Properties

Label 230.6.a.c.1.2
Level $230$
Weight $6$
Character 230.1
Self dual yes
Analytic conductor $36.888$
Analytic rank $1$
Dimension $3$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [230,6,Mod(1,230)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(230, 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("230.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 230 = 2 \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 230.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: \(36.8882785570\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.27980.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - 47x - 106 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2\cdot 3 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(7.78556\) of defining polynomial
Character \(\chi\) \(=\) 230.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-4.00000 q^{2} -10.5273 q^{3} +16.0000 q^{4} +25.0000 q^{5} +42.1092 q^{6} +156.388 q^{7} -64.0000 q^{8} -132.176 q^{9} +O(q^{10})\) \(q-4.00000 q^{2} -10.5273 q^{3} +16.0000 q^{4} +25.0000 q^{5} +42.1092 q^{6} +156.388 q^{7} -64.0000 q^{8} -132.176 q^{9} -100.000 q^{10} -397.468 q^{11} -168.437 q^{12} -71.8579 q^{13} -625.550 q^{14} -263.182 q^{15} +256.000 q^{16} +221.370 q^{17} +528.704 q^{18} -173.333 q^{19} +400.000 q^{20} -1646.34 q^{21} +1589.87 q^{22} +529.000 q^{23} +673.747 q^{24} +625.000 q^{25} +287.432 q^{26} +3949.59 q^{27} +2502.20 q^{28} +6966.29 q^{29} +1052.73 q^{30} -2009.47 q^{31} -1024.00 q^{32} +4184.26 q^{33} -885.481 q^{34} +3909.69 q^{35} -2114.82 q^{36} -12305.3 q^{37} +693.333 q^{38} +756.469 q^{39} -1600.00 q^{40} +420.431 q^{41} +6585.35 q^{42} -14261.3 q^{43} -6359.49 q^{44} -3304.40 q^{45} -2116.00 q^{46} +1706.64 q^{47} -2694.99 q^{48} +7650.08 q^{49} -2500.00 q^{50} -2330.43 q^{51} -1149.73 q^{52} -6102.11 q^{53} -15798.4 q^{54} -9936.70 q^{55} -10008.8 q^{56} +1824.73 q^{57} -27865.1 q^{58} +23220.2 q^{59} -4210.92 q^{60} -29339.9 q^{61} +8037.88 q^{62} -20670.7 q^{63} +4096.00 q^{64} -1796.45 q^{65} -16737.0 q^{66} -26398.9 q^{67} +3541.92 q^{68} -5568.94 q^{69} -15638.8 q^{70} -9484.52 q^{71} +8459.27 q^{72} -1769.79 q^{73} +49221.2 q^{74} -6579.56 q^{75} -2773.33 q^{76} -62159.1 q^{77} -3025.88 q^{78} -8643.34 q^{79} +6400.00 q^{80} -9459.68 q^{81} -1681.72 q^{82} -108289. q^{83} -26341.4 q^{84} +5534.26 q^{85} +57045.3 q^{86} -73336.1 q^{87} +25438.0 q^{88} -59654.8 q^{89} +13217.6 q^{90} -11237.7 q^{91} +8464.00 q^{92} +21154.3 q^{93} -6826.54 q^{94} -4333.33 q^{95} +10779.9 q^{96} -140826. q^{97} -30600.3 q^{98} +52535.8 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 12 q^{2} - 26 q^{3} + 48 q^{4} + 75 q^{5} + 104 q^{6} + q^{7} - 192 q^{8} - 71 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 12 q^{2} - 26 q^{3} + 48 q^{4} + 75 q^{5} + 104 q^{6} + q^{7} - 192 q^{8} - 71 q^{9} - 300 q^{10} + 178 q^{11} - 416 q^{12} - 236 q^{13} - 4 q^{14} - 650 q^{15} + 768 q^{16} - 1683 q^{17} + 284 q^{18} - 194 q^{19} + 1200 q^{20} + 504 q^{21} - 712 q^{22} + 1587 q^{23} + 1664 q^{24} + 1875 q^{25} + 944 q^{26} + 622 q^{27} + 16 q^{28} + 6901 q^{29} + 2600 q^{30} + 6119 q^{31} - 3072 q^{32} + 4792 q^{33} + 6732 q^{34} + 25 q^{35} - 1136 q^{36} + 5185 q^{37} + 776 q^{38} + 16234 q^{39} - 4800 q^{40} + 6727 q^{41} - 2016 q^{42} - 22988 q^{43} + 2848 q^{44} - 1775 q^{45} - 6348 q^{46} - 13730 q^{47} - 6656 q^{48} - 11788 q^{49} - 7500 q^{50} - 20140 q^{51} - 3776 q^{52} - 38027 q^{53} - 2488 q^{54} + 4450 q^{55} - 64 q^{56} - 62424 q^{57} - 27604 q^{58} - 43715 q^{59} - 10400 q^{60} - 51364 q^{61} - 24476 q^{62} - 40095 q^{63} + 12288 q^{64} - 5900 q^{65} - 19168 q^{66} - 51891 q^{67} - 26928 q^{68} - 13754 q^{69} - 100 q^{70} - 25667 q^{71} + 4544 q^{72} - 108532 q^{73} - 20740 q^{74} - 16250 q^{75} - 3104 q^{76} - 95646 q^{77} - 64936 q^{78} - 224 q^{79} + 19200 q^{80} - 38213 q^{81} - 26908 q^{82} - 61071 q^{83} + 8064 q^{84} - 42075 q^{85} + 91952 q^{86} - 82842 q^{87} - 11392 q^{88} - 93372 q^{89} + 7100 q^{90} + 33030 q^{91} + 25392 q^{92} - 73542 q^{93} + 54920 q^{94} - 4850 q^{95} + 26624 q^{96} - 260238 q^{97} + 47152 q^{98} - 8150 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −4.00000 −0.707107
\(3\) −10.5273 −0.675326 −0.337663 0.941267i \(-0.609636\pi\)
−0.337663 + 0.941267i \(0.609636\pi\)
\(4\) 16.0000 0.500000
\(5\) 25.0000 0.447214
\(6\) 42.1092 0.477528
\(7\) 156.388 1.20631 0.603153 0.797626i \(-0.293911\pi\)
0.603153 + 0.797626i \(0.293911\pi\)
\(8\) −64.0000 −0.353553
\(9\) −132.176 −0.543935
\(10\) −100.000 −0.316228
\(11\) −397.468 −0.990422 −0.495211 0.868773i \(-0.664909\pi\)
−0.495211 + 0.868773i \(0.664909\pi\)
\(12\) −168.437 −0.337663
\(13\) −71.8579 −0.117928 −0.0589639 0.998260i \(-0.518780\pi\)
−0.0589639 + 0.998260i \(0.518780\pi\)
\(14\) −625.550 −0.852987
\(15\) −263.182 −0.302015
\(16\) 256.000 0.250000
\(17\) 221.370 0.185779 0.0928896 0.995676i \(-0.470390\pi\)
0.0928896 + 0.995676i \(0.470390\pi\)
\(18\) 528.704 0.384620
\(19\) −173.333 −0.110153 −0.0550767 0.998482i \(-0.517540\pi\)
−0.0550767 + 0.998482i \(0.517540\pi\)
\(20\) 400.000 0.223607
\(21\) −1646.34 −0.814649
\(22\) 1589.87 0.700334
\(23\) 529.000 0.208514
\(24\) 673.747 0.238764
\(25\) 625.000 0.200000
\(26\) 287.432 0.0833875
\(27\) 3949.59 1.04266
\(28\) 2502.20 0.603153
\(29\) 6966.29 1.53818 0.769088 0.639142i \(-0.220711\pi\)
0.769088 + 0.639142i \(0.220711\pi\)
\(30\) 1052.73 0.213557
\(31\) −2009.47 −0.375558 −0.187779 0.982211i \(-0.560129\pi\)
