Properties

Label 6027.2.a.z.1.3
Level $6027$
Weight $2$
Character 6027.1
Self dual yes
Analytic conductor $48.126$
Analytic rank $1$
Dimension $8$
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] = [6027,2,Mod(1,6027)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6027, 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("6027.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6027 = 3 \cdot 7^{2} \cdot 41 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6027.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: \(48.1258372982\)
Analytic rank: \(1\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 2x^{7} - 8x^{6} + 14x^{5} + 18x^{4} - 24x^{3} - 10x^{2} + 10x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(1.17091\) of defining polynomial
Character \(\chi\) \(=\) 6027.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.17091 q^{2} -1.00000 q^{3} -0.628975 q^{4} +2.94759 q^{5} +1.17091 q^{6} +3.07829 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.17091 q^{2} -1.00000 q^{3} -0.628975 q^{4} +2.94759 q^{5} +1.17091 q^{6} +3.07829 q^{8} +1.00000 q^{9} -3.45135 q^{10} +0.735179 q^{11} +0.628975 q^{12} +4.56463 q^{13} -2.94759 q^{15} -2.34644 q^{16} -1.29990 q^{17} -1.17091 q^{18} -0.205172 q^{19} -1.85396 q^{20} -0.860827 q^{22} +7.81932 q^{23} -3.07829 q^{24} +3.68826 q^{25} -5.34477 q^{26} -1.00000 q^{27} -10.2205 q^{29} +3.45135 q^{30} -6.98437 q^{31} -3.40911 q^{32} -0.735179 q^{33} +1.52206 q^{34} -0.628975 q^{36} -5.74813 q^{37} +0.240237 q^{38} -4.56463 q^{39} +9.07351 q^{40} +1.00000 q^{41} -2.33978 q^{43} -0.462409 q^{44} +2.94759 q^{45} -9.15571 q^{46} -3.34825 q^{47} +2.34644 q^{48} -4.31861 q^{50} +1.29990 q^{51} -2.87104 q^{52} -5.83919 q^{53} +1.17091 q^{54} +2.16700 q^{55} +0.205172 q^{57} +11.9673 q^{58} -12.6741 q^{59} +1.85396 q^{60} -12.0452 q^{61} +8.17806 q^{62} +8.68463 q^{64} +13.4546 q^{65} +0.860827 q^{66} -0.636074 q^{67} +0.817601 q^{68} -7.81932 q^{69} -13.2397 q^{71} +3.07829 q^{72} -2.69801 q^{73} +6.73054 q^{74} -3.68826 q^{75} +0.129048 q^{76} +5.34477 q^{78} +10.7810 q^{79} -6.91634 q^{80} +1.00000 q^{81} -1.17091 q^{82} -2.11698 q^{83} -3.83155 q^{85} +2.73966 q^{86} +10.2205 q^{87} +2.26309 q^{88} -9.36156 q^{89} -3.45135 q^{90} -4.91815 q^{92} +6.98437 q^{93} +3.92049 q^{94} -0.604761 q^{95} +3.40911 q^{96} +0.379314 q^{97} +0.735179 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 2 q^{2} - 8 q^{3} + 4 q^{4} + 2 q^{5} + 2 q^{6} - 6 q^{8} + 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 2 q^{2} - 8 q^{3} + 4 q^{4} + 2 q^{5} + 2 q^{6} - 6 q^{8} + 8 q^{9} - 2 q^{10} - 2 q^{11} - 4 q^{12} + 4 q^{13} - 2 q^{15} + 8 q^{17} - 2 q^{18} + 6 q^{19} + 4 q^{20} - 14 q^{22} - 12 q^{23} + 6 q^{24} - 4 q^{25} + 4 q^{26} - 8 q^{27} - 4 q^{29} + 2 q^{30} - 10 q^{31} - 4 q^{32} + 2 q^{33} + 4 q^{34} + 4 q^{36} - 20 q^{37} - 18 q^{38} - 4 q^{39} + 12 q^{40} + 8 q^{41} - 8 q^{43} + 20 q^{44} + 2 q^{45} - 12 q^{46} + 24 q^{47} - 22 q^{50} - 8 q^{51} - 30 q^{52} - 36 q^{53} + 2 q^{54} + 4 q^{55} - 6 q^{57} + 14 q^{58} + 10 q^{59} - 4 q^{60} - 22 q^{61} + 30 q^{62} - 24 q^{64} + 8 q^{65} + 14 q^{66} - 14 q^{67} + 38 q^{68} + 12 q^{69} - 10 q^{71} - 6 q^{72} - 12 q^{73} - 2 q^{74} + 4 q^{75} + 32 q^{76} - 4 q^{78} + 16 q^{79} - 14 q^{80} + 8 q^{81} - 2 q^{82} + 24 q^{83} - 44 q^{85} + 36 q^{86} + 4 q^{87} - 34 q^{88} + 2 q^{89} - 2 q^{90} - 48 q^{92} + 10 q^{93} - 34 q^{94} - 24 q^{95} + 4 q^{96} + 16 q^{97} - 2 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.17091 −0.827957 −0.413978 0.910287i \(-0.635861\pi\)
−0.413978 + 0.910287i \(0.635861\pi\)
\(3\) −1.00000 −0.577350
\(4\) −0.628975 −0.314487
\(5\) 2.94759 1.31820 0.659100 0.752055i \(-0.270937\pi\)
0.659100 + 0.752055i \(0.270937\pi\)
\(6\) 1.17091 0.478021
\(7\) 0 0
\(8\) 3.07829 1.08834
\(9\) 1.00000 0.333333
\(10\) −3.45135 −1.09141
\(11\) 0.735179 0.221665 0.110832 0.993839i \(-0.464648\pi\)
0.110832 + 0.993839i \(0.464648\pi\)
\(12\) 0.628975 0.181569
\(13\) 4.56463 1.26600 0.633001 0.774151i \(-0.281823\pi\)
0.633001 + 0.774151i \(0.281823\pi\)
\(14\) 0 0
\(15\) −2.94759 −0.761063
\(16\) −2.34644 −0.586610
\(17\) −1.29990 −0.315271 −0.157635 0.987497i \(-0.550387\pi\)
−0.157635 + 0.987497i \(0.550387\pi\)
\(18\) −1.17091 −0.275986
\(19\) −0.205172 −0.0470696 −0.0235348 0.999723i \(-0.507492\pi\)
−0.0235348 + 0.999723i \(0.507492\pi\)
\(20\) −1.85396 −0.414557
\(21\) 0 0
\(22\) −0.860827 −0.183529
\(23\) 7.81932 1.63044 0.815221 0.579150i \(-0.196616\pi\)
0.815221 + 0.579150i \(0.196616\pi\)
\(24\) −3.07829 −0.628353
\(25\) 3.68826 0.737652
\(26\) −5.34477 −1.04819
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) −10.2205 −1.89791 −0.948954 0.315414i \(-0.897857\pi\)
−0.948954 + 0.315414i \(0.897857\pi\)
\(30\) 3.45135 0.630128
\(31\) −6.98437 −1.25443 −0.627215 0.778846i \(-0.715805\pi\)
−0.627215 + 0.778846i \(0.715805\pi\)
\(32\) −3.40911 −0.602651
\(33\) −0.735179 −0.127978
\(34\) 1.52206 0.261031
\(35\) 0 0
\(36\) −0.628975 −0.104829
