Properties

Label 7098.2.a.d.1.1
Level $7098$
Weight $2$
Character 7098.1
Self dual yes
Analytic conductor $56.678$
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] = [7098,2,Mod(1,7098)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(7098, 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("7098.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 7098 = 2 \cdot 3 \cdot 7 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7098.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: \(56.6778153547\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 546)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 7098.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{6} +1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{6} +1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} -1.00000 q^{11} -1.00000 q^{12} -1.00000 q^{14} -1.00000 q^{15} +1.00000 q^{16} -1.00000 q^{17} -1.00000 q^{18} -1.00000 q^{19} +1.00000 q^{20} -1.00000 q^{21} +1.00000 q^{22} -3.00000 q^{23} +1.00000 q^{24} -4.00000 q^{25} -1.00000 q^{27} +1.00000 q^{28} +9.00000 q^{29} +1.00000 q^{30} -4.00000 q^{31} -1.00000 q^{32} +1.00000 q^{33} +1.00000 q^{34} +1.00000 q^{35} +1.00000 q^{36} +9.00000 q^{37} +1.00000 q^{38} -1.00000 q^{40} +8.00000 q^{41} +1.00000 q^{42} -7.00000 q^{43} -1.00000 q^{44} +1.00000 q^{45} +3.00000 q^{46} -8.00000 q^{47} -1.00000 q^{48} +1.00000 q^{49} +4.00000 q^{50} +1.00000 q^{51} -10.0000 q^{53} +1.00000 q^{54} -1.00000 q^{55} -1.00000 q^{56} +1.00000 q^{57} -9.00000 q^{58} +6.00000 q^{59} -1.00000 q^{60} +11.0000 q^{61} +4.00000 q^{62} +1.00000 q^{63} +1.00000 q^{64} -1.00000 q^{66} -12.0000 q^{67} -1.00000 q^{68} +3.00000 q^{69} -1.00000 q^{70} +6.00000 q^{71} -1.00000 q^{72} -11.0000 q^{73} -9.00000 q^{74} +4.00000 q^{75} -1.00000 q^{76} -1.00000 q^{77} -12.0000 q^{79} +1.00000 q^{80} +1.00000 q^{81} -8.00000 q^{82} +6.00000 q^{83} -1.00000 q^{84} -1.00000 q^{85} +7.00000 q^{86} -9.00000 q^{87} +1.00000 q^{88} -12.0000 q^{89} -1.00000 q^{90} -3.00000 q^{92} +4.00000 q^{93} +8.00000 q^{94} -1.00000 q^{95} +1.00000 q^{96} -2.00000 q^{97} -1.00000 q^{98} -1.00000 q^{99} +O(q^{100})\)

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
\(3\) −1.00000 −0.577350
\(4\) 1.00000 0.500000
\(5\) 1.00000 0.447214 0.223607 0.974679i \(-0.428217\pi\)
0.223607 + 0.974679i \(0.428217\pi\)
\(6\) 1.00000 0.408248
\(7\) 1.00000 0.377964
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) −1.00000 −0.316228
\(11\) −1.00000 −0.301511 −0.150756 0.988571i \(-0.548171\pi\)
−0.150756 + 0.988571i \(0.548171\pi\)
\(12\) −1.00000 −0.288675
\(13\) 0 0
\(14\) −1.00000 −0.267261
\(15\) −1.00000 −0.258199
\(16\) 1.00000 0.250000
