Properties

Label 1216.4.a.x.1.1
Level $1216$
Weight $4$
Character 1216.1
Self dual yes
Analytic conductor $71.746$
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] = [1216,4,Mod(1,1216)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1216, 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("1216.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1216 = 2^{6} \cdot 19 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1216.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: \(71.7463225670\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.3221.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 9x + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 152)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(0.218090\) of defining polynomial
Character \(\chi\) \(=\) 1216.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-4.73435 q^{3} +18.0754 q^{5} +0.213413 q^{7} -4.58596 q^{9} +O(q^{10})\) \(q-4.73435 q^{3} +18.0754 q^{5} +0.213413 q^{7} -4.58596 q^{9} +3.39329 q^{11} +90.7156 q^{13} -85.5752 q^{15} -2.59392 q^{17} -19.0000 q^{19} -1.01037 q^{21} +26.6112 q^{23} +201.720 q^{25} +149.539 q^{27} -60.1034 q^{29} -176.070 q^{31} -16.0650 q^{33} +3.85753 q^{35} +154.115 q^{37} -429.479 q^{39} +434.137 q^{41} +365.511 q^{43} -82.8931 q^{45} +204.021 q^{47} -342.954 q^{49} +12.2805 q^{51} +135.726 q^{53} +61.3351 q^{55} +89.9526 q^{57} -759.895 q^{59} -284.941 q^{61} -0.978705 q^{63} +1639.72 q^{65} -590.922 q^{67} -125.986 q^{69} -972.291 q^{71} +368.462 q^{73} -955.013 q^{75} +0.724174 q^{77} +204.854 q^{79} -584.148 q^{81} +782.229 q^{83} -46.8861 q^{85} +284.550 q^{87} +213.620 q^{89} +19.3599 q^{91} +833.576 q^{93} -343.433 q^{95} -1219.54 q^{97} -15.5615 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 5 q^{3} - 2 q^{5} - 35 q^{7} + 48 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 5 q^{3} - 2 q^{5} - 35 q^{7} + 48 q^{9} + 28 q^{11} + 109 q^{13} - 228 q^{15} - 123 q^{17} - 57 q^{19} - 25 q^{21} - 193 q^{23} + 187 q^{25} + 719 q^{27} + 297 q^{29} - 140 q^{31} + 30 q^{33} + 246 q^{35} - 38 q^{37} + 57 q^{39} + 736 q^{41} + 514 q^{43} - 1046 q^{45} + 134 q^{47} - 42 q^{49} - 929 q^{51} - 311 q^{53} - 130 q^{55} - 95 q^{57} - 199 q^{59} - 56 q^{61} + 508 q^{63} + 1156 q^{65} + 509 q^{67} - 621 q^{69} - 874 q^{71} + 203 q^{73} - 129 q^{75} - 616 q^{77} + 242 q^{79} + 3531 q^{81} + 62 q^{83} + 1430 q^{85} + 3237 q^{87} + 1764 q^{89} + 687 q^{91} + 2592 q^{93} + 38 q^{95} - 2178 q^{97} - 86 q^{99}+O(q^{100}) \) 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
\(3\) −4.73435 −0.911125 −0.455563 0.890204i \(-0.650562\pi\)
−0.455563 + 0.890204i \(0.650562\pi\)
\(4\) 0 0
\(5\) 18.0754 1.61671 0.808356 0.588693i \(-0.200357\pi\)
0.808356 + 0.588693i \(0.200357\pi\)
\(6\) 0 0
\(7\) 0.213413 0.0115232 0.00576162 0.999983i \(-0.498166\pi\)
0.00576162 + 0.999983i \(0.498166\pi\)
\(8\) 0 0
\(9\) −4.58596 −0.169850
\(10\) 0 0
\(11\) 3.39329 0.0930106 0.0465053 0.998918i \(-0.485192\pi\)
0.0465053 + 0.998918i \(0.485192\pi\)
\(12\) 0 0
\(13\) 90.7156 1.93538 0.967692 0.252135i \(-0.0811328\pi\)
0.967692 + 0.252135i \(0.0811328\pi\)
\(14\) 0 0
\(15\) −85.5752 −1.47303
\(16\) 0 0
\(17\) −2.59392 −0.0370069 −0.0185035 0.999829i \(-0.505890\pi\)
−0.0185035 + 0.999829i \(0.505890\pi\)
\(18\) 0 0
\(19\) −19.0000 −0.229416
\(20\) 0 0
\(21\) −1.01037 −0.0104991
\(22\) 0 0
\(23\) 26.6112 0.241253 0.120626 0.992698i \(-0.461510\pi\)
0.120626 + 0.992698i \(0.461510\pi\)
\(24\) 0 0
\(25\) 201.720 1.61376
\(26\) 0 0
\(27\) 149.539 1.06588
\(28\) 0 0
\(29\) −60.1034 −0.384859 −0.192430 0.981311i \(-0.561637\pi\)
−0.192430 + 0.981311i \(0.561637\pi\)
\(30\) 0 0
\(31\) −176.070 −1.02010 −0.510050 0.860145i \(-0.670373\pi\)
−0.510050 + 0.860145i \(0.670373\pi\)
\(32\) 0 0
\(33\) −16.0650 −0.0847443
\(34\) 0 0
\(35\) 3.85753 0.0186298
\(36\) 0 0
\(37\) 154.115 0.684768 0.342384 0.939560i \(-0.388766\pi\)
0.342384 + 0.939560i \(0.388766\pi\)
\(38\) 0 0
\(39\) −429.479 −1.76338
\(40\) 0 0
\(41\) 434.137 1.65368 0.826839 0.562439i \(-0.190137\pi\)
0.826839 + 0.562439i \(0.190137\pi\)
\(42\) 0 0
\(43\) 365.511 1.29628 0.648138 0.761523i \(-0.275548\pi\)
0.648138 + 0.761523i \(0.275548\pi\)
\(44\) 0 0
\(45\) −82.8931 −0.274599
\(46\) 0 0
\(47\) 204.021 0.633180 0.316590 0.948562i \(-0.397462\pi\)