\(37\) −5.74813 −0.944987 −0.472494 0.881334i \(-0.656646\pi\)
−0.472494 + 0.881334i \(0.656646\pi\)
\(38\) 0.240237 0.0389716
\(39\) −4.56463 −0.730926
\(40\) 9.07351 1.43465
\(41\) 1.00000 0.156174
\(42\) 0 0
\(43\) −2.33978 −0.356813 −0.178406 0.983957i \(-0.557094\pi\)
−0.178406 + 0.983957i \(0.557094\pi\)
\(44\) −0.462409 −0.0697108
\(45\) 2.94759 0.439400
\(46\) −9.15571 −1.34994
\(47\) −3.34825 −0.488392 −0.244196 0.969726i \(-0.578524\pi\)
−0.244196 + 0.969726i \(0.578524\pi\)
\(48\) 2.34644 0.338680
\(49\) 0 0
\(50\) −4.31861 −0.610744
\(51\) 1.29990 0.182022
\(52\) −2.87104 −0.398141
\(53\) −5.83919 −0.802075 −0.401038 0.916062i \(-0.631350\pi\)
−0.401038 + 0.916062i \(0.631350\pi\)
\(54\) 1.17091 0.159340
\(55\) 2.16700 0.292199
\(56\) 0 0
\(57\) 0.205172 0.0271756
\(58\) 11.9673 1.57139
\(59\) −12.6741 −1.65003 −0.825016 0.565109i \(-0.808834\pi\)
−0.825016 + 0.565109i \(0.808834\pi\)
\(60\) 1.85396 0.239345
\(61\) −12.0452 −1.54224 −0.771118 0.636693i \(-0.780302\pi\)
−0.771118 + 0.636693i \(0.780302\pi\)
\(62\) 8.17806 1.03861
\(63\) 0 0
\(64\) 8.68463 1.08558
\(65\) 13.4546 1.66884
\(66\) 0.860827 0.105961
\(67\) −0.636074 −0.0777088 −0.0388544 0.999245i \(-0.512371\pi\)
−0.0388544 + 0.999245i \(0.512371\pi\)
\(68\) 0.817601 0.0991487
\(69\) −7.81932 −0.941336
\(70\) 0 0
\(71\) −13.2397 −1.57127 −0.785634 0.618691i \(-0.787663\pi\)
−0.785634 + 0.618691i \(0.787663\pi\)
\(72\) 3.07829 0.362780
\(73\) −2.69801 −0.315779 −0.157889 0.987457i \(-0.550469\pi\)
−0.157889 + 0.987457i \(0.550469\pi\)
\(74\) 6.73054 0.782409
\(75\) −3.68826 −0.425883
\(76\) 0.129048 0.0148028
\(77\) 0 0
\(78\) 5.34477 0.605176
\(79\) 10.7810 1.21296 0.606479 0.795099i \(-0.292581\pi\)
0.606479 + 0.795099i \(0.292581\pi\)
\(80\) −6.91634 −0.773270
\(81\) 1.00000 0.111111
\(82\) −1.17091 −0.129305
\(83\) −2.11698 −0.232369 −0.116185 0.993228i \(-0.537066\pi\)
−0.116185 + 0.993228i \(0.537066\pi\)
\(84\) 0 0
\(85\) −3.83155 −0.415590
\(86\) 2.73966 0.295426
\(87\) 10.2205 1.09576
\(88\) 2.26309 0.241247
\(89\) −9.36156 −0.992323 −0.496162 0.868230i \(-0.665258\pi\)
−0.496162 + 0.868230i \(0.665258\pi\)
\(90\) −3.45135 −0.363804
\(91\) 0 0
\(92\) −4.91815 −0.512753
\(93\) 6.98437 0.724246
\(94\) 3.92049 0.404368
\(95\) −0.604761 −0.0620471
\(96\) 3.40911 0.347941
\(97\) 0.379314 0.0385135 0.0192567 0.999815i \(-0.493870\pi\)
0.0192567 + 0.999815i \(0.493870\pi\)
\(98\) 0 0
\(99\) 0.735179 0.0738883
\(100\) −2.31982 −0.231982
\(101\) −11.1148 −1.10597 −0.552984 0.833192i \(-0.686511\pi\)
−0.552984 + 0.833192i \(0.686511\pi\)
\(102\) −1.52206 −0.150706
\(103\) 1.83093 0.180407 0.0902035 0.995923i \(-0.471248\pi\)
0.0902035 + 0.995923i \(0.471248\pi\)
\(104\) 14.0513 1.37784
\(105\) 0 0
\(106\) 6.83716 0.664084
\(107\) −12.0728 −1.16713 −0.583563 0.812068i \(-0.698342\pi\)
−0.583563 + 0.812068i \(0.698342\pi\)
\(108\) 0.628975 0.0605231
\(109\) −4.72109 −0.452198 −0.226099 0.974104i \(-0.572597\pi\)
−0.226099 + 0.974104i \(0.572597\pi\)
\(110\) −2.53736 −0.241928
\(111\) 5.74813 0.545589
\(112\) 0 0
\(113\) 0.290899 0.0273654 0.0136827 0.999906i \(-0.495645\pi\)
0.0136827 + 0.999906i \(0.495645\pi\)
\(114\) −0.240237 −0.0225003
\(115\) 23.0481 2.14925
\(116\) 6.42847 0.596868
\(117\) 4.56463 0.422001
\(118\) 14.8403 1.36616
\(119\) 0 0
\(120\) −9.07351 −0.828295
\(121\) −10.4595 −0.950865
\(122\) 14.1039 1.27690
\(123\) −1.00000 −0.0901670
\(124\) 4.39299 0.394502
\(125\) −3.86647 −0.345828
\(126\) 0 0
\(127\) −7.01519 −0.622497 −0.311249 0.950328i \(-0.600747\pi\)
−0.311249 + 0.950328i \(0.600747\pi\)
\(128\) −3.35069 −0.296162
\(129\) 2.33978 0.206006
\(130\) −15.7542 −1.38173
\(131\) 20.5015 1.79123 0.895613 0.444835i \(-0.146738\pi\)
0.895613 + 0.444835i \(0.146738\pi\)
\(132\) 0.462409 0.0402475
\(133\) 0 0
\(134\) 0.744784 0.0643395
\(135\) −2.94759 −0.253688
\(136\) −4.00145 −0.343122
\(137\) 16.3722 1.39877 0.699387 0.714743i \(-0.253456\pi\)
0.699387 + 0.714743i \(0.253456\pi\)
\(138\) 9.15571 0.779386
\(139\) −16.6768 −1.41451 −0.707253 0.706960i \(-0.750066\pi\)
−0.707253 + 0.706960i \(0.750066\pi\)
\(140\) 0 0
\(141\) 3.34825 0.281973
\(142\) 15.5025 1.30094
\(143\) 3.35583 0.280628
\(144\) −2.34644 −0.195537
\(145\) −30.1259 −2.50182
\(146\) 3.15913 0.261451
\(147\) 0 0
\(148\) 3.61543 0.297187
\(149\) −18.7722 −1.53788 −0.768941 0.639319i \(-0.779216\pi\)
−0.768941 + 0.639319i \(0.779216\pi\)
\(150\) 4.31861 0.352613
\(151\) 7.05428 0.574069 0.287034 0.957920i \(-0.407331\pi\)
0.287034 + 0.957920i \(0.407331\pi\)
\(152\) −0.631577 −0.0512277
\(153\) −1.29990 −0.105090
\(154\) 0 0
\(155\) −20.5870 −1.65359
\(156\) 2.87104 0.229867
\(157\) −9.41375 −0.751299 −0.375649 0.926762i \(-0.622580\pi\)
−0.375649 + 0.926762i \(0.622580\pi\)
\(158\) −12.6236 −1.00428
\(159\) 5.83919 0.463078
\(160\) −10.0486 −0.794414
\(161\) 0 0
\(162\) −1.17091 −0.0919952
\(163\) 13.1149 1.02724 0.513620 0.858018i \(-0.328304\pi\)
0.513620 + 0.858018i \(0.328304\pi\)
\(164\) −0.628975 −0.0491147
\(165\) −2.16700 −0.168701
\(166\) 2.47879 0.192392
