Properties

Label 304.7.e.f.113.9
Level $304$
Weight $7$
Character 304.113
Analytic conductor $69.936$
Analytic rank $0$
Dimension $30$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [304,7,Mod(113,304)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(304, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1]))
 
N = Newforms(chi, 7, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("304.113");
 
S:= CuspForms(chi, 7);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 304 = 2^{4} \cdot 19 \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 304.e (of order \(2\), degree \(1\), not minimal)

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: no
Analytic conductor: \(69.9364414204\)
Analytic rank: \(0\)
Dimension: \(30\)
Twist minimal: no (minimal twist has level 152)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 113.9
Character \(\chi\) \(=\) 304.113
Dual form 304.7.e.f.113.22

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-27.9511i q^{3} -141.368 q^{5} -302.893 q^{7} -52.2651 q^{9} +O(q^{10})\) \(q-27.9511i q^{3} -141.368 q^{5} -302.893 q^{7} -52.2651 q^{9} -1423.94 q^{11} -1109.38i q^{13} +3951.39i q^{15} -1818.90 q^{17} +(-6854.30 - 253.905i) q^{19} +8466.19i q^{21} -14409.0 q^{23} +4359.90 q^{25} -18915.5i q^{27} +6338.49i q^{29} +8988.56i q^{31} +39800.7i q^{33} +42819.3 q^{35} +40845.9i q^{37} -31008.5 q^{39} +46829.9i q^{41} +37009.5 q^{43} +7388.61 q^{45} +68823.4 q^{47} -25905.0 q^{49} +50840.4i q^{51} +9195.15i q^{53} +201299. q^{55} +(-7096.93 + 191585. i) q^{57} -154031. i q^{59} +312647. q^{61} +15830.7 q^{63} +156831. i q^{65} -327569. i q^{67} +402749. i q^{69} -601597. i q^{71} +277023. q^{73} -121864. i q^{75} +431301. q^{77} +139958. i q^{79} -566811. q^{81} -676886. q^{83} +257134. q^{85} +177168. q^{87} -412278. i q^{89} +336024. i q^{91} +251240. q^{93} +(968978. + 35894.0i) q^{95} -834434. i q^{97} +74422.4 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 30 q + 720 q^{7} - 8670 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 30 q + 720 q^{7} - 8670 q^{9} + 2524 q^{11} + 9700 q^{17} - 4014 q^{19} + 39376 q^{23} + 110742 q^{25} + 19976 q^{35} + 266500 q^{39} + 106788 q^{43} - 91360 q^{45} - 222756 q^{47} + 593586 q^{49} - 540936 q^{55} - 545972 q^{57} - 242640 q^{61} + 377716 q^{63} + 545964 q^{73} - 272356 q^{77} + 2189926 q^{81} - 1542652 q^{83} - 826908 q^{85} + 2729572 q^{87} - 2139912 q^{93} - 2142716 q^{95} + 293012 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/304\mathbb{Z}\right)^\times\).

\(n\) \(97\) \(191\) \(229\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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
\(3\) 27.9511i 1.03523i −0.855615 0.517613i \(-0.826821\pi\)
0.855615 0.517613i \(-0.173179\pi\)
\(4\) 0 0
\(5\) −141.368 −1.13094 −0.565472 0.824768i \(-0.691306\pi\)
−0.565472 + 0.824768i \(0.691306\pi\)
\(6\) 0 0
\(7\) −302.893 −0.883069 −0.441535 0.897244i \(-0.645566\pi\)
−0.441535 + 0.897244i \(0.645566\pi\)
\(8\) 0 0
\(9\) −52.2651 −0.0716943
\(10\) 0 0
\(11\) −1423.94 −1.06983 −0.534913 0.844907i \(-0.679656\pi\)
−0.534913 + 0.844907i \(0.679656\pi\)
\(12\) 0 0
\(13\) 1109.38i 0.504953i −0.967603 0.252477i \(-0.918755\pi\)
0.967603 0.252477i \(-0.0812451\pi\)
\(14\) 0 0
\(15\) 3951.39i 1.17078i
\(16\) 0 0
\(17\) −1818.90 −0.370222 −0.185111 0.982718i \(-0.559265\pi\)
−0.185111 + 0.982718i \(0.559265\pi\)
\(18\) 0 0
\(19\) −6854.30 253.905i −0.999315 0.0370178i
\(20\) 0 0
\(21\) 8466.19i 0.914177i
\(22\) 0 0
\(23\) −14409.0 −1.18427 −0.592136 0.805838i \(-0.701715\pi\)
−0.592136 + 0.805838i \(0.701715\pi\)
\(24\) 0 0
\(25\) 4359.90 0.279034
\(26\) 0 0
\(27\) 18915.5i 0.961007i
\(28\) 0 0
\(29\) 6338.49i 0.259891i 0.991521 + 0.129946i \(0.0414803\pi\)
−0.991521 + 0.129946i \(0.958520\pi\)
\(30\) 0 0
\(31\) 8988.56i 0.301721i 0.988555 + 0.150860i \(0.0482044\pi\)
−0.988555 + 0.150860i \(0.951796\pi\)
\(32\) 0 0
\(33\) 39800.7i 1.10751i
\(34\) 0 0
\(35\) 42819.3 0.998701
\(36\) 0 0
\(37\) 40845.9i 0.806386i 0.915115 + 0.403193i \(0.132100\pi\)
−0.915115 + 0.403193i \(0.867900\pi\)
\(38\) 0 0
\(39\) −31008.5 −0.522741
\(40\) 0 0
\(41\) 46829.9i 0.679472i 0.940521 + 0.339736i \(0.110338\pi\)
−0.940521 + 0.339736i \(0.889662\pi\)
\(42\) 0 0
\(43\) 37009.5 0.465487 0.232743 0.972538i \(-0.425230\pi\)
0.232743 + 0.972538i \(0.425230\pi\)
\(44\) 0 0
\(45\) 7388.61 0.0810822
\(46\) 0 0
\(47\) 68823.4 0.662892 0.331446 0.943474i \(-0.392464\pi\)
0.331446 + 0.943474i \(0.392464\pi\)
\(48\) 0 0
