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Properties

Label 3450.2.a.ba.1.1
Level $3450$
Weight $2$
Character 3450.1
Self dual yes
Analytic conductor $27.548$
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] = [3450,2,Mod(1,3450)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3450, 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("3450.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3450 = 2 \cdot 3 \cdot 5^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3450.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: \(27.5483886973\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 690)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 3450.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} +2.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} +2.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +6.00000 q^{11} +1.00000 q^{12} +2.00000 q^{13} +2.00000 q^{14} +1.00000 q^{16} +1.00000 q^{18} -4.00000 q^{19} +2.00000 q^{21} +6.00000 q^{22} +1.00000 q^{23} +1.00000 q^{24} +2.00000 q^{26} +1.00000 q^{27} +2.00000 q^{28} -2.00000 q^{29} -8.00000 q^{31} +1.00000 q^{32} +6.00000 q^{33} +1.00000 q^{36} +4.00000 q^{37} -4.00000 q^{38} +2.00000 q^{39} +2.00000 q^{41} +2.00000 q^{42} +8.00000 q^{43} +6.00000 q^{44} +1.00000 q^{46} +1.00000 q^{48} -3.00000 q^{49} +2.00000 q^{52} +2.00000 q^{53} +1.00000 q^{54} +2.00000 q^{56} -4.00000 q^{57} -2.00000 q^{58} -4.00000 q^{59} -8.00000 q^{62} +2.00000 q^{63} +1.00000 q^{64} +6.00000 q^{66} +1.00000 q^{69} -8.00000 q^{71} +1.00000 q^{72} -6.00000 q^{73} +4.00000 q^{74} -4.00000 q^{76} +12.0000 q^{77} +2.00000 q^{78} -14.0000 q^{79} +1.00000 q^{81} +2.00000 q^{82} +6.00000 q^{83} +2.00000 q^{84} +8.00000 q^{86} -2.00000 q^{87} +6.00000 q^{88} -16.0000 q^{89} +4.00000 q^{91} +1.00000 q^{92} -8.00000 q^{93} +1.00000 q^{96} -2.00000 q^{97} -3.00000 q^{98} +6.00000 q^{99} +O(q^{100})\)

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
\(5\) 0 0
\(6\) 1.00000 0.408248
\(7\) 2.00000 0.755929 0.377964 − 0.925820i \(-0.376624\pi\)
0.377964 + 0.925820i \(0.376624\pi\)
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 6.00000 1.80907 0.904534 − 0.426401i \(-0.140219\pi\)
0.904534 + 0.426401i \(0.140219\pi\)
\(12\) 1.00000 0.288675
\(13\) 2.00000 0.554700 0.277350 − 0.960769i \(-0.410544\pi\)
0.277350 + 0.960769i \(0.410544\pi\)
\(14\) 2.00000 0.534522
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(18\) 1.00000 0.235702
\(19\) −4.00000 −0.917663 −0.458831 − 0.888523i \(-0.651732\pi\)
−0.458831 + 0.888523i \(0.651732\pi\)
\(20\) 0 0
\(21\) 2.00000 0.436436
\(22\) 6.00000 1.27920
\(23\) 1.00000 0.208514
\(24\) 1.00000 0.204124
\(25\) 0 0
\(26\) 2.00000 0.392232
\(27\) 1.00000 0.192450
\(28\) 2.00000 0.377964
\(29\) −2.00000 −0.371391 −0.185695 − 0.982607i \(-0.559454\pi\)
−0.185695 + 0.982607i \(0.559454\pi\)
\(30\) 0 0
\(31\) −8.00000 −1.43684 −0.718421 − 0.695608i \(-0.755135\pi\)
−0.718421 + 0.695608i \(0.755135\pi\)
\(32\) 1.00000 0.176777
\(33\) 6.00000 1.04447
\(34\) 0 0
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) 4.00000 0.657596 0.328798 − 0.944400i \(-0.393356\pi\)
0.328798 + 0.944400i \(0.393356\pi\)
\(38\) −4.00000 −0.648886
\(39\) 2.00000 0.320256
\(40\) 0 0
\(41\) 2.00000 0.312348 0.156174 − 0.987730i \(-0.450084\pi\)
0.156174 + 0.987730i \(0.450084\pi\)
\(42\) 2.00000 0.308607
\(43\) 8.00000 1.21999 0.609994 − 0.792406i \(-0.291172\pi\)
0.609994 + 0.792406i \(0.291172\pi\)
\(44\) 6.00000 0.904534
\(45\) 0 0
\(46\) 1.00000 0.147442
\(47\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 1.00000 0.144338
\(49\) −3.00000 −0.428571
\(50\) 0 0
\(51\) 0 0
\(52\) 2.00000 0.277350
\(53\) 2.00000 0.274721 0.137361 − 0.990521i \(-0.456138\pi\)
0.137361 + 0.990521i \(0.456138\pi\)
\(54\) 1.00000 0.136083
\(55\) 0 0
\(56\) 2.00000 0.267261
\(57\) −4.00000 −0.529813
\(58\) −2.00000 −0.262613
\(59\) −4.00000 −0.520756 −0.260378 − 0.965507i \(-0.583847\pi\)
−0.260378 + 0.965507i \(0.583847\pi\)
\(60\) 0 0
\(61\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(62\) −8.00000 −1.01600
\(63\) 2.00000 0.251976
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 6.00000 0.738549
\(67\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(68\) 0 0
\(69\) 1.00000 0.120386
\(70\) 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 \(-0.614200\pi\)
−0.351123 + 0.936329i \(0.614200\pi\)
\(74\) 4.00000 0.464991
\(75\) 0 0
\(76\) −4.00000 −0.458831
\(77\) 12.0000 1.36753
\(78\) 2.00000 0.226455
