Properties

Label 5054.2.a.z.1.2
Level $5054$
Weight $2$
Character 5054.1
Self dual yes
Analytic conductor $40.356$
Analytic rank $1$
Dimension $6$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [5054,2,Mod(1,5054)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(5054, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("5054.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 5054 = 2 \cdot 7 \cdot 19^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5054.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: \(40.3563931816\)
Analytic rank: \(1\)
Dimension: \(6\)
Coefficient field: 6.6.36538000.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 13x^{4} - 4x^{3} + 41x^{2} + 16x - 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-2.14489\) of defining polynomial
Character \(\chi\) \(=\) 5054.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} -2.14489 q^{3} +1.00000 q^{4} +2.45305 q^{5} +2.14489 q^{6} +1.00000 q^{7} -1.00000 q^{8} +1.60057 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -2.14489 q^{3} +1.00000 q^{4} +2.45305 q^{5} +2.14489 q^{6} +1.00000 q^{7} -1.00000 q^{8} +1.60057 q^{9} -2.45305 q^{10} +4.66096 q^{11} -2.14489 q^{12} +1.96912 q^{13} -1.00000 q^{14} -5.26154 q^{15} +1.00000 q^{16} -1.67933 q^{17} -1.60057 q^{18} +2.45305 q^{20} -2.14489 q^{21} -4.66096 q^{22} +0.371174 q^{23} +2.14489 q^{24} +1.01746 q^{25} -1.96912 q^{26} +3.00162 q^{27} +1.00000 q^{28} -6.25075 q^{29} +5.26154 q^{30} -7.32289 q^{31} -1.00000 q^{32} -9.99728 q^{33} +1.67933 q^{34} +2.45305 q^{35} +1.60057 q^{36} +0.646287 q^{37} -4.22356 q^{39} -2.45305 q^{40} +1.10585 q^{41} +2.14489 q^{42} -9.92250 q^{43} +4.66096 q^{44} +3.92629 q^{45} -0.371174 q^{46} -12.6448 q^{47} -2.14489 q^{48} +1.00000 q^{49} -1.01746 q^{50} +3.60199 q^{51} +1.96912 q^{52} -5.55239 q^{53} -3.00162 q^{54} +11.4336 q^{55} -1.00000 q^{56} +6.25075 q^{58} -2.02042 q^{59} -5.26154 q^{60} -11.6961 q^{61} +7.32289 q^{62} +1.60057 q^{63} +1.00000 q^{64} +4.83035 q^{65} +9.99728 q^{66} +6.75247 q^{67} -1.67933 q^{68} -0.796130 q^{69} -2.45305 q^{70} -8.28979 q^{71} -1.60057 q^{72} -2.51209 q^{73} -0.646287 q^{74} -2.18235 q^{75} +4.66096 q^{77} +4.22356 q^{78} -9.58177 q^{79} +2.45305 q^{80} -11.2399 q^{81} -1.10585 q^{82} +9.73988 q^{83} -2.14489 q^{84} -4.11949 q^{85} +9.92250 q^{86} +13.4072 q^{87} -4.66096 q^{88} +1.96542 q^{89} -3.92629 q^{90} +1.96912 q^{91} +0.371174 q^{92} +15.7068 q^{93} +12.6448 q^{94} +2.14489 q^{96} -10.8945 q^{97} -1.00000 q^{98} +7.46021 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 6 q^{2} + 6 q^{4} - q^{5} + 6 q^{7} - 6 q^{8} + 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 6 q^{2} + 6 q^{4} - q^{5} + 6 q^{7} - 6 q^{8} + 8 q^{9} + q^{10} + 4 q^{11} - 15 q^{13} - 6 q^{14} - 6 q^{15} + 6 q^{16} - 9 q^{17} - 8 q^{18} - q^{20} - 4 q^{22} + 4 q^{23} + q^{25} + 15 q^{26} + 12 q^{27} + 6 q^{28} - 7 q^{29} + 6 q^{30} - 4 q^{31} - 6 q^{32} - 28 q^{33} + 9 q^{34} - q^{35} + 8 q^{36} - 3 q^{37} - 8 q^{39} + q^{40} - 11 q^{41} - 10 q^{43} + 4 q^{44} + 19 q^{45} - 4 q^{46} + 12 q^{47} + 6 q^{49} - q^{50} - 44 q^{51} - 15 q^{52} + 5 q^{53} - 12 q^{54} - 8 q^{55} - 6 q^{56} + 7 q^{58} - 4 q^{59} - 6 q^{60} - 21 q^{61} + 4 q^{62} + 8 q^{63} + 6 q^{64} + 20 q^{65} + 28 q^{66} - 14 q^{67} - 9 q^{68} + 24 q^{69} + q^{70} - 24 q^{71} - 8 q^{72} - 21 q^{73} + 3 q^{74} - 12 q^{75} + 4 q^{77} + 8 q^{78} - 58 q^{79} - q^{80} - 6 q^{81} + 11 q^{82} + 20 q^{83} - 4 q^{85} + 10 q^{86} - 8 q^{87} - 4 q^{88} - 7 q^{89} - 19 q^{90} - 15 q^{91} + 4 q^{92} - 22 q^{93} - 12 q^{94} - 7 q^{97} - 6 q^{98} - 26 q^{99}+O(q^{100}) \) 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
\(3\) −2.14489 −1.23836 −0.619178 0.785251i \(-0.712534\pi\)
−0.619178 + 0.785251i \(0.712534\pi\)
\(4\) 1.00000 0.500000
\(5\) 2.45305 1.09704 0.548519 0.836138i \(-0.315192\pi\)
0.548519 + 0.836138i \(0.315192\pi\)
\(6\) 2.14489 0.875650
\(7\) 1.00000 0.377964
\(8\) −1.00000 −0.353553
\(9\) 1.60057 0.533524
\(10\) −2.45305 −0.775723
\(11\) 4.66096 1.40533 0.702667 0.711519i \(-0.251993\pi\)
0.702667 + 0.711519i \(0.251993\pi\)
\(12\) −2.14489 −0.619178
\(13\) 1.96912 0.546136 0.273068 0.961995i \(-0.411962\pi\)
0.273068 + 0.961995i \(0.411962\pi\)
\(14\) −1.00000 −0.267261
\(15\) −5.26154 −1.35852
\(16\) 1.00000 0.250000
\(17\) −1.67933 −0.407298 −0.203649 0.979044i \(-0.565280\pi\)
−0.203649 + 0.979044i \(0.565280\pi\)
\(18\) −1.60057 −0.377259
\(19\) 0 0
\(20\) 2.45305 0.548519
\(21\) −2.14489 −0.468054
\(22\) −4.66096 −0.993721
\(23\) 0.371174 0.0773952 0.0386976 0.999251i \(-0.487679\pi\)
0.0386976 + 0.999251i \(0.487679\pi\)
\(24\) 2.14489 0.437825
\(25\) 1.01746 0.203492
\(26\) −1.96912 −0.386176
\(27\) 3.00162 0.577663
\(28\) 1.00000 0.188982
\(29\) −6.25075 −1.16073 −0.580367 0.814355i \(-0.697091\pi\)
−0.580367 + 0.814355i \(0.697091\pi\)
\(30\) 5.26154 0.960621
\(31\) −7.32289 −1.31523 −0.657615 0.753354i \(-0.728435\pi\)
−0.657615 + 0.753354i \(0.728435\pi\)
\(32\) −1.00000 −0.176777
\(33\) −9.99728 −1.74030
\(34\) 1.67933 0.288003
\(35\) 2.45305 0.414641
\(36\) 1.60057 0.266762
\(37\) 0.646287 0.106249 0.0531244 0.998588i \(-0.483082\pi\)
0.0531244 + 0.998588i \(0.483082\pi\)
