Properties

Label 5733.2.a.bu.1.3
Level $5733$
Weight $2$
Character 5733.1
Self dual yes
Analytic conductor $45.778$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [5733,2,Mod(1,5733)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(5733, 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("5733.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 5733 = 3^{2} \cdot 7^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5733.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: \(45.7782354788\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.6.4507648.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 2x^{5} - 5x^{4} + 8x^{3} + 7x^{2} - 6x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 637)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(1.90903\) of defining polynomial
Character \(\chi\) \(=\) 5733.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-0.264627 q^{2} -1.92997 q^{4} -1.43515 q^{5} +1.03998 q^{8} +O(q^{10})\) \(q-0.264627 q^{2} -1.92997 q^{4} -1.43515 q^{5} +1.03998 q^{8} +0.379780 q^{10} -5.50474 q^{11} -1.00000 q^{13} +3.58474 q^{16} +4.83072 q^{17} -2.82036 q^{19} +2.76981 q^{20} +1.45670 q^{22} +5.99956 q^{23} -2.94033 q^{25} +0.264627 q^{26} -1.04188 q^{29} -9.20895 q^{31} -3.02857 q^{32} -1.27834 q^{34} +0.612497 q^{37} +0.746342 q^{38} -1.49252 q^{40} -10.6196 q^{41} -8.43685 q^{43} +10.6240 q^{44} -1.58764 q^{46} +2.40922 q^{47} +0.778091 q^{50} +1.92997 q^{52} +1.82959 q^{53} +7.90015 q^{55} +0.275709 q^{58} -0.870914 q^{59} +3.33253 q^{61} +2.43693 q^{62} -6.36804 q^{64} +1.43515 q^{65} -6.62741 q^{67} -9.32316 q^{68} +6.85856 q^{71} -3.14147 q^{73} -0.162083 q^{74} +5.44322 q^{76} -17.5723 q^{79} -5.14465 q^{80} +2.81022 q^{82} +11.4525 q^{83} -6.93283 q^{85} +2.23261 q^{86} -5.72479 q^{88} -0.995318 q^{89} -11.5790 q^{92} -0.637545 q^{94} +4.04765 q^{95} -13.5090 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 4 q^{4} + 6 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 4 q^{4} + 6 q^{5} - 4 q^{10} - 4 q^{11} - 6 q^{13} + 16 q^{17} - 2 q^{19} + 16 q^{20} - 12 q^{22} + 6 q^{23} - 4 q^{25} + 6 q^{29} - 6 q^{31} + 20 q^{32} + 8 q^{38} - 4 q^{40} - 8 q^{41} + 2 q^{43} + 4 q^{44} + 8 q^{46} + 30 q^{47} - 8 q^{50} - 4 q^{52} + 14 q^{53} + 8 q^{55} - 8 q^{58} + 24 q^{59} + 28 q^{62} - 20 q^{64} - 6 q^{65} + 16 q^{67} + 28 q^{68} - 8 q^{71} + 6 q^{73} + 12 q^{74} + 16 q^{76} - 22 q^{79} - 28 q^{80} + 40 q^{82} + 50 q^{83} - 8 q^{85} + 16 q^{86} - 44 q^{88} + 26 q^{89} - 20 q^{92} + 32 q^{94} + 6 q^{95} + 14 q^{97}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.264627 −0.187119 −0.0935596 0.995614i \(-0.529825\pi\)
−0.0935596 + 0.995614i \(0.529825\pi\)
\(3\) 0 0
\(4\) −1.92997 −0.964986
\(5\) −1.43515 −0.641820 −0.320910 0.947110i \(-0.603989\pi\)
−0.320910 + 0.947110i \(0.603989\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 1.03998 0.367687
\(9\) 0 0
\(10\) 0.379780 0.120097
\(11\) −5.50474 −1.65974 −0.829871 0.557955i \(-0.811586\pi\)
−0.829871 + 0.557955i \(0.811586\pi\)
\(12\) 0 0
\(13\) −1.00000 −0.277350
\(14\) 0 0
\(15\) 0 0
\(16\) 3.58474 0.896185
\(17\) 4.83072 1.17162 0.585811 0.810448i \(-0.300776\pi\)
0.585811 + 0.810448i \(0.300776\pi\)
\(18\) 0 0
\(19\) −2.82036 −0.647035 −0.323518 0.946222i \(-0.604865\pi\)
−0.323518 + 0.946222i \(0.604865\pi\)
\(20\) 2.76981 0.619348
\(21\) 0 0
\(22\) 1.45670 0.310570
\(23\) 5.99956 1.25100 0.625498 0.780226i \(-0.284896\pi\)
0.625498 + 0.780226i \(0.284896\pi\)
\(24\) 0 0
\(25\) −2.94033 −0.588067
\(26\) 0.264627 0.0518975
\(27\) 0 0
\(28\) 0 0
\(29\) −1.04188 −0.193472 −0.0967361 0.995310i \(-0.530840\pi\)
−0.0967361 + 0.995310i \(0.530840\pi\)
\(30\) 0 0
\(31\) −9.20895 −1.65398 −0.826988 0.562219i \(-0.809948\pi\)
−0.826988 + 0.562219i \(0.809948\pi\)
\(32\) −3.02857 −0.535380
\(33\) 0 0
\(34\) −1.27834 −0.219233
\(35\) 0 0
\(36\) 0 0
\(37\) 0.612497 0.100694 0.0503470 0.998732i \(-0.483967\pi\)
0.0503470 + 0.998732i \(0.483967\pi\)
\(38\) 0.746342 0.121073
\(39\) 0 0
\(40\) −1.49252 −0.235989
\(41\) −10.6196 −1.65850 −0.829249 0.558879i \(-0.811232\pi\)
−0.829249 + 0.558879i \(0.811232\pi\)
\(42\) 0 0
\(43\) −8.43685 −1.28661 −0.643304 0.765611i \(-0.722437\pi\)
−0.643304 + 0.765611i \(0.722437\pi\)
\(44\) 10.6240 1.60163
\(45\) 0 0
\(46\) −1.58764 −0.234085
\(47\) 2.40922 0.351422 0.175711 0.984442i \(-0.443778\pi\)
0.175711 + 0.984442i \(0.443778\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0.778091 0.110039
\(51\) 0 0
\(52\) 1.92997 0.267639
\(53\) 1.82959 0.251313 0.125657 0.992074i \(-0.459896\pi\)
