Properties

Label 4004.2.a.i.1.1
Level $4004$
Weight $2$
Character 4004.1
Self dual yes
Analytic conductor $31.972$
Analytic rank $0$
Dimension $9$
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] = [4004,2,Mod(1,4004)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4004, 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("4004.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4004 = 2^{2} \cdot 7 \cdot 11 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4004.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: \(31.9721009693\)
Analytic rank: \(0\)
Dimension: \(9\)
Coefficient field: \(\mathbb{Q}[x]/(x^{9} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{9} - 4x^{8} - 12x^{7} + 60x^{6} + 15x^{5} - 233x^{4} + 74x^{3} + 271x^{2} - 67x - 87 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-3.11788\) of defining polynomial
Character \(\chi\) \(=\) 4004.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-3.11788 q^{3} +4.40203 q^{5} +1.00000 q^{7} +6.72118 q^{9} +O(q^{10})\) \(q-3.11788 q^{3} +4.40203 q^{5} +1.00000 q^{7} +6.72118 q^{9} -1.00000 q^{11} +1.00000 q^{13} -13.7250 q^{15} -1.59028 q^{17} +3.13343 q^{19} -3.11788 q^{21} -2.76292 q^{23} +14.3779 q^{25} -11.6022 q^{27} +1.82484 q^{29} +5.86167 q^{31} +3.11788 q^{33} +4.40203 q^{35} +2.94070 q^{37} -3.11788 q^{39} -6.96742 q^{41} +5.12559 q^{43} +29.5869 q^{45} +8.67005 q^{47} +1.00000 q^{49} +4.95831 q^{51} -0.705189 q^{53} -4.40203 q^{55} -9.76966 q^{57} -11.2123 q^{59} +6.91210 q^{61} +6.72118 q^{63} +4.40203 q^{65} +2.69117 q^{67} +8.61444 q^{69} -9.24458 q^{71} +13.2313 q^{73} -44.8285 q^{75} -1.00000 q^{77} +9.32396 q^{79} +16.0108 q^{81} -6.98352 q^{83} -7.00047 q^{85} -5.68963 q^{87} +0.583301 q^{89} +1.00000 q^{91} -18.2760 q^{93} +13.7935 q^{95} +0.299773 q^{97} -6.72118 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 9 q + 4 q^{3} + 4 q^{5} + 9 q^{7} + 13 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 9 q + 4 q^{3} + 4 q^{5} + 9 q^{7} + 13 q^{9} - 9 q^{11} + 9 q^{13} - 5 q^{17} + 17 q^{19} + 4 q^{21} - 8 q^{23} + 35 q^{25} + 4 q^{27} - 3 q^{29} + 9 q^{31} - 4 q^{33} + 4 q^{35} + 7 q^{37} + 4 q^{39} + 14 q^{41} + 21 q^{43} + 43 q^{45} + q^{47} + 9 q^{49} + 25 q^{51} - 8 q^{53} - 4 q^{55} + 8 q^{57} - 4 q^{59} + 30 q^{61} + 13 q^{63} + 4 q^{65} + 15 q^{67} + 9 q^{69} - q^{71} + 18 q^{73} - 32 q^{75} - 9 q^{77} + 13 q^{79} + 13 q^{81} + 2 q^{83} + 45 q^{85} + 21 q^{87} + 9 q^{91} - 2 q^{93} + 11 q^{95} + 29 q^{97} - 13 q^{99}+O(q^{100}) \) 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
\(3\) −3.11788 −1.80011 −0.900055 0.435777i \(-0.856474\pi\)
−0.900055 + 0.435777i \(0.856474\pi\)
\(4\) 0 0
\(5\) 4.40203 1.96865 0.984324 0.176368i \(-0.0564348\pi\)
0.984324 + 0.176368i \(0.0564348\pi\)
\(6\) 0 0
\(7\) 1.00000 0.377964
\(8\) 0 0
\(9\) 6.72118 2.24039
\(10\) 0 0
\(11\) −1.00000 −0.301511
\(12\) 0 0
\(13\) 1.00000 0.277350
\(14\) 0 0
\(15\) −13.7250 −3.54378
\(16\) 0 0
\(17\) −1.59028 −0.385700 −0.192850 0.981228i \(-0.561773\pi\)
−0.192850 + 0.981228i \(0.561773\pi\)
\(18\) 0 0
\(19\) 3.13343 0.718858 0.359429 0.933172i \(-0.382971\pi\)
0.359429 + 0.933172i \(0.382971\pi\)
\(20\) 0 0
\(21\) −3.11788 −0.680378
\(22\) 0 0
\(23\) −2.76292 −0.576108 −0.288054 0.957614i \(-0.593008\pi\)
−0.288054 + 0.957614i \(0.593008\pi\)
\(24\) 0 0
\(25\) 14.3779 2.87558
\(26\) 0 0
\(27\) −11.6022 −2.23285
\(28\) 0 0
\(29\) 1.82484 0.338864 0.169432 0.985542i \(-0.445807\pi\)
0.169432 + 0.985542i \(0.445807\pi\)
\(30\) 0 0
\(31\) 5.86167 1.05279 0.526394 0.850241i \(-0.323544\pi\)
0.526394 + 0.850241i \(0.323544\pi\)
\(32\) 0 0
\(33\) 3.11788 0.542753
\(34\) 0 0
\(35\) 4.40203 0.744079
\(36\) 0 0
\(37\) 2.94070 0.483448 0.241724 0.970345i \(-0.422287\pi\)
0.241724 + 0.970345i \(0.422287\pi\)
\(38\) 0 0
\(39\) −3.11788 −0.499261
\(40\) 0 0
\(41\) −6.96742 −1.08813 −0.544064 0.839044i \(-0.683115\pi\)
−0.544064 + 0.839044i \(0.683115\pi\)
\(42\) 0 0
\(43\) 5.12559 0.781645 0.390823 0.920466i \(-0.372191\pi\)
0.390823 + 0.920466i \(0.372191\pi\)
\(44\) 0 0
\(45\) 29.5869 4.41055
\(46\) 0 0
\(47\) 8.67005 1.26466 0.632329 0.774700i \(-0.282099\pi\)
0.632329 + 0.774700i \(0.282099\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 4.95831 0.694302
