Properties

Label 845.2.a.g.1.1
Level $845$
Weight $2$
Character 845.1
Self dual yes
Analytic conductor $6.747$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [845,2,Mod(1,845)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(845, 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("845.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 845 = 5 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 845.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: \(6.74735897080\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{8})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 65)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.41421\) of defining polynomial
Character \(\chi\) \(=\) 845.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-0.414214 q^{2} +1.41421 q^{3} -1.82843 q^{4} -1.00000 q^{5} -0.585786 q^{6} +0.828427 q^{7} +1.58579 q^{8} -1.00000 q^{9} +O(q^{10})\) \(q-0.414214 q^{2} +1.41421 q^{3} -1.82843 q^{4} -1.00000 q^{5} -0.585786 q^{6} +0.828427 q^{7} +1.58579 q^{8} -1.00000 q^{9} +0.414214 q^{10} -0.585786 q^{11} -2.58579 q^{12} -0.343146 q^{14} -1.41421 q^{15} +3.00000 q^{16} -4.82843 q^{17} +0.414214 q^{18} -3.41421 q^{19} +1.82843 q^{20} +1.17157 q^{21} +0.242641 q^{22} -1.41421 q^{23} +2.24264 q^{24} +1.00000 q^{25} -5.65685 q^{27} -1.51472 q^{28} +5.65685 q^{29} +0.585786 q^{30} -10.2426 q^{31} -4.41421 q^{32} -0.828427 q^{33} +2.00000 q^{34} -0.828427 q^{35} +1.82843 q^{36} -8.48528 q^{37} +1.41421 q^{38} -1.58579 q^{40} +8.82843 q^{41} -0.485281 q^{42} +3.07107 q^{43} +1.07107 q^{44} +1.00000 q^{45} +0.585786 q^{46} -0.828427 q^{47} +4.24264 q^{48} -6.31371 q^{49} -0.414214 q^{50} -6.82843 q^{51} -14.4853 q^{53} +2.34315 q^{54} +0.585786 q^{55} +1.31371 q^{56} -4.82843 q^{57} -2.34315 q^{58} -10.2426 q^{59} +2.58579 q^{60} -8.00000 q^{61} +4.24264 q^{62} -0.828427 q^{63} -4.17157 q^{64} +0.343146 q^{66} +2.00000 q^{67} +8.82843 q^{68} -2.00000 q^{69} +0.343146 q^{70} +7.89949 q^{71} -1.58579 q^{72} +8.48528 q^{73} +3.51472 q^{74} +1.41421 q^{75} +6.24264 q^{76} -0.485281 q^{77} +8.48528 q^{79} -3.00000 q^{80} -5.00000 q^{81} -3.65685 q^{82} +8.82843 q^{83} -2.14214 q^{84} +4.82843 q^{85} -1.27208 q^{86} +8.00000 q^{87} -0.928932 q^{88} -6.00000 q^{89} -0.414214 q^{90} +2.58579 q^{92} -14.4853 q^{93} +0.343146 q^{94} +3.41421 q^{95} -6.24264 q^{96} -3.65685 q^{97} +2.61522 q^{98} +0.585786 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} + 2 q^{4} - 2 q^{5} - 4 q^{6} - 4 q^{7} + 6 q^{8} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{2} + 2 q^{4} - 2 q^{5} - 4 q^{6} - 4 q^{7} + 6 q^{8} - 2 q^{9} - 2 q^{10} - 4 q^{11} - 8 q^{12} - 12 q^{14} + 6 q^{16} - 4 q^{17} - 2 q^{18} - 4 q^{19} - 2 q^{20} + 8 q^{21} - 8 q^{22} - 4 q^{24} + 2 q^{25} - 20 q^{28} + 4 q^{30} - 12 q^{31} - 6 q^{32} + 4 q^{33} + 4 q^{34} + 4 q^{35} - 2 q^{36} - 6 q^{40} + 12 q^{41} + 16 q^{42} - 8 q^{43} - 12 q^{44} + 2 q^{45} + 4 q^{46} + 4 q^{47} + 10 q^{49} + 2 q^{50} - 8 q^{51} - 12 q^{53} + 16 q^{54} + 4 q^{55} - 20 q^{56} - 4 q^{57} - 16 q^{58} - 12 q^{59} + 8 q^{60} - 16 q^{61} + 4 q^{63} - 14 q^{64} + 12 q^{66} + 4 q^{67} + 12 q^{68} - 4 q^{69} + 12 q^{70} - 4 q^{71} - 6 q^{72} + 24 q^{74} + 4 q^{76} + 16 q^{77} - 6 q^{80} - 10 q^{81} + 4 q^{82} + 12 q^{83} + 24 q^{84} + 4 q^{85} - 28 q^{86} + 16 q^{87} - 16 q^{88} - 12 q^{89} + 2 q^{90} + 8 q^{92} - 12 q^{93} + 12 q^{94} + 4 q^{95} - 4 q^{96} + 4 q^{97} + 42 q^{98} + 4 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.414214 −0.292893 −0.146447 0.989219i \(-0.546784\pi\)
−0.146447 + 0.989219i \(0.546784\pi\)
\(3\) 1.41421 0.816497 0.408248 0.912871i \(-0.366140\pi\)
0.408248 + 0.912871i \(0.366140\pi\)
\(4\) −1.82843 −0.914214
\(5\) −1.00000 −0.447214
\(6\) −0.585786 −0.239146
\(7\) 0.828427 0.313116 0.156558 0.987669i \(-0.449960\pi\)
0.156558 + 0.987669i \(0.449960\pi\)
\(8\) 1.58579 0.560660
\(9\) −1.00000 −0.333333
\(10\) 0.414214 0.130986
\(11\) −0.585786 −0.176621 −0.0883106 0.996093i \(-0.528147\pi\)
−0.0883106 + 0.996093i \(0.528147\pi\)
\(12\) −2.58579 −0.746452
\(13\) 0 0
\(14\) −0.343146 −0.0917096
\(15\) −1.41421 −0.365148
\(16\) 3.00000 0.750000
\(17\) −4.82843 −1.17107 −0.585533 0.810649i \(-0.699115\pi\)
−0.585533 + 0.810649i \(0.699115\pi\)
\(18\) 0.414214 0.0976311
\(19\) −3.41421 −0.783274 −0.391637 0.920120i \(-0.628091\pi\)
−0.391637 + 0.920120i \(0.628091\pi\)
\(20\) 1.82843 0.408849
\(21\) 1.17157 0.255658
\(22\) 0.242641 0.0517312
\(23\) −1.41421 −0.294884 −0.147442 0.989071i \(-0.547104\pi\)
−0.147442 + 0.989071i \(0.547104\pi\)
\(24\) 2.24264 0.457777
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) −5.65685 −1.08866
\(28\) −1.51472 −0.286255
\(29\) 5.65685 1.05045 0.525226 0.850963i \(-0.323981\pi\)
0.525226 + 0.850963i \(0.323981\pi\)
\(30\) 0.585786 0.106949
\(31\) −10.2426 −1.83963 −0.919816 0.392349i \(-0.871662\pi\)
−0.919816 + 0.392349i \(0.871662\pi\)
\(32\) −4.41421 −0.780330
\(33\) −0.828427 −0.144211
\(34\) 2.00000 0.342997
\(35\) −0.828427 −0.140030
\(36\) 1.82843 0.304738
\(37\) −8.48528 −1.39497 −0.697486 0.716599i \(-0.745698\pi\)
−0.697486 + 0.716599i \(0.745698\pi\)
\(38\) 1.41421 0.229416
\(39\) 0 0
\(40\) −1.58579 −0.250735
\(41\) 8.82843 1.37877 0.689384 0.724396i \(-0.257881\pi\)
0.689384 + 0.724396i \(0.257881\pi\)
\(42\) −0.485281 −0.0748805
