Properties

Label 3762.2.a.bb.1.1
Level $3762$
Weight $2$
Character 3762.1
Self dual yes
Analytic conductor $30.040$
Analytic rank $0$
Dimension $2$
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] = [3762,2,Mod(1,3762)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3762, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("3762.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3762 = 2 \cdot 3^{2} \cdot 11 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3762.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: \(30.0397212404\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{13}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 418)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.30278\) of defining polynomial
Character \(\chi\) \(=\) 3762.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{4} -1.30278 q^{5} -2.30278 q^{7} +1.00000 q^{8} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{4} -1.30278 q^{5} -2.30278 q^{7} +1.00000 q^{8} -1.30278 q^{10} -1.00000 q^{11} +0.302776 q^{13} -2.30278 q^{14} +1.00000 q^{16} -2.60555 q^{17} +1.00000 q^{19} -1.30278 q^{20} -1.00000 q^{22} +8.60555 q^{23} -3.30278 q^{25} +0.302776 q^{26} -2.30278 q^{28} +4.69722 q^{29} +0.302776 q^{31} +1.00000 q^{32} -2.60555 q^{34} +3.00000 q^{35} -9.21110 q^{37} +1.00000 q^{38} -1.30278 q^{40} +6.90833 q^{41} +11.9083 q^{43} -1.00000 q^{44} +8.60555 q^{46} +6.00000 q^{47} -1.69722 q^{49} -3.30278 q^{50} +0.302776 q^{52} +3.39445 q^{53} +1.30278 q^{55} -2.30278 q^{56} +4.69722 q^{58} -3.21110 q^{61} +0.302776 q^{62} +1.00000 q^{64} -0.394449 q^{65} +11.5139 q^{67} -2.60555 q^{68} +3.00000 q^{70} +13.3028 q^{71} +4.60555 q^{73} -9.21110 q^{74} +1.00000 q^{76} +2.30278 q^{77} -5.81665 q^{79} -1.30278 q^{80} +6.90833 q^{82} -10.6972 q^{83} +3.39445 q^{85} +11.9083 q^{86} -1.00000 q^{88} +2.60555 q^{89} -0.697224 q^{91} +8.60555 q^{92} +6.00000 q^{94} -1.30278 q^{95} +8.00000 q^{97} -1.69722 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} + 2 q^{4} + q^{5} - q^{7} + 2 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{2} + 2 q^{4} + q^{5} - q^{7} + 2 q^{8} + q^{10} - 2 q^{11} - 3 q^{13} - q^{14} + 2 q^{16} + 2 q^{17} + 2 q^{19} + q^{20} - 2 q^{22} + 10 q^{23} - 3 q^{25} - 3 q^{26} - q^{28} + 13 q^{29} - 3 q^{31} + 2 q^{32} + 2 q^{34} + 6 q^{35} - 4 q^{37} + 2 q^{38} + q^{40} + 3 q^{41} + 13 q^{43} - 2 q^{44} + 10 q^{46} + 12 q^{47} - 7 q^{49} - 3 q^{50} - 3 q^{52} + 14 q^{53} - q^{55} - q^{56} + 13 q^{58} + 8 q^{61} - 3 q^{62} + 2 q^{64} - 8 q^{65} + 5 q^{67} + 2 q^{68} + 6 q^{70} + 23 q^{71} + 2 q^{73} - 4 q^{74} + 2 q^{76} + q^{77} + 10 q^{79} + q^{80} + 3 q^{82} - 25 q^{83} + 14 q^{85} + 13 q^{86} - 2 q^{88} - 2 q^{89} - 5 q^{91} + 10 q^{92} + 12 q^{94} + q^{95} + 16 q^{97} - 7 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) 0 0
\(4\) 1.00000 0.500000
\(5\) −1.30278 −0.582619 −0.291309 0.956629i \(-0.594091\pi\)
−0.291309 + 0.956629i \(0.594091\pi\)
\(6\) 0 0
\(7\) −2.30278 −0.870367 −0.435184 0.900342i \(-0.643317\pi\)
−0.435184 + 0.900342i \(0.643317\pi\)
\(8\) 1.00000 0.353553
\(9\) 0 0
\(10\) −1.30278 −0.411974
\(11\) −1.00000 −0.301511
\(12\) 0 0
\(13\) 0.302776 0.0839749 0.0419874 0.999118i \(-0.486631\pi\)
0.0419874 + 0.999118i \(0.486631\pi\)
\(14\) −2.30278 −0.615443
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −2.60555 −0.631939 −0.315970 0.948769i \(-0.602330\pi\)
−0.315970 + 0.948769i \(0.602330\pi\)
\(18\) 0 0
\(19\) 1.00000 0.229416
\(20\) −1.30278 −0.291309
\(21\) 0 0
\(22\) −1.00000 −0.213201
\(23\) 8.60555 1.79438 0.897191 0.441643i \(-0.145604\pi\)
0.897191 + 0.441643i \(0.145604\pi\)
\(24\) 0 0
\(25\) −3.30278 −0.660555
\(26\) 0.302776 0.0593792
\(27\) 0 0
\(28\) −2.30278 −0.435184
\(29\) 4.69722 0.872253 0.436126 0.899885i \(-0.356350\pi\)
0.436126 + 0.899885i \(0.356350\pi\)
\(30\) 0 0
\(31\) 0.302776 0.0543801 0.0271901 0.999630i \(-0.491344\pi\)
0.0271901 + 0.999630i \(0.491344\pi\)
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) −2.60555 −0.446848
\(35\) 3.00000 0.507093
\(36\) 0 0
\(37\) −9.21110 −1.51430 −0.757148 0.653243i \(-0.773408\pi\)
−0.757148 + 0.653243i \(0.773408\pi\)
\(38\) 1.00000 0.162221
\(39\) 0 0
\(40\) −1.30278 −0.205987
\(41\) 6.90833 1.07890 0.539450 0.842018i \(-0.318632\pi\)
0.539450 + 0.842018i \(0.318632\pi\)
\(42\) 0 0
\(43\) 11.9083 1.81600 0.908001 0.418967i \(-0.137608\pi\)
0.908001 + 0.418967i \(0.137608\pi\)
\(44\) −1.00000 −0.150756
\(45\) 0 0
\(46\) 8.60555 1.26882
\(47\) 6.00000 0.875190 0.437595 0.899172i \(-0.355830\pi\)
0.437595 + 0.899172i \(0.355830\pi\)
