Properties

Label 4851.2.a.bm.1.1
Level $4851$
Weight $2$
Character 4851.1
Self dual yes
Analytic conductor $38.735$
Analytic rank $1$
Dimension $3$
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] = [4851,2,Mod(1,4851)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4851, 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("4851.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4851 = 3^{2} \cdot 7^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4851.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: \(38.7354300205\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.229.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - 4x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1617)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(2.11491\) of defining polynomial
Character \(\chi\) \(=\) 4851.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.11491 q^{2} +2.47283 q^{4} +1.64207 q^{5} -1.00000 q^{8} +O(q^{10})\) \(q-2.11491 q^{2} +2.47283 q^{4} +1.64207 q^{5} -1.00000 q^{8} -3.47283 q^{10} -1.00000 q^{11} -4.58774 q^{13} -2.83076 q^{16} +0.715853 q^{17} -1.64207 q^{19} +4.06058 q^{20} +2.11491 q^{22} -2.30359 q^{25} +9.70265 q^{26} +5.53341 q^{29} +5.17548 q^{31} +7.98680 q^{32} -1.51396 q^{34} -6.58774 q^{37} +3.47283 q^{38} -1.64207 q^{40} +5.17548 q^{41} +5.17548 q^{43} -2.47283 q^{44} +7.87189 q^{47} +4.87189 q^{50} -11.3447 q^{52} -2.71585 q^{53} -1.64207 q^{55} -11.7026 q^{58} -3.15604 q^{59} -13.1755 q^{61} -10.9457 q^{62} -11.2298 q^{64} -7.53341 q^{65} +4.21037 q^{67} +1.77018 q^{68} +1.85244 q^{71} +10.4791 q^{73} +13.9325 q^{74} -4.06058 q^{76} -12.4596 q^{79} -4.64832 q^{80} -10.9457 q^{82} -9.17548 q^{83} +1.17548 q^{85} -10.9457 q^{86} +1.00000 q^{88} +4.71585 q^{89} -16.6483 q^{94} -2.69641 q^{95} +13.6351 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 2 q^{4} + 4 q^{5} - 3 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 2 q^{4} + 4 q^{5} - 3 q^{8} - 5 q^{10} - 3 q^{11} - 2 q^{13} - 4 q^{16} + 4 q^{17} - 4 q^{19} - 5 q^{20} + 3 q^{25} + 11 q^{26} - 6 q^{29} - 8 q^{31} + 4 q^{32} + 10 q^{34} - 8 q^{37} + 5 q^{38} - 4 q^{40} - 8 q^{41} - 8 q^{43} - 2 q^{44} + 10 q^{47} + q^{50} - 15 q^{52} - 10 q^{53} - 4 q^{55} - 17 q^{58} + 6 q^{59} - 16 q^{61} - 22 q^{62} - 21 q^{64} + 8 q^{67} + 18 q^{68} - 2 q^{73} + 11 q^{74} + 5 q^{76} - 12 q^{79} + 15 q^{80} - 22 q^{82} - 4 q^{83} - 20 q^{85} - 22 q^{86} + 3 q^{88} + 16 q^{89} - 21 q^{94} - 18 q^{95} - 8 q^{97}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.11491 −1.49547 −0.747733 0.664000i \(-0.768858\pi\)
−0.747733 + 0.664000i \(0.768858\pi\)
\(3\) 0 0
\(4\) 2.47283 1.23642
\(5\) 1.64207 0.734358 0.367179 0.930150i \(-0.380324\pi\)
0.367179 + 0.930150i \(0.380324\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) −1.00000 −0.353553
\(9\) 0 0
\(10\) −3.47283 −1.09821
\(11\) −1.00000 −0.301511
\(12\) 0 0
\(13\) −4.58774 −1.27241 −0.636205 0.771520i \(-0.719497\pi\)
−0.636205 + 0.771520i \(0.719497\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) −2.83076 −0.707690
\(17\) 0.715853 0.173620 0.0868099 0.996225i \(-0.472333\pi\)
0.0868099 + 0.996225i \(0.472333\pi\)
\(18\) 0 0
\(19\) −1.64207 −0.376718 −0.188359 0.982100i \(-0.560317\pi\)
−0.188359 + 0.982100i \(0.560317\pi\)
\(20\) 4.06058 0.907972
\(21\) 0 0
\(22\) 2.11491 0.450900
\(23\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(24\) 0 0
\(25\) −2.30359 −0.460719
\(26\) 9.70265 1.90285
\(27\) 0 0
\(28\) 0 0
\(29\) 5.53341 1.02753 0.513764 0.857931i \(-0.328251\pi\)
0.513764 + 0.857931i \(0.328251\pi\)
\(30\) 0 0
\(31\) 5.17548 0.929544 0.464772 0.885430i \(-0.346136\pi\)
0.464772 + 0.885430i \(0.346136\pi\)
\(32\) 7.98680 1.41188
\(33\) 0 0
\(34\) −1.51396 −0.259642
\(35\) 0 0
\(36\) 0 0
\(37\) −6.58774 −1.08302 −0.541509 0.840695i \(-0.682147\pi\)
−0.541509 + 0.840695i \(0.682147\pi\)
\(38\) 3.47283 0.563368
\(39\) 0 0
\(40\) −1.64207 −0.259635
\(41\) 5.17548 0.808275 0.404137 0.914698i \(-0.367572\pi\)
0.404137 + 0.914698i \(0.367572\pi\)
\(42\) 0 0
\(43\) 5.17548 0.789254 0.394627 0.918841i \(-0.370874\pi\)
0.394627 + 0.918841i \(0.370874\pi\)
\(44\) −2.47283 −0.372794
\(45\) 0 0
\(46\) 0 0
\(47\) 7.87189 1.14823 0.574116 0.818774i \(-0.305346\pi\)
0.574116 + 0.818774i \(0.305346\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 4.87189 0.688989
\(51\) 0 0
\(52\) −11.3447 −1.57323
\(53\) −2.71585 −0.373051 −0.186526 0.982450i \(-0.559723\pi\)
