Properties

Label 75.10.a.g.1.2
Level $75$
Weight $10$
Character 75.1
Self dual yes
Analytic conductor $38.628$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [75,10,Mod(1,75)]
 
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, 0]))
 
N = Newforms(chi, 10, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("75.1");
 
S:= CuspForms(chi, 10);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 75 = 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 75.a (trivial)

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: yes
Analytic conductor: \(38.6276877123\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{4729}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 1182 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 15)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-33.8839\) of defining polynomial
Character \(\chi\) \(=\) 75.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+24.8839 q^{2} -81.0000 q^{3} +107.207 q^{4} -2015.59 q^{6} +4010.50 q^{7} -10072.8 q^{8} +6561.00 q^{9} +O(q^{10})\) \(q+24.8839 q^{2} -81.0000 q^{3} +107.207 q^{4} -2015.59 q^{6} +4010.50 q^{7} -10072.8 q^{8} +6561.00 q^{9} +84861.3 q^{11} -8683.74 q^{12} -119425. q^{13} +99796.8 q^{14} -305541. q^{16} -116934. q^{17} +163263. q^{18} -234932. q^{19} -324851. q^{21} +2.11168e6 q^{22} -2.34570e6 q^{23} +815899. q^{24} -2.97176e6 q^{26} -531441. q^{27} +429953. q^{28} -464196. q^{29} -5.11766e6 q^{31} -2.44574e6 q^{32} -6.87377e6 q^{33} -2.90976e6 q^{34} +703383. q^{36} -8.69354e6 q^{37} -5.84601e6 q^{38} +9.67345e6 q^{39} -9.05805e6 q^{41} -8.08354e6 q^{42} -8.63491e6 q^{43} +9.09769e6 q^{44} -5.83701e7 q^{46} -3.31511e7 q^{47} +2.47488e7 q^{48} -2.42695e7 q^{49} +9.47161e6 q^{51} -1.28032e7 q^{52} +6.41254e7 q^{53} -1.32243e7 q^{54} -4.03971e7 q^{56} +1.90295e7 q^{57} -1.15510e7 q^{58} +1.49407e8 q^{59} +1.54634e8 q^{61} -1.27347e8 q^{62} +2.63129e7 q^{63} +9.55772e7 q^{64} -1.71046e8 q^{66} -2.72755e8 q^{67} -1.25360e7 q^{68} +1.90002e8 q^{69} -3.56924e8 q^{71} -6.60878e7 q^{72} -2.06253e8 q^{73} -2.16329e8 q^{74} -2.51862e7 q^{76} +3.40337e8 q^{77} +2.40713e8 q^{78} -4.04380e8 q^{79} +4.30467e7 q^{81} -2.25399e8 q^{82} +5.17034e6 q^{83} -3.48262e7 q^{84} -2.14870e8 q^{86} +3.75999e7 q^{87} -8.54793e8 q^{88} +4.32242e8 q^{89} -4.78955e8 q^{91} -2.51475e8 q^{92} +4.14530e8 q^{93} -8.24928e8 q^{94} +1.98105e8 q^{96} +1.32066e9 q^{97} -6.03918e8 q^{98} +5.56775e8 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 19 q^{2} - 162 q^{3} + 1521 q^{4} + 1539 q^{6} + 11872 q^{7} - 49647 q^{8} + 13122 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 19 q^{2} - 162 q^{3} + 1521 q^{4} + 1539 q^{6} + 11872 q^{7} - 49647 q^{8} + 13122 q^{9} + 35488 q^{11} - 123201 q^{12} - 143676 q^{13} - 245196 q^{14} + 707265 q^{16} - 385156 q^{17} - 124659 q^{18} - 403296 q^{19} - 961632 q^{21} + 4278368 q^{22} - 223704 q^{23} + 4021407 q^{24} - 1907546 q^{26} - 1062882 q^{27} + 11544484 q^{28} - 74572 q^{29} - 5027128 q^{31} - 26629583 q^{32} - 2874528 q^{33} + 8860882 q^{34} + 9979281 q^{36} - 5373628 q^{37} + 1542476 q^{38} + 11637756 q^{39} + 14211332 q^{41} + 19860876 q^{42} - 27748920 q^{43} - 60705952 q^{44} - 151491648 q^{46} - 95966440 q^{47} - 57288465 q^{48} - 2819950 q^{49} + 31197636 q^{51} - 47088706 q^{52} + 64305596 q^{53} + 10097379 q^{54} - 351509340 q^{56} + 32666976 q^{57} - 28649198 q^{58} + 187863136 q^{59} + 154080060 q^{61} - 131320056 q^{62} + 77892192 q^{63} + 638301089 q^{64} - 346547808 q^{66} - 33592376 q^{67} - 391747238 q^{68} + 18120024 q^{69} - 228270976 q^{71} - 325733967 q^{72} + 33122316 q^{73} - 362019226 q^{74} - 263218724 q^{76} - 47811456 q^{77} + 154511226 q^{78} - 932406760 q^{79} + 86093442 q^{81} - 1246549646 q^{82} - 207040152 q^{83} - 935103204 q^{84} + 623926708 q^{86} + 6040332 q^{87} + 1099114848 q^{88} + 224518164 q^{89} - 669602528 q^{91} + 2748592992 q^{92} + 407197368 q^{93} + 1931650816 q^{94} + 2156996223 q^{96} - 387134596 q^{97} - 1545205739 q^{98} + 232836768 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 24.8839 1.09972 0.549861 0.835256i \(-0.314681\pi\)
0.549861 + 0.835256i \(0.314681\pi\)
\(3\) −81.0000 −0.577350
\(4\) 107.207 0.209388
\(5\) 0 0
\(6\) −2015.59 −0.634925
\(7\) 4010.50 0.631332 0.315666 0.948870i \(-0.397772\pi\)
0.315666 + 0.948870i \(0.397772\pi\)
\(8\) −10072.8 −0.869453
\(9\) 6561.00 0.333333
\(10\) 0 0
\(11\) 84861.3 1.74760 0.873801 0.486283i \(-0.161648\pi\)
0.873801 + 0.486283i \(0.161648\pi\)
\(12\) −8683.74 −0.120890
\(13\) −119425. −1.15971 −0.579857 0.814718i \(-0.696892\pi\)
−0.579857 + 0.814718i \(0.696892\pi\)
\(14\) 99796.8 0.694289
\(15\) 0 0
\(16\) −305541. −1.16554
\(17\) −116934. −0.339562 −0.169781 0.985482i \(-0.554306\pi\)
−0.169781 + 0.985482i \(0.554306\pi\)
\(18\) 163263. 0.366574
\(19\) −234932. −0.413571 −0.206786 0.978386i \(-0.566300\pi\)
−0.206786 + 0.978386i \(0.566300\pi\)
\(20\) 0 0
\(21\) −324851. −0.364500
\(22\) 2.11168e6 1.92188
\(23\) −2.34570e6 −1.74782 −0.873912 0.486084i \(-0.838425\pi\)
−0.873912 + 0.486084i \(0.838425\pi\)
\(24\) 815899. 0.501979
\(25\) 0 0
\(26\) −2.97176e6 −1.27536
\(27\) −531441. −0.192450
\(28\) 429953. 0.132193
\(29\) −464196. −0.121874 −0.0609369 0.998142i \(-0.519409\pi\)
−0.0609369 + 0.998142i \(0.519409\pi\)
\(30\) 0 0
