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Properties

Label 2898.2.a.h.1.1
Level $2898$
Weight $2$
Character 2898.1
Self dual yes
Analytic conductor $23.141$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

Related objects

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2898,2,Mod(1,2898)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2898, 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("2898.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2898 = 2 \cdot 3^{2} \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2898.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: \(23.1406465058\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 322)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 2898.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{4} +2.00000 q^{5} +1.00000 q^{7} -1.00000 q^{8} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{4} +2.00000 q^{5} +1.00000 q^{7} -1.00000 q^{8} -2.00000 q^{10} -6.00000 q^{11} -4.00000 q^{13} -1.00000 q^{14} +1.00000 q^{16} +2.00000 q^{17} +4.00000 q^{19} +2.00000 q^{20} +6.00000 q^{22} -1.00000 q^{23} -1.00000 q^{25} +4.00000 q^{26} +1.00000 q^{28} +10.0000 q^{29} -8.00000 q^{31} -1.00000 q^{32} -2.00000 q^{34} +2.00000 q^{35} -8.00000 q^{37} -4.00000 q^{38} -2.00000 q^{40} +2.00000 q^{41} +6.00000 q^{43} -6.00000 q^{44} +1.00000 q^{46} -12.0000 q^{47} +1.00000 q^{49} +1.00000 q^{50} -4.00000 q^{52} -12.0000 q^{53} -12.0000 q^{55} -1.00000 q^{56} -10.0000 q^{58} +6.00000 q^{59} -6.00000 q^{61} +8.00000 q^{62} +1.00000 q^{64} -8.00000 q^{65} -2.00000 q^{67} +2.00000 q^{68} -2.00000 q^{70} -16.0000 q^{71} +2.00000 q^{73} +8.00000 q^{74} +4.00000 q^{76} -6.00000 q^{77} +2.00000 q^{80} -2.00000 q^{82} -4.00000 q^{83} +4.00000 q^{85} -6.00000 q^{86} +6.00000 q^{88} +6.00000 q^{89} -4.00000 q^{91} -1.00000 q^{92} +12.0000 q^{94} +8.00000 q^{95} +2.00000 q^{97} -1.00000 q^{98} +O(q^{100})\)

Display 100 coefficients 10 coefficients

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) 0 0
\(4\) 1.00000 0.500000
\(5\) 2.00000 0.894427 0.447214 − 0.894427i \(-0.352416\pi\)
0.447214 + 0.894427i \(0.352416\pi\)
\(6\) 0 0
\(7\) 1.00000 0.377964
\(8\) −1.00000 −0.353553
\(9\) 0 0
\(10\) −2.00000 −0.632456
\(11\) −6.00000 −1.80907 −0.904534 − 0.426401i \(-0.859781\pi\)
−0.904534 + 0.426401i \(0.859781\pi\)
\(12\) 0 0
\(13\) −4.00000 −1.10940 −0.554700 − 0.832050i \(-0.687167\pi\)
−0.554700 + 0.832050i \(0.687167\pi\)
\(14\) −1.00000 −0.267261
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 2.00000 0.485071 0.242536 − 0.970143i \(-0.422021\pi\)
0.242536 + 0.970143i \(0.422021\pi\)
\(18\) 0 0
\(19\) 4.00000 0.917663 0.458831 − 0.888523i \(-0.348268\pi\)
0.458831 + 0.888523i \(0.348268\pi\)
\(20\) 2.00000 0.447214
\(21\) 0 0
\(22\) 6.00000 1.27920
\(23\) −1.00000 −0.208514
\(24\) 0 0
\(25\) −1.00000 −0.200000
\(26\) 4.00000 0.784465
\(27\) 0 0
\(28\) 1.00000 0.188982
\(29\) 10.0000 1.85695 0.928477 − 0.371391i \(-0.121119\pi\)
0.928477 + 0.371391i \(0.121119\pi\)
\(30\) 0 0
\(31\) −8.00000 −1.43684 −0.718421 − 0.695608i \(-0.755135\pi\)
−0.718421 + 0.695608i \(0.755135\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0 0
\(34\) −2.00000 −0.342997
\(35\) 2.00000 0.338062
\(36\) 0 0
\(37\) −8.00000 −1.31519 −0.657596 − 0.753371i \(-0.728427\pi\)
−0.657596 + 0.753371i \(0.728427\pi\)
\(38\) −4.00000 −0.648886
\(39\) 0 0
\(40\) −2.00000 −0.316228
\(41\) 2.00000 0.312348 0.156174 − 0.987730i \(-0.450084\pi\)
0.156174 + 0.987730i \(0.450084\pi\)
\(42\) 0 0
\(43\) 6.00000 0.914991 0.457496 − 0.889212i \(-0.348747\pi\)
0.457496 + 0.889212i \(0.348747\pi\)
\(44\) −6.00000 −0.904534
\(45\) 0 0
\(46\) 1.00000 0.147442
\(47\) −12.0000 −1.75038 −0.875190 − 0.483779i \(-0.839264\pi\)
−0.875190 + 0.483779i \(0.839264\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 1.00000 0.141421
\(51\) 0 0
\(52\) −4.00000 −0.554700
\(53\) −12.0000 −1.64833 −0.824163 − 0.566352i \(-0.808354\pi\)
−0.824163 + 0.566352i \(0.808354\pi\)
\(54\) 0 0
\(55\) −12.0000 −1.61808
\(56\) −1.00000 −0.133631
\(57\) 0 0
\(58\) −10.0000 −1.31306
\(59\) 6.00000 0.781133 0.390567 − 0.920575i \(-0.372279\pi\)
0.390567 + 0.920575i \(0.372279\pi\)
\(60\) 0 0
\(61\) −6.00000 −0.768221 −0.384111 − 0.923287i \(-0.625492\pi\)
−0.384111 + 0.923287i \(0.625492\pi\)
\(62\) 8.00000 1.01600
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) −8.00000 −0.992278
\(66\) 0 0
\(67\) −2.00000 −0.244339 −0.122169 − 0.992509i \(-0.538985\pi\)
