Properties

Label 6018.2.a.bc.1.2
Level $6018$
Weight $2$
Character 6018.1
Self dual yes
Analytic conductor $48.054$
Analytic rank $0$
Dimension $14$
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] = [6018,2,Mod(1,6018)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6018, 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("6018.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6018 = 2 \cdot 3 \cdot 17 \cdot 59 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6018.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: \(48.0539719364\)
Analytic rank: \(0\)
Dimension: \(14\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{14} - 2 x^{13} - 49 x^{12} + 79 x^{11} + 956 x^{10} - 1179 x^{9} - 9396 x^{8} + 8315 x^{7} + 48570 x^{6} - 28124 x^{5} - 125592 x^{4} + 40576 x^{3} + 138096 x^{2} + \cdots - 43744 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-3.15319\) of defining polynomial
Character \(\chi\) \(=\) 6018.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -3.15319 q^{5} -1.00000 q^{6} -5.11963 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -3.15319 q^{5} -1.00000 q^{6} -5.11963 q^{7} +1.00000 q^{8} +1.00000 q^{9} -3.15319 q^{10} +2.30228 q^{11} -1.00000 q^{12} -0.811206 q^{13} -5.11963 q^{14} +3.15319 q^{15} +1.00000 q^{16} +1.00000 q^{17} +1.00000 q^{18} -5.24117 q^{19} -3.15319 q^{20} +5.11963 q^{21} +2.30228 q^{22} -3.26815 q^{23} -1.00000 q^{24} +4.94258 q^{25} -0.811206 q^{26} -1.00000 q^{27} -5.11963 q^{28} -8.27378 q^{29} +3.15319 q^{30} -8.46791 q^{31} +1.00000 q^{32} -2.30228 q^{33} +1.00000 q^{34} +16.1432 q^{35} +1.00000 q^{36} -4.64496 q^{37} -5.24117 q^{38} +0.811206 q^{39} -3.15319 q^{40} -4.54510 q^{41} +5.11963 q^{42} +0.284370 q^{43} +2.30228 q^{44} -3.15319 q^{45} -3.26815 q^{46} -6.61944 q^{47} -1.00000 q^{48} +19.2106 q^{49} +4.94258 q^{50} -1.00000 q^{51} -0.811206 q^{52} +13.2381 q^{53} -1.00000 q^{54} -7.25952 q^{55} -5.11963 q^{56} +5.24117 q^{57} -8.27378 q^{58} +1.00000 q^{59} +3.15319 q^{60} -2.14012 q^{61} -8.46791 q^{62} -5.11963 q^{63} +1.00000 q^{64} +2.55788 q^{65} -2.30228 q^{66} -13.5658 q^{67} +1.00000 q^{68} +3.26815 q^{69} +16.1432 q^{70} -2.05889 q^{71} +1.00000 q^{72} +12.6553 q^{73} -4.64496 q^{74} -4.94258 q^{75} -5.24117 q^{76} -11.7868 q^{77} +0.811206 q^{78} +4.35553 q^{79} -3.15319 q^{80} +1.00000 q^{81} -4.54510 q^{82} +7.85392 q^{83} +5.11963 q^{84} -3.15319 q^{85} +0.284370 q^{86} +8.27378 q^{87} +2.30228 q^{88} +5.97840 q^{89} -3.15319 q^{90} +4.15308 q^{91} -3.26815 q^{92} +8.46791 q^{93} -6.61944 q^{94} +16.5264 q^{95} -1.00000 q^{96} +5.62903 q^{97} +19.2106 q^{98} +2.30228 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q + 14 q^{2} - 14 q^{3} + 14 q^{4} + 2 q^{5} - 14 q^{6} + q^{7} + 14 q^{8} + 14 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 14 q + 14 q^{2} - 14 q^{3} + 14 q^{4} + 2 q^{5} - 14 q^{6} + q^{7} + 14 q^{8} + 14 q^{9} + 2 q^{10} + 3 q^{11} - 14 q^{12} + 16 q^{13} + q^{14} - 2 q^{15} + 14 q^{16} + 14 q^{17} + 14 q^{18} + 13 q^{19} + 2 q^{20} - q^{21} + 3 q^{22} + 4 q^{23} - 14 q^{24} + 32 q^{25} + 16 q^{26} - 14 q^{27} + q^{28} - 2 q^{30} - 13 q^{31} + 14 q^{32} - 3 q^{33} + 14 q^{34} + 14 q^{36} + 12 q^{37} + 13 q^{38} - 16 q^{39} + 2 q^{40} - 18 q^{41} - q^{42} + 29 q^{43} + 3 q^{44} + 2 q^{45} + 4 q^{46} - 14 q^{48} + 49 q^{49} + 32 q^{50} - 14 q^{51} + 16 q^{52} + 24 q^{53} - 14 q^{54} + 15 q^{55} + q^{56} - 13 q^{57} + 14 q^{59} - 2 q^{60} + 29 q^{61} - 13 q^{62} + q^{63} + 14 q^{64} + 6 q^{65} - 3 q^{66} + 4 q^{67} + 14 q^{68} - 4 q^{69} - 10 q^{71} + 14 q^{72} + 18 q^{73} + 12 q^{74} - 32 q^{75} + 13 q^{76} + 20 q^{77} - 16 q^{78} + 7 q^{79} + 2 q^{80} + 14 q^{81} - 18 q^{82} + 28 q^{83} - q^{84} + 2 q^{85} + 29 q^{86} + 3 q^{88} + 23 q^{89} + 2 q^{90} + 9 q^{91} + 4 q^{92} + 13 q^{93} + 5 q^{95} - 14 q^{96} - 7 q^{97} + 49 q^{98} + 3 q^{99}+O(q^{100}) \) 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
\(3\) −1.00000 −0.577350
\(4\) 1.00000 0.500000
\(5\) −3.15319 −1.41015 −0.705074 0.709134i \(-0.749086\pi\)
−0.705074 + 0.709134i \(0.749086\pi\)
\(6\) −1.00000 −0.408248
\(7\) −5.11963 −1.93504 −0.967520 0.252796i \(-0.918650\pi\)
−0.967520 + 0.252796i \(0.918650\pi\)
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) −3.15319 −0.997125
\(11\) 2.30228 0.694164 0.347082 0.937835i \(-0.387173\pi\)
0.347082 + 0.937835i \(0.387173\pi\)
\(12\) −1.00000 −0.288675
\(13\) −0.811206 −0.224988 −0.112494 0.993652i \(-0.535884\pi\)
−0.112494 + 0.993652i \(0.535884\pi\)
\(14\) −5.11963 −1.36828
\(15\) 3.15319 0.814149
\(16\) 1.00000 0.250000
\(17\) 1.00000 0.242536
\(18\) 1.00000 0.235702
\(19\) −5.24117 −1.20241 −0.601203 0.799096i \(-0.705312\pi\)
−0.601203 + 0.799096i \(0.705312\pi\)
\(20\) −3.15319 −0.705074
\(21\) 5.11963 1.11720
\(22\) 2.30228 0.490848
\(23\) −3.26815 −0.681457 −0.340729 0.940162i \(-0.610674\pi\)
−0.340729 + 0.940162i \(0.610674\pi\)
\(24\) −1.00000 −0.204124
\(25\) 4.94258 0.988516
\(26\) −0.811206 −0.159091
\(27\) −1.00000 −0.192450
\(28\) −5.11963 −0.967520
\(29\) −8.27378 −1.53640 −0.768201 0.640209i \(-0.778848\pi\)
−0.768201 + 0.640209i \(0.778848\pi\)
\(30\) 3.15319 0.575690
\(31\) −8.46791 −1.52088 −0.760441 0.649407i \(-0.775017\pi\)
−0.760441 + 0.649407i \(0.775017\pi\)
\(32\) 1.00000 0.176777
