Properties

Label 3822.2.a.bv.1.1
Level $3822$
Weight $2$
Character 3822.1
Self dual yes
Analytic conductor $30.519$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3822,2,Mod(1,3822)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3822, 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("3822.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3822 = 2 \cdot 3 \cdot 7^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3822.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: \(30.5188236525\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.2700.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - 15x - 20 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 546)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(4.41883\) of defining polynomial
Character \(\chi\) \(=\) 3822.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -3.41883 q^{5} +1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -3.41883 q^{5} +1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} +3.41883 q^{10} -4.73042 q^{11} -1.00000 q^{12} +1.00000 q^{13} +3.41883 q^{15} +1.00000 q^{16} +7.79567 q^{17} -1.00000 q^{18} +2.68842 q^{19} -3.41883 q^{20} +4.73042 q^{22} -3.73042 q^{23} +1.00000 q^{24} +6.68842 q^{25} -1.00000 q^{26} -1.00000 q^{27} -3.00000 q^{29} -3.41883 q^{30} -5.68842 q^{31} -1.00000 q^{32} +4.73042 q^{33} -7.79567 q^{34} +1.00000 q^{36} -3.10725 q^{37} -2.68842 q^{38} -1.00000 q^{39} +3.41883 q^{40} +0.892750 q^{41} -12.8377 q^{43} -4.73042 q^{44} -3.41883 q^{45} +3.73042 q^{46} -10.1492 q^{47} -1.00000 q^{48} -6.68842 q^{50} -7.79567 q^{51} +1.00000 q^{52} -9.21450 q^{53} +1.00000 q^{54} +16.1725 q^{55} -2.68842 q^{57} +3.00000 q^{58} -6.10725 q^{59} +3.41883 q^{60} +3.04200 q^{61} +5.68842 q^{62} +1.00000 q^{64} -3.41883 q^{65} -4.73042 q^{66} -0.0652506 q^{67} +7.79567 q^{68} +3.73042 q^{69} -12.6884 q^{71} -1.00000 q^{72} +11.7304 q^{73} +3.10725 q^{74} -6.68842 q^{75} +2.68842 q^{76} +1.00000 q^{78} +14.5261 q^{79} -3.41883 q^{80} +1.00000 q^{81} -0.892750 q^{82} +0.934749 q^{83} -26.6521 q^{85} +12.8377 q^{86} +3.00000 q^{87} +4.73042 q^{88} +15.1072 q^{89} +3.41883 q^{90} -3.73042 q^{92} +5.68842 q^{93} +10.1492 q^{94} -9.19125 q^{95} +1.00000 q^{96} -4.79567 q^{97} -4.73042 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 3 q^{2} - 3 q^{3} + 3 q^{4} + 3 q^{5} + 3 q^{6} - 3 q^{8} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 3 q^{2} - 3 q^{3} + 3 q^{4} + 3 q^{5} + 3 q^{6} - 3 q^{8} + 3 q^{9} - 3 q^{10} - 3 q^{11} - 3 q^{12} + 3 q^{13} - 3 q^{15} + 3 q^{16} + 6 q^{17} - 3 q^{18} + 6 q^{19} + 3 q^{20} + 3 q^{22} + 3 q^{24} + 18 q^{25} - 3 q^{26} - 3 q^{27} - 9 q^{29} + 3 q^{30} - 15 q^{31} - 3 q^{32} + 3 q^{33} - 6 q^{34} + 3 q^{36} + 6 q^{37} - 6 q^{38} - 3 q^{39} - 3 q^{40} + 18 q^{41} - 12 q^{43} - 3 q^{44} + 3 q^{45} - 6 q^{47} - 3 q^{48} - 18 q^{50} - 6 q^{51} + 3 q^{52} + 3 q^{53} + 3 q^{54} + 27 q^{55} - 6 q^{57} + 9 q^{58} - 3 q^{59} - 3 q^{60} + 15 q^{62} + 3 q^{64} + 3 q^{65} - 3 q^{66} + 6 q^{67} + 6 q^{68} - 36 q^{71} - 3 q^{72} + 24 q^{73} - 6 q^{74} - 18 q^{75} + 6 q^{76} + 3 q^{78} + 15 q^{79} + 3 q^{80} + 3 q^{81} - 18 q^{82} + 9 q^{83} - 24 q^{85} + 12 q^{86} + 9 q^{87} + 3 q^{88} + 30 q^{89} - 3 q^{90} + 15 q^{93} + 6 q^{94} + 6 q^{95} + 3 q^{96} + 3 q^{97} - 3 q^{99}+O(q^{100}) \) 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
\(5\) −3.41883 −1.52895 −0.764474 0.644654i \(-0.777001\pi\)
−0.764474 + 0.644654i \(0.777001\pi\)
\(6\) 1.00000 0.408248
\(7\) 0 0
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) 3.41883 1.08113
\(11\) −4.73042 −1.42627 −0.713137 0.701025i \(-0.752726\pi\)
−0.713137 + 0.701025i \(0.752726\pi\)
\(12\) −1.00000 −0.288675
\(13\) 1.00000 0.277350
\(14\) 0 0
\(15\) 3.41883 0.882739
\(16\) 1.00000 0.250000
\(17\) 7.79567 1.89073 0.945363 0.326018i \(-0.105707\pi\)
0.945363 + 0.326018i \(0.105707\pi\)
\(18\) −1.00000 −0.235702
\(19\) 2.68842 0.616765 0.308383 0.951262i \(-0.400212\pi\)
0.308383 + 0.951262i \(0.400212\pi\)
\(20\) −3.41883 −0.764474
\(21\) 0 0
\(22\) 4.73042 1.00853
\(23\) −3.73042 −0.777845 −0.388923 0.921270i \(-0.627153\pi\)
−0.388923 + 0.921270i \(0.627153\pi\)
\(24\) 1.00000 0.204124
\(25\) 6.68842 1.33768
\(26\) −1.00000 −0.196116
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) −3.00000 −0.557086 −0.278543 0.960424i \(-0.589851\pi\)
−0.278543 + 0.960424i \(0.589851\pi\)
\(30\) −3.41883 −0.624191
\(31\) −5.68842 −1.02167 −0.510835 0.859679i \(-0.670664\pi\)
−0.510835 + 0.859679i \(0.670664\pi\)
\(32\) −1.00000 −0.176777
\(33\) 4.73042 0.823460
\(34\) −7.79567 −1.33695
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) −3.10725 −0.510829 −0.255414 0.966832i \(-0.582212\pi\)
−0.255414 + 0.966832i \(0.582212\pi\)
\(38\) −2.68842 −0.436119
\(39\) −1.00000 −0.160128
\(40\) 3.41883 0.540565
\(41\) 0.892750 0.139424 0.0697121 0.997567i \(-0.477792\pi\)
0.0697121 + 0.997567i \(0.477792\pi\)
\(42\) 0 0
\(43\) −12.8377 −1.95773 −0.978863 0.204518i \(-0.934437\pi\)
−0.978863 + 0.204518i \(0.934437\pi\)
