 0.940254i \(-0.389413\pi\)
0.340473 + 0.940254i \(0.389413\pi\)
\(480\) 0 0
\(481\) −10.7096 −0.488317
\(482\) 0 0
\(483\) 14.4603 0.657966
\(484\) 0 0
\(485\) −11.2192 −0.509440
\(486\) 0 0
\(487\) −34.7576 −1.57502 −0.787508 0.616304i \(-0.788629\pi\)
−0.787508 + 0.616304i \(0.788629\pi\)
\(488\) 0 0
\(489\) 46.8988 2.12084
\(490\) 0 0
\(491\) −31.1338 −1.40505 −0.702525 0.711659i \(-0.747944\pi\)
−0.702525 + 0.711659i \(0.747944\pi\)
\(492\) 0 0
\(493\) 71.6958 3.22902
\(494\) 0 0
\(495\) −5.52543 −0.248350
\(496\) 0 0
\(497\) 3.50129 0.157054
\(498\) 0 0
\(499\) −20.8015 −0.931203 −0.465602 0.884994i \(-0.654162\pi\)
−0.465602 + 0.884994i \(0.654162\pi\)
\(500\) 0 0
\(501\) 36.4242 1.62731
\(502\) 0 0
\(503\) 11.5986 0.517154 0.258577 0.965991i \(-0.416746\pi\)
0.258577 + 0.965991i \(0.416746\pi\)
\(504\) 0 0
\(505\) −0.963435 −0.0428723
\(506\) 0 0
\(507\) −24.9797 −1.10939
\(508\) 0 0
\(509\) −12.5018 −0.554131 −0.277066 0.960851i \(-0.589362\pi\)
−0.277066 + 0.960851i \(0.589362\pi\)
\(510\) 0 0
\(511\) −21.4193 −0.947533
\(512\) 0 0
\(513\) 2.42864 0.107227
\(514\) 0 0
\(515\) 17.8874 0.788213
\(516\) 0 0
\(517\) 12.4286 0.546611
\(518\) 0 0
\(519\) −43.9353 −1.92855
\(520\) 0 0
\(521\) −15.8064 −0.692492 −0.346246 0.938144i \(-0.612544\pi\)
−0.346246 + 0.938144i \(0.612544\pi\)
\(522\) 0 0
\(523\) 2.90967 0.127231 0.0636154 0.997974i \(-0.479737\pi\)
0.0636154 + 0.997974i \(0.479737\pi\)
\(524\) 0 0
\(525\) −3.37778 −0.147419
\(526\) 0 0
\(527\) −60.9403 −2.65460
\(528\) 0 0
\(529\) −4.67307 −0.203177
\(530\) 0 0
\(531\) 3.55262 0.154171
\(532\) 0 0
\(533\) 14.4889 0.627582
\(534\) 0 0
\(535\) −13.5002 −0.583666
\(536\) 0 0
\(537\) 45.8479 1.97848
\(538\) 0 0
\(539\) 13.5669 0.584368
\(540\) 0 0
\(541\) 4.18913 0.180105 0.0900524 0.995937i \(-0.471297\pi\)
0.0900524 + 0.995937i \(0.471297\pi\)
\(542\) 0 0
\(543\) −25.4893 −1.09385
\(544\) 0 0
\(545\) 10.5303 0.451071
\(546\) 0 0
\(547\) 18.2143 0.778788 0.389394 0.921071i \(-0.372684\pi\)
0.389394 + 0.921071i \(0.372684\pi\)
\(548\) 0 0
\(549\) 1.35059 0.0576417
\(550\) 0 0
\(551\) 9.18421 0.391260
\(552\) 0 0
\(553\) −9.35599 −0.397857
\(554\) 0 0
\(555\) 18.0874 0.767768
\(556\) 0 0
\(557\) −40.6548 −1.72260 −0.861300 0.508097i \(-0.830349\pi\)
−0.861300 + 0.508097i \(0.830349\pi\)
\(558\) 0 0
\(559\) 1.43801 0.0608212
\(560\) 0 0
\(561\) 50.1847 2.11880
\(562\) 0 0
\(563\) −21.4400 −0.903589 −0.451794 0.892122i \(-0.649216\pi\)
−0.451794 + 0.892122i \(0.649216\pi\)
\(564\) 0 0
\(565\) −18.5970 −0.782383
\(566\) 0 0
\(567\) 16.9131 0.710282
\(568\) 0 0
\(569\) 25.1941 1.05619 0.528095 0.849185i \(-0.322907\pi\)
0.528095 + 0.849185i \(0.322907\pi\)