\(49\) −25905.0 −0.220189
\(50\) 0 0
\(51\) 50840.4i 0.383264i
\(52\) 0 0
\(53\) 9195.15i 0.0617634i 0.999523 + 0.0308817i \(0.00983151\pi\)
−0.999523 + 0.0308817i \(0.990168\pi\)
\(54\) 0 0
\(55\) 201299. 1.20991
\(56\) 0 0
\(57\) −7096.93 + 191585.i −0.0383218 + 1.03452i
\(58\) 0 0
\(59\) 154031.i 0.749985i −0.927028 0.374992i \(-0.877645\pi\)
0.927028 0.374992i \(-0.122355\pi\)
\(60\) 0 0
\(61\) 312647. 1.37741 0.688707 0.725040i \(-0.258179\pi\)
0.688707 + 0.725040i \(0.258179\pi\)
\(62\) 0 0
\(63\) 15830.7 0.0633110
\(64\) 0 0
\(65\) 156831.i 0.571074i
\(66\) 0 0
\(67\) 327569.i 1.08913i −0.838720 0.544563i \(-0.816695\pi\)
0.838720 0.544563i \(-0.183305\pi\)
\(68\) 0 0
\(69\) 402749.i 1.22599i
\(70\) 0 0
\(71\) 601597.i 1.68086i −0.541922 0.840429i \(-0.682303\pi\)
0.541922 0.840429i \(-0.317697\pi\)
\(72\) 0 0
\(73\) 277023. 0.712109 0.356055 0.934465i \(-0.384122\pi\)
0.356055 + 0.934465i \(0.384122\pi\)
\(74\) 0 0
\(75\) 121864.i 0.288863i
\(76\) 0 0
\(77\) 431301. 0.944731
\(78\) 0 0
\(79\) 139958.i 0.283869i 0.989876 + 0.141934i \(0.0453322\pi\)
−0.989876 + 0.141934i \(0.954668\pi\)
\(80\) 0 0
\(81\) −566811. −1.06655
\(82\) 0 0
\(83\) −676886. −1.18381 −0.591904 0.806008i \(-0.701624\pi\)
−0.591904 + 0.806008i \(0.701624\pi\)
\(84\) 0 0
\(85\) 257134. 0.418701
\(86\) 0 0
\(87\) 177168. 0.269047
\(88\) 0 0
\(89\) 412278.i 0.584817i −0.956293 0.292409i \(-0.905543\pi\)
0.956293 0.292409i \(-0.0944567\pi\)
\(90\) 0 0
\(91\) 336024.i 0.445909i
\(92\) 0 0
\(93\) 251240. 0.312349
\(94\) 0 0
\(95\) 968978. + 35894.0i 1.13017 + 0.0418650i
\(96\) 0 0
\(97\) 834434.i 0.914275i −0.889396 0.457137i \(-0.848875\pi\)
0.889396 0.457137i \(-0.151125\pi\)
\(98\) 0 0
\(99\) 74422.4 0.0767004
\(100\) 0 0
\(101\) 729972. 0.708504 0.354252 0.935150i \(-0.384736\pi\)
0.354252 + 0.935150i \(0.384736\pi\)
\(102\) 0 0
\(103\) 252659.i 0.231219i 0.993295 + 0.115609i \(0.0368821\pi\)
−0.993295 + 0.115609i \(0.963118\pi\)
\(104\) 0 0
\(105\) 1.19685e6i 1.03388i
\(106\) 0 0
\(107\) 2.04228e6i 1.66711i −0.552435 0.833556i \(-0.686301\pi\)
0.552435 0.833556i \(-0.313699\pi\)
\(108\) 0 0
\(109\) 1.40931e6i 1.08824i −0.839006 0.544122i \(-0.816863\pi\)
0.839006 0.544122i \(-0.183137\pi\)
\(110\) 0 0
\(111\) 1.14169e6 0.834792
\(112\) 0 0
\(113\) 2.51884e6i 1.74568i 0.488003 + 0.872842i \(0.337725\pi\)
−0.488003 + 0.872842i \(0.662275\pi\)
\(114\) 0 0
\(115\) 2.03698e6 1.33934
\(116\) 0 0
\(117\) 57982.0i 0.0362023i
\(118\) 0 0
\(119\) 550932. 0.326932
\(120\) 0 0
\(121\) 256042. 0.144529
\(122\) 0 0
\(123\) 1.30895e6 0.703408
\(124\) 0 0
\(125\) 1.59252e6 0.815372
\(126\) 0 0
\(127\) 3.24362e6i 1.58350i 0.610842 + 0.791752i \(0.290831\pi\)
−0.610842 + 0.791752i \(0.709169\pi\)
\(128\) 0 0
\(129\) 1.03446e6i 0.481884i
\(130\) 0 0
\(131\) 468655. 0.208468 0.104234 0.994553i \(-0.466761\pi\)
0.104234 + 0.994553i \(0.466761\pi\)
\(132\) 0 0
\(133\) 2.07612e6 + 76905.9i 0.882464 + 0.0326892i
\(134\) 0 0
\(135\) 2.67405e6i 1.08684i
\(136\) 0 0
\(137\) −643914. −0.250418 −0.125209 0.992130i \(-0.539960\pi\)
−0.125209 + 0.992130i \(0.539960\pi\)
\(138\) 0 0
\(139\) 355772. 0.132473 0.0662364 0.997804i \(-0.478901\pi\)
0.0662364 + 0.997804i \(0.478901\pi\)
\(140\) 0 0
\(141\) 1.92369e6i 0.686244i
\(142\) 0 0
\(143\) 1.57969e6i 0.540212i
\(144\) 0 0
\(145\) 896060.i 0.293923i
\(146\) 0 0
\(147\) 724074.i 0.227945i
\(148\) 0 0
\(149\) −545081. −0.164779 −0.0823896 0.996600i \(-0.526255\pi\)
−0.0823896 + 0.996600i \(0.526255\pi\)
\(150\) 0 0
\(151\) 1.20603e6i 0.350291i 0.984543 + 0.175146i \(0.0560396\pi\)
−0.984543 + 0.175146i \(0.943960\pi\)
\(152\) 0 0
\(153\) 95065.2 0.0265428
\(154\) 0 0
\(155\) 1.27069e6i 0.341229i
\(156\) 0 0
\(157\) 5.46765e6 1.41287 0.706434 0.707779i \(-0.250303\pi\)
0.706434 + 0.707779i \(0.250303\pi\)
\(158\) 0 0
\(159\) 257015. 0.0639391
\(160\) 0 0
\(161\) 4.36439e6 1.04579
\(162\) 0 0
\(163\) 2.43164e6 0.561482 0.280741 0.959784i \(-0.409420\pi\)
0.280741 + 0.959784i \(0.409420\pi\)
\(164\) 0 0
\(165\) 5.62654e6i 1.25253i
\(166\) 0 0
\(167\) 8.60010e6i 1.84652i 0.384177 + 0.923260i \(0.374485\pi\)
−0.384177 + 0.923260i \(0.625515\pi\)
\(168\) 0 0
\(169\) 3.59608e6 0.745022
\(170\) 0 0
\(171\) 358241. + 13270.4i 0.0716451 + 0.00265396i
\(172\) 0 0
\(173\) 2.20708e6i 0.426265i 0.977023 + 0.213133i \(0.0683666\pi\)
−0.977023 + 0.213133i \(0.931633\pi\)
\(174\) 0 0
\(175\) −1.32058e6 −0.246406
\(176\) 0 0
\(177\) −4.30534e6 −0.776404
\(178\) 0 0
\(179\) 3.69883e6i 0.644919i 0.946583 + 0.322459i \(0.104510\pi\)