\(48\) 0 0
\(49\) −1.69722 −0.242461
\(50\) −3.30278 −0.467083
\(51\) 0 0
\(52\) 0.302776 0.0419874
\(53\) 3.39445 0.466263 0.233132 0.972445i \(-0.425103\pi\)
0.233132 + 0.972445i \(0.425103\pi\)
\(54\) 0 0
\(55\) 1.30278 0.175666
\(56\) −2.30278 −0.307721
\(57\) 0 0
\(58\) 4.69722 0.616776
\(59\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 0 0
\(61\) −3.21110 −0.411140 −0.205570 0.978642i \(-0.565905\pi\)
−0.205570 + 0.978642i \(0.565905\pi\)
\(62\) 0.302776 0.0384525
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) −0.394449 −0.0489253
\(66\) 0 0
\(67\) 11.5139 1.40664 0.703322 0.710871i \(-0.251699\pi\)
0.703322 + 0.710871i \(0.251699\pi\)
\(68\) −2.60555 −0.315970
\(69\) 0 0
\(70\) 3.00000 0.358569
\(71\) 13.3028 1.57875 0.789375 0.613912i \(-0.210405\pi\)
0.789375 + 0.613912i \(0.210405\pi\)
\(72\) 0 0
\(73\) 4.60555 0.539039 0.269520 0.962995i \(-0.413135\pi\)
0.269520 + 0.962995i \(0.413135\pi\)
\(74\) −9.21110 −1.07077
\(75\) 0 0
\(76\) 1.00000 0.114708
\(77\) 2.30278 0.262426
\(78\) 0 0
\(79\) −5.81665 −0.654425 −0.327212 0.944951i \(-0.606109\pi\)
−0.327212 + 0.944951i \(0.606109\pi\)
\(80\) −1.30278 −0.145655
\(81\) 0 0
\(82\) 6.90833 0.762897
\(83\) −10.6972 −1.17417 −0.587086 0.809524i \(-0.699725\pi\)
−0.587086 + 0.809524i \(0.699725\pi\)
\(84\) 0 0
\(85\) 3.39445 0.368180
\(86\) 11.9083 1.28411
\(87\) 0 0
\(88\) −1.00000 −0.106600
\(89\) 2.60555 0.276188 0.138094 0.990419i \(-0.455902\pi\)
0.138094 + 0.990419i \(0.455902\pi\)
\(90\) 0 0
\(91\) −0.697224 −0.0730890
\(92\) 8.60555 0.897191
\(93\) 0 0
\(94\) 6.00000 0.618853
\(95\) −1.30278 −0.133662
\(96\) 0 0
\(97\) 8.00000 0.812277 0.406138 0.913812i \(-0.366875\pi\)
0.406138 + 0.913812i \(0.366875\pi\)
\(98\) −1.69722 −0.171446
\(99\) 0 0
\(100\) −3.30278 −0.330278
\(101\) 13.8167 1.37481 0.687404 0.726275i \(-0.258750\pi\)
0.687404 + 0.726275i \(0.258750\pi\)
\(102\) 0 0
\(103\) −5.30278 −0.522498 −0.261249 0.965271i \(-0.584134\pi\)
−0.261249 + 0.965271i \(0.584134\pi\)
\(104\) 0.302776 0.0296896
\(105\) 0 0
\(106\) 3.39445 0.329698
\(107\) 11.2111 1.08382 0.541909 0.840437i \(-0.317702\pi\)
0.541909 + 0.840437i \(0.317702\pi\)
\(108\) 0 0
\(109\) 2.00000 0.191565 0.0957826 0.995402i \(-0.469465\pi\)
0.0957826 + 0.995402i \(0.469465\pi\)
\(110\) 1.30278 0.124215
\(111\) 0 0
\(112\) −2.30278 −0.217592
\(113\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(114\) 0 0
\(115\) −11.2111 −1.04544
\(116\) 4.69722 0.436126
\(117\) 0 0
\(118\) 0 0
\(119\) 6.00000 0.550019
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) −3.21110 −0.290720
\(123\) 0 0
\(124\) 0.302776 0.0271901
\(125\) 10.8167 0.967471
\(126\) 0 0
\(127\) −12.6056 −1.11856 −0.559281 0.828978i \(-0.688923\pi\)
−0.559281 + 0.828978i \(0.688923\pi\)
\(128\) 1.00000 0.0883883
\(129\) 0 0
\(130\) −0.394449 −0.0345954
\(131\) 6.11943 0.534657 0.267329 0.963605i \(-0.413859\pi\)
0.267329 + 0.963605i \(0.413859\pi\)
\(132\) 0 0
\(133\) −2.30278 −0.199676
\(134\) 11.5139 0.994648
\(135\) 0 0
\(136\) −2.60555 −0.223424
\(137\) 21.5139 1.83805 0.919027 0.394194i \(-0.128976\pi\)
0.919027 + 0.394194i \(0.128976\pi\)
\(138\) 0 0
\(139\) −2.69722 −0.228776 −0.114388 0.993436i \(-0.536491\pi\)
−0.114388 + 0.993436i \(0.536491\pi\)
\(140\) 3.00000 0.253546
\(141\) 0 0
\(142\) 13.3028 1.11634
\(143\) −0.302776 −0.0253194
\(144\) 0 0
\(145\) −6.11943 −0.508191
\(146\) 4.60555 0.381158
\(147\) 0 0
\(148\) −9.21110 −0.757148
\(149\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(150\) 0 0
\(151\) 12.4222 1.01090 0.505452 0.862855i \(-0.331326\pi\)
0.505452 + 0.862855i \(0.331326\pi\)
\(152\) 1.00000 0.0811107
\(153\) 0 0
\(154\) 2.30278 0.185563
\(155\) −0.394449 −0.0316829
\(156\) 0 0
\(157\) 11.9083 0.950388 0.475194 0.879881i \(-0.342378\pi\)
0.475194 + 0.879881i \(0.342378\pi\)
\(158\) −5.81665 −0.462748
\(159\) 0 0
\(160\) −1.30278 −0.102993
\(161\) −19.8167 −1.56177
\(162\) 0 0
\(163\) −18.6056 −1.45730 −0.728650 0.684887i \(-0.759852\pi\)
−0.728650 + 0.684887i \(0.759852\pi\)
\(164\) 6.90833 0.539450
\(165\) 0 0
\(166\) −10.6972 −0.830266
\(167\) −16.4222 −1.27079 −0.635394 0.772188i \(-0.719162\pi\)
−0.635394 + 0.772188i \(0.719162\pi\)
\(168\) 0 0
\(169\) −12.9083 −0.992948
\(170\) 3.39445 0.260342
\(171\) 0 0
\(172\) 11.9083 0.908001
\(173\) −20.3305 −1.54570 −0.772851 0.634588i \(-0.781170\pi\)
−0.772851 + 0.634588i \(0.781170\pi\)
\(174\) 0 0
\(175\) 7.60555 0.574926
\(176\) −1.00000 −0.0753778
\(177\) 0 0
\(178\) 2.60555 0.195294
\(179\) −12.1194 −0.905849 −0.452924 0.891549i \(-0.649619\pi\)
−0.452924 + 0.891549i \(0.649619\pi\)
