Properties

Label 1450.2.a.k.1.1
Level $1450$
Weight $2$
Character 1450.1
Self dual yes
Analytic conductor $11.578$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1450,2,Mod(1,1450)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1450, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1450.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1450 = 2 \cdot 5^{2} \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1450.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: \(11.5783082931\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{10})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 290)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-0.618034\) of defining polynomial
Character \(\chi\) \(=\) 1450.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} +0.381966 q^{3} +1.00000 q^{4} -0.381966 q^{6} +2.61803 q^{7} -1.00000 q^{8} -2.85410 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} +0.381966 q^{3} +1.00000 q^{4} -0.381966 q^{6} +2.61803 q^{7} -1.00000 q^{8} -2.85410 q^{9} +2.00000 q^{11} +0.381966 q^{12} +0.381966 q^{13} -2.61803 q^{14} +1.00000 q^{16} +3.38197 q^{17} +2.85410 q^{18} -2.00000 q^{19} +1.00000 q^{21} -2.00000 q^{22} +4.85410 q^{23} -0.381966 q^{24} -0.381966 q^{26} -2.23607 q^{27} +2.61803 q^{28} -1.00000 q^{29} -3.85410 q^{31} -1.00000 q^{32} +0.763932 q^{33} -3.38197 q^{34} -2.85410 q^{36} +8.94427 q^{37} +2.00000 q^{38} +0.145898 q^{39} +7.70820 q^{41} -1.00000 q^{42} +3.38197 q^{43} +2.00000 q^{44} -4.85410 q^{46} +1.52786 q^{47} +0.381966 q^{48} -0.145898 q^{49} +1.29180 q^{51} +0.381966 q^{52} +9.38197 q^{53} +2.23607 q^{54} -2.61803 q^{56} -0.763932 q^{57} +1.00000 q^{58} +2.85410 q^{59} -10.8541 q^{61} +3.85410 q^{62} -7.47214 q^{63} +1.00000 q^{64} -0.763932 q^{66} -8.94427 q^{67} +3.38197 q^{68} +1.85410 q^{69} -5.70820 q^{71} +2.85410 q^{72} -5.56231 q^{73} -8.94427 q^{74} -2.00000 q^{76} +5.23607 q^{77} -0.145898 q^{78} +6.56231 q^{79} +7.70820 q^{81} -7.70820 q^{82} +14.9443 q^{83} +1.00000 q^{84} -3.38197 q^{86} -0.381966 q^{87} -2.00000 q^{88} +2.00000 q^{89} +1.00000 q^{91} +4.85410 q^{92} -1.47214 q^{93} -1.52786 q^{94} -0.381966 q^{96} +7.14590 q^{97} +0.145898 q^{98} -5.70820 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} + 3 q^{3} + 2 q^{4} - 3 q^{6} + 3 q^{7} - 2 q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{2} + 3 q^{3} + 2 q^{4} - 3 q^{6} + 3 q^{7} - 2 q^{8} + q^{9} + 4 q^{11} + 3 q^{12} + 3 q^{13} - 3 q^{14} + 2 q^{16} + 9 q^{17} - q^{18} - 4 q^{19} + 2 q^{21} - 4 q^{22} + 3 q^{23} - 3 q^{24} - 3 q^{26} + 3 q^{28} - 2 q^{29} - q^{31} - 2 q^{32} + 6 q^{33} - 9 q^{34} + q^{36} + 4 q^{38} + 7 q^{39} + 2 q^{41} - 2 q^{42} + 9 q^{43} + 4 q^{44} - 3 q^{46} + 12 q^{47} + 3 q^{48} - 7 q^{49} + 16 q^{51} + 3 q^{52} + 21 q^{53} - 3 q^{56} - 6 q^{57} + 2 q^{58} - q^{59} - 15 q^{61} + q^{62} - 6 q^{63} + 2 q^{64} - 6 q^{66} + 9 q^{68} - 3 q^{69} + 2 q^{71} - q^{72} + 9 q^{73} - 4 q^{76} + 6 q^{77} - 7 q^{78} - 7 q^{79} + 2 q^{81} - 2 q^{82} + 12 q^{83} + 2 q^{84} - 9 q^{86} - 3 q^{87} - 4 q^{88} + 4 q^{89} + 2 q^{91} + 3 q^{92} + 6 q^{93} - 12 q^{94} - 3 q^{96} + 21 q^{97} + 7 q^{98} + 2 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) 0.381966 0.220528 0.110264 0.993902i \(-0.464830\pi\)
0.110264 + 0.993902i \(0.464830\pi\)
\(4\) 1.00000 0.500000
\(5\) 0 0
\(6\) −0.381966 −0.155937
\(7\) 2.61803 0.989524 0.494762 0.869029i \(-0.335255\pi\)
0.494762 + 0.869029i \(0.335255\pi\)
\(8\) −1.00000 −0.353553
\(9\) −2.85410 −0.951367
\(10\) 0 0
\(11\) 2.00000 0.603023 0.301511 0.953463i \(-0.402509\pi\)
0.301511 + 0.953463i \(0.402509\pi\)
\(12\) 0.381966 0.110264
\(13\) 0.381966 0.105938 0.0529692 0.998596i \(-0.483131\pi\)
0.0529692 + 0.998596i \(0.483131\pi\)
\(14\) −2.61803 −0.699699
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 3.38197 0.820247 0.410124 0.912030i \(-0.365486\pi\)
0.410124 + 0.912030i \(0.365486\pi\)
\(18\) 2.85410 0.672718
\(19\) −2.00000 −0.458831 −0.229416 0.973329i \(-0.573682\pi\)
−0.229416 + 0.973329i \(0.573682\pi\)
\(20\) 0 0
\(21\) 1.00000 0.218218
\(22\) −2.00000 −0.426401
\(23\) 4.85410 1.01215 0.506075 0.862489i \(-0.331096\pi\)
0.506075 + 0.862489i \(0.331096\pi\)
\(24\) −0.381966 −0.0779685
\(25\) 0 0
\(26\) −0.381966 −0.0749097
\(27\) −2.23607 −0.430331
\(28\) 2.61803 0.494762
\(29\) −1.00000 −0.185695
\(30\) 0 0
\(31\) −3.85410 −0.692217 −0.346109 0.938194i \(-0.612497\pi\)
−0.346109 + 0.938194i \(0.612497\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0.763932 0.132983
\(34\) −3.38197 −0.580002
\(35\) 0 0
\(36\) −2.85410 −0.475684
\(37\) 8.94427 1.47043 0.735215 0.677834i \(-0.237081\pi\)
0.735215 + 0.677834i \(0.237081\pi\)
\(38\) 2.00000 0.324443
\(39\) 0.145898 0.0233624
\(40\) 0 0
\(41\) 7.70820 1.20382 0.601910 0.798564i \(-0.294407\pi\)
0.601910 + 0.798564i \(0.294407\pi\)
\(42\) −1.00000 −0.154303
\(43\) 3.38197 0.515745 0.257872 0.966179i \(-0.416979\pi\)
