Properties

Label 177.14.a.c.1.28
Level $177$
Weight $14$
Character 177.1
Self dual yes
Analytic conductor $189.799$
Analytic rank $0$
Dimension $31$
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] = [177,14,Mod(1,177)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(177, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 14, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("177.1");
 
S:= CuspForms(chi, 14);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 177 = 3 \cdot 59 \)
Weight: \( k \) \(=\) \( 14 \)
Character orbit: \([\chi]\) \(=\) 177.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: \(189.798744245\)
Analytic rank: \(0\)
Dimension: \(31\)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.28
Character \(\chi\) \(=\) 177.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+152.658 q^{2} +729.000 q^{3} +15112.3 q^{4} -3728.44 q^{5} +111287. q^{6} +589623. q^{7} +1.05644e6 q^{8} +531441. q^{9} +O(q^{10})\) \(q+152.658 q^{2} +729.000 q^{3} +15112.3 q^{4} -3728.44 q^{5} +111287. q^{6} +589623. q^{7} +1.05644e6 q^{8} +531441. q^{9} -569175. q^{10} +5.43191e6 q^{11} +1.10169e7 q^{12} +2.28754e7 q^{13} +9.00105e7 q^{14} -2.71803e6 q^{15} +3.74734e7 q^{16} -1.09897e7 q^{17} +8.11285e7 q^{18} +2.75303e8 q^{19} -5.63454e7 q^{20} +4.29835e8 q^{21} +8.29222e8 q^{22} -7.85204e7 q^{23} +7.70145e8 q^{24} -1.20680e9 q^{25} +3.49211e9 q^{26} +3.87420e8 q^{27} +8.91058e9 q^{28} -3.24056e8 q^{29} -4.14929e8 q^{30} -7.36239e9 q^{31} -2.93376e9 q^{32} +3.95986e9 q^{33} -1.67766e9 q^{34} -2.19838e9 q^{35} +8.03131e9 q^{36} -1.97544e10 q^{37} +4.20271e10 q^{38} +1.66762e10 q^{39} -3.93888e9 q^{40} +3.27680e10 q^{41} +6.56176e10 q^{42} +1.31535e10 q^{43} +8.20888e10 q^{44} -1.98145e9 q^{45} -1.19867e10 q^{46} -1.06879e11 q^{47} +2.73181e10 q^{48} +2.50767e11 q^{49} -1.84227e11 q^{50} -8.01149e9 q^{51} +3.45701e11 q^{52} -1.42998e11 q^{53} +5.91427e10 q^{54} -2.02526e10 q^{55} +6.22902e11 q^{56} +2.00696e11 q^{57} -4.94695e10 q^{58} -4.21805e10 q^{59} -4.10758e10 q^{60} -2.06447e11 q^{61} -1.12392e12 q^{62} +3.13350e11 q^{63} -7.54843e11 q^{64} -8.52897e10 q^{65} +6.04503e11 q^{66} +3.95170e11 q^{67} -1.66080e11 q^{68} -5.72413e10 q^{69} -3.35599e11 q^{70} +9.27090e11 q^{71} +5.61436e11 q^{72} -1.26876e12 q^{73} -3.01566e12 q^{74} -8.79759e11 q^{75} +4.16047e12 q^{76} +3.20278e12 q^{77} +2.54575e12 q^{78} +1.43511e12 q^{79} -1.39717e11 q^{80} +2.82430e11 q^{81} +5.00228e12 q^{82} +2.65875e12 q^{83} +6.49581e12 q^{84} +4.09745e10 q^{85} +2.00797e12 q^{86} -2.36237e11 q^{87} +5.73849e12 q^{88} -4.12973e12 q^{89} -3.02483e11 q^{90} +1.34879e13 q^{91} -1.18663e12 q^{92} -5.36718e12 q^{93} -1.63159e13 q^{94} -1.02645e12 q^{95} -2.13871e12 q^{96} -8.48986e12 q^{97} +3.82814e13 q^{98} +2.88674e12 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 31 q + 310 q^{2} + 22599 q^{3} + 126886 q^{4} + 81008 q^{5} + 225990 q^{6} + 1002941 q^{7} + 4632723 q^{8} + 16474671 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 31 q + 310 q^{2} + 22599 q^{3} + 126886 q^{4} + 81008 q^{5} + 225990 q^{6} + 1002941 q^{7} + 4632723 q^{8} + 16474671 q^{9} + 4647481 q^{10} + 17937316 q^{11} + 92499894 q^{12} + 40664720 q^{13} + 139193613 q^{14} + 59054832 q^{15} + 370110498 q^{16} + 213442823 q^{17} + 164746710 q^{18} - 62592329 q^{19} + 1637085153 q^{20} + 731143989 q^{21} + 4142028314 q^{22} + 1873486387 q^{23} + 3377255067 q^{24} + 8307272395 q^{25} - 534777728 q^{26} + 12010035159 q^{27} + 766416778 q^{28} + 13765513563 q^{29} + 3388013649 q^{30} + 14274077235 q^{31} + 30574460156 q^{32} + 13076303364 q^{33} - 677551028 q^{34} + 36023610185 q^{35} + 67432422726 q^{36} - 18278838391 q^{37} - 23650502933 q^{38} + 29644580880 q^{39} + 10045447572 q^{40} + 34748006725 q^{41} + 101472143877 q^{42} + 40350158146 q^{43} + 163101196592 q^{44} + 43050972528 q^{45} + 296118466353 q^{46} + 233954631099 q^{47} + 269810553042 q^{48} + 324065402790 q^{49} - 102960745787 q^{50} + 155599817967 q^{51} + 668297695096 q^{52} + 500927963876 q^{53} + 120100351590 q^{54} + 884972340924 q^{55} + 1392234478810 q^{56} - 45629807841 q^{57} + 689262776200 q^{58} - 1307596542871 q^{59} + 1193435076537 q^{60} + 1716832157925 q^{61} + 1816094290366 q^{62} + 533003967981 q^{63} + 4381780009133 q^{64} + 1457007885906 q^{65} + 3019538640906 q^{66} + 1212131702006 q^{67} + 6552992665503 q^{68} + 1365771576123 q^{69} + 8806714081634 q^{70} + 6074000239936 q^{71} + 2462018943843 q^{72} + 3756145185973 q^{73} + 8066450143602 q^{74} + 6056001575955 q^{75} + 7913230001992 q^{76} + 6031241575915 q^{77} - 389852963712 q^{78} + 11377744190862 q^{79} + 16473302366969 q^{80} + 8755315630911 q^{81} + 10413363680159 q^{82} + 19915461517429 q^{83} + 558717831162 q^{84} + 15280981141573 q^{85} + 7573325358452 q^{86} + 10035059387427 q^{87} + 19271409121081 q^{88} + 14115863121241 q^{89} + 2469861950121 q^{90} + 18296287784699 q^{91} + 15158951168774 q^{92} + 10405802304315 q^{93} - 18637923572412 q^{94} - 2294034679397 q^{95} + 22288781453724 q^{96} + 38558536599054 q^{97} - 1998410212380 q^{98} + 9532625152356 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 152.658 1.68664 0.843322 0.537409i \(-0.180597\pi\)
0.843322 + 0.537409i \(0.180597\pi\)
\(3\) 729.000 0.577350
\(4\) 15112.3 1.84477
\(5\) −3728.44 −0.106714 −0.0533571 0.998575i \(-0.516992\pi\)
−0.0533571 + 0.998575i \(0.516992\pi\)
\(6\) 111287. 0.973784
\(7\) 589623. 1.89425 0.947125 0.320864i \(-0.103973\pi\)
0.947125 + 0.320864i \(0.103973\pi\)
\(8\) 1.05644e6 1.42482
\(9\) 531441. 0.333333
\(10\) −569175. −0.179989
\(11\) 5.43191e6 0.924486 0.462243 0.886753i \(-0.347045\pi\)
0.462243 + 0.886753i \(0.347045\pi\)
\(12\) 1.10169e7 1.06508
\(13\) 2.28754e7 1.31443 0.657215 0.753703i \(-0.271734\pi\)
0.657215 + 0.753703i \(0.271734\pi\)
\(14\) 9.00105e7 3.19493