0.125657 + 0.992074i \(0.459896\pi\)
\(54\) 0 0
\(55\) 7.90015 1.06526
\(56\) 0 0
\(57\) 0 0
\(58\) 0.275709 0.0362024
\(59\) −0.870914 −0.113383 −0.0566917 0.998392i \(-0.518055\pi\)
−0.0566917 + 0.998392i \(0.518055\pi\)
\(60\) 0 0
\(61\) 3.33253 0.426686 0.213343 0.976977i \(-0.431565\pi\)
0.213343 + 0.976977i \(0.431565\pi\)
\(62\) 2.43693 0.309491
\(63\) 0 0
\(64\) −6.36804 −0.796005
\(65\) 1.43515 0.178009
\(66\) 0 0
\(67\) −6.62741 −0.809667 −0.404833 0.914390i \(-0.632671\pi\)
−0.404833 + 0.914390i \(0.632671\pi\)
\(68\) −9.32316 −1.13060
\(69\) 0 0
\(70\) 0 0
\(71\) 6.85856 0.813961 0.406980 0.913437i \(-0.366582\pi\)
0.406980 + 0.913437i \(0.366582\pi\)
\(72\) 0 0
\(73\) −3.14147 −0.367682 −0.183841 0.982956i \(-0.558853\pi\)
−0.183841 + 0.982956i \(0.558853\pi\)
\(74\) −0.162083 −0.0188418
\(75\) 0 0
\(76\) 5.44322 0.624380
\(77\) 0 0
\(78\) 0 0
\(79\) −17.5723 −1.97704 −0.988518 0.151101i \(-0.951718\pi\)
−0.988518 + 0.151101i \(0.951718\pi\)
\(80\) −5.14465 −0.575190
\(81\) 0 0
\(82\) 2.81022 0.310337
\(83\) 11.4525 1.25708 0.628538 0.777779i \(-0.283654\pi\)
0.628538 + 0.777779i \(0.283654\pi\)
\(84\) 0 0
\(85\) −6.93283 −0.751971
\(86\) 2.23261 0.240749
\(87\) 0 0
\(88\) −5.72479 −0.610265
\(89\) −0.995318 −0.105503 −0.0527517 0.998608i \(-0.516799\pi\)
−0.0527517 + 0.998608i \(0.516799\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) −11.5790 −1.20719
\(93\) 0 0
\(94\) −0.637545 −0.0657577
\(95\) 4.04765 0.415280
\(96\) 0 0
\(97\) −13.5090 −1.37163 −0.685817 0.727774i \(-0.740555\pi\)
−0.685817 + 0.727774i \(0.740555\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 5.67477 0.567477
\(101\) −1.00807 −0.100306 −0.0501532 0.998742i \(-0.515971\pi\)
−0.0501532 + 0.998742i \(0.515971\pi\)
\(102\) 0 0
\(103\) 12.7754 1.25880 0.629401 0.777081i \(-0.283300\pi\)
0.629401 + 0.777081i \(0.283300\pi\)
\(104\) −1.03998 −0.101978
\(105\) 0 0
\(106\) −0.484157 −0.0470255
\(107\) 0.685495 0.0662693 0.0331347 0.999451i \(-0.489451\pi\)
0.0331347 + 0.999451i \(0.489451\pi\)
\(108\) 0 0
\(109\) −2.90344 −0.278099 −0.139050 0.990285i \(-0.544405\pi\)
−0.139050 + 0.990285i \(0.544405\pi\)
\(110\) −2.09059 −0.199330
\(111\) 0 0
\(112\) 0 0
\(113\) −12.0315 −1.13183 −0.565915 0.824464i \(-0.691477\pi\)
−0.565915 + 0.824464i \(0.691477\pi\)
\(114\) 0 0
\(115\) −8.61029 −0.802914
\(116\) 2.01080 0.186698
\(117\) 0 0
\(118\) 0.230467 0.0212162
\(119\) 0 0
\(120\) 0 0
\(121\) 19.3022 1.75474
\(122\) −0.881875 −0.0798412
\(123\) 0 0
\(124\) 17.7730 1.59606
\(125\) 11.3956 1.01925
\(126\) 0 0
\(127\) 15.6659 1.39012 0.695062 0.718950i \(-0.255377\pi\)
0.695062 + 0.718950i \(0.255377\pi\)
\(128\) 7.74229 0.684328
\(129\) 0 0
\(130\) −0.379780 −0.0333089
\(131\) 12.1273 1.05957 0.529784 0.848132i \(-0.322273\pi\)
0.529784 + 0.848132i \(0.322273\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 1.75379 0.151504
\(135\) 0 0
\(136\) 5.02383 0.430790
\(137\) −15.9375 −1.36163 −0.680815 0.732456i \(-0.738374\pi\)
−0.680815 + 0.732456i \(0.738374\pi\)
\(138\) 0 0
\(139\) 6.64088 0.563272 0.281636 0.959521i \(-0.409123\pi\)
0.281636 + 0.959521i \(0.409123\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) −1.81496 −0.152308
\(143\) 5.50474 0.460330
\(144\) 0 0
\(145\) 1.49526 0.124174
\(146\) 0.831317 0.0688003
\(147\) 0 0
\(148\) −1.18210 −0.0971683
\(149\) 19.5502 1.60162 0.800809 0.598920i \(-0.204403\pi\)
0.800809 + 0.598920i \(0.204403\pi\)
\(150\) 0 0
\(151\) 10.6880 0.869779 0.434890 0.900484i \(-0.356787\pi\)
0.434890 + 0.900484i \(0.356787\pi\)
\(152\) −2.93311 −0.237906
\(153\) 0 0
\(154\) 0 0
\(155\) 13.2163 1.06156
\(156\) 0 0
\(157\) 15.0734 1.20299 0.601496 0.798876i \(-0.294572\pi\)
0.601496 + 0.798876i \(0.294572\pi\)
\(158\) 4.65009 0.369942
\(159\) 0 0
\(160\) 4.34646 0.343618
\(161\) 0 0
\(162\) 0 0
\(163\) −23.8135 −1.86521 −0.932607 0.360894i \(-0.882472\pi\)
−0.932607 + 0.360894i \(0.882472\pi\)
\(164\) 20.4955 1.60043
\(165\) 0 0
\(166\) −3.03064 −0.235223
\(167\) −7.12371 −0.551249 −0.275625 0.961265i \(-0.588885\pi\)
−0.275625 + 0.961265i \(0.588885\pi\)
\(168\) 0 0
\(169\) 1.00000 0.0769231
\(170\) 1.83461 0.140708
\(171\) 0 0
\(172\) 16.2829 1.24156
\(173\) 11.2367 0.854309 0.427155 0.904179i \(-0.359516\pi\)
0.427155 + 0.904179i \(0.359516\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −19.7331 −1.48744
\(177\) 0 0
\(178\) 0.263387 0.0197417
\(179\) 13.1945 0.986204 0.493102 0.869972i \(-0.335863\pi\)
0.493102 + 0.869972i \(0.335863\pi\)
\(180\) 0 0
