Properties

Label 74.8.a.a.1.2
Level $74$
Weight $8$
Character 74.1
Self dual yes
Analytic conductor $23.116$
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] = [74,8,Mod(1,74)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(74, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("74.1");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 74 = 2 \cdot 37 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 74.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: \(23.1164918858\)
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: \( x^{4} - 2x^{3} - 405x^{2} - 2998x - 4396 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-13.3612\) of defining polynomial
Character \(\chi\) \(=\) 74.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-8.00000 q^{2} -36.0598 q^{3} +64.0000 q^{4} -19.7362 q^{5} +288.479 q^{6} -50.9272 q^{7} -512.000 q^{8} -886.690 q^{9} +O(q^{10})\) \(q-8.00000 q^{2} -36.0598 q^{3} +64.0000 q^{4} -19.7362 q^{5} +288.479 q^{6} -50.9272 q^{7} -512.000 q^{8} -886.690 q^{9} +157.889 q^{10} +5175.62 q^{11} -2307.83 q^{12} +4311.51 q^{13} +407.418 q^{14} +711.683 q^{15} +4096.00 q^{16} +14063.5 q^{17} +7093.52 q^{18} +26643.5 q^{19} -1263.12 q^{20} +1836.43 q^{21} -41405.0 q^{22} -77473.7 q^{23} +18462.6 q^{24} -77735.5 q^{25} -34492.1 q^{26} +110837. q^{27} -3259.34 q^{28} -33124.8 q^{29} -5693.46 q^{30} -172834. q^{31} -32768.0 q^{32} -186632. q^{33} -112508. q^{34} +1005.11 q^{35} -56748.1 q^{36} +50653.0 q^{37} -213148. q^{38} -155472. q^{39} +10104.9 q^{40} -846738. q^{41} -14691.4 q^{42} -307387. q^{43} +331240. q^{44} +17499.9 q^{45} +619790. q^{46} -322473. q^{47} -147701. q^{48} -820949. q^{49} +621884. q^{50} -507128. q^{51} +275937. q^{52} +1.83562e6 q^{53} -886693. q^{54} -102147. q^{55} +26074.7 q^{56} -960758. q^{57} +264998. q^{58} -797340. q^{59} +45547.7 q^{60} -684801. q^{61} +1.38267e6 q^{62} +45156.6 q^{63} +262144. q^{64} -85092.7 q^{65} +1.49306e6 q^{66} -3.79463e6 q^{67} +900066. q^{68} +2.79369e6 q^{69} -8040.87 q^{70} -342906. q^{71} +453985. q^{72} +2.64684e6 q^{73} -405224. q^{74} +2.80313e6 q^{75} +1.70518e6 q^{76} -263580. q^{77} +1.24378e6 q^{78} -2.44136e6 q^{79} -80839.4 q^{80} -2.05756e6 q^{81} +6.77391e6 q^{82} +5.61596e6 q^{83} +117531. q^{84} -277560. q^{85} +2.45910e6 q^{86} +1.19447e6 q^{87} -2.64992e6 q^{88} +8.43578e6 q^{89} -139999. q^{90} -219573. q^{91} -4.95832e6 q^{92} +6.23237e6 q^{93} +2.57979e6 q^{94} -525840. q^{95} +1.18161e6 q^{96} -3.40422e6 q^{97} +6.56760e6 q^{98} -4.58917e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 32 q^{2} - 53 q^{3} + 256 q^{4} + 111 q^{5} + 424 q^{6} - 1666 q^{7} - 2048 q^{8} + 4609 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 32 q^{2} - 53 q^{3} + 256 q^{4} + 111 q^{5} + 424 q^{6} - 1666 q^{7} - 2048 q^{8} + 4609 q^{9} - 888 q^{10} - 4593 q^{11} - 3392 q^{12} + 7847 q^{13} + 13328 q^{14} + 18900 q^{15} + 16384 q^{16} + 23172 q^{17} - 36872 q^{18} + 23696 q^{19} + 7104 q^{20} + 69416 q^{21} + 36744 q^{22} + 24105 q^{23} + 27136 q^{24} - 138149 q^{25} - 62776 q^{26} - 433646 q^{27} - 106624 q^{28} - 140949 q^{29} - 151200 q^{30} - 664609 q^{31} - 131072 q^{32} - 240450 q^{33} - 185376 q^{34} - 248544 q^{35} + 294976 q^{36} + 202612 q^{37} - 189568 q^{38} - 2288827 q^{39} - 56832 q^{40} - 709737 q^{41} - 555328 q^{42} - 128962 q^{43} - 293952 q^{44} - 1755342 q^{45} - 192840 q^{46} - 445842 q^{47} - 217088 q^{48} - 1602774 q^{49} + 1105192 q^{50} - 2883630 q^{51} + 502208 q^{52} - 975870 q^{53} + 3469168 q^{54} - 644145 q^{55} + 852992 q^{56} + 3494630 q^{57} + 1127592 q^{58} - 1812858 q^{59} + 1209600 q^{60} - 2955031 q^{61} + 5316872 q^{62} - 3362482 q^{63} + 1048576 q^{64} + 666 q^{65} + 1923600 q^{66} + 2737235 q^{67} + 1483008 q^{68} - 1781673 q^{69} + 1988352 q^{70} + 4958184 q^{71} - 2359808 q^{72} - 931591 q^{73} - 1620896 q^{74} + 4945810 q^{75} + 1516544 q^{76} + 4352514 q^{77} + 18310616 q^{78} + 5813561 q^{79} + 454656 q^{80} + 16394896 q^{81} + 5677896 q^{82} + 2120460 q^{83} + 4442624 q^{84} - 4845402 q^{85} + 1031696 q^{86} + 7965333 q^{87} + 2351616 q^{88} + 8833716 q^{89} + 14042736 q^{90} - 18886274 q^{91} + 1542720 q^{92} + 3024182 q^{93} + 3566736 q^{94} - 3151794 q^{95} + 1736704 q^{96} - 22666876 q^{97} + 12822192 q^{98} - 17931894 q^{99}+O(q^{100}) \) 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
\(3\) −36.0598 −0.771079 −0.385540 0.922691i \(-0.625985\pi\)
−0.385540 + 0.922691i \(0.625985\pi\)
\(4\) 64.0000 0.500000
\(5\) −19.7362 −0.0706103 −0.0353051 0.999377i \(-0.511240\pi\)
−0.0353051 + 0.999377i \(0.511240\pi\)
\(6\) 288.479 0.545236
\(7\) −50.9272 −0.0561186 −0.0280593 0.999606i \(-0.508933\pi\)
−0.0280593 + 0.999606i \(0.508933\pi\)
\(8\) −512.000 −0.353553
\(9\) −886.690 −0.405436
\(10\) 157.889 0.0499290
\(11\) 5175.62 1.17243 0.586217 0.810154i \(-0.300617\pi\)
0.586217 + 0.810154i \(0.300617\pi\)
\(12\) −2307.83 −0.385540
\(13\) 4311.51 0.544287 0.272143 0.962257i \(-0.412267\pi\)
0.272143 + 0.962257i \(0.412267\pi\)
\(14\) 407.418 0.0396818
\(15\) 711.683 0.0544461
\(16\) 4096.00 0.250000
\(17\) 14063.5 0.694262 0.347131 0.937817i \(-0.387156\pi\)
0.347131 + 0.937817i \(0.387156\pi\)
\(18\) 7093.52 0.286687
\(19\) 26643.5 0.891154 0.445577 0.895244i \(-0.352999\pi\)
0.445577 + 0.895244i \(0.352999\pi\)
\(20\) −1263.12 −0.0353051
\(21\) 1836.43 0.0432719
\(22\) −41405.0 −0.829036
\(23\) −77473.7 −1.32772 −0.663861 0.747856i \(-0.731083\pi\)
−0.663861 + 0.747856i \(0.731083\pi\)
\(24\) 18462.6 0.272618
\(25\) −77735.5 −0.995014
\(26\) −34492.1 −0.384869
\(27\) 110837. 1.08370
\(28\) −3259.34 −0.0280593
\(29\) −33124.8 −0.252209 −0.126104 0.992017i \(-0.540247\pi\)
