Properties

Label 1089.2.a.v.1.4
Level $1089$
Weight $2$
Character 1089.1
Self dual yes
Analytic conductor $8.696$
Analytic rank $1$
Dimension $4$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1089,2,Mod(1,1089)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1089, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("1089.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1089 = 3^{2} \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1089.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: \(8.69570878012\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: 4.4.4400.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 7x^{2} + 11 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 99)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(2.14896\) of defining polynomial
Character \(\chi\) \(=\) 1089.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+2.14896 q^{2} +2.61803 q^{4} -2.14896 q^{5} -4.23607 q^{7} +1.32813 q^{8} -4.61803 q^{10} +0.236068 q^{13} -9.10315 q^{14} -2.38197 q^{16} -6.13335 q^{17} -2.38197 q^{19} -5.62605 q^{20} +3.98439 q^{23} -0.381966 q^{25} +0.507301 q^{26} -11.0902 q^{28} +6.95418 q^{29} -4.61803 q^{31} -7.77501 q^{32} -13.1803 q^{34} +9.10315 q^{35} +1.76393 q^{37} -5.11875 q^{38} -2.85410 q^{40} -2.96979 q^{41} +2.70820 q^{43} +8.56231 q^{46} +3.47709 q^{47} +10.9443 q^{49} -0.820830 q^{50} +0.618034 q^{52} -1.83543 q^{53} -5.62605 q^{56} +14.9443 q^{58} -7.46149 q^{59} -11.3262 q^{61} -9.92398 q^{62} -11.9443 q^{64} -0.507301 q^{65} -2.85410 q^{67} -16.0573 q^{68} +19.5623 q^{70} +10.7448 q^{71} +2.47214 q^{73} +3.79062 q^{74} -6.23607 q^{76} -9.47214 q^{79} +5.11875 q^{80} -6.38197 q^{82} +8.28232 q^{83} +13.1803 q^{85} +5.81983 q^{86} +8.90937 q^{89} -1.00000 q^{91} +10.4313 q^{92} +7.47214 q^{94} +5.11875 q^{95} +11.7984 q^{97} +23.5188 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 6 q^{4} - 8 q^{7} - 14 q^{10} - 8 q^{13} - 14 q^{16} - 14 q^{19} - 6 q^{25} - 22 q^{28} - 14 q^{31} - 8 q^{34} + 16 q^{37} + 2 q^{40} - 16 q^{43} - 6 q^{46} + 8 q^{49} - 2 q^{52} + 24 q^{58} - 14 q^{61}+ \cdots - 2 q^{97}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.14896 1.51954 0.759772 0.650189i \(-0.225310\pi\)
0.759772 + 0.650189i \(0.225310\pi\)
\(3\) 0 0
\(4\) 2.61803 1.30902
\(5\) −2.14896 −0.961045 −0.480522 0.876982i \(-0.659553\pi\)
−0.480522 + 0.876982i \(0.659553\pi\)
\(6\) 0 0
\(7\) −4.23607 −1.60108 −0.800542 0.599277i \(-0.795455\pi\)
−0.800542 + 0.599277i \(0.795455\pi\)
\(8\) 1.32813 0.469565
\(9\) 0 0
\(10\) −4.61803 −1.46035
\(11\) 0 0
\(12\) 0 0
\(13\) 0.236068 0.0654735 0.0327367 0.999464i \(-0.489578\pi\)
0.0327367 + 0.999464i \(0.489578\pi\)
\(14\) −9.10315 −2.43292
\(15\) 0 0
\(16\) −2.38197 −0.595492
\(17\) −6.13335 −1.48756 −0.743778 0.668426i \(-0.766968\pi\)
−0.743778 + 0.668426i \(0.766968\pi\)
\(18\) 0 0
\(19\) −2.38197 −0.546460 −0.273230 0.961949i \(-0.588092\pi\)
−0.273230 + 0.961949i \(0.588092\pi\)
\(20\) −5.62605 −1.25802
\(21\) 0 0
\(22\) 0 0
\(23\) 3.98439 0.830803 0.415402 0.909638i \(-0.363641\pi\)
0.415402 + 0.909638i \(0.363641\pi\)
\(24\) 0 0
\(25\) −0.381966 −0.0763932
\(26\) 0.507301 0.0994899
\(27\) 0 0
\(28\) −11.0902 −2.09585
\(29\) 6.95418 1.29136 0.645680 0.763608i \(-0.276574\pi\)
0.645680 + 0.763608i \(0.276574\pi\)
\(30\) 0 0
\(31\) −4.61803 −0.829423 −0.414712 0.909953i \(-0.636118\pi\)
−0.414712 + 0.909953i \(0.636118\pi\)
\(32\) −7.77501 −1.37444
\(33\) 0 0
\(34\) −13.1803 −2.26041
\(35\) 9.10315 1.53871
\(36\) 0 0
\(37\) 1.76393 0.289989 0.144994 0.989432i \(-0.453684\pi\)
0.144994 + 0.989432i \(0.453684\pi\)
\(38\) −5.11875 −0.830371
\(39\) 0 0
\(40\) −2.85410 −0.451273
\(41\) −2.96979 −0.463803 −0.231902 0.972739i \(-0.574495\pi\)
−0.231902 + 0.972739i \(0.574495\pi\)
\(42\) 0 0
\(43\) 2.70820 0.412997 0.206499 0.978447i \(-0.433793\pi\)
0.206499 + 0.978447i \(0.433793\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 8.56231 1.26244
\(47\) 3.47709 0.507186 0.253593 0.967311i \(-0.418388\pi\)
0.253593 + 0.967311i \(0.418388\pi\)
\(48\) 0 0
\(49\) 10.9443 1.56347
\(50\) −0.820830 −0.116083
\(51\) 0 0
\(52\) 0.618034 0.0857059
\(53\) −1.83543 −0.252116 −0.126058 0.992023i \(-0.540233\pi\)
−0.126058 + 0.992023i \(0.540233\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −5.62605 −0.751813
\(57\) 0 0
\(58\) 14.9443 1.96228
\(59\) −7.46149 −0.971403 −0.485701 0.874125i \(-0.661436\pi\)
−0.485701 + 0.874125i \(0.661436\pi\)
\(60\) 0 0
\(61\) −11.3262 −1.45018 −0.725088 0.688656i \(-0.758201\pi\)
−0.725088 + 0.688656i \(0.758201\pi\)