−0.187779 + 0.982211i \(0.560129\pi\)
\(32\) −1024.00 −0.176777
\(33\) 4184.26 0.668858
\(34\) −885.481 −0.131366
\(35\) 3909.69 0.539476
\(36\) −2114.82 −0.271967
\(37\) −12305.3 −1.47771 −0.738853 0.673866i \(-0.764632\pi\)
−0.738853 + 0.673866i \(0.764632\pi\)
\(38\) 693.333 0.0778902
\(39\) 756.469 0.0796397
\(40\) −1600.00 −0.158114
\(41\) 420.431 0.0390603 0.0195301 0.999809i \(-0.493783\pi\)
0.0195301 + 0.999809i \(0.493783\pi\)
\(42\) 6585.35 0.576044
\(43\) −14261.3 −1.17622 −0.588110 0.808781i \(-0.700128\pi\)
−0.588110 + 0.808781i \(0.700128\pi\)
\(44\) −6359.49 −0.495211
\(45\) −3304.40 −0.243255
\(46\) −2116.00 −0.147442
\(47\) 1706.64 0.112693 0.0563464 0.998411i \(-0.482055\pi\)
0.0563464 + 0.998411i \(0.482055\pi\)
\(48\) −2694.99 −0.168832
\(49\) 7650.08 0.455172
\(50\) −2500.00 −0.141421
\(51\) −2330.43 −0.125462
\(52\) −1149.73 −0.0589639
\(53\) −6102.11 −0.298394 −0.149197 0.988807i \(-0.547669\pi\)
−0.149197 + 0.988807i \(0.547669\pi\)
\(54\) −15798.4 −0.737272
\(55\) −9936.70 −0.442930
\(56\) −10008.8 −0.426493
\(57\) 1824.73 0.0743894
\(58\) −27865.1 −1.08766
\(59\) 23220.2 0.868432 0.434216 0.900809i \(-0.357025\pi\)
0.434216 + 0.900809i \(0.357025\pi\)
\(60\) −4210.92 −0.151008
\(61\) −29339.9 −1.00957 −0.504783 0.863246i \(-0.668427\pi\)
−0.504783 + 0.863246i \(0.668427\pi\)
\(62\) 8037.88 0.265560
\(63\) −20670.7 −0.656151
\(64\) 4096.00 0.125000
\(65\) −1796.45 −0.0527389
\(66\) −16737.0 −0.472954
\(67\) −26398.9 −0.718454 −0.359227 0.933250i \(-0.616960\pi\)
−0.359227 + 0.933250i \(0.616960\pi\)
\(68\) 3541.92 0.0928896
\(69\) −5568.94 −0.140815
\(70\) −15638.8 −0.381467
\(71\) −9484.52 −0.223290 −0.111645 0.993748i \(-0.535612\pi\)
−0.111645 + 0.993748i \(0.535612\pi\)
\(72\) 8459.27 0.192310
\(73\) −1769.79 −0.0388700 −0.0194350 0.999811i \(-0.506187\pi\)
−0.0194350 + 0.999811i \(0.506187\pi\)
\(74\) 49221.2 1.04490
\(75\) −6579.56 −0.135065
\(76\) −2773.33 −0.0550767
\(77\) −62159.1 −1.19475
\(78\) −3025.88 −0.0563138
\(79\) −8643.34 −0.155817 −0.0779083 0.996961i \(-0.524824\pi\)
−0.0779083 + 0.996961i \(0.524824\pi\)
\(80\) 6400.00 0.111803
\(81\) −9459.68 −0.160201
\(82\) −1681.72 −0.0276198
\(83\) −108289. −1.72540 −0.862700 0.505717i \(-0.831228\pi\)
−0.862700 + 0.505717i \(0.831228\pi\)
\(84\) −26341.4 −0.407325
\(85\) 5534.26 0.0830830
\(86\) 57045.3 0.831713
\(87\) −73336.1 −1.03877
\(88\) 25438.0 0.350167
\(89\) −59654.8 −0.798308 −0.399154 0.916884i \(-0.630696\pi\)
−0.399154 + 0.916884i \(0.630696\pi\)
\(90\) 13217.6 0.172007
\(91\) −11237.7 −0.142257
\(92\) 8464.00 0.104257
\(93\) 21154.3 0.253624
\(94\) −6826.54 −0.0796858
\(95\) −4333.33 −0.0492621
\(96\) 10779.9 0.119382
\(97\) −140826. −1.51968 −0.759842 0.650108i \(-0.774724\pi\)
−0.759842 + 0.650108i \(0.774724\pi\)
\(98\) −30600.3 −0.321855
\(99\) 52535.8 0.538725
\(100\) 10000.0 0.100000
\(101\) 43468.1 0.424001 0.212001 0.977270i \(-0.432002\pi\)
0.212001 + 0.977270i \(0.432002\pi\)
\(102\) 9321.72 0.0887147
\(103\) −119511. −1.10998 −0.554990 0.831857i \(-0.687278\pi\)
−0.554990 + 0.831857i \(0.687278\pi\)
\(104\) 4598.91 0.0416938
\(105\) −41158.4 −0.364322
\(106\) 24408.4 0.210996
\(107\) −74071.6 −0.625450 −0.312725 0.949844i \(-0.601242\pi\)
−0.312725 + 0.949844i \(0.601242\pi\)
\(108\) 63193.4 0.521330
\(109\) 19868.2 0.160174 0.0800871 0.996788i \(-0.474480\pi\)
0.0800871 + 0.996788i \(0.474480\pi\)
\(110\) 39746.8 0.313199
\(111\) 129542. 0.997934
\(112\) 40035.2 0.301576
\(113\) 28794.8 0.212138 0.106069 0.994359i \(-0.466174\pi\)
0.106069 + 0.994359i \(0.466174\pi\)
\(114\) −7298.92 −0.0526013
\(115\) 13225.0 0.0932505
\(116\) 111461. 0.769088
\(117\) 9497.90 0.0641450
\(118\) −92880.8 −0.614074
\(119\) 34619.6 0.224106
\(120\) 16843.7 0.106778
\(121\) −3070.20 −0.0190635
\(122\) 117360. 0.713871
\(123\) −4426.00 −0.0263784
\(124\) −32151.5 −0.187779
\(125\) 15625.0 0.0894427
\(126\) 82682.8 0.463969
\(127\) −102984. −0.566581 −0.283291 0.959034i \(-0.591426\pi\)
−0.283291 + 0.959034i \(0.591426\pi\)
\(128\) −16384.0 −0.0883883
\(129\) 150133. 0.794332
\(130\) 7185.79 0.0372920
\(131\) 11416.3 0.0581228 0.0290614 0.999578i \(-0.490748\pi\)
0.0290614 + 0.999578i \(0.490748\pi\)
\(132\) 66948.2 0.334429
\(133\) −27107.2 −0.132879
\(134\) 105596. 0.508024
\(135\) 98739.7 0.466291
\(136\) −14167.7 −0.0656828
\(137\) −139487. −0.634938 −0.317469 0.948269i \(-0.602833\pi\)
−0.317469 + 0.948269i \(0.602833\pi\)
\(138\) 22275.8 0.0995714
\(139\) −46437.5 −0.203860 −0.101930 0.994792i \(-0.532502\pi\)
−0.101930 + 0.994792i \(0.532502\pi\)
\(140\) 62555.0 0.269738
\(141\) −17966.2 −0.0761044
\(142\) 37938.1 0.157890
\(143\) 28561.2 0.116798
\(144\) −33837.1 −0.135984
\(145\) 174157. 0.687894
\(146\) 7079.16 0.0274852
\(147\) −80534.6 −0.307390
\(148\) −196885. −0.738853
\(149\) −337323. −1.24475 −0.622373 0.782721i \(-0.713831\pi\)
−0.622373 + 0.782721i \(0.713831\pi\)
\(150\) 26318.2 0.0955055
\(151\) 372864. 1.33079 0.665393 0.746493i \(-0.268264\pi\)
0.665393 + 0.746493i \(0.268264\pi\)
\(152\) 11093.3 0.0389451
\(153\) −29259.9 −0.101052
\(154\) 248636. 0.844817
\(155\) −50236.8 −0.167955
\(156\) 12103.5 0.0398199
\(157\) −194815. −0.630772 −0.315386 0.948963i \(-0.602134\pi\)
−0.315386 + 0.948963i \(0.602134\pi\)
\(158\) 34573.4 0.110179
\(159\) 64238.7 0.201513
\(160\) −25600.0 −0.0790569
\(161\) 82729.0 0.251532
\(162\) 37838.7 0.113279
\(163\) 332950. 0.981543 0.490772 0.871288i \(-0.336715\pi\)
0.490772 + 0.871288i \(0.336715\pi\)
