Properties

Label 325.10.a.b.1.1
Level $325$
Weight $10$
Character 325.1
Self dual yes
Analytic conductor $167.387$
Analytic rank $1$
Dimension $5$
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] = [325,10,Mod(1,325)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(325, 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("325.1");
 
S:= CuspForms(chi, 10);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 325 = 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 325.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: \(167.386646753\)
Analytic rank: \(1\)
Dimension: \(5\)
Coefficient field: \(\mathbb{Q}[x]/(x^{5} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{5} - 1438x^{3} - 4164x^{2} + 396957x - 59580 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{2}\cdot 3 \)
Twist minimal: no (minimal twist has level 13)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(35.1685\) of defining polynomial
Character \(\chi\) \(=\) 325.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-38.1685 q^{2} -47.8784 q^{3} +944.833 q^{4} +1827.45 q^{6} -5947.44 q^{7} -16520.6 q^{8} -17390.7 q^{9} +O(q^{10})\) \(q-38.1685 q^{2} -47.8784 q^{3} +944.833 q^{4} +1827.45 q^{6} -5947.44 q^{7} -16520.6 q^{8} -17390.7 q^{9} -25205.7 q^{11} -45237.1 q^{12} -28561.0 q^{13} +227005. q^{14} +146811. q^{16} -109318. q^{17} +663775. q^{18} -904609. q^{19} +284754. q^{21} +962062. q^{22} +435749. q^{23} +790979. q^{24} +1.09013e6 q^{26} +1.77503e6 q^{27} -5.61934e6 q^{28} +6.44791e6 q^{29} +6.62308e6 q^{31} +2.85499e6 q^{32} +1.20681e6 q^{33} +4.17250e6 q^{34} -1.64313e7 q^{36} -4.14357e6 q^{37} +3.45275e7 q^{38} +1.36745e6 q^{39} +1.49568e7 q^{41} -1.08686e7 q^{42} -4.01789e7 q^{43} -2.38151e7 q^{44} -1.66319e7 q^{46} -6.30151e6 q^{47} -7.02907e6 q^{48} -4.98153e6 q^{49} +5.23397e6 q^{51} -2.69854e7 q^{52} -1.53111e7 q^{53} -6.77501e7 q^{54} +9.82552e7 q^{56} +4.33112e7 q^{57} -2.46107e8 q^{58} -1.52760e8 q^{59} +8.66321e7 q^{61} -2.52793e8 q^{62} +1.03430e8 q^{63} -1.84138e8 q^{64} -4.60620e7 q^{66} +1.01034e8 q^{67} -1.03287e8 q^{68} -2.08630e7 q^{69} +4.13122e8 q^{71} +2.87304e8 q^{72} +3.14453e8 q^{73} +1.58154e8 q^{74} -8.54704e8 q^{76} +1.49909e8 q^{77} -5.21937e7 q^{78} -2.00580e8 q^{79} +2.57315e8 q^{81} -5.70879e8 q^{82} -6.34578e7 q^{83} +2.69045e8 q^{84} +1.53357e9 q^{86} -3.08715e8 q^{87} +4.16412e8 q^{88} +3.47074e7 q^{89} +1.69865e8 q^{91} +4.11710e8 q^{92} -3.17102e8 q^{93} +2.40519e8 q^{94} -1.36692e8 q^{96} +1.25403e9 q^{97} +1.90137e8 q^{98} +4.38343e8 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q - 15 q^{2} - 161 q^{3} + 361 q^{4} + 5693 q^{6} - 10099 q^{7} - 23151 q^{8} + 61060 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 5 q - 15 q^{2} - 161 q^{3} + 361 q^{4} + 5693 q^{6} - 10099 q^{7} - 23151 q^{8} + 61060 q^{9} + 121746 q^{11} - 113389 q^{12} - 142805 q^{13} + 8475 q^{14} - 322463 q^{16} + 495669 q^{17} + 656228 q^{18} - 840738 q^{19} - 1599467 q^{21} + 2023594 q^{22} + 592152 q^{23} - 2295657 q^{24} + 428415 q^{26} - 6847883 q^{27} - 2587955 q^{28} + 10678182 q^{29} + 12885296 q^{31} - 3282927 q^{32} - 17278298 q^{33} - 9934079 q^{34} - 20483302 q^{36} - 7171823 q^{37} + 25568814 q^{38} + 4598321 q^{39} + 9294012 q^{41} + 69520457 q^{42} - 12831975 q^{43} - 41479074 q^{44} - 59319696 q^{46} - 43354215 q^{47} + 86874671 q^{48} + 25249488 q^{49} + 16905901 q^{51} - 10310521 q^{52} - 93231780 q^{53} + 58983719 q^{54} + 199599225 q^{56} - 90173382 q^{57} - 151020970 q^{58} + 246496182 q^{59} - 132232612 q^{61} - 158135724 q^{62} + 416955202 q^{63} + 91019105 q^{64} - 323733130 q^{66} + 369388534 q^{67} - 238172073 q^{68} - 579986760 q^{69} + 212150457 q^{71} + 415774278 q^{72} + 252729806 q^{73} + 192105957 q^{74} - 953775990 q^{76} - 449666118 q^{77} - 162597773 q^{78} - 1247271728 q^{79} - 317713115 q^{81} - 169559388 q^{82} - 1696894296 q^{83} + 1247983739 q^{84} + 3291621459 q^{86} + 614530466 q^{87} + 220227222 q^{88} - 753854382 q^{89} + 288437539 q^{91} - 13876128 q^{92} + 892784668 q^{93} + 272071215 q^{94} + 930612847 q^{96} - 3824606 q^{97} - 1570614816 q^{98} + 3016199848 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −38.1685 −1.68682 −0.843412 0.537267i \(-0.819457\pi\)
−0.843412 + 0.537267i \(0.819457\pi\)
\(3\) −47.8784 −0.341267 −0.170633 0.985335i \(-0.554581\pi\)
−0.170633 + 0.985335i \(0.554581\pi\)
\(4\) 944.833 1.84538
\(5\) 0 0
\(6\) 1827.45 0.575657
\(7\) −5947.44 −0.936244 −0.468122 0.883664i \(-0.655069\pi\)
−0.468122 + 0.883664i \(0.655069\pi\)
\(8\) −16520.6 −1.42600
\(9\) −17390.7 −0.883537
\(10\) 0 0
\(11\) −25205.7 −0.519076 −0.259538 0.965733i \(-0.583570\pi\)
−0.259538 + 0.965733i \(0.583570\pi\)
\(12\) −45237.1 −0.629766
\(13\) −28561.0 −0.277350
\(14\) 227005. 1.57928
\(15\) 0 0
\(16\) 146811. 0.560039
\(17\) −109318. −0.317447 −0.158724 0.987323i \(-0.550738\pi\)
−0.158724 + 0.987323i \(0.550738\pi\)
\(18\) 663775. 1.49037
\(19\) −904609. −1.59246 −0.796232 0.604992i \(-0.793176\pi\)
−0.796232 + 0.604992i \(0.793176\pi\)
\(20\) 0 0
\(21\) 284754. 0.319509
\(22\) 962062. 0.875590
\(23\) 435749. 0.324684 0.162342 0.986735i \(-0.448095\pi\)
0.162342 + 0.986735i \(0.448095\pi\)
\(24\) 790979. 0.486647
\(25\) 0 0
\(26\) 1.09013e6 0.467841
\(27\) 1.77503e6 0.642789
