Properties

Label 570.8.a.a.1.1
Level $570$
Weight $8$
Character 570.1
Self dual yes
Analytic conductor $178.059$
Analytic rank $1$
Dimension $4$
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] = [570,8,Mod(1,570)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(570, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("570.1");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 570 = 2 \cdot 3 \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 570.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: \(178.059464526\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: \(\mathbb{Q}[x]/(x^{4} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 2x^{3} - 4122x^{2} - 49773x + 620550 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{6}\cdot 3 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(7.65975\) of defining polynomial
Character \(\chi\) \(=\) 570.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+8.00000 q^{2} -27.0000 q^{3} +64.0000 q^{4} +125.000 q^{5} -216.000 q^{6} -1165.98 q^{7} +512.000 q^{8} +729.000 q^{9} +O(q^{10})\) \(q+8.00000 q^{2} -27.0000 q^{3} +64.0000 q^{4} +125.000 q^{5} -216.000 q^{6} -1165.98 q^{7} +512.000 q^{8} +729.000 q^{9} +1000.00 q^{10} -144.005 q^{11} -1728.00 q^{12} -3048.40 q^{13} -9327.86 q^{14} -3375.00 q^{15} +4096.00 q^{16} +6703.52 q^{17} +5832.00 q^{18} +6859.00 q^{19} +8000.00 q^{20} +31481.5 q^{21} -1152.04 q^{22} +15414.7 q^{23} -13824.0 q^{24} +15625.0 q^{25} -24387.2 q^{26} -19683.0 q^{27} -74622.9 q^{28} +180651. q^{29} -27000.0 q^{30} +173839. q^{31} +32768.0 q^{32} +3888.13 q^{33} +53628.1 q^{34} -145748. q^{35} +46656.0 q^{36} -452052. q^{37} +54872.0 q^{38} +82306.9 q^{39} +64000.0 q^{40} +338502. q^{41} +251852. q^{42} -357281. q^{43} -9216.30 q^{44} +91125.0 q^{45} +123318. q^{46} -486670. q^{47} -110592. q^{48} +535972. q^{49} +125000. q^{50} -180995. q^{51} -195098. q^{52} -1.80340e6 q^{53} -157464. q^{54} -18000.6 q^{55} -596983. q^{56} -185193. q^{57} +1.44521e6 q^{58} +1.20379e6 q^{59} -216000. q^{60} -614829. q^{61} +1.39071e6 q^{62} -850001. q^{63} +262144. q^{64} -381051. q^{65} +31105.0 q^{66} +2.28349e6 q^{67} +429025. q^{68} -416197. q^{69} -1.16598e6 q^{70} +2.30022e6 q^{71} +373248. q^{72} -1.41070e6 q^{73} -3.61641e6 q^{74} -421875. q^{75} +438976. q^{76} +167907. q^{77} +658455. q^{78} -5.82643e6 q^{79} +512000. q^{80} +531441. q^{81} +2.70802e6 q^{82} +5.65567e6 q^{83} +2.01482e6 q^{84} +837939. q^{85} -2.85825e6 q^{86} -4.87758e6 q^{87} -73730.4 q^{88} +5.01119e6 q^{89} +729000. q^{90} +3.55439e6 q^{91} +986540. q^{92} -4.69364e6 q^{93} -3.89336e6 q^{94} +857375. q^{95} -884736. q^{96} -5.52163e6 q^{97} +4.28778e6 q^{98} -104979. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 32 q^{2} - 108 q^{3} + 256 q^{4} + 500 q^{5} - 864 q^{6} - 1496 q^{7} + 2048 q^{8} + 2916 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 32 q^{2} - 108 q^{3} + 256 q^{4} + 500 q^{5} - 864 q^{6} - 1496 q^{7} + 2048 q^{8} + 2916 q^{9} + 4000 q^{10} - 2912 q^{11} - 6912 q^{12} - 3696 q^{13} - 11968 q^{14} - 13500 q^{15} + 16384 q^{16} + 16 q^{17} + 23328 q^{18} + 27436 q^{19} + 32000 q^{20} + 40392 q^{21} - 23296 q^{22} - 73016 q^{23} - 55296 q^{24} + 62500 q^{25} - 29568 q^{26} - 78732 q^{27} - 95744 q^{28} - 137784 q^{29} - 108000 q^{30} + 198072 q^{31} + 131072 q^{32} + 78624 q^{33} + 128 q^{34} - 187000 q^{35} + 186624 q^{36} + 207256 q^{37} + 219488 q^{38} + 99792 q^{39} + 256000 q^{40} - 504056 q^{41} + 323136 q^{42} - 250368 q^{43} - 186368 q^{44} + 364500 q^{45} - 584128 q^{46} - 1000376 q^{47} - 442368 q^{48} - 940908 q^{49} + 500000 q^{50} - 432 q^{51} - 236544 q^{52} - 2178688 q^{53} - 629856 q^{54} - 364000 q^{55} - 765952 q^{56} - 740772 q^{57} - 1102272 q^{58} + 327976 q^{59} - 864000 q^{60} + 572936 q^{61} + 1584576 q^{62} - 1090584 q^{63} + 1048576 q^{64} - 462000 q^{65} + 628992 q^{66} + 2017152 q^{67} + 1024 q^{68} + 1971432 q^{69} - 1496000 q^{70} + 2828960 q^{71} + 1492992 q^{72} - 132392 q^{73} + 1658048 q^{74} - 1687500 q^{75} + 1755904 q^{76} - 2304704 q^{77} + 798336 q^{78} + 3418408 q^{79} + 2048000 q^{80} + 2125764 q^{81} - 4032448 q^{82} - 3201760 q^{83} + 2585088 q^{84} + 2000 q^{85} - 2002944 q^{86} + 3720168 q^{87} - 1490944 q^{88} - 1389392 q^{89} + 2916000 q^{90} - 7865280 q^{91} - 4673024 q^{92} - 5347944 q^{93} - 8003008 q^{94} + 3429500 q^{95} - 3538944 q^{96} - 21061144 q^{97} - 7527264 q^{98} - 2122848 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 8.00000 0.707107
\(3\) −27.0000 −0.577350
\(4\) 64.0000 0.500000
\(5\) 125.000 0.447214
\(6\) −216.000 −0.408248
\(7\) −1165.98 −1.28484 −0.642420 0.766353i \(-0.722069\pi\)
−0.642420 + 0.766353i \(0.722069\pi\)
\(8\) 512.000 0.353553
\(9\) 729.000 0.333333
\(10\) 1000.00 0.316228
\(11\) −144.005 −0.0326214 −0.0163107 0.999867i \(-0.505192\pi\)
−0.0163107 + 0.999867i \(0.505192\pi\)
\(12\) −1728.00 −0.288675
\(13\) −3048.40 −0.384832 −0.192416 0.981313i \(-0.561632\pi\)
−0.192416 + 0.981313i \(0.561632\pi\)
\(14\) −9327.86 −0.908519
\(15\) −3375.00 −0.258199
\(16\) 4096.00 0.250000
\(17\) 6703.52 0.330926 0.165463 0.986216i \(-0.447088\pi\)
0.165463 + 0.986216i \(0.447088\pi\)
\(18\) 5832.00 0.235702
\(19\) 6859.00 0.229416
\(20\) 8000.00 0.223607
\(21\) 31481.5 0.741802
\(22\) −1152.04 −0.0230668
\(23\) 15414.7 0.264172 0.132086 0.991238i \(-0.457832\pi\)
0.132086 + 0.991238i \(0.457832\pi\)
\(24\) −13824.0 −0.204124
\(25\) 15625.0 0.200000
\(26\) −24387.2 −0.272117
\(27\) −19683.0 −0.192450
\(28\) −74622.9 −0.642420
\(29\) 180651. 1.37546 0.687730 0.725967i \(-0.258607\pi\)
0.687730 + 0.725967i \(0.258607\pi\)