−0.186526 + 0.982450i \(0.559723\pi\)
\(54\) 0 0
\(55\) −1.64207 −0.221417
\(56\) 0 0
\(57\) 0 0
\(58\) −11.7026 −1.53663
\(59\) −3.15604 −0.410881 −0.205440 0.978670i \(-0.565863\pi\)
−0.205440 + 0.978670i \(0.565863\pi\)
\(60\) 0 0
\(61\) −13.1755 −1.68695 −0.843474 0.537170i \(-0.819493\pi\)
−0.843474 + 0.537170i \(0.819493\pi\)
\(62\) −10.9457 −1.39010
\(63\) 0 0
\(64\) −11.2298 −1.40373
\(65\) −7.53341 −0.934404
\(66\) 0 0
\(67\) 4.21037 0.514378 0.257189 0.966361i \(-0.417204\pi\)
0.257189 + 0.966361i \(0.417204\pi\)
\(68\) 1.77018 0.214666
\(69\) 0 0
\(70\) 0 0
\(71\) 1.85244 0.219844 0.109922 0.993940i \(-0.464940\pi\)
0.109922 + 0.993940i \(0.464940\pi\)
\(72\) 0 0
\(73\) 10.4791 1.22648 0.613242 0.789895i \(-0.289865\pi\)
0.613242 + 0.789895i \(0.289865\pi\)
\(74\) 13.9325 1.61962
\(75\) 0 0
\(76\) −4.06058 −0.465780
\(77\) 0 0
\(78\) 0 0
\(79\) −12.4596 −1.40182 −0.700909 0.713251i \(-0.747222\pi\)
−0.700909 + 0.713251i \(0.747222\pi\)
\(80\) −4.64832 −0.519698
\(81\) 0 0
\(82\) −10.9457 −1.20875
\(83\) −9.17548 −1.00714 −0.503570 0.863954i \(-0.667980\pi\)
−0.503570 + 0.863954i \(0.667980\pi\)
\(84\) 0 0
\(85\) 1.17548 0.127499
\(86\) −10.9457 −1.18030
\(87\) 0 0
\(88\) 1.00000 0.106600
\(89\) 4.71585 0.499879 0.249940 0.968261i \(-0.419589\pi\)
0.249940 + 0.968261i \(0.419589\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) −16.6483 −1.71714
\(95\) −2.69641 −0.276645
\(96\) 0 0
\(97\) 13.6351 1.38444 0.692218 0.721688i \(-0.256634\pi\)
0.692218 + 0.721688i \(0.256634\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) −5.69641 −0.569641
\(101\) −14.6072 −1.45347 −0.726735 0.686918i \(-0.758963\pi\)
−0.726735 + 0.686918i \(0.758963\pi\)
\(102\) 0 0
\(103\) −17.8913 −1.76289 −0.881443 0.472291i \(-0.843427\pi\)
−0.881443 + 0.472291i \(0.843427\pi\)
\(104\) 4.58774 0.449865
\(105\) 0 0
\(106\) 5.74378 0.557885
\(107\) −11.1560 −1.07849 −0.539247 0.842147i \(-0.681291\pi\)
−0.539247 + 0.842147i \(0.681291\pi\)
\(108\) 0 0
\(109\) −0.108664 −0.0104082 −0.00520408 0.999986i \(-0.501657\pi\)
−0.00520408 + 0.999986i \(0.501657\pi\)
\(110\) 3.47283 0.331122
\(111\) 0 0
\(112\) 0 0
\(113\) 13.0668 1.22922 0.614611 0.788830i \(-0.289313\pi\)
0.614611 + 0.788830i \(0.289313\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 13.6832 1.27045
\(117\) 0 0
\(118\) 6.67472 0.614458
\(119\) 0 0
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 27.8649 2.52277
\(123\) 0 0
\(124\) 12.7981 1.14930
\(125\) −11.9930 −1.07269
\(126\) 0 0
\(127\) −20.4596 −1.81550 −0.907749 0.419513i \(-0.862201\pi\)
−0.907749 + 0.419513i \(0.862201\pi\)
\(128\) 7.77643 0.687346
\(129\) 0 0
\(130\) 15.9325 1.39737
\(131\) −15.0668 −1.31639 −0.658197 0.752846i \(-0.728681\pi\)
−0.658197 + 0.752846i \(0.728681\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) −8.90454 −0.769235
\(135\) 0 0
\(136\) −0.715853 −0.0613839
\(137\) 14.4596 1.23537 0.617685 0.786426i \(-0.288071\pi\)
0.617685 + 0.786426i \(0.288071\pi\)
\(138\) 0 0
\(139\) 5.13659 0.435680 0.217840 0.975985i \(-0.430099\pi\)
0.217840 + 0.975985i \(0.430099\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) −3.91774 −0.328770
\(143\) 4.58774 0.383646
\(144\) 0 0
\(145\) 9.08627 0.754573
\(146\) −22.1623 −1.83416
\(147\) 0 0
\(148\) −16.2904 −1.33906
\(149\) −16.8176 −1.37775 −0.688874 0.724881i \(-0.741895\pi\)
−0.688874 + 0.724881i \(0.741895\pi\)
\(150\) 0 0
\(151\) 4.71585 0.383771 0.191885 0.981417i \(-0.438540\pi\)
0.191885 + 0.981417i \(0.438540\pi\)
\(152\) 1.64207 0.133190
\(153\) 0 0
\(154\) 0 0
\(155\) 8.49852 0.682618
\(156\) 0 0
\(157\) 11.0279 0.880124 0.440062 0.897967i \(-0.354956\pi\)
0.440062 + 0.897967i \(0.354956\pi\)
\(158\) 26.3510 2.09637
\(159\) 0 0
\(160\) 13.1149 1.03682
\(161\) 0 0
\(162\) 0 0
\(163\) 5.64207 0.441921 0.220961 0.975283i \(-0.429081\pi\)
0.220961 + 0.975283i \(0.429081\pi\)
\(164\) 12.7981 0.999364
\(165\) 0 0
\(166\) 19.4053 1.50614
\(167\) 16.0000 1.23812 0.619059 0.785345i \(-0.287514\pi\)
0.619059 + 0.785345i \(0.287514\pi\)
\(168\) 0 0
\(169\) 8.04737 0.619029
\(170\) −2.48604 −0.190670
\(171\) 0 0
\(172\) 12.7981 0.975847
\(173\) −15.0279 −1.14255 −0.571276 0.820758i \(-0.693551\pi\)
−0.571276 + 0.820758i \(0.693551\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 2.83076 0.213377
\(177\) 0 0
\(178\) −9.97359 −0.747552
\(179\) −9.89134 −0.739313 −0.369657 0.929168i \(-0.620525\pi\)