\(17\) −1.00000 −0.242536 −0.121268 0.992620i \(-0.538696\pi\)
−0.121268 + 0.992620i \(0.538696\pi\)
\(18\) −1.00000 −0.235702
\(19\) −1.00000 −0.229416 −0.114708 0.993399i \(-0.536593\pi\)
−0.114708 + 0.993399i \(0.536593\pi\)
\(20\) 1.00000 0.223607
\(21\) −1.00000 −0.218218
\(22\) 1.00000 0.213201
\(23\) −3.00000 −0.625543 −0.312772 0.949828i \(-0.601257\pi\)
−0.312772 + 0.949828i \(0.601257\pi\)
\(24\) 1.00000 0.204124
\(25\) −4.00000 −0.800000
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 1.00000 0.188982
\(29\) 9.00000 1.67126 0.835629 0.549294i \(-0.185103\pi\)
0.835629 + 0.549294i \(0.185103\pi\)
\(30\) 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
\(33\) 1.00000 0.174078
\(34\) 1.00000 0.171499
\(35\) 1.00000 0.169031
\(36\) 1.00000 0.166667
\(37\) 9.00000 1.47959 0.739795 0.672832i \(-0.234922\pi\)
0.739795 + 0.672832i \(0.234922\pi\)
\(38\) 1.00000 0.162221
\(39\) 0 0
\(40\) −1.00000 −0.158114
\(41\) 8.00000 1.24939 0.624695 0.780869i \(-0.285223\pi\)
0.624695 + 0.780869i \(0.285223\pi\)
\(42\) 1.00000 0.154303
\(43\) −7.00000 −1.06749 −0.533745 0.845645i \(-0.679216\pi\)
−0.533745 + 0.845645i \(0.679216\pi\)
\(44\) −1.00000 −0.150756
\(45\) 1.00000 0.149071
\(46\) 3.00000 0.442326
\(47\) −8.00000 −1.16692 −0.583460 0.812142i \(-0.698301\pi\)
−0.583460 + 0.812142i \(0.698301\pi\)
\(48\) −1.00000 −0.144338
\(49\) 1.00000 0.142857
\(50\) 4.00000 0.565685
\(51\) 1.00000 0.140028
\(52\) 0 0
\(53\) −10.0000 −1.37361 −0.686803 0.726844i \(-0.740986\pi\)
−0.686803 + 0.726844i \(0.740986\pi\)
\(54\) 1.00000 0.136083
\(55\) −1.00000 −0.134840
\(56\) −1.00000 −0.133631
\(57\) 1.00000 0.132453
\(58\) −9.00000 −1.18176
\(59\) 6.00000 0.781133 0.390567 0.920575i \(-0.372279\pi\)
0.390567 + 0.920575i \(0.372279\pi\)
\(60\) −1.00000 −0.129099
\(61\) 11.0000 1.40841 0.704203 0.709999i \(-0.251305\pi\)
0.704203 + 0.709999i \(0.251305\pi\)
\(62\) 4.00000 0.508001
\(63\) 1.00000 0.125988
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) −1.00000 −0.123091
\(67\) −12.0000 −1.46603 −0.733017 0.680211i \(-0.761888\pi\)
−0.733017 + 0.680211i \(0.761888\pi\)
\(68\) −1.00000 −0.121268
\(69\) 3.00000 0.361158
\(70\) −1.00000 −0.119523
\(71\) 6.00000 0.712069 0.356034 0.934473i \(-0.384129\pi\)
0.356034 + 0.934473i \(0.384129\pi\)
\(72\) −1.00000 −0.117851
\(73\) −11.0000 −1.28745 −0.643726 0.765256i \(-0.722612\pi\)
−0.643726 + 0.765256i \(0.722612\pi\)
\(74\) −9.00000 −1.04623
\(75\) 4.00000 0.461880
\(76\) −1.00000 −0.114708
\(77\) −1.00000 −0.113961
\(78\) 0 0
\(79\) −12.0000 −1.35011 −0.675053 0.737769i \(-0.735879\pi\)
−0.675053 + 0.737769i \(0.735879\pi\)
\(80\) 1.00000 0.111803
\(81\) 1.00000 0.111111