−0.946583 + 0.322459i \(0.895490\pi\)
\(180\) 0 0
\(181\) 7.18691e6i 1.21201i 0.795461 + 0.606005i \(0.207229\pi\)
−0.795461 + 0.606005i \(0.792771\pi\)
\(182\) 0 0
\(183\) 8.73883e6i 1.42594i
\(184\) 0 0
\(185\) 5.77429e6i 0.911977i
\(186\) 0 0
\(187\) 2.59001e6 0.396074
\(188\) 0 0
\(189\) 5.72937e6i 0.848635i
\(190\) 0 0
\(191\) −1.52887e6 −0.219417 −0.109708 0.993964i \(-0.534992\pi\)
−0.109708 + 0.993964i \(0.534992\pi\)
\(192\) 0 0
\(193\) 9.01811e6i 1.25442i 0.778849 + 0.627211i \(0.215804\pi\)
−0.778849 + 0.627211i \(0.784196\pi\)
\(194\) 0 0
\(195\) 4.38360e6 0.591191
\(196\) 0 0
\(197\) −2.62332e6 −0.343125 −0.171562 0.985173i \(-0.554881\pi\)
−0.171562 + 0.985173i \(0.554881\pi\)
\(198\) 0 0
\(199\) −5.27787e6 −0.669729 −0.334864 0.942266i \(-0.608691\pi\)
−0.334864 + 0.942266i \(0.608691\pi\)
\(200\) 0 0
\(201\) −9.15592e6 −1.12749
\(202\) 0 0
\(203\) 1.91988e6i 0.229502i
\(204\) 0 0
\(205\) 6.62025e6i 0.768445i
\(206\) 0 0
\(207\) 753090. 0.0849055
\(208\) 0 0
\(209\) 9.76010e6 + 361545.i 1.06909 + 0.0396026i
\(210\) 0 0
\(211\) 1.58144e7i 1.68347i −0.539891 0.841735i \(-0.681535\pi\)
0.539891 0.841735i \(-0.318465\pi\)
\(212\) 0 0
\(213\) −1.68153e7 −1.74007
\(214\) 0 0
\(215\) −5.23195e6 −0.526439
\(216\) 0 0
\(217\) 2.72257e6i 0.266440i
\(218\) 0 0
\(219\) 7.74309e6i 0.737194i
\(220\) 0 0
\(221\) 2.01786e6i 0.186945i
\(222\) 0 0
\(223\) 6.14905e6i 0.554489i 0.960799 + 0.277245i \(0.0894212\pi\)
−0.960799 + 0.277245i \(0.910579\pi\)
\(224\) 0 0
\(225\) −227871. −0.0200051
\(226\) 0 0
\(227\) 8.95249e6i 0.765361i −0.923881 0.382680i \(-0.875001\pi\)
0.923881 0.382680i \(-0.124999\pi\)
\(228\) 0 0
\(229\) −1.14151e7 −0.950548 −0.475274 0.879838i \(-0.657651\pi\)
−0.475274 + 0.879838i \(0.657651\pi\)
\(230\) 0 0
\(231\) 1.20553e7i 0.978011i
\(232\) 0 0
\(233\) −8.75905e6 −0.692451 −0.346226 0.938151i \(-0.612537\pi\)
−0.346226 + 0.938151i \(0.612537\pi\)
\(234\) 0 0
\(235\) −9.72943e6 −0.749694
\(236\) 0 0
\(237\) 3.91199e6 0.293868
\(238\) 0 0
\(239\) 7.69135e6 0.563390 0.281695 0.959504i \(-0.409103\pi\)
0.281695 + 0.959504i \(0.409103\pi\)
\(240\) 0 0
\(241\) 5.03469e6i 0.359684i −0.983695 0.179842i \(-0.942441\pi\)
0.983695 0.179842i \(-0.0575587\pi\)
\(242\) 0 0
\(243\) 2.05359e6i 0.143119i
\(244\) 0 0
\(245\) 3.66214e6 0.249021
\(246\) 0 0
\(247\) −281678. + 7.60404e6i −0.0186922 + 0.504607i
\(248\) 0 0
\(249\) 1.89197e7i 1.22551i
\(250\) 0 0
\(251\) −2.72919e7 −1.72589 −0.862945 0.505297i \(-0.831383\pi\)
−0.862945 + 0.505297i \(0.831383\pi\)
\(252\) 0 0
\(253\) 2.05176e7 1.26697
\(254\) 0 0
\(255\) 7.18720e6i 0.433450i
\(256\) 0 0
\(257\) 3.97818e6i 0.234361i −0.993111 0.117180i \(-0.962614\pi\)
0.993111 0.117180i \(-0.0373856\pi\)
\(258\) 0 0
\(259\) 1.23719e7i 0.712094i
\(260\) 0 0
\(261\) 331282.i 0.0186327i
\(262\) 0 0
\(263\) 1.64161e7 0.902409 0.451205 0.892420i \(-0.350994\pi\)
0.451205 + 0.892420i \(0.350994\pi\)
\(264\) 0 0
\(265\) 1.29990e6i 0.0698509i
\(266\) 0 0
\(267\) −1.15236e7 −0.605418
\(268\) 0 0
\(269\) 3.00061e7i 1.54153i 0.637117 + 0.770767i \(0.280127\pi\)
−0.637117 + 0.770767i \(0.719873\pi\)
\(270\) 0 0
\(271\) 8.91185e6 0.447775 0.223888 0.974615i \(-0.428125\pi\)
0.223888 + 0.974615i \(0.428125\pi\)
\(272\) 0 0
\(273\) 9.39224e6 0.461616
\(274\) 0 0
\(275\) −6.20823e6 −0.298517
\(276\) 0 0
\(277\) −3.77222e6 −0.177484 −0.0887418 0.996055i \(-0.528285\pi\)
−0.0887418 + 0.996055i \(0.528285\pi\)
\(278\) 0 0
\(279\) 469788.i 0.0216316i
\(280\) 0 0
\(281\) 4.13268e6i 0.186257i −0.995654 0.0931286i \(-0.970313\pi\)
0.995654 0.0931286i \(-0.0296868\pi\)
\(282\) 0 0
\(283\) −9.57367e6 −0.422395 −0.211198 0.977443i \(-0.567736\pi\)
−0.211198 + 0.977443i \(0.567736\pi\)
\(284\) 0 0
\(285\) 1.00328e6 2.70840e7i 0.0433398 1.16998i
\(286\) 0 0
\(287\) 1.41844e7i 0.600021i
\(288\) 0 0
\(289\) −2.08292e7 −0.862935
\(290\) 0 0
\(291\) −2.33234e7 −0.946482
\(292\) 0 0
\(293\) 1.36722e7i 0.543544i −0.962362 0.271772i \(-0.912390\pi\)
0.962362 0.271772i \(-0.0876097\pi\)
\(294\) 0 0
\(295\) 2.17751e7i 0.848191i
\(296\) 0 0
\(297\) 2.69345e7i 1.02811i
\(298\) 0 0
\(299\) 1.59851e7i 0.598002i
\(300\) 0 0
\(301\) −1.12099e7 −0.411057
\(302\) 0 0
\(303\) 2.04035e7i 0.733462i
\(304\) 0 0
\(305\) −4.41982e7 −1.55778
\(306\) 0 0
\(307\) 3.56154e7i 1.23090i 0.788175 + 0.615451i \(0.211026\pi\)
−0.788175 + 0.615451i \(0.788974\pi\)
\(308\) 0 0
\(309\) 7.06211e6 0.239364
\(310\) 0 0
\(311\) −5.01481e7 −1.66714 −0.833572 0.552411i \(-0.813708\pi\)
−0.833572 + 0.552411i \(0.813708\pi\)
