Properties

Label 115.6.a.c.1.5
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 1.5
Root \(-4.26829\) 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)
 
\(f(q)\) \(=\) \(q+5.26829 q^{2} -11.8410 q^{3} -4.24511 q^{4} -25.0000 q^{5} -62.3818 q^{6} +72.8708 q^{7} -190.950 q^{8} -102.791 q^{9} +O(q^{10})\) \(q+5.26829 q^{2} -11.8410 q^{3} -4.24511 q^{4} -25.0000 q^{5} -62.3818 q^{6} +72.8708 q^{7} -190.950 q^{8} -102.791 q^{9} -131.707 q^{10} +732.810 q^{11} +50.2663 q^{12} +676.311 q^{13} +383.904 q^{14} +296.025 q^{15} -870.135 q^{16} +381.818 q^{17} -541.533 q^{18} +853.077 q^{19} +106.128 q^{20} -862.862 q^{21} +3860.66 q^{22} +529.000 q^{23} +2261.03 q^{24} +625.000 q^{25} +3563.00 q^{26} +4094.51 q^{27} -309.345 q^{28} -3162.45 q^{29} +1559.54 q^{30} +6803.69 q^{31} +1526.27 q^{32} -8677.20 q^{33} +2011.53 q^{34} -1821.77 q^{35} +436.359 q^{36} -2630.93 q^{37} +4494.26 q^{38} -8008.19 q^{39} +4773.74 q^{40} +15206.0 q^{41} -4545.81 q^{42} -6961.13 q^{43} -3110.86 q^{44} +2569.78 q^{45} +2786.93 q^{46} +8075.94 q^{47} +10303.3 q^{48} -11496.8 q^{49} +3292.68 q^{50} -4521.10 q^{51} -2871.01 q^{52} -1322.64 q^{53} +21571.1 q^{54} -18320.3 q^{55} -13914.7 q^{56} -10101.3 q^{57} -16660.7 q^{58} -2011.03 q^{59} -1256.66 q^{60} -19642.0 q^{61} +35843.8 q^{62} -7490.46 q^{63} +35885.2 q^{64} -16907.8 q^{65} -45714.0 q^{66} +23755.1 q^{67} -1620.86 q^{68} -6263.88 q^{69} -9597.61 q^{70} +44466.6 q^{71} +19627.9 q^{72} +9203.35 q^{73} -13860.5 q^{74} -7400.62 q^{75} -3621.41 q^{76} +53400.5 q^{77} -42189.5 q^{78} +27674.3 q^{79} +21753.4 q^{80} -23504.8 q^{81} +80109.8 q^{82} +32584.3 q^{83} +3662.94 q^{84} -9545.45 q^{85} -36673.3 q^{86} +37446.5 q^{87} -139930. q^{88} -117078. q^{89} +13538.3 q^{90} +49283.3 q^{91} -2245.66 q^{92} -80562.4 q^{93} +42546.4 q^{94} -21326.9 q^{95} -18072.5 q^{96} -28382.3 q^{97} -60568.7 q^{98} -75326.3 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 7 q + 4 q^{2} - 3 q^{3} + 178 q^{4} - 175 q^{5} - 381 q^{6} + 33 q^{7} + 546 q^{8} + 440 q^{9}+O(q^{10}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 5.26829 0.931311 0.465656 0.884966i \(-0.345819\pi\)
0.465656 + 0.884966i \(0.345819\pi\)
\(3\) −11.8410 −0.759600 −0.379800 0.925069i \(-0.624007\pi\)
−0.379800 + 0.925069i \(0.624007\pi\)
\(4\) −4.24511 −0.132660
\(5\) −25.0000 −0.447214
\(6\) −62.3818 −0.707424
\(7\) 72.8708 0.562093 0.281047 0.959694i \(-0.409318\pi\)
0.281047 + 0.959694i \(0.409318\pi\)
\(8\) −190.950 −1.05486
\(9\) −102.791 −0.423008
\(10\) −131.707 −0.416495
\(11\) 732.810 1.82604 0.913019 0.407917i \(-0.133745\pi\)
0.913019 + 0.407917i \(0.133745\pi\)
\(12\) 50.2663 0.100768
\(13\) 676.311 1.10991 0.554955 0.831880i \(-0.312735\pi\)
0.554955 + 0.831880i \(0.312735\pi\)
\(14\) 383.904 0.523484
\(15\) 296.025 0.339703
\(16\) −870.135 −0.849742
\(17\) 381.818 0.320431 0.160215 0.987082i \(-0.448781\pi\)
0.160215 + 0.987082i \(0.448781\pi\)
\(18\) −541.533 −0.393952
\(19\) 853.077 0.542131 0.271066 0.962561i \(-0.412624\pi\)
0.271066 + 0.962561i \(0.412624\pi\)
\(20\) 106.128 0.0593272
\(21\) −862.862 −0.426966
\(22\) 3860.66 1.70061
\(23\) 529.000 0.208514
\(24\) 2261.03 0.801270
\(25\) 625.000 0.200000
\(26\) 3563.00 1.03367
\(27\) 4094.51 1.08092
\(28\) −309.345 −0.0745671
\(29\) −3162.45 −0.698278 −0.349139 0.937071i \(-0.613526\pi\)
−0.349139 + 0.937071i \(0.613526\pi\)
\(30\) 1559.54 0.316369
\(31\) 6803.69 1.27157 0.635785 0.771866i \(-0.280677\pi\)
0.635785 + 0.771866i \(0.280677\pi\)
\(32\) 1526.27 0.263485
\(33\) −8677.20 −1.38706
\(34\) 2011.53 0.298421
\(35\) −1821.77 −0.251376
\(36\) 436.359 0.0561162
\(37\) −2630.93 −0.315940 −0.157970 0.987444i \(-0.550495\pi\)
−0.157970 + 0.987444i \(0.550495\pi\)
\(38\) 4494.26 0.504893
\(39\) −8008.19 −0.843087
\(40\) 4773.74 0.471747
\(41\) 15206.0 1.41272 0.706360 0.707852i \(-0.250336\pi\)
0.706360 + 0.707852i \(0.250336\pi\)
\(42\) −4545.81 −0.397638
\(43\) −6961.13 −0.574128 −0.287064 0.957911i \(-0.592679\pi\)
−0.287064 + 0.957911i \(0.592679\pi\)
\(44\) −3110.86 −0.242242
\(45\) 2569.78 0.189175
\(46\) 2786.93 0.194192
\(47\) 8075.94 0.533271 0.266636 0.963797i \(-0.414088\pi\)
0.266636 + 0.963797i \(0.414088\pi\)
\(48\) 10303.3 0.645464
\(49\) −11496.8 −0.684051
\(50\) 3292.68 0.186262
\(51\) −4521.10 −0.243399
\(52\) −2871.01 −0.147240
\(53\) −1322.64 −0.0646774 −0.0323387 0.999477i \(-0.510296\pi\)
−0.0323387 + 0.999477i \(0.510296\pi\)
\(54\) 21571.1 1.00667
\(55\) −18320.3 −0.816629
\(56\) −13914.7 −0.592929
\(57\) −10101.3 −0.411803
\(58\) −16660.7 −0.650314
\(59\) −2011.03 −0.0752122 −0.0376061 0.999293i \(-0.511973\pi\)
−0.0376061 + 0.999293i \(0.511973\pi\)
\(60\) −1256.66 −0.0450649
\(61\) −19642.0 −0.675866 −0.337933 0.941170i \(-0.609728\pi\)
−0.337933 + 0.941170i \(0.609728\pi\)
\(62\) 35843.8 1.18423
\(63\) −7490.46 −0.237770
\(64\) 35885.2 1.09513
\(65\) −16907.8 −0.496367
\(66\) −45714.0 −1.29178
\(67\) 23755.1 0.646502 0.323251 0.946313i \(-0.395224\pi\)
0.323251 + 0.946313i \(0.395224\pi\)
\(68\) −1620.86 −0.0425083
\(69\) −6263.88 −0.158387
\(70\) −9597.61 −0.234109
\(71\) 44466.6 1.04686 0.523429 0.852069i \(-0.324653\pi\)
0.523429 + 0.852069i \(0.324653\pi\)
\(72\) 19627.9 0.446214
\(73\) 9203.35 0.202134 0.101067 0.994880i \(-0.467774\pi\)