\(15\) −2.71803e6 −0.0616115
\(16\) 3.74734e7 0.558397
\(17\) −1.09897e7 −0.110425 −0.0552125 0.998475i \(-0.517584\pi\)
−0.0552125 + 0.998475i \(0.517584\pi\)
\(18\) 8.11285e7 0.562215
\(19\) 2.75303e8 1.34249 0.671247 0.741233i \(-0.265759\pi\)
0.671247 + 0.741233i \(0.265759\pi\)
\(20\) −5.63454e7 −0.196863
\(21\) 4.29835e8 1.09365
\(22\) 8.29222e8 1.55928
\(23\) −7.85204e7 −0.110599 −0.0552995 0.998470i \(-0.517611\pi\)
−0.0552995 + 0.998470i \(0.517611\pi\)
\(24\) 7.70145e8 0.822620
\(25\) −1.20680e9 −0.988612
\(26\) 3.49211e9 2.21697
\(27\) 3.87420e8 0.192450
\(28\) 8.91058e9 3.49445
\(29\) −3.24056e8 −0.101166 −0.0505828 0.998720i \(-0.516108\pi\)
−0.0505828 + 0.998720i \(0.516108\pi\)
\(30\) −4.14929e8 −0.103917
\(31\) −7.36239e9 −1.48994 −0.744968 0.667100i \(-0.767535\pi\)
−0.744968 + 0.667100i \(0.767535\pi\)
\(32\) −2.93376e9 −0.483003
\(33\) 3.95986e9 0.533752
\(34\) −1.67766e9 −0.186248
\(35\) −2.19838e9 −0.202144
\(36\) 8.03131e9 0.614922
\(37\) −1.97544e10 −1.26576 −0.632881 0.774249i \(-0.718128\pi\)
−0.632881 + 0.774249i \(0.718128\pi\)
\(38\) 4.20271e10 2.26431
\(39\) 1.66762e10 0.758886
\(40\) −3.93888e9 −0.152049
\(41\) 3.27680e10 1.07734 0.538672 0.842515i \(-0.318926\pi\)
0.538672 + 0.842515i \(0.318926\pi\)
\(42\) 6.56176e10 1.84459
\(43\) 1.31535e10 0.317318 0.158659 0.987333i \(-0.449283\pi\)
0.158659 + 0.987333i \(0.449283\pi\)
\(44\) 8.20888e10 1.70546
\(45\) −1.98145e9 −0.0355714
\(46\) −1.19867e10 −0.186541
\(47\) −1.06879e11 −1.44629 −0.723146 0.690695i \(-0.757305\pi\)
−0.723146 + 0.690695i \(0.757305\pi\)
\(48\) 2.73181e10 0.322391
\(49\) 2.50767e11 2.58818
\(50\) −1.84227e11 −1.66744
\(51\) −8.01149e9 −0.0637540
\(52\) 3.45701e11 2.42482
\(53\) −1.42998e11 −0.886210 −0.443105 0.896470i \(-0.646123\pi\)
−0.443105 + 0.896470i \(0.646123\pi\)
\(54\) 5.91427e10 0.324595
\(55\) −2.02526e10 −0.0986558
\(56\) 6.22902e11 2.69897
\(57\) 2.00696e11 0.775090
\(58\) −4.94695e10 −0.170630
\(59\) −4.21805e10 −0.130189
\(60\) −4.10758e10 −0.113659
\(61\) −2.06447e11 −0.513056 −0.256528 0.966537i \(-0.582579\pi\)
−0.256528 + 0.966537i \(0.582579\pi\)
\(62\) −1.12392e12 −2.51299
\(63\) 3.13350e11 0.631417
\(64\) −7.54843e11 −1.37305
\(65\) −8.52897e10 −0.140268
\(66\) 6.04503e11 0.900250
\(67\) 3.95170e11 0.533701 0.266850 0.963738i \(-0.414017\pi\)
0.266850 + 0.963738i \(0.414017\pi\)
\(68\) −1.66080e11 −0.203709
\(69\) −5.72413e10 −0.0638544
\(70\) −3.35599e11 −0.340944
\(71\) 9.27090e11 0.858900 0.429450 0.903091i \(-0.358707\pi\)
0.429450 + 0.903091i \(0.358707\pi\)
\(72\) 5.61436e11 0.474940
\(73\) −1.26876e12 −0.981254 −0.490627 0.871370i \(-0.663232\pi\)
−0.490627 + 0.871370i \(0.663232\pi\)
\(74\) −3.01566e12 −2.13489
\(75\) −8.79759e11 −0.570775
\(76\) 4.16047e12 2.47659
\(77\) 3.20278e12 1.75121
\(78\) 2.54575e12 1.27997
\(79\) 1.43511e12 0.664214 0.332107 0.943242i \(-0.392240\pi\)
0.332107 + 0.943242i \(0.392240\pi\)
\(80\) −1.39717e11 −0.0595890
\(81\) 2.82430e11 0.111111
\(82\) 5.00228e12 1.81710
\(83\) 2.65875e12 0.892626 0.446313 0.894877i \(-0.352737\pi\)
0.446313 + 0.894877i \(0.352737\pi\)
\(84\) 6.49581e12 2.01752
\(85\) 4.09745e10 0.0117839
\(86\) 2.00797e12 0.535203
\(87\) −2.36237e11 −0.0584079
\(88\) 5.73849e12 1.31723
\(89\) −4.12973e12 −0.880819 −0.440409 0.897797i \(-0.645167\pi\)
−0.440409 + 0.897797i \(0.645167\pi\)
\(90\) −3.02483e11 −0.0599963
\(91\) 1.34879e13 2.48986
\(92\) −1.18663e12 −0.204029
\(93\) −5.36718e12 −0.860215
\(94\) −1.63159e13 −2.43938
\(95\) −1.02645e12 −0.143263
\(96\) −2.13871e12 −0.278862
\(97\) −8.48986e12 −1.03487 −0.517433 0.855724i \(-0.673112\pi\)
−0.517433 + 0.855724i \(0.673112\pi\)
\(98\) 3.82814e13 4.36535
\(99\) 2.88674e12 0.308162
\(100\) −1.82376e13 −1.82376
\(101\) 5.35984e12 0.502415 0.251207 0.967933i \(-0.419172\pi\)
0.251207 + 0.967933i \(0.419172\pi\)
\(102\) −1.22301e12 −0.107530
\(103\) 5.96798e11 0.0492477 0.0246238 0.999697i \(-0.492161\pi\)
0.0246238 + 0.999697i \(0.492161\pi\)
\(104\) 2.41665e13 1.87283
\(105\) −1.60262e12 −0.116708
\(106\) −2.18297e13 −1.49472
\(107\) −4.76546e12 −0.306980 −0.153490 0.988150i \(-0.549051\pi\)
−0.153490 + 0.988150i \(0.549051\pi\)
\(108\) 5.85483e12 0.355026
\(109\) 1.72499e13 0.985177 0.492589 0.870262i \(-0.336051\pi\)
0.492589 + 0.870262i \(0.336051\pi\)
\(110\) −3.09171e12 −0.166397
\(111\) −1.44009e13 −0.730788
\(112\) 2.20952e13 1.05774
\(113\) −8.56566e12 −0.387036 −0.193518 0.981097i \(-0.561990\pi\)
−0.193518 + 0.981097i \(0.561990\pi\)
\(114\) 3.06378e13 1.30730
\(115\) 2.92759e11 0.0118025
\(116\) −4.89723e12 −0.186627
\(117\) 1.21569e13 0.438143
\(118\) −6.43918e12 −0.219582
\(119\) −6.47978e12 −0.209173
\(120\) −2.87144e12 −0.0877853
\(121\) −5.01704e12 −0.145326
\(122\) −3.15157e13 −0.865342
\(123\) 2.38879e13 0.622005
\(124\) −1.11263e14 −2.74858
\(125\) 9.05081e12 0.212213
\(126\) 4.78352e13 1.06498
\(127\) 8.94189e12 0.189106 0.0945530 0.995520i \(-0.469858\pi\)
0.0945530 + 0.995520i \(0.469858\pi\)
\(128\) −9.11991e13 −1.83285
\(129\) 9.58887e12 0.183204
\(130\) −1.30201e13 −0.236583
\(131\) 1.09136e14 1.88670 0.943350 0.331798i \(-0.107655\pi\)
0.943350 + 0.331798i \(0.107655\pi\)
\(132\) 5.98428e13 0.984648
\(133\) 1.62325e14 2.54302
\(134\) 6.03257e13 0.900163
\(135\) −1.44448e12 −0.0205372
\(136\) −1.16100e13 −0.157336
\(137\) 6.92853e13 0.895277 0.447639 0.894215i \(-0.352265\pi\)
0.447639 + 0.894215i \(0.352265\pi\)
\(138\) −8.73832e12 −0.107700
\(139\) −8.52271e13 −1.00226 −0.501131 0.865371i \(-0.667083\pi\)
−0.501131 + 0.865371i \(0.667083\pi\)
\(140\) −3.32226e13 −0.372908
\(141\) −7.79147e13 −0.835017
\(142\) 1.41527e14 1.44866
\(143\) 1.24257e14 1.21517
\(144\) 1.99149e13 0.186132
\(145\) 1.20822e12 0.0107958
\(146\) −1.93686e14 −1.65503
\(147\) 1.82809e14 1.49429
\(148\) −2.98535e14 −2.33504
