Properties

Label 207.8.a.e.1.6
Level $207$
Weight $8$
Character 207.1
Self dual yes
Analytic conductor $64.664$
Analytic rank $1$
Dimension $8$
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] = [207,8,Mod(1,207)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(207, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("207.1");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 207 = 3^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 207.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: \(64.6637002752\)
Analytic rank: \(1\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 757x^{6} - 1170x^{5} + 170343x^{4} + 424132x^{3} - 9973075x^{2} - 5161010x + 130545120 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{6}\cdot 3^{2} \)
Twist minimal: no (minimal twist has level 69)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.6
Root \(5.38924\) of defining polynomial
Character \(\chi\) \(=\) 207.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+2.38924 q^{2} -122.292 q^{4} +147.281 q^{5} -1223.93 q^{7} -598.006 q^{8} +O(q^{10})\) \(q+2.38924 q^{2} -122.292 q^{4} +147.281 q^{5} -1223.93 q^{7} -598.006 q^{8} +351.889 q^{10} +1637.33 q^{11} +12894.7 q^{13} -2924.26 q^{14} +14224.5 q^{16} +12052.3 q^{17} +27609.1 q^{19} -18011.2 q^{20} +3911.97 q^{22} -12167.0 q^{23} -56433.4 q^{25} +30808.6 q^{26} +149676. q^{28} -240721. q^{29} -128196. q^{31} +110531. q^{32} +28795.9 q^{34} -180261. q^{35} +179159. q^{37} +65964.8 q^{38} -88074.8 q^{40} -473457. q^{41} +936296. q^{43} -200232. q^{44} -29069.9 q^{46} -796871. q^{47} +674461. q^{49} -134833. q^{50} -1.57692e6 q^{52} +2.04568e6 q^{53} +241147. q^{55} +731918. q^{56} -575140. q^{58} -1.90819e6 q^{59} -3.32363e6 q^{61} -306291. q^{62} -1.55666e6 q^{64} +1.89915e6 q^{65} -285366. q^{67} -1.47390e6 q^{68} -430687. q^{70} -3.91888e6 q^{71} -3.52247e6 q^{73} +428053. q^{74} -3.37636e6 q^{76} -2.00398e6 q^{77} +1.10676e6 q^{79} +2.09500e6 q^{80} -1.13120e6 q^{82} +3.81576e6 q^{83} +1.77508e6 q^{85} +2.23703e6 q^{86} -979134. q^{88} -5.29119e6 q^{89} -1.57823e7 q^{91} +1.48792e6 q^{92} -1.90392e6 q^{94} +4.06629e6 q^{95} +6.41098e6 q^{97} +1.61145e6 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 24 q^{2} + 562 q^{4} - 378 q^{5} + 126 q^{7} - 4188 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 24 q^{2} + 562 q^{4} - 378 q^{5} + 126 q^{7} - 4188 q^{8} + 11720 q^{10} - 6932 q^{11} + 12404 q^{13} - 30222 q^{14} + 27058 q^{16} - 24434 q^{17} - 14682 q^{19} + 3760 q^{20} + 36294 q^{22} - 97336 q^{23} + 144644 q^{25} - 325840 q^{26} - 21566 q^{28} - 255356 q^{29} + 450764 q^{31} - 647588 q^{32} + 191822 q^{34} - 1022616 q^{35} + 206240 q^{37} - 737372 q^{38} + 590028 q^{40} - 1053344 q^{41} + 1587806 q^{43} - 589366 q^{44} + 292008 q^{46} - 443336 q^{47} + 1944828 q^{49} + 1556112 q^{50} - 614236 q^{52} + 375530 q^{53} + 407792 q^{55} + 1316922 q^{56} - 1413384 q^{58} - 624008 q^{59} - 2005568 q^{61} + 3908272 q^{62} - 5082310 q^{64} - 646124 q^{65} - 2712286 q^{67} + 2289698 q^{68} - 16499468 q^{70} + 6287176 q^{71} - 10358312 q^{73} + 2000150 q^{74} - 25107464 q^{76} + 2156840 q^{77} - 8800574 q^{79} - 2384344 q^{80} - 31799800 q^{82} - 384948 q^{83} - 17826684 q^{85} + 11563928 q^{86} - 25202782 q^{88} + 3445530 q^{89} - 16316740 q^{91} - 6837854 q^{92} - 24237616 q^{94} - 26164288 q^{95} - 28043764 q^{97} + 9998012 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.38924 0.211181 0.105590 0.994410i \(-0.466327\pi\)
0.105590 + 0.994410i \(0.466327\pi\)
\(3\) 0 0
\(4\) −122.292 −0.955403
\(5\) 147.281 0.526927 0.263464 0.964669i \(-0.415135\pi\)
0.263464 + 0.964669i \(0.415135\pi\)
\(6\) 0 0
\(7\) −1223.93 −1.34869 −0.674347 0.738415i \(-0.735575\pi\)
−0.674347 + 0.738415i \(0.735575\pi\)
\(8\) −598.006 −0.412944
\(9\) 0 0
\(10\) 351.889 0.111277
\(11\) 1637.33 0.370904 0.185452 0.982653i \(-0.440625\pi\)
0.185452 + 0.982653i \(0.440625\pi\)
\(12\) 0 0
\(13\) 12894.7 1.62784 0.813919 0.580978i \(-0.197330\pi\)
0.813919 + 0.580978i \(0.197330\pi\)
\(14\) −2924.26 −0.284818
\(15\) 0 0
\(16\) 14224.5 0.868197
\(17\) 12052.3 0.594977 0.297489 0.954725i \(-0.403851\pi\)
0.297489 + 0.954725i \(0.403851\pi\)
\(18\) 0 0
\(19\) 27609.1 0.923454 0.461727 0.887022i \(-0.347230\pi\)
0.461727 + 0.887022i \(0.347230\pi\)
\(20\) −18011.2 −0.503428
\(21\) 0 0
\(22\) 3911.97 0.0783279
\(23\) −12167.0 −0.208514
\(24\) 0 0
\(25\) −56433.4 −0.722348
\(26\) 30808.6 0.343768
\(27\) 0 0
\(28\) 149676. 1.28855
\(29\) −240721. −1.83283 −0.916413 0.400235i \(-0.868929\pi\)
−0.916413 + 0.400235i \(0.868929\pi\)
\(30\) 0 0
\(31\) −128196. −0.772874 −0.386437 0.922316i \(-0.626294\pi\)
−0.386437 + 0.922316i \(0.626294\pi\)
\(32\) 110531. 0.596290
\(33\) 0 0
\(34\) 28795.9 0.125648
\(35\) −180261. −0.710664
\(36\) 0 0
\(37\) 179159. 0.581476 0.290738 0.956803i \(-0.406099\pi\)
