Properties

Label 1520.2.a.r.1.3
Level $1520$
Weight $2$
Character 1520.1
Self dual yes
Analytic conductor $12.137$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1520,2,Mod(1,1520)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1520, 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("1520.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1520 = 2^{4} \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1520.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: \(12.1372611072\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.568.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 6x - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 760)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(3.12489\) of defining polynomial
Character \(\chi\) \(=\) 1520.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+3.12489 q^{3} -1.00000 q^{5} +1.51514 q^{7} +6.76491 q^{9} +O(q^{10})\) \(q+3.12489 q^{3} -1.00000 q^{5} +1.51514 q^{7} +6.76491 q^{9} -4.24977 q^{11} +4.15516 q^{13} -3.12489 q^{15} -3.51514 q^{17} +1.00000 q^{19} +4.73463 q^{21} +8.73463 q^{23} +1.00000 q^{25} +11.7649 q^{27} +1.45459 q^{29} +4.96972 q^{31} -13.2800 q^{33} -1.51514 q^{35} +7.60975 q^{37} +12.9844 q^{39} -9.21949 q^{41} +8.31032 q^{43} -6.76491 q^{45} -5.28005 q^{47} -4.70436 q^{49} -10.9844 q^{51} +0.155162 q^{53} +4.24977 q^{55} +3.12489 q^{57} +2.48486 q^{59} -4.49954 q^{61} +10.2498 q^{63} -4.15516 q^{65} -7.43521 q^{67} +27.2947 q^{69} -8.49954 q^{71} -15.0450 q^{73} +3.12489 q^{75} -6.43899 q^{77} +0.310323 q^{79} +16.4693 q^{81} +8.96972 q^{83} +3.51514 q^{85} +4.54541 q^{87} +0.719953 q^{89} +6.29564 q^{91} +15.5298 q^{93} -1.00000 q^{95} +17.3893 q^{97} -28.7493 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + q^{3} - 3 q^{5} + 5 q^{7} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + q^{3} - 3 q^{5} + 5 q^{7} + 4 q^{9} + 4 q^{11} + 5 q^{13} - q^{15} - 11 q^{17} + 3 q^{19} - 3 q^{21} + 9 q^{23} + 3 q^{25} + 19 q^{27} + 3 q^{29} + 14 q^{31} - 24 q^{33} - 5 q^{35} + 14 q^{37} + 5 q^{39} - 10 q^{41} + 10 q^{43} - 4 q^{45} + 4 q^{49} + q^{51} - 7 q^{53} - 4 q^{55} + q^{57} + 7 q^{59} + 20 q^{61} + 14 q^{63} - 5 q^{65} + q^{67} + 33 q^{69} + 8 q^{71} - 13 q^{73} + q^{75} + 16 q^{77} - 14 q^{79} + 15 q^{81} + 26 q^{83} + 11 q^{85} + 15 q^{87} + 18 q^{89} + 37 q^{91} + 14 q^{93} - 3 q^{95} - 6 q^{97} - 36 q^{99}+O(q^{100}) \) 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
\(3\) 3.12489 1.80415 0.902077 0.431576i \(-0.142042\pi\)
0.902077 + 0.431576i \(0.142042\pi\)
\(4\) 0 0
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) 1.51514 0.572668 0.286334 0.958130i \(-0.407563\pi\)
0.286334 + 0.958130i \(0.407563\pi\)
\(8\) 0 0
\(9\) 6.76491 2.25497
\(10\) 0 0
\(11\) −4.24977 −1.28135 −0.640677 0.767810i \(-0.721346\pi\)
−0.640677 + 0.767810i \(0.721346\pi\)
\(12\) 0 0
\(13\) 4.15516 1.15243 0.576217 0.817297i \(-0.304528\pi\)
0.576217 + 0.817297i \(0.304528\pi\)
\(14\) 0 0
\(15\) −3.12489 −0.806842
\(16\) 0 0
\(17\) −3.51514 −0.852546 −0.426273 0.904595i \(-0.640174\pi\)
−0.426273 + 0.904595i \(0.640174\pi\)
\(18\) 0 0
\(19\) 1.00000 0.229416
\(20\) 0 0
\(21\) 4.73463 1.03318
\(22\) 0 0
\(23\) 8.73463 1.82130 0.910648 0.413182i \(-0.135583\pi\)
0.910648 + 0.413182i \(0.135583\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 11.7649 2.26416
\(28\) 0 0
\(29\) 1.45459 0.270110 0.135055 0.990838i \(-0.456879\pi\)
0.135055 + 0.990838i \(0.456879\pi\)
\(30\) 0 0
\(31\) 4.96972 0.892589 0.446294 0.894886i \(-0.352743\pi\)
0.446294 + 0.894886i \(0.352743\pi\)
\(32\) 0 0
\(33\) −13.2800 −2.31176
\(34\) 0 0
\(35\) −1.51514 −0.256105
\(36\) 0 0
\(37\) 7.60975 1.25103 0.625517 0.780210i \(-0.284888\pi\)
0.625517 + 0.780210i \(0.284888\pi\)
\(38\) 0 0
\(39\) 12.9844 2.07917
\(40\) 0 0
\(41\) −9.21949 −1.43984 −0.719922 0.694055i \(-0.755822\pi\)
−0.719922 + 0.694055i \(0.755822\pi\)
\(42\) 0 0
\(43\) 8.31032 1.26731 0.633656 0.773615i \(-0.281553\pi\)
0.633656 + 0.773615i \(0.281553\pi\)
\(44\) 0 0
\(45\) −6.76491 −1.00845
\(46\) 0 0
\(47\) −5.28005 −0.770174 −0.385087 0.922880i \(-0.625829\pi\)
−0.385087 + 0.922880i \(0.625829\pi\)
\(48\) 0 0
\(49\) −4.70436 −0.672051
\(50\) 0 0
\(51\) −10.9844 −1.53812
\(52\) 0 0
\(53\) 0.155162 0.0213131 0.0106565 0.999943i \(-0.496608\pi\)
