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Properties

Label 4998.2.a.bq.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)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +2.00000 q^{5} +1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +2.00000 q^{5} +1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} +2.00000 q^{10} +4.00000 q^{11} +1.00000 q^{12} +2.00000 q^{13} +2.00000 q^{15} +1.00000 q^{16} -1.00000 q^{17} +1.00000 q^{18} -4.00000 q^{19} +2.00000 q^{20} +4.00000 q^{22} +8.00000 q^{23} +1.00000 q^{24} -1.00000 q^{25} +2.00000 q^{26} +1.00000 q^{27} +6.00000 q^{29} +2.00000 q^{30} +1.00000 q^{32} +4.00000 q^{33} -1.00000 q^{34} +1.00000 q^{36} -2.00000 q^{37} -4.00000 q^{38} +2.00000 q^{39} +2.00000 q^{40} -10.0000 q^{41} -4.00000 q^{43} +4.00000 q^{44} +2.00000 q^{45} +8.00000 q^{46} +1.00000 q^{48} -1.00000 q^{50} -1.00000 q^{51} +2.00000 q^{52} +6.00000 q^{53} +1.00000 q^{54} +8.00000 q^{55} -4.00000 q^{57} +6.00000 q^{58} +4.00000 q^{59} +2.00000 q^{60} -6.00000 q^{61} +1.00000 q^{64} +4.00000 q^{65} +4.00000 q^{66} -12.0000 q^{67} -1.00000 q^{68} +8.00000 q^{69} -8.00000 q^{71} +1.00000 q^{72} +6.00000 q^{73} -2.00000 q^{74} -1.00000 q^{75} -4.00000 q^{76} +2.00000 q^{78} +2.00000 q^{80} +1.00000 q^{81} -10.0000 q^{82} +12.0000 q^{83} -2.00000 q^{85} -4.00000 q^{86} +6.00000 q^{87} +4.00000 q^{88} +6.00000 q^{89} +2.00000 q^{90} +8.00000 q^{92} -8.00000 q^{95} +1.00000 q^{96} -2.00000 q^{97} +4.00000 q^{99} +O(q^{100})\)

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
\(11\) 4.00000 1.20605 0.603023 − 0.797724i \(-0.293963\pi\)
0.603023 + 0.797724i \(0.293963\pi\)
\(12\) 1.00000 0.288675
\(13\) 2.00000 0.554700 0.277350 − 0.960769i \(-0.410544\pi\)
0.277350 + 0.960769i \(0.410544\pi\)
\(14\) 0 0
\(15\) 2.00000 0.516398
\(16\) 1.00000 0.250000
\(17\) −1.00000 −0.242536
\(18\) 1.00000 0.235702
\(19\) −4.00000 −0.917663 −0.458831 − 0.888523i \(-0.651732\pi\)
−0.458831 + 0.888523i \(0.651732\pi\)
\(20\) 2.00000 0.447214
\(21\) 0 0
\(22\) 4.00000 0.852803
\(23\) 8.00000 1.66812 0.834058 − 0.551677i \(-0.186012\pi\)
0.834058 + 0.551677i \(0.186012\pi\)
\(24\) 1.00000 0.204124
\(25\) −1.00000 −0.200000
\(26\) 2.00000 0.392232
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) 6.00000 1.11417 0.557086 − 0.830455i \(-0.311919\pi\)
0.557086 + 0.830455i \(0.311919\pi\)
\(30\) 2.00000 0.365148
\(31\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) 1.00000 0.176777
\(33\) 4.00000 0.696311
\(34\) −1.00000 −0.171499
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) −2.00000 −0.328798 −0.164399 − 0.986394i \(-0.552568\pi\)
−0.164399 + 0.986394i \(0.552568\pi\)
\(38\) −4.00000 −0.648886
\(39\) 2.00000 0.320256
\(40\) 2.00000 0.316228
\(41\) −10.0000 −1.56174 −0.780869 − 0.624695i \(-0.785223\pi\)
−0.780869 + 0.624695i \(0.785223\pi\)
\(42\) 0 0
\(43\) −4.00000 −0.609994 −0.304997 − 0.952353i \(-0.598656\pi\)
−0.304997 + 0.952353i \(0.598656\pi\)
\(44\) 4.00000 0.603023
\(45\) 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
\(52\) 2.00000 0.277350
\(53\) 6.00000 0.824163 0.412082 − 0.911147i \(-0.364802\pi\)
0.412082 + 0.911147i \(0.364802\pi\)
\(54\) 1.00000 0.136083
\(55\) 8.00000 1.07872
\(56\) 0 0
\(57\) −4.00000 −0.529813
\(58\) 6.00000 0.787839
\(59\) 4.00000 0.520756 0.260378 − 0.965507i \(-0.416153\pi\)
0.260378 + 0.965507i \(0.416153\pi\)
\(60\) 2.00000 0.258199
\(61\) −6.00000 −0.768221 −0.384111 − 0.923287i \(-0.625492\pi\)
−0.384111 + 0.923287i \(0.625492\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 4.00000 0.496139
\(66\) 4.00000 0.492366
\(67\) −12.0000 −1.46603 −0.733017 − 0.680211i \(-0.761888\pi\)
−0.733017 + 0.680211i \(0.761888\pi\)
\(68\) −1.00000 −0.121268
\(69\) 8.00000 0.963087
\(70\) 0 0
\(71\) −8.00000 −0.949425 −0.474713 − 0.880141i \(-0.657448\pi\)
−0.474713 + 0.880141i \(0.657448\pi\)
\(72\) 1.00000 0.117851
\(73\) 6.00000 0.702247 0.351123 − 0.936329i \(-0.385800\pi\)
0.351123 + 0.936329i \(0.385800\pi\)
\(74\) −2.00000 −0.232495
\(75\) −1.00000 −0.115470
\(76\) −4.00000 −0.458831
\(77\) 0 0
\(78\) 2.00000 0.226455
\(79\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(80\) 2.00000 0.223607
\(81\) 1.00000 0.111111
\(82\) −10.0000 −1.10432
\(83\) 12.0000 1.31717 0.658586 − 0.752506i \(-0.271155\pi\)
0.658586 + 0.752506i \(0.271155\pi\)
\(84\) 0 0
\(85\) −2.00000 −0.216930
\(86\) −4.00000 −0.431331
\(87\) 6.00000 0.643268
\(88\) 4.00000 0.426401
\(89\) 6.00000 0.635999 0.317999 − 0.948091i \(-0.396989\pi\)
0.317999 + 0.948091i \(0.396989\pi\)
\(90\) 2.00000 0.210819