\(38\) 0 0
\(39\) −4.22356 −0.676310
\(40\) −2.45305 −0.387861
\(41\) 1.10585 0.172705 0.0863524 0.996265i \(-0.472479\pi\)
0.0863524 + 0.996265i \(0.472479\pi\)
\(42\) 2.14489 0.330964
\(43\) −9.92250 −1.51317 −0.756584 0.653897i \(-0.773133\pi\)
−0.756584 + 0.653897i \(0.773133\pi\)
\(44\) 4.66096 0.702667
\(45\) 3.92629 0.585296
\(46\) −0.371174 −0.0547267
\(47\) −12.6448 −1.84444 −0.922218 0.386669i \(-0.873625\pi\)
−0.922218 + 0.386669i \(0.873625\pi\)
\(48\) −2.14489 −0.309589
\(49\) 1.00000 0.142857
\(50\) −1.01746 −0.143891
\(51\) 3.60199 0.504379
\(52\) 1.96912 0.273068
\(53\) −5.55239 −0.762679 −0.381340 0.924435i \(-0.624537\pi\)
−0.381340 + 0.924435i \(0.624537\pi\)
\(54\) −3.00162 −0.408469
\(55\) 11.4336 1.54170
\(56\) −1.00000 −0.133631
\(57\) 0 0
\(58\) 6.25075 0.820763
\(59\) −2.02042 −0.263037 −0.131518 0.991314i \(-0.541985\pi\)
−0.131518 + 0.991314i \(0.541985\pi\)
\(60\) −5.26154 −0.679261
\(61\) −11.6961 −1.49753 −0.748764 0.662837i \(-0.769352\pi\)
−0.748764 + 0.662837i \(0.769352\pi\)
\(62\) 7.32289 0.930008
\(63\) 1.60057 0.201653
\(64\) 1.00000 0.125000
\(65\) 4.83035 0.599132
\(66\) 9.99728 1.23058
\(67\) 6.75247 0.824946 0.412473 0.910970i \(-0.364665\pi\)
0.412473 + 0.910970i \(0.364665\pi\)
\(68\) −1.67933 −0.203649
\(69\) −0.796130 −0.0958428
\(70\) −2.45305 −0.293196
\(71\) −8.28979 −0.983817 −0.491908 0.870647i \(-0.663701\pi\)
−0.491908 + 0.870647i \(0.663701\pi\)
\(72\) −1.60057 −0.188629
\(73\) −2.51209 −0.294017 −0.147009 0.989135i \(-0.546965\pi\)
−0.147009 + 0.989135i \(0.546965\pi\)
\(74\) −0.646287 −0.0751293
\(75\) −2.18235 −0.251996
\(76\) 0 0
\(77\) 4.66096 0.531166
\(78\) 4.22356 0.478224
\(79\) −9.58177 −1.07803 −0.539017 0.842295i \(-0.681204\pi\)
−0.539017 + 0.842295i \(0.681204\pi\)
\(80\) 2.45305 0.274259
\(81\) −11.2399 −1.24888
\(82\) −1.10585 −0.122121
\(83\) 9.73988 1.06909 0.534545 0.845140i \(-0.320483\pi\)
0.534545 + 0.845140i \(0.320483\pi\)
\(84\) −2.14489 −0.234027
\(85\) −4.11949 −0.446821
\(86\) 9.92250 1.06997
\(87\) 13.4072 1.43740
\(88\) −4.66096 −0.496860
\(89\) 1.96542 0.208334 0.104167 0.994560i \(-0.466782\pi\)
0.104167 + 0.994560i \(0.466782\pi\)
\(90\) −3.92629 −0.413867
\(91\) 1.96912 0.206420
\(92\) 0.371174 0.0386976
\(93\) 15.7068 1.62872
\(94\) 12.6448 1.30421
\(95\) 0 0
\(96\) 2.14489 0.218912
\(97\) −10.8945 −1.10617 −0.553086 0.833124i \(-0.686550\pi\)
−0.553086 + 0.833124i \(0.686550\pi\)
\(98\) −1.00000 −0.101015
\(99\) 7.46021 0.749780
\(100\) 1.01746 0.101746
\(101\) −18.0979 −1.80081 −0.900406 0.435051i \(-0.856730\pi\)
−0.900406 + 0.435051i \(0.856730\pi\)
\(102\) −3.60199 −0.356650
\(103\) 19.3790 1.90947 0.954733 0.297465i \(-0.0961413\pi\)
0.954733 + 0.297465i \(0.0961413\pi\)
\(104\) −1.96912 −0.193088
\(105\) −5.26154 −0.513473
\(106\) 5.55239 0.539296
\(107\) −20.1089 −1.94400 −0.971998 0.234989i \(-0.924495\pi\)
−0.971998 + 0.234989i \(0.924495\pi\)
\(108\) 3.00162 0.288831
\(109\) −12.3454 −1.18247 −0.591237 0.806498i \(-0.701360\pi\)
−0.591237 + 0.806498i \(0.701360\pi\)
\(110\) −11.4336 −1.09015
\(111\) −1.38622 −0.131574
\(112\) 1.00000 0.0944911
\(113\) −5.92051 −0.556954 −0.278477 0.960443i \(-0.589830\pi\)
−0.278477 + 0.960443i \(0.589830\pi\)
\(114\) 0 0
\(115\) 0.910510 0.0849055
\(116\) −6.25075 −0.580367
\(117\) 3.15172 0.291377
\(118\) 2.02042 0.185995
\(119\) −1.67933 −0.153944
\(120\) 5.26154 0.480310
\(121\) 10.7246 0.974962
\(122\) 11.6961 1.05891
\(123\) −2.37193 −0.213870
\(124\) −7.32289 −0.657615
\(125\) −9.76937 −0.873799
\(126\) −1.60057 −0.142590
\(127\) 18.0378 1.60060 0.800299 0.599600i \(-0.204674\pi\)
0.800299 + 0.599600i \(0.204674\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 21.2827 1.87384
\(130\) −4.83035 −0.423650
\(131\) 2.03649 0.177929 0.0889643 0.996035i \(-0.471644\pi\)
0.0889643 + 0.996035i \(0.471644\pi\)
\(132\) −9.99728 −0.870151
\(133\) 0 0
\(134\) −6.75247 −0.583325
\(135\) 7.36314 0.633718
\(136\) 1.67933 0.144001
\(137\) 9.02736 0.771259 0.385630 0.922654i \(-0.373984\pi\)
0.385630 + 0.922654i \(0.373984\pi\)
\(138\) 0.796130 0.0677711
\(139\) 4.89160 0.414900 0.207450 0.978246i \(-0.433484\pi\)
0.207450 + 0.978246i \(0.433484\pi\)
\(140\) 2.45305 0.207321
\(141\) 27.1218 2.28407
\(142\) 8.28979 0.695664
\(143\) 9.17800 0.767503
\(144\) 1.60057 0.133381
\(145\) −15.3334 −1.27337
\(146\) 2.51209 0.207902
\(147\) −2.14489 −0.176908
\(148\) 0.646287 0.0531244
\(149\) −1.97345 −0.161671 −0.0808356 0.996727i \(-0.525759\pi\)
−0.0808356 + 0.996727i \(0.525759\pi\)
\(150\) 2.18235 0.178188
\(151\) 5.21877 0.424697 0.212349 0.977194i \(-0.431889\pi\)
0.212349 + 0.977194i \(0.431889\pi\)
\(152\) 0 0
\(153\) −2.68789 −0.217303
\(154\) −4.66096 −0.375591
\(155\) −17.9634 −1.44286
\(156\) −4.22356 −0.338155
\(157\) 6.50205 0.518920 0.259460 0.965754i \(-0.416455\pi\)
0.259460 + 0.965754i \(0.416455\pi\)
\(158\) 9.58177 0.762285
\(159\) 11.9093 0.944468
\(160\) −2.45305 −0.193931
\(161\) 0.371174 0.0292526
\(162\) 11.2399 0.883089
\(163\) −20.9999 −1.64484 −0.822418 0.568884i \(-0.807375\pi\)
−0.822418 + 0.568884i \(0.807375\pi\)
\(164\) 1.10585 0.0863524
\(165\) −24.5238 −1.90918
\(166\) −9.73988 −0.755961
