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Properties

Label 1122.2.a.f.1.1
Level $1122$
Weight $2$
Character 1122.1
Self dual yes
Analytic conductor $8.959$
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] = [1122,2,Mod(1,1122)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1122, 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("1122.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1122 = 2 \cdot 3 \cdot 11 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1122.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: \(8.95921510679\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 1122.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -2.00000 q^{5} -1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -2.00000 q^{5} -1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} -2.00000 q^{10} +1.00000 q^{11} -1.00000 q^{12} -4.00000 q^{13} +2.00000 q^{15} +1.00000 q^{16} -1.00000 q^{17} +1.00000 q^{18} -4.00000 q^{19} -2.00000 q^{20} +1.00000 q^{22} -4.00000 q^{23} -1.00000 q^{24} -1.00000 q^{25} -4.00000 q^{26} -1.00000 q^{27} +2.00000 q^{30} +2.00000 q^{31} +1.00000 q^{32} -1.00000 q^{33} -1.00000 q^{34} +1.00000 q^{36} -4.00000 q^{38} +4.00000 q^{39} -2.00000 q^{40} +6.00000 q^{41} -12.0000 q^{43} +1.00000 q^{44} -2.00000 q^{45} -4.00000 q^{46} -10.0000 q^{47} -1.00000 q^{48} -7.00000 q^{49} -1.00000 q^{50} +1.00000 q^{51} -4.00000 q^{52} -4.00000 q^{53} -1.00000 q^{54} -2.00000 q^{55} +4.00000 q^{57} +4.00000 q^{59} +2.00000 q^{60} -2.00000 q^{61} +2.00000 q^{62} +1.00000 q^{64} +8.00000 q^{65} -1.00000 q^{66} -12.0000 q^{67} -1.00000 q^{68} +4.00000 q^{69} +8.00000 q^{71} +1.00000 q^{72} -10.0000 q^{73} +1.00000 q^{75} -4.00000 q^{76} +4.00000 q^{78} +8.00000 q^{79} -2.00000 q^{80} +1.00000 q^{81} +6.00000 q^{82} +4.00000 q^{83} +2.00000 q^{85} -12.0000 q^{86} +1.00000 q^{88} -6.00000 q^{89} -2.00000 q^{90} -4.00000 q^{92} -2.00000 q^{93} -10.0000 q^{94} +8.00000 q^{95} -1.00000 q^{96} -2.00000 q^{97} -7.00000 q^{98} +1.00000 q^{99} +O(q^{100})\)

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 \(-0.647584\pi\)
−0.447214 + 0.894427i \(0.647584\pi\)
\(6\) −1.00000 −0.408248
\(7\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) −2.00000 −0.632456
\(11\) 1.00000 0.301511
\(12\) −1.00000 −0.288675
\(13\) −4.00000 −1.10940 −0.554700 − 0.832050i \(-0.687167\pi\)
−0.554700 + 0.832050i \(0.687167\pi\)
\(14\) 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
\(22\) 1.00000 0.213201
\(23\) −4.00000 −0.834058 −0.417029 − 0.908893i \(-0.636929\pi\)
−0.417029 + 0.908893i \(0.636929\pi\)
\(24\) −1.00000 −0.204124
\(25\) −1.00000 −0.200000
\(26\) −4.00000 −0.784465
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(30\) 2.00000 0.365148
\(31\) 2.00000 0.359211 0.179605 − 0.983739i \(-0.442518\pi\)
0.179605 + 0.983739i \(0.442518\pi\)
\(32\) 1.00000 0.176777
\(33\) −1.00000 −0.174078
\(34\) −1.00000 −0.171499
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(38\) −4.00000 −0.648886
\(39\) 4.00000 0.640513
\(40\) −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 \(-0.867803\pi\)
−0.914991 + 0.403473i \(0.867803\pi\)
\(44\) 1.00000 0.150756
\(45\) −2.00000 −0.298142
\(46\) −4.00000 −0.589768
\(47\) −10.0000 −1.45865 −0.729325 − 0.684167i \(-0.760166\pi\)
−0.729325 + 0.684167i \(0.760166\pi\)
\(48\) −1.00000 −0.144338
\(49\) −7.00000 −1.00000
\(50\) −1.00000 −0.141421
\(51\) 1.00000 0.140028
\(52\) −4.00000 −0.554700
\(53\) −4.00000 −0.549442 −0.274721 − 0.961524i \(-0.588586\pi\)
−0.274721 + 0.961524i \(0.588586\pi\)
\(54\) −1.00000 −0.136083
\(55\) −2.00000 −0.269680
\(56\) 0 0
\(57\) 4.00000 0.529813
\(58\) 0 0
\(59\) 4.00000 0.520756 0.260378 − 0.965507i \(-0.416153\pi\)
0.260378 + 0.965507i \(0.416153\pi\)
\(60\) 2.00000 0.258199
\(61\) −2.00000 −0.256074 −0.128037 − 0.991769i \(-0.540868\pi\)
−0.128037 + 0.991769i \(0.540868\pi\)
\(62\) 2.00000 0.254000
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 8.00000 0.992278
\(66\) −1.00000 −0.123091
\(67\) −12.0000 −1.46603 −0.733017 − 0.680211i \(-0.761888\pi\)
−0.733017 + 0.680211i \(0.761888\pi\)
\(68\) −1.00000 −0.121268
\(69\) 4.00000 0.481543
\(70\) 0 0
\(71\) 8.00000 0.949425 0.474713 − 0.880141i \(-0.342552\pi\)
0.474713 + 0.880141i \(0.342552\pi\)
\(72\) 1.00000 0.117851
\(73\) −10.0000 −1.17041 −0.585206 − 0.810885i \(-0.698986\pi\)
−0.585206 + 0.810885i \(0.698986\pi\)
\(74\) 0 0
\(75\) 1.00000 0.115470
\(76\) −4.00000 −0.458831
\(77\) 0 0
\(78\) 4.00000 0.452911
\(79\) 8.00000 0.900070 0.450035 − 0.893011i \(-0.351411\pi\)
0.450035 + 0.893011i \(0.351411\pi\)
\(80\) −2.00000 −0.223607
\(81\) 1.00000 0.111111
\(82\) 6.00000 0.662589
