Properties

Label 4030.2.a.d.1.2
Level $4030$
Weight $2$
Character 4030.1
Self dual yes
Analytic conductor $32.180$
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] = [4030,2,Mod(1,4030)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4030, 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("4030.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4030 = 2 \cdot 5 \cdot 13 \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4030.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: \(32.1797120146\)
Analytic rank: \(1\)
Dimension: \(6\)
Coefficient field: 6.6.3728437.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 2x^{5} - 5x^{4} + 6x^{3} + 7x^{2} - 3x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.58577\) of defining polynomial
Character \(\chi\) \(=\) 4030.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} -1.27014 q^{3} +1.00000 q^{4} +1.00000 q^{5} +1.27014 q^{6} -3.95315 q^{7} -1.00000 q^{8} -1.38673 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -1.27014 q^{3} +1.00000 q^{4} +1.00000 q^{5} +1.27014 q^{6} -3.95315 q^{7} -1.00000 q^{8} -1.38673 q^{9} -1.00000 q^{10} +4.66379 q^{11} -1.27014 q^{12} +1.00000 q^{13} +3.95315 q^{14} -1.27014 q^{15} +1.00000 q^{16} -1.48534 q^{17} +1.38673 q^{18} +3.80214 q^{19} +1.00000 q^{20} +5.02107 q^{21} -4.66379 q^{22} -7.81311 q^{23} +1.27014 q^{24} +1.00000 q^{25} -1.00000 q^{26} +5.57178 q^{27} -3.95315 q^{28} +0.373984 q^{29} +1.27014 q^{30} -1.00000 q^{31} -1.00000 q^{32} -5.92369 q^{33} +1.48534 q^{34} -3.95315 q^{35} -1.38673 q^{36} -3.50437 q^{37} -3.80214 q^{38} -1.27014 q^{39} -1.00000 q^{40} -6.09942 q^{41} -5.02107 q^{42} +3.20595 q^{43} +4.66379 q^{44} -1.38673 q^{45} +7.81311 q^{46} +6.40206 q^{47} -1.27014 q^{48} +8.62737 q^{49} -1.00000 q^{50} +1.88660 q^{51} +1.00000 q^{52} +9.25929 q^{53} -5.57178 q^{54} +4.66379 q^{55} +3.95315 q^{56} -4.82927 q^{57} -0.373984 q^{58} +6.57218 q^{59} -1.27014 q^{60} +4.49166 q^{61} +1.00000 q^{62} +5.48196 q^{63} +1.00000 q^{64} +1.00000 q^{65} +5.92369 q^{66} -9.82335 q^{67} -1.48534 q^{68} +9.92378 q^{69} +3.95315 q^{70} +9.44658 q^{71} +1.38673 q^{72} +3.09187 q^{73} +3.50437 q^{74} -1.27014 q^{75} +3.80214 q^{76} -18.4367 q^{77} +1.27014 q^{78} +0.814856 q^{79} +1.00000 q^{80} -2.91678 q^{81} +6.09942 q^{82} -7.28306 q^{83} +5.02107 q^{84} -1.48534 q^{85} -3.20595 q^{86} -0.475013 q^{87} -4.66379 q^{88} -6.30532 q^{89} +1.38673 q^{90} -3.95315 q^{91} -7.81311 q^{92} +1.27014 q^{93} -6.40206 q^{94} +3.80214 q^{95} +1.27014 q^{96} -12.8930 q^{97} -8.62737 q^{98} -6.46743 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 6 q^{2} - q^{3} + 6 q^{4} + 6 q^{5} + q^{6} - 2 q^{7} - 6 q^{8} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 6 q^{2} - q^{3} + 6 q^{4} + 6 q^{5} + q^{6} - 2 q^{7} - 6 q^{8} - q^{9} - 6 q^{10} - q^{12} + 6 q^{13} + 2 q^{14} - q^{15} + 6 q^{16} - 10 q^{17} + q^{18} + q^{19} + 6 q^{20} + 3 q^{21} - 13 q^{23} + q^{24} + 6 q^{25} - 6 q^{26} - 7 q^{27} - 2 q^{28} - 16 q^{29} + q^{30} - 6 q^{31} - 6 q^{32} - 22 q^{33} + 10 q^{34} - 2 q^{35} - q^{36} - 14 q^{37} - q^{38} - q^{39} - 6 q^{40} - 4 q^{41} - 3 q^{42} + 5 q^{43} - q^{45} + 13 q^{46} - 4 q^{47} - q^{48} + 2 q^{49} - 6 q^{50} + 11 q^{51} + 6 q^{52} - 14 q^{53} + 7 q^{54} + 2 q^{56} - 3 q^{57} + 16 q^{58} - 7 q^{59} - q^{60} - 3 q^{61} + 6 q^{62} - 3 q^{63} + 6 q^{64} + 6 q^{65} + 22 q^{66} - 4 q^{67} - 10 q^{68} - 4 q^{69} + 2 q^{70} + q^{72} - 5 q^{73} + 14 q^{74} - q^{75} + q^{76} - 23 q^{77} + q^{78} + 6 q^{79} + 6 q^{80} - 18 q^{81} + 4 q^{82} - 2 q^{83} + 3 q^{84} - 10 q^{85} - 5 q^{86} - 21 q^{87} - 14 q^{89} + q^{90} - 2 q^{91} - 13 q^{92} + q^{93} + 4 q^{94} + q^{95} + q^{96} - 35 q^{97} - 2 q^{98} + 3 q^{99}+O(q^{100}) \) 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
\(3\) −1.27014 −0.733318 −0.366659 0.930355i \(-0.619498\pi\)
−0.366659 + 0.930355i \(0.619498\pi\)
\(4\) 1.00000 0.500000
\(5\) 1.00000 0.447214
\(6\) 1.27014 0.518534
\(7\) −3.95315 −1.49415 −0.747074 0.664740i \(-0.768542\pi\)
−0.747074 + 0.664740i \(0.768542\pi\)
\(8\) −1.00000 −0.353553
\(9\) −1.38673 −0.462244
\(10\) −1.00000 −0.316228
\(11\) 4.66379 1.40619 0.703093 0.711098i \(-0.251802\pi\)
0.703093 + 0.711098i \(0.251802\pi\)
\(12\) −1.27014 −0.366659
\(13\) 1.00000 0.277350
\(14\) 3.95315 1.05652
\(15\) −1.27014 −0.327950
\(16\) 1.00000 0.250000
\(17\) −1.48534 −0.360249 −0.180124 0.983644i \(-0.557650\pi\)
−0.180124 + 0.983644i \(0.557650\pi\)
\(18\) 1.38673 0.326856
\(19\) 3.80214 0.872272 0.436136 0.899881i \(-0.356347\pi\)
0.436136 + 0.899881i \(0.356347\pi\)
\(20\) 1.00000 0.223607
\(21\) 5.02107 1.09569
\(22\) −4.66379 −0.994324
\(23\) −7.81311 −1.62915 −0.814573 0.580061i \(-0.803029\pi\)
−0.814573 + 0.580061i \(0.803029\pi\)
\(24\) 1.27014 0.259267
\(25\) 1.00000 0.200000
\(26\) −1.00000 −0.196116
\(27\) 5.57178 1.07229
\(28\) −3.95315 −0.747074
\(29\) 0.373984 0.0694470 0.0347235 0.999397i \(-0.488945\pi\)
0.0347235 + 0.999397i \(0.488945\pi\)
\(30\) 1.27014 0.231896
\(31\) −1.00000 −0.179605
\(32\) −1.00000 −0.176777
\(33\) −5.92369 −1.03118
\(34\) 1.48534 0.254734
\(35\) −3.95315 −0.668204
\(36\) −1.38673 −0.231122
\(37\) −3.50437 −0.576115 −0.288058 0.957613i \(-0.593009\pi\)
−0.288058 + 0.957613i \(0.593009\pi\)
\(38\) −3.80214 −0.616789
