Properties

Label 384.6.c.a.383.16
Level $384$
Weight $6$
Character 384.383
Analytic conductor $61.587$
Analytic rank $0$
Dimension $20$
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] = [384,6,Mod(383,384)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(384, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 1]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("384.383");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 384 = 2^{7} \cdot 3 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 384.c (of order \(2\), degree \(1\), 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: \(61.5873868082\)
Analytic rank: \(0\)
Dimension: \(20\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} + 306 x^{18} + 37827 x^{16} + 2442168 x^{14} + 88368509 x^{12} + 1774000974 x^{10} + 18093172325 x^{8} + 74958811500 x^{6} + 79355888475 x^{4} + \cdots + 2870280625 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{88}\cdot 3^{14}\cdot 41^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 383.16
Root \(-4.52328i\) of defining polynomial
Character \(\chi\) \(=\) 384.383
Dual form 384.6.c.a.383.15

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(10.5848 + 11.4438i) q^{3} +101.915i q^{5} +13.9460i q^{7} +(-18.9231 + 242.262i) q^{9} +O(q^{10})\) \(q+(10.5848 + 11.4438i) q^{3} +101.915i q^{5} +13.9460i q^{7} +(-18.9231 + 242.262i) q^{9} -445.507 q^{11} +217.527 q^{13} +(-1166.30 + 1078.76i) q^{15} -1436.17i q^{17} +2611.15i q^{19} +(-159.596 + 147.616i) q^{21} -696.672 q^{23} -7261.72 q^{25} +(-2972.71 + 2347.75i) q^{27} +4602.14i q^{29} -8335.61i q^{31} +(-4715.61 - 5098.31i) q^{33} -1421.31 q^{35} +6925.77 q^{37} +(2302.48 + 2489.34i) q^{39} +11600.9i q^{41} +6050.27i q^{43} +(-24690.2 - 1928.55i) q^{45} -13047.5 q^{47} +16612.5 q^{49} +(16435.3 - 15201.6i) q^{51} -37847.2i q^{53} -45403.9i q^{55} +(-29881.6 + 27638.6i) q^{57} -4116.26 q^{59} +203.748 q^{61} +(-3378.59 - 263.901i) q^{63} +22169.3i q^{65} +1289.39i q^{67} +(-7374.15 - 7972.61i) q^{69} +45445.3 q^{71} +42464.1 q^{73} +(-76864.1 - 83102.0i) q^{75} -6213.04i q^{77} -28560.5i q^{79} +(-58332.8 - 9168.68i) q^{81} -7615.76 q^{83} +146368. q^{85} +(-52666.2 + 48712.9i) q^{87} +15207.1i q^{89} +3033.63i q^{91} +(95391.4 - 88230.9i) q^{93} -266116. q^{95} -87803.8 q^{97} +(8430.35 - 107929. i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q - 2 q^{3}+O(q^{10}) \) Copy content Toggle raw display \( 20 q - 2 q^{3} - 948 q^{11} - 852 q^{15} - 1640 q^{21} - 328 q^{23} - 12500 q^{25} - 2030 q^{27} + 2836 q^{33} + 7184 q^{35} - 15056 q^{37} + 12980 q^{39} - 11800 q^{45} - 36640 q^{47} - 33388 q^{49} - 1936 q^{51} + 15404 q^{57} - 62908 q^{59} - 73264 q^{61} - 23608 q^{63} + 84024 q^{69} - 34888 q^{71} + 52568 q^{73} - 115698 q^{75} + 55444 q^{81} + 225172 q^{83} + 30112 q^{85} + 225700 q^{87} + 148016 q^{93} - 418616 q^{95} + 7600 q^{97} - 378260 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/384\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(133\) \(257\)
\(\chi(n)\) \(-1\) \(1\) \(-1\)

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
\(3\) 10.5848 + 11.4438i 0.679017 + 0.734123i
\(4\) 0 0
\(5\) 101.915i 1.82312i 0.411172 + 0.911558i \(0.365120\pi\)
−0.411172 + 0.911558i \(0.634880\pi\)
\(6\) 0 0
\(7\) 13.9460i 0.107573i 0.998552 + 0.0537867i \(0.0171291\pi\)
−0.998552 + 0.0537867i \(0.982871\pi\)
\(8\) 0 0
\(9\) −18.9231 + 242.262i −0.0778727 + 0.996963i
\(10\) 0 0
\(11\) −445.507 −1.11013 −0.555063 0.831808i \(-0.687306\pi\)
−0.555063 + 0.831808i \(0.687306\pi\)
\(12\) 0 0
\(13\) 217.527 0.356988 0.178494 0.983941i \(-0.442877\pi\)
0.178494 + 0.983941i \(0.442877\pi\)
\(14\) 0 0
\(15\) −1166.30 + 1078.76i −1.33839 + 1.23793i
\(16\) 0 0
\(17\) 1436.17i 1.20527i −0.798018 0.602634i \(-0.794118\pi\)
0.798018 0.602634i \(-0.205882\pi\)
\(18\) 0 0
\(19\) 2611.15i 1.65939i 0.558219 + 0.829694i \(0.311485\pi\)
−0.558219 + 0.829694i \(0.688515\pi\)
\(20\) 0 0
\(21\) −159.596 + 147.616i −0.0789721 + 0.0730442i
\(22\) 0 0
\(23\) −696.672 −0.274605 −0.137303 0.990529i \(-0.543843\pi\)
−0.137303 + 0.990529i \(0.543843\pi\)
\(24\) 0 0
\(25\) −7261.72 −2.32375
\(26\) 0 0
\(27\) −2972.71 + 2347.75i −0.784770 + 0.619787i
\(28\) 0 0
\(29\) 4602.14i 1.01617i 0.861308 + 0.508083i \(0.169646\pi\)
−0.861308 + 0.508083i \(0.830354\pi\)
\(30\) 0 0
\(31\) 8335.61i 1.55788i −0.627101 0.778938i \(-0.715759\pi\)
0.627101 0.778938i \(-0.284241\pi\)
\(32\) 0 0
\(33\) −4715.61 5098.31i −0.753794 0.814969i
\(34\) 0 0
\(35\) −1421.31 −0.196119
\(36\) 0 0
\(37\) 6925.77 0.831694 0.415847 0.909435i \(-0.363485\pi\)
0.415847 + 0.909435i \(0.363485\pi\)
\(38\) 0 0
\(39\) 2302.48 + 2489.34i 0.242401 + 0.262073i
\(40\) 0 0
\(41\) 11600.9i 1.07778i 0.842375 + 0.538891i \(0.181157\pi\)
−0.842375 + 0.538891i \(0.818843\pi\)
\(42\) 0 0
\(43\) 6050.27i 0.499004i 0.968374 + 0.249502i \(0.0802669\pi\)
−0.968374 + 0.249502i \(0.919733\pi\)
\(44\) 0 0
\(45\) −24690.2 1928.55i −1.81758 0.141971i