0.290738 + 0.956803i \(0.406099\pi\)
\(38\) 65964.8 0.195016
\(39\) 0 0
\(40\) −88074.8 −0.217591
\(41\) −473457. −1.07285 −0.536423 0.843949i \(-0.680225\pi\)
−0.536423 + 0.843949i \(0.680225\pi\)
\(42\) 0 0
\(43\) 936296. 1.79586 0.897932 0.440135i \(-0.145069\pi\)
0.897932 + 0.440135i \(0.145069\pi\)
\(44\) −200232. −0.354363
\(45\) 0 0
\(46\) −29069.9 −0.0440343
\(47\) −796871. −1.11956 −0.559778 0.828643i \(-0.689113\pi\)
−0.559778 + 0.828643i \(0.689113\pi\)
\(48\) 0 0
\(49\) 674461. 0.818975
\(50\) −134833. −0.152546
\(51\) 0 0
\(52\) −1.57692e6 −1.55524
\(53\) 2.04568e6 1.88743 0.943717 0.330755i \(-0.107303\pi\)
0.943717 + 0.330755i \(0.107303\pi\)
\(54\) 0 0
\(55\) 241147. 0.195440
\(56\) 731918. 0.556935
\(57\) 0 0
\(58\) −575140. −0.387058
\(59\) −1.90819e6 −1.20959 −0.604796 0.796380i \(-0.706746\pi\)
−0.604796 + 0.796380i \(0.706746\pi\)
\(60\) 0 0
\(61\) −3.32363e6 −1.87481 −0.937407 0.348236i \(-0.886781\pi\)
−0.937407 + 0.348236i \(0.886781\pi\)
\(62\) −306291. −0.163216
\(63\) 0 0
\(64\) −1.55666e6 −0.742272
\(65\) 1.89915e6 0.857752
\(66\) 0 0
\(67\) −285366. −0.115915 −0.0579576 0.998319i \(-0.518459\pi\)
−0.0579576 + 0.998319i \(0.518459\pi\)
\(68\) −1.47390e6 −0.568443
\(69\) 0 0
\(70\) −430687. −0.150079
\(71\) −3.91888e6 −1.29944 −0.649722 0.760172i \(-0.725115\pi\)
−0.649722 + 0.760172i \(0.725115\pi\)
\(72\) 0 0
\(73\) −3.52247e6 −1.05978 −0.529891 0.848065i \(-0.677767\pi\)
−0.529891 + 0.848065i \(0.677767\pi\)
\(74\) 428053. 0.122797
\(75\) 0 0
\(76\) −3.37636e6 −0.882270
\(77\) −2.00398e6 −0.500236
\(78\) 0 0
\(79\) 1.10676e6 0.252556 0.126278 0.991995i \(-0.459697\pi\)
0.126278 + 0.991995i \(0.459697\pi\)
\(80\) 2.09500e6 0.457477
\(81\) 0 0
\(82\) −1.13120e6 −0.226565
\(83\) 3.81576e6 0.732501 0.366250 0.930516i \(-0.380641\pi\)
0.366250 + 0.930516i \(0.380641\pi\)
\(84\) 0 0
\(85\) 1.77508e6 0.313510
\(86\) 2.23703e6 0.379252
\(87\) 0 0
\(88\) −979134. −0.153163
\(89\) −5.29119e6 −0.795588 −0.397794 0.917475i \(-0.630224\pi\)
−0.397794 + 0.917475i \(0.630224\pi\)
\(90\) 0 0
\(91\) −1.57823e7 −2.19546
\(92\) 1.48792e6 0.199215
\(93\) 0 0
\(94\) −1.90392e6 −0.236429
\(95\) 4.06629e6 0.486593
\(96\) 0 0
\(97\) 6.41098e6 0.713220 0.356610 0.934253i \(-0.383932\pi\)
0.356610 + 0.934253i \(0.383932\pi\)
\(98\) 1.61145e6 0.172952
\(99\) 0 0
\(100\) 6.90133e6 0.690133
\(101\) −7.39048e6 −0.713752 −0.356876 0.934152i \(-0.616158\pi\)
−0.356876 + 0.934152i \(0.616158\pi\)
\(102\) 0 0
\(103\) 6.97336e6 0.628799 0.314400 0.949291i \(-0.398197\pi\)
0.314400 + 0.949291i \(0.398197\pi\)
\(104\) −7.71114e6 −0.672205
\(105\) 0 0
\(106\) 4.88761e6 0.398590
\(107\) 1.70173e7 1.34291 0.671456 0.741045i \(-0.265669\pi\)
0.671456 + 0.741045i \(0.265669\pi\)
\(108\) 0 0
\(109\) −4.11126e6 −0.304076 −0.152038 0.988375i \(-0.548584\pi\)
−0.152038 + 0.988375i \(0.548584\pi\)
\(110\) 576158. 0.0412731
\(111\) 0 0
\(112\) −1.74098e7 −1.17093
\(113\) −2.09127e7 −1.36344 −0.681721 0.731613i \(-0.738768\pi\)
−0.681721 + 0.731613i \(0.738768\pi\)
\(114\) 0 0
\(115\) −1.79196e6 −0.109872
\(116\) 2.94381e7 1.75109
\(117\) 0 0
\(118\) −4.55912e6 −0.255443
\(119\) −1.47512e7 −0.802442
\(120\) 0 0
\(121\) −1.68063e7 −0.862430
\(122\) −7.94095e6 −0.395925
\(123\) 0 0
\(124\) 1.56773e7 0.738406
\(125\) −1.98178e7 −0.907552
\(126\) 0 0
\(127\) 2.31258e7 1.00181 0.500903 0.865503i \(-0.333001\pi\)
0.500903 + 0.865503i \(0.333001\pi\)
\(128\) −1.78671e7 −0.753044
\(129\) 0 0
\(130\) 4.53752e6 0.181141
\(131\) −4.03840e7 −1.56950 −0.784748 0.619815i \(-0.787208\pi\)
−0.784748 + 0.619815i \(0.787208\pi\)
\(132\) 0 0
\(133\) −3.37917e7 −1.24546
\(134\) −681808. −0.0244791
\(135\) 0 0
\(136\) −7.20738e6 −0.245692
\(137\) −4.05545e7 −1.34746 −0.673732 0.738976i \(-0.735310\pi\)
−0.673732 + 0.738976i \(0.735310\pi\)
\(138\) 0 0
\(139\) −2.68652e7 −0.848474 −0.424237 0.905551i \(-0.639458\pi\)
−0.424237 + 0.905551i \(0.639458\pi\)
\(140\) 2.20444e7 0.678970
\(141\) 0 0
\(142\) −9.36314e6 −0.274418
\(143\) 2.11130e7 0.603772
\(144\) 0 0
\(145\) −3.54536e7 −0.965766
\(146\) −8.41602e6 −0.223806
\(147\) 0 0
\(148\) −2.19096e7 −0.555544
\(149\) 1.26633e7 0.313614 0.156807 0.987629i \(-0.449880\pi\)
0.156807 + 0.987629i \(0.449880\pi\)
\(150\) 0 0
\(151\) 840099. 0.0198569 0.00992844 0.999951i \(-0.496840\pi\)
0.00992844 + 0.999951i \(0.496840\pi\)
\(152\) −1.65104e7 −0.381335
\(153\) 0 0
\(154\) −4.78798e6 −0.105640
\(155\) −1.88808e7 −0.407248
\(156\) 0 0
\(157\) −960571. −0.0198098 −0.00990491 0.999951i \(-0.503153\pi\)
−0.00990491 + 0.999951i \(0.503153\pi\)
\(158\) 2.64431e6 0.0533350
\(159\) 0 0
\(160\) 1.62790e7 0.314202
\(161\) 1.48916e7 0.281222
\(162\) 0 0
\(163\) 3.15848e7 0.571245 0.285622 0.958342i \(-0.407800\pi\)
0.285622 + 0.958342i \(0.407800\pi\)
\(164\) 5.78998e7 1.02500
\(165\) 0 0
