Properties

Label 304.7.e.c.113.6
Level $304$
Weight $7$
Character 304.113
Analytic conductor $69.936$
Analytic rank $0$
Dimension $8$
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: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 5090x^{6} + 8905881x^{4} + 5831691048x^{2} + 827887219200 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{8}\cdot 3\cdot 5 \)
Twist minimal: no (minimal twist has level 76)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 113.6
Root \(38.1507i\) of defining polynomial
Character \(\chi\) \(=\) 304.113
Dual form 304.7.e.c.113.3

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+38.1507i q^{3} +102.472 q^{5} -512.609 q^{7} -726.473 q^{9} +O(q^{10})\) \(q+38.1507i q^{3} +102.472 q^{5} -512.609 q^{7} -726.473 q^{9} -2416.13 q^{11} +3771.60i q^{13} +3909.38i q^{15} +1567.21 q^{17} +(2736.35 + 6289.54i) q^{19} -19556.4i q^{21} +626.209 q^{23} -5124.47 q^{25} +96.4222i q^{27} -15705.3i q^{29} -26199.8i q^{31} -92177.0i q^{33} -52528.1 q^{35} -49488.8i q^{37} -143889. q^{39} +119758. i q^{41} +13335.4 q^{43} -74443.2 q^{45} +186343. q^{47} +145119. q^{49} +59790.3i q^{51} -204595. i q^{53} -247586. q^{55} +(-239950. + 104393. i) q^{57} -18264.5i q^{59} -353270. q^{61} +372396. q^{63} +386484. i q^{65} -519557. i q^{67} +23890.3i q^{69} -268077. i q^{71} +58589.5 q^{73} -195502. i q^{75} +1.23853e6 q^{77} +473406. i q^{79} -533277. q^{81} -127056. q^{83} +160596. q^{85} +599166. q^{87} -607216. i q^{89} -1.93336e6i q^{91} +999541. q^{93} +(280399. + 644502. i) q^{95} +282195. i q^{97} +1.75525e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 2 q^{5} - 362 q^{7} - 4348 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 2 q^{5} - 362 q^{7} - 4348 q^{9} - 902 q^{11} + 1550 q^{17} - 6232 q^{19} + 18820 q^{23} - 12158 q^{25} + 101762 q^{35} - 167028 q^{39} + 335042 q^{43} - 57230 q^{45} + 570394 q^{47} + 448182 q^{49} - 1089198 q^{55} + 341316 q^{57} - 632014 q^{61} - 328174 q^{63} - 852938 q^{73} + 1850530 q^{77} - 1819456 q^{81} - 441200 q^{83} - 1828374 q^{85} - 1483380 q^{87} + 2131176 q^{93} - 627950 q^{95} + 865394 q^{99}+O(q^{100}) \) 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
\(3\) 38.1507i 1.41299i 0.707719 + 0.706494i \(0.249724\pi\)
−0.707719 + 0.706494i \(0.750276\pi\)
\(4\) 0 0
\(5\) 102.472 0.819777 0.409888 0.912136i \(-0.365568\pi\)
0.409888 + 0.912136i \(0.365568\pi\)
\(6\) 0 0
\(7\) −512.609 −1.49449 −0.747243 0.664551i \(-0.768623\pi\)
−0.747243 + 0.664551i \(0.768623\pi\)
\(8\) 0 0
\(9\) −726.473 −0.996533
\(10\) 0 0
\(11\) −2416.13 −1.81528 −0.907638 0.419754i \(-0.862116\pi\)
−0.907638 + 0.419754i \(0.862116\pi\)
\(12\) 0 0
\(13\) 3771.60i 1.71671i 0.513059 + 0.858353i \(0.328512\pi\)
−0.513059 + 0.858353i \(0.671488\pi\)
\(14\) 0 0
\(15\) 3909.38i 1.15833i
\(16\) 0 0
\(17\) 1567.21 0.318993 0.159497 0.987198i \(-0.449013\pi\)
0.159497 + 0.987198i \(0.449013\pi\)
\(18\) 0 0
\(19\) 2736.35 + 6289.54i 0.398942 + 0.916976i
\(20\) 0 0
\(21\) 19556.4i 2.11169i
\(22\) 0 0
\(23\) 626.209 0.0514678 0.0257339 0.999669i \(-0.491808\pi\)
0.0257339 + 0.999669i \(0.491808\pi\)
\(24\) 0 0
\(25\) −5124.47 −0.327966
\(26\) 0 0
\(27\) 96.4222i 0.00489875i
\(28\) 0 0
\(29\) 15705.3i 0.643948i −0.946748 0.321974i \(-0.895654\pi\)
0.946748 0.321974i \(-0.104346\pi\)
\(30\) 0 0
\(31\) 26199.8i 0.879454i −0.898131 0.439727i \(-0.855075\pi\)
0.898131 0.439727i \(-0.144925\pi\)
\(32\) 0 0
\(33\) 92177.0i 2.56496i
\(34\) 0 0
\(35\) −52528.1 −1.22515
\(36\) 0 0
\(37\) 49488.8i 0.977015i −0.872560 0.488508i \(-0.837541\pi\)
0.872560 0.488508i \(-0.162459\pi\)
\(38\) 0 0
\(39\) −143889. −2.42568
\(40\) 0 0
\(41\) 119758.i 1.73761i 0.495155 + 0.868805i \(0.335111\pi\)
−0.495155 + 0.868805i \(0.664889\pi\)
\(42\) 0 0
\(43\) 13335.4 0.167726 0.0838628 0.996477i \(-0.473274\pi\)
0.0838628 + 0.996477i \(0.473274\pi\)
\(44\) 0 0
\(45\) −74443.2 −0.816935
\(46\) 0 0
\(47\) 186343. 1.79482 0.897408 0.441201i \(-0.145447\pi\)
0.897408 + 0.441201i \(0.145447\pi\)
\(48\) 0 0
\(49\) 145119. 1.23349
\(50\) 0 0
\(51\) 59790.3i 0.450734i
\(52\) 0 0
\(53\) 204595.i 1.37425i −0.726537 0.687127i \(-0.758872\pi\)
0.726537 0.687127i \(-0.241128\pi\)
\(54\) 0 0
\(55\) −247586. −1.48812
\(56\) 0 0
\(57\) −239950. + 104393.i −1.29568 + 0.563701i
\(58\) 0 0
\(59\) 18264.5i 0.0889306i −0.999011 0.0444653i \(-0.985842\pi\)
0.999011 0.0444653i \(-0.0141584\pi\)
\(60\) 0 0
\(61\) −353270. −1.55639 −0.778193 0.628025i \(-0.783864\pi\)