\(149\) 1.19681e14 0.896010 0.448005 0.894031i \(-0.352135\pi\)
0.448005 + 0.894031i \(0.352135\pi\)
\(150\) −1.34302e14 −0.962695
\(151\) −1.12308e14 −0.771010 −0.385505 0.922706i \(-0.625973\pi\)
−0.385505 + 0.922706i \(0.625973\pi\)
\(152\) 2.90841e14 1.91281
\(153\) −5.84038e12 −0.0368084
\(154\) 4.88929e14 2.95366
\(155\) 2.74502e13 0.158997
\(156\) 2.52016e14 1.39997
\(157\) −4.74552e12 −0.0252893 −0.0126446 0.999920i \(-0.504025\pi\)
−0.0126446 + 0.999920i \(0.504025\pi\)
\(158\) 2.19080e14 1.12029
\(159\) −1.04245e14 −0.511653
\(160\) 1.09384e13 0.0515433
\(161\) −4.62974e13 −0.209502
\(162\) 4.31150e13 0.187405
\(163\) −3.13861e14 −1.31074 −0.655372 0.755307i \(-0.727488\pi\)
−0.655372 + 0.755307i \(0.727488\pi\)
\(164\) 4.95201e14 1.98745
\(165\) −1.47641e13 −0.0569590
\(166\) 4.05878e14 1.50554
\(167\) 2.18550e14 0.779639 0.389820 0.920891i \(-0.372537\pi\)
0.389820 + 0.920891i \(0.372537\pi\)
\(168\) 4.54095e14 1.55825
\(169\) 2.20410e14 0.727725
\(170\) 6.25506e12 0.0198753
\(171\) 1.46307e14 0.447498
\(172\) 1.98779e14 0.585378
\(173\) −6.49174e13 −0.184103 −0.0920516 0.995754i \(-0.529342\pi\)
−0.0920516 + 0.995754i \(0.529342\pi\)
\(174\) −3.60633e13 −0.0985134
\(175\) −7.11559e14 −1.87268
\(176\) 2.03552e14 0.516230
\(177\) −3.07496e13 −0.0751646
\(178\) −6.30434e14 −1.48563
\(179\) 2.14196e14 0.486706 0.243353 0.969938i \(-0.421753\pi\)
0.243353 + 0.969938i \(0.421753\pi\)
\(180\) −2.99443e13 −0.0656210
\(181\) −5.05300e14 −1.06816 −0.534082 0.845432i \(-0.679343\pi\)
−0.534082 + 0.845432i \(0.679343\pi\)
\(182\) 2.05903e15 4.19950
\(183\) −1.50500e14 −0.296213
\(184\) −8.29521e13 −0.157584
\(185\) 7.36531e13 0.135075
\(186\) −8.19340e14 −1.45088
\(187\) −5.96951e13 −0.102086
\(188\) −1.61519e15 −2.66807
\(189\) 2.28432e14 0.364549
\(190\) −1.56696e14 −0.241634
\(191\) 3.23683e13 0.0482394 0.0241197 0.999709i \(-0.492322\pi\)
0.0241197 + 0.999709i \(0.492322\pi\)
\(192\) −5.50280e14 −0.792731
\(193\) −8.88242e14 −1.23711 −0.618555 0.785741i \(-0.712282\pi\)
−0.618555 + 0.785741i \(0.712282\pi\)
\(194\) −1.29604e15 −1.74545
\(195\) −6.21762e13 −0.0809840
\(196\) 3.78967e15 4.77460
\(197\) 7.18560e14 0.875855 0.437928 0.899010i \(-0.355713\pi\)
0.437928 + 0.899010i \(0.355713\pi\)
\(198\) 4.40683e14 0.519759
\(199\) −6.24778e14 −0.713149 −0.356575 0.934267i \(-0.616055\pi\)
−0.356575 + 0.934267i \(0.616055\pi\)
\(200\) −1.27491e15 −1.40859
\(201\) 2.88079e14 0.308132
\(202\) 8.18219e14 0.847395
\(203\) −1.91071e14 −0.191633
\(204\) −1.21072e14 −0.117611
\(205\) −1.22174e14 −0.114968
\(206\) 9.11058e13 0.0830633
\(207\) −4.17289e13 −0.0368663
\(208\) 8.57220e14 0.733974
\(209\) 1.49542e15 1.24112
\(210\) −2.44652e14 −0.196844
\(211\) 2.02198e15 1.57740 0.788698 0.614781i \(-0.210755\pi\)
0.788698 + 0.614781i \(0.210755\pi\)
\(212\) −2.16103e15 −1.63485
\(213\) 6.75849e14 0.495886
\(214\) −7.27484e14 −0.517766
\(215\) −4.90419e13 −0.0338624
\(216\) 4.09287e14 0.274207
\(217\) −4.34103e15 −2.82231
\(218\) 2.63333e15 1.66164
\(219\) −9.24927e14 −0.566527
\(220\) −3.06064e14 −0.181997
\(221\) −2.51394e14 −0.145146
\(222\) −2.19841e15 −1.23258
\(223\) −2.84503e15 −1.54919 −0.774597 0.632455i \(-0.782047\pi\)
−0.774597 + 0.632455i \(0.782047\pi\)
\(224\) −1.72981e15 −0.914929
\(225\) −6.41344e14 −0.329537
\(226\) −1.30761e15 −0.652791
\(227\) 1.84605e14 0.0895522 0.0447761 0.998997i \(-0.485743\pi\)
0.0447761 + 0.998997i \(0.485743\pi\)
\(228\) 3.03298e15 1.42986
\(229\) −1.29996e15 −0.595660 −0.297830 0.954619i \(-0.596263\pi\)
−0.297830 + 0.954619i \(0.596263\pi\)
\(230\) 4.46918e13 0.0199066
\(231\) 2.33483e15 1.01106
\(232\) −3.42345e14 −0.144143
\(233\) −4.49509e15 −1.84045 −0.920227 0.391385i \(-0.871996\pi\)
−0.920227 + 0.391385i \(0.871996\pi\)
\(234\) 1.85585e15 0.738991
\(235\) 3.98492e14 0.154340
\(236\) −6.37446e14 −0.240168
\(237\) 1.04619e15 0.383484
\(238\) −9.89188e14 −0.352800
\(239\) 3.10807e15 1.07871 0.539355 0.842078i \(-0.318668\pi\)
0.539355 + 0.842078i \(0.318668\pi\)
\(240\) −1.01854e14 −0.0344037
\(241\) 1.98521e15 0.652673 0.326337 0.945254i \(-0.394186\pi\)
0.326337 + 0.945254i \(0.394186\pi\)
\(242\) −7.65890e14 −0.245113
\(243\) 2.05891e14 0.0641500
\(244\) −3.11989e15 −0.946468
\(245\) −9.34969e14 −0.276196
\(246\) 3.64666e15 1.04910
\(247\) 6.29767e15 1.76461
\(248\) −7.77792e15 −2.12289
\(249\) 1.93823e15 0.515358
\(250\) 1.38167e15 0.357928
\(251\) −7.19917e15 −1.81720 −0.908601 0.417666i \(-0.862848\pi\)
−0.908601 + 0.417666i \(0.862848\pi\)
\(252\) 4.73545e15 1.16482
\(253\) −4.26516e14 −0.102247
\(254\) 1.36505e15 0.318954
\(255\) 2.98704e13 0.00680346
\(256\) −7.73856e15 −1.71831
\(257\) 6.27503e15 1.35847 0.679236 0.733920i \(-0.262311\pi\)
0.679236 + 0.733920i \(0.262311\pi\)
\(258\) 1.46381e15 0.308999
\(259\) −1.16476e16 −2.39767
\(260\) −1.28893e15 −0.258762
\(261\) −1.72216e14 −0.0337218
\(262\) 1.66604e16 3.18219
\(263\) 3.04483e15 0.567350 0.283675 0.958920i \(-0.408446\pi\)
0.283675 + 0.958920i \(0.408446\pi\)
\(264\) 4.18336e15 0.760501
\(265\) 5.33159e14 0.0945712
\(266\) 2.47802e16 4.28917
\(267\) −3.01057e15 −0.508541
\(268\) 5.97194e15 0.984553
\(269\) −4.95500e15 −0.797359 −0.398680 0.917090i \(-0.630531\pi\)
−0.398680 + 0.917090i \(0.630531\pi\)
\(270\) −2.20510e14 −0.0346389
\(271\) −4.41996e15 −0.677825 −0.338913 0.940818i \(-0.610059\pi\)
−0.338913 + 0.940818i \(0.610059\pi\)
\(272\) −4.11821e14 −0.0616611
\(273\) 9.83267e15 1.43752
\(274\) 1.05769e16 1.51001
\(275\) −6.55524e15 −0.913958
\(276\) −8.65050e14 −0.117796
\(277\) 2.65957e15 0.353747 0.176874 0.984234i \(-0.443402\pi\)
0.176874 + 0.984234i \(0.443402\pi\)
\(278\) −1.30106e16 −1.69046
\(279\) −3.91267e15 −0.496645
\(280\) −2.32245e15 −0.288018
\(281\) 1.37797e16 1.66974 0.834872 0.550445i \(-0.185542\pi\)
0.834872 + 0.550445i \(0.185542\pi\)
\(282\) −1.18943e16 −1.40838