\(30\) −27000.0 −0.182574
\(31\) 173839. 1.04805 0.524023 0.851704i \(-0.324431\pi\)
0.524023 + 0.851704i \(0.324431\pi\)
\(32\) 32768.0 0.176777
\(33\) 3888.13 0.0188340
\(34\) 53628.1 0.234000
\(35\) −145748. −0.574598
\(36\) 46656.0 0.166667
\(37\) −452052. −1.46718 −0.733588 0.679595i \(-0.762156\pi\)
−0.733588 + 0.679595i \(0.762156\pi\)
\(38\) 54872.0 0.162221
\(39\) 82306.9 0.222183
\(40\) 64000.0 0.158114
\(41\) 338502. 0.767040 0.383520 0.923533i \(-0.374712\pi\)
0.383520 + 0.923533i \(0.374712\pi\)
\(42\) 251852. 0.524534
\(43\) −357281. −0.685283 −0.342642 0.939466i \(-0.611322\pi\)
−0.342642 + 0.939466i \(0.611322\pi\)
\(44\) −9216.30 −0.0163107
\(45\) 91125.0 0.149071
\(46\) 123318. 0.186798
\(47\) −486670. −0.683741 −0.341871 0.939747i \(-0.611060\pi\)
−0.341871 + 0.939747i \(0.611060\pi\)
\(48\) −110592. −0.144338
\(49\) 535972. 0.650813
\(50\) 125000. 0.141421
\(51\) −180995. −0.191060
\(52\) −195098. −0.192416
\(53\) −1.80340e6 −1.66389 −0.831947 0.554856i \(-0.812773\pi\)
−0.831947 + 0.554856i \(0.812773\pi\)
\(54\) −157464. −0.136083
\(55\) −18000.6 −0.0145887
\(56\) −596983. −0.454259
\(57\) −185193. −0.132453
\(58\) 1.44521e6 0.972597
\(59\) 1.20379e6 0.763076 0.381538 0.924353i \(-0.375395\pi\)
0.381538 + 0.924353i \(0.375395\pi\)
\(60\) −216000. −0.129099
\(61\) −614829. −0.346817 −0.173408 0.984850i \(-0.555478\pi\)
−0.173408 + 0.984850i \(0.555478\pi\)
\(62\) 1.39071e6 0.741080
\(63\) −850001. −0.428280
\(64\) 262144. 0.125000
\(65\) −381051. −0.172102
\(66\) 31105.0 0.0133176
\(67\) 2.28349e6 0.927552 0.463776 0.885953i \(-0.346494\pi\)
0.463776 + 0.885953i \(0.346494\pi\)
\(68\) 429025. 0.165463
\(69\) −416197. −0.152520
\(70\) −1.16598e6 −0.406302
\(71\) 2.30022e6 0.762721 0.381361 0.924426i \(-0.375456\pi\)
0.381361 + 0.924426i \(0.375456\pi\)
\(72\) 373248. 0.117851
\(73\) −1.41070e6 −0.424427 −0.212214 0.977223i \(-0.568067\pi\)
−0.212214 + 0.977223i \(0.568067\pi\)
\(74\) −3.61641e6 −1.03745
\(75\) −421875. −0.115470
\(76\) 438976. 0.114708
\(77\) 167907. 0.0419133
\(78\) 658455. 0.157107
\(79\) −5.82643e6 −1.32956 −0.664779 0.747040i \(-0.731474\pi\)
−0.664779 + 0.747040i \(0.731474\pi\)
\(80\) 512000. 0.111803
\(81\) 531441. 0.111111
\(82\) 2.70802e6 0.542379
\(83\) 5.65567e6 1.08570 0.542851 0.839829i \(-0.317345\pi\)
0.542851 + 0.839829i \(0.317345\pi\)
\(84\) 2.01482e6 0.370901
\(85\) 837939. 0.147995
\(86\) −2.85825e6 −0.484569
\(87\) −4.87758e6 −0.794122
\(88\) −73730.4 −0.0115334
\(89\) 5.01119e6 0.753487 0.376743 0.926318i \(-0.377044\pi\)
0.376743 + 0.926318i \(0.377044\pi\)
\(90\) 729000. 0.105409
\(91\) 3.55439e6 0.494447
\(92\) 986540. 0.132086
\(93\) −4.69364e6 −0.605090
\(94\) −3.89336e6 −0.483478
\(95\) 857375. 0.102598
\(96\) −884736. −0.102062
\(97\) −5.52163e6 −0.614280 −0.307140 0.951664i \(-0.599372\pi\)
−0.307140 + 0.951664i \(0.599372\pi\)
\(98\) 4.28778e6 0.460194
\(99\) −104979. −0.0108738
\(100\) 1.00000e6 0.100000
\(101\) −1.91291e7 −1.84743 −0.923717 0.383076i \(-0.874865\pi\)
−0.923717 + 0.383076i \(0.874865\pi\)
\(102\) −1.44796e6 −0.135100
\(103\) −1.88635e7 −1.70095 −0.850476 0.526013i \(-0.823686\pi\)
−0.850476 + 0.526013i \(0.823686\pi\)
\(104\) −1.56078e6 −0.136059
\(105\) 3.93519e6 0.331744
\(106\) −1.44272e7 −1.17655
\(107\) −7.28717e6 −0.575063 −0.287532 0.957771i \(-0.592835\pi\)
−0.287532 + 0.957771i \(0.592835\pi\)
\(108\) −1.25971e6 −0.0962250
\(109\) −6.73832e6 −0.498378 −0.249189 0.968455i \(-0.580164\pi\)
−0.249189 + 0.968455i \(0.580164\pi\)
\(110\) −144005. −0.0103158
\(111\) 1.22054e7 0.847074
\(112\) −4.77586e6 −0.321210
\(113\) −1.27998e7 −0.834507 −0.417254 0.908790i \(-0.637007\pi\)
−0.417254 + 0.908790i \(0.637007\pi\)
\(114\) −1.48154e6 −0.0936586
\(115\) 1.92684e6 0.118141
\(116\) 1.15617e7 0.687730
\(117\) −2.22229e6 −0.128277
\(118\) 9.63030e6 0.539576
\(119\) −7.81618e6 −0.425187
\(120\) −1.72800e6 −0.0912871
\(121\) −1.94664e7 −0.998936
\(122\) −4.91864e6 −0.245237
\(123\) −9.13956e6 −0.442851
\(124\) 1.11257e7 0.524023
\(125\) 1.95312e6 0.0894427
\(126\) −6.80001e6 −0.302840
\(127\) 2.56926e7 1.11300 0.556501 0.830847i \(-0.312144\pi\)
0.556501 + 0.830847i \(0.312144\pi\)
\(128\) 2.09715e6 0.0883883
\(129\) 9.64659e6 0.395649
\(130\) −3.04840e6 −0.121694
\(131\) 2.74701e7 1.06760 0.533802 0.845609i \(-0.320763\pi\)
0.533802 + 0.845609i \(0.320763\pi\)
\(132\) 248840. 0.00941698
\(133\) −7.99747e6 −0.294762
\(134\) 1.82680e7 0.655878
\(135\) −2.46038e6 −0.0860663
\(136\) 3.43220e6 0.117000
\(137\) 5.09858e7 1.69405 0.847026 0.531551i \(-0.178391\pi\)
0.847026 + 0.531551i \(0.178391\pi\)
\(138\) −3.32957e6 −0.107848
\(139\) −3.63943e7 −1.14943 −0.574714 0.818354i \(-0.694887\pi\)
−0.574714 + 0.818354i \(0.694887\pi\)
\(140\) −9.32786e6 −0.287299
\(141\) 1.31401e7 0.394758
\(142\) 1.84018e7 0.539325
\(143\) 438985. 0.0125537
\(144\) 2.98598e6 0.0833333
\(145\) 2.25814e7 0.615124
\(146\) −1.12856e7 −0.300115
\(147\) −1.44713e7 −0.375747
\(148\) −2.89313e7 −0.733588
\(149\) 6.18264e7 1.53117 0.765583 0.643337i \(-0.222451\pi\)
0.765583 + 0.643337i \(0.222451\pi\)
\(150\) −3.37500e6 −0.0816497
\(151\) −5.50315e7 −1.30074 −0.650372 0.759616i \(-0.725387\pi\)
−0.650372 + 0.759616i \(0.725387\pi\)
\(152\) 3.51181e6 0.0811107
\(153\) 4.88686e6 0.110309
\(154\) 1.34326e6 0.0296371
\(155\) 2.17298e7 0.468700
\(156\) 5.26764e6 0.111091
\(157\) −8.57283e7 −1.76797 −0.883986 0.467513i \(-0.845150\pi\)
−0.883986 + 0.467513i \(0.845150\pi\)
\(158\) −4.66114e7 −0.940140
\(159\) 4.86917e7 0.960649
\(160\) 4.09600e6 0.0790569
\(161\) −1.79733e7 −0.339419
\(162\) 4.25153e6 0.0785674
\(163\) 8.87150e7 1.60450 0.802251 0.596987i \(-0.203636\pi\)
