Properties

Label 75.10.b.e.49.2
Level $75$
Weight $10$
Character 75.49
Analytic conductor $38.628$
Analytic rank $0$
Dimension $4$
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] = [75,10,Mod(49,75)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(75, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 10, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("75.49");
 
S:= CuspForms(chi, 10);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 75 = 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 75.b (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: \(38.6276877123\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{4729})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 2365x^{2} + 1397124 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 15)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 49.2
Root \(-33.8839i\) of defining polynomial
Character \(\chi\) \(=\) 75.49
Dual form 75.10.b.e.49.3

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-24.8839i q^{2} -81.0000i q^{3} -107.207 q^{4} -2015.59 q^{6} -4010.50i q^{7} -10072.8i q^{8} -6561.00 q^{9} +O(q^{10})\) \(q-24.8839i q^{2} -81.0000i q^{3} -107.207 q^{4} -2015.59 q^{6} -4010.50i q^{7} -10072.8i q^{8} -6561.00 q^{9} +84861.3 q^{11} +8683.74i q^{12} -119425. i q^{13} -99796.8 q^{14} -305541. q^{16} +116934. i q^{17} +163263. i q^{18} +234932. q^{19} -324851. q^{21} -2.11168e6i q^{22} -2.34570e6i q^{23} -815899. q^{24} -2.97176e6 q^{26} +531441. i q^{27} +429953. i q^{28} +464196. q^{29} -5.11766e6 q^{31} +2.44574e6i q^{32} -6.87377e6i q^{33} +2.90976e6 q^{34} +703383. q^{36} +8.69354e6i q^{37} -5.84601e6i q^{38} -9.67345e6 q^{39} -9.05805e6 q^{41} +8.08354e6i q^{42} -8.63491e6i q^{43} -9.09769e6 q^{44} -5.83701e7 q^{46} +3.31511e7i q^{47} +2.47488e7i q^{48} +2.42695e7 q^{49} +9.47161e6 q^{51} +1.28032e7i q^{52} +6.41254e7i q^{53} +1.32243e7 q^{54} -4.03971e7 q^{56} -1.90295e7i q^{57} -1.15510e7i q^{58} -1.49407e8 q^{59} +1.54634e8 q^{61} +1.27347e8i q^{62} +2.63129e7i q^{63} -9.55772e7 q^{64} -1.71046e8 q^{66} +2.72755e8i q^{67} -1.25360e7i q^{68} -1.90002e8 q^{69} -3.56924e8 q^{71} +6.60878e7i q^{72} -2.06253e8i q^{73} +2.16329e8 q^{74} -2.51862e7 q^{76} -3.40337e8i q^{77} +2.40713e8i q^{78} +4.04380e8 q^{79} +4.30467e7 q^{81} +2.25399e8i q^{82} +5.17034e6i q^{83} +3.48262e7 q^{84} -2.14870e8 q^{86} -3.75999e7i q^{87} -8.54793e8i q^{88} -4.32242e8 q^{89} -4.78955e8 q^{91} +2.51475e8i q^{92} +4.14530e8i q^{93} +8.24928e8 q^{94} +1.98105e8 q^{96} -1.32066e9i q^{97} -6.03918e8i q^{98} -5.56775e8 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 3042 q^{4} + 3078 q^{6} - 26244 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 3042 q^{4} + 3078 q^{6} - 26244 q^{9} + 70976 q^{11} + 490392 q^{14} + 1414530 q^{16} + 806592 q^{19} - 1923264 q^{21} - 8042814 q^{24} - 3815092 q^{26} + 149144 q^{29} - 10054256 q^{31} - 17721764 q^{34} + 19958562 q^{36} - 23275512 q^{39} + 28422664 q^{41} + 121411904 q^{44} - 302983296 q^{46} + 5639900 q^{49} + 62395272 q^{51} - 20194758 q^{54} - 703018680 q^{56} - 375726272 q^{59} + 308160120 q^{61} - 1276602178 q^{64} - 693095616 q^{66} - 36240048 q^{69} - 456541952 q^{71} + 724038452 q^{74} - 526437448 q^{76} + 1864813520 q^{79} + 172186884 q^{81} + 1870206408 q^{84} + 1247853416 q^{86} - 449036328 q^{89} - 1339205056 q^{91} - 3863301632 q^{94} + 4313992446 q^{96} - 465673536 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/75\mathbb{Z}\right)^\times\).

\(n\) \(26\) \(52\)
\(\chi(n)\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 24.8839i − 1.09972i −0.835256 0.549861i \(-0.814681\pi\)
0.835256 0.549861i \(-0.185319\pi\)
\(3\) − 81.0000i − 0.577350i
\(4\) −107.207 −0.209388
\(5\) 0 0
\(6\) −2015.59 −0.634925
\(7\) − 4010.50i − 0.631332i −0.948870 0.315666i \(-0.897772\pi\)
0.948870 0.315666i \(-0.102228\pi\)
\(8\) − 10072.8i − 0.869453i
\(9\) −6561.00 −0.333333
\(10\) 0 0
\(11\) 84861.3 1.74760 0.873801 0.486283i \(-0.161648\pi\)
0.873801 + 0.486283i \(0.161648\pi\)
\(12\) 8683.74i 0.120890i
\(13\) − 119425.i − 1.15971i −0.814718 0.579857i \(-0.803108\pi\)
0.814718 0.579857i \(-0.196892\pi\)
\(14\) −99796.8 −0.694289
\(15\) 0 0
\(16\) −305541. −1.16554
\(17\) 116934.i 0.339562i 0.985482 + 0.169781i \(0.0543060\pi\)
−0.985482 + 0.169781i \(0.945694\pi\)
\(18\) 163263.i 0.366574i
\(19\) 234932. 0.413571 0.206786 0.978386i \(-0.433700\pi\)
0.206786 + 0.978386i \(0.433700\pi\)
\(20\) 0 0
\(21\) −324851. −0.364500
\(22\) − 2.11168e6i − 1.92188i
\(23\) − 2.34570e6i − 1.74782i −0.486084 0.873912i \(-0.661575\pi\)
0.486084 0.873912i \(-0.338425\pi\)
\(24\) −815899. −0.501979
\(25\) 0 0
\(26\) −2.97176e6 −1.27536
\(27\) 531441.i 0.192450i
\(28\) 429953.i 0.132193i
\(29\) 464196. 0.121874 0.0609369 0.998142i \(-0.480591\pi\)
0.0609369 + 0.998142i \(0.480591\pi\)
\(30\) 0 0
\(31\) −5.11766e6 −0.995277 −0.497638 0.867385i \(-0.665799\pi\)
−0.497638 + 0.867385i \(0.665799\pi\)
\(32\) 2.44574e6i 0.412321i
\(33\) − 6.87377e6i − 1.00898i
\(34\) 2.90976e6 0.373423
\(35\) 0 0
\(36\) 703383. 0.0697960
\(37\) 8.69354e6i 0.762586i 0.924454 + 0.381293i \(0.124521\pi\)
−0.924454 + 0.381293i \(0.875479\pi\)
\(38\) − 5.84601e6i − 0.454813i
\(39\) −9.67345e6 −0.669562
\(40\) 0 0
\(41\) −9.05805e6 −0.500619 −0.250310 0.968166i \(-0.580532\pi\)
−0.250310 + 0.968166i \(0.580532\pi\)
\(42\) 8.08354e6i 0.400848i
\(43\) − 8.63491e6i − 0.385168i −0.981281 0.192584i \(-0.938313\pi\)