0.316590 + 0.948562i \(0.397462\pi\)
\(48\) 0 0
\(49\) −342.954 −0.999867
\(50\) 0 0
\(51\) 12.2805 0.0337180
\(52\) 0 0
\(53\) 135.726 0.351763 0.175881 0.984411i \(-0.443722\pi\)
0.175881 + 0.984411i \(0.443722\pi\)
\(54\) 0 0
\(55\) 61.3351 0.150371
\(56\) 0 0
\(57\) 89.9526 0.209027
\(58\) 0 0
\(59\) −759.895 −1.67678 −0.838389 0.545072i \(-0.816502\pi\)
−0.838389 + 0.545072i \(0.816502\pi\)
\(60\) 0 0
\(61\) −284.941 −0.598082 −0.299041 0.954240i \(-0.596667\pi\)
−0.299041 + 0.954240i \(0.596667\pi\)
\(62\) 0 0
\(63\) −0.978705 −0.00195723
\(64\) 0 0
\(65\) 1639.72 3.12896
\(66\) 0 0
\(67\) −590.922 −1.07750 −0.538750 0.842465i \(-0.681103\pi\)
−0.538750 + 0.842465i \(0.681103\pi\)
\(68\) 0 0
\(69\) −125.986 −0.219811
\(70\) 0 0
\(71\) −972.291 −1.62521 −0.812604 0.582817i \(-0.801950\pi\)
−0.812604 + 0.582817i \(0.801950\pi\)
\(72\) 0 0
\(73\) 368.462 0.590756 0.295378 0.955380i \(-0.404554\pi\)
0.295378 + 0.955380i \(0.404554\pi\)
\(74\) 0 0
\(75\) −955.013 −1.47034
\(76\) 0 0
\(77\) 0.724174 0.00107178
\(78\) 0 0
\(79\) 204.854 0.291745 0.145873 0.989303i \(-0.453401\pi\)
0.145873 + 0.989303i \(0.453401\pi\)
\(80\) 0 0
\(81\) −584.148 −0.801300
\(82\) 0 0
\(83\) 782.229 1.03447 0.517234 0.855844i \(-0.326962\pi\)
0.517234 + 0.855844i \(0.326962\pi\)
\(84\) 0 0
\(85\) −46.8861 −0.0598296
\(86\) 0 0
\(87\) 284.550 0.350655
\(88\) 0 0
\(89\) 213.620 0.254423 0.127211 0.991876i \(-0.459397\pi\)
0.127211 + 0.991876i \(0.459397\pi\)
\(90\) 0 0
\(91\) 19.3599 0.0223019
\(92\) 0 0
\(93\) 833.576 0.929438
\(94\) 0 0
\(95\) −343.433 −0.370899
\(96\) 0 0
\(97\) −1219.54 −1.27655 −0.638274 0.769809i \(-0.720351\pi\)
−0.638274 + 0.769809i \(0.720351\pi\)
\(98\) 0 0
\(99\) −15.5615 −0.0157979
\(100\) 0 0
\(101\) 53.9766 0.0531770 0.0265885 0.999646i \(-0.491536\pi\)
0.0265885 + 0.999646i \(0.491536\pi\)
\(102\) 0 0
\(103\) 1987.50 1.90130 0.950652 0.310260i \(-0.100416\pi\)
0.950652 + 0.310260i \(0.100416\pi\)
\(104\) 0 0
\(105\) −18.2629 −0.0169741
\(106\) 0 0
\(107\) 1076.92 0.972991 0.486496 0.873683i \(-0.338275\pi\)
0.486496 + 0.873683i \(0.338275\pi\)
\(108\) 0 0
\(109\) 1200.57 1.05499 0.527494 0.849559i \(-0.323132\pi\)
0.527494 + 0.849559i \(0.323132\pi\)
\(110\) 0 0
\(111\) −729.636 −0.623910
\(112\) 0 0
\(113\) 1402.72 1.16776 0.583880 0.811840i \(-0.301534\pi\)
0.583880 + 0.811840i \(0.301534\pi\)
\(114\) 0 0
\(115\) 481.007 0.390036
\(116\) 0 0
\(117\) −416.018 −0.328726
\(118\) 0 0
\(119\) −0.553577 −0.000426440 0
\(120\) 0 0
\(121\) −1319.49 −0.991349
\(122\) 0 0
\(123\) −2055.35 −1.50671
\(124\) 0 0
\(125\) 1386.75 0.992275
\(126\) 0 0
\(127\) −1055.22 −0.737289 −0.368644 0.929570i \(-0.620178\pi\)
−0.368644 + 0.929570i \(0.620178\pi\)
\(128\) 0 0
\(129\) −1730.45 −1.18107
\(130\) 0 0
\(131\) 1026.98 0.684944 0.342472 0.939528i \(-0.388736\pi\)
0.342472 + 0.939528i \(0.388736\pi\)
\(132\) 0 0
\(133\) −4.05485 −0.00264361
\(134\) 0 0
\(135\) 2702.98 1.72322
\(136\) 0 0
\(137\) −1728.02 −1.07763 −0.538813 0.842425i \(-0.681127\pi\)
−0.538813 + 0.842425i \(0.681127\pi\)
\(138\) 0 0
\(139\) −624.188 −0.380885 −0.190442 0.981698i \(-0.560992\pi\)
−0.190442 + 0.981698i \(0.560992\pi\)
\(140\) 0 0
\(141\) −965.904 −0.576906
\(142\) 0 0
\(143\) 307.825 0.180011
\(144\) 0 0
\(145\) −1086.39 −0.622207
\(146\) 0 0
\(147\) 1623.67 0.911004
\(148\) 0 0
\(149\) −57.5590 −0.0316471 −0.0158235 0.999875i \(-0.505037\pi\)
−0.0158235 + 0.999875i \(0.505037\pi\)
\(150\) 0 0
\(151\) −2567.35 −1.38363 −0.691814 0.722076i \(-0.743188\pi\)
−0.691814 + 0.722076i \(0.743188\pi\)
\(152\) 0 0
\(153\) 11.8956 0.00628564
\(154\) 0 0
\(155\) −3182.53 −1.64921
\(156\) 0 0
\(157\) 3015.67 1.53297 0.766486 0.642261i \(-0.222003\pi\)
0.766486 + 0.642261i \(0.222003\pi\)
\(158\) 0 0
\(159\) −642.575 −0.320500
\(160\) 0 0
\(161\) 5.67918 0.00278001
\(162\) 0 0
\(163\) 3331.49 1.60088 0.800438 0.599416i \(-0.204600\pi\)
0.800438 + 0.599416i \(0.204600\pi\)
\(164\) 0 0
\(165\) −290.382 −0.137007
\(166\) 0 0
\(167\) 2465.56 1.14246 0.571229 0.820791i \(-0.306467\pi\)
0.571229 + 0.820791i \(0.306467\pi\)
\(168\) 0 0
\(169\) 6032.33 2.74571
\(170\) 0 0
\(171\) 87.1333 0.0389664
\(172\) 0 0
\(173\) 549.111 0.241319 0.120659 0.992694i \(-0.461499\pi\)
0.120659 + 0.992694i \(0.461499\pi\)
\(174\) 0 0
\(175\) 43.0498 0.0185958