\(31\) −5.11766e6 −0.995277 −0.497638 0.867385i \(-0.665799\pi\)
−0.497638 + 0.867385i \(0.665799\pi\)
\(32\) −2.44574e6 −0.412321
\(33\) −6.87377e6 −1.00898
\(34\) −2.90976e6 −0.373423
\(35\) 0 0
\(36\) 703383. 0.0697960
\(37\) −8.69354e6 −0.762586 −0.381293 0.924454i \(-0.624521\pi\)
−0.381293 + 0.924454i \(0.624521\pi\)
\(38\) −5.84601e6 −0.454813
\(39\) 9.67345e6 0.669562
\(40\) 0 0
\(41\) −9.05805e6 −0.500619 −0.250310 0.968166i \(-0.580532\pi\)
−0.250310 + 0.968166i \(0.580532\pi\)
\(42\) −8.08354e6 −0.400848
\(43\) −8.63491e6 −0.385168 −0.192584 0.981281i \(-0.561687\pi\)
−0.192584 + 0.981281i \(0.561687\pi\)
\(44\) 9.09769e6 0.365927
\(45\) 0 0
\(46\) −5.83701e7 −1.92212
\(47\) −3.31511e7 −0.990964 −0.495482 0.868618i \(-0.665009\pi\)
−0.495482 + 0.868618i \(0.665009\pi\)
\(48\) 2.47488e7 0.672927
\(49\) −2.42695e7 −0.601420
\(50\) 0 0
\(51\) 9.47161e6 0.196046
\(52\) −1.28032e7 −0.242830
\(53\) 6.41254e7 1.11632 0.558160 0.829733i \(-0.311508\pi\)
0.558160 + 0.829733i \(0.311508\pi\)
\(54\) −1.32243e7 −0.211642
\(55\) 0 0
\(56\) −4.03971e7 −0.548914
\(57\) 1.90295e7 0.238775
\(58\) −1.15510e7 −0.134027
\(59\) 1.49407e8 1.60523 0.802613 0.596500i \(-0.203442\pi\)
0.802613 + 0.596500i \(0.203442\pi\)
\(60\) 0 0
\(61\) 1.54634e8 1.42995 0.714973 0.699152i \(-0.246439\pi\)
0.714973 + 0.699152i \(0.246439\pi\)
\(62\) −1.27347e8 −1.09453
\(63\) 2.63129e7 0.210444
\(64\) 9.55772e7 0.712106
\(65\) 0 0
\(66\) −1.71046e8 −1.10960
\(67\) −2.72755e8 −1.65362 −0.826811 0.562479i \(-0.809848\pi\)
−0.826811 + 0.562479i \(0.809848\pi\)
\(68\) −1.25360e7 −0.0711001
\(69\) 1.90002e8 1.00911
\(70\) 0 0
\(71\) −3.56924e8 −1.66691 −0.833457 0.552584i \(-0.813642\pi\)
−0.833457 + 0.552584i \(0.813642\pi\)
\(72\) −6.60878e7 −0.289818
\(73\) −2.06253e8 −0.850057 −0.425029 0.905180i \(-0.639736\pi\)
−0.425029 + 0.905180i \(0.639736\pi\)
\(74\) −2.16329e8 −0.838632
\(75\) 0 0
\(76\) −2.51862e7 −0.0865968
\(77\) 3.40337e8 1.10332
\(78\) 2.40713e8 0.736331
\(79\) −4.04380e8 −1.16807 −0.584033 0.811730i \(-0.698526\pi\)
−0.584033 + 0.811730i \(0.698526\pi\)
\(80\) 0 0
\(81\) 4.30467e7 0.111111
\(82\) −2.25399e8 −0.550542
\(83\) 5.17034e6 0.0119582 0.00597912 0.999982i \(-0.498097\pi\)
0.00597912 + 0.999982i \(0.498097\pi\)
\(84\) −3.48262e7 −0.0763218
\(85\) 0 0
\(86\) −2.14870e8 −0.423577
\(87\) 3.75999e7 0.0703639
\(88\) −8.54793e8 −1.51946
\(89\) 4.32242e8 0.730250 0.365125 0.930958i \(-0.381026\pi\)
0.365125 + 0.930958i \(0.381026\pi\)
\(90\) 0 0
\(91\) −4.78955e8 −0.732165
\(92\) −2.51475e8 −0.365973
\(93\) 4.14530e8 0.574623
\(94\) −8.24928e8 −1.08978
\(95\) 0 0
\(96\) 1.98105e8 0.238054
\(97\) 1.32066e9 1.51468 0.757338 0.653023i \(-0.226499\pi\)
0.757338 + 0.653023i \(0.226499\pi\)
\(98\) −6.03918e8 −0.661395
\(99\) 5.56775e8 0.582534
\(100\) 0 0
\(101\) 5.30458e8 0.507230 0.253615 0.967305i \(-0.418380\pi\)
0.253615 + 0.967305i \(0.418380\pi\)
\(102\) 2.35690e8 0.215596
\(103\) 6.07207e7 0.0531580 0.0265790 0.999647i \(-0.491539\pi\)
0.0265790 + 0.999647i \(0.491539\pi\)
\(104\) 1.20295e9 1.00832
\(105\) 0 0
\(106\) 1.59569e9 1.22764
\(107\) 1.00828e9 0.743625 0.371812 0.928308i \(-0.378736\pi\)
0.371812 + 0.928308i \(0.378736\pi\)
\(108\) −5.69740e7 −0.0402967
\(109\) −1.77424e9 −1.20391 −0.601954 0.798531i \(-0.705611\pi\)
−0.601954 + 0.798531i \(0.705611\pi\)
\(110\) 0 0
\(111\) 7.04177e8 0.440279
\(112\) −1.22537e9 −0.735846
\(113\) −9.45495e8 −0.545514 −0.272757 0.962083i \(-0.587936\pi\)
−0.272757 + 0.962083i \(0.587936\pi\)
\(114\) 4.73526e8 0.262586
\(115\) 0 0
\(116\) −4.97649e7 −0.0255189
\(117\) −7.83549e8 −0.386572
\(118\) 3.71782e9 1.76530
\(119\) −4.68962e8 −0.214376
\(120\) 0 0
\(121\) 4.84349e9 2.05411
\(122\) 3.84788e9 1.57254
\(123\) 7.33702e8 0.289033
\(124\) −5.48647e8 −0.208399
\(125\) 0 0
\(126\) 6.54767e8 0.231430
\(127\) 5.19758e9 1.77290 0.886450 0.462825i \(-0.153164\pi\)
0.886450 + 0.462825i \(0.153164\pi\)
\(128\) 3.63055e9 1.19544
\(129\) 6.99428e8 0.222377
\(130\) 0 0
\(131\) 5.28408e8 0.156765 0.0783824 0.996923i \(-0.475024\pi\)
0.0783824 + 0.996923i \(0.475024\pi\)
\(132\) −7.36913e8 −0.211268
\(133\) −9.42194e8 −0.261101
\(134\) −6.78720e9 −1.81852
\(135\) 0 0
\(136\) 1.17785e9 0.295233
\(137\) −5.01761e9 −1.21690 −0.608449 0.793593i \(-0.708208\pi\)
−0.608449 + 0.793593i \(0.708208\pi\)
\(138\) 4.72798e9 1.10974
\(139\) 3.51872e9 0.799499 0.399750 0.916624i \(-0.369097\pi\)
0.399750 + 0.916624i \(0.369097\pi\)
\(140\) 0 0
\(141\) 2.68524e9 0.572133
\(142\) −8.88165e9 −1.83314
\(143\) −1.01346e10 −2.02672
\(144\) −2.00465e9 −0.388515
\(145\) 0 0
\(146\) −5.13238e9 −0.934827
\(147\) 1.96583e9 0.347230
\(148\) −9.32005e8 −0.159676
\(149\) 4.32815e9 0.719390 0.359695 0.933070i \(-0.382881\pi\)
0.359695 + 0.933070i \(0.382881\pi\)
\(150\) 0 0
\(151\) −5.61832e9 −0.879448 −0.439724 0.898133i \(-0.644924\pi\)
−0.439724 + 0.898133i \(0.644924\pi\)
\(152\) 2.36642e9 0.359581
\(153\) −7.67201e8 −0.113187
\(154\) 8.46889e9 1.21334
\(155\) 0 0
\(156\) 1.03706e9 0.140198
\(157\) −1.55603e9 −0.204394 −0.102197 0.994764i \(-0.532587\pi\)
−0.102197 + 0.994764i \(0.532587\pi\)
\(158\) −1.00625e10 −1.28455
\(159\) −5.19416e9 −0.644507
\(160\) 0 0
\(161\) −9.40745e9 −1.10346
\(162\) 1.07117e9 0.122191
\(163\) 1.15580e10 1.28245 0.641224 0.767354i \(-0.278427\pi\)
