Properties

Label 2205.4.a.cb.1.3
Level $2205$
Weight $4$
Character 2205.1
Self dual yes
Analytic conductor $130.099$
Analytic rank $1$
Dimension $8$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2205,4,Mod(1,2205)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2205, 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("2205.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2205 = 3^{2} \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 2205.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: \(130.099211563\)
Analytic rank: \(1\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 4x^{7} - 45x^{6} + 134x^{5} + 641x^{4} - 1130x^{3} - 2877x^{2} + 2584x + 3696 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 315)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-2.02408\) of defining polynomial
Character \(\chi\) \(=\) 2205.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-3.02408 q^{2} +1.14504 q^{4} -5.00000 q^{5} +20.7299 q^{8} +O(q^{10})\) \(q-3.02408 q^{2} +1.14504 q^{4} -5.00000 q^{5} +20.7299 q^{8} +15.1204 q^{10} +22.1595 q^{11} +35.9983 q^{13} -71.8492 q^{16} -85.8421 q^{17} +97.0029 q^{19} -5.72522 q^{20} -67.0121 q^{22} +98.2844 q^{23} +25.0000 q^{25} -108.862 q^{26} -200.052 q^{29} -296.976 q^{31} +51.4382 q^{32} +259.593 q^{34} -185.337 q^{37} -293.344 q^{38} -103.650 q^{40} +291.601 q^{41} +173.597 q^{43} +25.3736 q^{44} -297.220 q^{46} -52.9175 q^{47} -75.6019 q^{50} +41.2196 q^{52} -294.704 q^{53} -110.798 q^{55} +604.973 q^{58} +197.462 q^{59} +430.880 q^{61} +898.078 q^{62} +419.241 q^{64} -179.991 q^{65} -519.033 q^{67} -98.2929 q^{68} -980.318 q^{71} -55.7062 q^{73} +560.473 q^{74} +111.072 q^{76} +830.967 q^{79} +359.246 q^{80} -881.824 q^{82} -1280.48 q^{83} +429.210 q^{85} -524.969 q^{86} +459.365 q^{88} +506.166 q^{89} +112.540 q^{92} +160.026 q^{94} -485.014 q^{95} +1041.60 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 4 q^{2} + 42 q^{4} - 40 q^{5} - 48 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 4 q^{2} + 42 q^{4} - 40 q^{5} - 48 q^{8} + 20 q^{10} - 100 q^{11} + 102 q^{13} + 266 q^{16} + 56 q^{17} - 210 q^{20} - 70 q^{22} - 190 q^{23} + 200 q^{25} + 60 q^{26} - 296 q^{29} - 42 q^{31} - 718 q^{32} + 488 q^{34} + 314 q^{37} - 514 q^{38} + 240 q^{40} - 28 q^{41} + 714 q^{43} - 410 q^{44} - 650 q^{46} + 326 q^{47} - 100 q^{50} + 794 q^{52} - 1282 q^{53} + 500 q^{55} + 942 q^{58} - 924 q^{59} + 536 q^{61} + 50 q^{62} + 1902 q^{64} - 510 q^{65} + 2 q^{67} + 2690 q^{68} - 1516 q^{71} - 86 q^{73} - 4754 q^{74} - 30 q^{76} - 42 q^{79} - 1330 q^{80} + 3602 q^{82} + 572 q^{83} - 280 q^{85} - 1758 q^{86} - 4312 q^{88} + 940 q^{89} - 3844 q^{92} - 2866 q^{94} + 1720 q^{97}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −3.02408 −1.06917 −0.534586 0.845114i \(-0.679533\pi\)
−0.534586 + 0.845114i \(0.679533\pi\)
\(3\) 0 0
\(4\) 1.14504 0.143130
\(5\) −5.00000 −0.447214
\(6\) 0 0
\(7\) 0 0
\(8\) 20.7299 0.916142
\(9\) 0 0
\(10\) 15.1204 0.478149
\(11\) 22.1595 0.607395 0.303698 0.952769i \(-0.401779\pi\)
0.303698 + 0.952769i \(0.401779\pi\)
\(12\) 0 0
\(13\) 35.9983 0.768010 0.384005 0.923331i \(-0.374545\pi\)
0.384005 + 0.923331i \(0.374545\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) −71.8492 −1.12264
\(17\) −85.8421 −1.22469 −0.612346 0.790590i \(-0.709774\pi\)
−0.612346 + 0.790590i \(0.709774\pi\)
\(18\) 0 0
\(19\) 97.0029 1.17126 0.585631 0.810578i \(-0.300847\pi\)
0.585631 + 0.810578i \(0.300847\pi\)
\(20\) −5.72522 −0.0640099
\(21\) 0 0
\(22\) −67.0121 −0.649410
\(23\) 98.2844 0.891031 0.445516 0.895274i \(-0.353020\pi\)
0.445516 + 0.895274i \(0.353020\pi\)
\(24\) 0 0
\(25\) 25.0000 0.200000
\(26\) −108.862 −0.821135
\(27\) 0 0
\(28\) 0 0
\(29\) −200.052 −1.28099 −0.640496 0.767962i \(-0.721271\pi\)
−0.640496 + 0.767962i \(0.721271\pi\)
\(30\) 0 0
\(31\) −296.976 −1.72059 −0.860297 0.509793i \(-0.829722\pi\)
−0.860297 + 0.509793i \(0.829722\pi\)
\(32\) 51.4382 0.284159
\(33\) 0 0
\(34\) 259.593 1.30941
\(35\) 0 0
\(36\) 0 0
\(37\) −185.337 −0.823492 −0.411746 0.911299i \(-0.635081\pi\)
−0.411746 + 0.911299i \(0.635081\pi\)
\(38\) −293.344 −1.25228
\(39\) 0 0
\(40\) −103.650 −0.409711
\(41\) 291.601 1.11074 0.555371 0.831603i \(-0.312576\pi\)
0.555371 + 0.831603i \(0.312576\pi\)
\(42\) 0 0
\(43\) 173.597 0.615656 0.307828 0.951442i \(-0.400398\pi\)
0.307828 + 0.951442i \(0.400398\pi\)
\(44\) 25.3736 0.0869367
\(45\) 0 0
\(46\) −297.220 −0.952666
\(47\) −52.9175 −0.164230 −0.0821150 0.996623i \(-0.526167\pi\)
