Properties

Label 6021.2.a.l.1.3
Level $6021$
Weight $2$
Character 6021.1
Self dual yes
Analytic conductor $48.078$
Analytic rank $0$
Dimension $10$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [6021,2,Mod(1,6021)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6021, 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("6021.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6021 = 3^{3} \cdot 223 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6021.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: \(48.0779270570\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - 20x^{8} + 139x^{6} - 384x^{4} + 331x^{2} - 63 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-2.31657\) of defining polynomial
Character \(\chi\) \(=\) 6021.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.31657 q^{2} +3.36649 q^{4} -1.78959 q^{5} +0.694189 q^{7} -3.16557 q^{8} +O(q^{10})\) \(q-2.31657 q^{2} +3.36649 q^{4} -1.78959 q^{5} +0.694189 q^{7} -3.16557 q^{8} +4.14570 q^{10} -5.07293 q^{11} +4.67900 q^{13} -1.60814 q^{14} +0.600282 q^{16} +0.374011 q^{17} +3.67230 q^{19} -6.02463 q^{20} +11.7518 q^{22} +9.24219 q^{23} -1.79738 q^{25} -10.8392 q^{26} +2.33698 q^{28} -9.13260 q^{29} +0.920410 q^{31} +4.94055 q^{32} -0.866423 q^{34} -1.24231 q^{35} +4.16978 q^{37} -8.50714 q^{38} +5.66507 q^{40} +5.53413 q^{41} +1.34832 q^{43} -17.0780 q^{44} -21.4102 q^{46} +11.4169 q^{47} -6.51810 q^{49} +4.16375 q^{50} +15.7518 q^{52} +2.04525 q^{53} +9.07845 q^{55} -2.19751 q^{56} +21.1563 q^{58} -12.3267 q^{59} +3.40231 q^{61} -2.13219 q^{62} -12.6457 q^{64} -8.37347 q^{65} -1.22157 q^{67} +1.25911 q^{68} +2.87790 q^{70} -12.3816 q^{71} -5.34539 q^{73} -9.65958 q^{74} +12.3628 q^{76} -3.52157 q^{77} -3.37909 q^{79} -1.07426 q^{80} -12.8202 q^{82} +11.4450 q^{83} -0.669325 q^{85} -3.12348 q^{86} +16.0587 q^{88} +3.41759 q^{89} +3.24811 q^{91} +31.1138 q^{92} -26.4480 q^{94} -6.57190 q^{95} -9.83863 q^{97} +15.0996 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q + 20 q^{4} + 2 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 10 q + 20 q^{4} + 2 q^{7} - 10 q^{10} + 2 q^{13} + 44 q^{16} + 28 q^{19} - 42 q^{22} + 22 q^{25} + 40 q^{28} - 18 q^{31} + 36 q^{34} + 20 q^{37} - 4 q^{40} + 2 q^{43} - 30 q^{46} - 32 q^{49} - 2 q^{52} + 52 q^{55} + 84 q^{58} + 40 q^{61} + 64 q^{64} + 18 q^{67} + 18 q^{70} + 32 q^{73} + 104 q^{76} - 16 q^{79} - 94 q^{82} - 40 q^{85} - 32 q^{88} + 14 q^{91} - 56 q^{94} - 18 q^{97}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.31657 −1.63806 −0.819031 0.573750i \(-0.805488\pi\)
−0.819031 + 0.573750i \(0.805488\pi\)
\(3\) 0 0
\(4\) 3.36649 1.68325
\(5\) −1.78959 −0.800327 −0.400164 0.916444i \(-0.631047\pi\)
−0.400164 + 0.916444i \(0.631047\pi\)
\(6\) 0 0
\(7\) 0.694189 0.262379 0.131189 0.991357i \(-0.458120\pi\)
0.131189 + 0.991357i \(0.458120\pi\)
\(8\) −3.16557 −1.11920
\(9\) 0 0
\(10\) 4.14570 1.31099
\(11\) −5.07293 −1.52955 −0.764773 0.644299i \(-0.777149\pi\)
−0.764773 + 0.644299i \(0.777149\pi\)
\(12\) 0 0
\(13\) 4.67900 1.29772 0.648860 0.760908i \(-0.275246\pi\)
0.648860 + 0.760908i \(0.275246\pi\)
\(14\) −1.60814 −0.429793
\(15\) 0 0
\(16\) 0.600282 0.150071
\(17\) 0.374011 0.0907110 0.0453555 0.998971i \(-0.485558\pi\)
0.0453555 + 0.998971i \(0.485558\pi\)
\(18\) 0 0
\(19\) 3.67230 0.842484 0.421242 0.906948i \(-0.361594\pi\)
0.421242 + 0.906948i \(0.361594\pi\)
\(20\) −6.02463 −1.34715
\(21\) 0 0
\(22\) 11.7518 2.50549
\(23\) 9.24219 1.92713 0.963565 0.267474i \(-0.0861891\pi\)
0.963565 + 0.267474i \(0.0861891\pi\)
\(24\) 0 0
\(25\) −1.79738 −0.359476
\(26\) −10.8392 −2.12575
\(27\) 0 0
\(28\) 2.33698 0.441648
\(29\) −9.13260 −1.69588 −0.847941 0.530091i \(-0.822158\pi\)
−0.847941 + 0.530091i \(0.822158\pi\)
\(30\) 0 0
\(31\) 0.920410 0.165311 0.0826553 0.996578i \(-0.473660\pi\)
0.0826553 + 0.996578i \(0.473660\pi\)
\(32\) 4.94055 0.873374
\(33\) 0 0
\(34\) −0.866423 −0.148590
\(35\) −1.24231 −0.209989
\(36\) 0 0
\(37\) 4.16978 0.685508 0.342754 0.939425i \(-0.388640\pi\)
0.342754 + 0.939425i \(0.388640\pi\)
\(38\) −8.50714 −1.38004
\(39\) 0 0
\(40\) 5.66507 0.895725
\(41\) 5.53413 0.864286 0.432143 0.901805i \(-0.357758\pi\)
0.432143 + 0.901805i \(0.357758\pi\)
\(42\) 0 0
\(43\) 1.34832 0.205617 0.102809 0.994701i \(-0.467217\pi\)
0.102809 + 0.994701i \(0.467217\pi\)
\(44\) −17.0780 −2.57460
\(45\) 0 0
\(46\) −21.4102 −3.15676
\(47\) 11.4169 1.66533 0.832663 0.553780i \(-0.186815\pi\)
0.832663 + 0.553780i \(0.186815\pi\)
\(48\) 0 0
\(49\) −6.51810 −0.931157
\(50\) 4.16375 0.588844