0.981281 0.192584i \(-0.0616867\pi\)
\(44\) −9.09769e6 −0.365927
\(45\) 0 0
\(46\) −5.83701e7 −1.92212
\(47\) 3.31511e7i 0.990964i 0.868618 + 0.495482i \(0.165009\pi\)
−0.868618 + 0.495482i \(0.834991\pi\)
\(48\) 2.47488e7i 0.672927i
\(49\) 2.42695e7 0.601420
\(50\) 0 0
\(51\) 9.47161e6 0.196046
\(52\) 1.28032e7i 0.242830i
\(53\) 6.41254e7i 1.11632i 0.829733 + 0.558160i \(0.188492\pi\)
−0.829733 + 0.558160i \(0.811508\pi\)
\(54\) 1.32243e7 0.211642
\(55\) 0 0
\(56\) −4.03971e7 −0.548914
\(57\) − 1.90295e7i − 0.238775i
\(58\) − 1.15510e7i − 0.134027i
\(59\) −1.49407e8 −1.60523 −0.802613 0.596500i \(-0.796558\pi\)
−0.802613 + 0.596500i \(0.796558\pi\)
\(60\) 0 0
\(61\) 1.54634e8 1.42995 0.714973 0.699152i \(-0.246439\pi\)
0.714973 + 0.699152i \(0.246439\pi\)
\(62\) 1.27347e8i 1.09453i
\(63\) 2.63129e7i 0.210444i
\(64\) −9.55772e7 −0.712106
\(65\) 0 0
\(66\) −1.71046e8 −1.10960
\(67\) 2.72755e8i 1.65362i 0.562479 + 0.826811i \(0.309848\pi\)
−0.562479 + 0.826811i \(0.690152\pi\)
\(68\) − 1.25360e7i − 0.0711001i
\(69\) −1.90002e8 −1.00911
\(70\) 0 0
\(71\) −3.56924e8 −1.66691 −0.833457 0.552584i \(-0.813642\pi\)
−0.833457 + 0.552584i \(0.813642\pi\)
\(72\) 6.60878e7i 0.289818i
\(73\) − 2.06253e8i − 0.850057i −0.905180 0.425029i \(-0.860264\pi\)
0.905180 0.425029i \(-0.139736\pi\)
\(74\) 2.16329e8 0.838632
\(75\) 0 0
\(76\) −2.51862e7 −0.0865968
\(77\) − 3.40337e8i − 1.10332i
\(78\) 2.40713e8i 0.736331i
\(79\) 4.04380e8 1.16807 0.584033 0.811730i \(-0.301474\pi\)
0.584033 + 0.811730i \(0.301474\pi\)
\(80\) 0 0
\(81\) 4.30467e7 0.111111
\(82\) 2.25399e8i 0.550542i
\(83\) 5.17034e6i 0.0119582i 0.999982 + 0.00597912i \(0.00190323\pi\)
−0.999982 + 0.00597912i \(0.998097\pi\)
\(84\) 3.48262e7 0.0763218
\(85\) 0 0
\(86\) −2.14870e8 −0.423577
\(87\) − 3.75999e7i − 0.0703639i
\(88\) − 8.54793e8i − 1.51946i
\(89\) −4.32242e8 −0.730250 −0.365125 0.930958i \(-0.618974\pi\)
−0.365125 + 0.930958i \(0.618974\pi\)
\(90\) 0 0
\(91\) −4.78955e8 −0.732165
\(92\) 2.51475e8i 0.365973i
\(93\) 4.14530e8i 0.574623i
\(94\) 8.24928e8 1.08978
\(95\) 0 0
\(96\) 1.98105e8 0.238054
\(97\) − 1.32066e9i − 1.51468i −0.653023 0.757338i \(-0.726499\pi\)
0.653023 0.757338i \(-0.273501\pi\)
\(98\) − 6.03918e8i − 0.661395i
\(99\) −5.56775e8 −0.582534
\(100\) 0 0
\(101\) 5.30458e8 0.507230 0.253615 0.967305i \(-0.418380\pi\)
0.253615 + 0.967305i \(0.418380\pi\)
\(102\) − 2.35690e8i − 0.215596i
\(103\) 6.07207e7i 0.0531580i 0.999647 + 0.0265790i \(0.00846136\pi\)
−0.999647 + 0.0265790i \(0.991539\pi\)
\(104\) −1.20295e9 −1.00832
\(105\) 0 0
\(106\) 1.59569e9 1.22764
\(107\) − 1.00828e9i − 0.743625i −0.928308 0.371812i \(-0.878736\pi\)
0.928308 0.371812i \(-0.121264\pi\)
\(108\) − 5.69740e7i − 0.0402967i
\(109\) 1.77424e9 1.20391 0.601954 0.798531i \(-0.294389\pi\)
0.601954 + 0.798531i \(0.294389\pi\)
\(110\) 0 0
\(111\) 7.04177e8 0.440279
\(112\) 1.22537e9i 0.735846i
\(113\) − 9.45495e8i − 0.545514i −0.962083 0.272757i \(-0.912064\pi\)
0.962083 0.272757i \(-0.0879356\pi\)
\(114\) −4.73526e8 −0.262586
\(115\) 0 0
\(116\) −4.97649e7 −0.0255189
\(117\) 7.83549e8i 0.386572i
\(118\) 3.71782e9i 1.76530i
\(119\) 4.68962e8 0.214376
\(120\) 0 0
\(121\) 4.84349e9 2.05411
\(122\) − 3.84788e9i − 1.57254i
\(123\) 7.33702e8i 0.289033i
\(124\) 5.48647e8 0.208399
\(125\) 0 0
\(126\) 6.54767e8 0.231430
\(127\) − 5.19758e9i − 1.77290i −0.462825 0.886450i \(-0.653164\pi\)
0.462825 0.886450i \(-0.346836\pi\)
\(128\) 3.63055e9i 1.19544i
\(129\) −6.99428e8 −0.222377
\(130\) 0 0
\(131\) 5.28408e8 0.156765 0.0783824 0.996923i \(-0.475024\pi\)
0.0783824 + 0.996923i \(0.475024\pi\)
\(132\) 7.36913e8i 0.211268i
\(133\) − 9.42194e8i − 0.261101i
\(134\) 6.78720e9 1.81852
\(135\) 0 0
\(136\) 1.17785e9 0.295233
\(137\) 5.01761e9i 1.21690i 0.793593 + 0.608449i \(0.208208\pi\)
−0.793593 + 0.608449i \(0.791792\pi\)
\(138\) 4.72798e9i 1.10974i
\(139\) −3.51872e9 −0.799499 −0.399750 0.916624i \(-0.630903\pi\)
−0.399750 + 0.916624i \(0.630903\pi\)
\(140\) 0 0
\(141\) 2.68524e9 0.572133
\(142\) 8.88165e9i 1.83314i
\(143\) − 1.01346e10i − 2.02672i
\(144\) 2.00465e9 0.388515
\(145\) 0 0
\(146\) −5.13238e9 −0.934827
\(147\) − 1.96583e9i − 0.347230i
\(148\) − 9.32005e8i − 0.159676i
\(149\) −4.32815e9 −0.719390 −0.359695 0.933070i \(-0.617119\pi\)
−0.359695 + 0.933070i \(0.617119\pi\)
\(150\) 0 0
\(151\) −5.61832e9 −0.879448 −0.439724 0.898133i \(-0.644924\pi\)
−0.439724 + 0.898133i \(0.644924\pi\)
\(152\) − 2.36642e9i − 0.359581i
\(153\) − 7.67201e8i − 0.113187i
\(154\) −8.46889e9 −1.21334
\(155\) 0 0
\(156\) 1.03706e9 0.140198
\(157\) 1.55603e9i 0.204394i 0.994764 + 0.102197i \(0.0325873\pi\)
−0.994764 + 0.102197i \(0.967413\pi\)
\(158\) − 1.00625e10i − 1.28455i
\(159\) 5.19416e9 0.644507
\(160\) 0 0
\(161\) −9.40745e9 −1.10346
\(162\) − 1.07117e9i − 0.122191i
\(163\) 1.15580e10i 1.28245i 0.767354 + 0.641224i \(0.221573\pi\)
−0.767354 + 0.641224i \(0.778427\pi\)
\(164\) 9.71083e8 0.104824
\(165\) 0 0
\(166\) 1.28658e8 0.0131507
\(167\) − 1.50486e10i − 1.49717i −0.663039 0.748585i \(-0.730734\pi\)
0.663039 0.748585i \(-0.269266\pi\)
\(168\) 3.27216e9i 0.316915i
\(169\) −3.65789e9 −0.344938
\(170\) 0 0
\(171\) −1.54139e9 −0.137857
\(172\) 9.25719e8i 0.0806494i
\(173\) − 2.23157e10i − 1.89410i −0.321081 0.947052i \(-0.604046\pi\)
0.321081 0.947052i \(-0.395954\pi\)
\(174\) −9.35630e8 −0.0773807
\(175\) 0 0
\(176\) −2.59286e10 −2.03691