0.0106565 + 0.999943i \(0.496608\pi\)
\(54\) 0 0
\(55\) 4.24977 0.573039
\(56\) 0 0
\(57\) 3.12489 0.413901
\(58\) 0 0
\(59\) 2.48486 0.323501 0.161751 0.986832i \(-0.448286\pi\)
0.161751 + 0.986832i \(0.448286\pi\)
\(60\) 0 0
\(61\) −4.49954 −0.576107 −0.288054 0.957614i \(-0.593008\pi\)
−0.288054 + 0.957614i \(0.593008\pi\)
\(62\) 0 0
\(63\) 10.2498 1.29135
\(64\) 0 0
\(65\) −4.15516 −0.515384
\(66\) 0 0
\(67\) −7.43521 −0.908355 −0.454178 0.890911i \(-0.650067\pi\)
−0.454178 + 0.890911i \(0.650067\pi\)
\(68\) 0 0
\(69\) 27.2947 3.28590
\(70\) 0 0
\(71\) −8.49954 −1.00871 −0.504355 0.863496i \(-0.668270\pi\)
−0.504355 + 0.863496i \(0.668270\pi\)
\(72\) 0 0
\(73\) −15.0450 −1.76088 −0.880439 0.474159i \(-0.842752\pi\)
−0.880439 + 0.474159i \(0.842752\pi\)
\(74\) 0 0
\(75\) 3.12489 0.360831
\(76\) 0 0
\(77\) −6.43899 −0.733791
\(78\) 0 0
\(79\) 0.310323 0.0349141 0.0174570 0.999848i \(-0.494443\pi\)
0.0174570 + 0.999848i \(0.494443\pi\)
\(80\) 0 0
\(81\) 16.4693 1.82992
\(82\) 0 0
\(83\) 8.96972 0.984555 0.492278 0.870438i \(-0.336165\pi\)
0.492278 + 0.870438i \(0.336165\pi\)
\(84\) 0 0
\(85\) 3.51514 0.381270
\(86\) 0 0
\(87\) 4.54541 0.487320
\(88\) 0 0
\(89\) 0.719953 0.0763149 0.0381574 0.999272i \(-0.487851\pi\)
0.0381574 + 0.999272i \(0.487851\pi\)
\(90\) 0 0
\(91\) 6.29564 0.659963
\(92\) 0 0
\(93\) 15.5298 1.61037
\(94\) 0 0
\(95\) −1.00000 −0.102598
\(96\) 0 0
\(97\) 17.3893 1.76562 0.882810 0.469731i \(-0.155649\pi\)
0.882810 + 0.469731i \(0.155649\pi\)
\(98\) 0 0
\(99\) −28.7493 −2.88941
\(100\) 0 0
\(101\) −12.4995 −1.24375 −0.621875 0.783116i \(-0.713629\pi\)
−0.621875 + 0.783116i \(0.713629\pi\)
\(102\) 0 0
\(103\) 10.8898 1.07300 0.536502 0.843899i \(-0.319746\pi\)
0.536502 + 0.843899i \(0.319746\pi\)
\(104\) 0 0
\(105\) −4.73463 −0.462053
\(106\) 0 0
\(107\) 7.43521 0.718789 0.359394 0.933186i \(-0.382983\pi\)
0.359394 + 0.933186i \(0.382983\pi\)
\(108\) 0 0
\(109\) −19.3553 −1.85390 −0.926950 0.375186i \(-0.877579\pi\)
−0.926950 + 0.375186i \(0.877579\pi\)
\(110\) 0 0
\(111\) 23.7796 2.25706
\(112\) 0 0
\(113\) −17.8595 −1.68008 −0.840041 0.542523i \(-0.817469\pi\)
−0.840041 + 0.542523i \(0.817469\pi\)
\(114\) 0 0
\(115\) −8.73463 −0.814509
\(116\) 0 0
\(117\) 28.1093 2.59870
\(118\) 0 0
\(119\) −5.32592 −0.488226
\(120\) 0 0
\(121\) 7.06055 0.641868
\(122\) 0 0
\(123\) −28.8099 −2.59770
\(124\) 0 0
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) −2.70058 −0.239637 −0.119819 0.992796i \(-0.538231\pi\)
−0.119819 + 0.992796i \(0.538231\pi\)
\(128\) 0 0
\(129\) 25.9688 2.28643
\(130\) 0 0
\(131\) 4.00000 0.349482 0.174741 0.984614i \(-0.444091\pi\)
0.174741 + 0.984614i \(0.444091\pi\)
\(132\) 0 0
\(133\) 1.51514 0.131379
\(134\) 0 0
\(135\) −11.7649 −1.01256
\(136\) 0 0
\(137\) −7.51514 −0.642062 −0.321031 0.947069i \(-0.604029\pi\)
−0.321031 + 0.947069i \(0.604029\pi\)
\(138\) 0 0
\(139\) −16.7493 −1.42066 −0.710329 0.703870i \(-0.751454\pi\)
−0.710329 + 0.703870i \(0.751454\pi\)
\(140\) 0 0
\(141\) −16.4995 −1.38951
\(142\) 0 0
\(143\) −17.6585 −1.47668
\(144\) 0 0
\(145\) −1.45459 −0.120797
\(146\) 0 0
\(147\) −14.7006 −1.21248
\(148\) 0 0
\(149\) −12.4995 −1.02400 −0.512001 0.858985i \(-0.671096\pi\)
−0.512001 + 0.858985i \(0.671096\pi\)
\(150\) 0 0
\(151\) −14.2498 −1.15963 −0.579815 0.814748i \(-0.696875\pi\)
−0.579815 + 0.814748i \(0.696875\pi\)
\(152\) 0 0
\(153\) −23.7796 −1.92247
\(154\) 0 0
\(155\) −4.96972 −0.399178
\(156\) 0 0
\(157\) 4.71995 0.376693 0.188347 0.982103i \(-0.439687\pi\)
0.188347 + 0.982103i \(0.439687\pi\)
\(158\) 0 0
\(159\) 0.484862 0.0384521
\(160\) 0 0
\(161\) 13.2342 1.04300
\(162\) 0 0
\(163\) −10.7493 −0.841951 −0.420976 0.907072i \(-0.638312\pi\)
−0.420976 + 0.907072i \(0.638312\pi\)
\(164\) 0 0
\(165\) 13.2800 1.03385
\(166\) 0 0
\(167\) 10.7006 0.828035 0.414018 0.910269i \(-0.364125\pi\)
0.414018 + 0.910269i \(0.364125\pi\)
\(168\) 0 0
\(169\) 4.26537 0.328105
\(170\) 0 0
\(171\) 6.76491 0.517326
\(172\) 0 0
\(173\) −4.39025 −0.333785 −0.166892 0.985975i \(-0.553373\pi\)
−0.166892 + 0.985975i \(0.553373\pi\)
\(174\) 0 0
\(175\) 1.51514 0.114534
\(176\) 0 0
\(177\) 7.76491 0.583646
\(178\) 0 0
\(179\) 3.21949 0.240636 0.120318 0.992735i \(-0.461609\pi\)
0.120318 + 0.992735i \(0.461609\pi\)