\(52\) 0 0
\(53\) −0.705189 −0.0968651 −0.0484326 0.998826i \(-0.515423\pi\)
−0.0484326 + 0.998826i \(0.515423\pi\)
\(54\) 0 0
\(55\) −4.40203 −0.593570
\(56\) 0 0
\(57\) −9.76966 −1.29402
\(58\) 0 0
\(59\) −11.2123 −1.45972 −0.729861 0.683595i \(-0.760415\pi\)
−0.729861 + 0.683595i \(0.760415\pi\)
\(60\) 0 0
\(61\) 6.91210 0.885004 0.442502 0.896768i \(-0.354091\pi\)
0.442502 + 0.896768i \(0.354091\pi\)
\(62\) 0 0
\(63\) 6.72118 0.846790
\(64\) 0 0
\(65\) 4.40203 0.546005
\(66\) 0 0
\(67\) 2.69117 0.328779 0.164389 0.986396i \(-0.447435\pi\)
0.164389 + 0.986396i \(0.447435\pi\)
\(68\) 0 0
\(69\) 8.61444 1.03706
\(70\) 0 0
\(71\) −9.24458 −1.09713 −0.548565 0.836108i \(-0.684826\pi\)
−0.548565 + 0.836108i \(0.684826\pi\)
\(72\) 0 0
\(73\) 13.2313 1.54861 0.774305 0.632813i \(-0.218100\pi\)
0.774305 + 0.632813i \(0.218100\pi\)
\(74\) 0 0
\(75\) −44.8285 −5.17635
\(76\) 0 0
\(77\) −1.00000 −0.113961
\(78\) 0 0
\(79\) 9.32396 1.04903 0.524514 0.851402i \(-0.324247\pi\)
0.524514 + 0.851402i \(0.324247\pi\)
\(80\) 0 0
\(81\) 16.0108 1.77897
\(82\) 0 0
\(83\) −6.98352 −0.766541 −0.383270 0.923636i \(-0.625202\pi\)
−0.383270 + 0.923636i \(0.625202\pi\)
\(84\) 0 0
\(85\) −7.00047 −0.759308
\(86\) 0 0
\(87\) −5.68963 −0.609992
\(88\) 0 0
\(89\) 0.583301 0.0618298 0.0309149 0.999522i \(-0.490158\pi\)
0.0309149 + 0.999522i \(0.490158\pi\)
\(90\) 0 0
\(91\) 1.00000 0.104828
\(92\) 0 0
\(93\) −18.2760 −1.89513
\(94\) 0 0
\(95\) 13.7935 1.41518
\(96\) 0 0
\(97\) 0.299773 0.0304373 0.0152187 0.999884i \(-0.495156\pi\)
0.0152187 + 0.999884i \(0.495156\pi\)
\(98\) 0 0
\(99\) −6.72118 −0.675504
\(100\) 0 0
\(101\) −5.94277 −0.591327 −0.295664 0.955292i \(-0.595541\pi\)
−0.295664 + 0.955292i \(0.595541\pi\)
\(102\) 0 0
\(103\) −17.0401 −1.67901 −0.839505 0.543351i \(-0.817155\pi\)
−0.839505 + 0.543351i \(0.817155\pi\)
\(104\) 0 0
\(105\) −13.7250 −1.33942
\(106\) 0 0
\(107\) −8.97928 −0.868060 −0.434030 0.900898i \(-0.642909\pi\)
−0.434030 + 0.900898i \(0.642909\pi\)
\(108\) 0 0
\(109\) 12.6494 1.21159 0.605795 0.795620i \(-0.292855\pi\)
0.605795 + 0.795620i \(0.292855\pi\)
\(110\) 0 0
\(111\) −9.16876 −0.870260
\(112\) 0 0
\(113\) −8.30123 −0.780914 −0.390457 0.920621i \(-0.627683\pi\)
−0.390457 + 0.920621i \(0.627683\pi\)
\(114\) 0 0
\(115\) −12.1624 −1.13415
\(116\) 0 0
\(117\) 6.72118 0.621374
\(118\) 0 0
\(119\) −1.59028 −0.145781
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 0 0
\(123\) 21.7236 1.95875
\(124\) 0 0
\(125\) 41.2818 3.69235
\(126\) 0 0
\(127\) −21.7835 −1.93297 −0.966486 0.256721i \(-0.917358\pi\)
−0.966486 + 0.256721i \(0.917358\pi\)
\(128\) 0 0
\(129\) −15.9810 −1.40705
\(130\) 0 0
\(131\) 5.17119 0.451809 0.225905 0.974149i \(-0.427466\pi\)
0.225905 + 0.974149i \(0.427466\pi\)
\(132\) 0 0
\(133\) 3.13343 0.271703
\(134\) 0 0
\(135\) −51.0733 −4.39569
\(136\) 0 0
\(137\) 11.1209 0.950121 0.475061 0.879953i \(-0.342426\pi\)
0.475061 + 0.879953i \(0.342426\pi\)
\(138\) 0 0
\(139\) 20.3566 1.72662 0.863312 0.504671i \(-0.168386\pi\)
0.863312 + 0.504671i \(0.168386\pi\)
\(140\) 0 0
\(141\) −27.0322 −2.27652
\(142\) 0 0
\(143\) −1.00000 −0.0836242
\(144\) 0 0
\(145\) 8.03300 0.667104
\(146\) 0 0
\(147\) −3.11788 −0.257159
\(148\) 0 0
\(149\) 3.34391 0.273944 0.136972 0.990575i \(-0.456263\pi\)
0.136972 + 0.990575i \(0.456263\pi\)
\(150\) 0 0
\(151\) −16.3852 −1.33341 −0.666704 0.745322i \(-0.732296\pi\)
−0.666704 + 0.745322i \(0.732296\pi\)
\(152\) 0 0
\(153\) −10.6886 −0.864120
\(154\) 0 0
\(155\) 25.8033 2.07257
\(156\) 0 0
\(157\) −16.4456 −1.31250 −0.656252 0.754541i \(-0.727859\pi\)
−0.656252 + 0.754541i \(0.727859\pi\)
\(158\) 0 0
\(159\) 2.19870 0.174368
\(160\) 0 0
\(161\) −2.76292 −0.217748
\(162\) 0 0
\(163\) −9.60889 −0.752626 −0.376313 0.926493i \(-0.622808\pi\)
−0.376313 + 0.926493i \(0.622808\pi\)
\(164\) 0 0
\(165\) 13.7250 1.06849
\(166\) 0 0
\(167\) 12.0173 0.929925 0.464962 0.885330i \(-0.346068\pi\)
0.464962 + 0.885330i \(0.346068\pi\)
\(168\) 0 0
\(169\) 1.00000 0.0769231
\(170\) 0 0
\(171\) 21.0604 1.61053
\(172\) 0 0
\(173\) −20.1081 −1.52879 −0.764396 0.644747i \(-0.776963\pi\)
−0.764396 + 0.644747i \(0.776963\pi\)
\(174\) 0 0
\(175\) 14.3779 1.08687
\(176\) 0 0