0.641224 + 0.767354i \(0.278427\pi\)
\(164\) −9.71083e8 −0.104824
\(165\) 0 0
\(166\) 1.28658e8 0.0131507
\(167\) 1.50486e10 1.49717 0.748585 0.663039i \(-0.230734\pi\)
0.748585 + 0.663039i \(0.230734\pi\)
\(168\) 3.27216e9 0.316915
\(169\) 3.65789e9 0.344938
\(170\) 0 0
\(171\) −1.54139e9 −0.137857
\(172\) −9.25719e8 −0.0806494
\(173\) −2.23157e10 −1.89410 −0.947052 0.321081i \(-0.895954\pi\)
−0.947052 + 0.321081i \(0.895954\pi\)
\(174\) 9.35630e8 0.0773807
\(175\) 0 0
\(176\) −2.59286e10 −2.03691
\(177\) −1.21019e10 −0.926778
\(178\) 1.07558e10 0.803072
\(179\) −1.73543e10 −1.26348 −0.631742 0.775179i \(-0.717660\pi\)
−0.631742 + 0.775179i \(0.717660\pi\)
\(180\) 0 0
\(181\) −1.34040e10 −0.928282 −0.464141 0.885761i \(-0.653637\pi\)
−0.464141 + 0.885761i \(0.653637\pi\)
\(182\) −1.19183e10 −0.805178
\(183\) −1.25253e10 −0.825580
\(184\) 2.36278e10 1.51965
\(185\) 0 0
\(186\) 1.03151e10 0.631926
\(187\) −9.92313e9 −0.593419
\(188\) −3.55402e9 −0.207496
\(189\) −2.13135e9 −0.121500
\(190\) 0 0
\(191\) −6.01312e9 −0.326926 −0.163463 0.986549i \(-0.552266\pi\)
−0.163463 + 0.986549i \(0.552266\pi\)
\(192\) −7.74175e9 −0.411134
\(193\) 6.91844e9 0.358922 0.179461 0.983765i \(-0.442565\pi\)
0.179461 + 0.983765i \(0.442565\pi\)
\(194\) 3.28632e10 1.66572
\(195\) 0 0
\(196\) −2.60185e9 −0.125930
\(197\) −1.66139e10 −0.785909 −0.392955 0.919558i \(-0.628547\pi\)
−0.392955 + 0.919558i \(0.628547\pi\)
\(198\) 1.38547e10 0.640625
\(199\) −3.06711e10 −1.38641 −0.693204 0.720742i \(-0.743801\pi\)
−0.693204 + 0.720742i \(0.743801\pi\)
\(200\) 0 0
\(201\) 2.20932e10 0.954719
\(202\) 1.31998e10 0.557812
\(203\) −1.86166e9 −0.0769428
\(204\) 1.01542e9 0.0410497
\(205\) 0 0
\(206\) 1.51096e9 0.0584591
\(207\) −1.53902e10 −0.582608
\(208\) 3.64893e10 1.35170
\(209\) −1.99366e10 −0.722758
\(210\) 0 0
\(211\) 3.85598e10 1.33926 0.669628 0.742697i \(-0.266454\pi\)
0.669628 + 0.742697i \(0.266454\pi\)
\(212\) 6.87467e9 0.233744
\(213\) 2.89108e10 0.962393
\(214\) 2.50899e10 0.817780
\(215\) 0 0
\(216\) 5.35311e9 0.167326
\(217\) −2.05244e10 −0.628350
\(218\) −4.41500e10 −1.32396
\(219\) 1.67065e10 0.490781
\(220\) 0 0
\(221\) 1.39648e10 0.393795
\(222\) 1.75226e10 0.484185
\(223\) 3.11357e10 0.843115 0.421558 0.906802i \(-0.361484\pi\)
0.421558 + 0.906802i \(0.361484\pi\)
\(224\) −9.80866e9 −0.260312
\(225\) 0 0
\(226\) −2.35276e10 −0.599914
\(227\) 1.14589e10 0.286436 0.143218 0.989691i \(-0.454255\pi\)
0.143218 + 0.989691i \(0.454255\pi\)
\(228\) 2.04008e9 0.0499967
\(229\) −3.04556e10 −0.731825 −0.365913 0.930649i \(-0.619243\pi\)
−0.365913 + 0.930649i \(0.619243\pi\)
\(230\) 0 0
\(231\) −2.75673e10 −0.637000
\(232\) 4.67576e9 0.105964
\(233\) −2.83630e9 −0.0630451 −0.0315225 0.999503i \(-0.510036\pi\)
−0.0315225 + 0.999503i \(0.510036\pi\)
\(234\) −1.94977e10 −0.425121
\(235\) 0 0
\(236\) 1.60174e10 0.336115
\(237\) 3.27548e10 0.674384
\(238\) −1.16696e10 −0.235754
\(239\) −6.25862e10 −1.24076 −0.620381 0.784301i \(-0.713022\pi\)
−0.620381 + 0.784301i \(0.713022\pi\)
\(240\) 0 0
\(241\) 7.24015e10 1.38252 0.691259 0.722607i \(-0.257056\pi\)
0.691259 + 0.722607i \(0.257056\pi\)
\(242\) 1.20525e11 2.25895
\(243\) −3.48678e9 −0.0641500
\(244\) 1.65778e10 0.299414
\(245\) 0 0
\(246\) 1.82573e10 0.317855
\(247\) 2.80568e10 0.479624
\(248\) 5.15493e10 0.865347
\(249\) −4.18797e8 −0.00690410
\(250\) 0 0
\(251\) −5.65927e10 −0.899971 −0.449986 0.893036i \(-0.648571\pi\)
−0.449986 + 0.893036i \(0.648571\pi\)
\(252\) 2.82092e9 0.0440644
\(253\) −1.99059e11 −3.05450
\(254\) 1.29336e11 1.94970
\(255\) 0 0
\(256\) 4.14066e10 0.602545
\(257\) −4.95688e10 −0.708777 −0.354388 0.935098i \(-0.615311\pi\)
−0.354388 + 0.935098i \(0.615311\pi\)
\(258\) 1.74045e10 0.244552
\(259\) −3.48655e10 −0.481445
\(260\) 0 0
\(261\) −3.04559e9 −0.0406246
\(262\) 1.31488e10 0.172398
\(263\) −3.59498e10 −0.463336 −0.231668 0.972795i \(-0.574418\pi\)
−0.231668 + 0.972795i \(0.574418\pi\)
\(264\) 6.92382e10 0.877260
\(265\) 0 0
\(266\) −2.34454e10 −0.287138
\(267\) −3.50116e10 −0.421610
\(268\) −2.92412e10 −0.346249
\(269\) −7.09650e10 −0.826340 −0.413170 0.910654i \(-0.635579\pi\)
−0.413170 + 0.910654i \(0.635579\pi\)
\(270\) 0 0
\(271\) 1.47201e11 1.65786 0.828930 0.559352i \(-0.188950\pi\)
0.828930 + 0.559352i \(0.188950\pi\)
\(272\) 3.57279e10 0.395774
\(273\) 3.87954e10 0.422716
\(274\) −1.24857e11 −1.33825
\(275\) 0 0
\(276\) 2.03695e10 0.211295
\(277\) 2.87336e10 0.293245 0.146622 0.989193i \(-0.453160\pi\)
0.146622 + 0.989193i \(0.453160\pi\)
\(278\) 8.75593e10 0.879227
\(279\) −3.35770e10 −0.331759
\(280\) 0 0
\(281\) 5.17071e10 0.494734 0.247367 0.968922i \(-0.420435\pi\)
0.247367 + 0.968922i \(0.420435\pi\)
\(282\) 6.68192e10 0.629187
\(283\) 3.22434e9 0.0298814 0.0149407 0.999888i \(-0.495244\pi\)
0.0149407 + 0.999888i \(0.495244\pi\)
\(284\) −3.82646e10 −0.349032
\(285\) 0 0
\(286\) −2.52188e11 −2.22883
\(287\) −3.63273e10 −0.316057
\(288\) −1.60465e10 −0.137440
\(289\) −1.04914e11 −0.884698
\(290\) 0 0
\(291\) −1.06974e11 −0.874499
\(292\) −2.21117e10 −0.177992
\(293\) 1.13591e11 0.900409 0.450204 0.892926i \(-0.351351\pi\)
0.450204 + 0.892926i \(0.351351\pi\)
\(294\) 4.89174e10 0.381856
\(295\) 0 0
\(296\) 8.75685e10 0.663033
\(297\) −4.50988e10 −0.336326
\(298\) 1.07701e11 0.791129
\(299\) 2.80136e11 2.02698