\(62\) −9.92398 −1.26035
\(63\) 0 0
\(64\) −11.9443 −1.49303
\(65\) −0.507301 −0.0629229
\(66\) 0 0
\(67\) −2.85410 −0.348684 −0.174342 0.984685i \(-0.555780\pi\)
−0.174342 + 0.984685i \(0.555780\pi\)
\(68\) −16.0573 −1.94724
\(69\) 0 0
\(70\) 19.5623 2.33814
\(71\) 10.7448 1.27517 0.637587 0.770378i \(-0.279933\pi\)
0.637587 + 0.770378i \(0.279933\pi\)
\(72\) 0 0
\(73\) 2.47214 0.289342 0.144671 0.989480i \(-0.453788\pi\)
0.144671 + 0.989480i \(0.453788\pi\)
\(74\) 3.79062 0.440651
\(75\) 0 0
\(76\) −6.23607 −0.715326
\(77\) 0 0
\(78\) 0 0
\(79\) −9.47214 −1.06570 −0.532849 0.846210i \(-0.678879\pi\)
−0.532849 + 0.846210i \(0.678879\pi\)
\(80\) 5.11875 0.572294
\(81\) 0 0
\(82\) −6.38197 −0.704770
\(83\) 8.28232 0.909102 0.454551 0.890721i \(-0.349800\pi\)
0.454551 + 0.890721i \(0.349800\pi\)
\(84\) 0 0
\(85\) 13.1803 1.42961
\(86\) 5.81983 0.627568
\(87\) 0 0
\(88\) 0 0
\(89\) 8.90937 0.944392 0.472196 0.881494i \(-0.343462\pi\)
0.472196 + 0.881494i \(0.343462\pi\)
\(90\) 0 0
\(91\) −1.00000 −0.104828
\(92\) 10.4313 1.08754
\(93\) 0 0
\(94\) 7.47214 0.770692
\(95\) 5.11875 0.525173
\(96\) 0 0
\(97\) 11.7984 1.19794 0.598972 0.800770i \(-0.295576\pi\)
0.598972 + 0.800770i \(0.295576\pi\)
\(98\) 23.5188 2.37576
\(99\) 0 0
\(100\) −1.00000 −0.100000
\(101\) −0.313529 −0.0311973 −0.0155987 0.999878i \(-0.504965\pi\)
−0.0155987 + 0.999878i \(0.504965\pi\)
\(102\) 0 0
\(103\) −10.9443 −1.07837 −0.539186 0.842187i \(-0.681268\pi\)
−0.539186 + 0.842187i \(0.681268\pi\)
\(104\) 0.313529 0.0307441
\(105\) 0 0
\(106\) −3.94427 −0.383102
\(107\) −10.9386 −1.05747 −0.528736 0.848786i \(-0.677334\pi\)
−0.528736 + 0.848786i \(0.677334\pi\)
\(108\) 0 0
\(109\) −13.7082 −1.31301 −0.656504 0.754323i \(-0.727965\pi\)
−0.656504 + 0.754323i \(0.727965\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 10.0902 0.953431
\(113\) 17.8928 1.68321 0.841605 0.540094i \(-0.181611\pi\)
0.841605 + 0.540094i \(0.181611\pi\)
\(114\) 0 0
\(115\) −8.56231 −0.798439
\(116\) 18.2063 1.69041
\(117\) 0 0
\(118\) −16.0344 −1.47609
\(119\) 25.9813 2.38170
\(120\) 0 0
\(121\) 0 0
\(122\) −24.3396 −2.20361
\(123\) 0 0
\(124\) −12.0902 −1.08573
\(125\) 11.5656 1.03446
\(126\) 0 0
\(127\) −2.76393 −0.245259 −0.122630 0.992453i \(-0.539133\pi\)
−0.122630 + 0.992453i \(0.539133\pi\)
\(128\) −10.1177 −0.894291
\(129\) 0 0
\(130\) −1.09017 −0.0956142
\(131\) 2.46249 0.215149 0.107574 0.994197i \(-0.465692\pi\)
0.107574 + 0.994197i \(0.465692\pi\)
\(132\) 0 0
\(133\) 10.0902 0.874929
\(134\) −6.13335 −0.529841
\(135\) 0 0
\(136\) −8.14590 −0.698505
\(137\) 0.313529 0.0267866 0.0133933 0.999910i \(-0.495737\pi\)
0.0133933 + 0.999910i \(0.495737\pi\)
\(138\) 0 0
\(139\) −11.3820 −0.965406 −0.482703 0.875784i \(-0.660345\pi\)
−0.482703 + 0.875784i \(0.660345\pi\)
\(140\) 23.8323 2.01420
\(141\) 0 0
\(142\) 23.0902 1.93768
\(143\) 0 0
\(144\) 0 0
\(145\) −14.9443 −1.24105
\(146\) 5.31252 0.439668
\(147\) 0 0
\(148\) 4.61803 0.379600
\(149\) −4.61145 −0.377785 −0.188892 0.981998i \(-0.560490\pi\)
−0.188892 + 0.981998i \(0.560490\pi\)
\(150\) 0 0
\(151\) −13.2361 −1.07714 −0.538568 0.842582i \(-0.681034\pi\)
−0.538568 + 0.842582i \(0.681034\pi\)
\(152\) −3.16356 −0.256599
\(153\) 0 0
\(154\) 0 0
\(155\) 9.92398 0.797113
\(156\) 0 0
\(157\) −20.6525 −1.64825 −0.824124 0.566410i \(-0.808332\pi\)
−0.824124 + 0.566410i \(0.808332\pi\)
\(158\) −20.3553 −1.61938
\(159\) 0 0
\(160\) 16.7082 1.32090
\(161\) −16.8782 −1.33019
\(162\) 0 0
\(163\) 1.38197 0.108244 0.0541220 0.998534i \(-0.482764\pi\)
0.0541220 + 0.998534i \(0.482764\pi\)
\(164\) −7.77501 −0.607127
\(165\) 0 0
\(166\) 17.7984 1.38142
\(167\) −10.7448 −0.831458 −0.415729 0.909489i \(-0.636474\pi\)
−0.415729 + 0.909489i \(0.636474\pi\)
\(168\) 0 0
\(169\) −12.9443 −0.995713
\(170\) 28.3240 2.17235
\(171\) 0 0
\(172\) 7.09017 0.540620
\(173\) −13.4011 −1.01886 −0.509432 0.860511i \(-0.670145\pi\)
−0.509432 + 0.860511i \(0.670145\pi\)
\(174\) 0 0
\(175\) 1.61803 0.122312
\(176\) 0 0
\(177\) 0 0
\(178\) 19.1459 1.43505
\(179\) 1.95519 0.146138 0.0730689 0.997327i \(-0.476721\pi\)
0.0730689 + 0.997327i \(0.476721\pi\)
\(180\) 0 0
\(181\) 12.2361 0.909500 0.454750 0.890619i \(-0.349729\pi\)
0.454750 + 0.890619i \(0.349729\pi\)
\(182\) −2.14896 −0.159292
\(183\) 0 0
\(184\) 5.29180 0.390116
\(185\) −3.79062 −0.278692
\(186\) 0 0
\(187\) 0 0
\(188\) 9.10315 0.663915
\(189\) 0 0
\(190\) 11.0000 0.798024
\(191\) −9.92398 −0.718074 −0.359037 0.933323i \(-0.616895\pi\)
−0.359037 + 0.933323i \(0.616895\pi\)
\(192\) 0 0