−0.122169 + 0.992509i \(0.538985\pi\)
\(68\) 2.00000 0.242536
\(69\) 0 0
\(70\) −2.00000 −0.239046
\(71\) −16.0000 −1.89885 −0.949425 − 0.313993i \(-0.898333\pi\)
−0.949425 + 0.313993i \(0.898333\pi\)
\(72\) 0 0
\(73\) 2.00000 0.234082 0.117041 − 0.993127i \(-0.462659\pi\)
0.117041 + 0.993127i \(0.462659\pi\)
\(74\) 8.00000 0.929981
\(75\) 0 0
\(76\) 4.00000 0.458831
\(77\) −6.00000 −0.683763
\(78\) 0 0
\(79\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(80\) 2.00000 0.223607
\(81\) 0 0
\(82\) −2.00000 −0.220863
\(83\) −4.00000 −0.439057 −0.219529 − 0.975606i \(-0.570452\pi\)
−0.219529 + 0.975606i \(0.570452\pi\)
\(84\) 0 0
\(85\) 4.00000 0.433861
\(86\) −6.00000 −0.646997
\(87\) 0 0
\(88\) 6.00000 0.639602
\(89\) 6.00000 0.635999 0.317999 − 0.948091i \(-0.396989\pi\)
0.317999 + 0.948091i \(0.396989\pi\)
\(90\) 0 0
\(91\) −4.00000 −0.419314
\(92\) −1.00000 −0.104257
\(93\) 0 0
\(94\) 12.0000 1.23771
\(95\) 8.00000 0.820783
\(96\) 0 0
\(97\) 2.00000 0.203069 0.101535 − 0.994832i \(-0.467625\pi\)
0.101535 + 0.994832i \(0.467625\pi\)
\(98\) −1.00000 −0.101015
\(99\) 0 0
\(100\) −1.00000 −0.100000
\(101\) −8.00000 −0.796030 −0.398015 − 0.917379i \(-0.630301\pi\)
−0.398015 + 0.917379i \(0.630301\pi\)
\(102\) 0 0
\(103\) −8.00000 −0.788263 −0.394132 − 0.919054i \(-0.628955\pi\)
−0.394132 + 0.919054i \(0.628955\pi\)
\(104\) 4.00000 0.392232
\(105\) 0 0
\(106\) 12.0000 1.16554
\(107\) −6.00000 −0.580042 −0.290021 − 0.957020i \(-0.593662\pi\)
−0.290021 + 0.957020i \(0.593662\pi\)
\(108\) 0 0
\(109\) 4.00000 0.383131 0.191565 − 0.981480i \(-0.438644\pi\)
0.191565 + 0.981480i \(0.438644\pi\)
\(110\) 12.0000 1.14416
\(111\) 0 0
\(112\) 1.00000 0.0944911
\(113\) 14.0000 1.31701 0.658505 − 0.752577i \(-0.271189\pi\)
0.658505 + 0.752577i \(0.271189\pi\)
\(114\) 0 0
\(115\) −2.00000 −0.186501
\(116\) 10.0000 0.928477
\(117\) 0 0
\(118\) −6.00000 −0.552345
\(119\) 2.00000 0.183340
\(120\) 0 0
\(121\) 25.0000 2.27273
\(122\) 6.00000 0.543214
\(123\) 0 0
\(124\) −8.00000 −0.718421
\(125\) −12.0000 −1.07331
\(126\) 0 0
\(127\) 8.00000 0.709885 0.354943 − 0.934888i \(-0.384500\pi\)
0.354943 + 0.934888i \(0.384500\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 0 0
\(130\) 8.00000 0.701646
\(131\) −10.0000 −0.873704 −0.436852 − 0.899533i \(-0.643907\pi\)
−0.436852 + 0.899533i \(0.643907\pi\)
\(132\) 0 0
\(133\) 4.00000 0.346844
\(134\) 2.00000 0.172774
\(135\) 0 0
\(136\) −2.00000 −0.171499
\(137\) −6.00000 −0.512615 −0.256307 − 0.966595i \(-0.582506\pi\)
−0.256307 + 0.966595i \(0.582506\pi\)
\(138\) 0 0
\(139\) −10.0000 −0.848189 −0.424094 − 0.905618i \(-0.639408\pi\)
−0.424094 + 0.905618i \(0.639408\pi\)
\(140\) 2.00000 0.169031
\(141\) 0 0
\(142\) 16.0000 1.34269
\(143\) 24.0000 2.00698
\(144\) 0 0
\(145\) 20.0000 1.66091
\(146\) −2.00000 −0.165521
\(147\) 0 0
\(148\) −8.00000 −0.657596
\(149\) 4.00000 0.327693 0.163846 − 0.986486i \(-0.447610\pi\)
0.163846 + 0.986486i \(0.447610\pi\)
\(150\) 0 0
\(151\) −8.00000 −0.651031 −0.325515 − 0.945537i \(-0.605538\pi\)
−0.325515 + 0.945537i \(0.605538\pi\)
\(152\) −4.00000 −0.324443
\(153\) 0 0
\(154\) 6.00000 0.483494
\(155\) −16.0000 −1.28515
\(156\) 0 0
\(157\) −6.00000 −0.478852 −0.239426 − 0.970915i \(-0.576959\pi\)
−0.239426 + 0.970915i \(0.576959\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) −2.00000 −0.158114
\(161\) −1.00000 −0.0788110
\(162\) 0 0
\(163\) 20.0000 1.56652 0.783260 − 0.621694i \(-0.213555\pi\)
0.783260 + 0.621694i \(0.213555\pi\)
\(164\) 2.00000 0.156174
\(165\) 0 0
\(166\) 4.00000 0.310460
\(167\) 12.0000 0.928588 0.464294 − 0.885681i \(-0.346308\pi\)
0.464294 + 0.885681i \(0.346308\pi\)
\(168\) 0 0
\(169\) 3.00000 0.230769
\(170\) −4.00000 −0.306786
\(171\) 0 0
\(172\) 6.00000 0.457496
\(173\) −24.0000 −1.82469 −0.912343 − 0.409426i \(-0.865729\pi\)
−0.912343 + 0.409426i \(0.865729\pi\)
\(174\) 0 0
\(175\) −1.00000 −0.0755929
\(176\) −6.00000 −0.452267
\(177\) 0 0
\(178\) −6.00000 −0.449719
\(179\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(180\) 0 0
\(181\) −2.00000 −0.148659 −0.0743294 − 0.997234i \(-0.523682\pi\)
−0.0743294 + 0.997234i \(0.523682\pi\)
\(182\) 4.00000 0.296500
\(183\) 0 0
\(184\) 1.00000 0.0737210
\(185\) −16.0000 −1.17634
\(186\) 0 0
\(187\) −12.0000 −0.877527
\(188\) −12.0000 −0.875190
\(189\) 0 0
\(190\) −8.00000 −0.580381
\(191\) −16.0000 −1.15772 −0.578860 − 0.815427i \(-0.696502\pi\)
−0.578860 + 0.815427i \(0.696502\pi\)
\(192\) 0 0
\(193\) −2.00000 −0.143963 −0.0719816 − 0.997406i \(-0.522932\pi\)
