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Properties

Label 5775.2.a.ba.1.1
Level $5775$
Weight $2$
Character 5775.1
Self dual yes
Analytic conductor $46.114$
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] = [5775,2,Mod(1,5775)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(5775, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("5775.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 5775 = 3 \cdot 5^{2} \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5775.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: \(46.1136071673\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1155)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 5775.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+2.00000 q^{2} +1.00000 q^{3} +2.00000 q^{4} +2.00000 q^{6} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q+2.00000 q^{2} +1.00000 q^{3} +2.00000 q^{4} +2.00000 q^{6} -1.00000 q^{7} +1.00000 q^{9} -1.00000 q^{11} +2.00000 q^{12} +2.00000 q^{13} -2.00000 q^{14} -4.00000 q^{16} +3.00000 q^{17} +2.00000 q^{18} +1.00000 q^{19} -1.00000 q^{21} -2.00000 q^{22} +7.00000 q^{23} +4.00000 q^{26} +1.00000 q^{27} -2.00000 q^{28} +1.00000 q^{29} +8.00000 q^{31} -8.00000 q^{32} -1.00000 q^{33} +6.00000 q^{34} +2.00000 q^{36} +2.00000 q^{37} +2.00000 q^{38} +2.00000 q^{39} -8.00000 q^{41} -2.00000 q^{42} -9.00000 q^{43} -2.00000 q^{44} +14.0000 q^{46} +12.0000 q^{47} -4.00000 q^{48} +1.00000 q^{49} +3.00000 q^{51} +4.00000 q^{52} +5.00000 q^{53} +2.00000 q^{54} +1.00000 q^{57} +2.00000 q^{58} +3.00000 q^{59} +13.0000 q^{61} +16.0000 q^{62} -1.00000 q^{63} -8.00000 q^{64} -2.00000 q^{66} +14.0000 q^{67} +6.00000 q^{68} +7.00000 q^{69} +8.00000 q^{71} +8.00000 q^{73} +4.00000 q^{74} +2.00000 q^{76} +1.00000 q^{77} +4.00000 q^{78} -14.0000 q^{79} +1.00000 q^{81} -16.0000 q^{82} -11.0000 q^{83} -2.00000 q^{84} -18.0000 q^{86} +1.00000 q^{87} -9.00000 q^{89} -2.00000 q^{91} +14.0000 q^{92} +8.00000 q^{93} +24.0000 q^{94} -8.00000 q^{96} -5.00000 q^{97} +2.00000 q^{98} -1.00000 q^{99} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.00000 1.41421 0.707107 − 0.707107i \(-0.250000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(3\) 1.00000 0.577350
\(4\) 2.00000 1.00000
\(5\) 0 0
\(6\) 2.00000 0.816497
\(7\) −1.00000 −0.377964
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −1.00000 −0.301511
\(12\) 2.00000 0.577350
\(13\) 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
\(16\) −4.00000 −1.00000
\(17\) 3.00000 0.727607 0.363803 − 0.931476i \(-0.381478\pi\)
0.363803 + 0.931476i \(0.381478\pi\)
\(18\) 2.00000 0.471405
\(19\) 1.00000 0.229416 0.114708 − 0.993399i \(-0.463407\pi\)
0.114708 + 0.993399i \(0.463407\pi\)
\(20\) 0 0
\(21\) −1.00000 −0.218218
\(22\) −2.00000 −0.426401
\(23\) 7.00000 1.45960 0.729800 − 0.683660i \(-0.239613\pi\)
0.729800 + 0.683660i \(0.239613\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 4.00000 0.784465
\(27\) 1.00000 0.192450
\(28\) −2.00000 −0.377964
\(29\) 1.00000 0.185695 0.0928477 − 0.995680i \(-0.470403\pi\)
0.0928477 + 0.995680i \(0.470403\pi\)
\(30\) 0 0
\(31\) 8.00000 1.43684 0.718421 − 0.695608i \(-0.244865\pi\)
0.718421 + 0.695608i \(0.244865\pi\)
\(32\) −8.00000 −1.41421
\(33\) −1.00000 −0.174078
\(34\) 6.00000 1.02899
\(35\) 0 0
\(36\) 2.00000 0.333333
\(37\) 2.00000 0.328798 0.164399 − 0.986394i \(-0.447432\pi\)
0.164399 + 0.986394i \(0.447432\pi\)
\(38\) 2.00000 0.324443
\(39\) 2.00000 0.320256
\(40\) 0 0
\(41\) −8.00000 −1.24939 −0.624695 − 0.780869i \(-0.714777\pi\)
−0.624695 + 0.780869i \(0.714777\pi\)
\(42\) −2.00000 −0.308607
\(43\) −9.00000 −1.37249 −0.686244 − 0.727372i \(-0.740742\pi\)
−0.686244 + 0.727372i \(0.740742\pi\)
\(44\) −2.00000 −0.301511
\(45\) 0 0
\(46\) 14.0000 2.06419
\(47\) 12.0000 1.75038 0.875190 − 0.483779i \(-0.160736\pi\)
0.875190 + 0.483779i \(0.160736\pi\)
\(48\) −4.00000 −0.577350
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 3.00000 0.420084
\(52\) 4.00000 0.554700
\(53\) 5.00000 0.686803 0.343401 − 0.939189i \(-0.388421\pi\)
0.343401 + 0.939189i \(0.388421\pi\)
\(54\) 2.00000 0.272166
\(55\) 0 0
\(56\) 0 0
\(57\) 1.00000 0.132453
\(58\) 2.00000 0.262613
\(59\) 3.00000 0.390567 0.195283 − 0.980747i \(-0.437437\pi\)
0.195283 + 0.980747i \(0.437437\pi\)
\(60\) 0 0
\(61\) 13.0000 1.66448 0.832240 − 0.554416i \(-0.187058\pi\)
0.832240 + 0.554416i \(0.187058\pi\)
\(62\) 16.0000 2.03200
\(63\) −1.00000 −0.125988
\(64\) −8.00000 −1.00000
\(65\) 0 0
\(66\) −2.00000 −0.246183
\(67\) 14.0000 1.71037 0.855186 − 0.518321i \(-0.173443\pi\)
0.855186 + 0.518321i \(0.173443\pi\)
\(68\) 6.00000 0.727607
\(69\) 7.00000 0.842701
