Properties

Label 304.5.e.e.113.4
Level $304$
Weight $5$
Character 304.113
Analytic conductor $31.424$
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] = [304,5,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, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("304.113");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 304 = 2^{4} \cdot 19 \)
Weight: \( k \) \(=\) \( 5 \)
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: \(31.4244687775\)
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: \( x^{8} + 450x^{6} + 68229x^{4} + 4001228x^{2} + 77475204 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{7}\cdot 3^{2} \)
Twist minimal: no (minimal twist has level 38)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 113.4
Root \(-8.07810i\) of defining polynomial
Character \(\chi\) \(=\) 304.113
Dual form 304.5.e.e.113.5

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-6.66389i q^{3} +26.7027 q^{5} -51.4469 q^{7} +36.5926 q^{9} +O(q^{10})\) \(q-6.66389i q^{3} +26.7027 q^{5} -51.4469 q^{7} +36.5926 q^{9} +25.0891 q^{11} -53.2531i q^{13} -177.944i q^{15} -24.4913 q^{17} +(78.0101 - 352.470i) q^{19} +342.837i q^{21} +612.242 q^{23} +88.0343 q^{25} -783.624i q^{27} -1345.96i q^{29} +912.680i q^{31} -167.191i q^{33} -1373.77 q^{35} +325.059i q^{37} -354.872 q^{39} -51.7494i q^{41} +2545.26 q^{43} +977.121 q^{45} -2998.19 q^{47} +245.788 q^{49} +163.208i q^{51} -4067.86i q^{53} +669.948 q^{55} +(-2348.82 - 519.851i) q^{57} -1078.88i q^{59} -6530.89 q^{61} -1882.58 q^{63} -1422.00i q^{65} -7386.18i q^{67} -4079.92i q^{69} -5806.02i q^{71} +3629.56 q^{73} -586.651i q^{75} -1290.76 q^{77} -3903.16i q^{79} -2257.99 q^{81} +997.967 q^{83} -653.985 q^{85} -8969.34 q^{87} -6152.85i q^{89} +2739.71i q^{91} +6082.00 q^{93} +(2083.08 - 9411.91i) q^{95} +5023.58i q^{97} +918.076 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 18 q^{5} + 162 q^{7} - 268 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 18 q^{5} + 162 q^{7} - 268 q^{9} + 6 q^{11} + 510 q^{17} + 12 q^{19} + 396 q^{23} + 3458 q^{25} - 1002 q^{35} + 6588 q^{39} + 8654 q^{43} - 10334 q^{45} - 3210 q^{47} + 9222 q^{49} - 17146 q^{55} - 14076 q^{57} + 1314 q^{61} - 29938 q^{63} + 23398 q^{73} - 44622 q^{77} - 20368 q^{81} + 10440 q^{83} + 21274 q^{85} + 14316 q^{87} + 19416 q^{93} + 34686 q^{95} + 56798 q^{99}+O(q^{100}) \) 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
\(3\) 6.66389i 0.740432i −0.928946 0.370216i \(-0.879284\pi\)
0.928946 0.370216i \(-0.120716\pi\)
\(4\) 0 0
\(5\) 26.7027 1.06811 0.534054 0.845450i \(-0.320668\pi\)
0.534054 + 0.845450i \(0.320668\pi\)
\(6\) 0 0
\(7\) −51.4469 −1.04994 −0.524969 0.851121i \(-0.675923\pi\)
−0.524969 + 0.851121i \(0.675923\pi\)
\(8\) 0 0
\(9\) 36.5926 0.451760
\(10\) 0 0
\(11\) 25.0891 0.207348 0.103674 0.994611i \(-0.466940\pi\)
0.103674 + 0.994611i \(0.466940\pi\)
\(12\) 0 0
\(13\) 53.2531i 0.315107i −0.987510 0.157553i \(-0.949639\pi\)
0.987510 0.157553i \(-0.0503606\pi\)
\(14\) 0 0
\(15\) 177.944i 0.790862i
\(16\) 0 0
\(17\) −24.4913 −0.0847451 −0.0423726 0.999102i \(-0.513492\pi\)
−0.0423726 + 0.999102i \(0.513492\pi\)
\(18\) 0 0
\(19\) 78.0101 352.470i 0.216094 0.976372i
\(20\) 0 0
\(21\) 342.837i 0.777408i
\(22\) 0 0
\(23\) 612.242 1.15736 0.578679 0.815555i \(-0.303568\pi\)
0.578679 + 0.815555i \(0.303568\pi\)
\(24\) 0 0
\(25\) 88.0343 0.140855
\(26\) 0 0
\(27\) 783.624i 1.07493i
\(28\) 0 0
\(29\) 1345.96i 1.60043i −0.599713 0.800215i \(-0.704718\pi\)
0.599713 0.800215i \(-0.295282\pi\)
\(30\) 0 0
\(31\) 912.680i 0.949719i 0.880062 + 0.474860i \(0.157501\pi\)
−0.880062 + 0.474860i \(0.842499\pi\)
\(32\) 0 0
\(33\) 167.191i 0.153527i
\(34\) 0 0
\(35\) −1373.77 −1.12145
\(36\) 0 0
\(37\) 325.059i 0.237443i 0.992928 + 0.118721i \(0.0378795\pi\)
−0.992928 + 0.118721i \(0.962120\pi\)
\(38\) 0 0
\(39\) −354.872 −0.233315
\(40\) 0 0
\(41\) 51.7494i 0.0307849i −0.999882 0.0153924i \(-0.995100\pi\)
0.999882 0.0153924i \(-0.00489976\pi\)
\(42\) 0 0
\(43\) 2545.26 1.37656 0.688280 0.725445i \(-0.258366\pi\)
0.688280 + 0.725445i \(0.258366\pi\)
\(44\) 0 0
\(45\) 977.121 0.482529
\(46\) 0 0
\(47\) −2998.19 −1.35726 −0.678630 0.734480i \(-0.737426\pi\)
−0.678630 + 0.734480i \(0.737426\pi\)
\(48\) 0 0
\(49\) 245.788 0.102369
\(50\) 0 0
\(51\) 163.208i 0.0627480i
\(52\) 0 0
\(53\) 4067.86i 1.44815i −0.689720 0.724077i \(-0.742266\pi\)