\(51\) 0 0
\(52\) 15.7518 2.18438
\(53\) 2.04525 0.280937 0.140468 0.990085i \(-0.455139\pi\)
0.140468 + 0.990085i \(0.455139\pi\)
\(54\) 0 0
\(55\) 9.07845 1.22414
\(56\) −2.19751 −0.293654
\(57\) 0 0
\(58\) 21.1563 2.77796
\(59\) −12.3267 −1.60479 −0.802397 0.596791i \(-0.796442\pi\)
−0.802397 + 0.596791i \(0.796442\pi\)
\(60\) 0 0
\(61\) 3.40231 0.435621 0.217810 0.975991i \(-0.430109\pi\)
0.217810 + 0.975991i \(0.430109\pi\)
\(62\) −2.13219 −0.270789
\(63\) 0 0
\(64\) −12.6457 −1.58071
\(65\) −8.37347 −1.03860
\(66\) 0 0
\(67\) −1.22157 −0.149239 −0.0746196 0.997212i \(-0.523774\pi\)
−0.0746196 + 0.997212i \(0.523774\pi\)
\(68\) 1.25911 0.152689
\(69\) 0 0
\(70\) 2.87790 0.343975
\(71\) −12.3816 −1.46942 −0.734711 0.678380i \(-0.762682\pi\)
−0.734711 + 0.678380i \(0.762682\pi\)
\(72\) 0 0
\(73\) −5.34539 −0.625630 −0.312815 0.949814i \(-0.601272\pi\)
−0.312815 + 0.949814i \(0.601272\pi\)
\(74\) −9.65958 −1.12290
\(75\) 0 0
\(76\) 12.3628 1.41811
\(77\) −3.52157 −0.401321
\(78\) 0 0
\(79\) −3.37909 −0.380178 −0.190089 0.981767i \(-0.560878\pi\)
−0.190089 + 0.981767i \(0.560878\pi\)
\(80\) −1.07426 −0.120106
\(81\) 0 0
\(82\) −12.8202 −1.41575
\(83\) 11.4450 1.25625 0.628124 0.778114i \(-0.283823\pi\)
0.628124 + 0.778114i \(0.283823\pi\)
\(84\) 0 0
\(85\) −0.669325 −0.0725985
\(86\) −3.12348 −0.336813
\(87\) 0 0
\(88\) 16.0587 1.71187
\(89\) 3.41759 0.362264 0.181132 0.983459i \(-0.442024\pi\)
0.181132 + 0.983459i \(0.442024\pi\)
\(90\) 0 0
\(91\) 3.24811 0.340494
\(92\) 31.1138 3.24383
\(93\) 0 0
\(94\) −26.4480 −2.72791
\(95\) −6.57190 −0.674263
\(96\) 0 0
\(97\) −9.83863 −0.998961 −0.499481 0.866325i \(-0.666476\pi\)
−0.499481 + 0.866325i \(0.666476\pi\)
\(98\) 15.0996 1.52529
\(99\) 0 0
\(100\) −6.05086 −0.605086
\(101\) 0.412606 0.0410558 0.0205279 0.999789i \(-0.493465\pi\)
0.0205279 + 0.999789i \(0.493465\pi\)
\(102\) 0 0
\(103\) 6.70719 0.660879 0.330439 0.943827i \(-0.392803\pi\)
0.330439 + 0.943827i \(0.392803\pi\)
\(104\) −14.8117 −1.45241
\(105\) 0 0
\(106\) −4.73796 −0.460192
\(107\) −2.67071 −0.258187 −0.129094 0.991632i \(-0.541207\pi\)
−0.129094 + 0.991632i \(0.541207\pi\)
\(108\) 0 0
\(109\) 7.77178 0.744401 0.372201 0.928152i \(-0.378603\pi\)
0.372201 + 0.928152i \(0.378603\pi\)
\(110\) −21.0309 −2.00521
\(111\) 0 0
\(112\) 0.416710 0.0393753
\(113\) 16.3605 1.53907 0.769534 0.638605i \(-0.220488\pi\)
0.769534 + 0.638605i \(0.220488\pi\)
\(114\) 0 0
\(115\) −16.5397 −1.54234
\(116\) −30.7448 −2.85459
\(117\) 0 0
\(118\) 28.5555 2.62875
\(119\) 0.259634 0.0238006
\(120\) 0 0
\(121\) 14.7347 1.33951
\(122\) −7.88168 −0.713574
\(123\) 0 0
\(124\) 3.09855 0.278258
\(125\) 12.1645 1.08803
\(126\) 0 0
\(127\) −9.45416 −0.838921 −0.419460 0.907774i \(-0.637781\pi\)
−0.419460 + 0.907774i \(0.637781\pi\)
\(128\) 19.4135 1.71593
\(129\) 0 0
\(130\) 19.3977 1.70129
\(131\) −14.9905 −1.30972 −0.654862 0.755749i \(-0.727273\pi\)
−0.654862 + 0.755749i \(0.727273\pi\)
\(132\) 0 0
\(133\) 2.54927 0.221050
\(134\) 2.82986 0.244463
\(135\) 0 0
\(136\) −1.18396 −0.101524
\(137\) −9.22476 −0.788124 −0.394062 0.919084i \(-0.628930\pi\)
−0.394062 + 0.919084i \(0.628930\pi\)
\(138\) 0 0
\(139\) −4.70631 −0.399184 −0.199592 0.979879i \(-0.563962\pi\)
−0.199592 + 0.979879i \(0.563962\pi\)
\(140\) −4.18223 −0.353463
\(141\) 0 0
\(142\) 28.6828 2.40700
\(143\) −23.7362 −1.98492
\(144\) 0 0
\(145\) 16.3436 1.35726
\(146\) 12.3830 1.02482
\(147\) 0 0
\(148\) 14.0375 1.15388
\(149\) −0.872272 −0.0714593 −0.0357296 0.999361i \(-0.511376\pi\)
−0.0357296 + 0.999361i \(0.511376\pi\)
\(150\) 0 0
\(151\) 0.663895 0.0540270 0.0270135 0.999635i \(-0.491400\pi\)
0.0270135 + 0.999635i \(0.491400\pi\)
\(152\) −11.6249 −0.942907
\(153\) 0 0
\(154\) 8.15797 0.657388
\(155\) −1.64715 −0.132303
\(156\) 0 0
\(157\) 9.60531 0.766588 0.383294 0.923626i \(-0.374790\pi\)
0.383294 + 0.923626i \(0.374790\pi\)
\(158\) 7.82790 0.622755
\(159\) 0 0
\(160\) −8.84154 −0.698985
\(161\) 6.41583 0.505638
\(162\) 0 0
\(163\) −11.6592 −0.913216 −0.456608 0.889668i \(-0.650936\pi\)
−0.456608 + 0.889668i \(0.650936\pi\)
\(164\) 18.6306 1.45481
\(165\) 0 0
\(166\) −26.5130 −2.05781
\(167\) −15.0364 −1.16355 −0.581775 0.813350i \(-0.697642\pi\)
−0.581775 + 0.813350i \(0.697642\pi\)
\(168\) 0 0
\(169\) 8.89300 0.684077
\(170\) 1.55054 0.118921
\(171\) 0 0
\(172\) 4.53911 0.346104
\(173\) 11.7444 0.892912 0.446456 0.894806i \(-0.352686\pi\)
0.446456 + 0.894806i \(0.352686\pi\)
\(174\) 0 0
\(175\) −1.24772 −0.0943189
\(176\) −3.04519 −0.229540
\(177\) 0 0
