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Properties

Label 4998.2.a.bp.1.1
Level $4998$
Weight $2$
Character 4998.1
Self dual yes
Analytic conductor $39.909$
Analytic rank $0$
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] = [4998,2,Mod(1,4998)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4998, 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("4998.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4998 = 2 \cdot 3 \cdot 7^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4998.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: \(39.9092309302\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 714)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 4998.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +2.00000 q^{5} +1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +2.00000 q^{5} +1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} +2.00000 q^{10} +1.00000 q^{12} +6.00000 q^{13} +2.00000 q^{15} +1.00000 q^{16} -1.00000 q^{17} +1.00000 q^{18} +2.00000 q^{20} -8.00000 q^{23} +1.00000 q^{24} -1.00000 q^{25} +6.00000 q^{26} +1.00000 q^{27} -6.00000 q^{29} +2.00000 q^{30} +8.00000 q^{31} +1.00000 q^{32} -1.00000 q^{34} +1.00000 q^{36} +10.0000 q^{37} +6.00000 q^{39} +2.00000 q^{40} +6.00000 q^{41} +12.0000 q^{43} +2.00000 q^{45} -8.00000 q^{46} +1.00000 q^{48} -1.00000 q^{50} -1.00000 q^{51} +6.00000 q^{52} -10.0000 q^{53} +1.00000 q^{54} -6.00000 q^{58} +8.00000 q^{59} +2.00000 q^{60} -6.00000 q^{61} +8.00000 q^{62} +1.00000 q^{64} +12.0000 q^{65} +12.0000 q^{67} -1.00000 q^{68} -8.00000 q^{69} +1.00000 q^{72} +6.00000 q^{73} +10.0000 q^{74} -1.00000 q^{75} +6.00000 q^{78} -8.00000 q^{79} +2.00000 q^{80} +1.00000 q^{81} +6.00000 q^{82} -16.0000 q^{83} -2.00000 q^{85} +12.0000 q^{86} -6.00000 q^{87} -2.00000 q^{89} +2.00000 q^{90} -8.00000 q^{92} +8.00000 q^{93} +1.00000 q^{96} -2.00000 q^{97} +O(q^{100})\)

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
\(3\) 1.00000 0.577350
\(4\) 1.00000 0.500000
\(5\) 2.00000 0.894427 0.447214 − 0.894427i \(-0.352416\pi\)
0.447214 + 0.894427i \(0.352416\pi\)
\(6\) 1.00000 0.408248
\(7\) 0 0
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) 2.00000 0.632456
\(11\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) 1.00000 0.288675
\(13\) 6.00000 1.66410 0.832050 − 0.554700i \(-0.187167\pi\)
0.832050 + 0.554700i \(0.187167\pi\)
\(14\) 0 0
\(15\) 2.00000 0.516398
\(16\) 1.00000 0.250000
\(17\) −1.00000 −0.242536
\(18\) 1.00000 0.235702
\(19\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(20\) 2.00000 0.447214
\(21\) 0 0
\(22\) 0 0
\(23\) −8.00000 −1.66812 −0.834058 − 0.551677i \(-0.813988\pi\)
−0.834058 + 0.551677i \(0.813988\pi\)
\(24\) 1.00000 0.204124
\(25\) −1.00000 −0.200000
\(26\) 6.00000 1.17670
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) −6.00000 −1.11417 −0.557086 − 0.830455i \(-0.688081\pi\)
−0.557086 + 0.830455i \(0.688081\pi\)
\(30\) 2.00000 0.365148
\(31\) 8.00000 1.43684 0.718421 − 0.695608i \(-0.244865\pi\)
0.718421 + 0.695608i \(0.244865\pi\)
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) −1.00000 −0.171499
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) 10.0000 1.64399 0.821995 − 0.569495i \(-0.192861\pi\)
0.821995 + 0.569495i \(0.192861\pi\)
\(38\) 0 0
\(39\) 6.00000 0.960769
\(40\) 2.00000 0.316228
\(41\) 6.00000 0.937043 0.468521 − 0.883452i \(-0.344787\pi\)
0.468521 + 0.883452i \(0.344787\pi\)
\(42\) 0 0
\(43\) 12.0000 1.82998 0.914991 − 0.403473i \(-0.132197\pi\)
0.914991 + 0.403473i \(0.132197\pi\)
\(44\) 0 0
\(45\) 2.00000 0.298142
\(46\) −8.00000 −1.17954
\(47\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 1.00000 0.144338
\(49\) 0 0
\(50\) −1.00000 −0.141421
\(51\) −1.00000 −0.140028
\(52\) 6.00000 0.832050
\(53\) −10.0000 −1.37361 −0.686803 − 0.726844i \(-0.740986\pi\)
−0.686803 + 0.726844i \(0.740986\pi\)
\(54\) 1.00000 0.136083
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) −6.00000 −0.787839
\(59\) 8.00000 1.04151 0.520756 − 0.853706i \(-0.325650\pi\)
0.520756 + 0.853706i \(0.325650\pi\)
\(60\) 2.00000 0.258199
\(61\) −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
\(65\) 12.0000 1.48842
\(66\) 0 0
\(67\) 12.0000 1.46603 0.733017 − 0.680211i \(-0.238112\pi\)
0.733017 + 0.680211i \(0.238112\pi\)
\(68\) −1.00000 −0.121268
\(69\) −8.00000 −0.963087
\(70\) 0 0
\(71\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 1.00000 0.117851
\(73\) 6.00000 0.702247 0.351123 − 0.936329i \(-0.385800\pi\)
0.351123 + 0.936329i \(0.385800\pi\)