\(44\) −4.73042 −0.713137
\(45\) −3.41883 −0.509649
\(46\) 3.73042 0.550020
\(47\) −10.1492 −1.48042 −0.740210 0.672376i \(-0.765274\pi\)
−0.740210 + 0.672376i \(0.765274\pi\)
\(48\) −1.00000 −0.144338
\(49\) 0 0
\(50\) −6.68842 −0.945885
\(51\) −7.79567 −1.09161
\(52\) 1.00000 0.138675
\(53\) −9.21450 −1.26571 −0.632854 0.774271i \(-0.718117\pi\)
−0.632854 + 0.774271i \(0.718117\pi\)
\(54\) 1.00000 0.136083
\(55\) 16.1725 2.18070
\(56\) 0 0
\(57\) −2.68842 −0.356090
\(58\) 3.00000 0.393919
\(59\) −6.10725 −0.795096 −0.397548 0.917581i \(-0.630139\pi\)
−0.397548 + 0.917581i \(0.630139\pi\)
\(60\) 3.41883 0.441369
\(61\) 3.04200 0.389488 0.194744 0.980854i \(-0.437612\pi\)
0.194744 + 0.980854i \(0.437612\pi\)
\(62\) 5.68842 0.722430
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) −3.41883 −0.424054
\(66\) −4.73042 −0.582274
\(67\) −0.0652506 −0.00797163 −0.00398581 0.999992i \(-0.501269\pi\)
−0.00398581 + 0.999992i \(0.501269\pi\)
\(68\) 7.79567 0.945363
\(69\) 3.73042 0.449089
\(70\) 0 0
\(71\) −12.6884 −1.50584 −0.752919 0.658113i \(-0.771355\pi\)
−0.752919 + 0.658113i \(0.771355\pi\)
\(72\) −1.00000 −0.117851
\(73\) 11.7304 1.37294 0.686471 0.727158i \(-0.259159\pi\)
0.686471 + 0.727158i \(0.259159\pi\)
\(74\) 3.10725 0.361210
\(75\) −6.68842 −0.772312
\(76\) 2.68842 0.308383
\(77\) 0 0
\(78\) 1.00000 0.113228
\(79\) 14.5261 1.63431 0.817156 0.576417i \(-0.195549\pi\)
0.817156 + 0.576417i \(0.195549\pi\)
\(80\) −3.41883 −0.382237
\(81\) 1.00000 0.111111
\(82\) −0.892750 −0.0985878
\(83\) 0.934749 0.102602 0.0513010 0.998683i \(-0.483663\pi\)
0.0513010 + 0.998683i \(0.483663\pi\)
\(84\) 0 0
\(85\) −26.6521 −2.89082
\(86\) 12.8377 1.38432
\(87\) 3.00000 0.321634
\(88\) 4.73042 0.504264
\(89\) 15.1072 1.60137 0.800683 0.599089i \(-0.204470\pi\)
0.800683 + 0.599089i \(0.204470\pi\)
\(90\) 3.41883 0.360377
\(91\) 0 0
\(92\) −3.73042 −0.388923
\(93\) 5.68842 0.589861
\(94\) 10.1492 1.04682
\(95\) −9.19125 −0.943002
\(96\) 1.00000 0.102062
\(97\) −4.79567 −0.486926 −0.243463 0.969910i \(-0.578283\pi\)
−0.243463 + 0.969910i \(0.578283\pi\)
\(98\) 0 0
\(99\) −4.73042 −0.475425
\(100\) 6.68842 0.668842
\(101\) 0.623166 0.0620074 0.0310037 0.999519i \(-0.490130\pi\)
0.0310037 + 0.999519i \(0.490130\pi\)
\(102\) 7.79567 0.771886
\(103\) 8.83767 0.870801 0.435401 0.900237i \(-0.356607\pi\)
0.435401 + 0.900237i \(0.356607\pi\)
\(104\) −1.00000 −0.0980581
\(105\) 0 0
\(106\) 9.21450 0.894991
\(107\) 5.77241 0.558040 0.279020 0.960285i \(-0.409990\pi\)
0.279020 + 0.960285i \(0.409990\pi\)
\(108\) −1.00000 −0.0962250
\(109\) 4.00000 0.383131 0.191565 0.981480i \(-0.438644\pi\)
0.191565 + 0.981480i \(0.438644\pi\)
\(110\) −16.1725 −1.54199
\(111\) 3.10725 0.294927
\(112\) 0 0
\(113\) 10.6333 1.00030 0.500150 0.865939i \(-0.333278\pi\)
0.500150 + 0.865939i \(0.333278\pi\)
\(114\) 2.68842 0.251793
\(115\) 12.7537 1.18929
\(116\) −3.00000 −0.278543
\(117\) 1.00000 0.0924500
\(118\) 6.10725 0.562218
\(119\) 0 0
\(120\) −3.41883 −0.312095
\(121\) 11.3768 1.03426
\(122\) −3.04200 −0.275410
\(123\) −0.892750 −0.0804966
\(124\) −5.68842 −0.510835
\(125\) −5.77241 −0.516300
\(126\) 0 0
\(127\) −6.17250 −0.547721 −0.273860 0.961769i \(-0.588301\pi\)
−0.273860 + 0.961769i \(0.588301\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 12.8377 1.13029
\(130\) 3.41883 0.299851
\(131\) −20.4710 −1.78856 −0.894280 0.447509i \(-0.852311\pi\)
−0.894280 + 0.447509i \(0.852311\pi\)
\(132\) 4.73042 0.411730
\(133\) 0 0
\(134\) 0.0652506 0.00563679
\(135\) 3.41883 0.294246
\(136\) −7.79567 −0.668473
\(137\) −11.1072 −0.948956 −0.474478 0.880267i \(-0.657363\pi\)
−0.474478 + 0.880267i \(0.657363\pi\)
\(138\) −3.73042 −0.317554
\(139\) −2.26958 −0.192504 −0.0962518 0.995357i \(-0.530685\pi\)
−0.0962518 + 0.995357i \(0.530685\pi\)
\(140\) 0 0
\(141\) 10.1492 0.854721
\(142\) 12.6884 1.06479
\(143\) −4.73042 −0.395577
\(144\) 1.00000 0.0833333
\(145\) 10.2565 0.851756
\(146\) −11.7304 −0.970816
\(147\) 0 0
\(148\) −3.10725 −0.255414
\(149\) −8.26958 −0.677471 −0.338735 0.940882i \(-0.609999\pi\)
−0.338735 + 0.940882i \(0.609999\pi\)
\(150\) 6.68842 0.546107
\(151\) −5.26958 −0.428833 −0.214416 0.976742i \(-0.568785\pi\)
−0.214416 + 0.976742i \(0.568785\pi\)
\(152\) −2.68842 −0.218059
\(153\) 7.79567 0.630242
\(154\) 0 0
\(155\) 19.4477 1.56208
\(156\) −1.00000 −0.0800641
\(157\) −11.2565 −0.898366 −0.449183 0.893440i \(-0.648285\pi\)
−0.449183 + 0.893440i \(0.648285\pi\)
\(158\) −14.5261 −1.15563
\(159\) 9.21450 0.730757
\(160\) 3.41883 0.270282
\(161\) 0 0
\(162\) −1.00000 −0.0785674
\(163\) 16.3637 1.28171 0.640854 0.767663i \(-0.278580\pi\)
0.640854 + 0.767663i \(0.278580\pi\)
\(164\) 0.892750 0.0697121
\(165\) −16.1725 −1.25903
\(166\) −0.934749 −0.0725506
\(167\) 22.9869 1.77878 0.889390 0.457148i \(-0.151129\pi\)
0.889390 + 0.457148i \(0.151129\pi\)
\(168\) 0 0
\(169\) 1.00000 0.0769231
\(170\) 26.6521 2.04412
\(171\) 2.68842 0.205588
\(172\) −12.8377 −0.978863
\(173\) 10.7724 0.819012 0.409506 0.912308i \(-0.365701\pi\)