\(166\) 9.11677e6 0.154690
\(167\) 7.78579e7 1.29358 0.646792 0.762666i \(-0.276110\pi\)
0.646792 + 0.762666i \(0.276110\pi\)
\(168\) 0 0
\(169\) 1.03526e8 1.64986
\(170\) 4.24108e6 0.0662072
\(171\) 0 0
\(172\) −1.14501e8 −1.71577
\(173\) −5.01041e7 −0.735719 −0.367860 0.929881i \(-0.619909\pi\)
−0.367860 + 0.929881i \(0.619909\pi\)
\(174\) 0 0
\(175\) 6.90705e7 0.974226
\(176\) 2.32903e7 0.322018
\(177\) 0 0
\(178\) −1.26419e7 −0.168013
\(179\) 3.80878e7 0.496365 0.248182 0.968713i \(-0.420167\pi\)
0.248182 + 0.968713i \(0.420167\pi\)
\(180\) 0 0
\(181\) 3.63781e7 0.456000 0.228000 0.973661i \(-0.426781\pi\)
0.228000 + 0.973661i \(0.426781\pi\)
\(182\) −3.77076e7 −0.463638
\(183\) 0 0
\(184\) 7.27594e6 0.0861047
\(185\) 2.63866e7 0.306396
\(186\) 0 0
\(187\) 1.97337e7 0.220679
\(188\) 9.74506e7 1.06963
\(189\) 0 0
\(190\) 9.71534e6 0.102759
\(191\) −9.38592e7 −0.974676 −0.487338 0.873214i \(-0.662032\pi\)
−0.487338 + 0.873214i \(0.662032\pi\)
\(192\) 0 0
\(193\) −9.36357e7 −0.937543 −0.468771 0.883320i \(-0.655303\pi\)
−0.468771 + 0.883320i \(0.655303\pi\)
\(194\) 1.53174e7 0.150618
\(195\) 0 0
\(196\) −8.24809e7 −0.782451
\(197\) −6.82715e7 −0.636220 −0.318110 0.948054i \(-0.603048\pi\)
−0.318110 + 0.948054i \(0.603048\pi\)
\(198\) 0 0
\(199\) 6.37165e7 0.573147 0.286573 0.958058i \(-0.407484\pi\)
0.286573 + 0.958058i \(0.407484\pi\)
\(200\) 3.37475e7 0.298289
\(201\) 0 0
\(202\) −1.76576e7 −0.150731
\(203\) 2.94626e8 2.47192
\(204\) 0 0
\(205\) −6.97311e7 −0.565312
\(206\) 1.66610e7 0.132790
\(207\) 0 0
\(208\) 1.83422e8 1.41328
\(209\) 4.52053e7 0.342513
\(210\) 0 0
\(211\) −3.67268e7 −0.269150 −0.134575 0.990903i \(-0.542967\pi\)
−0.134575 + 0.990903i \(0.542967\pi\)
\(212\) −2.50169e8 −1.80326
\(213\) 0 0
\(214\) 4.06584e7 0.283597
\(215\) 1.37898e8 0.946289
\(216\) 0 0
\(217\) 1.56903e8 1.04237
\(218\) −9.82278e6 −0.0642150
\(219\) 0 0
\(220\) −2.94902e7 −0.186723
\(221\) 1.55412e8 0.968526
\(222\) 0 0
\(223\) 1.94555e8 1.17483 0.587415 0.809286i \(-0.300146\pi\)
0.587415 + 0.809286i \(0.300146\pi\)
\(224\) −1.35282e8 −0.804213
\(225\) 0 0
\(226\) −4.99655e7 −0.287933
\(227\) −2.74851e8 −1.55958 −0.779789 0.626043i \(-0.784674\pi\)
−0.779789 + 0.626043i \(0.784674\pi\)
\(228\) 0 0
\(229\) −1.43628e8 −0.790344 −0.395172 0.918607i \(-0.629315\pi\)
−0.395172 + 0.918607i \(0.629315\pi\)
\(230\) −4.28143e6 −0.0232029
\(231\) 0 0
\(232\) 1.43953e8 0.756854
\(233\) −3.20512e8 −1.65996 −0.829982 0.557790i \(-0.811649\pi\)
−0.829982 + 0.557790i \(0.811649\pi\)
\(234\) 0 0
\(235\) −1.17364e8 −0.589924
\(236\) 2.33355e8 1.15565
\(237\) 0 0
\(238\) −3.52442e7 −0.169460
\(239\) 4.41622e7 0.209247 0.104623 0.994512i \(-0.466636\pi\)
0.104623 + 0.994512i \(0.466636\pi\)
\(240\) 0 0
\(241\) 7.53980e7 0.346977 0.173488 0.984836i \(-0.444496\pi\)
0.173488 + 0.984836i \(0.444496\pi\)
\(242\) −4.01543e7 −0.182129
\(243\) 0 0
\(244\) 4.06452e8 1.79120
\(245\) 9.93351e7 0.431540
\(246\) 0 0
\(247\) 3.56013e8 1.50323
\(248\) 7.66620e7 0.319153
\(249\) 0 0
\(250\) −4.73496e7 −0.191658
\(251\) −7.67520e7 −0.306360 −0.153180 0.988198i \(-0.548951\pi\)
−0.153180 + 0.988198i \(0.548951\pi\)
\(252\) 0 0
\(253\) −1.99214e7 −0.0773389
\(254\) 5.52530e7 0.211562
\(255\) 0 0
\(256\) 1.56563e8 0.583243
\(257\) −8.08367e6 −0.0297059 −0.0148529 0.999890i \(-0.504728\pi\)
−0.0148529 + 0.999890i \(0.504728\pi\)
\(258\) 0 0
\(259\) −2.19278e8 −0.784234
\(260\) −2.32250e8 −0.819499
\(261\) 0 0
\(262\) −9.64871e7 −0.331448
\(263\) −2.78979e8 −0.945640 −0.472820 0.881159i \(-0.656764\pi\)
−0.472820 + 0.881159i \(0.656764\pi\)
\(264\) 0 0
\(265\) 3.01289e8 0.994540
\(266\) −8.07364e7 −0.263017
\(267\) 0 0
\(268\) 3.48978e7 0.110746
\(269\) 3.28037e8 1.02752 0.513760 0.857934i \(-0.328252\pi\)
0.513760 + 0.857934i \(0.328252\pi\)
\(270\) 0 0
\(271\) −1.91549e8 −0.584639 −0.292319 0.956321i \(-0.594427\pi\)
−0.292319 + 0.956321i \(0.594427\pi\)
\(272\) 1.71439e8 0.516557
\(273\) 0 0
\(274\) −9.68944e7 −0.284558
\(275\) −9.24001e7 −0.267922
\(276\) 0 0
\(277\) 1.80215e8 0.509463 0.254731 0.967012i \(-0.418013\pi\)
0.254731 + 0.967012i \(0.418013\pi\)
\(278\) −6.41874e7 −0.179181
\(279\) 0 0
\(280\) 1.07797e8 0.293464
\(281\) 1.57503e8 0.423465 0.211733 0.977328i \(-0.432089\pi\)
0.211733 + 0.977328i \(0.432089\pi\)
\(282\) 0 0
\(283\) 6.94865e8 1.82242 0.911208 0.411946i \(-0.135151\pi\)
0.911208 + 0.411946i \(0.135151\pi\)
\(284\) 4.79246e8 1.24149
\(285\) 0 0
\(286\) 5.04439e7 0.127505
\(287\) 5.79479e8 1.44694
\(288\) 0 0
\(289\) −2.65080e8 −0.646002
\(290\) −8.47070e7 −0.203951
\(291\) 0 0
\(292\) 4.30768e8 1.01252
\(293\) 3.61524e8 0.839653 0.419827 0.907604i \(-0.362091\pi\)
0.419827 + 0.907604i \(0.362091\pi\)
\(294\) 0 0
\(295\) −2.81039e8 −0.637367
\(296\) −1.07138e8 −0.240117
\(297\) 0 0
\(298\) 3.02557e7 0.0662293
\(299\) −1.56890e8 −0.339428
\(300\) 0 0