\(570\) 0 0
\(571\) −5.27163 −0.220611 −0.110305 0.993898i \(-0.535183\pi\)
−0.110305 + 0.993898i \(0.535183\pi\)
\(572\) 0 0
\(573\) −22.3051 −0.931810
\(574\) 0 0
\(575\) −4.28100 −0.178530
\(576\) 0 0
\(577\) 23.3145 0.970595 0.485298 0.874349i \(-0.338711\pi\)
0.485298 + 0.874349i \(0.338711\pi\)
\(578\) 0 0
\(579\) −28.1891 −1.17150
\(580\) 0 0
\(581\) −17.9398 −0.744267
\(582\) 0 0
\(583\) −16.1017 −0.666865
\(584\) 0 0
\(585\) −2.49532 −0.103169
\(586\) 0 0
\(587\) 28.9131 1.19337 0.596685 0.802476i \(-0.296484\pi\)
0.596685 + 0.802476i \(0.296484\pi\)
\(588\) 0 0
\(589\) −7.80642 −0.321658
\(590\) 0 0
\(591\) 27.0923 1.11443
\(592\) 0 0
\(593\) −17.6128 −0.723273 −0.361636 0.932319i \(-0.617782\pi\)
−0.361636 + 0.932319i \(0.617782\pi\)
\(594\) 0 0
\(595\) 11.9081 0.488186
\(596\) 0 0
\(597\) 11.6128 0.475282
\(598\) 0 0
\(599\) 30.4286 1.24328 0.621640 0.783303i \(-0.286467\pi\)
0.621640 + 0.783303i \(0.286467\pi\)
\(600\) 0 0
\(601\) −17.4479 −0.711713 −0.355856 0.934541i \(-0.615811\pi\)
−0.355856 + 0.934541i \(0.615811\pi\)
\(602\) 0 0
\(603\) −2.77631 −0.113060
\(604\) 0 0
\(605\) −2.57136 −0.104541
\(606\) 0 0
\(607\) 24.4592 0.992769 0.496385 0.868103i \(-0.334661\pi\)
0.496385 + 0.868103i \(0.334661\pi\)
\(608\) 0 0
\(609\) 31.0223 1.25709
\(610\) 0 0
\(611\) 5.61285 0.227072
\(612\) 0 0
\(613\) −12.1432 −0.490459 −0.245230 0.969465i \(-0.578863\pi\)
−0.245230 + 0.969465i \(0.578863\pi\)
\(614\) 0 0
\(615\) −24.4701 −0.986731
\(616\) 0 0
\(617\) 25.3876 1.02207 0.511034 0.859561i \(-0.329263\pi\)
0.511034 + 0.859561i \(0.329263\pi\)
\(618\) 0 0
\(619\) −20.4429 −0.821671 −0.410835 0.911710i \(-0.634763\pi\)
−0.410835 + 0.911710i \(0.634763\pi\)
\(620\) 0 0
\(621\) 10.3970 0.417217
\(622\) 0 0
\(623\) −5.60300 −0.224480
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 6.42864 0.256735
\(628\) 0 0
\(629\) −63.7659 −2.54251
\(630\) 0 0
\(631\) −5.65878 −0.225273 −0.112636 0.993636i \(-0.535929\pi\)
−0.112636 + 0.993636i \(0.535929\pi\)
\(632\) 0 0
\(633\) −37.0321 −1.47189
\(634\) 0 0
\(635\) −15.2050 −0.603390
\(636\) 0 0
\(637\) 6.12690 0.242757
\(638\) 0 0
\(639\) −4.36842 −0.172812
\(640\) 0 0
\(641\) 8.20342 0.324016 0.162008 0.986789i \(-0.448203\pi\)
0.162008 + 0.986789i \(0.448203\pi\)
\(642\) 0 0
\(643\) −10.0874 −0.397809 −0.198905 0.980019i \(-0.563738\pi\)
−0.198905 + 0.980019i \(0.563738\pi\)
\(644\) 0 0
\(645\) −2.42864 −0.0956276
\(646\) 0 0
\(647\) 31.3733 1.23341 0.616707 0.787193i \(-0.288466\pi\)
0.616707 + 0.787193i \(0.288466\pi\)
\(648\) 0 0
\(649\) −5.41927 −0.212725
\(650\) 0 0
\(651\) −26.3684 −1.03346
\(652\) 0 0