\(164\) 6726.90 0.0195301
\(165\) 104607. 0.299122
\(166\) 433157. 1.22004
\(167\) 289233. 0.802522 0.401261 0.915964i \(-0.368572\pi\)
0.401261 + 0.915964i \(0.368572\pi\)
\(168\) 105366. 0.288022
\(169\) −366129. −0.986093
\(170\) −22137.0 −0.0587485
\(171\) 22910.5 0.0599162
\(172\) −228181. −0.588110
\(173\) −120295. −0.305584 −0.152792 0.988258i \(-0.548827\pi\)
−0.152792 + 0.988258i \(0.548827\pi\)
\(174\) 293344. 0.734522
\(175\) 97742.2 0.241261
\(176\) −101752. −0.247606
\(177\) −244446. −0.586475
\(178\) 238619. 0.564489
\(179\) 93474.3 0.218052 0.109026 0.994039i \(-0.465227\pi\)
0.109026 + 0.994039i \(0.465227\pi\)
\(180\) −52870.4 −0.121627
\(181\) −578671. −1.31291 −0.656456 0.754364i \(-0.727945\pi\)
−0.656456 + 0.754364i \(0.727945\pi\)
\(182\) 44950.7 0.100591
\(183\) 308870. 0.681786
\(184\) −33856.0 −0.0737210
\(185\) −307633. −0.660851
\(186\) −84617.1 −0.179339
\(187\) −87987.6 −0.184000
\(188\) 27306.2 0.0563464
\(189\) 617667. 1.25777
\(190\) 17333.3 0.0348336
\(191\) 334102. 0.662668 0.331334 0.943513i \(-0.392501\pi\)
0.331334 + 0.943513i \(0.392501\pi\)
\(192\) −43119.8 −0.0844158
\(193\) −357766. −0.691362 −0.345681 0.938352i \(-0.612352\pi\)
−0.345681 + 0.938352i \(0.612352\pi\)
\(194\) 563304. 1.07458
\(195\) 18911.7 0.0356160
\(196\) 122401. 0.227586
\(197\) 205858. 0.377923 0.188961 0.981985i \(-0.439488\pi\)
0.188961 + 0.981985i \(0.439488\pi\)
\(198\) −210143. −0.380936
\(199\) 992810. 1.77719 0.888594 0.458695i \(-0.151683\pi\)
0.888594 + 0.458695i \(0.151683\pi\)
\(200\) −40000.0 −0.0707107
\(201\) 277909. 0.485191
\(202\) −173872. −0.299814
\(203\) 1.08944e6 1.85551
\(204\) −37286.9 −0.0627308
\(205\) 10510.8 0.0174683
\(206\) 478044. 0.784875
\(207\) −69921.2 −0.113418
\(208\) −18395.6 −0.0294819
\(209\) 68894.4 0.109098
\(210\) 164634. 0.257615
\(211\) −784498. −1.21307 −0.606535 0.795057i \(-0.707441\pi\)
−0.606535 + 0.795057i \(0.707441\pi\)
\(212\) −97633.7 −0.149197
\(213\) 99846.3 0.150794
\(214\) 296287. 0.442260
\(215\) −356533. −0.526021
\(216\) −252774. −0.368636
\(217\) −314256. −0.453038
\(218\) −79472.8 −0.113260
\(219\) 18631.1 0.0262499
\(220\) −158987. −0.221465
\(221\) −15907.2 −0.0219085
\(222\) −518166. −0.705646
\(223\) 1.25999e6 1.69670 0.848349 0.529438i \(-0.177597\pi\)
0.848349 + 0.529438i \(0.177597\pi\)
\(224\) −160141. −0.213247
\(225\) −82610.1 −0.108787
\(226\) −115179. −0.150004
\(227\) −1.43471e6 −1.84799 −0.923997 0.382399i \(-0.875098\pi\)
−0.923997 + 0.382399i \(0.875098\pi\)
\(228\) 29195.7 0.0371947
\(229\) −206190. −0.259824 −0.129912 0.991526i \(-0.541470\pi\)
−0.129912 + 0.991526i \(0.541470\pi\)
\(230\) −52900.0 −0.0659380
\(231\) 654367. 0.806847
\(232\) −445842. −0.543828
\(233\) 874544. 1.05534 0.527669 0.849450i \(-0.323066\pi\)
0.527669 + 0.849450i \(0.323066\pi\)
\(234\) −37991.6 −0.0453574
\(235\) 42665.9 0.0503977
\(236\) 371523. 0.434216
\(237\) 90990.9 0.105227
\(238\) −138478. −0.158467
\(239\) −410321. −0.464653 −0.232326 0.972638i \(-0.574634\pi\)
−0.232326 + 0.972638i \(0.574634\pi\)
\(240\) −67374.7 −0.0755038
\(241\) −1.23413e6 −1.36874 −0.684368 0.729137i \(-0.739922\pi\)
−0.684368 + 0.729137i \(0.739922\pi\)
\(242\) 12280.8 0.0134800
\(243\) −860165. −0.934472
\(244\) −469439. −0.504783
\(245\) 191252. 0.203559
\(246\) 17704.0 0.0186524
\(247\) 12455.4 0.0129901
\(248\) 128606. 0.132780
\(249\) 1.13999e6 1.16521
\(250\) −62500.0 −0.0632456
\(251\) 102403. 0.102595 0.0512977 0.998683i \(-0.483664\pi\)
0.0512977 + 0.998683i \(0.483664\pi\)
\(252\) −330731. −0.328076
\(253\) −210261. −0.206517
\(254\) 411938. 0.400633
\(255\) −58260.7 −0.0561081
\(256\) 65536.0 0.0625000
\(257\) −1.73380e6 −1.63744 −0.818720 0.574192i \(-0.805316\pi\)
−0.818720 + 0.574192i \(0.805316\pi\)
\(258\) −600532. −0.561678
\(259\) −1.92440e6 −1.78257
\(260\) −28743.2 −0.0263695
\(261\) −920776. −0.836668
\(262\) −45665.1 −0.0410990
\(263\) −1598.98 −0.00142545 −0.000712726 1.00000i \(-0.500227\pi\)
−0.000712726 1.00000i \(0.500227\pi\)
\(264\) −267793. −0.236477
\(265\) −152553. −0.133446
\(266\) 108429. 0.0939593
\(267\) 628004. 0.539118
\(268\) −422382. −0.359227
\(269\) −179772. −0.151476 −0.0757378 0.997128i \(-0.524131\pi\)
−0.0757378 + 0.997128i \(0.524131\pi\)
\(270\) −394959. −0.329718
\(271\) 651012. 0.538475 0.269238 0.963074i \(-0.413228\pi\)
0.269238 + 0.963074i \(0.413228\pi\)
\(272\) 56670.8 0.0464448
\(273\) 118302. 0.0960698
\(274\) 557947. 0.448969
\(275\) −248417. −0.198084
\(276\) −89103.0 −0.0704076
\(277\) 104670. 0.0819643 0.0409821 0.999160i \(-0.486951\pi\)
0.0409821 + 0.999160i \(0.486951\pi\)
\(278\) 185750. 0.144151
\(279\) 265604. 0.204279
\(280\) −250220. −0.190734
\(281\) 951050. 0.718518 0.359259 0.933238i \(-0.383030\pi\)
0.359259 + 0.933238i \(0.383030\pi\)
\(282\) 71865.0 0.0538139
\(283\) 46711.8 0.0346705 0.0173353 0.999850i \(-0.494482\pi\)
0.0173353 + 0.999850i \(0.494482\pi\)
\(284\) −151752. −0.111645
\(285\) 45618.2 0.0332680
\(286\) −114245. −0.0825889
\(287\) 65750.2 0.0471186
\(288\) 135348. 0.0961550
\(289\) −1.37085e6 −0.965486
\(290\) −696629. −0.486414
\(291\) 1.48252e6 1.02628
\(292\) −28316.6 −0.0194350
\(293\) 539801. 0.367337 0.183669 0.982988i \(-0.441203\pi\)
0.183669 + 0.982988i \(0.441203\pi\)
\(294\) 322139. 0.217357
\(295\) 580505. 0.388375
\(296\) 787540. 0.522448
\(297\) −1.56984e6 −1.03267
\(298\) 1.34929e6 0.880168
\(299\) −38012.8 −0.0245896
\(300\) −105273. −0.0675326
\(301\) −2.23029e6 −1.41888