0.802251 + 0.596987i \(0.203636\pi\)
\(164\) 2.16641e7 0.383520
\(165\) 486016. 0.00842281
\(166\) 4.52453e7 0.767707
\(167\) −1.07059e8 −1.77875 −0.889377 0.457174i \(-0.848862\pi\)
−0.889377 + 0.457174i \(0.848862\pi\)
\(168\) 1.61185e7 0.262267
\(169\) −5.34557e7 −0.851905
\(170\) 6.70352e6 0.104648
\(171\) 5.00021e6 0.0764719
\(172\) −2.28660e7 −0.342642
\(173\) −1.80227e7 −0.264642 −0.132321 0.991207i \(-0.542243\pi\)
−0.132321 + 0.991207i \(0.542243\pi\)
\(174\) −3.90207e7 −0.561529
\(175\) −1.82185e7 −0.256968
\(176\) −589844. −0.00815535
\(177\) −3.25022e7 −0.440562
\(178\) 4.00895e7 0.532796
\(179\) 1.05848e8 1.37942 0.689711 0.724085i \(-0.257738\pi\)
0.689711 + 0.724085i \(0.257738\pi\)
\(180\) 5.83200e6 0.0745356
\(181\) −1.14320e8 −1.43300 −0.716502 0.697585i \(-0.754258\pi\)
−0.716502 + 0.697585i \(0.754258\pi\)
\(182\) 2.84351e7 0.349627
\(183\) 1.66004e7 0.200235
\(184\) 7.89232e6 0.0933991
\(185\) −5.65065e7 −0.656141
\(186\) −3.75491e7 −0.427863
\(187\) −965338. −0.0107953
\(188\) −3.11469e7 −0.341871
\(189\) 2.29500e7 0.247267
\(190\) 6.85900e6 0.0725476
\(191\) −2.93960e7 −0.305261 −0.152631 0.988283i \(-0.548775\pi\)
−0.152631 + 0.988283i \(0.548775\pi\)
\(192\) −7.07789e6 −0.0721688
\(193\) −4.92637e7 −0.493260 −0.246630 0.969110i \(-0.579323\pi\)
−0.246630 + 0.969110i \(0.579323\pi\)
\(194\) −4.41730e7 −0.434361
\(195\) 1.02884e7 0.0993631
\(196\) 3.43022e7 0.325406
\(197\) −6.14818e7 −0.572948 −0.286474 0.958088i \(-0.592483\pi\)
−0.286474 + 0.958088i \(0.592483\pi\)
\(198\) −839836. −0.00768894
\(199\) 6.76104e7 0.608174 0.304087 0.952644i \(-0.401649\pi\)
0.304087 + 0.952644i \(0.401649\pi\)
\(200\) 8.00000e6 0.0707107
\(201\) −6.16544e7 −0.535522
\(202\) −1.53033e8 −1.30633
\(203\) −2.10636e8 −1.76724
\(204\) −1.15837e7 −0.0955302
\(205\) 4.23128e7 0.343031
\(206\) −1.50908e8 −1.20276
\(207\) 1.12373e7 0.0880575
\(208\) −1.24863e7 −0.0962079
\(209\) −987729. −0.00748386
\(210\) 3.14815e7 0.234579
\(211\) −6.49827e7 −0.476221 −0.238111 0.971238i \(-0.576528\pi\)
−0.238111 + 0.971238i \(0.576528\pi\)
\(212\) −1.15417e8 −0.831947
\(213\) −6.21060e7 −0.440357
\(214\) −5.82973e7 −0.406631
\(215\) −4.46601e7 −0.306468
\(216\) −1.00777e7 −0.0680414
\(217\) −2.02693e8 −1.34657
\(218\) −5.39065e7 −0.352406
\(219\) 3.80888e7 0.245043
\(220\) −1.15204e6 −0.00729436
\(221\) −2.04350e7 −0.127351
\(222\) 9.76432e7 0.598972
\(223\) −1.45234e8 −0.877001 −0.438501 0.898731i \(-0.644490\pi\)
−0.438501 + 0.898731i \(0.644490\pi\)
\(224\) −3.82069e7 −0.227130
\(225\) 1.13906e7 0.0666667
\(226\) −1.02399e8 −0.590086
\(227\) −1.49446e8 −0.847997 −0.423998 0.905663i \(-0.639374\pi\)
−0.423998 + 0.905663i \(0.639374\pi\)
\(228\) −1.18524e7 −0.0662266
\(229\) −7.16910e7 −0.394494 −0.197247 0.980354i \(-0.563200\pi\)
−0.197247 + 0.980354i \(0.563200\pi\)
\(230\) 1.54147e7 0.0835387
\(231\) −4.53349e6 −0.0241986
\(232\) 9.24934e7 0.486298
\(233\) 8.34184e7 0.432032 0.216016 0.976390i \(-0.430694\pi\)
0.216016 + 0.976390i \(0.430694\pi\)
\(234\) −1.77783e7 −0.0907057
\(235\) −6.08337e7 −0.305778
\(236\) 7.70424e7 0.381538
\(237\) 1.57314e8 0.767621
\(238\) −6.25295e7 −0.300653
\(239\) −2.70927e8 −1.28369 −0.641845 0.766834i \(-0.721831\pi\)
−0.641845 + 0.766834i \(0.721831\pi\)
\(240\) −1.38240e7 −0.0645497
\(241\) −1.55946e8 −0.717652 −0.358826 0.933405i \(-0.616823\pi\)
−0.358826 + 0.933405i \(0.616823\pi\)
\(242\) −1.55731e8 −0.706354
\(243\) −1.43489e7 −0.0641500
\(244\) −3.93491e7 −0.173408
\(245\) 6.69965e7 0.291052
\(246\) −7.31165e7 −0.313143
\(247\) −2.09090e7 −0.0882865
\(248\) 8.90054e7 0.370540
\(249\) −1.52703e8 −0.626830
\(250\) 1.56250e7 0.0632456
\(251\) 4.06626e7 0.162307 0.0811535 0.996702i \(-0.474140\pi\)
0.0811535 + 0.996702i \(0.474140\pi\)
\(252\) −5.44001e7 −0.214140
\(253\) −2.21979e6 −0.00861767
\(254\) 2.05541e8 0.787011
\(255\) −2.26244e7 −0.0854448
\(256\) 1.67772e7 0.0625000
\(257\) −8.75166e7 −0.321606 −0.160803 0.986987i \(-0.551408\pi\)
−0.160803 + 0.986987i \(0.551408\pi\)
\(258\) 7.71727e7 0.279766
\(259\) 5.27084e8 1.88509
\(260\) −2.43872e7 −0.0860510
\(261\) 1.31695e8 0.458487
\(262\) 2.19761e8 0.754910
\(263\) −2.53858e7 −0.0860490 −0.0430245 0.999074i \(-0.513699\pi\)
−0.0430245 + 0.999074i \(0.513699\pi\)
\(264\) 1.99072e6 0.00665881
\(265\) −2.25424e8 −0.744116
\(266\) −6.39798e7 −0.208429
\(267\) −1.35302e8 −0.435026
\(268\) 1.46144e8 0.463776
\(269\) −1.27680e8 −0.399934 −0.199967 0.979803i \(-0.564084\pi\)
−0.199967 + 0.979803i \(0.564084\pi\)
\(270\) −1.96830e7 −0.0608581
\(271\) −2.27746e8 −0.695118 −0.347559 0.937658i \(-0.612989\pi\)
−0.347559 + 0.937658i \(0.612989\pi\)
\(272\) 2.74576e7 0.0827316
\(273\) −9.59684e7 −0.285469
\(274\) 4.07886e8 1.19788
\(275\) −2.25007e6 −0.00652428
\(276\) −2.66366e7 −0.0762600
\(277\) −2.02190e7 −0.0571583 −0.0285792 0.999592i \(-0.509098\pi\)
−0.0285792 + 0.999592i \(0.509098\pi\)
\(278\) −2.91155e8 −0.812769
\(279\) 1.26728e8 0.349349
\(280\) −7.46229e7 −0.203151
\(281\) −5.73631e8 −1.54227 −0.771135 0.636671i \(-0.780311\pi\)
−0.771135 + 0.636671i \(0.780311\pi\)
\(282\) 1.05121e8 0.279136
\(283\) −5.12585e8 −1.34435 −0.672176 0.740391i \(-0.734640\pi\)
−0.672176 + 0.740391i \(0.734640\pi\)
\(284\) 1.47214e8 0.381361
\(285\) −2.31491e7 −0.0592349
\(286\) 3.51188e6 0.00887684
\(287\) −3.94688e8 −0.985523
\(288\) 2.38879e7 0.0589256
\(289\) −3.65402e8 −0.890488
\(290\) 1.80651e8 0.434958
\(291\) 1.49084e8 0.354654
\(292\) −9.02845e7 −0.212214
\(293\) 1.24773e8 0.289791 0.144896 0.989447i \(-0.453715\pi\)
0.144896 + 0.989447i \(0.453715\pi\)
\(294\) −1.15770e8 −0.265693
\(295\) 1.50473e8 0.341258