\(167\) 17.8255 1.37938 0.689689 0.724105i \(-0.257747\pi\)
0.689689 + 0.724105i \(0.257747\pi\)
\(168\) 0 0
\(169\) 7.83588 0.602760
\(170\) 4.48640 0.344091
\(171\) −0.205172 −0.0156899
\(172\) 1.47166 0.112213
\(173\) 19.7655 1.50274 0.751371 0.659880i \(-0.229393\pi\)
0.751371 + 0.659880i \(0.229393\pi\)
\(174\) −11.9673 −0.907240
\(175\) 0 0
\(176\) −1.72506 −0.130031
\(177\) 12.6741 0.952647
\(178\) 10.9615 0.821601
\(179\) −13.4411 −1.00463 −0.502316 0.864684i \(-0.667519\pi\)
−0.502316 + 0.864684i \(0.667519\pi\)
\(180\) −1.85396 −0.138186
\(181\) −8.21899 −0.610912 −0.305456 0.952206i \(-0.598809\pi\)
−0.305456 + 0.952206i \(0.598809\pi\)
\(182\) 0 0
\(183\) 12.0452 0.890410
\(184\) 24.0701 1.77447
\(185\) −16.9431 −1.24568
\(186\) −8.17806 −0.599644
\(187\) −0.955657 −0.0698845
\(188\) 2.10596 0.153593
\(189\) 0 0
\(190\) 0.708119 0.0513724
\(191\) 3.28155 0.237445 0.118722 0.992927i \(-0.462120\pi\)
0.118722 + 0.992927i \(0.462120\pi\)
\(192\) −8.68463 −0.626759
\(193\) 13.7930 0.992841 0.496420 0.868082i \(-0.334648\pi\)
0.496420 + 0.868082i \(0.334648\pi\)
\(194\) −0.444142 −0.0318875
\(195\) −13.4546 −0.963507
\(196\) 0 0
\(197\) −23.8854 −1.70177 −0.850883 0.525355i \(-0.823933\pi\)
−0.850883 + 0.525355i \(0.823933\pi\)
\(198\) −0.860827 −0.0611763
\(199\) 25.3686 1.79833 0.899166 0.437608i \(-0.144174\pi\)
0.899166 + 0.437608i \(0.144174\pi\)
\(200\) 11.3535 0.802815
\(201\) 0.636074 0.0448652
\(202\) 13.0144 0.915693
\(203\) 0 0
\(204\) −0.817601 −0.0572435
\(205\) 2.94759 0.205868
\(206\) −2.14385 −0.149369
\(207\) 7.81932 0.543480
\(208\) −10.7106 −0.742650
\(209\) −0.150838 −0.0104337
\(210\) 0 0
\(211\) −2.29547 −0.158027 −0.0790134 0.996874i \(-0.525177\pi\)
−0.0790134 + 0.996874i \(0.525177\pi\)
\(212\) 3.67270 0.252242
\(213\) 13.2397 0.907172
\(214\) 14.1362 0.966330
\(215\) −6.89669 −0.470351
\(216\) −3.07829 −0.209451
\(217\) 0 0
\(218\) 5.52796 0.374401
\(219\) 2.69801 0.182315
\(220\) −1.36299 −0.0918928
\(221\) −5.93355 −0.399134
\(222\) −6.73054 −0.451724
\(223\) 9.29813 0.622649 0.311325 0.950304i \(-0.399227\pi\)
0.311325 + 0.950304i \(0.399227\pi\)
\(224\) 0 0
\(225\) 3.68826 0.245884
\(226\) −0.340616 −0.0226574
\(227\) 6.59863 0.437966 0.218983 0.975729i \(-0.429726\pi\)
0.218983 + 0.975729i \(0.429726\pi\)
\(228\) −0.129048 −0.00854639
\(229\) −9.69155 −0.640436 −0.320218 0.947344i \(-0.603756\pi\)
−0.320218 + 0.947344i \(0.603756\pi\)
\(230\) −26.9872 −1.77948
\(231\) 0 0
\(232\) −31.4618 −2.06557
\(233\) 24.0753 1.57723 0.788613 0.614890i \(-0.210799\pi\)
0.788613 + 0.614890i \(0.210799\pi\)
\(234\) −5.34477 −0.349398
\(235\) −9.86925 −0.643799
\(236\) 7.97171 0.518914
\(237\) −10.7810 −0.700302
\(238\) 0 0
\(239\) −5.05867 −0.327218 −0.163609 0.986525i \(-0.552314\pi\)
−0.163609 + 0.986525i \(0.552314\pi\)
\(240\) 6.91634 0.446448
\(241\) −3.21287 −0.206959 −0.103480 0.994632i \(-0.532998\pi\)
−0.103480 + 0.994632i \(0.532998\pi\)
\(242\) 12.2471 0.787275
\(243\) −1.00000 −0.0641500
\(244\) 7.57615 0.485013
\(245\) 0 0
\(246\) 1.17091 0.0746544
\(247\) −0.936533 −0.0595902
\(248\) −21.4999 −1.36525
\(249\) 2.11698 0.134158
\(250\) 4.52728 0.286330
\(251\) 28.9183 1.82531 0.912653 0.408736i \(-0.134030\pi\)
0.912653 + 0.408736i \(0.134030\pi\)
\(252\) 0 0
\(253\) 5.74861 0.361412
\(254\) 8.21414 0.515401
\(255\) 3.83155 0.239941
\(256\) −13.4459 −0.840370
\(257\) 7.74014 0.482816 0.241408 0.970424i \(-0.422391\pi\)
0.241408 + 0.970424i \(0.422391\pi\)
\(258\) −2.73966 −0.170564
\(259\) 0 0
\(260\) −8.46263 −0.524830
\(261\) −10.2205 −0.632636
\(262\) −24.0054 −1.48306
\(263\) −13.8041 −0.851199 −0.425599 0.904912i \(-0.639937\pi\)
−0.425599 + 0.904912i \(0.639937\pi\)
\(264\) −2.26309 −0.139284
\(265\) −17.2115 −1.05730
\(266\) 0 0
\(267\) 9.36156 0.572918
\(268\) 0.400074 0.0244384
\(269\) 9.88416 0.602648 0.301324 0.953522i \(-0.402571\pi\)
0.301324 + 0.953522i \(0.402571\pi\)
\(270\) 3.45135 0.210043
\(271\) 9.17765 0.557502 0.278751 0.960363i \(-0.410080\pi\)
0.278751 + 0.960363i \(0.410080\pi\)
\(272\) 3.05013 0.184941
\(273\) 0 0
\(274\) −19.1704 −1.15813
\(275\) 2.71153 0.163511
\(276\) 4.91815 0.296038
\(277\) −13.6038 −0.817373 −0.408687 0.912675i \(-0.634013\pi\)
−0.408687 + 0.912675i \(0.634013\pi\)
\(278\) 19.5270 1.17115
\(279\) −6.98437 −0.418144
\(280\) 0 0
\(281\) 15.9950 0.954181 0.477090 0.878854i \(-0.341691\pi\)
0.477090 + 0.878854i \(0.341691\pi\)
\(282\) −3.92049 −0.233462
\(283\) −19.9521 −1.18603 −0.593014 0.805192i \(-0.702062\pi\)
−0.593014 + 0.805192i \(0.702062\pi\)
\(284\) 8.32746 0.494144
\(285\) 0.604761 0.0358229
\(286\) −3.92936 −0.232348
\(287\) 0 0
\(288\) −3.40911 −0.200884
\(289\) −15.3103 −0.900604
\(290\) 35.2747 2.07140
\(291\) −0.379314 −0.0222358
\(292\) 1.69698 0.0993084
\(293\) −5.62507 −0.328620 −0.164310 0.986409i \(-0.552540\pi\)
−0.164310 + 0.986409i \(0.552540\pi\)
\(294\) 0 0
\(295\) −37.3581 −2.17507
\(296\) −17.6944 −1.02847
\(297\) −0.735179 −0.0426594
\(298\) 21.9806 1.27330
\(299\) 35.6923 2.06414