\(33\) −2.30228 −0.400776
\(34\) 1.00000 0.171499
\(35\) 16.1432 2.72869
\(36\) 1.00000 0.166667
\(37\) −4.64496 −0.763627 −0.381814 0.924239i \(-0.624700\pi\)
−0.381814 + 0.924239i \(0.624700\pi\)
\(38\) −5.24117 −0.850230
\(39\) 0.811206 0.129897
\(40\) −3.15319 −0.498562
\(41\) −4.54510 −0.709825 −0.354912 0.934900i \(-0.615489\pi\)
−0.354912 + 0.934900i \(0.615489\pi\)
\(42\) 5.11963 0.789977
\(43\) 0.284370 0.0433659 0.0216830 0.999765i \(-0.493098\pi\)
0.0216830 + 0.999765i \(0.493098\pi\)
\(44\) 2.30228 0.347082
\(45\) −3.15319 −0.470049
\(46\) −3.26815 −0.481863
\(47\) −6.61944 −0.965544 −0.482772 0.875746i \(-0.660370\pi\)
−0.482772 + 0.875746i \(0.660370\pi\)
\(48\) −1.00000 −0.144338
\(49\) 19.2106 2.74438
\(50\) 4.94258 0.698986
\(51\) −1.00000 −0.140028
\(52\) −0.811206 −0.112494
\(53\) 13.2381 1.81840 0.909199 0.416362i \(-0.136695\pi\)
0.909199 + 0.416362i \(0.136695\pi\)
\(54\) −1.00000 −0.136083
\(55\) −7.25952 −0.978873
\(56\) −5.11963 −0.684140
\(57\) 5.24117 0.694210
\(58\) −8.27378 −1.08640
\(59\) 1.00000 0.130189
\(60\) 3.15319 0.407074
\(61\) −2.14012 −0.274014 −0.137007 0.990570i \(-0.543748\pi\)
−0.137007 + 0.990570i \(0.543748\pi\)
\(62\) −8.46791 −1.07543
\(63\) −5.11963 −0.645013
\(64\) 1.00000 0.125000
\(65\) 2.55788 0.317266
\(66\) −2.30228 −0.283391
\(67\) −13.5658 −1.65733 −0.828666 0.559744i \(-0.810900\pi\)
−0.828666 + 0.559744i \(0.810900\pi\)
\(68\) 1.00000 0.121268
\(69\) 3.26815 0.393440
\(70\) 16.1432 1.92948
\(71\) −2.05889 −0.244345 −0.122172 0.992509i \(-0.538986\pi\)
−0.122172 + 0.992509i \(0.538986\pi\)
\(72\) 1.00000 0.117851
\(73\) 12.6553 1.48119 0.740594 0.671953i \(-0.234544\pi\)
0.740594 + 0.671953i \(0.234544\pi\)
\(74\) −4.64496 −0.539966
\(75\) −4.94258 −0.570720
\(76\) −5.24117 −0.601203
\(77\) −11.7868 −1.34323
\(78\) 0.811206 0.0918510
\(79\) 4.35553 0.490035 0.245018 0.969519i \(-0.421206\pi\)
0.245018 + 0.969519i \(0.421206\pi\)
\(80\) −3.15319 −0.352537
\(81\) 1.00000 0.111111
\(82\) −4.54510 −0.501922
\(83\) 7.85392 0.862080 0.431040 0.902333i \(-0.358147\pi\)
0.431040 + 0.902333i \(0.358147\pi\)
\(84\) 5.11963 0.558598
\(85\) −3.15319 −0.342011
\(86\) 0.284370 0.0306644
\(87\) 8.27378 0.887042
\(88\) 2.30228 0.245424
\(89\) 5.97840 0.633709 0.316854 0.948474i \(-0.397373\pi\)
0.316854 + 0.948474i \(0.397373\pi\)
\(90\) −3.15319 −0.332375
\(91\) 4.15308 0.435361
\(92\) −3.26815 −0.340729
\(93\) 8.46791 0.878081
\(94\) −6.61944 −0.682743
\(95\) 16.5264 1.69557
\(96\) −1.00000 −0.102062
\(97\) 5.62903 0.571542 0.285771 0.958298i \(-0.407750\pi\)
0.285771 + 0.958298i \(0.407750\pi\)
\(98\) 19.2106 1.94057
\(99\) 2.30228 0.231388
\(100\) 4.94258 0.494258
\(101\) −9.49591 −0.944878 −0.472439 0.881363i \(-0.656626\pi\)
−0.472439 + 0.881363i \(0.656626\pi\)
\(102\) −1.00000 −0.0990148
\(103\) −13.8577 −1.36544 −0.682719 0.730681i \(-0.739203\pi\)
−0.682719 + 0.730681i \(0.739203\pi\)
\(104\) −0.811206 −0.0795453
\(105\) −16.1432 −1.57541
\(106\) 13.2381 1.28580
\(107\) 16.6567 1.61026 0.805130 0.593098i \(-0.202095\pi\)
0.805130 + 0.593098i \(0.202095\pi\)
\(108\) −1.00000 −0.0962250
\(109\) −3.89060 −0.372652 −0.186326 0.982488i \(-0.559658\pi\)
−0.186326 + 0.982488i \(0.559658\pi\)
\(110\) −7.25952 −0.692168
\(111\) 4.64496 0.440880
\(112\) −5.11963 −0.483760
\(113\) −8.95400 −0.842322 −0.421161 0.906986i \(-0.638377\pi\)
−0.421161 + 0.906986i \(0.638377\pi\)
\(114\) 5.24117 0.490880
\(115\) 10.3051 0.960955
\(116\) −8.27378 −0.768201
\(117\) −0.811206 −0.0749960
\(118\) 1.00000 0.0920575
\(119\) −5.11963 −0.469316
\(120\) 3.15319 0.287845
\(121\) −5.69951 −0.518137
\(122\) −2.14012 −0.193757
\(123\) 4.54510 0.409818
\(124\) −8.46791 −0.760441
\(125\) 0.181060 0.0161945
\(126\) −5.11963 −0.456093
\(127\) 8.72754 0.774444 0.387222 0.921987i \(-0.373435\pi\)
0.387222 + 0.921987i \(0.373435\pi\)
\(128\) 1.00000 0.0883883
\(129\) −0.284370 −0.0250373
\(130\) 2.55788 0.224341
\(131\) −12.3287 −1.07716 −0.538580 0.842574i \(-0.681039\pi\)
−0.538580 + 0.842574i \(0.681039\pi\)
\(132\) −2.30228 −0.200388
\(133\) 26.8329 2.32670
\(134\) −13.5658 −1.17191
\(135\) 3.15319 0.271383
\(136\) 1.00000 0.0857493
\(137\) 4.23705 0.361995 0.180998 0.983484i \(-0.442067\pi\)
0.180998 + 0.983484i \(0.442067\pi\)
\(138\) 3.26815 0.278204
\(139\) −5.15360 −0.437122 −0.218561 0.975823i \(-0.570136\pi\)
−0.218561 + 0.975823i \(0.570136\pi\)
\(140\) 16.1432 1.36435
\(141\) 6.61944 0.557457
\(142\) −2.05889 −0.172778
\(143\) −1.86762 −0.156179
\(144\) 1.00000 0.0833333
\(145\) 26.0887 2.16655
\(146\) 12.6553 1.04736
\(147\) −19.2106 −1.58447
\(148\) −4.64496 −0.381814
\(149\) 20.1215 1.64842 0.824208 0.566288i \(-0.191621\pi\)
0.824208 + 0.566288i \(0.191621\pi\)
\(150\) −4.94258 −0.403560
\(151\) 5.88208 0.478677 0.239339 0.970936i \(-0.423069\pi\)
0.239339 + 0.970936i \(0.423069\pi\)
\(152\) −5.24117 −0.425115
\(153\) 1.00000 0.0808452
\(154\) −11.7868 −0.949810
\(155\) 26.7009 2.14467
\(156\) 0.811206 0.0649485
\(157\) −4.50079 −0.359202 −0.179601 0.983740i \(-0.557481\pi\)
−0.179601 + 0.983740i \(0.557481\pi\)
\(158\) 4.35553 0.346507
\(159\) −13.2381 −1.04985
\(160\) −3.15319 −0.249281
\(161\) 16.7318 1.31865
\(162\) 1.00000 0.0785674
\(163\) 12.6561 0.991300 0.495650 0.868522i \(-0.334930\pi\)
0.495650 + 0.868522i \(0.334930\pi\)