\(177\) 34.9588 2.62766
\(178\) 0 0
\(179\) 14.3689 1.07398 0.536992 0.843588i \(-0.319561\pi\)
0.536992 + 0.843588i \(0.319561\pi\)
\(180\) 0 0
\(181\) 10.5788 0.786313 0.393157 0.919472i \(-0.371383\pi\)
0.393157 + 0.919472i \(0.371383\pi\)
\(182\) 0 0
\(183\) −21.5511 −1.59310
\(184\) 0 0
\(185\) 12.9451 0.951740
\(186\) 0 0
\(187\) 1.59028 0.116293
\(188\) 0 0
\(189\) −11.6022 −0.843937
\(190\) 0 0
\(191\) −16.7043 −1.20868 −0.604339 0.796727i \(-0.706563\pi\)
−0.604339 + 0.796727i \(0.706563\pi\)
\(192\) 0 0
\(193\) −2.33097 −0.167787 −0.0838936 0.996475i \(-0.526736\pi\)
−0.0838936 + 0.996475i \(0.526736\pi\)
\(194\) 0 0
\(195\) −13.7250 −0.982869
\(196\) 0 0
\(197\) −9.05196 −0.644925 −0.322463 0.946582i \(-0.604511\pi\)
−0.322463 + 0.946582i \(0.604511\pi\)
\(198\) 0 0
\(199\) 20.9671 1.48632 0.743161 0.669113i \(-0.233326\pi\)
0.743161 + 0.669113i \(0.233326\pi\)
\(200\) 0 0
\(201\) −8.39075 −0.591838
\(202\) 0 0
\(203\) 1.82484 0.128079
\(204\) 0 0
\(205\) −30.6708 −2.14214
\(206\) 0 0
\(207\) −18.5701 −1.29071
\(208\) 0 0
\(209\) −3.13343 −0.216744
\(210\) 0 0
\(211\) 10.6891 0.735869 0.367935 0.929852i \(-0.380065\pi\)
0.367935 + 0.929852i \(0.380065\pi\)
\(212\) 0 0
\(213\) 28.8235 1.97495
\(214\) 0 0
\(215\) 22.5630 1.53879
\(216\) 0 0
\(217\) 5.86167 0.397916
\(218\) 0 0
\(219\) −41.2537 −2.78767
\(220\) 0 0
\(221\) −1.59028 −0.106974
\(222\) 0 0
\(223\) 17.4147 1.16617 0.583086 0.812410i \(-0.301845\pi\)
0.583086 + 0.812410i \(0.301845\pi\)
\(224\) 0 0
\(225\) 96.6364 6.44243
\(226\) 0 0
\(227\) 2.25306 0.149541 0.0747706 0.997201i \(-0.476178\pi\)
0.0747706 + 0.997201i \(0.476178\pi\)
\(228\) 0 0
\(229\) 9.95142 0.657608 0.328804 0.944398i \(-0.393354\pi\)
0.328804 + 0.944398i \(0.393354\pi\)
\(230\) 0 0
\(231\) 3.11788 0.205142
\(232\) 0 0
\(233\) 27.0633 1.77297 0.886486 0.462754i \(-0.153139\pi\)
0.886486 + 0.462754i \(0.153139\pi\)
\(234\) 0 0
\(235\) 38.1659 2.48967
\(236\) 0 0
\(237\) −29.0710 −1.88836
\(238\) 0 0
\(239\) 16.6316 1.07581 0.537906 0.843005i \(-0.319216\pi\)
0.537906 + 0.843005i \(0.319216\pi\)
\(240\) 0 0
\(241\) 22.1457 1.42653 0.713266 0.700893i \(-0.247215\pi\)
0.713266 + 0.700893i \(0.247215\pi\)
\(242\) 0 0
\(243\) −15.1130 −0.969502
\(244\) 0 0
\(245\) 4.40203 0.281236
\(246\) 0 0
\(247\) 3.13343 0.199375
\(248\) 0 0
\(249\) 21.7738 1.37986
\(250\) 0 0
\(251\) 19.6199 1.23839 0.619197 0.785236i \(-0.287458\pi\)
0.619197 + 0.785236i \(0.287458\pi\)
\(252\) 0 0
\(253\) 2.76292 0.173703
\(254\) 0 0
\(255\) 21.8266 1.36684
\(256\) 0 0
\(257\) −6.91710 −0.431477 −0.215738 0.976451i \(-0.569216\pi\)
−0.215738 + 0.976451i \(0.569216\pi\)
\(258\) 0 0
\(259\) 2.94070 0.182726
\(260\) 0 0
\(261\) 12.2651 0.759189
\(262\) 0 0
\(263\) 24.3498 1.50147 0.750737 0.660602i \(-0.229699\pi\)
0.750737 + 0.660602i \(0.229699\pi\)
\(264\) 0 0
\(265\) −3.10426 −0.190693
\(266\) 0 0
\(267\) −1.81866 −0.111300
\(268\) 0 0
\(269\) 7.53061 0.459149 0.229575 0.973291i \(-0.426266\pi\)
0.229575 + 0.973291i \(0.426266\pi\)
\(270\) 0 0
\(271\) 20.1877 1.22632 0.613158 0.789960i \(-0.289899\pi\)
0.613158 + 0.789960i \(0.289899\pi\)
\(272\) 0 0
\(273\) −3.11788 −0.188703
\(274\) 0 0
\(275\) −14.3779 −0.867019
\(276\) 0 0
\(277\) 2.11132 0.126857 0.0634286 0.997986i \(-0.479796\pi\)
0.0634286 + 0.997986i \(0.479796\pi\)
\(278\) 0 0
\(279\) 39.3974 2.35866
\(280\) 0 0
\(281\) −10.7278 −0.639969 −0.319984 0.947423i \(-0.603678\pi\)
−0.319984 + 0.947423i \(0.603678\pi\)
\(282\) 0 0
\(283\) 32.3124 1.92077 0.960385 0.278675i \(-0.0898952\pi\)
0.960385 + 0.278675i \(0.0898952\pi\)
\(284\) 0 0
\(285\) −43.0064 −2.54748
\(286\) 0 0
\(287\) −6.96742 −0.411274
\(288\) 0 0
\(289\) −14.4710 −0.851236
\(290\) 0 0
\(291\) −0.934656 −0.0547905
\(292\) 0 0
\(293\) 24.0659 1.40594 0.702972 0.711217i \(-0.251856\pi\)
0.702972 + 0.711217i \(0.251856\pi\)
\(294\) 0 0
\(295\) −49.3571 −2.87368
\(296\) 0 0
\(297\) 11.6022 0.673229
\(298\) 0 0
\(299\) −2.76292 −0.159784
\(300\) 0 0
\(301\) 5.12559 0.295434
\(302\) 0 0
\(303\) 18.5288 1.06445
\(304\) 0 0
\(305\) 30.4273 1.74226
\(306\) 0 0
\(307\) −13.0355 −0.743978 −0.371989 0.928237i \(-0.621324\pi\)
−0.371989 + 0.928237i \(0.621324\pi\)
\(308\) 0 0
\(309\) 53.1290 3.02240
\(310\) 0 0