\(39\) −1.27014 −0.203386
\(40\) −1.00000 −0.158114
\(41\) −6.09942 −0.952569 −0.476284 0.879291i \(-0.658017\pi\)
−0.476284 + 0.879291i \(0.658017\pi\)
\(42\) −5.02107 −0.774768
\(43\) 3.20595 0.488903 0.244451 0.969662i \(-0.421392\pi\)
0.244451 + 0.969662i \(0.421392\pi\)
\(44\) 4.66379 0.703093
\(45\) −1.38673 −0.206722
\(46\) 7.81311 1.15198
\(47\) 6.40206 0.933837 0.466918 0.884300i \(-0.345364\pi\)
0.466918 + 0.884300i \(0.345364\pi\)
\(48\) −1.27014 −0.183330
\(49\) 8.62737 1.23248
\(50\) −1.00000 −0.141421
\(51\) 1.88660 0.264177
\(52\) 1.00000 0.138675
\(53\) 9.25929 1.27186 0.635931 0.771746i \(-0.280616\pi\)
0.635931 + 0.771746i \(0.280616\pi\)
\(54\) −5.57178 −0.758224
\(55\) 4.66379 0.628866
\(56\) 3.95315 0.528261
\(57\) −4.82927 −0.639653
\(58\) −0.373984 −0.0491065
\(59\) 6.57218 0.855625 0.427812 0.903868i \(-0.359284\pi\)
0.427812 + 0.903868i \(0.359284\pi\)
\(60\) −1.27014 −0.163975
\(61\) 4.49166 0.575098 0.287549 0.957766i \(-0.407160\pi\)
0.287549 + 0.957766i \(0.407160\pi\)
\(62\) 1.00000 0.127000
\(63\) 5.48196 0.690662
\(64\) 1.00000 0.125000
\(65\) 1.00000 0.124035
\(66\) 5.92369 0.729156
\(67\) −9.82335 −1.20011 −0.600057 0.799957i \(-0.704855\pi\)
−0.600057 + 0.799957i \(0.704855\pi\)
\(68\) −1.48534 −0.180124
\(69\) 9.92378 1.19468
\(70\) 3.95315 0.472491
\(71\) 9.44658 1.12110 0.560552 0.828120i \(-0.310589\pi\)
0.560552 + 0.828120i \(0.310589\pi\)
\(72\) 1.38673 0.163428
\(73\) 3.09187 0.361876 0.180938 0.983495i \(-0.442087\pi\)
0.180938 + 0.983495i \(0.442087\pi\)
\(74\) 3.50437 0.407375
\(75\) −1.27014 −0.146664
\(76\) 3.80214 0.436136
\(77\) −18.4367 −2.10105
\(78\) 1.27014 0.143816
\(79\) 0.814856 0.0916785 0.0458392 0.998949i \(-0.485404\pi\)
0.0458392 + 0.998949i \(0.485404\pi\)
\(80\) 1.00000 0.111803
\(81\) −2.91678 −0.324086
\(82\) 6.09942 0.673568
\(83\) −7.28306 −0.799420 −0.399710 0.916642i \(-0.630889\pi\)
−0.399710 + 0.916642i \(0.630889\pi\)
\(84\) 5.02107 0.547843
\(85\) −1.48534 −0.161108
\(86\) −3.20595 −0.345706
\(87\) −0.475013 −0.0509268
\(88\) −4.66379 −0.497162
\(89\) −6.30532 −0.668363 −0.334181 0.942509i \(-0.608460\pi\)
−0.334181 + 0.942509i \(0.608460\pi\)
\(90\) 1.38673 0.146174
\(91\) −3.95315 −0.414402
\(92\) −7.81311 −0.814573
\(93\) 1.27014 0.131708
\(94\) −6.40206 −0.660322
\(95\) 3.80214 0.390092
\(96\) 1.27014 0.129634
\(97\) −12.8930 −1.30909 −0.654543 0.756025i \(-0.727139\pi\)
−0.654543 + 0.756025i \(0.727139\pi\)
\(98\) −8.62737 −0.871496
\(99\) −6.46743 −0.650001
\(100\) 1.00000 0.100000
\(101\) −12.0537 −1.19939 −0.599696 0.800228i \(-0.704712\pi\)
−0.599696 + 0.800228i \(0.704712\pi\)
\(102\) −1.88660 −0.186801
\(103\) −11.1145 −1.09515 −0.547573 0.836758i \(-0.684448\pi\)
−0.547573 + 0.836758i \(0.684448\pi\)
\(104\) −1.00000 −0.0980581
\(105\) 5.02107 0.490006
\(106\) −9.25929 −0.899342
\(107\) −4.49281 −0.434336 −0.217168 0.976134i \(-0.569682\pi\)
−0.217168 + 0.976134i \(0.569682\pi\)
\(108\) 5.57178 0.536145
\(109\) −14.8468 −1.42207 −0.711033 0.703158i \(-0.751772\pi\)
−0.711033 + 0.703158i \(0.751772\pi\)
\(110\) −4.66379 −0.444675
\(111\) 4.45106 0.422476
\(112\) −3.95315 −0.373537
\(113\) 17.2057 1.61858 0.809288 0.587412i \(-0.199853\pi\)
0.809288 + 0.587412i \(0.199853\pi\)
\(114\) 4.82927 0.452303
\(115\) −7.81311 −0.728576
\(116\) 0.373984 0.0347235
\(117\) −1.38673 −0.128203
\(118\) −6.57218 −0.605018
\(119\) 5.87178 0.538265
\(120\) 1.27014 0.115948
\(121\) 10.7510 0.977359
\(122\) −4.49166 −0.406656
\(123\) 7.74714 0.698536
\(124\) −1.00000 −0.0898027
\(125\) 1.00000 0.0894427
\(126\) −5.48196 −0.488371
\(127\) 13.0366 1.15681 0.578407 0.815749i \(-0.303675\pi\)
0.578407 + 0.815749i \(0.303675\pi\)
\(128\) −1.00000 −0.0883883
\(129\) −4.07202 −0.358521
\(130\) −1.00000 −0.0877058
\(131\) 20.8577 1.82234 0.911171 0.412029i \(-0.135180\pi\)
0.911171 + 0.412029i \(0.135180\pi\)
\(132\) −5.92369 −0.515591
\(133\) −15.0304 −1.30330
\(134\) 9.82335 0.848609
\(135\) 5.57178 0.479543
\(136\) 1.48534 0.127367
\(137\) −8.09473 −0.691580 −0.345790 0.938312i \(-0.612389\pi\)
−0.345790 + 0.938312i \(0.612389\pi\)
\(138\) −9.92378 −0.844768
\(139\) −8.76155 −0.743145 −0.371572 0.928404i \(-0.621181\pi\)
−0.371572 + 0.928404i \(0.621181\pi\)
\(140\) −3.95315 −0.334102
\(141\) −8.13155 −0.684800
\(142\) −9.44658 −0.792740
\(143\) 4.66379 0.390006
\(144\) −1.38673 −0.115561
\(145\) 0.373984 0.0310577
\(146\) −3.09187 −0.255885
\(147\) −10.9580 −0.903801
\(148\) −3.50437 −0.288058
\(149\) −2.23059 −0.182737 −0.0913686 0.995817i \(-0.529124\pi\)
−0.0913686 + 0.995817i \(0.529124\pi\)
\(150\) 1.27014 0.103707
\(151\) 7.29600 0.593740 0.296870 0.954918i \(-0.404057\pi\)
0.296870 + 0.954918i \(0.404057\pi\)
\(152\) −3.80214 −0.308395
\(153\) 2.05977 0.166523
\(154\) 18.4367 1.48567
\(155\) −1.00000 −0.0803219
\(156\) −1.27014 −0.101693
\(157\) −17.2073 −1.37329 −0.686644 0.726994i \(-0.740917\pi\)
−0.686644 + 0.726994i \(0.740917\pi\)
\(158\) −0.814856 −0.0648265
\(159\) −11.7606 −0.932679
\(160\) −1.00000 −0.0790569
\(161\) 30.8864 2.43419
\(162\) 2.91678 0.229164
\(163\) 11.3738 0.890865 0.445433 0.895315i \(-0.353050\pi\)
0.445433 + 0.895315i \(0.353050\pi\)
\(164\) −6.09942 −0.476284
\(165\) −5.92369 −0.461159
\(166\) 7.28306 0.565275
\(167\) −2.35504 −0.182238 −0.0911191 0.995840i \(-0.529044\pi\)