\(74\) 10.0000 1.16248
\(75\) −1.00000 −0.115470
\(76\) 0 0
\(77\) 0 0
\(78\) 6.00000 0.679366
\(79\) −8.00000 −0.900070 −0.450035 − 0.893011i \(-0.648589\pi\)
−0.450035 + 0.893011i \(0.648589\pi\)
\(80\) 2.00000 0.223607
\(81\) 1.00000 0.111111
\(82\) 6.00000 0.662589
\(83\) −16.0000 −1.75623 −0.878114 − 0.478451i \(-0.841198\pi\)
−0.878114 + 0.478451i \(0.841198\pi\)
\(84\) 0 0
\(85\) −2.00000 −0.216930
\(86\) 12.0000 1.29399
\(87\) −6.00000 −0.643268
\(88\) 0 0
\(89\) −2.00000 −0.212000 −0.106000 − 0.994366i \(-0.533804\pi\)
−0.106000 + 0.994366i \(0.533804\pi\)
\(90\) 2.00000 0.210819
\(91\) 0 0
\(92\) −8.00000 −0.834058
\(93\) 8.00000 0.829561
\(94\) 0 0
\(95\) 0 0
\(96\) 1.00000 0.102062
\(97\) −2.00000 −0.203069 −0.101535 − 0.994832i \(-0.532375\pi\)
−0.101535 + 0.994832i \(0.532375\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) −1.00000 −0.100000
\(101\) 6.00000 0.597022 0.298511 − 0.954406i \(-0.403510\pi\)
0.298511 + 0.954406i \(0.403510\pi\)
\(102\) −1.00000 −0.0990148
\(103\) −16.0000 −1.57653 −0.788263 − 0.615338i \(-0.789020\pi\)
−0.788263 + 0.615338i \(0.789020\pi\)
\(104\) 6.00000 0.588348
\(105\) 0 0
\(106\) −10.0000 −0.971286
\(107\) 16.0000 1.54678 0.773389 − 0.633932i \(-0.218560\pi\)
0.773389 + 0.633932i \(0.218560\pi\)
\(108\) 1.00000 0.0962250
\(109\) −6.00000 −0.574696 −0.287348 − 0.957826i \(-0.592774\pi\)
−0.287348 + 0.957826i \(0.592774\pi\)
\(110\) 0 0
\(111\) 10.0000 0.949158
\(112\) 0 0
\(113\) −6.00000 −0.564433 −0.282216 − 0.959351i \(-0.591070\pi\)
−0.282216 + 0.959351i \(0.591070\pi\)
\(114\) 0 0
\(115\) −16.0000 −1.49201
\(116\) −6.00000 −0.557086
\(117\) 6.00000 0.554700
\(118\) 8.00000 0.736460
\(119\) 0 0
\(120\) 2.00000 0.182574
\(121\) −11.0000 −1.00000
\(122\) −6.00000 −0.543214
\(123\) 6.00000 0.541002
\(124\) 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
\(129\) 12.0000 1.05654
\(130\) 12.0000 1.05247
\(131\) 4.00000 0.349482 0.174741 − 0.984614i \(-0.444091\pi\)
0.174741 + 0.984614i \(0.444091\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 12.0000 1.03664
\(135\) 2.00000 0.172133
\(136\) −1.00000 −0.0857493
\(137\) −22.0000 −1.87959 −0.939793 − 0.341743i \(-0.888983\pi\)
−0.939793 + 0.341743i \(0.888983\pi\)
\(138\) −8.00000 −0.681005
\(139\) 4.00000 0.339276 0.169638 − 0.985506i \(-0.445740\pi\)
0.169638 + 0.985506i \(0.445740\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 1.00000 0.0833333
\(145\) −12.0000 −0.996546
\(146\) 6.00000 0.496564
\(147\) 0 0
\(148\) 10.0000 0.821995
\(149\) −18.0000 −1.47462 −0.737309 − 0.675556i \(-0.763904\pi\)
−0.737309 + 0.675556i \(0.763904\pi\)
\(150\) −1.00000 −0.0816497
\(151\) −16.0000 −1.30206 −0.651031 − 0.759051i \(-0.725663\pi\)
−0.651031 + 0.759051i \(0.725663\pi\)
\(152\) 0 0
\(153\) −1.00000 −0.0808452
\(154\) 0 0
\(155\) 16.0000 1.28515
\(156\) 6.00000 0.480384
\(157\) −18.0000 −1.43656 −0.718278 − 0.695756i \(-0.755069\pi\)
−0.718278 + 0.695756i \(0.755069\pi\)
\(158\) −8.00000 −0.636446
\(159\) −10.0000 −0.793052
\(160\) 2.00000 0.158114
\(161\) 0 0
\(162\) 1.00000 0.0785674
\(163\) 8.00000 0.626608 0.313304 − 0.949653i \(-0.398564\pi\)
0.313304 + 0.949653i \(0.398564\pi\)
\(164\) 6.00000 0.468521
\(165\) 0 0
\(166\) −16.0000 −1.24184
\(167\) 8.00000 0.619059 0.309529 − 0.950890i \(-0.399829\pi\)
0.309529 + 0.950890i \(0.399829\pi\)
\(168\) 0 0
\(169\) 23.0000 1.76923
\(170\) −2.00000 −0.153393
\(171\) 0 0
\(172\) 12.0000 0.914991
\(173\) 2.00000 0.152057 0.0760286 − 0.997106i \(-0.475776\pi\)
0.0760286 + 0.997106i \(0.475776\pi\)
\(174\) −6.00000 −0.454859
\(175\) 0 0
\(176\) 0 0
\(177\) 8.00000 0.601317
\(178\) −2.00000 −0.149906
\(179\) 4.00000 0.298974 0.149487 − 0.988764i \(-0.452238\pi\)
0.149487 + 0.988764i \(0.452238\pi\)
\(180\) 2.00000 0.149071
\(181\) 10.0000 0.743294 0.371647 − 0.928374i \(-0.378793\pi\)
0.371647 + 0.928374i \(0.378793\pi\)
\(182\) 0 0
\(183\) −6.00000 −0.443533
\(184\) −8.00000 −0.589768
\(185\) 20.0000 1.47043
\(186\) 8.00000 0.586588
\(187\) 0 0
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 16.0000 1.15772 0.578860 − 0.815427i \(-0.303498\pi\)
0.578860 + 0.815427i \(0.303498\pi\)
\(192\) 1.00000 0.0721688
\(193\) −6.00000 −0.431889 −0.215945 − 0.976406i \(-0.569283\pi\)
−0.215945 + 0.976406i \(0.569283\pi\)
\(194\) −2.00000 −0.143592
\(195\) 12.0000 0.859338
\(196\) 0 0
\(197\) 18.0000 1.28245 0.641223 − 0.767354i \(-0.278427\pi\)
0.641223 + 0.767354i \(0.278427\pi\)
\(198\) 0 0
\(199\) 24.0000 1.70131 0.850657 − 0.525720i \(-0.176204\pi\)
0.850657 + 0.525720i \(0.176204\pi\)