\(83\) 4.00000 0.439057 0.219529 − 0.975606i \(-0.429548\pi\)
0.219529 + 0.975606i \(0.429548\pi\)
\(84\) 0 0
\(85\) 2.00000 0.216930
\(86\) −12.0000 −1.29399
\(87\) 0 0
\(88\) 1.00000 0.106600
\(89\) −6.00000 −0.635999 −0.317999 − 0.948091i \(-0.603011\pi\)
−0.317999 + 0.948091i \(0.603011\pi\)
\(90\) −2.00000 −0.210819
\(91\) 0 0
\(92\) −4.00000 −0.417029
\(93\) −2.00000 −0.207390
\(94\) −10.0000 −1.03142
\(95\) 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 \(0.532375\pi\)
\(98\) −7.00000 −0.707107
\(99\) 1.00000 0.100504
\(100\) −1.00000 −0.100000
\(101\) −2.00000 −0.199007 −0.0995037 − 0.995037i \(-0.531726\pi\)
−0.0995037 + 0.995037i \(0.531726\pi\)
\(102\) 1.00000 0.0990148
\(103\) 8.00000 0.788263 0.394132 − 0.919054i \(-0.371045\pi\)
0.394132 + 0.919054i \(0.371045\pi\)
\(104\) −4.00000 −0.392232
\(105\) 0 0
\(106\) −4.00000 −0.388514
\(107\) 20.0000 1.93347 0.966736 − 0.255774i \(-0.0823304\pi\)
0.966736 + 0.255774i \(0.0823304\pi\)
\(108\) −1.00000 −0.0962250
\(109\) −2.00000 −0.191565 −0.0957826 − 0.995402i \(-0.530535\pi\)
−0.0957826 + 0.995402i \(0.530535\pi\)
\(110\) −2.00000 −0.190693
\(111\) 0 0
\(112\) 0 0
\(113\) 6.00000 0.564433 0.282216 − 0.959351i \(-0.408930\pi\)
0.282216 + 0.959351i \(0.408930\pi\)
\(114\) 4.00000 0.374634
\(115\) 8.00000 0.746004
\(116\) 0 0
\(117\) −4.00000 −0.369800
\(118\) 4.00000 0.368230
\(119\) 0 0
\(120\) 2.00000 0.182574
\(121\) 1.00000 0.0909091
\(122\) −2.00000 −0.181071
\(123\) −6.00000 −0.541002
\(124\) 2.00000 0.179605
\(125\) 12.0000 1.07331
\(126\) 0 0
\(127\) 22.0000 1.95218 0.976092 − 0.217357i \(-0.0697436\pi\)
0.976092 + 0.217357i \(0.0697436\pi\)
\(128\) 1.00000 0.0883883
\(129\) 12.0000 1.05654
\(130\) 8.00000 0.701646
\(131\) −8.00000 −0.698963 −0.349482 − 0.936943i \(-0.613642\pi\)
−0.349482 + 0.936943i \(0.613642\pi\)
\(132\) −1.00000 −0.0870388
\(133\) 0 0
\(134\) −12.0000 −1.03664
\(135\) 2.00000 0.172133
\(136\) −1.00000 −0.0857493
\(137\) 6.00000 0.512615 0.256307 − 0.966595i \(-0.417494\pi\)
0.256307 + 0.966595i \(0.417494\pi\)
\(138\) 4.00000 0.340503
\(139\) −20.0000 −1.69638 −0.848189 − 0.529694i \(-0.822307\pi\)
−0.848189 + 0.529694i \(0.822307\pi\)
\(140\) 0 0
\(141\) 10.0000 0.842152
\(142\) 8.00000 0.671345
\(143\) −4.00000 −0.334497
\(144\) 1.00000 0.0833333
\(145\) 0 0
\(146\) −10.0000 −0.827606
\(147\) 7.00000 0.577350
\(148\) 0 0
\(149\) −10.0000 −0.819232 −0.409616 − 0.912258i \(-0.634337\pi\)
−0.409616 + 0.912258i \(0.634337\pi\)
\(150\) 1.00000 0.0816497
\(151\) 22.0000 1.79033 0.895167 − 0.445730i \(-0.147056\pi\)
0.895167 + 0.445730i \(0.147056\pi\)
\(152\) −4.00000 −0.324443
\(153\) −1.00000 −0.0808452
\(154\) 0 0
\(155\) −4.00000 −0.321288
\(156\) 4.00000 0.320256
\(157\) −14.0000 −1.11732 −0.558661 − 0.829396i \(-0.688685\pi\)
−0.558661 + 0.829396i \(0.688685\pi\)
\(158\) 8.00000 0.636446
\(159\) 4.00000 0.317221
\(160\) −2.00000 −0.158114
\(161\) 0 0
\(162\) 1.00000 0.0785674
\(163\) 20.0000 1.56652 0.783260 − 0.621694i \(-0.213555\pi\)
0.783260 + 0.621694i \(0.213555\pi\)
\(164\) 6.00000 0.468521
\(165\) 2.00000 0.155700
\(166\) 4.00000 0.310460
\(167\) 2.00000 0.154765 0.0773823 − 0.997001i \(-0.475344\pi\)
0.0773823 + 0.997001i \(0.475344\pi\)
\(168\) 0 0
\(169\) 3.00000 0.230769
\(170\) 2.00000 0.153393
\(171\) −4.00000 −0.305888
\(172\) −12.0000 −0.914991
\(173\) 8.00000 0.608229 0.304114 − 0.952636i \(-0.401639\pi\)
0.304114 + 0.952636i \(0.401639\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 1.00000 0.0753778
\(177\) −4.00000 −0.300658
\(178\) −6.00000 −0.449719
\(179\) 16.0000 1.19590 0.597948 − 0.801535i \(-0.295983\pi\)
0.597948 + 0.801535i \(0.295983\pi\)
\(180\) −2.00000 −0.149071
\(181\) 4.00000 0.297318 0.148659 − 0.988889i \(-0.452504\pi\)
0.148659 + 0.988889i \(0.452504\pi\)
\(182\) 0 0
\(183\) 2.00000 0.147844
\(184\) −4.00000 −0.294884
\(185\) 0 0
\(186\) −2.00000 −0.146647
\(187\) −1.00000 −0.0731272
\(188\) −10.0000 −0.729325
\(189\) 0 0
\(190\) 8.00000 0.580381
\(191\) 6.00000 0.434145 0.217072 − 0.976156i \(-0.430349\pi\)
0.217072 + 0.976156i \(0.430349\pi\)
\(192\) −1.00000 −0.0721688
\(193\) −22.0000 −1.58359 −0.791797 − 0.610784i \(-0.790854\pi\)
−0.791797 + 0.610784i \(0.790854\pi\)
\(194\) −2.00000 −0.143592
\(195\) −8.00000 −0.572892
\(196\) −7.00000 −0.500000
\(197\) −4.00000 −0.284988 −0.142494 − 0.989796i \(-0.545512\pi\)
−0.142494 + 0.989796i \(0.545512\pi\)
\(198\) 1.00000 0.0710669
\(199\) −14.0000 −0.992434 −0.496217 − 0.868199i \(-0.665278\pi\)
−0.496217 + 0.868199i \(0.665278\pi\)
\(200\) −1.00000 −0.0707107
\(201\) 12.0000 0.846415
\(202\) −2.00000 −0.140720
\(203\) 0 0
\(204\) 1.00000 0.0700140
\(205\) −12.0000 −0.838116
\(206\) 8.00000 0.557386
\(207\) −4.00000 −0.278019