\(181\) 13.7414 1.02139 0.510696 0.859761i \(-0.329388\pi\)
0.510696 + 0.859761i \(0.329388\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 6.23939 0.459974
\(185\) −0.879028 −0.0646274
\(186\) 0 0
\(187\) −26.5919 −1.94459
\(188\) −4.64974 −0.339117
\(189\) 0 0
\(190\) −1.07112 −0.0777069
\(191\) 16.3307 1.18165 0.590824 0.806800i \(-0.298803\pi\)
0.590824 + 0.806800i \(0.298803\pi\)
\(192\) 0 0
\(193\) −14.0533 −1.01158 −0.505790 0.862656i \(-0.668799\pi\)
−0.505790 + 0.862656i \(0.668799\pi\)
\(194\) 3.57485 0.256659
\(195\) 0 0
\(196\) 0 0
\(197\) 1.46898 0.104660 0.0523302 0.998630i \(-0.483335\pi\)
0.0523302 + 0.998630i \(0.483335\pi\)
\(198\) 0 0
\(199\) 13.3772 0.948285 0.474142 0.880448i \(-0.342758\pi\)
0.474142 + 0.880448i \(0.342758\pi\)
\(200\) −3.05787 −0.216224
\(201\) 0 0
\(202\) 0.266761 0.0187692
\(203\) 0 0
\(204\) 0 0
\(205\) 15.2407 1.06446
\(206\) −3.38072 −0.235546
\(207\) 0 0
\(208\) −3.58474 −0.248557
\(209\) 15.5254 1.07391
\(210\) 0 0
\(211\) 3.47044 0.238915 0.119457 0.992839i \(-0.461885\pi\)
0.119457 + 0.992839i \(0.461885\pi\)
\(212\) −3.53105 −0.242514
\(213\) 0 0
\(214\) −0.181400 −0.0124003
\(215\) 12.1082 0.825771
\(216\) 0 0
\(217\) 0 0
\(218\) 0.768328 0.0520378
\(219\) 0 0
\(220\) −15.2471 −1.02796
\(221\) −4.83072 −0.324950
\(222\) 0 0
\(223\) 9.91318 0.663836 0.331918 0.943308i \(-0.392304\pi\)
0.331918 + 0.943308i \(0.392304\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 3.18386 0.211787
\(227\) 12.0727 0.801292 0.400646 0.916233i \(-0.368786\pi\)
0.400646 + 0.916233i \(0.368786\pi\)
\(228\) 0 0
\(229\) −4.05171 −0.267745 −0.133872 0.990999i \(-0.542741\pi\)
−0.133872 + 0.990999i \(0.542741\pi\)
\(230\) 2.27851 0.150241
\(231\) 0 0
\(232\) −1.08353 −0.0711372
\(233\) −12.5450 −0.821850 −0.410925 0.911669i \(-0.634794\pi\)
−0.410925 + 0.911669i \(0.634794\pi\)
\(234\) 0 0
\(235\) −3.45761 −0.225549
\(236\) 1.68084 0.109413
\(237\) 0 0
\(238\) 0 0
\(239\) −13.3463 −0.863299 −0.431649 0.902042i \(-0.642068\pi\)
−0.431649 + 0.902042i \(0.642068\pi\)
\(240\) 0 0
\(241\) −20.3854 −1.31314 −0.656568 0.754267i \(-0.727993\pi\)
−0.656568 + 0.754267i \(0.727993\pi\)
\(242\) −5.10787 −0.328346
\(243\) 0 0
\(244\) −6.43169 −0.411747
\(245\) 0 0
\(246\) 0 0
\(247\) 2.82036 0.179455
\(248\) −9.57708 −0.608145
\(249\) 0 0
\(250\) −3.01558 −0.190722
\(251\) 17.1921 1.08515 0.542577 0.840006i \(-0.317449\pi\)
0.542577 + 0.840006i \(0.317449\pi\)
\(252\) 0 0
\(253\) −33.0260 −2.07633
\(254\) −4.14561 −0.260119
\(255\) 0 0
\(256\) 10.6873 0.667954
\(257\) −7.64695 −0.477004 −0.238502 0.971142i \(-0.576656\pi\)
−0.238502 + 0.971142i \(0.576656\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) −2.76981 −0.171776
\(261\) 0 0
\(262\) −3.20921 −0.198266
\(263\) 0.101037 0.00623022 0.00311511 0.999995i \(-0.499008\pi\)
0.00311511 + 0.999995i \(0.499008\pi\)
\(264\) 0 0
\(265\) −2.62574 −0.161298
\(266\) 0 0
\(267\) 0 0
\(268\) 12.7907 0.781318
\(269\) −7.56852 −0.461461 −0.230730 0.973018i \(-0.574111\pi\)
−0.230730 + 0.973018i \(0.574111\pi\)
\(270\) 0 0
\(271\) 13.8554 0.841653 0.420826 0.907141i \(-0.361740\pi\)
0.420826 + 0.907141i \(0.361740\pi\)
\(272\) 17.3169 1.04999
\(273\) 0 0
\(274\) 4.21748 0.254787
\(275\) 16.1858 0.976039
\(276\) 0 0
\(277\) 0.552935 0.0332226 0.0166113 0.999862i \(-0.494712\pi\)
0.0166113 + 0.999862i \(0.494712\pi\)
\(278\) −1.75735 −0.105399
\(279\) 0 0
\(280\) 0 0
\(281\) −1.14667 −0.0684043 −0.0342022 0.999415i \(-0.510889\pi\)
−0.0342022 + 0.999415i \(0.510889\pi\)
\(282\) 0 0
\(283\) 4.05396 0.240983 0.120491 0.992714i \(-0.461553\pi\)
0.120491 + 0.992714i \(0.461553\pi\)
\(284\) −13.2368 −0.785461
\(285\) 0 0
\(286\) −1.45670 −0.0861365
\(287\) 0 0
\(288\) 0 0
\(289\) 6.33588 0.372699
\(290\) −0.395685 −0.0232354
\(291\) 0 0
\(292\) 6.06296 0.354808
\(293\) 15.0649 0.880102 0.440051 0.897973i \(-0.354960\pi\)
0.440051 + 0.897973i \(0.354960\pi\)
\(294\) 0 0
\(295\) 1.24990 0.0727718
\(296\) 0.636982 0.0370238
\(297\) 0 0
\(298\) −5.17351 −0.299694
\(299\) −5.99956 −0.346964
\(300\) 0 0
\(301\) 0 0
\(302\) −2.82834 −0.162752
\(303\) 0 0
\(304\) −10.1103 −0.579863
\(305\) −4.78269 −0.273856
\(306\) 0 0
\(307\) 19.9408 1.13808 0.569040 0.822310i \(-0.307315\pi\)
0.569040 + 0.822310i \(0.307315\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) −3.49737 −0.198637
\(311\) −10.8956 −0.617833 −0.308916 0.951089i \(-0.599966\pi\)
−0.308916 + 0.951089i \(0.599966\pi\)
\(312\) 0 0
\(313\) 0.0519190 0.00293464 0.00146732 0.999999i \(-0.499533\pi\)
