Properties

Label 3850.2.a.bx.1.1
Level $3850$
Weight $2$
Character 3850.1
Self dual yes
Analytic conductor $30.742$
Analytic rank $1$
Dimension $4$
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] = [3850,2,Mod(1,3850)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3850, 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("3850.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3850 = 2 \cdot 5^{2} \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3850.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: \(30.7424047782\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: 4.4.10304.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 2x^{3} - 7x^{2} + 8x + 8 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 770)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-2.16053\) of defining polynomial
Character \(\chi\) \(=\) 3850.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} -1.41421 q^{3} +1.00000 q^{4} +1.41421 q^{6} +1.00000 q^{7} -1.00000 q^{8} -1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -1.41421 q^{3} +1.00000 q^{4} +1.41421 q^{6} +1.00000 q^{7} -1.00000 q^{8} -1.00000 q^{9} -1.00000 q^{11} -1.41421 q^{12} +1.56282 q^{13} -1.00000 q^{14} +1.00000 q^{16} +0.210157 q^{17} +1.00000 q^{18} -6.24264 q^{19} -1.41421 q^{21} +1.00000 q^{22} +8.79073 q^{23} +1.41421 q^{24} -1.56282 q^{26} +5.65685 q^{27} +1.00000 q^{28} -9.96230 q^{29} +6.93933 q^{31} -1.00000 q^{32} +1.41421 q^{33} -0.210157 q^{34} -1.00000 q^{36} +0.358761 q^{37} +6.24264 q^{38} -2.21016 q^{39} -4.76687 q^{41} +1.41421 q^{42} +5.44670 q^{43} -1.00000 q^{44} -8.79073 q^{46} -3.65685 q^{47} -1.41421 q^{48} +1.00000 q^{49} -0.297207 q^{51} +1.56282 q^{52} -2.76687 q^{53} -5.65685 q^{54} -1.00000 q^{56} +8.82843 q^{57} +9.96230 q^{58} -2.11612 q^{59} +6.71231 q^{61} -6.93933 q^{62} -1.00000 q^{63} +1.00000 q^{64} -1.41421 q^{66} -11.9779 q^{67} +0.210157 q^{68} -12.4320 q^{69} +10.2751 q^{71} +1.00000 q^{72} -6.02386 q^{73} -0.358761 q^{74} -6.24264 q^{76} -1.00000 q^{77} +2.21016 q^{78} -9.59530 q^{79} -5.00000 q^{81} +4.76687 q^{82} +1.56282 q^{83} -1.41421 q^{84} -5.44670 q^{86} +14.0888 q^{87} +1.00000 q^{88} +8.59619 q^{89} +1.56282 q^{91} +8.79073 q^{92} -9.81370 q^{93} +3.65685 q^{94} +1.41421 q^{96} +14.1725 q^{97} -1.00000 q^{98} +1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{2} + 4 q^{4} + 4 q^{7} - 4 q^{8} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{2} + 4 q^{4} + 4 q^{7} - 4 q^{8} - 4 q^{9} - 4 q^{11} + 4 q^{13} - 4 q^{14} + 4 q^{16} + 4 q^{17} + 4 q^{18} - 8 q^{19} + 4 q^{22} - 4 q^{23} - 4 q^{26} + 4 q^{28} - 12 q^{29} - 8 q^{31} - 4 q^{32} - 4 q^{34} - 4 q^{36} + 8 q^{37} + 8 q^{38} - 12 q^{39} - 8 q^{41} - 4 q^{43} - 4 q^{44} + 4 q^{46} + 8 q^{47} + 4 q^{49} - 8 q^{51} + 4 q^{52} - 4 q^{56} + 24 q^{57} + 12 q^{58} - 32 q^{59} - 8 q^{61} + 8 q^{62} - 4 q^{63} + 4 q^{64} - 4 q^{67} + 4 q^{68} - 4 q^{69} + 4 q^{71} + 4 q^{72} + 4 q^{73} - 8 q^{74} - 8 q^{76} - 4 q^{77} + 12 q^{78} - 16 q^{79} - 20 q^{81} + 8 q^{82} + 4 q^{83} + 4 q^{86} - 12 q^{87} + 4 q^{88} - 24 q^{89} + 4 q^{91} - 4 q^{92} - 8 q^{93} - 8 q^{94} + 32 q^{97} - 4 q^{98} + 4 q^{99}+O(q^{100}) \) 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
\(3\) −1.41421 −0.816497 −0.408248 0.912871i \(-0.633860\pi\)
−0.408248 + 0.912871i \(0.633860\pi\)
\(4\) 1.00000 0.500000
\(5\) 0 0
\(6\) 1.41421 0.577350
\(7\) 1.00000 0.377964
\(8\) −1.00000 −0.353553
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) −1.00000 −0.301511
\(12\) −1.41421 −0.408248
\(13\) 1.56282 0.433447 0.216724 0.976233i \(-0.430463\pi\)
0.216724 + 0.976233i \(0.430463\pi\)
\(14\) −1.00000 −0.267261
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 0.210157 0.0509706 0.0254853 0.999675i \(-0.491887\pi\)
0.0254853 + 0.999675i \(0.491887\pi\)
\(18\) 1.00000 0.235702
\(19\) −6.24264 −1.43216 −0.716080 0.698018i \(-0.754065\pi\)
−0.716080 + 0.698018i \(0.754065\pi\)
\(20\) 0 0
\(21\) −1.41421 −0.308607
\(22\) 1.00000 0.213201
\(23\) 8.79073 1.83299 0.916497 0.400042i \(-0.131004\pi\)
0.916497 + 0.400042i \(0.131004\pi\)
\(24\) 1.41421 0.288675
\(25\) 0 0
\(26\) −1.56282 −0.306494
\(27\) 5.65685 1.08866
\(28\) 1.00000 0.188982
\(29\) −9.96230 −1.84995 −0.924977 0.380024i \(-0.875916\pi\)
−0.924977 + 0.380024i \(0.875916\pi\)
\(30\) 0 0
\(31\) 6.93933 1.24634 0.623170 0.782086i \(-0.285844\pi\)
0.623170 + 0.782086i \(0.285844\pi\)
\(32\) −1.00000 −0.176777
\(33\) 1.41421 0.246183
\(34\) −0.210157 −0.0360417
\(35\) 0 0
\(36\) −1.00000 −0.166667
\(37\) 0.358761 0.0589799 0.0294900 0.999565i \(-0.490612\pi\)
0.0294900 + 0.999565i \(0.490612\pi\)
\(38\) 6.24264 1.01269
\(39\) −2.21016 −0.353908
\(40\) 0 0
\(41\) −4.76687 −0.744461 −0.372230 0.928140i \(-0.621407\pi\)
−0.372230 + 0.928140i \(0.621407\pi\)
\(42\) 1.41421 0.218218
\(43\) 5.44670 0.830614 0.415307 0.909681i \(-0.363674\pi\)
0.415307 + 0.909681i \(0.363674\pi\)