−0.0911191 + 0.995840i \(0.529044\pi\)
\(168\) −5.02107 −0.387384
\(169\) 1.00000 0.0769231
\(170\) 1.48534 0.113921
\(171\) −5.27256 −0.403202
\(172\) 3.20595 0.244451
\(173\) 20.9496 1.59277 0.796385 0.604791i \(-0.206743\pi\)
0.796385 + 0.604791i \(0.206743\pi\)
\(174\) 0.475013 0.0360107
\(175\) −3.95315 −0.298830
\(176\) 4.66379 0.351547
\(177\) −8.34762 −0.627445
\(178\) 6.30532 0.472604
\(179\) −19.4658 −1.45494 −0.727470 0.686140i \(-0.759304\pi\)
−0.727470 + 0.686140i \(0.759304\pi\)
\(180\) −1.38673 −0.103361
\(181\) −5.12519 −0.380952 −0.190476 0.981692i \(-0.561003\pi\)
−0.190476 + 0.981692i \(0.561003\pi\)
\(182\) 3.95315 0.293027
\(183\) −5.70506 −0.421730
\(184\) 7.81311 0.575990
\(185\) −3.50437 −0.257647
\(186\) −1.27014 −0.0931315
\(187\) −6.92733 −0.506577
\(188\) 6.40206 0.466918
\(189\) −22.0261 −1.60216
\(190\) −3.80214 −0.275836
\(191\) −4.28978 −0.310397 −0.155199 0.987883i \(-0.549602\pi\)
−0.155199 + 0.987883i \(0.549602\pi\)
\(192\) −1.27014 −0.0916648
\(193\) −10.9142 −0.785621 −0.392810 0.919619i \(-0.628497\pi\)
−0.392810 + 0.919619i \(0.628497\pi\)
\(194\) 12.8930 0.925664
\(195\) −1.27014 −0.0909570
\(196\) 8.62737 0.616240
\(197\) −23.8116 −1.69650 −0.848251 0.529594i \(-0.822344\pi\)
−0.848251 + 0.529594i \(0.822344\pi\)
\(198\) 6.46743 0.459620
\(199\) 21.3366 1.51251 0.756255 0.654277i \(-0.227027\pi\)
0.756255 + 0.654277i \(0.227027\pi\)
\(200\) −1.00000 −0.0707107
\(201\) 12.4771 0.880065
\(202\) 12.0537 0.848098
\(203\) −1.47841 −0.103764
\(204\) 1.88660 0.132088
\(205\) −6.09942 −0.426002
\(206\) 11.1145 0.774385
\(207\) 10.8347 0.753063
\(208\) 1.00000 0.0693375
\(209\) 17.7324 1.22658
\(210\) −5.02107 −0.346487
\(211\) −21.8827 −1.50647 −0.753233 0.657753i \(-0.771507\pi\)
−0.753233 + 0.657753i \(0.771507\pi\)
\(212\) 9.25929 0.635931
\(213\) −11.9985 −0.822125
\(214\) 4.49281 0.307122
\(215\) 3.20595 0.218644
\(216\) −5.57178 −0.379112
\(217\) 3.95315 0.268357
\(218\) 14.8468 1.00555
\(219\) −3.92712 −0.265370
\(220\) 4.66379 0.314433
\(221\) −1.48534 −0.0999150
\(222\) −4.45106 −0.298736
\(223\) 26.9945 1.80769 0.903843 0.427864i \(-0.140734\pi\)
0.903843 + 0.427864i \(0.140734\pi\)
\(224\) 3.95315 0.264131
\(225\) −1.38673 −0.0924488
\(226\) −17.2057 −1.14451
\(227\) −13.6541 −0.906257 −0.453128 0.891445i \(-0.649692\pi\)
−0.453128 + 0.891445i \(0.649692\pi\)
\(228\) −4.82927 −0.319826
\(229\) −4.50691 −0.297825 −0.148913 0.988850i \(-0.547577\pi\)
−0.148913 + 0.988850i \(0.547577\pi\)
\(230\) 7.81311 0.515181
\(231\) 23.4172 1.54074
\(232\) −0.373984 −0.0245532
\(233\) −4.53438 −0.297057 −0.148529 0.988908i \(-0.547454\pi\)
−0.148529 + 0.988908i \(0.547454\pi\)
\(234\) 1.38673 0.0906535
\(235\) 6.40206 0.417625
\(236\) 6.57218 0.427812
\(237\) −1.03499 −0.0672295
\(238\) −5.87178 −0.380611
\(239\) −19.0123 −1.22981 −0.614903 0.788603i \(-0.710805\pi\)
−0.614903 + 0.788603i \(0.710805\pi\)
\(240\) −1.27014 −0.0819875
\(241\) −26.6169 −1.71455 −0.857274 0.514861i \(-0.827844\pi\)
−0.857274 + 0.514861i \(0.827844\pi\)
\(242\) −10.7510 −0.691097
\(243\) −13.0106 −0.834632
\(244\) 4.49166 0.287549
\(245\) 8.62737 0.551182
\(246\) −7.74714 −0.493940
\(247\) 3.80214 0.241925
\(248\) 1.00000 0.0635001
\(249\) 9.25055 0.586230
\(250\) −1.00000 −0.0632456
\(251\) −22.7877 −1.43835 −0.719173 0.694831i \(-0.755479\pi\)
−0.719173 + 0.694831i \(0.755479\pi\)
\(252\) 5.48196 0.345331
\(253\) −36.4387 −2.29088
\(254\) −13.0366 −0.817991
\(255\) 1.88660 0.118144
\(256\) 1.00000 0.0625000
\(257\) −10.3418 −0.645101 −0.322550 0.946552i \(-0.604540\pi\)
−0.322550 + 0.946552i \(0.604540\pi\)
\(258\) 4.07202 0.253513
\(259\) 13.8533 0.860802
\(260\) 1.00000 0.0620174
\(261\) −0.518615 −0.0321015
\(262\) −20.8577 −1.28859
\(263\) −13.4039 −0.826521 −0.413260 0.910613i \(-0.635610\pi\)
−0.413260 + 0.910613i \(0.635610\pi\)
\(264\) 5.92369 0.364578
\(265\) 9.25929 0.568794
\(266\) 15.0304 0.921575
\(267\) 8.00867 0.490123
\(268\) −9.82335 −0.600057
\(269\) −30.8839 −1.88303 −0.941514 0.336974i \(-0.890596\pi\)
−0.941514 + 0.336974i \(0.890596\pi\)
\(270\) −5.57178 −0.339088
\(271\) 5.83820 0.354645 0.177323 0.984153i \(-0.443256\pi\)
0.177323 + 0.984153i \(0.443256\pi\)
\(272\) −1.48534 −0.0900622
\(273\) 5.02107 0.303889
\(274\) 8.09473 0.489021
\(275\) 4.66379 0.281237
\(276\) 9.92378 0.597341
\(277\) 21.3108 1.28044 0.640219 0.768192i \(-0.278843\pi\)
0.640219 + 0.768192i \(0.278843\pi\)
\(278\) 8.76155 0.525483
\(279\) 1.38673 0.0830215
\(280\) 3.95315 0.236246
\(281\) −20.7103 −1.23547 −0.617737 0.786385i \(-0.711950\pi\)
−0.617737 + 0.786385i \(0.711950\pi\)
\(282\) 8.13155 0.484227
\(283\) −2.82014 −0.167640 −0.0838199 0.996481i \(-0.526712\pi\)
−0.0838199 + 0.996481i \(0.526712\pi\)
\(284\) 9.44658 0.560552
\(285\) −4.82927 −0.286061
\(286\) −4.66379 −0.275776
\(287\) 24.1119 1.42328
\(288\) 1.38673 0.0817140
\(289\) −14.7938 −0.870221
\(290\) −0.373984 −0.0219611
\(291\) 16.3760 0.959977
\(292\) 3.09187 0.180938
\(293\) 15.1264 0.883693 0.441846 0.897091i \(-0.354324\pi\)
0.441846 + 0.897091i \(0.354324\pi\)
\(294\) 10.9580 0.639084
\(295\) 6.57218 0.382647
\(296\) 3.50437 0.203687
\(297\) 25.9856 1.50784
\(298\) 2.23059 0.129215
\(299\) −7.81311 −0.451844
\(300\) −1.27014 −0.0733318