\(302\) −1.49146e6 −0.941007
\(303\) −457601. −0.286339
\(304\) −44373.3 −0.0275383
\(305\) −733498. −0.451491
\(306\) 117039. 0.0714543
\(307\) −1.94725e6 −1.17917 −0.589584 0.807707i \(-0.700708\pi\)
−0.589584 + 0.807707i \(0.700708\pi\)
\(308\) −994545. −0.597376
\(309\) 1.25813e6 0.749599
\(310\) 200947. 0.118762
\(311\) 2.33189e6 1.36712 0.683560 0.729894i \(-0.260431\pi\)
0.683560 + 0.729894i \(0.260431\pi\)
\(312\) −48414.0 −0.0281569
\(313\) −441207. −0.254555 −0.127277 0.991867i \(-0.540624\pi\)
−0.127277 + 0.991867i \(0.540624\pi\)
\(314\) 779259. 0.446023
\(315\) −516768. −0.293440
\(316\) −138293. −0.0779083
\(317\) 815875. 0.456011 0.228005 0.973660i \(-0.426780\pi\)
0.228005 + 0.973660i \(0.426780\pi\)
\(318\) −256955. −0.142491
\(319\) −2.76888e6 −1.52344
\(320\) 102400. 0.0559017
\(321\) 779774. 0.422383
\(322\) −330916. −0.177860
\(323\) −38370.8 −0.0204642
\(324\) −151355. −0.0801003
\(325\) −44911.2 −0.0235856
\(326\) −1.33180e6 −0.694056
\(327\) −209158. −0.108170
\(328\) −26907.6 −0.0138099
\(329\) 266897. 0.135942
\(330\) −418426. −0.211512
\(331\) 2.11055e6 1.05883 0.529415 0.848363i \(-0.322412\pi\)
0.529415 + 0.848363i \(0.322412\pi\)
\(332\) −1.73263e6 −0.862700
\(333\) 1.62647e6 0.803776
\(334\) −1.15693e6 −0.567469
\(335\) −659973. −0.321302
\(336\) −421463. −0.203662
\(337\) −1.13820e6 −0.545938 −0.272969 0.962023i \(-0.588006\pi\)
−0.272969 + 0.962023i \(0.588006\pi\)
\(338\) 1.46452e6 0.697273
\(339\) −303131. −0.143262
\(340\) 88548.1 0.0415415
\(341\) 798700. 0.371961
\(342\) −91642.0 −0.0423672
\(343\) −1.43203e6 −0.657229
\(344\) 912724. 0.415856
\(345\) −139223. −0.0629745
\(346\) 481179. 0.216081
\(347\) −1.33256e6 −0.594103 −0.297051 0.954861i \(-0.596003\pi\)
−0.297051 + 0.954861i \(0.596003\pi\)
\(348\) −1.17338e6 −0.519386
\(349\) 2.72390e6 1.19709 0.598545 0.801089i \(-0.295746\pi\)
0.598545 + 0.801089i \(0.295746\pi\)
\(350\) −390969. −0.170597
\(351\) −283809. −0.122959
\(352\) 407007. 0.175084
\(353\) 4.36203e6 1.86317 0.931585 0.363524i \(-0.118427\pi\)
0.931585 + 0.363524i \(0.118427\pi\)
\(354\) 977783. 0.414700
\(355\) −237113. −0.0998584
\(356\) −954477. −0.399154
\(357\) −364450. −0.151345
\(358\) −373897. −0.154186
\(359\) 3.27980e6 1.34311 0.671554 0.740956i \(-0.265627\pi\)
0.671554 + 0.740956i \(0.265627\pi\)
\(360\) 211482. 0.0860036
\(361\) −2.44605e6 −0.987866
\(362\) 2.31469e6 0.928369
\(363\) 32320.9 0.0128741
\(364\) −179803. −0.0711285
\(365\) −44244.7 −0.0173832
\(366\) −1.23548e6 −0.482096
\(367\) −1.15790e6 −0.448750 −0.224375 0.974503i \(-0.572034\pi\)
−0.224375 + 0.974503i \(0.572034\pi\)
\(368\) 135424. 0.0521286
\(369\) −55570.9 −0.0212462
\(370\) 1.23053e6 0.467292
\(371\) −954294. −0.359954
\(372\) 338468. 0.126812
\(373\) 4.19142e6 1.55987 0.779936 0.625859i \(-0.215252\pi\)
0.779936 + 0.625859i \(0.215252\pi\)
\(374\) 351950. 0.130108
\(375\) −164489. −0.0604030
\(376\) −109225. −0.0398429
\(377\) −500583. −0.181394
\(378\) −2.47067e6 −0.889374
\(379\) 4.37682e6 1.56517 0.782583 0.622546i \(-0.213902\pi\)
0.782583 + 0.622546i \(0.213902\pi\)
\(380\) −69333.3 −0.0246310
\(381\) 1.08415e6 0.382627
\(382\) −1.33641e6 −0.468577
\(383\) 2.90888e6 1.01328 0.506639 0.862158i \(-0.330888\pi\)
0.506639 + 0.862158i \(0.330888\pi\)
\(384\) 172479. 0.0596910
\(385\) −1.55398e6 −0.534309
\(386\) 1.43106e6 0.488867
\(387\) 1.88501e6 0.639787
\(388\) −2.25321e6 −0.759842
\(389\) 723744. 0.242500 0.121250 0.992622i \(-0.461310\pi\)
0.121250 + 0.992622i \(0.461310\pi\)
\(390\) −75646.9 −0.0251843
\(391\) 117105. 0.0387376
\(392\) −489605. −0.160928
\(393\) −120183. −0.0392519
\(394\) −823434. −0.267232
\(395\) −216083. −0.0696833
\(396\) 840572. 0.269362
\(397\) −2.05819e6 −0.655405 −0.327702 0.944781i \(-0.606274\pi\)
−0.327702 + 0.944781i \(0.606274\pi\)
\(398\) −3.97124e6 −1.25666
\(399\) 285365. 0.0897364
\(400\) 160000. 0.0500000
\(401\) −417799. −0.129750 −0.0648748 0.997893i \(-0.520665\pi\)
−0.0648748 + 0.997893i \(0.520665\pi\)
\(402\) −1.11164e6 −0.343082
\(403\) 144396. 0.0442888
\(404\) 695489. 0.212001
\(405\) −236492. −0.0716439
\(406\) −4.35776e6 −1.31204
\(407\) 4.89097e6 1.46355
\(408\) 149147. 0.0443573
\(409\) 2.12284e6 0.627493 0.313747 0.949507i \(-0.398416\pi\)
0.313747 + 0.949507i \(0.398416\pi\)
\(410\) −42043.1 −0.0123519
\(411\) 1.46842e6 0.428790
\(412\) −1.91218e6 −0.554990
\(413\) 3.63135e6 1.04759
\(414\) 279685. 0.0801988
\(415\) −2.70723e6 −0.771622
\(416\) 73582.5 0.0208469
\(417\) 488861. 0.137672
\(418\) −275578. −0.0771442
\(419\) −1.59156e6 −0.442881 −0.221440 0.975174i \(-0.571076\pi\)
−0.221440 + 0.975174i \(0.571076\pi\)
\(420\) −658535. −0.182161
\(421\) −2.38860e6 −0.656808 −0.328404 0.944537i \(-0.606511\pi\)
−0.328404 + 0.944537i \(0.606511\pi\)
\(422\) 3.13799e6 0.857770
\(423\) −225576. −0.0612975
\(424\) 390535. 0.105498
\(425\) 138356. 0.0371558
\(426\) −399385. −0.106627
\(427\) −4.58840e6 −1.21784
\(428\) −1.18515e6 −0.312725
\(429\) −300672. −0.0788770
\(430\) 1.42613e6 0.371953
\(431\) 4.70317e6 1.21954 0.609772 0.792577i \(-0.291261\pi\)
0.609772 + 0.792577i \(0.291261\pi\)
\(432\) 1.01109e6 0.260665
\(433\) −5.44455e6 −1.39554 −0.697770 0.716322i \(-0.745824\pi\)
−0.697770 + 0.716322i \(0.745824\pi\)
\(434\) 1.25702e6 0.320346
\(435\) −1.83340e6 −0.464553
\(436\) 317891. 0.0800871
\(437\) −91693.3 −0.0229686
\(438\) −74524.4 −0.0185615
\(439\) 2.90009e6 0.718209 0.359105 0.933297i \(-0.383082\pi\)
0.359105 + 0.933297i \(0.383082\pi\)
\(440\) 635949. 0.156600
