Properties

Label 2166.4.a.n.1.2
Level $2166$
Weight $4$
Character 2166.1
Self dual yes
Analytic conductor $127.798$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2166,4,Mod(1,2166)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2166, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("2166.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2166 = 2 \cdot 3 \cdot 19^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 2166.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: \(127.798137072\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{17}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2\cdot 3 \)
Twist minimal: no (minimal twist has level 114)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(2.56155\) of defining polynomial
Character \(\chi\) \(=\) 2166.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.00000 q^{2} +3.00000 q^{3} +4.00000 q^{4} +21.3693 q^{5} -6.00000 q^{6} -22.7386 q^{7} -8.00000 q^{8} +9.00000 q^{9} +O(q^{10})\) \(q-2.00000 q^{2} +3.00000 q^{3} +4.00000 q^{4} +21.3693 q^{5} -6.00000 q^{6} -22.7386 q^{7} -8.00000 q^{8} +9.00000 q^{9} -42.7386 q^{10} +54.7386 q^{11} +12.0000 q^{12} +8.10795 q^{13} +45.4773 q^{14} +64.1080 q^{15} +16.0000 q^{16} +72.7386 q^{17} -18.0000 q^{18} +85.4773 q^{20} -68.2159 q^{21} -109.477 q^{22} -27.3693 q^{23} -24.0000 q^{24} +331.648 q^{25} -16.2159 q^{26} +27.0000 q^{27} -90.9545 q^{28} +224.955 q^{29} -128.216 q^{30} +305.062 q^{31} -32.0000 q^{32} +164.216 q^{33} -145.477 q^{34} -485.909 q^{35} +36.0000 q^{36} +165.585 q^{37} +24.3239 q^{39} -170.955 q^{40} -371.080 q^{41} +136.432 q^{42} -222.955 q^{43} +218.955 q^{44} +192.324 q^{45} +54.7386 q^{46} -541.062 q^{47} +48.0000 q^{48} +174.045 q^{49} -663.295 q^{50} +218.216 q^{51} +32.4318 q^{52} +452.773 q^{53} -54.0000 q^{54} +1169.73 q^{55} +181.909 q^{56} -449.909 q^{58} -341.727 q^{59} +256.432 q^{60} -254.250 q^{61} -610.125 q^{62} -204.648 q^{63} +64.0000 q^{64} +173.261 q^{65} -328.432 q^{66} -251.602 q^{67} +290.955 q^{68} -82.1080 q^{69} +971.818 q^{70} -353.727 q^{71} -72.0000 q^{72} +481.170 q^{73} -331.170 q^{74} +994.943 q^{75} -1244.68 q^{77} -48.6477 q^{78} -583.403 q^{79} +341.909 q^{80} +81.0000 q^{81} +742.159 q^{82} -523.386 q^{83} -272.864 q^{84} +1554.37 q^{85} +445.909 q^{86} +674.864 q^{87} -437.909 q^{88} +412.523 q^{89} -384.648 q^{90} -184.364 q^{91} -109.477 q^{92} +915.187 q^{93} +1082.12 q^{94} -96.0000 q^{96} +1390.10 q^{97} -348.091 q^{98} +492.648 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 4 q^{2} + 6 q^{3} + 8 q^{4} + 18 q^{5} - 12 q^{6} + 4 q^{7} - 16 q^{8} + 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 4 q^{2} + 6 q^{3} + 8 q^{4} + 18 q^{5} - 12 q^{6} + 4 q^{7} - 16 q^{8} + 18 q^{9} - 36 q^{10} + 60 q^{11} + 24 q^{12} - 58 q^{13} - 8 q^{14} + 54 q^{15} + 32 q^{16} + 96 q^{17} - 36 q^{18} + 72 q^{20} + 12 q^{21} - 120 q^{22} - 30 q^{23} - 48 q^{24} + 218 q^{25} + 116 q^{26} + 54 q^{27} + 16 q^{28} + 252 q^{29} - 108 q^{30} + 338 q^{31} - 64 q^{32} + 180 q^{33} - 192 q^{34} - 576 q^{35} + 72 q^{36} + 158 q^{37} - 174 q^{39} - 144 q^{40} - 24 q^{42} - 248 q^{43} + 240 q^{44} + 162 q^{45} + 60 q^{46} - 810 q^{47} + 96 q^{48} + 546 q^{49} - 436 q^{50} + 288 q^{51} - 232 q^{52} - 84 q^{53} - 108 q^{54} + 1152 q^{55} - 32 q^{56} - 504 q^{58} + 504 q^{59} + 216 q^{60} + 580 q^{61} - 676 q^{62} + 36 q^{63} + 128 q^{64} + 396 q^{65} - 360 q^{66} + 140 q^{67} + 384 q^{68} - 90 q^{69} + 1152 q^{70} + 480 q^{71} - 144 q^{72} + 616 q^{73} - 316 q^{74} + 654 q^{75} - 1104 q^{77} + 348 q^{78} - 202 q^{79} + 288 q^{80} + 162 q^{81} - 552 q^{83} + 48 q^{84} + 1476 q^{85} + 496 q^{86} + 756 q^{87} - 480 q^{88} + 924 q^{89} - 324 q^{90} - 1952 q^{91} - 120 q^{92} + 1014 q^{93} + 1620 q^{94} - 192 q^{96} - 40 q^{97} - 1092 q^{98} + 540 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.00000 −0.707107
\(3\) 3.00000 0.577350
\(4\) 4.00000 0.500000
\(5\) 21.3693 1.91133 0.955665 0.294456i \(-0.0951386\pi\)
0.955665 + 0.294456i \(0.0951386\pi\)
\(6\) −6.00000 −0.408248
\(7\) −22.7386 −1.22777 −0.613885 0.789395i \(-0.710394\pi\)
−0.613885 + 0.789395i \(0.710394\pi\)
\(8\) −8.00000 −0.353553
\(9\) 9.00000 0.333333
\(10\) −42.7386 −1.35151
\(11\) 54.7386 1.50039 0.750196 0.661215i \(-0.229959\pi\)
0.750196 + 0.661215i \(0.229959\pi\)
\(12\) 12.0000 0.288675
\(13\) 8.10795 0.172980 0.0864900 0.996253i \(-0.472435\pi\)
0.0864900 + 0.996253i \(0.472435\pi\)
\(14\) 45.4773 0.868165
\(15\) 64.1080 1.10351
\(16\) 16.0000 0.250000
\(17\) 72.7386 1.03775 0.518874 0.854851i \(-0.326351\pi\)
0.518874 + 0.854851i \(0.326351\pi\)
\(18\) −18.0000 −0.235702
\(19\) 0 0
\(20\) 85.4773 0.955665
\(21\) −68.2159 −0.708854
\(22\) −109.477 −1.06094
\(23\) −27.3693 −0.248126 −0.124063 0.992274i \(-0.539592\pi\)
−0.124063 + 0.992274i \(0.539592\pi\)
\(24\) −24.0000 −0.204124
\(25\) 331.648 2.65318
\(26\) −16.2159 −0.122315
\(27\) 27.0000 0.192450
\(28\) −90.9545 −0.613885
\(29\) 224.955 1.44045 0.720224 0.693741i \(-0.244039\pi\)
0.720224 + 0.693741i \(0.244039\pi\)
\(30\) −128.216 −0.780297
\(31\) 305.062 1.76745 0.883723 0.468010i \(-0.155029\pi\)
0.883723 + 0.468010i \(0.155029\pi\)
\(32\) −32.0000 −0.176777
\(33\) 164.216 0.866252