\(43\) 3.07107 0.468333 0.234167 0.972196i \(-0.424764\pi\)
0.234167 + 0.972196i \(0.424764\pi\)
\(44\) 1.07107 0.161470
\(45\) 1.00000 0.149071
\(46\) 0.585786 0.0863695
\(47\) −0.828427 −0.120839 −0.0604193 0.998173i \(-0.519244\pi\)
−0.0604193 + 0.998173i \(0.519244\pi\)
\(48\) 4.24264 0.612372
\(49\) −6.31371 −0.901958
\(50\) −0.414214 −0.0585786
\(51\) −6.82843 −0.956171
\(52\) 0 0
\(53\) −14.4853 −1.98971 −0.994853 0.101327i \(-0.967691\pi\)
−0.994853 + 0.101327i \(0.967691\pi\)
\(54\) 2.34315 0.318862
\(55\) 0.585786 0.0789874
\(56\) 1.31371 0.175552
\(57\) −4.82843 −0.639541
\(58\) −2.34315 −0.307670
\(59\) −10.2426 −1.33348 −0.666739 0.745291i \(-0.732310\pi\)
−0.666739 + 0.745291i \(0.732310\pi\)
\(60\) 2.58579 0.333824
\(61\) −8.00000 −1.02430 −0.512148 0.858898i \(-0.671150\pi\)
−0.512148 + 0.858898i \(0.671150\pi\)
\(62\) 4.24264 0.538816
\(63\) −0.828427 −0.104372
\(64\) −4.17157 −0.521447
\(65\) 0 0
\(66\) 0.343146 0.0422383
\(67\) 2.00000 0.244339 0.122169 0.992509i \(-0.461015\pi\)
0.122169 + 0.992509i \(0.461015\pi\)
\(68\) 8.82843 1.07060
\(69\) −2.00000 −0.240772
\(70\) 0.343146 0.0410138
\(71\) 7.89949 0.937498 0.468749 0.883332i \(-0.344705\pi\)
0.468749 + 0.883332i \(0.344705\pi\)
\(72\) −1.58579 −0.186887
\(73\) 8.48528 0.993127 0.496564 0.868000i \(-0.334595\pi\)
0.496564 + 0.868000i \(0.334595\pi\)
\(74\) 3.51472 0.408578
\(75\) 1.41421 0.163299
\(76\) 6.24264 0.716080
\(77\) −0.485281 −0.0553029
\(78\) 0 0
\(79\) 8.48528 0.954669 0.477334 0.878722i \(-0.341603\pi\)
0.477334 + 0.878722i \(0.341603\pi\)
\(80\) −3.00000 −0.335410
\(81\) −5.00000 −0.555556
\(82\) −3.65685 −0.403832
\(83\) 8.82843 0.969046 0.484523 0.874779i \(-0.338993\pi\)
0.484523 + 0.874779i \(0.338993\pi\)
\(84\) −2.14214 −0.233726
\(85\) 4.82843 0.523716
\(86\) −1.27208 −0.137172
\(87\) 8.00000 0.857690
\(88\) −0.928932 −0.0990245
\(89\) −6.00000 −0.635999 −0.317999 0.948091i \(-0.603011\pi\)
−0.317999 + 0.948091i \(0.603011\pi\)
\(90\) −0.414214 −0.0436619
\(91\) 0 0
\(92\) 2.58579 0.269587
\(93\) −14.4853 −1.50205
\(94\) 0.343146 0.0353928
\(95\) 3.41421 0.350291
\(96\) −6.24264 −0.637137
\(97\) −3.65685 −0.371297 −0.185649 0.982616i \(-0.559439\pi\)
−0.185649 + 0.982616i \(0.559439\pi\)
\(98\) 2.61522 0.264177
\(99\) 0.585786 0.0588738
\(100\) −1.82843 −0.182843
\(101\) 7.65685 0.761885 0.380943 0.924599i \(-0.375599\pi\)
0.380943 + 0.924599i \(0.375599\pi\)
\(102\) 2.82843 0.280056
\(103\) 17.4142 1.71587 0.857937 0.513755i \(-0.171746\pi\)
0.857937 + 0.513755i \(0.171746\pi\)
\(104\) 0 0
\(105\) −1.17157 −0.114334
\(106\) 6.00000 0.582772
\(107\) −6.58579 −0.636672 −0.318336 0.947978i \(-0.603124\pi\)
−0.318336 + 0.947978i \(0.603124\pi\)
\(108\) 10.3431 0.995270
\(109\) 2.00000 0.191565 0.0957826 0.995402i \(-0.469465\pi\)
0.0957826 + 0.995402i \(0.469465\pi\)
\(110\) −0.242641 −0.0231349
\(111\) −12.0000 −1.13899
\(112\) 2.48528 0.234837
\(113\) −3.17157 −0.298356 −0.149178 0.988810i \(-0.547663\pi\)
−0.149178 + 0.988810i \(0.547663\pi\)
\(114\) 2.00000 0.187317
\(115\) 1.41421 0.131876
\(116\) −10.3431 −0.960337
\(117\) 0 0
\(118\) 4.24264 0.390567
\(119\) −4.00000 −0.366679
\(120\) −2.24264 −0.204724
\(121\) −10.6569 −0.968805
\(122\) 3.31371 0.300009
\(123\) 12.4853 1.12576
\(124\) 18.7279 1.68182
\(125\) −1.00000 −0.0894427
\(126\) 0.343146 0.0305699
\(127\) −9.41421 −0.835376 −0.417688 0.908590i \(-0.637160\pi\)
−0.417688 + 0.908590i \(0.637160\pi\)
\(128\) 10.5563 0.933058
\(129\) 4.34315 0.382393
\(130\) 0 0
\(131\) 16.9706 1.48272 0.741362 0.671105i \(-0.234180\pi\)
0.741362 + 0.671105i \(0.234180\pi\)
\(132\) 1.51472 0.131839
\(133\) −2.82843 −0.245256
\(134\) −0.828427 −0.0715652
\(135\) 5.65685 0.486864
\(136\) −7.65685 −0.656570
\(137\) −5.31371 −0.453981 −0.226990 0.973897i \(-0.572889\pi\)
−0.226990 + 0.973897i \(0.572889\pi\)
\(138\) 0.828427 0.0705204
\(139\) −12.4853 −1.05899 −0.529494 0.848314i \(-0.677618\pi\)
−0.529494 + 0.848314i \(0.677618\pi\)
\(140\) 1.51472 0.128017
\(141\) −1.17157 −0.0986642
\(142\) −3.27208 −0.274587
\(143\) 0 0
\(144\) −3.00000 −0.250000
\(145\) −5.65685 −0.469776
\(146\) −3.51472 −0.290880
\(147\) −8.92893 −0.736446
\(148\) 15.5147 1.27530
\(149\) 0.343146 0.0281116 0.0140558 0.999901i \(-0.495526\pi\)
0.0140558 + 0.999901i \(0.495526\pi\)
\(150\) −0.585786 −0.0478293
\(151\) −18.2426 −1.48457 −0.742283 0.670087i \(-0.766257\pi\)
−0.742283 + 0.670087i \(0.766257\pi\)
\(152\) −5.41421 −0.439151
\(153\) 4.82843 0.390355
\(154\) 0.201010 0.0161979
\(155\) 10.2426 0.822709
\(156\) 0 0
\(157\) 18.0000 1.43656 0.718278 0.695756i \(-0.244931\pi\)
0.718278 + 0.695756i \(0.244931\pi\)
\(158\) −3.51472 −0.279616
\(159\) −20.4853 −1.62459
\(160\) 4.41421 0.348974
\(161\) −1.17157 −0.0923329
\(162\) 2.07107 0.162718
\(163\) 14.9706 1.17258 0.586292 0.810099i \(-0.300587\pi\)
0.586292 + 0.810099i \(0.300587\pi\)
\(164\) −16.1421 −1.26049
\(165\) 0.828427 0.0644930
\(166\) −3.65685 −0.283827
\(167\) 8.82843 0.683164 0.341582 0.939852i \(-0.389037\pi\)
0.341582 + 0.939852i \(0.389037\pi\)
\(168\) 1.85786 0.143337
\(169\) 0 0
\(170\) −2.00000 −0.153393
\(171\) 3.41421 0.261091
\(172\) −5.61522 −0.428157
\(173\) 11.1716 0.849359 0.424679 0.905344i \(-0.360387\pi\)
0.424679 + 0.905344i \(0.360387\pi\)
\(174\) −3.31371 −0.251212
\(175\) 0.828427 0.0626232
\(176\) −1.75736 −0.132466
\(177\) −14.4853 −1.08878
