Properties

Label 2850.2.a.bg.1.2
Level $2850$
Weight $2$
Character 2850.1
Self dual yes
Analytic conductor $22.757$
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] = [2850,2,Mod(1,2850)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2850, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("2850.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2850 = 2 \cdot 3 \cdot 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2850.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: \(22.7573645761\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{10}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 10 \) 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 1.2
Root \(3.16228\) of defining polynomial
Character \(\chi\) \(=\) 2850.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} +3.16228 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} +3.16228 q^{7} +1.00000 q^{8} +1.00000 q^{9} +4.16228 q^{11} -1.00000 q^{12} +5.16228 q^{13} +3.16228 q^{14} +1.00000 q^{16} +0.837722 q^{17} +1.00000 q^{18} -1.00000 q^{19} -3.16228 q^{21} +4.16228 q^{22} -5.32456 q^{23} -1.00000 q^{24} +5.16228 q^{26} -1.00000 q^{27} +3.16228 q^{28} -3.83772 q^{29} +5.32456 q^{31} +1.00000 q^{32} -4.16228 q^{33} +0.837722 q^{34} +1.00000 q^{36} +10.0000 q^{37} -1.00000 q^{38} -5.16228 q^{39} -6.32456 q^{41} -3.16228 q^{42} -9.16228 q^{43} +4.16228 q^{44} -5.32456 q^{46} +6.00000 q^{47} -1.00000 q^{48} +3.00000 q^{49} -0.837722 q^{51} +5.16228 q^{52} +1.83772 q^{53} -1.00000 q^{54} +3.16228 q^{56} +1.00000 q^{57} -3.83772 q^{58} -7.48683 q^{59} -4.16228 q^{61} +5.32456 q^{62} +3.16228 q^{63} +1.00000 q^{64} -4.16228 q^{66} -6.48683 q^{67} +0.837722 q^{68} +5.32456 q^{69} +1.16228 q^{71} +1.00000 q^{72} -1.00000 q^{73} +10.0000 q^{74} -1.00000 q^{76} +13.1623 q^{77} -5.16228 q^{78} -5.32456 q^{79} +1.00000 q^{81} -6.32456 q^{82} +6.48683 q^{83} -3.16228 q^{84} -9.16228 q^{86} +3.83772 q^{87} +4.16228 q^{88} -7.32456 q^{89} +16.3246 q^{91} -5.32456 q^{92} -5.32456 q^{93} +6.00000 q^{94} -1.00000 q^{96} +0.513167 q^{97} +3.00000 q^{98} +4.16228 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} - 2 q^{3} + 2 q^{4} - 2 q^{6} + 2 q^{8} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{2} - 2 q^{3} + 2 q^{4} - 2 q^{6} + 2 q^{8} + 2 q^{9} + 2 q^{11} - 2 q^{12} + 4 q^{13} + 2 q^{16} + 8 q^{17} + 2 q^{18} - 2 q^{19} + 2 q^{22} + 2 q^{23} - 2 q^{24} + 4 q^{26} - 2 q^{27} - 14 q^{29} - 2 q^{31} + 2 q^{32} - 2 q^{33} + 8 q^{34} + 2 q^{36} + 20 q^{37} - 2 q^{38} - 4 q^{39} - 12 q^{43} + 2 q^{44} + 2 q^{46} + 12 q^{47} - 2 q^{48} + 6 q^{49} - 8 q^{51} + 4 q^{52} + 10 q^{53} - 2 q^{54} + 2 q^{57} - 14 q^{58} + 4 q^{59} - 2 q^{61} - 2 q^{62} + 2 q^{64} - 2 q^{66} + 6 q^{67} + 8 q^{68} - 2 q^{69} - 4 q^{71} + 2 q^{72} - 2 q^{73} + 20 q^{74} - 2 q^{76} + 20 q^{77} - 4 q^{78} + 2 q^{79} + 2 q^{81} - 6 q^{83} - 12 q^{86} + 14 q^{87} + 2 q^{88} - 2 q^{89} + 20 q^{91} + 2 q^{92} + 2 q^{93} + 12 q^{94} - 2 q^{96} + 20 q^{97} + 6 q^{98} + 2 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) −1.00000 −0.577350
\(4\) 1.00000 0.500000
\(5\) 0 0
\(6\) −1.00000 −0.408248
\(7\) 3.16228 1.19523 0.597614 0.801784i \(-0.296115\pi\)
0.597614 + 0.801784i \(0.296115\pi\)
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 4.16228 1.25497 0.627487 0.778627i \(-0.284084\pi\)
0.627487 + 0.778627i \(0.284084\pi\)
\(12\) −1.00000 −0.288675
\(13\) 5.16228 1.43176 0.715879 0.698224i \(-0.246026\pi\)
0.715879 + 0.698224i \(0.246026\pi\)
\(14\) 3.16228 0.845154
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 0.837722 0.203178 0.101589 0.994826i \(-0.467607\pi\)
0.101589 + 0.994826i \(0.467607\pi\)
\(18\) 1.00000 0.235702
\(19\) −1.00000 −0.229416
\(20\) 0 0
\(21\) −3.16228 −0.690066
\(22\) 4.16228 0.887401
\(23\) −5.32456 −1.11025 −0.555123 0.831768i \(-0.687329\pi\)
−0.555123 + 0.831768i \(0.687329\pi\)
\(24\) −1.00000 −0.204124
\(25\) 0 0
\(26\) 5.16228 1.01241
\(27\) −1.00000 −0.192450
\(28\) 3.16228 0.597614
\(29\) −3.83772 −0.712647 −0.356324 0.934363i \(-0.615970\pi\)
−0.356324 + 0.934363i \(0.615970\pi\)
\(30\) 0 0
\(31\) 5.32456 0.956318 0.478159 0.878273i \(-0.341304\pi\)
0.478159 + 0.878273i \(0.341304\pi\)
\(32\) 1.00000 0.176777
\(33\) −4.16228 −0.724560
\(34\) 0.837722 0.143668
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) 10.0000 1.64399 0.821995 0.569495i \(-0.192861\pi\)
0.821995 + 0.569495i \(0.192861\pi\)
\(38\) −1.00000 −0.162221
\(39\) −5.16228 −0.826626
\(40\) 0 0
\(41\) −6.32456 −0.987730 −0.493865 0.869539i \(-0.664416\pi\)
−0.493865 + 0.869539i \(0.664416\pi\)
\(42\) −3.16228 −0.487950
\(43\) −9.16228 −1.39723 −0.698617 0.715496i \(-0.746201\pi\)
−0.698617 + 0.715496i \(0.746201\pi\)
\(44\) 4.16228 0.627487
\(45\) 0 0
\(46\) −5.32456 −0.785063
\(47\) 6.00000 0.875190 0.437595 0.899172i \(-0.355830\pi\)
0.437595 + 0.899172i \(0.355830\pi\)
