Properties

Label 3800.2.a.bb.1.3
Level $3800$
Weight $2$
Character 3800.1
Self dual yes
Analytic conductor $30.343$
Analytic rank $0$
Dimension $6$
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] = [3800,2,Mod(1,3800)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3800, 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("3800.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3800 = 2^{3} \cdot 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3800.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: \(30.3431527681\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 2x^{5} - 12x^{4} + 16x^{3} + 33x^{2} - 4x - 7 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(0.486697\) of defining polynomial
Character \(\chi\) \(=\) 3800.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-0.486697 q^{3} -3.63249 q^{7} -2.76313 q^{9} +O(q^{10})\) \(q-0.486697 q^{3} -3.63249 q^{7} -2.76313 q^{9} -2.79460 q^{11} -2.86457 q^{13} -1.17400 q^{17} +1.00000 q^{19} +1.76793 q^{21} +0.617328 q^{23} +2.80490 q^{27} -4.96700 q^{29} +0.745377 q^{31} +1.36012 q^{33} -8.23679 q^{37} +1.39418 q^{39} +9.98217 q^{41} -10.4955 q^{43} -5.07742 q^{47} +6.19502 q^{49} +0.571381 q^{51} +7.45370 q^{53} -0.486697 q^{57} +3.83310 q^{59} +11.2450 q^{61} +10.0370 q^{63} -6.10730 q^{67} -0.300452 q^{69} -9.40599 q^{71} +9.52367 q^{73} +10.1514 q^{77} -3.70094 q^{79} +6.92424 q^{81} +4.66397 q^{83} +2.41743 q^{87} +10.6888 q^{89} +10.4055 q^{91} -0.362773 q^{93} -0.629221 q^{97} +7.72183 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 2 q^{3} - 2 q^{7} + 10 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 2 q^{3} - 2 q^{7} + 10 q^{9} + 3 q^{11} + 3 q^{13} + 2 q^{17} + 6 q^{19} + 11 q^{21} - 4 q^{23} - 20 q^{27} + 7 q^{29} + 5 q^{31} + 16 q^{33} + 8 q^{39} + 11 q^{41} + 7 q^{43} - 20 q^{47} - 2 q^{49} + 13 q^{51} + 7 q^{53} - 2 q^{57} - 4 q^{59} + 13 q^{61} + q^{63} - 25 q^{67} + 7 q^{69} + 29 q^{71} + 19 q^{73} - 24 q^{77} + 28 q^{79} + 38 q^{81} + 15 q^{83} - 57 q^{87} - 12 q^{89} + 27 q^{93} + 13 q^{97} + 10 q^{99}+O(q^{100}) \) 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
\(3\) −0.486697 −0.280995 −0.140497 0.990081i \(-0.544870\pi\)
−0.140497 + 0.990081i \(0.544870\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) −3.63249 −1.37295 −0.686477 0.727152i \(-0.740844\pi\)
−0.686477 + 0.727152i \(0.740844\pi\)
\(8\) 0 0
\(9\) −2.76313 −0.921042
\(10\) 0 0
\(11\) −2.79460 −0.842603 −0.421302 0.906921i \(-0.638427\pi\)
−0.421302 + 0.906921i \(0.638427\pi\)
\(12\) 0 0
\(13\) −2.86457 −0.794489 −0.397244 0.917713i \(-0.630033\pi\)
−0.397244 + 0.917713i \(0.630033\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −1.17400 −0.284736 −0.142368 0.989814i \(-0.545472\pi\)
−0.142368 + 0.989814i \(0.545472\pi\)
\(18\) 0 0
\(19\) 1.00000 0.229416
\(20\) 0 0
\(21\) 1.76793 0.385793
\(22\) 0 0
\(23\) 0.617328 0.128722 0.0643609 0.997927i \(-0.479499\pi\)
0.0643609 + 0.997927i \(0.479499\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 2.80490 0.539803
\(28\) 0 0
\(29\) −4.96700 −0.922349 −0.461175 0.887309i \(-0.652572\pi\)
−0.461175 + 0.887309i \(0.652572\pi\)
\(30\) 0 0
\(31\) 0.745377 0.133874 0.0669369 0.997757i \(-0.478677\pi\)
0.0669369 + 0.997757i \(0.478677\pi\)
\(32\) 0 0
\(33\) 1.36012 0.236767
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −8.23679 −1.35412 −0.677060 0.735928i \(-0.736746\pi\)
−0.677060 + 0.735928i \(0.736746\pi\)
\(38\) 0 0
\(39\) 1.39418 0.223247
\(40\) 0 0
\(41\) 9.98217 1.55895 0.779476 0.626432i \(-0.215485\pi\)
0.779476 + 0.626432i \(0.215485\pi\)
\(42\) 0 0
\(43\) −10.4955 −1.60055 −0.800277 0.599630i \(-0.795314\pi\)
−0.800277 + 0.599630i \(0.795314\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −5.07742 −0.740618 −0.370309 0.928909i \(-0.620748\pi\)
−0.370309 + 0.928909i \(0.620748\pi\)
\(48\) 0 0
\(49\) 6.19502 0.885003
\(50\) 0 0
\(51\) 0.571381 0.0800093
\(52\) 0 0
\(53\) 7.45370 1.02384 0.511922 0.859032i \(-0.328934\pi\)
0.511922 + 0.859032i \(0.328934\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −0.486697 −0.0644646
\(58\) 0 0
\(59\) 3.83310 0.499027 0.249513 0.968371i \(-0.419729\pi\)
