Properties

Label 6046.2.a.g.1.14
Level $6046$
Weight $2$
Character 6046.1
Self dual yes
Analytic conductor $48.278$
Analytic rank $0$
Dimension $69$
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] = [6046,2,Mod(1,6046)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6046, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("6046.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6046 = 2 \cdot 3023 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6046.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: \(48.2775530621\)
Analytic rank: \(0\)
Dimension: \(69\)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.14
Character \(\chi\) \(=\) 6046.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} -2.29528 q^{3} +1.00000 q^{4} +2.67586 q^{5} +2.29528 q^{6} -0.797143 q^{7} -1.00000 q^{8} +2.26829 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -2.29528 q^{3} +1.00000 q^{4} +2.67586 q^{5} +2.29528 q^{6} -0.797143 q^{7} -1.00000 q^{8} +2.26829 q^{9} -2.67586 q^{10} -4.72849 q^{11} -2.29528 q^{12} +2.21536 q^{13} +0.797143 q^{14} -6.14183 q^{15} +1.00000 q^{16} +5.95149 q^{17} -2.26829 q^{18} -8.03117 q^{19} +2.67586 q^{20} +1.82966 q^{21} +4.72849 q^{22} -2.81923 q^{23} +2.29528 q^{24} +2.16022 q^{25} -2.21536 q^{26} +1.67947 q^{27} -0.797143 q^{28} -0.0257294 q^{29} +6.14183 q^{30} +2.78469 q^{31} -1.00000 q^{32} +10.8532 q^{33} -5.95149 q^{34} -2.13304 q^{35} +2.26829 q^{36} -10.1286 q^{37} +8.03117 q^{38} -5.08487 q^{39} -2.67586 q^{40} -6.45270 q^{41} -1.82966 q^{42} +5.94048 q^{43} -4.72849 q^{44} +6.06963 q^{45} +2.81923 q^{46} +6.16027 q^{47} -2.29528 q^{48} -6.36456 q^{49} -2.16022 q^{50} -13.6603 q^{51} +2.21536 q^{52} -3.53867 q^{53} -1.67947 q^{54} -12.6528 q^{55} +0.797143 q^{56} +18.4337 q^{57} +0.0257294 q^{58} -4.31537 q^{59} -6.14183 q^{60} +10.1809 q^{61} -2.78469 q^{62} -1.80815 q^{63} +1.00000 q^{64} +5.92800 q^{65} -10.8532 q^{66} +3.52973 q^{67} +5.95149 q^{68} +6.47091 q^{69} +2.13304 q^{70} -0.491838 q^{71} -2.26829 q^{72} +11.9382 q^{73} +10.1286 q^{74} -4.95830 q^{75} -8.03117 q^{76} +3.76928 q^{77} +5.08487 q^{78} +4.37177 q^{79} +2.67586 q^{80} -10.6597 q^{81} +6.45270 q^{82} -15.8166 q^{83} +1.82966 q^{84} +15.9253 q^{85} -5.94048 q^{86} +0.0590561 q^{87} +4.72849 q^{88} +13.0607 q^{89} -6.06963 q^{90} -1.76596 q^{91} -2.81923 q^{92} -6.39163 q^{93} -6.16027 q^{94} -21.4903 q^{95} +2.29528 q^{96} -17.5486 q^{97} +6.36456 q^{98} -10.7256 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 69 q - 69 q^{2} + 69 q^{4} + 13 q^{5} - 27 q^{7} - 69 q^{8} + 99 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 69 q - 69 q^{2} + 69 q^{4} + 13 q^{5} - 27 q^{7} - 69 q^{8} + 99 q^{9} - 13 q^{10} + 42 q^{11} - 5 q^{13} + 27 q^{14} + 18 q^{15} + 69 q^{16} + 24 q^{17} - 99 q^{18} + q^{19} + 13 q^{20} + 7 q^{21} - 42 q^{22} + 25 q^{23} + 100 q^{25} + 5 q^{26} + 15 q^{27} - 27 q^{28} + 87 q^{29} - 18 q^{30} + 5 q^{31} - 69 q^{32} + 28 q^{33} - 24 q^{34} + 33 q^{35} + 99 q^{36} - 5 q^{37} - q^{38} + 22 q^{39} - 13 q^{40} + 47 q^{41} - 7 q^{42} - 23 q^{43} + 42 q^{44} + 14 q^{45} - 25 q^{46} + 13 q^{47} + 106 q^{49} - 100 q^{50} + 2 q^{51} - 5 q^{52} + 51 q^{53} - 15 q^{54} - 11 q^{55} + 27 q^{56} + 52 q^{57} - 87 q^{58} + 73 q^{59} + 18 q^{60} + 4 q^{61} - 5 q^{62} - 86 q^{63} + 69 q^{64} + 70 q^{65} - 28 q^{66} - 24 q^{67} + 24 q^{68} + 56 q^{69} - 33 q^{70} + 84 q^{71} - 99 q^{72} + 27 q^{73} + 5 q^{74} + 27 q^{75} + q^{76} + 45 q^{77} - 22 q^{78} + 42 q^{79} + 13 q^{80} + 205 q^{81} - 47 q^{82} + q^{83} + 7 q^{84} - 18 q^{85} + 23 q^{86} - q^{87} - 42 q^{88} + 94 q^{89} - 14 q^{90} + 6 q^{91} + 25 q^{92} - 13 q^{93} - 13 q^{94} + 86 q^{95} + 35 q^{97} - 106 q^{98} + 83 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) −2.29528 −1.32518 −0.662589 0.748983i \(-0.730542\pi\)
−0.662589 + 0.748983i \(0.730542\pi\)
\(4\) 1.00000 0.500000
\(5\) 2.67586 1.19668 0.598340 0.801242i \(-0.295827\pi\)
0.598340 + 0.801242i \(0.295827\pi\)
\(6\) 2.29528 0.937042
\(7\) −0.797143 −0.301292 −0.150646 0.988588i \(-0.548135\pi\)
−0.150646 + 0.988588i \(0.548135\pi\)
\(8\) −1.00000 −0.353553
\(9\) 2.26829 0.756097
\(10\) −2.67586 −0.846181
\(11\) −4.72849 −1.42569 −0.712847 0.701320i \(-0.752595\pi\)
−0.712847 + 0.701320i \(0.752595\pi\)
\(12\) −2.29528 −0.662589
\(13\) 2.21536 0.614432 0.307216 0.951640i \(-0.400603\pi\)
0.307216 + 0.951640i \(0.400603\pi\)
\(14\) 0.797143 0.213045
\(15\) −6.14183 −1.58581
\(16\) 1.00000 0.250000
\(17\) 5.95149 1.44345 0.721724 0.692181i \(-0.243350\pi\)
0.721724 + 0.692181i \(0.243350\pi\)
\(18\) −2.26829 −0.534641
\(19\) −8.03117 −1.84248 −0.921238 0.389000i \(-0.872821\pi\)
−0.921238 + 0.389000i \(0.872821\pi\)
\(20\) 2.67586 0.598340
\(21\) 1.82966 0.399265
\(22\) 4.72849 1.00812
\(23\) −2.81923 −0.587850 −0.293925 0.955829i \(-0.594962\pi\)
−0.293925 + 0.955829i \(0.594962\pi\)
\(24\) 2.29528 0.468521
\(25\) 2.16022 0.432044
\(26\) −2.21536 −0.434469
\(27\) 1.67947 0.323215
\(28\) −0.797143 −0.150646
\(29\) −0.0257294 −0.00477784 −0.00238892 0.999997i \(-0.500760\pi\)
−0.00238892 + 0.999997i \(0.500760\pi\)
\(30\) 6.14183 1.12134
\(31\) 2.78469 0.500145 0.250073 0.968227i \(-0.419546\pi\)
0.250073 + 0.968227i \(0.419546\pi\)
\(32\) −1.00000 −0.176777
\(33\) 10.8532 1.88930
\(34\) −5.95149 −1.02067