−0.126104 + 0.992017i \(0.540247\pi\)
\(30\) −5693.46 −0.0384992
\(31\) −172834. −1.04199 −0.520995 0.853560i \(-0.674439\pi\)
−0.520995 + 0.853560i \(0.674439\pi\)
\(32\) −32768.0 −0.176777
\(33\) −186632. −0.904039
\(34\) −112508. −0.490917
\(35\) 1005.11 0.00396255
\(36\) −56748.1 −0.202718
\(37\) 50653.0 0.164399
\(38\) −213148. −0.630141
\(39\) −155472. −0.419688
\(40\) 10104.9 0.0249645
\(41\) −846738. −1.91869 −0.959347 0.282229i \(-0.908926\pi\)
−0.959347 + 0.282229i \(0.908926\pi\)
\(42\) −14691.4 −0.0305978
\(43\) −307387. −0.589585 −0.294792 0.955561i \(-0.595250\pi\)
−0.294792 + 0.955561i \(0.595250\pi\)
\(44\) 331240. 0.586217
\(45\) 17499.9 0.0286280
\(46\) 619790. 0.938841
\(47\) −322473. −0.453056 −0.226528 0.974005i \(-0.572737\pi\)
−0.226528 + 0.974005i \(0.572737\pi\)
\(48\) −147701. −0.192770
\(49\) −820949. −0.996851
\(50\) 621884. 0.703581
\(51\) −507128. −0.535331
\(52\) 275937. 0.272143
\(53\) 1.83562e6 1.69363 0.846814 0.531890i \(-0.178518\pi\)
0.846814 + 0.531890i \(0.178518\pi\)
\(54\) −886693. −0.766294
\(55\) −102147. −0.0827859
\(56\) 26074.7 0.0198409
\(57\) −960758. −0.687151
\(58\) 264998. 0.178338
\(59\) −797340. −0.505430 −0.252715 0.967541i \(-0.581324\pi\)
−0.252715 + 0.967541i \(0.581324\pi\)
\(60\) 45547.7 0.0272231
\(61\) −684801. −0.386287 −0.193144 0.981171i \(-0.561868\pi\)
−0.193144 + 0.981171i \(0.561868\pi\)
\(62\) 1.38267e6 0.736798
\(63\) 45156.6 0.0227525
\(64\) 262144. 0.125000
\(65\) −85092.7 −0.0384322
\(66\) 1.49306e6 0.639252
\(67\) −3.79463e6 −1.54137 −0.770687 0.637214i \(-0.780087\pi\)
−0.770687 + 0.637214i \(0.780087\pi\)
\(68\) 900066. 0.347131
\(69\) 2.79369e6 1.02378
\(70\) −8040.87 −0.00280195
\(71\) −342906. −0.113703 −0.0568514 0.998383i \(-0.518106\pi\)
−0.0568514 + 0.998383i \(0.518106\pi\)
\(72\) 453985. 0.143343
\(73\) 2.64684e6 0.796337 0.398168 0.917312i \(-0.369646\pi\)
0.398168 + 0.917312i \(0.369646\pi\)
\(74\) −405224. −0.116248
\(75\) 2.80313e6 0.767235
\(76\) 1.70518e6 0.445577
\(77\) −263580. −0.0657953
\(78\) 1.24378e6 0.296764
\(79\) −2.44136e6 −0.557104 −0.278552 0.960421i \(-0.589854\pi\)
−0.278552 + 0.960421i \(0.589854\pi\)
\(80\) −80839.4 −0.0176526
\(81\) −2.05756e6 −0.430185
\(82\) 6.77391e6 1.35672
\(83\) 5.61596e6 1.07808 0.539040 0.842280i \(-0.318787\pi\)
0.539040 + 0.842280i \(0.318787\pi\)
\(84\) 117531. 0.0216359
\(85\) −277560. −0.0490220
\(86\) 2.45910e6 0.416899
\(87\) 1.19447e6 0.194473
\(88\) −2.64992e6 −0.414518
\(89\) 8.43578e6 1.26841 0.634206 0.773164i \(-0.281327\pi\)
0.634206 + 0.773164i \(0.281327\pi\)
\(90\) −139999. −0.0202430
\(91\) −219573. −0.0305446
\(92\) −4.95832e6 −0.663861
\(93\) 6.23237e6 0.803457
\(94\) 2.57979e6 0.320359
\(95\) −525840. −0.0629246
\(96\) 1.18161e6 0.136309
\(97\) −3.40422e6 −0.378718 −0.189359 0.981908i \(-0.560641\pi\)
−0.189359 + 0.981908i \(0.560641\pi\)
\(98\) 6.56760e6 0.704880
\(99\) −4.58917e6 −0.475347
\(100\) −4.97507e6 −0.497507
\(101\) −1.62461e6 −0.156901 −0.0784504 0.996918i \(-0.524997\pi\)
−0.0784504 + 0.996918i \(0.524997\pi\)
\(102\) 4.05703e6 0.378536
\(103\) −7.21129e6 −0.650253 −0.325127 0.945671i \(-0.605407\pi\)
−0.325127 + 0.945671i \(0.605407\pi\)
\(104\) −2.20749e6 −0.192434
\(105\) −36244.0 −0.00305544
\(106\) −1.46850e7 −1.19758
\(107\) −1.38534e7 −1.09323 −0.546616 0.837383i \(-0.684084\pi\)
−0.546616 + 0.837383i \(0.684084\pi\)
\(108\) 7.09355e6 0.541852
\(109\) −5.56203e6 −0.411377 −0.205689 0.978617i \(-0.565943\pi\)
−0.205689 + 0.978617i \(0.565943\pi\)
\(110\) 817176. 0.0585384
\(111\) −1.82654e6 −0.126765
\(112\) −208598. −0.0140296
\(113\) −2.13570e7 −1.39241 −0.696203 0.717845i \(-0.745129\pi\)
−0.696203 + 0.717845i \(0.745129\pi\)
\(114\) 7.68606e6 0.485889
\(115\) 1.52903e6 0.0937508
\(116\) −2.11999e6 −0.126104
\(117\) −3.82297e6 −0.220674
\(118\) 6.37872e6 0.357393
\(119\) −716216. −0.0389610
\(120\) −364382. −0.0192496
\(121\) 7.29990e6 0.374600
\(122\) 5.47841e6 0.273146
\(123\) 3.05332e7 1.47947
\(124\) −1.10614e7 −0.520995
\(125\) 3.07609e6 0.140869
\(126\) −361253. −0.0160885
\(127\) −2.51759e7 −1.09062 −0.545308 0.838236i \(-0.683587\pi\)
−0.545308 + 0.838236i \(0.683587\pi\)
\(128\) −2.09715e6 −0.0883883
\(129\) 1.10843e7 0.454617
\(130\) 680742. 0.0271757
\(131\) −2.73598e7 −1.06332 −0.531659 0.846959i \(-0.678431\pi\)
−0.531659 + 0.846959i \(0.678431\pi\)
\(132\) −1.19444e7 −0.452020
\(133\) −1.35688e6 −0.0500103
\(134\) 3.03571e7 1.08992
\(135\) −2.18749e6 −0.0765206
\(136\) −7.20053e6 −0.245459
\(137\) 3.06260e7 1.01758 0.508789 0.860891i \(-0.330093\pi\)
0.508789 + 0.860891i \(0.330093\pi\)
\(138\) −2.23495e7 −0.723921
\(139\) −3.33288e7 −1.05261 −0.526305 0.850296i \(-0.676423\pi\)
−0.526305 + 0.850296i \(0.676423\pi\)
\(140\) 64326.9 0.00198127
\(141\) 1.16283e7 0.349342
\(142\) 2.74325e6 0.0804000
\(143\) 2.23148e7 0.638140
\(144\) −3.63188e6 −0.101359
\(145\) 653756. 0.0178085
\(146\) −2.11747e7 −0.563095
\(147\) 2.96033e7 0.768651
\(148\) 3.24179e6 0.0821995
\(149\) 7.15248e6 0.177135 0.0885675 0.996070i \(-0.471771\pi\)
0.0885675 + 0.996070i \(0.471771\pi\)
\(150\) −2.24250e7 −0.542517
\(151\) −4.28753e7 −1.01342 −0.506708 0.862118i \(-0.669138\pi\)
−0.506708 + 0.862118i \(0.669138\pi\)
\(152\) −1.36414e7 −0.315071
\(153\) −1.24700e7 −0.281479
\(154\) 2.10864e6 0.0465243
\(155\) 3.41108e6 0.0735752
\(156\) −9.95023e6 −0.209844
\(157\) 6.89941e7 1.42286 0.711431 0.702756i \(-0.248047\pi\)
0.711431 + 0.702756i \(0.248047\pi\)
\(158\) 1.95308e7 0.393932
\(159\) −6.61922e7 −1.30592
\(160\) 646715. 0.0124823
\(161\) 3.94552e6 0.0745098
\(162\) 1.64605e7 0.304187