\(177\) 1.21019e10i 0.926778i
\(178\) 1.07558e10i 0.803072i
\(179\) 1.73543e10 1.26348 0.631742 0.775179i \(-0.282340\pi\)
0.631742 + 0.775179i \(0.282340\pi\)
\(180\) 0 0
\(181\) −1.34040e10 −0.928282 −0.464141 0.885761i \(-0.653637\pi\)
−0.464141 + 0.885761i \(0.653637\pi\)
\(182\) 1.19183e10i 0.805178i
\(183\) − 1.25253e10i − 0.825580i
\(184\) −2.36278e10 −1.51965
\(185\) 0 0
\(186\) 1.03151e10 0.631926
\(187\) 9.92313e9i 0.593419i
\(188\) − 3.55402e9i − 0.207496i
\(189\) 2.13135e9 0.121500
\(190\) 0 0
\(191\) −6.01312e9 −0.326926 −0.163463 0.986549i \(-0.552266\pi\)
−0.163463 + 0.986549i \(0.552266\pi\)
\(192\) 7.74175e9i 0.411134i
\(193\) 6.91844e9i 0.358922i 0.983765 + 0.179461i \(0.0574354\pi\)
−0.983765 + 0.179461i \(0.942565\pi\)
\(194\) −3.28632e10 −1.66572
\(195\) 0 0
\(196\) −2.60185e9 −0.125930
\(197\) 1.66139e10i 0.785909i 0.919558 + 0.392955i \(0.128547\pi\)
−0.919558 + 0.392955i \(0.871453\pi\)
\(198\) 1.38547e10i 0.640625i
\(199\) 3.06711e10 1.38641 0.693204 0.720742i \(-0.256199\pi\)
0.693204 + 0.720742i \(0.256199\pi\)
\(200\) 0 0
\(201\) 2.20932e10 0.954719
\(202\) − 1.31998e10i − 0.557812i
\(203\) − 1.86166e9i − 0.0769428i
\(204\) −1.01542e9 −0.0410497
\(205\) 0 0
\(206\) 1.51096e9 0.0584591
\(207\) 1.53902e10i 0.582608i
\(208\) 3.64893e10i 1.35170i
\(209\) 1.99366e10 0.722758
\(210\) 0 0
\(211\) 3.85598e10 1.33926 0.669628 0.742697i \(-0.266454\pi\)
0.669628 + 0.742697i \(0.266454\pi\)
\(212\) − 6.87467e9i − 0.233744i
\(213\) 2.89108e10i 0.962393i
\(214\) −2.50899e10 −0.817780
\(215\) 0 0
\(216\) 5.35311e9 0.167326
\(217\) 2.05244e10i 0.628350i
\(218\) − 4.41500e10i − 1.32396i
\(219\) −1.67065e10 −0.490781
\(220\) 0 0
\(221\) 1.39648e10 0.393795
\(222\) − 1.75226e10i − 0.484185i
\(223\) 3.11357e10i 0.843115i 0.906802 + 0.421558i \(0.138516\pi\)
−0.906802 + 0.421558i \(0.861484\pi\)
\(224\) 9.80866e9 0.260312
\(225\) 0 0
\(226\) −2.35276e10 −0.599914
\(227\) − 1.14589e10i − 0.286436i −0.989691 0.143218i \(-0.954255\pi\)
0.989691 0.143218i \(-0.0457451\pi\)
\(228\) 2.04008e9i 0.0499967i
\(229\) 3.04556e10 0.731825 0.365913 0.930649i \(-0.380757\pi\)
0.365913 + 0.930649i \(0.380757\pi\)
\(230\) 0 0
\(231\) −2.75673e10 −0.637000
\(232\) − 4.67576e9i − 0.105964i
\(233\) − 2.83630e9i − 0.0630451i −0.999503 0.0315225i \(-0.989964\pi\)
0.999503 0.0315225i \(-0.0100356\pi\)
\(234\) 1.94977e10 0.425121
\(235\) 0 0
\(236\) 1.60174e10 0.336115
\(237\) − 3.27548e10i − 0.674384i
\(238\) − 1.16696e10i − 0.235754i
\(239\) 6.25862e10 1.24076 0.620381 0.784301i \(-0.286978\pi\)
0.620381 + 0.784301i \(0.286978\pi\)
\(240\) 0 0
\(241\) 7.24015e10 1.38252 0.691259 0.722607i \(-0.257056\pi\)
0.691259 + 0.722607i \(0.257056\pi\)
\(242\) − 1.20525e11i − 2.25895i
\(243\) − 3.48678e9i − 0.0641500i
\(244\) −1.65778e10 −0.299414
\(245\) 0 0
\(246\) 1.82573e10 0.317855
\(247\) − 2.80568e10i − 0.479624i
\(248\) 5.15493e10i 0.865347i
\(249\) 4.18797e8 0.00690410
\(250\) 0 0
\(251\) −5.65927e10 −0.899971 −0.449986 0.893036i \(-0.648571\pi\)
−0.449986 + 0.893036i \(0.648571\pi\)
\(252\) − 2.82092e9i − 0.0440644i
\(253\) − 1.99059e11i − 3.05450i
\(254\) −1.29336e11 −1.94970
\(255\) 0 0
\(256\) 4.14066e10 0.602545
\(257\) 4.95688e10i 0.708777i 0.935098 + 0.354388i \(0.115311\pi\)
−0.935098 + 0.354388i \(0.884689\pi\)
\(258\) 1.74045e10i 0.244552i
\(259\) 3.48655e10 0.481445
\(260\) 0 0
\(261\) −3.04559e9 −0.0406246
\(262\) − 1.31488e10i − 0.172398i
\(263\) − 3.59498e10i − 0.463336i −0.972795 0.231668i \(-0.925582\pi\)
0.972795 0.231668i \(-0.0744183\pi\)
\(264\) −6.92382e10 −0.877260
\(265\) 0 0
\(266\) −2.34454e10 −0.287138
\(267\) 3.50116e10i 0.421610i
\(268\) − 2.92412e10i − 0.346249i
\(269\) 7.09650e10 0.826340 0.413170 0.910654i \(-0.364421\pi\)
0.413170 + 0.910654i \(0.364421\pi\)
\(270\) 0 0
\(271\) 1.47201e11 1.65786 0.828930 0.559352i \(-0.188950\pi\)
0.828930 + 0.559352i \(0.188950\pi\)
\(272\) − 3.57279e10i − 0.395774i
\(273\) 3.87954e10i 0.422716i
\(274\) 1.24857e11 1.33825
\(275\) 0 0
\(276\) 2.03695e10 0.211295
\(277\) − 2.87336e10i − 0.293245i −0.989193 0.146622i \(-0.953160\pi\)
0.989193 0.146622i \(-0.0468402\pi\)
\(278\) 8.75593e10i 0.879227i
\(279\) 3.35770e10 0.331759
\(280\) 0 0
\(281\) 5.17071e10 0.494734 0.247367 0.968922i \(-0.420435\pi\)
0.247367 + 0.968922i \(0.420435\pi\)
\(282\) − 6.68192e10i − 0.629187i
\(283\) 3.22434e9i 0.0298814i 0.999888 + 0.0149407i \(0.00475595\pi\)
−0.999888 + 0.0149407i \(0.995244\pi\)
\(284\) 3.82646e10 0.349032
\(285\) 0 0
\(286\) −2.52188e11 −2.22883
\(287\) 3.63273e10i 0.316057i
\(288\) − 1.60465e10i − 0.137440i
\(289\) 1.04914e11 0.884698
\(290\) 0 0
\(291\) −1.06974e11 −0.874499
\(292\) 2.21117e10i 0.177992i
\(293\) 1.13591e11i 0.900409i 0.892926 + 0.450204i \(0.148649\pi\)
−0.892926 + 0.450204i \(0.851351\pi\)
\(294\) −4.89174e10 −0.381856
\(295\) 0 0
\(296\) 8.75685e10 0.663033
\(297\) 4.50988e10i 0.336326i
\(298\) 1.07701e11i 0.791129i
\(299\) −2.80136e11 −2.02698
\(300\) 0 0
\(301\) −3.46303e10 −0.243169
\(302\) 1.39805e11i 0.967148i
\(303\) − 4.29671e10i − 0.292849i
\(304\) −7.17811e10 −0.482036
\(305\) 0 0
\(306\) −1.90909e10 −0.124474
\(307\) − 2.30543e10i − 0.148125i −0.997254 0.0740627i \(-0.976403\pi\)
0.997254 0.0740627i \(-0.0235965\pi\)
\(308\) 3.64863e10i 0.231021i
\(309\) 4.91837e9 0.0306908
\(310\) 0 0
\(311\) 7.71709e10 0.467769 0.233885 0.972264i \(-0.424856\pi\)
0.233885 + 0.972264i \(0.424856\pi\)