\(300\) 0 0
\(301\) −3.46303e10 −0.243169
\(302\) −1.39805e11 −0.967148
\(303\) −4.29671e10 −0.292849
\(304\) 7.17811e10 0.482036
\(305\) 0 0
\(306\) −1.90909e10 −0.124474
\(307\) 2.30543e10 0.148125 0.0740627 0.997254i \(-0.476403\pi\)
0.0740627 + 0.997254i \(0.476403\pi\)
\(308\) 3.64863e10 0.231021
\(309\) −4.91837e9 −0.0306908
\(310\) 0 0
\(311\) 7.71709e10 0.467769 0.233885 0.972264i \(-0.424856\pi\)
0.233885 + 0.972264i \(0.424856\pi\)
\(312\) −9.74389e10 −0.582152
\(313\) 4.68832e10 0.276101 0.138050 0.990425i \(-0.455916\pi\)
0.138050 + 0.990425i \(0.455916\pi\)
\(314\) −3.87200e10 −0.224777
\(315\) 0 0
\(316\) −4.33522e10 −0.244579
\(317\) −2.19420e11 −1.22042 −0.610210 0.792240i \(-0.708915\pi\)
−0.610210 + 0.792240i \(0.708915\pi\)
\(318\) −1.29251e11 −0.708779
\(319\) −3.93923e10 −0.212987
\(320\) 0 0
\(321\) −8.16706e10 −0.429332
\(322\) −2.34094e11 −1.21350
\(323\) 2.74714e10 0.140433
\(324\) 4.61489e9 0.0232653
\(325\) 0 0
\(326\) 2.87609e11 1.41034
\(327\) 1.43714e11 0.695077
\(328\) 9.12401e10 0.435265
\(329\) −1.32953e11 −0.625627
\(330\) 0 0
\(331\) 3.89075e9 0.0178159 0.00890794 0.999960i \(-0.497164\pi\)
0.00890794 + 0.999960i \(0.497164\pi\)
\(332\) 5.54295e8 0.00250391
\(333\) −5.70383e10 −0.254195
\(334\) 3.74466e11 1.64647
\(335\) 0 0
\(336\) 9.92551e10 0.424841
\(337\) 1.48259e11 0.626163 0.313082 0.949726i \(-0.398639\pi\)
0.313082 + 0.949726i \(0.398639\pi\)
\(338\) 9.10225e10 0.379336
\(339\) 7.65851e10 0.314953
\(340\) 0 0
\(341\) −4.34291e11 −1.73935
\(342\) −3.83556e10 −0.151604
\(343\) −2.59171e11 −1.01103
\(344\) 8.69779e10 0.334885
\(345\) 0 0
\(346\) −5.55302e11 −2.08299
\(347\) 1.63695e11 0.606114 0.303057 0.952972i \(-0.401993\pi\)
0.303057 + 0.952972i \(0.401993\pi\)
\(348\) 4.03096e9 0.0147333
\(349\) 7.02057e10 0.253313 0.126657 0.991947i \(-0.459575\pi\)
0.126657 + 0.991947i \(0.459575\pi\)
\(350\) 0 0
\(351\) 6.34675e10 0.223187
\(352\) −2.07549e11 −0.720574
\(353\) 1.55727e10 0.0533798 0.0266899 0.999644i \(-0.491503\pi\)
0.0266899 + 0.999644i \(0.491503\pi\)
\(354\) −3.01143e11 −1.01920
\(355\) 0 0
\(356\) 4.63392e10 0.152906
\(357\) 3.79859e10 0.123770
\(358\) −4.31843e11 −1.38948
\(359\) 3.28525e10 0.104386 0.0521931 0.998637i \(-0.483379\pi\)
0.0521931 + 0.998637i \(0.483379\pi\)
\(360\) 0 0
\(361\) −2.67495e11 −0.828959
\(362\) −3.33543e11 −1.02085
\(363\) −3.92323e11 −1.18594
\(364\) −5.13472e10 −0.153306
\(365\) 0 0
\(366\) −3.11679e11 −0.907908
\(367\) 4.76030e11 1.36974 0.684868 0.728667i \(-0.259860\pi\)
0.684868 + 0.728667i \(0.259860\pi\)
\(368\) 7.16707e11 2.03717
\(369\) −5.94299e10 −0.166873
\(370\) 0 0
\(371\) 2.57175e11 0.704768
\(372\) 4.44404e10 0.120319
\(373\) 9.72745e10 0.260201 0.130101 0.991501i \(-0.458470\pi\)
0.130101 + 0.991501i \(0.458470\pi\)
\(374\) −2.46926e11 −0.652596
\(375\) 0 0
\(376\) 3.33925e11 0.861597
\(377\) 5.54367e10 0.141339
\(378\) −5.30361e10 −0.133616
\(379\) 4.45171e11 1.10828 0.554141 0.832423i \(-0.313047\pi\)
0.554141 + 0.832423i \(0.313047\pi\)
\(380\) 0 0
\(381\) −4.21004e11 −1.02358
\(382\) −1.49630e11 −0.359528
\(383\) −2.19414e11 −0.521039 −0.260519 0.965469i \(-0.583894\pi\)
−0.260519 + 0.965469i \(0.583894\pi\)
\(384\) −2.94075e11 −0.690187
\(385\) 0 0
\(386\) 1.72157e11 0.394714
\(387\) −5.66536e10 −0.128389
\(388\) 1.41584e11 0.317155
\(389\) −3.71956e11 −0.823603 −0.411801 0.911274i \(-0.635100\pi\)
−0.411801 + 0.911274i \(0.635100\pi\)
\(390\) 0 0
\(391\) 2.74291e11 0.593494
\(392\) 2.44462e11 0.522907
\(393\) −4.28011e10 −0.0905082
\(394\) −4.13417e11 −0.864281
\(395\) 0 0
\(396\) 5.96900e10 0.121976
\(397\) 1.38786e11 0.280406 0.140203 0.990123i \(-0.455224\pi\)
0.140203 + 0.990123i \(0.455224\pi\)
\(398\) −7.63216e11 −1.52466
\(399\) 7.63177e10 0.150747
\(400\) 0 0
\(401\) 6.95127e11 1.34250 0.671251 0.741231i \(-0.265757\pi\)
0.671251 + 0.741231i \(0.265757\pi\)
\(402\) 5.49763e11 1.04993
\(403\) 6.11178e11 1.15424
\(404\) 5.68686e10 0.106208
\(405\) 0 0
\(406\) −4.63253e10 −0.0846157
\(407\) −7.37745e11 −1.33270
\(408\) −9.54059e10 −0.170453
\(409\) 9.10800e11 1.60942 0.804708 0.593671i \(-0.202322\pi\)
0.804708 + 0.593671i \(0.202322\pi\)
\(410\) 0 0
\(411\) 4.06426e11 0.702577
\(412\) 6.50966e9 0.0111307
\(413\) 5.99196e11 1.01343
\(414\) −3.82967e11 −0.640707
\(415\) 0 0
\(416\) 2.92084e11 0.478175
\(417\) −2.85016e11 −0.461591
\(418\) −4.96100e11 −0.794832
\(419\) 3.11152e11 0.493184 0.246592 0.969119i \(-0.420689\pi\)
0.246592 + 0.969119i \(0.420689\pi\)
\(420\) 0 0
\(421\) 7.63941e11 1.18520 0.592598 0.805498i \(-0.298102\pi\)
0.592598 + 0.805498i \(0.298102\pi\)
\(422\) 9.59516e11 1.47281
\(423\) −2.17505e11 −0.330321
\(424\) −6.45924e11 −0.970588
\(425\) 0 0
\(426\) 7.19414e11 1.05836
\(427\) 6.20159e11 0.902771
\(428\) 1.08094e11 0.155706
\(429\) 8.20901e11 1.17013
\(430\) 0 0
\(431\) 6.54629e10 0.0913792 0.0456896 0.998956i \(-0.485451\pi\)
0.0456896 + 0.998956i \(0.485451\pi\)
\(432\) 1.62377e11 0.224309
\(433\) 7.87434e11 1.07651 0.538256 0.842781i \(-0.319083\pi\)
0.538256 + 0.842781i \(0.319083\pi\)
\(434\) −5.10726e11 −0.691010
\(435\) 0 0
\(436\) −1.90210e11 −0.252084
\(437\) 5.51080e11 0.722850
\(438\) 4.15723e11 0.539722
\(439\) −8.39211e11 −1.07840 −0.539201 0.842177i \(-0.681274\pi\)
−0.539201 + 0.842177i \(0.681274\pi\)