\(82\) −8.00000 −0.883452
\(83\) 6.00000 0.658586 0.329293 0.944228i \(-0.393190\pi\)
0.329293 + 0.944228i \(0.393190\pi\)
\(84\) −1.00000 −0.109109
\(85\) −1.00000 −0.108465
\(86\) 7.00000 0.754829
\(87\) −9.00000 −0.964901
\(88\) 1.00000 0.106600
\(89\) −12.0000 −1.27200 −0.635999 0.771690i \(-0.719412\pi\)
−0.635999 + 0.771690i \(0.719412\pi\)
\(90\) −1.00000 −0.105409
\(91\) 0 0
\(92\) −3.00000 −0.312772
\(93\) 4.00000 0.414781
\(94\) 8.00000 0.825137
\(95\) −1.00000 −0.102598
\(96\) 1.00000 0.102062
\(97\) −2.00000 −0.203069 −0.101535 0.994832i \(-0.532375\pi\)
−0.101535 + 0.994832i \(0.532375\pi\)
\(98\) −1.00000 −0.101015
\(99\) −1.00000 −0.100504
\(100\) −4.00000 −0.400000
\(101\) 2.00000 0.199007 0.0995037 0.995037i \(-0.468274\pi\)
0.0995037 + 0.995037i \(0.468274\pi\)
\(102\) −1.00000 −0.0990148
\(103\) 5.00000 0.492665 0.246332 0.969185i \(-0.420775\pi\)
0.246332 + 0.969185i \(0.420775\pi\)
\(104\) 0 0
\(105\) −1.00000 −0.0975900
\(106\) 10.0000 0.971286
\(107\) −2.00000 −0.193347 −0.0966736 0.995316i \(-0.530820\pi\)
−0.0966736 + 0.995316i \(0.530820\pi\)
\(108\) −1.00000 −0.0962250
\(109\) 7.00000 0.670478 0.335239 0.942133i \(-0.391183\pi\)
0.335239 + 0.942133i \(0.391183\pi\)
\(110\) 1.00000 0.0953463
\(111\) −9.00000 −0.854242
\(112\) 1.00000 0.0944911
\(113\) −2.00000 −0.188144 −0.0940721 0.995565i \(-0.529988\pi\)
−0.0940721 + 0.995565i \(0.529988\pi\)
\(114\) −1.00000 −0.0936586
\(115\) −3.00000 −0.279751
\(116\) 9.00000 0.835629
\(117\) 0 0
\(118\) −6.00000 −0.552345
\(119\) −1.00000 −0.0916698
\(120\) 1.00000 0.0912871
\(121\) −10.0000 −0.909091
\(122\) −11.0000 −0.995893
\(123\) −8.00000 −0.721336
\(124\) −4.00000 −0.359211
\(125\) −9.00000 −0.804984
\(126\) −1.00000 −0.0890871
\(127\) −16.0000 −1.41977 −0.709885 0.704317i \(-0.751253\pi\)
−0.709885 + 0.704317i \(0.751253\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 7.00000 0.616316
\(130\) 0 0
\(131\) −3.00000 −0.262111 −0.131056 0.991375i \(-0.541837\pi\)
−0.131056 + 0.991375i \(0.541837\pi\)
\(132\) 1.00000 0.0870388
\(133\) −1.00000 −0.0867110
\(134\) 12.0000 1.03664
\(135\) −1.00000 −0.0860663
\(136\) 1.00000 0.0857493
\(137\) 15.0000 1.28154 0.640768 0.767734i \(-0.278616\pi\)
0.640768 + 0.767734i \(0.278616\pi\)
\(138\) −3.00000 −0.255377
\(139\) −12.0000 −1.01783 −0.508913 0.860818i \(-0.669953\pi\)
−0.508913 + 0.860818i \(0.669953\pi\)
\(140\) 1.00000 0.0845154
\(141\) 8.00000 0.673722
\(142\) −6.00000 −0.503509
\(143\) 0 0
\(144\) 1.00000 0.0833333
\(145\) 9.00000 0.747409
\(146\) 11.0000 0.910366
\(147\) −1.00000 −0.0824786
\(148\) 9.00000 0.739795
\(149\) 6.00000 0.491539 0.245770 0.969328i \(-0.420959\pi\)