0.409506 + 0.912308i \(0.365701\pi\)
\(174\) −3.00000 −0.227429
\(175\) 0 0
\(176\) −4.73042 −0.356569
\(177\) 6.10725 0.459049
\(178\) −15.1072 −1.13234
\(179\) 19.0522 1.42403 0.712013 0.702166i \(-0.247784\pi\)
0.712013 + 0.702166i \(0.247784\pi\)
\(180\) −3.41883 −0.254825
\(181\) 13.7957 1.02542 0.512712 0.858561i \(-0.328641\pi\)
0.512712 + 0.858561i \(0.328641\pi\)
\(182\) 0 0
\(183\) −3.04200 −0.224871
\(184\) 3.73042 0.275010
\(185\) 10.6232 0.781031
\(186\) −5.68842 −0.417095
\(187\) −36.8767 −2.69669
\(188\) −10.1492 −0.740210
\(189\) 0 0
\(190\) 9.19125 0.666803
\(191\) 10.2985 0.745173 0.372587 0.927997i \(-0.378471\pi\)
0.372587 + 0.927997i \(0.378471\pi\)
\(192\) −1.00000 −0.0721688
\(193\) 17.6333 1.26927 0.634637 0.772810i \(-0.281149\pi\)
0.634637 + 0.772810i \(0.281149\pi\)
\(194\) 4.79567 0.344309
\(195\) 3.41883 0.244828
\(196\) 0 0
\(197\) −8.26958 −0.589183 −0.294592 0.955623i \(-0.595184\pi\)
−0.294592 + 0.955623i \(0.595184\pi\)
\(198\) 4.73042 0.336176
\(199\) −5.02325 −0.356089 −0.178044 0.984022i \(-0.556977\pi\)
−0.178044 + 0.984022i \(0.556977\pi\)
\(200\) −6.68842 −0.472942
\(201\) 0.0652506 0.00460242
\(202\) −0.623166 −0.0438458
\(203\) 0 0
\(204\) −7.79567 −0.545806
\(205\) −3.05216 −0.213172
\(206\) −8.83767 −0.615749
\(207\) −3.73042 −0.259282
\(208\) 1.00000 0.0693375
\(209\) −12.7173 −0.879676
\(210\) 0 0
\(211\) 14.5681 1.00291 0.501454 0.865184i \(-0.332799\pi\)
0.501454 + 0.865184i \(0.332799\pi\)
\(212\) −9.21450 −0.632854
\(213\) 12.6884 0.869396
\(214\) −5.77241 −0.394594
\(215\) 43.8898 2.99326
\(216\) 1.00000 0.0680414
\(217\) 0 0
\(218\) −4.00000 −0.270914
\(219\) −11.7304 −0.792668
\(220\) 16.1725 1.09035
\(221\) 7.79567 0.524393
\(222\) −3.10725 −0.208545
\(223\) −10.3217 −0.691195 −0.345598 0.938383i \(-0.612324\pi\)
−0.345598 + 0.938383i \(0.612324\pi\)
\(224\) 0 0
\(225\) 6.68842 0.445894
\(226\) −10.6333 −0.707319
\(227\) 4.44208 0.294831 0.147416 0.989075i \(-0.452904\pi\)
0.147416 + 0.989075i \(0.452904\pi\)
\(228\) −2.68842 −0.178045
\(229\) −21.4608 −1.41817 −0.709086 0.705122i \(-0.750892\pi\)
−0.709086 + 0.705122i \(0.750892\pi\)
\(230\) −12.7537 −0.840952
\(231\) 0 0
\(232\) 3.00000 0.196960
\(233\) −7.66517 −0.502162 −0.251081 0.967966i \(-0.580786\pi\)
−0.251081 + 0.967966i \(0.580786\pi\)
\(234\) −1.00000 −0.0653720
\(235\) 34.6986 2.26349
\(236\) −6.10725 −0.397548
\(237\) −14.5261 −0.943570
\(238\) 0 0
\(239\) −9.52608 −0.616191 −0.308096 0.951355i \(-0.599692\pi\)
−0.308096 + 0.951355i \(0.599692\pi\)
\(240\) 3.41883 0.220685
\(241\) −7.36375 −0.474341 −0.237170 0.971468i \(-0.576220\pi\)
−0.237170 + 0.971468i \(0.576220\pi\)
\(242\) −11.3768 −0.731331
\(243\) −1.00000 −0.0641500
\(244\) 3.04200 0.194744
\(245\) 0 0
\(246\) 0.892750 0.0569197
\(247\) 2.68842 0.171060
\(248\) 5.68842 0.361215
\(249\) −0.934749 −0.0592373
\(250\) 5.77241 0.365080
\(251\) 12.7406 0.804178 0.402089 0.915601i \(-0.368284\pi\)
0.402089 + 0.915601i \(0.368284\pi\)
\(252\) 0 0
\(253\) 17.6464 1.10942
\(254\) 6.17250 0.387297
\(255\) 26.6521 1.66902
\(256\) 1.00000 0.0625000
\(257\) −13.4608 −0.839664 −0.419832 0.907602i \(-0.637911\pi\)
−0.419832 + 0.907602i \(0.637911\pi\)
\(258\) −12.8377 −0.799238
\(259\) 0 0
\(260\) −3.41883 −0.212027
\(261\) −3.00000 −0.185695
\(262\) 20.4710 1.26470
\(263\) 24.0289 1.48169 0.740843 0.671678i \(-0.234426\pi\)
0.740843 + 0.671678i \(0.234426\pi\)
\(264\) −4.73042 −0.291137
\(265\) 31.5028 1.93520
\(266\) 0 0
\(267\) −15.1072 −0.924549
\(268\) −0.0652506 −0.00398581
\(269\) 15.9217 0.970761 0.485380 0.874303i \(-0.338681\pi\)
0.485380 + 0.874303i \(0.338681\pi\)
\(270\) −3.41883 −0.208064
\(271\) −19.6521 −1.19378 −0.596889 0.802324i \(-0.703597\pi\)
−0.596889 + 0.802324i \(0.703597\pi\)
\(272\) 7.79567 0.472682
\(273\) 0 0
\(274\) 11.1072 0.671013
\(275\) −31.6390 −1.90790
\(276\) 3.73042 0.224545
\(277\) −5.17250 −0.310785 −0.155393 0.987853i \(-0.549664\pi\)
−0.155393 + 0.987853i \(0.549664\pi\)
\(278\) 2.26958 0.136121
\(279\) −5.68842 −0.340557
\(280\) 0 0
\(281\) −24.4290 −1.45731 −0.728656 0.684880i \(-0.759855\pi\)
−0.728656 + 0.684880i \(0.759855\pi\)
\(282\) −10.1492 −0.604379
\(283\) −3.19125 −0.189700 −0.0948500 0.995492i \(-0.530237\pi\)
−0.0948500 + 0.995492i \(0.530237\pi\)
\(284\) −12.6884 −0.752919
\(285\) 9.19125 0.544443
\(286\) 4.73042 0.279715
\(287\) 0 0
\(288\) −1.00000 −0.0589256
\(289\) 43.7724 2.57485
\(290\) −10.2565 −0.602282
\(291\) 4.79567 0.281127
\(292\) 11.7304 0.686471
\(293\) 2.58117 0.150793 0.0753967 0.997154i \(-0.475978\pi\)
0.0753967 + 0.997154i \(0.475978\pi\)
\(294\) 0 0
\(295\) 20.8797 1.21566
\(296\) 3.10725 0.180605
\(297\) 4.73042 0.274487
\(298\) 8.26958 0.479044
\(299\) −3.73042 −0.215736
\(300\) −6.68842 −0.386156
\(301\) 0 0
\(302\) 5.26958 0.303230
\(303\) −0.623166 −0.0358000
\(304\) 2.68842 0.154191
\(305\) −10.4001 −0.595507
\(306\) −7.79567 −0.445649
\(307\) 23.3116 1.33046 0.665231 0.746637i \(-0.268333\pi\)
0.665231 + 0.746637i \(0.268333\pi\)