\(34\) −145.477 −0.733798
\(35\) −485.909 −2.34667
\(36\) 36.0000 0.166667
\(37\) 165.585 0.735731 0.367865 0.929879i \(-0.380089\pi\)
0.367865 + 0.929879i \(0.380089\pi\)
\(38\) 0 0
\(39\) 24.3239 0.0998701
\(40\) −170.955 −0.675757
\(41\) −371.080 −1.41348 −0.706742 0.707471i \(-0.749836\pi\)
−0.706742 + 0.707471i \(0.749836\pi\)
\(42\) 136.432 0.501235
\(43\) −222.955 −0.790703 −0.395352 0.918530i \(-0.629377\pi\)
−0.395352 + 0.918530i \(0.629377\pi\)
\(44\) 218.955 0.750196
\(45\) 192.324 0.637110
\(46\) 54.7386 0.175452
\(47\) −541.062 −1.67919 −0.839597 0.543211i \(-0.817209\pi\)
−0.839597 + 0.543211i \(0.817209\pi\)
\(48\) 48.0000 0.144338
\(49\) 174.045 0.507421
\(50\) −663.295 −1.87608
\(51\) 218.216 0.599144
\(52\) 32.4318 0.0864900
\(53\) 452.773 1.17345 0.586727 0.809784i \(-0.300416\pi\)
0.586727 + 0.809784i \(0.300416\pi\)
\(54\) −54.0000 −0.136083
\(55\) 1169.73 2.86775
\(56\) 181.909 0.434083
\(57\) 0 0
\(58\) −449.909 −1.01855
\(59\) −341.727 −0.754052 −0.377026 0.926203i \(-0.623053\pi\)
−0.377026 + 0.926203i \(0.623053\pi\)
\(60\) 256.432 0.551753
\(61\) −254.250 −0.533662 −0.266831 0.963743i \(-0.585977\pi\)
−0.266831 + 0.963743i \(0.585977\pi\)
\(62\) −610.125 −1.24977
\(63\) −204.648 −0.409257
\(64\) 64.0000 0.125000
\(65\) 173.261 0.330622
\(66\) −328.432 −0.612533
\(67\) −251.602 −0.458778 −0.229389 0.973335i \(-0.573673\pi\)
−0.229389 + 0.973335i \(0.573673\pi\)
\(68\) 290.955 0.518874
\(69\) −82.1080 −0.143256
\(70\) 971.818 1.65935
\(71\) −353.727 −0.591263 −0.295632 0.955302i \(-0.595530\pi\)
−0.295632 + 0.955302i \(0.595530\pi\)
\(72\) −72.0000 −0.117851
\(73\) 481.170 0.771462 0.385731 0.922611i \(-0.373949\pi\)
0.385731 + 0.922611i \(0.373949\pi\)
\(74\) −331.170 −0.520240
\(75\) 994.943 1.53182
\(76\) 0 0
\(77\) −1244.68 −1.84214
\(78\) −48.6477 −0.0706188
\(79\) −583.403 −0.830861 −0.415430 0.909625i \(-0.636369\pi\)
−0.415430 + 0.909625i \(0.636369\pi\)
\(80\) 341.909 0.477832
\(81\) 81.0000 0.111111
\(82\) 742.159 0.999485
\(83\) −523.386 −0.692158 −0.346079 0.938205i \(-0.612487\pi\)
−0.346079 + 0.938205i \(0.612487\pi\)
\(84\) −272.864 −0.354427
\(85\) 1554.37 1.98348
\(86\) 445.909 0.559112
\(87\) 674.864 0.831643
\(88\) −437.909 −0.530469
\(89\) 412.523 0.491318 0.245659 0.969356i \(-0.420996\pi\)
0.245659 + 0.969356i \(0.420996\pi\)
\(90\) −384.648 −0.450505
\(91\) −184.364 −0.212380
\(92\) −109.477 −0.124063
\(93\) 915.187 1.02044
\(94\) 1082.12 1.18737
\(95\) 0 0
\(96\) −96.0000 −0.102062
\(97\) 1390.10 1.45509 0.727544 0.686061i \(-0.240662\pi\)
0.727544 + 0.686061i \(0.240662\pi\)
\(98\) −348.091 −0.358801
\(99\) 492.648 0.500131
\(100\) 1326.59 1.32659
\(101\) 336.688 0.331700 0.165850 0.986151i \(-0.446963\pi\)
0.165850 + 0.986151i \(0.446963\pi\)
\(102\) −436.432 −0.423659
\(103\) 257.528 0.246359 0.123180 0.992384i \(-0.460691\pi\)
0.123180 + 0.992384i \(0.460691\pi\)
\(104\) −64.8636 −0.0611577
\(105\) −1457.73 −1.35485
\(106\) −905.545 −0.829758
\(107\) 382.523 0.345606 0.172803 0.984956i \(-0.444718\pi\)
0.172803 + 0.984956i \(0.444718\pi\)
\(108\) 108.000 0.0962250
\(109\) 284.756 0.250226 0.125113 0.992142i \(-0.460071\pi\)
0.125113 + 0.992142i \(0.460071\pi\)
\(110\) −2339.45 −2.02780
\(111\) 496.756 0.424774
\(112\) −363.818 −0.306943
\(113\) −1249.85 −1.04050 −0.520249 0.854015i \(-0.674161\pi\)
−0.520249 + 0.854015i \(0.674161\pi\)
\(114\) 0 0
\(115\) −584.864 −0.474251
\(116\) 899.818 0.720224
\(117\) 72.9716 0.0576600
\(118\) 683.454 0.533196
\(119\) −1653.98 −1.27412
\(120\) −512.864 −0.390149
\(121\) 1665.32 1.25118
\(122\) 508.500 0.377356
\(123\) −1113.24 −0.816076
\(124\) 1220.25 0.883723
\(125\) 4415.92 3.15978
\(126\) 409.295 0.289388
\(127\) −414.210 −0.289411 −0.144706 0.989475i \(-0.546223\pi\)
−0.144706 + 0.989475i \(0.546223\pi\)
\(128\) −128.000 −0.0883883
\(129\) −668.864 −0.456513
\(130\) −346.523 −0.233785
\(131\) 2042.95 1.36255 0.681274 0.732029i \(-0.261426\pi\)
0.681274 + 0.732029i \(0.261426\pi\)
\(132\) 656.864 0.433126
\(133\) 0 0
\(134\) 503.204 0.324405
\(135\) 576.972 0.367836
\(136\) −581.909 −0.366899
\(137\) 564.193 0.351841 0.175921 0.984404i \(-0.443710\pi\)
0.175921 + 0.984404i \(0.443710\pi\)
\(138\) 164.216 0.101297
\(139\) −749.659 −0.457448 −0.228724 0.973491i \(-0.573455\pi\)
−0.228724 + 0.973491i \(0.573455\pi\)
\(140\) −1943.64 −1.17334
\(141\) −1623.19 −0.969483
\(142\) 707.454 0.418086
\(143\) 443.818 0.259538
\(144\) 144.000 0.0833333
\(145\) 4807.12 2.75317
\(146\) −962.341 −0.545506
\(147\) 522.136 0.292960
\(148\) 662.341 0.367865
\(149\) −659.960 −0.362859 −0.181430 0.983404i \(-0.558072\pi\)
−0.181430 + 0.983404i \(0.558072\pi\)
\(150\) −1989.89 −1.08316
\(151\) 870.438 0.469107 0.234554 0.972103i \(-0.424637\pi\)
0.234554 + 0.972103i \(0.424637\pi\)
\(152\) 0 0
\(153\) 654.648 0.345916
\(154\) 2489.36 1.30259
\(155\) 6518.98 3.37817
\(156\) 97.2954 0.0499350
\(157\) 340.875 0.173279 0.0866394 0.996240i \(-0.472387\pi\)
0.0866394 + 0.996240i \(0.472387\pi\)
\(158\) 1166.81 0.587507
\(159\) 1358.32 0.677495
\(160\) −683.818 −0.337879
\(161\) 622.341 0.304642
\(162\) −162.000 −0.0785674
\(163\) 447.386 0.214982 0.107491 0.994206i \(-0.465718\pi\)
0.107491 + 0.994206i \(0.465718\pi\)