0.689720 0.724077i \(-0.257734\pi\)
\(54\) 0 0
\(55\) 669.948 0.221470
\(56\) 0 0
\(57\) −2348.82 519.851i −0.722938 0.160003i
\(58\) 0 0
\(59\) 1078.88i 0.309933i −0.987920 0.154966i \(-0.950473\pi\)
0.987920 0.154966i \(-0.0495269\pi\)
\(60\) 0 0
\(61\) −6530.89 −1.75514 −0.877572 0.479445i \(-0.840838\pi\)
−0.877572 + 0.479445i \(0.840838\pi\)
\(62\) 0 0
\(63\) −1882.58 −0.474320
\(64\) 0 0
\(65\) 1422.00i 0.336568i
\(66\) 0 0
\(67\) 7386.18i 1.64540i −0.568478 0.822698i \(-0.692468\pi\)
0.568478 0.822698i \(-0.307532\pi\)
\(68\) 0 0
\(69\) 4079.92i 0.856945i
\(70\) 0 0
\(71\) 5806.02i 1.15176i −0.817535 0.575879i \(-0.804660\pi\)
0.817535 0.575879i \(-0.195340\pi\)
\(72\) 0 0
\(73\) 3629.56 0.681096 0.340548 0.940227i \(-0.389387\pi\)
0.340548 + 0.940227i \(0.389387\pi\)
\(74\) 0 0
\(75\) 586.651i 0.104293i
\(76\) 0 0
\(77\) −1290.76 −0.217703
\(78\) 0 0
\(79\) 3903.16i 0.625407i −0.949851 0.312703i \(-0.898765\pi\)
0.949851 0.312703i \(-0.101235\pi\)
\(80\) 0 0
\(81\) −2257.99 −0.344153
\(82\) 0 0
\(83\) 997.967 0.144864 0.0724319 0.997373i \(-0.476924\pi\)
0.0724319 + 0.997373i \(0.476924\pi\)
\(84\) 0 0
\(85\) −653.985 −0.0905169
\(86\) 0 0
\(87\) −8969.34 −1.18501
\(88\) 0 0
\(89\) 6152.85i 0.776777i −0.921496 0.388389i \(-0.873032\pi\)
0.921496 0.388389i \(-0.126968\pi\)
\(90\) 0 0
\(91\) 2739.71i 0.330843i
\(92\) 0 0
\(93\) 6082.00 0.703203
\(94\) 0 0
\(95\) 2083.08 9411.91i 0.230812 1.04287i
\(96\) 0 0
\(97\) 5023.58i 0.533912i 0.963709 + 0.266956i \(0.0860178\pi\)
−0.963709 + 0.266956i \(0.913982\pi\)
\(98\) 0 0
\(99\) 918.076 0.0936717
\(100\) 0 0
\(101\) 15793.7 1.54825 0.774123 0.633035i \(-0.218191\pi\)
0.774123 + 0.633035i \(0.218191\pi\)
\(102\) 0 0
\(103\) 13068.7i 1.23185i 0.787806 + 0.615924i \(0.211217\pi\)
−0.787806 + 0.615924i \(0.788783\pi\)
\(104\) 0 0
\(105\) 9154.67i 0.830355i
\(106\) 0 0
\(107\) 8273.09i 0.722604i 0.932449 + 0.361302i \(0.117668\pi\)
−0.932449 + 0.361302i \(0.882332\pi\)
\(108\) 0 0
\(109\) 14219.6i 1.19683i 0.801186 + 0.598416i \(0.204203\pi\)
−0.801186 + 0.598416i \(0.795797\pi\)
\(110\) 0 0
\(111\) 2166.16 0.175810
\(112\) 0 0
\(113\) 20747.4i 1.62483i −0.583080 0.812414i \(-0.698153\pi\)
0.583080 0.812414i \(-0.301847\pi\)
\(114\) 0 0
\(115\) 16348.5 1.23618
\(116\) 0 0
\(117\) 1948.67i 0.142353i
\(118\) 0 0
\(119\) 1260.00 0.0889771
\(120\) 0 0
\(121\) −14011.5 −0.957007
\(122\) 0 0
\(123\) −344.852 −0.0227941
\(124\) 0 0
\(125\) −14338.4 −0.917660
\(126\) 0 0
\(127\) 11117.8i 0.689305i 0.938730 + 0.344653i \(0.112003\pi\)
−0.938730 + 0.344653i \(0.887997\pi\)
\(128\) 0 0
\(129\) 16961.3i 1.01925i
\(130\) 0 0
\(131\) −223.317 −0.0130130 −0.00650652 0.999979i \(-0.502071\pi\)
−0.00650652 + 0.999979i \(0.502071\pi\)
\(132\) 0 0
\(133\) −4013.38 + 18133.5i −0.226886 + 1.02513i
\(134\) 0 0
\(135\) 20924.9i 1.14814i
\(136\) 0 0
\(137\) 18953.1 1.00981 0.504903 0.863176i \(-0.331528\pi\)
0.504903 + 0.863176i \(0.331528\pi\)
\(138\) 0 0
\(139\) 11115.5 0.575309 0.287655 0.957734i \(-0.407125\pi\)
0.287655 + 0.957734i \(0.407125\pi\)
\(140\) 0 0
\(141\) 19979.6i 1.00496i
\(142\) 0 0
\(143\) 1336.07i 0.0653368i
\(144\) 0 0
\(145\) 35940.8i 1.70943i
\(146\) 0 0
\(147\) 1637.91i 0.0757975i
\(148\) 0 0
\(149\) −19727.7 −0.888597 −0.444299 0.895879i \(-0.646547\pi\)
−0.444299 + 0.895879i \(0.646547\pi\)
\(150\) 0 0
\(151\) 770.565i 0.0337952i −0.999857 0.0168976i \(-0.994621\pi\)
0.999857 0.0168976i \(-0.00537894\pi\)
\(152\) 0 0
\(153\) −896.201 −0.0382845
\(154\) 0 0
\(155\) 24371.0i 1.01440i
\(156\) 0 0
\(157\) 46244.5 1.87612 0.938061 0.346470i \(-0.112620\pi\)
0.938061 + 0.346470i \(0.112620\pi\)
\(158\) 0 0
\(159\) −27107.8 −1.07226
\(160\) 0 0
\(161\) −31498.0 −1.21515
\(162\) 0 0
\(163\) −32951.9 −1.24024 −0.620119 0.784507i \(-0.712916\pi\)
−0.620119 + 0.784507i \(0.712916\pi\)
\(164\) 0 0
\(165\) 4464.46i 0.163984i
\(166\) 0 0
\(167\) 48049.1i 1.72287i 0.507870 + 0.861434i \(0.330433\pi\)
−0.507870 + 0.861434i \(0.669567\pi\)
\(168\) 0 0
\(169\) 25725.1 0.900708
\(170\) 0 0
\(171\) 2854.59 12897.8i 0.0976228 0.441086i
\(172\) 0 0
\(173\) 46219.6i 1.54431i 0.635436 + 0.772153i \(0.280820\pi\)
−0.635436 + 0.772153i \(0.719180\pi\)
\(174\) 0 0
\(175\) −4529.10 −0.147889
\(176\) 0 0
\(177\) −7189.51 −0.229484
\(178\) 0 0
\(179\) 39826.5i 1.24299i −0.783420 0.621493i \(-0.786527\pi\)
0.783420 0.621493i \(-0.213473\pi\)
\(180\) 0 0