\(176\) 0 0
\(177\) 3597.61 1.52776
\(178\) 0 0
\(179\) 4186.74 1.74822 0.874110 0.485728i \(-0.161445\pi\)
0.874110 + 0.485728i \(0.161445\pi\)
\(180\) 0 0
\(181\) 3954.07 1.62378 0.811890 0.583811i \(-0.198439\pi\)
0.811890 + 0.583811i \(0.198439\pi\)
\(182\) 0 0
\(183\) 1349.01 0.544928
\(184\) 0 0
\(185\) 2785.70 1.10707
\(186\) 0 0
\(187\) −8.80193 −0.00344204
\(188\) 0 0
\(189\) 31.9136 0.0122824
\(190\) 0 0
\(191\) −1623.62 −0.615084 −0.307542 0.951535i \(-0.599506\pi\)
−0.307542 + 0.951535i \(0.599506\pi\)
\(192\) 0 0
\(193\) 1817.44 0.677833 0.338917 0.940816i \(-0.389940\pi\)
0.338917 + 0.940816i \(0.389940\pi\)
\(194\) 0 0
\(195\) −7763.01 −2.85088
\(196\) 0 0
\(197\) −151.771 −0.0548894 −0.0274447 0.999623i \(-0.508737\pi\)
−0.0274447 + 0.999623i \(0.508737\pi\)
\(198\) 0 0
\(199\) 1229.91 0.438119 0.219060 0.975711i \(-0.429701\pi\)
0.219060 + 0.975711i \(0.429701\pi\)
\(200\) 0 0
\(201\) 2797.63 0.981738
\(202\) 0 0
\(203\) −12.8269 −0.00443483
\(204\) 0 0
\(205\) 7847.20 2.67352
\(206\) 0 0
\(207\) −122.038 −0.0409769
\(208\) 0 0
\(209\) −64.4726 −0.0213381
\(210\) 0 0
\(211\) 2998.12 0.978194 0.489097 0.872229i \(-0.337326\pi\)
0.489097 + 0.872229i \(0.337326\pi\)
\(212\) 0 0
\(213\) 4603.16 1.48077
\(214\) 0 0
\(215\) 6606.75 2.09571
\(216\) 0 0
\(217\) −37.5757 −0.0117549
\(218\) 0 0
\(219\) −1744.43 −0.538253
\(220\) 0 0
\(221\) −235.309 −0.0716226
\(222\) 0 0
\(223\) −781.297 −0.234617 −0.117308 0.993096i \(-0.537427\pi\)
−0.117308 + 0.993096i \(0.537427\pi\)
\(224\) 0 0
\(225\) −925.080 −0.274098
\(226\) 0 0
\(227\) 2695.16 0.788035 0.394017 0.919103i \(-0.371085\pi\)
0.394017 + 0.919103i \(0.371085\pi\)
\(228\) 0 0
\(229\) −3952.95 −1.14069 −0.570346 0.821405i \(-0.693191\pi\)
−0.570346 + 0.821405i \(0.693191\pi\)
\(230\) 0 0
\(231\) −3.42849 −0.000976529 0
\(232\) 0 0
\(233\) 4315.28 1.21332 0.606659 0.794962i \(-0.292509\pi\)
0.606659 + 0.794962i \(0.292509\pi\)
\(234\) 0 0
\(235\) 3687.75 1.02367
\(236\) 0 0
\(237\) −969.849 −0.265816
\(238\) 0 0
\(239\) 3808.38 1.03073 0.515363 0.856972i \(-0.327657\pi\)
0.515363 + 0.856972i \(0.327657\pi\)
\(240\) 0 0
\(241\) 4274.22 1.14243 0.571217 0.820799i \(-0.306471\pi\)
0.571217 + 0.820799i \(0.306471\pi\)
\(242\) 0 0
\(243\) −1271.99 −0.335795
\(244\) 0 0
\(245\) −6199.04 −1.61650
\(246\) 0 0
\(247\) −1723.60 −0.444008
\(248\) 0 0
\(249\) −3703.35 −0.942530
\(250\) 0 0
\(251\) −2552.20 −0.641806 −0.320903 0.947112i \(-0.603986\pi\)
−0.320903 + 0.947112i \(0.603986\pi\)
\(252\) 0 0
\(253\) 90.2995 0.0224390
\(254\) 0 0
\(255\) 221.975 0.0545123
\(256\) 0 0
\(257\) 2099.29 0.509533 0.254766 0.967003i \(-0.418001\pi\)
0.254766 + 0.967003i \(0.418001\pi\)
\(258\) 0 0
\(259\) 32.8903 0.00789075
\(260\) 0 0
\(261\) 275.632 0.0653685
\(262\) 0 0
\(263\) −7691.71 −1.80339 −0.901695 0.432374i \(-0.857676\pi\)
−0.901695 + 0.432374i \(0.857676\pi\)
\(264\) 0 0
\(265\) 2453.31 0.568700
\(266\) 0 0
\(267\) −1011.35 −0.231811
\(268\) 0 0
\(269\) −1460.77 −0.331096 −0.165548 0.986202i \(-0.552939\pi\)
−0.165548 + 0.986202i \(0.552939\pi\)
\(270\) 0 0
\(271\) 3180.10 0.712832 0.356416 0.934327i \(-0.383999\pi\)
0.356416 + 0.934327i \(0.383999\pi\)
\(272\) 0 0
\(273\) −91.6566 −0.0203198
\(274\) 0 0
\(275\) 684.495 0.150097
\(276\) 0 0
\(277\) 1227.66 0.266293 0.133146 0.991096i \(-0.457492\pi\)
0.133146 + 0.991096i \(0.457492\pi\)
\(278\) 0 0
\(279\) 807.449 0.173264
\(280\) 0 0
\(281\) 6497.13 1.37931 0.689655 0.724138i \(-0.257762\pi\)
0.689655 + 0.724138i \(0.257762\pi\)
\(282\) 0 0
\(283\) 48.1723 0.0101185 0.00505927 0.999987i \(-0.498390\pi\)
0.00505927 + 0.999987i \(0.498390\pi\)
\(284\) 0 0
\(285\) 1625.93 0.337936
\(286\) 0 0
\(287\) 92.6506 0.0190557
\(288\) 0 0
\(289\) −4906.27 −0.998630
\(290\) 0 0
\(291\) 5773.71 1.16310
\(292\) 0 0
\(293\) −1140.74 −0.227449 −0.113724 0.993512i \(-0.536278\pi\)
−0.113724 + 0.993512i \(0.536278\pi\)
\(294\) 0 0
\(295\) −13735.4 −2.71087
\(296\) 0 0
\(297\) 507.429 0.0991382
\(298\) 0 0
\(299\) 2414.05 0.466916
\(300\) 0 0
\(301\) 78.0049 0.0149373
\(302\) 0 0
\(303\) −255.544 −0.0484509
\(304\) 0 0
\(305\) −5150.43 −0.966927
\(306\) 0 0
\(307\) 6728.53 1.25087 0.625435 0.780276i \(-0.284921\pi\)
0.625435 + 0.780276i \(0.284921\pi\)
\(308\) 0 0
\(309\) −9409.52 −1.73233
\(310\) 0 0