\(296\) −2.31450e8 −0.518725
\(297\) 2.83445e6 0.00627799
\(298\) 4.94611e8 1.08270
\(299\) −4.69902e7 −0.101662
\(300\) −2.70000e7 −0.0577350
\(301\) 4.16583e8 0.880479
\(302\) −4.40252e8 −0.919765
\(303\) 5.16485e8 1.06662
\(304\) 2.80945e7 0.0573539
\(305\) −7.68537e7 −0.155101
\(306\) 3.90949e7 0.0780001
\(307\) 7.43416e8 1.46638 0.733191 0.680022i \(-0.238030\pi\)
0.733191 + 0.680022i \(0.238030\pi\)
\(308\) 1.07461e7 0.0209566
\(309\) 5.09315e8 0.982046
\(310\) 1.73839e8 0.331421
\(311\) 7.14985e8 1.34783 0.673915 0.738809i \(-0.264611\pi\)
0.673915 + 0.738809i \(0.264611\pi\)
\(312\) 4.21411e7 0.0785535
\(313\) −4.67452e8 −0.861652 −0.430826 0.902435i \(-0.641778\pi\)
−0.430826 + 0.902435i \(0.641778\pi\)
\(314\) −6.85827e8 −1.25015
\(315\) −1.06250e8 −0.191533
\(316\) −3.72891e8 −0.664779
\(317\) 4.40096e7 0.0775961 0.0387981 0.999247i \(-0.487647\pi\)
0.0387981 + 0.999247i \(0.487647\pi\)
\(318\) 3.89533e8 0.679282
\(319\) −2.60146e7 −0.0448694
\(320\) 3.27680e7 0.0559017
\(321\) 1.96754e8 0.332013
\(322\) −1.43786e8 −0.240006
\(323\) 4.59794e7 0.0759197
\(324\) 3.40122e7 0.0555556
\(325\) −4.76313e7 −0.0769664
\(326\) 7.09720e8 1.13455
\(327\) 1.81935e8 0.287739
\(328\) 1.73313e8 0.271189
\(329\) 5.67448e8 0.878498
\(330\) 3.88813e6 0.00595582
\(331\) 1.96813e8 0.298302 0.149151 0.988814i \(-0.452346\pi\)
0.149151 + 0.988814i \(0.452346\pi\)
\(332\) 3.61963e8 0.542851
\(333\) −3.29546e8 −0.489059
\(334\) −8.56473e8 −1.25777
\(335\) 2.85437e8 0.414814
\(336\) 1.28948e8 0.185451
\(337\) −9.32816e8 −1.32767 −0.663837 0.747878i \(-0.731073\pi\)
−0.663837 + 0.747878i \(0.731073\pi\)
\(338\) −4.27646e8 −0.602387
\(339\) 3.45596e8 0.481803
\(340\) 5.36281e7 0.0739974
\(341\) −2.50336e7 −0.0341887
\(342\) 4.00017e7 0.0540738
\(343\) 3.35302e8 0.448650
\(344\) −1.82928e8 −0.242284
\(345\) −5.20246e7 −0.0682090
\(346\) −1.44182e8 −0.187130
\(347\) −7.33690e8 −0.942669 −0.471335 0.881955i \(-0.656228\pi\)
−0.471335 + 0.881955i \(0.656228\pi\)
\(348\) −3.12165e8 −0.397061
\(349\) −1.05594e9 −1.32969 −0.664843 0.746983i \(-0.731502\pi\)
−0.664843 + 0.746983i \(0.731502\pi\)
\(350\) −1.45748e8 −0.181704
\(351\) 6.00018e7 0.0740609
\(352\) −4.71875e6 −0.00576670
\(353\) −2.18673e8 −0.264596 −0.132298 0.991210i \(-0.542236\pi\)
−0.132298 + 0.991210i \(0.542236\pi\)
\(354\) −2.60018e8 −0.311524
\(355\) 2.87528e8 0.341099
\(356\) 3.20716e8 0.376743
\(357\) 2.11037e8 0.245482
\(358\) 8.46784e8 0.975398
\(359\) 1.43158e8 0.163300 0.0816500 0.996661i \(-0.473981\pi\)
0.0816500 + 0.996661i \(0.473981\pi\)
\(360\) 4.66560e7 0.0527046
\(361\) 4.70459e7 0.0526316
\(362\) −9.14561e8 −1.01329
\(363\) 5.25594e8 0.576736
\(364\) 2.27481e8 0.247224
\(365\) −1.76337e8 −0.189810
\(366\) 1.32803e8 0.141587
\(367\) 3.29394e7 0.0347844 0.0173922 0.999849i \(-0.494464\pi\)
0.0173922 + 0.999849i \(0.494464\pi\)
\(368\) 6.31386e7 0.0660431
\(369\) 2.46768e8 0.255680
\(370\) −4.52052e8 −0.463962
\(371\) 2.10273e9 2.13784
\(372\) −3.00393e8 −0.302545
\(373\) −4.14192e8 −0.413258 −0.206629 0.978419i \(-0.566249\pi\)
−0.206629 + 0.978419i \(0.566249\pi\)
\(374\) −7.72271e6 −0.00763341
\(375\) −5.27344e7 −0.0516398
\(376\) −2.49175e8 −0.241739
\(377\) −5.50698e8 −0.529321
\(378\) 1.83600e8 0.174845
\(379\) −3.32558e8 −0.313783 −0.156892 0.987616i \(-0.550147\pi\)
−0.156892 + 0.987616i \(0.550147\pi\)
\(380\) 5.48720e7 0.0512989
\(381\) −6.93701e8 −0.642591
\(382\) −2.35168e8 −0.215852
\(383\) 1.62226e9 1.47545 0.737724 0.675102i \(-0.235900\pi\)
0.737724 + 0.675102i \(0.235900\pi\)
\(384\) −5.66231e7 −0.0510310
\(385\) 2.09884e7 0.0187442
\(386\) −3.94109e8 −0.348788
\(387\) −2.60458e8 −0.228428
\(388\) −3.53384e8 −0.307140
\(389\) −1.56167e9 −1.34513 −0.672566 0.740037i \(-0.734808\pi\)
−0.672566 + 0.740037i \(0.734808\pi\)
\(390\) 8.23069e7 0.0702603
\(391\) 1.03333e8 0.0874216
\(392\) 2.74418e8 0.230097
\(393\) −7.41692e8 −0.616382
\(394\) −4.91854e8 −0.405135
\(395\) −7.28303e8 −0.594597
\(396\) −6.71869e6 −0.00543690
\(397\) 1.02018e9 0.818297 0.409149 0.912468i \(-0.365826\pi\)
0.409149 + 0.912468i \(0.365826\pi\)
\(398\) 5.40883e8 0.430044
\(399\) 2.15932e8 0.170181
\(400\) 6.40000e7 0.0500000
\(401\) 1.24502e9 0.964207 0.482104 0.876114i \(-0.339873\pi\)
0.482104 + 0.876114i \(0.339873\pi\)
\(402\) −4.93235e8 −0.378672
\(403\) −5.29930e8 −0.403321
\(404\) −1.22426e9 −0.923717
\(405\) 6.64301e7 0.0496904
\(406\) −1.68509e9 −1.24963
\(407\) 6.50976e7 0.0478613
\(408\) −9.26694e7 −0.0675501
\(409\) 3.36075e8 0.242887 0.121444 0.992598i \(-0.461248\pi\)
0.121444 + 0.992598i \(0.461248\pi\)
\(410\) 3.38502e8 0.242559
\(411\) −1.37662e9 −0.978062
\(412\) −1.20726e9 −0.850476
\(413\) −1.40359e9 −0.980430
\(414\) 8.98985e7 0.0622660
\(415\) 7.06958e8 0.485540
\(416\) −9.98901e7 −0.0680293
\(417\) 9.82647e8 0.663623
\(418\) −7.90183e6 −0.00529189
\(419\) −1.63065e9 −1.08296 −0.541478 0.840715i \(-0.682135\pi\)
−0.541478 + 0.840715i \(0.682135\pi\)
\(420\) 2.51852e8 0.165872
\(421\) 4.71104e8 0.307702 0.153851 0.988094i \(-0.450832\pi\)
0.153851 + 0.988094i \(0.450832\pi\)
\(422\) −5.19861e8 −0.336739
\(423\) −3.54782e8 −0.227914
\(424\) −9.23339e8 −0.588275
\(425\) 1.04742e8 0.0661853
\(426\) −4.96848e8 −0.311380
\(427\) 7.16880e8 0.445604
\(428\) −4.66379e8 −0.287532
\(429\) −1.18526e7 −0.00724791
\(430\) −3.57281e8 −0.216706
\(431\) 3.16809e9 1.90602 0.953008 0.302944i \(-0.0979695\pi\)
0.953008 + 0.302944i \(0.0979695\pi\)
\(432\) −8.06216e7 −0.0481125
\(433\) 1.05873e9 0.626723 0.313362 0.949634i \(-0.398545\pi\)
0.313362 + 0.949634i \(0.398545\pi\)
\(434\) −1.62154e9 −0.952169
\(435\) −6.09698e8 −0.355142