\(312\) 0 0
\(313\) −8.40051e6 −0.273951 −0.136975 0.990574i \(-0.543738\pi\)
−0.136975 + 0.990574i \(0.543738\pi\)
\(314\) 0 0
\(315\) −2.23796e6 −0.0716012
\(316\) 0 0
\(317\) 2.76287e7i 0.867325i 0.901075 + 0.433663i \(0.142779\pi\)
−0.901075 + 0.433663i \(0.857221\pi\)
\(318\) 0 0
\(319\) 9.02563e6i 0.278039i
\(320\) 0 0
\(321\) −5.70841e7 −1.72584
\(322\) 0 0
\(323\) 1.24673e7 + 461828.i 0.369969 + 0.0137048i
\(324\) 0 0
\(325\) 4.83679e6i 0.140899i
\(326\) 0 0
\(327\) −3.93917e7 −1.12658
\(328\) 0 0
\(329\) −2.08461e7 −0.585380
\(330\) 0 0
\(331\) 6.72197e7i 1.85358i 0.375575 + 0.926792i \(0.377445\pi\)
−0.375575 + 0.926792i \(0.622555\pi\)
\(332\) 0 0
\(333\) 2.13481e6i 0.0578132i
\(334\) 0 0
\(335\) 4.63077e7i 1.23174i
\(336\) 0 0
\(337\) 1.50842e7i 0.394124i −0.980391 0.197062i \(-0.936860\pi\)
0.980391 0.197062i \(-0.0631400\pi\)
\(338\) 0 0
\(339\) 7.04045e7 1.80718
\(340\) 0 0
\(341\) 1.27992e7i 0.322789i
\(342\) 0 0
\(343\) 4.34815e7 1.07751
\(344\) 0 0
\(345\) 5.69358e7i 1.38653i
\(346\) 0 0
\(347\) 8.85858e6 0.212019 0.106010 0.994365i \(-0.466193\pi\)
0.106010 + 0.994365i \(0.466193\pi\)
\(348\) 0 0
\(349\) 7.01758e7 1.65086 0.825432 0.564502i \(-0.190932\pi\)
0.825432 + 0.564502i \(0.190932\pi\)
\(350\) 0 0
\(351\) −2.09845e7 −0.485263
\(352\) 0 0
\(353\) −7.94479e7 −1.80617 −0.903084 0.429463i \(-0.858703\pi\)
−0.903084 + 0.429463i \(0.858703\pi\)
\(354\) 0 0
\(355\) 8.50466e7i 1.90096i
\(356\) 0 0
\(357\) 1.53992e7i 0.338449i
\(358\) 0 0
\(359\) −7.55216e7 −1.63225 −0.816127 0.577872i \(-0.803883\pi\)
−0.816127 + 0.577872i \(0.803883\pi\)
\(360\) 0 0
\(361\) 4.69169e7 + 3.48068e6i 0.997259 + 0.0739848i
\(362\) 0 0
\(363\) 7.15665e6i 0.149620i
\(364\) 0 0
\(365\) −3.91621e7 −0.805355
\(366\) 0 0
\(367\) 1.35966e7 0.275062 0.137531 0.990497i \(-0.456083\pi\)
0.137531 + 0.990497i \(0.456083\pi\)
\(368\) 0 0
\(369\) 2.44757e6i 0.0487143i
\(370\) 0 0
\(371\) 2.78514e6i 0.0545413i
\(372\) 0 0
\(373\) 3.44978e7i 0.664759i −0.943146 0.332380i \(-0.892148\pi\)
0.943146 0.332380i \(-0.107852\pi\)
\(374\) 0 0
\(375\) 4.45128e7i 0.844095i
\(376\) 0 0
\(377\) 7.03181e6 0.131233
\(378\) 0 0
\(379\) 3.57185e7i 0.656108i −0.944659 0.328054i \(-0.893607\pi\)
0.944659 0.328054i \(-0.106393\pi\)
\(380\) 0 0
\(381\) 9.06629e7 1.63929
\(382\) 0 0
\(383\) 3.94898e6i 0.0702893i 0.999382 + 0.0351446i \(0.0111892\pi\)
−0.999382 + 0.0351446i \(0.988811\pi\)
\(384\) 0 0
\(385\) −6.09721e7 −1.06844
\(386\) 0 0
\(387\) −1.93430e6 −0.0333727
\(388\) 0 0
\(389\) 9.60746e7 1.63215 0.816074 0.577947i \(-0.196146\pi\)
0.816074 + 0.577947i \(0.196146\pi\)
\(390\) 0 0
\(391\) 2.62086e7 0.438444
\(392\) 0 0
\(393\) 1.30994e7i 0.215812i
\(394\) 0 0
\(395\) 1.97856e7i 0.321040i
\(396\) 0 0
\(397\) 2.60152e7 0.415773 0.207886 0.978153i \(-0.433342\pi\)
0.207886 + 0.978153i \(0.433342\pi\)
\(398\) 0 0
\(399\) 2.14961e6 5.80298e7i 0.0338408 0.913550i
\(400\) 0 0
\(401\) 1.00439e8i 1.55765i 0.627241 + 0.778825i \(0.284184\pi\)
−0.627241 + 0.778825i \(0.715816\pi\)
\(402\) 0 0
\(403\) 9.97175e6 0.152355
\(404\) 0 0
\(405\) 8.01289e7 1.20621
\(406\) 0 0
\(407\) 5.81620e7i 0.862693i
\(408\) 0 0
\(409\) 8.41310e7i 1.22966i −0.788658 0.614832i \(-0.789224\pi\)
0.788658 0.614832i \(-0.210776\pi\)
\(410\) 0 0
\(411\) 1.79981e7i 0.259240i
\(412\) 0 0
\(413\) 4.66549e7i 0.662289i
\(414\) 0 0
\(415\) 9.56900e7 1.33882
\(416\) 0 0
\(417\) 9.94422e6i 0.137139i
\(418\) 0 0
\(419\) 1.09172e8 1.48412 0.742058 0.670336i \(-0.233850\pi\)
0.742058 + 0.670336i \(0.233850\pi\)
\(420\) 0 0
\(421\) 1.23025e8i 1.64873i −0.566061 0.824363i \(-0.691533\pi\)
0.566061 0.824363i \(-0.308467\pi\)
\(422\) 0 0
\(423\) −3.59707e6 −0.0475256
\(424\) 0 0
\(425\) −7.93023e6 −0.103304
\(426\) 0 0
\(427\) −9.46984e7 −1.21635
\(428\) 0 0
\(429\) 4.41542e7 0.559242
\(430\) 0 0
\(431\) 4.81803e7i 0.601780i −0.953659 0.300890i \(-0.902716\pi\)
0.953659 0.300890i \(-0.0972838\pi\)
\(432\) 0 0
\(433\) 5.03495e7i 0.620200i 0.950704 + 0.310100i \(0.100363\pi\)
−0.950704 + 0.310100i \(0.899637\pi\)
\(434\) 0 0
\(435\) −2.50459e7 −0.304276
\(436\) 0 0
\(437\) 9.87638e7 + 3.65852e6i 1.18346 + 0.0438391i
\(438\) 0 0
\(439\) 1.27207e8i 1.50355i 0.659420 + 0.751775i \(0.270802\pi\)
−0.659420 + 0.751775i \(0.729198\pi\)
\(440\) 0 0
\(441\) 1.35393e6 0.0157863
\(442\) 0 0
\(443\) 4.16056e6 0.0478565 0.0239282 0.999714i \(-0.492383\pi\)
0.0239282 + 0.999714i \(0.492383\pi\)
\(444\) 0 0
\(445\) 5.82829e7i 0.661395i
\(446\) 0 0
\(447\) 1.52356e7i 0.170584i
\(448\) 0 0
