Properties

Label 4640.2.a.g.1.2
Level $4640$
Weight $2$
Character 4640.1
Self dual yes
Analytic conductor $37.051$
Analytic rank $0$
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] = [4640,2,Mod(1,4640)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4640, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 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("4640.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4640 = 2^{5} \cdot 5 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4640.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: \(37.0505865379\)
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: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-0.618034\) of defining polynomial
Character \(\chi\) \(=\) 4640.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+0.618034 q^{3} -1.00000 q^{5} -2.85410 q^{7} -2.61803 q^{9} -3.23607 q^{11} -5.09017 q^{13} -0.618034 q^{15} -3.61803 q^{17} +2.76393 q^{19} -1.76393 q^{21} +5.85410 q^{23} +1.00000 q^{25} -3.47214 q^{27} +1.00000 q^{29} -1.61803 q^{31} -2.00000 q^{33} +2.85410 q^{35} +9.23607 q^{37} -3.14590 q^{39} -3.70820 q^{41} -7.61803 q^{43} +2.61803 q^{45} -8.00000 q^{47} +1.14590 q^{49} -2.23607 q^{51} +9.32624 q^{53} +3.23607 q^{55} +1.70820 q^{57} -9.38197 q^{59} +2.14590 q^{61} +7.47214 q^{63} +5.09017 q^{65} -2.47214 q^{67} +3.61803 q^{69} +12.9443 q^{71} +1.90983 q^{73} +0.618034 q^{75} +9.23607 q^{77} -2.90983 q^{79} +5.70820 q^{81} +15.7082 q^{83} +3.61803 q^{85} +0.618034 q^{87} -1.70820 q^{89} +14.5279 q^{91} -1.00000 q^{93} -2.76393 q^{95} +8.79837 q^{97} +8.47214 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{3} - 2 q^{5} + q^{7} - 3 q^{9} - 2 q^{11} + q^{13} + q^{15} - 5 q^{17} + 10 q^{19} - 8 q^{21} + 5 q^{23} + 2 q^{25} + 2 q^{27} + 2 q^{29} - q^{31} - 4 q^{33} - q^{35} + 14 q^{37} - 13 q^{39}+ \cdots + 8 q^{99}+O(q^{100}) \) 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 0
\(3\) 0.618034 0.356822 0.178411 0.983956i \(-0.442904\pi\)
0.178411 + 0.983956i \(0.442904\pi\)
\(4\) 0 0
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) −2.85410 −1.07875 −0.539375 0.842066i \(-0.681339\pi\)
−0.539375 + 0.842066i \(0.681339\pi\)
\(8\) 0 0
\(9\) −2.61803 −0.872678
\(10\) 0 0
\(11\) −3.23607 −0.975711 −0.487856 0.872924i \(-0.662221\pi\)
−0.487856 + 0.872924i \(0.662221\pi\)
\(12\) 0 0
\(13\) −5.09017 −1.41176 −0.705880 0.708332i \(-0.749448\pi\)
−0.705880 + 0.708332i \(0.749448\pi\)
\(14\) 0 0
\(15\) −0.618034 −0.159576
\(16\) 0 0
\(17\) −3.61803 −0.877502 −0.438751 0.898609i \(-0.644579\pi\)
−0.438751 + 0.898609i \(0.644579\pi\)
\(18\) 0 0
\(19\) 2.76393 0.634089 0.317045 0.948411i \(-0.397309\pi\)
0.317045 + 0.948411i \(0.397309\pi\)
\(20\) 0 0
\(21\) −1.76393 −0.384922
\(22\) 0 0
\(23\) 5.85410 1.22066 0.610332 0.792145i \(-0.291036\pi\)
0.610332 + 0.792145i \(0.291036\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) −3.47214 −0.668213
\(28\) 0 0
\(29\) 1.00000 0.185695
\(30\) 0 0
\(31\) −1.61803 −0.290607 −0.145304 0.989387i \(-0.546416\pi\)
−0.145304 + 0.989387i \(0.546416\pi\)
\(32\) 0 0
\(33\) −2.00000 −0.348155
\(34\) 0 0
\(35\) 2.85410 0.482431
\(36\) 0 0
\(37\) 9.23607 1.51840 0.759200 0.650857i \(-0.225590\pi\)
0.759200 + 0.650857i \(0.225590\pi\)
\(38\) 0 0
\(39\) −3.14590 −0.503747
\(40\) 0 0
\(41\) −3.70820 −0.579124 −0.289562 0.957159i \(-0.593510\pi\)
−0.289562 + 0.957159i \(0.593510\pi\)
\(42\) 0 0
\(43\) −7.61803 −1.16174 −0.580870 0.813997i \(-0.697287\pi\)
−0.580870 + 0.813997i \(0.697287\pi\)
\(44\) 0 0
\(45\) 2.61803 0.390273
\(46\) 0 0
\(47\) −8.00000 −1.16692 −0.583460 0.812142i \(-0.698301\pi\)
−0.583460 + 0.812142i \(0.698301\pi\)
\(48\) 0 0
\(49\) 1.14590 0.163700
\(50\) 0 0
\(51\) −2.23607 −0.313112
\(52\) 0 0
\(53\) 9.32624 1.28106 0.640529 0.767934i \(-0.278715\pi\)
0.640529 + 0.767934i \(0.278715\pi\)
\(54\) 0 0
\(55\) 3.23607 0.436351
\(56\) 0 0
\(57\) 1.70820 0.226257
\(58\) 0 0
\(59\) −9.38197 −1.22143 −0.610714 0.791851i \(-0.709117\pi\)
−0.610714 + 0.791851i \(0.709117\pi\)
\(60\) 0 0
\(61\) 2.14590 0.274754 0.137377 0.990519i \(-0.456133\pi\)
0.137377 + 0.990519i \(0.456133\pi\)
\(62\) 0 0
\(63\) 7.47214 0.941401
\(64\) 0 0
\(65\) 5.09017 0.631358