−0.0821150 + 0.996623i \(0.526167\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) −75.6019 −0.213835
\(51\) 0 0
\(52\) 41.2196 0.109926
\(53\) −294.704 −0.763788 −0.381894 0.924206i \(-0.624728\pi\)
−0.381894 + 0.924206i \(0.624728\pi\)
\(54\) 0 0
\(55\) −110.798 −0.271635
\(56\) 0 0
\(57\) 0 0
\(58\) 604.973 1.36960
\(59\) 197.462 0.435719 0.217859 0.975980i \(-0.430093\pi\)
0.217859 + 0.975980i \(0.430093\pi\)
\(60\) 0 0
\(61\) 430.880 0.904401 0.452201 0.891916i \(-0.350639\pi\)
0.452201 + 0.891916i \(0.350639\pi\)
\(62\) 898.078 1.83961
\(63\) 0 0
\(64\) 419.241 0.818829
\(65\) −179.991 −0.343464
\(66\) 0 0
\(67\) −519.033 −0.946417 −0.473209 0.880950i \(-0.656904\pi\)
−0.473209 + 0.880950i \(0.656904\pi\)
\(68\) −98.2929 −0.175291
\(69\) 0 0
\(70\) 0 0
\(71\) −980.318 −1.63862 −0.819312 0.573347i \(-0.805645\pi\)
−0.819312 + 0.573347i \(0.805645\pi\)
\(72\) 0 0
\(73\) −55.7062 −0.0893140 −0.0446570 0.999002i \(-0.514219\pi\)
−0.0446570 + 0.999002i \(0.514219\pi\)
\(74\) 560.473 0.880455
\(75\) 0 0
\(76\) 111.072 0.167643
\(77\) 0 0
\(78\) 0 0
\(79\) 830.967 1.18343 0.591716 0.806147i \(-0.298451\pi\)
0.591716 + 0.806147i \(0.298451\pi\)
\(80\) 359.246 0.502062
\(81\) 0 0
\(82\) −881.824 −1.18757
\(83\) −1280.48 −1.69338 −0.846690 0.532087i \(-0.821408\pi\)
−0.846690 + 0.532087i \(0.821408\pi\)
\(84\) 0 0
\(85\) 429.210 0.547699
\(86\) −524.969 −0.658243
\(87\) 0 0
\(88\) 459.365 0.556460
\(89\) 506.166 0.602848 0.301424 0.953490i \(-0.402538\pi\)
0.301424 + 0.953490i \(0.402538\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 112.540 0.127534
\(93\) 0 0
\(94\) 160.026 0.175590
\(95\) −485.014 −0.523804
\(96\) 0 0
\(97\) 1041.60 1.09030 0.545149 0.838339i \(-0.316473\pi\)
0.545149 + 0.838339i \(0.316473\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 28.6261 0.0286261
\(101\) −249.341 −0.245647 −0.122824 0.992429i \(-0.539195\pi\)
−0.122824 + 0.992429i \(0.539195\pi\)
\(102\) 0 0
\(103\) 1788.97 1.71138 0.855692 0.517485i \(-0.173132\pi\)
0.855692 + 0.517485i \(0.173132\pi\)
\(104\) 746.241 0.703606
\(105\) 0 0
\(106\) 891.208 0.816621
\(107\) 327.084 0.295517 0.147759 0.989023i \(-0.452794\pi\)
0.147759 + 0.989023i \(0.452794\pi\)
\(108\) 0 0
\(109\) 1148.95 1.00963 0.504816 0.863227i \(-0.331560\pi\)
0.504816 + 0.863227i \(0.331560\pi\)
\(110\) 335.060 0.290425
\(111\) 0 0
\(112\) 0 0
\(113\) 1417.50 1.18006 0.590032 0.807380i \(-0.299115\pi\)
0.590032 + 0.807380i \(0.299115\pi\)
\(114\) 0 0
\(115\) −491.422 −0.398481
\(116\) −229.068 −0.183349
\(117\) 0 0
\(118\) −597.142 −0.465859
\(119\) 0 0
\(120\) 0 0
\(121\) −839.956 −0.631071
\(122\) −1303.01 −0.966961
\(123\) 0 0
\(124\) −340.050 −0.246269
\(125\) −125.000 −0.0894427
\(126\) 0 0
\(127\) 2413.19 1.68611 0.843055 0.537828i \(-0.180755\pi\)
0.843055 + 0.537828i \(0.180755\pi\)
\(128\) −1679.32 −1.15963
\(129\) 0 0
\(130\) 544.308 0.367223
\(131\) −2248.54 −1.49966 −0.749832 0.661628i \(-0.769866\pi\)
−0.749832 + 0.661628i \(0.769866\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 1569.60 1.01188
\(135\) 0 0
\(136\) −1779.50 −1.12199
\(137\) −1788.25 −1.11519 −0.557595 0.830113i \(-0.688276\pi\)
−0.557595 + 0.830113i \(0.688276\pi\)
\(138\) 0 0
\(139\) 874.354 0.533538 0.266769 0.963761i \(-0.414044\pi\)
0.266769 + 0.963761i \(0.414044\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 2964.56 1.75197
\(143\) 797.704 0.466485
\(144\) 0 0
\(145\) 1000.26 0.572877
\(146\) 168.460 0.0954921
\(147\) 0 0
\(148\) −212.219 −0.117867
\(149\) −2678.32 −1.47260 −0.736298 0.676657i \(-0.763428\pi\)
−0.736298 + 0.676657i \(0.763428\pi\)
\(150\) 0 0
\(151\) 2304.52 1.24198 0.620990 0.783819i \(-0.286731\pi\)
0.620990 + 0.783819i \(0.286731\pi\)
\(152\) 2010.86 1.07304
\(153\) 0 0
\(154\) 0 0
\(155\) 1484.88 0.769473
\(156\) 0 0
\(157\) −2417.93 −1.22912 −0.614561 0.788870i \(-0.710667\pi\)
−0.614561 + 0.788870i \(0.710667\pi\)
\(158\) −2512.91 −1.26529
\(159\) 0 0
\(160\) −257.191 −0.127080
\(161\) 0 0
\(162\) 0 0
\(163\) 1025.39 0.492727 0.246363 0.969178i \(-0.420764\pi\)
0.246363 + 0.969178i \(0.420764\pi\)
\(164\) 333.896 0.158981
\(165\) 0 0
\(166\) 3872.26 1.81051
\(167\) 756.612 0.350589 0.175295 0.984516i \(-0.443912\pi\)
0.175295 + 0.984516i \(0.443912\pi\)
\(168\) 0 0
\(169\) −901.124 −0.410161
\(170\) −1297.97 −0.585585
\(171\) 0 0
\(172\) 198.776 0.0881191
\(173\) 493.967 0.217084 0.108542 0.994092i \(-0.465382\pi\)
0.108542 + 0.994092i \(0.465382\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −1592.14 −0.681888