−0.778193 + 0.628025i \(0.783864\pi\)
\(62\) 0 0
\(63\) 372396. 1.48930
\(64\) 0 0
\(65\) 386484.i 1.40732i
\(66\) 0 0
\(67\) 519557.i 1.72746i −0.503952 0.863732i \(-0.668121\pi\)
0.503952 0.863732i \(-0.331879\pi\)
\(68\) 0 0
\(69\) 23890.3i 0.0727234i
\(70\) 0 0
\(71\) 268077.i 0.749004i −0.927226 0.374502i \(-0.877814\pi\)
0.927226 0.374502i \(-0.122186\pi\)
\(72\) 0 0
\(73\) 58589.5 0.150609 0.0753045 0.997161i \(-0.476007\pi\)
0.0753045 + 0.997161i \(0.476007\pi\)
\(74\) 0 0
\(75\) 195502.i 0.463412i
\(76\) 0 0
\(77\) 1.23853e6 2.71290
\(78\) 0 0
\(79\) 473406.i 0.960179i 0.877219 + 0.480090i \(0.159396\pi\)
−0.877219 + 0.480090i \(0.840604\pi\)
\(80\) 0 0
\(81\) −533277. −1.00345
\(82\) 0 0
\(83\) −127056. −0.222208 −0.111104 0.993809i \(-0.535439\pi\)
−0.111104 + 0.993809i \(0.535439\pi\)
\(84\) 0 0
\(85\) 160596. 0.261503
\(86\) 0 0
\(87\) 599166. 0.909891
\(88\) 0 0
\(89\) 607216.i 0.861337i −0.902510 0.430669i \(-0.858278\pi\)
0.902510 0.430669i \(-0.141722\pi\)
\(90\) 0 0
\(91\) 1.93336e6i 2.56559i
\(92\) 0 0
\(93\) 999541. 1.24266
\(94\) 0 0
\(95\) 280399. + 644502.i 0.327044 + 0.751716i
\(96\) 0 0
\(97\) 282195.i 0.309196i 0.987977 + 0.154598i \(0.0494083\pi\)
−0.987977 + 0.154598i \(0.950592\pi\)
\(98\) 0 0
\(99\) 1.75525e6 1.80898
\(100\) 0 0
\(101\) 443149. 0.430116 0.215058 0.976601i \(-0.431006\pi\)
0.215058 + 0.976601i \(0.431006\pi\)
\(102\) 0 0
\(103\) 836042.i 0.765097i 0.923935 + 0.382549i \(0.124954\pi\)
−0.923935 + 0.382549i \(0.875046\pi\)
\(104\) 0 0
\(105\) 2.00398e6i 1.73111i
\(106\) 0 0
\(107\) 617387.i 0.503972i −0.967731 0.251986i \(-0.918916\pi\)
0.967731 0.251986i \(-0.0810836\pi\)
\(108\) 0 0
\(109\) 955640.i 0.737929i −0.929443 0.368965i \(-0.879712\pi\)
0.929443 0.368965i \(-0.120288\pi\)
\(110\) 0 0
\(111\) 1.88803e6 1.38051
\(112\) 0 0
\(113\) 262850.i 0.182168i −0.995843 0.0910842i \(-0.970967\pi\)
0.995843 0.0910842i \(-0.0290333\pi\)
\(114\) 0 0
\(115\) 64169.0 0.0421921
\(116\) 0 0
\(117\) 2.73997e6i 1.71075i
\(118\) 0 0
\(119\) −803368. −0.476731
\(120\) 0 0
\(121\) 4.06613e6 2.29523
\(122\) 0 0
\(123\) −4.56884e6 −2.45522
\(124\) 0 0
\(125\) −2.12624e6 −1.08864
\(126\) 0 0
\(127\) 933464.i 0.455708i 0.973695 + 0.227854i \(0.0731709\pi\)
−0.973695 + 0.227854i \(0.926829\pi\)
\(128\) 0 0
\(129\) 508753.i 0.236994i
\(130\) 0 0
\(131\) 1.46381e6 0.651135 0.325568 0.945519i \(-0.394445\pi\)
0.325568 + 0.945519i \(0.394445\pi\)
\(132\) 0 0
\(133\) −1.40267e6 3.22407e6i −0.596214 1.37041i
\(134\) 0 0
\(135\) 9880.58i 0.00401589i
\(136\) 0 0
\(137\) 1.07931e6 0.419743 0.209871 0.977729i \(-0.432695\pi\)
0.209871 + 0.977729i \(0.432695\pi\)
\(138\) 0 0
\(139\) −2.79160e6 −1.03946 −0.519731 0.854330i \(-0.673968\pi\)
−0.519731 + 0.854330i \(0.673968\pi\)
\(140\) 0 0
\(141\) 7.10912e6i 2.53605i
\(142\) 0 0
\(143\) 9.11270e6i 3.11630i
\(144\) 0 0
\(145\) 1.60935e6i 0.527894i
\(146\) 0 0
\(147\) 5.53638e6i 1.74290i
\(148\) 0 0
\(149\) −1.47423e6 −0.445662 −0.222831 0.974857i \(-0.571530\pi\)
−0.222831 + 0.974857i \(0.571530\pi\)
\(150\) 0 0
\(151\) 3.70643e6i 1.07653i 0.842776 + 0.538264i \(0.180920\pi\)
−0.842776 + 0.538264i \(0.819080\pi\)
\(152\) 0 0
\(153\) −1.13854e6 −0.317887
\(154\) 0 0
\(155\) 2.68475e6i 0.720956i
\(156\) 0 0
\(157\) 517298. 0.133673 0.0668363 0.997764i \(-0.478709\pi\)
0.0668363 + 0.997764i \(0.478709\pi\)
\(158\) 0 0
\(159\) 7.80543e6 1.94180
\(160\) 0 0
\(161\) −321000. −0.0769180
\(162\) 0 0
\(163\) −3.07611e6 −0.710296 −0.355148 0.934810i \(-0.615570\pi\)
−0.355148 + 0.934810i \(0.615570\pi\)
\(164\) 0 0
\(165\) 9.44557e6i 2.10270i
\(166\) 0 0
\(167\) 3.20302e6i 0.687717i −0.939021 0.343859i \(-0.888266\pi\)
0.939021 0.343859i \(-0.111734\pi\)
\(168\) 0 0
\(169\) −9.39819e6 −1.94708
\(170\) 0 0
\(171\) −1.98788e6 4.56918e6i −0.397559 0.913797i
\(172\) 0 0
\(173\) 2.52360e6i 0.487395i 0.969851 + 0.243698i \(0.0783605\pi\)
−0.969851 + 0.243698i \(0.921640\pi\)
\(174\) 0 0
\(175\) 2.62685e6 0.490141
\(176\) 0 0
\(177\) 696802. 0.125658
\(178\) 0 0
\(179\) 7.68706e6i 1.34030i −0.742227 0.670149i \(-0.766230\pi\)
0.742227 0.670149i \(-0.233770\pi\)
\(180\) 0 0
\(181\) 3.05055e6i 0.514449i 0.966352 + 0.257225i \(0.0828081\pi\)
−0.966352 + 0.257225i \(0.917192\pi\)
\(182\) 0 0
\(183\) 1.34775e7i 2.19915i
\(184\) 0 0
\(185\) 5.07122e6i 0.800934i
\(186\) 0 0
\(187\) −3.78660e6 −0.579061
\(188\) 0 0
\(189\) 49426.9i 0.00732112i
\(190\) 0 0