\(164\) −1484.32 −0.706742
\(165\) 3509.18 1.65569
\(166\) 1046.77 0.489430
\(167\) −2165.08 −1.00323 −0.501613 0.865092i \(-0.667260\pi\)
−0.501613 + 0.865092i \(0.667260\pi\)
\(168\) 545.727 0.250618
\(169\) −2131.26 −0.970078
\(170\) −3108.75 −1.40253
\(171\) 0 0
\(172\) −891.818 −0.395352
\(173\) −614.784 −0.270180 −0.135090 0.990833i \(-0.543132\pi\)
−0.135090 + 0.990833i \(0.543132\pi\)
\(174\) −1349.73 −0.588061
\(175\) −7541.22 −3.25750
\(176\) 875.818 0.375098
\(177\) −1025.18 −0.435352
\(178\) −825.045 −0.347414
\(179\) 2425.11 1.01263 0.506317 0.862347i \(-0.331007\pi\)
0.506317 + 0.862347i \(0.331007\pi\)
\(180\) 769.295 0.318555
\(181\) −3139.68 −1.28934 −0.644669 0.764462i \(-0.723005\pi\)
−0.644669 + 0.764462i \(0.723005\pi\)
\(182\) 368.727 0.150175
\(183\) −762.750 −0.308110
\(184\) 218.955 0.0877258
\(185\) 3538.44 1.40622
\(186\) −1830.37 −0.721557
\(187\) 3981.61 1.55703
\(188\) −2164.25 −0.839597
\(189\) −613.943 −0.236285
\(190\) 0 0
\(191\) 4714.68 1.78608 0.893042 0.449974i \(-0.148567\pi\)
0.893042 + 0.449974i \(0.148567\pi\)
\(192\) 192.000 0.0721688
\(193\) 1606.83 0.599286 0.299643 0.954051i \(-0.403132\pi\)
0.299643 + 0.954051i \(0.403132\pi\)
\(194\) −2780.20 −1.02890
\(195\) 519.784 0.190885
\(196\) 696.182 0.253711
\(197\) −4161.01 −1.50487 −0.752435 0.658667i \(-0.771121\pi\)
−0.752435 + 0.658667i \(0.771121\pi\)
\(198\) −985.295 −0.353646
\(199\) −2200.10 −0.783724 −0.391862 0.920024i \(-0.628169\pi\)
−0.391862 + 0.920024i \(0.628169\pi\)
\(200\) −2653.18 −0.938041
\(201\) −754.807 −0.264875
\(202\) −673.375 −0.234547
\(203\) −5115.16 −1.76854
\(204\) 872.864 0.299572
\(205\) −7929.72 −2.70164
\(206\) −515.057 −0.174202
\(207\) −246.324 −0.0827087
\(208\) 129.727 0.0432450
\(209\) 0 0
\(210\) 2915.45 0.958026
\(211\) 1739.57 0.567568 0.283784 0.958888i \(-0.408410\pi\)
0.283784 + 0.958888i \(0.408410\pi\)
\(212\) 1811.09 0.586727
\(213\) −1061.18 −0.341366
\(214\) −765.045 −0.244381
\(215\) −4764.39 −1.51130
\(216\) −216.000 −0.0680414
\(217\) −6936.70 −2.17002
\(218\) −569.511 −0.176937
\(219\) 1443.51 0.445404
\(220\) 4678.91 1.43387
\(221\) 589.761 0.179510
\(222\) −993.511 −0.300361
\(223\) 94.0399 0.0282394 0.0141197 0.999900i \(-0.495505\pi\)
0.0141197 + 0.999900i \(0.495505\pi\)
\(224\) 727.636 0.217041
\(225\) 2984.83 0.884394
\(226\) 2499.70 0.735743
\(227\) −1410.48 −0.412408 −0.206204 0.978509i \(-0.566111\pi\)
−0.206204 + 0.978509i \(0.566111\pi\)
\(228\) 0 0
\(229\) 3651.11 1.05359 0.526796 0.849992i \(-0.323393\pi\)
0.526796 + 0.849992i \(0.323393\pi\)
\(230\) 1169.73 0.335346
\(231\) −3734.05 −1.06356
\(232\) −1799.64 −0.509275
\(233\) −7015.86 −1.97264 −0.986319 0.164850i \(-0.947286\pi\)
−0.986319 + 0.164850i \(0.947286\pi\)
\(234\) −145.943 −0.0407718
\(235\) −11562.1 −3.20949
\(236\) −1366.91 −0.377026
\(237\) −1750.21 −0.479698
\(238\) 3307.95 0.900936
\(239\) 316.460 0.0856491 0.0428245 0.999083i \(-0.486364\pi\)
0.0428245 + 0.999083i \(0.486364\pi\)
\(240\) 1025.73 0.275877
\(241\) −3681.10 −0.983903 −0.491952 0.870623i \(-0.663716\pi\)
−0.491952 + 0.870623i \(0.663716\pi\)
\(242\) −3330.64 −0.884717
\(243\) 243.000 0.0641500
\(244\) −1017.00 −0.266831
\(245\) 3719.23 0.969849
\(246\) 2226.48 0.577053
\(247\) 0 0
\(248\) −2440.50 −0.624887
\(249\) −1570.16 −0.399618
\(250\) −8831.84 −2.23430
\(251\) 7650.66 1.92393 0.961963 0.273181i \(-0.0880757\pi\)
0.961963 + 0.273181i \(0.0880757\pi\)
\(252\) −818.591 −0.204628
\(253\) −1498.16 −0.372286
\(254\) 828.420 0.204645
\(255\) 4663.12 1.14516
\(256\) 256.000 0.0625000
\(257\) −1979.70 −0.480508 −0.240254 0.970710i \(-0.577231\pi\)
−0.240254 + 0.970710i \(0.577231\pi\)
\(258\) 1337.73 0.322803
\(259\) −3765.18 −0.903309
\(260\) 693.045 0.165311
\(261\) 2024.59 0.480150
\(262\) −4085.91 −0.963467
\(263\) −1982.93 −0.464914 −0.232457 0.972607i \(-0.574677\pi\)
−0.232457 + 0.972607i \(0.574677\pi\)
\(264\) −1313.73 −0.306266
\(265\) 9675.44 2.24286
\(266\) 0 0
\(267\) 1237.57 0.283663
\(268\) −1006.41 −0.229389
\(269\) 1255.56 0.284582 0.142291 0.989825i \(-0.454553\pi\)
0.142291 + 0.989825i \(0.454553\pi\)
\(270\) −1153.94 −0.260099
\(271\) −18.4887 −0.00414431 −0.00207215 0.999998i \(-0.500660\pi\)
−0.00207215 + 0.999998i \(0.500660\pi\)
\(272\) 1163.82 0.259437
\(273\) −553.091 −0.122618
\(274\) −1128.39 −0.248789
\(275\) 18153.9 3.98081
\(276\) −328.432 −0.0716278
\(277\) 1266.98 0.274821 0.137410 0.990514i \(-0.456122\pi\)
0.137410 + 0.990514i \(0.456122\pi\)
\(278\) 1499.32 0.323465
\(279\) 2745.56 0.589149
\(280\) 3887.27 0.829675
\(281\) −2698.15 −0.572804 −0.286402 0.958109i \(-0.592459\pi\)
−0.286402 + 0.958109i \(0.592459\pi\)
\(282\) 3246.37 0.685528
\(283\) 3548.00 0.745253 0.372627 0.927981i \(-0.378457\pi\)
0.372627 + 0.927981i \(0.378457\pi\)
\(284\) −1414.91 −0.295632
\(285\) 0 0
\(286\) −887.636 −0.183521
\(287\) 8437.84 1.73544
\(288\) −288.000 −0.0589256
\(289\) 377.909 0.0769202
\(290\) −9614.25 −1.94679
\(291\) 4170.31 0.840095
\(292\) 1924.68 0.385731
\(293\) 4325.53 0.862459 0.431229 0.902242i \(-0.358080\pi\)
0.431229 + 0.902242i \(0.358080\pi\)
\(294\) −1044.27 −0.207154
\(295\) −7302.48 −1.44124
\(296\) −1324.68 −0.260120
\(297\) 1477.94 0.288751