0.245770 + 0.969328i \(0.420959\pi\)
\(150\) −4.00000 −0.326599
\(151\) −1.00000 −0.0813788 −0.0406894 0.999172i \(-0.512955\pi\)
−0.0406894 + 0.999172i \(0.512955\pi\)
\(152\) 1.00000 0.0811107
\(153\) −1.00000 −0.0808452
\(154\) 1.00000 0.0805823
\(155\) −4.00000 −0.321288
\(156\) 0 0
\(157\) −21.0000 −1.67598 −0.837991 0.545684i \(-0.816270\pi\)
−0.837991 + 0.545684i \(0.816270\pi\)
\(158\) 12.0000 0.954669
\(159\) 10.0000 0.793052
\(160\) −1.00000 −0.0790569
\(161\) −3.00000 −0.236433
\(162\) −1.00000 −0.0785674
\(163\) 20.0000 1.56652 0.783260 0.621694i \(-0.213555\pi\)
0.783260 + 0.621694i \(0.213555\pi\)
\(164\) 8.00000 0.624695
\(165\) 1.00000 0.0778499
\(166\) −6.00000 −0.465690
\(167\) 11.0000 0.851206 0.425603 0.904910i \(-0.360062\pi\)
0.425603 + 0.904910i \(0.360062\pi\)
\(168\) 1.00000 0.0771517
\(169\) 0 0
\(170\) 1.00000 0.0766965
\(171\) −1.00000 −0.0764719
\(172\) −7.00000 −0.533745
\(173\) −14.0000 −1.06440 −0.532200 0.846619i \(-0.678635\pi\)
−0.532200 + 0.846619i \(0.678635\pi\)
\(174\) 9.00000 0.682288
\(175\) −4.00000 −0.302372
\(176\) −1.00000 −0.0753778
\(177\) −6.00000 −0.450988
\(178\) 12.0000 0.899438
\(179\) 18.0000 1.34538 0.672692 0.739923i \(-0.265138\pi\)
0.672692 + 0.739923i \(0.265138\pi\)
\(180\) 1.00000 0.0745356
\(181\) 2.00000 0.148659 0.0743294 0.997234i \(-0.476318\pi\)
0.0743294 + 0.997234i \(0.476318\pi\)
\(182\) 0 0
\(183\) −11.0000 −0.813143
\(184\) 3.00000 0.221163
\(185\) 9.00000 0.661693
\(186\) −4.00000 −0.293294
\(187\) 1.00000 0.0731272
\(188\) −8.00000 −0.583460
\(189\) −1.00000 −0.0727393
\(190\) 1.00000 0.0725476
\(191\) −11.0000 −0.795932 −0.397966 0.917400i \(-0.630284\pi\)
−0.397966 + 0.917400i \(0.630284\pi\)
\(192\) −1.00000 −0.0721688
\(193\) 12.0000 0.863779 0.431889 0.901927i \(-0.357847\pi\)
0.431889 + 0.901927i \(0.357847\pi\)
\(194\) 2.00000 0.143592
\(195\) 0 0
\(196\) 1.00000 0.0714286
\(197\) −2.00000 −0.142494 −0.0712470 0.997459i \(-0.522698\pi\)
−0.0712470 + 0.997459i \(0.522698\pi\)
\(198\) 1.00000 0.0710669
\(199\) −5.00000 −0.354441 −0.177220 0.984171i \(-0.556711\pi\)
−0.177220 + 0.984171i \(0.556711\pi\)
\(200\) 4.00000 0.282843
\(201\) 12.0000 0.846415
\(202\) −2.00000 −0.140720
\(203\) 9.00000 0.631676
\(204\) 1.00000 0.0700140
\(205\) 8.00000 0.558744
\(206\) −5.00000 −0.348367
\(207\) −3.00000 −0.208514
\(208\) 0 0
\(209\) 1.00000 0.0691714
\(210\) 1.00000 0.0690066
\(211\) −5.00000 −0.344214 −0.172107 0.985078i \(-0.555058\pi\)
−0.172107 + 0.985078i \(0.555058\pi\)
\(212\) −10.0000 −0.686803
\(213\) −6.00000 −0.411113
\(214\) 2.00000 0.136717
\(215\) −7.00000 −0.477396
\(216\) 1.00000 0.0680414
\(217\) −4.00000 −0.271538
\(218\) −7.00000 −0.474100
