Properties

Label 115.6.a.c.1.7
Level $115$
Weight $6$
Character 115.1
Self dual yes
Analytic conductor $18.444$
Analytic rank $0$
Dimension $7$
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] = [115,6,Mod(1,115)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(115, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("115.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 115 = 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 115.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: \(18.4441392785\)
Analytic rank: \(0\)
Dimension: \(7\)
Coefficient field: \(\mathbb{Q}[x]/(x^{7} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{7} - 3x^{6} - 196x^{5} + 464x^{4} + 11003x^{3} - 21041x^{2} - 142416x + 243340 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.7
Root \(-9.58627\) of defining polynomial
Character \(\chi\) \(=\) 115.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+10.5863 q^{2} -30.8137 q^{3} +80.0691 q^{4} -25.0000 q^{5} -326.202 q^{6} -29.6892 q^{7} +508.872 q^{8} +706.483 q^{9} +O(q^{10})\) \(q+10.5863 q^{2} -30.8137 q^{3} +80.0691 q^{4} -25.0000 q^{5} -326.202 q^{6} -29.6892 q^{7} +508.872 q^{8} +706.483 q^{9} -264.657 q^{10} +121.358 q^{11} -2467.22 q^{12} +795.368 q^{13} -314.298 q^{14} +770.342 q^{15} +2824.85 q^{16} +1892.06 q^{17} +7479.02 q^{18} -892.045 q^{19} -2001.73 q^{20} +914.834 q^{21} +1284.73 q^{22} +529.000 q^{23} -15680.2 q^{24} +625.000 q^{25} +8419.98 q^{26} -14281.6 q^{27} -2377.19 q^{28} +428.602 q^{29} +8155.05 q^{30} +369.935 q^{31} +13620.7 q^{32} -3739.50 q^{33} +20029.9 q^{34} +742.230 q^{35} +56567.5 q^{36} +10904.5 q^{37} -9443.43 q^{38} -24508.2 q^{39} -12721.8 q^{40} -7277.25 q^{41} +9684.67 q^{42} +22031.8 q^{43} +9717.05 q^{44} -17662.1 q^{45} +5600.14 q^{46} -15837.3 q^{47} -87044.0 q^{48} -15925.6 q^{49} +6616.42 q^{50} -58301.4 q^{51} +63684.4 q^{52} +16331.8 q^{53} -151189. q^{54} -3033.96 q^{55} -15108.0 q^{56} +27487.2 q^{57} +4537.29 q^{58} +21055.8 q^{59} +61680.6 q^{60} -26448.1 q^{61} +3916.23 q^{62} -20974.9 q^{63} +53797.1 q^{64} -19884.2 q^{65} -39587.3 q^{66} -8916.95 q^{67} +151496. q^{68} -16300.4 q^{69} +7857.45 q^{70} -25809.1 q^{71} +359510. q^{72} -20568.3 q^{73} +115438. q^{74} -19258.6 q^{75} -71425.3 q^{76} -3603.03 q^{77} -259451. q^{78} -65352.9 q^{79} -70621.2 q^{80} +268394. q^{81} -77038.9 q^{82} +41110.6 q^{83} +73249.9 q^{84} -47301.6 q^{85} +233235. q^{86} -13206.8 q^{87} +61755.9 q^{88} -46352.9 q^{89} -186976. q^{90} -23613.8 q^{91} +42356.5 q^{92} -11399.0 q^{93} -167657. q^{94} +22301.1 q^{95} -419704. q^{96} +93669.5 q^{97} -168592. q^{98} +85737.7 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 7 q + 4 q^{2} - 3 q^{3} + 178 q^{4} - 175 q^{5} - 381 q^{6} + 33 q^{7} + 546 q^{8} + 440 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 7 q + 4 q^{2} - 3 q^{3} + 178 q^{4} - 175 q^{5} - 381 q^{6} + 33 q^{7} + 546 q^{8} + 440 q^{9} - 100 q^{10} + 1373 q^{11} - 285 q^{12} + 605 q^{13} + 1317 q^{14} + 75 q^{15} + 3770 q^{16} + 2505 q^{17} + 7971 q^{18} - 115 q^{19} - 4450 q^{20} + 608 q^{21} + 2977 q^{22} + 3703 q^{23} - 12447 q^{24} + 4375 q^{25} + 9379 q^{26} - 12276 q^{27} + 5777 q^{28} + 2440 q^{29} + 9525 q^{30} + 13565 q^{31} + 14086 q^{32} + 10519 q^{33} + 26997 q^{34} - 825 q^{35} + 79889 q^{36} + 9414 q^{37} + 28717 q^{38} - 21738 q^{39} - 13650 q^{40} + 13725 q^{41} + 12426 q^{42} + 76694 q^{43} + 55203 q^{44} - 11000 q^{45} + 2116 q^{46} + 59692 q^{47} - 32985 q^{48} - 53608 q^{49} + 2500 q^{50} - 24725 q^{51} + 61195 q^{52} + 49536 q^{53} - 156168 q^{54} - 34325 q^{55} - 54461 q^{56} - 7580 q^{57} - 95562 q^{58} + 44536 q^{59} + 7125 q^{60} - 49097 q^{61} - 25763 q^{62} - 3578 q^{63} - 18654 q^{64} - 15125 q^{65} - 201873 q^{66} + 788 q^{67} + 163845 q^{68} - 1587 q^{69} - 32925 q^{70} + 49521 q^{71} + 328503 q^{72} - 3760 q^{73} + 88170 q^{74} - 1875 q^{75} - 411465 q^{76} + 77728 q^{77} - 389832 q^{78} + 918 q^{79} - 94250 q^{80} + 121235 q^{81} - 227459 q^{82} + 99202 q^{83} + 336602 q^{84} - 62625 q^{85} + 24584 q^{86} - 38666 q^{87} - 201275 q^{88} - 141676 q^{89} - 199275 q^{90} - 223605 q^{91} + 94162 q^{92} + 51412 q^{93} - 354292 q^{94} + 2875 q^{95} - 592095 q^{96} + 28731 q^{97} - 149557 q^{98} + 237333 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 10.5863 1.87141 0.935703 0.352789i \(-0.114767\pi\)
0.935703 + 0.352789i \(0.114767\pi\)
\(3\) −30.8137 −1.97670 −0.988350 0.152201i \(-0.951364\pi\)
−0.988350 + 0.152201i \(0.951364\pi\)
\(4\) 80.0691 2.50216
\(5\) −25.0000 −0.447214
\(6\) −326.202 −3.69921
\(7\) −29.6892 −0.229009 −0.114505 0.993423i \(-0.536528\pi\)
−0.114505 + 0.993423i \(0.536528\pi\)
\(8\) 508.872 2.81115
\(9\) 706.483 2.90734
\(10\) −264.657 −0.836918
\(11\) 121.358 0.302404 0.151202 0.988503i \(-0.451686\pi\)
0.151202 + 0.988503i \(0.451686\pi\)
\(12\) −2467.22 −4.94602
\(13\) 795.368 1.30530 0.652649 0.757660i \(-0.273658\pi\)
0.652649 + 0.757660i \(0.273658\pi\)
\(14\) −314.298 −0.428569
\(15\) 770.342 0.884007
\(16\) 2824.85 2.75864
\(17\) 1892.06 1.58786 0.793932 0.608007i \(-0.208031\pi\)
0.793932 + 0.608007i \(0.208031\pi\)
\(18\) 7479.02 5.44081
\(19\) −892.045 −0.566895 −0.283448 0.958988i \(-0.591478\pi\)
−0.283448 + 0.958988i \(0.591478\pi\)
\(20\) −2001.73 −1.11900
\(21\) 914.834 0.452683
\(22\) 1284.73 0.565921
\(23\) 529.000 0.208514
\(24\) −15680.2 −5.55679
\(25\) 625.000 0.200000
\(26\) 8419.98 2.44274
\(27\) −14281.6 −3.77024
\(28\) −2377.19 −0.573018