\(44\) −1.00000 −0.150756
\(45\) 0 0
\(46\) −8.79073 −1.29612
\(47\) −3.65685 −0.533407 −0.266704 0.963779i \(-0.585934\pi\)
−0.266704 + 0.963779i \(0.585934\pi\)
\(48\) −1.41421 −0.204124
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) −0.297207 −0.0416173
\(52\) 1.56282 0.216724
\(53\) −2.76687 −0.380059 −0.190030 0.981778i \(-0.560858\pi\)
−0.190030 + 0.981778i \(0.560858\pi\)
\(54\) −5.65685 −0.769800
\(55\) 0 0
\(56\) −1.00000 −0.133631
\(57\) 8.82843 1.16935
\(58\) 9.96230 1.30811
\(59\) −2.11612 −0.275495 −0.137748 0.990467i \(-0.543986\pi\)
−0.137748 + 0.990467i \(0.543986\pi\)
\(60\) 0 0
\(61\) 6.71231 0.859423 0.429711 0.902966i \(-0.358615\pi\)
0.429711 + 0.902966i \(0.358615\pi\)
\(62\) −6.93933 −0.881296
\(63\) −1.00000 −0.125988
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) −1.41421 −0.174078
\(67\) −11.9779 −1.46334 −0.731668 0.681661i \(-0.761258\pi\)
−0.731668 + 0.681661i \(0.761258\pi\)
\(68\) 0.210157 0.0254853
\(69\) −12.4320 −1.49663
\(70\) 0 0
\(71\) 10.2751 1.21943 0.609716 0.792620i \(-0.291283\pi\)
0.609716 + 0.792620i \(0.291283\pi\)
\(72\) 1.00000 0.117851
\(73\) −6.02386 −0.705039 −0.352519 0.935804i \(-0.614675\pi\)
−0.352519 + 0.935804i \(0.614675\pi\)
\(74\) −0.358761 −0.0417051
\(75\) 0 0
\(76\) −6.24264 −0.716080
\(77\) −1.00000 −0.113961
\(78\) 2.21016 0.250251
\(79\) −9.59530 −1.07956 −0.539778 0.841808i \(-0.681492\pi\)
−0.539778 + 0.841808i \(0.681492\pi\)
\(80\) 0 0
\(81\) −5.00000 −0.555556
\(82\) 4.76687 0.526413
\(83\) 1.56282 0.171541 0.0857707 0.996315i \(-0.472665\pi\)
0.0857707 + 0.996315i \(0.472665\pi\)
\(84\) −1.41421 −0.154303
\(85\) 0 0
\(86\) −5.44670 −0.587332
\(87\) 14.0888 1.51048
\(88\) 1.00000 0.106600
\(89\) 8.59619 0.911194 0.455597 0.890186i \(-0.349426\pi\)
0.455597 + 0.890186i \(0.349426\pi\)
\(90\) 0 0
\(91\) 1.56282 0.163828
\(92\) 8.79073 0.916497
\(93\) −9.81370 −1.01763
\(94\) 3.65685 0.377176
\(95\) 0 0
\(96\) 1.41421 0.144338
\(97\) 14.1725 1.43900 0.719498 0.694495i \(-0.244372\pi\)
0.719498 + 0.694495i \(0.244372\pi\)
\(98\) −1.00000 −0.101015
\(99\) 1.00000 0.100504
\(100\) 0 0
\(101\) −9.92982 −0.988054 −0.494027 0.869447i \(-0.664476\pi\)
−0.494027 + 0.869447i \(0.664476\pi\)
\(102\) 0.297207 0.0294279
\(103\) −10.9393 −1.07788 −0.538942 0.842343i \(-0.681176\pi\)
−0.538942 + 0.842343i \(0.681176\pi\)
\(104\) −1.56282 −0.153247
\(105\) 0 0
\(106\) 2.76687 0.268743
\(107\) −12.8523 −1.24248 −0.621239 0.783622i \(-0.713370\pi\)
−0.621239 + 0.783622i \(0.713370\pi\)
\(108\) 5.65685 0.544331
\(109\) 14.4476 1.38383 0.691914 0.721980i \(-0.256768\pi\)
0.691914 + 0.721980i \(0.256768\pi\)
\(110\) 0 0
\(111\) −0.507364 −0.0481569
\(112\) 1.00000 0.0944911
\(113\) 5.55760 0.522815 0.261408 0.965228i \(-0.415813\pi\)
0.261408 + 0.965228i \(0.415813\pi\)
\(114\) −8.82843 −0.826858
\(115\) 0 0
\(116\) −9.96230 −0.924977
\(117\) −1.56282 −0.144482
\(118\) 2.11612 0.194805
\(119\) 0.210157 0.0192651
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) −6.71231 −0.607704
\(123\) 6.74138 0.607850
\(124\) 6.93933 0.623170
\(125\) 0 0
\(126\) 1.00000 0.0890871
\(127\) −1.07232 −0.0951531 −0.0475766 0.998868i \(-0.515150\pi\)
−0.0475766 + 0.998868i \(0.515150\pi\)
\(128\) −1.00000 −0.0883883
\(129\) −7.70279 −0.678193
\(130\) 0 0
\(131\) −21.8700 −1.91079 −0.955397 0.295324i \(-0.904572\pi\)
−0.955397 + 0.295324i \(0.904572\pi\)
\(132\) 1.41421 0.123091
\(133\) −6.24264 −0.541306
\(134\) 11.9779 1.03473
\(135\) 0 0
\(136\) −0.210157 −0.0180208
\(137\) 8.22181 0.702437 0.351218 0.936294i \(-0.385767\pi\)
0.351218 + 0.936294i \(0.385767\pi\)
\(138\) 12.4320 1.05828
\(139\) 9.71319 0.823862 0.411931 0.911215i \(-0.364854\pi\)
0.411931 + 0.911215i \(0.364854\pi\)
\(140\) 0 0
\(141\) 5.17157 0.435525
\(142\) −10.2751 −0.862269
\(143\) −1.56282 −0.130689
\(144\) −1.00000 −0.0833333
\(145\) 0 0
\(146\) 6.02386 0.498538
\(147\) −1.41421 −0.116642
\(148\) 0.358761 0.0294900
\(149\) −5.02297 −0.411498 −0.205749 0.978605i \(-0.565963\pi\)
−0.205749 + 0.978605i \(0.565963\pi\)
\(150\) 0 0
\(151\) 2.56892 0.209056 0.104528 0.994522i \(-0.466667\pi\)
0.104528 + 0.994522i \(0.466667\pi\)
\(152\) 6.24264 0.506345
\(153\) −0.210157 −0.0169902
\(154\) 1.00000 0.0805823
\(155\) 0 0
\(156\) −2.21016 −0.176954
\(157\) 16.8692 1.34630 0.673152 0.739504i \(-0.264940\pi\)
0.673152 + 0.739504i \(0.264940\pi\)
\(158\) 9.59530 0.763361
\(159\) 3.91295 0.310317
\(160\) 0 0
\(161\) 8.79073 0.692807
\(162\) 5.00000 0.392837
\(163\) −12.5723 −0.984741 −0.492370 0.870386i \(-0.663869\pi\)
−0.492370 + 0.870386i \(0.663869\pi\)
\(164\) −4.76687 −0.372230
\(165\) 0 0
\(166\) −1.56282 −0.121298
\(167\) 13.0962 1.01341 0.506706 0.862119i \(-0.330863\pi\)
0.506706 + 0.862119i \(0.330863\pi\)
\(168\) 1.41421 0.109109
\(169\) −10.5576 −0.812123
\(170\) 0 0
\(171\) 6.24264 0.477387
\(172\) 5.44670 0.415307
