Properties

Label 20.7.f.a.13.1
Level $20$
Weight $7$
Character 20.13
Analytic conductor $4.601$
Analytic rank $0$
Dimension $6$
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] = [20,7,Mod(13,20)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(20, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 3]))
 
N = Newforms(chi, 7, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("20.13");
 
S:= CuspForms(chi, 7);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 20 = 2^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 20.f (of order \(4\), degree \(2\), 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: \(4.60108167240\)
Analytic rank: \(0\)
Dimension: \(6\)
Relative dimension: \(3\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 2x^{5} + 2x^{4} - 450x^{3} + 23409x^{2} - 115668x + 285768 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 2^{7}\cdot 5 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 13.1
Root \(2.61494 + 2.61494i\) of defining polynomial
Character \(\chi\) \(=\) 20.13
Dual form 20.7.f.a.17.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-16.7249 - 16.7249i) q^{3} +(-22.1946 + 123.014i) q^{5} +(-422.022 + 422.022i) q^{7} -169.553i q^{9} +O(q^{10})\) \(q+(-16.7249 - 16.7249i) q^{3} +(-22.1946 + 123.014i) q^{5} +(-422.022 + 422.022i) q^{7} -169.553i q^{9} -1359.94 q^{11} +(1248.94 + 1248.94i) q^{13} +(2428.60 - 1686.20i) q^{15} +(-648.372 + 648.372i) q^{17} -9632.63i q^{19} +14116.6 q^{21} +(886.911 + 886.911i) q^{23} +(-14639.8 - 5460.48i) q^{25} +(-15028.2 + 15028.2i) q^{27} +26789.3i q^{29} -6880.94 q^{31} +(22744.9 + 22744.9i) q^{33} +(-42547.9 - 61281.1i) q^{35} +(-42367.5 + 42367.5i) q^{37} -41777.0i q^{39} +132500. q^{41} +(20888.6 + 20888.6i) q^{43} +(20857.4 + 3763.16i) q^{45} +(-31606.0 + 31606.0i) q^{47} -238556. i q^{49} +21688.0 q^{51} +(80106.6 + 80106.6i) q^{53} +(30183.3 - 167291. i) q^{55} +(-161105. + 161105. i) q^{57} +290136. i q^{59} -253971. q^{61} +(71555.1 + 71555.1i) q^{63} +(-181357. + 125918. i) q^{65} +(11368.7 - 11368.7i) q^{67} -29667.0i q^{69} +179758. q^{71} +(-414484. - 414484. i) q^{73} +(153524. + 336176. i) q^{75} +(573924. - 573924. i) q^{77} +145313. i q^{79} +379089. q^{81} +(-63129.1 - 63129.1i) q^{83} +(-65368.3 - 94149.0i) q^{85} +(448049. - 448049. i) q^{87} +333388. i q^{89} -1.05416e6 q^{91} +(115083. + 115083. i) q^{93} +(1.18495e6 + 213792. i) q^{95} +(519696. - 519696. i) q^{97} +230582. i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 32 q^{3} - 156 q^{5} - 264 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 32 q^{3} - 156 q^{5} - 264 q^{7} + 2200 q^{11} + 858 q^{13} - 7768 q^{15} - 3278 q^{17} + 33176 q^{21} + 19984 q^{23} - 24174 q^{25} - 115528 q^{27} + 104976 q^{31} + 177320 q^{33} - 116072 q^{35} - 241554 q^{37} + 351736 q^{41} + 60720 q^{43} - 287846 q^{45} - 355248 q^{47} + 641872 q^{51} + 346526 q^{53} - 310200 q^{55} - 112816 q^{57} - 492888 q^{61} + 2288 q^{63} - 5082 q^{65} - 230304 q^{67} + 174128 q^{71} - 332442 q^{73} + 1855048 q^{75} + 1618760 q^{77} - 3085166 q^{81} - 2190936 q^{83} - 164934 q^{85} + 2614304 q^{87} - 2186976 q^{91} + 242072 q^{93} + 3484184 q^{95} + 3338406 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/20\mathbb{Z}\right)^\times\).

\(n\) \(11\) \(17\)
\(\chi(n)\) \(1\) \(e\left(\frac{3}{4}\right)\)

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
\(3\) −16.7249 16.7249i −0.619442 0.619442i 0.325946 0.945388i \(-0.394317\pi\)
−0.945388 + 0.325946i \(0.894317\pi\)
\(4\) 0 0
\(5\) −22.1946 + 123.014i −0.177557 + 0.984111i
\(6\) 0 0
\(7\) −422.022 + 422.022i −1.23038 + 1.23038i −0.266568 + 0.963816i \(0.585890\pi\)
−0.963816 + 0.266568i \(0.914110\pi\)
\(8\) 0 0
\(9\) 169.553i 0.232583i
\(10\) 0 0
\(11\) −1359.94 −1.02174 −0.510872 0.859657i \(-0.670677\pi\)
−0.510872 + 0.859657i \(0.670677\pi\)
\(12\) 0 0
\(13\) 1248.94 + 1248.94i 0.568477 + 0.568477i 0.931702 0.363224i \(-0.118324\pi\)
−0.363224 + 0.931702i \(0.618324\pi\)
\(14\) 0 0
\(15\) 2428.60 1686.20i 0.719586 0.499614i
\(16\) 0 0
\(17\) −648.372 + 648.372i −0.131971 + 0.131971i −0.770007 0.638036i \(-0.779747\pi\)
0.638036 + 0.770007i \(0.279747\pi\)
\(18\) 0 0
\(19\) 9632.63i 1.40438i −0.711990 0.702189i \(-0.752206\pi\)
0.711990 0.702189i \(-0.247794\pi\)
\(20\) 0 0
\(21\) 14116.6 1.52430
\(22\) 0 0
\(23\) 886.911 + 886.911i 0.0728948 + 0.0728948i 0.742614 0.669719i \(-0.233586\pi\)
−0.669719 + 0.742614i \(0.733586\pi\)
\(24\) 0 0
\(25\) −14639.8 5460.48i −0.936947 0.349471i
\(26\) 0 0
\(27\) −15028.2 + 15028.2i −0.763514 + 0.763514i
\(28\) 0 0
\(29\) 26789.3i 1.09842i 0.835685 + 0.549209i \(0.185071\pi\)
−0.835685 + 0.549209i \(0.814929\pi\)
\(30\) 0 0
\(31\) −6880.94 −0.230974 −0.115487 0.993309i \(-0.536843\pi\)
−0.115487 + 0.993309i \(0.536843\pi\)
\(32\) 0 0
\(33\) 22744.9 + 22744.9i 0.632911 + 0.632911i
\(34\) 0 0
\(35\) −42547.9 61281.1i −0.992371 1.42930i
\(36\) 0 0
\(37\) −42367.5 + 42367.5i −0.836426 + 0.836426i −0.988387 0.151960i \(-0.951441\pi\)
0.151960 + 0.988387i \(0.451441\pi\)
\(38\) 0 0
\(39\) 41777.0i 0.704278i
\(40\) 0 0
\(41\) 132500. 1.92250 0.961248 0.275684i \(-0.0889042\pi\)
0.961248 + 0.275684i \(0.0889042\pi\)
\(42\) 0 0
\(43\) 20888.6 + 20888.6i 0.262727 + 0.262727i 0.826161 0.563434i \(-0.190520\pi\)
−0.563434 + 0.826161i \(0.690520\pi\)
\(44\) 0 0
\(45\) 20857.4 + 3763.16i 0.228887 + 0.0412966i
\(46\) 0 0