\(70\) 0 0
\(71\) 8.00000 0.949425 0.474713 − 0.880141i \(-0.342552\pi\)
0.474713 + 0.880141i \(0.342552\pi\)
\(72\) 0 0
\(73\) 8.00000 0.936329 0.468165 − 0.883641i \(-0.344915\pi\)
0.468165 + 0.883641i \(0.344915\pi\)
\(74\) 4.00000 0.464991
\(75\) 0 0
\(76\) 2.00000 0.229416
\(77\) 1.00000 0.113961
\(78\) 4.00000 0.452911
\(79\) −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
\(82\) −16.0000 −1.76690
\(83\) −11.0000 −1.20741 −0.603703 − 0.797209i \(-0.706309\pi\)
−0.603703 + 0.797209i \(0.706309\pi\)
\(84\) −2.00000 −0.218218
\(85\) 0 0
\(86\) −18.0000 −1.94099
\(87\) 1.00000 0.107211
\(88\) 0 0
\(89\) −9.00000 −0.953998 −0.476999 − 0.878904i \(-0.658275\pi\)
−0.476999 + 0.878904i \(0.658275\pi\)
\(90\) 0 0
\(91\) −2.00000 −0.209657
\(92\) 14.0000 1.45960
\(93\) 8.00000 0.829561
\(94\) 24.0000 2.47541
\(95\) 0 0
\(96\) −8.00000 −0.816497
\(97\) −5.00000 −0.507673 −0.253837 − 0.967247i \(-0.581693\pi\)
−0.253837 + 0.967247i \(0.581693\pi\)
\(98\) 2.00000 0.202031
\(99\) −1.00000 −0.100504
\(100\) 0 0
\(101\) −14.0000 −1.39305 −0.696526 − 0.717532i \(-0.745272\pi\)
−0.696526 + 0.717532i \(0.745272\pi\)
\(102\) 6.00000 0.594089
\(103\) 7.00000 0.689730 0.344865 − 0.938652i \(-0.387925\pi\)
0.344865 + 0.938652i \(0.387925\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 10.0000 0.971286
\(107\) 8.00000 0.773389 0.386695 − 0.922208i \(-0.373617\pi\)
0.386695 + 0.922208i \(0.373617\pi\)
\(108\) 2.00000 0.192450
\(109\) −16.0000 −1.53252 −0.766261 − 0.642529i \(-0.777885\pi\)
−0.766261 + 0.642529i \(0.777885\pi\)
\(110\) 0 0
\(111\) 2.00000 0.189832
\(112\) 4.00000 0.377964
\(113\) 9.00000 0.846649 0.423324 − 0.905978i \(-0.360863\pi\)
0.423324 + 0.905978i \(0.360863\pi\)
\(114\) 2.00000 0.187317
\(115\) 0 0
\(116\) 2.00000 0.185695
\(117\) 2.00000 0.184900
\(118\) 6.00000 0.552345
\(119\) −3.00000 −0.275010
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 26.0000 2.35393
\(123\) −8.00000 −0.721336
\(124\) 16.0000 1.43684
\(125\) 0 0
\(126\) −2.00000 −0.178174
\(127\) −5.00000 −0.443678 −0.221839 − 0.975083i \(-0.571206\pi\)
−0.221839 + 0.975083i \(0.571206\pi\)
\(128\) 0 0
\(129\) −9.00000 −0.792406
\(130\) 0 0
\(131\) 2.00000 0.174741 0.0873704 − 0.996176i \(-0.472154\pi\)
0.0873704 + 0.996176i \(0.472154\pi\)
\(132\) −2.00000 −0.174078
\(133\) −1.00000 −0.0867110
\(134\) 28.0000 2.41883
\(135\) 0 0
\(136\) 0 0
\(137\) 10.0000 0.854358 0.427179 − 0.904167i \(-0.359507\pi\)
0.427179 + 0.904167i \(0.359507\pi\)
\(138\) 14.0000 1.19176
\(139\) 8.00000 0.678551 0.339276 − 0.940687i \(-0.389818\pi\)
0.339276 + 0.940687i \(0.389818\pi\)
\(140\) 0 0
\(141\) 12.0000 1.01058
\(142\) 16.0000 1.34269
\(143\) −2.00000 −0.167248
\(144\) −4.00000 −0.333333
\(145\) 0 0
\(146\) 16.0000 1.32417
\(147\) 1.00000 0.0824786
\(148\) 4.00000 0.328798
\(149\) −6.00000 −0.491539 −0.245770 − 0.969328i \(-0.579041\pi\)
−0.245770 + 0.969328i \(0.579041\pi\)
\(150\) 0 0
\(151\) 14.0000 1.13930 0.569652 − 0.821886i \(-0.307078\pi\)
0.569652 + 0.821886i \(0.307078\pi\)
\(152\) 0 0
\(153\) 3.00000 0.242536
\(154\) 2.00000 0.161165
\(155\) 0 0
\(156\) 4.00000 0.320256
\(157\) 23.0000 1.83560 0.917800 − 0.397043i \(-0.129964\pi\)
0.917800 + 0.397043i \(0.129964\pi\)
\(158\) −28.0000 −2.22756
\(159\) 5.00000 0.396526
\(160\) 0 0
\(161\) −7.00000 −0.551677
\(162\) 2.00000 0.157135
\(163\) 14.0000 1.09656 0.548282 − 0.836293i \(-0.315282\pi\)
0.548282 + 0.836293i \(0.315282\pi\)
\(164\) −16.0000 −1.24939
\(165\) 0 0
\(166\) −22.0000 −1.70753
\(167\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(168\) 0 0
\(169\) −9.00000 −0.692308
\(170\) 0 0
\(171\) 1.00000 0.0764719
\(172\) −18.0000 −1.37249
\(173\) −2.00000 −0.152057 −0.0760286 − 0.997106i \(-0.524224\pi\)
−0.0760286 + 0.997106i \(0.524224\pi\)
\(174\) 2.00000 0.151620
\(175\) 0 0
\(176\) 4.00000 0.301511
\(177\) 3.00000 0.225494
\(178\) −18.0000 −1.34916
\(179\) −4.00000 −0.298974 −0.149487 − 0.988764i \(-0.547762\pi\)
−0.149487 + 0.988764i \(0.547762\pi\)
\(180\) 0 0
\(181\) 2.00000 0.148659 0.0743294 − 0.997234i \(-0.476318\pi\)
0.0743294 + 0.997234i \(0.476318\pi\)
\(182\) −4.00000 −0.296500
\(183\) 13.0000 0.960988
\(184\) 0 0
\(185\) 0 0
\(186\) 16.0000 1.17318
\(187\) −3.00000 −0.219382
\(188\) 24.0000 1.75038
\(189\) −1.00000 −0.0727393
\(190\) 0 0
\(191\) 4.00000 0.289430 0.144715 − 0.989473i \(-0.453773\pi\)
0.144715 + 0.989473i \(0.453773\pi\)
\(192\) −8.00000 −0.577350
\(193\) −26.0000 −1.87152 −0.935760 − 0.352636i \(-0.885285\pi\)
−0.935760 + 0.352636i \(0.885285\pi\)
\(194\) −10.0000 −0.717958
\(195\) 0 0
\(196\) 2.00000 0.142857
\(197\) 16.0000 1.13995 0.569976 − 0.821661i \(-0.306952\pi\)