\(35\) −2.13304 −0.360550
\(36\) 2.26829 0.378048
\(37\) −10.1286 −1.66513 −0.832566 0.553926i \(-0.813129\pi\)
−0.832566 + 0.553926i \(0.813129\pi\)
\(38\) 8.03117 1.30283
\(39\) −5.08487 −0.814231
\(40\) −2.67586 −0.423090
\(41\) −6.45270 −1.00774 −0.503871 0.863779i \(-0.668091\pi\)
−0.503871 + 0.863779i \(0.668091\pi\)
\(42\) −1.82966 −0.282323
\(43\) 5.94048 0.905915 0.452957 0.891532i \(-0.350369\pi\)
0.452957 + 0.891532i \(0.350369\pi\)
\(44\) −4.72849 −0.712847
\(45\) 6.06963 0.904806
\(46\) 2.81923 0.415673
\(47\) 6.16027 0.898568 0.449284 0.893389i \(-0.351679\pi\)
0.449284 + 0.893389i \(0.351679\pi\)
\(48\) −2.29528 −0.331295
\(49\) −6.36456 −0.909223
\(50\) −2.16022 −0.305501
\(51\) −13.6603 −1.91283
\(52\) 2.21536 0.307216
\(53\) −3.53867 −0.486074 −0.243037 0.970017i \(-0.578144\pi\)
−0.243037 + 0.970017i \(0.578144\pi\)
\(54\) −1.67947 −0.228548
\(55\) −12.6528 −1.70610
\(56\) 0.797143 0.106523
\(57\) 18.4337 2.44161
\(58\) 0.0257294 0.00337844
\(59\) −4.31537 −0.561814 −0.280907 0.959735i \(-0.590635\pi\)
−0.280907 + 0.959735i \(0.590635\pi\)
\(60\) −6.14183 −0.792907
\(61\) 10.1809 1.30352 0.651762 0.758423i \(-0.274030\pi\)
0.651762 + 0.758423i \(0.274030\pi\)
\(62\) −2.78469 −0.353656
\(63\) −1.80815 −0.227806
\(64\) 1.00000 0.125000
\(65\) 5.92800 0.735278
\(66\) −10.8532 −1.33594
\(67\) 3.52973 0.431226 0.215613 0.976479i \(-0.430825\pi\)
0.215613 + 0.976479i \(0.430825\pi\)
\(68\) 5.95149 0.721724
\(69\) 6.47091 0.779006
\(70\) 2.13304 0.254947
\(71\) −0.491838 −0.0583704 −0.0291852 0.999574i \(-0.509291\pi\)
−0.0291852 + 0.999574i \(0.509291\pi\)
\(72\) −2.26829 −0.267321
\(73\) 11.9382 1.39727 0.698633 0.715481i \(-0.253792\pi\)
0.698633 + 0.715481i \(0.253792\pi\)
\(74\) 10.1286 1.17743
\(75\) −4.95830 −0.572535
\(76\) −8.03117 −0.921238
\(77\) 3.76928 0.429550
\(78\) 5.08487 0.575749
\(79\) 4.37177 0.491862 0.245931 0.969287i \(-0.420906\pi\)
0.245931 + 0.969287i \(0.420906\pi\)
\(80\) 2.67586 0.299170
\(81\) −10.6597 −1.18441
\(82\) 6.45270 0.712582
\(83\) −15.8166 −1.73610 −0.868048 0.496480i \(-0.834626\pi\)
−0.868048 + 0.496480i \(0.834626\pi\)
\(84\) 1.82966 0.199633
\(85\) 15.9253 1.72735
\(86\) −5.94048 −0.640578
\(87\) 0.0590561 0.00633148
\(88\) 4.72849 0.504059
\(89\) 13.0607 1.38443 0.692215 0.721692i \(-0.256635\pi\)
0.692215 + 0.721692i \(0.256635\pi\)
\(90\) −6.06963 −0.639795
\(91\) −1.76596 −0.185123
\(92\) −2.81923 −0.293925
\(93\) −6.39163 −0.662781
\(94\) −6.16027 −0.635383
\(95\) −21.4903 −2.20485
\(96\) 2.29528 0.234261
\(97\) −17.5486 −1.78179 −0.890893 0.454213i \(-0.849921\pi\)
−0.890893 + 0.454213i \(0.849921\pi\)
\(98\) 6.36456 0.642918
\(99\) −10.7256 −1.07796
\(100\) 2.16022 0.216022
\(101\) 0.211788 0.0210737 0.0105368 0.999944i \(-0.496646\pi\)
0.0105368 + 0.999944i \(0.496646\pi\)
\(102\) 13.6603 1.35257
\(103\) 15.0701 1.48490 0.742450 0.669901i \(-0.233664\pi\)
0.742450 + 0.669901i \(0.233664\pi\)
\(104\) −2.21536 −0.217234
\(105\) 4.89592 0.477793
\(106\) 3.53867 0.343706
\(107\) 9.01855 0.871856 0.435928 0.899982i \(-0.356420\pi\)
0.435928 + 0.899982i \(0.356420\pi\)
\(108\) 1.67947 0.161608
\(109\) 12.2861 1.17680 0.588398 0.808572i \(-0.299759\pi\)
0.588398 + 0.808572i \(0.299759\pi\)
\(110\) 12.6528 1.20639
\(111\) 23.2479 2.20660
\(112\) −0.797143 −0.0753229
\(113\) 19.4534 1.83002 0.915011 0.403429i \(-0.132182\pi\)
0.915011 + 0.403429i \(0.132182\pi\)
\(114\) −18.4337 −1.72648
\(115\) −7.54386 −0.703468
\(116\) −0.0257294 −0.00238892
\(117\) 5.02509 0.464570
\(118\) 4.31537 0.397262
\(119\) −4.74419 −0.434899
\(120\) 6.14183 0.560670
\(121\) 11.3586 1.03260
\(122\) −10.1809 −0.921731
\(123\) 14.8107 1.33544
\(124\) 2.78469 0.250073
\(125\) −7.59885 −0.679662
\(126\) 1.80815 0.161083
\(127\) −5.89558 −0.523149 −0.261574 0.965183i \(-0.584242\pi\)
−0.261574 + 0.965183i \(0.584242\pi\)
\(128\) −1.00000 −0.0883883
\(129\) −13.6350 −1.20050
\(130\) −5.92800 −0.519920
\(131\) −3.79936 −0.331952 −0.165976 0.986130i \(-0.553077\pi\)
−0.165976 + 0.986130i \(0.553077\pi\)
\(132\) 10.8532 0.944649
\(133\) 6.40199 0.555123
\(134\) −3.52973 −0.304923
\(135\) 4.49404 0.386785
\(136\) −5.95149 −0.510336
\(137\) 8.27956 0.707371 0.353685 0.935364i \(-0.384928\pi\)
0.353685 + 0.935364i \(0.384928\pi\)
\(138\) −6.47091 −0.550840
\(139\) 7.07166 0.599810 0.299905 0.953969i \(-0.403045\pi\)
0.299905 + 0.953969i \(0.403045\pi\)
\(140\) −2.13304 −0.180275
\(141\) −14.1395 −1.19076
\(142\) 0.491838 0.0412741
\(143\) −10.4753 −0.875991
\(144\) 2.26829 0.189024
\(145\) −0.0688483 −0.00571754
\(146\) −11.9382 −0.988016
\(147\) 14.6084 1.20488
\(148\) −10.1286 −0.832566
\(149\) −5.35489 −0.438690 −0.219345 0.975647i \(-0.570392\pi\)
−0.219345 + 0.975647i \(0.570392\pi\)
\(150\) 4.95830 0.404844
\(151\) −2.19805 −0.178875 −0.0894375 0.995992i \(-0.528507\pi\)
−0.0894375 + 0.995992i \(0.528507\pi\)
\(152\) 8.03117 0.651414
\(153\) 13.4997 1.09139
\(154\) −3.76928 −0.303737
\(155\) 7.45144 0.598514
\(156\) −5.08487 −0.407116
\(157\) 16.0486 1.28081 0.640407 0.768035i \(-0.278766\pi\)
0.640407 + 0.768035i \(0.278766\pi\)
\(158\) −4.37177 −0.347799
\(159\) 8.12223 0.644135
\(160\) −2.67586 −0.211545
\(161\) 2.24733 0.177114
\(162\) 10.6597 0.837507
\(163\) −21.3215 −1.67003 −0.835014 0.550229i \(-0.814541\pi\)
−0.835014 + 0.550229i \(0.814541\pi\)