\(46\) 0 0
\(47\) −13047.5 −0.861553 −0.430776 0.902459i \(-0.641760\pi\)
−0.430776 + 0.902459i \(0.641760\pi\)
\(48\) 0 0
\(49\) 16612.5 0.988428
\(50\) 0 0
\(51\) 16435.3 15201.6i 0.884815 0.818397i
\(52\) 0 0
\(53\) 37847.2i 1.85073i −0.379073 0.925367i \(-0.623757\pi\)
0.379073 0.925367i \(-0.376243\pi\)
\(54\) 0 0
\(55\) 45403.9i 2.02389i
\(56\) 0 0
\(57\) −29881.6 + 27638.6i −1.21819 + 1.12675i
\(58\) 0 0
\(59\) −4116.26 −0.153947 −0.0769737 0.997033i \(-0.524526\pi\)
−0.0769737 + 0.997033i \(0.524526\pi\)
\(60\) 0 0
\(61\) 203.748 0.00701083 0.00350541 0.999994i \(-0.498884\pi\)
0.00350541 + 0.999994i \(0.498884\pi\)
\(62\) 0 0
\(63\) −3378.59 263.901i −0.107247 0.00837703i
\(64\) 0 0
\(65\) 22169.3i 0.650831i
\(66\) 0 0
\(67\) 1289.39i 0.0350912i 0.999846 + 0.0175456i \(0.00558522\pi\)
−0.999846 + 0.0175456i \(0.994415\pi\)
\(68\) 0 0
\(69\) −7374.15 7972.61i −0.186462 0.201594i
\(70\) 0 0
\(71\) 45445.3 1.06990 0.534950 0.844884i \(-0.320330\pi\)
0.534950 + 0.844884i \(0.320330\pi\)
\(72\) 0 0
\(73\) 42464.1 0.932642 0.466321 0.884616i \(-0.345579\pi\)
0.466321 + 0.884616i \(0.345579\pi\)
\(74\) 0 0
\(75\) −76864.1 83102.0i −1.57787 1.70592i
\(76\) 0 0
\(77\) 6213.04i 0.119420i
\(78\) 0 0
\(79\) 28560.5i 0.514871i −0.966295 0.257435i \(-0.917123\pi\)
0.966295 0.257435i \(-0.0828775\pi\)
\(80\) 0 0
\(81\) −58332.8 9168.68i −0.987872 0.155272i
\(82\) 0 0
\(83\) −7615.76 −0.121344 −0.0606719 0.998158i \(-0.519324\pi\)
−0.0606719 + 0.998158i \(0.519324\pi\)
\(84\) 0 0
\(85\) 146368. 2.19734
\(86\) 0 0
\(87\) −52666.2 + 48712.9i −0.745991 + 0.689994i
\(88\) 0 0
\(89\) 15207.1i 0.203504i 0.994810 + 0.101752i \(0.0324448\pi\)
−0.994810 + 0.101752i \(0.967555\pi\)
\(90\) 0 0
\(91\) 3033.63i 0.0384024i
\(92\) 0 0
\(93\) 95391.4 88230.9i 1.14367 1.05782i
\(94\) 0 0
\(95\) −266116. −3.02526
\(96\) 0 0
\(97\) −87803.8 −0.947510 −0.473755 0.880657i \(-0.657102\pi\)
−0.473755 + 0.880657i \(0.657102\pi\)
\(98\) 0 0
\(99\) 8430.35 107929.i 0.0864485 1.10676i
\(100\) 0 0
\(101\) 90132.6i 0.879182i 0.898198 + 0.439591i \(0.144877\pi\)
−0.898198 + 0.439591i \(0.855123\pi\)
\(102\) 0 0
\(103\) 35966.5i 0.334045i 0.985953 + 0.167022i \(0.0534152\pi\)
−0.985953 + 0.167022i \(0.946585\pi\)
\(104\) 0 0
\(105\) −15044.3 16265.3i −0.133168 0.143975i
\(106\) 0 0
\(107\) −162071. −1.36850 −0.684250 0.729247i \(-0.739870\pi\)
−0.684250 + 0.729247i \(0.739870\pi\)
\(108\) 0 0
\(109\) 209727. 1.69079 0.845393 0.534145i \(-0.179366\pi\)
0.845393 + 0.534145i \(0.179366\pi\)
\(110\) 0 0
\(111\) 73308.0 + 79257.4i 0.564734 + 0.610566i
\(112\) 0 0
\(113\) 97384.1i 0.717450i 0.933443 + 0.358725i \(0.116788\pi\)
−0.933443 + 0.358725i \(0.883212\pi\)
\(114\) 0 0
\(115\) 71001.6i 0.500637i
\(116\) 0 0
\(117\) −4116.27 + 52698.4i −0.0277996 + 0.355904i
\(118\) 0 0
\(119\) 20028.9 0.129655
\(120\) 0 0
\(121\) 37425.1 0.232381
\(122\) 0 0
\(123\) −132759. + 122793.i −0.791225 + 0.731832i
\(124\) 0 0
\(125\) 421595.i 2.41335i
\(126\) 0 0
\(127\) 136193.i 0.749284i −0.927169 0.374642i \(-0.877766\pi\)
0.927169 0.374642i \(-0.122234\pi\)
\(128\) 0 0
\(129\) −69238.4 + 64041.1i −0.366330 + 0.338832i
\(130\) 0 0
\(131\) −319348. −1.62587 −0.812936 0.582353i \(-0.802132\pi\)
−0.812936 + 0.582353i \(0.802132\pi\)
\(132\) 0 0
\(133\) −36415.1 −0.178506
\(134\) 0 0
\(135\) −239271. 302964.i −1.12994 1.43073i
\(136\) 0 0
\(137\) 340623.i 1.55050i 0.631652 + 0.775252i \(0.282377\pi\)
−0.631652 + 0.775252i \(0.717623\pi\)
\(138\) 0 0
\(139\) 434345.i 1.90677i −0.301760 0.953384i \(-0.597574\pi\)
0.301760 0.953384i \(-0.402426\pi\)
\(140\) 0 0
\(141\) −138105. 149313.i −0.585009 0.632485i
\(142\) 0 0
\(143\) −96909.5 −0.396302
\(144\) 0 0
\(145\) −469029. −1.85259
\(146\) 0 0
\(147\) 175840. + 190111.i 0.671159 + 0.725628i
\(148\) 0 0
\(149\) 109904.i 0.405555i 0.979225 + 0.202777i \(0.0649967\pi\)
−0.979225 + 0.202777i \(0.935003\pi\)
\(150\) 0 0
\(151\) 281346.i 1.00415i −0.864825 0.502074i \(-0.832571\pi\)
0.864825 0.502074i \(-0.167429\pi\)
\(152\) 0 0
\(153\) 347930. + 27176.7i 1.20161 + 0.0938574i
\(154\) 0 0
\(155\) 849526. 2.84019
\(156\) 0 0
\(157\) −486230. −1.57432 −0.787160 0.616749i \(-0.788449\pi\)
−0.787160 + 0.616749i \(0.788449\pi\)
\(158\) 0 0
\(159\) 433117. 400606.i 1.35867 1.25668i
\(160\) 0 0
\(161\) 9715.80i 0.0295402i
\(162\) 0 0
\(163\) 435448.i 1.28371i −0.766826 0.641855i \(-0.778165\pi\)
0.766826 0.641855i \(-0.221835\pi\)
\(164\) 0 0
\(165\) 519595. 480593.i 1.48578 1.37425i
\(166\) 0 0
\(167\) −156653. −0.434657 −0.217328 0.976099i \(-0.569734\pi\)
−0.217328 + 0.976099i \(0.569734\pi\)
\(168\) 0 0
\(169\) −323975. −0.872559
\(170\) 0 0
\(171\) −632583. 49411.0i −1.65435 0.129221i
\(172\) 0 0
\(173\) 285352.i 0.724879i 0.932007 + 0.362440i \(0.118056\pi\)
−0.932007 + 0.362440i \(0.881944\pi\)
\(174\) 0 0
\(175\) 101272.i 0.249974i
\(176\) 0 0
\(177\) −43569.9 47105.8i −0.104533 0.113016i
\(178\) 0 0
\(179\) −277875. −0.648211 −0.324105 0.946021i \(-0.605063\pi\)
