Properties

Label 3920.2.a.bu.1.1
Level $3920$
Weight $2$
Character 3920.1
Self dual yes
Analytic conductor $31.301$
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] = [3920,2,Mod(1,3920)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3920, 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("3920.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3920 = 2^{4} \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3920.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: \(31.3013575923\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{17}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 280)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.56155\) of defining polynomial
Character \(\chi\) \(=\) 3920.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.56155 q^{3} -1.00000 q^{5} -0.561553 q^{9} +O(q^{10})\) \(q-1.56155 q^{3} -1.00000 q^{5} -0.561553 q^{9} -1.56155 q^{11} -6.68466 q^{13} +1.56155 q^{15} -7.56155 q^{17} -7.12311 q^{19} -3.12311 q^{23} +1.00000 q^{25} +5.56155 q^{27} +0.438447 q^{29} +6.24621 q^{31} +2.43845 q^{33} -8.24621 q^{37} +10.4384 q^{39} +1.12311 q^{41} +7.12311 q^{43} +0.561553 q^{45} +2.43845 q^{47} +11.8078 q^{51} -13.1231 q^{53} +1.56155 q^{55} +11.1231 q^{57} -4.00000 q^{59} +6.87689 q^{61} +6.68466 q^{65} -2.24621 q^{67} +4.87689 q^{69} +4.24621 q^{73} -1.56155 q^{75} -0.684658 q^{79} -7.00000 q^{81} +12.0000 q^{83} +7.56155 q^{85} -0.684658 q^{87} -5.12311 q^{89} -9.75379 q^{93} +7.12311 q^{95} -1.31534 q^{97} +0.876894 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{3} - 2 q^{5} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + q^{3} - 2 q^{5} + 3 q^{9} + q^{11} - q^{13} - q^{15} - 11 q^{17} - 6 q^{19} + 2 q^{23} + 2 q^{25} + 7 q^{27} + 5 q^{29} - 4 q^{31} + 9 q^{33} + 25 q^{39} - 6 q^{41} + 6 q^{43} - 3 q^{45} + 9 q^{47} + 3 q^{51} - 18 q^{53} - q^{55} + 14 q^{57} - 8 q^{59} + 22 q^{61} + q^{65} + 12 q^{67} + 18 q^{69} - 8 q^{73} + q^{75} + 11 q^{79} - 14 q^{81} + 24 q^{83} + 11 q^{85} + 11 q^{87} - 2 q^{89} - 36 q^{93} + 6 q^{95} - 15 q^{97} + 10 q^{99}+O(q^{100}) \) 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
\(3\) −1.56155 −0.901563 −0.450781 0.892634i \(-0.648855\pi\)
−0.450781 + 0.892634i \(0.648855\pi\)
\(4\) 0 0
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) −0.561553 −0.187184
\(10\) 0 0
\(11\) −1.56155 −0.470826 −0.235413 0.971895i \(-0.575644\pi\)
−0.235413 + 0.971895i \(0.575644\pi\)
\(12\) 0 0
\(13\) −6.68466 −1.85399 −0.926995 0.375073i \(-0.877618\pi\)
−0.926995 + 0.375073i \(0.877618\pi\)
\(14\) 0 0
\(15\) 1.56155 0.403191
\(16\) 0 0
\(17\) −7.56155 −1.83395 −0.916973 0.398949i \(-0.869375\pi\)
−0.916973 + 0.398949i \(0.869375\pi\)
\(18\) 0 0
\(19\) −7.12311 −1.63415 −0.817076 0.576530i \(-0.804407\pi\)
−0.817076 + 0.576530i \(0.804407\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −3.12311 −0.651213 −0.325606 0.945505i \(-0.605568\pi\)
−0.325606 + 0.945505i \(0.605568\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 5.56155 1.07032
\(28\) 0 0
\(29\) 0.438447 0.0814176 0.0407088 0.999171i \(-0.487038\pi\)
0.0407088 + 0.999171i \(0.487038\pi\)
\(30\) 0 0
\(31\) 6.24621 1.12185 0.560926 0.827866i \(-0.310445\pi\)
0.560926 + 0.827866i \(0.310445\pi\)
\(32\) 0 0
\(33\) 2.43845 0.424479
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −8.24621 −1.35567 −0.677834 0.735215i \(-0.737081\pi\)
−0.677834 + 0.735215i \(0.737081\pi\)
\(38\) 0 0
\(39\) 10.4384 1.67149
\(40\) 0 0
\(41\) 1.12311 0.175400 0.0876998 0.996147i \(-0.472048\pi\)
0.0876998 + 0.996147i \(0.472048\pi\)
\(42\) 0 0
\(43\) 7.12311 1.08626 0.543132 0.839648i \(-0.317238\pi\)
0.543132 + 0.839648i \(0.317238\pi\)
\(44\) 0 0
\(45\) 0.561553 0.0837114
\(46\) 0 0
\(47\) 2.43845 0.355684 0.177842 0.984059i \(-0.443088\pi\)
0.177842 + 0.984059i \(0.443088\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) 11.8078 1.65342
\(52\) 0 0
\(53\) −13.1231 −1.80260 −0.901299 0.433198i \(-0.857385\pi\)
−0.901299 + 0.433198i \(0.857385\pi\)
\(54\) 0 0
\(55\) 1.56155 0.210560
\(56\) 0 0
\(57\) 11.1231 1.47329
\(58\) 0 0
\(59\) −4.00000 −0.520756 −0.260378 0.965507i \(-0.583847\pi\)
−0.260378 + 0.965507i \(0.583847\pi\)
\(60\) 0 0
\(61\) 6.87689 0.880496 0.440248 0.897876i \(-0.354891\pi\)