\(301\) −12.6736 −0.730493
\(302\) −7.29600 −0.419837
\(303\) 15.3100 0.879536
\(304\) 3.80214 0.218068
\(305\) 4.49166 0.257192
\(306\) −2.05977 −0.117749
\(307\) 11.7199 0.668889 0.334445 0.942415i \(-0.391451\pi\)
0.334445 + 0.942415i \(0.391451\pi\)
\(308\) −18.4367 −1.05053
\(309\) 14.1171 0.803091
\(310\) 1.00000 0.0567962
\(311\) −10.8823 −0.617078 −0.308539 0.951212i \(-0.599840\pi\)
−0.308539 + 0.951212i \(0.599840\pi\)
\(312\) 1.27014 0.0719078
\(313\) −21.9056 −1.23818 −0.619090 0.785320i \(-0.712498\pi\)
−0.619090 + 0.785320i \(0.712498\pi\)
\(314\) 17.2073 0.971061
\(315\) 5.48196 0.308873
\(316\) 0.814856 0.0458392
\(317\) 9.03617 0.507522 0.253761 0.967267i \(-0.418332\pi\)
0.253761 + 0.967267i \(0.418332\pi\)
\(318\) 11.7606 0.659504
\(319\) 1.74418 0.0976555
\(320\) 1.00000 0.0559017
\(321\) 5.70652 0.318507
\(322\) −30.8864 −1.72123
\(323\) −5.64749 −0.314235
\(324\) −2.91678 −0.162043
\(325\) 1.00000 0.0554700
\(326\) −11.3738 −0.629937
\(327\) 18.8576 1.04283
\(328\) 6.09942 0.336784
\(329\) −25.3083 −1.39529
\(330\) 5.92369 0.326088
\(331\) −2.78825 −0.153256 −0.0766280 0.997060i \(-0.524415\pi\)
−0.0766280 + 0.997060i \(0.524415\pi\)
\(332\) −7.28306 −0.399710
\(333\) 4.85963 0.266306
\(334\) 2.35504 0.128862
\(335\) −9.82335 −0.536707
\(336\) 5.02107 0.273922
\(337\) −3.80184 −0.207099 −0.103550 0.994624i \(-0.533020\pi\)
−0.103550 + 0.994624i \(0.533020\pi\)
\(338\) −1.00000 −0.0543928
\(339\) −21.8537 −1.18693
\(340\) −1.48534 −0.0805541
\(341\) −4.66379 −0.252558
\(342\) 5.27256 0.285107
\(343\) −6.43321 −0.347361
\(344\) −3.20595 −0.172853
\(345\) 9.92378 0.534278
\(346\) −20.9496 −1.12626
\(347\) −32.7754 −1.75947 −0.879737 0.475462i \(-0.842281\pi\)
−0.879737 + 0.475462i \(0.842281\pi\)
\(348\) −0.475013 −0.0254634
\(349\) 23.5163 1.25880 0.629398 0.777083i \(-0.283301\pi\)
0.629398 + 0.777083i \(0.283301\pi\)
\(350\) 3.95315 0.211305
\(351\) 5.57178 0.297400
\(352\) −4.66379 −0.248581
\(353\) −18.9683 −1.00958 −0.504792 0.863241i \(-0.668431\pi\)
−0.504792 + 0.863241i \(0.668431\pi\)
\(354\) 8.34762 0.443671
\(355\) 9.44658 0.501373
\(356\) −6.30532 −0.334181
\(357\) −7.45801 −0.394720
\(358\) 19.4658 1.02880
\(359\) 17.0190 0.898227 0.449113 0.893475i \(-0.351740\pi\)
0.449113 + 0.893475i \(0.351740\pi\)
\(360\) 1.38673 0.0730872
\(361\) −4.54370 −0.239142
\(362\) 5.12519 0.269374
\(363\) −13.6553 −0.716715
\(364\) −3.95315 −0.207201
\(365\) 3.09187 0.161836
\(366\) 5.70506 0.298208
\(367\) −27.4066 −1.43061 −0.715305 0.698812i \(-0.753712\pi\)
−0.715305 + 0.698812i \(0.753712\pi\)
\(368\) −7.81311 −0.407286
\(369\) 8.45826 0.440319
\(370\) 3.50437 0.182184
\(371\) −36.6033 −1.90035
\(372\) 1.27014 0.0658539
\(373\) −14.4935 −0.750447 −0.375224 0.926934i \(-0.622434\pi\)
−0.375224 + 0.926934i \(0.622434\pi\)
\(374\) 6.92733 0.358204
\(375\) −1.27014 −0.0655900
\(376\) −6.40206 −0.330161
\(377\) 0.373984 0.0192611
\(378\) 22.0261 1.13290
\(379\) 5.12722 0.263368 0.131684 0.991292i \(-0.457962\pi\)
0.131684 + 0.991292i \(0.457962\pi\)
\(380\) 3.80214 0.195046
\(381\) −16.5584 −0.848313
\(382\) 4.28978 0.219484
\(383\) −4.34716 −0.222130 −0.111065 0.993813i \(-0.535426\pi\)
−0.111065 + 0.993813i \(0.535426\pi\)
\(384\) 1.27014 0.0648168
\(385\) −18.4367 −0.939619
\(386\) 10.9142 0.555518
\(387\) −4.44579 −0.225992
\(388\) −12.8930 −0.654543
\(389\) 16.4912 0.836135 0.418068 0.908416i \(-0.362708\pi\)
0.418068 + 0.908416i \(0.362708\pi\)
\(390\) 1.27014 0.0643163
\(391\) 11.6051 0.586898
\(392\) −8.62737 −0.435748
\(393\) −26.4922 −1.33636
\(394\) 23.8116 1.19961
\(395\) 0.814856 0.0409999
\(396\) −6.46743 −0.325001
\(397\) −9.92346 −0.498044 −0.249022 0.968498i \(-0.580109\pi\)
−0.249022 + 0.968498i \(0.580109\pi\)
\(398\) −21.3366 −1.06951
\(399\) 19.0908 0.955736
\(400\) 1.00000 0.0500000
\(401\) 29.1905 1.45770 0.728852 0.684671i \(-0.240054\pi\)
0.728852 + 0.684671i \(0.240054\pi\)
\(402\) −12.4771 −0.622300
\(403\) −1.00000 −0.0498135
\(404\) −12.0537 −0.599696
\(405\) −2.91678 −0.144936
\(406\) 1.47841 0.0733724
\(407\) −16.3437 −0.810125
\(408\) −1.88660 −0.0934007
\(409\) −13.9128 −0.687946 −0.343973 0.938980i \(-0.611773\pi\)
−0.343973 + 0.938980i \(0.611773\pi\)
\(410\) 6.09942 0.301229
\(411\) 10.2815 0.507148
\(412\) −11.1145 −0.547573
\(413\) −25.9808 −1.27843
\(414\) −10.8347 −0.532496
\(415\) −7.28306 −0.357512
\(416\) −1.00000 −0.0490290
\(417\) 11.1284 0.544962
\(418\) −17.7324 −0.867320
\(419\) −19.0311 −0.929729 −0.464864 0.885382i \(-0.653897\pi\)
−0.464864 + 0.885382i \(0.653897\pi\)
\(420\) 5.02107 0.245003
\(421\) 11.0709 0.539561 0.269781 0.962922i \(-0.413049\pi\)
0.269781 + 0.962922i \(0.413049\pi\)
\(422\) 21.8827 1.06523
\(423\) −8.87795 −0.431661
\(424\) −9.25929 −0.449671
\(425\) −1.48534 −0.0720497
\(426\) 11.9985 0.581330
\(427\) −17.7562 −0.859282
\(428\) −4.49281 −0.217168
\(429\) −5.92369 −0.285998
\(430\) −3.20595 −0.154605
\(431\) 28.2467 1.36059 0.680297 0.732936i \(-0.261851\pi\)
0.680297 + 0.732936i \(0.261851\pi\)
\(432\) 5.57178 0.268073
\(433\) −10.7500 −0.516612 −0.258306 0.966063i \(-0.583164\pi\)
−0.258306 + 0.966063i \(0.583164\pi\)
\(434\) −3.95315 −0.189757
\(435\) −0.475013 −0.0227752
\(436\) −14.8468 −0.711033
\(437\) −29.7066 −1.42106
\(438\) 3.92712 0.187645