\(48\) −1.00000 −0.144338
\(49\) 3.00000 0.428571
\(50\) 0 0
\(51\) −0.837722 −0.117305
\(52\) 5.16228 0.715879
\(53\) 1.83772 0.252431 0.126215 0.992003i \(-0.459717\pi\)
0.126215 + 0.992003i \(0.459717\pi\)
\(54\) −1.00000 −0.136083
\(55\) 0 0
\(56\) 3.16228 0.422577
\(57\) 1.00000 0.132453
\(58\) −3.83772 −0.503918
\(59\) −7.48683 −0.974703 −0.487351 0.873206i \(-0.662037\pi\)
−0.487351 + 0.873206i \(0.662037\pi\)
\(60\) 0 0
\(61\) −4.16228 −0.532925 −0.266463 0.963845i \(-0.585855\pi\)
−0.266463 + 0.963845i \(0.585855\pi\)
\(62\) 5.32456 0.676219
\(63\) 3.16228 0.398410
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) −4.16228 −0.512341
\(67\) −6.48683 −0.792493 −0.396246 0.918144i \(-0.629687\pi\)
−0.396246 + 0.918144i \(0.629687\pi\)
\(68\) 0.837722 0.101589
\(69\) 5.32456 0.641001
\(70\) 0 0
\(71\) 1.16228 0.137937 0.0689685 0.997619i \(-0.478029\pi\)
0.0689685 + 0.997619i \(0.478029\pi\)
\(72\) 1.00000 0.117851
\(73\) −1.00000 −0.117041 −0.0585206 0.998286i \(-0.518638\pi\)
−0.0585206 + 0.998286i \(0.518638\pi\)
\(74\) 10.0000 1.16248
\(75\) 0 0
\(76\) −1.00000 −0.114708
\(77\) 13.1623 1.49998
\(78\) −5.16228 −0.584513
\(79\) −5.32456 −0.599059 −0.299530 0.954087i \(-0.596830\pi\)
−0.299530 + 0.954087i \(0.596830\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) −6.32456 −0.698430
\(83\) 6.48683 0.712022 0.356011 0.934482i \(-0.384136\pi\)
0.356011 + 0.934482i \(0.384136\pi\)
\(84\) −3.16228 −0.345033
\(85\) 0 0
\(86\) −9.16228 −0.987994
\(87\) 3.83772 0.411447
\(88\) 4.16228 0.443700
\(89\) −7.32456 −0.776401 −0.388201 0.921575i \(-0.626903\pi\)
−0.388201 + 0.921575i \(0.626903\pi\)
\(90\) 0 0
\(91\) 16.3246 1.71128
\(92\) −5.32456 −0.555123
\(93\) −5.32456 −0.552131
\(94\) 6.00000 0.618853
\(95\) 0 0
\(96\) −1.00000 −0.102062
\(97\) 0.513167 0.0521042 0.0260521 0.999661i \(-0.491706\pi\)
0.0260521 + 0.999661i \(0.491706\pi\)
\(98\) 3.00000 0.303046
\(99\) 4.16228 0.418325
\(100\) 0 0
\(101\) −13.8114 −1.37428 −0.687142 0.726523i \(-0.741135\pi\)
−0.687142 + 0.726523i \(0.741135\pi\)
\(102\) −0.837722 −0.0829469
\(103\) 15.6491 1.54195 0.770976 0.636864i \(-0.219769\pi\)
0.770976 + 0.636864i \(0.219769\pi\)
\(104\) 5.16228 0.506203
\(105\) 0 0
\(106\) 1.83772 0.178495
\(107\) 12.8377 1.24107 0.620535 0.784179i \(-0.286916\pi\)
0.620535 + 0.784179i \(0.286916\pi\)
\(108\) −1.00000 −0.0962250
\(109\) 7.67544 0.735174 0.367587 0.929989i \(-0.380184\pi\)
0.367587 + 0.929989i \(0.380184\pi\)
\(110\) 0 0
\(111\) −10.0000 −0.949158
\(112\) 3.16228 0.298807
\(113\) 9.64911 0.907712 0.453856 0.891075i \(-0.350048\pi\)
0.453856 + 0.891075i \(0.350048\pi\)
\(114\) 1.00000 0.0936586
\(115\) 0 0
\(116\) −3.83772 −0.356324
\(117\) 5.16228 0.477253
\(118\) −7.48683 −0.689219
\(119\) 2.64911 0.242844
\(120\) 0 0
\(121\) 6.32456 0.574960
\(122\) −4.16228 −0.376835
\(123\) 6.32456 0.570266
\(124\) 5.32456 0.478159
\(125\) 0 0
\(126\) 3.16228 0.281718
\(127\) 15.6491 1.38863 0.694317 0.719669i \(-0.255707\pi\)
0.694317 + 0.719669i \(0.255707\pi\)
\(128\) 1.00000 0.0883883
\(129\) 9.16228 0.806694
\(130\) 0 0
\(131\) −12.4868 −1.09098 −0.545490 0.838117i \(-0.683656\pi\)
−0.545490 + 0.838117i \(0.683656\pi\)
\(132\) −4.16228 −0.362280
\(133\) −3.16228 −0.274204
\(134\) −6.48683 −0.560377
\(135\) 0 0
\(136\) 0.837722 0.0718341
\(137\) 2.32456 0.198600 0.0993001 0.995058i \(-0.468340\pi\)
0.0993001 + 0.995058i \(0.468340\pi\)
\(138\) 5.32456 0.453256
\(139\) 21.4868 1.82249 0.911245 0.411865i \(-0.135123\pi\)
0.911245 + 0.411865i \(0.135123\pi\)
\(140\) 0 0
\(141\) −6.00000 −0.505291
\(142\) 1.16228 0.0975362
\(143\) 21.4868 1.79682
\(144\) 1.00000 0.0833333
\(145\) 0 0
\(146\) −1.00000 −0.0827606
\(147\) −3.00000 −0.247436
\(148\) 10.0000 0.821995
\(149\) 16.6491 1.36395 0.681974 0.731376i \(-0.261122\pi\)
0.681974 + 0.731376i \(0.261122\pi\)
\(150\) 0 0
\(151\) −10.0000 −0.813788 −0.406894 0.913475i \(-0.633388\pi\)
−0.406894 + 0.913475i \(0.633388\pi\)
\(152\) −1.00000 −0.0811107
\(153\) 0.837722 0.0677258
\(154\) 13.1623 1.06065
\(155\) 0 0
\(156\) −5.16228 −0.413313
\(157\) 18.3246 1.46246 0.731229 0.682132i \(-0.238947\pi\)
0.731229 + 0.682132i \(0.238947\pi\)
\(158\) −5.32456 −0.423599
\(159\) −1.83772 −0.145741
\(160\) 0 0
\(161\) −16.8377 −1.32700
\(162\) 1.00000 0.0785674
\(163\) −22.6491 −1.77402 −0.887008 0.461755i \(-0.847220\pi\)
−0.887008 + 0.461755i \(0.847220\pi\)
\(164\) −6.32456 −0.493865
\(165\) 0 0
\(166\) 6.48683 0.503476
\(167\) 2.64911 0.204994 0.102497 0.994733i \(-0.467317\pi\)
0.102497 + 0.994733i \(0.467317\pi\)
\(168\) −3.16228 −0.243975
\(169\) 13.6491 1.04993
\(170\) 0 0
\(171\) −1.00000 −0.0764719
\(172\) −9.16228 −0.698617
\(173\) −2.16228 −0.164395 −0.0821975 0.996616i \(-0.526194\pi\)
−0.0821975 + 0.996616i \(0.526194\pi\)
\(174\) 3.83772 0.290937
\(175\) 0 0
\(176\) 4.16228 0.313743
\(177\) 7.48683 0.562745