\(91\) 0 0
\(92\) 8.00000 0.834058
\(93\) 0 0
\(94\) 0 0
\(95\) −8.00000 −0.820783
\(96\) 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
\(99\) 4.00000 0.402015
\(100\) −1.00000 −0.100000
\(101\) 10.0000 0.995037 0.497519 − 0.867453i \(-0.334245\pi\)
0.497519 + 0.867453i \(0.334245\pi\)
\(102\) −1.00000 −0.0990148
\(103\) −8.00000 −0.788263 −0.394132 − 0.919054i \(-0.628955\pi\)
−0.394132 + 0.919054i \(0.628955\pi\)
\(104\) 2.00000 0.196116
\(105\) 0 0
\(106\) 6.00000 0.582772
\(107\) −12.0000 −1.16008 −0.580042 − 0.814587i \(-0.696964\pi\)
−0.580042 + 0.814587i \(0.696964\pi\)
\(108\) 1.00000 0.0962250
\(109\) −10.0000 −0.957826 −0.478913 − 0.877862i \(-0.658969\pi\)
−0.478913 + 0.877862i \(0.658969\pi\)
\(110\) 8.00000 0.762770
\(111\) −2.00000 −0.189832
\(112\) 0 0
\(113\) 18.0000 1.69330 0.846649 − 0.532152i \(-0.178617\pi\)
0.846649 + 0.532152i \(0.178617\pi\)
\(114\) −4.00000 −0.374634
\(115\) 16.0000 1.49201
\(116\) 6.00000 0.557086
\(117\) 2.00000 0.184900
\(118\) 4.00000 0.368230
\(119\) 0 0
\(120\) 2.00000 0.182574
\(121\) 5.00000 0.454545
\(122\) −6.00000 −0.543214
\(123\) −10.0000 −0.901670
\(124\) 0 0
\(125\) −12.0000 −1.07331
\(126\) 0 0
\(127\) 16.0000 1.41977 0.709885 − 0.704317i \(-0.248747\pi\)
0.709885 + 0.704317i \(0.248747\pi\)
\(128\) 1.00000 0.0883883
\(129\) −4.00000 −0.352180
\(130\) 4.00000 0.350823
\(131\) −12.0000 −1.04844 −0.524222 − 0.851581i \(-0.675644\pi\)
−0.524222 + 0.851581i \(0.675644\pi\)
\(132\) 4.00000 0.348155
\(133\) 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
\(139\) 12.0000 1.01783 0.508913 − 0.860818i \(-0.330047\pi\)
0.508913 + 0.860818i \(0.330047\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) −8.00000 −0.671345
\(143\) 8.00000 0.668994
\(144\) 1.00000 0.0833333
\(145\) 12.0000 0.996546
\(146\) 6.00000 0.496564
\(147\) 0 0
\(148\) −2.00000 −0.164399
\(149\) 22.0000 1.80231 0.901155 − 0.433497i \(-0.142720\pi\)
0.901155 + 0.433497i \(0.142720\pi\)
\(150\) −1.00000 −0.0816497
\(151\) 8.00000 0.651031 0.325515 − 0.945537i \(-0.394462\pi\)
0.325515 + 0.945537i \(0.394462\pi\)
\(152\) −4.00000 −0.324443
\(153\) −1.00000 −0.0808452
\(154\) 0 0
\(155\) 0 0
\(156\) 2.00000 0.160128
\(157\) 2.00000 0.159617 0.0798087 − 0.996810i \(-0.474569\pi\)
0.0798087 + 0.996810i \(0.474569\pi\)
\(158\) 0 0
\(159\) 6.00000 0.475831
\(160\) 2.00000 0.158114
\(161\) 0 0
\(162\) 1.00000 0.0785674
\(163\) −20.0000 −1.56652 −0.783260 − 0.621694i \(-0.786445\pi\)
−0.783260 + 0.621694i \(0.786445\pi\)
\(164\) −10.0000 −0.780869
\(165\) 8.00000 0.622799
\(166\) 12.0000 0.931381
\(167\) −8.00000 −0.619059 −0.309529 − 0.950890i \(-0.600171\pi\)
−0.309529 + 0.950890i \(0.600171\pi\)
\(168\) 0 0
\(169\) −9.00000 −0.692308
\(170\) −2.00000 −0.153393
\(171\) −4.00000 −0.305888
\(172\) −4.00000 −0.304997
\(173\) −6.00000 −0.456172 −0.228086 − 0.973641i \(-0.573247\pi\)
−0.228086 + 0.973641i \(0.573247\pi\)
\(174\) 6.00000 0.454859
\(175\) 0 0
\(176\) 4.00000 0.301511
\(177\) 4.00000 0.300658
\(178\) 6.00000 0.449719
\(179\) −12.0000 −0.896922 −0.448461 − 0.893802i \(-0.648028\pi\)
−0.448461 + 0.893802i \(0.648028\pi\)
\(180\) 2.00000 0.149071
\(181\) 2.00000 0.148659 0.0743294 − 0.997234i \(-0.476318\pi\)
0.0743294 + 0.997234i \(0.476318\pi\)
\(182\) 0 0
\(183\) −6.00000 −0.443533
\(184\) 8.00000 0.589768
\(185\) −4.00000 −0.294086
\(186\) 0 0
\(187\) −4.00000 −0.292509
\(188\) 0 0
\(189\) 0 0
\(190\) −8.00000 −0.580381
\(191\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(192\) 1.00000 0.0721688
\(193\) −14.0000 −1.00774 −0.503871 − 0.863779i \(-0.668091\pi\)
−0.503871 + 0.863779i \(0.668091\pi\)
\(194\) −2.00000 −0.143592
\(195\) 4.00000 0.286446
\(196\) 0 0
\(197\) −2.00000 −0.142494 −0.0712470 − 0.997459i \(-0.522698\pi\)
−0.0712470 + 0.997459i \(0.522698\pi\)
\(198\) 4.00000 0.284268
\(199\) 8.00000 0.567105 0.283552 − 0.958957i \(-0.408487\pi\)
0.283552 + 0.958957i \(0.408487\pi\)
\(200\) −1.00000 −0.0707107
\(201\) −12.0000 −0.846415
\(202\) 10.0000 0.703598
\(203\) 0 0
\(204\) −1.00000 −0.0700140
\(205\) −20.0000 −1.39686
\(206\) −8.00000 −0.557386
\(207\) 8.00000 0.556038
\(208\) 2.00000 0.138675
\(209\) −16.0000 −1.10674
\(210\) 0 0
\(211\) 12.0000 0.826114 0.413057 − 0.910705i \(-0.364461\pi\)
0.413057 + 0.910705i \(0.364461\pi\)
\(212\) 6.00000 0.412082
\(213\) −8.00000 −0.548151
\(214\) −12.0000 −0.820303
\(215\) −8.00000 −0.545595
\(216\) 1.00000 0.0680414
\(217\) 0 0
\(218\) −10.0000 −0.677285
\(219\) 6.00000 0.405442
\(220\) 8.00000 0.539360
\(221\) −2.00000 −0.134535