\(177\) 0 0
\(178\) −1530.69 −0.644549
\(179\) −875.260 −0.365475 −0.182737 0.983162i \(-0.558496\pi\)
−0.182737 + 0.983162i \(0.558496\pi\)
\(180\) 0 0
\(181\) −2385.85 −0.979771 −0.489886 0.871787i \(-0.662961\pi\)
−0.489886 + 0.871787i \(0.662961\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 2037.43 0.816311
\(185\) 926.685 0.368277
\(186\) 0 0
\(187\) −1902.22 −0.743872
\(188\) −60.5928 −0.0235063
\(189\) 0 0
\(190\) 1466.72 0.560037
\(191\) −282.807 −0.107137 −0.0535685 0.998564i \(-0.517060\pi\)
−0.0535685 + 0.998564i \(0.517060\pi\)
\(192\) 0 0
\(193\) −4348.60 −1.62186 −0.810930 0.585143i \(-0.801038\pi\)
−0.810930 + 0.585143i \(0.801038\pi\)
\(194\) −3149.89 −1.16572
\(195\) 0 0
\(196\) 0 0
\(197\) −1145.52 −0.414289 −0.207145 0.978310i \(-0.566417\pi\)
−0.207145 + 0.978310i \(0.566417\pi\)
\(198\) 0 0
\(199\) −345.518 −0.123081 −0.0615405 0.998105i \(-0.519601\pi\)
−0.0615405 + 0.998105i \(0.519601\pi\)
\(200\) 518.248 0.183228
\(201\) 0 0
\(202\) 754.027 0.262640
\(203\) 0 0
\(204\) 0 0
\(205\) −1458.00 −0.496739
\(206\) −5409.99 −1.82977
\(207\) 0 0
\(208\) −2586.45 −0.862202
\(209\) 2149.54 0.711419
\(210\) 0 0
\(211\) 4921.63 1.60578 0.802888 0.596129i \(-0.203295\pi\)
0.802888 + 0.596129i \(0.203295\pi\)
\(212\) −337.449 −0.109321
\(213\) 0 0
\(214\) −989.126 −0.315959
\(215\) −867.983 −0.275330
\(216\) 0 0
\(217\) 0 0
\(218\) −3474.53 −1.07947
\(219\) 0 0
\(220\) −126.868 −0.0388793
\(221\) −3090.17 −0.940575
\(222\) 0 0
\(223\) 2413.56 0.724771 0.362386 0.932028i \(-0.381962\pi\)
0.362386 + 0.932028i \(0.381962\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) −4286.63 −1.26169
\(227\) 2918.57 0.853358 0.426679 0.904403i \(-0.359683\pi\)
0.426679 + 0.904403i \(0.359683\pi\)
\(228\) 0 0
\(229\) −2376.86 −0.685883 −0.342942 0.939357i \(-0.611423\pi\)
−0.342942 + 0.939357i \(0.611423\pi\)
\(230\) 1486.10 0.426045
\(231\) 0 0
\(232\) −4147.07 −1.17357
\(233\) 90.9088 0.0255607 0.0127803 0.999918i \(-0.495932\pi\)
0.0127803 + 0.999918i \(0.495932\pi\)
\(234\) 0 0
\(235\) 264.587 0.0734458
\(236\) 226.103 0.0623646
\(237\) 0 0
\(238\) 0 0
\(239\) −4650.97 −1.25877 −0.629385 0.777093i \(-0.716693\pi\)
−0.629385 + 0.777093i \(0.716693\pi\)
\(240\) 0 0
\(241\) −1764.47 −0.471615 −0.235808 0.971800i \(-0.575774\pi\)
−0.235808 + 0.971800i \(0.575774\pi\)
\(242\) 2540.09 0.674724
\(243\) 0 0
\(244\) 493.376 0.129447
\(245\) 0 0
\(246\) 0 0
\(247\) 3491.94 0.899541
\(248\) −6156.28 −1.57631
\(249\) 0 0
\(250\) 378.010 0.0956297
\(251\) 3136.94 0.788853 0.394427 0.918927i \(-0.370943\pi\)
0.394427 + 0.918927i \(0.370943\pi\)
\(252\) 0 0
\(253\) 2177.93 0.541208
\(254\) −7297.67 −1.80274
\(255\) 0 0
\(256\) 1724.47 0.421014
\(257\) −4213.67 −1.02273 −0.511364 0.859364i \(-0.670860\pi\)
−0.511364 + 0.859364i \(0.670860\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) −206.098 −0.0491602
\(261\) 0 0
\(262\) 6799.76 1.60340
\(263\) 2704.90 0.634188 0.317094 0.948394i \(-0.397293\pi\)
0.317094 + 0.948394i \(0.397293\pi\)
\(264\) 0 0
\(265\) 1473.52 0.341576
\(266\) 0 0
\(267\) 0 0
\(268\) −594.315 −0.135461
\(269\) 1717.93 0.389384 0.194692 0.980864i \(-0.437629\pi\)
0.194692 + 0.980864i \(0.437629\pi\)
\(270\) 0 0
\(271\) 3950.61 0.885543 0.442772 0.896634i \(-0.353995\pi\)
0.442772 + 0.896634i \(0.353995\pi\)
\(272\) 6167.69 1.37489
\(273\) 0 0
\(274\) 5407.82 1.19233
\(275\) 553.988 0.121479
\(276\) 0 0
\(277\) −488.734 −0.106011 −0.0530057 0.998594i \(-0.516880\pi\)
−0.0530057 + 0.998594i \(0.516880\pi\)
\(278\) −2644.11 −0.570444
\(279\) 0 0
\(280\) 0 0
\(281\) 8404.58 1.78425 0.892127 0.451786i \(-0.149213\pi\)
0.892127 + 0.451786i \(0.149213\pi\)
\(282\) 0 0
\(283\) −2370.21 −0.497859 −0.248929 0.968522i \(-0.580079\pi\)
−0.248929 + 0.968522i \(0.580079\pi\)
\(284\) −1122.51 −0.234537
\(285\) 0 0
\(286\) −2412.32 −0.498753
\(287\) 0 0
\(288\) 0 0
\(289\) 2455.86 0.499870
\(290\) −3024.87 −0.612504
\(291\) 0 0
\(292\) −63.7861 −0.0127835
\(293\) 9033.59 1.80119 0.900593 0.434663i \(-0.143132\pi\)
0.900593 + 0.434663i \(0.143132\pi\)
\(294\) 0 0
\(295\) −987.312 −0.194859
\(296\) −3842.02 −0.754435
\(297\) 0 0
\(298\) 8099.46 1.57446
\(299\) 3538.07 0.684321
\(300\) 0 0
\(301\) 0 0
\(302\) −6969.04 −1.32789
\(303\) 0 0
\(304\) −6969.58 −1.31491
\(305\) −2154.40 −0.404460
\(306\) 0 0
\(307\) 3800.19 0.706477 0.353239 0.935533i \(-0.385080\pi\)
0.353239 + 0.935533i \(0.385080\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) −4490.39 −0.822700
\(311\) −2934.67 −0.535080 −0.267540 0.963547i \(-0.586211\pi\)