\(311\) −9758.17 −1.77921 −0.889606 0.456728i \(-0.849021\pi\)
−0.889606 + 0.456728i \(0.849021\pi\)
\(312\) 0 0
\(313\) −1660.82 −0.299920 −0.149960 0.988692i \(-0.547915\pi\)
−0.149960 + 0.988692i \(0.547915\pi\)
\(314\) 0 0
\(315\) −17.6905 −0.00316427
\(316\) 0 0
\(317\) −6578.84 −1.16563 −0.582814 0.812605i \(-0.698049\pi\)
−0.582814 + 0.812605i \(0.698049\pi\)
\(318\) 0 0
\(319\) −203.948 −0.0357960
\(320\) 0 0
\(321\) −5098.53 −0.886517
\(322\) 0 0
\(323\) 49.2845 0.00848997
\(324\) 0 0
\(325\) 18299.2 3.12325
\(326\) 0 0
\(327\) −5683.91 −0.961226
\(328\) 0 0
\(329\) 43.5407 0.00729629
\(330\) 0 0
\(331\) 5522.33 0.917024 0.458512 0.888688i \(-0.348383\pi\)
0.458512 + 0.888688i \(0.348383\pi\)
\(332\) 0 0
\(333\) −706.767 −0.116308
\(334\) 0 0
\(335\) −10681.1 −1.74201
\(336\) 0 0
\(337\) 1350.63 0.218319 0.109159 0.994024i \(-0.465184\pi\)
0.109159 + 0.994024i \(0.465184\pi\)
\(338\) 0 0
\(339\) −6640.97 −1.06398
\(340\) 0 0
\(341\) −597.457 −0.0948800
\(342\) 0 0
\(343\) −146.392 −0.0230450
\(344\) 0 0
\(345\) −2277.26 −0.355372
\(346\) 0 0
\(347\) −7266.15 −1.12411 −0.562057 0.827099i \(-0.689990\pi\)
−0.562057 + 0.827099i \(0.689990\pi\)
\(348\) 0 0
\(349\) −10360.4 −1.58905 −0.794524 0.607233i \(-0.792279\pi\)
−0.794524 + 0.607233i \(0.792279\pi\)
\(350\) 0 0
\(351\) 13565.5 2.06289
\(352\) 0 0
\(353\) −4014.20 −0.605253 −0.302626 0.953109i \(-0.597863\pi\)
−0.302626 + 0.953109i \(0.597863\pi\)
\(354\) 0 0
\(355\) −17574.5 −2.62749
\(356\) 0 0
\(357\) 2.62083 0.000388540 0
\(358\) 0 0
\(359\) 7120.01 1.04674 0.523370 0.852106i \(-0.324675\pi\)
0.523370 + 0.852106i \(0.324675\pi\)
\(360\) 0 0
\(361\) 361.000 0.0526316
\(362\) 0 0
\(363\) 6246.90 0.903243
\(364\) 0 0
\(365\) 6660.09 0.955083
\(366\) 0 0
\(367\) 12447.9 1.77050 0.885251 0.465114i \(-0.153987\pi\)
0.885251 + 0.465114i \(0.153987\pi\)
\(368\) 0 0
\(369\) −1990.93 −0.280878
\(370\) 0 0
\(371\) 28.9658 0.00405345
\(372\) 0 0
\(373\) 2379.11 0.330257 0.165128 0.986272i \(-0.447196\pi\)
0.165128 + 0.986272i \(0.447196\pi\)
\(374\) 0 0
\(375\) −6565.34 −0.904087
\(376\) 0 0
\(377\) −5452.32 −0.744850
\(378\) 0 0
\(379\) −10559.0 −1.43108 −0.715541 0.698571i \(-0.753820\pi\)
−0.715541 + 0.698571i \(0.753820\pi\)
\(380\) 0 0
\(381\) 4995.78 0.671763
\(382\) 0 0
\(383\) 3074.48 0.410180 0.205090 0.978743i \(-0.434251\pi\)
0.205090 + 0.978743i \(0.434251\pi\)
\(384\) 0 0
\(385\) 13.0897 0.00173277
\(386\) 0 0
\(387\) −1676.22 −0.220173
\(388\) 0 0
\(389\) −7437.27 −0.969369 −0.484684 0.874689i \(-0.661066\pi\)
−0.484684 + 0.874689i \(0.661066\pi\)
\(390\) 0 0
\(391\) −69.0272 −0.00892802
\(392\) 0 0
\(393\) −4862.08 −0.624070
\(394\) 0 0
\(395\) 3702.82 0.471668
\(396\) 0 0
\(397\) 10690.7 1.35151 0.675756 0.737125i \(-0.263817\pi\)
0.675756 + 0.737125i \(0.263817\pi\)
\(398\) 0 0
\(399\) 19.1971 0.00240866
\(400\) 0 0
\(401\) −8536.44 −1.06307 −0.531533 0.847038i \(-0.678384\pi\)
−0.531533 + 0.847038i \(0.678384\pi\)
\(402\) 0 0
\(403\) −15972.3 −1.97428
\(404\) 0 0
\(405\) −10558.7 −1.29547
\(406\) 0 0
\(407\) 522.959 0.0636907
\(408\) 0 0
\(409\) 5072.21 0.613214 0.306607 0.951836i \(-0.400806\pi\)
0.306607 + 0.951836i \(0.400806\pi\)
\(410\) 0 0
\(411\) 8181.05 0.981852
\(412\) 0 0
\(413\) −162.172 −0.0193219
\(414\) 0 0
\(415\) 14139.1 1.67244
\(416\) 0 0
\(417\) 2955.12 0.347034
\(418\) 0 0
\(419\) 13968.1 1.62861 0.814306 0.580436i \(-0.197118\pi\)
0.814306 + 0.580436i \(0.197118\pi\)
\(420\) 0 0
\(421\) −14983.2 −1.73453 −0.867264 0.497849i \(-0.834123\pi\)
−0.867264 + 0.497849i \(0.834123\pi\)
\(422\) 0 0
\(423\) −935.630 −0.107546
\(424\) 0 0
\(425\) −523.246 −0.0597203
\(426\) 0 0
\(427\) −60.8103 −0.00689184
\(428\) 0 0
\(429\) −1457.35 −0.164013
\(430\) 0 0
\(431\) −9101.07 −1.01713 −0.508565 0.861023i \(-0.669824\pi\)
−0.508565 + 0.861023i \(0.669824\pi\)
\(432\) 0 0
\(433\) −8406.65 −0.933020 −0.466510 0.884516i \(-0.654489\pi\)
−0.466510 + 0.884516i \(0.654489\pi\)
\(434\) 0 0
\(435\) 5143.36 0.566909
\(436\) 0 0
\(437\) −505.612 −0.0553472
\(438\) 0 0
\(439\) 4687.92 0.509664 0.254832 0.966985i \(-0.417980\pi\)
0.254832 + 0.966985i \(0.417980\pi\)
\(440\) 0 0
\(441\) 1572.78 0.169828
\(442\) 0 0
\(443\) −9503.30 −1.01922 −0.509611 0.860405i \(-0.670211\pi\)
−0.509611 + 0.860405i \(0.670211\pi\)
\(444\) 0 0