\(283\) 9.87686e15 1.14290 0.571448 0.820638i \(-0.306382\pi\)
0.571448 + 0.820638i \(0.306382\pi\)
\(284\) 1.40105e16 1.58447
\(285\) −7.48283e14 −0.0827131
\(286\) 1.89688e16 2.04956
\(287\) 1.93208e16 2.04076
\(288\) −1.55912e15 −0.161001
\(289\) −9.78380e15 −0.987806
\(290\) 1.84444e14 0.0182087
\(291\) −6.18911e15 −0.597481
\(292\) −1.91739e16 −1.81018
\(293\) −8.99561e15 −0.830598 −0.415299 0.909685i \(-0.636323\pi\)
−0.415299 + 0.909685i \(0.636323\pi\)
\(294\) 2.79072e16 2.52033
\(295\) 1.57268e14 0.0138930
\(296\) −2.08693e16 −1.80348
\(297\) 2.10443e15 0.177917
\(298\) 1.82701e16 1.51125
\(299\) −1.79619e15 −0.145375
\(300\) −1.32952e16 −1.05295
\(301\) 7.75559e15 0.601080
\(302\) −1.71446e16 −1.30042
\(303\) 3.90732e15 0.290069
\(304\) 1.03165e16 0.749646
\(305\) 7.69726e14 0.0547504
\(306\) −8.91578e14 −0.0620826
\(307\) −1.50563e16 −1.02641 −0.513204 0.858267i \(-0.671542\pi\)
−0.513204 + 0.858267i \(0.671542\pi\)
\(308\) 4.84015e16 3.23057
\(309\) 4.35066e14 0.0284332
\(310\) 4.19049e15 0.268172
\(311\) −1.08453e16 −0.679672 −0.339836 0.940485i \(-0.610372\pi\)
−0.339836 + 0.940485i \(0.610372\pi\)
\(312\) 1.76174e16 1.08128
\(313\) 1.05865e16 0.636375 0.318187 0.948028i \(-0.396926\pi\)
0.318187 + 0.948028i \(0.396926\pi\)
\(314\) −7.24439e14 −0.0426540
\(315\) −1.16831e15 −0.0673812
\(316\) 2.16878e16 1.22532
\(317\) 3.53509e16 1.95666 0.978330 0.207052i \(-0.0663869\pi\)
0.978330 + 0.207052i \(0.0663869\pi\)
\(318\) −1.59139e16 −0.862977
\(319\) −1.76024e15 −0.0935261
\(320\) 2.81439e15 0.146524
\(321\) −3.47402e15 −0.177235
\(322\) −7.06765e15 −0.353356
\(323\) −3.02550e15 −0.148245
\(324\) 4.26817e15 0.204974
\(325\) −2.76061e16 −1.29946
\(326\) −4.79132e16 −2.21076
\(327\) 1.25752e16 0.568792
\(328\) 3.46174e16 1.53502
\(329\) −6.30183e16 −2.73964
\(330\) −2.25386e15 −0.0960695
\(331\) 1.03400e16 0.432152 0.216076 0.976377i \(-0.430674\pi\)
0.216076 + 0.976377i \(0.430674\pi\)
\(332\) 4.01799e16 1.64669
\(333\) −1.04983e16 −0.421921
\(334\) 3.33633e16 1.31497
\(335\) −1.47337e15 −0.0569535
\(336\) 1.61074e16 0.610689
\(337\) −5.59310e15 −0.207998 −0.103999 0.994577i \(-0.533164\pi\)
−0.103999 + 0.994577i \(0.533164\pi\)
\(338\) 3.36472e16 1.22741
\(339\) −6.24437e15 −0.223455
\(340\) 6.19219e14 0.0217386
\(341\) −3.99918e16 −1.37742
\(342\) 2.23349e16 0.754770
\(343\) 9.07299e16 3.00842
\(344\) 1.38958e16 0.452121
\(345\) 2.13421e14 0.00681417
\(346\) −9.91013e15 −0.310516
\(347\) 4.98969e16 1.53438 0.767189 0.641422i \(-0.221655\pi\)
0.767189 + 0.641422i \(0.221655\pi\)
\(348\) −3.57008e15 −0.107749
\(349\) 5.12688e16 1.51876 0.759378 0.650650i \(-0.225503\pi\)
0.759378 + 0.650650i \(0.225503\pi\)
\(350\) −1.08625e17 −3.15854
\(351\) 8.86241e15 0.252962
\(352\) −1.59359e16 −0.446529
\(353\) −4.97761e16 −1.36926 −0.684629 0.728892i \(-0.740036\pi\)
−0.684629 + 0.728892i \(0.740036\pi\)
\(354\) −4.69416e15 −0.126776
\(355\) −3.45660e15 −0.0916569
\(356\) −6.24098e16 −1.62491
\(357\) −4.72376e15 −0.120766
\(358\) 3.26986e16 0.820899
\(359\) −4.26935e13 −0.00105256 −0.000526281 1.00000i \(-0.500168\pi\)
−0.000526281 1.00000i \(0.500168\pi\)
\(360\) −2.09328e15 −0.0506829
\(361\) 3.37388e16 0.802293
\(362\) −7.71378e16 −1.80161
\(363\) −3.65742e15 −0.0839039
\(364\) 2.03833e17 4.59321
\(365\) 4.73050e15 0.104714
\(366\) −2.29749e16 −0.499606
\(367\) 5.34119e16 1.14106 0.570530 0.821277i \(-0.306738\pi\)
0.570530 + 0.821277i \(0.306738\pi\)
\(368\) −2.94243e15 −0.0617582
\(369\) 1.74143e16 0.359115
\(370\) 1.12437e16 0.227823
\(371\) −8.43149e16 −1.67870
\(372\) −8.11106e16 −1.58690
\(373\) 8.78818e16 1.68963 0.844815 0.535059i \(-0.179710\pi\)
0.844815 + 0.535059i \(0.179710\pi\)
\(374\) −9.11290e15 −0.172183
\(375\) 6.59804e15 0.122521
\(376\) −1.12911e17 −2.06071
\(377\) −7.41291e15 −0.132975
\(378\) 3.48719e16 0.614864
\(379\) −8.28194e16 −1.43541 −0.717707 0.696345i \(-0.754808\pi\)
−0.717707 + 0.696345i \(0.754808\pi\)
\(380\) −1.55121e16 −0.264287
\(381\) 6.51864e15 0.109180
\(382\) 4.94126e15 0.0813628
\(383\) 1.06814e17 1.72916 0.864582 0.502491i \(-0.167583\pi\)
0.864582 + 0.502491i \(0.167583\pi\)
\(384\) −6.64841e16 −1.05819
\(385\) −1.19414e16 −0.186879
\(386\) −1.35597e17 −2.08657
\(387\) 6.99029e15 0.105773
\(388\) −1.28302e17 −1.90909
\(389\) 1.75112e16 0.256239 0.128119 0.991759i \(-0.459106\pi\)
0.128119 + 0.991759i \(0.459106\pi\)
\(390\) −9.49166e15 −0.136591
\(391\) 8.62915e14 0.0122129
\(392\) 2.64920e17 3.68770
\(393\) 7.95599e16 1.08929
\(394\) 1.09694e17 1.47726
\(395\) −5.35071e15 −0.0708811
\(396\) 4.36254e16 0.568487
\(397\) −2.57669e16 −0.330311 −0.165156 0.986268i \(-0.552813\pi\)
−0.165156 + 0.986268i \(0.552813\pi\)
\(398\) −9.53770e16 −1.20283
\(399\) 1.18335e17 1.46821
\(400\) −4.52230e16 −0.552038
\(401\) −2.00205e16 −0.240456 −0.120228 0.992746i \(-0.538363\pi\)
−0.120228 + 0.992746i \(0.538363\pi\)
\(402\) 4.39774e16 0.519709
\(403\) −1.68418e17 −1.95842
\(404\) 8.09996e16 0.926838
\(405\) −1.05302e15 −0.0118571
\(406\) −2.91684e16 −0.323216
\(407\) −1.07304e17 −1.17018
\(408\) −8.46366e15 −0.0908379
\(409\) 4.44128e16 0.469144 0.234572 0.972099i \(-0.424631\pi\)
0.234572 + 0.972099i \(0.424631\pi\)
\(410\) −1.86507e16 −0.193910
\(411\) 5.05090e16 0.516889
\(412\) 9.01901e15 0.0908505
\(413\) −2.48706e16 −0.246610
\(414\) −6.37024e15 −0.0621804
\(415\) −9.91299e15 −0.0952560
\(416\) −6.71110e16 −0.634873
\(417\) −6.21306e16 −0.578657
\(418\) 2.28287e17 2.09332
\(419\) −1.91747e17 −1.73116 −0.865581 0.500770i \(-0.833050\pi\)
−0.865581 + 0.500770i \(0.833050\pi\)
\(420\) −2.42193e16 −0.215298
\(421\) 8.96505e16 0.784728 0.392364 0.919810i \(-0.371657\pi\)
0.392364 + 0.919810i \(0.371657\pi\)
\(422\) 3.08670e17 2.66051
\(423\) −5.67998e16 −0.482097
\(424\) −1.51069e17 −1.26269
\(425\) 1.32624e16 0.109168
\(426\) 1.03173e17 0.836383