\(164\) −6.45270 −0.503871
\(165\) 29.0416 2.26089
\(166\) 15.8166 1.22761
\(167\) −5.39808 −0.417716 −0.208858 0.977946i \(-0.566975\pi\)
−0.208858 + 0.977946i \(0.566975\pi\)
\(168\) −1.82966 −0.141162
\(169\) −8.09216 −0.622474
\(170\) −15.9253 −1.22142
\(171\) −18.2170 −1.39309
\(172\) 5.94048 0.452957
\(173\) 6.45502 0.490766 0.245383 0.969426i \(-0.421086\pi\)
0.245383 + 0.969426i \(0.421086\pi\)
\(174\) −0.0590561 −0.00447703
\(175\) −1.72200 −0.130171
\(176\) −4.72849 −0.356423
\(177\) 9.90497 0.744503
\(178\) −13.0607 −0.978939
\(179\) −13.0398 −0.974637 −0.487318 0.873224i \(-0.662025\pi\)
−0.487318 + 0.873224i \(0.662025\pi\)
\(180\) 6.06963 0.452403
\(181\) 17.9465 1.33395 0.666977 0.745078i \(-0.267588\pi\)
0.666977 + 0.745078i \(0.267588\pi\)
\(182\) 1.76596 0.130902
\(183\) −23.3679 −1.72740
\(184\) 2.81923 0.207836
\(185\) −27.1027 −1.99263
\(186\) 6.39163 0.468657
\(187\) −28.1416 −2.05792
\(188\) 6.16027 0.449284
\(189\) −1.33878 −0.0973820
\(190\) 21.4903 1.55907
\(191\) −4.33850 −0.313923 −0.156961 0.987605i \(-0.550170\pi\)
−0.156961 + 0.987605i \(0.550170\pi\)
\(192\) −2.29528 −0.165647
\(193\) −22.0936 −1.59033 −0.795167 0.606391i \(-0.792617\pi\)
−0.795167 + 0.606391i \(0.792617\pi\)
\(194\) 17.5486 1.25991
\(195\) −13.6064 −0.974375
\(196\) −6.36456 −0.454612
\(197\) −21.1403 −1.50618 −0.753092 0.657916i \(-0.771438\pi\)
−0.753092 + 0.657916i \(0.771438\pi\)
\(198\) 10.7256 0.762235
\(199\) 6.95972 0.493362 0.246681 0.969097i \(-0.420660\pi\)
0.246681 + 0.969097i \(0.420660\pi\)
\(200\) −2.16022 −0.152751
\(201\) −8.10171 −0.571451
\(202\) −0.211788 −0.0149013
\(203\) 0.0205100 0.00143952
\(204\) −13.6603 −0.956413
\(205\) −17.2665 −1.20595
\(206\) −15.0701 −1.04998
\(207\) −6.39483 −0.444471
\(208\) 2.21536 0.153608
\(209\) 37.9753 2.62681
\(210\) −4.89592 −0.337850
\(211\) −11.6737 −0.803652 −0.401826 0.915716i \(-0.631624\pi\)
−0.401826 + 0.915716i \(0.631624\pi\)
\(212\) −3.53867 −0.243037
\(213\) 1.12890 0.0773511
\(214\) −9.01855 −0.616495
\(215\) 15.8959 1.08409
\(216\) −1.67947 −0.114274
\(217\) −2.21980 −0.150690
\(218\) −12.2861 −0.832120
\(219\) −27.4016 −1.85163
\(220\) −12.6528 −0.853050
\(221\) 13.1847 0.886901
\(222\) −23.2479 −1.56030
\(223\) −1.81507 −0.121546 −0.0607731 0.998152i \(-0.519357\pi\)
−0.0607731 + 0.998152i \(0.519357\pi\)
\(224\) 0.797143 0.0532613
\(225\) 4.90001 0.326667
\(226\) −19.4534 −1.29402
\(227\) 16.1950 1.07490 0.537450 0.843295i \(-0.319387\pi\)
0.537450 + 0.843295i \(0.319387\pi\)
\(228\) 18.4337 1.22080
\(229\) −8.80688 −0.581975 −0.290988 0.956727i \(-0.593984\pi\)
−0.290988 + 0.956727i \(0.593984\pi\)
\(230\) 7.54386 0.497427
\(231\) −8.65154 −0.569230
\(232\) 0.0257294 0.00168922
\(233\) 6.09380 0.399218 0.199609 0.979876i \(-0.436033\pi\)
0.199609 + 0.979876i \(0.436033\pi\)
\(234\) −5.02509 −0.328500
\(235\) 16.4840 1.07530
\(236\) −4.31537 −0.280907
\(237\) −10.0344 −0.651805
\(238\) 4.74419 0.307520
\(239\) 17.5768 1.13695 0.568476 0.822700i \(-0.307533\pi\)
0.568476 + 0.822700i \(0.307533\pi\)
\(240\) −6.14183 −0.396454
\(241\) 1.56163 0.100594 0.0502969 0.998734i \(-0.483983\pi\)
0.0502969 + 0.998734i \(0.483983\pi\)
\(242\) −11.3586 −0.730160
\(243\) 19.4286 1.24634
\(244\) 10.1809 0.651762
\(245\) −17.0307 −1.08805
\(246\) −14.8107 −0.944298
\(247\) −17.7920 −1.13208
\(248\) −2.78469 −0.176828
\(249\) 36.3034 2.30064
\(250\) 7.59885 0.480593
\(251\) 22.2420 1.40390 0.701952 0.712224i \(-0.252312\pi\)
0.701952 + 0.712224i \(0.252312\pi\)
\(252\) −1.80815 −0.113903
\(253\) 13.3307 0.838094
\(254\) 5.89558 0.369922
\(255\) −36.5531 −2.28904
\(256\) 1.00000 0.0625000
\(257\) 21.4026 1.33506 0.667529 0.744584i \(-0.267352\pi\)
0.667529 + 0.744584i \(0.267352\pi\)
\(258\) 13.6350 0.848880
\(259\) 8.07394 0.501690
\(260\) 5.92800 0.367639
\(261\) −0.0583618 −0.00361251
\(262\) 3.79936 0.234725
\(263\) 1.93215 0.119141 0.0595706 0.998224i \(-0.481027\pi\)
0.0595706 + 0.998224i \(0.481027\pi\)
\(264\) −10.8532 −0.667968
\(265\) −9.46899 −0.581675
\(266\) −6.40199 −0.392531
\(267\) −29.9779 −1.83462
\(268\) 3.52973 0.215613
\(269\) 6.89306 0.420277 0.210139 0.977672i \(-0.432608\pi\)
0.210139 + 0.977672i \(0.432608\pi\)
\(270\) −4.49404 −0.273498
\(271\) −11.7295 −0.712516 −0.356258 0.934388i \(-0.615948\pi\)
−0.356258 + 0.934388i \(0.615948\pi\)
\(272\) 5.95149 0.360862
\(273\) 4.05337 0.245321
\(274\) −8.27956 −0.500187
\(275\) −10.2146 −0.615962
\(276\) 6.47091 0.389503
\(277\) −30.6706 −1.84282 −0.921409 0.388594i \(-0.872961\pi\)
−0.921409 + 0.388594i \(0.872961\pi\)
\(278\) −7.07166 −0.424130
\(279\) 6.31649 0.378158
\(280\) 2.13304 0.127474
\(281\) 24.4405 1.45800 0.728998 0.684516i \(-0.239986\pi\)
0.728998 + 0.684516i \(0.239986\pi\)
\(282\) 14.1395 0.841996
\(283\) 15.8345 0.941266 0.470633 0.882329i \(-0.344026\pi\)
0.470633 + 0.882329i \(0.344026\pi\)
\(284\) −0.491838 −0.0291852
\(285\) 49.3261 2.92182
\(286\) 10.4753 0.619419
\(287\) 5.14372 0.303624
\(288\) −2.26829 −0.133660
\(289\) 18.4202 1.08354
\(290\) 0.0688483 0.00404291
\(291\) 40.2788 2.36118
\(292\) 11.9382 0.698633
\(293\) 6.24103 0.364605 0.182302 0.983243i \(-0.441645\pi\)
0.182302 + 0.983243i \(0.441645\pi\)
\(294\) −14.6084 −0.851981
\(295\) −11.5473 −0.672311
\(296\) 10.1286 0.588713