\(79\) −14.0000 −1.57512 −0.787562 − 0.616236i \(-0.788657\pi\)
−0.787562 + 0.616236i \(0.788657\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 2.00000 0.220863
\(83\) 6.00000 0.658586 0.329293 − 0.944228i \(-0.393190\pi\)
0.329293 + 0.944228i \(0.393190\pi\)
\(84\) 2.00000 0.218218
\(85\) 0 0
\(86\) 8.00000 0.862662
\(87\) −2.00000 −0.214423
\(88\) 6.00000 0.639602
\(89\) −16.0000 −1.69600 −0.847998 − 0.529999i \(-0.822192\pi\)
−0.847998 + 0.529999i \(0.822192\pi\)
\(90\) 0 0
\(91\) 4.00000 0.419314
\(92\) 1.00000 0.104257
\(93\) −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 \(0.532375\pi\)
\(98\) −3.00000 −0.303046
\(99\) 6.00000 0.603023
\(100\) 0 0
\(101\) 2.00000 0.199007 0.0995037 − 0.995037i \(-0.468274\pi\)
0.0995037 + 0.995037i \(0.468274\pi\)
\(102\) 0 0
\(103\) 10.0000 0.985329 0.492665 − 0.870219i \(-0.336023\pi\)
0.492665 + 0.870219i \(0.336023\pi\)
\(104\) 2.00000 0.196116
\(105\) 0 0
\(106\) 2.00000 0.194257
\(107\) 6.00000 0.580042 0.290021 − 0.957020i \(-0.406338\pi\)
0.290021 + 0.957020i \(0.406338\pi\)
\(108\) 1.00000 0.0962250
\(109\) 12.0000 1.14939 0.574696 − 0.818367i \(-0.305120\pi\)
0.574696 + 0.818367i \(0.305120\pi\)
\(110\) 0 0
\(111\) 4.00000 0.379663
\(112\) 2.00000 0.188982
\(113\) −16.0000 −1.50515 −0.752577 − 0.658505i \(-0.771189\pi\)
−0.752577 + 0.658505i \(0.771189\pi\)
\(114\) −4.00000 −0.374634
\(115\) 0 0
\(116\) −2.00000 −0.185695
\(117\) 2.00000 0.184900
\(118\) −4.00000 −0.368230
\(119\) 0 0
\(120\) 0 0
\(121\) 25.0000 2.27273
\(122\) 0 0
\(123\) 2.00000 0.180334
\(124\) −8.00000 −0.718421
\(125\) 0 0
\(126\) 2.00000 0.178174
\(127\) 12.0000 1.06483 0.532414 − 0.846484i \(-0.321285\pi\)
0.532414 + 0.846484i \(0.321285\pi\)
\(128\) 1.00000 0.0883883
\(129\) 8.00000 0.704361
\(130\) 0 0
\(131\) 8.00000 0.698963 0.349482 − 0.936943i \(-0.386358\pi\)
0.349482 + 0.936943i \(0.386358\pi\)
\(132\) 6.00000 0.522233
\(133\) −8.00000 −0.693688
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −20.0000 −1.70872 −0.854358 − 0.519685i \(-0.826049\pi\)
−0.854358 + 0.519685i \(0.826049\pi\)
\(138\) 1.00000 0.0851257
\(139\) −4.00000 −0.339276 −0.169638 − 0.985506i \(-0.554260\pi\)
−0.169638 + 0.985506i \(0.554260\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) −8.00000 −0.671345
\(143\) 12.0000 1.00349
\(144\) 1.00000 0.0833333
\(145\) 0 0
\(146\) −6.00000 −0.496564
\(147\) −3.00000 −0.247436
\(148\) 4.00000 0.328798
\(149\) 10.0000 0.819232 0.409616 − 0.912258i \(-0.365663\pi\)
0.409616 + 0.912258i \(0.365663\pi\)
\(150\) 0 0
\(151\) 20.0000 1.62758 0.813788 − 0.581161i \(-0.197401\pi\)
0.813788 + 0.581161i \(0.197401\pi\)
\(152\) −4.00000 −0.324443
\(153\) 0 0
\(154\) 12.0000 0.966988
\(155\) 0 0
\(156\) 2.00000 0.160128
\(157\) 4.00000 0.319235 0.159617 − 0.987179i \(-0.448974\pi\)
0.159617 + 0.987179i \(0.448974\pi\)
\(158\) −14.0000 −1.11378
\(159\) 2.00000 0.158610
\(160\) 0 0
\(161\) 2.00000 0.157622
\(162\) 1.00000 0.0785674
\(163\) −12.0000 −0.939913 −0.469956 − 0.882690i \(-0.655730\pi\)
−0.469956 + 0.882690i \(0.655730\pi\)
\(164\) 2.00000 0.156174
\(165\) 0 0
\(166\) 6.00000 0.465690
\(167\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(168\) 2.00000 0.154303
\(169\) −9.00000 −0.692308
\(170\) 0 0
\(171\) −4.00000 −0.305888
\(172\) 8.00000 0.609994
\(173\) −6.00000 −0.456172 −0.228086 − 0.973641i \(-0.573247\pi\)
−0.228086 + 0.973641i \(0.573247\pi\)
\(174\) −2.00000 −0.151620
\(175\) 0 0
\(176\) 6.00000 0.452267
\(177\) −4.00000 −0.300658
\(178\) −16.0000 −1.19925
\(179\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(180\) 0 0
\(181\) 4.00000 0.297318 0.148659 − 0.988889i \(-0.452504\pi\)
0.148659 + 0.988889i \(0.452504\pi\)
\(182\) 4.00000 0.296500
\(183\) 0 0
\(184\) 1.00000 0.0737210
\(185\) 0 0
\(186\) −8.00000 −0.586588
\(187\) 0 0
\(188\) 0 0
\(189\) 2.00000 0.145479
\(190\) 0 0
\(191\) 8.00000 0.578860 0.289430 − 0.957199i \(-0.406534\pi\)
0.289430 + 0.957199i \(0.406534\pi\)
\(192\) 1.00000 0.0721688
\(193\) 18.0000 1.29567 0.647834 − 0.761781i \(-0.275675\pi\)
0.647834 + 0.761781i \(0.275675\pi\)
\(194\) −2.00000 −0.143592
\(195\) 0 0
\(196\) −3.00000 −0.214286
\(197\) 22.0000 1.56744 0.783718 − 0.621117i \(-0.213321\pi\)
0.783718 + 0.621117i \(0.213321\pi\)
\(198\) 6.00000 0.426401
\(199\) −22.0000 −1.55954 −0.779769 − 0.626067i \(-0.784664\pi\)
−0.779769 + 0.626067i \(0.784664\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 2.00000 0.140720
\(203\) −4.00000 −0.280745
\(204\) 0 0
\(205\) 0 0
\(206\) 10.0000 0.696733