\(191\) 1.01067e7 1.45047 0.725234 0.688503i \(-0.241732\pi\)
0.725234 + 0.688503i \(0.241732\pi\)
\(192\) 0 0
\(193\) 2.57566e6i 0.358275i 0.983824 + 0.179138i \(0.0573307\pi\)
−0.983824 + 0.179138i \(0.942669\pi\)
\(194\) 0 0
\(195\) −1.47446e7 −1.98852
\(196\) 0 0
\(197\) −333309. −0.0435962 −0.0217981 0.999762i \(-0.506939\pi\)
−0.0217981 + 0.999762i \(0.506939\pi\)
\(198\) 0 0
\(199\) 9.52584e6 1.20877 0.604385 0.796692i \(-0.293419\pi\)
0.604385 + 0.796692i \(0.293419\pi\)
\(200\) 0 0
\(201\) 1.98214e7 2.44088
\(202\) 0 0
\(203\) 8.05065e6i 0.962372i
\(204\) 0 0
\(205\) 1.22718e7i 1.42445i
\(206\) 0 0
\(207\) −454924. −0.0512894
\(208\) 0 0
\(209\) −6.61137e6 1.51964e7i −0.724191 1.66456i
\(210\) 0 0
\(211\) 7.87152e6i 0.837937i 0.908001 + 0.418969i \(0.137608\pi\)
−0.908001 + 0.418969i \(0.862392\pi\)
\(212\) 0 0
\(213\) 1.02273e7 1.05833
\(214\) 0 0
\(215\) 1.36650e6 0.137498
\(216\) 0 0
\(217\) 1.34303e7i 1.31433i
\(218\) 0 0
\(219\) 2.23523e6i 0.212809i
\(220\) 0 0
\(221\) 5.91091e6i 0.547618i
\(222\) 0 0
\(223\) 1.47262e7i 1.32793i 0.747762 + 0.663967i \(0.231128\pi\)
−0.747762 + 0.663967i \(0.768872\pi\)
\(224\) 0 0
\(225\) 3.72279e6 0.326829
\(226\) 0 0
\(227\) 7.49958e6i 0.641150i −0.947223 0.320575i \(-0.896124\pi\)
0.947223 0.320575i \(-0.103876\pi\)
\(228\) 0 0
\(229\) 5.68862e6 0.473697 0.236848 0.971547i \(-0.423886\pi\)
0.236848 + 0.971547i \(0.423886\pi\)
\(230\) 0 0
\(231\) 4.72508e7i 3.83330i
\(232\) 0 0
\(233\) −1.88368e7 −1.48916 −0.744578 0.667536i \(-0.767349\pi\)
−0.744578 + 0.667536i \(0.767349\pi\)
\(234\) 0 0
\(235\) 1.90950e7 1.47135
\(236\) 0 0
\(237\) −1.80607e7 −1.35672
\(238\) 0 0
\(239\) −8.34352e6 −0.611161 −0.305581 0.952166i \(-0.598851\pi\)
−0.305581 + 0.952166i \(0.598851\pi\)
\(240\) 0 0
\(241\) 1.05537e7i 0.753970i −0.926219 0.376985i \(-0.876961\pi\)
0.926219 0.376985i \(-0.123039\pi\)
\(242\) 0 0
\(243\) 2.02746e7i 1.41297i
\(244\) 0 0
\(245\) 1.48706e7 1.01119
\(246\) 0 0
\(247\) −2.37217e7 + 1.03204e7i −1.57418 + 0.684867i
\(248\) 0 0
\(249\) 4.84726e6i 0.313977i
\(250\) 0 0
\(251\) 1.91869e6 0.121335 0.0606673 0.998158i \(-0.480677\pi\)
0.0606673 + 0.998158i \(0.480677\pi\)
\(252\) 0 0
\(253\) −1.51300e6 −0.0934283
\(254\) 0 0
\(255\) 6.12683e6i 0.369501i
\(256\) 0 0
\(257\) 1.62596e7i 0.957877i 0.877848 + 0.478938i \(0.158978\pi\)
−0.877848 + 0.478938i \(0.841022\pi\)
\(258\) 0 0
\(259\) 2.53684e7i 1.46014i
\(260\) 0 0
\(261\) 1.14094e7i 0.641716i
\(262\) 0 0
\(263\) 6.66631e6 0.366453 0.183226 0.983071i \(-0.441346\pi\)
0.183226 + 0.983071i \(0.441346\pi\)
\(264\) 0 0
\(265\) 2.09653e7i 1.12658i
\(266\) 0 0
\(267\) 2.31657e7 1.21706
\(268\) 0 0
\(269\) 2.25023e7i 1.15603i 0.816025 + 0.578016i \(0.196173\pi\)
−0.816025 + 0.578016i \(0.803827\pi\)
\(270\) 0 0
\(271\) −2.63688e7 −1.32490 −0.662448 0.749108i \(-0.730483\pi\)
−0.662448 + 0.749108i \(0.730483\pi\)
\(272\) 0 0
\(273\) 7.37589e7 3.62515
\(274\) 0 0
\(275\) 1.23814e7 0.595349
\(276\) 0 0
\(277\) −2.95860e7 −1.39203 −0.696013 0.718030i \(-0.745044\pi\)
−0.696013 + 0.718030i \(0.745044\pi\)
\(278\) 0 0
\(279\) 1.90335e7i 0.876405i
\(280\) 0 0
\(281\) 1.14941e7i 0.518029i 0.965873 + 0.259015i \(0.0833978\pi\)
−0.965873 + 0.259015i \(0.916602\pi\)
\(282\) 0 0
\(283\) −3.92088e7 −1.72991 −0.864957 0.501846i \(-0.832655\pi\)
−0.864957 + 0.501846i \(0.832655\pi\)
\(284\) 0 0
\(285\) −2.45882e7 + 1.06974e7i −1.06216 + 0.462109i
\(286\) 0 0
\(287\) 6.13889e7i 2.59683i
\(288\) 0 0
\(289\) −2.16814e7 −0.898243
\(290\) 0 0
\(291\) −1.07659e7 −0.436891
\(292\) 0 0
\(293\) 5.81821e6i 0.231306i 0.993290 + 0.115653i \(0.0368960\pi\)
−0.993290 + 0.115653i \(0.963104\pi\)
\(294\) 0 0
\(295\) 1.87160e6i 0.0729033i
\(296\) 0 0
\(297\) 232969.i 0.00889259i
\(298\) 0 0
\(299\) 2.36181e6i 0.0883552i
\(300\) 0 0
\(301\) −6.83583e6 −0.250664
\(302\) 0 0
\(303\) 1.69064e7i 0.607748i
\(304\) 0 0
\(305\) −3.62003e7 −1.27589
\(306\) 0 0
\(307\) 2.09554e7i 0.724236i −0.932132 0.362118i \(-0.882054\pi\)
0.932132 0.362118i \(-0.117946\pi\)
\(308\) 0 0
\(309\) −3.18956e7 −1.08107
\(310\) 0 0
\(311\) −2.79502e7 −0.929189 −0.464594 0.885524i \(-0.653800\pi\)
−0.464594 + 0.885524i \(0.653800\pi\)
\(312\) 0 0
\(313\) 1.47351e7 0.480528 0.240264 0.970708i \(-0.422766\pi\)
0.240264 + 0.970708i \(0.422766\pi\)
\(314\) 0 0
\(315\) 3.81602e7 1.22090
\(316\) 0 0
\(317\) 3.55764e7i 1.11682i 0.829564 + 0.558411i \(0.188589\pi\)
−0.829564 + 0.558411i \(0.811411\pi\)
\(318\) 0 0
\(319\) 3.79460e7i 1.16894i