−0.0719816 + 0.997406i \(0.522932\pi\)
\(194\) −2.00000 −0.143592
\(195\) 0 0
\(196\) 1.00000 0.0714286
\(197\) −2.00000 −0.142494 −0.0712470 − 0.997459i \(-0.522698\pi\)
−0.0712470 + 0.997459i \(0.522698\pi\)
\(198\) 0 0
\(199\) −16.0000 −1.13421 −0.567105 − 0.823646i \(-0.691937\pi\)
−0.567105 + 0.823646i \(0.691937\pi\)
\(200\) 1.00000 0.0707107
\(201\) 0 0
\(202\) 8.00000 0.562878
\(203\) 10.0000 0.701862
\(204\) 0 0
\(205\) 4.00000 0.279372
\(206\) 8.00000 0.557386
\(207\) 0 0
\(208\) −4.00000 −0.277350
\(209\) −24.0000 −1.66011
\(210\) 0 0
\(211\) −12.0000 −0.826114 −0.413057 − 0.910705i \(-0.635539\pi\)
−0.413057 + 0.910705i \(0.635539\pi\)
\(212\) −12.0000 −0.824163
\(213\) 0 0
\(214\) 6.00000 0.410152
\(215\) 12.0000 0.818393
\(216\) 0 0
\(217\) −8.00000 −0.543075
\(218\) −4.00000 −0.270914
\(219\) 0 0
\(220\) −12.0000 −0.809040
\(221\) −8.00000 −0.538138
\(222\) 0 0
\(223\) 4.00000 0.267860 0.133930 − 0.990991i \(-0.457240\pi\)
0.133930 + 0.990991i \(0.457240\pi\)
\(224\) −1.00000 −0.0668153
\(225\) 0 0
\(226\) −14.0000 −0.931266
\(227\) −24.0000 −1.59294 −0.796468 − 0.604681i \(-0.793301\pi\)
−0.796468 + 0.604681i \(0.793301\pi\)
\(228\) 0 0
\(229\) 22.0000 1.45380 0.726900 − 0.686743i \(-0.240960\pi\)
0.726900 + 0.686743i \(0.240960\pi\)
\(230\) 2.00000 0.131876
\(231\) 0 0
\(232\) −10.0000 −0.656532
\(233\) 6.00000 0.393073 0.196537 − 0.980497i \(-0.437031\pi\)
0.196537 + 0.980497i \(0.437031\pi\)
\(234\) 0 0
\(235\) −24.0000 −1.56559
\(236\) 6.00000 0.390567
\(237\) 0 0
\(238\) −2.00000 −0.129641
\(239\) 24.0000 1.55243 0.776215 − 0.630468i \(-0.217137\pi\)
0.776215 + 0.630468i \(0.217137\pi\)
\(240\) 0 0
\(241\) −6.00000 −0.386494 −0.193247 − 0.981150i \(-0.561902\pi\)
−0.193247 + 0.981150i \(0.561902\pi\)
\(242\) −25.0000 −1.60706
\(243\) 0 0
\(244\) −6.00000 −0.384111
\(245\) 2.00000 0.127775
\(246\) 0 0
\(247\) −16.0000 −1.01806
\(248\) 8.00000 0.508001
\(249\) 0 0
\(250\) 12.0000 0.758947
\(251\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(252\) 0 0
\(253\) 6.00000 0.377217
\(254\) −8.00000 −0.501965
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −2.00000 −0.124757 −0.0623783 − 0.998053i \(-0.519869\pi\)
−0.0623783 + 0.998053i \(0.519869\pi\)
\(258\) 0 0
\(259\) −8.00000 −0.497096
\(260\) −8.00000 −0.496139
\(261\) 0 0
\(262\) 10.0000 0.617802
\(263\) −28.0000 −1.72655 −0.863277 − 0.504730i \(-0.831592\pi\)
−0.863277 + 0.504730i \(0.831592\pi\)
\(264\) 0 0
\(265\) −24.0000 −1.47431
\(266\) −4.00000 −0.245256
\(267\) 0 0
\(268\) −2.00000 −0.122169
\(269\) 24.0000 1.46331 0.731653 − 0.681677i \(-0.238749\pi\)
0.731653 + 0.681677i \(0.238749\pi\)
\(270\) 0 0
\(271\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(272\) 2.00000 0.121268
\(273\) 0 0
\(274\) 6.00000 0.362473
\(275\) 6.00000 0.361814
\(276\) 0 0
\(277\) −14.0000 −0.841178 −0.420589 − 0.907251i \(-0.638177\pi\)
−0.420589 + 0.907251i \(0.638177\pi\)
\(278\) 10.0000 0.599760
\(279\) 0 0
\(280\) −2.00000 −0.119523
\(281\) −2.00000 −0.119310 −0.0596550 − 0.998219i \(-0.519000\pi\)
−0.0596550 + 0.998219i \(0.519000\pi\)
\(282\) 0 0
\(283\) 24.0000 1.42665 0.713326 − 0.700832i \(-0.247188\pi\)
0.713326 + 0.700832i \(0.247188\pi\)
\(284\) −16.0000 −0.949425
\(285\) 0 0
\(286\) −24.0000 −1.41915
\(287\) 2.00000 0.118056
\(288\) 0 0
\(289\) −13.0000 −0.764706
\(290\) −20.0000 −1.17444
\(291\) 0 0
\(292\) 2.00000 0.117041
\(293\) 26.0000 1.51894 0.759468 − 0.650545i \(-0.225459\pi\)
0.759468 + 0.650545i \(0.225459\pi\)
\(294\) 0 0
\(295\) 12.0000 0.698667
\(296\) 8.00000 0.464991
\(297\) 0 0
\(298\) −4.00000 −0.231714
\(299\) 4.00000 0.231326
\(300\) 0 0
\(301\) 6.00000 0.345834
\(302\) 8.00000 0.460348
\(303\) 0 0
\(304\) 4.00000 0.229416
\(305\) −12.0000 −0.687118
\(306\) 0 0
\(307\) −2.00000 −0.114146 −0.0570730 − 0.998370i \(-0.518177\pi\)
−0.0570730 + 0.998370i \(0.518177\pi\)
\(308\) −6.00000 −0.341882
\(309\) 0 0
\(310\) 16.0000 0.908739
\(311\) −28.0000 −1.58773 −0.793867 − 0.608091i \(-0.791935\pi\)
−0.793867 + 0.608091i \(0.791935\pi\)
\(312\) 0 0
\(313\) 2.00000 0.113047 0.0565233 − 0.998401i \(-0.481998\pi\)
0.0565233 + 0.998401i \(0.481998\pi\)
\(314\) 6.00000 0.338600
\(315\) 0 0
\(316\) 0 0
\(317\) 22.0000 1.23564 0.617822 − 0.786318i \(-0.288015\pi\)
0.617822 + 0.786318i \(0.288015\pi\)
\(318\) 0 0
\(319\) −60.0000 −3.35936
\(320\) 2.00000 0.111803
\(321\) 0 0
\(322\) 1.00000 0.0557278
\(323\) 8.00000 0.445132
\(324\) 0 0
\(325\) 4.00000 0.221880
