Properties

Label 592.8.a.c.1.3
Level $592$
Weight $8$
Character 592.1
Self dual yes
Analytic conductor $184.932$
Analytic rank $1$
Dimension $6$
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] = [592,8,Mod(1,592)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(592, 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("592.1");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 592 = 2^{4} \cdot 37 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 592.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: \(184.931935087\)
Analytic rank: \(1\)
Dimension: \(6\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 2x^{5} - 10621x^{4} + 102052x^{3} + 31004503x^{2} - 305547358x - 22608804936 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: no (minimal twist has level 74)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-26.0193\) of defining polynomial
Character \(\chi\) \(=\) 592.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-31.0193 q^{3} -544.510 q^{5} +1685.44 q^{7} -1224.80 q^{9} +O(q^{10})\) \(q-31.0193 q^{3} -544.510 q^{5} +1685.44 q^{7} -1224.80 q^{9} -5629.72 q^{11} +5228.81 q^{13} +16890.3 q^{15} -12575.5 q^{17} -18367.1 q^{19} -52281.0 q^{21} -40252.9 q^{23} +218366. q^{25} +105832. q^{27} +18705.8 q^{29} +169281. q^{31} +174630. q^{33} -917737. q^{35} -50653.0 q^{37} -162194. q^{39} -26437.9 q^{41} -406992. q^{43} +666918. q^{45} -175342. q^{47} +2.01715e6 q^{49} +390084. q^{51} -1.82314e6 q^{53} +3.06544e6 q^{55} +569735. q^{57} +1.24173e6 q^{59} +3.25092e6 q^{61} -2.06433e6 q^{63} -2.84714e6 q^{65} +1.66755e6 q^{67} +1.24862e6 q^{69} -262493. q^{71} +2.68612e6 q^{73} -6.77356e6 q^{75} -9.48853e6 q^{77} -363284. q^{79} -604180. q^{81} +6.71044e6 q^{83} +6.84750e6 q^{85} -580239. q^{87} +1.16083e7 q^{89} +8.81283e6 q^{91} -5.25097e6 q^{93} +1.00011e7 q^{95} +1.63488e7 q^{97} +6.89530e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 28 q^{3} - 14 q^{5} + 980 q^{7} + 8254 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 28 q^{3} - 14 q^{5} + 980 q^{7} + 8254 q^{9} - 2956 q^{11} + 2394 q^{13} + 28820 q^{15} - 45108 q^{17} - 11764 q^{19} - 135378 q^{21} - 21052 q^{23} + 194744 q^{25} - 439240 q^{27} + 288454 q^{29} - 578868 q^{31} + 980174 q^{33} - 1243052 q^{35} - 303918 q^{37} - 1735296 q^{39} + 1176840 q^{41} - 2669236 q^{43} + 2560692 q^{45} + 131044 q^{47} + 2460856 q^{49} - 2899732 q^{51} + 983190 q^{53} + 1200168 q^{55} - 163216 q^{57} + 1215568 q^{59} + 3136358 q^{61} + 1444880 q^{63} - 1302836 q^{65} - 2179276 q^{67} - 929514 q^{69} - 325164 q^{71} + 5011444 q^{73} + 9374520 q^{75} - 26500426 q^{77} - 3173032 q^{79} - 2565226 q^{81} + 22567048 q^{83} + 1486476 q^{85} + 157228 q^{87} + 26836996 q^{89} - 17942380 q^{91} + 16734948 q^{93} + 4252048 q^{95} + 295792 q^{97} - 25990712 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −31.0193 −0.663296 −0.331648 0.943403i \(-0.607605\pi\)
−0.331648 + 0.943403i \(0.607605\pi\)
\(4\) 0 0
\(5\) −544.510 −1.94810 −0.974049 0.226337i \(-0.927325\pi\)
−0.974049 + 0.226337i \(0.927325\pi\)
\(6\) 0 0
\(7\) 1685.44 1.85724 0.928622 0.371026i \(-0.120994\pi\)
0.928622 + 0.371026i \(0.120994\pi\)
\(8\) 0 0
\(9\) −1224.80 −0.560038
\(10\) 0 0
\(11\) −5629.72 −1.27530 −0.637650 0.770327i \(-0.720093\pi\)
−0.637650 + 0.770327i \(0.720093\pi\)
\(12\) 0 0
\(13\) 5228.81 0.660087 0.330044 0.943966i \(-0.392937\pi\)
0.330044 + 0.943966i \(0.392937\pi\)
\(14\) 0 0
\(15\) 16890.3 1.29217
\(16\) 0 0
\(17\) −12575.5 −0.620805 −0.310402 0.950605i \(-0.600464\pi\)
−0.310402 + 0.950605i \(0.600464\pi\)
\(18\) 0 0
\(19\) −18367.1 −0.614332 −0.307166 0.951656i \(-0.599381\pi\)
−0.307166 + 0.951656i \(0.599381\pi\)
\(20\) 0 0
\(21\) −52281.0 −1.23190
\(22\) 0 0
\(23\) −40252.9 −0.689843 −0.344921 0.938632i \(-0.612094\pi\)
−0.344921 + 0.938632i \(0.612094\pi\)
\(24\) 0 0
\(25\) 218366. 2.79509
\(26\) 0 0
\(27\) 105832. 1.03477
\(28\) 0 0
\(29\) 18705.8 0.142424 0.0712118 0.997461i \(-0.477313\pi\)
0.0712118 + 0.997461i \(0.477313\pi\)
\(30\) 0 0
\(31\) 169281. 1.02057 0.510284 0.860006i \(-0.329540\pi\)
0.510284 + 0.860006i \(0.329540\pi\)
\(32\) 0 0
\(33\) 174630. 0.845901
\(34\) 0 0
\(35\) −917737. −3.61809
\(36\) 0 0
\(37\) −50653.0 −0.164399
\(38\) 0 0
\(39\) −162194. −0.437833
\(40\) 0 0
\(41\) −26437.9 −0.0599078 −0.0299539 0.999551i \(-0.509536\pi\)
−0.0299539 + 0.999551i \(0.509536\pi\)
\(42\) 0 0