0.00146732 + 0.999999i \(0.499533\pi\)
\(314\) −3.98883 −0.225103
\(315\) 0 0
\(316\) 33.9140 1.90781
\(317\) −16.1010 −0.904321 −0.452161 0.891937i \(-0.649347\pi\)
−0.452161 + 0.891937i \(0.649347\pi\)
\(318\) 0 0
\(319\) 5.73528 0.321114
\(320\) 9.13912 0.510892
\(321\) 0 0
\(322\) 0 0
\(323\) −13.6244 −0.758081
\(324\) 0 0
\(325\) 2.94033 0.163100
\(326\) 6.30167 0.349017
\(327\) 0 0
\(328\) −11.0441 −0.609808
\(329\) 0 0
\(330\) 0 0
\(331\) 30.7862 1.69216 0.846081 0.533054i \(-0.178956\pi\)
0.846081 + 0.533054i \(0.178956\pi\)
\(332\) −22.1030 −1.21306
\(333\) 0 0
\(334\) 1.88512 0.103149
\(335\) 9.51135 0.519661
\(336\) 0 0
\(337\) −2.41842 −0.131740 −0.0658700 0.997828i \(-0.520982\pi\)
−0.0658700 + 0.997828i \(0.520982\pi\)
\(338\) −0.264627 −0.0143938
\(339\) 0 0
\(340\) 13.3802 0.725642
\(341\) 50.6929 2.74517
\(342\) 0 0
\(343\) 0 0
\(344\) −8.77411 −0.473069
\(345\) 0 0
\(346\) −2.97353 −0.159858
\(347\) 0.492527 0.0264403 0.0132201 0.999913i \(-0.495792\pi\)
0.0132201 + 0.999913i \(0.495792\pi\)
\(348\) 0 0
\(349\) 11.9442 0.639356 0.319678 0.947526i \(-0.396425\pi\)
0.319678 + 0.947526i \(0.396425\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 16.6715 0.888593
\(353\) 15.5299 0.826575 0.413288 0.910601i \(-0.364380\pi\)
0.413288 + 0.910601i \(0.364380\pi\)
\(354\) 0 0
\(355\) −9.84308 −0.522417
\(356\) 1.92094 0.101809
\(357\) 0 0
\(358\) −3.49162 −0.184538
\(359\) 8.50709 0.448987 0.224493 0.974476i \(-0.427927\pi\)
0.224493 + 0.974476i \(0.427927\pi\)
\(360\) 0 0
\(361\) −11.0456 −0.581345
\(362\) −3.63635 −0.191122
\(363\) 0 0
\(364\) 0 0
\(365\) 4.50850 0.235985
\(366\) 0 0
\(367\) −5.19084 −0.270960 −0.135480 0.990780i \(-0.543258\pi\)
−0.135480 + 0.990780i \(0.543258\pi\)
\(368\) 21.5069 1.12112
\(369\) 0 0
\(370\) 0.232614 0.0120930
\(371\) 0 0
\(372\) 0 0
\(373\) −10.1427 −0.525169 −0.262585 0.964909i \(-0.584575\pi\)
−0.262585 + 0.964909i \(0.584575\pi\)
\(374\) 7.03692 0.363870
\(375\) 0 0
\(376\) 2.50553 0.129213
\(377\) 1.04188 0.0536595
\(378\) 0 0
\(379\) −3.63670 −0.186805 −0.0934024 0.995628i \(-0.529774\pi\)
−0.0934024 + 0.995628i \(0.529774\pi\)
\(380\) −7.81186 −0.400740
\(381\) 0 0
\(382\) −4.32154 −0.221109
\(383\) 4.60281 0.235192 0.117596 0.993061i \(-0.462481\pi\)
0.117596 + 0.993061i \(0.462481\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 3.71888 0.189286
\(387\) 0 0
\(388\) 26.0721 1.32361
\(389\) −19.6104 −0.994286 −0.497143 0.867669i \(-0.665618\pi\)
−0.497143 + 0.867669i \(0.665618\pi\)
\(390\) 0 0
\(391\) 28.9822 1.46569
\(392\) 0 0
\(393\) 0 0
\(394\) −0.388731 −0.0195840
\(395\) 25.2189 1.26890
\(396\) 0 0
\(397\) 19.8635 0.996919 0.498459 0.866913i \(-0.333899\pi\)
0.498459 + 0.866913i \(0.333899\pi\)
\(398\) −3.53996 −0.177442
\(399\) 0 0
\(400\) −10.5403 −0.527017
\(401\) 15.1117 0.754644 0.377322 0.926082i \(-0.376845\pi\)
0.377322 + 0.926082i \(0.376845\pi\)
\(402\) 0 0
\(403\) 9.20895 0.458731
\(404\) 1.94554 0.0967943
\(405\) 0 0
\(406\) 0 0
\(407\) −3.37164 −0.167126
\(408\) 0 0
\(409\) 35.2443 1.74272 0.871360 0.490644i \(-0.163238\pi\)
0.871360 + 0.490644i \(0.163238\pi\)
\(410\) −4.03310 −0.199181
\(411\) 0 0
\(412\) −24.6563 −1.21473
\(413\) 0 0
\(414\) 0 0
\(415\) −16.4361 −0.806816
\(416\) 3.02857 0.148488
\(417\) 0 0
\(418\) −4.10842 −0.200950
\(419\) 1.50468 0.0735084 0.0367542 0.999324i \(-0.488298\pi\)
0.0367542 + 0.999324i \(0.488298\pi\)
\(420\) 0 0
\(421\) −24.5079 −1.19444 −0.597221 0.802077i \(-0.703728\pi\)
−0.597221 + 0.802077i \(0.703728\pi\)
\(422\) −0.918370 −0.0447056
\(423\) 0 0
\(424\) 1.90272 0.0924045
\(425\) −14.2039 −0.688992
\(426\) 0 0
\(427\) 0 0
\(428\) −1.32299 −0.0639490
\(429\) 0 0
\(430\) −3.20414 −0.154518
\(431\) −41.0655 −1.97805 −0.989027 0.147732i \(-0.952803\pi\)
−0.989027 + 0.147732i \(0.952803\pi\)
\(432\) 0 0
\(433\) −6.65603 −0.319869 −0.159934 0.987128i \(-0.551128\pi\)
−0.159934 + 0.987128i \(0.551128\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 5.60357 0.268362
\(437\) −16.9209 −0.809438
\(438\) 0 0
\(439\) −8.22990 −0.392792 −0.196396 0.980525i \(-0.562924\pi\)
−0.196396 + 0.980525i \(0.562924\pi\)
\(440\) 8.21596 0.391681
\(441\) 0 0
\(442\) 1.27834 0.0608043
\(443\) −17.6856 −0.840266 −0.420133 0.907463i \(-0.638017\pi\)
−0.420133 + 0.907463i \(0.638017\pi\)
\(444\) 0 0
\(445\) 1.42843 0.0677143
\(446\) −2.62329 −0.124216
\(447\) 0 0
\(448\) 0 0
\(449\) 14.5250 0.685477 0.342738 0.939431i \(-0.388646\pi\)
0.342738 + 0.939431i \(0.388646\pi\)
\(450\) 0 0
\(451\) 58.4580 2.75268
\(452\) 23.2205 1.09220