\(66\) 0 0
\(67\) −2.47214 −0.302019 −0.151010 0.988532i \(-0.548252\pi\)
−0.151010 + 0.988532i \(0.548252\pi\)
\(68\) 0 0
\(69\) 3.61803 0.435560
\(70\) 0 0
\(71\) 12.9443 1.53620 0.768101 0.640328i \(-0.221202\pi\)
0.768101 + 0.640328i \(0.221202\pi\)
\(72\) 0 0
\(73\) 1.90983 0.223529 0.111764 0.993735i \(-0.464350\pi\)
0.111764 + 0.993735i \(0.464350\pi\)
\(74\) 0 0
\(75\) 0.618034 0.0713644
\(76\) 0 0
\(77\) 9.23607 1.05255
\(78\) 0 0
\(79\) −2.90983 −0.327381 −0.163691 0.986512i \(-0.552340\pi\)
−0.163691 + 0.986512i \(0.552340\pi\)
\(80\) 0 0
\(81\) 5.70820 0.634245
\(82\) 0 0
\(83\) 15.7082 1.72420 0.862100 0.506739i \(-0.169149\pi\)
0.862100 + 0.506739i \(0.169149\pi\)
\(84\) 0 0
\(85\) 3.61803 0.392431
\(86\) 0 0
\(87\) 0.618034 0.0662602
\(88\) 0 0
\(89\) −1.70820 −0.181069 −0.0905346 0.995893i \(-0.528858\pi\)
−0.0905346 + 0.995893i \(0.528858\pi\)
\(90\) 0 0
\(91\) 14.5279 1.52293
\(92\) 0 0
\(93\) −1.00000 −0.103695
\(94\) 0 0
\(95\) −2.76393 −0.283573
\(96\) 0 0
\(97\) 8.79837 0.893340 0.446670 0.894699i \(-0.352610\pi\)
0.446670 + 0.894699i \(0.352610\pi\)
\(98\) 0 0
\(99\) 8.47214 0.851482
\(100\) 0 0
\(101\) −7.09017 −0.705498 −0.352749 0.935718i \(-0.614753\pi\)
−0.352749 + 0.935718i \(0.614753\pi\)
\(102\) 0 0
\(103\) 8.94427 0.881305 0.440653 0.897678i \(-0.354747\pi\)
0.440653 + 0.897678i \(0.354747\pi\)
\(104\) 0 0
\(105\) 1.76393 0.172142
\(106\) 0 0
\(107\) 0.291796 0.0282090 0.0141045 0.999901i \(-0.495510\pi\)
0.0141045 + 0.999901i \(0.495510\pi\)
\(108\) 0 0
\(109\) 11.7082 1.12144 0.560721 0.828005i \(-0.310524\pi\)
0.560721 + 0.828005i \(0.310524\pi\)
\(110\) 0 0
\(111\) 5.70820 0.541799
\(112\) 0 0
\(113\) 8.79837 0.827681 0.413841 0.910349i \(-0.364187\pi\)
0.413841 + 0.910349i \(0.364187\pi\)
\(114\) 0 0
\(115\) −5.85410 −0.545898
\(116\) 0 0
\(117\) 13.3262 1.23201
\(118\) 0 0
\(119\) 10.3262 0.946605
\(120\) 0 0
\(121\) −0.527864 −0.0479876
\(122\) 0 0
\(123\) −2.29180 −0.206644
\(124\) 0 0
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) −0.944272 −0.0837906 −0.0418953 0.999122i \(-0.513340\pi\)
−0.0418953 + 0.999122i \(0.513340\pi\)
\(128\) 0 0
\(129\) −4.70820 −0.414534
\(130\) 0 0
\(131\) −18.9443 −1.65517 −0.827584 0.561341i \(-0.810285\pi\)
−0.827584 + 0.561341i \(0.810285\pi\)
\(132\) 0 0
\(133\) −7.88854 −0.684023
\(134\) 0 0
\(135\) 3.47214 0.298834
\(136\) 0 0
\(137\) −0.145898 −0.0124649 −0.00623246 0.999981i \(-0.501984\pi\)
−0.00623246 + 0.999981i \(0.501984\pi\)
\(138\) 0 0
\(139\) 5.61803 0.476515 0.238258 0.971202i \(-0.423424\pi\)
0.238258 + 0.971202i \(0.423424\pi\)
\(140\) 0 0
\(141\) −4.94427 −0.416383
\(142\) 0 0
\(143\) 16.4721 1.37747
\(144\) 0 0
\(145\) −1.00000 −0.0830455
\(146\) 0 0
\(147\) 0.708204 0.0584117
\(148\) 0 0
\(149\) 16.1803 1.32555 0.662773 0.748821i \(-0.269380\pi\)
0.662773 + 0.748821i \(0.269380\pi\)
\(150\) 0 0
\(151\) 1.23607 0.100590 0.0502949 0.998734i \(-0.483984\pi\)
0.0502949 + 0.998734i \(0.483984\pi\)
\(152\) 0 0
\(153\) 9.47214 0.765777
\(154\) 0 0
\(155\) 1.61803 0.129964
\(156\) 0 0
\(157\) 5.05573 0.403491 0.201746 0.979438i \(-0.435339\pi\)
0.201746 + 0.979438i \(0.435339\pi\)
\(158\) 0 0
\(159\) 5.76393 0.457110
\(160\) 0 0
\(161\) −16.7082 −1.31679
\(162\) 0 0
\(163\) −13.5279 −1.05958 −0.529792 0.848128i \(-0.677730\pi\)
−0.529792 + 0.848128i \(0.677730\pi\)
\(164\) 0 0
\(165\) 2.00000 0.155700
\(166\) 0 0
\(167\) −4.38197 −0.339087 −0.169543 0.985523i \(-0.554229\pi\)
−0.169543 + 0.985523i \(0.554229\pi\)
\(168\) 0 0
\(169\) 12.9098 0.993064
\(170\) 0 0
\(171\) −7.23607 −0.553356
\(172\) 0 0
\(173\) −4.79837 −0.364814 −0.182407 0.983223i \(-0.558389\pi\)
−0.182407 + 0.983223i \(0.558389\pi\)
\(174\) 0 0
\(175\) −2.85410 −0.215750
\(176\) 0 0
\(177\) −5.79837 −0.435832
\(178\) 0 0
\(179\) −1.09017 −0.0814831 −0.0407416 0.999170i \(-0.512972\pi\)
−0.0407416 + 0.999170i \(0.512972\pi\)
\(180\) 0 0
\(181\) 0.291796 0.0216890 0.0108445 0.999941i \(-0.496548\pi\)
0.0108445 + 0.999941i \(0.496548\pi\)
\(182\) 0 0
\(183\) 1.32624 0.0980383
\(184\) 0 0
\(185\) −9.23607 −0.679049
\(186\) 0 0
\(187\) 11.7082 0.856189
\(188\) 0 0
\(189\) 9.90983 0.720834
\(190\) 0 0
\(191\) −20.8541 −1.50895 −0.754475 0.656329i \(-0.772108\pi\)