\(28\) −5.61934e6 −1.72772
\(29\) 6.44791e6 1.69289 0.846443 0.532479i \(-0.178740\pi\)
0.846443 + 0.532479i \(0.178740\pi\)
\(30\) 0 0
\(31\) 6.62308e6 1.28805 0.644024 0.765005i \(-0.277264\pi\)
0.644024 + 0.765005i \(0.277264\pi\)
\(32\) 2.85499e6 0.481315
\(33\) 1.20681e6 0.177143
\(34\) 4.17250e6 0.535477
\(35\) 0 0
\(36\) −1.64313e7 −1.63046
\(37\) −4.14357e6 −0.363469 −0.181734 0.983348i \(-0.558171\pi\)
−0.181734 + 0.983348i \(0.558171\pi\)
\(38\) 3.45275e7 2.68621
\(39\) 1.36745e6 0.0946504
\(40\) 0 0
\(41\) 1.49568e7 0.826632 0.413316 0.910588i \(-0.364370\pi\)
0.413316 + 0.910588i \(0.364370\pi\)
\(42\) −1.08686e7 −0.538956
\(43\) −4.01789e7 −1.79221 −0.896107 0.443838i \(-0.853616\pi\)
−0.896107 + 0.443838i \(0.853616\pi\)
\(44\) −2.38151e7 −0.957891
\(45\) 0 0
\(46\) −1.66319e7 −0.547685
\(47\) −6.30151e6 −0.188367 −0.0941834 0.995555i \(-0.530024\pi\)
−0.0941834 + 0.995555i \(0.530024\pi\)
\(48\) −7.02907e6 −0.191123
\(49\) −4.98153e6 −0.123447
\(50\) 0 0
\(51\) 5.23397e6 0.108334
\(52\) −2.69854e7 −0.511815
\(53\) −1.53111e7 −0.266542 −0.133271 0.991080i \(-0.542548\pi\)
−0.133271 + 0.991080i \(0.542548\pi\)
\(54\) −6.77501e7 −1.08427
\(55\) 0 0
\(56\) 9.82552e7 1.33509
\(57\) 4.33112e7 0.543455
\(58\) −2.46107e8 −2.85560
\(59\) −1.52760e8 −1.64126 −0.820629 0.571462i \(-0.806376\pi\)
−0.820629 + 0.571462i \(0.806376\pi\)
\(60\) 0 0
\(61\) 8.66321e7 0.801114 0.400557 0.916272i \(-0.368817\pi\)
0.400557 + 0.916272i \(0.368817\pi\)
\(62\) −2.52793e8 −2.17271
\(63\) 1.03430e8 0.827206
\(64\) −1.84138e8 −1.37193
\(65\) 0 0
\(66\) −4.60620e7 −0.298810
\(67\) 1.01034e8 0.612537 0.306268 0.951945i \(-0.400920\pi\)
0.306268 + 0.951945i \(0.400920\pi\)
\(68\) −1.03287e8 −0.585809
\(69\) −2.08630e7 −0.110804
\(70\) 0 0
\(71\) 4.13122e8 1.92937 0.964685 0.263406i \(-0.0848458\pi\)
0.964685 + 0.263406i \(0.0848458\pi\)
\(72\) 2.87304e8 1.25993
\(73\) 3.14453e8 1.29599 0.647997 0.761643i \(-0.275607\pi\)
0.647997 + 0.761643i \(0.275607\pi\)
\(74\) 1.58154e8 0.613108
\(75\) 0 0
\(76\) −8.54704e8 −2.93870
\(77\) 1.49909e8 0.485982
\(78\) −5.21937e7 −0.159659
\(79\) −2.00580e8 −0.579383 −0.289692 0.957120i \(-0.593553\pi\)
−0.289692 + 0.957120i \(0.593553\pi\)
\(80\) 0 0
\(81\) 2.57315e8 0.664175
\(82\) −5.70879e8 −1.39438
\(83\) −6.34578e7 −0.146769 −0.0733843 0.997304i \(-0.523380\pi\)
−0.0733843 + 0.997304i \(0.523380\pi\)
\(84\) 2.69045e8 0.589614
\(85\) 0 0
\(86\) 1.53357e9 3.02315
\(87\) −3.08715e8 −0.577726
\(88\) 4.16412e8 0.740204
\(89\) 3.47074e7 0.0586364 0.0293182 0.999570i \(-0.490666\pi\)
0.0293182 + 0.999570i \(0.490666\pi\)
\(90\) 0 0
\(91\) 1.69865e8 0.259667
\(92\) 4.11710e8 0.599164
\(93\) −3.17102e8 −0.439568
\(94\) 2.40519e8 0.317742
\(95\) 0 0
\(96\) −1.36692e8 −0.164257
\(97\) 1.25403e9 1.43825 0.719127 0.694879i \(-0.244542\pi\)
0.719127 + 0.694879i \(0.244542\pi\)
\(98\) 1.90137e8 0.208233
\(99\) 4.38343e8 0.458623
\(100\) 0 0
\(101\) −9.06459e8 −0.866766 −0.433383 0.901210i \(-0.642680\pi\)
−0.433383 + 0.901210i \(0.642680\pi\)
\(102\) −1.99773e8 −0.182741
\(103\) 4.17013e8 0.365075 0.182537 0.983199i \(-0.441569\pi\)
0.182537 + 0.983199i \(0.441569\pi\)
\(104\) 4.71844e8 0.395502
\(105\) 0 0
\(106\) 5.84402e8 0.449610
\(107\) −6.71636e8 −0.495344 −0.247672 0.968844i \(-0.579666\pi\)
−0.247672 + 0.968844i \(0.579666\pi\)
\(108\) 1.67710e9 1.18619
\(109\) 1.66748e9 1.13147 0.565734 0.824588i \(-0.308593\pi\)
0.565734 + 0.824588i \(0.308593\pi\)
\(110\) 0 0
\(111\) 1.98387e8 0.124040
\(112\) −8.73149e8 −0.524333
\(113\) 1.84580e9 1.06496 0.532478 0.846444i \(-0.321261\pi\)
0.532478 + 0.846444i \(0.321261\pi\)
\(114\) −1.65312e9 −0.916713
\(115\) 0 0
\(116\) 6.09219e9 3.12401
\(117\) 4.96695e8 0.245049
\(118\) 5.83063e9 2.76851
\(119\) 6.50162e8 0.297208
\(120\) 0 0
\(121\) −1.72262e9 −0.730560
\(122\) −3.30661e9 −1.35134
\(123\) −7.16109e8 −0.282102
\(124\) 6.25770e9 2.37694
\(125\) 0 0
\(126\) −3.94776e9 −1.39535
\(127\) −1.87922e9 −0.641006 −0.320503 0.947248i \(-0.603852\pi\)
−0.320503 + 0.947248i \(0.603852\pi\)
\(128\) 5.56650e9 1.83290
\(129\) 1.92370e9 0.611623
\(130\) 0 0
\(131\) 3.46045e9 1.02663 0.513313 0.858201i \(-0.328418\pi\)
0.513313 + 0.858201i \(0.328418\pi\)
\(132\) 1.14023e9 0.326896
\(133\) 5.38011e9 1.49093
\(134\) −3.85632e9 −1.03324
\(135\) 0 0
\(136\) 1.80600e9 0.452680
\(137\) −5.04786e9 −1.22424 −0.612118 0.790766i \(-0.709682\pi\)
−0.612118 + 0.790766i \(0.709682\pi\)
\(138\) 7.96307e8 0.186907
\(139\) 3.59716e9 0.817322 0.408661 0.912686i \(-0.365996\pi\)
0.408661 + 0.912686i \(0.365996\pi\)
\(140\) 0 0
\(141\) 3.01706e8 0.0642833
\(142\) −1.57682e10 −3.25451
\(143\) 7.19899e8 0.143966
\(144\) −2.55314e9 −0.494815
\(145\) 0 0
\(146\) −1.20022e10 −2.18611
\(147\) 2.38508e8 0.0421283
\(148\) −3.91498e9 −0.670736
\(149\) 1.27561e9 0.212022 0.106011 0.994365i \(-0.466192\pi\)
0.106011 + 0.994365i \(0.466192\pi\)
\(150\) 0 0
\(151\) 4.35273e9 0.681342 0.340671 0.940183i \(-0.389346\pi\)
0.340671 + 0.940183i \(0.389346\pi\)
\(152\) 1.49447e10 2.27086
\(153\) 1.90111e9 0.280476
\(154\) −5.72181e9 −0.819766
\(155\) 0 0
\(156\) 1.29202e9 0.174666
\(157\) 1.41002e10 1.85215 0.926075 0.377339i \(-0.123161\pi\)
0.926075 + 0.377339i \(0.123161\pi\)
\(158\) 7.65584e9 0.977318
\(159\) 7.33072e8 0.0909619
\(160\) 0 0
