Properties

Label 624.4.a.n.1.1
Level $624$
Weight $4$
Character 624.1
Self dual yes
Analytic conductor $36.817$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [624,4,Mod(1,624)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(624, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("624.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 624 = 2^{4} \cdot 3 \cdot 13 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 624.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: \(36.8171918436\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{55}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 55 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 312)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-7.41620\) of defining polynomial
Character \(\chi\) \(=\) 624.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+3.00000 q^{3} -16.8324 q^{5} +4.83240 q^{7} +9.00000 q^{9} +O(q^{10})\) \(q+3.00000 q^{3} -16.8324 q^{5} +4.83240 q^{7} +9.00000 q^{9} +39.6648 q^{11} -13.0000 q^{13} -50.4972 q^{15} -4.33521 q^{17} -28.8324 q^{19} +14.4972 q^{21} -0.670412 q^{23} +158.330 q^{25} +27.0000 q^{27} -6.00000 q^{29} -274.497 q^{31} +118.994 q^{33} -81.3408 q^{35} +84.3352 q^{37} -39.0000 q^{39} +209.156 q^{41} -364.324 q^{43} -151.492 q^{45} -239.341 q^{47} -319.648 q^{49} -13.0056 q^{51} -415.318 q^{53} -667.654 q^{55} -86.4972 q^{57} -78.0000 q^{59} -813.955 q^{61} +43.4916 q^{63} +218.821 q^{65} -858.452 q^{67} -2.01124 q^{69} +213.006 q^{71} +568.313 q^{73} +474.989 q^{75} +191.676 q^{77} -583.352 q^{79} +81.0000 q^{81} -162.648 q^{83} +72.9719 q^{85} -18.0000 q^{87} +1417.41 q^{89} -62.8212 q^{91} -823.492 q^{93} +485.318 q^{95} -798.949 q^{97} +356.983 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 6 q^{3} - 4 q^{5} - 20 q^{7} + 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 6 q^{3} - 4 q^{5} - 20 q^{7} + 18 q^{9} + 20 q^{11} - 26 q^{13} - 12 q^{15} - 68 q^{17} - 28 q^{19} - 60 q^{21} - 120 q^{23} + 198 q^{25} + 54 q^{27} - 12 q^{29} - 460 q^{31} + 60 q^{33} - 400 q^{35} + 228 q^{37} - 78 q^{39} + 92 q^{41} - 432 q^{43} - 36 q^{45} - 716 q^{47} - 46 q^{49} - 204 q^{51} - 356 q^{53} - 920 q^{55} - 84 q^{57} - 156 q^{59} - 204 q^{61} - 180 q^{63} + 52 q^{65} - 204 q^{67} - 360 q^{69} + 604 q^{71} + 484 q^{73} + 594 q^{75} + 680 q^{77} - 1760 q^{79} + 162 q^{81} + 268 q^{83} - 744 q^{85} - 36 q^{87} + 76 q^{89} + 260 q^{91} - 1380 q^{93} + 496 q^{95} + 4 q^{97} + 180 q^{99}+O(q^{100}) \) 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
\(3\) 3.00000 0.577350
\(4\) 0 0
\(5\) −16.8324 −1.50554 −0.752768 0.658286i \(-0.771282\pi\)
−0.752768 + 0.658286i \(0.771282\pi\)
\(6\) 0 0
\(7\) 4.83240 0.260925 0.130462 0.991453i \(-0.458354\pi\)
0.130462 + 0.991453i \(0.458354\pi\)
\(8\) 0 0
\(9\) 9.00000 0.333333
\(10\) 0 0
\(11\) 39.6648 1.08722 0.543608 0.839339i \(-0.317058\pi\)
0.543608 + 0.839339i \(0.317058\pi\)
\(12\) 0 0
\(13\) −13.0000 −0.277350
\(14\) 0 0
\(15\) −50.4972 −0.869221
\(16\) 0 0
\(17\) −4.33521 −0.0618495 −0.0309248 0.999522i \(-0.509845\pi\)
−0.0309248 + 0.999522i \(0.509845\pi\)
\(18\) 0 0
\(19\) −28.8324 −0.348137 −0.174069 0.984734i \(-0.555691\pi\)
−0.174069 + 0.984734i \(0.555691\pi\)
\(20\) 0 0
\(21\) 14.4972 0.150645
\(22\) 0 0
\(23\) −0.670412 −0.00607785 −0.00303893 0.999995i \(-0.500967\pi\)
−0.00303893 + 0.999995i \(0.500967\pi\)
\(24\) 0 0
\(25\) 158.330 1.26664
\(26\) 0 0
\(27\) 27.0000 0.192450
\(28\) 0 0
\(29\) −6.00000 −0.0384197 −0.0192099 0.999815i \(-0.506115\pi\)
−0.0192099 + 0.999815i \(0.506115\pi\)
\(30\) 0 0
\(31\) −274.497 −1.59036 −0.795180 0.606374i \(-0.792624\pi\)
−0.795180 + 0.606374i \(0.792624\pi\)
\(32\) 0 0
\(33\) 118.994 0.627705
\(34\) 0 0
\(35\) −81.3408 −0.392832
\(36\) 0 0
\(37\) 84.3352 0.374720 0.187360 0.982291i \(-0.440007\pi\)
0.187360 + 0.982291i \(0.440007\pi\)
\(38\) 0 0
\(39\) −39.0000 −0.160128
\(40\) 0 0
\(41\) 209.156 0.796701 0.398350 0.917233i \(-0.369583\pi\)
0.398350 + 0.917233i \(0.369583\pi\)
\(42\) 0 0
\(43\) −364.324 −1.29207 −0.646034 0.763309i \(-0.723573\pi\)
−0.646034 + 0.763309i \(0.723573\pi\)
\(44\) 0 0
\(45\) −151.492 −0.501845
\(46\) 0 0
\(47\) −239.341 −0.742797 −0.371398 0.928474i \(-0.621122\pi\)
−0.371398 + 0.928474i \(0.621122\pi\)
\(48\) 0 0
\(49\) −319.648 −0.931918