\(163\) −2.76360e7 −0.499826 −0.249913 0.968268i \(-0.580402\pi\)
−0.249913 + 0.968268i \(0.580402\pi\)
\(164\) −5.41912e7 −0.959347
\(165\) 3.68340e6 0.0638345
\(166\) −4.49277e7 −0.762317
\(167\) 4.33284e7 0.719888 0.359944 0.932974i \(-0.382796\pi\)
0.359944 + 0.932974i \(0.382796\pi\)
\(168\) −940250. −0.0152989
\(169\) −4.41594e7 −0.703752
\(170\) 2.22048e6 0.0346638
\(171\) −2.36245e7 −0.361306
\(172\) −1.96728e7 −0.294792
\(173\) 3.02184e7 0.443722 0.221861 0.975078i \(-0.428787\pi\)
0.221861 + 0.975078i \(0.428787\pi\)
\(174\) −9.55579e6 −0.137513
\(175\) 3.95885e6 0.0558388
\(176\) 2.11994e7 0.293108
\(177\) 2.87519e7 0.389727
\(178\) −6.74862e7 −0.896903
\(179\) 1.27627e8 1.66325 0.831624 0.555340i \(-0.187412\pi\)
0.831624 + 0.555340i \(0.187412\pi\)
\(180\) 1.11999e6 0.0143140
\(181\) 4.76747e7 0.597603 0.298802 0.954315i \(-0.403413\pi\)
0.298802 + 0.954315i \(0.403413\pi\)
\(182\) 1.75659e6 0.0215983
\(183\) 2.46938e7 0.297858
\(184\) 3.96665e7 0.469420
\(185\) −999696. −0.0116083
\(186\) −4.98589e7 −0.568130
\(187\) 7.27875e7 0.813976
\(188\) −2.06383e7 −0.226528
\(189\) −5.64460e6 −0.0608159
\(190\) 4.20672e6 0.0444944
\(191\) 8.01658e7 0.832477 0.416239 0.909255i \(-0.363348\pi\)
0.416239 + 0.909255i \(0.363348\pi\)
\(192\) −9.45286e6 −0.0963849
\(193\) 1.63480e6 0.0163687 0.00818436 0.999967i \(-0.497395\pi\)
0.00818436 + 0.999967i \(0.497395\pi\)
\(194\) 2.72337e7 0.267794
\(195\) 3.06843e6 0.0296343
\(196\) −5.25408e7 −0.498425
\(197\) −1.06595e8 −0.993356 −0.496678 0.867935i \(-0.665447\pi\)
−0.496678 + 0.867935i \(0.665447\pi\)
\(198\) 3.67134e7 0.336121
\(199\) 2.24394e7 0.201848 0.100924 0.994894i \(-0.467820\pi\)
0.100924 + 0.994894i \(0.467820\pi\)
\(200\) 3.98006e7 0.351791
\(201\) 1.36834e8 1.18852
\(202\) 1.29969e7 0.110946
\(203\) 1.68695e6 0.0141536
\(204\) −3.24562e7 −0.267665
\(205\) 1.67114e7 0.135480
\(206\) 5.76903e7 0.459798
\(207\) 6.86951e7 0.538307
\(208\) 1.76599e7 0.136072
\(209\) 1.37896e8 1.04482
\(210\) 289952. 0.00216052
\(211\) −2.68426e7 −0.196714 −0.0983571 0.995151i \(-0.531359\pi\)
−0.0983571 + 0.995151i \(0.531359\pi\)
\(212\) 1.17480e8 0.846814
\(213\) 1.23651e7 0.0876739
\(214\) 1.10827e8 0.773032
\(215\) 6.06665e6 0.0416307
\(216\) −5.67484e7 −0.383147
\(217\) 8.80196e6 0.0584750
\(218\) 4.44962e7 0.290888
\(219\) −9.54444e7 −0.614039
\(220\) −6.53741e6 −0.0413929
\(221\) 6.06351e7 0.377877
\(222\) 1.46123e7 0.0896362
\(223\) −1.08818e7 −0.0657102 −0.0328551 0.999460i \(-0.510460\pi\)
−0.0328551 + 0.999460i \(0.510460\pi\)
\(224\) 1.66878e6 0.00992046
\(225\) 6.89272e7 0.403415
\(226\) 1.70856e8 0.984579
\(227\) −1.56786e8 −0.889642 −0.444821 0.895619i \(-0.646733\pi\)
−0.444821 + 0.895619i \(0.646733\pi\)
\(228\) −6.14885e7 −0.343575
\(229\) −1.95778e8 −1.07731 −0.538653 0.842528i \(-0.681067\pi\)
−0.538653 + 0.842528i \(0.681067\pi\)
\(230\) −1.22323e7 −0.0662918
\(231\) 9.50465e6 0.0507334
\(232\) 1.69599e7 0.0891692
\(233\) 1.45843e8 0.755337 0.377668 0.925941i \(-0.376726\pi\)
0.377668 + 0.925941i \(0.376726\pi\)
\(234\) 3.05838e7 0.156040
\(235\) 6.36439e6 0.0319904
\(236\) −5.10297e7 −0.252715
\(237\) 8.80348e7 0.429571
\(238\) 5.72973e6 0.0275496
\(239\) −3.52653e8 −1.67092 −0.835458 0.549555i \(-0.814797\pi\)
−0.835458 + 0.549555i \(0.814797\pi\)
\(240\) 2.91505e6 0.0136115
\(241\) −3.09151e8 −1.42269 −0.711346 0.702842i \(-0.751914\pi\)
−0.711346 + 0.702842i \(0.751914\pi\)
\(242\) −5.83992e7 −0.264882
\(243\) −1.68205e8 −0.751997
\(244\) −4.38273e7 −0.193144
\(245\) 1.62024e7 0.0703879
\(246\) −2.44266e8 −1.04614
\(247\) 1.14874e8 0.485043
\(248\) 8.84911e7 0.368399
\(249\) −2.02511e8 −0.831285
\(250\) −2.46087e7 −0.0996091
\(251\) −1.96899e8 −0.785932 −0.392966 0.919553i \(-0.628551\pi\)
−0.392966 + 0.919553i \(0.628551\pi\)
\(252\) 2.89002e6 0.0113763
\(253\) −4.00975e8 −1.55666
\(254\) 2.01407e8 0.771182
\(255\) 1.00088e7 0.0377999
\(256\) 1.67772e7 0.0625000
\(257\) 3.14559e8 1.15594 0.577972 0.816057i \(-0.303844\pi\)
0.577972 + 0.816057i \(0.303844\pi\)
\(258\) −8.86746e7 −0.321463
\(259\) −2.57962e6 −0.00922584
\(260\) −5.44593e6 −0.0192161
\(261\) 2.93714e7 0.102255
\(262\) 2.18878e8 0.751879
\(263\) 5.69257e7 0.192958 0.0964792 0.995335i \(-0.469242\pi\)
0.0964792 + 0.995335i \(0.469242\pi\)
\(264\) 9.55556e7 0.319626
\(265\) −3.62282e7 −0.119587
\(266\) 1.08550e7 0.0353626
\(267\) −3.04193e8 −0.978046
\(268\) −2.42857e8 −0.770687
\(269\) −3.56640e7 −0.111711 −0.0558557 0.998439i \(-0.517789\pi\)
−0.0558557 + 0.998439i \(0.517789\pi\)
\(270\) 1.74999e7 0.0541082
\(271\) −5.25367e8 −1.60350 −0.801752 0.597657i \(-0.796099\pi\)
−0.801752 + 0.597657i \(0.796099\pi\)
\(272\) 5.76042e7 0.173565
\(273\) 7.91777e6 0.0235523
\(274\) −2.45008e8 −0.719536
\(275\) −4.02330e8 −1.16659
\(276\) 1.78796e8 0.511889
\(277\) −6.45650e8 −1.82523 −0.912616 0.408818i \(-0.865941\pi\)
−0.912616 + 0.408818i \(0.865941\pi\)
\(278\) 2.66630e8 0.744308
\(279\) 1.53250e8 0.422461
\(280\) −514615. −0.00140097
\(281\) 4.38689e8 1.17946 0.589731 0.807599i \(-0.299234\pi\)
0.589731 + 0.807599i \(0.299234\pi\)
\(282\) −9.30267e7 −0.247022
\(283\) 1.91165e8 0.501367 0.250683 0.968069i \(-0.419345\pi\)
0.250683 + 0.968069i \(0.419345\pi\)
\(284\) −2.19460e7 −0.0568514
\(285\) 1.89617e7 0.0485199
\(286\) −1.78518e8 −0.451233
\(287\) 4.31220e7 0.107674
\(288\) 2.90550e7 0.0716717
\(289\) −2.12556e8 −0.518001
\(290\) −5.23005e6 −0.0125925
\(291\) 1.22755e8 0.292022
\(292\) 1.69397e8 0.398168
\(293\) 6.78441e8 1.57571 0.787854 0.615862i \(-0.211192\pi\)
0.787854 + 0.615862i \(0.211192\pi\)
\(294\) −2.36826e8 −0.543518
\(295\) 1.57364e7 0.0356886
\(296\) −2.59343e7 −0.0581238
\(297\) 5.73649e8 1.27057