0.257872 + 0.966179i \(0.416979\pi\)
\(44\) 2.00000 0.301511
\(45\) 0 0
\(46\) −4.85410 −0.715698
\(47\) 1.52786 0.222862 0.111431 0.993772i \(-0.464457\pi\)
0.111431 + 0.993772i \(0.464457\pi\)
\(48\) 0.381966 0.0551320
\(49\) −0.145898 −0.0208426
\(50\) 0 0
\(51\) 1.29180 0.180888
\(52\) 0.381966 0.0529692
\(53\) 9.38197 1.28871 0.644356 0.764726i \(-0.277125\pi\)
0.644356 + 0.764726i \(0.277125\pi\)
\(54\) 2.23607 0.304290
\(55\) 0 0
\(56\) −2.61803 −0.349850
\(57\) −0.763932 −0.101185
\(58\) 1.00000 0.131306
\(59\) 2.85410 0.371572 0.185786 0.982590i \(-0.440517\pi\)
0.185786 + 0.982590i \(0.440517\pi\)
\(60\) 0 0
\(61\) −10.8541 −1.38973 −0.694863 0.719142i \(-0.744535\pi\)
−0.694863 + 0.719142i \(0.744535\pi\)
\(62\) 3.85410 0.489471
\(63\) −7.47214 −0.941401
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) −0.763932 −0.0940335
\(67\) −8.94427 −1.09272 −0.546358 0.837552i \(-0.683986\pi\)
−0.546358 + 0.837552i \(0.683986\pi\)
\(68\) 3.38197 0.410124
\(69\) 1.85410 0.223208
\(70\) 0 0
\(71\) −5.70820 −0.677439 −0.338720 0.940887i \(-0.609994\pi\)
−0.338720 + 0.940887i \(0.609994\pi\)
\(72\) 2.85410 0.336359
\(73\) −5.56231 −0.651019 −0.325509 0.945539i \(-0.605536\pi\)
−0.325509 + 0.945539i \(0.605536\pi\)
\(74\) −8.94427 −1.03975
\(75\) 0 0
\(76\) −2.00000 −0.229416
\(77\) 5.23607 0.596705
\(78\) −0.145898 −0.0165197
\(79\) 6.56231 0.738317 0.369159 0.929366i \(-0.379646\pi\)
0.369159 + 0.929366i \(0.379646\pi\)
\(80\) 0 0
\(81\) 7.70820 0.856467
\(82\) −7.70820 −0.851229
\(83\) 14.9443 1.64035 0.820173 0.572115i \(-0.193877\pi\)
0.820173 + 0.572115i \(0.193877\pi\)
\(84\) 1.00000 0.109109
\(85\) 0 0
\(86\) −3.38197 −0.364687
\(87\) −0.381966 −0.0409511
\(88\) −2.00000 −0.213201
\(89\) 2.00000 0.212000 0.106000 0.994366i \(-0.466196\pi\)
0.106000 + 0.994366i \(0.466196\pi\)
\(90\) 0 0
\(91\) 1.00000 0.104828
\(92\) 4.85410 0.506075
\(93\) −1.47214 −0.152653
\(94\) −1.52786 −0.157587
\(95\) 0 0
\(96\) −0.381966 −0.0389842
\(97\) 7.14590 0.725556 0.362778 0.931876i \(-0.381828\pi\)
0.362778 + 0.931876i \(0.381828\pi\)
\(98\) 0.145898 0.0147379
\(99\) −5.70820 −0.573696
\(100\) 0 0
\(101\) 10.8541 1.08002 0.540012 0.841657i \(-0.318420\pi\)
0.540012 + 0.841657i \(0.318420\pi\)
\(102\) −1.29180 −0.127907
\(103\) 7.41641 0.730760 0.365380 0.930858i \(-0.380939\pi\)
0.365380 + 0.930858i \(0.380939\pi\)
\(104\) −0.381966 −0.0374548
\(105\) 0 0
\(106\) −9.38197 −0.911257
\(107\) −14.1803 −1.37087 −0.685433 0.728136i \(-0.740387\pi\)
−0.685433 + 0.728136i \(0.740387\pi\)
\(108\) −2.23607 −0.215166
\(109\) 19.4164 1.85975 0.929877 0.367870i \(-0.119913\pi\)
0.929877 + 0.367870i \(0.119913\pi\)
\(110\) 0 0
\(111\) 3.41641 0.324271
\(112\) 2.61803 0.247381
\(113\) 4.09017 0.384771 0.192385 0.981319i \(-0.438378\pi\)
0.192385 + 0.981319i \(0.438378\pi\)
\(114\) 0.763932 0.0715488
\(115\) 0 0
\(116\) −1.00000 −0.0928477
\(117\) −1.09017 −0.100786
\(118\) −2.85410 −0.262741
\(119\) 8.85410 0.811654
\(120\) 0 0
\(121\) −7.00000 −0.636364
\(122\) 10.8541 0.982684
\(123\) 2.94427 0.265476
\(124\) −3.85410 −0.346109
\(125\) 0 0
\(126\) 7.47214 0.665671
\(127\) −8.29180 −0.735778 −0.367889 0.929870i \(-0.619919\pi\)
−0.367889 + 0.929870i \(0.619919\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 1.29180 0.113736
\(130\) 0 0
\(131\) 21.4164 1.87116 0.935580 0.353114i \(-0.114877\pi\)
0.935580 + 0.353114i \(0.114877\pi\)
\(132\) 0.763932 0.0664917
\(133\) −5.23607 −0.454025
\(134\) 8.94427 0.772667
\(135\) 0 0
\(136\) −3.38197 −0.290001
\(137\) −4.85410 −0.414714 −0.207357 0.978265i \(-0.566486\pi\)
−0.207357 + 0.978265i \(0.566486\pi\)
\(138\) −1.85410 −0.157832
\(139\) −5.56231 −0.471789 −0.235894 0.971779i \(-0.575802\pi\)
−0.235894 + 0.971779i \(0.575802\pi\)
\(140\) 0 0
\(141\) 0.583592 0.0491473
\(142\) 5.70820 0.479022
\(143\) 0.763932 0.0638832
\(144\) −2.85410 −0.237842
\(145\) 0 0
\(146\) 5.56231 0.460340
\(147\) −0.0557281 −0.00459638
\(148\) 8.94427 0.735215
\(149\) −21.4164 −1.75450 −0.877250 0.480033i \(-0.840625\pi\)
−0.877250 + 0.480033i \(0.840625\pi\)
\(150\) 0 0
\(151\) 4.29180 0.349261 0.174631 0.984634i \(-0.444127\pi\)
0.174631 + 0.984634i \(0.444127\pi\)
\(152\) 2.00000 0.162221
\(153\) −9.65248 −0.780356
\(154\) −5.23607 −0.421934
\(155\) 0 0
\(156\) 0.145898 0.0116812
\(157\) 9.70820 0.774799 0.387400 0.921912i \(-0.373373\pi\)
0.387400 + 0.921912i \(0.373373\pi\)
\(158\) −6.56231 −0.522069
\(159\) 3.58359 0.284197
\(160\) 0 0
\(161\) 12.7082 1.00155
\(162\) −7.70820 −0.605614
\(163\) 7.41641 0.580898 0.290449 0.956890i \(-0.406195\pi\)
0.290449 + 0.956890i \(0.406195\pi\)
\(164\) 7.70820 0.601910
\(165\) 0 0
\(166\) −14.9443 −1.15990
\(167\) 2.61803 0.202590 0.101295 0.994856i \(-0.467701\pi\)
0.101295 + 0.994856i \(0.467701\pi\)
\(168\) −1.00000 −0.0771517
\(169\) −12.8541 −0.988777
\(170\) 0 0
\(171\) 5.70820 0.436517
\(172\) 3.38197 0.257872
\(173\) 17.5066 1.33100 0.665500 0.746398i \(-0.268218\pi\)