0.569976 + 0.821661i \(0.306952\pi\)
\(198\) −2.00000 −0.142134
\(199\) −20.0000 −1.41776 −0.708881 − 0.705328i \(-0.750800\pi\)
−0.708881 + 0.705328i \(0.750800\pi\)
\(200\) 0 0
\(201\) 14.0000 0.987484
\(202\) −28.0000 −1.97007
\(203\) −1.00000 −0.0701862
\(204\) 6.00000 0.420084
\(205\) 0 0
\(206\) 14.0000 0.975426
\(207\) 7.00000 0.486534
\(208\) −8.00000 −0.554700
\(209\) −1.00000 −0.0691714
\(210\) 0 0
\(211\) −22.0000 −1.51454 −0.757271 − 0.653101i \(-0.773468\pi\)
−0.757271 + 0.653101i \(0.773468\pi\)
\(212\) 10.0000 0.686803
\(213\) 8.00000 0.548151
\(214\) 16.0000 1.09374
\(215\) 0 0
\(216\) 0 0
\(217\) −8.00000 −0.543075
\(218\) −32.0000 −2.16731
\(219\) 8.00000 0.540590
\(220\) 0 0
\(221\) 6.00000 0.403604
\(222\) 4.00000 0.268462
\(223\) −15.0000 −1.00447 −0.502237 − 0.864730i \(-0.667490\pi\)
−0.502237 + 0.864730i \(0.667490\pi\)
\(224\) 8.00000 0.534522
\(225\) 0 0
\(226\) 18.0000 1.19734
\(227\) −11.0000 −0.730096 −0.365048 − 0.930989i \(-0.618947\pi\)
−0.365048 + 0.930989i \(0.618947\pi\)
\(228\) 2.00000 0.132453
\(229\) −4.00000 −0.264327 −0.132164 − 0.991228i \(-0.542192\pi\)
−0.132164 + 0.991228i \(0.542192\pi\)
\(230\) 0 0
\(231\) 1.00000 0.0657952
\(232\) 0 0
\(233\) −10.0000 −0.655122 −0.327561 − 0.944830i \(-0.606227\pi\)
−0.327561 + 0.944830i \(0.606227\pi\)
\(234\) 4.00000 0.261488
\(235\) 0 0
\(236\) 6.00000 0.390567
\(237\) −14.0000 −0.909398
\(238\) −6.00000 −0.388922
\(239\) 15.0000 0.970269 0.485135 − 0.874439i \(-0.338771\pi\)
0.485135 + 0.874439i \(0.338771\pi\)
\(240\) 0 0
\(241\) 2.00000 0.128831 0.0644157 − 0.997923i \(-0.479482\pi\)
0.0644157 + 0.997923i \(0.479482\pi\)
\(242\) 2.00000 0.128565
\(243\) 1.00000 0.0641500
\(244\) 26.0000 1.66448
\(245\) 0 0
\(246\) −16.0000 −1.02012
\(247\) 2.00000 0.127257
\(248\) 0 0
\(249\) −11.0000 −0.697097
\(250\) 0 0
\(251\) −12.0000 −0.757433 −0.378717 − 0.925513i \(-0.623635\pi\)
−0.378717 + 0.925513i \(0.623635\pi\)
\(252\) −2.00000 −0.125988
\(253\) −7.00000 −0.440086
\(254\) −10.0000 −0.627456
\(255\) 0 0
\(256\) 16.0000 1.00000
\(257\) 14.0000 0.873296 0.436648 − 0.899632i \(-0.356166\pi\)
0.436648 + 0.899632i \(0.356166\pi\)
\(258\) −18.0000 −1.12063
\(259\) −2.00000 −0.124274
\(260\) 0 0
\(261\) 1.00000 0.0618984
\(262\) 4.00000 0.247121
\(263\) −12.0000 −0.739952 −0.369976 − 0.929041i \(-0.620634\pi\)
−0.369976 + 0.929041i \(0.620634\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) −2.00000 −0.122628
\(267\) −9.00000 −0.550791
\(268\) 28.0000 1.71037
\(269\) −17.0000 −1.03651 −0.518254 − 0.855227i \(-0.673418\pi\)
−0.518254 + 0.855227i \(0.673418\pi\)
\(270\) 0 0
\(271\) 7.00000 0.425220 0.212610 − 0.977137i \(-0.431804\pi\)
0.212610 + 0.977137i \(0.431804\pi\)
\(272\) −12.0000 −0.727607
\(273\) −2.00000 −0.121046
\(274\) 20.0000 1.20824
\(275\) 0 0
\(276\) 14.0000 0.842701
\(277\) 22.0000 1.32185 0.660926 − 0.750451i \(-0.270164\pi\)
0.660926 + 0.750451i \(0.270164\pi\)
\(278\) 16.0000 0.959616
\(279\) 8.00000 0.478947
\(280\) 0 0
\(281\) 18.0000 1.07379 0.536895 − 0.843649i \(-0.319597\pi\)
0.536895 + 0.843649i \(0.319597\pi\)
\(282\) 24.0000 1.42918
\(283\) −14.0000 −0.832214 −0.416107 − 0.909316i \(-0.636606\pi\)
−0.416107 + 0.909316i \(0.636606\pi\)
\(284\) 16.0000 0.949425
\(285\) 0 0
\(286\) −4.00000 −0.236525
\(287\) 8.00000 0.472225
\(288\) −8.00000 −0.471405
\(289\) −8.00000 −0.470588
\(290\) 0 0
\(291\) −5.00000 −0.293105
\(292\) 16.0000 0.936329
\(293\) −9.00000 −0.525786 −0.262893 − 0.964825i \(-0.584677\pi\)
−0.262893 + 0.964825i \(0.584677\pi\)
\(294\) 2.00000 0.116642
\(295\) 0 0
\(296\) 0 0
\(297\) −1.00000 −0.0580259
\(298\) −12.0000 −0.695141
\(299\) 14.0000 0.809641
\(300\) 0 0
\(301\) 9.00000 0.518751
\(302\) 28.0000 1.61122
\(303\) −14.0000 −0.804279
\(304\) −4.00000 −0.229416
\(305\) 0 0
\(306\) 6.00000 0.342997
\(307\) −30.0000 −1.71219 −0.856095 − 0.516818i \(-0.827116\pi\)
−0.856095 + 0.516818i \(0.827116\pi\)
\(308\) 2.00000 0.113961
\(309\) 7.00000 0.398216
\(310\) 0 0
\(311\) −4.00000 −0.226819 −0.113410 − 0.993548i \(-0.536177\pi\)
−0.113410 + 0.993548i \(0.536177\pi\)
\(312\) 0 0
\(313\) −15.0000 −0.847850 −0.423925 − 0.905697i \(-0.639348\pi\)
−0.423925 + 0.905697i \(0.639348\pi\)
\(314\) 46.0000 2.59593
\(315\) 0 0
\(316\) −28.0000 −1.57512
\(317\) −30.0000 −1.68497 −0.842484 − 0.538721i \(-0.818908\pi\)
−0.842484 + 0.538721i \(0.818908\pi\)
\(318\) 10.0000 0.560772
\(319\) −1.00000 −0.0559893
\(320\) 0 0
\(321\) 8.00000 0.446516
\(322\) −14.0000 −0.780189
\(323\) 3.00000 0.166924
\(324\) 2.00000 0.111111
\(325\) 0 0
\(326\) 28.0000 1.55078
\(327\) −16.0000 −0.884802
