Properties

Label 400.6.a.y.1.1
Level $400$
Weight $6$
Character 400.1
Self dual yes
Analytic conductor $64.154$
Analytic rank $1$
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] = [400,6,Mod(1,400)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(400, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("400.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 400 = 2^{4} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 400.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: \(64.1535279252\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.47217.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 38x - 24 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{4}\cdot 5 \)
Twist minimal: no (minimal twist has level 200)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-5.30760\) of defining polynomial
Character \(\chi\) \(=\) 400.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-19.0934 q^{3} -210.117 q^{7} +121.557 q^{9} +O(q^{10})\) \(q-19.0934 q^{3} -210.117 q^{7} +121.557 q^{9} +18.2768 q^{11} +834.918 q^{13} -292.543 q^{17} +2287.27 q^{19} +4011.84 q^{21} -3788.91 q^{23} +2318.76 q^{27} +4554.90 q^{29} +1458.49 q^{31} -348.965 q^{33} +5509.69 q^{37} -15941.4 q^{39} -616.780 q^{41} -15866.5 q^{43} +29157.2 q^{47} +27342.2 q^{49} +5585.64 q^{51} -36944.9 q^{53} -43671.6 q^{57} -18713.3 q^{59} +33063.9 q^{61} -25541.2 q^{63} -5592.37 q^{67} +72343.0 q^{69} -31534.1 q^{71} +86147.7 q^{73} -3840.26 q^{77} +34527.0 q^{79} -73811.2 q^{81} -89370.3 q^{83} -86968.5 q^{87} +45489.4 q^{89} -175431. q^{91} -27847.6 q^{93} -133312. q^{97} +2221.67 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + q^{3} - 70 q^{7} + 154 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + q^{3} - 70 q^{7} + 154 q^{9} + 19 q^{11} + 196 q^{13} - 1223 q^{17} - 1221 q^{19} + 958 q^{21} - 2490 q^{23} + 4123 q^{27} + 11912 q^{29} - 7442 q^{31} + 6969 q^{33} - 14766 q^{37} - 27396 q^{39} + 3223 q^{41} - 41060 q^{43} + 29188 q^{47} + 66423 q^{49} - 43165 q^{51} + 12878 q^{53} - 77791 q^{57} - 64912 q^{59} + 22478 q^{61} - 112916 q^{63} + 26499 q^{67} + 178642 q^{69} - 86676 q^{71} - 8305 q^{73} - 106822 q^{77} - 21982 q^{79} - 71453 q^{81} - 213353 q^{83} + 14952 q^{87} + 182381 q^{89} - 149224 q^{91} - 196470 q^{93} - 76342 q^{97} + 149130 q^{99}+O(q^{100}) \) 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
\(3\) −19.0934 −1.22484 −0.612420 0.790532i \(-0.709804\pi\)
−0.612420 + 0.790532i \(0.709804\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) −210.117 −1.62075 −0.810376 0.585911i \(-0.800737\pi\)
−0.810376 + 0.585911i \(0.800737\pi\)
\(8\) 0 0
\(9\) 121.557 0.500234
\(10\) 0 0
\(11\) 18.2768 0.0455426 0.0227713 0.999741i \(-0.492751\pi\)
0.0227713 + 0.999741i \(0.492751\pi\)
\(12\) 0 0
\(13\) 834.918 1.37021 0.685103 0.728447i \(-0.259757\pi\)
0.685103 + 0.728447i \(0.259757\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −292.543 −0.245509 −0.122755 0.992437i \(-0.539173\pi\)
−0.122755 + 0.992437i \(0.539173\pi\)
\(18\) 0 0
\(19\) 2287.27 1.45356 0.726779 0.686871i \(-0.241016\pi\)
0.726779 + 0.686871i \(0.241016\pi\)
\(20\) 0 0
\(21\) 4011.84 1.98516
\(22\) 0 0
\(23\) −3788.91 −1.49346 −0.746731 0.665126i \(-0.768378\pi\)
−0.746731 + 0.665126i \(0.768378\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 2318.76 0.612133
\(28\) 0 0
\(29\) 4554.90 1.00574 0.502868 0.864363i \(-0.332278\pi\)
0.502868 + 0.864363i \(0.332278\pi\)
\(30\) 0 0
\(31\) 1458.49 0.272584 0.136292 0.990669i \(-0.456481\pi\)
0.136292 + 0.990669i \(0.456481\pi\)
\(32\) 0 0
\(33\) −348.965 −0.0557824
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 5509.69 0.661642 0.330821 0.943693i \(-0.392674\pi\)
0.330821 + 0.943693i \(0.392674\pi\)
\(38\) 0 0
\(39\) −15941.4 −1.67828
\(40\) 0 0
\(41\) −616.780 −0.0573021 −0.0286511 0.999589i \(-0.509121\pi\)
−0.0286511 + 0.999589i \(0.509121\pi\)
\(42\) 0 0
\(43\) −15866.5 −1.30861 −0.654303 0.756232i \(-0.727038\pi\)
−0.654303 + 0.756232i \(0.727038\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 29157.2 1.92531 0.962655 0.270730i \(-0.0872650\pi\)
0.962655 + 0.270730i \(0.0872650\pi\)
\(48\) 0 0
\(49\) 27342.2 1.62683
\(50\) 0 0
\(51\) 5585.64 0.300710
\(52\) 0 0
\(53\) −36944.9 −1.80661 −0.903305 0.428998i \(-0.858867\pi\)