0.665500 + 0.746398i \(0.268218\pi\)
\(174\) 0.381966 0.0289568
\(175\) 0 0
\(176\) 2.00000 0.150756
\(177\) 1.09017 0.0819422
\(178\) −2.00000 −0.149906
\(179\) 9.14590 0.683597 0.341798 0.939773i \(-0.388964\pi\)
0.341798 + 0.939773i \(0.388964\pi\)
\(180\) 0 0
\(181\) −10.0000 −0.743294 −0.371647 0.928374i \(-0.621207\pi\)
−0.371647 + 0.928374i \(0.621207\pi\)
\(182\) −1.00000 −0.0741249
\(183\) −4.14590 −0.306474
\(184\) −4.85410 −0.357849
\(185\) 0 0
\(186\) 1.47214 0.107942
\(187\) 6.76393 0.494628
\(188\) 1.52786 0.111431
\(189\) −5.85410 −0.425823
\(190\) 0 0
\(191\) 23.2705 1.68379 0.841897 0.539637i \(-0.181439\pi\)
0.841897 + 0.539637i \(0.181439\pi\)
\(192\) 0.381966 0.0275660
\(193\) −2.56231 −0.184439 −0.0922194 0.995739i \(-0.529396\pi\)
−0.0922194 + 0.995739i \(0.529396\pi\)
\(194\) −7.14590 −0.513046
\(195\) 0 0
\(196\) −0.145898 −0.0104213
\(197\) 9.38197 0.668437 0.334219 0.942496i \(-0.391528\pi\)
0.334219 + 0.942496i \(0.391528\pi\)
\(198\) 5.70820 0.405664
\(199\) 19.1246 1.35571 0.677854 0.735197i \(-0.262910\pi\)
0.677854 + 0.735197i \(0.262910\pi\)
\(200\) 0 0
\(201\) −3.41641 −0.240975
\(202\) −10.8541 −0.763692
\(203\) −2.61803 −0.183750
\(204\) 1.29180 0.0904438
\(205\) 0 0
\(206\) −7.41641 −0.516726
\(207\) −13.8541 −0.962927
\(208\) 0.381966 0.0264846
\(209\) −4.00000 −0.276686
\(210\) 0 0
\(211\) −11.4164 −0.785938 −0.392969 0.919552i \(-0.628552\pi\)
−0.392969 + 0.919552i \(0.628552\pi\)
\(212\) 9.38197 0.644356
\(213\) −2.18034 −0.149394
\(214\) 14.1803 0.969348
\(215\) 0 0
\(216\) 2.23607 0.152145
\(217\) −10.0902 −0.684965
\(218\) −19.4164 −1.31505
\(219\) −2.12461 −0.143568
\(220\) 0 0
\(221\) 1.29180 0.0868956
\(222\) −3.41641 −0.229294
\(223\) −26.5623 −1.77874 −0.889372 0.457185i \(-0.848858\pi\)
−0.889372 + 0.457185i \(0.848858\pi\)
\(224\) −2.61803 −0.174925
\(225\) 0 0
\(226\) −4.09017 −0.272074
\(227\) 16.4721 1.09329 0.546647 0.837363i \(-0.315904\pi\)
0.546647 + 0.837363i \(0.315904\pi\)
\(228\) −0.763932 −0.0505926
\(229\) −8.14590 −0.538296 −0.269148 0.963099i \(-0.586742\pi\)
−0.269148 + 0.963099i \(0.586742\pi\)
\(230\) 0 0
\(231\) 2.00000 0.131590
\(232\) 1.00000 0.0656532
\(233\) −11.2361 −0.736099 −0.368050 0.929806i \(-0.619974\pi\)
−0.368050 + 0.929806i \(0.619974\pi\)
\(234\) 1.09017 0.0712666
\(235\) 0 0
\(236\) 2.85410 0.185786
\(237\) 2.50658 0.162820
\(238\) −8.85410 −0.573926
\(239\) 5.70820 0.369233 0.184617 0.982811i \(-0.440896\pi\)
0.184617 + 0.982811i \(0.440896\pi\)
\(240\) 0 0
\(241\) −8.85410 −0.570343 −0.285171 0.958477i \(-0.592051\pi\)
−0.285171 + 0.958477i \(0.592051\pi\)
\(242\) 7.00000 0.449977
\(243\) 9.65248 0.619207
\(244\) −10.8541 −0.694863
\(245\) 0 0
\(246\) −2.94427 −0.187720
\(247\) −0.763932 −0.0486078
\(248\) 3.85410 0.244736
\(249\) 5.70820 0.361743
\(250\) 0 0
\(251\) −29.1246 −1.83833 −0.919165 0.393874i \(-0.871135\pi\)
−0.919165 + 0.393874i \(0.871135\pi\)
\(252\) −7.47214 −0.470700
\(253\) 9.70820 0.610350
\(254\) 8.29180 0.520274
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 2.29180 0.142958 0.0714792 0.997442i \(-0.477228\pi\)
0.0714792 + 0.997442i \(0.477228\pi\)
\(258\) −1.29180 −0.0804237
\(259\) 23.4164 1.45502
\(260\) 0 0
\(261\) 2.85410 0.176664
\(262\) −21.4164 −1.32311
\(263\) −21.5967 −1.33171 −0.665856 0.746080i \(-0.731934\pi\)
−0.665856 + 0.746080i \(0.731934\pi\)
\(264\) −0.763932 −0.0470168
\(265\) 0 0
\(266\) 5.23607 0.321044
\(267\) 0.763932 0.0467519
\(268\) −8.94427 −0.546358
\(269\) 3.27051 0.199407 0.0997033 0.995017i \(-0.468211\pi\)
0.0997033 + 0.995017i \(0.468211\pi\)
\(270\) 0 0
\(271\) 23.4164 1.42245 0.711223 0.702967i \(-0.248142\pi\)
0.711223 + 0.702967i \(0.248142\pi\)
\(272\) 3.38197 0.205062
\(273\) 0.381966 0.0231176
\(274\) 4.85410 0.293247
\(275\) 0 0
\(276\) 1.85410 0.111604
\(277\) −30.0000 −1.80253 −0.901263 0.433273i \(-0.857359\pi\)
−0.901263 + 0.433273i \(0.857359\pi\)
\(278\) 5.56231 0.333605
\(279\) 11.0000 0.658553
\(280\) 0 0
\(281\) −17.5623 −1.04768 −0.523840 0.851817i \(-0.675501\pi\)
−0.523840 + 0.851817i \(0.675501\pi\)
\(282\) −0.583592 −0.0347524
\(283\) −6.76393 −0.402074 −0.201037 0.979584i \(-0.564431\pi\)
−0.201037 + 0.979584i \(0.564431\pi\)
\(284\) −5.70820 −0.338720
\(285\) 0 0
\(286\) −0.763932 −0.0451722
\(287\) 20.1803 1.19121
\(288\) 2.85410 0.168180
\(289\) −5.56231 −0.327194
\(290\) 0 0
\(291\) 2.72949 0.160006
\(292\) −5.56231 −0.325509
\(293\) −19.4164 −1.13432 −0.567159 0.823608i \(-0.691958\pi\)
−0.567159 + 0.823608i \(0.691958\pi\)
\(294\) 0.0557281 0.00325013
\(295\) 0 0
\(296\) −8.94427 −0.519875
\(297\) −4.47214 −0.259500
\(298\) 21.4164 1.24062
\(299\) 1.85410 0.107225
\(300\) 0 0
\(301\) 8.85410 0.510342
\(302\) −4.29180 −0.246965
\(303\) 4.14590 0.238176
\(304\) −2.00000 −0.114708
\(305\) 0 0
\(306\) 9.65248 0.551795
\(307\) −24.0000 −1.36975 −0.684876 0.728659i \(-0.740144\pi\)
−0.684876 + 0.728659i \(0.740144\pi\)
\(308\) 5.23607 0.298353
