Properties

Label 3040.2.a.l.1.1
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 1.1
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 \(-0.720742\pi\)
−0.639219 + 0.769025i \(0.720742\pi\)
\(4\) 0 0
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) 1.52543 0.576557 0.288279 0.957547i \(-0.406917\pi\)
0.288279 + 0.957547i \(0.406917\pi\)
\(8\) 0 0
\(9\) 1.90321 0.634404
\(10\) 0 0
\(11\) 2.90321 0.875351 0.437676 0.899133i \(-0.355802\pi\)
0.437676 + 0.899133i \(0.355802\pi\)
\(12\) 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 \(-0.352733\pi\)
0.446325 + 0.894871i \(0.352733\pi\)
\(24\) 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 \(-0.747279\pi\)
−0.701038 + 0.713124i \(0.747279\pi\)
\(32\) 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 \(-0.473349\pi\)
0.0836293 + 0.996497i \(0.473349\pi\)
\(44\) 0 0
\(45\) 1.90321 0.283714
\(46\) 0 0
\(47\) 4.28100 0.624447 0.312224 0.950009i \(-0.398926\pi\)
0.312224 + 0.950009i \(0.398926\pi\)
\(48\) 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 \(-0.538773\pi\)
−0.121508 + 0.992590i \(0.538773\pi\)
\(60\) 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 \(-0.471599\pi\)
0.0891074 + 0.996022i \(0.471599\pi\)
\(68\) 0 0
\(69\) −9.47949 −1.14120
\(70\) 0 0
\(71\) 2.29529 0.272400 0.136200 0.990681i \(-0.456511\pi\)
0.136200 + 0.990681i \(0.456511\pi\)
\(72\) 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 \(-0.612131\pi\)
−0.345028 + 0.938592i \(0.612131\pi\)
\(80\) 0 0
\(81\) −11.0874 −1.23194
\(82\) 0 0
\(83\) −11.7605 −1.29088 −0.645441 0.763810i \(-0.723326\pi\)
−0.645441 + 0.763810i \(0.723326\pi\)
\(84\) 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 \(-0.843296\pi\)
−0.881248 + 0.472653i \(0.843296\pi\)
\(104\) 0 0
\(105\) −3.37778 −0.329638
\(106\) 0 0
\(107\) 13.5002 1.30512 0.652559 0.757738i \(-0.273696\pi\)
0.652559 + 0.757738i \(0.273696\pi\)
\(108\) 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 \(-0.264312\pi\)
0.674611 + 0.738174i \(0.264312\pi\)
\(128\) 0 0
\(129\) −2.42864 −0.213830
\(130\) 0 0
\(131\) −11.0923 −0.969142 −0.484571 0.874752i \(-0.661024\pi\)
−0.484571 + 0.874752i \(0.661024\pi\)
\(132\) 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 \(-0.355487\pi\)
0.438565 + 0.898699i \(0.355487\pi\)
\(140\) 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 \(-0.654699\pi\)
−0.467094 + 0.884208i \(0.654699\pi\)
\(152\) 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 \(-0.811354\pi\)
−0.829464 + 0.558561i \(0.811354\pi\)
\(164\) 0 0
\(165\) −6.42864 −0.500469
\(166\) 0 0
\(167\) −16.4494 −1.27289 −0.636446 0.771321i \(-0.719596\pi\)
−0.636446 + 0.771321i \(0.719596\pi\)
\(168\) 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 \(-0.781641\pi\)
−0.773789 + 0.633443i \(0.781641\pi\)
\(180\) 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 \(-0.381263\pi\)
0.364433 + 0.931230i \(0.381263\pi\)
\(192\) 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 \(-0.559515\pi\)
−0.185884 + 0.982572i \(0.559515\pi\)
\(200\) 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 \(-0.304745\pi\)
0.575661 + 0.817688i \(0.304745\pi\)
\(212\) 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 \(-0.872443\pi\)
−0.920776 + 0.390092i \(0.872443\pi\)
\(224\) 0 0
\(225\) 1.90321 0.126881
\(226\) 0 0
\(227\) 19.3067 1.28143 0.640714 0.767780i \(-0.278638\pi\)
0.640714 + 0.767780i \(0.278638\pi\)
\(228\) 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 \(-0.624359\pi\)
−0.380824 + 0.924648i \(0.624359\pi\)
\(240\) 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 \(-0.613398\pi\)
−0.348762 + 0.937211i \(0.613398\pi\)
\(252\) 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 \(-0.725953\pi\)