\(50\) 0 0
\(51\) −13.0056 −0.0357088
\(52\) 0 0
\(53\) −415.318 −1.07638 −0.538192 0.842822i \(-0.680893\pi\)
−0.538192 + 0.842822i \(0.680893\pi\)
\(54\) 0 0
\(55\) −667.654 −1.63684
\(56\) 0 0
\(57\) −86.4972 −0.200997
\(58\) 0 0
\(59\) −78.0000 −0.172114 −0.0860571 0.996290i \(-0.527427\pi\)
−0.0860571 + 0.996290i \(0.527427\pi\)
\(60\) 0 0
\(61\) −813.955 −1.70846 −0.854232 0.519893i \(-0.825972\pi\)
−0.854232 + 0.519893i \(0.825972\pi\)
\(62\) 0 0
\(63\) 43.4916 0.0869750
\(64\) 0 0
\(65\) 218.821 0.417560
\(66\) 0 0
\(67\) −858.452 −1.56532 −0.782661 0.622448i \(-0.786138\pi\)
−0.782661 + 0.622448i \(0.786138\pi\)
\(68\) 0 0
\(69\) −2.01124 −0.00350905
\(70\) 0 0
\(71\) 213.006 0.356044 0.178022 0.984027i \(-0.443030\pi\)
0.178022 + 0.984027i \(0.443030\pi\)
\(72\) 0 0
\(73\) 568.313 0.911178 0.455589 0.890190i \(-0.349429\pi\)
0.455589 + 0.890190i \(0.349429\pi\)
\(74\) 0 0
\(75\) 474.989 0.731293
\(76\) 0 0
\(77\) 191.676 0.283682
\(78\) 0 0
\(79\) −583.352 −0.830788 −0.415394 0.909642i \(-0.636356\pi\)
−0.415394 + 0.909642i \(0.636356\pi\)
\(80\) 0 0
\(81\) 81.0000 0.111111
\(82\) 0 0
\(83\) −162.648 −0.215096 −0.107548 0.994200i \(-0.534300\pi\)
−0.107548 + 0.994200i \(0.534300\pi\)
\(84\) 0 0
\(85\) 72.9719 0.0931166
\(86\) 0 0
\(87\) −18.0000 −0.0221816
\(88\) 0 0
\(89\) 1417.41 1.68815 0.844076 0.536224i \(-0.180150\pi\)
0.844076 + 0.536224i \(0.180150\pi\)
\(90\) 0 0
\(91\) −62.8212 −0.0723675
\(92\) 0 0
\(93\) −823.492 −0.918195
\(94\) 0 0
\(95\) 485.318 0.524133
\(96\) 0 0
\(97\) −798.949 −0.836299 −0.418150 0.908378i \(-0.637321\pi\)
−0.418150 + 0.908378i \(0.637321\pi\)
\(98\) 0 0
\(99\) 356.983 0.362406
\(100\) 0 0
\(101\) −1629.63 −1.60549 −0.802744 0.596323i \(-0.796628\pi\)
−0.802744 + 0.596323i \(0.796628\pi\)
\(102\) 0 0
\(103\) 384.927 0.368233 0.184116 0.982904i \(-0.441058\pi\)
0.184116 + 0.982904i \(0.441058\pi\)
\(104\) 0 0
\(105\) −244.022 −0.226801
\(106\) 0 0
\(107\) 336.045 0.303614 0.151807 0.988410i \(-0.451491\pi\)
0.151807 + 0.988410i \(0.451491\pi\)
\(108\) 0 0
\(109\) 777.307 0.683051 0.341525 0.939873i \(-0.389056\pi\)
0.341525 + 0.939873i \(0.389056\pi\)
\(110\) 0 0
\(111\) 253.006 0.216344
\(112\) 0 0
\(113\) −338.648 −0.281923 −0.140962 0.990015i \(-0.545019\pi\)
−0.140962 + 0.990015i \(0.545019\pi\)
\(114\) 0 0
\(115\) 11.2846 0.00915042
\(116\) 0 0
\(117\) −117.000 −0.0924500
\(118\) 0 0
\(119\) −20.9494 −0.0161381
\(120\) 0 0
\(121\) 242.296 0.182040
\(122\) 0 0
\(123\) 627.469 0.459975
\(124\) 0 0
\(125\) −561.017 −0.401431
\(126\) 0 0
\(127\) 1389.60 0.970920 0.485460 0.874259i \(-0.338652\pi\)
0.485460 + 0.874259i \(0.338652\pi\)
\(128\) 0 0
\(129\) −1092.97 −0.745975
\(130\) 0 0
\(131\) 701.966 0.468176 0.234088 0.972215i \(-0.424790\pi\)
0.234088 + 0.972215i \(0.424790\pi\)
\(132\) 0 0
\(133\) −139.330 −0.0908377
\(134\) 0 0
\(135\) −454.475 −0.289740
\(136\) 0 0
\(137\) −1044.21 −0.651187 −0.325593 0.945510i \(-0.605564\pi\)
−0.325593 + 0.945510i \(0.605564\pi\)
\(138\) 0 0
\(139\) −1935.33 −1.18095 −0.590477 0.807055i \(-0.701060\pi\)
−0.590477 + 0.807055i \(0.701060\pi\)
\(140\) 0 0
\(141\) −718.022 −0.428854
\(142\) 0 0
\(143\) −515.642 −0.301540
\(144\) 0 0
\(145\) 100.994 0.0578423
\(146\) 0 0
\(147\) −958.944 −0.538043
\(148\) 0 0
\(149\) −2252.37 −1.23840 −0.619200 0.785233i \(-0.712543\pi\)
−0.619200 + 0.785233i \(0.712543\pi\)
\(150\) 0 0
\(151\) −626.128 −0.337441 −0.168721 0.985664i \(-0.553964\pi\)
−0.168721 + 0.985664i \(0.553964\pi\)
\(152\) 0 0
\(153\) −39.0169 −0.0206165
\(154\) 0 0
\(155\) 4620.45 2.39434
\(156\) 0 0
\(157\) 1102.56 0.560470 0.280235 0.959931i \(-0.409588\pi\)
0.280235 + 0.959931i \(0.409588\pi\)
\(158\) 0 0
\(159\) −1245.96 −0.621451
\(160\) 0 0
\(161\) −3.23970 −0.00158586
\(162\) 0 0
\(163\) −695.145 −0.334037 −0.167018 0.985954i \(-0.553414\pi\)
−0.167018 + 0.985954i \(0.553414\pi\)
\(164\) 0 0
\(165\) −2002.96 −0.945032
\(166\) 0 0
\(167\) −1984.50 −0.919553 −0.459777 0.888035i \(-0.652070\pi\)
−0.459777 + 0.888035i \(0.652070\pi\)
\(168\) 0 0
\(169\) 169.000 0.0769231
\(170\) 0 0
\(171\) −259.492 −0.116046
\(172\) 0 0
\(173\) −1100.85 −0.483792 −0.241896 0.970302i \(-0.577769\pi\)
−0.241896 + 0.970302i \(0.577769\pi\)
\(174\) 0 0
\(175\) 765.111 0.330497
\(176\) 0 0
\(177\) −234.000 −0.0993702