\(300\) 2.31982 0.133935
\(301\) 0 0
\(302\) −8.25991 −0.475304
\(303\) 11.1148 0.638530
\(304\) 0.481423 0.0276115
\(305\) −35.5044 −2.03297
\(306\) 1.52206 0.0870103
\(307\) 1.07877 0.0615685 0.0307842 0.999526i \(-0.490200\pi\)
0.0307842 + 0.999526i \(0.490200\pi\)
\(308\) 0 0
\(309\) −1.83093 −0.104158
\(310\) 24.1055 1.36910
\(311\) 9.73481 0.552010 0.276005 0.961156i \(-0.410989\pi\)
0.276005 + 0.961156i \(0.410989\pi\)
\(312\) −14.0513 −0.795496
\(313\) 30.4329 1.72017 0.860084 0.510152i \(-0.170411\pi\)
0.860084 + 0.510152i \(0.170411\pi\)
\(314\) 11.0226 0.622043
\(315\) 0 0
\(316\) −6.78098 −0.381460
\(317\) −21.8406 −1.22669 −0.613345 0.789815i \(-0.710177\pi\)
−0.613345 + 0.789815i \(0.710177\pi\)
\(318\) −6.83716 −0.383409
\(319\) −7.51394 −0.420700
\(320\) 25.5987 1.43101
\(321\) 12.0728 0.673840
\(322\) 0 0
\(323\) 0.266702 0.0148397
\(324\) −0.628975 −0.0349430
\(325\) 16.8355 0.933868
\(326\) −15.3564 −0.850510
\(327\) 4.72109 0.261077
\(328\) 3.07829 0.169970
\(329\) 0 0
\(330\) 2.53736 0.139677
\(331\) −20.4051 −1.12157 −0.560783 0.827963i \(-0.689500\pi\)
−0.560783 + 0.827963i \(0.689500\pi\)
\(332\) 1.33153 0.0730772
\(333\) −5.74813 −0.314996
\(334\) −20.8720 −1.14207
\(335\) −1.87488 −0.102436
\(336\) 0 0
\(337\) 15.1472 0.825119 0.412559 0.910931i \(-0.364635\pi\)
0.412559 + 0.910931i \(0.364635\pi\)
\(338\) −9.17510 −0.499060
\(339\) −0.290899 −0.0157994
\(340\) 2.40995 0.130698
\(341\) −5.13477 −0.278063
\(342\) 0.240237 0.0129905
\(343\) 0 0
\(344\) −7.20251 −0.388333
\(345\) −23.0481 −1.24087
\(346\) −23.1436 −1.24421
\(347\) −10.2650 −0.551054 −0.275527 0.961293i \(-0.588852\pi\)
−0.275527 + 0.961293i \(0.588852\pi\)
\(348\) −6.42847 −0.344602
\(349\) −11.4058 −0.610540 −0.305270 0.952266i \(-0.598747\pi\)
−0.305270 + 0.952266i \(0.598747\pi\)
\(350\) 0 0
\(351\) −4.56463 −0.243642
\(352\) −2.50631 −0.133587
\(353\) 29.5367 1.57208 0.786041 0.618175i \(-0.212128\pi\)
0.786041 + 0.618175i \(0.212128\pi\)
\(354\) −14.8403 −0.788750
\(355\) −39.0253 −2.07125
\(356\) 5.88818 0.312073
\(357\) 0 0
\(358\) 15.7382 0.831792
\(359\) 7.96542 0.420399 0.210199 0.977659i \(-0.432589\pi\)
0.210199 + 0.977659i \(0.432589\pi\)
\(360\) 9.07351 0.478216
\(361\) −18.9579 −0.997784
\(362\) 9.62368 0.505809
\(363\) 10.4595 0.548982
\(364\) 0 0
\(365\) −7.95263 −0.416260
\(366\) −14.1039 −0.737221
\(367\) −2.82624 −0.147529 −0.0737644 0.997276i \(-0.523501\pi\)
−0.0737644 + 0.997276i \(0.523501\pi\)
\(368\) −18.3476 −0.956434
\(369\) 1.00000 0.0520579
\(370\) 19.8388 1.03137
\(371\) 0 0
\(372\) −4.39299 −0.227766
\(373\) 6.18083 0.320031 0.160016 0.987114i \(-0.448846\pi\)
0.160016 + 0.987114i \(0.448846\pi\)
\(374\) 1.11899 0.0578614
\(375\) 3.86647 0.199664
\(376\) −10.3069 −0.531536
\(377\) −46.6531 −2.40276
\(378\) 0 0
\(379\) 13.4202 0.689348 0.344674 0.938722i \(-0.387989\pi\)
0.344674 + 0.938722i \(0.387989\pi\)
\(380\) 0.380379 0.0195130
\(381\) 7.01519 0.359399
\(382\) −3.84240 −0.196594
\(383\) −6.04337 −0.308802 −0.154401 0.988008i \(-0.549345\pi\)
−0.154401 + 0.988008i \(0.549345\pi\)
\(384\) 3.35069 0.170989
\(385\) 0 0
\(386\) −16.1503 −0.822029
\(387\) −2.33978 −0.118938
\(388\) −0.238579 −0.0121120
\(389\) 23.2716 1.17992 0.589959 0.807433i \(-0.299144\pi\)
0.589959 + 0.807433i \(0.299144\pi\)
\(390\) 15.7542 0.797743
\(391\) −10.1643 −0.514031
\(392\) 0 0
\(393\) −20.5015 −1.03416
\(394\) 27.9676 1.40899
\(395\) 31.7779 1.59892
\(396\) −0.462409 −0.0232369
\(397\) −11.1219 −0.558191 −0.279095 0.960263i \(-0.590035\pi\)
−0.279095 + 0.960263i \(0.590035\pi\)
\(398\) −29.7043 −1.48894
\(399\) 0 0
\(400\) −8.65428 −0.432714
\(401\) −2.50482 −0.125085 −0.0625424 0.998042i \(-0.519921\pi\)
−0.0625424 + 0.998042i \(0.519921\pi\)
\(402\) −0.744784 −0.0371464
\(403\) −31.8811 −1.58811
\(404\) 6.99095 0.347813
\(405\) 2.94759 0.146467
\(406\) 0 0
\(407\) −4.22591 −0.209471
\(408\) 4.00145 0.198101
\(409\) −9.26167 −0.457960 −0.228980 0.973431i \(-0.573539\pi\)
−0.228980 + 0.973431i \(0.573539\pi\)
\(410\) −3.45135 −0.170450
\(411\) −16.3722 −0.807583
\(412\) −1.15161 −0.0567357
\(413\) 0 0
\(414\) −9.15571 −0.449978
\(415\) −6.23999 −0.306309
\(416\) −15.5613 −0.762957
\(417\) 16.6768 0.816666
\(418\) 0.176617 0.00863864
\(419\) 26.5761 1.29833 0.649163 0.760649i \(-0.275119\pi\)
0.649163 + 0.760649i \(0.275119\pi\)
\(420\) 0 0
\(421\) −33.0969 −1.61304 −0.806521 0.591205i \(-0.798652\pi\)
−0.806521 + 0.591205i \(0.798652\pi\)
\(422\) 2.68779 0.130839
\(423\) −3.34825 −0.162797
\(424\) −17.9747 −0.872930
\(425\) −4.79435 −0.232560
\(426\) −15.5025 −0.751100
\(427\) 0 0
\(428\) 7.59351 0.367046
\(429\) −3.35583 −0.162021
\(430\) 8.07539 0.389430
\(431\) −22.0997 −1.06450 −0.532252 0.846586i \(-0.678654\pi\)
−0.532252 + 0.846586i \(0.678654\pi\)
\(432\) 2.34644 0.112893
\(433\) 11.0547 0.531253 0.265626 0.964076i \(-0.414421\pi\)
0.265626 + 0.964076i \(0.414421\pi\)
\(434\) 0 0
\(435\) 30.1259 1.44443
\(436\) 2.96945 0.142211
\(437\) −1.60430 −0.0767442
\(438\) −3.15913 −0.150949