\(180\) 0 0
\(181\) 22.0000 1.63525 0.817624 0.575753i \(-0.195291\pi\)
0.817624 + 0.575753i \(0.195291\pi\)
\(182\) 0 0
\(183\) −14.0606 −1.03939
\(184\) 0 0
\(185\) −7.60975 −0.559480
\(186\) 0 0
\(187\) 14.9385 1.09241
\(188\) 0 0
\(189\) 17.8255 1.29661
\(190\) 0 0
\(191\) −13.2342 −0.957591 −0.478796 0.877926i \(-0.658927\pi\)
−0.478796 + 0.877926i \(0.658927\pi\)
\(192\) 0 0
\(193\) 17.3893 1.25171 0.625856 0.779939i \(-0.284750\pi\)
0.625856 + 0.779939i \(0.284750\pi\)
\(194\) 0 0
\(195\) −12.9844 −0.929832
\(196\) 0 0
\(197\) −2.00000 −0.142494 −0.0712470 0.997459i \(-0.522698\pi\)
−0.0712470 + 0.997459i \(0.522698\pi\)
\(198\) 0 0
\(199\) 14.3250 1.01547 0.507736 0.861513i \(-0.330482\pi\)
0.507736 + 0.861513i \(0.330482\pi\)
\(200\) 0 0
\(201\) −23.2342 −1.63881
\(202\) 0 0
\(203\) 2.20390 0.154683
\(204\) 0 0
\(205\) 9.21949 0.643917
\(206\) 0 0
\(207\) 59.0890 4.10697
\(208\) 0 0
\(209\) −4.24977 −0.293963
\(210\) 0 0
\(211\) 6.98440 0.480826 0.240413 0.970671i \(-0.422717\pi\)
0.240413 + 0.970671i \(0.422717\pi\)
\(212\) 0 0
\(213\) −26.5601 −1.81987
\(214\) 0 0
\(215\) −8.31032 −0.566759
\(216\) 0 0
\(217\) 7.52982 0.511157
\(218\) 0 0
\(219\) −47.0138 −3.17690
\(220\) 0 0
\(221\) −14.6060 −0.982504
\(222\) 0 0
\(223\) −24.4802 −1.63931 −0.819657 0.572855i \(-0.805836\pi\)
−0.819657 + 0.572855i \(0.805836\pi\)
\(224\) 0 0
\(225\) 6.76491 0.450994
\(226\) 0 0
\(227\) 24.9045 1.65297 0.826484 0.562960i \(-0.190338\pi\)
0.826484 + 0.562960i \(0.190338\pi\)
\(228\) 0 0
\(229\) 4.47018 0.295398 0.147699 0.989032i \(-0.452813\pi\)
0.147699 + 0.989032i \(0.452813\pi\)
\(230\) 0 0
\(231\) −20.1211 −1.32387
\(232\) 0 0
\(233\) −3.93945 −0.258082 −0.129041 0.991639i \(-0.541190\pi\)
−0.129041 + 0.991639i \(0.541190\pi\)
\(234\) 0 0
\(235\) 5.28005 0.344432
\(236\) 0 0
\(237\) 0.969724 0.0629903
\(238\) 0 0
\(239\) 17.8255 1.15303 0.576517 0.817085i \(-0.304412\pi\)
0.576517 + 0.817085i \(0.304412\pi\)
\(240\) 0 0
\(241\) −27.3094 −1.75915 −0.879577 0.475757i \(-0.842174\pi\)
−0.879577 + 0.475757i \(0.842174\pi\)
\(242\) 0 0
\(243\) 16.1698 1.03730
\(244\) 0 0
\(245\) 4.70436 0.300550
\(246\) 0 0
\(247\) 4.15516 0.264387
\(248\) 0 0
\(249\) 28.0294 1.77629
\(250\) 0 0
\(251\) 18.4390 1.16386 0.581929 0.813239i \(-0.302298\pi\)
0.581929 + 0.813239i \(0.302298\pi\)
\(252\) 0 0
\(253\) −37.1202 −2.33373
\(254\) 0 0
\(255\) 10.9844 0.687870
\(256\) 0 0
\(257\) −0.889794 −0.0555038 −0.0277519 0.999615i \(-0.508835\pi\)
−0.0277519 + 0.999615i \(0.508835\pi\)
\(258\) 0 0
\(259\) 11.5298 0.716428
\(260\) 0 0
\(261\) 9.84014 0.609089
\(262\) 0 0
\(263\) 12.6206 0.778222 0.389111 0.921191i \(-0.372782\pi\)
0.389111 + 0.921191i \(0.372782\pi\)
\(264\) 0 0
\(265\) −0.155162 −0.00953150
\(266\) 0 0
\(267\) 2.24977 0.137684
\(268\) 0 0
\(269\) −16.4390 −1.00230 −0.501151 0.865360i \(-0.667090\pi\)
−0.501151 + 0.865360i \(0.667090\pi\)
\(270\) 0 0
\(271\) −25.6050 −1.55540 −0.777698 0.628638i \(-0.783613\pi\)
−0.777698 + 0.628638i \(0.783613\pi\)
\(272\) 0 0
\(273\) 19.6732 1.19067
\(274\) 0 0
\(275\) −4.24977 −0.256271
\(276\) 0 0
\(277\) 0.749313 0.0450218 0.0225109 0.999747i \(-0.492834\pi\)
0.0225109 + 0.999747i \(0.492834\pi\)
\(278\) 0 0
\(279\) 33.6197 2.01276
\(280\) 0 0
\(281\) 3.77959 0.225471 0.112736 0.993625i \(-0.464039\pi\)
0.112736 + 0.993625i \(0.464039\pi\)
\(282\) 0 0
\(283\) 9.46927 0.562889 0.281445 0.959577i \(-0.409186\pi\)
0.281445 + 0.959577i \(0.409186\pi\)
\(284\) 0 0
\(285\) −3.12489 −0.185102
\(286\) 0 0
\(287\) −13.9688 −0.824553
\(288\) 0 0
\(289\) −4.64380 −0.273165
\(290\) 0 0
\(291\) 54.3397 3.18545
\(292\) 0 0
\(293\) 24.6841 1.44206 0.721029 0.692905i \(-0.243669\pi\)
0.721029 + 0.692905i \(0.243669\pi\)
\(294\) 0 0
\(295\) −2.48486 −0.144674
\(296\) 0 0
\(297\) −49.9982 −2.90119
\(298\) 0 0
\(299\) 36.2938 2.09893
\(300\) 0 0
\(301\) 12.5913 0.725750
\(302\) 0 0
\(303\) −39.0596 −2.24392
\(304\) 0 0
\(305\) 4.49954 0.257643
\(306\) 0 0
\(307\) −13.3288 −0.760714 −0.380357 0.924840i \(-0.624199\pi\)
−0.380357 + 0.924840i \(0.624199\pi\)
\(308\) 0 0
\(309\) 34.0294 1.93586
\(310\) 0 0
\(311\) 13.4546 0.762940 0.381470 0.924381i \(-0.375418\pi\)
0.381470 + 0.924381i \(0.375418\pi\)
\(312\) 0 0
\(313\) −21.7649 −1.23023 −0.615113 0.788439i \(-0.710889\pi\)