\(653\) 28.4099 1.11177 0.555883 0.831261i \(-0.312380\pi\)
0.555883 + 0.831261i \(0.312380\pi\)
\(654\) 0 0
\(655\) 11.0923 0.433414
\(656\) 0 0
\(657\) 26.7239 1.04260
\(658\) 0 0
\(659\) 9.74620 0.379658 0.189829 0.981817i \(-0.439207\pi\)
0.189829 + 0.981817i \(0.439207\pi\)
\(660\) 0 0
\(661\) 15.6128 0.607269 0.303635 0.952789i \(-0.401800\pi\)
0.303635 + 0.952789i \(0.401800\pi\)
\(662\) 0 0
\(663\) 22.6637 0.880185
\(664\) 0 0
\(665\) 1.52543 0.0591535
\(666\) 0 0
\(667\) 39.3176 1.52238
\(668\) 0 0
\(669\) 60.8943 2.35431
\(670\) 0 0
\(671\) −2.06022 −0.0795340
\(672\) 0 0
\(673\) 22.0350 0.849388 0.424694 0.905337i \(-0.360382\pi\)
0.424694 + 0.905337i \(0.360382\pi\)
\(674\) 0 0
\(675\) −2.42864 −0.0934784
\(676\) 0 0
\(677\) 30.4953 1.17203 0.586015 0.810300i \(-0.300696\pi\)
0.586015 + 0.810300i \(0.300696\pi\)
\(678\) 0 0
\(679\) 17.1141 0.656780
\(680\) 0 0
\(681\) −42.7511 −1.63823
\(682\) 0 0
\(683\) 11.2968 0.432261 0.216130 0.976365i \(-0.430656\pi\)
0.216130 + 0.976365i \(0.430656\pi\)
\(684\) 0 0
\(685\) 12.1017 0.462383
\(686\) 0 0
\(687\) −42.6035 −1.62542
\(688\) 0 0
\(689\) −7.27163 −0.277027
\(690\) 0 0
\(691\) −18.5161 −0.704384 −0.352192 0.935928i \(-0.614564\pi\)
−0.352192 + 0.935928i \(0.614564\pi\)
\(692\) 0 0
\(693\) 8.42864 0.320178
\(694\) 0 0
\(695\) −10.3412 −0.392265
\(696\) 0 0
\(697\) 86.2677 3.26762
\(698\) 0 0
\(699\) −33.1941 −1.25551
\(700\) 0 0
\(701\) −27.0781 −1.02272 −0.511362 0.859365i \(-0.670859\pi\)
−0.511362 + 0.859365i \(0.670859\pi\)
\(702\) 0 0
\(703\) −8.16839 −0.308077
\(704\) 0 0
\(705\) −9.47949 −0.357018
\(706\) 0 0
\(707\) 1.46965 0.0552719
\(708\) 0 0
\(709\) −7.01921 −0.263612 −0.131806 0.991276i \(-0.542078\pi\)
−0.131806 + 0.991276i \(0.542078\pi\)
\(710\) 0 0
\(711\) 11.6731 0.437774
\(712\) 0 0
\(713\) −33.4193 −1.25156
\(714\) 0 0
\(715\) 3.80642 0.142352
\(716\) 0 0
\(717\) 26.0731 0.973719
\(718\) 0 0
\(719\) −36.0272 −1.34359 −0.671794 0.740738i \(-0.734476\pi\)
−0.671794 + 0.740738i \(0.734476\pi\)
\(720\) 0 0
\(721\) −27.2859 −1.01618
\(722\) 0 0
\(723\) −23.2257 −0.863773
\(724\) 0 0
\(725\) −9.18421 −0.341093
\(726\) 0 0
\(727\) −15.9442 −0.591338 −0.295669 0.955290i \(-0.595543\pi\)
−0.295669 + 0.955290i \(0.595543\pi\)
\(728\) 0 0
\(729\) −4.96836 −0.184013
\(730\) 0 0
\(731\) 8.56199 0.316677
\(732\) 0 0
\(733\) −10.9621 −0.404893 −0.202446 0.979293i \(-0.564889\pi\)
−0.202446 + 0.979293i \(0.564889\pi\)
\(734\) 0 0
\(735\) −10.3477 −0.381680
\(736\) 0 0
\(737\) 4.23506 0.156001
\(738\) 0 0
\(739\) 13.7146 0.504498 0.252249 0.967662i \(-0.418830\pi\)
0.252249 + 0.967662i \(0.418830\pi\)
\(740\) 0 0
\(741\) 2.90321 0.106652
\(742\) 0 0
