Properties

Label 5265.2.a.bd.1.3
Level $5265$
Weight $2$
Character 5265.1
Self dual yes
Analytic conductor $42.041$
Analytic rank $1$
Dimension $8$
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] = [5265,2,Mod(1,5265)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(5265, 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("5265.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 5265 = 3^{4} \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5265.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: \(42.0412366642\)
Analytic rank: \(1\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 11x^{6} - 2x^{5} + 38x^{4} + 14x^{3} - 39x^{2} - 22x - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(1.46324\) of defining polynomial
Character \(\chi\) \(=\) 5265.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.46324 q^{2} +0.141066 q^{4} +1.00000 q^{5} -4.95241 q^{7} +2.72006 q^{8} +O(q^{10})\) \(q-1.46324 q^{2} +0.141066 q^{4} +1.00000 q^{5} -4.95241 q^{7} +2.72006 q^{8} -1.46324 q^{10} +1.59098 q^{11} -1.00000 q^{13} +7.24655 q^{14} -4.26223 q^{16} +5.55131 q^{17} -4.74454 q^{19} +0.141066 q^{20} -2.32799 q^{22} -0.0764816 q^{23} +1.00000 q^{25} +1.46324 q^{26} -0.698616 q^{28} -3.53848 q^{29} +1.59217 q^{31} +0.796535 q^{32} -8.12288 q^{34} -4.95241 q^{35} +4.13079 q^{37} +6.94240 q^{38} +2.72006 q^{40} +4.90631 q^{41} +4.71684 q^{43} +0.224434 q^{44} +0.111911 q^{46} -11.5366 q^{47} +17.5264 q^{49} -1.46324 q^{50} -0.141066 q^{52} +1.29315 q^{53} +1.59098 q^{55} -13.4709 q^{56} +5.17764 q^{58} -4.66770 q^{59} -10.6244 q^{61} -2.32972 q^{62} +7.35894 q^{64} -1.00000 q^{65} -1.55688 q^{67} +0.783100 q^{68} +7.24655 q^{70} +2.51438 q^{71} +2.90174 q^{73} -6.04434 q^{74} -0.669293 q^{76} -7.87921 q^{77} +3.32589 q^{79} -4.26223 q^{80} -7.17911 q^{82} +5.35683 q^{83} +5.55131 q^{85} -6.90186 q^{86} +4.32758 q^{88} +0.225989 q^{89} +4.95241 q^{91} -0.0107889 q^{92} +16.8808 q^{94} -4.74454 q^{95} +18.8550 q^{97} -25.6452 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 6 q^{4} + 8 q^{5} + 4 q^{7} - 6 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 6 q^{4} + 8 q^{5} + 4 q^{7} - 6 q^{8} - 6 q^{11} - 8 q^{13} - 4 q^{14} - 10 q^{16} + 2 q^{17} - 10 q^{19} + 6 q^{20} - 30 q^{23} + 8 q^{25} + 2 q^{28} - 8 q^{29} - 10 q^{31} + 8 q^{32} - 4 q^{34} + 4 q^{35} - 8 q^{37} - 24 q^{38} - 6 q^{40} - 6 q^{41} - 34 q^{44} + 16 q^{46} - 18 q^{47} + 26 q^{49} - 6 q^{52} - 14 q^{53} - 6 q^{55} - 28 q^{56} - 30 q^{59} - 18 q^{61} - 10 q^{62} - 36 q^{64} - 8 q^{65} - 6 q^{67} + 8 q^{68} - 4 q^{70} - 16 q^{71} + 12 q^{73} - 20 q^{74} - 2 q^{76} - 8 q^{77} - 30 q^{79} - 10 q^{80} + 20 q^{82} - 26 q^{83} + 2 q^{85} - 30 q^{86} + 10 q^{88} - 14 q^{89} - 4 q^{91} - 26 q^{92} - 16 q^{94} - 10 q^{95} + 44 q^{97} - 60 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.46324 −1.03467 −0.517333 0.855784i \(-0.673075\pi\)
−0.517333 + 0.855784i \(0.673075\pi\)
\(3\) 0 0
\(4\) 0.141066 0.0705330
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) −4.95241 −1.87183 −0.935917 0.352220i \(-0.885427\pi\)
−0.935917 + 0.352220i \(0.885427\pi\)
\(8\) 2.72006 0.961688
\(9\) 0 0
\(10\) −1.46324 −0.462717
\(11\) 1.59098 0.479700 0.239850 0.970810i \(-0.422902\pi\)
0.239850 + 0.970810i \(0.422902\pi\)
\(12\) 0 0
\(13\) −1.00000 −0.277350
\(14\) 7.24655 1.93672
\(15\) 0 0
\(16\) −4.26223 −1.06556
\(17\) 5.55131 1.34639 0.673195 0.739465i \(-0.264922\pi\)
0.673195 + 0.739465i \(0.264922\pi\)
\(18\) 0 0
\(19\) −4.74454 −1.08847 −0.544236 0.838932i \(-0.683181\pi\)
−0.544236 + 0.838932i \(0.683181\pi\)
\(20\) 0.141066 0.0315433
\(21\) 0 0
\(22\) −2.32799 −0.496329
\(23\) −0.0764816 −0.0159475 −0.00797376 0.999968i \(-0.502538\pi\)
−0.00797376 + 0.999968i \(0.502538\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 1.46324 0.286965
\(27\) 0 0
\(28\) −0.698616 −0.132026
\(29\) −3.53848 −0.657079 −0.328539 0.944490i \(-0.606556\pi\)
−0.328539 + 0.944490i \(0.606556\pi\)
\(30\) 0 0
\(31\) 1.59217 0.285962 0.142981 0.989725i \(-0.454331\pi\)
0.142981 + 0.989725i \(0.454331\pi\)
\(32\) 0.796535 0.140809
\(33\) 0 0
\(34\) −8.12288 −1.39306
\(35\) −4.95241 −0.837110
\(36\) 0 0
\(37\) 4.13079 0.679099 0.339549 0.940588i \(-0.389725\pi\)
0.339549 + 0.940588i \(0.389725\pi\)
\(38\) 6.94240 1.12621
\(39\) 0 0
\(40\) 2.72006 0.430080
\(41\) 4.90631 0.766237 0.383119 0.923699i \(-0.374850\pi\)
0.383119 + 0.923699i \(0.374850\pi\)
\(42\) 0 0
\(43\) 4.71684 0.719311 0.359656 0.933085i \(-0.382894\pi\)
0.359656 + 0.933085i \(0.382894\pi\)
\(44\) 0.224434 0.0338347
\(45\) 0 0
\(46\) 0.111911 0.0165003