\(180\) 0 0
\(181\) 21.0278 1.56298 0.781490 0.623917i \(-0.214460\pi\)
0.781490 + 0.623917i \(0.214460\pi\)
\(182\) −0.697224 −0.0516817
\(183\) 0 0
\(184\) 8.60555 0.634410
\(185\) 12.0000 0.882258
\(186\) 0 0
\(187\) 2.60555 0.190537
\(188\) 6.00000 0.437595
\(189\) 0 0
\(190\) −1.30278 −0.0945133
\(191\) 6.00000 0.434145 0.217072 0.976156i \(-0.430349\pi\)
0.217072 + 0.976156i \(0.430349\pi\)
\(192\) 0 0
\(193\) 7.72498 0.556056 0.278028 0.960573i \(-0.410319\pi\)
0.278028 + 0.960573i \(0.410319\pi\)
\(194\) 8.00000 0.574367
\(195\) 0 0
\(196\) −1.69722 −0.121230
\(197\) 21.6333 1.54131 0.770655 0.637253i \(-0.219929\pi\)
0.770655 + 0.637253i \(0.219929\pi\)
\(198\) 0 0
\(199\) −23.8167 −1.68832 −0.844159 0.536093i \(-0.819900\pi\)
−0.844159 + 0.536093i \(0.819900\pi\)
\(200\) −3.30278 −0.233542
\(201\) 0 0
\(202\) 13.8167 0.972136
\(203\) −10.8167 −0.759180
\(204\) 0 0
\(205\) −9.00000 −0.628587
\(206\) −5.30278 −0.369462
\(207\) 0 0
\(208\) 0.302776 0.0209937
\(209\) −1.00000 −0.0691714
\(210\) 0 0
\(211\) 17.3944 1.19748 0.598742 0.800942i \(-0.295668\pi\)
0.598742 + 0.800942i \(0.295668\pi\)
\(212\) 3.39445 0.233132
\(213\) 0 0
\(214\) 11.2111 0.766375
\(215\) −15.5139 −1.05804
\(216\) 0 0
\(217\) −0.697224 −0.0473307
\(218\) 2.00000 0.135457
\(219\) 0 0
\(220\) 1.30278 0.0878331
\(221\) −0.788897 −0.0530670
\(222\) 0 0
\(223\) −10.7889 −0.722478 −0.361239 0.932473i \(-0.617646\pi\)
−0.361239 + 0.932473i \(0.617646\pi\)
\(224\) −2.30278 −0.153861
\(225\) 0 0
\(226\) 0 0
\(227\) 19.8167 1.31528 0.657639 0.753333i \(-0.271555\pi\)
0.657639 + 0.753333i \(0.271555\pi\)
\(228\) 0 0
\(229\) 1.09167 0.0721398 0.0360699 0.999349i \(-0.488516\pi\)
0.0360699 + 0.999349i \(0.488516\pi\)
\(230\) −11.2111 −0.739238
\(231\) 0 0
\(232\) 4.69722 0.308388
\(233\) 5.21110 0.341391 0.170695 0.985324i \(-0.445399\pi\)
0.170695 + 0.985324i \(0.445399\pi\)
\(234\) 0 0
\(235\) −7.81665 −0.509902
\(236\) 0 0
\(237\) 0 0
\(238\) 6.00000 0.388922
\(239\) −22.9361 −1.48361 −0.741806 0.670615i \(-0.766030\pi\)
−0.741806 + 0.670615i \(0.766030\pi\)
\(240\) 0 0
\(241\) 23.9083 1.54007 0.770035 0.638001i \(-0.220239\pi\)
0.770035 + 0.638001i \(0.220239\pi\)
\(242\) 1.00000 0.0642824
\(243\) 0 0
\(244\) −3.21110 −0.205570
\(245\) 2.21110 0.141262
\(246\) 0 0
\(247\) 0.302776 0.0192652
\(248\) 0.302776 0.0192263
\(249\) 0 0
\(250\) 10.8167 0.684105
\(251\) −24.0000 −1.51487 −0.757433 0.652913i \(-0.773547\pi\)
−0.757433 + 0.652913i \(0.773547\pi\)
\(252\) 0 0
\(253\) −8.60555 −0.541026
\(254\) −12.6056 −0.790943
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −2.60555 −0.162530 −0.0812649 0.996693i \(-0.525896\pi\)
−0.0812649 + 0.996693i \(0.525896\pi\)
\(258\) 0 0
\(259\) 21.2111 1.31799
\(260\) −0.394449 −0.0244627
\(261\) 0 0
\(262\) 6.11943 0.378060
\(263\) 16.6972 1.02959 0.514797 0.857312i \(-0.327867\pi\)
0.514797 + 0.857312i \(0.327867\pi\)
\(264\) 0 0
\(265\) −4.42221 −0.271654
\(266\) −2.30278 −0.141192
\(267\) 0 0
\(268\) 11.5139 0.703322
\(269\) −1.81665 −0.110763 −0.0553817 0.998465i \(-0.517638\pi\)
−0.0553817 + 0.998465i \(0.517638\pi\)
\(270\) 0 0
\(271\) 11.5139 0.699418 0.349709 0.936858i \(-0.386280\pi\)
0.349709 + 0.936858i \(0.386280\pi\)
\(272\) −2.60555 −0.157985
\(273\) 0 0
\(274\) 21.5139 1.29970
\(275\) 3.30278 0.199165
\(276\) 0 0
\(277\) −10.0000 −0.600842 −0.300421 0.953807i \(-0.597127\pi\)
−0.300421 + 0.953807i \(0.597127\pi\)
\(278\) −2.69722 −0.161769
\(279\) 0 0
\(280\) 3.00000 0.179284
\(281\) 18.5139 1.10445 0.552223 0.833697i \(-0.313780\pi\)
0.552223 + 0.833697i \(0.313780\pi\)
\(282\) 0 0
\(283\) 27.9361 1.66063 0.830314 0.557296i \(-0.188161\pi\)
0.830314 + 0.557296i \(0.188161\pi\)
\(284\) 13.3028 0.789375
\(285\) 0 0
\(286\) −0.302776 −0.0179035
\(287\) −15.9083 −0.939039
\(288\) 0 0
\(289\) −10.2111 −0.600653
\(290\) −6.11943 −0.359345
\(291\) 0 0
\(292\) 4.60555 0.269520
\(293\) −28.5416 −1.66742 −0.833710 0.552202i \(-0.813788\pi\)
−0.833710 + 0.552202i \(0.813788\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) −9.21110 −0.535384
\(297\) 0 0
\(298\) 0 0
\(299\) 2.60555 0.150683
\(300\) 0 0
\(301\) −27.4222 −1.58059
\(302\) 12.4222 0.714818
\(303\) 0 0
\(304\) 1.00000 0.0573539
\(305\) 4.18335 0.239538
\(306\) 0 0
\(307\) −23.8167 −1.35929 −0.679644 0.733542i \(-0.737866\pi\)
−0.679644 + 0.733542i \(0.737866\pi\)
\(308\) 2.30278 0.131213
\(309\) 0 0
\(310\) −0.394449 −0.0224032
\(311\) −4.18335 −0.237216 −0.118608 0.992941i \(-0.537843\pi\)
−0.118608 + 0.992941i \(0.537843\pi\)
\(312\) 0 0
