Properties

Label 9702.2.a.cg.1.1
Level $9702$
Weight $2$
Character 9702.1
Self dual yes
Analytic conductor $77.471$
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] = [9702,2,Mod(1,9702)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(9702, 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("9702.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 9702 = 2 \cdot 3^{2} \cdot 7^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9702.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: \(77.4708600410\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 154)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 9702.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{4} +4.00000 q^{5} +1.00000 q^{8} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{4} +4.00000 q^{5} +1.00000 q^{8} +4.00000 q^{10} +1.00000 q^{11} -1.00000 q^{13} +1.00000 q^{16} -2.00000 q^{17} +6.00000 q^{19} +4.00000 q^{20} +1.00000 q^{22} +2.00000 q^{23} +11.0000 q^{25} -1.00000 q^{26} -1.00000 q^{29} +4.00000 q^{31} +1.00000 q^{32} -2.00000 q^{34} -2.00000 q^{37} +6.00000 q^{38} +4.00000 q^{40} +2.00000 q^{41} +4.00000 q^{43} +1.00000 q^{44} +2.00000 q^{46} -2.00000 q^{47} +11.0000 q^{50} -1.00000 q^{52} +12.0000 q^{53} +4.00000 q^{55} -1.00000 q^{58} -9.00000 q^{59} -5.00000 q^{61} +4.00000 q^{62} +1.00000 q^{64} -4.00000 q^{65} -9.00000 q^{67} -2.00000 q^{68} -4.00000 q^{71} -2.00000 q^{73} -2.00000 q^{74} +6.00000 q^{76} -15.0000 q^{79} +4.00000 q^{80} +2.00000 q^{82} +6.00000 q^{83} -8.00000 q^{85} +4.00000 q^{86} +1.00000 q^{88} -6.00000 q^{89} +2.00000 q^{92} -2.00000 q^{94} +24.0000 q^{95} -5.00000 q^{97} +O(q^{100})\)

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
\(3\) 0 0
\(4\) 1.00000 0.500000
\(5\) 4.00000 1.78885 0.894427 0.447214i \(-0.147584\pi\)
0.894427 + 0.447214i \(0.147584\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 1.00000 0.353553
\(9\) 0 0
\(10\) 4.00000 1.26491
\(11\) 1.00000 0.301511
\(12\) 0 0
\(13\) −1.00000 −0.277350 −0.138675 0.990338i \(-0.544284\pi\)
−0.138675 + 0.990338i \(0.544284\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −2.00000 −0.485071 −0.242536 0.970143i \(-0.577979\pi\)
−0.242536 + 0.970143i \(0.577979\pi\)
\(18\) 0 0
\(19\) 6.00000 1.37649 0.688247 0.725476i \(-0.258380\pi\)
0.688247 + 0.725476i \(0.258380\pi\)
\(20\) 4.00000 0.894427
\(21\) 0 0
\(22\) 1.00000 0.213201
\(23\) 2.00000 0.417029 0.208514 0.978019i \(-0.433137\pi\)
0.208514 + 0.978019i \(0.433137\pi\)
\(24\) 0 0
\(25\) 11.0000 2.20000
\(26\) −1.00000 −0.196116
\(27\) 0 0
\(28\) 0 0
\(29\) −1.00000 −0.185695 −0.0928477 0.995680i \(-0.529597\pi\)
−0.0928477 + 0.995680i \(0.529597\pi\)
\(30\) 0 0
\(31\) 4.00000 0.718421 0.359211 0.933257i \(-0.383046\pi\)
0.359211 + 0.933257i \(0.383046\pi\)
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) −2.00000 −0.342997
\(35\) 0 0
\(36\) 0 0
\(37\) −2.00000 −0.328798 −0.164399 0.986394i \(-0.552568\pi\)
−0.164399 + 0.986394i \(0.552568\pi\)
\(38\) 6.00000 0.973329
\(39\) 0 0
\(40\) 4.00000 0.632456
\(41\) 2.00000 0.312348 0.156174 0.987730i \(-0.450084\pi\)
0.156174 + 0.987730i \(0.450084\pi\)
\(42\) 0 0
\(43\) 4.00000 0.609994 0.304997 0.952353i \(-0.401344\pi\)
0.304997 + 0.952353i \(0.401344\pi\)
\(44\) 1.00000 0.150756
\(45\) 0 0
\(46\) 2.00000 0.294884
\(47\) −2.00000 −0.291730 −0.145865 0.989305i \(-0.546597\pi\)
−0.145865 + 0.989305i \(0.546597\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 11.0000 1.55563
\(51\) 0 0
\(52\) −1.00000 −0.138675
\(53\) 12.0000 1.64833 0.824163 0.566352i \(-0.191646\pi\)
0.824163 + 0.566352i \(0.191646\pi\)
\(54\) 0 0
\(55\) 4.00000 0.539360
\(56\) 0 0
\(57\) 0 0
\(58\) −1.00000 −0.131306
\(59\) −9.00000 −1.17170 −0.585850 0.810419i \(-0.699239\pi\)
−0.585850 + 0.810419i \(0.699239\pi\)
\(60\) 0 0
\(61\) −5.00000 −0.640184 −0.320092 0.947386i \(-0.603714\pi\)
−0.320092 + 0.947386i \(0.603714\pi\)
\(62\) 4.00000 0.508001
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) −4.00000 −0.496139
\(66\) 0 0
\(67\) −9.00000 −1.09952 −0.549762 0.835321i \(-0.685282\pi\)
−0.549762 + 0.835321i \(0.685282\pi\)
\(68\) −2.00000 −0.242536
\(69\) 0 0
\(70\) 0 0
\(71\) −4.00000 −0.474713 −0.237356 0.971423i \(-0.576281\pi\)
−0.237356 + 0.971423i \(0.576281\pi\)
\(72\) 0 0
\(73\) −2.00000 −0.234082 −0.117041 0.993127i \(-0.537341\pi\)
−0.117041 + 0.993127i \(0.537341\pi\)
\(74\) −2.00000 −0.232495