\(219\) 11.0000 0.743311
\(220\) −1.00000 −0.0674200
\(221\) 0 0
\(222\) 9.00000 0.604040
\(223\) −8.00000 −0.535720 −0.267860 0.963458i \(-0.586316\pi\)
−0.267860 + 0.963458i \(0.586316\pi\)
\(224\) −1.00000 −0.0668153
\(225\) −4.00000 −0.266667
\(226\) 2.00000 0.133038
\(227\) −12.0000 −0.796468 −0.398234 0.917284i \(-0.630377\pi\)
−0.398234 + 0.917284i \(0.630377\pi\)
\(228\) 1.00000 0.0662266
\(229\) −2.00000 −0.132164 −0.0660819 0.997814i \(-0.521050\pi\)
−0.0660819 + 0.997814i \(0.521050\pi\)
\(230\) 3.00000 0.197814
\(231\) 1.00000 0.0657952
\(232\) −9.00000 −0.590879
\(233\) 18.0000 1.17922 0.589610 0.807688i \(-0.299282\pi\)
0.589610 + 0.807688i \(0.299282\pi\)
\(234\) 0 0
\(235\) −8.00000 −0.521862
\(236\) 6.00000 0.390567
\(237\) 12.0000 0.779484
\(238\) 1.00000 0.0648204
\(239\) −8.00000 −0.517477 −0.258738 0.965947i \(-0.583307\pi\)
−0.258738 + 0.965947i \(0.583307\pi\)
\(240\) −1.00000 −0.0645497
\(241\) −10.0000 −0.644157 −0.322078 0.946713i \(-0.604381\pi\)
−0.322078 + 0.946713i \(0.604381\pi\)
\(242\) 10.0000 0.642824
\(243\) −1.00000 −0.0641500
\(244\) 11.0000 0.704203
\(245\) 1.00000 0.0638877
\(246\) 8.00000 0.510061
\(247\) 0 0
\(248\) 4.00000 0.254000
\(249\) −6.00000 −0.380235
\(250\) 9.00000 0.569210
\(251\) −11.0000 −0.694314 −0.347157 0.937807i \(-0.612853\pi\)
−0.347157 + 0.937807i \(0.612853\pi\)
\(252\) 1.00000 0.0629941
\(253\) 3.00000 0.188608
\(254\) 16.0000 1.00393
\(255\) 1.00000 0.0626224
\(256\) 1.00000 0.0625000
\(257\) −14.0000 −0.873296 −0.436648 0.899632i \(-0.643834\pi\)
−0.436648 + 0.899632i \(0.643834\pi\)
\(258\) −7.00000 −0.435801
\(259\) 9.00000 0.559233
\(260\) 0 0
\(261\) 9.00000 0.557086
\(262\) 3.00000 0.185341
\(263\) 16.0000 0.986602 0.493301 0.869859i \(-0.335790\pi\)
0.493301 + 0.869859i \(0.335790\pi\)
\(264\) −1.00000 −0.0615457
\(265\) −10.0000 −0.614295
\(266\) 1.00000 0.0613139
\(267\) 12.0000 0.734388
\(268\) −12.0000 −0.733017
\(269\) −20.0000 −1.21942 −0.609711 0.792624i \(-0.708714\pi\)
−0.609711 + 0.792624i \(0.708714\pi\)
\(270\) 1.00000 0.0608581
\(271\) 4.00000 0.242983 0.121491 0.992592i \(-0.461232\pi\)
0.121491 + 0.992592i \(0.461232\pi\)
\(272\) −1.00000 −0.0606339
\(273\) 0 0
\(274\) −15.0000 −0.906183
\(275\) 4.00000 0.241209
\(276\) 3.00000 0.180579
\(277\) −16.0000 −0.961347 −0.480673 0.876900i \(-0.659608\pi\)
−0.480673 + 0.876900i \(0.659608\pi\)
\(278\) 12.0000 0.719712
\(279\) −4.00000 −0.239474
\(280\) −1.00000 −0.0597614
\(281\) 22.0000 1.31241 0.656205 0.754583i \(-0.272161\pi\)
0.656205 + 0.754583i \(0.272161\pi\)
\(282\) −8.00000 −0.476393
\(283\) 14.0000 0.832214 0.416107 0.909316i \(-0.363394\pi\)
0.416107 + 0.909316i \(0.363394\pi\)