\(47\) −31606.0 + 31606.0i −0.304422 + 0.304422i −0.842741 0.538319i \(-0.819059\pi\)
0.538319 + 0.842741i \(0.319059\pi\)
\(48\) 0 0
\(49\) 238556.i 2.02769i
\(50\) 0 0
\(51\) 21688.0 0.163496
\(52\) 0 0
\(53\) 80106.6 + 80106.6i 0.538073 + 0.538073i 0.922962 0.384890i \(-0.125761\pi\)
−0.384890 + 0.922962i \(0.625761\pi\)
\(54\) 0 0
\(55\) 30183.3 167291.i 0.181417 1.00551i
\(56\) 0 0
\(57\) −161105. + 161105.i −0.869931 + 0.869931i
\(58\) 0 0
\(59\) 290136.i 1.41268i 0.707871 + 0.706342i \(0.249656\pi\)
−0.707871 + 0.706342i \(0.750344\pi\)
\(60\) 0 0
\(61\) −253971. −1.11891 −0.559454 0.828862i \(-0.688989\pi\)
−0.559454 + 0.828862i \(0.688989\pi\)
\(62\) 0 0
\(63\) 71555.1 + 71555.1i 0.286166 + 0.286166i
\(64\) 0 0
\(65\) −181357. + 125918.i −0.660381 + 0.458508i
\(66\) 0 0
\(67\) 11368.7 11368.7i 0.0377997 0.0377997i −0.687954 0.725754i \(-0.741491\pi\)
0.725754 + 0.687954i \(0.241491\pi\)
\(68\) 0 0
\(69\) 29667.0i 0.0903082i
\(70\) 0 0
\(71\) 179758. 0.502243 0.251122 0.967956i \(-0.419201\pi\)
0.251122 + 0.967956i \(0.419201\pi\)
\(72\) 0 0
\(73\) −414484. 414484.i −1.06547 1.06547i −0.997701 0.0677642i \(-0.978413\pi\)
−0.0677642 0.997701i \(-0.521587\pi\)
\(74\) 0 0
\(75\) 153524. + 336176.i 0.363908 + 0.796861i
\(76\) 0 0
\(77\) 573924. 573924.i 1.25714 1.25714i
\(78\) 0 0
\(79\) 145313.i 0.294729i 0.989082 + 0.147364i \(0.0470790\pi\)
−0.989082 + 0.147364i \(0.952921\pi\)
\(80\) 0 0
\(81\) 379089. 0.713322
\(82\) 0 0
\(83\) −63129.1 63129.1i −0.110407 0.110407i 0.649745 0.760152i \(-0.274876\pi\)
−0.760152 + 0.649745i \(0.774876\pi\)
\(84\) 0 0
\(85\) −65368.3 94149.0i −0.106441 0.153306i
\(86\) 0 0
\(87\) 448049. 448049.i 0.680406 0.680406i
\(88\) 0 0
\(89\) 333388.i 0.472912i 0.971642 + 0.236456i \(0.0759859\pi\)
−0.971642 + 0.236456i \(0.924014\pi\)
\(90\) 0 0
\(91\) −1.05416e6 −1.39889
\(92\) 0 0
\(93\) 115083. + 115083.i 0.143075 + 0.143075i
\(94\) 0 0
\(95\) 1.18495e6 + 213792.i 1.38206 + 0.249357i
\(96\) 0 0
\(97\) 519696. 519696.i 0.569422 0.569422i −0.362545 0.931966i \(-0.618092\pi\)
0.931966 + 0.362545i \(0.118092\pi\)
\(98\) 0 0
\(99\) 230582.i 0.237640i
\(100\) 0 0
\(101\) −1.97076e6 −1.91280 −0.956399 0.292065i \(-0.905658\pi\)
−0.956399 + 0.292065i \(0.905658\pi\)
\(102\) 0 0
\(103\) −862587. 862587.i −0.789389 0.789389i 0.192005 0.981394i \(-0.438501\pi\)
−0.981394 + 0.192005i \(0.938501\pi\)
\(104\) 0 0
\(105\) −313311. + 1.73653e6i −0.270650 + 1.50008i
\(106\) 0 0
\(107\) −1.28226e6 + 1.28226e6i −1.04671 + 1.04671i −0.0478536 + 0.998854i \(0.515238\pi\)
−0.998854 + 0.0478536i \(0.984762\pi\)
\(108\) 0 0
\(109\) 186075.i 0.143684i −0.997416 0.0718419i \(-0.977112\pi\)
0.997416 0.0718419i \(-0.0228877\pi\)
\(110\) 0 0
\(111\) 1.41719e6 1.03624
\(112\) 0 0
\(113\) 1.01659e6 + 1.01659e6i 0.704545 + 0.704545i 0.965383 0.260838i \(-0.0839987\pi\)
−0.260838 + 0.965383i \(0.583999\pi\)
\(114\) 0 0
\(115\) −128787. + 89417.7i −0.0846795 + 0.0587936i
\(116\) 0 0
\(117\) 211762. 211762.i 0.132218 0.132218i
\(118\) 0 0
\(119\) 547254.i 0.324749i
\(120\) 0 0
\(121\) 77877.5 0.0439598
\(122\) 0 0
\(123\) −2.21606e6 2.21606e6i −1.19088 1.19088i
\(124\) 0 0
\(125\) 996638. 1.67971e6i 0.510279 0.860009i
\(126\) 0 0
\(127\) 1.52424e6 1.52424e6i 0.744120 0.744120i −0.229248 0.973368i \(-0.573627\pi\)
0.973368 + 0.229248i \(0.0736266\pi\)
\(128\) 0 0
\(129\) 698721.i 0.325488i
\(130\) 0 0
\(131\) 1.10323e6 0.490739 0.245370 0.969430i \(-0.421091\pi\)
0.245370 + 0.969430i \(0.421091\pi\)
\(132\) 0 0
\(133\) 4.06518e6 + 4.06518e6i 1.72792 + 1.72792i
\(134\) 0 0
\(135\) −1.51514e6 2.18223e6i −0.615815 0.886949i
\(136\) 0 0
\(137\) 720618. 720618.i 0.280248 0.280248i −0.552960 0.833208i \(-0.686502\pi\)
0.833208 + 0.552960i \(0.186502\pi\)
\(138\) 0 0
\(139\) 2.89720e6i 1.07878i 0.842055 + 0.539391i \(0.181346\pi\)
−0.842055 + 0.539391i \(0.818654\pi\)
\(140\) 0 0
\(141\) 1.05722e6 0.377143
\(142\) 0 0
\(143\) −1.69849e6 1.69849e6i −0.580838 0.580838i
\(144\) 0 0
\(145\) −3.29545e6 594577.i −1.08096 0.195031i
\(146\) 0 0
\(147\) −3.98983e6 + 3.98983e6i −1.25604 + 1.25604i
\(148\) 0 0
\(149\) 1.98382e6i 0.599714i −0.953984 0.299857i \(-0.903061\pi\)
0.953984 0.299857i \(-0.0969389\pi\)
\(150\) 0 0
\(151\) 4.02159e6 1.16807 0.584033 0.811730i \(-0.301474\pi\)
0.584033 + 0.811730i \(0.301474\pi\)
\(152\) 0 0
\(153\) 109933. + 109933.i 0.0306941 + 0.0306941i
\(154\) 0 0
\(155\) 152719. 846450.i 0.0410109 0.227304i
\(156\) 0 0
\(157\) −3.96237e6 + 3.96237e6i −1.02390 + 1.02390i −0.0241894 + 0.999707i \(0.507700\pi\)
−0.999707 + 0.0241894i \(0.992300\pi\)
\(158\) 0 0
\(159\) 2.67956e6i 0.666610i
\(160\) 0 0
\(161\) −748591. −0.179377
\(162\) 0 0
\(163\) 2.80620e6 + 2.80620e6i 0.647971 + 0.647971i 0.952502 0.304531i \(-0.0984998\pi\)
−0.304531 + 0.952502i \(0.598500\pi\)
\(164\) 0 0
\(165\) −3.30275e6 + 2.29313e6i −0.735232 + 0.510477i
\(166\) 0 0
\(167\) −4.05536e6 + 4.05536e6i −0.870724 + 0.870724i −0.992551 0.121827i \(-0.961125\pi\)
0.121827 + 0.992551i \(0.461125\pi\)
\(168\) 0 0
\(169\) 1.70708e6i 0.353667i
\(170\) 0 0
\(171\) −1.63324e6 −0.326635
\(172\) 0 0
\(173\) −593924. 593924.i −0.114708 0.114708i 0.647423 0.762131i \(-0.275847\pi\)
−0.762131 + 0.647423i \(0.775847\pi\)
\(174\) 0 0
\(175\) 8.48275e6 3.87387e6i 1.58279 0.722822i
\(176\) 0 0
\(177\) 4.85250e6 4.85250e6i 0.875076 0.875076i
\(178\) 0 0
\(179\) 3.48674e6i 0.607939i 0.952682 + 0.303970i \(0.0983122\pi\)