0.101067 + 0.994880i \(0.467774\pi\)
\(74\) −13860.5 −0.294238
\(75\) −7400.62 −0.151920
\(76\) −3621.41 −0.0719190
\(77\) 53400.5 1.02640
\(78\) −42189.5 −0.785177
\(79\) 27674.3 0.498895 0.249447 0.968388i \(-0.419751\pi\)
0.249447 + 0.968388i \(0.419751\pi\)
\(80\) 21753.4 0.380016
\(81\) −23504.8 −0.398056
\(82\) 80109.8 1.31568
\(83\) 32584.3 0.519175 0.259587 0.965720i \(-0.416413\pi\)
0.259587 + 0.965720i \(0.416413\pi\)
\(84\) 3662.94 0.0566412
\(85\) −9545.45 −0.143301
\(86\) −36673.3 −0.534692
\(87\) 37446.5 0.530412
\(88\) −139930. −1.92621
\(89\) −117078. −1.56675 −0.783377 0.621547i \(-0.786505\pi\)
−0.783377 + 0.621547i \(0.786505\pi\)
\(90\) 13538.3 0.176181
\(91\) 49283.3 0.623873
\(92\) −2245.66 −0.0276615
\(93\) −80562.4 −0.965884
\(94\) 42546.4 0.496642
\(95\) −21326.9 −0.242448
\(96\) −18072.5 −0.200143
\(97\) −28382.3 −0.306280 −0.153140 0.988204i \(-0.548939\pi\)
−0.153140 + 0.988204i \(0.548939\pi\)
\(98\) −60568.7 −0.637064
\(99\) −75326.3 −0.772429
\(100\) −2653.19 −0.0265319
\(101\) −55582.3 −0.542167 −0.271083 0.962556i \(-0.587382\pi\)
−0.271083 + 0.962556i \(0.587382\pi\)
\(102\) −23818.5 −0.226680
\(103\) −52035.7 −0.483290 −0.241645 0.970365i \(-0.577687\pi\)
−0.241645 + 0.970365i \(0.577687\pi\)
\(104\) −129141. −1.17080
\(105\) 21571.6 0.190945
\(106\) −6968.06 −0.0602348
\(107\) −5725.28 −0.0483434 −0.0241717 0.999708i \(-0.507695\pi\)
−0.0241717 + 0.999708i \(0.507695\pi\)
\(108\) −17381.6 −0.143394
\(109\) 188458. 1.51932 0.759660 0.650321i \(-0.225366\pi\)
0.759660 + 0.650321i \(0.225366\pi\)
\(110\) −96516.4 −0.760536
\(111\) 31152.8 0.239988
\(112\) −63407.5 −0.477634
\(113\) 117240. 0.863735 0.431868 0.901937i \(-0.357855\pi\)
0.431868 + 0.901937i \(0.357855\pi\)
\(114\) −53216.5 −0.383516
\(115\) −13225.0 −0.0932505
\(116\) 13424.9 0.0926334
\(117\) −69518.6 −0.469501
\(118\) −10594.7 −0.0700459
\(119\) 27823.4 0.180112
\(120\) −56525.9 −0.358339
\(121\) 375960. 2.33442
\(122\) −103480. −0.629441
\(123\) −180054. −1.07310
\(124\) −28882.4 −0.168686
\(125\) −15625.0 −0.0894427
\(126\) −39461.9 −0.221438
\(127\) 130123. 0.715887 0.357944 0.933743i \(-0.383478\pi\)
0.357944 + 0.933743i \(0.383478\pi\)
\(128\) 140213. 0.756420
\(129\) 82426.7 0.436107
\(130\) −89075.0 −0.462272
\(131\) −336278. −1.71206 −0.856032 0.516923i \(-0.827077\pi\)
−0.856032 + 0.516923i \(0.827077\pi\)
\(132\) 36835.7 0.184007
\(133\) 62164.4 0.304728
\(134\) 125149. 0.602095
\(135\) −102363. −0.483401
\(136\) −72908.1 −0.338009
\(137\) 158649. 0.722166 0.361083 0.932534i \(-0.382407\pi\)
0.361083 + 0.932534i \(0.382407\pi\)
\(138\) −33000.0 −0.147508
\(139\) 343153. 1.50644 0.753218 0.657771i \(-0.228501\pi\)
0.753218 + 0.657771i \(0.228501\pi\)
\(140\) 7733.61 0.0333474
\(141\) −95627.1 −0.405073
\(142\) 234263. 0.974950
\(143\) 495607. 2.02674
\(144\) 89442.1 0.359448
\(145\) 79061.2 0.312280
\(146\) 48485.9 0.188249
\(147\) 136134. 0.519605
\(148\) 11168.6 0.0419125
\(149\) 343414. 1.26722 0.633611 0.773652i \(-0.281572\pi\)
0.633611 + 0.773652i \(0.281572\pi\)
\(150\) −38988.6 −0.141485
\(151\) −518178. −1.84943 −0.924713 0.380666i \(-0.875695\pi\)
−0.924713 + 0.380666i \(0.875695\pi\)
\(152\) −162895. −0.571872
\(153\) −39247.5 −0.135545
\(154\) 281329. 0.955901
\(155\) −170092. −0.568663
\(156\) 33995.6 0.111844
\(157\) 96581.7 0.312713 0.156356 0.987701i \(-0.450025\pi\)
0.156356 + 0.987701i \(0.450025\pi\)
\(158\) 145796. 0.464626
\(159\) 15661.4 0.0491290
\(160\) −38156.7 −0.117834
\(161\) 38548.6 0.117205
\(162\) −123830. −0.370714
\(163\) 218234. 0.643360 0.321680 0.946848i \(-0.395752\pi\)
0.321680 + 0.946848i \(0.395752\pi\)
\(164\) −64551.3 −0.187411
\(165\) 216930. 0.620311
\(166\) 171664. 0.483513
\(167\) 633975. 1.75906 0.879530 0.475843i \(-0.157857\pi\)
0.879530 + 0.475843i \(0.157857\pi\)
\(168\) 164763. 0.450389
\(169\) 86103.1 0.231901
\(170\) −50288.2 −0.133458
\(171\) −87688.7 −0.229326
\(172\) 29550.8 0.0761637
\(173\) 423316. 1.07535 0.537674 0.843153i \(-0.319303\pi\)
0.537674 + 0.843153i \(0.319303\pi\)
\(174\) 197279. 0.493979
\(175\) 45544.2 0.112419
\(176\) −637644. −1.55166
\(177\) 23812.6 0.0571312
\(178\) −616802. −1.45914
\(179\) −411280. −0.959411 −0.479705 0.877430i \(-0.659256\pi\)
−0.479705 + 0.877430i \(0.659256\pi\)
\(180\) −10909.0 −0.0250959
\(181\) −3600.89 −0.00816984 −0.00408492 0.999992i \(-0.501300\pi\)
−0.00408492 + 0.999992i \(0.501300\pi\)
\(182\) 259639. 0.581020
\(183\) 232580. 0.513387
\(184\) −101012. −0.219953
\(185\) 65773.2 0.141293
\(186\) −424426. −0.899539
\(187\) 279800. 0.585119
\(188\) −34283.2 −0.0707436
\(189\) 298370. 0.607576
\(190\) −112356. −0.225795
\(191\) −111906. −0.221957 −0.110979 0.993823i \(-0.535399\pi\)
−0.110979 + 0.993823i \(0.535399\pi\)
\(192\) −424916. −0.831859
\(193\) 498666. 0.963643 0.481822 0.876269i \(-0.339975\pi\)
0.481822 + 0.876269i \(0.339975\pi\)
\(194\) −149526. −0.285242
\(195\) 200205. 0.377040
\(196\) 48805.4 0.0907460
\(197\) −352505. −0.647142 −0.323571 0.946204i \(-0.604883\pi\)
−0.323571 + 0.946204i \(0.604883\pi\)
\(198\) −396841. −0.719372
\(199\) −619579. −1.10908 −0.554542 0.832156i \(-0.687106\pi\)
−0.554542 + 0.832156i \(0.687106\pi\)
\(200\) −119344. −0.210972