\(313\) −7.11943 −0.402414 −0.201207 0.979549i \(-0.564486\pi\)
−0.201207 + 0.979549i \(0.564486\pi\)
\(314\) 11.9083 0.672026
\(315\) 0 0
\(316\) −5.81665 −0.327212
\(317\) −7.81665 −0.439027 −0.219514 0.975609i \(-0.570447\pi\)
−0.219514 + 0.975609i \(0.570447\pi\)
\(318\) 0 0
\(319\) −4.69722 −0.262994
\(320\) −1.30278 −0.0728274
\(321\) 0 0
\(322\) −19.8167 −1.10434
\(323\) −2.60555 −0.144977
\(324\) 0 0
\(325\) −1.00000 −0.0554700
\(326\) −18.6056 −1.03047
\(327\) 0 0
\(328\) 6.90833 0.381449
\(329\) −13.8167 −0.761737
\(330\) 0 0
\(331\) −9.72498 −0.534533 −0.267267 0.963623i \(-0.586120\pi\)
−0.267267 + 0.963623i \(0.586120\pi\)
\(332\) −10.6972 −0.587086
\(333\) 0 0
\(334\) −16.4222 −0.898583
\(335\) −15.0000 −0.819538
\(336\) 0 0
\(337\) −5.69722 −0.310348 −0.155174 0.987887i \(-0.549594\pi\)
−0.155174 + 0.987887i \(0.549594\pi\)
\(338\) −12.9083 −0.702120
\(339\) 0 0
\(340\) 3.39445 0.184090
\(341\) −0.302776 −0.0163962
\(342\) 0 0
\(343\) 20.0278 1.08140
\(344\) 11.9083 0.642054
\(345\) 0 0
\(346\) −20.3305 −1.09298
\(347\) −17.2111 −0.923940 −0.461970 0.886895i \(-0.652857\pi\)
−0.461970 + 0.886895i \(0.652857\pi\)
\(348\) 0 0
\(349\) 3.57779 0.191515 0.0957575 0.995405i \(-0.469473\pi\)
0.0957575 + 0.995405i \(0.469473\pi\)
\(350\) 7.60555 0.406534
\(351\) 0 0
\(352\) −1.00000 −0.0533002
\(353\) 0.788897 0.0419888 0.0209944 0.999780i \(-0.493317\pi\)
0.0209944 + 0.999780i \(0.493317\pi\)
\(354\) 0 0
\(355\) −17.3305 −0.919809
\(356\) 2.60555 0.138094
\(357\) 0 0
\(358\) −12.1194 −0.640532
\(359\) 27.5139 1.45213 0.726063 0.687628i \(-0.241348\pi\)
0.726063 + 0.687628i \(0.241348\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 21.0278 1.10519
\(363\) 0 0
\(364\) −0.697224 −0.0365445
\(365\) −6.00000 −0.314054
\(366\) 0 0
\(367\) 7.21110 0.376416 0.188208 0.982129i \(-0.439732\pi\)
0.188208 + 0.982129i \(0.439732\pi\)
\(368\) 8.60555 0.448595
\(369\) 0 0
\(370\) 12.0000 0.623850
\(371\) −7.81665 −0.405820
\(372\) 0 0
\(373\) 17.9083 0.927258 0.463629 0.886029i \(-0.346547\pi\)
0.463629 + 0.886029i \(0.346547\pi\)
\(374\) 2.60555 0.134730
\(375\) 0 0
\(376\) 6.00000 0.309426
\(377\) 1.42221 0.0732473
\(378\) 0 0
\(379\) −31.5139 −1.61876 −0.809380 0.587286i \(-0.800196\pi\)
−0.809380 + 0.587286i \(0.800196\pi\)
\(380\) −1.30278 −0.0668310
\(381\) 0 0
\(382\) 6.00000 0.306987
\(383\) −21.5139 −1.09931 −0.549654 0.835392i \(-0.685240\pi\)
−0.549654 + 0.835392i \(0.685240\pi\)
\(384\) 0 0
\(385\) −3.00000 −0.152894
\(386\) 7.72498 0.393191
\(387\) 0 0
\(388\) 8.00000 0.406138
\(389\) 19.5416 0.990800 0.495400 0.868665i \(-0.335021\pi\)
0.495400 + 0.868665i \(0.335021\pi\)
\(390\) 0 0
\(391\) −22.4222 −1.13394
\(392\) −1.69722 −0.0857228
\(393\) 0 0
\(394\) 21.6333 1.08987
\(395\) 7.57779 0.381280
\(396\) 0 0
\(397\) 35.7527 1.79438 0.897189 0.441646i \(-0.145605\pi\)
0.897189 + 0.441646i \(0.145605\pi\)
\(398\) −23.8167 −1.19382
\(399\) 0 0
\(400\) −3.30278 −0.165139
\(401\) −15.3944 −0.768762 −0.384381 0.923175i \(-0.625585\pi\)
−0.384381 + 0.923175i \(0.625585\pi\)
\(402\) 0 0
\(403\) 0.0916731 0.00456656
\(404\) 13.8167 0.687404
\(405\) 0 0
\(406\) −10.8167 −0.536822
\(407\) 9.21110 0.456577
\(408\) 0 0
\(409\) −38.1472 −1.88626 −0.943128 0.332428i \(-0.892132\pi\)
−0.943128 + 0.332428i \(0.892132\pi\)
\(410\) −9.00000 −0.444478
\(411\) 0 0
\(412\) −5.30278 −0.261249
\(413\) 0 0
\(414\) 0 0
\(415\) 13.9361 0.684095
\(416\) 0.302776 0.0148448
\(417\) 0 0
\(418\) −1.00000 −0.0489116
\(419\) −23.2111 −1.13394 −0.566968 0.823740i \(-0.691884\pi\)
−0.566968 + 0.823740i \(0.691884\pi\)
\(420\) 0 0
\(421\) 20.2389 0.986382 0.493191 0.869921i \(-0.335830\pi\)
0.493191 + 0.869921i \(0.335830\pi\)
\(422\) 17.3944 0.846749
\(423\) 0 0
\(424\) 3.39445 0.164849
\(425\) 8.60555 0.417431
\(426\) 0 0
\(427\) 7.39445 0.357842
\(428\) 11.2111 0.541909
\(429\) 0 0
\(430\) −15.5139 −0.748146
\(431\) −17.2111 −0.829030 −0.414515 0.910043i \(-0.636049\pi\)
−0.414515 + 0.910043i \(0.636049\pi\)
\(432\) 0 0
\(433\) 26.2389 1.26096 0.630480 0.776206i \(-0.282858\pi\)
0.630480 + 0.776206i \(0.282858\pi\)
\(434\) −0.697224 −0.0334678
\(435\) 0 0
\(436\) 2.00000 0.0957826
\(437\) 8.60555 0.411659
\(438\) 0 0
\(439\) −10.7889 −0.514926 −0.257463 0.966288i \(-0.582887\pi\)
−0.257463 + 0.966288i \(0.582887\pi\)
\(440\) 1.30278 0.0621074
\(441\) 0 0
\(442\) −0.788897 −0.0375240
\(443\) 3.63331 0.172624 0.0863118 0.996268i \(-0.472492\pi\)
0.0863118 + 0.996268i \(0.472492\pi\)
\(444\) 0 0
\(445\) −3.39445 −0.160912
\(446\) −10.7889 −0.510869
\(447\) 0 0
\(448\) −2.30278 −0.108796