−0.952682 + 0.303970i \(0.901688\pi\)
\(180\) 0 0
\(181\) −1.41422e6 −0.238496 −0.119248 0.992864i \(-0.538048\pi\)
−0.119248 + 0.992864i \(0.538048\pi\)
\(182\) 0 0
\(183\) 4.24764e6 + 4.24764e6i 0.693098 + 0.693098i
\(184\) 0 0
\(185\) −4.27146e6 6.15212e6i −0.674623 0.971649i
\(186\) 0 0
\(187\) 881747. 881747.i 0.134840 0.134840i
\(188\) 0 0
\(189\) 1.26845e7i 1.87883i
\(190\) 0 0
\(191\) 4.51204e6 0.647549 0.323775 0.946134i \(-0.395048\pi\)
0.323775 + 0.946134i \(0.395048\pi\)
\(192\) 0 0
\(193\) 1.70789e6 + 1.70789e6i 0.237569 + 0.237569i 0.815843 0.578274i \(-0.196274\pi\)
−0.578274 + 0.815843i \(0.696274\pi\)
\(194\) 0 0
\(195\) 5.13915e6 + 927223.i 0.693087 + 0.125049i
\(196\) 0 0
\(197\) 2.88696e6 2.88696e6i 0.377608 0.377608i −0.492630 0.870239i \(-0.663964\pi\)
0.870239 + 0.492630i \(0.163964\pi\)
\(198\) 0 0
\(199\) 4.21379e6i 0.534704i 0.963599 + 0.267352i \(0.0861487\pi\)
−0.963599 + 0.267352i \(0.913851\pi\)
\(200\) 0 0
\(201\) −380283. −0.0468294
\(202\) 0 0
\(203\) −1.13057e7 1.13057e7i −1.35148 1.35148i
\(204\) 0 0
\(205\) −2.94079e6 + 1.62994e7i −0.341352 + 1.89195i
\(206\) 0 0
\(207\) 150378. 150378.i 0.0169541 0.0169541i
\(208\) 0 0
\(209\) 1.30998e7i 1.43491i
\(210\) 0 0
\(211\) −3.89683e6 −0.414825 −0.207412 0.978254i \(-0.566504\pi\)
−0.207412 + 0.978254i \(0.566504\pi\)
\(212\) 0 0
\(213\) −3.00645e6 3.00645e6i −0.311111 0.311111i
\(214\) 0 0
\(215\) −3.03320e6 + 2.10597e6i −0.305201 + 0.211903i
\(216\) 0 0
\(217\) 2.90391e6 2.90391e6i 0.284186 0.284186i
\(218\) 0 0
\(219\) 1.38644e7i 1.31999i
\(220\) 0 0
\(221\) −1.61956e6 −0.150045
\(222\) 0 0
\(223\) −3.41658e6 3.41658e6i −0.308090 0.308090i 0.536078 0.844168i \(-0.319905\pi\)
−0.844168 + 0.536078i \(0.819905\pi\)
\(224\) 0 0
\(225\) −925841. + 2.48222e6i −0.0812809 + 0.217918i
\(226\) 0 0
\(227\) 7.67846e6 7.67846e6i 0.656442 0.656442i −0.298094 0.954537i \(-0.596351\pi\)
0.954537 + 0.298094i \(0.0963509\pi\)
\(228\) 0 0
\(229\) 7.64574e6i 0.636668i −0.947979 0.318334i \(-0.896877\pi\)
0.947979 0.318334i \(-0.103123\pi\)
\(230\) 0 0
\(231\) −1.91977e7 −1.55745
\(232\) 0 0
\(233\) 1.32288e6 + 1.32288e6i 0.104581 + 0.104581i 0.757461 0.652880i \(-0.226440\pi\)
−0.652880 + 0.757461i \(0.726440\pi\)
\(234\) 0 0
\(235\) −3.18649e6 4.58946e6i −0.245533 0.353637i
\(236\) 0 0
\(237\) 2.43035e6 2.43035e6i 0.182568 0.182568i
\(238\) 0 0
\(239\) 1.04557e7i 0.765875i 0.923774 + 0.382937i \(0.125087\pi\)
−0.923774 + 0.382937i \(0.874913\pi\)
\(240\) 0 0
\(241\) −7.42256e6 −0.530277 −0.265138 0.964210i \(-0.585418\pi\)
−0.265138 + 0.964210i \(0.585418\pi\)
\(242\) 0 0
\(243\) 4.61536e6 + 4.61536e6i 0.321652 + 0.321652i
\(244\) 0 0
\(245\) 2.93456e7 + 5.29464e6i 1.99547 + 0.360030i
\(246\) 0 0
\(247\) 1.20306e7 1.20306e7i 0.798357 0.798357i
\(248\) 0 0
\(249\) 2.11166e6i 0.136781i
\(250\) 0 0
\(251\) 1.25795e7 0.795505 0.397752 0.917493i \(-0.369790\pi\)
0.397752 + 0.917493i \(0.369790\pi\)
\(252\) 0 0
\(253\) −1.20615e6 1.20615e6i −0.0744798 0.0744798i
\(254\) 0 0
\(255\) −481355. + 2.66792e6i −0.0290298 + 0.160898i
\(256\) 0 0
\(257\) −3.35530e6 + 3.35530e6i −0.197666 + 0.197666i −0.798999 0.601333i \(-0.794637\pi\)
0.601333 + 0.798999i \(0.294637\pi\)
\(258\) 0 0
\(259\) 3.57600e7i 2.05825i
\(260\) 0 0
\(261\) 4.54221e6 0.255473
\(262\) 0 0
\(263\) −1.97354e7 1.97354e7i −1.08487 1.08487i −0.996047 0.0888280i \(-0.971688\pi\)
−0.0888280 0.996047i \(-0.528312\pi\)
\(264\) 0 0
\(265\) −1.16322e7 + 8.07629e6i −0.625061 + 0.433985i
\(266\) 0 0
\(267\) 5.57590e6 5.57590e6i 0.292942 0.292942i
\(268\) 0 0
\(269\) 3.82612e6i 0.196563i −0.995159 0.0982816i \(-0.968665\pi\)
0.995159 0.0982816i \(-0.0313346\pi\)
\(270\) 0 0
\(271\) 238579. 0.0119874 0.00599368 0.999982i \(-0.498092\pi\)
0.00599368 + 0.999982i \(0.498092\pi\)
\(272\) 0 0
\(273\) 1.76308e7 + 1.76308e7i 0.866532 + 0.866532i
\(274\) 0 0
\(275\) 1.99093e7 + 7.42593e6i 0.957320 + 0.357069i
\(276\) 0 0
\(277\) −8.83629e6 + 8.83629e6i −0.415749 + 0.415749i −0.883735 0.467987i \(-0.844979\pi\)
0.467987 + 0.883735i \(0.344979\pi\)
\(278\) 0 0
\(279\) 1.16668e6i 0.0537206i
\(280\) 0 0
\(281\) 9.44266e6 0.425574 0.212787 0.977099i \(-0.431746\pi\)
0.212787 + 0.977099i \(0.431746\pi\)
\(282\) 0 0
\(283\) 3.88228e6 + 3.88228e6i 0.171288 + 0.171288i 0.787545 0.616257i \(-0.211352\pi\)
−0.616257 + 0.787545i \(0.711352\pi\)
\(284\) 0 0
\(285\) −1.62425e7 2.33938e7i −0.701646 1.01057i
\(286\) 0 0
\(287\) −5.59180e7 + 5.59180e7i −2.36541 + 2.36541i
\(288\) 0 0
\(289\) 2.32968e7i 0.965168i
\(290\) 0 0
\(291\) −1.73838e7 −0.705448
\(292\) 0 0
\(293\) 4.41027e6 + 4.41027e6i 0.175332 + 0.175332i 0.789318 0.613985i \(-0.210435\pi\)
−0.613985 + 0.789318i \(0.710435\pi\)
\(294\) 0 0
\(295\) −3.56907e7 6.43944e6i −1.39024 0.250831i
\(296\) 0 0
\(297\) 2.04375e7 2.04375e7i 0.780115 0.780115i
\(298\) 0 0
\(299\) 2.21540e6i 0.0828780i
\(300\) 0 0
\(301\) −1.76309e7 −0.646509
\(302\) 0 0
\(303\) 3.29608e7 + 3.29608e7i 1.18487 + 1.18487i
\(304\) 0 0
\(305\) 5.63677e6 3.12419e7i 0.198669 1.10113i
\(306\) 0 0
\(307\) 2.02549e7 2.02549e7i 0.700026 0.700026i −0.264390 0.964416i \(-0.585170\pi\)
0.964416 + 0.264390i \(0.0851704\pi\)
\(308\) 0 0
\(309\) 2.88534e7i 0.977962i
\(310\) 0 0
\(311\) 4.11216e7 1.36707 0.683533 0.729920i \(-0.260443\pi\)
0.683533 + 0.729920i \(0.260443\pi\)
\(312\) 0 0
\(313\) −3.75640e7 3.75640e7i −1.22501 1.22501i −0.965830 0.259178i \(-0.916548\pi\)