\(301\) −1.14596e9 −2.42207
\(302\) 2.00720e6 0.00419339
\(303\) 0 0
\(304\) 3.92727e8 0.801740
\(305\) −4.89506e8 −0.987891
\(306\) 0 0
\(307\) −3.33653e8 −0.658129 −0.329065 0.944307i \(-0.606733\pi\)
−0.329065 + 0.944307i \(0.606733\pi\)
\(308\) 2.45069e8 0.477927
\(309\) 0 0
\(310\) −4.51107e7 −0.0860031
\(311\) −2.96226e8 −0.558421 −0.279211 0.960230i \(-0.590073\pi\)
−0.279211 + 0.960230i \(0.590073\pi\)
\(312\) 0 0
\(313\) −3.47401e8 −0.640363 −0.320181 0.947356i \(-0.603744\pi\)
−0.320181 + 0.947356i \(0.603744\pi\)
\(314\) −2.29503e6 −0.00418346
\(315\) 0 0
\(316\) −1.35347e8 −0.241293
\(317\) 2.12080e8 0.373931 0.186966 0.982366i \(-0.440135\pi\)
0.186966 + 0.982366i \(0.440135\pi\)
\(318\) 0 0
\(319\) −3.94140e8 −0.679803
\(320\) −2.29265e8 −0.391123
\(321\) 0 0
\(322\) 3.55795e7 0.0593887
\(323\) 3.32755e8 0.549434
\(324\) 0 0
\(325\) −7.27695e8 −1.17586
\(326\) 7.54638e7 0.120636
\(327\) 0 0
\(328\) 2.83130e8 0.443025
\(329\) 9.75314e8 1.50994
\(330\) 0 0
\(331\) 3.92104e8 0.594296 0.297148 0.954831i \(-0.403964\pi\)
0.297148 + 0.954831i \(0.403964\pi\)
\(332\) −4.66636e8 −0.699833
\(333\) 0 0
\(334\) 1.86021e8 0.273180
\(335\) −4.20289e7 −0.0610789
\(336\) 0 0
\(337\) −9.07353e8 −1.29143 −0.645716 0.763578i \(-0.723441\pi\)
−0.645716 + 0.763578i \(0.723441\pi\)
\(338\) 2.47348e8 0.348418
\(339\) 0 0
\(340\) −2.17077e8 −0.299528
\(341\) −2.09899e8 −0.286662
\(342\) 0 0
\(343\) 1.82465e8 0.244147
\(344\) −5.59911e8 −0.741591
\(345\) 0 0
\(346\) −1.19711e8 −0.155370
\(347\) 3.73740e8 0.480194 0.240097 0.970749i \(-0.422821\pi\)
0.240097 + 0.970749i \(0.422821\pi\)
\(348\) 0 0
\(349\) −1.88713e8 −0.237636 −0.118818 0.992916i \(-0.537910\pi\)
−0.118818 + 0.992916i \(0.537910\pi\)
\(350\) 1.65026e8 0.205738
\(351\) 0 0
\(352\) 1.80975e8 0.221167
\(353\) −1.22544e9 −1.48279 −0.741395 0.671069i \(-0.765836\pi\)
−0.741395 + 0.671069i \(0.765836\pi\)
\(354\) 0 0
\(355\) −5.77175e8 −0.684712
\(356\) 6.47067e8 0.760107
\(357\) 0 0
\(358\) 9.10010e7 0.104823
\(359\) 4.27509e8 0.487657 0.243829 0.969818i \(-0.421597\pi\)
0.243829 + 0.969818i \(0.421597\pi\)
\(360\) 0 0
\(361\) −1.31607e8 −0.147232
\(362\) 8.69159e7 0.0962984
\(363\) 0 0
\(364\) 1.93004e9 2.09754
\(365\) −5.18791e8 −0.558428
\(366\) 0 0
\(367\) −1.58489e9 −1.67366 −0.836831 0.547462i \(-0.815594\pi\)
−0.836831 + 0.547462i \(0.815594\pi\)
\(368\) −1.73070e8 −0.181032
\(369\) 0 0
\(370\) 6.30439e7 0.0647049
\(371\) −2.50377e9 −2.54557
\(372\) 0 0
\(373\) −1.23598e9 −1.23320 −0.616598 0.787278i \(-0.711490\pi\)
−0.616598 + 0.787278i \(0.711490\pi\)
\(374\) 4.71484e7 0.0466033
\(375\) 0 0
\(376\) 4.76534e8 0.462313
\(377\) −3.10404e9 −2.98354
\(378\) 0 0
\(379\) 6.22899e8 0.587733 0.293867 0.955846i \(-0.405058\pi\)
0.293867 + 0.955846i \(0.405058\pi\)
\(380\) −4.97273e8 −0.464892
\(381\) 0 0
\(382\) −2.24252e8 −0.205833
\(383\) 1.18281e9 1.07577 0.537884 0.843019i \(-0.319224\pi\)
0.537884 + 0.843019i \(0.319224\pi\)
\(384\) 0 0
\(385\) −2.95147e8 −0.263588
\(386\) −2.23718e8 −0.197991
\(387\) 0 0
\(388\) −7.84009e8 −0.681412
\(389\) −7.89553e8 −0.680077 −0.340038 0.940412i \(-0.610440\pi\)
−0.340038 + 0.940412i \(0.610440\pi\)
\(390\) 0 0
\(391\) −1.46641e8 −0.124061
\(392\) −4.03332e8 −0.338191
\(393\) 0 0
\(394\) −1.63117e8 −0.134358
\(395\) 1.63004e8 0.133079
\(396\) 0 0
\(397\) −1.58576e9 −1.27195 −0.635977 0.771708i \(-0.719403\pi\)
−0.635977 + 0.771708i \(0.719403\pi\)
\(398\) 1.52234e8 0.121038
\(399\) 0 0
\(400\) −8.02739e8 −0.627140
\(401\) 9.63924e8 0.746513 0.373256 0.927728i \(-0.378241\pi\)
0.373256 + 0.927728i \(0.378241\pi\)
\(402\) 0 0
\(403\) −1.65306e9 −1.25811
\(404\) 9.03793e8 0.681921
\(405\) 0 0
\(406\) 7.03931e8 0.522022
\(407\) 2.93342e8 0.215672
\(408\) 0 0
\(409\) 1.37917e9 0.996752 0.498376 0.866961i \(-0.333930\pi\)
0.498376 + 0.866961i \(0.333930\pi\)
\(410\) −1.66604e8 −0.119383
\(411\) 0 0
\(412\) −8.52783e8 −0.600756
\(413\) 2.33549e9 1.63137
\(414\) 0 0
\(415\) 5.61988e8 0.385975
\(416\) 1.42526e9 0.970664
\(417\) 0 0
\(418\) 1.08006e8 0.0723322
\(419\) −2.43158e9 −1.61487 −0.807437 0.589953i \(-0.799146\pi\)
−0.807437 + 0.589953i \(0.799146\pi\)
\(420\) 0 0
\(421\) 2.63767e7 0.0172279 0.00861396 0.999963i \(-0.497258\pi\)
0.00861396 + 0.999963i \(0.497258\pi\)
\(422\) −8.77491e7 −0.0568393
\(423\) 0 0
\(424\) −1.22333e9 −0.779404
\(425\) −6.80155e8 −0.429780
\(426\) 0 0
\(427\) 4.06789e9 2.52855
\(428\) −2.08107e9 −1.28302
\(429\) 0 0
\(430\) 3.29472e8 0.199838
\(431\) 2.18762e9 1.31614 0.658070 0.752957i \(-0.271373\pi\)
0.658070 + 0.752957i \(0.271373\pi\)
\(432\) 0 0
\(433\) −1.05899e6 −0.000626880 0 −0.000313440 1.00000i \(-0.500100\pi\)
−0.000313440 1.00000i \(0.500100\pi\)
\(434\) 3.74879e8 0.220129
\(435\) 0 0
\(436\) 5.02772e8 0.290515
\(437\) −3.35920e8 −0.192553
\(438\) 0 0