\(178\) 2.48528 0.186280
\(179\) −5.65685 −0.422813 −0.211407 0.977398i \(-0.567804\pi\)
−0.211407 + 0.977398i \(0.567804\pi\)
\(180\) −1.82843 −0.136283
\(181\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(182\) 0 0
\(183\) −11.3137 −0.836333
\(184\) −2.24264 −0.165330
\(185\) 8.48528 0.623850
\(186\) 6.00000 0.439941
\(187\) 2.82843 0.206835
\(188\) 1.51472 0.110472
\(189\) −4.68629 −0.340878
\(190\) −1.41421 −0.102598
\(191\) 13.6569 0.988175 0.494088 0.869412i \(-0.335502\pi\)
0.494088 + 0.869412i \(0.335502\pi\)
\(192\) −5.89949 −0.425759
\(193\) −15.6569 −1.12701 −0.563503 0.826114i \(-0.690546\pi\)
−0.563503 + 0.826114i \(0.690546\pi\)
\(194\) 1.51472 0.108750
\(195\) 0 0
\(196\) 11.5442 0.824583
\(197\) 22.9706 1.63658 0.818292 0.574802i \(-0.194921\pi\)
0.818292 + 0.574802i \(0.194921\pi\)
\(198\) −0.242641 −0.0172437
\(199\) 4.00000 0.283552 0.141776 0.989899i \(-0.454719\pi\)
0.141776 + 0.989899i \(0.454719\pi\)
\(200\) 1.58579 0.112132
\(201\) 2.82843 0.199502
\(202\) −3.17157 −0.223151
\(203\) 4.68629 0.328913
\(204\) 12.4853 0.874145
\(205\) −8.82843 −0.616604
\(206\) −7.21320 −0.502568
\(207\) 1.41421 0.0982946
\(208\) 0 0
\(209\) 2.00000 0.138343
\(210\) 0.485281 0.0334876
\(211\) −19.3137 −1.32961 −0.664805 0.747017i \(-0.731485\pi\)
−0.664805 + 0.747017i \(0.731485\pi\)
\(212\) 26.4853 1.81902
\(213\) 11.1716 0.765464
\(214\) 2.72792 0.186477
\(215\) −3.07107 −0.209445
\(216\) −8.97056 −0.610369
\(217\) −8.48528 −0.576018
\(218\) −0.828427 −0.0561082
\(219\) 12.0000 0.810885
\(220\) −1.07107 −0.0722114
\(221\) 0 0
\(222\) 4.97056 0.333602
\(223\) −26.4853 −1.77359 −0.886793 0.462167i \(-0.847072\pi\)
−0.886793 + 0.462167i \(0.847072\pi\)
\(224\) −3.65685 −0.244334
\(225\) −1.00000 −0.0666667
\(226\) 1.31371 0.0873866
\(227\) 27.6569 1.83565 0.917825 0.396985i \(-0.129944\pi\)
0.917825 + 0.396985i \(0.129944\pi\)
\(228\) 8.82843 0.584677
\(229\) −0.828427 −0.0547440 −0.0273720 0.999625i \(-0.508714\pi\)
−0.0273720 + 0.999625i \(0.508714\pi\)
\(230\) −0.585786 −0.0386256
\(231\) −0.686292 −0.0451547
\(232\) 8.97056 0.588946
\(233\) 24.6274 1.61340 0.806698 0.590964i \(-0.201253\pi\)
0.806698 + 0.590964i \(0.201253\pi\)
\(234\) 0 0
\(235\) 0.828427 0.0540406
\(236\) 18.7279 1.21908
\(237\) 12.0000 0.779484
\(238\) 1.65685 0.107398
\(239\) 0.585786 0.0378914 0.0189457 0.999821i \(-0.493969\pi\)
0.0189457 + 0.999821i \(0.493969\pi\)
\(240\) −4.24264 −0.273861
\(241\) −2.48528 −0.160091 −0.0800455 0.996791i \(-0.525507\pi\)
−0.0800455 + 0.996791i \(0.525507\pi\)
\(242\) 4.41421 0.283756
\(243\) 9.89949 0.635053
\(244\) 14.6274 0.936424
\(245\) 6.31371 0.403368
\(246\) −5.17157 −0.329727
\(247\) 0 0
\(248\) −16.2426 −1.03141
\(249\) 12.4853 0.791223
\(250\) 0.414214 0.0261972
\(251\) −19.7990 −1.24970 −0.624851 0.780744i \(-0.714840\pi\)
−0.624851 + 0.780744i \(0.714840\pi\)
\(252\) 1.51472 0.0954183
\(253\) 0.828427 0.0520828
\(254\) 3.89949 0.244676
\(255\) 6.82843 0.427613
\(256\) 3.97056 0.248160
\(257\) 16.3431 1.01946 0.509729 0.860335i \(-0.329746\pi\)
0.509729 + 0.860335i \(0.329746\pi\)
\(258\) −1.79899 −0.112000
\(259\) −7.02944 −0.436788
\(260\) 0 0
\(261\) −5.65685 −0.350150
\(262\) −7.02944 −0.434280
\(263\) −13.4142 −0.827156 −0.413578 0.910469i \(-0.635721\pi\)
−0.413578 + 0.910469i \(0.635721\pi\)
\(264\) −1.31371 −0.0808532
\(265\) 14.4853 0.889824
\(266\) 1.17157 0.0718337
\(267\) −8.48528 −0.519291
\(268\) −3.65685 −0.223378
\(269\) −2.68629 −0.163786 −0.0818930 0.996641i \(-0.526097\pi\)
−0.0818930 + 0.996641i \(0.526097\pi\)
\(270\) −2.34315 −0.142599
\(271\) −1.27208 −0.0772732 −0.0386366 0.999253i \(-0.512301\pi\)
−0.0386366 + 0.999253i \(0.512301\pi\)
\(272\) −14.4853 −0.878299
\(273\) 0 0
\(274\) 2.20101 0.132968
\(275\) −0.585786 −0.0353243
\(276\) 3.65685 0.220117
\(277\) −7.17157 −0.430898 −0.215449 0.976515i \(-0.569122\pi\)
−0.215449 + 0.976515i \(0.569122\pi\)
\(278\) 5.17157 0.310170
\(279\) 10.2426 0.613211
\(280\) −1.31371 −0.0785091
\(281\) 17.7990 1.06180 0.530899 0.847435i \(-0.321854\pi\)
0.530899 + 0.847435i \(0.321854\pi\)
\(282\) 0.485281 0.0288981
\(283\) 8.72792 0.518821 0.259411 0.965767i \(-0.416472\pi\)
0.259411 + 0.965767i \(0.416472\pi\)
\(284\) −14.4437 −0.857073
\(285\) 4.82843 0.286011
\(286\) 0 0
\(287\) 7.31371 0.431715
\(288\) 4.41421 0.260110
\(289\) 6.31371 0.371395
\(290\) 2.34315 0.137594
\(291\) −5.17157 −0.303163
\(292\) −15.5147 −0.907930
\(293\) 2.14214 0.125145 0.0625724 0.998040i \(-0.480070\pi\)
0.0625724 + 0.998040i \(0.480070\pi\)
\(294\) 3.69848 0.215700
\(295\) 10.2426 0.596350
\(296\) −13.4558 −0.782105
\(297\) 3.31371 0.192281
\(298\) −0.142136 −0.00823370
\(299\) 0 0
\(300\) −2.58579 −0.149290
\(301\) 2.54416 0.146643
\(302\) 7.55635 0.434819
\(303\) 10.8284 0.622077
\(304\) −10.2426 −0.587456
\(305\) 8.00000 0.458079
\(306\) −2.00000 −0.114332
\(307\) −19.1716 −1.09418 −0.547090 0.837074i \(-0.684264\pi\)
−0.547090 + 0.837074i \(0.684264\pi\)
\(308\) 0.887302 0.0505587
\(309\) 24.6274 1.40100
\(310\) −4.24264 −0.240966
\(311\) −8.48528 −0.481156 −0.240578 0.970630i \(-0.577337\pi\)
−0.240578 + 0.970630i \(0.577337\pi\)
\(312\) 0 0
\(313\) 0.828427 0.0468255 0.0234127 0.999726i \(-0.492547\pi\)
0.0234127 + 0.999726i \(0.492547\pi\)
\(314\) −7.45584 −0.420758
\(315\) 0.828427 0.0466766
\(316\) −15.5147 −0.872771