−0.369657 + 0.929168i \(0.620525\pi\)
\(180\) 0 0
\(181\) −6.82452 −0.507262 −0.253631 0.967301i \(-0.581625\pi\)
−0.253631 + 0.967301i \(0.581625\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −10.8176 −0.795323
\(186\) 0 0
\(187\) −0.715853 −0.0523483
\(188\) 19.4659 1.41969
\(189\) 0 0
\(190\) 5.70265 0.413714
\(191\) −11.0279 −0.797953 −0.398976 0.916961i \(-0.630634\pi\)
−0.398976 + 0.916961i \(0.630634\pi\)
\(192\) 0 0
\(193\) −7.85244 −0.565231 −0.282616 0.959233i \(-0.591202\pi\)
−0.282616 + 0.959233i \(0.591202\pi\)
\(194\) −28.8370 −2.07038
\(195\) 0 0
\(196\) 0 0
\(197\) −19.8913 −1.41720 −0.708599 0.705611i \(-0.750673\pi\)
−0.708599 + 0.705611i \(0.750673\pi\)
\(198\) 0 0
\(199\) −23.5264 −1.66775 −0.833873 0.551957i \(-0.813881\pi\)
−0.833873 + 0.551957i \(0.813881\pi\)
\(200\) 2.30359 0.162889
\(201\) 0 0
\(202\) 30.8929 2.17361
\(203\) 0 0
\(204\) 0 0
\(205\) 8.49852 0.593563
\(206\) 37.8385 2.63633
\(207\) 0 0
\(208\) 12.9868 0.900472
\(209\) 1.64207 0.113585
\(210\) 0 0
\(211\) −4.25622 −0.293010 −0.146505 0.989210i \(-0.546803\pi\)
−0.146505 + 0.989210i \(0.546803\pi\)
\(212\) −6.71585 −0.461247
\(213\) 0 0
\(214\) 23.5940 1.61285
\(215\) 8.49852 0.579595
\(216\) 0 0
\(217\) 0 0
\(218\) 0.229815 0.0155650
\(219\) 0 0
\(220\) −4.06058 −0.273764
\(221\) −3.28415 −0.220916
\(222\) 0 0
\(223\) 3.74378 0.250702 0.125351 0.992112i \(-0.459994\pi\)
0.125351 + 0.992112i \(0.459994\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) −27.6351 −1.83826
\(227\) −15.4876 −1.02795 −0.513973 0.857807i \(-0.671827\pi\)
−0.513973 + 0.857807i \(0.671827\pi\)
\(228\) 0 0
\(229\) 8.00000 0.528655 0.264327 0.964433i \(-0.414850\pi\)
0.264327 + 0.964433i \(0.414850\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) −5.53341 −0.363286
\(233\) 5.54037 0.362962 0.181481 0.983394i \(-0.441911\pi\)
0.181481 + 0.983394i \(0.441911\pi\)
\(234\) 0 0
\(235\) 12.9262 0.843214
\(236\) −7.80435 −0.508020
\(237\) 0 0
\(238\) 0 0
\(239\) 1.26470 0.0818067 0.0409033 0.999163i \(-0.486976\pi\)
0.0409033 + 0.999163i \(0.486976\pi\)
\(240\) 0 0
\(241\) 23.8719 1.53772 0.768862 0.639415i \(-0.220823\pi\)
0.768862 + 0.639415i \(0.220823\pi\)
\(242\) −2.11491 −0.135951
\(243\) 0 0
\(244\) −32.5808 −2.08577
\(245\) 0 0
\(246\) 0 0
\(247\) 7.53341 0.479339
\(248\) −5.17548 −0.328643
\(249\) 0 0
\(250\) 25.3642 1.60417
\(251\) 22.0194 1.38986 0.694928 0.719080i \(-0.255436\pi\)
0.694928 + 0.719080i \(0.255436\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 43.2702 2.71502
\(255\) 0 0
\(256\) 6.01320 0.375825
\(257\) 27.3161 1.70393 0.851965 0.523599i \(-0.175411\pi\)
0.851965 + 0.523599i \(0.175411\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) −18.6289 −1.15531
\(261\) 0 0
\(262\) 31.8649 1.96862
\(263\) 11.8719 0.732052 0.366026 0.930605i \(-0.380718\pi\)
0.366026 + 0.930605i \(0.380718\pi\)
\(264\) 0 0
\(265\) −4.45963 −0.273953
\(266\) 0 0
\(267\) 0 0
\(268\) 10.4115 0.635986
\(269\) 5.63511 0.343579 0.171789 0.985134i \(-0.445045\pi\)
0.171789 + 0.985134i \(0.445045\pi\)
\(270\) 0 0
\(271\) 5.68097 0.345094 0.172547 0.985001i \(-0.444800\pi\)
0.172547 + 0.985001i \(0.444800\pi\)
\(272\) −2.02641 −0.122869
\(273\) 0 0
\(274\) −30.5808 −1.84745
\(275\) 2.30359 0.138912
\(276\) 0 0
\(277\) −25.4876 −1.53140 −0.765699 0.643199i \(-0.777607\pi\)
−0.765699 + 0.643199i \(0.777607\pi\)
\(278\) −10.8634 −0.651544
\(279\) 0 0
\(280\) 0 0
\(281\) −17.5723 −1.04828 −0.524138 0.851633i \(-0.675612\pi\)
−0.524138 + 0.851633i \(0.675612\pi\)
\(282\) 0 0
\(283\) −25.1296 −1.49380 −0.746901 0.664936i \(-0.768459\pi\)
−0.746901 + 0.664936i \(0.768459\pi\)
\(284\) 4.58078 0.271819
\(285\) 0 0
\(286\) −9.70265 −0.573730
\(287\) 0 0
\(288\) 0 0
\(289\) −16.4876 −0.969856
\(290\) −19.2166 −1.12844
\(291\) 0 0
\(292\) 25.9130 1.51644
\(293\) −14.3510 −0.838392 −0.419196 0.907896i \(-0.637688\pi\)
−0.419196 + 0.907896i \(0.637688\pi\)
\(294\) 0 0
\(295\) −5.18244 −0.301734
\(296\) 6.58774 0.382905
\(297\) 0 0
\(298\) 35.5676 2.06037
\(299\) 0 0
\(300\) 0 0
\(301\) 0 0
\(302\) −9.97359 −0.573916
\(303\) 0 0
\(304\) 4.64832 0.266599
\(305\) −21.6351 −1.23882
\(306\) 0 0
\(307\) −13.1366 −0.749745 −0.374872 0.927076i \(-0.622313\pi\)
−0.374872 + 0.927076i \(0.622313\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) −17.9736 −1.02083
\(311\) −14.3510 −0.813769 −0.406884 0.913480i \(-0.633385\pi\)