\(207\) 1.00000 0.0695048
\(208\) 2.00000 0.138675
\(209\) −24.0000 −1.66011
\(210\) 0 0
\(211\) 28.0000 1.92760 0.963800 − 0.266627i \(-0.0859092\pi\)
0.963800 + 0.266627i \(0.0859092\pi\)
\(212\) 2.00000 0.137361
\(213\) −8.00000 −0.548151
\(214\) 6.00000 0.410152
\(215\) 0 0
\(216\) 1.00000 0.0680414
\(217\) −16.0000 −1.08615
\(218\) 12.0000 0.812743
\(219\) −6.00000 −0.405442
\(220\) 0 0
\(221\) 0 0
\(222\) 4.00000 0.268462
\(223\) 28.0000 1.87502 0.937509 − 0.347960i \(-0.113126\pi\)
0.937509 + 0.347960i \(0.113126\pi\)
\(224\) 2.00000 0.133631
\(225\) 0 0
\(226\) −16.0000 −1.06430
\(227\) −10.0000 −0.663723 −0.331862 − 0.943328i \(-0.607677\pi\)
−0.331862 + 0.943328i \(0.607677\pi\)
\(228\) −4.00000 −0.264906
\(229\) 4.00000 0.264327 0.132164 − 0.991228i \(-0.457808\pi\)
0.132164 + 0.991228i \(0.457808\pi\)
\(230\) 0 0
\(231\) 12.0000 0.789542
\(232\) −2.00000 −0.131306
\(233\) −2.00000 −0.131024 −0.0655122 − 0.997852i \(-0.520868\pi\)
−0.0655122 + 0.997852i \(0.520868\pi\)
\(234\) 2.00000 0.130744
\(235\) 0 0
\(236\) −4.00000 −0.260378
\(237\) −14.0000 −0.909398
\(238\) 0 0
\(239\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(240\) 0 0
\(241\) 18.0000 1.15948 0.579741 − 0.814801i \(-0.303154\pi\)
0.579741 + 0.814801i \(0.303154\pi\)
\(242\) 25.0000 1.60706
\(243\) 1.00000 0.0641500
\(244\) 0 0
\(245\) 0 0
\(246\) 2.00000 0.127515
\(247\) −8.00000 −0.509028
\(248\) −8.00000 −0.508001
\(249\) 6.00000 0.380235
\(250\) 0 0
\(251\) −18.0000 −1.13615 −0.568075 − 0.822977i \(-0.692312\pi\)
−0.568075 + 0.822977i \(0.692312\pi\)
\(252\) 2.00000 0.125988
\(253\) 6.00000 0.377217
\(254\) 12.0000 0.752947
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 6.00000 0.374270 0.187135 − 0.982334i \(-0.440080\pi\)
0.187135 + 0.982334i \(0.440080\pi\)
\(258\) 8.00000 0.498058
\(259\) 8.00000 0.497096
\(260\) 0 0
\(261\) −2.00000 −0.123797
\(262\) 8.00000 0.494242
\(263\) −4.00000 −0.246651 −0.123325 − 0.992366i \(-0.539356\pi\)
−0.123325 + 0.992366i \(0.539356\pi\)
\(264\) 6.00000 0.369274
\(265\) 0 0
\(266\) −8.00000 −0.490511
\(267\) −16.0000 −0.979184
\(268\) 0 0
\(269\) −30.0000 −1.82913 −0.914566 − 0.404436i \(-0.867468\pi\)
−0.914566 + 0.404436i \(0.867468\pi\)
\(270\) 0 0
\(271\) −4.00000 −0.242983 −0.121491 − 0.992592i \(-0.538768\pi\)
−0.121491 + 0.992592i \(0.538768\pi\)
\(272\) 0 0
\(273\) 4.00000 0.242091
\(274\) −20.0000 −1.20824
\(275\) 0 0
\(276\) 1.00000 0.0601929
\(277\) −10.0000 −0.600842 −0.300421 − 0.953807i \(-0.597127\pi\)
−0.300421 + 0.953807i \(0.597127\pi\)
\(278\) −4.00000 −0.239904
\(279\) −8.00000 −0.478947
\(280\) 0 0
\(281\) −16.0000 −0.954480 −0.477240 − 0.878773i \(-0.658363\pi\)
−0.477240 + 0.878773i \(0.658363\pi\)
\(282\) 0 0
\(283\) −28.0000 −1.66443 −0.832214 − 0.554455i \(-0.812927\pi\)
−0.832214 + 0.554455i \(0.812927\pi\)
\(284\) −8.00000 −0.474713
\(285\) 0 0
\(286\) 12.0000 0.709575
\(287\) 4.00000 0.236113
\(288\) 1.00000 0.0589256
\(289\) −17.0000 −1.00000
\(290\) 0 0
\(291\) −2.00000 −0.117242
\(292\) −6.00000 −0.351123
\(293\) −22.0000 −1.28525 −0.642627 − 0.766179i \(-0.722155\pi\)
−0.642627 + 0.766179i \(0.722155\pi\)
\(294\) −3.00000 −0.174964
\(295\) 0 0
\(296\) 4.00000 0.232495
\(297\) 6.00000 0.348155
\(298\) 10.0000 0.579284
\(299\) 2.00000 0.115663
\(300\) 0 0
\(301\) 16.0000 0.922225
\(302\) 20.0000 1.15087
\(303\) 2.00000 0.114897
\(304\) −4.00000 −0.229416
\(305\) 0 0
\(306\) 0 0
\(307\) 12.0000 0.684876 0.342438 − 0.939540i \(-0.388747\pi\)
0.342438 + 0.939540i \(0.388747\pi\)
\(308\) 12.0000 0.683763
\(309\) 10.0000 0.568880
\(310\) 0 0
\(311\) −32.0000 −1.81455 −0.907277 − 0.420534i \(-0.861843\pi\)
−0.907277 + 0.420534i \(0.861843\pi\)
\(312\) 2.00000 0.113228
\(313\) 18.0000 1.01742 0.508710 − 0.860938i \(-0.330123\pi\)
0.508710 + 0.860938i \(0.330123\pi\)
\(314\) 4.00000 0.225733
\(315\) 0 0
\(316\) −14.0000 −0.787562
\(317\) −30.0000 −1.68497 −0.842484 − 0.538721i \(-0.818908\pi\)
−0.842484 + 0.538721i \(0.818908\pi\)
\(318\) 2.00000 0.112154
\(319\) −12.0000 −0.671871
\(320\) 0 0
\(321\) 6.00000 0.334887
\(322\) 2.00000 0.111456
\(323\) 0 0
\(324\) 1.00000 0.0555556
\(325\) 0 0
\(326\) −12.0000 −0.664619
\(327\) 12.0000 0.663602
\(328\) 2.00000 0.110432
\(329\) 0 0
\(330\) 0 0
\(331\) −20.0000 −1.09930 −0.549650 − 0.835395i \(-0.685239\pi\)
−0.549650 + 0.835395i \(0.685239\pi\)
\(332\) 6.00000 0.329293
\(333\) 4.00000 0.219199
\(334\) 0 0
\(335\) 0 0
\(336\) 2.00000 0.109109
\(337\) −22.0000 −1.19842 −0.599208 − 0.800593i \(-0.704518\pi\)