\(222\) −2.00000 −0.134231
\(223\) 16.0000 1.07144 0.535720 − 0.844396i \(-0.320040\pi\)
0.535720 + 0.844396i \(0.320040\pi\)
\(224\) 0 0
\(225\) −1.00000 −0.0666667
\(226\) 18.0000 1.19734
\(227\) 20.0000 1.32745 0.663723 − 0.747978i \(-0.268975\pi\)
0.663723 + 0.747978i \(0.268975\pi\)
\(228\) −4.00000 −0.264906
\(229\) −6.00000 −0.396491 −0.198246 − 0.980152i \(-0.563524\pi\)
−0.198246 + 0.980152i \(0.563524\pi\)
\(230\) 16.0000 1.05501
\(231\) 0 0
\(232\) 6.00000 0.393919
\(233\) 10.0000 0.655122 0.327561 − 0.944830i \(-0.393773\pi\)
0.327561 + 0.944830i \(0.393773\pi\)
\(234\) 2.00000 0.130744
\(235\) 0 0
\(236\) 4.00000 0.260378
\(237\) 0 0
\(238\) 0 0
\(239\) 16.0000 1.03495 0.517477 − 0.855697i \(-0.326871\pi\)
0.517477 + 0.855697i \(0.326871\pi\)
\(240\) 2.00000 0.129099
\(241\) −18.0000 −1.15948 −0.579741 − 0.814801i \(-0.696846\pi\)
−0.579741 + 0.814801i \(0.696846\pi\)
\(242\) 5.00000 0.321412
\(243\) 1.00000 0.0641500
\(244\) −6.00000 −0.384111
\(245\) 0 0
\(246\) −10.0000 −0.637577
\(247\) −8.00000 −0.509028
\(248\) 0 0
\(249\) 12.0000 0.760469
\(250\) −12.0000 −0.758947
\(251\) 4.00000 0.252478 0.126239 − 0.992000i \(-0.459709\pi\)
0.126239 + 0.992000i \(0.459709\pi\)
\(252\) 0 0
\(253\) 32.0000 2.01182
\(254\) 16.0000 1.00393
\(255\) −2.00000 −0.125245
\(256\) 1.00000 0.0625000
\(257\) −2.00000 −0.124757 −0.0623783 − 0.998053i \(-0.519869\pi\)
−0.0623783 + 0.998053i \(0.519869\pi\)
\(258\) −4.00000 −0.249029
\(259\) 0 0
\(260\) 4.00000 0.248069
\(261\) 6.00000 0.371391
\(262\) −12.0000 −0.741362
\(263\) −8.00000 −0.493301 −0.246651 − 0.969104i \(-0.579330\pi\)
−0.246651 + 0.969104i \(0.579330\pi\)
\(264\) 4.00000 0.246183
\(265\) 12.0000 0.737154
\(266\) 0 0
\(267\) 6.00000 0.367194
\(268\) −12.0000 −0.733017
\(269\) 26.0000 1.58525 0.792624 − 0.609711i \(-0.208714\pi\)
0.792624 + 0.609711i \(0.208714\pi\)
\(270\) 2.00000 0.121716
\(271\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(272\) −1.00000 −0.0606339
\(273\) 0 0
\(274\) −22.0000 −1.32907
\(275\) −4.00000 −0.241209
\(276\) 8.00000 0.481543
\(277\) −2.00000 −0.120168 −0.0600842 − 0.998193i \(-0.519137\pi\)
−0.0600842 + 0.998193i \(0.519137\pi\)
\(278\) 12.0000 0.719712
\(279\) 0 0
\(280\) 0 0
\(281\) 26.0000 1.55103 0.775515 − 0.631329i \(-0.217490\pi\)
0.775515 + 0.631329i \(0.217490\pi\)
\(282\) 0 0
\(283\) −20.0000 −1.18888 −0.594438 − 0.804141i \(-0.702626\pi\)
−0.594438 + 0.804141i \(0.702626\pi\)
\(284\) −8.00000 −0.474713
\(285\) −8.00000 −0.473879
\(286\) 8.00000 0.473050
\(287\) 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
\(293\) −6.00000 −0.350524 −0.175262 − 0.984522i \(-0.556077\pi\)
−0.175262 + 0.984522i \(0.556077\pi\)
\(294\) 0 0
\(295\) 8.00000 0.465778
\(296\) −2.00000 −0.116248
\(297\) 4.00000 0.232104
\(298\) 22.0000 1.27443
\(299\) 16.0000 0.925304
\(300\) −1.00000 −0.0577350
\(301\) 0 0
\(302\) 8.00000 0.460348
\(303\) 10.0000 0.574485
\(304\) −4.00000 −0.229416
\(305\) −12.0000 −0.687118
\(306\) −1.00000 −0.0571662
\(307\) 28.0000 1.59804 0.799022 − 0.601302i \(-0.205351\pi\)
0.799022 + 0.601302i \(0.205351\pi\)
\(308\) 0 0
\(309\) −8.00000 −0.455104
\(310\) 0 0
\(311\) −24.0000 −1.36092 −0.680458 − 0.732787i \(-0.738219\pi\)
−0.680458 + 0.732787i \(0.738219\pi\)
\(312\) 2.00000 0.113228
\(313\) −26.0000 −1.46961 −0.734803 − 0.678280i \(-0.762726\pi\)
−0.734803 + 0.678280i \(0.762726\pi\)
\(314\) 2.00000 0.112867
\(315\) 0 0
\(316\) 0 0
\(317\) 22.0000 1.23564 0.617822 − 0.786318i \(-0.288015\pi\)
0.617822 + 0.786318i \(0.288015\pi\)
\(318\) 6.00000 0.336463
\(319\) 24.0000 1.34374
\(320\) 2.00000 0.111803
\(321\) −12.0000 −0.669775
\(322\) 0 0
\(323\) 4.00000 0.222566
\(324\) 1.00000 0.0555556
\(325\) −2.00000 −0.110940
\(326\) −20.0000 −1.10770
\(327\) −10.0000 −0.553001
\(328\) −10.0000 −0.552158
\(329\) 0 0
\(330\) 8.00000 0.440386
\(331\) −4.00000 −0.219860 −0.109930 − 0.993939i \(-0.535063\pi\)
−0.109930 + 0.993939i \(0.535063\pi\)
\(332\) 12.0000 0.658586
\(333\) −2.00000 −0.109599
\(334\) −8.00000 −0.437741
\(335\) −24.0000 −1.31126
\(336\) 0 0
\(337\) −14.0000 −0.762629 −0.381314 − 0.924445i \(-0.624528\pi\)
−0.381314 + 0.924445i \(0.624528\pi\)
\(338\) −9.00000 −0.489535
\(339\) 18.0000 0.977626
\(340\) −2.00000 −0.108465
\(341\) 0 0
\(342\) −4.00000 −0.216295
\(343\) 0 0
\(344\) −4.00000 −0.215666
\(345\) 16.0000 0.861411
\(346\) −6.00000 −0.322562
\(347\) −28.0000 −1.50312 −0.751559 − 0.659665i \(-0.770698\pi\)
−0.751559 + 0.659665i \(0.770698\pi\)
\(348\) 6.00000 0.321634
\(349\) 18.0000 0.963518 0.481759 − 0.876304i \(-0.339998\pi\)