−0.903305 + 0.428998i \(0.858867\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −43671.6 −1.78038
\(58\) 0 0
\(59\) −18713.3 −0.699874 −0.349937 0.936773i \(-0.613797\pi\)
−0.349937 + 0.936773i \(0.613797\pi\)
\(60\) 0 0
\(61\) 33063.9 1.13770 0.568852 0.822440i \(-0.307388\pi\)
0.568852 + 0.822440i \(0.307388\pi\)
\(62\) 0 0
\(63\) −25541.2 −0.810755
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −5592.37 −0.152198 −0.0760990 0.997100i \(-0.524246\pi\)
−0.0760990 + 0.997100i \(0.524246\pi\)
\(68\) 0 0
\(69\) 72343.0 1.82925
\(70\) 0 0
\(71\) −31534.1 −0.742394 −0.371197 0.928554i \(-0.621053\pi\)
−0.371197 + 0.928554i \(0.621053\pi\)
\(72\) 0 0
\(73\) 86147.7 1.89207 0.946034 0.324067i \(-0.105050\pi\)
0.946034 + 0.324067i \(0.105050\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −3840.26 −0.0738132
\(78\) 0 0
\(79\) 34527.0 0.622430 0.311215 0.950340i \(-0.399264\pi\)
0.311215 + 0.950340i \(0.399264\pi\)
\(80\) 0 0
\(81\) −73811.2 −1.25000
\(82\) 0 0
\(83\) −89370.3 −1.42396 −0.711980 0.702199i \(-0.752201\pi\)
−0.711980 + 0.702199i \(0.752201\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) −86968.5 −1.23187
\(88\) 0 0
\(89\) 45489.4 0.608745 0.304373 0.952553i \(-0.401553\pi\)
0.304373 + 0.952553i \(0.401553\pi\)
\(90\) 0 0
\(91\) −175431. −2.22076
\(92\) 0 0
\(93\) −27847.6 −0.333872
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −133312. −1.43860 −0.719302 0.694697i \(-0.755538\pi\)
−0.719302 + 0.694697i \(0.755538\pi\)
\(98\) 0 0
\(99\) 2221.67 0.0227820
\(100\) 0 0
\(101\) −67742.9 −0.660785 −0.330393 0.943844i \(-0.607181\pi\)
−0.330393 + 0.943844i \(0.607181\pi\)
\(102\) 0 0
\(103\) −62032.7 −0.576139 −0.288070 0.957609i \(-0.593013\pi\)
−0.288070 + 0.957609i \(0.593013\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 22529.0 0.190231 0.0951156 0.995466i \(-0.469678\pi\)
0.0951156 + 0.995466i \(0.469678\pi\)
\(108\) 0 0
\(109\) −212316. −1.71166 −0.855829 0.517259i \(-0.826953\pi\)
−0.855829 + 0.517259i \(0.826953\pi\)
\(110\) 0 0
\(111\) −105199. −0.810406
\(112\) 0 0
\(113\) −198742. −1.46417 −0.732087 0.681211i \(-0.761454\pi\)
−0.732087 + 0.681211i \(0.761454\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 101490. 0.685423
\(118\) 0 0
\(119\) 61468.4 0.397910
\(120\) 0 0
\(121\) −160717. −0.997926
\(122\) 0 0
\(123\) 11776.4 0.0701860
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) −84802.8 −0.466553 −0.233276 0.972410i \(-0.574945\pi\)
−0.233276 + 0.972410i \(0.574945\pi\)
\(128\) 0 0
\(129\) 302944. 1.60283
\(130\) 0 0
\(131\) 220540. 1.12282 0.561410 0.827538i \(-0.310259\pi\)
0.561410 + 0.827538i \(0.310259\pi\)
\(132\) 0 0
\(133\) −480594. −2.35586
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −46832.2 −0.213178 −0.106589 0.994303i \(-0.533993\pi\)
−0.106589 + 0.994303i \(0.533993\pi\)
\(138\) 0 0
\(139\) 169151. 0.742569 0.371284 0.928519i \(-0.378917\pi\)
0.371284 + 0.928519i \(0.378917\pi\)
\(140\) 0 0
\(141\) −556709. −2.35820
\(142\) 0 0
\(143\) 15259.6 0.0624027
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) −522055. −1.99261
\(148\) 0 0
\(149\) −280036. −1.03335 −0.516675 0.856181i \(-0.672831\pi\)
−0.516675 + 0.856181i \(0.672831\pi\)
\(150\) 0 0
\(151\) −98970.7 −0.353235 −0.176618 0.984280i \(-0.556516\pi\)
−0.176618 + 0.984280i \(0.556516\pi\)
\(152\) 0 0
\(153\) −35560.7 −0.122812
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 31033.4 0.100480 0.0502401 0.998737i \(-0.484001\pi\)
0.0502401 + 0.998737i \(0.484001\pi\)
\(158\) 0 0
\(159\) 705402. 2.21281
\(160\) 0 0
\(161\) 796114. 2.42053
\(162\) 0 0
\(163\) 98784.1 0.291218 0.145609 0.989342i \(-0.453486\pi\)
0.145609 + 0.989342i \(0.453486\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 418435. 1.16101 0.580506 0.814256i \(-0.302855\pi\)
0.580506 + 0.814256i \(0.302855\pi\)
\(168\) 0 0
\(169\) 325796. 0.877462
\(170\) 0 0
\(171\) 278033. 0.727120
\(172\) 0 0
\(173\) −284193. −0.721934 −0.360967 0.932579i \(-0.617553\pi\)
−0.360967 + 0.932579i \(0.617553\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 357299. 0.857233
\(178\) 0 0
\(179\) 783123. 1.82683 0.913414 0.407032i \(-0.133436\pi\)
0.913414 + 0.407032i \(0.133436\pi\)
\(180\) 0 0
\(181\) −64283.0 −0.145848 −0.0729238 0.997338i \(-0.523233\pi\)