\(173\) 0.461038 0.0350521 0.0175260 0.999846i \(-0.494421\pi\)
0.0175260 + 0.999846i \(0.494421\pi\)
\(174\) −14.0888 −1.06807
\(175\) 0 0
\(176\) −1.00000 −0.0753778
\(177\) 2.99265 0.224941
\(178\) −8.59619 −0.644311
\(179\) −4.69544 −0.350954 −0.175477 0.984484i \(-0.556147\pi\)
−0.175477 + 0.984484i \(0.556147\pi\)
\(180\) 0 0
\(181\) −15.4537 −1.14866 −0.574332 0.818623i \(-0.694738\pi\)
−0.574332 + 0.818623i \(0.694738\pi\)
\(182\) −1.56282 −0.115844
\(183\) −9.49264 −0.701716
\(184\) −8.79073 −0.648061
\(185\) 0 0
\(186\) 9.81370 0.719575
\(187\) −0.210157 −0.0153682
\(188\) −3.65685 −0.266704
\(189\) 5.65685 0.411476
\(190\) 0 0
\(191\) −14.6954 −1.06332 −0.531662 0.846956i \(-0.678432\pi\)
−0.531662 + 0.846956i \(0.678432\pi\)
\(192\) −1.41421 −0.102062
\(193\) 18.2751 1.31547 0.657736 0.753248i \(-0.271514\pi\)
0.657736 + 0.753248i \(0.271514\pi\)
\(194\) −14.1725 −1.01752
\(195\) 0 0
\(196\) 1.00000 0.0714286
\(197\) −19.8787 −1.41630 −0.708148 0.706064i \(-0.750469\pi\)
−0.708148 + 0.706064i \(0.750469\pi\)
\(198\) −1.00000 −0.0710669
\(199\) −22.7941 −1.61583 −0.807917 0.589296i \(-0.799405\pi\)
−0.807917 + 0.589296i \(0.799405\pi\)
\(200\) 0 0
\(201\) 16.9393 1.19481
\(202\) 9.92982 0.698660
\(203\) −9.96230 −0.699217
\(204\) −0.297207 −0.0208087
\(205\) 0 0
\(206\) 10.9393 0.762179
\(207\) −8.79073 −0.610998
\(208\) 1.56282 0.108362
\(209\) 6.24264 0.431812
\(210\) 0 0
\(211\) −13.8449 −0.953124 −0.476562 0.879141i \(-0.658117\pi\)
−0.476562 + 0.879141i \(0.658117\pi\)
\(212\) −2.76687 −0.190030
\(213\) −14.5312 −0.995663
\(214\) 12.8523 0.878564
\(215\) 0 0
\(216\) −5.65685 −0.384900
\(217\) 6.93933 0.471073
\(218\) −14.4476 −0.978514
\(219\) 8.51902 0.575662
\(220\) 0 0
\(221\) 0.328437 0.0220931
\(222\) 0.507364 0.0340521
\(223\) −11.1274 −0.745146 −0.372573 0.928003i \(-0.621524\pi\)
−0.372573 + 0.928003i \(0.621524\pi\)
\(224\) −1.00000 −0.0668153
\(225\) 0 0
\(226\) −5.55760 −0.369686
\(227\) −17.0021 −1.12847 −0.564236 0.825614i \(-0.690829\pi\)
−0.564236 + 0.825614i \(0.690829\pi\)
\(228\) 8.82843 0.584677
\(229\) 14.7895 0.977316 0.488658 0.872475i \(-0.337487\pi\)
0.488658 + 0.872475i \(0.337487\pi\)
\(230\) 0 0
\(231\) 1.41421 0.0930484
\(232\) 9.96230 0.654057
\(233\) −22.0888 −1.44709 −0.723543 0.690279i \(-0.757488\pi\)
−0.723543 + 0.690279i \(0.757488\pi\)
\(234\) 1.56282 0.102165
\(235\) 0 0
\(236\) −2.11612 −0.137748
\(237\) 13.5698 0.881454
\(238\) −0.210157 −0.0136225
\(239\) −12.3489 −0.798783 −0.399391 0.916781i \(-0.630779\pi\)
−0.399391 + 0.916781i \(0.630779\pi\)
\(240\) 0 0
\(241\) 11.0659 0.712814 0.356407 0.934331i \(-0.384002\pi\)
0.356407 + 0.934331i \(0.384002\pi\)
\(242\) −1.00000 −0.0642824
\(243\) −9.89949 −0.635053
\(244\) 6.71231 0.429711
\(245\) 0 0
\(246\) −6.74138 −0.429815
\(247\) −9.75611 −0.620766
\(248\) −6.93933 −0.440648
\(249\) −2.21016 −0.140063
\(250\) 0 0
\(251\) 21.9488 1.38540 0.692699 0.721226i \(-0.256421\pi\)
0.692699 + 0.721226i \(0.256421\pi\)
\(252\) −1.00000 −0.0629941
\(253\) −8.79073 −0.552668
\(254\) 1.07232 0.0672834
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 3.93845 0.245674 0.122837 0.992427i \(-0.460801\pi\)
0.122837 + 0.992427i \(0.460801\pi\)
\(258\) 7.70279 0.479555
\(259\) 0.358761 0.0222923
\(260\) 0 0
\(261\) 9.96230 0.616651
\(262\) 21.8700 1.35114
\(263\) 1.77996 0.109757 0.0548786 0.998493i \(-0.482523\pi\)
0.0548786 + 0.998493i \(0.482523\pi\)
\(264\) −1.41421 −0.0870388
\(265\) 0 0
\(266\) 6.24264 0.382761
\(267\) −12.1568 −0.743987
\(268\) −11.9779 −0.731668
\(269\) −19.0555 −1.16183 −0.580916 0.813964i \(-0.697305\pi\)
−0.580916 + 0.813964i \(0.697305\pi\)
\(270\) 0 0
\(271\) −12.5190 −0.760476 −0.380238 0.924889i \(-0.624158\pi\)
−0.380238 + 0.924889i \(0.624158\pi\)
\(272\) 0.210157 0.0127427
\(273\) −2.21016 −0.133765
\(274\) −8.22181 −0.496698
\(275\) 0 0
\(276\) −12.4320 −0.748317
\(277\) 2.75126 0.165307 0.0826535 0.996578i \(-0.473661\pi\)
0.0826535 + 0.996578i \(0.473661\pi\)
\(278\) −9.71319 −0.582559
\(279\) −6.93933 −0.415447
\(280\) 0 0
\(281\) 9.75050 0.581666 0.290833 0.956774i \(-0.406068\pi\)
0.290833 + 0.956774i \(0.406068\pi\)
\(282\) −5.17157 −0.307963
\(283\) 26.1608 1.55510 0.777548 0.628823i \(-0.216463\pi\)
0.777548 + 0.628823i \(0.216463\pi\)
\(284\) 10.2751 0.609716
\(285\) 0 0
\(286\) 1.56282 0.0924113
\(287\) −4.76687 −0.281380
\(288\) 1.00000 0.0589256
\(289\) −16.9558 −0.997402
\(290\) 0 0
\(291\) −20.0429 −1.17493
\(292\) −6.02386 −0.352519
\(293\) 0.814065 0.0475582 0.0237791 0.999717i \(-0.492430\pi\)
0.0237791 + 0.999717i \(0.492430\pi\)
\(294\) 1.41421 0.0824786
\(295\) 0 0
\(296\) −0.358761 −0.0208525
\(297\) −5.65685 −0.328244
\(298\) 5.02297 0.290973
\(299\) 13.7383 0.794507
\(300\) 0 0
\(301\) 5.44670 0.313942
\(302\) −2.56892 −0.147825
\(303\) 14.0429 0.806743
\(304\) −6.24264 −0.358040
\(305\) 0 0
\(306\) 0.210157 0.0120139
\(307\) −24.8593 −1.41879 −0.709397 0.704809i \(-0.751033\pi\)