\(193\) 7.38197 0.531366 0.265683 0.964061i \(-0.414403\pi\)
0.265683 + 0.964061i \(0.414403\pi\)
\(194\) 25.3542 1.82033
\(195\) 0 0
\(196\) 28.6525 2.04661
\(197\) −2.46249 −0.175445 −0.0877226 0.996145i \(-0.527959\pi\)
−0.0877226 + 0.996145i \(0.527959\pi\)
\(198\) 0 0
\(199\) 0.416408 0.0295184 0.0147592 0.999891i \(-0.495302\pi\)
0.0147592 + 0.999891i \(0.495302\pi\)
\(200\) −0.507301 −0.0358716
\(201\) 0 0
\(202\) −0.673762 −0.0474057
\(203\) −29.4584 −2.06757
\(204\) 0 0
\(205\) 6.38197 0.445736
\(206\) −23.5188 −1.63863
\(207\) 0 0
\(208\) −0.562306 −0.0389889
\(209\) 0 0
\(210\) 0 0
\(211\) −18.0344 −1.24154 −0.620771 0.783992i \(-0.713180\pi\)
−0.620771 + 0.783992i \(0.713180\pi\)
\(212\) −4.80522 −0.330024
\(213\) 0 0
\(214\) −23.5066 −1.60688
\(215\) −5.81983 −0.396909
\(216\) 0 0
\(217\) 19.5623 1.32798
\(218\) −29.4584 −1.99517
\(219\) 0 0
\(220\) 0 0
\(221\) −1.44789 −0.0973955
\(222\) 0 0
\(223\) 3.18034 0.212971 0.106486 0.994314i \(-0.466040\pi\)
0.106486 + 0.994314i \(0.466040\pi\)
\(224\) 32.9355 2.20059
\(225\) 0 0
\(226\) 38.4508 2.55771
\(227\) 10.9386 0.726019 0.363009 0.931785i \(-0.381749\pi\)
0.363009 + 0.931785i \(0.381749\pi\)
\(228\) 0 0
\(229\) −18.4721 −1.22067 −0.610337 0.792142i \(-0.708966\pi\)
−0.610337 + 0.792142i \(0.708966\pi\)
\(230\) −18.4001 −1.21326
\(231\) 0 0
\(232\) 9.23607 0.606378
\(233\) 23.0115 1.50753 0.753767 0.657142i \(-0.228235\pi\)
0.753767 + 0.657142i \(0.228235\pi\)
\(234\) 0 0
\(235\) −7.47214 −0.487428
\(236\) −19.5344 −1.27158
\(237\) 0 0
\(238\) 55.8328 3.61910
\(239\) 11.4459 0.740372 0.370186 0.928958i \(-0.379294\pi\)
0.370186 + 0.928958i \(0.379294\pi\)
\(240\) 0 0
\(241\) 14.0000 0.901819 0.450910 0.892570i \(-0.351100\pi\)
0.450910 + 0.892570i \(0.351100\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) −29.6525 −1.89831
\(245\) −23.5188 −1.50256
\(246\) 0 0
\(247\) −0.562306 −0.0357787
\(248\) −6.13335 −0.389468
\(249\) 0 0
\(250\) 24.8541 1.57191
\(251\) −22.6980 −1.43268 −0.716342 0.697749i \(-0.754185\pi\)
−0.716342 + 0.697749i \(0.754185\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) −5.93958 −0.372683
\(255\) 0 0
\(256\) 2.14590 0.134119
\(257\) −12.3865 −0.772647 −0.386323 0.922363i \(-0.626255\pi\)
−0.386323 + 0.922363i \(0.626255\pi\)
\(258\) 0 0
\(259\) −7.47214 −0.464296
\(260\) −1.32813 −0.0823672
\(261\) 0 0
\(262\) 5.29180 0.326928
\(263\) −6.44688 −0.397532 −0.198766 0.980047i \(-0.563693\pi\)
−0.198766 + 0.980047i \(0.563693\pi\)
\(264\) 0 0
\(265\) 3.94427 0.242295
\(266\) 21.6834 1.32949
\(267\) 0 0
\(268\) −7.47214 −0.456433
\(269\) 20.2355 1.23378 0.616890 0.787049i \(-0.288392\pi\)
0.616890 + 0.787049i \(0.288392\pi\)
\(270\) 0 0
\(271\) −0.965558 −0.0586535 −0.0293267 0.999570i \(-0.509336\pi\)
−0.0293267 + 0.999570i \(0.509336\pi\)
\(272\) 14.6094 0.885827
\(273\) 0 0
\(274\) 0.673762 0.0407035
\(275\) 0 0
\(276\) 0 0
\(277\) 19.3820 1.16455 0.582275 0.812992i \(-0.302163\pi\)
0.582275 + 0.812992i \(0.302163\pi\)
\(278\) −24.4594 −1.46698
\(279\) 0 0
\(280\) 12.0902 0.722526
\(281\) −16.5646 −0.988163 −0.494082 0.869416i \(-0.664496\pi\)
−0.494082 + 0.869416i \(0.664496\pi\)
\(282\) 0 0
\(283\) −10.9443 −0.650569 −0.325285 0.945616i \(-0.605460\pi\)
−0.325285 + 0.945616i \(0.605460\pi\)
\(284\) 28.1303 1.66922
\(285\) 0 0
\(286\) 0 0
\(287\) 12.5802 0.742588
\(288\) 0 0
\(289\) 20.6180 1.21283
\(290\) −32.1147 −1.88584
\(291\) 0 0
\(292\) 6.47214 0.378753
\(293\) −4.99899 −0.292044 −0.146022 0.989281i \(-0.546647\pi\)
−0.146022 + 0.989281i \(0.546647\pi\)
\(294\) 0 0
\(295\) 16.0344 0.933561
\(296\) 2.34273 0.136169
\(297\) 0 0
\(298\) −9.90983 −0.574061
\(299\) 0.940588 0.0543956
\(300\) 0 0
\(301\) −11.4721 −0.661243
\(302\) −28.4438 −1.63676
\(303\) 0 0
\(304\) 5.67376 0.325413
\(305\) 24.3396 1.39368
\(306\) 0 0
\(307\) 17.2705 0.985680 0.492840 0.870120i \(-0.335959\pi\)
0.492840 + 0.870120i \(0.335959\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 21.3262 1.21125
\(311\) −28.7573 −1.63068 −0.815339 0.578984i \(-0.803449\pi\)
−0.815339 + 0.578984i \(0.803449\pi\)
\(312\) 0 0
\(313\) 31.6525 1.78910 0.894552 0.446964i \(-0.147495\pi\)
0.894552 + 0.446964i \(0.147495\pi\)
\(314\) −44.3814 −2.50459
\(315\) 0 0
\(316\) −24.7984 −1.39502
\(317\) −0.313529 −0.0176096 −0.00880478 0.999961i \(-0.502803\pi\)
−0.00880478 + 0.999961i \(0.502803\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 25.6678 1.43487
\(321\) 0 0
\(322\) −36.2705 −2.02128
\(323\) 14.6094 0.812891
\(324\) 0 0
\(325\) −0.0901699 −0.00500173