\(309\) 2.83282 0.161153
\(310\) 0 0
\(311\) −34.5623 −1.95985 −0.979924 0.199370i \(-0.936110\pi\)
−0.979924 + 0.199370i \(0.936110\pi\)
\(312\) −0.145898 −0.00825985
\(313\) −12.7639 −0.721460 −0.360730 0.932670i \(-0.617472\pi\)
−0.360730 + 0.932670i \(0.617472\pi\)
\(314\) −9.70820 −0.547866
\(315\) 0 0
\(316\) 6.56231 0.369159
\(317\) −20.1803 −1.13344 −0.566720 0.823910i \(-0.691788\pi\)
−0.566720 + 0.823910i \(0.691788\pi\)
\(318\) −3.58359 −0.200958
\(319\) −2.00000 −0.111979
\(320\) 0 0
\(321\) −5.41641 −0.302314
\(322\) −12.7082 −0.708201
\(323\) −6.76393 −0.376355
\(324\) 7.70820 0.428234
\(325\) 0 0
\(326\) −7.41641 −0.410757
\(327\) 7.41641 0.410128
\(328\) −7.70820 −0.425614
\(329\) 4.00000 0.220527
\(330\) 0 0
\(331\) 27.1246 1.49090 0.745452 0.666560i \(-0.232234\pi\)
0.745452 + 0.666560i \(0.232234\pi\)
\(332\) 14.9443 0.820173
\(333\) −25.5279 −1.39892
\(334\) −2.61803 −0.143252
\(335\) 0 0
\(336\) 1.00000 0.0545545
\(337\) 8.61803 0.469454 0.234727 0.972061i \(-0.424580\pi\)
0.234727 + 0.972061i \(0.424580\pi\)
\(338\) 12.8541 0.699171
\(339\) 1.56231 0.0848528
\(340\) 0 0
\(341\) −7.70820 −0.417423
\(342\) −5.70820 −0.308664
\(343\) −18.7082 −1.01015
\(344\) −3.38197 −0.182343
\(345\) 0 0
\(346\) −17.5066 −0.941159
\(347\) −6.00000 −0.322097 −0.161048 0.986947i \(-0.551488\pi\)
−0.161048 + 0.986947i \(0.551488\pi\)
\(348\) −0.381966 −0.0204755
\(349\) −24.8328 −1.32927 −0.664635 0.747168i \(-0.731413\pi\)
−0.664635 + 0.747168i \(0.731413\pi\)
\(350\) 0 0
\(351\) −0.854102 −0.0455886
\(352\) −2.00000 −0.106600
\(353\) −3.05573 −0.162640 −0.0813200 0.996688i \(-0.525914\pi\)
−0.0813200 + 0.996688i \(0.525914\pi\)
\(354\) −1.09017 −0.0579419
\(355\) 0 0
\(356\) 2.00000 0.106000
\(357\) 3.38197 0.178993
\(358\) −9.14590 −0.483376
\(359\) 0.270510 0.0142770 0.00713848 0.999975i \(-0.497728\pi\)
0.00713848 + 0.999975i \(0.497728\pi\)
\(360\) 0 0
\(361\) −15.0000 −0.789474
\(362\) 10.0000 0.525588
\(363\) −2.67376 −0.140336
\(364\) 1.00000 0.0524142
\(365\) 0 0
\(366\) 4.14590 0.216710
\(367\) 35.1246 1.83349 0.916745 0.399473i \(-0.130807\pi\)
0.916745 + 0.399473i \(0.130807\pi\)
\(368\) 4.85410 0.253038
\(369\) −22.0000 −1.14527
\(370\) 0 0
\(371\) 24.5623 1.27521
\(372\) −1.47214 −0.0763267
\(373\) 15.2705 0.790677 0.395339 0.918535i \(-0.370627\pi\)
0.395339 + 0.918535i \(0.370627\pi\)
\(374\) −6.76393 −0.349755
\(375\) 0 0
\(376\) −1.52786 −0.0787936
\(377\) −0.381966 −0.0196723
\(378\) 5.85410 0.301103
\(379\) 26.0000 1.33553 0.667765 0.744372i \(-0.267251\pi\)
0.667765 + 0.744372i \(0.267251\pi\)
\(380\) 0 0
\(381\) −3.16718 −0.162260
\(382\) −23.2705 −1.19062
\(383\) −7.85410 −0.401326 −0.200663 0.979660i \(-0.564310\pi\)
−0.200663 + 0.979660i \(0.564310\pi\)
\(384\) −0.381966 −0.0194921
\(385\) 0 0
\(386\) 2.56231 0.130418
\(387\) −9.65248 −0.490663
\(388\) 7.14590 0.362778
\(389\) −20.8328 −1.05627 −0.528133 0.849162i \(-0.677108\pi\)
−0.528133 + 0.849162i \(0.677108\pi\)
\(390\) 0 0
\(391\) 16.4164 0.830213
\(392\) 0.145898 0.00736896
\(393\) 8.18034 0.412644
\(394\) −9.38197 −0.472657
\(395\) 0 0
\(396\) −5.70820 −0.286848
\(397\) −1.96556 −0.0986485 −0.0493243 0.998783i \(-0.515707\pi\)
−0.0493243 + 0.998783i \(0.515707\pi\)
\(398\) −19.1246 −0.958630
\(399\) −2.00000 −0.100125
\(400\) 0 0
\(401\) 21.9787 1.09756 0.548782 0.835965i \(-0.315092\pi\)
0.548782 + 0.835965i \(0.315092\pi\)
\(402\) 3.41641 0.170395
\(403\) −1.47214 −0.0733323
\(404\) 10.8541 0.540012
\(405\) 0 0
\(406\) 2.61803 0.129931
\(407\) 17.8885 0.886702
\(408\) −1.29180 −0.0639534
\(409\) −7.12461 −0.352289 −0.176145 0.984364i \(-0.556363\pi\)
−0.176145 + 0.984364i \(0.556363\pi\)
\(410\) 0 0
\(411\) −1.85410 −0.0914561
\(412\) 7.41641 0.365380
\(413\) 7.47214 0.367680
\(414\) 13.8541 0.680892
\(415\) 0 0
\(416\) −0.381966 −0.0187274
\(417\) −2.12461 −0.104043
\(418\) 4.00000 0.195646
\(419\) −21.8541 −1.06764 −0.533821 0.845597i \(-0.679245\pi\)
−0.533821 + 0.845597i \(0.679245\pi\)
\(420\) 0 0
\(421\) −10.0000 −0.487370 −0.243685 0.969854i \(-0.578356\pi\)
−0.243685 + 0.969854i \(0.578356\pi\)
\(422\) 11.4164 0.555742
\(423\) −4.36068 −0.212024
\(424\) −9.38197 −0.455629
\(425\) 0 0
\(426\) 2.18034 0.105638
\(427\) −28.4164 −1.37517
\(428\) −14.1803 −0.685433
\(429\) 0.291796 0.0140880
\(430\) 0 0
\(431\) −33.4164 −1.60961 −0.804806 0.593538i \(-0.797731\pi\)
−0.804806 + 0.593538i \(0.797731\pi\)
\(432\) −2.23607 −0.107583
\(433\) −16.4721 −0.791600 −0.395800 0.918337i \(-0.629533\pi\)
−0.395800 + 0.918337i \(0.629533\pi\)
\(434\) 10.0902 0.484344
\(435\) 0 0
\(436\) 19.4164 0.929877
\(437\) −9.70820 −0.464406
\(438\) 2.12461 0.101518
\(439\) −37.1246 −1.77186 −0.885931 0.463818i \(-0.846479\pi\)
−0.885931 + 0.463818i \(0.846479\pi\)
\(440\) 0 0
\(441\) 0.416408 0.0198289
\(442\) −1.29180 −0.0614445
\(443\) −1.90983 −0.0907388 −0.0453694 0.998970i \(-0.514446\pi\)
−0.0453694 + 0.998970i \(0.514446\pi\)
\(444\) 3.41641 0.162136