\(320\) 0 0
\(321\) 2.35537e7 0.712106
\(322\) 0 0
\(323\) 4.28844e6 + 9.85706e6i 0.127260 + 0.292509i
\(324\) 0 0
\(325\) 1.93275e7i 0.563021i
\(326\) 0 0
\(327\) 3.64583e7 1.04268
\(328\) 0 0
\(329\) −9.55212e7 −2.68233
\(330\) 0 0
\(331\) 4.77022e7i 1.31539i −0.753285 0.657694i \(-0.771532\pi\)
0.753285 0.657694i \(-0.228468\pi\)
\(332\) 0 0
\(333\) 3.59522e7i 0.973628i
\(334\) 0 0
\(335\) 5.32401e7i 1.41613i
\(336\) 0 0
\(337\) 4.64803e7i 1.21445i −0.794530 0.607225i \(-0.792283\pi\)
0.794530 0.607225i \(-0.207717\pi\)
\(338\) 0 0
\(339\) 1.00279e7 0.257402
\(340\) 0 0
\(341\) 6.33022e7i 1.59645i
\(342\) 0 0
\(343\) −1.40812e7 −0.348946
\(344\) 0 0
\(345\) 2.44809e6i 0.0596170i
\(346\) 0 0
\(347\) −5.43540e7 −1.30090 −0.650448 0.759550i \(-0.725419\pi\)
−0.650448 + 0.759550i \(0.725419\pi\)
\(348\) 0 0
\(349\) 3.19672e7 0.752018 0.376009 0.926616i \(-0.377296\pi\)
0.376009 + 0.926616i \(0.377296\pi\)
\(350\) 0 0
\(351\) −363666. −0.00840972
\(352\) 0 0
\(353\) 1.94618e7 0.442444 0.221222 0.975224i \(-0.428995\pi\)
0.221222 + 0.975224i \(0.428995\pi\)
\(354\) 0 0
\(355\) 2.74704e7i 0.614016i
\(356\) 0 0
\(357\) 3.06490e7i 0.673615i
\(358\) 0 0
\(359\) −8.12905e6 −0.175694 −0.0878469 0.996134i \(-0.527999\pi\)
−0.0878469 + 0.996134i \(0.527999\pi\)
\(360\) 0 0
\(361\) −3.20707e7 + 3.44207e7i −0.681690 + 0.731641i
\(362\) 0 0
\(363\) 1.55126e8i 3.24313i
\(364\) 0 0
\(365\) 6.00379e6 0.123466
\(366\) 0 0
\(367\) −9.45918e7 −1.91362 −0.956809 0.290716i \(-0.906107\pi\)
−0.956809 + 0.290716i \(0.906107\pi\)
\(368\) 0 0
\(369\) 8.70008e7i 1.73159i
\(370\) 0 0
\(371\) 1.04877e8i 2.05380i
\(372\) 0 0
\(373\) 1.41683e7i 0.273018i −0.990639 0.136509i \(-0.956412\pi\)
0.990639 0.136509i \(-0.0435882\pi\)
\(374\) 0 0
\(375\) 8.11175e7i 1.53823i
\(376\) 0 0
\(377\) 5.92340e7 1.10547
\(378\) 0 0
\(379\) 5.66625e6i 0.104083i −0.998645 0.0520413i \(-0.983427\pi\)
0.998645 0.0520413i \(-0.0165727\pi\)
\(380\) 0 0
\(381\) −3.56123e7 −0.643909
\(382\) 0 0
\(383\) 8.42138e7i 1.49895i −0.662034 0.749474i \(-0.730306\pi\)
0.662034 0.749474i \(-0.269694\pi\)
\(384\) 0 0
\(385\) 1.26915e8 2.22398
\(386\) 0 0
\(387\) −9.68778e6 −0.167144
\(388\) 0 0
\(389\) 1.89837e7 0.322501 0.161251 0.986913i \(-0.448447\pi\)
0.161251 + 0.986913i \(0.448447\pi\)
\(390\) 0 0
\(391\) 981404. 0.0164179
\(392\) 0 0
\(393\) 5.58454e7i 0.920046i
\(394\) 0 0
\(395\) 4.85109e7i 0.787133i
\(396\) 0 0
\(397\) −2.79727e7 −0.447057 −0.223528 0.974697i \(-0.571758\pi\)
−0.223528 + 0.974697i \(0.571758\pi\)
\(398\) 0 0
\(399\) 1.23000e8 5.35130e7i 1.93637 0.842443i
\(400\) 0 0
\(401\) 9.08568e7i 1.40904i 0.709682 + 0.704522i \(0.248839\pi\)
−0.709682 + 0.704522i \(0.751161\pi\)
\(402\) 0 0
\(403\) 9.88154e7 1.50977
\(404\) 0 0
\(405\) −5.46460e7 −0.822609
\(406\) 0 0
\(407\) 1.19571e8i 1.77355i
\(408\) 0 0
\(409\) 9.73970e7i 1.42356i −0.702403 0.711780i \(-0.747889\pi\)
0.702403 0.711780i \(-0.252111\pi\)
\(410\) 0 0
\(411\) 4.11763e7i 0.593091i
\(412\) 0 0
\(413\) 9.36253e6i 0.132906i
\(414\) 0 0
\(415\) −1.30197e7 −0.182161
\(416\) 0 0
\(417\) 1.06501e8i 1.46875i
\(418\) 0 0
\(419\) 4.44767e7 0.604631 0.302316 0.953208i \(-0.402240\pi\)
0.302316 + 0.953208i \(0.402240\pi\)
\(420\) 0 0
\(421\) 8.49965e7i 1.13908i −0.821963 0.569541i \(-0.807121\pi\)
0.821963 0.569541i \(-0.192879\pi\)
\(422\) 0 0
\(423\) −1.35373e8 −1.78859
\(424\) 0 0
\(425\) −8.03114e6 −0.104619
\(426\) 0 0
\(427\) 1.81089e8 2.32600
\(428\) 0 0
\(429\) 3.47655e8 4.40329
\(430\) 0 0
\(431\) 1.20089e8i 1.49993i 0.661476 + 0.749966i \(0.269930\pi\)
−0.661476 + 0.749966i \(0.730070\pi\)
\(432\) 0 0
\(433\) 5.72220e7i 0.704854i −0.935839 0.352427i \(-0.885356\pi\)
0.935839 0.352427i \(-0.114644\pi\)
\(434\) 0 0
\(435\) 6.13978e7 0.745907
\(436\) 0 0
\(437\) 1.71352e6 + 3.93857e6i 0.0205327 + 0.0471948i
\(438\) 0 0
\(439\) 4.39003e7i 0.518889i −0.965758 0.259444i \(-0.916461\pi\)
0.965758 0.259444i \(-0.0835394\pi\)
\(440\) 0 0
\(441\) −1.05425e8 −1.22921
\(442\) 0 0
\(443\) −7.04729e7 −0.810608 −0.405304 0.914182i \(-0.632834\pi\)
−0.405304 + 0.914182i \(0.632834\pi\)
\(444\) 0 0
\(445\) 6.22227e7i 0.706105i
\(446\) 0 0
\(447\) 5.62428e7i 0.629715i
\(448\) 0 0
\(449\) 1.26837e8i 1.40122i −0.713543 0.700611i \(-0.752911\pi\)
0.713543 0.700611i \(-0.247089\pi\)
\(450\) 0 0
\(451\) 2.89351e8i 3.15424i
\(452\) 0 0
\(453\) −1.41403e8 −1.52112
\(454\) 0 0
\(455\) 1.98115e8i 2.10321i
\(456\) 0 0