−0.259178 0.965830i \(-0.583452\pi\)
\(314\) 0 0
\(315\) −1.03904e7 + 7.21413e6i −0.332430 + 0.230809i
\(316\) 0 0
\(317\) −2.01898e7 + 2.01898e7i −0.633803 + 0.633803i −0.949020 0.315217i \(-0.897923\pi\)
0.315217 + 0.949020i \(0.397923\pi\)
\(318\) 0 0
\(319\) 3.64319e7i 1.12230i
\(320\) 0 0
\(321\) 4.28915e7 1.29675
\(322\) 0 0
\(323\) 6.24553e6 + 6.24553e6i 0.185337 + 0.185337i
\(324\) 0 0
\(325\) −1.14645e7 2.51041e7i −0.333967 0.731299i
\(326\) 0 0
\(327\) −3.11209e6 + 3.11209e6i −0.0890038 + 0.0890038i
\(328\) 0 0
\(329\) 2.66768e7i 0.749112i
\(330\) 0 0
\(331\) 5.35648e7 1.47705 0.738526 0.674225i \(-0.235522\pi\)
0.738526 + 0.674225i \(0.235522\pi\)
\(332\) 0 0
\(333\) 7.18354e6 + 7.18354e6i 0.194539 + 0.194539i
\(334\) 0 0
\(335\) 1.14619e6 + 1.65084e6i 0.0304875 + 0.0439107i
\(336\) 0 0
\(337\) −3.35730e7 + 3.35730e7i −0.877203 + 0.877203i −0.993244 0.116041i \(-0.962980\pi\)
0.116041 + 0.993244i \(0.462980\pi\)
\(338\) 0 0
\(339\) 3.40047e7i 0.872850i
\(340\) 0 0
\(341\) 9.35767e6 0.235996
\(342\) 0 0
\(343\) 5.10252e7 + 5.10252e7i 1.26445 + 1.26445i
\(344\) 0 0
\(345\) 3.64946e6 + 658447.i 0.0888732 + 0.0160348i
\(346\) 0 0
\(347\) 3.02313e7 3.02313e7i 0.723550 0.723550i −0.245776 0.969327i \(-0.579043\pi\)
0.969327 + 0.245776i \(0.0790428\pi\)
\(348\) 0 0
\(349\) 2.12050e7i 0.498842i −0.968395 0.249421i \(-0.919760\pi\)
0.968395 0.249421i \(-0.0802402\pi\)
\(350\) 0 0
\(351\) −3.75389e7 −0.868080
\(352\) 0 0
\(353\) −5.10632e7 5.10632e7i −1.16087 1.16087i −0.984285 0.176586i \(-0.943495\pi\)
−0.176586 0.984285i \(-0.556505\pi\)
\(354\) 0 0
\(355\) −3.98966e6 + 2.21128e7i −0.0891766 + 0.494263i
\(356\) 0 0
\(357\) −9.15279e6 + 9.15279e6i −0.201163 + 0.201163i
\(358\) 0 0
\(359\) 6.68160e7i 1.44410i 0.691841 + 0.722050i \(0.256800\pi\)
−0.691841 + 0.722050i \(0.743200\pi\)
\(360\) 0 0
\(361\) −4.57417e7 −0.972279
\(362\) 0 0
\(363\) −1.30250e6 1.30250e6i −0.0272306 0.0272306i
\(364\) 0 0
\(365\) 6.01866e7 4.17880e7i 1.23772 0.859356i
\(366\) 0 0
\(367\) −5.04382e6 + 5.04382e6i −0.102038 + 0.102038i −0.756283 0.654245i \(-0.772987\pi\)
0.654245 + 0.756283i \(0.272987\pi\)
\(368\) 0 0
\(369\) 2.24658e7i 0.447140i
\(370\) 0 0
\(371\) −6.76135e7 −1.32407
\(372\) 0 0
\(373\) 5.06926e7 + 5.06926e7i 0.976828 + 0.976828i 0.999738 0.0229094i \(-0.00729292\pi\)
−0.0229094 + 0.999738i \(0.507293\pi\)
\(374\) 0 0
\(375\) −4.47617e7 + 1.14242e7i −0.848814 + 0.216638i
\(376\) 0 0
\(377\) −3.34584e7 + 3.34584e7i −0.624425 + 0.624425i
\(378\) 0 0
\(379\) 2.97376e7i 0.546245i 0.961979 + 0.273123i \(0.0880565\pi\)
−0.961979 + 0.273123i \(0.911944\pi\)
\(380\) 0 0
\(381\) −5.09857e7 −0.921879
\(382\) 0 0
\(383\) −2.82121e7 2.82121e7i −0.502157 0.502157i 0.409951 0.912108i \(-0.365546\pi\)
−0.912108 + 0.409951i \(0.865546\pi\)
\(384\) 0 0
\(385\) 5.78626e7 + 8.33387e7i 1.01395 + 1.46037i
\(386\) 0 0
\(387\) 3.54173e6 3.54173e6i 0.0611058 0.0611058i
\(388\) 0 0
\(389\) 8.32404e7i 1.41412i 0.707155 + 0.707059i \(0.249978\pi\)
−0.707155 + 0.707059i \(0.750022\pi\)
\(390\) 0 0
\(391\) −1.15010e6 −0.0192399
\(392\) 0 0
\(393\) −1.84514e7 1.84514e7i −0.303984 0.303984i
\(394\) 0 0
\(395\) −1.78755e7 3.22516e6i −0.290046 0.0523311i
\(396\) 0 0
\(397\) 4.98898e7 4.98898e7i 0.797333 0.797333i −0.185341 0.982674i \(-0.559339\pi\)
0.982674 + 0.185341i \(0.0593390\pi\)
\(398\) 0 0
\(399\) 1.35980e8i 2.14070i
\(400\) 0 0
\(401\) 1.12175e8 1.73966 0.869829 0.493353i \(-0.164229\pi\)
0.869829 + 0.493353i \(0.164229\pi\)
\(402\) 0 0
\(403\) −8.59391e6 8.59391e6i −0.131303 0.131303i
\(404\) 0 0
\(405\) −8.41371e6 + 4.66331e7i −0.126655 + 0.701988i
\(406\) 0 0
\(407\) 5.76173e7 5.76173e7i 0.854613 0.854613i
\(408\) 0 0
\(409\) 1.66273e7i 0.243025i 0.992590 + 0.121513i \(0.0387745\pi\)
−0.992590 + 0.121513i \(0.961226\pi\)
\(410\) 0 0
\(411\) −2.41046e7 −0.347195
\(412\) 0 0
\(413\) −1.22444e8 1.22444e8i −1.73814 1.73814i
\(414\) 0 0
\(415\) 9.16688e6 6.36463e6i 0.128256 0.0890490i
\(416\) 0 0
\(417\) 4.84555e7 4.84555e7i 0.668244 0.668244i
\(418\) 0 0
\(419\) 7.77132e7i 1.05646i 0.849101 + 0.528230i \(0.177144\pi\)
−0.849101 + 0.528230i \(0.822856\pi\)
\(420\) 0 0
\(421\) 4.97347e7 0.666520 0.333260 0.942835i \(-0.391851\pi\)
0.333260 + 0.942835i \(0.391851\pi\)
\(422\) 0 0
\(423\) 5.35889e6 + 5.35889e6i 0.0708034 + 0.0708034i
\(424\) 0 0
\(425\) 1.30325e7 5.95161e6i 0.169769 0.0775297i
\(426\) 0 0
\(427\) 1.07181e8 1.07181e8i 1.37669 1.37669i
\(428\) 0 0
\(429\) 5.68143e7i 0.719591i
\(430\) 0 0
\(431\) −6.70972e7 −0.838056 −0.419028 0.907973i \(-0.637629\pi\)
−0.419028 + 0.907973i \(0.637629\pi\)
\(432\) 0 0
\(433\) 9.05301e7 + 9.05301e7i 1.11514 + 1.11514i 0.992444 + 0.122696i \(0.0391539\pi\)
0.122696 + 0.992444i \(0.460846\pi\)
\(434\) 0 0
\(435\) 4.51720e7 + 6.50605e7i 0.548784 + 0.790405i
\(436\) 0 0
\(437\) 8.54328e6 8.54328e6i 0.102372 0.102372i
\(438\) 0 0
\(439\) 231381.i 0.00273485i −0.999999 0.00136743i \(-0.999565\pi\)
0.999999 0.00136743i \(-0.000435265\pi\)
\(440\) 0 0
\(441\) −4.04478e7 −0.471606
\(442\) 0 0
\(443\) −2.08251e7 2.08251e7i −0.239538 0.239538i 0.577121 0.816659i \(-0.304176\pi\)
−0.816659 + 0.577121i \(0.804176\pi\)
\(444\) 0 0
\(445\) −4.10114e7 7.39941e6i −0.465398 0.0839686i
\(446\) 0 0
\(447\) −3.31793e7 + 3.31793e7i −0.371488 + 0.371488i
\(448\) 0 0
\(449\) 6.49546e7i 0.717581i 0.933418 + 0.358790i \(0.116811\pi\)
−0.933418 + 0.358790i \(0.883189\pi\)