\(439\) −21.1957 −1.01161 −0.505807 0.862647i \(-0.668805\pi\)
−0.505807 + 0.862647i \(0.668805\pi\)
\(440\) 6.67066 0.318011
\(441\) 0 0
\(442\) 6.94764 0.330465
\(443\) −15.7890 −0.750160 −0.375080 0.926992i \(-0.622385\pi\)
−0.375080 + 0.926992i \(0.622385\pi\)
\(444\) −3.61543 −0.171581
\(445\) −27.5940 −1.30808
\(446\) −10.8873 −0.515527
\(447\) 18.7722 0.887897
\(448\) 0 0
\(449\) −28.0956 −1.32592 −0.662958 0.748657i \(-0.730699\pi\)
−0.662958 + 0.748657i \(0.730699\pi\)
\(450\) −4.31861 −0.203581
\(451\) 0.735179 0.0346182
\(452\) −0.182968 −0.00860608
\(453\) −7.05428 −0.331439
\(454\) −7.72638 −0.362617
\(455\) 0 0
\(456\) 0.631577 0.0295763
\(457\) 3.55168 0.166140 0.0830702 0.996544i \(-0.473527\pi\)
0.0830702 + 0.996544i \(0.473527\pi\)
\(458\) 11.3479 0.530253
\(459\) 1.29990 0.0606739
\(460\) −14.4967 −0.675911
\(461\) 7.68730 0.358033 0.179017 0.983846i \(-0.442708\pi\)
0.179017 + 0.983846i \(0.442708\pi\)
\(462\) 0 0
\(463\) −25.3808 −1.17955 −0.589774 0.807568i \(-0.700783\pi\)
−0.589774 + 0.807568i \(0.700783\pi\)
\(464\) 23.9819 1.11333
\(465\) 20.5870 0.954701
\(466\) −28.1900 −1.30588
\(467\) 17.3365 0.802237 0.401118 0.916026i \(-0.368622\pi\)
0.401118 + 0.916026i \(0.368622\pi\)
\(468\) −2.87104 −0.132714
\(469\) 0 0
\(470\) 11.5560 0.533038
\(471\) 9.41375 0.433763
\(472\) −39.0146 −1.79579
\(473\) −1.72016 −0.0790929
\(474\) 12.6236 0.579820
\(475\) −0.756726 −0.0347210
\(476\) 0 0
\(477\) −5.83919 −0.267358
\(478\) 5.92323 0.270922
\(479\) 2.91060 0.132989 0.0664944 0.997787i \(-0.478819\pi\)
0.0664944 + 0.997787i \(0.478819\pi\)
\(480\) 10.0486 0.458655
\(481\) −26.2381 −1.19636
\(482\) 3.76198 0.171353
\(483\) 0 0
\(484\) 6.57877 0.299035
\(485\) 1.11806 0.0507685
\(486\) 1.17091 0.0531135
\(487\) −31.1136 −1.40989 −0.704946 0.709261i \(-0.749029\pi\)
−0.704946 + 0.709261i \(0.749029\pi\)
\(488\) −37.0787 −1.67847
\(489\) −13.1149 −0.593077
\(490\) 0 0
\(491\) 33.0456 1.49133 0.745663 0.666323i \(-0.232133\pi\)
0.745663 + 0.666323i \(0.232133\pi\)
\(492\) 0.628975 0.0283564
\(493\) 13.2856 0.598355
\(494\) 1.09659 0.0493381
\(495\) 2.16700 0.0973996
\(496\) 16.3884 0.735862
\(497\) 0 0
\(498\) −2.47879 −0.111077
\(499\) −17.9770 −0.804762 −0.402381 0.915472i \(-0.631817\pi\)
−0.402381 + 0.915472i \(0.631817\pi\)
\(500\) 2.43191 0.108758
\(501\) −17.8255 −0.796385
\(502\) −33.8606 −1.51127
\(503\) −11.7380 −0.523373 −0.261686 0.965153i \(-0.584279\pi\)
−0.261686 + 0.965153i \(0.584279\pi\)
\(504\) 0 0
\(505\) −32.7619 −1.45789
\(506\) −6.73109 −0.299233
\(507\) −7.83588 −0.348004
\(508\) 4.41238 0.195767
\(509\) −1.29563 −0.0574278 −0.0287139 0.999588i \(-0.509141\pi\)
−0.0287139 + 0.999588i \(0.509141\pi\)
\(510\) −4.48640 −0.198661
\(511\) 0 0
\(512\) 22.4453 0.991952
\(513\) 0.205172 0.00905855
\(514\) −9.06299 −0.399751
\(515\) 5.39682 0.237813
\(516\) −1.47166 −0.0647862
\(517\) −2.46156 −0.108259
\(518\) 0 0
\(519\) −19.7655 −0.867609
\(520\) 41.4173 1.81627
\(521\) −20.7146 −0.907524 −0.453762 0.891123i \(-0.649918\pi\)
−0.453762 + 0.891123i \(0.649918\pi\)
\(522\) 11.9673 0.523795
\(523\) −11.0745 −0.484255 −0.242128 0.970244i \(-0.577845\pi\)
−0.242128 + 0.970244i \(0.577845\pi\)
\(524\) −12.8949 −0.563318
\(525\) 0 0
\(526\) 16.1634 0.704756
\(527\) 9.07896 0.395486
\(528\) 1.72506 0.0750734
\(529\) 38.1418 1.65834
\(530\) 20.1531 0.875395
\(531\) −12.6741 −0.550011
\(532\) 0 0
\(533\) 4.56463 0.197716
\(534\) −10.9615 −0.474352
\(535\) −35.5857 −1.53851
\(536\) −1.95802 −0.0845735
\(537\) 13.4411 0.580024
\(538\) −11.5734 −0.498967
\(539\) 0 0
\(540\) 1.85396 0.0797816
\(541\) −10.7336 −0.461472 −0.230736 0.973016i \(-0.574113\pi\)
−0.230736 + 0.973016i \(0.574113\pi\)
\(542\) −10.7462 −0.461588
\(543\) 8.21899 0.352710
\(544\) 4.43148 0.189998
\(545\) −13.9158 −0.596088
\(546\) 0 0
\(547\) 26.2180 1.12100 0.560500 0.828155i \(-0.310609\pi\)
0.560500 + 0.828155i \(0.310609\pi\)
\(548\) −10.2977 −0.439897
\(549\) −12.0452 −0.514078
\(550\) −3.17495 −0.135380
\(551\) 2.09697 0.0893338
\(552\) −24.0701 −1.02449
\(553\) 0 0
\(554\) 15.9288 0.676750
\(555\) 16.9431 0.719195
\(556\) 10.4893 0.444844
\(557\) 3.82486 0.162064 0.0810322 0.996711i \(-0.474178\pi\)
0.0810322 + 0.996711i \(0.474178\pi\)
\(558\) 8.17806 0.346205
\(559\) −10.6802 −0.451726
\(560\) 0 0
\(561\) 0.955657 0.0403478
\(562\) −18.7286 −0.790020
\(563\) 34.6211 1.45910 0.729552 0.683926i \(-0.239729\pi\)
0.729552 + 0.683926i \(0.239729\pi\)
\(564\) −2.10596 −0.0886770
\(565\) 0.857449 0.0360731
\(566\) 23.3620 0.981980
\(567\) 0 0
\(568\) −40.7557 −1.71007
\(569\) −11.6212 −0.487185 −0.243592 0.969878i \(-0.578326\pi\)
−0.243592 + 0.969878i \(0.578326\pi\)
\(570\) −0.708119 −0.0296598
\(571\) 15.0096 0.628132 0.314066 0.949401i \(-0.398309\pi\)
0.314066 + 0.949401i \(0.398309\pi\)
\(572\) −2.11073 −0.0882540
\(573\) −3.28155 −0.137089
\(574\) 0 0
\(575\) 28.8397 1.20270
\(576\) 8.68463 0.361860
\(577\) 23.3891 0.973700 0.486850 0.873486i \(-0.338146\pi\)
0.486850 + 0.873486i \(0.338146\pi\)
\(578\) 17.9269 0.745662