\(328\) 0 0
\(329\) −12.0000 −0.661581
\(330\) 0 0
\(331\) −27.0000 −1.48405 −0.742027 − 0.670370i \(-0.766135\pi\)
−0.742027 + 0.670370i \(0.766135\pi\)
\(332\) −22.0000 −1.20741
\(333\) 2.00000 0.109599
\(334\) 0 0
\(335\) 0 0
\(336\) 4.00000 0.218218
\(337\) 23.0000 1.25289 0.626445 − 0.779466i \(-0.284509\pi\)
0.626445 + 0.779466i \(0.284509\pi\)
\(338\) −18.0000 −0.979071
\(339\) 9.00000 0.488813
\(340\) 0 0
\(341\) −8.00000 −0.433224
\(342\) 2.00000 0.108148
\(343\) −1.00000 −0.0539949
\(344\) 0 0
\(345\) 0 0
\(346\) −4.00000 −0.215041
\(347\) 28.0000 1.50312 0.751559 − 0.659665i \(-0.229302\pi\)
0.751559 + 0.659665i \(0.229302\pi\)
\(348\) 2.00000 0.107211
\(349\) −27.0000 −1.44528 −0.722638 − 0.691226i \(-0.757071\pi\)
−0.722638 + 0.691226i \(0.757071\pi\)
\(350\) 0 0
\(351\) 2.00000 0.106752
\(352\) 8.00000 0.426401
\(353\) −30.0000 −1.59674 −0.798369 − 0.602168i \(-0.794304\pi\)
−0.798369 + 0.602168i \(0.794304\pi\)
\(354\) 6.00000 0.318896
\(355\) 0 0
\(356\) −18.0000 −0.953998
\(357\) −3.00000 −0.158777
\(358\) −8.00000 −0.422813
\(359\) 3.00000 0.158334 0.0791670 − 0.996861i \(-0.474774\pi\)
0.0791670 + 0.996861i \(0.474774\pi\)
\(360\) 0 0
\(361\) −18.0000 −0.947368
\(362\) 4.00000 0.210235
\(363\) 1.00000 0.0524864
\(364\) −4.00000 −0.209657
\(365\) 0 0
\(366\) 26.0000 1.35904
\(367\) −25.0000 −1.30499 −0.652495 − 0.757793i \(-0.726278\pi\)
−0.652495 + 0.757793i \(0.726278\pi\)
\(368\) −28.0000 −1.45960
\(369\) −8.00000 −0.416463
\(370\) 0 0
\(371\) −5.00000 −0.259587
\(372\) 16.0000 0.829561
\(373\) 15.0000 0.776671 0.388335 − 0.921518i \(-0.373050\pi\)
0.388335 + 0.921518i \(0.373050\pi\)
\(374\) −6.00000 −0.310253
\(375\) 0 0
\(376\) 0 0
\(377\) 2.00000 0.103005
\(378\) −2.00000 −0.102869
\(379\) 13.0000 0.667765 0.333883 − 0.942615i \(-0.391641\pi\)
0.333883 + 0.942615i \(0.391641\pi\)
\(380\) 0 0
\(381\) −5.00000 −0.256158
\(382\) 8.00000 0.409316
\(383\) 14.0000 0.715367 0.357683 − 0.933843i \(-0.383567\pi\)
0.357683 + 0.933843i \(0.383567\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −52.0000 −2.64673
\(387\) −9.00000 −0.457496
\(388\) −10.0000 −0.507673
\(389\) −2.00000 −0.101404 −0.0507020 − 0.998714i \(-0.516146\pi\)
−0.0507020 + 0.998714i \(0.516146\pi\)
\(390\) 0 0
\(391\) 21.0000 1.06202
\(392\) 0 0
\(393\) 2.00000 0.100887
\(394\) 32.0000 1.61214
\(395\) 0 0
\(396\) −2.00000 −0.100504
\(397\) 30.0000 1.50566 0.752828 − 0.658217i \(-0.228689\pi\)
0.752828 + 0.658217i \(0.228689\pi\)
\(398\) −40.0000 −2.00502
\(399\) −1.00000 −0.0500626
\(400\) 0 0
\(401\) −12.0000 −0.599251 −0.299626 − 0.954057i \(-0.596862\pi\)
−0.299626 + 0.954057i \(0.596862\pi\)
\(402\) 28.0000 1.39651
\(403\) 16.0000 0.797017
\(404\) −28.0000 −1.39305
\(405\) 0 0
\(406\) −2.00000 −0.0992583
\(407\) −2.00000 −0.0991363
\(408\) 0 0
\(409\) 26.0000 1.28562 0.642809 − 0.766027i \(-0.277769\pi\)
0.642809 + 0.766027i \(0.277769\pi\)
\(410\) 0 0
\(411\) 10.0000 0.493264
\(412\) 14.0000 0.689730
\(413\) −3.00000 −0.147620
\(414\) 14.0000 0.688062
\(415\) 0 0
\(416\) −16.0000 −0.784465
\(417\) 8.00000 0.391762
\(418\) −2.00000 −0.0978232
\(419\) 1.00000 0.0488532 0.0244266 − 0.999702i \(-0.492224\pi\)
0.0244266 + 0.999702i \(0.492224\pi\)
\(420\) 0 0
\(421\) 9.00000 0.438633 0.219317 − 0.975654i \(-0.429617\pi\)
0.219317 + 0.975654i \(0.429617\pi\)
\(422\) −44.0000 −2.14189
\(423\) 12.0000 0.583460
\(424\) 0 0
\(425\) 0 0
\(426\) 16.0000 0.775203
\(427\) −13.0000 −0.629114
\(428\) 16.0000 0.773389
\(429\) −2.00000 −0.0965609
\(430\) 0 0
\(431\) 16.0000 0.770693 0.385346 − 0.922772i \(-0.374082\pi\)
0.385346 + 0.922772i \(0.374082\pi\)
\(432\) −4.00000 −0.192450
\(433\) 38.0000 1.82616 0.913082 − 0.407777i \(-0.133696\pi\)
0.913082 + 0.407777i \(0.133696\pi\)
\(434\) −16.0000 −0.768025
\(435\) 0 0
\(436\) −32.0000 −1.53252
\(437\) 7.00000 0.334855
\(438\) 16.0000 0.764510
\(439\) 1.00000 0.0477274 0.0238637 − 0.999715i \(-0.492403\pi\)
0.0238637 + 0.999715i \(0.492403\pi\)
\(440\) 0 0
\(441\) 1.00000 0.0476190
\(442\) 12.0000 0.570782
\(443\) −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
\(446\) −30.0000 −1.42054
\(447\) −6.00000 −0.283790
\(448\) 8.00000 0.377964
\(449\) −12.0000 −0.566315 −0.283158 − 0.959073i \(-0.591382\pi\)
−0.283158 + 0.959073i \(0.591382\pi\)
\(450\) 0 0
\(451\) 8.00000 0.376705
\(452\) 18.0000 0.846649
\(453\) 14.0000 0.657777
\(454\) −22.0000 −1.03251
\(455\) 0 0
\(456\) 0 0
\(457\) 3.00000 0.140334 0.0701670 − 0.997535i \(-0.477647\pi\)
0.0701670 + 0.997535i \(0.477647\pi\)
\(458\) −8.00000 −0.373815
\(459\) 3.00000 0.140028
\(460\) 0 0
\(461\) 6.00000 0.279448 0.139724 − 0.990190i \(-0.455378\pi\)