−0.709397 + 0.704809i \(0.751033\pi\)
\(308\) −1.00000 −0.0569803
\(309\) 15.4706 0.880089
\(310\) 0 0
\(311\) −13.6447 −0.773717 −0.386859 0.922139i \(-0.626440\pi\)
−0.386859 + 0.922139i \(0.626440\pi\)
\(312\) 2.21016 0.125126
\(313\) −22.3943 −1.26580 −0.632899 0.774234i \(-0.718135\pi\)
−0.632899 + 0.774234i \(0.718135\pi\)
\(314\) −16.8692 −0.951981
\(315\) 0 0
\(316\) −9.59530 −0.539778
\(317\) −19.3753 −1.08822 −0.544112 0.839013i \(-0.683133\pi\)
−0.544112 + 0.839013i \(0.683133\pi\)
\(318\) −3.91295 −0.219427
\(319\) 9.96230 0.557782
\(320\) 0 0
\(321\) 18.1759 1.01448
\(322\) −8.79073 −0.489888
\(323\) −1.31194 −0.0729981
\(324\) −5.00000 −0.277778
\(325\) 0 0
\(326\) 12.5723 0.696317
\(327\) −20.4320 −1.12989
\(328\) 4.76687 0.263207
\(329\) −3.65685 −0.201609
\(330\) 0 0
\(331\) −33.1391 −1.82149 −0.910744 0.412972i \(-0.864491\pi\)
−0.910744 + 0.412972i \(0.864491\pi\)
\(332\) 1.56282 0.0857707
\(333\) −0.358761 −0.0196600
\(334\) −13.0962 −0.716591
\(335\) 0 0
\(336\) −1.41421 −0.0771517
\(337\) 9.62740 0.524438 0.262219 0.965008i \(-0.415546\pi\)
0.262219 + 0.965008i \(0.415546\pi\)
\(338\) 10.5576 0.574258
\(339\) −7.85964 −0.426877
\(340\) 0 0
\(341\) −6.93933 −0.375786
\(342\) −6.24264 −0.337563
\(343\) 1.00000 0.0539949
\(344\) −5.44670 −0.293666
\(345\) 0 0
\(346\) −0.461038 −0.0247856
\(347\) −29.9147 −1.60591 −0.802953 0.596042i \(-0.796739\pi\)
−0.802953 + 0.596042i \(0.796739\pi\)
\(348\) 14.0888 0.755240
\(349\) −8.89861 −0.476332 −0.238166 0.971225i \(-0.576546\pi\)
−0.238166 + 0.971225i \(0.576546\pi\)
\(350\) 0 0
\(351\) 8.84063 0.471878
\(352\) 1.00000 0.0533002
\(353\) −13.6265 −0.725267 −0.362634 0.931932i \(-0.618122\pi\)
−0.362634 + 0.931932i \(0.618122\pi\)
\(354\) −2.99265 −0.159057
\(355\) 0 0
\(356\) 8.59619 0.455597
\(357\) −0.297207 −0.0157299
\(358\) 4.69544 0.248162
\(359\) −23.2110 −1.22503 −0.612516 0.790458i \(-0.709843\pi\)
−0.612516 + 0.790458i \(0.709843\pi\)
\(360\) 0 0
\(361\) 19.9706 1.05108
\(362\) 15.4537 0.812228
\(363\) −1.41421 −0.0742270
\(364\) 1.56282 0.0819139
\(365\) 0 0
\(366\) 9.49264 0.496188
\(367\) −9.84316 −0.513809 −0.256904 0.966437i \(-0.582703\pi\)
−0.256904 + 0.966437i \(0.582703\pi\)
\(368\) 8.79073 0.458248
\(369\) 4.76687 0.248154
\(370\) 0 0
\(371\) −2.76687 −0.143649
\(372\) −9.81370 −0.508817
\(373\) 4.71929 0.244356 0.122178 0.992508i \(-0.461012\pi\)
0.122178 + 0.992508i \(0.461012\pi\)
\(374\) 0.210157 0.0108670
\(375\) 0 0
\(376\) 3.65685 0.188588
\(377\) −15.5693 −0.801857
\(378\) −5.65685 −0.290957
\(379\) 21.6906 1.11417 0.557085 0.830455i \(-0.311920\pi\)
0.557085 + 0.830455i \(0.311920\pi\)
\(380\) 0 0
\(381\) 1.51649 0.0776922
\(382\) 14.6954 0.751884
\(383\) 20.1759 1.03094 0.515469 0.856908i \(-0.327618\pi\)
0.515469 + 0.856908i \(0.327618\pi\)
\(384\) 1.41421 0.0721688
\(385\) 0 0
\(386\) −18.2751 −0.930179
\(387\) −5.44670 −0.276871
\(388\) 14.1725 0.719498
\(389\) −15.3769 −0.779640 −0.389820 0.920891i \(-0.627463\pi\)
−0.389820 + 0.920891i \(0.627463\pi\)
\(390\) 0 0
\(391\) 1.84744 0.0934288
\(392\) −1.00000 −0.0505076
\(393\) 30.9289 1.56016
\(394\) 19.8787 1.00147
\(395\) 0 0
\(396\) 1.00000 0.0502519
\(397\) −1.42983 −0.0717610 −0.0358805 0.999356i \(-0.511424\pi\)
−0.0358805 + 0.999356i \(0.511424\pi\)
\(398\) 22.7941 1.14257
\(399\) 8.82843 0.441974
\(400\) 0 0
\(401\) −32.3761 −1.61679 −0.808394 0.588642i \(-0.799663\pi\)
−0.808394 + 0.588642i \(0.799663\pi\)
\(402\) −16.9393 −0.844857
\(403\) 10.8449 0.540223
\(404\) −9.92982 −0.494027
\(405\) 0 0
\(406\) 9.96230 0.494421
\(407\) −0.358761 −0.0177831
\(408\) 0.297207 0.0147139
\(409\) −19.1100 −0.944930 −0.472465 0.881350i \(-0.656636\pi\)
−0.472465 + 0.881350i \(0.656636\pi\)
\(410\) 0 0
\(411\) −11.6274 −0.573537
\(412\) −10.9393 −0.538942
\(413\) −2.11612 −0.104127
\(414\) 8.79073 0.432041
\(415\) 0 0
\(416\) −1.56282 −0.0766234
\(417\) −13.7365 −0.672681
\(418\) −6.24264 −0.305338
\(419\) −34.7453 −1.69742 −0.848710 0.528859i \(-0.822620\pi\)
−0.848710 + 0.528859i \(0.822620\pi\)
\(420\) 0 0
\(421\) 22.3787 1.09067 0.545334 0.838219i \(-0.316403\pi\)
0.545334 + 0.838219i \(0.316403\pi\)
\(422\) 13.8449 0.673961
\(423\) 3.65685 0.177802
\(424\) 2.76687 0.134371
\(425\) 0 0
\(426\) 14.5312 0.704040
\(427\) 6.71231 0.324831
\(428\) −12.8523 −0.621239
\(429\) 2.21016 0.106707
\(430\) 0 0
\(431\) −12.6971 −0.611597 −0.305798 0.952096i \(-0.598923\pi\)
−0.305798 + 0.952096i \(0.598923\pi\)
\(432\) 5.65685 0.272166
\(433\) 15.0131 0.721483 0.360741 0.932666i \(-0.382524\pi\)
0.360741 + 0.932666i \(0.382524\pi\)
\(434\) −6.93933 −0.333099
\(435\) 0 0
\(436\) 14.4476 0.691914
\(437\) −54.8774 −2.62514
\(438\) −8.51902 −0.407054
\(439\) 4.29721 0.205095 0.102547 0.994728i \(-0.467301\pi\)
0.102547 + 0.994728i \(0.467301\pi\)
\(440\) 0 0
\(441\) −1.00000 −0.0476190
\(442\) −0.328437 −0.0156222