\(29\) 428.602 0.0946366 0.0473183 0.998880i \(-0.484932\pi\)
0.0473183 + 0.998880i \(0.484932\pi\)
\(30\) 8155.05 1.65434
\(31\) 369.935 0.0691386 0.0345693 0.999402i \(-0.488994\pi\)
0.0345693 + 0.999402i \(0.488994\pi\)
\(32\) 13620.7 2.35139
\(33\) −3739.50 −0.597762
\(34\) 20029.9 2.97154
\(35\) 742.230 0.102416
\(36\) 56567.5 7.27462
\(37\) 10904.5 1.30949 0.654745 0.755850i \(-0.272776\pi\)
0.654745 + 0.755850i \(0.272776\pi\)
\(38\) −9443.43 −1.06089
\(39\) −24508.2 −2.58018
\(40\) −12721.8 −1.25718
\(41\) −7277.25 −0.676095 −0.338047 0.941129i \(-0.609766\pi\)
−0.338047 + 0.941129i \(0.609766\pi\)
\(42\) 9684.67 0.847153
\(43\) 22031.8 1.81710 0.908552 0.417772i \(-0.137189\pi\)
0.908552 + 0.417772i \(0.137189\pi\)
\(44\) 9717.05 0.756664
\(45\) −17662.1 −1.30020
\(46\) 5600.14 0.390215
\(47\) −15837.3 −1.04577 −0.522884 0.852404i \(-0.675144\pi\)
−0.522884 + 0.852404i \(0.675144\pi\)
\(48\) −87044.0 −5.45300
\(49\) −15925.6 −0.947555
\(50\) 6616.42 0.374281
\(51\) −58301.4 −3.13873
\(52\) 63684.4 3.26606
\(53\) 16331.8 0.798630 0.399315 0.916814i \(-0.369248\pi\)
0.399315 + 0.916814i \(0.369248\pi\)
\(54\) −151189. −7.05564
\(55\) −3033.96 −0.135239
\(56\) −15108.0 −0.643779
\(57\) 27487.2 1.12058
\(58\) 4537.29 0.177103
\(59\) 21055.8 0.787482 0.393741 0.919221i \(-0.371181\pi\)
0.393741 + 0.919221i \(0.371181\pi\)
\(60\) 61680.6 2.21193
\(61\) −26448.1 −0.910058 −0.455029 0.890477i \(-0.650371\pi\)
−0.455029 + 0.890477i \(0.650371\pi\)
\(62\) 3916.23 0.129386
\(63\) −20974.9 −0.665808
\(64\) 53797.1 1.64176
\(65\) −19884.2 −0.583747
\(66\) −39587.3 −1.11866
\(67\) −8916.95 −0.242677 −0.121339 0.992611i \(-0.538719\pi\)
−0.121339 + 0.992611i \(0.538719\pi\)
\(68\) 151496. 3.97309
\(69\) −16300.4 −0.412170
\(70\) 7857.45 0.191662
\(71\) −25809.1 −0.607612 −0.303806 0.952734i \(-0.598257\pi\)
−0.303806 + 0.952734i \(0.598257\pi\)
\(72\) 359510. 8.17296
\(73\) −20568.3 −0.451743 −0.225871 0.974157i \(-0.572523\pi\)
−0.225871 + 0.974157i \(0.572523\pi\)
\(74\) 115438. 2.45059
\(75\) −19258.6 −0.395340
\(76\) −71425.3 −1.41846
\(77\) −3603.03 −0.0692534
\(78\) −259451. −4.82857
\(79\) −65352.9 −1.17814 −0.589070 0.808082i \(-0.700506\pi\)
−0.589070 + 0.808082i \(0.700506\pi\)
\(80\) −70621.2 −1.23370
\(81\) 268394. 4.54528
\(82\) −77038.9 −1.26525
\(83\) 41110.6 0.655025 0.327513 0.944847i \(-0.393790\pi\)
0.327513 + 0.944847i \(0.393790\pi\)
\(84\) 73249.9 1.13268
\(85\) −47301.6 −0.710114
\(86\) 233235. 3.40054
\(87\) −13206.8 −0.187068
\(88\) 61755.9 0.850103
\(89\) −46352.9 −0.620300 −0.310150 0.950688i \(-0.600379\pi\)
−0.310150 + 0.950688i \(0.600379\pi\)
\(90\) −186976. −2.43320
\(91\) −23613.8 −0.298925
\(92\) 42356.5 0.521736
\(93\) −11399.0 −0.136666
\(94\) −167657. −1.95706
\(95\) 22301.1 0.253523
\(96\) −419704. −4.64798
\(97\) 93669.5 1.01081 0.505404 0.862883i \(-0.331343\pi\)
0.505404 + 0.862883i \(0.331343\pi\)
\(98\) −168592. −1.77326
\(99\) 85737.7 0.879192
\(100\) 50043.2 0.500432
\(101\) 88353.7 0.861830 0.430915 0.902393i \(-0.358191\pi\)
0.430915 + 0.902393i \(0.358191\pi\)
\(102\) −617194. −5.87383
\(103\) 91968.3 0.854171 0.427086 0.904211i \(-0.359540\pi\)
0.427086 + 0.904211i \(0.359540\pi\)
\(104\) 404741. 3.66939
\(105\) −22870.8 −0.202446
\(106\) 172893. 1.49456
\(107\) 124183. 1.04858 0.524291 0.851539i \(-0.324330\pi\)
0.524291 + 0.851539i \(0.324330\pi\)
\(108\) −1.14352e6 −9.43373
\(109\) 146604. 1.18190 0.590949 0.806709i \(-0.298753\pi\)
0.590949 + 0.806709i \(0.298753\pi\)
\(110\) −32118.3 −0.253088
\(111\) −336008. −2.58847
\(112\) −83867.5 −0.631755
\(113\) −129446. −0.953661 −0.476831 0.878995i \(-0.658214\pi\)
−0.476831 + 0.878995i \(0.658214\pi\)
\(114\) 290987. 2.09706
\(115\) −13225.0 −0.0932505
\(116\) 34317.8 0.236796
\(117\) 561914. 3.79494
\(118\) 222902. 1.47370
\(119\) −56173.8 −0.363636
\(120\) 392006. 2.48507
\(121\) −146323. −0.908552
\(122\) −279986. −1.70309
\(123\) 224239. 1.33644
\(124\) 29620.3 0.172996
\(125\) −15625.0 −0.0894427
\(126\) −222046. −1.24600
\(127\) −196601. −1.08163 −0.540813 0.841143i \(-0.681883\pi\)
−0.540813 + 0.841143i \(0.681883\pi\)
\(128\) 133649. 0.721007
\(129\) −678883. −3.59187
\(130\) −210499. −1.09243
\(131\) 203953. 1.03837 0.519184 0.854662i \(-0.326236\pi\)
0.519184 + 0.854662i \(0.326236\pi\)
\(132\) −299418. −1.49570
\(133\) 26484.1 0.129824
\(134\) −94397.2 −0.454148
\(135\) 357041. 1.68610
\(136\) 962818. 4.46372
\(137\) −179636. −0.817695 −0.408847 0.912603i \(-0.634069\pi\)
−0.408847 + 0.912603i \(0.634069\pi\)
\(138\) −172561. −0.771338
\(139\) 33143.8 0.145501 0.0727504 0.997350i \(-0.476822\pi\)
0.0727504 + 0.997350i \(0.476822\pi\)
\(140\) 59429.7 0.256261
\(141\) 488004. 2.06717
\(142\) −273222. −1.13709
\(143\) 96524.5 0.394728
\(144\) 1.99571e6 8.02030
\(145\) −10715.0 −0.0423228
\(146\) −217742. −0.845394
\(147\) 490725. 1.87303
\(148\) 873115. 3.27655
\(149\) 2520.75 0.00930175 0.00465088 0.999989i \(-0.498520\pi\)
0.00465088 + 0.999989i \(0.498520\pi\)
\(150\) −203876. −0.739841
\(151\) 262891. 0.938281 0.469141 0.883123i \(-0.344564\pi\)
0.469141 + 0.883123i \(0.344564\pi\)
\(152\) −453937. −1.59363
\(153\) 1.33671e6 4.61646
\(154\) −38142.7 −0.129601
\(155\) −9248.36 −0.0309197
\(156\) −1.96235e6 −6.45602
\(157\) −452310. −1.46449 −0.732247 0.681039i \(-0.761528\pi\)
−0.732247 + 0.681039i \(0.761528\pi\)
\(158\) −691844. −2.20478
\(159\) −503244. −1.57865
\(160\) −340517. −1.05157
\(161\) −15705.6 −0.0477518
\(162\) 2.84129e6 8.50607