\(298\) −5.72198e7 −0.125253
\(299\) −3.34029e8 −0.722661
\(300\) 1.79400e8 0.383618
\(301\) 1.56544e7 0.0330867
\(302\) 3.43002e8 0.716593
\(303\) 5.85833e7 0.120983
\(304\) 1.09132e8 0.222789
\(305\) 1.35154e7 0.0272758
\(306\) 9.97599e7 0.199036
\(307\) −4.40928e8 −0.869728 −0.434864 0.900496i \(-0.643203\pi\)
−0.434864 + 0.900496i \(0.643203\pi\)
\(308\) −1.68691e7 −0.0328977
\(309\) 2.60038e8 0.501397
\(310\) −2.72887e7 −0.0520255
\(311\) −6.01820e8 −1.13450 −0.567251 0.823545i \(-0.691993\pi\)
−0.567251 + 0.823545i \(0.691993\pi\)
\(312\) 7.96018e7 0.148382
\(313\) 6.38537e8 1.17701 0.588506 0.808493i \(-0.299716\pi\)
0.588506 + 0.808493i \(0.299716\pi\)
\(314\) −5.51953e8 −1.00612
\(315\) −891219. −0.00160656
\(316\) −1.56247e8 −0.278552
\(317\) 6.62098e8 1.16739 0.583693 0.811974i \(-0.301607\pi\)
0.583693 + 0.811974i \(0.301607\pi\)
\(318\) 5.29538e8 0.923426
\(319\) −1.71441e8 −0.295698
\(320\) −5.17372e6 −0.00882628
\(321\) 4.99550e8 0.842969
\(322\) −3.15642e7 −0.0526864
\(323\) 3.74701e8 0.618694
\(324\) −1.31684e8 −0.215092
\(325\) −3.35157e8 −0.541573
\(326\) 2.21088e8 0.353430
\(327\) 2.00566e8 0.317205
\(328\) 4.33530e8 0.678361
\(329\) 1.64227e7 0.0254248
\(330\) −2.94672e7 −0.0451378
\(331\) −4.62478e8 −0.700960 −0.350480 0.936570i \(-0.613982\pi\)
−0.350480 + 0.936570i \(0.613982\pi\)
\(332\) 3.59422e8 0.539040
\(333\) −4.49135e7 −0.0666533
\(334\) −3.46627e8 −0.509037
\(335\) 7.48915e7 0.108837
\(336\) 7.52200e6 0.0108180
\(337\) 6.65983e8 0.947891 0.473946 0.880554i \(-0.342829\pi\)
0.473946 + 0.880554i \(0.342829\pi\)
\(338\) 3.53275e8 0.497628
\(339\) 7.70130e8 1.07366
\(340\) −1.77639e7 −0.0245110
\(341\) −8.94524e8 −1.22166
\(342\) 1.88996e8 0.255482
\(343\) 8.37494e7 0.112060
\(344\) 1.57382e8 0.208450
\(345\) −5.51367e7 −0.0722893
\(346\) −2.41747e8 −0.313759
\(347\) 3.49930e8 0.449601 0.224801 0.974405i \(-0.427827\pi\)
0.224801 + 0.974405i \(0.427827\pi\)
\(348\) 7.64463e7 0.0972364
\(349\) 9.83350e8 1.23828 0.619140 0.785280i \(-0.287481\pi\)
0.619140 + 0.785280i \(0.287481\pi\)
\(350\) −3.16708e7 −0.0394840
\(351\) 4.77874e8 0.589845
\(352\) −1.69595e8 −0.207259
\(353\) −6.25026e8 −0.756287 −0.378144 0.925747i \(-0.623437\pi\)
−0.378144 + 0.925747i \(0.623437\pi\)
\(354\) −2.30015e8 −0.275579
\(355\) 6.76766e6 0.00802859
\(356\) 5.39890e8 0.634206
\(357\) 2.58266e7 0.0300420
\(358\) −1.02102e9 −1.17609
\(359\) −6.83539e8 −0.779709 −0.389855 0.920876i \(-0.627475\pi\)
−0.389855 + 0.920876i \(0.627475\pi\)
\(360\) −8.95993e6 −0.0101215
\(361\) −1.83998e8 −0.205844
\(362\) −3.81398e8 −0.422569
\(363\) −2.63233e8 −0.288847
\(364\) −1.40527e7 −0.0152723
\(365\) −5.22384e7 −0.0562296
\(366\) −1.97551e8 −0.210617
\(367\) −5.50838e8 −0.581691 −0.290846 0.956770i \(-0.593937\pi\)
−0.290846 + 0.956770i \(0.593937\pi\)
\(368\) −3.17332e8 −0.331930
\(369\) 7.50794e8 0.777908
\(370\) 7.99757e6 0.00820828
\(371\) −9.34831e7 −0.0950440
\(372\) 3.98872e8 0.401729
\(373\) 1.03057e9 1.02824 0.514121 0.857717i \(-0.328118\pi\)
0.514121 + 0.857717i \(0.328118\pi\)
\(374\) −5.82300e8 −0.575568
\(375\) −1.10923e8 −0.108621
\(376\) 1.65106e8 0.160179
\(377\) −1.42818e8 −0.137274
\(378\) 4.51568e7 0.0430033
\(379\) −1.53319e8 −0.144663 −0.0723317 0.997381i \(-0.523044\pi\)
−0.0723317 + 0.997381i \(0.523044\pi\)
\(380\) −3.36537e7 −0.0314623
\(381\) 9.07839e8 0.840952
\(382\) −6.41327e8 −0.588650
\(383\) −1.20085e9 −1.09218 −0.546089 0.837727i \(-0.683884\pi\)
−0.546089 + 0.837727i \(0.683884\pi\)
\(384\) 7.56229e7 0.0681544
\(385\) 5.20206e6 0.00464583
\(386\) −1.30784e7 −0.0115744
\(387\) 2.72557e8 0.239039
\(388\) −2.17870e8 −0.189359
\(389\) 9.04660e8 0.779223 0.389611 0.920979i \(-0.372609\pi\)
0.389611 + 0.920979i \(0.372609\pi\)
\(390\) −2.45474e7 −0.0209546
\(391\) −1.08955e9 −0.921786
\(392\) 4.20326e8 0.352440
\(393\) 9.86589e8 0.819903
\(394\) 8.52760e8 0.702409
\(395\) 4.81830e7 0.0393373
\(396\) −2.93707e8 −0.237674
\(397\) 5.60100e8 0.449261 0.224630 0.974444i \(-0.427882\pi\)
0.224630 + 0.974444i \(0.427882\pi\)
\(398\) −1.79515e8 −0.142728
\(399\) 4.89287e7 0.0385619
\(400\) −3.18405e8 −0.248754
\(401\) 1.75894e9 1.36222 0.681108 0.732183i \(-0.261498\pi\)
0.681108 + 0.732183i \(0.261498\pi\)
\(402\) −1.09467e9 −0.840412
\(403\) −7.45176e8 −0.567141
\(404\) −1.03975e8 −0.0784504
\(405\) 4.06084e7 0.0303755
\(406\) −1.34956e7 −0.0100081
\(407\) 2.62161e8 0.192747
\(408\) 2.59650e8 0.189268
\(409\) 3.48105e8 0.251582 0.125791 0.992057i \(-0.459853\pi\)
0.125791 + 0.992057i \(0.459853\pi\)
\(410\) −1.33691e8 −0.0957985
\(411\) −1.10437e9 −0.784633
\(412\) −4.61522e8 −0.325127
\(413\) 4.06063e7 0.0283640
\(414\) −5.49561e8 −0.380640
\(415\) −1.10838e8 −0.0761235
\(416\) −1.41280e8 −0.0962172
\(417\) 1.20183e9 0.811647
\(418\) −1.10317e9 −0.738799
\(419\) −9.10860e8 −0.604927 −0.302463 0.953161i \(-0.597809\pi\)
−0.302463 + 0.953161i \(0.597809\pi\)
\(420\) −2.31962e6 −0.00152772
\(421\) −2.03192e9 −1.32715 −0.663574 0.748111i \(-0.730961\pi\)
−0.663574 + 0.748111i \(0.730961\pi\)
\(422\) 2.14741e8 0.139098
\(423\) 2.85934e8 0.183685
\(424\) −9.39839e8 −0.598788
\(425\) −1.09324e9 −0.690800
\(426\) −9.89211e7 −0.0619948
\(427\) 3.48750e7 0.0216779
\(428\) −8.86616e8 −0.546616
\(429\) −8.04666e8 −0.492057
\(430\) −4.85332e7 −0.0294374
\(431\) 1.21720e9 0.732304 0.366152 0.930555i \(-0.380675\pi\)
0.366152 + 0.930555i \(0.380675\pi\)
\(432\) 4.53987e8 0.270926
\(433\) −1.40845e9 −0.833745 −0.416872 0.908965i \(-0.636874\pi\)
−0.416872 + 0.908965i \(0.636874\pi\)
\(434\) −7.04157e7 −0.0413481
\(435\) −2.35743e7 −0.0137318
\(436\) −3.55970e8 −0.205689
\(437\) −2.06417e9 −1.18320