\(178\) 0 0
\(179\) 3792.63 1.58365 0.791827 0.610745i \(-0.209130\pi\)
0.791827 + 0.610745i \(0.209130\pi\)
\(180\) 0 0
\(181\) 1926.47 0.791123 0.395561 0.918440i \(-0.370550\pi\)
0.395561 + 0.918440i \(0.370550\pi\)
\(182\) 0 0
\(183\) −2441.87 −0.986382
\(184\) 0 0
\(185\) −1419.56 −0.564153
\(186\) 0 0
\(187\) −171.955 −0.0672438
\(188\) 0 0
\(189\) 130.475 0.0502150
\(190\) 0 0
\(191\) −2048.51 −0.776048 −0.388024 0.921649i \(-0.626842\pi\)
−0.388024 + 0.921649i \(0.626842\pi\)
\(192\) 0 0
\(193\) 4134.67 1.54207 0.771036 0.636791i \(-0.219739\pi\)
0.771036 + 0.636791i \(0.219739\pi\)
\(194\) 0 0
\(195\) 656.463 0.241079
\(196\) 0 0
\(197\) 1856.98 0.671594 0.335797 0.941934i \(-0.390994\pi\)
0.335797 + 0.941934i \(0.390994\pi\)
\(198\) 0 0
\(199\) −5515.56 −1.96476 −0.982382 0.186885i \(-0.940161\pi\)
−0.982382 + 0.186885i \(0.940161\pi\)
\(200\) 0 0
\(201\) −2575.36 −0.903740
\(202\) 0 0
\(203\) −28.9944 −0.0100247
\(204\) 0 0
\(205\) −3520.60 −1.19946
\(206\) 0 0
\(207\) −6.03371 −0.00202595
\(208\) 0 0
\(209\) −1143.63 −0.378501
\(210\) 0 0
\(211\) −4035.73 −1.31674 −0.658368 0.752697i \(-0.728753\pi\)
−0.658368 + 0.752697i \(0.728753\pi\)
\(212\) 0 0
\(213\) 639.017 0.205562
\(214\) 0 0
\(215\) 6132.45 1.94525
\(216\) 0 0
\(217\) −1326.48 −0.414964
\(218\) 0 0
\(219\) 1704.94 0.526069
\(220\) 0 0
\(221\) 56.3577 0.0171540
\(222\) 0 0
\(223\) 4452.72 1.33711 0.668556 0.743661i \(-0.266912\pi\)
0.668556 + 0.743661i \(0.266912\pi\)
\(224\) 0 0
\(225\) 1424.97 0.422212
\(226\) 0 0
\(227\) −5482.68 −1.60308 −0.801538 0.597944i \(-0.795985\pi\)
−0.801538 + 0.597944i \(0.795985\pi\)
\(228\) 0 0
\(229\) −1406.74 −0.405938 −0.202969 0.979185i \(-0.565059\pi\)
−0.202969 + 0.979185i \(0.565059\pi\)
\(230\) 0 0
\(231\) 575.028 0.163784
\(232\) 0 0
\(233\) 6420.16 1.80514 0.902572 0.430540i \(-0.141677\pi\)
0.902572 + 0.430540i \(0.141677\pi\)
\(234\) 0 0
\(235\) 4028.68 1.11831
\(236\) 0 0
\(237\) −1750.06 −0.479656
\(238\) 0 0
\(239\) 4899.44 1.32602 0.663009 0.748611i \(-0.269279\pi\)
0.663009 + 0.748611i \(0.269279\pi\)
\(240\) 0 0
\(241\) 259.129 0.0692613 0.0346307 0.999400i \(-0.488975\pi\)
0.0346307 + 0.999400i \(0.488975\pi\)
\(242\) 0 0
\(243\) 243.000 0.0641500
\(244\) 0 0
\(245\) 5380.44 1.40304
\(246\) 0 0
\(247\) 374.821 0.0965559
\(248\) 0 0
\(249\) −487.944 −0.124185
\(250\) 0 0
\(251\) −884.760 −0.222492 −0.111246 0.993793i \(-0.535484\pi\)
−0.111246 + 0.993793i \(0.535484\pi\)
\(252\) 0 0
\(253\) −26.5918 −0.00660794
\(254\) 0 0
\(255\) 218.916 0.0537609
\(256\) 0 0
\(257\) 2913.01 0.707036 0.353518 0.935428i \(-0.384985\pi\)
0.353518 + 0.935428i \(0.384985\pi\)
\(258\) 0 0
\(259\) 407.541 0.0977737
\(260\) 0 0
\(261\) −54.0000 −0.0128066
\(262\) 0 0
\(263\) 127.330 0.0298535 0.0149268 0.999889i \(-0.495248\pi\)
0.0149268 + 0.999889i \(0.495248\pi\)
\(264\) 0 0
\(265\) 6990.80 1.62053
\(266\) 0 0
\(267\) 4252.24 0.974655
\(268\) 0 0
\(269\) 8177.51 1.85350 0.926750 0.375679i \(-0.122591\pi\)
0.926750 + 0.375679i \(0.122591\pi\)
\(270\) 0 0
\(271\) 4113.84 0.922132 0.461066 0.887366i \(-0.347467\pi\)
0.461066 + 0.887366i \(0.347467\pi\)
\(272\) 0 0
\(273\) −188.463 −0.0417814
\(274\) 0 0
\(275\) 6280.11 1.37711
\(276\) 0 0
\(277\) 587.543 0.127444 0.0637221 0.997968i \(-0.479703\pi\)
0.0637221 + 0.997968i \(0.479703\pi\)
\(278\) 0 0
\(279\) −2470.47 −0.530120
\(280\) 0 0
\(281\) 3925.13 0.833288 0.416644 0.909070i \(-0.363206\pi\)
0.416644 + 0.909070i \(0.363206\pi\)
\(282\) 0 0
\(283\) −6461.26 −1.35718 −0.678590 0.734517i \(-0.737409\pi\)
−0.678590 + 0.734517i \(0.737409\pi\)
\(284\) 0 0
\(285\) 1455.96 0.302608
\(286\) 0 0
\(287\) 1010.73 0.207879
\(288\) 0 0
\(289\) −4894.21 −0.996175
\(290\) 0 0
\(291\) −2396.85 −0.482838
\(292\) 0 0
\(293\) 9603.82 1.91489 0.957443 0.288623i \(-0.0931975\pi\)
0.957443 + 0.288623i \(0.0931975\pi\)
\(294\) 0 0
\(295\) 1312.93 0.259124
\(296\) 0 0
\(297\) 1070.95 0.209235
\(298\) 0 0
\(299\) 8.71536 0.00168569
\(300\) 0 0
\(301\) −1760.56 −0.337132
\(302\) 0 0
\(303\) −4888.89 −0.926929
\(304\) 0 0
\(305\) 13700.8 2.57215
\(306\) 0 0
\(307\) −5313.21 −0.987756 −0.493878 0.869531i \(-0.664421\pi\)
−0.493878 + 0.869531i \(0.664421\pi\)
\(308\) 0 0
\(309\) 1154.78 0.212599
\(310\) 0 0
\(311\) −6662.88 −1.21485 −0.607423 0.794378i \(-0.707797\pi\)