0.249513 + 0.968371i \(0.419729\pi\)
\(60\) 0 0
\(61\) 11.2450 1.43978 0.719889 0.694089i \(-0.244193\pi\)
0.719889 + 0.694089i \(0.244193\pi\)
\(62\) 0 0
\(63\) 10.0370 1.26455
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −6.10730 −0.746125 −0.373063 0.927806i \(-0.621692\pi\)
−0.373063 + 0.927806i \(0.621692\pi\)
\(68\) 0 0
\(69\) −0.300452 −0.0361702
\(70\) 0 0
\(71\) −9.40599 −1.11629 −0.558143 0.829745i \(-0.688486\pi\)
−0.558143 + 0.829745i \(0.688486\pi\)
\(72\) 0 0
\(73\) 9.52367 1.11466 0.557331 0.830291i \(-0.311826\pi\)
0.557331 + 0.830291i \(0.311826\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 10.1514 1.15686
\(78\) 0 0
\(79\) −3.70094 −0.416388 −0.208194 0.978088i \(-0.566759\pi\)
−0.208194 + 0.978088i \(0.566759\pi\)
\(80\) 0 0
\(81\) 6.92424 0.769360
\(82\) 0 0
\(83\) 4.66397 0.511937 0.255968 0.966685i \(-0.417606\pi\)
0.255968 + 0.966685i \(0.417606\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 2.41743 0.259175
\(88\) 0 0
\(89\) 10.6888 1.13301 0.566506 0.824058i \(-0.308295\pi\)
0.566506 + 0.824058i \(0.308295\pi\)
\(90\) 0 0
\(91\) 10.4055 1.09080
\(92\) 0 0
\(93\) −0.362773 −0.0376178
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −0.629221 −0.0638877 −0.0319439 0.999490i \(-0.510170\pi\)
−0.0319439 + 0.999490i \(0.510170\pi\)
\(98\) 0 0
\(99\) 7.72183 0.776073
\(100\) 0 0
\(101\) 3.82531 0.380633 0.190316 0.981723i \(-0.439049\pi\)
0.190316 + 0.981723i \(0.439049\pi\)
\(102\) 0 0
\(103\) 10.0702 0.992251 0.496125 0.868251i \(-0.334756\pi\)
0.496125 + 0.868251i \(0.334756\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −4.53412 −0.438330 −0.219165 0.975688i \(-0.570333\pi\)
−0.219165 + 0.975688i \(0.570333\pi\)
\(108\) 0 0
\(109\) 15.9974 1.53227 0.766137 0.642678i \(-0.222177\pi\)
0.766137 + 0.642678i \(0.222177\pi\)
\(110\) 0 0
\(111\) 4.00882 0.380501
\(112\) 0 0
\(113\) 2.34828 0.220908 0.110454 0.993881i \(-0.464770\pi\)
0.110454 + 0.993881i \(0.464770\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 7.91517 0.731757
\(118\) 0 0
\(119\) 4.26454 0.390929
\(120\) 0 0
\(121\) −3.19022 −0.290020
\(122\) 0 0
\(123\) −4.85829 −0.438058
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) −17.4542 −1.54881 −0.774403 0.632693i \(-0.781950\pi\)
−0.774403 + 0.632693i \(0.781950\pi\)
\(128\) 0 0
\(129\) 5.10815 0.449748
\(130\) 0 0
\(131\) 9.00877 0.787100 0.393550 0.919303i \(-0.371247\pi\)
0.393550 + 0.919303i \(0.371247\pi\)
\(132\) 0 0
\(133\) −3.63249 −0.314977
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −15.4369 −1.31887 −0.659433 0.751764i \(-0.729203\pi\)
−0.659433 + 0.751764i \(0.729203\pi\)
\(138\) 0 0
\(139\) 15.1144 1.28199 0.640993 0.767547i \(-0.278523\pi\)
0.640993 + 0.767547i \(0.278523\pi\)
\(140\) 0 0
\(141\) 2.47117 0.208110
\(142\) 0 0
\(143\) 8.00532 0.669439
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) −3.01510 −0.248681
\(148\) 0 0
\(149\) −11.8422 −0.970153 −0.485076 0.874472i \(-0.661208\pi\)
−0.485076 + 0.874472i \(0.661208\pi\)
\(150\) 0 0
\(151\) −8.31262 −0.676471 −0.338236 0.941061i \(-0.609830\pi\)
−0.338236 + 0.941061i \(0.609830\pi\)
\(152\) 0 0
\(153\) 3.24390 0.262254
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 10.6650 0.851162 0.425581 0.904920i \(-0.360070\pi\)
0.425581 + 0.904920i \(0.360070\pi\)
\(158\) 0 0
\(159\) −3.62770 −0.287695
\(160\) 0 0
\(161\) −2.24244 −0.176729
\(162\) 0 0
\(163\) 1.65583 0.129694 0.0648472 0.997895i \(-0.479344\pi\)
0.0648472 + 0.997895i \(0.479344\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −9.83568 −0.761108 −0.380554 0.924759i \(-0.624267\pi\)
−0.380554 + 0.924759i \(0.624267\pi\)
\(168\) 0 0
\(169\) −4.79424 −0.368788
\(170\) 0 0
\(171\) −2.76313 −0.211302
\(172\) 0 0
\(173\) 8.05586 0.612476 0.306238 0.951955i \(-0.400930\pi\)
0.306238 + 0.951955i \(0.400930\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) −1.86556 −0.140224
\(178\) 0 0
\(179\) −14.0058 −1.04685 −0.523423 0.852073i \(-0.675345\pi\)
−0.523423 + 0.852073i \(0.675345\pi\)
\(180\) 0 0
\(181\) 2.43741 0.181171 0.0905856 0.995889i \(-0.471126\pi\)
0.0905856 + 0.995889i \(0.471126\pi\)
\(182\) 0 0
\(183\) −5.47292 −0.404570