\(161\) −2.59159e9 −0.303983
\(162\) −9.82132e9 −1.12035
\(163\) −7.24812e8 −0.0804231 −0.0402116 0.999191i \(-0.512803\pi\)
−0.0402116 + 0.999191i \(0.512803\pi\)
\(164\) 1.41317e10 1.52545
\(165\) 0 0
\(166\) 2.42209e9 0.247573
\(167\) 8.33031e8 0.0828776 0.0414388 0.999141i \(-0.486806\pi\)
0.0414388 + 0.999141i \(0.486806\pi\)
\(168\) −4.70430e9 −0.455621
\(169\) 8.15731e8 0.0769231
\(170\) 0 0
\(171\) 1.57317e10 1.40700
\(172\) −3.79623e10 −3.30731
\(173\) −5.77955e8 −0.0490553 −0.0245277 0.999699i \(-0.507808\pi\)
−0.0245277 + 0.999699i \(0.507808\pi\)
\(174\) 1.17832e10 0.974522
\(175\) 0 0
\(176\) −3.70046e9 −0.290703
\(177\) 7.31392e9 0.560106
\(178\) −1.32473e9 −0.0989094
\(179\) −1.56569e10 −1.13990 −0.569952 0.821678i \(-0.693038\pi\)
−0.569952 + 0.821678i \(0.693038\pi\)
\(180\) 0 0
\(181\) −2.18552e10 −1.51356 −0.756781 0.653668i \(-0.773229\pi\)
−0.756781 + 0.653668i \(0.773229\pi\)
\(182\) −6.48349e9 −0.438013
\(183\) −4.14780e9 −0.273394
\(184\) −7.19882e9 −0.463000
\(185\) 0 0
\(186\) 1.21033e10 0.741474
\(187\) 2.75543e9 0.164779
\(188\) −5.95388e9 −0.347608
\(189\) −1.05569e10 −0.601807
\(190\) 0 0
\(191\) 1.67784e10 0.912222 0.456111 0.889923i \(-0.349242\pi\)
0.456111 + 0.889923i \(0.349242\pi\)
\(192\) 8.81622e9 0.468195
\(193\) 2.70036e10 1.40092 0.700462 0.713690i \(-0.252977\pi\)
0.700462 + 0.713690i \(0.252977\pi\)
\(194\) −4.78645e10 −2.42608
\(195\) 0 0
\(196\) −4.70671e9 −0.227806
\(197\) −1.58277e10 −0.748720 −0.374360 0.927283i \(-0.622138\pi\)
−0.374360 + 0.927283i \(0.622138\pi\)
\(198\) −1.67309e10 −0.773616
\(199\) 8.80397e9 0.397960 0.198980 0.980004i \(-0.436237\pi\)
0.198980 + 0.980004i \(0.436237\pi\)
\(200\) 0 0
\(201\) −4.83736e9 −0.209038
\(202\) 3.45982e10 1.46208
\(203\) −3.83485e10 −1.58495
\(204\) 4.94523e9 0.199917
\(205\) 0 0
\(206\) −1.59167e10 −0.615817
\(207\) −7.57796e9 −0.286870
\(208\) −4.19306e9 −0.155327
\(209\) 2.28013e10 0.826610
\(210\) 0 0
\(211\) −1.82054e10 −0.632308 −0.316154 0.948708i \(-0.602392\pi\)
−0.316154 + 0.948708i \(0.602392\pi\)
\(212\) −1.44665e10 −0.491870
\(213\) −1.97796e10 −0.658430
\(214\) 2.56353e10 0.835558
\(215\) 0 0
\(216\) −2.93245e10 −0.916618
\(217\) −3.93904e10 −1.20593
\(218\) −6.36453e10 −1.90859
\(219\) −1.50555e10 −0.442280
\(220\) 0 0
\(221\) 3.12223e9 0.0880440
\(222\) −7.57215e9 −0.209233
\(223\) −1.53511e10 −0.415688 −0.207844 0.978162i \(-0.566645\pi\)
−0.207844 + 0.978162i \(0.566645\pi\)
\(224\) −1.69799e10 −0.450628
\(225\) 0 0
\(226\) −7.04514e10 −1.79639
\(227\) −4.20620e10 −1.05141 −0.525707 0.850666i \(-0.676199\pi\)
−0.525707 + 0.850666i \(0.676199\pi\)
\(228\) 4.09219e10 1.00288
\(229\) −6.68760e10 −1.60698 −0.803490 0.595318i \(-0.797026\pi\)
−0.803490 + 0.595318i \(0.797026\pi\)
\(230\) 0 0
\(231\) −7.17741e9 −0.165849
\(232\) −1.06523e11 −2.41406
\(233\) 5.19268e10 1.15422 0.577112 0.816665i \(-0.304180\pi\)
0.577112 + 0.816665i \(0.304180\pi\)
\(234\) −1.89581e10 −0.413355
\(235\) 0 0
\(236\) −1.44333e11 −3.02874
\(237\) 9.60345e9 0.197724
\(238\) −2.48157e10 −0.501338
\(239\) 7.45881e10 1.47870 0.739348 0.673323i \(-0.235134\pi\)
0.739348 + 0.673323i \(0.235134\pi\)
\(240\) 0 0
\(241\) 5.74852e10 1.09769 0.548845 0.835924i \(-0.315068\pi\)
0.548845 + 0.835924i \(0.315068\pi\)
\(242\) 6.57499e10 1.23233
\(243\) −4.72577e10 −0.869449
\(244\) 8.18528e10 1.47836
\(245\) 0 0
\(246\) 2.73328e10 0.475857
\(247\) 2.58365e10 0.441670
\(248\) −1.09417e11 −1.83676
\(249\) 3.03826e9 0.0500873
\(250\) 0 0
\(251\) −1.07873e11 −1.71546 −0.857732 0.514097i \(-0.828127\pi\)
−0.857732 + 0.514097i \(0.828127\pi\)
\(252\) 9.77240e10 1.52651
\(253\) −1.09833e10 −0.168536
\(254\) 7.17272e10 1.08126
\(255\) 0 0
\(256\) −1.18186e11 −1.71984
\(257\) 6.64074e10 0.949550 0.474775 0.880107i \(-0.342530\pi\)
0.474775 + 0.880107i \(0.342530\pi\)
\(258\) −7.34247e10 −1.03170
\(259\) 2.46436e10 0.340295
\(260\) 0 0
\(261\) −1.12133e11 −1.49573
\(262\) −1.32080e11 −1.73174
\(263\) 8.15356e10 1.05086 0.525432 0.850836i \(-0.323904\pi\)
0.525432 + 0.850836i \(0.323904\pi\)
\(264\) −1.99371e10 −0.252607
\(265\) 0 0
\(266\) −2.05351e11 −2.51494
\(267\) −1.66174e9 −0.0200107
\(268\) 9.54605e10 1.13036
\(269\) 1.00568e11 1.17105 0.585523 0.810656i \(-0.300889\pi\)
0.585523 + 0.810656i \(0.300889\pi\)
\(270\) 0 0
\(271\) −2.98757e10 −0.336477 −0.168239 0.985746i \(-0.553808\pi\)
−0.168239 + 0.985746i \(0.553808\pi\)
\(272\) −1.60491e10 −0.177783
\(273\) −8.13286e9 −0.0886158
\(274\) 1.92669e11 2.06507
\(275\) 0 0
\(276\) −1.97120e10 −0.204475
\(277\) 3.17195e10 0.323718 0.161859 0.986814i \(-0.448251\pi\)
0.161859 + 0.986814i \(0.448251\pi\)
\(278\) −1.37298e11 −1.37868
\(279\) −1.15180e11 −1.13804
\(280\) 0 0
\(281\) 9.86953e10 0.944318 0.472159 0.881513i \(-0.343475\pi\)
0.472159 + 0.881513i \(0.343475\pi\)
\(282\) −1.15157e10 −0.108435
\(283\) 1.06164e11 0.983871 0.491935 0.870632i \(-0.336290\pi\)
0.491935 + 0.870632i \(0.336290\pi\)
\(284\) 3.90331e11 3.56042
\(285\) 0 0
\(286\) −2.74774e10 −0.242845
\(287\) −8.89549e10 −0.773929
\(288\) −4.96501e10 −0.425260
\(289\) −1.06637e11 −0.899227
\(290\) 0 0
\(291\) −6.00410e10 −0.490828
\(292\) 2.97106e11 2.39160
\(293\) −2.26355e11 −1.79426 −0.897132 0.441763i \(-0.854353\pi\)
−0.897132 + 0.441763i \(0.854353\pi\)