−0.599208 + 0.800593i \(0.704518\pi\)
\(338\) −9.00000 −0.489535
\(339\) −16.0000 −0.869001
\(340\) 0 0
\(341\) −48.0000 −2.59935
\(342\) −4.00000 −0.216295
\(343\) −20.0000 −1.07990
\(344\) 8.00000 0.431331
\(345\) 0 0
\(346\) −6.00000 −0.322562
\(347\) 32.0000 1.71785 0.858925 − 0.512101i \(-0.171133\pi\)
0.858925 + 0.512101i \(0.171133\pi\)
\(348\) −2.00000 −0.107211
\(349\) 10.0000 0.535288 0.267644 − 0.963518i \(-0.413755\pi\)
0.267644 + 0.963518i \(0.413755\pi\)
\(350\) 0 0
\(351\) 2.00000 0.106752
\(352\) 6.00000 0.319801
\(353\) −6.00000 −0.319348 −0.159674 − 0.987170i \(-0.551044\pi\)
−0.159674 + 0.987170i \(0.551044\pi\)
\(354\) −4.00000 −0.212598
\(355\) 0 0
\(356\) −16.0000 −0.847998
\(357\) 0 0
\(358\) 0 0
\(359\) 32.0000 1.68890 0.844448 − 0.535638i \(-0.179929\pi\)
0.844448 + 0.535638i \(0.179929\pi\)
\(360\) 0 0
\(361\) −3.00000 −0.157895
\(362\) 4.00000 0.210235
\(363\) 25.0000 1.31216
\(364\) 4.00000 0.209657
\(365\) 0 0
\(366\) 0 0
\(367\) −2.00000 −0.104399 −0.0521996 − 0.998637i \(-0.516623\pi\)
−0.0521996 + 0.998637i \(0.516623\pi\)
\(368\) 1.00000 0.0521286
\(369\) 2.00000 0.104116
\(370\) 0 0
\(371\) 4.00000 0.207670
\(372\) −8.00000 −0.414781
\(373\) 4.00000 0.207112 0.103556 − 0.994624i \(-0.466978\pi\)
0.103556 + 0.994624i \(0.466978\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −4.00000 −0.206010
\(378\) 2.00000 0.102869
\(379\) −36.0000 −1.84920 −0.924598 − 0.380945i \(-0.875599\pi\)
−0.924598 + 0.380945i \(0.875599\pi\)
\(380\) 0 0
\(381\) 12.0000 0.614779
\(382\) 8.00000 0.409316
\(383\) 12.0000 0.613171 0.306586 − 0.951843i \(-0.400813\pi\)
0.306586 + 0.951843i \(0.400813\pi\)
\(384\) 1.00000 0.0510310
\(385\) 0 0
\(386\) 18.0000 0.916176
\(387\) 8.00000 0.406663
\(388\) −2.00000 −0.101535
\(389\) 10.0000 0.507020 0.253510 − 0.967333i \(-0.418415\pi\)
0.253510 + 0.967333i \(0.418415\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) −3.00000 −0.151523
\(393\) 8.00000 0.403547
\(394\) 22.0000 1.10834
\(395\) 0 0
\(396\) 6.00000 0.301511
\(397\) −14.0000 −0.702640 −0.351320 − 0.936255i \(-0.614267\pi\)
−0.351320 + 0.936255i \(0.614267\pi\)
\(398\) −22.0000 −1.10276
\(399\) −8.00000 −0.400501
\(400\) 0 0
\(401\) 8.00000 0.399501 0.199750 − 0.979847i \(-0.435987\pi\)
0.199750 + 0.979847i \(0.435987\pi\)
\(402\) 0 0
\(403\) −16.0000 −0.797017
\(404\) 2.00000 0.0995037
\(405\) 0 0
\(406\) −4.00000 −0.198517
\(407\) 24.0000 1.18964
\(408\) 0 0
\(409\) −10.0000 −0.494468 −0.247234 − 0.968956i \(-0.579522\pi\)
−0.247234 + 0.968956i \(0.579522\pi\)
\(410\) 0 0
\(411\) −20.0000 −0.986527
\(412\) 10.0000 0.492665
\(413\) −8.00000 −0.393654
\(414\) 1.00000 0.0491473
\(415\) 0 0
\(416\) 2.00000 0.0980581
\(417\) −4.00000 −0.195881
\(418\) −24.0000 −1.17388
\(419\) 26.0000 1.27018 0.635092 − 0.772437i \(-0.280962\pi\)
0.635092 + 0.772437i \(0.280962\pi\)
\(420\) 0 0
\(421\) 24.0000 1.16969 0.584844 − 0.811146i \(-0.301156\pi\)
0.584844 + 0.811146i \(0.301156\pi\)
\(422\) 28.0000 1.36302
\(423\) 0 0
\(424\) 2.00000 0.0971286
\(425\) 0 0
\(426\) −8.00000 −0.387601
\(427\) 0 0
\(428\) 6.00000 0.290021
\(429\) 12.0000 0.579365
\(430\) 0 0
\(431\) −36.0000 −1.73406 −0.867029 − 0.498257i \(-0.833974\pi\)
−0.867029 + 0.498257i \(0.833974\pi\)
\(432\) 1.00000 0.0481125
\(433\) 26.0000 1.24948 0.624740 − 0.780833i \(-0.285205\pi\)
0.624740 + 0.780833i \(0.285205\pi\)
\(434\) −16.0000 −0.768025
\(435\) 0 0
\(436\) 12.0000 0.574696
\(437\) −4.00000 −0.191346
\(438\) −6.00000 −0.286691
\(439\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(440\) 0 0
\(441\) −3.00000 −0.142857
\(442\) 0 0
\(443\) −4.00000 −0.190046 −0.0950229 − 0.995475i \(-0.530292\pi\)
−0.0950229 + 0.995475i \(0.530292\pi\)
\(444\) 4.00000 0.189832
\(445\) 0 0
\(446\) 28.0000 1.32584
\(447\) 10.0000 0.472984
\(448\) 2.00000 0.0944911
\(449\) 10.0000 0.471929 0.235965 − 0.971762i \(-0.424175\pi\)
0.235965 + 0.971762i \(0.424175\pi\)
\(450\) 0 0
\(451\) 12.0000 0.565058
\(452\) −16.0000 −0.752577
\(453\) 20.0000 0.939682
\(454\) −10.0000 −0.469323
\(455\) 0 0
\(456\) −4.00000 −0.187317
\(457\) −18.0000 −0.842004 −0.421002 − 0.907060i \(-0.638322\pi\)
−0.421002 + 0.907060i \(0.638322\pi\)
\(458\) 4.00000 0.186908
\(459\) 0 0
\(460\) 0 0
\(461\) −14.0000 −0.652045 −0.326023 − 0.945362i \(-0.605709\pi\)
−0.326023 + 0.945362i \(0.605709\pi\)
\(462\) 12.0000 0.558291
\(463\) −32.0000 −1.48717 −0.743583 − 0.668644i \(-0.766875\pi\)
−0.743583 + 0.668644i \(0.766875\pi\)
\(464\) −2.00000 −0.0928477
\(465\) 0 0
\(466\) −2.00000 −0.0926482