\(312\) 9.74389e10i 0.582152i
\(313\) 4.68832e10i 0.276101i 0.990425 + 0.138050i \(0.0440836\pi\)
−0.990425 + 0.138050i \(0.955916\pi\)
\(314\) 3.87200e10 0.224777
\(315\) 0 0
\(316\) −4.33522e10 −0.244579
\(317\) 2.19420e11i 1.22042i 0.792240 + 0.610210i \(0.208915\pi\)
−0.792240 + 0.610210i \(0.791085\pi\)
\(318\) − 1.29251e11i − 0.708779i
\(319\) 3.93923e10 0.212987
\(320\) 0 0
\(321\) −8.16706e10 −0.429332
\(322\) 2.34094e11i 1.21350i
\(323\) 2.74714e10i 0.140433i
\(324\) −4.61489e9 −0.0232653
\(325\) 0 0
\(326\) 2.87609e11 1.41034
\(327\) − 1.43714e11i − 0.695077i
\(328\) 9.12401e10i 0.435265i
\(329\) 1.32953e11 0.625627
\(330\) 0 0
\(331\) 3.89075e9 0.0178159 0.00890794 0.999960i \(-0.497164\pi\)
0.00890794 + 0.999960i \(0.497164\pi\)
\(332\) − 5.54295e8i − 0.00250391i
\(333\) − 5.70383e10i − 0.254195i
\(334\) −3.74466e11 −1.64647
\(335\) 0 0
\(336\) 9.92551e10 0.424841
\(337\) − 1.48259e11i − 0.626163i −0.949726 0.313082i \(-0.898639\pi\)
0.949726 0.313082i \(-0.101361\pi\)
\(338\) 9.10225e10i 0.379336i
\(339\) −7.65851e10 −0.314953
\(340\) 0 0
\(341\) −4.34291e11 −1.73935
\(342\) 3.83556e10i 0.151604i
\(343\) − 2.59171e11i − 1.01103i
\(344\) −8.69779e10 −0.334885
\(345\) 0 0
\(346\) −5.55302e11 −2.08299
\(347\) − 1.63695e11i − 0.606114i −0.952972 0.303057i \(-0.901993\pi\)
0.952972 0.303057i \(-0.0980072\pi\)
\(348\) 4.03096e9i 0.0147333i
\(349\) −7.02057e10 −0.253313 −0.126657 0.991947i \(-0.540425\pi\)
−0.126657 + 0.991947i \(0.540425\pi\)
\(350\) 0 0
\(351\) 6.34675e10 0.223187
\(352\) 2.07549e11i 0.720574i
\(353\) 1.55727e10i 0.0533798i 0.999644 + 0.0266899i \(0.00849667\pi\)
−0.999644 + 0.0266899i \(0.991503\pi\)
\(354\) 3.01143e11 1.01920
\(355\) 0 0
\(356\) 4.63392e10 0.152906
\(357\) − 3.79859e10i − 0.123770i
\(358\) − 4.31843e11i − 1.38948i
\(359\) −3.28525e10 −0.104386 −0.0521931 0.998637i \(-0.516621\pi\)
−0.0521931 + 0.998637i \(0.516621\pi\)
\(360\) 0 0
\(361\) −2.67495e11 −0.828959
\(362\) 3.33543e11i 1.02085i
\(363\) − 3.92323e11i − 1.18594i
\(364\) 5.13472e10 0.153306
\(365\) 0 0
\(366\) −3.11679e11 −0.907908
\(367\) − 4.76030e11i − 1.36974i −0.728667 0.684868i \(-0.759860\pi\)
0.728667 0.684868i \(-0.240140\pi\)
\(368\) 7.16707e11i 2.03717i
\(369\) 5.94299e10 0.166873
\(370\) 0 0
\(371\) 2.57175e11 0.704768
\(372\) − 4.44404e10i − 0.120319i
\(373\) 9.72745e10i 0.260201i 0.991501 + 0.130101i \(0.0415300\pi\)
−0.991501 + 0.130101i \(0.958470\pi\)
\(374\) 2.46926e11 0.652596
\(375\) 0 0
\(376\) 3.33925e11 0.861597
\(377\) − 5.54367e10i − 0.141339i
\(378\) − 5.30361e10i − 0.133616i
\(379\) −4.45171e11 −1.10828 −0.554141 0.832423i \(-0.686953\pi\)
−0.554141 + 0.832423i \(0.686953\pi\)
\(380\) 0 0
\(381\) −4.21004e11 −1.02358
\(382\) 1.49630e11i 0.359528i
\(383\) − 2.19414e11i − 0.521039i −0.965469 0.260519i \(-0.916106\pi\)
0.965469 0.260519i \(-0.0838938\pi\)
\(384\) 2.94075e11 0.690187
\(385\) 0 0
\(386\) 1.72157e11 0.394714
\(387\) 5.66536e10i 0.128389i
\(388\) 1.41584e11i 0.317155i
\(389\) 3.71956e11 0.823603 0.411801 0.911274i \(-0.364900\pi\)
0.411801 + 0.911274i \(0.364900\pi\)
\(390\) 0 0
\(391\) 2.74291e11 0.593494
\(392\) − 2.44462e11i − 0.522907i
\(393\) − 4.28011e10i − 0.0905082i
\(394\) 4.13417e11 0.864281
\(395\) 0 0
\(396\) 5.96900e10 0.121976
\(397\) − 1.38786e11i − 0.280406i −0.990123 0.140203i \(-0.955224\pi\)
0.990123 0.140203i \(-0.0447756\pi\)
\(398\) − 7.63216e11i − 1.52466i
\(399\) −7.63177e10 −0.150747
\(400\) 0 0
\(401\) 6.95127e11 1.34250 0.671251 0.741231i \(-0.265757\pi\)
0.671251 + 0.741231i \(0.265757\pi\)
\(402\) − 5.49763e11i − 1.04993i
\(403\) 6.11178e11i 1.15424i
\(404\) −5.68686e10 −0.106208
\(405\) 0 0
\(406\) −4.63253e10 −0.0846157
\(407\) 7.37745e11i 1.33270i
\(408\) − 9.54059e10i − 0.170453i
\(409\) −9.10800e11 −1.60942 −0.804708 0.593671i \(-0.797678\pi\)
−0.804708 + 0.593671i \(0.797678\pi\)
\(410\) 0 0
\(411\) 4.06426e11 0.702577
\(412\) − 6.50966e9i − 0.0111307i
\(413\) 5.99196e11i 1.01343i
\(414\) 3.82967e11 0.640707
\(415\) 0 0
\(416\) 2.92084e11 0.478175
\(417\) 2.85016e11i 0.461591i
\(418\) − 4.96100e11i − 0.794832i
\(419\) −3.11152e11 −0.493184 −0.246592 0.969119i \(-0.579311\pi\)
−0.246592 + 0.969119i \(0.579311\pi\)
\(420\) 0 0
\(421\) 7.63941e11 1.18520 0.592598 0.805498i \(-0.298102\pi\)
0.592598 + 0.805498i \(0.298102\pi\)
\(422\) − 9.59516e11i − 1.47281i
\(423\) − 2.17505e11i − 0.330321i
\(424\) 6.45924e11 0.970588
\(425\) 0 0
\(426\) 7.19414e11 1.05836
\(427\) − 6.20159e11i − 0.902771i
\(428\) 1.08094e11i 0.155706i
\(429\) −8.20901e11 −1.17013
\(430\) 0 0
\(431\) 6.54629e10 0.0913792 0.0456896 0.998956i \(-0.485451\pi\)
0.0456896 + 0.998956i \(0.485451\pi\)
\(432\) − 1.62377e11i − 0.224309i
\(433\) 7.87434e11i 1.07651i 0.842781 + 0.538256i \(0.180917\pi\)
−0.842781 + 0.538256i \(0.819083\pi\)
\(434\) 5.10726e11 0.691010
\(435\) 0 0
\(436\) −1.90210e11 −0.252084
\(437\) − 5.51080e11i − 0.722850i
\(438\) 4.15723e11i 0.539722i
\(439\) 8.39211e11 1.07840 0.539201 0.842177i \(-0.318726\pi\)
0.539201 + 0.842177i \(0.318726\pi\)
\(440\) 0 0
\(441\) −1.59232e11 −0.200473
\(442\) − 3.47499e11i − 0.433065i
\(443\) 1.28252e12i 1.58215i 0.611720 + 0.791074i \(0.290478\pi\)
−0.611720 + 0.791074i \(0.709522\pi\)
\(444\) −7.54924e10 −0.0921891
\(445\) 0 0
\(446\) 7.74777e11 0.927192
\(447\) 3.50581e11i 0.415340i
\(448\) 3.83313e11i 0.449575i
\(449\) −8.02030e10 −0.0931284 −0.0465642 0.998915i \(-0.514827\pi\)