\(167\) 20.1931 1.56259 0.781295 0.624162i \(-0.214560\pi\)
0.781295 + 0.624162i \(0.214560\pi\)
\(168\) 2.14489 0.165482
\(169\) −9.12256 −0.701736
\(170\) 4.11949 0.315950
\(171\) 0 0
\(172\) −9.92250 −0.756584
\(173\) 15.0110 1.14127 0.570634 0.821204i \(-0.306697\pi\)
0.570634 + 0.821204i \(0.306697\pi\)
\(174\) −13.4072 −1.01640
\(175\) 1.01746 0.0769128
\(176\) 4.66096 0.351333
\(177\) 4.33359 0.325733
\(178\) −1.96542 −0.147315
\(179\) 20.4827 1.53095 0.765476 0.643464i \(-0.222504\pi\)
0.765476 + 0.643464i \(0.222504\pi\)
\(180\) 3.92629 0.292648
\(181\) −9.69046 −0.720286 −0.360143 0.932897i \(-0.617272\pi\)
−0.360143 + 0.932897i \(0.617272\pi\)
\(182\) −1.96912 −0.145961
\(183\) 25.0868 1.85447
\(184\) −0.371174 −0.0273633
\(185\) 1.58537 0.116559
\(186\) −15.7068 −1.15168
\(187\) −7.82730 −0.572389
\(188\) −12.6448 −0.922218
\(189\) 3.00162 0.218336
\(190\) 0 0
\(191\) −7.42834 −0.537496 −0.268748 0.963211i \(-0.586610\pi\)
−0.268748 + 0.963211i \(0.586610\pi\)
\(192\) −2.14489 −0.154794
\(193\) 15.8766 1.14282 0.571411 0.820664i \(-0.306396\pi\)
0.571411 + 0.820664i \(0.306396\pi\)
\(194\) 10.8945 0.782182
\(195\) −10.3606 −0.741938
\(196\) 1.00000 0.0714286
\(197\) 19.4013 1.38229 0.691144 0.722717i \(-0.257107\pi\)
0.691144 + 0.722717i \(0.257107\pi\)
\(198\) −7.46021 −0.530174
\(199\) 16.8874 1.19711 0.598557 0.801080i \(-0.295741\pi\)
0.598557 + 0.801080i \(0.295741\pi\)
\(200\) −1.01746 −0.0719454
\(201\) −14.4833 −1.02158
\(202\) 18.0979 1.27337
\(203\) −6.25075 −0.438716
\(204\) 3.60199 0.252190
\(205\) 2.71271 0.189464
\(206\) −19.3790 −1.35020
\(207\) 0.594092 0.0412922
\(208\) 1.96912 0.136534
\(209\) 0 0
\(210\) 5.26154 0.363081
\(211\) −13.9533 −0.960588 −0.480294 0.877108i \(-0.659470\pi\)
−0.480294 + 0.877108i \(0.659470\pi\)
\(212\) −5.55239 −0.381340
\(213\) 17.7807 1.21832
\(214\) 20.1089 1.37461
\(215\) −24.3404 −1.66000
\(216\) −3.00162 −0.204235
\(217\) −7.32289 −0.497110
\(218\) 12.3454 0.836136
\(219\) 5.38816 0.364098
\(220\) 11.4336 0.770852
\(221\) −3.30681 −0.222440
\(222\) 1.38622 0.0930368
\(223\) 9.61825 0.644086 0.322043 0.946725i \(-0.395630\pi\)
0.322043 + 0.946725i \(0.395630\pi\)
\(224\) −1.00000 −0.0668153
\(225\) 1.62852 0.108568
\(226\) 5.92051 0.393826
\(227\) −17.5348 −1.16382 −0.581912 0.813251i \(-0.697695\pi\)
−0.581912 + 0.813251i \(0.697695\pi\)
\(228\) 0 0
\(229\) −3.06018 −0.202223 −0.101111 0.994875i \(-0.532240\pi\)
−0.101111 + 0.994875i \(0.532240\pi\)
\(230\) −0.910510 −0.0600373
\(231\) −9.99728 −0.657772
\(232\) 6.25075 0.410382
\(233\) 26.3535 1.72648 0.863238 0.504798i \(-0.168433\pi\)
0.863238 + 0.504798i \(0.168433\pi\)
\(234\) −3.15172 −0.206034
\(235\) −31.0184 −2.02342
\(236\) −2.02042 −0.131518
\(237\) 20.5519 1.33499
\(238\) 1.67933 0.108855
\(239\) 27.4395 1.77491 0.887456 0.460893i \(-0.152471\pi\)
0.887456 + 0.460893i \(0.152471\pi\)
\(240\) −5.26154 −0.339631
\(241\) −14.9775 −0.964786 −0.482393 0.875955i \(-0.660232\pi\)
−0.482393 + 0.875955i \(0.660232\pi\)
\(242\) −10.7246 −0.689402
\(243\) 15.1035 0.968890
\(244\) −11.6961 −0.748764
\(245\) 2.45305 0.156720
\(246\) 2.37193 0.151229
\(247\) 0 0
\(248\) 7.32289 0.465004
\(249\) −20.8910 −1.32391
\(250\) 9.76937 0.617869
\(251\) 14.1051 0.890307 0.445154 0.895454i \(-0.353149\pi\)
0.445154 + 0.895454i \(0.353149\pi\)
\(252\) 1.60057 0.100827
\(253\) 1.73003 0.108766
\(254\) −18.0378 −1.13179
\(255\) 8.83586 0.553323
\(256\) 1.00000 0.0625000
\(257\) −10.0917 −0.629505 −0.314752 0.949174i \(-0.601921\pi\)
−0.314752 + 0.949174i \(0.601921\pi\)
\(258\) −21.2827 −1.32500
\(259\) 0.646287 0.0401583
\(260\) 4.83035 0.299566
\(261\) −10.0048 −0.619280
\(262\) −2.03649 −0.125815
\(263\) −23.8534 −1.47086 −0.735432 0.677598i \(-0.763021\pi\)
−0.735432 + 0.677598i \(0.763021\pi\)
\(264\) 9.99728 0.615290
\(265\) −13.6203 −0.836688
\(266\) 0 0
\(267\) −4.21563 −0.257992
\(268\) 6.75247 0.412473
\(269\) −8.77050 −0.534747 −0.267374 0.963593i \(-0.586156\pi\)
−0.267374 + 0.963593i \(0.586156\pi\)
\(270\) −7.36314 −0.448106
\(271\) 6.08145 0.369422 0.184711 0.982793i \(-0.440865\pi\)
0.184711 + 0.982793i \(0.440865\pi\)
\(272\) −1.67933 −0.101824
\(273\) −4.22356 −0.255621
\(274\) −9.02736 −0.545363
\(275\) 4.74235 0.285974
\(276\) −0.796130 −0.0479214
\(277\) −25.0478 −1.50498 −0.752488 0.658606i \(-0.771147\pi\)
−0.752488 + 0.658606i \(0.771147\pi\)
\(278\) −4.89160 −0.293379
\(279\) −11.7208 −0.701707
\(280\) −2.45305 −0.146598
\(281\) −8.07617 −0.481784 −0.240892 0.970552i \(-0.577440\pi\)
−0.240892 + 0.970552i \(0.577440\pi\)
\(282\) −27.1218 −1.61508
\(283\) −9.35289 −0.555971 −0.277986 0.960585i \(-0.589667\pi\)
−0.277986 + 0.960585i \(0.589667\pi\)
\(284\) −8.28979 −0.491908
\(285\) 0 0
\(286\) −9.17800 −0.542706
\(287\) 1.10585 0.0652763
\(288\) −1.60057 −0.0943147
\(289\) −14.1798 −0.834109
\(290\) 15.3334 0.900408
\(291\) 23.3676 1.36983
\(292\) −2.51209 −0.147009
\(293\) −13.5630 −0.792360 −0.396180 0.918173i \(-0.629664\pi\)
−0.396180 + 0.918173i \(0.629664\pi\)
\(294\) 2.14489 0.125093
\(295\) −4.95620 −0.288561
\(296\) −0.646287 −0.0375646
\(297\) 13.9905 0.811809
\(298\) 1.97345 0.114319
\(299\) 0.730887 0.0422683