\(436\) −4.31252e8 −0.249189
\(437\) 1.05729e8 0.0606053
\(438\) 3.04710e8 0.173272
\(439\) −4.96163e8 −0.279897 −0.139949 0.990159i \(-0.544694\pi\)
−0.139949 + 0.990159i \(0.544694\pi\)
\(440\) −9.21630e6 −0.00515789
\(441\) 3.90724e8 0.216938
\(442\) −1.63480e8 −0.0900507
\(443\) 1.64604e9 0.899556 0.449778 0.893140i \(-0.351503\pi\)
0.449778 + 0.893140i \(0.351503\pi\)
\(444\) 7.81145e8 0.423537
\(445\) 6.26398e8 0.336969
\(446\) −1.16187e9 −0.620134
\(447\) −1.66931e9 −0.884019
\(448\) −3.05655e8 −0.160605
\(449\) 2.58049e9 1.34536 0.672682 0.739932i \(-0.265142\pi\)
0.672682 + 0.739932i \(0.265142\pi\)
\(450\) 9.11250e7 0.0471405
\(451\) −4.87459e7 −0.0250219
\(452\) −8.19190e8 −0.417254
\(453\) 1.48585e9 0.750985
\(454\) −1.19557e9 −0.599624
\(455\) 4.44298e8 0.221123
\(456\) −9.48188e7 −0.0468293
\(457\) −3.45832e8 −0.169496 −0.0847479 0.996402i \(-0.527008\pi\)
−0.0847479 + 0.996402i \(0.527008\pi\)
\(458\) −5.73528e8 −0.278949
\(459\) −1.31945e8 −0.0636868
\(460\) 1.23318e8 0.0590707
\(461\) −4.77825e8 −0.227152 −0.113576 0.993529i \(-0.536230\pi\)
−0.113576 + 0.993529i \(0.536230\pi\)
\(462\) −3.62679e7 −0.0171110
\(463\) −1.44249e9 −0.675428 −0.337714 0.941249i \(-0.609654\pi\)
−0.337714 + 0.941249i \(0.609654\pi\)
\(464\) 7.39947e8 0.343865
\(465\) −5.86705e8 −0.270604
\(466\) 6.67347e8 0.305493
\(467\) 1.72538e9 0.783928 0.391964 0.919980i \(-0.371796\pi\)
0.391964 + 0.919980i \(0.371796\pi\)
\(468\) −1.42226e8 −0.0641386
\(469\) −2.66252e9 −1.19176
\(470\) −4.86670e8 −0.216218
\(471\) 2.31466e9 1.02074
\(472\) 6.16339e8 0.269788
\(473\) 5.14502e7 0.0223549
\(474\) 1.25851e9 0.542790
\(475\) 1.07172e8 0.0458831
\(476\) −5.00236e8 −0.212594
\(477\) −1.31468e9 −0.554631
\(478\) −2.16742e9 −0.907706
\(479\) 2.28041e9 0.948067 0.474034 0.880507i \(-0.342798\pi\)
0.474034 + 0.880507i \(0.342798\pi\)
\(480\) −1.10592e8 −0.0456435
\(481\) 1.37804e9 0.564616
\(482\) −1.24757e9 −0.507456
\(483\) 4.85278e8 0.195964
\(484\) −1.24585e9 −0.499468
\(485\) −6.90204e8 −0.274714
\(486\) −1.14791e8 −0.0453609
\(487\) 1.27569e9 0.500490 0.250245 0.968183i \(-0.419489\pi\)
0.250245 + 0.968183i \(0.419489\pi\)
\(488\) −3.14793e8 −0.122618
\(489\) −2.39530e9 −0.926360
\(490\) 5.35972e8 0.205805
\(491\) 1.99246e9 0.759635 0.379818 0.925061i \(-0.375987\pi\)
0.379818 + 0.925061i \(0.375987\pi\)
\(492\) −5.84932e8 −0.221425
\(493\) 1.21100e9 0.455176
\(494\) −1.67272e8 −0.0624280
\(495\) −1.31224e7 −0.00486291
\(496\) 7.12043e8 0.262011
\(497\) −2.68202e9 −0.979974
\(498\) −1.22162e9 −0.443236
\(499\) −2.86016e9 −1.03048 −0.515239 0.857046i \(-0.672297\pi\)
−0.515239 + 0.857046i \(0.672297\pi\)
\(500\) 1.25000e8 0.0447214
\(501\) 2.89060e9 1.02696
\(502\) 3.25301e8 0.114768
\(503\) −1.76061e9 −0.616845 −0.308423 0.951249i \(-0.599801\pi\)
−0.308423 + 0.951249i \(0.599801\pi\)
\(504\) −4.35201e8 −0.151420
\(505\) −2.39113e9 −0.826198
\(506\) −1.77583e7 −0.00609361
\(507\) 1.44331e9 0.491847
\(508\) 1.64433e9 0.556501
\(509\) 4.26460e9 1.43340 0.716698 0.697384i \(-0.245653\pi\)
0.716698 + 0.697384i \(0.245653\pi\)
\(510\) −1.80995e8 −0.0604186
\(511\) 1.64485e9 0.545321
\(512\) 1.34218e8 0.0441942
\(513\) −1.35006e8 −0.0441511
\(514\) −7.00133e8 −0.227410
\(515\) −2.35794e9 −0.760689
\(516\) 6.17381e8 0.197824
\(517\) 7.00827e7 0.0223046
\(518\) 4.21668e9 1.33296
\(519\) 4.86613e8 0.152791
\(520\) −1.95098e8 −0.0608472
\(521\) −1.50718e8 −0.0466910 −0.0233455 0.999727i \(-0.507432\pi\)
−0.0233455 + 0.999727i \(0.507432\pi\)
\(522\) 1.05356e9 0.324199
\(523\) −1.88541e9 −0.576301 −0.288150 0.957585i \(-0.593040\pi\)
−0.288150 + 0.957585i \(0.593040\pi\)
\(524\) 1.75809e9 0.533802
\(525\) 4.91899e8 0.148360
\(526\) −2.03086e8 −0.0608458
\(527\) 1.16533e9 0.346826
\(528\) 1.59258e7 0.00470849
\(529\) −3.16721e9 −0.930213
\(530\) −1.80340e9 −0.526169
\(531\) 8.77561e8 0.254359
\(532\) −5.11838e8 −0.147381
\(533\) −1.03189e9 −0.295181
\(534\) −1.08242e9 −0.307610
\(535\) −9.10896e8 −0.257176
\(536\) 1.16915e9 0.327939
\(537\) −2.85789e9 −0.796409
\(538\) −1.02144e9 −0.282796
\(539\) −7.71826e7 −0.0212304
\(540\) −1.57464e8 −0.0430331
\(541\) −2.25346e8 −0.0611870 −0.0305935 0.999532i \(-0.509740\pi\)
−0.0305935 + 0.999532i \(0.509740\pi\)
\(542\) −1.82197e9 −0.491522
\(543\) 3.08664e9 0.827345
\(544\) 2.19661e8 0.0585001
\(545\) −8.42290e8 −0.222881
\(546\) −7.67748e8 −0.201857
\(547\) −7.09512e9 −1.85355 −0.926775 0.375617i \(-0.877431\pi\)
−0.926775 + 0.375617i \(0.877431\pi\)
\(548\) 3.26309e9 0.847026
\(549\) −4.48211e8 −0.115606
\(550\) −1.80006e7 −0.00461336
\(551\) 1.23909e9 0.315552
\(552\) −2.13093e8 −0.0539240
\(553\) 6.79351e9 1.70827
\(554\) −1.61752e8 −0.0404170
\(555\) 1.52567e9 0.378823
\(556\) −2.32924e9 −0.574714
\(557\) 2.69068e7 0.00659735 0.00329868 0.999995i \(-0.498950\pi\)
0.00329868 + 0.999995i \(0.498950\pi\)
\(558\) 1.01383e9 0.247027
\(559\) 1.08914e9 0.263719
\(560\) −5.96983e8 −0.143649
\(561\) 2.60641e7 0.00623266
\(562\) −4.58905e9 −1.09055
\(563\) −1.81117e9 −0.427739 −0.213870 0.976862i \(-0.568607\pi\)
−0.213870 + 0.976862i \(0.568607\pi\)
\(564\) 8.40965e8 0.197379
\(565\) −1.59998e9 −0.373203
\(566\) −4.10068e9 −0.950600
\(567\) −6.19651e8 −0.142760
\(568\) 1.17771e9 0.269663
\(569\) −4.50289e9 −1.02470 −0.512352 0.858775i \(-0.671226\pi\)
−0.512352 + 0.858775i \(0.671226\pi\)
\(570\) −1.85193e8 −0.0418854
\(571\) 8.04967e9 1.80947 0.904736 0.425973i \(-0.140068\pi\)
0.904736 + 0.425973i \(0.140068\pi\)
\(572\) 2.80950e7 0.00627687
\(573\) 7.93693e8 0.176243
\(574\) −3.15750e9 −0.696870
\(575\) 2.40855e8 0.0528345
\(576\) 1.91103e8 0.0416667
\(577\) 8.84269e9 1.91633 0.958163 0.286225i \(-0.0924003\pi\)