\(438\) 7.63555e8 0.434191
\(439\) 1.96037e9 1.10589 0.552946 0.833217i \(-0.313504\pi\)
0.552946 + 0.833217i \(0.313504\pi\)
\(440\) 5.22993e7 0.0292692
\(441\) 7.27927e8 0.404160
\(442\) −4.85081e8 −0.267200
\(443\) 2.51266e9 1.37316 0.686578 0.727056i \(-0.259112\pi\)
0.686578 + 0.727056i \(0.259112\pi\)
\(444\) −1.16898e8 −0.0633823
\(445\) −1.66490e8 −0.0895629
\(446\) 8.70543e7 0.0464641
\(447\) −2.57917e8 −0.136585
\(448\) −1.33503e7 −0.00701482
\(449\) −1.67892e9 −0.875324 −0.437662 0.899140i \(-0.644193\pi\)
−0.437662 + 0.899140i \(0.644193\pi\)
\(450\) −5.51418e8 −0.285257
\(451\) −4.38240e9 −2.24954
\(452\) −1.36685e9 −0.696203
\(453\) 1.54607e9 0.781424
\(454\) 1.25428e9 0.629072
\(455\) 4.33354e6 0.00215676
\(456\) 4.91908e8 0.242944
\(457\) −6.46678e8 −0.316943 −0.158472 0.987364i \(-0.550657\pi\)
−0.158472 + 0.987364i \(0.550657\pi\)
\(458\) 1.56622e9 0.761770
\(459\) 1.55876e9 0.752373
\(460\) 9.78582e7 0.0468754
\(461\) 1.32962e9 0.632083 0.316041 0.948745i \(-0.397646\pi\)
0.316041 + 0.948745i \(0.397646\pi\)
\(462\) −7.60372e7 −0.0358739
\(463\) 3.34991e9 1.56856 0.784278 0.620409i \(-0.213033\pi\)
0.784278 + 0.620409i \(0.213033\pi\)
\(464\) −1.35679e8 −0.0630522
\(465\) −1.23003e8 −0.0567323
\(466\) −1.16675e9 −0.534104
\(467\) 6.13048e8 0.278539 0.139269 0.990255i \(-0.455525\pi\)
0.139269 + 0.990255i \(0.455525\pi\)
\(468\) −2.44670e8 −0.110337
\(469\) 1.93250e8 0.0864997
\(470\) −5.09151e7 −0.0226206
\(471\) −2.48791e9 −1.09714
\(472\) 4.08238e8 0.178697
\(473\) −1.59092e9 −0.691249
\(474\) −7.04279e8 −0.303753
\(475\) −2.07114e9 −0.886711
\(476\) −4.58379e7 −0.0194805
\(477\) −1.62763e9 −0.686658
\(478\) 2.82122e9 1.18152
\(479\) −3.20691e9 −1.33325 −0.666627 0.745392i \(-0.732262\pi\)
−0.666627 + 0.745392i \(0.732262\pi\)
\(480\) −2.33204e7 −0.00962481
\(481\) 2.18391e8 0.0894802
\(482\) 2.47321e9 1.00599
\(483\) −1.42275e8 −0.0574530
\(484\) 4.67194e8 0.187300
\(485\) 6.71862e7 0.0267414
\(486\) 1.34564e9 0.531742
\(487\) 2.67973e9 1.05133 0.525665 0.850692i \(-0.323817\pi\)
0.525665 + 0.850692i \(0.323817\pi\)
\(488\) 3.50618e8 0.136573
\(489\) 9.96550e8 0.385406
\(490\) −1.29619e8 −0.0497718
\(491\) 1.74917e9 0.666879 0.333439 0.942772i \(-0.391791\pi\)
0.333439 + 0.942772i \(0.391791\pi\)
\(492\) 1.95413e9 0.739733
\(493\) −4.65851e8 −0.175099
\(494\) −9.18988e8 −0.342978
\(495\) 9.05727e7 0.0335644
\(496\) −7.07929e8 −0.260498
\(497\) 1.74633e7 0.00638084
\(498\) 1.62008e9 0.587807
\(499\) −2.21759e9 −0.798967 −0.399484 0.916740i \(-0.630811\pi\)
−0.399484 + 0.916740i \(0.630811\pi\)
\(500\) 1.96870e8 0.0704343
\(501\) −1.56241e9 −0.555091
\(502\) 1.57519e9 0.555738
\(503\) 4.21833e9 1.47792 0.738962 0.673747i \(-0.235316\pi\)
0.738962 + 0.673747i \(0.235316\pi\)
\(504\) −2.31202e7 −0.00804423
\(505\) 3.20637e7 0.0110788
\(506\) 3.20780e9 1.10073
\(507\) 1.59238e9 0.542649
\(508\) −1.61126e9 −0.545308
\(509\) −4.92334e9 −1.65481 −0.827404 0.561607i \(-0.810183\pi\)
−0.827404 + 0.561607i \(0.810183\pi\)
\(510\) −8.00702e7 −0.0267285
\(511\) −1.34796e8 −0.0446893
\(512\) −1.34218e8 −0.0441942
\(513\) 2.95307e9 0.965747
\(514\) −2.51647e9 −0.817376
\(515\) 1.42323e8 0.0459146
\(516\) 7.09397e8 0.227308
\(517\) −1.66900e9 −0.531178
\(518\) 2.06369e7 0.00652365
\(519\) −1.08967e9 −0.342145
\(520\) 4.35675e7 0.0135878
\(521\) 3.66580e9 1.13563 0.567814 0.823157i \(-0.307789\pi\)
0.567814 + 0.823157i \(0.307789\pi\)
\(522\) −2.34971e8 −0.0723049
\(523\) 5.07711e9 1.55189 0.775945 0.630800i \(-0.217273\pi\)
0.775945 + 0.630800i \(0.217273\pi\)
\(524\) −1.75103e9 −0.531659
\(525\) −1.42755e8 −0.0430561
\(526\) −4.55406e8 −0.136442
\(527\) −2.43066e9 −0.723414
\(528\) −7.64445e8 −0.226010
\(529\) 2.59735e9 0.762843
\(530\) 2.89825e8 0.0845611
\(531\) 7.06993e8 0.204920
\(532\) −8.68401e7 −0.0250052
\(533\) −3.65072e9 −1.04432
\(534\) 2.43354e9 0.691583
\(535\) 2.73413e8 0.0771934
\(536\) 1.94285e9 0.544958
\(537\) −4.60221e9 −1.28250
\(538\) 2.85312e8 0.0789919
\(539\) −4.24892e9 −1.16874
\(540\) −1.39999e8 −0.0382603
\(541\) 5.97271e9 1.62174 0.810870 0.585226i \(-0.198994\pi\)
0.810870 + 0.585226i \(0.198994\pi\)
\(542\) 4.20293e9 1.13385
\(543\) −1.71914e9 −0.460800
\(544\) −4.60834e8 −0.122729
\(545\) 1.09773e8 0.0290475
\(546\) −6.33422e7 −0.0166540
\(547\) −4.63928e9 −1.21198 −0.605990 0.795473i \(-0.707223\pi\)
−0.605990 + 0.795473i \(0.707223\pi\)
\(548\) 1.96006e9 0.508789
\(549\) 6.07206e8 0.156615
\(550\) 3.21864e9 0.824902
\(551\) −8.82558e8 −0.224757
\(552\) −1.43037e9 −0.361960
\(553\) 1.24331e8 0.0312639
\(554\) 5.16520e9 1.29063
\(555\) 3.60489e7 0.00895089
\(556\) −2.13304e9 −0.526305
\(557\) −2.87674e9 −0.705354 −0.352677 0.935745i \(-0.614729\pi\)
−0.352677 + 0.935745i \(0.614729\pi\)
\(558\) −1.22600e9 −0.298725
\(559\) −1.32530e9 −0.320903
\(560\) 4.11692e6 0.000990637 0
\(561\) −2.62471e9 −0.627640
\(562\) −3.50951e9 −0.834006
\(563\) 5.48798e9 1.29608 0.648042 0.761604i \(-0.275588\pi\)
0.648042 + 0.761604i \(0.275588\pi\)
\(564\) 7.44213e8 0.174671
\(565\) 4.21505e8 0.0983181
\(566\) −1.52932e9 −0.354520
\(567\) 1.04786e8 0.0241414
\(568\) 1.75568e8 0.0402000
\(569\) 5.47367e9 1.24562 0.622810 0.782373i \(-0.285991\pi\)
0.622810 + 0.782373i \(0.285991\pi\)
\(570\) −1.51693e8 −0.0343088
\(571\) 1.73348e9 0.389665 0.194833 0.980837i \(-0.437584\pi\)
0.194833 + 0.980837i \(0.437584\pi\)
\(572\) 1.42814e9 0.319070
\(573\) −2.89076e9 −0.641906
\(574\) −3.44976e8 −0.0761373
\(575\) 6.02246e9 1.32110
\(576\) −2.32440e8 −0.0506796
\(577\) −2.24479e8 −0.0486475 −0.0243238 0.999704i \(-0.507743\pi\)
−0.0243238 + 0.999704i \(0.507743\pi\)
\(578\) 1.70045e9 0.366282