−0.406884 + 0.913480i \(0.633385\pi\)
\(312\) 0 0
\(313\) −23.7438 −1.34208 −0.671039 0.741422i \(-0.734152\pi\)
−0.671039 + 0.741422i \(0.734152\pi\)
\(314\) −23.3230 −1.31620
\(315\) 0 0
\(316\) −30.8106 −1.73323
\(317\) 6.45963 0.362809 0.181404 0.983409i \(-0.441936\pi\)
0.181404 + 0.983409i \(0.441936\pi\)
\(318\) 0 0
\(319\) −5.53341 −0.309811
\(320\) −18.4402 −1.03084
\(321\) 0 0
\(322\) 0 0
\(323\) −1.17548 −0.0654056
\(324\) 0 0
\(325\) 10.5683 0.586224
\(326\) −11.9325 −0.660878
\(327\) 0 0
\(328\) −5.17548 −0.285768
\(329\) 0 0
\(330\) 0 0
\(331\) 31.2702 1.71877 0.859384 0.511332i \(-0.170848\pi\)
0.859384 + 0.511332i \(0.170848\pi\)
\(332\) −22.6894 −1.24525
\(333\) 0 0
\(334\) −33.8385 −1.85156
\(335\) 6.91373 0.377738
\(336\) 0 0
\(337\) −16.3121 −0.888575 −0.444288 0.895884i \(-0.646543\pi\)
−0.444288 + 0.895884i \(0.646543\pi\)
\(338\) −17.0194 −0.925736
\(339\) 0 0
\(340\) 2.90677 0.157642
\(341\) −5.17548 −0.280268
\(342\) 0 0
\(343\) 0 0
\(344\) −5.17548 −0.279043
\(345\) 0 0
\(346\) 31.7827 1.70865
\(347\) 6.86341 0.368447 0.184224 0.982884i \(-0.441023\pi\)
0.184224 + 0.982884i \(0.441023\pi\)
\(348\) 0 0
\(349\) −23.8580 −1.27709 −0.638544 0.769585i \(-0.720463\pi\)
−0.638544 + 0.769585i \(0.720463\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) −7.98680 −0.425698
\(353\) 11.7368 0.624688 0.312344 0.949969i \(-0.398886\pi\)
0.312344 + 0.949969i \(0.398886\pi\)
\(354\) 0 0
\(355\) 3.04185 0.161444
\(356\) 11.6615 0.618059
\(357\) 0 0
\(358\) 20.9193 1.10562
\(359\) 27.7827 1.46631 0.733157 0.680060i \(-0.238046\pi\)
0.733157 + 0.680060i \(0.238046\pi\)
\(360\) 0 0
\(361\) −16.3036 −0.858084
\(362\) 14.4332 0.758593
\(363\) 0 0
\(364\) 0 0
\(365\) 17.2074 0.900677
\(366\) 0 0
\(367\) 26.5544 1.38613 0.693064 0.720877i \(-0.256261\pi\)
0.693064 + 0.720877i \(0.256261\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 22.8781 1.18938
\(371\) 0 0
\(372\) 0 0
\(373\) 5.06682 0.262350 0.131175 0.991359i \(-0.458125\pi\)
0.131175 + 0.991359i \(0.458125\pi\)
\(374\) 1.51396 0.0782851
\(375\) 0 0
\(376\) −7.87189 −0.405962
\(377\) −25.3859 −1.30744
\(378\) 0 0
\(379\) −30.5613 −1.56983 −0.784915 0.619603i \(-0.787294\pi\)
−0.784915 + 0.619603i \(0.787294\pi\)
\(380\) −6.66776 −0.342049
\(381\) 0 0
\(382\) 23.3230 1.19331
\(383\) 9.13659 0.466858 0.233429 0.972374i \(-0.425005\pi\)
0.233429 + 0.972374i \(0.425005\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 16.6072 0.845284
\(387\) 0 0
\(388\) 33.7174 1.71174
\(389\) −21.2702 −1.07844 −0.539222 0.842164i \(-0.681281\pi\)
−0.539222 + 0.842164i \(0.681281\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) 42.0683 2.11937
\(395\) −20.4596 −1.02944
\(396\) 0 0
\(397\) −28.9582 −1.45337 −0.726684 0.686972i \(-0.758940\pi\)
−0.726684 + 0.686972i \(0.758940\pi\)
\(398\) 49.7563 2.49406
\(399\) 0 0
\(400\) 6.52092 0.326046
\(401\) −32.3510 −1.61553 −0.807765 0.589505i \(-0.799323\pi\)
−0.807765 + 0.589505i \(0.799323\pi\)
\(402\) 0 0
\(403\) −23.7438 −1.18276
\(404\) −36.1212 −1.79709
\(405\) 0 0
\(406\) 0 0
\(407\) 6.58774 0.326542
\(408\) 0 0
\(409\) −2.31207 −0.114325 −0.0571623 0.998365i \(-0.518205\pi\)
−0.0571623 + 0.998365i \(0.518205\pi\)
\(410\) −17.9736 −0.887652
\(411\) 0 0
\(412\) −44.2423 −2.17966
\(413\) 0 0
\(414\) 0 0
\(415\) −15.0668 −0.739601
\(416\) −36.6414 −1.79649
\(417\) 0 0
\(418\) −3.47283 −0.169862
\(419\) 38.4263 1.87725 0.938623 0.344945i \(-0.112102\pi\)
0.938623 + 0.344945i \(0.112102\pi\)
\(420\) 0 0
\(421\) 15.5070 0.755765 0.377883 0.925854i \(-0.376652\pi\)
0.377883 + 0.925854i \(0.376652\pi\)
\(422\) 9.00152 0.438187
\(423\) 0 0
\(424\) 2.71585 0.131893
\(425\) −1.64903 −0.0799899
\(426\) 0 0
\(427\) 0 0
\(428\) −27.5870 −1.33347
\(429\) 0 0
\(430\) −17.9736 −0.866764
\(431\) −37.9666 −1.82879 −0.914394 0.404825i \(-0.867332\pi\)
−0.914394 + 0.404825i \(0.867332\pi\)
\(432\) 0 0
\(433\) −10.1087 −0.485791 −0.242896 0.970052i \(-0.578097\pi\)
−0.242896 + 0.970052i \(0.578097\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −0.268709 −0.0128688
\(437\) 0 0
\(438\) 0 0
\(439\) 22.5224 1.07494 0.537469 0.843284i \(-0.319381\pi\)
0.537469 + 0.843284i \(0.319381\pi\)
\(440\) 1.64207 0.0782828
\(441\) 0 0
\(442\) 6.94567 0.330372
\(443\) −20.6630 −0.981731 −0.490865 0.871235i \(-0.663319\pi\)
−0.490865 + 0.871235i \(0.663319\pi\)
\(444\) 0 0
\(445\) 7.74378 0.367090
\(446\) −7.91774 −0.374916
\(447\) 0 0