\(311\) −27.6464 −1.56769 −0.783843 0.620959i \(-0.786743\pi\)
−0.783843 + 0.620959i \(0.786743\pi\)
\(312\) 0 0
\(313\) −24.3230 −1.37482 −0.687409 0.726270i \(-0.741252\pi\)
−0.687409 + 0.726270i \(0.741252\pi\)
\(314\) 0 0
\(315\) 29.5869 1.66703
\(316\) 0 0
\(317\) 1.18317 0.0664537 0.0332269 0.999448i \(-0.489422\pi\)
0.0332269 + 0.999448i \(0.489422\pi\)
\(318\) 0 0
\(319\) −1.82484 −0.102171
\(320\) 0 0
\(321\) 27.9963 1.56260
\(322\) 0 0
\(323\) −4.98304 −0.277264
\(324\) 0 0
\(325\) 14.3779 0.797542
\(326\) 0 0
\(327\) −39.4393 −2.18100
\(328\) 0 0
\(329\) 8.67005 0.477995
\(330\) 0 0
\(331\) 18.0890 0.994264 0.497132 0.867675i \(-0.334386\pi\)
0.497132 + 0.867675i \(0.334386\pi\)
\(332\) 0 0
\(333\) 19.7650 1.08312
\(334\) 0 0
\(335\) 11.8466 0.647250
\(336\) 0 0
\(337\) −2.59644 −0.141437 −0.0707185 0.997496i \(-0.522529\pi\)
−0.0707185 + 0.997496i \(0.522529\pi\)
\(338\) 0 0
\(339\) 25.8822 1.40573
\(340\) 0 0
\(341\) −5.86167 −0.317427
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) 0 0
\(345\) 37.9211 2.04160
\(346\) 0 0
\(347\) −25.8501 −1.38771 −0.693854 0.720116i \(-0.744089\pi\)
−0.693854 + 0.720116i \(0.744089\pi\)
\(348\) 0 0
\(349\) −28.2982 −1.51477 −0.757384 0.652970i \(-0.773523\pi\)
−0.757384 + 0.652970i \(0.773523\pi\)
\(350\) 0 0
\(351\) −11.6022 −0.619280
\(352\) 0 0
\(353\) −36.2423 −1.92898 −0.964492 0.264113i \(-0.914921\pi\)
−0.964492 + 0.264113i \(0.914921\pi\)
\(354\) 0 0
\(355\) −40.6949 −2.15986
\(356\) 0 0
\(357\) 4.95831 0.262422
\(358\) 0 0
\(359\) −8.96447 −0.473127 −0.236563 0.971616i \(-0.576021\pi\)
−0.236563 + 0.971616i \(0.576021\pi\)
\(360\) 0 0
\(361\) −9.18162 −0.483243
\(362\) 0 0
\(363\) −3.11788 −0.163646
\(364\) 0 0
\(365\) 58.2447 3.04867
\(366\) 0 0
\(367\) 0.558020 0.0291284 0.0145642 0.999894i \(-0.495364\pi\)
0.0145642 + 0.999894i \(0.495364\pi\)
\(368\) 0 0
\(369\) −46.8293 −2.43784
\(370\) 0 0
\(371\) −0.705189 −0.0366116
\(372\) 0 0
\(373\) −10.9607 −0.567525 −0.283763 0.958895i \(-0.591583\pi\)
−0.283763 + 0.958895i \(0.591583\pi\)
\(374\) 0 0
\(375\) −128.712 −6.64664
\(376\) 0 0
\(377\) 1.82484 0.0939840
\(378\) 0 0
\(379\) −22.9617 −1.17946 −0.589731 0.807600i \(-0.700766\pi\)
−0.589731 + 0.807600i \(0.700766\pi\)
\(380\) 0 0
\(381\) 67.9183 3.47956
\(382\) 0 0
\(383\) 28.8734 1.47536 0.737681 0.675149i \(-0.235921\pi\)
0.737681 + 0.675149i \(0.235921\pi\)
\(384\) 0 0
\(385\) −4.40203 −0.224348
\(386\) 0 0
\(387\) 34.4500 1.75119
\(388\) 0 0
\(389\) −20.5087 −1.03983 −0.519915 0.854218i \(-0.674036\pi\)
−0.519915 + 0.854218i \(0.674036\pi\)
\(390\) 0 0
\(391\) 4.39381 0.222205
\(392\) 0 0
\(393\) −16.1232 −0.813306
\(394\) 0 0
\(395\) 41.0444 2.06517
\(396\) 0 0
\(397\) −6.42381 −0.322402 −0.161201 0.986922i \(-0.551537\pi\)
−0.161201 + 0.986922i \(0.551537\pi\)
\(398\) 0 0
\(399\) −9.76966 −0.489095
\(400\) 0 0
\(401\) 11.5374 0.576148 0.288074 0.957608i \(-0.406985\pi\)
0.288074 + 0.957608i \(0.406985\pi\)
\(402\) 0 0
\(403\) 5.86167 0.291991
\(404\) 0 0
\(405\) 70.4799 3.50218
\(406\) 0 0
\(407\) −2.94070 −0.145765
\(408\) 0 0
\(409\) 1.08137 0.0534702 0.0267351 0.999643i \(-0.491489\pi\)
0.0267351 + 0.999643i \(0.491489\pi\)
\(410\) 0 0
\(411\) −34.6736 −1.71032
\(412\) 0 0
\(413\) −11.2123 −0.551723
\(414\) 0 0
\(415\) −30.7417 −1.50905
\(416\) 0 0
\(417\) −63.4694 −3.10811
\(418\) 0 0
\(419\) 11.8080 0.576860 0.288430 0.957501i \(-0.406867\pi\)
0.288430 + 0.957501i \(0.406867\pi\)
\(420\) 0 0
\(421\) 15.7001 0.765174 0.382587 0.923919i \(-0.375033\pi\)
0.382587 + 0.923919i \(0.375033\pi\)
\(422\) 0 0
\(423\) 58.2730 2.83333
\(424\) 0 0
\(425\) −22.8649 −1.10911
\(426\) 0 0
\(427\) 6.91210 0.334500
\(428\) 0 0
\(429\) 3.11788 0.150533
\(430\) 0 0
\(431\) 38.8166 1.86973 0.934865 0.355003i \(-0.115520\pi\)
0.934865 + 0.355003i \(0.115520\pi\)
\(432\) 0 0
\(433\) −8.07206 −0.387919 −0.193959 0.981010i \(-0.562133\pi\)
−0.193959 + 0.981010i \(0.562133\pi\)
\(434\) 0 0
\(435\) −25.0459 −1.20086
\(436\) 0 0
\(437\) −8.65740 −0.414140
\(438\) 0 0
\(439\) 20.6070 0.983518 0.491759 0.870731i \(-0.336354\pi\)
0.491759 + 0.870731i \(0.336354\pi\)
\(440\) 0 0
\(441\) 6.72118 0.320056
\(442\) 0 0
\(443\) −4.64130 −0.220515 −0.110257 0.993903i \(-0.535167\pi\)