\(75\) 0 0
\(76\) 6.00000 0.688247
\(77\) 0 0
\(78\) 0 0
\(79\) −15.0000 −1.68763 −0.843816 0.536633i \(-0.819696\pi\)
−0.843816 + 0.536633i \(0.819696\pi\)
\(80\) 4.00000 0.447214
\(81\) 0 0
\(82\) 2.00000 0.220863
\(83\) 6.00000 0.658586 0.329293 0.944228i \(-0.393190\pi\)
0.329293 + 0.944228i \(0.393190\pi\)
\(84\) 0 0
\(85\) −8.00000 −0.867722
\(86\) 4.00000 0.431331
\(87\) 0 0
\(88\) 1.00000 0.106600
\(89\) −6.00000 −0.635999 −0.317999 0.948091i \(-0.603011\pi\)
−0.317999 + 0.948091i \(0.603011\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 2.00000 0.208514
\(93\) 0 0
\(94\) −2.00000 −0.206284
\(95\) 24.0000 2.46235
\(96\) 0 0
\(97\) −5.00000 −0.507673 −0.253837 0.967247i \(-0.581693\pi\)
−0.253837 + 0.967247i \(0.581693\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 11.0000 1.10000
\(101\) 15.0000 1.49256 0.746278 0.665635i \(-0.231839\pi\)
0.746278 + 0.665635i \(0.231839\pi\)
\(102\) 0 0
\(103\) 12.0000 1.18240 0.591198 0.806527i \(-0.298655\pi\)
0.591198 + 0.806527i \(0.298655\pi\)
\(104\) −1.00000 −0.0980581
\(105\) 0 0
\(106\) 12.0000 1.16554
\(107\) −8.00000 −0.773389 −0.386695 0.922208i \(-0.626383\pi\)
−0.386695 + 0.922208i \(0.626383\pi\)
\(108\) 0 0
\(109\) 10.0000 0.957826 0.478913 0.877862i \(-0.341031\pi\)
0.478913 + 0.877862i \(0.341031\pi\)
\(110\) 4.00000 0.381385
\(111\) 0 0
\(112\) 0 0
\(113\) −17.0000 −1.59923 −0.799613 0.600516i \(-0.794962\pi\)
−0.799613 + 0.600516i \(0.794962\pi\)
\(114\) 0 0
\(115\) 8.00000 0.746004
\(116\) −1.00000 −0.0928477
\(117\) 0 0
\(118\) −9.00000 −0.828517
\(119\) 0 0
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) −5.00000 −0.452679
\(123\) 0 0
\(124\) 4.00000 0.359211
\(125\) 24.0000 2.14663
\(126\) 0 0
\(127\) 5.00000 0.443678 0.221839 0.975083i \(-0.428794\pi\)
0.221839 + 0.975083i \(0.428794\pi\)
\(128\) 1.00000 0.0883883
\(129\) 0 0
\(130\) −4.00000 −0.350823
\(131\) 18.0000 1.57267 0.786334 0.617802i \(-0.211977\pi\)
0.786334 + 0.617802i \(0.211977\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) −9.00000 −0.777482
\(135\) 0 0
\(136\) −2.00000 −0.171499
\(137\) 9.00000 0.768922 0.384461 0.923141i \(-0.374387\pi\)
0.384461 + 0.923141i \(0.374387\pi\)
\(138\) 0 0
\(139\) 8.00000 0.678551 0.339276 0.940687i \(-0.389818\pi\)
0.339276 + 0.940687i \(0.389818\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) −4.00000 −0.335673
\(143\) −1.00000 −0.0836242
\(144\) 0 0
\(145\) −4.00000 −0.332182
\(146\) −2.00000 −0.165521
\(147\) 0 0
\(148\) −2.00000 −0.164399
\(149\) 10.0000 0.819232 0.409616 0.912258i \(-0.365663\pi\)
0.409616 + 0.912258i \(0.365663\pi\)
\(150\) 0 0
\(151\) 9.00000 0.732410 0.366205 0.930534i \(-0.380657\pi\)
0.366205 + 0.930534i \(0.380657\pi\)
\(152\) 6.00000 0.486664
\(153\) 0 0
\(154\) 0 0
\(155\) 16.0000 1.28515
\(156\) 0 0
\(157\) 4.00000 0.319235 0.159617 0.987179i \(-0.448974\pi\)
0.159617 + 0.987179i \(0.448974\pi\)
\(158\) −15.0000 −1.19334
\(159\) 0 0
\(160\) 4.00000 0.316228
\(161\) 0 0
\(162\) 0 0
\(163\) 13.0000 1.01824 0.509119 0.860696i \(-0.329971\pi\)
0.509119 + 0.860696i \(0.329971\pi\)
\(164\) 2.00000 0.156174
\(165\) 0 0
\(166\) 6.00000 0.465690
\(167\) 17.0000 1.31550 0.657750 0.753237i \(-0.271508\pi\)
0.657750 + 0.753237i \(0.271508\pi\)
\(168\) 0 0
\(169\) −12.0000 −0.923077
\(170\) −8.00000 −0.613572
\(171\) 0 0
\(172\) 4.00000 0.304997
\(173\) 5.00000 0.380143 0.190071 0.981770i \(-0.439128\pi\)
0.190071 + 0.981770i \(0.439128\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 1.00000 0.0753778
\(177\) 0 0
\(178\) −6.00000 −0.449719
\(179\) −13.0000 −0.971666 −0.485833 0.874052i \(-0.661484\pi\)
−0.485833 + 0.874052i \(0.661484\pi\)
\(180\) 0 0
\(181\) −22.0000 −1.63525 −0.817624 0.575753i \(-0.804709\pi\)
−0.817624 + 0.575753i \(0.804709\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 2.00000 0.147442
\(185\) −8.00000 −0.588172
\(186\) 0 0
\(187\) −2.00000 −0.146254
\(188\) −2.00000 −0.145865
\(189\) 0 0
\(190\) 24.0000 1.74114
\(191\) −14.0000 −1.01300 −0.506502 0.862239i \(-0.669062\pi\)
−0.506502 + 0.862239i \(0.669062\pi\)
\(192\) 0 0
\(193\) −22.0000 −1.58359 −0.791797 0.610784i \(-0.790854\pi\)
−0.791797 + 0.610784i \(0.790854\pi\)
\(194\) −5.00000 −0.358979
\(195\) 0 0
\(196\) 0 0
\(197\) 3.00000 0.213741 0.106871 0.994273i \(-0.465917\pi\)
0.106871 + 0.994273i \(0.465917\pi\)
\(198\) 0 0
\(199\) 10.0000 0.708881 0.354441 0.935079i \(-0.384671\pi\)