\(208\) −4.00000 −0.277350
\(209\) −4.00000 −0.276686
\(210\) 0 0
\(211\) 4.00000 0.275371 0.137686 − 0.990476i \(-0.456034\pi\)
0.137686 + 0.990476i \(0.456034\pi\)
\(212\) −4.00000 −0.274721
\(213\) −8.00000 −0.548151
\(214\) 20.0000 1.36717
\(215\) 24.0000 1.63679
\(216\) −1.00000 −0.0680414
\(217\) 0 0
\(218\) −2.00000 −0.135457
\(219\) 10.0000 0.675737
\(220\) −2.00000 −0.134840
\(221\) 4.00000 0.269069
\(222\) 0 0
\(223\) 12.0000 0.803579 0.401790 − 0.915732i \(-0.368388\pi\)
0.401790 + 0.915732i \(0.368388\pi\)
\(224\) 0 0
\(225\) −1.00000 −0.0666667
\(226\) 6.00000 0.399114
\(227\) −16.0000 −1.06196 −0.530979 − 0.847385i \(-0.678176\pi\)
−0.530979 + 0.847385i \(0.678176\pi\)
\(228\) 4.00000 0.264906
\(229\) −14.0000 −0.925146 −0.462573 − 0.886581i \(-0.653074\pi\)
−0.462573 + 0.886581i \(0.653074\pi\)
\(230\) 8.00000 0.527504
\(231\) 0 0
\(232\) 0 0
\(233\) −6.00000 −0.393073 −0.196537 − 0.980497i \(-0.562969\pi\)
−0.196537 + 0.980497i \(0.562969\pi\)
\(234\) −4.00000 −0.261488
\(235\) 20.0000 1.30466
\(236\) 4.00000 0.260378
\(237\) −8.00000 −0.519656
\(238\) 0 0
\(239\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(240\) 2.00000 0.129099
\(241\) −6.00000 −0.386494 −0.193247 − 0.981150i \(-0.561902\pi\)
−0.193247 + 0.981150i \(0.561902\pi\)
\(242\) 1.00000 0.0642824
\(243\) −1.00000 −0.0641500
\(244\) −2.00000 −0.128037
\(245\) 14.0000 0.894427
\(246\) −6.00000 −0.382546
\(247\) 16.0000 1.01806
\(248\) 2.00000 0.127000
\(249\) −4.00000 −0.253490
\(250\) 12.0000 0.758947
\(251\) −16.0000 −1.00991 −0.504956 − 0.863145i \(-0.668491\pi\)
−0.504956 + 0.863145i \(0.668491\pi\)
\(252\) 0 0
\(253\) −4.00000 −0.251478
\(254\) 22.0000 1.38040
\(255\) −2.00000 −0.125245
\(256\) 1.00000 0.0625000
\(257\) −14.0000 −0.873296 −0.436648 − 0.899632i \(-0.643834\pi\)
−0.436648 + 0.899632i \(0.643834\pi\)
\(258\) 12.0000 0.747087
\(259\) 0 0
\(260\) 8.00000 0.496139
\(261\) 0 0
\(262\) −8.00000 −0.494242
\(263\) 12.0000 0.739952 0.369976 − 0.929041i \(-0.379366\pi\)
0.369976 + 0.929041i \(0.379366\pi\)
\(264\) −1.00000 −0.0615457
\(265\) 8.00000 0.491436
\(266\) 0 0
\(267\) 6.00000 0.367194
\(268\) −12.0000 −0.733017
\(269\) 14.0000 0.853595 0.426798 − 0.904347i \(-0.359642\pi\)
0.426798 + 0.904347i \(0.359642\pi\)
\(270\) 2.00000 0.121716
\(271\) −6.00000 −0.364474 −0.182237 − 0.983255i \(-0.558334\pi\)
−0.182237 + 0.983255i \(0.558334\pi\)
\(272\) −1.00000 −0.0606339
\(273\) 0 0
\(274\) 6.00000 0.362473
\(275\) −1.00000 −0.0603023
\(276\) 4.00000 0.240772
\(277\) 10.0000 0.600842 0.300421 − 0.953807i \(-0.402873\pi\)
0.300421 + 0.953807i \(0.402873\pi\)
\(278\) −20.0000 −1.19952
\(279\) 2.00000 0.119737
\(280\) 0 0
\(281\) 30.0000 1.78965 0.894825 − 0.446417i \(-0.147300\pi\)
0.894825 + 0.446417i \(0.147300\pi\)
\(282\) 10.0000 0.595491
\(283\) −28.0000 −1.66443 −0.832214 − 0.554455i \(-0.812927\pi\)
−0.832214 + 0.554455i \(0.812927\pi\)
\(284\) 8.00000 0.474713
\(285\) −8.00000 −0.473879
\(286\) −4.00000 −0.236525
\(287\) 0 0
\(288\) 1.00000 0.0589256
\(289\) 1.00000 0.0588235
\(290\) 0 0
\(291\) 2.00000 0.117242
\(292\) −10.0000 −0.585206
\(293\) 6.00000 0.350524 0.175262 − 0.984522i \(-0.443923\pi\)
0.175262 + 0.984522i \(0.443923\pi\)
\(294\) 7.00000 0.408248
\(295\) −8.00000 −0.465778
\(296\) 0 0
\(297\) −1.00000 −0.0580259
\(298\) −10.0000 −0.579284
\(299\) 16.0000 0.925304
\(300\) 1.00000 0.0577350
\(301\) 0 0
\(302\) 22.0000 1.26596
\(303\) 2.00000 0.114897
\(304\) −4.00000 −0.229416
\(305\) 4.00000 0.229039
\(306\) −1.00000 −0.0571662
\(307\) 8.00000 0.456584 0.228292 − 0.973593i \(-0.426686\pi\)
0.228292 + 0.973593i \(0.426686\pi\)
\(308\) 0 0
\(309\) −8.00000 −0.455104
\(310\) −4.00000 −0.227185
\(311\) 20.0000 1.13410 0.567048 − 0.823685i \(-0.308085\pi\)
0.567048 + 0.823685i \(0.308085\pi\)
\(312\) 4.00000 0.226455
\(313\) −10.0000 −0.565233 −0.282617 − 0.959233i \(-0.591202\pi\)
−0.282617 + 0.959233i \(0.591202\pi\)
\(314\) −14.0000 −0.790066
\(315\) 0 0
\(316\) 8.00000 0.450035
\(317\) −18.0000 −1.01098 −0.505490 − 0.862832i \(-0.668688\pi\)
−0.505490 + 0.862832i \(0.668688\pi\)
\(318\) 4.00000 0.224309
\(319\) 0 0
\(320\) −2.00000 −0.111803
\(321\) −20.0000 −1.11629
\(322\) 0 0
\(323\) 4.00000 0.222566
\(324\) 1.00000 0.0555556
\(325\) 4.00000 0.221880
\(326\) 20.0000 1.10770
\(327\) 2.00000 0.110600
\(328\) 6.00000 0.331295
\(329\) 0 0
\(330\) 2.00000 0.110096
\(331\) −4.00000 −0.219860 −0.109930 − 0.993939i \(-0.535063\pi\)
−0.109930 + 0.993939i \(0.535063\pi\)
\(332\) 4.00000 0.219529
\(333\) 0 0
\(334\) 2.00000 0.109435
\(335\) 24.0000 1.31126
\(336\) 0 0
\(337\) 2.00000 0.108947 0.0544735 − 0.998515i \(-0.482652\pi\)
0.0544735 + 0.998515i \(0.482652\pi\)