\(43\) −406992. −0.780632 −0.390316 0.920681i \(-0.627634\pi\)
−0.390316 + 0.920681i \(0.627634\pi\)
\(44\) 0 0
\(45\) 666918. 1.09101
\(46\) 0 0
\(47\) −175342. −0.246345 −0.123173 0.992385i \(-0.539307\pi\)
−0.123173 + 0.992385i \(0.539307\pi\)
\(48\) 0 0
\(49\) 2.01715e6 2.44936
\(50\) 0 0
\(51\) 390084. 0.411778
\(52\) 0 0
\(53\) −1.82314e6 −1.68211 −0.841056 0.540949i \(-0.818065\pi\)
−0.841056 + 0.540949i \(0.818065\pi\)
\(54\) 0 0
\(55\) 3.06544e6 2.48441
\(56\) 0 0
\(57\) 569735. 0.407484
\(58\) 0 0
\(59\) 1.24173e6 0.787131 0.393565 0.919297i \(-0.371242\pi\)
0.393565 + 0.919297i \(0.371242\pi\)
\(60\) 0 0
\(61\) 3.25092e6 1.83380 0.916899 0.399120i \(-0.130684\pi\)
0.916899 + 0.399120i \(0.130684\pi\)
\(62\) 0 0
\(63\) −2.06433e6 −1.04013
\(64\) 0 0
\(65\) −2.84714e6 −1.28592
\(66\) 0 0
\(67\) 1.66755e6 0.677357 0.338678 0.940902i \(-0.390020\pi\)
0.338678 + 0.940902i \(0.390020\pi\)
\(68\) 0 0
\(69\) 1.24862e6 0.457570
\(70\) 0 0
\(71\) −262493. −0.0870388 −0.0435194 0.999053i \(-0.513857\pi\)
−0.0435194 + 0.999053i \(0.513857\pi\)
\(72\) 0 0
\(73\) 2.68612e6 0.808157 0.404078 0.914724i \(-0.367592\pi\)
0.404078 + 0.914724i \(0.367592\pi\)
\(74\) 0 0
\(75\) −6.77356e6 −1.85397
\(76\) 0 0
\(77\) −9.48853e6 −2.36854
\(78\) 0 0
\(79\) −363284. −0.0828993 −0.0414497 0.999141i \(-0.513198\pi\)
−0.0414497 + 0.999141i \(0.513198\pi\)
\(80\) 0 0
\(81\) −604180. −0.126319
\(82\) 0 0
\(83\) 6.71044e6 1.28818 0.644091 0.764948i \(-0.277236\pi\)
0.644091 + 0.764948i \(0.277236\pi\)
\(84\) 0 0
\(85\) 6.84750e6 1.20939
\(86\) 0 0
\(87\) −580239. −0.0944691
\(88\) 0 0
\(89\) 1.16083e7 1.74543 0.872715 0.488229i \(-0.162357\pi\)
0.872715 + 0.488229i \(0.162357\pi\)
\(90\) 0 0
\(91\) 8.81283e6 1.22594
\(92\) 0 0
\(93\) −5.25097e6 −0.676939
\(94\) 0 0
\(95\) 1.00011e7 1.19678
\(96\) 0 0
\(97\) 1.63488e7 1.81880 0.909400 0.415923i \(-0.136541\pi\)
0.909400 + 0.415923i \(0.136541\pi\)
\(98\) 0 0
\(99\) 6.89530e6 0.714216
\(100\) 0 0
\(101\) 7.79452e6 0.752774 0.376387 0.926462i \(-0.377166\pi\)
0.376387 + 0.926462i \(0.377166\pi\)
\(102\) 0 0
\(103\) −8.12543e6 −0.732683 −0.366341 0.930481i \(-0.619390\pi\)
−0.366341 + 0.930481i \(0.619390\pi\)
\(104\) 0 0
\(105\) 2.84675e7 2.39987
\(106\) 0 0
\(107\) −1.15348e6 −0.0910264 −0.0455132 0.998964i \(-0.514492\pi\)
−0.0455132 + 0.998964i \(0.514492\pi\)
\(108\) 0 0
\(109\) −8.60358e6 −0.636336 −0.318168 0.948034i \(-0.603068\pi\)
−0.318168 + 0.948034i \(0.603068\pi\)
\(110\) 0 0
\(111\) 1.57122e6 0.109045
\(112\) 0 0
\(113\) −7.76780e6 −0.506435 −0.253217 0.967409i \(-0.581489\pi\)
−0.253217 + 0.967409i \(0.581489\pi\)
\(114\) 0 0
\(115\) 2.19181e7 1.34388
\(116\) 0 0
\(117\) −6.40427e6 −0.369674
\(118\) 0 0
\(119\) −2.11953e7 −1.15299
\(120\) 0 0
\(121\) 1.22065e7 0.626388
\(122\) 0 0
\(123\) 820084. 0.0397366
\(124\) 0 0
\(125\) −7.63627e7 −3.49700
\(126\) 0 0
\(127\) −4.49536e6 −0.194738 −0.0973691 0.995248i \(-0.531043\pi\)
−0.0973691 + 0.995248i \(0.531043\pi\)
\(128\) 0 0
\(129\) 1.26246e7 0.517791
\(130\) 0 0
\(131\) 1.57640e7 0.612656 0.306328 0.951926i \(-0.400900\pi\)
0.306328 + 0.951926i \(0.400900\pi\)
\(132\) 0 0
\(133\) −3.09566e7 −1.14097
\(134\) 0 0
\(135\) −5.76264e7 −2.01583
\(136\) 0 0
\(137\) −9.27787e6 −0.308267 −0.154133 0.988050i \(-0.549259\pi\)
−0.154133 + 0.988050i \(0.549259\pi\)
\(138\) 0 0
\(139\) −5.30404e7 −1.67515 −0.837577 0.546319i \(-0.816028\pi\)
−0.837577 + 0.546319i \(0.816028\pi\)
\(140\) 0 0
\(141\) 5.43899e6 0.163400
\(142\) 0 0
\(143\) −2.94367e7 −0.841809
\(144\) 0 0
\(145\) −1.01855e7 −0.277455
\(146\) 0 0
\(147\) −6.25706e7 −1.62465
\(148\) 0 0
\(149\) 2.61358e7 0.647267 0.323633 0.946183i \(-0.395096\pi\)
0.323633 + 0.946183i \(0.395096\pi\)
\(150\) 0 0
\(151\) 3.32760e7 0.786523 0.393261 0.919427i \(-0.371347\pi\)
0.393261 + 0.919427i \(0.371347\pi\)
\(152\) 0 0
\(153\) 1.54026e7 0.347674
\(154\) 0 0
\(155\) −9.21751e7 −1.98817
\(156\) 0 0
\(157\) −2.08133e7 −0.429232 −0.214616 0.976698i \(-0.568850\pi\)
−0.214616 + 0.976698i \(0.568850\pi\)
\(158\) 0 0
\(159\) 5.65525e7 1.11574
\(160\) 0 0
\(161\) −6.78438e7 −1.28121
\(162\) 0 0
\(163\) −4.00702e7 −0.724711 −0.362355 0.932040i \(-0.618027\pi\)
−0.362355 + 0.932040i \(0.618027\pi\)
\(164\) 0 0
\(165\) −9.50877e7 −1.64790
\(166\) 0 0
\(167\) −4.26523e7 −0.708656 −0.354328 0.935121i \(-0.615290\pi\)
−0.354328 + 0.935121i \(0.615290\pi\)
\(168\) 0 0
\(169\) −3.54080e7 −0.564285
\(170\) 0 0
\(171\) 2.24961e7 0.344050