\(427\) −1.21726e17 −0.971856
\(428\) −7.20172e16 −0.566307
\(429\) 9.05836e16 0.701580
\(430\) −7.48662e15 −0.0571138
\(431\) −4.39883e16 −0.330548 −0.165274 0.986248i \(-0.552851\pi\)
−0.165274 + 0.986248i \(0.552851\pi\)
\(432\) 1.45180e16 0.107464
\(433\) −1.69538e17 −1.23622 −0.618111 0.786091i \(-0.712102\pi\)
−0.618111 + 0.786091i \(0.712102\pi\)
\(434\) −6.62692e17 −4.76023
\(435\) 8.80794e14 0.00623296
\(436\) 2.60686e17 1.81742
\(437\) −2.16169e16 −0.148479
\(438\) −1.41197e17 −0.955529
\(439\) −7.60767e16 −0.507262 −0.253631 0.967301i \(-0.581625\pi\)
−0.253631 + 0.967301i \(0.581625\pi\)
\(440\) −2.13956e16 −0.140567
\(441\) 1.33268e17 0.862728
\(442\) −3.83772e16 −0.244810
\(443\) −1.35028e17 −0.848791 −0.424395 0.905477i \(-0.639513\pi\)
−0.424395 + 0.905477i \(0.639513\pi\)
\(444\) −2.17632e17 −1.34813
\(445\) 1.53975e16 0.0939959
\(446\) −4.34316e17 −2.61294
\(447\) 8.72471e16 0.517312
\(448\) −4.45073e17 −2.60090
\(449\) −2.93046e17 −1.68785 −0.843925 0.536460i \(-0.819761\pi\)
−0.843925 + 0.536460i \(0.819761\pi\)
\(450\) −9.79060e16 −0.555812
\(451\) 1.77993e17 0.995990
\(452\) −1.29447e17 −0.713990
\(453\) −8.18724e16 −0.445143
\(454\) 2.81814e16 0.151043
\(455\) −5.02888e16 −0.265703
\(456\) 2.12023e17 1.10436
\(457\) −1.25854e17 −0.646268 −0.323134 0.946353i \(-0.604736\pi\)
−0.323134 + 0.946353i \(0.604736\pi\)
\(458\) −1.98448e17 −1.00467
\(459\) −4.25763e15 −0.0212513
\(460\) 4.42427e15 0.0217728
\(461\) 3.67569e16 0.178354 0.0891769 0.996016i \(-0.471576\pi\)
0.0891769 + 0.996016i \(0.471576\pi\)
\(462\) 3.56429e17 1.70530
\(463\) −1.80159e17 −0.849923 −0.424962 0.905211i \(-0.639712\pi\)
−0.424962 + 0.905211i \(0.639712\pi\)
\(464\) −1.21435e16 −0.0564906
\(465\) 2.00112e16 0.0917972
\(466\) −6.86210e17 −3.10419
\(467\) −3.35310e15 −0.0149585 −0.00747923 0.999972i \(-0.502381\pi\)
−0.00747923 + 0.999972i \(0.502381\pi\)
\(468\) 1.83720e17 0.808272
\(469\) 2.33001e17 1.01096
\(470\) 6.08328e16 0.260317
\(471\) −3.45948e15 −0.0146008
\(472\) −4.45612e16 −0.185496
\(473\) 7.14484e16 0.293356
\(474\) 1.59709e17 0.646801
\(475\) −3.32236e17 −1.32721
\(476\) −9.79246e16 −0.385875
\(477\) −7.59949e16 −0.295403
\(478\) 4.74471e17 1.81940
\(479\) −3.44522e17 −1.30328 −0.651638 0.758530i \(-0.725918\pi\)
−0.651638 + 0.758530i \(0.725918\pi\)
\(480\) 7.97406e15 0.0297585
\(481\) −4.51890e17 −1.66376
\(482\) 3.03057e17 1.10083
\(483\) −3.37508e16 −0.120956
\(484\) −7.58192e16 −0.268092
\(485\) 3.16540e16 0.110435
\(486\) 3.14308e16 0.108198
\(487\) −5.87042e17 −1.99403 −0.997014 0.0772206i \(-0.975395\pi\)
−0.997014 + 0.0772206i \(0.975395\pi\)
\(488\) −2.18099e17 −0.731012
\(489\) −2.28804e17 −0.756758
\(490\) −1.42730e17 −0.465845
\(491\) −4.10341e17 −1.32165 −0.660823 0.750542i \(-0.729793\pi\)
−0.660823 + 0.750542i \(0.729793\pi\)
\(492\) 3.61001e17 1.14745
\(493\) 3.56127e15 0.0111712
\(494\) 9.61387e17 2.97628
\(495\) −1.07630e16 −0.0328853
\(496\) −2.75894e17 −0.831976
\(497\) 5.46634e17 1.62697
\(498\) 2.95885e17 0.869225
\(499\) −2.56903e17 −0.744930 −0.372465 0.928046i \(-0.621487\pi\)
−0.372465 + 0.928046i \(0.621487\pi\)
\(500\) 1.36779e17 0.391484
\(501\) 1.59323e17 0.450125
\(502\) −1.09901e18 −3.06497
\(503\) 3.08442e17 0.849143 0.424572 0.905394i \(-0.360425\pi\)
0.424572 + 0.905394i \(0.360425\pi\)
\(504\) 3.31036e17 0.899655
\(505\) −1.99838e16 −0.0536148
\(506\) −6.51108e16 −0.172455
\(507\) 1.60679e17 0.420152
\(508\) 1.35133e17 0.348856
\(509\) 4.12094e17 1.05034 0.525171 0.850997i \(-0.324001\pi\)
0.525171 + 0.850997i \(0.324001\pi\)
\(510\) 4.55994e15 0.0114750
\(511\) −7.48091e17 −1.85874
\(512\) −4.34246e17 −1.06532
\(513\) 1.06658e17 0.258363
\(514\) 9.57931e17 2.29126
\(515\) −2.22513e15 −0.00525543
\(516\) 1.44910e17 0.337968
\(517\) −5.80557e17 −1.33708
\(518\) −1.77810e18 −4.04402
\(519\) −4.73248e16 −0.106292
\(520\) −9.01035e16 −0.199857
\(521\) −3.15331e17 −0.690751 −0.345376 0.938465i \(-0.612248\pi\)
−0.345376 + 0.938465i \(0.612248\pi\)
\(522\) −2.62901e16 −0.0568767
\(523\) −7.00011e17 −1.49570 −0.747849 0.663869i \(-0.768913\pi\)
−0.747849 + 0.663869i \(0.768913\pi\)
\(524\) 1.64929e18 3.48052
\(525\) −5.18726e17 −1.08119
\(526\) 4.64817e17 0.956917
\(527\) 8.09104e16 0.164526
\(528\) 1.48390e17 0.298046
\(529\) −4.97871e17 −0.987768
\(530\) 8.13908e16 0.159508
\(531\) −2.24165e16 −0.0433963
\(532\) 2.45311e18 4.69128
\(533\) 7.49582e17 1.41609
\(534\) −4.59587e17 −0.857727
\(535\) 1.77678e16 0.0327592
\(536\) 4.17474e17 0.760428
\(537\) 1.56149e17 0.281000
\(538\) −7.56418e17 −1.34486
\(539\) 1.36214e18 2.39274
\(540\) −2.18294e16 −0.0378863
\(541\) 1.10711e18 1.89849 0.949243 0.314543i \(-0.101851\pi\)
0.949243 + 0.314543i \(0.101851\pi\)
\(542\) −6.74740e17 −1.14325
\(543\) −3.68363e17 −0.616705
\(544\) 3.22411e16 0.0533356
\(545\) −6.43153e16 −0.105132
\(546\) 1.50103e18 2.42459
\(547\) 1.04012e18 1.66022 0.830112 0.557597i \(-0.188277\pi\)
0.830112 + 0.557597i \(0.188277\pi\)
\(548\) 1.04706e18 1.65158
\(549\) −1.09714e17 −0.171019
\(550\) −1.00071e18 −1.54152
\(551\) −8.92135e16 −0.135814
\(552\) −6.04721e16 −0.0909810
\(553\) 8.46172e17 1.25819
\(554\) 4.06003e17 0.596645
\(555\) 5.36931e16 0.0779855
\(556\) −1.28798e18 −1.84894
\(557\) −6.27117e17 −0.889794 −0.444897 0.895582i \(-0.646760\pi\)
−0.444897 + 0.895582i \(0.646760\pi\)
\(558\) −5.97299e17 −0.837664
\(559\) 3.00891e17 0.417092
\(560\) −8.23807e16 −0.112876
\(561\) −4.35177e16 −0.0589396
\(562\) 2.10358e18 2.81626
\(563\) 2.96789e17 0.392775 0.196388 0.980526i \(-0.437079\pi\)
0.196388 + 0.980526i \(0.437079\pi\)
\(564\) −1.17747e18 −1.54041
\(565\) 3.19366e16 0.0413022
\(566\) 1.50778e18 1.92766
\(567\) 1.66527e17 0.210472
\(568\) 9.79415e17 1.22378
\(569\) 6.18879e17 0.764497 0.382248 0.924060i \(-0.375150\pi\)
0.382248 + 0.924060i \(0.375150\pi\)