\(178\) −7.32456 −0.548999
\(179\) −20.6491 −1.54339 −0.771693 0.635995i \(-0.780590\pi\)
−0.771693 + 0.635995i \(0.780590\pi\)
\(180\) 0 0
\(181\) 9.16228 0.681027 0.340513 0.940240i \(-0.389399\pi\)
0.340513 + 0.940240i \(0.389399\pi\)
\(182\) 16.3246 1.21006
\(183\) 4.16228 0.307684
\(184\) −5.32456 −0.392531
\(185\) 0 0
\(186\) −5.32456 −0.390415
\(187\) 3.48683 0.254982
\(188\) 6.00000 0.437595
\(189\) −3.16228 −0.230022
\(190\) 0 0
\(191\) 21.6491 1.56647 0.783237 0.621723i \(-0.213567\pi\)
0.783237 + 0.621723i \(0.213567\pi\)
\(192\) −1.00000 −0.0721688
\(193\) −9.16228 −0.659515 −0.329758 0.944066i \(-0.606967\pi\)
−0.329758 + 0.944066i \(0.606967\pi\)
\(194\) 0.513167 0.0368432
\(195\) 0 0
\(196\) 3.00000 0.214286
\(197\) −24.1359 −1.71961 −0.859807 0.510618i \(-0.829416\pi\)
−0.859807 + 0.510618i \(0.829416\pi\)
\(198\) 4.16228 0.295800
\(199\) −18.6491 −1.32200 −0.661000 0.750386i \(-0.729868\pi\)
−0.661000 + 0.750386i \(0.729868\pi\)
\(200\) 0 0
\(201\) 6.48683 0.457546
\(202\) −13.8114 −0.971766
\(203\) −12.1359 −0.851776
\(204\) −0.837722 −0.0586523
\(205\) 0 0
\(206\) 15.6491 1.09033
\(207\) −5.32456 −0.370082
\(208\) 5.16228 0.357940
\(209\) −4.16228 −0.287911
\(210\) 0 0
\(211\) −18.8114 −1.29503 −0.647515 0.762053i \(-0.724192\pi\)
−0.647515 + 0.762053i \(0.724192\pi\)
\(212\) 1.83772 0.126215
\(213\) −1.16228 −0.0796380
\(214\) 12.8377 0.877569
\(215\) 0 0
\(216\) −1.00000 −0.0680414
\(217\) 16.8377 1.14302
\(218\) 7.67544 0.519847
\(219\) 1.00000 0.0675737
\(220\) 0 0
\(221\) 4.32456 0.290901
\(222\) −10.0000 −0.671156
\(223\) −25.6491 −1.71759 −0.858796 0.512318i \(-0.828787\pi\)
−0.858796 + 0.512318i \(0.828787\pi\)
\(224\) 3.16228 0.211289
\(225\) 0 0
\(226\) 9.64911 0.641849
\(227\) −2.00000 −0.132745 −0.0663723 0.997795i \(-0.521143\pi\)
−0.0663723 + 0.997795i \(0.521143\pi\)
\(228\) 1.00000 0.0662266
\(229\) 8.16228 0.539378 0.269689 0.962947i \(-0.413079\pi\)
0.269689 + 0.962947i \(0.413079\pi\)
\(230\) 0 0
\(231\) −13.1623 −0.866014
\(232\) −3.83772 −0.251959
\(233\) −11.8114 −0.773790 −0.386895 0.922124i \(-0.626452\pi\)
−0.386895 + 0.922124i \(0.626452\pi\)
\(234\) 5.16228 0.337469
\(235\) 0 0
\(236\) −7.48683 −0.487351
\(237\) 5.32456 0.345867
\(238\) 2.64911 0.171716
\(239\) −24.3246 −1.57342 −0.786712 0.617320i \(-0.788218\pi\)
−0.786712 + 0.617320i \(0.788218\pi\)
\(240\) 0 0
\(241\) 10.5132 0.677213 0.338606 0.940928i \(-0.390045\pi\)
0.338606 + 0.940928i \(0.390045\pi\)
\(242\) 6.32456 0.406558
\(243\) −1.00000 −0.0641500
\(244\) −4.16228 −0.266463
\(245\) 0 0
\(246\) 6.32456 0.403239
\(247\) −5.16228 −0.328468
\(248\) 5.32456 0.338110
\(249\) −6.48683 −0.411086
\(250\) 0 0
\(251\) −10.6491 −0.672166 −0.336083 0.941832i \(-0.609102\pi\)
−0.336083 + 0.941832i \(0.609102\pi\)
\(252\) 3.16228 0.199205
\(253\) −22.1623 −1.39333
\(254\) 15.6491 0.981913
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −19.3246 −1.20543 −0.602716 0.797956i \(-0.705915\pi\)
−0.602716 + 0.797956i \(0.705915\pi\)
\(258\) 9.16228 0.570418
\(259\) 31.6228 1.96494
\(260\) 0 0
\(261\) −3.83772 −0.237549
\(262\) −12.4868 −0.771439
\(263\) 31.9737 1.97158 0.985790 0.167980i \(-0.0537245\pi\)
0.985790 + 0.167980i \(0.0537245\pi\)
\(264\) −4.16228 −0.256170
\(265\) 0 0
\(266\) −3.16228 −0.193892
\(267\) 7.32456 0.448256
\(268\) −6.48683 −0.396246
\(269\) 12.3246 0.751441 0.375721 0.926733i \(-0.377395\pi\)
0.375721 + 0.926733i \(0.377395\pi\)
\(270\) 0 0
\(271\) −1.16228 −0.0706033 −0.0353017 0.999377i \(-0.511239\pi\)
−0.0353017 + 0.999377i \(0.511239\pi\)
\(272\) 0.837722 0.0507944
\(273\) −16.3246 −0.988007
\(274\) 2.32456 0.140432
\(275\) 0 0
\(276\) 5.32456 0.320501
\(277\) 24.8114 1.49077 0.745386 0.666633i \(-0.232265\pi\)
0.745386 + 0.666633i \(0.232265\pi\)
\(278\) 21.4868 1.28869
\(279\) 5.32456 0.318773
\(280\) 0 0
\(281\) 7.64911 0.456308 0.228154 0.973625i \(-0.426731\pi\)
0.228154 + 0.973625i \(0.426731\pi\)
\(282\) −6.00000 −0.357295
\(283\) 24.6491 1.46524 0.732619 0.680639i \(-0.238298\pi\)
0.732619 + 0.680639i \(0.238298\pi\)
\(284\) 1.16228 0.0689685
\(285\) 0 0
\(286\) 21.4868 1.27054
\(287\) −20.0000 −1.18056
\(288\) 1.00000 0.0589256
\(289\) −16.2982 −0.958719
\(290\) 0 0
\(291\) −0.513167 −0.0300824
\(292\) −1.00000 −0.0585206
\(293\) 16.4868 0.963171 0.481586 0.876399i \(-0.340061\pi\)
0.481586 + 0.876399i \(0.340061\pi\)
\(294\) −3.00000 −0.174964
\(295\) 0 0
\(296\) 10.0000 0.581238
\(297\) −4.16228 −0.241520
\(298\) 16.6491 0.964457
\(299\) −27.4868 −1.58960
\(300\) 0 0
\(301\) −28.9737 −1.67001
\(302\) −10.0000 −0.575435
\(303\) 13.8114 0.793444
\(304\) −1.00000 −0.0573539
\(305\) 0 0
\(306\) 0.837722 0.0478894
\(307\) 13.5132 0.771237 0.385619 0.922658i \(-0.373988\pi\)
0.385619 + 0.922658i \(0.373988\pi\)
\(308\) 13.1623 0.749990
\(309\) −15.6491 −0.890247
\(310\) 0 0
\(311\) −26.0000 −1.47432 −0.737162 0.675716i \(-0.763835\pi\)
