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Properties

Label 4998.2.a.q.1.1
Level $4998$
Weight $2$
Character 4998.1
Self dual yes
Analytic conductor $39.909$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

Related objects

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [4998,2,Mod(1,4998)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4998, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("4998.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4998 = 2 \cdot 3 \cdot 7^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4998.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(39.9092309302\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 714)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 4998.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{10} +3.00000 q^{11} +1.00000 q^{12} +3.00000 q^{13} -1.00000 q^{15} +1.00000 q^{16} -1.00000 q^{17} -1.00000 q^{18} -6.00000 q^{19} -1.00000 q^{20} -3.00000 q^{22} -2.00000 q^{23} -1.00000 q^{24} -4.00000 q^{25} -3.00000 q^{26} +1.00000 q^{27} +6.00000 q^{29} +1.00000 q^{30} -4.00000 q^{31} -1.00000 q^{32} +3.00000 q^{33} +1.00000 q^{34} +1.00000 q^{36} -11.0000 q^{37} +6.00000 q^{38} +3.00000 q^{39} +1.00000 q^{40} -12.0000 q^{41} +3.00000 q^{43} +3.00000 q^{44} -1.00000 q^{45} +2.00000 q^{46} -12.0000 q^{47} +1.00000 q^{48} +4.00000 q^{50} -1.00000 q^{51} +3.00000 q^{52} +5.00000 q^{53} -1.00000 q^{54} -3.00000 q^{55} -6.00000 q^{57} -6.00000 q^{58} -4.00000 q^{59} -1.00000 q^{60} +6.00000 q^{61} +4.00000 q^{62} +1.00000 q^{64} -3.00000 q^{65} -3.00000 q^{66} +9.00000 q^{67} -1.00000 q^{68} -2.00000 q^{69} +12.0000 q^{71} -1.00000 q^{72} -15.0000 q^{73} +11.0000 q^{74} -4.00000 q^{75} -6.00000 q^{76} -3.00000 q^{78} -11.0000 q^{79} -1.00000 q^{80} +1.00000 q^{81} +12.0000 q^{82} -7.00000 q^{83} +1.00000 q^{85} -3.00000 q^{86} +6.00000 q^{87} -3.00000 q^{88} +13.0000 q^{89} +1.00000 q^{90} -2.00000 q^{92} -4.00000 q^{93} +12.0000 q^{94} +6.00000 q^{95} -1.00000 q^{96} -11.0000 q^{97} +3.00000 q^{99} +O(q^{100})\)

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
\(5\) −1.00000 −0.447214 −0.223607 − 0.974679i \(-0.571783\pi\)
−0.223607 + 0.974679i \(0.571783\pi\)
\(6\) −1.00000 −0.408248
\(7\) 0 0
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) 1.00000 0.316228
\(11\) 3.00000 0.904534 0.452267 − 0.891883i \(-0.350615\pi\)
0.452267 + 0.891883i \(0.350615\pi\)
\(12\) 1.00000 0.288675
\(13\) 3.00000 0.832050 0.416025 − 0.909353i \(-0.363423\pi\)
0.416025 + 0.909353i \(0.363423\pi\)
\(14\) 0 0
\(15\) −1.00000 −0.258199
\(16\) 1.00000 0.250000
\(17\) −1.00000 −0.242536
\(18\) −1.00000 −0.235702
\(19\) −6.00000 −1.37649 −0.688247 − 0.725476i \(-0.741620\pi\)
−0.688247 + 0.725476i \(0.741620\pi\)
\(20\) −1.00000 −0.223607
\(21\) 0 0
\(22\) −3.00000 −0.639602
\(23\) −2.00000 −0.417029 −0.208514 − 0.978019i \(-0.566863\pi\)
−0.208514 + 0.978019i \(0.566863\pi\)
\(24\) −1.00000 −0.204124
\(25\) −4.00000 −0.800000
\(26\) −3.00000 −0.588348
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) 6.00000 1.11417 0.557086 − 0.830455i \(-0.311919\pi\)
0.557086 + 0.830455i \(0.311919\pi\)
\(30\) 1.00000 0.182574
\(31\) −4.00000 −0.718421 −0.359211 − 0.933257i \(-0.616954\pi\)
−0.359211 + 0.933257i \(0.616954\pi\)
\(32\) −1.00000 −0.176777
\(33\) 3.00000 0.522233
\(34\) 1.00000 0.171499
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) −11.0000 −1.80839 −0.904194 − 0.427121i \(-0.859528\pi\)
−0.904194 + 0.427121i \(0.859528\pi\)
\(38\) 6.00000 0.973329
\(39\) 3.00000 0.480384
\(40\) 1.00000 0.158114
\(41\) −12.0000 −1.87409 −0.937043 − 0.349215i \(-0.886448\pi\)
−0.937043 + 0.349215i \(0.886448\pi\)
\(42\) 0 0
\(43\) 3.00000 0.457496 0.228748 − 0.973486i \(-0.426537\pi\)
0.228748 + 0.973486i \(0.426537\pi\)
\(44\) 3.00000 0.452267
\(45\) −1.00000 −0.149071
\(46\) 2.00000 0.294884
\(47\) −12.0000 −1.75038 −0.875190 − 0.483779i \(-0.839264\pi\)
−0.875190 + 0.483779i \(0.839264\pi\)
\(48\) 1.00000 0.144338
\(49\) 0 0
\(50\) 4.00000 0.565685
\(51\) −1.00000 −0.140028
\(52\) 3.00000 0.416025
\(53\) 5.00000 0.686803 0.343401 − 0.939189i \(-0.388421\pi\)
0.343401 + 0.939189i \(0.388421\pi\)
\(54\) −1.00000 −0.136083
\(55\) −3.00000 −0.404520
\(56\) 0 0
\(57\) −6.00000 −0.794719
\(58\) −6.00000 −0.787839
\(59\) −4.00000 −0.520756 −0.260378 − 0.965507i \(-0.583847\pi\)
−0.260378 + 0.965507i \(0.583847\pi\)
\(60\) −1.00000 −0.129099
\(61\) 6.00000 0.768221 0.384111 − 0.923287i \(-0.374508\pi\)
0.384111 + 0.923287i \(0.374508\pi\)
\(62\) 4.00000 0.508001
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) −3.00000 −0.372104
\(66\) −3.00000 −0.369274
\(67\) 9.00000 1.09952 0.549762 − 0.835321i \(-0.314718\pi\)
0.549762 + 0.835321i \(0.314718\pi\)
\(68\) −1.00000 −0.121268
\(69\) −2.00000 −0.240772
\(70\) 0 0