\(439\) −17.5758 −0.838846 −0.419423 0.907791i \(-0.637768\pi\)
−0.419423 + 0.907791i \(0.637768\pi\)
\(440\) −4.66379 −0.222338
\(441\) −11.9638 −0.569707
\(442\) 1.48534 0.0706506
\(443\) −9.28428 −0.441109 −0.220555 0.975375i \(-0.570787\pi\)
−0.220555 + 0.975375i \(0.570787\pi\)
\(444\) 4.45106 0.211238
\(445\) −6.30532 −0.298901
\(446\) −26.9945 −1.27823
\(447\) 2.83317 0.134005
\(448\) −3.95315 −0.186769
\(449\) −8.32815 −0.393030 −0.196515 0.980501i \(-0.562962\pi\)
−0.196515 + 0.980501i \(0.562962\pi\)
\(450\) 1.38673 0.0653712
\(451\) −28.4464 −1.33949
\(452\) 17.2057 0.809288
\(453\) −9.26697 −0.435400
\(454\) 13.6541 0.640820
\(455\) −3.95315 −0.185326
\(456\) 4.82927 0.226151
\(457\) −41.2452 −1.92937 −0.964685 0.263405i \(-0.915154\pi\)
−0.964685 + 0.263405i \(0.915154\pi\)
\(458\) 4.50691 0.210594
\(459\) −8.27601 −0.386291
\(460\) −7.81311 −0.364288
\(461\) −24.8401 −1.15692 −0.578460 0.815711i \(-0.696346\pi\)
−0.578460 + 0.815711i \(0.696346\pi\)
\(462\) −23.4172 −1.08947
\(463\) 21.8812 1.01691 0.508453 0.861090i \(-0.330217\pi\)
0.508453 + 0.861090i \(0.330217\pi\)
\(464\) 0.373984 0.0173618
\(465\) 1.27014 0.0589015
\(466\) 4.53438 0.210051
\(467\) 2.00619 0.0928355 0.0464177 0.998922i \(-0.485219\pi\)
0.0464177 + 0.998922i \(0.485219\pi\)
\(468\) −1.38673 −0.0641017
\(469\) 38.8332 1.79315
\(470\) −6.40206 −0.295305
\(471\) 21.8557 1.00706
\(472\) −6.57218 −0.302509
\(473\) 14.9519 0.687488
\(474\) 1.03499 0.0475384
\(475\) 3.80214 0.174454
\(476\) 5.87178 0.269133
\(477\) −12.8402 −0.587910
\(478\) 19.0123 0.869604
\(479\) 11.5411 0.527328 0.263664 0.964615i \(-0.415069\pi\)
0.263664 + 0.964615i \(0.415069\pi\)
\(480\) 1.27014 0.0579739
\(481\) −3.50437 −0.159786
\(482\) 26.6169 1.21237
\(483\) −39.2301 −1.78503
\(484\) 10.7510 0.488680
\(485\) −12.8930 −0.585441
\(486\) 13.0106 0.590174
\(487\) 21.5414 0.976132 0.488066 0.872807i \(-0.337703\pi\)
0.488066 + 0.872807i \(0.337703\pi\)
\(488\) −4.49166 −0.203328
\(489\) −14.4464 −0.653288
\(490\) −8.62737 −0.389745
\(491\) 7.25724 0.327515 0.163757 0.986501i \(-0.447639\pi\)
0.163757 + 0.986501i \(0.447639\pi\)
\(492\) 7.74714 0.349268
\(493\) −0.555494 −0.0250182
\(494\) −3.80214 −0.171067
\(495\) −6.46743 −0.290689
\(496\) −1.00000 −0.0449013
\(497\) −37.3437 −1.67509
\(498\) −9.25055 −0.414527
\(499\) 14.8212 0.663488 0.331744 0.943369i \(-0.392363\pi\)
0.331744 + 0.943369i \(0.392363\pi\)
\(500\) 1.00000 0.0447214
\(501\) 2.99124 0.133639
\(502\) 22.7877 1.01706
\(503\) 20.3670 0.908122 0.454061 0.890971i \(-0.349975\pi\)
0.454061 + 0.890971i \(0.349975\pi\)
\(504\) −5.48196 −0.244186
\(505\) −12.0537 −0.536384
\(506\) 36.4387 1.61990
\(507\) −1.27014 −0.0564091
\(508\) 13.0366 0.578407
\(509\) −3.57242 −0.158345 −0.0791723 0.996861i \(-0.525228\pi\)
−0.0791723 + 0.996861i \(0.525228\pi\)
\(510\) −1.88660 −0.0835401
\(511\) −12.2226 −0.540696
\(512\) −1.00000 −0.0441942
\(513\) 21.1847 0.935328
\(514\) 10.3418 0.456155
\(515\) −11.1145 −0.489764
\(516\) −4.07202 −0.179261
\(517\) 29.8579 1.31315
\(518\) −13.8533 −0.608679
\(519\) −26.6090 −1.16801
\(520\) −1.00000 −0.0438529
\(521\) −37.6544 −1.64967 −0.824834 0.565375i \(-0.808731\pi\)
−0.824834 + 0.565375i \(0.808731\pi\)
\(522\) 0.518615 0.0226992
\(523\) 33.1986 1.45167 0.725837 0.687867i \(-0.241453\pi\)
0.725837 + 0.687867i \(0.241453\pi\)
\(524\) 20.8577 0.911171
\(525\) 5.02107 0.219137
\(526\) 13.4039 0.584439
\(527\) 1.48534 0.0647026
\(528\) −5.92369 −0.257796
\(529\) 38.0447 1.65412
\(530\) −9.25929 −0.402198
\(531\) −9.11385 −0.395508
\(532\) −15.0304 −0.651652
\(533\) −6.09942 −0.264195
\(534\) −8.00867 −0.346569
\(535\) −4.49281 −0.194241
\(536\) 9.82335 0.424304
\(537\) 24.7243 1.06693
\(538\) 30.8839 1.33150
\(539\) 40.2362 1.73310
\(540\) 5.57178 0.239771
\(541\) 2.77824 0.119446 0.0597229 0.998215i \(-0.480978\pi\)
0.0597229 + 0.998215i \(0.480978\pi\)
\(542\) −5.83820 −0.250772
\(543\) 6.50973 0.279359
\(544\) 1.48534 0.0636836
\(545\) −14.8468 −0.635967
\(546\) −5.02107 −0.214882
\(547\) −17.7341 −0.758254 −0.379127 0.925345i \(-0.623776\pi\)
−0.379127 + 0.925345i \(0.623776\pi\)
\(548\) −8.09473 −0.345790
\(549\) −6.22873 −0.265836
\(550\) −4.66379 −0.198865
\(551\) 1.42194 0.0605767
\(552\) −9.92378 −0.422384
\(553\) −3.22125 −0.136981
\(554\) −21.3108 −0.905407
\(555\) 4.45106 0.188937
\(556\) −8.76155 −0.371572
\(557\) −18.7270 −0.793487 −0.396744 0.917929i \(-0.629860\pi\)
−0.396744 + 0.917929i \(0.629860\pi\)
\(558\) −1.38673 −0.0587051
\(559\) 3.20595 0.135597
\(560\) −3.95315 −0.167051
\(561\) 8.79871 0.371482
\(562\) 20.7103 0.873612
\(563\) −7.49091 −0.315704 −0.157852 0.987463i \(-0.550457\pi\)
−0.157852 + 0.987463i \(0.550457\pi\)
\(564\) −8.13155 −0.342400
\(565\) 17.2057 0.723849
\(566\) 2.82014 0.118539
\(567\) 11.5304 0.484233
\(568\) −9.44658 −0.396370
\(569\) −26.4785 −1.11004 −0.555018 0.831838i \(-0.687289\pi\)
−0.555018 + 0.831838i \(0.687289\pi\)
\(570\) 4.82927 0.202276
\(571\) 17.3569 0.726362 0.363181 0.931719i \(-0.381691\pi\)
0.363181 + 0.931719i \(0.381691\pi\)
\(572\) 4.66379 0.195003
\(573\) 5.44864 0.227620
\(574\) −24.1119 −1.00641
\(575\) −7.81311 −0.325829
\(576\) −1.38673 −0.0577805
\(577\) −6.32021 −0.263114 −0.131557 0.991309i \(-0.541998\pi\)
−0.131557 + 0.991309i \(0.541998\pi\)