\(445\) 3861.26 0.411329
\(446\) 0 0
\(447\) 272.504 0.0288345
\(448\) 0 0
\(449\) −2589.35 −0.272159 −0.136079 0.990698i \(-0.543450\pi\)
−0.136079 + 0.990698i \(0.543450\pi\)
\(450\) 0 0
\(451\) 1473.15 0.153810
\(452\) 0 0
\(453\) 12154.7 1.26066
\(454\) 0 0
\(455\) 349.939 0.0360558
\(456\) 0 0
\(457\) −11145.8 −1.14087 −0.570436 0.821342i \(-0.693226\pi\)
−0.570436 + 0.821342i \(0.693226\pi\)
\(458\) 0 0
\(459\) −387.892 −0.0394450
\(460\) 0 0
\(461\) −4634.89 −0.468261 −0.234131 0.972205i \(-0.575224\pi\)
−0.234131 + 0.972205i \(0.575224\pi\)
\(462\) 0 0
\(463\) 8329.82 0.836112 0.418056 0.908421i \(-0.362712\pi\)
0.418056 + 0.908421i \(0.362712\pi\)
\(464\) 0 0
\(465\) 15067.2 1.50264
\(466\) 0 0
\(467\) −13878.1 −1.37516 −0.687582 0.726107i \(-0.741328\pi\)
−0.687582 + 0.726107i \(0.741328\pi\)
\(468\) 0 0
\(469\) −126.111 −0.0124163
\(470\) 0 0
\(471\) −14277.2 −1.39673
\(472\) 0 0
\(473\) 1240.29 0.120567
\(474\) 0 0
\(475\) −3832.68 −0.370222
\(476\) 0 0
\(477\) −622.435 −0.0597471
\(478\) 0 0
\(479\) 15896.9 1.51639 0.758194 0.652029i \(-0.226082\pi\)
0.758194 + 0.652029i \(0.226082\pi\)
\(480\) 0 0
\(481\) 13980.7 1.32529
\(482\) 0 0
\(483\) −26.8872 −0.00253294
\(484\) 0 0
\(485\) −22043.6 −2.06381
\(486\) 0 0
\(487\) −10987.2 −1.02233 −0.511166 0.859482i \(-0.670786\pi\)
−0.511166 + 0.859482i \(0.670786\pi\)
\(488\) 0 0
\(489\) −15772.4 −1.45860
\(490\) 0 0
\(491\) −4422.00 −0.406440 −0.203220 0.979133i \(-0.565141\pi\)
−0.203220 + 0.979133i \(0.565141\pi\)
\(492\) 0 0
\(493\) 155.903 0.0142425
\(494\) 0 0
\(495\) −281.281 −0.0255406
\(496\) 0 0
\(497\) −207.500 −0.0187277
\(498\) 0 0
\(499\) −9203.68 −0.825678 −0.412839 0.910804i \(-0.635463\pi\)
−0.412839 + 0.910804i \(0.635463\pi\)
\(500\) 0 0
\(501\) −11672.8 −1.04092
\(502\) 0 0
\(503\) −5394.92 −0.478226 −0.239113 0.970992i \(-0.576857\pi\)
−0.239113 + 0.970992i \(0.576857\pi\)
\(504\) 0 0
\(505\) 975.649 0.0859719
\(506\) 0 0
\(507\) −28559.1 −2.50169
\(508\) 0 0
\(509\) −14409.8 −1.25482 −0.627410 0.778689i \(-0.715885\pi\)
−0.627410 + 0.778689i \(0.715885\pi\)
\(510\) 0 0
\(511\) 78.6347 0.00680742
\(512\) 0 0
\(513\) −2841.24 −0.244530
\(514\) 0 0
\(515\) 35924.9 3.07386
\(516\) 0 0
\(517\) 692.302 0.0588924
\(518\) 0 0
\(519\) −2599.68 −0.219872
\(520\) 0 0
\(521\) −2249.98 −0.189201 −0.0946003 0.995515i \(-0.530157\pi\)
−0.0946003 + 0.995515i \(0.530157\pi\)
\(522\) 0 0
\(523\) 19298.1 1.61348 0.806738 0.590909i \(-0.201231\pi\)
0.806738 + 0.590909i \(0.201231\pi\)
\(524\) 0 0
\(525\) −203.813 −0.0169431
\(526\) 0 0
\(527\) 456.711 0.0377507
\(528\) 0 0
\(529\) −11458.8 −0.941797
\(530\) 0 0
\(531\) 3484.85 0.284801
\(532\) 0 0
\(533\) 39383.0 3.20050
\(534\) 0 0
\(535\) 19465.8 1.57305
\(536\) 0 0
\(537\) −19821.5 −1.59285
\(538\) 0 0
\(539\) −1163.75 −0.0929982
\(540\) 0 0
\(541\) 307.192 0.0244126 0.0122063 0.999926i \(-0.496115\pi\)
0.0122063 + 0.999926i \(0.496115\pi\)
\(542\) 0 0
\(543\) −18720.0 −1.47947
\(544\) 0 0
\(545\) 21700.8 1.70561
\(546\) 0 0
\(547\) 13468.0 1.05274 0.526371 0.850255i \(-0.323552\pi\)
0.526371 + 0.850255i \(0.323552\pi\)
\(548\) 0 0
\(549\) 1306.73 0.101584
\(550\) 0 0
\(551\) 1141.96 0.0882928
\(552\) 0 0
\(553\) 43.7186 0.00336185
\(554\) 0 0
\(555\) −13188.5 −1.00868
\(556\) 0 0
\(557\) −3808.21 −0.289693 −0.144846 0.989454i \(-0.546269\pi\)
−0.144846 + 0.989454i \(0.546269\pi\)
\(558\) 0 0
\(559\) 33157.5 2.50879
\(560\) 0 0
\(561\) 41.6714 0.00313613
\(562\) 0 0
\(563\) 22451.9 1.68070 0.840351 0.542042i \(-0.182349\pi\)
0.840351 + 0.542042i \(0.182349\pi\)
\(564\) 0 0
\(565\) 25354.7 1.88793
\(566\) 0 0
\(567\) −124.665 −0.00923358
\(568\) 0 0
\(569\) 21111.7 1.55545 0.777723 0.628607i \(-0.216375\pi\)
0.777723 + 0.628607i \(0.216375\pi\)
\(570\) 0 0
\(571\) 7839.51 0.574559 0.287279 0.957847i \(-0.407249\pi\)
0.287279 + 0.957847i \(0.407249\pi\)
\(572\) 0 0
\(573\) 7686.78 0.560419
\(574\) 0 0
\(575\) 5368.01 0.389324
\(576\) 0 0
\(577\) −2092.09 −0.150944 −0.0754720 0.997148i \(-0.524046\pi\)
−0.0754720 + 0.997148i \(0.524046\pi\)
\(578\) 0 0
\(579\) −8604.37 −0.617591
\(580\) 0 0
\(581\) 166.938 0.0119204
\(582\) 0 0
\(583\) 460.559 0.0327177
\(584\) 0 0
\(585\) −7519.70 −0.531455
\(586\) 0 0
\(587\) 19301.8 1.35719 0.678595 0.734513i \(-0.262589\pi\)
0.678595 + 0.734513i \(0.262589\pi\)