\(445\) 0 0
\(446\) 26.5623 1.25776
\(447\) −8.18034 −0.386917
\(448\) 2.61803 0.123690
\(449\) −4.58359 −0.216313 −0.108157 0.994134i \(-0.534495\pi\)
−0.108157 + 0.994134i \(0.534495\pi\)
\(450\) 0 0
\(451\) 15.4164 0.725930
\(452\) 4.09017 0.192385
\(453\) 1.63932 0.0770220
\(454\) −16.4721 −0.773076
\(455\) 0 0
\(456\) 0.763932 0.0357744
\(457\) −14.2918 −0.668542 −0.334271 0.942477i \(-0.608490\pi\)
−0.334271 + 0.942477i \(0.608490\pi\)
\(458\) 8.14590 0.380633
\(459\) −7.56231 −0.352978
\(460\) 0 0
\(461\) −24.1459 −1.12459 −0.562293 0.826938i \(-0.690081\pi\)
−0.562293 + 0.826938i \(0.690081\pi\)
\(462\) −2.00000 −0.0930484
\(463\) −17.8885 −0.831351 −0.415676 0.909513i \(-0.636455\pi\)
−0.415676 + 0.909513i \(0.636455\pi\)
\(464\) −1.00000 −0.0464238
\(465\) 0 0
\(466\) 11.2361 0.520501
\(467\) 34.7984 1.61028 0.805138 0.593087i \(-0.202091\pi\)
0.805138 + 0.593087i \(0.202091\pi\)
\(468\) −1.09017 −0.0503931
\(469\) −23.4164 −1.08127
\(470\) 0 0
\(471\) 3.70820 0.170865
\(472\) −2.85410 −0.131371
\(473\) 6.76393 0.311006
\(474\) −2.50658 −0.115131
\(475\) 0 0
\(476\) 8.85410 0.405827
\(477\) −26.7771 −1.22604
\(478\) −5.70820 −0.261087
\(479\) −18.9787 −0.867160 −0.433580 0.901115i \(-0.642750\pi\)
−0.433580 + 0.901115i \(0.642750\pi\)
\(480\) 0 0
\(481\) 3.41641 0.155775
\(482\) 8.85410 0.403293
\(483\) 4.85410 0.220869
\(484\) −7.00000 −0.318182
\(485\) 0 0
\(486\) −9.65248 −0.437845
\(487\) 38.5623 1.74742 0.873712 0.486443i \(-0.161706\pi\)
0.873712 + 0.486443i \(0.161706\pi\)
\(488\) 10.8541 0.491342
\(489\) 2.83282 0.128104
\(490\) 0 0
\(491\) −6.29180 −0.283945 −0.141972 0.989871i \(-0.545344\pi\)
−0.141972 + 0.989871i \(0.545344\pi\)
\(492\) 2.94427 0.132738
\(493\) −3.38197 −0.152316
\(494\) 0.763932 0.0343709
\(495\) 0 0
\(496\) −3.85410 −0.173054
\(497\) −14.9443 −0.670342
\(498\) −5.70820 −0.255791
\(499\) 28.5623 1.27862 0.639312 0.768947i \(-0.279219\pi\)
0.639312 + 0.768947i \(0.279219\pi\)
\(500\) 0 0
\(501\) 1.00000 0.0446767
\(502\) 29.1246 1.29990
\(503\) 32.9443 1.46891 0.734456 0.678656i \(-0.237437\pi\)
0.734456 + 0.678656i \(0.237437\pi\)
\(504\) 7.47214 0.332835
\(505\) 0 0
\(506\) −9.70820 −0.431582
\(507\) −4.90983 −0.218053
\(508\) −8.29180 −0.367889
\(509\) −21.1246 −0.936332 −0.468166 0.883641i \(-0.655085\pi\)
−0.468166 + 0.883641i \(0.655085\pi\)
\(510\) 0 0
\(511\) −14.5623 −0.644198
\(512\) −1.00000 −0.0441942
\(513\) 4.47214 0.197450
\(514\) −2.29180 −0.101087
\(515\) 0 0
\(516\) 1.29180 0.0568682
\(517\) 3.05573 0.134391
\(518\) −23.4164 −1.02886
\(519\) 6.68692 0.293523
\(520\) 0 0
\(521\) −36.3951 −1.59450 −0.797250 0.603650i \(-0.793713\pi\)
−0.797250 + 0.603650i \(0.793713\pi\)
\(522\) −2.85410 −0.124921
\(523\) −11.1246 −0.486445 −0.243223 0.969970i \(-0.578205\pi\)
−0.243223 + 0.969970i \(0.578205\pi\)
\(524\) 21.4164 0.935580
\(525\) 0 0
\(526\) 21.5967 0.941663
\(527\) −13.0344 −0.567789
\(528\) 0.763932 0.0332459
\(529\) 0.562306 0.0244481
\(530\) 0 0
\(531\) −8.14590 −0.353502
\(532\) −5.23607 −0.227012
\(533\) 2.94427 0.127531
\(534\) −0.763932 −0.0330586
\(535\) 0 0
\(536\) 8.94427 0.386334
\(537\) 3.49342 0.150752
\(538\) −3.27051 −0.141002
\(539\) −0.291796 −0.0125685
\(540\) 0 0
\(541\) −27.5623 −1.18500 −0.592498 0.805572i \(-0.701858\pi\)
−0.592498 + 0.805572i \(0.701858\pi\)
\(542\) −23.4164 −1.00582
\(543\) −3.81966 −0.163917
\(544\) −3.38197 −0.145001
\(545\) 0 0
\(546\) −0.381966 −0.0163466
\(547\) −14.8328 −0.634205 −0.317103 0.948391i \(-0.602710\pi\)
−0.317103 + 0.948391i \(0.602710\pi\)
\(548\) −4.85410 −0.207357
\(549\) 30.9787 1.32214
\(550\) 0 0
\(551\) 2.00000 0.0852029
\(552\) −1.85410 −0.0789158
\(553\) 17.1803 0.730582
\(554\) 30.0000 1.27458
\(555\) 0 0
\(556\) −5.56231 −0.235894
\(557\) −7.79837 −0.330428 −0.165214 0.986258i \(-0.552831\pi\)
−0.165214 + 0.986258i \(0.552831\pi\)
\(558\) −11.0000 −0.465667
\(559\) 1.29180 0.0546372
\(560\) 0 0
\(561\) 2.58359 0.109079
\(562\) 17.5623 0.740821
\(563\) 33.3262 1.40453 0.702267 0.711914i \(-0.252171\pi\)
0.702267 + 0.711914i \(0.252171\pi\)
\(564\) 0.583592 0.0245737
\(565\) 0 0
\(566\) 6.76393 0.284309
\(567\) 20.1803 0.847495
\(568\) 5.70820 0.239511
\(569\) 12.0000 0.503066 0.251533 0.967849i \(-0.419065\pi\)
0.251533 + 0.967849i \(0.419065\pi\)
\(570\) 0 0
\(571\) −6.85410 −0.286835 −0.143418 0.989662i \(-0.545809\pi\)
−0.143418 + 0.989662i \(0.545809\pi\)
\(572\) 0.763932 0.0319416
\(573\) 8.88854 0.371324
\(574\) −20.1803 −0.842311
\(575\) 0 0
\(576\) −2.85410 −0.118921
\(577\) −28.0344 −1.16709 −0.583545 0.812081i \(-0.698335\pi\)
−0.583545 + 0.812081i \(0.698335\pi\)
\(578\) 5.56231 0.231361
\(579\) −0.978714 −0.0406740
\(580\) 0 0
\(581\) 39.1246 1.62316
\(582\) −2.72949 −0.113141
\(583\) 18.7639 0.777123
\(584\) 5.56231 0.230170
\(585\) 0 0
\(586\) 19.4164 0.802084
\(587\) 12.6525 0.522224 0.261112 0.965309i \(-0.415911\pi\)
0.261112 + 0.965309i \(0.415911\pi\)