\(300\) −2.18235 −0.125998
\(301\) −9.92250 −0.571923
\(302\) −5.21877 −0.300306
\(303\) 38.8182 2.23004
\(304\) 0 0
\(305\) −28.6910 −1.64284
\(306\) 2.68789 0.153657
\(307\) −0.739222 −0.0421896 −0.0210948 0.999777i \(-0.506715\pi\)
−0.0210948 + 0.999777i \(0.506715\pi\)
\(308\) 4.66096 0.265583
\(309\) −41.5658 −2.36460
\(310\) 17.9634 1.02025
\(311\) 0.944861 0.0535782 0.0267891 0.999641i \(-0.491472\pi\)
0.0267891 + 0.999641i \(0.491472\pi\)
\(312\) 4.22356 0.239112
\(313\) −16.8338 −0.951504 −0.475752 0.879579i \(-0.657824\pi\)
−0.475752 + 0.879579i \(0.657824\pi\)
\(314\) −6.50205 −0.366932
\(315\) 3.92629 0.221221
\(316\) −9.58177 −0.539017
\(317\) 4.26863 0.239750 0.119875 0.992789i \(-0.461751\pi\)
0.119875 + 0.992789i \(0.461751\pi\)
\(318\) −11.9093 −0.667840
\(319\) −29.1345 −1.63122
\(320\) 2.45305 0.137130
\(321\) 43.1314 2.40736
\(322\) −0.371174 −0.0206847
\(323\) 0 0
\(324\) −11.2399 −0.624438
\(325\) 2.00350 0.111134
\(326\) 20.9999 1.16307
\(327\) 26.4796 1.46432
\(328\) −1.10585 −0.0610604
\(329\) −12.6448 −0.697132
\(330\) 24.5238 1.34999
\(331\) 25.9974 1.42895 0.714473 0.699663i \(-0.246667\pi\)
0.714473 + 0.699663i \(0.246667\pi\)
\(332\) 9.73988 0.534545
\(333\) 1.03443 0.0566863
\(334\) −20.1931 −1.10492
\(335\) 16.5642 0.904997
\(336\) −2.14489 −0.117014
\(337\) 11.6954 0.637087 0.318544 0.947908i \(-0.396806\pi\)
0.318544 + 0.947908i \(0.396806\pi\)
\(338\) 9.12256 0.496202
\(339\) 12.6989 0.689707
\(340\) −4.11949 −0.223410
\(341\) −34.1317 −1.84834
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) 9.92250 0.534985
\(345\) −1.95295 −0.105143
\(346\) −15.0110 −0.806999
\(347\) −19.4270 −1.04290 −0.521449 0.853282i \(-0.674608\pi\)
−0.521449 + 0.853282i \(0.674608\pi\)
\(348\) 13.4072 0.718701
\(349\) 6.79008 0.363465 0.181732 0.983348i \(-0.441830\pi\)
0.181732 + 0.983348i \(0.441830\pi\)
\(350\) −1.01746 −0.0543856
\(351\) 5.91056 0.315482
\(352\) −4.66096 −0.248430
\(353\) 2.61595 0.139233 0.0696164 0.997574i \(-0.477822\pi\)
0.0696164 + 0.997574i \(0.477822\pi\)
\(354\) −4.33359 −0.230328
\(355\) −20.3353 −1.07928
\(356\) 1.96542 0.104167
\(357\) 3.60199 0.190637
\(358\) −20.4827 −1.08255
\(359\) −19.7876 −1.04435 −0.522174 0.852839i \(-0.674879\pi\)
−0.522174 + 0.852839i \(0.674879\pi\)
\(360\) −3.92629 −0.206934
\(361\) 0 0
\(362\) 9.69046 0.509319
\(363\) −23.0031 −1.20735
\(364\) 1.96912 0.103210
\(365\) −6.16228 −0.322548
\(366\) −25.0868 −1.31131
\(367\) −29.0819 −1.51806 −0.759032 0.651054i \(-0.774327\pi\)
−0.759032 + 0.651054i \(0.774327\pi\)
\(368\) 0.371174 0.0193488
\(369\) 1.76999 0.0921422
\(370\) −1.58537 −0.0824197
\(371\) −5.55239 −0.288266
\(372\) 15.7068 0.814361
\(373\) −6.65147 −0.344400 −0.172200 0.985062i \(-0.555088\pi\)
−0.172200 + 0.985062i \(0.555088\pi\)
\(374\) 7.82730 0.404740
\(375\) 20.9543 1.08207
\(376\) 12.6448 0.652107
\(377\) −12.3085 −0.633918
\(378\) −3.00162 −0.154387
\(379\) 4.62364 0.237500 0.118750 0.992924i \(-0.462111\pi\)
0.118750 + 0.992924i \(0.462111\pi\)
\(380\) 0 0
\(381\) −38.6893 −1.98211
\(382\) 7.42834 0.380067
\(383\) 22.5475 1.15212 0.576061 0.817407i \(-0.304589\pi\)
0.576061 + 0.817407i \(0.304589\pi\)
\(384\) 2.14489 0.109456
\(385\) 11.4336 0.582709
\(386\) −15.8766 −0.808098
\(387\) −15.8817 −0.807311
\(388\) −10.8945 −0.553086
\(389\) −15.1594 −0.768614 −0.384307 0.923205i \(-0.625560\pi\)
−0.384307 + 0.923205i \(0.625560\pi\)
\(390\) 10.3606 0.524629
\(391\) −0.623325 −0.0315229
\(392\) −1.00000 −0.0505076
\(393\) −4.36805 −0.220339
\(394\) −19.4013 −0.977426
\(395\) −23.5046 −1.18264
\(396\) 7.46021 0.374890
\(397\) 38.5487 1.93470 0.967351 0.253440i \(-0.0815622\pi\)
0.967351 + 0.253440i \(0.0815622\pi\)
\(398\) −16.8874 −0.846487
\(399\) 0 0
\(400\) 1.01746 0.0508731
\(401\) −21.9495 −1.09611 −0.548054 0.836443i \(-0.684631\pi\)
−0.548054 + 0.836443i \(0.684631\pi\)
\(402\) 14.4833 0.722363
\(403\) −14.4197 −0.718294
\(404\) −18.0979 −0.900406
\(405\) −27.5720 −1.37006
\(406\) 6.25075 0.310219
\(407\) 3.01232 0.149315
\(408\) −3.60199 −0.178325
\(409\) −26.2379 −1.29738 −0.648691 0.761052i \(-0.724683\pi\)
−0.648691 + 0.761052i \(0.724683\pi\)
\(410\) −2.71271 −0.133971
\(411\) −19.3627 −0.955093
\(412\) 19.3790 0.954733
\(413\) −2.02042 −0.0994185
\(414\) −0.594092 −0.0291980
\(415\) 23.8924 1.17283
\(416\) −1.96912 −0.0965441
\(417\) −10.4920 −0.513794
\(418\) 0 0
\(419\) −14.7648 −0.721306 −0.360653 0.932700i \(-0.617446\pi\)
−0.360653 + 0.932700i \(0.617446\pi\)
\(420\) −5.26154 −0.256737
\(421\) −9.35190 −0.455784 −0.227892 0.973686i \(-0.573183\pi\)
−0.227892 + 0.973686i \(0.573183\pi\)
\(422\) 13.9533 0.679238
\(423\) −20.2390 −0.984052
\(424\) 5.55239 0.269648
\(425\) −1.70865 −0.0828819
\(426\) −17.7807 −0.861479
\(427\) −11.6961 −0.566012
\(428\) −20.1089 −0.971998
\(429\) −19.6858 −0.950441
\(430\) 24.3404 1.17380
\(431\) −23.8387 −1.14827 −0.574134 0.818761i \(-0.694661\pi\)
−0.574134 + 0.818761i \(0.694661\pi\)
\(432\) 3.00162 0.144416
\(433\) 38.4316 1.84690 0.923452 0.383714i \(-0.125355\pi\)
0.923452 + 0.383714i \(0.125355\pi\)
\(434\) 7.32289 0.351510
\(435\) 32.8885 1.57688
\(436\) −12.3454 −0.591237
\(437\) 0 0
\(438\) −5.38816 −0.257456