\(163\) −209381. −0.617261 −0.308631 0.951182i \(-0.599871\pi\)
−0.308631 + 0.951182i \(0.599871\pi\)
\(164\) −582683. −1.69170
\(165\) 93487.5 0.267327
\(166\) 435207. 1.22582
\(167\) −653719. −1.81384 −0.906922 0.421298i \(-0.861575\pi\)
−0.906922 + 0.421298i \(0.861575\pi\)
\(168\) 465533. 1.27256
\(169\) 261317. 0.703802
\(170\) −500747. −1.32891
\(171\) −630215. −1.64816
\(172\) 1.76407e6 4.54668
\(173\) 176929. 0.449452 0.224726 0.974422i \(-0.427851\pi\)
0.224726 + 0.974422i \(0.427851\pi\)
\(174\) −139811. −0.350080
\(175\) −18555.7 −0.0458019
\(176\) 342819. 0.834225
\(177\) −648806. −1.55662
\(178\) −490704. −1.16083
\(179\) −696684. −1.62519 −0.812594 0.582831i \(-0.801945\pi\)
−0.812594 + 0.582831i \(0.801945\pi\)
\(180\) −1.41419e6 −3.25331
\(181\) −126126. −0.286160 −0.143080 0.989711i \(-0.545701\pi\)
−0.143080 + 0.989711i \(0.545701\pi\)
\(182\) −249982. −0.559411
\(183\) 814962. 1.79891
\(184\) 269193. 0.586165
\(185\) −272613. −0.585622
\(186\) −120673. −0.255758
\(187\) 229618. 0.480177
\(188\) −1.26807e6 −2.61668
\(189\) 424010. 0.863419
\(190\) 236086. 0.474445
\(191\) 324452. 0.643526 0.321763 0.946820i \(-0.395724\pi\)
0.321763 + 0.946820i \(0.395724\pi\)
\(192\) −1.65769e6 −3.24526
\(193\) 65147.3 0.125893 0.0629467 0.998017i \(-0.479950\pi\)
0.0629467 + 0.998017i \(0.479950\pi\)
\(194\) 991611. 1.89163
\(195\) 612705. 1.15389
\(196\) −1.27514e6 −2.37093
\(197\) −48652.6 −0.0893182 −0.0446591 0.999002i \(-0.514220\pi\)
−0.0446591 + 0.999002i \(0.514220\pi\)
\(198\) 907642. 1.64532
\(199\) −263903. −0.472403 −0.236201 0.971704i \(-0.575902\pi\)
−0.236201 + 0.971704i \(0.575902\pi\)
\(200\) 318045. 0.562230
\(201\) 274764. 0.479700
\(202\) 935336. 1.61283
\(203\) −12724.8 −0.0216727
\(204\) −4.66814e6 −7.85359
\(205\) 181931. 0.302359
\(206\) 973601. 1.59850
\(207\) 373730. 0.606222
\(208\) 2.24679e6 3.60085
\(209\) −108257. −0.171432
\(210\) −242117. −0.378858
\(211\) −1.03953e6 −1.60743 −0.803714 0.595015i \(-0.797146\pi\)
−0.803714 + 0.595015i \(0.797146\pi\)
\(212\) 1.30768e6 1.99830
\(213\) 795272. 1.20107
\(214\) 1.31463e6 1.96232
\(215\) −550796. −0.812634
\(216\) −7.26753e6 −10.5987
\(217\) −10983.1 −0.0158334
\(218\) 1.55199e6 2.21181
\(219\) 633785. 0.892960
\(220\) −242926. −0.338390
\(221\) 1.50489e6 2.07263
\(222\) −3.55708e6 −4.84407
\(223\) 806391. 1.08588 0.542942 0.839770i \(-0.317310\pi\)
0.542942 + 0.839770i \(0.317310\pi\)
\(224\) −404387. −0.538490
\(225\) 441552. 0.581468
\(226\) −1.37035e6 −1.78469
\(227\) −310344. −0.399741 −0.199870 0.979822i \(-0.564052\pi\)
−0.199870 + 0.979822i \(0.564052\pi\)
\(228\) 2.20088e6 2.80387
\(229\) 658501. 0.829789 0.414894 0.909870i \(-0.363819\pi\)
0.414894 + 0.909870i \(0.363819\pi\)
\(230\) −140003. −0.174509
\(231\) 111023. 0.136893
\(232\) 218104. 0.266038
\(233\) 484198. 0.584296 0.292148 0.956373i \(-0.405630\pi\)
0.292148 + 0.956373i \(0.405630\pi\)
\(234\) 5.94857e6 7.10188
\(235\) 395931. 0.467681
\(236\) 1.68592e6 1.97041
\(237\) 2.01376e6 2.32883
\(238\) −594671. −0.680510
\(239\) −378822. −0.428983 −0.214491 0.976726i \(-0.568809\pi\)
−0.214491 + 0.976726i \(0.568809\pi\)
\(240\) 2.17610e6 2.43866
\(241\) −1.22272e6 −1.35608 −0.678041 0.735024i \(-0.737171\pi\)
−0.678041 + 0.735024i \(0.737171\pi\)
\(242\) −1.54902e6 −1.70027
\(243\) −4.79978e6 −5.21442
\(244\) −2.11767e6 −2.27711
\(245\) 398139. 0.423759
\(246\) 2.37385e6 2.50101
\(247\) −709504. −0.739967
\(248\) 188249. 0.194359
\(249\) −1.26677e6 −1.29479
\(250\) −165410. −0.167384
\(251\) −820720. −0.822263 −0.411132 0.911576i \(-0.634866\pi\)
−0.411132 + 0.911576i \(0.634866\pi\)
\(252\) −1.67944e6 −1.66596
\(253\) 64198.6 0.0630557
\(254\) −2.08128e6 −2.02416
\(255\) 1.45754e6 1.40368
\(256\) −306667. −0.292460
\(257\) 560153. 0.529023 0.264511 0.964383i \(-0.414789\pi\)
0.264511 + 0.964383i \(0.414789\pi\)
\(258\) −7.18683e6 −6.72184
\(259\) −323746. −0.299886
\(260\) −1.59211e6 −1.46063
\(261\) 302800. 0.275141
\(262\) 2.15910e6 1.94321
\(263\) −1.51123e6 −1.34723 −0.673613 0.739084i \(-0.735259\pi\)
−0.673613 + 0.739084i \(0.735259\pi\)
\(264\) −1.90293e6 −1.68040
\(265\) −408296. −0.357158
\(266\) 280368. 0.242954
\(267\) 1.42830e6 1.22615
\(268\) −713972. −0.607217
\(269\) −701534. −0.591110 −0.295555 0.955326i \(-0.595505\pi\)
−0.295555 + 0.955326i \(0.595505\pi\)
\(270\) 3.77973e6 3.15538
\(271\) −473696. −0.391811 −0.195906 0.980623i \(-0.562765\pi\)
−0.195906 + 0.980623i \(0.562765\pi\)
\(272\) 5.34479e6 4.38034
\(273\) 727629. 0.590886
\(274\) −1.90167e6 −1.53024
\(275\) 75849.0 0.0604809
\(276\) −1.30516e6 −1.03132
\(277\) −349184. −0.273435 −0.136718 0.990610i \(-0.543655\pi\)
−0.136718 + 0.990610i \(0.543655\pi\)
\(278\) 350869. 0.272291
\(279\) 261353. 0.201009
\(280\) 377700. 0.287907
\(281\) 864133. 0.652852 0.326426 0.945223i \(-0.394156\pi\)
0.326426 + 0.945223i \(0.394156\pi\)
\(282\) 5.16614e6 3.86851
\(283\) −1.89459e6 −1.40621 −0.703105 0.711086i \(-0.748204\pi\)
−0.703105 + 0.711086i \(0.748204\pi\)
\(284\) −2.06651e6 −1.52034
\(285\) −687180. −0.501139
\(286\) 1.02183e6 0.738696
\(287\) 216056. 0.154832
\(288\) 9.62279e6 6.83628
\(289\) 2.16004e6 1.52131
\(290\) −113432. −0.0792031
\(291\) −2.88630e6 −1.99806
\(292\) −1.64689e6 −1.13033
\(293\) 108668. 0.0739488 0.0369744 0.999316i \(-0.488228\pi\)
0.0369744 + 0.999316i \(0.488228\pi\)
\(294\) 5.19495e6 3.50520
\(295\) −526394. −0.352173
\(296\) 5.54901e6 3.68117
\(297\) −1.73320e6 −1.14014
\(298\) 26685.4 0.0174073
\(299\) 420750. 0.272173
\(300\) −1.54201e6 −0.989203