\(338\) 3.00000 0.163178
\(339\) −6.00000 −0.325875
\(340\) 2.00000 0.108465
\(341\) 2.00000 0.108306
\(342\) −4.00000 −0.216295
\(343\) 0 0
\(344\) −12.0000 −0.646997
\(345\) −8.00000 −0.430706
\(346\) 8.00000 0.430083
\(347\) −4.00000 −0.214731 −0.107366 − 0.994220i \(-0.534242\pi\)
−0.107366 + 0.994220i \(0.534242\pi\)
\(348\) 0 0
\(349\) −28.0000 −1.49881 −0.749403 − 0.662114i \(-0.769659\pi\)
−0.749403 + 0.662114i \(0.769659\pi\)
\(350\) 0 0
\(351\) 4.00000 0.213504
\(352\) 1.00000 0.0533002
\(353\) −30.0000 −1.59674 −0.798369 − 0.602168i \(-0.794304\pi\)
−0.798369 + 0.602168i \(0.794304\pi\)
\(354\) −4.00000 −0.212598
\(355\) −16.0000 −0.849192
\(356\) −6.00000 −0.317999
\(357\) 0 0
\(358\) 16.0000 0.845626
\(359\) −20.0000 −1.05556 −0.527780 − 0.849381i \(-0.676975\pi\)
−0.527780 + 0.849381i \(0.676975\pi\)
\(360\) −2.00000 −0.105409
\(361\) −3.00000 −0.157895
\(362\) 4.00000 0.210235
\(363\) −1.00000 −0.0524864
\(364\) 0 0
\(365\) 20.0000 1.04685
\(366\) 2.00000 0.104542
\(367\) 26.0000 1.35719 0.678594 − 0.734513i \(-0.262589\pi\)
0.678594 + 0.734513i \(0.262589\pi\)
\(368\) −4.00000 −0.208514
\(369\) 6.00000 0.312348
\(370\) 0 0
\(371\) 0 0
\(372\) −2.00000 −0.103695
\(373\) −12.0000 −0.621336 −0.310668 − 0.950518i \(-0.600553\pi\)
−0.310668 + 0.950518i \(0.600553\pi\)
\(374\) −1.00000 −0.0517088
\(375\) −12.0000 −0.619677
\(376\) −10.0000 −0.515711
\(377\) 0 0
\(378\) 0 0
\(379\) −20.0000 −1.02733 −0.513665 − 0.857991i \(-0.671713\pi\)
−0.513665 + 0.857991i \(0.671713\pi\)
\(380\) 8.00000 0.410391
\(381\) −22.0000 −1.12709
\(382\) 6.00000 0.306987
\(383\) −6.00000 −0.306586 −0.153293 − 0.988181i \(-0.548988\pi\)
−0.153293 + 0.988181i \(0.548988\pi\)
\(384\) −1.00000 −0.0510310
\(385\) 0 0
\(386\) −22.0000 −1.11977
\(387\) −12.0000 −0.609994
\(388\) −2.00000 −0.101535
\(389\) 24.0000 1.21685 0.608424 − 0.793612i \(-0.291802\pi\)
0.608424 + 0.793612i \(0.291802\pi\)
\(390\) −8.00000 −0.405096
\(391\) 4.00000 0.202289
\(392\) −7.00000 −0.353553
\(393\) 8.00000 0.403547
\(394\) −4.00000 −0.201517
\(395\) −16.0000 −0.805047
\(396\) 1.00000 0.0502519
\(397\) −28.0000 −1.40528 −0.702640 − 0.711546i \(-0.747995\pi\)
−0.702640 + 0.711546i \(0.747995\pi\)
\(398\) −14.0000 −0.701757
\(399\) 0 0
\(400\) −1.00000 −0.0500000
\(401\) 26.0000 1.29838 0.649189 − 0.760627i \(-0.275108\pi\)
0.649189 + 0.760627i \(0.275108\pi\)
\(402\) 12.0000 0.598506
\(403\) −8.00000 −0.398508
\(404\) −2.00000 −0.0995037
\(405\) −2.00000 −0.0993808
\(406\) 0 0
\(407\) 0 0
\(408\) 1.00000 0.0495074
\(409\) 14.0000 0.692255 0.346128 − 0.938187i \(-0.387496\pi\)
0.346128 + 0.938187i \(0.387496\pi\)
\(410\) −12.0000 −0.592638
\(411\) −6.00000 −0.295958
\(412\) 8.00000 0.394132
\(413\) 0 0
\(414\) −4.00000 −0.196589
\(415\) −8.00000 −0.392705
\(416\) −4.00000 −0.196116
\(417\) 20.0000 0.979404
\(418\) −4.00000 −0.195646
\(419\) 20.0000 0.977064 0.488532 − 0.872546i \(-0.337533\pi\)
0.488532 + 0.872546i \(0.337533\pi\)
\(420\) 0 0
\(421\) −22.0000 −1.07221 −0.536107 − 0.844150i \(-0.680106\pi\)
−0.536107 + 0.844150i \(0.680106\pi\)
\(422\) 4.00000 0.194717
\(423\) −10.0000 −0.486217
\(424\) −4.00000 −0.194257
\(425\) 1.00000 0.0485071
\(426\) −8.00000 −0.387601
\(427\) 0 0
\(428\) 20.0000 0.966736
\(429\) 4.00000 0.193122
\(430\) 24.0000 1.15738
\(431\) −10.0000 −0.481683 −0.240842 − 0.970564i \(-0.577423\pi\)
−0.240842 + 0.970564i \(0.577423\pi\)
\(432\) −1.00000 −0.0481125
\(433\) −2.00000 −0.0961139 −0.0480569 − 0.998845i \(-0.515303\pi\)
−0.0480569 + 0.998845i \(0.515303\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −2.00000 −0.0957826
\(437\) 16.0000 0.765384
\(438\) 10.0000 0.477818
\(439\) 28.0000 1.33637 0.668184 − 0.743996i \(-0.267072\pi\)
0.668184 + 0.743996i \(0.267072\pi\)
\(440\) −2.00000 −0.0953463
\(441\) −7.00000 −0.333333
\(442\) 4.00000 0.190261
\(443\) −24.0000 −1.14027 −0.570137 − 0.821549i \(-0.693110\pi\)
−0.570137 + 0.821549i \(0.693110\pi\)
\(444\) 0 0
\(445\) 12.0000 0.568855
\(446\) 12.0000 0.568216
\(447\) 10.0000 0.472984
\(448\) 0 0
\(449\) −14.0000 −0.660701 −0.330350 − 0.943858i \(-0.607167\pi\)
−0.330350 + 0.943858i \(0.607167\pi\)
\(450\) −1.00000 −0.0471405
\(451\) 6.00000 0.282529
\(452\) 6.00000 0.282216
\(453\) −22.0000 −1.03365
\(454\) −16.0000 −0.750917
\(455\) 0 0
\(456\) 4.00000 0.187317
\(457\) −26.0000 −1.21623 −0.608114 − 0.793849i \(-0.708074\pi\)
−0.608114 + 0.793849i \(0.708074\pi\)
\(458\) −14.0000 −0.654177
\(459\) 1.00000 0.0466760
\(460\) 8.00000 0.373002
\(461\) 18.0000 0.838344 0.419172 − 0.907907i \(-0.362320\pi\)
0.419172 + 0.907907i \(0.362320\pi\)
\(462\) 0 0
\(463\) −8.00000 −0.371792 −0.185896 − 0.982569i \(-0.559519\pi\)