−0.754475 + 0.656329i \(0.772108\pi\)
\(192\) 0 0
\(193\) 0.145898 0.0105020 0.00525099 0.999986i \(-0.498329\pi\)
0.00525099 + 0.999986i \(0.498329\pi\)
\(194\) 0 0
\(195\) 3.14590 0.225282
\(196\) 0 0
\(197\) 18.8541 1.34330 0.671650 0.740869i \(-0.265586\pi\)
0.671650 + 0.740869i \(0.265586\pi\)
\(198\) 0 0
\(199\) −8.76393 −0.621259 −0.310629 0.950531i \(-0.600540\pi\)
−0.310629 + 0.950531i \(0.600540\pi\)
\(200\) 0 0
\(201\) −1.52786 −0.107767
\(202\) 0 0
\(203\) −2.85410 −0.200319
\(204\) 0 0
\(205\) 3.70820 0.258992
\(206\) 0 0
\(207\) −15.3262 −1.06525
\(208\) 0 0
\(209\) −8.94427 −0.618688
\(210\) 0 0
\(211\) 15.2361 1.04889 0.524447 0.851443i \(-0.324272\pi\)
0.524447 + 0.851443i \(0.324272\pi\)
\(212\) 0 0
\(213\) 8.00000 0.548151
\(214\) 0 0
\(215\) 7.61803 0.519546
\(216\) 0 0
\(217\) 4.61803 0.313493
\(218\) 0 0
\(219\) 1.18034 0.0797600
\(220\) 0 0
\(221\) 18.4164 1.23882
\(222\) 0 0
\(223\) −24.9787 −1.67270 −0.836349 0.548197i \(-0.815314\pi\)
−0.836349 + 0.548197i \(0.815314\pi\)
\(224\) 0 0
\(225\) −2.61803 −0.174536
\(226\) 0 0
\(227\) 2.00000 0.132745 0.0663723 0.997795i \(-0.478857\pi\)
0.0663723 + 0.997795i \(0.478857\pi\)
\(228\) 0 0
\(229\) 4.09017 0.270286 0.135143 0.990826i \(-0.456851\pi\)
0.135143 + 0.990826i \(0.456851\pi\)
\(230\) 0 0
\(231\) 5.70820 0.375572
\(232\) 0 0
\(233\) 16.4721 1.07913 0.539563 0.841945i \(-0.318590\pi\)
0.539563 + 0.841945i \(0.318590\pi\)
\(234\) 0 0
\(235\) 8.00000 0.521862
\(236\) 0 0
\(237\) −1.79837 −0.116817
\(238\) 0 0
\(239\) 11.5279 0.745676 0.372838 0.927897i \(-0.378385\pi\)
0.372838 + 0.927897i \(0.378385\pi\)
\(240\) 0 0
\(241\) −1.20163 −0.0774035 −0.0387018 0.999251i \(-0.512322\pi\)
−0.0387018 + 0.999251i \(0.512322\pi\)
\(242\) 0 0
\(243\) 13.9443 0.894525
\(244\) 0 0
\(245\) −1.14590 −0.0732087
\(246\) 0 0
\(247\) −14.0689 −0.895182
\(248\) 0 0
\(249\) 9.70820 0.615232
\(250\) 0 0
\(251\) −18.1803 −1.14753 −0.573766 0.819019i \(-0.694518\pi\)
−0.573766 + 0.819019i \(0.694518\pi\)
\(252\) 0 0
\(253\) −18.9443 −1.19102
\(254\) 0 0
\(255\) 2.23607 0.140028
\(256\) 0 0
\(257\) −28.0689 −1.75089 −0.875444 0.483319i \(-0.839431\pi\)
−0.875444 + 0.483319i \(0.839431\pi\)
\(258\) 0 0
\(259\) −26.3607 −1.63797
\(260\) 0 0
\(261\) −2.61803 −0.162052
\(262\) 0 0
\(263\) −17.5279 −1.08081 −0.540407 0.841404i \(-0.681730\pi\)
−0.540407 + 0.841404i \(0.681730\pi\)
\(264\) 0 0
\(265\) −9.32624 −0.572906
\(266\) 0 0
\(267\) −1.05573 −0.0646095
\(268\) 0 0
\(269\) 26.5623 1.61953 0.809766 0.586753i \(-0.199594\pi\)
0.809766 + 0.586753i \(0.199594\pi\)
\(270\) 0 0
\(271\) −15.4164 −0.936480 −0.468240 0.883601i \(-0.655112\pi\)
−0.468240 + 0.883601i \(0.655112\pi\)
\(272\) 0 0
\(273\) 8.97871 0.543416
\(274\) 0 0
\(275\) −3.23607 −0.195142
\(276\) 0 0
\(277\) 30.0000 1.80253 0.901263 0.433273i \(-0.142641\pi\)
0.901263 + 0.433273i \(0.142641\pi\)
\(278\) 0 0
\(279\) 4.23607 0.253607
\(280\) 0 0
\(281\) −4.85410 −0.289571 −0.144786 0.989463i \(-0.546249\pi\)
−0.144786 + 0.989463i \(0.546249\pi\)
\(282\) 0 0
\(283\) 22.1803 1.31848 0.659242 0.751931i \(-0.270877\pi\)
0.659242 + 0.751931i \(0.270877\pi\)
\(284\) 0 0
\(285\) −1.70820 −0.101185
\(286\) 0 0
\(287\) 10.5836 0.624730
\(288\) 0 0
\(289\) −3.90983 −0.229990
\(290\) 0 0
\(291\) 5.43769 0.318763
\(292\) 0 0
\(293\) 14.1803 0.828424 0.414212 0.910180i \(-0.364057\pi\)
0.414212 + 0.910180i \(0.364057\pi\)
\(294\) 0 0
\(295\) 9.38197 0.546239
\(296\) 0 0
\(297\) 11.2361 0.651983
\(298\) 0 0
\(299\) −29.7984 −1.72328
\(300\) 0 0
\(301\) 21.7426 1.25323
\(302\) 0 0
\(303\) −4.38197 −0.251737
\(304\) 0 0
\(305\) −2.14590 −0.122874
\(306\) 0 0
\(307\) −9.52786 −0.543784 −0.271892 0.962328i \(-0.587649\pi\)
−0.271892 + 0.962328i \(0.587649\pi\)
\(308\) 0 0
\(309\) 5.52786 0.314469
\(310\) 0 0
\(311\) 6.32624 0.358728 0.179364 0.983783i \(-0.442596\pi\)
0.179364 + 0.983783i \(0.442596\pi\)
\(312\) 0 0
\(313\) −3.88854 −0.219793 −0.109897 0.993943i \(-0.535052\pi\)
−0.109897 + 0.993943i \(0.535052\pi\)
\(314\) 0 0
\(315\) −7.47214 −0.421007
\(316\) 0 0
\(317\) −4.76393 −0.267569 −0.133785 0.991010i \(-0.542713\pi\)
−0.133785 + 0.991010i \(0.542713\pi\)
\(318\) 0 0