\(181\) 51844.5i 1.58251i 0.611488 + 0.791254i \(0.290571\pi\)
−0.611488 + 0.791254i \(0.709429\pi\)
\(182\) 0 0
\(183\) 43521.1i 1.29957i
\(184\) 0 0
\(185\) 8679.95i 0.253614i
\(186\) 0 0
\(187\) −614.467 −0.0175718
\(188\) 0 0
\(189\) 40315.1i 1.12861i
\(190\) 0 0
\(191\) 38958.8 1.06792 0.533960 0.845509i \(-0.320703\pi\)
0.533960 + 0.845509i \(0.320703\pi\)
\(192\) 0 0
\(193\) 43058.1i 1.15595i 0.816053 + 0.577977i \(0.196158\pi\)
−0.816053 + 0.577977i \(0.803842\pi\)
\(194\) 0 0
\(195\) −9476.05 −0.249206
\(196\) 0 0
\(197\) 26311.0 0.677962 0.338981 0.940793i \(-0.389918\pi\)
0.338981 + 0.940793i \(0.389918\pi\)
\(198\) 0 0
\(199\) 11773.1 0.297293 0.148647 0.988890i \(-0.452508\pi\)
0.148647 + 0.988890i \(0.452508\pi\)
\(200\) 0 0
\(201\) −49220.7 −1.21830
\(202\) 0 0
\(203\) 69245.7i 1.68035i
\(204\) 0 0
\(205\) 1381.85i 0.0328816i
\(206\) 0 0
\(207\) 22403.5 0.522848
\(208\) 0 0
\(209\) 1957.21 8843.18i 0.0448068 0.202449i
\(210\) 0 0
\(211\) 12495.7i 0.280670i 0.990104 + 0.140335i \(0.0448179\pi\)
−0.990104 + 0.140335i \(0.955182\pi\)
\(212\) 0 0
\(213\) −38690.7 −0.852799
\(214\) 0 0
\(215\) 67965.3 1.47032
\(216\) 0 0
\(217\) 46954.6i 0.997146i
\(218\) 0 0
\(219\) 24187.0i 0.504305i
\(220\) 0 0
\(221\) 1304.24i 0.0267038i
\(222\) 0 0
\(223\) 59706.5i 1.20064i −0.799761 0.600319i \(-0.795040\pi\)
0.799761 0.600319i \(-0.204960\pi\)
\(224\) 0 0
\(225\) 3221.40 0.0636326
\(226\) 0 0
\(227\) 57964.2i 1.12489i 0.826836 + 0.562443i \(0.190138\pi\)
−0.826836 + 0.562443i \(0.809862\pi\)
\(228\) 0 0
\(229\) 40157.9 0.765773 0.382886 0.923795i \(-0.374930\pi\)
0.382886 + 0.923795i \(0.374930\pi\)
\(230\) 0 0
\(231\) 8601.48i 0.161194i
\(232\) 0 0
\(233\) 71740.1 1.32145 0.660724 0.750629i \(-0.270249\pi\)
0.660724 + 0.750629i \(0.270249\pi\)
\(234\) 0 0
\(235\) −80059.7 −1.44970
\(236\) 0 0
\(237\) −26010.2 −0.463071
\(238\) 0 0
\(239\) −44178.4 −0.773417 −0.386709 0.922202i \(-0.626388\pi\)
−0.386709 + 0.922202i \(0.626388\pi\)
\(240\) 0 0
\(241\) 33823.7i 0.582353i −0.956669 0.291177i \(-0.905953\pi\)
0.956669 0.291177i \(-0.0940467\pi\)
\(242\) 0 0
\(243\) 48426.6i 0.820108i
\(244\) 0 0
\(245\) 6563.22 0.109341
\(246\) 0 0
\(247\) −18770.1 4154.27i −0.307662 0.0680928i
\(248\) 0 0
\(249\) 6650.34i 0.107262i
\(250\) 0 0
\(251\) 64952.8 1.03098 0.515490 0.856896i \(-0.327610\pi\)
0.515490 + 0.856896i \(0.327610\pi\)
\(252\) 0 0
\(253\) 15360.6 0.239976
\(254\) 0 0
\(255\) 4358.08i 0.0670217i
\(256\) 0 0
\(257\) 1155.59i 0.0174959i 0.999962 + 0.00874796i \(0.00278460\pi\)
−0.999962 + 0.00874796i \(0.997215\pi\)
\(258\) 0 0
\(259\) 16723.3i 0.249300i
\(260\) 0 0
\(261\) 49252.2i 0.723011i
\(262\) 0 0
\(263\) −6732.94 −0.0973404 −0.0486702 0.998815i \(-0.515498\pi\)
−0.0486702 + 0.998815i \(0.515498\pi\)
\(264\) 0 0
\(265\) 108623.i 1.54678i
\(266\) 0 0
\(267\) −41001.9 −0.575151
\(268\) 0 0
\(269\) 22231.5i 0.307230i −0.988131 0.153615i \(-0.950908\pi\)
0.988131 0.153615i \(-0.0490915\pi\)
\(270\) 0 0
\(271\) 46597.6 0.634490 0.317245 0.948344i \(-0.397242\pi\)
0.317245 + 0.948344i \(0.397242\pi\)
\(272\) 0 0
\(273\) 18257.1 0.244966
\(274\) 0 0
\(275\) 2208.70 0.0292060
\(276\) 0 0
\(277\) 105817. 1.37910 0.689548 0.724240i \(-0.257809\pi\)
0.689548 + 0.724240i \(0.257809\pi\)
\(278\) 0 0
\(279\) 33397.3i 0.429045i
\(280\) 0 0
\(281\) 20298.2i 0.257067i 0.991705 + 0.128533i \(0.0410269\pi\)
−0.991705 + 0.128533i \(0.958973\pi\)
\(282\) 0 0
\(283\) 55710.6 0.695609 0.347804 0.937567i \(-0.386927\pi\)
0.347804 + 0.937567i \(0.386927\pi\)
\(284\) 0 0
\(285\) −62720.0 13881.4i −0.772176 0.170901i
\(286\) 0 0
\(287\) 2662.35i 0.0323222i
\(288\) 0 0
\(289\) −82921.2 −0.992818
\(290\) 0 0
\(291\) 33476.6 0.395326
\(292\) 0 0
\(293\) 84053.1i 0.979081i 0.871980 + 0.489541i \(0.162836\pi\)
−0.871980 + 0.489541i \(0.837164\pi\)
\(294\) 0 0
\(295\) 28808.9i 0.331042i
\(296\) 0 0
\(297\) 19660.4i 0.222885i
\(298\) 0 0
\(299\) 32603.8i 0.364691i
\(300\) 0 0
\(301\) −130946. −1.44530
\(302\) 0 0
\(303\) 105247.i 1.14637i
\(304\) 0 0
\(305\) −174392. −1.87468
\(306\) 0 0
\(307\) 10396.7i 0.110311i 0.998478 + 0.0551555i \(0.0175655\pi\)
−0.998478 + 0.0551555i \(0.982435\pi\)
\(308\) 0 0
\(309\) 87088.2 0.912100
\(310\) 0 0
\(311\) −118107. −1.22111 −0.610557 0.791973i \(-0.709054\pi\)
−0.610557 + 0.791973i \(0.709054\pi\)
\(312\) 0 0
\(313\) −8391.17 −0.0856513 −0.0428257 0.999083i \(-0.513636\pi\)