−0.607423 + 0.794378i \(0.707797\pi\)
\(312\) 0 0
\(313\) 1556.70 0.281119 0.140559 0.990072i \(-0.455110\pi\)
0.140559 + 0.990072i \(0.455110\pi\)
\(314\) 0 0
\(315\) −732.067 −0.130944
\(316\) 0 0
\(317\) −662.664 −0.117410 −0.0587049 0.998275i \(-0.518697\pi\)
−0.0587049 + 0.998275i \(0.518697\pi\)
\(318\) 0 0
\(319\) −237.989 −0.0417706
\(320\) 0 0
\(321\) 1008.13 0.175292
\(322\) 0 0
\(323\) 124.994 0.0215321
\(324\) 0 0
\(325\) −2058.28 −0.351302
\(326\) 0 0
\(327\) 2331.92 0.394359
\(328\) 0 0
\(329\) −1156.59 −0.193814
\(330\) 0 0
\(331\) −1378.83 −0.228965 −0.114482 0.993425i \(-0.536521\pi\)
−0.114482 + 0.993425i \(0.536521\pi\)
\(332\) 0 0
\(333\) 759.017 0.124907
\(334\) 0 0
\(335\) 14449.8 2.35665
\(336\) 0 0
\(337\) 8180.86 1.32237 0.661187 0.750222i \(-0.270053\pi\)
0.661187 + 0.750222i \(0.270053\pi\)
\(338\) 0 0
\(339\) −1015.94 −0.162768
\(340\) 0 0
\(341\) −10887.9 −1.72907
\(342\) 0 0
\(343\) −3202.18 −0.504086
\(344\) 0 0
\(345\) 33.8539 0.00528300
\(346\) 0 0
\(347\) −2712.49 −0.419637 −0.209819 0.977740i \(-0.567287\pi\)
−0.209819 + 0.977740i \(0.567287\pi\)
\(348\) 0 0
\(349\) −6789.20 −1.04131 −0.520656 0.853767i \(-0.674313\pi\)
−0.520656 + 0.853767i \(0.674313\pi\)
\(350\) 0 0
\(351\) −351.000 −0.0533761
\(352\) 0 0
\(353\) 228.018 0.0343801 0.0171900 0.999852i \(-0.494528\pi\)
0.0171900 + 0.999852i \(0.494528\pi\)
\(354\) 0 0
\(355\) −3585.40 −0.536037
\(356\) 0 0
\(357\) −62.8483 −0.00931733
\(358\) 0 0
\(359\) 507.161 0.0745597 0.0372799 0.999305i \(-0.488131\pi\)
0.0372799 + 0.999305i \(0.488131\pi\)
\(360\) 0 0
\(361\) −6027.69 −0.878801
\(362\) 0 0
\(363\) 726.888 0.105101
\(364\) 0 0
\(365\) −9566.07 −1.37181
\(366\) 0 0
\(367\) 9627.62 1.36937 0.684683 0.728841i \(-0.259941\pi\)
0.684683 + 0.728841i \(0.259941\pi\)
\(368\) 0 0
\(369\) 1882.41 0.265567
\(370\) 0 0
\(371\) −2006.98 −0.280855
\(372\) 0 0
\(373\) −7342.00 −1.01918 −0.509590 0.860417i \(-0.670203\pi\)
−0.509590 + 0.860417i \(0.670203\pi\)
\(374\) 0 0
\(375\) −1683.05 −0.231766
\(376\) 0 0
\(377\) 78.0000 0.0106557
\(378\) 0 0
\(379\) 11735.7 1.59055 0.795277 0.606246i \(-0.207325\pi\)
0.795277 + 0.606246i \(0.207325\pi\)
\(380\) 0 0
\(381\) 4168.79 0.560561
\(382\) 0 0
\(383\) −6739.67 −0.899168 −0.449584 0.893238i \(-0.648428\pi\)
−0.449584 + 0.893238i \(0.648428\pi\)
\(384\) 0 0
\(385\) −3226.37 −0.427093
\(386\) 0 0
\(387\) −3278.92 −0.430689
\(388\) 0 0
\(389\) −13234.7 −1.72500 −0.862498 0.506060i \(-0.831102\pi\)
−0.862498 + 0.506060i \(0.831102\pi\)
\(390\) 0 0
\(391\) 2.90637 0.000375912 0
\(392\) 0 0
\(393\) 2105.90 0.270302
\(394\) 0 0
\(395\) 9819.21 1.25078
\(396\) 0 0
\(397\) −14292.8 −1.80689 −0.903446 0.428701i \(-0.858971\pi\)
−0.903446 + 0.428701i \(0.858971\pi\)
\(398\) 0 0
\(399\) −417.989 −0.0524451
\(400\) 0 0
\(401\) 11779.5 1.46693 0.733466 0.679726i \(-0.237901\pi\)
0.733466 + 0.679726i \(0.237901\pi\)
\(402\) 0 0
\(403\) 3568.46 0.441086
\(404\) 0 0
\(405\) −1363.42 −0.167282
\(406\) 0 0
\(407\) 3345.14 0.407401
\(408\) 0 0
\(409\) 1039.22 0.125638 0.0628189 0.998025i \(-0.479991\pi\)
0.0628189 + 0.998025i \(0.479991\pi\)
\(410\) 0 0
\(411\) −3132.62 −0.375963
\(412\) 0 0
\(413\) −376.927 −0.0449089
\(414\) 0 0
\(415\) 2737.75 0.323834
\(416\) 0 0
\(417\) −5805.99 −0.681824
\(418\) 0 0
\(419\) −6085.74 −0.709565 −0.354783 0.934949i \(-0.615445\pi\)
−0.354783 + 0.934949i \(0.615445\pi\)
\(420\) 0 0
\(421\) 8309.91 0.961995 0.480998 0.876722i \(-0.340275\pi\)
0.480998 + 0.876722i \(0.340275\pi\)
\(422\) 0 0
\(423\) −2154.07 −0.247599
\(424\) 0 0
\(425\) −686.391 −0.0783409
\(426\) 0 0
\(427\) −3933.35 −0.445781
\(428\) 0 0
\(429\) −1546.93 −0.174094
\(430\) 0 0
\(431\) 15601.0 1.74356 0.871778 0.489902i \(-0.162967\pi\)
0.871778 + 0.489902i \(0.162967\pi\)
\(432\) 0 0
\(433\) 7887.07 0.875354 0.437677 0.899132i \(-0.355801\pi\)
0.437677 + 0.899132i \(0.355801\pi\)
\(434\) 0 0
\(435\) 302.983 0.0333952
\(436\) 0 0
\(437\) 19.3296 0.00211593
\(438\) 0 0
\(439\) −2105.13 −0.228866 −0.114433 0.993431i \(-0.536505\pi\)
−0.114433 + 0.993431i \(0.536505\pi\)
\(440\) 0 0
\(441\) −2876.83 −0.310639
\(442\) 0 0
\(443\) 1387.00 0.148754 0.0743772 0.997230i \(-0.476303\pi\)
0.0743772 + 0.997230i \(0.476303\pi\)
\(444\) 0 0
\(445\) −23858.5 −2.54157
\(446\) 0 0