\(326\) −20.0000 −1.10770
\(327\) 0 0
\(328\) −2.00000 −0.110432
\(329\) −12.0000 −0.661581
\(330\) 0 0
\(331\) −8.00000 −0.439720 −0.219860 − 0.975531i \(-0.570560\pi\)
−0.219860 + 0.975531i \(0.570560\pi\)
\(332\) −4.00000 −0.219529
\(333\) 0 0
\(334\) −12.0000 −0.656611
\(335\) −4.00000 −0.218543
\(336\) 0 0
\(337\) −6.00000 −0.326841 −0.163420 − 0.986557i \(-0.552253\pi\)
−0.163420 + 0.986557i \(0.552253\pi\)
\(338\) −3.00000 −0.163178
\(339\) 0 0
\(340\) 4.00000 0.216930
\(341\) 48.0000 2.59935
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) −6.00000 −0.323498
\(345\) 0 0
\(346\) 24.0000 1.29025
\(347\) 12.0000 0.644194 0.322097 − 0.946707i \(-0.395612\pi\)
0.322097 + 0.946707i \(0.395612\pi\)
\(348\) 0 0
\(349\) −8.00000 −0.428230 −0.214115 − 0.976808i \(-0.568687\pi\)
−0.214115 + 0.976808i \(0.568687\pi\)
\(350\) 1.00000 0.0534522
\(351\) 0 0
\(352\) 6.00000 0.319801
\(353\) 14.0000 0.745145 0.372572 − 0.928003i \(-0.378476\pi\)
0.372572 + 0.928003i \(0.378476\pi\)
\(354\) 0 0
\(355\) −32.0000 −1.69838
\(356\) 6.00000 0.317999
\(357\) 0 0
\(358\) 0 0
\(359\) 12.0000 0.633336 0.316668 − 0.948536i \(-0.397436\pi\)
0.316668 + 0.948536i \(0.397436\pi\)
\(360\) 0 0
\(361\) −3.00000 −0.157895
\(362\) 2.00000 0.105118
\(363\) 0 0
\(364\) −4.00000 −0.209657
\(365\) 4.00000 0.209370
\(366\) 0 0
\(367\) −8.00000 −0.417597 −0.208798 − 0.977959i \(-0.566955\pi\)
−0.208798 + 0.977959i \(0.566955\pi\)
\(368\) −1.00000 −0.0521286
\(369\) 0 0
\(370\) 16.0000 0.831800
\(371\) −12.0000 −0.623009
\(372\) 0 0
\(373\) −4.00000 −0.207112 −0.103556 − 0.994624i \(-0.533022\pi\)
−0.103556 + 0.994624i \(0.533022\pi\)
\(374\) 12.0000 0.620505
\(375\) 0 0
\(376\) 12.0000 0.618853
\(377\) −40.0000 −2.06010
\(378\) 0 0
\(379\) 26.0000 1.33553 0.667765 − 0.744372i \(-0.267251\pi\)
0.667765 + 0.744372i \(0.267251\pi\)
\(380\) 8.00000 0.410391
\(381\) 0 0
\(382\) 16.0000 0.818631
\(383\) −8.00000 −0.408781 −0.204390 − 0.978889i \(-0.565521\pi\)
−0.204390 + 0.978889i \(0.565521\pi\)
\(384\) 0 0
\(385\) −12.0000 −0.611577
\(386\) 2.00000 0.101797
\(387\) 0 0
\(388\) 2.00000 0.101535
\(389\) 4.00000 0.202808 0.101404 − 0.994845i \(-0.467667\pi\)
0.101404 + 0.994845i \(0.467667\pi\)
\(390\) 0 0
\(391\) −2.00000 −0.101144
\(392\) −1.00000 −0.0505076
\(393\) 0 0
\(394\) 2.00000 0.100759
\(395\) 0 0
\(396\) 0 0
\(397\) 28.0000 1.40528 0.702640 − 0.711546i \(-0.252005\pi\)
0.702640 + 0.711546i \(0.252005\pi\)
\(398\) 16.0000 0.802008
\(399\) 0 0
\(400\) −1.00000 −0.0500000
\(401\) 26.0000 1.29838 0.649189 − 0.760627i \(-0.275108\pi\)
0.649189 + 0.760627i \(0.275108\pi\)
\(402\) 0 0
\(403\) 32.0000 1.59403
\(404\) −8.00000 −0.398015
\(405\) 0 0
\(406\) −10.0000 −0.496292
\(407\) 48.0000 2.37927
\(408\) 0 0
\(409\) 22.0000 1.08783 0.543915 − 0.839140i \(-0.316941\pi\)
0.543915 + 0.839140i \(0.316941\pi\)
\(410\) −4.00000 −0.197546
\(411\) 0 0
\(412\) −8.00000 −0.394132
\(413\) 6.00000 0.295241
\(414\) 0 0
\(415\) −8.00000 −0.392705
\(416\) 4.00000 0.196116
\(417\) 0 0
\(418\) 24.0000 1.17388
\(419\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(420\) 0 0
\(421\) −12.0000 −0.584844 −0.292422 − 0.956289i \(-0.594461\pi\)
−0.292422 + 0.956289i \(0.594461\pi\)
\(422\) 12.0000 0.584151
\(423\) 0 0
\(424\) 12.0000 0.582772
\(425\) −2.00000 −0.0970143
\(426\) 0 0
\(427\) −6.00000 −0.290360
\(428\) −6.00000 −0.290021
\(429\) 0 0
\(430\) −12.0000 −0.578691
\(431\) −12.0000 −0.578020 −0.289010 − 0.957326i \(-0.593326\pi\)
−0.289010 + 0.957326i \(0.593326\pi\)
\(432\) 0 0
\(433\) −34.0000 −1.63394 −0.816968 − 0.576683i \(-0.804347\pi\)
−0.816968 + 0.576683i \(0.804347\pi\)
\(434\) 8.00000 0.384012
\(435\) 0 0
\(436\) 4.00000 0.191565
\(437\) −4.00000 −0.191346
\(438\) 0 0
\(439\) 8.00000 0.381819 0.190910 − 0.981608i \(-0.438856\pi\)
0.190910 + 0.981608i \(0.438856\pi\)
\(440\) 12.0000 0.572078
\(441\) 0 0
\(442\) 8.00000 0.380521
\(443\) −16.0000 −0.760183 −0.380091 − 0.924949i \(-0.624107\pi\)
−0.380091 + 0.924949i \(0.624107\pi\)
\(444\) 0 0
\(445\) 12.0000 0.568855
\(446\) −4.00000 −0.189405
\(447\) 0 0
\(448\) 1.00000 0.0472456
\(449\) 18.0000 0.849473 0.424736 − 0.905317i \(-0.360367\pi\)
0.424736 + 0.905317i \(0.360367\pi\)
\(450\) 0 0
\(451\) −12.0000 −0.565058
\(452\) 14.0000 0.658505
\(453\) 0 0
\(454\) 24.0000 1.12638
\(455\) −8.00000 −0.375046
\(456\) 0 0
\(457\) −26.0000 −1.21623 −0.608114 − 0.793849i \(-0.708074\pi\)
−0.608114 + 0.793849i \(0.708074\pi\)
\(458\) −22.0000 −1.02799
\(459\) 0 0