0.354441 + 0.935079i \(0.384671\pi\)
\(200\) 11.0000 0.777817
\(201\) 0 0
\(202\) 15.0000 1.05540
\(203\) 0 0
\(204\) 0 0
\(205\) 8.00000 0.558744
\(206\) 12.0000 0.836080
\(207\) 0 0
\(208\) −1.00000 −0.0693375
\(209\) 6.00000 0.415029
\(210\) 0 0
\(211\) −14.0000 −0.963800 −0.481900 0.876226i \(-0.660053\pi\)
−0.481900 + 0.876226i \(0.660053\pi\)
\(212\) 12.0000 0.824163
\(213\) 0 0
\(214\) −8.00000 −0.546869
\(215\) 16.0000 1.09119
\(216\) 0 0
\(217\) 0 0
\(218\) 10.0000 0.677285
\(219\) 0 0
\(220\) 4.00000 0.269680
\(221\) 2.00000 0.134535
\(222\) 0 0
\(223\) −26.0000 −1.74109 −0.870544 0.492090i \(-0.836233\pi\)
−0.870544 + 0.492090i \(0.836233\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) −17.0000 −1.13082
\(227\) −10.0000 −0.663723 −0.331862 0.943328i \(-0.607677\pi\)
−0.331862 + 0.943328i \(0.607677\pi\)
\(228\) 0 0
\(229\) −16.0000 −1.05731 −0.528655 0.848837i \(-0.677303\pi\)
−0.528655 + 0.848837i \(0.677303\pi\)
\(230\) 8.00000 0.527504
\(231\) 0 0
\(232\) −1.00000 −0.0656532
\(233\) −6.00000 −0.393073 −0.196537 0.980497i \(-0.562969\pi\)
−0.196537 + 0.980497i \(0.562969\pi\)
\(234\) 0 0
\(235\) −8.00000 −0.521862
\(236\) −9.00000 −0.585850
\(237\) 0 0
\(238\) 0 0
\(239\) −19.0000 −1.22901 −0.614504 0.788914i \(-0.710644\pi\)
−0.614504 + 0.788914i \(0.710644\pi\)
\(240\) 0 0
\(241\) 30.0000 1.93247 0.966235 0.257663i \(-0.0829523\pi\)
0.966235 + 0.257663i \(0.0829523\pi\)
\(242\) 1.00000 0.0642824
\(243\) 0 0
\(244\) −5.00000 −0.320092
\(245\) 0 0
\(246\) 0 0
\(247\) −6.00000 −0.381771
\(248\) 4.00000 0.254000
\(249\) 0 0
\(250\) 24.0000 1.51789
\(251\) −12.0000 −0.757433 −0.378717 0.925513i \(-0.623635\pi\)
−0.378717 + 0.925513i \(0.623635\pi\)
\(252\) 0 0
\(253\) 2.00000 0.125739
\(254\) 5.00000 0.313728
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 21.0000 1.30994 0.654972 0.755653i \(-0.272680\pi\)
0.654972 + 0.755653i \(0.272680\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) −4.00000 −0.248069
\(261\) 0 0
\(262\) 18.0000 1.11204
\(263\) −3.00000 −0.184988 −0.0924940 0.995713i \(-0.529484\pi\)
−0.0924940 + 0.995713i \(0.529484\pi\)
\(264\) 0 0
\(265\) 48.0000 2.94862
\(266\) 0 0
\(267\) 0 0
\(268\) −9.00000 −0.549762
\(269\) −12.0000 −0.731653 −0.365826 0.930683i \(-0.619214\pi\)
−0.365826 + 0.930683i \(0.619214\pi\)
\(270\) 0 0
\(271\) 25.0000 1.51864 0.759321 0.650716i \(-0.225531\pi\)
0.759321 + 0.650716i \(0.225531\pi\)
\(272\) −2.00000 −0.121268
\(273\) 0 0
\(274\) 9.00000 0.543710
\(275\) 11.0000 0.663325
\(276\) 0 0
\(277\) −3.00000 −0.180253 −0.0901263 0.995930i \(-0.528727\pi\)
−0.0901263 + 0.995930i \(0.528727\pi\)
\(278\) 8.00000 0.479808
\(279\) 0 0
\(280\) 0 0
\(281\) 10.0000 0.596550 0.298275 0.954480i \(-0.403589\pi\)
0.298275 + 0.954480i \(0.403589\pi\)
\(282\) 0 0
\(283\) −6.00000 −0.356663 −0.178331 0.983970i \(-0.557070\pi\)
−0.178331 + 0.983970i \(0.557070\pi\)
\(284\) −4.00000 −0.237356
\(285\) 0 0
\(286\) −1.00000 −0.0591312
\(287\) 0 0
\(288\) 0 0
\(289\) −13.0000 −0.764706
\(290\) −4.00000 −0.234888
\(291\) 0 0
\(292\) −2.00000 −0.117041
\(293\) 6.00000 0.350524 0.175262 0.984522i \(-0.443923\pi\)
0.175262 + 0.984522i \(0.443923\pi\)
\(294\) 0 0
\(295\) −36.0000 −2.09600
\(296\) −2.00000 −0.116248
\(297\) 0 0
\(298\) 10.0000 0.579284
\(299\) −2.00000 −0.115663
\(300\) 0 0
\(301\) 0 0
\(302\) 9.00000 0.517892
\(303\) 0 0
\(304\) 6.00000 0.344124
\(305\) −20.0000 −1.14520
\(306\) 0 0
\(307\) 32.0000 1.82634 0.913168 0.407583i \(-0.133628\pi\)
0.913168 + 0.407583i \(0.133628\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 16.0000 0.908739
\(311\) 28.0000 1.58773 0.793867 0.608091i \(-0.208065\pi\)
0.793867 + 0.608091i \(0.208065\pi\)
\(312\) 0 0
\(313\) 1.00000 0.0565233 0.0282617 0.999601i \(-0.491003\pi\)
0.0282617 + 0.999601i \(0.491003\pi\)
\(314\) 4.00000 0.225733
\(315\) 0 0
\(316\) −15.0000 −0.843816
\(317\) 12.0000 0.673987 0.336994 0.941507i \(-0.390590\pi\)
0.336994 + 0.941507i \(0.390590\pi\)
\(318\) 0 0
\(319\) −1.00000 −0.0559893
\(320\) 4.00000 0.223607
\(321\) 0 0
\(322\) 0 0
\(323\) −12.0000 −0.667698
\(324\) 0 0
\(325\) −11.0000 −0.610170
\(326\) 13.0000 0.720003
\(327\) 0 0
\(328\) 2.00000 0.110432
\(329\) 0 0
\(330\) 0 0
\(331\) 7.00000 0.384755 0.192377 0.981321i \(-0.438380\pi\)
0.192377 + 0.981321i \(0.438380\pi\)
\(332\) 6.00000 0.329293
\(333\) 0 0
\(334\) 17.0000 0.930199