\(467\) −30.0000 −1.38823 −0.694117 − 0.719862i \(-0.744205\pi\)
−0.694117 + 0.719862i \(0.744205\pi\)
\(468\) 2.00000 0.0924500
\(469\) 0 0
\(470\) 0 0
\(471\) 4.00000 0.184310
\(472\) −4.00000 −0.184115
\(473\) 48.0000 2.20704
\(474\) −14.0000 −0.643041
\(475\) 0 0
\(476\) 0 0
\(477\) 2.00000 0.0915737
\(478\) 0 0
\(479\) −40.0000 −1.82765 −0.913823 − 0.406112i \(-0.866884\pi\)
−0.913823 + 0.406112i \(0.866884\pi\)
\(480\) 0 0
\(481\) 8.00000 0.364769
\(482\) 18.0000 0.819878
\(483\) 2.00000 0.0910032
\(484\) 25.0000 1.13636
\(485\) 0 0
\(486\) 1.00000 0.0453609
\(487\) 32.0000 1.45006 0.725029 − 0.688718i \(-0.241826\pi\)
0.725029 + 0.688718i \(0.241826\pi\)
\(488\) 0 0
\(489\) −12.0000 −0.542659
\(490\) 0 0
\(491\) −32.0000 −1.44414 −0.722070 − 0.691820i \(-0.756809\pi\)
−0.722070 + 0.691820i \(0.756809\pi\)
\(492\) 2.00000 0.0901670
\(493\) 0 0
\(494\) −8.00000 −0.359937
\(495\) 0 0
\(496\) −8.00000 −0.359211
\(497\) −16.0000 −0.717698
\(498\) 6.00000 0.268866
\(499\) 36.0000 1.61158 0.805791 − 0.592200i \(-0.201741\pi\)
0.805791 + 0.592200i \(0.201741\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) −18.0000 −0.803379
\(503\) −28.0000 −1.24846 −0.624229 − 0.781241i \(-0.714587\pi\)
−0.624229 + 0.781241i \(0.714587\pi\)
\(504\) 2.00000 0.0890871
\(505\) 0 0
\(506\) 6.00000 0.266733
\(507\) −9.00000 −0.399704
\(508\) 12.0000 0.532414
\(509\) 30.0000 1.32973 0.664863 − 0.746965i \(-0.268490\pi\)
0.664863 + 0.746965i \(0.268490\pi\)
\(510\) 0 0
\(511\) −12.0000 −0.530849
\(512\) 1.00000 0.0441942
\(513\) −4.00000 −0.176604
\(514\) 6.00000 0.264649
\(515\) 0 0
\(516\) 8.00000 0.352180
\(517\) 0 0
\(518\) 8.00000 0.351500
\(519\) −6.00000 −0.263371
\(520\) 0 0
\(521\) −4.00000 −0.175243 −0.0876216 − 0.996154i \(-0.527927\pi\)
−0.0876216 + 0.996154i \(0.527927\pi\)
\(522\) −2.00000 −0.0875376
\(523\) −40.0000 −1.74908 −0.874539 − 0.484955i \(-0.838836\pi\)
−0.874539 + 0.484955i \(0.838836\pi\)
\(524\) 8.00000 0.349482
\(525\) 0 0
\(526\) −4.00000 −0.174408
\(527\) 0 0
\(528\) 6.00000 0.261116
\(529\) 1.00000 0.0434783
\(530\) 0 0
\(531\) −4.00000 −0.173585
\(532\) −8.00000 −0.346844
\(533\) 4.00000 0.173259
\(534\) −16.0000 −0.692388
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) −30.0000 −1.29339
\(539\) −18.0000 −0.775315
\(540\) 0 0
\(541\) −6.00000 −0.257960 −0.128980 − 0.991647i \(-0.541170\pi\)
−0.128980 + 0.991647i \(0.541170\pi\)
\(542\) −4.00000 −0.171815
\(543\) 4.00000 0.171656
\(544\) 0 0
\(545\) 0 0
\(546\) 4.00000 0.171184
\(547\) 12.0000 0.513083 0.256541 − 0.966533i \(-0.417417\pi\)
0.256541 + 0.966533i \(0.417417\pi\)
\(548\) −20.0000 −0.854358
\(549\) 0 0
\(550\) 0 0
\(551\) 8.00000 0.340811
\(552\) 1.00000 0.0425628
\(553\) −28.0000 −1.19068
\(554\) −10.0000 −0.424859
\(555\) 0 0
\(556\) −4.00000 −0.169638
\(557\) −10.0000 −0.423714 −0.211857 − 0.977301i \(-0.567951\pi\)
−0.211857 + 0.977301i \(0.567951\pi\)
\(558\) −8.00000 −0.338667
\(559\) 16.0000 0.676728
\(560\) 0 0
\(561\) 0 0
\(562\) −16.0000 −0.674919
\(563\) −18.0000 −0.758610 −0.379305 − 0.925272i \(-0.623837\pi\)
−0.379305 + 0.925272i \(0.623837\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) −28.0000 −1.17693
\(567\) 2.00000 0.0839921
\(568\) −8.00000 −0.335673
\(569\) 24.0000 1.00613 0.503066 − 0.864248i \(-0.332205\pi\)
0.503066 + 0.864248i \(0.332205\pi\)
\(570\) 0 0
\(571\) −32.0000 −1.33916 −0.669579 − 0.742741i \(-0.733526\pi\)
−0.669579 + 0.742741i \(0.733526\pi\)
\(572\) 12.0000 0.501745
\(573\) 8.00000 0.334205
\(574\) 4.00000 0.166957
\(575\) 0 0
\(576\) 1.00000 0.0416667
\(577\) 14.0000 0.582828 0.291414 − 0.956597i \(-0.405874\pi\)
0.291414 + 0.956597i \(0.405874\pi\)
\(578\) −17.0000 −0.707107
\(579\) 18.0000 0.748054
\(580\) 0 0
\(581\) 12.0000 0.497844
\(582\) −2.00000 −0.0829027
\(583\) 12.0000 0.496989
\(584\) −6.00000 −0.248282
\(585\) 0 0
\(586\) −22.0000 −0.908812
\(587\) −24.0000 −0.990586 −0.495293 − 0.868726i \(-0.664939\pi\)
−0.495293 + 0.868726i \(0.664939\pi\)
\(588\) −3.00000 −0.123718
\(589\) 32.0000 1.31854
\(590\) 0 0
\(591\) 22.0000 0.904959
\(592\) 4.00000 0.164399
\(593\) 6.00000 0.246390 0.123195 − 0.992382i \(-0.460686\pi\)
0.123195 + 0.992382i \(0.460686\pi\)
\(594\) 6.00000 0.246183
\(595\) 0 0
\(596\) 10.0000 0.409616
\(597\) −22.0000 −0.900400
\(598\) 2.00000 0.0817861
\(599\) 16.0000 0.653742 0.326871 − 0.945069i \(-0.394006\pi\)
0.326871 + 0.945069i \(0.394006\pi\)
\(600\) 0 0
\(601\) −38.0000 −1.55005 −0.775026 − 0.631929i \(-0.782263\pi\)