\(588\) 0 0
\(589\) 3345.33 0.234027
\(590\) 0 0
\(591\) 718.535 0.0500111
\(592\) 0 0
\(593\) −22422.7 −1.55276 −0.776381 0.630264i \(-0.782947\pi\)
−0.776381 + 0.630264i \(0.782947\pi\)
\(594\) 0 0
\(595\) −10.0061 −0.000689431 0
\(596\) 0 0
\(597\) −5822.80 −0.399182
\(598\) 0 0
\(599\) −6779.52 −0.462443 −0.231222 0.972901i \(-0.574272\pi\)
−0.231222 + 0.972901i \(0.574272\pi\)
\(600\) 0 0
\(601\) 13065.1 0.886746 0.443373 0.896337i \(-0.353782\pi\)
0.443373 + 0.896337i \(0.353782\pi\)
\(602\) 0 0
\(603\) 2709.94 0.183014
\(604\) 0 0
\(605\) −23850.2 −1.60273
\(606\) 0 0
\(607\) −17590.3 −1.17622 −0.588112 0.808780i \(-0.700128\pi\)
−0.588112 + 0.808780i \(0.700128\pi\)
\(608\) 0 0
\(609\) 60.7268 0.00404068
\(610\) 0 0
\(611\) 18507.9 1.22545
\(612\) 0 0
\(613\) −10723.1 −0.706530 −0.353265 0.935523i \(-0.614929\pi\)
−0.353265 + 0.935523i \(0.614929\pi\)
\(614\) 0 0
\(615\) −37151.4 −2.43591
\(616\) 0 0
\(617\) −2406.12 −0.156997 −0.0784983 0.996914i \(-0.525013\pi\)
−0.0784983 + 0.996914i \(0.525013\pi\)
\(618\) 0 0
\(619\) −5375.50 −0.349046 −0.174523 0.984653i \(-0.555838\pi\)
−0.174523 + 0.984653i \(0.555838\pi\)
\(620\) 0 0
\(621\) 3979.40 0.257146
\(622\) 0 0
\(623\) 45.5893 0.00293178
\(624\) 0 0
\(625\) −149.015 −0.00953698
\(626\) 0 0
\(627\) 305.236 0.0194417
\(628\) 0 0
\(629\) −399.763 −0.0253412
\(630\) 0 0
\(631\) −16557.2 −1.04458 −0.522291 0.852767i \(-0.674923\pi\)
−0.522291 + 0.852767i \(0.674923\pi\)
\(632\) 0 0
\(633\) −14194.1 −0.891258
\(634\) 0 0
\(635\) −19073.5 −1.19198
\(636\) 0 0
\(637\) −31111.3 −1.93513
\(638\) 0 0
\(639\) 4458.89 0.276042
\(640\) 0 0
\(641\) −1184.92 −0.0730131 −0.0365066 0.999333i \(-0.511623\pi\)
−0.0365066 + 0.999333i \(0.511623\pi\)
\(642\) 0 0
\(643\) 8126.38 0.498403 0.249202 0.968452i \(-0.419832\pi\)
0.249202 + 0.968452i \(0.419832\pi\)
\(644\) 0 0
\(645\) −31278.7 −1.90945
\(646\) 0 0
\(647\) 16438.6 0.998868 0.499434 0.866352i \(-0.333541\pi\)
0.499434 + 0.866352i \(0.333541\pi\)
\(648\) 0 0
\(649\) −2578.55 −0.155958
\(650\) 0 0
\(651\) 177.896 0.0107101
\(652\) 0 0
\(653\) 10561.8 0.632947 0.316473 0.948601i \(-0.397501\pi\)
0.316473 + 0.948601i \(0.397501\pi\)
\(654\) 0 0
\(655\) 18563.1 1.10736
\(656\) 0 0
\(657\) −1689.75 −0.100340
\(658\) 0 0
\(659\) 28057.4 1.65851 0.829257 0.558868i \(-0.188764\pi\)
0.829257 + 0.558868i \(0.188764\pi\)
\(660\) 0 0
\(661\) −16123.6 −0.948769 −0.474385 0.880318i \(-0.657329\pi\)
−0.474385 + 0.880318i \(0.657329\pi\)
\(662\) 0 0
\(663\) 1114.03 0.0652572
\(664\) 0 0
\(665\) −73.2931 −0.00427396
\(666\) 0 0
\(667\) −1599.42 −0.0928483
\(668\) 0 0
\(669\) 3698.93 0.213765
\(670\) 0 0
\(671\) −966.889 −0.0556279
\(672\) 0 0
\(673\) −31833.0 −1.82329 −0.911643 0.410982i \(-0.865186\pi\)
−0.911643 + 0.410982i \(0.865186\pi\)
\(674\) 0 0
\(675\) 30165.0 1.72008
\(676\) 0 0
\(677\) 18505.4 1.05055 0.525273 0.850934i \(-0.323963\pi\)
0.525273 + 0.850934i \(0.323963\pi\)
\(678\) 0 0
\(679\) −260.266 −0.0147100
\(680\) 0 0
\(681\) −12759.8 −0.717999
\(682\) 0 0
\(683\) −2660.37 −0.149043 −0.0745213 0.997219i \(-0.523743\pi\)
−0.0745213 + 0.997219i \(0.523743\pi\)
\(684\) 0 0
\(685\) −31234.7 −1.74221
\(686\) 0 0
\(687\) 18714.6 1.03931
\(688\) 0 0
\(689\) 12312.5 0.680796
\(690\) 0 0
\(691\) −23883.5 −1.31487 −0.657433 0.753513i \(-0.728358\pi\)
−0.657433 + 0.753513i \(0.728358\pi\)
\(692\) 0 0
\(693\) −3.32103 −0.000182043 0
\(694\) 0 0
\(695\) −11282.5 −0.615781
\(696\) 0 0
\(697\) −1126.12 −0.0611975
\(698\) 0 0
\(699\) −20430.0 −1.10549
\(700\) 0 0
\(701\) −25949.8 −1.39816 −0.699079 0.715044i \(-0.746406\pi\)
−0.699079 + 0.715044i \(0.746406\pi\)
\(702\) 0 0
\(703\) −2928.19 −0.157097
\(704\) 0 0
\(705\) −17459.1 −0.932692
\(706\) 0 0
\(707\) 11.5193 0.000612771 0
\(708\) 0 0
\(709\) −12320.8 −0.652633 −0.326316 0.945261i \(-0.605807\pi\)
−0.326316 + 0.945261i \(0.605807\pi\)
\(710\) 0 0
\(711\) −939.452 −0.0495530
\(712\) 0 0
\(713\) −4685.42 −0.246102
\(714\) 0 0
\(715\) 5564.06 0.291026
\(716\) 0 0
\(717\) −18030.2 −0.939120
\(718\) 0 0
\(719\) −27252.6 −1.41356 −0.706781 0.707432i \(-0.749854\pi\)
−0.706781 + 0.707432i \(0.749854\pi\)
\(720\) 0 0
\(721\) 424.159 0.0219092
\(722\) 0 0
\(723\) −20235.6 −1.04090
\(724\) 0 0
\(725\) −12124.1 −0.621071
\(726\) 0 0