\(71\) 12.0000 1.42414 0.712069 − 0.702109i \(-0.247758\pi\)
0.712069 + 0.702109i \(0.247758\pi\)
\(72\) −1.00000 −0.117851
\(73\) −15.0000 −1.75562 −0.877809 − 0.479012i \(-0.840995\pi\)
−0.877809 + 0.479012i \(0.840995\pi\)
\(74\) 11.0000 1.27872
\(75\) −4.00000 −0.461880
\(76\) −6.00000 −0.688247
\(77\) 0 0
\(78\) −3.00000 −0.339683
\(79\) −11.0000 −1.23760 −0.618798 − 0.785550i \(-0.712380\pi\)
−0.618798 + 0.785550i \(0.712380\pi\)
\(80\) −1.00000 −0.111803
\(81\) 1.00000 0.111111
\(82\) 12.0000 1.32518
\(83\) −7.00000 −0.768350 −0.384175 − 0.923260i \(-0.625514\pi\)
−0.384175 + 0.923260i \(0.625514\pi\)
\(84\) 0 0
\(85\) 1.00000 0.108465
\(86\) −3.00000 −0.323498
\(87\) 6.00000 0.643268
\(88\) −3.00000 −0.319801
\(89\) 13.0000 1.37800 0.688999 − 0.724763i \(-0.258051\pi\)
0.688999 + 0.724763i \(0.258051\pi\)
\(90\) 1.00000 0.105409
\(91\) 0 0
\(92\) −2.00000 −0.208514
\(93\) −4.00000 −0.414781
\(94\) 12.0000 1.23771
\(95\) 6.00000 0.615587
\(96\) −1.00000 −0.102062
\(97\) −11.0000 −1.11688 −0.558440 − 0.829545i \(-0.688600\pi\)
−0.558440 + 0.829545i \(0.688600\pi\)
\(98\) 0 0
\(99\) 3.00000 0.301511
\(100\) −4.00000 −0.400000
\(101\) 12.0000 1.19404 0.597022 − 0.802225i \(-0.296350\pi\)
0.597022 + 0.802225i \(0.296350\pi\)
\(102\) 1.00000 0.0990148
\(103\) −1.00000 −0.0985329 −0.0492665 − 0.998786i \(-0.515688\pi\)
−0.0492665 + 0.998786i \(0.515688\pi\)
\(104\) −3.00000 −0.294174
\(105\) 0 0
\(106\) −5.00000 −0.485643
\(107\) 16.0000 1.54678 0.773389 − 0.633932i \(-0.218560\pi\)
0.773389 + 0.633932i \(0.218560\pi\)
\(108\) 1.00000 0.0962250
\(109\) 6.00000 0.574696 0.287348 − 0.957826i \(-0.407226\pi\)
0.287348 + 0.957826i \(0.407226\pi\)
\(110\) 3.00000 0.286039
\(111\) −11.0000 −1.04407
\(112\) 0 0
\(113\) −15.0000 −1.41108 −0.705541 − 0.708669i \(-0.749296\pi\)
−0.705541 + 0.708669i \(0.749296\pi\)
\(114\) 6.00000 0.561951
\(115\) 2.00000 0.186501
\(116\) 6.00000 0.557086
\(117\) 3.00000 0.277350
\(118\) 4.00000 0.368230
\(119\) 0 0
\(120\) 1.00000 0.0912871
\(121\) −2.00000 −0.181818
\(122\) −6.00000 −0.543214
\(123\) −12.0000 −1.08200
\(124\) −4.00000 −0.359211
\(125\) 9.00000 0.804984
\(126\) 0 0
\(127\) 2.00000 0.177471 0.0887357 − 0.996055i \(-0.471717\pi\)
0.0887357 + 0.996055i \(0.471717\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 3.00000 0.264135
\(130\) 3.00000 0.263117
\(131\) 10.0000 0.873704 0.436852 − 0.899533i \(-0.356093\pi\)
0.436852 + 0.899533i \(0.356093\pi\)
\(132\) 3.00000 0.261116
\(133\) 0 0
\(134\) −9.00000 −0.777482
\(135\) −1.00000 −0.0860663
\(136\) 1.00000 0.0857493
\(137\) 8.00000 0.683486 0.341743 − 0.939793i \(-0.388983\pi\)
0.341743 + 0.939793i \(0.388983\pi\)
\(138\) 2.00000 0.170251
\(139\) 1.00000 0.0848189 0.0424094 − 0.999100i \(-0.486497\pi\)
0.0424094 + 0.999100i \(0.486497\pi\)
\(140\) 0 0
\(141\) −12.0000 −1.01058
\(142\) −12.0000 −1.00702
\(143\) 9.00000 0.752618
\(144\) 1.00000 0.0833333
\(145\) −6.00000 −0.498273
\(146\) 15.0000 1.24141
\(147\) 0 0
\(148\) −11.0000 −0.904194
\(149\) −15.0000 −1.22885 −0.614424 − 0.788976i \(-0.710612\pi\)
−0.614424 + 0.788976i \(0.710612\pi\)
\(150\) 4.00000 0.326599
\(151\) −16.0000 −1.30206 −0.651031 − 0.759051i \(-0.725663\pi\)
−0.651031 + 0.759051i \(0.725663\pi\)
\(152\) 6.00000 0.486664
\(153\) −1.00000 −0.0808452
\(154\) 0 0
\(155\) 4.00000 0.321288
\(156\) 3.00000 0.240192
\(157\) 18.0000 1.43656 0.718278 − 0.695756i \(-0.244931\pi\)
0.718278 + 0.695756i \(0.244931\pi\)
\(158\) 11.0000 0.875113
\(159\) 5.00000 0.396526
\(160\) 1.00000 0.0790569
\(161\) 0 0
\(162\) −1.00000 −0.0785674
\(163\) −22.0000 −1.72317 −0.861586 − 0.507611i \(-0.830529\pi\)
−0.861586 + 0.507611i \(0.830529\pi\)
\(164\) −12.0000 −0.937043
\(165\) −3.00000 −0.233550
\(166\) 7.00000 0.543305
\(167\) 5.00000 0.386912 0.193456 − 0.981109i \(-0.438030\pi\)
0.193456 + 0.981109i \(0.438030\pi\)
\(168\) 0 0
\(169\) −4.00000 −0.307692
\(170\) −1.00000 −0.0766965
\(171\) −6.00000 −0.458831
\(172\) 3.00000 0.228748
\(173\) −10.0000 −0.760286 −0.380143 − 0.924928i \(-0.624125\pi\)
−0.380143 + 0.924928i \(0.624125\pi\)
\(174\) −6.00000 −0.454859
\(175\) 0 0
\(176\) 3.00000 0.226134
\(177\) −4.00000 −0.300658
\(178\) −13.0000 −0.974391
\(179\) 10.0000 0.747435 0.373718 − 0.927543i \(-0.378083\pi\)
0.373718 + 0.927543i \(0.378083\pi\)
\(180\) −1.00000 −0.0745356
\(181\) −20.0000 −1.48659 −0.743294 − 0.668965i \(-0.766738\pi\)
−0.743294 + 0.668965i \(0.766738\pi\)
\(182\) 0 0
\(183\) 6.00000 0.443533
\(184\) 2.00000 0.147442
\(185\) 11.0000 0.808736
\(186\) 4.00000 0.293294
\(187\) −3.00000 −0.219382
\(188\) −12.0000 −0.875190
\(189\) 0 0
\(190\) −6.00000 −0.435286
\(191\) −11.0000 −0.795932 −0.397966 − 0.917400i \(-0.630284\pi\)
−0.397966 + 0.917400i \(0.630284\pi\)
\(192\) 1.00000 0.0721688
\(193\) −24.0000 −1.72756 −0.863779 − 0.503871i \(-0.831909\pi\)