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 3.28085 0.239919
\(188\) 0 0
\(189\) −10.1888 −0.741124
\(190\) 0 0
\(191\) 14.3568 1.03882 0.519412 0.854524i \(-0.326151\pi\)
0.519412 + 0.854524i \(0.326151\pi\)
\(192\) 0 0
\(193\) 10.3628 0.745929 0.372964 0.927846i \(-0.378341\pi\)
0.372964 + 0.927846i \(0.378341\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 12.9168 0.920286 0.460143 0.887845i \(-0.347798\pi\)
0.460143 + 0.887845i \(0.347798\pi\)
\(198\) 0 0
\(199\) 14.9400 1.05907 0.529536 0.848288i \(-0.322366\pi\)
0.529536 + 0.848288i \(0.322366\pi\)
\(200\) 0 0
\(201\) 2.97241 0.209657
\(202\) 0 0
\(203\) 18.0426 1.26634
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) −1.70576 −0.118558
\(208\) 0 0
\(209\) −2.79460 −0.193306
\(210\) 0 0
\(211\) 5.28130 0.363579 0.181790 0.983337i \(-0.441811\pi\)
0.181790 + 0.983337i \(0.441811\pi\)
\(212\) 0 0
\(213\) 4.57787 0.313670
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) −2.70758 −0.183802
\(218\) 0 0
\(219\) −4.63514 −0.313214
\(220\) 0 0
\(221\) 3.36299 0.226219
\(222\) 0 0
\(223\) −11.5273 −0.771926 −0.385963 0.922514i \(-0.626131\pi\)
−0.385963 + 0.922514i \(0.626131\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −29.1127 −1.93228 −0.966139 0.258021i \(-0.916930\pi\)
−0.966139 + 0.258021i \(0.916930\pi\)
\(228\) 0 0
\(229\) 2.60133 0.171901 0.0859503 0.996299i \(-0.472607\pi\)
0.0859503 + 0.996299i \(0.472607\pi\)
\(230\) 0 0
\(231\) −4.94064 −0.325070
\(232\) 0 0
\(233\) 11.2634 0.737890 0.368945 0.929451i \(-0.379719\pi\)
0.368945 + 0.929451i \(0.379719\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 1.80124 0.117003
\(238\) 0 0
\(239\) −22.6347 −1.46412 −0.732059 0.681242i \(-0.761440\pi\)
−0.732059 + 0.681242i \(0.761440\pi\)
\(240\) 0 0
\(241\) −13.2940 −0.856339 −0.428169 0.903698i \(-0.640841\pi\)
−0.428169 + 0.903698i \(0.640841\pi\)
\(242\) 0 0
\(243\) −11.7847 −0.755989
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −2.86457 −0.182268
\(248\) 0 0
\(249\) −2.26994 −0.143852
\(250\) 0 0
\(251\) 16.2433 1.02527 0.512633 0.858608i \(-0.328670\pi\)
0.512633 + 0.858608i \(0.328670\pi\)
\(252\) 0 0
\(253\) −1.72518 −0.108461
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 27.4124 1.70994 0.854968 0.518680i \(-0.173577\pi\)
0.854968 + 0.518680i \(0.173577\pi\)
\(258\) 0 0
\(259\) 29.9201 1.85914
\(260\) 0 0
\(261\) 13.7244 0.849522
\(262\) 0 0
\(263\) 24.0460 1.48274 0.741370 0.671097i \(-0.234176\pi\)
0.741370 + 0.671097i \(0.234176\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) −5.20221 −0.318370
\(268\) 0 0
\(269\) −12.5398 −0.764565 −0.382283 0.924045i \(-0.624862\pi\)
−0.382283 + 0.924045i \(0.624862\pi\)
\(270\) 0 0
\(271\) 17.5270 1.06469 0.532344 0.846528i \(-0.321311\pi\)
0.532344 + 0.846528i \(0.321311\pi\)
\(272\) 0 0
\(273\) −5.06435 −0.306508
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −5.79301 −0.348068 −0.174034 0.984740i \(-0.555680\pi\)
−0.174034 + 0.984740i \(0.555680\pi\)
\(278\) 0 0
\(279\) −2.05957 −0.123303
\(280\) 0 0
\(281\) 19.2228 1.14673 0.573367 0.819298i \(-0.305637\pi\)
0.573367 + 0.819298i \(0.305637\pi\)
\(282\) 0 0
\(283\) −28.3864 −1.68740 −0.843698 0.536818i \(-0.819626\pi\)
−0.843698 + 0.536818i \(0.819626\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −36.2602 −2.14037
\(288\) 0 0
\(289\) −15.6217 −0.918925
\(290\) 0 0
\(291\) 0.306240 0.0179521
\(292\) 0 0
\(293\) −19.8463 −1.15943 −0.579717 0.814818i \(-0.696837\pi\)
−0.579717 + 0.814818i \(0.696837\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) −7.83856 −0.454840
\(298\) 0 0
\(299\) −1.76838 −0.102268
\(300\) 0 0
\(301\) 38.1250 2.19749
\(302\) 0 0
\(303\) −1.86177 −0.106956
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 6.35995 0.362982 0.181491 0.983393i \(-0.441908\pi\)
0.181491 + 0.983393i \(0.441908\pi\)
\(308\) 0 0
\(309\) −4.90116 −0.278817
\(310\) 0 0
\(311\) 23.4371 1.32899 0.664497 0.747291i \(-0.268646\pi\)
0.664497 + 0.747291i \(0.268646\pi\)
\(312\) 0 0
\(313\) 3.38952 0.191587 0.0957934 0.995401i \(-0.469461\pi\)
0.0957934 + 0.995401i \(0.469461\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 5.37856 0.302090 0.151045 0.988527i \(-0.451736\pi\)
