Properties

Label 608.4.a.j.1.2
Level $608$
Weight $4$
Character 608.1
Self dual yes
Analytic conductor $35.873$
Analytic rank $1$
Dimension $7$
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] = [608,4,Mod(1,608)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(608, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("608.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 608 = 2^{5} \cdot 19 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 608.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: \(35.8731612835\)
Analytic rank: \(1\)
Dimension: \(7\)
Coefficient field: \(\mathbb{Q}[x]/(x^{7} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{7} - 3x^{6} - 121x^{5} + 402x^{4} + 4234x^{3} - 14542x^{2} - 40996x + 141664 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(7.37394\) of defining polynomial
Character \(\chi\) \(=\) 608.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-6.88542 q^{3} +4.69330 q^{5} -3.62949 q^{7} +20.4091 q^{9} +O(q^{10})\) \(q-6.88542 q^{3} +4.69330 q^{5} -3.62949 q^{7} +20.4091 q^{9} -43.4975 q^{11} +75.0368 q^{13} -32.3154 q^{15} -13.1508 q^{17} +19.0000 q^{19} +24.9906 q^{21} +96.5591 q^{23} -102.973 q^{25} +45.3813 q^{27} +40.5580 q^{29} -138.828 q^{31} +299.499 q^{33} -17.0343 q^{35} +348.686 q^{37} -516.660 q^{39} -44.3243 q^{41} -214.385 q^{43} +95.7860 q^{45} +334.307 q^{47} -329.827 q^{49} +90.5487 q^{51} +311.122 q^{53} -204.147 q^{55} -130.823 q^{57} -502.875 q^{59} -54.7193 q^{61} -74.0746 q^{63} +352.170 q^{65} -312.959 q^{67} -664.850 q^{69} -642.657 q^{71} -837.348 q^{73} +709.012 q^{75} +157.874 q^{77} -349.595 q^{79} -863.515 q^{81} -521.399 q^{83} -61.7206 q^{85} -279.259 q^{87} +63.4094 q^{89} -272.345 q^{91} +955.887 q^{93} +89.1728 q^{95} -1229.92 q^{97} -887.745 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 7 q - 3 q^{3} - 17 q^{5} - 42 q^{7} + 86 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 7 q - 3 q^{3} - 17 q^{5} - 42 q^{7} + 86 q^{9} + 33 q^{11} - 35 q^{13} - 120 q^{15} + 66 q^{17} + 133 q^{19} + 33 q^{21} - 389 q^{23} + 44 q^{25} + 39 q^{27} + 233 q^{29} - 158 q^{31} - 206 q^{33} - 123 q^{35} - 436 q^{37} - 807 q^{39} - 94 q^{41} - 645 q^{43} + 103 q^{45} - 1451 q^{47} + 93 q^{49} - 1741 q^{51} + 3 q^{53} - 1971 q^{55} - 57 q^{57} - 297 q^{59} + 93 q^{61} - 2999 q^{63} - 788 q^{65} - 1641 q^{67} + 945 q^{69} - 2392 q^{71} + 324 q^{73} - 1909 q^{75} - 711 q^{77} - 2492 q^{79} + 143 q^{81} - 310 q^{83} + 2353 q^{85} - 4795 q^{87} - 440 q^{89} + 107 q^{91} + 900 q^{93} - 323 q^{95} - 532 q^{97} + 1591 q^{99}+O(q^{100}) \) 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
\(3\) −6.88542 −1.32510 −0.662550 0.749017i \(-0.730526\pi\)
−0.662550 + 0.749017i \(0.730526\pi\)
\(4\) 0 0
\(5\) 4.69330 0.419782 0.209891 0.977725i \(-0.432689\pi\)
0.209891 + 0.977725i \(0.432689\pi\)
\(6\) 0 0
\(7\) −3.62949 −0.195974 −0.0979871 0.995188i \(-0.531240\pi\)
−0.0979871 + 0.995188i \(0.531240\pi\)
\(8\) 0 0
\(9\) 20.4091 0.755892
\(10\) 0 0
\(11\) −43.4975 −1.19227 −0.596137 0.802883i \(-0.703298\pi\)
−0.596137 + 0.802883i \(0.703298\pi\)
\(12\) 0 0
\(13\) 75.0368 1.60088 0.800441 0.599412i \(-0.204599\pi\)
0.800441 + 0.599412i \(0.204599\pi\)
\(14\) 0 0
\(15\) −32.3154 −0.556253
\(16\) 0 0
\(17\) −13.1508 −0.187620 −0.0938098 0.995590i \(-0.529905\pi\)
−0.0938098 + 0.995590i \(0.529905\pi\)
\(18\) 0 0
\(19\) 19.0000 0.229416
\(20\) 0 0
\(21\) 24.9906 0.259686
\(22\) 0 0
\(23\) 96.5591 0.875390 0.437695 0.899124i \(-0.355795\pi\)
0.437695 + 0.899124i \(0.355795\pi\)
\(24\) 0 0
\(25\) −102.973 −0.823783
\(26\) 0 0
\(27\) 45.3813 0.323468
\(28\) 0 0
\(29\) 40.5580 0.259704 0.129852 0.991533i \(-0.458550\pi\)
0.129852 + 0.991533i \(0.458550\pi\)
\(30\) 0 0
\(31\) −138.828 −0.804328 −0.402164 0.915568i \(-0.631742\pi\)
−0.402164 + 0.915568i \(0.631742\pi\)
\(32\) 0 0
\(33\) 299.499 1.57988
\(34\) 0 0
\(35\) −17.0343 −0.0822664
\(36\) 0 0
\(37\) 348.686 1.54929 0.774644 0.632398i \(-0.217929\pi\)
0.774644 + 0.632398i \(0.217929\pi\)
\(38\) 0 0
\(39\) −516.660 −2.12133
\(40\) 0 0
\(41\) −44.3243 −0.168836 −0.0844182 0.996430i \(-0.526903\pi\)
−0.0844182 + 0.996430i \(0.526903\pi\)
\(42\) 0 0
\(43\) −214.385 −0.760313 −0.380157 0.924922i \(-0.624130\pi\)
−0.380157 + 0.924922i \(0.624130\pi\)
\(44\) 0 0