−0.737162 + 0.675716i \(0.763835\pi\)
\(312\) −5.16228 −0.292256
\(313\) −10.6754 −0.603412 −0.301706 0.953401i \(-0.597556\pi\)
−0.301706 + 0.953401i \(0.597556\pi\)
\(314\) 18.3246 1.03411
\(315\) 0 0
\(316\) −5.32456 −0.299530
\(317\) 4.16228 0.233777 0.116888 0.993145i \(-0.462708\pi\)
0.116888 + 0.993145i \(0.462708\pi\)
\(318\) −1.83772 −0.103054
\(319\) −15.9737 −0.894354
\(320\) 0 0
\(321\) −12.8377 −0.716532
\(322\) −16.8377 −0.938330
\(323\) −0.837722 −0.0466121
\(324\) 1.00000 0.0555556
\(325\) 0 0
\(326\) −22.6491 −1.25442
\(327\) −7.67544 −0.424453
\(328\) −6.32456 −0.349215
\(329\) 18.9737 1.04605
\(330\) 0 0
\(331\) −3.18861 −0.175262 −0.0876310 0.996153i \(-0.527930\pi\)
−0.0876310 + 0.996153i \(0.527930\pi\)
\(332\) 6.48683 0.356011
\(333\) 10.0000 0.547997
\(334\) 2.64911 0.144953
\(335\) 0 0
\(336\) −3.16228 −0.172516
\(337\) −2.97367 −0.161986 −0.0809930 0.996715i \(-0.525809\pi\)
−0.0809930 + 0.996715i \(0.525809\pi\)
\(338\) 13.6491 0.742414
\(339\) −9.64911 −0.524068
\(340\) 0 0
\(341\) 22.1623 1.20015
\(342\) −1.00000 −0.0540738
\(343\) −12.6491 −0.682988
\(344\) −9.16228 −0.493997
\(345\) 0 0
\(346\) −2.16228 −0.116245
\(347\) −20.0000 −1.07366 −0.536828 0.843692i \(-0.680378\pi\)
−0.536828 + 0.843692i \(0.680378\pi\)
\(348\) 3.83772 0.205724
\(349\) −17.8377 −0.954831 −0.477416 0.878678i \(-0.658426\pi\)
−0.477416 + 0.878678i \(0.658426\pi\)
\(350\) 0 0
\(351\) −5.16228 −0.275542
\(352\) 4.16228 0.221850
\(353\) 23.8114 1.26735 0.633676 0.773598i \(-0.281545\pi\)
0.633676 + 0.773598i \(0.281545\pi\)
\(354\) 7.48683 0.397921
\(355\) 0 0
\(356\) −7.32456 −0.388201
\(357\) −2.64911 −0.140206
\(358\) −20.6491 −1.09134
\(359\) −11.6754 −0.616206 −0.308103 0.951353i \(-0.599694\pi\)
−0.308103 + 0.951353i \(0.599694\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 9.16228 0.481559
\(363\) −6.32456 −0.331953
\(364\) 16.3246 0.855639
\(365\) 0 0
\(366\) 4.16228 0.217566
\(367\) −7.48683 −0.390810 −0.195405 0.980723i \(-0.562602\pi\)
−0.195405 + 0.980723i \(0.562602\pi\)
\(368\) −5.32456 −0.277562
\(369\) −6.32456 −0.329243
\(370\) 0 0
\(371\) 5.81139 0.301712
\(372\) −5.32456 −0.276065
\(373\) −16.1359 −0.835487 −0.417744 0.908565i \(-0.637179\pi\)
−0.417744 + 0.908565i \(0.637179\pi\)
\(374\) 3.48683 0.180300
\(375\) 0 0
\(376\) 6.00000 0.309426
\(377\) −19.8114 −1.02034
\(378\) −3.16228 −0.162650
\(379\) 22.9737 1.18008 0.590039 0.807375i \(-0.299112\pi\)
0.590039 + 0.807375i \(0.299112\pi\)
\(380\) 0 0
\(381\) −15.6491 −0.801728
\(382\) 21.6491 1.10766
\(383\) −3.48683 −0.178169 −0.0890844 0.996024i \(-0.528394\pi\)
−0.0890844 + 0.996024i \(0.528394\pi\)
\(384\) −1.00000 −0.0510310
\(385\) 0 0
\(386\) −9.16228 −0.466348
\(387\) −9.16228 −0.465745
\(388\) 0.513167 0.0260521
\(389\) −34.6491 −1.75678 −0.878390 0.477945i \(-0.841382\pi\)
−0.878390 + 0.477945i \(0.841382\pi\)
\(390\) 0 0
\(391\) −4.46050 −0.225577
\(392\) 3.00000 0.151523
\(393\) 12.4868 0.629877
\(394\) −24.1359 −1.21595
\(395\) 0 0
\(396\) 4.16228 0.209162
\(397\) 17.1359 0.860028 0.430014 0.902822i \(-0.358509\pi\)
0.430014 + 0.902822i \(0.358509\pi\)
\(398\) −18.6491 −0.934795
\(399\) 3.16228 0.158312
\(400\) 0 0
\(401\) 13.6491 0.681604 0.340802 0.940135i \(-0.389301\pi\)
0.340802 + 0.940135i \(0.389301\pi\)
\(402\) 6.48683 0.323534
\(403\) 27.4868 1.36922
\(404\) −13.8114 −0.687142
\(405\) 0 0
\(406\) −12.1359 −0.602297
\(407\) 41.6228 2.06316
\(408\) −0.837722 −0.0414734
\(409\) −7.67544 −0.379526 −0.189763 0.981830i \(-0.560772\pi\)
−0.189763 + 0.981830i \(0.560772\pi\)
\(410\) 0 0
\(411\) −2.32456 −0.114662
\(412\) 15.6491 0.770976
\(413\) −23.6754 −1.16499
\(414\) −5.32456 −0.261688
\(415\) 0 0
\(416\) 5.16228 0.253101
\(417\) −21.4868 −1.05221
\(418\) −4.16228 −0.203584
\(419\) 8.00000 0.390826 0.195413 0.980721i \(-0.437395\pi\)
0.195413 + 0.980721i \(0.437395\pi\)
\(420\) 0 0
\(421\) 18.6491 0.908902 0.454451 0.890772i \(-0.349835\pi\)
0.454451 + 0.890772i \(0.349835\pi\)
\(422\) −18.8114 −0.915724
\(423\) 6.00000 0.291730
\(424\) 1.83772 0.0892477
\(425\) 0 0
\(426\) −1.16228 −0.0563125
\(427\) −13.1623 −0.636967
\(428\) 12.8377 0.620535
\(429\) −21.4868 −1.03739
\(430\) 0 0
\(431\) −0.324555 −0.0156333 −0.00781664 0.999969i \(-0.502488\pi\)
−0.00781664 + 0.999969i \(0.502488\pi\)
\(432\) −1.00000 −0.0481125
\(433\) 28.1359 1.35213 0.676064 0.736843i \(-0.263684\pi\)
0.676064 + 0.736843i \(0.263684\pi\)
\(434\) 16.8377 0.808237
\(435\) 0 0
\(436\) 7.67544 0.367587
\(437\) 5.32456 0.254708
\(438\) 1.00000 0.0477818
\(439\) 13.6491 0.651437 0.325718 0.945467i \(-0.394394\pi\)
0.325718 + 0.945467i \(0.394394\pi\)
\(440\) 0 0
\(441\) 3.00000 0.142857
\(442\) 4.32456 0.205698
\(443\) −11.8377 −0.562427 −0.281214 0.959645i \(-0.590737\pi\)
−0.281214 + 0.959645i \(0.590737\pi\)
\(444\) −10.0000 −0.474579
\(445\) 0 0
\(446\) −25.6491 −1.21452