−0.0729238 + 0.997338i \(0.523233\pi\)
\(182\) 0 0
\(183\) −631302. −1.39351
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) −5346.75 −0.0111811
\(188\) 0 0
\(189\) −487211. −0.992116
\(190\) 0 0
\(191\) 112334. 0.222806 0.111403 0.993775i \(-0.464465\pi\)
0.111403 + 0.993775i \(0.464465\pi\)
\(192\) 0 0
\(193\) −161036. −0.311193 −0.155596 0.987821i \(-0.549730\pi\)
−0.155596 + 0.987821i \(0.549730\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 456071. 0.837273 0.418636 0.908154i \(-0.362508\pi\)
0.418636 + 0.908154i \(0.362508\pi\)
\(198\) 0 0
\(199\) −569030. −1.01860 −0.509299 0.860590i \(-0.670095\pi\)
−0.509299 + 0.860590i \(0.670095\pi\)
\(200\) 0 0
\(201\) 106777. 0.186418
\(202\) 0 0
\(203\) −957064. −1.63005
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) −460568. −0.747081
\(208\) 0 0
\(209\) 41803.8 0.0661988
\(210\) 0 0
\(211\) −402226. −0.621963 −0.310981 0.950416i \(-0.600658\pi\)
−0.310981 + 0.950416i \(0.600658\pi\)
\(212\) 0 0
\(213\) 602092. 0.909314
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) −306455. −0.441791
\(218\) 0 0
\(219\) −1.64485e6 −2.31748
\(220\) 0 0
\(221\) −244250. −0.336398
\(222\) 0 0
\(223\) 1.24567e6 1.67742 0.838709 0.544579i \(-0.183311\pi\)
0.838709 + 0.544579i \(0.183311\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 475220. 0.612111 0.306055 0.952014i \(-0.400991\pi\)
0.306055 + 0.952014i \(0.400991\pi\)
\(228\) 0 0
\(229\) −399176. −0.503009 −0.251505 0.967856i \(-0.580925\pi\)
−0.251505 + 0.967856i \(0.580925\pi\)
\(230\) 0 0
\(231\) 73323.5 0.0904094
\(232\) 0 0
\(233\) 280117. 0.338026 0.169013 0.985614i \(-0.445942\pi\)
0.169013 + 0.985614i \(0.445942\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) −659236. −0.762378
\(238\) 0 0
\(239\) −190860. −0.216133 −0.108067 0.994144i \(-0.534466\pi\)
−0.108067 + 0.994144i \(0.534466\pi\)
\(240\) 0 0
\(241\) −31402.5 −0.0348275 −0.0174137 0.999848i \(-0.505543\pi\)
−0.0174137 + 0.999848i \(0.505543\pi\)
\(242\) 0 0
\(243\) 845847. 0.918917
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 1.90968e6 1.99167
\(248\) 0 0
\(249\) 1.70638e6 1.74412
\(250\) 0 0
\(251\) −47989.7 −0.0480799 −0.0240400 0.999711i \(-0.507653\pi\)
−0.0240400 + 0.999711i \(0.507653\pi\)
\(252\) 0 0
\(253\) −69248.9 −0.0680161
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −1.79852e6 −1.69857 −0.849284 0.527937i \(-0.822966\pi\)
−0.849284 + 0.527937i \(0.822966\pi\)
\(258\) 0 0
\(259\) −1.15768e6 −1.07236
\(260\) 0 0
\(261\) 553680. 0.503104
\(262\) 0 0
\(263\) 967118. 0.862164 0.431082 0.902313i \(-0.358132\pi\)
0.431082 + 0.902313i \(0.358132\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) −868547. −0.745616
\(268\) 0 0
\(269\) 878552. 0.740264 0.370132 0.928979i \(-0.379312\pi\)
0.370132 + 0.928979i \(0.379312\pi\)
\(270\) 0 0
\(271\) −1.19503e6 −0.988455 −0.494228 0.869333i \(-0.664549\pi\)
−0.494228 + 0.869333i \(0.664549\pi\)
\(272\) 0 0
\(273\) 3.34956e6 2.72008
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 850774. 0.666215 0.333108 0.942889i \(-0.391903\pi\)
0.333108 + 0.942889i \(0.391903\pi\)
\(278\) 0 0
\(279\) 177290. 0.136356
\(280\) 0 0
\(281\) −1.05422e6 −0.796462 −0.398231 0.917285i \(-0.630376\pi\)
−0.398231 + 0.917285i \(0.630376\pi\)
\(282\) 0 0
\(283\) −226845. −0.168369 −0.0841846 0.996450i \(-0.526829\pi\)
−0.0841846 + 0.996450i \(0.526829\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 129596. 0.0928725
\(288\) 0 0
\(289\) −1.33428e6 −0.939725
\(290\) 0 0
\(291\) 2.54538e6 1.76206
\(292\) 0 0
\(293\) 1.64442e6 1.11904 0.559519 0.828818i \(-0.310986\pi\)
0.559519 + 0.828818i \(0.310986\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 42379.4 0.0278781
\(298\) 0 0
\(299\) −3.16343e6 −2.04635
\(300\) 0 0
\(301\) 3.33382e6 2.12093
\(302\) 0 0
\(303\) 1.29344e6 0.809356
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) −2.09665e6 −1.26964 −0.634820 0.772660i \(-0.718926\pi\)
−0.634820 + 0.772660i \(0.718926\pi\)
\(308\) 0 0
\(309\) 1.18441e6 0.705678
\(310\) 0 0
\(311\) −1.46148e6 −0.856825 −0.428413 0.903583i \(-0.640927\pi\)
−0.428413 + 0.903583i \(0.640927\pi\)
\(312\) 0 0
\(313\) 1.03377e6 0.596437 0.298219 0.954498i \(-0.403608\pi\)