\(439\) 8.78467 0.419269 0.209635 0.977780i \(-0.432773\pi\)
0.209635 + 0.977780i \(0.432773\pi\)
\(440\) −11.4336 −0.545075
\(441\) 1.60057 0.0762178
\(442\) 3.30681 0.157289
\(443\) 10.6376 0.505409 0.252704 0.967544i \(-0.418680\pi\)
0.252704 + 0.967544i \(0.418680\pi\)
\(444\) −1.38622 −0.0657869
\(445\) 4.82128 0.228551
\(446\) −9.61825 −0.455437
\(447\) 4.23284 0.200207
\(448\) 1.00000 0.0472456
\(449\) 9.25065 0.436565 0.218282 0.975886i \(-0.429955\pi\)
0.218282 + 0.975886i \(0.429955\pi\)
\(450\) −1.62852 −0.0767692
\(451\) 5.15433 0.242708
\(452\) −5.92051 −0.278477
\(453\) −11.1937 −0.525926
\(454\) 17.5348 0.822948
\(455\) 4.83035 0.226450
\(456\) 0 0
\(457\) −14.2859 −0.668265 −0.334132 0.942526i \(-0.608443\pi\)
−0.334132 + 0.942526i \(0.608443\pi\)
\(458\) 3.06018 0.142993
\(459\) −5.04072 −0.235281
\(460\) 0.910510 0.0424528
\(461\) −3.02462 −0.140871 −0.0704354 0.997516i \(-0.522439\pi\)
−0.0704354 + 0.997516i \(0.522439\pi\)
\(462\) 9.99728 0.465115
\(463\) 4.98327 0.231592 0.115796 0.993273i \(-0.463058\pi\)
0.115796 + 0.993273i \(0.463058\pi\)
\(464\) −6.25075 −0.290184
\(465\) 38.5297 1.78677
\(466\) −26.3535 −1.22080
\(467\) 18.1672 0.840679 0.420340 0.907367i \(-0.361911\pi\)
0.420340 + 0.907367i \(0.361911\pi\)
\(468\) 3.15172 0.145688
\(469\) 6.75247 0.311800
\(470\) 31.0184 1.43077
\(471\) −13.9462 −0.642607
\(472\) 2.02042 0.0929975
\(473\) −46.2484 −2.12650
\(474\) −20.5519 −0.943980
\(475\) 0 0
\(476\) −1.67933 −0.0769720
\(477\) −8.88700 −0.406908
\(478\) −27.4395 −1.25505
\(479\) 26.4885 1.21029 0.605144 0.796116i \(-0.293115\pi\)
0.605144 + 0.796116i \(0.293115\pi\)
\(480\) 5.26154 0.240155
\(481\) 1.27262 0.0580263
\(482\) 14.9775 0.682207
\(483\) −0.796130 −0.0362252
\(484\) 10.7246 0.487481
\(485\) −26.7248 −1.21351
\(486\) −15.1035 −0.685109
\(487\) −34.4343 −1.56037 −0.780184 0.625550i \(-0.784874\pi\)
−0.780184 + 0.625550i \(0.784874\pi\)
\(488\) 11.6961 0.529456
\(489\) 45.0425 2.03689
\(490\) −2.45305 −0.110818
\(491\) 38.9864 1.75943 0.879715 0.475502i \(-0.157733\pi\)
0.879715 + 0.475502i \(0.157733\pi\)
\(492\) −2.37193 −0.106935
\(493\) 10.4971 0.472764
\(494\) 0 0
\(495\) 18.3003 0.822537
\(496\) −7.32289 −0.328808
\(497\) −8.28979 −0.371848
\(498\) 20.8910 0.936149
\(499\) 20.0894 0.899325 0.449662 0.893199i \(-0.351544\pi\)
0.449662 + 0.893199i \(0.351544\pi\)
\(500\) −9.76937 −0.436900
\(501\) −43.3121 −1.93504
\(502\) −14.1051 −0.629542
\(503\) 20.7208 0.923895 0.461948 0.886907i \(-0.347151\pi\)
0.461948 + 0.886907i \(0.347151\pi\)
\(504\) −1.60057 −0.0712952
\(505\) −44.3952 −1.97556
\(506\) −1.73003 −0.0769093
\(507\) 19.5669 0.868998
\(508\) 18.0378 0.800299
\(509\) 12.0371 0.533535 0.266767 0.963761i \(-0.414045\pi\)
0.266767 + 0.963761i \(0.414045\pi\)
\(510\) −8.83586 −0.391259
\(511\) −2.51209 −0.111128
\(512\) −1.00000 −0.0441942
\(513\) 0 0
\(514\) 10.0917 0.445127
\(515\) 47.5376 2.09476
\(516\) 21.2827 0.936919
\(517\) −58.9371 −2.59205
\(518\) −0.646287 −0.0283962
\(519\) −32.1971 −1.41330
\(520\) −4.83035 −0.211825
\(521\) −12.7325 −0.557822 −0.278911 0.960317i \(-0.589973\pi\)
−0.278911 + 0.960317i \(0.589973\pi\)
\(522\) 10.0048 0.437897
\(523\) 17.3287 0.757731 0.378865 0.925452i \(-0.376314\pi\)
0.378865 + 0.925452i \(0.376314\pi\)
\(524\) 2.03649 0.0889643
\(525\) −2.18235 −0.0952454
\(526\) 23.8534 1.04006
\(527\) 12.2976 0.535690
\(528\) −9.99728 −0.435076
\(529\) −22.8622 −0.994010
\(530\) 13.6203 0.591628
\(531\) −3.23383 −0.140336
\(532\) 0 0
\(533\) 2.17755 0.0943203
\(534\) 4.21563 0.182428
\(535\) −49.3280 −2.13264
\(536\) −6.75247 −0.291662
\(537\) −43.9333 −1.89586
\(538\) 8.77050 0.378123
\(539\) 4.66096 0.200762
\(540\) 7.36314 0.316859
\(541\) −22.4788 −0.966438 −0.483219 0.875500i \(-0.660533\pi\)
−0.483219 + 0.875500i \(0.660533\pi\)
\(542\) −6.08145 −0.261221
\(543\) 20.7850 0.891970
\(544\) 1.67933 0.0720007
\(545\) −30.2839 −1.29722
\(546\) 4.22356 0.180752
\(547\) −28.7786 −1.23048 −0.615242 0.788338i \(-0.710942\pi\)
−0.615242 + 0.788338i \(0.710942\pi\)
\(548\) 9.02736 0.385630
\(549\) −18.7204 −0.798967
\(550\) −4.74235 −0.202214
\(551\) 0 0
\(552\) 0.796130 0.0338856
\(553\) −9.58177 −0.407458
\(554\) 25.0478 1.06418
\(555\) −3.40046 −0.144342
\(556\) 4.89160 0.207450
\(557\) 19.4872 0.825698 0.412849 0.910800i \(-0.364534\pi\)
0.412849 + 0.910800i \(0.364534\pi\)
\(558\) 11.7208 0.496182
\(559\) −19.5386 −0.826395
\(560\) 2.45305 0.103660
\(561\) 16.7887 0.708821
\(562\) 8.07617 0.340673
\(563\) 30.7775 1.29711 0.648557 0.761166i \(-0.275373\pi\)
0.648557 + 0.761166i \(0.275373\pi\)
\(564\) 27.1218 1.14203
\(565\) −14.5233 −0.611000
\(566\) 9.35289 0.393131
\(567\) −11.2399 −0.472031
\(568\) 8.28979 0.347832
\(569\) 5.30686 0.222475 0.111237 0.993794i \(-0.464519\pi\)
0.111237 + 0.993794i \(0.464519\pi\)
\(570\) 0 0
\(571\) −33.0139 −1.38159 −0.690794 0.723052i \(-0.742739\pi\)
−0.690794 + 0.723052i \(0.742739\pi\)
\(572\) 9.17800 0.383751
\(573\) 15.9330 0.665611
\(574\) −1.10585 −0.0461573
\(575\) 0.377656 0.0157493
\(576\) 1.60057 0.0666905
\(577\) −32.6599 −1.35965 −0.679824 0.733375i \(-0.737944\pi\)
−0.679824 + 0.733375i \(0.737944\pi\)
\(578\) 14.1798 0.589804