−0.267540 + 0.963547i \(0.586211\pi\)
\(312\) 0 0
\(313\) −237.718 −0.0429285 −0.0214643 0.999770i \(-0.506833\pi\)
−0.0214643 + 0.999770i \(0.506833\pi\)
\(314\) 7312.02 1.31414
\(315\) 0 0
\(316\) 951.493 0.169385
\(317\) −7181.26 −1.27236 −0.636182 0.771539i \(-0.719487\pi\)
−0.636182 + 0.771539i \(0.719487\pi\)
\(318\) 0 0
\(319\) −4433.06 −0.778068
\(320\) −2096.20 −0.366192
\(321\) 0 0
\(322\) 0 0
\(323\) −8326.93 −1.43444
\(324\) 0 0
\(325\) 899.957 0.153602
\(326\) −3100.85 −0.526810
\(327\) 0 0
\(328\) 6044.86 1.01760
\(329\) 0 0
\(330\) 0 0
\(331\) −807.118 −0.134028 −0.0670139 0.997752i \(-0.521347\pi\)
−0.0670139 + 0.997752i \(0.521347\pi\)
\(332\) −1466.20 −0.242374
\(333\) 0 0
\(334\) −2288.05 −0.374840
\(335\) 2595.16 0.423251
\(336\) 0 0
\(337\) −7760.21 −1.25438 −0.627189 0.778867i \(-0.715795\pi\)
−0.627189 + 0.778867i \(0.715795\pi\)
\(338\) 2725.07 0.438533
\(339\) 0 0
\(340\) 491.464 0.0783924
\(341\) −6580.84 −1.04508
\(342\) 0 0
\(343\) 0 0
\(344\) 3598.64 0.564028
\(345\) 0 0
\(346\) −1493.79 −0.232101
\(347\) −8225.96 −1.27260 −0.636301 0.771441i \(-0.719536\pi\)
−0.636301 + 0.771441i \(0.719536\pi\)
\(348\) 0 0
\(349\) −3056.70 −0.468829 −0.234415 0.972137i \(-0.575317\pi\)
−0.234415 + 0.972137i \(0.575317\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 1139.85 0.172597
\(353\) −7141.73 −1.07682 −0.538408 0.842684i \(-0.680974\pi\)
−0.538408 + 0.842684i \(0.680974\pi\)
\(354\) 0 0
\(355\) 4901.59 0.732815
\(356\) 579.582 0.0862859
\(357\) 0 0
\(358\) 2646.85 0.390756
\(359\) −991.981 −0.145835 −0.0729175 0.997338i \(-0.523231\pi\)
−0.0729175 + 0.997338i \(0.523231\pi\)
\(360\) 0 0
\(361\) 2550.55 0.371855
\(362\) 7214.98 1.04754
\(363\) 0 0
\(364\) 0 0
\(365\) 278.531 0.0399424
\(366\) 0 0
\(367\) 9246.88 1.31521 0.657606 0.753362i \(-0.271569\pi\)
0.657606 + 0.753362i \(0.271569\pi\)
\(368\) −7061.66 −1.00031
\(369\) 0 0
\(370\) −2802.37 −0.393752
\(371\) 0 0
\(372\) 0 0
\(373\) −2729.78 −0.378935 −0.189468 0.981887i \(-0.560676\pi\)
−0.189468 + 0.981887i \(0.560676\pi\)
\(374\) 5752.46 0.795327
\(375\) 0 0
\(376\) −1096.97 −0.150458
\(377\) −7201.53 −0.983814
\(378\) 0 0
\(379\) −12382.1 −1.67817 −0.839084 0.544001i \(-0.816909\pi\)
−0.839084 + 0.544001i \(0.816909\pi\)
\(380\) −555.362 −0.0749723
\(381\) 0 0
\(382\) 855.230 0.114548
\(383\) −8448.60 −1.12716 −0.563581 0.826061i \(-0.690577\pi\)
−0.563581 + 0.826061i \(0.690577\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 13150.5 1.73405
\(387\) 0 0
\(388\) 1192.68 0.156055
\(389\) 556.013 0.0724704 0.0362352 0.999343i \(-0.488463\pi\)
0.0362352 + 0.999343i \(0.488463\pi\)
\(390\) 0 0
\(391\) −8436.94 −1.09124
\(392\) 0 0
\(393\) 0 0
\(394\) 3464.14 0.442947
\(395\) −4154.84 −0.529247
\(396\) 0 0
\(397\) 1490.84 0.188471 0.0942357 0.995550i \(-0.469959\pi\)
0.0942357 + 0.995550i \(0.469959\pi\)
\(398\) 1044.87 0.131595
\(399\) 0 0
\(400\) −1796.23 −0.224529
\(401\) −13949.1 −1.73711 −0.868557 0.495589i \(-0.834952\pi\)
−0.868557 + 0.495589i \(0.834952\pi\)
\(402\) 0 0
\(403\) −10690.6 −1.32143
\(404\) −285.507 −0.0351596
\(405\) 0 0
\(406\) 0 0
\(407\) −4106.98 −0.500185
\(408\) 0 0
\(409\) 2437.58 0.294696 0.147348 0.989085i \(-0.452926\pi\)
0.147348 + 0.989085i \(0.452926\pi\)
\(410\) 4409.12 0.531100
\(411\) 0 0
\(412\) 2048.45 0.244951
\(413\) 0 0
\(414\) 0 0
\(415\) 6402.38 0.757302
\(416\) 1851.69 0.218237
\(417\) 0 0
\(418\) −6500.36 −0.760630
\(419\) −409.228 −0.0477139 −0.0238569 0.999715i \(-0.507595\pi\)
−0.0238569 + 0.999715i \(0.507595\pi\)
\(420\) 0 0
\(421\) −6585.66 −0.762388 −0.381194 0.924495i \(-0.624487\pi\)
−0.381194 + 0.924495i \(0.624487\pi\)
\(422\) −14883.4 −1.71685
\(423\) 0 0
\(424\) −6109.20 −0.699738
\(425\) −2146.05 −0.244938
\(426\) 0 0
\(427\) 0 0
\(428\) 374.525 0.0422975
\(429\) 0 0
\(430\) 2624.85 0.294375
\(431\) −13490.9 −1.50773 −0.753866 0.657028i \(-0.771813\pi\)
−0.753866 + 0.657028i \(0.771813\pi\)
\(432\) 0 0
\(433\) −12067.5 −1.33933 −0.669663 0.742666i \(-0.733561\pi\)
−0.669663 + 0.742666i \(0.733561\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 1315.60 0.144509
\(437\) 9533.87 1.04363
\(438\) 0 0
\(439\) 3668.25 0.398807 0.199403 0.979918i \(-0.436100\pi\)
0.199403 + 0.979918i \(0.436100\pi\)
\(440\) −2296.82 −0.248856
\(441\) 0 0
\(442\) 9344.90 1.00564
\(443\) −12294.1 −1.31853 −0.659264 0.751911i \(-0.729132\pi\)
−0.659264 + 0.751911i \(0.729132\pi\)
\(444\) 0 0
\(445\) −2530.83 −0.269602
\(446\) −7298.80 −0.774906
\(447\) 0 0