\(308\) 0 0
\(309\) −8.83767 −0.502757
\(310\) −19.4477 −1.10456
\(311\) 10.7826 0.611424 0.305712 0.952124i \(-0.401106\pi\)
0.305712 + 0.952124i \(0.401106\pi\)
\(312\) 1.00000 0.0566139
\(313\) 14.8508 0.839414 0.419707 0.907660i \(-0.362133\pi\)
0.419707 + 0.907660i \(0.362133\pi\)
\(314\) 11.2565 0.635241
\(315\) 0 0
\(316\) 14.5261 0.817156
\(317\) 23.0653 1.29547 0.647737 0.761864i \(-0.275716\pi\)
0.647737 + 0.761864i \(0.275716\pi\)
\(318\) −9.21450 −0.516723
\(319\) 14.1912 0.794557
\(320\) −3.41883 −0.191119
\(321\) −5.77241 −0.322185
\(322\) 0 0
\(323\) 20.9580 1.16613
\(324\) 1.00000 0.0555556
\(325\) 6.68842 0.371007
\(326\) −16.3637 −0.906304
\(327\) −4.00000 −0.221201
\(328\) −0.892750 −0.0492939
\(329\) 0 0
\(330\) 16.1725 0.890267
\(331\) 0.130501 0.00717299 0.00358650 0.999994i \(-0.498858\pi\)
0.00358650 + 0.999994i \(0.498858\pi\)
\(332\) 0.934749 0.0513010
\(333\) −3.10725 −0.170276
\(334\) −22.9869 −1.25779
\(335\) 0.223081 0.0121882
\(336\) 0 0
\(337\) 16.0522 0.874417 0.437209 0.899360i \(-0.355967\pi\)
0.437209 + 0.899360i \(0.355967\pi\)
\(338\) −1.00000 −0.0543928
\(339\) −10.6333 −0.577523
\(340\) −26.6521 −1.44541
\(341\) 26.9086 1.45718
\(342\) −2.68842 −0.145373
\(343\) 0 0
\(344\) 12.8377 0.692161
\(345\) −12.7537 −0.686634
\(346\) −10.7724 −0.579129
\(347\) 21.3217 1.14461 0.572306 0.820040i \(-0.306049\pi\)
0.572306 + 0.820040i \(0.306049\pi\)
\(348\) 3.00000 0.160817
\(349\) 6.35358 0.340099 0.170050 0.985435i \(-0.445607\pi\)
0.170050 + 0.985435i \(0.445607\pi\)
\(350\) 0 0
\(351\) −1.00000 −0.0533761
\(352\) 4.73042 0.252132
\(353\) 30.4290 1.61957 0.809786 0.586725i \(-0.199583\pi\)
0.809786 + 0.586725i \(0.199583\pi\)
\(354\) −6.10725 −0.324597
\(355\) 43.3796 2.30235
\(356\) 15.1072 0.800683
\(357\) 0 0
\(358\) −19.0522 −1.00694
\(359\) −18.0000 −0.950004 −0.475002 0.879985i \(-0.657553\pi\)
−0.475002 + 0.879985i \(0.657553\pi\)
\(360\) 3.41883 0.180188
\(361\) −11.7724 −0.619601
\(362\) −13.7957 −0.725084
\(363\) −11.3768 −0.597129
\(364\) 0 0
\(365\) −40.1043 −2.09916
\(366\) 3.04200 0.159008
\(367\) 4.22759 0.220678 0.110339 0.993894i \(-0.464806\pi\)
0.110339 + 0.993894i \(0.464806\pi\)
\(368\) −3.73042 −0.194461
\(369\) 0.892750 0.0464747
\(370\) −10.6232 −0.552272
\(371\) 0 0
\(372\) 5.68842 0.294931
\(373\) −35.6855 −1.84772 −0.923862 0.382725i \(-0.874986\pi\)
−0.923862 + 0.382725i \(0.874986\pi\)
\(374\) 36.8767 1.90685
\(375\) 5.77241 0.298086
\(376\) 10.1492 0.523408
\(377\) −3.00000 −0.154508
\(378\) 0 0
\(379\) 9.67533 0.496988 0.248494 0.968633i \(-0.420064\pi\)
0.248494 + 0.968633i \(0.420064\pi\)
\(380\) −9.19125 −0.471501
\(381\) 6.17250 0.316227
\(382\) −10.2985 −0.526917
\(383\) −3.67533 −0.187801 −0.0939003 0.995582i \(-0.529933\pi\)
−0.0939003 + 0.995582i \(0.529933\pi\)
\(384\) 1.00000 0.0510310
\(385\) 0 0
\(386\) −17.6333 −0.897513
\(387\) −12.8377 −0.652575
\(388\) −4.79567 −0.243463
\(389\) 6.47392 0.328241 0.164120 0.986440i \(-0.447521\pi\)
0.164120 + 0.986440i \(0.447521\pi\)
\(390\) −3.41883 −0.173119
\(391\) −29.0811 −1.47069
\(392\) 0 0
\(393\) 20.4710 1.03263
\(394\) 8.26958 0.416616
\(395\) −49.6622 −2.49878
\(396\) −4.73042 −0.237712
\(397\) −7.86092 −0.394528 −0.197264 0.980350i \(-0.563206\pi\)
−0.197264 + 0.980350i \(0.563206\pi\)
\(398\) 5.02325 0.251793
\(399\) 0 0
\(400\) 6.68842 0.334421
\(401\) −5.64642 −0.281969 −0.140984 0.990012i \(-0.545027\pi\)
−0.140984 + 0.990012i \(0.545027\pi\)
\(402\) −0.0652506 −0.00325440
\(403\) −5.68842 −0.283360
\(404\) 0.623166 0.0310037
\(405\) −3.41883 −0.169883
\(406\) 0 0
\(407\) 14.6986 0.728582
\(408\) 7.79567 0.385943
\(409\) 25.7173 1.27164 0.635820 0.771837i \(-0.280662\pi\)
0.635820 + 0.771837i \(0.280662\pi\)
\(410\) 3.05216 0.150736
\(411\) 11.1072 0.547880
\(412\) 8.83767 0.435401
\(413\) 0 0
\(414\) 3.73042 0.183340
\(415\) −3.19575 −0.156873
\(416\) −1.00000 −0.0490290
\(417\) 2.26958 0.111142
\(418\) 12.7173 0.622025
\(419\) 8.83767 0.431748 0.215874 0.976421i \(-0.430740\pi\)
0.215874 + 0.976421i \(0.430740\pi\)
\(420\) 0 0
\(421\) −10.1594 −0.495140 −0.247570 0.968870i \(-0.579632\pi\)
−0.247570 + 0.968870i \(0.579632\pi\)
\(422\) −14.5681 −0.709163
\(423\) −10.1492 −0.493473
\(424\) 9.21450 0.447496
\(425\) 52.1407 2.52919
\(426\) −12.6884 −0.614756
\(427\) 0 0
\(428\) 5.77241 0.279020
\(429\) 4.73042 0.228387
\(430\) −43.8898 −2.11656
\(431\) −11.6753 −0.562381 −0.281190 0.959652i \(-0.590729\pi\)
−0.281190 + 0.959652i \(0.590729\pi\)
\(432\) −1.00000 −0.0481125
\(433\) 6.60442 0.317388 0.158694 0.987328i \(-0.449272\pi\)
0.158694 + 0.987328i \(0.449272\pi\)
\(434\) 0 0
\(435\) −10.2565 −0.491761
\(436\) 4.00000 0.191565
\(437\) −10.0289 −0.479748
\(438\) 11.7304 0.560501
\(439\) −25.0102 −1.19367 −0.596835 0.802364i \(-0.703575\pi\)
−0.596835 + 0.802364i \(0.703575\pi\)
\(440\) −16.1725 −0.770994
\(441\) 0 0
\(442\) −7.79567 −0.370802
\(443\) −16.8797 −0.801977 −0.400989 0.916083i \(-0.631333\pi\)
−0.400989 + 0.916083i \(0.631333\pi\)