\(460\) −2.00000 −0.0932505
\(461\) −8.00000 −0.372597 −0.186299 − 0.982493i \(-0.559649\pi\)
−0.186299 + 0.982493i \(0.559649\pi\)
\(462\) 0 0
\(463\) −32.0000 −1.48717 −0.743583 − 0.668644i \(-0.766875\pi\)
−0.743583 + 0.668644i \(0.766875\pi\)
\(464\) 10.0000 0.464238
\(465\) 0 0
\(466\) −6.00000 −0.277945
\(467\) −28.0000 −1.29569 −0.647843 − 0.761774i \(-0.724329\pi\)
−0.647843 + 0.761774i \(0.724329\pi\)
\(468\) 0 0
\(469\) −2.00000 −0.0923514
\(470\) 24.0000 1.10704
\(471\) 0 0
\(472\) −6.00000 −0.276172
\(473\) −36.0000 −1.65528
\(474\) 0 0
\(475\) −4.00000 −0.183533
\(476\) 2.00000 0.0916698
\(477\) 0 0
\(478\) −24.0000 −1.09773
\(479\) 32.0000 1.46212 0.731059 − 0.682315i \(-0.239027\pi\)
0.731059 + 0.682315i \(0.239027\pi\)
\(480\) 0 0
\(481\) 32.0000 1.45907
\(482\) 6.00000 0.273293
\(483\) 0 0
\(484\) 25.0000 1.13636
\(485\) 4.00000 0.181631
\(486\) 0 0
\(487\) −24.0000 −1.08754 −0.543772 − 0.839233i \(-0.683004\pi\)
−0.543772 + 0.839233i \(0.683004\pi\)
\(488\) 6.00000 0.271607
\(489\) 0 0
\(490\) −2.00000 −0.0903508
\(491\) −16.0000 −0.722070 −0.361035 − 0.932552i \(-0.617576\pi\)
−0.361035 + 0.932552i \(0.617576\pi\)
\(492\) 0 0
\(493\) 20.0000 0.900755
\(494\) 16.0000 0.719874
\(495\) 0 0
\(496\) −8.00000 −0.359211
\(497\) −16.0000 −0.717698
\(498\) 0 0
\(499\) 24.0000 1.07439 0.537194 − 0.843459i \(-0.319484\pi\)
0.537194 + 0.843459i \(0.319484\pi\)
\(500\) −12.0000 −0.536656
\(501\) 0 0
\(502\) 0 0
\(503\) −32.0000 −1.42681 −0.713405 − 0.700752i \(-0.752848\pi\)
−0.713405 + 0.700752i \(0.752848\pi\)
\(504\) 0 0
\(505\) −16.0000 −0.711991
\(506\) −6.00000 −0.266733
\(507\) 0 0
\(508\) 8.00000 0.354943
\(509\) 16.0000 0.709188 0.354594 − 0.935020i \(-0.384619\pi\)
0.354594 + 0.935020i \(0.384619\pi\)
\(510\) 0 0
\(511\) 2.00000 0.0884748
\(512\) −1.00000 −0.0441942
\(513\) 0 0
\(514\) 2.00000 0.0882162
\(515\) −16.0000 −0.705044
\(516\) 0 0
\(517\) 72.0000 3.16656
\(518\) 8.00000 0.351500
\(519\) 0 0
\(520\) 8.00000 0.350823
\(521\) −6.00000 −0.262865 −0.131432 − 0.991325i \(-0.541958\pi\)
−0.131432 + 0.991325i \(0.541958\pi\)
\(522\) 0 0
\(523\) 36.0000 1.57417 0.787085 − 0.616844i \(-0.211589\pi\)
0.787085 + 0.616844i \(0.211589\pi\)
\(524\) −10.0000 −0.436852
\(525\) 0 0
\(526\) 28.0000 1.22086
\(527\) −16.0000 −0.696971
\(528\) 0 0
\(529\) 1.00000 0.0434783
\(530\) 24.0000 1.04249
\(531\) 0 0
\(532\) 4.00000 0.173422
\(533\) −8.00000 −0.346518
\(534\) 0 0
\(535\) −12.0000 −0.518805
\(536\) 2.00000 0.0863868
\(537\) 0 0
\(538\) −24.0000 −1.03471
\(539\) −6.00000 −0.258438
\(540\) 0 0
\(541\) −14.0000 −0.601907 −0.300954 − 0.953639i \(-0.597305\pi\)
−0.300954 + 0.953639i \(0.597305\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) −2.00000 −0.0857493
\(545\) 8.00000 0.342682
\(546\) 0 0
\(547\) −32.0000 −1.36822 −0.684111 − 0.729378i \(-0.739809\pi\)
−0.684111 + 0.729378i \(0.739809\pi\)
\(548\) −6.00000 −0.256307
\(549\) 0 0
\(550\) −6.00000 −0.255841
\(551\) 40.0000 1.70406
\(552\) 0 0
\(553\) 0 0
\(554\) 14.0000 0.594803
\(555\) 0 0
\(556\) −10.0000 −0.424094
\(557\) −24.0000 −1.01691 −0.508456 − 0.861088i \(-0.669784\pi\)
−0.508456 + 0.861088i \(0.669784\pi\)
\(558\) 0 0
\(559\) −24.0000 −1.01509
\(560\) 2.00000 0.0845154
\(561\) 0 0
\(562\) 2.00000 0.0843649
\(563\) −36.0000 −1.51722 −0.758610 − 0.651546i \(-0.774121\pi\)
−0.758610 + 0.651546i \(0.774121\pi\)
\(564\) 0 0
\(565\) 28.0000 1.17797
\(566\) −24.0000 −1.00880
\(567\) 0 0
\(568\) 16.0000 0.671345
\(569\) −6.00000 −0.251533 −0.125767 − 0.992060i \(-0.540139\pi\)
−0.125767 + 0.992060i \(0.540139\pi\)
\(570\) 0 0
\(571\) 26.0000 1.08807 0.544033 − 0.839064i \(-0.316897\pi\)
0.544033 + 0.839064i \(0.316897\pi\)
\(572\) 24.0000 1.00349
\(573\) 0 0
\(574\) −2.00000 −0.0834784
\(575\) 1.00000 0.0417029
\(576\) 0 0
\(577\) −6.00000 −0.249783 −0.124892 − 0.992170i \(-0.539858\pi\)
−0.124892 + 0.992170i \(0.539858\pi\)
\(578\) 13.0000 0.540729
\(579\) 0 0
\(580\) 20.0000 0.830455
\(581\) −4.00000 −0.165948
\(582\) 0 0
\(583\) 72.0000 2.98194
\(584\) −2.00000 −0.0827606
\(585\) 0 0
\(586\) −26.0000 −1.07405
\(587\) 42.0000 1.73353 0.866763 − 0.498721i \(-0.166197\pi\)
0.866763 + 0.498721i \(0.166197\pi\)
\(588\) 0 0
\(589\) −32.0000 −1.31854
\(590\) −12.0000 −0.494032
\(591\) 0 0
\(592\) −8.00000 −0.328798
\(593\) 30.0000 1.23195 0.615976 − 0.787765i \(-0.288762\pi\)
0.615976 + 0.787765i \(0.288762\pi\)
\(594\) 0 0
\(595\) 4.00000 0.163984
\(596\) 4.00000 0.163846
\(597\) 0 0
\(598\) −4.00000 −0.163572