−0.615113 + 0.788439i \(0.710889\pi\)
\(314\) 0 0
\(315\) −10.2498 −0.577509
\(316\) 0 0
\(317\) 23.5639 1.32348 0.661740 0.749734i \(-0.269818\pi\)
0.661740 + 0.749734i \(0.269818\pi\)
\(318\) 0 0
\(319\) −6.18166 −0.346106
\(320\) 0 0
\(321\) 23.2342 1.29681
\(322\) 0 0
\(323\) −3.51514 −0.195588
\(324\) 0 0
\(325\) 4.15516 0.230487
\(326\) 0 0
\(327\) −60.4830 −3.34472
\(328\) 0 0
\(329\) −8.00000 −0.441054
\(330\) 0 0
\(331\) 11.9541 0.657058 0.328529 0.944494i \(-0.393447\pi\)
0.328529 + 0.944494i \(0.393447\pi\)
\(332\) 0 0
\(333\) 51.4792 2.82105
\(334\) 0 0
\(335\) 7.43521 0.406229
\(336\) 0 0
\(337\) −22.8292 −1.24359 −0.621794 0.783181i \(-0.713596\pi\)
−0.621794 + 0.783181i \(0.713596\pi\)
\(338\) 0 0
\(339\) −55.8089 −3.03113
\(340\) 0 0
\(341\) −21.1202 −1.14372
\(342\) 0 0
\(343\) −17.7337 −0.957531
\(344\) 0 0
\(345\) −27.2947 −1.46950
\(346\) 0 0
\(347\) −29.4693 −1.58199 −0.790997 0.611821i \(-0.790437\pi\)
−0.790997 + 0.611821i \(0.790437\pi\)
\(348\) 0 0
\(349\) 4.56009 0.244096 0.122048 0.992524i \(-0.461054\pi\)
0.122048 + 0.992524i \(0.461054\pi\)
\(350\) 0 0
\(351\) 48.8851 2.60929
\(352\) 0 0
\(353\) −24.4849 −1.30320 −0.651599 0.758564i \(-0.725901\pi\)
−0.651599 + 0.758564i \(0.725901\pi\)
\(354\) 0 0
\(355\) 8.49954 0.451109
\(356\) 0 0
\(357\) −16.6429 −0.880835
\(358\) 0 0
\(359\) −12.0147 −0.634111 −0.317055 0.948407i \(-0.602694\pi\)
−0.317055 + 0.948407i \(0.602694\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 0 0
\(363\) 22.0634 1.15803
\(364\) 0 0
\(365\) 15.0450 0.787489
\(366\) 0 0
\(367\) 25.1589 1.31329 0.656643 0.754201i \(-0.271976\pi\)
0.656643 + 0.754201i \(0.271976\pi\)
\(368\) 0 0
\(369\) −62.3690 −3.24680
\(370\) 0 0
\(371\) 0.235091 0.0122053
\(372\) 0 0
\(373\) 9.62443 0.498334 0.249167 0.968461i \(-0.419843\pi\)
0.249167 + 0.968461i \(0.419843\pi\)
\(374\) 0 0
\(375\) −3.12489 −0.161368
\(376\) 0 0
\(377\) 6.04404 0.311284
\(378\) 0 0
\(379\) −19.7943 −1.01676 −0.508382 0.861132i \(-0.669756\pi\)
−0.508382 + 0.861132i \(0.669756\pi\)
\(380\) 0 0
\(381\) −8.43899 −0.432343
\(382\) 0 0
\(383\) −33.3288 −1.70302 −0.851511 0.524337i \(-0.824313\pi\)
−0.851511 + 0.524337i \(0.824313\pi\)
\(384\) 0 0
\(385\) 6.43899 0.328161
\(386\) 0 0
\(387\) 56.2186 2.85775
\(388\) 0 0
\(389\) 11.5904 0.587655 0.293828 0.955858i \(-0.405071\pi\)
0.293828 + 0.955858i \(0.405071\pi\)
\(390\) 0 0
\(391\) −30.7034 −1.55274
\(392\) 0 0
\(393\) 12.4995 0.630518
\(394\) 0 0
\(395\) −0.310323 −0.0156140
\(396\) 0 0
\(397\) −8.74931 −0.439115 −0.219558 0.975600i \(-0.570461\pi\)
−0.219558 + 0.975600i \(0.570461\pi\)
\(398\) 0 0
\(399\) 4.73463 0.237028
\(400\) 0 0
\(401\) −18.6206 −0.929871 −0.464935 0.885345i \(-0.653922\pi\)
−0.464935 + 0.885345i \(0.653922\pi\)
\(402\) 0 0
\(403\) 20.6500 1.02865
\(404\) 0 0
\(405\) −16.4693 −0.818364
\(406\) 0 0
\(407\) −32.3397 −1.60302
\(408\) 0 0
\(409\) 7.93945 0.392580 0.196290 0.980546i \(-0.437111\pi\)
0.196290 + 0.980546i \(0.437111\pi\)
\(410\) 0 0
\(411\) −23.4839 −1.15838
\(412\) 0 0
\(413\) 3.76491 0.185259
\(414\) 0 0
\(415\) −8.96972 −0.440306
\(416\) 0 0
\(417\) −52.3397 −2.56309
\(418\) 0 0
\(419\) 13.0908 0.639529 0.319764 0.947497i \(-0.396396\pi\)
0.319764 + 0.947497i \(0.396396\pi\)
\(420\) 0 0
\(421\) 19.8936 0.969554 0.484777 0.874638i \(-0.338901\pi\)
0.484777 + 0.874638i \(0.338901\pi\)
\(422\) 0 0
\(423\) −35.7190 −1.73672
\(424\) 0 0
\(425\) −3.51514 −0.170509
\(426\) 0 0
\(427\) −6.81743 −0.329918
\(428\) 0 0
\(429\) −55.1807 −2.66415
\(430\) 0 0
\(431\) 33.7796 1.62711 0.813553 0.581491i \(-0.197530\pi\)
0.813553 + 0.581491i \(0.197530\pi\)
\(432\) 0 0
\(433\) 9.07901 0.436310 0.218155 0.975914i \(-0.429996\pi\)
0.218155 + 0.975914i \(0.429996\pi\)
\(434\) 0 0
\(435\) −4.54541 −0.217936
\(436\) 0 0
\(437\) 8.73463 0.417834
\(438\) 0 0
\(439\) 12.3103 0.587540 0.293770 0.955876i \(-0.405090\pi\)
0.293770 + 0.955876i \(0.405090\pi\)
\(440\) 0 0
\(441\) −31.8245 −1.51545
\(442\) 0 0
\(443\) −2.56009 −0.121634 −0.0608169 0.998149i \(-0.519371\pi\)
−0.0608169 + 0.998149i \(0.519371\pi\)
\(444\) 0 0
\(445\) −0.719953 −0.0341290
\(446\) 0 0
\(447\) −39.0596 −1.84746
\(448\) 0 0
\(449\) 22.9991 1.08539 0.542697 0.839929i \(-0.317403\pi\)
0.542697 + 0.839929i \(0.317403\pi\)