\(447\) −6757.12 −0.714991
\(448\) 0 0
\(449\) 12563.5 1.32050 0.660252 0.751044i \(-0.270449\pi\)
0.660252 + 0.751044i \(0.270449\pi\)
\(450\) 0 0
\(451\) 8296.14 0.866187
\(452\) 0 0
\(453\) −1878.38 −0.194822
\(454\) 0 0
\(455\) 1057.43 0.108952
\(456\) 0 0
\(457\) −12331.4 −1.26222 −0.631112 0.775692i \(-0.717401\pi\)
−0.631112 + 0.775692i \(0.717401\pi\)
\(458\) 0 0
\(459\) −117.051 −0.0119029
\(460\) 0 0
\(461\) 1825.26 0.184406 0.0922029 0.995740i \(-0.470609\pi\)
0.0922029 + 0.995740i \(0.470609\pi\)
\(462\) 0 0
\(463\) −8172.00 −0.820270 −0.410135 0.912025i \(-0.634518\pi\)
−0.410135 + 0.912025i \(0.634518\pi\)
\(464\) 0 0
\(465\) 13861.3 1.38237
\(466\) 0 0
\(467\) −9959.22 −0.986847 −0.493424 0.869789i \(-0.664255\pi\)
−0.493424 + 0.869789i \(0.664255\pi\)
\(468\) 0 0
\(469\) −4148.38 −0.408432
\(470\) 0 0
\(471\) 3307.67 0.323587
\(472\) 0 0
\(473\) −14450.8 −1.40476
\(474\) 0 0
\(475\) −4565.02 −0.440963
\(476\) 0 0
\(477\) −3737.87 −0.358795
\(478\) 0 0
\(479\) 2793.32 0.266451 0.133226 0.991086i \(-0.457467\pi\)
0.133226 + 0.991086i \(0.457467\pi\)
\(480\) 0 0
\(481\) −1096.36 −0.103928
\(482\) 0 0
\(483\) −9.71909 −0.000915598 0
\(484\) 0 0
\(485\) 13448.2 1.25908
\(486\) 0 0
\(487\) 3870.17 0.360111 0.180056 0.983656i \(-0.442372\pi\)
0.180056 + 0.983656i \(0.442372\pi\)
\(488\) 0 0
\(489\) −2085.44 −0.192856
\(490\) 0 0
\(491\) 1888.65 0.173592 0.0867958 0.996226i \(-0.472337\pi\)
0.0867958 + 0.996226i \(0.472337\pi\)
\(492\) 0 0
\(493\) 26.0112 0.00237624
\(494\) 0 0
\(495\) −6008.88 −0.545614
\(496\) 0 0
\(497\) 1029.33 0.0929007
\(498\) 0 0
\(499\) 15879.0 1.42453 0.712266 0.701910i \(-0.247669\pi\)
0.712266 + 0.701910i \(0.247669\pi\)
\(500\) 0 0
\(501\) −5953.51 −0.530904
\(502\) 0 0
\(503\) −16338.7 −1.44833 −0.724164 0.689628i \(-0.757774\pi\)
−0.724164 + 0.689628i \(0.757774\pi\)
\(504\) 0 0
\(505\) 27430.6 2.41712
\(506\) 0 0
\(507\) 507.000 0.0444116
\(508\) 0 0
\(509\) −11875.7 −1.03415 −0.517076 0.855940i \(-0.672979\pi\)
−0.517076 + 0.855940i \(0.672979\pi\)
\(510\) 0 0
\(511\) 2746.31 0.237749
\(512\) 0 0
\(513\) −778.475 −0.0669990
\(514\) 0 0
\(515\) −6479.24 −0.554388
\(516\) 0 0
\(517\) −9493.40 −0.807581
\(518\) 0 0
\(519\) −3302.54 −0.279317
\(520\) 0 0
\(521\) 4971.92 0.418088 0.209044 0.977906i \(-0.432965\pi\)
0.209044 + 0.977906i \(0.432965\pi\)
\(522\) 0 0
\(523\) −1131.76 −0.0946238 −0.0473119 0.998880i \(-0.515065\pi\)
−0.0473119 + 0.998880i \(0.515065\pi\)
\(524\) 0 0
\(525\) 2295.33 0.190813
\(526\) 0 0
\(527\) 1190.00 0.0983630
\(528\) 0 0
\(529\) −12166.6 −0.999963
\(530\) 0 0
\(531\) −702.000 −0.0573714
\(532\) 0 0
\(533\) −2719.03 −0.220965
\(534\) 0 0
\(535\) −5656.44 −0.457102
\(536\) 0 0
\(537\) 11377.9 0.914323
\(538\) 0 0
\(539\) −12678.8 −1.01320
\(540\) 0 0
\(541\) −10514.6 −0.835593 −0.417797 0.908541i \(-0.637198\pi\)
−0.417797 + 0.908541i \(0.637198\pi\)
\(542\) 0 0
\(543\) 5779.40 0.456755
\(544\) 0 0
\(545\) −13083.9 −1.02836
\(546\) 0 0
\(547\) 21257.4 1.66161 0.830806 0.556562i \(-0.187880\pi\)
0.830806 + 0.556562i \(0.187880\pi\)
\(548\) 0 0
\(549\) −7325.60 −0.569488
\(550\) 0 0
\(551\) 172.994 0.0133753
\(552\) 0 0
\(553\) −2818.99 −0.216773
\(554\) 0 0
\(555\) −4258.69 −0.325714
\(556\) 0 0
\(557\) 2350.31 0.178789 0.0893947 0.995996i \(-0.471507\pi\)
0.0893947 + 0.995996i \(0.471507\pi\)
\(558\) 0 0
\(559\) 4736.21 0.358355
\(560\) 0 0
\(561\) −515.865 −0.0388233
\(562\) 0 0
\(563\) −19787.9 −1.48128 −0.740639 0.671904i \(-0.765477\pi\)
−0.740639 + 0.671904i \(0.765477\pi\)
\(564\) 0 0
\(565\) 5700.26 0.424445
\(566\) 0 0
\(567\) 391.424 0.0289917
\(568\) 0 0
\(569\) −20580.0 −1.51627 −0.758136 0.652097i \(-0.773890\pi\)
−0.758136 + 0.652097i \(0.773890\pi\)
\(570\) 0 0
\(571\) 13301.9 0.974897 0.487449 0.873152i \(-0.337928\pi\)
0.487449 + 0.873152i \(0.337928\pi\)
\(572\) 0 0
\(573\) −6145.54 −0.448052
\(574\) 0 0
\(575\) −106.146 −0.00769843
\(576\) 0 0
\(577\) −6423.54 −0.463458 −0.231729 0.972780i \(-0.574438\pi\)
−0.231729 + 0.972780i \(0.574438\pi\)
\(578\) 0 0
\(579\) 12404.0 0.890316
\(580\) 0 0
\(581\) −785.979 −0.0561238
\(582\) 0 0
\(583\) −16473.5 −1.17026
\(584\) 0 0
\(585\) 1969.39 0.139187
\(586\) 0 0
\(587\) −18600.5 −1.30788 −0.653941 0.756546i \(-0.726885\pi\)
−0.653941 + 0.756546i \(0.726885\pi\)