0.298219 + 0.954498i \(0.403608\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 148727. 0.0831272 0.0415636 0.999136i \(-0.486766\pi\)
0.0415636 + 0.999136i \(0.486766\pi\)
\(318\) 0 0
\(319\) 83248.9 0.0458038
\(320\) 0 0
\(321\) −430154. −0.233003
\(322\) 0 0
\(323\) −669125. −0.356862
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 4.05383e6 2.09651
\(328\) 0 0
\(329\) −6.12642e6 −3.12045
\(330\) 0 0
\(331\) −1.54319e6 −0.774194 −0.387097 0.922039i \(-0.626522\pi\)
−0.387097 + 0.922039i \(0.626522\pi\)
\(332\) 0 0
\(333\) 669741. 0.330976
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 3.22154e6 1.54522 0.772608 0.634883i \(-0.218952\pi\)
0.772608 + 0.634883i \(0.218952\pi\)
\(338\) 0 0
\(339\) 3.79465e6 1.79338
\(340\) 0 0
\(341\) 26656.6 0.0124142
\(342\) 0 0
\(343\) −2.21363e6 −1.01594
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 217036. 0.0967626 0.0483813 0.998829i \(-0.484594\pi\)
0.0483813 + 0.998829i \(0.484594\pi\)
\(348\) 0 0
\(349\) −37149.9 −0.0163265 −0.00816326 0.999967i \(-0.502598\pi\)
−0.00816326 + 0.999967i \(0.502598\pi\)
\(350\) 0 0
\(351\) 1.93597e6 0.838748
\(352\) 0 0
\(353\) −2.92411e6 −1.24898 −0.624492 0.781031i \(-0.714694\pi\)
−0.624492 + 0.781031i \(0.714694\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) −1.17364e6 −0.487376
\(358\) 0 0
\(359\) −1.55303e6 −0.635980 −0.317990 0.948094i \(-0.603008\pi\)
−0.317990 + 0.948094i \(0.603008\pi\)
\(360\) 0 0
\(361\) 2.75549e6 1.11283
\(362\) 0 0
\(363\) 3.06863e6 1.22230
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 413541. 0.160271 0.0801353 0.996784i \(-0.474465\pi\)
0.0801353 + 0.996784i \(0.474465\pi\)
\(368\) 0 0
\(369\) −74973.9 −0.0286645
\(370\) 0 0
\(371\) 7.76275e6 2.92807
\(372\) 0 0
\(373\) −2.06546e6 −0.768677 −0.384338 0.923192i \(-0.625570\pi\)
−0.384338 + 0.923192i \(0.625570\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 3.80297e6 1.37807
\(378\) 0 0
\(379\) −2.27742e6 −0.814412 −0.407206 0.913336i \(-0.633497\pi\)
−0.407206 + 0.913336i \(0.633497\pi\)
\(380\) 0 0
\(381\) 1.61917e6 0.571453
\(382\) 0 0
\(383\) 2.88095e6 1.00355 0.501774 0.864999i \(-0.332681\pi\)
0.501774 + 0.864999i \(0.332681\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −1.92868e6 −0.654610
\(388\) 0 0
\(389\) 634299. 0.212530 0.106265 0.994338i \(-0.466111\pi\)
0.106265 + 0.994338i \(0.466111\pi\)
\(390\) 0 0
\(391\) 1.10842e6 0.366659
\(392\) 0 0
\(393\) −4.21086e6 −1.37527
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 4.36800e6 1.39093 0.695466 0.718559i \(-0.255198\pi\)
0.695466 + 0.718559i \(0.255198\pi\)
\(398\) 0 0
\(399\) 9.17616e6 2.88555
\(400\) 0 0
\(401\) −5.50000e6 −1.70806 −0.854028 0.520228i \(-0.825847\pi\)
−0.854028 + 0.520228i \(0.825847\pi\)
\(402\) 0 0
\(403\) 1.21772e6 0.373496
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 100699. 0.0301329
\(408\) 0 0
\(409\) 3.21295e6 0.949721 0.474860 0.880061i \(-0.342499\pi\)
0.474860 + 0.880061i \(0.342499\pi\)
\(410\) 0 0
\(411\) 894184. 0.261109
\(412\) 0 0
\(413\) 3.93198e6 1.13432
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) −3.22966e6 −0.909528
\(418\) 0 0
\(419\) −890973. −0.247930 −0.123965 0.992287i \(-0.539561\pi\)
−0.123965 + 0.992287i \(0.539561\pi\)
\(420\) 0 0
\(421\) −2.87922e6 −0.791716 −0.395858 0.918312i \(-0.629553\pi\)
−0.395858 + 0.918312i \(0.629553\pi\)
\(422\) 0 0
\(423\) 3.54426e6 0.963106
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) −6.94729e6 −1.84394
\(428\) 0 0
\(429\) −291357. −0.0764333
\(430\) 0 0
\(431\) −6.29450e6 −1.63218 −0.816090 0.577924i \(-0.803863\pi\)
−0.816090 + 0.577924i \(0.803863\pi\)
\(432\) 0 0
\(433\) −5.96375e6 −1.52862 −0.764311 0.644848i \(-0.776921\pi\)
−0.764311 + 0.644848i \(0.776921\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −8.66623e6 −2.17084
\(438\) 0 0
\(439\) 4.91648e6 1.21757 0.608784 0.793336i \(-0.291658\pi\)
0.608784 + 0.793336i \(0.291658\pi\)
\(440\) 0 0
\(441\) 3.32363e6 0.813798
\(442\) 0 0
\(443\) −5.53513e6 −1.34004 −0.670021 0.742342i \(-0.733715\pi\)
−0.670021 + 0.742342i \(0.733715\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 5.34683e6 1.26569
\(448\) 0 0
\(449\) −7.20215e6 −1.68596 −0.842978 0.537948i \(-0.819200\pi\)
−0.842978 + 0.537948i \(0.819200\pi\)
\(450\) 0 0
\(451\) −11272.7 −0.00260969
\(452\) 0 0
\(453\) 1.88968e6 0.432657