\(297\) −7.94138 −0.460806
\(298\) 5.35489 0.310201
\(299\) −6.24562 −0.361194
\(300\) −4.95830 −0.286268
\(301\) −4.73541 −0.272945
\(302\) 2.19805 0.126484
\(303\) −0.486111 −0.0279263
\(304\) −8.03117 −0.460619
\(305\) 27.2425 1.55990
\(306\) −13.4997 −0.771727
\(307\) 2.64951 0.151215 0.0756077 0.997138i \(-0.475910\pi\)
0.0756077 + 0.997138i \(0.475910\pi\)
\(308\) 3.76928 0.214775
\(309\) −34.5900 −1.96776
\(310\) −7.45144 −0.423213
\(311\) 0.0266046 0.00150861 0.000754305 1.00000i \(-0.499760\pi\)
0.000754305 1.00000i \(0.499760\pi\)
\(312\) 5.08487 0.287874
\(313\) 13.9540 0.788725 0.394362 0.918955i \(-0.370966\pi\)
0.394362 + 0.918955i \(0.370966\pi\)
\(314\) −16.0486 −0.905673
\(315\) −4.83836 −0.272611
\(316\) 4.37177 0.245931
\(317\) −9.84385 −0.552886 −0.276443 0.961030i \(-0.589156\pi\)
−0.276443 + 0.961030i \(0.589156\pi\)
\(318\) −8.12223 −0.455472
\(319\) 0.121661 0.00681173
\(320\) 2.67586 0.149585
\(321\) −20.7001 −1.15536
\(322\) −2.24733 −0.125239
\(323\) −47.7974 −2.65952
\(324\) −10.6597 −0.592207
\(325\) 4.78568 0.265462
\(326\) 21.3215 1.18089
\(327\) −28.2000 −1.55946
\(328\) 6.45270 0.356291
\(329\) −4.91062 −0.270731
\(330\) −29.0416 −1.59869
\(331\) −4.67627 −0.257031 −0.128515 0.991708i \(-0.541021\pi\)
−0.128515 + 0.991708i \(0.541021\pi\)
\(332\) −15.8166 −0.868048
\(333\) −22.9746 −1.25900
\(334\) 5.39808 0.295370
\(335\) 9.44507 0.516039
\(336\) 1.82966 0.0998163
\(337\) −11.8810 −0.647197 −0.323599 0.946194i \(-0.604893\pi\)
−0.323599 + 0.946194i \(0.604893\pi\)
\(338\) 8.09216 0.440155
\(339\) −44.6509 −2.42510
\(340\) 15.9253 0.863673
\(341\) −13.1674 −0.713054
\(342\) 18.2170 0.985063
\(343\) 10.6535 0.575233
\(344\) −5.94048 −0.320289
\(345\) 17.3152 0.932221
\(346\) −6.45502 −0.347024
\(347\) −35.8859 −1.92646 −0.963228 0.268686i \(-0.913411\pi\)
−0.963228 + 0.268686i \(0.913411\pi\)
\(348\) 0.0590561 0.00316574
\(349\) 24.0051 1.28496 0.642481 0.766302i \(-0.277905\pi\)
0.642481 + 0.766302i \(0.277905\pi\)
\(350\) 1.72200 0.0920450
\(351\) 3.72065 0.198594
\(352\) 4.72849 0.252029
\(353\) 20.7832 1.10618 0.553090 0.833122i \(-0.313449\pi\)
0.553090 + 0.833122i \(0.313449\pi\)
\(354\) −9.90497 −0.526443
\(355\) −1.31609 −0.0698507
\(356\) 13.0607 0.692215
\(357\) 10.8892 0.576319
\(358\) 13.0398 0.689172
\(359\) 25.8606 1.36487 0.682436 0.730945i \(-0.260921\pi\)
0.682436 + 0.730945i \(0.260921\pi\)
\(360\) −6.06963 −0.319897
\(361\) 45.4996 2.39472
\(362\) −17.9465 −0.943248
\(363\) −26.0712 −1.36838
\(364\) −1.76596 −0.0925616
\(365\) 31.9450 1.67208
\(366\) 23.3679 1.22146
\(367\) 5.57150 0.290830 0.145415 0.989371i \(-0.453548\pi\)
0.145415 + 0.989371i \(0.453548\pi\)
\(368\) −2.81923 −0.146962
\(369\) −14.6366 −0.761951
\(370\) 27.1027 1.40900
\(371\) 2.82083 0.146450
\(372\) −6.39163 −0.331391
\(373\) −2.72743 −0.141221 −0.0706105 0.997504i \(-0.522495\pi\)
−0.0706105 + 0.997504i \(0.522495\pi\)
\(374\) 28.1416 1.45517
\(375\) 17.4415 0.900673
\(376\) −6.16027 −0.317692
\(377\) −0.0570001 −0.00293565
\(378\) 1.33878 0.0688595
\(379\) 31.0762 1.59628 0.798139 0.602473i \(-0.205818\pi\)
0.798139 + 0.602473i \(0.205818\pi\)
\(380\) −21.4903 −1.10243
\(381\) 13.5320 0.693265
\(382\) 4.33850 0.221977
\(383\) −34.7199 −1.77410 −0.887052 0.461669i \(-0.847251\pi\)
−0.887052 + 0.461669i \(0.847251\pi\)
\(384\) 2.29528 0.117130
\(385\) 10.0861 0.514034
\(386\) 22.0936 1.12454
\(387\) 13.4747 0.684959
\(388\) −17.5486 −0.890893
\(389\) 17.2467 0.874444 0.437222 0.899354i \(-0.355962\pi\)
0.437222 + 0.899354i \(0.355962\pi\)
\(390\) 13.6064 0.688987
\(391\) −16.7786 −0.848531
\(392\) 6.36456 0.321459
\(393\) 8.72058 0.439895
\(394\) 21.1403 1.06503
\(395\) 11.6982 0.588602
\(396\) −10.7256 −0.538981
\(397\) 21.1110 1.05953 0.529764 0.848145i \(-0.322280\pi\)
0.529764 + 0.848145i \(0.322280\pi\)
\(398\) −6.95972 −0.348859
\(399\) −14.6943 −0.735636
\(400\) 2.16022 0.108011
\(401\) 28.8448 1.44044 0.720219 0.693746i \(-0.244041\pi\)
0.720219 + 0.693746i \(0.244041\pi\)
\(402\) 8.10171 0.404077
\(403\) 6.16910 0.307305
\(404\) 0.211788 0.0105368
\(405\) −28.5239 −1.41737
\(406\) −0.0205100 −0.00101790
\(407\) 47.8930 2.37397
\(408\) 13.6603 0.676286
\(409\) −12.5644 −0.621272 −0.310636 0.950529i \(-0.600542\pi\)
−0.310636 + 0.950529i \(0.600542\pi\)
\(410\) 17.2665 0.852732
\(411\) −19.0039 −0.937392
\(412\) 15.0701 0.742450
\(413\) 3.43997 0.169270
\(414\) 6.39483 0.314289
\(415\) −42.3230 −2.07755
\(416\) −2.21536 −0.108617
\(417\) −16.2314 −0.794855
\(418\) −37.9753 −1.85743
\(419\) −22.1539 −1.08229 −0.541145 0.840929i \(-0.682009\pi\)
−0.541145 + 0.840929i \(0.682009\pi\)
\(420\) 4.89592 0.238896
\(421\) −34.6616 −1.68930 −0.844651 0.535318i \(-0.820192\pi\)
−0.844651 + 0.535318i \(0.820192\pi\)
\(422\) 11.6737 0.568267
\(423\) 13.9733 0.679404
\(424\) 3.53867 0.171853
\(425\) 12.8565 0.623633
\(426\) −1.12890 −0.0546955
\(427\) −8.11559 −0.392741
\(428\) 9.01855 0.435928
\(429\) 24.0438 1.16084
\(430\) −15.8959 −0.766568
\(431\) 5.67464 0.273338 0.136669 0.990617i \(-0.456360\pi\)
0.136669 + 0.990617i \(0.456360\pi\)
\(432\) 1.67947 0.0808038
\(433\) 22.5861 1.08542 0.542710 0.839920i \(-0.317398\pi\)
0.542710 + 0.839920i \(0.317398\pi\)
\(434\) 2.21980 0.106554
\(435\) 0.158026 0.00757676
\(436\) 12.2861 0.588398
\(437\) 22.6417 1.08310