\(579\) −34.0536 −1.41522
\(580\) −15.3334 −0.636685
\(581\) 9.73988 0.404078
\(582\) −23.3676 −0.968619
\(583\) −25.8795 −1.07182
\(584\) 2.51209 0.103951
\(585\) 7.73133 0.319651
\(586\) 13.5630 0.560283
\(587\) 0.00708371 0.000292376 0 0.000146188 1.00000i \(-0.499953\pi\)
0.000146188 1.00000i \(0.499953\pi\)
\(588\) −2.14489 −0.0884540
\(589\) 0 0
\(590\) 4.95620 0.204044
\(591\) −41.6138 −1.71176
\(592\) 0.646287 0.0265622
\(593\) −19.6830 −0.808285 −0.404143 0.914696i \(-0.632430\pi\)
−0.404143 + 0.914696i \(0.632430\pi\)
\(594\) −13.9905 −0.574035
\(595\) −4.11949 −0.168882
\(596\) −1.97345 −0.0808356
\(597\) −36.2216 −1.48245
\(598\) −0.730887 −0.0298882
\(599\) −44.7592 −1.82881 −0.914405 0.404801i \(-0.867341\pi\)
−0.914405 + 0.404801i \(0.867341\pi\)
\(600\) 2.18235 0.0890939
\(601\) −22.6915 −0.925604 −0.462802 0.886462i \(-0.653156\pi\)
−0.462802 + 0.886462i \(0.653156\pi\)
\(602\) 9.92250 0.404411
\(603\) 10.8078 0.440129
\(604\) 5.21877 0.212349
\(605\) 26.3080 1.06957
\(606\) −38.8182 −1.57688
\(607\) 1.23495 0.0501250 0.0250625 0.999686i \(-0.492022\pi\)
0.0250625 + 0.999686i \(0.492022\pi\)
\(608\) 0 0
\(609\) 13.4072 0.543287
\(610\) 28.6910 1.16167
\(611\) −24.8992 −1.00731
\(612\) −2.68789 −0.108652
\(613\) 32.0569 1.29477 0.647383 0.762165i \(-0.275864\pi\)
0.647383 + 0.762165i \(0.275864\pi\)
\(614\) 0.739222 0.0298326
\(615\) −5.81847 −0.234623
\(616\) −4.66096 −0.187796
\(617\) −23.9413 −0.963839 −0.481919 0.876216i \(-0.660060\pi\)
−0.481919 + 0.876216i \(0.660060\pi\)
\(618\) 41.5658 1.67202
\(619\) −23.3629 −0.939033 −0.469516 0.882924i \(-0.655572\pi\)
−0.469516 + 0.882924i \(0.655572\pi\)
\(620\) −17.9634 −0.721429
\(621\) 1.11413 0.0447083
\(622\) −0.944861 −0.0378855
\(623\) 1.96542 0.0787430
\(624\) −4.22356 −0.169078
\(625\) −29.0521 −1.16208
\(626\) 16.8338 0.672815
\(627\) 0 0
\(628\) 6.50205 0.259460
\(629\) −1.08533 −0.0432749
\(630\) −3.92629 −0.156427
\(631\) 32.4789 1.29296 0.646482 0.762929i \(-0.276240\pi\)
0.646482 + 0.762929i \(0.276240\pi\)
\(632\) 9.58177 0.381142
\(633\) 29.9285 1.18955
\(634\) −4.26863 −0.169529
\(635\) 44.2477 1.75592
\(636\) 11.9093 0.472234
\(637\) 1.96912 0.0780194
\(638\) 29.1345 1.15345
\(639\) −13.2684 −0.524890
\(640\) −2.45305 −0.0969654
\(641\) −7.57108 −0.299040 −0.149520 0.988759i \(-0.547773\pi\)
−0.149520 + 0.988759i \(0.547773\pi\)
\(642\) −43.1314 −1.70226
\(643\) 10.3612 0.408605 0.204303 0.978908i \(-0.434507\pi\)
0.204303 + 0.978908i \(0.434507\pi\)
\(644\) 0.371174 0.0146263
\(645\) 52.2076 2.05567
\(646\) 0 0
\(647\) 34.0826 1.33993 0.669963 0.742395i \(-0.266310\pi\)
0.669963 + 0.742395i \(0.266310\pi\)
\(648\) 11.2399 0.441544
\(649\) −9.41712 −0.369654
\(650\) −2.00350 −0.0785839
\(651\) 15.7068 0.615599
\(652\) −20.9999 −0.822418
\(653\) −14.6817 −0.574537 −0.287269 0.957850i \(-0.592747\pi\)
−0.287269 + 0.957850i \(0.592747\pi\)
\(654\) −26.4796 −1.03543
\(655\) 4.99560 0.195194
\(656\) 1.10585 0.0431762
\(657\) −4.02078 −0.156865
\(658\) 12.6448 0.492946
\(659\) 26.3608 1.02687 0.513435 0.858128i \(-0.328373\pi\)
0.513435 + 0.858128i \(0.328373\pi\)
\(660\) −24.5238 −0.954589
\(661\) 36.6051 1.42378 0.711888 0.702294i \(-0.247841\pi\)
0.711888 + 0.702294i \(0.247841\pi\)
\(662\) −25.9974 −1.01042
\(663\) 7.09275 0.275460
\(664\) −9.73988 −0.377981
\(665\) 0 0
\(666\) −1.03443 −0.0400833
\(667\) −2.32012 −0.0898353
\(668\) 20.1931 0.781295
\(669\) −20.6301 −0.797607
\(670\) −16.5642 −0.639929
\(671\) −54.5149 −2.10453
\(672\) 2.14489 0.0827411
\(673\) −28.9120 −1.11448 −0.557238 0.830353i \(-0.688139\pi\)
−0.557238 + 0.830353i \(0.688139\pi\)
\(674\) −11.6954 −0.450489
\(675\) 3.05404 0.117550
\(676\) −9.12256 −0.350868
\(677\) −26.1788 −1.00613 −0.503067 0.864248i \(-0.667795\pi\)
−0.503067 + 0.864248i \(0.667795\pi\)
\(678\) −12.6989 −0.487697
\(679\) −10.8945 −0.418094
\(680\) 4.11949 0.157975
\(681\) 37.6103 1.44123
\(682\) 34.1317 1.30697
\(683\) 31.7764 1.21589 0.607946 0.793979i \(-0.291994\pi\)
0.607946 + 0.793979i \(0.291994\pi\)
\(684\) 0 0
\(685\) 22.1446 0.846101
\(686\) −1.00000 −0.0381802
\(687\) 6.56377 0.250424
\(688\) −9.92250 −0.378292
\(689\) −10.9333 −0.416526
\(690\) 1.95295 0.0743475
\(691\) −23.4349 −0.891504 −0.445752 0.895156i \(-0.647064\pi\)
−0.445752 + 0.895156i \(0.647064\pi\)
\(692\) 15.0110 0.570634
\(693\) 7.46021 0.283390
\(694\) 19.4270 0.737441
\(695\) 11.9994 0.455161
\(696\) −13.4072 −0.508198
\(697\) −1.85709 −0.0703423
\(698\) −6.79008 −0.257008
\(699\) −56.5255 −2.13799
\(700\) 1.01746 0.0384564
\(701\) 13.9672 0.527535 0.263767 0.964586i \(-0.415035\pi\)
0.263767 + 0.964586i \(0.415035\pi\)
\(702\) −5.91056 −0.223080
\(703\) 0 0
\(704\) 4.66096 0.175667
\(705\) 66.5312 2.50571
\(706\) −2.61595 −0.0984525
\(707\) −18.0979 −0.680643
\(708\) 4.33359 0.162866
\(709\) 15.1140 0.567619 0.283810 0.958881i \(-0.408402\pi\)
0.283810 + 0.958881i \(0.408402\pi\)
\(710\) 20.3353 0.763169
\(711\) −15.3363 −0.575157
\(712\) −1.96542 −0.0736573
\(713\) −2.71807 −0.101793
\(714\) −3.60199 −0.134801
\(715\) 22.5141 0.841980
\(716\) 20.4827 0.765476
\(717\) −58.8548 −2.19797
\(718\) 19.7876 0.738466
\(719\) 9.56757 0.356810 0.178405 0.983957i \(-0.442906\pi\)