−0.651721 + 0.758459i \(0.725953\pi\)
\(264\) 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 \(-0.279771\pi\)
0.637979 + 0.770054i \(0.279771\pi\)
\(272\) 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 \(-0.191122\pi\)
0.825095 + 0.564994i \(0.191122\pi\)
\(284\) 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 \(-0.231806\pi\)
0.746347 + 0.665558i \(0.231806\pi\)
\(308\) 0 0
\(309\) 39.6084 2.25324
\(310\) 0 0
\(311\) 14.1289 0.801177 0.400588 0.916258i \(-0.368806\pi\)
0.400588 + 0.916258i \(0.368806\pi\)
\(312\) 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 \(-0.610959\pi\)
−0.341570 + 0.939856i \(0.610959\pi\)
\(332\) 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 \(-0.530794\pi\)
−0.0965903 + 0.995324i \(0.530794\pi\)
\(348\) 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 \(-0.537164\pi\)
−0.116490 + 0.993192i \(0.537164\pi\)
\(360\) 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 \(-0.654109\pi\)
−0.465454 + 0.885072i \(0.654109\pi\)
\(368\) 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 \(-0.228498\pi\)
0.753224 + 0.657764i \(0.228498\pi\)
\(380\) 0 0
\(381\) −33.6686 −1.72490
\(382\) 0 0
\(383\) 26.7447 1.36659 0.683294 0.730143i \(-0.260547\pi\)
0.683294 + 0.730143i \(0.260547\pi\)
\(384\) 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 \(-0.498741\pi\)
0.00395548 + 0.999992i \(0.498741\pi\)
\(420\) 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 \(-0.419992\pi\)
0.248715 + 0.968577i \(0.419992\pi\)
\(432\) 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 \(-0.604982\pi\)
−0.323863 + 0.946104i \(0.604982\pi\)
\(440\) 0 0
\(441\) −8.89384 −0.423516
\(442\) 0 0
\(443\) 0.280996 0.0133505 0.00667527 0.999978i \(-0.497875\pi\)
0.00667527 + 0.999978i \(0.497875\pi\)
\(444\) 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 \(-0.321805\pi\)
0.531031 + 0.847353i \(0.321805\pi\)
\(464\) 0 0
\(465\) 17.2859 0.801615
\(466\) 0 0
\(467\) 10.6494 0.492796 0.246398 0.969169i \(-0.420753\pi\)
0.246398 + 0.969169i \(0.420753\pi\)
\(468\) 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 \(-0.610587\pi\)
−0.340473 + 0.940254i \(0.610587\pi\)
\(480\) 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 \(-0.211371\pi\)
0.787508 + 0.616304i \(0.211371\pi\)
\(488\) 0 0
\(489\) 46.8988 2.12084
\(490\) 0 0
\(491\) 31.1338 1.40505 0.702525 0.711659i \(-0.252056\pi\)
0.702525 + 0.711659i \(0.252056\pi\)
\(492\) 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 \(-0.345838\pi\)
0.465602 + 0.884994i \(0.345838\pi\)
\(500\) 0 0
\(501\) 36.4242 1.62731
\(502\) 0 0
\(503\) −11.5986 −0.517154 −0.258577 0.965991i \(-0.583254\pi\)
−0.258577 + 0.965991i \(0.583254\pi\)
\(504\) 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 \(-0.520263\pi\)
−0.0636154 + 0.997974i \(0.520263\pi\)
\(524\) 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 \(-0.627316\pi\)
−0.389394 + 0.921071i \(0.627316\pi\)
\(548\) 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 \(-0.350784\pi\)
0.451794 + 0.892122i \(0.350784\pi\)
\(564\) 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 \(-0.464817\pi\)
0.110305 + 0.993898i \(0.464817\pi\)
\(572\) 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 \(-0.703516\pi\)
−0.596685 + 0.802476i \(0.703516\pi\)
\(588\) 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 \(-0.713533\pi\)
−0.621640 + 0.783303i \(0.713533\pi\)
\(600\) 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 \(-0.665339\pi\)
−0.496385 + 0.868103i \(0.665339\pi\)
\(608\) 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 \(-0.365237\pi\)
0.410835 + 0.911710i \(0.365237\pi\)
\(620\) 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 \(-0.464071\pi\)
0.112636 + 0.993636i \(0.464071\pi\)