−0.863779 + 0.503871i \(0.831909\pi\)
\(194\) 11.0000 0.789754
\(195\) −3.00000 −0.214834
\(196\) 0 0
\(197\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(198\) −3.00000 −0.213201
\(199\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(200\) 4.00000 0.282843
\(201\) 9.00000 0.634811
\(202\) −12.0000 −0.844317
\(203\) 0 0
\(204\) −1.00000 −0.0700140
\(205\) 12.0000 0.838116
\(206\) 1.00000 0.0696733
\(207\) −2.00000 −0.139010
\(208\) 3.00000 0.208013
\(209\) −18.0000 −1.24509
\(210\) 0 0
\(211\) 14.0000 0.963800 0.481900 − 0.876226i \(-0.339947\pi\)
0.481900 + 0.876226i \(0.339947\pi\)
\(212\) 5.00000 0.343401
\(213\) 12.0000 0.822226
\(214\) −16.0000 −1.09374
\(215\) −3.00000 −0.204598
\(216\) −1.00000 −0.0680414
\(217\) 0 0
\(218\) −6.00000 −0.406371
\(219\) −15.0000 −1.01361
\(220\) −3.00000 −0.202260
\(221\) −3.00000 −0.201802
\(222\) 11.0000 0.738272
\(223\) −16.0000 −1.07144 −0.535720 − 0.844396i \(-0.679960\pi\)
−0.535720 + 0.844396i \(0.679960\pi\)
\(224\) 0 0
\(225\) −4.00000 −0.266667
\(226\) 15.0000 0.997785
\(227\) −8.00000 −0.530979 −0.265489 − 0.964114i \(-0.585534\pi\)
−0.265489 + 0.964114i \(0.585534\pi\)
\(228\) −6.00000 −0.397360
\(229\) −19.0000 −1.25556 −0.627778 − 0.778393i \(-0.716035\pi\)
−0.627778 + 0.778393i \(0.716035\pi\)
\(230\) −2.00000 −0.131876
\(231\) 0 0
\(232\) −6.00000 −0.393919
\(233\) −3.00000 −0.196537 −0.0982683 − 0.995160i \(-0.531330\pi\)
−0.0982683 + 0.995160i \(0.531330\pi\)
\(234\) −3.00000 −0.196116
\(235\) 12.0000 0.782794
\(236\) −4.00000 −0.260378
\(237\) −11.0000 −0.714527
\(238\) 0 0
\(239\) −21.0000 −1.35838 −0.679189 − 0.733964i \(-0.737668\pi\)
−0.679189 + 0.733964i \(0.737668\pi\)
\(240\) −1.00000 −0.0645497
\(241\) −14.0000 −0.901819 −0.450910 − 0.892570i \(-0.648900\pi\)
−0.450910 + 0.892570i \(0.648900\pi\)
\(242\) 2.00000 0.128565
\(243\) 1.00000 0.0641500
\(244\) 6.00000 0.384111
\(245\) 0 0
\(246\) 12.0000 0.765092
\(247\) −18.0000 −1.14531
\(248\) 4.00000 0.254000
\(249\) −7.00000 −0.443607
\(250\) −9.00000 −0.569210
\(251\) −9.00000 −0.568075 −0.284037 − 0.958813i \(-0.591674\pi\)
−0.284037 + 0.958813i \(0.591674\pi\)
\(252\) 0 0
\(253\) −6.00000 −0.377217
\(254\) −2.00000 −0.125491
\(255\) 1.00000 0.0626224
\(256\) 1.00000 0.0625000
\(257\) 27.0000 1.68421 0.842107 − 0.539311i \(-0.181315\pi\)
0.842107 + 0.539311i \(0.181315\pi\)
\(258\) −3.00000 −0.186772
\(259\) 0 0
\(260\) −3.00000 −0.186052
\(261\) 6.00000 0.371391
\(262\) −10.0000 −0.617802
\(263\) −19.0000 −1.17159 −0.585795 − 0.810459i \(-0.699218\pi\)
−0.585795 + 0.810459i \(0.699218\pi\)
\(264\) −3.00000 −0.184637
\(265\) −5.00000 −0.307148
\(266\) 0 0
\(267\) 13.0000 0.795587
\(268\) 9.00000 0.549762
\(269\) −18.0000 −1.09748 −0.548740 − 0.835993i \(-0.684892\pi\)
−0.548740 + 0.835993i \(0.684892\pi\)
\(270\) 1.00000 0.0608581
\(271\) 9.00000 0.546711 0.273356 − 0.961913i \(-0.411866\pi\)
0.273356 + 0.961913i \(0.411866\pi\)
\(272\) −1.00000 −0.0606339
\(273\) 0 0
\(274\) −8.00000 −0.483298
\(275\) −12.0000 −0.723627
\(276\) −2.00000 −0.120386
\(277\) 18.0000 1.08152 0.540758 − 0.841178i \(-0.318138\pi\)
0.540758 + 0.841178i \(0.318138\pi\)
\(278\) −1.00000 −0.0599760
\(279\) −4.00000 −0.239474
\(280\) 0 0
\(281\) −22.0000 −1.31241 −0.656205 − 0.754583i \(-0.727839\pi\)
−0.656205 + 0.754583i \(0.727839\pi\)
\(282\) 12.0000 0.714590
\(283\) 13.0000 0.772770 0.386385 − 0.922338i \(-0.373724\pi\)
0.386385 + 0.922338i \(0.373724\pi\)
\(284\) 12.0000 0.712069
\(285\) 6.00000 0.355409
\(286\) −9.00000 −0.532181
\(287\) 0 0
\(288\) −1.00000 −0.0589256
\(289\) 1.00000 0.0588235
\(290\) 6.00000 0.352332
\(291\) −11.0000 −0.644831
\(292\) −15.0000 −0.877809
\(293\) 8.00000 0.467365 0.233682 − 0.972313i \(-0.424922\pi\)
0.233682 + 0.972313i \(0.424922\pi\)
\(294\) 0 0
\(295\) 4.00000 0.232889
\(296\) 11.0000 0.639362
\(297\) 3.00000 0.174078
\(298\) 15.0000 0.868927
\(299\) −6.00000 −0.346989
\(300\) −4.00000 −0.230940
\(301\) 0 0
\(302\) 16.0000 0.920697
\(303\) 12.0000 0.689382
\(304\) −6.00000 −0.344124
\(305\) −6.00000 −0.343559
\(306\) 1.00000 0.0571662
\(307\) 8.00000 0.456584 0.228292 − 0.973593i \(-0.426686\pi\)
0.228292 + 0.973593i \(0.426686\pi\)
\(308\) 0 0
\(309\) −1.00000 −0.0568880
\(310\) −4.00000 −0.227185
\(311\) −3.00000 −0.170114 −0.0850572 − 0.996376i \(-0.527107\pi\)
−0.0850572 + 0.996376i \(0.527107\pi\)
\(312\) −3.00000 −0.169842
\(313\) 22.0000 1.24351 0.621757 − 0.783210i \(-0.286419\pi\)
0.621757 + 0.783210i \(0.286419\pi\)
\(314\) −18.0000 −1.01580
\(315\) 0 0
\(316\) −11.0000 −0.618798
\(317\) 18.0000 1.01098 0.505490 − 0.862832i \(-0.331312\pi\)
0.505490 + 0.862832i \(0.331312\pi\)
\(318\) −5.00000 −0.280386
\(319\) 18.0000 1.00781
\(320\) −1.00000 −0.0559017
\(321\) 16.0000 0.893033
\(322\) 0 0
\(323\) 6.00000 0.333849
\(324\) 1.00000 0.0555556
\(325\) −12.0000 −0.665640