−0.110257 + 0.993903i \(0.535167\pi\)
\(444\) 0 0
\(445\) 2.56771 0.121721
\(446\) 0 0
\(447\) −10.4259 −0.493129
\(448\) 0 0
\(449\) −17.3457 −0.818594 −0.409297 0.912401i \(-0.634226\pi\)
−0.409297 + 0.912401i \(0.634226\pi\)
\(450\) 0 0
\(451\) 6.96742 0.328083
\(452\) 0 0
\(453\) 51.0871 2.40028
\(454\) 0 0
\(455\) 4.40203 0.206370
\(456\) 0 0
\(457\) −23.6146 −1.10464 −0.552321 0.833631i \(-0.686258\pi\)
−0.552321 + 0.833631i \(0.686258\pi\)
\(458\) 0 0
\(459\) 18.4508 0.861209
\(460\) 0 0
\(461\) 14.5369 0.677052 0.338526 0.940957i \(-0.390072\pi\)
0.338526 + 0.940957i \(0.390072\pi\)
\(462\) 0 0
\(463\) −36.0894 −1.67722 −0.838609 0.544734i \(-0.816631\pi\)
−0.838609 + 0.544734i \(0.816631\pi\)
\(464\) 0 0
\(465\) −80.4515 −3.73085
\(466\) 0 0
\(467\) 38.4901 1.78111 0.890555 0.454876i \(-0.150316\pi\)
0.890555 + 0.454876i \(0.150316\pi\)
\(468\) 0 0
\(469\) 2.69117 0.124267
\(470\) 0 0
\(471\) 51.2755 2.36265
\(472\) 0 0
\(473\) −5.12559 −0.235675
\(474\) 0 0
\(475\) 45.0521 2.06713
\(476\) 0 0
\(477\) −4.73970 −0.217016
\(478\) 0 0
\(479\) −4.54964 −0.207878 −0.103939 0.994584i \(-0.533145\pi\)
−0.103939 + 0.994584i \(0.533145\pi\)
\(480\) 0 0
\(481\) 2.94070 0.134084
\(482\) 0 0
\(483\) 8.61444 0.391971
\(484\) 0 0
\(485\) 1.31961 0.0599204
\(486\) 0 0
\(487\) −29.7711 −1.34906 −0.674528 0.738250i \(-0.735653\pi\)
−0.674528 + 0.738250i \(0.735653\pi\)
\(488\) 0 0
\(489\) 29.9594 1.35481
\(490\) 0 0
\(491\) −10.8353 −0.488990 −0.244495 0.969651i \(-0.578622\pi\)
−0.244495 + 0.969651i \(0.578622\pi\)
\(492\) 0 0
\(493\) −2.90201 −0.130700
\(494\) 0 0
\(495\) −29.5869 −1.32983
\(496\) 0 0
\(497\) −9.24458 −0.414676
\(498\) 0 0
\(499\) 16.4447 0.736167 0.368083 0.929793i \(-0.380014\pi\)
0.368083 + 0.929793i \(0.380014\pi\)
\(500\) 0 0
\(501\) −37.4684 −1.67397
\(502\) 0 0
\(503\) 11.7689 0.524751 0.262375 0.964966i \(-0.415494\pi\)
0.262375 + 0.964966i \(0.415494\pi\)
\(504\) 0 0
\(505\) −26.1602 −1.16412
\(506\) 0 0
\(507\) −3.11788 −0.138470
\(508\) 0 0
\(509\) 2.65928 0.117870 0.0589352 0.998262i \(-0.481229\pi\)
0.0589352 + 0.998262i \(0.481229\pi\)
\(510\) 0 0
\(511\) 13.2313 0.585320
\(512\) 0 0
\(513\) −36.3547 −1.60510
\(514\) 0 0
\(515\) −75.0111 −3.30538
\(516\) 0 0
\(517\) −8.67005 −0.381308
\(518\) 0 0
\(519\) 62.6948 2.75199
\(520\) 0 0
\(521\) 16.6894 0.731176 0.365588 0.930777i \(-0.380868\pi\)
0.365588 + 0.930777i \(0.380868\pi\)
\(522\) 0 0
\(523\) 45.2942 1.98058 0.990289 0.139021i \(-0.0443957\pi\)
0.990289 + 0.139021i \(0.0443957\pi\)
\(524\) 0 0
\(525\) −44.8285 −1.95648
\(526\) 0 0
\(527\) −9.32171 −0.406060
\(528\) 0 0
\(529\) −15.3663 −0.668100
\(530\) 0 0
\(531\) −75.3602 −3.27036
\(532\) 0 0
\(533\) −6.96742 −0.301793
\(534\) 0 0
\(535\) −39.5271 −1.70891
\(536\) 0 0
\(537\) −44.8006 −1.93329
\(538\) 0 0
\(539\) −1.00000 −0.0430730
\(540\) 0 0
\(541\) 3.93737 0.169281 0.0846405 0.996412i \(-0.473026\pi\)
0.0846405 + 0.996412i \(0.473026\pi\)
\(542\) 0 0
\(543\) −32.9833 −1.41545
\(544\) 0 0
\(545\) 55.6830 2.38520
\(546\) 0 0
\(547\) −19.1443 −0.818550 −0.409275 0.912411i \(-0.634218\pi\)
−0.409275 + 0.912411i \(0.634218\pi\)
\(548\) 0 0
\(549\) 46.4575 1.98276
\(550\) 0 0
\(551\) 5.71800 0.243595
\(552\) 0 0
\(553\) 9.32396 0.396495
\(554\) 0 0
\(555\) −40.3612 −1.71324
\(556\) 0 0
\(557\) −16.4472 −0.696889 −0.348445 0.937329i \(-0.613290\pi\)
−0.348445 + 0.937329i \(0.613290\pi\)
\(558\) 0 0
\(559\) 5.12559 0.216789
\(560\) 0 0
\(561\) −4.95831 −0.209340
\(562\) 0 0
\(563\) −11.6079 −0.489214 −0.244607 0.969622i \(-0.578659\pi\)
−0.244607 + 0.969622i \(0.578659\pi\)
\(564\) 0 0
\(565\) −36.5423 −1.53735
\(566\) 0 0
\(567\) 16.0108 0.672389
\(568\) 0 0
\(569\) 13.9489 0.584768 0.292384 0.956301i \(-0.405551\pi\)
0.292384 + 0.956301i \(0.405551\pi\)
\(570\) 0 0
\(571\) 6.24914 0.261518 0.130759 0.991414i \(-0.458259\pi\)
0.130759 + 0.991414i \(0.458259\pi\)
\(572\) 0 0
\(573\) 52.0819 2.17575
\(574\) 0 0
\(575\) −39.7249 −1.65664
\(576\) 0 0
\(577\) 10.3523 0.430970 0.215485 0.976507i \(-0.430867\pi\)
0.215485 + 0.976507i \(0.430867\pi\)
\(578\) 0 0
\(579\) 7.26770 0.302035
\(580\) 0 0
\(581\) −6.98352 −0.289725
\(582\) 0 0
\(583\) 0.705189 0.0292059
\(584\) 0 0
\(585\) 29.5869 1.22327