\(301\) −654108. −0.416134
\(302\) 2.78303e6 1.75591
\(303\) −2.72250e6 −1.70358
\(304\) −2.51989e6 −1.56386
\(305\) 661202. 0.406990
\(306\) 1.41508e7 8.63926
\(307\) 944344. 0.571853 0.285926 0.958252i \(-0.407699\pi\)
0.285926 + 0.958252i \(0.407699\pi\)
\(308\) −288491. −0.173283
\(309\) −2.83388e6 −1.68844
\(310\) −97905.7 −0.0578633
\(311\) 1.23478e6 0.723916 0.361958 0.932194i \(-0.382108\pi\)
0.361958 + 0.932194i \(0.382108\pi\)
\(312\) −1.24716e7 −7.25327
\(313\) 651465. 0.375863 0.187932 0.982182i \(-0.439822\pi\)
0.187932 + 0.982182i \(0.439822\pi\)
\(314\) −4.78828e6 −2.74066
\(315\) 524373. 0.297758
\(316\) −5.23275e6 −2.94790
\(317\) 182350. 0.101920 0.0509599 0.998701i \(-0.483772\pi\)
0.0509599 + 0.998701i \(0.483772\pi\)
\(318\) −5.32748e6 −2.95430
\(319\) 52014.4 0.0286185
\(320\) −1.34493e6 −0.734216
\(321\) −3.82654e6 −2.07273
\(322\) −166264. −0.0893629
\(323\) −1.68781e6 −0.900152
\(324\) 2.14901e7 11.3730
\(325\) 497105. 0.261060
\(326\) −2.21657e6 −1.15515
\(327\) −4.51742e6 −2.33626
\(328\) −3.70319e6 −1.90060
\(329\) 470195. 0.239491
\(330\) 989683. 0.500278
\(331\) −111410. −0.0558927 −0.0279464 0.999609i \(-0.508897\pi\)
−0.0279464 + 0.999609i \(0.508897\pi\)
\(332\) 3.29168e6 1.63898
\(333\) 7.70386e6 3.80713
\(334\) −6.92045e6 −3.39444
\(335\) 222924. 0.108529
\(336\) 2.58427e6 1.24879
\(337\) 2.11461e6 1.01427 0.507137 0.861865i \(-0.330704\pi\)
0.507137 + 0.861865i \(0.330704\pi\)
\(338\) 2.76637e6 1.31710
\(339\) 3.98872e6 1.88510
\(340\) −3.78739e6 −1.77682
\(341\) 44894.6 0.0209078
\(342\) −6.67163e6 −3.08437
\(343\) 971803. 0.446008
\(344\) 1.12114e7 5.10815
\(345\) 407511. 0.184328
\(346\) 1.87301e6 0.841106
\(347\) −1.72358e6 −0.768436 −0.384218 0.923242i \(-0.625529\pi\)
−0.384218 + 0.923242i \(0.625529\pi\)
\(348\) −1.05746e6 −0.468074
\(349\) −1.93915e6 −0.852210 −0.426105 0.904674i \(-0.640115\pi\)
−0.426105 + 0.904674i \(0.640115\pi\)
\(350\) −196436. −0.0857139
\(351\) −1.13592e7 −4.92128
\(352\) 1.65298e6 0.711069
\(353\) −1.60735e6 −0.686554 −0.343277 0.939234i \(-0.611537\pi\)
−0.343277 + 0.939234i \(0.611537\pi\)
\(354\) −6.86843e6 −2.91306
\(355\) 645226. 0.271732
\(356\) −3.71143e6 −1.55209
\(357\) 1.73092e6 0.718798
\(358\) −7.37529e6 −3.04138
\(359\) 41269.4 0.0169002 0.00845010 0.999964i \(-0.497310\pi\)
0.00845010 + 0.999964i \(0.497310\pi\)
\(360\) −8.98775e6 −3.65506
\(361\) −1.68035e6 −0.678630
\(362\) −1.33521e6 −0.535522
\(363\) 4.50876e6 1.79593
\(364\) −1.89074e6 −0.747959
\(365\) 514208. 0.202026
\(366\) 8.62741e6 3.36649
\(367\) 2.04233e6 0.791516 0.395758 0.918355i \(-0.370482\pi\)
0.395758 + 0.918355i \(0.370482\pi\)
\(368\) 1.49434e6 0.575216
\(369\) −5.14126e6 −1.96564
\(370\) −2.88595e6 −1.09594
\(371\) −484879. −0.182894
\(372\) −912711. −0.341961
\(373\) 3.09250e6 1.15090 0.575450 0.817837i \(-0.304827\pi\)
0.575450 + 0.817837i \(0.304827\pi\)
\(374\) 2.43079e6 0.898605
\(375\) 481464. 0.176801
\(376\) −8.05914e6 −2.93981
\(377\) 340896. 0.123529
\(378\) 4.48869e6 1.61581
\(379\) −2.35878e6 −0.843510 −0.421755 0.906710i \(-0.638586\pi\)
−0.421755 + 0.906710i \(0.638586\pi\)
\(380\) 1.78563e6 0.634356
\(381\) 6.05801e6 2.13805
\(382\) 3.43473e6 1.20430
\(383\) 3.92180e6 1.36612 0.683060 0.730362i \(-0.260649\pi\)
0.683060 + 0.730362i \(0.260649\pi\)
\(384\) −4.11821e6 −1.42521
\(385\) 90075.8 0.0309711
\(386\) 689667. 0.235598
\(387\) 1.55651e7 5.28294
\(388\) 7.50003e6 2.52920
\(389\) −2.16806e6 −0.726436 −0.363218 0.931704i \(-0.618322\pi\)
−0.363218 + 0.931704i \(0.618322\pi\)
\(390\) 6.48626e6 2.15940
\(391\) 1.00090e6 0.331092
\(392\) −8.10407e6 −2.66372
\(393\) −6.28454e6 −2.05254
\(394\) −515049. −0.167151
\(395\) 1.63382e6 0.526881
\(396\) 6.86494e6 2.19988
\(397\) −1.54195e6 −0.491013 −0.245506 0.969395i \(-0.578954\pi\)
−0.245506 + 0.969395i \(0.578954\pi\)
\(398\) −2.79375e6 −0.884057
\(399\) −816073. −0.256624
\(400\) 1.76553e6 0.551728
\(401\) 6.02060e6 1.86973 0.934865 0.355004i \(-0.115521\pi\)
0.934865 + 0.355004i \(0.115521\pi\)
\(402\) 2.90873e6 0.897713
\(403\) 294234. 0.0902465
\(404\) 7.07440e6 2.15644
\(405\) −6.70986e6 −2.03271
\(406\) −134709. −0.0405584
\(407\) 1.32335e6 0.395995
\(408\) −2.96680e7 −8.82343
\(409\) −3.80839e6 −1.12573 −0.562864 0.826549i \(-0.690301\pi\)
−0.562864 + 0.826549i \(0.690301\pi\)
\(410\) 1.92597e6 0.565836
\(411\) 5.53524e6 1.61634
\(412\) 7.36381e6 2.13727
\(413\) −625128. −0.180341
\(414\) 3.95640e6 1.13449
\(415\) −1.02776e6 −0.292936
\(416\) 1.08335e7 3.06926
\(417\) −1.02128e6 −0.287611
\(418\) −1.14604e6 −0.320818
\(419\) 3.69823e6 1.02910 0.514552 0.857459i \(-0.327958\pi\)
0.514552 + 0.857459i \(0.327958\pi\)
\(420\) −1.83125e6 −0.506552
\(421\) −5.33638e6 −1.46738 −0.733689 0.679486i \(-0.762203\pi\)
−0.733689 + 0.679486i \(0.762203\pi\)
\(422\) −1.10048e7 −3.00815
\(423\) −1.11888e7 −3.04040
\(424\) 8.31082e6 2.24507
\(425\) 1.18254e6 0.317573
\(426\) 8.41897e6 2.24768
\(427\) 785222. 0.208412
\(428\) 9.94322e6 2.62372
\(429\) −2.97428e6 −0.780258
\(430\) −5.83088e6 −1.52077
\(431\) 587147. 0.152249 0.0761244 0.997098i \(-0.475745\pi\)
0.0761244 + 0.997098i \(0.475745\pi\)
\(432\) −4.03434e7 −10.4007
\(433\) −1.72787e6 −0.442886 −0.221443 0.975173i \(-0.571077\pi\)
−0.221443 + 0.975173i \(0.571077\pi\)
\(434\) −116270. −0.0296307
\(435\) 330170. 0.0836594
\(436\) 1.17385e7 2.95730
\(437\) −471892. −0.118206
\(438\) 6.70942e6 1.67109
\(439\) 2.14062e6 0.530124 0.265062 0.964231i \(-0.414608\pi\)