\(457\) −1.93303e7 −0.202530 −0.101265 0.994859i \(-0.532289\pi\)
−0.101265 + 0.994859i \(0.532289\pi\)
\(458\) 0 0
\(459\) 151114.i 0.00156267i
\(460\) 0 0
\(461\) −6.62945e7 −0.676667 −0.338333 0.941026i \(-0.609863\pi\)
−0.338333 + 0.941026i \(0.609863\pi\)
\(462\) 0 0
\(463\) 2.57206e7 0.259142 0.129571 0.991570i \(-0.458640\pi\)
0.129571 + 0.991570i \(0.458640\pi\)
\(464\) 0 0
\(465\) 1.02425e8 1.01870
\(466\) 0 0
\(467\) −6.14641e7 −0.603491 −0.301745 0.953389i \(-0.597569\pi\)
−0.301745 + 0.953389i \(0.597569\pi\)
\(468\) 0 0
\(469\) 2.66330e8i 2.58167i
\(470\) 0 0
\(471\) 1.97353e7i 0.188878i
\(472\) 0 0
\(473\) −3.22200e7 −0.304468
\(474\) 0 0
\(475\) −1.40223e7 3.22305e7i −0.130840 0.300737i
\(476\) 0 0
\(477\) 1.48633e8i 1.36949i
\(478\) 0 0
\(479\) −1.28933e8 −1.17316 −0.586581 0.809890i \(-0.699527\pi\)
−0.586581 + 0.809890i \(0.699527\pi\)
\(480\) 0 0
\(481\) 1.86652e8 1.67725
\(482\) 0 0
\(483\) 1.22464e7i 0.108684i
\(484\) 0 0
\(485\) 2.89171e7i 0.253472i
\(486\) 0 0
\(487\) 1.30778e8i 1.13227i 0.824314 + 0.566133i \(0.191561\pi\)
−0.824314 + 0.566133i \(0.808439\pi\)
\(488\) 0 0
\(489\) 1.17356e8i 1.00364i
\(490\) 0 0
\(491\) 1.59776e8 1.34980 0.674898 0.737911i \(-0.264188\pi\)
0.674898 + 0.737911i \(0.264188\pi\)
\(492\) 0 0
\(493\) 2.46135e7i 0.205415i
\(494\) 0 0
\(495\) 1.79865e8 1.48296
\(496\) 0 0
\(497\) 1.37419e8i 1.11938i
\(498\) 0 0
\(499\) −9.75200e7 −0.784860 −0.392430 0.919782i \(-0.628365\pi\)
−0.392430 + 0.919782i \(0.628365\pi\)
\(500\) 0 0
\(501\) 1.22197e8 0.971736
\(502\) 0 0
\(503\) 1.63836e8 1.28738 0.643690 0.765286i \(-0.277403\pi\)
0.643690 + 0.765286i \(0.277403\pi\)
\(504\) 0 0
\(505\) 4.54104e7 0.352599
\(506\) 0 0
\(507\) 3.58547e8i 2.75120i
\(508\) 0 0
\(509\) 1.33348e8i 1.01119i −0.862771 0.505594i \(-0.831273\pi\)
0.862771 0.505594i \(-0.168727\pi\)
\(510\) 0 0
\(511\) −3.00335e7 −0.225083
\(512\) 0 0
\(513\) −606451. + 263844.i −0.00449204 + 0.00195432i
\(514\) 0 0
\(515\) 8.56710e7i 0.627209i
\(516\) 0 0
\(517\) −4.50230e8 −3.25809
\(518\) 0 0
\(519\) −9.62768e7 −0.688684
\(520\) 0 0
\(521\) 1.78512e8i 1.26228i −0.775669 0.631139i \(-0.782588\pi\)
0.775669 0.631139i \(-0.217412\pi\)
\(522\) 0 0
\(523\) 1.67427e8i 1.17036i 0.810902 + 0.585181i \(0.198977\pi\)
−0.810902 + 0.585181i \(0.801023\pi\)
\(524\) 0 0
\(525\) 1.00216e8i 0.692562i
\(526\) 0 0
\(527\) 4.10607e7i 0.280540i
\(528\) 0 0
\(529\) −1.47644e8 −0.997351
\(530\) 0 0
\(531\) 1.32686e7i 0.0886223i
\(532\) 0 0
\(533\) −4.51679e8 −2.98297
\(534\) 0 0
\(535\) 6.32649e7i 0.413144i
\(536\) 0 0
\(537\) 2.93266e8 1.89382
\(538\) 0 0
\(539\) −3.50626e8 −2.23912
\(540\) 0 0
\(541\) −1.01782e8 −0.642808 −0.321404 0.946942i \(-0.604155\pi\)
−0.321404 + 0.946942i \(0.604155\pi\)
\(542\) 0 0
\(543\) −1.16381e8 −0.726911
\(544\) 0 0
\(545\) 9.79264e7i 0.604937i
\(546\) 0 0
\(547\) 2.32682e8i 1.42168i −0.703354 0.710840i \(-0.748315\pi\)
0.703354 0.710840i \(-0.251685\pi\)
\(548\) 0 0
\(549\) 2.56641e8 1.55099
\(550\) 0 0
\(551\) 9.87788e7 4.29750e7i 0.590485 0.256898i
\(552\) 0 0
\(553\) 2.42672e8i 1.43497i
\(554\) 0 0
\(555\) 1.93470e8 1.13171
\(556\) 0 0
\(557\) −9.88543e7 −0.572045 −0.286022 0.958223i \(-0.592333\pi\)
−0.286022 + 0.958223i \(0.592333\pi\)
\(558\) 0 0
\(559\) 5.02957e7i 0.287936i
\(560\) 0 0
\(561\) 1.44461e8i 0.818206i
\(562\) 0 0
\(563\) 2.96849e8i 1.66345i −0.555187 0.831726i \(-0.687353\pi\)
0.555187 0.831726i \(-0.312647\pi\)
\(564\) 0 0
\(565\) 2.69348e7i 0.149338i
\(566\) 0 0
\(567\) 2.73363e8 1.49965
\(568\) 0 0
\(569\) 2.25877e8i 1.22613i −0.790034 0.613063i \(-0.789937\pi\)
0.790034 0.613063i \(-0.210063\pi\)
\(570\) 0 0
\(571\) −1.34657e8 −0.723303 −0.361652 0.932313i \(-0.617787\pi\)
−0.361652 + 0.932313i \(0.617787\pi\)
\(572\) 0 0
\(573\) 3.85576e8i 2.04949i
\(574\) 0 0
\(575\) −3.20899e6 −0.0168797
\(576\) 0 0
\(577\) 1.22386e8 0.637097 0.318549 0.947907i \(-0.396805\pi\)
0.318549 + 0.947907i \(0.396805\pi\)
\(578\) 0 0
\(579\) −9.82631e7 −0.506238
\(580\) 0 0
\(581\) 6.51298e7 0.332087
\(582\) 0 0
\(583\) 4.94328e8i 2.49465i
\(584\) 0 0
\(585\) 2.80770e8i 1.40244i
\(586\) 0 0
\(587\) −3.54883e8 −1.75457 −0.877285 0.479970i \(-0.840647\pi\)
−0.877285 + 0.479970i \(0.840647\pi\)
\(588\) 0 0
\(589\) 1.64785e8 7.16918e7i 0.806439 0.350852i
\(590\) 0 0
\(591\) 1.27160e7i 0.0616009i
\(592\) 0 0
\(593\) 3.36642e8 1.61437 0.807186 0.590297i \(-0.200989\pi\)
0.807186 + 0.590297i \(0.200989\pi\)
\(594\) 0 0
\(595\) −8.23228e7 −0.390813
\(596\) 0 0