\(448\) 0 0
\(449\) −40.8106 −1.92597 −0.962986 0.269553i \(-0.913124\pi\)
−0.962986 + 0.269553i \(0.913124\pi\)
\(450\) 0 0
\(451\) −5.17548 −0.243704
\(452\) 32.3121 1.51983
\(453\) 0 0
\(454\) 32.7547 1.53726
\(455\) 0 0
\(456\) 0 0
\(457\) −28.3899 −1.32802 −0.664011 0.747723i \(-0.731147\pi\)
−0.664011 + 0.747723i \(0.731147\pi\)
\(458\) −16.9193 −0.790585
\(459\) 0 0
\(460\) 0 0
\(461\) 21.4178 0.997526 0.498763 0.866739i \(-0.333788\pi\)
0.498763 + 0.866739i \(0.333788\pi\)
\(462\) 0 0
\(463\) 20.7478 0.964231 0.482116 0.876108i \(-0.339868\pi\)
0.482116 + 0.876108i \(0.339868\pi\)
\(464\) −15.6638 −0.727172
\(465\) 0 0
\(466\) −11.7174 −0.542797
\(467\) 26.8998 1.24477 0.622387 0.782709i \(-0.286163\pi\)
0.622387 + 0.782709i \(0.286163\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) −27.3378 −1.26100
\(471\) 0 0
\(472\) 3.15604 0.145268
\(473\) −5.17548 −0.237969
\(474\) 0 0
\(475\) 3.78267 0.173561
\(476\) 0 0
\(477\) 0 0
\(478\) −2.67472 −0.122339
\(479\) 30.8106 1.40777 0.703886 0.710313i \(-0.251447\pi\)
0.703886 + 0.710313i \(0.251447\pi\)
\(480\) 0 0
\(481\) 30.2229 1.37804
\(482\) −50.4868 −2.29961
\(483\) 0 0
\(484\) 2.47283 0.112402
\(485\) 22.3899 1.01667
\(486\) 0 0
\(487\) −23.4876 −1.06432 −0.532161 0.846643i \(-0.678620\pi\)
−0.532161 + 0.846643i \(0.678620\pi\)
\(488\) 13.1755 0.596426
\(489\) 0 0
\(490\) 0 0
\(491\) 16.6266 0.750350 0.375175 0.926954i \(-0.377583\pi\)
0.375175 + 0.926954i \(0.377583\pi\)
\(492\) 0 0
\(493\) 3.96111 0.178399
\(494\) −15.9325 −0.716835
\(495\) 0 0
\(496\) −14.6506 −0.657829
\(497\) 0 0
\(498\) 0 0
\(499\) −7.99304 −0.357818 −0.178909 0.983866i \(-0.557257\pi\)
−0.178909 + 0.983866i \(0.557257\pi\)
\(500\) −29.6568 −1.32629
\(501\) 0 0
\(502\) −46.5691 −2.07848
\(503\) 24.6630 1.09967 0.549835 0.835273i \(-0.314690\pi\)
0.549835 + 0.835273i \(0.314690\pi\)
\(504\) 0 0
\(505\) −23.9861 −1.06737
\(506\) 0 0
\(507\) 0 0
\(508\) −50.5933 −2.24471
\(509\) 15.0668 0.667825 0.333912 0.942604i \(-0.391631\pi\)
0.333912 + 0.942604i \(0.391631\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) −28.2702 −1.24938
\(513\) 0 0
\(514\) −57.7710 −2.54817
\(515\) −29.3789 −1.29459
\(516\) 0 0
\(517\) −7.87189 −0.346205
\(518\) 0 0
\(519\) 0 0
\(520\) 7.53341 0.330362
\(521\) −42.3051 −1.85342 −0.926710 0.375777i \(-0.877376\pi\)
−0.926710 + 0.375777i \(0.877376\pi\)
\(522\) 0 0
\(523\) −5.84548 −0.255605 −0.127803 0.991800i \(-0.540792\pi\)
−0.127803 + 0.991800i \(0.540792\pi\)
\(524\) −37.2577 −1.62761
\(525\) 0 0
\(526\) −25.1079 −1.09476
\(527\) 3.70488 0.161387
\(528\) 0 0
\(529\) −23.0000 −1.00000
\(530\) 9.43171 0.409687
\(531\) 0 0
\(532\) 0 0
\(533\) −23.7438 −1.02846
\(534\) 0 0
\(535\) −18.3190 −0.792001
\(536\) −4.21037 −0.181860
\(537\) 0 0
\(538\) −11.9177 −0.513810
\(539\) 0 0
\(540\) 0 0
\(541\) −40.2982 −1.73255 −0.866276 0.499565i \(-0.833493\pi\)
−0.866276 + 0.499565i \(0.833493\pi\)
\(542\) −12.0147 −0.516076
\(543\) 0 0
\(544\) 5.71737 0.245130
\(545\) −0.178435 −0.00764331
\(546\) 0 0
\(547\) 2.98903 0.127802 0.0639009 0.997956i \(-0.479646\pi\)
0.0639009 + 0.997956i \(0.479646\pi\)
\(548\) 35.7563 1.52743
\(549\) 0 0
\(550\) −4.87189 −0.207738
\(551\) −9.08627 −0.387088
\(552\) 0 0
\(553\) 0 0
\(554\) 53.9038 2.29015
\(555\) 0 0
\(556\) 12.7019 0.538682
\(557\) −21.1127 −0.894573 −0.447286 0.894391i \(-0.647610\pi\)
−0.447286 + 0.894391i \(0.647610\pi\)
\(558\) 0 0
\(559\) −23.7438 −1.00425
\(560\) 0 0
\(561\) 0 0
\(562\) 37.1638 1.56766
\(563\) 29.8774 1.25918 0.629591 0.776926i \(-0.283222\pi\)
0.629591 + 0.776926i \(0.283222\pi\)
\(564\) 0 0
\(565\) 21.4567 0.902689
\(566\) 53.1468 2.23393
\(567\) 0 0
\(568\) −1.85244 −0.0777267
\(569\) −0.186452 −0.00781648 −0.00390824 0.999992i \(-0.501244\pi\)
−0.00390824 + 0.999992i \(0.501244\pi\)
\(570\) 0 0
\(571\) 3.06682 0.128342 0.0641712 0.997939i \(-0.479560\pi\)
0.0641712 + 0.997939i \(0.479560\pi\)
\(572\) 11.3447 0.474347
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) −18.8495 −0.784715 −0.392357 0.919813i \(-0.628340\pi\)
−0.392357 + 0.919813i \(0.628340\pi\)
\(578\) 34.8697 1.45039
\(579\) 0 0
\(580\) 22.4688 0.932967
\(581\) 0 0
\(582\) 0 0
\(583\) 2.71585 0.112479
\(584\) −10.4791 −0.433627
\(585\) 0 0
\(586\) 30.3510 1.25379
\(587\) −8.11419 −0.334908 −0.167454 0.985880i \(-0.553555\pi\)
−0.167454 + 0.985880i \(0.553555\pi\)
\(588\) 0 0
\(589\) −8.49852 −0.350176