\(45\) 95.7860 0.317310
\(46\) 0 0
\(47\) 334.307 1.03752 0.518762 0.854919i \(-0.326393\pi\)
0.518762 + 0.854919i \(0.326393\pi\)
\(48\) 0 0
\(49\) −329.827 −0.961594
\(50\) 0 0
\(51\) 90.5487 0.248615
\(52\) 0 0
\(53\) 311.122 0.806338 0.403169 0.915126i \(-0.367909\pi\)
0.403169 + 0.915126i \(0.367909\pi\)
\(54\) 0 0
\(55\) −204.147 −0.500495
\(56\) 0 0
\(57\) −130.823 −0.303999
\(58\) 0 0
\(59\) −502.875 −1.10964 −0.554820 0.831971i \(-0.687213\pi\)
−0.554820 + 0.831971i \(0.687213\pi\)
\(60\) 0 0
\(61\) −54.7193 −0.114854 −0.0574270 0.998350i \(-0.518290\pi\)
−0.0574270 + 0.998350i \(0.518290\pi\)
\(62\) 0 0
\(63\) −74.0746 −0.148135
\(64\) 0 0
\(65\) 352.170 0.672021
\(66\) 0 0
\(67\) −312.959 −0.570658 −0.285329 0.958430i \(-0.592103\pi\)
−0.285329 + 0.958430i \(0.592103\pi\)
\(68\) 0 0
\(69\) −664.850 −1.15998
\(70\) 0 0
\(71\) −642.657 −1.07422 −0.537108 0.843513i \(-0.680483\pi\)
−0.537108 + 0.843513i \(0.680483\pi\)
\(72\) 0 0
\(73\) −837.348 −1.34252 −0.671261 0.741221i \(-0.734247\pi\)
−0.671261 + 0.741221i \(0.734247\pi\)
\(74\) 0 0
\(75\) 709.012 1.09160
\(76\) 0 0
\(77\) 157.874 0.233655
\(78\) 0 0
\(79\) −349.595 −0.497880 −0.248940 0.968519i \(-0.580082\pi\)
−0.248940 + 0.968519i \(0.580082\pi\)
\(80\) 0 0
\(81\) −863.515 −1.18452
\(82\) 0 0
\(83\) −521.399 −0.689530 −0.344765 0.938689i \(-0.612041\pi\)
−0.344765 + 0.938689i \(0.612041\pi\)
\(84\) 0 0
\(85\) −61.7206 −0.0787593
\(86\) 0 0
\(87\) −279.259 −0.344135
\(88\) 0 0
\(89\) 63.4094 0.0755211 0.0377606 0.999287i \(-0.487978\pi\)
0.0377606 + 0.999287i \(0.487978\pi\)
\(90\) 0 0
\(91\) −272.345 −0.313731
\(92\) 0 0
\(93\) 955.887 1.06582
\(94\) 0 0
\(95\) 89.1728 0.0963046
\(96\) 0 0
\(97\) −1229.92 −1.28742 −0.643711 0.765269i \(-0.722606\pi\)
−0.643711 + 0.765269i \(0.722606\pi\)
\(98\) 0 0
\(99\) −887.745 −0.901229
\(100\) 0 0
\(101\) −1077.89 −1.06192 −0.530962 0.847396i \(-0.678169\pi\)
−0.530962 + 0.847396i \(0.678169\pi\)
\(102\) 0 0
\(103\) −1896.77 −1.81451 −0.907255 0.420580i \(-0.861826\pi\)
−0.907255 + 0.420580i \(0.861826\pi\)
\(104\) 0 0
\(105\) 117.288 0.109011
\(106\) 0 0
\(107\) 480.564 0.434186 0.217093 0.976151i \(-0.430343\pi\)
0.217093 + 0.976151i \(0.430343\pi\)
\(108\) 0 0
\(109\) 124.669 0.109552 0.0547758 0.998499i \(-0.482556\pi\)
0.0547758 + 0.998499i \(0.482556\pi\)
\(110\) 0 0
\(111\) −2400.85 −2.05296
\(112\) 0 0
\(113\) −1778.75 −1.48080 −0.740400 0.672166i \(-0.765364\pi\)
−0.740400 + 0.672166i \(0.765364\pi\)
\(114\) 0 0
\(115\) 453.181 0.367473
\(116\) 0 0
\(117\) 1531.43 1.21009
\(118\) 0 0
\(119\) 47.7307 0.0367686
\(120\) 0 0
\(121\) 561.036 0.421515
\(122\) 0 0
\(123\) 305.191 0.223725
\(124\) 0 0
\(125\) −1069.95 −0.765591
\(126\) 0 0
\(127\) −1077.80 −0.753068 −0.376534 0.926403i \(-0.622884\pi\)
−0.376534 + 0.926403i \(0.622884\pi\)
\(128\) 0 0
\(129\) 1476.13 1.00749
\(130\) 0 0
\(131\) 1092.79 0.728835 0.364417 0.931236i \(-0.381268\pi\)
0.364417 + 0.931236i \(0.381268\pi\)
\(132\) 0 0
\(133\) −68.9604 −0.0449596
\(134\) 0 0
\(135\) 212.988 0.135786
\(136\) 0 0
\(137\) 3052.21 1.90341 0.951706 0.307010i \(-0.0993286\pi\)
0.951706 + 0.307010i \(0.0993286\pi\)
\(138\) 0 0
\(139\) 1428.73 0.871821 0.435911 0.899990i \(-0.356426\pi\)
0.435911 + 0.899990i \(0.356426\pi\)
\(140\) 0 0
\(141\) −2301.84 −1.37482
\(142\) 0 0
\(143\) −3263.92 −1.90869
\(144\) 0 0
\(145\) 190.351 0.109019
\(146\) 0 0
\(147\) 2271.00 1.27421
\(148\) 0 0
\(149\) 571.022 0.313959 0.156980 0.987602i \(-0.449824\pi\)
0.156980 + 0.987602i \(0.449824\pi\)
\(150\) 0 0
\(151\) 997.782 0.537737 0.268869 0.963177i \(-0.413350\pi\)
0.268869 + 0.963177i \(0.413350\pi\)
\(152\) 0 0
\(153\) −268.395 −0.141820
\(154\) 0 0
\(155\) −651.560 −0.337642
\(156\) 0 0
\(157\) −2171.23 −1.10371 −0.551857 0.833939i \(-0.686080\pi\)
−0.551857 + 0.833939i \(0.686080\pi\)
\(158\) 0 0
\(159\) −2142.21 −1.06848
\(160\) 0 0
\(161\) −350.461 −0.171554
\(162\) 0 0
\(163\) −305.488 −0.146795 −0.0733977 0.997303i \(-0.523384\pi\)
−0.0733977 + 0.997303i \(0.523384\pi\)
\(164\) 0 0
\(165\) 1405.64 0.663206
\(166\) 0 0
\(167\) 136.673 0.0633300 0.0316650 0.999499i \(-0.489919\pi\)
0.0316650 + 0.999499i \(0.489919\pi\)
\(168\) 0 0
\(169\) 3433.52 1.56282
\(170\) 0 0
\(171\) 387.772 0.173413
\(172\) 0 0