0.139724 + 0.990190i \(0.455378\pi\)
\(462\) 2.00000 0.0930484
\(463\) 2.00000 0.0929479 0.0464739 − 0.998920i \(-0.485202\pi\)
0.0464739 + 0.998920i \(0.485202\pi\)
\(464\) −4.00000 −0.185695
\(465\) 0 0
\(466\) −20.0000 −0.926482
\(467\) −32.0000 −1.48078 −0.740392 − 0.672176i \(-0.765360\pi\)
−0.740392 + 0.672176i \(0.765360\pi\)
\(468\) 4.00000 0.184900
\(469\) −14.0000 −0.646460
\(470\) 0 0
\(471\) 23.0000 1.05978
\(472\) 0 0
\(473\) 9.00000 0.413820
\(474\) −28.0000 −1.28608
\(475\) 0 0
\(476\) −6.00000 −0.275010
\(477\) 5.00000 0.228934
\(478\) 30.0000 1.37217
\(479\) 10.0000 0.456912 0.228456 − 0.973554i \(-0.426632\pi\)
0.228456 + 0.973554i \(0.426632\pi\)
\(480\) 0 0
\(481\) 4.00000 0.182384
\(482\) 4.00000 0.182195
\(483\) −7.00000 −0.318511
\(484\) 2.00000 0.0909091
\(485\) 0 0
\(486\) 2.00000 0.0907218
\(487\) −34.0000 −1.54069 −0.770344 − 0.637629i \(-0.779915\pi\)
−0.770344 + 0.637629i \(0.779915\pi\)
\(488\) 0 0
\(489\) 14.0000 0.633102
\(490\) 0 0
\(491\) −15.0000 −0.676941 −0.338470 − 0.940977i \(-0.609909\pi\)
−0.338470 + 0.940977i \(0.609909\pi\)
\(492\) −16.0000 −0.721336
\(493\) 3.00000 0.135113
\(494\) 4.00000 0.179969
\(495\) 0 0
\(496\) −32.0000 −1.43684
\(497\) −8.00000 −0.358849
\(498\) −22.0000 −0.985844
\(499\) −41.0000 −1.83541 −0.917706 − 0.397260i \(-0.869961\pi\)
−0.917706 + 0.397260i \(0.869961\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) −24.0000 −1.07117
\(503\) −25.0000 −1.11469 −0.557347 − 0.830279i \(-0.688181\pi\)
−0.557347 + 0.830279i \(0.688181\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) −14.0000 −0.622376
\(507\) −9.00000 −0.399704
\(508\) −10.0000 −0.443678
\(509\) −3.00000 −0.132973 −0.0664863 − 0.997787i \(-0.521179\pi\)
−0.0664863 + 0.997787i \(0.521179\pi\)
\(510\) 0 0
\(511\) −8.00000 −0.353899
\(512\) 32.0000 1.41421
\(513\) 1.00000 0.0441511
\(514\) 28.0000 1.23503
\(515\) 0 0
\(516\) −18.0000 −0.792406
\(517\) −12.0000 −0.527759
\(518\) −4.00000 −0.175750
\(519\) −2.00000 −0.0877903
\(520\) 0 0
\(521\) 1.00000 0.0438108 0.0219054 − 0.999760i \(-0.493027\pi\)
0.0219054 + 0.999760i \(0.493027\pi\)
\(522\) 2.00000 0.0875376
\(523\) −8.00000 −0.349816 −0.174908 − 0.984585i \(-0.555963\pi\)
−0.174908 + 0.984585i \(0.555963\pi\)
\(524\) 4.00000 0.174741
\(525\) 0 0
\(526\) −24.0000 −1.04645
\(527\) 24.0000 1.04546
\(528\) 4.00000 0.174078
\(529\) 26.0000 1.13043
\(530\) 0 0
\(531\) 3.00000 0.130189
\(532\) −2.00000 −0.0867110
\(533\) −16.0000 −0.693037
\(534\) −18.0000 −0.778936
\(535\) 0 0
\(536\) 0 0
\(537\) −4.00000 −0.172613
\(538\) −34.0000 −1.46584
\(539\) −1.00000 −0.0430730
\(540\) 0 0
\(541\) −12.0000 −0.515920 −0.257960 − 0.966156i \(-0.583050\pi\)
−0.257960 + 0.966156i \(0.583050\pi\)
\(542\) 14.0000 0.601351
\(543\) 2.00000 0.0858282
\(544\) −24.0000 −1.02899
\(545\) 0 0
\(546\) −4.00000 −0.171184
\(547\) 25.0000 1.06892 0.534461 − 0.845193i \(-0.320514\pi\)
0.534461 + 0.845193i \(0.320514\pi\)
\(548\) 20.0000 0.854358
\(549\) 13.0000 0.554826
\(550\) 0 0
\(551\) 1.00000 0.0426014
\(552\) 0 0
\(553\) 14.0000 0.595341
\(554\) 44.0000 1.86938
\(555\) 0 0
\(556\) 16.0000 0.678551
\(557\) 42.0000 1.77960 0.889799 − 0.456354i \(-0.150845\pi\)
0.889799 + 0.456354i \(0.150845\pi\)
\(558\) 16.0000 0.677334
\(559\) −18.0000 −0.761319
\(560\) 0 0
\(561\) −3.00000 −0.126660
\(562\) 36.0000 1.51857
\(563\) −12.0000 −0.505740 −0.252870 − 0.967500i \(-0.581374\pi\)
−0.252870 + 0.967500i \(0.581374\pi\)
\(564\) 24.0000 1.01058
\(565\) 0 0
\(566\) −28.0000 −1.17693
\(567\) −1.00000 −0.0419961
\(568\) 0 0
\(569\) 37.0000 1.55112 0.775560 − 0.631273i \(-0.217467\pi\)
0.775560 + 0.631273i \(0.217467\pi\)
\(570\) 0 0
\(571\) 2.00000 0.0836974 0.0418487 − 0.999124i \(-0.486675\pi\)
0.0418487 + 0.999124i \(0.486675\pi\)
\(572\) −4.00000 −0.167248
\(573\) 4.00000 0.167102
\(574\) 16.0000 0.667827
\(575\) 0 0
\(576\) −8.00000 −0.333333
\(577\) −34.0000 −1.41544 −0.707719 − 0.706494i \(-0.750276\pi\)
−0.707719 + 0.706494i \(0.750276\pi\)
\(578\) −16.0000 −0.665512
\(579\) −26.0000 −1.08052
\(580\) 0 0
\(581\) 11.0000 0.456357
\(582\) −10.0000 −0.414513
\(583\) −5.00000 −0.207079
\(584\) 0 0
\(585\) 0 0
\(586\) −18.0000 −0.743573
\(587\) −18.0000 −0.742940 −0.371470 − 0.928445i \(-0.621146\pi\)
−0.371470 + 0.928445i \(0.621146\pi\)
\(588\) 2.00000 0.0824786
\(589\) 8.00000 0.329634
\(590\) 0 0
\(591\) 16.0000 0.658152
\(592\) −8.00000 −0.328798
\(593\) 14.0000 0.574911 0.287456 − 0.957794i \(-0.407191\pi\)
0.287456 + 0.957794i \(0.407191\pi\)
\(594\) −2.00000 −0.0820610
\(595\) 0 0
\(596\) −12.0000 −0.491539
\(597\) −20.0000 −0.818546
\(598\) 28.0000 1.14501