\(326\) 2.96979 0.164482
\(327\) 0 0
\(328\) −3.94427 −0.217786
\(329\) −14.7292 −0.812047
\(330\) 0 0
\(331\) 7.29180 0.400793 0.200397 0.979715i \(-0.435777\pi\)
0.200397 + 0.979715i \(0.435777\pi\)
\(332\) 21.6834 1.19003
\(333\) 0 0
\(334\) −23.0902 −1.26344
\(335\) 6.13335 0.335101
\(336\) 0 0
\(337\) −26.6525 −1.45185 −0.725926 0.687772i \(-0.758589\pi\)
−0.725926 + 0.687772i \(0.758589\pi\)
\(338\) −27.8167 −1.51303
\(339\) 0 0
\(340\) 34.5066 1.87138
\(341\) 0 0
\(342\) 0 0
\(343\) −16.7082 −0.902158
\(344\) 3.59685 0.193929
\(345\) 0 0
\(346\) −28.7984 −1.54821
\(347\) 25.1605 1.35069 0.675343 0.737504i \(-0.263996\pi\)
0.675343 + 0.737504i \(0.263996\pi\)
\(348\) 0 0
\(349\) 9.18034 0.491412 0.245706 0.969344i \(-0.420980\pi\)
0.245706 + 0.969344i \(0.420980\pi\)
\(350\) 3.47709 0.185858
\(351\) 0 0
\(352\) 0 0
\(353\) −33.7563 −1.79667 −0.898334 0.439314i \(-0.855222\pi\)
−0.898334 + 0.439314i \(0.855222\pi\)
\(354\) 0 0
\(355\) −23.0902 −1.22550
\(356\) 23.3250 1.23622
\(357\) 0 0
\(358\) 4.20163 0.222063
\(359\) 2.02920 0.107097 0.0535486 0.998565i \(-0.482947\pi\)
0.0535486 + 0.998565i \(0.482947\pi\)
\(360\) 0 0
\(361\) −13.3262 −0.701381
\(362\) 26.2948 1.38203
\(363\) 0 0
\(364\) −2.61803 −0.137222
\(365\) −5.31252 −0.278070
\(366\) 0 0
\(367\) 23.0344 1.20239 0.601194 0.799103i \(-0.294692\pi\)
0.601194 + 0.799103i \(0.294692\pi\)
\(368\) −9.49069 −0.494736
\(369\) 0 0
\(370\) −8.14590 −0.423485
\(371\) 7.77501 0.403659
\(372\) 0 0
\(373\) −14.4164 −0.746453 −0.373227 0.927740i \(-0.621749\pi\)
−0.373227 + 0.927740i \(0.621749\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 4.61803 0.238157
\(377\) 1.64166 0.0845498
\(378\) 0 0
\(379\) 6.23607 0.320325 0.160163 0.987091i \(-0.448798\pi\)
0.160163 + 0.987091i \(0.448798\pi\)
\(380\) 13.4011 0.687460
\(381\) 0 0
\(382\) −21.3262 −1.09115
\(383\) −34.0698 −1.74089 −0.870444 0.492267i \(-0.836168\pi\)
−0.870444 + 0.492267i \(0.836168\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 15.8636 0.807434
\(387\) 0 0
\(388\) 30.8885 1.56813
\(389\) 22.3104 1.13118 0.565592 0.824685i \(-0.308648\pi\)
0.565592 + 0.824685i \(0.308648\pi\)
\(390\) 0 0
\(391\) −24.4377 −1.23587
\(392\) 14.5354 0.734150
\(393\) 0 0
\(394\) −5.29180 −0.266597
\(395\) 20.3553 1.02418
\(396\) 0 0
\(397\) 24.4164 1.22542 0.612712 0.790306i \(-0.290078\pi\)
0.612712 + 0.790306i \(0.290078\pi\)
\(398\) 0.894844 0.0448545
\(399\) 0 0
\(400\) 0.909830 0.0454915
\(401\) −19.0271 −0.950169 −0.475085 0.879940i \(-0.657583\pi\)
−0.475085 + 0.879940i \(0.657583\pi\)
\(402\) 0 0
\(403\) −1.09017 −0.0543052
\(404\) −0.820830 −0.0408378
\(405\) 0 0
\(406\) −63.3050 −3.14177
\(407\) 0 0
\(408\) 0 0
\(409\) −5.05573 −0.249990 −0.124995 0.992157i \(-0.539891\pi\)
−0.124995 + 0.992157i \(0.539891\pi\)
\(410\) 13.7146 0.677316
\(411\) 0 0
\(412\) −28.6525 −1.41161
\(413\) 31.6074 1.55530
\(414\) 0 0
\(415\) −17.7984 −0.873688
\(416\) −1.83543 −0.0899895
\(417\) 0 0
\(418\) 0 0
\(419\) −35.2782 −1.72345 −0.861727 0.507372i \(-0.830617\pi\)
−0.861727 + 0.507372i \(0.830617\pi\)
\(420\) 0 0
\(421\) −29.2148 −1.42384 −0.711921 0.702260i \(-0.752174\pi\)
−0.711921 + 0.702260i \(0.752174\pi\)
\(422\) −38.7553 −1.88658
\(423\) 0 0
\(424\) −2.43769 −0.118385
\(425\) 2.34273 0.113639
\(426\) 0 0
\(427\) 47.9787 2.32185
\(428\) −28.6376 −1.38425
\(429\) 0 0
\(430\) −12.5066 −0.603121
\(431\) −35.5918 −1.71439 −0.857197 0.514988i \(-0.827796\pi\)
−0.857197 + 0.514988i \(0.827796\pi\)
\(432\) 0 0
\(433\) −2.65248 −0.127470 −0.0637349 0.997967i \(-0.520301\pi\)
−0.0637349 + 0.997967i \(0.520301\pi\)
\(434\) 42.0386 2.01792
\(435\) 0 0
\(436\) −35.8885 −1.71875
\(437\) −9.49069 −0.454001
\(438\) 0 0
\(439\) −25.0000 −1.19318 −0.596592 0.802544i \(-0.703479\pi\)
−0.596592 + 0.802544i \(0.703479\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) −3.11146 −0.147997
\(443\) 21.8772 1.03941 0.519707 0.854344i \(-0.326041\pi\)
0.519707 + 0.854344i \(0.326041\pi\)
\(444\) 0 0
\(445\) −19.1459 −0.907603
\(446\) 6.83443 0.323619
\(447\) 0 0
\(448\) 50.5967 2.39047
\(449\) 29.4584 1.39023 0.695114 0.718900i \(-0.255354\pi\)
0.695114 + 0.718900i \(0.255354\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 46.8439 2.20335
\(453\) 0 0
\(454\) 23.5066 1.10322
\(455\) 2.14896 0.100745
\(456\) 0 0
\(457\) 35.0902 1.64145 0.820724 0.571324i \(-0.193570\pi\)
0.820724 + 0.571324i \(0.193570\pi\)
\(458\) −39.6959 −1.85487
\(459\) 0 0
\(460\) −22.4164 −1.04517
\(461\) 1.52190 0.0708821 0.0354410 0.999372i \(-0.488716\pi\)