\(173\) −1508.96 −0.663144 −0.331572 0.943430i \(-0.607579\pi\)
−0.331572 + 0.943430i \(0.607579\pi\)
\(174\) 0 0
\(175\) 373.739 0.161440
\(176\) 0 0
\(177\) 3462.51 1.47038
\(178\) 0 0
\(179\) −770.548 −0.321751 −0.160875 0.986975i \(-0.551432\pi\)
−0.160875 + 0.986975i \(0.551432\pi\)
\(180\) 0 0
\(181\) 2292.62 0.941486 0.470743 0.882270i \(-0.343986\pi\)
0.470743 + 0.882270i \(0.343986\pi\)
\(182\) 0 0
\(183\) 376.766 0.152193
\(184\) 0 0
\(185\) 1636.49 0.650363
\(186\) 0 0
\(187\) 572.027 0.223694
\(188\) 0 0
\(189\) −164.711 −0.0633914
\(190\) 0 0
\(191\) 1670.39 0.632803 0.316402 0.948625i \(-0.397525\pi\)
0.316402 + 0.948625i \(0.397525\pi\)
\(192\) 0 0
\(193\) −663.849 −0.247590 −0.123795 0.992308i \(-0.539507\pi\)
−0.123795 + 0.992308i \(0.539507\pi\)
\(194\) 0 0
\(195\) −2424.84 −0.890495
\(196\) 0 0
\(197\) −2470.86 −0.893611 −0.446806 0.894631i \(-0.647438\pi\)
−0.446806 + 0.894631i \(0.647438\pi\)
\(198\) 0 0
\(199\) 4348.26 1.54894 0.774472 0.632608i \(-0.218016\pi\)
0.774472 + 0.632608i \(0.218016\pi\)
\(200\) 0 0
\(201\) 2154.86 0.756179
\(202\) 0 0
\(203\) −147.205 −0.0508954
\(204\) 0 0
\(205\) −208.027 −0.0708744
\(206\) 0 0
\(207\) 1970.68 0.661700
\(208\) 0 0
\(209\) −826.453 −0.273526
\(210\) 0 0
\(211\) 3910.79 1.27597 0.637986 0.770048i \(-0.279768\pi\)
0.637986 + 0.770048i \(0.279768\pi\)
\(212\) 0 0
\(213\) 4424.97 1.42344
\(214\) 0 0
\(215\) −1006.18 −0.319166
\(216\) 0 0
\(217\) 503.874 0.157628
\(218\) 0 0
\(219\) 5765.50 1.77898
\(220\) 0 0
\(221\) −986.792 −0.300357
\(222\) 0 0
\(223\) 70.6739 0.0212228 0.0106114 0.999944i \(-0.496622\pi\)
0.0106114 + 0.999944i \(0.496622\pi\)
\(224\) 0 0
\(225\) −2101.58 −0.622691
\(226\) 0 0
\(227\) 1132.37 0.331091 0.165546 0.986202i \(-0.447061\pi\)
0.165546 + 0.986202i \(0.447061\pi\)
\(228\) 0 0
\(229\) 2922.00 0.843194 0.421597 0.906783i \(-0.361470\pi\)
0.421597 + 0.906783i \(0.361470\pi\)
\(230\) 0 0
\(231\) −1087.03 −0.309616
\(232\) 0 0
\(233\) −926.052 −0.260376 −0.130188 0.991489i \(-0.541558\pi\)
−0.130188 + 0.991489i \(0.541558\pi\)
\(234\) 0 0
\(235\) 1569.00 0.435534
\(236\) 0 0
\(237\) 2407.11 0.659740
\(238\) 0 0
\(239\) −4376.11 −1.18438 −0.592190 0.805798i \(-0.701737\pi\)
−0.592190 + 0.805798i \(0.701737\pi\)
\(240\) 0 0
\(241\) −3964.69 −1.05970 −0.529850 0.848091i \(-0.677752\pi\)
−0.529850 + 0.848091i \(0.677752\pi\)
\(242\) 0 0
\(243\) 4720.37 1.24614
\(244\) 0 0
\(245\) −1547.98 −0.403660
\(246\) 0 0
\(247\) 1425.70 0.367267
\(248\) 0 0
\(249\) 3590.05 0.913697
\(250\) 0 0
\(251\) 219.854 0.0552870 0.0276435 0.999618i \(-0.491200\pi\)
0.0276435 + 0.999618i \(0.491200\pi\)
\(252\) 0 0
\(253\) −4200.08 −1.04370
\(254\) 0 0
\(255\) 424.973 0.104364
\(256\) 0 0
\(257\) −5582.44 −1.35495 −0.677477 0.735544i \(-0.736926\pi\)
−0.677477 + 0.735544i \(0.736926\pi\)
\(258\) 0 0
\(259\) −1265.55 −0.303620
\(260\) 0 0
\(261\) 827.751 0.196308
\(262\) 0 0
\(263\) 2971.76 0.696755 0.348378 0.937354i \(-0.386733\pi\)
0.348378 + 0.937354i \(0.386733\pi\)
\(264\) 0 0
\(265\) 1460.19 0.338486
\(266\) 0 0
\(267\) −436.601 −0.100073
\(268\) 0 0
\(269\) −5969.51 −1.35304 −0.676519 0.736425i \(-0.736513\pi\)
−0.676519 + 0.736425i \(0.736513\pi\)
\(270\) 0 0
\(271\) −6355.51 −1.42461 −0.712306 0.701869i \(-0.752349\pi\)
−0.712306 + 0.701869i \(0.752349\pi\)
\(272\) 0 0
\(273\) 1875.21 0.415726
\(274\) 0 0
\(275\) 4479.07 0.982174
\(276\) 0 0
\(277\) −8038.80 −1.74370 −0.871849 0.489774i \(-0.837079\pi\)
−0.871849 + 0.489774i \(0.837079\pi\)
\(278\) 0 0
\(279\) −2833.34 −0.607985
\(280\) 0 0
\(281\) −4478.81 −0.950831 −0.475416 0.879761i \(-0.657702\pi\)
−0.475416 + 0.879761i \(0.657702\pi\)
\(282\) 0 0
\(283\) −8812.45 −1.85105 −0.925523 0.378691i \(-0.876374\pi\)
−0.925523 + 0.378691i \(0.876374\pi\)
\(284\) 0 0
\(285\) −613.992 −0.127613
\(286\) 0 0
\(287\) 160.875 0.0330876
\(288\) 0 0
\(289\) −4740.06 −0.964799
\(290\) 0 0
\(291\) 8468.55 1.70596
\(292\) 0 0
\(293\) 942.016 0.187826 0.0939132 0.995580i \(-0.470062\pi\)
0.0939132 + 0.995580i \(0.470062\pi\)
\(294\) 0 0
\(295\) −2360.14 −0.465807
\(296\) 0 0
\(297\) −1973.98 −0.385662
\(298\) 0 0
\(299\) 7245.48 1.40140
\(300\) 0 0
\(301\) 778.110 0.149002
\(302\) 0 0
\(303\) 7421.74 1.40715
\(304\) 0 0
\(305\) −256.815 −0.0482136
\(306\) 0 0
\(307\) 5371.60 0.998610 0.499305 0.866426i \(-0.333589\pi\)