−0.324105 + 0.946021i \(0.605063\pi\)
\(180\) 0 0
\(181\) 74483.2 0.168990 0.0844951 0.996424i \(-0.473072\pi\)
0.0844951 + 0.996424i \(0.473072\pi\)
\(182\) 0 0
\(183\) 2156.64 + 2331.66i 0.00476047 + 0.00514681i
\(184\) 0 0
\(185\) 705842.i 1.51627i
\(186\) 0 0
\(187\) 639823.i 1.33800i
\(188\) 0 0
\(189\) −32741.7 41457.4i −0.0666726 0.0844204i
\(190\) 0 0
\(191\) −307683. −0.610268 −0.305134 0.952309i \(-0.598701\pi\)
−0.305134 + 0.952309i \(0.598701\pi\)
\(192\) 0 0
\(193\) −293837. −0.567824 −0.283912 0.958850i \(-0.591632\pi\)
−0.283912 + 0.958850i \(0.591632\pi\)
\(194\) 0 0
\(195\) −253702. + 234658.i −0.477790 + 0.441925i
\(196\) 0 0
\(197\) 291128.i 0.534464i 0.963632 + 0.267232i \(0.0861091\pi\)
−0.963632 + 0.267232i \(0.913891\pi\)
\(198\) 0 0
\(199\) 700006.i 1.25305i 0.779400 + 0.626526i \(0.215524\pi\)
−0.779400 + 0.626526i \(0.784476\pi\)
\(200\) 0 0
\(201\) −14755.6 + 13648.0i −0.0257612 + 0.0238275i
\(202\) 0 0
\(203\) −64181.5 −0.109313
\(204\) 0 0
\(205\) −1.18231e6 −1.96492
\(206\) 0 0
\(207\) 13183.2 168777.i 0.0213843 0.273771i
\(208\) 0 0
\(209\) 1.16328e6i 1.84213i
\(210\) 0 0
\(211\) 933534.i 1.44352i 0.692142 + 0.721762i \(0.256667\pi\)
−0.692142 + 0.721762i \(0.743333\pi\)
\(212\) 0 0
\(213\) 481031. + 520069.i 0.726480 + 0.785438i
\(214\) 0 0
\(215\) −616615. −0.909742
\(216\) 0 0
\(217\) 116249. 0.167586
\(218\) 0 0
\(219\) 449475. + 485953.i 0.633279 + 0.684674i
\(220\) 0 0
\(221\) 312405.i 0.430266i
\(222\) 0 0
\(223\) 1.03626e6i 1.39542i 0.716379 + 0.697711i \(0.245798\pi\)
−0.716379 + 0.697711i \(0.754202\pi\)
\(224\) 0 0
\(225\) 137414. 1.75924e6i 0.180957 2.31670i
\(226\) 0 0
\(227\) −98404.6 −0.126751 −0.0633754 0.997990i \(-0.520187\pi\)
−0.0633754 + 0.997990i \(0.520187\pi\)
\(228\) 0 0
\(229\) 1.43694e6 1.81072 0.905360 0.424646i \(-0.139601\pi\)
0.905360 + 0.424646i \(0.139601\pi\)
\(230\) 0 0
\(231\) 71101.1 65763.9i 0.0876690 0.0810882i
\(232\) 0 0
\(233\) 677501.i 0.817561i 0.912633 + 0.408780i \(0.134046\pi\)
−0.912633 + 0.408780i \(0.865954\pi\)
\(234\) 0 0
\(235\) 1.32974e6i 1.57071i
\(236\) 0 0
\(237\) 326842. 302308.i 0.377978 0.349606i
\(238\) 0 0
\(239\) 510742. 0.578371 0.289186 0.957273i \(-0.406615\pi\)
0.289186 + 0.957273i \(0.406615\pi\)
\(240\) 0 0
\(241\) 385443. 0.427481 0.213741 0.976890i \(-0.431435\pi\)
0.213741 + 0.976890i \(0.431435\pi\)
\(242\) 0 0
\(243\) −512518. 764601.i −0.556792 0.830652i
\(244\) 0 0
\(245\) 1.69307e6i 1.80202i
\(246\) 0 0
\(247\) 567995.i 0.592382i
\(248\) 0 0
\(249\) −80611.4 87153.5i −0.0823945 0.0890813i
\(250\) 0 0
\(251\) 1.33377e6 1.33628 0.668139 0.744037i \(-0.267091\pi\)
0.668139 + 0.744037i \(0.267091\pi\)
\(252\) 0 0
\(253\) 310372. 0.304847
\(254\) 0 0
\(255\) 1.54928e6 + 1.67501e6i 1.49203 + 1.61312i
\(256\) 0 0
\(257\) 803514.i 0.758858i 0.925221 + 0.379429i \(0.123880\pi\)
−0.925221 + 0.379429i \(0.876120\pi\)
\(258\) 0 0
\(259\) 96586.9i 0.0894682i
\(260\) 0 0
\(261\) −1.11492e6 87086.6i −1.01308 0.0791316i
\(262\) 0 0
\(263\) 312904. 0.278947 0.139474 0.990226i \(-0.455459\pi\)
0.139474 + 0.990226i \(0.455459\pi\)
\(264\) 0 0
\(265\) 3.85721e6 3.37410
\(266\) 0 0
\(267\) −174028. + 160965.i −0.149397 + 0.138182i
\(268\) 0 0
\(269\) 1.65266e6i 1.39253i 0.717786 + 0.696264i \(0.245156\pi\)
−0.717786 + 0.696264i \(0.754844\pi\)
\(270\) 0 0
\(271\) 969577.i 0.801971i −0.916084 0.400986i \(-0.868668\pi\)
0.916084 0.400986i \(-0.131332\pi\)
\(272\) 0 0
\(273\) −34716.4 + 32110.4i −0.0281921 + 0.0260759i
\(274\) 0 0
\(275\) 3.23515e6 2.57966
\(276\) 0 0
\(277\) −279272. −0.218689 −0.109345 0.994004i \(-0.534875\pi\)
−0.109345 + 0.994004i \(0.534875\pi\)
\(278\) 0 0
\(279\) 2.01940e6 + 157735.i 1.55315 + 0.121316i
\(280\) 0 0
\(281\) 388827.i 0.293759i −0.989154 0.146879i \(-0.953077\pi\)
0.989154 0.146879i \(-0.0469230\pi\)
\(282\) 0 0
\(283\) 1.06951e6i 0.793814i 0.917859 + 0.396907i \(0.129916\pi\)
−0.917859 + 0.396907i \(0.870084\pi\)
\(284\) 0 0
\(285\) −2.81679e6 3.04539e6i −2.05420 2.22091i
\(286\) 0 0
\(287\) −161786. −0.115941
\(288\) 0 0
\(289\) −642728. −0.452671
\(290\) 0 0
\(291\) −929387. 1.00481e6i −0.643375 0.695589i
\(292\) 0 0
\(293\) 1.18695e6i 0.807725i 0.914820 + 0.403863i \(0.132333\pi\)
−0.914820 + 0.403863i \(0.867667\pi\)
\(294\) 0 0
\(295\) 419510.i 0.280664i
\(296\) 0 0
\(297\) 1.32436e6 1.04594e6i 0.871194 0.688041i
\(298\) 0 0
\(299\) −151545. −0.0980309
\(300\) 0 0
\(301\) −84377.2 −0.0536796
\(302\) 0 0
\(303\) −1.03146e6 + 954038.i −0.645427 + 0.596979i
\(304\) 0 0
\(305\) 20765.1i 0.0127816i
\(306\) 0 0
\(307\) 782074.i 0.473589i 0.971560 + 0.236795i \(0.0760969\pi\)
−0.971560 + 0.236795i \(0.923903\pi\)
\(308\) 0 0
\(309\) −411595. + 380699.i −0.245230 + 0.226822i
\(310\) 0 0
\(311\) 1.00278e6 0.587902 0.293951 0.955821i \(-0.405030\pi\)
0.293951 + 0.955821i \(0.405030\pi\)
\(312\) 0 0
\(313\) 1.62413e6 0.937047 0.468523 0.883451i \(-0.344786\pi\)
0.468523 + 0.883451i \(0.344786\pi\)