0.440248 + 0.897876i \(0.354891\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 6.68466 0.829130
\(66\) 0 0
\(67\) −2.24621 −0.274418 −0.137209 0.990542i \(-0.543813\pi\)
−0.137209 + 0.990542i \(0.543813\pi\)
\(68\) 0 0
\(69\) 4.87689 0.587109
\(70\) 0 0
\(71\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) 4.24621 0.496981 0.248491 0.968634i \(-0.420065\pi\)
0.248491 + 0.968634i \(0.420065\pi\)
\(74\) 0 0
\(75\) −1.56155 −0.180313
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −0.684658 −0.0770301 −0.0385150 0.999258i \(-0.512263\pi\)
−0.0385150 + 0.999258i \(0.512263\pi\)
\(80\) 0 0
\(81\) −7.00000 −0.777778
\(82\) 0 0
\(83\) 12.0000 1.31717 0.658586 0.752506i \(-0.271155\pi\)
0.658586 + 0.752506i \(0.271155\pi\)
\(84\) 0 0
\(85\) 7.56155 0.820166
\(86\) 0 0
\(87\) −0.684658 −0.0734031
\(88\) 0 0
\(89\) −5.12311 −0.543048 −0.271524 0.962432i \(-0.587528\pi\)
−0.271524 + 0.962432i \(0.587528\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) −9.75379 −1.01142
\(94\) 0 0
\(95\) 7.12311 0.730815
\(96\) 0 0
\(97\) −1.31534 −0.133553 −0.0667764 0.997768i \(-0.521271\pi\)
−0.0667764 + 0.997768i \(0.521271\pi\)
\(98\) 0 0
\(99\) 0.876894 0.0881312
\(100\) 0 0
\(101\) −6.00000 −0.597022 −0.298511 0.954406i \(-0.596490\pi\)
−0.298511 + 0.954406i \(0.596490\pi\)
\(102\) 0 0
\(103\) −11.8078 −1.16345 −0.581727 0.813384i \(-0.697623\pi\)
−0.581727 + 0.813384i \(0.697623\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −15.1231 −1.46201 −0.731003 0.682374i \(-0.760947\pi\)
−0.731003 + 0.682374i \(0.760947\pi\)
\(108\) 0 0
\(109\) −4.43845 −0.425126 −0.212563 0.977147i \(-0.568181\pi\)
−0.212563 + 0.977147i \(0.568181\pi\)
\(110\) 0 0
\(111\) 12.8769 1.22222
\(112\) 0 0
\(113\) 8.24621 0.775738 0.387869 0.921714i \(-0.373211\pi\)
0.387869 + 0.921714i \(0.373211\pi\)
\(114\) 0 0
\(115\) 3.12311 0.291231
\(116\) 0 0
\(117\) 3.75379 0.347038
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −8.56155 −0.778323
\(122\) 0 0
\(123\) −1.75379 −0.158134
\(124\) 0 0
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) 6.24621 0.554262 0.277131 0.960832i \(-0.410616\pi\)
0.277131 + 0.960832i \(0.410616\pi\)
\(128\) 0 0
\(129\) −11.1231 −0.979335
\(130\) 0 0
\(131\) 15.1231 1.32131 0.660656 0.750689i \(-0.270278\pi\)
0.660656 + 0.750689i \(0.270278\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) −5.56155 −0.478662
\(136\) 0 0
\(137\) −7.36932 −0.629603 −0.314802 0.949157i \(-0.601938\pi\)
−0.314802 + 0.949157i \(0.601938\pi\)
\(138\) 0 0
\(139\) −21.3693 −1.81252 −0.906261 0.422719i \(-0.861076\pi\)
−0.906261 + 0.422719i \(0.861076\pi\)
\(140\) 0 0
\(141\) −3.80776 −0.320672
\(142\) 0 0
\(143\) 10.4384 0.872907
\(144\) 0 0
\(145\) −0.438447 −0.0364111
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −0.246211 −0.0201704 −0.0100852 0.999949i \(-0.503210\pi\)
−0.0100852 + 0.999949i \(0.503210\pi\)
\(150\) 0 0
\(151\) 19.8078 1.61193 0.805966 0.591961i \(-0.201646\pi\)
0.805966 + 0.591961i \(0.201646\pi\)
\(152\) 0 0
\(153\) 4.24621 0.343286
\(154\) 0 0
\(155\) −6.24621 −0.501708
\(156\) 0 0
\(157\) −4.24621 −0.338885 −0.169442 0.985540i \(-0.554197\pi\)
−0.169442 + 0.985540i \(0.554197\pi\)
\(158\) 0 0
\(159\) 20.4924 1.62515
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) −19.6155 −1.53641 −0.768203 0.640206i \(-0.778849\pi\)
−0.768203 + 0.640206i \(0.778849\pi\)
\(164\) 0 0
\(165\) −2.43845 −0.189833
\(166\) 0 0
\(167\) 4.19224 0.324405 0.162202 0.986757i \(-0.448140\pi\)
0.162202 + 0.986757i \(0.448140\pi\)
\(168\) 0 0
\(169\) 31.6847 2.43728
\(170\) 0 0
\(171\) 4.00000 0.305888
\(172\) 0 0
\(173\) 23.1771 1.76212 0.881060 0.473004i \(-0.156830\pi\)
0.881060 + 0.473004i \(0.156830\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 6.24621 0.469494
\(178\) 0 0
\(179\) 4.00000 0.298974 0.149487 0.988764i \(-0.452238\pi\)
0.149487 + 0.988764i \(0.452238\pi\)
\(180\) 0 0
\(181\) 5.12311 0.380797 0.190399 0.981707i \(-0.439022\pi\)
0.190399 + 0.981707i \(0.439022\pi\)
\(182\) 0 0
\(183\) −10.7386 −0.793823
\(184\) 0 0
\(185\) 8.24621 0.606274
\(186\) 0 0
\(187\) 11.8078 0.863469
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 0.684658 0.0495401 0.0247701 0.999693i \(-0.492115\pi\)
0.0247701 + 0.999693i \(0.492115\pi\)
\(192\) 0 0