\(570\) −1.14231e17 −0.139508
\(571\) 2.66375e17 0.321632 0.160816 0.986984i \(-0.448588\pi\)
0.160816 + 0.986984i \(0.448588\pi\)
\(572\) 1.87782e18 2.24171
\(573\) 2.35965e16 0.0278511
\(574\) 2.94946e18 3.44204
\(575\) 9.47585e16 0.109340
\(576\) −4.01154e17 −0.457684
\(577\) −1.03017e18 −1.16216 −0.581081 0.813846i \(-0.697370\pi\)
−0.581081 + 0.813846i \(0.697370\pi\)
\(578\) −1.49357e18 −1.66608
\(579\) −6.47528e17 −0.714246
\(580\) 1.82591e16 0.0199157
\(581\) 1.56766e18 1.69086
\(582\) −9.44814e17 −1.00774
\(583\) −7.76752e17 −0.819288
\(584\) −1.34037e18 −1.39811
\(585\) −4.53264e16 −0.0467561
\(586\) −1.37325e18 −1.40092
\(587\) 4.28882e17 0.432703 0.216352 0.976315i \(-0.430584\pi\)
0.216352 + 0.976315i \(0.430584\pi\)
\(588\) 2.76267e18 2.75662
\(589\) −2.02689e18 −2.00023
\(590\) 2.40081e16 0.0234326
\(591\) 5.23830e17 0.505675
\(592\) −7.40264e17 −0.706798
\(593\) 1.50466e18 1.42097 0.710484 0.703713i \(-0.248476\pi\)
0.710484 + 0.703713i \(0.248476\pi\)
\(594\) 3.21258e17 0.300083
\(595\) 2.41595e16 0.0223217
\(596\) 1.80865e18 1.65293
\(597\) −4.55463e17 −0.411737
\(598\) −2.74201e17 −0.245195
\(599\) −1.65999e18 −1.46835 −0.734177 0.678958i \(-0.762432\pi\)
−0.734177 + 0.678958i \(0.762432\pi\)
\(600\) −9.29412e17 −0.813252
\(601\) 1.36382e18 1.18052 0.590261 0.807213i \(-0.299025\pi\)
0.590261 + 0.807213i \(0.299025\pi\)
\(602\) 1.18395e18 1.01381
\(603\) 2.10010e17 0.177900
\(604\) −1.69723e18 −1.42233
\(605\) 1.87058e16 0.0155083
\(606\) 5.96482e17 0.489244
\(607\) −9.40198e17 −0.762944 −0.381472 0.924380i \(-0.624583\pi\)
−0.381472 + 0.924380i \(0.624583\pi\)
\(608\) −8.07673e17 −0.648429
\(609\) −1.39291e17 −0.110639
\(610\) 1.17504e17 0.0923443
\(611\) −2.44490e18 −1.90105
\(612\) −8.82617e16 −0.0679028
\(613\) −1.67828e18 −1.27753 −0.638765 0.769402i \(-0.720555\pi\)
−0.638765 + 0.769402i \(0.720555\pi\)
\(614\) −2.29846e18 −1.73118
\(615\) −8.90645e16 −0.0663768
\(616\) 3.38355e18 2.49516
\(617\) −2.35454e18 −1.71812 −0.859059 0.511876i \(-0.828951\pi\)
−0.859059 + 0.511876i \(0.828951\pi\)
\(618\) 6.64161e16 0.0479566
\(619\) 1.89405e18 1.35332 0.676662 0.736294i \(-0.263426\pi\)
0.676662 + 0.736294i \(0.263426\pi\)
\(620\) 4.14837e17 0.293313
\(621\) −3.04204e16 −0.0212848
\(622\) −1.65562e18 −1.14636
\(623\) −2.43499e18 −1.66849
\(624\) 6.24913e17 0.423760
\(625\) 1.43940e18 0.965966
\(626\) 1.61610e18 1.07334
\(627\) 1.09016e18 0.716560
\(628\) −7.17159e16 −0.0466528
\(629\) 2.17095e17 0.139772
\(630\) −1.78351e17 −0.113648
\(631\) −2.34490e18 −1.47888 −0.739440 0.673223i \(-0.764910\pi\)
−0.739440 + 0.673223i \(0.764910\pi\)
\(632\) 1.51610e18 0.946386
\(633\) 1.47402e18 0.910710
\(634\) 5.39658e18 3.30019
\(635\) −3.33393e16 −0.0201803
\(636\) −1.57539e18 −0.943881
\(637\) 5.73639e18 3.40199
\(638\) −2.68714e17 −0.157745
\(639\) 4.92694e17 0.286300
\(640\) 3.40031e17 0.195591
\(641\) −2.94203e17 −0.167521 −0.0837606 0.996486i \(-0.526693\pi\)
−0.0837606 + 0.996486i \(0.526693\pi\)
\(642\) −5.30336e17 −0.298933
\(643\) 1.10539e18 0.616801 0.308400 0.951257i \(-0.400206\pi\)
0.308400 + 0.951257i \(0.400206\pi\)
\(644\) −6.99662e17 −0.386483
\(645\) −3.57516e16 −0.0195505
\(646\) −4.61865e17 −0.250037
\(647\) −2.06742e18 −1.10803 −0.554013 0.832508i \(-0.686904\pi\)
−0.554013 + 0.832508i \(0.686904\pi\)
\(648\) 2.98370e17 0.158313
\(649\) −2.29121e17 −0.120358
\(650\) −4.21428e18 −2.19173
\(651\) −3.16461e18 −1.62946
\(652\) −4.74317e18 −2.41802
\(653\) 1.93951e18 0.978940 0.489470 0.872020i \(-0.337190\pi\)
0.489470 + 0.872020i \(0.337190\pi\)
\(654\) 1.91970e18 0.959350
\(655\) −4.06906e17 −0.201338
\(656\) 1.22793e18 0.601586
\(657\) −6.74272e17 −0.327085
\(658\) −9.62022e18 −4.62080
\(659\) 1.27523e18 0.606505 0.303253 0.952910i \(-0.401927\pi\)
0.303253 + 0.952910i \(0.401927\pi\)
\(660\) −2.23120e17 −0.105076
\(661\) −2.10805e18 −0.983042 −0.491521 0.870866i \(-0.663559\pi\)
−0.491521 + 0.870866i \(0.663559\pi\)
\(662\) 1.57847e18 0.728887
\(663\) −1.83266e17 −0.0838001
\(664\) 2.80881e18 1.27183
\(665\) −6.05220e17 −0.271377
\(666\) −1.60264e18 −0.711630
\(667\) 2.54450e16 0.0111888
\(668\) 3.30280e18 1.43825
\(669\) −2.07403e18 −0.894428
\(670\) −2.24921e17 −0.0960602
\(671\) −1.12140e18 −0.474313
\(672\) −1.26103e18 −0.528234
\(673\) 2.24879e18 0.932933 0.466467 0.884539i \(-0.345527\pi\)
0.466467 + 0.884539i \(0.345527\pi\)
\(674\) −8.53829e17 −0.350818
\(675\) −4.67540e17 −0.190258
\(676\) 3.33091e18 1.34248
\(677\) 3.03040e18 1.20969 0.604844 0.796344i \(-0.293236\pi\)
0.604844 + 0.796344i \(0.293236\pi\)
\(678\) −9.53250e17 −0.376889
\(679\) −5.00582e18 −1.96030
\(680\) 4.32871e16 0.0167900
\(681\) 1.34577e17 0.0517030
\(682\) −6.10506e18 −2.32322
\(683\) −1.63290e18 −0.615494 −0.307747 0.951468i \(-0.599575\pi\)
−0.307747 + 0.951468i \(0.599575\pi\)
\(684\) 2.21104e18 0.825530
\(685\) −2.58326e17 −0.0955389
\(686\) 1.38506e19 5.07413
\(687\) −9.47669e17 −0.343904
\(688\) 4.92905e17 0.177190
\(689\) −3.27114e18 −1.16486
\(690\) 3.25803e16 0.0114931
\(691\) −9.29015e16 −0.0324650 −0.0162325 0.999868i \(-0.505167\pi\)
−0.0162325 + 0.999868i \(0.505167\pi\)
\(692\) −9.81053e17 −0.339627
\(693\) 1.70209e18 0.583736
\(694\) 7.61714e18 2.58795
\(695\) 3.17764e17 0.106956
\(696\) −2.49570e17 −0.0832208
\(697\) −3.60110e17 −0.118966
\(698\) 7.82657e18 2.56160
\(699\) −3.27692e18 −1.06259
\(700\) −1.07533e19 −3.45466
\(701\) 1.37308e17 0.0437046 0.0218523 0.999761i \(-0.493044\pi\)
0.0218523 + 0.999761i \(0.493044\pi\)
\(702\) 1.35291e18 0.426657
\(703\) −5.43844e18 −1.69928
\(704\) −4.10024e18 −1.26937
\(705\) 2.90501e17 0.0891082
\(706\) −7.59869e18 −2.30945
\(707\) 3.16028e18 0.951700
\(708\) −4.64698e17 −0.138661
\(709\) −1.89261e18 −0.559577 −0.279788 0.960062i \(-0.590264\pi\)
−0.279788 + 0.960062i \(0.590264\pi\)