−0.185896 + 0.982569i \(0.559519\pi\)
\(464\) 0 0
\(465\) 4.00000 0.185496
\(466\) −6.00000 −0.277945
\(467\) −28.0000 −1.29569 −0.647843 − 0.761774i \(-0.724329\pi\)
−0.647843 + 0.761774i \(0.724329\pi\)
\(468\) −4.00000 −0.184900
\(469\) 0 0
\(470\) 20.0000 0.922531
\(471\) 14.0000 0.645086
\(472\) 4.00000 0.184115
\(473\) −12.0000 −0.551761
\(474\) −8.00000 −0.367452
\(475\) 4.00000 0.183533
\(476\) 0 0
\(477\) −4.00000 −0.183147
\(478\) 0 0
\(479\) 10.0000 0.456912 0.228456 − 0.973554i \(-0.426632\pi\)
0.228456 + 0.973554i \(0.426632\pi\)
\(480\) 2.00000 0.0912871
\(481\) 0 0
\(482\) −6.00000 −0.273293
\(483\) 0 0
\(484\) 1.00000 0.0454545
\(485\) 4.00000 0.181631
\(486\) −1.00000 −0.0453609
\(487\) 14.0000 0.634401 0.317200 − 0.948359i \(-0.397257\pi\)
0.317200 + 0.948359i \(0.397257\pi\)
\(488\) −2.00000 −0.0905357
\(489\) −20.0000 −0.904431
\(490\) 14.0000 0.632456
\(491\) 12.0000 0.541552 0.270776 − 0.962642i \(-0.412720\pi\)
0.270776 + 0.962642i \(0.412720\pi\)
\(492\) −6.00000 −0.270501
\(493\) 0 0
\(494\) 16.0000 0.719874
\(495\) −2.00000 −0.0898933
\(496\) 2.00000 0.0898027
\(497\) 0 0
\(498\) −4.00000 −0.179244
\(499\) −24.0000 −1.07439 −0.537194 − 0.843459i \(-0.680516\pi\)
−0.537194 + 0.843459i \(0.680516\pi\)
\(500\) 12.0000 0.536656
\(501\) −2.00000 −0.0893534
\(502\) −16.0000 −0.714115
\(503\) −6.00000 −0.267527 −0.133763 − 0.991013i \(-0.542706\pi\)
−0.133763 + 0.991013i \(0.542706\pi\)
\(504\) 0 0
\(505\) 4.00000 0.177998
\(506\) −4.00000 −0.177822
\(507\) −3.00000 −0.133235
\(508\) 22.0000 0.976092
\(509\) −40.0000 −1.77297 −0.886484 − 0.462758i \(-0.846860\pi\)
−0.886484 + 0.462758i \(0.846860\pi\)
\(510\) −2.00000 −0.0885615
\(511\) 0 0
\(512\) 1.00000 0.0441942
\(513\) 4.00000 0.176604
\(514\) −14.0000 −0.617514
\(515\) −16.0000 −0.705044
\(516\) 12.0000 0.528271
\(517\) −10.0000 −0.439799
\(518\) 0 0
\(519\) −8.00000 −0.351161
\(520\) 8.00000 0.350823
\(521\) −30.0000 −1.31432 −0.657162 − 0.753749i \(-0.728243\pi\)
−0.657162 + 0.753749i \(0.728243\pi\)
\(522\) 0 0
\(523\) 24.0000 1.04945 0.524723 − 0.851273i \(-0.324169\pi\)
0.524723 + 0.851273i \(0.324169\pi\)
\(524\) −8.00000 −0.349482
\(525\) 0 0
\(526\) 12.0000 0.523225
\(527\) −2.00000 −0.0871214
\(528\) −1.00000 −0.0435194
\(529\) −7.00000 −0.304348
\(530\) 8.00000 0.347498
\(531\) 4.00000 0.173585
\(532\) 0 0
\(533\) −24.0000 −1.03956
\(534\) 6.00000 0.259645
\(535\) −40.0000 −1.72935
\(536\) −12.0000 −0.518321
\(537\) −16.0000 −0.690451
\(538\) 14.0000 0.603583
\(539\) −7.00000 −0.301511
\(540\) 2.00000 0.0860663
\(541\) 18.0000 0.773880 0.386940 − 0.922105i \(-0.373532\pi\)
0.386940 + 0.922105i \(0.373532\pi\)
\(542\) −6.00000 −0.257722
\(543\) −4.00000 −0.171656
\(544\) −1.00000 −0.0428746
\(545\) 4.00000 0.171341
\(546\) 0 0
\(547\) 28.0000 1.19719 0.598597 − 0.801050i \(-0.295725\pi\)
0.598597 + 0.801050i \(0.295725\pi\)
\(548\) 6.00000 0.256307
\(549\) −2.00000 −0.0853579
\(550\) −1.00000 −0.0426401
\(551\) 0 0
\(552\) 4.00000 0.170251
\(553\) 0 0
\(554\) 10.0000 0.424859
\(555\) 0 0
\(556\) −20.0000 −0.848189
\(557\) 18.0000 0.762684 0.381342 − 0.924434i \(-0.375462\pi\)
0.381342 + 0.924434i \(0.375462\pi\)
\(558\) 2.00000 0.0846668
\(559\) 48.0000 2.03018
\(560\) 0 0
\(561\) 1.00000 0.0422200
\(562\) 30.0000 1.26547
\(563\) 20.0000 0.842900 0.421450 − 0.906852i \(-0.361521\pi\)
0.421450 + 0.906852i \(0.361521\pi\)
\(564\) 10.0000 0.421076
\(565\) −12.0000 −0.504844
\(566\) −28.0000 −1.17693
\(567\) 0 0
\(568\) 8.00000 0.335673
\(569\) 18.0000 0.754599 0.377300 − 0.926091i \(-0.376853\pi\)
0.377300 + 0.926091i \(0.376853\pi\)
\(570\) −8.00000 −0.335083
\(571\) −28.0000 −1.17176 −0.585882 − 0.810397i \(-0.699252\pi\)
−0.585882 + 0.810397i \(0.699252\pi\)
\(572\) −4.00000 −0.167248
\(573\) −6.00000 −0.250654
\(574\) 0 0
\(575\) 4.00000 0.166812
\(576\) 1.00000 0.0416667
\(577\) 22.0000 0.915872 0.457936 − 0.888985i \(-0.348589\pi\)
0.457936 + 0.888985i \(0.348589\pi\)
\(578\) 1.00000 0.0415945
\(579\) 22.0000 0.914289
\(580\) 0 0
\(581\) 0 0
\(582\) 2.00000 0.0829027
\(583\) −4.00000 −0.165663
\(584\) −10.0000 −0.413803
\(585\) 8.00000 0.330759
\(586\) 6.00000 0.247858
\(587\) 8.00000 0.330195 0.165098 − 0.986277i \(-0.447206\pi\)
0.165098 + 0.986277i \(0.447206\pi\)
\(588\) 7.00000 0.288675
\(589\) −8.00000 −0.329634
\(590\) −8.00000 −0.329355
\(591\) 4.00000 0.164538
\(592\) 0 0
\(593\) −14.0000 −0.574911 −0.287456 − 0.957794i \(-0.592809\pi\)
−0.287456 + 0.957794i \(0.592809\pi\)
\(594\) −1.00000 −0.0410305
\(595\) 0 0
\(596\) −10.0000 −0.409616
\(597\) 14.0000 0.572982
\(598\) 16.0000 0.654289
\(599\) −18.0000 −0.735460 −0.367730 − 0.929933i \(-0.619865\pi\)
−0.367730 + 0.929933i \(0.619865\pi\)