\(47\) −11.5366 −1.68279 −0.841395 0.540421i \(-0.818265\pi\)
−0.841395 + 0.540421i \(0.818265\pi\)
\(48\) 0 0
\(49\) 17.5264 2.50377
\(50\) −1.46324 −0.206933
\(51\) 0 0
\(52\) −0.141066 −0.0195623
\(53\) 1.29315 0.177628 0.0888139 0.996048i \(-0.471692\pi\)
0.0888139 + 0.996048i \(0.471692\pi\)
\(54\) 0 0
\(55\) 1.59098 0.214528
\(56\) −13.4709 −1.80012
\(57\) 0 0
\(58\) 5.17764 0.679857
\(59\) −4.66770 −0.607683 −0.303841 0.952723i \(-0.598269\pi\)
−0.303841 + 0.952723i \(0.598269\pi\)
\(60\) 0 0
\(61\) −10.6244 −1.36032 −0.680160 0.733063i \(-0.738090\pi\)
−0.680160 + 0.733063i \(0.738090\pi\)
\(62\) −2.32972 −0.295875
\(63\) 0 0
\(64\) 7.35894 0.919868
\(65\) −1.00000 −0.124035
\(66\) 0 0
\(67\) −1.55688 −0.190203 −0.0951014 0.995468i \(-0.530318\pi\)
−0.0951014 + 0.995468i \(0.530318\pi\)
\(68\) 0.783100 0.0949649
\(69\) 0 0
\(70\) 7.24655 0.866129
\(71\) 2.51438 0.298402 0.149201 0.988807i \(-0.452330\pi\)
0.149201 + 0.988807i \(0.452330\pi\)
\(72\) 0 0
\(73\) 2.90174 0.339624 0.169812 0.985477i \(-0.445684\pi\)
0.169812 + 0.985477i \(0.445684\pi\)
\(74\) −6.04434 −0.702640
\(75\) 0 0
\(76\) −0.669293 −0.0767732
\(77\) −7.87921 −0.897919
\(78\) 0 0
\(79\) 3.32589 0.374192 0.187096 0.982342i \(-0.440092\pi\)
0.187096 + 0.982342i \(0.440092\pi\)
\(80\) −4.26223 −0.476532
\(81\) 0 0
\(82\) −7.17911 −0.792800
\(83\) 5.35683 0.587988 0.293994 0.955807i \(-0.405015\pi\)
0.293994 + 0.955807i \(0.405015\pi\)
\(84\) 0 0
\(85\) 5.55131 0.602124
\(86\) −6.90186 −0.744247
\(87\) 0 0
\(88\) 4.32758 0.461322
\(89\) 0.225989 0.0239548 0.0119774 0.999928i \(-0.496187\pi\)
0.0119774 + 0.999928i \(0.496187\pi\)
\(90\) 0 0
\(91\) 4.95241 0.519154
\(92\) −0.0107889 −0.00112483
\(93\) 0 0
\(94\) 16.8808 1.74112
\(95\) −4.74454 −0.486780
\(96\) 0 0
\(97\) 18.8550 1.91444 0.957219 0.289363i \(-0.0934435\pi\)
0.957219 + 0.289363i \(0.0934435\pi\)
\(98\) −25.6452 −2.59056
\(99\) 0 0
\(100\) 0.141066 0.0141066
\(101\) 10.3079 1.02567 0.512837 0.858486i \(-0.328594\pi\)
0.512837 + 0.858486i \(0.328594\pi\)
\(102\) 0 0
\(103\) −2.65848 −0.261948 −0.130974 0.991386i \(-0.541810\pi\)
−0.130974 + 0.991386i \(0.541810\pi\)
\(104\) −2.72006 −0.266724
\(105\) 0 0
\(106\) −1.89219 −0.183785
\(107\) 17.5939 1.70087 0.850433 0.526084i \(-0.176340\pi\)
0.850433 + 0.526084i \(0.176340\pi\)
\(108\) 0 0
\(109\) −17.2053 −1.64797 −0.823985 0.566612i \(-0.808254\pi\)
−0.823985 + 0.566612i \(0.808254\pi\)
\(110\) −2.32799 −0.221965
\(111\) 0 0
\(112\) 21.1083 1.99455
\(113\) 16.3236 1.53560 0.767799 0.640690i \(-0.221352\pi\)
0.767799 + 0.640690i \(0.221352\pi\)
\(114\) 0 0
\(115\) −0.0764816 −0.00713195
\(116\) −0.499159 −0.0463457
\(117\) 0 0
\(118\) 6.82996 0.628748
\(119\) −27.4923 −2.52022
\(120\) 0 0
\(121\) −8.46877 −0.769888
\(122\) 15.5461 1.40748
\(123\) 0 0
\(124\) 0.224601 0.0201698
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) −11.1380 −0.988342 −0.494171 0.869365i \(-0.664528\pi\)
−0.494171 + 0.869365i \(0.664528\pi\)
\(128\) −12.3610 −1.09256
\(129\) 0 0
\(130\) 1.46324 0.128334
\(131\) −9.15691 −0.800043 −0.400022 0.916506i \(-0.630997\pi\)
−0.400022 + 0.916506i \(0.630997\pi\)
\(132\) 0 0
\(133\) 23.4969 2.03744
\(134\) 2.27808 0.196796
\(135\) 0 0
\(136\) 15.0999 1.29481
\(137\) −14.3061 −1.22225 −0.611125 0.791534i \(-0.709283\pi\)
−0.611125 + 0.791534i \(0.709283\pi\)
\(138\) 0 0
\(139\) 2.68341 0.227604 0.113802 0.993503i \(-0.463697\pi\)
0.113802 + 0.993503i \(0.463697\pi\)
\(140\) −0.698616 −0.0590439
\(141\) 0 0
\(142\) −3.67913 −0.308746
\(143\) −1.59098 −0.133045
\(144\) 0 0
\(145\) −3.53848 −0.293855
\(146\) −4.24594 −0.351397
\(147\) 0 0
\(148\) 0.582715 0.0478988
\(149\) 8.10911 0.664324 0.332162 0.943222i \(-0.392222\pi\)
0.332162 + 0.943222i \(0.392222\pi\)
\(150\) 0 0
\(151\) −9.49289 −0.772520 −0.386260 0.922390i \(-0.626233\pi\)
−0.386260 + 0.922390i \(0.626233\pi\)
\(152\) −12.9055 −1.04677
\(153\) 0 0
\(154\) 11.5292 0.929046
\(155\) 1.59217 0.127886
\(156\) 0 0
\(157\) −1.37156 −0.109463 −0.0547314 0.998501i \(-0.517430\pi\)
−0.0547314 + 0.998501i \(0.517430\pi\)
\(158\) −4.86657 −0.387164
\(159\) 0 0
\(160\) 0.796535 0.0629716
\(161\) 0.378768 0.0298511
\(162\) 0 0
\(163\) −8.03430 −0.629295 −0.314648 0.949209i \(-0.601886\pi\)
−0.314648 + 0.949209i \(0.601886\pi\)
\(164\) 0.692114 0.0540450
\(165\) 0 0
\(166\) −7.83832 −0.608371
\(167\) −23.4981 −1.81834 −0.909171 0.416424i \(-0.863283\pi\)
−0.909171 + 0.416424i \(0.863283\pi\)
\(168\) 0 0
\(169\) 1.00000 0.0769231
\(170\) −8.12288 −0.622997
\(171\) 0 0
\(172\) 0.665386 0.0507352
\(173\) −1.84099 −0.139968 −0.0699839 0.997548i \(-0.522295\pi\)