−0.0428257 + 0.999083i \(0.513636\pi\)
\(314\) 0 0
\(315\) −50269.9 −0.506625
\(316\) 0 0
\(317\) 143474.i 1.42776i 0.700270 + 0.713878i \(0.253063\pi\)
−0.700270 + 0.713878i \(0.746937\pi\)
\(318\) 0 0
\(319\) 33769.0i 0.331847i
\(320\) 0 0
\(321\) 55131.0 0.535039
\(322\) 0 0
\(323\) −1910.57 + 8632.47i −0.0183129 + 0.0827428i
\(324\) 0 0
\(325\) 4688.09i 0.0443843i
\(326\) 0 0
\(327\) 94757.6 0.886173
\(328\) 0 0
\(329\) 154248. 1.42504
\(330\) 0 0
\(331\) 16943.3i 0.154647i 0.997006 + 0.0773237i \(0.0246375\pi\)
−0.997006 + 0.0773237i \(0.975363\pi\)
\(332\) 0 0
\(333\) 11894.7i 0.107267i
\(334\) 0 0
\(335\) 197231.i 1.75746i
\(336\) 0 0
\(337\) 217168.i 1.91221i −0.293021 0.956106i \(-0.594661\pi\)
0.293021 0.956106i \(-0.405339\pi\)
\(338\) 0 0
\(339\) −138259. −1.20308
\(340\) 0 0
\(341\) 22898.4i 0.196923i
\(342\) 0 0
\(343\) 110879. 0.942456
\(344\) 0 0
\(345\) 108945.i 0.915310i
\(346\) 0 0
\(347\) −64555.0 −0.536131 −0.268066 0.963401i \(-0.586384\pi\)
−0.268066 + 0.963401i \(0.586384\pi\)
\(348\) 0 0
\(349\) −113275. −0.930001 −0.465001 0.885310i \(-0.653946\pi\)
−0.465001 + 0.885310i \(0.653946\pi\)
\(350\) 0 0
\(351\) −41730.4 −0.338718
\(352\) 0 0
\(353\) 27685.3 0.222178 0.111089 0.993810i \(-0.464566\pi\)
0.111089 + 0.993810i \(0.464566\pi\)
\(354\) 0 0
\(355\) 155036.i 1.23020i
\(356\) 0 0
\(357\) 8396.53i 0.0658815i
\(358\) 0 0
\(359\) −65336.2 −0.506950 −0.253475 0.967342i \(-0.581574\pi\)
−0.253475 + 0.967342i \(0.581574\pi\)
\(360\) 0 0
\(361\) −118150. 54992.5i −0.906606 0.421977i
\(362\) 0 0
\(363\) 93371.3i 0.708599i
\(364\) 0 0
\(365\) 96919.1 0.727484
\(366\) 0 0
\(367\) −177298. −1.31635 −0.658176 0.752864i \(-0.728672\pi\)
−0.658176 + 0.752864i \(0.728672\pi\)
\(368\) 0 0
\(369\) 1893.64i 0.0139074i
\(370\) 0 0
\(371\) 209279.i 1.52047i
\(372\) 0 0
\(373\) 15381.6i 0.110557i 0.998471 + 0.0552783i \(0.0176046\pi\)
−0.998471 + 0.0552783i \(0.982395\pi\)
\(374\) 0 0
\(375\) 95549.8i 0.679465i
\(376\) 0 0
\(377\) −71676.6 −0.504307
\(378\) 0 0
\(379\) 232626.i 1.61949i −0.586780 0.809746i \(-0.699605\pi\)
0.586780 0.809746i \(-0.300395\pi\)
\(380\) 0 0
\(381\) 74087.8 0.510384
\(382\) 0 0
\(383\) 52033.6i 0.354721i 0.984146 + 0.177360i \(0.0567558\pi\)
−0.984146 + 0.177360i \(0.943244\pi\)
\(384\) 0 0
\(385\) −34466.8 −0.232530
\(386\) 0 0
\(387\) 93137.6 0.621875
\(388\) 0 0
\(389\) 57687.6 0.381227 0.190613 0.981665i \(-0.438952\pi\)
0.190613 + 0.981665i \(0.438952\pi\)
\(390\) 0 0
\(391\) −14994.6 −0.0980804
\(392\) 0 0
\(393\) 1488.16i 0.00963527i
\(394\) 0 0
\(395\) 104225.i 0.668002i
\(396\) 0 0
\(397\) 16914.2 0.107317 0.0536586 0.998559i \(-0.482912\pi\)
0.0536586 + 0.998559i \(0.482912\pi\)
\(398\) 0 0
\(399\) 120840. + 26744.7i 0.759039 + 0.167993i
\(400\) 0 0
\(401\) 23481.6i 0.146029i −0.997331 0.0730144i \(-0.976738\pi\)
0.997331 0.0730144i \(-0.0232619\pi\)
\(402\) 0 0
\(403\) 48603.0 0.299263
\(404\) 0 0
\(405\) −60294.3 −0.367592
\(406\) 0 0
\(407\) 8155.45i 0.0492333i
\(408\) 0 0
\(409\) 310776.i 1.85781i −0.370321 0.928904i \(-0.620752\pi\)
0.370321 0.928904i \(-0.379248\pi\)
\(410\) 0 0
\(411\) 126301.i 0.747693i
\(412\) 0 0
\(413\) 55504.9i 0.325410i
\(414\) 0 0
\(415\) 26648.4 0.154730
\(416\) 0 0
\(417\) 74072.8i 0.425977i
\(418\) 0 0
\(419\) 271562. 1.54682 0.773411 0.633905i \(-0.218549\pi\)
0.773411 + 0.633905i \(0.218549\pi\)
\(420\) 0 0
\(421\) 322467.i 1.81937i 0.415299 + 0.909685i \(0.363677\pi\)
−0.415299 + 0.909685i \(0.636323\pi\)
\(422\) 0 0
\(423\) −109711. −0.613156
\(424\) 0 0
\(425\) −2156.08 −0.0119368
\(426\) 0 0
\(427\) 335994. 1.84279
\(428\) 0 0
\(429\) −8903.44 −0.0483775
\(430\) 0 0
\(431\) 157951.i 0.850290i 0.905125 + 0.425145i \(0.139777\pi\)
−0.905125 + 0.425145i \(0.860223\pi\)
\(432\) 0 0
\(433\) 36946.7i 0.197061i 0.995134 + 0.0985304i \(0.0314142\pi\)
−0.995134 + 0.0985304i \(0.968586\pi\)
\(434\) 0 0
\(435\) −239506. −1.26572
\(436\) 0 0
\(437\) 47761.1 215797.i 0.250099 1.13001i
\(438\) 0 0
\(439\) 194377.i 1.00859i 0.863531 + 0.504296i \(0.168248\pi\)
−0.863531 + 0.504296i \(0.831752\pi\)
\(440\) 0 0
\(441\) 8994.03 0.0462463
\(442\) 0 0
\(443\) −30671.7 −0.156290 −0.0781449 0.996942i \(-0.524900\pi\)
−0.0781449 + 0.996942i \(0.524900\pi\)
\(444\) 0 0
\(445\) 164298.i 0.829682i
\(446\) 0 0
\(447\) 131464.i 0.657946i
\(448\) 0 0
\(449\) 236846.i 1.17483i 0.809287 + 0.587413i \(0.199854\pi\)
−0.809287 + 0.587413i \(0.800146\pi\)