\(200\) −1.00000 −0.0707107
\(201\) 12.0000 0.846415
\(202\) 6.00000 0.422159
\(203\) 0 0
\(204\) −1.00000 −0.0700140
\(205\) 12.0000 0.838116
\(206\) −16.0000 −1.11477
\(207\) −8.00000 −0.556038
\(208\) 6.00000 0.416025
\(209\) 0 0
\(210\) 0 0
\(211\) 8.00000 0.550743 0.275371 − 0.961338i \(-0.411199\pi\)
0.275371 + 0.961338i \(0.411199\pi\)
\(212\) −10.0000 −0.686803
\(213\) 0 0
\(214\) 16.0000 1.09374
\(215\) 24.0000 1.63679
\(216\) 1.00000 0.0680414
\(217\) 0 0
\(218\) −6.00000 −0.406371
\(219\) 6.00000 0.405442
\(220\) 0 0
\(221\) −6.00000 −0.403604
\(222\) 10.0000 0.671156
\(223\) 8.00000 0.535720 0.267860 − 0.963458i \(-0.413684\pi\)
0.267860 + 0.963458i \(0.413684\pi\)
\(224\) 0 0
\(225\) −1.00000 −0.0666667
\(226\) −6.00000 −0.399114
\(227\) 4.00000 0.265489 0.132745 − 0.991150i \(-0.457621\pi\)
0.132745 + 0.991150i \(0.457621\pi\)
\(228\) 0 0
\(229\) −10.0000 −0.660819 −0.330409 − 0.943838i \(-0.607187\pi\)
−0.330409 + 0.943838i \(0.607187\pi\)
\(230\) −16.0000 −1.05501
\(231\) 0 0
\(232\) −6.00000 −0.393919
\(233\) −30.0000 −1.96537 −0.982683 − 0.185296i \(-0.940675\pi\)
−0.982683 + 0.185296i \(0.940675\pi\)
\(234\) 6.00000 0.392232
\(235\) 0 0
\(236\) 8.00000 0.520756
\(237\) −8.00000 −0.519656
\(238\) 0 0
\(239\) 24.0000 1.55243 0.776215 − 0.630468i \(-0.217137\pi\)
0.776215 + 0.630468i \(0.217137\pi\)
\(240\) 2.00000 0.129099
\(241\) −2.00000 −0.128831 −0.0644157 − 0.997923i \(-0.520518\pi\)
−0.0644157 + 0.997923i \(0.520518\pi\)
\(242\) −11.0000 −0.707107
\(243\) 1.00000 0.0641500
\(244\) −6.00000 −0.384111
\(245\) 0 0
\(246\) 6.00000 0.382546
\(247\) 0 0
\(248\) 8.00000 0.508001
\(249\) −16.0000 −1.01396
\(250\) −12.0000 −0.758947
\(251\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 8.00000 0.501965
\(255\) −2.00000 −0.125245
\(256\) 1.00000 0.0625000
\(257\) 6.00000 0.374270 0.187135 − 0.982334i \(-0.440080\pi\)
0.187135 + 0.982334i \(0.440080\pi\)
\(258\) 12.0000 0.747087
\(259\) 0 0
\(260\) 12.0000 0.744208
\(261\) −6.00000 −0.371391
\(262\) 4.00000 0.247121
\(263\) −16.0000 −0.986602 −0.493301 − 0.869859i \(-0.664210\pi\)
−0.493301 + 0.869859i \(0.664210\pi\)
\(264\) 0 0
\(265\) −20.0000 −1.22859
\(266\) 0 0
\(267\) −2.00000 −0.122398
\(268\) 12.0000 0.733017
\(269\) 18.0000 1.09748 0.548740 − 0.835993i \(-0.315108\pi\)
0.548740 + 0.835993i \(0.315108\pi\)
\(270\) 2.00000 0.121716
\(271\) −24.0000 −1.45790 −0.728948 − 0.684569i \(-0.759990\pi\)
−0.728948 + 0.684569i \(0.759990\pi\)
\(272\) −1.00000 −0.0606339
\(273\) 0 0
\(274\) −22.0000 −1.32907
\(275\) 0 0
\(276\) −8.00000 −0.481543
\(277\) −6.00000 −0.360505 −0.180253 − 0.983620i \(-0.557691\pi\)
−0.180253 + 0.983620i \(0.557691\pi\)
\(278\) 4.00000 0.239904
\(279\) 8.00000 0.478947
\(280\) 0 0
\(281\) −22.0000 −1.31241 −0.656205 − 0.754583i \(-0.727839\pi\)
−0.656205 + 0.754583i \(0.727839\pi\)
\(282\) 0 0
\(283\) 4.00000 0.237775 0.118888 − 0.992908i \(-0.462067\pi\)
0.118888 + 0.992908i \(0.462067\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 1.00000 0.0589256
\(289\) 1.00000 0.0588235
\(290\) −12.0000 −0.704664
\(291\) −2.00000 −0.117242
\(292\) 6.00000 0.351123
\(293\) 14.0000 0.817889 0.408944 − 0.912559i \(-0.365897\pi\)
0.408944 + 0.912559i \(0.365897\pi\)
\(294\) 0 0
\(295\) 16.0000 0.931556
\(296\) 10.0000 0.581238
\(297\) 0 0
\(298\) −18.0000 −1.04271
\(299\) −48.0000 −2.77591
\(300\) −1.00000 −0.0577350
\(301\) 0 0
\(302\) −16.0000 −0.920697
\(303\) 6.00000 0.344691
\(304\) 0 0
\(305\) −12.0000 −0.687118
\(306\) −1.00000 −0.0571662
\(307\) 32.0000 1.82634 0.913168 − 0.407583i \(-0.133628\pi\)
0.913168 + 0.407583i \(0.133628\pi\)
\(308\) 0 0
\(309\) −16.0000 −0.910208
\(310\) 16.0000 0.908739
\(311\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(312\) 6.00000 0.339683
\(313\) −26.0000 −1.46961 −0.734803 − 0.678280i \(-0.762726\pi\)
−0.734803 + 0.678280i \(0.762726\pi\)
\(314\) −18.0000 −1.01580
\(315\) 0 0
\(316\) −8.00000 −0.450035
\(317\) −6.00000 −0.336994 −0.168497 − 0.985702i \(-0.553891\pi\)
−0.168497 + 0.985702i \(0.553891\pi\)
\(318\) −10.0000 −0.560772
\(319\) 0 0
\(320\) 2.00000 0.111803
\(321\) 16.0000 0.893033
\(322\) 0 0
\(323\) 0 0
\(324\) 1.00000 0.0555556
\(325\) −6.00000 −0.332820
\(326\) 8.00000 0.443079
\(327\) −6.00000 −0.331801
\(328\) 6.00000 0.331295
\(329\) 0 0
\(330\) 0 0
\(331\) −36.0000 −1.97874 −0.989369 − 0.145424i \(-0.953545\pi\)
−0.989369 + 0.145424i \(0.953545\pi\)
\(332\) −16.0000 −0.878114
\(333\) 10.0000 0.547997
\(334\) 8.00000 0.437741
\(335\) 24.0000 1.31126
\(336\) 0 0