−0.0465642 + 0.998915i \(0.514827\pi\)
\(450\) 0 0
\(451\) −7.68678e11 −0.874883
\(452\) 1.01363e11i 0.114224i
\(453\) 4.55084e11i 0.507749i
\(454\) −2.85143e11 −0.315000
\(455\) 0 0
\(456\) −1.91680e11 −0.207604
\(457\) 1.27085e11i 0.136292i 0.997675 + 0.0681461i \(0.0217084\pi\)
−0.997675 + 0.0681461i \(0.978292\pi\)
\(458\) − 7.57853e11i − 0.804804i
\(459\) −6.21433e10 −0.0653487
\(460\) 0 0
\(461\) −7.39880e11 −0.762970 −0.381485 0.924375i \(-0.624587\pi\)
−0.381485 + 0.924375i \(0.624587\pi\)
\(462\) 6.85980e11i 0.700523i
\(463\) − 5.71025e11i − 0.577485i −0.957407 0.288742i \(-0.906763\pi\)
0.957407 0.288742i \(-0.0932371\pi\)
\(464\) −1.41831e11 −0.142049
\(465\) 0 0
\(466\) −7.05782e10 −0.0693320
\(467\) 1.19075e12i 1.15850i 0.815150 + 0.579249i \(0.196654\pi\)
−0.815150 + 0.579249i \(0.803346\pi\)
\(468\) − 8.40017e10i − 0.0809434i
\(469\) 1.09389e12 1.04398
\(470\) 0 0
\(471\) 1.26038e11 0.118007
\(472\) 1.50495e12i 1.39567i
\(473\) − 7.32770e11i − 0.673120i
\(474\) −8.15065e11 −0.741634
\(475\) 0 0
\(476\) −5.02759e10 −0.0448878
\(477\) − 4.20727e11i − 0.372107i
\(478\) − 1.55739e12i − 1.36449i
\(479\) −1.46074e12 −1.26784 −0.633919 0.773400i \(-0.718555\pi\)
−0.633919 + 0.773400i \(0.718555\pi\)
\(480\) 0 0
\(481\) 1.03823e12 0.884382
\(482\) − 1.80163e12i − 1.52038i
\(483\) 7.62003e11i 0.637081i
\(484\) −5.19254e11 −0.430107
\(485\) 0 0
\(486\) −8.67647e10 −0.0705472
\(487\) − 1.46931e12i − 1.18368i −0.806056 0.591839i \(-0.798402\pi\)
0.806056 0.591839i \(-0.201598\pi\)
\(488\) − 1.55760e12i − 1.24327i
\(489\) 9.36201e11 0.740422
\(490\) 0 0
\(491\) −1.76138e12 −1.36769 −0.683843 0.729629i \(-0.739693\pi\)
−0.683843 + 0.729629i \(0.739693\pi\)
\(492\) − 7.86577e10i − 0.0605199i
\(493\) 5.42801e10i 0.0413837i
\(494\) −6.98161e11 −0.527453
\(495\) 0 0
\(496\) 1.56365e12 1.16004
\(497\) 1.43145e12i 1.05238i
\(498\) − 1.04213e10i − 0.00759259i
\(499\) 1.63016e12 1.17700 0.588502 0.808496i \(-0.299718\pi\)
0.588502 + 0.808496i \(0.299718\pi\)
\(500\) 0 0
\(501\) −1.21893e12 −0.864391
\(502\) 1.40825e12i 0.989718i
\(503\) − 9.47625e11i − 0.660056i −0.943971 0.330028i \(-0.892942\pi\)
0.943971 0.330028i \(-0.107058\pi\)
\(504\) 2.65045e11 0.182971
\(505\) 0 0
\(506\) −4.95337e12 −3.35910
\(507\) 2.96289e11i 0.199150i
\(508\) 5.57214e11i 0.371224i
\(509\) 2.90709e11 0.191968 0.0959838 0.995383i \(-0.469400\pi\)
0.0959838 + 0.995383i \(0.469400\pi\)
\(510\) 0 0
\(511\) −8.27180e11 −0.536668
\(512\) 8.28486e11i 0.532808i
\(513\) 1.24852e11i 0.0795918i
\(514\) 1.23346e12 0.779457
\(515\) 0 0
\(516\) 7.49833e10 0.0465630
\(517\) 2.81325e12i 1.73181i
\(518\) − 8.67587e11i − 0.529455i
\(519\) −1.80757e12 −1.09356
\(520\) 0 0
\(521\) −2.04811e12 −1.21782 −0.608910 0.793240i \(-0.708393\pi\)
−0.608910 + 0.793240i \(0.708393\pi\)
\(522\) 7.57860e10i 0.0446758i
\(523\) − 8.49299e10i − 0.0496367i −0.999692 0.0248184i \(-0.992099\pi\)
0.999692 0.0248184i \(-0.00790074\pi\)
\(524\) −5.66488e10 −0.0328247
\(525\) 0 0
\(526\) −8.94570e11 −0.509540
\(527\) − 5.98426e11i − 0.337958i
\(528\) 2.10021e12i 1.17601i
\(529\) −3.70117e12 −2.05489
\(530\) 0 0
\(531\) 9.80258e11 0.535075
\(532\) 1.01009e11i 0.0546713i
\(533\) 1.08176e12i 0.580575i
\(534\) 8.71224e11 0.463654
\(535\) 0 0
\(536\) 2.74741e12 1.43775
\(537\) − 1.40570e12i − 0.729472i
\(538\) − 1.76588e12i − 0.908745i
\(539\) 2.05954e12 1.05104
\(540\) 0 0
\(541\) −1.29101e12 −0.647949 −0.323974 0.946066i \(-0.605019\pi\)
−0.323974 + 0.946066i \(0.605019\pi\)
\(542\) − 3.66292e12i − 1.82318i
\(543\) 1.08572e12i 0.535944i
\(544\) −2.85989e11 −0.140009
\(545\) 0 0
\(546\) 9.65379e11 0.464870
\(547\) − 5.30222e11i − 0.253230i −0.991952 0.126615i \(-0.959589\pi\)
0.991952 0.126615i \(-0.0404112\pi\)
\(548\) − 5.37921e11i − 0.254804i
\(549\) −1.01455e12 −0.476649
\(550\) 0 0
\(551\) 1.09054e11 0.0504035
\(552\) 1.91386e12i 0.877371i
\(553\) − 1.62177e12i − 0.737438i
\(554\) −7.15002e11 −0.322488
\(555\) 0 0
\(556\) 3.77230e11 0.167405
\(557\) 8.17386e11i 0.359815i 0.983684 + 0.179907i \(0.0575798\pi\)
−0.983684 + 0.179907i \(0.942420\pi\)
\(558\) − 8.35525e11i − 0.364842i
\(559\) −1.03123e12 −0.446684
\(560\) 0 0
\(561\) 8.03773e11 0.342611
\(562\) − 1.28667e12i − 0.544069i
\(563\) 1.42197e12i 0.596488i 0.954490 + 0.298244i \(0.0964009\pi\)
−0.954490 + 0.298244i \(0.903599\pi\)
\(564\) −2.87876e11 −0.119798
\(565\) 0 0
\(566\) 8.02339e10 0.0328613
\(567\) − 1.72639e11i − 0.0701480i
\(568\) 3.59523e12i 1.44930i
\(569\) 1.04614e12 0.418394 0.209197 0.977874i \(-0.432915\pi\)
0.209197 + 0.977874i \(0.432915\pi\)
\(570\) 0 0
\(571\) 2.09508e12 0.824780 0.412390 0.911007i \(-0.364694\pi\)
0.412390 + 0.911007i \(0.364694\pi\)
\(572\) 1.08649e12i 0.424371i
\(573\) 4.87063e11i 0.188751i
\(574\) 9.03965e11 0.347575
\(575\) 0 0
\(576\) 6.27082e11 0.237369
\(577\) 2.68760e12i 1.00942i 0.863288 + 0.504711i \(0.168401\pi\)
−0.863288 + 0.504711i \(0.831599\pi\)
\(578\) − 2.61068e12i − 0.972921i
\(579\) 5.60394e11 0.207224
\(580\) 0 0
\(581\) 2.07357e10 0.00754962
\(582\) 2.66192e12i 0.961705i
\(583\) 5.44176e12i 1.95088i
\(584\) −2.07755e12 −0.739085
\(585\) 0 0
\(586\) 2.82658e12 0.990199
\(587\) 1.61925e12i 0.562913i 0.959574 + 0.281457i \(0.0908176\pi\)
−0.959574 + 0.281457i \(0.909182\pi\)
\(588\) 2.10750e11i 0.0727058i
\(589\) −1.20230e12 −0.411618
\(590\) 0 0
\(591\) 1.34572e12 0.453745
\(592\) − 2.65623e12i − 0.888828i