\(178\) −7.91709 −0.593411
\(179\) 9.06429 0.677497 0.338748 0.940877i \(-0.389996\pi\)
0.338748 + 0.940877i \(0.389996\pi\)
\(180\) 0 0
\(181\) 12.0468 0.895433 0.447716 0.894176i \(-0.352237\pi\)
0.447716 + 0.894176i \(0.352237\pi\)
\(182\) −7.52447 −0.557750
\(183\) 0 0
\(184\) −29.2568 −2.15684
\(185\) −7.46218 −0.548631
\(186\) 0 0
\(187\) −1.89733 −0.138747
\(188\) 38.4349 2.80315
\(189\) 0 0
\(190\) 15.2243 1.10448
\(191\) −8.99893 −0.651140 −0.325570 0.945518i \(-0.605556\pi\)
−0.325570 + 0.945518i \(0.605556\pi\)
\(192\) 0 0
\(193\) 1.51054 0.108731 0.0543656 0.998521i \(-0.482686\pi\)
0.0543656 + 0.998521i \(0.482686\pi\)
\(194\) 22.7919 1.63636
\(195\) 0 0
\(196\) −21.9431 −1.56737
\(197\) 22.6144 1.61121 0.805604 0.592455i \(-0.201841\pi\)
0.805604 + 0.592455i \(0.201841\pi\)
\(198\) 0 0
\(199\) 14.5611 1.03221 0.516103 0.856526i \(-0.327382\pi\)
0.516103 + 0.856526i \(0.327382\pi\)
\(200\) 5.68973 0.402325
\(201\) 0 0
\(202\) −0.955829 −0.0672519
\(203\) −6.33975 −0.444963
\(204\) 0 0
\(205\) −9.90381 −0.691712
\(206\) −15.5377 −1.08256
\(207\) 0 0
\(208\) 2.80872 0.194750
\(209\) −18.6293 −1.28862
\(210\) 0 0
\(211\) 22.0830 1.52025 0.760127 0.649775i \(-0.225137\pi\)
0.760127 + 0.649775i \(0.225137\pi\)
\(212\) 6.88532 0.472885
\(213\) 0 0
\(214\) 6.18688 0.422927
\(215\) −2.41294 −0.164561
\(216\) 0 0
\(217\) 0.638939 0.0433740
\(218\) −18.0039 −1.21938
\(219\) 0 0
\(220\) 30.5625 2.06053
\(221\) 1.75000 0.117718
\(222\) 0 0
\(223\) −1.00000 −0.0669650
\(224\) 3.42967 0.229155
\(225\) 0 0
\(226\) −37.9003 −2.52109
\(227\) 17.6360 1.17054 0.585270 0.810839i \(-0.300989\pi\)
0.585270 + 0.810839i \(0.300989\pi\)
\(228\) 0 0
\(229\) −8.09312 −0.534809 −0.267404 0.963584i \(-0.586166\pi\)
−0.267404 + 0.963584i \(0.586166\pi\)
\(230\) 38.3154 2.52644
\(231\) 0 0
\(232\) 28.9099 1.89803
\(233\) −11.4724 −0.751584 −0.375792 0.926704i \(-0.622629\pi\)
−0.375792 + 0.926704i \(0.622629\pi\)
\(234\) 0 0
\(235\) −20.4315 −1.33281
\(236\) −41.4976 −2.70126
\(237\) 0 0
\(238\) −0.601461 −0.0389869
\(239\) −15.0997 −0.976721 −0.488360 0.872642i \(-0.662405\pi\)
−0.488360 + 0.872642i \(0.662405\pi\)
\(240\) 0 0
\(241\) 3.00743 0.193726 0.0968629 0.995298i \(-0.469119\pi\)
0.0968629 + 0.995298i \(0.469119\pi\)
\(242\) −34.1338 −2.19421
\(243\) 0 0
\(244\) 11.4538 0.733257
\(245\) 11.6647 0.745231
\(246\) 0 0
\(247\) 17.1827 1.09331
\(248\) −2.91362 −0.185015
\(249\) 0 0
\(250\) −28.1799 −1.78225
\(251\) −12.9343 −0.816407 −0.408204 0.912891i \(-0.633845\pi\)
−0.408204 + 0.912891i \(0.633845\pi\)
\(252\) 0 0
\(253\) −46.8850 −2.94764
\(254\) 21.9012 1.37420
\(255\) 0 0
\(256\) −19.6814 −1.23008
\(257\) 12.9167 0.805724 0.402862 0.915261i \(-0.368015\pi\)
0.402862 + 0.915261i \(0.368015\pi\)
\(258\) 0 0
\(259\) 2.89462 0.179863
\(260\) −28.1892 −1.74822
\(261\) 0 0
\(262\) 34.7264 2.14541
\(263\) −14.7541 −0.909777 −0.454888 0.890548i \(-0.650321\pi\)
−0.454888 + 0.890548i \(0.650321\pi\)
\(264\) 0 0
\(265\) −3.66015 −0.224841
\(266\) −5.90556 −0.362093
\(267\) 0 0
\(268\) −4.11242 −0.251206
\(269\) −5.78037 −0.352435 −0.176218 0.984351i \(-0.556386\pi\)
−0.176218 + 0.984351i \(0.556386\pi\)
\(270\) 0 0
\(271\) −5.11744 −0.310862 −0.155431 0.987847i \(-0.549677\pi\)
−0.155431 + 0.987847i \(0.549677\pi\)
\(272\) 0.224512 0.0136131
\(273\) 0 0
\(274\) 21.3698 1.29100
\(275\) 9.11799 0.549835
\(276\) 0 0
\(277\) 19.9271 1.19730 0.598652 0.801009i \(-0.295703\pi\)
0.598652 + 0.801009i \(0.295703\pi\)
\(278\) 10.9025 0.653888
\(279\) 0 0
\(280\) 3.93263 0.235019
\(281\) −16.6653 −0.994166 −0.497083 0.867703i \(-0.665595\pi\)
−0.497083 + 0.867703i \(0.665595\pi\)
\(282\) 0 0
\(283\) −28.2883 −1.68157 −0.840783 0.541372i \(-0.817905\pi\)
−0.840783 + 0.541372i \(0.817905\pi\)
\(284\) −41.6824 −2.47340
\(285\) 0 0
\(286\) 54.9866 3.25143
\(287\) 3.84173 0.226770
\(288\) 0 0
\(289\) −16.8601 −0.991772
\(290\) −37.8610 −2.22328
\(291\) 0 0
\(292\) −17.9952 −1.05309
\(293\) −18.9266 −1.10570 −0.552852 0.833279i \(-0.686461\pi\)
−0.552852 + 0.833279i \(0.686461\pi\)
\(294\) 0 0
\(295\) 22.0596 1.28436
\(296\) −13.1997 −0.767219
\(297\) 0 0
\(298\) 2.02068 0.117055
\(299\) 43.2442 2.50087
\(300\) 0 0
\(301\) 0.935990 0.0539495
\(302\) −1.53796 −0.0884996
\(303\) 0 0
\(304\) 2.20442 0.126432
\(305\) −6.08873 −0.348639
\(306\) 0 0
\(307\) 12.6545 0.722232 0.361116 0.932521i \(-0.382396\pi\)
0.361116 + 0.932521i \(0.382396\pi\)
\(308\) −11.8554 −0.675521
\(309\) 0 0
\(310\) 3.81574 0.216720
\(311\) 32.7411 1.85658 0.928288 0.371861i \(-0.121280\pi\)