\(326\) 22.0000 1.21847
\(327\) 6.00000 0.331801
\(328\) 12.0000 0.662589
\(329\) 0 0
\(330\) 3.00000 0.165145
\(331\) −3.00000 −0.164895 −0.0824475 − 0.996595i \(-0.526274\pi\)
−0.0824475 + 0.996595i \(0.526274\pi\)
\(332\) −7.00000 −0.384175
\(333\) −11.0000 −0.602796
\(334\) −5.00000 −0.273588
\(335\) −9.00000 −0.491723
\(336\) 0 0
\(337\) 14.0000 0.762629 0.381314 − 0.924445i \(-0.375472\pi\)
0.381314 + 0.924445i \(0.375472\pi\)
\(338\) 4.00000 0.217571
\(339\) −15.0000 −0.814688
\(340\) 1.00000 0.0542326
\(341\) −12.0000 −0.649836
\(342\) 6.00000 0.324443
\(343\) 0 0
\(344\) −3.00000 −0.161749
\(345\) 2.00000 0.107676
\(346\) 10.0000 0.537603
\(347\) 9.00000 0.483145 0.241573 − 0.970383i \(-0.422337\pi\)
0.241573 + 0.970383i \(0.422337\pi\)
\(348\) 6.00000 0.321634
\(349\) 1.00000 0.0535288 0.0267644 − 0.999642i \(-0.491480\pi\)
0.0267644 + 0.999642i \(0.491480\pi\)
\(350\) 0 0
\(351\) 3.00000 0.160128
\(352\) −3.00000 −0.159901
\(353\) 9.00000 0.479022 0.239511 − 0.970894i \(-0.423013\pi\)
0.239511 + 0.970894i \(0.423013\pi\)
\(354\) 4.00000 0.212598
\(355\) −12.0000 −0.636894
\(356\) 13.0000 0.688999
\(357\) 0 0
\(358\) −10.0000 −0.528516
\(359\) 16.0000 0.844448 0.422224 − 0.906492i \(-0.361250\pi\)
0.422224 + 0.906492i \(0.361250\pi\)
\(360\) 1.00000 0.0527046
\(361\) 17.0000 0.894737
\(362\) 20.0000 1.05118
\(363\) −2.00000 −0.104973
\(364\) 0 0
\(365\) 15.0000 0.785136
\(366\) −6.00000 −0.313625
\(367\) −20.0000 −1.04399 −0.521996 − 0.852948i \(-0.674812\pi\)
−0.521996 + 0.852948i \(0.674812\pi\)
\(368\) −2.00000 −0.104257
\(369\) −12.0000 −0.624695
\(370\) −11.0000 −0.571863
\(371\) 0 0
\(372\) −4.00000 −0.207390
\(373\) −20.0000 −1.03556 −0.517780 − 0.855514i \(-0.673242\pi\)
−0.517780 + 0.855514i \(0.673242\pi\)
\(374\) 3.00000 0.155126
\(375\) 9.00000 0.464758
\(376\) 12.0000 0.618853
\(377\) 18.0000 0.927047
\(378\) 0 0
\(379\) 22.0000 1.13006 0.565032 − 0.825069i \(-0.308864\pi\)
0.565032 + 0.825069i \(0.308864\pi\)
\(380\) 6.00000 0.307794
\(381\) 2.00000 0.102463
\(382\) 11.0000 0.562809
\(383\) −20.0000 −1.02195 −0.510976 − 0.859595i \(-0.670716\pi\)
−0.510976 + 0.859595i \(0.670716\pi\)
\(384\) −1.00000 −0.0510310
\(385\) 0 0
\(386\) 24.0000 1.22157
\(387\) 3.00000 0.152499
\(388\) −11.0000 −0.558440
\(389\) 6.00000 0.304212 0.152106 − 0.988364i \(-0.451394\pi\)
0.152106 + 0.988364i \(0.451394\pi\)
\(390\) 3.00000 0.151911
\(391\) 2.00000 0.101144
\(392\) 0 0
\(393\) 10.0000 0.504433
\(394\) 0 0
\(395\) 11.0000 0.553470
\(396\) 3.00000 0.150756
\(397\) −12.0000 −0.602263 −0.301131 − 0.953583i \(-0.597364\pi\)
−0.301131 + 0.953583i \(0.597364\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) −4.00000 −0.200000
\(401\) 14.0000 0.699127 0.349563 − 0.936913i \(-0.386330\pi\)
0.349563 + 0.936913i \(0.386330\pi\)
\(402\) −9.00000 −0.448879
\(403\) −12.0000 −0.597763
\(404\) 12.0000 0.597022
\(405\) −1.00000 −0.0496904
\(406\) 0 0
\(407\) −33.0000 −1.63575
\(408\) 1.00000 0.0495074
\(409\) 14.0000 0.692255 0.346128 − 0.938187i \(-0.387496\pi\)
0.346128 + 0.938187i \(0.387496\pi\)
\(410\) −12.0000 −0.592638
\(411\) 8.00000 0.394611
\(412\) −1.00000 −0.0492665
\(413\) 0 0
\(414\) 2.00000 0.0982946
\(415\) 7.00000 0.343616
\(416\) −3.00000 −0.147087
\(417\) 1.00000 0.0489702
\(418\) 18.0000 0.880409
\(419\) 6.00000 0.293119 0.146560 − 0.989202i \(-0.453180\pi\)
0.146560 + 0.989202i \(0.453180\pi\)
\(420\) 0 0
\(421\) −10.0000 −0.487370 −0.243685 − 0.969854i \(-0.578356\pi\)
−0.243685 + 0.969854i \(0.578356\pi\)
\(422\) −14.0000 −0.681509
\(423\) −12.0000 −0.583460
\(424\) −5.00000 −0.242821
\(425\) 4.00000 0.194029
\(426\) −12.0000 −0.581402
\(427\) 0 0
\(428\) 16.0000 0.773389
\(429\) 9.00000 0.434524
\(430\) 3.00000 0.144673
\(431\) −4.00000 −0.192673 −0.0963366 − 0.995349i \(-0.530713\pi\)
−0.0963366 + 0.995349i \(0.530713\pi\)
\(432\) 1.00000 0.0481125
\(433\) −32.0000 −1.53782 −0.768911 − 0.639356i \(-0.779201\pi\)
−0.768911 + 0.639356i \(0.779201\pi\)
\(434\) 0 0
\(435\) −6.00000 −0.287678
\(436\) 6.00000 0.287348
\(437\) 12.0000 0.574038
\(438\) 15.0000 0.716728
\(439\) 32.0000 1.52728 0.763638 − 0.645644i \(-0.223411\pi\)
0.763638 + 0.645644i \(0.223411\pi\)
\(440\) 3.00000 0.143019
\(441\) 0 0
\(442\) 3.00000 0.142695
\(443\) 4.00000 0.190046 0.0950229 − 0.995475i \(-0.469708\pi\)
0.0950229 + 0.995475i \(0.469708\pi\)
\(444\) −11.0000 −0.522037
\(445\) −13.0000 −0.616259
\(446\) 16.0000 0.757622
\(447\) −15.0000 −0.709476
\(448\) 0 0
\(449\) 1.00000 0.0471929 0.0235965 − 0.999722i \(-0.492488\pi\)
0.0235965 + 0.999722i \(0.492488\pi\)
\(450\) 4.00000 0.188562
\(451\) −36.0000 −1.69517
\(452\) −15.0000 −0.705541
\(453\) −16.0000 −0.751746
\(454\) 8.00000 0.375459
\(455\) 0 0
\(456\) 6.00000 0.280976
\(457\) −17.0000 −0.795226 −0.397613 − 0.917553i \(-0.630161\pi\)