\(727\) 24073.8 1.22813 0.614063 0.789257i \(-0.289534\pi\)
0.614063 + 0.789257i \(0.289534\pi\)
\(728\) 0 0
\(729\) 21794.0 1.10725
\(730\) 0 0
\(731\) −948.105 −0.0479712
\(732\) 0 0
\(733\) −10232.2 −0.515600 −0.257800 0.966198i \(-0.582998\pi\)
−0.257800 + 0.966198i \(0.582998\pi\)
\(734\) 0 0
\(735\) 29348.4 1.47283
\(736\) 0 0
\(737\) −2005.17 −0.100219
\(738\) 0 0
\(739\) −5327.77 −0.265203 −0.132602 0.991169i \(-0.542333\pi\)
−0.132602 + 0.991169i \(0.542333\pi\)
\(740\) 0 0
\(741\) 8160.11 0.404547
\(742\) 0 0
\(743\) 20416.6 1.00809 0.504046 0.863677i \(-0.331844\pi\)
0.504046 + 0.863677i \(0.331844\pi\)
\(744\) 0 0
\(745\) −1040.40 −0.0511642
\(746\) 0 0
\(747\) −3587.27 −0.175705
\(748\) 0 0
\(749\) 229.830 0.0112120
\(750\) 0 0
\(751\) 11444.8 0.556092 0.278046 0.960568i \(-0.410313\pi\)
0.278046 + 0.960568i \(0.410313\pi\)
\(752\) 0 0
\(753\) 12083.0 0.584766
\(754\) 0 0
\(755\) −46405.8 −2.23693
\(756\) 0 0
\(757\) −31158.8 −1.49602 −0.748009 0.663689i \(-0.768990\pi\)
−0.748009 + 0.663689i \(0.768990\pi\)
\(758\) 0 0
\(759\) −427.509 −0.0204448
\(760\) 0 0
\(761\) −8933.92 −0.425564 −0.212782 0.977100i \(-0.568252\pi\)
−0.212782 + 0.977100i \(0.568252\pi\)
\(762\) 0 0
\(763\) 256.217 0.0121569
\(764\) 0 0
\(765\) 215.018 0.0101621
\(766\) 0 0
\(767\) −68934.4 −3.24521
\(768\) 0 0
\(769\) 29688.6 1.39220 0.696099 0.717946i \(-0.254918\pi\)
0.696099 + 0.717946i \(0.254918\pi\)
\(770\) 0 0
\(771\) −9938.75 −0.464248
\(772\) 0 0
\(773\) 35704.1 1.66130 0.830651 0.556793i \(-0.187968\pi\)
0.830651 + 0.556793i \(0.187968\pi\)
\(774\) 0 0
\(775\) −35516.8 −1.64620
\(776\) 0 0
\(777\) −155.714 −0.00718946
\(778\) 0 0
\(779\) −8248.60 −0.379380
\(780\) 0 0
\(781\) −3299.27 −0.151161
\(782\) 0 0
\(783\) −8987.79 −0.410214
\(784\) 0 0
\(785\) 54509.4 2.47838
\(786\) 0 0
\(787\) 26828.3 1.21515 0.607577 0.794261i \(-0.292142\pi\)
0.607577 + 0.794261i \(0.292142\pi\)
\(788\) 0 0
\(789\) 36415.2 1.64311
\(790\) 0 0
\(791\) 299.360 0.0134564
\(792\) 0 0
\(793\) −25848.6 −1.15752
\(794\) 0 0
\(795\) −11614.8 −0.518157
\(796\) 0 0
\(797\) 42689.8 1.89730 0.948650 0.316327i \(-0.102450\pi\)
0.948650 + 0.316327i \(0.102450\pi\)
\(798\) 0 0
\(799\) −529.213 −0.0234320
\(800\) 0 0
\(801\) −979.651 −0.0432138
\(802\) 0 0
\(803\) 1250.30 0.0549465
\(804\) 0 0
\(805\) 102.653 0.00449448
\(806\) 0 0
\(807\) 6915.81 0.301670
\(808\) 0 0
\(809\) −21486.1 −0.933759 −0.466880 0.884321i \(-0.654622\pi\)
−0.466880 + 0.884321i \(0.654622\pi\)
\(810\) 0 0
\(811\) −3074.32 −0.133112 −0.0665560 0.997783i \(-0.521201\pi\)
−0.0665560 + 0.997783i \(0.521201\pi\)
\(812\) 0 0
\(813\) −15055.7 −0.649480
\(814\) 0 0
\(815\) 60218.1 2.58816
\(816\) 0 0
\(817\) −6944.70 −0.297386
\(818\) 0 0
\(819\) −88.7839 −0.00378799
\(820\) 0 0
\(821\) 32460.2 1.37987 0.689933 0.723873i \(-0.257640\pi\)
0.689933 + 0.723873i \(0.257640\pi\)
\(822\) 0 0
\(823\) −25382.1 −1.07505 −0.537523 0.843249i \(-0.680640\pi\)
−0.537523 + 0.843249i \(0.680640\pi\)
\(824\) 0 0
\(825\) −3240.64 −0.136757
\(826\) 0 0
\(827\) 9830.51 0.413350 0.206675 0.978410i \(-0.433736\pi\)
0.206675 + 0.978410i \(0.433736\pi\)
\(828\) 0 0
\(829\) −47474.0 −1.98895 −0.994476 0.104968i \(-0.966526\pi\)
−0.994476 + 0.104968i \(0.966526\pi\)
\(830\) 0 0
\(831\) −5812.18 −0.242626
\(832\) 0 0
\(833\) 889.596 0.0370020
\(834\) 0 0
\(835\) 44565.9 1.84703
\(836\) 0 0
\(837\) −26329.3 −1.08730
\(838\) 0 0
\(839\) 29050.3 1.19539 0.597693 0.801725i \(-0.296084\pi\)
0.597693 + 0.801725i \(0.296084\pi\)
\(840\) 0 0
\(841\) −20776.6 −0.851883
\(842\) 0 0
\(843\) −30759.7 −1.25673
\(844\) 0 0
\(845\) 109037. 4.43903
\(846\) 0 0
\(847\) −281.596 −0.0114236
\(848\) 0 0
\(849\) −228.064 −0.00921926
\(850\) 0 0
\(851\) 4101.19 0.165202
\(852\) 0 0
\(853\) −21804.2 −0.875217 −0.437609 0.899166i \(-0.644174\pi\)
−0.437609 + 0.899166i \(0.644174\pi\)
\(854\) 0 0
\(855\) 1574.97 0.0629974
\(856\) 0 0
\(857\) 30879.5 1.23083 0.615416 0.788203i \(-0.288988\pi\)
0.615416 + 0.788203i \(0.288988\pi\)
\(858\) 0 0
\(859\) −6065.94 −0.240940 −0.120470 0.992717i \(-0.538440\pi\)
−0.120470 + 0.992717i \(0.538440\pi\)
\(860\) 0 0
\(861\) −438.640 −0.0173622
\(862\) 0 0
\(863\) −1527.39 −0.0602467 −0.0301233 0.999546i \(-0.509590\pi\)
−0.0301233 + 0.999546i \(0.509590\pi\)
\(864\) 0 0
\(865\) 9925.40 0.390143
\(866\) 0 0