0.151045 + 0.988527i \(0.451736\pi\)
\(318\) 0 0
\(319\) 13.8808 0.777174
\(320\) 0 0
\(321\) 2.20674 0.123168
\(322\) 0 0
\(323\) −1.17400 −0.0653229
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) −7.78589 −0.430561
\(328\) 0 0
\(329\) 18.4437 1.01683
\(330\) 0 0
\(331\) −3.80034 −0.208885 −0.104443 0.994531i \(-0.533306\pi\)
−0.104443 + 0.994531i \(0.533306\pi\)
\(332\) 0 0
\(333\) 22.7593 1.24720
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 28.2975 1.54146 0.770732 0.637159i \(-0.219891\pi\)
0.770732 + 0.637159i \(0.219891\pi\)
\(338\) 0 0
\(339\) −1.14290 −0.0620739
\(340\) 0 0
\(341\) −2.08303 −0.112802
\(342\) 0 0
\(343\) 2.92409 0.157886
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 14.0367 0.753530 0.376765 0.926309i \(-0.377036\pi\)
0.376765 + 0.926309i \(0.377036\pi\)
\(348\) 0 0
\(349\) −31.9225 −1.70877 −0.854387 0.519637i \(-0.826067\pi\)
−0.854387 + 0.519637i \(0.826067\pi\)
\(350\) 0 0
\(351\) −8.03482 −0.428867
\(352\) 0 0
\(353\) 2.65209 0.141157 0.0705783 0.997506i \(-0.477516\pi\)
0.0705783 + 0.997506i \(0.477516\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) −2.07554 −0.109849
\(358\) 0 0
\(359\) 23.7750 1.25480 0.627398 0.778699i \(-0.284120\pi\)
0.627398 + 0.778699i \(0.284120\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 0 0
\(363\) 1.55267 0.0814941
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) −17.3923 −0.907869 −0.453934 0.891035i \(-0.649980\pi\)
−0.453934 + 0.891035i \(0.649980\pi\)
\(368\) 0 0
\(369\) −27.5820 −1.43586
\(370\) 0 0
\(371\) −27.0755 −1.40569
\(372\) 0 0
\(373\) −17.8099 −0.922164 −0.461082 0.887358i \(-0.652539\pi\)
−0.461082 + 0.887358i \(0.652539\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 14.2283 0.732796
\(378\) 0 0
\(379\) 25.7998 1.32525 0.662624 0.748952i \(-0.269443\pi\)
0.662624 + 0.748952i \(0.269443\pi\)
\(380\) 0 0
\(381\) 8.49489 0.435206
\(382\) 0 0
\(383\) −2.08964 −0.106776 −0.0533879 0.998574i \(-0.517002\pi\)
−0.0533879 + 0.998574i \(0.517002\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 29.0005 1.47418
\(388\) 0 0
\(389\) −17.8701 −0.906052 −0.453026 0.891497i \(-0.649656\pi\)
−0.453026 + 0.891497i \(0.649656\pi\)
\(390\) 0 0
\(391\) −0.724741 −0.0366517
\(392\) 0 0
\(393\) −4.38455 −0.221171
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) −28.4105 −1.42588 −0.712941 0.701224i \(-0.752637\pi\)
−0.712941 + 0.701224i \(0.752637\pi\)
\(398\) 0 0
\(399\) 1.76793 0.0885070
\(400\) 0 0
\(401\) −20.7047 −1.03394 −0.516971 0.856003i \(-0.672940\pi\)
−0.516971 + 0.856003i \(0.672940\pi\)
\(402\) 0 0
\(403\) −2.13518 −0.106361
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 23.0185 1.14099
\(408\) 0 0
\(409\) 29.1270 1.44024 0.720119 0.693850i \(-0.244087\pi\)
0.720119 + 0.693850i \(0.244087\pi\)
\(410\) 0 0
\(411\) 7.51311 0.370594
\(412\) 0 0
\(413\) −13.9237 −0.685141
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) −7.35613 −0.360231
\(418\) 0 0
\(419\) 16.3711 0.799782 0.399891 0.916563i \(-0.369048\pi\)
0.399891 + 0.916563i \(0.369048\pi\)
\(420\) 0 0
\(421\) −8.82857 −0.430278 −0.215139 0.976583i \(-0.569020\pi\)
−0.215139 + 0.976583i \(0.569020\pi\)
\(422\) 0 0
\(423\) 14.0296 0.682140
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) −40.8475 −1.97675
\(428\) 0 0
\(429\) −3.89617 −0.188109
\(430\) 0 0
\(431\) −24.8989 −1.19934 −0.599668 0.800249i \(-0.704701\pi\)
−0.599668 + 0.800249i \(0.704701\pi\)
\(432\) 0 0
\(433\) 18.9901 0.912608 0.456304 0.889824i \(-0.349173\pi\)
0.456304 + 0.889824i \(0.349173\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0.617328 0.0295308
\(438\) 0 0
\(439\) −13.3540 −0.637352 −0.318676 0.947864i \(-0.603238\pi\)
−0.318676 + 0.947864i \(0.603238\pi\)
\(440\) 0 0
\(441\) −17.1176 −0.815125
\(442\) 0 0
\(443\) −8.67208 −0.412023 −0.206012 0.978550i \(-0.566048\pi\)
−0.206012 + 0.978550i \(0.566048\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 5.76358 0.272608
\(448\) 0 0
\(449\) 40.5277 1.91262 0.956310 0.292354i \(-0.0944386\pi\)
0.956310 + 0.292354i \(0.0944386\pi\)
\(450\) 0 0
\(451\) −27.8962 −1.31358
\(452\) 0 0
\(453\) 4.04573 0.190085