\(193\) 13.1231 0.944622 0.472311 0.881432i \(-0.343420\pi\)
0.472311 + 0.881432i \(0.343420\pi\)
\(194\) 0 0
\(195\) −10.4384 −0.747513
\(196\) 0 0
\(197\) −13.1231 −0.934983 −0.467491 0.883998i \(-0.654842\pi\)
−0.467491 + 0.883998i \(0.654842\pi\)
\(198\) 0 0
\(199\) −14.2462 −1.00989 −0.504944 0.863152i \(-0.668487\pi\)
−0.504944 + 0.863152i \(0.668487\pi\)
\(200\) 0 0
\(201\) 3.50758 0.247405
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) −1.12311 −0.0784411
\(206\) 0 0
\(207\) 1.75379 0.121897
\(208\) 0 0
\(209\) 11.1231 0.769401
\(210\) 0 0
\(211\) 17.5616 1.20899 0.604494 0.796610i \(-0.293376\pi\)
0.604494 + 0.796610i \(0.293376\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −7.12311 −0.485792
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) −6.63068 −0.448060
\(220\) 0 0
\(221\) 50.5464 3.40012
\(222\) 0 0
\(223\) 24.6847 1.65301 0.826503 0.562932i \(-0.190327\pi\)
0.826503 + 0.562932i \(0.190327\pi\)
\(224\) 0 0
\(225\) −0.561553 −0.0374369
\(226\) 0 0
\(227\) −11.3153 −0.751026 −0.375513 0.926817i \(-0.622533\pi\)
−0.375513 + 0.926817i \(0.622533\pi\)
\(228\) 0 0
\(229\) 11.3693 0.751306 0.375653 0.926760i \(-0.377419\pi\)
0.375653 + 0.926760i \(0.377419\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −10.8769 −0.712569 −0.356285 0.934378i \(-0.615957\pi\)
−0.356285 + 0.934378i \(0.615957\pi\)
\(234\) 0 0
\(235\) −2.43845 −0.159067
\(236\) 0 0
\(237\) 1.06913 0.0694475
\(238\) 0 0
\(239\) 18.0540 1.16781 0.583907 0.811820i \(-0.301523\pi\)
0.583907 + 0.811820i \(0.301523\pi\)
\(240\) 0 0
\(241\) −2.00000 −0.128831 −0.0644157 0.997923i \(-0.520518\pi\)
−0.0644157 + 0.997923i \(0.520518\pi\)
\(242\) 0 0
\(243\) −5.75379 −0.369106
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 47.6155 3.02970
\(248\) 0 0
\(249\) −18.7386 −1.18751
\(250\) 0 0
\(251\) 13.3693 0.843864 0.421932 0.906628i \(-0.361352\pi\)
0.421932 + 0.906628i \(0.361352\pi\)
\(252\) 0 0
\(253\) 4.87689 0.306608
\(254\) 0 0
\(255\) −11.8078 −0.739431
\(256\) 0 0
\(257\) 18.4924 1.15353 0.576763 0.816912i \(-0.304316\pi\)
0.576763 + 0.816912i \(0.304316\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) −0.246211 −0.0152401
\(262\) 0 0
\(263\) −9.36932 −0.577737 −0.288868 0.957369i \(-0.593279\pi\)
−0.288868 + 0.957369i \(0.593279\pi\)
\(264\) 0 0
\(265\) 13.1231 0.806146
\(266\) 0 0
\(267\) 8.00000 0.489592
\(268\) 0 0
\(269\) −4.24621 −0.258896 −0.129448 0.991586i \(-0.541321\pi\)
−0.129448 + 0.991586i \(0.541321\pi\)
\(270\) 0 0
\(271\) −6.24621 −0.379430 −0.189715 0.981839i \(-0.560756\pi\)
−0.189715 + 0.981839i \(0.560756\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −1.56155 −0.0941652
\(276\) 0 0
\(277\) −8.24621 −0.495467 −0.247733 0.968828i \(-0.579686\pi\)
−0.247733 + 0.968828i \(0.579686\pi\)
\(278\) 0 0
\(279\) −3.50758 −0.209993
\(280\) 0 0
\(281\) −19.5616 −1.16694 −0.583472 0.812133i \(-0.698306\pi\)
−0.583472 + 0.812133i \(0.698306\pi\)
\(282\) 0 0
\(283\) −4.68466 −0.278474 −0.139237 0.990259i \(-0.544465\pi\)
−0.139237 + 0.990259i \(0.544465\pi\)
\(284\) 0 0
\(285\) −11.1231 −0.658876
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 40.1771 2.36336
\(290\) 0 0
\(291\) 2.05398 0.120406
\(292\) 0 0
\(293\) −32.0540 −1.87261 −0.936307 0.351184i \(-0.885779\pi\)
−0.936307 + 0.351184i \(0.885779\pi\)
\(294\) 0 0
\(295\) 4.00000 0.232889
\(296\) 0 0
\(297\) −8.68466 −0.503935
\(298\) 0 0
\(299\) 20.8769 1.20734
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 9.36932 0.538253
\(304\) 0 0
\(305\) −6.87689 −0.393770
\(306\) 0 0
\(307\) −28.6847 −1.63712 −0.818560 0.574421i \(-0.805227\pi\)
−0.818560 + 0.574421i \(0.805227\pi\)
\(308\) 0 0
\(309\) 18.4384 1.04893
\(310\) 0 0
\(311\) −12.8769 −0.730182 −0.365091 0.930972i \(-0.618962\pi\)
−0.365091 + 0.930972i \(0.618962\pi\)
\(312\) 0 0
\(313\) −26.6847 −1.50831 −0.754153 0.656699i \(-0.771952\pi\)
−0.754153 + 0.656699i \(0.771952\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −10.0000 −0.561656 −0.280828 0.959758i \(-0.590609\pi\)
−0.280828 + 0.959758i \(0.590609\pi\)
\(318\) 0 0
\(319\) −0.684658 −0.0383335
\(320\) 0 0
\(321\) 23.6155 1.31809
\(322\) 0 0
\(323\) 53.8617 2.99695
\(324\) 0 0
\(325\) −6.68466 −0.370798
\(326\) 0 0
\(327\) 6.93087 0.383278