0.0354410 + 0.999372i \(0.488716\pi\)
\(462\) 0 0
\(463\) −21.5623 −1.00209 −0.501043 0.865423i \(-0.667050\pi\)
−0.501043 + 0.865423i \(0.667050\pi\)
\(464\) −16.5646 −0.768994
\(465\) 0 0
\(466\) 49.4508 2.29077
\(467\) 21.1761 0.979912 0.489956 0.871747i \(-0.337013\pi\)
0.489956 + 0.871747i \(0.337013\pi\)
\(468\) 0 0
\(469\) 12.0902 0.558272
\(470\) −16.0573 −0.740669
\(471\) 0 0
\(472\) −9.90983 −0.456137
\(473\) 0 0
\(474\) 0 0
\(475\) 0.909830 0.0417459
\(476\) 68.0199 3.11769
\(477\) 0 0
\(478\) 24.5967 1.12503
\(479\) −36.8459 −1.68353 −0.841765 0.539844i \(-0.818483\pi\)
−0.841765 + 0.539844i \(0.818483\pi\)
\(480\) 0 0
\(481\) 0.416408 0.0189866
\(482\) 30.0855 1.37035
\(483\) 0 0
\(484\) 0 0
\(485\) −25.3542 −1.15128
\(486\) 0 0
\(487\) 18.8885 0.855922 0.427961 0.903797i \(-0.359232\pi\)
0.427961 + 0.903797i \(0.359232\pi\)
\(488\) −15.0427 −0.680952
\(489\) 0 0
\(490\) −50.5410 −2.28321
\(491\) −19.4147 −0.876172 −0.438086 0.898933i \(-0.644343\pi\)
−0.438086 + 0.898933i \(0.644343\pi\)
\(492\) 0 0
\(493\) −42.6525 −1.92097
\(494\) −1.20837 −0.0543673
\(495\) 0 0
\(496\) 11.0000 0.493915
\(497\) −45.5157 −2.04166
\(498\) 0 0
\(499\) 32.7426 1.46576 0.732881 0.680357i \(-0.238175\pi\)
0.732881 + 0.680357i \(0.238175\pi\)
\(500\) 30.2792 1.35413
\(501\) 0 0
\(502\) −48.7771 −2.17703
\(503\) 5.93958 0.264833 0.132416 0.991194i \(-0.457726\pi\)
0.132416 + 0.991194i \(0.457726\pi\)
\(504\) 0 0
\(505\) 0.673762 0.0299820
\(506\) 0 0
\(507\) 0 0
\(508\) −7.23607 −0.321049
\(509\) 16.3709 0.725626 0.362813 0.931862i \(-0.381816\pi\)
0.362813 + 0.931862i \(0.381816\pi\)
\(510\) 0 0
\(511\) −10.4721 −0.463260
\(512\) 24.8469 1.09809
\(513\) 0 0
\(514\) −26.6180 −1.17407
\(515\) 23.5188 1.03636
\(516\) 0 0
\(517\) 0 0
\(518\) −16.0573 −0.705519
\(519\) 0 0
\(520\) −0.673762 −0.0295464
\(521\) −26.8021 −1.17422 −0.587111 0.809506i \(-0.699735\pi\)
−0.587111 + 0.809506i \(0.699735\pi\)
\(522\) 0 0
\(523\) −37.4508 −1.63761 −0.818806 0.574071i \(-0.805363\pi\)
−0.818806 + 0.574071i \(0.805363\pi\)
\(524\) 6.44688 0.281633
\(525\) 0 0
\(526\) −13.8541 −0.604068
\(527\) 28.3240 1.23381
\(528\) 0 0
\(529\) −7.12461 −0.309766
\(530\) 8.47609 0.368178
\(531\) 0 0
\(532\) 26.4164 1.14530
\(533\) −0.701073 −0.0303668
\(534\) 0 0
\(535\) 23.5066 1.01628
\(536\) −3.79062 −0.163730
\(537\) 0 0
\(538\) 43.4853 1.87478
\(539\) 0 0
\(540\) 0 0
\(541\) 38.6312 1.66088 0.830442 0.557105i \(-0.188088\pi\)
0.830442 + 0.557105i \(0.188088\pi\)
\(542\) −2.07495 −0.0891266
\(543\) 0 0
\(544\) 47.6869 2.04456
\(545\) 29.4584 1.26186
\(546\) 0 0
\(547\) 20.0344 0.856611 0.428305 0.903634i \(-0.359111\pi\)
0.428305 + 0.903634i \(0.359111\pi\)
\(548\) 0.820830 0.0350641
\(549\) 0 0
\(550\) 0 0
\(551\) −16.5646 −0.705677
\(552\) 0 0
\(553\) 40.1246 1.70627
\(554\) 41.6511 1.76959
\(555\) 0 0
\(556\) −29.7984 −1.26373
\(557\) 5.43228 0.230173 0.115087 0.993355i \(-0.463285\pi\)
0.115087 + 0.993355i \(0.463285\pi\)
\(558\) 0 0
\(559\) 0.639320 0.0270404
\(560\) −21.6834 −0.916290
\(561\) 0 0
\(562\) −35.5967 −1.50156
\(563\) 15.6240 0.658475 0.329237 0.944247i \(-0.393208\pi\)
0.329237 + 0.944247i \(0.393208\pi\)
\(564\) 0 0
\(565\) −38.4508 −1.61764
\(566\) −23.5188 −0.988570
\(567\) 0 0
\(568\) 14.2705 0.598777
\(569\) 31.7271 1.33007 0.665035 0.746812i \(-0.268417\pi\)
0.665035 + 0.746812i \(0.268417\pi\)
\(570\) 0 0
\(571\) 11.9787 0.501294 0.250647 0.968079i \(-0.419357\pi\)
0.250647 + 0.968079i \(0.419357\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 27.0344 1.12840
\(575\) −1.52190 −0.0634677
\(576\) 0 0
\(577\) −1.34752 −0.0560982 −0.0280491 0.999607i \(-0.508929\pi\)
−0.0280491 + 0.999607i \(0.508929\pi\)
\(578\) 44.3074 1.84294
\(579\) 0 0
\(580\) −39.1246 −1.62456
\(581\) −35.0845 −1.45555
\(582\) 0 0
\(583\) 0 0
\(584\) 3.28332 0.135865
\(585\) 0 0
\(586\) −10.7426 −0.443775
\(587\) 36.7261 1.51585 0.757924 0.652342i \(-0.226214\pi\)
0.757924 + 0.652342i \(0.226214\pi\)
\(588\) 0 0
\(589\) 11.0000 0.453247
\(590\) 34.4574 1.41859
\(591\) 0 0
\(592\) −4.20163 −0.172686
\(593\) 27.3094 1.12146 0.560732 0.827997i \(-0.310520\pi\)
0.560732 + 0.827997i \(0.310520\pi\)
\(594\) 0 0
\(595\) −55.8328 −2.28892
\(596\) −12.0729 −0.494527
\(597\) 0 0
\(598\) 2.02129 0.0826565
\(599\) 17.1917 0.702433 0.351217 0.936294i \(-0.385768\pi\)
0.351217 + 0.936294i \(0.385768\pi\)
\(600\) 0 0
\(601\) −9.36068 −0.381830 −0.190915 0.981607i \(-0.561146\pi\)
−0.190915 + 0.981607i \(0.561146\pi\)
\(602\) −24.6532 −1.00479
\(603\) 0 0
\(604\) −34.6525 −1.40999
\(605\) 0 0