\(337\) 2.00000 0.108947 0.0544735 − 0.998515i \(-0.482652\pi\)
0.0544735 + 0.998515i \(0.482652\pi\)
\(338\) 23.0000 1.25104
\(339\) −6.00000 −0.325875
\(340\) −2.00000 −0.108465
\(341\) 0 0
\(342\) 0 0
\(343\) 0 0
\(344\) 12.0000 0.646997
\(345\) −16.0000 −0.861411
\(346\) 2.00000 0.107521
\(347\) −24.0000 −1.28839 −0.644194 − 0.764862i \(-0.722807\pi\)
−0.644194 + 0.764862i \(0.722807\pi\)
\(348\) −6.00000 −0.321634
\(349\) 22.0000 1.17763 0.588817 − 0.808267i \(-0.299594\pi\)
0.588817 + 0.808267i \(0.299594\pi\)
\(350\) 0 0
\(351\) 6.00000 0.320256
\(352\) 0 0
\(353\) 6.00000 0.319348 0.159674 − 0.987170i \(-0.448956\pi\)
0.159674 + 0.987170i \(0.448956\pi\)
\(354\) 8.00000 0.425195
\(355\) 0 0
\(356\) −2.00000 −0.106000
\(357\) 0 0
\(358\) 4.00000 0.211407
\(359\) 16.0000 0.844448 0.422224 − 0.906492i \(-0.361250\pi\)
0.422224 + 0.906492i \(0.361250\pi\)
\(360\) 2.00000 0.105409
\(361\) −19.0000 −1.00000
\(362\) 10.0000 0.525588
\(363\) −11.0000 −0.577350
\(364\) 0 0
\(365\) 12.0000 0.628109
\(366\) −6.00000 −0.313625
\(367\) −8.00000 −0.417597 −0.208798 − 0.977959i \(-0.566955\pi\)
−0.208798 + 0.977959i \(0.566955\pi\)
\(368\) −8.00000 −0.417029
\(369\) 6.00000 0.312348
\(370\) 20.0000 1.03975
\(371\) 0 0
\(372\) 8.00000 0.414781
\(373\) −2.00000 −0.103556 −0.0517780 − 0.998659i \(-0.516489\pi\)
−0.0517780 + 0.998659i \(0.516489\pi\)
\(374\) 0 0
\(375\) −12.0000 −0.619677
\(376\) 0 0
\(377\) −36.0000 −1.85409
\(378\) 0 0
\(379\) −8.00000 −0.410932 −0.205466 − 0.978664i \(-0.565871\pi\)
−0.205466 + 0.978664i \(0.565871\pi\)
\(380\) 0 0
\(381\) 8.00000 0.409852
\(382\) 16.0000 0.818631
\(383\) −8.00000 −0.408781 −0.204390 − 0.978889i \(-0.565521\pi\)
−0.204390 + 0.978889i \(0.565521\pi\)
\(384\) 1.00000 0.0510310
\(385\) 0 0
\(386\) −6.00000 −0.305392
\(387\) 12.0000 0.609994
\(388\) −2.00000 −0.101535
\(389\) 6.00000 0.304212 0.152106 − 0.988364i \(-0.451394\pi\)
0.152106 + 0.988364i \(0.451394\pi\)
\(390\) 12.0000 0.607644
\(391\) 8.00000 0.404577
\(392\) 0 0
\(393\) 4.00000 0.201773
\(394\) 18.0000 0.906827
\(395\) −16.0000 −0.805047
\(396\) 0 0
\(397\) 18.0000 0.903394 0.451697 − 0.892171i \(-0.350819\pi\)
0.451697 + 0.892171i \(0.350819\pi\)
\(398\) 24.0000 1.20301
\(399\) 0 0
\(400\) −1.00000 −0.0500000
\(401\) 2.00000 0.0998752 0.0499376 − 0.998752i \(-0.484098\pi\)
0.0499376 + 0.998752i \(0.484098\pi\)
\(402\) 12.0000 0.598506
\(403\) 48.0000 2.39105
\(404\) 6.00000 0.298511
\(405\) 2.00000 0.0993808
\(406\) 0 0
\(407\) 0 0
\(408\) −1.00000 −0.0495074
\(409\) −34.0000 −1.68119 −0.840596 − 0.541663i \(-0.817795\pi\)
−0.840596 + 0.541663i \(0.817795\pi\)
\(410\) 12.0000 0.592638
\(411\) −22.0000 −1.08518
\(412\) −16.0000 −0.788263
\(413\) 0 0
\(414\) −8.00000 −0.393179
\(415\) −32.0000 −1.57082
\(416\) 6.00000 0.294174
\(417\) 4.00000 0.195881
\(418\) 0 0
\(419\) 12.0000 0.586238 0.293119 − 0.956076i \(-0.405307\pi\)
0.293119 + 0.956076i \(0.405307\pi\)
\(420\) 0 0
\(421\) −34.0000 −1.65706 −0.828529 − 0.559946i \(-0.810822\pi\)
−0.828529 + 0.559946i \(0.810822\pi\)
\(422\) 8.00000 0.389434
\(423\) 0 0
\(424\) −10.0000 −0.485643
\(425\) 1.00000 0.0485071
\(426\) 0 0
\(427\) 0 0
\(428\) 16.0000 0.773389
\(429\) 0 0
\(430\) 24.0000 1.15738
\(431\) −16.0000 −0.770693 −0.385346 − 0.922772i \(-0.625918\pi\)
−0.385346 + 0.922772i \(0.625918\pi\)
\(432\) 1.00000 0.0481125
\(433\) 22.0000 1.05725 0.528626 − 0.848855i \(-0.322707\pi\)
0.528626 + 0.848855i \(0.322707\pi\)
\(434\) 0 0
\(435\) −12.0000 −0.575356
\(436\) −6.00000 −0.287348
\(437\) 0 0
\(438\) 6.00000 0.286691
\(439\) −16.0000 −0.763638 −0.381819 − 0.924237i \(-0.624702\pi\)
−0.381819 + 0.924237i \(0.624702\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) −6.00000 −0.285391
\(443\) 4.00000 0.190046 0.0950229 − 0.995475i \(-0.469708\pi\)
0.0950229 + 0.995475i \(0.469708\pi\)
\(444\) 10.0000 0.474579
\(445\) −4.00000 −0.189618
\(446\) 8.00000 0.378811
\(447\) −18.0000 −0.851371
\(448\) 0 0
\(449\) 34.0000 1.60456 0.802280 − 0.596948i \(-0.203620\pi\)
0.802280 + 0.596948i \(0.203620\pi\)
\(450\) −1.00000 −0.0471405
\(451\) 0 0
\(452\) −6.00000 −0.282216
\(453\) −16.0000 −0.751746
\(454\) 4.00000 0.187729
\(455\) 0 0
\(456\) 0 0
\(457\) −38.0000 −1.77757 −0.888783 − 0.458329i \(-0.848448\pi\)
−0.888783 + 0.458329i \(0.848448\pi\)
\(458\) −10.0000 −0.467269
\(459\) −1.00000 −0.0466760
\(460\) −16.0000 −0.746004
\(461\) 6.00000 0.279448 0.139724 − 0.990190i \(-0.455378\pi\)
0.139724 + 0.990190i \(0.455378\pi\)
\(462\) 0 0
\(463\) −16.0000 −0.743583 −0.371792 − 0.928316i \(-0.621256\pi\)