−0.0699839 + 0.997548i \(0.522295\pi\)
\(174\) 0 0
\(175\) −4.95241 −0.374367
\(176\) −6.78115 −0.511148
\(177\) 0 0
\(178\) −0.330676 −0.0247852
\(179\) 12.4946 0.933888 0.466944 0.884287i \(-0.345355\pi\)
0.466944 + 0.884287i \(0.345355\pi\)
\(180\) 0 0
\(181\) −23.3250 −1.73374 −0.866868 0.498538i \(-0.833870\pi\)
−0.866868 + 0.498538i \(0.833870\pi\)
\(182\) −7.24655 −0.537150
\(183\) 0 0
\(184\) −0.208035 −0.0153365
\(185\) 4.13079 0.303702
\(186\) 0 0
\(187\) 8.83204 0.645863
\(188\) −1.62742 −0.118692
\(189\) 0 0
\(190\) 6.94240 0.503654
\(191\) −19.5264 −1.41288 −0.706439 0.707774i \(-0.749700\pi\)
−0.706439 + 0.707774i \(0.749700\pi\)
\(192\) 0 0
\(193\) 23.3207 1.67866 0.839332 0.543620i \(-0.182947\pi\)
0.839332 + 0.543620i \(0.182947\pi\)
\(194\) −27.5894 −1.98080
\(195\) 0 0
\(196\) 2.47237 0.176598
\(197\) −9.68456 −0.689996 −0.344998 0.938603i \(-0.612120\pi\)
−0.344998 + 0.938603i \(0.612120\pi\)
\(198\) 0 0
\(199\) −12.9358 −0.916993 −0.458497 0.888696i \(-0.651612\pi\)
−0.458497 + 0.888696i \(0.651612\pi\)
\(200\) 2.72006 0.192338
\(201\) 0 0
\(202\) −15.0829 −1.06123
\(203\) 17.5240 1.22994
\(204\) 0 0
\(205\) 4.90631 0.342672
\(206\) 3.88999 0.271028
\(207\) 0 0
\(208\) 4.26223 0.295533
\(209\) −7.54850 −0.522140
\(210\) 0 0
\(211\) 17.0435 1.17332 0.586661 0.809833i \(-0.300442\pi\)
0.586661 + 0.809833i \(0.300442\pi\)
\(212\) 0.182419 0.0125286
\(213\) 0 0
\(214\) −25.7441 −1.75983
\(215\) 4.71684 0.321686
\(216\) 0 0
\(217\) −7.88508 −0.535274
\(218\) 25.1755 1.70510
\(219\) 0 0
\(220\) 0.224434 0.0151313
\(221\) −5.55131 −0.373421
\(222\) 0 0
\(223\) 10.2204 0.684407 0.342203 0.939626i \(-0.388827\pi\)
0.342203 + 0.939626i \(0.388827\pi\)
\(224\) −3.94476 −0.263571
\(225\) 0 0
\(226\) −23.8854 −1.58883
\(227\) −21.9465 −1.45664 −0.728319 0.685238i \(-0.759698\pi\)
−0.728319 + 0.685238i \(0.759698\pi\)
\(228\) 0 0
\(229\) −23.2117 −1.53387 −0.766937 0.641722i \(-0.778220\pi\)
−0.766937 + 0.641722i \(0.778220\pi\)
\(230\) 0.111911 0.00737918
\(231\) 0 0
\(232\) −9.62488 −0.631905
\(233\) 14.7926 0.969096 0.484548 0.874765i \(-0.338984\pi\)
0.484548 + 0.874765i \(0.338984\pi\)
\(234\) 0 0
\(235\) −11.5366 −0.752566
\(236\) −0.658453 −0.0428617
\(237\) 0 0
\(238\) 40.2278 2.60758
\(239\) 2.62092 0.169533 0.0847666 0.996401i \(-0.472986\pi\)
0.0847666 + 0.996401i \(0.472986\pi\)
\(240\) 0 0
\(241\) −4.32645 −0.278691 −0.139346 0.990244i \(-0.544500\pi\)
−0.139346 + 0.990244i \(0.544500\pi\)
\(242\) 12.3918 0.796577
\(243\) 0 0
\(244\) −1.49875 −0.0959475
\(245\) 17.5264 1.11972
\(246\) 0 0
\(247\) 4.74454 0.301888
\(248\) 4.33081 0.275006
\(249\) 0 0
\(250\) −1.46324 −0.0925433
\(251\) −10.8607 −0.685520 −0.342760 0.939423i \(-0.611362\pi\)
−0.342760 + 0.939423i \(0.611362\pi\)
\(252\) 0 0
\(253\) −0.121681 −0.00765002
\(254\) 16.2976 1.02260
\(255\) 0 0
\(256\) 3.36914 0.210571
\(257\) −7.01308 −0.437464 −0.218732 0.975785i \(-0.570192\pi\)
−0.218732 + 0.975785i \(0.570192\pi\)
\(258\) 0 0
\(259\) −20.4574 −1.27116
\(260\) −0.141066 −0.00874854
\(261\) 0 0
\(262\) 13.3987 0.827777
\(263\) 8.84407 0.545349 0.272674 0.962106i \(-0.412092\pi\)
0.272674 + 0.962106i \(0.412092\pi\)
\(264\) 0 0
\(265\) 1.29315 0.0794375
\(266\) −34.3816 −2.10807
\(267\) 0 0
\(268\) −0.219622 −0.0134156
\(269\) −25.3530 −1.54580 −0.772899 0.634529i \(-0.781194\pi\)
−0.772899 + 0.634529i \(0.781194\pi\)
\(270\) 0 0
\(271\) 20.7057 1.25778 0.628892 0.777493i \(-0.283509\pi\)
0.628892 + 0.777493i \(0.283509\pi\)
\(272\) −23.6610 −1.43466
\(273\) 0 0
\(274\) 20.9332 1.26462
\(275\) 1.59098 0.0959400
\(276\) 0 0
\(277\) 20.1257 1.20924 0.604620 0.796514i \(-0.293325\pi\)
0.604620 + 0.796514i \(0.293325\pi\)
\(278\) −3.92647 −0.235494
\(279\) 0 0
\(280\) −13.4709 −0.805038
\(281\) −14.2406 −0.849523 −0.424762 0.905305i \(-0.639642\pi\)
−0.424762 + 0.905305i \(0.639642\pi\)
\(282\) 0 0
\(283\) 26.4221 1.57063 0.785315 0.619096i \(-0.212501\pi\)
0.785315 + 0.619096i \(0.212501\pi\)
\(284\) 0.354693 0.0210472
\(285\) 0 0
\(286\) 2.32799 0.137657
\(287\) −24.2981 −1.43427
\(288\) 0 0
\(289\) 13.8170 0.812765
\(290\) 5.17764 0.304041
\(291\) 0 0
\(292\) 0.409337 0.0239547
\(293\) 17.0177 0.994183 0.497091 0.867698i \(-0.334401\pi\)
0.497091 + 0.867698i \(0.334401\pi\)
\(294\) 0 0
\(295\) −4.66770 −0.271764
\(296\) 11.2360 0.653081
\(297\) 0 0
\(298\) −11.8656 −0.687354
\(299\) 0.0764816 0.00442304
\(300\) 0 0
\(301\) −23.3597 −1.34643
\(302\) 13.8904 0.799300
\(303\) 0 0
\(304\) 20.2223 1.15983
\(305\) −10.6244 −0.608354
\(306\) 0 0
\(307\) 17.3829 0.992095 0.496048 0.868295i \(-0.334784\pi\)
0.496048 + 0.868295i \(0.334784\pi\)