−0.775026 + 0.631929i \(0.782263\pi\)
\(602\) 16.0000 0.652111
\(603\) 0 0
\(604\) 20.0000 0.813788
\(605\) 0 0
\(606\) 2.00000 0.0812444
\(607\) −4.00000 −0.162355 −0.0811775 − 0.996700i \(-0.525868\pi\)
−0.0811775 + 0.996700i \(0.525868\pi\)
\(608\) −4.00000 −0.162221
\(609\) −4.00000 −0.162088
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 16.0000 0.646234 0.323117 − 0.946359i \(-0.395269\pi\)
0.323117 + 0.946359i \(0.395269\pi\)
\(614\) 12.0000 0.484281
\(615\) 0 0
\(616\) 12.0000 0.483494
\(617\) 40.0000 1.61034 0.805170 − 0.593045i \(-0.202074\pi\)
0.805170 + 0.593045i \(0.202074\pi\)
\(618\) 10.0000 0.402259
\(619\) 4.00000 0.160774 0.0803868 − 0.996764i \(-0.474384\pi\)
0.0803868 + 0.996764i \(0.474384\pi\)
\(620\) 0 0
\(621\) 1.00000 0.0401286
\(622\) −32.0000 −1.28308
\(623\) −32.0000 −1.28205
\(624\) 2.00000 0.0800641
\(625\) 0 0
\(626\) 18.0000 0.719425
\(627\) −24.0000 −0.958468
\(628\) 4.00000 0.159617
\(629\) 0 0
\(630\) 0 0
\(631\) −34.0000 −1.35352 −0.676759 − 0.736204i \(-0.736616\pi\)
−0.676759 + 0.736204i \(0.736616\pi\)
\(632\) −14.0000 −0.556890
\(633\) 28.0000 1.11290
\(634\) −30.0000 −1.19145
\(635\) 0 0
\(636\) 2.00000 0.0793052
\(637\) −6.00000 −0.237729
\(638\) −12.0000 −0.475085
\(639\) −8.00000 −0.316475
\(640\) 0 0
\(641\) −24.0000 −0.947943 −0.473972 − 0.880540i \(-0.657180\pi\)
−0.473972 + 0.880540i \(0.657180\pi\)
\(642\) 6.00000 0.236801
\(643\) 16.0000 0.630978 0.315489 − 0.948929i \(-0.397831\pi\)
0.315489 + 0.948929i \(0.397831\pi\)
\(644\) 2.00000 0.0788110
\(645\) 0 0
\(646\) 0 0
\(647\) −24.0000 −0.943537 −0.471769 − 0.881722i \(-0.656384\pi\)
−0.471769 + 0.881722i \(0.656384\pi\)
\(648\) 1.00000 0.0392837
\(649\) −24.0000 −0.942082
\(650\) 0 0
\(651\) −16.0000 −0.627089
\(652\) −12.0000 −0.469956
\(653\) −34.0000 −1.33052 −0.665261 − 0.746611i \(-0.731680\pi\)
−0.665261 + 0.746611i \(0.731680\pi\)
\(654\) 12.0000 0.469237
\(655\) 0 0
\(656\) 2.00000 0.0780869
\(657\) −6.00000 −0.234082
\(658\) 0 0
\(659\) 30.0000 1.16863 0.584317 − 0.811525i \(-0.301362\pi\)
0.584317 + 0.811525i \(0.301362\pi\)
\(660\) 0 0
\(661\) 40.0000 1.55582 0.777910 − 0.628376i \(-0.216280\pi\)
0.777910 + 0.628376i \(0.216280\pi\)
\(662\) −20.0000 −0.777322
\(663\) 0 0
\(664\) 6.00000 0.232845
\(665\) 0 0
\(666\) 4.00000 0.154997
\(667\) −2.00000 −0.0774403
\(668\) 0 0
\(669\) 28.0000 1.08254
\(670\) 0 0
\(671\) 0 0
\(672\) 2.00000 0.0771517
\(673\) 30.0000 1.15642 0.578208 − 0.815890i \(-0.303752\pi\)
0.578208 + 0.815890i \(0.303752\pi\)
\(674\) −22.0000 −0.847408
\(675\) 0 0
\(676\) −9.00000 −0.346154
\(677\) −18.0000 −0.691796 −0.345898 − 0.938272i \(-0.612426\pi\)
−0.345898 + 0.938272i \(0.612426\pi\)
\(678\) −16.0000 −0.614476
\(679\) −4.00000 −0.153506
\(680\) 0 0
\(681\) −10.0000 −0.383201
\(682\) −48.0000 −1.83801
\(683\) −16.0000 −0.612223 −0.306111 − 0.951996i \(-0.599028\pi\)
−0.306111 + 0.951996i \(0.599028\pi\)
\(684\) −4.00000 −0.152944
\(685\) 0 0
\(686\) −20.0000 −0.763604
\(687\) 4.00000 0.152610
\(688\) 8.00000 0.304997
\(689\) 4.00000 0.152388
\(690\) 0 0
\(691\) 20.0000 0.760836 0.380418 − 0.924815i \(-0.375780\pi\)
0.380418 + 0.924815i \(0.375780\pi\)
\(692\) −6.00000 −0.228086
\(693\) 12.0000 0.455842
\(694\) 32.0000 1.21470
\(695\) 0 0
\(696\) −2.00000 −0.0758098
\(697\) 0 0
\(698\) 10.0000 0.378506
\(699\) −2.00000 −0.0756469
\(700\) 0 0
\(701\) 6.00000 0.226617 0.113308 − 0.993560i \(-0.463855\pi\)
0.113308 + 0.993560i \(0.463855\pi\)
\(702\) 2.00000 0.0754851
\(703\) −16.0000 −0.603451
\(704\) 6.00000 0.226134
\(705\) 0 0
\(706\) −6.00000 −0.225813
\(707\) 4.00000 0.150435
\(708\) −4.00000 −0.150329
\(709\) 24.0000 0.901339 0.450669 − 0.892691i \(-0.351185\pi\)
0.450669 + 0.892691i \(0.351185\pi\)
\(710\) 0 0
\(711\) −14.0000 −0.525041
\(712\) −16.0000 −0.599625
\(713\) −8.00000 −0.299602
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 32.0000 1.19423
\(719\) 32.0000 1.19340 0.596699 − 0.802465i \(-0.296479\pi\)
0.596699 + 0.802465i \(0.296479\pi\)
\(720\) 0 0
\(721\) 20.0000 0.744839
\(722\) −3.00000 −0.111648
\(723\) 18.0000 0.669427
\(724\) 4.00000 0.148659
\(725\) 0 0
\(726\) 25.0000 0.927837
\(727\) 2.00000 0.0741759 0.0370879 − 0.999312i \(-0.488192\pi\)
0.0370879 + 0.999312i \(0.488192\pi\)
\(728\) 4.00000 0.148250
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 48.0000 1.77292 0.886460 − 0.462805i \(-0.153157\pi\)
0.886460 + 0.462805i \(0.153157\pi\)
\(734\) −2.00000 −0.0738213
\(735\) 0 0
\(736\) 1.00000 0.0368605
\(737\) 0 0
\(738\) 2.00000 0.0736210