\(597\) 3.63417e8i 1.70798i
\(598\) 0 0
\(599\) 5.74144e7i 0.267141i −0.991039 0.133571i \(-0.957356\pi\)
0.991039 0.133571i \(-0.0426443\pi\)
\(600\) 0 0
\(601\) 1.19611e7i 0.0550994i −0.999620 0.0275497i \(-0.991230\pi\)
0.999620 0.0275497i \(-0.00877045\pi\)
\(602\) 0 0
\(603\) 3.77444e8i 1.72147i
\(604\) 0 0
\(605\) 4.16665e8 1.88157
\(606\) 0 0
\(607\) 2.53084e8i 1.13161i −0.824538 0.565807i \(-0.808565\pi\)
0.824538 0.565807i \(-0.191435\pi\)
\(608\) 0 0
\(609\) −3.07138e8 −1.35982
\(610\) 0 0
\(611\) 7.02813e8i 3.08117i
\(612\) 0 0
\(613\) 4.80487e7 0.208593 0.104297 0.994546i \(-0.466741\pi\)
0.104297 + 0.994546i \(0.466741\pi\)
\(614\) 0 0
\(615\) −4.68179e8 −2.01273
\(616\) 0 0
\(617\) −1.50367e8 −0.640174 −0.320087 0.947388i \(-0.603712\pi\)
−0.320087 + 0.947388i \(0.603712\pi\)
\(618\) 0 0
\(619\) 2.30454e8 0.971657 0.485828 0.874054i \(-0.338518\pi\)
0.485828 + 0.874054i \(0.338518\pi\)
\(620\) 0 0
\(621\) 60380.5i 0.000252128i
\(622\) 0 0
\(623\) 3.11264e8i 1.28726i
\(624\) 0 0
\(625\) −1.37811e8 −0.564472
\(626\) 0 0
\(627\) 5.79751e8 2.52228e8i 2.35201 1.02327i
\(628\) 0 0
\(629\) 7.75595e7i 0.311661i
\(630\) 0 0
\(631\) 2.64740e8 1.05374 0.526868 0.849947i \(-0.323366\pi\)
0.526868 + 0.849947i \(0.323366\pi\)
\(632\) 0 0
\(633\) −3.00304e8 −1.18399
\(634\) 0 0
\(635\) 9.56540e7i 0.373579i
\(636\) 0 0
\(637\) 5.47331e8i 2.11754i
\(638\) 0 0
\(639\) 1.94751e8i 0.746408i
\(640\) 0 0
\(641\) 2.39436e8i 0.909108i 0.890719 + 0.454554i \(0.150201\pi\)
−0.890719 + 0.454554i \(0.849799\pi\)
\(642\) 0 0
\(643\) −3.28913e8 −1.23722 −0.618611 0.785697i \(-0.712304\pi\)
−0.618611 + 0.785697i \(0.712304\pi\)
\(644\) 0 0
\(645\) 5.21330e7i 0.194282i
\(646\) 0 0
\(647\) 2.24926e8 0.830476 0.415238 0.909713i \(-0.363698\pi\)
0.415238 + 0.909713i \(0.363698\pi\)
\(648\) 0 0
\(649\) 4.41294e7i 0.161434i
\(650\) 0 0
\(651\) −5.12373e8 −1.85714
\(652\) 0 0
\(653\) 2.25975e8 0.811560 0.405780 0.913971i \(-0.367000\pi\)
0.405780 + 0.913971i \(0.367000\pi\)
\(654\) 0 0
\(655\) 1.50000e8 0.533786
\(656\) 0 0
\(657\) −4.25637e7 −0.150087
\(658\) 0 0
\(659\) 3.30068e8i 1.15331i 0.816987 + 0.576656i \(0.195643\pi\)
−0.816987 + 0.576656i \(0.804357\pi\)
\(660\) 0 0
\(661\) 3.67503e8i 1.27250i 0.771484 + 0.636248i \(0.219515\pi\)
−0.771484 + 0.636248i \(0.780485\pi\)
\(662\) 0 0
\(663\) −2.25505e8 −0.773777
\(664\) 0 0
\(665\) −1.43735e8 3.30378e8i −0.488762 1.12343i
\(666\) 0 0
\(667\) 9.83477e6i 0.0331426i
\(668\) 0 0
\(669\) −5.61815e8 −1.87635
\(670\) 0 0
\(671\) 8.53547e8 2.82527
\(672\) 0 0
\(673\) 3.21489e8i 1.05468i 0.849654 + 0.527340i \(0.176811\pi\)
−0.849654 + 0.527340i \(0.823189\pi\)
\(674\) 0 0
\(675\) 494112.i 0.00160662i
\(676\) 0 0
\(677\) 3.83480e8i 1.23588i −0.786225 0.617941i \(-0.787967\pi\)
0.786225 0.617941i \(-0.212033\pi\)
\(678\) 0 0
\(679\) 1.44656e8i 0.462090i
\(680\) 0 0
\(681\) 2.86114e8 0.905936
\(682\) 0 0
\(683\) 8.83834e7i 0.277401i 0.990334 + 0.138701i \(0.0442926\pi\)
−0.990334 + 0.138701i \(0.955707\pi\)
\(684\) 0 0
\(685\) 1.10599e8 0.344095
\(686\) 0 0
\(687\) 2.17025e8i 0.669328i
\(688\) 0 0
\(689\) 7.71651e8 2.35919
\(690\) 0 0
\(691\) 1.24144e8 0.376262 0.188131 0.982144i \(-0.439757\pi\)
0.188131 + 0.982144i \(0.439757\pi\)
\(692\) 0 0
\(693\) −8.99759e8 −2.70350
\(694\) 0 0
\(695\) −2.86061e8 −0.852127
\(696\) 0 0
\(697\) 1.87686e8i 0.554286i
\(698\) 0 0
\(699\) 7.18637e8i 2.10416i
\(700\) 0 0
\(701\) 2.26176e8 0.656588 0.328294 0.944576i \(-0.393526\pi\)
0.328294 + 0.944576i \(0.393526\pi\)
\(702\) 0 0
\(703\) 3.11261e8 1.35418e8i 0.895900 0.389773i
\(704\) 0 0
\(705\) 7.28486e8i 2.07900i
\(706\) 0 0
\(707\) −2.27162e8 −0.642802
\(708\) 0 0
\(709\) 4.83198e8 1.35577 0.677886 0.735167i \(-0.262896\pi\)
0.677886 + 0.735167i \(0.262896\pi\)
\(710\) 0 0
\(711\) 3.43916e8i 0.956851i
\(712\) 0 0
\(713\) 1.64066e7i 0.0452636i
\(714\) 0 0
\(715\) 9.33797e8i 2.55467i
\(716\) 0 0
\(717\) 3.18311e8i 0.863563i
\(718\) 0 0
\(719\) −3.11946e8 −0.839253 −0.419627 0.907697i \(-0.637839\pi\)
−0.419627 + 0.907697i \(0.637839\pi\)
\(720\) 0 0
\(721\) 4.28563e8i 1.14343i
\(722\) 0 0
\(723\) 4.02631e8 1.06535
\(724\) 0 0
\(725\) 8.04811e7i 0.211193i
\(726\) 0 0
\(727\) 1.30234e8 0.338938 0.169469 0.985535i \(-0.445795\pi\)
0.169469 + 0.985535i \(0.445795\pi\)
\(728\) 0 0
\(729\) 3.84730e8 0.993054
\(730\) 0 0
\(731\) 2.08994e7 0.0535034
\(732\) 0 0
\(733\) 1.90792e8 0.484448 0.242224 0.970220i \(-0.422123\pi\)