\(335\) −36.0000 −1.96689
\(336\) 0 0
\(337\) −30.0000 −1.63420 −0.817102 0.576493i \(-0.804421\pi\)
−0.817102 + 0.576493i \(0.804421\pi\)
\(338\) −12.0000 −0.652714
\(339\) 0 0
\(340\) −8.00000 −0.433861
\(341\) 4.00000 0.216612
\(342\) 0 0
\(343\) 0 0
\(344\) 4.00000 0.215666
\(345\) 0 0
\(346\) 5.00000 0.268802
\(347\) 28.0000 1.50312 0.751559 0.659665i \(-0.229302\pi\)
0.751559 + 0.659665i \(0.229302\pi\)
\(348\) 0 0
\(349\) 2.00000 0.107058 0.0535288 0.998566i \(-0.482953\pi\)
0.0535288 + 0.998566i \(0.482953\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 1.00000 0.0533002
\(353\) 18.0000 0.958043 0.479022 0.877803i \(-0.340992\pi\)
0.479022 + 0.877803i \(0.340992\pi\)
\(354\) 0 0
\(355\) −16.0000 −0.849192
\(356\) −6.00000 −0.317999
\(357\) 0 0
\(358\) −13.0000 −0.687071
\(359\) 31.0000 1.63612 0.818059 0.575135i \(-0.195050\pi\)
0.818059 + 0.575135i \(0.195050\pi\)
\(360\) 0 0
\(361\) 17.0000 0.894737
\(362\) −22.0000 −1.15629
\(363\) 0 0
\(364\) 0 0
\(365\) −8.00000 −0.418739
\(366\) 0 0
\(367\) −14.0000 −0.730794 −0.365397 0.930852i \(-0.619067\pi\)
−0.365397 + 0.930852i \(0.619067\pi\)
\(368\) 2.00000 0.104257
\(369\) 0 0
\(370\) −8.00000 −0.415900
\(371\) 0 0
\(372\) 0 0
\(373\) −7.00000 −0.362446 −0.181223 0.983442i \(-0.558006\pi\)
−0.181223 + 0.983442i \(0.558006\pi\)
\(374\) −2.00000 −0.103418
\(375\) 0 0
\(376\) −2.00000 −0.103142
\(377\) 1.00000 0.0515026
\(378\) 0 0
\(379\) −29.0000 −1.48963 −0.744815 0.667271i \(-0.767462\pi\)
−0.744815 + 0.667271i \(0.767462\pi\)
\(380\) 24.0000 1.23117
\(381\) 0 0
\(382\) −14.0000 −0.716302
\(383\) −8.00000 −0.408781 −0.204390 0.978889i \(-0.565521\pi\)
−0.204390 + 0.978889i \(0.565521\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −22.0000 −1.11977
\(387\) 0 0
\(388\) −5.00000 −0.253837
\(389\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(390\) 0 0
\(391\) −4.00000 −0.202289
\(392\) 0 0
\(393\) 0 0
\(394\) 3.00000 0.151138
\(395\) −60.0000 −3.01893
\(396\) 0 0
\(397\) −30.0000 −1.50566 −0.752828 0.658217i \(-0.771311\pi\)
−0.752828 + 0.658217i \(0.771311\pi\)
\(398\) 10.0000 0.501255
\(399\) 0 0
\(400\) 11.0000 0.550000
\(401\) −15.0000 −0.749064 −0.374532 0.927214i \(-0.622197\pi\)
−0.374532 + 0.927214i \(0.622197\pi\)
\(402\) 0 0
\(403\) −4.00000 −0.199254
\(404\) 15.0000 0.746278
\(405\) 0 0
\(406\) 0 0
\(407\) −2.00000 −0.0991363
\(408\) 0 0
\(409\) −32.0000 −1.58230 −0.791149 0.611623i \(-0.790517\pi\)
−0.791149 + 0.611623i \(0.790517\pi\)
\(410\) 8.00000 0.395092
\(411\) 0 0
\(412\) 12.0000 0.591198
\(413\) 0 0
\(414\) 0 0
\(415\) 24.0000 1.17811
\(416\) −1.00000 −0.0490290
\(417\) 0 0
\(418\) 6.00000 0.293470
\(419\) 20.0000 0.977064 0.488532 0.872546i \(-0.337533\pi\)
0.488532 + 0.872546i \(0.337533\pi\)
\(420\) 0 0
\(421\) −20.0000 −0.974740 −0.487370 0.873195i \(-0.662044\pi\)
−0.487370 + 0.873195i \(0.662044\pi\)
\(422\) −14.0000 −0.681509
\(423\) 0 0
\(424\) 12.0000 0.582772
\(425\) −22.0000 −1.06716
\(426\) 0 0
\(427\) 0 0
\(428\) −8.00000 −0.386695
\(429\) 0 0
\(430\) 16.0000 0.771589
\(431\) −1.00000 −0.0481683 −0.0240842 0.999710i \(-0.507667\pi\)
−0.0240842 + 0.999710i \(0.507667\pi\)
\(432\) 0 0
\(433\) −10.0000 −0.480569 −0.240285 0.970702i \(-0.577241\pi\)
−0.240285 + 0.970702i \(0.577241\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 10.0000 0.478913
\(437\) 12.0000 0.574038
\(438\) 0 0
\(439\) −5.00000 −0.238637 −0.119318 0.992856i \(-0.538071\pi\)
−0.119318 + 0.992856i \(0.538071\pi\)
\(440\) 4.00000 0.190693
\(441\) 0 0
\(442\) 2.00000 0.0951303
\(443\) 12.0000 0.570137 0.285069 0.958507i \(-0.407984\pi\)
0.285069 + 0.958507i \(0.407984\pi\)
\(444\) 0 0
\(445\) −24.0000 −1.13771
\(446\) −26.0000 −1.23114
\(447\) 0 0
\(448\) 0 0
\(449\) −30.0000 −1.41579 −0.707894 0.706319i \(-0.750354\pi\)
−0.707894 + 0.706319i \(0.750354\pi\)
\(450\) 0 0
\(451\) 2.00000 0.0941763
\(452\) −17.0000 −0.799613
\(453\) 0 0
\(454\) −10.0000 −0.469323
\(455\) 0 0
\(456\) 0 0
\(457\) 2.00000 0.0935561 0.0467780 0.998905i \(-0.485105\pi\)
0.0467780 + 0.998905i \(0.485105\pi\)
\(458\) −16.0000 −0.747631
\(459\) 0 0
\(460\) 8.00000 0.373002
\(461\) 3.00000 0.139724 0.0698620 0.997557i \(-0.477744\pi\)
0.0698620 + 0.997557i \(0.477744\pi\)
\(462\) 0 0
\(463\) −14.0000 −0.650635 −0.325318 0.945605i \(-0.605471\pi\)
−0.325318 + 0.945605i \(0.605471\pi\)
\(464\) −1.00000 −0.0464238
\(465\) 0 0
\(466\) −6.00000 −0.277945