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 5.49984e6 1.23185 0.615927 0.787803i \(-0.288782\pi\)
0.615927 + 0.787803i \(0.288782\pi\)
\(458\) 0 0
\(459\) −678338. −0.150285
\(460\) 0 0
\(461\) −3.79223e6 −0.831078 −0.415539 0.909575i \(-0.636407\pi\)
−0.415539 + 0.909575i \(0.636407\pi\)
\(462\) 0 0
\(463\) 4.78947e6 1.03833 0.519164 0.854674i \(-0.326243\pi\)
0.519164 + 0.854674i \(0.326243\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −5.62497e6 −1.19352 −0.596758 0.802422i \(-0.703545\pi\)
−0.596758 + 0.802422i \(0.703545\pi\)
\(468\) 0 0
\(469\) 1.17505e6 0.246675
\(470\) 0 0
\(471\) −592532. −0.123072
\(472\) 0 0
\(473\) −289988. −0.0595973
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) −4.49090e6 −0.903728
\(478\) 0 0
\(479\) −760544. −0.151456 −0.0757278 0.997129i \(-0.524128\pi\)
−0.0757278 + 0.997129i \(0.524128\pi\)
\(480\) 0 0
\(481\) 4.60014e6 0.906586
\(482\) 0 0
\(483\) −1.52005e7 −2.96476
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 2.52969e6 0.483332 0.241666 0.970360i \(-0.422306\pi\)
0.241666 + 0.970360i \(0.422306\pi\)
\(488\) 0 0
\(489\) −1.88612e6 −0.356695
\(490\) 0 0
\(491\) −4.83619e6 −0.905316 −0.452658 0.891684i \(-0.649524\pi\)
−0.452658 + 0.891684i \(0.649524\pi\)
\(492\) 0 0
\(493\) −1.33251e6 −0.246918
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 6.62585e6 1.20324
\(498\) 0 0
\(499\) −9.17357e6 −1.64925 −0.824626 0.565678i \(-0.808615\pi\)
−0.824626 + 0.565678i \(0.808615\pi\)
\(500\) 0 0
\(501\) −7.98933e6 −1.42205
\(502\) 0 0
\(503\) −1.02119e7 −1.79964 −0.899820 0.436262i \(-0.856302\pi\)
−0.899820 + 0.436262i \(0.856302\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) −6.22054e6 −1.07475
\(508\) 0 0
\(509\) 6.61475e6 1.13167 0.565834 0.824519i \(-0.308554\pi\)
0.565834 + 0.824519i \(0.308554\pi\)
\(510\) 0 0
\(511\) −1.81011e7 −3.06657
\(512\) 0 0
\(513\) 5.30362e6 0.889772
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 532899. 0.0876836
\(518\) 0 0
\(519\) 5.42620e6 0.884254
\(520\) 0 0
\(521\) −9.91327e6 −1.60001 −0.800005 0.599993i \(-0.795170\pi\)
−0.800005 + 0.599993i \(0.795170\pi\)
\(522\) 0 0
\(523\) 4.66698e6 0.746074 0.373037 0.927816i \(-0.378316\pi\)
0.373037 + 0.927816i \(0.378316\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −426673. −0.0669220
\(528\) 0 0
\(529\) 7.91946e6 1.23043
\(530\) 0 0
\(531\) −2.27473e6 −0.350101
\(532\) 0 0
\(533\) −514961. −0.0785157
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) −1.49525e7 −2.23757
\(538\) 0 0
\(539\) 499727. 0.0740902
\(540\) 0 0
\(541\) −1.47136e6 −0.216136 −0.108068 0.994144i \(-0.534466\pi\)
−0.108068 + 0.994144i \(0.534466\pi\)
\(542\) 0 0
\(543\) 1.22738e6 0.178640
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 4.55704e6 0.651200 0.325600 0.945508i \(-0.394434\pi\)
0.325600 + 0.945508i \(0.394434\pi\)
\(548\) 0 0
\(549\) 4.01915e6 0.569119
\(550\) 0 0
\(551\) 1.04183e7 1.46190
\(552\) 0 0
\(553\) −7.25470e6 −1.00880
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −1.05967e7 −1.44721 −0.723605 0.690215i \(-0.757516\pi\)
−0.723605 + 0.690215i \(0.757516\pi\)
\(558\) 0 0
\(559\) −1.32472e7 −1.79306
\(560\) 0 0
\(561\) 102087. 0.0136951
\(562\) 0 0
\(563\) −2.17451e6 −0.289129 −0.144564 0.989495i \(-0.546178\pi\)
−0.144564 + 0.989495i \(0.546178\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 1.55090e7 2.02594
\(568\) 0 0
\(569\) 9.36030e6 1.21202 0.606009 0.795458i \(-0.292769\pi\)
0.606009 + 0.795458i \(0.292769\pi\)
\(570\) 0 0
\(571\) −1.64209e6 −0.210770 −0.105385 0.994432i \(-0.533607\pi\)
−0.105385 + 0.994432i \(0.533607\pi\)
\(572\) 0 0
\(573\) −2.14484e6 −0.272902
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) −5.20742e6 −0.651153 −0.325576 0.945516i \(-0.605558\pi\)
−0.325576 + 0.945516i \(0.605558\pi\)
\(578\) 0 0
\(579\) 3.07472e6 0.381161
\(580\) 0 0
\(581\) 1.87782e7 2.30789
\(582\) 0 0
\(583\) −675233. −0.0822777
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −1.06067e7 −1.27053 −0.635267 0.772292i \(-0.719110\pi\)
−0.635267 + 0.772292i \(0.719110\pi\)
\(588\) 0 0
\(589\) 3.33596e6 0.396217
\(590\) 0 0
\(591\) −8.70794e6 −1.02553
\(592\) 0 0
\(593\) 1.99180e6 0.232600 0.116300 0.993214i \(-0.462897\pi\)