\(164\) −4.54510 −0.354912
\(165\) 7.25952 0.565153
\(166\) 7.85392 0.609583
\(167\) 1.56094 0.120789 0.0603946 0.998175i \(-0.480764\pi\)
0.0603946 + 0.998175i \(0.480764\pi\)
\(168\) 5.11963 0.394988
\(169\) −12.3419 −0.949380
\(170\) −3.15319 −0.241838
\(171\) −5.24117 −0.400802
\(172\) 0.284370 0.0216830
\(173\) −18.3308 −1.39366 −0.696831 0.717235i \(-0.745407\pi\)
−0.696831 + 0.717235i \(0.745407\pi\)
\(174\) 8.27378 0.627233
\(175\) −25.3042 −1.91282
\(176\) 2.30228 0.173541
\(177\) −1.00000 −0.0751646
\(178\) 5.97840 0.448100
\(179\) 4.45467 0.332958 0.166479 0.986045i \(-0.446760\pi\)
0.166479 + 0.986045i \(0.446760\pi\)
\(180\) −3.15319 −0.235025
\(181\) 4.16845 0.309838 0.154919 0.987927i \(-0.450488\pi\)
0.154919 + 0.987927i \(0.450488\pi\)
\(182\) 4.15308 0.307847
\(183\) 2.14012 0.158202
\(184\) −3.26815 −0.240932
\(185\) 14.6464 1.07683
\(186\) 8.46791 0.620897
\(187\) 2.30228 0.168359
\(188\) −6.61944 −0.482772
\(189\) 5.11963 0.372399
\(190\) 16.5264 1.19895
\(191\) 19.3340 1.39896 0.699480 0.714652i \(-0.253415\pi\)
0.699480 + 0.714652i \(0.253415\pi\)
\(192\) −1.00000 −0.0721688
\(193\) 2.51412 0.180970 0.0904852 0.995898i \(-0.471158\pi\)
0.0904852 + 0.995898i \(0.471158\pi\)
\(194\) 5.62903 0.404141
\(195\) −2.55788 −0.183174
\(196\) 19.2106 1.37219
\(197\) 10.0295 0.714570 0.357285 0.933995i \(-0.383703\pi\)
0.357285 + 0.933995i \(0.383703\pi\)
\(198\) 2.30228 0.163616
\(199\) 19.8206 1.40504 0.702521 0.711663i \(-0.252058\pi\)
0.702521 + 0.711663i \(0.252058\pi\)
\(200\) 4.94258 0.349493
\(201\) 13.5658 0.956861
\(202\) −9.49591 −0.668130
\(203\) 42.3587 2.97300
\(204\) −1.00000 −0.0700140
\(205\) 14.3315 1.00096
\(206\) −13.8577 −0.965510
\(207\) −3.26815 −0.227152
\(208\) −0.811206 −0.0562470
\(209\) −12.0666 −0.834667
\(210\) −16.1432 −1.11398
\(211\) 12.0737 0.831188 0.415594 0.909550i \(-0.363574\pi\)
0.415594 + 0.909550i \(0.363574\pi\)
\(212\) 13.2381 0.909199
\(213\) 2.05889 0.141073
\(214\) 16.6567 1.13863
\(215\) −0.896670 −0.0611524
\(216\) −1.00000 −0.0680414
\(217\) 43.3526 2.94297
\(218\) −3.89060 −0.263505
\(219\) −12.6553 −0.855164
\(220\) −7.25952 −0.489436
\(221\) −0.811206 −0.0545676
\(222\) 4.64496 0.311749
\(223\) −16.0016 −1.07155 −0.535775 0.844361i \(-0.679980\pi\)
−0.535775 + 0.844361i \(0.679980\pi\)
\(224\) −5.11963 −0.342070
\(225\) 4.94258 0.329505
\(226\) −8.95400 −0.595612
\(227\) 16.1709 1.07330 0.536650 0.843805i \(-0.319690\pi\)
0.536650 + 0.843805i \(0.319690\pi\)
\(228\) 5.24117 0.347105
\(229\) 9.83530 0.649935 0.324967 0.945725i \(-0.394647\pi\)
0.324967 + 0.945725i \(0.394647\pi\)
\(230\) 10.3051 0.679498
\(231\) 11.7868 0.775516
\(232\) −8.27378 −0.543200
\(233\) −11.5963 −0.759697 −0.379848 0.925049i \(-0.624024\pi\)
−0.379848 + 0.925049i \(0.624024\pi\)
\(234\) −0.811206 −0.0530302
\(235\) 20.8723 1.36156
\(236\) 1.00000 0.0650945
\(237\) −4.35553 −0.282922
\(238\) −5.11963 −0.331857
\(239\) 8.56045 0.553730 0.276865 0.960909i \(-0.410705\pi\)
0.276865 + 0.960909i \(0.410705\pi\)
\(240\) 3.15319 0.203537
\(241\) 0.644767 0.0415331 0.0207666 0.999784i \(-0.493389\pi\)
0.0207666 + 0.999784i \(0.493389\pi\)
\(242\) −5.69951 −0.366378
\(243\) −1.00000 −0.0641500
\(244\) −2.14012 −0.137007
\(245\) −60.5747 −3.86998
\(246\) 4.54510 0.289785
\(247\) 4.25167 0.270527
\(248\) −8.46791 −0.537713
\(249\) −7.85392 −0.497722
\(250\) 0.181060 0.0114512
\(251\) −17.2624 −1.08959 −0.544796 0.838569i \(-0.683393\pi\)
−0.544796 + 0.838569i \(0.683393\pi\)
\(252\) −5.11963 −0.322507
\(253\) −7.52421 −0.473043
\(254\) 8.72754 0.547615
\(255\) 3.15319 0.197460
\(256\) 1.00000 0.0625000
\(257\) 23.2591 1.45086 0.725431 0.688295i \(-0.241641\pi\)
0.725431 + 0.688295i \(0.241641\pi\)
\(258\) −0.284370 −0.0177041
\(259\) 23.7805 1.47765
\(260\) 2.55788 0.158633
\(261\) −8.27378 −0.512134
\(262\) −12.3287 −0.761667
\(263\) −17.8743 −1.10218 −0.551089 0.834447i \(-0.685787\pi\)
−0.551089 + 0.834447i \(0.685787\pi\)
\(264\) −2.30228 −0.141696
\(265\) −41.7423 −2.56421
\(266\) 26.8329 1.64523
\(267\) −5.97840 −0.365872
\(268\) −13.5658 −0.828666
\(269\) 8.34596 0.508862 0.254431 0.967091i \(-0.418112\pi\)
0.254431 + 0.967091i \(0.418112\pi\)
\(270\) 3.15319 0.191897
\(271\) −27.4856 −1.66963 −0.834815 0.550531i \(-0.814425\pi\)
−0.834815 + 0.550531i \(0.814425\pi\)
\(272\) 1.00000 0.0606339
\(273\) −4.15308 −0.251356
\(274\) 4.23705 0.255969
\(275\) 11.3792 0.686192
\(276\) 3.26815 0.196720
\(277\) 21.1091 1.26832 0.634161 0.773201i \(-0.281346\pi\)
0.634161 + 0.773201i \(0.281346\pi\)
\(278\) −5.15360 −0.309092
\(279\) −8.46791 −0.506961
\(280\) 16.1432 0.964738
\(281\) −6.47421 −0.386219 −0.193109 0.981177i \(-0.561857\pi\)
−0.193109 + 0.981177i \(0.561857\pi\)
\(282\) 6.61944 0.394182
\(283\) 7.80340 0.463864 0.231932 0.972732i \(-0.425495\pi\)
0.231932 + 0.972732i \(0.425495\pi\)
\(284\) −2.05889 −0.122172
\(285\) −16.5264 −0.978938
\(286\) −1.86762 −0.110435
\(287\) 23.2692 1.37354
\(288\) 1.00000 0.0589256
\(289\) 1.00000 0.0588235
\(290\) 26.0887 1.53198
\(291\) −5.62903 −0.329980
\(292\) 12.6553 0.740594
\(293\) 2.54573 0.148723 0.0743615 0.997231i \(-0.476308\pi\)
0.0743615 + 0.997231i \(0.476308\pi\)
\(294\) −19.2106 −1.12039
\(295\) −3.15319 −0.183586
\(296\) −4.64496 −0.269983
\(297\) −2.30228 −0.133592