0.499305 + 0.866426i \(0.333589\pi\)
\(308\) 0 0
\(309\) 13060.1 2.40441
\(310\) 0 0
\(311\) 8180.21 1.49150 0.745751 0.666225i \(-0.232091\pi\)
0.745751 + 0.666225i \(0.232091\pi\)
\(312\) 0 0
\(313\) −621.565 −0.112246 −0.0561229 0.998424i \(-0.517874\pi\)
−0.0561229 + 0.998424i \(0.517874\pi\)
\(314\) 0 0
\(315\) −347.655 −0.0621845
\(316\) 0 0
\(317\) −2641.47 −0.468012 −0.234006 0.972235i \(-0.575184\pi\)
−0.234006 + 0.972235i \(0.575184\pi\)
\(318\) 0 0
\(319\) −1764.17 −0.309639
\(320\) 0 0
\(321\) −3308.89 −0.575340
\(322\) 0 0
\(323\) −249.865 −0.0430429
\(324\) 0 0
\(325\) −7726.76 −1.31878
\(326\) 0 0
\(327\) −858.398 −0.145167
\(328\) 0 0
\(329\) −1213.36 −0.203328
\(330\) 0 0
\(331\) −7450.56 −1.23722 −0.618610 0.785698i \(-0.712304\pi\)
−0.618610 + 0.785698i \(0.712304\pi\)
\(332\) 0 0
\(333\) 7116.36 1.17109
\(334\) 0 0
\(335\) −1468.81 −0.239552
\(336\) 0 0
\(337\) 121.084 0.0195722 0.00978612 0.999952i \(-0.496885\pi\)
0.00978612 + 0.999952i \(0.496885\pi\)
\(338\) 0 0
\(339\) 12247.4 1.96221
\(340\) 0 0
\(341\) 6038.66 0.958978
\(342\) 0 0
\(343\) 2442.02 0.384422
\(344\) 0 0
\(345\) −3120.35 −0.486938
\(346\) 0 0
\(347\) 9376.43 1.45059 0.725293 0.688441i \(-0.241704\pi\)
0.725293 + 0.688441i \(0.241704\pi\)
\(348\) 0 0
\(349\) −4225.67 −0.648124 −0.324062 0.946036i \(-0.605049\pi\)
−0.324062 + 0.946036i \(0.605049\pi\)
\(350\) 0 0
\(351\) 3405.27 0.517834
\(352\) 0 0
\(353\) 10833.3 1.63343 0.816714 0.577043i \(-0.195794\pi\)
0.816714 + 0.577043i \(0.195794\pi\)
\(354\) 0 0
\(355\) −3016.19 −0.450937
\(356\) 0 0
\(357\) −328.646 −0.0487221
\(358\) 0 0
\(359\) 1002.24 0.147343 0.0736715 0.997283i \(-0.476528\pi\)
0.0736715 + 0.997283i \(0.476528\pi\)
\(360\) 0 0
\(361\) 361.000 0.0526316
\(362\) 0 0
\(363\) −3862.97 −0.558550
\(364\) 0 0
\(365\) −3929.93 −0.563567
\(366\) 0 0
\(367\) 4868.42 0.692451 0.346225 0.938151i \(-0.387463\pi\)
0.346225 + 0.938151i \(0.387463\pi\)
\(368\) 0 0
\(369\) −904.617 −0.127622
\(370\) 0 0
\(371\) −1129.22 −0.158021
\(372\) 0 0
\(373\) 10466.0 1.45284 0.726421 0.687250i \(-0.241182\pi\)
0.726421 + 0.687250i \(0.241182\pi\)
\(374\) 0 0
\(375\) 7367.03 1.01449
\(376\) 0 0
\(377\) 3043.34 0.415756
\(378\) 0 0
\(379\) −11355.1 −1.53898 −0.769490 0.638659i \(-0.779490\pi\)
−0.769490 + 0.638659i \(0.779490\pi\)
\(380\) 0 0
\(381\) 7421.14 0.997891
\(382\) 0 0
\(383\) 8723.90 1.16389 0.581946 0.813228i \(-0.302292\pi\)
0.581946 + 0.813228i \(0.302292\pi\)
\(384\) 0 0
\(385\) 740.951 0.0980840
\(386\) 0 0
\(387\) −4375.41 −0.574714
\(388\) 0 0
\(389\) 2155.33 0.280924 0.140462 0.990086i \(-0.455141\pi\)
0.140462 + 0.990086i \(0.455141\pi\)
\(390\) 0 0
\(391\) −1269.83 −0.164240
\(392\) 0 0
\(393\) −7524.31 −0.965780
\(394\) 0 0
\(395\) −1640.75 −0.209001
\(396\) 0 0
\(397\) −4708.96 −0.595305 −0.297652 0.954674i \(-0.596204\pi\)
−0.297652 + 0.954674i \(0.596204\pi\)
\(398\) 0 0
\(399\) 474.821 0.0595759
\(400\) 0 0
\(401\) −1051.63 −0.130962 −0.0654812 0.997854i \(-0.520858\pi\)
−0.0654812 + 0.997854i \(0.520858\pi\)
\(402\) 0 0
\(403\) −10417.2 −1.28763
\(404\) 0 0
\(405\) −4052.74 −0.497240
\(406\) 0 0
\(407\) −15167.0 −1.84717
\(408\) 0 0
\(409\) 3346.05 0.404527 0.202263 0.979331i \(-0.435170\pi\)
0.202263 + 0.979331i \(0.435170\pi\)
\(410\) 0 0
\(411\) −21015.7 −2.52221
\(412\) 0 0
\(413\) 1825.18 0.217461
\(414\) 0 0
\(415\) −2447.08 −0.289452
\(416\) 0 0
\(417\) −9837.40 −1.15525
\(418\) 0 0
\(419\) 15743.9 1.83565 0.917827 0.396982i \(-0.129942\pi\)
0.917827 + 0.396982i \(0.129942\pi\)
\(420\) 0 0
\(421\) 3755.30 0.434732 0.217366 0.976090i \(-0.430254\pi\)
0.217366 + 0.976090i \(0.430254\pi\)
\(422\) 0 0
\(423\) 6822.89 0.784255
\(424\) 0 0
\(425\) 1354.17 0.154558
\(426\) 0 0
\(427\) 198.603 0.0225084
\(428\) 0 0
\(429\) 22473.4 2.52920
\(430\) 0 0
\(431\) −9670.09 −1.08072 −0.540362 0.841433i \(-0.681713\pi\)
−0.540362 + 0.841433i \(0.681713\pi\)
\(432\) 0 0
\(433\) −10656.7 −1.18274 −0.591370 0.806401i \(-0.701413\pi\)
−0.591370 + 0.806401i \(0.701413\pi\)
\(434\) 0 0
\(435\) −1310.65 −0.144461
\(436\) 0 0
\(437\) 1834.62 0.200828
\(438\) 0 0
\(439\) −5113.47 −0.555929 −0.277964 0.960591i \(-0.589660\pi\)
−0.277964 + 0.960591i \(0.589660\pi\)
\(440\) 0 0
\(441\) −6731.46 −0.726861
\(442\) 0 0