\(743\) 33.6523 1.23458 0.617292 0.786734i \(-0.288230\pi\)
0.617292 + 0.786734i \(0.288230\pi\)
\(744\) 0 0
\(745\) −5.65878 −0.207322
\(746\) 0 0
\(747\) 22.3827 0.818940
\(748\) 0 0
\(749\) 20.5936 0.752475
\(750\) 0 0
\(751\) −36.7150 −1.33975 −0.669875 0.742474i \(-0.733652\pi\)
−0.669875 + 0.742474i \(0.733652\pi\)
\(752\) 0 0
\(753\) 24.4701 0.891741
\(754\) 0 0
\(755\) 11.4795 0.417782
\(756\) 0 0
\(757\) 40.6222 1.47644 0.738220 0.674560i \(-0.235667\pi\)
0.738220 + 0.674560i \(0.235667\pi\)
\(758\) 0 0
\(759\) 27.5210 0.998948
\(760\) 0 0
\(761\) −35.1481 −1.27412 −0.637059 0.770815i \(-0.719849\pi\)
−0.637059 + 0.770815i \(0.719849\pi\)
\(762\) 0 0
\(763\) −16.0633 −0.581530
\(764\) 0 0
\(765\) −14.8573 −0.537166
\(766\) 0 0
\(767\) −2.44738 −0.0883696
\(768\) 0 0
\(769\) 41.1383 1.48348 0.741742 0.670685i \(-0.234000\pi\)
0.741742 + 0.670685i \(0.234000\pi\)
\(770\) 0 0
\(771\) 39.1481 1.40989
\(772\) 0 0
\(773\) 45.0484 1.62028 0.810139 0.586237i \(-0.199391\pi\)
0.810139 + 0.586237i \(0.199391\pi\)
\(774\) 0 0
\(775\) 7.80642 0.280415
\(776\) 0 0
\(777\) −27.5910 −0.989823
\(778\) 0 0
\(779\) 11.0509 0.395938
\(780\) 0 0
\(781\) 6.66370 0.238446
\(782\) 0 0
\(783\) 22.3051 0.797120
\(784\) 0 0
\(785\) −0.488863 −0.0174483
\(786\) 0 0
\(787\) 25.4400 0.906839 0.453419 0.891297i \(-0.350204\pi\)
0.453419 + 0.891297i \(0.350204\pi\)
\(788\) 0 0
\(789\) 46.8069 1.66637
\(790\) 0 0
\(791\) 28.3684 1.00866
\(792\) 0 0
\(793\) −0.930409 −0.0330398
\(794\) 0 0
\(795\) 12.2810 0.435562
\(796\) 0 0
\(797\) 20.5684 0.728572 0.364286 0.931287i \(-0.381313\pi\)
0.364286 + 0.931287i \(0.381313\pi\)
\(798\) 0 0
\(799\) 33.4193 1.18229
\(800\) 0 0
\(801\) 6.99063 0.247002
\(802\) 0 0
\(803\) −40.7654 −1.43858
\(804\) 0 0
\(805\) 6.53035 0.230165
\(806\) 0 0
\(807\) −38.9304 −1.37042
\(808\) 0 0
\(809\) 53.2039 1.87055 0.935275 0.353923i \(-0.115152\pi\)
0.935275 + 0.353923i \(0.115152\pi\)
\(810\) 0 0
\(811\) 24.7467 0.868973 0.434487 0.900678i \(-0.356930\pi\)
0.434487 + 0.900678i \(0.356930\pi\)
\(812\) 0 0
\(813\) −46.5116 −1.63123
\(814\) 0 0
\(815\) 21.1798 0.741895
\(816\) 0 0
\(817\) 1.09679 0.0383718
\(818\) 0 0
\(819\) 3.80642 0.133007
\(820\) 0 0
\(821\) −6.61237 −0.230773 −0.115387 0.993321i \(-0.536811\pi\)
−0.115387 + 0.993321i \(0.536811\pi\)
\(822\) 0 0
\(823\) 37.7418 1.31559 0.657797 0.753195i \(-0.271488\pi\)
0.657797 + 0.753195i \(0.271488\pi\)
\(824\) 0 0
\(825\) −6.42864 −0.223816
\(826\) 0 0
\(827\) 25.1318 0.873919 0.436960 0.899481i \(-0.356055\pi\)
0.436960 + 0.899481i \(0.356055\pi\)
\(828\) 0 0
\(829\) −12.5116 −0.434546 −0.217273 0.976111i \(-0.569716\pi\)