0.481759 + 0.876304i \(0.339998\pi\)
\(350\) 0 0
\(351\) 2.00000 0.106752
\(352\) 4.00000 0.213201
\(353\) −34.0000 −1.80964 −0.904819 − 0.425797i \(-0.859994\pi\)
−0.904819 + 0.425797i \(0.859994\pi\)
\(354\) 4.00000 0.212598
\(355\) −16.0000 −0.849192
\(356\) 6.00000 0.317999
\(357\) 0 0
\(358\) −12.0000 −0.634220
\(359\) −24.0000 −1.26667 −0.633336 − 0.773877i \(-0.718315\pi\)
−0.633336 + 0.773877i \(0.718315\pi\)
\(360\) 2.00000 0.105409
\(361\) −3.00000 −0.157895
\(362\) 2.00000 0.105118
\(363\) 5.00000 0.262432
\(364\) 0 0
\(365\) 12.0000 0.628109
\(366\) −6.00000 −0.313625
\(367\) 32.0000 1.67039 0.835193 − 0.549957i \(-0.185356\pi\)
0.835193 + 0.549957i \(0.185356\pi\)
\(368\) 8.00000 0.417029
\(369\) −10.0000 −0.520579
\(370\) −4.00000 −0.207950
\(371\) 0 0
\(372\) 0 0
\(373\) −26.0000 −1.34623 −0.673114 − 0.739538i \(-0.735044\pi\)
−0.673114 + 0.739538i \(0.735044\pi\)
\(374\) −4.00000 −0.206835
\(375\) −12.0000 −0.619677
\(376\) 0 0
\(377\) 12.0000 0.618031
\(378\) 0 0
\(379\) 20.0000 1.02733 0.513665 − 0.857991i \(-0.328287\pi\)
0.513665 + 0.857991i \(0.328287\pi\)
\(380\) −8.00000 −0.410391
\(381\) 16.0000 0.819705
\(382\) 0 0
\(383\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(384\) 1.00000 0.0510310
\(385\) 0 0
\(386\) −14.0000 −0.712581
\(387\) −4.00000 −0.203331
\(388\) −2.00000 −0.101535
\(389\) −26.0000 −1.31825 −0.659126 − 0.752032i \(-0.729074\pi\)
−0.659126 + 0.752032i \(0.729074\pi\)
\(390\) 4.00000 0.202548
\(391\) −8.00000 −0.404577
\(392\) 0 0
\(393\) −12.0000 −0.605320
\(394\) −2.00000 −0.100759
\(395\) 0 0
\(396\) 4.00000 0.201008
\(397\) −22.0000 −1.10415 −0.552074 − 0.833795i \(-0.686163\pi\)
−0.552074 + 0.833795i \(0.686163\pi\)
\(398\) 8.00000 0.401004
\(399\) 0 0
\(400\) −1.00000 −0.0500000
\(401\) −30.0000 −1.49813 −0.749064 − 0.662497i \(-0.769497\pi\)
−0.749064 + 0.662497i \(0.769497\pi\)
\(402\) −12.0000 −0.598506
\(403\) 0 0
\(404\) 10.0000 0.497519
\(405\) 2.00000 0.0993808
\(406\) 0 0
\(407\) −8.00000 −0.396545
\(408\) −1.00000 −0.0495074
\(409\) −26.0000 −1.28562 −0.642809 − 0.766027i \(-0.722231\pi\)
−0.642809 + 0.766027i \(0.722231\pi\)
\(410\) −20.0000 −0.987730
\(411\) −22.0000 −1.08518
\(412\) −8.00000 −0.394132
\(413\) 0 0
\(414\) 8.00000 0.393179
\(415\) 24.0000 1.17811
\(416\) 2.00000 0.0980581
\(417\) 12.0000 0.587643
\(418\) −16.0000 −0.782586
\(419\) 20.0000 0.977064 0.488532 − 0.872546i \(-0.337533\pi\)
0.488532 + 0.872546i \(0.337533\pi\)
\(420\) 0 0
\(421\) −10.0000 −0.487370 −0.243685 − 0.969854i \(-0.578356\pi\)
−0.243685 + 0.969854i \(0.578356\pi\)
\(422\) 12.0000 0.584151
\(423\) 0 0
\(424\) 6.00000 0.291386
\(425\) 1.00000 0.0485071
\(426\) −8.00000 −0.387601
\(427\) 0 0
\(428\) −12.0000 −0.580042
\(429\) 8.00000 0.386244
\(430\) −8.00000 −0.385794
\(431\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(432\) 1.00000 0.0481125
\(433\) 14.0000 0.672797 0.336399 − 0.941720i \(-0.390791\pi\)
0.336399 + 0.941720i \(0.390791\pi\)
\(434\) 0 0
\(435\) 12.0000 0.575356
\(436\) −10.0000 −0.478913
\(437\) −32.0000 −1.53077
\(438\) 6.00000 0.286691
\(439\) −8.00000 −0.381819 −0.190910 − 0.981608i \(-0.561144\pi\)
−0.190910 + 0.981608i \(0.561144\pi\)
\(440\) 8.00000 0.381385
\(441\) 0 0
\(442\) −2.00000 −0.0951303
\(443\) 12.0000 0.570137 0.285069 − 0.958507i \(-0.407984\pi\)
0.285069 + 0.958507i \(0.407984\pi\)
\(444\) −2.00000 −0.0949158
\(445\) 12.0000 0.568855
\(446\) 16.0000 0.757622
\(447\) 22.0000 1.04056
\(448\) 0 0
\(449\) 18.0000 0.849473 0.424736 − 0.905317i \(-0.360367\pi\)
0.424736 + 0.905317i \(0.360367\pi\)
\(450\) −1.00000 −0.0471405
\(451\) −40.0000 −1.88353
\(452\) 18.0000 0.846649
\(453\) 8.00000 0.375873
\(454\) 20.0000 0.938647
\(455\) 0 0
\(456\) −4.00000 −0.187317
\(457\) 10.0000 0.467780 0.233890 − 0.972263i \(-0.424854\pi\)
0.233890 + 0.972263i \(0.424854\pi\)
\(458\) −6.00000 −0.280362
\(459\) −1.00000 −0.0466760
\(460\) 16.0000 0.746004
\(461\) 2.00000 0.0931493 0.0465746 − 0.998915i \(-0.485169\pi\)
0.0465746 + 0.998915i \(0.485169\pi\)
\(462\) 0 0
\(463\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(464\) 6.00000 0.278543
\(465\) 0 0
\(466\) 10.0000 0.463241
\(467\) −36.0000 −1.66588 −0.832941 − 0.553362i \(-0.813345\pi\)
−0.832941 + 0.553362i \(0.813345\pi\)
\(468\) 2.00000 0.0924500
\(469\) 0 0
\(470\) 0 0
\(471\) 2.00000 0.0921551
\(472\) 4.00000 0.184115
\(473\) −16.0000 −0.735681
\(474\) 0 0
\(475\) 4.00000 0.183533
\(476\) 0 0
\(477\) 6.00000 0.274721
\(478\) 16.0000 0.731823
\(479\) 16.0000 0.731059 0.365529 − 0.930800i \(-0.380888\pi\)