0.958163 + 0.286225i \(0.0924003\pi\)
\(578\) −2.92321e9 −0.629670
\(579\) 1.33012e9 0.284784
\(580\) 1.44521e9 0.307562
\(581\) −6.59441e9 −1.39495
\(582\) 1.19267e9 0.250779
\(583\) 2.59698e8 0.0542785
\(584\) −7.22276e8 −0.150058
\(585\) −2.77786e8 −0.0573673
\(586\) 9.98186e8 0.204913
\(587\) 1.89609e9 0.386924 0.193462 0.981108i \(-0.438028\pi\)
0.193462 + 0.981108i \(0.438028\pi\)
\(588\) −9.26160e8 −0.187873
\(589\) 1.19236e9 0.240438
\(590\) 1.20379e9 0.241306
\(591\) 1.66001e9 0.330791
\(592\) −1.85160e9 −0.366794
\(593\) 1.38074e9 0.271906 0.135953 0.990715i \(-0.456590\pi\)
0.135953 + 0.990715i \(0.456590\pi\)
\(594\) 2.26756e7 0.00443921
\(595\) −9.77023e8 −0.190150
\(596\) 3.95689e9 0.765583
\(597\) −1.82548e9 −0.351129
\(598\) −3.75922e8 −0.0718858
\(599\) −7.22264e9 −1.37310 −0.686550 0.727083i \(-0.740876\pi\)
−0.686550 + 0.727083i \(0.740876\pi\)
\(600\) −2.16000e8 −0.0408248
\(601\) 9.06698e9 1.70373 0.851867 0.523758i \(-0.175470\pi\)
0.851867 + 0.523758i \(0.175470\pi\)
\(602\) 3.33267e9 0.622593
\(603\) 1.66467e9 0.309184
\(604\) −3.52202e9 −0.650372
\(605\) −2.43330e9 −0.446738
\(606\) 4.13188e9 0.754212
\(607\) 6.09436e9 1.10603 0.553016 0.833170i \(-0.313477\pi\)
0.553016 + 0.833170i \(0.313477\pi\)
\(608\) 2.24756e8 0.0405554
\(609\) 5.68718e9 1.02032
\(610\) −6.14829e8 −0.109673
\(611\) 1.48357e9 0.263125
\(612\) 3.12759e8 0.0551544
\(613\) −9.22110e9 −1.61686 −0.808428 0.588596i \(-0.799681\pi\)
−0.808428 + 0.588596i \(0.799681\pi\)
\(614\) 5.94733e9 1.03689
\(615\) −1.14244e9 −0.198049
\(616\) 8.59684e7 0.0148186
\(617\) −1.98507e9 −0.340234 −0.170117 0.985424i \(-0.554415\pi\)
−0.170117 + 0.985424i \(0.554415\pi\)
\(618\) 4.07452e9 0.694411
\(619\) 7.48090e9 1.26776 0.633880 0.773432i \(-0.281461\pi\)
0.633880 + 0.773432i \(0.281461\pi\)
\(620\) 1.39071e9 0.234350
\(621\) −3.03407e8 −0.0508400
\(622\) 5.71988e9 0.953060
\(623\) −5.84296e9 −0.968110
\(624\) 3.37129e8 0.0555457
\(625\) 2.44141e8 0.0400000
\(626\) −3.73961e9 −0.609280
\(627\) 2.66687e7 0.00432081
\(628\) −5.48661e9 −0.883986
\(629\) −3.03034e9 −0.485527
\(630\) −8.50001e8 −0.135434
\(631\) 1.88610e9 0.298856 0.149428 0.988773i \(-0.452257\pi\)
0.149428 + 0.988773i \(0.452257\pi\)
\(632\) −2.98313e9 −0.470070
\(633\) 1.75453e9 0.274947
\(634\) 3.52077e8 0.0548688
\(635\) 3.21158e9 0.497749
\(636\) 3.11627e9 0.480325
\(637\) −1.63386e9 −0.250453
\(638\) −2.08117e8 −0.0317275
\(639\) 1.67686e9 0.254240
\(640\) 2.62144e8 0.0395285
\(641\) 4.81042e9 0.721406 0.360703 0.932681i \(-0.382537\pi\)
0.360703 + 0.932681i \(0.382537\pi\)
\(642\) 1.57403e9 0.234769
\(643\) 1.32954e10 1.97225 0.986125 0.166003i \(-0.0530860\pi\)
0.986125 + 0.166003i \(0.0530860\pi\)
\(644\) −1.15029e9 −0.169710
\(645\) 1.20582e9 0.176939
\(646\) 3.67835e8 0.0536833
\(647\) −6.99878e9 −1.01592 −0.507958 0.861382i \(-0.669599\pi\)
−0.507958 + 0.861382i \(0.669599\pi\)
\(648\) 2.72098e8 0.0392837
\(649\) −1.73351e8 −0.0248926
\(650\) −3.81051e8 −0.0544234
\(651\) 5.47271e9 0.777443
\(652\) 5.67776e9 0.802251
\(653\) 9.55199e9 1.34245 0.671224 0.741254i \(-0.265769\pi\)
0.671224 + 0.741254i \(0.265769\pi\)
\(654\) 1.45548e9 0.203462
\(655\) 3.43376e9 0.477447
\(656\) 1.38650e9 0.191760
\(657\) −1.02840e9 −0.141476
\(658\) 4.53959e9 0.621192
\(659\) −1.11552e9 −0.151837 −0.0759183 0.997114i \(-0.524189\pi\)
−0.0759183 + 0.997114i \(0.524189\pi\)
\(660\) 3.11050e7 0.00421140
\(661\) −7.86694e9 −1.05950 −0.529749 0.848154i \(-0.677714\pi\)
−0.529749 + 0.848154i \(0.677714\pi\)
\(662\) 1.57451e9 0.210931
\(663\) 5.51746e8 0.0735261
\(664\) 2.89570e9 0.383853
\(665\) −9.99684e8 −0.131822
\(666\) −2.63637e9 −0.345817
\(667\) 2.78468e9 0.363358
\(668\) −6.85178e9 −0.889377
\(669\) 3.92131e9 0.506337
\(670\) 2.28349e9 0.293318
\(671\) 8.85384e7 0.0113136
\(672\) 1.03159e9 0.131133
\(673\) 6.46874e9 0.818026 0.409013 0.912529i \(-0.365873\pi\)
0.409013 + 0.912529i \(0.365873\pi\)
\(674\) −7.46253e9 −0.938807
\(675\) −3.07547e8 −0.0384900
\(676\) −3.42117e9 −0.425952
\(677\) 3.31062e9 0.410061 0.205031 0.978756i \(-0.434271\pi\)
0.205031 + 0.978756i \(0.434271\pi\)
\(678\) 2.76477e9 0.340686
\(679\) 6.43812e9 0.789251
\(680\) 4.29025e8 0.0523241
\(681\) 4.03505e9 0.489591
\(682\) −2.00269e8 −0.0241751
\(683\) 2.70297e9 0.324615 0.162308 0.986740i \(-0.448106\pi\)
0.162308 + 0.986740i \(0.448106\pi\)
\(684\) 3.20014e8 0.0382360
\(685\) 6.37322e9 0.757603
\(686\) 2.68242e9 0.317243
\(687\) 1.93566e9 0.227761
\(688\) −1.46342e9 −0.171321
\(689\) 5.49748e9 0.640319
\(690\) −4.16197e8 −0.0482311
\(691\) −1.40352e10 −1.61825 −0.809126 0.587636i \(-0.800059\pi\)
−0.809126 + 0.587636i \(0.800059\pi\)
\(692\) −1.15345e9 −0.132321
\(693\) 1.22404e8 0.0139711
\(694\) −5.86952e9 −0.666568
\(695\) −4.54929e9 −0.514040
\(696\) −2.49732e9 −0.280764
\(697\) 2.26915e9 0.253834
\(698\) −8.44750e9 −0.940230
\(699\) −2.25230e9 −0.249434
\(700\) −1.16598e9 −0.128484
\(701\) 3.94768e9 0.432841 0.216421 0.976300i \(-0.430562\pi\)
0.216421 + 0.976300i \(0.430562\pi\)
\(702\) 4.80014e8 0.0523690
\(703\) −3.10062e9 −0.336593
\(704\) −3.77500e7 −0.00407767
\(705\) 1.64251e9 0.176541
\(706\) −1.74938e9 −0.187098
\(707\) 2.23042e10 2.37366
\(708\) −2.08014e9 −0.220281
\(709\) 6.98988e9 0.736559 0.368280 0.929715i \(-0.379947\pi\)
0.368280 + 0.929715i \(0.379947\pi\)
\(710\) 2.30022e9 0.241194
\(711\) −4.24747e9 −0.443186
\(712\) 2.56573e9 0.266398
\(713\) 2.67967e9 0.276865
\(714\) 1.68830e9 0.173582
\(715\) 5.48731e7 0.00561421
\(716\) 6.77427e9 0.689711
\(717\) 7.31504e9 0.741139
\(718\) 1.14527e9 0.115471
\(719\) −8.57873e9 −0.860740 −0.430370 0.902653i \(-0.641617\pi\)