\(298\) 1319.92 0.256580
\(299\) −221.909 −0.0429208
\(300\) 3979.77 0.765908
\(301\) 5069.68 0.970803
\(302\) −1740.88 −0.331709
\(303\) 1010.06 0.191507
\(304\) 0 0
\(305\) −5433.15 −1.02000
\(306\) −1309.30 −0.244599
\(307\) −4108.76 −0.763842 −0.381921 0.924195i \(-0.624737\pi\)
−0.381921 + 0.924195i \(0.624737\pi\)
\(308\) −4978.73 −0.921069
\(309\) 772.585 0.142236
\(310\) −13038.0 −2.38873
\(311\) −10394.8 −1.89529 −0.947644 0.319330i \(-0.896542\pi\)
−0.947644 + 0.319330i \(0.896542\pi\)
\(312\) −194.591 −0.0353094
\(313\) 2629.15 0.474787 0.237393 0.971414i \(-0.423707\pi\)
0.237393 + 0.971414i \(0.423707\pi\)
\(314\) −681.750 −0.122527
\(315\) −4373.18 −0.782225
\(316\) −2333.61 −0.415430
\(317\) 8890.03 1.57512 0.787562 0.616236i \(-0.211343\pi\)
0.787562 + 0.616236i \(0.211343\pi\)
\(318\) −2716.64 −0.479061
\(319\) 12313.7 2.16124
\(320\) 1367.64 0.238916
\(321\) 1147.57 0.199536
\(322\) −1244.68 −0.215414
\(323\) 0 0
\(324\) 324.000 0.0555556
\(325\) 2688.98 0.458948
\(326\) −894.773 −0.152015
\(327\) 854.267 0.144468
\(328\) 2968.64 0.499742
\(329\) 12303.0 2.06166
\(330\) −7018.36 −1.17075
\(331\) −302.716 −0.0502682 −0.0251341 0.999684i \(-0.508001\pi\)
−0.0251341 + 0.999684i \(0.508001\pi\)
\(332\) −2093.55 −0.346079
\(333\) 1490.27 0.245244
\(334\) 4330.16 0.709389
\(335\) −5376.57 −0.876875
\(336\) −1091.45 −0.177213
\(337\) 5927.86 0.958194 0.479097 0.877762i \(-0.340964\pi\)
0.479097 + 0.877762i \(0.340964\pi\)
\(338\) 4262.52 0.685949
\(339\) −3749.56 −0.600731
\(340\) 6217.50 0.991739
\(341\) 16698.7 2.65186
\(342\) 0 0
\(343\) 3841.80 0.604774
\(344\) 1783.64 0.279556
\(345\) −1754.59 −0.273809
\(346\) 1229.57 0.191046
\(347\) 12248.1 1.89485 0.947423 0.319985i \(-0.103678\pi\)
0.947423 + 0.319985i \(0.103678\pi\)
\(348\) 2699.45 0.415822
\(349\) 11493.4 1.76282 0.881411 0.472350i \(-0.156594\pi\)
0.881411 + 0.472350i \(0.156594\pi\)
\(350\) 15082.4 2.30340
\(351\) 218.915 0.0332900
\(352\) −1751.64 −0.265234
\(353\) 1653.89 0.249370 0.124685 0.992196i \(-0.460208\pi\)
0.124685 + 0.992196i \(0.460208\pi\)
\(354\) 2050.36 0.307841
\(355\) −7558.91 −1.13010
\(356\) 1650.09 0.245659
\(357\) −4961.93 −0.735611
\(358\) −4850.23 −0.716040
\(359\) 998.722 0.146826 0.0734130 0.997302i \(-0.476611\pi\)
0.0734130 + 0.997302i \(0.476611\pi\)
\(360\) −1538.59 −0.225252
\(361\) 0 0
\(362\) 6279.35 0.911700
\(363\) 4995.95 0.722368
\(364\) −737.455 −0.106190
\(365\) 10282.3 1.47452
\(366\) 1525.50 0.217867
\(367\) 4653.99 0.661951 0.330976 0.943639i \(-0.392622\pi\)
0.330976 + 0.943639i \(0.392622\pi\)
\(368\) −437.909 −0.0620315
\(369\) −3339.72 −0.471162
\(370\) −7076.89 −0.994351
\(371\) −10295.4 −1.44073
\(372\) 3660.75 0.510218
\(373\) −11333.1 −1.57320 −0.786599 0.617464i \(-0.788160\pi\)
−0.786599 + 0.617464i \(0.788160\pi\)
\(374\) −7963.23 −1.10099
\(375\) 13247.8 1.82430
\(376\) 4328.50 0.593684
\(377\) 1823.92 0.249169
\(378\) 1227.89 0.167078
\(379\) 11569.9 1.56809 0.784043 0.620707i \(-0.213154\pi\)
0.784043 + 0.620707i \(0.213154\pi\)
\(380\) 0 0
\(381\) −1242.63 −0.167092
\(382\) −9429.35 −1.26295
\(383\) −6334.91 −0.845166 −0.422583 0.906324i \(-0.638877\pi\)
−0.422583 + 0.906324i \(0.638877\pi\)
\(384\) −384.000 −0.0510310
\(385\) −26598.0 −3.52093
\(386\) −3213.66 −0.423759
\(387\) −2006.59 −0.263568
\(388\) 5560.41 0.727544
\(389\) −9212.97 −1.20081 −0.600406 0.799695i \(-0.704994\pi\)
−0.600406 + 0.799695i \(0.704994\pi\)
\(390\) −1039.57 −0.134976
\(391\) −1990.81 −0.257492
\(392\) −1392.36 −0.179400
\(393\) 6128.86 0.786667
\(394\) 8322.01 1.06410
\(395\) −12466.9 −1.58805
\(396\) 1970.59 0.250065
\(397\) −2805.03 −0.354611 −0.177306 0.984156i \(-0.556738\pi\)
−0.177306 + 0.984156i \(0.556738\pi\)
\(398\) 4400.20 0.554177
\(399\) 0 0
\(400\) 5306.36 0.663295
\(401\) 13872.4 1.72757 0.863785 0.503861i \(-0.168087\pi\)
0.863785 + 0.503861i \(0.168087\pi\)
\(402\) 1509.61 0.187295
\(403\) 2473.43 0.305733
\(404\) 1346.75 0.165850
\(405\) 1730.91 0.212370
\(406\) 10230.3 1.25055
\(407\) 9063.91 1.10389
\(408\) −1745.73 −0.211829
\(409\) −8128.35 −0.982692 −0.491346 0.870964i \(-0.663495\pi\)
−0.491346 + 0.870964i \(0.663495\pi\)
\(410\) 15859.4 1.91035
\(411\) 1692.58 0.203136
\(412\) 1030.11 0.123180
\(413\) 7770.41 0.925804
\(414\) 492.648 0.0584838
\(415\) −11184.4 −1.32294
\(416\) −259.454 −0.0305788
\(417\) −2248.98 −0.264108
\(418\) 0 0
\(419\) −1031.78 −0.120300 −0.0601502 0.998189i \(-0.519158\pi\)
−0.0601502 + 0.998189i \(0.519158\pi\)
\(420\) −5830.91 −0.677427
\(421\) 843.335 0.0976286 0.0488143 0.998808i \(-0.484456\pi\)
0.0488143 + 0.998808i \(0.484456\pi\)
\(422\) −3479.14 −0.401331
\(423\) −4869.56 −0.559731
\(424\) −3622.18 −0.414879
\(425\) 24123.6 2.75333
\(426\) 2122.36 0.241382
\(427\) 5781.30 0.655214
\(428\) 1530.09 0.172803
\(429\) 1331.45 0.149844
\(430\) 9528.77 1.06865
\(431\) −10461.4 −1.16916 −0.584580 0.811336i \(-0.698741\pi\)
−0.584580 + 0.811336i \(0.698741\pi\)
\(432\) 432.000 0.0481125
\(433\) 7440.65 0.825808 0.412904 0.910775i \(-0.364515\pi\)
0.412904 + 0.910775i \(0.364515\pi\)
\(434\) 13873.4 1.53444
\(435\) 14421.4 1.58954
\(436\) 1139.02 0.125113
\(437\) 0 0
\(438\) −2887.02 −0.314948