\(294\) −9.10348e9 −0.0710631
\(295\) 0 0
\(296\) 6.84541e10 0.518307
\(297\) −4.47407e10 −0.333656
\(298\) −4.86881e10 −0.357643
\(299\) −1.24454e10 −0.0900511
\(300\) 0 0
\(301\) 2.38962e11 1.67795
\(302\) −1.66137e11 −1.14930
\(303\) 4.33998e10 0.295798
\(304\) −1.32806e11 −0.891841
\(305\) 0 0
\(306\) −7.25625e10 −0.473114
\(307\) 2.23009e10 0.143285 0.0716424 0.997430i \(-0.477176\pi\)
0.0716424 + 0.997430i \(0.477176\pi\)
\(308\) 1.41639e11 0.896820
\(309\) −1.99659e10 −0.124588
\(310\) 0 0
\(311\) −2.71805e11 −1.64754 −0.823770 0.566925i \(-0.808133\pi\)
−0.823770 + 0.566925i \(0.808133\pi\)
\(312\) −2.25911e10 −0.134972
\(313\) 7.90775e10 0.465697 0.232849 0.972513i \(-0.425195\pi\)
0.232849 + 0.972513i \(0.425195\pi\)
\(314\) −5.38183e11 −3.12425
\(315\) 0 0
\(316\) −1.89515e11 −1.06918
\(317\) −2.03032e11 −1.12927 −0.564636 0.825340i \(-0.690983\pi\)
−0.564636 + 0.825340i \(0.690983\pi\)
\(318\) −2.79802e10 −0.153437
\(319\) −1.62524e11 −0.878736
\(320\) 0 0
\(321\) 3.21568e10 0.169044
\(322\) 9.89171e10 0.512767
\(323\) 9.88899e10 0.505523
\(324\) 2.43120e11 1.22565
\(325\) 0 0
\(326\) 2.76650e10 0.135660
\(327\) −7.98365e10 −0.386132
\(328\) −2.47095e11 −1.17878
\(329\) 3.74779e10 0.176357
\(330\) 0 0
\(331\) 3.24137e11 1.48424 0.742118 0.670270i \(-0.233822\pi\)
0.742118 + 0.670270i \(0.233822\pi\)
\(332\) −5.99570e10 −0.270844
\(333\) 7.20594e10 0.321138
\(334\) −3.17955e10 −0.139800
\(335\) 0 0
\(336\) 4.18050e10 0.178937
\(337\) −2.68510e11 −1.13403 −0.567017 0.823706i \(-0.691903\pi\)
−0.567017 + 0.823706i \(0.691903\pi\)
\(338\) −3.11352e10 −0.129756
\(339\) −8.83739e10 −0.363434
\(340\) 0 0
\(341\) −1.66939e11 −0.668595
\(342\) −6.00457e11 −2.37336
\(343\) 2.69628e11 1.05182
\(344\) 6.63778e11 2.55570
\(345\) 0 0
\(346\) 2.20597e10 0.0827478
\(347\) −2.74715e11 −1.01719 −0.508593 0.861007i \(-0.669834\pi\)
−0.508593 + 0.861007i \(0.669834\pi\)
\(348\) −2.91684e11 −1.06612
\(349\) −3.54006e11 −1.27731 −0.638655 0.769494i \(-0.720509\pi\)
−0.638655 + 0.769494i \(0.720509\pi\)
\(350\) 0 0
\(351\) −5.06966e10 −0.178277
\(352\) −7.19619e10 −0.249839
\(353\) 2.25650e11 0.773481 0.386741 0.922189i \(-0.373601\pi\)
0.386741 + 0.922189i \(0.373601\pi\)
\(354\) −2.79161e11 −0.944801
\(355\) 0 0
\(356\) 3.27927e10 0.108206
\(357\) −3.11287e10 −0.101427
\(358\) 5.97602e11 1.92282
\(359\) 3.39022e10 0.107722 0.0538608 0.998548i \(-0.482847\pi\)
0.0538608 + 0.998548i \(0.482847\pi\)
\(360\) 0 0
\(361\) 4.95629e11 1.53594
\(362\) 8.34178e11 2.55312
\(363\) 8.24764e10 0.249316
\(364\) 1.60494e11 0.479184
\(365\) 0 0
\(366\) 1.58315e11 0.461167
\(367\) 7.74387e10 0.222823 0.111412 0.993774i \(-0.464463\pi\)
0.111412 + 0.993774i \(0.464463\pi\)
\(368\) 6.39726e10 0.181836
\(369\) −2.60109e11 −0.730360
\(370\) 0 0
\(371\) 9.10620e10 0.249548
\(372\) −2.99609e11 −0.811169
\(373\) −6.50821e11 −1.74089 −0.870446 0.492264i \(-0.836169\pi\)
−0.870446 + 0.492264i \(0.836169\pi\)
\(374\) −1.05171e11 −0.277954
\(375\) 0 0
\(376\) 1.04105e11 0.268612
\(377\) −1.84159e11 −0.469522
\(378\) 4.02940e11 1.01514
\(379\) 1.93989e11 0.482948 0.241474 0.970407i \(-0.422369\pi\)
0.241474 + 0.970407i \(0.422369\pi\)
\(380\) 0 0
\(381\) 8.99743e10 0.218754
\(382\) −6.40406e11 −1.53876
\(383\) −4.44645e11 −1.05589 −0.527946 0.849278i \(-0.677037\pi\)
−0.527946 + 0.849278i \(0.677037\pi\)
\(384\) −2.66515e11 −0.625506
\(385\) 0 0
\(386\) −1.03069e12 −2.36311
\(387\) 6.98737e11 1.58349
\(388\) 1.18485e12 2.65412
\(389\) −1.69623e11 −0.375589 −0.187794 0.982208i \(-0.560134\pi\)
−0.187794 + 0.982208i \(0.560134\pi\)
\(390\) 0 0
\(391\) −4.76352e10 −0.103070
\(392\) 8.22978e10 0.176036
\(393\) −1.65681e11 −0.350353
\(394\) 6.04119e11 1.26296
\(395\) 0 0
\(396\) 4.14161e11 0.846332
\(397\) −2.94468e11 −0.594951 −0.297476 0.954729i \(-0.596145\pi\)
−0.297476 + 0.954729i \(0.596145\pi\)
\(398\) −3.36034e11 −0.671289
\(399\) −2.57591e11 −0.508806
\(400\) 0 0
\(401\) 4.43362e11 0.856265 0.428133 0.903716i \(-0.359172\pi\)
0.428133 + 0.903716i \(0.359172\pi\)
\(402\) 1.84635e11 0.352611
\(403\) −1.89162e11 −0.357240
\(404\) −8.56452e11 −1.59951
\(405\) 0 0
\(406\) 1.46371e12 2.67354
\(407\) 1.04441e11 0.188668
\(408\) −8.64682e10 −0.154485
\(409\) −1.01253e11 −0.178918 −0.0894592 0.995990i \(-0.528514\pi\)
−0.0894592 + 0.995990i \(0.528514\pi\)
\(410\) 0 0
\(411\) 2.41684e11 0.417791
\(412\) 3.94007e11 0.673700
\(413\) 9.08533e11 1.53662
\(414\) 2.89239e11 0.483900
\(415\) 0 0
\(416\) −8.15413e10 −0.133493
\(417\) −1.72226e11 −0.278925
\(418\) −8.70289e11 −1.39435
\(419\) 7.93982e10 0.125848 0.0629242 0.998018i \(-0.479957\pi\)
0.0629242 + 0.998018i \(0.479957\pi\)
\(420\) 0 0
\(421\) −6.06765e11 −0.941350 −0.470675 0.882307i \(-0.655990\pi\)
−0.470675 + 0.882307i \(0.655990\pi\)
\(422\) 6.94871e11 1.06659
\(423\) 1.09587e11 0.166429
\(424\) 2.52949e11 0.380089
\(425\) 0 0
\(426\) 7.54958e11 1.11066
\(427\) −5.15239e11 −0.750038
\(428\) −6.34583e11 −0.914096
\(429\) −3.44676e10 −0.0491307
\(430\) 0 0
\(431\) −2.72544e11 −0.380442 −0.190221 0.981741i \(-0.560920\pi\)
−0.190221 + 0.981741i \(0.560920\pi\)
\(432\) 2.60593e11 0.359987
\(433\) 1.15522e12 1.57931 0.789655 0.613551i \(-0.210260\pi\)
0.789655 + 0.613551i \(0.210260\pi\)
\(434\) 1.50347e12 2.03419
\(435\) 0 0
\(436\) 1.57549e12 2.08799