−0.397613 + 0.917553i \(0.630161\pi\)
\(458\) 19.0000 0.887812
\(459\) −1.00000 −0.0466760
\(460\) 2.00000 0.0932505
\(461\) 30.0000 1.39724 0.698620 − 0.715493i \(-0.253798\pi\)
0.698620 + 0.715493i \(0.253798\pi\)
\(462\) 0 0
\(463\) −16.0000 −0.743583 −0.371792 − 0.928316i \(-0.621256\pi\)
−0.371792 + 0.928316i \(0.621256\pi\)
\(464\) 6.00000 0.278543
\(465\) 4.00000 0.185496
\(466\) 3.00000 0.138972
\(467\) −32.0000 −1.48078 −0.740392 − 0.672176i \(-0.765360\pi\)
−0.740392 + 0.672176i \(0.765360\pi\)
\(468\) 3.00000 0.138675
\(469\) 0 0
\(470\) −12.0000 −0.553519
\(471\) 18.0000 0.829396
\(472\) 4.00000 0.184115
\(473\) 9.00000 0.413820
\(474\) 11.0000 0.505247
\(475\) 24.0000 1.10120
\(476\) 0 0
\(477\) 5.00000 0.228934
\(478\) 21.0000 0.960518
\(479\) 11.0000 0.502603 0.251301 − 0.967909i \(-0.419141\pi\)
0.251301 + 0.967909i \(0.419141\pi\)
\(480\) 1.00000 0.0456435
\(481\) −33.0000 −1.50467
\(482\) 14.0000 0.637683
\(483\) 0 0
\(484\) −2.00000 −0.0909091
\(485\) 11.0000 0.499484
\(486\) −1.00000 −0.0453609
\(487\) −13.0000 −0.589086 −0.294543 − 0.955638i \(-0.595167\pi\)
−0.294543 + 0.955638i \(0.595167\pi\)
\(488\) −6.00000 −0.271607
\(489\) −22.0000 −0.994874
\(490\) 0 0
\(491\) −14.0000 −0.631811 −0.315906 − 0.948791i \(-0.602308\pi\)
−0.315906 + 0.948791i \(0.602308\pi\)
\(492\) −12.0000 −0.541002
\(493\) −6.00000 −0.270226
\(494\) 18.0000 0.809858
\(495\) −3.00000 −0.134840
\(496\) −4.00000 −0.179605
\(497\) 0 0
\(498\) 7.00000 0.313678
\(499\) 20.0000 0.895323 0.447661 − 0.894203i \(-0.352257\pi\)
0.447661 + 0.894203i \(0.352257\pi\)
\(500\) 9.00000 0.402492
\(501\) 5.00000 0.223384
\(502\) 9.00000 0.401690
\(503\) −24.0000 −1.07011 −0.535054 − 0.844818i \(-0.679709\pi\)
−0.535054 + 0.844818i \(0.679709\pi\)
\(504\) 0 0
\(505\) −12.0000 −0.533993
\(506\) 6.00000 0.266733
\(507\) −4.00000 −0.177646
\(508\) 2.00000 0.0887357
\(509\) −8.00000 −0.354594 −0.177297 − 0.984157i \(-0.556735\pi\)
−0.177297 + 0.984157i \(0.556735\pi\)
\(510\) −1.00000 −0.0442807
\(511\) 0 0
\(512\) −1.00000 −0.0441942
\(513\) −6.00000 −0.264906
\(514\) −27.0000 −1.19092
\(515\) 1.00000 0.0440653
\(516\) 3.00000 0.132068
\(517\) −36.0000 −1.58328
\(518\) 0 0
\(519\) −10.0000 −0.438951
\(520\) 3.00000 0.131559
\(521\) 20.0000 0.876216 0.438108 − 0.898922i \(-0.355649\pi\)
0.438108 + 0.898922i \(0.355649\pi\)
\(522\) −6.00000 −0.262613
\(523\) 4.00000 0.174908 0.0874539 − 0.996169i \(-0.472127\pi\)
0.0874539 + 0.996169i \(0.472127\pi\)
\(524\) 10.0000 0.436852
\(525\) 0 0
\(526\) 19.0000 0.828439
\(527\) 4.00000 0.174243
\(528\) 3.00000 0.130558
\(529\) −19.0000 −0.826087
\(530\) 5.00000 0.217186
\(531\) −4.00000 −0.173585
\(532\) 0 0
\(533\) −36.0000 −1.55933
\(534\) −13.0000 −0.562565
\(535\) −16.0000 −0.691740
\(536\) −9.00000 −0.388741
\(537\) 10.0000 0.431532
\(538\) 18.0000 0.776035
\(539\) 0 0
\(540\) −1.00000 −0.0430331
\(541\) −31.0000 −1.33279 −0.666397 − 0.745597i \(-0.732164\pi\)
−0.666397 + 0.745597i \(0.732164\pi\)
\(542\) −9.00000 −0.386583
\(543\) −20.0000 −0.858282
\(544\) 1.00000 0.0428746
\(545\) −6.00000 −0.257012
\(546\) 0 0
\(547\) −38.0000 −1.62476 −0.812381 − 0.583127i \(-0.801829\pi\)
−0.812381 + 0.583127i \(0.801829\pi\)
\(548\) 8.00000 0.341743
\(549\) 6.00000 0.256074
\(550\) 12.0000 0.511682
\(551\) −36.0000 −1.53365
\(552\) 2.00000 0.0851257
\(553\) 0 0
\(554\) −18.0000 −0.764747
\(555\) 11.0000 0.466924
\(556\) 1.00000 0.0424094
\(557\) 42.0000 1.77960 0.889799 − 0.456354i \(-0.150845\pi\)
0.889799 + 0.456354i \(0.150845\pi\)
\(558\) 4.00000 0.169334
\(559\) 9.00000 0.380659
\(560\) 0 0
\(561\) −3.00000 −0.126660
\(562\) 22.0000 0.928014
\(563\) 21.0000 0.885044 0.442522 − 0.896758i \(-0.354084\pi\)
0.442522 + 0.896758i \(0.354084\pi\)
\(564\) −12.0000 −0.505291
\(565\) 15.0000 0.631055
\(566\) −13.0000 −0.546431
\(567\) 0 0
\(568\) −12.0000 −0.503509
\(569\) −42.0000 −1.76073 −0.880366 − 0.474295i \(-0.842703\pi\)
−0.880366 + 0.474295i \(0.842703\pi\)
\(570\) −6.00000 −0.251312
\(571\) −16.0000 −0.669579 −0.334790 − 0.942293i \(-0.608665\pi\)
−0.334790 + 0.942293i \(0.608665\pi\)
\(572\) 9.00000 0.376309
\(573\) −11.0000 −0.459532
\(574\) 0 0
\(575\) 8.00000 0.333623
\(576\) 1.00000 0.0416667
\(577\) −30.0000 −1.24892 −0.624458 − 0.781058i \(-0.714680\pi\)
−0.624458 + 0.781058i \(0.714680\pi\)
\(578\) −1.00000 −0.0415945
\(579\) −24.0000 −0.997406
\(580\) −6.00000 −0.249136
\(581\) 0 0
\(582\) 11.0000 0.455965
\(583\) 15.0000 0.621237
\(584\) 15.0000 0.620704
\(585\) −3.00000 −0.124035
\(586\) −8.00000 −0.330477
\(587\) 17.0000 0.701665 0.350833 − 0.936438i \(-0.385899\pi\)
0.350833 + 0.936438i \(0.385899\pi\)
\(588\) 0 0
\(589\) 24.0000 0.988903
\(590\) −4.00000 −0.164677
\(591\) 0 0
\(592\) −11.0000 −0.452097
\(593\) 34.0000 1.39621 0.698106 − 0.715994i \(-0.254026\pi\)
0.698106 + 0.715994i \(0.254026\pi\)