\(600\) 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
\(604\) 22.0000 0.895167
\(605\) −2.00000 −0.0813116
\(606\) 2.00000 0.0812444
\(607\) −16.0000 −0.649420 −0.324710 − 0.945814i \(-0.605267\pi\)
−0.324710 + 0.945814i \(0.605267\pi\)
\(608\) −4.00000 −0.162221
\(609\) 0 0
\(610\) 4.00000 0.161955
\(611\) 40.0000 1.61823
\(612\) −1.00000 −0.0404226
\(613\) 12.0000 0.484675 0.242338 − 0.970192i \(-0.422086\pi\)
0.242338 + 0.970192i \(0.422086\pi\)
\(614\) 8.00000 0.322854
\(615\) 12.0000 0.483887
\(616\) 0 0
\(617\) −6.00000 −0.241551 −0.120775 − 0.992680i \(-0.538538\pi\)
−0.120775 + 0.992680i \(0.538538\pi\)
\(618\) −8.00000 −0.321807
\(619\) 40.0000 1.60774 0.803868 − 0.594808i \(-0.202772\pi\)
0.803868 + 0.594808i \(0.202772\pi\)
\(620\) −4.00000 −0.160644
\(621\) 4.00000 0.160514
\(622\) 20.0000 0.801927
\(623\) 0 0
\(624\) 4.00000 0.160128
\(625\) −19.0000 −0.760000
\(626\) −10.0000 −0.399680
\(627\) 4.00000 0.159745
\(628\) −14.0000 −0.558661
\(629\) 0 0
\(630\) 0 0
\(631\) −4.00000 −0.159237 −0.0796187 − 0.996825i \(-0.525370\pi\)
−0.0796187 + 0.996825i \(0.525370\pi\)
\(632\) 8.00000 0.318223
\(633\) −4.00000 −0.158986
\(634\) −18.0000 −0.714871
\(635\) −44.0000 −1.74609
\(636\) 4.00000 0.158610
\(637\) 28.0000 1.10940
\(638\) 0 0
\(639\) 8.00000 0.316475
\(640\) −2.00000 −0.0790569
\(641\) −2.00000 −0.0789953 −0.0394976 − 0.999220i \(-0.512576\pi\)
−0.0394976 + 0.999220i \(0.512576\pi\)
\(642\) −20.0000 −0.789337
\(643\) 16.0000 0.630978 0.315489 − 0.948929i \(-0.397831\pi\)
0.315489 + 0.948929i \(0.397831\pi\)
\(644\) 0 0
\(645\) −24.0000 −0.944999
\(646\) 4.00000 0.157378
\(647\) 18.0000 0.707653 0.353827 − 0.935311i \(-0.384880\pi\)
0.353827 + 0.935311i \(0.384880\pi\)
\(648\) 1.00000 0.0392837
\(649\) 4.00000 0.157014
\(650\) 4.00000 0.156893
\(651\) 0 0
\(652\) 20.0000 0.783260
\(653\) −6.00000 −0.234798 −0.117399 − 0.993085i \(-0.537456\pi\)
−0.117399 + 0.993085i \(0.537456\pi\)
\(654\) 2.00000 0.0782062
\(655\) 16.0000 0.625172
\(656\) 6.00000 0.234261
\(657\) −10.0000 −0.390137
\(658\) 0 0
\(659\) −20.0000 −0.779089 −0.389545 − 0.921008i \(-0.627368\pi\)
−0.389545 + 0.921008i \(0.627368\pi\)
\(660\) 2.00000 0.0778499
\(661\) −18.0000 −0.700119 −0.350059 − 0.936727i \(-0.613839\pi\)
−0.350059 + 0.936727i \(0.613839\pi\)
\(662\) −4.00000 −0.155464
\(663\) −4.00000 −0.155347
\(664\) 4.00000 0.155230
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 2.00000 0.0773823
\(669\) −12.0000 −0.463947
\(670\) 24.0000 0.927201
\(671\) −2.00000 −0.0772091
\(672\) 0 0
\(673\) 10.0000 0.385472 0.192736 − 0.981251i \(-0.438264\pi\)
0.192736 + 0.981251i \(0.438264\pi\)
\(674\) 2.00000 0.0770371
\(675\) 1.00000 0.0384900
\(676\) 3.00000 0.115385
\(677\) −4.00000 −0.153732 −0.0768662 − 0.997041i \(-0.524491\pi\)
−0.0768662 + 0.997041i \(0.524491\pi\)
\(678\) −6.00000 −0.230429
\(679\) 0 0
\(680\) 2.00000 0.0766965
\(681\) 16.0000 0.613121
\(682\) 2.00000 0.0765840
\(683\) −44.0000 −1.68361 −0.841807 − 0.539779i \(-0.818508\pi\)
−0.841807 + 0.539779i \(0.818508\pi\)
\(684\) −4.00000 −0.152944
\(685\) −12.0000 −0.458496
\(686\) 0 0
\(687\) 14.0000 0.534133
\(688\) −12.0000 −0.457496
\(689\) 16.0000 0.609551
\(690\) −8.00000 −0.304555
\(691\) 44.0000 1.67384 0.836919 − 0.547326i \(-0.184354\pi\)
0.836919 + 0.547326i \(0.184354\pi\)
\(692\) 8.00000 0.304114
\(693\) 0 0
\(694\) −4.00000 −0.151838
\(695\) 40.0000 1.51729
\(696\) 0 0
\(697\) −6.00000 −0.227266
\(698\) −28.0000 −1.05982
\(699\) 6.00000 0.226941
\(700\) 0 0
\(701\) −30.0000 −1.13308 −0.566542 − 0.824033i \(-0.691719\pi\)
−0.566542 + 0.824033i \(0.691719\pi\)
\(702\) 4.00000 0.150970
\(703\) 0 0
\(704\) 1.00000 0.0376889
\(705\) −20.0000 −0.753244
\(706\) −30.0000 −1.12906
\(707\) 0 0
\(708\) −4.00000 −0.150329
\(709\) −48.0000 −1.80268 −0.901339 − 0.433114i \(-0.857415\pi\)
−0.901339 + 0.433114i \(0.857415\pi\)
\(710\) −16.0000 −0.600469
\(711\) 8.00000 0.300023
\(712\) −6.00000 −0.224860
\(713\) −8.00000 −0.299602
\(714\) 0 0
\(715\) 8.00000 0.299183
\(716\) 16.0000 0.597948
\(717\) 0 0
\(718\) −20.0000 −0.746393
\(719\) −16.0000 −0.596699 −0.298350 − 0.954457i \(-0.596436\pi\)
−0.298350 + 0.954457i \(0.596436\pi\)
\(720\) −2.00000 −0.0745356
\(721\) 0 0
\(722\) −3.00000 −0.111648
\(723\) 6.00000 0.223142
\(724\) 4.00000 0.148659
\(725\) 0 0
\(726\) −1.00000 −0.0371135
\(727\) −28.0000 −1.03846 −0.519231 − 0.854634i \(-0.673782\pi\)
−0.519231 + 0.854634i \(0.673782\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 20.0000 0.740233
\(731\) 12.0000 0.443836
\(732\) 2.00000 0.0739221
\(733\) −28.0000 −1.03420 −0.517102 − 0.855924i \(-0.672989\pi\)
−0.517102 + 0.855924i \(0.672989\pi\)
\(734\) 26.0000 0.959678