\(201\) −281284. −0.491083
\(202\) −292824. −0.504926
\(203\) −230450. −0.392497
\(204\) 19192.6 0.0322893
\(205\) −380151. −0.631788
\(206\) −274139. −0.450094
\(207\) −54376.4 −0.0882033
\(208\) −588482. −0.943137
\(209\) 625144. 0.989952
\(210\) 113645. 0.177829
\(211\) −1.28079e6 −1.98048 −0.990241 0.139366i \(-0.955494\pi\)
−0.990241 + 0.139366i \(0.955494\pi\)
\(212\) 5614.76 0.00858009
\(213\) −526528. −0.795193
\(214\) −30162.4 −0.0450227
\(215\) 174028. 0.256758
\(216\) −781845. −1.14021
\(217\) 495790. 0.714741
\(218\) 992853. 1.41496
\(219\) −108977. −0.153541
\(220\) 77771.5 0.108334
\(221\) 258228. 0.355649
\(222\) 164122. 0.223503
\(223\) 156448. 0.210673 0.105336 0.994437i \(-0.466408\pi\)
0.105336 + 0.994437i \(0.466408\pi\)
\(224\) 111220. 0.148103
\(225\) −64244.4 −0.0846017
\(226\) 617656. 0.804406
\(227\) 212664. 0.273924 0.136962 0.990576i \(-0.456266\pi\)
0.136962 + 0.990576i \(0.456266\pi\)
\(228\) 42881.0 0.0546296
\(229\) −719349. −0.906465 −0.453232 0.891392i \(-0.649729\pi\)
−0.453232 + 0.891392i \(0.649729\pi\)
\(230\) −69673.1 −0.0868452
\(231\) −632314. −0.779656
\(232\) 603869. 0.736585
\(233\) 1.61407e6 1.94774 0.973871 0.227103i \(-0.0729256\pi\)
0.973871 + 0.227103i \(0.0729256\pi\)
\(234\) −366244. −0.437252
\(235\) −201898. −0.238486
\(236\) 8537.04 0.00997763
\(237\) −327691. −0.378960
\(238\) 146582. 0.167740
\(239\) −1.00977e6 −1.14347 −0.571737 0.820437i \(-0.693730\pi\)
−0.571737 + 0.820437i \(0.693730\pi\)
\(240\) −257582. −0.288660
\(241\) −995482. −1.10406 −0.552028 0.833826i \(-0.686146\pi\)
−0.552028 + 0.833826i \(0.686146\pi\)
\(242\) 1.98067e6 2.17407
\(243\) −716645. −0.778554
\(244\) 83382.3 0.0896601
\(245\) 287421. 0.305917
\(246\) −948579. −0.999392
\(247\) 576945. 0.601717
\(248\) −1.29916e6 −1.34133
\(249\) −385831. −0.394365
\(250\) −82317.0 −0.0832990
\(251\) −599825. −0.600953 −0.300476 0.953789i \(-0.597146\pi\)
−0.300476 + 0.953789i \(0.597146\pi\)
\(252\) 31797.8 0.0315425
\(253\) 387657. 0.380755
\(254\) 685525. 0.666713
\(255\) 113028. 0.108851
\(256\) −409643. −0.390666
\(257\) −1.57463e6 −1.48712 −0.743559 0.668670i \(-0.766864\pi\)
−0.743559 + 0.668670i \(0.766864\pi\)
\(258\) 434248. 0.406152
\(259\) −191718. −0.177588
\(260\) 71775.3 0.0658479
\(261\) 325071. 0.295377
\(262\) −1.77161e6 −1.59446
\(263\) −1.47167e6 −1.31196 −0.655982 0.754777i \(-0.727745\pi\)
−0.655982 + 0.754777i \(0.727745\pi\)
\(264\) 1.65691e6 1.46315
\(265\) 33066.1 0.0289246
\(266\) 327500. 0.283797
\(267\) 1.38632e6 1.19011
\(268\) −100843. −0.0857648
\(269\) −135287. −0.113992 −0.0569959 0.998374i \(-0.518152\pi\)
−0.0569959 + 0.998374i \(0.518152\pi\)
\(270\) −539276. −0.450196
\(271\) 876933. 0.725342 0.362671 0.931917i \(-0.381865\pi\)
0.362671 + 0.931917i \(0.381865\pi\)
\(272\) −332234. −0.272283
\(273\) −583563. −0.473894
\(274\) 835811. 0.672561
\(275\) 458006. 0.365208
\(276\) 26590.9 0.0210116
\(277\) −329285. −0.257853 −0.128927 0.991654i \(-0.541153\pi\)
−0.128927 + 0.991654i \(0.541153\pi\)
\(278\) 1.80783e6 1.40296
\(279\) −699358. −0.537885
\(280\) 347867. 0.265166
\(281\) 413432. 0.312348 0.156174 0.987730i \(-0.450084\pi\)
0.156174 + 0.987730i \(0.450084\pi\)
\(282\) −503791. −0.377249
\(283\) 1.93000e6 1.43249 0.716246 0.697848i \(-0.245859\pi\)
0.716246 + 0.697848i \(0.245859\pi\)
\(284\) −188766. −0.138876
\(285\) 252532. 0.184164
\(286\) 2.61100e6 1.88752
\(287\) 1.10808e6 0.794081
\(288\) −156886. −0.111456
\(289\) −1.27407e6 −0.897324
\(290\) 416518. 0.290829
\(291\) 336075. 0.232650
\(292\) −39069.2 −0.0268150
\(293\) −2.02419e6 −1.37747 −0.688736 0.725013i \(-0.741834\pi\)
−0.688736 + 0.725013i \(0.741834\pi\)
\(294\) 717194. 0.483914
\(295\) 50275.7 0.0336359
\(296\) 502375. 0.333272
\(297\) 3.00050e6 1.97380
\(298\) 1.80921e6 1.18018
\(299\) 357768. 0.231432
\(300\) 31416.4 0.0201537
\(301\) −507263. −0.322713
\(302\) −2.72991e6 −1.72239
\(303\) 658149. 0.411830
\(304\) −742293. −0.460672
\(305\) 491049. 0.302256
\(306\) −206767. −0.126234
\(307\) −2.44745e6 −1.48207 −0.741034 0.671467i \(-0.765664\pi\)
−0.741034 + 0.671467i \(0.765664\pi\)
\(308\) −226691. −0.136162
\(309\) 616154. 0.367107
\(310\) −896095. −0.529602
\(311\) −962776. −0.564448 −0.282224 0.959348i \(-0.591072\pi\)
−0.282224 + 0.959348i \(0.591072\pi\)
\(312\) 1.52916e6 0.889338
\(313\) 1.33248e6 0.768774 0.384387 0.923172i \(-0.374413\pi\)
0.384387 + 0.923172i \(0.374413\pi\)
\(314\) 508821. 0.291233
\(315\) 187262. 0.106334
\(316\) −117480. −0.0661832
\(317\) 2.64829e6 1.48019 0.740096 0.672501i \(-0.234780\pi\)
0.740096 + 0.672501i \(0.234780\pi\)
\(318\) 82508.8 0.0457543
\(319\) −2.31748e6 −1.27508
\(320\) −897129. −0.489756
\(321\) 67793.0 0.0367216
\(322\) 203085. 0.109154
\(323\) 325720. 0.173716
\(324\) 99780.5 0.0528060
\(325\) 422694. 0.221982
\(326\) 1.14972e6 0.599168
\(327\) −2.23153e6 −1.15407
\(328\) −2.90359e6 −1.49022
\(329\) 588500. 0.299748
\(330\) 1.14285e6 0.577703
\(331\) −3.03269e6 −1.52145 −0.760725 0.649075i \(-0.775156\pi\)
−0.760725 + 0.649075i \(0.775156\pi\)
\(332\) −138324. −0.0688735
\(333\) 270436. 0.133645
\(334\) 3.33996e6 1.63823
\(335\) −593878. −0.289125
\(336\) 750807. 0.362811
\(337\) −3.06223e6 −1.46880 −0.734402 0.678715i \(-0.762537\pi\)