\(444\) 3.10725 0.147464
\(445\) −51.6492 −2.44840
\(446\) 10.3217 0.488749
\(447\) 8.26958 0.391138
\(448\) 0 0
\(449\) −20.1680 −0.951787 −0.475893 0.879503i \(-0.657875\pi\)
−0.475893 + 0.879503i \(0.657875\pi\)
\(450\) −6.68842 −0.315295
\(451\) −4.22308 −0.198857
\(452\) 10.6333 0.500150
\(453\) 5.26958 0.247587
\(454\) −4.44208 −0.208477
\(455\) 0 0
\(456\) 2.68842 0.125897
\(457\) −16.2565 −0.760447 −0.380223 0.924895i \(-0.624153\pi\)
−0.380223 + 0.924895i \(0.624153\pi\)
\(458\) 21.4608 1.00280
\(459\) −7.79567 −0.363871
\(460\) 12.7537 0.594643
\(461\) −16.0840 −0.749106 −0.374553 0.927205i \(-0.622204\pi\)
−0.374553 + 0.927205i \(0.622204\pi\)
\(462\) 0 0
\(463\) 1.67533 0.0778592 0.0389296 0.999242i \(-0.487605\pi\)
0.0389296 + 0.999242i \(0.487605\pi\)
\(464\) −3.00000 −0.139272
\(465\) −19.4477 −0.901868
\(466\) 7.66517 0.355082
\(467\) 31.9449 1.47823 0.739117 0.673577i \(-0.235243\pi\)
0.739117 + 0.673577i \(0.235243\pi\)
\(468\) 1.00000 0.0462250
\(469\) 0 0
\(470\) −34.6986 −1.60053
\(471\) 11.2565 0.518672
\(472\) 6.10725 0.281109
\(473\) 60.7275 2.79225
\(474\) 14.5261 0.667205
\(475\) 17.9813 0.825036
\(476\) 0 0
\(477\) −9.21450 −0.421903
\(478\) 9.52608 0.435713
\(479\) 37.6941 1.72229 0.861143 0.508362i \(-0.169749\pi\)
0.861143 + 0.508362i \(0.169749\pi\)
\(480\) −3.41883 −0.156048
\(481\) −3.10725 −0.141678
\(482\) 7.36375 0.335410
\(483\) 0 0
\(484\) 11.3768 0.517129
\(485\) 16.3956 0.744485
\(486\) 1.00000 0.0453609
\(487\) −4.94491 −0.224075 −0.112038 0.993704i \(-0.535738\pi\)
−0.112038 + 0.993704i \(0.535738\pi\)
\(488\) −3.04200 −0.137705
\(489\) −16.3637 −0.739994
\(490\) 0 0
\(491\) 28.2014 1.27271 0.636356 0.771396i \(-0.280441\pi\)
0.636356 + 0.771396i \(0.280441\pi\)
\(492\) −0.892750 −0.0402483
\(493\) −23.3870 −1.05330
\(494\) −2.68842 −0.120958
\(495\) 16.1725 0.726900
\(496\) −5.68842 −0.255417
\(497\) 0 0
\(498\) 0.934749 0.0418871
\(499\) 19.7593 0.884549 0.442275 0.896880i \(-0.354172\pi\)
0.442275 + 0.896880i \(0.354172\pi\)
\(500\) −5.77241 −0.258150
\(501\) −22.9869 −1.02698
\(502\) −12.7406 −0.568640
\(503\) −2.40867 −0.107397 −0.0536986 0.998557i \(-0.517101\pi\)
−0.0536986 + 0.998557i \(0.517101\pi\)
\(504\) 0 0
\(505\) −2.13050 −0.0948061
\(506\) −17.6464 −0.784479
\(507\) −1.00000 −0.0444116
\(508\) −6.17250 −0.273860
\(509\) 25.5782 1.13374 0.566868 0.823809i \(-0.308155\pi\)
0.566868 + 0.823809i \(0.308155\pi\)
\(510\) −26.6521 −1.18017
\(511\) 0 0
\(512\) −1.00000 −0.0441942
\(513\) −2.68842 −0.118697
\(514\) 13.4608 0.593732
\(515\) −30.2145 −1.33141
\(516\) 12.8377 0.565147
\(517\) 48.0102 2.11148
\(518\) 0 0
\(519\) −10.7724 −0.472857
\(520\) 3.41883 0.149926
\(521\) 14.7912 0.648013 0.324006 0.946055i \(-0.394970\pi\)
0.324006 + 0.946055i \(0.394970\pi\)
\(522\) 3.00000 0.131306
\(523\) −22.5130 −0.984425 −0.492212 0.870475i \(-0.663812\pi\)
−0.492212 + 0.870475i \(0.663812\pi\)
\(524\) −20.4710 −0.894280
\(525\) 0 0
\(526\) −24.0289 −1.04771
\(527\) −44.3450 −1.93170
\(528\) 4.73042 0.205865
\(529\) −9.08400 −0.394956
\(530\) −31.5028 −1.36840
\(531\) −6.10725 −0.265032
\(532\) 0 0
\(533\) 0.892750 0.0386693
\(534\) 15.1072 0.653755
\(535\) −19.7349 −0.853215
\(536\) 0.0652506 0.00281840
\(537\) −19.0522 −0.822162
\(538\) −15.9217 −0.686432
\(539\) 0 0
\(540\) 3.41883 0.147123
\(541\) 29.3217 1.26064 0.630320 0.776335i \(-0.282924\pi\)
0.630320 + 0.776335i \(0.282924\pi\)
\(542\) 19.6521 0.844129
\(543\) −13.7957 −0.592029
\(544\) −7.79567 −0.334236
\(545\) −13.6753 −0.585787
\(546\) 0 0
\(547\) 7.64642 0.326937 0.163469 0.986549i \(-0.447732\pi\)
0.163469 + 0.986549i \(0.447732\pi\)
\(548\) −11.1072 −0.474478
\(549\) 3.04200 0.129829
\(550\) 31.6390 1.34909
\(551\) −8.06525 −0.343591
\(552\) −3.73042 −0.158777
\(553\) 0 0
\(554\) 5.17250 0.219758
\(555\) −10.6232 −0.450928
\(556\) −2.26958 −0.0962518
\(557\) 12.2276 0.518099 0.259050 0.965864i \(-0.416591\pi\)
0.259050 + 0.965864i \(0.416591\pi\)
\(558\) 5.68842 0.240810
\(559\) −12.8377 −0.542975
\(560\) 0 0
\(561\) 36.8767 1.55694
\(562\) 24.4290 1.03048
\(563\) −30.3116 −1.27748 −0.638740 0.769422i \(-0.720544\pi\)
−0.638740 + 0.769422i \(0.720544\pi\)
\(564\) 10.1492 0.427360
\(565\) −36.3536 −1.52941
\(566\) 3.19125 0.134138
\(567\) 0 0
\(568\) 12.6884 0.532394
\(569\) 18.0942 0.758547 0.379273 0.925285i \(-0.376174\pi\)
0.379273 + 0.925285i \(0.376174\pi\)
\(570\) −9.19125 −0.384979
\(571\) −12.4290 −0.520137 −0.260069 0.965590i \(-0.583745\pi\)
−0.260069 + 0.965590i \(0.583745\pi\)
\(572\) −4.73042 −0.197789
\(573\) −10.2985 −0.430226
\(574\) 0 0
\(575\) −24.9506 −1.04051
\(576\) 1.00000 0.0416667
\(577\) −2.09708 −0.0873028 −0.0436514 0.999047i \(-0.513899\pi\)
−0.0436514 + 0.999047i \(0.513899\pi\)
\(578\) −43.7724 −1.82069
\(579\) −17.6333 −0.732816
\(580\) 10.2565 0.425878
\(581\) 0 0
\(582\) −4.79567 −0.198787
\(583\) 43.5884 1.80525
\(584\) −11.7304 −0.485408
\(585\) −3.41883 −0.141351
\(586\) −2.58117 −0.106627
\(587\) 47.7826 1.97220 0.986099 0.166158i \(-0.0531363\pi\)