\(314\) 0 0
\(315\) 26895.6 344330.i 0.0152723 0.195523i
\(316\) 0 0
\(317\) 2.05573e6i 1.14899i −0.818507 0.574496i \(-0.805198\pi\)
0.818507 0.574496i \(-0.194802\pi\)
\(318\) 0 0
\(319\) 2.05028e6i 1.12807i
\(320\) 0 0
\(321\) −1.71549e6 1.85471e6i −0.929235 1.00465i
\(322\) 0 0
\(323\) 3.75006e6 2.00001
\(324\) 0 0
\(325\) −1.57962e6 −0.829552
\(326\) 0 0
\(327\) 2.21993e6 + 2.40009e6i 1.14807 + 1.24124i
\(328\) 0 0
\(329\) 181960.i 0.0926802i
\(330\) 0 0
\(331\) 2.97905e6i 1.49454i 0.664521 + 0.747269i \(0.268635\pi\)
−0.664521 + 0.747269i \(0.731365\pi\)
\(332\) 0 0
\(333\) −131057. + 1.67785e6i −0.0647662 + 0.829168i
\(334\) 0 0
\(335\) −131409. −0.0639753
\(336\) 0 0
\(337\) 25986.7 0.0124646 0.00623228 0.999981i \(-0.498016\pi\)
0.00623228 + 0.999981i \(0.498016\pi\)
\(338\) 0 0
\(339\) −1.11445e6 + 1.03079e6i −0.526697 + 0.487161i
\(340\) 0 0
\(341\) 3.71357e6i 1.72944i
\(342\) 0 0
\(343\) 466069.i 0.213902i
\(344\) 0 0
\(345\) 812531. 751539.i 0.367529 0.339941i
\(346\) 0 0
\(347\) −2.98374e6 −1.33026 −0.665131 0.746727i \(-0.731624\pi\)
−0.665131 + 0.746727i \(0.731624\pi\)
\(348\) 0 0
\(349\) −593510. −0.260834 −0.130417 0.991459i \(-0.541632\pi\)
−0.130417 + 0.991459i \(0.541632\pi\)
\(350\) 0 0
\(351\) −646643. + 510698.i −0.280154 + 0.221257i
\(352\) 0 0
\(353\) 1.49836e6i 0.640000i 0.947418 + 0.320000i \(0.103683\pi\)
−0.947418 + 0.320000i \(0.896317\pi\)
\(354\) 0 0
\(355\) 4.63157e6i 1.95055i
\(356\) 0 0
\(357\) 212002. + 229207.i 0.0880378 + 0.0951826i
\(358\) 0 0
\(359\) −4.69341e6 −1.92200 −0.960998 0.276555i \(-0.910807\pi\)
−0.960998 + 0.276555i \(0.910807\pi\)
\(360\) 0 0
\(361\) −4.34201e6 −1.75357
\(362\) 0 0
\(363\) 396138. + 428287.i 0.157790 + 0.170596i
\(364\) 0 0
\(365\) 4.32774e6i 1.70031i
\(366\) 0 0
\(367\) 2.22654e6i 0.862910i 0.902135 + 0.431455i \(0.142000\pi\)
−0.902135 + 0.431455i \(0.858000\pi\)
\(368\) 0 0
\(369\) −2.81045e6 219524.i −1.07451 0.0839298i
\(370\) 0 0
\(371\) 527817. 0.199090
\(372\) 0 0
\(373\) 2.53599e6 0.943789 0.471895 0.881655i \(-0.343570\pi\)
0.471895 + 0.881655i \(0.343570\pi\)
\(374\) 0 0
\(375\) 4.82467e6 4.46251e6i 1.77170 1.63871i
\(376\) 0 0
\(377\) 1.00109e6i 0.362760i
\(378\) 0 0
\(379\) 3.18712e6i 1.13973i −0.821740 0.569863i \(-0.806996\pi\)
0.821740 0.569863i \(-0.193004\pi\)
\(380\) 0 0
\(381\) 1.55858e6 1.44158e6i 0.550067 0.508777i
\(382\) 0 0
\(383\) −2.61862e6 −0.912171 −0.456086 0.889936i \(-0.650749\pi\)
−0.456086 + 0.889936i \(0.650749\pi\)
\(384\) 0 0
\(385\) 633204. 0.217717
\(386\) 0 0
\(387\) −1.46575e6 114490.i −0.497488 0.0388588i
\(388\) 0 0
\(389\) 226309.i 0.0758276i 0.999281 + 0.0379138i \(0.0120712\pi\)
−0.999281 + 0.0379138i \(0.987929\pi\)
\(390\) 0 0
\(391\) 1.00054e6i 0.330973i
\(392\) 0 0
\(393\) −3.38024e6 3.65457e6i −1.10399 1.19359i
\(394\) 0 0
\(395\) 2.91075e6 0.938669
\(396\) 0 0
\(397\) −1.69515e6 −0.539800 −0.269900 0.962888i \(-0.586991\pi\)
−0.269900 + 0.962888i \(0.586991\pi\)
\(398\) 0 0
\(399\) −385448. 416729.i −0.121209 0.131045i
\(400\) 0 0
\(401\) 3.96084e6i 1.23006i −0.788504 0.615030i \(-0.789144\pi\)
0.788504 0.615030i \(-0.210856\pi\)
\(402\) 0 0
\(403\) 1.81322e6i 0.556144i
\(404\) 0 0
\(405\) 934429. 5.94501e6i 0.283080 1.80100i
\(406\) 0 0
\(407\) −3.08548e6 −0.923286
\(408\) 0 0
\(409\) 794687. 0.234902 0.117451 0.993079i \(-0.462528\pi\)
0.117451 + 0.993079i \(0.462528\pi\)
\(410\) 0 0
\(411\) −3.89804e6 + 3.60544e6i −1.13826 + 1.05282i
\(412\) 0 0
\(413\) 57405.4i 0.0165607i
\(414\) 0 0
\(415\) 776162.i 0.221224i
\(416\) 0 0
\(417\) 4.97058e6 4.59746e6i 1.39980 1.29473i
\(418\) 0 0
\(419\) 2.09860e6 0.583976 0.291988 0.956422i \(-0.405683\pi\)
0.291988 + 0.956422i \(0.405683\pi\)
\(420\) 0 0
\(421\) −2.09920e6 −0.577228 −0.288614 0.957446i \(-0.593194\pi\)
−0.288614 + 0.957446i \(0.593194\pi\)
\(422\) 0 0
\(423\) 246898. 3.16091e6i 0.0670914 0.858936i
\(424\) 0 0
\(425\) 1.04291e7i 2.80074i
\(426\) 0 0
\(427\) 2841.48i 0.000754179i
\(428\) 0 0
\(429\) −1.02577e6 1.10902e6i −0.269096 0.290934i
\(430\) 0 0
\(431\) 5.13075e6 1.33042 0.665208 0.746658i \(-0.268343\pi\)
0.665208 + 0.746658i \(0.268343\pi\)
\(432\) 0 0
\(433\) −276424. −0.0708527 −0.0354264 0.999372i \(-0.511279\pi\)
−0.0354264 + 0.999372i \(0.511279\pi\)
\(434\) 0 0
\(435\) −4.96458e6 5.36749e6i −1.25794 1.36003i
\(436\) 0 0
\(437\) 1.81912e6i 0.455677i
\(438\) 0 0
\(439\) 4.00849e6i 0.992703i 0.868122 + 0.496351i \(0.165327\pi\)
−0.868122 + 0.496351i \(0.834673\pi\)
\(440\) 0 0
\(441\) −314359. + 4.02458e6i −0.0769715 + 0.985426i
\(442\) 0 0
\(443\) −484754. −0.117358 −0.0586790 0.998277i \(-0.518689\pi\)
−0.0586790 + 0.998277i \(0.518689\pi\)
\(444\) 0 0
\(445\) −1.54984e6 −0.371011
\(446\) 0 0
\(447\) −1.25773e6 + 1.16332e6i −0.297727 + 0.275378i
\(448\) 0 0
\(449\) 7.23618e6i 1.69392i 0.531655 + 0.846961i \(0.321570\pi\)
−0.531655 + 0.846961i \(0.678430\pi\)
\(450\) 0 0
\(451\) 5.16827e6i 1.19647i
\(452\) 0 0