\(593\) − 5.82677e12i − 1.93500i −0.252863 0.967502i \(-0.581372\pi\)
0.252863 0.967502i \(-0.418628\pi\)
\(594\) 1.12223e12 0.369865
\(595\) 0 0
\(596\) 4.64007e11 0.150632
\(597\) − 2.48436e12i − 0.800442i
\(598\) 6.97087e12i 2.22911i
\(599\) 3.89086e12 1.23488 0.617441 0.786617i \(-0.288169\pi\)
0.617441 + 0.786617i \(0.288169\pi\)
\(600\) 0 0
\(601\) 2.13051e12 0.666115 0.333058 0.942907i \(-0.391920\pi\)
0.333058 + 0.942907i \(0.391920\pi\)
\(602\) 8.61737e11i 0.267418i
\(603\) − 1.78955e12i − 0.551207i
\(604\) 6.02321e11 0.184146
\(605\) 0 0
\(606\) −1.06919e12 −0.322053
\(607\) − 2.52733e12i − 0.755637i −0.925880 0.377818i \(-0.876674\pi\)
0.925880 0.377818i \(-0.123326\pi\)
\(608\) 5.74582e11i 0.170524i
\(609\) −1.50794e11 −0.0444230
\(610\) 0 0
\(611\) 3.95908e12 1.14924
\(612\) 8.22490e10i 0.0237000i
\(613\) − 1.89788e12i − 0.542870i −0.962457 0.271435i \(-0.912502\pi\)
0.962457 0.271435i \(-0.0874983\pi\)
\(614\) −5.73681e11 −0.162897
\(615\) 0 0
\(616\) −3.42815e12 −0.959283
\(617\) − 1.40480e12i − 0.390240i −0.980779 0.195120i \(-0.937490\pi\)
0.980779 0.195120i \(-0.0625096\pi\)
\(618\) − 1.22388e11i − 0.0337514i
\(619\) 2.81250e12 0.769989 0.384995 0.922919i \(-0.374203\pi\)
0.384995 + 0.922919i \(0.374203\pi\)
\(620\) 0 0
\(621\) 1.24660e12 0.336369
\(622\) − 1.92031e12i − 0.514416i
\(623\) 1.73351e12i 0.461030i
\(624\) 2.95563e12 0.780404
\(625\) 0 0
\(626\) 1.16664e12 0.303634
\(627\) − 1.61486e12i − 0.417284i
\(628\) − 1.66817e11i − 0.0427977i
\(629\) −1.01657e12 −0.258945
\(630\) 0 0
\(631\) −1.92653e12 −0.483775 −0.241888 0.970304i \(-0.577767\pi\)
−0.241888 + 0.970304i \(0.577767\pi\)
\(632\) − 4.07325e12i − 1.01558i
\(633\) − 3.12334e12i − 0.773219i
\(634\) 5.46001e12 1.34212
\(635\) 0 0
\(636\) −5.56848e11 −0.134952
\(637\) − 2.89839e12i − 0.697476i
\(638\) − 9.80232e11i − 0.234226i
\(639\) 2.34178e12 0.555638
\(640\) 0 0
\(641\) −7.10318e12 −1.66185 −0.830924 0.556385i \(-0.812188\pi\)
−0.830924 + 0.556385i \(0.812188\pi\)
\(642\) 2.03228e12i 0.472146i
\(643\) − 3.40733e12i − 0.786077i −0.919522 0.393038i \(-0.871424\pi\)
0.919522 0.393038i \(-0.128576\pi\)
\(644\) 1.00854e12 0.231051
\(645\) 0 0
\(646\) 6.83594e11 0.154437
\(647\) − 4.31347e12i − 0.967737i −0.875141 0.483868i \(-0.839231\pi\)
0.875141 0.483868i \(-0.160769\pi\)
\(648\) − 4.33602e11i − 0.0966059i
\(649\) −1.26789e13 −2.80530
\(650\) 0 0
\(651\) 1.66248e12 0.362778
\(652\) − 1.23910e12i − 0.268529i
\(653\) 2.53169e12i 0.544880i 0.962173 + 0.272440i \(0.0878306\pi\)
−0.962173 + 0.272440i \(0.912169\pi\)
\(654\) −3.57615e12 −0.764391
\(655\) 0 0
\(656\) 2.76760e12 0.583494
\(657\) 1.35323e12i 0.283352i
\(658\) − 3.30838e12i − 0.688016i
\(659\) 4.43263e12 0.915540 0.457770 0.889071i \(-0.348648\pi\)
0.457770 + 0.889071i \(0.348648\pi\)
\(660\) 0 0
\(661\) 9.79827e11 0.199638 0.0998189 0.995006i \(-0.468174\pi\)
0.0998189 + 0.995006i \(0.468174\pi\)
\(662\) − 9.68168e10i − 0.0195925i
\(663\) − 1.13115e12i − 0.227358i
\(664\) 5.20799e10 0.0103971
\(665\) 0 0
\(666\) −1.41933e12 −0.279544
\(667\) − 1.08887e12i − 0.213014i
\(668\) 1.61331e12i 0.313489i
\(669\) 2.52199e12 0.486773
\(670\) 0 0
\(671\) 1.31224e13 2.49898
\(672\) − 7.94502e11i − 0.150291i
\(673\) − 7.69434e12i − 1.44578i −0.690961 0.722892i \(-0.742812\pi\)
0.690961 0.722892i \(-0.257188\pi\)
\(674\) −3.68927e12 −0.688605
\(675\) 0 0
\(676\) 3.92151e11 0.0722258
\(677\) 2.45111e12i 0.448451i 0.974537 + 0.224225i \(0.0719852\pi\)
−0.974537 + 0.224225i \(0.928015\pi\)
\(678\) 1.90573e12i 0.346360i
\(679\) −5.29653e12 −0.956264
\(680\) 0 0
\(681\) −9.28174e11 −0.165374
\(682\) 1.08068e13i 1.91280i
\(683\) − 3.13070e12i − 0.550488i −0.961374 0.275244i \(-0.911241\pi\)
0.961374 0.275244i \(-0.0887587\pi\)
\(684\) 1.65247e11 0.0288656
\(685\) 0 0
\(686\) −6.44918e12 −1.11185
\(687\) − 2.46690e12i − 0.422519i
\(688\) 2.63831e12i 0.448930i
\(689\) 7.65819e12 1.29461
\(690\) 0 0
\(691\) −8.92815e12 −1.48974 −0.744870 0.667210i \(-0.767488\pi\)
−0.744870 + 0.667210i \(0.767488\pi\)
\(692\) 2.39239e12i 0.396602i
\(693\) 2.23295e12i 0.367772i
\(694\) −4.07338e12 −0.666556
\(695\) 0 0
\(696\) −3.78737e11 −0.0611781
\(697\) − 1.05919e12i − 0.169991i
\(698\) 1.74699e12i 0.278574i
\(699\) −2.29741e11 −0.0363991
\(700\) 0 0
\(701\) −1.44295e12 −0.225695 −0.112847 0.993612i \(-0.535997\pi\)
−0.112847 + 0.993612i \(0.535997\pi\)
\(702\) − 1.57932e12i − 0.245444i
\(703\) 2.04239e12i 0.315383i
\(704\) −8.11081e12 −1.24448
\(705\) 0 0
\(706\) 3.87509e11 0.0587030
\(707\) − 2.12740e12i − 0.320230i
\(708\) − 1.29741e12i − 0.194056i
\(709\) 7.40550e12 1.10064 0.550321 0.834953i \(-0.314505\pi\)
0.550321 + 0.834953i \(0.314505\pi\)
\(710\) 0 0
\(711\) −2.65314e12 −0.389355
\(712\) 4.35390e12i 0.634919i
\(713\) 1.20045e13i 1.73957i
\(714\) −9.45237e11 −0.136113
\(715\) 0 0
\(716\) −1.86050e12 −0.264558
\(717\) − 5.06949e12i − 0.716354i
\(718\) 8.17496e11i 0.114796i
\(719\) 3.90289e12 0.544636 0.272318 0.962207i \(-0.412210\pi\)
0.272318 + 0.962207i \(0.412210\pi\)
\(720\) 0 0
\(721\) 2.43520e11 0.0335604
\(722\) 6.65631e12i 0.911624i
\(723\) − 5.86452e12i − 0.798197i
\(724\) 1.43699e12 0.194371
\(725\) 0 0
\(726\) −9.76251e12 −1.30421
\(727\) 1.33674e12i 0.177477i 0.996055 + 0.0887384i \(0.0282835\pi\)
−0.996055 + 0.0887384i \(0.971716\pi\)
\(728\) 4.82443e12i 0.636583i
\(729\) −2.82430e11 −0.0370370
\(730\) 0 0
\(731\) 1.00971e12 0.130788