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 12.0000 0.659580 0.329790 0.944054i \(-0.393022\pi\)
0.329790 + 0.944054i \(0.393022\pi\)
\(332\) 0 0
\(333\) 4.63068 0.253760
\(334\) 0 0
\(335\) 2.24621 0.122724
\(336\) 0 0
\(337\) 2.00000 0.108947 0.0544735 0.998515i \(-0.482652\pi\)
0.0544735 + 0.998515i \(0.482652\pi\)
\(338\) 0 0
\(339\) −12.8769 −0.699377
\(340\) 0 0
\(341\) −9.75379 −0.528197
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) −4.87689 −0.262563
\(346\) 0 0
\(347\) −15.1231 −0.811851 −0.405925 0.913906i \(-0.633051\pi\)
−0.405925 + 0.913906i \(0.633051\pi\)
\(348\) 0 0
\(349\) 11.7538 0.629166 0.314583 0.949230i \(-0.398135\pi\)
0.314583 + 0.949230i \(0.398135\pi\)
\(350\) 0 0
\(351\) −37.1771 −1.98437
\(352\) 0 0
\(353\) 2.19224 0.116681 0.0583405 0.998297i \(-0.481419\pi\)
0.0583405 + 0.998297i \(0.481419\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −32.0000 −1.68890 −0.844448 0.535638i \(-0.820071\pi\)
−0.844448 + 0.535638i \(0.820071\pi\)
\(360\) 0 0
\(361\) 31.7386 1.67045
\(362\) 0 0
\(363\) 13.3693 0.701707
\(364\) 0 0
\(365\) −4.24621 −0.222257
\(366\) 0 0
\(367\) 14.9309 0.779385 0.389693 0.920945i \(-0.372581\pi\)
0.389693 + 0.920945i \(0.372581\pi\)
\(368\) 0 0
\(369\) −0.630683 −0.0328321
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 15.3693 0.795793 0.397897 0.917430i \(-0.369740\pi\)
0.397897 + 0.917430i \(0.369740\pi\)
\(374\) 0 0
\(375\) 1.56155 0.0806382
\(376\) 0 0
\(377\) −2.93087 −0.150947
\(378\) 0 0
\(379\) −32.4924 −1.66902 −0.834512 0.550990i \(-0.814250\pi\)
−0.834512 + 0.550990i \(0.814250\pi\)
\(380\) 0 0
\(381\) −9.75379 −0.499702
\(382\) 0 0
\(383\) −9.75379 −0.498395 −0.249198 0.968453i \(-0.580167\pi\)
−0.249198 + 0.968453i \(0.580167\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −4.00000 −0.203331
\(388\) 0 0
\(389\) 22.3002 1.13066 0.565332 0.824863i \(-0.308748\pi\)
0.565332 + 0.824863i \(0.308748\pi\)
\(390\) 0 0
\(391\) 23.6155 1.19429
\(392\) 0 0
\(393\) −23.6155 −1.19125
\(394\) 0 0
\(395\) 0.684658 0.0344489
\(396\) 0 0
\(397\) 23.1771 1.16322 0.581612 0.813466i \(-0.302422\pi\)
0.581612 + 0.813466i \(0.302422\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −12.9309 −0.645737 −0.322868 0.946444i \(-0.604647\pi\)
−0.322868 + 0.946444i \(0.604647\pi\)
\(402\) 0 0
\(403\) −41.7538 −2.07990
\(404\) 0 0
\(405\) 7.00000 0.347833
\(406\) 0 0
\(407\) 12.8769 0.638284
\(408\) 0 0
\(409\) 18.4924 0.914391 0.457196 0.889366i \(-0.348854\pi\)
0.457196 + 0.889366i \(0.348854\pi\)
\(410\) 0 0
\(411\) 11.5076 0.567627
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) −12.0000 −0.589057
\(416\) 0 0
\(417\) 33.3693 1.63410
\(418\) 0 0
\(419\) −18.2462 −0.891386 −0.445693 0.895186i \(-0.647043\pi\)
−0.445693 + 0.895186i \(0.647043\pi\)
\(420\) 0 0
\(421\) 3.56155 0.173579 0.0867897 0.996227i \(-0.472339\pi\)
0.0867897 + 0.996227i \(0.472339\pi\)
\(422\) 0 0
\(423\) −1.36932 −0.0665785
\(424\) 0 0
\(425\) −7.56155 −0.366789
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) −16.3002 −0.786980
\(430\) 0 0
\(431\) 22.9309 1.10454 0.552271 0.833665i \(-0.313762\pi\)
0.552271 + 0.833665i \(0.313762\pi\)
\(432\) 0 0
\(433\) −19.7538 −0.949307 −0.474653 0.880173i \(-0.657427\pi\)
−0.474653 + 0.880173i \(0.657427\pi\)
\(434\) 0 0
\(435\) 0.684658 0.0328269
\(436\) 0 0
\(437\) 22.2462 1.06418
\(438\) 0 0
\(439\) 19.1231 0.912696 0.456348 0.889801i \(-0.349157\pi\)
0.456348 + 0.889801i \(0.349157\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 19.6155 0.931962 0.465981 0.884795i \(-0.345702\pi\)
0.465981 + 0.884795i \(0.345702\pi\)
\(444\) 0 0
\(445\) 5.12311 0.242858
\(446\) 0 0
\(447\) 0.384472 0.0181849
\(448\) 0 0
\(449\) −21.3153 −1.00593 −0.502967 0.864306i \(-0.667758\pi\)
−0.502967 + 0.864306i \(0.667758\pi\)
\(450\) 0 0
\(451\) −1.75379 −0.0825827
\(452\) 0 0
\(453\) −30.9309 −1.45326
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 8.63068 0.403726 0.201863 0.979414i \(-0.435300\pi\)
0.201863 + 0.979414i \(0.435300\pi\)
\(458\) 0 0
\(459\) −42.0540 −1.96291
\(460\) 0 0
\(461\) −18.8769 −0.879185 −0.439592 0.898197i \(-0.644877\pi\)
−0.439592 + 0.898197i \(0.644877\pi\)
\(462\) 0 0
\(463\) 6.24621 0.290286 0.145143 0.989411i \(-0.453636\pi\)
0.145143 + 0.989411i \(0.453636\pi\)