−0.734402 + 0.678715i \(0.762537\pi\)
\(338\) 453616. 0.215972
\(339\) −1.38824e6 −0.656093
\(340\) 40521.5 0.0190103
\(341\) 4.98581e6 2.32194
\(342\) −461969. −0.213574
\(343\) −2.06252e6 −0.946594
\(344\) 1.32923e6 0.605624
\(345\) 156597. 0.0708330
\(346\) 2.23015e6 1.00148
\(347\) 3.47125e6 1.54761 0.773806 0.633423i \(-0.218351\pi\)
0.773806 + 0.633423i \(0.218351\pi\)
\(348\) −158965. −0.0703643
\(349\) −607747. −0.267091 −0.133546 0.991043i \(-0.542636\pi\)
−0.133546 + 0.991043i \(0.542636\pi\)
\(350\) 239940. 0.104697
\(351\) 2.76916e6 1.19972
\(352\) 1.11846e6 0.481133
\(353\) 4.43515e6 1.89440 0.947201 0.320641i \(-0.103898\pi\)
0.947201 + 0.320641i \(0.103898\pi\)
\(354\) 125452. 0.0532069
\(355\) −1.11166e6 −0.468169
\(356\) 497010. 0.207845
\(357\) −329456. −0.136813
\(358\) −2.16674e6 −0.893510
\(359\) 979250. 0.401012 0.200506 0.979692i \(-0.435741\pi\)
0.200506 + 0.979692i \(0.435741\pi\)
\(360\) −490698. −0.199553
\(361\) −1.74836e6 −0.706094
\(362\) −18970.5 −0.00760866
\(363\) −4.45174e6 −1.77322
\(364\) −209213. −0.0827628
\(365\) −230084. −0.0903969
\(366\) 1.22530e6 0.478123
\(367\) 578609. 0.224244 0.112122 0.993694i \(-0.464235\pi\)
0.112122 + 0.993694i \(0.464235\pi\)
\(368\) −460302. −0.177183
\(369\) −1.56304e6 −0.597592
\(370\) 346512. 0.131587
\(371\) −96382.0 −0.0363547
\(372\) 341996. 0.128134
\(373\) −5.24299e6 −1.95122 −0.975612 0.219504i \(-0.929556\pi\)
−0.975612 + 0.219504i \(0.929556\pi\)
\(374\) 1.47407e6 0.544928
\(375\) 185015. 0.0679407
\(376\) −1.54210e6 −0.562526
\(377\) −2.13880e6 −0.775026
\(378\) 1.57190e6 0.565842
\(379\) −1.60569e6 −0.574202 −0.287101 0.957900i \(-0.592692\pi\)
−0.287101 + 0.957900i \(0.592692\pi\)
\(380\) 90535.2 0.0321631
\(381\) −1.54078e6 −0.543788
\(382\) −589553. −0.206711
\(383\) −4.63165e6 −1.61339 −0.806694 0.590969i \(-0.798745\pi\)
−0.806694 + 0.590969i \(0.798745\pi\)
\(384\) −1.66026e6 −0.574576
\(385\) −1.33501e6 −0.459022
\(386\) 2.62712e6 0.897452
\(387\) 715542. 0.242861
\(388\) 120486. 0.0406310
\(389\) 4.24906e6 1.42370 0.711850 0.702331i \(-0.247857\pi\)
0.711850 + 0.702331i \(0.247857\pi\)
\(390\) 1.05474e6 0.351142
\(391\) 201982. 0.0668144
\(392\) 2.19532e6 0.721577
\(393\) 3.98186e6 1.30048
\(394\) −1.85710e6 −0.602690
\(395\) −691857. −0.223112
\(396\) 319769. 0.102470
\(397\) −735094. −0.234081 −0.117041 0.993127i \(-0.537341\pi\)
−0.117041 + 0.993127i \(0.537341\pi\)
\(398\) −3.26413e6 −1.03290
\(399\) −736088. −0.231472
\(400\) −543835. −0.169948
\(401\) 3.02966e6 0.940879 0.470439 0.882432i \(-0.344095\pi\)
0.470439 + 0.882432i \(0.344095\pi\)
\(402\) −1.48189e6 −0.457351
\(403\) 4.60141e6 1.41133
\(404\) 235953. 0.0719237
\(405\) 587620. 0.178016
\(406\) −1.21408e6 −0.365537
\(407\) −1.92797e6 −0.576919
\(408\) 863304. 0.256752
\(409\) −4.45759e6 −1.31762 −0.658812 0.752307i \(-0.728941\pi\)
−0.658812 + 0.752307i \(0.728941\pi\)
\(410\) −2.00275e6 −0.588391
\(411\) −1.87856e6 −0.548557
\(412\) 220897. 0.0641132
\(413\) −146545. −0.0422763
\(414\) −286471. −0.0821447
\(415\) −814608. −0.232182
\(416\) 1.03223e6 0.292444
\(417\) −4.06327e6 −1.14429
\(418\) 3.29344e6 0.921954
\(419\) 6.56774e6 1.82760 0.913799 0.406166i \(-0.133135\pi\)
0.913799 + 0.406166i \(0.133135\pi\)
\(420\) −91573.6 −0.0253307
\(421\) 5.39612e6 1.48380 0.741902 0.670508i \(-0.233924\pi\)
0.741902 + 0.670508i \(0.233924\pi\)
\(422\) −6.74756e6 −1.84444
\(423\) −830134. −0.225578
\(424\) 252558. 0.0682255
\(425\) 238636. 0.0640862
\(426\) −2.77390e6 −0.740572
\(427\) −1.43133e6 −0.379899
\(428\) 24304.4 0.00641322
\(429\) −5.86848e6 −1.53951
\(430\) 916832. 0.239121
\(431\) 2.83429e6 0.734938 0.367469 0.930036i \(-0.380224\pi\)
0.367469 + 0.930036i \(0.380224\pi\)
\(432\) −3.56278e6 −0.918500
\(433\) −6.24204e6 −1.59995 −0.799975 0.600033i \(-0.795154\pi\)
−0.799975 + 0.600033i \(0.795154\pi\)
\(434\) 2.61197e6 0.665646
\(435\) −936163. −0.237207
\(436\) −800026. −0.201552
\(437\) 451278. 0.113042
\(438\) −574121. −0.142994
\(439\) −3.28139e6 −0.812637 −0.406319 0.913731i \(-0.633188\pi\)
−0.406319 + 0.913731i \(0.633188\pi\)
\(440\) 3.49825e6 0.861428
\(441\) 1.18177e6 0.289359
\(442\) 1.36042e6 0.331220
\(443\) −1.20220e6 −0.291050 −0.145525 0.989355i \(-0.546487\pi\)
−0.145525 + 0.989355i \(0.546487\pi\)
\(444\) −132247. −0.0318367
\(445\) 2.92695e6 0.700674
\(446\) 824215. 0.196202
\(447\) −4.06637e6 −0.962582
\(448\) 2.61498e6 0.615564
\(449\) 889806. 0.208295 0.104148 0.994562i \(-0.466789\pi\)
0.104148 + 0.994562i \(0.466789\pi\)
\(450\) −338458. −0.0787905
\(451\) 1.11431e7 2.57968
\(452\) −497698. −0.114583
\(453\) 6.13574e6 1.40482
\(454\) 1.12038e6 0.255108
\(455\) −1.23208e6 −0.279004
\(456\) 1.92884e6 0.434394
\(457\) −1.58331e6 −0.354631 −0.177315 0.984154i \(-0.556741\pi\)
−0.177315 + 0.984154i \(0.556741\pi\)
\(458\) −3.78974e6 −0.844200
\(459\) 1.56336e6 0.346359
\(460\) 56141.6 0.0123706
\(461\) 2.11854e6 0.464284 0.232142 0.972682i \(-0.425427\pi\)
0.232142 + 0.972682i \(0.425427\pi\)
\(462\) −3.33122e6 −0.726102
\(463\) −3.20820e6 −0.695520 −0.347760 0.937584i \(-0.613058\pi\)
−0.347760 + 0.937584i \(0.613058\pi\)
\(464\) 2.75176e6 0.593356
\(465\) 2.01406e6 0.431957
\(466\) 8.50337e6 1.81395