\(732\) 1.34280e12i 0.172866i
\(733\) 5.98863e11i 0.0766231i 0.999266 + 0.0383116i \(0.0121979\pi\)
−0.999266 + 0.0383116i \(0.987802\pi\)
\(734\) −1.18455e13 −1.50633
\(735\) 0 0
\(736\) 5.73699e12 0.720665
\(737\) 2.31464e13i 2.88987i
\(738\) − 1.47884e12i − 0.183514i
\(739\) 1.40697e13 1.73534 0.867671 0.497138i \(-0.165616\pi\)
0.867671 + 0.497138i \(0.165616\pi\)
\(740\) 0 0
\(741\) −2.27260e12 −0.276911
\(742\) − 6.39951e12i − 0.775049i
\(743\) 2.52604e12i 0.304082i 0.988374 + 0.152041i \(0.0485846\pi\)
−0.988374 + 0.152041i \(0.951415\pi\)
\(744\) 4.17549e12 0.499608
\(745\) 0 0
\(746\) 2.42056e12 0.286149
\(747\) − 3.39226e10i − 0.00398608i
\(748\) − 1.06383e12i − 0.124255i
\(749\) −4.04371e12 −0.469474
\(750\) 0 0
\(751\) 3.78704e12 0.434430 0.217215 0.976124i \(-0.430303\pi\)
0.217215 + 0.976124i \(0.430303\pi\)
\(752\) − 1.01290e13i − 1.15501i
\(753\) 4.58401e12i 0.519599i
\(754\) −1.37948e12 −0.155433
\(755\) 0 0
\(756\) −2.28494e11 −0.0254406
\(757\) 3.55646e12i 0.393629i 0.980441 + 0.196814i \(0.0630596\pi\)
−0.980441 + 0.196814i \(0.936940\pi\)
\(758\) 1.10776e13i 1.21880i
\(759\) −1.61238e13 −1.76352
\(760\) 0 0
\(761\) 1.67100e13 1.80612 0.903058 0.429519i \(-0.141317\pi\)
0.903058 + 0.429519i \(0.141317\pi\)
\(762\) 1.04762e13i 1.12566i
\(763\) − 7.11560e12i − 0.760066i
\(764\) 6.44647e11 0.0684544
\(765\) 0 0
\(766\) −5.45987e12 −0.572997
\(767\) 1.78429e13i 1.86160i
\(768\) − 3.35393e12i − 0.347880i
\(769\) 7.51512e12 0.774939 0.387469 0.921883i \(-0.373349\pi\)
0.387469 + 0.921883i \(0.373349\pi\)
\(770\) 0 0
\(771\) 4.01507e12 0.409213
\(772\) − 7.41702e11i − 0.0751540i
\(773\) 8.52580e12i 0.858870i 0.903098 + 0.429435i \(0.141287\pi\)
−0.903098 + 0.429435i \(0.858713\pi\)
\(774\) 1.40976e12 0.141192
\(775\) 0 0
\(776\) −1.33028e13 −1.31694
\(777\) − 2.82410e12i − 0.277962i
\(778\) − 9.25569e12i − 0.905734i
\(779\) −2.12802e12 −0.207042
\(780\) 0 0
\(781\) −3.02890e13 −2.91310
\(782\) − 6.82543e12i − 0.652679i
\(783\) 2.46693e11i 0.0234546i
\(784\) −7.41531e12 −0.700982
\(785\) 0 0
\(786\) −1.06506e12 −0.0995338
\(787\) 3.20742e12i 0.298037i 0.988834 + 0.149018i \(0.0476114\pi\)
−0.988834 + 0.149018i \(0.952389\pi\)
\(788\) − 1.78112e12i − 0.164560i
\(789\) −2.91194e12 −0.267507
\(790\) 0 0
\(791\) −3.79191e12 −0.344401
\(792\) 5.60830e12i 0.506486i
\(793\) − 1.84672e13i − 1.65833i
\(794\) −3.45353e12 −0.308369
\(795\) 0 0
\(796\) −3.28815e12 −0.290297
\(797\) − 1.83036e13i − 1.60684i −0.595412 0.803421i \(-0.703011\pi\)
0.595412 0.803421i \(-0.296989\pi\)
\(798\) 1.89908e12i 0.165779i
\(799\) −3.87648e12 −0.336494
\(800\) 0 0
\(801\) 2.83594e12 0.243417
\(802\) − 1.72975e13i − 1.47638i
\(803\) − 1.75029e13i − 1.48556i
\(804\) −2.36853e12 −0.199907
\(805\) 0 0
\(806\) 1.52085e13 1.26934
\(807\) − 5.74817e12i − 0.477088i
\(808\) − 5.34321e12i − 0.441013i
\(809\) −9.38919e12 −0.770655 −0.385327 0.922780i \(-0.625911\pi\)
−0.385327 + 0.922780i \(0.625911\pi\)
\(810\) 0 0
\(811\) 2.03949e13 1.65549 0.827746 0.561103i \(-0.189623\pi\)
0.827746 + 0.561103i \(0.189623\pi\)
\(812\) 1.99582e11i 0.0161109i
\(813\) − 1.19232e13i − 0.957166i
\(814\) 1.83579e13 1.46560
\(815\) 0 0
\(816\) −2.89396e12 −0.228500
\(817\) − 2.02861e12i − 0.159294i
\(818\) 2.26642e13i 1.76991i
\(819\) 3.14243e12 0.244055
\(820\) 0 0
\(821\) −1.52259e13 −1.16961 −0.584803 0.811176i \(-0.698828\pi\)
−0.584803 + 0.811176i \(0.698828\pi\)
\(822\) − 1.01135e13i − 0.772639i
\(823\) 2.70705e12i 0.205682i 0.994698 + 0.102841i \(0.0327933\pi\)
−0.994698 + 0.102841i \(0.967207\pi\)
\(824\) 6.11629e11 0.0462184
\(825\) 0 0
\(826\) 1.49103e13 1.11449
\(827\) 5.67203e10i 0.00421661i 0.999998 + 0.00210831i \(0.000671095\pi\)
−0.999998 + 0.00210831i \(0.999329\pi\)
\(828\) − 1.64993e12i − 0.121991i
\(829\) 1.43206e13 1.05309 0.526547 0.850146i \(-0.323487\pi\)
0.526547 + 0.850146i \(0.323487\pi\)
\(830\) 0 0
\(831\) −2.32742e12 −0.169305
\(832\) 1.14143e13i 0.825839i
\(833\) 2.83791e12i 0.204219i
\(834\) 7.09231e12 0.507622
\(835\) 0 0
\(836\) −2.13734e12 −0.151337
\(837\) − 2.71973e12i − 0.191541i
\(838\) 7.74265e12i 0.542365i
\(839\) 2.13693e13 1.48889 0.744443 0.667686i \(-0.232715\pi\)
0.744443 + 0.667686i \(0.232715\pi\)
\(840\) 0 0
\(841\) −1.42917e13 −0.985147
\(842\) − 1.90098e13i − 1.30339i
\(843\) − 4.18827e12i − 0.285635i
\(844\) −4.13386e12 −0.280424
\(845\) 0 0
\(846\) −5.41235e12 −0.363262
\(847\) − 1.94248e13i − 1.29683i
\(848\) − 1.95929e13i − 1.30112i
\(849\) 2.61171e11 0.0172521
\(850\) 0 0
\(851\) 2.03925e13 1.33287
\(852\) − 3.09943e12i − 0.201514i
\(853\) 1.18123e13i 0.763948i 0.924173 + 0.381974i \(0.124756\pi\)
−0.924173 + 0.381974i \(0.875244\pi\)
\(854\) −1.54319e13 −0.992797
\(855\) 0 0
\(856\) −1.01562e13 −0.646547
\(857\) 2.69459e13i 1.70640i 0.521587 + 0.853198i \(0.325340\pi\)
−0.521587 + 0.853198i \(0.674660\pi\)
\(858\) 2.04272e13i 1.28681i
\(859\) −9.66669e12 −0.605771 −0.302885 0.953027i \(-0.597950\pi\)
−0.302885 + 0.953027i \(0.597950\pi\)
\(860\) 0 0
\(861\) 2.94251e12 0.182475
\(862\) − 1.62897e12i − 0.100492i
\(863\) − 2.80982e12i − 0.172437i −0.996276 0.0862185i \(-0.972522\pi\)
0.996276 0.0862185i \(-0.0274783\pi\)
\(864\) −1.29977e12 −0.0793513
\(865\) 0 0
\(866\) 1.95944e13 1.18386
\(867\) − 8.49807e12i − 0.510781i
\(868\) − 2.20035e12i − 0.131569i
\(869\) 3.43162e13 2.04132
\(870\) 0 0
\(871\) 3.25739e13 1.91773
\(872\) − 1.78716e13i − 1.04674i