\(464\) 0 0
\(465\) 9.75379 0.452321
\(466\) 0 0
\(467\) 25.5616 1.18285 0.591424 0.806361i \(-0.298566\pi\)
0.591424 + 0.806361i \(0.298566\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 6.63068 0.305526
\(472\) 0 0
\(473\) −11.1231 −0.511441
\(474\) 0 0
\(475\) −7.12311 −0.326831
\(476\) 0 0
\(477\) 7.36932 0.337418
\(478\) 0 0
\(479\) 17.3693 0.793624 0.396812 0.917900i \(-0.370116\pi\)
0.396812 + 0.917900i \(0.370116\pi\)
\(480\) 0 0
\(481\) 55.1231 2.51340
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 1.31534 0.0597266
\(486\) 0 0
\(487\) −3.12311 −0.141521 −0.0707607 0.997493i \(-0.522543\pi\)
−0.0707607 + 0.997493i \(0.522543\pi\)
\(488\) 0 0
\(489\) 30.6307 1.38517
\(490\) 0 0
\(491\) 3.31534 0.149619 0.0748096 0.997198i \(-0.476165\pi\)
0.0748096 + 0.997198i \(0.476165\pi\)
\(492\) 0 0
\(493\) −3.31534 −0.149315
\(494\) 0 0
\(495\) −0.876894 −0.0394135
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 0.192236 0.00860566 0.00430283 0.999991i \(-0.498630\pi\)
0.00430283 + 0.999991i \(0.498630\pi\)
\(500\) 0 0
\(501\) −6.54640 −0.292471
\(502\) 0 0
\(503\) −29.1771 −1.30094 −0.650471 0.759531i \(-0.725428\pi\)
−0.650471 + 0.759531i \(0.725428\pi\)
\(504\) 0 0
\(505\) 6.00000 0.266996
\(506\) 0 0
\(507\) −49.4773 −2.19736
\(508\) 0 0
\(509\) −32.7386 −1.45111 −0.725557 0.688162i \(-0.758418\pi\)
−0.725557 + 0.688162i \(0.758418\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) −39.6155 −1.74907
\(514\) 0 0
\(515\) 11.8078 0.520312
\(516\) 0 0
\(517\) −3.80776 −0.167465
\(518\) 0 0
\(519\) −36.1922 −1.58866
\(520\) 0 0
\(521\) 12.2462 0.536516 0.268258 0.963347i \(-0.413552\pi\)
0.268258 + 0.963347i \(0.413552\pi\)
\(522\) 0 0
\(523\) −12.0000 −0.524723 −0.262362 0.964970i \(-0.584501\pi\)
−0.262362 + 0.964970i \(0.584501\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −47.2311 −2.05742
\(528\) 0 0
\(529\) −13.2462 −0.575922
\(530\) 0 0
\(531\) 2.24621 0.0974773
\(532\) 0 0
\(533\) −7.50758 −0.325189
\(534\) 0 0
\(535\) 15.1231 0.653829
\(536\) 0 0
\(537\) −6.24621 −0.269544
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 41.4233 1.78093 0.890463 0.455055i \(-0.150380\pi\)
0.890463 + 0.455055i \(0.150380\pi\)
\(542\) 0 0
\(543\) −8.00000 −0.343313
\(544\) 0 0
\(545\) 4.43845 0.190122
\(546\) 0 0
\(547\) −12.0000 −0.513083 −0.256541 0.966533i \(-0.582583\pi\)
−0.256541 + 0.966533i \(0.582583\pi\)
\(548\) 0 0
\(549\) −3.86174 −0.164815
\(550\) 0 0
\(551\) −3.12311 −0.133049
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) −12.8769 −0.546594
\(556\) 0 0
\(557\) −17.6155 −0.746394 −0.373197 0.927752i \(-0.621738\pi\)
−0.373197 + 0.927752i \(0.621738\pi\)
\(558\) 0 0
\(559\) −47.6155 −2.01392
\(560\) 0 0
\(561\) −18.4384 −0.778472
\(562\) 0 0
\(563\) −7.50758 −0.316407 −0.158203 0.987407i \(-0.550570\pi\)
−0.158203 + 0.987407i \(0.550570\pi\)
\(564\) 0 0
\(565\) −8.24621 −0.346921
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 18.9848 0.795886 0.397943 0.917410i \(-0.369724\pi\)
0.397943 + 0.917410i \(0.369724\pi\)
\(570\) 0 0
\(571\) −20.0000 −0.836974 −0.418487 0.908223i \(-0.637439\pi\)
−0.418487 + 0.908223i \(0.637439\pi\)
\(572\) 0 0
\(573\) −1.06913 −0.0446636
\(574\) 0 0
\(575\) −3.12311 −0.130243
\(576\) 0 0
\(577\) 3.56155 0.148269 0.0741347 0.997248i \(-0.476381\pi\)
0.0741347 + 0.997248i \(0.476381\pi\)
\(578\) 0 0
\(579\) −20.4924 −0.851636
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 20.4924 0.848709
\(584\) 0 0
\(585\) −3.75379 −0.155200
\(586\) 0 0
\(587\) 10.2462 0.422906 0.211453 0.977388i \(-0.432180\pi\)
0.211453 + 0.977388i \(0.432180\pi\)
\(588\) 0 0
\(589\) −44.4924 −1.83328
\(590\) 0 0
\(591\) 20.4924 0.842946
\(592\) 0 0
\(593\) −37.4233 −1.53679 −0.768395 0.639976i \(-0.778944\pi\)
−0.768395 + 0.639976i \(0.778944\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 22.2462 0.910477
\(598\) 0 0
\(599\) 46.9309 1.91754 0.958772 0.284178i \(-0.0917205\pi\)
0.958772 + 0.284178i \(0.0917205\pi\)
\(600\) 0 0
\(601\) 22.0000 0.897399 0.448699 0.893683i \(-0.351887\pi\)
0.448699 + 0.893683i \(0.351887\pi\)
\(602\) 0 0
\(603\) 1.26137 0.0513668
\(604\) 0 0
\(605\) 8.56155 0.348077
\(606\) 0 0
\(607\) −8.68466 −0.352499 −0.176250 0.984345i \(-0.556397\pi\)