\(586\) 0 0
\(587\) 44.0542 1.81831 0.909155 0.416457i \(-0.136728\pi\)
0.909155 + 0.416457i \(0.136728\pi\)
\(588\) 0 0
\(589\) 18.3671 0.756805
\(590\) 0 0
\(591\) 28.2229 1.16094
\(592\) 0 0
\(593\) −38.7471 −1.59115 −0.795577 0.605853i \(-0.792832\pi\)
−0.795577 + 0.605853i \(0.792832\pi\)
\(594\) 0 0
\(595\) −7.00047 −0.286991
\(596\) 0 0
\(597\) −65.3731 −2.67554
\(598\) 0 0
\(599\) 21.3148 0.870899 0.435450 0.900213i \(-0.356589\pi\)
0.435450 + 0.900213i \(0.356589\pi\)
\(600\) 0 0
\(601\) 39.1302 1.59615 0.798076 0.602556i \(-0.205851\pi\)
0.798076 + 0.602556i \(0.205851\pi\)
\(602\) 0 0
\(603\) 18.0879 0.736594
\(604\) 0 0
\(605\) 4.40203 0.178968
\(606\) 0 0
\(607\) 8.12471 0.329772 0.164886 0.986313i \(-0.447274\pi\)
0.164886 + 0.986313i \(0.447274\pi\)
\(608\) 0 0
\(609\) −5.68963 −0.230555
\(610\) 0 0
\(611\) 8.67005 0.350753
\(612\) 0 0
\(613\) 16.8590 0.680928 0.340464 0.940258i \(-0.389416\pi\)
0.340464 + 0.940258i \(0.389416\pi\)
\(614\) 0 0
\(615\) 95.6280 3.85609
\(616\) 0 0
\(617\) −29.3547 −1.18177 −0.590887 0.806754i \(-0.701222\pi\)
−0.590887 + 0.806754i \(0.701222\pi\)
\(618\) 0 0
\(619\) 12.1330 0.487665 0.243832 0.969817i \(-0.421595\pi\)
0.243832 + 0.969817i \(0.421595\pi\)
\(620\) 0 0
\(621\) 32.0559 1.28636
\(622\) 0 0
\(623\) 0.583301 0.0233695
\(624\) 0 0
\(625\) 109.834 4.39337
\(626\) 0 0
\(627\) 9.76966 0.390163
\(628\) 0 0
\(629\) −4.67654 −0.186466
\(630\) 0 0
\(631\) 9.12676 0.363331 0.181665 0.983360i \(-0.441851\pi\)
0.181665 + 0.983360i \(0.441851\pi\)
\(632\) 0 0
\(633\) −33.3274 −1.32465
\(634\) 0 0
\(635\) −95.8916 −3.80534
\(636\) 0 0
\(637\) 1.00000 0.0396214
\(638\) 0 0
\(639\) −62.1345 −2.45800
\(640\) 0 0
\(641\) −23.5552 −0.930373 −0.465186 0.885213i \(-0.654013\pi\)
−0.465186 + 0.885213i \(0.654013\pi\)
\(642\) 0 0
\(643\) −28.3696 −1.11879 −0.559394 0.828902i \(-0.688966\pi\)
−0.559394 + 0.828902i \(0.688966\pi\)
\(644\) 0 0
\(645\) −70.3488 −2.76998
\(646\) 0 0
\(647\) −19.8445 −0.780169 −0.390085 0.920779i \(-0.627554\pi\)
−0.390085 + 0.920779i \(0.627554\pi\)
\(648\) 0 0
\(649\) 11.2123 0.440123
\(650\) 0 0
\(651\) −18.2760 −0.716293
\(652\) 0 0
\(653\) 48.6537 1.90397 0.951984 0.306149i \(-0.0990406\pi\)
0.951984 + 0.306149i \(0.0990406\pi\)
\(654\) 0 0
\(655\) 22.7638 0.889454
\(656\) 0 0
\(657\) 88.9302 3.46950
\(658\) 0 0
\(659\) 32.4053 1.26233 0.631166 0.775648i \(-0.282577\pi\)
0.631166 + 0.775648i \(0.282577\pi\)
\(660\) 0 0
\(661\) 8.89016 0.345787 0.172894 0.984941i \(-0.444688\pi\)
0.172894 + 0.984941i \(0.444688\pi\)
\(662\) 0 0
\(663\) 4.95831 0.192565
\(664\) 0 0
\(665\) 13.7935 0.534887
\(666\) 0 0
\(667\) −5.04188 −0.195222
\(668\) 0 0
\(669\) −54.2969 −2.09924
\(670\) 0 0
\(671\) −6.91210 −0.266839
\(672\) 0 0
\(673\) −48.8341 −1.88242 −0.941209 0.337825i \(-0.890309\pi\)
−0.941209 + 0.337825i \(0.890309\pi\)
\(674\) 0 0
\(675\) −166.815 −6.42072
\(676\) 0 0
\(677\) 50.9265 1.95726 0.978632 0.205620i \(-0.0659211\pi\)
0.978632 + 0.205620i \(0.0659211\pi\)
\(678\) 0 0
\(679\) 0.299773 0.0115042
\(680\) 0 0
\(681\) −7.02479 −0.269190
\(682\) 0 0
\(683\) −2.95801 −0.113185 −0.0565925 0.998397i \(-0.518024\pi\)
−0.0565925 + 0.998397i \(0.518024\pi\)
\(684\) 0 0
\(685\) 48.9545 1.87046
\(686\) 0 0
\(687\) −31.0273 −1.18377
\(688\) 0 0
\(689\) −0.705189 −0.0268656
\(690\) 0 0
\(691\) 34.7041 1.32021 0.660103 0.751176i \(-0.270513\pi\)
0.660103 + 0.751176i \(0.270513\pi\)
\(692\) 0 0
\(693\) −6.72118 −0.255317
\(694\) 0 0
\(695\) 89.6103 3.39911
\(696\) 0 0
\(697\) 11.0802 0.419691
\(698\) 0 0
\(699\) −84.3800 −3.19155
\(700\) 0 0
\(701\) 14.2432 0.537960 0.268980 0.963146i \(-0.413313\pi\)
0.268980 + 0.963146i \(0.413313\pi\)
\(702\) 0 0
\(703\) 9.21448 0.347531
\(704\) 0 0
\(705\) −118.997 −4.48167
\(706\) 0 0
\(707\) −5.94277 −0.223501
\(708\) 0 0
\(709\) −35.5130 −1.33372 −0.666860 0.745183i \(-0.732362\pi\)
−0.666860 + 0.745183i \(0.732362\pi\)
\(710\) 0 0
\(711\) 62.6680 2.35024
\(712\) 0 0
\(713\) −16.1953 −0.606519
\(714\) 0 0
\(715\) −4.40203 −0.164627
\(716\) 0 0
\(717\) −51.8555 −1.93658
\(718\) 0 0
\(719\) −15.8106 −0.589635 −0.294817 0.955554i \(-0.595259\pi\)
−0.294817 + 0.955554i \(0.595259\pi\)
\(720\) 0 0
\(721\) −17.0401 −0.634606
\(722\) 0 0