0.365529 + 0.930800i \(0.380888\pi\)
\(480\) 2.00000 0.0912871
\(481\) −4.00000 −0.182384
\(482\) −18.0000 −0.819878
\(483\) 0 0
\(484\) 5.00000 0.227273
\(485\) −4.00000 −0.181631
\(486\) 1.00000 0.0453609
\(487\) 24.0000 1.08754 0.543772 − 0.839233i \(-0.316996\pi\)
0.543772 + 0.839233i \(0.316996\pi\)
\(488\) −6.00000 −0.271607
\(489\) −20.0000 −0.904431
\(490\) 0 0
\(491\) −20.0000 −0.902587 −0.451294 − 0.892375i \(-0.649037\pi\)
−0.451294 + 0.892375i \(0.649037\pi\)
\(492\) −10.0000 −0.450835
\(493\) −6.00000 −0.270226
\(494\) −8.00000 −0.359937
\(495\) 8.00000 0.359573
\(496\) 0 0
\(497\) 0 0
\(498\) 12.0000 0.537733
\(499\) −20.0000 −0.895323 −0.447661 − 0.894203i \(-0.647743\pi\)
−0.447661 + 0.894203i \(0.647743\pi\)
\(500\) −12.0000 −0.536656
\(501\) −8.00000 −0.357414
\(502\) 4.00000 0.178529
\(503\) 24.0000 1.07011 0.535054 − 0.844818i \(-0.320291\pi\)
0.535054 + 0.844818i \(0.320291\pi\)
\(504\) 0 0
\(505\) 20.0000 0.889988
\(506\) 32.0000 1.42257
\(507\) −9.00000 −0.399704
\(508\) 16.0000 0.709885
\(509\) 2.00000 0.0886484 0.0443242 − 0.999017i \(-0.485887\pi\)
0.0443242 + 0.999017i \(0.485887\pi\)
\(510\) −2.00000 −0.0885615
\(511\) 0 0
\(512\) 1.00000 0.0441942
\(513\) −4.00000 −0.176604
\(514\) −2.00000 −0.0882162
\(515\) −16.0000 −0.705044
\(516\) −4.00000 −0.176090
\(517\) 0 0
\(518\) 0 0
\(519\) −6.00000 −0.263371
\(520\) 4.00000 0.175412
\(521\) −10.0000 −0.438108 −0.219054 − 0.975713i \(-0.570297\pi\)
−0.219054 + 0.975713i \(0.570297\pi\)
\(522\) 6.00000 0.262613
\(523\) −44.0000 −1.92399 −0.961993 − 0.273075i \(-0.911959\pi\)
−0.961993 + 0.273075i \(0.911959\pi\)
\(524\) −12.0000 −0.524222
\(525\) 0 0
\(526\) −8.00000 −0.348817
\(527\) 0 0
\(528\) 4.00000 0.174078
\(529\) 41.0000 1.78261
\(530\) 12.0000 0.521247
\(531\) 4.00000 0.173585
\(532\) 0 0
\(533\) −20.0000 −0.866296
\(534\) 6.00000 0.259645
\(535\) −24.0000 −1.03761
\(536\) −12.0000 −0.518321
\(537\) −12.0000 −0.517838
\(538\) 26.0000 1.12094
\(539\) 0 0
\(540\) 2.00000 0.0860663
\(541\) 22.0000 0.945854 0.472927 − 0.881102i \(-0.343197\pi\)
0.472927 + 0.881102i \(0.343197\pi\)
\(542\) 0 0
\(543\) 2.00000 0.0858282
\(544\) −1.00000 −0.0428746
\(545\) −20.0000 −0.856706
\(546\) 0 0
\(547\) −20.0000 −0.855138 −0.427569 − 0.903983i \(-0.640630\pi\)
−0.427569 + 0.903983i \(0.640630\pi\)
\(548\) −22.0000 −0.939793
\(549\) −6.00000 −0.256074
\(550\) −4.00000 −0.170561
\(551\) −24.0000 −1.02243
\(552\) 8.00000 0.340503
\(553\) 0 0
\(554\) −2.00000 −0.0849719
\(555\) −4.00000 −0.169791
\(556\) 12.0000 0.508913
\(557\) −2.00000 −0.0847427 −0.0423714 − 0.999102i \(-0.513491\pi\)
−0.0423714 + 0.999102i \(0.513491\pi\)
\(558\) 0 0
\(559\) −8.00000 −0.338364
\(560\) 0 0
\(561\) −4.00000 −0.168880
\(562\) 26.0000 1.09674
\(563\) 12.0000 0.505740 0.252870 − 0.967500i \(-0.418626\pi\)
0.252870 + 0.967500i \(0.418626\pi\)
\(564\) 0 0
\(565\) 36.0000 1.51453
\(566\) −20.0000 −0.840663
\(567\) 0 0
\(568\) −8.00000 −0.335673
\(569\) −38.0000 −1.59304 −0.796521 − 0.604610i \(-0.793329\pi\)
−0.796521 + 0.604610i \(0.793329\pi\)
\(570\) −8.00000 −0.335083
\(571\) 36.0000 1.50655 0.753277 − 0.657704i \(-0.228472\pi\)
0.753277 + 0.657704i \(0.228472\pi\)
\(572\) 8.00000 0.334497
\(573\) 0 0
\(574\) 0 0
\(575\) −8.00000 −0.333623
\(576\) 1.00000 0.0416667
\(577\) 30.0000 1.24892 0.624458 − 0.781058i \(-0.285320\pi\)
0.624458 + 0.781058i \(0.285320\pi\)
\(578\) 1.00000 0.0415945
\(579\) −14.0000 −0.581820
\(580\) 12.0000 0.498273
\(581\) 0 0
\(582\) −2.00000 −0.0829027
\(583\) 24.0000 0.993978
\(584\) 6.00000 0.248282
\(585\) 4.00000 0.165380
\(586\) −6.00000 −0.247858
\(587\) 36.0000 1.48588 0.742940 − 0.669359i \(-0.233431\pi\)
0.742940 + 0.669359i \(0.233431\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 8.00000 0.329355
\(591\) −2.00000 −0.0822690
\(592\) −2.00000 −0.0821995
\(593\) 14.0000 0.574911 0.287456 − 0.957794i \(-0.407191\pi\)
0.287456 + 0.957794i \(0.407191\pi\)
\(594\) 4.00000 0.164122
\(595\) 0 0
\(596\) 22.0000 0.901155
\(597\) 8.00000 0.327418
\(598\) 16.0000 0.654289
\(599\) 8.00000 0.326871 0.163436 − 0.986554i \(-0.447742\pi\)
0.163436 + 0.986554i \(0.447742\pi\)
\(600\) −1.00000 −0.0408248
\(601\) −10.0000 −0.407909 −0.203954 − 0.978980i \(-0.565379\pi\)
−0.203954 + 0.978980i \(0.565379\pi\)
\(602\) 0 0
\(603\) −12.0000 −0.488678
\(604\) 8.00000 0.325515
\(605\) 10.0000 0.406558
\(606\) 10.0000 0.406222
\(607\) 32.0000 1.29884 0.649420 − 0.760430i \(-0.275012\pi\)
0.649420 + 0.760430i \(0.275012\pi\)
\(608\) −4.00000 −0.162221
\(609\) 0 0