\(441\) −1.01116e6 −0.247584
\(442\) 63628.8 0.0154917
\(443\) 1.97143e6 0.477278 0.238639 0.971108i \(-0.423299\pi\)
0.238639 + 0.971108i \(0.423299\pi\)
\(444\) 2.07267e6 0.498967
\(445\) −1.49137e6 −0.357014
\(446\) −5.03995e6 −1.19975
\(447\) 3.55110e6 0.840609
\(448\) 640564. 0.150788
\(449\) 2.31589e6 0.542128 0.271064 0.962561i \(-0.412624\pi\)
0.271064 + 0.962561i \(0.412624\pi\)
\(450\) 330440. 0.0769240
\(451\) −167108. −0.0386862
\(452\) 460717. 0.106069
\(453\) −3.92525e6 −0.898714
\(454\) 5.73886e6 1.30673
\(455\) −280942. −0.0636192
\(456\) −116783. −0.0263006
\(457\) −4.45559e6 −0.997964 −0.498982 0.866612i \(-0.666293\pi\)
−0.498982 + 0.866612i \(0.666293\pi\)
\(458\) 824762. 0.183724
\(459\) 874321. 0.193704
\(460\) 211600. 0.0466252
\(461\) −7.02079e6 −1.53863 −0.769314 0.638871i \(-0.779402\pi\)
−0.769314 + 0.638871i \(0.779402\pi\)
\(462\) −2.61747e6 −0.570527
\(463\) 3.69813e6 0.801733 0.400867 0.916136i \(-0.368709\pi\)
0.400867 + 0.916136i \(0.368709\pi\)
\(464\) 1.78337e6 0.384544
\(465\) 528857. 0.113424
\(466\) −3.49817e6 −0.746237
\(467\) −1.73226e6 −0.367553 −0.183776 0.982968i \(-0.558832\pi\)
−0.183776 + 0.982968i \(0.558832\pi\)
\(468\) 151966. 0.0320725
\(469\) −4.12846e6 −0.866675
\(470\) −170664. −0.0356366
\(471\) 2.05087e6 0.425977
\(472\) −1.48609e6 −0.307037
\(473\) 5.66842e6 1.16495
\(474\) −363964. −0.0744067
\(475\) −108333. −0.0220307
\(476\) 553913. 0.112053
\(477\) 806553. 0.162307
\(478\) 1.64128e6 0.328559
\(479\) 2.57714e6 0.513215 0.256607 0.966516i \(-0.417395\pi\)
0.256607 + 0.966516i \(0.417395\pi\)
\(480\) 269499. 0.0533892
\(481\) 884234. 0.174263
\(482\) 4.93654e6 0.967843
\(483\) −870913. −0.169866
\(484\) −49123.2 −0.00953177
\(485\) −3.52065e6 −0.679623
\(486\) 3.44066e6 0.660771
\(487\) 7.56663e6 1.44571 0.722854 0.691001i \(-0.242830\pi\)
0.722854 + 0.691001i \(0.242830\pi\)
\(488\) 1.87776e6 0.356935
\(489\) −3.50506e6 −0.662862
\(490\) −765008. −0.143938
\(491\) −522784. −0.0978630 −0.0489315 0.998802i \(-0.515582\pi\)
−0.0489315 + 0.998802i \(0.515582\pi\)
\(492\) −70816.0 −0.0131892
\(493\) 1.54213e6 0.285761
\(494\) −49821.4 −0.00918542
\(495\) 1.31339e6 0.240925
\(496\) −514424. −0.0938896
\(497\) −1.48326e6 −0.269356
\(498\) −4.55997e6 −0.823926
\(499\) 9.66184e6 1.73703 0.868517 0.495659i \(-0.165073\pi\)
0.868517 + 0.495659i \(0.165073\pi\)
\(500\) 250000. 0.0447214
\(501\) −3.04484e6 −0.541964
\(502\) −409612. −0.0725459
\(503\) −3.77794e6 −0.665787 −0.332893 0.942964i \(-0.608025\pi\)
−0.332893 + 0.942964i \(0.608025\pi\)
\(504\) 1.32293e6 0.231984
\(505\) 1.08670e6 0.189619
\(506\) 841042. 0.146030
\(507\) 3.85435e6 0.665934
\(508\) −1.64775e6 −0.283291
\(509\) 3.05552e6 0.522747 0.261373 0.965238i \(-0.415825\pi\)
0.261373 + 0.965238i \(0.415825\pi\)
\(510\) 233043. 0.0396744
\(511\) −276773. −0.0468891
\(512\) −262144. −0.0441942
\(513\) −684595. −0.114852
\(514\) 6.93519e6 1.15785
\(515\) −2.98778e6 −0.496398
\(516\) 2.40213e6 0.397166
\(517\) −678333. −0.111613
\(518\) 7.69759e6 1.26046
\(519\) 1.26638e6 0.206369
\(520\) 114973. 0.0186460
\(521\) 1.06262e7 1.71507 0.857537 0.514422i \(-0.171993\pi\)
0.857537 + 0.514422i \(0.171993\pi\)
\(522\) 3.68311e6 0.591613
\(523\) −6.76874e6 −1.08207 −0.541033 0.841002i \(-0.681967\pi\)
−0.541033 + 0.841002i \(0.681967\pi\)
\(524\) 182661. 0.0290614
\(525\) −1.02896e6 −0.162930
\(526\) 6395.91 0.00100795
\(527\) −444837. −0.0697709
\(528\) 1.07117e6 0.167215
\(529\) 279841. 0.0434783
\(530\) 610211. 0.0943605
\(531\) −3.06916e6 −0.472370
\(532\) −433715. −0.0664393
\(533\) −30211.3 −0.00460629
\(534\) −2.51202e6 −0.381214
\(535\) −1.85179e6 −0.279710
\(536\) 1.68953e6 0.254012
\(537\) −984031. −0.147256
\(538\) 719090. 0.107109
\(539\) −3.04066e6 −0.450813
\(540\) 1.57984e6 0.233146
\(541\) 1.40510e6 0.206402 0.103201 0.994661i \(-0.467092\pi\)
0.103201 + 0.994661i \(0.467092\pi\)
\(542\) −2.60405e6 −0.380759
\(543\) 6.09184e6 0.886644
\(544\) −226683. −0.0328414
\(545\) 496705. 0.0716320
\(546\) −473210. −0.0679316
\(547\) 5.89133e6 0.841870 0.420935 0.907091i \(-0.361702\pi\)
0.420935 + 0.907091i \(0.361702\pi\)
\(548\) −2.23179e6 −0.317469
\(549\) 3.87804e6 0.549138
\(550\) 993670. 0.140067
\(551\) −1.20749e6 −0.169435
\(552\) 356412. 0.0497857
\(553\) −1.35171e6 −0.187962
\(554\) −418682. −0.0579575
\(555\) 3.23854e6 0.446290
\(556\) −743000. −0.101930
\(557\) 1.63738e6 0.223621 0.111810 0.993730i \(-0.464335\pi\)
0.111810 + 0.993730i \(0.464335\pi\)
\(558\) −1.06242e6 −0.144447
\(559\) 1.02479e6 0.138709
\(560\) 1.00088e6 0.134869
\(561\) 926271. 0.124260
\(562\) −3.80420e6 −0.508069
\(563\) 7.94633e6 1.05656 0.528282 0.849069i \(-0.322837\pi\)
0.528282 + 0.849069i \(0.322837\pi\)
\(564\) −287460. −0.0380522
\(565\) 719870. 0.0948708
\(566\) −186847. −0.0245157
\(567\) −1.47938e6 −0.193251
\(568\) 607009. 0.0789450
\(569\) −6.68375e6 −0.865445 −0.432722 0.901527i \(-0.642447\pi\)
−0.432722 + 0.901527i \(0.642447\pi\)
\(570\) −182473. −0.0235240
\(571\) 5.30348e6 0.680723 0.340362 0.940295i \(-0.389451\pi\)
0.340362 + 0.940295i \(0.389451\pi\)
\(572\) 456979. 0.0583992
\(573\) −3.51719e6 −0.447517
\(574\) −263001. −0.0333179
\(575\) 330625. 0.0417029
\(576\) −541393. −0.0679918
\(577\) −1.06295e7 −1.32915 −0.664573 0.747223i \(-0.731387\pi\)
−0.664573 + 0.747223i \(0.731387\pi\)
\(578\) 5.48341e6 0.682702
\(579\) 3.76630e6 0.466895
\(580\) 2.78651e6 0.343947
\(581\) −1.69351e7 −2.08136
\(582\) −5.93006e6 −0.725691