0.986099 + 0.166158i \(0.0531363\pi\)
\(588\) 0 0
\(589\) −15.2928 −0.630130
\(590\) −20.8797 −0.859602
\(591\) 8.26958 0.340165
\(592\) −3.10725 −0.127707
\(593\) −9.32175 −0.382798 −0.191399 0.981512i \(-0.561302\pi\)
−0.191399 + 0.981512i \(0.561302\pi\)
\(594\) −4.73042 −0.194091
\(595\) 0 0
\(596\) −8.26958 −0.338735
\(597\) 5.02325 0.205588
\(598\) 3.73042 0.152548
\(599\) −5.64642 −0.230706 −0.115353 0.993325i \(-0.536800\pi\)
−0.115353 + 0.993325i \(0.536800\pi\)
\(600\) 6.68842 0.273053
\(601\) −19.3450 −0.789099 −0.394550 0.918875i \(-0.629099\pi\)
−0.394550 + 0.918875i \(0.629099\pi\)
\(602\) 0 0
\(603\) −0.0652506 −0.00265721
\(604\) −5.26958 −0.214416
\(605\) −38.8955 −1.58133
\(606\) 0.623166 0.0253144
\(607\) 42.9637 1.74384 0.871921 0.489647i \(-0.162874\pi\)
0.871921 + 0.489647i \(0.162874\pi\)
\(608\) −2.68842 −0.109030
\(609\) 0 0
\(610\) 10.4001 0.421087
\(611\) −10.1492 −0.410595
\(612\) 7.79567 0.315121
\(613\) 27.8898 1.12646 0.563230 0.826300i \(-0.309559\pi\)
0.563230 + 0.826300i \(0.309559\pi\)
\(614\) −23.3116 −0.940779
\(615\) 3.05216 0.123075
\(616\) 0 0
\(617\) 41.4347 1.66810 0.834048 0.551691i \(-0.186017\pi\)
0.834048 + 0.551691i \(0.186017\pi\)
\(618\) 8.83767 0.355503
\(619\) −30.6232 −1.23085 −0.615424 0.788196i \(-0.711015\pi\)
−0.615424 + 0.788196i \(0.711015\pi\)
\(620\) 19.4477 0.781040
\(621\) 3.73042 0.149696
\(622\) −10.7826 −0.432342
\(623\) 0 0
\(624\) −1.00000 −0.0400320
\(625\) −13.7072 −0.548287
\(626\) −14.8508 −0.593555
\(627\) 12.7173 0.507881
\(628\) −11.2565 −0.449183
\(629\) −24.2231 −0.965837
\(630\) 0 0
\(631\) −12.9347 −0.514924 −0.257462 0.966288i \(-0.582886\pi\)
−0.257462 + 0.966288i \(0.582886\pi\)
\(632\) −14.5261 −0.577817
\(633\) −14.5681 −0.579029
\(634\) −23.0653 −0.916038
\(635\) 21.1027 0.837437
\(636\) 9.21450 0.365379
\(637\) 0 0
\(638\) −14.1912 −0.561837
\(639\) −12.6884 −0.501946
\(640\) 3.41883 0.135141
\(641\) −6.92166 −0.273389 −0.136695 0.990613i \(-0.543648\pi\)
−0.136695 + 0.990613i \(0.543648\pi\)
\(642\) 5.77241 0.227819
\(643\) −19.2276 −0.758262 −0.379131 0.925343i \(-0.623777\pi\)
−0.379131 + 0.925343i \(0.623777\pi\)
\(644\) 0 0
\(645\) −43.8898 −1.72816
\(646\) −20.9580 −0.824582
\(647\) 7.05216 0.277249 0.138625 0.990345i \(-0.455732\pi\)
0.138625 + 0.990345i \(0.455732\pi\)
\(648\) −1.00000 −0.0392837
\(649\) 28.8898 1.13403
\(650\) −6.68842 −0.262341
\(651\) 0 0
\(652\) 16.3637 0.640854
\(653\) −3.01875 −0.118133 −0.0590664 0.998254i \(-0.518812\pi\)
−0.0590664 + 0.998254i \(0.518812\pi\)
\(654\) 4.00000 0.156412
\(655\) 69.9869 2.73462
\(656\) 0.892750 0.0348560
\(657\) 11.7304 0.457647
\(658\) 0 0
\(659\) −12.3536 −0.481227 −0.240614 0.970621i \(-0.577349\pi\)
−0.240614 + 0.970621i \(0.577349\pi\)
\(660\) −16.1725 −0.629514
\(661\) 40.5884 1.57871 0.789353 0.613939i \(-0.210416\pi\)
0.789353 + 0.613939i \(0.210416\pi\)
\(662\) −0.130501 −0.00507207
\(663\) −7.79567 −0.302759
\(664\) −0.934749 −0.0362753
\(665\) 0 0
\(666\) 3.10725 0.120403
\(667\) 11.1912 0.433327
\(668\) 22.9869 0.889390
\(669\) 10.3217 0.399062
\(670\) −0.223081 −0.00861837
\(671\) −14.3899 −0.555517
\(672\) 0 0
\(673\) 28.5261 1.09960 0.549800 0.835296i \(-0.314704\pi\)
0.549800 + 0.835296i \(0.314704\pi\)
\(674\) −16.0522 −0.618306
\(675\) −6.68842 −0.257437
\(676\) 1.00000 0.0384615
\(677\) −18.1362 −0.697029 −0.348515 0.937303i \(-0.613314\pi\)
−0.348515 + 0.937303i \(0.613314\pi\)
\(678\) 10.6333 0.408371
\(679\) 0 0
\(680\) 26.6521 1.02206
\(681\) −4.44208 −0.170221
\(682\) −26.9086 −1.03038
\(683\) 11.1958 0.428394 0.214197 0.976791i \(-0.431287\pi\)
0.214197 + 0.976791i \(0.431287\pi\)
\(684\) 2.68842 0.102794
\(685\) 37.9738 1.45091
\(686\) 0 0
\(687\) 21.4608 0.818782
\(688\) −12.8377 −0.489431
\(689\) −9.21450 −0.351044
\(690\) 12.7537 0.485524
\(691\) −0.641914 −0.0244195 −0.0122098 0.999925i \(-0.503887\pi\)
−0.0122098 + 0.999925i \(0.503887\pi\)
\(692\) 10.7724 0.409506
\(693\) 0 0
\(694\) −21.3217 −0.809363
\(695\) 7.75933 0.294328
\(696\) −3.00000 −0.113715
\(697\) 6.95958 0.263613
\(698\) −6.35358 −0.240487
\(699\) 7.66517 0.289923
\(700\) 0 0
\(701\) 30.0334 1.13435 0.567173 0.823599i \(-0.308037\pi\)
0.567173 + 0.823599i \(0.308037\pi\)
\(702\) 1.00000 0.0377426
\(703\) −8.35358 −0.315061
\(704\) −4.73042 −0.178284
\(705\) −34.6986 −1.30682
\(706\) −30.4290 −1.14521
\(707\) 0 0
\(708\) 6.10725 0.229524
\(709\) 30.2434 1.13582 0.567908 0.823092i \(-0.307753\pi\)
0.567908 + 0.823092i \(0.307753\pi\)
\(710\) −43.3796 −1.62801
\(711\) 14.5261 0.544771
\(712\) −15.1072 −0.566168
\(713\) 21.2202 0.794701
\(714\) 0 0
\(715\) 16.1725 0.604817
\(716\) 19.0522 0.712013
\(717\) 9.52608 0.355758
\(718\) 18.0000 0.671754
\(719\) 42.5970 1.58860 0.794300 0.607526i \(-0.207838\pi\)
0.794300 + 0.607526i \(0.207838\pi\)
\(720\) −3.41883 −0.127412
\(721\) 0 0
\(722\) 11.7724 0.438124
\(723\) 7.36375 0.273861
\(724\) 13.7957 0.512712
\(725\) −20.0653 −0.745205
\(726\) 11.3768 0.422234
\(727\) 19.3637 0.718162 0.359081 0.933306i \(-0.383090\pi\)