\(873\) 8.66488e12i 0.504892i
\(874\) −1.37130e13 −0.794933
\(875\) 0 0
\(876\) 1.79105e12 0.102764
\(877\) 3.37045e13i 1.92393i 0.273169 + 0.961966i \(0.411928\pi\)
−0.273169 + 0.961966i \(0.588072\pi\)
\(878\) − 2.08828e13i − 1.18594i
\(879\) 9.20087e12 0.519851
\(880\) 0 0
\(881\) −1.20571e13 −0.674297 −0.337148 0.941452i \(-0.609462\pi\)
−0.337148 + 0.941452i \(0.609462\pi\)
\(882\) 3.96231e12i 0.220465i
\(883\) 1.91595e13i 1.06062i 0.847803 + 0.530311i \(0.177925\pi\)
−0.847803 + 0.530311i \(0.822075\pi\)
\(884\) −1.49712e12 −0.0824559
\(885\) 0 0
\(886\) 3.19140e13 1.73992
\(887\) − 2.24991e13i − 1.22042i −0.792241 0.610209i \(-0.791086\pi\)
0.792241 0.610209i \(-0.208914\pi\)
\(888\) − 7.09305e12i − 0.382802i
\(889\) −2.08449e13 −1.11929
\(890\) 0 0
\(891\) 3.65300e12 0.194178
\(892\) − 3.33795e12i − 0.176538i
\(893\) 7.78825e12i 0.409834i
\(894\) 8.72380e12 0.456759
\(895\) 0 0
\(896\) 1.45603e13 0.754719
\(897\) 2.26910e13i 1.17028i
\(898\) 1.99576e12i 0.102415i
\(899\) −2.37560e12 −0.121298
\(900\) 0 0
\(901\) −7.49841e12 −0.379060
\(902\) 1.91277e13i 0.962128i
\(903\) 2.80506e12i 0.140393i
\(904\) −9.52380e12 −0.474299
\(905\) 0 0
\(906\) 1.13242e13 0.558383
\(907\) − 2.39724e13i − 1.17619i −0.808791 0.588096i \(-0.799878\pi\)
0.808791 0.588096i \(-0.200122\pi\)
\(908\) 1.22847e12i 0.0599763i
\(909\) −3.48033e12 −0.169077
\(910\) 0 0
\(911\) −6.54470e12 −0.314816 −0.157408 0.987534i \(-0.550314\pi\)
−0.157408 + 0.987534i \(0.550314\pi\)
\(912\) 5.81427e12i 0.278303i
\(913\) 4.38762e11i 0.0208983i
\(914\) 3.16236e12 0.149884
\(915\) 0 0
\(916\) −3.26504e12 −0.153235
\(917\) − 2.11918e12i − 0.0989706i
\(918\) 1.54636e12i 0.0718654i
\(919\) −2.63348e12 −0.121789 −0.0608947 0.998144i \(-0.519395\pi\)
−0.0608947 + 0.998144i \(0.519395\pi\)
\(920\) 0 0
\(921\) −1.86740e12 −0.0855203
\(922\) 1.84111e13i 0.839055i
\(923\) 4.26258e13i 1.93314i
\(924\) 2.95539e12 0.133380
\(925\) 0 0
\(926\) −1.42093e13 −0.635073
\(927\) − 3.98388e11i − 0.0177193i
\(928\) 1.13530e12i 0.0502512i
\(929\) −2.94271e13 −1.29621 −0.648106 0.761550i \(-0.724439\pi\)
−0.648106 + 0.761550i \(0.724439\pi\)
\(930\) 0 0
\(931\) 5.70166e12 0.248730
\(932\) 3.04070e11i 0.0132009i
\(933\) − 6.25084e12i − 0.270067i
\(934\) 2.96305e13 1.27403
\(935\) 0 0
\(936\) 7.89255e12 0.336106
\(937\) − 1.71769e13i − 0.727974i −0.931404 0.363987i \(-0.881415\pi\)
0.931404 0.363987i \(-0.118585\pi\)
\(938\) − 2.72201e13i − 1.14809i
\(939\) 3.79754e12 0.159407
\(940\) 0 0
\(941\) −2.46944e13 −1.02670 −0.513352 0.858178i \(-0.671596\pi\)
−0.513352 + 0.858178i \(0.671596\pi\)
\(942\) − 3.13632e12i − 0.129775i
\(943\) 2.12475e13i 0.874994i
\(944\) 4.56498e13 1.87096
\(945\) 0 0
\(946\) −1.82341e13 −0.740244
\(947\) − 1.19428e12i − 0.0482537i −0.999709 0.0241268i \(-0.992319\pi\)
0.999709 0.0241268i \(-0.00768056\pi\)
\(948\) 3.51153e12i 0.141208i
\(949\) −2.46319e13 −0.985824
\(950\) 0 0
\(951\) 1.77730e13 0.704610
\(952\) − 4.72377e12i − 0.186390i
\(953\) − 1.47694e13i − 0.580023i −0.957023 0.290011i \(-0.906341\pi\)
0.957023 0.290011i \(-0.0936591\pi\)
\(954\) −1.04693e13 −0.409214
\(955\) 0 0
\(956\) −6.70966e12 −0.259800
\(957\) − 3.19077e12i − 0.122968i
\(958\) 3.63489e13i 1.39427i
\(959\) 2.01231e13 0.768267
\(960\) 0 0
\(961\) −2.49176e11 −0.00942435
\(962\) − 2.58351e13i − 0.972574i
\(963\) 6.61532e12i 0.247875i
\(964\) −7.76192e12 −0.289483
\(965\) 0 0
\(966\) 1.89616e13 0.700612
\(967\) − 8.95584e12i − 0.329372i −0.986346 0.164686i \(-0.947339\pi\)
0.986346 0.164686i \(-0.0526611\pi\)
\(968\) − 4.87876e13i − 1.78596i
\(969\) 2.22518e12 0.0810790
\(970\) 0 0
\(971\) −2.20587e13 −0.796330 −0.398165 0.917314i \(-0.630353\pi\)
−0.398165 + 0.917314i \(0.630353\pi\)
\(972\) 3.73806e11i 0.0134322i
\(973\) 1.41118e13i 0.504749i
\(974\) −3.65621e13 −1.30172
\(975\) 0 0
\(976\) −4.72469e13 −1.66667
\(977\) 3.56972e13i 1.25345i 0.779239 + 0.626727i \(0.215606\pi\)
−0.779239 + 0.626727i \(0.784394\pi\)
\(978\) − 2.32963e13i − 0.814258i
\(979\) −3.66806e13 −1.27619
\(980\) 0 0
\(981\) −1.16408e13 −0.401303
\(982\) 4.38300e13i 1.50408i
\(983\) − 3.19525e13i − 1.09147i −0.837956 0.545737i \(-0.816250\pi\)
0.837956 0.545737i \(-0.183750\pi\)
\(984\) 7.39045e12 0.251300
\(985\) 0 0
\(986\) 1.35070e12 0.0455105
\(987\) − 1.07692e13i − 0.361206i
\(988\) 3.00787e12i 0.100428i
\(989\) −2.02549e13 −0.673205
\(990\) 0 0
\(991\) 1.56883e13 0.516706 0.258353 0.966051i \(-0.416820\pi\)
0.258353 + 0.966051i \(0.416820\pi\)
\(992\) − 1.25165e13i − 0.410374i
\(993\) − 3.15151e11i − 0.0102860i
\(994\) 3.56199e13 1.15732
\(995\) 0 0
\(996\) −4.48979e10 −0.00144563
\(997\) 6.52145e12i 0.209033i 0.994523 + 0.104517i \(0.0333296\pi\)
−0.994523 + 0.104517i \(0.966670\pi\)
\(998\) − 4.05647e13i − 1.29438i
\(999\) −4.62010e12 −0.146760
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 75.10.b.e.49.2 4
3.2 odd 2 225.10.b.g.199.3 4
5.2 odd 4 75.10.a.g.1.2 2
5.3 odd 4 15.10.a.c.1.1 2
5.4 even 2 inner 75.10.b.e.49.3 4
15.2 even 4 225.10.a.j.1.1 2
15.8 even 4 45.10.a.e.1.2 2
15.14 odd 2 225.10.b.g.199.2 4
20.3 even 4 240.10.a.m.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
15.10.a.c.1.1 2 5.3 odd 4
45.10.a.e.1.2 2 15.8 even 4
75.10.a.g.1.2 2 5.2 odd 4
75.10.b.e.49.2 4 1.1 even 1 trivial
75.10.b.e.49.3 4 5.4 even 2 inner
225.10.a.j.1.1 2 15.2 even 4
225.10.b.g.199.2 4 15.14 odd 2
225.10.b.g.199.3 4 3.2 odd 2
240.10.a.m.1.1 2 20.3 even 4