\(437\) −3.94182e11 −0.517047
\(438\) 5.74646e11 0.746048
\(439\) 1.78053e11 0.228801 0.114401 0.993435i \(-0.463505\pi\)
0.114401 + 0.993435i \(0.463505\pi\)
\(440\) 0 0
\(441\) 8.66321e10 0.109070
\(442\) −1.19171e11 −0.148515
\(443\) 6.55260e10 0.0808345 0.0404173 0.999183i \(-0.487131\pi\)
0.0404173 + 0.999183i \(0.487131\pi\)
\(444\) 1.87443e11 0.228900
\(445\) 0 0
\(446\) 5.85928e11 0.701192
\(447\) −6.10742e10 −0.0723559
\(448\) 1.09515e12 1.28446
\(449\) −7.98150e11 −0.926779 −0.463390 0.886155i \(-0.653367\pi\)
−0.463390 + 0.886155i \(0.653367\pi\)
\(450\) 0 0
\(451\) −3.76997e11 −0.429085
\(452\) 1.74397e12 1.96525
\(453\) −2.08402e11 −0.232519
\(454\) 1.60544e12 1.77355
\(455\) 0 0
\(456\) −7.15526e11 −0.774968
\(457\) 8.92736e11 0.957415 0.478708 0.877974i \(-0.341105\pi\)
0.478708 + 0.877974i \(0.341105\pi\)
\(458\) 2.55256e12 2.71069
\(459\) −1.94042e11 −0.204051
\(460\) 0 0
\(461\) 2.32490e11 0.239745 0.119872 0.992789i \(-0.461751\pi\)
0.119872 + 0.992789i \(0.461751\pi\)
\(462\) 2.73951e11 0.279759
\(463\) 1.54421e12 1.56168 0.780841 0.624730i \(-0.214791\pi\)
0.780841 + 0.624730i \(0.214791\pi\)
\(464\) 9.46622e11 0.948082
\(465\) 0 0
\(466\) −1.98197e12 −1.94697
\(467\) −1.63317e12 −1.58893 −0.794465 0.607310i \(-0.792249\pi\)
−0.794465 + 0.607310i \(0.792249\pi\)
\(468\) 4.69293e11 0.452208
\(469\) −6.00895e11 −0.573484
\(470\) 0 0
\(471\) −6.75094e11 −0.632077
\(472\) 2.52369e12 2.34044
\(473\) 1.01274e12 0.930295
\(474\) −3.66549e11 −0.333526
\(475\) 0 0
\(476\) 6.14295e11 0.548461
\(477\) 2.66271e11 0.235500
\(478\) −2.84692e12 −2.49430
\(479\) 1.54414e12 1.34022 0.670112 0.742260i \(-0.266246\pi\)
0.670112 + 0.742260i \(0.266246\pi\)
\(480\) 0 0
\(481\) 1.18344e11 0.100808
\(482\) −2.19412e12 −1.85161
\(483\) 1.24081e11 0.103739
\(484\) −1.62759e12 −1.34816
\(485\) 0 0
\(486\) 1.80375e12 1.46661
\(487\) −2.55439e11 −0.205782 −0.102891 0.994693i \(-0.532809\pi\)
−0.102891 + 0.994693i \(0.532809\pi\)
\(488\) −1.43121e12 −1.14239
\(489\) 3.47028e10 0.0274457
\(490\) 0 0
\(491\) 1.58407e12 1.23000 0.615002 0.788526i \(-0.289155\pi\)
0.615002 + 0.788526i \(0.289155\pi\)
\(492\) −6.76603e11 −0.520584
\(493\) −7.04872e11 −0.537402
\(494\) −9.86141e11 −0.745020
\(495\) 0 0
\(496\) 9.72340e11 0.721357
\(497\) −2.45702e12 −1.80636
\(498\) −1.15966e11 −0.0844884
\(499\) −6.29948e11 −0.454833 −0.227417 0.973798i \(-0.573028\pi\)
−0.227417 + 0.973798i \(0.573028\pi\)
\(500\) 0 0
\(501\) −3.98842e10 −0.0282834
\(502\) 4.11736e12 2.89369
\(503\) −5.49263e11 −0.382582 −0.191291 0.981533i \(-0.561267\pi\)
−0.191291 + 0.981533i \(0.561267\pi\)
\(504\) −1.70872e12 −1.17960
\(505\) 0 0
\(506\) 4.19217e11 0.284290
\(507\) −3.90559e10 −0.0262513
\(508\) −1.77555e12 −1.18290
\(509\) 1.36923e11 0.0904165 0.0452083 0.998978i \(-0.485605\pi\)
0.0452083 + 0.998978i \(0.485605\pi\)
\(510\) 0 0
\(511\) −1.87019e12 −1.21337
\(512\) 1.66095e12 1.06817
\(513\) −1.60570e12 −1.02362
\(514\) −2.53467e12 −1.60172
\(515\) 0 0
\(516\) 1.81758e12 1.12867
\(517\) 1.58834e11 0.0977767
\(518\) −9.40610e11 −0.574018
\(519\) 2.76715e10 0.0167410
\(520\) 0 0
\(521\) −1.66627e12 −0.990779 −0.495389 0.868671i \(-0.664975\pi\)
−0.495389 + 0.868671i \(0.664975\pi\)
\(522\) 4.27996e12 2.52303
\(523\) −1.97187e12 −1.15244 −0.576222 0.817293i \(-0.695474\pi\)
−0.576222 + 0.817293i \(0.695474\pi\)
\(524\) 3.26955e12 1.89451
\(525\) 0 0
\(526\) −3.11209e12 −1.77262
\(527\) −7.24021e11 −0.408887
\(528\) 1.77172e11 0.0992072
\(529\) −1.61128e12 −0.894580
\(530\) 0 0
\(531\) 2.65660e12 1.45011
\(532\) 5.08330e12 2.75134
\(533\) −4.27182e11 −0.229266
\(534\) 6.34260e10 0.0337545
\(535\) 0 0
\(536\) −1.66914e12 −0.873479
\(537\) 7.49629e11 0.389011
\(538\) −3.83852e12 −1.97535
\(539\) 1.25563e11 0.0640784
\(540\) 0 0
\(541\) 1.54916e12 0.777513 0.388756 0.921341i \(-0.372905\pi\)
0.388756 + 0.921341i \(0.372905\pi\)
\(542\) 1.14031e12 0.567578
\(543\) 1.04639e12 0.516529
\(544\) −3.12101e11 −0.152792
\(545\) 0 0
\(546\) 3.10419e11 0.149479
\(547\) −1.98052e12 −0.945879 −0.472939 0.881095i \(-0.656807\pi\)
−0.472939 + 0.881095i \(0.656807\pi\)
\(548\) −4.76939e12 −2.25918
\(549\) −1.50659e12 −0.707814
\(550\) 0 0
\(551\) −5.83283e12 −2.69586
\(552\) 3.44668e11 0.158007
\(553\) 1.19294e12 0.542444
\(554\) −1.21068e12 −0.546056
\(555\) 0 0
\(556\) 3.39871e12 1.50827
\(557\) −8.86577e11 −0.390273 −0.195136 0.980776i \(-0.562515\pi\)
−0.195136 + 0.980776i \(0.562515\pi\)
\(558\) 4.39623e12 1.91967
\(559\) 1.14755e12 0.497071
\(560\) 0 0
\(561\) −1.31926e11 −0.0562336
\(562\) −3.76705e12 −1.59290
\(563\) −3.89786e12 −1.63508 −0.817539 0.575873i \(-0.804662\pi\)
−0.817539 + 0.575873i \(0.804662\pi\)
\(564\) 2.85062e11 0.118627
\(565\) 0 0
\(566\) −4.05212e12 −1.65962
\(567\) −1.53037e12 −0.621830
\(568\) −6.82501e12 −2.75129
\(569\) −2.26590e12 −0.906223 −0.453112 0.891454i \(-0.649686\pi\)
−0.453112 + 0.891454i \(0.649686\pi\)
\(570\) 0 0
\(571\) −3.09546e12 −1.21860 −0.609302 0.792938i \(-0.708550\pi\)
−0.609302 + 0.792938i \(0.708550\pi\)
\(572\) 6.80184e11 0.265671
\(573\) −8.03323e11 −0.311311
\(574\) 3.39527e12 1.30548
\(575\) 0 0
\(576\) 3.20228e12 1.21215
\(577\) 3.23415e12 1.21470 0.607350 0.794434i \(-0.292233\pi\)
0.607350 + 0.794434i \(0.292233\pi\)