−0.430370 + 0.902653i \(0.641617\pi\)
\(720\) 3.73248e8 0.0372678
\(721\) 2.19945e10 2.18545
\(722\) 3.76367e8 0.0372161
\(723\) 4.21053e9 0.414336
\(724\) −7.31648e9 −0.716502
\(725\) 2.82267e9 0.275092
\(726\) 4.20475e9 0.407814
\(727\) 1.74014e10 1.67963 0.839815 0.542873i \(-0.182664\pi\)
0.839815 + 0.542873i \(0.182664\pi\)
\(728\) 1.81985e9 0.174813
\(729\) 3.87420e8 0.0370370
\(730\) −1.41070e9 −0.134216
\(731\) −2.39504e9 −0.226778
\(732\) 1.06243e9 0.100117
\(733\) −2.57715e9 −0.241700 −0.120850 0.992671i \(-0.538562\pi\)
−0.120850 + 0.992671i \(0.538562\pi\)
\(734\) 2.63515e8 0.0245963
\(735\) −1.80891e9 −0.168039
\(736\) 5.05109e8 0.0466995
\(737\) −3.28834e8 −0.0302580
\(738\) 1.97414e9 0.180793
\(739\) 6.14971e9 0.560531 0.280265 0.959923i \(-0.409578\pi\)
0.280265 + 0.959923i \(0.409578\pi\)
\(740\) −3.61641e9 −0.328070
\(741\) 5.64543e8 0.0509722
\(742\) 1.68218e10 1.51168
\(743\) −1.88648e10 −1.68729 −0.843646 0.536899i \(-0.819595\pi\)
−0.843646 + 0.536899i \(0.819595\pi\)
\(744\) −2.40314e9 −0.213931
\(745\) 7.72830e9 0.684758
\(746\) −3.31354e9 −0.292217
\(747\) 4.12298e9 0.361900
\(748\) −6.17816e7 −0.00539764
\(749\) 8.49671e9 0.738864
\(750\) −4.21875e8 −0.0365148
\(751\) −4.86541e9 −0.419160 −0.209580 0.977792i \(-0.567210\pi\)
−0.209580 + 0.977792i \(0.567210\pi\)
\(752\) −1.99340e9 −0.170935
\(753\) −1.09789e9 −0.0937080
\(754\) −4.40558e9 −0.374286
\(755\) −6.87894e9 −0.581711
\(756\) 1.46880e9 0.123634
\(757\) 7.18694e9 0.602155 0.301077 0.953600i \(-0.402654\pi\)
0.301077 + 0.953600i \(0.402654\pi\)
\(758\) −2.66046e9 −0.221878
\(759\) 5.99343e7 0.00497541
\(760\) 4.38976e8 0.0362738
\(761\) 1.80224e10 1.48240 0.741202 0.671282i \(-0.234256\pi\)
0.741202 + 0.671282i \(0.234256\pi\)
\(762\) −5.54961e9 −0.454381
\(763\) 7.85676e9 0.640335
\(764\) −1.88135e9 −0.152631
\(765\) 6.10858e8 0.0493316
\(766\) 1.29781e10 1.04330
\(767\) −3.66963e9 −0.293656
\(768\) −4.52985e8 −0.0360844
\(769\) −6.91004e8 −0.0547947 −0.0273973 0.999625i \(-0.508722\pi\)
−0.0273973 + 0.999625i \(0.508722\pi\)
\(770\) 1.67907e8 0.0132541
\(771\) 2.36295e9 0.185679
\(772\) −3.15288e9 −0.246630
\(773\) 5.49161e9 0.427633 0.213816 0.976874i \(-0.431411\pi\)
0.213816 + 0.976874i \(0.431411\pi\)
\(774\) −2.08366e9 −0.161523
\(775\) 2.71623e9 0.209609
\(776\) −2.82707e9 −0.217181
\(777\) −1.42313e10 −1.08835
\(778\) −1.24933e10 −0.951152
\(779\) 2.32179e9 0.175971
\(780\) 6.58455e8 0.0496816
\(781\) −3.31243e8 −0.0248810
\(782\) 8.26661e8 0.0618164
\(783\) −3.55576e9 −0.264707
\(784\) 2.19534e9 0.162703
\(785\) −1.07160e10 −0.790661
\(786\) −5.93354e9 −0.435848
\(787\) 1.30151e10 0.951779 0.475889 0.879505i \(-0.342126\pi\)
0.475889 + 0.879505i \(0.342126\pi\)
\(788\) −3.93484e9 −0.286474
\(789\) 6.85417e8 0.0496804
\(790\) −5.82643e9 −0.420443
\(791\) 1.49244e10 1.07221
\(792\) −5.37495e7 −0.00384447
\(793\) 1.87425e9 0.133466
\(794\) 8.16146e9 0.578623
\(795\) 6.08646e9 0.429615
\(796\) 4.32707e9 0.304087
\(797\) −1.41341e10 −0.988930 −0.494465 0.869198i \(-0.664636\pi\)
−0.494465 + 0.869198i \(0.664636\pi\)
\(798\) 1.72745e9 0.120336
\(799\) −3.26240e9 −0.226268
\(800\) 5.12000e8 0.0353553
\(801\) 3.65316e9 0.251162
\(802\) 9.96014e9 0.681797
\(803\) 2.03147e8 0.0138454
\(804\) −3.94588e9 −0.267761
\(805\) −2.24666e9 −0.151793
\(806\) −4.23944e9 −0.285191
\(807\) 3.44735e9 0.230902
\(808\) −9.79408e9 −0.653166
\(809\) −2.02199e10 −1.34264 −0.671318 0.741169i \(-0.734272\pi\)
−0.671318 + 0.741169i \(0.734272\pi\)
\(810\) 5.31441e8 0.0351364
\(811\) 1.48939e10 0.980474 0.490237 0.871589i \(-0.336910\pi\)
0.490237 + 0.871589i \(0.336910\pi\)
\(812\) −1.34807e10 −0.883622
\(813\) 6.14914e9 0.401326
\(814\) 5.20781e8 0.0338431
\(815\) 1.10894e10 0.717555
\(816\) −7.41355e8 −0.0477651
\(817\) −2.45059e9 −0.157215
\(818\) 2.68860e9 0.171747
\(819\) 2.59115e9 0.164816
\(820\) 2.70802e9 0.171515
\(821\) −2.34070e10 −1.47620 −0.738100 0.674691i \(-0.764277\pi\)
−0.738100 + 0.674691i \(0.764277\pi\)
\(822\) −1.10129e10 −0.691594
\(823\) −1.69046e10 −1.05707 −0.528537 0.848911i \(-0.677259\pi\)
−0.528537 + 0.848911i \(0.677259\pi\)
\(824\) −9.65812e9 −0.601378
\(825\) 6.07520e7 0.00376679
\(826\) −1.12288e10 −0.693269
\(827\) −1.99843e10 −1.22862 −0.614312 0.789063i \(-0.710566\pi\)
−0.614312 + 0.789063i \(0.710566\pi\)
\(828\) 7.19188e8 0.0440287
\(829\) −7.50348e9 −0.457427 −0.228714 0.973494i \(-0.573452\pi\)
−0.228714 + 0.973494i \(0.573452\pi\)
\(830\) 5.65567e9 0.343329
\(831\) 5.45912e8 0.0330004
\(832\) −7.99121e8 −0.0481040
\(833\) 3.59290e9 0.215371
\(834\) 7.86118e9 0.469252
\(835\) −1.33824e10 −0.795483
\(836\) −6.32146e7 −0.00374193
\(837\) −3.42167e9 −0.201697
\(838\) −1.30452e10 −0.765766
\(839\) −1.35129e10 −0.789920 −0.394960 0.918698i \(-0.629242\pi\)
−0.394960 + 0.918698i \(0.629242\pi\)
\(840\) 2.01482e9 0.117289
\(841\) 1.53850e10 0.891889
\(842\) 3.76883e9 0.217578
\(843\) 1.54880e10 0.890430
\(844\) −4.15889e9 −0.238111
\(845\) −6.68197e9 −0.380983
\(846\) −2.83826e9 −0.161159
\(847\) 2.26975e10 1.28347
\(848\) −7.38671e9 −0.415973
\(849\) 1.38398e10 0.776162
\(850\) 8.37939e8 0.0468001
\(851\) −6.96824e9 −0.387587
\(852\) −3.97479e9 −0.220179
\(853\) 2.07977e10 1.14734 0.573671 0.819086i \(-0.305519\pi\)
0.573671 + 0.819086i \(0.305519\pi\)
\(854\) 5.73504e9 0.315090
\(855\) 6.25026e8 0.0341993
\(856\) −3.73103e9 −0.203316
\(857\) −1.07072e9 −0.0581091 −0.0290546 0.999578i \(-0.509250\pi\)
−0.0290546 + 0.999578i \(0.509250\pi\)
\(858\) −9.48207e7 −0.00512505
\(859\) −5.19338e9 −0.279559 −0.139780 0.990183i \(-0.544639\pi\)
−0.139780 + 0.990183i \(0.544639\pi\)