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 16.9859 0.794566 0.397283 0.917696i \(-0.369953\pi\)
0.397283 + 0.917696i \(0.369953\pi\)
\(458\) 0 0
\(459\) −3.29294 −0.153701
\(460\) 0 0
\(461\) 33.1374 1.54336 0.771681 0.636010i \(-0.219416\pi\)
0.771681 + 0.636010i \(0.219416\pi\)
\(462\) 0 0
\(463\) −8.11961 −0.377350 −0.188675 0.982040i \(-0.560419\pi\)
−0.188675 + 0.982040i \(0.560419\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −16.1221 −0.746042 −0.373021 0.927823i \(-0.621678\pi\)
−0.373021 + 0.927823i \(0.621678\pi\)
\(468\) 0 0
\(469\) 22.1847 1.02440
\(470\) 0 0
\(471\) −5.19064 −0.239172
\(472\) 0 0
\(473\) 29.3308 1.34863
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) −20.5955 −0.943003
\(478\) 0 0
\(479\) −5.30225 −0.242266 −0.121133 0.992636i \(-0.538653\pi\)
−0.121133 + 0.992636i \(0.538653\pi\)
\(480\) 0 0
\(481\) 23.5949 1.07583
\(482\) 0 0
\(483\) 1.09139 0.0496600
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 20.9188 0.947922 0.473961 0.880546i \(-0.342824\pi\)
0.473961 + 0.880546i \(0.342824\pi\)
\(488\) 0 0
\(489\) −0.805886 −0.0364434
\(490\) 0 0
\(491\) −22.1370 −0.999029 −0.499515 0.866305i \(-0.666488\pi\)
−0.499515 + 0.866305i \(0.666488\pi\)
\(492\) 0 0
\(493\) 5.83124 0.262626
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 34.1672 1.53261
\(498\) 0 0
\(499\) 5.44182 0.243609 0.121805 0.992554i \(-0.461132\pi\)
0.121805 + 0.992554i \(0.461132\pi\)
\(500\) 0 0
\(501\) 4.78700 0.213867
\(502\) 0 0
\(503\) −17.9666 −0.801092 −0.400546 0.916277i \(-0.631180\pi\)
−0.400546 + 0.916277i \(0.631180\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 2.33334 0.103627
\(508\) 0 0
\(509\) 0.433419 0.0192110 0.00960548 0.999954i \(-0.496942\pi\)
0.00960548 + 0.999954i \(0.496942\pi\)
\(510\) 0 0
\(511\) −34.5947 −1.53038
\(512\) 0 0
\(513\) 2.80490 0.123839
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 14.1894 0.624047
\(518\) 0 0
\(519\) −3.92077 −0.172103
\(520\) 0 0
\(521\) −1.29105 −0.0565621 −0.0282811 0.999600i \(-0.509003\pi\)
−0.0282811 + 0.999600i \(0.509003\pi\)
\(522\) 0 0
\(523\) 1.17153 0.0512275 0.0256138 0.999672i \(-0.491846\pi\)
0.0256138 + 0.999672i \(0.491846\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −0.875070 −0.0381187
\(528\) 0 0
\(529\) −22.6189 −0.983431
\(530\) 0 0
\(531\) −10.5913 −0.459624
\(532\) 0 0
\(533\) −28.5946 −1.23857
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 6.81660 0.294158
\(538\) 0 0
\(539\) −17.3126 −0.745706
\(540\) 0 0
\(541\) −21.1522 −0.909402 −0.454701 0.890644i \(-0.650254\pi\)
−0.454701 + 0.890644i \(0.650254\pi\)
\(542\) 0 0
\(543\) −1.18628 −0.0509082
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 39.5096 1.68931 0.844655 0.535311i \(-0.179806\pi\)
0.844655 + 0.535311i \(0.179806\pi\)
\(548\) 0 0
\(549\) −31.0714 −1.32610
\(550\) 0 0
\(551\) −4.96700 −0.211601
\(552\) 0 0
\(553\) 13.4436 0.571682
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 7.90886 0.335109 0.167554 0.985863i \(-0.446413\pi\)
0.167554 + 0.985863i \(0.446413\pi\)
\(558\) 0 0
\(559\) 30.0652 1.27162
\(560\) 0 0
\(561\) −1.59678 −0.0674161
\(562\) 0 0
\(563\) 5.40347 0.227729 0.113865 0.993496i \(-0.463677\pi\)
0.113865 + 0.993496i \(0.463677\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) −25.1523 −1.05630
\(568\) 0 0
\(569\) −4.60607 −0.193096 −0.0965482 0.995328i \(-0.530780\pi\)
−0.0965482 + 0.995328i \(0.530780\pi\)
\(570\) 0 0
\(571\) −38.6010 −1.61540 −0.807701 0.589593i \(-0.799288\pi\)
−0.807701 + 0.589593i \(0.799288\pi\)
\(572\) 0 0
\(573\) −6.98743 −0.291904
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 30.0011 1.24896 0.624480 0.781040i \(-0.285311\pi\)
0.624480 + 0.781040i \(0.285311\pi\)
\(578\) 0 0
\(579\) −5.04353 −0.209602
\(580\) 0 0
\(581\) −16.9418 −0.702866
\(582\) 0 0
\(583\) −20.8301 −0.862694
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 1.90807 0.0787545 0.0393772 0.999224i \(-0.487463\pi\)
0.0393772 + 0.999224i \(0.487463\pi\)
\(588\) 0 0
\(589\) 0.745377 0.0307127
\(590\) 0 0
\(591\) −6.28658 −0.258596
\(592\) 0 0
\(593\) 11.5819 0.475610 0.237805 0.971313i \(-0.423572\pi\)