\(599\) 24.0000 0.980613 0.490307 − 0.871550i \(-0.336885\pi\)
0.490307 + 0.871550i \(0.336885\pi\)
\(600\) 0 0
\(601\) 2.00000 0.0815817 0.0407909 − 0.999168i \(-0.487012\pi\)
0.0407909 + 0.999168i \(0.487012\pi\)
\(602\) −6.00000 −0.244542
\(603\) 0 0
\(604\) −8.00000 −0.325515
\(605\) 50.0000 2.03279
\(606\) 0 0
\(607\) 36.0000 1.46119 0.730597 − 0.682808i \(-0.239242\pi\)
0.730597 + 0.682808i \(0.239242\pi\)
\(608\) −4.00000 −0.162221
\(609\) 0 0
\(610\) 12.0000 0.485866
\(611\) 48.0000 1.94187
\(612\) 0 0
\(613\) 36.0000 1.45403 0.727013 − 0.686624i \(-0.240908\pi\)
0.727013 + 0.686624i \(0.240908\pi\)
\(614\) 2.00000 0.0807134
\(615\) 0 0
\(616\) 6.00000 0.241747
\(617\) 30.0000 1.20775 0.603877 − 0.797077i \(-0.293622\pi\)
0.603877 + 0.797077i \(0.293622\pi\)
\(618\) 0 0
\(619\) −40.0000 −1.60774 −0.803868 − 0.594808i \(-0.797228\pi\)
−0.803868 + 0.594808i \(0.797228\pi\)
\(620\) −16.0000 −0.642575
\(621\) 0 0
\(622\) 28.0000 1.12270
\(623\) 6.00000 0.240385
\(624\) 0 0
\(625\) −19.0000 −0.760000
\(626\) −2.00000 −0.0799361
\(627\) 0 0
\(628\) −6.00000 −0.239426
\(629\) −16.0000 −0.637962
\(630\) 0 0
\(631\) 4.00000 0.159237 0.0796187 − 0.996825i \(-0.474630\pi\)
0.0796187 + 0.996825i \(0.474630\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) −22.0000 −0.873732
\(635\) 16.0000 0.634941
\(636\) 0 0
\(637\) −4.00000 −0.158486
\(638\) 60.0000 2.37542
\(639\) 0 0
\(640\) −2.00000 −0.0790569
\(641\) 18.0000 0.710957 0.355479 − 0.934684i \(-0.384318\pi\)
0.355479 + 0.934684i \(0.384318\pi\)
\(642\) 0 0
\(643\) 4.00000 0.157745 0.0788723 − 0.996885i \(-0.474868\pi\)
0.0788723 + 0.996885i \(0.474868\pi\)
\(644\) −1.00000 −0.0394055
\(645\) 0 0
\(646\) −8.00000 −0.314756
\(647\) −32.0000 −1.25805 −0.629025 − 0.777385i \(-0.716546\pi\)
−0.629025 + 0.777385i \(0.716546\pi\)
\(648\) 0 0
\(649\) −36.0000 −1.41312
\(650\) −4.00000 −0.156893
\(651\) 0 0
\(652\) 20.0000 0.783260
\(653\) −26.0000 −1.01746 −0.508729 − 0.860927i \(-0.669885\pi\)
−0.508729 + 0.860927i \(0.669885\pi\)
\(654\) 0 0
\(655\) −20.0000 −0.781465
\(656\) 2.00000 0.0780869
\(657\) 0 0
\(658\) 12.0000 0.467809
\(659\) −10.0000 −0.389545 −0.194772 − 0.980848i \(-0.562397\pi\)
−0.194772 + 0.980848i \(0.562397\pi\)
\(660\) 0 0
\(661\) 26.0000 1.01128 0.505641 − 0.862744i \(-0.331256\pi\)
0.505641 + 0.862744i \(0.331256\pi\)
\(662\) 8.00000 0.310929
\(663\) 0 0
\(664\) 4.00000 0.155230
\(665\) 8.00000 0.310227
\(666\) 0 0
\(667\) −10.0000 −0.387202
\(668\) 12.0000 0.464294
\(669\) 0 0
\(670\) 4.00000 0.154533
\(671\) 36.0000 1.38976
\(672\) 0 0
\(673\) −50.0000 −1.92736 −0.963679 − 0.267063i \(-0.913947\pi\)
−0.963679 + 0.267063i \(0.913947\pi\)
\(674\) 6.00000 0.231111
\(675\) 0 0
\(676\) 3.00000 0.115385
\(677\) 14.0000 0.538064 0.269032 − 0.963131i \(-0.413296\pi\)
0.269032 + 0.963131i \(0.413296\pi\)
\(678\) 0 0
\(679\) 2.00000 0.0767530
\(680\) −4.00000 −0.153393
\(681\) 0 0
\(682\) −48.0000 −1.83801
\(683\) 44.0000 1.68361 0.841807 − 0.539779i \(-0.181492\pi\)
0.841807 + 0.539779i \(0.181492\pi\)
\(684\) 0 0
\(685\) −12.0000 −0.458496
\(686\) −1.00000 −0.0381802
\(687\) 0 0
\(688\) 6.00000 0.228748
\(689\) 48.0000 1.82865
\(690\) 0 0
\(691\) −22.0000 −0.836919 −0.418460 − 0.908235i \(-0.637430\pi\)
−0.418460 + 0.908235i \(0.637430\pi\)
\(692\) −24.0000 −0.912343
\(693\) 0 0
\(694\) −12.0000 −0.455514
\(695\) −20.0000 −0.758643
\(696\) 0 0
\(697\) 4.00000 0.151511
\(698\) 8.00000 0.302804
\(699\) 0 0
\(700\) −1.00000 −0.0377964
\(701\) 4.00000 0.151078 0.0755390 − 0.997143i \(-0.475932\pi\)
0.0755390 + 0.997143i \(0.475932\pi\)
\(702\) 0 0
\(703\) −32.0000 −1.20690
\(704\) −6.00000 −0.226134
\(705\) 0 0
\(706\) −14.0000 −0.526897
\(707\) −8.00000 −0.300871
\(708\) 0 0
\(709\) 32.0000 1.20179 0.600893 − 0.799330i \(-0.294812\pi\)
0.600893 + 0.799330i \(0.294812\pi\)
\(710\) 32.0000 1.20094
\(711\) 0 0
\(712\) −6.00000 −0.224860
\(713\) 8.00000 0.299602
\(714\) 0 0
\(715\) 48.0000 1.79510
\(716\) 0 0
\(717\) 0 0
\(718\) −12.0000 −0.447836
\(719\) 48.0000 1.79010 0.895049 − 0.445968i \(-0.147140\pi\)
0.895049 + 0.445968i \(0.147140\pi\)
\(720\) 0 0
\(721\) −8.00000 −0.297936
\(722\) 3.00000 0.111648
\(723\) 0 0
\(724\) −2.00000 −0.0743294
\(725\) −10.0000 −0.371391
\(726\) 0 0
\(727\) 32.0000 1.18681 0.593407 − 0.804902i \(-0.297782\pi\)
0.593407 + 0.804902i \(0.297782\pi\)
\(728\) 4.00000 0.148250
\(729\) 0 0
\(730\) −4.00000 −0.148047
\(731\) 12.0000 0.443836
\(732\) 0 0