\(449\) 6.00799e7i 0.663728i −0.943327 0.331864i \(-0.892322\pi\)
0.943327 0.331864i \(-0.107678\pi\)
\(450\) 0 0
\(451\) 6.66829e7i 0.726917i
\(452\) 0 0
\(453\) 3.37100e7 0.362631
\(454\) 0 0
\(455\) 4.75030e7i 0.504297i
\(456\) 0 0
\(457\) −1.25241e8 −1.31219 −0.656097 0.754677i \(-0.727794\pi\)
−0.656097 + 0.754677i \(0.727794\pi\)
\(458\) 0 0
\(459\) 3.44054e7i 0.355786i
\(460\) 0 0
\(461\) −5.99428e7 −0.611835 −0.305917 0.952058i \(-0.598963\pi\)
−0.305917 + 0.952058i \(0.598963\pi\)
\(462\) 0 0
\(463\) 1.41095e8 1.42157 0.710784 0.703411i \(-0.248341\pi\)
0.710784 + 0.703411i \(0.248341\pi\)
\(464\) 0 0
\(465\) −3.55173e7 −0.353249
\(466\) 0 0
\(467\) 9.37901e7 0.920887 0.460444 0.887689i \(-0.347690\pi\)
0.460444 + 0.887689i \(0.347690\pi\)
\(468\) 0 0
\(469\) 9.92182e7i 0.961774i
\(470\) 0 0
\(471\) 1.52827e8i 1.46264i
\(472\) 0 0
\(473\) −5.26992e7 −0.497990
\(474\) 0 0
\(475\) −2.98840e7 1.10700e6i −0.278842 0.0103292i
\(476\) 0 0
\(477\) 480586.i 0.00442808i
\(478\) 0 0
\(479\) 9.04641e6 0.0823133 0.0411566 0.999153i \(-0.486896\pi\)
0.0411566 + 0.999153i \(0.486896\pi\)
\(480\) 0 0
\(481\) 4.53137e7 0.407187
\(482\) 0 0
\(483\) 1.21990e8i 1.08263i
\(484\) 0 0
\(485\) 1.17962e8i 1.03399i
\(486\) 0 0
\(487\) 6.31773e7i 0.546983i −0.961874 0.273492i \(-0.911821\pi\)
0.961874 0.273492i \(-0.0881786\pi\)
\(488\) 0 0
\(489\) 6.79670e7i 0.581262i
\(490\) 0 0
\(491\) 1.12129e8 0.947269 0.473635 0.880721i \(-0.342942\pi\)
0.473635 + 0.880721i \(0.342942\pi\)
\(492\) 0 0
\(493\) 1.15291e7i 0.0962176i
\(494\) 0 0
\(495\) −1.05209e7 −0.0867439
\(496\) 0 0
\(497\) 1.82219e8i 1.48431i
\(498\) 0 0
\(499\) −5.47216e7 −0.440410 −0.220205 0.975454i \(-0.570673\pi\)
−0.220205 + 0.975454i \(0.570673\pi\)
\(500\) 0 0
\(501\) 2.40382e8 1.91157
\(502\) 0 0
\(503\) −8.65203e7 −0.679851 −0.339926 0.940452i \(-0.610402\pi\)
−0.339926 + 0.940452i \(0.610402\pi\)
\(504\) 0 0
\(505\) −1.03195e8 −0.801278
\(506\) 0 0
\(507\) 1.00514e8i 0.771267i
\(508\) 0 0
\(509\) 1.09132e7i 0.0827557i −0.999144 0.0413778i \(-0.986825\pi\)
0.999144 0.0413778i \(-0.0131747\pi\)
\(510\) 0 0
\(511\) −8.39081e7 −0.628842
\(512\) 0 0
\(513\) −4.80274e6 + 1.29652e8i −0.0355743 + 0.960348i
\(514\) 0 0
\(515\) 3.57179e7i 0.261496i
\(516\) 0 0
\(517\) −9.80004e7 −0.709180
\(518\) 0 0
\(519\) 6.16904e7 0.441281
\(520\) 0 0
\(521\) 1.60201e8i 1.13280i 0.824131 + 0.566399i \(0.191664\pi\)
−0.824131 + 0.566399i \(0.808336\pi\)
\(522\) 0 0
\(523\) 1.49611e8i 1.04582i 0.852388 + 0.522910i \(0.175154\pi\)
−0.852388 + 0.522910i \(0.824846\pi\)
\(524\) 0 0
\(525\) 3.69117e7i 0.255086i
\(526\) 0 0
\(527\) 1.63493e7i 0.111704i
\(528\) 0 0
\(529\) 5.95844e7 0.402500
\(530\) 0 0
\(531\) 8.05046e6i 0.0537696i
\(532\) 0 0
\(533\) 5.19523e7 0.343102
\(534\) 0 0
\(535\) 2.88713e8i 1.88541i
\(536\) 0 0
\(537\) 1.03386e8 0.667637
\(538\) 0 0
\(539\) 3.68872e7 0.235564
\(540\) 0 0
\(541\) 6.87285e7 0.434055 0.217028 0.976165i \(-0.430364\pi\)
0.217028 + 0.976165i \(0.430364\pi\)
\(542\) 0 0
\(543\) 2.00882e8 1.25471
\(544\) 0 0
\(545\) 1.99231e8i 1.23074i
\(546\) 0 0
\(547\) 1.72965e8i 1.05681i 0.848992 + 0.528405i \(0.177210\pi\)
−0.848992 + 0.528405i \(0.822790\pi\)
\(548\) 0 0
\(549\) −1.63405e7 −0.0987527
\(550\) 0 0
\(551\) 1.60937e6 4.34459e7i 0.00962060 0.259713i
\(552\) 0 0
\(553\) 4.23924e7i 0.250676i
\(554\) 0 0
\(555\) −1.61398e8 −0.944103
\(556\) 0 0
\(557\) 2.16158e8 1.25085 0.625425 0.780284i \(-0.284926\pi\)
0.625425 + 0.780284i \(0.284926\pi\)
\(558\) 0 0
\(559\) 4.10576e7i 0.235049i
\(560\) 0 0
\(561\) 7.23936e7i 0.410026i
\(562\) 0 0
\(563\) 3.61156e7i 0.202381i −0.994867 0.101191i \(-0.967735\pi\)
0.994867 0.101191i \(-0.0322652\pi\)
\(564\) 0 0
\(565\) 3.56084e8i 1.97427i
\(566\) 0 0
\(567\) 1.71683e8 0.941841
\(568\) 0 0
\(569\) 3.09877e8i 1.68210i −0.540955 0.841052i \(-0.681937\pi\)
0.540955 0.841052i \(-0.318063\pi\)
\(570\) 0 0
\(571\) −1.07573e7 −0.0577824 −0.0288912 0.999583i \(-0.509198\pi\)
−0.0288912 + 0.999583i \(0.509198\pi\)
\(572\) 0 0
\(573\) 4.27336e7i 0.227146i
\(574\) 0 0
\(575\) −6.28219e7 −0.330452
\(576\) 0 0
\(577\) 3.31948e8 1.72800 0.863998 0.503496i \(-0.167953\pi\)
0.863998 + 0.503496i \(0.167953\pi\)
\(578\) 0 0
\(579\) 2.52066e8 1.29861
\(580\) 0 0
\(581\) 2.05024e8 1.04538
\(582\) 0 0
\(583\) 1.30933e7i 0.0660761i
\(584\) 0 0
\(585\) 8.19680e6i 0.0409427i
\(586\) 0 0
\(587\) −7.23251e7 −0.357581 −0.178791 0.983887i \(-0.557218\pi\)
−0.178791 + 0.983887i \(0.557218\pi\)
\(588\) 0 0
\(589\) 2.28224e6 6.16103e7i 0.0111690 0.301514i