0.359081 + 0.933306i \(0.383090\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 40.1043 1.48433
\(731\) −100.078 −3.70152
\(732\) −3.04200 −0.112436
\(733\) 26.0754 0.963117 0.481559 0.876414i \(-0.340071\pi\)
0.481559 + 0.876414i \(0.340071\pi\)
\(734\) −4.22759 −0.156043
\(735\) 0 0
\(736\) 3.73042 0.137505
\(737\) 0.308662 0.0113697
\(738\) −0.892750 −0.0328626
\(739\) −20.6435 −0.759383 −0.379692 0.925113i \(-0.623970\pi\)
−0.379692 + 0.925113i \(0.623970\pi\)
\(740\) 10.6232 0.390515
\(741\) −2.68842 −0.0987615
\(742\) 0 0
\(743\) −2.81892 −0.103416 −0.0517080 0.998662i \(-0.516467\pi\)
−0.0517080 + 0.998662i \(0.516467\pi\)
\(744\) −5.68842 −0.208547
\(745\) 28.2723 1.03582
\(746\) 35.6855 1.30654
\(747\) 0.934749 0.0342007
\(748\) −36.8767 −1.34835
\(749\) 0 0
\(750\) −5.77241 −0.210779
\(751\) −16.0709 −0.586436 −0.293218 0.956046i \(-0.594726\pi\)
−0.293218 + 0.956046i \(0.594726\pi\)
\(752\) −10.1492 −0.370105
\(753\) −12.7406 −0.464293
\(754\) 3.00000 0.109254
\(755\) 18.0158 0.655663
\(756\) 0 0
\(757\) 5.34050 0.194104 0.0970518 0.995279i \(-0.469059\pi\)
0.0970518 + 0.995279i \(0.469059\pi\)
\(758\) −9.67533 −0.351424
\(759\) −17.6464 −0.640524
\(760\) 9.19125 0.333402
\(761\) −36.3188 −1.31656 −0.658278 0.752775i \(-0.728715\pi\)
−0.658278 + 0.752775i \(0.728715\pi\)
\(762\) −6.17250 −0.223606
\(763\) 0 0
\(764\) 10.2985 0.372587
\(765\) −26.6521 −0.963608
\(766\) 3.67533 0.132795
\(767\) −6.10725 −0.220520
\(768\) −1.00000 −0.0360844
\(769\) 39.9783 1.44166 0.720828 0.693114i \(-0.243762\pi\)
0.720828 + 0.693114i \(0.243762\pi\)
\(770\) 0 0
\(771\) 13.4608 0.484780
\(772\) 17.6333 0.634637
\(773\) 18.5970 0.668887 0.334444 0.942416i \(-0.391452\pi\)
0.334444 + 0.942416i \(0.391452\pi\)
\(774\) 12.8377 0.461440
\(775\) −38.0465 −1.36667
\(776\) 4.79567 0.172154
\(777\) 0 0
\(778\) −6.47392 −0.232101
\(779\) 2.40009 0.0859920
\(780\) 3.41883 0.122414
\(781\) 60.0215 2.14774
\(782\) 29.0811 1.03994
\(783\) 3.00000 0.107211
\(784\) 0 0
\(785\) 38.4841 1.37356
\(786\) −20.4710 −0.730176
\(787\) 0.772415 0.0275336 0.0137668 0.999905i \(-0.495618\pi\)
0.0137668 + 0.999905i \(0.495618\pi\)
\(788\) −8.26958 −0.294592
\(789\) −24.0289 −0.855452
\(790\) 49.6622 1.76690
\(791\) 0 0
\(792\) 4.73042 0.168088
\(793\) 3.04200 0.108025
\(794\) 7.86092 0.278974
\(795\) −31.5028 −1.11729
\(796\) −5.02325 −0.178044
\(797\) 7.10275 0.251592 0.125796 0.992056i \(-0.459851\pi\)
0.125796 + 0.992056i \(0.459851\pi\)
\(798\) 0 0
\(799\) −79.1202 −2.79907
\(800\) −6.68842 −0.236471
\(801\) 15.1072 0.533788
\(802\) 5.64642 0.199382
\(803\) −55.4897 −1.95819
\(804\) 0.0652506 0.00230121
\(805\) 0 0
\(806\) 5.68842 0.200366
\(807\) −15.9217 −0.560469
\(808\) −0.623166 −0.0219229
\(809\) −20.6798 −0.727064 −0.363532 0.931582i \(-0.618429\pi\)
−0.363532 + 0.931582i \(0.618429\pi\)
\(810\) 3.41883 0.120126
\(811\) 36.2985 1.27461 0.637306 0.770611i \(-0.280049\pi\)
0.637306 + 0.770611i \(0.280049\pi\)
\(812\) 0 0
\(813\) 19.6521 0.689229
\(814\) −14.6986 −0.515185
\(815\) −55.9449 −1.95966
\(816\) −7.79567 −0.272903
\(817\) −34.5130 −1.20746
\(818\) −25.7173 −0.899185
\(819\) 0 0
\(820\) −3.05216 −0.106586
\(821\) −52.1928 −1.82154 −0.910771 0.412911i \(-0.864512\pi\)
−0.910771 + 0.412911i \(0.864512\pi\)
\(822\) −11.1072 −0.387410
\(823\) 39.1072 1.36319 0.681597 0.731728i \(-0.261286\pi\)
0.681597 + 0.731728i \(0.261286\pi\)
\(824\) −8.83767 −0.307875
\(825\) 31.6390 1.10153
\(826\) 0 0
\(827\) −51.6724 −1.79683 −0.898413 0.439152i \(-0.855279\pi\)
−0.898413 + 0.439152i \(0.855279\pi\)
\(828\) −3.73042 −0.129641
\(829\) −31.9840 −1.11085 −0.555425 0.831567i \(-0.687444\pi\)
−0.555425 + 0.831567i \(0.687444\pi\)
\(830\) 3.19575 0.110926
\(831\) 5.17250 0.179432
\(832\) 1.00000 0.0346688
\(833\) 0 0
\(834\) −2.26958 −0.0785893
\(835\) −78.5884 −2.71966
\(836\) −12.7173 −0.439838
\(837\) 5.68842 0.196620
\(838\) −8.83767 −0.305292
\(839\) −18.0187 −0.622076 −0.311038 0.950397i \(-0.600677\pi\)
−0.311038 + 0.950397i \(0.600677\pi\)
\(840\) 0 0
\(841\) −20.0000 −0.689655
\(842\) 10.1594 0.350117
\(843\) 24.4290 0.841379
\(844\) 14.5681 0.501454
\(845\) −3.41883 −0.117611
\(846\) 10.1492 0.348938
\(847\) 0 0
\(848\) −9.21450 −0.316427
\(849\) 3.19125 0.109523
\(850\) −52.1407 −1.78841
\(851\) 11.5913 0.397346
\(852\) 12.6884 0.434698
\(853\) 49.1651 1.68338 0.841690 0.539961i \(-0.181561\pi\)
0.841690 + 0.539961i \(0.181561\pi\)
\(854\) 0 0
\(855\) −9.19125 −0.314334
\(856\) −5.77241 −0.197297
\(857\) 37.6015 1.28444 0.642221 0.766519i \(-0.278013\pi\)
0.642221 + 0.766519i \(0.278013\pi\)
\(858\) −4.73042 −0.161494
\(859\) −5.78550 −0.197399 −0.0986994 0.995117i \(-0.531468\pi\)
−0.0986994 + 0.995117i \(0.531468\pi\)
\(860\) 43.8898 1.49663
\(861\) 0 0
\(862\) 11.6753 0.397663
\(863\) 18.9217 0.644101 0.322050 0.946723i \(-0.395628\pi\)
0.322050 + 0.946723i \(0.395628\pi\)
\(864\) 1.00000 0.0340207
\(865\) −36.8291 −1.25223
\(866\) −6.60442 −0.224427
\(867\) −43.7724 −1.48659