−0.176250 + 0.984345i \(0.556397\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −16.3002 −0.659435
\(612\) 0 0
\(613\) 16.7386 0.676067 0.338034 0.941134i \(-0.390238\pi\)
0.338034 + 0.941134i \(0.390238\pi\)
\(614\) 0 0
\(615\) 1.75379 0.0707196
\(616\) 0 0
\(617\) −15.7538 −0.634224 −0.317112 0.948388i \(-0.602713\pi\)
−0.317112 + 0.948388i \(0.602713\pi\)
\(618\) 0 0
\(619\) 10.6307 0.427283 0.213642 0.976912i \(-0.431468\pi\)
0.213642 + 0.976912i \(0.431468\pi\)
\(620\) 0 0
\(621\) −17.3693 −0.697007
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) −17.3693 −0.693664
\(628\) 0 0
\(629\) 62.3542 2.48622
\(630\) 0 0
\(631\) 27.4233 1.09170 0.545852 0.837882i \(-0.316206\pi\)
0.545852 + 0.837882i \(0.316206\pi\)
\(632\) 0 0
\(633\) −27.4233 −1.08998
\(634\) 0 0
\(635\) −6.24621 −0.247873
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −22.9848 −0.907847 −0.453923 0.891041i \(-0.649976\pi\)
−0.453923 + 0.891041i \(0.649976\pi\)
\(642\) 0 0
\(643\) −30.0540 −1.18521 −0.592607 0.805492i \(-0.701901\pi\)
−0.592607 + 0.805492i \(0.701901\pi\)
\(644\) 0 0
\(645\) 11.1231 0.437972
\(646\) 0 0
\(647\) 8.00000 0.314512 0.157256 0.987558i \(-0.449735\pi\)
0.157256 + 0.987558i \(0.449735\pi\)
\(648\) 0 0
\(649\) 6.24621 0.245185
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −38.4924 −1.50632 −0.753162 0.657835i \(-0.771473\pi\)
−0.753162 + 0.657835i \(0.771473\pi\)
\(654\) 0 0
\(655\) −15.1231 −0.590909
\(656\) 0 0
\(657\) −2.38447 −0.0930271
\(658\) 0 0
\(659\) 0.192236 0.00748845 0.00374422 0.999993i \(-0.498808\pi\)
0.00374422 + 0.999993i \(0.498808\pi\)
\(660\) 0 0
\(661\) 17.6155 0.685165 0.342582 0.939488i \(-0.388698\pi\)
0.342582 + 0.939488i \(0.388698\pi\)
\(662\) 0 0
\(663\) −78.9309 −3.06542
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −1.36932 −0.0530202
\(668\) 0 0
\(669\) −38.5464 −1.49029
\(670\) 0 0
\(671\) −10.7386 −0.414560
\(672\) 0 0
\(673\) 41.6155 1.60416 0.802080 0.597216i \(-0.203727\pi\)
0.802080 + 0.597216i \(0.203727\pi\)
\(674\) 0 0
\(675\) 5.56155 0.214064
\(676\) 0 0
\(677\) 1.31534 0.0505527 0.0252763 0.999681i \(-0.491953\pi\)
0.0252763 + 0.999681i \(0.491953\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 17.6695 0.677097
\(682\) 0 0
\(683\) −44.9848 −1.72130 −0.860649 0.509199i \(-0.829942\pi\)
−0.860649 + 0.509199i \(0.829942\pi\)
\(684\) 0 0
\(685\) 7.36932 0.281567
\(686\) 0 0
\(687\) −17.7538 −0.677349
\(688\) 0 0
\(689\) 87.7235 3.34200
\(690\) 0 0
\(691\) 16.4924 0.627401 0.313701 0.949522i \(-0.398431\pi\)
0.313701 + 0.949522i \(0.398431\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 21.3693 0.810584
\(696\) 0 0
\(697\) −8.49242 −0.321673
\(698\) 0 0
\(699\) 16.9848 0.642426
\(700\) 0 0
\(701\) −9.31534 −0.351836 −0.175918 0.984405i \(-0.556289\pi\)
−0.175918 + 0.984405i \(0.556289\pi\)
\(702\) 0 0
\(703\) 58.7386 2.21537
\(704\) 0 0
\(705\) 3.80776 0.143409
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −4.05398 −0.152250 −0.0761251 0.997098i \(-0.524255\pi\)
−0.0761251 + 0.997098i \(0.524255\pi\)
\(710\) 0 0
\(711\) 0.384472 0.0144188
\(712\) 0 0
\(713\) −19.5076 −0.730565
\(714\) 0 0
\(715\) −10.4384 −0.390376
\(716\) 0 0
\(717\) −28.1922 −1.05286
\(718\) 0 0
\(719\) −23.6155 −0.880711 −0.440355 0.897824i \(-0.645148\pi\)
−0.440355 + 0.897824i \(0.645148\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 3.12311 0.116150
\(724\) 0 0
\(725\) 0.438447 0.0162835
\(726\) 0 0
\(727\) 48.9848 1.81675 0.908374 0.418159i \(-0.137325\pi\)
0.908374 + 0.418159i \(0.137325\pi\)
\(728\) 0 0
\(729\) 29.9848 1.11055
\(730\) 0 0
\(731\) −53.8617 −1.99215
\(732\) 0 0
\(733\) −16.4384 −0.607168 −0.303584 0.952805i \(-0.598183\pi\)
−0.303584 + 0.952805i \(0.598183\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 3.50758 0.129203
\(738\) 0 0
\(739\) 6.43845 0.236842 0.118421 0.992963i \(-0.462217\pi\)
0.118421 + 0.992963i \(0.462217\pi\)
\(740\) 0 0
\(741\) −74.3542 −2.73147
\(742\) 0 0
\(743\) 24.0000 0.880475 0.440237 0.897881i \(-0.354894\pi\)
0.440237 + 0.897881i \(0.354894\pi\)
\(744\) 0 0
\(745\) 0.246211 0.00902048
\(746\) 0 0
\(747\) −6.73863 −0.246554
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 31.3153 1.14271 0.571357 0.820702i \(-0.306417\pi\)
0.571357 + 0.820702i \(0.306417\pi\)