0.178405 + 0.983957i \(0.442906\pi\)
\(720\) 3.92629 0.146324
\(721\) 19.3790 0.721710
\(722\) 0 0
\(723\) 32.1252 1.19475
\(724\) −9.69046 −0.360143
\(725\) −6.35989 −0.236200
\(726\) 23.0031 0.853725
\(727\) 45.2628 1.67870 0.839351 0.543589i \(-0.182935\pi\)
0.839351 + 0.543589i \(0.182935\pi\)
\(728\) −1.96912 −0.0729805
\(729\) 1.32424 0.0490460
\(730\) 6.16228 0.228076
\(731\) 16.6632 0.616309
\(732\) 25.0868 0.927236
\(733\) 15.6509 0.578079 0.289039 0.957317i \(-0.406664\pi\)
0.289039 + 0.957317i \(0.406664\pi\)
\(734\) 29.0819 1.07343
\(735\) −5.26154 −0.194075
\(736\) −0.371174 −0.0136817
\(737\) 31.4730 1.15932
\(738\) −1.76999 −0.0651544
\(739\) −29.8506 −1.09807 −0.549035 0.835799i \(-0.685005\pi\)
−0.549035 + 0.835799i \(0.685005\pi\)
\(740\) 1.58537 0.0582795
\(741\) 0 0
\(742\) 5.55239 0.203835
\(743\) −19.4553 −0.713746 −0.356873 0.934153i \(-0.616157\pi\)
−0.356873 + 0.934153i \(0.616157\pi\)
\(744\) −15.7068 −0.575841
\(745\) −4.84097 −0.177360
\(746\) 6.65147 0.243528
\(747\) 15.5894 0.570386
\(748\) −7.82730 −0.286195
\(749\) −20.1089 −0.734761
\(750\) −20.9543 −0.765142
\(751\) −10.4124 −0.379953 −0.189977 0.981789i \(-0.560841\pi\)
−0.189977 + 0.981789i \(0.560841\pi\)
\(752\) −12.6448 −0.461109
\(753\) −30.2540 −1.10252
\(754\) 12.3085 0.448248
\(755\) 12.8019 0.465909
\(756\) 3.00162 0.109168
\(757\) −13.6379 −0.495678 −0.247839 0.968801i \(-0.579720\pi\)
−0.247839 + 0.968801i \(0.579720\pi\)
\(758\) −4.62364 −0.167938
\(759\) −3.71073 −0.134691
\(760\) 0 0
\(761\) 3.29787 0.119548 0.0597738 0.998212i \(-0.480962\pi\)
0.0597738 + 0.998212i \(0.480962\pi\)
\(762\) 38.6893 1.40156
\(763\) −12.3454 −0.446934
\(764\) −7.42834 −0.268748
\(765\) −6.59354 −0.238390
\(766\) −22.5475 −0.814673
\(767\) −3.97846 −0.143654
\(768\) −2.14489 −0.0773972
\(769\) 2.62924 0.0948129 0.0474064 0.998876i \(-0.484904\pi\)
0.0474064 + 0.998876i \(0.484904\pi\)
\(770\) −11.4336 −0.412038
\(771\) 21.6457 0.779550
\(772\) 15.8766 0.571411
\(773\) −8.76330 −0.315194 −0.157597 0.987504i \(-0.550375\pi\)
−0.157597 + 0.987504i \(0.550375\pi\)
\(774\) 15.8817 0.570855
\(775\) −7.45076 −0.267639
\(776\) 10.8945 0.391091
\(777\) −1.38622 −0.0497302
\(778\) 15.1594 0.543492
\(779\) 0 0
\(780\) −10.3606 −0.370969
\(781\) −38.6384 −1.38259
\(782\) 0.623325 0.0222901
\(783\) −18.7624 −0.670513
\(784\) 1.00000 0.0357143
\(785\) 15.9499 0.569275
\(786\) 4.36805 0.155803
\(787\) −29.4525 −1.04987 −0.524934 0.851143i \(-0.675910\pi\)
−0.524934 + 0.851143i \(0.675910\pi\)
\(788\) 19.4013 0.691144
\(789\) 51.1631 1.82145
\(790\) 23.5046 0.836256
\(791\) −5.92051 −0.210509
\(792\) −7.46021 −0.265087
\(793\) −23.0310 −0.817853
\(794\) −38.5487 −1.36804
\(795\) 29.2141 1.03612
\(796\) 16.8874 0.598557
\(797\) 31.9299 1.13102 0.565508 0.824743i \(-0.308680\pi\)
0.565508 + 0.824743i \(0.308680\pi\)
\(798\) 0 0
\(799\) 21.2348 0.751235
\(800\) −1.01746 −0.0359727
\(801\) 3.14580 0.111151
\(802\) 21.9495 0.775065
\(803\) −11.7087 −0.413193
\(804\) −14.4833 −0.510788
\(805\) 0.910510 0.0320913
\(806\) 14.4197 0.507911
\(807\) 18.8118 0.662207
\(808\) 18.0979 0.636683
\(809\) −10.5294 −0.370193 −0.185097 0.982720i \(-0.559260\pi\)
−0.185097 + 0.982720i \(0.559260\pi\)
\(810\) 27.5720 0.968782
\(811\) −17.2804 −0.606795 −0.303398 0.952864i \(-0.598121\pi\)
−0.303398 + 0.952864i \(0.598121\pi\)
\(812\) −6.25075 −0.219358
\(813\) −13.0441 −0.457476
\(814\) −3.01232 −0.105582
\(815\) −51.5137 −1.80445
\(816\) 3.60199 0.126095
\(817\) 0 0
\(818\) 26.2379 0.917388
\(819\) 3.15172 0.110130
\(820\) 2.71271 0.0947319
\(821\) 55.1719 1.92551 0.962757 0.270368i \(-0.0871454\pi\)
0.962757 + 0.270368i \(0.0871454\pi\)
\(822\) 19.3627 0.675353
\(823\) 22.7550 0.793190 0.396595 0.917994i \(-0.370192\pi\)
0.396595 + 0.917994i \(0.370192\pi\)
\(824\) −19.3790 −0.675098
\(825\) −10.1718 −0.354138
\(826\) 2.02042 0.0702995
\(827\) −11.7599 −0.408931 −0.204465 0.978874i \(-0.565546\pi\)
−0.204465 + 0.978874i \(0.565546\pi\)
\(828\) 0.594092 0.0206461
\(829\) 36.5744 1.27028 0.635141 0.772397i \(-0.280942\pi\)
0.635141 + 0.772397i \(0.280942\pi\)
\(830\) −23.8924 −0.829318
\(831\) 53.7249 1.86370
\(832\) 1.96912 0.0682670
\(833\) −1.67933 −0.0581854
\(834\) 10.4920 0.363307
\(835\) 49.5347 1.71422
\(836\) 0 0
\(837\) −21.9806 −0.759760
\(838\) 14.7648 0.510040
\(839\) 21.4530 0.740640 0.370320 0.928904i \(-0.379248\pi\)
0.370320 + 0.928904i \(0.379248\pi\)
\(840\) 5.26154 0.181540
\(841\) 10.0718 0.347304
\(842\) 9.35190 0.322288
\(843\) 17.3225 0.596620
\(844\) −13.9533 −0.480294
\(845\) −22.3781 −0.769831
\(846\) 20.2390 0.695830
\(847\) 10.7246 0.368501
\(848\) −5.55239 −0.190670
\(849\) 20.0610 0.688490
\(850\) 1.70865 0.0586064
\(851\) 0.239885 0.00822315
\(852\) 17.7807 0.609158
\(853\) −8.29792 −0.284115 −0.142058 0.989858i \(-0.545372\pi\)
−0.142058 + 0.989858i \(0.545372\pi\)
\(854\) 11.6961 0.400231
\(855\) 0 0
\(856\) 20.1089 0.687306
\(857\) 17.1840 0.586995 0.293498 0.955960i \(-0.405181\pi\)
0.293498 + 0.955960i \(0.405181\pi\)
\(858\) 19.6858 0.672064
\(859\) 27.7559 0.947019 0.473510 0.880789i \(-0.342987\pi\)
0.473510 + 0.880789i \(0.342987\pi\)
\(860\) −24.3404 −0.830001