\(723\) −69.0478 −2.56792
\(724\) 0 0
\(725\) 26.2373 0.974430
\(726\) 0 0
\(727\) −23.3117 −0.864583 −0.432292 0.901734i \(-0.642295\pi\)
−0.432292 + 0.901734i \(0.642295\pi\)
\(728\) 0 0
\(729\) −0.911641 −0.0337645
\(730\) 0 0
\(731\) −8.15113 −0.301481
\(732\) 0 0
\(733\) −23.0355 −0.850835 −0.425418 0.904997i \(-0.639873\pi\)
−0.425418 + 0.904997i \(0.639873\pi\)
\(734\) 0 0
\(735\) −13.7250 −0.506255
\(736\) 0 0
\(737\) −2.69117 −0.0991305
\(738\) 0 0
\(739\) −17.1229 −0.629875 −0.314938 0.949112i \(-0.601984\pi\)
−0.314938 + 0.949112i \(0.601984\pi\)
\(740\) 0 0
\(741\) −9.76966 −0.358898
\(742\) 0 0
\(743\) −25.4339 −0.933080 −0.466540 0.884500i \(-0.654500\pi\)
−0.466540 + 0.884500i \(0.654500\pi\)
\(744\) 0 0
\(745\) 14.7200 0.539299
\(746\) 0 0
\(747\) −46.9375 −1.71735
\(748\) 0 0
\(749\) −8.97928 −0.328096
\(750\) 0 0
\(751\) −2.27585 −0.0830468 −0.0415234 0.999138i \(-0.513221\pi\)
−0.0415234 + 0.999138i \(0.513221\pi\)
\(752\) 0 0
\(753\) −61.1724 −2.22925
\(754\) 0 0
\(755\) −72.1282 −2.62501
\(756\) 0 0
\(757\) 45.0726 1.63819 0.819096 0.573656i \(-0.194475\pi\)
0.819096 + 0.573656i \(0.194475\pi\)
\(758\) 0 0
\(759\) −8.61444 −0.312685
\(760\) 0 0
\(761\) −17.8818 −0.648216 −0.324108 0.946020i \(-0.605064\pi\)
−0.324108 + 0.946020i \(0.605064\pi\)
\(762\) 0 0
\(763\) 12.6494 0.457938
\(764\) 0 0
\(765\) −47.0515 −1.70115
\(766\) 0 0
\(767\) −11.2123 −0.404854
\(768\) 0 0
\(769\) −31.1844 −1.12454 −0.562269 0.826954i \(-0.690072\pi\)
−0.562269 + 0.826954i \(0.690072\pi\)
\(770\) 0 0
\(771\) 21.5667 0.776705
\(772\) 0 0
\(773\) 1.87795 0.0675450 0.0337725 0.999430i \(-0.489248\pi\)
0.0337725 + 0.999430i \(0.489248\pi\)
\(774\) 0 0
\(775\) 84.2785 3.02737
\(776\) 0 0
\(777\) −9.16876 −0.328927
\(778\) 0 0
\(779\) −21.8319 −0.782210
\(780\) 0 0
\(781\) 9.24458 0.330797
\(782\) 0 0
\(783\) −21.1722 −0.756631
\(784\) 0 0
\(785\) −72.3942 −2.58386
\(786\) 0 0
\(787\) −17.7663 −0.633300 −0.316650 0.948542i \(-0.602558\pi\)
−0.316650 + 0.948542i \(0.602558\pi\)
\(788\) 0 0
\(789\) −75.9198 −2.70282
\(790\) 0 0
\(791\) −8.30123 −0.295158
\(792\) 0 0
\(793\) 6.91210 0.245456
\(794\) 0 0
\(795\) 9.67873 0.343269
\(796\) 0 0
\(797\) 5.28933 0.187358 0.0936789 0.995602i \(-0.470137\pi\)
0.0936789 + 0.995602i \(0.470137\pi\)
\(798\) 0 0
\(799\) −13.7878 −0.487778
\(800\) 0 0
\(801\) 3.92048 0.138523
\(802\) 0 0
\(803\) −13.2313 −0.466923
\(804\) 0 0
\(805\) −12.1624 −0.428670
\(806\) 0 0
\(807\) −23.4795 −0.826519
\(808\) 0 0
\(809\) 41.7900 1.46926 0.734629 0.678470i \(-0.237356\pi\)
0.734629 + 0.678470i \(0.237356\pi\)
\(810\) 0 0
\(811\) −12.8870 −0.452522 −0.226261 0.974067i \(-0.572650\pi\)
−0.226261 + 0.974067i \(0.572650\pi\)
\(812\) 0 0
\(813\) −62.9429 −2.20750
\(814\) 0 0
\(815\) −42.2986 −1.48166
\(816\) 0 0
\(817\) 16.0607 0.561892
\(818\) 0 0
\(819\) 6.72118 0.234857
\(820\) 0 0
\(821\) −19.9826 −0.697396 −0.348698 0.937235i \(-0.613376\pi\)
−0.348698 + 0.937235i \(0.613376\pi\)
\(822\) 0 0
\(823\) 38.6836 1.34843 0.674213 0.738537i \(-0.264483\pi\)
0.674213 + 0.738537i \(0.264483\pi\)
\(824\) 0 0
\(825\) 44.8285 1.56073
\(826\) 0 0
\(827\) −43.7188 −1.52025 −0.760126 0.649775i \(-0.774863\pi\)
−0.760126 + 0.649775i \(0.774863\pi\)
\(828\) 0 0
\(829\) −9.45743 −0.328470 −0.164235 0.986421i \(-0.552516\pi\)
−0.164235 + 0.986421i \(0.552516\pi\)
\(830\) 0 0
\(831\) −6.58286 −0.228357
\(832\) 0 0
\(833\) −1.59028 −0.0551000
\(834\) 0 0
\(835\) 52.9004 1.83070
\(836\) 0 0
\(837\) −68.0084 −2.35071
\(838\) 0 0
\(839\) −17.3236 −0.598078 −0.299039 0.954241i \(-0.596666\pi\)
−0.299039 + 0.954241i \(0.596666\pi\)
\(840\) 0 0
\(841\) −25.6700 −0.885171
\(842\) 0 0
\(843\) 33.4481 1.15201
\(844\) 0 0
\(845\) 4.40203 0.151435
\(846\) 0 0
\(847\) 1.00000 0.0343604
\(848\) 0 0
\(849\) −100.746 −3.45760
\(850\) 0 0
\(851\) −8.12491 −0.278518
\(852\) 0 0
\(853\) −1.53061 −0.0524069 −0.0262035 0.999657i \(-0.508342\pi\)
−0.0262035 + 0.999657i \(0.508342\pi\)
\(854\) 0 0
\(855\) 92.7084 3.17056
\(856\) 0 0
\(857\) 14.8561 0.507474 0.253737 0.967273i \(-0.418340\pi\)
0.253737 + 0.967273i \(0.418340\pi\)
\(858\) 0 0
\(859\) 49.9383 1.70387 0.851936 0.523645i \(-0.175428\pi\)
0.851936 + 0.523645i \(0.175428\pi\)
\(860\) 0 0