\(298\) 20.1215 1.16561
\(299\) 2.65115 0.153320
\(300\) −4.94258 −0.285360
\(301\) −1.45587 −0.0839148
\(302\) 5.88208 0.338476
\(303\) 9.49591 0.545526
\(304\) −5.24117 −0.300602
\(305\) 6.74819 0.386400
\(306\) 1.00000 0.0571662
\(307\) 4.59402 0.262195 0.131097 0.991369i \(-0.458150\pi\)
0.131097 + 0.991369i \(0.458150\pi\)
\(308\) −11.7868 −0.671617
\(309\) 13.8577 0.788336
\(310\) 26.7009 1.51651
\(311\) −6.04953 −0.343038 −0.171519 0.985181i \(-0.554867\pi\)
−0.171519 + 0.985181i \(0.554867\pi\)
\(312\) 0.811206 0.0459255
\(313\) 3.18784 0.180187 0.0900936 0.995933i \(-0.471283\pi\)
0.0900936 + 0.995933i \(0.471283\pi\)
\(314\) −4.50079 −0.253994
\(315\) 16.1432 0.909564
\(316\) 4.35553 0.245018
\(317\) −8.16915 −0.458825 −0.229413 0.973329i \(-0.573680\pi\)
−0.229413 + 0.973329i \(0.573680\pi\)
\(318\) −13.2381 −0.742358
\(319\) −19.0485 −1.06651
\(320\) −3.15319 −0.176268
\(321\) −16.6567 −0.929684
\(322\) 16.7318 0.932424
\(323\) −5.24117 −0.291626
\(324\) 1.00000 0.0555556
\(325\) −4.00945 −0.222404
\(326\) 12.6561 0.700955
\(327\) 3.89060 0.215151
\(328\) −4.54510 −0.250961
\(329\) 33.8891 1.86837
\(330\) 7.25952 0.399623
\(331\) 20.3364 1.11779 0.558896 0.829238i \(-0.311225\pi\)
0.558896 + 0.829238i \(0.311225\pi\)
\(332\) 7.85392 0.431040
\(333\) −4.64496 −0.254542
\(334\) 1.56094 0.0854109
\(335\) 42.7756 2.33708
\(336\) 5.11963 0.279299
\(337\) 19.9935 1.08912 0.544558 0.838723i \(-0.316698\pi\)
0.544558 + 0.838723i \(0.316698\pi\)
\(338\) −12.3419 −0.671313
\(339\) 8.95400 0.486315
\(340\) −3.15319 −0.171005
\(341\) −19.4955 −1.05574
\(342\) −5.24117 −0.283410
\(343\) −62.5140 −3.37544
\(344\) 0.284370 0.0153322
\(345\) −10.3051 −0.554808
\(346\) −18.3308 −0.985468
\(347\) 5.46345 0.293294 0.146647 0.989189i \(-0.453152\pi\)
0.146647 + 0.989189i \(0.453152\pi\)
\(348\) 8.27378 0.443521
\(349\) 18.1267 0.970301 0.485150 0.874431i \(-0.338765\pi\)
0.485150 + 0.874431i \(0.338765\pi\)
\(350\) −25.3042 −1.35257
\(351\) 0.811206 0.0432990
\(352\) 2.30228 0.122712
\(353\) 22.8832 1.21795 0.608975 0.793189i \(-0.291581\pi\)
0.608975 + 0.793189i \(0.291581\pi\)
\(354\) −1.00000 −0.0531494
\(355\) 6.49205 0.344562
\(356\) 5.97840 0.316854
\(357\) 5.11963 0.270960
\(358\) 4.45467 0.235437
\(359\) −36.1346 −1.90711 −0.953554 0.301221i \(-0.902606\pi\)
−0.953554 + 0.301221i \(0.902606\pi\)
\(360\) −3.15319 −0.166187
\(361\) 8.46984 0.445781
\(362\) 4.16845 0.219089
\(363\) 5.69951 0.299146
\(364\) 4.15308 0.217680
\(365\) −39.9044 −2.08869
\(366\) 2.14012 0.111866
\(367\) −8.33428 −0.435046 −0.217523 0.976055i \(-0.569798\pi\)
−0.217523 + 0.976055i \(0.569798\pi\)
\(368\) −3.26815 −0.170364
\(369\) −4.54510 −0.236608
\(370\) 14.6464 0.761432
\(371\) −67.7744 −3.51867
\(372\) 8.46791 0.439041
\(373\) 31.3538 1.62344 0.811719 0.584048i \(-0.198532\pi\)
0.811719 + 0.584048i \(0.198532\pi\)
\(374\) 2.30228 0.119048
\(375\) −0.181060 −0.00934990
\(376\) −6.61944 −0.341371
\(377\) 6.71174 0.345672
\(378\) 5.11963 0.263326
\(379\) −19.0407 −0.978055 −0.489028 0.872268i \(-0.662648\pi\)
−0.489028 + 0.872268i \(0.662648\pi\)
\(380\) 16.5264 0.847785
\(381\) −8.72754 −0.447125
\(382\) 19.3340 0.989214
\(383\) −33.1369 −1.69321 −0.846607 0.532218i \(-0.821359\pi\)
−0.846607 + 0.532218i \(0.821359\pi\)
\(384\) −1.00000 −0.0510310
\(385\) 37.1661 1.89416
\(386\) 2.51412 0.127965
\(387\) 0.284370 0.0144553
\(388\) 5.62903 0.285771
\(389\) −19.1532 −0.971104 −0.485552 0.874208i \(-0.661381\pi\)
−0.485552 + 0.874208i \(0.661381\pi\)
\(390\) −2.55788 −0.129523
\(391\) −3.26815 −0.165278
\(392\) 19.2106 0.970284
\(393\) 12.3287 0.621898
\(394\) 10.0295 0.505277
\(395\) −13.7338 −0.691022
\(396\) 2.30228 0.115694
\(397\) −13.1806 −0.661514 −0.330757 0.943716i \(-0.607304\pi\)
−0.330757 + 0.943716i \(0.607304\pi\)
\(398\) 19.8206 0.993515
\(399\) −26.8329 −1.34332
\(400\) 4.94258 0.247129
\(401\) −25.2188 −1.25937 −0.629684 0.776851i \(-0.716816\pi\)
−0.629684 + 0.776851i \(0.716816\pi\)
\(402\) 13.5658 0.676603
\(403\) 6.86922 0.342180
\(404\) −9.49591 −0.472439
\(405\) −3.15319 −0.156683
\(406\) 42.3587 2.10223
\(407\) −10.6940 −0.530082
\(408\) −1.00000 −0.0495074
\(409\) 22.9408 1.13435 0.567175 0.823597i \(-0.308036\pi\)
0.567175 + 0.823597i \(0.308036\pi\)
\(410\) 14.3315 0.707784
\(411\) −4.23705 −0.208998
\(412\) −13.8577 −0.682719
\(413\) −5.11963 −0.251921
\(414\) −3.26815 −0.160621
\(415\) −24.7649 −1.21566
\(416\) −0.811206 −0.0397726
\(417\) 5.15360 0.252373
\(418\) −12.0666 −0.590198
\(419\) −32.8647 −1.60554 −0.802772 0.596285i \(-0.796643\pi\)
−0.802772 + 0.596285i \(0.796643\pi\)
\(420\) −16.1432 −0.787705
\(421\) 12.3415 0.601488 0.300744 0.953705i \(-0.402765\pi\)
0.300744 + 0.953705i \(0.402765\pi\)
\(422\) 12.0737 0.587739
\(423\) −6.61944 −0.321848
\(424\) 13.2381 0.642901
\(425\) 4.94258 0.239750
\(426\) 2.05889 0.0997534
\(427\) 10.9566 0.530228
\(428\) 16.6567 0.805130
\(429\) 1.86762 0.0901697
\(430\) −0.896670 −0.0432413
\(431\) 0.409194 0.0197102 0.00985508 0.999951i \(-0.496863\pi\)
0.00985508 + 0.999951i \(0.496863\pi\)
\(432\) −1.00000 −0.0481125
\(433\) 30.9599 1.48784 0.743918 0.668270i \(-0.232965\pi\)
0.743918 + 0.668270i \(0.232965\pi\)
\(434\) 43.3526 2.08099
\(435\) −26.0887 −1.25086
\(436\) −3.89060 −0.186326
\(437\) 17.1289 0.819389