\(319\) −3.23607 −0.181185
\(320\) 0 0
\(321\) 0.180340 0.0100656
\(322\) 0 0
\(323\) −10.0000 −0.556415
\(324\) 0 0
\(325\) −5.09017 −0.282352
\(326\) 0 0
\(327\) 7.23607 0.400155
\(328\) 0 0
\(329\) 22.8328 1.25881
\(330\) 0 0
\(331\) −10.2918 −0.565688 −0.282844 0.959166i \(-0.591278\pi\)
−0.282844 + 0.959166i \(0.591278\pi\)
\(332\) 0 0
\(333\) −24.1803 −1.32507
\(334\) 0 0
\(335\) 2.47214 0.135067
\(336\) 0 0
\(337\) 17.1459 0.933997 0.466998 0.884258i \(-0.345335\pi\)
0.466998 + 0.884258i \(0.345335\pi\)
\(338\) 0 0
\(339\) 5.43769 0.295335
\(340\) 0 0
\(341\) 5.23607 0.283549
\(342\) 0 0
\(343\) 16.7082 0.902158
\(344\) 0 0
\(345\) −3.61803 −0.194788
\(346\) 0 0
\(347\) 30.7639 1.65149 0.825747 0.564040i \(-0.190754\pi\)
0.825747 + 0.564040i \(0.190754\pi\)
\(348\) 0 0
\(349\) −12.9443 −0.692891 −0.346445 0.938070i \(-0.612611\pi\)
−0.346445 + 0.938070i \(0.612611\pi\)
\(350\) 0 0
\(351\) 17.6738 0.943356
\(352\) 0 0
\(353\) −18.4721 −0.983173 −0.491586 0.870829i \(-0.663583\pi\)
−0.491586 + 0.870829i \(0.663583\pi\)
\(354\) 0 0
\(355\) −12.9443 −0.687011
\(356\) 0 0
\(357\) 6.38197 0.337769
\(358\) 0 0
\(359\) 14.9098 0.786911 0.393455 0.919344i \(-0.371280\pi\)
0.393455 + 0.919344i \(0.371280\pi\)
\(360\) 0 0
\(361\) −11.3607 −0.597931
\(362\) 0 0
\(363\) −0.326238 −0.0171231
\(364\) 0 0
\(365\) −1.90983 −0.0999651
\(366\) 0 0
\(367\) −18.1803 −0.949006 −0.474503 0.880254i \(-0.657372\pi\)
−0.474503 + 0.880254i \(0.657372\pi\)
\(368\) 0 0
\(369\) 9.70820 0.505389
\(370\) 0 0
\(371\) −26.6180 −1.38194
\(372\) 0 0
\(373\) 26.2705 1.36024 0.680118 0.733103i \(-0.261929\pi\)
0.680118 + 0.733103i \(0.261929\pi\)
\(374\) 0 0
\(375\) −0.618034 −0.0319151
\(376\) 0 0
\(377\) −5.09017 −0.262157
\(378\) 0 0
\(379\) 38.0689 1.95547 0.977734 0.209850i \(-0.0672975\pi\)
0.977734 + 0.209850i \(0.0672975\pi\)
\(380\) 0 0
\(381\) −0.583592 −0.0298983
\(382\) 0 0
\(383\) −26.2705 −1.34236 −0.671180 0.741294i \(-0.734212\pi\)
−0.671180 + 0.741294i \(0.734212\pi\)
\(384\) 0 0
\(385\) −9.23607 −0.470714
\(386\) 0 0
\(387\) 19.9443 1.01382
\(388\) 0 0
\(389\) −18.0000 −0.912636 −0.456318 0.889817i \(-0.650832\pi\)
−0.456318 + 0.889817i \(0.650832\pi\)
\(390\) 0 0
\(391\) −21.1803 −1.07114
\(392\) 0 0
\(393\) −11.7082 −0.590601
\(394\) 0 0
\(395\) 2.90983 0.146409
\(396\) 0 0
\(397\) 5.14590 0.258265 0.129133 0.991627i \(-0.458781\pi\)
0.129133 + 0.991627i \(0.458781\pi\)
\(398\) 0 0
\(399\) −4.87539 −0.244075
\(400\) 0 0
\(401\) 8.32624 0.415792 0.207896 0.978151i \(-0.433338\pi\)
0.207896 + 0.978151i \(0.433338\pi\)
\(402\) 0 0
\(403\) 8.23607 0.410268
\(404\) 0 0
\(405\) −5.70820 −0.283643
\(406\) 0 0
\(407\) −29.8885 −1.48152
\(408\) 0 0
\(409\) 7.41641 0.366718 0.183359 0.983046i \(-0.441303\pi\)
0.183359 + 0.983046i \(0.441303\pi\)
\(410\) 0 0
\(411\) −0.0901699 −0.00444776
\(412\) 0 0
\(413\) 26.7771 1.31761
\(414\) 0 0
\(415\) −15.7082 −0.771085
\(416\) 0 0
\(417\) 3.47214 0.170031
\(418\) 0 0
\(419\) 3.90983 0.191008 0.0955038 0.995429i \(-0.469554\pi\)
0.0955038 + 0.995429i \(0.469554\pi\)
\(420\) 0 0
\(421\) 7.52786 0.366886 0.183443 0.983030i \(-0.441276\pi\)
0.183443 + 0.983030i \(0.441276\pi\)
\(422\) 0 0
\(423\) 20.9443 1.01835
\(424\) 0 0
\(425\) −3.61803 −0.175500
\(426\) 0 0
\(427\) −6.12461 −0.296391
\(428\) 0 0
\(429\) 10.1803 0.491511
\(430\) 0 0
\(431\) 23.7082 1.14198 0.570992 0.820956i \(-0.306559\pi\)
0.570992 + 0.820956i \(0.306559\pi\)
\(432\) 0 0
\(433\) 4.47214 0.214917 0.107459 0.994210i \(-0.465729\pi\)
0.107459 + 0.994210i \(0.465729\pi\)
\(434\) 0 0
\(435\) −0.618034 −0.0296325
\(436\) 0 0
\(437\) 16.1803 0.774011
\(438\) 0 0
\(439\) 6.00000 0.286364 0.143182 0.989696i \(-0.454267\pi\)
0.143182 + 0.989696i \(0.454267\pi\)
\(440\) 0 0
\(441\) −3.00000 −0.142857
\(442\) 0 0
\(443\) 25.2705 1.20064 0.600319 0.799761i \(-0.295040\pi\)
0.600319 + 0.799761i \(0.295040\pi\)
\(444\) 0 0
\(445\) 1.70820 0.0809766
\(446\) 0 0
\(447\) 10.0000 0.472984
\(448\) 0 0
\(449\) −24.0689 −1.13588 −0.567940 0.823070i \(-0.692260\pi\)
−0.567940 + 0.823070i \(0.692260\pi\)
\(450\) 0 0
\(451\) 12.0000 0.565058
\(452\) 0 0
\(453\) 0.763932 0.0358927
\(454\) 0 0
\(455\) −14.5279 −0.681077
\(456\) 0 0