\(453\) 3.21967e6 2.97799e6i 0.737168 0.681833i
\(454\) 0 0
\(455\) −309173. −0.0700121
\(456\) 0 0
\(457\) −3.18803e6 −0.714055 −0.357027 0.934094i \(-0.616210\pi\)
−0.357027 + 0.934094i \(0.616210\pi\)
\(458\) 0 0
\(459\) 3.37177e6 + 4.26931e6i 0.747009 + 0.945859i
\(460\) 0 0
\(461\) 208368.i 0.0456645i −0.999739 0.0228323i \(-0.992732\pi\)
0.999739 0.0228323i \(-0.00726837\pi\)
\(462\) 0 0
\(463\) 7.76931e6i 1.68434i −0.539211 0.842171i \(-0.681278\pi\)
0.539211 0.842171i \(-0.318722\pi\)
\(464\) 0 0
\(465\) 8.99208e6 + 9.72184e6i 1.92854 + 2.08505i
\(466\) 0 0
\(467\) −3.32855e6 −0.706258 −0.353129 0.935575i \(-0.614882\pi\)
−0.353129 + 0.935575i \(0.614882\pi\)
\(468\) 0 0
\(469\) −17981.9 −0.00377488
\(470\) 0 0
\(471\) −5.14666e6 5.56434e6i −1.06899 1.15574i
\(472\) 0 0
\(473\) 2.69544e6i 0.553957i
\(474\) 0 0
\(475\) 1.89615e7i 3.85601i
\(476\) 0 0
\(477\) 9.16894e6 + 716185.i 1.84511 + 0.144122i
\(478\) 0 0
\(479\) −5.46336e6 −1.08798 −0.543990 0.839092i \(-0.683087\pi\)
−0.543990 + 0.839092i \(0.683087\pi\)
\(480\) 0 0
\(481\) 1.50654e6 0.296905
\(482\) 0 0
\(483\) 111186. 102840.i 0.0216862 0.0200583i
\(484\) 0 0
\(485\) 8.94855e6i 1.72742i
\(486\) 0 0
\(487\) 7.36280e6i 1.40676i 0.710813 + 0.703381i \(0.248327\pi\)
−0.710813 + 0.703381i \(0.751673\pi\)
\(488\) 0 0
\(489\) 4.98319e6 4.60914e6i 0.942401 0.871660i
\(490\) 0 0
\(491\) 5.98040e6 1.11951 0.559753 0.828660i \(-0.310896\pi\)
0.559753 + 0.828660i \(0.310896\pi\)
\(492\) 0 0
\(493\) 6.60946e6 1.22475
\(494\) 0 0
\(495\) 1.09997e7 + 859181.i 2.01774 + 0.157606i
\(496\) 0 0
\(497\) 633781.i 0.115093i
\(498\) 0 0
\(499\) 1.54433e6i 0.277645i 0.990317 + 0.138822i \(0.0443317\pi\)
−0.990317 + 0.138822i \(0.955668\pi\)
\(500\) 0 0
\(501\) −1.65814e6 1.79271e6i −0.295139 0.319092i
\(502\) 0 0
\(503\) −161847. −0.0285222 −0.0142611 0.999898i \(-0.504540\pi\)
−0.0142611 + 0.999898i \(0.504540\pi\)
\(504\) 0 0
\(505\) −9.18589e6 −1.60285
\(506\) 0 0
\(507\) −3.42922e6 3.70752e6i −0.592482 0.640566i
\(508\) 0 0
\(509\) 3.22871e6i 0.552375i 0.961104 + 0.276188i \(0.0890711\pi\)
−0.961104 + 0.276188i \(0.910929\pi\)
\(510\) 0 0
\(511\) 592205.i 0.100327i
\(512\) 0 0
\(513\) −6.13033e6 7.76218e6i −1.02847 1.30224i
\(514\) 0 0
\(515\) −3.66553e6 −0.609003
\(516\) 0 0
\(517\) 5.81274e6 0.956432
\(518\) 0 0
\(519\) −3.26552e6 + 3.02040e6i −0.532150 + 0.492205i
\(520\) 0 0
\(521\) 5.06042e6i 0.816755i −0.912813 0.408378i \(-0.866095\pi\)
0.912813 0.408378i \(-0.133905\pi\)
\(522\) 0 0
\(523\) 1.82857e6i 0.292320i −0.989261 0.146160i \(-0.953309\pi\)
0.989261 0.146160i \(-0.0466914\pi\)
\(524\) 0 0
\(525\) 1.15894e6 1.07195e6i 0.183512 0.169736i
\(526\) 0 0
\(527\) −1.19714e7 −1.87766
\(528\) 0 0
\(529\) −5.95099e6 −0.924592
\(530\) 0 0
\(531\) 77892.2 997213.i 0.0119883 0.153480i
\(532\) 0 0
\(533\) 2.52350e6i 0.384756i
\(534\) 0 0
\(535\) 1.65175e7i 2.49494i
\(536\) 0 0
\(537\) −2.94125e6 3.17995e6i −0.440146 0.475866i
\(538\) 0 0
\(539\) −7.40098e6 −1.09728
\(540\) 0 0
\(541\) 3.75554e6 0.551670 0.275835 0.961205i \(-0.411046\pi\)
0.275835 + 0.961205i \(0.411046\pi\)
\(542\) 0 0
\(543\) 788391. + 852374.i 0.114747 + 0.124060i
\(544\) 0 0
\(545\) 2.13744e7i 3.08250i
\(546\) 0 0
\(547\) 5.56727e6i 0.795562i 0.917480 + 0.397781i \(0.130220\pi\)
−0.917480 + 0.397781i \(0.869780\pi\)
\(548\) 0 0
\(549\) −3855.54 + 49360.5i −0.000545952 + 0.00698954i
\(550\) 0 0
\(551\) −1.20169e7 −1.68621
\(552\) 0 0
\(553\) 398305. 0.0553864
\(554\) 0 0
\(555\) −8.07754e6 + 7.47121e6i −1.11313 + 1.02958i
\(556\) 0 0
\(557\) 356674.i 0.0487117i 0.999703 + 0.0243559i \(0.00775348\pi\)
−0.999703 + 0.0243559i \(0.992247\pi\)
\(558\) 0 0
\(559\) 1.31610e6i 0.178138i
\(560\) 0 0
\(561\) −7.32204e6 + 6.77242e6i −0.982256 + 0.908524i
\(562\) 0 0
\(563\) −1.13052e7 −1.50317 −0.751586 0.659635i \(-0.770711\pi\)
−0.751586 + 0.659635i \(0.770711\pi\)
\(564\) 0 0
\(565\) −9.92493e6 −1.30800
\(566\) 0 0
\(567\) 127867. 813511.i 0.0167032 0.106269i
\(568\) 0 0
\(569\) 7.39855e6i 0.958001i −0.877815 0.479001i \(-0.840999\pi\)
0.877815 0.479001i \(-0.159001\pi\)
\(570\) 0 0
\(571\) 1.47941e7i 1.89888i 0.313953 + 0.949439i \(0.398347\pi\)
−0.313953 + 0.949439i \(0.601653\pi\)
\(572\) 0 0
\(573\) −3.25677e6 3.52108e6i −0.414382 0.448012i
\(574\) 0 0
\(575\) 5.05904e6 0.638115
\(576\) 0 0
\(577\) 408152. 0.0510366 0.0255183 0.999674i \(-0.491876\pi\)
0.0255183 + 0.999674i \(0.491876\pi\)
\(578\) 0 0
\(579\) −3.11021e6 3.36263e6i −0.385562 0.416852i
\(580\) 0 0
\(581\) 106209.i 0.0130534i
\(582\) 0 0
\(583\) 1.68612e7i 2.05455i
\(584\) 0 0
\(585\) −5.37078e6 419511.i −0.648855 0.0506819i
\(586\) 0 0
\(587\) 9.26473e6 1.10978 0.554891 0.831923i \(-0.312760\pi\)
0.554891 + 0.831923i \(0.312760\pi\)
\(588\) 0 0
\(589\) 2.17655e7 2.58512
\(590\) 0 0
\(591\) −3.33163e6 + 3.08154e6i −0.392363 + 0.362910i
\(592\) 0 0
\(593\) 5.46033e6i 0.637650i −0.947814 0.318825i \(-0.896712\pi\)
0.947814 0.318825i \(-0.103288\pi\)
\(594\) 0 0
\(595\) 2.04125e6i 0.236376i