\(439\) −755.392 −0.0821250 −0.0410625 0.999157i \(-0.513074\pi\)
−0.0410625 + 0.999157i \(0.513074\pi\)
\(440\) −9357.82 −1.01390
\(441\) 1566.41 0.169140
\(442\) −1179.52 −0.126932
\(443\) −15130.1 −1.62270 −0.811349 0.584562i \(-0.801266\pi\)
−0.811349 + 0.584562i \(0.801266\pi\)
\(444\) 1987.02 0.212387
\(445\) 8815.33 0.939071
\(446\) −188.080 −0.0199682
\(447\) −1979.88 −0.209497
\(448\) −1455.27 −0.153471
\(449\) 1225.61 0.128820 0.0644101 0.997924i \(-0.479483\pi\)
0.0644101 + 0.997924i \(0.479483\pi\)
\(450\) −5969.66 −0.625361
\(451\) −20312.4 −2.12078
\(452\) −4999.41 −0.520249
\(453\) 2611.31 0.270839
\(454\) 2820.95 0.291617
\(455\) −3939.73 −0.405928
\(456\) 0 0
\(457\) 6333.39 0.648279 0.324139 0.946009i \(-0.394925\pi\)
0.324139 + 0.946009i \(0.394925\pi\)
\(458\) −7302.23 −0.745001
\(459\) 1963.94 0.199715
\(460\) −2339.45 −0.237125
\(461\) 14291.6 1.44388 0.721938 0.691958i \(-0.243252\pi\)
0.721938 + 0.691958i \(0.243252\pi\)
\(462\) 7468.09 0.752050
\(463\) −7529.19 −0.755748 −0.377874 0.925857i \(-0.623345\pi\)
−0.377874 + 0.925857i \(0.623345\pi\)
\(464\) 3599.27 0.360112
\(465\) 19556.9 1.95039
\(466\) 14031.7 1.39487
\(467\) −592.636 −0.0587236 −0.0293618 0.999569i \(-0.509347\pi\)
−0.0293618 + 0.999569i \(0.509347\pi\)
\(468\) 291.886 0.0288300
\(469\) 5721.09 0.563274
\(470\) 23124.3 2.26945
\(471\) 1022.62 0.100043
\(472\) 2733.82 0.266598
\(473\) −12204.2 −1.18637
\(474\) 3500.42 0.339198
\(475\) 0 0
\(476\) −6615.91 −0.637058
\(477\) 4074.95 0.391152
\(478\) −632.921 −0.0605630
\(479\) 13979.2 1.33346 0.666729 0.745301i \(-0.267694\pi\)
0.666729 + 0.745301i \(0.267694\pi\)
\(480\) −2051.45 −0.195074
\(481\) 1342.56 0.127267
\(482\) 7362.20 0.695725
\(483\) 1867.02 0.175885
\(484\) 6661.27 0.625589
\(485\) 29705.5 2.78115
\(486\) −486.000 −0.0453609
\(487\) −10744.9 −0.999788 −0.499894 0.866086i \(-0.666628\pi\)
−0.499894 + 0.866086i \(0.666628\pi\)
\(488\) 2034.00 0.188678
\(489\) 1342.16 0.124120
\(490\) −7438.47 −0.685787
\(491\) 5322.05 0.489166 0.244583 0.969628i \(-0.421349\pi\)
0.244583 + 0.969628i \(0.421349\pi\)
\(492\) −4452.95 −0.408038
\(493\) 16362.9 1.49482
\(494\) 0 0
\(495\) 10527.5 0.955915
\(496\) 4881.00 0.441862
\(497\) 8043.27 0.725936
\(498\) 3140.32 0.282572
\(499\) 43.7958 0.00392899 0.00196450 0.999998i \(-0.499375\pi\)
0.00196450 + 0.999998i \(0.499375\pi\)
\(500\) 17663.7 1.57989
\(501\) −6495.24 −0.579213
\(502\) −15301.3 −1.36042
\(503\) −3382.49 −0.299837 −0.149918 0.988698i \(-0.547901\pi\)
−0.149918 + 0.988698i \(0.547901\pi\)
\(504\) 1637.18 0.144694
\(505\) 7194.78 0.633987
\(506\) 2996.32 0.263246
\(507\) −6393.78 −0.560075
\(508\) −1656.84 −0.144706
\(509\) 13475.5 1.17346 0.586731 0.809782i \(-0.300414\pi\)
0.586731 + 0.809782i \(0.300414\pi\)
\(510\) −9326.25 −0.809752
\(511\) −10941.2 −0.947179
\(512\) −512.000 −0.0441942
\(513\) 0 0
\(514\) 3959.41 0.339771
\(515\) 5503.20 0.470874
\(516\) −2675.45 −0.228256
\(517\) −29617.0 −2.51945
\(518\) 7530.36 0.638736
\(519\) −1844.35 −0.155989
\(520\) −1386.09 −0.116893
\(521\) −16318.7 −1.37224 −0.686119 0.727489i \(-0.740687\pi\)
−0.686119 + 0.727489i \(0.740687\pi\)
\(522\) −4049.18 −0.339517
\(523\) −13301.2 −1.11209 −0.556044 0.831153i \(-0.687681\pi\)
−0.556044 + 0.831153i \(0.687681\pi\)
\(524\) 8171.82 0.681274
\(525\) −22623.6 −1.88072
\(526\) 3965.85 0.328744
\(527\) 22189.8 1.83416
\(528\) 2627.45 0.216563
\(529\) −11417.9 −0.938434
\(530\) −19350.9 −1.58594
\(531\) −3075.54 −0.251351
\(532\) 0 0
\(533\) −3008.69 −0.244505
\(534\) −2475.14 −0.200580
\(535\) 8174.25 0.660568
\(536\) 2012.82 0.162202
\(537\) 7275.34 0.584645
\(538\) −2511.11 −0.201230
\(539\) 9527.01 0.761331
\(540\) 2307.89 0.183918
\(541\) 23091.4 1.83508 0.917539 0.397645i \(-0.130172\pi\)
0.917539 + 0.397645i \(0.130172\pi\)
\(542\) 36.9774 0.00293047
\(543\) −9419.03 −0.744400
\(544\) −2327.64 −0.183450
\(545\) 6085.03 0.478265
\(546\) 1106.18 0.0867037
\(547\) 8357.56 0.653278 0.326639 0.945149i \(-0.394084\pi\)
0.326639 + 0.945149i \(0.394084\pi\)
\(548\) 2256.77 0.175921
\(549\) −2288.25 −0.177887
\(550\) −36307.9 −2.81486
\(551\) 0 0
\(552\) 656.864 0.0506485
\(553\) 13265.8 1.02011
\(554\) −2533.95 −0.194327
\(555\) 10615.3 0.811884
\(556\) −2998.64 −0.228724
\(557\) −17353.4 −1.32009 −0.660044 0.751227i \(-0.729462\pi\)
−0.660044 + 0.751227i \(0.729462\pi\)
\(558\) −5491.12 −0.416591
\(559\) −1807.70 −0.136776
\(560\) −7774.55 −0.586669
\(561\) 11944.8 0.898951
\(562\) 5396.30 0.405034
\(563\) −5675.64 −0.424866 −0.212433 0.977176i \(-0.568139\pi\)
−0.212433 + 0.977176i \(0.568139\pi\)
\(564\) −6492.75 −0.484741
\(565\) −26708.5 −1.98873
\(566\) −7096.00 −0.526974
\(567\) −1841.83 −0.136419
\(568\) 2829.82 0.209043
\(569\) 1674.57 0.123377 0.0616885 0.998095i \(-0.480351\pi\)
0.0616885 + 0.998095i \(0.480351\pi\)
\(570\) 0 0
\(571\) 1490.84 0.109264 0.0546320 0.998507i \(-0.482601\pi\)
0.0546320 + 0.998507i \(0.482601\pi\)
\(572\) 1775.27 0.129769
\(573\) 14144.0 1.03120
\(574\) −16875.7 −1.22714
\(575\) −9076.97 −0.658323
\(576\) 576.000 0.0416667
\(577\) −7929.18 −0.572090 −0.286045 0.958216i \(-0.592341\pi\)
−0.286045 + 0.958216i \(0.592341\pi\)
\(578\) −755.818 −0.0543908
\(579\) 4820.49 0.345998