\(457\) 22.0000 1.02912 0.514558 0.857455i \(-0.327956\pi\)
0.514558 + 0.857455i \(0.327956\pi\)
\(458\) 0 0
\(459\) 12.5623 0.586358
\(460\) 0 0
\(461\) −19.3262 −0.900113 −0.450056 0.893000i \(-0.648596\pi\)
−0.450056 + 0.893000i \(0.648596\pi\)
\(462\) 0 0
\(463\) −38.8328 −1.80471 −0.902357 0.430989i \(-0.858165\pi\)
−0.902357 + 0.430989i \(0.858165\pi\)
\(464\) 0 0
\(465\) 1.00000 0.0463739
\(466\) 0 0
\(467\) −22.6738 −1.04922 −0.524608 0.851344i \(-0.675788\pi\)
−0.524608 + 0.851344i \(0.675788\pi\)
\(468\) 0 0
\(469\) 7.05573 0.325803
\(470\) 0 0
\(471\) 3.12461 0.143975
\(472\) 0 0
\(473\) 24.6525 1.13352
\(474\) 0 0
\(475\) 2.76393 0.126818
\(476\) 0 0
\(477\) −24.4164 −1.11795
\(478\) 0 0
\(479\) 0.673762 0.0307850 0.0153925 0.999882i \(-0.495100\pi\)
0.0153925 + 0.999882i \(0.495100\pi\)
\(480\) 0 0
\(481\) −47.0132 −2.14362
\(482\) 0 0
\(483\) −10.3262 −0.469860
\(484\) 0 0
\(485\) −8.79837 −0.399514
\(486\) 0 0
\(487\) −21.2705 −0.963859 −0.481929 0.876210i \(-0.660064\pi\)
−0.481929 + 0.876210i \(0.660064\pi\)
\(488\) 0 0
\(489\) −8.36068 −0.378083
\(490\) 0 0
\(491\) −41.3050 −1.86407 −0.932033 0.362373i \(-0.881967\pi\)
−0.932033 + 0.362373i \(0.881967\pi\)
\(492\) 0 0
\(493\) −3.61803 −0.162948
\(494\) 0 0
\(495\) −8.47214 −0.380794
\(496\) 0 0
\(497\) −36.9443 −1.65718
\(498\) 0 0
\(499\) −1.67376 −0.0749279 −0.0374639 0.999298i \(-0.511928\pi\)
−0.0374639 + 0.999298i \(0.511928\pi\)
\(500\) 0 0
\(501\) −2.70820 −0.120994
\(502\) 0 0
\(503\) 20.0000 0.891756 0.445878 0.895094i \(-0.352892\pi\)
0.445878 + 0.895094i \(0.352892\pi\)
\(504\) 0 0
\(505\) 7.09017 0.315508
\(506\) 0 0
\(507\) 7.97871 0.354347
\(508\) 0 0
\(509\) −3.70820 −0.164363 −0.0821816 0.996617i \(-0.526189\pi\)
−0.0821816 + 0.996617i \(0.526189\pi\)
\(510\) 0 0
\(511\) −5.45085 −0.241131
\(512\) 0 0
\(513\) −9.59675 −0.423707
\(514\) 0 0
\(515\) −8.94427 −0.394132
\(516\) 0 0
\(517\) 25.8885 1.13858
\(518\) 0 0
\(519\) −2.96556 −0.130174
\(520\) 0 0
\(521\) −35.3262 −1.54767 −0.773835 0.633387i \(-0.781664\pi\)
−0.773835 + 0.633387i \(0.781664\pi\)
\(522\) 0 0
\(523\) −10.1803 −0.445155 −0.222578 0.974915i \(-0.571447\pi\)
−0.222578 + 0.974915i \(0.571447\pi\)
\(524\) 0 0
\(525\) −1.76393 −0.0769843
\(526\) 0 0
\(527\) 5.85410 0.255009
\(528\) 0 0
\(529\) 11.2705 0.490022
\(530\) 0 0
\(531\) 24.5623 1.06591
\(532\) 0 0
\(533\) 18.8754 0.817584
\(534\) 0 0
\(535\) −0.291796 −0.0126154
\(536\) 0 0
\(537\) −0.673762 −0.0290750
\(538\) 0 0
\(539\) −3.70820 −0.159724
\(540\) 0 0
\(541\) 5.03444 0.216448 0.108224 0.994127i \(-0.465484\pi\)
0.108224 + 0.994127i \(0.465484\pi\)
\(542\) 0 0
\(543\) 0.180340 0.00773913
\(544\) 0 0
\(545\) −11.7082 −0.501524
\(546\) 0 0
\(547\) 19.1246 0.817709 0.408855 0.912600i \(-0.365928\pi\)
0.408855 + 0.912600i \(0.365928\pi\)
\(548\) 0 0
\(549\) −5.61803 −0.239772
\(550\) 0 0
\(551\) 2.76393 0.117747
\(552\) 0 0
\(553\) 8.30495 0.353162
\(554\) 0 0
\(555\) −5.70820 −0.242300
\(556\) 0 0
\(557\) 24.1459 1.02309 0.511547 0.859255i \(-0.329073\pi\)
0.511547 + 0.859255i \(0.329073\pi\)
\(558\) 0 0
\(559\) 38.7771 1.64010
\(560\) 0 0
\(561\) 7.23607 0.305507
\(562\) 0 0
\(563\) 28.3951 1.19671 0.598356 0.801230i \(-0.295821\pi\)
0.598356 + 0.801230i \(0.295821\pi\)
\(564\) 0 0
\(565\) −8.79837 −0.370150
\(566\) 0 0
\(567\) −16.2918 −0.684191
\(568\) 0 0
\(569\) 32.3607 1.35663 0.678315 0.734771i \(-0.262710\pi\)
0.678315 + 0.734771i \(0.262710\pi\)
\(570\) 0 0
\(571\) −20.1459 −0.843080 −0.421540 0.906810i \(-0.638510\pi\)
−0.421540 + 0.906810i \(0.638510\pi\)
\(572\) 0 0
\(573\) −12.8885 −0.538427
\(574\) 0 0
\(575\) 5.85410 0.244133
\(576\) 0 0
\(577\) 12.7426 0.530483 0.265242 0.964182i \(-0.414548\pi\)
0.265242 + 0.964182i \(0.414548\pi\)
\(578\) 0 0
\(579\) 0.0901699 0.00374733
\(580\) 0 0
\(581\) −44.8328 −1.85998
\(582\) 0 0
\(583\) −30.1803 −1.24994
\(584\) 0 0
\(585\) −13.3262 −0.550972
\(586\) 0 0
\(587\) −8.83282 −0.364569 −0.182285 0.983246i \(-0.558349\pi\)
−0.182285 + 0.983246i \(0.558349\pi\)
\(588\) 0 0
\(589\) −4.47214 −0.184271
\(590\) 0 0
\(591\) 11.6525 0.479319
\(592\) 0 0
\(593\) −6.65248 −0.273184 −0.136592 0.990627i \(-0.543615\pi\)
−0.136592 + 0.990627i \(0.543615\pi\)