−0.371792 + 0.928316i \(0.621256\pi\)
\(464\) −6.00000 −0.278543
\(465\) 16.0000 0.741982
\(466\) −30.0000 −1.38972
\(467\) −8.00000 −0.370196 −0.185098 − 0.982720i \(-0.559260\pi\)
−0.185098 + 0.982720i \(0.559260\pi\)
\(468\) 6.00000 0.277350
\(469\) 0 0
\(470\) 0 0
\(471\) −18.0000 −0.829396
\(472\) 8.00000 0.368230
\(473\) 0 0
\(474\) −8.00000 −0.367452
\(475\) 0 0
\(476\) 0 0
\(477\) −10.0000 −0.457869
\(478\) 24.0000 1.09773
\(479\) −40.0000 −1.82765 −0.913823 − 0.406112i \(-0.866884\pi\)
−0.913823 + 0.406112i \(0.866884\pi\)
\(480\) 2.00000 0.0912871
\(481\) 60.0000 2.73576
\(482\) −2.00000 −0.0910975
\(483\) 0 0
\(484\) −11.0000 −0.500000
\(485\) −4.00000 −0.181631
\(486\) 1.00000 0.0453609
\(487\) 8.00000 0.362515 0.181257 − 0.983436i \(-0.441983\pi\)
0.181257 + 0.983436i \(0.441983\pi\)
\(488\) −6.00000 −0.271607
\(489\) 8.00000 0.361773
\(490\) 0 0
\(491\) −20.0000 −0.902587 −0.451294 − 0.892375i \(-0.649037\pi\)
−0.451294 + 0.892375i \(0.649037\pi\)
\(492\) 6.00000 0.270501
\(493\) 6.00000 0.270226
\(494\) 0 0
\(495\) 0 0
\(496\) 8.00000 0.359211
\(497\) 0 0
\(498\) −16.0000 −0.716977
\(499\) 8.00000 0.358129 0.179065 − 0.983837i \(-0.442693\pi\)
0.179065 + 0.983837i \(0.442693\pi\)
\(500\) −12.0000 −0.536656
\(501\) 8.00000 0.357414
\(502\) 0 0
\(503\) −24.0000 −1.07011 −0.535054 − 0.844818i \(-0.679709\pi\)
−0.535054 + 0.844818i \(0.679709\pi\)
\(504\) 0 0
\(505\) 12.0000 0.533993
\(506\) 0 0
\(507\) 23.0000 1.02147
\(508\) 8.00000 0.354943
\(509\) −2.00000 −0.0886484 −0.0443242 − 0.999017i \(-0.514113\pi\)
−0.0443242 + 0.999017i \(0.514113\pi\)
\(510\) −2.00000 −0.0885615
\(511\) 0 0
\(512\) 1.00000 0.0441942
\(513\) 0 0
\(514\) 6.00000 0.264649
\(515\) −32.0000 −1.41009
\(516\) 12.0000 0.528271
\(517\) 0 0
\(518\) 0 0
\(519\) 2.00000 0.0877903
\(520\) 12.0000 0.526235
\(521\) −10.0000 −0.438108 −0.219054 − 0.975713i \(-0.570297\pi\)
−0.219054 + 0.975713i \(0.570297\pi\)
\(522\) −6.00000 −0.262613
\(523\) 16.0000 0.699631 0.349816 − 0.936819i \(-0.386244\pi\)
0.349816 + 0.936819i \(0.386244\pi\)
\(524\) 4.00000 0.174741
\(525\) 0 0
\(526\) −16.0000 −0.697633
\(527\) −8.00000 −0.348485
\(528\) 0 0
\(529\) 41.0000 1.78261
\(530\) −20.0000 −0.868744
\(531\) 8.00000 0.347170
\(532\) 0 0
\(533\) 36.0000 1.55933
\(534\) −2.00000 −0.0865485
\(535\) 32.0000 1.38348
\(536\) 12.0000 0.518321
\(537\) 4.00000 0.172613
\(538\) 18.0000 0.776035
\(539\) 0 0
\(540\) 2.00000 0.0860663
\(541\) 26.0000 1.11783 0.558914 − 0.829226i \(-0.311218\pi\)
0.558914 + 0.829226i \(0.311218\pi\)
\(542\) −24.0000 −1.03089
\(543\) 10.0000 0.429141
\(544\) −1.00000 −0.0428746
\(545\) −12.0000 −0.514024
\(546\) 0 0
\(547\) −8.00000 −0.342055 −0.171028 − 0.985266i \(-0.554709\pi\)
−0.171028 + 0.985266i \(0.554709\pi\)
\(548\) −22.0000 −0.939793
\(549\) −6.00000 −0.256074
\(550\) 0 0
\(551\) 0 0
\(552\) −8.00000 −0.340503
\(553\) 0 0
\(554\) −6.00000 −0.254916
\(555\) 20.0000 0.848953
\(556\) 4.00000 0.169638
\(557\) −18.0000 −0.762684 −0.381342 − 0.924434i \(-0.624538\pi\)
−0.381342 + 0.924434i \(0.624538\pi\)
\(558\) 8.00000 0.338667
\(559\) 72.0000 3.04528
\(560\) 0 0
\(561\) 0 0
\(562\) −22.0000 −0.928014
\(563\) −24.0000 −1.01148 −0.505740 − 0.862686i \(-0.668780\pi\)
−0.505740 + 0.862686i \(0.668780\pi\)
\(564\) 0 0
\(565\) −12.0000 −0.504844
\(566\) 4.00000 0.168133
\(567\) 0 0
\(568\) 0 0
\(569\) −6.00000 −0.251533 −0.125767 − 0.992060i \(-0.540139\pi\)
−0.125767 + 0.992060i \(0.540139\pi\)
\(570\) 0 0
\(571\) 32.0000 1.33916 0.669579 − 0.742741i \(-0.266474\pi\)
0.669579 + 0.742741i \(0.266474\pi\)
\(572\) 0 0
\(573\) 16.0000 0.668410
\(574\) 0 0
\(575\) 8.00000 0.333623
\(576\) 1.00000 0.0416667
\(577\) −18.0000 −0.749350 −0.374675 − 0.927156i \(-0.622246\pi\)
−0.374675 + 0.927156i \(0.622246\pi\)
\(578\) 1.00000 0.0415945
\(579\) −6.00000 −0.249351
\(580\) −12.0000 −0.498273
\(581\) 0 0
\(582\) −2.00000 −0.0829027
\(583\) 0 0
\(584\) 6.00000 0.248282
\(585\) 12.0000 0.496139
\(586\) 14.0000 0.578335
\(587\) 8.00000 0.330195 0.165098 − 0.986277i \(-0.447206\pi\)
0.165098 + 0.986277i \(0.447206\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 16.0000 0.658710
\(591\) 18.0000 0.740421
\(592\) 10.0000 0.410997
\(593\) −26.0000 −1.06769 −0.533846 − 0.845582i \(-0.679254\pi\)
−0.533846 + 0.845582i \(0.679254\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −18.0000 −0.737309
\(597\) 24.0000 0.982255
\(598\) −48.0000 −1.96287
\(599\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(600\) −1.00000 −0.0408248