\(590\) 0 0
\(591\) 7.33246e7i 0.355212i
\(592\) 0 0
\(593\) −1.90059e8 −0.911430 −0.455715 0.890126i \(-0.650616\pi\)
−0.455715 + 0.890126i \(0.650616\pi\)
\(594\) 0 0
\(595\) −7.78842e7 −0.369742
\(596\) 0 0
\(597\) 1.47522e8i 0.693321i
\(598\) 0 0
\(599\) 2.64594e8i 1.23112i 0.788092 + 0.615558i \(0.211069\pi\)
−0.788092 + 0.615558i \(0.788931\pi\)
\(600\) 0 0
\(601\) 3.02137e8i 1.39181i −0.718133 0.695906i \(-0.755003\pi\)
0.718133 0.695906i \(-0.244997\pi\)
\(602\) 0 0
\(603\) 1.71204e7i 0.0780841i
\(604\) 0 0
\(605\) −3.61961e7 −0.163454
\(606\) 0 0
\(607\) 2.11107e8i 0.943921i 0.881620 + 0.471960i \(0.156453\pi\)
−0.881620 + 0.471960i \(0.843547\pi\)
\(608\) 0 0
\(609\) −5.36629e7 −0.237587
\(610\) 0 0
\(611\) 7.63515e7i 0.334729i
\(612\) 0 0
\(613\) −3.67466e8 −1.59528 −0.797639 0.603135i \(-0.793918\pi\)
−0.797639 + 0.603135i \(0.793918\pi\)
\(614\) 0 0
\(615\) −1.85043e8 −0.795515
\(616\) 0 0
\(617\) −1.32514e8 −0.564166 −0.282083 0.959390i \(-0.591025\pi\)
−0.282083 + 0.959390i \(0.591025\pi\)
\(618\) 0 0
\(619\) −7.35214e7 −0.309986 −0.154993 0.987916i \(-0.549535\pi\)
−0.154993 + 0.987916i \(0.549535\pi\)
\(620\) 0 0
\(621\) 2.72554e8i 1.13809i
\(622\) 0 0
\(623\) 1.24876e8i 0.516434i
\(624\) 0 0
\(625\) −2.93255e8 −1.20117
\(626\) 0 0
\(627\) 1.01056e7 2.72806e8i 0.0409977 1.10675i
\(628\) 0 0
\(629\) 7.42946e7i 0.298542i
\(630\) 0 0
\(631\) −5.74752e7 −0.228766 −0.114383 0.993437i \(-0.536489\pi\)
−0.114383 + 0.993437i \(0.536489\pi\)
\(632\) 0 0
\(633\) −4.42030e8 −1.74277
\(634\) 0 0
\(635\) 4.58544e8i 1.79085i
\(636\) 0 0
\(637\) 2.87386e7i 0.111185i
\(638\) 0 0
\(639\) 3.14426e7i 0.120508i
\(640\) 0 0
\(641\) 3.65121e8i 1.38632i −0.720786 0.693158i \(-0.756219\pi\)
0.720786 0.693158i \(-0.243781\pi\)
\(642\) 0 0
\(643\) −2.80157e8 −1.05383 −0.526913 0.849920i \(-0.676650\pi\)
−0.526913 + 0.849920i \(0.676650\pi\)
\(644\) 0 0
\(645\) 1.46239e8i 0.544984i
\(646\) 0 0
\(647\) −2.49771e7 −0.0922208 −0.0461104 0.998936i \(-0.514683\pi\)
−0.0461104 + 0.998936i \(0.514683\pi\)
\(648\) 0 0
\(649\) 2.19331e8i 0.802354i
\(650\) 0 0
\(651\) −7.60988e7 −0.275826
\(652\) 0 0
\(653\) −2.88241e8 −1.03518 −0.517591 0.855628i \(-0.673171\pi\)
−0.517591 + 0.855628i \(0.673171\pi\)
\(654\) 0 0
\(655\) −6.62528e7 −0.235766
\(656\) 0 0
\(657\) −1.44786e7 −0.0510542
\(658\) 0 0
\(659\) 4.65298e8i 1.62583i 0.582383 + 0.812915i \(0.302120\pi\)
−0.582383 + 0.812915i \(0.697880\pi\)
\(660\) 0 0
\(661\) 4.86400e8i 1.68418i −0.539336 0.842091i \(-0.681325\pi\)
0.539336 0.842091i \(-0.318675\pi\)
\(662\) 0 0
\(663\) 5.64014e7 0.193530
\(664\) 0 0
\(665\) −2.93496e8 1.08720e7i −0.998017 0.0369697i
\(666\) 0 0
\(667\) 9.13316e7i 0.307782i
\(668\) 0 0
\(669\) 1.71873e8 0.574022
\(670\) 0 0
\(671\) −4.45190e8 −1.47359
\(672\) 0 0
\(673\) 1.58465e8i 0.519862i −0.965627 0.259931i \(-0.916300\pi\)
0.965627 0.259931i \(-0.0836997\pi\)
\(674\) 0 0
\(675\) 8.24697e7i 0.268153i
\(676\) 0 0
\(677\) 5.27476e8i 1.69995i −0.526822 0.849976i \(-0.676617\pi\)
0.526822 0.849976i \(-0.323383\pi\)
\(678\) 0 0
\(679\) 2.52744e8i 0.807368i
\(680\) 0 0
\(681\) −2.50232e8 −0.792322
\(682\) 0 0
\(683\) 2.10229e8i 0.659827i −0.944011 0.329914i \(-0.892980\pi\)
0.944011 0.329914i \(-0.107020\pi\)
\(684\) 0 0
\(685\) 9.10288e7 0.283209
\(686\) 0 0
\(687\) 3.19065e8i 0.984032i
\(688\) 0 0
\(689\) 1.02009e7 0.0311876
\(690\) 0 0
\(691\) 4.00384e8 1.21351 0.606754 0.794890i \(-0.292471\pi\)
0.606754 + 0.794890i \(0.292471\pi\)
\(692\) 0 0
\(693\) −2.25420e7 −0.0677318
\(694\) 0 0
\(695\) −5.02947e7 −0.149819
\(696\) 0 0
\(697\) 8.51790e7i 0.251556i
\(698\) 0 0
\(699\) 2.44825e8i 0.716844i
\(700\) 0 0
\(701\) 1.82964e8 0.531145 0.265572 0.964091i \(-0.414439\pi\)
0.265572 + 0.964091i \(0.414439\pi\)
\(702\) 0 0
\(703\) 1.03710e7 2.79970e8i 0.0298506 0.805833i
\(704\) 0 0
\(705\) 2.71948e8i 0.776103i
\(706\) 0 0
\(707\) −2.21103e8 −0.625658
\(708\) 0 0
\(709\) −7.49438e7 −0.210280 −0.105140 0.994457i \(-0.533529\pi\)
−0.105140 + 0.994457i \(0.533529\pi\)
\(710\) 0 0
\(711\) 7.31494e6i 0.0203518i
\(712\) 0 0
\(713\) 1.29516e8i 0.357319i
\(714\) 0 0
\(715\) 2.23318e8i 0.610950i
\(716\) 0 0
\(717\) 2.14982e8i 0.583236i
\(718\) 0 0
\(719\) −3.87906e8 −1.04361 −0.521807 0.853064i \(-0.674742\pi\)
−0.521807 + 0.853064i \(0.674742\pi\)
\(720\) 0 0
\(721\) 7.65286e7i 0.204182i
\(722\) 0 0
\(723\) −1.40725e8 −0.372355
\(724\) 0 0
\(725\) 2.76352e7i 0.0725184i
\(726\) 0 0
\(727\) 5.28037e7 0.137423 0.0687117 0.997637i \(-0.478111\pi\)