\(580\) 19228.5 1.37659
\(581\) 11901.1 0.849811
\(582\) −8340.61 −0.594037
\(583\) 24784.2 1.76064
\(584\) −3849.36 −0.272753
\(585\) 1559.35 0.110207
\(586\) −8651.07 −0.609850
\(587\) −13353.5 −0.938939 −0.469470 0.882949i \(-0.655555\pi\)
−0.469470 + 0.882949i \(0.655555\pi\)
\(588\) 2088.55 0.146480
\(589\) 0 0
\(590\) 14605.0 1.01911
\(591\) −12483.0 −0.868837
\(592\) 2649.36 0.183933
\(593\) −9165.02 −0.634675 −0.317338 0.948313i \(-0.602789\pi\)
−0.317338 + 0.948313i \(0.602789\pi\)
\(594\) −2955.89 −0.204178
\(595\) −35344.4 −2.43526
\(596\) −2639.84 −0.181430
\(597\) −6600.31 −0.452483
\(598\) 443.818 0.0303496
\(599\) −13524.6 −0.922536 −0.461268 0.887261i \(-0.652605\pi\)
−0.461268 + 0.887261i \(0.652605\pi\)
\(600\) −7959.54 −0.541578
\(601\) 26518.9 1.79988 0.899941 0.436012i \(-0.143610\pi\)
0.899941 + 0.436012i \(0.143610\pi\)
\(602\) −10139.4 −0.686461
\(603\) −2264.42 −0.152926
\(604\) 3481.75 0.234554
\(605\) 35586.7 2.39141
\(606\) −2020.13 −0.135416
\(607\) −13559.2 −0.906674 −0.453337 0.891339i \(-0.649767\pi\)
−0.453337 + 0.891339i \(0.649767\pi\)
\(608\) 0 0
\(609\) −15345.5 −1.02107
\(610\) 10866.3 0.721252
\(611\) −4386.91 −0.290467
\(612\) 2618.59 0.172958
\(613\) −25283.6 −1.66590 −0.832949 0.553350i \(-0.813349\pi\)
−0.832949 + 0.553350i \(0.813349\pi\)
\(614\) 8217.52 0.540118
\(615\) −23789.1 −1.55979
\(616\) 9957.45 0.651294
\(617\) 7900.16 0.515476 0.257738 0.966215i \(-0.417023\pi\)
0.257738 + 0.966215i \(0.417023\pi\)
\(618\) −1545.17 −0.100576
\(619\) −8291.45 −0.538387 −0.269194 0.963086i \(-0.586757\pi\)
−0.269194 + 0.963086i \(0.586757\pi\)
\(620\) 26075.9 1.68909
\(621\) −738.972 −0.0477519
\(622\) 20789.6 1.34017
\(623\) −9380.20 −0.603226
\(624\) 389.182 0.0249675
\(625\) 52909.2 3.38619
\(626\) −5258.30 −0.335725
\(627\) 0 0
\(628\) 1363.50 0.0866394
\(629\) 12044.4 0.763503
\(630\) 8746.36 0.553117
\(631\) 20749.3 1.30906 0.654529 0.756037i \(-0.272867\pi\)
0.654529 + 0.756037i \(0.272867\pi\)
\(632\) 4667.23 0.293754
\(633\) 5218.70 0.327685
\(634\) −17780.1 −1.11378
\(635\) −8851.39 −0.553160
\(636\) 5433.27 0.338747
\(637\) 1411.15 0.0877738
\(638\) −24627.4 −1.52823
\(639\) −3183.54 −0.197088
\(640\) −2735.27 −0.168939
\(641\) −10132.5 −0.624352 −0.312176 0.950024i \(-0.601058\pi\)
−0.312176 + 0.950024i \(0.601058\pi\)
\(642\) −2295.14 −0.141093
\(643\) −1371.80 −0.0841343 −0.0420671 0.999115i \(-0.513394\pi\)
−0.0420671 + 0.999115i \(0.513394\pi\)
\(644\) 2489.36 0.152321
\(645\) −14293.2 −0.872547
\(646\) 0 0
\(647\) 16558.2 1.00614 0.503069 0.864246i \(-0.332204\pi\)
0.503069 + 0.864246i \(0.332204\pi\)
\(648\) −648.000 −0.0392837
\(649\) −18705.7 −1.13137
\(650\) −5377.97 −0.324525
\(651\) −20810.1 −1.25286
\(652\) 1789.55 0.107491
\(653\) 14690.4 0.880370 0.440185 0.897907i \(-0.354913\pi\)
0.440185 + 0.897907i \(0.354913\pi\)
\(654\) −1708.53 −0.102154
\(655\) 43656.5 2.60428
\(656\) −5937.27 −0.353371
\(657\) 4330.53 0.257154
\(658\) −24606.0 −1.45782
\(659\) 8334.23 0.492648 0.246324 0.969187i \(-0.420777\pi\)
0.246324 + 0.969187i \(0.420777\pi\)
\(660\) 14036.7 0.827847
\(661\) −20324.2 −1.19594 −0.597972 0.801517i \(-0.704027\pi\)
−0.597972 + 0.801517i \(0.704027\pi\)
\(662\) 605.432 0.0355450
\(663\) 1769.28 0.103640
\(664\) 4187.09 0.244715
\(665\) 0 0
\(666\) −2980.53 −0.173413
\(667\) −6156.85 −0.357413
\(668\) −8660.32 −0.501613
\(669\) 282.120 0.0163040
\(670\) 10753.1 0.620045
\(671\) −13917.3 −0.800702
\(672\) 2182.91 0.125309
\(673\) −31410.9 −1.79911 −0.899556 0.436805i \(-0.856110\pi\)
−0.899556 + 0.436805i \(0.856110\pi\)
\(674\) −11855.7 −0.677545
\(675\) 8954.49 0.510605
\(676\) −8525.04 −0.485039
\(677\) 91.4773 0.00519314 0.00259657 0.999997i \(-0.499173\pi\)
0.00259657 + 0.999997i \(0.499173\pi\)
\(678\) 7499.11 0.424781
\(679\) −31609.0 −1.78651
\(680\) −12435.0 −0.701265
\(681\) −4231.43 −0.238104
\(682\) −33397.4 −1.87515
\(683\) 25918.6 1.45205 0.726024 0.687670i \(-0.241366\pi\)
0.726024 + 0.687670i \(0.241366\pi\)
\(684\) 0 0
\(685\) 12056.4 0.672485
\(686\) −7683.59 −0.427640
\(687\) 10953.3 0.608291
\(688\) −3567.27 −0.197676
\(689\) 3671.06 0.202984
\(690\) 3509.18 0.193612
\(691\) 17339.8 0.954613 0.477307 0.878737i \(-0.341613\pi\)
0.477307 + 0.878737i \(0.341613\pi\)
\(692\) −2459.14 −0.135090
\(693\) −11202.1 −0.614046
\(694\) −24496.2 −1.33986
\(695\) −16019.7 −0.874334
\(696\) −5398.91 −0.294030
\(697\) −26991.8 −1.46684
\(698\) −22986.7 −1.24650
\(699\) −21047.6 −1.13890
\(700\) −30164.9 −1.62875
\(701\) 21860.1 1.17781 0.588905 0.808202i \(-0.299559\pi\)
0.588905 + 0.808202i \(0.299559\pi\)
\(702\) −437.829 −0.0235396
\(703\) 0 0
\(704\) 3503.27 0.187549
\(705\) −34686.4 −1.85300
\(706\) −3307.77 −0.176331
\(707\) −7655.82 −0.407251
\(708\) −4100.73 −0.217676
\(709\) 27875.5 1.47657 0.738285 0.674489i \(-0.235636\pi\)
0.738285 + 0.674489i \(0.235636\pi\)
\(710\) 15117.8 0.799101
\(711\) −5250.63 −0.276954
\(712\) −3300.18 −0.173707
\(713\) −8349.35 −0.438549
\(714\) 9923.86 0.520156
\(715\) 9484.09 0.496063
\(716\) 9700.45 0.506317
\(717\) 949.381 0.0494495
\(718\) −1997.44 −0.103822
\(719\) 3333.87 0.172924 0.0864620 0.996255i \(-0.472444\pi\)
0.0864620 + 0.996255i \(0.472444\pi\)