\(450\) 0 0
\(451\) 1298.35i 0.00638319i
\(452\) 0 0
\(453\) −5134.96 −0.0250231
\(454\) 0 0
\(455\) 73157.6i 0.353376i
\(456\) 0 0
\(457\) −82060.7 −0.392919 −0.196459 0.980512i \(-0.562944\pi\)
−0.196459 + 0.980512i \(0.562944\pi\)
\(458\) 0 0
\(459\) 19192.0i 0.0910951i
\(460\) 0 0
\(461\) −16781.0 −0.0789614 −0.0394807 0.999220i \(-0.512570\pi\)
−0.0394807 + 0.999220i \(0.512570\pi\)
\(462\) 0 0
\(463\) −100169. −0.467274 −0.233637 0.972324i \(-0.575063\pi\)
−0.233637 + 0.972324i \(0.575063\pi\)
\(464\) 0 0
\(465\) 162406. 0.751097
\(466\) 0 0
\(467\) 216772. 0.993960 0.496980 0.867762i \(-0.334442\pi\)
0.496980 + 0.867762i \(0.334442\pi\)
\(468\) 0 0
\(469\) 379997.i 1.72756i
\(470\) 0 0
\(471\) 308168.i 1.38914i
\(472\) 0 0
\(473\) 63858.4 0.285427
\(474\) 0 0
\(475\) 6867.56 31029.5i 0.0304379 0.137527i
\(476\) 0 0
\(477\) 148854.i 0.654218i
\(478\) 0 0
\(479\) −91676.2 −0.399563 −0.199782 0.979840i \(-0.564023\pi\)
−0.199782 + 0.979840i \(0.564023\pi\)
\(480\) 0 0
\(481\) 17310.4 0.0748198
\(482\) 0 0
\(483\) 209899.i 0.899739i
\(484\) 0 0
\(485\) 134143.i 0.570276i
\(486\) 0 0
\(487\) 107326.i 0.452528i 0.974066 + 0.226264i \(0.0726512\pi\)
−0.974066 + 0.226264i \(0.927349\pi\)
\(488\) 0 0
\(489\) 219588.i 0.918313i
\(490\) 0 0
\(491\) −211818. −0.878619 −0.439310 0.898336i \(-0.644777\pi\)
−0.439310 + 0.898336i \(0.644777\pi\)
\(492\) 0 0
\(493\) 32964.4i 0.135629i
\(494\) 0 0
\(495\) 24515.1 0.100051
\(496\) 0 0
\(497\) 298702.i 1.20928i
\(498\) 0 0
\(499\) −145489. −0.584291 −0.292146 0.956374i \(-0.594369\pi\)
−0.292146 + 0.956374i \(0.594369\pi\)
\(500\) 0 0
\(501\) 320194. 1.27567
\(502\) 0 0
\(503\) 33735.6 0.133338 0.0666689 0.997775i \(-0.478763\pi\)
0.0666689 + 0.997775i \(0.478763\pi\)
\(504\) 0 0
\(505\) 421733. 1.65369
\(506\) 0 0
\(507\) 171429.i 0.666913i
\(508\) 0 0
\(509\) 128247.i 0.495009i 0.968887 + 0.247505i \(0.0796105\pi\)
−0.968887 + 0.247505i \(0.920389\pi\)
\(510\) 0 0
\(511\) −186730. −0.715108
\(512\) 0 0
\(513\) −276204. 61130.6i −1.04953 0.232286i
\(514\) 0 0
\(515\) 348969.i 1.31575i
\(516\) 0 0
\(517\) −75222.0 −0.281426
\(518\) 0 0
\(519\) 308002. 1.14345
\(520\) 0 0
\(521\) 38092.9i 0.140336i 0.997535 + 0.0701680i \(0.0223535\pi\)
−0.997535 + 0.0701680i \(0.977646\pi\)
\(522\) 0 0
\(523\) 96329.4i 0.352173i 0.984375 + 0.176086i \(0.0563438\pi\)
−0.984375 + 0.176086i \(0.943656\pi\)
\(524\) 0 0
\(525\) 30181.4i 0.109502i
\(526\) 0 0
\(527\) 22352.8i 0.0804841i
\(528\) 0 0
\(529\) 94999.8 0.339478
\(530\) 0 0
\(531\) 39478.8i 0.140015i
\(532\) 0 0
\(533\) −2755.81 −0.00970053
\(534\) 0 0
\(535\) 220914.i 0.771819i
\(536\) 0 0
\(537\) −265399. −0.920346
\(538\) 0 0
\(539\) 6166.62 0.0212261
\(540\) 0 0
\(541\) −163021. −0.556993 −0.278496 0.960437i \(-0.589836\pi\)
−0.278496 + 0.960437i \(0.589836\pi\)
\(542\) 0 0
\(543\) 345486. 1.17174
\(544\) 0 0
\(545\) 379701.i 1.27835i
\(546\) 0 0
\(547\) 450817.i 1.50670i 0.657621 + 0.753349i \(0.271563\pi\)
−0.657621 + 0.753349i \(0.728437\pi\)
\(548\) 0 0
\(549\) −238982. −0.792904
\(550\) 0 0
\(551\) −474412. 104999.i −1.56262 0.345844i
\(552\) 0 0
\(553\) 200806.i 0.656638i
\(554\) 0 0
\(555\) 57842.2 0.187784
\(556\) 0 0
\(557\) −270930. −0.873268 −0.436634 0.899639i \(-0.643829\pi\)
−0.436634 + 0.899639i \(0.643829\pi\)
\(558\) 0 0
\(559\) 135543.i 0.433764i
\(560\) 0 0
\(561\) 4094.74i 0.0130107i
\(562\) 0 0
\(563\) 59429.1i 0.187492i −0.995596 0.0937459i \(-0.970116\pi\)
0.995596 0.0937459i \(-0.0298841\pi\)
\(564\) 0 0
\(565\) 554013.i 1.73549i
\(566\) 0 0
\(567\) 116166. 0.361339
\(568\) 0 0
\(569\) 364145.i 1.12473i 0.826888 + 0.562367i \(0.190109\pi\)
−0.826888 + 0.562367i \(0.809891\pi\)
\(570\) 0 0
\(571\) −410057. −1.25769 −0.628843 0.777532i \(-0.716471\pi\)
−0.628843 + 0.777532i \(0.716471\pi\)
\(572\) 0 0
\(573\) 259617.i 0.790723i
\(574\) 0 0
\(575\) 53898.3 0.163020
\(576\) 0 0
\(577\) −44617.5 −0.134015 −0.0670075 0.997752i \(-0.521345\pi\)
−0.0670075 + 0.997752i \(0.521345\pi\)
\(578\) 0 0
\(579\) 286935. 0.855906
\(580\) 0 0
\(581\) −51342.3 −0.152098
\(582\) 0 0
\(583\) 102059.i 0.300272i
\(584\) 0 0
\(585\) 52034.7i 0.152048i
\(586\) 0 0
\(587\) 371422. 1.07793 0.538966 0.842328i \(-0.318815\pi\)
0.538966 + 0.842328i \(0.318815\pi\)
\(588\) 0 0
\(589\) 321693. + 71198.3i 0.927280 + 0.205229i
\(590\) 0 0
\(591\) 175334.i 0.501985i
\(592\) 0 0