\(632\) 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 \(-0.436262\pi\)
0.198905 + 0.980019i \(0.436262\pi\)
\(644\) 0 0
\(645\) −2.42864 −0.0956276
\(646\) 0 0
\(647\) −31.3733 −1.23341 −0.616707 0.787193i \(-0.711534\pi\)
−0.616707 + 0.787193i \(0.711534\pi\)
\(648\) 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 \(-0.560793\pi\)
−0.189829 + 0.981817i \(0.560793\pi\)
\(660\) 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 \(-0.569344\pi\)
−0.216130 + 0.976365i \(0.569344\pi\)
\(684\) 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 \(-0.385436\pi\)
0.352192 + 0.935928i \(0.385436\pi\)
\(692\) 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 \(-0.265524\pi\)
0.671794 + 0.740738i \(0.265524\pi\)
\(720\) 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 \(-0.404457\pi\)
0.295669 + 0.955290i \(0.404457\pi\)
\(728\) 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 \(-0.581170\pi\)
−0.252249 + 0.967662i \(0.581170\pi\)
\(740\) 0 0
\(741\) 2.90321 0.106652
\(742\) 0 0
\(743\) −33.6523 −1.23458 −0.617292 0.786734i \(-0.711770\pi\)
−0.617292 + 0.786734i \(0.711770\pi\)
\(744\) 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 \(-0.266348\pi\)
0.669875 + 0.742474i \(0.266348\pi\)
\(752\) 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 \(-0.649796\pi\)
−0.453419 + 0.891297i \(0.649796\pi\)
\(788\) 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 \(-0.643070\pi\)
−0.434487 + 0.900678i \(0.643070\pi\)
\(812\) 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 \(-0.728512\pi\)
−0.657797 + 0.753195i \(0.728512\pi\)
\(824\) 0 0
\(825\) −6.42864 −0.223816
\(826\) 0 0
\(827\) −25.1318 −0.873919 −0.436960 0.899481i \(-0.643945\pi\)
−0.436960 + 0.899481i \(0.643945\pi\)
\(828\) 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 \(-0.590943\pi\)
−0.281834 + 0.959463i \(0.590943\pi\)
\(840\) 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 \(-0.284107\pi\)
0.627430 + 0.778673i \(0.284107\pi\)
\(860\) 0 0
\(861\) 37.3274 1.27211
\(862\) 0 0
\(863\) 17.7857 0.605432 0.302716 0.953081i \(-0.402107\pi\)
0.302716 + 0.953081i \(0.402107\pi\)
\(864\) 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 \(-0.575007\pi\)
−0.233467 + 0.972365i \(0.575007\pi\)
\(884\) 0 0
\(885\) 4.13335 0.138941
\(886\) 0 0
\(887\) 26.1827 0.879128 0.439564 0.898211i \(-0.355133\pi\)
0.439564 + 0.898211i \(0.355133\pi\)
\(888\) 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 \(-0.509756\pi\)
−0.0306461 + 0.999530i \(0.509756\pi\)
\(908\) 0 0
\(909\) −1.83362 −0.0608174
\(910\) 0 0
\(911\) −47.1467 −1.56204 −0.781021 0.624505i \(-0.785301\pi\)
−0.781021 + 0.624505i \(0.785301\pi\)
\(912\) 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 \(-0.582023\pi\)
−0.254839 + 0.966983i \(0.582023\pi\)
\(920\) 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 \(-0.558792\pi\)
−0.183653 + 0.982991i \(0.558792\pi\)
\(948\) 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 \(-0.134781\pi\)
0.911687 + 0.410886i \(0.134781\pi\)
\(968\) 0 0
\(969\) 17.2859 0.555304
\(970\) 0 0
\(971\) −24.8760 −0.798309 −0.399155 0.916884i \(-0.630696\pi\)
−0.399155 + 0.916884i \(0.630696\pi\)
\(972\) 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 \(-0.657372\pi\)
−0.474503 + 0.880254i \(0.657372\pi\)
\(984\) 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 \(-0.287542\pi\)
0.618992 + 0.785398i \(0.287542\pi\)
\(992\) 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.l.1.1 3
4.3 odd 2 3040.2.a.m.1.3 yes 3
8.3 odd 2 6080.2.a.bu.1.1 3
8.5 even 2 6080.2.a.bt.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3040.2.a.l.1.1 3 1.1 even 1 trivial
3040.2.a.m.1.3 yes 3 4.3 odd 2
6080.2.a.bt.1.3 3 8.5 even 2
6080.2.a.bu.1.1 3 8.3 odd 2