\(172\) 0 0
\(173\) 7.13528e6 0.104773 0.0523865 0.998627i \(-0.483317\pi\)
0.0523865 + 0.998627i \(0.483317\pi\)
\(174\) 0 0
\(175\) 3.68042e8 5.19116
\(176\) 0 0
\(177\) −3.85177e7 −0.522101
\(178\) 0 0
\(179\) −1.39174e8 −1.81373 −0.906866 0.421419i \(-0.861532\pi\)
−0.906866 + 0.421419i \(0.861532\pi\)
\(180\) 0 0
\(181\) −6.11153e7 −0.766082 −0.383041 0.923731i \(-0.625123\pi\)
−0.383041 + 0.923731i \(0.625123\pi\)
\(182\) 0 0
\(183\) −1.00841e8 −1.21635
\(184\) 0 0
\(185\) 2.75811e7 0.320265
\(186\) 0 0
\(187\) 7.07967e7 0.791712
\(188\) 0 0
\(189\) 1.78373e8 1.92182
\(190\) 0 0
\(191\) 1.77318e8 1.84135 0.920676 0.390329i \(-0.127639\pi\)
0.920676 + 0.390329i \(0.127639\pi\)
\(192\) 0 0
\(193\) −6.75041e7 −0.675896 −0.337948 0.941165i \(-0.609733\pi\)
−0.337948 + 0.941165i \(0.609733\pi\)
\(194\) 0 0
\(195\) 8.83163e7 0.852943
\(196\) 0 0
\(197\) −1.03097e8 −0.960761 −0.480380 0.877060i \(-0.659501\pi\)
−0.480380 + 0.877060i \(0.659501\pi\)
\(198\) 0 0
\(199\) −1.78780e8 −1.60817 −0.804086 0.594512i \(-0.797345\pi\)
−0.804086 + 0.594512i \(0.797345\pi\)
\(200\) 0 0
\(201\) −5.17263e7 −0.449288
\(202\) 0 0
\(203\) 3.15273e7 0.264516
\(204\) 0 0
\(205\) 1.43957e7 0.116706
\(206\) 0 0
\(207\) 4.93019e7 0.386338
\(208\) 0 0
\(209\) 1.03402e8 0.783458
\(210\) 0 0
\(211\) −2.20832e8 −1.61836 −0.809178 0.587564i \(-0.800087\pi\)
−0.809178 + 0.587564i \(0.800087\pi\)
\(212\) 0 0
\(213\) 8.14233e6 0.0577325
\(214\) 0 0
\(215\) 2.21611e8 1.52075
\(216\) 0 0
\(217\) 2.85312e8 1.89544
\(218\) 0 0
\(219\) −8.33216e7 −0.536047
\(220\) 0 0
\(221\) −6.57551e7 −0.409786
\(222\) 0 0
\(223\) −2.27015e8 −1.37085 −0.685423 0.728146i \(-0.740382\pi\)
−0.685423 + 0.728146i \(0.740382\pi\)
\(224\) 0 0
\(225\) −2.67456e8 −1.56535
\(226\) 0 0
\(227\) 2.27056e8 1.28838 0.644188 0.764867i \(-0.277195\pi\)
0.644188 + 0.764867i \(0.277195\pi\)
\(228\) 0 0
\(229\) 1.74743e8 0.961557 0.480778 0.876842i \(-0.340354\pi\)
0.480778 + 0.876842i \(0.340354\pi\)
\(230\) 0 0
\(231\) 2.94327e8 1.57105
\(232\) 0 0
\(233\) 2.66617e8 1.38083 0.690417 0.723411i \(-0.257427\pi\)
0.690417 + 0.723411i \(0.257427\pi\)
\(234\) 0 0
\(235\) 9.54755e7 0.479904
\(236\) 0 0
\(237\) 1.12688e7 0.0549868
\(238\) 0 0
\(239\) −7.94730e7 −0.376554 −0.188277 0.982116i \(-0.560290\pi\)
−0.188277 + 0.982116i \(0.560290\pi\)
\(240\) 0 0
\(241\) 5.79710e7 0.266779 0.133389 0.991064i \(-0.457414\pi\)
0.133389 + 0.991064i \(0.457414\pi\)
\(242\) 0 0
\(243\) −2.12713e8 −0.950980
\(244\) 0 0
\(245\) −1.09836e9 −4.77159
\(246\) 0 0
\(247\) −9.60382e7 −0.405513
\(248\) 0 0
\(249\) −2.08153e8 −0.854447
\(250\) 0 0
\(251\) −3.14343e8 −1.25472 −0.627358 0.778731i \(-0.715864\pi\)
−0.627358 + 0.778731i \(0.715864\pi\)
\(252\) 0 0
\(253\) 2.26613e8 0.879756
\(254\) 0 0
\(255\) −2.12405e8 −0.802183
\(256\) 0 0
\(257\) 3.09191e8 1.13622 0.568108 0.822954i \(-0.307675\pi\)
0.568108 + 0.822954i \(0.307675\pi\)
\(258\) 0 0
\(259\) −8.53724e7 −0.305329
\(260\) 0 0
\(261\) −2.29109e7 −0.0797627
\(262\) 0 0
\(263\) 3.09539e8 1.04923 0.524614 0.851340i \(-0.324210\pi\)
0.524614 + 0.851340i \(0.324210\pi\)
\(264\) 0 0
\(265\) 9.92718e8 3.27692
\(266\) 0 0
\(267\) −3.60080e8 −1.15774
\(268\) 0 0
\(269\) −2.59451e8 −0.812686 −0.406343 0.913721i \(-0.633196\pi\)
−0.406343 + 0.913721i \(0.633196\pi\)
\(270\) 0 0
\(271\) −7.90601e7 −0.241304 −0.120652 0.992695i \(-0.538499\pi\)
−0.120652 + 0.992695i \(0.538499\pi\)
\(272\) 0 0
\(273\) −2.73368e8 −0.813164
\(274\) 0 0
\(275\) −1.22934e9 −3.56457
\(276\) 0 0
\(277\) 2.41348e8 0.682283 0.341142 0.940012i \(-0.389186\pi\)
0.341142 + 0.940012i \(0.389186\pi\)
\(278\) 0 0
\(279\) −2.07336e8 −0.571557
\(280\) 0 0
\(281\) −1.24454e8 −0.334609 −0.167304 0.985905i \(-0.553506\pi\)
−0.167304 + 0.985905i \(0.553506\pi\)
\(282\) 0 0
\(283\) −4.05747e8 −1.06415 −0.532074 0.846698i \(-0.678587\pi\)
−0.532074 + 0.846698i \(0.678587\pi\)
\(284\) 0 0
\(285\) −3.10226e8 −0.793819
\(286\) 0 0
\(287\) −4.45593e7 −0.111263
\(288\) 0 0
\(289\) −2.52195e8 −0.614601
\(290\) 0 0
\(291\) −5.07128e8 −1.20640
\(292\) 0 0
\(293\) 4.99201e8 1.15941 0.579707 0.814825i \(-0.303167\pi\)
0.579707 + 0.814825i \(0.303167\pi\)
\(294\) 0 0
\(295\) −6.76137e8 −1.53341
\(296\) 0 0
\(297\) −5.95803e8 −1.31964
\(298\) 0 0
\(299\) −2.10475e8 −0.455357
\(300\) 0 0
\(301\) −6.85959e8 −1.44983
\(302\) 0 0
\(303\) −2.41781e8 −0.499312
\(304\) 0 0
\(305\) −1.77016e9 −3.57242