0.237805 + 0.971313i \(0.423572\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −7.27128 −0.297594
\(598\) 0 0
\(599\) −35.9808 −1.47014 −0.735068 0.677993i \(-0.762850\pi\)
−0.735068 + 0.677993i \(0.762850\pi\)
\(600\) 0 0
\(601\) −6.25635 −0.255202 −0.127601 0.991826i \(-0.540728\pi\)
−0.127601 + 0.991826i \(0.540728\pi\)
\(602\) 0 0
\(603\) 16.8752 0.687213
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 3.98858 0.161891 0.0809457 0.996719i \(-0.474206\pi\)
0.0809457 + 0.996719i \(0.474206\pi\)
\(608\) 0 0
\(609\) −8.78129 −0.355836
\(610\) 0 0
\(611\) 14.5446 0.588412
\(612\) 0 0
\(613\) −26.1306 −1.05541 −0.527703 0.849429i \(-0.676947\pi\)
−0.527703 + 0.849429i \(0.676947\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −6.30081 −0.253661 −0.126831 0.991924i \(-0.540480\pi\)
−0.126831 + 0.991924i \(0.540480\pi\)
\(618\) 0 0
\(619\) 12.7231 0.511387 0.255693 0.966758i \(-0.417696\pi\)
0.255693 + 0.966758i \(0.417696\pi\)
\(620\) 0 0
\(621\) 1.73154 0.0694844
\(622\) 0 0
\(623\) −38.8270 −1.55557
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 1.36012 0.0543181
\(628\) 0 0
\(629\) 9.66996 0.385567
\(630\) 0 0
\(631\) 27.1455 1.08065 0.540323 0.841458i \(-0.318302\pi\)
0.540323 + 0.841458i \(0.318302\pi\)
\(632\) 0 0
\(633\) −2.57039 −0.102164
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) −17.7461 −0.703125
\(638\) 0 0
\(639\) 25.9899 1.02815
\(640\) 0 0
\(641\) −20.8345 −0.822914 −0.411457 0.911429i \(-0.634980\pi\)
−0.411457 + 0.911429i \(0.634980\pi\)
\(642\) 0 0
\(643\) 42.3146 1.66872 0.834362 0.551217i \(-0.185836\pi\)
0.834362 + 0.551217i \(0.185836\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 5.57122 0.219027 0.109514 0.993985i \(-0.465071\pi\)
0.109514 + 0.993985i \(0.465071\pi\)
\(648\) 0 0
\(649\) −10.7120 −0.420481
\(650\) 0 0
\(651\) 1.31777 0.0516475
\(652\) 0 0
\(653\) 44.6953 1.74906 0.874532 0.484968i \(-0.161169\pi\)
0.874532 + 0.484968i \(0.161169\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) −26.3151 −1.02665
\(658\) 0 0
\(659\) −37.0288 −1.44244 −0.721218 0.692708i \(-0.756417\pi\)
−0.721218 + 0.692708i \(0.756417\pi\)
\(660\) 0 0
\(661\) −6.16081 −0.239628 −0.119814 0.992796i \(-0.538230\pi\)
−0.119814 + 0.992796i \(0.538230\pi\)
\(662\) 0 0
\(663\) −1.63676 −0.0635665
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −3.06627 −0.118726
\(668\) 0 0
\(669\) 5.61031 0.216907
\(670\) 0 0
\(671\) −31.4253 −1.21316
\(672\) 0 0
\(673\) 13.5879 0.523773 0.261887 0.965099i \(-0.415655\pi\)
0.261887 + 0.965099i \(0.415655\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −17.0051 −0.653559 −0.326779 0.945101i \(-0.605963\pi\)
−0.326779 + 0.945101i \(0.605963\pi\)
\(678\) 0 0
\(679\) 2.28564 0.0877149
\(680\) 0 0
\(681\) 14.1691 0.542960
\(682\) 0 0
\(683\) −10.9418 −0.418677 −0.209338 0.977843i \(-0.567131\pi\)
−0.209338 + 0.977843i \(0.567131\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) −1.26606 −0.0483032
\(688\) 0 0
\(689\) −21.3516 −0.813433
\(690\) 0 0
\(691\) −21.4168 −0.814733 −0.407366 0.913265i \(-0.633553\pi\)
−0.407366 + 0.913265i \(0.633553\pi\)
\(692\) 0 0
\(693\) −28.0495 −1.06551
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) −11.7190 −0.443890
\(698\) 0 0
\(699\) −5.48186 −0.207343
\(700\) 0 0
\(701\) −27.2640 −1.02975 −0.514874 0.857266i \(-0.672161\pi\)
−0.514874 + 0.857266i \(0.672161\pi\)
\(702\) 0 0
\(703\) −8.23679 −0.310656
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −13.8954 −0.522591
\(708\) 0 0
\(709\) 25.9811 0.975740 0.487870 0.872916i \(-0.337774\pi\)
0.487870 + 0.872916i \(0.337774\pi\)
\(710\) 0 0
\(711\) 10.2262 0.383511
\(712\) 0 0
\(713\) 0.460142 0.0172325
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 11.0162 0.411409
\(718\) 0 0
\(719\) −32.4059 −1.20854 −0.604268 0.796781i \(-0.706535\pi\)
−0.604268 + 0.796781i \(0.706535\pi\)
\(720\) 0 0
\(721\) −36.5801 −1.36231
\(722\) 0 0
\(723\) 6.47013 0.240627
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 47.1313 1.74800 0.874001 0.485924i \(-0.161517\pi\)
0.874001 + 0.485924i \(0.161517\pi\)
\(728\) 0 0
\(729\) −15.0371 −0.556931
\(730\) 0 0
\(731\) 12.3217 0.455735
\(732\) 0 0