\(588\) 0 0
\(589\) 7914.41 0.553663
\(590\) 0 0
\(591\) 5570.93 0.387745
\(592\) 0 0
\(593\) 11942.3 0.827001 0.413501 0.910504i \(-0.364306\pi\)
0.413501 + 0.910504i \(0.364306\pi\)
\(594\) 0 0
\(595\) 352.629 0.0242965
\(596\) 0 0
\(597\) −16546.7 −1.13436
\(598\) 0 0
\(599\) −746.614 −0.0509279 −0.0254640 0.999676i \(-0.508106\pi\)
−0.0254640 + 0.999676i \(0.508106\pi\)
\(600\) 0 0
\(601\) 11287.4 0.766096 0.383048 0.923728i \(-0.374874\pi\)
0.383048 + 0.923728i \(0.374874\pi\)
\(602\) 0 0
\(603\) −7726.07 −0.521774
\(604\) 0 0
\(605\) −4078.42 −0.274068
\(606\) 0 0
\(607\) −7090.08 −0.474098 −0.237049 0.971498i \(-0.576180\pi\)
−0.237049 + 0.971498i \(0.576180\pi\)
\(608\) 0 0
\(609\) −86.9831 −0.00578774
\(610\) 0 0
\(611\) 3111.43 0.206015
\(612\) 0 0
\(613\) 21842.3 1.43915 0.719577 0.694413i \(-0.244336\pi\)
0.719577 + 0.694413i \(0.244336\pi\)
\(614\) 0 0
\(615\) −10561.8 −0.692509
\(616\) 0 0
\(617\) 7845.18 0.511888 0.255944 0.966692i \(-0.417614\pi\)
0.255944 + 0.966692i \(0.417614\pi\)
\(618\) 0 0
\(619\) −29667.4 −1.92639 −0.963193 0.268809i \(-0.913370\pi\)
−0.963193 + 0.268809i \(0.913370\pi\)
\(620\) 0 0
\(621\) −18.1011 −0.00116968
\(622\) 0 0
\(623\) 6849.50 0.440481
\(624\) 0 0
\(625\) −10347.9 −0.662268
\(626\) 0 0
\(627\) −3430.89 −0.218527
\(628\) 0 0
\(629\) −365.610 −0.0231762
\(630\) 0 0
\(631\) −24321.2 −1.53441 −0.767204 0.641404i \(-0.778352\pi\)
−0.767204 + 0.641404i \(0.778352\pi\)
\(632\) 0 0
\(633\) −12107.2 −0.760217
\(634\) 0 0
\(635\) −23390.3 −1.46175
\(636\) 0 0
\(637\) 4155.42 0.258468
\(638\) 0 0
\(639\) 1917.05 0.118681
\(640\) 0 0
\(641\) 16751.4 1.03220 0.516099 0.856529i \(-0.327384\pi\)
0.516099 + 0.856529i \(0.327384\pi\)
\(642\) 0 0
\(643\) 27260.5 1.67193 0.835964 0.548784i \(-0.184909\pi\)
0.835964 + 0.548784i \(0.184909\pi\)
\(644\) 0 0
\(645\) 18397.3 1.12309
\(646\) 0 0
\(647\) 9550.09 0.580298 0.290149 0.956981i \(-0.406295\pi\)
0.290149 + 0.956981i \(0.406295\pi\)
\(648\) 0 0
\(649\) −3093.85 −0.187125
\(650\) 0 0
\(651\) −3979.44 −0.239580
\(652\) 0 0
\(653\) −1093.89 −0.0655548 −0.0327774 0.999463i \(-0.510435\pi\)
−0.0327774 + 0.999463i \(0.510435\pi\)
\(654\) 0 0
\(655\) −11815.8 −0.704856
\(656\) 0 0
\(657\) 5114.81 0.303726
\(658\) 0 0
\(659\) 9129.47 0.539657 0.269828 0.962908i \(-0.413033\pi\)
0.269828 + 0.962908i \(0.413033\pi\)
\(660\) 0 0
\(661\) 10726.5 0.631181 0.315591 0.948895i \(-0.397797\pi\)
0.315591 + 0.948895i \(0.397797\pi\)
\(662\) 0 0
\(663\) 169.073 0.00990385
\(664\) 0 0
\(665\) 2345.25 0.136759
\(666\) 0 0
\(667\) 4.02247 0.000233509 0
\(668\) 0 0
\(669\) 13358.2 0.771983
\(670\) 0 0
\(671\) −32285.4 −1.85747
\(672\) 0 0
\(673\) −6797.04 −0.389311 −0.194656 0.980872i \(-0.562359\pi\)
−0.194656 + 0.980872i \(0.562359\pi\)
\(674\) 0 0
\(675\) 4274.90 0.243764
\(676\) 0 0
\(677\) −22167.0 −1.25841 −0.629207 0.777238i \(-0.716620\pi\)
−0.629207 + 0.777238i \(0.716620\pi\)
\(678\) 0 0
\(679\) −3860.84 −0.218211
\(680\) 0 0
\(681\) −16448.0 −0.925536
\(682\) 0 0
\(683\) −16756.8 −0.938773 −0.469387 0.882993i \(-0.655525\pi\)
−0.469387 + 0.882993i \(0.655525\pi\)
\(684\) 0 0
\(685\) 17576.5 0.980385
\(686\) 0 0
\(687\) −4220.21 −0.234369
\(688\) 0 0
\(689\) 5399.14 0.298535
\(690\) 0 0
\(691\) 1807.58 0.0995132 0.0497566 0.998761i \(-0.484155\pi\)
0.0497566 + 0.998761i \(0.484155\pi\)
\(692\) 0 0
\(693\) 1725.08 0.0945607
\(694\) 0 0
\(695\) 32576.2 1.77797
\(696\) 0 0
\(697\) −906.736 −0.0492756
\(698\) 0 0
\(699\) 19260.5 1.04220
\(700\) 0 0
\(701\) −26699.0 −1.43853 −0.719264 0.694737i \(-0.755521\pi\)
−0.719264 + 0.694737i \(0.755521\pi\)
\(702\) 0 0
\(703\) −2431.59 −0.130454
\(704\) 0 0
\(705\) 12086.0 0.645655
\(706\) 0 0
\(707\) −7875.02 −0.418912
\(708\) 0 0
\(709\) 19226.1 1.01841 0.509204 0.860646i \(-0.329940\pi\)
0.509204 + 0.860646i \(0.329940\pi\)
\(710\) 0 0
\(711\) −5250.17 −0.276929
\(712\) 0 0
\(713\) 184.026 0.00966597
\(714\) 0 0
\(715\) 8679.50 0.453979
\(716\) 0 0
\(717\) 14698.3 0.765577
\(718\) 0 0
\(719\) 14372.2 0.745470 0.372735 0.927938i \(-0.378420\pi\)
0.372735 + 0.927938i \(0.378420\pi\)
\(720\) 0 0
\(721\) 1860.12 0.0960811
\(722\) 0 0
\(723\) 777.388 0.0399880
\(724\) 0 0
\(725\) −949.978 −0.0486638
\(726\) 0 0
\(727\) −26347.2 −1.34410 −0.672052 0.740504i \(-0.734587\pi\)