\(588\) −0.0557281 −0.00229819
\(589\) 7.70820 0.317611
\(590\) 0 0
\(591\) 3.58359 0.147409
\(592\) 8.94427 0.367607
\(593\) 43.3050 1.77832 0.889161 0.457595i \(-0.151289\pi\)
0.889161 + 0.457595i \(0.151289\pi\)
\(594\) 4.47214 0.183494
\(595\) 0 0
\(596\) −21.4164 −0.877250
\(597\) 7.30495 0.298972
\(598\) −1.85410 −0.0758199
\(599\) 23.5623 0.962730 0.481365 0.876520i \(-0.340141\pi\)
0.481365 + 0.876520i \(0.340141\pi\)
\(600\) 0 0
\(601\) 5.70820 0.232842 0.116421 0.993200i \(-0.462858\pi\)
0.116421 + 0.993200i \(0.462858\pi\)
\(602\) −8.85410 −0.360866
\(603\) 25.5279 1.03957
\(604\) 4.29180 0.174631
\(605\) 0 0
\(606\) −4.14590 −0.168416
\(607\) 0.763932 0.0310070 0.0155035 0.999880i \(-0.495065\pi\)
0.0155035 + 0.999880i \(0.495065\pi\)
\(608\) 2.00000 0.0811107
\(609\) −1.00000 −0.0405220
\(610\) 0 0
\(611\) 0.583592 0.0236096
\(612\) −9.65248 −0.390178
\(613\) 13.7426 0.555060 0.277530 0.960717i \(-0.410484\pi\)
0.277530 + 0.960717i \(0.410484\pi\)
\(614\) 24.0000 0.968561
\(615\) 0 0
\(616\) −5.23607 −0.210967
\(617\) 32.6180 1.31315 0.656576 0.754260i \(-0.272004\pi\)
0.656576 + 0.754260i \(0.272004\pi\)
\(618\) −2.83282 −0.113953
\(619\) −7.41641 −0.298091 −0.149045 0.988830i \(-0.547620\pi\)
−0.149045 + 0.988830i \(0.547620\pi\)
\(620\) 0 0
\(621\) −10.8541 −0.435560
\(622\) 34.5623 1.38582
\(623\) 5.23607 0.209779
\(624\) 0.145898 0.00584060
\(625\) 0 0
\(626\) 12.7639 0.510149
\(627\) −1.52786 −0.0610170
\(628\) 9.70820 0.387400
\(629\) 30.2492 1.20612
\(630\) 0 0
\(631\) −34.8328 −1.38667 −0.693336 0.720614i \(-0.743860\pi\)
−0.693336 + 0.720614i \(0.743860\pi\)
\(632\) −6.56231 −0.261035
\(633\) −4.36068 −0.173321
\(634\) 20.1803 0.801464
\(635\) 0 0
\(636\) 3.58359 0.142099
\(637\) −0.0557281 −0.00220803
\(638\) 2.00000 0.0791808
\(639\) 16.2918 0.644493
\(640\) 0 0
\(641\) 0.291796 0.0115253 0.00576263 0.999983i \(-0.498166\pi\)
0.00576263 + 0.999983i \(0.498166\pi\)
\(642\) 5.41641 0.213769
\(643\) −16.5836 −0.653993 −0.326997 0.945026i \(-0.606037\pi\)
−0.326997 + 0.945026i \(0.606037\pi\)
\(644\) 12.7082 0.500773
\(645\) 0 0
\(646\) 6.76393 0.266123
\(647\) 13.5279 0.531835 0.265918 0.963996i \(-0.414325\pi\)
0.265918 + 0.963996i \(0.414325\pi\)
\(648\) −7.70820 −0.302807
\(649\) 5.70820 0.224067
\(650\) 0 0
\(651\) −3.85410 −0.151054
\(652\) 7.41641 0.290449
\(653\) 37.3050 1.45986 0.729928 0.683524i \(-0.239554\pi\)
0.729928 + 0.683524i \(0.239554\pi\)
\(654\) −7.41641 −0.290004
\(655\) 0 0
\(656\) 7.70820 0.300955
\(657\) 15.8754 0.619358
\(658\) −4.00000 −0.155936
\(659\) 40.5410 1.57925 0.789627 0.613587i \(-0.210274\pi\)
0.789627 + 0.613587i \(0.210274\pi\)
\(660\) 0 0
\(661\) 28.8328 1.12147 0.560733 0.827996i \(-0.310519\pi\)
0.560733 + 0.827996i \(0.310519\pi\)
\(662\) −27.1246 −1.05423
\(663\) 0.493422 0.0191629
\(664\) −14.9443 −0.579950
\(665\) 0 0
\(666\) 25.5279 0.989185
\(667\) −4.85410 −0.187952
\(668\) 2.61803 0.101295
\(669\) −10.1459 −0.392263
\(670\) 0 0
\(671\) −21.7082 −0.838036
\(672\) −1.00000 −0.0385758
\(673\) 26.9443 1.03863 0.519313 0.854584i \(-0.326188\pi\)
0.519313 + 0.854584i \(0.326188\pi\)
\(674\) −8.61803 −0.331954
\(675\) 0 0
\(676\) −12.8541 −0.494389
\(677\) 32.9443 1.26615 0.633076 0.774090i \(-0.281792\pi\)
0.633076 + 0.774090i \(0.281792\pi\)
\(678\) −1.56231 −0.0600000
\(679\) 18.7082 0.717955
\(680\) 0 0
\(681\) 6.29180 0.241102
\(682\) 7.70820 0.295162
\(683\) −23.7771 −0.909805 −0.454902 0.890541i \(-0.650326\pi\)
−0.454902 + 0.890541i \(0.650326\pi\)
\(684\) 5.70820 0.218259
\(685\) 0 0
\(686\) 18.7082 0.714283
\(687\) −3.11146 −0.118709
\(688\) 3.38197 0.128936
\(689\) 3.58359 0.136524
\(690\) 0 0
\(691\) −13.5623 −0.515934 −0.257967 0.966154i \(-0.583053\pi\)
−0.257967 + 0.966154i \(0.583053\pi\)
\(692\) 17.5066 0.665500
\(693\) −14.9443 −0.567686
\(694\) 6.00000 0.227757
\(695\) 0 0
\(696\) 0.381966 0.0144784
\(697\) 26.0689 0.987429
\(698\) 24.8328 0.939936
\(699\) −4.29180 −0.162331
\(700\) 0 0
\(701\) −9.12461 −0.344632 −0.172316 0.985042i \(-0.555125\pi\)
−0.172316 + 0.985042i \(0.555125\pi\)
\(702\) 0.854102 0.0322360
\(703\) −17.8885 −0.674679
\(704\) 2.00000 0.0753778
\(705\) 0 0
\(706\) 3.05573 0.115004
\(707\) 28.4164 1.06871
\(708\) 1.09017 0.0409711
\(709\) 12.2918 0.461628 0.230814 0.972998i \(-0.425861\pi\)
0.230814 + 0.972998i \(0.425861\pi\)
\(710\) 0 0
\(711\) −18.7295 −0.702411
\(712\) −2.00000 −0.0749532
\(713\) −18.7082 −0.700628
\(714\) −3.38197 −0.126567
\(715\) 0 0
\(716\) 9.14590 0.341798
\(717\) 2.18034 0.0814263
\(718\) −0.270510 −0.0100953
\(719\) 8.29180 0.309232 0.154616 0.987975i \(-0.450586\pi\)
0.154616 + 0.987975i \(0.450586\pi\)
\(720\) 0 0
\(721\) 19.4164 0.723105
\(722\) 15.0000 0.558242
\(723\) −3.38197 −0.125777
\(724\) −10.0000 −0.371647
\(725\) 0 0
\(726\) 2.67376 0.0992326
\(727\) −32.9443 −1.22184 −0.610918 0.791694i \(-0.709199\pi\)
−0.610918 + 0.791694i \(0.709199\pi\)
\(728\) −1.00000 −0.0370625