\(467\) 12.0000 0.555294 0.277647 0.960683i \(-0.410445\pi\)
0.277647 + 0.960683i \(0.410445\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) −8.00000 −0.369012
\(471\) 0 0
\(472\) −9.00000 −0.414259
\(473\) 4.00000 0.183920
\(474\) 0 0
\(475\) 66.0000 3.02829
\(476\) 0 0
\(477\) 0 0
\(478\) −19.0000 −0.869040
\(479\) 37.0000 1.69057 0.845287 0.534313i \(-0.179430\pi\)
0.845287 + 0.534313i \(0.179430\pi\)
\(480\) 0 0
\(481\) 2.00000 0.0911922
\(482\) 30.0000 1.36646
\(483\) 0 0
\(484\) 1.00000 0.0454545
\(485\) −20.0000 −0.908153
\(486\) 0 0
\(487\) −4.00000 −0.181257 −0.0906287 0.995885i \(-0.528888\pi\)
−0.0906287 + 0.995885i \(0.528888\pi\)
\(488\) −5.00000 −0.226339
\(489\) 0 0
\(490\) 0 0
\(491\) −18.0000 −0.812329 −0.406164 0.913800i \(-0.633134\pi\)
−0.406164 + 0.913800i \(0.633134\pi\)
\(492\) 0 0
\(493\) 2.00000 0.0900755
\(494\) −6.00000 −0.269953
\(495\) 0 0
\(496\) 4.00000 0.179605
\(497\) 0 0
\(498\) 0 0
\(499\) −16.0000 −0.716258 −0.358129 0.933672i \(-0.616585\pi\)
−0.358129 + 0.933672i \(0.616585\pi\)
\(500\) 24.0000 1.07331
\(501\) 0 0
\(502\) −12.0000 −0.535586
\(503\) 21.0000 0.936344 0.468172 0.883637i \(-0.344913\pi\)
0.468172 + 0.883637i \(0.344913\pi\)
\(504\) 0 0
\(505\) 60.0000 2.66996
\(506\) 2.00000 0.0889108
\(507\) 0 0
\(508\) 5.00000 0.221839
\(509\) −2.00000 −0.0886484 −0.0443242 0.999017i \(-0.514113\pi\)
−0.0443242 + 0.999017i \(0.514113\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 1.00000 0.0441942
\(513\) 0 0
\(514\) 21.0000 0.926270
\(515\) 48.0000 2.11513
\(516\) 0 0
\(517\) −2.00000 −0.0879599
\(518\) 0 0
\(519\) 0 0
\(520\) −4.00000 −0.175412
\(521\) 26.0000 1.13908 0.569540 0.821963i \(-0.307121\pi\)
0.569540 + 0.821963i \(0.307121\pi\)
\(522\) 0 0
\(523\) 20.0000 0.874539 0.437269 0.899331i \(-0.355946\pi\)
0.437269 + 0.899331i \(0.355946\pi\)
\(524\) 18.0000 0.786334
\(525\) 0 0
\(526\) −3.00000 −0.130806
\(527\) −8.00000 −0.348485
\(528\) 0 0
\(529\) −19.0000 −0.826087
\(530\) 48.0000 2.08499
\(531\) 0 0
\(532\) 0 0
\(533\) −2.00000 −0.0866296
\(534\) 0 0
\(535\) −32.0000 −1.38348
\(536\) −9.00000 −0.388741
\(537\) 0 0
\(538\) −12.0000 −0.517357
\(539\) 0 0
\(540\) 0 0
\(541\) −25.0000 −1.07483 −0.537417 0.843317i \(-0.680600\pi\)
−0.537417 + 0.843317i \(0.680600\pi\)
\(542\) 25.0000 1.07384
\(543\) 0 0
\(544\) −2.00000 −0.0857493
\(545\) 40.0000 1.71341
\(546\) 0 0
\(547\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(548\) 9.00000 0.384461
\(549\) 0 0
\(550\) 11.0000 0.469042
\(551\) −6.00000 −0.255609
\(552\) 0 0
\(553\) 0 0
\(554\) −3.00000 −0.127458
\(555\) 0 0
\(556\) 8.00000 0.339276
\(557\) −38.0000 −1.61011 −0.805056 0.593199i \(-0.797865\pi\)
−0.805056 + 0.593199i \(0.797865\pi\)
\(558\) 0 0
\(559\) −4.00000 −0.169182
\(560\) 0 0
\(561\) 0 0
\(562\) 10.0000 0.421825
\(563\) −26.0000 −1.09577 −0.547885 0.836554i \(-0.684567\pi\)
−0.547885 + 0.836554i \(0.684567\pi\)
\(564\) 0 0
\(565\) −68.0000 −2.86078
\(566\) −6.00000 −0.252199
\(567\) 0 0
\(568\) −4.00000 −0.167836
\(569\) −36.0000 −1.50920 −0.754599 0.656186i \(-0.772169\pi\)
−0.754599 + 0.656186i \(0.772169\pi\)
\(570\) 0 0
\(571\) −20.0000 −0.836974 −0.418487 0.908223i \(-0.637439\pi\)
−0.418487 + 0.908223i \(0.637439\pi\)
\(572\) −1.00000 −0.0418121
\(573\) 0 0
\(574\) 0 0
\(575\) 22.0000 0.917463
\(576\) 0 0
\(577\) −7.00000 −0.291414 −0.145707 0.989328i \(-0.546546\pi\)
−0.145707 + 0.989328i \(0.546546\pi\)
\(578\) −13.0000 −0.540729
\(579\) 0 0
\(580\) −4.00000 −0.166091
\(581\) 0 0
\(582\) 0 0
\(583\) 12.0000 0.496989
\(584\) −2.00000 −0.0827606
\(585\) 0 0
\(586\) 6.00000 0.247858
\(587\) 3.00000 0.123823 0.0619116 0.998082i \(-0.480280\pi\)
0.0619116 + 0.998082i \(0.480280\pi\)
\(588\) 0 0
\(589\) 24.0000 0.988903
\(590\) −36.0000 −1.48210
\(591\) 0 0
\(592\) −2.00000 −0.0821995
\(593\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 10.0000 0.409616
\(597\) 0 0
\(598\) −2.00000 −0.0817861
\(599\) 24.0000 0.980613 0.490307 0.871550i \(-0.336885\pi\)
0.490307 + 0.871550i \(0.336885\pi\)
\(600\) 0 0
\(601\) 26.0000 1.06056 0.530281 0.847822i \(-0.322086\pi\)
0.530281 + 0.847822i \(0.322086\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 9.00000 0.366205
\(605\) 4.00000 0.162623
\(606\) 0 0
\(607\) −16.0000 −0.649420 −0.324710 0.945814i \(-0.605267\pi\)
−0.324710 + 0.945814i \(0.605267\pi\)
\(608\) 6.00000 0.243332
\(609\) 0 0