\(733\) −12.7925 −0.472500 −0.236250 0.971692i \(-0.575919\pi\)
−0.236250 + 0.971692i \(0.575919\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 17.0675 0.628688
\(738\) 0 0
\(739\) −28.2097 −1.03771 −0.518856 0.854862i \(-0.673642\pi\)
−0.518856 + 0.854862i \(0.673642\pi\)
\(740\) 0 0
\(741\) 1.39418 0.0512164
\(742\) 0 0
\(743\) −16.0366 −0.588325 −0.294162 0.955755i \(-0.595041\pi\)
−0.294162 + 0.955755i \(0.595041\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) −12.8871 −0.471515
\(748\) 0 0
\(749\) 16.4702 0.601807
\(750\) 0 0
\(751\) 28.8165 1.05153 0.525765 0.850630i \(-0.323779\pi\)
0.525765 + 0.850630i \(0.323779\pi\)
\(752\) 0 0
\(753\) −7.90555 −0.288094
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 39.3944 1.43181 0.715906 0.698196i \(-0.246014\pi\)
0.715906 + 0.698196i \(0.246014\pi\)
\(758\) 0 0
\(759\) 0.839643 0.0304771
\(760\) 0 0
\(761\) 21.7891 0.789856 0.394928 0.918712i \(-0.370770\pi\)
0.394928 + 0.918712i \(0.370770\pi\)
\(762\) 0 0
\(763\) −58.1105 −2.10374
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −10.9802 −0.396471
\(768\) 0 0
\(769\) 15.7229 0.566983 0.283491 0.958975i \(-0.408507\pi\)
0.283491 + 0.958975i \(0.408507\pi\)
\(770\) 0 0
\(771\) −13.3415 −0.480483
\(772\) 0 0
\(773\) −26.6810 −0.959649 −0.479825 0.877364i \(-0.659300\pi\)
−0.479825 + 0.877364i \(0.659300\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) −14.5620 −0.522410
\(778\) 0 0
\(779\) 9.98217 0.357648
\(780\) 0 0
\(781\) 26.2860 0.940586
\(782\) 0 0
\(783\) −13.9319 −0.497887
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 38.7243 1.38037 0.690187 0.723631i \(-0.257528\pi\)
0.690187 + 0.723631i \(0.257528\pi\)
\(788\) 0 0
\(789\) −11.7031 −0.416642
\(790\) 0 0
\(791\) −8.53012 −0.303296
\(792\) 0 0
\(793\) −32.2121 −1.14389
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −33.5802 −1.18947 −0.594736 0.803921i \(-0.702744\pi\)
−0.594736 + 0.803921i \(0.702744\pi\)
\(798\) 0 0
\(799\) 5.96087 0.210881
\(800\) 0 0
\(801\) −29.5345 −1.04355
\(802\) 0 0
\(803\) −26.6148 −0.939217
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 6.10309 0.214839
\(808\) 0 0
\(809\) −25.5182 −0.897172 −0.448586 0.893740i \(-0.648072\pi\)
−0.448586 + 0.893740i \(0.648072\pi\)
\(810\) 0 0
\(811\) 9.88905 0.347252 0.173626 0.984812i \(-0.444452\pi\)
0.173626 + 0.984812i \(0.444452\pi\)
\(812\) 0 0
\(813\) −8.53033 −0.299172
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) −10.4955 −0.367192
\(818\) 0 0
\(819\) −28.7518 −1.00467
\(820\) 0 0
\(821\) −8.01417 −0.279696 −0.139848 0.990173i \(-0.544661\pi\)
−0.139848 + 0.990173i \(0.544661\pi\)
\(822\) 0 0
\(823\) 18.4568 0.643363 0.321681 0.946848i \(-0.395752\pi\)
0.321681 + 0.946848i \(0.395752\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −10.3135 −0.358637 −0.179318 0.983791i \(-0.557389\pi\)
−0.179318 + 0.983791i \(0.557389\pi\)
\(828\) 0 0
\(829\) −28.5805 −0.992640 −0.496320 0.868140i \(-0.665316\pi\)
−0.496320 + 0.868140i \(0.665316\pi\)
\(830\) 0 0
\(831\) 2.81944 0.0978053
\(832\) 0 0
\(833\) −7.27293 −0.251992
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 2.09071 0.0722654
\(838\) 0 0
\(839\) 5.06942 0.175016 0.0875079 0.996164i \(-0.472110\pi\)
0.0875079 + 0.996164i \(0.472110\pi\)
\(840\) 0 0
\(841\) −4.32890 −0.149272
\(842\) 0 0
\(843\) −9.35567 −0.322227
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 11.5885 0.398184
\(848\) 0 0
\(849\) 13.8156 0.474149
\(850\) 0 0
\(851\) −5.08480 −0.174305
\(852\) 0 0
\(853\) 17.1101 0.585839 0.292920 0.956137i \(-0.405373\pi\)
0.292920 + 0.956137i \(0.405373\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 31.4488 1.07427 0.537135 0.843496i \(-0.319506\pi\)
0.537135 + 0.843496i \(0.319506\pi\)
\(858\) 0 0
\(859\) 5.98445 0.204187 0.102093 0.994775i \(-0.467446\pi\)
0.102093 + 0.994775i \(0.467446\pi\)
\(860\) 0 0
\(861\) 17.6477 0.601433
\(862\) 0 0
\(863\) −20.8530 −0.709844 −0.354922 0.934896i \(-0.615493\pi\)
−0.354922 + 0.934896i \(0.615493\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 7.60306 0.258213
\(868\) 0 0
\(869\) 10.3426 0.350850
\(870\) 0 0
\(871\) 17.4948 0.592788
\(872\) 0 0
\(873\) 1.73862 0.0588432