\(610\) −12.0000 −0.485866
\(611\) 0 0
\(612\) −1.00000 −0.0404226
\(613\) −26.0000 −1.05013 −0.525065 − 0.851062i \(-0.675959\pi\)
−0.525065 + 0.851062i \(0.675959\pi\)
\(614\) 28.0000 1.12999
\(615\) −20.0000 −0.806478
\(616\) 0 0
\(617\) −22.0000 −0.885687 −0.442843 − 0.896599i \(-0.646030\pi\)
−0.442843 + 0.896599i \(0.646030\pi\)
\(618\) −8.00000 −0.321807
\(619\) 44.0000 1.76851 0.884255 − 0.467005i \(-0.154667\pi\)
0.884255 + 0.467005i \(0.154667\pi\)
\(620\) 0 0
\(621\) 8.00000 0.321029
\(622\) −24.0000 −0.962312
\(623\) 0 0
\(624\) 2.00000 0.0800641
\(625\) −19.0000 −0.760000
\(626\) −26.0000 −1.03917
\(627\) −16.0000 −0.638978
\(628\) 2.00000 0.0798087
\(629\) 2.00000 0.0797452
\(630\) 0 0
\(631\) −8.00000 −0.318475 −0.159237 − 0.987240i \(-0.550904\pi\)
−0.159237 + 0.987240i \(0.550904\pi\)
\(632\) 0 0
\(633\) 12.0000 0.476957
\(634\) 22.0000 0.873732
\(635\) 32.0000 1.26988
\(636\) 6.00000 0.237915
\(637\) 0 0
\(638\) 24.0000 0.950169
\(639\) −8.00000 −0.316475
\(640\) 2.00000 0.0790569
\(641\) 18.0000 0.710957 0.355479 − 0.934684i \(-0.384318\pi\)
0.355479 + 0.934684i \(0.384318\pi\)
\(642\) −12.0000 −0.473602
\(643\) 36.0000 1.41970 0.709851 − 0.704352i \(-0.248762\pi\)
0.709851 + 0.704352i \(0.248762\pi\)
\(644\) 0 0
\(645\) −8.00000 −0.315000
\(646\) 4.00000 0.157378
\(647\) −24.0000 −0.943537 −0.471769 − 0.881722i \(-0.656384\pi\)
−0.471769 + 0.881722i \(0.656384\pi\)
\(648\) 1.00000 0.0392837
\(649\) 16.0000 0.628055
\(650\) −2.00000 −0.0784465
\(651\) 0 0
\(652\) −20.0000 −0.783260
\(653\) −42.0000 −1.64359 −0.821794 − 0.569785i \(-0.807026\pi\)
−0.821794 + 0.569785i \(0.807026\pi\)
\(654\) −10.0000 −0.391031
\(655\) −24.0000 −0.937758
\(656\) −10.0000 −0.390434
\(657\) 6.00000 0.234082
\(658\) 0 0
\(659\) −12.0000 −0.467454 −0.233727 − 0.972302i \(-0.575092\pi\)
−0.233727 + 0.972302i \(0.575092\pi\)
\(660\) 8.00000 0.311400
\(661\) −38.0000 −1.47803 −0.739014 − 0.673690i \(-0.764708\pi\)
−0.739014 + 0.673690i \(0.764708\pi\)
\(662\) −4.00000 −0.155464
\(663\) −2.00000 −0.0776736
\(664\) 12.0000 0.465690
\(665\) 0 0
\(666\) −2.00000 −0.0774984
\(667\) 48.0000 1.85857
\(668\) −8.00000 −0.309529
\(669\) 16.0000 0.618596
\(670\) −24.0000 −0.927201
\(671\) −24.0000 −0.926510
\(672\) 0 0
\(673\) −46.0000 −1.77317 −0.886585 − 0.462566i \(-0.846929\pi\)
−0.886585 + 0.462566i \(0.846929\pi\)
\(674\) −14.0000 −0.539260
\(675\) −1.00000 −0.0384900
\(676\) −9.00000 −0.346154
\(677\) 18.0000 0.691796 0.345898 − 0.938272i \(-0.387574\pi\)
0.345898 + 0.938272i \(0.387574\pi\)
\(678\) 18.0000 0.691286
\(679\) 0 0
\(680\) −2.00000 −0.0766965
\(681\) 20.0000 0.766402
\(682\) 0 0
\(683\) −28.0000 −1.07139 −0.535695 − 0.844411i \(-0.679950\pi\)
−0.535695 + 0.844411i \(0.679950\pi\)
\(684\) −4.00000 −0.152944
\(685\) −44.0000 −1.68115
\(686\) 0 0
\(687\) −6.00000 −0.228914
\(688\) −4.00000 −0.152499
\(689\) 12.0000 0.457164
\(690\) 16.0000 0.609110
\(691\) 20.0000 0.760836 0.380418 − 0.924815i \(-0.375780\pi\)
0.380418 + 0.924815i \(0.375780\pi\)
\(692\) −6.00000 −0.228086
\(693\) 0 0
\(694\) −28.0000 −1.06287
\(695\) 24.0000 0.910372
\(696\) 6.00000 0.227429
\(697\) 10.0000 0.378777
\(698\) 18.0000 0.681310
\(699\) 10.0000 0.378235
\(700\) 0 0
\(701\) 14.0000 0.528773 0.264386 − 0.964417i \(-0.414831\pi\)
0.264386 + 0.964417i \(0.414831\pi\)
\(702\) 2.00000 0.0754851
\(703\) 8.00000 0.301726
\(704\) 4.00000 0.150756
\(705\) 0 0
\(706\) −34.0000 −1.27961
\(707\) 0 0
\(708\) 4.00000 0.150329
\(709\) −18.0000 −0.676004 −0.338002 − 0.941145i \(-0.609751\pi\)
−0.338002 + 0.941145i \(0.609751\pi\)
\(710\) −16.0000 −0.600469
\(711\) 0 0
\(712\) 6.00000 0.224860
\(713\) 0 0
\(714\) 0 0
\(715\) 16.0000 0.598366
\(716\) −12.0000 −0.448461
\(717\) 16.0000 0.597531
\(718\) −24.0000 −0.895672
\(719\) 32.0000 1.19340 0.596699 − 0.802465i \(-0.296479\pi\)
0.596699 + 0.802465i \(0.296479\pi\)
\(720\) 2.00000 0.0745356
\(721\) 0 0
\(722\) −3.00000 −0.111648
\(723\) −18.0000 −0.669427
\(724\) 2.00000 0.0743294
\(725\) −6.00000 −0.222834
\(726\) 5.00000 0.185567
\(727\) −8.00000 −0.296704 −0.148352 − 0.988935i \(-0.547397\pi\)
−0.148352 + 0.988935i \(0.547397\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 12.0000 0.444140
\(731\) 4.00000 0.147945
\(732\) −6.00000 −0.221766
\(733\) 50.0000 1.84679 0.923396 − 0.383849i \(-0.125402\pi\)
0.923396 + 0.383849i \(0.125402\pi\)
\(734\) 32.0000 1.18114
\(735\) 0 0
\(736\) 8.00000 0.294884
\(737\) −48.0000 −1.76810
\(738\) −10.0000 −0.368105
\(739\) 36.0000 1.32428 0.662141 − 0.749380i \(-0.269648\pi\)