−0.217273 + 0.976111i \(0.569716\pi\)
\(830\) 0 0
\(831\) −15.1842 −0.526734
\(832\) 0 0
\(833\) 36.4800 1.26396
\(834\) 0 0
\(835\) 16.4494 0.569254
\(836\) 0 0
\(837\) −18.9590 −0.655319
\(838\) 0 0
\(839\) 16.3269 0.563668 0.281834 0.959463i \(-0.409057\pi\)
0.281834 + 0.959463i \(0.409057\pi\)
\(840\) 0 0
\(841\) 55.3497 1.90861
\(842\) 0 0
\(843\) 37.7146 1.29896
\(844\) 0 0
\(845\) −11.2810 −0.388078
\(846\) 0 0
\(847\) 3.92242 0.134776
\(848\) 0 0
\(849\) −61.4706 −2.10967
\(850\) 0 0
\(851\) −34.9688 −1.19872
\(852\) 0 0
\(853\) 14.8287 0.507725 0.253862 0.967240i \(-0.418299\pi\)
0.253862 + 0.967240i \(0.418299\pi\)
\(854\) 0 0
\(855\) −1.90321 −0.0650885
\(856\) 0 0
\(857\) −10.0796 −0.344312 −0.172156 0.985070i \(-0.555073\pi\)
−0.172156 + 0.985070i \(0.555073\pi\)
\(858\) 0 0
\(859\) −36.7783 −1.25486 −0.627430 0.778673i \(-0.715893\pi\)
−0.627430 + 0.778673i \(0.715893\pi\)
\(860\) 0 0
\(861\) 37.3274 1.27211
\(862\) 0 0
\(863\) −17.7857 −0.605432 −0.302716 0.953081i \(-0.597893\pi\)
−0.302716 + 0.953081i \(0.597893\pi\)
\(864\) 0 0
\(865\) −19.8415 −0.674630
\(866\) 0 0
\(867\) 97.2978 3.30441
\(868\) 0 0
\(869\) −17.8064 −0.604042
\(870\) 0 0
\(871\) 1.91258 0.0648053
\(872\) 0 0
\(873\) −21.3526 −0.722676
\(874\) 0 0
\(875\) −1.52543 −0.0515689
\(876\) 0 0
\(877\) −26.8321 −0.906055 −0.453028 0.891496i \(-0.649656\pi\)
−0.453028 + 0.891496i \(0.649656\pi\)
\(878\) 0 0
\(879\) −71.6182 −2.41562
\(880\) 0 0
\(881\) −27.8524 −0.938370 −0.469185 0.883100i \(-0.655452\pi\)
−0.469185 + 0.883100i \(0.655452\pi\)
\(882\) 0 0
\(883\) 13.8751 0.466935 0.233467 0.972365i \(-0.424993\pi\)
0.233467 + 0.972365i \(0.424993\pi\)
\(884\) 0 0
\(885\) 4.13335 0.138941
\(886\) 0 0
\(887\) −26.1827 −0.879128 −0.439564 0.898211i \(-0.644867\pi\)
−0.439564 + 0.898211i \(0.644867\pi\)
\(888\) 0 0
\(889\) 23.1941 0.777904
\(890\) 0 0
\(891\) 32.1891 1.07838
\(892\) 0 0
\(893\) 4.28100 0.143258
\(894\) 0 0
\(895\) 20.7052 0.692098
\(896\) 0 0
\(897\) 12.4286 0.414980
\(898\) 0 0
\(899\) −71.6958 −2.39119
\(900\) 0 0
\(901\) −43.2958 −1.44239
\(902\) 0 0
\(903\) 3.70471 0.123285
\(904\) 0 0
\(905\) −11.5111 −0.382643
\(906\) 0 0
\(907\) 1.84590 0.0612922 0.0306461 0.999530i \(-0.490244\pi\)
0.0306461 + 0.999530i \(0.490244\pi\)
\(908\) 0 0
\(909\) −1.83362 −0.0608174
\(910\) 0 0
\(911\) 47.1467 1.56204 0.781021 0.624505i \(-0.214699\pi\)
0.781021 + 0.624505i \(0.214699\pi\)
\(912\) 0 0
\(913\) −34.1432 −1.12997
\(914\) 0 0
\(915\) 1.57136 0.0519476
\(916\) 0 0
\(917\) −16.9206 −0.558766
\(918\) 0 0
\(919\) 15.4509 0.509679 0.254839 0.966983i \(-0.417977\pi\)
0.254839 + 0.966983i \(0.417977\pi\)
\(920\) 0 0