\(448\) 0 0
\(449\) −7971.90 −0.837901 −0.418950 0.908009i \(-0.637602\pi\)
−0.418950 + 0.908009i \(0.637602\pi\)
\(450\) 0 0
\(451\) 6461.73 0.674659
\(452\) 1623.10 0.168903
\(453\) 0 0
\(454\) −8825.98 −0.912387
\(455\) 0 0
\(456\) 0 0
\(457\) −13781.4 −1.41065 −0.705324 0.708885i \(-0.749198\pi\)
−0.705324 + 0.708885i \(0.749198\pi\)
\(458\) 7187.80 0.733327
\(459\) 0 0
\(460\) −562.699 −0.0570348
\(461\) −10974.6 −1.10876 −0.554380 0.832263i \(-0.687045\pi\)
−0.554380 + 0.832263i \(0.687045\pi\)
\(462\) 0 0
\(463\) −8803.53 −0.883661 −0.441830 0.897099i \(-0.645671\pi\)
−0.441830 + 0.897099i \(0.645671\pi\)
\(464\) 14373.6 1.43810
\(465\) 0 0
\(466\) −274.915 −0.0273288
\(467\) 7378.30 0.731107 0.365554 0.930790i \(-0.380880\pi\)
0.365554 + 0.930790i \(0.380880\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) −800.132 −0.0785263
\(471\) 0 0
\(472\) 4093.38 0.399180
\(473\) 3846.81 0.373947
\(474\) 0 0
\(475\) 2425.07 0.234252
\(476\) 0 0
\(477\) 0 0
\(478\) 14064.9 1.34584
\(479\) −12014.4 −1.14603 −0.573017 0.819543i \(-0.694227\pi\)
−0.573017 + 0.819543i \(0.694227\pi\)
\(480\) 0 0
\(481\) −6671.81 −0.632450
\(482\) 5335.88 0.504238
\(483\) 0 0
\(484\) −961.786 −0.0903255
\(485\) −5208.02 −0.487596
\(486\) 0 0
\(487\) 2736.76 0.254650 0.127325 0.991861i \(-0.459361\pi\)
0.127325 + 0.991861i \(0.459361\pi\)
\(488\) 8932.10 0.828560
\(489\) 0 0
\(490\) 0 0
\(491\) −2154.33 −0.198012 −0.0990058 0.995087i \(-0.531566\pi\)
−0.0990058 + 0.995087i \(0.531566\pi\)
\(492\) 0 0
\(493\) 17172.9 1.56882
\(494\) −10559.9 −0.961764
\(495\) 0 0
\(496\) 21337.5 1.93162
\(497\) 0 0
\(498\) 0 0
\(499\) −14437.6 −1.29522 −0.647609 0.761973i \(-0.724231\pi\)
−0.647609 + 0.761973i \(0.724231\pi\)
\(500\) −143.130 −0.0128020
\(501\) 0 0
\(502\) −9486.36 −0.843420
\(503\) 10779.5 0.955532 0.477766 0.878487i \(-0.341447\pi\)
0.477766 + 0.878487i \(0.341447\pi\)
\(504\) 0 0
\(505\) 1246.71 0.109857
\(506\) −6586.24 −0.578645
\(507\) 0 0
\(508\) 2763.21 0.241334
\(509\) 20715.5 1.80393 0.901964 0.431811i \(-0.142125\pi\)
0.901964 + 0.431811i \(0.142125\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 8219.63 0.709492
\(513\) 0 0
\(514\) 12742.5 1.09347
\(515\) −8944.86 −0.765354
\(516\) 0 0
\(517\) −1172.63 −0.0997524
\(518\) 0 0
\(519\) 0 0
\(520\) −3731.21 −0.314662
\(521\) −16612.0 −1.39690 −0.698448 0.715661i \(-0.746126\pi\)
−0.698448 + 0.715661i \(0.746126\pi\)
\(522\) 0 0
\(523\) 10772.5 0.900667 0.450333 0.892860i \(-0.351305\pi\)
0.450333 + 0.892860i \(0.351305\pi\)
\(524\) −2574.68 −0.214648
\(525\) 0 0
\(526\) −8179.83 −0.678056
\(527\) 25493.0 2.10720
\(528\) 0 0
\(529\) −2507.18 −0.206064
\(530\) −4456.04 −0.365204
\(531\) 0 0
\(532\) 0 0
\(533\) 10497.1 0.853060
\(534\) 0 0
\(535\) −1635.42 −0.132159
\(536\) −10759.5 −0.867052
\(537\) 0 0
\(538\) −5195.16 −0.416318
\(539\) 0 0
\(540\) 0 0
\(541\) −12431.9 −0.987969 −0.493984 0.869471i \(-0.664460\pi\)
−0.493984 + 0.869471i \(0.664460\pi\)
\(542\) −11946.9 −0.946799
\(543\) 0 0
\(544\) −4415.57 −0.348007
\(545\) −5744.77 −0.451521
\(546\) 0 0
\(547\) 10016.4 0.782947 0.391473 0.920189i \(-0.371966\pi\)
0.391473 + 0.920189i \(0.371966\pi\)
\(548\) −2047.63 −0.159617
\(549\) 0 0
\(550\) −1675.30 −0.129882
\(551\) −19405.6 −1.50038
\(552\) 0 0
\(553\) 0 0
\(554\) 1477.97 0.113345
\(555\) 0 0
\(556\) 1001.17 0.0763655
\(557\) 23857.0 1.81481 0.907407 0.420252i \(-0.138058\pi\)
0.907407 + 0.420252i \(0.138058\pi\)
\(558\) 0 0
\(559\) 6249.18 0.472830
\(560\) 0 0
\(561\) 0 0
\(562\) −25416.1 −1.90767
\(563\) 15827.6 1.18482 0.592411 0.805636i \(-0.298176\pi\)
0.592411 + 0.805636i \(0.298176\pi\)
\(564\) 0 0
\(565\) −7087.51 −0.527741
\(566\) 7167.68 0.532297
\(567\) 0 0
\(568\) −20321.9 −1.50121
\(569\) −15189.3 −1.11910 −0.559552 0.828795i \(-0.689027\pi\)
−0.559552 + 0.828795i \(0.689027\pi\)
\(570\) 0 0
\(571\) −15482.1 −1.13469 −0.567345 0.823481i \(-0.692029\pi\)
−0.567345 + 0.823481i \(0.692029\pi\)
\(572\) 913.406 0.0667682
\(573\) 0 0
\(574\) 0 0
\(575\) 2457.11 0.178206
\(576\) 0 0
\(577\) −10789.8 −0.778481 −0.389241 0.921136i \(-0.627263\pi\)
−0.389241 + 0.921136i \(0.627263\pi\)
\(578\) −7426.72 −0.534447
\(579\) 0 0
\(580\) 1145.34 0.0819961
\(581\) 0 0
\(582\) 0 0
\(583\) −6530.50 −0.463921
\(584\) −1154.79 −0.0818243
\(585\) 0 0
\(586\) −27318.3 −1.92578
\(587\) 8755.42 0.615630 0.307815 0.951446i \(-0.400402\pi\)
0.307815 + 0.951446i \(0.400402\pi\)
\(588\) 0 0
\(589\) −28807.5 −2.01527
\(590\) 2985.71 0.208338
\(591\) 0 0