\(583\) 2.42539e6 0.295536
\(584\) 113267. 0.0137426
\(585\) 237447. 0.0286865
\(586\) −2.15920e6 −0.259747
\(587\) −560366. −0.0671238 −0.0335619 0.999437i \(-0.510685\pi\)
−0.0335619 + 0.999437i \(0.510685\pi\)
\(588\) −1.28855e6 −0.153695
\(589\) 348308. 0.0413690
\(590\) −2.32202e6 −0.274622
\(591\) −2.16713e6 −0.255221
\(592\) −3.15016e6 −0.369427
\(593\) 9.64020e6 1.12577 0.562885 0.826535i \(-0.309692\pi\)
0.562885 + 0.826535i \(0.309692\pi\)
\(594\) 6.27934e6 0.730210
\(595\) 865489. 0.100223
\(596\) −5.39717e6 −0.622373
\(597\) −1.04516e7 −1.20018
\(598\) 152051. 0.0173875
\(599\) −1.27940e7 −1.45693 −0.728465 0.685083i \(-0.759766\pi\)
−0.728465 + 0.685083i \(0.759766\pi\)
\(600\) 421092. 0.0477528
\(601\) −4.10663e6 −0.463767 −0.231883 0.972744i \(-0.574489\pi\)
−0.231883 + 0.972744i \(0.574489\pi\)
\(602\) 8.92117e6 1.00330
\(603\) 3.48930e6 0.390792
\(604\) 5.96582e6 0.665393
\(605\) −76755.1 −0.00852548
\(606\) 1.83040e6 0.202472
\(607\) −606045. −0.0667626 −0.0333813 0.999443i \(-0.510628\pi\)
−0.0333813 + 0.999443i \(0.510628\pi\)
\(608\) 177493. 0.0194725
\(609\) −1.14689e7 −1.25307
\(610\) 2.93399e6 0.319253
\(611\) −122635. −0.0132896
\(612\) −468158. −0.0505259
\(613\) −1.12941e7 −1.21395 −0.606976 0.794720i \(-0.707618\pi\)
−0.606976 + 0.794720i \(0.707618\pi\)
\(614\) 7.78900e6 0.833798
\(615\) −110650. −0.0117968
\(616\) 3.97818e6 0.422408
\(617\) −1.04308e7 −1.10308 −0.551539 0.834149i \(-0.685959\pi\)
−0.551539 + 0.834149i \(0.685959\pi\)
\(618\) −5.03251e6 −0.530046
\(619\) −6.00861e6 −0.630300 −0.315150 0.949042i \(-0.602055\pi\)
−0.315150 + 0.949042i \(0.602055\pi\)
\(620\) −803788. −0.0839774
\(621\) 2.08933e6 0.217410
\(622\) −9.32756e6 −0.966700
\(623\) −9.32928e6 −0.963003
\(624\) 193656. 0.0199099
\(625\) 390625. 0.0400000
\(626\) 1.76483e6 0.179997
\(627\) −725272. −0.0736770
\(628\) −3.11704e6 −0.315386
\(629\) −2.72403e6 −0.274527
\(630\) 2.06707e6 0.207493
\(631\) −1.10118e7 −1.10099 −0.550496 0.834838i \(-0.685561\pi\)
−0.550496 + 0.834838i \(0.685561\pi\)
\(632\) 553174. 0.0550895
\(633\) 8.25864e6 0.819218
\(634\) −3.26350e6 −0.322448
\(635\) −2.57461e6 −0.253383
\(636\) 1.02782e6 0.100757
\(637\) −549719. −0.0536775
\(638\) 1.10755e7 1.07724
\(639\) 1.25363e6 0.121455
\(640\) −409600. −0.0395285
\(641\) 5.54887e6 0.533408 0.266704 0.963778i \(-0.414065\pi\)
0.266704 + 0.963778i \(0.414065\pi\)
\(642\) −3.11909e6 −0.298670
\(643\) −3.29092e6 −0.313899 −0.156949 0.987607i \(-0.550166\pi\)
−0.156949 + 0.987607i \(0.550166\pi\)
\(644\) 1.32366e6 0.125766
\(645\) 3.75333e6 0.355236
\(646\) 153483. 0.0144704
\(647\) −5.30849e6 −0.498551 −0.249276 0.968433i \(-0.580193\pi\)
−0.249276 + 0.968433i \(0.580193\pi\)
\(648\) 605420. 0.0566394
\(649\) −9.22929e6 −0.860115
\(650\) 179645. 0.0166775
\(651\) 3.30827e6 0.305948
\(652\) 5.32719e6 0.490772
\(653\) −3.53529e6 −0.324445 −0.162223 0.986754i \(-0.551866\pi\)
−0.162223 + 0.986754i \(0.551866\pi\)
\(654\) 836634. 0.0764876
\(655\) 285407. 0.0259933
\(656\) 107630. 0.00976506
\(657\) 233924. 0.0211427
\(658\) −1.06759e6 −0.0961254
\(659\) −1.61264e7 −1.44652 −0.723260 0.690575i \(-0.757357\pi\)
−0.723260 + 0.690575i \(0.757357\pi\)
\(660\) 1.67370e6 0.149561
\(661\) 8.53766e6 0.760038 0.380019 0.924979i \(-0.375917\pi\)
0.380019 + 0.924979i \(0.375917\pi\)
\(662\) −8.44221e6 −0.748706
\(663\) 167460. 0.0147954
\(664\) 6.93050e6 0.610021
\(665\) −677679. −0.0594251
\(666\) −6.50587e6 −0.568355
\(667\) 3.68516e6 0.320732
\(668\) 4.62773e6 0.401261
\(669\) −1.32643e7 −1.14582
\(670\) 2.63989e6 0.227195
\(671\) 1.16617e7 0.999896
\(672\) 1.68585e6 0.144011
\(673\) −2.02365e7 −1.72226 −0.861128 0.508388i \(-0.830242\pi\)
−0.861128 + 0.508388i \(0.830242\pi\)
\(674\) 4.55279e6 0.386036
\(675\) 2.46849e6 0.208532
\(676\) −5.85807e6 −0.493047
\(677\) −1.31785e7 −1.10508 −0.552542 0.833485i \(-0.686342\pi\)
−0.552542 + 0.833485i \(0.686342\pi\)
\(678\) 1.21252e6 0.101302
\(679\) −2.20234e7 −1.83320
\(680\) −354192. −0.0293743
\(681\) 1.51037e7 1.24800
\(682\) −3.19480e6 −0.263016
\(683\) −1.61463e7 −1.32441 −0.662205 0.749323i \(-0.730379\pi\)
−0.662205 + 0.749323i \(0.730379\pi\)
\(684\) 366568. 0.0299581
\(685\) −3.48717e6 −0.283953
\(686\) 5.72811e6 0.464731
\(687\) 2.17063e6 0.175466
\(688\) −3.65090e6 −0.294055
\(689\) 438485. 0.0351890
\(690\) 556894. 0.0445297
\(691\) −7.39037e6 −0.588805 −0.294402 0.955682i \(-0.595121\pi\)
−0.294402 + 0.955682i \(0.595121\pi\)
\(692\) −1.92471e6 −0.152792
\(693\) 8.21594e6 0.649867
\(694\) 5.33022e6 0.420094
\(695\) −1.16094e6 −0.0911689
\(696\) 4.69351e6 0.367261
\(697\) 93070.9 0.00725658
\(698\) −1.08956e7 −0.846471
\(699\) −9.20658e6 −0.712698
\(700\) 1.56388e6 0.120631
\(701\) 2.20579e7 1.69538 0.847692 0.530488i \(-0.177991\pi\)
0.847692 + 0.530488i \(0.177991\pi\)
\(702\) 1.13524e6 0.0869448
\(703\) 2.13292e6 0.162774
\(704\) −1.62803e6 −0.123803
\(705\) −449156. −0.0340349
\(706\) −1.74481e7 −1.31746
\(707\) 6.79787e6 0.511475
\(708\) −3.91113e6 −0.293237
\(709\) −7.51982e6 −0.561813 −0.280906 0.959735i \(-0.590635\pi\)
−0.280906 + 0.959735i \(0.590635\pi\)
\(710\) 948452. 0.0706105
\(711\) 1.14244e6 0.0847540
\(712\) 3.81791e6 0.282245
\(713\) −1.06301e6 −0.0783093
\(714\) 1.45780e6 0.107017
\(715\) 714030. 0.0522338
\(716\) 1.49559e6 0.109026
\(717\) 4.31957e6 0.313792
\(718\) −1.31192e7 −0.949721
\(719\) −1.09180e7 −0.787628 −0.393814 0.919190i \(-0.628845\pi\)
−0.393814 + 0.919190i \(0.628845\pi\)
\(720\) −845927. −0.0608137