\(606\) 0 0
\(607\) −28.6180 −1.16157 −0.580785 0.814057i \(-0.697254\pi\)
−0.580785 + 0.814057i \(0.697254\pi\)
\(608\) 18.5198 0.751078
\(609\) 0 0
\(610\) 52.3050 2.11777
\(611\) 0.820830 0.0332072
\(612\) 0 0
\(613\) 25.0557 1.01199 0.505996 0.862536i \(-0.331125\pi\)
0.505996 + 0.862536i \(0.331125\pi\)
\(614\) 37.1137 1.49779
\(615\) 0 0
\(616\) 0 0
\(617\) 16.8782 0.679489 0.339745 0.940518i \(-0.389659\pi\)
0.339745 + 0.940518i \(0.389659\pi\)
\(618\) 0 0
\(619\) 6.88854 0.276874 0.138437 0.990371i \(-0.455792\pi\)
0.138437 + 0.990371i \(0.455792\pi\)
\(620\) 25.9813 1.04343
\(621\) 0 0
\(622\) −61.7984 −2.47789
\(623\) −37.7407 −1.51205
\(624\) 0 0
\(625\) −22.9443 −0.917771
\(626\) 68.0199 2.71862
\(627\) 0 0
\(628\) −54.0689 −2.15758
\(629\) −10.8188 −0.431375
\(630\) 0 0
\(631\) −17.3820 −0.691965 −0.345983 0.938241i \(-0.612454\pi\)
−0.345983 + 0.938241i \(0.612454\pi\)
\(632\) −12.5802 −0.500415
\(633\) 0 0
\(634\) −0.673762 −0.0267585
\(635\) 5.93958 0.235705
\(636\) 0 0
\(637\) 2.58359 0.102366
\(638\) 0 0
\(639\) 0 0
\(640\) 21.7426 0.859454
\(641\) −21.3699 −0.844058 −0.422029 0.906582i \(-0.638682\pi\)
−0.422029 + 0.906582i \(0.638682\pi\)
\(642\) 0 0
\(643\) −41.1591 −1.62315 −0.811577 0.584245i \(-0.801391\pi\)
−0.811577 + 0.584245i \(0.801391\pi\)
\(644\) −44.1876 −1.74124
\(645\) 0 0
\(646\) 31.3951 1.23522
\(647\) −11.6854 −0.459400 −0.229700 0.973261i \(-0.573775\pi\)
−0.229700 + 0.973261i \(0.573775\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) −0.193772 −0.00760035
\(651\) 0 0
\(652\) 3.61803 0.141693
\(653\) −34.1896 −1.33794 −0.668971 0.743288i \(-0.733265\pi\)
−0.668971 + 0.743288i \(0.733265\pi\)
\(654\) 0 0
\(655\) −5.29180 −0.206768
\(656\) 7.07394 0.276191
\(657\) 0 0
\(658\) −31.6525 −1.23394
\(659\) −33.7563 −1.31496 −0.657480 0.753472i \(-0.728377\pi\)
−0.657480 + 0.753472i \(0.728377\pi\)
\(660\) 0 0
\(661\) 2.43769 0.0948153 0.0474077 0.998876i \(-0.484904\pi\)
0.0474077 + 0.998876i \(0.484904\pi\)
\(662\) 15.6698 0.609024
\(663\) 0 0
\(664\) 11.0000 0.426883
\(665\) −21.6834 −0.840846
\(666\) 0 0
\(667\) 27.7082 1.07287
\(668\) −28.1303 −1.08839
\(669\) 0 0
\(670\) 13.1803 0.509201
\(671\) 0 0
\(672\) 0 0
\(673\) 18.2361 0.702949 0.351474 0.936198i \(-0.385680\pi\)
0.351474 + 0.936198i \(0.385680\pi\)
\(674\) −57.2751 −2.20616
\(675\) 0 0
\(676\) −33.8885 −1.30341
\(677\) −8.59584 −0.330365 −0.165183 0.986263i \(-0.552821\pi\)
−0.165183 + 0.986263i \(0.552821\pi\)
\(678\) 0 0
\(679\) −49.9787 −1.91801
\(680\) 17.5052 0.671294
\(681\) 0 0
\(682\) 0 0
\(683\) −0.940588 −0.0359906 −0.0179953 0.999838i \(-0.505728\pi\)
−0.0179953 + 0.999838i \(0.505728\pi\)
\(684\) 0 0
\(685\) −0.673762 −0.0257431
\(686\) −35.9053 −1.37087
\(687\) 0 0
\(688\) −6.45085 −0.245936
\(689\) −0.433287 −0.0165069
\(690\) 0 0
\(691\) 32.3607 1.23106 0.615529 0.788114i \(-0.288942\pi\)
0.615529 + 0.788114i \(0.288942\pi\)
\(692\) −35.0845 −1.33371
\(693\) 0 0
\(694\) 54.0689 2.05243
\(695\) 24.4594 0.927798
\(696\) 0 0
\(697\) 18.2148 0.689934
\(698\) 19.7282 0.746723
\(699\) 0 0
\(700\) 4.23607 0.160108
\(701\) 23.3991 0.883770 0.441885 0.897072i \(-0.354310\pi\)
0.441885 + 0.897072i \(0.354310\pi\)
\(702\) 0 0
\(703\) −4.20163 −0.158467
\(704\) 0 0
\(705\) 0 0
\(706\) −72.5410 −2.73012
\(707\) 1.32813 0.0499495
\(708\) 0 0
\(709\) −15.2016 −0.570909 −0.285455 0.958392i \(-0.592145\pi\)
−0.285455 + 0.958392i \(0.592145\pi\)
\(710\) −49.6199 −1.86220
\(711\) 0 0
\(712\) 11.8328 0.443454
\(713\) −18.4001 −0.689088
\(714\) 0 0
\(715\) 0 0
\(716\) 5.11875 0.191297
\(717\) 0 0
\(718\) 4.36068 0.162739
\(719\) −12.9678 −0.483617 −0.241808 0.970324i \(-0.577741\pi\)
−0.241808 + 0.970324i \(0.577741\pi\)
\(720\) 0 0
\(721\) 46.3607 1.72656
\(722\) −28.6376 −1.06578
\(723\) 0 0
\(724\) 32.0344 1.19055
\(725\) −2.65626 −0.0986511
\(726\) 0 0
\(727\) 19.5623 0.725526 0.362763 0.931881i \(-0.381833\pi\)
0.362763 + 0.931881i \(0.381833\pi\)
\(728\) −1.32813 −0.0492238
\(729\) 0 0
\(730\) −11.4164 −0.422540
\(731\) −16.6104 −0.614357
\(732\) 0 0
\(733\) −14.6525 −0.541202 −0.270601 0.962692i \(-0.587222\pi\)
−0.270601 + 0.962692i \(0.587222\pi\)
\(734\) 49.5001 1.82708
\(735\) 0 0
\(736\) −30.9787 −1.14189
\(737\) 0 0
\(738\) 0 0
\(739\) 18.2361 0.670825 0.335412 0.942071i \(-0.391124\pi\)
0.335412 + 0.942071i \(0.391124\pi\)
\(740\) −9.92398 −0.364813
\(741\) 0 0
\(742\) 16.7082 0.613377
\(743\) 18.1323 0.665209 0.332604 0.943066i \(-0.392073\pi\)
0.332604 + 0.943066i \(0.392073\pi\)
\(744\) 0 0
\(745\) 9.90983 0.363068