\(593\) −327963. −0.932644 −0.466322 0.884615i \(-0.654421\pi\)
−0.466322 + 0.884615i \(0.654421\pi\)
\(594\) 0 0
\(595\) 33645.5 0.0950372
\(596\) 0 0
\(597\) 78454.7i 0.220125i
\(598\) 0 0
\(599\) 81737.2i 0.227806i 0.993492 + 0.113903i \(0.0363354\pi\)
−0.993492 + 0.113903i \(0.963665\pi\)
\(600\) 0 0
\(601\) 75524.8i 0.209093i −0.994520 0.104547i \(-0.966661\pi\)
0.994520 0.104547i \(-0.0333392\pi\)
\(602\) 0 0
\(603\) 270279.i 0.743325i
\(604\) 0 0
\(605\) −374146. −1.02219
\(606\) 0 0
\(607\) 97397.5i 0.264345i 0.991227 + 0.132172i \(0.0421952\pi\)
−0.991227 + 0.132172i \(0.957805\pi\)
\(608\) 0 0
\(609\) 461445. 1.24419
\(610\) 0 0
\(611\) 159663.i 0.427682i
\(612\) 0 0
\(613\) 185283. 0.493078 0.246539 0.969133i \(-0.420707\pi\)
0.246539 + 0.969133i \(0.420707\pi\)
\(614\) 0 0
\(615\) −9208.49 −0.0243466
\(616\) 0 0
\(617\) 628408. 1.65071 0.825357 0.564612i \(-0.190974\pi\)
0.825357 + 0.564612i \(0.190974\pi\)
\(618\) 0 0
\(619\) −203791. −0.531867 −0.265933 0.963991i \(-0.585680\pi\)
−0.265933 + 0.963991i \(0.585680\pi\)
\(620\) 0 0
\(621\) 479768.i 1.24408i
\(622\) 0 0
\(623\) 316545.i 0.815568i
\(624\) 0 0
\(625\) −437896. −1.12101
\(626\) 0 0
\(627\) −58930.0 13042.6i −0.149900 0.0331764i
\(628\) 0 0
\(629\) 7961.13i 0.0201221i
\(630\) 0 0
\(631\) −496722. −1.24754 −0.623771 0.781607i \(-0.714400\pi\)
−0.623771 + 0.781607i \(0.714400\pi\)
\(632\) 0 0
\(633\) 83270.0 0.207817
\(634\) 0 0
\(635\) 296875.i 0.736252i
\(636\) 0 0
\(637\) 13089.0i 0.0322572i
\(638\) 0 0
\(639\) 212457.i 0.520319i
\(640\) 0 0
\(641\) 44345.5i 0.107928i −0.998543 0.0539639i \(-0.982814\pi\)
0.998543 0.0539639i \(-0.0171856\pi\)
\(642\) 0 0
\(643\) −139327. −0.336987 −0.168494 0.985703i \(-0.553890\pi\)
−0.168494 + 0.985703i \(0.553890\pi\)
\(644\) 0 0
\(645\) 452913.i 1.08867i
\(646\) 0 0
\(647\) 617612. 1.47539 0.737696 0.675133i \(-0.235914\pi\)
0.737696 + 0.675133i \(0.235914\pi\)
\(648\) 0 0
\(649\) 27068.1i 0.0642640i
\(650\) 0 0
\(651\) −312900. −0.738319
\(652\) 0 0
\(653\) 570062. 1.33689 0.668446 0.743761i \(-0.266960\pi\)
0.668446 + 0.743761i \(0.266960\pi\)
\(654\) 0 0
\(655\) −5963.16 −0.0138993
\(656\) 0 0
\(657\) 132815. 0.307692
\(658\) 0 0
\(659\) 716546.i 1.64996i 0.565162 + 0.824980i \(0.308814\pi\)
−0.565162 + 0.824980i \(0.691186\pi\)
\(660\) 0 0
\(661\) 187692.i 0.429579i −0.976660 0.214790i \(-0.931093\pi\)
0.976660 0.214790i \(-0.0689066\pi\)
\(662\) 0 0
\(663\) 8691.30 0.0197723
\(664\) 0 0
\(665\) −107168. + 484214.i −0.242338 + 1.09495i
\(666\) 0 0
\(667\) 824055.i 1.85227i
\(668\) 0 0
\(669\) −397877. −0.888990
\(670\) 0 0
\(671\) −163854. −0.363926
\(672\) 0 0
\(673\) 504263.i 1.11334i 0.830734 + 0.556669i \(0.187921\pi\)
−0.830734 + 0.556669i \(0.812079\pi\)
\(674\) 0 0
\(675\) 68985.8i 0.151409i
\(676\) 0 0
\(677\) 689926.i 1.50531i −0.658417 0.752653i \(-0.728774\pi\)
0.658417 0.752653i \(-0.271226\pi\)
\(678\) 0 0
\(679\) 258448.i 0.560574i
\(680\) 0 0
\(681\) 386267. 0.832901
\(682\) 0 0
\(683\) 226300.i 0.485113i −0.970137 0.242556i \(-0.922014\pi\)
0.970137 0.242556i \(-0.0779859\pi\)
\(684\) 0 0
\(685\) 506098. 1.07858
\(686\) 0 0
\(687\) 267608.i 0.567003i
\(688\) 0 0
\(689\) −216626. −0.456323
\(690\) 0 0
\(691\) 921891. 1.93074 0.965370 0.260886i \(-0.0840148\pi\)
0.965370 + 0.260886i \(0.0840148\pi\)
\(692\) 0 0
\(693\) −47232.2 −0.0983494
\(694\) 0 0
\(695\) 296815. 0.614492
\(696\) 0 0
\(697\) 1267.41i 0.00260887i
\(698\) 0 0
\(699\) 478068.i 0.978443i
\(700\) 0 0
\(701\) −561368. −1.14238 −0.571191 0.820817i \(-0.693519\pi\)
−0.571191 + 0.820817i \(0.693519\pi\)
\(702\) 0 0
\(703\) 114574. + 25357.9i 0.231832 + 0.0513100i
\(704\) 0 0
\(705\) 533509.i 1.07341i
\(706\) 0 0
\(707\) −812536. −1.62556
\(708\) 0 0
\(709\) −287728. −0.572386 −0.286193 0.958172i \(-0.592390\pi\)
−0.286193 + 0.958172i \(0.592390\pi\)
\(710\) 0 0
\(711\) 142827.i 0.282534i
\(712\) 0 0
\(713\) 558782.i 1.09917i
\(714\) 0 0
\(715\) 35676.8i 0.0697868i
\(716\) 0 0
\(717\) 294400.i 0.572663i
\(718\) 0 0
\(719\) 805860. 1.55884 0.779420 0.626502i \(-0.215514\pi\)
0.779420 + 0.626502i \(0.215514\pi\)
\(720\) 0 0
\(721\) 672344.i 1.29336i
\(722\) 0 0
\(723\) −225397. −0.431193
\(724\) 0 0
\(725\) 118491.i 0.225428i
\(726\) 0 0
\(727\) 348604. 0.659575 0.329787 0.944055i \(-0.393023\pi\)
0.329787 + 0.944055i \(0.393023\pi\)
\(728\) 0 0
\(729\) −505606. −0.951387
\(730\) 0 0
\(731\) −62336.8 −0.116657