\(610\) −20.0000 −0.809776
\(611\) 2.00000 0.0809113
\(612\) 0 0
\(613\) 22.0000 0.888572 0.444286 0.895885i \(-0.353457\pi\)
0.444286 + 0.895885i \(0.353457\pi\)
\(614\) 32.0000 1.29141
\(615\) 0 0
\(616\) 0 0
\(617\) 27.0000 1.08698 0.543490 0.839416i \(-0.317103\pi\)
0.543490 + 0.839416i \(0.317103\pi\)
\(618\) 0 0
\(619\) 20.0000 0.803868 0.401934 0.915669i \(-0.368338\pi\)
0.401934 + 0.915669i \(0.368338\pi\)
\(620\) 16.0000 0.642575
\(621\) 0 0
\(622\) 28.0000 1.12270
\(623\) 0 0
\(624\) 0 0
\(625\) 41.0000 1.64000
\(626\) 1.00000 0.0399680
\(627\) 0 0
\(628\) 4.00000 0.159617
\(629\) 4.00000 0.159490
\(630\) 0 0
\(631\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(632\) −15.0000 −0.596668
\(633\) 0 0
\(634\) 12.0000 0.476581
\(635\) 20.0000 0.793676
\(636\) 0 0
\(637\) 0 0
\(638\) −1.00000 −0.0395904
\(639\) 0 0
\(640\) 4.00000 0.158114
\(641\) −9.00000 −0.355479 −0.177739 0.984078i \(-0.556878\pi\)
−0.177739 + 0.984078i \(0.556878\pi\)
\(642\) 0 0
\(643\) 1.00000 0.0394362 0.0197181 0.999806i \(-0.493723\pi\)
0.0197181 + 0.999806i \(0.493723\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) −12.0000 −0.472134
\(647\) −24.0000 −0.943537 −0.471769 0.881722i \(-0.656384\pi\)
−0.471769 + 0.881722i \(0.656384\pi\)
\(648\) 0 0
\(649\) −9.00000 −0.353281
\(650\) −11.0000 −0.431455
\(651\) 0 0
\(652\) 13.0000 0.509119
\(653\) −22.0000 −0.860927 −0.430463 0.902608i \(-0.641650\pi\)
−0.430463 + 0.902608i \(0.641650\pi\)
\(654\) 0 0
\(655\) 72.0000 2.81327
\(656\) 2.00000 0.0780869
\(657\) 0 0
\(658\) 0 0
\(659\) 32.0000 1.24654 0.623272 0.782006i \(-0.285803\pi\)
0.623272 + 0.782006i \(0.285803\pi\)
\(660\) 0 0
\(661\) −10.0000 −0.388955 −0.194477 0.980907i \(-0.562301\pi\)
−0.194477 + 0.980907i \(0.562301\pi\)
\(662\) 7.00000 0.272063
\(663\) 0 0
\(664\) 6.00000 0.232845
\(665\) 0 0
\(666\) 0 0
\(667\) −2.00000 −0.0774403
\(668\) 17.0000 0.657750
\(669\) 0 0
\(670\) −36.0000 −1.39080
\(671\) −5.00000 −0.193023
\(672\) 0 0
\(673\) 16.0000 0.616755 0.308377 0.951264i \(-0.400214\pi\)
0.308377 + 0.951264i \(0.400214\pi\)
\(674\) −30.0000 −1.15556
\(675\) 0 0
\(676\) −12.0000 −0.461538
\(677\) −38.0000 −1.46046 −0.730229 0.683202i \(-0.760587\pi\)
−0.730229 + 0.683202i \(0.760587\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) −8.00000 −0.306786
\(681\) 0 0
\(682\) 4.00000 0.153168
\(683\) 33.0000 1.26271 0.631355 0.775494i \(-0.282499\pi\)
0.631355 + 0.775494i \(0.282499\pi\)
\(684\) 0 0
\(685\) 36.0000 1.37549
\(686\) 0 0
\(687\) 0 0
\(688\) 4.00000 0.152499
\(689\) −12.0000 −0.457164
\(690\) 0 0
\(691\) 15.0000 0.570627 0.285313 0.958434i \(-0.407902\pi\)
0.285313 + 0.958434i \(0.407902\pi\)
\(692\) 5.00000 0.190071
\(693\) 0 0
\(694\) 28.0000 1.06287
\(695\) 32.0000 1.21383
\(696\) 0 0
\(697\) −4.00000 −0.151511
\(698\) 2.00000 0.0757011
\(699\) 0 0
\(700\) 0 0
\(701\) −39.0000 −1.47301 −0.736505 0.676432i \(-0.763525\pi\)
−0.736505 + 0.676432i \(0.763525\pi\)
\(702\) 0 0
\(703\) −12.0000 −0.452589
\(704\) 1.00000 0.0376889
\(705\) 0 0
\(706\) 18.0000 0.677439
\(707\) 0 0
\(708\) 0 0
\(709\) 6.00000 0.225335 0.112667 0.993633i \(-0.464061\pi\)
0.112667 + 0.993633i \(0.464061\pi\)
\(710\) −16.0000 −0.600469
\(711\) 0 0
\(712\) −6.00000 −0.224860
\(713\) 8.00000 0.299602
\(714\) 0 0
\(715\) −4.00000 −0.149592
\(716\) −13.0000 −0.485833
\(717\) 0 0
\(718\) 31.0000 1.15691
\(719\) 26.0000 0.969636 0.484818 0.874615i \(-0.338886\pi\)
0.484818 + 0.874615i \(0.338886\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 17.0000 0.632674
\(723\) 0 0
\(724\) −22.0000 −0.817624
\(725\) −11.0000 −0.408530
\(726\) 0 0
\(727\) −34.0000 −1.26099 −0.630495 0.776193i \(-0.717148\pi\)
−0.630495 + 0.776193i \(0.717148\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) −8.00000 −0.296093
\(731\) −8.00000 −0.295891
\(732\) 0 0
\(733\) −1.00000 −0.0369358 −0.0184679 0.999829i \(-0.505879\pi\)
−0.0184679 + 0.999829i \(0.505879\pi\)
\(734\) −14.0000 −0.516749
\(735\) 0 0
\(736\) 2.00000 0.0737210
\(737\) −9.00000 −0.331519
\(738\) 0 0
\(739\) 18.0000 0.662141 0.331070 0.943606i \(-0.392590\pi\)
0.331070 + 0.943606i \(0.392590\pi\)
\(740\) −8.00000 −0.294086
\(741\) 0 0
\(742\) 0 0
\(743\) −36.0000 −1.32071 −0.660356 0.750953i \(-0.729595\pi\)
−0.660356 + 0.750953i \(0.729595\pi\)
\(744\) 0 0
\(745\) 40.0000 1.46549
\(746\) −7.00000 −0.256288
\(747\) 0 0
\(748\) −2.00000 −0.0731272
\(749\) 0 0