\(579\) −5.89507e7 −0.0126216
\(580\) 4.18404e7 0.00890426
\(581\) −2.86005e8 −0.0605003
\(582\) −9.82043e8 −0.206490
\(583\) 9.50049e9 1.98567
\(584\) −1.35518e9 −0.281548
\(585\) 7.54508e7 0.0155818
\(586\) −5.42753e9 −1.11419
\(587\) 2.44680e9 0.499305 0.249652 0.968336i \(-0.419684\pi\)
0.249652 + 0.968336i \(0.419684\pi\)
\(588\) 1.89461e9 0.384326
\(589\) −4.60490e9 −0.928574
\(590\) −1.25891e8 −0.0252356
\(591\) 3.84380e9 0.765956
\(592\) 2.07475e8 0.0410997
\(593\) −1.85980e9 −0.366248 −0.183124 0.983090i \(-0.558621\pi\)
−0.183124 + 0.983090i \(0.558621\pi\)
\(594\) −4.58919e9 −0.898429
\(595\) 1.41354e7 0.00275105
\(596\) 4.57759e8 0.0885675
\(597\) −8.09161e8 −0.155641
\(598\) 2.67223e9 0.510998
\(599\) −1.26622e8 −0.0240722 −0.0120361 0.999928i \(-0.503831\pi\)
−0.0120361 + 0.999928i \(0.503831\pi\)
\(600\) −1.43520e9 −0.271259
\(601\) −1.07228e9 −0.201488 −0.100744 0.994912i \(-0.532122\pi\)
−0.100744 + 0.994912i \(0.532122\pi\)
\(602\) −1.25235e8 −0.0233958
\(603\) 3.36466e9 0.624929
\(604\) −2.74402e9 −0.506708
\(605\) −1.44072e8 −0.0264506
\(606\) −4.68666e8 −0.0855479
\(607\) −5.73945e9 −1.04162 −0.520811 0.853672i \(-0.674370\pi\)
−0.520811 + 0.853672i \(0.674370\pi\)
\(608\) −8.73053e8 −0.157535
\(609\) −6.08312e7 −0.0109135
\(610\) −1.08123e8 −0.0192869
\(611\) −1.39035e9 −0.246592
\(612\) −7.98079e8 −0.140739
\(613\) 8.64474e8 0.151579 0.0757897 0.997124i \(-0.475852\pi\)
0.0757897 + 0.997124i \(0.475852\pi\)
\(614\) 3.52742e9 0.614990
\(615\) −6.02609e8 −0.104465
\(616\) 1.34953e8 0.0232622
\(617\) −8.52653e9 −1.46142 −0.730708 0.682690i \(-0.760810\pi\)
−0.730708 + 0.682690i \(0.760810\pi\)
\(618\) −2.08030e9 −0.354541
\(619\) 3.14258e9 0.532561 0.266281 0.963896i \(-0.414205\pi\)
0.266281 + 0.963896i \(0.414205\pi\)
\(620\) 2.18309e8 0.0367876
\(621\) −8.58693e9 −1.43886
\(622\) 4.81456e9 0.802214
\(623\) −4.29611e8 −0.0711815
\(624\) −6.36815e8 −0.104922
\(625\) 6.01237e9 0.985067
\(626\) −5.10830e9 −0.832273
\(627\) −4.97252e9 −0.805639
\(628\) 4.41562e9 0.711431
\(629\) 7.12360e8 0.114136
\(630\) 7.12975e6 0.00113601
\(631\) 4.61616e9 0.731439 0.365719 0.930725i \(-0.380823\pi\)
0.365719 + 0.930725i \(0.380823\pi\)
\(632\) 1.24997e9 0.196966
\(633\) 9.67939e8 0.151682
\(634\) −5.29678e9 −0.825467
\(635\) 4.96876e8 0.0770087
\(636\) −4.23630e9 −0.652961
\(637\) −3.53953e9 −0.542573
\(638\) 1.37153e9 0.209090
\(639\) 3.04051e8 0.0460993
\(640\) 4.13898e7 0.00624113
\(641\) −1.00510e10 −1.50732 −0.753658 0.657266i \(-0.771713\pi\)
−0.753658 + 0.657266i \(0.771713\pi\)
\(642\) −3.99640e9 −0.596069
\(643\) −3.42743e9 −0.508429 −0.254214 0.967148i \(-0.581817\pi\)
−0.254214 + 0.967148i \(0.581817\pi\)
\(644\) 2.52513e8 0.0372549
\(645\) −2.18762e8 −0.0321006
\(646\) −2.99761e9 −0.437483
\(647\) −1.28014e9 −0.185821 −0.0929103 0.995674i \(-0.529617\pi\)
−0.0929103 + 0.995674i \(0.529617\pi\)
\(648\) 1.05347e9 0.152093
\(649\) −4.12673e9 −0.592584
\(650\) 2.68126e9 0.382950
\(651\) −3.17397e8 −0.0450889
\(652\) −1.76870e9 −0.249913
\(653\) 6.73100e9 0.945983 0.472992 0.881067i \(-0.343174\pi\)
0.472992 + 0.881067i \(0.343174\pi\)
\(654\) −1.60453e9 −0.224298
\(655\) 5.39977e8 0.0750812
\(656\) −3.46824e9 −0.479673
\(657\) −2.34692e9 −0.322864
\(658\) −1.31381e8 −0.0179781
\(659\) −8.07982e9 −1.09977 −0.549886 0.835240i \(-0.685329\pi\)
−0.549886 + 0.835240i \(0.685329\pi\)
\(660\) 2.35738e8 0.0319172
\(661\) −1.58791e8 −0.0213856 −0.0106928 0.999943i \(-0.503404\pi\)
−0.0106928 + 0.999943i \(0.503404\pi\)
\(662\) 3.69982e9 0.495653
\(663\) −2.18649e9 −0.291374
\(664\) −2.87537e9 −0.381159
\(665\) 2.67796e7 0.00353124
\(666\) 3.59308e8 0.0471310
\(667\) 2.56630e9 0.334863
\(668\) 2.77302e9 0.359944
\(669\) 3.92395e8 0.0506678
\(670\) −5.99132e8 −0.0769593
\(671\) −3.54427e9 −0.452896
\(672\) −6.01760e7 −0.00764946
\(673\) −7.20617e8 −0.0911280 −0.0455640 0.998961i \(-0.514509\pi\)
−0.0455640 + 0.998961i \(0.514509\pi\)
\(674\) −5.32787e9 −0.670260
\(675\) −8.61594e9 −1.07830
\(676\) −2.82620e9 −0.351876
\(677\) −8.42691e9 −1.04378 −0.521889 0.853013i \(-0.674773\pi\)
−0.521889 + 0.853013i \(0.674773\pi\)
\(678\) −6.16104e9 −0.759189
\(679\) 1.73367e8 0.0212531
\(680\) 1.42111e8 0.0173319
\(681\) 5.65366e9 0.685985
\(682\) 7.15620e9 0.863847
\(683\) 5.36376e9 0.644164 0.322082 0.946712i \(-0.395617\pi\)
0.322082 + 0.946712i \(0.395617\pi\)
\(684\) −1.51197e9 −0.180653
\(685\) −6.04439e8 −0.0718515
\(686\) −6.69995e8 −0.0792387
\(687\) 7.05971e9 0.830688
\(688\) −1.25906e9 −0.147396
\(689\) 7.91431e9 0.921819
\(690\) 4.41094e8 0.0511162
\(691\) 1.48364e10 1.71063 0.855313 0.518112i \(-0.173365\pi\)
0.855313 + 0.518112i \(0.173365\pi\)
\(692\) 1.93398e9 0.221861
\(693\) 2.33714e8 0.0266758
\(694\) −2.79944e9 −0.317916
\(695\) 6.57783e8 0.0743252
\(696\) −6.11570e8 −0.0687565
\(697\) −1.19081e10 −1.33208
\(698\) −7.86680e9 −0.875597
\(699\) −5.25908e9 −0.582425
\(700\) 2.53366e8 0.0279194
\(701\) 6.52624e9 0.715566 0.357783 0.933805i \(-0.383533\pi\)
0.357783 + 0.933805i \(0.383533\pi\)
\(702\) −3.82299e9 −0.417084
\(703\) 1.34957e9 0.146505
\(704\) 1.35676e9 0.146554
\(705\) −2.29499e8 −0.0246671
\(706\) 5.00021e9 0.534776
\(707\) 8.27371e7 0.00880506
\(708\) 1.84012e9 0.194864
\(709\) 1.69533e10 1.78646 0.893228 0.449603i \(-0.148435\pi\)
0.893228 + 0.449603i \(0.148435\pi\)
\(710\) −5.41412e7 −0.00567707
\(711\) 2.16472e9 0.225870
\(712\) −4.31912e9 −0.448451
\(713\) 1.33901e10 1.38347
\(714\) −2.06613e8 −0.0212429
\(715\) −4.40408e8 −0.0450592
\(716\) 8.16813e9 0.831624
\(717\) 1.27166e10 1.28841
\(718\) 5.46831e9 0.551338
\(719\) −4.30200e9 −0.431637 −0.215819 0.976433i \(-0.569242\pi\)