\(578\) 14.7938 0.615339
\(579\) 13.8626 0.576110
\(580\) 0.373984 0.0155288
\(581\) 28.7910 1.19445
\(582\) −16.3760 −0.678806
\(583\) 43.1834 1.78847
\(584\) −3.09187 −0.127942
\(585\) −1.38673 −0.0573343
\(586\) −15.1264 −0.624865
\(587\) −19.0343 −0.785629 −0.392815 0.919618i \(-0.628499\pi\)
−0.392815 + 0.919618i \(0.628499\pi\)
\(588\) −10.9580 −0.451900
\(589\) −3.80214 −0.156665
\(590\) −6.57218 −0.270572
\(591\) 30.2441 1.24408
\(592\) −3.50437 −0.144029
\(593\) −34.4793 −1.41590 −0.707948 0.706265i \(-0.750379\pi\)
−0.707948 + 0.706265i \(0.750379\pi\)
\(594\) −25.9856 −1.06620
\(595\) 5.87178 0.240719
\(596\) −2.23059 −0.0913686
\(597\) −27.1005 −1.10915
\(598\) 7.81311 0.319502
\(599\) −0.626472 −0.0255969 −0.0127985 0.999918i \(-0.504074\pi\)
−0.0127985 + 0.999918i \(0.504074\pi\)
\(600\) 1.27014 0.0518534
\(601\) 1.01389 0.0413575 0.0206788 0.999786i \(-0.493417\pi\)
0.0206788 + 0.999786i \(0.493417\pi\)
\(602\) 12.6736 0.516537
\(603\) 13.6224 0.554746
\(604\) 7.29600 0.296870
\(605\) 10.7510 0.437088
\(606\) −15.3100 −0.621926
\(607\) 23.2688 0.944450 0.472225 0.881478i \(-0.343451\pi\)
0.472225 + 0.881478i \(0.343451\pi\)
\(608\) −3.80214 −0.154197
\(609\) 1.87780 0.0760922
\(610\) −4.49166 −0.181862
\(611\) 6.40206 0.259000
\(612\) 2.05977 0.0832614
\(613\) 10.7274 0.433276 0.216638 0.976252i \(-0.430491\pi\)
0.216638 + 0.976252i \(0.430491\pi\)
\(614\) −11.7199 −0.472976
\(615\) 7.74714 0.312395
\(616\) 18.4367 0.742834
\(617\) 21.4576 0.863852 0.431926 0.901909i \(-0.357834\pi\)
0.431926 + 0.901909i \(0.357834\pi\)
\(618\) −14.1171 −0.567871
\(619\) 0.293212 0.0117852 0.00589258 0.999983i \(-0.498124\pi\)
0.00589258 + 0.999983i \(0.498124\pi\)
\(620\) −1.00000 −0.0401610
\(621\) −43.5330 −1.74692
\(622\) 10.8823 0.436340
\(623\) 24.9259 0.998633
\(624\) −1.27014 −0.0508465
\(625\) 1.00000 0.0400000
\(626\) 21.9056 0.875525
\(627\) −22.5227 −0.899471
\(628\) −17.2073 −0.686644
\(629\) 5.20520 0.207545
\(630\) −5.48196 −0.218406
\(631\) −23.2622 −0.926053 −0.463027 0.886344i \(-0.653237\pi\)
−0.463027 + 0.886344i \(0.653237\pi\)
\(632\) −0.814856 −0.0324132
\(633\) 27.7942 1.10472
\(634\) −9.03617 −0.358872
\(635\) 13.0366 0.517343
\(636\) −11.7606 −0.466340
\(637\) 8.62737 0.341829
\(638\) −1.74418 −0.0690528
\(639\) −13.0999 −0.518223
\(640\) −1.00000 −0.0395285
\(641\) −12.2104 −0.482281 −0.241140 0.970490i \(-0.577521\pi\)
−0.241140 + 0.970490i \(0.577521\pi\)
\(642\) −5.70652 −0.225218
\(643\) 0.445546 0.0175706 0.00878531 0.999961i \(-0.497204\pi\)
0.00878531 + 0.999961i \(0.497204\pi\)
\(644\) 30.8864 1.21709
\(645\) −4.07202 −0.160336
\(646\) 5.64749 0.222197
\(647\) −32.2420 −1.26757 −0.633783 0.773511i \(-0.718499\pi\)
−0.633783 + 0.773511i \(0.718499\pi\)
\(648\) 2.91678 0.114582
\(649\) 30.6513 1.20317
\(650\) −1.00000 −0.0392232
\(651\) −5.02107 −0.196791
\(652\) 11.3738 0.445433
\(653\) −9.25671 −0.362243 −0.181121 0.983461i \(-0.557973\pi\)
−0.181121 + 0.983461i \(0.557973\pi\)
\(654\) −18.8576 −0.737390
\(655\) 20.8577 0.814976
\(656\) −6.09942 −0.238142
\(657\) −4.28759 −0.167275
\(658\) 25.3083 0.986620
\(659\) −38.8000 −1.51143 −0.755717 0.654899i \(-0.772711\pi\)
−0.755717 + 0.654899i \(0.772711\pi\)
\(660\) −5.92369 −0.230579
\(661\) 35.4376 1.37836 0.689181 0.724589i \(-0.257971\pi\)
0.689181 + 0.724589i \(0.257971\pi\)
\(662\) 2.78825 0.108368
\(663\) 1.88660 0.0732695
\(664\) 7.28306 0.282638
\(665\) −15.0304 −0.582855
\(666\) −4.85963 −0.188307
\(667\) −2.92198 −0.113139
\(668\) −2.35504 −0.0911191
\(669\) −34.2869 −1.32561
\(670\) 9.82335 0.379509
\(671\) 20.9482 0.808695
\(672\) −5.02107 −0.193692
\(673\) −23.2486 −0.896168 −0.448084 0.893991i \(-0.647894\pi\)
−0.448084 + 0.893991i \(0.647894\pi\)
\(674\) 3.80184 0.146441
\(675\) 5.57178 0.214458
\(676\) 1.00000 0.0384615
\(677\) −40.6547 −1.56249 −0.781243 0.624227i \(-0.785414\pi\)
−0.781243 + 0.624227i \(0.785414\pi\)
\(678\) 21.8537 0.839287
\(679\) 50.9679 1.95597
\(680\) 1.48534 0.0569603
\(681\) 17.3427 0.664575
\(682\) 4.66379 0.178586
\(683\) −2.20624 −0.0844194 −0.0422097 0.999109i \(-0.513440\pi\)
−0.0422097 + 0.999109i \(0.513440\pi\)
\(684\) −5.27256 −0.201601
\(685\) −8.09473 −0.309284
\(686\) 6.43321 0.245621
\(687\) 5.72443 0.218401
\(688\) 3.20595 0.122226
\(689\) 9.25929 0.352751
\(690\) −9.92378 −0.377792
\(691\) 42.3349 1.61050 0.805248 0.592938i \(-0.202032\pi\)
0.805248 + 0.592938i \(0.202032\pi\)
\(692\) 20.9496 0.796385
\(693\) 25.5667 0.971199
\(694\) 32.7754 1.24414
\(695\) −8.76155 −0.332344
\(696\) 0.475013 0.0180053
\(697\) 9.05973 0.343162
\(698\) −23.5163 −0.890104
\(699\) 5.75932 0.217838
\(700\) −3.95315 −0.149415
\(701\) −27.5771 −1.04157 −0.520786 0.853687i \(-0.674361\pi\)
−0.520786 + 0.853687i \(0.674361\pi\)
\(702\) −5.57178 −0.210293
\(703\) −13.3241 −0.502529
\(704\) 4.66379 0.175773
\(705\) −8.13155 −0.306252
\(706\) 18.9683 0.713883
\(707\) 47.6502 1.79207
\(708\) −8.34762 −0.313723
\(709\) 37.4283 1.40565 0.702824 0.711364i \(-0.251922\pi\)
0.702824 + 0.711364i \(0.251922\pi\)
\(710\) −9.44658 −0.354524
\(711\) −1.12999 −0.0423778
\(712\) 6.30532 0.236302
\(713\) 7.81311 0.292603
\(714\) 7.45801 0.279109
\(715\) 4.66379 0.174416
\(716\) −19.4658 −0.727470
\(717\) 24.1484 0.901839
\(718\) −17.0190 −0.635142