0.265062 + 0.964231i \(0.414608\pi\)
\(440\) −1.54390e6 −0.380178
\(441\) −1.12511e7 −2.75486
\(442\) 1.59311e7 3.87874
\(443\) 3.19273e6 0.772954 0.386477 0.922299i \(-0.373692\pi\)
0.386477 + 0.922299i \(0.373692\pi\)
\(444\) −2.69039e7 −6.47676
\(445\) 1.15882e6 0.277407
\(446\) 8.53667e6 2.03213
\(447\) −77673.7 −0.0183868
\(448\) −1.59719e6 −0.375978
\(449\) −987505. −0.231166 −0.115583 0.993298i \(-0.536874\pi\)
−0.115583 + 0.993298i \(0.536874\pi\)
\(450\) 4.67439e6 1.08816
\(451\) −883155. −0.204454
\(452\) −1.03647e7 −2.38621
\(453\) −8.10064e6 −1.85470
\(454\) −3.28538e6 −0.748077
\(455\) 590346. 0.133684
\(456\) 1.39875e7 3.15012
\(457\) −8.05946e6 −1.80516 −0.902579 0.430524i \(-0.858329\pi\)
−0.902579 + 0.430524i \(0.858329\pi\)
\(458\) 6.97106e6 1.55287
\(459\) −2.70217e7 −5.98662
\(460\) −1.05891e6 −0.233328
\(461\) 4.62735e6 1.01410 0.507049 0.861917i \(-0.330736\pi\)
0.507049 + 0.861917i \(0.330736\pi\)
\(462\) 1.17532e6 0.256183
\(463\) −3.05000e6 −0.661222 −0.330611 0.943767i \(-0.607255\pi\)
−0.330611 + 0.943767i \(0.607255\pi\)
\(464\) 1.21074e6 0.261068
\(465\) 284976. 0.0611190
\(466\) 5.12585e6 1.09345
\(467\) 7.57924e6 1.60818 0.804088 0.594510i \(-0.202654\pi\)
0.804088 + 0.594510i \(0.202654\pi\)
\(468\) 4.49920e7 9.49555
\(469\) 264737. 0.0555754
\(470\) 4.19144e6 0.875222
\(471\) 1.39374e7 2.89486
\(472\) 1.07147e7 2.21373
\(473\) 2.67375e6 0.549500
\(474\) 2.13183e7 4.35819
\(475\) −557528. −0.113379
\(476\) −4.49778e6 −0.909874
\(477\) 1.15382e7 2.32189
\(478\) −4.01031e6 −0.802801
\(479\) 4.66635e6 0.929263 0.464632 0.885504i \(-0.346187\pi\)
0.464632 + 0.885504i \(0.346187\pi\)
\(480\) 1.04926e7 2.07864
\(481\) 8.67310e6 1.70927
\(482\) −1.29441e7 −2.53778
\(483\) 483947. 0.0943909
\(484\) −1.17160e7 −2.27334
\(485\) −2.34174e6 −0.452047
\(486\) −5.08118e7 −9.75829
\(487\) −7.27213e6 −1.38944 −0.694719 0.719281i \(-0.744471\pi\)
−0.694719 + 0.719281i \(0.744471\pi\)
\(488\) −1.34587e7 −2.55831
\(489\) 6.45181e6 1.22014
\(490\) 4.21480e6 0.793026
\(491\) −1.82846e6 −0.342279 −0.171140 0.985247i \(-0.554745\pi\)
−0.171140 + 0.985247i \(0.554745\pi\)
\(492\) 1.79546e7 3.34398
\(493\) 810941. 0.150270
\(494\) −7.51100e6 −1.38478
\(495\) −2.14344e6 −0.393187
\(496\) 1.04501e6 0.190729
\(497\) 766250. 0.139149
\(498\) −1.34103e7 −2.42307
\(499\) −7.64155e6 −1.37382 −0.686910 0.726742i \(-0.741034\pi\)
−0.686910 + 0.726742i \(0.741034\pi\)
\(500\) −1.25108e6 −0.223800
\(501\) 2.01435e7 3.58543
\(502\) −8.68837e6 −1.53879
\(503\) 6.03402e6 1.06338 0.531688 0.846941i \(-0.321558\pi\)
0.531688 + 0.846941i \(0.321558\pi\)
\(504\) −1.06736e7 −1.87169
\(505\) −2.20884e6 −0.385422
\(506\) 679623. 0.118003
\(507\) −8.05214e6 −1.39121
\(508\) −1.57417e7 −2.70640
\(509\) 8.48552e6 1.45172 0.725862 0.687840i \(-0.241441\pi\)
0.725862 + 0.687840i \(0.241441\pi\)
\(510\) 1.54299e7 2.62686
\(511\) 610656. 0.103453
\(512\) −7.52321e6 −1.26832
\(513\) 1.27399e7 2.13733
\(514\) 5.92993e6 0.990016
\(515\) −2.29921e6 −0.381997
\(516\) −5.43575e7 −8.98742
\(517\) −1.92198e6 −0.316245
\(518\) −3.42727e6 −0.561208
\(519\) −5.45183e6 −0.888431
\(520\) −1.01185e7 −1.64100
\(521\) 51099.8 0.00824755 0.00412378 0.999991i \(-0.498687\pi\)
0.00412378 + 0.999991i \(0.498687\pi\)
\(522\) 3.20552e6 0.514900
\(523\) 8.31977e6 1.33002 0.665008 0.746836i \(-0.268428\pi\)
0.665008 + 0.746836i \(0.268428\pi\)
\(524\) 1.63303e7 2.59816
\(525\) 571771. 0.0905365
\(526\) −1.59983e7 −2.52121
\(527\) 699939. 0.109783
\(528\) −1.05635e7 −1.64901
\(529\) 279841. 0.0434783
\(530\) −4.32233e6 −0.668388
\(531\) 1.48755e7 2.28948
\(532\) 2.12056e6 0.324841
\(533\) −5.78809e6 −0.882505
\(534\) 1.51204e7 2.29462
\(535\) −3.10458e6 −0.468941
\(536\) −4.53759e6 −0.682202
\(537\) 2.14674e7 3.21251
\(538\) −7.42663e6 −1.10621
\(539\) −1.93270e6 −0.286545
\(540\) 2.85879e7 4.21889
\(541\) −2.52062e6 −0.370266 −0.185133 0.982713i \(-0.559272\pi\)
−0.185133 + 0.982713i \(0.559272\pi\)
\(542\) −5.01468e6 −0.733238
\(543\) 3.88642e6 0.565653
\(544\) 2.57712e7 3.73368
\(545\) −3.66511e6 −0.528561
\(546\) 7.70288e6 1.10579
\(547\) 5.19609e6 0.742520 0.371260 0.928529i \(-0.378926\pi\)
0.371260 + 0.928529i \(0.378926\pi\)
\(548\) −1.43833e7 −2.04600
\(549\) −1.86851e7 −2.64585
\(550\) 802958. 0.113184
\(551\) −382332. −0.0536491
\(552\) −8.29484e6 −1.15867
\(553\) 1.94028e6 0.269805
\(554\) −3.69655e6 −0.511709
\(555\) 8.40021e6 1.15760
\(556\) 2.65379e6 0.364066
\(557\) −5.17834e6 −0.707217 −0.353609 0.935394i \(-0.615045\pi\)
−0.353609 + 0.935394i \(0.615045\pi\)
\(558\) 2.76675e6 0.376170
\(559\) 1.75234e7 2.37186
\(560\) 2.09669e6 0.282529
\(561\) −7.07536e6 −0.949165
\(562\) 9.14794e6 1.22175
\(563\) 306055. 0.0406939 0.0203469 0.999793i \(-0.493523\pi\)
0.0203469 + 0.999793i \(0.493523\pi\)
\(564\) 3.90740e7 5.17238
\(565\) 3.23616e6 0.426490
\(566\) −2.00567e7 −2.63159
\(567\) −7.96841e6 −1.04091
\(568\) −1.31335e7 −1.70809
\(569\) 2.54129e6 0.329059 0.164529 0.986372i \(-0.447389\pi\)
0.164529 + 0.986372i \(0.447389\pi\)
\(570\) −7.27468e6 −0.937835
\(571\) 1.28506e7 1.64943 0.824717 0.565546i \(-0.191335\pi\)
0.824717 + 0.565546i \(0.191335\pi\)
\(572\) 7.72863e6 0.987671
\(573\) −9.99755e6 −1.27206
\(574\) 2.28722e6 0.289754
\(575\) 330625. 0.0417029
\(576\) 3.80068e7 4.77314
\(577\) −115717. −0.0144696 −0.00723479 0.999974i \(-0.502303\pi\)
−0.00723479 + 0.999974i \(0.502303\pi\)
\(578\) 2.28668e7 2.84699
\(579\) −2.00743e6 −0.248853
\(580\) −857944. −0.105898
\(581\) −1.22054e6 −0.150007