\(440\) 0 0
\(441\) −1.59232e11 −0.200473
\(442\) 3.47499e11 0.433065
\(443\) 1.28252e12 1.58215 0.791074 0.611720i \(-0.209522\pi\)
0.791074 + 0.611720i \(0.209522\pi\)
\(444\) 7.54924e10 0.0921891
\(445\) 0 0
\(446\) 7.74777e11 0.927192
\(447\) −3.50581e11 −0.415340
\(448\) 3.83313e11 0.449575
\(449\) 8.02030e10 0.0931284 0.0465642 0.998915i \(-0.485173\pi\)
0.0465642 + 0.998915i \(0.485173\pi\)
\(450\) 0 0
\(451\) −7.68678e11 −0.874883
\(452\) −1.01363e11 −0.114224
\(453\) 4.55084e11 0.507749
\(454\) 2.85143e11 0.315000
\(455\) 0 0
\(456\) −1.91680e11 −0.207604
\(457\) −1.27085e11 −0.136292 −0.0681461 0.997675i \(-0.521708\pi\)
−0.0681461 + 0.997675i \(0.521708\pi\)
\(458\) −7.57853e11 −0.804804
\(459\) 6.21433e10 0.0653487
\(460\) 0 0
\(461\) −7.39880e11 −0.762970 −0.381485 0.924375i \(-0.624587\pi\)
−0.381485 + 0.924375i \(0.624587\pi\)
\(462\) −6.85980e11 −0.700523
\(463\) −5.71025e11 −0.577485 −0.288742 0.957407i \(-0.593237\pi\)
−0.288742 + 0.957407i \(0.593237\pi\)
\(464\) 1.41831e11 0.142049
\(465\) 0 0
\(466\) −7.05782e10 −0.0693320
\(467\) −1.19075e12 −1.15850 −0.579249 0.815150i \(-0.696654\pi\)
−0.579249 + 0.815150i \(0.696654\pi\)
\(468\) −8.40017e10 −0.0809434
\(469\) −1.09389e12 −1.04398
\(470\) 0 0
\(471\) 1.26038e11 0.118007
\(472\) −1.50495e12 −1.39567
\(473\) −7.32770e11 −0.673120
\(474\) 8.15065e11 0.741634
\(475\) 0 0
\(476\) −5.02759e10 −0.0448878
\(477\) 4.20727e11 0.372107
\(478\) −1.55739e12 −1.36449
\(479\) 1.46074e12 1.26784 0.633919 0.773400i \(-0.281445\pi\)
0.633919 + 0.773400i \(0.281445\pi\)
\(480\) 0 0
\(481\) 1.03823e12 0.884382
\(482\) 1.80163e12 1.52038
\(483\) 7.62003e11 0.637081
\(484\) 5.19254e11 0.430107
\(485\) 0 0
\(486\) −8.67647e10 −0.0705472
\(487\) 1.46931e12 1.18368 0.591839 0.806056i \(-0.298402\pi\)
0.591839 + 0.806056i \(0.298402\pi\)
\(488\) −1.55760e12 −1.24327
\(489\) −9.36201e11 −0.740422
\(490\) 0 0
\(491\) −1.76138e12 −1.36769 −0.683843 0.729629i \(-0.739693\pi\)
−0.683843 + 0.729629i \(0.739693\pi\)
\(492\) 7.86577e10 0.0605199
\(493\) 5.42801e10 0.0413837
\(494\) 6.98161e11 0.527453
\(495\) 0 0
\(496\) 1.56365e12 1.16004
\(497\) −1.43145e12 −1.05238
\(498\) −1.04213e10 −0.00759259
\(499\) −1.63016e12 −1.17700 −0.588502 0.808496i \(-0.700282\pi\)
−0.588502 + 0.808496i \(0.700282\pi\)
\(500\) 0 0
\(501\) −1.21893e12 −0.864391
\(502\) −1.40825e12 −0.989718
\(503\) −9.47625e11 −0.660056 −0.330028 0.943971i \(-0.607058\pi\)
−0.330028 + 0.943971i \(0.607058\pi\)
\(504\) −2.65045e11 −0.182971
\(505\) 0 0
\(506\) −4.95337e12 −3.35910
\(507\) −2.96289e11 −0.199150
\(508\) 5.57214e11 0.371224
\(509\) −2.90709e11 −0.191968 −0.0959838 0.995383i \(-0.530600\pi\)
−0.0959838 + 0.995383i \(0.530600\pi\)
\(510\) 0 0
\(511\) −8.27180e11 −0.536668
\(512\) −8.28486e11 −0.532808
\(513\) 1.24852e11 0.0795918
\(514\) −1.23346e12 −0.779457
\(515\) 0 0
\(516\) 7.49833e10 0.0465630
\(517\) −2.81325e12 −1.73181
\(518\) −8.67587e11 −0.529455
\(519\) 1.80757e12 1.09356
\(520\) 0 0
\(521\) −2.04811e12 −1.21782 −0.608910 0.793240i \(-0.708393\pi\)
−0.608910 + 0.793240i \(0.708393\pi\)
\(522\) −7.57860e10 −0.0446758
\(523\) −8.49299e10 −0.0496367 −0.0248184 0.999692i \(-0.507901\pi\)
−0.0248184 + 0.999692i \(0.507901\pi\)
\(524\) 5.66488e10 0.0328247
\(525\) 0 0
\(526\) −8.94570e11 −0.509540
\(527\) 5.98426e11 0.337958
\(528\) 2.10021e12 1.17601
\(529\) 3.70117e12 2.05489
\(530\) 0 0
\(531\) 9.80258e11 0.535075
\(532\) −1.01009e11 −0.0546713
\(533\) 1.08176e12 0.580575
\(534\) −8.71224e11 −0.463654
\(535\) 0 0
\(536\) 2.74741e12 1.43775
\(537\) 1.40570e12 0.729472
\(538\) −1.76588e12 −0.908745
\(539\) −2.05954e12 −1.05104
\(540\) 0 0
\(541\) −1.29101e12 −0.647949 −0.323974 0.946066i \(-0.605019\pi\)
−0.323974 + 0.946066i \(0.605019\pi\)
\(542\) 3.66292e12 1.82318
\(543\) 1.08572e12 0.535944
\(544\) 2.85989e11 0.140009
\(545\) 0 0
\(546\) 9.65379e11 0.464870
\(547\) 5.30222e11 0.253230 0.126615 0.991952i \(-0.459589\pi\)
0.126615 + 0.991952i \(0.459589\pi\)
\(548\) −5.37921e11 −0.254804
\(549\) 1.01455e12 0.476649
\(550\) 0 0
\(551\) 1.09054e11 0.0504035
\(552\) −1.91386e12 −0.877371
\(553\) −1.62177e12 −0.737438
\(554\) 7.15002e11 0.322488
\(555\) 0 0
\(556\) 3.77230e11 0.167405
\(557\) −8.17386e11 −0.359815 −0.179907 0.983684i \(-0.557580\pi\)
−0.179907 + 0.983684i \(0.557580\pi\)
\(558\) −8.35525e11 −0.364842
\(559\) 1.03123e12 0.446684
\(560\) 0 0
\(561\) 8.03773e11 0.342611
\(562\) 1.28667e12 0.544069
\(563\) 1.42197e12 0.596488 0.298244 0.954490i \(-0.403599\pi\)
0.298244 + 0.954490i \(0.403599\pi\)
\(564\) 2.87876e11 0.119798
\(565\) 0 0
\(566\) 8.02339e10 0.0328613
\(567\) 1.72639e11 0.0701480
\(568\) 3.59523e12 1.44930
\(569\) −1.04614e12 −0.418394 −0.209197 0.977874i \(-0.567085\pi\)
−0.209197 + 0.977874i \(0.567085\pi\)
\(570\) 0 0
\(571\) 2.09508e12 0.824780 0.412390 0.911007i \(-0.364694\pi\)
0.412390 + 0.911007i \(0.364694\pi\)
\(572\) −1.08649e12 −0.424371
\(573\) 4.87063e11 0.188751
\(574\) −9.03965e11 −0.347575
\(575\) 0 0
\(576\) 6.27082e11 0.237369
\(577\) −2.68760e12 −1.00942 −0.504711 0.863288i \(-0.668401\pi\)
−0.504711 + 0.863288i \(0.668401\pi\)
\(578\) −2.61068e12 −0.972921
\(579\) −5.60394e11 −0.207224
\(580\) 0 0
\(581\) 2.07357e10 0.00754962
\(582\) −2.66192e12 −0.961705
\(583\) 5.44176e12 1.95088
\(584\) 2.07755e12 0.739085