\(450\) 0 0
\(451\) 39.1807 1.84495
\(452\) 0 0
\(453\) −44.5289 −2.09215
\(454\) 0 0
\(455\) −6.29564 −0.295144
\(456\) 0 0
\(457\) 20.6741 0.967093 0.483546 0.875319i \(-0.339348\pi\)
0.483546 + 0.875319i \(0.339348\pi\)
\(458\) 0 0
\(459\) −41.3553 −1.93030
\(460\) 0 0
\(461\) −5.87890 −0.273807 −0.136904 0.990584i \(-0.543715\pi\)
−0.136904 + 0.990584i \(0.543715\pi\)
\(462\) 0 0
\(463\) −35.5592 −1.65258 −0.826288 0.563249i \(-0.809551\pi\)
−0.826288 + 0.563249i \(0.809551\pi\)
\(464\) 0 0
\(465\) −15.5298 −0.720178
\(466\) 0 0
\(467\) −41.7408 −1.93154 −0.965768 0.259408i \(-0.916472\pi\)
−0.965768 + 0.259408i \(0.916472\pi\)
\(468\) 0 0
\(469\) −11.2654 −0.520186
\(470\) 0 0
\(471\) 14.7493 0.679612
\(472\) 0 0
\(473\) −35.3170 −1.62388
\(474\) 0 0
\(475\) 1.00000 0.0458831
\(476\) 0 0
\(477\) 1.04965 0.0480603
\(478\) 0 0
\(479\) −16.8704 −0.770829 −0.385415 0.922744i \(-0.625942\pi\)
−0.385415 + 0.922744i \(0.625942\pi\)
\(480\) 0 0
\(481\) 31.6197 1.44174
\(482\) 0 0
\(483\) 41.3553 1.88173
\(484\) 0 0
\(485\) −17.3893 −0.789609
\(486\) 0 0
\(487\) −3.64094 −0.164987 −0.0824934 0.996592i \(-0.526288\pi\)
−0.0824934 + 0.996592i \(0.526288\pi\)
\(488\) 0 0
\(489\) −33.5904 −1.51901
\(490\) 0 0
\(491\) 1.96881 0.0888510 0.0444255 0.999013i \(-0.485854\pi\)
0.0444255 + 0.999013i \(0.485854\pi\)
\(492\) 0 0
\(493\) −5.11307 −0.230281
\(494\) 0 0
\(495\) 28.7493 1.29219
\(496\) 0 0
\(497\) −12.8780 −0.577656
\(498\) 0 0
\(499\) 36.7787 1.64644 0.823220 0.567723i \(-0.192175\pi\)
0.823220 + 0.567723i \(0.192175\pi\)
\(500\) 0 0
\(501\) 33.4381 1.49390
\(502\) 0 0
\(503\) −15.8860 −0.708322 −0.354161 0.935184i \(-0.615234\pi\)
−0.354161 + 0.935184i \(0.615234\pi\)
\(504\) 0 0
\(505\) 12.4995 0.556222
\(506\) 0 0
\(507\) 13.3288 0.591952
\(508\) 0 0
\(509\) 20.2791 0.898857 0.449428 0.893316i \(-0.351628\pi\)
0.449428 + 0.893316i \(0.351628\pi\)
\(510\) 0 0
\(511\) −22.7952 −1.00840
\(512\) 0 0
\(513\) 11.7649 0.519433
\(514\) 0 0
\(515\) −10.8898 −0.479862
\(516\) 0 0
\(517\) 22.4390 0.986866
\(518\) 0 0
\(519\) −13.7190 −0.602199
\(520\) 0 0
\(521\) −23.4693 −1.02821 −0.514104 0.857728i \(-0.671875\pi\)
−0.514104 + 0.857728i \(0.671875\pi\)
\(522\) 0 0
\(523\) 26.3737 1.15324 0.576622 0.817011i \(-0.304371\pi\)
0.576622 + 0.817011i \(0.304371\pi\)
\(524\) 0 0
\(525\) 4.73463 0.206636
\(526\) 0 0
\(527\) −17.4693 −0.760973
\(528\) 0 0
\(529\) 53.2938 2.31712
\(530\) 0 0
\(531\) 16.8099 0.729486
\(532\) 0 0
\(533\) −38.3085 −1.65932
\(534\) 0 0
\(535\) −7.43521 −0.321452
\(536\) 0 0
\(537\) 10.0606 0.434145
\(538\) 0 0
\(539\) 19.9924 0.861135
\(540\) 0 0
\(541\) 22.5601 0.969934 0.484967 0.874532i \(-0.338832\pi\)
0.484967 + 0.874532i \(0.338832\pi\)
\(542\) 0 0
\(543\) 68.7475 2.95024
\(544\) 0 0
\(545\) 19.3553 0.829089
\(546\) 0 0
\(547\) −20.6694 −0.883759 −0.441879 0.897074i \(-0.645688\pi\)
−0.441879 + 0.897074i \(0.645688\pi\)
\(548\) 0 0
\(549\) −30.4390 −1.29910
\(550\) 0 0
\(551\) 1.45459 0.0619674
\(552\) 0 0
\(553\) 0.470182 0.0199942
\(554\) 0 0
\(555\) −23.7796 −1.00939
\(556\) 0 0
\(557\) −37.7484 −1.59945 −0.799725 0.600366i \(-0.795022\pi\)
−0.799725 + 0.600366i \(0.795022\pi\)
\(558\) 0 0
\(559\) 34.5307 1.46049
\(560\) 0 0
\(561\) 46.6812 1.97088
\(562\) 0 0
\(563\) 36.6694 1.54543 0.772715 0.634753i \(-0.218898\pi\)
0.772715 + 0.634753i \(0.218898\pi\)
\(564\) 0 0
\(565\) 17.8595 0.751356
\(566\) 0 0
\(567\) 24.9532 1.04794
\(568\) 0 0
\(569\) 2.84862 0.119420 0.0597102 0.998216i \(-0.480982\pi\)
0.0597102 + 0.998216i \(0.480982\pi\)
\(570\) 0 0
\(571\) −28.4002 −1.18851 −0.594256 0.804276i \(-0.702554\pi\)
−0.594256 + 0.804276i \(0.702554\pi\)
\(572\) 0 0
\(573\) −41.3553 −1.72764
\(574\) 0 0
\(575\) 8.73463 0.364259
\(576\) 0 0
\(577\) −29.2947 −1.21956 −0.609778 0.792572i \(-0.708741\pi\)
−0.609778 + 0.792572i \(0.708741\pi\)
\(578\) 0 0
\(579\) 54.3397 2.25828
\(580\) 0 0
\(581\) 13.5904 0.563824
\(582\) 0 0
\(583\) −0.659401 −0.0273096
\(584\) 0 0
\(585\) −28.1093 −1.16218
\(586\) 0 0
\(587\) −22.4683 −0.927368 −0.463684 0.886001i \(-0.653473\pi\)
−0.463684 + 0.886001i \(0.653473\pi\)
\(588\) 0 0
\(589\) 4.96972 0.204774
\(590\) 0 0
\(591\) −6.24977 −0.257081
\(592\) 0 0
\(593\) 16.0606 0.659528 0.329764 0.944063i \(-0.393031\pi\)