\(592\) 13316.3 0.924488
\(593\) 11281.5 0.781244 0.390622 0.920551i \(-0.372260\pi\)
0.390622 + 0.920551i \(0.372260\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −3066.80 −0.210773
\(597\) 0 0
\(598\) −10699.4 −0.731657
\(599\) −1099.29 −0.0749845 −0.0374922 0.999297i \(-0.511937\pi\)
−0.0374922 + 0.999297i \(0.511937\pi\)
\(600\) 0 0
\(601\) −19709.3 −1.33770 −0.668852 0.743396i \(-0.733214\pi\)
−0.668852 + 0.743396i \(0.733214\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 2638.77 0.177765
\(605\) 4199.78 0.282224
\(606\) 0 0
\(607\) 13968.7 0.934057 0.467028 0.884242i \(-0.345325\pi\)
0.467028 + 0.884242i \(0.345325\pi\)
\(608\) 4989.66 0.332825
\(609\) 0 0
\(610\) 6515.06 0.432438
\(611\) −1904.94 −0.126130
\(612\) 0 0
\(613\) 5751.23 0.378940 0.189470 0.981887i \(-0.439323\pi\)
0.189470 + 0.981887i \(0.439323\pi\)
\(614\) −11492.1 −0.755346
\(615\) 0 0
\(616\) 0 0
\(617\) −29541.3 −1.92753 −0.963767 0.266745i \(-0.914052\pi\)
−0.963767 + 0.266745i \(0.914052\pi\)
\(618\) 0 0
\(619\) −25408.0 −1.64981 −0.824907 0.565269i \(-0.808772\pi\)
−0.824907 + 0.565269i \(0.808772\pi\)
\(620\) 1700.25 0.110135
\(621\) 0 0
\(622\) 8874.66 0.572093
\(623\) 0 0
\(624\) 0 0
\(625\) 625.000 0.0400000
\(626\) 718.879 0.0458980
\(627\) 0 0
\(628\) −2768.64 −0.175925
\(629\) 15909.7 1.00852
\(630\) 0 0
\(631\) −23162.9 −1.46133 −0.730667 0.682734i \(-0.760791\pi\)
−0.730667 + 0.682734i \(0.760791\pi\)
\(632\) 17225.9 1.08419
\(633\) 0 0
\(634\) 21716.7 1.36038
\(635\) −12065.9 −0.754051
\(636\) 0 0
\(637\) 0 0
\(638\) 13405.9 0.831889
\(639\) 0 0
\(640\) 8396.61 0.518602
\(641\) 15953.6 0.983042 0.491521 0.870866i \(-0.336441\pi\)
0.491521 + 0.870866i \(0.336441\pi\)
\(642\) 0 0
\(643\) 1142.04 0.0700428 0.0350214 0.999387i \(-0.488850\pi\)
0.0350214 + 0.999387i \(0.488850\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 25181.3 1.53366
\(647\) 19269.2 1.17086 0.585432 0.810721i \(-0.300925\pi\)
0.585432 + 0.810721i \(0.300925\pi\)
\(648\) 0 0
\(649\) 4375.67 0.264653
\(650\) −2721.54 −0.164227
\(651\) 0 0
\(652\) 1174.11 0.0705242
\(653\) −21703.4 −1.30064 −0.650320 0.759660i \(-0.725365\pi\)
−0.650320 + 0.759660i \(0.725365\pi\)
\(654\) 0 0
\(655\) 11242.7 0.670670
\(656\) −20951.3 −1.24697
\(657\) 0 0
\(658\) 0 0
\(659\) −25654.1 −1.51645 −0.758225 0.651993i \(-0.773933\pi\)
−0.758225 + 0.651993i \(0.773933\pi\)
\(660\) 0 0
\(661\) 2403.21 0.141413 0.0707065 0.997497i \(-0.477475\pi\)
0.0707065 + 0.997497i \(0.477475\pi\)
\(662\) 2440.79 0.143299
\(663\) 0 0
\(664\) −26544.2 −1.55138
\(665\) 0 0
\(666\) 0 0
\(667\) −19662.0 −1.14140
\(668\) 866.353 0.0501800
\(669\) 0 0
\(670\) −7847.98 −0.452528
\(671\) 9548.08 0.549329
\(672\) 0 0
\(673\) 20175.3 1.15557 0.577787 0.816187i \(-0.303916\pi\)
0.577787 + 0.816187i \(0.303916\pi\)
\(674\) 23467.5 1.34115
\(675\) 0 0
\(676\) −1031.83 −0.0587065
\(677\) 27147.8 1.54117 0.770587 0.637335i \(-0.219963\pi\)
0.770587 + 0.637335i \(0.219963\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 8897.50 0.501770
\(681\) 0 0
\(682\) 19901.0 1.11737
\(683\) 16658.9 0.933287 0.466643 0.884446i \(-0.345463\pi\)
0.466643 + 0.884446i \(0.345463\pi\)
\(684\) 0 0
\(685\) 8941.27 0.498728
\(686\) 0 0
\(687\) 0 0
\(688\) −12472.8 −0.691163
\(689\) −10608.8 −0.586596
\(690\) 0 0
\(691\) −1496.62 −0.0823938 −0.0411969 0.999151i \(-0.513117\pi\)
−0.0411969 + 0.999151i \(0.513117\pi\)
\(692\) 565.613 0.0310714
\(693\) 0 0
\(694\) 24875.9 1.36063
\(695\) −4371.77 −0.238605
\(696\) 0 0
\(697\) −25031.6 −1.36032
\(698\) 9243.70 0.501260
\(699\) 0 0
\(700\) 0 0
\(701\) 1953.58 0.105258 0.0526290 0.998614i \(-0.483240\pi\)
0.0526290 + 0.998614i \(0.483240\pi\)
\(702\) 0 0
\(703\) −17978.2 −0.964525
\(704\) 9290.17 0.497353
\(705\) 0 0
\(706\) 21597.1 1.15130
\(707\) 0 0
\(708\) 0 0
\(709\) 23486.6 1.24409 0.622043 0.782983i \(-0.286303\pi\)
0.622043 + 0.782983i \(0.286303\pi\)
\(710\) −14822.8 −0.783506
\(711\) 0 0
\(712\) 10492.8 0.552294
\(713\) −29188.1 −1.53310
\(714\) 0 0
\(715\) −3988.52 −0.208619
\(716\) −1002.21 −0.0523106
\(717\) 0 0
\(718\) 2999.83 0.155923
\(719\) −20053.2 −1.04014 −0.520069 0.854124i \(-0.674094\pi\)
−0.520069 + 0.854124i \(0.674094\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) −7713.07 −0.397577
\(723\) 0 0
\(724\) −2731.90 −0.140235
\(725\) −5001.30 −0.256198
\(726\) 0 0
\(727\) −28882.5 −1.47344 −0.736721 0.676197i \(-0.763627\pi\)
−0.736721 + 0.676197i \(0.763627\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) −842.300 −0.0427054