\(447\) −16.6491 −0.787476
\(448\) 3.16228 0.149404
\(449\) −11.0000 −0.519122 −0.259561 0.965727i \(-0.583578\pi\)
−0.259561 + 0.965727i \(0.583578\pi\)
\(450\) 0 0
\(451\) −26.3246 −1.23957
\(452\) 9.64911 0.453856
\(453\) 10.0000 0.469841
\(454\) −2.00000 −0.0938647
\(455\) 0 0
\(456\) 1.00000 0.0468293
\(457\) 9.67544 0.452598 0.226299 0.974058i \(-0.427337\pi\)
0.226299 + 0.974058i \(0.427337\pi\)
\(458\) 8.16228 0.381398
\(459\) −0.837722 −0.0391015
\(460\) 0 0
\(461\) 12.0000 0.558896 0.279448 0.960161i \(-0.409849\pi\)
0.279448 + 0.960161i \(0.409849\pi\)
\(462\) −13.1623 −0.612365
\(463\) −21.6754 −1.00734 −0.503672 0.863895i \(-0.668018\pi\)
−0.503672 + 0.863895i \(0.668018\pi\)
\(464\) −3.83772 −0.178162
\(465\) 0 0
\(466\) −11.8114 −0.547152
\(467\) −37.1359 −1.71845 −0.859223 0.511601i \(-0.829053\pi\)
−0.859223 + 0.511601i \(0.829053\pi\)
\(468\) 5.16228 0.238626
\(469\) −20.5132 −0.947210
\(470\) 0 0
\(471\) −18.3246 −0.844351
\(472\) −7.48683 −0.344609
\(473\) −38.1359 −1.75349
\(474\) 5.32456 0.244565
\(475\) 0 0
\(476\) 2.64911 0.121422
\(477\) 1.83772 0.0841435
\(478\) −24.3246 −1.11258
\(479\) −39.9737 −1.82644 −0.913222 0.407463i \(-0.866414\pi\)
−0.913222 + 0.407463i \(0.866414\pi\)
\(480\) 0 0
\(481\) 51.6228 2.35380
\(482\) 10.5132 0.478862
\(483\) 16.8377 0.766143
\(484\) 6.32456 0.287480
\(485\) 0 0
\(486\) −1.00000 −0.0453609
\(487\) −28.6491 −1.29822 −0.649108 0.760697i \(-0.724857\pi\)
−0.649108 + 0.760697i \(0.724857\pi\)
\(488\) −4.16228 −0.188417
\(489\) 22.6491 1.02423
\(490\) 0 0
\(491\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(492\) 6.32456 0.285133
\(493\) −3.21495 −0.144794
\(494\) −5.16228 −0.232262
\(495\) 0 0
\(496\) 5.32456 0.239080
\(497\) 3.67544 0.164866
\(498\) −6.48683 −0.290682
\(499\) −5.81139 −0.260153 −0.130077 0.991504i \(-0.541522\pi\)
−0.130077 + 0.991504i \(0.541522\pi\)
\(500\) 0 0
\(501\) −2.64911 −0.118354
\(502\) −10.6491 −0.475293
\(503\) −29.2982 −1.30634 −0.653172 0.757210i \(-0.726562\pi\)
−0.653172 + 0.757210i \(0.726562\pi\)
\(504\) 3.16228 0.140859
\(505\) 0 0
\(506\) −22.1623 −0.985233
\(507\) −13.6491 −0.606178
\(508\) 15.6491 0.694317
\(509\) 16.8114 0.745152 0.372576 0.928002i \(-0.378475\pi\)
0.372576 + 0.928002i \(0.378475\pi\)
\(510\) 0 0
\(511\) −3.16228 −0.139891
\(512\) 1.00000 0.0441942
\(513\) 1.00000 0.0441511
\(514\) −19.3246 −0.852370
\(515\) 0 0
\(516\) 9.16228 0.403347
\(517\) 24.9737 1.09834
\(518\) 31.6228 1.38943
\(519\) 2.16228 0.0949135
\(520\) 0 0
\(521\) 19.9737 0.875062 0.437531 0.899203i \(-0.355853\pi\)
0.437531 + 0.899203i \(0.355853\pi\)
\(522\) −3.83772 −0.167973
\(523\) −16.6491 −0.728015 −0.364007 0.931396i \(-0.618592\pi\)
−0.364007 + 0.931396i \(0.618592\pi\)
\(524\) −12.4868 −0.545490
\(525\) 0 0
\(526\) 31.9737 1.39412
\(527\) 4.46050 0.194302
\(528\) −4.16228 −0.181140
\(529\) 5.35089 0.232647
\(530\) 0 0
\(531\) −7.48683 −0.324901
\(532\) −3.16228 −0.137102
\(533\) −32.6491 −1.41419
\(534\) 7.32456 0.316965
\(535\) 0 0
\(536\) −6.48683 −0.280189
\(537\) 20.6491 0.891075
\(538\) 12.3246 0.531349
\(539\) 12.4868 0.537846
\(540\) 0 0
\(541\) −0.162278 −0.00697686 −0.00348843 0.999994i \(-0.501110\pi\)
−0.00348843 + 0.999994i \(0.501110\pi\)
\(542\) −1.16228 −0.0499241
\(543\) −9.16228 −0.393191
\(544\) 0.837722 0.0359170
\(545\) 0 0
\(546\) −16.3246 −0.698626
\(547\) −16.8114 −0.718803 −0.359402 0.933183i \(-0.617019\pi\)
−0.359402 + 0.933183i \(0.617019\pi\)
\(548\) 2.32456 0.0993001
\(549\) −4.16228 −0.177642
\(550\) 0 0
\(551\) 3.83772 0.163492
\(552\) 5.32456 0.226628
\(553\) −16.8377 −0.716013
\(554\) 24.8114 1.05413
\(555\) 0 0
\(556\) 21.4868 0.911245
\(557\) −22.8377 −0.967665 −0.483833 0.875161i \(-0.660756\pi\)
−0.483833 + 0.875161i \(0.660756\pi\)
\(558\) 5.32456 0.225406
\(559\) −47.2982 −2.00050
\(560\) 0 0
\(561\) −3.48683 −0.147214
\(562\) 7.64911 0.322658
\(563\) 31.1623 1.31333 0.656667 0.754181i \(-0.271966\pi\)
0.656667 + 0.754181i \(0.271966\pi\)
\(564\) −6.00000 −0.252646
\(565\) 0 0
\(566\) 24.6491 1.03608
\(567\) 3.16228 0.132803
\(568\) 1.16228 0.0487681
\(569\) 22.0000 0.922288 0.461144 0.887325i \(-0.347439\pi\)
0.461144 + 0.887325i \(0.347439\pi\)
\(570\) 0 0
\(571\) −15.8114 −0.661686 −0.330843 0.943686i \(-0.607333\pi\)
−0.330843 + 0.943686i \(0.607333\pi\)
\(572\) 21.4868 0.898410
\(573\) −21.6491 −0.904405
\(574\) −20.0000 −0.834784
\(575\) 0 0
\(576\) 1.00000 0.0416667
\(577\) −16.6754 −0.694208 −0.347104 0.937827i \(-0.612835\pi\)
−0.347104 + 0.937827i \(0.612835\pi\)
\(578\) −16.2982 −0.677917
\(579\) 9.16228 0.380771
\(580\) 0 0
\(581\) 20.5132 0.851030
\(582\) −0.513167 −0.0212715
\(583\) 7.64911 0.316794
\(584\) −1.00000 −0.0413803
\(585\) 0 0
\(586\) 16.4868 0.681065
\(587\) −18.8114 −0.776429 −0.388215 0.921569i \(-0.626908\pi\)
−0.388215 + 0.921569i \(0.626908\pi\)
\(588\) −3.00000 −0.123718