\(443\) −2552.54 −0.273758 −0.136879 0.990588i \(-0.543707\pi\)
−0.136879 + 0.990588i \(0.543707\pi\)
\(444\) 0 0
\(445\) 297.600 0.0317024
\(446\) 0 0
\(447\) −3931.73 −0.416028
\(448\) 0 0
\(449\) 4504.62 0.473466 0.236733 0.971575i \(-0.423923\pi\)
0.236733 + 0.971575i \(0.423923\pi\)
\(450\) 0 0
\(451\) 1928.00 0.201299
\(452\) 0 0
\(453\) −6870.15 −0.712556
\(454\) 0 0
\(455\) −1278.20 −0.131699
\(456\) 0 0
\(457\) 9278.63 0.949751 0.474875 0.880053i \(-0.342493\pi\)
0.474875 + 0.880053i \(0.342493\pi\)
\(458\) 0 0
\(459\) −596.800 −0.0606890
\(460\) 0 0
\(461\) 3861.47 0.390123 0.195062 0.980791i \(-0.437509\pi\)
0.195062 + 0.980791i \(0.437509\pi\)
\(462\) 0 0
\(463\) 154.016 0.0154595 0.00772975 0.999970i \(-0.497540\pi\)
0.00772975 + 0.999970i \(0.497540\pi\)
\(464\) 0 0
\(465\) 4486.27 0.447410
\(466\) 0 0
\(467\) −16010.0 −1.58641 −0.793205 0.608954i \(-0.791589\pi\)
−0.793205 + 0.608954i \(0.791589\pi\)
\(468\) 0 0
\(469\) 1135.88 0.111834
\(470\) 0 0
\(471\) 14949.8 1.46253
\(472\) 0 0
\(473\) 9325.24 0.906501
\(474\) 0 0
\(475\) −1956.49 −0.188989
\(476\) 0 0
\(477\) 6349.71 0.609504
\(478\) 0 0
\(479\) −10467.1 −0.998441 −0.499220 0.866475i \(-0.666380\pi\)
−0.499220 + 0.866475i \(0.666380\pi\)
\(480\) 0 0
\(481\) 26164.3 2.48023
\(482\) 0 0
\(483\) 2413.07 0.227326
\(484\) 0 0
\(485\) −5772.41 −0.540436
\(486\) 0 0
\(487\) −5868.52 −0.546054 −0.273027 0.962006i \(-0.588025\pi\)
−0.273027 + 0.962006i \(0.588025\pi\)
\(488\) 0 0
\(489\) 2103.41 0.194519
\(490\) 0 0
\(491\) 20259.1 1.86208 0.931041 0.364915i \(-0.118902\pi\)
0.931041 + 0.364915i \(0.118902\pi\)
\(492\) 0 0
\(493\) −533.369 −0.0487257
\(494\) 0 0
\(495\) −4166.46 −0.378320
\(496\) 0 0
\(497\) 2332.52 0.210519
\(498\) 0 0
\(499\) −6557.19 −0.588256 −0.294128 0.955766i \(-0.595029\pi\)
−0.294128 + 0.955766i \(0.595029\pi\)
\(500\) 0 0
\(501\) −941.054 −0.0839186
\(502\) 0 0
\(503\) 20542.5 1.82097 0.910484 0.413545i \(-0.135709\pi\)
0.910484 + 0.413545i \(0.135709\pi\)
\(504\) 0 0
\(505\) −5058.87 −0.445776
\(506\) 0 0
\(507\) −23641.2 −2.07090
\(508\) 0 0
\(509\) −5464.15 −0.475823 −0.237912 0.971287i \(-0.576463\pi\)
−0.237912 + 0.971287i \(0.576463\pi\)
\(510\) 0 0
\(511\) 3039.15 0.263100
\(512\) 0 0
\(513\) 862.245 0.0742087
\(514\) 0 0
\(515\) −8902.13 −0.761699
\(516\) 0 0
\(517\) −14541.5 −1.23701
\(518\) 0 0
\(519\) 10389.8 0.878733
\(520\) 0 0
\(521\) −12113.9 −1.01865 −0.509327 0.860573i \(-0.670106\pi\)
−0.509327 + 0.860573i \(0.670106\pi\)
\(522\) 0 0
\(523\) −12706.6 −1.06237 −0.531185 0.847256i \(-0.678253\pi\)
−0.531185 + 0.847256i \(0.678253\pi\)
\(524\) 0 0
\(525\) −2573.35 −0.213925
\(526\) 0 0
\(527\) 1825.69 0.150908
\(528\) 0 0
\(529\) −2843.34 −0.233693
\(530\) 0 0
\(531\) −10263.2 −0.838767
\(532\) 0 0
\(533\) −3325.95 −0.270287
\(534\) 0 0
\(535\) 2255.43 0.182263
\(536\) 0 0
\(537\) 5305.55 0.426352
\(538\) 0 0
\(539\) 14346.7 1.14648
\(540\) 0 0
\(541\) −20011.9 −1.59035 −0.795173 0.606383i \(-0.792620\pi\)
−0.795173 + 0.606383i \(0.792620\pi\)
\(542\) 0 0
\(543\) −15785.6 −1.24756
\(544\) 0 0
\(545\) 585.109 0.0459877
\(546\) 0 0
\(547\) −14732.5 −1.15158 −0.575790 0.817597i \(-0.695306\pi\)
−0.575790 + 0.817597i \(0.695306\pi\)
\(548\) 0 0
\(549\) −1116.77 −0.0868172
\(550\) 0 0
\(551\) 770.602 0.0595803
\(552\) 0 0
\(553\) 1268.85 0.0975715
\(554\) 0 0
\(555\) −11267.9 −0.861796
\(556\) 0 0
\(557\) −2202.91 −0.167577 −0.0837883 0.996484i \(-0.526702\pi\)
−0.0837883 + 0.996484i \(0.526702\pi\)
\(558\) 0 0
\(559\) −16086.8 −1.21717
\(560\) 0 0
\(561\) −3938.65 −0.296417
\(562\) 0 0
\(563\) −14661.2 −1.09751 −0.548755 0.835983i \(-0.684898\pi\)
−0.548755 + 0.835983i \(0.684898\pi\)
\(564\) 0 0
\(565\) −8348.20 −0.621613
\(566\) 0 0
\(567\) 3134.12 0.232135
\(568\) 0 0
\(569\) 2252.50 0.165957 0.0829785 0.996551i \(-0.473557\pi\)
0.0829785 + 0.996551i \(0.473557\pi\)
\(570\) 0 0
\(571\) −13749.4 −1.00770 −0.503848 0.863793i \(-0.668083\pi\)
−0.503848 + 0.863793i \(0.668083\pi\)
\(572\) 0 0
\(573\) −11501.4 −0.838528
\(574\) 0 0
\(575\) −9942.97 −0.721131
\(576\) 0 0
\(577\) −16355.8 −1.18007 −0.590037 0.807376i \(-0.700887\pi\)
−0.590037 + 0.807376i \(0.700887\pi\)
\(578\) 0 0
\(579\) 4570.88 0.328082
\(580\) 0 0
\(581\) 1892.41 0.135130
\(582\) 0 0
\(583\) −13533.0 −0.961375
\(584\) 0 0