\(467\) −2.09653e6 −0.444844 −0.222422 0.974950i \(-0.571396\pi\)
−0.222422 + 0.974950i \(0.571396\pi\)
\(468\) 295114. 0.0622839
\(469\) 1.73105e6 0.363395
\(470\) −1.06366e6 −0.222105
\(471\) −1.14362e6 −0.237537
\(472\) 384005. 0.0793382
\(473\) −5.10119e6 −1.04838
\(474\) −1.72637e6 −0.352930
\(475\) 533173. 0.108426
\(476\) −118113. −0.0238936
\(477\) 135956. 0.0273591
\(478\) −5.31975e6 −1.06493
\(479\) −6.14042e6 −1.22281 −0.611406 0.791317i \(-0.709396\pi\)
−0.611406 + 0.791317i \(0.709396\pi\)
\(480\) 451813. 0.0895066
\(481\) −1.77932e6 −0.350665
\(482\) −5.24449e6 −1.02822
\(483\) −456454. −0.0890285
\(484\) −1.59599e6 −0.309683
\(485\) 709558. 0.136973
\(486\) −3.77550e6 −0.725076
\(487\) 6.29369e6 1.20249 0.601247 0.799064i \(-0.294671\pi\)
0.601247 + 0.799064i \(0.294671\pi\)
\(488\) 3.75063e6 0.712943
\(489\) −2.58411e6 −0.488696
\(490\) 1.51422e6 0.284904
\(491\) 3.02860e6 0.566941 0.283470 0.958981i \(-0.408514\pi\)
0.283470 + 0.958981i \(0.408514\pi\)
\(492\) 764351. 0.142357
\(493\) −1.20748e6 −0.223750
\(494\) 3.03952e6 0.560386
\(495\) 1.88316e6 0.345441
\(496\) −5.92013e6 −1.08051
\(497\) 3.24031e6 0.588432
\(498\) −2.03267e6 −0.367276
\(499\) −5.56440e6 −1.00038 −0.500192 0.865914i \(-0.666737\pi\)
−0.500192 + 0.865914i \(0.666737\pi\)
\(500\) 66329.9 0.0118654
\(501\) −7.50689e6 −1.33618
\(502\) −3.16005e6 −0.559674
\(503\) 4.17981e6 0.736608 0.368304 0.929705i \(-0.379938\pi\)
0.368304 + 0.929705i \(0.379938\pi\)
\(504\) 1.43030e6 0.250814
\(505\) 1.38956e6 0.242464
\(506\) 2.04229e6 0.354602
\(507\) −1.01955e6 −0.176152
\(508\) −552386. −0.0949694
\(509\) −2.36278e6 −0.404230 −0.202115 0.979362i \(-0.564782\pi\)
−0.202115 + 0.979362i \(0.564782\pi\)
\(510\) 595462. 0.101375
\(511\) 670655. 0.113618
\(512\) −6.64493e6 −1.12025
\(513\) 3.49293e6 0.585999
\(514\) −8.29560e6 −1.38497
\(515\) 1.30089e6 0.216134
\(516\) −349910. −0.0578539
\(517\) 5.91813e6 0.973774
\(518\) −1.01003e6 −0.165389
\(519\) −5.01247e6 −0.816834
\(520\) 3.22853e6 0.523597
\(521\) −7.34682e6 −1.18578 −0.592891 0.805282i \(-0.702014\pi\)
−0.592891 + 0.805282i \(0.702014\pi\)
\(522\) 1.71257e6 0.275088
\(523\) 4.81938e6 0.770436 0.385218 0.922826i \(-0.374126\pi\)
0.385218 + 0.922826i \(0.374126\pi\)
\(524\) 1.42754e6 0.227122
\(525\) −539289. −0.0853932
\(526\) −7.75320e6 −1.22185
\(527\) 2.59777e6 0.407450
\(528\) 7.55034e6 1.17864
\(529\) 279841. 0.0434783
\(530\) 174202. 0.0269378
\(531\) 206716. 0.0318154
\(532\) −263895. −0.0404252
\(533\) 1.02840e7 1.56799
\(534\) 7.30354e6 1.10836
\(535\) 143132. 0.0216198
\(536\) −4.53603e6 −0.681968
\(537\) 4.86996e6 0.728768
\(538\) −712729. −0.106162
\(539\) −8.42501e6 −1.24910
\(540\) 434541. 0.0641278
\(541\) 8.00206e6 1.17546 0.587731 0.809056i \(-0.300021\pi\)
0.587731 + 0.809056i \(0.300021\pi\)
\(542\) 4.61994e6 0.675519
\(543\) 42638.1 0.00620580
\(544\) 582756. 0.0844286
\(545\) −4.71146e6 −0.679460
\(546\) −3.07438e6 −0.441342
\(547\) 1.18832e7 1.69811 0.849057 0.528302i \(-0.177171\pi\)
0.849057 + 0.528302i \(0.177171\pi\)
\(548\) −673484. −0.0958023
\(549\) 2.01902e6 0.285897
\(550\) 2.41291e6 0.340122
\(551\) −2.69781e6 −0.378558
\(552\) 1.19609e6 0.167076
\(553\) 2.01665e6 0.280425
\(554\) −1.73477e6 −0.240142
\(555\) −778820. −0.107326
\(556\) −1.45672e6 −0.199843
\(557\) 1.13254e7 1.54673 0.773367 0.633959i \(-0.218571\pi\)
0.773367 + 0.633959i \(0.218571\pi\)
\(558\) −3.68442e6 −0.500938
\(559\) −4.70789e6 −0.637230
\(560\) 1.58519e6 0.213604
\(561\) −3.31311e6 −0.444456
\(562\) 2.17808e6 0.290893
\(563\) 1.13454e7 1.50852 0.754259 0.656578i \(-0.227997\pi\)
0.754259 + 0.656578i \(0.227997\pi\)
\(564\) 405948. 0.0537368
\(565\) −2.93101e6 −0.386274
\(566\) 1.01678e7 1.33409
\(567\) −1.71281e6 −0.223744
\(568\) −8.49088e6 −1.10429
\(569\) 5.30510e6 0.686930 0.343465 0.939165i \(-0.388399\pi\)
0.343465 + 0.939165i \(0.388399\pi\)
\(570\) 1.33041e6 0.171514
\(571\) 1.42416e7 1.82796 0.913981 0.405756i \(-0.132992\pi\)
0.913981 + 0.405756i \(0.132992\pi\)
\(572\) −2.10391e6 −0.268867
\(573\) 1.32508e6 0.168599
\(574\) 5.83766e6 0.739536
\(575\) 330625. 0.0417029
\(576\) −3.68867e6 −0.463248
\(577\) −567010. −0.0709008 −0.0354504 0.999371i \(-0.511287\pi\)
−0.0354504 + 0.999371i \(0.511287\pi\)
\(578\) −6.71218e6 −0.835688
\(579\) −5.90470e6 −0.731983
\(580\) −335624. −0.0414269
\(581\) 2.37444e6 0.291824
\(582\) 1.77054e6 0.216670
\(583\) −969246. −0.118103
\(584\) −1.75738e6 −0.213222
\(585\) 1.73797e6 0.209967
\(586\) −1.06640e7 −1.28285
\(587\) −2.37878e6 −0.284944 −0.142472 0.989799i \(-0.545505\pi\)
−0.142472 + 0.989799i \(0.545505\pi\)
\(588\) −577904. −0.0689307
\(589\) 5.80407e6 0.689358
\(590\) 264867. 0.0313255
\(591\) 4.17400e6 0.491569
\(592\) 2.28926e6 0.268467
\(593\) 1.40968e7 1.64621 0.823103 0.567892i \(-0.192241\pi\)
0.823103 + 0.567892i \(0.192241\pi\)
\(594\) 1.58075e7 1.83822
\(595\) −695585. −0.0805485
\(596\) −1.45783e6 −0.168109
\(597\) 7.33643e6 0.842460
\(598\) 1.88483e6 0.215535
\(599\) 1.02416e7 1.16628 0.583139 0.812372i \(-0.301824\pi\)
0.583139 + 0.812372i \(0.301824\pi\)
\(600\) 1.41315e6 0.160254
\(601\) −312364. −0.0352756 −0.0176378 0.999844i \(-0.505615\pi\)
−0.0176378 + 0.999844i \(0.505615\pi\)
\(602\) −2.67241e6 −0.300547
\(603\) −2.44181e6 −0.273476