\(868\) 0 0
\(869\) −68.7144 −2.33098
\(870\) 10.2565 0.347728
\(871\) −0.0652506 −0.00221093
\(872\) −4.00000 −0.135457
\(873\) −4.79567 −0.162309
\(874\) 10.0289 0.339233
\(875\) 0 0
\(876\) −11.7304 −0.396334
\(877\) −30.3739 −1.02565 −0.512827 0.858492i \(-0.671402\pi\)
−0.512827 + 0.858492i \(0.671402\pi\)
\(878\) 25.0102 0.844052
\(879\) −2.58117 −0.0870606
\(880\) 16.1725 0.545175
\(881\) −42.0578 −1.41696 −0.708482 0.705729i \(-0.750620\pi\)
−0.708482 + 0.705729i \(0.750620\pi\)
\(882\) 0 0
\(883\) 8.84625 0.297700 0.148850 0.988860i \(-0.452443\pi\)
0.148850 + 0.988860i \(0.452443\pi\)
\(884\) 7.79567 0.262197
\(885\) −20.8797 −0.701862
\(886\) 16.8797 0.567083
\(887\) 44.5595 1.49616 0.748081 0.663608i \(-0.230976\pi\)
0.748081 + 0.663608i \(0.230976\pi\)
\(888\) −3.10725 −0.104272
\(889\) 0 0
\(890\) 51.6492 1.73128
\(891\) −4.73042 −0.158475
\(892\) −10.3217 −0.345598
\(893\) −27.2854 −0.913071
\(894\) −8.26958 −0.276576
\(895\) −65.1362 −2.17726
\(896\) 0 0
\(897\) 3.73042 0.124555
\(898\) 20.1680 0.673015
\(899\) 17.0653 0.569158
\(900\) 6.68842 0.222947
\(901\) −71.8332 −2.39311
\(902\) 4.22308 0.140613
\(903\) 0 0
\(904\) −10.6333 −0.353659
\(905\) −47.1651 −1.56782
\(906\) −5.26958 −0.175070
\(907\) −42.0289 −1.39555 −0.697774 0.716318i \(-0.745826\pi\)
−0.697774 + 0.716318i \(0.745826\pi\)
\(908\) 4.44208 0.147416
\(909\) 0.623166 0.0206691
\(910\) 0 0
\(911\) 46.0000 1.52405 0.762024 0.647549i \(-0.224206\pi\)
0.762024 + 0.647549i \(0.224206\pi\)
\(912\) −2.68842 −0.0890224
\(913\) −4.42175 −0.146339
\(914\) 16.2565 0.537717
\(915\) 10.4001 0.343816
\(916\) −21.4608 −0.709086
\(917\) 0 0
\(918\) 7.79567 0.257295
\(919\) 9.91600 0.327099 0.163549 0.986535i \(-0.447706\pi\)
0.163549 + 0.986535i \(0.447706\pi\)
\(920\) −12.7537 −0.420476
\(921\) −23.3116 −0.768143
\(922\) 16.0840 0.529698
\(923\) −12.6884 −0.417644
\(924\) 0 0
\(925\) −20.7826 −0.683327
\(926\) −1.67533 −0.0550548
\(927\) 8.83767 0.290267
\(928\) 3.00000 0.0984798
\(929\) −20.2985 −0.665972 −0.332986 0.942932i \(-0.608056\pi\)
−0.332986 + 0.942932i \(0.608056\pi\)
\(930\) 19.4477 0.637717
\(931\) 0 0
\(932\) −7.66517 −0.251081
\(933\) −10.7826 −0.353006
\(934\) −31.9449 −1.04527
\(935\) 126.075 4.12311
\(936\) −1.00000 −0.0326860
\(937\) −34.2667 −1.11944 −0.559722 0.828681i \(-0.689092\pi\)
−0.559722 + 0.828681i \(0.689092\pi\)
\(938\) 0 0
\(939\) −14.8508 −0.484636
\(940\) 34.6986 1.13174
\(941\) −25.1492 −0.819842 −0.409921 0.912121i \(-0.634444\pi\)
−0.409921 + 0.912121i \(0.634444\pi\)
\(942\) −11.2565 −0.366757
\(943\) −3.33033 −0.108450
\(944\) −6.10725 −0.198774
\(945\) 0 0
\(946\) −60.7275 −1.97442
\(947\) 27.2100 0.884206 0.442103 0.896964i \(-0.354233\pi\)
0.442103 + 0.896964i \(0.354233\pi\)
\(948\) −14.5261 −0.471785
\(949\) 11.7304 0.380785
\(950\) −17.9813 −0.583389
\(951\) −23.0653 −0.747942
\(952\) 0 0
\(953\) 54.5028 1.76552 0.882760 0.469824i \(-0.155683\pi\)
0.882760 + 0.469824i \(0.155683\pi\)
\(954\) 9.21450 0.298330
\(955\) −35.2088 −1.13933
\(956\) −9.52608 −0.308096
\(957\) −14.1912 −0.458738
\(958\) −37.6941 −1.21784
\(959\) 0 0
\(960\) 3.41883 0.110342
\(961\) 1.35809 0.0438092
\(962\) 3.10725 0.100182
\(963\) 5.77241 0.186013
\(964\) −7.36375 −0.237170
\(965\) −60.2854 −1.94066
\(966\) 0 0
\(967\) −13.9767 −0.449462 −0.224731 0.974421i \(-0.572150\pi\)
−0.224731 + 0.974421i \(0.572150\pi\)
\(968\) −11.3768 −0.365665
\(969\) −20.9580 −0.673268
\(970\) −16.3956 −0.526430
\(971\) −35.9783 −1.15460 −0.577300 0.816532i \(-0.695894\pi\)
−0.577300 + 0.816532i \(0.695894\pi\)
\(972\) −1.00000 −0.0320750
\(973\) 0 0
\(974\) 4.94491 0.158445
\(975\) −6.68842 −0.214201
\(976\) 3.04200 0.0973720
\(977\) 13.4608 0.430650 0.215325 0.976542i \(-0.430919\pi\)
0.215325 + 0.976542i \(0.430919\pi\)
\(978\) 16.3637 0.523255
\(979\) −71.4636 −2.28399
\(980\) 0 0
\(981\) 4.00000 0.127710
\(982\) −28.2014 −0.899943
\(983\) 2.06525 0.0658713 0.0329356 0.999457i \(-0.489514\pi\)
0.0329356 + 0.999457i \(0.489514\pi\)
\(984\) 0.892750 0.0284598
\(985\) 28.2723 0.900831
\(986\) 23.3870 0.744794
\(987\) 0 0
\(988\) 2.68842 0.0855299
\(989\) 47.8898 1.52281
\(990\) −16.1725 −0.513996
\(991\) −23.0187 −0.731215 −0.365607 0.930769i \(-0.619139\pi\)
−0.365607 + 0.930769i \(0.619139\pi\)
\(992\) 5.68842 0.180607
\(993\) −0.130501 −0.00414133
\(994\) 0 0
\(995\) 17.1737 0.544442
\(996\) −0.934749 −0.0296187
\(997\) −26.3927 −0.835864 −0.417932 0.908478i \(-0.637245\pi\)
−0.417932 + 0.908478i \(0.637245\pi\)
\(998\) −19.7593 −0.625471
\(999\) 3.10725 0.0983090
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 3822.2.a.bv.1.1 3
7.2 even 3 546.2.i.k.235.3 yes 6
7.4 even 3 546.2.i.k.79.3 6
7.6 odd 2 3822.2.a.bw.1.3 3
21.2 odd 6 1638.2.j.q.235.1 6
21.11 odd 6 1638.2.j.q.1171.1 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
546.2.i.k.79.3 6 7.4 even 3
546.2.i.k.235.3 yes 6 7.2 even 3
1638.2.j.q.235.1 6 21.2 odd 6
1638.2.j.q.1171.1 6 21.11 odd 6
3822.2.a.bv.1.1 3 1.1 even 1 trivial
3822.2.a.bw.1.3 3 7.6 odd 2