\(601\) −26.0000 −1.06056 −0.530281 − 0.847822i \(-0.677914\pi\)
−0.530281 + 0.847822i \(0.677914\pi\)
\(602\) 0 0
\(603\) 12.0000 0.488678
\(604\) −16.0000 −0.651031
\(605\) −22.0000 −0.894427
\(606\) 6.00000 0.243733
\(607\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) −12.0000 −0.485866
\(611\) 0 0
\(612\) −1.00000 −0.0404226
\(613\) 14.0000 0.565455 0.282727 − 0.959200i \(-0.408761\pi\)
0.282727 + 0.959200i \(0.408761\pi\)
\(614\) 32.0000 1.29141
\(615\) 12.0000 0.483887
\(616\) 0 0
\(617\) −14.0000 −0.563619 −0.281809 − 0.959470i \(-0.590935\pi\)
−0.281809 + 0.959470i \(0.590935\pi\)
\(618\) −16.0000 −0.643614
\(619\) −4.00000 −0.160774 −0.0803868 − 0.996764i \(-0.525616\pi\)
−0.0803868 + 0.996764i \(0.525616\pi\)
\(620\) 16.0000 0.642575
\(621\) −8.00000 −0.321029
\(622\) 0 0
\(623\) 0 0
\(624\) 6.00000 0.240192
\(625\) −19.0000 −0.760000
\(626\) −26.0000 −1.03917
\(627\) 0 0
\(628\) −18.0000 −0.718278
\(629\) −10.0000 −0.398726
\(630\) 0 0
\(631\) −16.0000 −0.636950 −0.318475 − 0.947931i \(-0.603171\pi\)
−0.318475 + 0.947931i \(0.603171\pi\)
\(632\) −8.00000 −0.318223
\(633\) 8.00000 0.317971
\(634\) −6.00000 −0.238290
\(635\) 16.0000 0.634941
\(636\) −10.0000 −0.396526
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) 2.00000 0.0790569
\(641\) −30.0000 −1.18493 −0.592464 − 0.805597i \(-0.701845\pi\)
−0.592464 + 0.805597i \(0.701845\pi\)
\(642\) 16.0000 0.631470
\(643\) 36.0000 1.41970 0.709851 − 0.704352i \(-0.248762\pi\)
0.709851 + 0.704352i \(0.248762\pi\)
\(644\) 0 0
\(645\) 24.0000 0.944999
\(646\) 0 0
\(647\) 24.0000 0.943537 0.471769 − 0.881722i \(-0.343616\pi\)
0.471769 + 0.881722i \(0.343616\pi\)
\(648\) 1.00000 0.0392837
\(649\) 0 0
\(650\) −6.00000 −0.235339
\(651\) 0 0
\(652\) 8.00000 0.313304
\(653\) 42.0000 1.64359 0.821794 − 0.569785i \(-0.192974\pi\)
0.821794 + 0.569785i \(0.192974\pi\)
\(654\) −6.00000 −0.234619
\(655\) 8.00000 0.312586
\(656\) 6.00000 0.234261
\(657\) 6.00000 0.234082
\(658\) 0 0
\(659\) 12.0000 0.467454 0.233727 − 0.972302i \(-0.424908\pi\)
0.233727 + 0.972302i \(0.424908\pi\)
\(660\) 0 0
\(661\) −10.0000 −0.388955 −0.194477 − 0.980907i \(-0.562301\pi\)
−0.194477 + 0.980907i \(0.562301\pi\)
\(662\) −36.0000 −1.39918
\(663\) −6.00000 −0.233021
\(664\) −16.0000 −0.620920
\(665\) 0 0
\(666\) 10.0000 0.387492
\(667\) 48.0000 1.85857
\(668\) 8.00000 0.309529
\(669\) 8.00000 0.309298
\(670\) 24.0000 0.927201
\(671\) 0 0
\(672\) 0 0
\(673\) −6.00000 −0.231283 −0.115642 − 0.993291i \(-0.536892\pi\)
−0.115642 + 0.993291i \(0.536892\pi\)
\(674\) 2.00000 0.0770371
\(675\) −1.00000 −0.0384900
\(676\) 23.0000 0.884615
\(677\) −30.0000 −1.15299 −0.576497 − 0.817099i \(-0.695581\pi\)
−0.576497 + 0.817099i \(0.695581\pi\)
\(678\) −6.00000 −0.230429
\(679\) 0 0
\(680\) −2.00000 −0.0766965
\(681\) 4.00000 0.153280
\(682\) 0 0
\(683\) 24.0000 0.918334 0.459167 − 0.888350i \(-0.348148\pi\)
0.459167 + 0.888350i \(0.348148\pi\)
\(684\) 0 0
\(685\) −44.0000 −1.68115
\(686\) 0 0
\(687\) −10.0000 −0.381524
\(688\) 12.0000 0.457496
\(689\) −60.0000 −2.28582
\(690\) −16.0000 −0.609110
\(691\) −36.0000 −1.36950 −0.684752 − 0.728776i \(-0.740090\pi\)
−0.684752 + 0.728776i \(0.740090\pi\)
\(692\) 2.00000 0.0760286
\(693\) 0 0
\(694\) −24.0000 −0.911028
\(695\) 8.00000 0.303457
\(696\) −6.00000 −0.227429
\(697\) −6.00000 −0.227266
\(698\) 22.0000 0.832712
\(699\) −30.0000 −1.13470
\(700\) 0 0
\(701\) −10.0000 −0.377695 −0.188847 − 0.982006i \(-0.560475\pi\)
−0.188847 + 0.982006i \(0.560475\pi\)
\(702\) 6.00000 0.226455
\(703\) 0 0
\(704\) 0 0
\(705\) 0 0
\(706\) 6.00000 0.225813
\(707\) 0 0
\(708\) 8.00000 0.300658
\(709\) 50.0000 1.87779 0.938895 − 0.344204i \(-0.111851\pi\)
0.938895 + 0.344204i \(0.111851\pi\)
\(710\) 0 0
\(711\) −8.00000 −0.300023
\(712\) −2.00000 −0.0749532
\(713\) −64.0000 −2.39682
\(714\) 0 0
\(715\) 0 0
\(716\) 4.00000 0.149487
\(717\) 24.0000 0.896296
\(718\) 16.0000 0.597115
\(719\) 8.00000 0.298350 0.149175 − 0.988811i \(-0.452338\pi\)
0.149175 + 0.988811i \(0.452338\pi\)
\(720\) 2.00000 0.0745356
\(721\) 0 0
\(722\) −19.0000 −0.707107
\(723\) −2.00000 −0.0743808
\(724\) 10.0000 0.371647
\(725\) 6.00000 0.222834
\(726\) −11.0000 −0.408248
\(727\) −40.0000 −1.48352 −0.741759 − 0.670667i \(-0.766008\pi\)
−0.741759 + 0.670667i \(0.766008\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 12.0000 0.444140
\(731\) −12.0000 −0.443836
\(732\) −6.00000 −0.221766
\(733\) 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
\(736\) −8.00000 −0.294884
\(737\) 0 0