\(735\) −14.0000 −0.516398
\(736\) −4.00000 −0.147442
\(737\) −12.0000 −0.442026
\(738\) 6.00000 0.220863
\(739\) −16.0000 −0.588570 −0.294285 − 0.955718i \(-0.595081\pi\)
−0.294285 + 0.955718i \(0.595081\pi\)
\(740\) 0 0
\(741\) −16.0000 −0.587775
\(742\) 0 0
\(743\) −30.0000 −1.10059 −0.550297 − 0.834969i \(-0.685485\pi\)
−0.550297 + 0.834969i \(0.685485\pi\)
\(744\) −2.00000 −0.0733236
\(745\) 20.0000 0.732743
\(746\) −12.0000 −0.439351
\(747\) 4.00000 0.146352
\(748\) −1.00000 −0.0365636
\(749\) 0 0
\(750\) −12.0000 −0.438178
\(751\) −10.0000 −0.364905 −0.182453 − 0.983215i \(-0.558404\pi\)
−0.182453 + 0.983215i \(0.558404\pi\)
\(752\) −10.0000 −0.364662
\(753\) 16.0000 0.583072
\(754\) 0 0
\(755\) −44.0000 −1.60132
\(756\) 0 0
\(757\) 22.0000 0.799604 0.399802 − 0.916602i \(-0.369079\pi\)
0.399802 + 0.916602i \(0.369079\pi\)
\(758\) −20.0000 −0.726433
\(759\) 4.00000 0.145191
\(760\) 8.00000 0.290191
\(761\) −14.0000 −0.507500 −0.253750 − 0.967270i \(-0.581664\pi\)
−0.253750 + 0.967270i \(0.581664\pi\)
\(762\) −22.0000 −0.796976
\(763\) 0 0
\(764\) 6.00000 0.217072
\(765\) 2.00000 0.0723102
\(766\) −6.00000 −0.216789
\(767\) −16.0000 −0.577727
\(768\) −1.00000 −0.0360844
\(769\) 54.0000 1.94729 0.973645 − 0.228069i \(-0.0732413\pi\)
0.973645 + 0.228069i \(0.0732413\pi\)
\(770\) 0 0
\(771\) 14.0000 0.504198
\(772\) −22.0000 −0.791797
\(773\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(774\) −12.0000 −0.431331
\(775\) −2.00000 −0.0718421
\(776\) −2.00000 −0.0717958
\(777\) 0 0
\(778\) 24.0000 0.860442
\(779\) −24.0000 −0.859889
\(780\) −8.00000 −0.286446
\(781\) 8.00000 0.286263
\(782\) 4.00000 0.143040
\(783\) 0 0
\(784\) −7.00000 −0.250000
\(785\) 28.0000 0.999363
\(786\) 8.00000 0.285351
\(787\) 28.0000 0.998092 0.499046 − 0.866575i \(-0.333684\pi\)
0.499046 + 0.866575i \(0.333684\pi\)
\(788\) −4.00000 −0.142494
\(789\) −12.0000 −0.427211
\(790\) −16.0000 −0.569254
\(791\) 0 0
\(792\) 1.00000 0.0355335
\(793\) 8.00000 0.284088
\(794\) −28.0000 −0.993683
\(795\) −8.00000 −0.283731
\(796\) −14.0000 −0.496217
\(797\) 12.0000 0.425062 0.212531 − 0.977154i \(-0.431829\pi\)
0.212531 + 0.977154i \(0.431829\pi\)
\(798\) 0 0
\(799\) 10.0000 0.353775
\(800\) −1.00000 −0.0353553
\(801\) −6.00000 −0.212000
\(802\) 26.0000 0.918092
\(803\) −10.0000 −0.352892
\(804\) 12.0000 0.423207
\(805\) 0 0
\(806\) −8.00000 −0.281788
\(807\) −14.0000 −0.492823
\(808\) −2.00000 −0.0703598
\(809\) 14.0000 0.492214 0.246107 − 0.969243i \(-0.420849\pi\)
0.246107 + 0.969243i \(0.420849\pi\)
\(810\) −2.00000 −0.0702728
\(811\) −20.0000 −0.702295 −0.351147 − 0.936320i \(-0.614208\pi\)
−0.351147 + 0.936320i \(0.614208\pi\)
\(812\) 0 0
\(813\) 6.00000 0.210429
\(814\) 0 0
\(815\) −40.0000 −1.40114
\(816\) 1.00000 0.0350070
\(817\) 48.0000 1.67931
\(818\) 14.0000 0.489499
\(819\) 0 0
\(820\) −12.0000 −0.419058
\(821\) −36.0000 −1.25641 −0.628204 − 0.778048i \(-0.716210\pi\)
−0.628204 + 0.778048i \(0.716210\pi\)
\(822\) −6.00000 −0.209274
\(823\) −6.00000 −0.209147 −0.104573 − 0.994517i \(-0.533348\pi\)
−0.104573 + 0.994517i \(0.533348\pi\)
\(824\) 8.00000 0.278693
\(825\) 1.00000 0.0348155
\(826\) 0 0
\(827\) −32.0000 −1.11275 −0.556375 − 0.830932i \(-0.687808\pi\)
−0.556375 + 0.830932i \(0.687808\pi\)
\(828\) −4.00000 −0.139010
\(829\) 30.0000 1.04194 0.520972 − 0.853574i \(-0.325570\pi\)
0.520972 + 0.853574i \(0.325570\pi\)
\(830\) −8.00000 −0.277684
\(831\) −10.0000 −0.346896
\(832\) −4.00000 −0.138675
\(833\) 7.00000 0.242536
\(834\) 20.0000 0.692543
\(835\) −4.00000 −0.138426
\(836\) −4.00000 −0.138343
\(837\) −2.00000 −0.0691301
\(838\) 20.0000 0.690889
\(839\) 4.00000 0.138095 0.0690477 − 0.997613i \(-0.478004\pi\)
0.0690477 + 0.997613i \(0.478004\pi\)
\(840\) 0 0
\(841\) −29.0000 −1.00000
\(842\) −22.0000 −0.758170
\(843\) −30.0000 −1.03325
\(844\) 4.00000 0.137686
\(845\) −6.00000 −0.206406
\(846\) −10.0000 −0.343807
\(847\) 0 0
\(848\) −4.00000 −0.137361
\(849\) 28.0000 0.960958
\(850\) 1.00000 0.0342997
\(851\) 0 0
\(852\) −8.00000 −0.274075
\(853\) 26.0000 0.890223 0.445112 − 0.895475i \(-0.353164\pi\)
0.445112 + 0.895475i \(0.353164\pi\)
\(854\) 0 0
\(855\) 8.00000 0.273594
\(856\) 20.0000 0.683586
\(857\) −10.0000 −0.341593 −0.170797 − 0.985306i \(-0.554634\pi\)
−0.170797 + 0.985306i \(0.554634\pi\)
\(858\) 4.00000 0.136558
\(859\) −4.00000 −0.136478 −0.0682391 − 0.997669i \(-0.521738\pi\)
−0.0682391 + 0.997669i \(0.521738\pi\)
\(860\) 24.0000 0.818393
\(861\) 0 0
\(862\) −10.0000 −0.340601
\(863\) −54.0000 −1.83818 −0.919091 − 0.394046i \(-0.871075\pi\)
−0.919091 + 0.394046i \(0.871075\pi\)
\(864\) −1.00000 −0.0340207
\(865\) −16.0000 −0.544016
\(866\) −2.00000 −0.0679628
\(867\) −1.00000 −0.0339618