\(860\) −2.85825e9 −0.153234
\(861\) 1.06566e10 0.568992
\(862\) 2.53447e10 1.34776
\(863\) 1.63619e10 0.866555 0.433277 0.901261i \(-0.357357\pi\)
0.433277 + 0.901261i \(0.357357\pi\)
\(864\) −6.44973e8 −0.0340207
\(865\) −2.25284e9 −0.118352
\(866\) 8.46981e9 0.443160
\(867\) 9.86584e9 0.514123
\(868\) −1.29723e10 −0.673285
\(869\) 8.39033e8 0.0433721
\(870\) −4.87758e9 −0.251123
\(871\) −6.96102e9 −0.356951
\(872\) −3.45002e9 −0.176203
\(873\) −4.02527e9 −0.204760
\(874\) 8.45835e8 0.0428544
\(875\) −2.27731e9 −0.114920
\(876\) 2.43768e9 0.122522
\(877\) −1.72847e10 −0.865294 −0.432647 0.901563i \(-0.642420\pi\)
−0.432647 + 0.901563i \(0.642420\pi\)
\(878\) −3.96931e9 −0.197917
\(879\) −3.36888e9 −0.167311
\(880\) −7.37304e7 −0.00364718
\(881\) −3.03943e10 −1.49753 −0.748766 0.662834i \(-0.769353\pi\)
−0.748766 + 0.662834i \(0.769353\pi\)
\(882\) 3.12579e9 0.153398
\(883\) −3.74396e10 −1.83007 −0.915036 0.403372i \(-0.867838\pi\)
−0.915036 + 0.403372i \(0.867838\pi\)
\(884\) −1.30784e9 −0.0636755
\(885\) −4.06278e9 −0.197025
\(886\) 1.31684e10 0.636082
\(887\) −2.70761e10 −1.30273 −0.651364 0.758766i \(-0.725803\pi\)
−0.651364 + 0.758766i \(0.725803\pi\)
\(888\) 6.24916e9 0.299486
\(889\) −2.99572e10 −1.43003
\(890\) 5.01119e9 0.238273
\(891\) −7.65300e7 −0.00362460
\(892\) −9.29495e9 −0.438501
\(893\) −3.33807e9 −0.156861
\(894\) −1.33545e10 −0.625096
\(895\) 1.32310e10 0.616896
\(896\) −2.44524e9 −0.113565
\(897\) 1.26874e9 0.0586945
\(898\) 2.06439e10 0.951316
\(899\) 3.14041e10 1.44154
\(900\) 7.29000e8 0.0333333
\(901\) −1.20891e10 −0.550626
\(902\) −3.89967e8 −0.0176932
\(903\) −1.12478e10 −0.508345
\(904\) −6.55352e9 −0.295043
\(905\) −1.42900e10 −0.640859
\(906\) 1.18868e10 0.531027
\(907\) −1.02866e10 −0.457770 −0.228885 0.973454i \(-0.573508\pi\)
−0.228885 + 0.973454i \(0.573508\pi\)
\(908\) −9.56455e9 −0.423998
\(909\) −1.39451e10 −0.615811
\(910\) 3.55439e9 0.156358
\(911\) 1.93021e10 0.845844 0.422922 0.906166i \(-0.361004\pi\)
0.422922 + 0.906166i \(0.361004\pi\)
\(912\) −7.58551e8 −0.0331133
\(913\) −8.14443e8 −0.0354171
\(914\) −2.76666e9 −0.119852
\(915\) 2.07505e9 0.0895477
\(916\) −4.58822e9 −0.197247
\(917\) −3.20296e10 −1.37170
\(918\) −1.05556e9 −0.0450334
\(919\) 2.38844e10 1.01510 0.507551 0.861622i \(-0.330551\pi\)
0.507551 + 0.861622i \(0.330551\pi\)
\(920\) 9.86540e8 0.0417693
\(921\) −2.00722e10 −0.846617
\(922\) −3.82260e9 −0.160620
\(923\) −7.01201e9 −0.293519
\(924\) −2.90143e8 −0.0120993
\(925\) −7.06331e9 −0.293435
\(926\) −1.15399e10 −0.477600
\(927\) −1.37515e10 −0.566984
\(928\) 5.91958e9 0.243149
\(929\) −3.05958e9 −0.125201 −0.0626004 0.998039i \(-0.519939\pi\)
−0.0626004 + 0.998039i \(0.519939\pi\)
\(930\) −4.69364e9 −0.191346
\(931\) 3.67623e9 0.149307
\(932\) 5.33878e9 0.216016
\(933\) −1.93046e10 −0.778170
\(934\) 1.38031e10 0.554321
\(935\) −1.20667e8 −0.00482780
\(936\) −1.13781e9 −0.0453529
\(937\) 3.74978e10 1.48908 0.744538 0.667580i \(-0.232670\pi\)
0.744538 + 0.667580i \(0.232670\pi\)
\(938\) −2.13001e10 −0.842698
\(939\) 1.26212e10 0.497475
\(940\) −3.89336e9 −0.152889
\(941\) −8.71231e9 −0.340855 −0.170427 0.985370i \(-0.554515\pi\)
−0.170427 + 0.985370i \(0.554515\pi\)
\(942\) 1.85173e10 0.721772
\(943\) 5.21791e9 0.202631
\(944\) 4.93071e9 0.190769
\(945\) 2.86875e9 0.110581
\(946\) 4.11601e8 0.0158073
\(947\) 1.30219e10 0.498254 0.249127 0.968471i \(-0.419856\pi\)
0.249127 + 0.968471i \(0.419856\pi\)
\(948\) 1.00681e10 0.383811
\(949\) 4.30037e9 0.163333
\(950\) 8.57375e8 0.0324443
\(951\) −1.18826e9 −0.0448002
\(952\) −4.00189e9 −0.150326
\(953\) 7.85485e9 0.293977 0.146988 0.989138i \(-0.453042\pi\)
0.146988 + 0.989138i \(0.453042\pi\)
\(954\) −1.05174e10 −0.392183
\(955\) −3.67450e9 −0.136517
\(956\) −1.73394e10 −0.641845
\(957\) 7.02395e8 0.0259054
\(958\) 1.82433e10 0.670385
\(959\) −5.94485e10 −2.17659
\(960\) −8.84736e8 −0.0322749
\(961\) 2.70725e9 0.0984002
\(962\) 1.10243e10 0.399244
\(963\) −5.31235e9 −0.191688
\(964\) −9.98052e9 −0.358826
\(965\) −6.15796e9 −0.220593
\(966\) 3.88222e9 0.138567
\(967\) 1.72618e10 0.613893 0.306946 0.951727i \(-0.400693\pi\)
0.306946 + 0.951727i \(0.400693\pi\)
\(968\) −9.96681e9 −0.353177
\(969\) −1.24144e9 −0.0438323
\(970\) −5.52163e9 −0.194252
\(971\) −3.66194e10 −1.28364 −0.641821 0.766854i \(-0.721821\pi\)
−0.641821 + 0.766854i \(0.721821\pi\)
\(972\) −9.18330e8 −0.0320750
\(973\) 4.24352e10 1.47683
\(974\) 1.02056e10 0.353900
\(975\) 1.28605e9 0.0444365
\(976\) −2.51834e9 −0.0867042
\(977\) −3.24882e10 −1.11454 −0.557268 0.830333i \(-0.688150\pi\)
−0.557268 + 0.830333i \(0.688150\pi\)
\(978\) −1.91624e10 −0.655035
\(979\) −7.21635e8 −0.0245798
\(980\) 4.28778e9 0.145526
\(981\) −4.91223e9 −0.166126
\(982\) 1.59397e10 0.537143
\(983\) 5.28599e10 1.77496 0.887480 0.460846i \(-0.152454\pi\)
0.887480 + 0.460846i \(0.152454\pi\)
\(984\) −4.67945e9 −0.156571
\(985\) −7.68523e9 −0.256230
\(986\) 9.68798e9 0.321858
\(987\) −1.53211e10 −0.507201
\(988\) −1.33818e9 −0.0441432
\(989\) −5.50738e9 −0.181033
\(990\) −1.04979e8 −0.00343860
\(991\) −2.55077e10 −0.832555 −0.416277 0.909238i \(-0.636665\pi\)
−0.416277 + 0.909238i \(0.636665\pi\)
\(992\) 5.69634e9 0.185270
\(993\) −5.31396e9 −0.172225
\(994\) −2.14562e10 −0.692947
\(995\) 8.45130e9 0.271984
\(996\) −9.77299e9 −0.313415
\(997\) −8.42787e9 −0.269330 −0.134665 0.990891i \(-0.542996\pi\)
−0.134665 + 0.990891i \(0.542996\pi\)
\(998\) −2.28813e10 −0.728659
\(999\) 8.89773e9 0.282358
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 570.8.a.a.1.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
570.8.a.a.1.1 4 1.1 even 1 trivial