\(594\) −3.00000 −0.123091
\(595\) 0 0
\(596\) −15.0000 −0.614424
\(597\) 0 0
\(598\) 6.00000 0.245358
\(599\) −24.0000 −0.980613 −0.490307 − 0.871550i \(-0.663115\pi\)
−0.490307 + 0.871550i \(0.663115\pi\)
\(600\) 4.00000 0.163299
\(601\) 46.0000 1.87638 0.938190 − 0.346122i \(-0.112502\pi\)
0.938190 + 0.346122i \(0.112502\pi\)
\(602\) 0 0
\(603\) 9.00000 0.366508
\(604\) −16.0000 −0.651031
\(605\) 2.00000 0.0813116
\(606\) −12.0000 −0.487467
\(607\) −12.0000 −0.487065 −0.243532 − 0.969893i \(-0.578306\pi\)
−0.243532 + 0.969893i \(0.578306\pi\)
\(608\) 6.00000 0.243332
\(609\) 0 0
\(610\) 6.00000 0.242933
\(611\) −36.0000 −1.45640
\(612\) −1.00000 −0.0404226
\(613\) −28.0000 −1.13091 −0.565455 − 0.824779i \(-0.691299\pi\)
−0.565455 + 0.824779i \(0.691299\pi\)
\(614\) −8.00000 −0.322854
\(615\) 12.0000 0.483887
\(616\) 0 0
\(617\) 13.0000 0.523360 0.261680 − 0.965155i \(-0.415723\pi\)
0.261680 + 0.965155i \(0.415723\pi\)
\(618\) 1.00000 0.0402259
\(619\) 11.0000 0.442127 0.221064 − 0.975259i \(-0.429047\pi\)
0.221064 + 0.975259i \(0.429047\pi\)
\(620\) 4.00000 0.160644
\(621\) −2.00000 −0.0802572
\(622\) 3.00000 0.120289
\(623\) 0 0
\(624\) 3.00000 0.120096
\(625\) 11.0000 0.440000
\(626\) −22.0000 −0.879297
\(627\) −18.0000 −0.718851
\(628\) 18.0000 0.718278
\(629\) 11.0000 0.438599
\(630\) 0 0
\(631\) 32.0000 1.27390 0.636950 − 0.770905i \(-0.280196\pi\)
0.636950 + 0.770905i \(0.280196\pi\)
\(632\) 11.0000 0.437557
\(633\) 14.0000 0.556450
\(634\) −18.0000 −0.714871
\(635\) −2.00000 −0.0793676
\(636\) 5.00000 0.198263
\(637\) 0 0
\(638\) −18.0000 −0.712627
\(639\) 12.0000 0.474713
\(640\) 1.00000 0.0395285
\(641\) 33.0000 1.30342 0.651711 − 0.758468i \(-0.274052\pi\)
0.651711 + 0.758468i \(0.274052\pi\)
\(642\) −16.0000 −0.631470
\(643\) −3.00000 −0.118308 −0.0591542 − 0.998249i \(-0.518840\pi\)
−0.0591542 + 0.998249i \(0.518840\pi\)
\(644\) 0 0
\(645\) −3.00000 −0.118125
\(646\) −6.00000 −0.236067
\(647\) 12.0000 0.471769 0.235884 − 0.971781i \(-0.424201\pi\)
0.235884 + 0.971781i \(0.424201\pi\)
\(648\) −1.00000 −0.0392837
\(649\) −12.0000 −0.471041
\(650\) 12.0000 0.470679
\(651\) 0 0
\(652\) −22.0000 −0.861586
\(653\) −18.0000 −0.704394 −0.352197 − 0.935926i \(-0.614565\pi\)
−0.352197 + 0.935926i \(0.614565\pi\)
\(654\) −6.00000 −0.234619
\(655\) −10.0000 −0.390732
\(656\) −12.0000 −0.468521
\(657\) −15.0000 −0.585206
\(658\) 0 0
\(659\) 48.0000 1.86981 0.934907 − 0.354892i \(-0.115482\pi\)
0.934907 + 0.354892i \(0.115482\pi\)
\(660\) −3.00000 −0.116775
\(661\) −10.0000 −0.388955 −0.194477 − 0.980907i \(-0.562301\pi\)
−0.194477 + 0.980907i \(0.562301\pi\)
\(662\) 3.00000 0.116598
\(663\) −3.00000 −0.116510
\(664\) 7.00000 0.271653
\(665\) 0 0
\(666\) 11.0000 0.426241
\(667\) −12.0000 −0.464642
\(668\) 5.00000 0.193456
\(669\) −16.0000 −0.618596
\(670\) 9.00000 0.347700
\(671\) 18.0000 0.694882
\(672\) 0 0
\(673\) 48.0000 1.85026 0.925132 − 0.379646i \(-0.123954\pi\)
0.925132 + 0.379646i \(0.123954\pi\)
\(674\) −14.0000 −0.539260
\(675\) −4.00000 −0.153960
\(676\) −4.00000 −0.153846
\(677\) −33.0000 −1.26829 −0.634147 − 0.773213i \(-0.718648\pi\)
−0.634147 + 0.773213i \(0.718648\pi\)
\(678\) 15.0000 0.576072
\(679\) 0 0
\(680\) −1.00000 −0.0383482
\(681\) −8.00000 −0.306561
\(682\) 12.0000 0.459504
\(683\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(684\) −6.00000 −0.229416
\(685\) −8.00000 −0.305664
\(686\) 0 0
\(687\) −19.0000 −0.724895
\(688\) 3.00000 0.114374
\(689\) 15.0000 0.571454
\(690\) −2.00000 −0.0761387
\(691\) −21.0000 −0.798878 −0.399439 − 0.916760i \(-0.630795\pi\)
−0.399439 + 0.916760i \(0.630795\pi\)
\(692\) −10.0000 −0.380143
\(693\) 0 0
\(694\) −9.00000 −0.341635
\(695\) −1.00000 −0.0379322
\(696\) −6.00000 −0.227429
\(697\) 12.0000 0.454532
\(698\) −1.00000 −0.0378506
\(699\) −3.00000 −0.113470
\(700\) 0 0
\(701\) −1.00000 −0.0377695 −0.0188847 − 0.999822i \(-0.506012\pi\)
−0.0188847 + 0.999822i \(0.506012\pi\)
\(702\) −3.00000 −0.113228
\(703\) 66.0000 2.48924
\(704\) 3.00000 0.113067
\(705\) 12.0000 0.451946
\(706\) −9.00000 −0.338719
\(707\) 0 0
\(708\) −4.00000 −0.150329
\(709\) −19.0000 −0.713560 −0.356780 − 0.934188i \(-0.616125\pi\)
−0.356780 + 0.934188i \(0.616125\pi\)
\(710\) 12.0000 0.450352
\(711\) −11.0000 −0.412532
\(712\) −13.0000 −0.487196
\(713\) 8.00000 0.299602
\(714\) 0 0
\(715\) −9.00000 −0.336581
\(716\) 10.0000 0.373718
\(717\) −21.0000 −0.784259
\(718\) −16.0000 −0.597115
\(719\) −16.0000 −0.596699 −0.298350 − 0.954457i \(-0.596436\pi\)
−0.298350 + 0.954457i \(0.596436\pi\)
\(720\) −1.00000 −0.0372678
\(721\) 0 0
\(722\) −17.0000 −0.632674
\(723\) −14.0000 −0.520666
\(724\) −20.0000 −0.743294
\(725\) −24.0000 −0.891338
\(726\) 2.00000 0.0742270
\(727\) 47.0000 1.74313 0.871567 − 0.490277i \(-0.163104\pi\)