\(733\) 14.0000 0.517102 0.258551 − 0.965998i \(-0.416755\pi\)
0.258551 + 0.965998i \(0.416755\pi\)
\(734\) 8.00000 0.295285
\(735\) 0 0
\(736\) 1.00000 0.0368605
\(737\) 12.0000 0.442026
\(738\) 0 0
\(739\) 28.0000 1.03000 0.514998 − 0.857191i \(-0.327793\pi\)
0.514998 + 0.857191i \(0.327793\pi\)
\(740\) −16.0000 −0.588172
\(741\) 0 0
\(742\) 12.0000 0.440534
\(743\) −24.0000 −0.880475 −0.440237 − 0.897881i \(-0.645106\pi\)
−0.440237 + 0.897881i \(0.645106\pi\)
\(744\) 0 0
\(745\) 8.00000 0.293097
\(746\) 4.00000 0.146450
\(747\) 0 0
\(748\) −12.0000 −0.438763
\(749\) −6.00000 −0.219235
\(750\) 0 0
\(751\) 8.00000 0.291924 0.145962 − 0.989290i \(-0.453372\pi\)
0.145962 + 0.989290i \(0.453372\pi\)
\(752\) −12.0000 −0.437595
\(753\) 0 0
\(754\) 40.0000 1.45671
\(755\) −16.0000 −0.582300
\(756\) 0 0
\(757\) −28.0000 −1.01768 −0.508839 − 0.860862i \(-0.669925\pi\)
−0.508839 + 0.860862i \(0.669925\pi\)
\(758\) −26.0000 −0.944363
\(759\) 0 0
\(760\) −8.00000 −0.290191
\(761\) 42.0000 1.52250 0.761249 − 0.648459i \(-0.224586\pi\)
0.761249 + 0.648459i \(0.224586\pi\)
\(762\) 0 0
\(763\) 4.00000 0.144810
\(764\) −16.0000 −0.578860
\(765\) 0 0
\(766\) 8.00000 0.289052
\(767\) −24.0000 −0.866590
\(768\) 0 0
\(769\) 30.0000 1.08183 0.540914 − 0.841078i \(-0.318079\pi\)
0.540914 + 0.841078i \(0.318079\pi\)
\(770\) 12.0000 0.432450
\(771\) 0 0
\(772\) −2.00000 −0.0719816
\(773\) −34.0000 −1.22290 −0.611448 − 0.791285i \(-0.709412\pi\)
−0.611448 + 0.791285i \(0.709412\pi\)
\(774\) 0 0
\(775\) 8.00000 0.287368
\(776\) −2.00000 −0.0717958
\(777\) 0 0
\(778\) −4.00000 −0.143407
\(779\) 8.00000 0.286630
\(780\) 0 0
\(781\) 96.0000 3.43515
\(782\) 2.00000 0.0715199
\(783\) 0 0
\(784\) 1.00000 0.0357143
\(785\) −12.0000 −0.428298
\(786\) 0 0
\(787\) 40.0000 1.42585 0.712923 − 0.701242i \(-0.247371\pi\)
0.712923 + 0.701242i \(0.247371\pi\)
\(788\) −2.00000 −0.0712470
\(789\) 0 0
\(790\) 0 0
\(791\) 14.0000 0.497783
\(792\) 0 0
\(793\) 24.0000 0.852265
\(794\) −28.0000 −0.993683
\(795\) 0 0
\(796\) −16.0000 −0.567105
\(797\) 18.0000 0.637593 0.318796 − 0.947823i \(-0.396721\pi\)
0.318796 + 0.947823i \(0.396721\pi\)
\(798\) 0 0
\(799\) −24.0000 −0.849059
\(800\) 1.00000 0.0353553
\(801\) 0 0
\(802\) −26.0000 −0.918092
\(803\) −12.0000 −0.423471
\(804\) 0 0
\(805\) −2.00000 −0.0704907
\(806\) −32.0000 −1.12715
\(807\) 0 0
\(808\) 8.00000 0.281439
\(809\) −38.0000 −1.33601 −0.668004 − 0.744157i \(-0.732851\pi\)
−0.668004 + 0.744157i \(0.732851\pi\)
\(810\) 0 0
\(811\) −14.0000 −0.491606 −0.245803 − 0.969320i \(-0.579052\pi\)
−0.245803 + 0.969320i \(0.579052\pi\)
\(812\) 10.0000 0.350931
\(813\) 0 0
\(814\) −48.0000 −1.68240
\(815\) 40.0000 1.40114
\(816\) 0 0
\(817\) 24.0000 0.839654
\(818\) −22.0000 −0.769212
\(819\) 0 0
\(820\) 4.00000 0.139686
\(821\) −14.0000 −0.488603 −0.244302 − 0.969699i \(-0.578559\pi\)
−0.244302 + 0.969699i \(0.578559\pi\)
\(822\) 0 0
\(823\) −32.0000 −1.11545 −0.557725 − 0.830026i \(-0.688326\pi\)
−0.557725 + 0.830026i \(0.688326\pi\)
\(824\) 8.00000 0.278693
\(825\) 0 0
\(826\) −6.00000 −0.208767
\(827\) 10.0000 0.347734 0.173867 − 0.984769i \(-0.444374\pi\)
0.173867 + 0.984769i \(0.444374\pi\)
\(828\) 0 0
\(829\) −20.0000 −0.694629 −0.347314 − 0.937749i \(-0.612906\pi\)
−0.347314 + 0.937749i \(0.612906\pi\)
\(830\) 8.00000 0.277684
\(831\) 0 0
\(832\) −4.00000 −0.138675
\(833\) 2.00000 0.0692959
\(834\) 0 0
\(835\) 24.0000 0.830554
\(836\) −24.0000 −0.830057
\(837\) 0 0
\(838\) 0 0
\(839\) 32.0000 1.10476 0.552381 − 0.833592i \(-0.313719\pi\)
0.552381 + 0.833592i \(0.313719\pi\)
\(840\) 0 0
\(841\) 71.0000 2.44828
\(842\) 12.0000 0.413547
\(843\) 0 0
\(844\) −12.0000 −0.413057
\(845\) 6.00000 0.206406
\(846\) 0 0
\(847\) 25.0000 0.859010
\(848\) −12.0000 −0.412082
\(849\) 0 0
\(850\) 2.00000 0.0685994
\(851\) 8.00000 0.274236
\(852\) 0 0
\(853\) −12.0000 −0.410872 −0.205436 − 0.978671i \(-0.565861\pi\)
−0.205436 + 0.978671i \(0.565861\pi\)
\(854\) 6.00000 0.205316
\(855\) 0 0
\(856\) 6.00000 0.205076
\(857\) −22.0000 −0.751506 −0.375753 − 0.926720i \(-0.622616\pi\)
−0.375753 + 0.926720i \(0.622616\pi\)
\(858\) 0 0
\(859\) −14.0000 −0.477674 −0.238837 − 0.971060i \(-0.576766\pi\)
−0.238837 + 0.971060i \(0.576766\pi\)
\(860\) 12.0000 0.409197
\(861\) 0 0
\(862\) 12.0000 0.408722
\(863\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(864\) 0 0
\(865\) −48.0000 −1.63205
\(866\) 34.0000 1.15537
\(867\) 0 0
\(868\) −8.00000 −0.271538
\(869\) 0 0
\(870\) 0 0
\(871\) 8.00000 0.271070
\(872\) −4.00000 −0.135457