\(449\) 10.1833 0.480582 0.240291 0.970701i \(-0.422757\pi\)
0.240291 + 0.970701i \(0.422757\pi\)
\(450\) 0 0
\(451\) −6.90833 −0.325300
\(452\) 0 0
\(453\) 0 0
\(454\) 19.8167 0.930042
\(455\) 0.908327 0.0425830
\(456\) 0 0
\(457\) 0.183346 0.00857657 0.00428829 0.999991i \(-0.498635\pi\)
0.00428829 + 0.999991i \(0.498635\pi\)
\(458\) 1.09167 0.0510105
\(459\) 0 0
\(460\) −11.2111 −0.522720
\(461\) 22.4222 1.04431 0.522153 0.852852i \(-0.325129\pi\)
0.522153 + 0.852852i \(0.325129\pi\)
\(462\) 0 0
\(463\) 13.2111 0.613972 0.306986 0.951714i \(-0.400680\pi\)
0.306986 + 0.951714i \(0.400680\pi\)
\(464\) 4.69722 0.218063
\(465\) 0 0
\(466\) 5.21110 0.241400
\(467\) 15.3944 0.712370 0.356185 0.934415i \(-0.384077\pi\)
0.356185 + 0.934415i \(0.384077\pi\)
\(468\) 0 0
\(469\) −26.5139 −1.22430
\(470\) −7.81665 −0.360555
\(471\) 0 0
\(472\) 0 0
\(473\) −11.9083 −0.547545
\(474\) 0 0
\(475\) −3.30278 −0.151542
\(476\) 6.00000 0.275010
\(477\) 0 0
\(478\) −22.9361 −1.04907
\(479\) −3.27502 −0.149639 −0.0748197 0.997197i \(-0.523838\pi\)
−0.0748197 + 0.997197i \(0.523838\pi\)
\(480\) 0 0
\(481\) −2.78890 −0.127163
\(482\) 23.9083 1.08899
\(483\) 0 0
\(484\) 1.00000 0.0454545
\(485\) −10.4222 −0.473248
\(486\) 0 0
\(487\) 10.3305 0.468121 0.234061 0.972222i \(-0.424799\pi\)
0.234061 + 0.972222i \(0.424799\pi\)
\(488\) −3.21110 −0.145360
\(489\) 0 0
\(490\) 2.21110 0.0998874
\(491\) −31.9361 −1.44126 −0.720628 0.693322i \(-0.756146\pi\)
−0.720628 + 0.693322i \(0.756146\pi\)
\(492\) 0 0
\(493\) −12.2389 −0.551210
\(494\) 0.302776 0.0136225
\(495\) 0 0
\(496\) 0.302776 0.0135950
\(497\) −30.6333 −1.37409
\(498\) 0 0
\(499\) −34.0000 −1.52205 −0.761025 0.648723i \(-0.775303\pi\)
−0.761025 + 0.648723i \(0.775303\pi\)
\(500\) 10.8167 0.483735
\(501\) 0 0
\(502\) −24.0000 −1.07117
\(503\) 3.11943 0.139088 0.0695442 0.997579i \(-0.477845\pi\)
0.0695442 + 0.997579i \(0.477845\pi\)
\(504\) 0 0
\(505\) −18.0000 −0.800989
\(506\) −8.60555 −0.382563
\(507\) 0 0
\(508\) −12.6056 −0.559281
\(509\) −10.4222 −0.461956 −0.230978 0.972959i \(-0.574193\pi\)
−0.230978 + 0.972959i \(0.574193\pi\)
\(510\) 0 0
\(511\) −10.6056 −0.469162
\(512\) 1.00000 0.0441942
\(513\) 0 0
\(514\) −2.60555 −0.114926
\(515\) 6.90833 0.304417
\(516\) 0 0
\(517\) −6.00000 −0.263880
\(518\) 21.2111 0.931962
\(519\) 0 0
\(520\) −0.394449 −0.0172977
\(521\) −7.57779 −0.331989 −0.165995 0.986127i \(-0.553083\pi\)
−0.165995 + 0.986127i \(0.553083\pi\)
\(522\) 0 0
\(523\) −17.0278 −0.744572 −0.372286 0.928118i \(-0.621426\pi\)
−0.372286 + 0.928118i \(0.621426\pi\)
\(524\) 6.11943 0.267329
\(525\) 0 0
\(526\) 16.6972 0.728034
\(527\) −0.788897 −0.0343649
\(528\) 0 0
\(529\) 51.0555 2.21980
\(530\) −4.42221 −0.192088
\(531\) 0 0
\(532\) −2.30278 −0.0998380
\(533\) 2.09167 0.0906004
\(534\) 0 0
\(535\) −14.6056 −0.631453
\(536\) 11.5139 0.497324
\(537\) 0 0
\(538\) −1.81665 −0.0783215
\(539\) 1.69722 0.0731046
\(540\) 0 0
\(541\) −17.8167 −0.765998 −0.382999 0.923749i \(-0.625109\pi\)
−0.382999 + 0.923749i \(0.625109\pi\)
\(542\) 11.5139 0.494563
\(543\) 0 0
\(544\) −2.60555 −0.111712
\(545\) −2.60555 −0.111610
\(546\) 0 0
\(547\) −4.00000 −0.171028 −0.0855138 0.996337i \(-0.527253\pi\)
−0.0855138 + 0.996337i \(0.527253\pi\)
\(548\) 21.5139 0.919027
\(549\) 0 0
\(550\) 3.30278 0.140831
\(551\) 4.69722 0.200108
\(552\) 0 0
\(553\) 13.3944 0.569590
\(554\) −10.0000 −0.424859
\(555\) 0 0
\(556\) −2.69722 −0.114388
\(557\) −15.6333 −0.662405 −0.331202 0.943560i \(-0.607454\pi\)
−0.331202 + 0.943560i \(0.607454\pi\)
\(558\) 0 0
\(559\) 3.60555 0.152499
\(560\) 3.00000 0.126773
\(561\) 0 0
\(562\) 18.5139 0.780961
\(563\) 21.6333 0.911735 0.455868 0.890048i \(-0.349329\pi\)
0.455868 + 0.890048i \(0.349329\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 27.9361 1.17424
\(567\) 0 0
\(568\) 13.3028 0.558172
\(569\) −44.7250 −1.87497 −0.937484 0.348027i \(-0.886852\pi\)
−0.937484 + 0.348027i \(0.886852\pi\)
\(570\) 0 0
\(571\) 39.9361 1.67127 0.835637 0.549283i \(-0.185099\pi\)
0.835637 + 0.549283i \(0.185099\pi\)
\(572\) −0.302776 −0.0126597
\(573\) 0 0
\(574\) −15.9083 −0.664001
\(575\) −28.4222 −1.18529
\(576\) 0 0
\(577\) −28.9083 −1.20347 −0.601735 0.798696i \(-0.705524\pi\)
−0.601735 + 0.798696i \(0.705524\pi\)
\(578\) −10.2111 −0.424726
\(579\) 0 0
\(580\) −6.11943 −0.254095
\(581\) 24.6333 1.02196
\(582\) 0 0
\(583\) −3.39445 −0.140584
\(584\) 4.60555 0.190579
\(585\) 0 0
\(586\) −28.5416 −1.17904
\(587\) −12.0000 −0.495293 −0.247647 0.968850i \(-0.579657\pi\)
−0.247647 + 0.968850i \(0.579657\pi\)
\(588\) 0 0
\(589\) 0.302776 0.0124757