−0.215819 + 0.976433i \(0.569242\pi\)
\(720\) 7.16794e7 0.00715700
\(721\) 3.67251e8 0.0364913
\(722\) 1.47199e9 0.145554
\(723\) 1.11479e10 1.09701
\(724\) 3.05118e9 0.298802
\(725\) 2.57497e9 0.250951
\(726\) 2.10586e9 0.204245
\(727\) 1.60836e10 1.55243 0.776217 0.630465i \(-0.217136\pi\)
0.776217 + 0.630465i \(0.217136\pi\)
\(728\) 1.12422e8 0.0107991
\(729\) 1.05653e10 1.01003
\(730\) 4.17907e8 0.0397603
\(731\) −4.32295e9 −0.409326
\(732\) 1.58040e9 0.148929
\(733\) 1.61253e9 0.151232 0.0756162 0.997137i \(-0.475908\pi\)
0.0756162 + 0.997137i \(0.475908\pi\)
\(734\) 4.40670e9 0.411318
\(735\) −5.84256e8 −0.0542747
\(736\) 2.53866e9 0.234710
\(737\) −1.96396e10 −1.80716
\(738\) −6.00635e9 −0.550064
\(739\) 5.21771e9 0.475581 0.237790 0.971317i \(-0.423577\pi\)
0.237790 + 0.971317i \(0.423577\pi\)
\(740\) −6.39806e7 −0.00580413
\(741\) −4.14232e9 −0.374007
\(742\) 7.47865e8 0.0672062
\(743\) 3.55715e9 0.318157 0.159078 0.987266i \(-0.449148\pi\)
0.159078 + 0.987266i \(0.449148\pi\)
\(744\) −3.19097e9 −0.284065
\(745\) −1.41163e8 −0.0125076
\(746\) −8.24454e9 −0.727077
\(747\) −4.97961e9 −0.437093
\(748\) 4.65840e9 0.406988
\(749\) 7.05514e8 0.0613507
\(750\) 8.87386e8 0.0768065
\(751\) 2.02483e10 1.74441 0.872205 0.489140i \(-0.162689\pi\)
0.872205 + 0.489140i \(0.162689\pi\)
\(752\) −1.32085e9 −0.113264
\(753\) 7.10014e9 0.606016
\(754\) 1.14254e9 0.0970672
\(755\) 8.46194e8 0.0715576
\(756\) −3.61255e8 −0.0304079
\(757\) 1.87776e10 1.57327 0.786636 0.617417i \(-0.211821\pi\)
0.786636 + 0.617417i \(0.211821\pi\)
\(758\) 1.22655e9 0.102292
\(759\) 1.44591e10 1.20031
\(760\) 2.69230e8 0.0222472
\(761\) −9.48144e9 −0.779880 −0.389940 0.920840i \(-0.627504\pi\)
−0.389940 + 0.920840i \(0.627504\pi\)
\(762\) −7.26271e9 −0.594643
\(763\) 2.83259e8 0.0230859
\(764\) 5.13061e9 0.416239
\(765\) 2.46110e8 0.0198753
\(766\) 9.60680e9 0.772286
\(767\) −3.43774e9 −0.275099
\(768\) −6.04983e8 −0.0481925
\(769\) −7.67008e9 −0.608216 −0.304108 0.952638i \(-0.598358\pi\)
−0.304108 + 0.952638i \(0.598358\pi\)
\(770\) −4.16165e7 −0.00328509
\(771\) −1.13430e10 −0.891324
\(772\) 1.04627e8 0.00818436
\(773\) −7.33496e9 −0.571175 −0.285588 0.958353i \(-0.592189\pi\)
−0.285588 + 0.958353i \(0.592189\pi\)
\(774\) −2.18046e9 −0.169026
\(775\) 1.34353e10 1.03680
\(776\) 1.74296e9 0.133897
\(777\) 9.30205e7 0.00711386
\(778\) −7.23728e9 −0.550994
\(779\) −2.25600e10 −1.70985
\(780\) 1.96379e8 0.0148172
\(781\) −1.77475e9 −0.133309
\(782\) 8.71643e9 0.651801
\(783\) −3.67144e9 −0.273319
\(784\) −3.36261e9 −0.249213
\(785\) −1.36168e9 −0.100469
\(786\) −7.89271e9 −0.579759
\(787\) −3.05525e9 −0.223427 −0.111713 0.993740i \(-0.535634\pi\)
−0.111713 + 0.993740i \(0.535634\pi\)
\(788\) −6.82208e9 −0.496678
\(789\) −2.05273e9 −0.148786
\(790\) −3.85464e8 −0.0278156
\(791\) 1.08765e9 0.0781398
\(792\) 2.34966e9 0.168061
\(793\) −2.95253e9 −0.210251
\(794\) −4.48080e9 −0.317675
\(795\) 1.30638e9 0.0922115
\(796\) 1.43612e9 0.100924
\(797\) −1.30056e10 −0.909971 −0.454985 0.890499i \(-0.650355\pi\)
−0.454985 + 0.890499i \(0.650355\pi\)
\(798\) −3.91430e8 −0.0272674
\(799\) −4.53512e9 −0.314539
\(800\) 2.54724e9 0.175895
\(801\) −7.47992e9 −0.514260
\(802\) −1.40715e10 −0.963233
\(803\) 1.36990e10 0.933652
\(804\) 8.75736e9 0.594261
\(805\) −7.78695e7 −0.00526116
\(806\) 5.96141e9 0.401030
\(807\) 1.28604e9 0.0861383
\(808\) 8.31802e8 0.0554728
\(809\) −2.37212e10 −1.57513 −0.787565 0.616231i \(-0.788659\pi\)
−0.787565 + 0.616231i \(0.788659\pi\)
\(810\) −3.24867e8 −0.0214787
\(811\) −1.28296e10 −0.844577 −0.422288 0.906462i \(-0.638773\pi\)
−0.422288 + 0.906462i \(0.638773\pi\)
\(812\) 1.07965e8 0.00707680
\(813\) 1.89446e10 1.23643
\(814\) −2.09729e9 −0.136293
\(815\) 5.45429e8 0.0352929
\(816\) −2.07720e9 −0.133833
\(817\) −8.18986e9 −0.525411
\(818\) −2.78484e9 −0.177895
\(819\) 1.94693e8 0.0123839
\(820\) 1.06953e9 0.0677398
\(821\) −7.10512e9 −0.448096 −0.224048 0.974578i \(-0.571927\pi\)
−0.224048 + 0.974578i \(0.571927\pi\)
\(822\) 8.83493e9 0.554820
\(823\) 3.31606e9 0.207359 0.103680 0.994611i \(-0.466938\pi\)
0.103680 + 0.994611i \(0.466938\pi\)
\(824\) 3.69218e9 0.229899
\(825\) 1.45079e10 0.899532
\(826\) −3.24850e8 −0.0200564
\(827\) −2.02601e10 −1.24558 −0.622791 0.782388i \(-0.714001\pi\)
−0.622791 + 0.782388i \(0.714001\pi\)
\(828\) 4.39649e9 0.269153
\(829\) 3.28928e9 0.200521 0.100261 0.994961i \(-0.468032\pi\)
0.100261 + 0.994961i \(0.468032\pi\)
\(830\) 8.86701e8 0.0538274
\(831\) 2.32820e10 1.40740
\(832\) 1.13024e9 0.0680358
\(833\) −1.15454e10 −0.692075
\(834\) −9.61464e9 −0.573921
\(835\) −8.55136e8 −0.0508315
\(836\) 8.82537e9 0.522410
\(837\) −1.91564e10 −1.12921
\(838\) 7.28688e9 0.427748
\(839\) −1.33201e10 −0.778646 −0.389323 0.921101i \(-0.627291\pi\)
−0.389323 + 0.921101i \(0.627291\pi\)
\(840\) 1.85569e7 0.00108026
\(841\) −1.61526e10 −0.936391
\(842\) 1.62554e10 0.938435
\(843\) −1.58190e10 −0.909460
\(844\) −1.71793e9 −0.0983571
\(845\) 8.71537e8 0.0496921
\(846\) −2.28747e9 −0.129885
\(847\) −3.71764e8 −0.0210220
\(848\) 7.51871e9 0.423407
\(849\) −6.89337e9 −0.386593
\(850\) 8.74588e9 0.488469
\(851\) −3.92427e9 −0.218276
\(852\) 7.91369e8 0.0438369
\(853\) −2.73074e10 −1.50646 −0.753231 0.657756i \(-0.771506\pi\)
−0.753231 + 0.657756i \(0.771506\pi\)
\(854\) −2.79000e8 −0.0153286
\(855\) 4.66257e8 0.0255119
\(856\) 7.09293e9 0.386516
\(857\) 1.77629e10 0.964008 0.482004 0.876169i \(-0.339909\pi\)
0.482004 + 0.876169i \(0.339909\pi\)
\(858\) 6.43733e9 0.347937
\(859\) 3.55499e10 1.91365 0.956825 0.290664i \(-0.0938761\pi\)
0.956825 + 0.290664i \(0.0938761\pi\)
\(860\) 3.88266e8 0.0208154