\(306\) 0 0
\(307\) 3.21540e8 0.634235 0.317118 0.948386i \(-0.397285\pi\)
0.317118 + 0.948386i \(0.397285\pi\)
\(308\) 0 0
\(309\) 2.52045e8 0.485986
\(310\) 0 0
\(311\) 1.03297e8 0.194726 0.0973631 0.995249i \(-0.468959\pi\)
0.0973631 + 0.995249i \(0.468959\pi\)
\(312\) 0 0
\(313\) 5.25918e8 0.969422 0.484711 0.874674i \(-0.338925\pi\)
0.484711 + 0.874674i \(0.338925\pi\)
\(314\) 0 0
\(315\) 1.12405e9 2.02627
\(316\) 0 0
\(317\) −6.26026e7 −0.110379 −0.0551893 0.998476i \(-0.517576\pi\)
−0.0551893 + 0.998476i \(0.517576\pi\)
\(318\) 0 0
\(319\) −1.05308e8 −0.181633
\(320\) 0 0
\(321\) 3.57802e7 0.0603775
\(322\) 0 0
\(323\) 2.30976e8 0.381381
\(324\) 0 0
\(325\) 1.14180e9 1.84500
\(326\) 0 0
\(327\) 2.66877e8 0.422079
\(328\) 0 0
\(329\) −2.95528e8 −0.457523
\(330\) 0 0
\(331\) 3.10780e8 0.471037 0.235518 0.971870i \(-0.424321\pi\)
0.235518 + 0.971870i \(0.424321\pi\)
\(332\) 0 0
\(333\) 6.20400e7 0.0920697
\(334\) 0 0
\(335\) −9.07998e8 −1.31956
\(336\) 0 0
\(337\) −9.56000e8 −1.36067 −0.680335 0.732901i \(-0.738166\pi\)
−0.680335 + 0.732901i \(0.738166\pi\)
\(338\) 0 0
\(339\) 2.40952e8 0.335916
\(340\) 0 0
\(341\) −9.53003e8 −1.30153
\(342\) 0 0
\(343\) 2.01175e9 2.69181
\(344\) 0 0
\(345\) −6.79885e8 −0.891392
\(346\) 0 0
\(347\) −1.37536e9 −1.76711 −0.883555 0.468327i \(-0.844857\pi\)
−0.883555 + 0.468327i \(0.844857\pi\)
\(348\) 0 0
\(349\) −1.38134e9 −1.73945 −0.869725 0.493537i \(-0.835704\pi\)
−0.869725 + 0.493537i \(0.835704\pi\)
\(350\) 0 0
\(351\) 5.53374e8 0.683037
\(352\) 0 0
\(353\) −6.48672e8 −0.784899 −0.392450 0.919774i \(-0.628372\pi\)
−0.392450 + 0.919774i \(0.628372\pi\)
\(354\) 0 0
\(355\) 1.42930e8 0.169560
\(356\) 0 0
\(357\) 6.57462e8 0.764772
\(358\) 0 0
\(359\) −3.64068e8 −0.415291 −0.207645 0.978204i \(-0.566580\pi\)
−0.207645 + 0.978204i \(0.566580\pi\)
\(360\) 0 0
\(361\) −5.56521e8 −0.622596
\(362\) 0 0
\(363\) −3.78638e8 −0.415481
\(364\) 0 0
\(365\) −1.46262e9 −1.57437
\(366\) 0 0
\(367\) −3.90647e8 −0.412528 −0.206264 0.978496i \(-0.566131\pi\)
−0.206264 + 0.978496i \(0.566131\pi\)
\(368\) 0 0
\(369\) 3.23812e7 0.0335506
\(370\) 0 0
\(371\) −3.07279e9 −3.12409
\(372\) 0 0
\(373\) −8.58735e7 −0.0856797 −0.0428399 0.999082i \(-0.513641\pi\)
−0.0428399 + 0.999082i \(0.513641\pi\)
\(374\) 0 0
\(375\) 2.36872e9 2.31955
\(376\) 0 0
\(377\) 9.78089e7 0.0940121
\(378\) 0 0
\(379\) −3.53330e8 −0.333383 −0.166691 0.986009i \(-0.553308\pi\)
−0.166691 + 0.986009i \(0.553308\pi\)
\(380\) 0 0
\(381\) 1.39443e8 0.129169
\(382\) 0 0
\(383\) 4.90315e8 0.445943 0.222971 0.974825i \(-0.428424\pi\)
0.222971 + 0.974825i \(0.428424\pi\)
\(384\) 0 0
\(385\) 5.16660e9 4.61415
\(386\) 0 0
\(387\) 4.98486e8 0.437184
\(388\) 0 0
\(389\) 9.97474e7 0.0859168 0.0429584 0.999077i \(-0.486322\pi\)
0.0429584 + 0.999077i \(0.486322\pi\)
\(390\) 0 0
\(391\) 5.06202e8 0.428258
\(392\) 0 0
\(393\) −4.88988e8 −0.406372
\(394\) 0 0
\(395\) 1.97812e8 0.161496
\(396\) 0 0
\(397\) −2.09994e8 −0.168438 −0.0842189 0.996447i \(-0.526839\pi\)
−0.0842189 + 0.996447i \(0.526839\pi\)
\(398\) 0 0
\(399\) 9.60252e8 0.756798
\(400\) 0 0
\(401\) 5.55414e8 0.430141 0.215071 0.976598i \(-0.431002\pi\)
0.215071 + 0.976598i \(0.431002\pi\)
\(402\) 0 0
\(403\) 8.85138e8 0.673664
\(404\) 0 0
\(405\) 3.28982e8 0.246082
\(406\) 0 0
\(407\) 2.85162e8 0.209658
\(408\) 0 0
\(409\) 1.80096e9 1.30158 0.650792 0.759256i \(-0.274437\pi\)
0.650792 + 0.759256i \(0.274437\pi\)
\(410\) 0 0
\(411\) 2.87793e8 0.204472
\(412\) 0 0
\(413\) 2.09286e9 1.46189
\(414\) 0 0
\(415\) −3.65390e9 −2.50951
\(416\) 0 0
\(417\) 1.64527e9 1.11112
\(418\) 0 0
\(419\) 1.69929e7 0.0112854 0.00564272 0.999984i \(-0.498204\pi\)
0.00564272 + 0.999984i \(0.498204\pi\)
\(420\) 0 0
\(421\) 2.76374e9 1.80514 0.902568 0.430548i \(-0.141680\pi\)
0.902568 + 0.430548i \(0.141680\pi\)
\(422\) 0 0
\(423\) 2.14760e8 0.137963
\(424\) 0 0
\(425\) −2.74607e9 −1.73520
\(426\) 0 0
\(427\) 5.47921e9 3.40581
\(428\) 0 0
\(429\) 9.13107e8 0.558369
\(430\) 0 0
\(431\) 2.95133e9 1.77561 0.887804 0.460222i \(-0.152230\pi\)
0.887804 + 0.460222i \(0.152230\pi\)
\(432\) 0 0
\(433\) −2.29366e9 −1.35776 −0.678878 0.734251i \(-0.737533\pi\)
−0.678878 + 0.734251i \(0.737533\pi\)
\(434\) 0 0
\(435\) 3.15946e8 0.184035
\(436\) 0 0
\(437\) 7.39331e8 0.423793
\(438\) 0 0
\(439\) −2.21394e9 −1.24893 −0.624467 0.781052i \(-0.714684\pi\)
−0.624467 + 0.781052i \(0.714684\pi\)
\(440\) 0 0
\(441\) −2.47061e9 −1.37173