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 7.11382 0.240217 0.120108 0.992761i \(-0.461676\pi\)
0.120108 + 0.992761i \(0.461676\pi\)
\(878\) 0 0
\(879\) 9.65915 0.325795
\(880\) 0 0
\(881\) −42.4272 −1.42941 −0.714704 0.699427i \(-0.753438\pi\)
−0.714704 + 0.699427i \(0.753438\pi\)
\(882\) 0 0
\(883\) −9.15841 −0.308205 −0.154103 0.988055i \(-0.549249\pi\)
−0.154103 + 0.988055i \(0.549249\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −21.2435 −0.713286 −0.356643 0.934241i \(-0.616079\pi\)
−0.356643 + 0.934241i \(0.616079\pi\)
\(888\) 0 0
\(889\) 63.4021 2.12644
\(890\) 0 0
\(891\) −19.3505 −0.648265
\(892\) 0 0
\(893\) −5.07742 −0.169909
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0.860666 0.0287368
\(898\) 0 0
\(899\) −3.70229 −0.123478
\(900\) 0 0
\(901\) −8.75062 −0.291525
\(902\) 0 0
\(903\) −18.5553 −0.617483
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) −19.1139 −0.634667 −0.317333 0.948314i \(-0.602787\pi\)
−0.317333 + 0.948314i \(0.602787\pi\)
\(908\) 0 0
\(909\) −10.5698 −0.350579
\(910\) 0 0
\(911\) −19.9557 −0.661161 −0.330580 0.943778i \(-0.607244\pi\)
−0.330580 + 0.943778i \(0.607244\pi\)
\(912\) 0 0
\(913\) −13.0339 −0.431360
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −32.7243 −1.08065
\(918\) 0 0
\(919\) 54.3957 1.79435 0.897174 0.441677i \(-0.145616\pi\)
0.897174 + 0.441677i \(0.145616\pi\)
\(920\) 0 0
\(921\) −3.09537 −0.101996
\(922\) 0 0
\(923\) 26.9441 0.886876
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) −27.8253 −0.913904
\(928\) 0 0
\(929\) −50.0187 −1.64106 −0.820530 0.571603i \(-0.806322\pi\)
−0.820530 + 0.571603i \(0.806322\pi\)
\(930\) 0 0
\(931\) 6.19502 0.203034
\(932\) 0 0
\(933\) −11.4068 −0.373441
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 50.3138 1.64368 0.821840 0.569719i \(-0.192948\pi\)
0.821840 + 0.569719i \(0.192948\pi\)
\(938\) 0 0
\(939\) −1.64967 −0.0538349
\(940\) 0 0
\(941\) −24.6083 −0.802207 −0.401103 0.916033i \(-0.631373\pi\)
−0.401103 + 0.916033i \(0.631373\pi\)
\(942\) 0 0
\(943\) 6.16227 0.200671
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 0.628142 0.0204119 0.0102059 0.999948i \(-0.496751\pi\)
0.0102059 + 0.999948i \(0.496751\pi\)
\(948\) 0 0
\(949\) −27.2812 −0.885586
\(950\) 0 0
\(951\) −2.61773 −0.0848858
\(952\) 0 0
\(953\) 40.6251 1.31598 0.657988 0.753028i \(-0.271408\pi\)
0.657988 + 0.753028i \(0.271408\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) −6.75574 −0.218382
\(958\) 0 0
\(959\) 56.0745 1.81074
\(960\) 0 0
\(961\) −30.4444 −0.982078
\(962\) 0 0
\(963\) 12.5283 0.403720
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 25.2241 0.811153 0.405576 0.914061i \(-0.367071\pi\)
0.405576 + 0.914061i \(0.367071\pi\)
\(968\) 0 0
\(969\) 0.571381 0.0183554
\(970\) 0 0
\(971\) 2.82319 0.0906006 0.0453003 0.998973i \(-0.485576\pi\)
0.0453003 + 0.998973i \(0.485576\pi\)
\(972\) 0 0
\(973\) −54.9030 −1.76011
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 34.1606 1.09289 0.546447 0.837494i \(-0.315980\pi\)
0.546447 + 0.837494i \(0.315980\pi\)
\(978\) 0 0
\(979\) −29.8709 −0.954679
\(980\) 0 0
\(981\) −44.2028 −1.41129
\(982\) 0 0
\(983\) −46.0699 −1.46940 −0.734701 0.678391i \(-0.762677\pi\)
−0.734701 + 0.678391i \(0.762677\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) −8.97650 −0.285725
\(988\) 0 0
\(989\) −6.47919 −0.206026
\(990\) 0 0
\(991\) 56.9827 1.81011 0.905057 0.425290i \(-0.139828\pi\)
0.905057 + 0.425290i \(0.139828\pi\)
\(992\) 0 0
\(993\) 1.84961 0.0586957
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 50.3518 1.59466 0.797328 0.603546i \(-0.206246\pi\)
0.797328 + 0.603546i \(0.206246\pi\)
\(998\) 0 0
\(999\) −23.1034 −0.730958
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 3800.2.a.bb.1.3 6
4.3 odd 2 7600.2.a.cm.1.4 6
5.2 odd 4 3800.2.d.p.3649.8 12
5.3 odd 4 3800.2.d.p.3649.5 12
5.4 even 2 3800.2.a.bd.1.4 yes 6
20.19 odd 2 7600.2.a.ci.1.3 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3800.2.a.bb.1.3 6 1.1 even 1 trivial
3800.2.a.bd.1.4 yes 6 5.4 even 2
3800.2.d.p.3649.5 12 5.3 odd 4
3800.2.d.p.3649.8 12 5.2 odd 4
7600.2.a.ci.1.3 6 20.19 odd 2
7600.2.a.cm.1.4 6 4.3 odd 2