\(599\) −14.0000 −0.572024 −0.286012 − 0.958226i \(-0.592330\pi\)
−0.286012 + 0.958226i \(0.592330\pi\)
\(600\) 0 0
\(601\) −9.00000 −0.367118 −0.183559 − 0.983009i \(-0.558762\pi\)
−0.183559 + 0.983009i \(0.558762\pi\)
\(602\) 18.0000 0.733625
\(603\) 14.0000 0.570124
\(604\) 28.0000 1.13930
\(605\) 0 0
\(606\) −28.0000 −1.13742
\(607\) −22.0000 −0.892952 −0.446476 − 0.894795i \(-0.647321\pi\)
−0.446476 + 0.894795i \(0.647321\pi\)
\(608\) −8.00000 −0.324443
\(609\) −1.00000 −0.0405220
\(610\) 0 0
\(611\) 24.0000 0.970936
\(612\) 6.00000 0.242536
\(613\) −2.00000 −0.0807792 −0.0403896 − 0.999184i \(-0.512860\pi\)
−0.0403896 + 0.999184i \(0.512860\pi\)
\(614\) −60.0000 −2.42140
\(615\) 0 0
\(616\) 0 0
\(617\) 42.0000 1.69086 0.845428 − 0.534089i \(-0.179345\pi\)
0.845428 + 0.534089i \(0.179345\pi\)
\(618\) 14.0000 0.563163
\(619\) −6.00000 −0.241160 −0.120580 − 0.992704i \(-0.538475\pi\)
−0.120580 + 0.992704i \(0.538475\pi\)
\(620\) 0 0
\(621\) 7.00000 0.280900
\(622\) −8.00000 −0.320771
\(623\) 9.00000 0.360577
\(624\) −8.00000 −0.320256
\(625\) 0 0
\(626\) −30.0000 −1.19904
\(627\) −1.00000 −0.0399362
\(628\) 46.0000 1.83560
\(629\) 6.00000 0.239236
\(630\) 0 0
\(631\) 5.00000 0.199047 0.0995234 − 0.995035i \(-0.468268\pi\)
0.0995234 + 0.995035i \(0.468268\pi\)
\(632\) 0 0
\(633\) −22.0000 −0.874421
\(634\) −60.0000 −2.38290
\(635\) 0 0
\(636\) 10.0000 0.396526
\(637\) 2.00000 0.0792429
\(638\) −2.00000 −0.0791808
\(639\) 8.00000 0.316475
\(640\) 0 0
\(641\) 30.0000 1.18493 0.592464 − 0.805597i \(-0.298155\pi\)
0.592464 + 0.805597i \(0.298155\pi\)
\(642\) 16.0000 0.631470
\(643\) 3.00000 0.118308 0.0591542 − 0.998249i \(-0.481160\pi\)
0.0591542 + 0.998249i \(0.481160\pi\)
\(644\) −14.0000 −0.551677
\(645\) 0 0
\(646\) 6.00000 0.236067
\(647\) 12.0000 0.471769 0.235884 − 0.971781i \(-0.424201\pi\)
0.235884 + 0.971781i \(0.424201\pi\)
\(648\) 0 0
\(649\) −3.00000 −0.117760
\(650\) 0 0
\(651\) −8.00000 −0.313545
\(652\) 28.0000 1.09656
\(653\) −9.00000 −0.352197 −0.176099 − 0.984373i \(-0.556348\pi\)
−0.176099 + 0.984373i \(0.556348\pi\)
\(654\) −32.0000 −1.25130
\(655\) 0 0
\(656\) 32.0000 1.24939
\(657\) 8.00000 0.312110
\(658\) −24.0000 −0.935617
\(659\) −3.00000 −0.116863 −0.0584317 − 0.998291i \(-0.518610\pi\)
−0.0584317 + 0.998291i \(0.518610\pi\)
\(660\) 0 0
\(661\) 32.0000 1.24466 0.622328 − 0.782757i \(-0.286187\pi\)
0.622328 + 0.782757i \(0.286187\pi\)
\(662\) −54.0000 −2.09877
\(663\) 6.00000 0.233021
\(664\) 0 0
\(665\) 0 0
\(666\) 4.00000 0.154997
\(667\) 7.00000 0.271041
\(668\) 0 0
\(669\) −15.0000 −0.579934
\(670\) 0 0
\(671\) −13.0000 −0.501859
\(672\) 8.00000 0.308607
\(673\) −19.0000 −0.732396 −0.366198 − 0.930537i \(-0.619341\pi\)
−0.366198 + 0.930537i \(0.619341\pi\)
\(674\) 46.0000 1.77185
\(675\) 0 0
\(676\) −18.0000 −0.692308
\(677\) −9.00000 −0.345898 −0.172949 − 0.984931i \(-0.555330\pi\)
−0.172949 + 0.984931i \(0.555330\pi\)
\(678\) 18.0000 0.691286
\(679\) 5.00000 0.191882
\(680\) 0 0
\(681\) −11.0000 −0.421521
\(682\) −16.0000 −0.612672
\(683\) −44.0000 −1.68361 −0.841807 − 0.539779i \(-0.818508\pi\)
−0.841807 + 0.539779i \(0.818508\pi\)
\(684\) 2.00000 0.0764719
\(685\) 0 0
\(686\) −2.00000 −0.0763604
\(687\) −4.00000 −0.152610
\(688\) 36.0000 1.37249
\(689\) 10.0000 0.380970
\(690\) 0 0
\(691\) −2.00000 −0.0760836 −0.0380418 − 0.999276i \(-0.512112\pi\)
−0.0380418 + 0.999276i \(0.512112\pi\)
\(692\) −4.00000 −0.152057
\(693\) 1.00000 0.0379869
\(694\) 56.0000 2.12573
\(695\) 0 0
\(696\) 0 0
\(697\) −24.0000 −0.909065
\(698\) −54.0000 −2.04393
\(699\) −10.0000 −0.378235
\(700\) 0 0
\(701\) 43.0000 1.62409 0.812044 − 0.583597i \(-0.198355\pi\)
0.812044 + 0.583597i \(0.198355\pi\)
\(702\) 4.00000 0.150970
\(703\) 2.00000 0.0754314
\(704\) 8.00000 0.301511
\(705\) 0 0
\(706\) −60.0000 −2.25813
\(707\) 14.0000 0.526524
\(708\) 6.00000 0.225494
\(709\) 21.0000 0.788672 0.394336 − 0.918966i \(-0.370975\pi\)
0.394336 + 0.918966i \(0.370975\pi\)
\(710\) 0 0
\(711\) −14.0000 −0.525041
\(712\) 0 0
\(713\) 56.0000 2.09722
\(714\) −6.00000 −0.224544
\(715\) 0 0
\(716\) −8.00000 −0.298974
\(717\) 15.0000 0.560185
\(718\) 6.00000 0.223918
\(719\) 15.0000 0.559406 0.279703 − 0.960087i \(-0.409764\pi\)
0.279703 + 0.960087i \(0.409764\pi\)
\(720\) 0 0
\(721\) −7.00000 −0.260694
\(722\) −36.0000 −1.33978
\(723\) 2.00000 0.0743808
\(724\) 4.00000 0.148659
\(725\) 0 0
\(726\) 2.00000 0.0742270
\(727\) −31.0000 −1.14973 −0.574863 − 0.818250i \(-0.694945\pi\)
−0.574863 + 0.818250i \(0.694945\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −27.0000 −0.998631
\(732\) 26.0000 0.960988