\(582\) −3.05552e7 −3.73919
\(583\) 1.98201e6 0.241509
\(584\) −1.04666e7 −1.26992
\(585\) −1.40479e7 −1.69715
\(586\) 1.15038e6 0.138388
\(587\) 484218. 0.0580024 0.0290012 0.999579i \(-0.490767\pi\)
0.0290012 + 0.999579i \(0.490767\pi\)
\(588\) 3.92919e7 4.68662
\(589\) −329998. −0.0391944
\(590\) −5.57255e6 −0.659058
\(591\) 1.49916e6 0.176555
\(592\) 3.08036e7 3.61241
\(593\) 1.55727e7 1.81856 0.909280 0.416186i \(-0.136633\pi\)
0.909280 + 0.416186i \(0.136633\pi\)
\(594\) −1.83481e7 −2.13366
\(595\) 1.40434e6 0.162623
\(596\) 201834. 0.0232745
\(597\) 8.13184e6 0.933798
\(598\) 4.45417e6 0.509347
\(599\) −1.65082e7 −1.87989 −0.939943 0.341330i \(-0.889123\pi\)
−0.939943 + 0.341330i \(0.889123\pi\)
\(600\) −9.80014e6 −1.11136
\(601\) −4.12370e6 −0.465694 −0.232847 0.972513i \(-0.574804\pi\)
−0.232847 + 0.972513i \(0.574804\pi\)
\(602\) −6.92456e6 −0.778755
\(603\) −6.29968e6 −0.705545
\(604\) 2.10494e7 2.34773
\(605\) 3.65808e6 0.406317
\(606\) −2.88212e7 −3.18809
\(607\) 6.83649e6 0.753115 0.376558 0.926393i \(-0.377108\pi\)
0.376558 + 0.926393i \(0.377108\pi\)
\(608\) −1.21503e7 −1.33299
\(609\) 392099. 0.0428403
\(610\) 6.99966e6 0.761644
\(611\) −1.25964e7 −1.36504
\(612\) 1.07029e8 11.5511
\(613\) 3.84085e6 0.412835 0.206417 0.978464i \(-0.433820\pi\)
0.206417 + 0.978464i \(0.433820\pi\)
\(614\) 9.99708e6 1.07017
\(615\) −5.60597e6 −0.597672
\(616\) −1.83348e6 −0.194682
\(617\) 2.03396e6 0.215095 0.107547 0.994200i \(-0.465700\pi\)
0.107547 + 0.994200i \(0.465700\pi\)
\(618\) −3.00002e7 −3.15975
\(619\) 7.73183e6 0.811065 0.405532 0.914081i \(-0.367086\pi\)
0.405532 + 0.914081i \(0.367086\pi\)
\(620\) −740508. −0.0773661
\(621\) −7.55498e6 −0.786148
\(622\) 1.30717e7 1.35474
\(623\) 1.37618e6 0.142055
\(624\) −6.92320e7 −7.11779
\(625\) 390625. 0.0400000
\(626\) 6.89658e6 0.703393
\(627\) 3.33580e6 0.338869
\(628\) −3.62161e7 −3.66440
\(629\) 2.06320e7 2.07929
\(630\) 5.55115e6 0.557227
\(631\) −1.85966e7 −1.85935 −0.929674 0.368384i \(-0.879911\pi\)
−0.929674 + 0.368384i \(0.879911\pi\)
\(632\) −3.32563e7 −3.31193
\(633\) 3.20318e7 3.17740
\(634\) 1.93041e6 0.190733
\(635\) 4.91503e6 0.483718
\(636\) −4.02943e7 −3.95003
\(637\) −1.26667e7 −1.23684
\(638\) 550639. 0.0535568
\(639\) −1.82337e7 −1.76653
\(640\) −3.34122e6 −0.322444
\(641\) 7.87977e6 0.757476 0.378738 0.925504i \(-0.376358\pi\)
0.378738 + 0.925504i \(0.376358\pi\)
\(642\) −4.05087e7 −3.87892
\(643\) −5.01179e6 −0.478041 −0.239021 0.971015i \(-0.576826\pi\)
−0.239021 + 0.971015i \(0.576826\pi\)
\(644\) −1.25753e6 −0.119482
\(645\) 1.69721e7 1.60633
\(646\) −1.78676e7 −1.68455
\(647\) 2.15204e6 0.202111 0.101055 0.994881i \(-0.467778\pi\)
0.101055 + 0.994881i \(0.467778\pi\)
\(648\) 1.36578e8 12.7775
\(649\) 2.55529e6 0.238138
\(650\) 5.26249e6 0.488548
\(651\) 338429. 0.0312979
\(652\) −1.67650e7 −1.54449
\(653\) 1.01515e7 0.931639 0.465820 0.884880i \(-0.345760\pi\)
0.465820 + 0.884880i \(0.345760\pi\)
\(654\) −4.78226e7 −4.37209
\(655\) −5.09882e6 −0.464372
\(656\) −2.05571e7 −1.86510
\(657\) −1.45312e7 −1.31337
\(658\) 4.97761e6 0.448184
\(659\) −1.84779e6 −0.165745 −0.0828724 0.996560i \(-0.526409\pi\)
−0.0828724 + 0.996560i \(0.526409\pi\)
\(660\) 7.48546e6 0.668896
\(661\) 1.70784e7 1.52035 0.760174 0.649720i \(-0.225114\pi\)
0.760174 + 0.649720i \(0.225114\pi\)
\(662\) −1.17942e6 −0.104598
\(663\) −4.63711e7 −4.09697
\(664\) 2.09200e7 1.84137
\(665\) −662103. −0.0580592
\(666\) 8.15552e7 7.12469
\(667\) 226730. 0.0197331
\(668\) −5.23427e7 −4.53853
\(669\) −2.48479e7 −2.14647
\(670\) 2.35993e6 0.203101
\(671\) −3.20969e6 −0.275206
\(672\) 1.24607e7 1.06443
\(673\) −2.05878e7 −1.75216 −0.876078 0.482170i \(-0.839849\pi\)
−0.876078 + 0.482170i \(0.839849\pi\)
\(674\) 2.23858e7 1.89812
\(675\) −8.92602e6 −0.754047
\(676\) 2.09234e7 1.76103
\(677\) −1.46673e7 −1.22992 −0.614961 0.788558i \(-0.710828\pi\)
−0.614961 + 0.788558i \(0.710828\pi\)
\(678\) 4.22257e7 3.52779
\(679\) −2.78097e6 −0.231485
\(680\) −2.40704e7 −1.99624
\(681\) 9.56284e6 0.790167
\(682\) 475267. 0.0391270
\(683\) 1.54350e7 1.26606 0.633030 0.774127i \(-0.281811\pi\)
0.633030 + 0.774127i \(0.281811\pi\)
\(684\) −5.04608e7 −4.12395
\(685\) 4.49089e6 0.365684
\(686\) 1.02878e7 0.834663
\(687\) −2.02908e7 −1.64024
\(688\) 6.22366e7 5.01274
\(689\) 1.29898e7 1.04245
\(690\) 4.31402e6 0.344953
\(691\) −2.55084e6 −0.203230 −0.101615 0.994824i \(-0.532401\pi\)
−0.101615 + 0.994824i \(0.532401\pi\)
\(692\) 1.41665e7 1.12460
\(693\) −2.54548e6 −0.201343
\(694\) −1.82463e7 −1.43806
\(695\) −828595. −0.0650699
\(696\) −6.72058e6 −0.525876
\(697\) −1.37690e7 −1.07355
\(698\) −2.05283e7 −1.59483
\(699\) −1.49199e7 −1.15498
\(700\) −1.48574e6 −0.114604
\(701\) −8.85672e6 −0.680734 −0.340367 0.940293i \(-0.610551\pi\)
−0.340367 + 0.940293i \(0.610551\pi\)
\(702\) −1.20251e8 −9.20971
\(703\) −9.72733e6 −0.742344
\(704\) 6.52873e6 0.496474
\(705\) −1.22001e7 −0.924466
\(706\) −1.70159e7 −1.28482
\(707\) −2.62315e6 −0.197367
\(708\) −5.19493e7 −3.89490
\(709\) 9.63817e6 0.720077 0.360039 0.932937i \(-0.382764\pi\)
0.360039 + 0.932937i \(0.382764\pi\)
\(710\) 6.83054e6 0.508521
\(711\) −4.61708e7 −3.42526
\(712\) −2.35877e7 −1.74376
\(713\) 195695. 0.0144164
\(714\) 1.83240e7 1.34516
\(715\) −2.41311e6 −0.176528
\(716\) −5.57829e7 −4.06648
\(717\) 1.16729e7 0.847970
\(718\) 436889. 0.0316271
\(719\) −1.44217e7 −1.04038 −0.520191 0.854050i \(-0.674139\pi\)
−0.520191 + 0.854050i \(0.674139\pi\)
\(720\) −4.98927e7 −3.58679