\(450\) 0 0
\(451\) −1.80193e8 −1.96430
\(452\) 0 0
\(453\) −6.72609e7 6.72609e7i −0.723549 0.723549i
\(454\) 0 0
\(455\) 2.33967e7 1.29677e8i 0.248382 1.37666i
\(456\) 0 0
\(457\) −6.96439e7 + 6.96439e7i −0.729684 + 0.729684i −0.970557 0.240873i \(-0.922566\pi\)
0.240873 + 0.970557i \(0.422566\pi\)
\(458\) 0 0
\(459\) 1.94878e7i 0.201523i
\(460\) 0 0
\(461\) 8.07689e7 0.824407 0.412203 0.911092i \(-0.364759\pi\)
0.412203 + 0.911092i \(0.364759\pi\)
\(462\) 0 0
\(463\) 8.35152e7 + 8.35152e7i 0.841439 + 0.841439i 0.989046 0.147607i \(-0.0471572\pi\)
−0.147607 + 0.989046i \(0.547157\pi\)
\(464\) 0 0
\(465\) −1.67111e7 + 1.16026e7i −0.166205 + 0.115398i
\(466\) 0 0
\(467\) −8.17085e7 + 8.17085e7i −0.802262 + 0.802262i −0.983449 0.181186i \(-0.942006\pi\)
0.181186 + 0.983449i \(0.442006\pi\)
\(468\) 0 0
\(469\) 9.59572e6i 0.0930163i
\(470\) 0 0
\(471\) 1.32541e8 1.26849
\(472\) 0 0
\(473\) −2.84073e7 2.84073e7i −0.268439 0.268439i
\(474\) 0 0
\(475\) −5.25988e7 + 1.41020e8i −0.490789 + 1.31583i
\(476\) 0 0
\(477\) 1.35823e7 1.35823e7i 0.125147 0.125147i
\(478\) 0 0
\(479\) 1.07532e8i 0.978433i −0.872162 0.489216i \(-0.837283\pi\)
0.872162 0.489216i \(-0.162717\pi\)
\(480\) 0 0
\(481\) −1.05829e8 −0.950979
\(482\) 0 0
\(483\) 1.25201e7 + 1.25201e7i 0.111114 + 0.111114i
\(484\) 0 0
\(485\) 5.23953e7 + 7.54642e7i 0.459269 + 0.661478i
\(486\) 0 0
\(487\) −4.33036e7 + 4.33036e7i −0.374919 + 0.374919i −0.869265 0.494346i \(-0.835408\pi\)
0.494346 + 0.869265i \(0.335408\pi\)
\(488\) 0 0
\(489\) 9.38670e7i 0.802761i
\(490\) 0 0
\(491\) −6.23191e7 −0.526474 −0.263237 0.964731i \(-0.584790\pi\)
−0.263237 + 0.964731i \(0.584790\pi\)
\(492\) 0 0
\(493\) −1.73694e7 1.73694e7i −0.144959 0.144959i
\(494\) 0 0
\(495\) −2.83648e7 5.11767e6i −0.233864 0.0421946i
\(496\) 0 0
\(497\) −7.58619e7 + 7.58619e7i −0.617952 + 0.617952i
\(498\) 0 0
\(499\) 1.67847e6i 0.0135086i 0.999977 + 0.00675432i \(0.00214998\pi\)
−0.999977 + 0.00675432i \(0.997850\pi\)
\(500\) 0 0
\(501\) 1.35651e8 1.07873
\(502\) 0 0
\(503\) 5.94281e7 + 5.94281e7i 0.466969 + 0.466969i 0.900931 0.433962i \(-0.142885\pi\)
−0.433962 + 0.900931i \(0.642885\pi\)
\(504\) 0 0
\(505\) 4.37401e7 2.42430e8i 0.339630 1.88240i
\(506\) 0 0
\(507\) −2.85509e7 + 2.85509e7i −0.219076 + 0.219076i
\(508\) 0 0
\(509\) 1.54356e8i 1.17049i −0.810855 0.585247i \(-0.800997\pi\)
0.810855 0.585247i \(-0.199003\pi\)
\(510\) 0 0
\(511\) 3.49843e8 2.62186
\(512\) 0 0
\(513\) 1.44762e8 + 1.44762e8i 1.07226 + 1.07226i
\(514\) 0 0
\(515\) 1.25255e8 8.69654e7i 0.917008 0.636685i
\(516\) 0 0
\(517\) 4.29823e7 4.29823e7i 0.311041 0.311041i
\(518\) 0 0
\(519\) 1.98667e7i 0.142110i
\(520\) 0 0
\(521\) −1.71124e8 −1.21004 −0.605019 0.796211i \(-0.706834\pi\)
−0.605019 + 0.796211i \(0.706834\pi\)
\(522\) 0 0
\(523\) −1.28297e8 1.28297e8i −0.896833 0.896833i 0.0983219 0.995155i \(-0.468653\pi\)
−0.995155 + 0.0983219i \(0.968653\pi\)
\(524\) 0 0
\(525\) −2.06664e8 7.70832e7i −1.42819 0.532699i
\(526\) 0 0
\(527\) 4.46141e6 4.46141e6i 0.0304817 0.0304817i
\(528\) 0 0
\(529\) 1.46463e8i 0.989373i
\(530\) 0 0
\(531\) 4.91934e7 0.328566
\(532\) 0 0
\(533\) 1.65486e8 + 1.65486e8i 1.09290 + 1.09290i
\(534\) 0 0
\(535\) −1.29277e8 1.86195e8i −0.844226 1.21593i
\(536\) 0 0
\(537\) 5.83155e7 5.83155e7i 0.376583 0.376583i
\(538\) 0 0
\(539\) 3.24422e8i 2.07178i
\(540\) 0 0
\(541\) −1.12555e8 −0.710841 −0.355420 0.934707i \(-0.615662\pi\)
−0.355420 + 0.934707i \(0.615662\pi\)
\(542\) 0 0
\(543\) 2.36528e7 + 2.36528e7i 0.147735 + 0.147735i
\(544\) 0 0
\(545\) 2.28898e7 + 4.12985e6i 0.141401 + 0.0255120i
\(546\) 0 0
\(547\) −3.86772e7 + 3.86772e7i −0.236316 + 0.236316i −0.815323 0.579007i \(-0.803440\pi\)
0.579007 + 0.815323i \(0.303440\pi\)
\(548\) 0 0
\(549\) 4.30615e7i 0.260239i
\(550\) 0 0
\(551\) 2.58051e8 1.54259
\(552\) 0 0
\(553\) −6.13252e7 6.13252e7i −0.362630 0.362630i
\(554\) 0 0
\(555\) −3.14539e7 + 1.74334e8i −0.183990 + 1.01977i
\(556\) 0 0
\(557\) −1.77413e8 + 1.77413e8i −1.02664 + 1.02664i −0.0270085 + 0.999635i \(0.508598\pi\)
−0.999635 + 0.0270085i \(0.991402\pi\)
\(558\) 0 0
\(559\) 5.21774e7i 0.298708i
\(560\) 0 0
\(561\) −2.94943e7 −0.167051
\(562\) 0 0
\(563\) −2.61909e7 2.61909e7i −0.146766 0.146766i 0.629906 0.776672i \(-0.283094\pi\)
−0.776672 + 0.629906i \(0.783094\pi\)
\(564\) 0 0
\(565\) −1.47617e8 + 1.02491e8i −0.818447 + 0.568254i
\(566\) 0 0
\(567\) −1.59984e8 + 1.59984e8i −0.877660 + 0.877660i
\(568\) 0 0
\(569\) 9.04667e7i 0.491080i −0.969387 0.245540i \(-0.921035\pi\)
0.969387 0.245540i \(-0.0789652\pi\)
\(570\) 0 0
\(571\) −3.72767e7 −0.200230 −0.100115 0.994976i \(-0.531921\pi\)
−0.100115 + 0.994976i \(0.531921\pi\)
\(572\) 0 0
\(573\) −7.54636e7 7.54636e7i −0.401119 0.401119i
\(574\) 0 0
\(575\) −8.14124e6 1.78272e7i −0.0428240 0.0937731i
\(576\) 0 0
\(577\) 6.48027e7 6.48027e7i 0.337339 0.337339i −0.518026 0.855365i \(-0.673333\pi\)
0.855365 + 0.518026i \(0.173333\pi\)
\(578\) 0 0
\(579\) 5.71288e7i 0.294320i
\(580\) 0 0
\(581\) 5.32837e7 0.271685
\(582\) 0 0
\(583\) −1.08940e8 1.08940e8i −0.549772 0.549772i
\(584\) 0 0
\(585\) 2.13497e7 + 3.07497e7i 0.106641 + 0.153593i
\(586\) 0 0
\(587\) 1.02171e8 1.02171e8i 0.505144 0.505144i −0.407888 0.913032i \(-0.633735\pi\)
0.913032 + 0.407888i \(0.133735\pi\)
\(588\) 0 0
\(589\) 6.62815e7i 0.324375i
\(590\) 0 0
\(591\) −9.65684e7 −0.467813
\(592\) 0 0
\(593\) 1.09929e8 + 1.09929e8i 0.527166 + 0.527166i 0.919726 0.392560i \(-0.128410\pi\)