\(738\) 6.00000 0.220863
\(739\) −4.00000 −0.147142 −0.0735712 − 0.997290i \(-0.523440\pi\)
−0.0735712 + 0.997290i \(0.523440\pi\)
\(740\) 20.0000 0.735215
\(741\) 0 0
\(742\) 0 0
\(743\) −16.0000 −0.586983 −0.293492 − 0.955962i \(-0.594817\pi\)
−0.293492 + 0.955962i \(0.594817\pi\)
\(744\) 8.00000 0.293294
\(745\) −36.0000 −1.31894
\(746\) −2.00000 −0.0732252
\(747\) −16.0000 −0.585409
\(748\) 0 0
\(749\) 0 0
\(750\) −12.0000 −0.438178
\(751\) −32.0000 −1.16770 −0.583848 − 0.811863i \(-0.698454\pi\)
−0.583848 + 0.811863i \(0.698454\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) −36.0000 −1.31104
\(755\) −32.0000 −1.16460
\(756\) 0 0
\(757\) 30.0000 1.09037 0.545184 − 0.838316i \(-0.316460\pi\)
0.545184 + 0.838316i \(0.316460\pi\)
\(758\) −8.00000 −0.290573
\(759\) 0 0
\(760\) 0 0
\(761\) 22.0000 0.797499 0.398750 − 0.917060i \(-0.369444\pi\)
0.398750 + 0.917060i \(0.369444\pi\)
\(762\) 8.00000 0.289809
\(763\) 0 0
\(764\) 16.0000 0.578860
\(765\) −2.00000 −0.0723102
\(766\) −8.00000 −0.289052
\(767\) 48.0000 1.73318
\(768\) 1.00000 0.0360844
\(769\) −26.0000 −0.937584 −0.468792 − 0.883309i \(-0.655311\pi\)
−0.468792 + 0.883309i \(0.655311\pi\)
\(770\) 0 0
\(771\) 6.00000 0.216085
\(772\) −6.00000 −0.215945
\(773\) 14.0000 0.503545 0.251773 − 0.967786i \(-0.418987\pi\)
0.251773 + 0.967786i \(0.418987\pi\)
\(774\) 12.0000 0.431331
\(775\) −8.00000 −0.287368
\(776\) −2.00000 −0.0717958
\(777\) 0 0
\(778\) 6.00000 0.215110
\(779\) 0 0
\(780\) 12.0000 0.429669
\(781\) 0 0
\(782\) 8.00000 0.286079
\(783\) −6.00000 −0.214423
\(784\) 0 0
\(785\) −36.0000 −1.28490
\(786\) 4.00000 0.142675
\(787\) 4.00000 0.142585 0.0712923 − 0.997455i \(-0.477288\pi\)
0.0712923 + 0.997455i \(0.477288\pi\)
\(788\) 18.0000 0.641223
\(789\) −16.0000 −0.569615
\(790\) −16.0000 −0.569254
\(791\) 0 0
\(792\) 0 0
\(793\) −36.0000 −1.27840
\(794\) 18.0000 0.638796
\(795\) −20.0000 −0.709327
\(796\) 24.0000 0.850657
\(797\) −42.0000 −1.48772 −0.743858 − 0.668338i \(-0.767006\pi\)
−0.743858 + 0.668338i \(0.767006\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) −1.00000 −0.0353553
\(801\) −2.00000 −0.0706665
\(802\) 2.00000 0.0706225
\(803\) 0 0
\(804\) 12.0000 0.423207
\(805\) 0 0
\(806\) 48.0000 1.69073
\(807\) 18.0000 0.633630
\(808\) 6.00000 0.211079
\(809\) −6.00000 −0.210949 −0.105474 − 0.994422i \(-0.533636\pi\)
−0.105474 + 0.994422i \(0.533636\pi\)
\(810\) 2.00000 0.0702728
\(811\) −12.0000 −0.421377 −0.210688 − 0.977553i \(-0.567571\pi\)
−0.210688 + 0.977553i \(0.567571\pi\)
\(812\) 0 0
\(813\) −24.0000 −0.841717
\(814\) 0 0
\(815\) 16.0000 0.560456
\(816\) −1.00000 −0.0350070
\(817\) 0 0
\(818\) −34.0000 −1.18878
\(819\) 0 0
\(820\) 12.0000 0.419058
\(821\) 2.00000 0.0698005 0.0349002 − 0.999391i \(-0.488889\pi\)
0.0349002 + 0.999391i \(0.488889\pi\)
\(822\) −22.0000 −0.767338
\(823\) −16.0000 −0.557725 −0.278862 − 0.960331i \(-0.589957\pi\)
−0.278862 + 0.960331i \(0.589957\pi\)
\(824\) −16.0000 −0.557386
\(825\) 0 0
\(826\) 0 0
\(827\) −8.00000 −0.278187 −0.139094 − 0.990279i \(-0.544419\pi\)
−0.139094 + 0.990279i \(0.544419\pi\)
\(828\) −8.00000 −0.278019
\(829\) 38.0000 1.31979 0.659897 − 0.751356i \(-0.270600\pi\)
0.659897 + 0.751356i \(0.270600\pi\)
\(830\) −32.0000 −1.11074
\(831\) −6.00000 −0.208138
\(832\) 6.00000 0.208013
\(833\) 0 0
\(834\) 4.00000 0.138509
\(835\) 16.0000 0.553703
\(836\) 0 0
\(837\) 8.00000 0.276520
\(838\) 12.0000 0.414533
\(839\) −40.0000 −1.38095 −0.690477 − 0.723355i \(-0.742599\pi\)
−0.690477 + 0.723355i \(0.742599\pi\)
\(840\) 0 0
\(841\) 7.00000 0.241379
\(842\) −34.0000 −1.17172
\(843\) −22.0000 −0.757720
\(844\) 8.00000 0.275371
\(845\) 46.0000 1.58245
\(846\) 0 0
\(847\) 0 0
\(848\) −10.0000 −0.343401
\(849\) 4.00000 0.137280
\(850\) 1.00000 0.0342997
\(851\) −80.0000 −2.74236
\(852\) 0 0
\(853\) −14.0000 −0.479351 −0.239675 − 0.970853i \(-0.577041\pi\)
−0.239675 + 0.970853i \(0.577041\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 16.0000 0.546869
\(857\) 54.0000 1.84460 0.922302 − 0.386469i \(-0.126305\pi\)
0.922302 + 0.386469i \(0.126305\pi\)
\(858\) 0 0
\(859\) 56.0000 1.91070 0.955348 − 0.295484i \(-0.0954809\pi\)
0.955348 + 0.295484i \(0.0954809\pi\)
\(860\) 24.0000 0.818393
\(861\) 0 0
\(862\) −16.0000 −0.544962
\(863\) 8.00000 0.272323 0.136162 − 0.990687i \(-0.456523\pi\)
0.136162 + 0.990687i \(0.456523\pi\)
\(864\) 1.00000 0.0340207
\(865\) 4.00000 0.136004
\(866\) 22.0000 0.747590
\(867\) 1.00000 0.0339618
\(868\) 0 0
\(869\) 0 0
\(870\) −12.0000 −0.406838
\(871\) 72.0000 2.43963