\(739\) 12.0000 0.441427 0.220714 − 0.975339i \(-0.429161\pi\)
0.220714 + 0.975339i \(0.429161\pi\)
\(740\) 0 0
\(741\) −8.00000 −0.293887
\(742\) 4.00000 0.146845
\(743\) −52.0000 −1.90769 −0.953847 − 0.300291i \(-0.902916\pi\)
−0.953847 + 0.300291i \(0.902916\pi\)
\(744\) −8.00000 −0.293294
\(745\) 0 0
\(746\) 4.00000 0.146450
\(747\) 6.00000 0.219529
\(748\) 0 0
\(749\) 12.0000 0.438470
\(750\) 0 0
\(751\) 26.0000 0.948753 0.474377 − 0.880322i \(-0.342673\pi\)
0.474377 + 0.880322i \(0.342673\pi\)
\(752\) 0 0
\(753\) −18.0000 −0.655956
\(754\) −4.00000 −0.145671
\(755\) 0 0
\(756\) 2.00000 0.0727393
\(757\) −20.0000 −0.726912 −0.363456 − 0.931611i \(-0.618403\pi\)
−0.363456 + 0.931611i \(0.618403\pi\)
\(758\) −36.0000 −1.30758
\(759\) 6.00000 0.217786
\(760\) 0 0
\(761\) 30.0000 1.08750 0.543750 − 0.839248i \(-0.317004\pi\)
0.543750 + 0.839248i \(0.317004\pi\)
\(762\) 12.0000 0.434714
\(763\) 24.0000 0.868858
\(764\) 8.00000 0.289430
\(765\) 0 0
\(766\) 12.0000 0.433578
\(767\) −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
\(771\) 6.00000 0.216085
\(772\) 18.0000 0.647834
\(773\) 42.0000 1.51064 0.755318 − 0.655359i \(-0.227483\pi\)
0.755318 + 0.655359i \(0.227483\pi\)
\(774\) 8.00000 0.287554
\(775\) 0 0
\(776\) −2.00000 −0.0717958
\(777\) 8.00000 0.286998
\(778\) 10.0000 0.358517
\(779\) −8.00000 −0.286630
\(780\) 0 0
\(781\) −48.0000 −1.71758
\(782\) 0 0
\(783\) −2.00000 −0.0714742
\(784\) −3.00000 −0.107143
\(785\) 0 0
\(786\) 8.00000 0.285351
\(787\) −8.00000 −0.285169 −0.142585 − 0.989783i \(-0.545541\pi\)
−0.142585 + 0.989783i \(0.545541\pi\)
\(788\) 22.0000 0.783718
\(789\) −4.00000 −0.142404
\(790\) 0 0
\(791\) −32.0000 −1.13779
\(792\) 6.00000 0.213201
\(793\) 0 0
\(794\) −14.0000 −0.496841
\(795\) 0 0
\(796\) −22.0000 −0.779769
\(797\) 30.0000 1.06265 0.531327 − 0.847167i \(-0.321693\pi\)
0.531327 + 0.847167i \(0.321693\pi\)
\(798\) −8.00000 −0.283197
\(799\) 0 0
\(800\) 0 0
\(801\) −16.0000 −0.565332
\(802\) 8.00000 0.282490
\(803\) −36.0000 −1.27041
\(804\) 0 0
\(805\) 0 0
\(806\) −16.0000 −0.563576
\(807\) −30.0000 −1.05605
\(808\) 2.00000 0.0703598
\(809\) −50.0000 −1.75791 −0.878953 − 0.476908i \(-0.841757\pi\)
−0.878953 + 0.476908i \(0.841757\pi\)
\(810\) 0 0
\(811\) 36.0000 1.26413 0.632065 − 0.774915i \(-0.282207\pi\)
0.632065 + 0.774915i \(0.282207\pi\)
\(812\) −4.00000 −0.140372
\(813\) −4.00000 −0.140286
\(814\) 24.0000 0.841200
\(815\) 0 0
\(816\) 0 0
\(817\) −32.0000 −1.11954
\(818\) −10.0000 −0.349642
\(819\) 4.00000 0.139771
\(820\) 0 0
\(821\) 18.0000 0.628204 0.314102 − 0.949389i \(-0.398297\pi\)
0.314102 + 0.949389i \(0.398297\pi\)
\(822\) −20.0000 −0.697580
\(823\) 32.0000 1.11545 0.557725 − 0.830026i \(-0.311674\pi\)
0.557725 + 0.830026i \(0.311674\pi\)
\(824\) 10.0000 0.348367
\(825\) 0 0
\(826\) −8.00000 −0.278356
\(827\) −30.0000 −1.04320 −0.521601 − 0.853189i \(-0.674665\pi\)
−0.521601 + 0.853189i \(0.674665\pi\)
\(828\) 1.00000 0.0347524
\(829\) 26.0000 0.903017 0.451509 − 0.892267i \(-0.350886\pi\)
0.451509 + 0.892267i \(0.350886\pi\)
\(830\) 0 0
\(831\) −10.0000 −0.346896
\(832\) 2.00000 0.0693375
\(833\) 0 0
\(834\) −4.00000 −0.138509
\(835\) 0 0
\(836\) −24.0000 −0.830057
\(837\) −8.00000 −0.276520
\(838\) 26.0000 0.898155
\(839\) −44.0000 −1.51905 −0.759524 − 0.650479i \(-0.774568\pi\)
−0.759524 + 0.650479i \(0.774568\pi\)
\(840\) 0 0
\(841\) −25.0000 −0.862069
\(842\) 24.0000 0.827095
\(843\) −16.0000 −0.551069
\(844\) 28.0000 0.963800
\(845\) 0 0
\(846\) 0 0
\(847\) 50.0000 1.71802
\(848\) 2.00000 0.0686803
\(849\) −28.0000 −0.960958
\(850\) 0 0
\(851\) 4.00000 0.137118
\(852\) −8.00000 −0.274075
\(853\) 18.0000 0.616308 0.308154 − 0.951336i \(-0.400289\pi\)
0.308154 + 0.951336i \(0.400289\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 6.00000 0.205076
\(857\) −10.0000 −0.341593 −0.170797 − 0.985306i \(-0.554634\pi\)
−0.170797 + 0.985306i \(0.554634\pi\)
\(858\) 12.0000 0.409673
\(859\) −36.0000 −1.22830 −0.614152 − 0.789188i \(-0.710502\pi\)
−0.614152 + 0.789188i \(0.710502\pi\)
\(860\) 0 0
\(861\) 4.00000 0.136320
\(862\) −36.0000 −1.22616
\(863\) 48.0000 1.63394 0.816970 − 0.576681i \(-0.195652\pi\)
0.816970 + 0.576681i \(0.195652\pi\)
\(864\) 1.00000 0.0340207
\(865\) 0 0
\(866\) 26.0000 0.883516
\(867\) −17.0000 −0.577350
\(868\) −16.0000 −0.543075
\(869\) −84.0000 −2.84950
\(870\) 0 0
\(871\) 0 0
\(872\) 12.0000 0.406371
\(873\) −2.00000 −0.0676897
\(874\) −4.00000 −0.135302
\(875\) 0 0
\(876\) −6.00000 −0.202721