\(590\) 10.9604 0.451232
\(591\) 0 0
\(592\) 18.6483 0.766441
\(593\) 6.36489 0.261375 0.130687 0.991424i \(-0.458282\pi\)
0.130687 + 0.991424i \(0.458282\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −41.5870 −1.70347
\(597\) 0 0
\(598\) 0 0
\(599\) −0.676959 −0.0276598 −0.0138299 0.999904i \(-0.504402\pi\)
−0.0138299 + 0.999904i \(0.504402\pi\)
\(600\) 0 0
\(601\) 22.6964 0.925806 0.462903 0.886409i \(-0.346808\pi\)
0.462903 + 0.886409i \(0.346808\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 11.6615 0.474501
\(605\) 1.64207 0.0667598
\(606\) 0 0
\(607\) −14.6141 −0.593170 −0.296585 0.955006i \(-0.595848\pi\)
−0.296585 + 0.955006i \(0.595848\pi\)
\(608\) −13.1149 −0.531880
\(609\) 0 0
\(610\) 45.7563 1.85262
\(611\) −36.1142 −1.46102
\(612\) 0 0
\(613\) −28.3899 −1.14666 −0.573328 0.819326i \(-0.694348\pi\)
−0.573328 + 0.819326i \(0.694348\pi\)
\(614\) 27.7827 1.12122
\(615\) 0 0
\(616\) 0 0
\(617\) 21.5404 0.867183 0.433591 0.901110i \(-0.357246\pi\)
0.433591 + 0.901110i \(0.357246\pi\)
\(618\) 0 0
\(619\) −23.7827 −0.955906 −0.477953 0.878385i \(-0.658621\pi\)
−0.477953 + 0.878385i \(0.658621\pi\)
\(620\) 21.0154 0.844000
\(621\) 0 0
\(622\) 30.3510 1.21696
\(623\) 0 0
\(624\) 0 0
\(625\) −8.17548 −0.327019
\(626\) 50.2159 2.00703
\(627\) 0 0
\(628\) 27.2702 1.08820
\(629\) −4.71585 −0.188033
\(630\) 0 0
\(631\) −40.4068 −1.60857 −0.804285 0.594244i \(-0.797451\pi\)
−0.804285 + 0.594244i \(0.797451\pi\)
\(632\) 12.4596 0.495617
\(633\) 0 0
\(634\) −13.6615 −0.542568
\(635\) −33.5962 −1.33323
\(636\) 0 0
\(637\) 0 0
\(638\) 11.7026 0.463312
\(639\) 0 0
\(640\) 12.7695 0.504758
\(641\) −13.4876 −0.532726 −0.266363 0.963873i \(-0.585822\pi\)
−0.266363 + 0.963873i \(0.585822\pi\)
\(642\) 0 0
\(643\) −26.8245 −1.05786 −0.528928 0.848667i \(-0.677406\pi\)
−0.528928 + 0.848667i \(0.677406\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 2.48604 0.0978118
\(647\) 42.4791 1.67002 0.835012 0.550231i \(-0.185460\pi\)
0.835012 + 0.550231i \(0.185460\pi\)
\(648\) 0 0
\(649\) 3.15604 0.123885
\(650\) −22.3510 −0.876677
\(651\) 0 0
\(652\) 13.9519 0.546399
\(653\) −7.39281 −0.289303 −0.144652 0.989483i \(-0.546206\pi\)
−0.144652 + 0.989483i \(0.546206\pi\)
\(654\) 0 0
\(655\) −24.7408 −0.966704
\(656\) −14.6506 −0.572008
\(657\) 0 0
\(658\) 0 0
\(659\) −18.2368 −0.710404 −0.355202 0.934790i \(-0.615588\pi\)
−0.355202 + 0.934790i \(0.615588\pi\)
\(660\) 0 0
\(661\) −8.16451 −0.317563 −0.158781 0.987314i \(-0.550757\pi\)
−0.158781 + 0.987314i \(0.550757\pi\)
\(662\) −66.1336 −2.57036
\(663\) 0 0
\(664\) 9.17548 0.356078
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 39.5653 1.53083
\(669\) 0 0
\(670\) −14.6219 −0.564894
\(671\) 13.1755 0.508634
\(672\) 0 0
\(673\) 43.6212 1.68147 0.840737 0.541444i \(-0.182122\pi\)
0.840737 + 0.541444i \(0.182122\pi\)
\(674\) 34.4985 1.32883
\(675\) 0 0
\(676\) 19.8998 0.765377
\(677\) −23.1057 −0.888025 −0.444012 0.896021i \(-0.646445\pi\)
−0.444012 + 0.896021i \(0.646445\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) −1.17548 −0.0450777
\(681\) 0 0
\(682\) 10.9457 0.419131
\(683\) −27.5653 −1.05476 −0.527379 0.849630i \(-0.676825\pi\)
−0.527379 + 0.849630i \(0.676825\pi\)
\(684\) 0 0
\(685\) 23.7438 0.907203
\(686\) 0 0
\(687\) 0 0
\(688\) −14.6506 −0.558547
\(689\) 12.4596 0.474674
\(690\) 0 0
\(691\) 5.85244 0.222637 0.111319 0.993785i \(-0.464493\pi\)
0.111319 + 0.993785i \(0.464493\pi\)
\(692\) −37.1616 −1.41267
\(693\) 0 0
\(694\) −14.5155 −0.551000
\(695\) 8.43466 0.319945
\(696\) 0 0
\(697\) 3.70488 0.140332
\(698\) 50.4574 1.90984
\(699\) 0 0
\(700\) 0 0
\(701\) −9.32304 −0.352126 −0.176063 0.984379i \(-0.556336\pi\)
−0.176063 + 0.984379i \(0.556336\pi\)
\(702\) 0 0
\(703\) 10.8176 0.407992
\(704\) 11.2298 0.423240
\(705\) 0 0
\(706\) −24.8223 −0.934199
\(707\) 0 0
\(708\) 0 0
\(709\) 41.5629 1.56093 0.780463 0.625202i \(-0.214983\pi\)
0.780463 + 0.625202i \(0.214983\pi\)
\(710\) −6.43322 −0.241435
\(711\) 0 0
\(712\) −4.71585 −0.176734
\(713\) 0 0
\(714\) 0 0
\(715\) 7.53341 0.281734
\(716\) −24.4596 −0.914099
\(717\) 0 0
\(718\) −58.7578 −2.19282
\(719\) −31.4372 −1.17241 −0.586205 0.810162i \(-0.699379\pi\)
−0.586205 + 0.810162i \(0.699379\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 34.4806 1.28323
\(723\) 0 0
\(724\) −16.8759 −0.627188
\(725\) −12.7467 −0.473402
\(726\) 0 0
\(727\) 14.2732 0.529363 0.264681 0.964336i \(-0.414733\pi\)