\(746\) −30.9803 −1.13427
\(747\) 0 0
\(748\) 0 0
\(749\) 46.3366 1.69310
\(750\) 0 0
\(751\) −1.50658 −0.0549758 −0.0274879 0.999622i \(-0.508751\pi\)
−0.0274879 + 0.999622i \(0.508751\pi\)
\(752\) −8.28232 −0.302025
\(753\) 0 0
\(754\) 3.52786 0.128477
\(755\) 28.4438 1.03518
\(756\) 0 0
\(757\) −40.8885 −1.48612 −0.743060 0.669225i \(-0.766626\pi\)
−0.743060 + 0.669225i \(0.766626\pi\)
\(758\) 13.4011 0.486749
\(759\) 0 0
\(760\) 6.79837 0.246603
\(761\) 33.4428 1.21230 0.606150 0.795350i \(-0.292713\pi\)
0.606150 + 0.795350i \(0.292713\pi\)
\(762\) 0 0
\(763\) 58.0689 2.10223
\(764\) −25.9813 −0.939971
\(765\) 0 0
\(766\) −73.2148 −2.64536
\(767\) −1.76142 −0.0636011
\(768\) 0 0
\(769\) −22.2705 −0.803095 −0.401548 0.915838i \(-0.631528\pi\)
−0.401548 + 0.915838i \(0.631528\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 19.3262 0.695567
\(773\) −33.6823 −1.21147 −0.605734 0.795667i \(-0.707121\pi\)
−0.605734 + 0.795667i \(0.707121\pi\)
\(774\) 0 0
\(775\) 1.76393 0.0633623
\(776\) 15.6698 0.562513
\(777\) 0 0
\(778\) 47.9443 1.71889
\(779\) 7.07394 0.253450
\(780\) 0 0
\(781\) 0 0
\(782\) −52.5157 −1.87796
\(783\) 0 0
\(784\) −26.0689 −0.931032
\(785\) 44.3814 1.58404
\(786\) 0 0
\(787\) 27.7771 0.990146 0.495073 0.868851i \(-0.335141\pi\)
0.495073 + 0.868851i \(0.335141\pi\)
\(788\) −6.44688 −0.229661
\(789\) 0 0
\(790\) 43.7426 1.55629
\(791\) −75.7950 −2.69496
\(792\) 0 0
\(793\) −2.67376 −0.0949481
\(794\) 52.4699 1.86209
\(795\) 0 0
\(796\) 1.09017 0.0386400
\(797\) 4.29792 0.152240 0.0761201 0.997099i \(-0.475747\pi\)
0.0761201 + 0.997099i \(0.475747\pi\)
\(798\) 0 0
\(799\) −21.3262 −0.754468
\(800\) 2.96979 0.104998
\(801\) 0 0
\(802\) −40.8885 −1.44382
\(803\) 0 0
\(804\) 0 0
\(805\) 36.2705 1.27837
\(806\) −2.34273 −0.0825192
\(807\) 0 0
\(808\) −0.416408 −0.0146492
\(809\) −7.07394 −0.248707 −0.124353 0.992238i \(-0.539686\pi\)
−0.124353 + 0.992238i \(0.539686\pi\)
\(810\) 0 0
\(811\) −44.2148 −1.55259 −0.776295 0.630369i \(-0.782904\pi\)
−0.776295 + 0.630369i \(0.782904\pi\)
\(812\) −77.1231 −2.70649
\(813\) 0 0
\(814\) 0 0
\(815\) −2.96979 −0.104027
\(816\) 0 0
\(817\) −6.45085 −0.225687
\(818\) −10.8646 −0.379871
\(819\) 0 0
\(820\) 16.7082 0.583476
\(821\) 39.7699 1.38798 0.693990 0.719985i \(-0.255851\pi\)
0.693990 + 0.719985i \(0.255851\pi\)
\(822\) 0 0
\(823\) 19.1115 0.666183 0.333092 0.942894i \(-0.391908\pi\)
0.333092 + 0.942894i \(0.391908\pi\)
\(824\) −14.5354 −0.506366
\(825\) 0 0
\(826\) 67.9230 2.36334
\(827\) 11.6397 0.404750 0.202375 0.979308i \(-0.435134\pi\)
0.202375 + 0.979308i \(0.435134\pi\)
\(828\) 0 0
\(829\) 20.0344 0.695825 0.347912 0.937527i \(-0.386891\pi\)
0.347912 + 0.937527i \(0.386891\pi\)
\(830\) −38.2480 −1.32761
\(831\) 0 0
\(832\) −2.81966 −0.0977541
\(833\) −67.1251 −2.32575
\(834\) 0 0
\(835\) 23.0902 0.799068
\(836\) 0 0
\(837\) 0 0
\(838\) −75.8115 −2.61887
\(839\) −13.9824 −0.482725 −0.241363 0.970435i \(-0.577594\pi\)
−0.241363 + 0.970435i \(0.577594\pi\)
\(840\) 0 0
\(841\) 19.3607 0.667610
\(842\) −62.7814 −2.16359
\(843\) 0 0
\(844\) −47.2148 −1.62520
\(845\) 27.8167 0.956925
\(846\) 0 0
\(847\) 0 0
\(848\) 4.37194 0.150133
\(849\) 0 0
\(850\) 5.03444 0.172680
\(851\) 7.02820 0.240924
\(852\) 0 0
\(853\) −1.18034 −0.0404141 −0.0202070 0.999796i \(-0.506433\pi\)
−0.0202070 + 0.999796i \(0.506433\pi\)
\(854\) 103.104 3.52816
\(855\) 0 0
\(856\) −14.5279 −0.496552
\(857\) 13.8344 0.472573 0.236286 0.971683i \(-0.424070\pi\)
0.236286 + 0.971683i \(0.424070\pi\)
\(858\) 0 0
\(859\) 42.4164 1.44723 0.723615 0.690204i \(-0.242479\pi\)
0.723615 + 0.690204i \(0.242479\pi\)
\(860\) −15.2365 −0.519560
\(861\) 0 0
\(862\) −76.4853 −2.60510
\(863\) −34.3834 −1.17042 −0.585212 0.810880i \(-0.698989\pi\)
−0.585212 + 0.810880i \(0.698989\pi\)
\(864\) 0 0
\(865\) 28.7984 0.979174
\(866\) −5.70007 −0.193696
\(867\) 0 0
\(868\) 51.2148 1.73834
\(869\) 0 0
\(870\) 0 0
\(871\) −0.673762 −0.0228296
\(872\) −18.2063 −0.616543
\(873\) 0 0
\(874\) −20.3951 −0.689875
\(875\) −48.9928 −1.65626
\(876\) 0 0
\(877\) 11.4721 0.387387 0.193693 0.981062i \(-0.437953\pi\)
0.193693 + 0.981062i \(0.437953\pi\)
\(878\) −53.7240 −1.81310
\(879\) 0 0
\(880\) 0 0
\(881\) 34.3376 1.15686 0.578432 0.815730i \(-0.303665\pi\)
0.578432 + 0.815730i \(0.303665\pi\)
\(882\) 0 0
\(883\) −14.1115 −0.474888 −0.237444 0.971401i \(-0.576310\pi\)
−0.237444 + 0.971401i \(0.576310\pi\)
\(884\) −3.79062 −0.127492
\(885\) 0 0
\(886\) 47.0132 1.57944
\(887\) −56.5741 −1.89957 −0.949786 0.312902i \(-0.898699\pi\)