\(439\) 2.62942e9 1.48332 0.741660 0.670776i \(-0.234039\pi\)
0.741660 + 0.670776i \(0.234039\pi\)
\(440\) −1.44207e8 −0.0807055
\(441\) 0 0
\(442\) 3.71316e8 0.204534
\(443\) −6.92479e8 −0.378437 −0.189219 0.981935i \(-0.560595\pi\)
−0.189219 + 0.981935i \(0.560595\pi\)
\(444\) 0 0
\(445\) −7.79289e8 −0.419217
\(446\) 4.64838e8 0.248102
\(447\) 0 0
\(448\) 1.90524e9 1.00110
\(449\) 1.34070e9 0.698990 0.349495 0.936938i \(-0.386353\pi\)
0.349495 + 0.936938i \(0.386353\pi\)
\(450\) 0 0
\(451\) −7.75206e8 −0.397923
\(452\) 2.55745e9 1.30264
\(453\) 0 0
\(454\) −6.56685e8 −0.329353
\(455\) −2.32442e9 −1.15685
\(456\) 0 0
\(457\) 1.96222e8 0.0961701 0.0480851 0.998843i \(-0.484688\pi\)
0.0480851 + 0.998843i \(0.484688\pi\)
\(458\) −3.43162e8 −0.166905
\(459\) 0 0
\(460\) 2.19142e8 0.104972
\(461\) 2.83515e9 1.34779 0.673895 0.738827i \(-0.264620\pi\)
0.673895 + 0.738827i \(0.264620\pi\)
\(462\) 0 0
\(463\) −2.19801e9 −1.02919 −0.514595 0.857433i \(-0.672058\pi\)
−0.514595 + 0.857433i \(0.672058\pi\)
\(464\) −3.42415e9 −1.59125
\(465\) 0 0
\(466\) −7.65780e8 −0.350553
\(467\) −1.42423e9 −0.647100 −0.323550 0.946211i \(-0.604876\pi\)
−0.323550 + 0.946211i \(0.604876\pi\)
\(468\) 0 0
\(469\) 3.49268e8 0.156334
\(470\) −2.80410e8 −0.124581
\(471\) 0 0
\(472\) 1.14111e9 0.499494
\(473\) 1.53302e9 0.666093
\(474\) 0 0
\(475\) −1.55808e9 −0.667055
\(476\) 1.80395e9 0.766655
\(477\) 0 0
\(478\) 1.05514e8 0.0441889
\(479\) −3.08517e9 −1.28264 −0.641320 0.767274i \(-0.721613\pi\)
−0.641320 + 0.767274i \(0.721613\pi\)
\(480\) 0 0
\(481\) 2.31021e9 0.946549
\(482\) 1.80144e8 0.0732749
\(483\) 0 0
\(484\) 2.05527e9 0.823968
\(485\) 9.44214e8 0.375815
\(486\) 0 0
\(487\) −2.32140e9 −0.910748 −0.455374 0.890300i \(-0.650494\pi\)
−0.455374 + 0.890300i \(0.650494\pi\)
\(488\) 1.98755e9 0.774193
\(489\) 0 0
\(490\) 2.37335e8 0.0911331
\(491\) 3.13714e8 0.119605 0.0598024 0.998210i \(-0.480953\pi\)
0.0598024 + 0.998210i \(0.480953\pi\)
\(492\) 0 0
\(493\) −2.90125e9 −1.09049
\(494\) 8.50600e8 0.317454
\(495\) 0 0
\(496\) −1.82353e9 −0.671007
\(497\) 4.79643e9 1.75255
\(498\) 0 0
\(499\) 5.75078e8 0.207193 0.103596 0.994619i \(-0.466965\pi\)
0.103596 + 0.994619i \(0.466965\pi\)
\(500\) 2.42356e9 0.867077
\(501\) 0 0
\(502\) −1.83379e8 −0.0646973
\(503\) 1.87631e9 0.657379 0.328690 0.944438i \(-0.393393\pi\)
0.328690 + 0.944438i \(0.393393\pi\)
\(504\) 0 0
\(505\) −1.08847e9 −0.376096
\(506\) −4.75970e7 −0.0163325
\(507\) 0 0
\(508\) −2.82809e9 −0.957128
\(509\) −4.24575e9 −1.42706 −0.713529 0.700625i \(-0.752904\pi\)
−0.713529 + 0.700625i \(0.752904\pi\)
\(510\) 0 0
\(511\) 4.31125e9 1.42932
\(512\) 2.66106e9 0.876214
\(513\) 0 0
\(514\) −1.93138e7 −0.00627331
\(515\) 1.02704e9 0.331331
\(516\) 0 0
\(517\) −1.30474e9 −0.415248
\(518\) −5.23907e8 −0.165615
\(519\) 0 0
\(520\) −1.13570e9 −0.354203
\(521\) −2.59048e9 −0.802506 −0.401253 0.915967i \(-0.631425\pi\)
−0.401253 + 0.915967i \(0.631425\pi\)
\(522\) 0 0
\(523\) 6.54783e8 0.200144 0.100072 0.994980i \(-0.468093\pi\)
0.100072 + 0.994980i \(0.468093\pi\)
\(524\) 4.93863e9 1.49950
\(525\) 0 0
\(526\) −6.66547e8 −0.199701
\(527\) −1.54506e9 −0.459842
\(528\) 0 0
\(529\) 1.48036e8 0.0434783
\(530\) 7.19851e8 0.210028
\(531\) 0 0
\(532\) 4.13243e9 1.18991
\(533\) −6.10511e9 −1.74642
\(534\) 0 0
\(535\) 2.50632e9 0.707617
\(536\) 1.70651e8 0.0478665
\(537\) 0 0
\(538\) 7.83759e8 0.216992
\(539\) 1.10432e9 0.303761
\(540\) 0 0
\(541\) −4.31638e9 −1.17200 −0.586002 0.810310i \(-0.699299\pi\)
−0.586002 + 0.810310i \(0.699299\pi\)
\(542\) −4.57657e8 −0.123465
\(543\) 0 0
\(544\) 1.33215e9 0.354779
\(545\) −6.05509e8 −0.160226
\(546\) 0 0
\(547\) −5.33812e9 −1.39455 −0.697273 0.716806i \(-0.745603\pi\)
−0.697273 + 0.716806i \(0.745603\pi\)
\(548\) 4.95947e9 1.28737
\(549\) 0 0
\(550\) −2.20766e8 −0.0565800
\(551\) −6.64610e9 −1.69253
\(552\) 0 0
\(553\) −1.35459e9 −0.340621
\(554\) 4.30577e8 0.107589
\(555\) 0 0
\(556\) 3.28539e9 0.810634
\(557\) 5.69967e9 1.39751 0.698757 0.715359i \(-0.253737\pi\)
0.698757 + 0.715359i \(0.253737\pi\)
\(558\) 0 0
\(559\) 1.20733e10 2.92337
\(560\) −2.56413e9 −0.616996
\(561\) 0 0
\(562\) 3.76313e8 0.0894278
\(563\) −4.19873e9 −0.991605 −0.495802 0.868435i \(-0.665126\pi\)
−0.495802 + 0.868435i \(0.665126\pi\)
\(564\) 0 0
\(565\) −3.08004e9 −0.718434
\(566\) 1.66020e9 0.384860
\(567\) 0 0
\(568\) 2.34351e9 0.536597
\(569\) 1.69971e9 0.386795 0.193398 0.981120i \(-0.438049\pi\)
0.193398 + 0.981120i \(0.438049\pi\)
\(570\) 0 0
\(571\) 6.47824e9 1.45623 0.728116 0.685454i \(-0.240396\pi\)
0.728116 + 0.685454i \(0.240396\pi\)
\(572\) −2.58194e9 −0.576845
\(573\) 0 0
\(574\) 1.38451e9 0.305566
\(575\) 6.86625e8 0.150620
\(576\) 0 0
\(577\) 3.45826e9 0.749450 0.374725 0.927136i \(-0.377737\pi\)
0.374725 + 0.927136i \(0.377737\pi\)
\(578\) −6.33339e8 −0.136423
\(579\) 0 0
\(580\) 4.33567e9 0.922695