\(873\) 0 0
\(874\) 4.00000 0.135302
\(875\) −12.0000 −0.405674
\(876\) 0 0
\(877\) −10.0000 −0.337676 −0.168838 − 0.985644i \(-0.554001\pi\)
−0.168838 + 0.985644i \(0.554001\pi\)
\(878\) −8.00000 −0.269987
\(879\) 0 0
\(880\) −12.0000 −0.404520
\(881\) 42.0000 1.41502 0.707508 − 0.706705i \(-0.249819\pi\)
0.707508 + 0.706705i \(0.249819\pi\)
\(882\) 0 0
\(883\) 44.0000 1.48072 0.740359 − 0.672212i \(-0.234656\pi\)
0.740359 + 0.672212i \(0.234656\pi\)
\(884\) −8.00000 −0.269069
\(885\) 0 0
\(886\) 16.0000 0.537531
\(887\) −12.0000 −0.402921 −0.201460 − 0.979497i \(-0.564569\pi\)
−0.201460 + 0.979497i \(0.564569\pi\)
\(888\) 0 0
\(889\) 8.00000 0.268311
\(890\) −12.0000 −0.402241
\(891\) 0 0
\(892\) 4.00000 0.133930
\(893\) −48.0000 −1.60626
\(894\) 0 0
\(895\) 0 0
\(896\) −1.00000 −0.0334077
\(897\) 0 0
\(898\) −18.0000 −0.600668
\(899\) −80.0000 −2.66815
\(900\) 0 0
\(901\) −24.0000 −0.799556
\(902\) 12.0000 0.399556
\(903\) 0 0
\(904\) −14.0000 −0.465633
\(905\) −4.00000 −0.132964
\(906\) 0 0
\(907\) 10.0000 0.332045 0.166022 − 0.986122i \(-0.446908\pi\)
0.166022 + 0.986122i \(0.446908\pi\)
\(908\) −24.0000 −0.796468
\(909\) 0 0
\(910\) 8.00000 0.265197
\(911\) 20.0000 0.662630 0.331315 − 0.943520i \(-0.392508\pi\)
0.331315 + 0.943520i \(0.392508\pi\)
\(912\) 0 0
\(913\) 24.0000 0.794284
\(914\) 26.0000 0.860004
\(915\) 0 0
\(916\) 22.0000 0.726900
\(917\) −10.0000 −0.330229
\(918\) 0 0
\(919\) −40.0000 −1.31948 −0.659739 − 0.751495i \(-0.729333\pi\)
−0.659739 + 0.751495i \(0.729333\pi\)
\(920\) 2.00000 0.0659380
\(921\) 0 0
\(922\) 8.00000 0.263466
\(923\) 64.0000 2.10659
\(924\) 0 0
\(925\) 8.00000 0.263038
\(926\) 32.0000 1.05159
\(927\) 0 0
\(928\) −10.0000 −0.328266
\(929\) 10.0000 0.328089 0.164045 − 0.986453i \(-0.447546\pi\)
0.164045 + 0.986453i \(0.447546\pi\)
\(930\) 0 0
\(931\) 4.00000 0.131095
\(932\) 6.00000 0.196537
\(933\) 0 0
\(934\) 28.0000 0.916188
\(935\) −24.0000 −0.784884
\(936\) 0 0
\(937\) 26.0000 0.849383 0.424691 − 0.905338i \(-0.360383\pi\)
0.424691 + 0.905338i \(0.360383\pi\)
\(938\) 2.00000 0.0653023
\(939\) 0 0
\(940\) −24.0000 −0.782794
\(941\) 42.0000 1.36916 0.684580 − 0.728937i \(-0.259985\pi\)
0.684580 + 0.728937i \(0.259985\pi\)
\(942\) 0 0
\(943\) −2.00000 −0.0651290
\(944\) 6.00000 0.195283
\(945\) 0 0
\(946\) 36.0000 1.17046
\(947\) −36.0000 −1.16984 −0.584921 − 0.811090i \(-0.698875\pi\)
−0.584921 + 0.811090i \(0.698875\pi\)
\(948\) 0 0
\(949\) −8.00000 −0.259691
\(950\) 4.00000 0.129777
\(951\) 0 0
\(952\) −2.00000 −0.0648204
\(953\) 26.0000 0.842223 0.421111 − 0.907009i \(-0.361640\pi\)
0.421111 + 0.907009i \(0.361640\pi\)
\(954\) 0 0
\(955\) −32.0000 −1.03550
\(956\) 24.0000 0.776215
\(957\) 0 0
\(958\) −32.0000 −1.03387
\(959\) −6.00000 −0.193750
\(960\) 0 0
\(961\) 33.0000 1.06452
\(962\) −32.0000 −1.03172
\(963\) 0 0
\(964\) −6.00000 −0.193247
\(965\) −4.00000 −0.128765
\(966\) 0 0
\(967\) 8.00000 0.257263 0.128631 − 0.991692i \(-0.458942\pi\)
0.128631 + 0.991692i \(0.458942\pi\)
\(968\) −25.0000 −0.803530
\(969\) 0 0
\(970\) −4.00000 −0.128432
\(971\) 8.00000 0.256732 0.128366 − 0.991727i \(-0.459027\pi\)
0.128366 + 0.991727i \(0.459027\pi\)
\(972\) 0 0
\(973\) −10.0000 −0.320585
\(974\) 24.0000 0.769010
\(975\) 0 0
\(976\) −6.00000 −0.192055
\(977\) −6.00000 −0.191957 −0.0959785 − 0.995383i \(-0.530598\pi\)
−0.0959785 + 0.995383i \(0.530598\pi\)
\(978\) 0 0
\(979\) −36.0000 −1.15056
\(980\) 2.00000 0.0638877
\(981\) 0 0
\(982\) 16.0000 0.510581
\(983\) −40.0000 −1.27580 −0.637901 − 0.770118i \(-0.720197\pi\)
−0.637901 + 0.770118i \(0.720197\pi\)
\(984\) 0 0
\(985\) −4.00000 −0.127451
\(986\) −20.0000 −0.636930
\(987\) 0 0
\(988\) −16.0000 −0.509028
\(989\) −6.00000 −0.190789
\(990\) 0 0
\(991\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(992\) 8.00000 0.254000
\(993\) 0 0
\(994\) 16.0000 0.507489
\(995\) −32.0000 −1.01447
\(996\) 0 0
\(997\) 12.0000 0.380044 0.190022 − 0.981780i \(-0.439144\pi\)
0.190022 + 0.981780i \(0.439144\pi\)
\(998\) −24.0000 −0.759707
\(999\) 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 2898.2.a.h.1.1 1
3.2 odd 2 322.2.a.d.1.1 ✓ 1
12.11 even 2 2576.2.a.b.1.1 1
15.14 odd 2 8050.2.a.a.1.1 1
21.20 even 2 2254.2.a.f.1.1 1
69.68 even 2 7406.2.a.i.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
322.2.a.d.1.1 ✓ 1 3.2 odd 2
2254.2.a.f.1.1 1 21.20 even 2
2576.2.a.b.1.1 1 12.11 even 2
2898.2.a.h.1.1 1 1.1 even 1 trivial
7406.2.a.i.1.1 1 69.68 even 2
8050.2.a.a.1.1 1 15.14 odd 2