\(317\) −26.1421 −1.46829 −0.734144 0.678993i \(-0.762416\pi\)
−0.734144 + 0.678993i \(0.762416\pi\)
\(318\) 8.48528 0.475831
\(319\) −3.31371 −0.185532
\(320\) 4.17157 0.233198
\(321\) −9.31371 −0.519841
\(322\) 0.485281 0.0270437
\(323\) 16.4853 0.917266
\(324\) 9.14214 0.507896
\(325\) 0 0
\(326\) −6.20101 −0.343442
\(327\) 2.82843 0.156412
\(328\) 14.0000 0.773021
\(329\) −0.686292 −0.0378365
\(330\) −0.343146 −0.0188896
\(331\) −22.0416 −1.21152 −0.605759 0.795648i \(-0.707130\pi\)
−0.605759 + 0.795648i \(0.707130\pi\)
\(332\) −16.1421 −0.885915
\(333\) 8.48528 0.464991
\(334\) −3.65685 −0.200094
\(335\) −2.00000 −0.109272
\(336\) 3.51472 0.191744
\(337\) 7.17157 0.390660 0.195330 0.980738i \(-0.437422\pi\)
0.195330 + 0.980738i \(0.437422\pi\)
\(338\) 0 0
\(339\) −4.48528 −0.243607
\(340\) −8.82843 −0.478789
\(341\) 6.00000 0.324918
\(342\) −1.41421 −0.0764719
\(343\) −11.0294 −0.595534
\(344\) 4.87006 0.262576
\(345\) 2.00000 0.107676
\(346\) −4.62742 −0.248771
\(347\) 4.24264 0.227757 0.113878 0.993495i \(-0.463673\pi\)
0.113878 + 0.993495i \(0.463673\pi\)
\(348\) −14.6274 −0.784112
\(349\) −1.51472 −0.0810810 −0.0405405 0.999178i \(-0.512908\pi\)
−0.0405405 + 0.999178i \(0.512908\pi\)
\(350\) −0.343146 −0.0183419
\(351\) 0 0
\(352\) 2.58579 0.137823
\(353\) −9.17157 −0.488154 −0.244077 0.969756i \(-0.578485\pi\)
−0.244077 + 0.969756i \(0.578485\pi\)
\(354\) 6.00000 0.318896
\(355\) −7.89949 −0.419262
\(356\) 10.9706 0.581439
\(357\) −5.65685 −0.299392
\(358\) 2.34315 0.123839
\(359\) 27.8995 1.47248 0.736240 0.676721i \(-0.236600\pi\)
0.736240 + 0.676721i \(0.236600\pi\)
\(360\) 1.58579 0.0835783
\(361\) −7.34315 −0.386481
\(362\) 0 0
\(363\) −15.0711 −0.791026
\(364\) 0 0
\(365\) −8.48528 −0.444140
\(366\) 4.68629 0.244956
\(367\) 4.44365 0.231957 0.115978 0.993252i \(-0.463000\pi\)
0.115978 + 0.993252i \(0.463000\pi\)
\(368\) −4.24264 −0.221163
\(369\) −8.82843 −0.459590
\(370\) −3.51472 −0.182722
\(371\) −12.0000 −0.623009
\(372\) 26.4853 1.37320
\(373\) −25.3137 −1.31069 −0.655347 0.755328i \(-0.727478\pi\)
−0.655347 + 0.755328i \(0.727478\pi\)
\(374\) −1.17157 −0.0605806
\(375\) −1.41421 −0.0730297
\(376\) −1.31371 −0.0677493
\(377\) 0 0
\(378\) 1.94113 0.0998407
\(379\) −14.9289 −0.766848 −0.383424 0.923572i \(-0.625255\pi\)
−0.383424 + 0.923572i \(0.625255\pi\)
\(380\) −6.24264 −0.320241
\(381\) −13.3137 −0.682082
\(382\) −5.65685 −0.289430
\(383\) −33.1127 −1.69198 −0.845990 0.533199i \(-0.820990\pi\)
−0.845990 + 0.533199i \(0.820990\pi\)
\(384\) 14.9289 0.761839
\(385\) 0.485281 0.0247322
\(386\) 6.48528 0.330092
\(387\) −3.07107 −0.156111
\(388\) 6.68629 0.339445
\(389\) −16.6274 −0.843044 −0.421522 0.906818i \(-0.638504\pi\)
−0.421522 + 0.906818i \(0.638504\pi\)
\(390\) 0 0
\(391\) 6.82843 0.345328
\(392\) −10.0122 −0.505692
\(393\) 24.0000 1.21064
\(394\) −9.51472 −0.479345
\(395\) −8.48528 −0.426941
\(396\) −1.07107 −0.0538232
\(397\) 27.7990 1.39519 0.697596 0.716492i \(-0.254253\pi\)
0.697596 + 0.716492i \(0.254253\pi\)
\(398\) −1.65685 −0.0830506
\(399\) −4.00000 −0.200250
\(400\) 3.00000 0.150000
\(401\) −17.3137 −0.864605 −0.432303 0.901729i \(-0.642299\pi\)
−0.432303 + 0.901729i \(0.642299\pi\)
\(402\) −1.17157 −0.0584327
\(403\) 0 0
\(404\) −14.0000 −0.696526
\(405\) 5.00000 0.248452
\(406\) −1.94113 −0.0963364
\(407\) 4.97056 0.246382
\(408\) −10.8284 −0.536087
\(409\) −12.8284 −0.634325 −0.317162 0.948371i \(-0.602730\pi\)
−0.317162 + 0.948371i \(0.602730\pi\)
\(410\) 3.65685 0.180599
\(411\) −7.51472 −0.370674
\(412\) −31.8406 −1.56867
\(413\) −8.48528 −0.417533
\(414\) −0.585786 −0.0287898
\(415\) −8.82843 −0.433370
\(416\) 0 0
\(417\) −17.6569 −0.864660
\(418\) −0.828427 −0.0405197
\(419\) 5.17157 0.252648 0.126324 0.991989i \(-0.459682\pi\)
0.126324 + 0.991989i \(0.459682\pi\)
\(420\) 2.14214 0.104526
\(421\) 1.02944 0.0501717 0.0250859 0.999685i \(-0.492014\pi\)
0.0250859 + 0.999685i \(0.492014\pi\)
\(422\) 8.00000 0.389434
\(423\) 0.828427 0.0402795
\(424\) −22.9706 −1.11555
\(425\) −4.82843 −0.234213
\(426\) −4.62742 −0.224199
\(427\) −6.62742 −0.320723
\(428\) 12.0416 0.582054
\(429\) 0 0
\(430\) 1.27208 0.0613450
\(431\) −3.61522 −0.174139 −0.0870696 0.996202i \(-0.527750\pi\)
−0.0870696 + 0.996202i \(0.527750\pi\)
\(432\) −16.9706 −0.816497
\(433\) −3.65685 −0.175737 −0.0878686 0.996132i \(-0.528006\pi\)
−0.0878686 + 0.996132i \(0.528006\pi\)
\(434\) 3.51472 0.168712
\(435\) −8.00000 −0.383571
\(436\) −3.65685 −0.175132
\(437\) 4.82843 0.230975
\(438\) −4.97056 −0.237503
\(439\) −32.9706 −1.57360 −0.786800 0.617209i \(-0.788263\pi\)
−0.786800 + 0.617209i \(0.788263\pi\)
\(440\) 0.928932 0.0442851
\(441\) 6.31371 0.300653
\(442\) 0 0
\(443\) −6.58579 −0.312900 −0.156450 0.987686i \(-0.550005\pi\)
−0.156450 + 0.987686i \(0.550005\pi\)
\(444\) 21.9411 1.04128
\(445\) 6.00000 0.284427
\(446\) 10.9706 0.519471
\(447\) 0.485281 0.0229530
\(448\) −3.45584 −0.163273
\(449\) −29.1127 −1.37391 −0.686957 0.726698i \(-0.741054\pi\)
−0.686957 + 0.726698i \(0.741054\pi\)
\(450\) 0.414214 0.0195262
\(451\) −5.17157 −0.243520
\(452\) 5.79899 0.272762
\(453\) −25.7990 −1.21214
\(454\) −11.4558 −0.537649
\(455\) 0 0
\(456\) −7.65685 −0.358565
\(457\) 18.0000 0.842004 0.421002 0.907060i \(-0.361678\pi\)
0.421002 + 0.907060i \(0.361678\pi\)
\(458\) 0.343146 0.0160341
\(459\) 27.3137 1.27489