\(585\) 0 0
\(586\) 2.82658e12 0.990199
\(587\) −1.61925e12 −0.562913 −0.281457 0.959574i \(-0.590818\pi\)
−0.281457 + 0.959574i \(0.590818\pi\)
\(588\) 2.10750e11 0.0727058
\(589\) 1.20230e12 0.411618
\(590\) 0 0
\(591\) 1.34572e12 0.453745
\(592\) 2.65623e12 0.888828
\(593\) −5.82677e12 −1.93500 −0.967502 0.252863i \(-0.918628\pi\)
−0.967502 + 0.252863i \(0.918628\pi\)
\(594\) −1.12223e12 −0.369865
\(595\) 0 0
\(596\) 4.64007e11 0.150632
\(597\) 2.48436e12 0.800442
\(598\) 6.97087e12 2.22911
\(599\) −3.89086e12 −1.23488 −0.617441 0.786617i \(-0.711831\pi\)
−0.617441 + 0.786617i \(0.711831\pi\)
\(600\) 0 0
\(601\) 2.13051e12 0.666115 0.333058 0.942907i \(-0.391920\pi\)
0.333058 + 0.942907i \(0.391920\pi\)
\(602\) −8.61737e11 −0.267418
\(603\) −1.78955e12 −0.551207
\(604\) −6.02321e11 −0.184146
\(605\) 0 0
\(606\) −1.06919e12 −0.322053
\(607\) 2.52733e12 0.755637 0.377818 0.925880i \(-0.376674\pi\)
0.377818 + 0.925880i \(0.376674\pi\)
\(608\) 5.74582e11 0.170524
\(609\) 1.50794e11 0.0444230
\(610\) 0 0
\(611\) 3.95908e12 1.14924
\(612\) −8.22490e10 −0.0237000
\(613\) −1.89788e12 −0.542870 −0.271435 0.962457i \(-0.587498\pi\)
−0.271435 + 0.962457i \(0.587498\pi\)
\(614\) 5.73681e11 0.162897
\(615\) 0 0
\(616\) −3.42815e12 −0.959283
\(617\) 1.40480e12 0.390240 0.195120 0.980779i \(-0.437490\pi\)
0.195120 + 0.980779i \(0.437490\pi\)
\(618\) −1.22388e11 −0.0337514
\(619\) −2.81250e12 −0.769989 −0.384995 0.922919i \(-0.625797\pi\)
−0.384995 + 0.922919i \(0.625797\pi\)
\(620\) 0 0
\(621\) 1.24660e12 0.336369
\(622\) 1.92031e12 0.514416
\(623\) 1.73351e12 0.461030
\(624\) −2.95563e12 −0.780404
\(625\) 0 0
\(626\) 1.16664e12 0.303634
\(627\) 1.61486e12 0.417284
\(628\) −1.66817e11 −0.0427977
\(629\) 1.01657e12 0.258945
\(630\) 0 0
\(631\) −1.92653e12 −0.483775 −0.241888 0.970304i \(-0.577767\pi\)
−0.241888 + 0.970304i \(0.577767\pi\)
\(632\) 4.07325e12 1.01558
\(633\) −3.12334e12 −0.773219
\(634\) −5.46001e12 −1.34212
\(635\) 0 0
\(636\) −5.56848e11 −0.134952
\(637\) 2.89839e12 0.697476
\(638\) −9.80232e11 −0.234226
\(639\) −2.34178e12 −0.555638
\(640\) 0 0
\(641\) −7.10318e12 −1.66185 −0.830924 0.556385i \(-0.812188\pi\)
−0.830924 + 0.556385i \(0.812188\pi\)
\(642\) −2.03228e12 −0.472146
\(643\) −3.40733e12 −0.786077 −0.393038 0.919522i \(-0.628576\pi\)
−0.393038 + 0.919522i \(0.628576\pi\)
\(644\) −1.00854e12 −0.231051
\(645\) 0 0
\(646\) 6.83594e11 0.154437
\(647\) 4.31347e12 0.967737 0.483868 0.875141i \(-0.339231\pi\)
0.483868 + 0.875141i \(0.339231\pi\)
\(648\) −4.33602e11 −0.0966059
\(649\) 1.26789e13 2.80530
\(650\) 0 0
\(651\) 1.66248e12 0.362778
\(652\) 1.23910e12 0.268529
\(653\) 2.53169e12 0.544880 0.272440 0.962173i \(-0.412169\pi\)
0.272440 + 0.962173i \(0.412169\pi\)
\(654\) 3.57615e12 0.764391
\(655\) 0 0
\(656\) 2.76760e12 0.583494
\(657\) −1.35323e12 −0.283352
\(658\) −3.30838e12 −0.688016
\(659\) −4.43263e12 −0.915540 −0.457770 0.889071i \(-0.651352\pi\)
−0.457770 + 0.889071i \(0.651352\pi\)
\(660\) 0 0
\(661\) 9.79827e11 0.199638 0.0998189 0.995006i \(-0.468174\pi\)
0.0998189 + 0.995006i \(0.468174\pi\)
\(662\) 9.68168e10 0.0195925
\(663\) −1.13115e12 −0.227358
\(664\) −5.20799e10 −0.0103971
\(665\) 0 0
\(666\) −1.41933e12 −0.279544
\(667\) 1.08887e12 0.213014
\(668\) 1.61331e12 0.313489
\(669\) −2.52199e12 −0.486773
\(670\) 0 0
\(671\) 1.31224e13 2.49898
\(672\) 7.94502e11 0.150291
\(673\) −7.69434e12 −1.44578 −0.722892 0.690961i \(-0.757188\pi\)
−0.722892 + 0.690961i \(0.757188\pi\)
\(674\) 3.68927e12 0.688605
\(675\) 0 0
\(676\) 3.92151e11 0.0722258
\(677\) −2.45111e12 −0.448451 −0.224225 0.974537i \(-0.571985\pi\)
−0.224225 + 0.974537i \(0.571985\pi\)
\(678\) 1.90573e12 0.346360
\(679\) 5.29653e12 0.956264
\(680\) 0 0
\(681\) −9.28174e11 −0.165374
\(682\) −1.08068e13 −1.91280
\(683\) −3.13070e12 −0.550488 −0.275244 0.961374i \(-0.588759\pi\)
−0.275244 + 0.961374i \(0.588759\pi\)
\(684\) −1.65247e11 −0.0288656
\(685\) 0 0
\(686\) −6.44918e12 −1.11185
\(687\) 2.46690e12 0.422519
\(688\) 2.63831e12 0.448930
\(689\) −7.65819e12 −1.29461
\(690\) 0 0
\(691\) −8.92815e12 −1.48974 −0.744870 0.667210i \(-0.767488\pi\)
−0.744870 + 0.667210i \(0.767488\pi\)
\(692\) −2.39239e12 −0.396602
\(693\) 2.23295e12 0.367772
\(694\) 4.07338e12 0.666556
\(695\) 0 0
\(696\) −3.78737e11 −0.0611781
\(697\) 1.05919e12 0.169991
\(698\) 1.74699e12 0.278574
\(699\) 2.29741e11 0.0363991
\(700\) 0 0
\(701\) −1.44295e12 −0.225695 −0.112847 0.993612i \(-0.535997\pi\)
−0.112847 + 0.993612i \(0.535997\pi\)
\(702\) 1.57932e12 0.245444
\(703\) 2.04239e12 0.315383
\(704\) 8.11081e12 1.24448
\(705\) 0 0
\(706\) 3.87509e11 0.0587030
\(707\) 2.12740e12 0.320230
\(708\) −1.29741e12 −0.194056
\(709\) −7.40550e12 −1.10064 −0.550321 0.834953i \(-0.685495\pi\)
−0.550321 + 0.834953i \(0.685495\pi\)
\(710\) 0 0
\(711\) −2.65314e12 −0.389355
\(712\) −4.35390e12 −0.634919
\(713\) 1.20045e13 1.73957
\(714\) 9.45237e11 0.136113
\(715\) 0 0
\(716\) −1.86050e12 −0.264558
\(717\) 5.06949e12 0.716354
\(718\) 8.17496e11 0.114796
\(719\) −3.90289e12 −0.544636 −0.272318 0.962207i \(-0.587790\pi\)
−0.272318 + 0.962207i \(0.587790\pi\)
\(720\) 0 0
\(721\) 2.43520e11 0.0335604
\(722\) −6.65631e12 −0.911624
\(723\) −5.86452e12 −0.798197
\(724\) −1.43699e12 −0.194371
\(725\) 0 0
\(726\) −9.76251e12 −1.30421