\(453\) 0 0
\(454\) −3.19475 −0.149937
\(455\) 0 0
\(456\) 0 0
\(457\) 3.78919 0.177251 0.0886255 0.996065i \(-0.471753\pi\)
0.0886255 + 0.996065i \(0.471753\pi\)
\(458\) 1.07219 0.0501002
\(459\) 0 0
\(460\) 16.6176 0.774801
\(461\) 13.1107 0.610627 0.305314 0.952252i \(-0.401239\pi\)
0.305314 + 0.952252i \(0.401239\pi\)
\(462\) 0 0
\(463\) 15.3027 0.711176 0.355588 0.934643i \(-0.384281\pi\)
0.355588 + 0.934643i \(0.384281\pi\)
\(464\) −3.73487 −0.173387
\(465\) 0 0
\(466\) 3.31974 0.153784
\(467\) 30.7738 1.42404 0.712022 0.702158i \(-0.247780\pi\)
0.712022 + 0.702158i \(0.247780\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0.914975 0.0422046
\(471\) 0 0
\(472\) −0.905729 −0.0416896
\(473\) 46.4427 2.13544
\(474\) 0 0
\(475\) 8.29280 0.380500
\(476\) 0 0
\(477\) 0 0
\(478\) 3.53178 0.161540
\(479\) −1.71740 −0.0784699 −0.0392350 0.999230i \(-0.512492\pi\)
−0.0392350 + 0.999230i \(0.512492\pi\)
\(480\) 0 0
\(481\) −0.612497 −0.0279275
\(482\) 5.39451 0.245713
\(483\) 0 0
\(484\) −37.2527 −1.69330
\(485\) 19.3875 0.880342
\(486\) 0 0
\(487\) 22.6805 1.02775 0.513877 0.857864i \(-0.328209\pi\)
0.513877 + 0.857864i \(0.328209\pi\)
\(488\) 3.46575 0.156887
\(489\) 0 0
\(490\) 0 0
\(491\) 13.1366 0.592846 0.296423 0.955057i \(-0.404206\pi\)
0.296423 + 0.955057i \(0.404206\pi\)
\(492\) 0 0
\(493\) −5.03303 −0.226676
\(494\) −0.746342 −0.0335795
\(495\) 0 0
\(496\) −33.0117 −1.48227
\(497\) 0 0
\(498\) 0 0
\(499\) 14.0395 0.628495 0.314248 0.949341i \(-0.398248\pi\)
0.314248 + 0.949341i \(0.398248\pi\)
\(500\) −21.9932 −0.983566
\(501\) 0 0
\(502\) −4.54948 −0.203053
\(503\) 0.367865 0.0164023 0.00820114 0.999966i \(-0.497389\pi\)
0.00820114 + 0.999966i \(0.497389\pi\)
\(504\) 0 0
\(505\) 1.44673 0.0643786
\(506\) 8.73957 0.388521
\(507\) 0 0
\(508\) −30.2348 −1.34145
\(509\) 41.2319 1.82757 0.913787 0.406194i \(-0.133144\pi\)
0.913787 + 0.406194i \(0.133144\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) −18.3127 −0.809315
\(513\) 0 0
\(514\) 2.02359 0.0892566
\(515\) −18.3347 −0.807925
\(516\) 0 0
\(517\) −13.2622 −0.583269
\(518\) 0 0
\(519\) 0 0
\(520\) 1.49252 0.0654515
\(521\) 1.04099 0.0456065 0.0228032 0.999740i \(-0.492741\pi\)
0.0228032 + 0.999740i \(0.492741\pi\)
\(522\) 0 0
\(523\) 20.0209 0.875451 0.437726 0.899109i \(-0.355784\pi\)
0.437726 + 0.899109i \(0.355784\pi\)
\(524\) −23.4054 −1.02247
\(525\) 0 0
\(526\) −0.0267371 −0.00116579
\(527\) −44.4859 −1.93784
\(528\) 0 0
\(529\) 12.9947 0.564989
\(530\) 0.694840 0.0301819
\(531\) 0 0
\(532\) 0 0
\(533\) 10.6196 0.459985
\(534\) 0 0
\(535\) −0.983791 −0.0425330
\(536\) −6.89234 −0.297704
\(537\) 0 0
\(538\) 2.00283 0.0863481
\(539\) 0 0
\(540\) 0 0
\(541\) −9.78749 −0.420797 −0.210399 0.977616i \(-0.567476\pi\)
−0.210399 + 0.977616i \(0.567476\pi\)
\(542\) −3.66649 −0.157489
\(543\) 0 0
\(544\) −14.6302 −0.627263
\(545\) 4.16689 0.178490
\(546\) 0 0
\(547\) −2.56174 −0.109532 −0.0547660 0.998499i \(-0.517441\pi\)
−0.0547660 + 0.998499i \(0.517441\pi\)
\(548\) 30.7589 1.31395
\(549\) 0 0
\(550\) −4.28319 −0.182636
\(551\) 2.93848 0.125183
\(552\) 0 0
\(553\) 0 0
\(554\) −0.146321 −0.00621660
\(555\) 0 0
\(556\) −12.8167 −0.543550
\(557\) 27.4442 1.16285 0.581424 0.813601i \(-0.302496\pi\)
0.581424 + 0.813601i \(0.302496\pi\)
\(558\) 0 0
\(559\) 8.43685 0.356841
\(560\) 0 0
\(561\) 0 0
\(562\) 0.303438 0.0127998
\(563\) 0.162708 0.00685734 0.00342867 0.999994i \(-0.498909\pi\)
0.00342867 + 0.999994i \(0.498909\pi\)
\(564\) 0 0
\(565\) 17.2671 0.726431
\(566\) −1.07278 −0.0450925
\(567\) 0 0
\(568\) 7.13273 0.299283
\(569\) 12.3901 0.519419 0.259709 0.965687i \(-0.416373\pi\)
0.259709 + 0.965687i \(0.416373\pi\)
\(570\) 0 0
\(571\) −21.8122 −0.912810 −0.456405 0.889772i \(-0.650863\pi\)
−0.456405 + 0.889772i \(0.650863\pi\)
\(572\) −10.6240 −0.444212
\(573\) 0 0
\(574\) 0 0
\(575\) −17.6407 −0.735669
\(576\) 0 0
\(577\) −6.06583 −0.252524 −0.126262 0.991997i \(-0.540298\pi\)
−0.126262 + 0.991997i \(0.540298\pi\)
\(578\) −1.67664 −0.0697391
\(579\) 0 0
\(580\) −2.88581 −0.119827
\(581\) 0 0
\(582\) 0 0
\(583\) −10.0714 −0.417115
\(584\) −3.26705 −0.135192
\(585\) 0 0
\(586\) −3.98658 −0.164684
\(587\) 20.5820 0.849510 0.424755 0.905308i \(-0.360360\pi\)
0.424755 + 0.905308i \(0.360360\pi\)
\(588\) 0 0
\(589\) 25.9726 1.07018
\(590\) −0.330756 −0.0136170
\(591\) 0 0
\(592\) 2.19564 0.0902404
\(593\) −24.0397 −0.987190 −0.493595 0.869692i \(-0.664318\pi\)
−0.493595 + 0.869692i \(0.664318\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −37.7314 −1.54554