\(861\) −1.55497e9 −0.0830255
\(862\) −9.73760e9 −0.517817
\(863\) −1.54322e10 −0.817317 −0.408658 0.912687i \(-0.634003\pi\)
−0.408658 + 0.912687i \(0.634003\pi\)
\(864\) −3.63190e9 −0.191573
\(865\) −5.96396e8 −0.0313313
\(866\) 1.12676e10 0.589546
\(867\) 7.66472e9 0.399420
\(868\) 5.63326e8 0.0292375
\(869\) −1.26355e10 −0.653167
\(870\) 1.88595e8 0.00970984
\(871\) −1.63606e10 −0.838950
\(872\) 2.84776e9 0.145444
\(873\) 3.01848e9 0.153546
\(874\) 1.65133e10 0.836652
\(875\) −1.56657e8 −0.00790534
\(876\) −6.10844e9 −0.307020
\(877\) 2.45939e10 1.23120 0.615599 0.788059i \(-0.288914\pi\)
0.615599 + 0.788059i \(0.288914\pi\)
\(878\) −1.56830e10 −0.781984
\(879\) −2.44645e10 −1.21500
\(880\) −4.18394e8 −0.0206965
\(881\) −3.24805e10 −1.60032 −0.800159 0.599788i \(-0.795252\pi\)
−0.800159 + 0.599788i \(0.795252\pi\)
\(882\) −5.82342e9 −0.285784
\(883\) −1.24274e10 −0.607460 −0.303730 0.952758i \(-0.598232\pi\)
−0.303730 + 0.952758i \(0.598232\pi\)
\(884\) 3.88064e9 0.188939
\(885\) −5.67453e8 −0.0275187
\(886\) −2.01013e10 −0.970968
\(887\) −2.06132e10 −0.991774 −0.495887 0.868387i \(-0.665157\pi\)
−0.495887 + 0.868387i \(0.665157\pi\)
\(888\) 9.35187e8 0.0448181
\(889\) 1.28214e9 0.0612039
\(890\) 1.33192e9 0.0633305
\(891\) −1.06492e10 −0.504363
\(892\) −6.96434e8 −0.0328551
\(893\) −8.59181e9 −0.403742
\(894\) 2.06334e9 0.0965803
\(895\) −2.51887e9 −0.117442
\(896\) 1.06802e8 0.00496023
\(897\) 1.20450e10 0.557229
\(898\) 1.34314e10 0.618947
\(899\) 5.72509e9 0.262799
\(900\) 4.41134e9 0.201708
\(901\) 2.58153e10 1.17582
\(902\) 3.50592e10 1.59067
\(903\) −5.64494e8 −0.0255125
\(904\) 1.09348e10 0.492290
\(905\) −9.40916e8 −0.0421969
\(906\) −1.23686e10 −0.552550
\(907\) −2.15584e10 −0.959380 −0.479690 0.877438i \(-0.659251\pi\)
−0.479690 + 0.877438i \(0.659251\pi\)
\(908\) −1.00343e10 −0.444821
\(909\) 1.44053e9 0.0636133
\(910\) −3.46683e7 −0.00152506
\(911\) 2.47542e10 1.08476 0.542381 0.840132i \(-0.317523\pi\)
0.542381 + 0.840132i \(0.317523\pi\)
\(912\) −3.93526e9 −0.171788
\(913\) 2.90661e10 1.26398
\(914\) 5.17342e9 0.224113
\(915\) −4.87361e8 −0.0210318
\(916\) −1.25298e10 −0.538653
\(917\) 1.39336e9 0.0596719
\(918\) −1.24700e10 −0.532008
\(919\) 3.82049e10 1.62373 0.811866 0.583843i \(-0.198452\pi\)
0.811866 + 0.583843i \(0.198452\pi\)
\(920\) −7.82866e8 −0.0331459
\(921\) 1.58998e10 0.670629
\(922\) −1.06370e10 −0.446950
\(923\) −1.47844e9 −0.0618869
\(924\) 6.08297e8 0.0253667
\(925\) −3.93754e9 −0.163579
\(926\) −2.67993e10 −1.10914
\(927\) 6.39417e9 0.263636
\(928\) 1.08543e9 0.0445846
\(929\) −1.51912e10 −0.621636 −0.310818 0.950469i \(-0.600603\pi\)
−0.310818 + 0.950469i \(0.600603\pi\)
\(930\) 9.84025e8 0.0401158
\(931\) −2.18729e10 −0.888348
\(932\) 9.33397e9 0.377668
\(933\) 2.17015e10 0.874791
\(934\) −4.90438e9 −0.196957
\(935\) −1.43655e9 −0.0574750
\(936\) 1.95736e9 0.0780199
\(937\) −2.93618e10 −1.16599 −0.582995 0.812476i \(-0.698119\pi\)
−0.582995 + 0.812476i \(0.698119\pi\)
\(938\) −1.54600e9 −0.0611646
\(939\) −2.30255e10 −0.907570
\(940\) 4.07321e8 0.0159952
\(941\) −1.88409e10 −0.737119 −0.368559 0.929604i \(-0.620149\pi\)
−0.368559 + 0.929604i \(0.620149\pi\)
\(942\) 1.99033e10 0.775795
\(943\) 6.55999e10 2.54749
\(944\) −3.26590e9 −0.126358
\(945\) 1.11403e8 0.00429423
\(946\) 1.27274e10 0.488787
\(947\) −1.73098e10 −0.662319 −0.331159 0.943575i \(-0.607440\pi\)
−0.331159 + 0.943575i \(0.607440\pi\)
\(948\) 5.63423e9 0.214786
\(949\) 1.14119e10 0.433436
\(950\) 1.65691e10 0.626999
\(951\) −2.38751e10 −0.900148
\(952\) 3.66703e8 0.0137748
\(953\) 1.22640e10 0.458993 0.229497 0.973309i \(-0.426292\pi\)
0.229497 + 0.973309i \(0.426292\pi\)
\(954\) 1.30210e10 0.485541
\(955\) −1.58217e9 −0.0587815
\(956\) −2.25698e10 −0.835458
\(957\) 6.18214e9 0.228007
\(958\) 2.56553e10 0.942753
\(959\) −1.55969e9 −0.0571050
\(960\) 1.86563e8 0.00680577
\(961\) 2.35903e9 0.0857436
\(962\) −1.74713e9 −0.0632721
\(963\) 1.22836e10 0.443236
\(964\) −1.97857e10 −0.711346
\(965\) −3.22647e7 −0.00115580
\(966\) 1.13820e9 0.0406254
\(967\) −2.65054e10 −0.942632 −0.471316 0.881964i \(-0.656221\pi\)
−0.471316 + 0.881964i \(0.656221\pi\)
\(968\) −3.73755e9 −0.132441
\(969\) −1.35116e10 −0.477062
\(970\) −5.37489e8 −0.0189090
\(971\) −4.44819e10 −1.55925 −0.779626 0.626246i \(-0.784591\pi\)
−0.779626 + 0.626246i \(0.784591\pi\)
\(972\) −1.07651e10 −0.375998
\(973\) 1.69734e9 0.0590710
\(974\) −2.14378e10 −0.743403
\(975\) 1.20857e10 0.417596
\(976\) −2.80495e9 −0.0965718
\(977\) 2.66074e10 0.912792 0.456396 0.889777i \(-0.349140\pi\)
0.456396 + 0.889777i \(0.349140\pi\)
\(978\) −7.97240e9 −0.272523
\(979\) 4.36604e10 1.48713
\(980\) 1.03695e9 0.0351940
\(981\) 4.93179e9 0.166787
\(982\) −1.39934e10 −0.471554
\(983\) 3.98658e10 1.33864 0.669319 0.742975i \(-0.266586\pi\)
0.669319 + 0.742975i \(0.266586\pi\)
\(984\) −1.56330e10 −0.523070
\(985\) 2.10378e9 0.0701411
\(986\) 3.72681e9 0.123814
\(987\) −5.92199e8 −0.0196046
\(988\) 7.35191e9 0.242522
\(989\) 2.38144e10 0.782804
\(990\) −7.24581e8 −0.0237336
\(991\) −1.96883e10 −0.642613 −0.321306 0.946975i \(-0.604122\pi\)
−0.321306 + 0.946975i \(0.604122\pi\)
\(992\) 5.66343e9 0.184200
\(993\) 1.66769e10 0.540496
\(994\) −1.39706e8 −0.00451194
\(995\) −4.42868e8 −0.0142526
\(996\) −1.29607e10 −0.415643
\(997\) 7.78796e9 0.248880 0.124440 0.992227i \(-0.460287\pi\)
0.124440 + 0.992227i \(0.460287\pi\)
\(998\) 1.77407e10 0.564955
\(999\) 5.61421e9 0.178160
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 74.8.a.a.1.2 4
4.3 odd 2 592.8.a.b.1.3 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
74.8.a.a.1.2 4 1.1 even 1 trivial
592.8.a.b.1.3 4 4.3 odd 2