\(308\) −1.11149 −0.0633329
\(309\) 0 0
\(310\) −2.32972 −0.132319
\(311\) −13.6360 −0.773225 −0.386612 0.922242i \(-0.626355\pi\)
−0.386612 + 0.922242i \(0.626355\pi\)
\(312\) 0 0
\(313\) 8.12157 0.459058 0.229529 0.973302i \(-0.426281\pi\)
0.229529 + 0.973302i \(0.426281\pi\)
\(314\) 2.00693 0.113257
\(315\) 0 0
\(316\) 0.469170 0.0263929
\(317\) −18.6219 −1.04591 −0.522955 0.852361i \(-0.675170\pi\)
−0.522955 + 0.852361i \(0.675170\pi\)
\(318\) 0 0
\(319\) −5.62967 −0.315201
\(320\) 7.35894 0.411378
\(321\) 0 0
\(322\) −0.554228 −0.0308859
\(323\) −26.3384 −1.46551
\(324\) 0 0
\(325\) −1.00000 −0.0554700
\(326\) 11.7561 0.651110
\(327\) 0 0
\(328\) 13.3455 0.736881
\(329\) 57.1341 3.14990
\(330\) 0 0
\(331\) −28.7903 −1.58246 −0.791229 0.611519i \(-0.790559\pi\)
−0.791229 + 0.611519i \(0.790559\pi\)
\(332\) 0.755666 0.0414726
\(333\) 0 0
\(334\) 34.3834 1.88138
\(335\) −1.55688 −0.0850613
\(336\) 0 0
\(337\) −11.8343 −0.644658 −0.322329 0.946628i \(-0.604466\pi\)
−0.322329 + 0.946628i \(0.604466\pi\)
\(338\) −1.46324 −0.0795897
\(339\) 0 0
\(340\) 0.783100 0.0424696
\(341\) 2.53312 0.137176
\(342\) 0 0
\(343\) −52.1308 −2.81480
\(344\) 12.8301 0.691753
\(345\) 0 0
\(346\) 2.69380 0.144820
\(347\) −31.9068 −1.71285 −0.856423 0.516275i \(-0.827318\pi\)
−0.856423 + 0.516275i \(0.827318\pi\)
\(348\) 0 0
\(349\) 31.2577 1.67319 0.836594 0.547824i \(-0.184544\pi\)
0.836594 + 0.547824i \(0.184544\pi\)
\(350\) 7.24655 0.387345
\(351\) 0 0
\(352\) 1.26727 0.0675459
\(353\) −10.2084 −0.543336 −0.271668 0.962391i \(-0.587575\pi\)
−0.271668 + 0.962391i \(0.587575\pi\)
\(354\) 0 0
\(355\) 2.51438 0.133449
\(356\) 0.0318794 0.00168960
\(357\) 0 0
\(358\) −18.2825 −0.966261
\(359\) −8.03454 −0.424047 −0.212023 0.977265i \(-0.568005\pi\)
−0.212023 + 0.977265i \(0.568005\pi\)
\(360\) 0 0
\(361\) 3.51068 0.184773
\(362\) 34.1301 1.79384
\(363\) 0 0
\(364\) 0.698616 0.0366174
\(365\) 2.90174 0.151884
\(366\) 0 0
\(367\) −23.5639 −1.23002 −0.615012 0.788518i \(-0.710849\pi\)
−0.615012 + 0.788518i \(0.710849\pi\)
\(368\) 0.325982 0.0169930
\(369\) 0 0
\(370\) −6.04434 −0.314230
\(371\) −6.40420 −0.332490
\(372\) 0 0
\(373\) −12.3593 −0.639942 −0.319971 0.947427i \(-0.603673\pi\)
−0.319971 + 0.947427i \(0.603673\pi\)
\(374\) −12.9234 −0.668252
\(375\) 0 0
\(376\) −31.3803 −1.61832
\(377\) 3.53848 0.182241
\(378\) 0 0
\(379\) −30.6886 −1.57637 −0.788183 0.615442i \(-0.788978\pi\)
−0.788183 + 0.615442i \(0.788978\pi\)
\(380\) −0.669293 −0.0343340
\(381\) 0 0
\(382\) 28.5717 1.46186
\(383\) −23.2200 −1.18649 −0.593244 0.805023i \(-0.702153\pi\)
−0.593244 + 0.805023i \(0.702153\pi\)
\(384\) 0 0
\(385\) −7.87921 −0.401562
\(386\) −34.1238 −1.73686
\(387\) 0 0
\(388\) 2.65980 0.135031
\(389\) 23.0785 1.17013 0.585064 0.810987i \(-0.301069\pi\)
0.585064 + 0.810987i \(0.301069\pi\)
\(390\) 0 0
\(391\) −0.424573 −0.0214716
\(392\) 47.6728 2.40784
\(393\) 0 0
\(394\) 14.1708 0.713916
\(395\) 3.32589 0.167344
\(396\) 0 0
\(397\) 16.3847 0.822325 0.411163 0.911562i \(-0.365123\pi\)
0.411163 + 0.911562i \(0.365123\pi\)
\(398\) 18.9281 0.948781
\(399\) 0 0
\(400\) −4.26223 −0.213112
\(401\) −19.8915 −0.993335 −0.496668 0.867941i \(-0.665443\pi\)
−0.496668 + 0.867941i \(0.665443\pi\)
\(402\) 0 0
\(403\) −1.59217 −0.0793117
\(404\) 1.45409 0.0723439
\(405\) 0 0
\(406\) −25.6418 −1.27258
\(407\) 6.57203 0.325764
\(408\) 0 0
\(409\) −33.2385 −1.64354 −0.821769 0.569820i \(-0.807013\pi\)
−0.821769 + 0.569820i \(0.807013\pi\)
\(410\) −7.17911 −0.354551
\(411\) 0 0
\(412\) −0.375021 −0.0184759
\(413\) 23.1164 1.13748
\(414\) 0 0
\(415\) 5.35683 0.262956
\(416\) −0.796535 −0.0390533
\(417\) 0 0
\(418\) 11.0452 0.540241
\(419\) 7.92162 0.386996 0.193498 0.981101i \(-0.438017\pi\)
0.193498 + 0.981101i \(0.438017\pi\)
\(420\) 0 0
\(421\) 1.23413 0.0601477 0.0300738 0.999548i \(-0.490426\pi\)
0.0300738 + 0.999548i \(0.490426\pi\)
\(422\) −24.9387 −1.21400
\(423\) 0 0
\(424\) 3.51745 0.170822
\(425\) 5.55131 0.269278
\(426\) 0 0
\(427\) 52.6166 2.54630
\(428\) 2.48190 0.119967
\(429\) 0 0
\(430\) −6.90186 −0.332837
\(431\) 5.34263 0.257345 0.128673 0.991687i \(-0.458928\pi\)
0.128673 + 0.991687i \(0.458928\pi\)
\(432\) 0 0
\(433\) −21.4233 −1.02954 −0.514770 0.857328i \(-0.672123\pi\)
−0.514770 + 0.857328i \(0.672123\pi\)
\(434\) 11.5378 0.553830
\(435\) 0 0
\(436\) −2.42708 −0.116236
\(437\) 0.362870 0.0173584
\(438\) 0 0
\(439\) −18.8049 −0.897510 −0.448755 0.893655i \(-0.648132\pi\)
−0.448755 + 0.893655i \(0.648132\pi\)
\(440\) 4.32758 0.206309
\(441\) 0 0
\(442\) 8.12288 0.386366
\(443\) 15.0986 0.717355 0.358678 0.933461i \(-0.383228\pi\)