\(729\) −19.4377 −0.719915
\(730\) 0 0
\(731\) 11.4377 0.423038
\(732\) −4.14590 −0.153237
\(733\) 12.0000 0.443230 0.221615 0.975134i \(-0.428867\pi\)
0.221615 + 0.975134i \(0.428867\pi\)
\(734\) −35.1246 −1.29647
\(735\) 0 0
\(736\) −4.85410 −0.178925
\(737\) −17.8885 −0.658933
\(738\) 22.0000 0.809831
\(739\) 28.2918 1.04073 0.520365 0.853944i \(-0.325796\pi\)
0.520365 + 0.853944i \(0.325796\pi\)
\(740\) 0 0
\(741\) −0.291796 −0.0107194
\(742\) −24.5623 −0.901711
\(743\) −25.4164 −0.932438 −0.466219 0.884669i \(-0.654384\pi\)
−0.466219 + 0.884669i \(0.654384\pi\)
\(744\) 1.47214 0.0539711
\(745\) 0 0
\(746\) −15.2705 −0.559093
\(747\) −42.6525 −1.56057
\(748\) 6.76393 0.247314
\(749\) −37.1246 −1.35650
\(750\) 0 0
\(751\) −4.00000 −0.145962 −0.0729810 0.997333i \(-0.523251\pi\)
−0.0729810 + 0.997333i \(0.523251\pi\)
\(752\) 1.52786 0.0557155
\(753\) −11.1246 −0.405403
\(754\) 0.381966 0.0139104
\(755\) 0 0
\(756\) −5.85410 −0.212912
\(757\) 32.0689 1.16556 0.582782 0.812629i \(-0.301964\pi\)
0.582782 + 0.812629i \(0.301964\pi\)
\(758\) −26.0000 −0.944363
\(759\) 3.70820 0.134599
\(760\) 0 0
\(761\) −9.85410 −0.357211 −0.178605 0.983921i \(-0.557159\pi\)
−0.178605 + 0.983921i \(0.557159\pi\)
\(762\) 3.16718 0.114735
\(763\) 50.8328 1.84027
\(764\) 23.2705 0.841897
\(765\) 0 0
\(766\) 7.85410 0.283780
\(767\) 1.09017 0.0393638
\(768\) 0.381966 0.0137830
\(769\) −18.5836 −0.670141 −0.335071 0.942193i \(-0.608760\pi\)
−0.335071 + 0.942193i \(0.608760\pi\)
\(770\) 0 0
\(771\) 0.875388 0.0315263
\(772\) −2.56231 −0.0922194
\(773\) 6.76393 0.243282 0.121641 0.992574i \(-0.461184\pi\)
0.121641 + 0.992574i \(0.461184\pi\)
\(774\) 9.65248 0.346951
\(775\) 0 0
\(776\) −7.14590 −0.256523
\(777\) 8.94427 0.320874
\(778\) 20.8328 0.746893
\(779\) −15.4164 −0.552350
\(780\) 0 0
\(781\) −11.4164 −0.408511
\(782\) −16.4164 −0.587050
\(783\) 2.23607 0.0799106
\(784\) −0.145898 −0.00521064
\(785\) 0 0
\(786\) −8.18034 −0.291783
\(787\) −41.7771 −1.48919 −0.744596 0.667515i \(-0.767358\pi\)
−0.744596 + 0.667515i \(0.767358\pi\)
\(788\) 9.38197 0.334219
\(789\) −8.24922 −0.293680
\(790\) 0 0
\(791\) 10.7082 0.380740
\(792\) 5.70820 0.202832
\(793\) −4.14590 −0.147225
\(794\) 1.96556 0.0697550
\(795\) 0 0
\(796\) 19.1246 0.677854
\(797\) −35.2361 −1.24813 −0.624063 0.781374i \(-0.714519\pi\)
−0.624063 + 0.781374i \(0.714519\pi\)
\(798\) 2.00000 0.0707992
\(799\) 5.16718 0.182802
\(800\) 0 0
\(801\) −5.70820 −0.201689
\(802\) −21.9787 −0.776095
\(803\) −11.1246 −0.392579
\(804\) −3.41641 −0.120487
\(805\) 0 0
\(806\) 1.47214 0.0518538
\(807\) 1.24922 0.0439748
\(808\) −10.8541 −0.381846
\(809\) −39.4164 −1.38581 −0.692904 0.721030i \(-0.743669\pi\)
−0.692904 + 0.721030i \(0.743669\pi\)
\(810\) 0 0
\(811\) 48.3951 1.69938 0.849691 0.527280i \(-0.176788\pi\)
0.849691 + 0.527280i \(0.176788\pi\)
\(812\) −2.61803 −0.0918750
\(813\) 8.94427 0.313689
\(814\) −17.8885 −0.626993
\(815\) 0 0
\(816\) 1.29180 0.0452219
\(817\) −6.76393 −0.236640
\(818\) 7.12461 0.249106
\(819\) −2.85410 −0.0997304
\(820\) 0 0
\(821\) −8.00000 −0.279202 −0.139601 0.990208i \(-0.544582\pi\)
−0.139601 + 0.990208i \(0.544582\pi\)
\(822\) 1.85410 0.0646692
\(823\) 5.23607 0.182518 0.0912589 0.995827i \(-0.470911\pi\)
0.0912589 + 0.995827i \(0.470911\pi\)
\(824\) −7.41641 −0.258363
\(825\) 0 0
\(826\) −7.47214 −0.259989
\(827\) 25.2016 0.876346 0.438173 0.898891i \(-0.355626\pi\)
0.438173 + 0.898891i \(0.355626\pi\)
\(828\) −13.8541 −0.481463
\(829\) −0.729490 −0.0253362 −0.0126681 0.999920i \(-0.504032\pi\)
−0.0126681 + 0.999920i \(0.504032\pi\)
\(830\) 0 0
\(831\) −11.4590 −0.397508
\(832\) 0.381966 0.0132423
\(833\) −0.493422 −0.0170961
\(834\) 2.12461 0.0735693
\(835\) 0 0
\(836\) −4.00000 −0.138343
\(837\) 8.61803 0.297883
\(838\) 21.8541 0.754937
\(839\) −38.8328 −1.34066 −0.670329 0.742064i \(-0.733847\pi\)
−0.670329 + 0.742064i \(0.733847\pi\)
\(840\) 0 0
\(841\) 1.00000 0.0344828
\(842\) 10.0000 0.344623
\(843\) −6.70820 −0.231043
\(844\) −11.4164 −0.392969
\(845\) 0 0
\(846\) 4.36068 0.149923
\(847\) −18.3262 −0.629697
\(848\) 9.38197 0.322178
\(849\) −2.58359 −0.0886687
\(850\) 0 0
\(851\) 43.4164 1.48830
\(852\) −2.18034 −0.0746972
\(853\) −8.83282 −0.302430 −0.151215 0.988501i \(-0.548319\pi\)
−0.151215 + 0.988501i \(0.548319\pi\)
\(854\) 28.4164 0.972389
\(855\) 0 0
\(856\) 14.1803 0.484674
\(857\) 21.8197 0.745345 0.372673 0.927963i \(-0.378441\pi\)
0.372673 + 0.927963i \(0.378441\pi\)
\(858\) −0.291796 −0.00996175
\(859\) 27.7082 0.945392 0.472696 0.881226i \(-0.343281\pi\)
0.472696 + 0.881226i \(0.343281\pi\)
\(860\) 0 0
\(861\) 7.70820 0.262695
\(862\) 33.4164 1.13817
\(863\) 30.9230 1.05263 0.526315 0.850289i \(-0.323573\pi\)
0.526315 + 0.850289i \(0.323573\pi\)
\(864\) 2.23607 0.0760726
\(865\) 0 0
\(866\) 16.4721 0.559746
\(867\) −2.12461 −0.0721556
\(868\) −10.0902 −0.342483
\(869\) 13.1246 0.445222
\(870\) 0 0
\(871\) −3.41641 −0.115761
\(872\) −19.4164 −0.657523