\(604\) 2.19972e6 0.245344
\(605\) −9.39900e6 −1.04398
\(606\) 3.46732e6 0.383541
\(607\) −1.50992e7 −1.66335 −0.831673 0.555266i \(-0.812616\pi\)
−0.831673 + 0.555266i \(0.812616\pi\)
\(608\) 1.30202e6 0.142843
\(609\) 2.72876e6 0.298141
\(610\) 2.58699e6 0.281495
\(611\) 5.46184e6 0.591883
\(612\) 166610. 0.0179813
\(613\) 1.01867e7 1.09492 0.547460 0.836832i \(-0.315595\pi\)
0.547460 + 0.836832i \(0.315595\pi\)
\(614\) −1.28939e7 −1.38027
\(615\) 4.50136e6 0.479906
\(616\) −1.01968e7 −1.08271
\(617\) 3.28554e6 0.347452 0.173726 0.984794i \(-0.444419\pi\)
0.173726 + 0.984794i \(0.444419\pi\)
\(618\) 3.24608e6 0.341891
\(619\) −3.05545e6 −0.320515 −0.160257 0.987075i \(-0.551232\pi\)
−0.160257 + 0.987075i \(0.551232\pi\)
\(620\) 722060. 0.0754387
\(621\) 2.16599e6 0.225387
\(622\) −5.07218e6 −0.525677
\(623\) −8.53158e6 −0.880662
\(624\) 6.96821e6 0.716407
\(625\) 390625. 0.0400000
\(626\) 7.01987e6 0.715967
\(627\) −7.40232e6 −0.751968
\(628\) −410000. −0.0414844
\(629\) −1.00454e6 −0.101237
\(630\) 986548. 0.0990300
\(631\) −1.67806e7 −1.67777 −0.838887 0.544305i \(-0.816793\pi\)
−0.838887 + 0.544305i \(0.816793\pi\)
\(632\) −5.28440e6 −0.526263
\(633\) 1.51658e7 1.50437
\(634\) 1.39520e7 1.37852
\(635\) −3.25307e6 −0.320154
\(636\) −66484.3 −0.00651743
\(637\) −7.77544e6 −0.759235
\(638\) −1.22091e7 −1.18750
\(639\) −4.57076e6 −0.442830
\(640\) −3.50532e6 −0.338281
\(641\) 2.00724e6 0.192954 0.0964772 0.995335i \(-0.469243\pi\)
0.0964772 + 0.995335i \(0.469243\pi\)
\(642\) 357153. 0.0341993
\(643\) 188198. 0.0179509 0.00897547 0.999960i \(-0.497143\pi\)
0.00897547 + 0.999960i \(0.497143\pi\)
\(644\) −163643. −0.0155483
\(645\) −2.06067e6 −0.195033
\(646\) 1.71599e6 0.161783
\(647\) −1.32071e7 −1.24035 −0.620177 0.784462i \(-0.712939\pi\)
−0.620177 + 0.784462i \(0.712939\pi\)
\(648\) 4.48824e6 0.419893
\(649\) −1.47370e6 −0.137340
\(650\) 2.22688e6 0.206734
\(651\) −5.87065e6 −0.542917
\(652\) −926429. −0.0853479
\(653\) −5.31660e6 −0.487922 −0.243961 0.969785i \(-0.578447\pi\)
−0.243961 + 0.969785i \(0.578447\pi\)
\(654\) −1.17564e7 −1.07480
\(655\) 8.40694e6 0.765658
\(656\) −1.32313e7 −1.20045
\(657\) −946021. −0.0855042
\(658\) 3.10039e6 0.279159
\(659\) 5.38018e6 0.482596 0.241298 0.970451i \(-0.422427\pi\)
0.241298 + 0.970451i \(0.422427\pi\)
\(660\) −920892. −0.0822903
\(661\) 8.76312e6 0.780108 0.390054 0.920792i \(-0.372456\pi\)
0.390054 + 0.920792i \(0.372456\pi\)
\(662\) −1.59771e7 −1.41694
\(663\) −3.05767e6 −0.270151
\(664\) −6.22197e6 −0.547656
\(665\) −1.55411e6 −0.136279
\(666\) 1.42473e6 0.124465
\(667\) −1.67294e6 −0.145601
\(668\) −2.69129e6 −0.233356
\(669\) −1.85250e6 −0.160027
\(670\) −3.12872e6 −0.269265
\(671\) −1.43938e7 −1.23416
\(672\) −1.31696e6 −0.112499
\(673\) −1.18629e7 −1.00961 −0.504804 0.863234i \(-0.668435\pi\)
−0.504804 + 0.863234i \(0.668435\pi\)
\(674\) −1.61327e7 −1.36791
\(675\) 2.55907e6 0.216183
\(676\) −365517. −0.0307639
\(677\) −8.08201e6 −0.677716 −0.338858 0.940838i \(-0.610041\pi\)
−0.338858 + 0.940838i \(0.610041\pi\)
\(678\) −7.31365e6 −0.611027
\(679\) −2.06824e6 −0.172158
\(680\) 1.82270e6 0.151162
\(681\) −2.51815e6 −0.208072
\(682\) 2.62667e7 2.16244
\(683\) −7.94744e6 −0.651892 −0.325946 0.945388i \(-0.605683\pi\)
−0.325946 + 0.945388i \(0.605683\pi\)
\(684\) 372248. 0.0304223
\(685\) −3.96623e6 −0.322962
\(686\) −1.08660e7 −0.881573
\(687\) 8.51780e6 0.688550
\(688\) 6.05713e6 0.487860
\(689\) −894517. −0.0717861
\(690\) 824999. 0.0659676
\(691\) −7.05380e6 −0.561989 −0.280995 0.959709i \(-0.590664\pi\)
−0.280995 + 0.959709i \(0.590664\pi\)
\(692\) −1.79702e6 −0.142655
\(693\) −5.48909e6 −0.434177
\(694\) 1.82875e7 1.44131
\(695\) −8.57882e6 −0.673699
\(696\) −7.15040e6 −0.559510
\(697\) 5.80594e6 0.452679
\(698\) −3.20179e6 −0.248745
\(699\) −1.91121e7 −1.47950
\(700\) −193340. −0.0149134
\(701\) −1.85070e7 −1.42246 −0.711231 0.702958i \(-0.751862\pi\)
−0.711231 + 0.702958i \(0.751862\pi\)
\(702\) 1.45887e7 1.11731
\(703\) −2.24439e6 −0.171281
\(704\) 2.62970e7 1.99975
\(705\) 2.39068e6 0.181154
\(706\) 2.33657e7 1.76428
\(707\) −4.05032e6 −0.304748
\(708\) −101087. −0.00757900
\(709\) −2.58780e6 −0.193337 −0.0966686 0.995317i \(-0.530819\pi\)
−0.0966686 + 0.995317i \(0.530819\pi\)
\(710\) −5.85657e6 −0.436011
\(711\) −2.84467e6 −0.211037
\(712\) 2.23561e7 1.65270
\(713\) 3.59915e6 0.265141
\(714\) −1.73567e6 −0.127415
\(715\) −1.23902e7 −0.906385
\(716\) 1.74593e6 0.127275
\(717\) 1.19566e7 0.868583
\(718\) 5.15897e6 0.373467
\(719\) −4.54097e6 −0.327587 −0.163793 0.986495i \(-0.552373\pi\)
−0.163793 + 0.986495i \(0.552373\pi\)
\(720\) −2.23605e6 −0.160750
\(721\) −3.79188e6 −0.271654
\(722\) −9.21086e6 −0.657593
\(723\) 1.17875e7 0.838640
\(724\) 15286.2 0.00108381
\(725\) −1.97653e6 −0.139656
\(726\) −2.34530e7 −1.65142
\(727\) −3.73890e6 −0.262366 −0.131183 0.991358i \(-0.541878\pi\)
−0.131183 + 0.991358i \(0.541878\pi\)
\(728\) −9.41063e6 −0.658098
\(729\) 1.41975e7 0.989445
\(730\) −1.21215e6 −0.0841877
\(731\) −2.65789e6 −0.183968
\(732\) −987329. −0.0681058
\(733\) 2.81782e6 0.193710 0.0968552 0.995298i \(-0.469122\pi\)
0.0968552 + 0.995298i \(0.469122\pi\)
\(734\) 3.04828e6 0.208841
\(735\) −3.40335e6 −0.232374