0.662141 + 0.749380i \(0.269648\pi\)
\(740\) −4.00000 −0.147043
\(741\) −8.00000 −0.293887
\(742\) 0 0
\(743\) 8.00000 0.293492 0.146746 − 0.989174i \(-0.453120\pi\)
0.146746 + 0.989174i \(0.453120\pi\)
\(744\) 0 0
\(745\) 44.0000 1.61204
\(746\) −26.0000 −0.951928
\(747\) 12.0000 0.439057
\(748\) −4.00000 −0.146254
\(749\) 0 0
\(750\) −12.0000 −0.438178
\(751\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(752\) 0 0
\(753\) 4.00000 0.145768
\(754\) 12.0000 0.437014
\(755\) 16.0000 0.582300
\(756\) 0 0
\(757\) −42.0000 −1.52652 −0.763258 − 0.646094i \(-0.776401\pi\)
−0.763258 + 0.646094i \(0.776401\pi\)
\(758\) 20.0000 0.726433
\(759\) 32.0000 1.16153
\(760\) −8.00000 −0.290191
\(761\) 6.00000 0.217500 0.108750 − 0.994069i \(-0.465315\pi\)
0.108750 + 0.994069i \(0.465315\pi\)
\(762\) 16.0000 0.579619
\(763\) 0 0
\(764\) 0 0
\(765\) −2.00000 −0.0723102
\(766\) 0 0
\(767\) 8.00000 0.288863
\(768\) 1.00000 0.0360844
\(769\) −2.00000 −0.0721218 −0.0360609 − 0.999350i \(-0.511481\pi\)
−0.0360609 + 0.999350i \(0.511481\pi\)
\(770\) 0 0
\(771\) −2.00000 −0.0720282
\(772\) −14.0000 −0.503871
\(773\) 26.0000 0.935155 0.467578 − 0.883952i \(-0.345127\pi\)
0.467578 + 0.883952i \(0.345127\pi\)
\(774\) −4.00000 −0.143777
\(775\) 0 0
\(776\) −2.00000 −0.0717958
\(777\) 0 0
\(778\) −26.0000 −0.932145
\(779\) 40.0000 1.43315
\(780\) 4.00000 0.143223
\(781\) −32.0000 −1.14505
\(782\) −8.00000 −0.286079
\(783\) 6.00000 0.214423
\(784\) 0 0
\(785\) 4.00000 0.142766
\(786\) −12.0000 −0.428026
\(787\) −28.0000 −0.998092 −0.499046 − 0.866575i \(-0.666316\pi\)
−0.499046 + 0.866575i \(0.666316\pi\)
\(788\) −2.00000 −0.0712470
\(789\) −8.00000 −0.284808
\(790\) 0 0
\(791\) 0 0
\(792\) 4.00000 0.142134
\(793\) −12.0000 −0.426132
\(794\) −22.0000 −0.780751
\(795\) 12.0000 0.425596
\(796\) 8.00000 0.283552
\(797\) −46.0000 −1.62940 −0.814702 − 0.579880i \(-0.803099\pi\)
−0.814702 + 0.579880i \(0.803099\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) −1.00000 −0.0353553
\(801\) 6.00000 0.212000
\(802\) −30.0000 −1.05934
\(803\) 24.0000 0.846942
\(804\) −12.0000 −0.423207
\(805\) 0 0
\(806\) 0 0
\(807\) 26.0000 0.915243
\(808\) 10.0000 0.351799
\(809\) −38.0000 −1.33601 −0.668004 − 0.744157i \(-0.732851\pi\)
−0.668004 + 0.744157i \(0.732851\pi\)
\(810\) 2.00000 0.0702728
\(811\) 28.0000 0.983213 0.491606 − 0.870817i \(-0.336410\pi\)
0.491606 + 0.870817i \(0.336410\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) −8.00000 −0.280400
\(815\) −40.0000 −1.40114
\(816\) −1.00000 −0.0350070
\(817\) 16.0000 0.559769
\(818\) −26.0000 −0.909069
\(819\) 0 0
\(820\) −20.0000 −0.698430
\(821\) 46.0000 1.60541 0.802706 − 0.596376i \(-0.203393\pi\)
0.802706 + 0.596376i \(0.203393\pi\)
\(822\) −22.0000 −0.767338
\(823\) 8.00000 0.278862 0.139431 − 0.990232i \(-0.455473\pi\)
0.139431 + 0.990232i \(0.455473\pi\)
\(824\) −8.00000 −0.278693
\(825\) −4.00000 −0.139262
\(826\) 0 0
\(827\) 20.0000 0.695468 0.347734 − 0.937593i \(-0.386951\pi\)
0.347734 + 0.937593i \(0.386951\pi\)
\(828\) 8.00000 0.278019
\(829\) −30.0000 −1.04194 −0.520972 − 0.853574i \(-0.674430\pi\)
−0.520972 + 0.853574i \(0.674430\pi\)
\(830\) 24.0000 0.833052
\(831\) −2.00000 −0.0693792
\(832\) 2.00000 0.0693375
\(833\) 0 0
\(834\) 12.0000 0.415526
\(835\) −16.0000 −0.553703
\(836\) −16.0000 −0.553372
\(837\) 0 0
\(838\) 20.0000 0.690889
\(839\) −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
\(842\) −10.0000 −0.344623
\(843\) 26.0000 0.895488
\(844\) 12.0000 0.413057
\(845\) −18.0000 −0.619219
\(846\) 0 0
\(847\) 0 0
\(848\) 6.00000 0.206041
\(849\) −20.0000 −0.686398
\(850\) 1.00000 0.0342997
\(851\) −16.0000 −0.548473
\(852\) −8.00000 −0.274075
\(853\) −46.0000 −1.57501 −0.787505 − 0.616308i \(-0.788628\pi\)
−0.787505 + 0.616308i \(0.788628\pi\)
\(854\) 0 0
\(855\) −8.00000 −0.273594
\(856\) −12.0000 −0.410152
\(857\) 6.00000 0.204956 0.102478 − 0.994735i \(-0.467323\pi\)
0.102478 + 0.994735i \(0.467323\pi\)
\(858\) 8.00000 0.273115
\(859\) −28.0000 −0.955348 −0.477674 − 0.878537i \(-0.658520\pi\)
−0.477674 + 0.878537i \(0.658520\pi\)
\(860\) −8.00000 −0.272798
\(861\) 0 0
\(862\) 0 0
\(863\) −48.0000 −1.63394 −0.816970 − 0.576681i \(-0.804348\pi\)
−0.816970 + 0.576681i \(0.804348\pi\)
\(864\) 1.00000 0.0340207
\(865\) −12.0000 −0.408012
\(866\) 14.0000 0.475739
\(867\) 1.00000 0.0339618
\(868\) 0 0
\(869\) 0 0
\(870\) 12.0000 0.406838
\(871\) −24.0000 −0.813209
\(872\) −10.0000 −0.338643
\(873\) −2.00000 −0.0676897
\(874\) −32.0000 −1.08242
\(875\) 0 0
\(876\) 6.00000 0.202721
\(877\) 54.0000 1.82345 0.911725 − 0.410801i \(-0.134751\pi\)