\(710\) −5.27676e17 −0.154592
\(711\) 7.62675e17 0.221405
\(712\) −4.36281e18 −1.25501
\(713\) 5.78097e17 0.164785
\(714\) −7.21118e17 −0.203689
\(715\) −4.63286e17 −0.129676
\(716\) 3.23700e18 0.897858
\(717\) 2.26579e18 0.622794
\(718\) −6.51748e15 −0.00177530
\(719\) 4.89899e18 1.32242 0.661209 0.750202i \(-0.270044\pi\)
0.661209 + 0.750202i \(0.270044\pi\)
\(720\) −7.42516e16 −0.0198630
\(721\) 3.51886e17 0.0932874
\(722\) 5.15048e18 1.35318
\(723\) 1.44722e18 0.376821
\(724\) −7.63625e18 −1.97051
\(725\) 3.91071e17 0.100013
\(726\) −5.58333e17 −0.141516
\(727\) 2.01727e18 0.506746 0.253373 0.967369i \(-0.418460\pi\)
0.253373 + 0.967369i \(0.418460\pi\)
\(728\) 1.42491e19 3.54760
\(729\) 1.50095e17 0.0370370
\(730\) 7.22147e17 0.176615
\(731\) −1.44553e17 −0.0350399
\(732\) −2.27440e18 −0.546444
\(733\) 7.91301e18 1.88437 0.942184 0.335095i \(-0.108768\pi\)
0.942184 + 0.335095i \(0.108768\pi\)
\(734\) 8.15373e18 1.92456
\(735\) −6.81593e17 −0.159462
\(736\) 2.30360e17 0.0534197
\(737\) 2.14653e18 0.493399
\(738\) 2.65842e18 0.605699
\(739\) −3.52219e18 −0.795470 −0.397735 0.917500i \(-0.630204\pi\)
−0.397735 + 0.917500i \(0.630204\pi\)
\(740\) 1.11307e18 0.249182
\(741\) 4.59100e18 1.01880
\(742\) −1.28713e19 −2.83137
\(743\) 6.44962e18 1.40639 0.703196 0.710996i \(-0.251755\pi\)
0.703196 + 0.710996i \(0.251755\pi\)
\(744\) −5.67010e18 −1.22565
\(745\) −4.46222e17 −0.0956170
\(746\) 1.34158e19 2.84980
\(747\) 1.41297e18 0.297542
\(748\) −9.02132e17 −0.188326
\(749\) −2.80983e18 −0.581498
\(750\) 1.00724e18 0.206650
\(751\) −1.56414e18 −0.318139 −0.159070 0.987267i \(-0.550849\pi\)
−0.159070 + 0.987267i \(0.550849\pi\)
\(752\) −4.00512e18 −0.807606
\(753\) −5.24819e18 −1.04916
\(754\) −1.13164e18 −0.224281
\(755\) 4.18733e17 0.0822778
\(756\) 3.45214e18 0.672507
\(757\) −9.31417e18 −1.79896 −0.899478 0.436965i \(-0.856053\pi\)
−0.899478 + 0.436965i \(0.856053\pi\)
\(758\) −1.26430e19 −2.42103
\(759\) −3.10930e17 −0.0590325
\(760\) −1.08438e18 −0.204125
\(761\) −3.18294e18 −0.594057 −0.297029 0.954869i \(-0.595996\pi\)
−0.297029 + 0.954869i \(0.595996\pi\)
\(762\) 9.95120e17 0.184148
\(763\) 1.01709e19 1.86617
\(764\) 4.89160e17 0.0889905
\(765\) 2.17755e16 0.00392798
\(766\) 1.63060e19 2.91648
\(767\) −9.64897e17 −0.171124
\(768\) −5.64141e18 −0.992064
\(769\) 9.15451e18 1.59630 0.798149 0.602460i \(-0.205813\pi\)
0.798149 + 0.602460i \(0.205813\pi\)
\(770\) −1.82294e18 −0.315198
\(771\) 4.57450e18 0.784314
\(772\) −1.34234e19 −2.28218
\(773\) 2.02144e18 0.340796 0.170398 0.985375i \(-0.445495\pi\)
0.170398 + 0.985375i \(0.445495\pi\)
\(774\) 1.06712e18 0.178401
\(775\) 8.88494e18 1.47297
\(776\) −8.96903e18 −1.47450
\(777\) −8.49113e18 −1.38430
\(778\) 2.67322e18 0.432183
\(779\) 9.02113e18 1.44633
\(780\) −9.39627e17 −0.149397
\(781\) 5.03587e18 0.794041
\(782\) 1.31731e17 0.0205988
\(783\) −1.25546e17 −0.0194693
\(784\) 9.39708e18 1.44524
\(785\) 1.76934e16 0.00269872
\(786\) 1.21454e19 1.83724
\(787\) −9.55199e17 −0.143304 −0.0716520 0.997430i \(-0.522827\pi\)
−0.0716520 + 0.997430i \(0.522827\pi\)
\(788\) 1.08591e19 1.61575
\(789\) 2.21968e18 0.327560
\(790\) −8.16827e17 −0.119551
\(791\) −5.05051e18 −0.733142
\(792\) 3.04967e18 0.439075
\(793\) −4.72256e18 −0.674376
\(794\) −3.93351e18 −0.557118
\(795\) 3.88673e17 0.0546007
\(796\) −9.44184e18 −1.31559
\(797\) 2.63426e18 0.364065 0.182032 0.983293i \(-0.441732\pi\)
0.182032 + 0.983293i \(0.441732\pi\)
\(798\) 1.80647e19 2.47635
\(799\) 1.17457e18 0.159707
\(800\) 3.54047e18 0.477503
\(801\) −2.19471e18 −0.293606
\(802\) −3.05628e18 −0.405564
\(803\) −6.89180e18 −0.907155
\(804\) 4.35354e18 0.568432
\(805\) 1.72617e17 0.0223569
\(806\) −2.57102e19 −3.30315
\(807\) −3.61220e18 −0.460355
\(808\) 5.66235e18 0.715851
\(809\) −1.15455e19 −1.44793 −0.723964 0.689838i \(-0.757682\pi\)
−0.723964 + 0.689838i \(0.757682\pi\)
\(810\) −1.60752e17 −0.0199988
\(811\) 6.67328e18 0.823577 0.411788 0.911280i \(-0.364904\pi\)
0.411788 + 0.911280i \(0.364904\pi\)
\(812\) −2.88752e18 −0.353518
\(813\) −3.22215e18 −0.391342
\(814\) −1.63808e19 −1.97368
\(815\) 1.17021e18 0.139875
\(816\) −3.00218e17 −0.0356000
\(817\) 3.62119e18 0.425998
\(818\) 6.77995e18 0.791279
\(819\) 7.16801e18 0.829953
\(820\) −1.84633e18 −0.212089
\(821\) 8.25197e18 0.940432 0.470216 0.882551i \(-0.344176\pi\)
0.470216 + 0.882551i \(0.344176\pi\)
\(822\) 7.71058e18 0.871807
\(823\) 3.25044e18 0.364622 0.182311 0.983241i \(-0.441642\pi\)
0.182311 + 0.983241i \(0.441642\pi\)
\(824\) 6.30482e17 0.0701691
\(825\) −4.77877e18 −0.527674
\(826\) −3.79669e18 −0.415944
\(827\) 3.22113e18 0.350125 0.175062 0.984557i \(-0.443987\pi\)
0.175062 + 0.984557i \(0.443987\pi\)
\(828\) −6.30621e17 −0.0680098
\(829\) 2.07018e18 0.221515 0.110758 0.993847i \(-0.464672\pi\)
0.110758 + 0.993847i \(0.464672\pi\)
\(830\) −1.51329e18 −0.160663
\(831\) 1.93883e18 0.204236
\(832\) −1.72673e19 −1.80478
\(833\) −2.75585e18 −0.285801
\(834\) −9.48470e18 −0.975988
\(835\) −8.14851e17 −0.0831986
\(836\) 2.25993e19 2.28957
\(837\) −2.85234e18 −0.286738
\(838\) −2.92716e19 −2.91985
\(839\) 3.34739e18 0.331324 0.165662 0.986183i \(-0.447024\pi\)
0.165662 + 0.986183i \(0.447024\pi\)
\(840\) −1.69307e18 −0.166287
\(841\) −1.01556e19 −0.989766
\(842\) 1.36858e19 1.32356
\(843\) 1.00454e19 0.964027
\(844\) 3.05568e19 2.90993
\(845\) −8.21785e17 −0.0776586
\(846\) −8.67092e18 −0.813126
\(847\) −2.95817e18 −0.275284
\(848\) −5.35862e18 −0.494857
\(849\) 7.20023e18 0.659852
\(850\) 2.02460e18 0.184127
\(851\) 1.55112e18 0.139992
\(852\) 1.02136e19 0.914794
\(853\) 2.19915e19 1.95473 0.977363 0.211571i \(-0.0678580\pi\)
0.977363 + 0.211571i \(0.0678580\pi\)
\(854\) −1.85824e19 −1.63917
\(855\) −5.45498e17 −0.0477544
\(856\) −5.03443e18 −0.437392
\(857\) −1.22000e19 −1.05192 −0.525962 0.850508i \(-0.676294\pi\)