\(720\) 3077.18 0.159277
\(721\) −5855.84 −0.302473
\(722\) 0 0
\(723\) −11043.3 −0.568057
\(724\) −12558.7 −0.644669
\(725\) 74605.7 3.82177
\(726\) −9991.91 −0.510791
\(727\) −2521.55 −0.128637 −0.0643184 0.997929i \(-0.520487\pi\)
−0.0643184 + 0.997929i \(0.520487\pi\)
\(728\) 1474.91 0.0750876
\(729\) 729.000 0.0370370
\(730\) −20564.6 −1.04264
\(731\) −16217.4 −0.820551
\(732\) −3051.00 −0.154055
\(733\) 16822.1 0.847664 0.423832 0.905741i \(-0.360685\pi\)
0.423832 + 0.905741i \(0.360685\pi\)
\(734\) −9307.98 −0.468070
\(735\) 11157.7 0.559943
\(736\) 875.818 0.0438629
\(737\) −13772.4 −0.688347
\(738\) 6679.43 0.333162
\(739\) −23858.7 −1.18763 −0.593815 0.804602i \(-0.702379\pi\)
−0.593815 + 0.804602i \(0.702379\pi\)
\(740\) 14153.8 0.703112
\(741\) 0 0
\(742\) 20590.9 1.01875
\(743\) −22343.1 −1.10322 −0.551609 0.834103i \(-0.685986\pi\)
−0.551609 + 0.834103i \(0.685986\pi\)
\(744\) −7321.50 −0.360778
\(745\) −14102.9 −0.693544
\(746\) 22666.1 1.11242
\(747\) −4710.48 −0.230719
\(748\) 15926.5 0.778514
\(749\) −8698.04 −0.424325
\(750\) −26495.5 −1.28997
\(751\) 12535.9 0.609110 0.304555 0.952495i \(-0.401492\pi\)
0.304555 + 0.952495i \(0.401492\pi\)
\(752\) −8657.00 −0.419798
\(753\) 22952.0 1.11078
\(754\) −3647.84 −0.176189
\(755\) 18600.7 0.896619
\(756\) −2455.77 −0.118142
\(757\) −18795.1 −0.902404 −0.451202 0.892422i \(-0.649005\pi\)
−0.451202 + 0.892422i \(0.649005\pi\)
\(758\) −23139.8 −1.10880
\(759\) −4494.48 −0.214940
\(760\) 0 0
\(761\) −16941.9 −0.807023 −0.403511 0.914975i \(-0.632210\pi\)
−0.403511 + 0.914975i \(0.632210\pi\)
\(762\) 2485.26 0.118152
\(763\) −6474.95 −0.307220
\(764\) 18858.7 0.893042
\(765\) 13989.4 0.661159
\(766\) 12669.8 0.597623
\(767\) −2770.71 −0.130436
\(768\) 768.000 0.0360844
\(769\) −9643.99 −0.452238 −0.226119 0.974100i \(-0.572604\pi\)
−0.226119 + 0.974100i \(0.572604\pi\)
\(770\) 53196.0 2.48968
\(771\) −5939.11 −0.277422
\(772\) 6427.32 0.299643
\(773\) 2745.06 0.127727 0.0638634 0.997959i \(-0.479658\pi\)
0.0638634 + 0.997959i \(0.479658\pi\)
\(774\) 4013.18 0.186371
\(775\) 101173. 4.68936
\(776\) −11120.8 −0.514451
\(777\) −11295.5 −0.521526
\(778\) 18425.9 0.849103
\(779\) 0 0
\(780\) 2079.14 0.0954423
\(781\) −19362.5 −0.887127
\(782\) 3981.61 0.182074
\(783\) 6073.77 0.277214
\(784\) 2784.73 0.126855
\(785\) 7284.26 0.331193
\(786\) −12257.7 −0.556258
\(787\) −1312.23 −0.0594357 −0.0297178 0.999558i \(-0.509461\pi\)
−0.0297178 + 0.999558i \(0.509461\pi\)
\(788\) −16644.0 −0.752435
\(789\) −5948.78 −0.268418
\(790\) 24933.9 1.12292
\(791\) 28419.9 1.27749
\(792\) −3941.18 −0.176823
\(793\) −2061.45 −0.0923129
\(794\) 5610.07 0.250748
\(795\) 29026.3 1.29492
\(796\) −8800.41 −0.391862
\(797\) −26003.1 −1.15568 −0.577841 0.816150i \(-0.696104\pi\)
−0.577841 + 0.816150i \(0.696104\pi\)
\(798\) 0 0
\(799\) −39356.1 −1.74258
\(800\) −10612.7 −0.469021
\(801\) 3712.70 0.163773
\(802\) −27744.8 −1.22158
\(803\) 26338.6 1.15750
\(804\) −3019.23 −0.132438
\(805\) 13299.0 0.582271
\(806\) −4946.86 −0.216186
\(807\) 3766.67 0.164304
\(808\) −2693.50 −0.117274
\(809\) −20785.7 −0.903320 −0.451660 0.892190i \(-0.649168\pi\)
−0.451660 + 0.892190i \(0.649168\pi\)
\(810\) −3461.83 −0.150168
\(811\) −41004.6 −1.77542 −0.887710 0.460402i \(-0.847705\pi\)
−0.887710 + 0.460402i \(0.847705\pi\)
\(812\) −20460.6 −0.884270
\(813\) −55.4661 −0.00239272
\(814\) −18127.8 −0.780565
\(815\) 9560.34 0.410901
\(816\) 3491.45 0.149786
\(817\) 0 0
\(818\) 16256.7 0.694868
\(819\) −1659.27 −0.0707933
\(820\) −31718.9 −1.35082
\(821\) 32751.4 1.39224 0.696121 0.717924i \(-0.254908\pi\)
0.696121 + 0.717924i \(0.254908\pi\)
\(822\) −3385.16 −0.143639
\(823\) −1560.05 −0.0660751 −0.0330375 0.999454i \(-0.510518\pi\)
−0.0330375 + 0.999454i \(0.510518\pi\)
\(824\) −2060.23 −0.0871012
\(825\) 54461.8 2.29832
\(826\) −15540.8 −0.654642
\(827\) −32096.7 −1.34959 −0.674796 0.738004i \(-0.735769\pi\)
−0.674796 + 0.738004i \(0.735769\pi\)
\(828\) −985.295 −0.0413543
\(829\) 39115.0 1.63874 0.819372 0.573262i \(-0.194322\pi\)
0.819372 + 0.573262i \(0.194322\pi\)
\(830\) 22368.8 0.935461
\(831\) 3800.93 0.158668
\(832\) 518.909 0.0216225
\(833\) 12659.8 0.526575
\(834\) 4497.95 0.186752
\(835\) −46266.3 −1.91750
\(836\) 0 0
\(837\) 8236.69 0.340145
\(838\) 2063.57 0.0850653
\(839\) −25854.6 −1.06389 −0.531943 0.846780i \(-0.678538\pi\)
−0.531943 + 0.846780i \(0.678538\pi\)
\(840\) 11661.8 0.479013
\(841\) 26215.5 1.07489
\(842\) −1686.67 −0.0690338
\(843\) −8094.44 −0.330709
\(844\) 6958.27 0.283784
\(845\) −45543.6 −1.85414
\(846\) 9739.12 0.395790
\(847\) −37867.1 −1.53616
\(848\) 7244.36 0.293364
\(849\) 10644.0 0.430272
\(850\) −48247.2 −1.94690
\(851\) −4531.95 −0.182554
\(852\) −4244.73 −0.170683
\(853\) 21196.4 0.850822 0.425411 0.905000i \(-0.360130\pi\)
0.425411 + 0.905000i \(0.360130\pi\)
\(854\) −11562.6 −0.463307
\(855\) 0 0
\(856\) −3060.18 −0.122190
\(857\) 35760.2 1.42537 0.712686 0.701483i \(-0.247478\pi\)
0.712686 + 0.701483i \(0.247478\pi\)
\(858\) −2662.91 −0.105956
\(859\) −21145.9 −0.839918 −0.419959 0.907543i \(-0.637956\pi\)
−0.419959 + 0.907543i \(0.637956\pi\)
\(860\) −19057.5 −0.755648
\(861\) 25313.5 1.00195
\(862\) 20922.8 0.826721