0.0687117 + 0.997637i \(0.478111\pi\)
\(728\) 0 0
\(729\) −3.55805e8 −0.918394
\(730\) 0 0
\(731\) −6.73166e7 −0.172334
\(732\) 0 0
\(733\) 3.53898e8 0.898599 0.449299 0.893381i \(-0.351674\pi\)
0.449299 + 0.893381i \(0.351674\pi\)
\(734\) 0 0
\(735\) 1.02361e8i 0.257793i
\(736\) 0 0
\(737\) 4.66438e8i 1.16518i
\(738\) 0 0
\(739\) −4.41903e7 −0.109495 −0.0547474 0.998500i \(-0.517435\pi\)
−0.0547474 + 0.998500i \(0.517435\pi\)
\(740\) 0 0
\(741\) 2.12541e8 + 7.87320e6i 0.522383 + 0.0193507i
\(742\) 0 0
\(743\) 1.60498e8i 0.391295i −0.980674 0.195647i \(-0.937319\pi\)
0.980674 0.195647i \(-0.0626808\pi\)
\(744\) 0 0
\(745\) 7.70570e7 0.186356
\(746\) 0 0
\(747\) 3.53775e7 0.0848723
\(748\) 0 0
\(749\) 6.18593e8i 1.47217i
\(750\) 0 0
\(751\) 2.63368e8i 0.621790i 0.950444 + 0.310895i \(0.100629\pi\)
−0.950444 + 0.310895i \(0.899371\pi\)
\(752\) 0 0
\(753\) 7.62840e8i 1.78669i
\(754\) 0 0
\(755\) 1.70495e8i 0.396159i
\(756\) 0 0
\(757\) 3.66664e8 0.845241 0.422621 0.906307i \(-0.361110\pi\)
0.422621 + 0.906307i \(0.361110\pi\)
\(758\) 0 0
\(759\) 5.73490e8i 1.31160i
\(760\) 0 0
\(761\) −2.04652e7 −0.0464367 −0.0232184 0.999730i \(-0.507391\pi\)
−0.0232184 + 0.999730i \(0.507391\pi\)
\(762\) 0 0
\(763\) 4.26869e8i 0.960995i
\(764\) 0 0
\(765\) −1.34392e7 −0.0300184
\(766\) 0 0
\(767\) −1.70879e8 −0.378707
\(768\) 0 0
\(769\) 7.44380e8 1.63688 0.818438 0.574595i \(-0.194841\pi\)
0.818438 + 0.574595i \(0.194841\pi\)
\(770\) 0 0
\(771\) −1.11195e8 −0.242617
\(772\) 0 0
\(773\) 2.08872e8i 0.452212i 0.974103 + 0.226106i \(0.0725995\pi\)
−0.974103 + 0.226106i \(0.927400\pi\)
\(774\) 0 0
\(775\) 3.91892e7i 0.0841902i
\(776\) 0 0
\(777\) −3.45809e8 −0.737179
\(778\) 0 0
\(779\) 1.18903e7 3.20986e8i 0.0251525 0.679006i
\(780\) 0 0
\(781\) 8.56638e8i 1.79823i
\(782\) 0 0
\(783\) 1.19896e8 0.249757
\(784\) 0 0
\(785\) −7.72950e8 −1.59787
\(786\) 0 0
\(787\) 4.18061e8i 0.857660i 0.903385 + 0.428830i \(0.141074\pi\)
−0.903385 + 0.428830i \(0.858926\pi\)
\(788\) 0 0
\(789\) 4.58849e8i 0.934198i
\(790\) 0 0
\(791\) 7.62939e8i 1.54156i
\(792\) 0 0
\(793\) 3.46845e8i 0.695530i
\(794\) 0 0
\(795\) −3.63336e7 −0.0723115
\(796\) 0 0
\(797\) 5.79777e8i 1.14521i 0.819830 + 0.572607i \(0.194068\pi\)
−0.819830 + 0.572607i \(0.805932\pi\)
\(798\) 0 0
\(799\) −1.25183e8 −0.245417
\(800\) 0 0
\(801\) 2.15478e7i 0.0419280i
\(802\) 0 0
\(803\) −3.94463e8 −0.761833
\(804\) 0 0
\(805\) −6.16985e8 −1.18273
\(806\) 0 0
\(807\) 8.38705e8 1.59584
\(808\) 0 0
\(809\) 7.81512e7 0.147601 0.0738006 0.997273i \(-0.476487\pi\)
0.0738006 + 0.997273i \(0.476487\pi\)
\(810\) 0 0
\(811\) 2.19059e8i 0.410676i 0.978691 + 0.205338i \(0.0658293\pi\)
−0.978691 + 0.205338i \(0.934171\pi\)
\(812\) 0 0
\(813\) 2.49096e8i 0.463549i
\(814\) 0 0
\(815\) −3.43756e8 −0.635005
\(816\) 0 0
\(817\) −2.53674e8 9.39688e6i −0.465168 0.0172313i
\(818\) 0 0
\(819\) 1.75623e7i 0.0319691i
\(820\) 0 0
\(821\) −1.78671e8 −0.322867 −0.161434 0.986884i \(-0.551612\pi\)
−0.161434 + 0.986884i \(0.551612\pi\)
\(822\) 0 0
\(823\) 6.21889e8 1.11561 0.557806 0.829971i \(-0.311643\pi\)
0.557806 + 0.829971i \(0.311643\pi\)
\(824\) 0 0
\(825\) 1.73527e8i 0.309033i
\(826\) 0 0
\(827\) 7.91151e8i 1.39876i 0.714750 + 0.699380i \(0.246540\pi\)
−0.714750 + 0.699380i \(0.753460\pi\)
\(828\) 0 0
\(829\) 4.15868e8i 0.729948i 0.931018 + 0.364974i \(0.118922\pi\)
−0.931018 + 0.364974i \(0.881078\pi\)
\(830\) 0 0
\(831\) 1.05438e8i 0.183736i
\(832\) 0 0
\(833\) 4.71187e7 0.0815189
\(834\) 0 0
\(835\) 1.21578e9i 2.08831i
\(836\) 0 0
\(837\) 1.70023e8 0.289956
\(838\) 0 0
\(839\) 2.51724e8i 0.426225i 0.977028 + 0.213112i \(0.0683601\pi\)
−0.977028 + 0.213112i \(0.931640\pi\)
\(840\) 0 0
\(841\) 5.54647e8 0.932456
\(842\) 0 0
\(843\) −1.15513e8 −0.192818
\(844\) 0 0
\(845\) −5.08370e8 −0.842578
\(846\) 0 0
\(847\) −7.75532e7 −0.127629
\(848\) 0 0
\(849\) 2.67595e8i 0.437275i
\(850\) 0 0
\(851\) 5.88549e8i 0.954980i
\(852\) 0 0
\(853\) 8.47932e8 1.36620 0.683099 0.730325i \(-0.260632\pi\)
0.683099 + 0.730325i \(0.260632\pi\)
\(854\) 0 0
\(855\) −5.06438e7 1.87601e6i −0.0810266 0.00300148i
\(856\) 0 0
\(857\) 1.20056e9i 1.90740i 0.300760 + 0.953700i \(0.402760\pi\)
−0.300760 + 0.953700i \(0.597240\pi\)
\(858\) 0 0
\(859\) 9.74459e8 1.53739 0.768695 0.639615i \(-0.220906\pi\)
0.768695 + 0.639615i \(0.220906\pi\)
\(860\) 0 0
\(861\) −3.96471e8 −0.621158
\(862\) 0 0
\(863\) 2.53525e8i 0.394446i −0.980359 0.197223i \(-0.936808\pi\)
0.980359 0.197223i \(-0.0631924\pi\)
\(864\) 0 0
\(865\) 3.12010e8i 0.482082i