\(442\) 0 0
\(443\) 2.80148e9 1.53100 0.765498 0.643438i \(-0.222493\pi\)
0.765498 + 0.643438i \(0.222493\pi\)
\(444\) 0 0
\(445\) −6.32082e9 −3.40027
\(446\) 0 0
\(447\) −8.10713e8 −0.429330
\(448\) 0 0
\(449\) −1.24071e9 −0.646856 −0.323428 0.946253i \(-0.604835\pi\)
−0.323428 + 0.946253i \(0.604835\pi\)
\(450\) 0 0
\(451\) 1.48838e8 0.0764003
\(452\) 0 0
\(453\) −1.03220e9 −0.521698
\(454\) 0 0
\(455\) −4.79867e9 −2.38826
\(456\) 0 0
\(457\) 8.16764e8 0.400304 0.200152 0.979765i \(-0.435856\pi\)
0.200152 + 0.979765i \(0.435856\pi\)
\(458\) 0 0
\(459\) −1.33089e9 −0.642389
\(460\) 0 0
\(461\) 1.07694e9 0.511962 0.255981 0.966682i \(-0.417601\pi\)
0.255981 + 0.966682i \(0.417601\pi\)
\(462\) 0 0
\(463\) −3.16535e9 −1.48214 −0.741068 0.671430i \(-0.765680\pi\)
−0.741068 + 0.671430i \(0.765680\pi\)
\(464\) 0 0
\(465\) 2.85921e9 1.31874
\(466\) 0 0
\(467\) −2.37777e9 −1.08034 −0.540171 0.841555i \(-0.681640\pi\)
−0.540171 + 0.841555i \(0.681640\pi\)
\(468\) 0 0
\(469\) 2.81055e9 1.25802
\(470\) 0 0
\(471\) 6.45614e8 0.284708
\(472\) 0 0
\(473\) 2.29125e9 0.995540
\(474\) 0 0
\(475\) −4.01076e9 −1.71711
\(476\) 0 0
\(477\) 2.23299e9 0.942047
\(478\) 0 0
\(479\) −3.68467e9 −1.53188 −0.765939 0.642914i \(-0.777725\pi\)
−0.765939 + 0.642914i \(0.777725\pi\)
\(480\) 0 0
\(481\) −2.64855e8 −0.108518
\(482\) 0 0
\(483\) 2.10447e9 0.849820
\(484\) 0 0
\(485\) −8.90209e9 −3.54320
\(486\) 0 0
\(487\) 1.99019e9 0.780807 0.390404 0.920644i \(-0.372335\pi\)
0.390404 + 0.920644i \(0.372335\pi\)
\(488\) 0 0
\(489\) 1.24295e9 0.480698
\(490\) 0 0
\(491\) 1.03457e9 0.394433 0.197217 0.980360i \(-0.436810\pi\)
0.197217 + 0.980360i \(0.436810\pi\)
\(492\) 0 0
\(493\) −2.35235e8 −0.0884173
\(494\) 0 0
\(495\) −3.75456e9 −1.39136
\(496\) 0 0
\(497\) −4.42414e8 −0.161652
\(498\) 0 0
\(499\) 1.69821e9 0.611843 0.305922 0.952057i \(-0.401035\pi\)
0.305922 + 0.952057i \(0.401035\pi\)
\(500\) 0 0
\(501\) 1.32305e9 0.470049
\(502\) 0 0
\(503\) −9.98691e8 −0.349899 −0.174950 0.984577i \(-0.555976\pi\)
−0.174950 + 0.984577i \(0.555976\pi\)
\(504\) 0 0
\(505\) −4.24420e9 −1.46648
\(506\) 0 0
\(507\) 1.09833e9 0.374288
\(508\) 0 0
\(509\) 4.86581e9 1.63547 0.817735 0.575594i \(-0.195229\pi\)
0.817735 + 0.575594i \(0.195229\pi\)
\(510\) 0 0
\(511\) 4.52729e9 1.50094
\(512\) 0 0
\(513\) −1.94382e9 −0.635691
\(514\) 0 0
\(515\) 4.42438e9 1.42734
\(516\) 0 0
\(517\) 9.87127e8 0.314164
\(518\) 0 0
\(519\) −2.21331e8 −0.0694956
\(520\) 0 0
\(521\) −2.29363e9 −0.710544 −0.355272 0.934763i \(-0.615612\pi\)
−0.355272 + 0.934763i \(0.615612\pi\)
\(522\) 0 0
\(523\) 1.41986e8 0.0434000 0.0217000 0.999765i \(-0.493092\pi\)
0.0217000 + 0.999765i \(0.493092\pi\)
\(524\) 0 0
\(525\) −1.14164e10 −3.44328
\(526\) 0 0
\(527\) −2.12880e9 −0.633574
\(528\) 0 0
\(529\) −1.78453e9 −0.524117
\(530\) 0 0
\(531\) −1.52088e9 −0.440823
\(532\) 0 0
\(533\) −1.38239e8 −0.0395444
\(534\) 0 0
\(535\) 6.28082e8 0.177328
\(536\) 0 0
\(537\) 4.31709e9 1.20304
\(538\) 0 0
\(539\) −1.13560e10 −3.12366
\(540\) 0 0
\(541\) 6.73035e9 1.82746 0.913729 0.406324i \(-0.133190\pi\)
0.913729 + 0.406324i \(0.133190\pi\)
\(542\) 0 0
\(543\) 1.89575e9 0.508139
\(544\) 0 0
\(545\) 4.68474e9 1.23964
\(546\) 0 0
\(547\) 2.10936e9 0.551056 0.275528 0.961293i \(-0.411147\pi\)
0.275528 + 0.961293i \(0.411147\pi\)
\(548\) 0 0
\(549\) −3.98173e9 −1.02700
\(550\) 0 0
\(551\) −3.43571e8 −0.0874955
\(552\) 0 0
\(553\) −6.12291e8 −0.153964
\(554\) 0 0
\(555\) −8.55545e8 −0.212431
\(556\) 0 0
\(557\) −5.46338e9 −1.33958 −0.669790 0.742551i \(-0.733616\pi\)
−0.669790 + 0.742551i \(0.733616\pi\)
\(558\) 0 0
\(559\) −2.12809e9 −0.515286
\(560\) 0 0
\(561\) −2.19606e9 −0.525140
\(562\) 0 0
\(563\) 2.38438e9 0.563113 0.281557 0.959545i \(-0.409149\pi\)
0.281557 + 0.959545i \(0.409149\pi\)
\(564\) 0 0
\(565\) 4.22964e9 0.986584
\(566\) 0 0
\(567\) −1.01831e9 −0.234605
\(568\) 0 0
\(569\) 4.69886e9 1.06930 0.534650 0.845073i \(-0.320443\pi\)
0.534650 + 0.845073i \(0.320443\pi\)
\(570\) 0 0
\(571\) −2.04617e9 −0.459954 −0.229977 0.973196i \(-0.573865\pi\)
−0.229977 + 0.973196i \(0.573865\pi\)
\(572\) 0 0
\(573\) −5.50029e9 −1.22136
\(574\) 0 0
\(575\) −8.78988e9 −1.92817
\(576\) 0 0
\(577\) −6.13969e9 −1.33055 −0.665274 0.746599i \(-0.731685\pi\)
−0.665274 + 0.746599i \(0.731685\pi\)
\(578\) 0 0
\(579\) 2.09393e9 0.448319
\(580\) 0 0
\(581\) 1.13100e10 2.39247
\(582\) 0 0
\(583\) 1.02638e10 2.14520