\(732\) 0 0
\(733\) 238795. 0.444443 0.222222 0.974996i \(-0.428669\pi\)
0.222222 + 0.974996i \(0.428669\pi\)
\(734\) 0 0
\(735\) 43736.6i 0.0809599i
\(736\) 0 0
\(737\) 185313.i 0.341170i
\(738\) 0 0
\(739\) −502122. −0.919433 −0.459717 0.888066i \(-0.652049\pi\)
−0.459717 + 0.888066i \(0.652049\pi\)
\(740\) 0 0
\(741\) −27683.6 + 125082.i −0.0504181 + 0.227803i
\(742\) 0 0
\(743\) 411861.i 0.746058i 0.927820 + 0.373029i \(0.121681\pi\)
−0.927820 + 0.373029i \(0.878319\pi\)
\(744\) 0 0
\(745\) −526784. −0.949118
\(746\) 0 0
\(747\) 36518.2 0.0654437
\(748\) 0 0
\(749\) 425625.i 0.758689i
\(750\) 0 0
\(751\) 876407.i 1.55391i −0.629556 0.776955i \(-0.716763\pi\)
0.629556 0.776955i \(-0.283237\pi\)
\(752\) 0 0
\(753\) 432838.i 0.763371i
\(754\) 0 0
\(755\) 20576.2i 0.0360970i
\(756\) 0 0
\(757\) −447975. −0.781739 −0.390870 0.920446i \(-0.627826\pi\)
−0.390870 + 0.920446i \(0.627826\pi\)
\(758\) 0 0
\(759\) 102362.i 0.177686i
\(760\) 0 0
\(761\) 527400. 0.910690 0.455345 0.890315i \(-0.349516\pi\)
0.455345 + 0.890315i \(0.349516\pi\)
\(762\) 0 0
\(763\) 731553.i 1.25660i
\(764\) 0 0
\(765\) −23931.0 −0.0408919
\(766\) 0 0
\(767\) −57453.4 −0.0976619
\(768\) 0 0
\(769\) 549287. 0.928852 0.464426 0.885612i \(-0.346261\pi\)
0.464426 + 0.885612i \(0.346261\pi\)
\(770\) 0 0
\(771\) 7700.71 0.0129545
\(772\) 0 0
\(773\) 1.03569e6i 1.73329i −0.498927 0.866644i \(-0.666272\pi\)
0.498927 0.866644i \(-0.333728\pi\)
\(774\) 0 0
\(775\) 80347.2i 0.133773i
\(776\) 0 0
\(777\) −111442. −0.184590
\(778\) 0 0
\(779\) −18240.1 4036.98i −0.0300575 0.00665244i
\(780\) 0 0
\(781\) 145668.i 0.238815i
\(782\) 0 0
\(783\) −1.05473e6 −1.72035
\(784\) 0 0
\(785\) 1.23485e6 2.00390
\(786\) 0 0
\(787\) 1.04115e6i 1.68098i −0.541824 0.840492i \(-0.682266\pi\)
0.541824 0.840492i \(-0.317734\pi\)
\(788\) 0 0
\(789\) 44867.6i 0.0720740i
\(790\) 0 0
\(791\) 1.06739e6i 1.70597i
\(792\) 0 0
\(793\) 347790.i 0.553058i
\(794\) 0 0
\(795\) −723851. −1.14529
\(796\) 0 0
\(797\) 476576.i 0.750266i 0.926971 + 0.375133i \(0.122403\pi\)
−0.926971 + 0.375133i \(0.877597\pi\)
\(798\) 0 0
\(799\) 73429.7 0.115021
\(800\) 0 0
\(801\) 225149.i 0.350917i
\(802\) 0 0
\(803\) 91062.5 0.141224
\(804\) 0 0
\(805\) −841082. −1.29792
\(806\) 0 0
\(807\) −148148. −0.227483
\(808\) 0 0
\(809\) −1.00569e6 −1.53663 −0.768314 0.640074i \(-0.778904\pi\)
−0.768314 + 0.640074i \(0.778904\pi\)
\(810\) 0 0
\(811\) 257961.i 0.392205i 0.980583 + 0.196102i \(0.0628285\pi\)
−0.980583 + 0.196102i \(0.937171\pi\)
\(812\) 0 0
\(813\) 310521.i 0.469797i
\(814\) 0 0
\(815\) −879905. −1.32471
\(816\) 0 0
\(817\) 198556. 897129.i 0.297467 1.34404i
\(818\) 0 0
\(819\) 100253.i 0.149461i
\(820\) 0 0
\(821\) −65320.9 −0.0969094 −0.0484547 0.998825i \(-0.515430\pi\)
−0.0484547 + 0.998825i \(0.515430\pi\)
\(822\) 0 0
\(823\) 109324. 0.161404 0.0807020 0.996738i \(-0.474284\pi\)
0.0807020 + 0.996738i \(0.474284\pi\)
\(824\) 0 0
\(825\) 14718.6i 0.0216251i
\(826\) 0 0
\(827\) 988603.i 1.44548i −0.691122 0.722738i \(-0.742883\pi\)
0.691122 0.722738i \(-0.257117\pi\)
\(828\) 0 0
\(829\) 467829.i 0.680735i −0.940292 0.340368i \(-0.889448\pi\)
0.940292 0.340368i \(-0.110552\pi\)
\(830\) 0 0
\(831\) 705151.i 1.02113i
\(832\) 0 0
\(833\) −6019.69 −0.00867529
\(834\) 0 0
\(835\) 1.28304e6i 1.84021i
\(836\) 0 0
\(837\) 715198. 1.02088
\(838\) 0 0
\(839\) 657894.i 0.934613i 0.884095 + 0.467306i \(0.154776\pi\)
−0.884095 + 0.467306i \(0.845224\pi\)
\(840\) 0 0
\(841\) −1.10433e6 −1.56138
\(842\) 0 0
\(843\) 135265. 0.190340
\(844\) 0 0
\(845\) 686930. 0.962053
\(846\) 0 0
\(847\) 720851. 1.00480
\(848\) 0 0
\(849\) 371249.i 0.515051i
\(850\) 0 0
\(851\) 199015.i 0.274806i
\(852\) 0 0
\(853\) −226576. −0.311398 −0.155699 0.987805i \(-0.549763\pi\)
−0.155699 + 0.987805i \(0.549763\pi\)
\(854\) 0 0
\(855\) 76225.2 344406.i 0.104272 0.471128i
\(856\) 0 0
\(857\) 183273.i 0.249539i −0.992186 0.124769i \(-0.960181\pi\)
0.992186 0.124769i \(-0.0398191\pi\)
\(858\) 0 0
\(859\) −565839. −0.766843 −0.383421 0.923574i \(-0.625254\pi\)
−0.383421 + 0.923574i \(0.625254\pi\)
\(860\) 0 0
\(861\) 17741.6 0.0239324
\(862\) 0 0
\(863\) 247431.i 0.332225i −0.986107 0.166113i \(-0.946878\pi\)
0.986107 0.166113i \(-0.0531216\pi\)
\(864\) 0 0
\(865\) 1.23419e6i 1.64949i
\(866\) 0 0
\(867\) 552578.i 0.735115i
\(868\) 0 0
\(869\) 97927.0i 0.129677i
\(870\) 0 0
\(871\) −393337. −0.518476