\(733\) −38.0000 −1.40356 −0.701781 − 0.712393i \(-0.747612\pi\)
−0.701781 + 0.712393i \(0.747612\pi\)
\(734\) −50.0000 −1.84553
\(735\) 0 0
\(736\) −56.0000 −2.06419
\(737\) −14.0000 −0.515697
\(738\) −16.0000 −0.588968
\(739\) −6.00000 −0.220714 −0.110357 − 0.993892i \(-0.535199\pi\)
−0.110357 + 0.993892i \(0.535199\pi\)
\(740\) 0 0
\(741\) 2.00000 0.0734718
\(742\) −10.0000 −0.367112
\(743\) 18.0000 0.660356 0.330178 − 0.943919i \(-0.392891\pi\)
0.330178 + 0.943919i \(0.392891\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 30.0000 1.09838
\(747\) −11.0000 −0.402469
\(748\) −6.00000 −0.219382
\(749\) −8.00000 −0.292314
\(750\) 0 0
\(751\) 17.0000 0.620339 0.310169 − 0.950681i \(-0.399614\pi\)
0.310169 + 0.950681i \(0.399614\pi\)
\(752\) −48.0000 −1.75038
\(753\) −12.0000 −0.437304
\(754\) 4.00000 0.145671
\(755\) 0 0
\(756\) −2.00000 −0.0727393
\(757\) −6.00000 −0.218074 −0.109037 − 0.994038i \(-0.534777\pi\)
−0.109037 + 0.994038i \(0.534777\pi\)
\(758\) 26.0000 0.944363
\(759\) −7.00000 −0.254084
\(760\) 0 0
\(761\) −30.0000 −1.08750 −0.543750 − 0.839248i \(-0.682996\pi\)
−0.543750 + 0.839248i \(0.682996\pi\)
\(762\) −10.0000 −0.362262
\(763\) 16.0000 0.579239
\(764\) 8.00000 0.289430
\(765\) 0 0
\(766\) 28.0000 1.01168
\(767\) 6.00000 0.216647
\(768\) 16.0000 0.577350
\(769\) 9.00000 0.324548 0.162274 − 0.986746i \(-0.448117\pi\)
0.162274 + 0.986746i \(0.448117\pi\)
\(770\) 0 0
\(771\) 14.0000 0.504198
\(772\) −52.0000 −1.87152
\(773\) 42.0000 1.51064 0.755318 − 0.655359i \(-0.227483\pi\)
0.755318 + 0.655359i \(0.227483\pi\)
\(774\) −18.0000 −0.646997
\(775\) 0 0
\(776\) 0 0
\(777\) −2.00000 −0.0717496
\(778\) −4.00000 −0.143407
\(779\) −8.00000 −0.286630
\(780\) 0 0
\(781\) −8.00000 −0.286263
\(782\) 42.0000 1.50192
\(783\) 1.00000 0.0357371
\(784\) −4.00000 −0.142857
\(785\) 0 0
\(786\) 4.00000 0.142675
\(787\) −52.0000 −1.85360 −0.926800 − 0.375555i \(-0.877452\pi\)
−0.926800 + 0.375555i \(0.877452\pi\)
\(788\) 32.0000 1.13995
\(789\) −12.0000 −0.427211
\(790\) 0 0
\(791\) −9.00000 −0.320003
\(792\) 0 0
\(793\) 26.0000 0.923287
\(794\) 60.0000 2.12932
\(795\) 0 0
\(796\) −40.0000 −1.41776
\(797\) −38.0000 −1.34603 −0.673015 − 0.739629i \(-0.735001\pi\)
−0.673015 + 0.739629i \(0.735001\pi\)
\(798\) −2.00000 −0.0707992
\(799\) 36.0000 1.27359
\(800\) 0 0
\(801\) −9.00000 −0.317999
\(802\) −24.0000 −0.847469
\(803\) −8.00000 −0.282314
\(804\) 28.0000 0.987484
\(805\) 0 0
\(806\) 32.0000 1.12715
\(807\) −17.0000 −0.598428
\(808\) 0 0
\(809\) −6.00000 −0.210949 −0.105474 − 0.994422i \(-0.533636\pi\)
−0.105474 + 0.994422i \(0.533636\pi\)
\(810\) 0 0
\(811\) −28.0000 −0.983213 −0.491606 − 0.870817i \(-0.663590\pi\)
−0.491606 + 0.870817i \(0.663590\pi\)
\(812\) −2.00000 −0.0701862
\(813\) 7.00000 0.245501
\(814\) −4.00000 −0.140200
\(815\) 0 0
\(816\) −12.0000 −0.420084
\(817\) −9.00000 −0.314870
\(818\) 52.0000 1.81814
\(819\) −2.00000 −0.0698857
\(820\) 0 0
\(821\) 7.00000 0.244302 0.122151 − 0.992512i \(-0.461021\pi\)
0.122151 + 0.992512i \(0.461021\pi\)
\(822\) 20.0000 0.697580
\(823\) 50.0000 1.74289 0.871445 − 0.490493i \(-0.163183\pi\)
0.871445 + 0.490493i \(0.163183\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) −6.00000 −0.208767
\(827\) 20.0000 0.695468 0.347734 − 0.937593i \(-0.386951\pi\)
0.347734 + 0.937593i \(0.386951\pi\)
\(828\) 14.0000 0.486534
\(829\) 38.0000 1.31979 0.659897 − 0.751356i \(-0.270600\pi\)
0.659897 + 0.751356i \(0.270600\pi\)
\(830\) 0 0
\(831\) 22.0000 0.763172
\(832\) −16.0000 −0.554700
\(833\) 3.00000 0.103944
\(834\) 16.0000 0.554035
\(835\) 0 0
\(836\) −2.00000 −0.0691714
\(837\) 8.00000 0.276520
\(838\) 2.00000 0.0690889
\(839\) −9.00000 −0.310715 −0.155357 − 0.987858i \(-0.549653\pi\)
−0.155357 + 0.987858i \(0.549653\pi\)
\(840\) 0 0
\(841\) −28.0000 −0.965517
\(842\) 18.0000 0.620321
\(843\) 18.0000 0.619953
\(844\) −44.0000 −1.51454
\(845\) 0 0
\(846\) 24.0000 0.825137
\(847\) −1.00000 −0.0343604
\(848\) −20.0000 −0.686803
\(849\) −14.0000 −0.480479
\(850\) 0 0
\(851\) 14.0000 0.479914
\(852\) 16.0000 0.548151
\(853\) 16.0000 0.547830 0.273915 − 0.961754i \(-0.411681\pi\)
0.273915 + 0.961754i \(0.411681\pi\)
\(854\) −26.0000 −0.889702
\(855\) 0 0
\(856\) 0 0
\(857\) −22.0000 −0.751506 −0.375753 − 0.926720i \(-0.622616\pi\)
−0.375753 + 0.926720i \(0.622616\pi\)
\(858\) −4.00000 −0.136558
\(859\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(860\) 0 0
\(861\) 8.00000 0.272639
\(862\) 32.0000 1.08992
\(863\) −1.00000 −0.0340404 −0.0170202 − 0.999855i \(-0.505418\pi\)
−0.0170202 + 0.999855i \(0.505418\pi\)
\(864\) −8.00000 −0.272166
\(865\) 0 0
\(866\) 76.0000 2.58259
\(867\) −8.00000 −0.271694