\(752\) 0 0
\(753\) −20.8769 −0.760796
\(754\) 0 0
\(755\) −19.8078 −0.720878
\(756\) 0 0
\(757\) 15.3693 0.558607 0.279304 0.960203i \(-0.409896\pi\)
0.279304 + 0.960203i \(0.409896\pi\)
\(758\) 0 0
\(759\) −7.61553 −0.276426
\(760\) 0 0
\(761\) −26.0000 −0.942499 −0.471250 0.882000i \(-0.656197\pi\)
−0.471250 + 0.882000i \(0.656197\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) −4.24621 −0.153522
\(766\) 0 0
\(767\) 26.7386 0.965476
\(768\) 0 0
\(769\) −42.9848 −1.55007 −0.775037 0.631916i \(-0.782269\pi\)
−0.775037 + 0.631916i \(0.782269\pi\)
\(770\) 0 0
\(771\) −28.8769 −1.03998
\(772\) 0 0
\(773\) 29.8078 1.07211 0.536055 0.844183i \(-0.319914\pi\)
0.536055 + 0.844183i \(0.319914\pi\)
\(774\) 0 0
\(775\) 6.24621 0.224371
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −8.00000 −0.286630
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 2.43845 0.0871430
\(784\) 0 0
\(785\) 4.24621 0.151554
\(786\) 0 0
\(787\) 33.1771 1.18264 0.591318 0.806439i \(-0.298608\pi\)
0.591318 + 0.806439i \(0.298608\pi\)
\(788\) 0 0
\(789\) 14.6307 0.520866
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) −45.9697 −1.63243
\(794\) 0 0
\(795\) −20.4924 −0.726791
\(796\) 0 0
\(797\) −17.8078 −0.630783 −0.315392 0.948962i \(-0.602136\pi\)
−0.315392 + 0.948962i \(0.602136\pi\)
\(798\) 0 0
\(799\) −18.4384 −0.652305
\(800\) 0 0
\(801\) 2.87689 0.101650
\(802\) 0 0
\(803\) −6.63068 −0.233992
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 6.63068 0.233411
\(808\) 0 0
\(809\) 4.05398 0.142530 0.0712651 0.997457i \(-0.477296\pi\)
0.0712651 + 0.997457i \(0.477296\pi\)
\(810\) 0 0
\(811\) −27.6155 −0.969712 −0.484856 0.874594i \(-0.661128\pi\)
−0.484856 + 0.874594i \(0.661128\pi\)
\(812\) 0 0
\(813\) 9.75379 0.342080
\(814\) 0 0
\(815\) 19.6155 0.687102
\(816\) 0 0
\(817\) −50.7386 −1.77512
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 54.3002 1.89509 0.947545 0.319623i \(-0.103556\pi\)
0.947545 + 0.319623i \(0.103556\pi\)
\(822\) 0 0
\(823\) 17.7538 0.618858 0.309429 0.950923i \(-0.399862\pi\)
0.309429 + 0.950923i \(0.399862\pi\)
\(824\) 0 0
\(825\) 2.43845 0.0848958
\(826\) 0 0
\(827\) −24.8769 −0.865054 −0.432527 0.901621i \(-0.642378\pi\)
−0.432527 + 0.901621i \(0.642378\pi\)
\(828\) 0 0
\(829\) −5.61553 −0.195035 −0.0975177 0.995234i \(-0.531090\pi\)
−0.0975177 + 0.995234i \(0.531090\pi\)
\(830\) 0 0
\(831\) 12.8769 0.446695
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) −4.19224 −0.145078
\(836\) 0 0
\(837\) 34.7386 1.20074
\(838\) 0 0
\(839\) 19.1231 0.660203 0.330101 0.943945i \(-0.392917\pi\)
0.330101 + 0.943945i \(0.392917\pi\)
\(840\) 0 0
\(841\) −28.8078 −0.993371
\(842\) 0 0
\(843\) 30.5464 1.05207
\(844\) 0 0
\(845\) −31.6847 −1.08999
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) 7.31534 0.251062
\(850\) 0 0
\(851\) 25.7538 0.882829
\(852\) 0 0
\(853\) −15.7538 −0.539399 −0.269700 0.962944i \(-0.586924\pi\)
−0.269700 + 0.962944i \(0.586924\pi\)
\(854\) 0 0
\(855\) −4.00000 −0.136797
\(856\) 0 0
\(857\) −52.7386 −1.80152 −0.900759 0.434320i \(-0.856989\pi\)
−0.900759 + 0.434320i \(0.856989\pi\)
\(858\) 0 0
\(859\) 36.9848 1.26191 0.630953 0.775821i \(-0.282664\pi\)
0.630953 + 0.775821i \(0.282664\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −28.4924 −0.969893 −0.484947 0.874544i \(-0.661161\pi\)
−0.484947 + 0.874544i \(0.661161\pi\)
\(864\) 0 0
\(865\) −23.1771 −0.788044
\(866\) 0 0
\(867\) −62.7386 −2.13072
\(868\) 0 0
\(869\) 1.06913 0.0362678
\(870\) 0 0
\(871\) 15.0152 0.508769
\(872\) 0 0
\(873\) 0.738634 0.0249990
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −13.5076 −0.456118 −0.228059 0.973647i \(-0.573238\pi\)
−0.228059 + 0.973647i \(0.573238\pi\)
\(878\) 0 0
\(879\) 50.0540 1.68828
\(880\) 0 0
\(881\) −54.1080 −1.82294 −0.911472 0.411363i \(-0.865053\pi\)
−0.911472 + 0.411363i \(0.865053\pi\)
\(882\) 0 0
\(883\) −21.7538 −0.732073 −0.366037 0.930600i \(-0.619286\pi\)
−0.366037 + 0.930600i \(0.619286\pi\)
\(884\) 0 0
\(885\) −6.24621 −0.209964
\(886\) 0 0
\(887\) −36.4924 −1.22530 −0.612648 0.790356i \(-0.709896\pi\)
−0.612648 + 0.790356i \(0.709896\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 10.9309 0.366198
\(892\) 0 0
\(893\) −17.3693 −0.581242
\(894\) 0 0
\(895\) −4.00000 −0.133705