\(731\) −14901.9 −0.753989
\(732\) 0 0
\(733\) 1365.77 0.0688210 0.0344105 0.999408i \(-0.489045\pi\)
0.0344105 + 0.999408i \(0.489045\pi\)
\(734\) −27963.3 −1.40619
\(735\) 0 0
\(736\) 5055.58 0.253194
\(737\) −11501.5 −0.574849
\(738\) 0 0
\(739\) −8684.84 −0.432310 −0.216155 0.976359i \(-0.569352\pi\)
−0.216155 + 0.976359i \(0.569352\pi\)
\(740\) 1061.09 0.0527116
\(741\) 0 0
\(742\) 0 0
\(743\) −13165.8 −0.650078 −0.325039 0.945701i \(-0.605377\pi\)
−0.325039 + 0.945701i \(0.605377\pi\)
\(744\) 0 0
\(745\) 13391.6 0.658565
\(746\) 8255.08 0.405147
\(747\) 0 0
\(748\) −2178.12 −0.106471
\(749\) 0 0
\(750\) 0 0
\(751\) 26495.8 1.28741 0.643705 0.765274i \(-0.277396\pi\)
0.643705 + 0.765274i \(0.277396\pi\)
\(752\) 3802.08 0.184372
\(753\) 0 0
\(754\) 21778.0 1.05187
\(755\) −11522.6 −0.555430
\(756\) 0 0
\(757\) 4927.95 0.236604 0.118302 0.992978i \(-0.462255\pi\)
0.118302 + 0.992978i \(0.462255\pi\)
\(758\) 37444.4 1.79425
\(759\) 0 0
\(760\) −10054.3 −0.479879
\(761\) −31910.6 −1.52005 −0.760025 0.649894i \(-0.774813\pi\)
−0.760025 + 0.649894i \(0.774813\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) −323.826 −0.0153346
\(765\) 0 0
\(766\) 25549.2 1.20513
\(767\) 7108.31 0.334636
\(768\) 0 0
\(769\) 8265.63 0.387602 0.193801 0.981041i \(-0.437918\pi\)
0.193801 + 0.981041i \(0.437918\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −4979.33 −0.232137
\(773\) 27988.1 1.30228 0.651141 0.758957i \(-0.274291\pi\)
0.651141 + 0.758957i \(0.274291\pi\)
\(774\) 0 0
\(775\) −7424.40 −0.344119
\(776\) 21592.4 0.998868
\(777\) 0 0
\(778\) −1681.43 −0.0774834
\(779\) 28286.1 1.30097
\(780\) 0 0
\(781\) −21723.4 −0.995293
\(782\) 25514.0 1.16672
\(783\) 0 0
\(784\) 0 0
\(785\) 12089.7 0.549680
\(786\) 0 0
\(787\) 20804.8 0.942328 0.471164 0.882046i \(-0.343834\pi\)
0.471164 + 0.882046i \(0.343834\pi\)
\(788\) −1311.67 −0.0592974
\(789\) 0 0
\(790\) 12564.5 0.565856
\(791\) 0 0
\(792\) 0 0
\(793\) 15510.9 0.694589
\(794\) −4508.42 −0.201508
\(795\) 0 0
\(796\) −395.633 −0.0176166
\(797\) 18327.5 0.814545 0.407273 0.913307i \(-0.366480\pi\)
0.407273 + 0.913307i \(0.366480\pi\)
\(798\) 0 0
\(799\) 4542.54 0.201131
\(800\) 1285.96 0.0568318
\(801\) 0 0
\(802\) 42183.0 1.85728
\(803\) −1234.42 −0.0542489
\(804\) 0 0
\(805\) 0 0
\(806\) 32329.3 1.41284
\(807\) 0 0
\(808\) −5168.83 −0.225048
\(809\) 20786.5 0.903357 0.451679 0.892181i \(-0.350825\pi\)
0.451679 + 0.892181i \(0.350825\pi\)
\(810\) 0 0
\(811\) −2566.26 −0.111114 −0.0555571 0.998456i \(-0.517693\pi\)
−0.0555571 + 0.998456i \(0.517693\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 12419.8 0.534784
\(815\) −5126.93 −0.220354
\(816\) 0 0
\(817\) 16839.4 0.721095
\(818\) −7371.44 −0.315081
\(819\) 0 0
\(820\) −1669.48 −0.0710984
\(821\) 5471.54 0.232592 0.116296 0.993215i \(-0.462898\pi\)
0.116296 + 0.993215i \(0.462898\pi\)
\(822\) 0 0
\(823\) −36025.6 −1.52585 −0.762924 0.646489i \(-0.776237\pi\)
−0.762924 + 0.646489i \(0.776237\pi\)
\(824\) 37085.2 1.56787
\(825\) 0 0
\(826\) 0 0
\(827\) −17877.2 −0.751696 −0.375848 0.926681i \(-0.622649\pi\)
−0.375848 + 0.926681i \(0.622649\pi\)
\(828\) 0 0
\(829\) −40520.0 −1.69761 −0.848804 0.528707i \(-0.822677\pi\)
−0.848804 + 0.528707i \(0.822677\pi\)
\(830\) −19361.3 −0.809687
\(831\) 0 0
\(832\) 15091.9 0.628869
\(833\) 0 0
\(834\) 0 0
\(835\) −3783.06 −0.156788
\(836\) 2461.31 0.101826
\(837\) 0 0
\(838\) 1237.54 0.0510144
\(839\) −16699.4 −0.687162 −0.343581 0.939123i \(-0.611640\pi\)
−0.343581 + 0.939123i \(0.611640\pi\)
\(840\) 0 0
\(841\) 15631.9 0.640939
\(842\) 19915.5 0.815124
\(843\) 0 0
\(844\) 5635.48 0.229835
\(845\) 4505.62 0.183430
\(846\) 0 0
\(847\) 0 0
\(848\) 21174.3 0.857462
\(849\) 0 0
\(850\) 6489.83 0.261881
\(851\) −18215.7 −0.733757
\(852\) 0 0
\(853\) −9970.55 −0.400217 −0.200109 0.979774i \(-0.564130\pi\)
−0.200109 + 0.979774i \(0.564130\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 6780.42 0.270736
\(857\) 44760.9 1.78414 0.892068 0.451901i \(-0.149254\pi\)
0.892068 + 0.451901i \(0.149254\pi\)
\(858\) 0 0
\(859\) −33091.0 −1.31438 −0.657189 0.753726i \(-0.728255\pi\)
−0.657189 + 0.753726i \(0.728255\pi\)
\(860\) −993.878 −0.0394081
\(861\) 0 0
\(862\) 40797.5 1.61203
\(863\) 12939.6 0.510391 0.255196 0.966889i \(-0.417860\pi\)
0.255196 + 0.966889i \(0.417860\pi\)
\(864\) 0 0
\(865\) −2469.83 −0.0970831
\(866\) 36493.1 1.43197
\(867\) 0 0
\(868\) 0 0
\(869\) 18413.8 0.718811
\(870\) 0 0
\(871\) −18684.3 −0.726858
\(872\) 23817.7 0.924966
\(873\) 0 0