\(868\) −16.0000 −0.543075
\(869\) 14.0000 0.474917
\(870\) 0 0
\(871\) 28.0000 0.948744
\(872\) 0 0
\(873\) −5.00000 −0.169224
\(874\) 14.0000 0.473557
\(875\) 0 0
\(876\) 16.0000 0.540590
\(877\) −3.00000 −0.101303 −0.0506514 − 0.998716i \(-0.516130\pi\)
−0.0506514 + 0.998716i \(0.516130\pi\)
\(878\) 2.00000 0.0674967
\(879\) −9.00000 −0.303562
\(880\) 0 0
\(881\) 47.0000 1.58347 0.791735 − 0.610865i \(-0.209178\pi\)
0.791735 + 0.610865i \(0.209178\pi\)
\(882\) 2.00000 0.0673435
\(883\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(884\) 12.0000 0.403604
\(885\) 0 0
\(886\) −8.00000 −0.268765
\(887\) −55.0000 −1.84672 −0.923360 − 0.383936i \(-0.874568\pi\)
−0.923360 + 0.383936i \(0.874568\pi\)
\(888\) 0 0
\(889\) 5.00000 0.167695
\(890\) 0 0
\(891\) −1.00000 −0.0335013
\(892\) −30.0000 −1.00447
\(893\) 12.0000 0.401565
\(894\) −12.0000 −0.401340
\(895\) 0 0
\(896\) 0 0
\(897\) 14.0000 0.467446
\(898\) −24.0000 −0.800890
\(899\) 8.00000 0.266815
\(900\) 0 0
\(901\) 15.0000 0.499722
\(902\) 16.0000 0.532742
\(903\) 9.00000 0.299501
\(904\) 0 0
\(905\) 0 0
\(906\) 28.0000 0.930238
\(907\) 6.00000 0.199227 0.0996134 − 0.995026i \(-0.468239\pi\)
0.0996134 + 0.995026i \(0.468239\pi\)
\(908\) −22.0000 −0.730096
\(909\) −14.0000 −0.464351
\(910\) 0 0
\(911\) −36.0000 −1.19273 −0.596367 − 0.802712i \(-0.703390\pi\)
−0.596367 + 0.802712i \(0.703390\pi\)
\(912\) −4.00000 −0.132453
\(913\) 11.0000 0.364047
\(914\) 6.00000 0.198462
\(915\) 0 0
\(916\) −8.00000 −0.264327
\(917\) −2.00000 −0.0660458
\(918\) 6.00000 0.198030
\(919\) 26.0000 0.857661 0.428830 − 0.903385i \(-0.358926\pi\)
0.428830 + 0.903385i \(0.358926\pi\)
\(920\) 0 0
\(921\) −30.0000 −0.988534
\(922\) 12.0000 0.395199
\(923\) 16.0000 0.526646
\(924\) 2.00000 0.0657952
\(925\) 0 0
\(926\) 4.00000 0.131448
\(927\) 7.00000 0.229910
\(928\) −8.00000 −0.262613
\(929\) 54.0000 1.77168 0.885841 − 0.463988i \(-0.153582\pi\)
0.885841 + 0.463988i \(0.153582\pi\)
\(930\) 0 0
\(931\) 1.00000 0.0327737
\(932\) −20.0000 −0.655122
\(933\) −4.00000 −0.130954
\(934\) −64.0000 −2.09414
\(935\) 0 0
\(936\) 0 0
\(937\) −52.0000 −1.69877 −0.849383 − 0.527777i \(-0.823026\pi\)
−0.849383 + 0.527777i \(0.823026\pi\)
\(938\) −28.0000 −0.914232
\(939\) −15.0000 −0.489506
\(940\) 0 0
\(941\) −34.0000 −1.10837 −0.554184 − 0.832394i \(-0.686970\pi\)
−0.554184 + 0.832394i \(0.686970\pi\)
\(942\) 46.0000 1.49876
\(943\) −56.0000 −1.82361
\(944\) −12.0000 −0.390567
\(945\) 0 0
\(946\) 18.0000 0.585230
\(947\) −57.0000 −1.85225 −0.926126 − 0.377215i \(-0.876882\pi\)
−0.926126 + 0.377215i \(0.876882\pi\)
\(948\) −28.0000 −0.909398
\(949\) 16.0000 0.519382
\(950\) 0 0
\(951\) −30.0000 −0.972817
\(952\) 0 0
\(953\) −14.0000 −0.453504 −0.226752 − 0.973952i \(-0.572811\pi\)
−0.226752 + 0.973952i \(0.572811\pi\)
\(954\) 10.0000 0.323762
\(955\) 0 0
\(956\) 30.0000 0.970269
\(957\) −1.00000 −0.0323254
\(958\) 20.0000 0.646171
\(959\) −10.0000 −0.322917
\(960\) 0 0
\(961\) 33.0000 1.06452
\(962\) 8.00000 0.257930
\(963\) 8.00000 0.257796
\(964\) 4.00000 0.128831
\(965\) 0 0
\(966\) −14.0000 −0.450443
\(967\) −5.00000 −0.160789 −0.0803946 − 0.996763i \(-0.525618\pi\)
−0.0803946 + 0.996763i \(0.525618\pi\)
\(968\) 0 0
\(969\) 3.00000 0.0963739
\(970\) 0 0
\(971\) 43.0000 1.37994 0.689968 − 0.723840i \(-0.257625\pi\)
0.689968 + 0.723840i \(0.257625\pi\)
\(972\) 2.00000 0.0641500
\(973\) −8.00000 −0.256468
\(974\) −68.0000 −2.17886
\(975\) 0 0
\(976\) −52.0000 −1.66448
\(977\) 41.0000 1.31171 0.655853 − 0.754889i \(-0.272309\pi\)
0.655853 + 0.754889i \(0.272309\pi\)
\(978\) 28.0000 0.895341
\(979\) 9.00000 0.287641
\(980\) 0 0
\(981\) −16.0000 −0.510841
\(982\) −30.0000 −0.957338
\(983\) −22.0000 −0.701691 −0.350846 − 0.936433i \(-0.614106\pi\)
−0.350846 + 0.936433i \(0.614106\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 6.00000 0.191079
\(987\) −12.0000 −0.381964
\(988\) 4.00000 0.127257
\(989\) −63.0000 −2.00328
\(990\) 0 0
\(991\) 47.0000 1.49300 0.746502 − 0.665383i \(-0.231732\pi\)
0.746502 + 0.665383i \(0.231732\pi\)
\(992\) −64.0000 −2.03200
\(993\) −27.0000 −0.856819
\(994\) −16.0000 −0.507489
\(995\) 0 0
\(996\) −22.0000 −0.697097
\(997\) 38.0000 1.20347 0.601736 − 0.798695i \(-0.294476\pi\)
0.601736 + 0.798695i \(0.294476\pi\)
\(998\) −82.0000 −2.59566
\(999\) 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 5775.2.a.ba.1.1 1
5.4 even 2 1155.2.a.a.1.1 ✓ 1
15.14 odd 2 3465.2.a.s.1.1 1
35.34 odd 2 8085.2.a.d.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1155.2.a.a.1.1 ✓ 1 5.4 even 2
3465.2.a.s.1.1 1 15.14 odd 2
5775.2.a.ba.1.1 1 1.1 even 1 trivial
8085.2.a.d.1.1 1 35.34 odd 2