\(750\) 0 0
\(751\) −44.0000 −1.60558 −0.802791 0.596260i \(-0.796653\pi\)
−0.802791 + 0.596260i \(0.796653\pi\)
\(752\) −2.00000 −0.0729325
\(753\) 0 0
\(754\) 1.00000 0.0364179
\(755\) 36.0000 1.31017
\(756\) 0 0
\(757\) 26.0000 0.944986 0.472493 0.881334i \(-0.343354\pi\)
0.472493 + 0.881334i \(0.343354\pi\)
\(758\) −29.0000 −1.05333
\(759\) 0 0
\(760\) 24.0000 0.870572
\(761\) −54.0000 −1.95750 −0.978749 0.205061i \(-0.934261\pi\)
−0.978749 + 0.205061i \(0.934261\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) −14.0000 −0.506502
\(765\) 0 0
\(766\) −8.00000 −0.289052
\(767\) 9.00000 0.324971
\(768\) 0 0
\(769\) −38.0000 −1.37032 −0.685158 0.728395i \(-0.740267\pi\)
−0.685158 + 0.728395i \(0.740267\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −22.0000 −0.791797
\(773\) 24.0000 0.863220 0.431610 0.902060i \(-0.357946\pi\)
0.431610 + 0.902060i \(0.357946\pi\)
\(774\) 0 0
\(775\) 44.0000 1.58053
\(776\) −5.00000 −0.179490
\(777\) 0 0
\(778\) 0 0
\(779\) 12.0000 0.429945
\(780\) 0 0
\(781\) −4.00000 −0.143131
\(782\) −4.00000 −0.143040
\(783\) 0 0
\(784\) 0 0
\(785\) 16.0000 0.571064
\(786\) 0 0
\(787\) −22.0000 −0.784215 −0.392108 0.919919i \(-0.628254\pi\)
−0.392108 + 0.919919i \(0.628254\pi\)
\(788\) 3.00000 0.106871
\(789\) 0 0
\(790\) −60.0000 −2.13470
\(791\) 0 0
\(792\) 0 0
\(793\) 5.00000 0.177555
\(794\) −30.0000 −1.06466
\(795\) 0 0
\(796\) 10.0000 0.354441
\(797\) −38.0000 −1.34603 −0.673015 0.739629i \(-0.735001\pi\)
−0.673015 + 0.739629i \(0.735001\pi\)
\(798\) 0 0
\(799\) 4.00000 0.141510
\(800\) 11.0000 0.388909
\(801\) 0 0
\(802\) −15.0000 −0.529668
\(803\) −2.00000 −0.0705785
\(804\) 0 0
\(805\) 0 0
\(806\) −4.00000 −0.140894
\(807\) 0 0
\(808\) 15.0000 0.527698
\(809\) 48.0000 1.68759 0.843795 0.536666i \(-0.180316\pi\)
0.843795 + 0.536666i \(0.180316\pi\)
\(810\) 0 0
\(811\) −28.0000 −0.983213 −0.491606 0.870817i \(-0.663590\pi\)
−0.491606 + 0.870817i \(0.663590\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) −2.00000 −0.0701000
\(815\) 52.0000 1.82148
\(816\) 0 0
\(817\) 24.0000 0.839654
\(818\) −32.0000 −1.11885
\(819\) 0 0
\(820\) 8.00000 0.279372
\(821\) 15.0000 0.523504 0.261752 0.965135i \(-0.415700\pi\)
0.261752 + 0.965135i \(0.415700\pi\)
\(822\) 0 0
\(823\) 38.0000 1.32460 0.662298 0.749240i \(-0.269581\pi\)
0.662298 + 0.749240i \(0.269581\pi\)
\(824\) 12.0000 0.418040
\(825\) 0 0
\(826\) 0 0
\(827\) 22.0000 0.765015 0.382507 0.923952i \(-0.375061\pi\)
0.382507 + 0.923952i \(0.375061\pi\)
\(828\) 0 0
\(829\) 16.0000 0.555703 0.277851 0.960624i \(-0.410378\pi\)
0.277851 + 0.960624i \(0.410378\pi\)
\(830\) 24.0000 0.833052
\(831\) 0 0
\(832\) −1.00000 −0.0346688
\(833\) 0 0
\(834\) 0 0
\(835\) 68.0000 2.35324
\(836\) 6.00000 0.207514
\(837\) 0 0
\(838\) 20.0000 0.690889
\(839\) −54.0000 −1.86429 −0.932144 0.362089i \(-0.882064\pi\)
−0.932144 + 0.362089i \(0.882064\pi\)
\(840\) 0 0
\(841\) −28.0000 −0.965517
\(842\) −20.0000 −0.689246
\(843\) 0 0
\(844\) −14.0000 −0.481900
\(845\) −48.0000 −1.65125
\(846\) 0 0
\(847\) 0 0
\(848\) 12.0000 0.412082
\(849\) 0 0
\(850\) −22.0000 −0.754594
\(851\) −4.00000 −0.137118
\(852\) 0 0
\(853\) 50.0000 1.71197 0.855984 0.517003i \(-0.172952\pi\)
0.855984 + 0.517003i \(0.172952\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) −8.00000 −0.273434
\(857\) 4.00000 0.136637 0.0683187 0.997664i \(-0.478237\pi\)
0.0683187 + 0.997664i \(0.478237\pi\)
\(858\) 0 0
\(859\) −17.0000 −0.580033 −0.290016 0.957022i \(-0.593661\pi\)
−0.290016 + 0.957022i \(0.593661\pi\)
\(860\) 16.0000 0.545595
\(861\) 0 0
\(862\) −1.00000 −0.0340601
\(863\) −10.0000 −0.340404 −0.170202 0.985409i \(-0.554442\pi\)
−0.170202 + 0.985409i \(0.554442\pi\)
\(864\) 0 0
\(865\) 20.0000 0.680020
\(866\) −10.0000 −0.339814
\(867\) 0 0
\(868\) 0 0
\(869\) −15.0000 −0.508840
\(870\) 0 0
\(871\) 9.00000 0.304953
\(872\) 10.0000 0.338643
\(873\) 0 0
\(874\) 12.0000 0.405906
\(875\) 0 0
\(876\) 0 0
\(877\) 25.0000 0.844190 0.422095 0.906552i \(-0.361295\pi\)
0.422095 + 0.906552i \(0.361295\pi\)
\(878\) −5.00000 −0.168742
\(879\) 0 0
\(880\) 4.00000 0.134840
\(881\) −9.00000 −0.303218 −0.151609 0.988441i \(-0.548445\pi\)
−0.151609 + 0.988441i \(0.548445\pi\)
\(882\) 0 0
\(883\) 1.00000 0.0336527 0.0168263 0.999858i \(-0.494644\pi\)
0.0168263 + 0.999858i \(0.494644\pi\)
\(884\) 2.00000 0.0672673
\(885\) 0 0
\(886\) 12.0000 0.403148