\(594\) 0 0
\(595\) −10.3262 −0.423334
\(596\) 0 0
\(597\) −5.41641 −0.221679
\(598\) 0 0
\(599\) −0.673762 −0.0275292 −0.0137646 0.999905i \(-0.504382\pi\)
−0.0137646 + 0.999905i \(0.504382\pi\)
\(600\) 0 0
\(601\) 20.5410 0.837886 0.418943 0.908013i \(-0.362401\pi\)
0.418943 + 0.908013i \(0.362401\pi\)
\(602\) 0 0
\(603\) 6.47214 0.263566
\(604\) 0 0
\(605\) 0.527864 0.0214607
\(606\) 0 0
\(607\) −2.47214 −0.100341 −0.0501705 0.998741i \(-0.515976\pi\)
−0.0501705 + 0.998741i \(0.515976\pi\)
\(608\) 0 0
\(609\) −1.76393 −0.0714781
\(610\) 0 0
\(611\) 40.7214 1.64741
\(612\) 0 0
\(613\) 32.7426 1.32246 0.661232 0.750182i \(-0.270034\pi\)
0.661232 + 0.750182i \(0.270034\pi\)
\(614\) 0 0
\(615\) 2.29180 0.0924141
\(616\) 0 0
\(617\) −31.7984 −1.28015 −0.640077 0.768311i \(-0.721098\pi\)
−0.640077 + 0.768311i \(0.721098\pi\)
\(618\) 0 0
\(619\) 41.7771 1.67916 0.839581 0.543234i \(-0.182800\pi\)
0.839581 + 0.543234i \(0.182800\pi\)
\(620\) 0 0
\(621\) −20.3262 −0.815664
\(622\) 0 0
\(623\) 4.87539 0.195328
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) −5.52786 −0.220762
\(628\) 0 0
\(629\) −33.4164 −1.33240
\(630\) 0 0
\(631\) 38.3607 1.52711 0.763557 0.645740i \(-0.223451\pi\)
0.763557 + 0.645740i \(0.223451\pi\)
\(632\) 0 0
\(633\) 9.41641 0.374269
\(634\) 0 0
\(635\) 0.944272 0.0374723
\(636\) 0 0
\(637\) −5.83282 −0.231105
\(638\) 0 0
\(639\) −33.8885 −1.34061
\(640\) 0 0
\(641\) −38.7639 −1.53108 −0.765542 0.643386i \(-0.777529\pi\)
−0.765542 + 0.643386i \(0.777529\pi\)
\(642\) 0 0
\(643\) −25.7082 −1.01383 −0.506916 0.861995i \(-0.669215\pi\)
−0.506916 + 0.861995i \(0.669215\pi\)
\(644\) 0 0
\(645\) 4.70820 0.185385
\(646\) 0 0
\(647\) 37.3050 1.46661 0.733304 0.679900i \(-0.237977\pi\)
0.733304 + 0.679900i \(0.237977\pi\)
\(648\) 0 0
\(649\) 30.3607 1.19176
\(650\) 0 0
\(651\) 2.85410 0.111861
\(652\) 0 0
\(653\) 39.5967 1.54954 0.774770 0.632243i \(-0.217866\pi\)
0.774770 + 0.632243i \(0.217866\pi\)
\(654\) 0 0
\(655\) 18.9443 0.740214
\(656\) 0 0
\(657\) −5.00000 −0.195069
\(658\) 0 0
\(659\) −15.8197 −0.616246 −0.308123 0.951346i \(-0.599701\pi\)
−0.308123 + 0.951346i \(0.599701\pi\)
\(660\) 0 0
\(661\) 27.5967 1.07339 0.536695 0.843777i \(-0.319673\pi\)
0.536695 + 0.843777i \(0.319673\pi\)
\(662\) 0 0
\(663\) 11.3820 0.442039
\(664\) 0 0
\(665\) 7.88854 0.305905
\(666\) 0 0
\(667\) 5.85410 0.226672
\(668\) 0 0
\(669\) −15.4377 −0.596856
\(670\) 0 0
\(671\) −6.94427 −0.268081
\(672\) 0 0
\(673\) 18.7639 0.723296 0.361648 0.932315i \(-0.382214\pi\)
0.361648 + 0.932315i \(0.382214\pi\)
\(674\) 0 0
\(675\) −3.47214 −0.133643
\(676\) 0 0
\(677\) −13.8885 −0.533780 −0.266890 0.963727i \(-0.585996\pi\)
−0.266890 + 0.963727i \(0.585996\pi\)
\(678\) 0 0
\(679\) −25.1115 −0.963689
\(680\) 0 0
\(681\) 1.23607 0.0473662
\(682\) 0 0
\(683\) 15.3475 0.587257 0.293628 0.955920i \(-0.405137\pi\)
0.293628 + 0.955920i \(0.405137\pi\)
\(684\) 0 0
\(685\) 0.145898 0.00557448
\(686\) 0 0
\(687\) 2.52786 0.0964440
\(688\) 0 0
\(689\) −47.4721 −1.80854
\(690\) 0 0
\(691\) 13.0344 0.495854 0.247927 0.968779i \(-0.420251\pi\)
0.247927 + 0.968779i \(0.420251\pi\)
\(692\) 0 0
\(693\) −24.1803 −0.918535
\(694\) 0 0
\(695\) −5.61803 −0.213104
\(696\) 0 0
\(697\) 13.4164 0.508183
\(698\) 0 0
\(699\) 10.1803 0.385056
\(700\) 0 0
\(701\) −13.1246 −0.495710 −0.247855 0.968797i \(-0.579726\pi\)
−0.247855 + 0.968797i \(0.579726\pi\)
\(702\) 0 0
\(703\) 25.5279 0.962802
\(704\) 0 0
\(705\) 4.94427 0.186212
\(706\) 0 0
\(707\) 20.2361 0.761056
\(708\) 0 0
\(709\) −18.2918 −0.686963 −0.343481 0.939159i \(-0.611606\pi\)
−0.343481 + 0.939159i \(0.611606\pi\)
\(710\) 0 0
\(711\) 7.61803 0.285699
\(712\) 0 0
\(713\) −9.47214 −0.354734
\(714\) 0 0
\(715\) −16.4721 −0.616023
\(716\) 0 0
\(717\) 7.12461 0.266074
\(718\) 0 0
\(719\) 20.6525 0.770207 0.385104 0.922873i \(-0.374166\pi\)
0.385104 + 0.922873i \(0.374166\pi\)
\(720\) 0 0
\(721\) −25.5279 −0.950707
\(722\) 0 0
\(723\) −0.742646 −0.0276193
\(724\) 0 0
\(725\) 1.00000 0.0371391
\(726\) 0 0
\(727\) 44.7639 1.66020 0.830101 0.557613i \(-0.188283\pi\)
0.830101 + 0.557613i \(0.188283\pi\)
\(728\) 0 0
\(729\) −8.50658 −0.315058
\(730\) 0 0
\(731\) 27.5623 1.01943