\(438\) −12.6553 −0.604692
\(439\) 36.5792 1.74583 0.872915 0.487872i \(-0.162227\pi\)
0.872915 + 0.487872i \(0.162227\pi\)
\(440\) −7.25952 −0.346084
\(441\) 19.2106 0.914793
\(442\) −0.811206 −0.0385851
\(443\) −21.9137 −1.04115 −0.520576 0.853815i \(-0.674283\pi\)
−0.520576 + 0.853815i \(0.674283\pi\)
\(444\) 4.64496 0.220440
\(445\) −18.8510 −0.893623
\(446\) −16.0016 −0.757700
\(447\) −20.1215 −0.951713
\(448\) −5.11963 −0.241880
\(449\) −34.0550 −1.60716 −0.803578 0.595199i \(-0.797073\pi\)
−0.803578 + 0.595199i \(0.797073\pi\)
\(450\) 4.94258 0.232995
\(451\) −10.4641 −0.492735
\(452\) −8.95400 −0.421161
\(453\) −5.88208 −0.276364
\(454\) 16.1709 0.758938
\(455\) −13.0954 −0.613923
\(456\) 5.24117 0.245440
\(457\) −33.0570 −1.54634 −0.773171 0.634198i \(-0.781330\pi\)
−0.773171 + 0.634198i \(0.781330\pi\)
\(458\) 9.83530 0.459573
\(459\) −1.00000 −0.0466760
\(460\) 10.3051 0.480478
\(461\) −29.0470 −1.35285 −0.676427 0.736509i \(-0.736473\pi\)
−0.676427 + 0.736509i \(0.736473\pi\)
\(462\) 11.7868 0.548373
\(463\) 33.9259 1.57667 0.788336 0.615245i \(-0.210943\pi\)
0.788336 + 0.615245i \(0.210943\pi\)
\(464\) −8.27378 −0.384100
\(465\) −26.7009 −1.23822
\(466\) −11.5963 −0.537187
\(467\) 6.51149 0.301316 0.150658 0.988586i \(-0.451861\pi\)
0.150658 + 0.988586i \(0.451861\pi\)
\(468\) −0.811206 −0.0374980
\(469\) 69.4521 3.20700
\(470\) 20.8723 0.962768
\(471\) 4.50079 0.207385
\(472\) 1.00000 0.0460287
\(473\) 0.654698 0.0301031
\(474\) −4.35553 −0.200056
\(475\) −25.9049 −1.18860
\(476\) −5.11963 −0.234658
\(477\) 13.2381 0.606133
\(478\) 8.56045 0.391546
\(479\) 18.6065 0.850153 0.425077 0.905157i \(-0.360247\pi\)
0.425077 + 0.905157i \(0.360247\pi\)
\(480\) 3.15319 0.143923
\(481\) 3.76802 0.171807
\(482\) 0.644767 0.0293683
\(483\) −16.7318 −0.761321
\(484\) −5.69951 −0.259068
\(485\) −17.7494 −0.805958
\(486\) −1.00000 −0.0453609
\(487\) 16.9762 0.769267 0.384633 0.923069i \(-0.374328\pi\)
0.384633 + 0.923069i \(0.374328\pi\)
\(488\) −2.14012 −0.0968786
\(489\) −12.6561 −0.572327
\(490\) −60.5747 −2.73649
\(491\) −34.6125 −1.56204 −0.781020 0.624507i \(-0.785300\pi\)
−0.781020 + 0.624507i \(0.785300\pi\)
\(492\) 4.54510 0.204909
\(493\) −8.27378 −0.372632
\(494\) 4.25167 0.191292
\(495\) −7.25952 −0.326291
\(496\) −8.46791 −0.380220
\(497\) 10.5407 0.472817
\(498\) −7.85392 −0.351943
\(499\) 30.0368 1.34463 0.672317 0.740264i \(-0.265299\pi\)
0.672317 + 0.740264i \(0.265299\pi\)
\(500\) 0.181060 0.00809725
\(501\) −1.56094 −0.0697377
\(502\) −17.2624 −0.770458
\(503\) −36.5156 −1.62815 −0.814074 0.580761i \(-0.802755\pi\)
−0.814074 + 0.580761i \(0.802755\pi\)
\(504\) −5.11963 −0.228047
\(505\) 29.9424 1.33242
\(506\) −7.52421 −0.334492
\(507\) 12.3419 0.548125
\(508\) 8.72754 0.387222
\(509\) −13.9137 −0.616712 −0.308356 0.951271i \(-0.599779\pi\)
−0.308356 + 0.951271i \(0.599779\pi\)
\(510\) 3.15319 0.139625
\(511\) −64.7904 −2.86616
\(512\) 1.00000 0.0441942
\(513\) 5.24117 0.231403
\(514\) 23.2591 1.02591
\(515\) 43.6958 1.92547
\(516\) −0.284370 −0.0125187
\(517\) −15.2398 −0.670246
\(518\) 23.7805 1.04486
\(519\) 18.3308 0.804631
\(520\) 2.55788 0.112171
\(521\) 19.4571 0.852432 0.426216 0.904621i \(-0.359846\pi\)
0.426216 + 0.904621i \(0.359846\pi\)
\(522\) −8.27378 −0.362133
\(523\) 10.7728 0.471061 0.235531 0.971867i \(-0.424317\pi\)
0.235531 + 0.971867i \(0.424317\pi\)
\(524\) −12.3287 −0.538580
\(525\) 25.3042 1.10437
\(526\) −17.8743 −0.779357
\(527\) −8.46791 −0.368868
\(528\) −2.30228 −0.100194
\(529\) −12.3192 −0.535616
\(530\) −41.7423 −1.81317
\(531\) 1.00000 0.0433963
\(532\) 26.8329 1.16335
\(533\) 3.68701 0.159702
\(534\) −5.97840 −0.258711
\(535\) −52.5216 −2.27070
\(536\) −13.5658 −0.585955
\(537\) −4.45467 −0.192233
\(538\) 8.34596 0.359820
\(539\) 44.2283 1.90505
\(540\) 3.15319 0.135691
\(541\) −21.3313 −0.917104 −0.458552 0.888668i \(-0.651632\pi\)
−0.458552 + 0.888668i \(0.651632\pi\)
\(542\) −27.4856 −1.18061
\(543\) −4.16845 −0.178885
\(544\) 1.00000 0.0428746
\(545\) 12.2678 0.525494
\(546\) −4.15308 −0.177735
\(547\) −27.0982 −1.15863 −0.579317 0.815102i \(-0.696681\pi\)
−0.579317 + 0.815102i \(0.696681\pi\)
\(548\) 4.23705 0.180998
\(549\) −2.14012 −0.0913380
\(550\) 11.3792 0.485211
\(551\) 43.3642 1.84738
\(552\) 3.26815 0.139102
\(553\) −22.2987 −0.948237
\(554\) 21.1091 0.896839
\(555\) −14.6464 −0.621706
\(556\) −5.15360 −0.218561
\(557\) −19.2460 −0.815479 −0.407739 0.913098i \(-0.633683\pi\)
−0.407739 + 0.913098i \(0.633683\pi\)
\(558\) −8.46791 −0.358475
\(559\) −0.230682 −0.00975682
\(560\) 16.1432 0.682173
\(561\) −2.30228 −0.0972023
\(562\) −6.47421 −0.273098
\(563\) −29.5931 −1.24720 −0.623600 0.781743i \(-0.714331\pi\)
−0.623600 + 0.781743i \(0.714331\pi\)
\(564\) 6.61944 0.278729
\(565\) 28.2336 1.18780
\(566\) 7.80340 0.328001
\(567\) −5.11963 −0.215004
\(568\) −2.05889 −0.0863890
\(569\) 15.8532 0.664602 0.332301 0.943173i \(-0.392175\pi\)
0.332301 + 0.943173i \(0.392175\pi\)
\(570\) −16.5264 −0.692214
\(571\) 17.3922 0.727840 0.363920 0.931430i \(-0.381438\pi\)
0.363920 + 0.931430i \(0.381438\pi\)
\(572\) −1.86762 −0.0780893
\(573\) −19.3340 −0.807690
\(574\) 23.2692 0.971239
\(575\) −16.1531 −0.673631
\(576\) 1.00000 0.0416667
\(577\) −1.99984 −0.0832545 −0.0416273 0.999133i \(-0.513254\pi\)