\(443\) −41.8228 −1.98706 −0.993532 0.113555i \(-0.963776\pi\)
−0.993532 + 0.113555i \(0.963776\pi\)
\(444\) −0.507364 −0.0240784
\(445\) 0 0
\(446\) 11.1274 0.526898
\(447\) 7.10355 0.335986
\(448\) 1.00000 0.0472456
\(449\) 15.4804 0.730567 0.365283 0.930896i \(-0.380972\pi\)
0.365283 + 0.930896i \(0.380972\pi\)
\(450\) 0 0
\(451\) 4.76687 0.224463
\(452\) 5.55760 0.261408
\(453\) −3.63300 −0.170693
\(454\) 17.0021 0.797950
\(455\) 0 0
\(456\) −8.82843 −0.413429
\(457\) 34.5980 1.61842 0.809212 0.587517i \(-0.199894\pi\)
0.809212 + 0.587517i \(0.199894\pi\)
\(458\) −14.7895 −0.691067
\(459\) 1.18883 0.0554898
\(460\) 0 0
\(461\) −30.8250 −1.43566 −0.717831 0.696217i \(-0.754865\pi\)
−0.717831 + 0.696217i \(0.754865\pi\)
\(462\) −1.41421 −0.0657952
\(463\) −41.3648 −1.92239 −0.961193 0.275875i \(-0.911032\pi\)
−0.961193 + 0.275875i \(0.911032\pi\)
\(464\) −9.96230 −0.462488
\(465\) 0 0
\(466\) 22.0888 1.02324
\(467\) 8.74440 0.404643 0.202321 0.979319i \(-0.435151\pi\)
0.202321 + 0.979319i \(0.435151\pi\)
\(468\) −1.56282 −0.0722412
\(469\) −11.9779 −0.553089
\(470\) 0 0
\(471\) −23.8566 −1.09925
\(472\) 2.11612 0.0974023
\(473\) −5.44670 −0.250439
\(474\) −13.5698 −0.623282
\(475\) 0 0
\(476\) 0.210157 0.00963254
\(477\) 2.76687 0.126686
\(478\) 12.3489 0.564825
\(479\) −4.87437 −0.222715 −0.111358 0.993780i \(-0.535520\pi\)
−0.111358 + 0.993780i \(0.535520\pi\)
\(480\) 0 0
\(481\) 0.560678 0.0255647
\(482\) −11.0659 −0.504036
\(483\) −12.4320 −0.565674
\(484\) 1.00000 0.0454545
\(485\) 0 0
\(486\) 9.89949 0.449050
\(487\) −16.3704 −0.741814 −0.370907 0.928670i \(-0.620953\pi\)
−0.370907 + 0.928670i \(0.620953\pi\)
\(488\) −6.71231 −0.303852
\(489\) 17.7800 0.804038
\(490\) 0 0
\(491\) 9.39088 0.423804 0.211902 0.977291i \(-0.432034\pi\)
0.211902 + 0.977291i \(0.432034\pi\)
\(492\) 6.74138 0.303925
\(493\) −2.09365 −0.0942932
\(494\) 9.75611 0.438948
\(495\) 0 0
\(496\) 6.93933 0.311585
\(497\) 10.2751 0.460902
\(498\) 2.21016 0.0990395
\(499\) 35.6703 1.59682 0.798411 0.602113i \(-0.205675\pi\)
0.798411 + 0.602113i \(0.205675\pi\)
\(500\) 0 0
\(501\) −18.5208 −0.827448
\(502\) −21.9488 −0.979625
\(503\) −0.313688 −0.0139867 −0.00699333 0.999976i \(-0.502226\pi\)
−0.00699333 + 0.999976i \(0.502226\pi\)
\(504\) 1.00000 0.0445435
\(505\) 0 0
\(506\) 8.79073 0.390796
\(507\) 14.9307 0.663096
\(508\) −1.07232 −0.0475766
\(509\) 0.0922646 0.00408956 0.00204478 0.999998i \(-0.499349\pi\)
0.00204478 + 0.999998i \(0.499349\pi\)
\(510\) 0 0
\(511\) −6.02386 −0.266480
\(512\) −1.00000 −0.0441942
\(513\) −35.3137 −1.55914
\(514\) −3.93845 −0.173717
\(515\) 0 0
\(516\) −7.70279 −0.339097
\(517\) 3.65685 0.160828
\(518\) −0.358761 −0.0157630
\(519\) −0.652007 −0.0286199
\(520\) 0 0
\(521\) −12.2850 −0.538216 −0.269108 0.963110i \(-0.586729\pi\)
−0.269108 + 0.963110i \(0.586729\pi\)
\(522\) −9.96230 −0.436038
\(523\) 2.46841 0.107936 0.0539681 0.998543i \(-0.482813\pi\)
0.0539681 + 0.998543i \(0.482813\pi\)
\(524\) −21.8700 −0.955397
\(525\) 0 0
\(526\) −1.77996 −0.0776100
\(527\) 1.45835 0.0635268
\(528\) 1.41421 0.0615457
\(529\) 54.2769 2.35987
\(530\) 0 0
\(531\) 2.11612 0.0918318
\(532\) −6.24264 −0.270653
\(533\) −7.44975 −0.322685
\(534\) 12.1568 0.526078
\(535\) 0 0
\(536\) 11.9779 0.517367
\(537\) 6.64035 0.286552
\(538\) 19.0555 0.821539
\(539\) −1.00000 −0.0430730
\(540\) 0 0
\(541\) −38.8991 −1.67240 −0.836202 0.548421i \(-0.815229\pi\)
−0.836202 + 0.548421i \(0.815229\pi\)
\(542\) 12.5190 0.537738
\(543\) 21.8548 0.937880
\(544\) −0.210157 −0.00901042
\(545\) 0 0
\(546\) 2.21016 0.0945860
\(547\) 34.2548 1.46463 0.732315 0.680966i \(-0.238440\pi\)
0.732315 + 0.680966i \(0.238440\pi\)
\(548\) 8.22181 0.351218
\(549\) −6.71231 −0.286474
\(550\) 0 0
\(551\) 62.1911 2.64943
\(552\) 12.4320 0.529140
\(553\) −9.59530 −0.408034
\(554\) −2.75126 −0.116890
\(555\) 0 0
\(556\) 9.71319 0.411931
\(557\) −29.2383 −1.23887 −0.619434 0.785049i \(-0.712638\pi\)
−0.619434 + 0.785049i \(0.712638\pi\)
\(558\) 6.93933 0.293765
\(559\) 8.51219 0.360027
\(560\) 0 0
\(561\) 0.297207 0.0125481
\(562\) −9.75050 −0.411300
\(563\) 26.8783 1.13279 0.566393 0.824136i \(-0.308339\pi\)
0.566393 + 0.824136i \(0.308339\pi\)
\(564\) 5.17157 0.217763
\(565\) 0 0
\(566\) −26.1608 −1.09962
\(567\) −5.00000 −0.209980
\(568\) −10.2751 −0.431135
\(569\) 6.92945 0.290498 0.145249 0.989395i \(-0.453602\pi\)
0.145249 + 0.989395i \(0.453602\pi\)
\(570\) 0 0
\(571\) 29.3155 1.22681 0.613407 0.789767i \(-0.289798\pi\)
0.613407 + 0.789767i \(0.289798\pi\)
\(572\) −1.56282 −0.0653447
\(573\) 20.7825 0.868201
\(574\) 4.76687 0.198965
\(575\) 0 0
\(576\) −1.00000 −0.0416667
\(577\) 4.09278 0.170385 0.0851924 0.996365i \(-0.472850\pi\)
0.0851924 + 0.996365i \(0.472850\pi\)
\(578\) 16.9558 0.705270
\(579\) −25.8449 −1.07408
\(580\) 0 0
\(581\) 1.56282 0.0648366
\(582\) 20.0429 0.830804
\(583\) 2.76687 0.114592
\(584\) 6.02386 0.249269