−0.672052 + 0.740504i \(0.734587\pi\)
\(728\) 0 0
\(729\) 729.000 0.0370370
\(730\) 0 0
\(731\) 1579.42 0.0799137
\(732\) 0 0
\(733\) 27597.8 1.39065 0.695326 0.718694i \(-0.255260\pi\)
0.695326 + 0.718694i \(0.255260\pi\)
\(734\) 0 0
\(735\) 16141.3 0.810043
\(736\) 0 0
\(737\) −34050.3 −1.70185
\(738\) 0 0
\(739\) −9546.08 −0.475180 −0.237590 0.971365i \(-0.576358\pi\)
−0.237590 + 0.971365i \(0.576358\pi\)
\(740\) 0 0
\(741\) 1124.46 0.0557466
\(742\) 0 0
\(743\) 32976.0 1.62823 0.814114 0.580705i \(-0.197223\pi\)
0.814114 + 0.580705i \(0.197223\pi\)
\(744\) 0 0
\(745\) 37912.8 1.86446
\(746\) 0 0
\(747\) −1463.83 −0.0716985
\(748\) 0 0
\(749\) 1623.90 0.0792204
\(750\) 0 0
\(751\) 9781.24 0.475263 0.237631 0.971355i \(-0.423629\pi\)
0.237631 + 0.971355i \(0.423629\pi\)
\(752\) 0 0
\(753\) −2654.28 −0.128456
\(754\) 0 0
\(755\) 10539.2 0.508029
\(756\) 0 0
\(757\) 4414.55 0.211955 0.105977 0.994369i \(-0.466203\pi\)
0.105977 + 0.994369i \(0.466203\pi\)
\(758\) 0 0
\(759\) −79.7753 −0.00381510
\(760\) 0 0
\(761\) −33676.2 −1.60415 −0.802077 0.597220i \(-0.796272\pi\)
−0.802077 + 0.597220i \(0.796272\pi\)
\(762\) 0 0
\(763\) 3756.26 0.178225
\(764\) 0 0
\(765\) 656.747 0.0310389
\(766\) 0 0
\(767\) 1014.00 0.0477359
\(768\) 0 0
\(769\) −34795.0 −1.63165 −0.815827 0.578297i \(-0.803718\pi\)
−0.815827 + 0.578297i \(0.803718\pi\)
\(770\) 0 0
\(771\) 8739.02 0.408208
\(772\) 0 0
\(773\) 28152.1 1.30991 0.654954 0.755668i \(-0.272688\pi\)
0.654954 + 0.755668i \(0.272688\pi\)
\(774\) 0 0
\(775\) −43461.0 −2.01441
\(776\) 0 0
\(777\) 1222.62 0.0564496
\(778\) 0 0
\(779\) −6030.48 −0.277361
\(780\) 0 0
\(781\) 8448.82 0.387097
\(782\) 0 0
\(783\) −162.000 −0.00739388
\(784\) 0 0
\(785\) −18558.7 −0.843807
\(786\) 0 0
\(787\) −2659.15 −0.120443 −0.0602215 0.998185i \(-0.519181\pi\)
−0.0602215 + 0.998185i \(0.519181\pi\)
\(788\) 0 0
\(789\) 381.989 0.0172359
\(790\) 0 0
\(791\) −1636.48 −0.0735608
\(792\) 0 0
\(793\) 10581.4 0.473842
\(794\) 0 0
\(795\) 20972.4 0.935616
\(796\) 0 0
\(797\) 33193.5 1.47525 0.737625 0.675210i \(-0.235947\pi\)
0.737625 + 0.675210i \(0.235947\pi\)
\(798\) 0 0
\(799\) 1037.59 0.0459416
\(800\) 0 0
\(801\) 12756.7 0.562717
\(802\) 0 0
\(803\) 22542.0 0.990648
\(804\) 0 0
\(805\) 54.5319 0.00238757
\(806\) 0 0
\(807\) 24532.5 1.07012
\(808\) 0 0
\(809\) 9198.73 0.399765 0.199883 0.979820i \(-0.435944\pi\)
0.199883 + 0.979820i \(0.435944\pi\)
\(810\) 0 0
\(811\) −2435.11 −0.105436 −0.0527179 0.998609i \(-0.516788\pi\)
−0.0527179 + 0.998609i \(0.516788\pi\)
\(812\) 0 0
\(813\) 12341.5 0.532393
\(814\) 0 0
\(815\) 11701.0 0.502904
\(816\) 0 0
\(817\) 10504.3 0.449817
\(818\) 0 0
\(819\) −565.390 −0.0241225
\(820\) 0 0
\(821\) 17767.2 0.755275 0.377638 0.925954i \(-0.376737\pi\)
0.377638 + 0.925954i \(0.376737\pi\)
\(822\) 0 0
\(823\) 15112.0 0.640060 0.320030 0.947407i \(-0.396307\pi\)
0.320030 + 0.947407i \(0.396307\pi\)
\(824\) 0 0
\(825\) 18840.3 0.795074
\(826\) 0 0
\(827\) −32606.2 −1.37102 −0.685508 0.728065i \(-0.740420\pi\)
−0.685508 + 0.728065i \(0.740420\pi\)
\(828\) 0 0
\(829\) −26701.9 −1.11869 −0.559347 0.828934i \(-0.688948\pi\)
−0.559347 + 0.828934i \(0.688948\pi\)
\(830\) 0 0
\(831\) 1762.63 0.0735799
\(832\) 0 0
\(833\) 1385.74 0.0576387
\(834\) 0 0
\(835\) 33403.9 1.38442
\(836\) 0 0
\(837\) −7411.42 −0.306065
\(838\) 0 0
\(839\) 20747.5 0.853736 0.426868 0.904314i \(-0.359617\pi\)
0.426868 + 0.904314i \(0.359617\pi\)
\(840\) 0 0
\(841\) −24353.0 −0.998524
\(842\) 0 0
\(843\) 11775.4 0.481099
\(844\) 0 0
\(845\) −2844.68 −0.115810
\(846\) 0 0
\(847\) 1170.87 0.0474989
\(848\) 0 0
\(849\) −19383.8 −0.783568
\(850\) 0 0
\(851\) −56.5393 −0.00227749
\(852\) 0 0
\(853\) 4041.50 0.162225 0.0811127 0.996705i \(-0.474153\pi\)
0.0811127 + 0.996705i \(0.474153\pi\)
\(854\) 0 0
\(855\) 4367.87 0.174711
\(856\) 0 0
\(857\) −32067.4 −1.27818 −0.639091 0.769131i \(-0.720689\pi\)
−0.639091 + 0.769131i \(0.720689\pi\)
\(858\) 0 0
\(859\) 3739.80 0.148545 0.0742726 0.997238i \(-0.476336\pi\)
0.0742726 + 0.997238i \(0.476336\pi\)
\(860\) 0 0
\(861\) 3032.18 0.120019
\(862\) 0 0
\(863\) −38258.1 −1.50906 −0.754531 0.656265i \(-0.772135\pi\)
−0.754531 + 0.656265i \(0.772135\pi\)
\(864\) 0 0
\(865\) 18529.9 0.728365
\(866\) 0 0
\(867\) −14682.6 −0.575142
\(868\) 0 0