\(896\) 0 0
\(897\) −32.6004 −1.08849
\(898\) 0 0
\(899\) 2.73863 0.0913385
\(900\) 0 0
\(901\) 99.2311 3.30587
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −5.12311 −0.170298
\(906\) 0 0
\(907\) 48.1080 1.59740 0.798699 0.601731i \(-0.205522\pi\)
0.798699 + 0.601731i \(0.205522\pi\)
\(908\) 0 0
\(909\) 3.36932 0.111753
\(910\) 0 0
\(911\) −8.00000 −0.265052 −0.132526 0.991180i \(-0.542309\pi\)
−0.132526 + 0.991180i \(0.542309\pi\)
\(912\) 0 0
\(913\) −18.7386 −0.620158
\(914\) 0 0
\(915\) 10.7386 0.355008
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) −2.43845 −0.0804370 −0.0402185 0.999191i \(-0.512805\pi\)
−0.0402185 + 0.999191i \(0.512805\pi\)
\(920\) 0 0
\(921\) 44.7926 1.47597
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) −8.24621 −0.271134
\(926\) 0 0
\(927\) 6.63068 0.217780
\(928\) 0 0
\(929\) 2.87689 0.0943878 0.0471939 0.998886i \(-0.484972\pi\)
0.0471939 + 0.998886i \(0.484972\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 20.1080 0.658305
\(934\) 0 0
\(935\) −11.8078 −0.386155
\(936\) 0 0
\(937\) −37.8078 −1.23513 −0.617563 0.786521i \(-0.711880\pi\)
−0.617563 + 0.786521i \(0.711880\pi\)
\(938\) 0 0
\(939\) 41.6695 1.35983
\(940\) 0 0
\(941\) −28.6307 −0.933334 −0.466667 0.884433i \(-0.654545\pi\)
−0.466667 + 0.884433i \(0.654545\pi\)
\(942\) 0 0
\(943\) −3.50758 −0.114222
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 35.2311 1.14486 0.572428 0.819955i \(-0.306002\pi\)
0.572428 + 0.819955i \(0.306002\pi\)
\(948\) 0 0
\(949\) −28.3845 −0.921399
\(950\) 0 0
\(951\) 15.6155 0.506368
\(952\) 0 0
\(953\) −51.8617 −1.67997 −0.839983 0.542612i \(-0.817435\pi\)
−0.839983 + 0.542612i \(0.817435\pi\)
\(954\) 0 0
\(955\) −0.684658 −0.0221550
\(956\) 0 0
\(957\) 1.06913 0.0345601
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 8.01515 0.258553
\(962\) 0 0
\(963\) 8.49242 0.273664
\(964\) 0 0
\(965\) −13.1231 −0.422448
\(966\) 0 0
\(967\) −44.1080 −1.41842 −0.709208 0.704999i \(-0.750947\pi\)
−0.709208 + 0.704999i \(0.750947\pi\)
\(968\) 0 0
\(969\) −84.1080 −2.70194
\(970\) 0 0
\(971\) −22.7386 −0.729717 −0.364859 0.931063i \(-0.618883\pi\)
−0.364859 + 0.931063i \(0.618883\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 10.4384 0.334298
\(976\) 0 0
\(977\) 10.9848 0.351436 0.175718 0.984441i \(-0.443775\pi\)
0.175718 + 0.984441i \(0.443775\pi\)
\(978\) 0 0
\(979\) 8.00000 0.255681
\(980\) 0 0
\(981\) 2.49242 0.0795769
\(982\) 0 0
\(983\) −29.1771 −0.930604 −0.465302 0.885152i \(-0.654054\pi\)
−0.465302 + 0.885152i \(0.654054\pi\)
\(984\) 0 0
\(985\) 13.1231 0.418137
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −22.2462 −0.707388
\(990\) 0 0
\(991\) 20.4924 0.650963 0.325482 0.945548i \(-0.394474\pi\)
0.325482 + 0.945548i \(0.394474\pi\)
\(992\) 0 0
\(993\) −18.7386 −0.594653
\(994\) 0 0
\(995\) 14.2462 0.451635
\(996\) 0 0
\(997\) 56.9309 1.80302 0.901509 0.432760i \(-0.142460\pi\)
0.901509 + 0.432760i \(0.142460\pi\)
\(998\) 0 0
\(999\) −45.8617 −1.45100
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 3920.2.a.bu.1.1 2
4.3 odd 2 1960.2.a.r.1.2 2
7.6 odd 2 560.2.a.g.1.2 2
20.19 odd 2 9800.2.a.by.1.1 2
21.20 even 2 5040.2.a.bq.1.2 2
28.3 even 6 1960.2.q.s.961.2 4
28.11 odd 6 1960.2.q.u.961.1 4
28.19 even 6 1960.2.q.s.361.2 4
28.23 odd 6 1960.2.q.u.361.1 4
28.27 even 2 280.2.a.d.1.1 2
35.13 even 4 2800.2.g.u.449.3 4
35.27 even 4 2800.2.g.u.449.2 4
35.34 odd 2 2800.2.a.bn.1.1 2
56.13 odd 2 2240.2.a.bi.1.1 2
56.27 even 2 2240.2.a.be.1.2 2
84.83 odd 2 2520.2.a.w.1.1 2
140.27 odd 4 1400.2.g.k.449.3 4
140.83 odd 4 1400.2.g.k.449.2 4
140.139 even 2 1400.2.a.p.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
280.2.a.d.1.1 2 28.27 even 2
560.2.a.g.1.2 2 7.6 odd 2
1400.2.a.p.1.2 2 140.139 even 2
1400.2.g.k.449.2 4 140.83 odd 4
1400.2.g.k.449.3 4 140.27 odd 4
1960.2.a.r.1.2 2 4.3 odd 2
1960.2.q.s.361.2 4 28.19 even 6
1960.2.q.s.961.2 4 28.3 even 6
1960.2.q.u.361.1 4 28.23 odd 6
1960.2.q.u.961.1 4 28.11 odd 6
2240.2.a.be.1.2 2 56.27 even 2
2240.2.a.bi.1.1 2 56.13 odd 2
2520.2.a.w.1.1 2 84.83 odd 2
2800.2.a.bn.1.1 2 35.34 odd 2
2800.2.g.u.449.2 4 35.27 even 4
2800.2.g.u.449.3 4 35.13 even 4
3920.2.a.bu.1.1 2 1.1 even 1 trivial
5040.2.a.bq.1.2 2 21.20 even 2
9800.2.a.by.1.1 2 20.19 odd 2