\(866\) 0 0
\(867\) 5.82198e8i 0.893334i
\(868\) 0 0
\(869\) 1.99292e8i 0.303690i
\(870\) 0 0
\(871\) −3.63399e8 −0.549958
\(872\) 0 0
\(873\) 4.36118e7i 0.0655483i
\(874\) 0 0
\(875\) −4.82364e8 −0.720030
\(876\) 0 0
\(877\) 6.12474e8i 0.908007i −0.891000 0.454004i \(-0.849995\pi\)
0.891000 0.454004i \(-0.150005\pi\)
\(878\) 0 0
\(879\) −3.82153e8 −0.562692
\(880\) 0 0
\(881\) −6.68417e8 −0.977507 −0.488754 0.872422i \(-0.662548\pi\)
−0.488754 + 0.872422i \(0.662548\pi\)
\(882\) 0 0
\(883\) 1.24854e9 1.81351 0.906753 0.421661i \(-0.138553\pi\)
0.906753 + 0.421661i \(0.138553\pi\)
\(884\) 0 0
\(885\) 6.08638e8 0.878070
\(886\) 0 0
\(887\) 8.98396e8i 1.28735i 0.765298 + 0.643676i \(0.222591\pi\)
−0.765298 + 0.643676i \(0.777409\pi\)
\(888\) 0 0
\(889\) 9.82470e8i 1.39834i
\(890\) 0 0
\(891\) 8.07104e8 1.14103
\(892\) 0 0
\(893\) −4.71736e8 1.74746e7i −0.662438 0.0245388i
\(894\) 0 0
\(895\) 5.22896e8i 0.729367i
\(896\) 0 0
\(897\) 4.46802e8 0.619067
\(898\) 0 0
\(899\) −5.69739e7 −0.0784146
\(900\) 0 0
\(901\) 1.67251e7i 0.0228662i
\(902\) 0 0
\(903\) 3.13329e8i 0.425537i
\(904\) 0 0
\(905\) 1.01600e9i 1.37072i
\(906\) 0 0
\(907\) 2.08672e8i 0.279668i −0.990175 0.139834i \(-0.955343\pi\)
0.990175 0.139834i \(-0.0446569\pi\)
\(908\) 0 0
\(909\) −3.81521e7 −0.0507957
\(910\) 0 0
\(911\) 1.37374e9i 1.81697i −0.417913 0.908487i \(-0.637238\pi\)
0.417913 0.908487i \(-0.362762\pi\)
\(912\) 0 0
\(913\) 9.63845e8 1.26647
\(914\) 0 0
\(915\) 1.23539e9i 1.61265i
\(916\) 0 0
\(917\) −1.41952e8 −0.184092
\(918\) 0 0
\(919\) 1.38508e8 0.178455 0.0892276 0.996011i \(-0.471560\pi\)
0.0892276 + 0.996011i \(0.471560\pi\)
\(920\) 0 0
\(921\) 9.95492e8 1.27426
\(922\) 0 0
\(923\) −6.67402e8 −0.848755
\(924\) 0 0
\(925\) 1.78084e8i 0.225009i
\(926\) 0 0
\(927\) 1.32053e7i 0.0165771i
\(928\) 0 0
\(929\) −4.89511e8 −0.610542 −0.305271 0.952266i \(-0.598747\pi\)
−0.305271 + 0.952266i \(0.598747\pi\)
\(930\) 0 0
\(931\) 1.77561e8 + 6.57741e6i 0.220038 + 0.00815090i
\(932\) 0 0
\(933\) 1.40169e9i 1.72587i
\(934\) 0 0
\(935\) −3.66144e8 −0.447937
\(936\) 0 0
\(937\) 1.43426e9 1.74345 0.871724 0.489997i \(-0.163002\pi\)
0.871724 + 0.489997i \(0.163002\pi\)
\(938\) 0 0
\(939\) 2.34804e8i 0.283601i
\(940\) 0 0
\(941\) 1.10210e9i 1.32268i −0.750087 0.661339i \(-0.769989\pi\)
0.750087 0.661339i \(-0.230011\pi\)
\(942\) 0 0
\(943\) 6.74774e8i 0.804680i
\(944\) 0 0
\(945\) 8.09949e8i 0.959759i
\(946\) 0 0
\(947\) −1.54268e9 −1.81646 −0.908229 0.418473i \(-0.862565\pi\)
−0.908229 + 0.418473i \(0.862565\pi\)
\(948\) 0 0
\(949\) 3.07324e8i 0.359582i
\(950\) 0 0
\(951\) 7.72252e8 0.897878
\(952\) 0 0
\(953\) 1.20290e8i 0.138980i 0.997583 + 0.0694899i \(0.0221372\pi\)
−0.997583 + 0.0694899i \(0.977863\pi\)
\(954\) 0 0
\(955\) 2.16133e8 0.248148
\(956\) 0 0
\(957\) −2.52276e8 −0.287833
\(958\) 0 0
\(959\) 1.95037e8 0.221137
\(960\) 0 0
\(961\) 8.06710e8 0.908965
\(962\) 0 0
\(963\) 1.06740e8i 0.119522i
\(964\) 0 0
\(965\) 1.27487e9i 1.41868i
\(966\) 0 0
\(967\) 1.40392e9 1.55261 0.776306 0.630357i \(-0.217091\pi\)
0.776306 + 0.630357i \(0.217091\pi\)
\(968\) 0 0
\(969\) 1.29086e7 3.48475e8i 0.0141876 0.383001i
\(970\) 0 0
\(971\) 6.57153e8i 0.717808i −0.933374 0.358904i \(-0.883150\pi\)
0.933374 0.358904i \(-0.116850\pi\)
\(972\) 0 0
\(973\) −1.07761e8 −0.116983
\(974\) 0 0
\(975\) −1.35194e8 −0.145862
\(976\) 0 0
\(977\) 1.55873e8i 0.167143i −0.996502 0.0835714i \(-0.973367\pi\)
0.996502 0.0835714i \(-0.0266327\pi\)
\(978\) 0 0
\(979\) 5.87059e8i 0.625653i
\(980\) 0 0
\(981\) 7.36577e7i 0.0780209i
\(982\) 0 0
\(983\) 5.23668e7i 0.0551309i −0.999620 0.0275655i \(-0.991225\pi\)
0.999620 0.0275655i \(-0.00877547\pi\)
\(984\) 0 0
\(985\) 3.70853e8 0.388055
\(986\) 0 0
\(987\) 5.82672e8i 0.606001i
\(988\) 0 0
\(989\) −5.33271e8 −0.551263
\(990\) 0 0
\(991\) 1.96783e8i 0.202193i 0.994877 + 0.101097i \(0.0322351\pi\)
−0.994877 + 0.101097i \(0.967765\pi\)
\(992\) 0 0
\(993\) 1.87886e9 1.91888
\(994\) 0 0
\(995\) 7.46121e8 0.757426
\(996\) 0 0
\(997\) −1.85567e9 −1.87247 −0.936236 0.351371i \(-0.885715\pi\)
−0.936236 + 0.351371i \(0.885715\pi\)
\(998\) 0 0
\(999\) 7.72620e8 0.774942
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 304.7.e.f.113.9 30
4.3 odd 2 152.7.e.a.113.22 yes 30
19.18 odd 2 inner 304.7.e.f.113.22 30
76.75 even 2 152.7.e.a.113.9 30
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
152.7.e.a.113.9 30 76.75 even 2
152.7.e.a.113.22 yes 30 4.3 odd 2
304.7.e.f.113.9 30 1.1 even 1 trivial
304.7.e.f.113.22 30 19.18 odd 2 inner