\(589\) −5.32456 −0.219394
\(590\) 0 0
\(591\) 24.1359 0.992820
\(592\) 10.0000 0.410997
\(593\) −2.32456 −0.0954580 −0.0477290 0.998860i \(-0.515198\pi\)
−0.0477290 + 0.998860i \(0.515198\pi\)
\(594\) −4.16228 −0.170780
\(595\) 0 0
\(596\) 16.6491 0.681974
\(597\) 18.6491 0.763257
\(598\) −27.4868 −1.12402
\(599\) −4.18861 −0.171142 −0.0855710 0.996332i \(-0.527271\pi\)
−0.0855710 + 0.996332i \(0.527271\pi\)
\(600\) 0 0
\(601\) 45.8114 1.86869 0.934343 0.356376i \(-0.115988\pi\)
0.934343 + 0.356376i \(0.115988\pi\)
\(602\) −28.9737 −1.18088
\(603\) −6.48683 −0.264164
\(604\) −10.0000 −0.406894
\(605\) 0 0
\(606\) 13.8114 0.561049
\(607\) 2.02633 0.0822464 0.0411232 0.999154i \(-0.486906\pi\)
0.0411232 + 0.999154i \(0.486906\pi\)
\(608\) −1.00000 −0.0405554
\(609\) 12.1359 0.491773
\(610\) 0 0
\(611\) 30.9737 1.25306
\(612\) 0.837722 0.0338629
\(613\) −27.2982 −1.10256 −0.551282 0.834319i \(-0.685861\pi\)
−0.551282 + 0.834319i \(0.685861\pi\)
\(614\) 13.5132 0.545347
\(615\) 0 0
\(616\) 13.1623 0.530323
\(617\) −11.6754 −0.470036 −0.235018 0.971991i \(-0.575515\pi\)
−0.235018 + 0.971991i \(0.575515\pi\)
\(618\) −15.6491 −0.629500
\(619\) 10.8377 0.435605 0.217802 0.975993i \(-0.430111\pi\)
0.217802 + 0.975993i \(0.430111\pi\)
\(620\) 0 0
\(621\) 5.32456 0.213667
\(622\) −26.0000 −1.04251
\(623\) −23.1623 −0.927977
\(624\) −5.16228 −0.206656
\(625\) 0 0
\(626\) −10.6754 −0.426677
\(627\) 4.16228 0.166225
\(628\) 18.3246 0.731229
\(629\) 8.37722 0.334022
\(630\) 0 0
\(631\) 45.2982 1.80329 0.901647 0.432473i \(-0.142359\pi\)
0.901647 + 0.432473i \(0.142359\pi\)
\(632\) −5.32456 −0.211799
\(633\) 18.8114 0.747686
\(634\) 4.16228 0.165305
\(635\) 0 0
\(636\) −1.83772 −0.0728704
\(637\) 15.4868 0.613611
\(638\) −15.9737 −0.632403
\(639\) 1.16228 0.0459790
\(640\) 0 0
\(641\) −37.2982 −1.47319 −0.736596 0.676333i \(-0.763568\pi\)
−0.736596 + 0.676333i \(0.763568\pi\)
\(642\) −12.8377 −0.506664
\(643\) 6.64911 0.262215 0.131108 0.991368i \(-0.458147\pi\)
0.131108 + 0.991368i \(0.458147\pi\)
\(644\) −16.8377 −0.663499
\(645\) 0 0
\(646\) −0.837722 −0.0329597
\(647\) −9.00000 −0.353827 −0.176913 0.984226i \(-0.556611\pi\)
−0.176913 + 0.984226i \(0.556611\pi\)
\(648\) 1.00000 0.0392837
\(649\) −31.1623 −1.22323
\(650\) 0 0
\(651\) −16.8377 −0.659922
\(652\) −22.6491 −0.887008
\(653\) 6.97367 0.272901 0.136450 0.990647i \(-0.456431\pi\)
0.136450 + 0.990647i \(0.456431\pi\)
\(654\) −7.67544 −0.300134
\(655\) 0 0
\(656\) −6.32456 −0.246932
\(657\) −1.00000 −0.0390137
\(658\) 18.9737 0.739671
\(659\) −49.8114 −1.94038 −0.970188 0.242353i \(-0.922081\pi\)
−0.970188 + 0.242353i \(0.922081\pi\)
\(660\) 0 0
\(661\) 37.2982 1.45073 0.725366 0.688363i \(-0.241670\pi\)
0.725366 + 0.688363i \(0.241670\pi\)
\(662\) −3.18861 −0.123929
\(663\) −4.32456 −0.167952
\(664\) 6.48683 0.251738
\(665\) 0 0
\(666\) 10.0000 0.387492
\(667\) 20.4342 0.791214
\(668\) 2.64911 0.102497
\(669\) 25.6491 0.991652
\(670\) 0 0
\(671\) −17.3246 −0.668807
\(672\) −3.16228 −0.121988
\(673\) −32.4605 −1.25126 −0.625630 0.780120i \(-0.715158\pi\)
−0.625630 + 0.780120i \(0.715158\pi\)
\(674\) −2.97367 −0.114541
\(675\) 0 0
\(676\) 13.6491 0.524966
\(677\) 18.4868 0.710507 0.355253 0.934770i \(-0.384395\pi\)
0.355253 + 0.934770i \(0.384395\pi\)
\(678\) −9.64911 −0.370572
\(679\) 1.62278 0.0622765
\(680\) 0 0
\(681\) 2.00000 0.0766402
\(682\) 22.1623 0.848637
\(683\) −37.1096 −1.41996 −0.709980 0.704222i \(-0.751296\pi\)
−0.709980 + 0.704222i \(0.751296\pi\)
\(684\) −1.00000 −0.0382360
\(685\) 0 0
\(686\) −12.6491 −0.482945
\(687\) −8.16228 −0.311410
\(688\) −9.16228 −0.349309
\(689\) 9.48683 0.361420
\(690\) 0 0
\(691\) 13.4868 0.513063 0.256532 0.966536i \(-0.417420\pi\)
0.256532 + 0.966536i \(0.417420\pi\)
\(692\) −2.16228 −0.0821975
\(693\) 13.1623 0.499994
\(694\) −20.0000 −0.759190
\(695\) 0 0
\(696\) 3.83772 0.145468
\(697\) −5.29822 −0.200684
\(698\) −17.8377 −0.675168
\(699\) 11.8114 0.446748
\(700\) 0 0
\(701\) −30.9737 −1.16986 −0.584930 0.811084i \(-0.698878\pi\)
−0.584930 + 0.811084i \(0.698878\pi\)
\(702\) −5.16228 −0.194838
\(703\) −10.0000 −0.377157
\(704\) 4.16228 0.156872
\(705\) 0 0
\(706\) 23.8114 0.896153
\(707\) −43.6754 −1.64258
\(708\) 7.48683 0.281372
\(709\) 23.5132 0.883056 0.441528 0.897248i \(-0.354437\pi\)
0.441528 + 0.897248i \(0.354437\pi\)
\(710\) 0 0
\(711\) −5.32456 −0.199686
\(712\) −7.32456 −0.274499
\(713\) −28.3509 −1.06175
\(714\) −2.64911 −0.0991405
\(715\) 0 0
\(716\) −20.6491 −0.771693
\(717\) 24.3246 0.908417
\(718\) −11.6754 −0.435724
\(719\) 19.3246 0.720684 0.360342 0.932820i \(-0.382660\pi\)
0.360342 + 0.932820i \(0.382660\pi\)
\(720\) 0 0
\(721\) 49.4868 1.84299
\(722\) 1.00000 0.0372161
\(723\) −10.5132 −0.390989
\(724\) 9.16228 0.340513
\(725\) 0 0
\(726\) −6.32456 −0.234726
\(727\) −49.6228 −1.84041 −0.920203 0.391440i \(-0.871977\pi\)