\(727\) −1.33674e12 −0.177477 −0.0887384 0.996055i \(-0.528284\pi\)
−0.0887384 + 0.996055i \(0.528284\pi\)
\(728\) 4.82443e12 0.636583
\(729\) 2.82430e11 0.0370370
\(730\) 0 0
\(731\) 1.00971e12 0.130788
\(732\) −1.34280e12 −0.172866
\(733\) 5.98863e11 0.0766231 0.0383116 0.999266i \(-0.487802\pi\)
0.0383116 + 0.999266i \(0.487802\pi\)
\(734\) 1.18455e13 1.50633
\(735\) 0 0
\(736\) 5.73699e12 0.720665
\(737\) −2.31464e13 −2.88987
\(738\) −1.47884e12 −0.183514
\(739\) −1.40697e13 −1.73534 −0.867671 0.497138i \(-0.834384\pi\)
−0.867671 + 0.497138i \(0.834384\pi\)
\(740\) 0 0
\(741\) −2.27260e12 −0.276911
\(742\) 6.39951e12 0.775049
\(743\) 2.52604e12 0.304082 0.152041 0.988374i \(-0.451415\pi\)
0.152041 + 0.988374i \(0.451415\pi\)
\(744\) −4.17549e12 −0.499608
\(745\) 0 0
\(746\) 2.42056e12 0.286149
\(747\) 3.39226e10 0.00398608
\(748\) −1.06383e12 −0.124255
\(749\) 4.04371e12 0.469474
\(750\) 0 0
\(751\) 3.78704e12 0.434430 0.217215 0.976124i \(-0.430303\pi\)
0.217215 + 0.976124i \(0.430303\pi\)
\(752\) 1.01290e13 1.15501
\(753\) 4.58401e12 0.519599
\(754\) 1.37948e12 0.155433
\(755\) 0 0
\(756\) −2.28494e11 −0.0254406
\(757\) −3.55646e12 −0.393629 −0.196814 0.980441i \(-0.563060\pi\)
−0.196814 + 0.980441i \(0.563060\pi\)
\(758\) 1.10776e13 1.21880
\(759\) 1.61238e13 1.76352
\(760\) 0 0
\(761\) 1.67100e13 1.80612 0.903058 0.429519i \(-0.141317\pi\)
0.903058 + 0.429519i \(0.141317\pi\)
\(762\) −1.04762e13 −1.12566
\(763\) −7.11560e12 −0.760066
\(764\) −6.44647e11 −0.0684544
\(765\) 0 0
\(766\) −5.45987e12 −0.572997
\(767\) −1.78429e13 −1.86160
\(768\) −3.35393e12 −0.347880
\(769\) −7.51512e12 −0.774939 −0.387469 0.921883i \(-0.626651\pi\)
−0.387469 + 0.921883i \(0.626651\pi\)
\(770\) 0 0
\(771\) 4.01507e12 0.409213
\(772\) 7.41702e11 0.0751540
\(773\) 8.52580e12 0.858870 0.429435 0.903098i \(-0.358713\pi\)
0.429435 + 0.903098i \(0.358713\pi\)
\(774\) −1.40976e12 −0.141192
\(775\) 0 0
\(776\) −1.33028e13 −1.31694
\(777\) 2.82410e12 0.277962
\(778\) −9.25569e12 −0.905734
\(779\) 2.12802e12 0.207042
\(780\) 0 0
\(781\) −3.02890e13 −2.91310
\(782\) 6.82543e12 0.652679
\(783\) 2.46693e11 0.0234546
\(784\) 7.41531e12 0.700982
\(785\) 0 0
\(786\) −1.06506e12 −0.0995338
\(787\) −3.20742e12 −0.298037 −0.149018 0.988834i \(-0.547611\pi\)
−0.149018 + 0.988834i \(0.547611\pi\)
\(788\) −1.78112e12 −0.164560
\(789\) 2.91194e12 0.267507
\(790\) 0 0
\(791\) −3.79191e12 −0.344401
\(792\) −5.60830e12 −0.506486
\(793\) −1.84672e13 −1.65833
\(794\) 3.45353e12 0.308369
\(795\) 0 0
\(796\) −3.28815e12 −0.290297
\(797\) 1.83036e13 1.60684 0.803421 0.595412i \(-0.203011\pi\)
0.803421 + 0.595412i \(0.203011\pi\)
\(798\) 1.89908e12 0.165779
\(799\) 3.87648e12 0.336494
\(800\) 0 0
\(801\) 2.83594e12 0.243417
\(802\) 1.72975e13 1.47638
\(803\) −1.75029e13 −1.48556
\(804\) 2.36853e12 0.199907
\(805\) 0 0
\(806\) 1.52085e13 1.26934
\(807\) 5.74817e12 0.477088
\(808\) −5.34321e12 −0.441013
\(809\) 9.38919e12 0.770655 0.385327 0.922780i \(-0.374089\pi\)
0.385327 + 0.922780i \(0.374089\pi\)
\(810\) 0 0
\(811\) 2.03949e13 1.65549 0.827746 0.561103i \(-0.189623\pi\)
0.827746 + 0.561103i \(0.189623\pi\)
\(812\) −1.99582e11 −0.0161109
\(813\) −1.19232e13 −0.957166
\(814\) −1.83579e13 −1.46560
\(815\) 0 0
\(816\) −2.89396e12 −0.228500
\(817\) 2.02861e12 0.159294
\(818\) 2.26642e13 1.76991
\(819\) −3.14243e12 −0.244055
\(820\) 0 0
\(821\) −1.52259e13 −1.16961 −0.584803 0.811176i \(-0.698828\pi\)
−0.584803 + 0.811176i \(0.698828\pi\)
\(822\) 1.01135e13 0.772639
\(823\) 2.70705e12 0.205682 0.102841 0.994698i \(-0.467207\pi\)
0.102841 + 0.994698i \(0.467207\pi\)
\(824\) −6.11629e11 −0.0462184
\(825\) 0 0
\(826\) 1.49103e13 1.11449
\(827\) −5.67203e10 −0.00421661 −0.00210831 0.999998i \(-0.500671\pi\)
−0.00210831 + 0.999998i \(0.500671\pi\)
\(828\) −1.64993e12 −0.121991
\(829\) −1.43206e13 −1.05309 −0.526547 0.850146i \(-0.676513\pi\)
−0.526547 + 0.850146i \(0.676513\pi\)
\(830\) 0 0
\(831\) −2.32742e12 −0.169305
\(832\) −1.14143e13 −0.825839
\(833\) 2.83791e12 0.204219
\(834\) −7.09231e12 −0.507622
\(835\) 0 0
\(836\) −2.13734e12 −0.151337
\(837\) 2.71973e12 0.191541
\(838\) 7.74265e12 0.542365
\(839\) −2.13693e13 −1.48889 −0.744443 0.667686i \(-0.767285\pi\)
−0.744443 + 0.667686i \(0.767285\pi\)
\(840\) 0 0
\(841\) −1.42917e13 −0.985147
\(842\) 1.90098e13 1.30339
\(843\) −4.18827e12 −0.285635
\(844\) 4.13386e12 0.280424
\(845\) 0 0
\(846\) −5.41235e12 −0.363262
\(847\) 1.94248e13 1.29683
\(848\) −1.95929e13 −1.30112
\(849\) −2.61171e11 −0.0172521
\(850\) 0 0
\(851\) 2.03925e13 1.33287
\(852\) 3.09943e12 0.201514
\(853\) 1.18123e13 0.763948 0.381974 0.924173i \(-0.375244\pi\)
0.381974 + 0.924173i \(0.375244\pi\)
\(854\) 1.54319e13 0.992797
\(855\) 0 0
\(856\) −1.01562e13 −0.646547
\(857\) −2.69459e13 −1.70640 −0.853198 0.521587i \(-0.825340\pi\)
−0.853198 + 0.521587i \(0.825340\pi\)
\(858\) 2.04272e13 1.28681
\(859\) 9.66669e12 0.605771 0.302885 0.953027i \(-0.402050\pi\)
0.302885 + 0.953027i \(0.402050\pi\)
\(860\) 0 0
\(861\) 2.94251e12 0.182475
\(862\) 1.62897e12 0.100492
\(863\) −2.80982e12 −0.172437 −0.0862185 0.996276i \(-0.527478\pi\)
−0.0862185 + 0.996276i \(0.527478\pi\)
\(864\) 1.29977e12 0.0793513
\(865\) 0 0
\(866\) 1.95944e13 1.18386
\(867\) 8.49807e12 0.510781
\(868\) −2.20035e12 −0.131569