\(736\) 807395. 0.0549404
\(737\) 1.74080e7 1.18054
\(738\) −8.23457e6 −0.556544
\(739\) −7.31760e6 −0.492899 −0.246449 0.969156i \(-0.579264\pi\)
−0.246449 + 0.969156i \(0.579264\pi\)
\(740\) −279215. −0.0187438
\(741\) −6.83160e6 −0.457064
\(742\) −507768. −0.0338576
\(743\) −1.99679e7 −1.32697 −0.663484 0.748190i \(-0.730923\pi\)
−0.663484 + 0.748190i \(0.730923\pi\)
\(744\) 1.53834e7 1.01887
\(745\) −8.58536e6 −0.566719
\(746\) −2.76216e7 −1.81720
\(747\) −3.34937e6 −0.219615
\(748\) −1.18778e6 −0.0776217
\(749\) −417205. −0.0271735
\(750\) 974715. 0.0632739
\(751\) 2.20695e7 1.42788 0.713942 0.700205i \(-0.246908\pi\)
0.713942 + 0.700205i \(0.246908\pi\)
\(752\) −7.02716e6 −0.453143
\(753\) 7.10252e6 0.456484
\(754\) −1.12678e7 −0.721790
\(755\) 1.29545e7 0.827088
\(756\) −1.26661e6 −0.0806008
\(757\) 1.41851e7 0.899688 0.449844 0.893107i \(-0.351480\pi\)
0.449844 + 0.893107i \(0.351480\pi\)
\(758\) −8.45926e6 −0.534761
\(759\) −4.59024e6 −0.289222
\(760\) 4.07237e6 0.255749
\(761\) 1.21787e6 0.0762320 0.0381160 0.999273i \(-0.487864\pi\)
0.0381160 + 0.999273i \(0.487864\pi\)
\(762\) −8.11730e6 −0.506435
\(763\) 1.37331e7 0.853999
\(764\) 475053. 0.0294448
\(765\) 981187. 0.0606175
\(766\) −2.44009e7 −1.50257
\(767\) −1.36008e6 −0.0834788
\(768\) 4.85057e6 0.296749
\(769\) −5.12952e6 −0.312796 −0.156398 0.987694i \(-0.549988\pi\)
−0.156398 + 0.987694i \(0.549988\pi\)
\(770\) −7.03323e6 −0.427492
\(771\) 1.86452e7 1.12961
\(772\) −2.11689e6 −0.127837
\(773\) 2.48406e7 1.49525 0.747624 0.664122i \(-0.231194\pi\)
0.747624 + 0.664122i \(0.231194\pi\)
\(774\) 3.76968e6 0.226179
\(775\) 4.25231e6 0.254314
\(776\) 5.41960e6 0.323082
\(777\) 2.27013e6 0.134896
\(778\) 2.23853e7 1.32591
\(779\) 1.29719e7 0.765880
\(780\) −849891. −0.0500180
\(781\) 3.25856e7 1.91160
\(782\) 1.06410e6 0.0622250
\(783\) −1.29487e7 −0.754781
\(784\) 1.00038e7 0.581267
\(785\) −2.41454e6 −0.139849
\(786\) 2.09776e7 1.21115
\(787\) 1.56230e7 0.899140 0.449570 0.893245i \(-0.351577\pi\)
0.449570 + 0.893245i \(0.351577\pi\)
\(788\) 1.49642e6 0.0858496
\(789\) 1.74261e7 0.996567
\(790\) −3.64491e6 −0.207787
\(791\) 8.54339e6 0.485500
\(792\) 1.43835e7 0.814804
\(793\) −1.32841e7 −0.750150
\(794\) −3.87269e6 −0.218002
\(795\) −391535. −0.0219711
\(796\) 2.63018e6 0.147131
\(797\) −2.24931e7 −1.25431 −0.627153 0.778896i \(-0.715780\pi\)
−0.627153 + 0.778896i \(0.715780\pi\)
\(798\) −3.87793e6 −0.215572
\(799\) 3.08354e6 0.170877
\(800\) 953916. 0.0526969
\(801\) 1.20346e7 0.662750
\(802\) 1.59612e7 0.876251
\(803\) 6.74431e6 0.369104
\(804\) 1.19408e6 0.0651469
\(805\) −963716. −0.0524155
\(806\) 2.42416e7 1.31439
\(807\) 1.60193e6 0.0865882
\(808\) 1.06134e7 0.571909
\(809\) 3.22031e7 1.72992 0.864961 0.501838i \(-0.167343\pi\)
0.864961 + 0.501838i \(0.167343\pi\)
\(810\) 3.09575e6 0.165788
\(811\) −3.63596e7 −1.94119 −0.970593 0.240728i \(-0.922614\pi\)
−0.970593 + 0.240728i \(0.922614\pi\)
\(812\) 978286. 0.0520686
\(813\) −1.03837e7 −0.550970
\(814\) −1.01571e7 −0.537291
\(815\) −5.45586e6 −0.287719
\(816\) 3.93397e6 0.206826
\(817\) −5.93838e6 −0.311253
\(818\) −2.34839e7 −1.22712
\(819\) −5.06588e6 −0.263903
\(820\) 1.61378e6 0.0838128
\(821\) 1.58072e7 0.818457 0.409229 0.912432i \(-0.365798\pi\)
0.409229 + 0.912432i \(0.365798\pi\)
\(822\) −9.89683e6 −0.510877
\(823\) 5.26367e6 0.270888 0.135444 0.990785i \(-0.456754\pi\)
0.135444 + 0.990785i \(0.456754\pi\)
\(824\) 9.93620e6 0.509803
\(825\) −5.42325e6 −0.277412
\(826\) −772043. −0.0393723
\(827\) −7.34520e6 −0.373457 −0.186728 0.982412i \(-0.559788\pi\)
−0.186728 + 0.982412i \(0.559788\pi\)
\(828\) 230834. 0.0117010
\(829\) 5.06008e6 0.255724 0.127862 0.991792i \(-0.459189\pi\)
0.127862 + 0.991792i \(0.459189\pi\)
\(830\) −4.29159e6 −0.216234
\(831\) 3.89906e6 0.195865
\(832\) 2.42695e7 1.21549
\(833\) −4.38971e6 −0.219191
\(834\) −2.14065e7 −1.06569
\(835\) −1.58494e7 −0.786676
\(836\) −2.65381e6 −0.131327
\(837\) 2.78578e7 1.37446
\(838\) 3.46008e7 1.70206
\(839\) 3.39646e7 1.66580 0.832898 0.553426i \(-0.186680\pi\)
0.832898 + 0.553426i \(0.186680\pi\)
\(840\) −4.11908e6 −0.201420
\(841\) −1.05101e7 −0.512407
\(842\) 2.84283e7 1.38188
\(843\) −4.89544e6 −0.237259
\(844\) 5.43708e6 0.262730
\(845\) −2.15258e6 −0.103709
\(846\) −4.37339e6 −0.210083
\(847\) 2.73965e7 1.31216
\(848\) 1.15088e6 0.0549591
\(849\) −2.28531e7 −1.08812
\(850\) 1.25721e6 0.0596842
\(851\) −1.39176e6 −0.0658780
\(852\) 2.23517e6 0.105490
\(853\) −3.72237e7 −1.75165 −0.875824 0.482631i \(-0.839681\pi\)
−0.875824 + 0.482631i \(0.839681\pi\)
\(854\) −7.54064e6 −0.353805
\(855\) 2.19222e6 0.102558
\(856\) 1.09324e6 0.0509954
\(857\) −3.24074e7 −1.50728 −0.753638 0.657290i \(-0.771703\pi\)
−0.753638 + 0.657290i \(0.771703\pi\)
\(858\) −3.09169e7 −1.43376
\(859\) −2.08628e7 −0.964694 −0.482347 0.875980i \(-0.660216\pi\)
−0.482347 + 0.875980i \(0.660216\pi\)
\(860\) −738769. −0.0340614
\(861\) −1.31207e7 −0.603183
\(862\) 1.49319e7 0.684456
\(863\) 292318. 0.0133607 0.00668034 0.999978i \(-0.497874\pi\)
0.00668034 + 0.999978i \(0.497874\pi\)
\(864\) 6.24931e6 0.284805
\(865\) −1.05829e7 −0.480910
\(866\) −3.28849e7 −1.49005
\(867\) 1.50863e7 0.681607
\(868\) −2.10468e6 −0.0948173
\(869\) 2.02800e7 0.911000