\(721\) −2.73046e6 −0.195613
\(722\) −1.77887e7 −1.26999
\(723\) 3.76767e7 2.68057
\(724\) −1.00988e7 −0.716019
\(725\) 267876. 0.0189273
\(726\) 4.77309e7 3.36092
\(727\) 727166. 0.0510267 0.0255133 0.999674i \(-0.491878\pi\)
0.0255133 + 0.999674i \(0.491878\pi\)
\(728\) −1.20164e7 −0.840324
\(729\) 8.26792e7 5.76205
\(730\) 5.44354e6 0.378072
\(731\) 4.16856e7 2.88531
\(732\) 6.52533e7 4.50116
\(733\) −2.63977e7 −1.81471 −0.907354 0.420368i \(-0.861901\pi\)
−0.907354 + 0.420368i \(0.861901\pi\)
\(734\) 2.16206e7 1.48125
\(735\) −1.22681e7 −0.837645
\(736\) 7.20534e6 0.490298
\(737\) −1.08215e6 −0.0733867
\(738\) −5.44267e7 −3.67850
\(739\) 762321. 0.0513484 0.0256742 0.999670i \(-0.491827\pi\)
0.0256742 + 0.999670i \(0.491827\pi\)
\(740\) −2.18279e7 −1.46532
\(741\) 2.18624e7 1.46269
\(742\) −5.13306e6 −0.342268
\(743\) −1.06898e7 −0.710392 −0.355196 0.934792i \(-0.615586\pi\)
−0.355196 + 0.934792i \(0.615586\pi\)
\(744\) −5.80066e6 −0.384189
\(745\) −63018.8 −0.00415987
\(746\) 3.27380e7 2.15380
\(747\) 2.90439e7 1.90438
\(748\) 1.83853e7 1.20148
\(749\) −3.68689e6 −0.240135
\(750\) 5.09691e6 0.330867
\(751\) 1.17700e7 0.761511 0.380755 0.924676i \(-0.375664\pi\)
0.380755 + 0.924676i \(0.375664\pi\)
\(752\) −4.47378e7 −2.88490
\(753\) 2.52894e7 1.62537
\(754\) 3.60882e6 0.231173
\(755\) −6.57227e6 −0.419612
\(756\) 3.39501e7 2.16041
\(757\) −4.99442e6 −0.316771 −0.158386 0.987377i \(-0.550629\pi\)
−0.158386 + 0.987377i \(0.550629\pi\)
\(758\) −2.49707e7 −1.57855
\(759\) −1.97819e6 −0.124642
\(760\) 1.13484e7 0.712692
\(761\) 1.26992e7 0.794903 0.397451 0.917623i \(-0.369895\pi\)
0.397451 + 0.917623i \(0.369895\pi\)
\(762\) 6.41318e7 4.00116
\(763\) −4.35256e6 −0.270666
\(764\) 2.59785e7 1.61021
\(765\) −3.34178e7 −2.06454
\(766\) 4.15172e7 2.55656
\(767\) 1.67471e7 1.02790
\(768\) 9.44953e6 0.578106
\(769\) −2.37894e7 −1.45067 −0.725333 0.688398i \(-0.758314\pi\)
−0.725333 + 0.688398i \(0.758314\pi\)
\(770\) 953567. 0.0579594
\(771\) −1.72604e7 −1.04572
\(772\) 5.21628e6 0.315005
\(773\) −2.86402e7 −1.72396 −0.861982 0.506940i \(-0.830777\pi\)
−0.861982 + 0.506940i \(0.830777\pi\)
\(774\) 1.64777e8 9.88652
\(775\) 231209. 0.0138277
\(776\) 4.76658e7 2.84153
\(777\) 9.97582e6 0.592784
\(778\) −2.29517e7 −1.35946
\(779\) 6.49164e6 0.383275
\(780\) 4.90588e7 2.88722
\(781\) −3.13215e6 −0.183744
\(782\) 1.05958e7 0.619608
\(783\) −6.12114e6 −0.356802
\(784\) −4.49873e7 −2.61396
\(785\) 1.13078e7 0.654941
\(786\) −6.65298e7 −3.84114
\(787\) 4.91527e6 0.282885 0.141443 0.989946i \(-0.454826\pi\)
0.141443 + 0.989946i \(0.454826\pi\)
\(788\) −3.89557e6 −0.223488
\(789\) 4.65665e7 2.66306
\(790\) 1.72961e7 0.986007
\(791\) 3.84316e6 0.218397
\(792\) 4.36295e7 2.47154
\(793\) −2.10359e7 −1.18790
\(794\) −1.63235e7 −0.918884
\(795\) 1.25811e7 0.705994
\(796\) −2.11305e7 −1.18203
\(797\) −3.08045e6 −0.171778 −0.0858892 0.996305i \(-0.527373\pi\)
−0.0858892 + 0.996305i \(0.527373\pi\)
\(798\) −8.63917e6 −0.480247
\(799\) −2.99651e7 −1.66054
\(800\) 8.51293e6 0.470277
\(801\) −3.27475e7 −1.80342
\(802\) 6.37357e7 3.49902
\(803\) −2.49614e6 −0.136609
\(804\) 2.20001e7 1.20029
\(805\) 392640. 0.0213552
\(806\) 3.11484e6 0.168888
\(807\) 2.16169e7 1.16845
\(808\) 4.49608e7 2.42273
\(809\) 3.49284e7 1.87632 0.938162 0.346197i \(-0.112527\pi\)
0.938162 + 0.346197i \(0.112527\pi\)
\(810\) −7.10324e7 −3.80403
\(811\) 1.49242e7 0.796779 0.398390 0.917216i \(-0.369569\pi\)
0.398390 + 0.917216i \(0.369569\pi\)
\(812\) −1.01887e6 −0.0542285
\(813\) 1.45963e7 0.774493
\(814\) 1.40094e7 0.741068
\(815\) 5.23454e6 0.276048
\(816\) −1.64693e8 −8.65862
\(817\) −1.96534e7 −1.03011
\(818\) −4.03167e7 −2.10669
\(819\) −1.66828e7 −0.869078
\(820\) 1.45671e7 0.756550
\(821\) 2.34285e7 1.21307 0.606536 0.795056i \(-0.292559\pi\)
0.606536 + 0.795056i \(0.292559\pi\)
\(822\) 5.85975e7 3.02482
\(823\) −2.90734e6 −0.149622 −0.0748111 0.997198i \(-0.523835\pi\)
−0.0748111 + 0.997198i \(0.523835\pi\)
\(824\) 4.68001e7 2.40120
\(825\) −2.33719e6 −0.119552
\(826\) −6.61778e6 −0.337491
\(827\) 1.53801e7 0.781978 0.390989 0.920395i \(-0.372133\pi\)
0.390989 + 0.920395i \(0.372133\pi\)
\(828\) 2.99242e7 1.51686
\(829\) 1.00621e6 0.0508514 0.0254257 0.999677i \(-0.491906\pi\)
0.0254257 + 0.999677i \(0.491906\pi\)
\(830\) −1.08802e7 −0.548202
\(831\) 1.07596e7 0.540500
\(832\) 4.27885e7 2.14298
\(833\) −3.01321e7 −1.50459
\(834\) −1.08116e7 −0.538237
\(835\) 1.63430e7 0.811176
\(836\) −8.66805e6 −0.428949
\(837\) −5.28327e6 −0.260669
\(838\) 3.91505e7 1.92587
\(839\) −2.19584e7 −1.07695 −0.538476 0.842641i \(-0.681000\pi\)
−0.538476 + 0.842641i \(0.681000\pi\)
\(840\) −1.16383e7 −0.569105
\(841\) −2.03274e7 −0.991044
\(842\) −5.64924e7 −2.74606
\(843\) −2.66271e7 −1.29049
\(844\) −8.32344e7 −4.02204
\(845\) −6.53292e6 −0.314750
\(846\) −1.18447e8 −5.68982
\(847\) 4.34422e6 0.208067
\(848\) 4.61350e7 2.20313
\(849\) 5.83795e7 2.77965
\(850\) 1.25187e7 0.594307
\(851\) 5.76849e6 0.273048
\(852\) 6.36767e7 3.00526
\(853\) 1.71543e7 0.807236 0.403618 0.914928i \(-0.367752\pi\)
0.403618 + 0.914928i \(0.367752\pi\)
\(854\) 8.31257e6 0.390023
\(855\) 1.57554e7 0.737078
\(856\) 6.31933e7 2.94772
\(857\) −3.23694e6 −0.150551 −0.0752753 0.997163i \(-0.523984\pi\)
−0.0752753 + 0.997163i \(0.523984\pi\)
\(858\) −3.14865e7 −1.46018
\(859\) −1.70092e7 −0.786503 −0.393252 0.919431i \(-0.628650\pi\)
−0.393252 + 0.919431i \(0.628650\pi\)
\(860\) −4.41018e7 −2.03334
\(861\) −6.65747e6 −0.306056
\(862\) 6.21570e6 0.284919