\(719\) −10.3837 −0.387246 −0.193623 0.981076i \(-0.562024\pi\)
−0.193623 + 0.981076i \(0.562024\pi\)
\(720\) −1.38673 −0.0516805
\(721\) 43.9373 1.63631
\(722\) 4.54370 0.169099
\(723\) 33.8074 1.25731
\(724\) −5.12519 −0.190476
\(725\) 0.373984 0.0138894
\(726\) 13.6553 0.506794
\(727\) 46.3517 1.71909 0.859544 0.511061i \(-0.170747\pi\)
0.859544 + 0.511061i \(0.170747\pi\)
\(728\) 3.95315 0.146513
\(729\) 25.2757 0.936137
\(730\) −3.09187 −0.114435
\(731\) −4.76193 −0.176127
\(732\) −5.70506 −0.210865
\(733\) −37.4186 −1.38209 −0.691043 0.722814i \(-0.742849\pi\)
−0.691043 + 0.722814i \(0.742849\pi\)
\(734\) 27.4066 1.01159
\(735\) −10.9580 −0.404192
\(736\) 7.81311 0.287995
\(737\) −45.8141 −1.68758
\(738\) −8.45826 −0.311353
\(739\) −42.5966 −1.56694 −0.783471 0.621429i \(-0.786552\pi\)
−0.783471 + 0.621429i \(0.786552\pi\)
\(740\) −3.50437 −0.128823
\(741\) −4.82927 −0.177408
\(742\) 36.6033 1.34375
\(743\) 33.0581 1.21278 0.606392 0.795166i \(-0.292616\pi\)
0.606392 + 0.795166i \(0.292616\pi\)
\(744\) −1.27014 −0.0465658
\(745\) −2.23059 −0.0817226
\(746\) 14.4935 0.530646
\(747\) 10.0997 0.369527
\(748\) −6.92733 −0.253288
\(749\) 17.7607 0.648963
\(750\) 1.27014 0.0463791
\(751\) −12.7191 −0.464125 −0.232063 0.972701i \(-0.574547\pi\)
−0.232063 + 0.972701i \(0.574547\pi\)
\(752\) 6.40206 0.233459
\(753\) 28.9437 1.05477
\(754\) −0.373984 −0.0136197
\(755\) 7.29600 0.265528
\(756\) −22.0261 −0.801081
\(757\) 16.7647 0.609325 0.304662 0.952460i \(-0.401456\pi\)
0.304662 + 0.952460i \(0.401456\pi\)
\(758\) −5.12722 −0.186229
\(759\) 46.2824 1.67995
\(760\) −3.80214 −0.137918
\(761\) 39.7382 1.44051 0.720254 0.693710i \(-0.244025\pi\)
0.720254 + 0.693710i \(0.244025\pi\)
\(762\) 16.5584 0.599848
\(763\) 58.6916 2.12478
\(764\) −4.28978 −0.155199
\(765\) 2.05977 0.0744713
\(766\) 4.34716 0.157069
\(767\) 6.57218 0.237308
\(768\) −1.27014 −0.0458324
\(769\) 18.6585 0.672842 0.336421 0.941712i \(-0.390783\pi\)
0.336421 + 0.941712i \(0.390783\pi\)
\(770\) 18.4367 0.664411
\(771\) 13.1355 0.473064
\(772\) −10.9142 −0.392810
\(773\) 17.3426 0.623770 0.311885 0.950120i \(-0.399040\pi\)
0.311885 + 0.950120i \(0.399040\pi\)
\(774\) 4.44579 0.159801
\(775\) −1.00000 −0.0359211
\(776\) 12.8930 0.462832
\(777\) −17.5957 −0.631242
\(778\) −16.4912 −0.591237
\(779\) −23.1909 −0.830899
\(780\) −1.27014 −0.0454785
\(781\) 44.0569 1.57648
\(782\) −11.6051 −0.414999
\(783\) 2.08376 0.0744674
\(784\) 8.62737 0.308120
\(785\) −17.2073 −0.614153
\(786\) 26.4922 0.944947
\(787\) −10.9153 −0.389088 −0.194544 0.980894i \(-0.562323\pi\)
−0.194544 + 0.980894i \(0.562323\pi\)
\(788\) −23.8116 −0.848251
\(789\) 17.0249 0.606103
\(790\) −0.814856 −0.0289913
\(791\) −68.0166 −2.41839
\(792\) 6.46743 0.229810
\(793\) 4.49166 0.159503
\(794\) 9.92346 0.352170
\(795\) −11.7606 −0.417107
\(796\) 21.3366 0.756255
\(797\) −32.2537 −1.14249 −0.571243 0.820781i \(-0.693538\pi\)
−0.571243 + 0.820781i \(0.693538\pi\)
\(798\) −19.0908 −0.675808
\(799\) −9.50926 −0.336414
\(800\) −1.00000 −0.0353553
\(801\) 8.74379 0.308947
\(802\) −29.1905 −1.03075
\(803\) 14.4198 0.508865
\(804\) 12.4771 0.440033
\(805\) 30.8864 1.08860
\(806\) 1.00000 0.0352235
\(807\) 39.2271 1.38086
\(808\) 12.0537 0.424049
\(809\) −26.4704 −0.930650 −0.465325 0.885140i \(-0.654062\pi\)
−0.465325 + 0.885140i \(0.654062\pi\)
\(810\) 2.91678 0.102485
\(811\) 11.9476 0.419536 0.209768 0.977751i \(-0.432729\pi\)
0.209768 + 0.977751i \(0.432729\pi\)
\(812\) −1.47841 −0.0518821
\(813\) −7.41536 −0.260068
\(814\) 16.3437 0.572845
\(815\) 11.3738 0.398407
\(816\) 1.88660 0.0660442
\(817\) 12.1895 0.426456
\(818\) 13.9128 0.486451
\(819\) 5.48196 0.191555
\(820\) −6.09942 −0.213001
\(821\) 39.7075 1.38580 0.692901 0.721033i \(-0.256332\pi\)
0.692901 + 0.721033i \(0.256332\pi\)
\(822\) −10.2815 −0.358608
\(823\) 49.0554 1.70996 0.854982 0.518658i \(-0.173568\pi\)
0.854982 + 0.518658i \(0.173568\pi\)
\(824\) 11.1145 0.387193
\(825\) −5.92369 −0.206236
\(826\) 25.9808 0.903987
\(827\) 45.3421 1.57670 0.788349 0.615228i \(-0.210936\pi\)
0.788349 + 0.615228i \(0.210936\pi\)
\(828\) 10.8347 0.376532
\(829\) 28.1278 0.976918 0.488459 0.872587i \(-0.337559\pi\)
0.488459 + 0.872587i \(0.337559\pi\)
\(830\) 7.28306 0.252799
\(831\) −27.0677 −0.938969
\(832\) 1.00000 0.0346688
\(833\) −12.8146 −0.444000
\(834\) −11.1284 −0.385346
\(835\) −2.35504 −0.0814994
\(836\) 17.7324 0.613288
\(837\) −5.57178 −0.192589
\(838\) 19.0311 0.657417
\(839\) 23.0958 0.797354 0.398677 0.917091i \(-0.369469\pi\)
0.398677 + 0.917091i \(0.369469\pi\)
\(840\) −5.02107 −0.173243
\(841\) −28.8601 −0.995177
\(842\) −11.0709 −0.381527
\(843\) 26.3051 0.905995
\(844\) −21.8827 −0.753233
\(845\) 1.00000 0.0344010
\(846\) 8.87795 0.305230
\(847\) −42.5001 −1.46032
\(848\) 9.25929 0.317965
\(849\) 3.58199 0.122933
\(850\) 1.48534 0.0509469
\(851\) 27.3800 0.938576
\(852\) −11.9985 −0.411063
\(853\) −15.9738 −0.546932 −0.273466 0.961882i \(-0.588170\pi\)
−0.273466 + 0.961882i \(0.588170\pi\)
\(854\) 17.7562 0.607604
\(855\) −5.27256 −0.180318
\(856\) 4.49281 0.153561
\(857\) 25.0465 0.855572 0.427786 0.903880i \(-0.359294\pi\)
0.427786 + 0.903880i \(0.359294\pi\)
\(858\) 5.92369 0.202231
\(859\) 7.00985 0.239173 0.119586 0.992824i \(-0.461843\pi\)