0.242224 + 0.970220i \(0.422123\pi\)
\(734\) 0 0
\(735\) 5.67324e8i 1.42879i
\(736\) 0 0
\(737\) 1.25532e9i 3.13582i
\(738\) 0 0
\(739\) −4.48582e8 −1.11150 −0.555749 0.831350i \(-0.687568\pi\)
−0.555749 + 0.831350i \(0.687568\pi\)
\(740\) 0 0
\(741\) −3.93731e8 9.04997e8i −0.967708 2.22429i
\(742\) 0 0
\(743\) 2.16076e8i 0.526794i 0.964687 + 0.263397i \(0.0848430\pi\)
−0.964687 + 0.263397i \(0.915157\pi\)
\(744\) 0 0
\(745\) −1.51067e8 −0.365344
\(746\) 0 0
\(747\) 9.23024e7 0.221438
\(748\) 0 0
\(749\) 3.16478e8i 0.753179i
\(750\) 0 0
\(751\) 1.56034e8i 0.368383i −0.982890 0.184191i \(-0.941033\pi\)
0.982890 0.184191i \(-0.0589667\pi\)
\(752\) 0 0
\(753\) 7.31994e7i 0.171444i
\(754\) 0 0
\(755\) 3.79806e8i 0.882512i
\(756\) 0 0
\(757\) −4.85982e8 −1.12030 −0.560148 0.828393i \(-0.689256\pi\)
−0.560148 + 0.828393i \(0.689256\pi\)
\(758\) 0 0
\(759\) 5.77221e7i 0.132013i
\(760\) 0 0
\(761\) −4.11783e8 −0.934360 −0.467180 0.884162i \(-0.654730\pi\)
−0.467180 + 0.884162i \(0.654730\pi\)
\(762\) 0 0
\(763\) 4.89869e8i 1.10282i
\(764\) 0 0
\(765\) −1.16668e8 −0.260597
\(766\) 0 0
\(767\) 6.88864e7 0.152668
\(768\) 0 0
\(769\) 2.85308e7 0.0627386 0.0313693 0.999508i \(-0.490013\pi\)
0.0313693 + 0.999508i \(0.490013\pi\)
\(770\) 0 0
\(771\) −6.20313e8 −1.35347
\(772\) 0 0
\(773\) 6.67394e8i 1.44492i 0.691413 + 0.722460i \(0.256989\pi\)
−0.691413 + 0.722460i \(0.743011\pi\)
\(774\) 0 0
\(775\) 1.34260e8i 0.288431i
\(776\) 0 0
\(777\) −9.67820e8 −2.06315
\(778\) 0 0
\(779\) −7.53221e8 + 3.27699e8i −1.59335 + 0.693206i
\(780\) 0 0
\(781\) 6.47709e8i 1.35965i
\(782\) 0 0
\(783\) 1.51433e6 0.00315454
\(784\) 0 0
\(785\) 5.30087e7 0.109582
\(786\) 0 0
\(787\) 3.31663e8i 0.680413i 0.940351 + 0.340206i \(0.110497\pi\)
−0.940351 + 0.340206i \(0.889503\pi\)
\(788\) 0 0
\(789\) 2.54324e8i 0.517793i
\(790\) 0 0
\(791\) 1.34739e8i 0.272248i
\(792\) 0 0
\(793\) 1.33240e9i 2.67186i
\(794\) 0 0
\(795\) 7.99839e8 1.59185
\(796\) 0 0
\(797\) 1.09251e7i 0.0215799i −0.999942 0.0107900i \(-0.996565\pi\)
0.999942 0.0107900i \(-0.00343462\pi\)
\(798\) 0 0
\(799\) 2.92040e8 0.572535
\(800\) 0 0
\(801\) 4.41126e8i 0.858351i
\(802\) 0 0
\(803\) −1.41560e8 −0.273397
\(804\) 0 0
\(805\) −3.28936e7 −0.0630556
\(806\) 0 0
\(807\) −8.58478e8 −1.63346
\(808\) 0 0
\(809\) −7.74089e8 −1.46199 −0.730997 0.682381i \(-0.760945\pi\)
−0.730997 + 0.682381i \(0.760945\pi\)
\(810\) 0 0
\(811\) 3.16901e8i 0.594102i 0.954862 + 0.297051i \(0.0960031\pi\)
−0.954862 + 0.297051i \(0.903997\pi\)
\(812\) 0 0
\(813\) 1.00599e9i 1.87206i
\(814\) 0 0
\(815\) −3.15216e8 −0.582285
\(816\) 0 0
\(817\) 3.64902e7 + 8.38733e7i 0.0669129 + 0.153800i
\(818\) 0 0
\(819\) 1.40453e9i 2.55670i
\(820\) 0 0
\(821\) −1.01886e8 −0.184113 −0.0920567 0.995754i \(-0.529344\pi\)
−0.0920567 + 0.995754i \(0.529344\pi\)
\(822\) 0 0
\(823\) −1.89000e8 −0.339050 −0.169525 0.985526i \(-0.554223\pi\)
−0.169525 + 0.985526i \(0.554223\pi\)
\(824\) 0 0
\(825\) 4.72358e8i 0.841220i
\(826\) 0 0
\(827\) 1.96997e8i 0.348291i −0.984720 0.174146i \(-0.944284\pi\)
0.984720 0.174146i \(-0.0557163\pi\)
\(828\) 0 0
\(829\) 6.54314e8i 1.14848i −0.818688 0.574239i \(-0.805298\pi\)
0.818688 0.574239i \(-0.194702\pi\)
\(830\) 0 0
\(831\) 1.12873e9i 1.96691i
\(832\) 0 0
\(833\) 2.27432e8 0.393475
\(834\) 0 0
\(835\) 3.28220e8i 0.563775i
\(836\) 0 0
\(837\) 2.52624e6 0.00430823
\(838\) 0 0
\(839\) 9.25960e8i 1.56786i 0.620852 + 0.783928i \(0.286787\pi\)
−0.620852 + 0.783928i \(0.713213\pi\)
\(840\) 0 0
\(841\) 3.48168e8 0.585331
\(842\) 0 0
\(843\) −4.38506e8 −0.731969
\(844\) 0 0
\(845\) −9.63052e8 −1.59617
\(846\) 0 0
\(847\) −2.08434e9 −3.43018
\(848\) 0 0
\(849\) 1.49584e9i 2.44435i
\(850\) 0 0
\(851\) 3.09903e7i 0.0502849i
\(852\) 0 0
\(853\) 1.08485e9 1.74793 0.873963 0.485993i \(-0.161542\pi\)
0.873963 + 0.485993i \(0.161542\pi\)
\(854\) 0 0
\(855\) −2.03702e8 4.68213e8i −0.325910 0.749110i
\(856\) 0 0
\(857\) 5.87373e7i 0.0933192i −0.998911 0.0466596i \(-0.985142\pi\)
0.998911 0.0466596i \(-0.0148576\pi\)
\(858\) 0 0
\(859\) 1.00505e9 1.58566 0.792828 0.609446i \(-0.208608\pi\)
0.792828 + 0.609446i \(0.208608\pi\)
\(860\) 0 0
\(861\) 2.34203e9 3.66929
\(862\) 0 0
\(863\) 1.11029e9i 1.72745i −0.503963 0.863725i \(-0.668125\pi\)
0.503963 0.863725i \(-0.331875\pi\)
\(864\) 0 0
\(865\) 2.58598e8i 0.399556i
\(866\) 0 0
\(867\) 8.27160e8i 1.26921i
\(868\) 0 0
\(869\) 1.14381e9i 1.74299i
\(870\) 0 0
\(871\) 1.95956e9 2.96555
\(872\) 0 0
\(873\) 2.05007e8i 0.308124i