\(861\) 21.7236 0.740338
\(862\) 0 0
\(863\) 26.7598 0.910916 0.455458 0.890257i \(-0.349476\pi\)
0.455458 + 0.890257i \(0.349476\pi\)
\(864\) 0 0
\(865\) −88.5166 −3.00966
\(866\) 0 0
\(867\) 45.1189 1.53232
\(868\) 0 0
\(869\) −9.32396 −0.316294
\(870\) 0 0
\(871\) 2.69117 0.0911868
\(872\) 0 0
\(873\) 2.01483 0.0681916
\(874\) 0 0
\(875\) 41.2818 1.39558
\(876\) 0 0
\(877\) 38.5987 1.30339 0.651693 0.758483i \(-0.274059\pi\)
0.651693 + 0.758483i \(0.274059\pi\)
\(878\) 0 0
\(879\) −75.0346 −2.53085
\(880\) 0 0
\(881\) 12.1256 0.408523 0.204262 0.978916i \(-0.434521\pi\)
0.204262 + 0.978916i \(0.434521\pi\)
\(882\) 0 0
\(883\) 17.8167 0.599581 0.299790 0.954005i \(-0.403083\pi\)
0.299790 + 0.954005i \(0.403083\pi\)
\(884\) 0 0
\(885\) 153.890 5.17294
\(886\) 0 0
\(887\) 17.0049 0.570967 0.285484 0.958384i \(-0.407846\pi\)
0.285484 + 0.958384i \(0.407846\pi\)
\(888\) 0 0
\(889\) −21.7835 −0.730594
\(890\) 0 0
\(891\) −16.0108 −0.536381
\(892\) 0 0
\(893\) 27.1670 0.909109
\(894\) 0 0
\(895\) 63.2524 2.11430
\(896\) 0 0
\(897\) 8.61444 0.287628
\(898\) 0 0
\(899\) 10.6966 0.356752
\(900\) 0 0
\(901\) 1.12145 0.0373609
\(902\) 0 0
\(903\) −15.9810 −0.531814
\(904\) 0 0
\(905\) 46.5680 1.54797
\(906\) 0 0
\(907\) −8.82387 −0.292992 −0.146496 0.989211i \(-0.546800\pi\)
−0.146496 + 0.989211i \(0.546800\pi\)
\(908\) 0 0
\(909\) −39.9424 −1.32481
\(910\) 0 0
\(911\) −20.6827 −0.685248 −0.342624 0.939473i \(-0.611316\pi\)
−0.342624 + 0.939473i \(0.611316\pi\)
\(912\) 0 0
\(913\) 6.98352 0.231121
\(914\) 0 0
\(915\) −94.8687 −3.13626
\(916\) 0 0
\(917\) 5.17119 0.170768
\(918\) 0 0
\(919\) 20.7796 0.685457 0.342728 0.939435i \(-0.388649\pi\)
0.342728 + 0.939435i \(0.388649\pi\)
\(920\) 0 0
\(921\) 40.6433 1.33924
\(922\) 0 0
\(923\) −9.24458 −0.304289
\(924\) 0 0
\(925\) 42.2811 1.39019
\(926\) 0 0
\(927\) −114.530 −3.76165
\(928\) 0 0
\(929\) −3.65036 −0.119764 −0.0598821 0.998205i \(-0.519072\pi\)
−0.0598821 + 0.998205i \(0.519072\pi\)
\(930\) 0 0
\(931\) 3.13343 0.102694
\(932\) 0 0
\(933\) 86.1983 2.82201
\(934\) 0 0
\(935\) 7.00047 0.228940
\(936\) 0 0
\(937\) −48.2398 −1.57593 −0.787963 0.615722i \(-0.788864\pi\)
−0.787963 + 0.615722i \(0.788864\pi\)
\(938\) 0 0
\(939\) 75.8363 2.47482
\(940\) 0 0
\(941\) −22.0198 −0.717826 −0.358913 0.933371i \(-0.616853\pi\)
−0.358913 + 0.933371i \(0.616853\pi\)
\(942\) 0 0
\(943\) 19.2504 0.626879
\(944\) 0 0
\(945\) −51.0733 −1.66141
\(946\) 0 0
\(947\) −61.3455 −1.99346 −0.996731 0.0807952i \(-0.974254\pi\)
−0.996731 + 0.0807952i \(0.974254\pi\)
\(948\) 0 0
\(949\) 13.2313 0.429507
\(950\) 0 0
\(951\) −3.68900 −0.119624
\(952\) 0 0
\(953\) −11.5852 −0.375283 −0.187641 0.982238i \(-0.560084\pi\)
−0.187641 + 0.982238i \(0.560084\pi\)
\(954\) 0 0
\(955\) −73.5327 −2.37946
\(956\) 0 0
\(957\) 5.68963 0.183920
\(958\) 0 0
\(959\) 11.1209 0.359112
\(960\) 0 0
\(961\) 3.35920 0.108361
\(962\) 0 0
\(963\) −60.3514 −1.94480
\(964\) 0 0
\(965\) −10.2610 −0.330314
\(966\) 0 0
\(967\) −12.9422 −0.416194 −0.208097 0.978108i \(-0.566727\pi\)
−0.208097 + 0.978108i \(0.566727\pi\)
\(968\) 0 0
\(969\) 15.5365 0.499105
\(970\) 0 0
\(971\) −17.2607 −0.553924 −0.276962 0.960881i \(-0.589328\pi\)
−0.276962 + 0.960881i \(0.589328\pi\)
\(972\) 0 0
\(973\) 20.3566 0.652602
\(974\) 0 0
\(975\) −44.8285 −1.43566
\(976\) 0 0
\(977\) 14.1370 0.452282 0.226141 0.974095i \(-0.427389\pi\)
0.226141 + 0.974095i \(0.427389\pi\)
\(978\) 0 0
\(979\) −0.583301 −0.0186424
\(980\) 0 0
\(981\) 85.0188 2.71444
\(982\) 0 0
\(983\) −20.9143 −0.667063 −0.333531 0.942739i \(-0.608240\pi\)
−0.333531 + 0.942739i \(0.608240\pi\)
\(984\) 0 0
\(985\) −39.8470 −1.26963
\(986\) 0 0
\(987\) −27.0322 −0.860444
\(988\) 0 0
\(989\) −14.1616 −0.450312
\(990\) 0 0
\(991\) −44.4247 −1.41120 −0.705599 0.708611i \(-0.749322\pi\)
−0.705599 + 0.708611i \(0.749322\pi\)
\(992\) 0 0
\(993\) −56.3995 −1.78978
\(994\) 0 0
\(995\) 92.2981 2.92604
\(996\) 0 0
\(997\) 39.9041 1.26378 0.631888 0.775059i \(-0.282280\pi\)
0.631888 + 0.775059i \(0.282280\pi\)
\(998\) 0 0
\(999\) −34.1186 −1.07947
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 4004.2.a.i.1.1 9
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4004.2.a.i.1.1 9 1.1 even 1 trivial