\(584\) 0 0
\(585\) 3.48719e9 0.720161
\(586\) 0 0
\(587\) 1.29772e9 0.264819 0.132410 0.991195i \(-0.457729\pi\)
0.132410 + 0.991195i \(0.457729\pi\)
\(588\) 0 0
\(589\) −3.10920e9 −0.626968
\(590\) 0 0
\(591\) 3.19800e9 0.637269
\(592\) 0 0
\(593\) −1.01765e9 −0.200404 −0.100202 0.994967i \(-0.531949\pi\)
−0.100202 + 0.994967i \(0.531949\pi\)
\(594\) 0 0
\(595\) 1.15410e10 2.24613
\(596\) 0 0
\(597\) 5.54562e9 1.06670
\(598\) 0 0
\(599\) −1.36024e9 −0.258595 −0.129298 0.991606i \(-0.541272\pi\)
−0.129298 + 0.991606i \(0.541272\pi\)
\(600\) 0 0
\(601\) 5.83436e9 1.09631 0.548154 0.836378i \(-0.315331\pi\)
0.548154 + 0.836378i \(0.315331\pi\)
\(602\) 0 0
\(603\) −2.04242e9 −0.379346
\(604\) 0 0
\(605\) −6.64658e9 −1.22027
\(606\) 0 0
\(607\) −4.09613e9 −0.743384 −0.371692 0.928356i \(-0.621222\pi\)
−0.371692 + 0.928356i \(0.621222\pi\)
\(608\) 0 0
\(609\) −9.77956e8 −0.175452
\(610\) 0 0
\(611\) −9.16831e8 −0.162609
\(612\) 0 0
\(613\) −5.20044e8 −0.0911861 −0.0455930 0.998960i \(-0.514518\pi\)
−0.0455930 + 0.998960i \(0.514518\pi\)
\(614\) 0 0
\(615\) −4.46544e8 −0.0774108
\(616\) 0 0
\(617\) 3.87499e9 0.664160 0.332080 0.943251i \(-0.392250\pi\)
0.332080 + 0.943251i \(0.392250\pi\)
\(618\) 0 0
\(619\) 6.93110e9 1.17459 0.587293 0.809375i \(-0.300194\pi\)
0.587293 + 0.809375i \(0.300194\pi\)
\(620\) 0 0
\(621\) −4.26004e9 −0.713827
\(622\) 0 0
\(623\) 1.95650e10 3.24169
\(624\) 0 0
\(625\) 2.45204e10 4.01742
\(626\) 0 0
\(627\) −3.20745e9 −0.519665
\(628\) 0 0
\(629\) 6.36988e8 0.102060
\(630\) 0 0
\(631\) −1.27564e9 −0.202128 −0.101064 0.994880i \(-0.532225\pi\)
−0.101064 + 0.994880i \(0.532225\pi\)
\(632\) 0 0
\(633\) 6.85006e9 1.07345
\(634\) 0 0
\(635\) 2.44777e9 0.379369
\(636\) 0 0
\(637\) 1.05473e10 1.61679
\(638\) 0 0
\(639\) 3.21502e8 0.0487450
\(640\) 0 0
\(641\) 8.45969e9 1.26868 0.634339 0.773055i \(-0.281272\pi\)
0.634339 + 0.773055i \(0.281272\pi\)
\(642\) 0 0
\(643\) 2.53383e9 0.375871 0.187935 0.982181i \(-0.439820\pi\)
0.187935 + 0.982181i \(0.439820\pi\)
\(644\) 0 0
\(645\) −6.87423e9 −1.00871
\(646\) 0 0
\(647\) 9.30854e8 0.135119 0.0675595 0.997715i \(-0.478479\pi\)
0.0675595 + 0.997715i \(0.478479\pi\)
\(648\) 0 0
\(649\) −6.99061e9 −1.00383
\(650\) 0 0
\(651\) −8.85018e9 −1.25724
\(652\) 0 0
\(653\) 4.54702e9 0.639044 0.319522 0.947579i \(-0.396478\pi\)
0.319522 + 0.947579i \(0.396478\pi\)
\(654\) 0 0
\(655\) −8.58365e9 −1.19351
\(656\) 0 0
\(657\) −3.28997e9 −0.452599
\(658\) 0 0
\(659\) −5.26043e9 −0.716015 −0.358007 0.933719i \(-0.616544\pi\)
−0.358007 + 0.933719i \(0.616544\pi\)
\(660\) 0 0
\(661\) −9.97829e8 −0.134385 −0.0671925 0.997740i \(-0.521404\pi\)
−0.0671925 + 0.997740i \(0.521404\pi\)
\(662\) 0 0
\(663\) 2.03968e9 0.271809
\(664\) 0 0
\(665\) 1.68562e10 2.22271
\(666\) 0 0
\(667\) −7.52962e8 −0.0982500
\(668\) 0 0
\(669\) 7.04186e9 0.909276
\(670\) 0 0
\(671\) −1.83017e10 −2.33864
\(672\) 0 0
\(673\) −2.48723e9 −0.314530 −0.157265 0.987556i \(-0.550268\pi\)
−0.157265 + 0.987556i \(0.550268\pi\)
\(674\) 0 0
\(675\) 2.31101e10 2.89226
\(676\) 0 0
\(677\) −6.47115e9 −0.801532 −0.400766 0.916180i \(-0.631256\pi\)
−0.400766 + 0.916180i \(0.631256\pi\)
\(678\) 0 0
\(679\) 2.75549e10 3.37796
\(680\) 0 0
\(681\) −7.04312e9 −0.854575
\(682\) 0 0
\(683\) 4.86398e9 0.584143 0.292071 0.956397i \(-0.405655\pi\)
0.292071 + 0.956397i \(0.405655\pi\)
\(684\) 0 0
\(685\) 5.05189e9 0.600533
\(686\) 0 0
\(687\) −5.42040e9 −0.637797
\(688\) 0 0
\(689\) −9.53287e9 −1.11034
\(690\) 0 0
\(691\) −7.51546e9 −0.866527 −0.433264 0.901267i \(-0.642638\pi\)
−0.433264 + 0.901267i \(0.642638\pi\)
\(692\) 0 0
\(693\) 1.16216e10 1.32647
\(694\) 0 0
\(695\) 2.88810e10 3.26336
\(696\) 0 0
\(697\) 3.32470e8 0.0371910
\(698\) 0 0
\(699\) −8.27026e9 −0.915902
\(700\) 0 0
\(701\) −1.02509e10 −1.12396 −0.561978 0.827152i \(-0.689960\pi\)
−0.561978 + 0.827152i \(0.689960\pi\)
\(702\) 0 0
\(703\) 9.30350e8 0.100996
\(704\) 0 0
\(705\) −2.96158e9 −0.318319
\(706\) 0 0
\(707\) 1.31372e10 1.39809
\(708\) 0 0
\(709\) 5.01078e9 0.528012 0.264006 0.964521i \(-0.414956\pi\)
0.264006 + 0.964521i \(0.414956\pi\)
\(710\) 0 0
\(711\) 4.44951e8 0.0464268
\(712\) 0 0
\(713\) −6.81405e9 −0.704032
\(714\) 0 0
\(715\) 1.60286e10 1.63993
\(716\) 0 0
\(717\) 2.46520e9 0.249767
\(718\) 0 0
\(719\) −1.55089e10 −1.55607 −0.778035 0.628221i \(-0.783783\pi\)
−0.778035 + 0.628221i \(0.783783\pi\)
\(720\) 0 0
\(721\) −1.36949e10 −1.36077