0.928288 + 0.371861i \(0.121280\pi\)
\(312\) 0 0
\(313\) −20.7729 −1.17415 −0.587076 0.809532i \(-0.699721\pi\)
−0.587076 + 0.809532i \(0.699721\pi\)
\(314\) −22.2514 −1.25572
\(315\) 0 0
\(316\) −11.3757 −0.639933
\(317\) 26.1965 1.47134 0.735670 0.677340i \(-0.236867\pi\)
0.735670 + 0.677340i \(0.236867\pi\)
\(318\) 0 0
\(319\) 46.3291 2.59393
\(320\) 22.6305 1.26509
\(321\) 0 0
\(322\) −14.8627 −0.828266
\(323\) 1.37348 0.0764226
\(324\) 0 0
\(325\) −8.40993 −0.466499
\(326\) 27.0092 1.49590
\(327\) 0 0
\(328\) −17.5187 −0.967308
\(329\) 7.92549 0.436946
\(330\) 0 0
\(331\) 24.7778 1.36191 0.680955 0.732325i \(-0.261565\pi\)
0.680955 + 0.732325i \(0.261565\pi\)
\(332\) 38.5293 2.11457
\(333\) 0 0
\(334\) 34.8328 1.90597
\(335\) 2.18611 0.119440
\(336\) 0 0
\(337\) −5.55094 −0.302379 −0.151190 0.988505i \(-0.548310\pi\)
−0.151190 + 0.988505i \(0.548310\pi\)
\(338\) −20.6012 −1.12056
\(339\) 0 0
\(340\) −2.25328 −0.122201
\(341\) −4.66918 −0.252850
\(342\) 0 0
\(343\) −9.38412 −0.506695
\(344\) −4.26821 −0.230126
\(345\) 0 0
\(346\) −27.2068 −1.46264
\(347\) 32.4295 1.74090 0.870452 0.492252i \(-0.163826\pi\)
0.870452 + 0.492252i \(0.163826\pi\)
\(348\) 0 0
\(349\) 3.02169 0.161748 0.0808738 0.996724i \(-0.474229\pi\)
0.0808738 + 0.996724i \(0.474229\pi\)
\(350\) 2.89043 0.154500
\(351\) 0 0
\(352\) −25.0631 −1.33587
\(353\) −7.25508 −0.386149 −0.193074 0.981184i \(-0.561846\pi\)
−0.193074 + 0.981184i \(0.561846\pi\)
\(354\) 0 0
\(355\) 22.1579 1.17602
\(356\) 11.5053 0.609780
\(357\) 0 0
\(358\) −20.9981 −1.10978
\(359\) 3.29124 0.173705 0.0868525 0.996221i \(-0.472319\pi\)
0.0868525 + 0.996221i \(0.472319\pi\)
\(360\) 0 0
\(361\) −5.51419 −0.290221
\(362\) −27.9073 −1.46677
\(363\) 0 0
\(364\) 10.9347 0.573135
\(365\) 9.56604 0.500709
\(366\) 0 0
\(367\) 3.67488 0.191827 0.0959136 0.995390i \(-0.469423\pi\)
0.0959136 + 0.995390i \(0.469423\pi\)
\(368\) 5.54793 0.289206
\(369\) 0 0
\(370\) 17.2867 0.898691
\(371\) 1.41979 0.0737118
\(372\) 0 0
\(373\) 19.1309 0.990560 0.495280 0.868733i \(-0.335065\pi\)
0.495280 + 0.868733i \(0.335065\pi\)
\(374\) 4.39530 0.227276
\(375\) 0 0
\(376\) −36.1410 −1.86383
\(377\) −42.7314 −2.20078
\(378\) 0 0
\(379\) 9.62044 0.494169 0.247084 0.968994i \(-0.420528\pi\)
0.247084 + 0.968994i \(0.420528\pi\)
\(380\) −22.1243 −1.13495
\(381\) 0 0
\(382\) 20.8466 1.06661
\(383\) 5.99657 0.306411 0.153205 0.988194i \(-0.451040\pi\)
0.153205 + 0.988194i \(0.451040\pi\)
\(384\) 0 0
\(385\) 6.30216 0.321188
\(386\) −3.49928 −0.178109
\(387\) 0 0
\(388\) −33.1217 −1.68150
\(389\) 25.4826 1.29202 0.646009 0.763330i \(-0.276437\pi\)
0.646009 + 0.763330i \(0.276437\pi\)
\(390\) 0 0
\(391\) 3.45668 0.174812
\(392\) 20.6335 1.04215
\(393\) 0 0
\(394\) −52.3878 −2.63926
\(395\) 6.04718 0.304267
\(396\) 0 0
\(397\) 24.8521 1.24729 0.623645 0.781708i \(-0.285651\pi\)
0.623645 + 0.781708i \(0.285651\pi\)
\(398\) −33.7317 −1.69082
\(399\) 0 0
\(400\) −1.07894 −0.0539468
\(401\) 15.9426 0.796135 0.398067 0.917356i \(-0.369681\pi\)
0.398067 + 0.917356i \(0.369681\pi\)
\(402\) 0 0
\(403\) 4.30659 0.214527
\(404\) 1.38903 0.0691070
\(405\) 0 0
\(406\) 14.6865 0.728877
\(407\) −21.1530 −1.04852
\(408\) 0 0
\(409\) −16.1882 −0.800457 −0.400229 0.916415i \(-0.631069\pi\)
−0.400229 + 0.916415i \(0.631069\pi\)
\(410\) 22.9429 1.13307
\(411\) 0 0
\(412\) 22.5797 1.11242
\(413\) −8.55703 −0.421064
\(414\) 0 0
\(415\) −20.4817 −1.00541
\(416\) 23.1168 1.13339
\(417\) 0 0
\(418\) 43.1562 2.11084
\(419\) −21.4969 −1.05019 −0.525096 0.851043i \(-0.675970\pi\)
−0.525096 + 0.851043i \(0.675970\pi\)
\(420\) 0 0
\(421\) −32.1847 −1.56858 −0.784292 0.620392i \(-0.786974\pi\)
−0.784292 + 0.620392i \(0.786974\pi\)
\(422\) −51.1567 −2.49027
\(423\) 0 0
\(424\) −6.47439 −0.314424
\(425\) −0.672240 −0.0326084
\(426\) 0 0
\(427\) 2.36185 0.114298
\(428\) −8.99092 −0.434593
\(429\) 0 0
\(430\) 5.58974 0.269561
\(431\) 17.9042 0.862413 0.431207 0.902253i \(-0.358088\pi\)
0.431207 + 0.902253i \(0.358088\pi\)
\(432\) 0 0
\(433\) −31.9371 −1.53480 −0.767399 0.641170i \(-0.778450\pi\)
−0.767399 + 0.641170i \(0.778450\pi\)
\(434\) −1.48015 −0.0710492
\(435\) 0 0
\(436\) 26.1636 1.25301
\(437\) 33.9401 1.62358
\(438\) 0 0
\(439\) 4.81237 0.229682 0.114841 0.993384i \(-0.463364\pi\)
0.114841 + 0.993384i \(0.463364\pi\)
\(440\) −28.7385 −1.37005
\(441\) 0 0
\(442\) −4.05399 −0.192829
\(443\) 14.0761 0.668775 0.334387 0.942436i \(-0.391471\pi\)
0.334387 + 0.942436i \(0.391471\pi\)
\(444\) 0 0
\(445\) −6.11608 −0.289930
\(446\) 2.31657 0.109693