\(872\) −6.00000 −0.203186
\(873\) −2.00000 −0.0676897
\(874\) 0 0
\(875\) 0 0
\(876\) 6.00000 0.202721
\(877\) −22.0000 −0.742887 −0.371444 − 0.928456i \(-0.621137\pi\)
−0.371444 + 0.928456i \(0.621137\pi\)
\(878\) −16.0000 −0.539974
\(879\) 14.0000 0.472208
\(880\) 0 0
\(881\) 30.0000 1.01073 0.505363 − 0.862907i \(-0.331359\pi\)
0.505363 + 0.862907i \(0.331359\pi\)
\(882\) 0 0
\(883\) −4.00000 −0.134611 −0.0673054 − 0.997732i \(-0.521440\pi\)
−0.0673054 + 0.997732i \(0.521440\pi\)
\(884\) −6.00000 −0.201802
\(885\) 16.0000 0.537834
\(886\) 4.00000 0.134383
\(887\) −16.0000 −0.537227 −0.268614 − 0.963248i \(-0.586566\pi\)
−0.268614 + 0.963248i \(0.586566\pi\)
\(888\) 10.0000 0.335578
\(889\) 0 0
\(890\) −4.00000 −0.134080
\(891\) 0 0
\(892\) 8.00000 0.267860
\(893\) 0 0
\(894\) −18.0000 −0.602010
\(895\) 8.00000 0.267411
\(896\) 0 0
\(897\) −48.0000 −1.60267
\(898\) 34.0000 1.13459
\(899\) −48.0000 −1.60089
\(900\) −1.00000 −0.0333333
\(901\) 10.0000 0.333148
\(902\) 0 0
\(903\) 0 0
\(904\) −6.00000 −0.199557
\(905\) 20.0000 0.664822
\(906\) −16.0000 −0.531564
\(907\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(908\) 4.00000 0.132745
\(909\) 6.00000 0.199007
\(910\) 0 0
\(911\) −8.00000 −0.265052 −0.132526 − 0.991180i \(-0.542309\pi\)
−0.132526 + 0.991180i \(0.542309\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) −38.0000 −1.25693
\(915\) −12.0000 −0.396708
\(916\) −10.0000 −0.330409
\(917\) 0 0
\(918\) −1.00000 −0.0330049
\(919\) −16.0000 −0.527791 −0.263896 − 0.964551i \(-0.585007\pi\)
−0.263896 + 0.964551i \(0.585007\pi\)
\(920\) −16.0000 −0.527504
\(921\) 32.0000 1.05444
\(922\) 6.00000 0.197599
\(923\) 0 0
\(924\) 0 0
\(925\) −10.0000 −0.328798
\(926\) −16.0000 −0.525793
\(927\) −16.0000 −0.525509
\(928\) −6.00000 −0.196960
\(929\) 30.0000 0.984268 0.492134 − 0.870519i \(-0.336217\pi\)
0.492134 + 0.870519i \(0.336217\pi\)
\(930\) 16.0000 0.524661
\(931\) 0 0
\(932\) −30.0000 −0.982683
\(933\) 0 0
\(934\) −8.00000 −0.261768
\(935\) 0 0
\(936\) 6.00000 0.196116
\(937\) 22.0000 0.718709 0.359354 − 0.933201i \(-0.382997\pi\)
0.359354 + 0.933201i \(0.382997\pi\)
\(938\) 0 0
\(939\) −26.0000 −0.848478
\(940\) 0 0
\(941\) −30.0000 −0.977972 −0.488986 − 0.872292i \(-0.662633\pi\)
−0.488986 + 0.872292i \(0.662633\pi\)
\(942\) −18.0000 −0.586472
\(943\) −48.0000 −1.56310
\(944\) 8.00000 0.260378
\(945\) 0 0
\(946\) 0 0
\(947\) −32.0000 −1.03986 −0.519930 − 0.854209i \(-0.674042\pi\)
−0.519930 + 0.854209i \(0.674042\pi\)
\(948\) −8.00000 −0.259828
\(949\) 36.0000 1.16861
\(950\) 0 0
\(951\) −6.00000 −0.194563
\(952\) 0 0
\(953\) 42.0000 1.36051 0.680257 − 0.732974i \(-0.261868\pi\)
0.680257 + 0.732974i \(0.261868\pi\)
\(954\) −10.0000 −0.323762
\(955\) 32.0000 1.03550
\(956\) 24.0000 0.776215
\(957\) 0 0
\(958\) −40.0000 −1.29234
\(959\) 0 0
\(960\) 2.00000 0.0645497
\(961\) 33.0000 1.06452
\(962\) 60.0000 1.93448
\(963\) 16.0000 0.515593
\(964\) −2.00000 −0.0644157
\(965\) −12.0000 −0.386294
\(966\) 0 0
\(967\) −8.00000 −0.257263 −0.128631 − 0.991692i \(-0.541058\pi\)
−0.128631 + 0.991692i \(0.541058\pi\)
\(968\) −11.0000 −0.353553
\(969\) 0 0
\(970\) −4.00000 −0.128432
\(971\) 24.0000 0.770197 0.385098 − 0.922876i \(-0.374168\pi\)
0.385098 + 0.922876i \(0.374168\pi\)
\(972\) 1.00000 0.0320750
\(973\) 0 0
\(974\) 8.00000 0.256337
\(975\) −6.00000 −0.192154
\(976\) −6.00000 −0.192055
\(977\) −30.0000 −0.959785 −0.479893 − 0.877327i \(-0.659324\pi\)
−0.479893 + 0.877327i \(0.659324\pi\)
\(978\) 8.00000 0.255812
\(979\) 0 0
\(980\) 0 0
\(981\) −6.00000 −0.191565
\(982\) −20.0000 −0.638226
\(983\) 48.0000 1.53096 0.765481 − 0.643458i \(-0.222501\pi\)
0.765481 + 0.643458i \(0.222501\pi\)
\(984\) 6.00000 0.191273
\(985\) 36.0000 1.14706
\(986\) 6.00000 0.191079
\(987\) 0 0
\(988\) 0 0
\(989\) −96.0000 −3.05262
\(990\) 0 0
\(991\) −56.0000 −1.77890 −0.889449 − 0.457034i \(-0.848912\pi\)
−0.889449 + 0.457034i \(0.848912\pi\)
\(992\) 8.00000 0.254000
\(993\) −36.0000 −1.14243
\(994\) 0 0
\(995\) 48.0000 1.52170
\(996\) −16.0000 −0.506979
\(997\) −30.0000 −0.950110 −0.475055 − 0.879956i \(-0.657572\pi\)
−0.475055 + 0.879956i \(0.657572\pi\)
\(998\) 8.00000 0.253236
\(999\) 10.0000 0.316386
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 4998.2.a.bp.1.1 1
7.6 odd 2 714.2.a.e.1.1 ✓ 1
21.20 even 2 2142.2.a.g.1.1 1
28.27 even 2 5712.2.a.q.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
714.2.a.e.1.1 ✓ 1 7.6 odd 2
2142.2.a.g.1.1 1 21.20 even 2
4998.2.a.bp.1.1 1 1.1 even 1 trivial
5712.2.a.q.1.1 1 28.27 even 2