0.329764 + 0.944063i \(0.393031\pi\)
\(594\) 0 0
\(595\) 5.32592 0.218341
\(596\) 0 0
\(597\) 44.7640 1.83207
\(598\) 0 0
\(599\) 23.2489 0.949922 0.474961 0.880007i \(-0.342462\pi\)
0.474961 + 0.880007i \(0.342462\pi\)
\(600\) 0 0
\(601\) −0.841057 −0.0343074 −0.0171537 0.999853i \(-0.505460\pi\)
−0.0171537 + 0.999853i \(0.505460\pi\)
\(602\) 0 0
\(603\) −50.2985 −2.04831
\(604\) 0 0
\(605\) −7.06055 −0.287052
\(606\) 0 0
\(607\) 32.4802 1.31833 0.659165 0.751999i \(-0.270910\pi\)
0.659165 + 0.751999i \(0.270910\pi\)
\(608\) 0 0
\(609\) 6.88693 0.279072
\(610\) 0 0
\(611\) −21.9394 −0.887575
\(612\) 0 0
\(613\) −19.4305 −0.784791 −0.392395 0.919797i \(-0.628354\pi\)
−0.392395 + 0.919797i \(0.628354\pi\)
\(614\) 0 0
\(615\) 28.8099 1.16173
\(616\) 0 0
\(617\) 20.7787 0.836518 0.418259 0.908328i \(-0.362640\pi\)
0.418259 + 0.908328i \(0.362640\pi\)
\(618\) 0 0
\(619\) −18.9310 −0.760900 −0.380450 0.924802i \(-0.624231\pi\)
−0.380450 + 0.924802i \(0.624231\pi\)
\(620\) 0 0
\(621\) 102.762 4.12370
\(622\) 0 0
\(623\) 1.09083 0.0437031
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) −13.2800 −0.530354
\(628\) 0 0
\(629\) −26.7493 −1.06656
\(630\) 0 0
\(631\) 0.370875 0.0147643 0.00738215 0.999973i \(-0.497650\pi\)
0.00738215 + 0.999973i \(0.497650\pi\)
\(632\) 0 0
\(633\) 21.8255 0.867484
\(634\) 0 0
\(635\) 2.70058 0.107169
\(636\) 0 0
\(637\) −19.5474 −0.774495
\(638\) 0 0
\(639\) −57.4986 −2.27461
\(640\) 0 0
\(641\) 26.8099 1.05893 0.529463 0.848333i \(-0.322393\pi\)
0.529463 + 0.848333i \(0.322393\pi\)
\(642\) 0 0
\(643\) 0.969724 0.0382422 0.0191211 0.999817i \(-0.493913\pi\)
0.0191211 + 0.999817i \(0.493913\pi\)
\(644\) 0 0
\(645\) −25.9688 −1.02252
\(646\) 0 0
\(647\) −1.20482 −0.0473662 −0.0236831 0.999720i \(-0.507539\pi\)
−0.0236831 + 0.999720i \(0.507539\pi\)
\(648\) 0 0
\(649\) −10.5601 −0.414520
\(650\) 0 0
\(651\) 23.5298 0.922206
\(652\) 0 0
\(653\) −22.9310 −0.897358 −0.448679 0.893693i \(-0.648105\pi\)
−0.448679 + 0.893693i \(0.648105\pi\)
\(654\) 0 0
\(655\) −4.00000 −0.156293
\(656\) 0 0
\(657\) −101.778 −3.97073
\(658\) 0 0
\(659\) −15.1055 −0.588427 −0.294214 0.955740i \(-0.595058\pi\)
−0.294214 + 0.955740i \(0.595058\pi\)
\(660\) 0 0
\(661\) 19.0450 0.740763 0.370381 0.928880i \(-0.379227\pi\)
0.370381 + 0.928880i \(0.379227\pi\)
\(662\) 0 0
\(663\) −45.6420 −1.77259
\(664\) 0 0
\(665\) −1.51514 −0.0587545
\(666\) 0 0
\(667\) 12.7053 0.491950
\(668\) 0 0
\(669\) −76.4977 −2.95757
\(670\) 0 0
\(671\) 19.1220 0.738197
\(672\) 0 0
\(673\) 15.4205 0.594418 0.297209 0.954812i \(-0.403944\pi\)
0.297209 + 0.954812i \(0.403944\pi\)
\(674\) 0 0
\(675\) 11.7649 0.452832
\(676\) 0 0
\(677\) 8.62534 0.331499 0.165749 0.986168i \(-0.446996\pi\)
0.165749 + 0.986168i \(0.446996\pi\)
\(678\) 0 0
\(679\) 26.3472 1.01111
\(680\) 0 0
\(681\) 77.8236 2.98221
\(682\) 0 0
\(683\) −38.8898 −1.48808 −0.744038 0.668137i \(-0.767092\pi\)
−0.744038 + 0.668137i \(0.767092\pi\)
\(684\) 0 0
\(685\) 7.51514 0.287139
\(686\) 0 0
\(687\) 13.9688 0.532943
\(688\) 0 0
\(689\) 0.644721 0.0245619
\(690\) 0 0
\(691\) 21.5979 0.821624 0.410812 0.911720i \(-0.365245\pi\)
0.410812 + 0.911720i \(0.365245\pi\)
\(692\) 0 0
\(693\) −43.5592 −1.65468
\(694\) 0 0
\(695\) 16.7493 0.635338
\(696\) 0 0
\(697\) 32.4078 1.22753
\(698\) 0 0
\(699\) −12.3103 −0.465619
\(700\) 0 0
\(701\) 17.8183 0.672990 0.336495 0.941685i \(-0.390759\pi\)
0.336495 + 0.941685i \(0.390759\pi\)
\(702\) 0 0
\(703\) 7.60975 0.287007
\(704\) 0 0
\(705\) 16.4995 0.621409
\(706\) 0 0
\(707\) −18.9385 −0.712257
\(708\) 0 0
\(709\) 47.4986 1.78385 0.891924 0.452185i \(-0.149355\pi\)
0.891924 + 0.452185i \(0.149355\pi\)
\(710\) 0 0
\(711\) 2.09931 0.0787302
\(712\) 0 0
\(713\) 43.4087 1.62567
\(714\) 0 0
\(715\) 17.6585 0.660390
\(716\) 0 0
\(717\) 55.7025 2.08025
\(718\) 0 0
\(719\) 25.4546 0.949296 0.474648 0.880176i \(-0.342575\pi\)
0.474648 + 0.880176i \(0.342575\pi\)
\(720\) 0 0
\(721\) 16.4995 0.614475
\(722\) 0 0
\(723\) −85.3388 −3.17378
\(724\) 0 0
\(725\) 1.45459 0.0540220
\(726\) 0 0
\(727\) −21.5445 −0.799041 −0.399521 0.916724i \(-0.630823\pi\)
−0.399521 + 0.916724i \(0.630823\pi\)
\(728\) 0 0
\(729\) 1.12110 0.0415224
\(730\) 0 0
\(731\) −29.2119 −1.08044
\(732\) 0 0
\(733\) 24.2110 0.894254 0.447127 0.894470i \(-0.352447\pi\)