\(877\) 6.00000 0.202606 0.101303 − 0.994856i \(-0.467699\pi\)
0.101303 + 0.994856i \(0.467699\pi\)
\(878\) 0 0
\(879\) −22.0000 −0.742042
\(880\) 0 0
\(881\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(882\) −3.00000 −0.101015
\(883\) 20.0000 0.673054 0.336527 − 0.941674i \(-0.390748\pi\)
0.336527 + 0.941674i \(0.390748\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) −4.00000 −0.134383
\(887\) 40.0000 1.34307 0.671534 − 0.740973i \(-0.265636\pi\)
0.671534 + 0.740973i \(0.265636\pi\)
\(888\) 4.00000 0.134231
\(889\) 24.0000 0.804934
\(890\) 0 0
\(891\) 6.00000 0.201008
\(892\) 28.0000 0.937509
\(893\) 0 0
\(894\) 10.0000 0.334450
\(895\) 0 0
\(896\) 2.00000 0.0668153
\(897\) 2.00000 0.0667781
\(898\) 10.0000 0.333704
\(899\) 16.0000 0.533630
\(900\) 0 0
\(901\) 0 0
\(902\) 12.0000 0.399556
\(903\) 16.0000 0.532447
\(904\) −16.0000 −0.532152
\(905\) 0 0
\(906\) 20.0000 0.664455
\(907\) 44.0000 1.46100 0.730498 − 0.682915i \(-0.239288\pi\)
0.730498 + 0.682915i \(0.239288\pi\)
\(908\) −10.0000 −0.331862
\(909\) 2.00000 0.0663358
\(910\) 0 0
\(911\) −12.0000 −0.397578 −0.198789 − 0.980042i \(-0.563701\pi\)
−0.198789 + 0.980042i \(0.563701\pi\)
\(912\) −4.00000 −0.132453
\(913\) 36.0000 1.19143
\(914\) −18.0000 −0.595387
\(915\) 0 0
\(916\) 4.00000 0.132164
\(917\) 16.0000 0.528367
\(918\) 0 0
\(919\) −30.0000 −0.989609 −0.494804 − 0.869004i \(-0.664760\pi\)
−0.494804 + 0.869004i \(0.664760\pi\)
\(920\) 0 0
\(921\) 12.0000 0.395413
\(922\) −14.0000 −0.461065
\(923\) −16.0000 −0.526646
\(924\) 12.0000 0.394771
\(925\) 0 0
\(926\) −32.0000 −1.05159
\(927\) 10.0000 0.328443
\(928\) −2.00000 −0.0656532
\(929\) 2.00000 0.0656179 0.0328089 − 0.999462i \(-0.489555\pi\)
0.0328089 + 0.999462i \(0.489555\pi\)
\(930\) 0 0
\(931\) 12.0000 0.393284
\(932\) −2.00000 −0.0655122
\(933\) −32.0000 −1.04763
\(934\) −30.0000 −0.981630
\(935\) 0 0
\(936\) 2.00000 0.0653720
\(937\) −2.00000 −0.0653372 −0.0326686 − 0.999466i \(-0.510401\pi\)
−0.0326686 + 0.999466i \(0.510401\pi\)
\(938\) 0 0
\(939\) 18.0000 0.587408
\(940\) 0 0
\(941\) −30.0000 −0.977972 −0.488986 − 0.872292i \(-0.662633\pi\)
−0.488986 + 0.872292i \(0.662633\pi\)
\(942\) 4.00000 0.130327
\(943\) 2.00000 0.0651290
\(944\) −4.00000 −0.130189
\(945\) 0 0
\(946\) 48.0000 1.56061
\(947\) −16.0000 −0.519930 −0.259965 − 0.965618i \(-0.583711\pi\)
−0.259965 + 0.965618i \(0.583711\pi\)
\(948\) −14.0000 −0.454699
\(949\) −12.0000 −0.389536
\(950\) 0 0
\(951\) −30.0000 −0.972817
\(952\) 0 0
\(953\) −24.0000 −0.777436 −0.388718 − 0.921357i \(-0.627082\pi\)
−0.388718 + 0.921357i \(0.627082\pi\)
\(954\) 2.00000 0.0647524
\(955\) 0 0
\(956\) 0 0
\(957\) −12.0000 −0.387905
\(958\) −40.0000 −1.29234
\(959\) −40.0000 −1.29167
\(960\) 0 0
\(961\) 33.0000 1.06452
\(962\) 8.00000 0.257930
\(963\) 6.00000 0.193347
\(964\) 18.0000 0.579741
\(965\) 0 0
\(966\) 2.00000 0.0643489
\(967\) −32.0000 −1.02905 −0.514525 − 0.857475i \(-0.672032\pi\)
−0.514525 + 0.857475i \(0.672032\pi\)
\(968\) 25.0000 0.803530
\(969\) 0 0
\(970\) 0 0
\(971\) 50.0000 1.60458 0.802288 − 0.596937i \(-0.203616\pi\)
0.802288 + 0.596937i \(0.203616\pi\)
\(972\) 1.00000 0.0320750
\(973\) −8.00000 −0.256468
\(974\) 32.0000 1.02535
\(975\) 0 0
\(976\) 0 0
\(977\) −28.0000 −0.895799 −0.447900 − 0.894084i \(-0.647828\pi\)
−0.447900 + 0.894084i \(0.647828\pi\)
\(978\) −12.0000 −0.383718
\(979\) −96.0000 −3.06817
\(980\) 0 0
\(981\) 12.0000 0.383131
\(982\) −32.0000 −1.02116
\(983\) −28.0000 −0.893061 −0.446531 − 0.894768i \(-0.647341\pi\)
−0.446531 + 0.894768i \(0.647341\pi\)
\(984\) 2.00000 0.0637577
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) −8.00000 −0.254514
\(989\) 8.00000 0.254385
\(990\) 0 0
\(991\) 16.0000 0.508257 0.254128 − 0.967170i \(-0.418211\pi\)
0.254128 + 0.967170i \(0.418211\pi\)
\(992\) −8.00000 −0.254000
\(993\) −20.0000 −0.634681
\(994\) −16.0000 −0.507489
\(995\) 0 0
\(996\) 6.00000 0.190117
\(997\) −34.0000 −1.07679 −0.538395 − 0.842692i \(-0.680969\pi\)
−0.538395 + 0.842692i \(0.680969\pi\)
\(998\) 36.0000 1.13956
\(999\) 4.00000 0.126554
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 3450.2.a.ba.1.1 1
5.2 odd 4 3450.2.d.u.2899.2 2
5.3 odd 4 3450.2.d.u.2899.1 2
5.4 even 2 690.2.a.a.1.1 ✓ 1
15.14 odd 2 2070.2.a.q.1.1 1
20.19 odd 2 5520.2.a.y.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
690.2.a.a.1.1 ✓ 1 5.4 even 2
2070.2.a.q.1.1 1 15.14 odd 2
3450.2.a.ba.1.1 1 1.1 even 1 trivial
3450.2.d.u.2899.1 2 5.3 odd 4
3450.2.d.u.2899.2 2 5.2 odd 4
5520.2.a.y.1.1 1 20.19 odd 2