\(590\) 0 0
\(591\) 0 0
\(592\) −9.21110 −0.378574
\(593\) 42.0000 1.72473 0.862367 0.506284i \(-0.168981\pi\)
0.862367 + 0.506284i \(0.168981\pi\)
\(594\) 0 0
\(595\) −7.81665 −0.320452
\(596\) 0 0
\(597\) 0 0
\(598\) 2.60555 0.106549
\(599\) 45.1194 1.84353 0.921765 0.387749i \(-0.126747\pi\)
0.921765 + 0.387749i \(0.126747\pi\)
\(600\) 0 0
\(601\) 14.9083 0.608123 0.304062 0.952652i \(-0.401657\pi\)
0.304062 + 0.952652i \(0.401657\pi\)
\(602\) −27.4222 −1.11765
\(603\) 0 0
\(604\) 12.4222 0.505452
\(605\) −1.30278 −0.0529654
\(606\) 0 0
\(607\) −17.8167 −0.723156 −0.361578 0.932342i \(-0.617762\pi\)
−0.361578 + 0.932342i \(0.617762\pi\)
\(608\) 1.00000 0.0405554
\(609\) 0 0
\(610\) 4.18335 0.169379
\(611\) 1.81665 0.0734939
\(612\) 0 0
\(613\) 2.00000 0.0807792 0.0403896 0.999184i \(-0.487140\pi\)
0.0403896 + 0.999184i \(0.487140\pi\)
\(614\) −23.8167 −0.961162
\(615\) 0 0
\(616\) 2.30278 0.0927815
\(617\) −12.9083 −0.519670 −0.259835 0.965653i \(-0.583668\pi\)
−0.259835 + 0.965653i \(0.583668\pi\)
\(618\) 0 0
\(619\) 22.8444 0.918194 0.459097 0.888386i \(-0.348173\pi\)
0.459097 + 0.888386i \(0.348173\pi\)
\(620\) −0.394449 −0.0158414
\(621\) 0 0
\(622\) −4.18335 −0.167737
\(623\) −6.00000 −0.240385
\(624\) 0 0
\(625\) 2.42221 0.0968882
\(626\) −7.11943 −0.284550
\(627\) 0 0
\(628\) 11.9083 0.475194
\(629\) 24.0000 0.956943
\(630\) 0 0
\(631\) −8.97224 −0.357179 −0.178590 0.983924i \(-0.557153\pi\)
−0.178590 + 0.983924i \(0.557153\pi\)
\(632\) −5.81665 −0.231374
\(633\) 0 0
\(634\) −7.81665 −0.310439
\(635\) 16.4222 0.651695
\(636\) 0 0
\(637\) −0.513878 −0.0203606
\(638\) −4.69722 −0.185965
\(639\) 0 0
\(640\) −1.30278 −0.0514967
\(641\) 30.2389 1.19436 0.597182 0.802106i \(-0.296287\pi\)
0.597182 + 0.802106i \(0.296287\pi\)
\(642\) 0 0
\(643\) −2.42221 −0.0955224 −0.0477612 0.998859i \(-0.515209\pi\)
−0.0477612 + 0.998859i \(0.515209\pi\)
\(644\) −19.8167 −0.780886
\(645\) 0 0
\(646\) −2.60555 −0.102514
\(647\) −4.18335 −0.164464 −0.0822322 0.996613i \(-0.526205\pi\)
−0.0822322 + 0.996613i \(0.526205\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) −1.00000 −0.0392232
\(651\) 0 0
\(652\) −18.6056 −0.728650
\(653\) 17.3305 0.678196 0.339098 0.940751i \(-0.389878\pi\)
0.339098 + 0.940751i \(0.389878\pi\)
\(654\) 0 0
\(655\) −7.97224 −0.311501
\(656\) 6.90833 0.269725
\(657\) 0 0
\(658\) −13.8167 −0.538629
\(659\) 7.02776 0.273763 0.136881 0.990587i \(-0.456292\pi\)
0.136881 + 0.990587i \(0.456292\pi\)
\(660\) 0 0
\(661\) −19.6333 −0.763647 −0.381824 0.924235i \(-0.624704\pi\)
−0.381824 + 0.924235i \(0.624704\pi\)
\(662\) −9.72498 −0.377972
\(663\) 0 0
\(664\) −10.6972 −0.415133
\(665\) 3.00000 0.116335
\(666\) 0 0
\(667\) 40.4222 1.56515
\(668\) −16.4222 −0.635394
\(669\) 0 0
\(670\) −15.0000 −0.579501
\(671\) 3.21110 0.123963
\(672\) 0 0
\(673\) −2.30278 −0.0887655 −0.0443827 0.999015i \(-0.514132\pi\)
−0.0443827 + 0.999015i \(0.514132\pi\)
\(674\) −5.69722 −0.219449
\(675\) 0 0
\(676\) −12.9083 −0.496474
\(677\) 4.69722 0.180529 0.0902645 0.995918i \(-0.471229\pi\)
0.0902645 + 0.995918i \(0.471229\pi\)
\(678\) 0 0
\(679\) −18.4222 −0.706979
\(680\) 3.39445 0.130171
\(681\) 0 0
\(682\) −0.302776 −0.0115939
\(683\) 46.4222 1.77630 0.888148 0.459557i \(-0.151992\pi\)
0.888148 + 0.459557i \(0.151992\pi\)
\(684\) 0 0
\(685\) −28.0278 −1.07089
\(686\) 20.0278 0.764663
\(687\) 0 0
\(688\) 11.9083 0.454001
\(689\) 1.02776 0.0391544
\(690\) 0 0
\(691\) 3.57779 0.136106 0.0680529 0.997682i \(-0.478321\pi\)
0.0680529 + 0.997682i \(0.478321\pi\)
\(692\) −20.3305 −0.772851
\(693\) 0 0
\(694\) −17.2111 −0.653325
\(695\) 3.51388 0.133289
\(696\) 0 0
\(697\) −18.0000 −0.681799
\(698\) 3.57779 0.135422
\(699\) 0 0
\(700\) 7.60555 0.287463
\(701\) −15.6333 −0.590462 −0.295231 0.955426i \(-0.595397\pi\)
−0.295231 + 0.955426i \(0.595397\pi\)
\(702\) 0 0
\(703\) −9.21110 −0.347403
\(704\) −1.00000 −0.0376889
\(705\) 0 0
\(706\) 0.788897 0.0296905
\(707\) −31.8167 −1.19659
\(708\) 0 0
\(709\) 18.3028 0.687375 0.343688 0.939084i \(-0.388324\pi\)
0.343688 + 0.939084i \(0.388324\pi\)
\(710\) −17.3305 −0.650403
\(711\) 0 0
\(712\) 2.60555 0.0976472
\(713\) 2.60555 0.0975787
\(714\) 0 0
\(715\) 0.394449 0.0147515
\(716\) −12.1194 −0.452924
\(717\) 0 0
\(718\) 27.5139 1.02681
\(719\) 4.18335 0.156012 0.0780062 0.996953i \(-0.475145\pi\)
0.0780062 + 0.996953i \(0.475145\pi\)
\(720\) 0 0
\(721\) 12.2111 0.454765
\(722\) 1.00000 0.0372161
\(723\) 0 0
\(724\) 21.0278 0.781490
\(725\) −15.5139 −0.576171
\(726\) 0 0
\(727\) −43.6333 −1.61827 −0.809135 0.587623i \(-0.800064\pi\)