−0.525962 + 0.850508i \(0.676294\pi\)
\(858\) 1.38283e19 1.18331
\(859\) −5.56749e18 −0.472829 −0.236414 0.971652i \(-0.575972\pi\)
−0.236414 + 0.971652i \(0.575972\pi\)
\(860\) −7.41137e17 −0.0624682
\(861\) 1.40848e19 1.17823
\(862\) −6.71515e18 −0.557517
\(863\) 2.53657e17 0.0209015 0.0104507 0.999945i \(-0.496673\pi\)
0.0104507 + 0.999945i \(0.496673\pi\)
\(864\) −1.13660e18 −0.0929540
\(865\) 2.42041e17 0.0196464
\(866\) −2.58813e19 −2.08506
\(867\) −7.13239e18 −0.570310
\(868\) −6.56031e19 −5.20651
\(869\) 7.79537e18 0.614057
\(870\) 1.34460e17 0.0105128
\(871\) 9.03968e18 0.701512
\(872\) 1.82235e19 1.40370
\(873\) −4.51186e18 −0.344956
\(874\) −3.29998e18 −0.250431
\(875\) 5.33657e18 0.401985
\(876\) −1.39778e19 −1.04511
\(877\) −7.95224e18 −0.590191 −0.295095 0.955468i \(-0.595351\pi\)
−0.295095 + 0.955468i \(0.595351\pi\)
\(878\) −1.16137e19 −0.855570
\(879\) −6.55780e18 −0.479546
\(880\) −7.58933e17 −0.0550891
\(881\) 2.23727e19 1.61203 0.806017 0.591892i \(-0.201619\pi\)
0.806017 + 0.591892i \(0.201619\pi\)
\(882\) 2.03443e19 1.45512
\(883\) −7.26839e18 −0.516052 −0.258026 0.966138i \(-0.583072\pi\)
−0.258026 + 0.966138i \(0.583072\pi\)
\(884\) −3.79915e18 −0.267761
\(885\) 1.14648e17 0.00802113
\(886\) −2.06131e19 −1.43161
\(887\) −1.76105e19 −1.21414 −0.607068 0.794650i \(-0.707654\pi\)
−0.607068 + 0.794650i \(0.707654\pi\)
\(888\) −1.52137e19 −1.04124
\(889\) 5.27235e18 0.358214
\(890\) 2.35054e18 0.158538
\(891\) 1.53413e18 0.102721
\(892\) −4.29951e19 −2.85790
\(893\) −2.94241e19 −1.94164
\(894\) 1.33189e19 0.872520
\(895\) −7.98617e17 −0.0519384
\(896\) −5.37731e19 −3.47187
\(897\) −1.30942e18 −0.0839321
\(898\) −4.47356e19 −2.84680
\(899\) 2.38582e18 0.150730
\(900\) −9.69220e18 −0.607920
\(901\) 1.57150e18 0.0978598
\(902\) 2.71720e19 1.67988
\(903\) 5.65382e18 0.347034
\(904\) −9.04911e18 −0.551456
\(905\) 1.88398e18 0.113988
\(906\) −1.24984e19 −0.750797
\(907\) 4.16084e18 0.248161 0.124081 0.992272i \(-0.460402\pi\)
0.124081 + 0.992272i \(0.460402\pi\)
\(908\) 2.78982e18 0.165203
\(909\) 2.84844e18 0.167472
\(910\) −7.67696e18 −0.448147
\(911\) −1.26272e19 −0.731877 −0.365938 0.930639i \(-0.619252\pi\)
−0.365938 + 0.930639i \(0.619252\pi\)
\(912\) 7.52076e18 0.432808
\(913\) 1.44421e19 0.825221
\(914\) −1.92126e19 −1.09002
\(915\) 5.61130e17 0.0316101
\(916\) −1.96454e19 −1.09885
\(917\) 6.43489e19 3.57388
\(918\) −6.49960e17 −0.0358434
\(919\) 2.11333e19 1.15722 0.578611 0.815604i \(-0.303595\pi\)
0.578611 + 0.815604i \(0.303595\pi\)
\(920\) 3.09282e17 0.0168164
\(921\) −1.09761e19 −0.592597
\(922\) 5.61121e18 0.300819
\(923\) 2.12076e19 1.12896
\(924\) 3.52847e19 1.86517
\(925\) 2.38396e19 1.25135
\(926\) −2.75026e19 −1.43352
\(927\) 3.17163e17 0.0164159
\(928\) 9.50701e17 0.0488632
\(929\) 8.38667e18 0.428043 0.214022 0.976829i \(-0.431344\pi\)
0.214022 + 0.976829i \(0.431344\pi\)
\(930\) 3.05486e18 0.154829
\(931\) 6.90368e19 3.47463
\(932\) −6.79313e19 −3.39521
\(933\) −7.90623e18 −0.392409
\(934\) −5.11876e17 −0.0252296
\(935\) 2.22570e17 0.0108941
\(936\) 1.28431e19 0.624275
\(937\) 2.76916e19 1.33672 0.668359 0.743839i \(-0.266997\pi\)
0.668359 + 0.743839i \(0.266997\pi\)
\(938\) 3.55694e19 1.70513
\(939\) 7.71754e18 0.367411
\(940\) 6.02214e18 0.284721
\(941\) 3.92882e19 1.84472 0.922358 0.386337i \(-0.126260\pi\)
0.922358 + 0.386337i \(0.126260\pi\)
\(942\) −5.28116e17 −0.0246263
\(943\) −2.57295e18 −0.119153
\(944\) −1.58065e18 −0.0726971
\(945\) −8.51696e17 −0.0389025
\(946\) 1.09071e19 0.494787
\(947\) 3.81184e19 1.71735 0.858677 0.512517i \(-0.171287\pi\)
0.858677 + 0.512517i \(0.171287\pi\)
\(948\) 1.58104e19 0.707439
\(949\) −2.90235e19 −1.28979
\(950\) −5.07184e19 −2.23852
\(951\) 2.57708e19 1.12968
\(952\) −6.84550e18 −0.298034
\(953\) 3.75528e19 1.62382 0.811912 0.583780i \(-0.198427\pi\)
0.811912 + 0.583780i \(0.198427\pi\)
\(954\) −1.16012e19 −0.498240
\(955\) −1.20683e17 −0.00514784
\(956\) 4.69702e19 1.98997
\(957\) −1.28322e18 −0.0539973
\(958\) −5.25939e19 −2.19816
\(959\) 4.08523e19 1.69588
\(960\) 2.05169e18 0.0845957
\(961\) 2.97872e19 1.21991
\(962\) −6.89844e19 −2.80616
\(963\) −2.53256e18 −0.102327
\(964\) 3.00011e19 1.20403
\(965\) 3.31176e18 0.132017
\(966\) −5.15232e18 −0.204010
\(967\) 6.34636e17 0.0249605 0.0124802 0.999922i \(-0.496027\pi\)
0.0124802 + 0.999922i \(0.496027\pi\)
\(968\) −5.30021e18 −0.207063
\(969\) −2.20559e18 −0.0855894
\(970\) 4.83222e18 0.186265
\(971\) 4.95806e18 0.189839 0.0949197 0.995485i \(-0.469741\pi\)
0.0949197 + 0.995485i \(0.469741\pi\)
\(972\) 3.11149e18 0.118342
\(973\) −5.02519e19 −1.89854
\(974\) −8.96165e19 −3.36321
\(975\) −2.01248e19 −0.750244
\(976\) −7.73627e18 −0.286489
\(977\) 4.91389e19 1.80764 0.903818 0.427917i \(-0.140752\pi\)
0.903818 + 0.427917i \(0.140752\pi\)
\(978\) −3.49287e19 −1.27638
\(979\) −2.24323e19 −0.814305
\(980\) −1.41296e19 −0.509518
\(981\) 9.16730e18 0.328392
\(982\) −6.26417e19 −2.22915
\(983\) −5.14976e19 −1.82049 −0.910246 0.414068i \(-0.864108\pi\)
−0.910246 + 0.414068i \(0.864108\pi\)
\(984\) 2.52361e19 0.886246
\(985\) −2.67911e18 −0.0934662
\(986\) 5.43655e17 0.0188419
\(987\) −4.59403e19 −1.58173
\(988\) 9.51725e19 3.25530
\(989\) −1.03281e18 −0.0350951
\(990\) −1.64306e18 −0.0554657
\(991\) −6.03400e18 −0.202361 −0.101180 0.994868i \(-0.532262\pi\)
−0.101180 + 0.994868i \(0.532262\pi\)
\(992\) 2.15995e19 0.719643
\(993\) 7.53783e18 0.249503
\(994\) 8.34478e19 2.74412
\(995\) 2.32945e18 0.0761032
\(996\) 2.92911e19 0.950715
\(997\) 1.33645e19 0.430958 0.215479 0.976508i \(-0.430869\pi\)
0.215479 + 0.976508i \(0.430869\pi\)
\(998\) −3.92182e19 −1.25643
\(999\) −7.65325e18 −0.243596
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 177.14.a.c.1.28 31
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
177.14.a.c.1.28 31 1.1 even 1 trivial