0.264681 + 0.964336i \(0.414733\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) −36.3921 −1.34693
\(731\) 3.70488 0.137030
\(732\) 0 0
\(733\) 2.31207 0.0853983 0.0426992 0.999088i \(-0.486404\pi\)
0.0426992 + 0.999088i \(0.486404\pi\)
\(734\) −56.1600 −2.07291
\(735\) 0 0
\(736\) 0 0
\(737\) −4.21037 −0.155091
\(738\) 0 0
\(739\) −26.0558 −0.958480 −0.479240 0.877684i \(-0.659088\pi\)
−0.479240 + 0.877684i \(0.659088\pi\)
\(740\) −26.7500 −0.983350
\(741\) 0 0
\(742\) 0 0
\(743\) −5.04737 −0.185170 −0.0925851 0.995705i \(-0.529513\pi\)
−0.0925851 + 0.995705i \(0.529513\pi\)
\(744\) 0 0
\(745\) −27.6157 −1.01176
\(746\) −10.7159 −0.392335
\(747\) 0 0
\(748\) −1.77018 −0.0647244
\(749\) 0 0
\(750\) 0 0
\(751\) −40.4387 −1.47563 −0.737815 0.675002i \(-0.764143\pi\)
−0.737815 + 0.675002i \(0.764143\pi\)
\(752\) −22.2834 −0.812593
\(753\) 0 0
\(754\) 53.6887 1.95523
\(755\) 7.74378 0.281825
\(756\) 0 0
\(757\) −24.4263 −0.887788 −0.443894 0.896079i \(-0.646403\pi\)
−0.443894 + 0.896079i \(0.646403\pi\)
\(758\) 64.6344 2.34763
\(759\) 0 0
\(760\) 2.69641 0.0978089
\(761\) 5.18940 0.188116 0.0940579 0.995567i \(-0.470016\pi\)
0.0940579 + 0.995567i \(0.470016\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) −27.2702 −0.986602
\(765\) 0 0
\(766\) −19.3230 −0.698170
\(767\) 14.4791 0.522809
\(768\) 0 0
\(769\) 28.1670 1.01573 0.507864 0.861437i \(-0.330435\pi\)
0.507864 + 0.861437i \(0.330435\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −19.4178 −0.698861
\(773\) 38.2662 1.37634 0.688170 0.725549i \(-0.258414\pi\)
0.688170 + 0.725549i \(0.258414\pi\)
\(774\) 0 0
\(775\) −11.9222 −0.428259
\(776\) −13.6351 −0.489472
\(777\) 0 0
\(778\) 44.9846 1.61277
\(779\) −8.49852 −0.304491
\(780\) 0 0
\(781\) −1.85244 −0.0662856
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 18.1087 0.646326
\(786\) 0 0
\(787\) 13.4247 0.478540 0.239270 0.970953i \(-0.423092\pi\)
0.239270 + 0.970953i \(0.423092\pi\)
\(788\) −49.1880 −1.75225
\(789\) 0 0
\(790\) 43.2702 1.53949
\(791\) 0 0
\(792\) 0 0
\(793\) 60.4457 2.14649
\(794\) 61.2438 2.17346
\(795\) 0 0
\(796\) −58.1770 −2.06203
\(797\) −22.3440 −0.791465 −0.395733 0.918366i \(-0.629509\pi\)
−0.395733 + 0.918366i \(0.629509\pi\)
\(798\) 0 0
\(799\) 5.63511 0.199356
\(800\) −18.3983 −0.650479
\(801\) 0 0
\(802\) 68.4193 2.41597
\(803\) −10.4791 −0.369799
\(804\) 0 0
\(805\) 0 0
\(806\) 50.2159 1.76878
\(807\) 0 0
\(808\) 14.6072 0.513879
\(809\) −31.8455 −1.11963 −0.559814 0.828618i \(-0.689127\pi\)
−0.559814 + 0.828618i \(0.689127\pi\)
\(810\) 0 0
\(811\) −8.70889 −0.305811 −0.152905 0.988241i \(-0.548863\pi\)
−0.152905 + 0.988241i \(0.548863\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) −13.9325 −0.488333
\(815\) 9.26470 0.324528
\(816\) 0 0
\(817\) −8.49852 −0.297326
\(818\) 4.88982 0.170968
\(819\) 0 0
\(820\) 21.0154 0.733891
\(821\) −6.45267 −0.225200 −0.112600 0.993640i \(-0.535918\pi\)
−0.112600 + 0.993640i \(0.535918\pi\)
\(822\) 0 0
\(823\) 25.3719 0.884410 0.442205 0.896914i \(-0.354196\pi\)
0.442205 + 0.896914i \(0.354196\pi\)
\(824\) 17.8913 0.623274
\(825\) 0 0
\(826\) 0 0
\(827\) −14.0194 −0.487504 −0.243752 0.969838i \(-0.578378\pi\)
−0.243752 + 0.969838i \(0.578378\pi\)
\(828\) 0 0
\(829\) 48.9970 1.70174 0.850869 0.525378i \(-0.176076\pi\)
0.850869 + 0.525378i \(0.176076\pi\)
\(830\) 31.8649 1.10605
\(831\) 0 0
\(832\) 51.5195 1.78612
\(833\) 0 0
\(834\) 0 0
\(835\) 26.2732 0.909221
\(836\) 4.06058 0.140438
\(837\) 0 0
\(838\) −81.2680 −2.80736
\(839\) −13.2647 −0.457948 −0.228974 0.973433i \(-0.573537\pi\)
−0.228974 + 0.973433i \(0.573537\pi\)
\(840\) 0 0
\(841\) 1.61862 0.0558144
\(842\) −32.7959 −1.13022
\(843\) 0 0
\(844\) −10.5249 −0.362283
\(845\) 13.2144 0.454588
\(846\) 0 0
\(847\) 0 0
\(848\) 7.68793 0.264005
\(849\) 0 0
\(850\) 3.48755 0.119622
\(851\) 0 0
\(852\) 0 0
\(853\) 4.25622 0.145730 0.0728651 0.997342i \(-0.476786\pi\)
0.0728651 + 0.997342i \(0.476786\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 11.1560 0.381305
\(857\) −23.5264 −0.803648 −0.401824 0.915717i \(-0.631624\pi\)
−0.401824 + 0.915717i \(0.631624\pi\)
\(858\) 0 0
\(859\) 31.0279 1.05866 0.529330 0.848416i \(-0.322444\pi\)
0.529330 + 0.848416i \(0.322444\pi\)
\(860\) 21.0154 0.716620
\(861\) 0 0
\(862\) 80.2959 2.73489
\(863\) 46.3121 1.57648 0.788241 0.615367i \(-0.210992\pi\)
0.788241 + 0.615367i \(0.210992\pi\)
\(864\) 0 0