−0.392560 + 0.919726i \(0.628410\pi\)
\(594\) 0 0
\(595\) 6.73198e7 + 1.21461e7i 0.319589 + 0.0576613i
\(596\) 0 0
\(597\) 7.04754e7 7.04754e7i 0.331218 0.331218i
\(598\) 0 0
\(599\) 2.43420e8i 1.13260i −0.824200 0.566298i \(-0.808375\pi\)
0.824200 0.566298i \(-0.191625\pi\)
\(600\) 0 0
\(601\) −1.12016e7 −0.0516007 −0.0258004 0.999667i \(-0.508213\pi\)
−0.0258004 + 0.999667i \(0.508213\pi\)
\(602\) 0 0
\(603\) −1.92761e6 1.92761e6i −0.00879156 0.00879156i
\(604\) 0 0
\(605\) −1.72846e6 + 9.58001e6i −0.00780535 + 0.0432613i
\(606\) 0 0
\(607\) −1.50111e8 + 1.50111e8i −0.671192 + 0.671192i −0.957991 0.286799i \(-0.907409\pi\)
0.286799 + 0.957991i \(0.407409\pi\)
\(608\) 0 0
\(609\) 3.78173e8i 1.67432i
\(610\) 0 0
\(611\) −7.89483e7 −0.346114
\(612\) 0 0
\(613\) 1.74572e8 + 1.74572e8i 0.757868 + 0.757868i 0.975934 0.218066i \(-0.0699749\pi\)
−0.218066 + 0.975934i \(0.569975\pi\)
\(614\) 0 0
\(615\) 3.21791e8 2.23422e8i 1.38340 0.960505i
\(616\) 0 0
\(617\) 2.43703e8 2.43703e8i 1.03754 1.03754i 0.0382750 0.999267i \(-0.487814\pi\)
0.999267 0.0382750i \(-0.0121863\pi\)
\(618\) 0 0
\(619\) 1.67973e8i 0.708220i −0.935204 0.354110i \(-0.884784\pi\)
0.935204 0.354110i \(-0.115216\pi\)
\(620\) 0 0
\(621\) −2.66574e7 −0.111312
\(622\) 0 0
\(623\) −1.40697e8 1.40697e8i −0.581863 0.581863i
\(624\) 0 0
\(625\) 1.84507e8 + 1.59881e8i 0.755741 + 0.654871i
\(626\) 0 0
\(627\) 2.19093e8 2.19093e8i 0.888846 0.888846i
\(628\) 0 0
\(629\) 5.49398e7i 0.220767i
\(630\) 0 0
\(631\) −4.57816e8 −1.82223 −0.911114 0.412154i \(-0.864777\pi\)
−0.911114 + 0.412154i \(0.864777\pi\)
\(632\) 0 0
\(633\) 6.51743e7 + 6.51743e7i 0.256960 + 0.256960i
\(634\) 0 0
\(635\) 1.53673e8 + 2.21333e8i 0.600173 + 0.864420i
\(636\) 0 0
\(637\) 2.97943e8 2.97943e8i 1.15270 1.15270i
\(638\) 0 0
\(639\) 3.04786e7i 0.116813i
\(640\) 0 0
\(641\) −4.41603e7 −0.167671 −0.0838356 0.996480i \(-0.526717\pi\)
−0.0838356 + 0.996480i \(0.526717\pi\)
\(642\) 0 0
\(643\) −2.97415e8 2.97415e8i −1.11874 1.11874i −0.991926 0.126816i \(-0.959524\pi\)
−0.126816 0.991926i \(-0.540476\pi\)
\(644\) 0 0
\(645\) 8.59524e7 + 1.55078e7i 0.320316 + 0.0577925i
\(646\) 0 0
\(647\) −8.69716e7 + 8.69716e7i −0.321118 + 0.321118i −0.849196 0.528078i \(-0.822913\pi\)
0.528078 + 0.849196i \(0.322913\pi\)
\(648\) 0 0
\(649\) 3.94567e8i 1.44340i
\(650\) 0 0
\(651\) −9.71353e7 −0.352074
\(652\) 0 0
\(653\) 2.33636e7 + 2.33636e7i 0.0839074 + 0.0839074i 0.747815 0.663907i \(-0.231103\pi\)
−0.663907 + 0.747815i \(0.731103\pi\)
\(654\) 0 0
\(655\) −2.44856e7 + 1.35712e8i −0.0871339 + 0.482942i
\(656\) 0 0
\(657\) −7.02770e7 + 7.02770e7i −0.247809 + 0.247809i
\(658\) 0 0
\(659\) 4.40867e8i 1.54046i 0.637764 + 0.770232i \(0.279859\pi\)
−0.637764 + 0.770232i \(0.720141\pi\)
\(660\) 0 0
\(661\) 2.14414e8 0.742419 0.371209 0.928549i \(-0.378943\pi\)
0.371209 + 0.928549i \(0.378943\pi\)
\(662\) 0 0
\(663\) 2.70870e7 + 2.70870e7i 0.0929439 + 0.0929439i
\(664\) 0 0
\(665\) −5.90298e8 + 4.09848e8i −2.00727 + 1.39366i
\(666\) 0 0
\(667\) −2.37597e7 + 2.37597e7i −0.0800689 + 0.0800689i
\(668\) 0 0
\(669\) 1.14284e8i 0.381688i
\(670\) 0 0
\(671\) 3.45385e8 1.14324
\(672\) 0 0
\(673\) −7.41496e7 7.41496e7i −0.243256 0.243256i 0.574940 0.818196i \(-0.305026\pi\)
−0.818196 + 0.574940i \(0.805026\pi\)
\(674\) 0 0
\(675\) 3.02072e8 1.37949e8i 0.982198 0.448547i
\(676\) 0 0
\(677\) −1.19427e8 + 1.19427e8i −0.384890 + 0.384890i −0.872860 0.487970i \(-0.837737\pi\)
0.487970 + 0.872860i \(0.337737\pi\)
\(678\) 0 0
\(679\) 4.38646e8i 1.40121i
\(680\) 0 0
\(681\) −2.56844e8 −0.813256
\(682\) 0 0
\(683\) −3.67224e8 3.67224e8i −1.15257 1.15257i −0.986035 0.166540i \(-0.946741\pi\)
−0.166540 0.986035i \(-0.553259\pi\)
\(684\) 0 0
\(685\) 7.26521e7 + 1.04640e8i 0.226036 + 0.325555i
\(686\) 0 0
\(687\) −1.27875e8 + 1.27875e8i −0.394379 + 0.394379i
\(688\) 0 0
\(689\) 2.00097e8i 0.611764i
\(690\) 0 0
\(691\) −1.40286e8 −0.425187 −0.212594 0.977141i \(-0.568191\pi\)
−0.212594 + 0.977141i \(0.568191\pi\)
\(692\) 0 0
\(693\) −9.73106e7 9.73106e7i −0.292389 0.292389i
\(694\) 0 0
\(695\) −3.56396e8 6.43021e7i −1.06164 0.191545i
\(696\) 0 0
\(697\) −8.59095e7 + 8.59095e7i −0.253713 + 0.253713i
\(698\) 0 0
\(699\) 4.42502e7i 0.129564i
\(700\) 0 0
\(701\) −5.48014e8 −1.59088 −0.795440 0.606032i \(-0.792760\pi\)
−0.795440 + 0.606032i \(0.792760\pi\)
\(702\) 0 0
\(703\) 4.08111e8 + 4.08111e8i 1.17466 + 1.17466i
\(704\) 0 0
\(705\) −2.34645e7 + 1.30052e8i −0.0669643 + 0.371151i
\(706\) 0 0
\(707\) 8.31702e8 8.31702e8i 2.35348 2.35348i
\(708\) 0 0
\(709\) 6.10430e6i 0.0171276i 0.999963 + 0.00856382i \(0.00272598\pi\)
−0.999963 + 0.00856382i \(0.997274\pi\)
\(710\) 0 0
\(711\) 2.46382e7 0.0685489
\(712\) 0 0
\(713\) −6.10278e6 6.10278e6i −0.0168368 0.0168368i
\(714\) 0 0
\(715\) 2.46635e8 1.71241e8i 0.674740 0.468477i
\(716\) 0 0
\(717\) 1.74870e8 1.74870e8i 0.474415 0.474415i
\(718\) 0 0
\(719\) 2.94556e7i 0.0792467i −0.999215 0.0396233i \(-0.987384\pi\)
0.999215 0.0396233i \(-0.0126158\pi\)
\(720\) 0 0
\(721\) 7.28061e8 1.94250
\(722\) 0 0
\(723\) 1.24142e8 + 1.24142e8i 0.328476 + 0.328476i
\(724\) 0 0
\(725\) 1.46282e8 3.92190e8i 0.383865 1.02916i
\(726\) 0 0
\(727\) −5.16355e8 + 5.16355e8i −1.34383 + 1.34383i −0.451627 + 0.892207i \(0.649156\pi\)
−0.892207 + 0.451627i \(0.850844\pi\)
\(728\) 0 0
\(729\) 4.30739e8i 1.11181i
\(730\) 0 0
\(731\) −2.70872e7 −0.0693444