\(585\) 0 0
\(586\) −0.814065 −0.0336287
\(587\) 42.7157 1.76307 0.881533 0.472122i \(-0.156512\pi\)
0.881533 + 0.472122i \(0.156512\pi\)
\(588\) −1.41421 −0.0583212
\(589\) −43.3198 −1.78496
\(590\) 0 0
\(591\) 28.1127 1.15640
\(592\) 0.358761 0.0147450
\(593\) −1.86952 −0.0767719 −0.0383860 0.999263i \(-0.512222\pi\)
−0.0383860 + 0.999263i \(0.512222\pi\)
\(594\) 5.65685 0.232104
\(595\) 0 0
\(596\) −5.02297 −0.205749
\(597\) 32.2358 1.31932
\(598\) −13.7383 −0.561801
\(599\) −13.5264 −0.552673 −0.276336 0.961061i \(-0.589120\pi\)
−0.276336 + 0.961061i \(0.589120\pi\)
\(600\) 0 0
\(601\) −11.9714 −0.488326 −0.244163 0.969734i \(-0.578513\pi\)
−0.244163 + 0.969734i \(0.578513\pi\)
\(602\) −5.44670 −0.221991
\(603\) 11.9779 0.487778
\(604\) 2.56892 0.104528
\(605\) 0 0
\(606\) −14.0429 −0.570453
\(607\) 8.99265 0.365000 0.182500 0.983206i \(-0.441581\pi\)
0.182500 + 0.983206i \(0.441581\pi\)
\(608\) 6.24264 0.253173
\(609\) 14.0888 0.570908
\(610\) 0 0
\(611\) −5.71499 −0.231204
\(612\) −0.210157 −0.00849510
\(613\) −8.56245 −0.345834 −0.172917 0.984936i \(-0.555319\pi\)
−0.172917 + 0.984936i \(0.555319\pi\)
\(614\) 24.8593 1.00324
\(615\) 0 0
\(616\) 1.00000 0.0402911
\(617\) 3.97614 0.160074 0.0800368 0.996792i \(-0.474496\pi\)
0.0800368 + 0.996792i \(0.474496\pi\)
\(618\) −15.4706 −0.622317
\(619\) 39.0450 1.56935 0.784676 0.619906i \(-0.212829\pi\)
0.784676 + 0.619906i \(0.212829\pi\)
\(620\) 0 0
\(621\) 49.7279 1.99551
\(622\) 13.6447 0.547101
\(623\) 8.59619 0.344399
\(624\) −2.21016 −0.0884771
\(625\) 0 0
\(626\) 22.3943 0.895055
\(627\) −8.82843 −0.352573
\(628\) 16.8692 0.673152
\(629\) 0.0753962 0.00300624
\(630\) 0 0
\(631\) −28.2769 −1.12569 −0.562843 0.826564i \(-0.690292\pi\)
−0.562843 + 0.826564i \(0.690292\pi\)
\(632\) 9.59530 0.381681
\(633\) 19.5797 0.778223
\(634\) 19.3753 0.769490
\(635\) 0 0
\(636\) 3.91295 0.155159
\(637\) 1.56282 0.0619211
\(638\) −9.96230 −0.394411
\(639\) −10.2751 −0.406478
\(640\) 0 0
\(641\) −6.78909 −0.268153 −0.134076 0.990971i \(-0.542807\pi\)
−0.134076 + 0.990971i \(0.542807\pi\)
\(642\) −18.1759 −0.717344
\(643\) 12.7861 0.504233 0.252117 0.967697i \(-0.418873\pi\)
0.252117 + 0.967697i \(0.418873\pi\)
\(644\) 8.79073 0.346403
\(645\) 0 0
\(646\) 1.31194 0.0516174
\(647\) 34.0312 1.33791 0.668953 0.743305i \(-0.266743\pi\)
0.668953 + 0.743305i \(0.266743\pi\)
\(648\) 5.00000 0.196419
\(649\) 2.11612 0.0830650
\(650\) 0 0
\(651\) −9.81370 −0.384629
\(652\) −12.5723 −0.492370
\(653\) −6.01812 −0.235507 −0.117754 0.993043i \(-0.537569\pi\)
−0.117754 + 0.993043i \(0.537569\pi\)
\(654\) 20.4320 0.798953
\(655\) 0 0
\(656\) −4.76687 −0.186115
\(657\) 6.02386 0.235013
\(658\) 3.65685 0.142559
\(659\) 16.9498 0.660269 0.330134 0.943934i \(-0.392906\pi\)
0.330134 + 0.943934i \(0.392906\pi\)
\(660\) 0 0
\(661\) 40.0714 1.55860 0.779299 0.626653i \(-0.215576\pi\)
0.779299 + 0.626653i \(0.215576\pi\)
\(662\) 33.1391 1.28799
\(663\) −0.464480 −0.0180389
\(664\) −1.56282 −0.0606491
\(665\) 0 0
\(666\) 0.358761 0.0139017
\(667\) −87.5759 −3.39095
\(668\) 13.0962 0.506706
\(669\) 15.7365 0.608409
\(670\) 0 0
\(671\) −6.71231 −0.259126
\(672\) 1.41421 0.0545545
\(673\) −24.4797 −0.943622 −0.471811 0.881700i \(-0.656400\pi\)
−0.471811 + 0.881700i \(0.656400\pi\)
\(674\) −9.62740 −0.370833
\(675\) 0 0
\(676\) −10.5576 −0.406062
\(677\) 33.0788 1.27132 0.635660 0.771969i \(-0.280728\pi\)
0.635660 + 0.771969i \(0.280728\pi\)
\(678\) 7.85964 0.301848
\(679\) 14.1725 0.543889
\(680\) 0 0
\(681\) 24.0447 0.921393
\(682\) 6.93933 0.265721
\(683\) −33.6175 −1.28634 −0.643169 0.765724i \(-0.722381\pi\)
−0.643169 + 0.765724i \(0.722381\pi\)
\(684\) 6.24264 0.238693
\(685\) 0 0
\(686\) −1.00000 −0.0381802
\(687\) −20.9155 −0.797975
\(688\) 5.44670 0.207653
\(689\) −4.32412 −0.164736
\(690\) 0 0
\(691\) 16.7877 0.638634 0.319317 0.947648i \(-0.396546\pi\)
0.319317 + 0.947648i \(0.396546\pi\)
\(692\) 0.461038 0.0175260
\(693\) 1.00000 0.0379869
\(694\) 29.9147 1.13555
\(695\) 0 0
\(696\) −14.0888 −0.534035
\(697\) −1.00179 −0.0379456
\(698\) 8.89861 0.336817
\(699\) 31.2383 1.18154
\(700\) 0 0
\(701\) −4.99606 −0.188699 −0.0943493 0.995539i \(-0.530077\pi\)
−0.0943493 + 0.995539i \(0.530077\pi\)
\(702\) −8.84063 −0.333668
\(703\) −2.23961 −0.0844687
\(704\) −1.00000 −0.0376889
\(705\) 0 0
\(706\) 13.6265 0.512841
\(707\) −9.92982 −0.373449
\(708\) 2.99265 0.112471
\(709\) −13.3310 −0.500655 −0.250327 0.968161i \(-0.580538\pi\)
−0.250327 + 0.968161i \(0.580538\pi\)
\(710\) 0 0
\(711\) 9.59530 0.359852
\(712\) −8.59619 −0.322156
\(713\) 61.0018 2.28454
\(714\) 0.297207 0.0111227
\(715\) 0 0
\(716\) −4.69544 −0.175477
\(717\) 17.4640 0.652203
\(718\) 23.2110 0.866228
\(719\) −6.39823 −0.238614 −0.119307 0.992857i \(-0.538067\pi\)
−0.119307 + 0.992857i \(0.538067\pi\)
\(720\) 0 0
\(721\) −10.9393 −0.407402
\(722\) −19.9706 −0.743227
\(723\) −15.6495 −0.582010