\(460\) −2.58579 −0.120563
\(461\) −26.4853 −1.23354 −0.616771 0.787142i \(-0.711560\pi\)
−0.616771 + 0.787142i \(0.711560\pi\)
\(462\) 0.284271 0.0132255
\(463\) 15.6569 0.727636 0.363818 0.931470i \(-0.381473\pi\)
0.363818 + 0.931470i \(0.381473\pi\)
\(464\) 16.9706 0.787839
\(465\) 14.4853 0.671739
\(466\) −10.2010 −0.472553
\(467\) −10.5858 −0.489852 −0.244926 0.969542i \(-0.578764\pi\)
−0.244926 + 0.969542i \(0.578764\pi\)
\(468\) 0 0
\(469\) 1.65685 0.0765064
\(470\) −0.343146 −0.0158281
\(471\) 25.4558 1.17294
\(472\) −16.2426 −0.747628
\(473\) −1.79899 −0.0827176
\(474\) −4.97056 −0.228306
\(475\) −3.41421 −0.156655
\(476\) 7.31371 0.335223
\(477\) 14.4853 0.663235
\(478\) −0.242641 −0.0110981
\(479\) 5.27208 0.240887 0.120444 0.992720i \(-0.461568\pi\)
0.120444 + 0.992720i \(0.461568\pi\)
\(480\) 6.24264 0.284936
\(481\) 0 0
\(482\) 1.02944 0.0468896
\(483\) −1.65685 −0.0753895
\(484\) 19.4853 0.885695
\(485\) 3.65685 0.166049
\(486\) −4.10051 −0.186003
\(487\) −22.9706 −1.04090 −0.520448 0.853894i \(-0.674235\pi\)
−0.520448 + 0.853894i \(0.674235\pi\)
\(488\) −12.6863 −0.574281
\(489\) 21.1716 0.957412
\(490\) −2.61522 −0.118144
\(491\) 10.8284 0.488680 0.244340 0.969690i \(-0.421429\pi\)
0.244340 + 0.969690i \(0.421429\pi\)
\(492\) −22.8284 −1.02918
\(493\) −27.3137 −1.23015
\(494\) 0 0
\(495\) −0.585786 −0.0263291
\(496\) −30.7279 −1.37972
\(497\) 6.54416 0.293546
\(498\) −5.17157 −0.231744
\(499\) −10.4437 −0.467522 −0.233761 0.972294i \(-0.575103\pi\)
−0.233761 + 0.972294i \(0.575103\pi\)
\(500\) 1.82843 0.0817697
\(501\) 12.4853 0.557801
\(502\) 8.20101 0.366029
\(503\) 18.1005 0.807062 0.403531 0.914966i \(-0.367783\pi\)
0.403531 + 0.914966i \(0.367783\pi\)
\(504\) −1.31371 −0.0585172
\(505\) −7.65685 −0.340726
\(506\) −0.343146 −0.0152547
\(507\) 0 0
\(508\) 17.2132 0.763712
\(509\) 21.1127 0.935804 0.467902 0.883780i \(-0.345010\pi\)
0.467902 + 0.883780i \(0.345010\pi\)
\(510\) −2.82843 −0.125245
\(511\) 7.02944 0.310964
\(512\) −22.7574 −1.00574
\(513\) 19.3137 0.852721
\(514\) −6.76955 −0.298592
\(515\) −17.4142 −0.767362
\(516\) −7.94113 −0.349589
\(517\) 0.485281 0.0213427
\(518\) 2.91169 0.127932
\(519\) 15.7990 0.693499
\(520\) 0 0
\(521\) −6.34315 −0.277898 −0.138949 0.990300i \(-0.544372\pi\)
−0.138949 + 0.990300i \(0.544372\pi\)
\(522\) 2.34315 0.102557
\(523\) −28.2426 −1.23496 −0.617482 0.786585i \(-0.711847\pi\)
−0.617482 + 0.786585i \(0.711847\pi\)
\(524\) −31.0294 −1.35553
\(525\) 1.17157 0.0511316
\(526\) 5.55635 0.242268
\(527\) 49.4558 2.15433
\(528\) −2.48528 −0.108158
\(529\) −21.0000 −0.913043
\(530\) −6.00000 −0.260623
\(531\) 10.2426 0.444493
\(532\) 5.17157 0.224216
\(533\) 0 0
\(534\) 3.51472 0.152097
\(535\) 6.58579 0.284728
\(536\) 3.17157 0.136991
\(537\) −8.00000 −0.345225
\(538\) 1.11270 0.0479718
\(539\) 3.69848 0.159305
\(540\) −10.3431 −0.445098
\(541\) 12.8284 0.551537 0.275769 0.961224i \(-0.411068\pi\)
0.275769 + 0.961224i \(0.411068\pi\)
\(542\) 0.526912 0.0226328
\(543\) 0 0
\(544\) 21.3137 0.913818
\(545\) −2.00000 −0.0856706
\(546\) 0 0
\(547\) −29.2132 −1.24907 −0.624533 0.780998i \(-0.714711\pi\)
−0.624533 + 0.780998i \(0.714711\pi\)
\(548\) 9.71573 0.415035
\(549\) 8.00000 0.341432
\(550\) 0.242641 0.0103462
\(551\) −19.3137 −0.822792
\(552\) −3.17157 −0.134991
\(553\) 7.02944 0.298922
\(554\) 2.97056 0.126207
\(555\) 12.0000 0.509372
\(556\) 22.8284 0.968141
\(557\) −3.79899 −0.160968 −0.0804842 0.996756i \(-0.525647\pi\)
−0.0804842 + 0.996756i \(0.525647\pi\)
\(558\) −4.24264 −0.179605
\(559\) 0 0
\(560\) −2.48528 −0.105022
\(561\) 4.00000 0.168880
\(562\) −7.37258 −0.310994
\(563\) −16.2426 −0.684546 −0.342273 0.939601i \(-0.611197\pi\)
−0.342273 + 0.939601i \(0.611197\pi\)
\(564\) 2.14214 0.0902002
\(565\) 3.17157 0.133429
\(566\) −3.61522 −0.151959
\(567\) −4.14214 −0.173953
\(568\) 12.5269 0.525618
\(569\) −21.6569 −0.907903 −0.453951 0.891027i \(-0.649986\pi\)
−0.453951 + 0.891027i \(0.649986\pi\)
\(570\) −2.00000 −0.0837708
\(571\) −28.4853 −1.19207 −0.596036 0.802958i \(-0.703258\pi\)
−0.596036 + 0.802958i \(0.703258\pi\)
\(572\) 0 0
\(573\) 19.3137 0.806842
\(574\) −3.02944 −0.126446
\(575\) −1.41421 −0.0589768
\(576\) 4.17157 0.173816
\(577\) 29.1716 1.21443 0.607214 0.794538i \(-0.292287\pi\)
0.607214 + 0.794538i \(0.292287\pi\)
\(578\) −2.61522 −0.108779
\(579\) −22.1421 −0.920196
\(580\) 10.3431 0.429476
\(581\) 7.31371 0.303424
\(582\) 2.14214 0.0887944
\(583\) 8.48528 0.351424
\(584\) 13.4558 0.556807
\(585\) 0 0
\(586\) −0.887302 −0.0366541
\(587\) −31.6569 −1.30662 −0.653309 0.757091i \(-0.726620\pi\)
−0.653309 + 0.757091i \(0.726620\pi\)
\(588\) 16.3259 0.673269
\(589\) 34.9706 1.44094
\(590\) −4.24264 −0.174667
\(591\) 32.4853 1.33627
\(592\) −25.4558 −1.04623
\(593\) −20.6274 −0.847066 −0.423533 0.905881i \(-0.639210\pi\)
−0.423533 + 0.905881i \(0.639210\pi\)
\(594\) −1.37258 −0.0563178
\(595\) 4.00000 0.163984
\(596\) −0.627417 −0.0257000
\(597\) 5.65685 0.231520
\(598\) 0 0
\(599\) −25.4558 −1.04010 −0.520049 0.854137i \(-0.674086\pi\)
−0.520049 + 0.854137i \(0.674086\pi\)
\(600\) 2.24264 0.0915554
\(601\) 0.627417 0.0255929 0.0127964 0.999918i \(-0.495927\pi\)
0.0127964 + 0.999918i \(0.495927\pi\)
\(602\) −1.05382 −0.0429507
\(603\) −2.00000 −0.0814463
\(604\) 33.3553 1.35721
\(605\) 10.6569 0.433263
\(606\) −4.48528 −0.182202