0.116300 + 0.993214i \(0.462897\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 1.08647e7 1.24762
\(598\) 0 0
\(599\) 1.16108e7 1.32219 0.661095 0.750303i \(-0.270092\pi\)
0.661095 + 0.750303i \(0.270092\pi\)
\(600\) 0 0
\(601\) −628490. −0.0709761 −0.0354881 0.999370i \(-0.511299\pi\)
−0.0354881 + 0.999370i \(0.511299\pi\)
\(602\) 0 0
\(603\) −679791. −0.0761346
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) −707145. −0.0778998 −0.0389499 0.999241i \(-0.512401\pi\)
−0.0389499 + 0.999241i \(0.512401\pi\)
\(608\) 0 0
\(609\) 1.82736e7 1.99655
\(610\) 0 0
\(611\) 2.43439e7 2.63807
\(612\) 0 0
\(613\) −1.37228e7 −1.47500 −0.737500 0.675347i \(-0.763994\pi\)
−0.737500 + 0.675347i \(0.763994\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −1.47885e7 −1.56390 −0.781952 0.623339i \(-0.785776\pi\)
−0.781952 + 0.623339i \(0.785776\pi\)
\(618\) 0 0
\(619\) −2.72872e6 −0.286241 −0.143121 0.989705i \(-0.545714\pi\)
−0.143121 + 0.989705i \(0.545714\pi\)
\(620\) 0 0
\(621\) −8.78556e6 −0.914198
\(622\) 0 0
\(623\) −9.55811e6 −0.986624
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) −798176. −0.0810830
\(628\) 0 0
\(629\) −1.61183e6 −0.162439
\(630\) 0 0
\(631\) −5.75522e6 −0.575425 −0.287713 0.957717i \(-0.592895\pi\)
−0.287713 + 0.957717i \(0.592895\pi\)
\(632\) 0 0
\(633\) 7.67986e6 0.761805
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 2.28285e7 2.22910
\(638\) 0 0
\(639\) −3.83319e6 −0.371371
\(640\) 0 0
\(641\) 3.94835e6 0.379551 0.189776 0.981827i \(-0.439224\pi\)
0.189776 + 0.981827i \(0.439224\pi\)
\(642\) 0 0
\(643\) −1.12977e7 −1.07761 −0.538804 0.842431i \(-0.681124\pi\)
−0.538804 + 0.842431i \(0.681124\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 1.60010e7 1.50275 0.751376 0.659875i \(-0.229391\pi\)
0.751376 + 0.659875i \(0.229391\pi\)
\(648\) 0 0
\(649\) −342018. −0.0318741
\(650\) 0 0
\(651\) 5.85125e6 0.541124
\(652\) 0 0
\(653\) −8.51457e6 −0.781412 −0.390706 0.920516i \(-0.627769\pi\)
−0.390706 + 0.920516i \(0.627769\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 1.04719e7 0.946477
\(658\) 0 0
\(659\) −1.15442e7 −1.03550 −0.517751 0.855531i \(-0.673231\pi\)
−0.517751 + 0.855531i \(0.673231\pi\)
\(660\) 0 0
\(661\) 1.51328e7 1.34715 0.673576 0.739118i \(-0.264757\pi\)
0.673576 + 0.739118i \(0.264757\pi\)
\(662\) 0 0
\(663\) 4.66355e6 0.412034
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −1.72581e7 −1.50203
\(668\) 0 0
\(669\) −2.37841e7 −2.05457
\(670\) 0 0
\(671\) 604301. 0.0518140
\(672\) 0 0
\(673\) 1.82845e7 1.55613 0.778063 0.628187i \(-0.216203\pi\)
0.778063 + 0.628187i \(0.216203\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 3.67653e6 0.308295 0.154147 0.988048i \(-0.450737\pi\)
0.154147 + 0.988048i \(0.450737\pi\)
\(678\) 0 0
\(679\) 2.80112e7 2.33162
\(680\) 0 0
\(681\) −9.07355e6 −0.749738
\(682\) 0 0
\(683\) −1.19937e6 −0.0983784 −0.0491892 0.998789i \(-0.515664\pi\)
−0.0491892 + 0.998789i \(0.515664\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 7.62162e6 0.616106
\(688\) 0 0
\(689\) −3.08460e7 −2.47543
\(690\) 0 0
\(691\) −1.32762e7 −1.05774 −0.528868 0.848704i \(-0.677383\pi\)
−0.528868 + 0.848704i \(0.677383\pi\)
\(692\) 0 0
\(693\) −466810. −0.0369239
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 180435. 0.0140682
\(698\) 0 0
\(699\) −5.34839e6 −0.414028
\(700\) 0 0
\(701\) 9.25654e6 0.711466 0.355733 0.934588i \(-0.384231\pi\)
0.355733 + 0.934588i \(0.384231\pi\)
\(702\) 0 0
\(703\) 1.26021e7 0.961736
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 1.42339e7 1.07097
\(708\) 0 0
\(709\) −9.58869e6 −0.716381 −0.358190 0.933649i \(-0.616606\pi\)
−0.358190 + 0.933649i \(0.616606\pi\)
\(710\) 0 0
\(711\) 4.19699e6 0.311361
\(712\) 0 0
\(713\) −5.52610e6 −0.407094
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 3.64417e6 0.264728
\(718\) 0 0
\(719\) −1.17477e6 −0.0847483 −0.0423741 0.999102i \(-0.513492\pi\)
−0.0423741 + 0.999102i \(0.513492\pi\)
\(720\) 0 0
\(721\) 1.30341e7 0.933778
\(722\) 0 0
\(723\) 599581. 0.0426581
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) −1.68157e7 −1.17999 −0.589997 0.807405i \(-0.700871\pi\)
−0.589997 + 0.807405i \(0.700871\pi\)
\(728\) 0 0
\(729\) 1.78606e6 0.124473
\(730\) 0 0
\(731\) 4.64163e6 0.321275
\(732\) 0 0
\(733\) 5.18574e6 0.356493 0.178247 0.983986i \(-0.442958\pi\)