\(874\) 0 0
\(875\) 1.08993e9 1.62695
\(876\) 0 0
\(877\) 6.01362e7i 0.0891532i −0.999006 0.0445766i \(-0.985806\pi\)
0.999006 0.0445766i \(-0.0141939\pi\)
\(878\) 0 0
\(879\) −2.21969e8 −0.326832
\(880\) 0 0
\(881\) 7.07347e8 1.03444 0.517220 0.855853i \(-0.326967\pi\)
0.517220 + 0.855853i \(0.326967\pi\)
\(882\) 0 0
\(883\) −1.66827e8 −0.242317 −0.121159 0.992633i \(-0.538661\pi\)
−0.121159 + 0.992633i \(0.538661\pi\)
\(884\) 0 0
\(885\) 7.14028e7 0.103011
\(886\) 0 0
\(887\) 4.94226e8i 0.708198i 0.935208 + 0.354099i \(0.115212\pi\)
−0.935208 + 0.354099i \(0.884788\pi\)
\(888\) 0 0
\(889\) 4.78502e8i 0.681049i
\(890\) 0 0
\(891\) 1.28847e9 1.82155
\(892\) 0 0
\(893\) 5.09900e8 + 1.17201e9i 0.716028 + 1.64580i
\(894\) 0 0
\(895\) 7.87709e8i 1.09874i
\(896\) 0 0
\(897\) −9.01047e7 −0.124845
\(898\) 0 0
\(899\) −4.11475e8 −0.566323
\(900\) 0 0
\(901\) 3.20644e8i 0.438378i
\(902\) 0 0
\(903\) 2.60791e8i 0.354185i
\(904\) 0 0
\(905\) 3.12596e8i 0.421734i
\(906\) 0 0
\(907\) 1.07109e9i 1.43551i −0.696297 0.717754i \(-0.745170\pi\)
0.696297 0.717754i \(-0.254830\pi\)
\(908\) 0 0
\(909\) −3.21935e8 −0.428625
\(910\) 0 0
\(911\) 1.35204e9i 1.78828i 0.447788 + 0.894140i \(0.352212\pi\)
−0.447788 + 0.894140i \(0.647788\pi\)
\(912\) 0 0
\(913\) 3.06983e8 0.403369
\(914\) 0 0
\(915\) 1.38107e9i 1.80282i
\(916\) 0 0
\(917\) −7.50362e8 −0.973113
\(918\) 0 0
\(919\) 4.68790e8 0.603994 0.301997 0.953309i \(-0.402347\pi\)
0.301997 + 0.953309i \(0.402347\pi\)
\(920\) 0 0
\(921\) 7.99461e8 1.02334
\(922\) 0 0
\(923\) 1.01108e9 1.28582
\(924\) 0 0
\(925\) 2.53604e8i 0.320428i
\(926\) 0 0
\(927\) 6.07362e8i 0.762445i
\(928\) 0 0
\(929\) −8.00888e8 −0.998906 −0.499453 0.866341i \(-0.666466\pi\)
−0.499453 + 0.866341i \(0.666466\pi\)
\(930\) 0 0
\(931\) 3.97095e8 + 9.12730e8i 0.492091 + 1.13108i
\(932\) 0 0
\(933\) 1.06632e9i 1.31293i
\(934\) 0 0
\(935\) −3.88021e8 −0.474701
\(936\) 0 0
\(937\) −1.10661e9 −1.34516 −0.672582 0.740023i \(-0.734815\pi\)
−0.672582 + 0.740023i \(0.734815\pi\)
\(938\) 0 0
\(939\) 5.62152e8i 0.678980i
\(940\) 0 0
\(941\) 2.80204e8i 0.336284i −0.985763 0.168142i \(-0.946223\pi\)
0.985763 0.168142i \(-0.0537766\pi\)
\(942\) 0 0
\(943\) 7.49934e7i 0.0894310i
\(944\) 0 0
\(945\) 5.06487e6i 0.00600169i
\(946\) 0 0
\(947\) −6.65023e8 −0.783045 −0.391522 0.920169i \(-0.628051\pi\)
−0.391522 + 0.920169i \(0.628051\pi\)
\(948\) 0 0
\(949\) 2.20976e8i 0.258552i
\(950\) 0 0
\(951\) −1.35726e9 −1.57806
\(952\) 0 0
\(953\) 4.34859e8i 0.502423i 0.967932 + 0.251212i \(0.0808290\pi\)
−0.967932 + 0.251212i \(0.919171\pi\)
\(954\) 0 0
\(955\) 1.03565e9 1.18906
\(956\) 0 0
\(957\) −1.44766e9 −1.65170
\(958\) 0 0
\(959\) −5.53262e8 −0.627300
\(960\) 0 0
\(961\) 2.01073e8 0.226560
\(962\) 0 0
\(963\) 4.48515e8i 0.502224i
\(964\) 0 0
\(965\) 2.63933e8i 0.293706i
\(966\) 0 0
\(967\) 1.10495e9 1.22198 0.610991 0.791637i \(-0.290771\pi\)
0.610991 + 0.791637i \(0.290771\pi\)
\(968\) 0 0
\(969\) −3.76053e8 + 1.63607e8i −0.413312 + 0.179817i
\(970\) 0 0
\(971\) 6.04768e8i 0.660588i −0.943878 0.330294i \(-0.892852\pi\)
0.943878 0.330294i \(-0.107148\pi\)
\(972\) 0 0
\(973\) 1.43100e9 1.55346
\(974\) 0 0
\(975\) 7.37355e8 0.795542
\(976\) 0 0
\(977\) 7.39173e8i 0.792616i −0.918118 0.396308i \(-0.870291\pi\)
0.918118 0.396308i \(-0.129709\pi\)
\(978\) 0 0
\(979\) 1.46711e9i 1.56357i
\(980\) 0 0
\(981\) 6.94246e8i 0.735371i
\(982\) 0 0
\(983\) 1.51347e9i 1.59336i −0.604403 0.796679i \(-0.706588\pi\)
0.604403 0.796679i \(-0.293412\pi\)
\(984\) 0 0
\(985\) −3.41549e7 −0.0357392
\(986\) 0 0
\(987\) 3.64420e9i 3.79010i
\(988\) 0 0
\(989\) 8.35073e6 0.00863248
\(990\) 0 0
\(991\) 6.25442e8i 0.642638i 0.946971 + 0.321319i \(0.104126\pi\)
−0.946971 + 0.321319i \(0.895874\pi\)
\(992\) 0 0
\(993\) 1.81987e9 1.85863
\(994\) 0 0
\(995\) 9.76133e8 0.990922
\(996\) 0 0
\(997\) −3.76728e8 −0.380139 −0.190070 0.981771i \(-0.560871\pi\)
−0.190070 + 0.981771i \(0.560871\pi\)
\(998\) 0 0
\(999\) 4.77181e6 0.00478616
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.c.113.6 8
4.3 odd 2 76.7.c.b.37.3 8
12.11 even 2 684.7.h.c.37.4 8
19.18 odd 2 inner 304.7.e.c.113.3 8
76.75 even 2 76.7.c.b.37.6 yes 8
228.227 odd 2 684.7.h.c.37.3 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
76.7.c.b.37.3 8 4.3 odd 2
76.7.c.b.37.6 yes 8 76.75 even 2
304.7.e.c.113.3 8 19.18 odd 2 inner
304.7.e.c.113.6 8 1.1 even 1 trivial
684.7.h.c.37.3 8 228.227 odd 2
684.7.h.c.37.4 8 12.11 even 2