\(579\) −13.7930 −0.573217
\(580\) 18.9484 0.786791
\(581\) 0 0
\(582\) 0.444142 0.0184103
\(583\) −4.29286 −0.177792
\(584\) −8.30526 −0.343674
\(585\) 13.4546 0.556281
\(586\) 6.58643 0.272083
\(587\) 38.5946 1.59297 0.796484 0.604660i \(-0.206691\pi\)
0.796484 + 0.604660i \(0.206691\pi\)
\(588\) 0 0
\(589\) 1.43300 0.0590455
\(590\) 43.7429 1.80087
\(591\) 23.8854 0.982515
\(592\) 13.4877 0.554340
\(593\) 27.6583 1.13579 0.567895 0.823101i \(-0.307758\pi\)
0.567895 + 0.823101i \(0.307758\pi\)
\(594\) 0.860827 0.0353202
\(595\) 0 0
\(596\) 11.8073 0.483644
\(597\) −25.3686 −1.03827
\(598\) −41.7925 −1.70902
\(599\) 1.90407 0.0777980 0.0388990 0.999243i \(-0.487615\pi\)
0.0388990 + 0.999243i \(0.487615\pi\)
\(600\) −11.3535 −0.463505
\(601\) −7.15202 −0.291737 −0.145868 0.989304i \(-0.546598\pi\)
−0.145868 + 0.989304i \(0.546598\pi\)
\(602\) 0 0
\(603\) −0.636074 −0.0259029
\(604\) −4.43696 −0.180537
\(605\) −30.8303 −1.25343
\(606\) −13.0144 −0.528676
\(607\) −27.1091 −1.10033 −0.550163 0.835057i \(-0.685434\pi\)
−0.550163 + 0.835057i \(0.685434\pi\)
\(608\) 0.699452 0.0283665
\(609\) 0 0
\(610\) 41.5724 1.68322
\(611\) −15.2835 −0.618305
\(612\) 0.817601 0.0330496
\(613\) −44.8087 −1.80981 −0.904904 0.425616i \(-0.860057\pi\)
−0.904904 + 0.425616i \(0.860057\pi\)
\(614\) −1.26314 −0.0509761
\(615\) −2.94759 −0.118858
\(616\) 0 0
\(617\) −19.2364 −0.774430 −0.387215 0.921989i \(-0.626563\pi\)
−0.387215 + 0.921989i \(0.626563\pi\)
\(618\) 2.14385 0.0862384
\(619\) 17.4185 0.700111 0.350055 0.936729i \(-0.386163\pi\)
0.350055 + 0.936729i \(0.386163\pi\)
\(620\) 12.9487 0.520033
\(621\) −7.81932 −0.313779
\(622\) −11.3986 −0.457041
\(623\) 0 0
\(624\) 10.7106 0.428769
\(625\) −29.8380 −1.19352
\(626\) −35.6341 −1.42423
\(627\) 0.150838 0.00602389
\(628\) 5.92101 0.236274
\(629\) 7.47197 0.297927
\(630\) 0 0
\(631\) 14.1325 0.562606 0.281303 0.959619i \(-0.409233\pi\)
0.281303 + 0.959619i \(0.409233\pi\)
\(632\) 33.1870 1.32011
\(633\) 2.29547 0.0912368
\(634\) 25.5733 1.01565
\(635\) −20.6779 −0.820576
\(636\) −3.67270 −0.145632
\(637\) 0 0
\(638\) 8.79813 0.348321
\(639\) −13.2397 −0.523756
\(640\) −9.87646 −0.390401
\(641\) −18.2182 −0.719576 −0.359788 0.933034i \(-0.617151\pi\)
−0.359788 + 0.933034i \(0.617151\pi\)
\(642\) −14.1362 −0.557911
\(643\) −7.78639 −0.307065 −0.153533 0.988144i \(-0.549065\pi\)
−0.153533 + 0.988144i \(0.549065\pi\)
\(644\) 0 0
\(645\) 6.89669 0.271557
\(646\) −0.312283 −0.0122866
\(647\) −15.2676 −0.600233 −0.300116 0.953903i \(-0.597026\pi\)
−0.300116 + 0.953903i \(0.597026\pi\)
\(648\) 3.07829 0.120927
\(649\) −9.31777 −0.365754
\(650\) −19.7129 −0.773203
\(651\) 0 0
\(652\) −8.24895 −0.323054
\(653\) 36.0434 1.41049 0.705244 0.708965i \(-0.250838\pi\)
0.705244 + 0.708965i \(0.250838\pi\)
\(654\) −5.52796 −0.216160
\(655\) 60.4299 2.36119
\(656\) −2.34644 −0.0916132
\(657\) −2.69801 −0.105260
\(658\) 0 0
\(659\) 40.5528 1.57971 0.789857 0.613291i \(-0.210155\pi\)
0.789857 + 0.613291i \(0.210155\pi\)
\(660\) 1.36299 0.0530543
\(661\) 1.83785 0.0714842 0.0357421 0.999361i \(-0.488621\pi\)
0.0357421 + 0.999361i \(0.488621\pi\)
\(662\) 23.8925 0.928608
\(663\) 5.93355 0.230440
\(664\) −6.51669 −0.252896
\(665\) 0 0
\(666\) 6.73054 0.260803
\(667\) −79.9178 −3.09443
\(668\) −11.2118 −0.433797
\(669\) −9.29813 −0.359487
\(670\) 2.19531 0.0848123
\(671\) −8.85541 −0.341859
\(672\) 0 0
\(673\) 31.5015 1.21429 0.607146 0.794590i \(-0.292314\pi\)
0.607146 + 0.794590i \(0.292314\pi\)
\(674\) −17.7359 −0.683163
\(675\) −3.68826 −0.141961
\(676\) −4.92857 −0.189560
\(677\) −5.03952 −0.193684 −0.0968422 0.995300i \(-0.530874\pi\)
−0.0968422 + 0.995300i \(0.530874\pi\)
\(678\) 0.340616 0.0130813
\(679\) 0 0
\(680\) −11.7946 −0.452303
\(681\) −6.59863 −0.252860
\(682\) 6.01234 0.230224
\(683\) 9.21689 0.352675 0.176337 0.984330i \(-0.443575\pi\)
0.176337 + 0.984330i \(0.443575\pi\)
\(684\) 0.129048 0.00493426
\(685\) 48.2586 1.84386
\(686\) 0 0
\(687\) 9.69155 0.369756
\(688\) 5.49015 0.209310
\(689\) −26.6538 −1.01543
\(690\) 26.9872 1.02739
\(691\) −31.7005 −1.20595 −0.602973 0.797762i \(-0.706017\pi\)
−0.602973 + 0.797762i \(0.706017\pi\)
\(692\) −12.4320 −0.472593
\(693\) 0 0
\(694\) 12.0194 0.456249
\(695\) −49.1562 −1.86460
\(696\) 31.4618 1.19256
\(697\) −1.29990 −0.0492371
\(698\) 13.3552 0.505501
\(699\) −24.0753 −0.910612
\(700\) 0 0
\(701\) −43.8376 −1.65572 −0.827861 0.560933i \(-0.810443\pi\)
−0.827861 + 0.560933i \(0.810443\pi\)
\(702\) 5.34477 0.201725
\(703\) 1.17935 0.0444802
\(704\) 6.38476 0.240635
\(705\) 9.86925 0.371697
\(706\) −34.5848 −1.30162
\(707\) 0 0
\(708\) −7.97171 −0.299595
\(709\) −26.3574 −0.989874 −0.494937 0.868929i \(-0.664809\pi\)
−0.494937 + 0.868929i \(0.664809\pi\)
\(710\) 45.6950 1.71490
\(711\) 10.7810 0.404319
\(712\) −28.8176 −1.07998
\(713\) −54.6131 −2.04528
\(714\) 0 0
\(715\) 9.89158 0.369924
\(716\) 8.45408 0.315944
\(717\) 5.05867 0.188919
\(718\) −9.32677 −0.348072
\(719\) 38.6791 1.44249 0.721243 0.692682i \(-0.243571\pi\)