\(721\) −1.86901e7 −1.33897
\(722\) 9.78422e6 0.698527
\(723\) 1.29921e7 0.924344
\(724\) −9.25874e6 −0.656456
\(725\) 4.35393e6 0.307635
\(726\) −129284. −0.00910337
\(727\) 2.27274e7 1.59483 0.797414 0.603432i \(-0.206201\pi\)
0.797414 + 0.603432i \(0.206201\pi\)
\(728\) 719212. 0.0502954
\(729\) 1.13539e7 0.791274
\(730\) 176979. 0.0122918
\(731\) −3.15703e6 −0.218517
\(732\) 4.94192e6 0.340893
\(733\) −1.86812e7 −1.28423 −0.642117 0.766607i \(-0.721944\pi\)
−0.642117 + 0.766607i \(0.721944\pi\)
\(734\) 4.63159e6 0.317314
\(735\) −2.01337e6 −0.137469
\(736\) −541696. −0.0368605
\(737\) 1.04927e7 0.711573
\(738\) 222284. 0.0150234
\(739\) −2.04129e6 −0.137497 −0.0687486 0.997634i \(-0.521901\pi\)
−0.0687486 + 0.997634i \(0.521901\pi\)
\(740\) −4.92212e6 −0.330425
\(741\) −131121. −0.00877258
\(742\) 3.81718e6 0.254526
\(743\) −4.75711e6 −0.316134 −0.158067 0.987428i \(-0.550526\pi\)
−0.158067 + 0.987428i \(0.550526\pi\)
\(744\) −1.35387e6 −0.0896697
\(745\) −8.43308e6 −0.556667
\(746\) −1.67657e7 −1.10300
\(747\) 1.43132e7 0.938504
\(748\) −1.40780e6 −0.0919999
\(749\) −1.15839e7 −0.754483
\(750\) 657956. 0.0427114
\(751\) 1.92465e7 1.24524 0.622619 0.782525i \(-0.286069\pi\)
0.622619 + 0.782525i \(0.286069\pi\)
\(752\) 436899. 0.0281732
\(753\) −1.07803e6 −0.0692854
\(754\) 2.00233e6 0.128265
\(755\) 9.32160e6 0.595145
\(756\) 9.88267e6 0.628883
\(757\) 2.32757e7 1.47626 0.738131 0.674658i \(-0.235709\pi\)
0.738131 + 0.674658i \(0.235709\pi\)
\(758\) −1.75073e7 −1.10674
\(759\) 2.21347e6 0.139467
\(760\) 277333. 0.0174168
\(761\) −1.23815e7 −0.775019 −0.387509 0.921866i \(-0.626665\pi\)
−0.387509 + 0.921866i \(0.626665\pi\)
\(762\) −4.33659e6 −0.270558
\(763\) 3.10714e6 0.193219
\(764\) 5.34564e6 0.331334
\(765\) −731496. −0.0451917
\(766\) −1.16355e7 −0.716495
\(767\) −1.66855e6 −0.102412
\(768\) −689917. −0.0422079
\(769\) −3.09591e7 −1.88787 −0.943936 0.330128i \(-0.892908\pi\)
−0.943936 + 0.330128i \(0.892908\pi\)
\(770\) 6.21591e6 0.377814
\(771\) 1.82522e7 1.10581
\(772\) −5.72425e6 −0.345681
\(773\) 2.73438e7 1.64593 0.822963 0.568094i \(-0.192319\pi\)
0.822963 + 0.568094i \(0.192319\pi\)
\(774\) −7.54002e6 −0.452397
\(775\) −1.25592e6 −0.0751117
\(776\) 9.01286e6 0.537289
\(777\) 2.02587e7 1.20381
\(778\) −2.89498e6 −0.171473
\(779\) −72874.7 −0.00430262
\(780\) 302588. 0.0178080
\(781\) 3.76979e6 0.221152
\(782\) −468419. −0.0273916
\(783\) 2.75140e7 1.60379
\(784\) 1.95842e6 0.113793
\(785\) −4.87037e6 −0.282090
\(786\) 480730. 0.0277553
\(787\) 1.78586e7 1.02781 0.513903 0.857848i \(-0.328199\pi\)
0.513903 + 0.857848i \(0.328199\pi\)
\(788\) 3.29373e6 0.188961
\(789\) 16832.9 0.000962645 0
\(790\) 864334. 0.0492735
\(791\) 4.50315e6 0.255903
\(792\) −3.36229e6 −0.190468
\(793\) 2.10831e6 0.119056
\(794\) 8.23277e6 0.463441
\(795\) 1.60597e6 0.0901195
\(796\) 1.58850e7 0.888594
\(797\) −8.37462e6 −0.467003 −0.233501 0.972356i \(-0.575018\pi\)
−0.233501 + 0.972356i \(0.575018\pi\)
\(798\) −1.14146e6 −0.0634532
\(799\) 377798. 0.0209360
\(800\) −640000. −0.0353553
\(801\) 7.88494e6 0.434227
\(802\) 1.67119e6 0.0917468
\(803\) 703435. 0.0384977
\(804\) 4.44654e6 0.242595
\(805\) 2.06823e6 0.112489
\(806\) −577585. −0.0313169
\(807\) 1.89252e6 0.102295
\(808\) −2.78196e6 −0.149907
\(809\) −5.71291e6 −0.306892 −0.153446 0.988157i \(-0.549037\pi\)
−0.153446 + 0.988157i \(0.549037\pi\)
\(810\) 945968. 0.0506599
\(811\) 1.92032e6 0.102523 0.0512616 0.998685i \(-0.483676\pi\)
0.0512616 + 0.998685i \(0.483676\pi\)
\(812\) 1.74310e7 0.927755
\(813\) −6.85339e6 −0.363646
\(814\) −1.95639e7 −1.03489
\(815\) 8.32374e6 0.438959
\(816\) −596590. −0.0313654
\(817\) 2.47196e6 0.129565
\(818\) −8.49136e6 −0.443705
\(819\) 1.48535e6 0.0773785
\(820\) 168172. 0.00873414
\(821\) 8.63223e6 0.446956 0.223478 0.974709i \(-0.428259\pi\)
0.223478 + 0.974709i \(0.428259\pi\)
\(822\) −5.87367e6 −0.303201
\(823\) 2.49722e7 1.28516 0.642580 0.766218i \(-0.277864\pi\)
0.642580 + 0.766218i \(0.277864\pi\)
\(824\) 7.64871e6 0.392437
\(825\) 2.61516e6 0.133772
\(826\) −1.45254e7 −0.740761
\(827\) 2.77371e7 1.41025 0.705127 0.709081i \(-0.250890\pi\)
0.705127 + 0.709081i \(0.250890\pi\)
\(828\) −1.11874e6 −0.0567091
\(829\) 1.15754e6 0.0584991 0.0292496 0.999572i \(-0.490688\pi\)
0.0292496 + 0.999572i \(0.490688\pi\)
\(830\) 1.08289e7 0.545619
\(831\) −1.10190e6 −0.0553526
\(832\) −294330. −0.0147410
\(833\) 1.69350e6 0.0845615
\(834\) −1.95544e6 −0.0973488
\(835\) 7.23083e6 0.358899
\(836\) 1.10231e6 0.0545492
\(837\) −7.93658e6 −0.391579
\(838\) 6.36622e6 0.313164
\(839\) 1.00284e7 0.491846 0.245923 0.969289i \(-0.420909\pi\)
0.245923 + 0.969289i \(0.420909\pi\)
\(840\) 2.63414e6 0.128807
\(841\) 2.80180e7 1.36599
\(842\) 9.55440e6 0.464433
\(843\) −1.00120e7 −0.485234
\(844\) −1.25520e7 −0.606535
\(845\) −9.15324e6 −0.440994
\(846\) 902306. 0.0433439
\(847\) −480142. −0.0229965
\(848\) −1.56214e6 −0.0745985
\(849\) −491748. −0.0234139
\(850\) −553426. −0.0262731
\(851\) −6.50951e6 −0.308123
\(852\) 1.59754e6 0.0753968
\(853\) 5.61703e6 0.264323 0.132161 0.991228i \(-0.457808\pi\)
0.132161 + 0.991228i \(0.457808\pi\)
\(854\) 1.83536e7 0.861146
\(855\) 572763. 0.0267954
\(856\) 4.74058e6 0.221130
\(857\) 1.85480e7 0.862670 0.431335 0.902192i \(-0.358043\pi\)
0.431335 + 0.902192i \(0.358043\pi\)
\(858\) 1.20269e6 0.0557744
\(859\) −5.66092e6 −0.261761 −0.130880 0.991398i \(-0.541780\pi\)
−0.130880 + 0.991398i \(0.541780\pi\)
\(860\) −5.70453e6 −0.263011