\(732\) 0 0
\(733\) −1.26280e8 1.26280e8i −0.320643 0.320643i 0.528371 0.849014i \(-0.322803\pi\)
−0.849014 + 0.528371i \(0.822803\pi\)
\(734\) 0 0
\(735\) −4.02251e8 5.79357e8i −1.01306 1.45910i
\(736\) 0 0
\(737\) −1.54608e7 + 1.54608e7i −0.0386216 + 0.0386216i
\(738\) 0 0
\(739\) 6.71482e8i 1.66380i 0.554927 + 0.831899i \(0.312746\pi\)
−0.554927 + 0.831899i \(0.687254\pi\)
\(740\) 0 0
\(741\) −4.02423e8 −0.989072
\(742\) 0 0
\(743\) 3.38189e8 + 3.38189e8i 0.824505 + 0.824505i 0.986750 0.162246i \(-0.0518737\pi\)
−0.162246 + 0.986750i \(0.551874\pi\)
\(744\) 0 0
\(745\) 2.44038e8 + 4.40301e7i 0.590185 + 0.106483i
\(746\) 0 0
\(747\) −1.07037e7 + 1.07037e7i −0.0256787 + 0.0256787i
\(748\) 0 0
\(749\) 1.08228e9i 2.57571i
\(750\) 0 0
\(751\) 7.72625e8 1.82410 0.912050 0.410078i \(-0.134499\pi\)
0.912050 + 0.410078i \(0.134499\pi\)
\(752\) 0 0
\(753\) −2.10392e8 2.10392e8i −0.492769 0.492769i
\(754\) 0 0
\(755\) −8.92575e7 + 4.94711e8i −0.207398 + 1.14951i
\(756\) 0 0
\(757\) −6.84105e7 + 6.84105e7i −0.157701 + 0.157701i −0.781547 0.623846i \(-0.785569\pi\)
0.623846 + 0.781547i \(0.285569\pi\)
\(758\) 0 0
\(759\) 4.03454e7i 0.0922718i
\(760\) 0 0
\(761\) 7.07244e8 1.60478 0.802389 0.596801i \(-0.203562\pi\)
0.802389 + 0.596801i \(0.203562\pi\)
\(762\) 0 0
\(763\) 7.85276e7 + 7.85276e7i 0.176786 + 0.176786i
\(764\) 0 0
\(765\) −1.59632e7 + 1.10834e7i −0.0356564 + 0.0247565i
\(766\) 0 0
\(767\) −3.62363e8 + 3.62363e8i −0.803079 + 0.803079i
\(768\) 0 0
\(769\) 1.14644e8i 0.252100i −0.992024 0.126050i \(-0.959770\pi\)
0.992024 0.126050i \(-0.0402300\pi\)
\(770\) 0 0
\(771\) 1.12234e8 0.244885
\(772\) 0 0
\(773\) 6.12226e8 + 6.12226e8i 1.32548 + 1.32548i 0.909266 + 0.416215i \(0.136644\pi\)
0.416215 + 0.909266i \(0.363356\pi\)
\(774\) 0 0
\(775\) 1.00736e8 + 3.75732e7i 0.216410 + 0.0807185i
\(776\) 0 0
\(777\) −5.98084e8 + 5.98084e8i −1.27497 + 1.27497i
\(778\) 0 0
\(779\) 1.27633e9i 2.69991i
\(780\) 0 0
\(781\) −2.44461e8 −0.513164
\(782\) 0 0
\(783\) −4.02596e8 4.02596e8i −0.838657 0.838657i
\(784\) 0 0
\(785\) −3.99483e8 5.75370e8i −0.825828 1.18943i
\(786\) 0 0
\(787\) −7.26664e7 + 7.26664e7i −0.149077 + 0.149077i −0.777705 0.628629i \(-0.783616\pi\)
0.628629 + 0.777705i \(0.283616\pi\)
\(788\) 0 0
\(789\) 6.60148e8i 1.34403i
\(790\) 0 0
\(791\) −8.58043e8 −1.73372
\(792\) 0 0
\(793\) −3.17195e8 3.17195e8i −0.636073 0.636073i
\(794\) 0 0
\(795\) 3.29622e8 + 5.94716e7i 0.656017 + 0.118361i
\(796\) 0 0
\(797\) −4.52017e8 + 4.52017e8i −0.892853 + 0.892853i −0.994791 0.101938i \(-0.967496\pi\)
0.101938 + 0.994791i \(0.467496\pi\)
\(798\) 0 0
\(799\) 4.09849e7i 0.0803495i
\(800\) 0 0
\(801\) 5.65270e7 0.109991
\(802\) 0 0
\(803\) 5.63674e8 + 5.63674e8i 1.08863 + 1.08863i
\(804\) 0 0
\(805\) 1.66147e7 9.20871e7i 0.0318496 0.176527i
\(806\) 0 0
\(807\) −6.39917e7 + 6.39917e7i −0.121759 + 0.121759i
\(808\) 0 0
\(809\) 7.32339e8i 1.38314i 0.722309 + 0.691570i \(0.243081\pi\)
−0.722309 + 0.691570i \(0.756919\pi\)
\(810\) 0 0
\(811\) 3.74076e8 0.701290 0.350645 0.936509i \(-0.385962\pi\)
0.350645 + 0.936509i \(0.385962\pi\)
\(812\) 0 0
\(813\) −3.99021e6 3.99021e6i −0.00742548 0.00742548i
\(814\) 0 0
\(815\) −4.07484e8 + 2.82919e8i −0.752727 + 0.522624i
\(816\) 0 0
\(817\) 2.01212e8 2.01212e8i 0.368968 0.368968i
\(818\) 0 0
\(819\) 1.78737e8i 0.325358i
\(820\) 0 0
\(821\) −4.05908e8 −0.733496 −0.366748 0.930320i \(-0.619529\pi\)
−0.366748 + 0.930320i \(0.619529\pi\)
\(822\) 0 0
\(823\) 5.03821e8 + 5.03821e8i 0.903809 + 0.903809i 0.995763 0.0919544i \(-0.0293114\pi\)
−0.0919544 + 0.995763i \(0.529311\pi\)
\(824\) 0 0
\(825\) −2.08783e8 4.57179e8i −0.371820 0.814188i
\(826\) 0 0
\(827\) −1.75124e8 + 1.75124e8i −0.309620 + 0.309620i −0.844762 0.535142i \(-0.820258\pi\)
0.535142 + 0.844762i \(0.320258\pi\)
\(828\) 0 0
\(829\) 4.90749e8i 0.861381i 0.902500 + 0.430691i \(0.141730\pi\)
−0.902500 + 0.430691i \(0.858270\pi\)
\(830\) 0 0
\(831\) 2.95573e8 0.515064
\(832\) 0 0
\(833\) 1.54673e8 + 1.54673e8i 0.267595 + 0.267595i
\(834\) 0 0
\(835\) −4.08859e8 5.88873e8i −0.702286 1.01149i
\(836\) 0 0
\(837\) 1.03408e8 1.03408e8i 0.176352 0.176352i
\(838\) 0 0
\(839\) 6.07365e8i 1.02840i −0.857669 0.514202i \(-0.828088\pi\)
0.857669 0.514202i \(-0.171912\pi\)
\(840\) 0 0
\(841\) −1.22843e8 −0.206521
\(842\) 0 0
\(843\) −1.57928e8 1.57928e8i −0.263618 0.263618i
\(844\) 0 0
\(845\) 2.09995e8 + 3.78880e7i 0.348048 + 0.0627959i
\(846\) 0 0
\(847\) −3.28660e7 + 3.28660e7i −0.0540874 + 0.0540874i
\(848\) 0 0
\(849\) 1.29862e8i 0.212206i
\(850\) 0 0
\(851\) −7.51524e7 −0.121942
\(852\) 0 0
\(853\) −3.41127e8 3.41127e8i −0.549628 0.549628i 0.376705 0.926333i \(-0.377057\pi\)
−0.926333 + 0.376705i \(0.877057\pi\)
\(854\) 0 0
\(855\) 3.62491e7 2.00911e8i 0.0579961 0.321445i
\(856\) 0 0
\(857\) 7.87515e8 7.87515e8i 1.25117 1.25117i 0.295974 0.955196i \(-0.404356\pi\)
0.955196 0.295974i \(-0.0956441\pi\)
\(858\) 0 0
\(859\) 2.57007e8i 0.405476i −0.979233 0.202738i \(-0.935016\pi\)
0.979233 0.202738i \(-0.0649840\pi\)
\(860\) 0 0
\(861\) 1.87045e9 2.93047
\(862\) 0 0
\(863\) −3.70540e8 3.70540e8i −0.576504 0.576504i 0.357434 0.933938i \(-0.383652\pi\)
−0.933938 + 0.357434i \(0.883652\pi\)
\(864\) 0 0
\(865\) 8.62428e7 5.98790e7i 0.133252 0.0925180i
\(866\) 0 0
\(867\) 3.89637e8 3.89637e8i 0.597865 0.597865i
\(868\) 0 0
\(869\) 1.97617e8i 0.301137i
\(870\) 0 0
\(871\) 2.83979e7 0.0429765