\(868\) 0 0
\(869\) 8.00000 0.271381
\(870\) 0 0
\(871\) 48.0000 1.62642
\(872\) −2.00000 −0.0677285
\(873\) −2.00000 −0.0676897
\(874\) 16.0000 0.541208
\(875\) 0 0
\(876\) 10.0000 0.337869
\(877\) 46.0000 1.55331 0.776655 − 0.629926i \(-0.216915\pi\)
0.776655 + 0.629926i \(0.216915\pi\)
\(878\) 28.0000 0.944954
\(879\) −6.00000 −0.202375
\(880\) −2.00000 −0.0674200
\(881\) −14.0000 −0.471672 −0.235836 − 0.971793i \(-0.575783\pi\)
−0.235836 + 0.971793i \(0.575783\pi\)
\(882\) −7.00000 −0.235702
\(883\) −12.0000 −0.403832 −0.201916 − 0.979403i \(-0.564717\pi\)
−0.201916 + 0.979403i \(0.564717\pi\)
\(884\) 4.00000 0.134535
\(885\) 8.00000 0.268917
\(886\) −24.0000 −0.806296
\(887\) 54.0000 1.81314 0.906571 − 0.422053i \(-0.138690\pi\)
0.906571 + 0.422053i \(0.138690\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 12.0000 0.402241
\(891\) 1.00000 0.0335013
\(892\) 12.0000 0.401790
\(893\) 40.0000 1.33855
\(894\) 10.0000 0.334450
\(895\) −32.0000 −1.06964
\(896\) 0 0
\(897\) −16.0000 −0.534224
\(898\) −14.0000 −0.467186
\(899\) 0 0
\(900\) −1.00000 −0.0333333
\(901\) 4.00000 0.133259
\(902\) 6.00000 0.199778
\(903\) 0 0
\(904\) 6.00000 0.199557
\(905\) −8.00000 −0.265929
\(906\) −22.0000 −0.730901
\(907\) −36.0000 −1.19536 −0.597680 − 0.801735i \(-0.703911\pi\)
−0.597680 + 0.801735i \(0.703911\pi\)
\(908\) −16.0000 −0.530979
\(909\) −2.00000 −0.0663358
\(910\) 0 0
\(911\) 8.00000 0.265052 0.132526 − 0.991180i \(-0.457691\pi\)
0.132526 + 0.991180i \(0.457691\pi\)
\(912\) 4.00000 0.132453
\(913\) 4.00000 0.132381
\(914\) −26.0000 −0.860004
\(915\) −4.00000 −0.132236
\(916\) −14.0000 −0.462573
\(917\) 0 0
\(918\) 1.00000 0.0330049
\(919\) 26.0000 0.857661 0.428830 − 0.903385i \(-0.358926\pi\)
0.428830 + 0.903385i \(0.358926\pi\)
\(920\) 8.00000 0.263752
\(921\) −8.00000 −0.263609
\(922\) 18.0000 0.592798
\(923\) −32.0000 −1.05329
\(924\) 0 0
\(925\) 0 0
\(926\) −8.00000 −0.262896
\(927\) 8.00000 0.262754
\(928\) 0 0
\(929\) 46.0000 1.50921 0.754606 − 0.656179i \(-0.227828\pi\)
0.754606 + 0.656179i \(0.227828\pi\)
\(930\) 4.00000 0.131165
\(931\) 28.0000 0.917663
\(932\) −6.00000 −0.196537
\(933\) −20.0000 −0.654771
\(934\) −28.0000 −0.916188
\(935\) 2.00000 0.0654070
\(936\) −4.00000 −0.130744
\(937\) −14.0000 −0.457360 −0.228680 − 0.973502i \(-0.573441\pi\)
−0.228680 + 0.973502i \(0.573441\pi\)
\(938\) 0 0
\(939\) 10.0000 0.326338
\(940\) 20.0000 0.652328
\(941\) −12.0000 −0.391189 −0.195594 − 0.980685i \(-0.562664\pi\)
−0.195594 + 0.980685i \(0.562664\pi\)
\(942\) 14.0000 0.456145
\(943\) −24.0000 −0.781548
\(944\) 4.00000 0.130189
\(945\) 0 0
\(946\) −12.0000 −0.390154
\(947\) −12.0000 −0.389948 −0.194974 − 0.980808i \(-0.562462\pi\)
−0.194974 + 0.980808i \(0.562462\pi\)
\(948\) −8.00000 −0.259828
\(949\) 40.0000 1.29845
\(950\) 4.00000 0.129777
\(951\) 18.0000 0.583690
\(952\) 0 0
\(953\) 18.0000 0.583077 0.291539 − 0.956559i \(-0.405833\pi\)
0.291539 + 0.956559i \(0.405833\pi\)
\(954\) −4.00000 −0.129505
\(955\) −12.0000 −0.388311
\(956\) 0 0
\(957\) 0 0
\(958\) 10.0000 0.323085
\(959\) 0 0
\(960\) 2.00000 0.0645497
\(961\) −27.0000 −0.870968
\(962\) 0 0
\(963\) 20.0000 0.644491
\(964\) −6.00000 −0.193247
\(965\) 44.0000 1.41641
\(966\) 0 0
\(967\) 42.0000 1.35063 0.675314 − 0.737530i \(-0.264008\pi\)
0.675314 + 0.737530i \(0.264008\pi\)
\(968\) 1.00000 0.0321412
\(969\) −4.00000 −0.128499
\(970\) 4.00000 0.128432
\(971\) −28.0000 −0.898563 −0.449281 − 0.893390i \(-0.648320\pi\)
−0.449281 + 0.893390i \(0.648320\pi\)
\(972\) −1.00000 −0.0320750
\(973\) 0 0
\(974\) 14.0000 0.448589
\(975\) −4.00000 −0.128103
\(976\) −2.00000 −0.0640184
\(977\) −2.00000 −0.0639857 −0.0319928 − 0.999488i \(-0.510185\pi\)
−0.0319928 + 0.999488i \(0.510185\pi\)
\(978\) −20.0000 −0.639529
\(979\) −6.00000 −0.191761
\(980\) 14.0000 0.447214
\(981\) −2.00000 −0.0638551
\(982\) 12.0000 0.382935
\(983\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(984\) −6.00000 −0.191273
\(985\) 8.00000 0.254901
\(986\) 0 0
\(987\) 0 0
\(988\) 16.0000 0.509028
\(989\) 48.0000 1.52631
\(990\) −2.00000 −0.0635642
\(991\) −2.00000 −0.0635321 −0.0317660 − 0.999495i \(-0.510113\pi\)
−0.0317660 + 0.999495i \(0.510113\pi\)
\(992\) 2.00000 0.0635001
\(993\) 4.00000 0.126936
\(994\) 0 0
\(995\) 28.0000 0.887660
\(996\) −4.00000 −0.126745
\(997\) 18.0000 0.570066 0.285033 − 0.958518i \(-0.407995\pi\)
0.285033 + 0.958518i \(0.407995\pi\)
\(998\) −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 1122.2.a.f.1.1 ✓ 1
3.2 odd 2 3366.2.a.j.1.1 1
4.3 odd 2 8976.2.a.u.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1122.2.a.f.1.1 ✓ 1 1.1 even 1 trivial
3366.2.a.j.1.1 1 3.2 odd 2
8976.2.a.u.1.1 1 4.3 odd 2