\(921\) −57.9135 −1.90832
\(922\) 0 0
\(923\) 3.00937 0.0990546
\(924\) 0 0
\(925\) 8.16839 0.268575
\(926\) 0 0
\(927\) 34.0435 1.11814
\(928\) 0 0
\(929\) 17.8163 0.584533 0.292266 0.956337i \(-0.405591\pi\)
0.292266 + 0.956337i \(0.405591\pi\)
\(930\) 0 0
\(931\) 4.67307 0.153154
\(932\) 0 0
\(933\) −31.2859 −1.02425
\(934\) 0 0
\(935\) 22.6637 0.741182
\(936\) 0 0
\(937\) 30.6953 1.00277 0.501387 0.865223i \(-0.332823\pi\)
0.501387 + 0.865223i \(0.332823\pi\)
\(938\) 0 0
\(939\) −15.4795 −0.505154
\(940\) 0 0
\(941\) −11.0696 −0.360858 −0.180429 0.983588i \(-0.557749\pi\)
−0.180429 + 0.983588i \(0.557749\pi\)
\(942\) 0 0
\(943\) 47.3087 1.54058
\(944\) 0 0
\(945\) 3.70471 0.120514
\(946\) 0 0
\(947\) 11.3033 0.367307 0.183653 0.982991i \(-0.441208\pi\)
0.183653 + 0.982991i \(0.441208\pi\)
\(948\) 0 0
\(949\) −18.4099 −0.597611
\(950\) 0 0
\(951\) −19.1066 −0.619575
\(952\) 0 0
\(953\) −20.8637 −0.675843 −0.337921 0.941174i \(-0.609724\pi\)
−0.337921 + 0.941174i \(0.609724\pi\)
\(954\) 0 0
\(955\) −10.0731 −0.325959
\(956\) 0 0
\(957\) 59.0420 1.90856
\(958\) 0 0
\(959\) −18.4603 −0.596114
\(960\) 0 0
\(961\) 29.9403 0.965815
\(962\) 0 0
\(963\) −25.6938 −0.827972
\(964\) 0 0
\(965\) −12.7304 −0.409805
\(966\) 0 0
\(967\) −56.7007 −1.82337 −0.911687 0.410886i \(-0.865219\pi\)
−0.911687 + 0.410886i \(0.865219\pi\)
\(968\) 0 0
\(969\) 17.2859 0.555304
\(970\) 0 0
\(971\) 24.8760 0.798309 0.399155 0.916884i \(-0.369304\pi\)
0.399155 + 0.916884i \(0.369304\pi\)
\(972\) 0 0
\(973\) 15.7748 0.505716
\(974\) 0 0
\(975\) −2.90321 −0.0929772
\(976\) 0 0
\(977\) −21.9847 −0.703351 −0.351676 0.936122i \(-0.614388\pi\)
−0.351676 + 0.936122i \(0.614388\pi\)
\(978\) 0 0
\(979\) −10.6637 −0.340813
\(980\) 0 0
\(981\) 20.0415 0.639875
\(982\) 0 0
\(983\) 29.7540 0.949006 0.474503 0.880254i \(-0.342628\pi\)
0.474503 + 0.880254i \(0.342628\pi\)
\(984\) 0 0
\(985\) 12.2351 0.389842
\(986\) 0 0
\(987\) 14.4603 0.460276
\(988\) 0 0
\(989\) 4.69535 0.149303
\(990\) 0 0
\(991\) −38.9719 −1.23798 −0.618992 0.785398i \(-0.712458\pi\)
−0.618992 + 0.785398i \(0.712458\pi\)
\(992\) 0 0
\(993\) 27.5210 0.873352
\(994\) 0 0
\(995\) 5.24443 0.166260
\(996\) 0 0
\(997\) −18.8287 −0.596311 −0.298155 0.954517i \(-0.596371\pi\)
−0.298155 + 0.954517i \(0.596371\pi\)
\(998\) 0 0
\(999\) −19.8381 −0.627649
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 3040.2.a.m.1.3 yes 3
4.3 odd 2 3040.2.a.l.1.1 3
8.3 odd 2 6080.2.a.bt.1.3 3
8.5 even 2 6080.2.a.bu.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3040.2.a.l.1.1 3 4.3 odd 2
3040.2.a.m.1.3 yes 3 1.1 even 1 trivial
6080.2.a.bt.1.3 3 8.3 odd 2
6080.2.a.bu.1.1 3 8.5 even 2