\(438\) 27.4016 1.30930
\(439\) 31.3216 1.49490 0.747448 0.664320i \(-0.231279\pi\)
0.747448 + 0.664320i \(0.231279\pi\)
\(440\) 12.6528 0.603197
\(441\) −14.4367 −0.687461
\(442\) −13.1847 −0.627133
\(443\) −10.5878 −0.503044 −0.251522 0.967852i \(-0.580931\pi\)
−0.251522 + 0.967852i \(0.580931\pi\)
\(444\) 23.2479 1.10330
\(445\) 34.9485 1.65672
\(446\) 1.81507 0.0859461
\(447\) 12.2910 0.581342
\(448\) −0.797143 −0.0376615
\(449\) 19.7320 0.931211 0.465606 0.884992i \(-0.345837\pi\)
0.465606 + 0.884992i \(0.345837\pi\)
\(450\) −4.90001 −0.230989
\(451\) 30.5115 1.43673
\(452\) 19.4534 0.915011
\(453\) 5.04514 0.237041
\(454\) −16.1950 −0.760070
\(455\) −4.72547 −0.221533
\(456\) −18.4337 −0.863239
\(457\) 8.13562 0.380568 0.190284 0.981729i \(-0.439059\pi\)
0.190284 + 0.981729i \(0.439059\pi\)
\(458\) 8.80688 0.411519
\(459\) 9.99538 0.466544
\(460\) −7.54386 −0.351734
\(461\) 30.0171 1.39803 0.699017 0.715105i \(-0.253621\pi\)
0.699017 + 0.715105i \(0.253621\pi\)
\(462\) 8.65154 0.402506
\(463\) −14.1202 −0.656224 −0.328112 0.944639i \(-0.606412\pi\)
−0.328112 + 0.944639i \(0.606412\pi\)
\(464\) −0.0257294 −0.00119446
\(465\) −17.1031 −0.793137
\(466\) −6.09380 −0.282290
\(467\) −6.67297 −0.308788 −0.154394 0.988009i \(-0.549343\pi\)
−0.154394 + 0.988009i \(0.549343\pi\)
\(468\) 5.02509 0.232285
\(469\) −2.81370 −0.129925
\(470\) −16.4840 −0.760351
\(471\) −36.8359 −1.69731
\(472\) 4.31537 0.198631
\(473\) −28.0895 −1.29156
\(474\) 10.0344 0.460896
\(475\) −17.3491 −0.796031
\(476\) −4.74419 −0.217450
\(477\) −8.02674 −0.367519
\(478\) −17.5768 −0.803946
\(479\) 24.9404 1.13955 0.569777 0.821799i \(-0.307029\pi\)
0.569777 + 0.821799i \(0.307029\pi\)
\(480\) 6.14183 0.280335
\(481\) −22.4386 −1.02311
\(482\) −1.56163 −0.0711305
\(483\) −5.15824 −0.234708
\(484\) 11.3586 0.516301
\(485\) −46.9575 −2.13223
\(486\) −19.4286 −0.881299
\(487\) 9.93041 0.449990 0.224995 0.974360i \(-0.427763\pi\)
0.224995 + 0.974360i \(0.427763\pi\)
\(488\) −10.1809 −0.460866
\(489\) 48.9387 2.21308
\(490\) 17.0307 0.769367
\(491\) −3.07555 −0.138798 −0.0693988 0.997589i \(-0.522108\pi\)
−0.0693988 + 0.997589i \(0.522108\pi\)
\(492\) 14.8107 0.667719
\(493\) −0.153128 −0.00689656
\(494\) 17.7920 0.800498
\(495\) −28.7002 −1.28998
\(496\) 2.78469 0.125036
\(497\) 0.392065 0.0175865
\(498\) −36.3034 −1.62680
\(499\) 19.9045 0.891048 0.445524 0.895270i \(-0.353017\pi\)
0.445524 + 0.895270i \(0.353017\pi\)
\(500\) −7.59885 −0.339831
\(501\) 12.3901 0.553548
\(502\) −22.2420 −0.992710
\(503\) 40.4470 1.80344 0.901722 0.432316i \(-0.142304\pi\)
0.901722 + 0.432316i \(0.142304\pi\)
\(504\) 1.80815 0.0805415
\(505\) 0.566714 0.0252184
\(506\) −13.3307 −0.592622
\(507\) 18.5737 0.824888
\(508\) −5.89558 −0.261574
\(509\) −4.01983 −0.178176 −0.0890879 0.996024i \(-0.528395\pi\)
−0.0890879 + 0.996024i \(0.528395\pi\)
\(510\) 36.5531 1.61860
\(511\) −9.51648 −0.420984
\(512\) −1.00000 −0.0441942
\(513\) −13.4881 −0.595516
\(514\) −21.4026 −0.944028
\(515\) 40.3254 1.77695
\(516\) −13.6350 −0.600249
\(517\) −29.1288 −1.28108
\(518\) −8.07394 −0.354749
\(519\) −14.8160 −0.650352
\(520\) −5.92800 −0.259960
\(521\) 6.65665 0.291633 0.145817 0.989312i \(-0.453419\pi\)
0.145817 + 0.989312i \(0.453419\pi\)
\(522\) 0.0583618 0.00255443
\(523\) 35.5975 1.55657 0.778285 0.627911i \(-0.216090\pi\)
0.778285 + 0.627911i \(0.216090\pi\)
\(524\) −3.79936 −0.165976
\(525\) 3.95247 0.172500
\(526\) −1.93215 −0.0842456
\(527\) 16.5731 0.721934
\(528\) 10.8532 0.472325
\(529\) −15.0519 −0.654432
\(530\) 9.46899 0.411307
\(531\) −9.78852 −0.424785
\(532\) 6.40199 0.277561
\(533\) −14.2951 −0.619189
\(534\) 29.9779 1.29727
\(535\) 24.1324 1.04333
\(536\) −3.52973 −0.152461
\(537\) 29.9298 1.29157
\(538\) −6.89306 −0.297181
\(539\) 30.0948 1.29627
\(540\) 4.49404 0.193393
\(541\) 17.7852 0.764646 0.382323 0.924029i \(-0.375124\pi\)
0.382323 + 0.924029i \(0.375124\pi\)
\(542\) 11.7295 0.503825
\(543\) −41.1922 −1.76773
\(544\) −5.95149 −0.255168
\(545\) 32.8759 1.40825
\(546\) −4.05337 −0.173468
\(547\) 33.8882 1.44895 0.724477 0.689299i \(-0.242082\pi\)
0.724477 + 0.689299i \(0.242082\pi\)
\(548\) 8.27956 0.353685
\(549\) 23.0931 0.985591
\(550\) 10.2146 0.435551
\(551\) 0.206637 0.00880305
\(552\) −6.47091 −0.275420
\(553\) −3.48492 −0.148194
\(554\) 30.6706 1.30307
\(555\) 62.2082 2.64059
\(556\) 7.07166 0.299905
\(557\) −18.3176 −0.776140 −0.388070 0.921630i \(-0.626858\pi\)
−0.388070 + 0.921630i \(0.626858\pi\)
\(558\) −6.31649 −0.267398
\(559\) 13.1603 0.556623
\(560\) −2.13304 −0.0901375
\(561\) 64.5927 2.72710
\(562\) −24.4405 −1.03096
\(563\) 27.9324 1.17721 0.588605 0.808421i \(-0.299677\pi\)
0.588605 + 0.808421i \(0.299677\pi\)
\(564\) −14.1395 −0.595381
\(565\) 52.0545 2.18995
\(566\) −15.8345 −0.665576
\(567\) 8.49733 0.356854
\(568\) 0.491838 0.0206370
\(569\) 5.07084 0.212581 0.106290 0.994335i \(-0.466103\pi\)
0.106290 + 0.994335i \(0.466103\pi\)
\(570\) −49.3261 −2.06604
\(571\) −17.1596 −0.718107 −0.359054 0.933317i \(-0.616900\pi\)
−0.359054 + 0.933317i \(0.616900\pi\)
\(572\) −10.4753 −0.437996
\(573\) 9.95805 0.416004
\(574\) −5.14372 −0.214695
\(575\) −6.09015 −0.253977
\(576\) 2.26829 0.0945121
\(577\) −6.15940 −0.256419 −0.128210 0.991747i \(-0.540923\pi\)
−0.128210 + 0.991747i \(0.540923\pi\)