\(872\) 0 0
\(873\) −8.81160e7 8.81160e7i −0.132438 0.132438i
\(874\) 0 0
\(875\) 2.88269e8 + 1.12948e9i 0.430302 + 1.68598i
\(876\) 0 0
\(877\) −1.21352e8 + 1.21352e8i −0.179907 + 0.179907i −0.791315 0.611408i \(-0.790603\pi\)
0.611408 + 0.791315i \(0.290603\pi\)
\(878\) 0 0
\(879\) 1.47523e8i 0.217216i
\(880\) 0 0
\(881\) −4.29574e8 −0.628218 −0.314109 0.949387i \(-0.601706\pi\)
−0.314109 + 0.949387i \(0.601706\pi\)
\(882\) 0 0
\(883\) −8.67016e8 8.67016e8i −1.25935 1.25935i −0.951406 0.307939i \(-0.900361\pi\)
−0.307939 0.951406i \(-0.599639\pi\)
\(884\) 0 0
\(885\) 4.89226e8 + 7.04624e8i 0.705796 + 1.01655i
\(886\) 0 0
\(887\) 5.00413e8 5.00413e8i 0.717064 0.717064i −0.250939 0.968003i \(-0.580739\pi\)
0.968003 + 0.250939i \(0.0807394\pi\)
\(888\) 0 0
\(889\) 1.28653e9i 1.83111i
\(890\) 0 0
\(891\) −5.15538e8 −0.728832
\(892\) 0 0
\(893\) 3.04449e8 + 3.04449e8i 0.427524 + 0.427524i
\(894\) 0 0
\(895\) −4.28917e8 7.73867e7i −0.598280 0.107944i
\(896\) 0 0
\(897\) 3.70525e7 3.70525e7i 0.0513381 0.0513381i
\(898\) 0 0
\(899\) 1.84336e8i 0.253706i
\(900\) 0 0
\(901\) −1.03878e8 −0.142020
\(902\) 0 0
\(903\) 2.94876e8 + 2.94876e8i 0.400475 + 0.400475i
\(904\) 0 0
\(905\) 3.13881e7 1.73969e8i 0.0423466 0.234707i
\(906\) 0 0
\(907\) 2.77507e8 2.77507e8i 0.371922 0.371922i −0.496255 0.868177i \(-0.665292\pi\)
0.868177 + 0.496255i \(0.165292\pi\)
\(908\) 0 0
\(909\) 3.34148e8i 0.444884i
\(910\) 0 0
\(911\) 7.67626e7 0.101530 0.0507650 0.998711i \(-0.483834\pi\)
0.0507650 + 0.998711i \(0.483834\pi\)
\(912\) 0 0
\(913\) 8.58518e7 + 8.58518e7i 0.112807 + 0.112807i
\(914\) 0 0
\(915\) −6.16794e8 + 4.28244e8i −0.805149 + 0.559021i
\(916\) 0 0
\(917\) −4.65585e8 + 4.65585e8i −0.603798 + 0.603798i
\(918\) 0 0
\(919\) 7.93161e8i 1.02192i −0.859606 0.510958i \(-0.829291\pi\)
0.859606 0.510958i \(-0.170709\pi\)
\(920\) 0 0
\(921\) −6.77523e8 −0.867252
\(922\) 0 0
\(923\) 2.24508e8 + 2.24508e8i 0.285514 + 0.285514i
\(924\) 0 0
\(925\) 8.51599e8 3.88905e8i 1.07599 0.491381i
\(926\) 0 0
\(927\) −1.46254e8 + 1.46254e8i −0.183599 + 0.183599i
\(928\) 0 0
\(929\) 5.48777e8i 0.684461i 0.939616 + 0.342231i \(0.111182\pi\)
−0.939616 + 0.342231i \(0.888818\pi\)
\(930\) 0 0
\(931\) −2.29792e9 −2.84764
\(932\) 0 0
\(933\) −6.87757e8 6.87757e8i −0.846818 0.846818i
\(934\) 0 0
\(935\) 8.88971e7 + 1.28037e8i 0.108756 + 0.156639i
\(936\) 0 0
\(937\) −5.56840e7 + 5.56840e7i −0.0676880 + 0.0676880i −0.740140 0.672452i \(-0.765241\pi\)
0.672452 + 0.740140i \(0.265241\pi\)
\(938\) 0 0
\(939\) 1.25651e9i 1.51764i
\(940\) 0 0
\(941\) −8.88360e8 −1.06616 −0.533078 0.846066i \(-0.678965\pi\)
−0.533078 + 0.846066i \(0.678965\pi\)
\(942\) 0 0
\(943\) 1.17516e8 + 1.17516e8i 0.140140 + 0.140140i
\(944\) 0 0
\(945\) 1.56037e9 + 2.81527e8i 1.84898 + 0.333599i
\(946\) 0 0
\(947\) 6.61360e8 6.61360e8i 0.778732 0.778732i −0.200883 0.979615i \(-0.564381\pi\)
0.979615 + 0.200883i \(0.0643811\pi\)
\(948\) 0 0
\(949\) 1.03534e9i 1.21139i
\(950\) 0 0
\(951\) 6.75346e8 0.785208
\(952\) 0 0
\(953\) 2.59406e8 + 2.59406e8i 0.299710 + 0.299710i 0.840900 0.541190i \(-0.182026\pi\)
−0.541190 + 0.840900i \(0.682026\pi\)
\(954\) 0 0
\(955\) −1.00143e8 + 5.55043e8i −0.114977 + 0.637260i
\(956\) 0 0
\(957\) −6.09321e8 + 6.09321e8i −0.695200 + 0.695200i
\(958\) 0 0
\(959\) 6.08233e8i 0.689626i
\(960\) 0 0
\(961\) −8.40156e8 −0.946651
\(962\) 0 0
\(963\) 2.17411e8 + 2.17411e8i 0.243446 + 0.243446i
\(964\) 0 0
\(965\) −2.48001e8 + 1.72189e8i −0.275976 + 0.191612i
\(966\) 0 0
\(967\) 5.21710e8 5.21710e8i 0.576965 0.576965i −0.357101 0.934066i \(-0.616235\pi\)
0.934066 + 0.357101i \(0.116235\pi\)
\(968\) 0 0
\(969\) 2.08912e8i 0.229611i
\(970\) 0 0
\(971\) 4.50595e8 0.492186 0.246093 0.969246i \(-0.420853\pi\)
0.246093 + 0.969246i \(0.420853\pi\)
\(972\) 0 0
\(973\) −1.22268e9 1.22268e9i −1.32732 1.32732i
\(974\) 0 0
\(975\) −2.28123e8 + 6.11608e8i −0.246124 + 0.659871i
\(976\) 0 0
\(977\) 5.75242e8 5.75242e8i 0.616832 0.616832i −0.327886 0.944717i \(-0.606336\pi\)
0.944717 + 0.327886i \(0.106336\pi\)
\(978\) 0 0
\(979\) 4.53388e8i 0.483195i
\(980\) 0 0
\(981\) −3.15495e7 −0.0334184
\(982\) 0 0
\(983\) 9.06660e8 + 9.06660e8i 0.954518 + 0.954518i 0.999010 0.0444920i \(-0.0141669\pi\)
−0.0444920 + 0.999010i \(0.514167\pi\)
\(984\) 0 0
\(985\) 2.91061e8 + 4.19210e8i 0.304562 + 0.438655i
\(986\) 0 0
\(987\) −4.46168e8 + 4.46168e8i −0.464031 + 0.464031i
\(988\) 0 0
\(989\) 3.70527e7i 0.0383028i
\(990\) 0 0
\(991\) 1.01456e9 1.04245 0.521227 0.853418i \(-0.325474\pi\)
0.521227 + 0.853418i \(0.325474\pi\)
\(992\) 0 0
\(993\) −8.95868e8 8.95868e8i −0.914948 0.914948i
\(994\) 0 0
\(995\) −5.18354e8 9.35233e7i −0.526208 0.0949402i
\(996\) 0 0
\(997\) 2.49097e8 2.49097e8i 0.251353 0.251353i −0.570172 0.821525i \(-0.693124\pi\)
0.821525 + 0.570172i \(0.193124\pi\)
\(998\) 0 0
\(999\) 1.27342e9i 1.27725i
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 20.7.f.a.13.1 6
3.2 odd 2 180.7.l.a.73.2 6
4.3 odd 2 80.7.p.d.33.3 6
5.2 odd 4 inner 20.7.f.a.17.1 yes 6
5.3 odd 4 100.7.f.b.57.3 6
5.4 even 2 100.7.f.b.93.3 6
15.2 even 4 180.7.l.a.37.2 6
20.7 even 4 80.7.p.d.17.3 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
20.7.f.a.13.1 6 1.1 even 1 trivial
20.7.f.a.17.1 yes 6 5.2 odd 4 inner
80.7.p.d.17.3 6 20.7 even 4
80.7.p.d.33.3 6 4.3 odd 2
100.7.f.b.57.3 6 5.3 odd 4
100.7.f.b.93.3 6 5.4 even 2
180.7.l.a.37.2 6 15.2 even 4
180.7.l.a.73.2 6 3.2 odd 2