0.721243 + 0.692682i \(0.243571\pi\)
\(720\) −6.91634 −0.257757
\(721\) 0 0
\(722\) 22.1980 0.826123
\(723\) 3.21287 0.119488
\(724\) 5.16953 0.192124
\(725\) −37.6960 −1.40000
\(726\) −12.2471 −0.454533
\(727\) 42.3237 1.56970 0.784849 0.619687i \(-0.212741\pi\)
0.784849 + 0.619687i \(0.212741\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 9.31180 0.344645
\(731\) 3.04147 0.112493
\(732\) −7.57615 −0.280023
\(733\) −10.4618 −0.386416 −0.193208 0.981158i \(-0.561889\pi\)
−0.193208 + 0.981158i \(0.561889\pi\)
\(734\) 3.30927 0.122147
\(735\) 0 0
\(736\) −26.6569 −0.982587
\(737\) −0.467628 −0.0172253
\(738\) −1.17091 −0.0431017
\(739\) −20.7922 −0.764854 −0.382427 0.923986i \(-0.624912\pi\)
−0.382427 + 0.923986i \(0.624912\pi\)
\(740\) 10.6568 0.391751
\(741\) 0.936533 0.0344044
\(742\) 0 0
\(743\) 41.0163 1.50474 0.752370 0.658740i \(-0.228910\pi\)
0.752370 + 0.658740i \(0.228910\pi\)
\(744\) 21.4999 0.788225
\(745\) −55.3328 −2.02724
\(746\) −7.23719 −0.264972
\(747\) −2.11698 −0.0774564
\(748\) 0.601084 0.0219778
\(749\) 0 0
\(750\) −4.52728 −0.165313
\(751\) −25.9895 −0.948371 −0.474185 0.880425i \(-0.657257\pi\)
−0.474185 + 0.880425i \(0.657257\pi\)
\(752\) 7.85647 0.286496
\(753\) −28.9183 −1.05384
\(754\) 54.6264 1.98938
\(755\) 20.7931 0.756738
\(756\) 0 0
\(757\) −19.2370 −0.699182 −0.349591 0.936902i \(-0.613679\pi\)
−0.349591 + 0.936902i \(0.613679\pi\)
\(758\) −15.7138 −0.570751
\(759\) −5.74861 −0.208661
\(760\) −1.86163 −0.0675283
\(761\) −13.8501 −0.502066 −0.251033 0.967978i \(-0.580770\pi\)
−0.251033 + 0.967978i \(0.580770\pi\)
\(762\) −8.21414 −0.297567
\(763\) 0 0
\(764\) −2.06401 −0.0746734
\(765\) −3.83155 −0.138530
\(766\) 7.07623 0.255674
\(767\) −57.8528 −2.08894
\(768\) 13.4459 0.485188
\(769\) 19.8296 0.715074 0.357537 0.933899i \(-0.383617\pi\)
0.357537 + 0.933899i \(0.383617\pi\)
\(770\) 0 0
\(771\) −7.74014 −0.278754
\(772\) −8.67543 −0.312236
\(773\) −52.2103 −1.87787 −0.938936 0.344091i \(-0.888187\pi\)
−0.938936 + 0.344091i \(0.888187\pi\)
\(774\) 2.73966 0.0984752
\(775\) −25.7602 −0.925333
\(776\) 1.16764 0.0419157
\(777\) 0 0
\(778\) −27.2489 −0.976921
\(779\) −0.205172 −0.00735104
\(780\) 8.46263 0.303011
\(781\) −9.73359 −0.348295
\(782\) 11.9015 0.425595
\(783\) 10.2205 0.365253
\(784\) 0 0
\(785\) −27.7478 −0.990362
\(786\) 24.0054 0.856244
\(787\) −38.7058 −1.37971 −0.689857 0.723946i \(-0.742327\pi\)
−0.689857 + 0.723946i \(0.742327\pi\)
\(788\) 15.0233 0.535184
\(789\) 13.8041 0.491440
\(790\) −37.2090 −1.32384
\(791\) 0 0
\(792\) 2.26309 0.0804155
\(793\) −54.9821 −1.95247
\(794\) 13.0227 0.462158
\(795\) 17.2115 0.610430
\(796\) −15.9562 −0.565552
\(797\) 37.4303 1.32585 0.662924 0.748687i \(-0.269315\pi\)
0.662924 + 0.748687i \(0.269315\pi\)
\(798\) 0 0
\(799\) 4.35237 0.153976
\(800\) −12.5737 −0.444546
\(801\) −9.36156 −0.330774
\(802\) 2.93291 0.103565
\(803\) −1.98352 −0.0699971
\(804\) −0.400074 −0.0141095
\(805\) 0 0
\(806\) 37.3299 1.31489
\(807\) −9.88416 −0.347939
\(808\) −34.2146 −1.20367
\(809\) 38.5794 1.35638 0.678190 0.734886i \(-0.262765\pi\)
0.678190 + 0.734886i \(0.262765\pi\)
\(810\) −3.45135 −0.121268
\(811\) 43.7757 1.53717 0.768587 0.639745i \(-0.220960\pi\)
0.768587 + 0.639745i \(0.220960\pi\)
\(812\) 0 0
\(813\) −9.17765 −0.321874
\(814\) 4.94815 0.173433
\(815\) 38.6573 1.35411
\(816\) −3.05013 −0.106776
\(817\) 0.480056 0.0167950
\(818\) 10.8446 0.379171
\(819\) 0 0
\(820\) −1.85396 −0.0647430
\(821\) −9.99336 −0.348771 −0.174385 0.984677i \(-0.555794\pi\)
−0.174385 + 0.984677i \(0.555794\pi\)
\(822\) 19.1704 0.668644
\(823\) −14.6858 −0.511913 −0.255957 0.966688i \(-0.582390\pi\)
−0.255957 + 0.966688i \(0.582390\pi\)
\(824\) 5.63613 0.196344
\(825\) −2.71153 −0.0944034
\(826\) 0 0
\(827\) −33.2522 −1.15629 −0.578146 0.815933i \(-0.696224\pi\)
−0.578146 + 0.815933i \(0.696224\pi\)
\(828\) −4.91815 −0.170918
\(829\) −29.7741 −1.03410 −0.517048 0.855956i \(-0.672969\pi\)
−0.517048 + 0.855956i \(0.672969\pi\)
\(830\) 7.30646 0.253611
\(831\) 13.6038 0.471911
\(832\) 39.6422 1.37435
\(833\) 0 0
\(834\) −19.5270 −0.676164
\(835\) 52.5422 1.81830
\(836\) 0.0948732 0.00328126
\(837\) 6.98437 0.241415
\(838\) −31.1181 −1.07496
\(839\) 41.2929 1.42559 0.712795 0.701372i \(-0.247429\pi\)
0.712795 + 0.701372i \(0.247429\pi\)
\(840\) 0 0
\(841\) 75.4596 2.60206
\(842\) 38.7534 1.33553
\(843\) −15.9950 −0.550896
\(844\) 1.44379 0.0496974
\(845\) 23.0969 0.794559
\(846\) 3.92049 0.134789
\(847\) 0 0
\(848\) 13.7013 0.470506
\(849\) 19.9521 0.684753
\(850\) 5.61374 0.192550
\(851\) −44.9465 −1.54075
\(852\) −8.32746 −0.285294
\(853\) 4.00104 0.136993 0.0684964 0.997651i \(-0.478180\pi\)
0.0684964 + 0.997651i \(0.478180\pi\)
\(854\) 0 0
\(855\) −0.604761 −0.0206824
\(856\) −37.1637 −1.27023
\(857\) −41.6354 −1.42224 −0.711119 0.703071i \(-0.751811\pi\)
−0.711119 + 0.703071i \(0.751811\pi\)
\(858\) 3.92936 0.134146
\(859\) −6.29690 −0.214848 −0.107424 0.994213i \(-0.534260\pi\)
−0.107424 + 0.994213i \(0.534260\pi\)
\(860\) 4.33784 0.147919