\(447\) 0 0
\(448\) −8.77850 −0.414745
\(449\) −7.73708 −0.365135 −0.182568 0.983193i \(-0.558441\pi\)
−0.182568 + 0.983193i \(0.558441\pi\)
\(450\) 0 0
\(451\) −28.0743 −1.32197
\(452\) 55.0776 2.59063
\(453\) 0 0
\(454\) −40.8549 −1.91742
\(455\) −5.81277 −0.272507
\(456\) 0 0
\(457\) 2.42234 0.113312 0.0566562 0.998394i \(-0.481956\pi\)
0.0566562 + 0.998394i \(0.481956\pi\)
\(458\) 18.7483 0.876049
\(459\) 0 0
\(460\) −55.6808 −2.59613
\(461\) 3.14473 0.146465 0.0732324 0.997315i \(-0.476669\pi\)
0.0732324 + 0.997315i \(0.476669\pi\)
\(462\) 0 0
\(463\) 14.6304 0.679931 0.339966 0.940438i \(-0.389585\pi\)
0.339966 + 0.940438i \(0.389585\pi\)
\(464\) −5.48214 −0.254502
\(465\) 0 0
\(466\) 26.5767 1.23114
\(467\) −37.7571 −1.74719 −0.873594 0.486655i \(-0.838217\pi\)
−0.873594 + 0.486655i \(0.838217\pi\)
\(468\) 0 0
\(469\) −0.848004 −0.0391572
\(470\) 47.3311 2.18322
\(471\) 0 0
\(472\) 39.0209 1.79608
\(473\) −6.83994 −0.314501
\(474\) 0 0
\(475\) −6.60052 −0.302853
\(476\) 0.874057 0.0400623
\(477\) 0 0
\(478\) 34.9796 1.59993
\(479\) −11.2029 −0.511875 −0.255938 0.966693i \(-0.582384\pi\)
−0.255938 + 0.966693i \(0.582384\pi\)
\(480\) 0 0
\(481\) 19.5104 0.889597
\(482\) −6.96692 −0.317335
\(483\) 0 0
\(484\) 49.6041 2.25473
\(485\) 17.6071 0.799496
\(486\) 0 0
\(487\) 34.8101 1.57740 0.788698 0.614781i \(-0.210756\pi\)
0.788698 + 0.614781i \(0.210756\pi\)
\(488\) −10.7703 −0.487546
\(489\) 0 0
\(490\) −27.0221 −1.22073
\(491\) 31.4656 1.42002 0.710012 0.704190i \(-0.248690\pi\)
0.710012 + 0.704190i \(0.248690\pi\)
\(492\) 0 0
\(493\) −3.41569 −0.153835
\(494\) −39.8049 −1.79091
\(495\) 0 0
\(496\) 0.552506 0.0248083
\(497\) −8.59515 −0.385545
\(498\) 0 0
\(499\) −3.05596 −0.136804 −0.0684018 0.997658i \(-0.521790\pi\)
−0.0684018 + 0.997658i \(0.521790\pi\)
\(500\) 40.9517 1.83142
\(501\) 0 0
\(502\) 29.9632 1.33733
\(503\) 40.4444 1.80333 0.901664 0.432437i \(-0.142346\pi\)
0.901664 + 0.432437i \(0.142346\pi\)
\(504\) 0 0
\(505\) −0.738393 −0.0328581
\(506\) 108.612 4.82841
\(507\) 0 0
\(508\) −31.8273 −1.41211
\(509\) 41.1229 1.82274 0.911370 0.411589i \(-0.135026\pi\)
0.911370 + 0.411589i \(0.135026\pi\)
\(510\) 0 0
\(511\) −3.71071 −0.164152
\(512\) 6.76620 0.299027
\(513\) 0 0
\(514\) −29.9225 −1.31983
\(515\) −12.0031 −0.528920
\(516\) 0 0
\(517\) −57.9172 −2.54719
\(518\) −6.70558 −0.294626
\(519\) 0 0
\(520\) 26.5068 1.16240
\(521\) −20.0545 −0.878603 −0.439302 0.898340i \(-0.644774\pi\)
−0.439302 + 0.898340i \(0.644774\pi\)
\(522\) 0 0
\(523\) −4.05018 −0.177102 −0.0885510 0.996072i \(-0.528224\pi\)
−0.0885510 + 0.996072i \(0.528224\pi\)
\(524\) −50.4653 −2.20459
\(525\) 0 0
\(526\) 34.1789 1.49027
\(527\) 0.344244 0.0149955
\(528\) 0 0
\(529\) 62.4181 2.71383
\(530\) 8.47899 0.368304
\(531\) 0 0
\(532\) 8.58210 0.372081
\(533\) 25.8942 1.12160
\(534\) 0 0
\(535\) 4.77947 0.206634
\(536\) 3.86698 0.167028
\(537\) 0 0
\(538\) 13.3906 0.577311
\(539\) 33.0659 1.42425
\(540\) 0 0
\(541\) 44.7203 1.92268 0.961338 0.275372i \(-0.0888011\pi\)
0.961338 + 0.275372i \(0.0888011\pi\)
\(542\) 11.8549 0.509212
\(543\) 0 0
\(544\) 1.84782 0.0792246
\(545\) −13.9083 −0.595765
\(546\) 0 0
\(547\) −27.1122 −1.15923 −0.579617 0.814889i \(-0.696798\pi\)
−0.579617 + 0.814889i \(0.696798\pi\)
\(548\) −31.0551 −1.32661
\(549\) 0 0
\(550\) −21.1224 −0.900664
\(551\) −33.5377 −1.42875
\(552\) 0 0
\(553\) −2.34573 −0.0997506
\(554\) −46.1625 −1.96126
\(555\) 0 0
\(556\) −15.8438 −0.671925
\(557\) 39.8298 1.68764 0.843822 0.536624i \(-0.180301\pi\)
0.843822 + 0.536624i \(0.180301\pi\)
\(558\) 0 0
\(559\) 6.30879 0.266833
\(560\) −0.745738 −0.0315132
\(561\) 0 0
\(562\) 38.6062 1.62850
\(563\) 20.2614 0.853918 0.426959 0.904271i \(-0.359585\pi\)
0.426959 + 0.904271i \(0.359585\pi\)
\(564\) 0 0
\(565\) −29.2786 −1.23176
\(566\) 65.5319 2.75451
\(567\) 0 0
\(568\) 39.1947 1.64458
\(569\) −12.2600 −0.513966 −0.256983 0.966416i \(-0.582728\pi\)
−0.256983 + 0.966416i \(0.582728\pi\)
\(570\) 0 0
\(571\) −38.3009 −1.60284 −0.801422 0.598100i \(-0.795923\pi\)
−0.801422 + 0.598100i \(0.795923\pi\)
\(572\) −79.9078 −3.34111
\(573\) 0 0
\(574\) −8.89964 −0.371464
\(575\) −16.6117 −0.692757
\(576\) 0 0
\(577\) −7.10968 −0.295980 −0.147990 0.988989i \(-0.547280\pi\)
−0.147990 + 0.988989i \(0.547280\pi\)
\(578\) 39.0576 1.62458
\(579\) 0 0
\(580\) 55.0205 2.28460
\(581\) 7.94496 0.329613
\(582\) 0 0
\(583\) −10.3754 −0.429706
\(584\) 16.9212 0.700205
\(585\) 0 0
\(586\) 43.8448 1.81121
\(587\) 35.2035 1.45300 0.726502 0.687165i \(-0.241145\pi\)