\(578\) −18.4202 −0.766181
\(579\) 50.7109 2.10748
\(580\) −0.0688483 −0.00285877
\(581\) 12.6081 0.523071
\(582\) −40.2788 −1.66961
\(583\) 16.7326 0.692993
\(584\) −11.9382 −0.494008
\(585\) 13.4464 0.555942
\(586\) −6.24103 −0.257815
\(587\) 5.63608 0.232626 0.116313 0.993213i \(-0.462892\pi\)
0.116313 + 0.993213i \(0.462892\pi\)
\(588\) 14.6084 0.602441
\(589\) −22.3643 −0.921505
\(590\) 11.5473 0.475396
\(591\) 48.5228 1.99596
\(592\) −10.1286 −0.416283
\(593\) −44.3124 −1.81969 −0.909846 0.414947i \(-0.863800\pi\)
−0.909846 + 0.414947i \(0.863800\pi\)
\(594\) 7.94138 0.325839
\(595\) −12.6948 −0.520435
\(596\) −5.35489 −0.219345
\(597\) −15.9745 −0.653792
\(598\) 6.24562 0.255402
\(599\) 24.7389 1.01081 0.505403 0.862883i \(-0.331344\pi\)
0.505403 + 0.862883i \(0.331344\pi\)
\(600\) 4.95830 0.202422
\(601\) −22.5642 −0.920412 −0.460206 0.887812i \(-0.652225\pi\)
−0.460206 + 0.887812i \(0.652225\pi\)
\(602\) 4.73541 0.193001
\(603\) 8.00646 0.326048
\(604\) −2.19805 −0.0894375
\(605\) 30.3941 1.23570
\(606\) 0.486111 0.0197469
\(607\) 31.8366 1.29221 0.646103 0.763250i \(-0.276398\pi\)
0.646103 + 0.763250i \(0.276398\pi\)
\(608\) 8.03117 0.325707
\(609\) −0.0470762 −0.00190762
\(610\) −27.2425 −1.10302
\(611\) 13.6472 0.552109
\(612\) 13.4997 0.545693
\(613\) 9.83441 0.397208 0.198604 0.980080i \(-0.436359\pi\)
0.198604 + 0.980080i \(0.436359\pi\)
\(614\) −2.64951 −0.106925
\(615\) 39.6314 1.59809
\(616\) −3.76928 −0.151869
\(617\) −11.3662 −0.457587 −0.228793 0.973475i \(-0.573478\pi\)
−0.228793 + 0.973475i \(0.573478\pi\)
\(618\) 34.5900 1.39141
\(619\) 27.4330 1.10262 0.551312 0.834299i \(-0.314127\pi\)
0.551312 + 0.834299i \(0.314127\pi\)
\(620\) 7.45144 0.299257
\(621\) −4.73482 −0.190002
\(622\) −0.0266046 −0.00106675
\(623\) −10.4112 −0.417117
\(624\) −5.08487 −0.203558
\(625\) −31.1345 −1.24538
\(626\) −13.9540 −0.557713
\(627\) −87.1638 −3.48099
\(628\) 16.0486 0.640407
\(629\) −60.2803 −2.40353
\(630\) 4.83836 0.192765
\(631\) 0.801172 0.0318941 0.0159471 0.999873i \(-0.494924\pi\)
0.0159471 + 0.999873i \(0.494924\pi\)
\(632\) −4.37177 −0.173900
\(633\) 26.7944 1.06498
\(634\) 9.84385 0.390949
\(635\) −15.7758 −0.626042
\(636\) 8.12223 0.322067
\(637\) −14.0998 −0.558656
\(638\) −0.121661 −0.00481662
\(639\) −1.11563 −0.0441337
\(640\) −2.67586 −0.105773
\(641\) −14.3826 −0.568079 −0.284040 0.958813i \(-0.591675\pi\)
−0.284040 + 0.958813i \(0.591675\pi\)
\(642\) 20.7001 0.816966
\(643\) −21.1048 −0.832294 −0.416147 0.909297i \(-0.636620\pi\)
−0.416147 + 0.909297i \(0.636620\pi\)
\(644\) 2.24733 0.0885571
\(645\) −36.4854 −1.43661
\(646\) 47.7974 1.88056
\(647\) 8.58306 0.337435 0.168718 0.985664i \(-0.446037\pi\)
0.168718 + 0.985664i \(0.446037\pi\)
\(648\) 10.6597 0.418754
\(649\) 20.4052 0.800974
\(650\) −4.78568 −0.187710
\(651\) 5.09504 0.199690
\(652\) −21.3215 −0.835014
\(653\) 7.99908 0.313028 0.156514 0.987676i \(-0.449974\pi\)
0.156514 + 0.987676i \(0.449974\pi\)
\(654\) 28.2000 1.10271
\(655\) −10.1665 −0.397240
\(656\) −6.45270 −0.251936
\(657\) 27.0794 1.05647
\(658\) 4.91062 0.191436
\(659\) 2.50165 0.0974506 0.0487253 0.998812i \(-0.484484\pi\)
0.0487253 + 0.998812i \(0.484484\pi\)
\(660\) 29.0416 1.13044
\(661\) −14.1074 −0.548713 −0.274357 0.961628i \(-0.588465\pi\)
−0.274357 + 0.961628i \(0.588465\pi\)
\(662\) 4.67627 0.181748
\(663\) −30.2626 −1.17530
\(664\) 15.8166 0.613803
\(665\) 17.1308 0.664304
\(666\) 22.9746 0.890248
\(667\) 0.0725372 0.00280865
\(668\) −5.39808 −0.208858
\(669\) 4.16609 0.161070
\(670\) −9.44507 −0.364895
\(671\) −48.1401 −1.85843
\(672\) −1.82966 −0.0705808
\(673\) −1.29326 −0.0498513 −0.0249257 0.999689i \(-0.507935\pi\)
−0.0249257 + 0.999689i \(0.507935\pi\)
\(674\) 11.8810 0.457637
\(675\) 3.62804 0.139643
\(676\) −8.09216 −0.311237
\(677\) 37.9245 1.45756 0.728779 0.684749i \(-0.240088\pi\)
0.728779 + 0.684749i \(0.240088\pi\)
\(678\) 44.6509 1.71481
\(679\) 13.9887 0.536837
\(680\) −15.9253 −0.610709
\(681\) −37.1720 −1.42444
\(682\) 13.1674 0.504205
\(683\) 6.84832 0.262044 0.131022 0.991379i \(-0.458174\pi\)
0.131022 + 0.991379i \(0.458174\pi\)
\(684\) −18.2170 −0.696545
\(685\) 22.1549 0.846497
\(686\) −10.6535 −0.406751
\(687\) 20.2142 0.771221
\(688\) 5.94048 0.226479
\(689\) −7.83945 −0.298659
\(690\) −17.3152 −0.659180
\(691\) 30.0540 1.14331 0.571654 0.820495i \(-0.306302\pi\)
0.571654 + 0.820495i \(0.306302\pi\)
\(692\) 6.45502 0.245383
\(693\) 8.54983 0.324781
\(694\) 35.8859 1.36221
\(695\) 18.9228 0.717781
\(696\) −0.0590561 −0.00223852
\(697\) −38.4032 −1.45462
\(698\) −24.0051 −0.908605
\(699\) −13.9870 −0.529035
\(700\) −1.72200 −0.0650856
\(701\) 13.0611 0.493311 0.246656 0.969103i \(-0.420668\pi\)
0.246656 + 0.969103i \(0.420668\pi\)
\(702\) −3.72065 −0.140427
\(703\) 81.3445 3.06797
\(704\) −4.72849 −0.178212
\(705\) −37.8354 −1.42496
\(706\) −20.7832 −0.782187
\(707\) −0.168825 −0.00634932
\(708\) 9.90497 0.372252
\(709\) −13.1844 −0.495149 −0.247575 0.968869i \(-0.579634\pi\)
−0.247575 + 0.968869i \(0.579634\pi\)
\(710\) 1.31609 0.0493919
\(711\) 9.91644 0.371896
\(712\) −13.0607 −0.489470
\(713\) −7.85068 −0.294010
\(714\) −10.8892 −0.407519
\(715\) −28.0305 −1.04828
\(716\) −13.0398 −0.487318
\(717\) −40.3437 −1.50666
\(718\) −25.8606 −0.965110