\(861\) −692172. −0.0318204
\(862\) −1.88127e7 −0.862348
\(863\) 2.77686e7 1.26919 0.634596 0.772844i \(-0.281166\pi\)
0.634596 + 0.772844i \(0.281166\pi\)
\(864\) −4.04438e6 −0.184318
\(865\) −3.00737e6 −0.136661
\(866\) 2.17782e7 0.986795
\(867\) 1.44314e7 0.652018
\(868\) −5.02810e6 −0.226519
\(869\) 3.43545e6 0.154324
\(870\) 7.33361e6 0.328488
\(871\) 1.89697e6 0.0847257
\(872\) −1.27157e6 −0.0566301
\(873\) 1.86138e7 0.826609
\(874\) 366773. 0.0162412
\(875\) 2.44356e6 0.107895
\(876\) 298098. 0.0131250
\(877\) −2.09926e7 −0.921653 −0.460827 0.887490i \(-0.652447\pi\)
−0.460827 + 0.887490i \(0.652447\pi\)
\(878\) −1.16004e7 −0.507850
\(879\) −5.68264e6 −0.248072
\(880\) −2.54380e6 −0.110733
\(881\) −1.66100e7 −0.720992 −0.360496 0.932761i \(-0.617393\pi\)
−0.360496 + 0.932761i \(0.617393\pi\)
\(882\) 4.04463e6 0.175068
\(883\) −4.19946e7 −1.81256 −0.906280 0.422679i \(-0.861090\pi\)
−0.906280 + 0.422679i \(0.861090\pi\)
\(884\) −254515. −0.0109543
\(885\) −6.11115e6 −0.262280
\(886\) −7.88570e6 −0.337486
\(887\) −3.01036e7 −1.28472 −0.642361 0.766402i \(-0.722045\pi\)
−0.642361 + 0.766402i \(0.722045\pi\)
\(888\) −8.29066e6 −0.352823
\(889\) −1.61055e7 −0.683470
\(890\) 5.96548e6 0.252447
\(891\) 3.75992e6 0.158666
\(892\) 2.01598e7 0.848349
\(893\) −295817. −0.0124135
\(894\) −1.42044e7 −0.594401
\(895\) 2.33686e6 0.0975157
\(896\) −2.56225e6 −0.106623
\(897\) 400172. 0.0166060
\(898\) −9.26356e6 −0.383343
\(899\) −1.39985e7 −0.577675
\(900\) −1.32176e6 −0.0543935
\(901\) −1.35083e6 −0.0554354
\(902\) 668431. 0.0273552
\(903\) 2.34789e7 0.958207
\(904\) −1.84287e6 −0.0750020
\(905\) −1.44668e7 −0.587152
\(906\) 1.57010e7 0.635487
\(907\) −2.41762e7 −0.975818 −0.487909 0.872894i \(-0.662240\pi\)
−0.487909 + 0.872894i \(0.662240\pi\)
\(908\) −2.29554e7 −0.923997
\(909\) −5.74544e6 −0.230629
\(910\) 1.12377e6 0.0449856
\(911\) 2.73957e7 1.09367 0.546834 0.837241i \(-0.315833\pi\)
0.546834 + 0.837241i \(0.315833\pi\)
\(912\) 467131. 0.0185974
\(913\) 4.30415e7 1.70887
\(914\) 1.78224e7 0.705667
\(915\) 7.72175e6 0.304904
\(916\) −3.29905e6 −0.129912
\(917\) 1.78537e6 0.0701138
\(918\) −3.49729e6 −0.136970
\(919\) 2.81548e7 1.09967 0.549837 0.835272i \(-0.314690\pi\)
0.549837 + 0.835272i \(0.314690\pi\)
\(920\) −846400. −0.0329690
\(921\) 2.04993e7 0.796323
\(922\) 2.80831e7 1.08797
\(923\) 681538. 0.0263321
\(924\) 1.04699e7 0.403424
\(925\) −7.69082e6 −0.295541
\(926\) −1.47925e7 −0.566911
\(927\) 1.57965e7 0.603757
\(928\) −7.13348e6 −0.271914
\(929\) 2.61544e6 0.0994272 0.0497136 0.998764i \(-0.484169\pi\)
0.0497136 + 0.998764i \(0.484169\pi\)
\(930\) −2.11543e6 −0.0802031
\(931\) −1.32601e6 −0.0501388
\(932\) 1.39927e7 0.527669
\(933\) −2.45485e7 −0.923252
\(934\) 6.92903e6 0.259899
\(935\) −2.19969e6 −0.0822872
\(936\) −607866. −0.0226787
\(937\) 4.13839e7 1.53986 0.769932 0.638126i \(-0.220290\pi\)
0.769932 + 0.638126i \(0.220290\pi\)
\(938\) 1.65138e7 0.612831
\(939\) 4.64471e6 0.171908
\(940\) 682654. 0.0251989
\(941\) −3.00692e7 −1.10700 −0.553500 0.832849i \(-0.686708\pi\)
−0.553500 + 0.832849i \(0.686708\pi\)
\(942\) −8.20349e6 −0.301211
\(943\) 222408. 0.00814463
\(944\) 5.94437e6 0.217108
\(945\) 1.54417e7 0.562490
\(946\) −2.26737e7 −0.823747
\(947\) −1.33174e7 −0.482551 −0.241276 0.970457i \(-0.577566\pi\)
−0.241276 + 0.970457i \(0.577566\pi\)
\(948\) 1.45586e6 0.0526135
\(949\) 127173. 0.00458385
\(950\) 433333. 0.0155780
\(951\) −8.58895e6 −0.307956
\(952\) −2.21565e6 −0.0792336
\(953\) 3.89545e7 1.38939 0.694697 0.719302i \(-0.255538\pi\)
0.694697 + 0.719302i \(0.255538\pi\)
\(954\) −3.22621e6 −0.114768
\(955\) 8.35256e6 0.296354
\(956\) −6.56513e6 −0.232326
\(957\) 2.91488e7 1.02882
\(958\) −1.03086e7 −0.362898
\(959\) −2.18140e7 −0.765929
\(960\) −1.07799e6 −0.0377519
\(961\) −2.45912e7 −0.858956
\(962\) −3.53694e6 −0.123222
\(963\) 9.79050e6 0.340204
\(964\) −1.97462e7 −0.684368
\(965\) −8.94414e6 −0.309186
\(966\) 3.48365e6 0.120114
\(967\) −1.83987e7 −0.632734 −0.316367 0.948637i \(-0.602463\pi\)
−0.316367 + 0.948637i \(0.602463\pi\)
\(968\) 196493. 0.00673998
\(969\) 403941. 0.0138200
\(970\) 1.40826e7 0.480566
\(971\) 9.56759e6 0.325653 0.162826 0.986655i \(-0.447939\pi\)
0.162826 + 0.986655i \(0.447939\pi\)
\(972\) −1.37626e7 −0.467236
\(973\) −7.26225e6 −0.245917
\(974\) −3.02665e7 −1.02227
\(975\) 472793. 0.0159279
\(976\) −7.51102e6 −0.252391
\(977\) −2.84840e7 −0.954695 −0.477347 0.878715i \(-0.658402\pi\)
−0.477347 + 0.878715i \(0.658402\pi\)
\(978\) 1.40202e7 0.468714
\(979\) 2.37109e7 0.790662
\(980\) 3.06003e6 0.101780
\(981\) −2.62610e6 −0.0871242
\(982\) 2.09114e6 0.0691996
\(983\) −1.41641e7 −0.467527 −0.233763 0.972294i \(-0.575104\pi\)
−0.233763 + 0.972294i \(0.575104\pi\)
\(984\) 283264. 0.00932618
\(985\) 5.14646e6 0.169012
\(986\) −6.16851e6 −0.202064
\(987\) −2.80970e6 −0.0918051
\(988\) 199286. 0.00649507
\(989\) −7.54424e6 −0.245259
\(990\) −5.25358e6 −0.170360
\(991\) 9.12692e6 0.295216 0.147608 0.989046i \(-0.452843\pi\)
0.147608 + 0.989046i \(0.452843\pi\)
\(992\) 2.05770e6 0.0663899
\(993\) −2.22184e7 −0.715055
\(994\) 5.93304e6 0.190463
\(995\) 2.48202e7 0.794783
\(996\) 1.82399e7 0.582604
\(997\) −2.74140e7 −0.873442 −0.436721 0.899597i \(-0.643860\pi\)
−0.436721 + 0.899597i \(0.643860\pi\)
\(998\) −3.86474e7 −1.22827
\(999\) −4.86009e7 −1.54074
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 230.6.a.c.1.2 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
230.6.a.c.1.2 3 1.1 even 1 trivial