\(597\) 0 0
\(598\) 1.58764 0.0649236
\(599\) −32.2523 −1.31779 −0.658896 0.752234i \(-0.728976\pi\)
−0.658896 + 0.752234i \(0.728976\pi\)
\(600\) 0 0
\(601\) −5.21454 −0.212705 −0.106353 0.994328i \(-0.533917\pi\)
−0.106353 + 0.994328i \(0.533917\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) −20.6276 −0.839325
\(605\) −27.7016 −1.12623
\(606\) 0 0
\(607\) −9.07048 −0.368160 −0.184080 0.982911i \(-0.558930\pi\)
−0.184080 + 0.982911i \(0.558930\pi\)
\(608\) 8.54165 0.346410
\(609\) 0 0
\(610\) 1.26563 0.0512437
\(611\) −2.40922 −0.0974668
\(612\) 0 0
\(613\) 20.0920 0.811507 0.405754 0.913983i \(-0.367009\pi\)
0.405754 + 0.913983i \(0.367009\pi\)
\(614\) −5.27686 −0.212957
\(615\) 0 0
\(616\) 0 0
\(617\) 12.9556 0.521572 0.260786 0.965397i \(-0.416018\pi\)
0.260786 + 0.965397i \(0.416018\pi\)
\(618\) 0 0
\(619\) 44.3644 1.78316 0.891578 0.452866i \(-0.149598\pi\)
0.891578 + 0.452866i \(0.149598\pi\)
\(620\) −25.5070 −1.02439
\(621\) 0 0
\(622\) 2.88327 0.115608
\(623\) 0 0
\(624\) 0 0
\(625\) −1.65276 −0.0661105
\(626\) −0.0137392 −0.000549127 0
\(627\) 0 0
\(628\) −29.0913 −1.16087
\(629\) 2.95880 0.117975
\(630\) 0 0
\(631\) 6.61717 0.263426 0.131713 0.991288i \(-0.457952\pi\)
0.131713 + 0.991288i \(0.457952\pi\)
\(632\) −18.2747 −0.726930
\(633\) 0 0
\(634\) 4.26075 0.169216
\(635\) −22.4830 −0.892210
\(636\) 0 0
\(637\) 0 0
\(638\) −1.51771 −0.0600866
\(639\) 0 0
\(640\) −11.1114 −0.439216
\(641\) −18.9567 −0.748744 −0.374372 0.927279i \(-0.622142\pi\)
−0.374372 + 0.927279i \(0.622142\pi\)
\(642\) 0 0
\(643\) −13.4019 −0.528517 −0.264259 0.964452i \(-0.585127\pi\)
−0.264259 + 0.964452i \(0.585127\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 3.60537 0.141851
\(647\) 42.7588 1.68102 0.840511 0.541794i \(-0.182255\pi\)
0.840511 + 0.541794i \(0.182255\pi\)
\(648\) 0 0
\(649\) 4.79416 0.188187
\(650\) −0.778091 −0.0305192
\(651\) 0 0
\(652\) 45.9593 1.79991
\(653\) −10.9852 −0.429884 −0.214942 0.976627i \(-0.568956\pi\)
−0.214942 + 0.976627i \(0.568956\pi\)
\(654\) 0 0
\(655\) −17.4046 −0.680052
\(656\) −38.0684 −1.48632
\(657\) 0 0
\(658\) 0 0
\(659\) 17.7614 0.691884 0.345942 0.938256i \(-0.387559\pi\)
0.345942 + 0.938256i \(0.387559\pi\)
\(660\) 0 0
\(661\) −8.18255 −0.318264 −0.159132 0.987257i \(-0.550870\pi\)
−0.159132 + 0.987257i \(0.550870\pi\)
\(662\) −8.14685 −0.316636
\(663\) 0 0
\(664\) 11.9103 0.462210
\(665\) 0 0
\(666\) 0 0
\(667\) −6.25082 −0.242033
\(668\) 13.7486 0.531948
\(669\) 0 0
\(670\) −2.51696 −0.0972385
\(671\) −18.3447 −0.708189
\(672\) 0 0
\(673\) 9.30129 0.358539 0.179269 0.983800i \(-0.442627\pi\)
0.179269 + 0.983800i \(0.442627\pi\)
\(674\) 0.639979 0.0246511
\(675\) 0 0
\(676\) −1.92997 −0.0742297
\(677\) −41.1552 −1.58172 −0.790862 0.611994i \(-0.790367\pi\)
−0.790862 + 0.611994i \(0.790367\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) −7.20997 −0.276490
\(681\) 0 0
\(682\) −13.4147 −0.513675
\(683\) −39.2842 −1.50317 −0.751583 0.659638i \(-0.770709\pi\)
−0.751583 + 0.659638i \(0.770709\pi\)
\(684\) 0 0
\(685\) 22.8727 0.873921
\(686\) 0 0
\(687\) 0 0
\(688\) −30.2439 −1.15304
\(689\) −1.82959 −0.0697017
\(690\) 0 0
\(691\) 3.03355 0.115402 0.0577009 0.998334i \(-0.481623\pi\)
0.0577009 + 0.998334i \(0.481623\pi\)
\(692\) −21.6865 −0.824397
\(693\) 0 0
\(694\) −0.130336 −0.00494748
\(695\) −9.53069 −0.361520
\(696\) 0 0
\(697\) −51.3002 −1.94313
\(698\) −3.16074 −0.119636
\(699\) 0 0
\(700\) 0 0
\(701\) 26.2320 0.990767 0.495384 0.868674i \(-0.335028\pi\)
0.495384 + 0.868674i \(0.335028\pi\)
\(702\) 0 0
\(703\) −1.72746 −0.0651525
\(704\) 35.0544 1.32116
\(705\) 0 0
\(706\) −4.10963 −0.154668
\(707\) 0 0
\(708\) 0 0
\(709\) −7.87770 −0.295853 −0.147927 0.988998i \(-0.547260\pi\)
−0.147927 + 0.988998i \(0.547260\pi\)
\(710\) 2.60474 0.0977542
\(711\) 0 0
\(712\) −1.03511 −0.0387922
\(713\) −55.2497 −2.06912
\(714\) 0 0
\(715\) −7.90015 −0.295449
\(716\) −25.4650 −0.951673
\(717\) 0 0
\(718\) −2.25120 −0.0840141
\(719\) 45.6656 1.70304 0.851519 0.524323i \(-0.175682\pi\)
0.851519 + 0.524323i \(0.175682\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 2.92295 0.108781
\(723\) 0 0
\(724\) −26.5206 −0.985630
\(725\) 3.06348 0.113775
\(726\) 0 0
\(727\) −37.5947 −1.39431 −0.697155 0.716921i \(-0.745551\pi\)
−0.697155 + 0.716921i \(0.745551\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) −1.19307 −0.0441574
\(731\) −40.7561 −1.50742
\(732\) 0 0
\(733\) −53.1810 −1.96429 −0.982143 0.188138i \(-0.939755\pi\)
−0.982143 + 0.188138i \(0.939755\pi\)