0.447127 + 0.894470i \(0.352447\pi\)
\(734\) 0 0
\(735\) 14.7006 0.542239
\(736\) 0 0
\(737\) 31.5979 1.16392
\(738\) 0 0
\(739\) 37.4693 1.37833 0.689165 0.724605i \(-0.257978\pi\)
0.689165 + 0.724605i \(0.257978\pi\)
\(740\) 0 0
\(741\) 12.9844 0.476994
\(742\) 0 0
\(743\) 15.8595 0.581829 0.290915 0.956749i \(-0.406040\pi\)
0.290915 + 0.956749i \(0.406040\pi\)
\(744\) 0 0
\(745\) 12.4995 0.457948
\(746\) 0 0
\(747\) 60.6794 2.22014
\(748\) 0 0
\(749\) 11.2654 0.411628
\(750\) 0 0
\(751\) −5.12958 −0.187181 −0.0935906 0.995611i \(-0.529834\pi\)
−0.0935906 + 0.995611i \(0.529834\pi\)
\(752\) 0 0
\(753\) 57.6197 2.09978
\(754\) 0 0
\(755\) 14.2498 0.518602
\(756\) 0 0
\(757\) 37.4381 1.36071 0.680355 0.732883i \(-0.261826\pi\)
0.680355 + 0.732883i \(0.261826\pi\)
\(758\) 0 0
\(759\) −115.996 −4.21040
\(760\) 0 0
\(761\) 21.5833 0.782392 0.391196 0.920307i \(-0.372061\pi\)
0.391196 + 0.920307i \(0.372061\pi\)
\(762\) 0 0
\(763\) −29.3259 −1.06167
\(764\) 0 0
\(765\) 23.7796 0.859753
\(766\) 0 0
\(767\) 10.3250 0.372814
\(768\) 0 0
\(769\) −40.7640 −1.46999 −0.734994 0.678074i \(-0.762815\pi\)
−0.734994 + 0.678074i \(0.762815\pi\)
\(770\) 0 0
\(771\) −2.78051 −0.100137
\(772\) 0 0
\(773\) 48.1845 1.73308 0.866538 0.499110i \(-0.166340\pi\)
0.866538 + 0.499110i \(0.166340\pi\)
\(774\) 0 0
\(775\) 4.96972 0.178518
\(776\) 0 0
\(777\) 36.0294 1.29255
\(778\) 0 0
\(779\) −9.21949 −0.330323
\(780\) 0 0
\(781\) 36.1211 1.29251
\(782\) 0 0
\(783\) 17.1131 0.611571
\(784\) 0 0
\(785\) −4.71995 −0.168462
\(786\) 0 0
\(787\) −27.6925 −0.987132 −0.493566 0.869708i \(-0.664307\pi\)
−0.493566 + 0.869708i \(0.664307\pi\)
\(788\) 0 0
\(789\) 39.4381 1.40403
\(790\) 0 0
\(791\) −27.0596 −0.962130
\(792\) 0 0
\(793\) −18.6963 −0.663926
\(794\) 0 0
\(795\) −0.484862 −0.0171963
\(796\) 0 0
\(797\) 10.3368 0.366149 0.183074 0.983099i \(-0.441395\pi\)
0.183074 + 0.983099i \(0.441395\pi\)
\(798\) 0 0
\(799\) 18.5601 0.656609
\(800\) 0 0
\(801\) 4.87042 0.172088
\(802\) 0 0
\(803\) 63.9376 2.25631
\(804\) 0 0
\(805\) −13.2342 −0.466443
\(806\) 0 0
\(807\) −51.3700 −1.80831
\(808\) 0 0
\(809\) −23.2342 −0.816870 −0.408435 0.912787i \(-0.633925\pi\)
−0.408435 + 0.912787i \(0.633925\pi\)
\(810\) 0 0
\(811\) −32.1433 −1.12871 −0.564353 0.825534i \(-0.690874\pi\)
−0.564353 + 0.825534i \(0.690874\pi\)
\(812\) 0 0
\(813\) −80.0128 −2.80617
\(814\) 0 0
\(815\) 10.7493 0.376532
\(816\) 0 0
\(817\) 8.31032 0.290741
\(818\) 0 0
\(819\) 42.5895 1.48820
\(820\) 0 0
\(821\) −27.0908 −0.945476 −0.472738 0.881203i \(-0.656734\pi\)
−0.472738 + 0.881203i \(0.656734\pi\)
\(822\) 0 0
\(823\) 14.4849 0.504911 0.252455 0.967609i \(-0.418762\pi\)
0.252455 + 0.967609i \(0.418762\pi\)
\(824\) 0 0
\(825\) −13.2800 −0.462352
\(826\) 0 0
\(827\) 54.0247 1.87862 0.939311 0.343067i \(-0.111466\pi\)
0.939311 + 0.343067i \(0.111466\pi\)
\(828\) 0 0
\(829\) 18.7640 0.651700 0.325850 0.945421i \(-0.394349\pi\)
0.325850 + 0.945421i \(0.394349\pi\)
\(830\) 0 0
\(831\) 2.34152 0.0812263
\(832\) 0 0
\(833\) 16.5365 0.572954
\(834\) 0 0
\(835\) −10.7006 −0.370309
\(836\) 0 0
\(837\) 58.4683 2.02096
\(838\) 0 0
\(839\) 7.84014 0.270672 0.135336 0.990800i \(-0.456789\pi\)
0.135336 + 0.990800i \(0.456789\pi\)
\(840\) 0 0
\(841\) −26.8842 −0.927041
\(842\) 0 0
\(843\) 11.8108 0.406785
\(844\) 0 0
\(845\) −4.26537 −0.146733
\(846\) 0 0
\(847\) 10.6977 0.367578
\(848\) 0 0
\(849\) 29.5904 1.01554
\(850\) 0 0
\(851\) 66.4683 2.27851
\(852\) 0 0
\(853\) 36.8174 1.26060 0.630302 0.776350i \(-0.282931\pi\)
0.630302 + 0.776350i \(0.282931\pi\)
\(854\) 0 0
\(855\) −6.76491 −0.231355
\(856\) 0 0
\(857\) −5.23887 −0.178956 −0.0894782 0.995989i \(-0.528520\pi\)
−0.0894782 + 0.995989i \(0.528520\pi\)
\(858\) 0 0
\(859\) 29.6197 1.01061 0.505306 0.862940i \(-0.331380\pi\)
0.505306 + 0.862940i \(0.331380\pi\)
\(860\) 0 0
\(861\) −43.6509 −1.48762
\(862\) 0 0
\(863\) 35.8889 1.22167 0.610836 0.791757i \(-0.290834\pi\)
0.610836 + 0.791757i \(0.290834\pi\)
\(864\) 0 0
\(865\) 4.39025 0.149273
\(866\) 0 0
\(867\) −14.5114 −0.492832
\(868\) 0 0
\(869\) −1.31880 −0.0447373
\(870\) 0 0
\(871\) −30.8945 −1.04682
\(872\) 0 0
\(873\) 117.637 3.98142
\(874\) 0 0
\(875\) −1.51514 −0.0512210
\(876\) 0 0
\(877\) −29.8136 −1.00674 −0.503368 0.864072i \(-0.667906\pi\)