\(861\) 0 0
\(862\) 25.8767 0.881363
\(863\) 1.64503 0.0559976 0.0279988 0.999608i \(-0.491087\pi\)
0.0279988 + 0.999608i \(0.491087\pi\)
\(864\) 3.40911 0.115980
\(865\) 58.2605 1.98092
\(866\) −12.9440 −0.439855
\(867\) 15.3103 0.519964
\(868\) 0 0
\(869\) 7.92598 0.268870
\(870\) −35.2747 −1.19592
\(871\) −2.90344 −0.0983794
\(872\) −14.5329 −0.492145
\(873\) 0.379314 0.0128378
\(874\) 1.87849 0.0635409
\(875\) 0 0
\(876\) −1.69698 −0.0573357
\(877\) −37.6279 −1.27060 −0.635302 0.772263i \(-0.719125\pi\)
−0.635302 + 0.772263i \(0.719125\pi\)
\(878\) 24.8182 0.837572
\(879\) 5.62507 0.189729
\(880\) −5.08475 −0.171407
\(881\) 34.9336 1.17694 0.588472 0.808518i \(-0.299730\pi\)
0.588472 + 0.808518i \(0.299730\pi\)
\(882\) 0 0
\(883\) 58.1016 1.95527 0.977637 0.210301i \(-0.0674443\pi\)
0.977637 + 0.210301i \(0.0674443\pi\)
\(884\) 3.73205 0.125522
\(885\) 37.3581 1.25578
\(886\) 18.4875 0.621100
\(887\) 45.9321 1.54225 0.771124 0.636685i \(-0.219695\pi\)
0.771124 + 0.636685i \(0.219695\pi\)
\(888\) 17.6944 0.593785
\(889\) 0 0
\(890\) 32.3100 1.08303
\(891\) 0.735179 0.0246294
\(892\) −5.84829 −0.195815
\(893\) 0.686965 0.0229884
\(894\) −21.9806 −0.735140
\(895\) −39.6187 −1.32431
\(896\) 0 0
\(897\) −35.6923 −1.19173
\(898\) 32.8974 1.09780
\(899\) 71.3841 2.38079
\(900\) −2.31982 −0.0773273
\(901\) 7.59034 0.252871
\(902\) −0.860827 −0.0286624
\(903\) 0 0
\(904\) 0.895470 0.0297829
\(905\) −24.2262 −0.805305
\(906\) 8.25991 0.274417
\(907\) −5.87846 −0.195191 −0.0975955 0.995226i \(-0.531115\pi\)
−0.0975955 + 0.995226i \(0.531115\pi\)
\(908\) −4.15037 −0.137735
\(909\) −11.1148 −0.368656
\(910\) 0 0
\(911\) 59.8217 1.98198 0.990990 0.133935i \(-0.0427615\pi\)
0.990990 + 0.133935i \(0.0427615\pi\)
\(912\) −0.481423 −0.0159415
\(913\) −1.55636 −0.0515081
\(914\) −4.15869 −0.137557
\(915\) 35.5044 1.17374
\(916\) 6.09574 0.201409
\(917\) 0 0
\(918\) −1.52206 −0.0502354
\(919\) 36.5137 1.20448 0.602238 0.798317i \(-0.294276\pi\)
0.602238 + 0.798317i \(0.294276\pi\)
\(920\) 70.9487 2.33911
\(921\) −1.07877 −0.0355466
\(922\) −9.00112 −0.296436
\(923\) −60.4346 −1.98923
\(924\) 0 0
\(925\) −21.2006 −0.697072
\(926\) 29.7186 0.976615
\(927\) 1.83093 0.0601357
\(928\) 34.8429 1.14378
\(929\) −7.35496 −0.241308 −0.120654 0.992695i \(-0.538499\pi\)
−0.120654 + 0.992695i \(0.538499\pi\)
\(930\) −24.1055 −0.790451
\(931\) 0 0
\(932\) −15.1428 −0.496018
\(933\) −9.73481 −0.318703
\(934\) −20.2994 −0.664218
\(935\) −2.81688 −0.0921218
\(936\) 14.0513 0.459280
\(937\) −10.5834 −0.345744 −0.172872 0.984944i \(-0.555305\pi\)
−0.172872 + 0.984944i \(0.555305\pi\)
\(938\) 0 0
\(939\) −30.4329 −0.993140
\(940\) 6.20750 0.202466
\(941\) −12.9045 −0.420675 −0.210338 0.977629i \(-0.567456\pi\)
−0.210338 + 0.977629i \(0.567456\pi\)
\(942\) −11.0226 −0.359137
\(943\) 7.81932 0.254632
\(944\) 29.7391 0.967926
\(945\) 0 0
\(946\) 2.01414 0.0654855
\(947\) −20.4076 −0.663157 −0.331579 0.943428i \(-0.607581\pi\)
−0.331579 + 0.943428i \(0.607581\pi\)
\(948\) 6.78098 0.220236
\(949\) −12.3154 −0.399776
\(950\) 0.886056 0.0287475
\(951\) 21.8406 0.708230
\(952\) 0 0
\(953\) −45.4473 −1.47218 −0.736092 0.676882i \(-0.763331\pi\)
−0.736092 + 0.676882i \(0.763331\pi\)
\(954\) 6.83716 0.221361
\(955\) 9.67266 0.313000
\(956\) 3.18177 0.102906
\(957\) 7.51394 0.242891
\(958\) −3.40805 −0.110109
\(959\) 0 0
\(960\) −25.5987 −0.826194
\(961\) 17.7815 0.573596
\(962\) 30.7224 0.990531
\(963\) −12.0728 −0.389042
\(964\) 2.02081 0.0650861
\(965\) 40.6560 1.30876
\(966\) 0 0
\(967\) 16.9520 0.545141 0.272570 0.962136i \(-0.412126\pi\)
0.272570 + 0.962136i \(0.412126\pi\)
\(968\) −32.1974 −1.03486
\(969\) −0.266702 −0.00856769
\(970\) −1.30915 −0.0420341
\(971\) 42.3882 1.36030 0.680152 0.733071i \(-0.261914\pi\)
0.680152 + 0.733071i \(0.261914\pi\)
\(972\) 0.628975 0.0201744
\(973\) 0 0
\(974\) 36.4312 1.16733
\(975\) −16.8355 −0.539169
\(976\) 28.2635 0.904691
\(977\) −4.20581 −0.134556 −0.0672780 0.997734i \(-0.521431\pi\)
−0.0672780 + 0.997734i \(0.521431\pi\)
\(978\) 15.3564 0.491042
\(979\) −6.88243 −0.219963
\(980\) 0 0
\(981\) −4.72109 −0.150733
\(982\) −38.6934 −1.23475
\(983\) −37.1082 −1.18357 −0.591783 0.806097i \(-0.701576\pi\)
−0.591783 + 0.806097i \(0.701576\pi\)
\(984\) −3.07829 −0.0981322
\(985\) −70.4043 −2.24327
\(986\) −15.5563 −0.495413
\(987\) 0 0
\(988\) 0.589056 0.0187404
\(989\) −18.2955 −0.581762
\(990\) −2.53736 −0.0806427
\(991\) −2.05021 −0.0651270 −0.0325635 0.999470i \(-0.510367\pi\)
−0.0325635 + 0.999470i \(0.510367\pi\)
\(992\) 23.8105 0.755983
\(993\) 20.4051 0.647536
\(994\) 0 0
\(995\) 74.7761 2.37056
\(996\) −1.33153 −0.0421911
\(997\) 18.3658 0.581651 0.290826 0.956776i \(-0.406070\pi\)
0.290826 + 0.956776i \(0.406070\pi\)
\(998\) 21.0494 0.666308
\(999\) 5.74813 0.181863
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 6027.2.a.z.1.3 8
7.6 odd 2 6027.2.a.ba.1.3 yes 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
6027.2.a.z.1.3 8 1.1 even 1 trivial
6027.2.a.ba.1.3 yes 8 7.6 odd 2