\(719\) 35.9265 1.33983 0.669917 0.742436i \(-0.266330\pi\)
0.669917 + 0.742436i \(0.266330\pi\)
\(720\) 6.06963 0.226202
\(721\) −12.0130 −0.447388
\(722\) −45.4996 −1.69332
\(723\) −3.58438 −0.133305
\(724\) 17.9465 0.666977
\(725\) −0.0555812 −0.00206424
\(726\) 26.0712 0.967592
\(727\) 32.4093 1.20200 0.600998 0.799251i \(-0.294770\pi\)
0.600998 + 0.799251i \(0.294770\pi\)
\(728\) 1.76596 0.0654509
\(729\) −12.6148 −0.467214
\(730\) −31.9450 −1.18234
\(731\) 35.3547 1.30764
\(732\) −23.3679 −0.863701
\(733\) 51.0919 1.88712 0.943562 0.331197i \(-0.107452\pi\)
0.943562 + 0.331197i \(0.107452\pi\)
\(734\) −5.57150 −0.205648
\(735\) 39.0901 1.44186
\(736\) 2.81923 0.103918
\(737\) −16.6903 −0.614796
\(738\) 14.6366 0.538781
\(739\) −18.9364 −0.696585 −0.348293 0.937386i \(-0.613238\pi\)
−0.348293 + 0.937386i \(0.613238\pi\)
\(740\) −27.1027 −0.996316
\(741\) 40.8375 1.50020
\(742\) −2.82083 −0.103556
\(743\) 8.20488 0.301008 0.150504 0.988609i \(-0.451910\pi\)
0.150504 + 0.988609i \(0.451910\pi\)
\(744\) 6.39163 0.234329
\(745\) −14.3289 −0.524971
\(746\) 2.72743 0.0998583
\(747\) −35.8766 −1.31266
\(748\) −28.1416 −1.02896
\(749\) −7.18907 −0.262683
\(750\) −17.4415 −0.636872
\(751\) 22.2586 0.812227 0.406114 0.913823i \(-0.366884\pi\)
0.406114 + 0.913823i \(0.366884\pi\)
\(752\) 6.16027 0.224642
\(753\) −51.0516 −1.86042
\(754\) 0.0570001 0.00207582
\(755\) −5.88168 −0.214056
\(756\) −1.33878 −0.0486910
\(757\) −24.2329 −0.880760 −0.440380 0.897812i \(-0.645156\pi\)
−0.440380 + 0.897812i \(0.645156\pi\)
\(758\) −31.0762 −1.12874
\(759\) −30.5976 −1.11062
\(760\) 21.4903 0.779534
\(761\) 32.9970 1.19614 0.598070 0.801444i \(-0.295935\pi\)
0.598070 + 0.801444i \(0.295935\pi\)
\(762\) −13.5320 −0.490212
\(763\) −9.79378 −0.354559
\(764\) −4.33850 −0.156961
\(765\) 36.1233 1.30604
\(766\) 34.7199 1.25448
\(767\) −9.56012 −0.345196
\(768\) −2.29528 −0.0828236
\(769\) −8.08582 −0.291582 −0.145791 0.989315i \(-0.546573\pi\)
−0.145791 + 0.989315i \(0.546573\pi\)
\(770\) −10.0861 −0.363477
\(771\) −49.1249 −1.76919
\(772\) −22.0936 −0.795167
\(773\) 40.9890 1.47427 0.737136 0.675744i \(-0.236178\pi\)
0.737136 + 0.675744i \(0.236178\pi\)
\(774\) −13.4747 −0.484339
\(775\) 6.01554 0.216085
\(776\) 17.5486 0.629957
\(777\) −18.5319 −0.664829
\(778\) −17.2467 −0.618325
\(779\) 51.8227 1.85674
\(780\) −13.6064 −0.487187
\(781\) 2.32565 0.0832183
\(782\) 16.7786 0.600002
\(783\) −0.0432119 −0.00154427
\(784\) −6.36456 −0.227306
\(785\) 42.9437 1.53273
\(786\) −8.72058 −0.311053
\(787\) 13.3284 0.475108 0.237554 0.971374i \(-0.423654\pi\)
0.237554 + 0.971374i \(0.423654\pi\)
\(788\) −21.1403 −0.753092
\(789\) −4.43481 −0.157883
\(790\) −11.6982 −0.416204
\(791\) −15.5071 −0.551370
\(792\) 10.7256 0.381117
\(793\) 22.5543 0.800927
\(794\) −21.1110 −0.749200
\(795\) 21.7339 0.770823
\(796\) 6.95972 0.246681
\(797\) 27.5858 0.977140 0.488570 0.872525i \(-0.337519\pi\)
0.488570 + 0.872525i \(0.337519\pi\)
\(798\) 14.6943 0.520173
\(799\) 36.6628 1.29704
\(800\) −2.16022 −0.0763753
\(801\) 29.6254 1.04676
\(802\) −28.8448 −1.01854
\(803\) −56.4499 −1.99207
\(804\) −8.10171 −0.285725
\(805\) 6.01353 0.211949
\(806\) −6.16910 −0.217297
\(807\) −15.8215 −0.556942
\(808\) −0.211788 −0.00745066
\(809\) 36.6445 1.28835 0.644177 0.764877i \(-0.277200\pi\)
0.644177 + 0.764877i \(0.277200\pi\)
\(810\) 28.5239 1.00223
\(811\) −20.9697 −0.736346 −0.368173 0.929757i \(-0.620017\pi\)
−0.368173 + 0.929757i \(0.620017\pi\)
\(812\) 0.0205100 0.000719761 0
\(813\) 26.9224 0.944210
\(814\) −47.8930 −1.67865
\(815\) −57.0533 −1.99849
\(816\) −13.6603 −0.478207
\(817\) −47.7090 −1.66913
\(818\) 12.5644 0.439305
\(819\) −4.00572 −0.139971
\(820\) −17.2665 −0.602973
\(821\) 9.41005 0.328413 0.164207 0.986426i \(-0.447494\pi\)
0.164207 + 0.986426i \(0.447494\pi\)
\(822\) 19.0039 0.662836
\(823\) −49.8448 −1.73748 −0.868740 0.495269i \(-0.835070\pi\)
−0.868740 + 0.495269i \(0.835070\pi\)
\(824\) −15.0701 −0.524992
\(825\) 23.4453 0.816260
\(826\) −3.43997 −0.119692
\(827\) 30.5515 1.06238 0.531189 0.847253i \(-0.321745\pi\)
0.531189 + 0.847253i \(0.321745\pi\)
\(828\) −6.39483 −0.222236
\(829\) −6.62474 −0.230087 −0.115043 0.993360i \(-0.536701\pi\)
−0.115043 + 0.993360i \(0.536701\pi\)
\(830\) 42.3230 1.46905
\(831\) 70.3975 2.44206
\(832\) 2.21536 0.0768040
\(833\) −37.8786 −1.31242
\(834\) 16.2314 0.562048
\(835\) −14.4445 −0.499872
\(836\) 37.9753 1.31340
\(837\) 4.67682 0.161654
\(838\) 22.1539 0.765295
\(839\) −2.75911 −0.0952552 −0.0476276 0.998865i \(-0.515166\pi\)
−0.0476276 + 0.998865i \(0.515166\pi\)
\(840\) −4.89592 −0.168925
\(841\) −28.9993 −0.999977
\(842\) 34.6616 1.19452
\(843\) −56.0976 −1.93210
\(844\) −11.6737 −0.401826
\(845\) −21.6535 −0.744902
\(846\) −13.9733 −0.480411
\(847\) −9.05445 −0.311115
\(848\) −3.53867 −0.121519
\(849\) −36.3446 −1.24734
\(850\) −12.8565 −0.440975
\(851\) 28.5549 0.978848
\(852\) 1.12890 0.0386756
\(853\) −8.26838 −0.283104 −0.141552 0.989931i \(-0.545209\pi\)
−0.141552 + 0.989931i \(0.545209\pi\)
\(854\) 8.11559 0.277710
\(855\) −48.7462 −1.66708
\(856\) −9.01855 −0.308248
\(857\) −4.07156 −0.139082 −0.0695409 0.997579i \(-0.522153\pi\)
−0.0695409 + 0.997579i \(0.522153\pi\)
\(858\) −24.0438 −0.820841
\(859\) −21.6744 −0.739520 −0.369760 0.929127i \(-0.620560\pi\)