−0.809135 + 0.587623i \(0.800064\pi\)
\(728\) −0.697224 −0.0258409
\(729\) 0 0
\(730\) −6.00000 −0.222070
\(731\) −31.0278 −1.14760
\(732\) 0 0
\(733\) 7.21110 0.266348 0.133174 0.991093i \(-0.457483\pi\)
0.133174 + 0.991093i \(0.457483\pi\)
\(734\) 7.21110 0.266167
\(735\) 0 0
\(736\) 8.60555 0.317205
\(737\) −11.5139 −0.424119
\(738\) 0 0
\(739\) 35.3583 1.30068 0.650338 0.759645i \(-0.274627\pi\)
0.650338 + 0.759645i \(0.274627\pi\)
\(740\) 12.0000 0.441129
\(741\) 0 0
\(742\) −7.81665 −0.286958
\(743\) −45.6333 −1.67412 −0.837062 0.547108i \(-0.815729\pi\)
−0.837062 + 0.547108i \(0.815729\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 17.9083 0.655670
\(747\) 0 0
\(748\) 2.60555 0.0952684
\(749\) −25.8167 −0.943320
\(750\) 0 0
\(751\) 16.8444 0.614661 0.307331 0.951603i \(-0.400564\pi\)
0.307331 + 0.951603i \(0.400564\pi\)
\(752\) 6.00000 0.218797
\(753\) 0 0
\(754\) 1.42221 0.0517937
\(755\) −16.1833 −0.588972
\(756\) 0 0
\(757\) −23.5416 −0.855635 −0.427818 0.903865i \(-0.640717\pi\)
−0.427818 + 0.903865i \(0.640717\pi\)
\(758\) −31.5139 −1.14464
\(759\) 0 0
\(760\) −1.30278 −0.0472566
\(761\) 30.2389 1.09616 0.548079 0.836427i \(-0.315359\pi\)
0.548079 + 0.836427i \(0.315359\pi\)
\(762\) 0 0
\(763\) −4.60555 −0.166732
\(764\) 6.00000 0.217072
\(765\) 0 0
\(766\) −21.5139 −0.777328
\(767\) 0 0
\(768\) 0 0
\(769\) 5.63331 0.203142 0.101571 0.994828i \(-0.467613\pi\)
0.101571 + 0.994828i \(0.467613\pi\)
\(770\) −3.00000 −0.108112
\(771\) 0 0
\(772\) 7.72498 0.278028
\(773\) −19.5778 −0.704164 −0.352082 0.935969i \(-0.614526\pi\)
−0.352082 + 0.935969i \(0.614526\pi\)
\(774\) 0 0
\(775\) −1.00000 −0.0359211
\(776\) 8.00000 0.287183
\(777\) 0 0
\(778\) 19.5416 0.700602
\(779\) 6.90833 0.247516
\(780\) 0 0
\(781\) −13.3028 −0.476011
\(782\) −22.4222 −0.801816
\(783\) 0 0
\(784\) −1.69722 −0.0606152
\(785\) −15.5139 −0.553714
\(786\) 0 0
\(787\) 21.0278 0.749559 0.374779 0.927114i \(-0.377718\pi\)
0.374779 + 0.927114i \(0.377718\pi\)
\(788\) 21.6333 0.770655
\(789\) 0 0
\(790\) 7.57779 0.269606
\(791\) 0 0
\(792\) 0 0
\(793\) −0.972244 −0.0345254
\(794\) 35.7527 1.26882
\(795\) 0 0
\(796\) −23.8167 −0.844159
\(797\) −9.39445 −0.332768 −0.166384 0.986061i \(-0.553209\pi\)
−0.166384 + 0.986061i \(0.553209\pi\)
\(798\) 0 0
\(799\) −15.6333 −0.553067
\(800\) −3.30278 −0.116771
\(801\) 0 0
\(802\) −15.3944 −0.543597
\(803\) −4.60555 −0.162526
\(804\) 0 0
\(805\) 25.8167 0.909917
\(806\) 0.0916731 0.00322905
\(807\) 0 0
\(808\) 13.8167 0.486068
\(809\) −18.7889 −0.660582 −0.330291 0.943879i \(-0.607147\pi\)
−0.330291 + 0.943879i \(0.607147\pi\)
\(810\) 0 0
\(811\) −13.6333 −0.478730 −0.239365 0.970930i \(-0.576939\pi\)
−0.239365 + 0.970930i \(0.576939\pi\)
\(812\) −10.8167 −0.379590
\(813\) 0 0
\(814\) 9.21110 0.322849
\(815\) 24.2389 0.849050
\(816\) 0 0
\(817\) 11.9083 0.416620
\(818\) −38.1472 −1.33379
\(819\) 0 0
\(820\) −9.00000 −0.314294
\(821\) 27.3944 0.956073 0.478036 0.878340i \(-0.341349\pi\)
0.478036 + 0.878340i \(0.341349\pi\)
\(822\) 0 0
\(823\) 32.0000 1.11545 0.557725 0.830026i \(-0.311674\pi\)
0.557725 + 0.830026i \(0.311674\pi\)
\(824\) −5.30278 −0.184731
\(825\) 0 0
\(826\) 0 0
\(827\) 50.0555 1.74060 0.870300 0.492521i \(-0.163925\pi\)
0.870300 + 0.492521i \(0.163925\pi\)
\(828\) 0 0
\(829\) −56.4222 −1.95962 −0.979812 0.199921i \(-0.935932\pi\)
−0.979812 + 0.199921i \(0.935932\pi\)
\(830\) 13.9361 0.483729
\(831\) 0 0
\(832\) 0.302776 0.0104969
\(833\) 4.42221 0.153220
\(834\) 0 0
\(835\) 21.3944 0.740385
\(836\) −1.00000 −0.0345857
\(837\) 0 0
\(838\) −23.2111 −0.801814
\(839\) −45.1194 −1.55770 −0.778848 0.627213i \(-0.784196\pi\)
−0.778848 + 0.627213i \(0.784196\pi\)
\(840\) 0 0
\(841\) −6.93608 −0.239175
\(842\) 20.2389 0.697477
\(843\) 0 0
\(844\) 17.3944 0.598742
\(845\) 16.8167 0.578510
\(846\) 0 0
\(847\) −2.30278 −0.0791243
\(848\) 3.39445 0.116566
\(849\) 0 0
\(850\) 8.60555 0.295168
\(851\) −79.2666 −2.71722
\(852\) 0 0
\(853\) −17.5778 −0.601852 −0.300926 0.953647i \(-0.597296\pi\)
−0.300926 + 0.953647i \(0.597296\pi\)
\(854\) 7.39445 0.253033
\(855\) 0 0
\(856\) 11.2111 0.383188
\(857\) −10.5416 −0.360095 −0.180048 0.983658i \(-0.557625\pi\)
−0.180048 + 0.983658i \(0.557625\pi\)
\(858\) 0 0
\(859\) −48.8444 −1.66655 −0.833275 0.552859i \(-0.813537\pi\)
−0.833275 + 0.552859i \(0.813537\pi\)
\(860\) −15.5139 −0.529019
\(861\) 0 0
\(862\) −17.2111 −0.586212
\(863\) −14.7250 −0.501244 −0.250622 0.968085i \(-0.580635\pi\)
−0.250622 + 0.968085i \(0.580635\pi\)
\(864\) 0 0
\(865\) 26.4861 0.900555
\(866\) 26.2389 0.891633