−0.920203 + 0.391440i \(0.871977\pi\)
\(728\) 16.3246 0.605028
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −7.67544 −0.283887
\(732\) 4.16228 0.153842
\(733\) 50.4868 1.86477 0.932387 0.361462i \(-0.117722\pi\)
0.932387 + 0.361462i \(0.117722\pi\)
\(734\) −7.48683 −0.276344
\(735\) 0 0
\(736\) −5.32456 −0.196266
\(737\) −27.0000 −0.994558
\(738\) −6.32456 −0.232810
\(739\) 6.32456 0.232653 0.116326 0.993211i \(-0.462888\pi\)
0.116326 + 0.993211i \(0.462888\pi\)
\(740\) 0 0
\(741\) 5.16228 0.189641
\(742\) 5.81139 0.213343
\(743\) 4.64911 0.170559 0.0852797 0.996357i \(-0.472822\pi\)
0.0852797 + 0.996357i \(0.472822\pi\)
\(744\) −5.32456 −0.195208
\(745\) 0 0
\(746\) −16.1359 −0.590779
\(747\) 6.48683 0.237341
\(748\) 3.48683 0.127491
\(749\) 40.5964 1.48336
\(750\) 0 0
\(751\) −26.0000 −0.948753 −0.474377 0.880322i \(-0.657327\pi\)
−0.474377 + 0.880322i \(0.657327\pi\)
\(752\) 6.00000 0.218797
\(753\) 10.6491 0.388075
\(754\) −19.8114 −0.721488
\(755\) 0 0
\(756\) −3.16228 −0.115011
\(757\) −4.86406 −0.176787 −0.0883936 0.996086i \(-0.528173\pi\)
−0.0883936 + 0.996086i \(0.528173\pi\)
\(758\) 22.9737 0.834441
\(759\) 22.1623 0.804440
\(760\) 0 0
\(761\) −39.4868 −1.43140 −0.715698 0.698410i \(-0.753891\pi\)
−0.715698 + 0.698410i \(0.753891\pi\)
\(762\) −15.6491 −0.566907
\(763\) 24.2719 0.878701
\(764\) 21.6491 0.783237
\(765\) 0 0
\(766\) −3.48683 −0.125984
\(767\) −38.6491 −1.39554
\(768\) −1.00000 −0.0360844
\(769\) 34.6754 1.25043 0.625214 0.780453i \(-0.285012\pi\)
0.625214 + 0.780453i \(0.285012\pi\)
\(770\) 0 0
\(771\) 19.3246 0.695957
\(772\) −9.16228 −0.329758
\(773\) 8.97367 0.322760 0.161380 0.986892i \(-0.448405\pi\)
0.161380 + 0.986892i \(0.448405\pi\)
\(774\) −9.16228 −0.329331
\(775\) 0 0
\(776\) 0.513167 0.0184216
\(777\) −31.6228 −1.13446
\(778\) −34.6491 −1.24223
\(779\) 6.32456 0.226601
\(780\) 0 0
\(781\) 4.83772 0.173107
\(782\) −4.46050 −0.159507
\(783\) 3.83772 0.137149
\(784\) 3.00000 0.107143
\(785\) 0 0
\(786\) 12.4868 0.445391
\(787\) 34.4868 1.22932 0.614661 0.788791i \(-0.289293\pi\)
0.614661 + 0.788791i \(0.289293\pi\)
\(788\) −24.1359 −0.859807
\(789\) −31.9737 −1.13829
\(790\) 0 0
\(791\) 30.5132 1.08492
\(792\) 4.16228 0.147900
\(793\) −21.4868 −0.763020
\(794\) 17.1359 0.608132
\(795\) 0 0
\(796\) −18.6491 −0.661000
\(797\) 47.6228 1.68689 0.843443 0.537219i \(-0.180525\pi\)
0.843443 + 0.537219i \(0.180525\pi\)
\(798\) 3.16228 0.111943
\(799\) 5.02633 0.177819
\(800\) 0 0
\(801\) −7.32456 −0.258800
\(802\) 13.6491 0.481967
\(803\) −4.16228 −0.146884
\(804\) 6.48683 0.228773
\(805\) 0 0
\(806\) 27.4868 0.968182
\(807\) −12.3246 −0.433845
\(808\) −13.8114 −0.485883
\(809\) −23.1623 −0.814342 −0.407171 0.913352i \(-0.633485\pi\)
−0.407171 + 0.913352i \(0.633485\pi\)
\(810\) 0 0
\(811\) 11.1886 0.392885 0.196443 0.980515i \(-0.437061\pi\)
0.196443 + 0.980515i \(0.437061\pi\)
\(812\) −12.1359 −0.425888
\(813\) 1.16228 0.0407629
\(814\) 41.6228 1.45888
\(815\) 0 0
\(816\) −0.837722 −0.0293261
\(817\) 9.16228 0.320548
\(818\) −7.67544 −0.268366
\(819\) 16.3246 0.570426
\(820\) 0 0
\(821\) 36.8377 1.28565 0.642823 0.766015i \(-0.277763\pi\)
0.642823 + 0.766015i \(0.277763\pi\)
\(822\) −2.32456 −0.0810782
\(823\) −27.2982 −0.951556 −0.475778 0.879565i \(-0.657833\pi\)
−0.475778 + 0.879565i \(0.657833\pi\)
\(824\) 15.6491 0.545163
\(825\) 0 0
\(826\) −23.6754 −0.823774
\(827\) 5.16228 0.179510 0.0897550 0.995964i \(-0.471392\pi\)
0.0897550 + 0.995964i \(0.471392\pi\)
\(828\) −5.32456 −0.185041
\(829\) −22.7851 −0.791358 −0.395679 0.918389i \(-0.629491\pi\)
−0.395679 + 0.918389i \(0.629491\pi\)
\(830\) 0 0
\(831\) −24.8114 −0.860698
\(832\) 5.16228 0.178970
\(833\) 2.51317 0.0870761
\(834\) −21.4868 −0.744028
\(835\) 0 0
\(836\) −4.16228 −0.143955
\(837\) −5.32456 −0.184044
\(838\) 8.00000 0.276355
\(839\) −12.8377 −0.443207 −0.221604 0.975137i \(-0.571129\pi\)
−0.221604 + 0.975137i \(0.571129\pi\)
\(840\) 0 0
\(841\) −14.2719 −0.492134
\(842\) 18.6491 0.642691
\(843\) −7.64911 −0.263449
\(844\) −18.8114 −0.647515
\(845\) 0 0
\(846\) 6.00000 0.206284
\(847\) 20.0000 0.687208
\(848\) 1.83772 0.0631076
\(849\) −24.6491 −0.845955
\(850\) 0 0
\(851\) −53.2456 −1.82523
\(852\) −1.16228 −0.0398190
\(853\) 32.3246 1.10677 0.553386 0.832925i \(-0.313336\pi\)
0.553386 + 0.832925i \(0.313336\pi\)
\(854\) −13.1623 −0.450404
\(855\) 0 0
\(856\) 12.8377 0.438784
\(857\) 25.6754 0.877056 0.438528 0.898717i \(-0.355500\pi\)
0.438528 + 0.898717i \(0.355500\pi\)
\(858\) −21.4868 −0.733548
\(859\) −5.35089 −0.182570 −0.0912850 0.995825i \(-0.529097\pi\)
−0.0912850 + 0.995825i \(0.529097\pi\)
\(860\) 0 0
\(861\) 20.0000 0.681598
\(862\) −0.324555 −0.0110544
\(863\) 54.9737 1.87133 0.935663 0.352896i \(-0.114803\pi\)
0.935663 + 0.352896i \(0.114803\pi\)
\(864\) −1.00000 −0.0340207
\(865\) 0 0
\(866\) 28.1359 0.956098
\(867\) 16.2982 0.553517