\(887\) 17.0000 0.570804 0.285402 0.958408i \(-0.407873\pi\)
0.285402 + 0.958408i \(0.407873\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) −24.0000 −0.804482
\(891\) 0 0
\(892\) −26.0000 −0.870544
\(893\) −12.0000 −0.401565
\(894\) 0 0
\(895\) −52.0000 −1.73817
\(896\) 0 0
\(897\) 0 0
\(898\) −30.0000 −1.00111
\(899\) −4.00000 −0.133407
\(900\) 0 0
\(901\) −24.0000 −0.799556
\(902\) 2.00000 0.0665927
\(903\) 0 0
\(904\) −17.0000 −0.565412
\(905\) −88.0000 −2.92522
\(906\) 0 0
\(907\) 44.0000 1.46100 0.730498 0.682915i \(-0.239288\pi\)
0.730498 + 0.682915i \(0.239288\pi\)
\(908\) −10.0000 −0.331862
\(909\) 0 0
\(910\) 0 0
\(911\) 54.0000 1.78910 0.894550 0.446968i \(-0.147496\pi\)
0.894550 + 0.446968i \(0.147496\pi\)
\(912\) 0 0
\(913\) 6.00000 0.198571
\(914\) 2.00000 0.0661541
\(915\) 0 0
\(916\) −16.0000 −0.528655
\(917\) 0 0
\(918\) 0 0
\(919\) −24.0000 −0.791687 −0.395843 0.918318i \(-0.629548\pi\)
−0.395843 + 0.918318i \(0.629548\pi\)
\(920\) 8.00000 0.263752
\(921\) 0 0
\(922\) 3.00000 0.0987997
\(923\) 4.00000 0.131662
\(924\) 0 0
\(925\) −22.0000 −0.723356
\(926\) −14.0000 −0.460069
\(927\) 0 0
\(928\) −1.00000 −0.0328266
\(929\) 33.0000 1.08269 0.541347 0.840799i \(-0.317914\pi\)
0.541347 + 0.840799i \(0.317914\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) −6.00000 −0.196537
\(933\) 0 0
\(934\) 12.0000 0.392652
\(935\) −8.00000 −0.261628
\(936\) 0 0
\(937\) −12.0000 −0.392023 −0.196011 0.980602i \(-0.562799\pi\)
−0.196011 + 0.980602i \(0.562799\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) −8.00000 −0.260931
\(941\) −55.0000 −1.79295 −0.896474 0.443096i \(-0.853880\pi\)
−0.896474 + 0.443096i \(0.853880\pi\)
\(942\) 0 0
\(943\) 4.00000 0.130258
\(944\) −9.00000 −0.292925
\(945\) 0 0
\(946\) 4.00000 0.130051
\(947\) 16.0000 0.519930 0.259965 0.965618i \(-0.416289\pi\)
0.259965 + 0.965618i \(0.416289\pi\)
\(948\) 0 0
\(949\) 2.00000 0.0649227
\(950\) 66.0000 2.14132
\(951\) 0 0
\(952\) 0 0
\(953\) −56.0000 −1.81402 −0.907009 0.421111i \(-0.861640\pi\)
−0.907009 + 0.421111i \(0.861640\pi\)
\(954\) 0 0
\(955\) −56.0000 −1.81212
\(956\) −19.0000 −0.614504
\(957\) 0 0
\(958\) 37.0000 1.19542
\(959\) 0 0
\(960\) 0 0
\(961\) −15.0000 −0.483871
\(962\) 2.00000 0.0644826
\(963\) 0 0
\(964\) 30.0000 0.966235
\(965\) −88.0000 −2.83282
\(966\) 0 0
\(967\) 56.0000 1.80084 0.900419 0.435023i \(-0.143260\pi\)
0.900419 + 0.435023i \(0.143260\pi\)
\(968\) 1.00000 0.0321412
\(969\) 0 0
\(970\) −20.0000 −0.642161
\(971\) 13.0000 0.417190 0.208595 0.978002i \(-0.433111\pi\)
0.208595 + 0.978002i \(0.433111\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) −4.00000 −0.128168
\(975\) 0 0
\(976\) −5.00000 −0.160046
\(977\) −14.0000 −0.447900 −0.223950 0.974601i \(-0.571895\pi\)
−0.223950 + 0.974601i \(0.571895\pi\)
\(978\) 0 0
\(979\) −6.00000 −0.191761
\(980\) 0 0
\(981\) 0 0
\(982\) −18.0000 −0.574403
\(983\) −18.0000 −0.574111 −0.287055 0.957914i \(-0.592676\pi\)
−0.287055 + 0.957914i \(0.592676\pi\)
\(984\) 0 0
\(985\) 12.0000 0.382352
\(986\) 2.00000 0.0636930
\(987\) 0 0
\(988\) −6.00000 −0.190885
\(989\) 8.00000 0.254385
\(990\) 0 0
\(991\) −20.0000 −0.635321 −0.317660 0.948205i \(-0.602897\pi\)
−0.317660 + 0.948205i \(0.602897\pi\)
\(992\) 4.00000 0.127000
\(993\) 0 0
\(994\) 0 0
\(995\) 40.0000 1.26809
\(996\) 0 0
\(997\) −42.0000 −1.33015 −0.665077 0.746775i \(-0.731601\pi\)
−0.665077 + 0.746775i \(0.731601\pi\)
\(998\) −16.0000 −0.506471
\(999\) 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 9702.2.a.cg.1.1 1
3.2 odd 2 1078.2.a.a.1.1 1
7.2 even 3 1386.2.k.a.991.1 2
7.4 even 3 1386.2.k.a.793.1 2
7.6 odd 2 9702.2.a.bb.1.1 1
12.11 even 2 8624.2.a.bd.1.1 1
21.2 odd 6 154.2.e.d.67.1 yes 2
21.5 even 6 1078.2.e.g.67.1 2
21.11 odd 6 154.2.e.d.23.1 2
21.17 even 6 1078.2.e.g.177.1 2
21.20 even 2 1078.2.a.f.1.1 1
84.11 even 6 1232.2.q.a.177.1 2
84.23 even 6 1232.2.q.a.529.1 2
84.83 odd 2 8624.2.a.d.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
154.2.e.d.23.1 2 21.11 odd 6
154.2.e.d.67.1 yes 2 21.2 odd 6
1078.2.a.a.1.1 1 3.2 odd 2
1078.2.a.f.1.1 1 21.20 even 2
1078.2.e.g.67.1 2 21.5 even 6
1078.2.e.g.177.1 2 21.17 even 6
1232.2.q.a.177.1 2 84.11 even 6
1232.2.q.a.529.1 2 84.23 even 6
1386.2.k.a.793.1 2 7.4 even 3
1386.2.k.a.991.1 2 7.2 even 3
8624.2.a.d.1.1 1 84.83 odd 2
8624.2.a.bd.1.1 1 12.11 even 2
9702.2.a.bb.1.1 1 7.6 odd 2
9702.2.a.cg.1.1 1 1.1 even 1 trivial