\(578\) 4.07019e12 1.51684
\(579\) −1.29289e12 −0.478089
\(580\) 0 0
\(581\) 3.77411e11 0.137411
\(582\) 2.29167e12 0.827941
\(583\) 3.85927e11 0.138356
\(584\) −5.19495e12 −1.84809
\(585\) 0 0
\(586\) 8.63964e12 3.02661
\(587\) 8.78491e11 0.305398 0.152699 0.988273i \(-0.451204\pi\)
0.152699 + 0.988273i \(0.451204\pi\)
\(588\) 2.25350e11 0.0777427
\(589\) −5.99129e12 −2.05117
\(590\) 0 0
\(591\) 7.57805e11 0.255513
\(592\) −6.08321e11 −0.203556
\(593\) 1.91730e12 0.636712 0.318356 0.947971i \(-0.396869\pi\)
0.318356 + 0.947971i \(0.396869\pi\)
\(594\) 1.70769e12 0.562819
\(595\) 0 0
\(596\) 1.20524e12 0.391260
\(597\) −4.21520e11 −0.135811
\(598\) 4.75023e11 0.151900
\(599\) −5.10464e12 −1.62011 −0.810054 0.586355i \(-0.800562\pi\)
−0.810054 + 0.586355i \(0.800562\pi\)
\(600\) 0 0
\(601\) 3.00419e12 0.939273 0.469636 0.882860i \(-0.344385\pi\)
0.469636 + 0.882860i \(0.344385\pi\)
\(602\) −9.12080e12 −2.83041
\(603\) −1.75705e12 −0.541199
\(604\) 4.11260e12 1.25733
\(605\) 0 0
\(606\) −1.65650e12 −0.498960
\(607\) −1.18434e12 −0.354101 −0.177051 0.984202i \(-0.556656\pi\)
−0.177051 + 0.984202i \(0.556656\pi\)
\(608\) −2.58265e12 −0.766477
\(609\) 1.83607e12 0.540892
\(610\) 0 0
\(611\) 1.79978e11 0.0522436
\(612\) 1.79623e12 0.517584
\(613\) −6.42597e11 −0.183809 −0.0919045 0.995768i \(-0.529295\pi\)
−0.0919045 + 0.995768i \(0.529295\pi\)
\(614\) −8.51193e11 −0.241696
\(615\) 0 0
\(616\) −2.47659e12 −0.693011
\(617\) −5.05474e12 −1.40416 −0.702078 0.712100i \(-0.747744\pi\)
−0.702078 + 0.712100i \(0.747744\pi\)
\(618\) 7.62068e11 0.210158
\(619\) −3.09492e11 −0.0847308 −0.0423654 0.999102i \(-0.513489\pi\)
−0.0423654 + 0.999102i \(0.513489\pi\)
\(620\) 0 0
\(621\) 7.73466e11 0.208703
\(622\) 1.03744e13 2.77911
\(623\) −2.06421e11 −0.0548980
\(624\) 2.00757e11 0.0530079
\(625\) 0 0
\(626\) −3.01827e12 −0.785549
\(627\) −1.09169e12 −0.282094
\(628\) 1.33223e13 3.41792
\(629\) 4.52966e11 0.115382
\(630\) 0 0
\(631\) −7.94237e11 −0.199443 −0.0997214 0.995015i \(-0.531795\pi\)
−0.0997214 + 0.995015i \(0.531795\pi\)
\(632\) 3.31370e12 0.826202
\(633\) 8.71644e11 0.215786
\(634\) 7.74944e12 1.90488
\(635\) 0 0
\(636\) 6.92631e11 0.167859
\(637\) 1.42278e11 0.0342380
\(638\) 6.20328e12 1.48227
\(639\) −7.18446e12 −1.70467
\(640\) 0 0
\(641\) 2.30298e12 0.538802 0.269401 0.963028i \(-0.413174\pi\)
0.269401 + 0.963028i \(0.413174\pi\)
\(642\) −1.22738e12 −0.285148
\(643\) 2.54593e12 0.587350 0.293675 0.955905i \(-0.405122\pi\)
0.293675 + 0.955905i \(0.405122\pi\)
\(644\) −2.44862e12 −0.560964
\(645\) 0 0
\(646\) −3.77448e12 −0.852728
\(647\) 2.74568e11 0.0616001 0.0308000 0.999526i \(-0.490194\pi\)
0.0308000 + 0.999526i \(0.490194\pi\)
\(648\) −4.25099e12 −0.947115
\(649\) 3.85043e12 0.851937
\(650\) 0 0
\(651\) 1.88595e12 0.411543
\(652\) −6.84826e11 −0.148411
\(653\) 6.05719e12 1.30365 0.651826 0.758369i \(-0.274003\pi\)
0.651826 + 0.758369i \(0.274003\pi\)
\(654\) 3.04724e12 0.651338
\(655\) 0 0
\(656\) 2.19582e12 0.462946
\(657\) −5.46855e12 −1.14506
\(658\) −1.43047e12 −0.297484
\(659\) 4.97718e12 1.02801 0.514007 0.857786i \(-0.328160\pi\)
0.514007 + 0.857786i \(0.328160\pi\)
\(660\) 0 0
\(661\) 1.79834e12 0.366409 0.183204 0.983075i \(-0.441353\pi\)
0.183204 + 0.983075i \(0.441353\pi\)
\(662\) −1.23718e13 −2.50364
\(663\) −1.49487e11 −0.0300465
\(664\) 1.04836e12 0.209292
\(665\) 0 0
\(666\) −2.75040e12 −0.541703
\(667\) 2.80967e12 0.549653
\(668\) 7.87076e11 0.152940
\(669\) 7.34986e11 0.141860
\(670\) 0 0
\(671\) −2.18362e12 −0.415839
\(672\) 8.12969e11 0.153784
\(673\) −1.37693e12 −0.258728 −0.129364 0.991597i \(-0.541294\pi\)
−0.129364 + 0.991597i \(0.541294\pi\)
\(674\) 1.02486e13 1.91292
\(675\) 0 0
\(676\) 7.70729e11 0.141952
\(677\) −6.49642e11 −0.118857 −0.0594285 0.998233i \(-0.518928\pi\)
−0.0594285 + 0.998233i \(0.518928\pi\)
\(678\) 3.37310e12 0.613049
\(679\) −7.45828e12 −1.34656
\(680\) 0 0
\(681\) 2.01386e12 0.358813
\(682\) 6.37181e12 1.12780
\(683\) 1.10011e12 0.193438 0.0967190 0.995312i \(-0.469165\pi\)
0.0967190 + 0.995312i \(0.469165\pi\)
\(684\) 1.48639e13 2.59645
\(685\) 0 0
\(686\) −1.02913e13 −1.77424
\(687\) 3.20192e12 0.548409
\(688\) −5.89869e12 −1.00371
\(689\) 4.37301e11 0.0739254
\(690\) 0 0
\(691\) 7.72289e12 1.28863 0.644315 0.764760i \(-0.277142\pi\)
0.644315 + 0.764760i \(0.277142\pi\)
\(692\) −5.46071e11 −0.0905256
\(693\) −2.60702e12 −0.429383
\(694\) 1.04855e13 1.71581
\(695\) 0 0
\(696\) 5.10016e12 0.823838
\(697\) −1.63505e12 −0.262412
\(698\) 1.35119e13 2.15460
\(699\) −2.48617e12 −0.393898
\(700\) 0 0
\(701\) 1.40436e12 0.219658 0.109829 0.993950i \(-0.464970\pi\)
0.109829 + 0.993950i \(0.464970\pi\)
\(702\) 1.93501e12 0.300723
\(703\) 3.74831e12 0.578810
\(704\) 4.64131e12 0.712137
\(705\) 0 0
\(706\) −8.61273e12 −1.30473
\(707\) 5.39111e12 0.811505
\(708\) 6.91043e12 1.03361
\(709\) 5.19764e12 0.772499 0.386250 0.922394i \(-0.373770\pi\)
0.386250 + 0.922394i \(0.373770\pi\)
\(710\) 0 0
\(711\) 3.48822e12 0.511907
\(712\) −5.73387e11 −0.0836157
\(713\) 2.88600e12 0.418209
\(714\) 1.18814e12 0.171090
\(715\) 0 0
\(716\) −1.47932e13 −2.10355
\(717\) −3.57116e12 −0.504630
\(718\) −1.29400e12 −0.181707
\(719\) −1.27042e13 −1.77284 −0.886418 0.462886i \(-0.846814\pi\)
−0.886418 + 0.462886i \(0.846814\pi\)
\(720\) 0 0