\(596\) 0 0
\(597\) −8.01076e6 + 7.40944e6i −0.919894 + 0.850843i
\(598\) 0 0
\(599\) 1.26892e7 1.44500 0.722502 0.691369i \(-0.242992\pi\)
0.722502 + 0.691369i \(0.242992\pi\)
\(600\) 0 0
\(601\) −4.19942e6 −0.474245 −0.237122 0.971480i \(-0.576204\pi\)
−0.237122 + 0.971480i \(0.576204\pi\)
\(602\) 0 0
\(603\) −312371. 24399.2i −0.0349846 0.00273264i
\(604\) 0 0
\(605\) 3.81419e6i 0.423657i
\(606\) 0 0
\(607\) 1.11137e7i 1.22430i −0.790741 0.612150i \(-0.790305\pi\)
0.790741 0.612150i \(-0.209695\pi\)
\(608\) 0 0
\(609\) −679350. 734483.i −0.0742250 0.0802488i
\(610\) 0 0
\(611\) −2.83817e6 −0.307564
\(612\) 0 0
\(613\) −1.84572e7 −1.98387 −0.991937 0.126734i \(-0.959550\pi\)
−0.991937 + 0.126734i \(0.959550\pi\)
\(614\) 0 0
\(615\) −1.25145e7 1.35301e7i −1.33422 1.44249i
\(616\) 0 0
\(617\) 6.43968e6i 0.681006i −0.940243 0.340503i \(-0.889403\pi\)
0.940243 0.340503i \(-0.110597\pi\)
\(618\) 0 0
\(619\) 7.60963e6i 0.798246i −0.916897 0.399123i \(-0.869315\pi\)
0.916897 0.399123i \(-0.130685\pi\)
\(620\) 0 0
\(621\) 2.07100e6 1.63561e6i 0.215502 0.170197i
\(622\) 0 0
\(623\) −212079. −0.0218916
\(624\) 0 0
\(625\) 2.02741e7 2.07607
\(626\) 0 0
\(627\) 1.33124e7 1.23132e7i 1.35235 1.25084i
\(628\) 0 0
\(629\) 9.94658e6i 1.00241i
\(630\) 0 0
\(631\) 122913.i 0.0122892i −0.999981 0.00614461i \(-0.998044\pi\)
0.999981 0.00614461i \(-0.00195590\pi\)
\(632\) 0 0
\(633\) −1.06832e7 + 9.88129e6i −1.05972 + 0.980176i
\(634\) 0 0
\(635\) 1.38802e7 1.36603
\(636\) 0 0
\(637\) 3.61366e6 0.352857
\(638\) 0 0
\(639\) −859964. + 1.10097e7i −0.0833160 + 1.06665i
\(640\) 0 0
\(641\) 1.49218e6i 0.143441i 0.997425 + 0.0717207i \(0.0228490\pi\)
−0.997425 + 0.0717207i \(0.977151\pi\)
\(642\) 0 0
\(643\) 1.69300e7i 1.61484i 0.589980 + 0.807418i \(0.299136\pi\)
−0.589980 + 0.807418i \(0.700864\pi\)
\(644\) 0 0
\(645\) −6.52676e6 7.05645e6i −0.617730 0.667862i
\(646\) 0 0
\(647\) 2.24710e6 0.211038 0.105519 0.994417i \(-0.466350\pi\)
0.105519 + 0.994417i \(0.466350\pi\)
\(648\) 0 0
\(649\) 1.83382e6 0.170901
\(650\) 0 0
\(651\) 1.23047e6 + 1.33033e6i 0.113794 + 0.123029i
\(652\) 0 0
\(653\) 6.12204e6i 0.561840i −0.959731 0.280920i \(-0.909360\pi\)
0.959731 0.280920i \(-0.0906396\pi\)
\(654\) 0 0
\(655\) 3.25465e7i 2.96415i
\(656\) 0 0
\(657\) −803551. + 1.02874e7i −0.0726273 + 0.929809i
\(658\) 0 0
\(659\) 3.29399e6 0.295467 0.147734 0.989027i \(-0.452802\pi\)
0.147734 + 0.989027i \(0.452802\pi\)
\(660\) 0 0
\(661\) 3.48328e6 0.310088 0.155044 0.987908i \(-0.450448\pi\)
0.155044 + 0.987908i \(0.450448\pi\)
\(662\) 0 0
\(663\) 3.57512e6 3.30675e6i 0.315868 0.292158i
\(664\) 0 0
\(665\) 3.71126e6i 0.325437i
\(666\) 0 0
\(667\) 3.20618e6i 0.279045i
\(668\) 0 0
\(669\) −1.18588e7 + 1.09686e7i −1.02441 + 0.947515i
\(670\) 0 0
\(671\) −90771.2 −0.00778291
\(672\) 0 0
\(673\) 3.87190e6 0.329523 0.164762 0.986333i \(-0.447315\pi\)
0.164762 + 0.986333i \(0.447315\pi\)
\(674\) 0 0
\(675\) 2.15870e7 1.70487e7i 1.82361 1.44023i
\(676\) 0 0
\(677\) 8.13996e6i 0.682575i 0.939959 + 0.341288i \(0.110863\pi\)
−0.939959 + 0.341288i \(0.889137\pi\)
\(678\) 0 0
\(679\) 1.22451e6i 0.101927i
\(680\) 0 0
\(681\) −1.04160e6 1.12613e6i −0.0860659 0.0930507i
\(682\) 0 0
\(683\) −1.71913e7 −1.41012 −0.705061 0.709147i \(-0.749080\pi\)
−0.705061 + 0.709147i \(0.749080\pi\)
\(684\) 0 0
\(685\) −3.47147e7 −2.82675
\(686\) 0 0
\(687\) 1.52098e7 + 1.64442e7i 1.22951 + 1.32929i
\(688\) 0 0
\(689\) 8.23277e6i 0.660690i
\(690\) 0 0
\(691\) 1.90803e7i 1.52016i 0.649827 + 0.760082i \(0.274841\pi\)
−0.649827 + 0.760082i \(0.725159\pi\)
\(692\) 0 0
\(693\) 1.50518e6 + 117570.i 0.119057 + 0.00929956i
\(694\) 0 0
\(695\) 4.42664e7 3.47626
\(696\) 0 0
\(697\) 1.66608e7 1.29902
\(698\) 0 0
\(699\) −7.75321e6 + 7.17122e6i −0.600190 + 0.555137i
\(700\) 0 0
\(701\) 1.88806e7i 1.45118i 0.688128 + 0.725590i \(0.258433\pi\)
−0.688128 + 0.725590i \(0.741567\pi\)
\(702\) 0 0
\(703\) 1.80842e7i 1.38010i
\(704\) 0 0
\(705\) 1.52173e7 1.40750e7i 1.15309 1.06654i
\(706\) 0 0
\(707\) −1.25699e6 −0.0945766
\(708\) 0 0
\(709\) 1.30901e7 0.977975 0.488988 0.872291i \(-0.337366\pi\)
0.488988 + 0.872291i \(0.337366\pi\)
\(710\) 0 0
\(711\) 6.91913e6 + 540452.i 0.513307 + 0.0400944i
\(712\) 0 0
\(713\) 5.80719e6i 0.427801i
\(714\) 0 0
\(715\) 9.87656e6i 0.722505i
\(716\) 0 0
\(717\) 5.40611e6 + 5.84485e6i 0.392724 + 0.424596i
\(718\) 0 0
\(719\) −6.98750e6 −0.504080 −0.252040 0.967717i \(-0.581101\pi\)
−0.252040 + 0.967717i \(0.581101\pi\)
\(720\) 0 0
\(721\) −501589. −0.0359344
\(722\) 0 0
\(723\) 4.07984e6 + 4.41095e6i 0.290267 + 0.313824i
\(724\) 0 0
\(725\) 3.34195e7i 2.36132i
\(726\) 0 0
\(727\) 481177.i 0.0337652i 0.999857 + 0.0168826i \(0.00537415\pi\)
−0.999857 + 0.0168826i \(0.994626\pi\)
\(728\) 0 0
\(729\) 3.32506e6 1.39583e7i 0.231729 0.972780i
\(730\) 0 0
\(731\) 8.68922e6 0.601433
\(732\) 0 0
\(733\) 2.08328e7 1.43215 0.716073 0.698025i \(-0.245938\pi\)
0.716073 + 0.698025i \(0.245938\pi\)