\(868\) 16.8377 0.571510
\(869\) −22.1623 −0.751804
\(870\) 0 0
\(871\) −33.4868 −1.13466
\(872\) 7.67544 0.259923
\(873\) 0.513167 0.0173681
\(874\) 5.32456 0.180106
\(875\) 0 0
\(876\) 1.00000 0.0337869
\(877\) 45.6228 1.54057 0.770286 0.637699i \(-0.220114\pi\)
0.770286 + 0.637699i \(0.220114\pi\)
\(878\) 13.6491 0.460635
\(879\) −16.4868 −0.556087
\(880\) 0 0
\(881\) 14.3246 0.482607 0.241303 0.970450i \(-0.422425\pi\)
0.241303 + 0.970450i \(0.422425\pi\)
\(882\) 3.00000 0.101015
\(883\) −29.4868 −0.992311 −0.496155 0.868234i \(-0.665255\pi\)
−0.496155 + 0.868234i \(0.665255\pi\)
\(884\) 4.32456 0.145451
\(885\) 0 0
\(886\) −11.8377 −0.397696
\(887\) 21.6754 0.727790 0.363895 0.931440i \(-0.381447\pi\)
0.363895 + 0.931440i \(0.381447\pi\)
\(888\) −10.0000 −0.335578
\(889\) 49.4868 1.65974
\(890\) 0 0
\(891\) 4.16228 0.139442
\(892\) −25.6491 −0.858796
\(893\) −6.00000 −0.200782
\(894\) −16.6491 −0.556830
\(895\) 0 0
\(896\) 3.16228 0.105644
\(897\) 27.4868 0.917759
\(898\) −11.0000 −0.367075
\(899\) −20.4342 −0.681518
\(900\) 0 0
\(901\) 1.53950 0.0512882
\(902\) −26.3246 −0.876512
\(903\) 28.9737 0.964183
\(904\) 9.64911 0.320925
\(905\) 0 0
\(906\) 10.0000 0.332228
\(907\) −38.6491 −1.28332 −0.641661 0.766988i \(-0.721755\pi\)
−0.641661 + 0.766988i \(0.721755\pi\)
\(908\) −2.00000 −0.0663723
\(909\) −13.8114 −0.458095
\(910\) 0 0
\(911\) −7.02633 −0.232793 −0.116396 0.993203i \(-0.537134\pi\)
−0.116396 + 0.993203i \(0.537134\pi\)
\(912\) 1.00000 0.0331133
\(913\) 27.0000 0.893570
\(914\) 9.67544 0.320035
\(915\) 0 0
\(916\) 8.16228 0.269689
\(917\) −39.4868 −1.30397
\(918\) −0.837722 −0.0276490
\(919\) −18.8377 −0.621399 −0.310700 0.950508i \(-0.600563\pi\)
−0.310700 + 0.950508i \(0.600563\pi\)
\(920\) 0 0
\(921\) −13.5132 −0.445274
\(922\) 12.0000 0.395199
\(923\) 6.00000 0.197492
\(924\) −13.1623 −0.433007
\(925\) 0 0
\(926\) −21.6754 −0.712299
\(927\) 15.6491 0.513984
\(928\) −3.83772 −0.125979
\(929\) −37.2982 −1.22371 −0.611857 0.790968i \(-0.709577\pi\)
−0.611857 + 0.790968i \(0.709577\pi\)
\(930\) 0 0
\(931\) −3.00000 −0.0983210
\(932\) −11.8114 −0.386895
\(933\) 26.0000 0.851202
\(934\) −37.1359 −1.21513
\(935\) 0 0
\(936\) 5.16228 0.168734
\(937\) −26.6491 −0.870588 −0.435294 0.900288i \(-0.643356\pi\)
−0.435294 + 0.900288i \(0.643356\pi\)
\(938\) −20.5132 −0.669779
\(939\) 10.6754 0.348380
\(940\) 0 0
\(941\) 32.8114 1.06962 0.534810 0.844972i \(-0.320383\pi\)
0.534810 + 0.844972i \(0.320383\pi\)
\(942\) −18.3246 −0.597046
\(943\) 33.6754 1.09662
\(944\) −7.48683 −0.243676
\(945\) 0 0
\(946\) −38.1359 −1.23991
\(947\) −48.0000 −1.55979 −0.779895 0.625910i \(-0.784728\pi\)
−0.779895 + 0.625910i \(0.784728\pi\)
\(948\) 5.32456 0.172934
\(949\) −5.16228 −0.167575
\(950\) 0 0
\(951\) −4.16228 −0.134971
\(952\) 2.64911 0.0858582
\(953\) −40.6754 −1.31761 −0.658803 0.752315i \(-0.728937\pi\)
−0.658803 + 0.752315i \(0.728937\pi\)
\(954\) 1.83772 0.0594985
\(955\) 0 0
\(956\) −24.3246 −0.786712
\(957\) 15.9737 0.516355
\(958\) −39.9737 −1.29149
\(959\) 7.35089 0.237373
\(960\) 0 0
\(961\) −2.64911 −0.0854552
\(962\) 51.6228 1.66439
\(963\) 12.8377 0.413690
\(964\) 10.5132 0.338606
\(965\) 0 0
\(966\) 16.8377 0.541745
\(967\) 24.3246 0.782225 0.391112 0.920343i \(-0.372090\pi\)
0.391112 + 0.920343i \(0.372090\pi\)
\(968\) 6.32456 0.203279
\(969\) 0.837722 0.0269115
\(970\) 0 0
\(971\) 12.5132 0.401567 0.200783 0.979636i \(-0.435651\pi\)
0.200783 + 0.979636i \(0.435651\pi\)
\(972\) −1.00000 −0.0320750
\(973\) 67.9473 2.17829
\(974\) −28.6491 −0.917977
\(975\) 0 0
\(976\) −4.16228 −0.133231
\(977\) −53.9473 −1.72593 −0.862964 0.505265i \(-0.831395\pi\)
−0.862964 + 0.505265i \(0.831395\pi\)
\(978\) 22.6491 0.724239
\(979\) −30.4868 −0.974363
\(980\) 0 0
\(981\) 7.67544 0.245058
\(982\) 0 0
\(983\) −36.1359 −1.15256 −0.576279 0.817253i \(-0.695496\pi\)
−0.576279 + 0.817253i \(0.695496\pi\)
\(984\) 6.32456 0.201619
\(985\) 0 0
\(986\) −3.21495 −0.102385
\(987\) −18.9737 −0.603938
\(988\) −5.16228 −0.164234
\(989\) 48.7851 1.55127
\(990\) 0 0
\(991\) 24.2982 0.771858 0.385929 0.922528i \(-0.373881\pi\)
0.385929 + 0.922528i \(0.373881\pi\)
\(992\) 5.32456 0.169055
\(993\) 3.18861 0.101188
\(994\) 3.67544 0.116578
\(995\) 0 0
\(996\) −6.48683 −0.205543
\(997\) 13.5132 0.427966 0.213983 0.976837i \(-0.431356\pi\)
0.213983 + 0.976837i \(0.431356\pi\)
\(998\) −5.81139 −0.183956
\(999\) −10.0000 −0.316386
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 2850.2.a.bg.1.2 yes 2
3.2 odd 2 8550.2.a.bq.1.2 2
5.2 odd 4 2850.2.d.v.799.4 4
5.3 odd 4 2850.2.d.v.799.1 4
5.4 even 2 2850.2.a.bf.1.1 2
15.14 odd 2 8550.2.a.ca.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2850.2.a.bf.1.1 2 5.4 even 2
2850.2.a.bg.1.2 yes 2 1.1 even 1 trivial
2850.2.d.v.799.1 4 5.3 odd 4
2850.2.d.v.799.4 4 5.2 odd 4
8550.2.a.bq.1.2 2 3.2 odd 2
8550.2.a.ca.1.1 2 15.14 odd 2