\(867\) 0 0
\(868\) −0.697224 −0.0236653
\(869\) 5.81665 0.197316
\(870\) 0 0
\(871\) 3.48612 0.118123
\(872\) 2.00000 0.0677285
\(873\) 0 0
\(874\) 8.60555 0.291087
\(875\) −24.9083 −0.842055
\(876\) 0 0
\(877\) −42.5694 −1.43747 −0.718733 0.695286i \(-0.755278\pi\)
−0.718733 + 0.695286i \(0.755278\pi\)
\(878\) −10.7889 −0.364108
\(879\) 0 0
\(880\) 1.30278 0.0439166
\(881\) 23.0917 0.777978 0.388989 0.921242i \(-0.372824\pi\)
0.388989 + 0.921242i \(0.372824\pi\)
\(882\) 0 0
\(883\) −2.97224 −0.100024 −0.0500120 0.998749i \(-0.515926\pi\)
−0.0500120 + 0.998749i \(0.515926\pi\)
\(884\) −0.788897 −0.0265335
\(885\) 0 0
\(886\) 3.63331 0.122063
\(887\) 32.8444 1.10281 0.551404 0.834239i \(-0.314092\pi\)
0.551404 + 0.834239i \(0.314092\pi\)
\(888\) 0 0
\(889\) 29.0278 0.973560
\(890\) −3.39445 −0.113782
\(891\) 0 0
\(892\) −10.7889 −0.361239
\(893\) 6.00000 0.200782
\(894\) 0 0
\(895\) 15.7889 0.527765
\(896\) −2.30278 −0.0769303
\(897\) 0 0
\(898\) 10.1833 0.339823
\(899\) 1.42221 0.0474332
\(900\) 0 0
\(901\) −8.84441 −0.294650
\(902\) −6.90833 −0.230022
\(903\) 0 0
\(904\) 0 0
\(905\) −27.3944 −0.910622
\(906\) 0 0
\(907\) 52.8444 1.75467 0.877335 0.479879i \(-0.159319\pi\)
0.877335 + 0.479879i \(0.159319\pi\)
\(908\) 19.8167 0.657639
\(909\) 0 0
\(910\) 0.908327 0.0301107
\(911\) 15.6333 0.517955 0.258977 0.965883i \(-0.416615\pi\)
0.258977 + 0.965883i \(0.416615\pi\)
\(912\) 0 0
\(913\) 10.6972 0.354026
\(914\) 0.183346 0.00606455
\(915\) 0 0
\(916\) 1.09167 0.0360699
\(917\) −14.0917 −0.465348
\(918\) 0 0
\(919\) −19.1194 −0.630692 −0.315346 0.948977i \(-0.602121\pi\)
−0.315346 + 0.948977i \(0.602121\pi\)
\(920\) −11.2111 −0.369619
\(921\) 0 0
\(922\) 22.4222 0.738436
\(923\) 4.02776 0.132575
\(924\) 0 0
\(925\) 30.4222 1.00028
\(926\) 13.2111 0.434144
\(927\) 0 0
\(928\) 4.69722 0.154194
\(929\) −9.51388 −0.312140 −0.156070 0.987746i \(-0.549883\pi\)
−0.156070 + 0.987746i \(0.549883\pi\)
\(930\) 0 0
\(931\) −1.69722 −0.0556243
\(932\) 5.21110 0.170695
\(933\) 0 0
\(934\) 15.3944 0.503722
\(935\) −3.39445 −0.111010
\(936\) 0 0
\(937\) 58.8444 1.92236 0.961182 0.275917i \(-0.0889814\pi\)
0.961182 + 0.275917i \(0.0889814\pi\)
\(938\) −26.5139 −0.865709
\(939\) 0 0
\(940\) −7.81665 −0.254951
\(941\) 14.3667 0.468341 0.234170 0.972196i \(-0.424763\pi\)
0.234170 + 0.972196i \(0.424763\pi\)
\(942\) 0 0
\(943\) 59.4500 1.93596
\(944\) 0 0
\(945\) 0 0
\(946\) −11.9083 −0.387173
\(947\) −4.18335 −0.135940 −0.0679702 0.997687i \(-0.521652\pi\)
−0.0679702 + 0.997687i \(0.521652\pi\)
\(948\) 0 0
\(949\) 1.39445 0.0452657
\(950\) −3.30278 −0.107156
\(951\) 0 0
\(952\) 6.00000 0.194461
\(953\) −21.6333 −0.700772 −0.350386 0.936605i \(-0.613950\pi\)
−0.350386 + 0.936605i \(0.613950\pi\)
\(954\) 0 0
\(955\) −7.81665 −0.252941
\(956\) −22.9361 −0.741806
\(957\) 0 0
\(958\) −3.27502 −0.105811
\(959\) −49.5416 −1.59978
\(960\) 0 0
\(961\) −30.9083 −0.997043
\(962\) −2.78890 −0.0899177
\(963\) 0 0
\(964\) 23.9083 0.770035
\(965\) −10.0639 −0.323969
\(966\) 0 0
\(967\) −19.6333 −0.631365 −0.315682 0.948865i \(-0.602233\pi\)
−0.315682 + 0.948865i \(0.602233\pi\)
\(968\) 1.00000 0.0321412
\(969\) 0 0
\(970\) −10.4222 −0.334637
\(971\) −22.9361 −0.736054 −0.368027 0.929815i \(-0.619967\pi\)
−0.368027 + 0.929815i \(0.619967\pi\)
\(972\) 0 0
\(973\) 6.21110 0.199119
\(974\) 10.3305 0.331012
\(975\) 0 0
\(976\) −3.21110 −0.102785
\(977\) 15.6333 0.500154 0.250077 0.968226i \(-0.419544\pi\)
0.250077 + 0.968226i \(0.419544\pi\)
\(978\) 0 0
\(979\) −2.60555 −0.0832738
\(980\) 2.21110 0.0706311
\(981\) 0 0
\(982\) −31.9361 −1.01912
\(983\) −19.6972 −0.628244 −0.314122 0.949383i \(-0.601710\pi\)
−0.314122 + 0.949383i \(0.601710\pi\)
\(984\) 0 0
\(985\) −28.1833 −0.897996
\(986\) −12.2389 −0.389765
\(987\) 0 0
\(988\) 0.302776 0.00963258
\(989\) 102.478 3.25860
\(990\) 0 0
\(991\) −15.0917 −0.479403 −0.239701 0.970847i \(-0.577050\pi\)
−0.239701 + 0.970847i \(0.577050\pi\)
\(992\) 0.302776 0.00961314
\(993\) 0 0
\(994\) −30.6333 −0.971630
\(995\) 31.0278 0.983646
\(996\) 0 0
\(997\) 21.8167 0.690940 0.345470 0.938430i \(-0.387719\pi\)
0.345470 + 0.938430i \(0.387719\pi\)
\(998\) −34.0000 −1.07625
\(999\) 0 0
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 3762.2.a.bb.1.1 2
3.2 odd 2 418.2.a.d.1.1 2
12.11 even 2 3344.2.a.o.1.2 2
33.32 even 2 4598.2.a.bc.1.1 2
57.56 even 2 7942.2.a.bb.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
418.2.a.d.1.1 2 3.2 odd 2
3344.2.a.o.1.2 2 12.11 even 2
3762.2.a.bb.1.1 2 1.1 even 1 trivial
4598.2.a.bc.1.1 2 33.32 even 2
7942.2.a.bb.1.2 2 57.56 even 2