\(724\) −15.4537 −0.574332
\(725\) 0 0
\(726\) 1.41421 0.0524864
\(727\) 2.00000 0.0741759 0.0370879 0.999312i \(-0.488192\pi\)
0.0370879 + 0.999312i \(0.488192\pi\)
\(728\) −1.56282 −0.0579219
\(729\) 29.0000 1.07407
\(730\) 0 0
\(731\) 1.14466 0.0423369
\(732\) −9.49264 −0.350858
\(733\) −22.4463 −0.829074 −0.414537 0.910032i \(-0.636056\pi\)
−0.414537 + 0.910032i \(0.636056\pi\)
\(734\) 9.84316 0.363318
\(735\) 0 0
\(736\) −8.79073 −0.324031
\(737\) 11.9779 0.441212
\(738\) −4.76687 −0.175471
\(739\) −37.6127 −1.38361 −0.691803 0.722087i \(-0.743183\pi\)
−0.691803 + 0.722087i \(0.743183\pi\)
\(740\) 0 0
\(741\) 13.7972 0.506853
\(742\) 2.76687 0.101575
\(743\) −23.4779 −0.861322 −0.430661 0.902514i \(-0.641719\pi\)
−0.430661 + 0.902514i \(0.641719\pi\)
\(744\) 9.81370 0.359788
\(745\) 0 0
\(746\) −4.71929 −0.172786
\(747\) −1.56282 −0.0571805
\(748\) −0.210157 −0.00768411
\(749\) −12.8523 −0.469612
\(750\) 0 0
\(751\) 10.9540 0.399719 0.199859 0.979825i \(-0.435951\pi\)
0.199859 + 0.979825i \(0.435951\pi\)
\(752\) −3.65685 −0.133352
\(753\) −31.0404 −1.13117
\(754\) 15.5693 0.566999
\(755\) 0 0
\(756\) 5.65685 0.205738
\(757\) 4.62223 0.167998 0.0839989 0.996466i \(-0.473231\pi\)
0.0839989 + 0.996466i \(0.473231\pi\)
\(758\) −21.6906 −0.787838
\(759\) 12.4320 0.451252
\(760\) 0 0
\(761\) −42.6932 −1.54763 −0.773815 0.633412i \(-0.781654\pi\)
−0.773815 + 0.633412i \(0.781654\pi\)
\(762\) −1.51649 −0.0549367
\(763\) 14.4476 0.523038
\(764\) −14.6954 −0.531662
\(765\) 0 0
\(766\) −20.1759 −0.728984
\(767\) −3.30711 −0.119413
\(768\) −1.41421 −0.0510310
\(769\) 48.9133 1.76386 0.881929 0.471382i \(-0.156245\pi\)
0.881929 + 0.471382i \(0.156245\pi\)
\(770\) 0 0
\(771\) −5.56980 −0.200592
\(772\) 18.2751 0.657736
\(773\) 48.4801 1.74371 0.871853 0.489767i \(-0.162918\pi\)
0.871853 + 0.489767i \(0.162918\pi\)
\(774\) 5.44670 0.195777
\(775\) 0 0
\(776\) −14.1725 −0.508762
\(777\) −0.507364 −0.0182016
\(778\) 15.3769 0.551289
\(779\) 29.7579 1.06619
\(780\) 0 0
\(781\) −10.2751 −0.367673
\(782\) −1.84744 −0.0660641
\(783\) −56.3553 −2.01397
\(784\) 1.00000 0.0357143
\(785\) 0 0
\(786\) −30.9289 −1.10320
\(787\) −10.5021 −0.374361 −0.187181 0.982326i \(-0.559935\pi\)
−0.187181 + 0.982326i \(0.559935\pi\)
\(788\) −19.8787 −0.708148
\(789\) −2.51725 −0.0896163
\(790\) 0 0
\(791\) 5.55760 0.197606
\(792\) −1.00000 −0.0355335
\(793\) 10.4901 0.372515
\(794\) 1.42983 0.0507427
\(795\) 0 0
\(796\) −22.7941 −0.807917
\(797\) 37.1872 1.31724 0.658618 0.752477i \(-0.271141\pi\)
0.658618 + 0.752477i \(0.271141\pi\)
\(798\) −8.82843 −0.312523
\(799\) −0.768514 −0.0271881
\(800\) 0 0
\(801\) −8.59619 −0.303731
\(802\) 32.3761 1.14324
\(803\) 6.02386 0.212577
\(804\) 16.9393 0.597404
\(805\) 0 0
\(806\) −10.8449 −0.381996
\(807\) 26.9485 0.948631
\(808\) 9.92982 0.349330
\(809\) 43.5403 1.53080 0.765399 0.643557i \(-0.222542\pi\)
0.765399 + 0.643557i \(0.222542\pi\)
\(810\) 0 0
\(811\) −47.0147 −1.65091 −0.825455 0.564468i \(-0.809081\pi\)
−0.825455 + 0.564468i \(0.809081\pi\)
\(812\) −9.96230 −0.349608
\(813\) 17.7046 0.620926
\(814\) 0.358761 0.0125746
\(815\) 0 0
\(816\) −0.297207 −0.0104043
\(817\) −34.0018 −1.18957
\(818\) 19.1100 0.668166
\(819\) −1.56282 −0.0546093
\(820\) 0 0
\(821\) −42.5126 −1.48370 −0.741849 0.670567i \(-0.766051\pi\)
−0.741849 + 0.670567i \(0.766051\pi\)
\(822\) 11.6274 0.405552
\(823\) 23.9262 0.834016 0.417008 0.908903i \(-0.363079\pi\)
0.417008 + 0.908903i \(0.363079\pi\)
\(824\) 10.9393 0.381090
\(825\) 0 0
\(826\) 2.11612 0.0736292
\(827\) −12.5944 −0.437951 −0.218975 0.975730i \(-0.570271\pi\)
−0.218975 + 0.975730i \(0.570271\pi\)
\(828\) −8.79073 −0.305499
\(829\) 11.6836 0.405788 0.202894 0.979201i \(-0.434965\pi\)
0.202894 + 0.979201i \(0.434965\pi\)
\(830\) 0 0
\(831\) −3.89087 −0.134973
\(832\) 1.56282 0.0541809
\(833\) 0.210157 0.00728152
\(834\) 13.7365 0.475657
\(835\) 0 0
\(836\) 6.24264 0.215906
\(837\) 39.2548 1.35684
\(838\) 34.7453 1.20026
\(839\) 40.3302 1.39235 0.696177 0.717870i \(-0.254883\pi\)
0.696177 + 0.717870i \(0.254883\pi\)
\(840\) 0 0
\(841\) 70.2475 2.42233
\(842\) −22.3787 −0.771219
\(843\) −13.7893 −0.474929
\(844\) −13.8449 −0.476562
\(845\) 0 0
\(846\) −3.65685 −0.125725
\(847\) 1.00000 0.0343604
\(848\) −2.76687 −0.0950148
\(849\) −36.9969 −1.26973
\(850\) 0 0
\(851\) 3.15377 0.108110
\(852\) −14.5312 −0.497831
\(853\) 30.2920 1.03718 0.518589 0.855024i \(-0.326457\pi\)
0.518589 + 0.855024i \(0.326457\pi\)
\(854\) −6.71231 −0.229690
\(855\) 0 0
\(856\) 12.8523 0.439282
\(857\) 34.0134 1.16188 0.580938 0.813948i \(-0.302686\pi\)
0.580938 + 0.813948i \(0.302686\pi\)
\(858\) −2.21016 −0.0754535
\(859\) −19.3250 −0.659360 −0.329680 0.944093i \(-0.606941\pi\)
−0.329680 + 0.944093i \(0.606941\pi\)
\(860\) 0 0
\(861\) 6.74138 0.229746
\(862\) 12.6971 0.432464
\(863\) −23.2331 −0.790865 −0.395432 0.918495i \(-0.629405\pi\)
−0.395432 + 0.918495i \(0.629405\pi\)