0.871567 + 0.490277i \(0.163104\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) −15.0000 −0.555175
\(731\) −3.00000 −0.110959
\(732\) 6.00000 0.221766
\(733\) 38.0000 1.40356 0.701781 − 0.712393i \(-0.252388\pi\)
0.701781 + 0.712393i \(0.252388\pi\)
\(734\) 20.0000 0.738213
\(735\) 0 0
\(736\) 2.00000 0.0737210
\(737\) 27.0000 0.994558
\(738\) 12.0000 0.441726
\(739\) 20.0000 0.735712 0.367856 − 0.929883i \(-0.380092\pi\)
0.367856 + 0.929883i \(0.380092\pi\)
\(740\) 11.0000 0.404368
\(741\) −18.0000 −0.661247
\(742\) 0 0
\(743\) −46.0000 −1.68758 −0.843788 − 0.536676i \(-0.819680\pi\)
−0.843788 + 0.536676i \(0.819680\pi\)
\(744\) 4.00000 0.146647
\(745\) 15.0000 0.549557
\(746\) 20.0000 0.732252
\(747\) −7.00000 −0.256117
\(748\) −3.00000 −0.109691
\(749\) 0 0
\(750\) −9.00000 −0.328634
\(751\) −20.0000 −0.729810 −0.364905 − 0.931045i \(-0.618899\pi\)
−0.364905 + 0.931045i \(0.618899\pi\)
\(752\) −12.0000 −0.437595
\(753\) −9.00000 −0.327978
\(754\) −18.0000 −0.655521
\(755\) 16.0000 0.582300
\(756\) 0 0
\(757\) −12.0000 −0.436147 −0.218074 − 0.975932i \(-0.569977\pi\)
−0.218074 + 0.975932i \(0.569977\pi\)
\(758\) −22.0000 −0.799076
\(759\) −6.00000 −0.217786
\(760\) −6.00000 −0.217643
\(761\) −41.0000 −1.48625 −0.743124 − 0.669153i \(-0.766657\pi\)
−0.743124 + 0.669153i \(0.766657\pi\)
\(762\) −2.00000 −0.0724524
\(763\) 0 0
\(764\) −11.0000 −0.397966
\(765\) 1.00000 0.0361551
\(766\) 20.0000 0.722629
\(767\) −12.0000 −0.433295
\(768\) 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
\(771\) 27.0000 0.972381
\(772\) −24.0000 −0.863779
\(773\) −4.00000 −0.143870 −0.0719350 − 0.997409i \(-0.522917\pi\)
−0.0719350 + 0.997409i \(0.522917\pi\)
\(774\) −3.00000 −0.107833
\(775\) 16.0000 0.574737
\(776\) 11.0000 0.394877
\(777\) 0 0
\(778\) −6.00000 −0.215110
\(779\) 72.0000 2.57967
\(780\) −3.00000 −0.107417
\(781\) 36.0000 1.28818
\(782\) −2.00000 −0.0715199
\(783\) 6.00000 0.214423
\(784\) 0 0
\(785\) −18.0000 −0.642448
\(786\) −10.0000 −0.356688
\(787\) −53.0000 −1.88925 −0.944623 − 0.328158i \(-0.893572\pi\)
−0.944623 + 0.328158i \(0.893572\pi\)
\(788\) 0 0
\(789\) −19.0000 −0.676418
\(790\) −11.0000 −0.391362
\(791\) 0 0
\(792\) −3.00000 −0.106600
\(793\) 18.0000 0.639199
\(794\) 12.0000 0.425864
\(795\) −5.00000 −0.177332
\(796\) 0 0
\(797\) −6.00000 −0.212531 −0.106265 − 0.994338i \(-0.533889\pi\)
−0.106265 + 0.994338i \(0.533889\pi\)
\(798\) 0 0
\(799\) 12.0000 0.424529
\(800\) 4.00000 0.141421
\(801\) 13.0000 0.459332
\(802\) −14.0000 −0.494357
\(803\) −45.0000 −1.58802
\(804\) 9.00000 0.317406
\(805\) 0 0
\(806\) 12.0000 0.422682
\(807\) −18.0000 −0.633630
\(808\) −12.0000 −0.422159
\(809\) −30.0000 −1.05474 −0.527372 − 0.849635i \(-0.676823\pi\)
−0.527372 + 0.849635i \(0.676823\pi\)
\(810\) 1.00000 0.0351364
\(811\) 21.0000 0.737410 0.368705 − 0.929547i \(-0.379801\pi\)
0.368705 + 0.929547i \(0.379801\pi\)
\(812\) 0 0
\(813\) 9.00000 0.315644
\(814\) 33.0000 1.15665
\(815\) 22.0000 0.770626
\(816\) −1.00000 −0.0350070
\(817\) −18.0000 −0.629740
\(818\) −14.0000 −0.489499
\(819\) 0 0
\(820\) 12.0000 0.419058
\(821\) 50.0000 1.74501 0.872506 − 0.488603i \(-0.162493\pi\)
0.872506 + 0.488603i \(0.162493\pi\)
\(822\) −8.00000 −0.279032
\(823\) −43.0000 −1.49889 −0.749443 − 0.662069i \(-0.769679\pi\)
−0.749443 + 0.662069i \(0.769679\pi\)
\(824\) 1.00000 0.0348367
\(825\) −12.0000 −0.417786
\(826\) 0 0
\(827\) 7.00000 0.243414 0.121707 − 0.992566i \(-0.461163\pi\)
0.121707 + 0.992566i \(0.461163\pi\)
\(828\) −2.00000 −0.0695048
\(829\) −37.0000 −1.28506 −0.642532 − 0.766259i \(-0.722116\pi\)
−0.642532 + 0.766259i \(0.722116\pi\)
\(830\) −7.00000 −0.242974
\(831\) 18.0000 0.624413
\(832\) 3.00000 0.104006
\(833\) 0 0
\(834\) −1.00000 −0.0346272
\(835\) −5.00000 −0.173032
\(836\) −18.0000 −0.622543
\(837\) −4.00000 −0.138260
\(838\) −6.00000 −0.207267
\(839\) 35.0000 1.20833 0.604167 − 0.796858i \(-0.293506\pi\)
0.604167 + 0.796858i \(0.293506\pi\)
\(840\) 0 0
\(841\) 7.00000 0.241379
\(842\) 10.0000 0.344623
\(843\) −22.0000 −0.757720
\(844\) 14.0000 0.481900
\(845\) 4.00000 0.137604
\(846\) 12.0000 0.412568
\(847\) 0 0
\(848\) 5.00000 0.171701
\(849\) 13.0000 0.446159
\(850\) −4.00000 −0.137199
\(851\) 22.0000 0.754150
\(852\) 12.0000 0.411113
\(853\) 34.0000 1.16414 0.582069 − 0.813139i \(-0.302243\pi\)
0.582069 + 0.813139i \(0.302243\pi\)
\(854\) 0 0
\(855\) 6.00000 0.205196
\(856\) −16.0000 −0.546869
\(857\) −12.0000 −0.409912 −0.204956 − 0.978771i \(-0.565705\pi\)
−0.204956 + 0.978771i \(0.565705\pi\)
\(858\) −9.00000 −0.307255
\(859\) 38.0000 1.29654 0.648272 − 0.761409i \(-0.275492\pi\)
0.648272 + 0.761409i \(0.275492\pi\)
\(860\) −3.00000 −0.102299
\(861\) 0 0
\(862\) 4.00000 0.136241
\(863\) −7.00000 −0.238283 −0.119141 − 0.992877i \(-0.538014\pi\)
−0.119141 + 0.992877i \(0.538014\pi\)
\(864\) −1.00000 −0.0340207