\(581\) −4.67023e9 −0.987920
\(582\) 0 0
\(583\) 3.34945e9 0.700057
\(584\) 2.10646e9 0.437631
\(585\) 0 0
\(586\) 8.63766e8 0.177319
\(587\) −1.31138e9 −0.267606 −0.133803 0.991008i \(-0.542719\pi\)
−0.133803 + 0.991008i \(0.542719\pi\)
\(588\) 0 0
\(589\) −3.53938e9 −0.713714
\(590\) −6.71470e8 −0.134600
\(591\) 0 0
\(592\) 2.54845e9 0.504836
\(593\) 9.08126e9 1.78836 0.894179 0.447709i \(-0.147760\pi\)
0.894179 + 0.447709i \(0.147760\pi\)
\(594\) 0 0
\(595\) −2.17257e9 −0.422828
\(596\) −1.54862e9 −0.299628
\(597\) 0 0
\(598\) −3.74849e8 −0.0716806
\(599\) 6.26416e8 0.119088 0.0595441 0.998226i \(-0.481035\pi\)
0.0595441 + 0.998226i \(0.481035\pi\)
\(600\) 0 0
\(601\) 9.92718e9 1.86537 0.932686 0.360690i \(-0.117459\pi\)
0.932686 + 0.360690i \(0.117459\pi\)
\(602\) −2.73797e9 −0.511495
\(603\) 0 0
\(604\) −1.02737e8 −0.0189713
\(605\) −2.47525e9 −0.454438
\(606\) 0 0
\(607\) 1.65614e9 0.300563 0.150282 0.988643i \(-0.451982\pi\)
0.150282 + 0.988643i \(0.451982\pi\)
\(608\) 3.05166e9 0.550647
\(609\) 0 0
\(610\) −1.16955e9 −0.208624
\(611\) −1.02755e10 −1.82245
\(612\) 0 0
\(613\) 4.45864e8 0.0781791 0.0390895 0.999236i \(-0.487554\pi\)
0.0390895 + 0.999236i \(0.487554\pi\)
\(614\) −7.97178e8 −0.138984
\(615\) 0 0
\(616\) 1.19839e9 0.206569
\(617\) 6.91965e9 1.18600 0.593002 0.805201i \(-0.297943\pi\)
0.593002 + 0.805201i \(0.297943\pi\)
\(618\) 0 0
\(619\) −6.02751e9 −1.02146 −0.510729 0.859742i \(-0.670624\pi\)
−0.510729 + 0.859742i \(0.670624\pi\)
\(620\) 2.30896e9 0.389086
\(621\) 0 0
\(622\) −7.07755e8 −0.117928
\(623\) 6.47604e9 1.07300
\(624\) 0 0
\(625\) 1.49008e9 0.244134
\(626\) −8.30025e8 −0.135232
\(627\) 0 0
\(628\) 1.17470e8 0.0189264
\(629\) 2.15928e9 0.345965
\(630\) 0 0
\(631\) −8.87449e9 −1.40618 −0.703089 0.711102i \(-0.748197\pi\)
−0.703089 + 0.711102i \(0.748197\pi\)
\(632\) −6.61848e8 −0.104291
\(633\) 0 0
\(634\) 5.06709e8 0.0789671
\(635\) 3.40598e9 0.527879
\(636\) 0 0
\(637\) 8.69701e9 1.33316
\(638\) −9.41694e8 −0.143561
\(639\) 0 0
\(640\) −2.63148e9 −0.396799
\(641\) −6.81517e9 −1.02205 −0.511027 0.859565i \(-0.670735\pi\)
−0.511027 + 0.859565i \(0.670735\pi\)
\(642\) 0 0
\(643\) −8.35764e9 −1.23978 −0.619891 0.784688i \(-0.712823\pi\)
−0.619891 + 0.784688i \(0.712823\pi\)
\(644\) −1.82111e9 −0.268680
\(645\) 0 0
\(646\) 7.95031e8 0.116030
\(647\) −3.48454e9 −0.505802 −0.252901 0.967492i \(-0.581385\pi\)
−0.252901 + 0.967492i \(0.581385\pi\)
\(648\) 0 0
\(649\) −3.12433e9 −0.448643
\(650\) −1.73864e9 −0.248320
\(651\) 0 0
\(652\) −3.86256e9 −0.545769
\(653\) 1.41407e10 1.98736 0.993678 0.112267i \(-0.0358110\pi\)
0.993678 + 0.112267i \(0.0358110\pi\)
\(654\) 0 0
\(655\) −5.94779e9 −0.827010
\(656\) −6.73471e9 −0.931441
\(657\) 0 0
\(658\) 2.33026e9 0.318870
\(659\) −4.91129e9 −0.668492 −0.334246 0.942486i \(-0.608482\pi\)
−0.334246 + 0.942486i \(0.608482\pi\)
\(660\) 0 0
\(661\) 6.95658e8 0.0936894 0.0468447 0.998902i \(-0.485083\pi\)
0.0468447 + 0.998902i \(0.485083\pi\)
\(662\) 9.36830e8 0.125504
\(663\) 0 0
\(664\) −2.28185e9 −0.302482
\(665\) −4.97686e9 −0.656265
\(666\) 0 0
\(667\) 2.92885e9 0.382171
\(668\) −9.52136e9 −1.23589
\(669\) 0 0
\(670\) −1.00417e8 −0.0128987
\(671\) −5.44188e9 −0.695376
\(672\) 0 0
\(673\) −2.86105e9 −0.361804 −0.180902 0.983501i \(-0.557902\pi\)
−0.180902 + 0.983501i \(0.557902\pi\)
\(674\) −2.16788e9 −0.272726
\(675\) 0 0
\(676\) −1.26604e10 −1.57628
\(677\) 5.03345e9 0.623455 0.311728 0.950171i \(-0.399092\pi\)
0.311728 + 0.950171i \(0.399092\pi\)
\(678\) 0 0
\(679\) −7.84659e9 −0.961916
\(680\) −1.06151e9 −0.129462
\(681\) 0 0
\(682\) −5.01499e8 −0.0605376
\(683\) −7.71371e9 −0.926383 −0.463192 0.886258i \(-0.653296\pi\)
−0.463192 + 0.886258i \(0.653296\pi\)
\(684\) 0 0
\(685\) −5.97289e9 −0.710015
\(686\) 4.35953e8 0.0515591
\(687\) 0 0
\(688\) 1.33184e10 1.55916
\(689\) 2.63785e10 3.07244
\(690\) 0 0
\(691\) −1.62102e10 −1.86903 −0.934515 0.355923i \(-0.884167\pi\)
−0.934515 + 0.355923i \(0.884167\pi\)
\(692\) 6.12731e9 0.702908
\(693\) 0 0
\(694\) 8.92954e8 0.101408
\(695\) −3.95672e9 −0.447084
\(696\) 0 0
\(697\) −5.70627e9 −0.638319
\(698\) −4.50879e8 −0.0501841
\(699\) 0 0
\(700\) −8.44674e9 −0.930778
\(701\) −1.56994e10 −1.72135 −0.860674 0.509156i \(-0.829958\pi\)
−0.860674 + 0.509156i \(0.829958\pi\)
\(702\) 0 0
\(703\) 4.94642e9 0.536967
\(704\) −2.54876e9 −0.275312
\(705\) 0 0
\(706\) −2.92786e9 −0.313137
\(707\) 9.04543e9 0.962634
\(708\) 0 0
\(709\) 8.60054e6 0.000906283 0 0.000453141 1.00000i \(-0.499856\pi\)
0.000453141 1.00000i \(0.499856\pi\)
\(710\) −1.37901e9 −0.144598
\(711\) 0 0
\(712\) 3.16416e9 0.328533
\(713\) 1.55976e9 0.161155
\(714\) 0 0
\(715\) 3.10953e9 0.318144
\(716\) −4.65782e9 −0.474228
\(717\) 0 0
\(718\) 1.02142e9 0.102984
\(719\) 1.69098e10 1.69663 0.848314 0.529493i \(-0.177618\pi\)