\(864\) −5.65685 −0.192450
\(865\) 0 0
\(866\) −15.0131 −0.510165
\(867\) 23.9792 0.814375
\(868\) 6.93933 0.235536
\(869\) 9.59530 0.325498
\(870\) 0 0
\(871\) −18.7193 −0.634279
\(872\) −14.4476 −0.489257
\(873\) −14.1725 −0.479665
\(874\) 54.8774 1.85625
\(875\) 0 0
\(876\) 8.51902 0.287831
\(877\) 2.37260 0.0801171 0.0400586 0.999197i \(-0.487246\pi\)
0.0400586 + 0.999197i \(0.487246\pi\)
\(878\) −4.29721 −0.145024
\(879\) −1.15126 −0.0388311
\(880\) 0 0
\(881\) 10.6665 0.359364 0.179682 0.983725i \(-0.442493\pi\)
0.179682 + 0.983725i \(0.442493\pi\)
\(882\) 1.00000 0.0336718
\(883\) −33.1018 −1.11396 −0.556982 0.830525i \(-0.688041\pi\)
−0.556982 + 0.830525i \(0.688041\pi\)
\(884\) 0.328437 0.0110465
\(885\) 0 0
\(886\) 41.8228 1.40507
\(887\) −18.4246 −0.618637 −0.309319 0.950958i \(-0.600101\pi\)
−0.309319 + 0.950958i \(0.600101\pi\)
\(888\) 0.507364 0.0170260
\(889\) −1.07232 −0.0359645
\(890\) 0 0
\(891\) 5.00000 0.167506
\(892\) −11.1274 −0.372573
\(893\) 22.8284 0.763924
\(894\) −7.10355 −0.237578
\(895\) 0 0
\(896\) −1.00000 −0.0334077
\(897\) −19.4289 −0.648712
\(898\) −15.4804 −0.516589
\(899\) −69.1317 −2.30567
\(900\) 0 0
\(901\) −0.581478 −0.0193719
\(902\) −4.76687 −0.158720
\(903\) −7.70279 −0.256333
\(904\) −5.55760 −0.184843
\(905\) 0 0
\(906\) 3.63300 0.120698
\(907\) 2.13048 0.0707415 0.0353707 0.999374i \(-0.488739\pi\)
0.0353707 + 0.999374i \(0.488739\pi\)
\(908\) −17.0021 −0.564236
\(909\) 9.92982 0.329351
\(910\) 0 0
\(911\) −18.0330 −0.597460 −0.298730 0.954338i \(-0.596563\pi\)
−0.298730 + 0.954338i \(0.596563\pi\)
\(912\) 8.82843 0.292338
\(913\) −1.56282 −0.0517217
\(914\) −34.5980 −1.14440
\(915\) 0 0
\(916\) 14.7895 0.488658
\(917\) −21.8700 −0.722212
\(918\) −1.18883 −0.0392372
\(919\) −31.1326 −1.02697 −0.513485 0.858099i \(-0.671646\pi\)
−0.513485 + 0.858099i \(0.671646\pi\)
\(920\) 0 0
\(921\) 35.1563 1.15844
\(922\) 30.8250 1.01517
\(923\) 16.0581 0.528560
\(924\) 1.41421 0.0465242
\(925\) 0 0
\(926\) 41.3648 1.35933
\(927\) 10.9393 0.359295
\(928\) 9.96230 0.327029
\(929\) 40.0911 1.31535 0.657674 0.753303i \(-0.271541\pi\)
0.657674 + 0.753303i \(0.271541\pi\)
\(930\) 0 0
\(931\) −6.24264 −0.204594
\(932\) −22.0888 −0.723543
\(933\) 19.2965 0.631737
\(934\) −8.74440 −0.286126
\(935\) 0 0
\(936\) 1.56282 0.0510823
\(937\) 2.48834 0.0812904 0.0406452 0.999174i \(-0.487059\pi\)
0.0406452 + 0.999174i \(0.487059\pi\)
\(938\) 11.9779 0.391093
\(939\) 31.6703 1.03352
\(940\) 0 0
\(941\) −23.3300 −0.760537 −0.380269 0.924876i \(-0.624168\pi\)
−0.380269 + 0.924876i \(0.624168\pi\)
\(942\) 23.8566 0.777289
\(943\) −41.9043 −1.36459
\(944\) −2.11612 −0.0688738
\(945\) 0 0
\(946\) 5.44670 0.177087
\(947\) −25.6556 −0.833694 −0.416847 0.908977i \(-0.636865\pi\)
−0.416847 + 0.908977i \(0.636865\pi\)
\(948\) 13.5698 0.440727
\(949\) −9.41418 −0.305597
\(950\) 0 0
\(951\) 27.4008 0.888530
\(952\) −0.210157 −0.00681123
\(953\) −9.07970 −0.294120 −0.147060 0.989128i \(-0.546981\pi\)
−0.147060 + 0.989128i \(0.546981\pi\)
\(954\) −2.76687 −0.0895808
\(955\) 0 0
\(956\) −12.3489 −0.399391
\(957\) −14.0888 −0.455427
\(958\) 4.87437 0.157484
\(959\) 8.22181 0.265496
\(960\) 0 0
\(961\) 17.1543 0.553366
\(962\) −0.560678 −0.0180770
\(963\) 12.8523 0.414159
\(964\) 11.0659 0.356407
\(965\) 0 0
\(966\) 12.4320 0.399992
\(967\) −52.3419 −1.68320 −0.841600 0.540101i \(-0.818386\pi\)
−0.841600 + 0.540101i \(0.818386\pi\)
\(968\) −1.00000 −0.0321412
\(969\) 1.85536 0.0596027
\(970\) 0 0
\(971\) 23.2600 0.746449 0.373225 0.927741i \(-0.378252\pi\)
0.373225 + 0.927741i \(0.378252\pi\)
\(972\) −9.89949 −0.317526
\(973\) 9.71319 0.311391
\(974\) 16.3704 0.524542
\(975\) 0 0
\(976\) 6.71231 0.214856
\(977\) 48.4436 1.54985 0.774924 0.632054i \(-0.217788\pi\)
0.774924 + 0.632054i \(0.217788\pi\)
\(978\) −17.7800 −0.568540
\(979\) −8.59619 −0.274735
\(980\) 0 0
\(981\) −14.4476 −0.461276
\(982\) −9.39088 −0.299675
\(983\) −2.25304 −0.0718609 −0.0359304 0.999354i \(-0.511439\pi\)
−0.0359304 + 0.999354i \(0.511439\pi\)
\(984\) −6.74138 −0.214907
\(985\) 0 0
\(986\) 2.09365 0.0666754
\(987\) 5.17157 0.164613
\(988\) −9.75611 −0.310383
\(989\) 47.8804 1.52251
\(990\) 0 0
\(991\) 27.8254 0.883901 0.441951 0.897039i \(-0.354287\pi\)
0.441951 + 0.897039i \(0.354287\pi\)
\(992\) −6.93933 −0.220324
\(993\) 46.8657 1.48724
\(994\) −10.2751 −0.325907
\(995\) 0 0
\(996\) −2.21016 −0.0700315
\(997\) 39.9966 1.26670 0.633352 0.773864i \(-0.281679\pi\)
0.633352 + 0.773864i \(0.281679\pi\)
\(998\) −35.6703 −1.12912
\(999\) 2.02946 0.0642092
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 3850.2.a.bx.1.1 4
5.2 odd 4 770.2.c.e.309.3 8
5.3 odd 4 770.2.c.e.309.5 yes 8
5.4 even 2 3850.2.a.by.1.4 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
770.2.c.e.309.3 8 5.2 odd 4
770.2.c.e.309.5 yes 8 5.3 odd 4
3850.2.a.bx.1.1 4 1.1 even 1 trivial
3850.2.a.by.1.4 4 5.4 even 2