\(863\) 3.09259e7 1.41350 0.706749 0.707465i \(-0.250161\pi\)
0.706749 + 0.707465i \(0.250161\pi\)
\(864\) −1.94526e8 −8.86528
\(865\) −4.42322e6 −0.201001
\(866\) −1.82917e7 −0.828818
\(867\) −6.65588e7 −3.00717
\(868\) −879403. −0.0396177
\(869\) −7.93112e6 −0.356275
\(870\) 3.49527e6 0.156561
\(871\) −7.09225e6 −0.316766
\(872\) 7.46028e7 3.32249
\(873\) 6.61760e7 2.93876
\(874\) −4.99558e6 −0.221211
\(875\) 463894. 0.0204832
\(876\) 5.07466e7 2.23433
\(877\) −2.19762e7 −0.964837 −0.482419 0.875941i \(-0.660242\pi\)
−0.482419 + 0.875941i \(0.660242\pi\)
\(878\) 2.26611e7 0.992077
\(879\) −3.34845e6 −0.146175
\(880\) −8.57047e6 −0.373077
\(881\) −3.17297e7 −1.37729 −0.688646 0.725098i \(-0.741794\pi\)
−0.688646 + 0.725098i \(0.741794\pi\)
\(882\) −1.19108e8 −5.15547
\(883\) 9.54705e6 0.412067 0.206033 0.978545i \(-0.433944\pi\)
0.206033 + 0.978545i \(0.433944\pi\)
\(884\) 1.20495e8 5.18606
\(885\) 1.62201e7 0.696140
\(886\) 3.37991e7 1.44651
\(887\) −4.01212e7 −1.71224 −0.856121 0.516776i \(-0.827132\pi\)
−0.856121 + 0.516776i \(0.827132\pi\)
\(888\) −1.70985e8 −7.27657
\(889\) 5.83694e6 0.247703
\(890\) 1.22676e7 0.519140
\(891\) 3.25719e7 1.37451
\(892\) 6.45670e7 2.71705
\(893\) 1.41275e7 0.592841
\(894\) −822275. −0.0344091
\(895\) 1.74171e7 0.726806
\(896\) −3.96792e6 −0.165117
\(897\) −1.29648e7 −0.538005
\(898\) −1.04540e7 −0.432605
\(899\) 158555. 0.00654304
\(900\) 3.53547e7 1.45492
\(901\) 3.09009e7 1.26811
\(902\) −9.34932e6 −0.382616
\(903\) 2.01555e7 0.822572
\(904\) −6.58717e7 −2.68088
\(905\) 3.15316e6 0.127975
\(906\) −8.57555e7 −3.47090
\(907\) −3.99289e7 −1.61165 −0.805823 0.592157i \(-0.798277\pi\)
−0.805823 + 0.592157i \(0.798277\pi\)
\(908\) −2.48489e7 −1.00022
\(909\) 6.24205e7 2.50563
\(910\) 6.24956e6 0.250176
\(911\) −1.20333e7 −0.480383 −0.240192 0.970725i \(-0.577210\pi\)
−0.240192 + 0.970725i \(0.577210\pi\)
\(912\) 7.76472e7 3.09128
\(913\) 4.98911e6 0.198082
\(914\) −8.53196e7 −3.37818
\(915\) −2.03741e7 −0.804498
\(916\) 5.27255e7 2.07626
\(917\) −6.05519e6 −0.237796
\(918\) −2.86059e8 −11.2034
\(919\) −4.70215e7 −1.83657 −0.918285 0.395919i \(-0.870426\pi\)
−0.918285 + 0.395919i \(0.870426\pi\)
\(920\) −6.72984e6 −0.262141
\(921\) −2.90987e7 −1.13038
\(922\) 4.89864e7 1.89779
\(923\) −2.05277e7 −0.793115
\(924\) 8.88949e6 0.342529
\(925\) 6.81532e6 0.261898
\(926\) −3.22881e7 −1.23741
\(927\) 6.49740e7 2.48337
\(928\) 5.83785e6 0.222527
\(929\) −2.53672e7 −0.964346 −0.482173 0.876076i \(-0.660152\pi\)
−0.482173 + 0.876076i \(0.660152\pi\)
\(930\) 3.01683e6 0.114378
\(931\) 1.42063e7 0.537164
\(932\) 3.87693e7 1.46200
\(933\) −3.80481e7 −1.43096
\(934\) 8.02359e7 3.00955
\(935\) −5.74044e6 −0.214742
\(936\) 2.85943e8 10.6682
\(937\) 173653. 0.00646152 0.00323076 0.999995i \(-0.498972\pi\)
0.00323076 + 0.999995i \(0.498972\pi\)
\(938\) 2.80258e6 0.104004
\(939\) −2.00740e7 −0.742969
\(940\) 3.17019e7 1.17021
\(941\) −1.22110e7 −0.449550 −0.224775 0.974411i \(-0.572165\pi\)
−0.224775 + 0.974411i \(0.572165\pi\)
\(942\) 1.47545e8 5.41746
\(943\) −3.84967e6 −0.140976
\(944\) 5.94793e7 2.17238
\(945\) −1.06003e7 −0.386133
\(946\) 2.83050e7 1.02834
\(947\) 7.25108e6 0.262741 0.131370 0.991333i \(-0.458062\pi\)
0.131370 + 0.991333i \(0.458062\pi\)
\(948\) 1.61240e8 5.82710
\(949\) −1.63594e7 −0.589659
\(950\) −5.90215e6 −0.212178
\(951\) −5.61889e6 −0.201465
\(952\) −2.85853e7 −1.02223
\(953\) 5.10793e7 1.82185 0.910924 0.412573i \(-0.135370\pi\)
0.910924 + 0.412573i \(0.135370\pi\)
\(954\) 1.22146e8 4.34519
\(955\) −8.11129e6 −0.287794
\(956\) −3.03319e7 −1.07338
\(957\) −1.60276e6 −0.0565702
\(958\) 4.93993e7 1.73903
\(959\) 5.33324e6 0.187260
\(960\) 4.14422e7 1.45132
\(961\) −2.84923e7 −0.995220
\(962\) 9.18158e7 3.19875
\(963\) 8.77332e7 3.04859
\(964\) −9.79024e7 −3.39313
\(965\) −1.62868e6 −0.0563012
\(966\) 5.12319e6 0.176644
\(967\) 1.14223e6 0.0392816 0.0196408 0.999807i \(-0.493748\pi\)
0.0196408 + 0.999807i \(0.493748\pi\)
\(968\) −7.44598e7 −2.55407
\(969\) 5.20075e7 1.77933
\(970\) −2.47903e7 −0.845964
\(971\) −1.92597e7 −0.655542 −0.327771 0.944757i \(-0.606297\pi\)
−0.327771 + 0.944757i \(0.606297\pi\)
\(972\) −3.84314e8 −13.0473
\(973\) −984013. −0.0333210
\(974\) −7.69847e7 −2.60020
\(975\) −1.53176e7 −0.516036
\(976\) −7.47117e7 −2.51052
\(977\) −5.18143e7 −1.73665 −0.868327 0.495993i \(-0.834804\pi\)
−0.868327 + 0.495993i \(0.834804\pi\)
\(978\) 6.83006e7 2.28338
\(979\) −5.62531e6 −0.187581
\(980\) 3.18786e7 1.06031
\(981\) 1.03573e8 3.43618
\(982\) −1.93565e7 −0.640543
\(983\) 4.26426e7 1.40754 0.703769 0.710429i \(-0.251499\pi\)
0.703769 + 0.710429i \(0.251499\pi\)
\(984\) 1.14109e8 3.75692
\(985\) 1.21631e6 0.0399443
\(986\) 8.58484e6 0.281216
\(987\) −1.44885e7 −0.473401
\(988\) −5.68094e7 −1.85152
\(989\) 1.16548e7 0.378892
\(990\) −2.26910e7 −0.735812
\(991\) −2.07817e7 −0.672198 −0.336099 0.941827i \(-0.609108\pi\)
−0.336099 + 0.941827i \(0.609108\pi\)
\(992\) 5.03876e6 0.162572
\(993\) 3.43296e6 0.110483
\(994\) 8.11173e6 0.260404
\(995\) 6.59758e6 0.211265
\(996\) −1.01429e8 −3.23977
\(997\) −6.06820e7 −1.93340 −0.966701 0.255910i \(-0.917625\pi\)
−0.966701 + 0.255910i \(0.917625\pi\)
\(998\) −8.08955e7 −2.57098
\(999\) −1.55734e8 −4.93709
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 115.6.a.c.1.7 7
3.2 odd 2 1035.6.a.b.1.1 7
5.4 even 2 575.6.a.d.1.1 7
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
115.6.a.c.1.7 7 1.1 even 1 trivial
575.6.a.d.1.1 7 5.4 even 2
1035.6.a.b.1.1 7 3.2 odd 2