\(585\) 7187.47 0.507975
\(586\) 0 0
\(587\) −12704.1 −0.893279 −0.446640 0.894714i \(-0.647379\pi\)
−0.446640 + 0.894714i \(0.647379\pi\)
\(588\) 0 0
\(589\) −2637.72 −0.184525
\(590\) 0 0
\(591\) 17012.9 1.18412
\(592\) 0 0
\(593\) 7442.67 0.515403 0.257701 0.966225i \(-0.417035\pi\)
0.257701 + 0.966225i \(0.417035\pi\)
\(594\) 0 0
\(595\) 224.015 0.0154348
\(596\) 0 0
\(597\) −29939.6 −2.05251
\(598\) 0 0
\(599\) 17522.7 1.19525 0.597627 0.801774i \(-0.296110\pi\)
0.597627 + 0.801774i \(0.296110\pi\)
\(600\) 0 0
\(601\) 14983.9 1.01698 0.508490 0.861068i \(-0.330204\pi\)
0.508490 + 0.861068i \(0.330204\pi\)
\(602\) 0 0
\(603\) −6387.21 −0.431355
\(604\) 0 0
\(605\) 2633.11 0.176944
\(606\) 0 0
\(607\) 18937.9 1.26634 0.633169 0.774013i \(-0.281754\pi\)
0.633169 + 0.774013i \(0.281754\pi\)
\(608\) 0 0
\(609\) 1013.57 0.0674415
\(610\) 0 0
\(611\) 25085.3 1.66095
\(612\) 0 0
\(613\) 7209.14 0.474999 0.237500 0.971388i \(-0.423672\pi\)
0.237500 + 0.971388i \(0.423672\pi\)
\(614\) 0 0
\(615\) 1432.36 0.0939158
\(616\) 0 0
\(617\) −25907.8 −1.69045 −0.845226 0.534408i \(-0.820534\pi\)
−0.845226 + 0.534408i \(0.820534\pi\)
\(618\) 0 0
\(619\) −26596.0 −1.72695 −0.863475 0.504391i \(-0.831717\pi\)
−0.863475 + 0.504391i \(0.831717\pi\)
\(620\) 0 0
\(621\) 4381.98 0.283161
\(622\) 0 0
\(623\) −230.144 −0.0148002
\(624\) 0 0
\(625\) 7850.03 0.502402
\(626\) 0 0
\(627\) 5690.48 0.362450
\(628\) 0 0
\(629\) −4585.50 −0.290677
\(630\) 0 0
\(631\) 231.639 0.0146140 0.00730699 0.999973i \(-0.497674\pi\)
0.00730699 + 0.999973i \(0.497674\pi\)
\(632\) 0 0
\(633\) −26927.5 −1.69079
\(634\) 0 0
\(635\) −5058.46 −0.316124
\(636\) 0 0
\(637\) −24749.1 −1.53940
\(638\) 0 0
\(639\) −13116.0 −0.811991
\(640\) 0 0
\(641\) 9194.84 0.566574 0.283287 0.959035i \(-0.408575\pi\)
0.283287 + 0.959035i \(0.408575\pi\)
\(642\) 0 0
\(643\) 12339.1 0.756775 0.378387 0.925647i \(-0.376479\pi\)
0.378387 + 0.925647i \(0.376479\pi\)
\(644\) 0 0
\(645\) 6927.95 0.422927
\(646\) 0 0
\(647\) 10852.4 0.659428 0.329714 0.944081i \(-0.393048\pi\)
0.329714 + 0.944081i \(0.393048\pi\)
\(648\) 0 0
\(649\) 21873.8 1.32299
\(650\) 0 0
\(651\) −3469.38 −0.208872
\(652\) 0 0
\(653\) 23310.7 1.39696 0.698482 0.715627i \(-0.253859\pi\)
0.698482 + 0.715627i \(0.253859\pi\)
\(654\) 0 0
\(655\) 5128.79 0.305952
\(656\) 0 0
\(657\) −17089.5 −1.01480
\(658\) 0 0
\(659\) −21318.0 −1.26014 −0.630069 0.776539i \(-0.716973\pi\)
−0.630069 + 0.776539i \(0.716973\pi\)
\(660\) 0 0
\(661\) 26645.4 1.56791 0.783954 0.620818i \(-0.213200\pi\)
0.783954 + 0.620818i \(0.213200\pi\)
\(662\) 0 0
\(663\) 6794.49 0.398003
\(664\) 0 0
\(665\) −323.652 −0.0188732
\(666\) 0 0
\(667\) 3916.24 0.227343
\(668\) 0 0
\(669\) −486.620 −0.0281223
\(670\) 0 0
\(671\) 2380.16 0.136937
\(672\) 0 0
\(673\) 12426.1 0.711728 0.355864 0.934538i \(-0.384187\pi\)
0.355864 + 0.934538i \(0.384187\pi\)
\(674\) 0 0
\(675\) −4673.05 −0.266468
\(676\) 0 0
\(677\) −28485.7 −1.61712 −0.808562 0.588411i \(-0.799754\pi\)
−0.808562 + 0.588411i \(0.799754\pi\)
\(678\) 0 0
\(679\) 4464.00 0.252301
\(680\) 0 0
\(681\) −7796.81 −0.438729
\(682\) 0 0
\(683\) 1899.14 0.106396 0.0531981 0.998584i \(-0.483059\pi\)
0.0531981 + 0.998584i \(0.483059\pi\)
\(684\) 0 0
\(685\) 14324.9 0.799018
\(686\) 0 0
\(687\) −20119.2 −1.11732
\(688\) 0 0
\(689\) 23345.6 1.29085
\(690\) 0 0
\(691\) 29568.6 1.62785 0.813923 0.580973i \(-0.197328\pi\)
0.813923 + 0.580973i \(0.197328\pi\)
\(692\) 0 0
\(693\) 3222.06 0.176618
\(694\) 0 0
\(695\) 6705.46 0.365975
\(696\) 0 0
\(697\) 582.899 0.0316770
\(698\) 0 0
\(699\) 6376.26 0.345025
\(700\) 0 0
\(701\) −6906.35 −0.372110 −0.186055 0.982539i \(-0.559570\pi\)
−0.186055 + 0.982539i \(0.559570\pi\)
\(702\) 0 0
\(703\) 6625.04 0.355431
\(704\) 0 0
\(705\) −10803.2 −0.577126
\(706\) 0 0
\(707\) 3912.20 0.208110
\(708\) 0 0
\(709\) 14410.0 0.763301 0.381650 0.924307i \(-0.375356\pi\)
0.381650 + 0.924307i \(0.375356\pi\)
\(710\) 0 0
\(711\) −7134.90 −0.376343
\(712\) 0 0
\(713\) −13405.1 −0.704100
\(714\) 0 0
\(715\) −15318.6 −0.801233
\(716\) 0 0
\(717\) 30131.4 1.56942
\(718\) 0 0
\(719\) 11170.9 0.579419 0.289710 0.957115i \(-0.406441\pi\)
0.289710 + 0.957115i \(0.406441\pi\)
\(720\) 0 0
\(721\) 6884.32 0.355597
\(722\) 0 0
\(723\) 27298.6 1.40421
\(724\) 0 0