\(721\) −2.48016e12 −0.341799
\(722\) −1.89174e13 −2.59086
\(723\) −2.75230e12 −0.374605
\(724\) −2.06495e13 −2.79309
\(725\) 0 0
\(726\) −3.14800e12 −0.420552
\(727\) −9.91998e11 −0.131706 −0.0658531 0.997829i \(-0.520977\pi\)
−0.0658531 + 0.997829i \(0.520977\pi\)
\(728\) −2.80627e12 −0.370286
\(729\) −2.80211e12 −0.367461
\(730\) 0 0
\(731\) 4.39227e12 0.568933
\(732\) −3.91898e12 −0.504514
\(733\) −5.54849e12 −0.709916 −0.354958 0.934882i \(-0.615505\pi\)
−0.354958 + 0.934882i \(0.615505\pi\)
\(734\) −2.95572e12 −0.375864
\(735\) 0 0
\(736\) 1.24406e12 0.156275
\(737\) −2.54663e12 −0.317953
\(738\) 9.92797e12 1.23199
\(739\) 8.78597e12 1.08365 0.541826 0.840491i \(-0.317733\pi\)
0.541826 + 0.840491i \(0.317733\pi\)
\(740\) 0 0
\(741\) −1.23701e12 −0.150727
\(742\) −3.47570e12 −0.420944
\(743\) −4.04326e12 −0.486724 −0.243362 0.969936i \(-0.578250\pi\)
−0.243362 + 0.969936i \(0.578250\pi\)
\(744\) 5.23871e12 0.626825
\(745\) 0 0
\(746\) 2.48408e13 2.93658
\(747\) 1.10357e12 0.129676
\(748\) 2.60342e12 0.304080
\(749\) 3.99451e12 0.463763
\(750\) 0 0
\(751\) −2.52936e11 −0.0290156 −0.0145078 0.999895i \(-0.504618\pi\)
−0.0145078 + 0.999895i \(0.504618\pi\)
\(752\) −9.25130e11 −0.105493
\(753\) 5.16479e12 0.585431
\(754\) 7.02906e12 0.792001
\(755\) 0 0
\(756\) −9.97448e12 −1.11056
\(757\) −9.99638e12 −1.10640 −0.553199 0.833049i \(-0.686593\pi\)
−0.553199 + 0.833049i \(0.686593\pi\)
\(758\) −7.40426e12 −0.814649
\(759\) 5.25864e11 0.0575156
\(760\) 0 0
\(761\) −2.98848e12 −0.323013 −0.161507 0.986872i \(-0.551635\pi\)
−0.161507 + 0.986872i \(0.551635\pi\)
\(762\) −3.43418e12 −0.369000
\(763\) −9.91726e12 −1.05933
\(764\) 1.58528e13 1.68339
\(765\) 0 0
\(766\) 1.69714e13 1.78110
\(767\) 4.36299e12 0.455203
\(768\) 5.65858e12 0.586924
\(769\) 1.54887e13 1.59715 0.798576 0.601894i \(-0.205587\pi\)
0.798576 + 0.601894i \(0.205587\pi\)
\(770\) 0 0
\(771\) −3.17948e12 −0.324050
\(772\) 2.55139e13 2.58523
\(773\) −1.07589e13 −1.08383 −0.541915 0.840434i \(-0.682300\pi\)
−0.541915 + 0.840434i \(0.682300\pi\)
\(774\) −2.66697e13 −2.67106
\(775\) 0 0
\(776\) −2.07173e13 −2.05095
\(777\) −1.17990e12 −0.116131
\(778\) 6.47427e12 0.633552
\(779\) −1.35301e13 −1.31638
\(780\) 0 0
\(781\) −1.04130e13 −1.00149
\(782\) 1.81816e12 0.173861
\(783\) 1.14452e13 1.08817
\(784\) −7.31343e11 −0.0691351
\(785\) 0 0
\(786\) 6.32379e12 0.590985
\(787\) −1.29304e13 −1.20151 −0.600755 0.799433i \(-0.705133\pi\)
−0.600755 + 0.799433i \(0.705133\pi\)
\(788\) −1.49545e13 −1.38167
\(789\) −3.90379e12 −0.358625
\(790\) 0 0
\(791\) −1.09778e13 −0.997059
\(792\) −7.24168e12 −0.653997
\(793\) −2.47430e12 −0.222189
\(794\) 1.12394e13 1.00358
\(795\) 0 0
\(796\) 8.31828e12 0.734387
\(797\) −1.24713e13 −1.09484 −0.547418 0.836859i \(-0.684389\pi\)
−0.547418 + 0.836859i \(0.684389\pi\)
\(798\) 9.83185e12 0.858267
\(799\) 6.88868e11 0.0597965
\(800\) 0 0
\(801\) −6.03585e11 −0.0518075
\(802\) −1.69224e13 −1.44437
\(803\) −7.92600e12 −0.672719
\(804\) −4.57049e12 −0.385755
\(805\) 0 0
\(806\) 7.22002e12 0.602602
\(807\) −4.81502e12 −0.399639
\(808\) 1.49752e13 1.23601
\(809\) −2.40763e13 −1.97616 −0.988079 0.153948i \(-0.950801\pi\)
−0.988079 + 0.153948i \(0.950801\pi\)
\(810\) 0 0
\(811\) 1.14686e13 0.930928 0.465464 0.885067i \(-0.345888\pi\)
0.465464 + 0.885067i \(0.345888\pi\)
\(812\) −3.62330e13 −2.92484
\(813\) 1.43040e12 0.114829
\(814\) −3.98637e12 −0.318249
\(815\) 0 0
\(816\) 7.68403e11 0.0606713
\(817\) 3.63462e13 2.85403
\(818\) 3.86469e12 0.301804
\(819\) −2.95406e12 −0.229426
\(820\) 0 0
\(821\) −2.50113e13 −1.92129 −0.960643 0.277786i \(-0.910399\pi\)
−0.960643 + 0.277786i \(0.910399\pi\)
\(822\) −9.22470e12 −0.704740
\(823\) −1.29351e13 −0.982808 −0.491404 0.870932i \(-0.663516\pi\)
−0.491404 + 0.870932i \(0.663516\pi\)
\(824\) −6.88929e12 −0.520597
\(825\) 0 0
\(826\) −3.46773e13 −2.59200
\(827\) −1.00464e13 −0.746851 −0.373425 0.927660i \(-0.621817\pi\)
−0.373425 + 0.927660i \(0.621817\pi\)
\(828\) −7.15990e12 −0.529384
\(829\) 8.77304e12 0.645141 0.322570 0.946545i \(-0.395453\pi\)
0.322570 + 0.946545i \(0.395453\pi\)
\(830\) 0 0
\(831\) −1.51868e12 −0.110474
\(832\) 5.25916e12 0.380506
\(833\) 5.44571e11 0.0391879
\(834\) 6.57361e12 0.470497
\(835\) 0 0
\(836\) 2.15434e13 1.52541
\(837\) 1.17561e13 0.827943
\(838\) −3.03051e12 −0.212284
\(839\) −1.54259e13 −1.07478 −0.537392 0.843332i \(-0.680590\pi\)
−0.537392 + 0.843332i \(0.680590\pi\)
\(840\) 0 0
\(841\) 2.70683e13 1.86586
\(842\) 2.31593e13 1.58789
\(843\) −4.72537e12 −0.322264
\(844\) −1.72010e13 −1.16685
\(845\) 0 0
\(846\) −4.18279e12 −0.280737
\(847\) 1.02452e13 0.683983
\(848\) −2.24784e12 −0.149274
\(849\) −5.08296e12 −0.335762
\(850\) 0 0
\(851\) −1.80556e12 −0.118012
\(852\) −1.86884e13 −1.21505
\(853\) 1.84928e12 0.119600 0.0598002 0.998210i \(-0.480954\pi\)
0.0598002 + 0.998210i \(0.480954\pi\)
\(854\) 1.96659e13 1.26518
\(855\) 0 0
\(856\) 1.10958e13 0.706361
\(857\) 2.34715e13 1.48637 0.743185 0.669086i \(-0.233314\pi\)
0.743185 + 0.669086i \(0.233314\pi\)
\(858\) 1.31558e12 0.0828749
\(859\) 1.25121e12 0.0784079 0.0392039 0.999231i \(-0.487518\pi\)
0.0392039 + 0.999231i \(0.487518\pi\)
\(860\) 0 0
\(861\) 4.25902e12 0.264116
\(862\) 1.04026e13 0.641739
\(863\) −2.38217e13 −1.46192 −0.730961 0.682419i \(-0.760928\pi\)