0.358678 + 0.933461i \(0.383228\pi\)
\(444\) 0 0
\(445\) 0.225989 0.0107129
\(446\) −14.9548 −0.708132
\(447\) 0 0
\(448\) −36.4445 −1.72184
\(449\) 6.11019 0.288358 0.144179 0.989552i \(-0.453946\pi\)
0.144179 + 0.989552i \(0.453946\pi\)
\(450\) 0 0
\(451\) 7.80587 0.367564
\(452\) 2.30271 0.108310
\(453\) 0 0
\(454\) 32.1129 1.50713
\(455\) 4.95241 0.232173
\(456\) 0 0
\(457\) −36.2218 −1.69438 −0.847192 0.531287i \(-0.821709\pi\)
−0.847192 + 0.531287i \(0.821709\pi\)
\(458\) 33.9643 1.58705
\(459\) 0 0
\(460\) −0.0107889 −0.000503037 0
\(461\) 15.2538 0.710441 0.355220 0.934783i \(-0.384406\pi\)
0.355220 + 0.934783i \(0.384406\pi\)
\(462\) 0 0
\(463\) 7.34655 0.341423 0.170712 0.985321i \(-0.445393\pi\)
0.170712 + 0.985321i \(0.445393\pi\)
\(464\) 15.0818 0.700156
\(465\) 0 0
\(466\) −21.6451 −1.00269
\(467\) 2.11495 0.0978682 0.0489341 0.998802i \(-0.484418\pi\)
0.0489341 + 0.998802i \(0.484418\pi\)
\(468\) 0 0
\(469\) 7.71029 0.356028
\(470\) 16.8808 0.778655
\(471\) 0 0
\(472\) −12.6964 −0.584401
\(473\) 7.50442 0.345054
\(474\) 0 0
\(475\) −4.74454 −0.217695
\(476\) −3.87823 −0.177759
\(477\) 0 0
\(478\) −3.83503 −0.175410
\(479\) −19.6744 −0.898944 −0.449472 0.893294i \(-0.648388\pi\)
−0.449472 + 0.893294i \(0.648388\pi\)
\(480\) 0 0
\(481\) −4.13079 −0.188348
\(482\) 6.33063 0.288352
\(483\) 0 0
\(484\) −1.19465 −0.0543025
\(485\) 18.8550 0.856163
\(486\) 0 0
\(487\) 19.4574 0.881698 0.440849 0.897581i \(-0.354677\pi\)
0.440849 + 0.897581i \(0.354677\pi\)
\(488\) −28.8992 −1.30820
\(489\) 0 0
\(490\) −25.6452 −1.15853
\(491\) 14.0281 0.633077 0.316539 0.948580i \(-0.397479\pi\)
0.316539 + 0.948580i \(0.397479\pi\)
\(492\) 0 0
\(493\) −19.6432 −0.884684
\(494\) −6.94240 −0.312353
\(495\) 0 0
\(496\) −6.78620 −0.304709
\(497\) −12.4522 −0.558559
\(498\) 0 0
\(499\) −6.41203 −0.287042 −0.143521 0.989647i \(-0.545842\pi\)
−0.143521 + 0.989647i \(0.545842\pi\)
\(500\) 0.141066 0.00630866
\(501\) 0 0
\(502\) 15.8918 0.709284
\(503\) −1.58905 −0.0708521 −0.0354260 0.999372i \(-0.511279\pi\)
−0.0354260 + 0.999372i \(0.511279\pi\)
\(504\) 0 0
\(505\) 10.3079 0.458696
\(506\) 0.178048 0.00791522
\(507\) 0 0
\(508\) −1.57120 −0.0697107
\(509\) −40.4369 −1.79233 −0.896167 0.443717i \(-0.853659\pi\)
−0.896167 + 0.443717i \(0.853659\pi\)
\(510\) 0 0
\(511\) −14.3706 −0.635719
\(512\) 19.7921 0.874694
\(513\) 0 0
\(514\) 10.2618 0.452629
\(515\) −2.65848 −0.117147
\(516\) 0 0
\(517\) −18.3546 −0.807234
\(518\) 29.9340 1.31523
\(519\) 0 0
\(520\) −2.72006 −0.119283
\(521\) −20.3148 −0.890007 −0.445003 0.895529i \(-0.646797\pi\)
−0.445003 + 0.895529i \(0.646797\pi\)
\(522\) 0 0
\(523\) −9.84446 −0.430468 −0.215234 0.976563i \(-0.569051\pi\)
−0.215234 + 0.976563i \(0.569051\pi\)
\(524\) −1.29173 −0.0564294
\(525\) 0 0
\(526\) −12.9410 −0.564254
\(527\) 8.83863 0.385017
\(528\) 0 0
\(529\) −22.9942 −0.999746
\(530\) −1.89219 −0.0821913
\(531\) 0 0
\(532\) 3.31461 0.143707
\(533\) −4.90631 −0.212516
\(534\) 0 0
\(535\) 17.5939 0.760650
\(536\) −4.23480 −0.182916
\(537\) 0 0
\(538\) 37.0974 1.59938
\(539\) 27.8842 1.20106
\(540\) 0 0
\(541\) −19.7146 −0.847596 −0.423798 0.905757i \(-0.639303\pi\)
−0.423798 + 0.905757i \(0.639303\pi\)
\(542\) −30.2974 −1.30139
\(543\) 0 0
\(544\) 4.42181 0.189583
\(545\) −17.2053 −0.736994
\(546\) 0 0
\(547\) 15.7344 0.672756 0.336378 0.941727i \(-0.390798\pi\)
0.336378 + 0.941727i \(0.390798\pi\)
\(548\) −2.01810 −0.0862090
\(549\) 0 0
\(550\) −2.32799 −0.0992658
\(551\) 16.7885 0.715212
\(552\) 0 0
\(553\) −16.4712 −0.700426
\(554\) −29.4488 −1.25116
\(555\) 0 0
\(556\) 0.378538 0.0160536
\(557\) −8.29587 −0.351507 −0.175754 0.984434i \(-0.556236\pi\)
−0.175754 + 0.984434i \(0.556236\pi\)
\(558\) 0 0
\(559\) −4.71684 −0.199501
\(560\) 21.1083 0.891989
\(561\) 0 0
\(562\) 20.8374 0.878973
\(563\) −42.1827 −1.77779 −0.888893 0.458114i \(-0.848525\pi\)
−0.888893 + 0.458114i \(0.848525\pi\)
\(564\) 0 0
\(565\) 16.3236 0.686741
\(566\) −38.6618 −1.62508
\(567\) 0 0
\(568\) 6.83927 0.286969
\(569\) 7.79477 0.326774 0.163387 0.986562i \(-0.447758\pi\)
0.163387 + 0.986562i \(0.447758\pi\)
\(570\) 0 0
\(571\) −29.3684 −1.22903 −0.614516 0.788905i \(-0.710649\pi\)
−0.614516 + 0.788905i \(0.710649\pi\)
\(572\) −0.224434 −0.00938405
\(573\) 0 0
\(574\) 35.5539 1.48399
\(575\) −0.0764816 −0.00318950
\(576\) 0 0
\(577\) −40.7191 −1.69516 −0.847580 0.530668i \(-0.821941\pi\)
−0.847580 + 0.530668i \(0.821941\pi\)
\(578\) −20.2176 −0.840940
\(579\) 0 0
\(580\) −0.499159 −0.0207264
\(581\) −26.5292 −1.10062
\(582\) 0 0
\(583\) 2.05738 0.0852080
\(584\) 7.89293 0.326612