\(869\) −23138.5 −0.903246
\(870\) 0 0
\(871\) 11159.9 0.434142
\(872\) 0 0
\(873\) −7190.54 −0.278766
\(874\) 0 0
\(875\) −2711.06 −0.104743
\(876\) 0 0
\(877\) −21528.7 −0.828932 −0.414466 0.910065i \(-0.636032\pi\)
−0.414466 + 0.910065i \(0.636032\pi\)
\(878\) 0 0
\(879\) 28811.5 1.10556
\(880\) 0 0
\(881\) 25995.3 0.994101 0.497051 0.867722i \(-0.334416\pi\)
0.497051 + 0.867722i \(0.334416\pi\)
\(882\) 0 0
\(883\) −9586.18 −0.365346 −0.182673 0.983174i \(-0.558475\pi\)
−0.182673 + 0.983174i \(0.558475\pi\)
\(884\) 0 0
\(885\) 3938.78 0.149605
\(886\) 0 0
\(887\) −20879.1 −0.790361 −0.395180 0.918604i \(-0.629318\pi\)
−0.395180 + 0.918604i \(0.629318\pi\)
\(888\) 0 0
\(889\) 6715.09 0.253337
\(890\) 0 0
\(891\) 3212.85 0.120802
\(892\) 0 0
\(893\) 6900.77 0.258595
\(894\) 0 0
\(895\) −63839.0 −2.38425
\(896\) 0 0
\(897\) 26.1461 0.000973235 0
\(898\) 0 0
\(899\) 1646.98 0.0611012
\(900\) 0 0
\(901\) 1800.49 0.0665739
\(902\) 0 0
\(903\) −5281.67 −0.194644
\(904\) 0 0
\(905\) −32427.1 −1.19106
\(906\) 0 0
\(907\) −1094.00 −0.0400503 −0.0200251 0.999799i \(-0.506375\pi\)
−0.0200251 + 0.999799i \(0.506375\pi\)
\(908\) 0 0
\(909\) −14666.7 −0.535163
\(910\) 0 0
\(911\) 29816.7 1.08438 0.542190 0.840256i \(-0.317595\pi\)
0.542190 + 0.840256i \(0.317595\pi\)
\(912\) 0 0
\(913\) −6451.40 −0.233856
\(914\) 0 0
\(915\) 41102.4 1.48503
\(916\) 0 0
\(917\) 3392.18 0.122159
\(918\) 0 0
\(919\) 40744.8 1.46251 0.731255 0.682104i \(-0.238935\pi\)
0.731255 + 0.682104i \(0.238935\pi\)
\(920\) 0 0
\(921\) −15939.6 −0.570281
\(922\) 0 0
\(923\) −2769.07 −0.0987488
\(924\) 0 0
\(925\) 13352.8 0.474634
\(926\) 0 0
\(927\) 3464.34 0.122744
\(928\) 0 0
\(929\) −40223.7 −1.42056 −0.710278 0.703921i \(-0.751431\pi\)
−0.710278 + 0.703921i \(0.751431\pi\)
\(930\) 0 0
\(931\) 9216.22 0.324435
\(932\) 0 0
\(933\) −19988.6 −0.701392
\(934\) 0 0
\(935\) 2894.42 0.101238
\(936\) 0 0
\(937\) −511.273 −0.0178256 −0.00891279 0.999960i \(-0.502837\pi\)
−0.00891279 + 0.999960i \(0.502837\pi\)
\(938\) 0 0
\(939\) 4670.11 0.162304
\(940\) 0 0
\(941\) 24696.7 0.855569 0.427785 0.903881i \(-0.359294\pi\)
0.427785 + 0.903881i \(0.359294\pi\)
\(942\) 0 0
\(943\) −140.221 −0.00484223
\(944\) 0 0
\(945\) −2196.20 −0.0756005
\(946\) 0 0
\(947\) 11177.7 0.383555 0.191778 0.981438i \(-0.438575\pi\)
0.191778 + 0.981438i \(0.438575\pi\)
\(948\) 0 0
\(949\) −7388.07 −0.252715
\(950\) 0 0
\(951\) −1987.99 −0.0677866
\(952\) 0 0
\(953\) 31891.5 1.08402 0.542008 0.840374i \(-0.317664\pi\)
0.542008 + 0.840374i \(0.317664\pi\)
\(954\) 0 0
\(955\) 34481.4 1.16837
\(956\) 0 0
\(957\) −713.966 −0.0241163
\(958\) 0 0
\(959\) −5046.02 −0.169911
\(960\) 0 0
\(961\) 45557.7 1.52924
\(962\) 0 0
\(963\) 3024.40 0.101205
\(964\) 0 0
\(965\) −69596.4 −2.32165
\(966\) 0 0
\(967\) 46487.2 1.54594 0.772971 0.634441i \(-0.218770\pi\)
0.772971 + 0.634441i \(0.218770\pi\)
\(968\) 0 0
\(969\) 374.983 0.0124316
\(970\) 0 0
\(971\) −45959.6 −1.51897 −0.759483 0.650528i \(-0.774548\pi\)
−0.759483 + 0.650528i \(0.774548\pi\)
\(972\) 0 0
\(973\) −9352.28 −0.308140
\(974\) 0 0
\(975\) −6174.85 −0.202824
\(976\) 0 0
\(977\) 3859.76 0.126392 0.0631959 0.998001i \(-0.479871\pi\)
0.0631959 + 0.998001i \(0.479871\pi\)
\(978\) 0 0
\(979\) 56221.4 1.83539
\(980\) 0 0
\(981\) 6995.76 0.227684
\(982\) 0 0
\(983\) 12285.9 0.398637 0.199318 0.979935i \(-0.436127\pi\)
0.199318 + 0.979935i \(0.436127\pi\)
\(984\) 0 0
\(985\) −31257.4 −1.01111
\(986\) 0 0
\(987\) −3469.77 −0.111899
\(988\) 0 0
\(989\) 244.247 0.00785299
\(990\) 0 0
\(991\) −5559.34 −0.178202 −0.0891011 0.996023i \(-0.528399\pi\)
−0.0891011 + 0.996023i \(0.528399\pi\)
\(992\) 0 0
\(993\) −4136.49 −0.132193
\(994\) 0 0
\(995\) 92840.2 2.95802
\(996\) 0 0
\(997\) 35491.1 1.12740 0.563698 0.825981i \(-0.309378\pi\)
0.563698 + 0.825981i \(0.309378\pi\)
\(998\) 0 0
\(999\) 2277.05 0.0721148
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 624.4.a.n.1.1 2
3.2 odd 2 1872.4.a.bd.1.2 2
4.3 odd 2 312.4.a.b.1.1 2
8.3 odd 2 2496.4.a.bi.1.2 2
8.5 even 2 2496.4.a.x.1.2 2
12.11 even 2 936.4.a.h.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
312.4.a.b.1.1 2 4.3 odd 2
624.4.a.n.1.1 2 1.1 even 1 trivial
936.4.a.h.1.2 2 12.11 even 2
1872.4.a.bd.1.2 2 3.2 odd 2
2496.4.a.x.1.2 2 8.5 even 2
2496.4.a.bi.1.2 2 8.3 odd 2