0.848314 + 0.529493i \(0.177618\pi\)
\(720\) 0 0
\(721\) −8.53491e9 −0.848058
\(722\) −3.14441e8 −0.0310927
\(723\) 0 0
\(724\) −4.44873e9 −0.435663
\(725\) 1.35847e10 1.32394
\(726\) 0 0
\(727\) 1.35812e10 1.31089 0.655447 0.755242i \(-0.272480\pi\)
0.655447 + 0.755242i \(0.272480\pi\)
\(728\) 9.43790e9 0.906599
\(729\) 0 0
\(730\) −1.23952e9 −0.117929
\(731\) 1.12846e10 1.06850
\(732\) 0 0
\(733\) −7.43282e9 −0.697092 −0.348546 0.937292i \(-0.613324\pi\)
−0.348546 + 0.937292i \(0.613324\pi\)
\(734\) −3.78668e9 −0.353445
\(735\) 0 0
\(736\) −1.34483e9 −0.124335
\(737\) −4.67238e8 −0.0429934
\(738\) 0 0
\(739\) 5.64451e9 0.514483 0.257242 0.966347i \(-0.417186\pi\)
0.257242 + 0.966347i \(0.417186\pi\)
\(740\) −3.22686e9 −0.292731
\(741\) 0 0
\(742\) −5.98210e9 −0.537576
\(743\) 9.78408e9 0.875103 0.437552 0.899193i \(-0.355846\pi\)
0.437552 + 0.899193i \(0.355846\pi\)
\(744\) 0 0
\(745\) 1.86506e9 0.165252
\(746\) −2.95306e9 −0.260428
\(747\) 0 0
\(748\) −2.41326e9 −0.210838
\(749\) −2.08280e10 −1.81118
\(750\) 0 0
\(751\) −1.20646e10 −1.03937 −0.519687 0.854356i \(-0.673952\pi\)
−0.519687 + 0.854356i \(0.673952\pi\)
\(752\) −1.13351e10 −0.971994
\(753\) 0 0
\(754\) −7.41629e9 −0.630067
\(755\) 1.23730e8 0.0104631
\(756\) 0 0
\(757\) 1.24335e10 1.04174 0.520869 0.853637i \(-0.325608\pi\)
0.520869 + 0.853637i \(0.325608\pi\)
\(758\) 1.48825e9 0.124118
\(759\) 0 0
\(760\) −2.43167e9 −0.200936
\(761\) −1.74989e10 −1.43935 −0.719673 0.694313i \(-0.755708\pi\)
−0.719673 + 0.694313i \(0.755708\pi\)
\(762\) 0 0
\(763\) 5.03189e9 0.410105
\(764\) 1.14782e10 0.931208
\(765\) 0 0
\(766\) 2.82601e9 0.227182
\(767\) −2.46056e10 −1.96902
\(768\) 0 0
\(769\) −8.61237e9 −0.682937 −0.341469 0.939893i \(-0.610924\pi\)
−0.341469 + 0.939893i \(0.610924\pi\)
\(770\) −7.05177e8 −0.0556648
\(771\) 0 0
\(772\) 1.14509e10 0.895731
\(773\) 3.27403e9 0.254950 0.127475 0.991842i \(-0.459313\pi\)
0.127475 + 0.991842i \(0.459313\pi\)
\(774\) 0 0
\(775\) 7.23454e9 0.558284
\(776\) −3.83381e9 −0.294520
\(777\) 0 0
\(778\) −1.88643e9 −0.143619
\(779\) −1.30717e10 −0.990724
\(780\) 0 0
\(781\) −6.41649e9 −0.481969
\(782\) −3.50360e8 −0.0261994
\(783\) 0 0
\(784\) 9.59390e9 0.711032
\(785\) −1.41473e8 −0.0104383
\(786\) 0 0
\(787\) 1.90391e10 1.39231 0.696154 0.717893i \(-0.254893\pi\)
0.696154 + 0.717893i \(0.254893\pi\)
\(788\) 8.34902e9 0.607846
\(789\) 0 0
\(790\) 3.89456e8 0.0281037
\(791\) 2.55957e10 1.83887
\(792\) 0 0
\(793\) −4.28574e10 −3.05189
\(794\) −3.78876e9 −0.268612
\(795\) 0 0
\(796\) −7.79198e9 −0.547586
\(797\) 1.09654e10 0.767222 0.383611 0.923495i \(-0.374680\pi\)
0.383611 + 0.923495i \(0.374680\pi\)
\(798\) 0 0
\(799\) −9.60416e9 −0.666110
\(800\) −6.23762e9 −0.430729
\(801\) 0 0
\(802\) 2.30304e9 0.157649
\(803\) −5.76744e9 −0.393078
\(804\) 0 0
\(805\) 2.19324e9 0.148184
\(806\) −3.94955e9 −0.265690
\(807\) 0 0
\(808\) 4.41955e9 0.294740
\(809\) −2.54612e10 −1.69067 −0.845336 0.534235i \(-0.820600\pi\)
−0.845336 + 0.534235i \(0.820600\pi\)
\(810\) 0 0
\(811\) 1.38636e10 0.912645 0.456323 0.889814i \(-0.349166\pi\)
0.456323 + 0.889814i \(0.349166\pi\)
\(812\) −3.60302e10 −2.36168
\(813\) 0 0
\(814\) 7.00864e8 0.0455458
\(815\) 4.65184e9 0.301004
\(816\) 0 0
\(817\) 2.58503e10 1.65840
\(818\) 3.29517e9 0.210495
\(819\) 0 0
\(820\) 8.52752e9 0.540100
\(821\) −1.57708e10 −0.994607 −0.497304 0.867577i \(-0.665677\pi\)
−0.497304 + 0.867577i \(0.665677\pi\)
\(822\) 0 0
\(823\) −5.42493e7 −0.00339231 −0.00169615 0.999999i \(-0.500540\pi\)
−0.00169615 + 0.999999i \(0.500540\pi\)
\(824\) −4.17011e9 −0.259659
\(825\) 0 0
\(826\) 5.58004e9 0.344514
\(827\) 2.41236e10 1.48311 0.741554 0.670893i \(-0.234089\pi\)
0.741554 + 0.670893i \(0.234089\pi\)
\(828\) 0 0
\(829\) 3.13296e7 0.00190991 0.000954956 1.00000i \(-0.499696\pi\)
0.000954956 1.00000i \(0.499696\pi\)
\(830\) 1.34272e9 0.0815105
\(831\) 0 0
\(832\) −2.00727e10 −1.20830
\(833\) 8.12884e9 0.487272
\(834\) 0 0
\(835\) 1.14670e10 0.681625
\(836\) −5.52822e9 −0.327238
\(837\) 0 0
\(838\) −5.80962e9 −0.341031
\(839\) 2.29764e8 0.0134312 0.00671559 0.999977i \(-0.497862\pi\)
0.00671559 + 0.999977i \(0.497862\pi\)
\(840\) 0 0
\(841\) 4.06968e10 2.35925
\(842\) 6.30202e7 0.00363821
\(843\) 0 0
\(844\) 4.49137e9 0.257147
\(845\) 1.52474e10 0.869354
\(846\) 0 0
\(847\) 2.05698e10 1.16315
\(848\) 2.90988e10 1.63866
\(849\) 0 0
\(850\) −1.62505e9 −0.0907614
\(851\) −2.17982e9 −0.121246
\(852\) 0 0
\(853\) −3.52100e10 −1.94243 −0.971213 0.238213i \(-0.923439\pi\)
−0.971213 + 0.238213i \(0.923439\pi\)
\(854\) 9.71916e9 0.533982
\(855\) 0 0
\(856\) −1.01765e10 −0.554547
\(857\) −2.17887e10 −1.18249 −0.591246 0.806491i \(-0.701364\pi\)
−0.591246 + 0.806491i \(0.701364\pi\)
\(858\) 0 0
\(859\) 2.04776e10 1.10231 0.551155 0.834403i \(-0.314187\pi\)
0.551155 + 0.834403i \(0.314187\pi\)
\(860\) −1.68638e10 −0.904087