\(607\) 40.2426 1.63340 0.816699 0.577064i \(-0.195802\pi\)
0.816699 + 0.577064i \(0.195802\pi\)
\(608\) 15.0711 0.611213
\(609\) 6.62742 0.268556
\(610\) −3.31371 −0.134168
\(611\) 0 0
\(612\) −8.82843 −0.356868
\(613\) 37.3137 1.50709 0.753543 0.657398i \(-0.228343\pi\)
0.753543 + 0.657398i \(0.228343\pi\)
\(614\) 7.94113 0.320478
\(615\) −12.4853 −0.503455
\(616\) −0.769553 −0.0310062
\(617\) 22.9706 0.924760 0.462380 0.886682i \(-0.346996\pi\)
0.462380 + 0.886682i \(0.346996\pi\)
\(618\) −10.2010 −0.410345
\(619\) −10.2426 −0.411686 −0.205843 0.978585i \(-0.565994\pi\)
−0.205843 + 0.978585i \(0.565994\pi\)
\(620\) −18.7279 −0.752131
\(621\) 8.00000 0.321029
\(622\) 3.51472 0.140927
\(623\) −4.97056 −0.199141
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) −0.343146 −0.0137149
\(627\) 2.82843 0.112956
\(628\) −32.9117 −1.31332
\(629\) 40.9706 1.63360
\(630\) −0.343146 −0.0136713
\(631\) 18.2426 0.726228 0.363114 0.931745i \(-0.381714\pi\)
0.363114 + 0.931745i \(0.381714\pi\)
\(632\) 13.4558 0.535245
\(633\) −27.3137 −1.08562
\(634\) 10.8284 0.430052
\(635\) 9.41421 0.373592
\(636\) 37.4558 1.48522
\(637\) 0 0
\(638\) 1.37258 0.0543411
\(639\) −7.89949 −0.312499
\(640\) −10.5563 −0.417276
\(641\) 36.3431 1.43547 0.717734 0.696317i \(-0.245179\pi\)
0.717734 + 0.696317i \(0.245179\pi\)
\(642\) 3.85786 0.152258
\(643\) −26.4853 −1.04448 −0.522239 0.852799i \(-0.674903\pi\)
−0.522239 + 0.852799i \(0.674903\pi\)
\(644\) 2.14214 0.0844120
\(645\) −4.34315 −0.171011
\(646\) −6.82843 −0.268661
\(647\) 6.58579 0.258914 0.129457 0.991585i \(-0.458677\pi\)
0.129457 + 0.991585i \(0.458677\pi\)
\(648\) −7.92893 −0.311478
\(649\) 6.00000 0.235521
\(650\) 0 0
\(651\) −12.0000 −0.470317
\(652\) −27.3726 −1.07199
\(653\) 13.0294 0.509881 0.254941 0.966957i \(-0.417944\pi\)
0.254941 + 0.966957i \(0.417944\pi\)
\(654\) −1.17157 −0.0458121
\(655\) −16.9706 −0.663095
\(656\) 26.4853 1.03408
\(657\) −8.48528 −0.331042
\(658\) 0.284271 0.0110820
\(659\) 46.1421 1.79744 0.898721 0.438520i \(-0.144497\pi\)
0.898721 + 0.438520i \(0.144497\pi\)
\(660\) −1.51472 −0.0589603
\(661\) 49.5980 1.92914 0.964569 0.263831i \(-0.0849861\pi\)
0.964569 + 0.263831i \(0.0849861\pi\)
\(662\) 9.12994 0.354845
\(663\) 0 0
\(664\) 14.0000 0.543305
\(665\) 2.82843 0.109682
\(666\) −3.51472 −0.136193
\(667\) −8.00000 −0.309761
\(668\) −16.1421 −0.624558
\(669\) −37.4558 −1.44813
\(670\) 0.828427 0.0320049
\(671\) 4.68629 0.180912
\(672\) −5.17157 −0.199498
\(673\) −10.4853 −0.404178 −0.202089 0.979367i \(-0.564773\pi\)
−0.202089 + 0.979367i \(0.564773\pi\)
\(674\) −2.97056 −0.114422
\(675\) −5.65685 −0.217732
\(676\) 0 0
\(677\) 8.14214 0.312928 0.156464 0.987684i \(-0.449991\pi\)
0.156464 + 0.987684i \(0.449991\pi\)
\(678\) 1.85786 0.0713509
\(679\) −3.02944 −0.116259
\(680\) 7.65685 0.293627
\(681\) 39.1127 1.49880
\(682\) −2.48528 −0.0951663
\(683\) −33.3137 −1.27471 −0.637357 0.770569i \(-0.719972\pi\)
−0.637357 + 0.770569i \(0.719972\pi\)
\(684\) −6.24264 −0.238693
\(685\) 5.31371 0.203026
\(686\) 4.56854 0.174428
\(687\) −1.17157 −0.0446983
\(688\) 9.21320 0.351250
\(689\) 0 0
\(690\) −0.828427 −0.0315377
\(691\) −21.0711 −0.801581 −0.400791 0.916170i \(-0.631265\pi\)
−0.400791 + 0.916170i \(0.631265\pi\)
\(692\) −20.4264 −0.776495
\(693\) 0.485281 0.0184343
\(694\) −1.75736 −0.0667084
\(695\) 12.4853 0.473594
\(696\) 12.6863 0.480873
\(697\) −42.6274 −1.61463
\(698\) 0.627417 0.0237481
\(699\) 34.8284 1.31733
\(700\) −1.51472 −0.0572510
\(701\) 37.3137 1.40932 0.704660 0.709545i \(-0.251100\pi\)
0.704660 + 0.709545i \(0.251100\pi\)
\(702\) 0 0
\(703\) 28.9706 1.09265
\(704\) 2.44365 0.0920986
\(705\) 1.17157 0.0441240
\(706\) 3.79899 0.142977
\(707\) 6.34315 0.238559
\(708\) 26.4853 0.995378
\(709\) 17.1127 0.642681 0.321340 0.946964i \(-0.395867\pi\)
0.321340 + 0.946964i \(0.395867\pi\)
\(710\) 3.27208 0.122799
\(711\) −8.48528 −0.318223
\(712\) −9.51472 −0.356579
\(713\) 14.4853 0.542478
\(714\) 2.34315 0.0876900
\(715\) 0 0
\(716\) 10.3431 0.386542
\(717\) 0.828427 0.0309382
\(718\) −11.5563 −0.431279
\(719\) −4.97056 −0.185371 −0.0926854 0.995695i \(-0.529545\pi\)
−0.0926854 + 0.995695i \(0.529545\pi\)
\(720\) 3.00000 0.111803
\(721\) 14.4264 0.537267
\(722\) 3.04163 0.113198
\(723\) −3.51472 −0.130714
\(724\) 0 0
\(725\) 5.65685 0.210090
\(726\) 6.24264 0.231686
\(727\) 19.3553 0.717850 0.358925 0.933366i \(-0.383143\pi\)
0.358925 + 0.933366i \(0.383143\pi\)
\(728\) 0 0
\(729\) 29.0000 1.07407
\(730\) 3.51472 0.130086
\(731\) −14.8284 −0.548449
\(732\) 20.6863 0.764587
\(733\) 1.31371 0.0485229 0.0242615 0.999706i \(-0.492277\pi\)
0.0242615 + 0.999706i \(0.492277\pi\)
\(734\) −1.84062 −0.0679385
\(735\) 8.92893 0.329349
\(736\) 6.24264 0.230107
\(737\) −1.17157 −0.0431554
\(738\) 3.65685 0.134611
\(739\) −30.7279 −1.13034 −0.565172 0.824973i \(-0.691190\pi\)
−0.565172 + 0.824973i \(0.691190\pi\)
\(740\) −15.5147 −0.570332
\(741\) 0 0
\(742\) 4.97056 0.182475
\(743\) 38.4853 1.41189 0.705944 0.708268i \(-0.250523\pi\)
0.705944 + 0.708268i \(0.250523\pi\)
\(744\) −22.9706 −0.842142
\(745\) −0.343146 −0.0125719
\(746\) 10.4853 0.383893
\(747\) −8.82843 −0.323015
\(748\) −5.17157 −0.189091
\(749\) −5.45584 −0.199352
\(750\) 0.585786 0.0213899
\(751\) −44.4853 −1.62329 −0.811645 0.584150i \(-0.801428\pi\)
−0.811645 + 0.584150i \(0.801428\pi\)