\(873\) −20.3951 −0.690270
\(874\) 9.70820 0.328385
\(875\) 0 0
\(876\) −2.12461 −0.0717840
\(877\) −42.2705 −1.42737 −0.713687 0.700465i \(-0.752976\pi\)
−0.713687 + 0.700465i \(0.752976\pi\)
\(878\) 37.1246 1.25289
\(879\) −7.41641 −0.250149
\(880\) 0 0
\(881\) −14.5836 −0.491334 −0.245667 0.969354i \(-0.579007\pi\)
−0.245667 + 0.969354i \(0.579007\pi\)
\(882\) −0.416408 −0.0140212
\(883\) −33.7082 −1.13437 −0.567186 0.823590i \(-0.691968\pi\)
−0.567186 + 0.823590i \(0.691968\pi\)
\(884\) 1.29180 0.0434478
\(885\) 0 0
\(886\) 1.90983 0.0641620
\(887\) 24.0000 0.805841 0.402921 0.915235i \(-0.367995\pi\)
0.402921 + 0.915235i \(0.367995\pi\)
\(888\) −3.41641 −0.114647
\(889\) −21.7082 −0.728070
\(890\) 0 0
\(891\) 15.4164 0.516469
\(892\) −26.5623 −0.889372
\(893\) −3.05573 −0.102256
\(894\) 8.18034 0.273591
\(895\) 0 0
\(896\) −2.61803 −0.0874624
\(897\) 0.708204 0.0236462
\(898\) 4.58359 0.152956
\(899\) 3.85410 0.128541
\(900\) 0 0
\(901\) 31.7295 1.05706
\(902\) −15.4164 −0.513310
\(903\) 3.38197 0.112545
\(904\) −4.09017 −0.136037
\(905\) 0 0
\(906\) −1.63932 −0.0544628
\(907\) −7.74265 −0.257090 −0.128545 0.991704i \(-0.541031\pi\)
−0.128545 + 0.991704i \(0.541031\pi\)
\(908\) 16.4721 0.546647
\(909\) −30.9787 −1.02750
\(910\) 0 0
\(911\) 12.8541 0.425875 0.212938 0.977066i \(-0.431697\pi\)
0.212938 + 0.977066i \(0.431697\pi\)
\(912\) −0.763932 −0.0252963
\(913\) 29.8885 0.989166
\(914\) 14.2918 0.472731
\(915\) 0 0
\(916\) −8.14590 −0.269148
\(917\) 56.0689 1.85156
\(918\) 7.56231 0.249593
\(919\) −52.5410 −1.73317 −0.866584 0.499031i \(-0.833689\pi\)
−0.866584 + 0.499031i \(0.833689\pi\)
\(920\) 0 0
\(921\) −9.16718 −0.302069
\(922\) 24.1459 0.795203
\(923\) −2.18034 −0.0717668
\(924\) 2.00000 0.0657952
\(925\) 0 0
\(926\) 17.8885 0.587854
\(927\) −21.1672 −0.695222
\(928\) 1.00000 0.0328266
\(929\) 34.1459 1.12029 0.560145 0.828394i \(-0.310745\pi\)
0.560145 + 0.828394i \(0.310745\pi\)
\(930\) 0 0
\(931\) 0.291796 0.00956323
\(932\) −11.2361 −0.368050
\(933\) −13.2016 −0.432202
\(934\) −34.7984 −1.13864
\(935\) 0 0
\(936\) 1.09017 0.0356333
\(937\) 44.0689 1.43967 0.719834 0.694146i \(-0.244218\pi\)
0.719834 + 0.694146i \(0.244218\pi\)
\(938\) 23.4164 0.764573
\(939\) −4.87539 −0.159102
\(940\) 0 0
\(941\) −21.7082 −0.707667 −0.353834 0.935308i \(-0.615122\pi\)
−0.353834 + 0.935308i \(0.615122\pi\)
\(942\) −3.70820 −0.120820
\(943\) 37.4164 1.21845
\(944\) 2.85410 0.0928931
\(945\) 0 0
\(946\) −6.76393 −0.219914
\(947\) 60.1591 1.95491 0.977453 0.211152i \(-0.0677214\pi\)
0.977453 + 0.211152i \(0.0677214\pi\)
\(948\) 2.50658 0.0814099
\(949\) −2.12461 −0.0689678
\(950\) 0 0
\(951\) −7.70820 −0.249956
\(952\) −8.85410 −0.286963
\(953\) −2.83282 −0.0917639 −0.0458820 0.998947i \(-0.514610\pi\)
−0.0458820 + 0.998947i \(0.514610\pi\)
\(954\) 26.7771 0.866940
\(955\) 0 0
\(956\) 5.70820 0.184617
\(957\) −0.763932 −0.0246944
\(958\) 18.9787 0.613174
\(959\) −12.7082 −0.410369
\(960\) 0 0
\(961\) −16.1459 −0.520835
\(962\) −3.41641 −0.110149
\(963\) 40.4721 1.30420
\(964\) −8.85410 −0.285171
\(965\) 0 0
\(966\) −4.85410 −0.156178
\(967\) 33.7082 1.08398 0.541991 0.840384i \(-0.317671\pi\)
0.541991 + 0.840384i \(0.317671\pi\)
\(968\) 7.00000 0.224989
\(969\) −2.58359 −0.0829969
\(970\) 0 0
\(971\) 1.70820 0.0548189 0.0274094 0.999624i \(-0.491274\pi\)
0.0274094 + 0.999624i \(0.491274\pi\)
\(972\) 9.65248 0.309603
\(973\) −14.5623 −0.466846
\(974\) −38.5623 −1.23562
\(975\) 0 0
\(976\) −10.8541 −0.347431
\(977\) −33.0557 −1.05755 −0.528773 0.848763i \(-0.677348\pi\)
−0.528773 + 0.848763i \(0.677348\pi\)
\(978\) −2.83282 −0.0905835
\(979\) 4.00000 0.127841
\(980\) 0 0
\(981\) −55.4164 −1.76931
\(982\) 6.29180 0.200779
\(983\) −3.59675 −0.114718 −0.0573592 0.998354i \(-0.518268\pi\)
−0.0573592 + 0.998354i \(0.518268\pi\)
\(984\) −2.94427 −0.0938600
\(985\) 0 0
\(986\) 3.38197 0.107704
\(987\) 1.52786 0.0486324
\(988\) −0.763932 −0.0243039
\(989\) 16.4164 0.522011
\(990\) 0 0
\(991\) −41.4164 −1.31564 −0.657818 0.753177i \(-0.728520\pi\)
−0.657818 + 0.753177i \(0.728520\pi\)
\(992\) 3.85410 0.122368
\(993\) 10.3607 0.328786
\(994\) 14.9443 0.474004
\(995\) 0 0
\(996\) 5.70820 0.180871
\(997\) 17.1246 0.542342 0.271171 0.962531i \(-0.412589\pi\)
0.271171 + 0.962531i \(0.412589\pi\)
\(998\) −28.5623 −0.904124
\(999\) −20.0000 −0.632772
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 1450.2.a.k.1.1 2
5.2 odd 4 290.2.b.a.59.2 4
5.3 odd 4 290.2.b.a.59.3 yes 4
5.4 even 2 1450.2.a.l.1.2 2
15.2 even 4 2610.2.e.e.2089.3 4
15.8 even 4 2610.2.e.e.2089.2 4
20.3 even 4 2320.2.d.d.929.2 4
20.7 even 4 2320.2.d.d.929.3 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
290.2.b.a.59.2 4 5.2 odd 4
290.2.b.a.59.3 yes 4 5.3 odd 4
1450.2.a.k.1.1 2 1.1 even 1 trivial
1450.2.a.l.1.2 2 5.4 even 2
2320.2.d.d.929.2 4 20.3 even 4
2320.2.d.d.929.3 4 20.7 even 4
2610.2.e.e.2089.2 4 15.8 even 4
2610.2.e.e.2089.3 4 15.2 even 4