\(722\) 0 0
\(723\) −1.79822e9 −0.176953
\(724\) 0 0
\(725\) 4.08470e9 0.398086
\(726\) 0 0
\(727\) 9.09964e8 0.0878322 0.0439161 0.999035i \(-0.486017\pi\)
0.0439161 + 0.999035i \(0.486017\pi\)
\(728\) 0 0
\(729\) 7.91954e9 0.757101
\(730\) 0 0
\(731\) 5.11814e9 0.484620
\(732\) 0 0
\(733\) −6.80412e9 −0.638128 −0.319064 0.947733i \(-0.603369\pi\)
−0.319064 + 0.947733i \(0.603369\pi\)
\(734\) 0 0
\(735\) 3.40703e10 3.16498
\(736\) 0 0
\(737\) −9.38784e9 −0.863832
\(738\) 0 0
\(739\) −9.88530e9 −0.901020 −0.450510 0.892771i \(-0.648758\pi\)
−0.450510 + 0.892771i \(0.648758\pi\)
\(740\) 0 0
\(741\) 2.97904e9 0.268975
\(742\) 0 0
\(743\) 1.40609e10 1.25763 0.628814 0.777556i \(-0.283541\pi\)
0.628814 + 0.777556i \(0.283541\pi\)
\(744\) 0 0
\(745\) −1.42312e10 −1.26094
\(746\) 0 0
\(747\) −8.21897e9 −0.721432
\(748\) 0 0
\(749\) −1.94412e9 −0.169058
\(750\) 0 0
\(751\) −1.89107e10 −1.62918 −0.814590 0.580038i \(-0.803038\pi\)
−0.814590 + 0.580038i \(0.803038\pi\)
\(752\) 0 0
\(753\) 9.75069e9 0.832248
\(754\) 0 0
\(755\) −1.81191e10 −1.53222
\(756\) 0 0
\(757\) −1.68703e9 −0.141348 −0.0706738 0.997499i \(-0.522515\pi\)
−0.0706738 + 0.997499i \(0.522515\pi\)
\(758\) 0 0
\(759\) −7.02936e9 −0.583539
\(760\) 0 0
\(761\) 1.35093e10 1.11118 0.555591 0.831456i \(-0.312492\pi\)
0.555591 + 0.831456i \(0.312492\pi\)
\(762\) 0 0
\(763\) −1.45008e10 −1.18183
\(764\) 0 0
\(765\) −8.38685e9 −0.677304
\(766\) 0 0
\(767\) 6.49280e9 0.519575
\(768\) 0 0
\(769\) −2.04375e10 −1.62064 −0.810318 0.585990i \(-0.800706\pi\)
−0.810318 + 0.585990i \(0.800706\pi\)
\(770\) 0 0
\(771\) −9.59089e9 −0.753648
\(772\) 0 0
\(773\) −3.22437e9 −0.251083 −0.125541 0.992088i \(-0.540067\pi\)
−0.125541 + 0.992088i \(0.540067\pi\)
\(774\) 0 0
\(775\) 3.69652e10 2.85258
\(776\) 0 0
\(777\) 2.64819e9 0.202524
\(778\) 0 0
\(779\) 4.85588e8 0.0368033
\(780\) 0 0
\(781\) 1.47776e9 0.111000
\(782\) 0 0
\(783\) 1.97966e9 0.147375
\(784\) 0 0
\(785\) 1.13331e10 0.836187
\(786\) 0 0
\(787\) 1.34167e10 0.981147 0.490574 0.871400i \(-0.336787\pi\)
0.490574 + 0.871400i \(0.336787\pi\)
\(788\) 0 0
\(789\) −9.60167e9 −0.695949
\(790\) 0 0
\(791\) −1.30921e10 −0.940573
\(792\) 0 0
\(793\) 1.69984e10 1.21047
\(794\) 0 0
\(795\) −3.07934e10 −2.17357
\(796\) 0 0
\(797\) 1.04838e10 0.733528 0.366764 0.930314i \(-0.380466\pi\)
0.366764 + 0.930314i \(0.380466\pi\)
\(798\) 0 0
\(799\) 2.20502e9 0.152932
\(800\) 0 0
\(801\) −1.42179e10 −0.977508
\(802\) 0 0
\(803\) −1.51221e10 −1.03064
\(804\) 0 0
\(805\) 3.69416e10 2.49592
\(806\) 0 0
\(807\) 8.04800e9 0.539052
\(808\) 0 0
\(809\) 8.66031e9 0.575060 0.287530 0.957772i \(-0.407166\pi\)
0.287530 + 0.957772i \(0.407166\pi\)
\(810\) 0 0
\(811\) −7.78352e9 −0.512393 −0.256196 0.966625i \(-0.582469\pi\)
−0.256196 + 0.966625i \(0.582469\pi\)
\(812\) 0 0
\(813\) 2.45239e9 0.160056
\(814\) 0 0
\(815\) 2.18186e10 1.41181
\(816\) 0 0
\(817\) 7.47528e9 0.479568
\(818\) 0 0
\(819\) −1.07940e10 −0.686575
\(820\) 0 0
\(821\) 7.91243e9 0.499009 0.249505 0.968374i \(-0.419732\pi\)
0.249505 + 0.968374i \(0.419732\pi\)
\(822\) 0 0
\(823\) 4.48091e9 0.280199 0.140100 0.990137i \(-0.455258\pi\)
0.140100 + 0.990137i \(0.455258\pi\)
\(824\) 0 0
\(825\) 3.81332e10 2.36437
\(826\) 0 0
\(827\) 7.15443e8 0.0439851 0.0219926 0.999758i \(-0.492999\pi\)
0.0219926 + 0.999758i \(0.492999\pi\)
\(828\) 0 0
\(829\) −2.36866e10 −1.44398 −0.721990 0.691903i \(-0.756773\pi\)
−0.721990 + 0.691903i \(0.756773\pi\)
\(830\) 0 0
\(831\) −7.48645e9 −0.452556
\(832\) 0 0
\(833\) −2.53668e10 −1.52057
\(834\) 0 0
\(835\) 2.32246e10 1.38053
\(836\) 0 0
\(837\) 1.79153e10 1.05605
\(838\) 0 0
\(839\) −2.37629e10 −1.38909 −0.694547 0.719447i \(-0.744395\pi\)
−0.694547 + 0.719447i \(0.744395\pi\)
\(840\) 0 0
\(841\) −1.69000e10 −0.979715
\(842\) 0 0
\(843\) 3.86048e9 0.221945
\(844\) 0 0
\(845\) 1.92800e10 1.09928
\(846\) 0 0
\(847\) 2.05733e10 1.16336
\(848\) 0 0
\(849\) 1.25860e10 0.705846
\(850\) 0 0
\(851\) 2.03893e9 0.113409
\(852\) 0 0
\(853\) −3.28503e10 −1.81225 −0.906126 0.423009i \(-0.860974\pi\)
−0.906126 + 0.423009i \(0.860974\pi\)
\(854\) 0 0
\(855\) −1.22494e10 −0.670242
\(856\) 0 0
\(857\) −5.67832e9 −0.308168 −0.154084 0.988058i \(-0.549243\pi\)
−0.154084 + 0.988058i \(0.549243\pi\)
\(858\) 0 0
\(859\) 1.22964e10 0.661914 0.330957 0.943646i \(-0.392628\pi\)
0.330957 + 0.943646i \(0.392628\pi\)
\(860\) 0 0
\(861\) 1.38220e9 0.0738006
\(862\) 0 0