−0.503368 + 0.864072i \(0.667906\pi\)
\(878\) 0 0
\(879\) 77.1349 2.60169
\(880\) 0 0
\(881\) −38.9679 −1.31286 −0.656431 0.754386i \(-0.727935\pi\)
−0.656431 + 0.754386i \(0.727935\pi\)
\(882\) 0 0
\(883\) 28.5677 0.961378 0.480689 0.876891i \(-0.340387\pi\)
0.480689 + 0.876891i \(0.340387\pi\)
\(884\) 0 0
\(885\) −7.76491 −0.261015
\(886\) 0 0
\(887\) −27.5104 −0.923710 −0.461855 0.886955i \(-0.652816\pi\)
−0.461855 + 0.886955i \(0.652816\pi\)
\(888\) 0 0
\(889\) −4.09174 −0.137233
\(890\) 0 0
\(891\) −69.9906 −2.34477
\(892\) 0 0
\(893\) −5.28005 −0.176690
\(894\) 0 0
\(895\) −3.21949 −0.107616
\(896\) 0 0
\(897\) 113.414 3.78678
\(898\) 0 0
\(899\) 7.22889 0.241097
\(900\) 0 0
\(901\) −0.545414 −0.0181704
\(902\) 0 0
\(903\) 39.3463 1.30936
\(904\) 0 0
\(905\) −22.0000 −0.731305
\(906\) 0 0
\(907\) 55.4939 1.84265 0.921323 0.388799i \(-0.127110\pi\)
0.921323 + 0.388799i \(0.127110\pi\)
\(908\) 0 0
\(909\) −84.5583 −2.80462
\(910\) 0 0
\(911\) −6.84106 −0.226654 −0.113327 0.993558i \(-0.536151\pi\)
−0.113327 + 0.993558i \(0.536151\pi\)
\(912\) 0 0
\(913\) −38.1193 −1.26156
\(914\) 0 0
\(915\) 14.0606 0.464828
\(916\) 0 0
\(917\) 6.06055 0.200137
\(918\) 0 0
\(919\) 6.17454 0.203679 0.101840 0.994801i \(-0.467527\pi\)
0.101840 + 0.994801i \(0.467527\pi\)
\(920\) 0 0
\(921\) −41.6509 −1.37244
\(922\) 0 0
\(923\) −35.3170 −1.16247
\(924\) 0 0
\(925\) 7.60975 0.250207
\(926\) 0 0
\(927\) 73.6685 2.41959
\(928\) 0 0
\(929\) 44.7034 1.46667 0.733336 0.679866i \(-0.237962\pi\)
0.733336 + 0.679866i \(0.237962\pi\)
\(930\) 0 0
\(931\) −4.70436 −0.154179
\(932\) 0 0
\(933\) 42.0440 1.37646
\(934\) 0 0
\(935\) −14.9385 −0.488542
\(936\) 0 0
\(937\) −12.1064 −0.395500 −0.197750 0.980253i \(-0.563363\pi\)
−0.197750 + 0.980253i \(0.563363\pi\)
\(938\) 0 0
\(939\) −68.0128 −2.21952
\(940\) 0 0
\(941\) −26.2645 −0.856197 −0.428098 0.903732i \(-0.640816\pi\)
−0.428098 + 0.903732i \(0.640816\pi\)
\(942\) 0 0
\(943\) −80.5289 −2.62238
\(944\) 0 0
\(945\) −17.8255 −0.579862
\(946\) 0 0
\(947\) −7.37935 −0.239797 −0.119898 0.992786i \(-0.538257\pi\)
−0.119898 + 0.992786i \(0.538257\pi\)
\(948\) 0 0
\(949\) −62.5142 −2.02930
\(950\) 0 0
\(951\) 73.6344 2.38776
\(952\) 0 0
\(953\) −21.5398 −0.697743 −0.348871 0.937171i \(-0.613435\pi\)
−0.348871 + 0.937171i \(0.613435\pi\)
\(954\) 0 0
\(955\) 13.2342 0.428248
\(956\) 0 0
\(957\) −19.3170 −0.624429
\(958\) 0 0
\(959\) −11.3865 −0.367688
\(960\) 0 0
\(961\) −6.30184 −0.203285
\(962\) 0 0
\(963\) 50.2985 1.62085
\(964\) 0 0
\(965\) −17.3893 −0.559783
\(966\) 0 0
\(967\) 39.8789 1.28242 0.641209 0.767366i \(-0.278433\pi\)
0.641209 + 0.767366i \(0.278433\pi\)
\(968\) 0 0
\(969\) −10.9844 −0.352870
\(970\) 0 0
\(971\) 29.7408 0.954429 0.477214 0.878787i \(-0.341647\pi\)
0.477214 + 0.878787i \(0.341647\pi\)
\(972\) 0 0
\(973\) −25.3775 −0.813566
\(974\) 0 0
\(975\) 12.9844 0.415834
\(976\) 0 0
\(977\) −58.6694 −1.87700 −0.938500 0.345279i \(-0.887784\pi\)
−0.938500 + 0.345279i \(0.887784\pi\)
\(978\) 0 0
\(979\) −3.05964 −0.0977864
\(980\) 0 0
\(981\) −130.937 −4.18049
\(982\) 0 0
\(983\) −11.7384 −0.374397 −0.187199 0.982322i \(-0.559941\pi\)
−0.187199 + 0.982322i \(0.559941\pi\)
\(984\) 0 0
\(985\) 2.00000 0.0637253
\(986\) 0 0
\(987\) −24.9991 −0.795730
\(988\) 0 0
\(989\) 72.5876 2.30815
\(990\) 0 0
\(991\) 26.7493 0.849720 0.424860 0.905259i \(-0.360323\pi\)
0.424860 + 0.905259i \(0.360323\pi\)
\(992\) 0 0
\(993\) 37.3553 1.18543
\(994\) 0 0
\(995\) −14.3250 −0.454133
\(996\) 0 0
\(997\) 45.8401 1.45177 0.725886 0.687815i \(-0.241430\pi\)
0.725886 + 0.687815i \(0.241430\pi\)
\(998\) 0 0
\(999\) 89.5280 2.83254
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 1520.2.a.r.1.3 3
4.3 odd 2 760.2.a.h.1.1 3
5.4 even 2 7600.2.a.bo.1.1 3
8.3 odd 2 6080.2.a.bw.1.3 3
8.5 even 2 6080.2.a.bs.1.1 3
12.11 even 2 6840.2.a.bj.1.2 3
20.3 even 4 3800.2.d.k.3649.1 6
20.7 even 4 3800.2.d.k.3649.6 6
20.19 odd 2 3800.2.a.y.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
760.2.a.h.1.1 3 4.3 odd 2
1520.2.a.r.1.3 3 1.1 even 1 trivial
3800.2.a.y.1.3 3 20.19 odd 2
3800.2.d.k.3649.1 6 20.3 even 4
3800.2.d.k.3649.6 6 20.7 even 4
6080.2.a.bs.1.1 3 8.5 even 2
6080.2.a.bw.1.3 3 8.3 odd 2
6840.2.a.bj.1.2 3 12.11 even 2
7600.2.a.bo.1.1 3 5.4 even 2