−0.949786 + 0.312902i \(0.898699\pi\)
\(888\) 0 0
\(889\) 11.7082 0.392681
\(890\) −41.1438 −1.37914
\(891\) 0 0
\(892\) 8.32624 0.278783
\(893\) −8.28232 −0.277157
\(894\) 0 0
\(895\) −4.20163 −0.140445
\(896\) 42.8595 1.43183
\(897\) 0 0
\(898\) 63.3050 2.11251
\(899\) −32.1147 −1.07108
\(900\) 0 0
\(901\) 11.2574 0.375037
\(902\) 0 0
\(903\) 0 0
\(904\) 23.7639 0.790377
\(905\) −26.2948 −0.874070
\(906\) 0 0
\(907\) −35.8673 −1.19095 −0.595476 0.803373i \(-0.703037\pi\)
−0.595476 + 0.803373i \(0.703037\pi\)
\(908\) 28.6376 0.950371
\(909\) 0 0
\(910\) 4.61803 0.153086
\(911\) 9.53643 0.315956 0.157978 0.987443i \(-0.449502\pi\)
0.157978 + 0.987443i \(0.449502\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 75.4074 2.49426
\(915\) 0 0
\(916\) −48.3607 −1.59788
\(917\) −10.4313 −0.344471
\(918\) 0 0
\(919\) −25.1803 −0.830623 −0.415311 0.909679i \(-0.636327\pi\)
−0.415311 + 0.909679i \(0.636327\pi\)
\(920\) −11.3719 −0.374919
\(921\) 0 0
\(922\) 3.27051 0.107709
\(923\) 2.53650 0.0834901
\(924\) 0 0
\(925\) −0.673762 −0.0221532
\(926\) −46.3366 −1.52271
\(927\) 0 0
\(928\) −54.0689 −1.77490
\(929\) 42.0386 1.37924 0.689621 0.724170i \(-0.257777\pi\)
0.689621 + 0.724170i \(0.257777\pi\)
\(930\) 0 0
\(931\) −26.0689 −0.854373
\(932\) 60.2449 1.97339
\(933\) 0 0
\(934\) 45.5066 1.48902
\(935\) 0 0
\(936\) 0 0
\(937\) 10.8197 0.353463 0.176731 0.984259i \(-0.443448\pi\)
0.176731 + 0.984259i \(0.443448\pi\)
\(938\) 25.9813 0.848320
\(939\) 0 0
\(940\) −19.5623 −0.638052
\(941\) −9.17716 −0.299167 −0.149583 0.988749i \(-0.547793\pi\)
−0.149583 + 0.988749i \(0.547793\pi\)
\(942\) 0 0
\(943\) −11.8328 −0.385329
\(944\) 17.7730 0.578462
\(945\) 0 0
\(946\) 0 0
\(947\) −22.3845 −0.727397 −0.363699 0.931517i \(-0.618486\pi\)
−0.363699 + 0.931517i \(0.618486\pi\)
\(948\) 0 0
\(949\) 0.583592 0.0189442
\(950\) 1.95519 0.0634347
\(951\) 0 0
\(952\) 34.5066 1.11836
\(953\) 16.8041 0.544340 0.272170 0.962249i \(-0.412259\pi\)
0.272170 + 0.962249i \(0.412259\pi\)
\(954\) 0 0
\(955\) 21.3262 0.690101
\(956\) 29.9657 0.969160
\(957\) 0 0
\(958\) −79.1803 −2.55820
\(959\) −1.32813 −0.0428876
\(960\) 0 0
\(961\) −9.67376 −0.312057
\(962\) 0.894844 0.0288509
\(963\) 0 0
\(964\) 36.6525 1.18050
\(965\) −15.8636 −0.510666
\(966\) 0 0
\(967\) −17.6869 −0.568773 −0.284386 0.958710i \(-0.591790\pi\)
−0.284386 + 0.958710i \(0.591790\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) −54.4853 −1.74942
\(971\) 1.01460 0.0325601 0.0162801 0.999867i \(-0.494818\pi\)
0.0162801 + 0.999867i \(0.494818\pi\)
\(972\) 0 0
\(973\) 48.2148 1.54569
\(974\) 40.5907 1.30061
\(975\) 0 0
\(976\) 26.9787 0.863568
\(977\) −3.35733 −0.107411 −0.0537053 0.998557i \(-0.517103\pi\)
−0.0537053 + 0.998557i \(0.517103\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) −61.5731 −1.96688
\(981\) 0 0
\(982\) −41.7214 −1.33138
\(983\) −50.7085 −1.61735 −0.808675 0.588256i \(-0.799815\pi\)
−0.808675 + 0.588256i \(0.799815\pi\)
\(984\) 0 0
\(985\) 5.29180 0.168611
\(986\) −91.6585 −2.91900
\(987\) 0 0
\(988\) −1.47214 −0.0468349
\(989\) 10.7905 0.343119
\(990\) 0 0
\(991\) −58.2705 −1.85102 −0.925512 0.378719i \(-0.876365\pi\)
−0.925512 + 0.378719i \(0.876365\pi\)
\(992\) 35.9053 1.13999
\(993\) 0 0
\(994\) −97.8115 −3.10239
\(995\) −0.894844 −0.0283685
\(996\) 0 0
\(997\) 38.7426 1.22699 0.613496 0.789698i \(-0.289763\pi\)
0.613496 + 0.789698i \(0.289763\pi\)
\(998\) 70.3627 2.22729
\(999\) 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 1089.2.a.v.1.4 4
3.2 odd 2 inner 1089.2.a.v.1.1 4
11.5 even 5 99.2.f.c.91.1 yes 8
11.9 even 5 99.2.f.c.37.1 8
11.10 odd 2 1089.2.a.w.1.1 4
33.5 odd 10 99.2.f.c.91.2 yes 8
33.20 odd 10 99.2.f.c.37.2 yes 8
33.32 even 2 1089.2.a.w.1.4 4
99.5 odd 30 891.2.n.e.784.2 16
99.16 even 15 891.2.n.e.190.2 16
99.20 odd 30 891.2.n.e.433.2 16
99.31 even 15 891.2.n.e.136.2 16
99.38 odd 30 891.2.n.e.190.1 16
99.49 even 15 891.2.n.e.784.1 16
99.86 odd 30 891.2.n.e.136.1 16
99.97 even 15 891.2.n.e.433.1 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
99.2.f.c.37.1 8 11.9 even 5
99.2.f.c.37.2 yes 8 33.20 odd 10
99.2.f.c.91.1 yes 8 11.5 even 5
99.2.f.c.91.2 yes 8 33.5 odd 10
891.2.n.e.136.1 16 99.86 odd 30
891.2.n.e.136.2 16 99.31 even 15
891.2.n.e.190.1 16 99.38 odd 30
891.2.n.e.190.2 16 99.16 even 15
891.2.n.e.433.1 16 99.97 even 15
891.2.n.e.433.2 16 99.20 odd 30
891.2.n.e.784.1 16 99.49 even 15
891.2.n.e.784.2 16 99.5 odd 30
1089.2.a.v.1.1 4 3.2 odd 2 inner
1089.2.a.v.1.4 4 1.1 even 1 trivial
1089.2.a.w.1.1 4 11.10 odd 2
1089.2.a.w.1.4 4 33.32 even 2