\(732\) 0 0
\(733\) 35.7771 1.32146 0.660728 0.750625i \(-0.270247\pi\)
0.660728 + 0.750625i \(0.270247\pi\)
\(734\) 0 0
\(735\) −0.708204 −0.0261225
\(736\) 0 0
\(737\) 8.00000 0.294684
\(738\) 0 0
\(739\) −3.41641 −0.125675 −0.0628373 0.998024i \(-0.520015\pi\)
−0.0628373 + 0.998024i \(0.520015\pi\)
\(740\) 0 0
\(741\) −8.69505 −0.319421
\(742\) 0 0
\(743\) −7.41641 −0.272082 −0.136041 0.990703i \(-0.543438\pi\)
−0.136041 + 0.990703i \(0.543438\pi\)
\(744\) 0 0
\(745\) −16.1803 −0.592802
\(746\) 0 0
\(747\) −41.1246 −1.50467
\(748\) 0 0
\(749\) −0.832816 −0.0304304
\(750\) 0 0
\(751\) −40.0000 −1.45962 −0.729810 0.683650i \(-0.760392\pi\)
−0.729810 + 0.683650i \(0.760392\pi\)
\(752\) 0 0
\(753\) −11.2361 −0.409465
\(754\) 0 0
\(755\) −1.23607 −0.0449851
\(756\) 0 0
\(757\) 22.1803 0.806158 0.403079 0.915165i \(-0.367940\pi\)
0.403079 + 0.915165i \(0.367940\pi\)
\(758\) 0 0
\(759\) −11.7082 −0.424981
\(760\) 0 0
\(761\) −25.0344 −0.907498 −0.453749 0.891130i \(-0.649914\pi\)
−0.453749 + 0.891130i \(0.649914\pi\)
\(762\) 0 0
\(763\) −33.4164 −1.20976
\(764\) 0 0
\(765\) −9.47214 −0.342466
\(766\) 0 0
\(767\) 47.7558 1.72436
\(768\) 0 0
\(769\) 50.6525 1.82657 0.913287 0.407316i \(-0.133535\pi\)
0.913287 + 0.407316i \(0.133535\pi\)
\(770\) 0 0
\(771\) −17.3475 −0.624756
\(772\) 0 0
\(773\) −52.8328 −1.90026 −0.950132 0.311848i \(-0.899052\pi\)
−0.950132 + 0.311848i \(0.899052\pi\)
\(774\) 0 0
\(775\) −1.61803 −0.0581215
\(776\) 0 0
\(777\) −16.2918 −0.584465
\(778\) 0 0
\(779\) −10.2492 −0.367217
\(780\) 0 0
\(781\) −41.8885 −1.49889
\(782\) 0 0
\(783\) −3.47214 −0.124084
\(784\) 0 0
\(785\) −5.05573 −0.180447
\(786\) 0 0
\(787\) −44.6525 −1.59169 −0.795844 0.605501i \(-0.792973\pi\)
−0.795844 + 0.605501i \(0.792973\pi\)
\(788\) 0 0
\(789\) −10.8328 −0.385658
\(790\) 0 0
\(791\) −25.1115 −0.892861
\(792\) 0 0
\(793\) −10.9230 −0.387887
\(794\) 0 0
\(795\) −5.76393 −0.204426
\(796\) 0 0
\(797\) 30.8328 1.09215 0.546077 0.837735i \(-0.316121\pi\)
0.546077 + 0.837735i \(0.316121\pi\)
\(798\) 0 0
\(799\) 28.9443 1.02397
\(800\) 0 0
\(801\) 4.47214 0.158015
\(802\) 0 0
\(803\) −6.18034 −0.218099
\(804\) 0 0
\(805\) 16.7082 0.588887
\(806\) 0 0
\(807\) 16.4164 0.577885
\(808\) 0 0
\(809\) 11.4164 0.401380 0.200690 0.979655i \(-0.435682\pi\)
0.200690 + 0.979655i \(0.435682\pi\)
\(810\) 0 0
\(811\) −0.673762 −0.0236590 −0.0118295 0.999930i \(-0.503766\pi\)
−0.0118295 + 0.999930i \(0.503766\pi\)
\(812\) 0 0
\(813\) −9.52786 −0.334157
\(814\) 0 0
\(815\) 13.5279 0.473860
\(816\) 0 0
\(817\) −21.0557 −0.736647
\(818\) 0 0
\(819\) −38.0344 −1.32903
\(820\) 0 0
\(821\) −34.0689 −1.18901 −0.594506 0.804091i \(-0.702652\pi\)
−0.594506 + 0.804091i \(0.702652\pi\)
\(822\) 0 0
\(823\) −45.7771 −1.59569 −0.797844 0.602863i \(-0.794026\pi\)
−0.797844 + 0.602863i \(0.794026\pi\)
\(824\) 0 0
\(825\) −2.00000 −0.0696311
\(826\) 0 0
\(827\) −1.90983 −0.0664113 −0.0332056 0.999449i \(-0.510572\pi\)
−0.0332056 + 0.999449i \(0.510572\pi\)
\(828\) 0 0
\(829\) −45.7984 −1.59064 −0.795322 0.606188i \(-0.792698\pi\)
−0.795322 + 0.606188i \(0.792698\pi\)
\(830\) 0 0
\(831\) 18.5410 0.643181
\(832\) 0 0
\(833\) −4.14590 −0.143647
\(834\) 0 0
\(835\) 4.38197 0.151644
\(836\) 0 0
\(837\) 5.61803 0.194188
\(838\) 0 0
\(839\) −28.3607 −0.979119 −0.489560 0.871970i \(-0.662842\pi\)
−0.489560 + 0.871970i \(0.662842\pi\)
\(840\) 0 0
\(841\) 1.00000 0.0344828
\(842\) 0 0
\(843\) −3.00000 −0.103325
\(844\) 0 0
\(845\) −12.9098 −0.444112
\(846\) 0 0
\(847\) 1.50658 0.0517666
\(848\) 0 0
\(849\) 13.7082 0.470464
\(850\) 0 0
\(851\) 54.0689 1.85346
\(852\) 0 0
\(853\) 1.81966 0.0623040 0.0311520 0.999515i \(-0.490082\pi\)
0.0311520 + 0.999515i \(0.490082\pi\)
\(854\) 0 0
\(855\) 7.23607 0.247468
\(856\) 0 0
\(857\) 33.7082 1.15145 0.575725 0.817643i \(-0.304720\pi\)
0.575725 + 0.817643i \(0.304720\pi\)
\(858\) 0 0
\(859\) −16.6525 −0.568175 −0.284088 0.958798i \(-0.591691\pi\)
−0.284088 + 0.958798i \(0.591691\pi\)
\(860\) 0 0
\(861\) 6.54102 0.222917
\(862\) 0 0
\(863\) 4.03444 0.137334 0.0686670 0.997640i \(-0.478125\pi\)
0.0686670 + 0.997640i \(0.478125\pi\)
\(864\) 0 0
\(865\) 4.79837 0.163150
\(866\) 0 0
\(867\) −2.41641 −0.0820655