\(752\) −2.48528 −0.0906289
\(753\) −28.0000 −1.02038
\(754\) 0 0
\(755\) 18.2426 0.663918
\(756\) 8.56854 0.311635
\(757\) 4.14214 0.150548 0.0752742 0.997163i \(-0.476017\pi\)
0.0752742 + 0.997163i \(0.476017\pi\)
\(758\) 6.18377 0.224605
\(759\) 1.17157 0.0425254
\(760\) 5.41421 0.196394
\(761\) 36.6274 1.32774 0.663871 0.747847i \(-0.268912\pi\)
0.663871 + 0.747847i \(0.268912\pi\)
\(762\) 5.51472 0.199777
\(763\) 1.65685 0.0599822
\(764\) −24.9706 −0.903403
\(765\) −4.82843 −0.174572
\(766\) 13.7157 0.495569
\(767\) 0 0
\(768\) 5.61522 0.202622
\(769\) −10.9706 −0.395609 −0.197804 0.980242i \(-0.563381\pi\)
−0.197804 + 0.980242i \(0.563381\pi\)
\(770\) −0.201010 −0.00724390
\(771\) 23.1127 0.832383
\(772\) 28.6274 1.03032
\(773\) −6.14214 −0.220917 −0.110459 0.993881i \(-0.535232\pi\)
−0.110459 + 0.993881i \(0.535232\pi\)
\(774\) 1.27208 0.0457239
\(775\) −10.2426 −0.367927
\(776\) −5.79899 −0.208172
\(777\) −9.94113 −0.356636
\(778\) 6.88730 0.246922
\(779\) −30.1421 −1.07995
\(780\) 0 0
\(781\) −4.62742 −0.165582
\(782\) −2.82843 −0.101144
\(783\) −32.0000 −1.14359
\(784\) −18.9411 −0.676469
\(785\) −18.0000 −0.642448
\(786\) −9.94113 −0.354588
\(787\) −5.51472 −0.196578 −0.0982892 0.995158i \(-0.531337\pi\)
−0.0982892 + 0.995158i \(0.531337\pi\)
\(788\) −42.0000 −1.49619
\(789\) −18.9706 −0.675370
\(790\) 3.51472 0.125048
\(791\) −2.62742 −0.0934202
\(792\) 0.928932 0.0330082
\(793\) 0 0
\(794\) −11.5147 −0.408642
\(795\) 20.4853 0.726538
\(796\) −7.31371 −0.259228
\(797\) 10.9706 0.388597 0.194299 0.980942i \(-0.437757\pi\)
0.194299 + 0.980942i \(0.437757\pi\)
\(798\) 1.65685 0.0586520
\(799\) 4.00000 0.141510
\(800\) −4.41421 −0.156066
\(801\) 6.00000 0.212000
\(802\) 7.17157 0.253237
\(803\) −4.97056 −0.175407
\(804\) −5.17157 −0.182387
\(805\) 1.17157 0.0412925
\(806\) 0 0
\(807\) −3.79899 −0.133731
\(808\) 12.1421 0.427159
\(809\) −45.2548 −1.59108 −0.795538 0.605904i \(-0.792811\pi\)
−0.795538 + 0.605904i \(0.792811\pi\)
\(810\) −2.07107 −0.0727699
\(811\) −8.38478 −0.294429 −0.147215 0.989105i \(-0.547031\pi\)
−0.147215 + 0.989105i \(0.547031\pi\)
\(812\) −8.56854 −0.300697
\(813\) −1.79899 −0.0630933
\(814\) −2.05887 −0.0721635
\(815\) −14.9706 −0.524396
\(816\) −20.4853 −0.717128
\(817\) −10.4853 −0.366834
\(818\) 5.31371 0.185789
\(819\) 0 0
\(820\) 16.1421 0.563708
\(821\) −39.2548 −1.37000 −0.685002 0.728542i \(-0.740199\pi\)
−0.685002 + 0.728542i \(0.740199\pi\)
\(822\) 3.11270 0.108568
\(823\) 34.3848 1.19858 0.599289 0.800533i \(-0.295450\pi\)
0.599289 + 0.800533i \(0.295450\pi\)
\(824\) 27.6152 0.962022
\(825\) −0.828427 −0.0288421
\(826\) 3.51472 0.122293
\(827\) −27.8579 −0.968713 −0.484356 0.874871i \(-0.660946\pi\)
−0.484356 + 0.874871i \(0.660946\pi\)
\(828\) −2.58579 −0.0898623
\(829\) 7.02944 0.244142 0.122071 0.992521i \(-0.461046\pi\)
0.122071 + 0.992521i \(0.461046\pi\)
\(830\) 3.65685 0.126931
\(831\) −10.1421 −0.351827
\(832\) 0 0
\(833\) 30.4853 1.05625
\(834\) 7.31371 0.253253
\(835\) −8.82843 −0.305520
\(836\) −3.65685 −0.126475
\(837\) 57.9411 2.00274
\(838\) −2.14214 −0.0739988
\(839\) 18.7279 0.646560 0.323280 0.946303i \(-0.395214\pi\)
0.323280 + 0.946303i \(0.395214\pi\)
\(840\) −1.85786 −0.0641024
\(841\) 3.00000 0.103448
\(842\) −0.426407 −0.0146950
\(843\) 25.1716 0.866955
\(844\) 35.3137 1.21555
\(845\) 0 0
\(846\) −0.343146 −0.0117976
\(847\) −8.82843 −0.303348
\(848\) −43.4558 −1.49228
\(849\) 12.3431 0.423616
\(850\) 2.00000 0.0685994
\(851\) 12.0000 0.411355
\(852\) −20.4264 −0.699797
\(853\) −37.4558 −1.28246 −0.641232 0.767347i \(-0.721576\pi\)
−0.641232 + 0.767347i \(0.721576\pi\)
\(854\) 2.74517 0.0939376
\(855\) −3.41421 −0.116764
\(856\) −10.4437 −0.356957
\(857\) 0.343146 0.0117216 0.00586082 0.999983i \(-0.498134\pi\)
0.00586082 + 0.999983i \(0.498134\pi\)
\(858\) 0 0
\(859\) 11.7990 0.402576 0.201288 0.979532i \(-0.435487\pi\)
0.201288 + 0.979532i \(0.435487\pi\)
\(860\) 5.61522 0.191478
\(861\) 10.3431 0.352493
\(862\) 1.49747 0.0510042
\(863\) −19.4558 −0.662285 −0.331142 0.943581i \(-0.607434\pi\)
−0.331142 + 0.943581i \(0.607434\pi\)
\(864\) 24.9706 0.849516
\(865\) −11.1716 −0.379845
\(866\) 1.51472 0.0514722
\(867\) 8.92893 0.303242
\(868\) 15.5147 0.526604
\(869\) −4.97056 −0.168615
\(870\) 3.31371 0.112345
\(871\) 0 0
\(872\) 3.17157 0.107403
\(873\) 3.65685 0.123766
\(874\) −2.00000 −0.0676510
\(875\) −0.828427 −0.0280059
\(876\) −21.9411 −0.741322
\(877\) −2.68629 −0.0907096 −0.0453548 0.998971i \(-0.514442\pi\)
−0.0453548 + 0.998971i \(0.514442\pi\)
\(878\) 13.6569 0.460897
\(879\) 3.02944 0.102180
\(880\) 1.75736 0.0592406
\(881\) 52.9706 1.78462 0.892312 0.451420i \(-0.149082\pi\)
0.892312 + 0.451420i \(0.149082\pi\)
\(882\) −2.61522 −0.0880592
\(883\) −32.2426 −1.08505 −0.542526 0.840039i \(-0.682532\pi\)
−0.542526 + 0.840039i \(0.682532\pi\)
\(884\) 0 0
\(885\) 14.4853 0.486917
\(886\) 2.72792 0.0916463
\(887\) 14.3848 0.482994 0.241497 0.970402i \(-0.422362\pi\)
0.241497 + 0.970402i \(0.422362\pi\)
\(888\) −19.0294 −0.638586
\(889\) −7.79899 −0.261570
\(890\) −2.48528 −0.0833068
\(891\) 2.92893 0.0981229
\(892\) 48.4264 1.62144
\(893\) 2.82843 0.0946497
\(894\) −0.201010 −0.00672278
\(895\) 5.65685 0.189088
\(896\) 8.74517 0.292155
\(897\) 0 0
\(898\) 12.0589 0.402410
\(899\) −57.9411 −1.93244
\(900\) 1.82843 0.0609476