−0.0416273 + 0.999133i \(0.513254\pi\)
\(578\) 1.00000 0.0415945
\(579\) −2.51412 −0.104483
\(580\) 26.0887 1.08328
\(581\) −40.2092 −1.66816
\(582\) −5.62903 −0.233331
\(583\) 30.4779 1.26227
\(584\) 12.6553 0.523679
\(585\) 2.55788 0.105755
\(586\) 2.54573 0.105163
\(587\) −15.0836 −0.622567 −0.311283 0.950317i \(-0.600759\pi\)
−0.311283 + 0.950317i \(0.600759\pi\)
\(588\) −19.2106 −0.792234
\(589\) 44.3817 1.82872
\(590\) −3.15319 −0.129815
\(591\) −10.0295 −0.412557
\(592\) −4.64496 −0.190907
\(593\) −15.5987 −0.640564 −0.320282 0.947322i \(-0.603778\pi\)
−0.320282 + 0.947322i \(0.603778\pi\)
\(594\) −2.30228 −0.0944637
\(595\) 16.1432 0.661805
\(596\) 20.1215 0.824208
\(597\) −19.8206 −0.811201
\(598\) 2.65115 0.108413
\(599\) 19.3460 0.790455 0.395228 0.918583i \(-0.370666\pi\)
0.395228 + 0.918583i \(0.370666\pi\)
\(600\) −4.94258 −0.201780
\(601\) −38.9533 −1.58894 −0.794470 0.607304i \(-0.792251\pi\)
−0.794470 + 0.607304i \(0.792251\pi\)
\(602\) −1.45587 −0.0593367
\(603\) −13.5658 −0.552444
\(604\) 5.88208 0.239339
\(605\) 17.9716 0.730649
\(606\) 9.49591 0.385745
\(607\) 12.5488 0.509341 0.254671 0.967028i \(-0.418033\pi\)
0.254671 + 0.967028i \(0.418033\pi\)
\(608\) −5.24117 −0.212557
\(609\) −42.3587 −1.71646
\(610\) 6.74819 0.273226
\(611\) 5.36973 0.217236
\(612\) 1.00000 0.0404226
\(613\) −45.0873 −1.82106 −0.910530 0.413443i \(-0.864326\pi\)
−0.910530 + 0.413443i \(0.864326\pi\)
\(614\) 4.59402 0.185400
\(615\) −14.3315 −0.577903
\(616\) −11.7868 −0.474905
\(617\) 44.3400 1.78506 0.892530 0.450988i \(-0.148928\pi\)
0.892530 + 0.450988i \(0.148928\pi\)
\(618\) 13.8577 0.557438
\(619\) −31.9344 −1.28355 −0.641775 0.766893i \(-0.721802\pi\)
−0.641775 + 0.766893i \(0.721802\pi\)
\(620\) 26.7009 1.07233
\(621\) 3.26815 0.131147
\(622\) −6.04953 −0.242564
\(623\) −30.6072 −1.22625
\(624\) 0.811206 0.0324742
\(625\) −25.2838 −1.01135
\(626\) 3.18784 0.127412
\(627\) 12.0666 0.481895
\(628\) −4.50079 −0.179601
\(629\) −4.64496 −0.185207
\(630\) 16.1432 0.643159
\(631\) 26.8686 1.06962 0.534812 0.844971i \(-0.320382\pi\)
0.534812 + 0.844971i \(0.320382\pi\)
\(632\) 4.35553 0.173254
\(633\) −12.0737 −0.479887
\(634\) −8.16915 −0.324438
\(635\) −27.5196 −1.09208
\(636\) −13.2381 −0.524926
\(637\) −15.5838 −0.617452
\(638\) −19.0485 −0.754139
\(639\) −2.05889 −0.0814483
\(640\) −3.15319 −0.124641
\(641\) −31.0172 −1.22511 −0.612553 0.790429i \(-0.709858\pi\)
−0.612553 + 0.790429i \(0.709858\pi\)
\(642\) −16.6567 −0.657386
\(643\) −17.8412 −0.703588 −0.351794 0.936077i \(-0.614428\pi\)
−0.351794 + 0.936077i \(0.614428\pi\)
\(644\) 16.7318 0.659323
\(645\) 0.896670 0.0353063
\(646\) −5.24117 −0.206211
\(647\) 35.0191 1.37674 0.688372 0.725358i \(-0.258326\pi\)
0.688372 + 0.725358i \(0.258326\pi\)
\(648\) 1.00000 0.0392837
\(649\) 2.30228 0.0903724
\(650\) −4.00945 −0.157264
\(651\) −43.3526 −1.69912
\(652\) 12.6561 0.495650
\(653\) −8.73770 −0.341933 −0.170966 0.985277i \(-0.554689\pi\)
−0.170966 + 0.985277i \(0.554689\pi\)
\(654\) 3.89060 0.152134
\(655\) 38.8745 1.51895
\(656\) −4.54510 −0.177456
\(657\) 12.6553 0.493729
\(658\) 33.8891 1.32113
\(659\) −15.0202 −0.585103 −0.292552 0.956250i \(-0.594504\pi\)
−0.292552 + 0.956250i \(0.594504\pi\)
\(660\) 7.25952 0.282576
\(661\) −6.26283 −0.243596 −0.121798 0.992555i \(-0.538866\pi\)
−0.121798 + 0.992555i \(0.538866\pi\)
\(662\) 20.3364 0.790398
\(663\) 0.811206 0.0315046
\(664\) 7.85392 0.304791
\(665\) −84.6090 −3.28100
\(666\) −4.64496 −0.179989
\(667\) 27.0400 1.04699
\(668\) 1.56094 0.0603946
\(669\) 16.0016 0.618659
\(670\) 42.7756 1.65257
\(671\) −4.92715 −0.190211
\(672\) 5.11963 0.197494
\(673\) 27.5408 1.06162 0.530811 0.847490i \(-0.321888\pi\)
0.530811 + 0.847490i \(0.321888\pi\)
\(674\) 19.9935 0.770121
\(675\) −4.94258 −0.190240
\(676\) −12.3419 −0.474690
\(677\) −9.22129 −0.354403 −0.177201 0.984175i \(-0.556704\pi\)
−0.177201 + 0.984175i \(0.556704\pi\)
\(678\) 8.95400 0.343876
\(679\) −28.8186 −1.10596
\(680\) −3.15319 −0.120919
\(681\) −16.1709 −0.619670
\(682\) −19.4955 −0.746521
\(683\) 5.55368 0.212506 0.106253 0.994339i \(-0.466115\pi\)
0.106253 + 0.994339i \(0.466115\pi\)
\(684\) −5.24117 −0.200401
\(685\) −13.3602 −0.510467
\(686\) −62.5140 −2.38680
\(687\) −9.83530 −0.375240
\(688\) 0.284370 0.0108415
\(689\) −10.7389 −0.409118
\(690\) −10.3051 −0.392308
\(691\) 7.82478 0.297668 0.148834 0.988862i \(-0.452448\pi\)
0.148834 + 0.988862i \(0.452448\pi\)
\(692\) −18.3308 −0.696831
\(693\) −11.7868 −0.447745
\(694\) 5.46345 0.207390
\(695\) 16.2502 0.616407
\(696\) 8.27378 0.313617
\(697\) −4.54510 −0.172158
\(698\) 18.1267 0.686106
\(699\) 11.5963 0.438611
\(700\) −25.3042 −0.956408
\(701\) 29.1820 1.10219 0.551095 0.834442i \(-0.314210\pi\)
0.551095 + 0.834442i \(0.314210\pi\)
\(702\) 0.811206 0.0306170
\(703\) 24.3450 0.918190
\(704\) 2.30228 0.0867704
\(705\) −20.8723 −0.786097
\(706\) 22.8832 0.861221
\(707\) 48.6156 1.82838
\(708\) −1.00000 −0.0375823
\(709\) 17.7849 0.667926 0.333963 0.942586i \(-0.391614\pi\)
0.333963 + 0.942586i \(0.391614\pi\)
\(710\) 6.49205 0.243642
\(711\) 4.35553 0.163345
\(712\) 5.97840 0.224050
\(713\) 27.6744 1.03642
\(714\) 5.11963 0.191597
\(715\) 5.88896 0.220235
\(716\) 4.45467 0.166479
\(717\) −8.56045 −0.319696
\(718\) −36.1346 −1.34853