\(725\) −4176.37 −0.213940
\(726\) 0 0
\(727\) −29676.9 −1.51397 −0.756985 0.653433i \(-0.773328\pi\)
−0.756985 + 0.653433i \(0.773328\pi\)
\(728\) 0 0
\(729\) −9186.85 −0.466740
\(730\) 0 0
\(731\) 2819.34 0.142650
\(732\) 0 0
\(733\) −252.315 −0.0127142 −0.00635708 0.999980i \(-0.502024\pi\)
−0.00635708 + 0.999980i \(0.502024\pi\)
\(734\) 0 0
\(735\) 10658.5 0.534890
\(736\) 0 0
\(737\) 13613.0 0.680380
\(738\) 0 0
\(739\) −1702.08 −0.0847252 −0.0423626 0.999102i \(-0.513488\pi\)
−0.0423626 + 0.999102i \(0.513488\pi\)
\(740\) 0 0
\(741\) −9816.54 −0.486666
\(742\) 0 0
\(743\) 11210.7 0.553539 0.276769 0.960936i \(-0.410736\pi\)
0.276769 + 0.960936i \(0.410736\pi\)
\(744\) 0 0
\(745\) 2679.98 0.131794
\(746\) 0 0
\(747\) −10641.3 −0.521210
\(748\) 0 0
\(749\) −1744.20 −0.0850892
\(750\) 0 0
\(751\) −37318.2 −1.81326 −0.906631 0.421925i \(-0.861354\pi\)
−0.906631 + 0.421925i \(0.861354\pi\)
\(752\) 0 0
\(753\) −1513.79 −0.0732609
\(754\) 0 0
\(755\) 4682.90 0.225732
\(756\) 0 0
\(757\) −13356.3 −0.641274 −0.320637 0.947202i \(-0.603897\pi\)
−0.320637 + 0.947202i \(0.603897\pi\)
\(758\) 0 0
\(759\) 28919.4 1.38301
\(760\) 0 0
\(761\) 37538.5 1.78814 0.894068 0.447932i \(-0.147839\pi\)
0.894068 + 0.447932i \(0.147839\pi\)
\(762\) 0 0
\(763\) −452.485 −0.0214693
\(764\) 0 0
\(765\) −1259.66 −0.0595335
\(766\) 0 0
\(767\) −37734.1 −1.77640
\(768\) 0 0
\(769\) 422.829 0.0198278 0.00991391 0.999951i \(-0.496844\pi\)
0.00991391 + 0.999951i \(0.496844\pi\)
\(770\) 0 0
\(771\) 38437.5 1.79545
\(772\) 0 0
\(773\) 23033.6 1.07175 0.535873 0.844298i \(-0.319982\pi\)
0.535873 + 0.844298i \(0.319982\pi\)
\(774\) 0 0
\(775\) 14295.5 0.662592
\(776\) 0 0
\(777\) 8713.88 0.402328
\(778\) 0 0
\(779\) −842.161 −0.0387337
\(780\) 0 0
\(781\) 27954.0 1.28076
\(782\) 0 0
\(783\) 1840.58 0.0840061
\(784\) 0 0
\(785\) −10190.2 −0.463319
\(786\) 0 0
\(787\) −37994.2 −1.72090 −0.860448 0.509538i \(-0.829816\pi\)
−0.860448 + 0.509538i \(0.829816\pi\)
\(788\) 0 0
\(789\) −20461.8 −0.923271
\(790\) 0 0
\(791\) 6455.95 0.290199
\(792\) 0 0
\(793\) −4105.96 −0.183868
\(794\) 0 0
\(795\) −10054.0 −0.448528
\(796\) 0 0
\(797\) −23765.7 −1.05624 −0.528122 0.849169i \(-0.677103\pi\)
−0.528122 + 0.849169i \(0.677103\pi\)
\(798\) 0 0
\(799\) −4396.39 −0.194660
\(800\) 0 0
\(801\) 1294.13 0.0570858
\(802\) 0 0
\(803\) 36422.6 1.60065
\(804\) 0 0
\(805\) −1644.82 −0.0720152
\(806\) 0 0
\(807\) 41102.6 1.79291
\(808\) 0 0
\(809\) −7019.58 −0.305062 −0.152531 0.988299i \(-0.548742\pi\)
−0.152531 + 0.988299i \(0.548742\pi\)
\(810\) 0 0
\(811\) −35061.1 −1.51808 −0.759039 0.651045i \(-0.774331\pi\)
−0.759039 + 0.651045i \(0.774331\pi\)
\(812\) 0 0
\(813\) 43760.4 1.88775
\(814\) 0 0
\(815\) −1433.75 −0.0616220
\(816\) 0 0
\(817\) −4073.32 −0.174428
\(818\) 0 0
\(819\) −5558.32 −0.237147
\(820\) 0 0
\(821\) −306.783 −0.0130412 −0.00652059 0.999979i \(-0.502076\pi\)
−0.00652059 + 0.999979i \(0.502076\pi\)
\(822\) 0 0
\(823\) 23239.1 0.984281 0.492140 0.870516i \(-0.336215\pi\)
0.492140 + 0.870516i \(0.336215\pi\)
\(824\) 0 0
\(825\) −30840.3 −1.30148
\(826\) 0 0
\(827\) 24374.9 1.02491 0.512455 0.858714i \(-0.328736\pi\)
0.512455 + 0.858714i \(0.328736\pi\)
\(828\) 0 0
\(829\) 27129.6 1.13661 0.568306 0.822817i \(-0.307599\pi\)
0.568306 + 0.822817i \(0.307599\pi\)
\(830\) 0 0
\(831\) 55350.5 2.31058
\(832\) 0 0
\(833\) 4337.48 0.180414
\(834\) 0 0
\(835\) 641.450 0.0265848
\(836\) 0 0
\(837\) −6300.18 −0.260174
\(838\) 0 0
\(839\) 4376.25 0.180077 0.0900386 0.995938i \(-0.471301\pi\)
0.0900386 + 0.995938i \(0.471301\pi\)
\(840\) 0 0
\(841\) −22744.0 −0.932554
\(842\) 0 0
\(843\) 30838.5 1.25995
\(844\) 0 0
\(845\) 16114.6 0.656044
\(846\) 0 0
\(847\) −2036.28 −0.0826061
\(848\) 0 0
\(849\) 60677.5 2.45282
\(850\) 0 0
\(851\) 33668.8 1.35623
\(852\) 0 0
\(853\) 46278.7 1.85762 0.928812 0.370550i \(-0.120831\pi\)
0.928812 + 0.370550i \(0.120831\pi\)
\(854\) 0 0
\(855\) 1819.93 0.0727958
\(856\) 0 0
\(857\) −5108.69 −0.203628 −0.101814 0.994803i \(-0.532465\pi\)
−0.101814 + 0.994803i \(0.532465\pi\)
\(858\) 0 0
\(859\) −12440.5 −0.494137 −0.247069 0.968998i \(-0.579467\pi\)
−0.247069 + 0.968998i \(0.579467\pi\)
\(860\) 0 0
\(861\) −1107.69 −0.0438444
\(862\) 0 0
\(863\) −33694.7 −1.32906 −0.664531 0.747260i \(-0.731369\pi\)