\(865\) 10.0000 0.340010
\(866\) 32.0000 1.08740
\(867\) 1.00000 0.0339618
\(868\) 0 0
\(869\) −33.0000 −1.11945
\(870\) 6.00000 0.203419
\(871\) 27.0000 0.914860
\(872\) −6.00000 −0.203186
\(873\) −11.0000 −0.372294
\(874\) −12.0000 −0.405906
\(875\) 0 0
\(876\) −15.0000 −0.506803
\(877\) −1.00000 −0.0337676 −0.0168838 − 0.999857i \(-0.505375\pi\)
−0.0168838 + 0.999857i \(0.505375\pi\)
\(878\) −32.0000 −1.07995
\(879\) 8.00000 0.269833
\(880\) −3.00000 −0.101130
\(881\) −42.0000 −1.41502 −0.707508 − 0.706705i \(-0.750181\pi\)
−0.707508 + 0.706705i \(0.750181\pi\)
\(882\) 0 0
\(883\) 20.0000 0.673054 0.336527 − 0.941674i \(-0.390748\pi\)
0.336527 + 0.941674i \(0.390748\pi\)
\(884\) −3.00000 −0.100901
\(885\) 4.00000 0.134459
\(886\) −4.00000 −0.134383
\(887\) −13.0000 −0.436497 −0.218249 − 0.975893i \(-0.570034\pi\)
−0.218249 + 0.975893i \(0.570034\pi\)
\(888\) 11.0000 0.369136
\(889\) 0 0
\(890\) 13.0000 0.435761
\(891\) 3.00000 0.100504
\(892\) −16.0000 −0.535720
\(893\) 72.0000 2.40939
\(894\) 15.0000 0.501675
\(895\) −10.0000 −0.334263
\(896\) 0 0
\(897\) −6.00000 −0.200334
\(898\) −1.00000 −0.0333704
\(899\) −24.0000 −0.800445
\(900\) −4.00000 −0.133333
\(901\) −5.00000 −0.166574
\(902\) 36.0000 1.19867
\(903\) 0 0
\(904\) 15.0000 0.498893
\(905\) 20.0000 0.664822
\(906\) 16.0000 0.531564
\(907\) 36.0000 1.19536 0.597680 − 0.801735i \(-0.296089\pi\)
0.597680 + 0.801735i \(0.296089\pi\)
\(908\) −8.00000 −0.265489
\(909\) 12.0000 0.398015
\(910\) 0 0
\(911\) 40.0000 1.32526 0.662630 − 0.748947i \(-0.269440\pi\)
0.662630 + 0.748947i \(0.269440\pi\)
\(912\) −6.00000 −0.198680
\(913\) −21.0000 −0.694999
\(914\) 17.0000 0.562310
\(915\) −6.00000 −0.198354
\(916\) −19.0000 −0.627778
\(917\) 0 0
\(918\) 1.00000 0.0330049
\(919\) −16.0000 −0.527791 −0.263896 − 0.964551i \(-0.585007\pi\)
−0.263896 + 0.964551i \(0.585007\pi\)
\(920\) −2.00000 −0.0659380
\(921\) 8.00000 0.263609
\(922\) −30.0000 −0.987997
\(923\) 36.0000 1.18495
\(924\) 0 0
\(925\) 44.0000 1.44671
\(926\) 16.0000 0.525793
\(927\) −1.00000 −0.0328443
\(928\) −6.00000 −0.196960
\(929\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(930\) −4.00000 −0.131165
\(931\) 0 0
\(932\) −3.00000 −0.0982683
\(933\) −3.00000 −0.0982156
\(934\) 32.0000 1.04707
\(935\) 3.00000 0.0981105
\(936\) −3.00000 −0.0980581
\(937\) −8.00000 −0.261349 −0.130674 − 0.991425i \(-0.541714\pi\)
−0.130674 + 0.991425i \(0.541714\pi\)
\(938\) 0 0
\(939\) 22.0000 0.717943
\(940\) 12.0000 0.391397
\(941\) 21.0000 0.684580 0.342290 − 0.939594i \(-0.388797\pi\)
0.342290 + 0.939594i \(0.388797\pi\)
\(942\) −18.0000 −0.586472
\(943\) 24.0000 0.781548
\(944\) −4.00000 −0.130189
\(945\) 0 0
\(946\) −9.00000 −0.292615
\(947\) 28.0000 0.909878 0.454939 − 0.890523i \(-0.349661\pi\)
0.454939 + 0.890523i \(0.349661\pi\)
\(948\) −11.0000 −0.357263
\(949\) −45.0000 −1.46076
\(950\) −24.0000 −0.778663
\(951\) 18.0000 0.583690
\(952\) 0 0
\(953\) −6.00000 −0.194359 −0.0971795 − 0.995267i \(-0.530982\pi\)
−0.0971795 + 0.995267i \(0.530982\pi\)
\(954\) −5.00000 −0.161881
\(955\) 11.0000 0.355952
\(956\) −21.0000 −0.679189
\(957\) 18.0000 0.581857
\(958\) −11.0000 −0.355394
\(959\) 0 0
\(960\) −1.00000 −0.0322749
\(961\) −15.0000 −0.483871
\(962\) 33.0000 1.06396
\(963\) 16.0000 0.515593
\(964\) −14.0000 −0.450910
\(965\) 24.0000 0.772587
\(966\) 0 0
\(967\) 16.0000 0.514525 0.257263 − 0.966342i \(-0.417179\pi\)
0.257263 + 0.966342i \(0.417179\pi\)
\(968\) 2.00000 0.0642824
\(969\) 6.00000 0.192748
\(970\) −11.0000 −0.353189
\(971\) −3.00000 −0.0962746 −0.0481373 − 0.998841i \(-0.515328\pi\)
−0.0481373 + 0.998841i \(0.515328\pi\)
\(972\) 1.00000 0.0320750
\(973\) 0 0
\(974\) 13.0000 0.416547
\(975\) −12.0000 −0.384308
\(976\) 6.00000 0.192055
\(977\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(978\) 22.0000 0.703482
\(979\) 39.0000 1.24645
\(980\) 0 0
\(981\) 6.00000 0.191565
\(982\) 14.0000 0.446758
\(983\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(984\) 12.0000 0.382546
\(985\) 0 0
\(986\) 6.00000 0.191079
\(987\) 0 0
\(988\) −18.0000 −0.572656
\(989\) −6.00000 −0.190789
\(990\) 3.00000 0.0953463
\(991\) 13.0000 0.412959 0.206479 − 0.978451i \(-0.433799\pi\)
0.206479 + 0.978451i \(0.433799\pi\)
\(992\) 4.00000 0.127000
\(993\) −3.00000 −0.0952021
\(994\) 0 0
\(995\) 0 0
\(996\) −7.00000 −0.221803
\(997\) −60.0000 −1.90022 −0.950110 − 0.311916i \(-0.899029\pi\)
−0.950110 + 0.311916i \(0.899029\pi\)
\(998\) −20.0000 −0.633089
\(999\) −11.0000 −0.348025
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 4998.2.a.q.1.1 1
7.6 odd 2 714.2.a.b.1.1 ✓ 1
21.20 even 2 2142.2.a.o.1.1 1
28.27 even 2 5712.2.a.v.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
714.2.a.b.1.1 ✓ 1 7.6 odd 2
2142.2.a.o.1.1 1 21.20 even 2
4998.2.a.q.1.1 1 1.1 even 1 trivial
5712.2.a.v.1.1 1 28.27 even 2