\(867\) 23228.0 0.909878
\(868\) 0 0
\(869\) 695.129 0.0271354
\(870\) 0 0
\(871\) −53605.8 −2.08538
\(872\) 0 0
\(873\) 5592.75 0.216822
\(874\) 0 0
\(875\) 295.950 0.0114342
\(876\) 0 0
\(877\) 565.469 0.0217725 0.0108863 0.999941i \(-0.496535\pi\)
0.0108863 + 0.999941i \(0.496535\pi\)
\(878\) 0 0
\(879\) 5400.64 0.207234
\(880\) 0 0
\(881\) −28897.8 −1.10510 −0.552549 0.833481i \(-0.686345\pi\)
−0.552549 + 0.833481i \(0.686345\pi\)
\(882\) 0 0
\(883\) −26073.0 −0.993687 −0.496844 0.867840i \(-0.665508\pi\)
−0.496844 + 0.867840i \(0.665508\pi\)
\(884\) 0 0
\(885\) 65028.2 2.46994
\(886\) 0 0
\(887\) −859.344 −0.0325298 −0.0162649 0.999868i \(-0.505178\pi\)
−0.0162649 + 0.999868i \(0.505178\pi\)
\(888\) 0 0
\(889\) −225.198 −0.00849596
\(890\) 0 0
\(891\) −1982.19 −0.0745294
\(892\) 0 0
\(893\) −3876.39 −0.145261
\(894\) 0 0
\(895\) 75676.9 2.82637
\(896\) 0 0
\(897\) −11428.9 −0.425419
\(898\) 0 0
\(899\) 10582.4 0.392595
\(900\) 0 0
\(901\) −352.063 −0.0130177
\(902\) 0 0
\(903\) −369.302 −0.0136098
\(904\) 0 0
\(905\) 71471.5 2.62519
\(906\) 0 0
\(907\) −36272.4 −1.32790 −0.663951 0.747776i \(-0.731122\pi\)
−0.663951 + 0.747776i \(0.731122\pi\)
\(908\) 0 0
\(909\) −247.535 −0.00903213
\(910\) 0 0
\(911\) −20484.7 −0.744993 −0.372497 0.928034i \(-0.621498\pi\)
−0.372497 + 0.928034i \(0.621498\pi\)
\(912\) 0 0
\(913\) 2654.33 0.0962165
\(914\) 0 0
\(915\) 24383.9 0.880992
\(916\) 0 0
\(917\) 219.171 0.00789277
\(918\) 0 0
\(919\) 6354.25 0.228082 0.114041 0.993476i \(-0.463620\pi\)
0.114041 + 0.993476i \(0.463620\pi\)
\(920\) 0 0
\(921\) −31855.2 −1.13970
\(922\) 0 0
\(923\) −88202.0 −3.14540
\(924\) 0 0
\(925\) 31088.2 1.10505
\(926\) 0 0
\(927\) −9114.60 −0.322937
\(928\) 0 0
\(929\) 33316.3 1.17661 0.588307 0.808638i \(-0.299795\pi\)
0.588307 + 0.808638i \(0.299795\pi\)
\(930\) 0 0
\(931\) 6516.13 0.229385
\(932\) 0 0
\(933\) 46198.6 1.62109
\(934\) 0 0
\(935\) −159.098 −0.00556478
\(936\) 0 0
\(937\) −6181.13 −0.215506 −0.107753 0.994178i \(-0.534365\pi\)
−0.107753 + 0.994178i \(0.534365\pi\)
\(938\) 0 0
\(939\) 7862.89 0.273265
\(940\) 0 0
\(941\) −609.315 −0.0211085 −0.0105543 0.999944i \(-0.503360\pi\)
−0.0105543 + 0.999944i \(0.503360\pi\)
\(942\) 0 0
\(943\) 11552.9 0.398954
\(944\) 0 0
\(945\) 576.851 0.0198571
\(946\) 0 0
\(947\) 9348.71 0.320794 0.160397 0.987053i \(-0.448722\pi\)
0.160397 + 0.987053i \(0.448722\pi\)
\(948\) 0 0
\(949\) 33425.2 1.14334
\(950\) 0 0
\(951\) 31146.5 1.06203
\(952\) 0 0
\(953\) −36990.5 −1.25734 −0.628668 0.777674i \(-0.716399\pi\)
−0.628668 + 0.777674i \(0.716399\pi\)
\(954\) 0 0
\(955\) −29347.6 −0.994414
\(956\) 0 0
\(957\) 965.563 0.0326146
\(958\) 0 0
\(959\) −368.783 −0.0124177
\(960\) 0 0
\(961\) 1209.59 0.0406027
\(962\) 0 0
\(963\) −4938.73 −0.165263
\(964\) 0 0
\(965\) 32850.9 1.09586
\(966\) 0 0
\(967\) −48806.2 −1.62306 −0.811532 0.584308i \(-0.801366\pi\)
−0.811532 + 0.584308i \(0.801366\pi\)
\(968\) 0 0
\(969\) −233.330 −0.00773543
\(970\) 0 0
\(971\) −401.490 −0.0132692 −0.00663462 0.999978i \(-0.502112\pi\)
−0.00663462 + 0.999978i \(0.502112\pi\)
\(972\) 0 0
\(973\) −133.210 −0.00438903
\(974\) 0 0
\(975\) −86634.6 −2.84567
\(976\) 0 0
\(977\) 6786.27 0.222223 0.111112 0.993808i \(-0.464559\pi\)
0.111112 + 0.993808i \(0.464559\pi\)
\(978\) 0 0
\(979\) 724.874 0.0236640
\(980\) 0 0
\(981\) −5505.76 −0.179190
\(982\) 0 0
\(983\) −17163.1 −0.556884 −0.278442 0.960453i \(-0.589818\pi\)
−0.278442 + 0.960453i \(0.589818\pi\)
\(984\) 0 0
\(985\) −2743.31 −0.0887404
\(986\) 0 0
\(987\) −206.137 −0.00664783
\(988\) 0 0
\(989\) 9726.67 0.312730
\(990\) 0 0
\(991\) 9323.61 0.298864 0.149432 0.988772i \(-0.452255\pi\)
0.149432 + 0.988772i \(0.452255\pi\)
\(992\) 0 0
\(993\) −26144.6 −0.835524
\(994\) 0 0
\(995\) 22231.0 0.708313
\(996\) 0 0
\(997\) −14764.7 −0.469011 −0.234506 0.972115i \(-0.575347\pi\)
−0.234506 + 0.972115i \(0.575347\pi\)
\(998\) 0 0
\(999\) 23046.3 0.729881
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 1216.4.a.x.1.1 3
4.3 odd 2 1216.4.a.q.1.3 3
8.3 odd 2 304.4.a.j.1.1 3
8.5 even 2 152.4.a.b.1.3 3
24.5 odd 2 1368.4.a.e.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
152.4.a.b.1.3 3 8.5 even 2
304.4.a.j.1.1 3 8.3 odd 2
1216.4.a.q.1.3 3 4.3 odd 2
1216.4.a.x.1.1 3 1.1 even 1 trivial
1368.4.a.e.1.3 3 24.5 odd 2