0.119586 + 0.992824i \(0.461843\pi\)
\(860\) 3.20595 0.109322
\(861\) −30.6256 −1.04372
\(862\) −28.2467 −0.962086
\(863\) 18.1111 0.616508 0.308254 0.951304i \(-0.400255\pi\)
0.308254 + 0.951304i \(0.400255\pi\)
\(864\) −5.57178 −0.189556
\(865\) 20.9496 0.712308
\(866\) 10.7500 0.365300
\(867\) 18.7902 0.638149
\(868\) 3.95315 0.134179
\(869\) 3.80032 0.128917
\(870\) 0.475013 0.0161045
\(871\) −9.82335 −0.332852
\(872\) 14.8468 0.502776
\(873\) 17.8792 0.605118
\(874\) 29.7066 1.00484
\(875\) −3.95315 −0.133641
\(876\) −3.92712 −0.132685
\(877\) −30.0882 −1.01601 −0.508003 0.861355i \(-0.669616\pi\)
−0.508003 + 0.861355i \(0.669616\pi\)
\(878\) 17.5758 0.593154
\(879\) −19.2127 −0.648028
\(880\) 4.66379 0.157216
\(881\) −20.0590 −0.675804 −0.337902 0.941181i \(-0.609717\pi\)
−0.337902 + 0.941181i \(0.609717\pi\)
\(882\) 11.9638 0.402844
\(883\) 53.5511 1.80214 0.901070 0.433675i \(-0.142783\pi\)
0.901070 + 0.433675i \(0.142783\pi\)
\(884\) −1.48534 −0.0499575
\(885\) −8.34762 −0.280602
\(886\) 9.28428 0.311911
\(887\) −55.5046 −1.86366 −0.931831 0.362891i \(-0.881790\pi\)
−0.931831 + 0.362891i \(0.881790\pi\)
\(888\) −4.45106 −0.149368
\(889\) −51.5357 −1.72845
\(890\) 6.30532 0.211355
\(891\) −13.6032 −0.455725
\(892\) 26.9945 0.903843
\(893\) 24.3416 0.814559
\(894\) −2.83317 −0.0947555
\(895\) −19.4658 −0.650669
\(896\) 3.95315 0.132065
\(897\) 9.92378 0.331345
\(898\) 8.32815 0.277914
\(899\) −0.373984 −0.0124731
\(900\) −1.38673 −0.0462244
\(901\) −13.7532 −0.458186
\(902\) 28.4464 0.947162
\(903\) 16.0973 0.535684
\(904\) −17.2057 −0.572253
\(905\) −5.12519 −0.170367
\(906\) 9.26697 0.307874
\(907\) 48.5593 1.61239 0.806193 0.591652i \(-0.201524\pi\)
0.806193 + 0.591652i \(0.201524\pi\)
\(908\) −13.6541 −0.453128
\(909\) 16.7153 0.554412
\(910\) 3.95315 0.131046
\(911\) 28.9992 0.960785 0.480393 0.877054i \(-0.340494\pi\)
0.480393 + 0.877054i \(0.340494\pi\)
\(912\) −4.82927 −0.159913
\(913\) −33.9667 −1.12413
\(914\) 41.2452 1.36427
\(915\) −5.70506 −0.188603
\(916\) −4.50691 −0.148913
\(917\) −82.4534 −2.72285
\(918\) 8.27601 0.273149
\(919\) −25.7246 −0.848577 −0.424289 0.905527i \(-0.639476\pi\)
−0.424289 + 0.905527i \(0.639476\pi\)
\(920\) 7.81311 0.257591
\(921\) −14.8859 −0.490509
\(922\) 24.8401 0.818066
\(923\) 9.44658 0.310938
\(924\) 23.4172 0.770370
\(925\) −3.50437 −0.115223
\(926\) −21.8812 −0.719061
\(927\) 15.4129 0.506225
\(928\) −0.373984 −0.0122766
\(929\) −5.02671 −0.164921 −0.0824605 0.996594i \(-0.526278\pi\)
−0.0824605 + 0.996594i \(0.526278\pi\)
\(930\) −1.27014 −0.0416497
\(931\) 32.8025 1.07506
\(932\) −4.53438 −0.148529
\(933\) 13.8221 0.452515
\(934\) −2.00619 −0.0656446
\(935\) −6.92733 −0.226548
\(936\) 1.38673 0.0453268
\(937\) 36.0203 1.17673 0.588366 0.808595i \(-0.299771\pi\)
0.588366 + 0.808595i \(0.299771\pi\)
\(938\) −38.8332 −1.26795
\(939\) 27.8233 0.907979
\(940\) 6.40206 0.208812
\(941\) 37.8548 1.23403 0.617015 0.786951i \(-0.288342\pi\)
0.617015 + 0.786951i \(0.288342\pi\)
\(942\) −21.8557 −0.712097
\(943\) 47.6554 1.55187
\(944\) 6.57218 0.213906
\(945\) −22.0261 −0.716508
\(946\) −14.9519 −0.486127
\(947\) −3.56214 −0.115754 −0.0578770 0.998324i \(-0.518433\pi\)
−0.0578770 + 0.998324i \(0.518433\pi\)
\(948\) −1.03499 −0.0336148
\(949\) 3.09187 0.100366
\(950\) −3.80214 −0.123358
\(951\) −11.4772 −0.372175
\(952\) −5.87178 −0.190305
\(953\) 27.6781 0.896583 0.448291 0.893887i \(-0.352033\pi\)
0.448291 + 0.893887i \(0.352033\pi\)
\(954\) 12.8402 0.415715
\(955\) −4.28978 −0.138814
\(956\) −19.0123 −0.614903
\(957\) −2.21536 −0.0716125
\(958\) −11.5411 −0.372877
\(959\) 31.9997 1.03332
\(960\) −1.27014 −0.0409937
\(961\) 1.00000 0.0322581
\(962\) 3.50437 0.112985
\(963\) 6.23033 0.200769
\(964\) −26.6169 −0.857274
\(965\) −10.9142 −0.351340
\(966\) 39.2301 1.26221
\(967\) −26.6772 −0.857880 −0.428940 0.903333i \(-0.641113\pi\)
−0.428940 + 0.903333i \(0.641113\pi\)
\(968\) −10.7510 −0.345549
\(969\) 7.17313 0.230434
\(970\) 12.8930 0.413970
\(971\) −29.3861 −0.943043 −0.471522 0.881854i \(-0.656295\pi\)
−0.471522 + 0.881854i \(0.656295\pi\)
\(972\) −13.0106 −0.417316
\(973\) 34.6357 1.11037
\(974\) −21.5414 −0.690230
\(975\) −1.27014 −0.0406772
\(976\) 4.49166 0.143775
\(977\) −28.5134 −0.912224 −0.456112 0.889922i \(-0.650758\pi\)
−0.456112 + 0.889922i \(0.650758\pi\)
\(978\) 14.4464 0.461944
\(979\) −29.4067 −0.939842
\(980\) 8.62737 0.275591
\(981\) 20.5886 0.657342
\(982\) −7.25724 −0.231588
\(983\) −49.5439 −1.58021 −0.790103 0.612974i \(-0.789973\pi\)
−0.790103 + 0.612974i \(0.789973\pi\)
\(984\) −7.74714 −0.246970
\(985\) −23.8116 −0.758699
\(986\) 0.555494 0.0176905
\(987\) 32.1452 1.02319
\(988\) 3.80214 0.120962
\(989\) −25.0484 −0.796494
\(990\) 6.46743 0.205548
\(991\) 2.03355 0.0645979 0.0322989 0.999478i \(-0.489717\pi\)
0.0322989 + 0.999478i \(0.489717\pi\)
\(992\) 1.00000 0.0317500
\(993\) 3.54148 0.112385
\(994\) 37.3437 1.18447
\(995\) 21.3366 0.676415
\(996\) 9.25055 0.293115
\(997\) 9.50217 0.300937 0.150468 0.988615i \(-0.451922\pi\)
0.150468 + 0.988615i \(0.451922\pi\)
\(998\) −14.8212 −0.469157
\(999\) −19.5256 −0.617763
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 4030.2.a.d.1.2 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4030.2.a.d.1.2 6 1.1 even 1 trivial