−0.664531 + 0.747260i \(0.731369\pi\)
\(864\) 0 0
\(865\) −7082.00 −0.278376
\(866\) 0 0
\(867\) 32637.3 1.27846
\(868\) 0 0
\(869\) 15206.5 0.593608
\(870\) 0 0
\(871\) −23483.5 −0.913555
\(872\) 0 0
\(873\) −25101.6 −0.973151
\(874\) 0 0
\(875\) 3883.36 0.150036
\(876\) 0 0
\(877\) 6266.75 0.241292 0.120646 0.992696i \(-0.461503\pi\)
0.120646 + 0.992696i \(0.461503\pi\)
\(878\) 0 0
\(879\) −6486.18 −0.248889
\(880\) 0 0
\(881\) −18875.2 −0.721818 −0.360909 0.932601i \(-0.617534\pi\)
−0.360909 + 0.932601i \(0.617534\pi\)
\(882\) 0 0
\(883\) 24369.5 0.928765 0.464383 0.885635i \(-0.346276\pi\)
0.464383 + 0.885635i \(0.346276\pi\)
\(884\) 0 0
\(885\) 16250.6 0.617241
\(886\) 0 0
\(887\) 46836.8 1.77297 0.886486 0.462754i \(-0.153139\pi\)
0.886486 + 0.462754i \(0.153139\pi\)
\(888\) 0 0
\(889\) 3911.88 0.147582
\(890\) 0 0
\(891\) 37560.8 1.41227
\(892\) 0 0
\(893\) 6351.82 0.238024
\(894\) 0 0
\(895\) −3616.41 −0.135065
\(896\) 0 0
\(897\) −49888.2 −1.85699
\(898\) 0 0
\(899\) −5630.57 −0.208888
\(900\) 0 0
\(901\) −4091.50 −0.151285
\(902\) 0 0
\(903\) −5357.62 −0.197442
\(904\) 0 0
\(905\) 10760.0 0.395219
\(906\) 0 0
\(907\) 19015.8 0.696150 0.348075 0.937467i \(-0.386836\pi\)
0.348075 + 0.937467i \(0.386836\pi\)
\(908\) 0 0
\(909\) −21998.8 −0.802699
\(910\) 0 0
\(911\) 30625.3 1.11379 0.556894 0.830583i \(-0.311993\pi\)
0.556894 + 0.830583i \(0.311993\pi\)
\(912\) 0 0
\(913\) 22679.6 0.822108
\(914\) 0 0
\(915\) 1768.28 0.0638879
\(916\) 0 0
\(917\) −3966.27 −0.142833
\(918\) 0 0
\(919\) −48642.7 −1.74600 −0.873001 0.487719i \(-0.837829\pi\)
−0.873001 + 0.487719i \(0.837829\pi\)
\(920\) 0 0
\(921\) −36985.7 −1.32326
\(922\) 0 0
\(923\) −48222.9 −1.71969
\(924\) 0 0
\(925\) −35905.2 −1.27628
\(926\) 0 0
\(927\) −38711.4 −1.37157
\(928\) 0 0
\(929\) 42194.6 1.49016 0.745080 0.666975i \(-0.232411\pi\)
0.745080 + 0.666975i \(0.232411\pi\)
\(930\) 0 0
\(931\) −6266.71 −0.220605
\(932\) 0 0
\(933\) −56324.2 −1.97639
\(934\) 0 0
\(935\) 2684.70 0.0939026
\(936\) 0 0
\(937\) −28438.1 −0.991498 −0.495749 0.868466i \(-0.665106\pi\)
−0.495749 + 0.868466i \(0.665106\pi\)
\(938\) 0 0
\(939\) 4279.74 0.148737
\(940\) 0 0
\(941\) 29883.2 1.03524 0.517622 0.855610i \(-0.326818\pi\)
0.517622 + 0.855610i \(0.326818\pi\)
\(942\) 0 0
\(943\) −4279.91 −0.147798
\(944\) 0 0
\(945\) −773.040 −0.0266106
\(946\) 0 0
\(947\) 46897.4 1.60925 0.804626 0.593782i \(-0.202366\pi\)
0.804626 + 0.593782i \(0.202366\pi\)
\(948\) 0 0
\(949\) −62831.9 −2.14922
\(950\) 0 0
\(951\) 18187.7 0.620163
\(952\) 0 0
\(953\) −20401.1 −0.693449 −0.346724 0.937967i \(-0.612706\pi\)
−0.346724 + 0.937967i \(0.612706\pi\)
\(954\) 0 0
\(955\) 7839.66 0.265639
\(956\) 0 0
\(957\) 12147.1 0.410302
\(958\) 0 0
\(959\) −11078.0 −0.373020
\(960\) 0 0
\(961\) −10517.9 −0.353057
\(962\) 0 0
\(963\) 9807.87 0.328197
\(964\) 0 0
\(965\) −3115.65 −0.103934
\(966\) 0 0
\(967\) 23598.5 0.784775 0.392388 0.919800i \(-0.371649\pi\)
0.392388 + 0.919800i \(0.371649\pi\)
\(968\) 0 0
\(969\) 1720.43 0.0570362
\(970\) 0 0
\(971\) 17149.1 0.566776 0.283388 0.959005i \(-0.408542\pi\)
0.283388 + 0.959005i \(0.408542\pi\)
\(972\) 0 0
\(973\) −5185.56 −0.170854
\(974\) 0 0
\(975\) 53202.0 1.74752
\(976\) 0 0
\(977\) −56653.2 −1.85517 −0.927583 0.373618i \(-0.878117\pi\)
−0.927583 + 0.373618i \(0.878117\pi\)
\(978\) 0 0
\(979\) −2758.15 −0.0900418
\(980\) 0 0
\(981\) 2544.38 0.0828091
\(982\) 0 0
\(983\) 7895.00 0.256166 0.128083 0.991763i \(-0.459118\pi\)
0.128083 + 0.991763i \(0.459118\pi\)
\(984\) 0 0
\(985\) −11596.5 −0.375122
\(986\) 0 0
\(987\) 8354.52 0.269430
\(988\) 0 0
\(989\) −20700.9 −0.665570
\(990\) 0 0
\(991\) −48458.9 −1.55333 −0.776664 0.629915i \(-0.783090\pi\)
−0.776664 + 0.629915i \(0.783090\pi\)
\(992\) 0 0
\(993\) 51300.3 1.63944
\(994\) 0 0
\(995\) 20407.7 0.650219
\(996\) 0 0
\(997\) 40325.2 1.28095 0.640477 0.767977i \(-0.278737\pi\)
0.640477 + 0.767977i \(0.278737\pi\)
\(998\) 0 0
\(999\) 15823.8 0.501145
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 608.4.a.j.1.2 7
4.3 odd 2 608.4.a.k.1.6 yes 7
8.3 odd 2 1216.4.a.bf.1.2 7
8.5 even 2 1216.4.a.bg.1.6 7
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
608.4.a.j.1.2 7 1.1 even 1 trivial
608.4.a.k.1.6 yes 7 4.3 odd 2
1216.4.a.bf.1.2 7 8.3 odd 2
1216.4.a.bg.1.6 7 8.5 even 2