0.178247 + 0.983986i \(0.442958\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −102210. −0.00693149
\(738\) 0 0
\(739\) 1.07382e7 0.723302 0.361651 0.932313i \(-0.382213\pi\)
0.361651 + 0.932313i \(0.382213\pi\)
\(740\) 0 0
\(741\) −3.64622e7 −2.43948
\(742\) 0 0
\(743\) 1.88845e7 1.25497 0.627487 0.778627i \(-0.284084\pi\)
0.627487 + 0.778627i \(0.284084\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) −1.08636e7 −0.712314
\(748\) 0 0
\(749\) −4.73372e6 −0.308318
\(750\) 0 0
\(751\) −7.40163e6 −0.478881 −0.239440 0.970911i \(-0.576964\pi\)
−0.239440 + 0.970911i \(0.576964\pi\)
\(752\) 0 0
\(753\) 916286. 0.0588903
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 7.24822e6 0.459718 0.229859 0.973224i \(-0.426173\pi\)
0.229859 + 0.973224i \(0.426173\pi\)
\(758\) 0 0
\(759\) 1.32220e6 0.0833089
\(760\) 0 0
\(761\) 5.50511e6 0.344591 0.172296 0.985045i \(-0.444882\pi\)
0.172296 + 0.985045i \(0.444882\pi\)
\(762\) 0 0
\(763\) 4.46113e7 2.77417
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −1.56241e7 −0.958970
\(768\) 0 0
\(769\) 1.81612e7 1.10746 0.553730 0.832696i \(-0.313204\pi\)
0.553730 + 0.832696i \(0.313204\pi\)
\(770\) 0 0
\(771\) 3.43398e7 2.08047
\(772\) 0 0
\(773\) −7.39969e6 −0.445415 −0.222707 0.974885i \(-0.571489\pi\)
−0.222707 + 0.974885i \(0.571489\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 2.21040e7 1.31347
\(778\) 0 0
\(779\) −1.41074e6 −0.0832920
\(780\) 0 0
\(781\) −576341. −0.0338105
\(782\) 0 0
\(783\) 1.05617e7 0.615645
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 1.85122e7 1.06542 0.532709 0.846298i \(-0.321174\pi\)
0.532709 + 0.846298i \(0.321174\pi\)
\(788\) 0 0
\(789\) −1.84655e7 −1.05601
\(790\) 0 0
\(791\) 4.17590e7 2.37306
\(792\) 0 0
\(793\) 2.76057e7 1.55889
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −1.68656e7 −0.940493 −0.470247 0.882535i \(-0.655835\pi\)
−0.470247 + 0.882535i \(0.655835\pi\)
\(798\) 0 0
\(799\) −8.52974e6 −0.472682
\(800\) 0 0
\(801\) 5.52955e6 0.304515
\(802\) 0 0
\(803\) 1.57450e6 0.0861697
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) −1.67745e7 −0.906706
\(808\) 0 0
\(809\) −5.53056e6 −0.297097 −0.148548 0.988905i \(-0.547460\pi\)
−0.148548 + 0.988905i \(0.547460\pi\)
\(810\) 0 0
\(811\) −2.91422e7 −1.55586 −0.777929 0.628353i \(-0.783730\pi\)
−0.777929 + 0.628353i \(0.783730\pi\)
\(812\) 0 0
\(813\) 2.28172e7 1.21070
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) −3.62908e7 −1.90214
\(818\) 0 0
\(819\) −2.13248e7 −1.11090
\(820\) 0 0
\(821\) −1.44892e7 −0.750216 −0.375108 0.926981i \(-0.622394\pi\)
−0.375108 + 0.926981i \(0.622394\pi\)
\(822\) 0 0
\(823\) 2.71404e7 1.39674 0.698371 0.715736i \(-0.253908\pi\)
0.698371 + 0.715736i \(0.253908\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −3.01025e7 −1.53052 −0.765259 0.643722i \(-0.777389\pi\)
−0.765259 + 0.643722i \(0.777389\pi\)
\(828\) 0 0
\(829\) −2.47022e7 −1.24839 −0.624194 0.781269i \(-0.714573\pi\)
−0.624194 + 0.781269i \(0.714573\pi\)
\(830\) 0 0
\(831\) −1.62441e7 −0.816008
\(832\) 0 0
\(833\) −7.99878e6 −0.399403
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 3.38190e6 0.166858
\(838\) 0 0
\(839\) −1.23647e7 −0.606428 −0.303214 0.952922i \(-0.598060\pi\)
−0.303214 + 0.952922i \(0.598060\pi\)
\(840\) 0 0
\(841\) 236010. 0.0115064
\(842\) 0 0
\(843\) 2.01286e7 0.975539
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 3.37694e7 1.61739
\(848\) 0 0
\(849\) 4.33123e6 0.206225
\(850\) 0 0
\(851\) −2.08757e7 −0.988137
\(852\) 0 0
\(853\) −1.20993e7 −0.569363 −0.284681 0.958622i \(-0.591888\pi\)
−0.284681 + 0.958622i \(0.591888\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −6.86289e6 −0.319194 −0.159597 0.987182i \(-0.551020\pi\)
−0.159597 + 0.987182i \(0.551020\pi\)
\(858\) 0 0
\(859\) 2.52569e7 1.16788 0.583938 0.811798i \(-0.301511\pi\)
0.583938 + 0.811798i \(0.301511\pi\)
\(860\) 0 0
\(861\) −2.47443e6 −0.113754
\(862\) 0 0
\(863\) 2.45114e6 0.112032 0.0560159 0.998430i \(-0.482160\pi\)
0.0560159 + 0.998430i \(0.482160\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 2.54758e7 1.15101
\(868\) 0 0
\(869\) 631041. 0.0283471
\(870\) 0 0
\(871\) −4.66917e6 −0.208542
\(872\) 0 0
\(873\) −1.62050e7 −0.719639
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 2.14652e7 0.942401 0.471201 0.882026i \(-0.343821\pi\)