−0.730961 + 0.682419i \(0.760928\pi\)
\(864\) 5.06768e12 0.309384
\(865\) 0 0
\(866\) −4.40928e13 −2.66402
\(867\) 5.10563e12 0.306876
\(868\) −3.72173e13 −2.22539
\(869\) 5.05575e12 0.300744
\(870\) 0 0
\(871\) −2.88564e12 −0.169887
\(872\) −2.75478e13 −1.61348
\(873\) −2.18084e13 −1.27075
\(874\) 1.50453e13 0.872168
\(875\) 0 0
\(876\) −1.42249e13 −0.816173
\(877\) 1.35717e13 0.774707 0.387353 0.921931i \(-0.373389\pi\)
0.387353 + 0.921931i \(0.373389\pi\)
\(878\) −6.79601e12 −0.385948
\(879\) 1.08375e13 0.612323
\(880\) 0 0
\(881\) 2.38436e12 0.133346 0.0666730 0.997775i \(-0.478762\pi\)
0.0666730 + 0.997775i \(0.478762\pi\)
\(882\) −3.30662e12 −0.183982
\(883\) 5.99779e11 0.0332023 0.0166011 0.999862i \(-0.494715\pi\)
0.0166011 + 0.999862i \(0.494715\pi\)
\(884\) 2.94999e12 0.162474
\(885\) 0 0
\(886\) −2.50103e12 −0.136354
\(887\) −1.02483e13 −0.555897 −0.277949 0.960596i \(-0.589654\pi\)
−0.277949 + 0.960596i \(0.589654\pi\)
\(888\) −3.27747e12 −0.176881
\(889\) 1.11766e13 0.600138
\(890\) 0 0
\(891\) −6.48579e12 −0.344757
\(892\) −1.45042e13 −0.767101
\(893\) 5.70040e12 0.299967
\(894\) 2.33111e12 0.122052
\(895\) 0 0
\(896\) −3.31065e13 −1.71604
\(897\) 5.95867e11 0.0307315
\(898\) 3.04642e13 1.56331
\(899\) 4.27050e13 2.18052
\(900\) 0 0
\(901\) 1.67378e12 0.0846130
\(902\) 1.43894e13 0.723791
\(903\) −1.14411e13 −0.572628
\(904\) −3.04937e13 −1.51863
\(905\) 0 0
\(906\) 7.95437e12 0.392219
\(907\) 2.43855e13 1.19646 0.598231 0.801323i \(-0.295870\pi\)
0.598231 + 0.801323i \(0.295870\pi\)
\(908\) −3.97416e13 −1.94026
\(909\) 1.57639e13 0.765820
\(910\) 0 0
\(911\) 2.90386e13 1.39683 0.698415 0.715693i \(-0.253889\pi\)
0.698415 + 0.715693i \(0.253889\pi\)
\(912\) 6.35855e12 0.304356
\(913\) 1.59949e12 0.0761841
\(914\) −3.40744e13 −1.61499
\(915\) 0 0
\(916\) −6.31867e13 −2.96549
\(917\) −2.05809e13 −0.961173
\(918\) 7.40630e12 0.344199
\(919\) −1.41793e13 −0.655746 −0.327873 0.944722i \(-0.606332\pi\)
−0.327873 + 0.944722i \(0.606332\pi\)
\(920\) 0 0
\(921\) −1.06773e12 −0.0488984
\(922\) −8.87378e12 −0.404408
\(923\) −1.17992e13 −0.535111
\(924\) −6.78146e12 −0.306055
\(925\) 0 0
\(926\) −5.89402e13 −2.63428
\(927\) −7.25212e12 −0.322557
\(928\) 1.84087e13 0.814812
\(929\) −3.56246e13 −1.56920 −0.784601 0.620001i \(-0.787132\pi\)
−0.784601 + 0.620001i \(0.787132\pi\)
\(930\) 0 0
\(931\) 4.50634e12 0.196585
\(932\) 4.90621e13 2.12998
\(933\) 1.30136e13 0.562250
\(934\) 6.23355e13 2.68025
\(935\) 0 0
\(936\) −8.20568e12 −0.349441
\(937\) −9.35575e12 −0.396506 −0.198253 0.980151i \(-0.563527\pi\)
−0.198253 + 0.980151i \(0.563527\pi\)
\(938\) 2.29353e13 0.967366
\(939\) −3.78610e12 −0.158927
\(940\) 0 0
\(941\) 1.23413e13 0.513106 0.256553 0.966530i \(-0.417413\pi\)
0.256553 + 0.966530i \(0.417413\pi\)
\(942\) 2.57673e13 1.06620
\(943\) 6.51742e12 0.268394
\(944\) −2.24269e13 −0.919168
\(945\) 0 0
\(946\) −3.86546e13 −1.56924
\(947\) −2.93722e12 −0.118676 −0.0593379 0.998238i \(-0.518899\pi\)
−0.0593379 + 0.998238i \(0.518899\pi\)
\(948\) 9.07366e12 0.364876
\(949\) −8.98110e12 −0.359444
\(950\) 0 0
\(951\) 9.72086e12 0.385383
\(952\) −1.07411e13 −0.423819
\(953\) 3.90544e12 0.153374 0.0766870 0.997055i \(-0.475566\pi\)
0.0766870 + 0.997055i \(0.475566\pi\)
\(954\) −1.01631e13 −0.397247
\(955\) 0 0
\(956\) 7.04733e13 2.72875
\(957\) 7.78137e12 0.299883
\(958\) −5.89375e13 −2.26072
\(959\) 3.00219e13 1.14618
\(960\) 0 0
\(961\) 1.74255e13 0.659069
\(962\) −4.51703e12 −0.170045
\(963\) 1.16802e13 0.437655
\(964\) 5.43139e13 2.02565
\(965\) 0 0
\(966\) −4.73599e12 −0.174990
\(967\) 1.06697e13 0.392405 0.196203 0.980563i \(-0.437139\pi\)
0.196203 + 0.980563i \(0.437139\pi\)
\(968\) 2.84587e13 1.04178
\(969\) −4.73469e12 −0.172518
\(970\) 0 0
\(971\) −3.10264e12 −0.112007 −0.0560034 0.998431i \(-0.517836\pi\)
−0.0560034 + 0.998431i \(0.517836\pi\)
\(972\) −4.46506e13 −1.60446
\(973\) −2.13939e13 −0.765213
\(974\) 9.74972e12 0.347118
\(975\) 0 0
\(976\) 1.27185e13 0.448655
\(977\) 4.66167e13 1.63688 0.818438 0.574594i \(-0.194840\pi\)
0.818438 + 0.574594i \(0.194840\pi\)
\(978\) −1.32455e12 −0.0462961
\(979\) −8.74824e11 −0.0304368
\(980\) 0 0
\(981\) −2.89986e13 −0.999694
\(982\) −6.04614e13 −2.07480
\(983\) 3.22316e13 1.10101 0.550505 0.834832i \(-0.314435\pi\)
0.550505 + 0.834832i \(0.314435\pi\)
\(984\) 1.18305e13 0.402278
\(985\) 0 0
\(986\) 2.69039e13 0.906502
\(987\) −1.79438e12 −0.0601849
\(988\) 2.44112e13 0.815047
\(989\) −1.75079e13 −0.581903
\(990\) 0 0
\(991\) −2.25008e13 −0.741081 −0.370540 0.928816i \(-0.620828\pi\)
−0.370540 + 0.928816i \(0.620828\pi\)
\(992\) 1.89088e13 0.619957
\(993\) −1.55192e13 −0.506520
\(994\) 9.37807e13 3.04701
\(995\) 0 0
\(996\) 2.87064e12 0.0924299
\(997\) 3.13756e13 1.00569 0.502844 0.864377i \(-0.332287\pi\)
0.502844 + 0.864377i \(0.332287\pi\)
\(998\) 2.40442e13 0.767224
\(999\) −7.35495e12 −0.233633
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 325.10.a.b.1.1 5
5.4 even 2 13.10.a.b.1.5 5
15.14 odd 2 117.10.a.e.1.1 5
20.19 odd 2 208.10.a.h.1.3 5
65.64 even 2 169.10.a.b.1.1 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
13.10.a.b.1.5 5 5.4 even 2
117.10.a.e.1.1 5 15.14 odd 2
169.10.a.b.1.1 5 65.64 even 2
208.10.a.h.1.3 5 20.19 odd 2
325.10.a.b.1.1 5 1.1 even 1 trivial