\(863\) −1.75550e10 −0.929743 −0.464871 0.885378i \(-0.653899\pi\)
−0.464871 + 0.885378i \(0.653899\pi\)
\(864\) 0 0
\(865\) −3.88523e9 −0.204108
\(866\) 0 0
\(867\) 7.82290e9 0.407663
\(868\) 0 0
\(869\) 2.04518e9 0.105721
\(870\) 0 0
\(871\) 8.71931e9 0.447115
\(872\) 0 0
\(873\) −2.00241e10 −1.01860
\(874\) 0 0
\(875\) −1.28704e11 −6.49479
\(876\) 0 0
\(877\) −9.75144e7 −0.00488169 −0.00244084 0.999997i \(-0.500777\pi\)
−0.00244084 + 0.999997i \(0.500777\pi\)
\(878\) 0 0
\(879\) −1.54848e10 −0.769035
\(880\) 0 0
\(881\) −1.68618e10 −0.830786 −0.415393 0.909642i \(-0.636356\pi\)
−0.415393 + 0.909642i \(0.636356\pi\)
\(882\) 0 0
\(883\) −2.81287e10 −1.37495 −0.687476 0.726207i \(-0.741281\pi\)
−0.687476 + 0.726207i \(0.741281\pi\)
\(884\) 0 0
\(885\) 2.09733e10 1.01710
\(886\) 0 0
\(887\) −1.49073e10 −0.717241 −0.358621 0.933483i \(-0.616753\pi\)
−0.358621 + 0.933483i \(0.616753\pi\)
\(888\) 0 0
\(889\) −7.57664e9 −0.361676
\(890\) 0 0
\(891\) 3.40136e9 0.161095
\(892\) 0 0
\(893\) 3.22053e9 0.151338
\(894\) 0 0
\(895\) 7.57817e10 3.53333
\(896\) 0 0
\(897\) 6.52879e9 0.302036
\(898\) 0 0
\(899\) 3.16653e9 0.145353
\(900\) 0 0
\(901\) 2.29270e10 1.04426
\(902\) 0 0
\(903\) 2.12780e10 0.961664
\(904\) 0 0
\(905\) 3.32779e10 1.49240
\(906\) 0 0
\(907\) −2.00171e10 −0.890790 −0.445395 0.895334i \(-0.646937\pi\)
−0.445395 + 0.895334i \(0.646937\pi\)
\(908\) 0 0
\(909\) −9.54676e9 −0.421582
\(910\) 0 0
\(911\) 1.55429e10 0.681113 0.340556 0.940224i \(-0.389385\pi\)
0.340556 + 0.940224i \(0.389385\pi\)
\(912\) 0 0
\(913\) −3.77779e10 −1.64282
\(914\) 0 0
\(915\) 5.49090e10 2.36957
\(916\) 0 0
\(917\) 2.65692e10 1.13785
\(918\) 0 0
\(919\) 7.07850e9 0.300841 0.150420 0.988622i \(-0.451937\pi\)
0.150420 + 0.988622i \(0.451937\pi\)
\(920\) 0 0
\(921\) −9.97394e9 −0.420686
\(922\) 0 0
\(923\) −1.37252e9 −0.0574532
\(924\) 0 0
\(925\) −1.10609e10 −0.459509
\(926\) 0 0
\(927\) 9.95205e9 0.410330
\(928\) 0 0
\(929\) −9.26783e9 −0.379248 −0.189624 0.981857i \(-0.560727\pi\)
−0.189624 + 0.981857i \(0.560727\pi\)
\(930\) 0 0
\(931\) −3.70493e10 −1.50472
\(932\) 0 0
\(933\) −3.20419e9 −0.129161
\(934\) 0 0
\(935\) −3.85495e10 −1.54233
\(936\) 0 0
\(937\) −2.95508e10 −1.17349 −0.586746 0.809771i \(-0.699591\pi\)
−0.586746 + 0.809771i \(0.699591\pi\)
\(938\) 0 0
\(939\) −1.63136e10 −0.643014
\(940\) 0 0
\(941\) 1.39298e10 0.544979 0.272489 0.962159i \(-0.412153\pi\)
0.272489 + 0.962159i \(0.412153\pi\)
\(942\) 0 0
\(943\) 1.06420e9 0.0413269
\(944\) 0 0
\(945\) −9.71257e10 −3.74389
\(946\) 0 0
\(947\) 2.48316e10 0.950122 0.475061 0.879953i \(-0.342426\pi\)
0.475061 + 0.879953i \(0.342426\pi\)
\(948\) 0 0
\(949\) 1.40452e10 0.533454
\(950\) 0 0
\(951\) 1.94189e9 0.0732137
\(952\) 0 0
\(953\) −3.01791e10 −1.12949 −0.564744 0.825266i \(-0.691025\pi\)
−0.564744 + 0.825266i \(0.691025\pi\)
\(954\) 0 0
\(955\) −9.65515e10 −3.58713
\(956\) 0 0
\(957\) 3.26658e9 0.120476
\(958\) 0 0
\(959\) −1.56373e10 −0.572526
\(960\) 0 0
\(961\) 1.14339e9 0.0415589
\(962\) 0 0
\(963\) 1.41279e9 0.0509783
\(964\) 0 0
\(965\) 3.67567e10 1.31671
\(966\) 0 0
\(967\) 2.93661e10 1.04437 0.522184 0.852833i \(-0.325118\pi\)
0.522184 + 0.852833i \(0.325118\pi\)
\(968\) 0 0
\(969\) −7.16472e9 −0.252968
\(970\) 0 0
\(971\) 2.39542e10 0.839683 0.419841 0.907598i \(-0.362086\pi\)
0.419841 + 0.907598i \(0.362086\pi\)
\(972\) 0 0
\(973\) −8.93961e10 −3.11117
\(974\) 0 0
\(975\) −3.54177e10 −1.22378
\(976\) 0 0
\(977\) 2.91663e10 1.00058 0.500289 0.865859i \(-0.333227\pi\)
0.500289 + 0.865859i \(0.333227\pi\)
\(978\) 0 0
\(979\) −6.53513e10 −2.22595
\(980\) 0 0
\(981\) 1.05377e10 0.356372
\(982\) 0 0
\(983\) −4.44637e9 −0.149303 −0.0746514 0.997210i \(-0.523784\pi\)
−0.0746514 + 0.997210i \(0.523784\pi\)
\(984\) 0 0
\(985\) 5.61375e10 1.87166
\(986\) 0 0
\(987\) 9.16707e9 0.303473
\(988\) 0 0
\(989\) 1.63826e10 0.538514
\(990\) 0 0
\(991\) −3.36917e10 −1.09968 −0.549838 0.835271i \(-0.685311\pi\)
−0.549838 + 0.835271i \(0.685311\pi\)
\(992\) 0 0
\(993\) −9.64017e9 −0.312437
\(994\) 0 0
\(995\) 9.73474e10 3.13288
\(996\) 0 0
\(997\) −7.98360e9 −0.255132 −0.127566 0.991830i \(-0.540717\pi\)
−0.127566 + 0.991830i \(0.540717\pi\)
\(998\) 0 0
\(999\) −5.36069e9 −0.170115
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 592.8.a.c.1.3 6
4.3 odd 2 74.8.a.c.1.4 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
74.8.a.c.1.4 6 4.3 odd 2
592.8.a.c.1.3 6 1.1 even 1 trivial