0.726502 + 0.687165i \(0.241145\pi\)
\(588\) 0 0
\(589\) 3.38002 0.139271
\(590\) −51.1026 −2.10386
\(591\) 0 0
\(592\) 2.50305 0.102875
\(593\) −40.5533 −1.66533 −0.832663 0.553780i \(-0.813185\pi\)
−0.832663 + 0.553780i \(0.813185\pi\)
\(594\) 0 0
\(595\) −0.464638 −0.0190483
\(596\) −2.93649 −0.120284
\(597\) 0 0
\(598\) −100.178 −4.09659
\(599\) 30.7944 1.25822 0.629112 0.777314i \(-0.283419\pi\)
0.629112 + 0.777314i \(0.283419\pi\)
\(600\) 0 0
\(601\) −23.1844 −0.945711 −0.472855 0.881140i \(-0.656777\pi\)
−0.472855 + 0.881140i \(0.656777\pi\)
\(602\) −2.16828 −0.0883727
\(603\) 0 0
\(604\) 2.23500 0.0909408
\(605\) −26.3689 −1.07205
\(606\) 0 0
\(607\) −9.78635 −0.397216 −0.198608 0.980079i \(-0.563642\pi\)
−0.198608 + 0.980079i \(0.563642\pi\)
\(608\) 18.1432 0.735803
\(609\) 0 0
\(610\) 14.1050 0.571093
\(611\) 53.4196 2.16113
\(612\) 0 0
\(613\) −8.39778 −0.339183 −0.169592 0.985514i \(-0.554245\pi\)
−0.169592 + 0.985514i \(0.554245\pi\)
\(614\) −29.3151 −1.18306
\(615\) 0 0
\(616\) 11.1478 0.449158
\(617\) 35.2055 1.41732 0.708660 0.705550i \(-0.249300\pi\)
0.708660 + 0.705550i \(0.249300\pi\)
\(618\) 0 0
\(619\) −25.5728 −1.02786 −0.513928 0.857833i \(-0.671810\pi\)
−0.513928 + 0.857833i \(0.671810\pi\)
\(620\) −5.54513 −0.222698
\(621\) 0 0
\(622\) −75.8470 −3.04119
\(623\) 2.37246 0.0950505
\(624\) 0 0
\(625\) −12.7825 −0.511301
\(626\) 48.1218 1.92333
\(627\) 0 0
\(628\) 32.3362 1.29036
\(629\) 1.55954 0.0621831
\(630\) 0 0
\(631\) 46.2488 1.84114 0.920568 0.390583i \(-0.127726\pi\)
0.920568 + 0.390583i \(0.127726\pi\)
\(632\) 10.6968 0.425494
\(633\) 0 0
\(634\) −60.6859 −2.41015
\(635\) 16.9190 0.671411
\(636\) 0 0
\(637\) −30.4982 −1.20838
\(638\) −107.325 −4.24902
\(639\) 0 0
\(640\) −34.7421 −1.37330
\(641\) 5.89743 0.232934 0.116467 0.993195i \(-0.462843\pi\)
0.116467 + 0.993195i \(0.462843\pi\)
\(642\) 0 0
\(643\) −0.507148 −0.0200000 −0.00999999 0.999950i \(-0.503183\pi\)
−0.00999999 + 0.999950i \(0.503183\pi\)
\(644\) 21.5988 0.851113
\(645\) 0 0
\(646\) −3.18177 −0.125185
\(647\) 46.6012 1.83208 0.916041 0.401085i \(-0.131367\pi\)
0.916041 + 0.401085i \(0.131367\pi\)
\(648\) 0 0
\(649\) 62.5323 2.45461
\(650\) 19.4822 0.764154
\(651\) 0 0
\(652\) −39.2505 −1.53717
\(653\) 28.1513 1.10165 0.550823 0.834622i \(-0.314314\pi\)
0.550823 + 0.834622i \(0.314314\pi\)
\(654\) 0 0
\(655\) 26.8267 1.04821
\(656\) 3.32204 0.129704
\(657\) 0 0
\(658\) −18.3599 −0.715745
\(659\) 15.5857 0.607134 0.303567 0.952810i \(-0.401822\pi\)
0.303567 + 0.952810i \(0.401822\pi\)
\(660\) 0 0
\(661\) −2.81367 −0.109439 −0.0547195 0.998502i \(-0.517426\pi\)
−0.0547195 + 0.998502i \(0.517426\pi\)
\(662\) −57.3995 −2.23089
\(663\) 0 0
\(664\) −36.2298 −1.40599
\(665\) −4.56214 −0.176912
\(666\) 0 0
\(667\) −84.4052 −3.26818
\(668\) −50.6198 −1.95854
\(669\) 0 0
\(670\) −5.06428 −0.195650
\(671\) −17.2597 −0.666303
\(672\) 0 0
\(673\) 16.0542 0.618842 0.309421 0.950925i \(-0.399865\pi\)
0.309421 + 0.950925i \(0.399865\pi\)
\(674\) 12.8591 0.495315
\(675\) 0 0
\(676\) 29.9382 1.15147
\(677\) 31.0430 1.19308 0.596540 0.802583i \(-0.296542\pi\)
0.596540 + 0.802583i \(0.296542\pi\)
\(678\) 0 0
\(679\) −6.82987 −0.262106
\(680\) 2.11880 0.0812522
\(681\) 0 0
\(682\) 10.8165 0.414184
\(683\) −12.0994 −0.462970 −0.231485 0.972838i \(-0.574358\pi\)
−0.231485 + 0.972838i \(0.574358\pi\)
\(684\) 0 0
\(685\) 16.5085 0.630757
\(686\) 21.7390 0.829997
\(687\) 0 0
\(688\) 0.809373 0.0308571
\(689\) 9.56971 0.364577
\(690\) 0 0
\(691\) 35.9787 1.36869 0.684346 0.729157i \(-0.260088\pi\)
0.684346 + 0.729157i \(0.260088\pi\)
\(692\) 39.5375 1.50299
\(693\) 0 0
\(694\) −75.1251 −2.85171
\(695\) 8.42235 0.319478
\(696\) 0 0
\(697\) 2.06983 0.0784003
\(698\) −6.99996 −0.264953
\(699\) 0 0
\(700\) −4.20044 −0.158762
\(701\) −17.6047 −0.664919 −0.332459 0.943118i \(-0.607878\pi\)
−0.332459 + 0.943118i \(0.607878\pi\)
\(702\) 0 0
\(703\) 15.3127 0.577529
\(704\) 64.1507 2.41777
\(705\) 0 0
\(706\) 16.8069 0.632535
\(707\) 0.286426 0.0107722
\(708\) 0 0
\(709\) 7.06306 0.265259 0.132629 0.991166i \(-0.457658\pi\)
0.132629 + 0.991166i \(0.457658\pi\)
\(710\) −51.3303 −1.92639
\(711\) 0 0
\(712\) −10.8186 −0.405446
\(713\) 8.50661 0.318575
\(714\) 0 0
\(715\) 42.4780 1.58859
\(716\) 30.5149 1.14039
\(717\) 0 0
\(718\) −7.62438 −0.284539
\(719\) 0.00798484 0.000297784 0 0.000148892 1.00000i \(-0.499953\pi\)
0.000148892 1.00000i \(0.499953\pi\)
\(720\) 0 0
\(721\) 4.65606 0.173401
\(722\) 12.7740 0.475399
\(723\) 0 0
\(724\) 40.5555 1.50723