\(734\) 1.37363 0.0507018
\(735\) 0 0
\(736\) −18.1701 −0.669758
\(737\) 36.4822 1.34384
\(738\) 0 0
\(739\) 41.9633 1.54364 0.771822 0.635839i \(-0.219346\pi\)
0.771822 + 0.635839i \(0.219346\pi\)
\(740\) 1.69650 0.0623646
\(741\) 0 0
\(742\) 0 0
\(743\) 38.5424 1.41398 0.706991 0.707222i \(-0.250052\pi\)
0.706991 + 0.707222i \(0.250052\pi\)
\(744\) 0 0
\(745\) −28.0576 −1.02795
\(746\) 2.68403 0.0982693
\(747\) 0 0
\(748\) 51.3216 1.87650
\(749\) 0 0
\(750\) 0 0
\(751\) 36.2434 1.32254 0.661270 0.750148i \(-0.270018\pi\)
0.661270 + 0.750148i \(0.270018\pi\)
\(752\) 8.63644 0.314939
\(753\) 0 0
\(754\) −0.275709 −0.0100407
\(755\) −15.3390 −0.558242
\(756\) 0 0
\(757\) 19.4752 0.707837 0.353919 0.935276i \(-0.384849\pi\)
0.353919 + 0.935276i \(0.384849\pi\)
\(758\) 0.962368 0.0349548
\(759\) 0 0
\(760\) 4.20946 0.152693
\(761\) −51.9059 −1.88159 −0.940793 0.338981i \(-0.889918\pi\)
−0.940793 + 0.338981i \(0.889918\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) −31.5178 −1.14028
\(765\) 0 0
\(766\) −1.21803 −0.0440090
\(767\) 0.870914 0.0314469
\(768\) 0 0
\(769\) 7.31376 0.263741 0.131870 0.991267i \(-0.457902\pi\)
0.131870 + 0.991267i \(0.457902\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 27.1225 0.976161
\(773\) −14.1844 −0.510178 −0.255089 0.966918i \(-0.582105\pi\)
−0.255089 + 0.966918i \(0.582105\pi\)
\(774\) 0 0
\(775\) 27.0774 0.972649
\(776\) −14.0491 −0.504332
\(777\) 0 0
\(778\) 5.18943 0.186050
\(779\) 29.9510 1.07311
\(780\) 0 0
\(781\) −37.7546 −1.35097
\(782\) −7.66946 −0.274259
\(783\) 0 0
\(784\) 0 0
\(785\) −21.6327 −0.772104
\(786\) 0 0
\(787\) −31.2777 −1.11493 −0.557465 0.830201i \(-0.688226\pi\)
−0.557465 + 0.830201i \(0.688226\pi\)
\(788\) −2.83509 −0.100996
\(789\) 0 0
\(790\) −6.67360 −0.237436
\(791\) 0 0
\(792\) 0 0
\(793\) −3.33253 −0.118342
\(794\) −5.25640 −0.186543
\(795\) 0 0
\(796\) −25.8176 −0.915082
\(797\) 20.2422 0.717017 0.358509 0.933526i \(-0.383285\pi\)
0.358509 + 0.933526i \(0.383285\pi\)
\(798\) 0 0
\(799\) 11.6383 0.411733
\(800\) 8.90500 0.314839
\(801\) 0 0
\(802\) −3.99897 −0.141208
\(803\) 17.2930 0.610257
\(804\) 0 0
\(805\) 0 0
\(806\) −2.43693 −0.0858373
\(807\) 0 0
\(808\) −1.04836 −0.0368813
\(809\) −7.88265 −0.277139 −0.138570 0.990353i \(-0.544250\pi\)
−0.138570 + 0.990353i \(0.544250\pi\)
\(810\) 0 0
\(811\) 5.99962 0.210675 0.105338 0.994437i \(-0.466408\pi\)
0.105338 + 0.994437i \(0.466408\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0.892226 0.0312725
\(815\) 34.1760 1.19713
\(816\) 0 0
\(817\) 23.7950 0.832480
\(818\) −9.32659 −0.326096
\(819\) 0 0
\(820\) −29.4142 −1.02719
\(821\) 19.1692 0.669011 0.334505 0.942394i \(-0.391431\pi\)
0.334505 + 0.942394i \(0.391431\pi\)
\(822\) 0 0
\(823\) −30.3735 −1.05875 −0.529376 0.848387i \(-0.677574\pi\)
−0.529376 + 0.848387i \(0.677574\pi\)
\(824\) 13.2861 0.462845
\(825\) 0 0
\(826\) 0 0
\(827\) 14.6870 0.510717 0.255359 0.966846i \(-0.417807\pi\)
0.255359 + 0.966846i \(0.417807\pi\)
\(828\) 0 0
\(829\) 34.9985 1.21555 0.607774 0.794110i \(-0.292062\pi\)
0.607774 + 0.794110i \(0.292062\pi\)
\(830\) 4.34943 0.150971
\(831\) 0 0
\(832\) 6.36804 0.220772
\(833\) 0 0
\(834\) 0 0
\(835\) 10.2236 0.353803
\(836\) −29.9635 −1.03631
\(837\) 0 0
\(838\) −0.398178 −0.0137548
\(839\) 27.6333 0.954008 0.477004 0.878901i \(-0.341723\pi\)
0.477004 + 0.878901i \(0.341723\pi\)
\(840\) 0 0
\(841\) −27.9145 −0.962568
\(842\) 6.48544 0.223503
\(843\) 0 0
\(844\) −6.69785 −0.230550
\(845\) −1.43515 −0.0493708
\(846\) 0 0
\(847\) 0 0
\(848\) 6.55859 0.225223
\(849\) 0 0
\(850\) 3.75874 0.128924
\(851\) 3.67472 0.125968
\(852\) 0 0
\(853\) −32.6336 −1.11735 −0.558676 0.829386i \(-0.688690\pi\)
−0.558676 + 0.829386i \(0.688690\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0.712898 0.0243664
\(857\) −18.8742 −0.644730 −0.322365 0.946616i \(-0.604478\pi\)
−0.322365 + 0.946616i \(0.604478\pi\)
\(858\) 0 0
\(859\) −15.8242 −0.539915 −0.269957 0.962872i \(-0.587010\pi\)
−0.269957 + 0.962872i \(0.587010\pi\)
\(860\) −23.3684 −0.796857
\(861\) 0 0
\(862\) 10.8670 0.370132
\(863\) −52.3212 −1.78104 −0.890518 0.454948i \(-0.849658\pi\)
−0.890518 + 0.454948i \(0.849658\pi\)
\(864\) 0 0
\(865\) −16.1264 −0.548313
\(866\) 1.76136 0.0598536
\(867\) 0 0
\(868\) 0 0
\(869\) 96.7309 3.28137
\(870\) 0 0
\(871\) 6.62741 0.224561
\(872\) −3.01951 −0.102253
\(873\) 0 0
\(874\) 4.47773 0.151461
\(875\) 0 0
\(876\) 0 0
\(877\) −54.0162 −1.82400 −0.911999 0.410193i \(-0.865461\pi\)
−0.911999 + 0.410193i \(0.865461\pi\)