0.911725 + 0.410801i \(0.134751\pi\)
\(878\) −8.00000 −0.269987
\(879\) −6.00000 −0.202375
\(880\) 8.00000 0.269680
\(881\) −34.0000 −1.14549 −0.572745 − 0.819734i \(-0.694121\pi\)
−0.572745 + 0.819734i \(0.694121\pi\)
\(882\) 0 0
\(883\) 4.00000 0.134611 0.0673054 − 0.997732i \(-0.478560\pi\)
0.0673054 + 0.997732i \(0.478560\pi\)
\(884\) −2.00000 −0.0672673
\(885\) 8.00000 0.268917
\(886\) 12.0000 0.403148
\(887\) −40.0000 −1.34307 −0.671534 − 0.740973i \(-0.734364\pi\)
−0.671534 + 0.740973i \(0.734364\pi\)
\(888\) −2.00000 −0.0671156
\(889\) 0 0
\(890\) 12.0000 0.402241
\(891\) 4.00000 0.134005
\(892\) 16.0000 0.535720
\(893\) 0 0
\(894\) 22.0000 0.735790
\(895\) −24.0000 −0.802232
\(896\) 0 0
\(897\) 16.0000 0.534224
\(898\) 18.0000 0.600668
\(899\) 0 0
\(900\) −1.00000 −0.0333333
\(901\) −6.00000 −0.199889
\(902\) −40.0000 −1.33185
\(903\) 0 0
\(904\) 18.0000 0.598671
\(905\) 4.00000 0.132964
\(906\) 8.00000 0.265782
\(907\) −28.0000 −0.929725 −0.464862 − 0.885383i \(-0.653896\pi\)
−0.464862 + 0.885383i \(0.653896\pi\)
\(908\) 20.0000 0.663723
\(909\) 10.0000 0.331679
\(910\) 0 0
\(911\) −48.0000 −1.59031 −0.795155 − 0.606406i \(-0.792611\pi\)
−0.795155 + 0.606406i \(0.792611\pi\)
\(912\) −4.00000 −0.132453
\(913\) 48.0000 1.58857
\(914\) 10.0000 0.330771
\(915\) −12.0000 −0.396708
\(916\) −6.00000 −0.198246
\(917\) 0 0
\(918\) −1.00000 −0.0330049
\(919\) −8.00000 −0.263896 −0.131948 − 0.991257i \(-0.542123\pi\)
−0.131948 + 0.991257i \(0.542123\pi\)
\(920\) 16.0000 0.527504
\(921\) 28.0000 0.922631
\(922\) 2.00000 0.0658665
\(923\) −16.0000 −0.526646
\(924\) 0 0
\(925\) 2.00000 0.0657596
\(926\) 0 0
\(927\) −8.00000 −0.262754
\(928\) 6.00000 0.196960
\(929\) −34.0000 −1.11550 −0.557752 − 0.830008i \(-0.688336\pi\)
−0.557752 + 0.830008i \(0.688336\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 10.0000 0.327561
\(933\) −24.0000 −0.785725
\(934\) −36.0000 −1.17796
\(935\) −8.00000 −0.261628
\(936\) 2.00000 0.0653720
\(937\) −10.0000 −0.326686 −0.163343 − 0.986569i \(-0.552228\pi\)
−0.163343 + 0.986569i \(0.552228\pi\)
\(938\) 0 0
\(939\) −26.0000 −0.848478
\(940\) 0 0
\(941\) −22.0000 −0.717180 −0.358590 − 0.933495i \(-0.616742\pi\)
−0.358590 + 0.933495i \(0.616742\pi\)
\(942\) 2.00000 0.0651635
\(943\) −80.0000 −2.60516
\(944\) 4.00000 0.130189
\(945\) 0 0
\(946\) −16.0000 −0.520205
\(947\) −4.00000 −0.129983 −0.0649913 − 0.997886i \(-0.520702\pi\)
−0.0649913 + 0.997886i \(0.520702\pi\)
\(948\) 0 0
\(949\) 12.0000 0.389536
\(950\) 4.00000 0.129777
\(951\) 22.0000 0.713399
\(952\) 0 0
\(953\) 26.0000 0.842223 0.421111 − 0.907009i \(-0.361640\pi\)
0.421111 + 0.907009i \(0.361640\pi\)
\(954\) 6.00000 0.194257
\(955\) 0 0
\(956\) 16.0000 0.517477
\(957\) 24.0000 0.775810
\(958\) 16.0000 0.516937
\(959\) 0 0
\(960\) 2.00000 0.0645497
\(961\) −31.0000 −1.00000
\(962\) −4.00000 −0.128965
\(963\) −12.0000 −0.386695
\(964\) −18.0000 −0.579741
\(965\) −28.0000 −0.901352
\(966\) 0 0
\(967\) 8.00000 0.257263 0.128631 − 0.991692i \(-0.458942\pi\)
0.128631 + 0.991692i \(0.458942\pi\)
\(968\) 5.00000 0.160706
\(969\) 4.00000 0.128499
\(970\) −4.00000 −0.128432
\(971\) 52.0000 1.66876 0.834380 − 0.551190i \(-0.185826\pi\)
0.834380 + 0.551190i \(0.185826\pi\)
\(972\) 1.00000 0.0320750
\(973\) 0 0
\(974\) 24.0000 0.769010
\(975\) −2.00000 −0.0640513
\(976\) −6.00000 −0.192055
\(977\) 50.0000 1.59964 0.799821 − 0.600239i \(-0.204928\pi\)
0.799821 + 0.600239i \(0.204928\pi\)
\(978\) −20.0000 −0.639529
\(979\) 24.0000 0.767043
\(980\) 0 0
\(981\) −10.0000 −0.319275
\(982\) −20.0000 −0.638226
\(983\) 24.0000 0.765481 0.382741 − 0.923856i \(-0.374980\pi\)
0.382741 + 0.923856i \(0.374980\pi\)
\(984\) −10.0000 −0.318788
\(985\) −4.00000 −0.127451
\(986\) −6.00000 −0.191079
\(987\) 0 0
\(988\) −8.00000 −0.254514
\(989\) −32.0000 −1.01754
\(990\) 8.00000 0.254257
\(991\) 16.0000 0.508257 0.254128 − 0.967170i \(-0.418211\pi\)
0.254128 + 0.967170i \(0.418211\pi\)
\(992\) 0 0
\(993\) −4.00000 −0.126936
\(994\) 0 0
\(995\) 16.0000 0.507234
\(996\) 12.0000 0.380235
\(997\) 18.0000 0.570066 0.285033 − 0.958518i \(-0.407995\pi\)
0.285033 + 0.958518i \(0.407995\pi\)
\(998\) −20.0000 −0.633089
\(999\) −2.00000 −0.0632772
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.bq.1.1 1
7.6 odd 2 714.2.a.f.1.1 ✓ 1
21.20 even 2 2142.2.a.h.1.1 1
28.27 even 2 5712.2.a.o.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
714.2.a.f.1.1 ✓ 1 7.6 odd 2
2142.2.a.h.1.1 1 21.20 even 2
4998.2.a.bq.1.1 1 1.1 even 1 trivial
5712.2.a.o.1.1 1 28.27 even 2