\(865\) −24.6770 −0.839042
\(866\) 21.3789 0.726484
\(867\) 0 0
\(868\) 0 0
\(869\) 12.4596 0.422664
\(870\) 0 0
\(871\) −19.3161 −0.654500
\(872\) 0.108664 0.00367984
\(873\) 0 0
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −12.8245 −0.433053 −0.216527 0.976277i \(-0.569473\pi\)
−0.216527 + 0.976277i \(0.569473\pi\)
\(878\) −47.6329 −1.60753
\(879\) 0 0
\(880\) 4.64832 0.156695
\(881\) −56.9512 −1.91873 −0.959367 0.282160i \(-0.908949\pi\)
−0.959367 + 0.282160i \(0.908949\pi\)
\(882\) 0 0
\(883\) 8.00696 0.269456 0.134728 0.990883i \(-0.456984\pi\)
0.134728 + 0.990883i \(0.456984\pi\)
\(884\) −8.12115 −0.273144
\(885\) 0 0
\(886\) 43.7004 1.46814
\(887\) 35.7827 1.20146 0.600732 0.799450i \(-0.294876\pi\)
0.600732 + 0.799450i \(0.294876\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) −16.3774 −0.548971
\(891\) 0 0
\(892\) 9.25774 0.309972
\(893\) −12.9262 −0.432559
\(894\) 0 0
\(895\) −16.2423 −0.542920
\(896\) 0 0
\(897\) 0 0
\(898\) 86.3106 2.88022
\(899\) 28.6381 0.955133
\(900\) 0 0
\(901\) −1.94415 −0.0647690
\(902\) 10.9457 0.364451
\(903\) 0 0
\(904\) −13.0668 −0.434596
\(905\) −11.2064 −0.372512
\(906\) 0 0
\(907\) 5.43171 0.180357 0.0901784 0.995926i \(-0.471256\pi\)
0.0901784 + 0.995926i \(0.471256\pi\)
\(908\) −38.2982 −1.27097
\(909\) 0 0
\(910\) 0 0
\(911\) −16.6770 −0.552532 −0.276266 0.961081i \(-0.589097\pi\)
−0.276266 + 0.961081i \(0.589097\pi\)
\(912\) 0 0
\(913\) 9.17548 0.303664
\(914\) 60.0419 1.98601
\(915\) 0 0
\(916\) 19.7827 0.653638
\(917\) 0 0
\(918\) 0 0
\(919\) 14.4038 0.475137 0.237568 0.971371i \(-0.423650\pi\)
0.237568 + 0.971371i \(0.423650\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) −45.2966 −1.49177
\(923\) −8.49852 −0.279732
\(924\) 0 0
\(925\) 15.1755 0.498967
\(926\) −43.8796 −1.44197
\(927\) 0 0
\(928\) 44.1942 1.45075
\(929\) 1.89830 0.0622811 0.0311405 0.999515i \(-0.490086\pi\)
0.0311405 + 0.999515i \(0.490086\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 13.7004 0.448772
\(933\) 0 0
\(934\) −56.8906 −1.86152
\(935\) −1.17548 −0.0384424
\(936\) 0 0
\(937\) 13.1755 0.430424 0.215212 0.976567i \(-0.430956\pi\)
0.215212 + 0.976567i \(0.430956\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 31.9644 1.04256
\(941\) 0.381842 0.0124477 0.00622385 0.999981i \(-0.498019\pi\)
0.00622385 + 0.999981i \(0.498019\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 8.93398 0.290776
\(945\) 0 0
\(946\) 10.9457 0.355874
\(947\) 52.2284 1.69719 0.848597 0.529040i \(-0.177448\pi\)
0.848597 + 0.529040i \(0.177448\pi\)
\(948\) 0 0
\(949\) −48.0753 −1.56059
\(950\) −8.00000 −0.259554
\(951\) 0 0
\(952\) 0 0
\(953\) 45.7757 1.48282 0.741410 0.671052i \(-0.234157\pi\)
0.741410 + 0.671052i \(0.234157\pi\)
\(954\) 0 0
\(955\) −18.1087 −0.585983
\(956\) 3.12739 0.101147
\(957\) 0 0
\(958\) −65.1616 −2.10527
\(959\) 0 0
\(960\) 0 0
\(961\) −4.21438 −0.135948
\(962\) −63.9185 −2.06082
\(963\) 0 0
\(964\) 59.0312 1.90127
\(965\) −12.8943 −0.415082
\(966\) 0 0
\(967\) −15.3230 −0.492756 −0.246378 0.969174i \(-0.579240\pi\)
−0.246378 + 0.969174i \(0.579240\pi\)
\(968\) −1.00000 −0.0321412
\(969\) 0 0
\(970\) −47.3525 −1.52040
\(971\) 12.0753 0.387515 0.193757 0.981049i \(-0.437933\pi\)
0.193757 + 0.981049i \(0.437933\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 49.6740 1.59166
\(975\) 0 0
\(976\) 37.2966 1.19384
\(977\) 39.3650 1.25940 0.629698 0.776840i \(-0.283178\pi\)
0.629698 + 0.776840i \(0.283178\pi\)
\(978\) 0 0
\(979\) −4.71585 −0.150719
\(980\) 0 0
\(981\) 0 0
\(982\) −35.1638 −1.12212
\(983\) 10.5683 0.337076 0.168538 0.985695i \(-0.446095\pi\)
0.168538 + 0.985695i \(0.446095\pi\)
\(984\) 0 0
\(985\) −32.6630 −1.04073
\(986\) −8.37737 −0.266790
\(987\) 0 0
\(988\) 18.6289 0.592663
\(989\) 0 0
\(990\) 0 0
\(991\) 42.8036 1.35970 0.679851 0.733350i \(-0.262044\pi\)
0.679851 + 0.733350i \(0.262044\pi\)
\(992\) 41.3355 1.31240
\(993\) 0 0
\(994\) 0 0
\(995\) −38.6322 −1.22472
\(996\) 0 0
\(997\) 10.3121 0.326587 0.163293 0.986578i \(-0.447788\pi\)
0.163293 + 0.986578i \(0.447788\pi\)
\(998\) 16.9045 0.535104
\(999\) 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 4851.2.a.bm.1.1 3
3.2 odd 2 1617.2.a.q.1.3 3
7.6 odd 2 4851.2.a.bl.1.1 3
21.20 even 2 1617.2.a.r.1.3 yes 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1617.2.a.q.1.3 3 3.2 odd 2
1617.2.a.r.1.3 yes 3 21.20 even 2
4851.2.a.bl.1.1 3 7.6 odd 2
4851.2.a.bm.1.1 3 1.1 even 1 trivial