\(585\) 0 0
\(586\) −24.9009 −1.02865
\(587\) −25.7170 −1.06145 −0.530726 0.847543i \(-0.678081\pi\)
−0.530726 + 0.847543i \(0.678081\pi\)
\(588\) 0 0
\(589\) −7.55412 −0.311262
\(590\) 6.82996 0.281185
\(591\) 0 0
\(592\) −17.6064 −0.723619
\(593\) 10.7998 0.443497 0.221748 0.975104i \(-0.428824\pi\)
0.221748 + 0.975104i \(0.428824\pi\)
\(594\) 0 0
\(595\) −27.4923 −1.12708
\(596\) 1.14392 0.0468568
\(597\) 0 0
\(598\) −0.111911 −0.00457637
\(599\) −14.7325 −0.601954 −0.300977 0.953631i \(-0.597313\pi\)
−0.300977 + 0.953631i \(0.597313\pi\)
\(600\) 0 0
\(601\) −17.1542 −0.699734 −0.349867 0.936799i \(-0.613773\pi\)
−0.349867 + 0.936799i \(0.613773\pi\)
\(602\) 34.1808 1.39311
\(603\) 0 0
\(604\) −1.33912 −0.0544882
\(605\) −8.46877 −0.344304
\(606\) 0 0
\(607\) −32.2003 −1.30697 −0.653486 0.756939i \(-0.726694\pi\)
−0.653486 + 0.756939i \(0.726694\pi\)
\(608\) −3.77919 −0.153266
\(609\) 0 0
\(610\) 15.5461 0.629443
\(611\) 11.5366 0.466722
\(612\) 0 0
\(613\) 14.3120 0.578058 0.289029 0.957320i \(-0.406668\pi\)
0.289029 + 0.957320i \(0.406668\pi\)
\(614\) −25.4353 −1.02649
\(615\) 0 0
\(616\) −21.4319 −0.863518
\(617\) −44.8862 −1.80705 −0.903525 0.428536i \(-0.859030\pi\)
−0.903525 + 0.428536i \(0.859030\pi\)
\(618\) 0 0
\(619\) 1.99365 0.0801317 0.0400659 0.999197i \(-0.487243\pi\)
0.0400659 + 0.999197i \(0.487243\pi\)
\(620\) 0.224601 0.00902020
\(621\) 0 0
\(622\) 19.9527 0.800029
\(623\) −1.11919 −0.0448394
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) −11.8838 −0.474972
\(627\) 0 0
\(628\) −0.193481 −0.00772073
\(629\) 22.9313 0.914331
\(630\) 0 0
\(631\) 33.9679 1.35224 0.676121 0.736791i \(-0.263660\pi\)
0.676121 + 0.736791i \(0.263660\pi\)
\(632\) 9.04664 0.359856
\(633\) 0 0
\(634\) 27.2482 1.08217
\(635\) −11.1380 −0.442000
\(636\) 0 0
\(637\) −17.5264 −0.694420
\(638\) 8.23754 0.326127
\(639\) 0 0
\(640\) −12.3610 −0.488610
\(641\) 46.5675 1.83931 0.919653 0.392732i \(-0.128470\pi\)
0.919653 + 0.392732i \(0.128470\pi\)
\(642\) 0 0
\(643\) 1.04675 0.0412799 0.0206400 0.999787i \(-0.493430\pi\)
0.0206400 + 0.999787i \(0.493430\pi\)
\(644\) 0.0534313 0.00210549
\(645\) 0 0
\(646\) 38.5394 1.51631
\(647\) −38.3293 −1.50688 −0.753440 0.657517i \(-0.771607\pi\)
−0.753440 + 0.657517i \(0.771607\pi\)
\(648\) 0 0
\(649\) −7.42624 −0.291505
\(650\) 1.46324 0.0573929
\(651\) 0 0
\(652\) −1.13337 −0.0443861
\(653\) −13.8726 −0.542878 −0.271439 0.962456i \(-0.587500\pi\)
−0.271439 + 0.962456i \(0.587500\pi\)
\(654\) 0 0
\(655\) −9.15691 −0.357790
\(656\) −20.9118 −0.816471
\(657\) 0 0
\(658\) −83.6008 −3.25910
\(659\) −4.74266 −0.184748 −0.0923740 0.995724i \(-0.529446\pi\)
−0.0923740 + 0.995724i \(0.529446\pi\)
\(660\) 0 0
\(661\) −6.74129 −0.262206 −0.131103 0.991369i \(-0.541852\pi\)
−0.131103 + 0.991369i \(0.541852\pi\)
\(662\) 42.1271 1.63732
\(663\) 0 0
\(664\) 14.5709 0.565461
\(665\) 23.4969 0.911171
\(666\) 0 0
\(667\) 0.270628 0.0104788
\(668\) −3.31479 −0.128253
\(669\) 0 0
\(670\) 2.27808 0.0880100
\(671\) −16.9033 −0.652546
\(672\) 0 0
\(673\) −2.58340 −0.0995828 −0.0497914 0.998760i \(-0.515856\pi\)
−0.0497914 + 0.998760i \(0.515856\pi\)
\(674\) 17.3165 0.667005
\(675\) 0 0
\(676\) 0.141066 0.00542561
\(677\) 42.7696 1.64377 0.821884 0.569654i \(-0.192923\pi\)
0.821884 + 0.569654i \(0.192923\pi\)
\(678\) 0 0
\(679\) −93.3779 −3.58351
\(680\) 15.0999 0.579055
\(681\) 0 0
\(682\) −3.70656 −0.141931
\(683\) −14.8065 −0.566553 −0.283277 0.959038i \(-0.591421\pi\)
−0.283277 + 0.959038i \(0.591421\pi\)
\(684\) 0 0
\(685\) −14.3061 −0.546607
\(686\) 76.2798 2.91238
\(687\) 0 0
\(688\) −20.1043 −0.766468
\(689\) −1.29315 −0.0492651
\(690\) 0 0
\(691\) 15.7360 0.598624 0.299312 0.954155i \(-0.403243\pi\)
0.299312 + 0.954155i \(0.403243\pi\)
\(692\) −0.259701 −0.00987234
\(693\) 0 0
\(694\) 46.6872 1.77222
\(695\) 2.68341 0.101788
\(696\) 0 0
\(697\) 27.2364 1.03165
\(698\) −45.7375 −1.73119
\(699\) 0 0
\(700\) −0.698616 −0.0264052
\(701\) −31.5607 −1.19203 −0.596015 0.802973i \(-0.703250\pi\)
−0.596015 + 0.802973i \(0.703250\pi\)
\(702\) 0 0
\(703\) −19.5987 −0.739180
\(704\) 11.7080 0.441261
\(705\) 0 0
\(706\) 14.9373 0.562172
\(707\) −51.0490 −1.91989
\(708\) 0 0
\(709\) 40.4938 1.52078 0.760388 0.649469i \(-0.225009\pi\)
0.760388 + 0.649469i \(0.225009\pi\)
\(710\) −3.67913 −0.138075
\(711\) 0 0
\(712\) 0.614705 0.0230370
\(713\) −0.121772 −0.00456039
\(714\) 0 0
\(715\) −1.59098 −0.0594995
\(716\) 1.76256 0.0658699
\(717\) 0 0
\(718\) 11.7564 0.438747
\(719\) −34.6714 −1.29302 −0.646512 0.762904i \(-0.723773\pi\)
−0.646512 + 0.762904i \(0.723773\pi\)
\(720\) 0 0