\(863\) 36704.3 1.44777 0.723887 0.689919i \(-0.242354\pi\)
0.723887 + 0.689919i \(0.242354\pi\)
\(864\) −864.000 −0.0340207
\(865\) −13137.5 −0.516403
\(866\) −14881.3 −0.583934
\(867\) 1133.73 0.0444099
\(868\) −27746.8 −1.08501
\(869\) −31934.7 −1.24662
\(870\) −28842.7 −1.12398
\(871\) −2039.98 −0.0793594
\(872\) −2278.05 −0.0884683
\(873\) 12510.9 0.485029
\(874\) 0 0
\(875\) −100412. −3.87948
\(876\) 5774.05 0.222702
\(877\) −24118.9 −0.928664 −0.464332 0.885661i \(-0.653706\pi\)
−0.464332 + 0.885661i \(0.653706\pi\)
\(878\) 1510.78 0.0580712
\(879\) 12976.6 0.497941
\(880\) 18715.6 0.716936
\(881\) −14331.9 −0.548073 −0.274037 0.961719i \(-0.588359\pi\)
−0.274037 + 0.961719i \(0.588359\pi\)
\(882\) −3132.82 −0.119600
\(883\) −51819.2 −1.97492 −0.987460 0.157870i \(-0.949537\pi\)
−0.987460 + 0.157870i \(0.949537\pi\)
\(884\) 2359.05 0.0897548
\(885\) −21907.4 −0.832102
\(886\) 30260.3 1.14742
\(887\) −8915.24 −0.337480 −0.168740 0.985661i \(-0.553970\pi\)
−0.168740 + 0.985661i \(0.553970\pi\)
\(888\) −3974.05 −0.150180
\(889\) 9418.57 0.355330
\(890\) −17630.7 −0.664024
\(891\) 4433.83 0.166710
\(892\) 376.159 0.0141197
\(893\) 0 0
\(894\) 3959.76 0.148137
\(895\) 51823.0 1.93548
\(896\) 2910.55 0.108521
\(897\) −665.727 −0.0247804
\(898\) −2451.23 −0.0910896
\(899\) 68625.2 2.54592
\(900\) 11939.3 0.442197
\(901\) 32934.1 1.21775
\(902\) 40624.8 1.49962
\(903\) 15209.0 0.560493
\(904\) 9998.82 0.367871
\(905\) −67092.7 −2.46435
\(906\) −5222.63 −0.191512
\(907\) −31147.5 −1.14028 −0.570141 0.821547i \(-0.693111\pi\)
−0.570141 + 0.821547i \(0.693111\pi\)
\(908\) −5641.91 −0.206204
\(909\) 3030.19 0.110567
\(910\) 7879.45 0.287034
\(911\) 49926.6 1.81574 0.907872 0.419247i \(-0.137706\pi\)
0.907872 + 0.419247i \(0.137706\pi\)
\(912\) 0 0
\(913\) −28649.5 −1.03851
\(914\) −12666.8 −0.458402
\(915\) −16299.4 −0.588899
\(916\) 14604.5 0.526796
\(917\) −46454.0 −1.67290
\(918\) −3927.89 −0.141220
\(919\) 134.681 0.00483430 0.00241715 0.999997i \(-0.499231\pi\)
0.00241715 + 0.999997i \(0.499231\pi\)
\(920\) 4678.91 0.167673
\(921\) −12326.3 −0.441004
\(922\) −28583.2 −1.02097
\(923\) −2868.00 −0.102277
\(924\) −14936.2 −0.531780
\(925\) 54916.0 1.95203
\(926\) 15058.4 0.534394
\(927\) 2317.75 0.0821198
\(928\) −7198.55 −0.254638
\(929\) 23279.6 0.822152 0.411076 0.911601i \(-0.365153\pi\)
0.411076 + 0.911601i \(0.365153\pi\)
\(930\) −39113.9 −1.37913
\(931\) 0 0
\(932\) −28063.5 −0.986319
\(933\) −31184.4 −1.09424
\(934\) 1185.27 0.0415239
\(935\) 85084.4 2.97600
\(936\) −583.772 −0.0203859
\(937\) −46692.7 −1.62794 −0.813972 0.580904i \(-0.802699\pi\)
−0.813972 + 0.580904i \(0.802699\pi\)
\(938\) −11442.2 −0.398295
\(939\) 7887.44 0.274118
\(940\) −46248.5 −1.60475
\(941\) −32111.4 −1.11244 −0.556218 0.831037i \(-0.687748\pi\)
−0.556218 + 0.831037i \(0.687748\pi\)
\(942\) −2045.25 −0.0707408
\(943\) 10156.2 0.350722
\(944\) −5467.64 −0.188513
\(945\) −13119.5 −0.451618
\(946\) 24408.5 0.838887
\(947\) 23641.5 0.811240 0.405620 0.914042i \(-0.367056\pi\)
0.405620 + 0.914042i \(0.367056\pi\)
\(948\) −7000.84 −0.239849
\(949\) 3901.31 0.133448
\(950\) 0 0
\(951\) 26670.1 0.909398
\(952\) 13231.8 0.450468
\(953\) 7361.42 0.250220 0.125110 0.992143i \(-0.460072\pi\)
0.125110 + 0.992143i \(0.460072\pi\)
\(954\) −8149.91 −0.276586
\(955\) 100749. 3.41379
\(956\) 1265.84 0.0428245
\(957\) 36941.1 1.24779
\(958\) −27958.4 −0.942896
\(959\) −12829.0 −0.431981
\(960\) 4102.91 0.137938
\(961\) 63272.1 2.12387
\(962\) −2685.11 −0.0899912
\(963\) 3442.70 0.115202
\(964\) −14724.4 −0.491952
\(965\) 34336.9 1.14543
\(966\) −3734.05 −0.124369
\(967\) 24365.6 0.810284 0.405142 0.914254i \(-0.367222\pi\)
0.405142 + 0.914254i \(0.367222\pi\)
\(968\) −13322.5 −0.442358
\(969\) 0 0
\(970\) −59411.1 −1.96657
\(971\) −32522.6 −1.07487 −0.537436 0.843305i \(-0.680607\pi\)
−0.537436 + 0.843305i \(0.680607\pi\)
\(972\) 972.000 0.0320750
\(973\) 17046.2 0.561641
\(974\) 21489.8 0.706957
\(975\) 8066.95 0.264973
\(976\) −4068.00 −0.133415
\(977\) 30882.3 1.01127 0.505636 0.862747i \(-0.331258\pi\)
0.505636 + 0.862747i \(0.331258\pi\)
\(978\) −2684.32 −0.0877659
\(979\) 22580.9 0.737170
\(980\) 14876.9 0.484925
\(981\) 2562.80 0.0834087
\(982\) −10644.1 −0.345893
\(983\) −33962.7 −1.10198 −0.550988 0.834513i \(-0.685749\pi\)
−0.550988 + 0.834513i \(0.685749\pi\)
\(984\) 8905.91 0.288526
\(985\) −88917.8 −2.87630
\(986\) −32725.8 −1.05700
\(987\) 36909.1 1.19030
\(988\) 0 0
\(989\) 6102.11 0.196194
\(990\) −21055.1 −0.675934
\(991\) 26955.6 0.864050 0.432025 0.901862i \(-0.357799\pi\)
0.432025 + 0.901862i \(0.357799\pi\)
\(992\) −9762.00 −0.312443
\(993\) −908.149 −0.0290224
\(994\) −16086.5 −0.513314
\(995\) −47014.7 −1.49796
\(996\) −6280.64 −0.199809
\(997\) 7081.56 0.224950 0.112475 0.993655i \(-0.464122\pi\)
0.112475 + 0.993655i \(0.464122\pi\)
\(998\) −87.5915 −0.00277822
\(999\) 4470.80 0.141591
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 2166.4.a.n.1.2 2
19.18 odd 2 114.4.a.e.1.2 2
57.56 even 2 342.4.a.f.1.1 2
76.75 even 2 912.4.a.m.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
114.4.a.e.1.2 2 19.18 odd 2
342.4.a.f.1.1 2 57.56 even 2
912.4.a.m.1.2 2 76.75 even 2
2166.4.a.n.1.2 2 1.1 even 1 trivial