\(901\) 69.9411 2.33008
\(902\) 2.14214 0.0713253
\(903\) 3.59798 0.119733
\(904\) −5.02944 −0.167277
\(905\) 0 0
\(906\) 10.6863 0.355028
\(907\) −33.2132 −1.10283 −0.551413 0.834232i \(-0.685911\pi\)
−0.551413 + 0.834232i \(0.685911\pi\)
\(908\) −50.5685 −1.67818
\(909\) −7.65685 −0.253962
\(910\) 0 0
\(911\) −12.0000 −0.397578 −0.198789 0.980042i \(-0.563701\pi\)
−0.198789 + 0.980042i \(0.563701\pi\)
\(912\) −14.4853 −0.479656
\(913\) −5.17157 −0.171154
\(914\) −7.45584 −0.246617
\(915\) 11.3137 0.374020
\(916\) 1.51472 0.0500477
\(917\) 14.0589 0.464265
\(918\) −11.3137 −0.373408
\(919\) −16.4853 −0.543799 −0.271900 0.962326i \(-0.587652\pi\)
−0.271900 + 0.962326i \(0.587652\pi\)
\(920\) 2.24264 0.0739377
\(921\) −27.1127 −0.893394
\(922\) 10.9706 0.361296
\(923\) 0 0
\(924\) 1.25483 0.0412810
\(925\) −8.48528 −0.278994
\(926\) −6.48528 −0.213120
\(927\) −17.4142 −0.571958
\(928\) −24.9706 −0.819699
\(929\) −11.1716 −0.366527 −0.183264 0.983064i \(-0.558666\pi\)
−0.183264 + 0.983064i \(0.558666\pi\)
\(930\) −6.00000 −0.196748
\(931\) 21.5563 0.706481
\(932\) −45.0294 −1.47499
\(933\) −12.0000 −0.392862
\(934\) 4.38478 0.143474
\(935\) −2.82843 −0.0924995
\(936\) 0 0
\(937\) 10.9706 0.358393 0.179196 0.983813i \(-0.442650\pi\)
0.179196 + 0.983813i \(0.442650\pi\)
\(938\) −0.686292 −0.0224082
\(939\) 1.17157 0.0382328
\(940\) −1.51472 −0.0494047
\(941\) 54.7696 1.78544 0.892718 0.450615i \(-0.148795\pi\)
0.892718 + 0.450615i \(0.148795\pi\)
\(942\) −10.5442 −0.343547
\(943\) −12.4853 −0.406577
\(944\) −30.7279 −1.00011
\(945\) 4.68629 0.152445
\(946\) 0.745166 0.0242274
\(947\) 45.1127 1.46597 0.732983 0.680247i \(-0.238128\pi\)
0.732983 + 0.680247i \(0.238128\pi\)
\(948\) −21.9411 −0.712615
\(949\) 0 0
\(950\) 1.41421 0.0458831
\(951\) −36.9706 −1.19885
\(952\) −6.34315 −0.205583
\(953\) −55.2548 −1.78988 −0.894940 0.446187i \(-0.852782\pi\)
−0.894940 + 0.446187i \(0.852782\pi\)
\(954\) −6.00000 −0.194257
\(955\) −13.6569 −0.441925
\(956\) −1.07107 −0.0346408
\(957\) −4.68629 −0.151486
\(958\) −2.18377 −0.0705543
\(959\) −4.40202 −0.142149
\(960\) 5.89949 0.190405
\(961\) 73.9117 2.38425
\(962\) 0 0
\(963\) 6.58579 0.212224
\(964\) 4.54416 0.146357
\(965\) 15.6569 0.504012
\(966\) 0.686292 0.0220811
\(967\) 19.9411 0.641263 0.320632 0.947204i \(-0.396105\pi\)
0.320632 + 0.947204i \(0.396105\pi\)
\(968\) −16.8995 −0.543170
\(969\) 23.3137 0.748944
\(970\) −1.51472 −0.0486347
\(971\) −12.2843 −0.394221 −0.197111 0.980381i \(-0.563156\pi\)
−0.197111 + 0.980381i \(0.563156\pi\)
\(972\) −18.1005 −0.580574
\(973\) −10.3431 −0.331586
\(974\) 9.51472 0.304871
\(975\) 0 0
\(976\) −24.0000 −0.768221
\(977\) 56.4853 1.80712 0.903562 0.428457i \(-0.140943\pi\)
0.903562 + 0.428457i \(0.140943\pi\)
\(978\) −8.76955 −0.280419
\(979\) 3.51472 0.112331
\(980\) −11.5442 −0.368765
\(981\) −2.00000 −0.0638551
\(982\) −4.48528 −0.143131
\(983\) 34.9706 1.11539 0.557694 0.830047i \(-0.311686\pi\)
0.557694 + 0.830047i \(0.311686\pi\)
\(984\) 19.7990 0.631169
\(985\) −22.9706 −0.731903
\(986\) 11.3137 0.360302
\(987\) −0.970563 −0.0308934
\(988\) 0 0
\(989\) −4.34315 −0.138104
\(990\) 0.242641 0.00771163
\(991\) 15.0294 0.477426 0.238713 0.971090i \(-0.423275\pi\)
0.238713 + 0.971090i \(0.423275\pi\)
\(992\) 45.2132 1.43552
\(993\) −31.1716 −0.989200
\(994\) −2.71068 −0.0859775
\(995\) −4.00000 −0.126809
\(996\) −22.8284 −0.723346
\(997\) 23.1716 0.733851 0.366926 0.930250i \(-0.380410\pi\)
0.366926 + 0.930250i \(0.380410\pi\)
\(998\) 4.32590 0.136934
\(999\) 48.0000 1.51865
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 845.2.a.g.1.1 2
3.2 odd 2 7605.2.a.x.1.2 2
5.4 even 2 4225.2.a.r.1.2 2
13.2 odd 12 845.2.m.f.316.2 8
13.3 even 3 845.2.e.c.191.2 4
13.4 even 6 845.2.e.h.146.1 4
13.5 odd 4 845.2.c.b.506.3 4
13.6 odd 12 845.2.m.f.361.3 8
13.7 odd 12 845.2.m.f.361.2 8
13.8 odd 4 845.2.c.b.506.2 4
13.9 even 3 845.2.e.c.146.2 4
13.10 even 6 845.2.e.h.191.1 4
13.11 odd 12 845.2.m.f.316.3 8
13.12 even 2 65.2.a.b.1.2 2
39.38 odd 2 585.2.a.m.1.1 2
52.51 odd 2 1040.2.a.j.1.1 2
65.12 odd 4 325.2.b.f.274.3 4
65.38 odd 4 325.2.b.f.274.2 4
65.64 even 2 325.2.a.i.1.1 2
91.90 odd 2 3185.2.a.j.1.2 2
104.51 odd 2 4160.2.a.z.1.2 2
104.77 even 2 4160.2.a.bf.1.1 2
143.142 odd 2 7865.2.a.j.1.1 2
156.155 even 2 9360.2.a.cd.1.2 2
195.38 even 4 2925.2.c.r.2224.3 4
195.77 even 4 2925.2.c.r.2224.2 4
195.194 odd 2 2925.2.a.u.1.2 2
260.259 odd 2 5200.2.a.bu.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
65.2.a.b.1.2 2 13.12 even 2
325.2.a.i.1.1 2 65.64 even 2
325.2.b.f.274.2 4 65.38 odd 4
325.2.b.f.274.3 4 65.12 odd 4
585.2.a.m.1.1 2 39.38 odd 2
845.2.a.g.1.1 2 1.1 even 1 trivial
845.2.c.b.506.2 4 13.8 odd 4
845.2.c.b.506.3 4 13.5 odd 4
845.2.e.c.146.2 4 13.9 even 3
845.2.e.c.191.2 4 13.3 even 3
845.2.e.h.146.1 4 13.4 even 6
845.2.e.h.191.1 4 13.10 even 6
845.2.m.f.316.2 8 13.2 odd 12
845.2.m.f.316.3 8 13.11 odd 12
845.2.m.f.361.2 8 13.7 odd 12
845.2.m.f.361.3 8 13.6 odd 12
1040.2.a.j.1.1 2 52.51 odd 2
2925.2.a.u.1.2 2 195.194 odd 2
2925.2.c.r.2224.2 4 195.77 even 4
2925.2.c.r.2224.3 4 195.38 even 4
3185.2.a.j.1.2 2 91.90 odd 2
4160.2.a.z.1.2 2 104.51 odd 2
4160.2.a.bf.1.1 2 104.77 even 2
4225.2.a.r.1.2 2 5.4 even 2
5200.2.a.bu.1.2 2 260.259 odd 2
7605.2.a.x.1.2 2 3.2 odd 2
7865.2.a.j.1.1 2 143.142 odd 2
9360.2.a.cd.1.2 2 156.155 even 2