\(734\) 0 0
\(735\) −1.93752e7 + 1.79208e7i −1.32290 + 1.22360i
\(736\) 0 0
\(737\) 574433.i 0.0389556i
\(738\) 0 0
\(739\) 5.87155e6i 0.395496i −0.980253 0.197748i \(-0.936637\pi\)
0.980253 0.197748i \(-0.0633627\pi\)
\(740\) 0 0
\(741\) −6.50004e6 + 6.01212e6i −0.434881 + 0.402237i
\(742\) 0 0
\(743\) −1.44391e7 −0.959550 −0.479775 0.877392i \(-0.659282\pi\)
−0.479775 + 0.877392i \(0.659282\pi\)
\(744\) 0 0
\(745\) −1.12009e7 −0.739373
\(746\) 0 0
\(747\) 144113. 1.84501e6i 0.00944937 0.120975i
\(748\) 0 0
\(749\) 2.26024e6i 0.147214i
\(750\) 0 0
\(751\) 4.23423e6i 0.273952i 0.990574 + 0.136976i \(0.0437384\pi\)
−0.990574 + 0.136976i \(0.956262\pi\)
\(752\) 0 0
\(753\) 1.41177e7 + 1.52634e7i 0.907354 + 0.980991i
\(754\) 0 0
\(755\) 2.86734e7 1.83068
\(756\) 0 0
\(757\) 1.34919e7 0.855723 0.427862 0.903844i \(-0.359267\pi\)
0.427862 + 0.903844i \(0.359267\pi\)
\(758\) 0 0
\(759\) 3.28523e6 + 3.55185e6i 0.206996 + 0.223795i
\(760\) 0 0
\(761\) 1.34987e7i 0.844948i −0.906375 0.422474i \(-0.861162\pi\)
0.906375 0.422474i \(-0.138838\pi\)
\(762\) 0 0
\(763\) 2.92486e6i 0.181884i
\(764\) 0 0
\(765\) −2.76972e6 + 3.54593e7i −0.171113 + 2.19067i
\(766\) 0 0
\(767\) −895395. −0.0549574
\(768\) 0 0
\(769\) 2.22121e7 1.35449 0.677243 0.735760i \(-0.263175\pi\)
0.677243 + 0.735760i \(0.263175\pi\)
\(770\) 0 0
\(771\) −9.19529e6 + 8.50505e6i −0.557095 + 0.515277i
\(772\) 0 0
\(773\) 1.78526e7i 1.07462i 0.843386 + 0.537308i \(0.180559\pi\)
−0.843386 + 0.537308i \(0.819441\pi\)
\(774\) 0 0
\(775\) 6.05309e7i 3.62012i
\(776\) 0 0
\(777\) −1.10532e6 + 1.02235e6i −0.0656806 + 0.0607504i
\(778\) 0 0
\(779\) −3.02916e7 −1.78846
\(780\) 0 0
\(781\) −2.02462e7 −1.18772
\(782\) 0 0
\(783\) −1.08047e7 1.36808e7i −0.629807 0.797458i
\(784\) 0 0
\(785\) 4.95543e7i 2.87017i
\(786\) 0 0
\(787\) 4.18718e6i 0.240982i 0.992714 + 0.120491i \(0.0384469\pi\)
−0.992714 + 0.120491i \(0.961553\pi\)
\(788\) 0 0
\(789\) 3.31204e6 + 3.58083e6i 0.189410 + 0.204782i
\(790\) 0 0
\(791\) −1.35812e6 −0.0771786
\(792\) 0 0
\(793\) 44320.7 0.00250278
\(794\) 0 0
\(795\) 4.08278e7 + 4.41413e7i 2.29107 + 2.47701i
\(796\) 0 0
\(797\) 7.76856e6i 0.433206i 0.976260 + 0.216603i \(0.0694978\pi\)
−0.976260 + 0.216603i \(0.930502\pi\)
\(798\) 0 0
\(799\) 1.87384e7i 1.03840i
\(800\) 0 0
\(801\) −3.68411e6 287766.i −0.202886 0.0158474i
\(802\) 0 0
\(803\) −1.89180e7 −1.03535
\(804\) 0 0
\(805\) 990189. 0.0538553
\(806\) 0 0
\(807\) −1.89128e7 + 1.74932e7i −1.02229 + 0.945550i
\(808\) 0 0
\(809\) 5.07918e6i 0.272849i −0.990650 0.136424i \(-0.956439\pi\)
0.990650 0.136424i \(-0.0435611\pi\)
\(810\) 0 0
\(811\) 1.12586e7i 0.601080i −0.953769 0.300540i \(-0.902833\pi\)
0.953769 0.300540i \(-0.0971669\pi\)
\(812\) 0 0
\(813\) 1.10957e7 1.02628e7i 0.588746 0.544552i
\(814\) 0 0
\(815\) 4.43788e7 2.34035
\(816\) 0 0
\(817\) −1.57982e7 −0.828041
\(818\) 0 0
\(819\) −734933. 57405.5i −0.0382858 0.00299050i
\(820\) 0 0
\(821\) 1.80009e7i 0.932043i 0.884773 + 0.466022i \(0.154313\pi\)
−0.884773 + 0.466022i \(0.845687\pi\)
\(822\) 0 0
\(823\) 8.19550e6i 0.421770i −0.977511 0.210885i \(-0.932365\pi\)
0.977511 0.210885i \(-0.0676346\pi\)
\(824\) 0 0
\(825\) 3.42434e7 + 3.70225e7i 1.75163 + 1.89379i
\(826\) 0 0
\(827\) −7.82150e6 −0.397673 −0.198837 0.980033i \(-0.563716\pi\)
−0.198837 + 0.980033i \(0.563716\pi\)
\(828\) 0 0
\(829\) −2.69630e6 −0.136264 −0.0681322 0.997676i \(-0.521704\pi\)
−0.0681322 + 0.997676i \(0.521704\pi\)
\(830\) 0 0
\(831\) −2.95604e6 3.19594e6i −0.148494 0.160545i
\(832\) 0 0
\(833\) 2.38584e7i 1.19132i
\(834\) 0 0
\(835\) 1.59653e7i 0.792430i
\(836\) 0 0
\(837\) 1.95699e7 + 2.47793e7i 0.965551 + 1.22258i
\(838\) 0 0
\(839\) 1.48635e7 0.728979 0.364490 0.931207i \(-0.381243\pi\)
0.364490 + 0.931207i \(0.381243\pi\)
\(840\) 0 0
\(841\) −668562. −0.0325951
\(842\) 0 0
\(843\) 4.44968e6 4.11567e6i 0.215655 0.199467i
\(844\) 0 0
\(845\) 3.30180e7i 1.59078i
\(846\) 0 0
\(847\) 521931.i 0.0249980i
\(848\) 0 0
\(849\) −1.22393e7 + 1.13206e7i −0.582757 + 0.539013i
\(850\) 0 0
\(851\) −4.82499e6 −0.228388
\(852\) 0 0
\(853\) −2.77927e7 −1.30785 −0.653924 0.756560i \(-0.726879\pi\)
−0.653924 + 0.756560i \(0.726879\pi\)
\(854\) 0 0
\(855\) 5.03573e6 6.44698e7i 0.235585 3.01607i
\(856\) 0 0
\(857\) 1.73505e7i 0.806974i −0.914985 0.403487i \(-0.867798\pi\)
0.914985 0.403487i \(-0.132202\pi\)
\(858\) 0 0
\(859\) 5.23238e6i 0.241945i 0.992656 + 0.120972i \(0.0386013\pi\)
−0.992656 + 0.120972i \(0.961399\pi\)
\(860\) 0 0
\(861\) −1.71248e6 1.85145e6i −0.0787257 0.0851148i
\(862\) 0 0
\(863\) 2.61326e7 1.19442 0.597208 0.802086i \(-0.296277\pi\)
0.597208 + 0.802086i \(0.296277\pi\)
\(864\) 0 0
\(865\) −2.90817e7 −1.32154
\(866\) 0 0
\(867\) −6.80317e6 7.35528e6i −0.307371 0.332316i
\(868\) 0 0
\(869\) 1.27239e7i 0.571572i
\(870\) 0 0
\(871\) 280477.i 0.0125271i
\(872\) 0 0
\(873\) 1.66152e6 2.12715e7i 0.0737851 0.944633i
\(874\) 0 0
\(875\) 5.87958e6 0.259613
\(876\) 0 0