\(874\) −28831.2 −1.11582
\(875\) 0 0
\(876\) 0 0
\(877\) −40107.6 −1.54429 −0.772143 0.635449i \(-0.780815\pi\)
−0.772143 + 0.635449i \(0.780815\pi\)
\(878\) −11093.1 −0.426393
\(879\) 0 0
\(880\) 7960.72 0.304950
\(881\) −52262.8 −1.99861 −0.999307 0.0372129i \(-0.988152\pi\)
−0.999307 + 0.0372129i \(0.988152\pi\)
\(882\) 0 0
\(883\) −10356.1 −0.394688 −0.197344 0.980334i \(-0.563232\pi\)
−0.197344 + 0.980334i \(0.563232\pi\)
\(884\) −3538.38 −0.134625
\(885\) 0 0
\(886\) 37178.2 1.40973
\(887\) 26596.6 1.00679 0.503397 0.864056i \(-0.332084\pi\)
0.503397 + 0.864056i \(0.332084\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 7653.43 0.288251
\(891\) 0 0
\(892\) 2763.63 0.103737
\(893\) −5133.14 −0.192356
\(894\) 0 0
\(895\) 4376.30 0.163445
\(896\) 0 0
\(897\) 0 0
\(898\) 24107.7 0.895861
\(899\) 59410.7 2.20407
\(900\) 0 0
\(901\) 25298.0 0.935404
\(902\) −19540.8 −0.721327
\(903\) 0 0
\(904\) 29384.7 1.08111
\(905\) 11929.2 0.438167
\(906\) 0 0
\(907\) 30836.3 1.12889 0.564445 0.825471i \(-0.309090\pi\)
0.564445 + 0.825471i \(0.309090\pi\)
\(908\) 3341.89 0.122141
\(909\) 0 0
\(910\) 0 0
\(911\) 50063.0 1.82070 0.910352 0.413835i \(-0.135811\pi\)
0.910352 + 0.413835i \(0.135811\pi\)
\(912\) 0 0
\(913\) −28374.7 −1.02855
\(914\) 41676.0 1.50823
\(915\) 0 0
\(916\) −2721.61 −0.0981707
\(917\) 0 0
\(918\) 0 0
\(919\) 41903.1 1.50409 0.752043 0.659115i \(-0.229069\pi\)
0.752043 + 0.659115i \(0.229069\pi\)
\(920\) −10187.1 −0.365065
\(921\) 0 0
\(922\) 33188.1 1.18546
\(923\) −35289.8 −1.25848
\(924\) 0 0
\(925\) −4633.42 −0.164698
\(926\) 26622.6 0.944786
\(927\) 0 0
\(928\) −10290.3 −0.364005
\(929\) 26257.8 0.927332 0.463666 0.886010i \(-0.346534\pi\)
0.463666 + 0.886010i \(0.346534\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 104.095 0.00365851
\(933\) 0 0
\(934\) −22312.6 −0.781680
\(935\) 9511.09 0.332670
\(936\) 0 0
\(937\) 42191.5 1.47101 0.735504 0.677520i \(-0.236945\pi\)
0.735504 + 0.677520i \(0.236945\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 302.964 0.0105123
\(941\) −28028.2 −0.970981 −0.485491 0.874242i \(-0.661359\pi\)
−0.485491 + 0.874242i \(0.661359\pi\)
\(942\) 0 0
\(943\) 28659.8 0.989705
\(944\) −14187.5 −0.489157
\(945\) 0 0
\(946\) −11633.1 −0.399813
\(947\) 27776.0 0.953115 0.476558 0.879143i \(-0.341884\pi\)
0.476558 + 0.879143i \(0.341884\pi\)
\(948\) 0 0
\(949\) −2005.33 −0.0685940
\(950\) −7333.60 −0.250456
\(951\) 0 0
\(952\) 0 0
\(953\) −26183.7 −0.890004 −0.445002 0.895530i \(-0.646797\pi\)
−0.445002 + 0.895530i \(0.646797\pi\)
\(954\) 0 0
\(955\) 1414.03 0.0479132
\(956\) −5325.56 −0.180168
\(957\) 0 0
\(958\) 36332.4 1.22531
\(959\) 0 0
\(960\) 0 0
\(961\) 58403.6 1.96045
\(962\) 20176.1 0.676198
\(963\) 0 0
\(964\) −2020.39 −0.0675025
\(965\) 21743.0 0.725318
\(966\) 0 0
\(967\) 12347.8 0.410630 0.205315 0.978696i \(-0.434178\pi\)
0.205315 + 0.978696i \(0.434178\pi\)
\(968\) −17412.2 −0.578151
\(969\) 0 0
\(970\) 15749.5 0.521325
\(971\) −59130.7 −1.95427 −0.977135 0.212620i \(-0.931800\pi\)
−0.977135 + 0.212620i \(0.931800\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) −8276.18 −0.272265
\(975\) 0 0
\(976\) −30958.4 −1.01532
\(977\) −39916.1 −1.30709 −0.653546 0.756887i \(-0.726719\pi\)
−0.653546 + 0.756887i \(0.726719\pi\)
\(978\) 0 0
\(979\) 11216.4 0.366167
\(980\) 0 0
\(981\) 0 0
\(982\) 6514.87 0.211709
\(983\) 13689.2 0.444167 0.222084 0.975028i \(-0.428714\pi\)
0.222084 + 0.975028i \(0.428714\pi\)
\(984\) 0 0
\(985\) 5727.60 0.185276
\(986\) −51932.2 −1.67734
\(987\) 0 0
\(988\) 3998.42 0.128752
\(989\) 17061.8 0.548569
\(990\) 0 0
\(991\) 9050.75 0.290117 0.145059 0.989423i \(-0.453663\pi\)
0.145059 + 0.989423i \(0.453663\pi\)
\(992\) −15275.9 −0.488922
\(993\) 0 0
\(994\) 0 0
\(995\) 1727.59 0.0550435
\(996\) 0 0
\(997\) 12171.9 0.386648 0.193324 0.981135i \(-0.438073\pi\)
0.193324 + 0.981135i \(0.438073\pi\)
\(998\) 43660.3 1.38481
\(999\) 0 0
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 2205.4.a.cb.1.3 8
3.2 odd 2 2205.4.a.cg.1.6 8
7.3 odd 6 315.4.j.j.226.6 yes 16
7.5 odd 6 315.4.j.j.46.6 yes 16
7.6 odd 2 2205.4.a.cc.1.3 8
21.5 even 6 315.4.j.i.46.3 16
21.17 even 6 315.4.j.i.226.3 yes 16
21.20 even 2 2205.4.a.cf.1.6 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
315.4.j.i.46.3 16 21.5 even 6
315.4.j.i.226.3 yes 16 21.17 even 6
315.4.j.j.46.6 yes 16 7.5 odd 6
315.4.j.j.226.6 yes 16 7.3 odd 6
2205.4.a.cb.1.3 8 1.1 even 1 trivial
2205.4.a.cc.1.3 8 7.6 odd 2
2205.4.a.cf.1.6 8 21.20 even 2
2205.4.a.cg.1.6 8 3.2 odd 2