\(872\) 0 0
\(873\) 183826.i 0.241200i
\(874\) 0 0
\(875\) 737669. 0.963486
\(876\) 0 0
\(877\) 58891.6i 0.0765692i 0.999267 + 0.0382846i \(0.0121893\pi\)
−0.999267 + 0.0382846i \(0.987811\pi\)
\(878\) 0 0
\(879\) 560121. 0.724943
\(880\) 0 0
\(881\) −844161. −1.08761 −0.543806 0.839211i \(-0.683017\pi\)
−0.543806 + 0.839211i \(0.683017\pi\)
\(882\) 0 0
\(883\) 1.34541e6 1.72558 0.862788 0.505565i \(-0.168716\pi\)
0.862788 + 0.505565i \(0.168716\pi\)
\(884\) 0 0
\(885\) −191979. −0.245114
\(886\) 0 0
\(887\) 268542.i 0.341323i 0.985330 + 0.170661i \(0.0545904\pi\)
−0.985330 + 0.170661i \(0.945410\pi\)
\(888\) 0 0
\(889\) 571977.i 0.723727i
\(890\) 0 0
\(891\) −56650.9 −0.0713594
\(892\) 0 0
\(893\) −233889. + 1.05677e6i −0.293296 + 1.32519i
\(894\) 0 0
\(895\) 1.06348e6i 1.32764i
\(896\) 0 0
\(897\) −217268. −0.270029
\(898\) 0 0
\(899\) 1.22843e6 1.51996
\(900\) 0 0
\(901\) 99627.4i 0.122724i
\(902\) 0 0
\(903\) 872609.i 1.07015i
\(904\) 0 0
\(905\) 1.38439e6i 1.69029i
\(906\) 0 0
\(907\) 1.24035e6i 1.50775i 0.657017 + 0.753876i \(0.271818\pi\)
−0.657017 + 0.753876i \(0.728182\pi\)
\(908\) 0 0
\(909\) 577931. 0.699436
\(910\) 0 0
\(911\) 191069.i 0.230226i −0.993352 0.115113i \(-0.963277\pi\)
0.993352 0.115113i \(-0.0367229\pi\)
\(912\) 0 0
\(913\) 25038.1 0.0300372
\(914\) 0 0
\(915\) 1.16213e6i 1.38808i
\(916\) 0 0
\(917\) 11489.0 0.0136629
\(918\) 0 0
\(919\) −830092. −0.982868 −0.491434 0.870915i \(-0.663527\pi\)
−0.491434 + 0.870915i \(0.663527\pi\)
\(920\) 0 0
\(921\) 69282.5 0.0816779
\(922\) 0 0
\(923\) −309188. −0.362927
\(924\) 0 0
\(925\) 28616.3i 0.0334449i
\(926\) 0 0
\(927\) 478216.i 0.556500i
\(928\) 0 0
\(929\) 1.18305e6 1.37080 0.685399 0.728168i \(-0.259628\pi\)
0.685399 + 0.728168i \(0.259628\pi\)
\(930\) 0 0
\(931\) 19174.0 86633.2i 0.0221214 0.0999505i
\(932\) 0 0
\(933\) 787054.i 0.904152i
\(934\) 0 0
\(935\) −16407.9 −0.0187685
\(936\) 0 0
\(937\) 1.14337e6 1.30229 0.651144 0.758954i \(-0.274289\pi\)
0.651144 + 0.758954i \(0.274289\pi\)
\(938\) 0 0
\(939\) 55917.9i 0.0634190i
\(940\) 0 0
\(941\) 1.12716e6i 1.27294i −0.771302 0.636469i \(-0.780394\pi\)
0.771302 0.636469i \(-0.219606\pi\)
\(942\) 0 0
\(943\) 31683.2i 0.0356291i
\(944\) 0 0
\(945\) 1.07652e6i 1.20548i
\(946\) 0 0
\(947\) −1.01859e6 −1.13579 −0.567897 0.823099i \(-0.692243\pi\)
−0.567897 + 0.823099i \(0.692243\pi\)
\(948\) 0 0
\(949\) 193285.i 0.214618i
\(950\) 0 0
\(951\) 956093. 1.05716
\(952\) 0 0
\(953\) 18352.7i 0.0202076i −0.999949 0.0101038i \(-0.996784\pi\)
0.999949 0.0101038i \(-0.00321619\pi\)
\(954\) 0 0
\(955\) 1.04031e6 1.14066
\(956\) 0 0
\(957\) −225033. −0.245710
\(958\) 0 0
\(959\) −975077. −1.06023
\(960\) 0 0
\(961\) 90535.6 0.0980331
\(962\) 0 0
\(963\) 302734.i 0.326444i
\(964\) 0 0
\(965\) 1.14977e6i 1.23468i
\(966\) 0 0
\(967\) −301406. −0.322329 −0.161164 0.986928i \(-0.551525\pi\)
−0.161164 + 0.986928i \(0.551525\pi\)
\(968\) 0 0
\(969\) 57525.9 + 12731.8i 0.0612654 + 0.0135595i
\(970\) 0 0
\(971\) 796081.i 0.844342i −0.906516 0.422171i \(-0.861268\pi\)
0.906516 0.422171i \(-0.138732\pi\)
\(972\) 0 0
\(973\) −571861. −0.604039
\(974\) 0 0
\(975\) −31240.9 −0.0328636
\(976\) 0 0
\(977\) 664866.i 0.696538i 0.937395 + 0.348269i \(0.113230\pi\)
−0.937395 + 0.348269i \(0.886770\pi\)
\(978\) 0 0
\(979\) 154370.i 0.161063i
\(980\) 0 0
\(981\) 520330.i 0.540681i
\(982\) 0 0
\(983\) 1.05287e6i 1.08960i 0.838566 + 0.544801i \(0.183395\pi\)
−0.838566 + 0.544801i \(0.816605\pi\)
\(984\) 0 0
\(985\) 702576. 0.724137
\(986\) 0 0
\(987\) 1.02789e6i 1.05514i
\(988\) 0 0
\(989\) 1.55832e6 1.59317
\(990\) 0 0
\(991\) 1.85377e6i 1.88759i −0.330529 0.943796i \(-0.607227\pi\)
0.330529 0.943796i \(-0.392773\pi\)
\(992\) 0 0
\(993\) 112908. 0.114506
\(994\) 0 0
\(995\) 314374. 0.317541
\(996\) 0 0
\(997\) 494751. 0.497733 0.248867 0.968538i \(-0.419942\pi\)
0.248867 + 0.968538i \(0.419942\pi\)
\(998\) 0 0
\(999\) 254724. 0.255234
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 304.5.e.e.113.4 8
4.3 odd 2 38.5.b.a.37.3 8
12.11 even 2 342.5.d.a.37.6 8
19.18 odd 2 inner 304.5.e.e.113.5 8
76.75 even 2 38.5.b.a.37.6 yes 8
228.227 odd 2 342.5.d.a.37.2 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
38.5.b.a.37.3 8 4.3 odd 2
38.5.b.a.37.6 yes 8 76.75 even 2
304.5.e.e.113.4 8 1.1 even 1 trivial
304.5.e.e.113.5 8 19.18 odd 2 inner
342.5.d.a.37.2 8 228.227 odd 2
342.5.d.a.37.6 8 12.11 even 2