0.471201 + 0.882026i \(0.343821\pi\)
\(878\) 0 0
\(879\) −3.13976e7 −1.37064
\(880\) 0 0
\(881\) 1.68815e7 0.732775 0.366388 0.930462i \(-0.380594\pi\)
0.366388 + 0.930462i \(0.380594\pi\)
\(882\) 0 0
\(883\) −3.37649e7 −1.45735 −0.728676 0.684859i \(-0.759864\pi\)
−0.728676 + 0.684859i \(0.759864\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 3.20696e7 1.36862 0.684312 0.729189i \(-0.260103\pi\)
0.684312 + 0.729189i \(0.260103\pi\)
\(888\) 0 0
\(889\) 1.78185e7 0.756166
\(890\) 0 0
\(891\) −1.34903e6 −0.0569282
\(892\) 0 0
\(893\) 6.66902e7 2.79855
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 6.04005e7 2.50645
\(898\) 0 0
\(899\) 6.64330e6 0.274148
\(900\) 0 0
\(901\) 1.08080e7 0.443540
\(902\) 0 0
\(903\) −6.36538e7 −2.59780
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 1.29290e7 0.521853 0.260926 0.965359i \(-0.415972\pi\)
0.260926 + 0.965359i \(0.415972\pi\)
\(908\) 0 0
\(909\) −8.23461e6 −0.330547
\(910\) 0 0
\(911\) −4.00076e6 −0.159715 −0.0798577 0.996806i \(-0.525447\pi\)
−0.0798577 + 0.996806i \(0.525447\pi\)
\(912\) 0 0
\(913\) −1.63340e6 −0.0648508
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −4.63393e7 −1.81981
\(918\) 0 0
\(919\) −2.55409e7 −0.997581 −0.498790 0.866723i \(-0.666222\pi\)
−0.498790 + 0.866723i \(0.666222\pi\)
\(920\) 0 0
\(921\) 4.00321e7 1.55511
\(922\) 0 0
\(923\) −2.63284e7 −1.01723
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) −7.54050e6 −0.288204
\(928\) 0 0
\(929\) −2.98080e7 −1.13316 −0.566582 0.824005i \(-0.691735\pi\)
−0.566582 + 0.824005i \(0.691735\pi\)
\(930\) 0 0
\(931\) 6.25389e7 2.36470
\(932\) 0 0
\(933\) 2.79046e7 1.04947
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 3.52937e7 1.31325 0.656627 0.754216i \(-0.271983\pi\)
0.656627 + 0.754216i \(0.271983\pi\)
\(938\) 0 0
\(939\) −1.97382e7 −0.730540
\(940\) 0 0
\(941\) 4.71557e7 1.73604 0.868020 0.496529i \(-0.165392\pi\)
0.868020 + 0.496529i \(0.165392\pi\)
\(942\) 0 0
\(943\) 2.33692e6 0.0855785
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −4.56893e7 −1.65554 −0.827770 0.561067i \(-0.810391\pi\)
−0.827770 + 0.561067i \(0.810391\pi\)
\(948\) 0 0
\(949\) 7.19263e7 2.59252
\(950\) 0 0
\(951\) −2.83971e6 −0.101818
\(952\) 0 0
\(953\) −2.47162e7 −0.881556 −0.440778 0.897616i \(-0.645297\pi\)
−0.440778 + 0.897616i \(0.645297\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) −1.58950e6 −0.0561024
\(958\) 0 0
\(959\) 9.84024e6 0.345509
\(960\) 0 0
\(961\) −2.65019e7 −0.925698
\(962\) 0 0
\(963\) 2.73855e6 0.0951602
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) −2.56789e6 −0.0883102 −0.0441551 0.999025i \(-0.514060\pi\)
−0.0441551 + 0.999025i \(0.514060\pi\)
\(968\) 0 0
\(969\) 1.27758e7 0.437099
\(970\) 0 0
\(971\) −3.46122e7 −1.17810 −0.589049 0.808097i \(-0.700498\pi\)
−0.589049 + 0.808097i \(0.700498\pi\)
\(972\) 0 0
\(973\) −3.55414e7 −1.20352
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −5.51384e7 −1.84807 −0.924034 0.382309i \(-0.875129\pi\)
−0.924034 + 0.382309i \(0.875129\pi\)
\(978\) 0 0
\(979\) 831400. 0.0277238
\(980\) 0 0
\(981\) −2.58085e7 −0.856230
\(982\) 0 0
\(983\) −3.37599e7 −1.11434 −0.557169 0.830399i \(-0.688113\pi\)
−0.557169 + 0.830399i \(0.688113\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 1.16974e8 3.82205
\(988\) 0 0
\(989\) 6.01166e7 1.95435
\(990\) 0 0
\(991\) 1.25256e7 0.405150 0.202575 0.979267i \(-0.435069\pi\)
0.202575 + 0.979267i \(0.435069\pi\)
\(992\) 0 0
\(993\) 2.94647e7 0.948265
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) −3.02324e7 −0.963240 −0.481620 0.876380i \(-0.659952\pi\)
−0.481620 + 0.876380i \(0.659952\pi\)
\(998\) 0 0
\(999\) 1.27757e7 0.405013
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 400.6.a.y.1.1 3
4.3 odd 2 200.6.a.h.1.3 3
5.2 odd 4 400.6.c.p.49.5 6
5.3 odd 4 400.6.c.p.49.2 6
5.4 even 2 400.6.a.x.1.3 3
20.3 even 4 200.6.c.g.49.5 6
20.7 even 4 200.6.c.g.49.2 6
20.19 odd 2 200.6.a.i.1.1 yes 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
200.6.a.h.1.3 3 4.3 odd 2
200.6.a.i.1.1 yes 3 20.19 odd 2
200.6.c.g.49.2 6 20.7 even 4
200.6.c.g.49.5 6 20.3 even 4
400.6.a.x.1.3 3 5.4 even 2
400.6.a.y.1.1 3 1.1 even 1 trivial
400.6.c.p.49.2 6 5.3 odd 4
400.6.c.p.49.5 6 5.2 odd 4