\(861\) −2.37193 −0.0808353
\(862\) 23.8387 0.811948
\(863\) 28.5248 0.970995 0.485497 0.874238i \(-0.338639\pi\)
0.485497 + 0.874238i \(0.338639\pi\)
\(864\) −3.00162 −0.102117
\(865\) 36.8229 1.25202
\(866\) −38.4316 −1.30596
\(867\) 30.4143 1.03292
\(868\) −7.32289 −0.248555
\(869\) −44.6603 −1.51500
\(870\) −32.8885 −1.11503
\(871\) 13.2964 0.450532
\(872\) 12.3454 0.418068
\(873\) −17.4375 −0.590170
\(874\) 0 0
\(875\) −9.76937 −0.330265
\(876\) 5.38816 0.182049
\(877\) 26.2838 0.887542 0.443771 0.896140i \(-0.353640\pi\)
0.443771 + 0.896140i \(0.353640\pi\)
\(878\) −8.78467 −0.296468
\(879\) 29.0913 0.981224
\(880\) 11.4336 0.385426
\(881\) 31.7390 1.06931 0.534657 0.845069i \(-0.320441\pi\)
0.534657 + 0.845069i \(0.320441\pi\)
\(882\) −1.60057 −0.0538941
\(883\) −37.9601 −1.27746 −0.638729 0.769432i \(-0.720540\pi\)
−0.638729 + 0.769432i \(0.720540\pi\)
\(884\) −3.30681 −0.111220
\(885\) 10.6305 0.357341
\(886\) −10.6376 −0.357378
\(887\) 53.7619 1.80515 0.902574 0.430535i \(-0.141675\pi\)
0.902574 + 0.430535i \(0.141675\pi\)
\(888\) 1.38622 0.0465184
\(889\) 18.0378 0.604970
\(890\) −4.82128 −0.161610
\(891\) −52.3887 −1.75509
\(892\) 9.61825 0.322043
\(893\) 0 0
\(894\) −4.23284 −0.141567
\(895\) 50.2452 1.67951
\(896\) −1.00000 −0.0334077
\(897\) −1.56768 −0.0523432
\(898\) −9.25065 −0.308698
\(899\) 45.7735 1.52663
\(900\) 1.62852 0.0542840
\(901\) 9.32430 0.310637
\(902\) −5.15433 −0.171620
\(903\) 21.2827 0.708245
\(904\) 5.92051 0.196913
\(905\) −23.7712 −0.790181
\(906\) 11.1937 0.371886
\(907\) −26.3458 −0.874798 −0.437399 0.899268i \(-0.644100\pi\)
−0.437399 + 0.899268i \(0.644100\pi\)
\(908\) −17.5348 −0.581912
\(909\) −28.9671 −0.960777
\(910\) −4.83035 −0.160125
\(911\) −11.1591 −0.369719 −0.184859 0.982765i \(-0.559183\pi\)
−0.184859 + 0.982765i \(0.559183\pi\)
\(912\) 0 0
\(913\) 45.3972 1.50243
\(914\) 14.2859 0.472535
\(915\) 61.5393 2.03443
\(916\) −3.06018 −0.101111
\(917\) 2.03649 0.0672507
\(918\) 5.04072 0.166369
\(919\) −4.71602 −0.155567 −0.0777835 0.996970i \(-0.524784\pi\)
−0.0777835 + 0.996970i \(0.524784\pi\)
\(920\) −0.910510 −0.0300186
\(921\) 1.58555 0.0522457
\(922\) 3.02462 0.0996107
\(923\) −16.3236 −0.537298
\(924\) −9.99728 −0.328886
\(925\) 0.657571 0.0216208
\(926\) −4.98327 −0.163761
\(927\) 31.0174 1.01875
\(928\) 6.25075 0.205191
\(929\) −51.0261 −1.67411 −0.837056 0.547117i \(-0.815725\pi\)
−0.837056 + 0.547117i \(0.815725\pi\)
\(930\) −38.5297 −1.26344
\(931\) 0 0
\(932\) 26.3535 0.863238
\(933\) −2.02663 −0.0663488
\(934\) −18.1672 −0.594450
\(935\) −19.2008 −0.627932
\(936\) −3.15172 −0.103017
\(937\) −51.8709 −1.69455 −0.847274 0.531156i \(-0.821758\pi\)
−0.847274 + 0.531156i \(0.821758\pi\)
\(938\) −6.75247 −0.220476
\(939\) 36.1068 1.17830
\(940\) −31.0184 −1.01171
\(941\) 1.79295 0.0584486 0.0292243 0.999573i \(-0.490696\pi\)
0.0292243 + 0.999573i \(0.490696\pi\)
\(942\) 13.9462 0.454392
\(943\) 0.410464 0.0133665
\(944\) −2.02042 −0.0657592
\(945\) 7.36314 0.239523
\(946\) 46.2484 1.50367
\(947\) 6.46733 0.210160 0.105080 0.994464i \(-0.466490\pi\)
0.105080 + 0.994464i \(0.466490\pi\)
\(948\) 20.5519 0.667495
\(949\) −4.94660 −0.160573
\(950\) 0 0
\(951\) −9.15576 −0.296896
\(952\) 1.67933 0.0544274
\(953\) 30.6939 0.994274 0.497137 0.867672i \(-0.334385\pi\)
0.497137 + 0.867672i \(0.334385\pi\)
\(954\) 8.88700 0.287727
\(955\) −18.2221 −0.589653
\(956\) 27.4395 0.887456
\(957\) 62.4904 2.02003
\(958\) −26.4885 −0.855803
\(959\) 9.02736 0.291509
\(960\) −5.26154 −0.169815
\(961\) 22.6248 0.729831
\(962\) −1.27262 −0.0410308
\(963\) −32.1857 −1.03717
\(964\) −14.9775 −0.482393
\(965\) 38.9461 1.25372
\(966\) 0.796130 0.0256151
\(967\) 17.7358 0.570345 0.285172 0.958476i \(-0.407949\pi\)
0.285172 + 0.958476i \(0.407949\pi\)
\(968\) −10.7246 −0.344701
\(969\) 0 0
\(970\) 26.7248 0.858083
\(971\) 33.8991 1.08788 0.543938 0.839126i \(-0.316933\pi\)
0.543938 + 0.839126i \(0.316933\pi\)
\(972\) 15.1035 0.484445
\(973\) 4.89160 0.156818
\(974\) 34.4343 1.10335
\(975\) −4.29730 −0.137624
\(976\) −11.6961 −0.374382
\(977\) −31.7370 −1.01536 −0.507678 0.861547i \(-0.669496\pi\)
−0.507678 + 0.861547i \(0.669496\pi\)
\(978\) −45.0425 −1.44030
\(979\) 9.16077 0.292779
\(980\) 2.45305 0.0783599
\(981\) −19.7597 −0.630879
\(982\) −38.9864 −1.24410
\(983\) 38.9896 1.24358 0.621788 0.783186i \(-0.286407\pi\)
0.621788 + 0.783186i \(0.286407\pi\)
\(984\) 2.37193 0.0756145
\(985\) 47.5925 1.51642
\(986\) −10.4971 −0.334295
\(987\) 27.1218 0.863297
\(988\) 0 0
\(989\) −3.68298 −0.117112
\(990\) −18.3003 −0.581621
\(991\) 3.37997 0.107368 0.0536841 0.998558i \(-0.482904\pi\)
0.0536841 + 0.998558i \(0.482904\pi\)
\(992\) 7.32289 0.232502
\(993\) −55.7616 −1.76954
\(994\) 8.28979 0.262936
\(995\) 41.4256 1.31328
\(996\) −20.8910 −0.661957
\(997\) −38.2557 −1.21157 −0.605785 0.795628i \(-0.707141\pi\)
−0.605785 + 0.795628i \(0.707141\pi\)
\(998\) −20.0894 −0.635919
\(999\) 1.93991 0.0613760
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 5054.2.a.z.1.2 6
19.18 odd 2 5054.2.a.be.1.5 yes 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
5054.2.a.z.1.2 6 1.1 even 1 trivial
5054.2.a.be.1.5 yes 6 19.18 odd 2