\(721\) 13.1659 0.490323
\(722\) −5.13697 −0.191178
\(723\) 0 0
\(724\) −3.29037 −0.122286
\(725\) −3.53848 −0.131416
\(726\) 0 0
\(727\) 36.1748 1.34165 0.670824 0.741616i \(-0.265940\pi\)
0.670824 + 0.741616i \(0.265940\pi\)
\(728\) 13.4709 0.499264
\(729\) 0 0
\(730\) −4.24594 −0.157149
\(731\) 26.1846 0.968473
\(732\) 0 0
\(733\) −47.2468 −1.74510 −0.872549 0.488526i \(-0.837535\pi\)
−0.872549 + 0.488526i \(0.837535\pi\)
\(734\) 34.4796 1.27266
\(735\) 0 0
\(736\) −0.0609202 −0.00224555
\(737\) −2.47697 −0.0912403
\(738\) 0 0
\(739\) 23.4117 0.861215 0.430607 0.902539i \(-0.358299\pi\)
0.430607 + 0.902539i \(0.358299\pi\)
\(740\) 0.582715 0.0214210
\(741\) 0 0
\(742\) 9.37088 0.344016
\(743\) −53.8155 −1.97430 −0.987150 0.159797i \(-0.948916\pi\)
−0.987150 + 0.159797i \(0.948916\pi\)
\(744\) 0 0
\(745\) 8.10911 0.297095
\(746\) 18.0847 0.662126
\(747\) 0 0
\(748\) 1.24590 0.0455546
\(749\) −87.1321 −3.18374
\(750\) 0 0
\(751\) 35.6661 1.30148 0.650738 0.759302i \(-0.274460\pi\)
0.650738 + 0.759302i \(0.274460\pi\)
\(752\) 49.1718 1.79311
\(753\) 0 0
\(754\) −5.17764 −0.188558
\(755\) −9.49289 −0.345482
\(756\) 0 0
\(757\) 46.6061 1.69393 0.846963 0.531651i \(-0.178428\pi\)
0.846963 + 0.531651i \(0.178428\pi\)
\(758\) 44.9047 1.63101
\(759\) 0 0
\(760\) −12.9055 −0.468130
\(761\) −18.2267 −0.660716 −0.330358 0.943856i \(-0.607169\pi\)
−0.330358 + 0.943856i \(0.607169\pi\)
\(762\) 0 0
\(763\) 85.2077 3.08473
\(764\) −2.75451 −0.0996545
\(765\) 0 0
\(766\) 33.9764 1.22762
\(767\) 4.66770 0.168541
\(768\) 0 0
\(769\) 10.2917 0.371128 0.185564 0.982632i \(-0.440589\pi\)
0.185564 + 0.982632i \(0.440589\pi\)
\(770\) 11.5292 0.415482
\(771\) 0 0
\(772\) 3.28976 0.118401
\(773\) 33.1268 1.19149 0.595745 0.803174i \(-0.296857\pi\)
0.595745 + 0.803174i \(0.296857\pi\)
\(774\) 0 0
\(775\) 1.59217 0.0571925
\(776\) 51.2869 1.84109
\(777\) 0 0
\(778\) −33.7694 −1.21069
\(779\) −23.2782 −0.834029
\(780\) 0 0
\(781\) 4.00034 0.143143
\(782\) 0.621251 0.0222159
\(783\) 0 0
\(784\) −74.7014 −2.66791
\(785\) −1.37156 −0.0489532
\(786\) 0 0
\(787\) 4.11339 0.146626 0.0733132 0.997309i \(-0.476643\pi\)
0.0733132 + 0.997309i \(0.476643\pi\)
\(788\) −1.36616 −0.0486675
\(789\) 0 0
\(790\) −4.86657 −0.173145
\(791\) −80.8413 −2.87439
\(792\) 0 0
\(793\) 10.6244 0.377285
\(794\) −23.9747 −0.850832
\(795\) 0 0
\(796\) −1.82480 −0.0646783
\(797\) 31.2433 1.10670 0.553348 0.832950i \(-0.313350\pi\)
0.553348 + 0.832950i \(0.313350\pi\)
\(798\) 0 0
\(799\) −64.0433 −2.26569
\(800\) 0.796535 0.0281617
\(801\) 0 0
\(802\) 29.1060 1.02777
\(803\) 4.61663 0.162917
\(804\) 0 0
\(805\) 0.378768 0.0133498
\(806\) 2.32972 0.0820611
\(807\) 0 0
\(808\) 28.0382 0.986379
\(809\) −24.4812 −0.860712 −0.430356 0.902659i \(-0.641612\pi\)
−0.430356 + 0.902659i \(0.641612\pi\)
\(810\) 0 0
\(811\) 27.5036 0.965782 0.482891 0.875681i \(-0.339587\pi\)
0.482891 + 0.875681i \(0.339587\pi\)
\(812\) 2.47204 0.0867516
\(813\) 0 0
\(814\) −9.61645 −0.337056
\(815\) −8.03430 −0.281429
\(816\) 0 0
\(817\) −22.3793 −0.782951
\(818\) 48.6359 1.70051
\(819\) 0 0
\(820\) 0.692114 0.0241697
\(821\) −19.2162 −0.670649 −0.335325 0.942103i \(-0.608846\pi\)
−0.335325 + 0.942103i \(0.608846\pi\)
\(822\) 0 0
\(823\) −50.8253 −1.77166 −0.885829 0.464012i \(-0.846409\pi\)
−0.885829 + 0.464012i \(0.846409\pi\)
\(824\) −7.23123 −0.251912
\(825\) 0 0
\(826\) −33.8247 −1.17691
\(827\) 56.5580 1.96671 0.983357 0.181684i \(-0.0581547\pi\)
0.983357 + 0.181684i \(0.0581547\pi\)
\(828\) 0 0
\(829\) 25.9253 0.900424 0.450212 0.892922i \(-0.351348\pi\)
0.450212 + 0.892922i \(0.351348\pi\)
\(830\) −7.83832 −0.272072
\(831\) 0 0
\(832\) −7.35894 −0.255126
\(833\) 97.2942 3.37104
\(834\) 0 0
\(835\) −23.4981 −0.813187
\(836\) −1.06484 −0.0368281
\(837\) 0 0
\(838\) −11.5912 −0.400412
\(839\) 2.90284 0.100217 0.0501086 0.998744i \(-0.484043\pi\)
0.0501086 + 0.998744i \(0.484043\pi\)
\(840\) 0 0
\(841\) −16.4792 −0.568247
\(842\) −1.80582 −0.0622327
\(843\) 0 0
\(844\) 2.40425 0.0827578
\(845\) 1.00000 0.0344010
\(846\) 0 0
\(847\) 41.9408 1.44110
\(848\) −5.51170 −0.189273
\(849\) 0 0
\(850\) −8.12288 −0.278613
\(851\) −0.315930 −0.0108299
\(852\) 0 0
\(853\) 33.4101 1.14394 0.571971 0.820274i \(-0.306179\pi\)
0.571971 + 0.820274i \(0.306179\pi\)
\(854\) −76.9906 −2.63456
\(855\) 0 0
\(856\) 47.8565 1.63570
\(857\) −17.5740 −0.600316 −0.300158 0.953890i \(-0.597039\pi\)
−0.300158 + 0.953890i \(0.597039\pi\)
\(858\) 0 0
\(859\) −1.25578 −0.0428466 −0.0214233 0.999770i \(-0.506820\pi\)
−0.0214233 + 0.999770i \(0.506820\pi\)
\(860\) 0.665386 0.0226895
\(861\) 0 0
\(862\) −7.81754 −0.266266