\(725\) 16.4148 0.609629
\(726\) 0 0
\(727\) 51.1360 1.89653 0.948265 0.317479i \(-0.102836\pi\)
0.948265 + 0.317479i \(0.102836\pi\)
\(728\) −10.2821 −0.381081
\(729\) 0 0
\(730\) −22.1604 −0.820193
\(731\) 0.504287 0.0186517
\(732\) 0 0
\(733\) 29.8058 1.10090 0.550451 0.834868i \(-0.314456\pi\)
0.550451 + 0.834868i \(0.314456\pi\)
\(734\) −8.51312 −0.314225
\(735\) 0 0
\(736\) 45.6615 1.68310
\(737\) 6.19697 0.228268
\(738\) 0 0
\(739\) 43.3189 1.59351 0.796756 0.604302i \(-0.206548\pi\)
0.796756 + 0.604302i \(0.206548\pi\)
\(740\) −25.1214 −0.923480
\(741\) 0 0
\(742\) −3.28904 −0.120745
\(743\) 21.6495 0.794244 0.397122 0.917766i \(-0.370009\pi\)
0.397122 + 0.917766i \(0.370009\pi\)
\(744\) 0 0
\(745\) 1.56101 0.0571908
\(746\) −44.3180 −1.62260
\(747\) 0 0
\(748\) −6.38736 −0.233545
\(749\) −1.85398 −0.0677429
\(750\) 0 0
\(751\) −6.21926 −0.226944 −0.113472 0.993541i \(-0.536197\pi\)
−0.113472 + 0.993541i \(0.536197\pi\)
\(752\) 6.85337 0.249917
\(753\) 0 0
\(754\) 98.9902 3.60501
\(755\) −1.18810 −0.0432393
\(756\) 0 0
\(757\) 33.3786 1.21317 0.606583 0.795020i \(-0.292540\pi\)
0.606583 + 0.795020i \(0.292540\pi\)
\(758\) −22.2864 −0.809479
\(759\) 0 0
\(760\) 20.8038 0.754634
\(761\) −15.1807 −0.550301 −0.275151 0.961401i \(-0.588728\pi\)
−0.275151 + 0.961401i \(0.588728\pi\)
\(762\) 0 0
\(763\) 5.39508 0.195315
\(764\) −30.2948 −1.09603
\(765\) 0 0
\(766\) −13.8915 −0.501919
\(767\) −57.6764 −2.08257
\(768\) 0 0
\(769\) 27.5453 0.993310 0.496655 0.867948i \(-0.334561\pi\)
0.496655 + 0.867948i \(0.334561\pi\)
\(770\) −14.5994 −0.526126
\(771\) 0 0
\(772\) 5.08523 0.183021
\(773\) −31.5503 −1.13479 −0.567393 0.823447i \(-0.692048\pi\)
−0.567393 + 0.823447i \(0.692048\pi\)
\(774\) 0 0
\(775\) −1.65433 −0.0594252
\(776\) 31.1449 1.11804
\(777\) 0 0
\(778\) −59.0322 −2.11641
\(779\) 20.3230 0.728147
\(780\) 0 0
\(781\) 62.8109 2.24755
\(782\) −8.00764 −0.286353
\(783\) 0 0
\(784\) −3.91270 −0.139739
\(785\) −17.1895 −0.613521
\(786\) 0 0
\(787\) −0.566008 −0.0201760 −0.0100880 0.999949i \(-0.503211\pi\)
−0.0100880 + 0.999949i \(0.503211\pi\)
\(788\) 76.1311 2.71206
\(789\) 0 0
\(790\) −14.0087 −0.498408
\(791\) 11.3573 0.403819
\(792\) 0 0
\(793\) 15.9194 0.565314
\(794\) −57.5716 −2.04314
\(795\) 0 0
\(796\) 49.0197 1.73746
\(797\) 26.6443 0.943789 0.471894 0.881655i \(-0.343570\pi\)
0.471894 + 0.881655i \(0.343570\pi\)
\(798\) 0 0
\(799\) 4.27005 0.151063
\(800\) −8.88004 −0.313957
\(801\) 0 0
\(802\) −36.9321 −1.30412
\(803\) 27.1168 0.956931
\(804\) 0 0
\(805\) −11.4817 −0.404676
\(806\) −9.97652 −0.351408
\(807\) 0 0
\(808\) −1.30613 −0.0459496
\(809\) 35.8890 1.26179 0.630895 0.775868i \(-0.282688\pi\)
0.630895 + 0.775868i \(0.282688\pi\)
\(810\) 0 0
\(811\) 32.5695 1.14367 0.571835 0.820369i \(-0.306232\pi\)
0.571835 + 0.820369i \(0.306232\pi\)
\(812\) −21.3427 −0.748983
\(813\) 0 0
\(814\) 49.0024 1.71753
\(815\) 20.8651 0.730872
\(816\) 0 0
\(817\) 4.95144 0.173229
\(818\) 37.5012 1.31120
\(819\) 0 0
\(820\) −33.3411 −1.16432
\(821\) −0.800013 −0.0279206 −0.0139603 0.999903i \(-0.504444\pi\)
−0.0139603 + 0.999903i \(0.504444\pi\)
\(822\) 0 0
\(823\) −25.4706 −0.887850 −0.443925 0.896064i \(-0.646414\pi\)
−0.443925 + 0.896064i \(0.646414\pi\)
\(824\) −21.2321 −0.739655
\(825\) 0 0
\(826\) 19.8229 0.689728
\(827\) 19.3649 0.673385 0.336692 0.941615i \(-0.390692\pi\)
0.336692 + 0.941615i \(0.390692\pi\)
\(828\) 0 0
\(829\) −31.7068 −1.10122 −0.550612 0.834761i \(-0.685606\pi\)
−0.550612 + 0.834761i \(0.685606\pi\)
\(830\) 47.4474 1.64692
\(831\) 0 0
\(832\) −59.1691 −2.05132
\(833\) −2.43784 −0.0844662
\(834\) 0 0
\(835\) 26.9089 0.931221
\(836\) −62.7155 −2.16906
\(837\) 0 0
\(838\) 49.7990 1.72028
\(839\) −33.4750 −1.15568 −0.577842 0.816149i \(-0.696105\pi\)
−0.577842 + 0.816149i \(0.696105\pi\)
\(840\) 0 0
\(841\) 54.4044 1.87601
\(842\) 74.5580 2.56944
\(843\) 0 0
\(844\) 74.3421 2.55896
\(845\) −15.9148 −0.547486
\(846\) 0 0
\(847\) 10.2286 0.351460
\(848\) 1.22773 0.0421603
\(849\) 0 0
\(850\) 1.55729 0.0534146
\(851\) 38.5379 1.32106
\(852\) 0 0
\(853\) 22.7066 0.777460 0.388730 0.921352i \(-0.372914\pi\)
0.388730 + 0.921352i \(0.372914\pi\)
\(854\) −5.47138 −0.187227
\(855\) 0 0
\(856\) 8.45432 0.288963
\(857\) 35.4417 1.21067 0.605333 0.795973i \(-0.293040\pi\)
0.605333 + 0.795973i \(0.293040\pi\)
\(858\) 0 0
\(859\) 53.3038 1.81870 0.909351 0.416029i \(-0.136579\pi\)
0.909351 + 0.416029i \(0.136579\pi\)
\(860\) −8.12313 −0.276997
\(861\) 0 0
\(862\) −41.4762 −1.41269
\(863\) 33.0046 1.12349 0.561745 0.827310i \(-0.310130\pi\)