\(869\) −3.43162e13 −2.04132
\(870\) 0 0
\(871\) 3.25739e13 1.91773
\(872\) 1.78716e13 1.04674
\(873\) 8.66488e12 0.504892
\(874\) 1.37130e13 0.794933
\(875\) 0 0
\(876\) 1.79105e12 0.102764
\(877\) −3.37045e13 −1.92393 −0.961966 0.273169i \(-0.911928\pi\)
−0.961966 + 0.273169i \(0.911928\pi\)
\(878\) −2.08828e13 −1.18594
\(879\) −9.20087e12 −0.519851
\(880\) 0 0
\(881\) −1.20571e13 −0.674297 −0.337148 0.941452i \(-0.609462\pi\)
−0.337148 + 0.941452i \(0.609462\pi\)
\(882\) −3.96231e12 −0.220465
\(883\) 1.91595e13 1.06062 0.530311 0.847803i \(-0.322075\pi\)
0.530311 + 0.847803i \(0.322075\pi\)
\(884\) 1.49712e12 0.0824559
\(885\) 0 0
\(886\) 3.19140e13 1.73992
\(887\) 2.24991e13 1.22042 0.610209 0.792241i \(-0.291086\pi\)
0.610209 + 0.792241i \(0.291086\pi\)
\(888\) −7.09305e12 −0.382802
\(889\) 2.08449e13 1.11929
\(890\) 0 0
\(891\) 3.65300e12 0.194178
\(892\) 3.33795e12 0.176538
\(893\) 7.78825e12 0.409834
\(894\) −8.72380e12 −0.456759
\(895\) 0 0
\(896\) 1.45603e13 0.754719
\(897\) −2.26910e13 −1.17028
\(898\) 1.99576e12 0.102415
\(899\) 2.37560e12 0.121298
\(900\) 0 0
\(901\) −7.49841e12 −0.379060
\(902\) −1.91277e13 −0.962128
\(903\) 2.80506e12 0.140393
\(904\) 9.52380e12 0.474299
\(905\) 0 0
\(906\) 1.13242e13 0.558383
\(907\) 2.39724e13 1.17619 0.588096 0.808791i \(-0.299878\pi\)
0.588096 + 0.808791i \(0.299878\pi\)
\(908\) 1.22847e12 0.0599763
\(909\) 3.48033e12 0.169077
\(910\) 0 0
\(911\) −6.54470e12 −0.314816 −0.157408 0.987534i \(-0.550314\pi\)
−0.157408 + 0.987534i \(0.550314\pi\)
\(912\) −5.81427e12 −0.278303
\(913\) 4.38762e11 0.0208983
\(914\) −3.16236e12 −0.149884
\(915\) 0 0
\(916\) −3.26504e12 −0.153235
\(917\) 2.11918e12 0.0989706
\(918\) 1.54636e12 0.0718654
\(919\) 2.63348e12 0.121789 0.0608947 0.998144i \(-0.480605\pi\)
0.0608947 + 0.998144i \(0.480605\pi\)
\(920\) 0 0
\(921\) −1.86740e12 −0.0855203
\(922\) −1.84111e13 −0.839055
\(923\) 4.26258e13 1.93314
\(924\) −2.95539e12 −0.133380
\(925\) 0 0
\(926\) −1.42093e13 −0.635073
\(927\) 3.98388e11 0.0177193
\(928\) 1.13530e12 0.0502512
\(929\) 2.94271e13 1.29621 0.648106 0.761550i \(-0.275561\pi\)
0.648106 + 0.761550i \(0.275561\pi\)
\(930\) 0 0
\(931\) 5.70166e12 0.248730
\(932\) −3.04070e11 −0.0132009
\(933\) −6.25084e12 −0.270067
\(934\) −2.96305e13 −1.27403
\(935\) 0 0
\(936\) 7.89255e12 0.336106
\(937\) 1.71769e13 0.727974 0.363987 0.931404i \(-0.381415\pi\)
0.363987 + 0.931404i \(0.381415\pi\)
\(938\) −2.72201e13 −1.14809
\(939\) −3.79754e12 −0.159407
\(940\) 0 0
\(941\) −2.46944e13 −1.02670 −0.513352 0.858178i \(-0.671596\pi\)
−0.513352 + 0.858178i \(0.671596\pi\)
\(942\) 3.13632e12 0.129775
\(943\) 2.12475e13 0.874994
\(944\) −4.56498e13 −1.87096
\(945\) 0 0
\(946\) −1.82341e13 −0.740244
\(947\) 1.19428e12 0.0482537 0.0241268 0.999709i \(-0.492319\pi\)
0.0241268 + 0.999709i \(0.492319\pi\)
\(948\) 3.51153e12 0.141208
\(949\) 2.46319e13 0.985824
\(950\) 0 0
\(951\) 1.77730e13 0.704610
\(952\) 4.72377e12 0.186390
\(953\) −1.47694e13 −0.580023 −0.290011 0.957023i \(-0.593659\pi\)
−0.290011 + 0.957023i \(0.593659\pi\)
\(954\) 1.04693e13 0.409214
\(955\) 0 0
\(956\) −6.70966e12 −0.259800
\(957\) 3.19077e12 0.122968
\(958\) 3.63489e13 1.39427
\(959\) −2.01231e13 −0.768267
\(960\) 0 0
\(961\) −2.49176e11 −0.00942435
\(962\) 2.58351e13 0.972574
\(963\) 6.61532e12 0.247875
\(964\) 7.76192e12 0.289483
\(965\) 0 0
\(966\) 1.89616e13 0.700612
\(967\) 8.95584e12 0.329372 0.164686 0.986346i \(-0.447339\pi\)
0.164686 + 0.986346i \(0.447339\pi\)
\(968\) −4.87876e13 −1.78596
\(969\) −2.22518e12 −0.0810790
\(970\) 0 0
\(971\) −2.20587e13 −0.796330 −0.398165 0.917314i \(-0.630353\pi\)
−0.398165 + 0.917314i \(0.630353\pi\)
\(972\) −3.73806e11 −0.0134322
\(973\) 1.41118e13 0.504749
\(974\) 3.65621e13 1.30172
\(975\) 0 0
\(976\) −4.72469e13 −1.66667
\(977\) −3.56972e13 −1.25345 −0.626727 0.779239i \(-0.715606\pi\)
−0.626727 + 0.779239i \(0.715606\pi\)
\(978\) −2.32963e13 −0.814258
\(979\) 3.66806e13 1.27619
\(980\) 0 0
\(981\) −1.16408e13 −0.401303
\(982\) −4.38300e13 −1.50408
\(983\) −3.19525e13 −1.09147 −0.545737 0.837956i \(-0.683750\pi\)
−0.545737 + 0.837956i \(0.683750\pi\)
\(984\) −7.39045e12 −0.251300
\(985\) 0 0
\(986\) 1.35070e12 0.0455105
\(987\) 1.07692e13 0.361206
\(988\) 3.00787e12 0.100428
\(989\) 2.02549e13 0.673205
\(990\) 0 0
\(991\) 1.56883e13 0.516706 0.258353 0.966051i \(-0.416820\pi\)
0.258353 + 0.966051i \(0.416820\pi\)
\(992\) 1.25165e13 0.410374
\(993\) −3.15151e11 −0.0102860
\(994\) −3.56199e13 −1.15732
\(995\) 0 0
\(996\) −4.48979e10 −0.00144563
\(997\) −6.52145e12 −0.209033 −0.104517 0.994523i \(-0.533330\pi\)
−0.104517 + 0.994523i \(0.533330\pi\)
\(998\) −4.05647e13 −1.29438
\(999\) 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.a.g.1.2 2
3.2 odd 2 225.10.a.j.1.1 2
5.2 odd 4 75.10.b.e.49.3 4
5.3 odd 4 75.10.b.e.49.2 4
5.4 even 2 15.10.a.c.1.1 2
15.2 even 4 225.10.b.g.199.2 4
15.8 even 4 225.10.b.g.199.3 4
15.14 odd 2 45.10.a.e.1.2 2
20.19 odd 2 240.10.a.m.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
15.10.a.c.1.1 2 5.4 even 2
45.10.a.e.1.2 2 15.14 odd 2
75.10.a.g.1.2 2 1.1 even 1 trivial
75.10.b.e.49.2 4 5.3 odd 4
75.10.b.e.49.3 4 5.2 odd 4
225.10.a.j.1.1 2 3.2 odd 2
225.10.b.g.199.2 4 15.2 even 4
225.10.b.g.199.3 4 15.8 even 4
240.10.a.m.1.1 2 20.19 odd 2