\(868\) 0 0
\(869\) 9.41641 0.319430
\(870\) 0 0
\(871\) 12.5836 0.426379
\(872\) 0 0
\(873\) −23.0344 −0.779598
\(874\) 0 0
\(875\) 2.85410 0.0964863
\(876\) 0 0
\(877\) 50.6180 1.70925 0.854625 0.519246i \(-0.173787\pi\)
0.854625 + 0.519246i \(0.173787\pi\)
\(878\) 0 0
\(879\) 8.76393 0.295600
\(880\) 0 0
\(881\) 10.1803 0.342984 0.171492 0.985185i \(-0.445141\pi\)
0.171492 + 0.985185i \(0.445141\pi\)
\(882\) 0 0
\(883\) 21.5967 0.726788 0.363394 0.931635i \(-0.381618\pi\)
0.363394 + 0.931635i \(0.381618\pi\)
\(884\) 0 0
\(885\) 5.79837 0.194910
\(886\) 0 0
\(887\) 31.4164 1.05486 0.527430 0.849599i \(-0.323156\pi\)
0.527430 + 0.849599i \(0.323156\pi\)
\(888\) 0 0
\(889\) 2.69505 0.0903890
\(890\) 0 0
\(891\) −18.4721 −0.618840
\(892\) 0 0
\(893\) −22.1115 −0.739932
\(894\) 0 0
\(895\) 1.09017 0.0364404
\(896\) 0 0
\(897\) −18.4164 −0.614906
\(898\) 0 0
\(899\) −1.61803 −0.0539645
\(900\) 0 0
\(901\) −33.7426 −1.12413
\(902\) 0 0
\(903\) 13.4377 0.447178
\(904\) 0 0
\(905\) −0.291796 −0.00969963
\(906\) 0 0
\(907\) −25.9098 −0.860322 −0.430161 0.902752i \(-0.641543\pi\)
−0.430161 + 0.902752i \(0.641543\pi\)
\(908\) 0 0
\(909\) 18.5623 0.615673
\(910\) 0 0
\(911\) 39.2705 1.30109 0.650545 0.759468i \(-0.274541\pi\)
0.650545 + 0.759468i \(0.274541\pi\)
\(912\) 0 0
\(913\) −50.8328 −1.68232
\(914\) 0 0
\(915\) −1.32624 −0.0438441
\(916\) 0 0
\(917\) 54.0689 1.78551
\(918\) 0 0
\(919\) 26.4721 0.873235 0.436618 0.899647i \(-0.356176\pi\)
0.436618 + 0.899647i \(0.356176\pi\)
\(920\) 0 0
\(921\) −5.88854 −0.194034
\(922\) 0 0
\(923\) −65.8885 −2.16875
\(924\) 0 0
\(925\) 9.23607 0.303680
\(926\) 0 0
\(927\) −23.4164 −0.769096
\(928\) 0 0
\(929\) −44.4508 −1.45839 −0.729193 0.684309i \(-0.760104\pi\)
−0.729193 + 0.684309i \(0.760104\pi\)
\(930\) 0 0
\(931\) 3.16718 0.103800
\(932\) 0 0
\(933\) 3.90983 0.128002
\(934\) 0 0
\(935\) −11.7082 −0.382899
\(936\) 0 0
\(937\) 43.1246 1.40882 0.704410 0.709793i \(-0.251212\pi\)
0.704410 + 0.709793i \(0.251212\pi\)
\(938\) 0 0
\(939\) −2.40325 −0.0784272
\(940\) 0 0
\(941\) 32.3607 1.05493 0.527464 0.849577i \(-0.323143\pi\)
0.527464 + 0.849577i \(0.323143\pi\)
\(942\) 0 0
\(943\) −21.7082 −0.706916
\(944\) 0 0
\(945\) −9.90983 −0.322367
\(946\) 0 0
\(947\) 55.0902 1.79019 0.895095 0.445875i \(-0.147108\pi\)
0.895095 + 0.445875i \(0.147108\pi\)
\(948\) 0 0
\(949\) −9.72136 −0.315569
\(950\) 0 0
\(951\) −2.94427 −0.0954746
\(952\) 0 0
\(953\) 12.8328 0.415696 0.207848 0.978161i \(-0.433354\pi\)
0.207848 + 0.978161i \(0.433354\pi\)
\(954\) 0 0
\(955\) 20.8541 0.674823
\(956\) 0 0
\(957\) −2.00000 −0.0646508
\(958\) 0 0
\(959\) 0.416408 0.0134465
\(960\) 0 0
\(961\) −28.3820 −0.915547
\(962\) 0 0
\(963\) −0.763932 −0.0246174
\(964\) 0 0
\(965\) −0.145898 −0.00469662
\(966\) 0 0
\(967\) −23.5967 −0.758820 −0.379410 0.925229i \(-0.623873\pi\)
−0.379410 + 0.925229i \(0.623873\pi\)
\(968\) 0 0
\(969\) −6.18034 −0.198541
\(970\) 0 0
\(971\) 52.0689 1.67097 0.835485 0.549513i \(-0.185187\pi\)
0.835485 + 0.549513i \(0.185187\pi\)
\(972\) 0 0
\(973\) −16.0344 −0.514041
\(974\) 0 0
\(975\) −3.14590 −0.100749
\(976\) 0 0
\(977\) 10.6525 0.340803 0.170401 0.985375i \(-0.445494\pi\)
0.170401 + 0.985375i \(0.445494\pi\)
\(978\) 0 0
\(979\) 5.52786 0.176671
\(980\) 0 0
\(981\) −30.6525 −0.978658
\(982\) 0 0
\(983\) −13.0557 −0.416413 −0.208207 0.978085i \(-0.566763\pi\)
−0.208207 + 0.978085i \(0.566763\pi\)
\(984\) 0 0
\(985\) −18.8541 −0.600742
\(986\) 0 0
\(987\) 14.1115 0.449173
\(988\) 0 0
\(989\) −44.5967 −1.41809
\(990\) 0 0
\(991\) −10.0689 −0.319849 −0.159924 0.987129i \(-0.551125\pi\)
−0.159924 + 0.987129i \(0.551125\pi\)
\(992\) 0 0
\(993\) −6.36068 −0.201850
\(994\) 0 0
\(995\) 8.76393 0.277835
\(996\) 0 0
\(997\) 44.1803 1.39921 0.699603 0.714532i \(-0.253360\pi\)
0.699603 + 0.714532i \(0.253360\pi\)
\(998\) 0 0
\(999\) −32.0689 −1.01461
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 4640.2.a.g.1.2 2
4.3 odd 2 4640.2.a.i.1.1 yes 2
8.3 odd 2 9280.2.a.y.1.2 2
8.5 even 2 9280.2.a.bd.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4640.2.a.g.1.2 2 1.1 even 1 trivial
4640.2.a.i.1.1 yes 2 4.3 odd 2
9280.2.a.y.1.2 2 8.3 odd 2
9280.2.a.bd.1.1 2 8.5 even 2