\(861\) 0 0
\(862\) 5.22676e9 0.277944
\(863\) −2.39053e9 −0.126606 −0.0633032 0.997994i \(-0.520164\pi\)
−0.0633032 + 0.997994i \(0.520164\pi\)
\(864\) 0 0
\(865\) −7.37937e9 −0.387670
\(866\) −2.53018e6 −0.000132385 0
\(867\) 0 0
\(868\) −1.91879e10 −0.995883
\(869\) 1.81213e9 0.0936741
\(870\) 0 0
\(871\) −3.67972e9 −0.188691
\(872\) 2.45856e9 0.125566
\(873\) 0 0
\(874\) −8.02594e8 −0.0406636
\(875\) 2.42557e10 1.22401
\(876\) 0 0
\(877\) −5.33337e9 −0.266995 −0.133497 0.991049i \(-0.542621\pi\)
−0.133497 + 0.991049i \(0.542621\pi\)
\(878\) 6.28232e9 0.313249
\(879\) 0 0
\(880\) 3.43020e9 0.169680
\(881\) 7.74528e9 0.381612 0.190806 0.981628i \(-0.438890\pi\)
0.190806 + 0.981628i \(0.438890\pi\)
\(882\) 0 0
\(883\) −7.35743e9 −0.359636 −0.179818 0.983700i \(-0.557551\pi\)
−0.179818 + 0.983700i \(0.557551\pi\)
\(884\) −1.90056e10 −0.925332
\(885\) 0 0
\(886\) −1.65450e9 −0.0799187
\(887\) −2.53512e10 −1.21974 −0.609869 0.792502i \(-0.708778\pi\)
−0.609869 + 0.792502i \(0.708778\pi\)
\(888\) 0 0
\(889\) −2.83043e10 −1.35113
\(890\) −1.86191e9 −0.0885306
\(891\) 0 0
\(892\) −2.37924e10 −1.12244
\(893\) −2.20009e10 −1.03386
\(894\) 0 0
\(895\) 5.60960e9 0.261548
\(896\) 2.18681e10 1.01563
\(897\) 0 0
\(898\) 3.20326e9 0.147613
\(899\) 3.08595e10 1.41654
\(900\) 0 0
\(901\) 2.46552e10 1.12298
\(902\) −1.85215e9 −0.0840337
\(903\) 0 0
\(904\) 1.25060e10 0.563025
\(905\) 5.35778e9 0.240279
\(906\) 0 0
\(907\) 1.17598e10 0.523329 0.261665 0.965159i \(-0.415729\pi\)
0.261665 + 0.965159i \(0.415729\pi\)
\(908\) 3.36120e10 1.49002
\(909\) 0 0
\(910\) −5.55360e9 −0.244304
\(911\) −1.28379e10 −0.562572 −0.281286 0.959624i \(-0.590761\pi\)
−0.281286 + 0.959624i \(0.590761\pi\)
\(912\) 0 0
\(913\) 6.24766e9 0.271688
\(914\) 4.68820e8 0.0203093
\(915\) 0 0
\(916\) 1.75645e10 0.755096
\(917\) 4.94272e10 2.11677
\(918\) 0 0
\(919\) 2.54812e10 1.08297 0.541485 0.840711i \(-0.317862\pi\)
0.541485 + 0.840711i \(0.317862\pi\)
\(920\) 1.07161e9 0.0453709
\(921\) 0 0
\(922\) 6.77384e9 0.284627
\(923\) −5.05329e10 −2.11528
\(924\) 0 0
\(925\) −1.01105e10 −0.420028
\(926\) −5.25156e9 −0.217345
\(927\) 0 0
\(928\) −2.66071e10 −1.09290
\(929\) 4.93247e9 0.201841 0.100921 0.994894i \(-0.467821\pi\)
0.100921 + 0.994894i \(0.467821\pi\)
\(930\) 0 0
\(931\) 1.86213e10 0.756286
\(932\) 3.91959e10 1.58593
\(933\) 0 0
\(934\) −3.40283e9 −0.136655
\(935\) 2.90639e9 0.116282
\(936\) 0 0
\(937\) 8.62092e9 0.342346 0.171173 0.985241i \(-0.445244\pi\)
0.171173 + 0.985241i \(0.445244\pi\)
\(938\) 8.34485e8 0.0330148
\(939\) 0 0
\(940\) 1.43526e10 0.563615
\(941\) 2.08832e10 0.817022 0.408511 0.912753i \(-0.366048\pi\)
0.408511 + 0.912753i \(0.366048\pi\)
\(942\) 0 0
\(943\) 5.76055e9 0.223704
\(944\) −2.71431e10 −1.05016
\(945\) 0 0
\(946\) 3.66276e9 0.140666
\(947\) −4.19780e10 −1.60619 −0.803095 0.595851i \(-0.796815\pi\)
−0.803095 + 0.595851i \(0.796815\pi\)
\(948\) 0 0
\(949\) −4.54213e10 −1.72515
\(950\) −3.72262e9 −0.140869
\(951\) 0 0
\(952\) 8.82132e9 0.331363
\(953\) 3.26688e10 1.22267 0.611333 0.791373i \(-0.290634\pi\)
0.611333 + 0.791373i \(0.290634\pi\)
\(954\) 0 0
\(955\) −1.38236e10 −0.513583
\(956\) −5.40067e9 −0.199915
\(957\) 0 0
\(958\) −7.37121e9 −0.270869
\(959\) 4.96358e10 1.81732
\(960\) 0 0
\(961\) −1.10784e10 −0.402666
\(962\) 5.51964e9 0.199893
\(963\) 0 0
\(964\) −9.22054e9 −0.331502
\(965\) −1.37907e10 −0.494017
\(966\) 0 0
\(967\) 5.39963e10 1.92031 0.960155 0.279467i \(-0.0901576\pi\)
0.960155 + 0.279467i \(0.0901576\pi\)
\(968\) 1.00503e10 0.356135
\(969\) 0 0
\(970\) 2.25595e9 0.0793650
\(971\) −1.25581e10 −0.440207 −0.220104 0.975476i \(-0.570640\pi\)
−0.220104 + 0.975476i \(0.570640\pi\)
\(972\) 0 0
\(973\) 3.28811e10 1.14433
\(974\) −5.54638e9 −0.192333
\(975\) 0 0
\(976\) −4.72771e10 −1.62771
\(977\) 1.93161e10 0.662657 0.331329 0.943515i \(-0.392503\pi\)
0.331329 + 0.943515i \(0.392503\pi\)
\(978\) 0 0
\(979\) −8.66342e9 −0.295087
\(980\) −1.21478e10 −0.412295
\(981\) 0 0
\(982\) 7.49538e8 0.0252583
\(983\) 3.38434e10 1.13642 0.568208 0.822885i \(-0.307637\pi\)
0.568208 + 0.822885i \(0.307637\pi\)
\(984\) 0 0
\(985\) −1.00551e10 −0.335242
\(986\) −6.93179e9 −0.230290
\(987\) 0 0
\(988\) −4.35374e10 −1.43619
\(989\) −1.13919e10 −0.374463
\(990\) 0 0
\(991\) 2.92468e10 0.954599 0.477300 0.878741i \(-0.341616\pi\)
0.477300 + 0.878741i \(0.341616\pi\)
\(992\) −1.41696e10 −0.460857
\(993\) 0 0
\(994\) 1.14598e10 0.370105
\(995\) 9.38420e9 0.302007
\(996\) 0 0
\(997\) −2.05938e9 −0.0658117 −0.0329059 0.999458i \(-0.510476\pi\)
−0.0329059 + 0.999458i \(0.510476\pi\)
\(998\) 1.37400e9 0.0437552
\(999\) 0 0
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 207.8.a.e.1.6 8
3.2 odd 2 69.8.a.d.1.3 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
69.8.a.d.1.3 8 3.2 odd 2
207.8.a.e.1.6 8 1.1 even 1 trivial