−0.369760 + 0.929127i \(0.620560\pi\)
\(860\) 15.8959 0.542045
\(861\) −11.8063 −0.402356
\(862\) −5.67464 −0.193279
\(863\) −23.3811 −0.795901 −0.397951 0.917407i \(-0.630278\pi\)
−0.397951 + 0.917407i \(0.630278\pi\)
\(864\) −1.67947 −0.0571369
\(865\) 17.2727 0.587290
\(866\) −22.5861 −0.767507
\(867\) −42.2795 −1.43589
\(868\) −2.21980 −0.0753448
\(869\) −20.6719 −0.701245
\(870\) −0.158026 −0.00535758
\(871\) 7.81965 0.264959
\(872\) −12.2861 −0.416060
\(873\) −39.8052 −1.34720
\(874\) −22.6417 −0.765867
\(875\) 6.05737 0.204776
\(876\) −27.4016 −0.925813
\(877\) 10.6455 0.359473 0.179736 0.983715i \(-0.442476\pi\)
0.179736 + 0.983715i \(0.442476\pi\)
\(878\) −31.3216 −1.05705
\(879\) −14.3249 −0.483166
\(880\) −12.6528 −0.426525
\(881\) −29.5271 −0.994793 −0.497396 0.867523i \(-0.665711\pi\)
−0.497396 + 0.867523i \(0.665711\pi\)
\(882\) 14.4367 0.486108
\(883\) −41.6942 −1.40312 −0.701561 0.712609i \(-0.747513\pi\)
−0.701561 + 0.712609i \(0.747513\pi\)
\(884\) 13.1847 0.443450
\(885\) 26.5043 0.890932
\(886\) 10.5878 0.355706
\(887\) 24.2635 0.814687 0.407344 0.913275i \(-0.366455\pi\)
0.407344 + 0.913275i \(0.366455\pi\)
\(888\) −23.2479 −0.780150
\(889\) 4.69962 0.157620
\(890\) −34.9485 −1.17148
\(891\) 50.4044 1.68861
\(892\) −1.81507 −0.0607731
\(893\) −49.4742 −1.65559
\(894\) −12.2910 −0.411071
\(895\) −34.8925 −1.16633
\(896\) 0.797143 0.0266307
\(897\) 14.3354 0.478646
\(898\) −19.7320 −0.658466
\(899\) −0.0716485 −0.00238961
\(900\) 4.90001 0.163334
\(901\) −21.0604 −0.701623
\(902\) −30.5115 −1.01592
\(903\) 10.8691 0.361700
\(904\) −19.4534 −0.647010
\(905\) 48.0223 1.59632
\(906\) −5.04514 −0.167613
\(907\) −0.554591 −0.0184149 −0.00920745 0.999958i \(-0.502931\pi\)
−0.00920745 + 0.999958i \(0.502931\pi\)
\(908\) 16.1950 0.537450
\(909\) 0.480396 0.0159337
\(910\) 4.72547 0.156648
\(911\) 17.0232 0.564003 0.282002 0.959414i \(-0.409002\pi\)
0.282002 + 0.959414i \(0.409002\pi\)
\(912\) 18.4337 0.610402
\(913\) 74.7886 2.47514
\(914\) −8.13562 −0.269102
\(915\) −62.5291 −2.06715
\(916\) −8.80688 −0.290988
\(917\) 3.02863 0.100014
\(918\) −9.99538 −0.329897
\(919\) 35.1128 1.15826 0.579132 0.815234i \(-0.303392\pi\)
0.579132 + 0.815234i \(0.303392\pi\)
\(920\) 7.54386 0.248714
\(921\) −6.08135 −0.200387
\(922\) −30.0171 −0.988559
\(923\) −1.08960 −0.0358646
\(924\) −8.65154 −0.284615
\(925\) −21.8800 −0.719410
\(926\) 14.1202 0.464020
\(927\) 34.1834 1.12273
\(928\) 0.0257294 0.000844610 0
\(929\) 43.5449 1.42866 0.714330 0.699809i \(-0.246731\pi\)
0.714330 + 0.699809i \(0.246731\pi\)
\(930\) 17.1031 0.560833
\(931\) 51.1149 1.67522
\(932\) 6.09380 0.199609
\(933\) −0.0610650 −0.00199918
\(934\) 6.67297 0.218346
\(935\) −75.3029 −2.46267
\(936\) −5.02509 −0.164250
\(937\) −18.9860 −0.620244 −0.310122 0.950697i \(-0.600370\pi\)
−0.310122 + 0.950697i \(0.600370\pi\)
\(938\) 2.81370 0.0918706
\(939\) −32.0282 −1.04520
\(940\) 16.4840 0.537649
\(941\) −38.2794 −1.24787 −0.623937 0.781475i \(-0.714468\pi\)
−0.623937 + 0.781475i \(0.714468\pi\)
\(942\) 36.8359 1.20018
\(943\) 18.1916 0.592401
\(944\) −4.31537 −0.140453
\(945\) −3.58239 −0.116535
\(946\) 28.0895 0.913269
\(947\) −39.6711 −1.28914 −0.644569 0.764546i \(-0.722963\pi\)
−0.644569 + 0.764546i \(0.722963\pi\)
\(948\) −10.0344 −0.325903
\(949\) 26.4476 0.858524
\(950\) 17.3491 0.562879
\(951\) 22.5944 0.732672
\(952\) 4.74419 0.153760
\(953\) 1.41878 0.0459588 0.0229794 0.999736i \(-0.492685\pi\)
0.0229794 + 0.999736i \(0.492685\pi\)
\(954\) 8.02674 0.259875
\(955\) −11.6092 −0.375665
\(956\) 17.5768 0.568476
\(957\) −0.279246 −0.00902676
\(958\) −24.9404 −0.805787
\(959\) −6.59999 −0.213125
\(960\) −6.14183 −0.198227
\(961\) −23.2455 −0.749855
\(962\) 22.4386 0.723448
\(963\) 20.4567 0.659208
\(964\) 1.56163 0.0502969
\(965\) −59.1194 −1.90312
\(966\) 5.15824 0.165964
\(967\) −21.7303 −0.698800 −0.349400 0.936974i \(-0.613615\pi\)
−0.349400 + 0.936974i \(0.613615\pi\)
\(968\) −11.3586 −0.365080
\(969\) 109.708 3.52434
\(970\) 46.9575 1.50771
\(971\) 38.7192 1.24256 0.621279 0.783590i \(-0.286614\pi\)
0.621279 + 0.783590i \(0.286614\pi\)
\(972\) 19.4286 0.623172
\(973\) −5.63712 −0.180718
\(974\) −9.93041 −0.318191
\(975\) −10.9844 −0.351784
\(976\) 10.1809 0.325881
\(977\) −42.9478 −1.37402 −0.687011 0.726647i \(-0.741078\pi\)
−0.687011 + 0.726647i \(0.741078\pi\)
\(978\) −48.9387 −1.56489
\(979\) −61.7573 −1.97377
\(980\) −17.0307 −0.544025
\(981\) 27.8685 0.889772
\(982\) 3.07555 0.0981447
\(983\) −19.2726 −0.614700 −0.307350 0.951597i \(-0.599442\pi\)
−0.307350 + 0.951597i \(0.599442\pi\)
\(984\) −14.8107 −0.472149
\(985\) −56.5684 −1.80242
\(986\) 0.153128 0.00487660
\(987\) 11.2712 0.358767
\(988\) −17.7920 −0.566038
\(989\) −16.7476 −0.532542
\(990\) 28.7002 0.912151
\(991\) −54.4445 −1.72949 −0.864743 0.502214i \(-0.832519\pi\)
−0.864743 + 0.502214i \(0.832519\pi\)
\(992\) −2.78469 −0.0884140
\(993\) 10.7333 0.340612
\(994\) −0.392065 −0.0124355
\(995\) 18.6232 0.590396
\(996\) 36.3034 1.15032
\(997\) 8.21388 0.260136 0.130068 0.991505i \(-0.458480\pi\)
0.130068 + 0.991505i \(0.458480\pi\)
\(998\) −19.9045 −0.630066
\(999\) −17.0107 −0.538196
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 6046.2.a.g.1.14 69
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
6046.2.a.g.1.14 69 1.1 even 1 trivial