\(877\) 1.01424e7 0.445287 0.222644 0.974900i \(-0.428531\pi\)
0.222644 + 0.974900i \(0.428531\pi\)
\(878\) 0 0
\(879\) −1.35833e7 + 1.25637e7i −0.592970 + 0.548459i
\(880\) 0 0
\(881\) 2.18718e7i 0.949390i 0.880150 + 0.474695i \(0.157442\pi\)
−0.880150 + 0.474695i \(0.842558\pi\)
\(882\) 0 0
\(883\) 2.05850e7i 0.888481i 0.895908 + 0.444241i \(0.146526\pi\)
−0.895908 + 0.444241i \(0.853474\pi\)
\(884\) 0 0
\(885\) 4.80080e6 4.44043e6i 0.206042 0.190576i
\(886\) 0 0
\(887\) 1.39584e7 0.595698 0.297849 0.954613i \(-0.403731\pi\)
0.297849 + 0.954613i \(0.403731\pi\)
\(888\) 0 0
\(889\) 1.89935e6 0.0806031
\(890\) 0 0
\(891\) 2.59877e7 + 4.08471e6i 1.09666 + 0.172372i
\(892\) 0 0
\(893\) 3.40689e7i 1.42965i
\(894\) 0 0
\(895\) 2.83197e7i 1.18176i
\(896\) 0 0
\(897\) −1.60407e6 1.73425e6i −0.0665646 0.0719667i
\(898\) 0 0
\(899\) 3.83616e7 1.58306
\(900\) 0 0
\(901\) −5.43550e7 −2.23063
\(902\) 0 0
\(903\) −893118. 965599.i −0.0364493 0.0394074i
\(904\) 0 0
\(905\) 7.59097e6i 0.308089i
\(906\) 0 0
\(907\) 3.01421e7i 1.21662i −0.793699 0.608310i \(-0.791848\pi\)
0.793699 0.608310i \(-0.208152\pi\)
\(908\) 0 0
\(909\) −2.18357e7 1.70558e6i −0.876512 0.0684642i
\(910\) 0 0
\(911\) 6.18404e6 0.246874 0.123437 0.992352i \(-0.460608\pi\)
0.123437 + 0.992352i \(0.460608\pi\)
\(912\) 0 0
\(913\) 3.39287e6 0.134707
\(914\) 0 0
\(915\) −237632. + 219795.i −0.00938323 + 0.00867889i
\(916\) 0 0
\(917\) 4.45364e6i 0.174901i
\(918\) 0 0
\(919\) 1.31915e7i 0.515237i −0.966247 0.257618i \(-0.917062\pi\)
0.966247 0.257618i \(-0.0829377\pi\)
\(920\) 0 0
\(921\) −8.94993e6 + 8.27812e6i −0.347673 + 0.321575i
\(922\) 0 0
\(923\) 9.88556e6 0.381942
\(924\) 0 0
\(925\) −5.02930e7 −1.93265
\(926\) 0 0
\(927\) −8.71332e6 680596.i −0.333031 0.0260130i
\(928\) 0 0
\(929\) 3.61366e7i 1.37375i 0.726776 + 0.686875i \(0.241018\pi\)
−0.726776 + 0.686875i \(0.758982\pi\)
\(930\) 0 0
\(931\) 4.33778e7i 1.64019i
\(932\) 0 0
\(933\) 1.06143e7 + 1.14757e7i 0.399195 + 0.431592i
\(934\) 0 0
\(935\) −6.52078e7 −2.43933
\(936\) 0 0
\(937\) 4.72568e7 1.75839 0.879195 0.476462i \(-0.158081\pi\)
0.879195 + 0.476462i \(0.158081\pi\)
\(938\) 0 0
\(939\) 1.71912e7 + 1.85863e7i 0.636270 + 0.687907i
\(940\) 0 0
\(941\) 3.04002e7i 1.11919i −0.828768 0.559593i \(-0.810958\pi\)
0.828768 0.559593i \(-0.189042\pi\)
\(942\) 0 0
\(943\) 8.08201e6i 0.295965i
\(944\) 0 0
\(945\) 4.22514e6 3.33688e6i 0.153908 0.121552i
\(946\) 0 0
\(947\) −3.23838e7 −1.17342 −0.586709 0.809798i \(-0.699577\pi\)
−0.586709 + 0.809798i \(0.699577\pi\)
\(948\) 0 0
\(949\) 9.23707e6 0.332942
\(950\) 0 0
\(951\) 2.35254e7 2.17595e7i 0.843501 0.780185i
\(952\) 0 0
\(953\) 1.93465e7i 0.690032i 0.938597 + 0.345016i \(0.112127\pi\)
−0.938597 + 0.345016i \(0.887873\pi\)
\(954\) 0 0
\(955\) 3.13576e7i 1.11259i
\(956\) 0 0
\(957\) 2.34631e7 2.17019e7i 0.828145 0.765981i
\(958\) 0 0
\(959\) −4.75034e6 −0.166793
\(960\) 0 0
\(961\) −4.08532e7 −1.42698
\(962\) 0 0
\(963\) 3.06687e6 3.92636e7i 0.106569 1.36434i
\(964\) 0 0
\(965\) 2.99465e7i 1.03521i
\(966\) 0 0
\(967\) 3.67206e7i 1.26282i 0.775447 + 0.631412i \(0.217524\pi\)
−0.775447 + 0.631412i \(0.782476\pi\)
\(968\) 0 0
\(969\) 3.96937e7 + 4.29151e7i 1.35804 + 1.46825i
\(970\) 0 0
\(971\) −5.16753e7 −1.75887 −0.879437 0.476015i \(-0.842081\pi\)
−0.879437 + 0.476015i \(0.842081\pi\)
\(972\) 0 0
\(973\) 6.05738e6 0.205118
\(974\) 0 0
\(975\) −1.67200e7 1.80769e7i −0.563280 0.608993i
\(976\) 0 0
\(977\) 8.88406e6i 0.297766i −0.988855 0.148883i \(-0.952432\pi\)
0.988855 0.148883i \(-0.0475678\pi\)
\(978\) 0 0
\(979\) 6.77488e6i 0.225915i
\(980\) 0 0
\(981\) −3.96868e6 + 5.08090e7i −0.131666 + 1.68565i
\(982\) 0 0
\(983\) −6.34925e6 −0.209575 −0.104787 0.994495i \(-0.533416\pi\)
−0.104787 + 0.994495i \(0.533416\pi\)
\(984\) 0 0
\(985\) −2.96704e7 −0.974391
\(986\) 0 0
\(987\) 2.08232e6 1.92602e6i 0.0680386 0.0629314i
\(988\) 0 0
\(989\) 4.21506e6i 0.137029i
\(990\) 0 0
\(991\) 4.45077e7i 1.43963i −0.694166 0.719815i \(-0.744226\pi\)
0.694166 0.719815i \(-0.255774\pi\)
\(992\) 0 0
\(993\) −3.40917e7 + 3.15327e7i −1.09717 + 1.01482i
\(994\) 0 0
\(995\) −7.13413e7 −2.28446
\(996\) 0 0
\(997\) 6.05424e7 1.92895 0.964477 0.264166i \(-0.0850965\pi\)
0.964477 + 0.264166i \(0.0850965\pi\)
\(998\) 0 0
\(999\) −2.05883e7 + 1.62600e7i −0.652689 + 0.515473i
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 384.6.c.a.383.16 yes 20
3.2 odd 2 384.6.c.d.383.6 yes 20
4.3 odd 2 384.6.c.d.383.5 yes 20
8.3 odd 2 384.6.c.b.383.16 yes 20
8.5 even 2 384.6.c.c.383.5 yes 20
12.11 even 2 inner 384.6.c.a.383.15 20
24.5 odd 2 384.6.c.b.383.15 yes 20
24.11 even 2 384.6.c.c.383.6 yes 20
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
384.6.c.a.383.15 20 12.11 even 2 inner
384.6.c.a.383.16 yes 20 1.1 even 1 trivial
384.6.c.b.383.15 yes 20 24.5 odd 2
384.6.c.b.383.16 yes 20 8.3 odd 2
384.6.c.c.383.5 yes 20 8.5 even 2
384.6.c.c.383.6 yes 20 24.11 even 2
384.6.c.d.383.5 yes 20 4.3 odd 2
384.6.c.d.383.6 yes 20 3.2 odd 2