\(870\) −4.93198e6 −0.220914
\(871\) 1.60658e7 0.717559
\(872\) −3.59861e7 −1.60267
\(873\) 2.91745e6 0.129559
\(874\) 2.37746e6 0.105277
\(875\) −1.13861e6 −0.0502751
\(876\) 462618. 0.0203687
\(877\) −2.37558e6 −0.104297 −0.0521484 0.998639i \(-0.516607\pi\)
−0.0521484 + 0.998639i \(0.516607\pi\)
\(878\) −1.72873e7 −0.756818
\(879\) 2.39684e7 1.04633
\(880\) 1.59411e7 0.693924
\(881\) −371982. −0.0161466 −0.00807331 0.999967i \(-0.502570\pi\)
−0.00807331 + 0.999967i \(0.502570\pi\)
\(882\) 6.22592e6 0.269484
\(883\) −2.97561e7 −1.28432 −0.642162 0.766569i \(-0.721963\pi\)
−0.642162 + 0.766569i \(0.721963\pi\)
\(884\) −1.09621e6 −0.0471804
\(885\) −595314. −0.0255498
\(886\) −6.33354e6 −0.271058
\(887\) −6.56407e6 −0.280133 −0.140066 0.990142i \(-0.544732\pi\)
−0.140066 + 0.990142i \(0.544732\pi\)
\(888\) −5.94862e6 −0.253153
\(889\) 9.48216e6 0.402395
\(890\) 1.54200e7 0.652545
\(891\) −1.72246e7 −0.726865
\(892\) −664140. −0.0279478
\(893\) 6.88940e6 0.289103
\(894\) −2.14228e7 −0.896463
\(895\) 1.02820e7 0.429061
\(896\) 1.02174e7 0.425179
\(897\) −4.23633e6 −0.175796
\(898\) 4.68776e6 0.193988
\(899\) −2.15163e7 −0.887910
\(900\) 272724. 0.0112232
\(901\) −505009. −0.0207246
\(902\) 5.87053e7 2.40249
\(903\) 6.00650e6 0.245133
\(904\) −2.23870e7 −0.911118
\(905\) 90022.2 0.00365366
\(906\) 3.23249e7 1.30833
\(907\) 2.06805e6 0.0834726 0.0417363 0.999129i \(-0.486711\pi\)
0.0417363 + 0.999129i \(0.486711\pi\)
\(908\) −902782. −0.0363386
\(909\) 5.71336e6 0.229341
\(910\) −6.49097e6 −0.259840
\(911\) −4.17228e7 −1.66563 −0.832813 0.553555i \(-0.813271\pi\)
−0.832813 + 0.553555i \(0.813271\pi\)
\(912\) 8.78948e6 0.349926
\(913\) 2.38781e7 0.948033
\(914\) −8.34136e6 −0.330272
\(915\) −5.81451e6 −0.229594
\(916\) 3.05372e6 0.120251
\(917\) −2.45048e7 −0.962339
\(918\) 8.23622e6 0.322568
\(919\) −8.00374e6 −0.312611 −0.156305 0.987709i \(-0.549958\pi\)
−0.156305 + 0.987709i \(0.549958\pi\)
\(920\) 2.52531e6 0.0983661
\(921\) 2.89802e7 1.12578
\(922\) 1.11611e7 0.432393
\(923\) 3.00732e7 1.16192
\(924\) 2.68424e6 0.103429
\(925\) −1.64433e6 −0.0631880
\(926\) −1.69017e7 −0.647745
\(927\) 5.34880e6 0.204436
\(928\) −4.82674e6 −0.183986
\(929\) −1.54596e7 −0.587704 −0.293852 0.955851i \(-0.594937\pi\)
−0.293852 + 0.955851i \(0.594937\pi\)
\(930\) 1.06107e7 0.402286
\(931\) −9.80770e6 −0.370846
\(932\) −6.85189e6 −0.258387
\(933\) 1.14002e7 0.428755
\(934\) −1.10451e7 −0.414289
\(935\) −6.99501e6 −0.261673
\(936\) 1.32746e7 0.495257
\(937\) −1.21380e7 −0.451645 −0.225822 0.974169i \(-0.572507\pi\)
−0.225822 + 0.974169i \(0.572507\pi\)
\(938\) 9.11969e6 0.338433
\(939\) −1.57778e7 −0.583960
\(940\) 857081. 0.0316375
\(941\) −3.01783e7 −1.11102 −0.555509 0.831510i \(-0.687477\pi\)
−0.555509 + 0.831510i \(0.687477\pi\)
\(942\) −6.02494e6 −0.221221
\(943\) 8.04399e6 0.294573
\(944\) 1.74987e6 0.0639109
\(945\) −7.45925e6 −0.271716
\(946\) −2.68745e7 −0.976368
\(947\) 2.58524e7 0.936753 0.468377 0.883529i \(-0.344839\pi\)
0.468377 + 0.883529i \(0.344839\pi\)
\(948\) 1.39108e6 0.0502727
\(949\) 6.22432e6 0.224350
\(950\) 2.80891e6 0.100979
\(951\) −3.13584e7 −1.12435
\(952\) −5.31287e6 −0.189993
\(953\) 2.11978e7 0.756063 0.378031 0.925793i \(-0.376601\pi\)
0.378031 + 0.925793i \(0.376601\pi\)
\(954\) 716254. 0.0254798
\(955\) 2.79765e6 0.0992623
\(956\) 4.28657e6 0.151693
\(957\) 2.74412e7 0.968553
\(958\) −3.23495e7 −1.13882
\(959\) 1.15609e7 0.405924
\(960\) 1.06229e7 0.372019
\(961\) 1.76610e7 0.616890
\(962\) −9.37400e6 −0.326578
\(963\) 588507. 0.0204497
\(964\) 4.22593e6 0.146464
\(965\) −1.24666e7 −0.430954
\(966\) −2.40473e6 −0.0829133
\(967\) −3.49861e7 −1.20318 −0.601588 0.798806i \(-0.705465\pi\)
−0.601588 + 0.798806i \(0.705465\pi\)
\(968\) −7.17895e7 −2.46248
\(969\) −3.85685e6 −0.131954
\(970\) 3.73816e6 0.127564
\(971\) 3.23066e7 1.09962 0.549810 0.835289i \(-0.314700\pi\)
0.549810 + 0.835289i \(0.314700\pi\)
\(972\) 3.04224e6 0.103283
\(973\) 2.50058e7 0.846757
\(974\) 3.31570e7 1.11990
\(975\) −5.00512e6 −0.168617
\(976\) 1.70912e7 0.574311
\(977\) −5.73644e7 −1.92268 −0.961338 0.275371i \(-0.911199\pi\)
−0.961338 + 0.275371i \(0.911199\pi\)
\(978\) −1.36138e7 −0.455128
\(979\) −8.57961e7 −2.86095
\(980\) −1.22013e6 −0.0405829
\(981\) −1.93718e7 −0.642684
\(982\) 1.59555e7 0.527998
\(983\) 3.81083e7 1.25787 0.628935 0.777458i \(-0.283491\pi\)
0.628935 + 0.777458i \(0.283491\pi\)
\(984\) 3.43814e7 1.13197
\(985\) 8.81262e6 0.289411
\(986\) −6.36136e6 −0.208381
\(987\) −6.96842e6 −0.227689
\(988\) −2.44920e6 −0.0798236
\(989\) −3.68244e6 −0.119714
\(990\) 9.92102e6 0.321713
\(991\) 1.93897e7 0.627171 0.313586 0.949560i \(-0.398470\pi\)
0.313586 + 0.949560i \(0.398470\pi\)
\(992\) 1.03842e7 0.335039
\(993\) 3.59100e7 1.15569
\(994\) 1.70709e7 0.548013
\(995\) 1.54895e7 0.495997
\(996\) 1.63789e6 0.0523163
\(997\) −3.27681e7 −1.04403 −0.522015 0.852936i \(-0.674820\pi\)
−0.522015 + 0.852936i \(0.674820\pi\)
\(998\) −2.93149e7 −0.931669
\(999\) −1.07724e7 −0.341505
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.1.5 7
3.2 odd 2 1035.6.a.b.1.3 7
5.4 even 2 575.6.a.d.1.3 7
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
115.6.a.c.1.5 7 1.1 even 1 trivial
575.6.a.d.1.3 7 5.4 even 2
1035.6.a.b.1.3 7 3.2 odd 2