\(863\) 16.8692 0.574235 0.287118 0.957895i \(-0.407303\pi\)
0.287118 + 0.957895i \(0.407303\pi\)
\(864\) 0 0
\(865\) −1.84099 −0.0625955
\(866\) 31.3474 1.06523
\(867\) 0 0
\(868\) −1.11232 −0.0377545
\(869\) 5.29145 0.179500
\(870\) 0 0
\(871\) 1.55688 0.0527528
\(872\) −46.7995 −1.58483
\(873\) 0 0
\(874\) −0.530965 −0.0179602
\(875\) −4.95241 −0.167422
\(876\) 0 0
\(877\) −17.9517 −0.606186 −0.303093 0.952961i \(-0.598019\pi\)
−0.303093 + 0.952961i \(0.598019\pi\)
\(878\) 27.5161 0.928623
\(879\) 0 0
\(880\) −6.78115 −0.228592
\(881\) −33.1498 −1.11685 −0.558423 0.829556i \(-0.688593\pi\)
−0.558423 + 0.829556i \(0.688593\pi\)
\(882\) 0 0
\(883\) −24.9328 −0.839055 −0.419528 0.907743i \(-0.637804\pi\)
−0.419528 + 0.907743i \(0.637804\pi\)
\(884\) −0.783100 −0.0263385
\(885\) 0 0
\(886\) −22.0928 −0.742223
\(887\) 54.6710 1.83567 0.917835 0.396961i \(-0.129935\pi\)
0.917835 + 0.396961i \(0.129935\pi\)
\(888\) 0 0
\(889\) 55.1602 1.85001
\(890\) −0.330676 −0.0110843
\(891\) 0 0
\(892\) 1.44175 0.0482732
\(893\) 54.7360 1.83167
\(894\) 0 0
\(895\) 12.4946 0.417647
\(896\) 61.2165 2.04510
\(897\) 0 0
\(898\) −8.94067 −0.298354
\(899\) −5.63386 −0.187900
\(900\) 0 0
\(901\) 7.17867 0.239156
\(902\) −11.4218 −0.380306
\(903\) 0 0
\(904\) 44.4013 1.47677
\(905\) −23.3250 −0.775350
\(906\) 0 0
\(907\) 50.8770 1.68934 0.844671 0.535286i \(-0.179796\pi\)
0.844671 + 0.535286i \(0.179796\pi\)
\(908\) −3.09590 −0.102741
\(909\) 0 0
\(910\) −7.24655 −0.240221
\(911\) 37.0508 1.22755 0.613773 0.789482i \(-0.289651\pi\)
0.613773 + 0.789482i \(0.289651\pi\)
\(912\) 0 0
\(913\) 8.52263 0.282058
\(914\) 53.0011 1.75312
\(915\) 0 0
\(916\) −3.27439 −0.108189
\(917\) 45.3488 1.49755
\(918\) 0 0
\(919\) 3.88820 0.128260 0.0641300 0.997942i \(-0.479573\pi\)
0.0641300 + 0.997942i \(0.479573\pi\)
\(920\) −0.208035 −0.00685870
\(921\) 0 0
\(922\) −22.3200 −0.735069
\(923\) −2.51438 −0.0827618
\(924\) 0 0
\(925\) 4.13079 0.135820
\(926\) −10.7498 −0.353259
\(927\) 0 0
\(928\) −2.81852 −0.0925225
\(929\) 10.2003 0.334661 0.167330 0.985901i \(-0.446485\pi\)
0.167330 + 0.985901i \(0.446485\pi\)
\(930\) 0 0
\(931\) −83.1545 −2.72528
\(932\) 2.08673 0.0683532
\(933\) 0 0
\(934\) −3.09467 −0.101261
\(935\) 8.83204 0.288839
\(936\) 0 0
\(937\) 48.7823 1.59365 0.796825 0.604211i \(-0.206511\pi\)
0.796825 + 0.604211i \(0.206511\pi\)
\(938\) −11.2820 −0.368370
\(939\) 0 0
\(940\) −1.62742 −0.0530807
\(941\) −45.7324 −1.49083 −0.745416 0.666599i \(-0.767749\pi\)
−0.745416 + 0.666599i \(0.767749\pi\)
\(942\) 0 0
\(943\) −0.375243 −0.0122196
\(944\) 19.8948 0.647521
\(945\) 0 0
\(946\) −10.9808 −0.357015
\(947\) −14.7312 −0.478699 −0.239350 0.970933i \(-0.576934\pi\)
−0.239350 + 0.970933i \(0.576934\pi\)
\(948\) 0 0
\(949\) −2.90174 −0.0941946
\(950\) 6.94240 0.225241
\(951\) 0 0
\(952\) −74.7809 −2.42366
\(953\) 57.2024 1.85297 0.926484 0.376334i \(-0.122816\pi\)
0.926484 + 0.376334i \(0.122816\pi\)
\(954\) 0 0
\(955\) −19.5264 −0.631858
\(956\) 0.369722 0.0119577
\(957\) 0 0
\(958\) 28.7883 0.930107
\(959\) 70.8495 2.28785
\(960\) 0 0
\(961\) −28.4650 −0.918226
\(962\) 6.04434 0.194877
\(963\) 0 0
\(964\) −0.610315 −0.0196569
\(965\) 23.3207 0.750721
\(966\) 0 0
\(967\) 10.2741 0.330391 0.165196 0.986261i \(-0.447174\pi\)
0.165196 + 0.986261i \(0.447174\pi\)
\(968\) −23.0356 −0.740392
\(969\) 0 0
\(970\) −27.5894 −0.885843
\(971\) −1.15531 −0.0370757 −0.0185378 0.999828i \(-0.505901\pi\)
−0.0185378 + 0.999828i \(0.505901\pi\)
\(972\) 0 0
\(973\) −13.2893 −0.426037
\(974\) −28.4708 −0.912263
\(975\) 0 0
\(976\) 45.2839 1.44950
\(977\) −20.7840 −0.664940 −0.332470 0.943114i \(-0.607882\pi\)
−0.332470 + 0.943114i \(0.607882\pi\)
\(978\) 0 0
\(979\) 0.359545 0.0114911
\(980\) 2.47237 0.0789770
\(981\) 0 0
\(982\) −20.5264 −0.655023
\(983\) −44.1555 −1.40834 −0.704171 0.710030i \(-0.748681\pi\)
−0.704171 + 0.710030i \(0.748681\pi\)
\(984\) 0 0
\(985\) −9.68456 −0.308576
\(986\) 28.7426 0.915352
\(987\) 0 0
\(988\) 0.669293 0.0212931
\(989\) −0.360752 −0.0114712
\(990\) 0 0
\(991\) −3.86612 −0.122811 −0.0614056 0.998113i \(-0.519558\pi\)
−0.0614056 + 0.998113i \(0.519558\pi\)
\(992\) 1.26822 0.0402660
\(993\) 0 0
\(994\) 18.2206 0.577922
\(995\) −12.9358 −0.410092
\(996\) 0 0
\(997\) 50.3098 1.59333 0.796663 0.604424i \(-0.206597\pi\)
0.796663 + 0.604424i \(0.206597\pi\)
\(998\) 9.38233 0.296993
\(999\) 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 5265.2.a.bd.1.3 yes 8
3.2 odd 2 5265.2.a.bc.1.6 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
5265.2.a.bc.1.6 8 3.2 odd 2
5265.2.a.bd.1.3 yes 8 1.1 even 1 trivial