\(878\) 2.17785 0.0734989
\(879\) 0 0
\(880\) 28.3200 0.954667
\(881\) −42.0823 −1.41779 −0.708895 0.705314i \(-0.750806\pi\)
−0.708895 + 0.705314i \(0.750806\pi\)
\(882\) 0 0
\(883\) 36.5314 1.22938 0.614689 0.788769i \(-0.289281\pi\)
0.614689 + 0.788769i \(0.289281\pi\)
\(884\) 9.32316 0.313572
\(885\) 0 0
\(886\) 4.68007 0.157230
\(887\) −11.4648 −0.384950 −0.192475 0.981302i \(-0.561651\pi\)
−0.192475 + 0.981302i \(0.561651\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) −0.378001 −0.0126706
\(891\) 0 0
\(892\) −19.1322 −0.640592
\(893\) −6.79488 −0.227382
\(894\) 0 0
\(895\) −18.9361 −0.632965
\(896\) 0 0
\(897\) 0 0
\(898\) −3.84370 −0.128266
\(899\) 9.59462 0.319999
\(900\) 0 0
\(901\) 8.83823 0.294444
\(902\) −15.4695 −0.515079
\(903\) 0 0
\(904\) −12.5125 −0.416159
\(905\) −19.7211 −0.655550
\(906\) 0 0
\(907\) −5.04665 −0.167571 −0.0837856 0.996484i \(-0.526701\pi\)
−0.0837856 + 0.996484i \(0.526701\pi\)
\(908\) −23.2999 −0.773236
\(909\) 0 0
\(910\) 0 0
\(911\) −47.5236 −1.57453 −0.787263 0.616618i \(-0.788503\pi\)
−0.787263 + 0.616618i \(0.788503\pi\)
\(912\) 0 0
\(913\) −63.0431 −2.08642
\(914\) −1.00272 −0.0331671
\(915\) 0 0
\(916\) 7.81969 0.258370
\(917\) 0 0
\(918\) 0 0
\(919\) −45.6698 −1.50651 −0.753254 0.657730i \(-0.771517\pi\)
−0.753254 + 0.657730i \(0.771517\pi\)
\(920\) −8.95449 −0.295221
\(921\) 0 0
\(922\) −3.46945 −0.114260
\(923\) −6.85856 −0.225752
\(924\) 0 0
\(925\) −1.80095 −0.0592148
\(926\) −4.04950 −0.133075
\(927\) 0 0
\(928\) 3.15540 0.103581
\(929\) −44.4449 −1.45819 −0.729095 0.684412i \(-0.760059\pi\)
−0.729095 + 0.684412i \(0.760059\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 24.2115 0.793074
\(933\) 0 0
\(934\) −8.14357 −0.266466
\(935\) 38.1634 1.24808
\(936\) 0 0
\(937\) −46.9796 −1.53476 −0.767379 0.641194i \(-0.778439\pi\)
−0.767379 + 0.641194i \(0.778439\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 6.67309 0.217652
\(941\) 35.5654 1.15940 0.579699 0.814830i \(-0.303170\pi\)
0.579699 + 0.814830i \(0.303170\pi\)
\(942\) 0 0
\(943\) −63.7128 −2.07477
\(944\) −3.12200 −0.101613
\(945\) 0 0
\(946\) −12.2900 −0.399581
\(947\) 22.3592 0.726576 0.363288 0.931677i \(-0.381654\pi\)
0.363288 + 0.931677i \(0.381654\pi\)
\(948\) 0 0
\(949\) 3.14147 0.101977
\(950\) −2.19450 −0.0711989
\(951\) 0 0
\(952\) 0 0
\(953\) 46.7684 1.51498 0.757488 0.652849i \(-0.226426\pi\)
0.757488 + 0.652849i \(0.226426\pi\)
\(954\) 0 0
\(955\) −23.4371 −0.758406
\(956\) 25.7579 0.833071
\(957\) 0 0
\(958\) 0.454469 0.0146832
\(959\) 0 0
\(960\) 0 0
\(961\) 53.8048 1.73564
\(962\) 0.162083 0.00522577
\(963\) 0 0
\(964\) 39.3432 1.26716
\(965\) 20.1687 0.649253
\(966\) 0 0
\(967\) 8.22976 0.264651 0.132326 0.991206i \(-0.457756\pi\)
0.132326 + 0.991206i \(0.457756\pi\)
\(968\) 20.0738 0.645196
\(969\) 0 0
\(970\) −5.13046 −0.164729
\(971\) −23.6326 −0.758407 −0.379204 0.925313i \(-0.623802\pi\)
−0.379204 + 0.925313i \(0.623802\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) −6.00187 −0.192312
\(975\) 0 0
\(976\) 11.9462 0.382390
\(977\) 26.6428 0.852379 0.426190 0.904634i \(-0.359856\pi\)
0.426190 + 0.904634i \(0.359856\pi\)
\(978\) 0 0
\(979\) 5.47897 0.175109
\(980\) 0 0
\(981\) 0 0
\(982\) −3.47629 −0.110933
\(983\) 43.6302 1.39159 0.695793 0.718242i \(-0.255053\pi\)
0.695793 + 0.718242i \(0.255053\pi\)
\(984\) 0 0
\(985\) −2.10821 −0.0671732
\(986\) 1.33187 0.0424155
\(987\) 0 0
\(988\) −5.44322 −0.173172
\(989\) −50.6174 −1.60954
\(990\) 0 0
\(991\) −7.60816 −0.241681 −0.120841 0.992672i \(-0.538559\pi\)
−0.120841 + 0.992672i \(0.538559\pi\)
\(992\) 27.8899 0.885506
\(993\) 0 0
\(994\) 0 0
\(995\) −19.1983 −0.608628
\(996\) 0 0
\(997\) −19.4356 −0.615532 −0.307766 0.951462i \(-0.599581\pi\)
−0.307766 + 0.951462i \(0.599581\pi\)
\(998\) −3.71523 −0.117604
\(999\) 0 0
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 5733.2.a.bu.1.3 6
3.2 odd 2 637.2.a.m.1.4 6
7.6 odd 2 5733.2.a.br.1.3 6
21.2 odd 6 637.2.e.o.508.3 12
21.5 even 6 637.2.e.n.508.3 12
21.11 odd 6 637.2.e.o.79.3 12
21.17 even 6 637.2.e.n.79.3 12
21.20 even 2 637.2.a.n.1.4 yes 6
39.38 odd 2 8281.2.a.cc.1.3 6
273.272 even 2 8281.2.a.cd.1.3 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
637.2.a.m.1.4 6 3.2 odd 2
637.2.a.n.1.4 yes 6 21.20 even 2
637.2.e.n.79.3 12 21.17 even 6
637.2.e.n.508.3 12 21.5 even 6
637.2.e.o.79.3 12 21.11 odd 6
637.2.e.o.508.3 12 21.2 odd 6
5733.2.a.br.1.3 6 7.6 odd 2
5733.2.a.bu.1.3 6 1.1 even 1 trivial
8281.2.a.cc.1.3 6 39.38 odd 2
8281.2.a.cd.1.3 6 273.272 even 2