\(719\) −19.3139 −0.720287 −0.360143 0.932897i \(-0.617272\pi\)
−0.360143 + 0.932897i \(0.617272\pi\)
\(720\) −3.15319 −0.117512
\(721\) 70.9462 2.64218
\(722\) 8.46984 0.315215
\(723\) −0.644767 −0.0239791
\(724\) 4.16845 0.154919
\(725\) −40.8938 −1.51876
\(726\) 5.69951 0.211529
\(727\) −41.9027 −1.55409 −0.777043 0.629448i \(-0.783281\pi\)
−0.777043 + 0.629448i \(0.783281\pi\)
\(728\) 4.15308 0.153923
\(729\) 1.00000 0.0370370
\(730\) −39.9044 −1.47693
\(731\) 0.284370 0.0105178
\(732\) 2.14012 0.0791010
\(733\) −36.7993 −1.35921 −0.679607 0.733577i \(-0.737850\pi\)
−0.679607 + 0.733577i \(0.737850\pi\)
\(734\) −8.33428 −0.307624
\(735\) 60.5747 2.23433
\(736\) −3.26815 −0.120466
\(737\) −31.2324 −1.15046
\(738\) −4.54510 −0.167307
\(739\) 26.7651 0.984569 0.492285 0.870434i \(-0.336162\pi\)
0.492285 + 0.870434i \(0.336162\pi\)
\(740\) 14.6464 0.538413
\(741\) −4.25167 −0.156189
\(742\) −67.7744 −2.48808
\(743\) −11.0891 −0.406820 −0.203410 0.979094i \(-0.565202\pi\)
−0.203410 + 0.979094i \(0.565202\pi\)
\(744\) 8.46791 0.310449
\(745\) −63.4467 −2.32451
\(746\) 31.3538 1.14794
\(747\) 7.85392 0.287360
\(748\) 2.30228 0.0841797
\(749\) −85.2760 −3.11592
\(750\) −0.181060 −0.00661138
\(751\) 6.55245 0.239102 0.119551 0.992828i \(-0.461854\pi\)
0.119551 + 0.992828i \(0.461854\pi\)
\(752\) −6.61944 −0.241386
\(753\) 17.2624 0.629076
\(754\) 6.71174 0.244427
\(755\) −18.5473 −0.675005
\(756\) 5.11963 0.186199
\(757\) 33.3712 1.21290 0.606449 0.795122i \(-0.292593\pi\)
0.606449 + 0.795122i \(0.292593\pi\)
\(758\) −19.0407 −0.691590
\(759\) 7.52421 0.273111
\(760\) 16.5264 0.599475
\(761\) −23.3228 −0.845450 −0.422725 0.906258i \(-0.638926\pi\)
−0.422725 + 0.906258i \(0.638926\pi\)
\(762\) −8.72754 −0.316165
\(763\) 19.9184 0.721096
\(764\) 19.3340 0.699480
\(765\) −3.15319 −0.114004
\(766\) −33.1369 −1.19728
\(767\) −0.811206 −0.0292909
\(768\) −1.00000 −0.0360844
\(769\) −35.0349 −1.26339 −0.631696 0.775216i \(-0.717641\pi\)
−0.631696 + 0.775216i \(0.717641\pi\)
\(770\) 37.1661 1.33937
\(771\) −23.2591 −0.837655
\(772\) 2.51412 0.0904852
\(773\) −34.7894 −1.25129 −0.625644 0.780109i \(-0.715164\pi\)
−0.625644 + 0.780109i \(0.715164\pi\)
\(774\) 0.284370 0.0102215
\(775\) −41.8533 −1.50342
\(776\) 5.62903 0.202070
\(777\) −23.7805 −0.853121
\(778\) −19.1532 −0.686674
\(779\) 23.8216 0.853498
\(780\) −2.55788 −0.0915869
\(781\) −4.74013 −0.169615
\(782\) −3.26815 −0.116869
\(783\) 8.27378 0.295681
\(784\) 19.2106 0.686094
\(785\) 14.1918 0.506528
\(786\) 12.3287 0.439749
\(787\) −1.13792 −0.0405625 −0.0202813 0.999794i \(-0.506456\pi\)
−0.0202813 + 0.999794i \(0.506456\pi\)
\(788\) 10.0295 0.357285
\(789\) 17.8743 0.636343
\(790\) −13.7338 −0.488626
\(791\) 45.8412 1.62993
\(792\) 2.30228 0.0818080
\(793\) 1.73608 0.0616499
\(794\) −13.1806 −0.467761
\(795\) 41.7423 1.48045
\(796\) 19.8206 0.702521
\(797\) 39.6660 1.40504 0.702521 0.711664i \(-0.252058\pi\)
0.702521 + 0.711664i \(0.252058\pi\)
\(798\) −26.8329 −0.949873
\(799\) −6.61944 −0.234179
\(800\) 4.94258 0.174747
\(801\) 5.97840 0.211236
\(802\) −25.2188 −0.890508
\(803\) 29.1360 1.02819
\(804\) 13.5658 0.478430
\(805\) −52.7583 −1.85949
\(806\) 6.86922 0.241958
\(807\) −8.34596 −0.293792
\(808\) −9.49591 −0.334065
\(809\) −6.23422 −0.219184 −0.109592 0.993977i \(-0.534954\pi\)
−0.109592 + 0.993977i \(0.534954\pi\)
\(810\) −3.15319 −0.110792
\(811\) 31.5603 1.10823 0.554116 0.832439i \(-0.313056\pi\)
0.554116 + 0.832439i \(0.313056\pi\)
\(812\) 42.3587 1.48650
\(813\) 27.4856 0.963961
\(814\) −10.6940 −0.374825
\(815\) −39.9069 −1.39788
\(816\) −1.00000 −0.0350070
\(817\) −1.49043 −0.0521435
\(818\) 22.9408 0.802107
\(819\) 4.15308 0.145120
\(820\) 14.3315 0.500479
\(821\) 14.8908 0.519692 0.259846 0.965650i \(-0.416328\pi\)
0.259846 + 0.965650i \(0.416328\pi\)
\(822\) −4.23705 −0.147784
\(823\) 41.6072 1.45034 0.725168 0.688572i \(-0.241762\pi\)
0.725168 + 0.688572i \(0.241762\pi\)
\(824\) −13.8577 −0.482755
\(825\) −11.3792 −0.396173
\(826\) −5.11963 −0.178135
\(827\) −34.6335 −1.20432 −0.602162 0.798374i \(-0.705694\pi\)
−0.602162 + 0.798374i \(0.705694\pi\)
\(828\) −3.26815 −0.113576
\(829\) 37.4839 1.30187 0.650935 0.759134i \(-0.274377\pi\)
0.650935 + 0.759134i \(0.274377\pi\)
\(830\) −24.7649 −0.859601
\(831\) −21.1091 −0.732266
\(832\) −0.811206 −0.0281235
\(833\) 19.2106 0.665609
\(834\) 5.15360 0.178454
\(835\) −4.92194 −0.170331
\(836\) −12.0666 −0.417333
\(837\) 8.46791 0.292694
\(838\) −32.8647 −1.13529
\(839\) −29.7976 −1.02873 −0.514363 0.857572i \(-0.671972\pi\)
−0.514363 + 0.857572i \(0.671972\pi\)
\(840\) −16.1432 −0.556992
\(841\) 39.4554 1.36053
\(842\) 12.3415 0.425316
\(843\) 6.47421 0.222984
\(844\) 12.0737 0.415594
\(845\) 38.9164 1.33877
\(846\) −6.61944 −0.227581
\(847\) 29.1794 1.00262
\(848\) 13.2381 0.454599
\(849\) −7.80340 −0.267812
\(850\) 4.94258 0.169529
\(851\) 15.1805 0.520379
\(852\) 2.05889 0.0705363
\(853\) 2.96095 0.101381 0.0506905 0.998714i \(-0.483858\pi\)
0.0506905 + 0.998714i \(0.483858\pi\)
\(854\) 10.9566 0.374928
\(855\) 16.5264 0.565190
\(856\) 16.6567 0.569313
\(857\) 27.8680 0.951954 0.475977 0.879458i \(-0.342095\pi\)
0.475977 + 0.879458i \(0.342095\pi\)
\(858\) 1.86762 0.0637596
\(859\) −23.7399 −0.809995 −0.404997 0.914318i \(-0.632728\pi\)