0.561745 + 0.827310i \(0.310130\pi\)
\(864\) 0 0
\(865\) −21.0177 −0.714622
\(866\) 73.9845 2.51409
\(867\) 0 0
\(868\) 2.15098 0.0730091
\(869\) 17.1419 0.581500
\(870\) 0 0
\(871\) −5.71574 −0.193671
\(872\) −24.6021 −0.833133
\(873\) 0 0
\(874\) −78.6246 −2.65952
\(875\) 8.44446 0.285475
\(876\) 0 0
\(877\) −0.0251686 −0.000849883 0 −0.000424942 1.00000i \(-0.500135\pi\)
−0.000424942 1.00000i \(0.500135\pi\)
\(878\) −11.1482 −0.376233
\(879\) 0 0
\(880\) 5.44964 0.183707
\(881\) 8.28965 0.279286 0.139643 0.990202i \(-0.455405\pi\)
0.139643 + 0.990202i \(0.455405\pi\)
\(882\) 0 0
\(883\) 53.4759 1.79961 0.899803 0.436296i \(-0.143710\pi\)
0.899803 + 0.436296i \(0.143710\pi\)
\(884\) 5.89135 0.198147
\(885\) 0 0
\(886\) −32.6082 −1.09549
\(887\) 17.2857 0.580399 0.290199 0.956966i \(-0.406278\pi\)
0.290199 + 0.956966i \(0.406278\pi\)
\(888\) 0 0
\(889\) −6.56297 −0.220115
\(890\) 14.1683 0.474923
\(891\) 0 0
\(892\) −3.36649 −0.112718
\(893\) 41.9263 1.40301
\(894\) 0 0
\(895\) −16.2213 −0.542219
\(896\) 13.4766 0.450223
\(897\) 0 0
\(898\) 17.9235 0.598114
\(899\) −8.40574 −0.280347
\(900\) 0 0
\(901\) 0.764946 0.0254841
\(902\) 65.0360 2.16546
\(903\) 0 0
\(904\) −51.7904 −1.72252
\(905\) −21.5588 −0.716639
\(906\) 0 0
\(907\) 21.0950 0.700447 0.350223 0.936666i \(-0.386106\pi\)
0.350223 + 0.936666i \(0.386106\pi\)
\(908\) 59.3713 1.97031
\(909\) 0 0
\(910\) 13.4657 0.446383
\(911\) −38.3792 −1.27156 −0.635780 0.771870i \(-0.719322\pi\)
−0.635780 + 0.771870i \(0.719322\pi\)
\(912\) 0 0
\(913\) −58.0595 −1.92149
\(914\) −5.61152 −0.185613
\(915\) 0 0
\(916\) −27.2454 −0.900214
\(917\) −10.4062 −0.343644
\(918\) 0 0
\(919\) −50.6007 −1.66916 −0.834582 0.550884i \(-0.814291\pi\)
−0.834582 + 0.550884i \(0.814291\pi\)
\(920\) 52.3576 1.72618
\(921\) 0 0
\(922\) −7.28499 −0.239918
\(923\) −57.9333 −1.90690
\(924\) 0 0
\(925\) −7.49468 −0.246424
\(926\) −33.8923 −1.11377
\(927\) 0 0
\(928\) −45.1201 −1.48114
\(929\) −42.1910 −1.38424 −0.692121 0.721781i \(-0.743324\pi\)
−0.692121 + 0.721781i \(0.743324\pi\)
\(930\) 0 0
\(931\) −23.9364 −0.784485
\(932\) −38.6219 −1.26510
\(933\) 0 0
\(934\) 87.4668 2.86200
\(935\) 3.39544 0.111043
\(936\) 0 0
\(937\) 24.8880 0.813056 0.406528 0.913638i \(-0.366739\pi\)
0.406528 + 0.913638i \(0.366739\pi\)
\(938\) 1.96446 0.0641419
\(939\) 0 0
\(940\) −68.7826 −2.24344
\(941\) 55.5293 1.81020 0.905101 0.425197i \(-0.139795\pi\)
0.905101 + 0.425197i \(0.139795\pi\)
\(942\) 0 0
\(943\) 51.1475 1.66559
\(944\) −7.39947 −0.240832
\(945\) 0 0
\(946\) 15.8452 0.515172
\(947\) −51.7979 −1.68321 −0.841603 0.540096i \(-0.818388\pi\)
−0.841603 + 0.540096i \(0.818388\pi\)
\(948\) 0 0
\(949\) −25.0111 −0.811893
\(950\) 15.2906 0.496091
\(951\) 0 0
\(952\) −0.821891 −0.0266377
\(953\) −45.4715 −1.47297 −0.736484 0.676455i \(-0.763515\pi\)
−0.736484 + 0.676455i \(0.763515\pi\)
\(954\) 0 0
\(955\) 16.1044 0.521125
\(956\) −50.8331 −1.64406
\(957\) 0 0
\(958\) 25.9524 0.838483
\(959\) −6.40372 −0.206787
\(960\) 0 0
\(961\) −30.1528 −0.972672
\(962\) −45.1972 −1.45721
\(963\) 0 0
\(964\) 10.1245 0.326088
\(965\) −2.70325 −0.0870206
\(966\) 0 0
\(967\) 2.52027 0.0810463 0.0405232 0.999179i \(-0.487098\pi\)
0.0405232 + 0.999179i \(0.487098\pi\)
\(968\) −46.6436 −1.49918
\(969\) 0 0
\(970\) −40.7880 −1.30962
\(971\) 37.2035 1.19392 0.596958 0.802273i \(-0.296376\pi\)
0.596958 + 0.802273i \(0.296376\pi\)
\(972\) 0 0
\(973\) −3.26707 −0.104737
\(974\) −80.6400 −2.58387
\(975\) 0 0
\(976\) 2.04235 0.0653739
\(977\) 31.3860 1.00413 0.502064 0.864830i \(-0.332574\pi\)
0.502064 + 0.864830i \(0.332574\pi\)
\(978\) 0 0
\(979\) −17.3372 −0.554100
\(980\) 39.2691 1.25441
\(981\) 0 0
\(982\) −72.8923 −2.32609
\(983\) 23.1176 0.737337 0.368668 0.929561i \(-0.379814\pi\)
0.368668 + 0.929561i \(0.379814\pi\)
\(984\) 0 0
\(985\) −40.4704 −1.28949
\(986\) 7.91269 0.251991
\(987\) 0 0
\(988\) 57.8454 1.84031
\(989\) 12.4614 0.396251
\(990\) 0 0
\(991\) 20.6114 0.654743 0.327371 0.944896i \(-0.393837\pi\)
0.327371 + 0.944896i \(0.393837\pi\)
\(992\) 4.54733 0.144378
\(993\) 0 0
\(994\) 19.9113 0.631547
\(995\) −26.0583 −0.826103
\(996\) 0 0
\(997\) 52.3446 1.65777 0.828885 0.559419i \(-0.188976\pi\)
0.828885 + 0.559419i \(0.188976\pi\)
\(998\) 7.07935 0.224093
\(999\) 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 6021.2.a.l.1.3 10
3.2 odd 2 inner 6021.2.a.l.1.8 yes 10
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
6021.2.a.l.1.3 10 1.1 even 1 trivial
6021.2.a.l.1.8 yes 10 3.2 odd 2 inner