−0.404997 + 0.914318i \(0.632728\pi\)
\(860\) −0.896670 −0.0305762
\(861\) −23.2692 −0.793013
\(862\) 0.409194 0.0139372
\(863\) 24.3687 0.829520 0.414760 0.909931i \(-0.363866\pi\)
0.414760 + 0.909931i \(0.363866\pi\)
\(864\) −1.00000 −0.0340207
\(865\) 57.8003 1.96527
\(866\) 30.9599 1.05206
\(867\) −1.00000 −0.0339618
\(868\) 43.3526 1.47148
\(869\) 10.0276 0.340165
\(870\) −26.0887 −0.884491
\(871\) 11.0047 0.372880
\(872\) −3.89060 −0.131752
\(873\) 5.62903 0.190514
\(874\) 17.1289 0.579395
\(875\) −0.926961 −0.0313370
\(876\) −12.6553 −0.427582
\(877\) 15.0748 0.509041 0.254520 0.967067i \(-0.418082\pi\)
0.254520 + 0.967067i \(0.418082\pi\)
\(878\) 36.5792 1.23449
\(879\) −2.54573 −0.0858652
\(880\) −7.25952 −0.244718
\(881\) −3.14534 −0.105969 −0.0529846 0.998595i \(-0.516873\pi\)
−0.0529846 + 0.998595i \(0.516873\pi\)
\(882\) 19.2106 0.646856
\(883\) −4.13667 −0.139210 −0.0696051 0.997575i \(-0.522174\pi\)
−0.0696051 + 0.997575i \(0.522174\pi\)
\(884\) −0.811206 −0.0272838
\(885\) 3.15319 0.105993
\(886\) −21.9137 −0.736206
\(887\) 35.4633 1.19074 0.595370 0.803452i \(-0.297005\pi\)
0.595370 + 0.803452i \(0.297005\pi\)
\(888\) 4.64496 0.155875
\(889\) −44.6818 −1.49858
\(890\) −18.8510 −0.631887
\(891\) 2.30228 0.0771293
\(892\) −16.0016 −0.535775
\(893\) 34.6936 1.16098
\(894\) −20.1215 −0.672963
\(895\) −14.0464 −0.469520
\(896\) −5.11963 −0.171035
\(897\) −2.65115 −0.0885192
\(898\) −34.0550 −1.13643
\(899\) 70.0616 2.33668
\(900\) 4.94258 0.164753
\(901\) 13.2381 0.441026
\(902\) −10.4641 −0.348416
\(903\) 1.45587 0.0484482
\(904\) −8.95400 −0.297806
\(905\) −13.1439 −0.436918
\(906\) −5.88208 −0.195419
\(907\) −16.8760 −0.560359 −0.280179 0.959948i \(-0.590394\pi\)
−0.280179 + 0.959948i \(0.590394\pi\)
\(908\) 16.1709 0.536650
\(909\) −9.49591 −0.314959
\(910\) −13.0954 −0.434109
\(911\) 12.6083 0.417732 0.208866 0.977944i \(-0.433023\pi\)
0.208866 + 0.977944i \(0.433023\pi\)
\(912\) 5.24117 0.173552
\(913\) 18.0819 0.598425
\(914\) −33.0570 −1.09343
\(915\) −6.74819 −0.223088
\(916\) 9.83530 0.324967
\(917\) 63.1182 2.08435
\(918\) −1.00000 −0.0330049
\(919\) −35.9069 −1.18446 −0.592230 0.805769i \(-0.701752\pi\)
−0.592230 + 0.805769i \(0.701752\pi\)
\(920\) 10.3051 0.339749
\(921\) −4.59402 −0.151378
\(922\) −29.0470 −0.956613
\(923\) 1.67018 0.0549747
\(924\) 11.7868 0.387758
\(925\) −22.9581 −0.754857
\(926\) 33.9259 1.11488
\(927\) −13.8577 −0.455146
\(928\) −8.27378 −0.271600
\(929\) 11.0955 0.364031 0.182015 0.983296i \(-0.441738\pi\)
0.182015 + 0.983296i \(0.441738\pi\)
\(930\) −26.7009 −0.875557
\(931\) −100.686 −3.29986
\(932\) −11.5963 −0.379848
\(933\) 6.04953 0.198053
\(934\) 6.51149 0.213063
\(935\) −7.25952 −0.237412
\(936\) −0.811206 −0.0265151
\(937\) −1.43815 −0.0469823 −0.0234912 0.999724i \(-0.507478\pi\)
−0.0234912 + 0.999724i \(0.507478\pi\)
\(938\) 69.4521 2.26769
\(939\) −3.18784 −0.104031
\(940\) 20.8723 0.680780
\(941\) −8.61313 −0.280780 −0.140390 0.990096i \(-0.544836\pi\)
−0.140390 + 0.990096i \(0.544836\pi\)
\(942\) 4.50079 0.146644
\(943\) 14.8541 0.483715
\(944\) 1.00000 0.0325472
\(945\) −16.1432 −0.525137
\(946\) 0.654698 0.0212861
\(947\) 51.1785 1.66308 0.831539 0.555466i \(-0.187460\pi\)
0.831539 + 0.555466i \(0.187460\pi\)
\(948\) −4.35553 −0.141461
\(949\) −10.2660 −0.333250
\(950\) −25.9049 −0.840465
\(951\) 8.16915 0.264903
\(952\) −5.11963 −0.165928
\(953\) −4.65755 −0.150873 −0.0754364 0.997151i \(-0.524035\pi\)
−0.0754364 + 0.997151i \(0.524035\pi\)
\(954\) 13.2381 0.428600
\(955\) −60.9637 −1.97274
\(956\) 8.56045 0.276865
\(957\) 19.0485 0.615752
\(958\) 18.6065 0.601149
\(959\) −21.6921 −0.700475
\(960\) 3.15319 0.101769
\(961\) 40.7055 1.31308
\(962\) 3.76802 0.121486
\(963\) 16.6567 0.536754
\(964\) 0.644767 0.0207666
\(965\) −7.92749 −0.255195
\(966\) −16.7318 −0.538335
\(967\) 32.4044 1.04205 0.521027 0.853540i \(-0.325549\pi\)
0.521027 + 0.853540i \(0.325549\pi\)
\(968\) −5.69951 −0.183189
\(969\) 5.24117 0.168371
\(970\) −17.7494 −0.569898
\(971\) 15.8345 0.508152 0.254076 0.967184i \(-0.418229\pi\)
0.254076 + 0.967184i \(0.418229\pi\)
\(972\) −1.00000 −0.0320750
\(973\) 26.3845 0.845849
\(974\) 16.9762 0.543954
\(975\) 4.00945 0.128405
\(976\) −2.14012 −0.0685035
\(977\) 51.5831 1.65029 0.825145 0.564922i \(-0.191094\pi\)
0.825145 + 0.564922i \(0.191094\pi\)
\(978\) −12.6561 −0.404696
\(979\) 13.7639 0.439898
\(980\) −60.5747 −1.93499
\(981\) −3.89060 −0.124217
\(982\) −34.6125 −1.10453
\(983\) 27.4354 0.875054 0.437527 0.899205i \(-0.355854\pi\)
0.437527 + 0.899205i \(0.355854\pi\)
\(984\) 4.54510 0.144892
\(985\) −31.6248 −1.00765
\(986\) −8.27378 −0.263491
\(987\) −33.8891 −1.07870
\(988\) 4.25167 0.135264
\(989\) −0.929364 −0.0295520
\(990\) −7.25952 −0.230723
\(991\) −18.6656 −0.592932 −0.296466 0.955043i \(-0.595808\pi\)
−0.296466 + 0.955043i \(0.595808\pi\)
\(992\) −8.46791 −0.268856
\(993\) −20.3364 −0.645357
\(994\) 10.5407 0.334332
\(995\) −62.4979 −1.98132
\(996\) −7.85392 −0.248861
\(997\) 39.8729 1.26279 0.631394 0.775462i \(-0.282483\pi\)
0.631394 + 0.775462i \(0.282483\pi\)
\(998\) 30.0368 0.950799
\(999\) 4.64496 0.146960
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 6018.2.a.bc.1.2 14
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
6018.2.a.bc.1.2 14 1.1 even 1 trivial