\(284\) 6.00000 0.356034
\(285\) 1.00000 0.0592349
\(286\) 0 0
\(287\) 8.00000 0.472225
\(288\) −1.00000 −0.0589256
\(289\) −16.0000 −0.941176
\(290\) −9.00000 −0.528498
\(291\) 2.00000 0.117242
\(292\) −11.0000 −0.643726
\(293\) 26.0000 1.51894 0.759468 0.650545i \(-0.225459\pi\)
0.759468 + 0.650545i \(0.225459\pi\)
\(294\) 1.00000 0.0583212
\(295\) 6.00000 0.349334
\(296\) −9.00000 −0.523114
\(297\) 1.00000 0.0580259
\(298\) −6.00000 −0.347571
\(299\) 0 0
\(300\) 4.00000 0.230940
\(301\) −7.00000 −0.403473
\(302\) 1.00000 0.0575435
\(303\) −2.00000 −0.114897
\(304\) −1.00000 −0.0573539
\(305\) 11.0000 0.629858
\(306\) 1.00000 0.0571662
\(307\) 8.00000 0.456584 0.228292 0.973593i \(-0.426686\pi\)
0.228292 + 0.973593i \(0.426686\pi\)
\(308\) −1.00000 −0.0569803
\(309\) −5.00000 −0.284440
\(310\) 4.00000 0.227185
\(311\) −8.00000 −0.453638 −0.226819 0.973937i \(-0.572833\pi\)
−0.226819 + 0.973937i \(0.572833\pi\)
\(312\) 0 0
\(313\) −14.0000 −0.791327 −0.395663 0.918396i \(-0.629485\pi\)
−0.395663 + 0.918396i \(0.629485\pi\)
\(314\) 21.0000 1.18510
\(315\) 1.00000 0.0563436
\(316\) −12.0000 −0.675053
\(317\) −12.0000 −0.673987 −0.336994 0.941507i \(-0.609410\pi\)
−0.336994 + 0.941507i \(0.609410\pi\)
\(318\) −10.0000 −0.560772
\(319\) −9.00000 −0.503903
\(320\) 1.00000 0.0559017
\(321\) 2.00000 0.111629
\(322\) 3.00000 0.167183
\(323\) 1.00000 0.0556415
\(324\) 1.00000 0.0555556
\(325\) 0 0
\(326\) −20.0000 −1.10770
\(327\) −7.00000 −0.387101
\(328\) −8.00000 −0.441726
\(329\) −8.00000 −0.441054
\(330\) −1.00000 −0.0550482
\(331\) −8.00000 −0.439720 −0.219860 0.975531i \(-0.570560\pi\)
−0.219860 + 0.975531i \(0.570560\pi\)
\(332\) 6.00000 0.329293
\(333\) 9.00000 0.493197
\(334\) −11.0000 −0.601893
\(335\) −12.0000 −0.655630
\(336\) −1.00000 −0.0545545
\(337\) −17.0000 −0.926049 −0.463025 0.886345i \(-0.653236\pi\)
−0.463025 + 0.886345i \(0.653236\pi\)
\(338\) 0 0
\(339\) 2.00000 0.108625
\(340\) −1.00000 −0.0542326
\(341\) 4.00000 0.216612
\(342\) 1.00000 0.0540738
\(343\) 1.00000 0.0539949
\(344\) 7.00000 0.377415
\(345\) 3.00000 0.161515
\(346\) 14.0000 0.752645
\(347\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(348\) −9.00000 −0.482451
\(349\) 4.00000 0.214115 0.107058 0.994253i \(-0.465857\pi\)
0.107058 + 0.994253i \(0.465857\pi\)
\(350\) 4.00000 0.213809
\(351\) 0 0
\(352\) 1.00000 0.0533002
\(353\) −16.0000 −0.851594 −0.425797 0.904819i \(-0.640006\pi\)
−0.425797 + 0.904819i \(0.640006\pi\)
\(354\) 6.00000 0.318896
\(355\) 6.00000 0.318447
\(356\) −12.0000 −0.635999
\(357\) 1.00000 0.0529256
\(358\) −18.0000 −0.951330
\(359\) −30.0000 −1.58334 −0.791670 0.610949i \(-0.790788\pi\)
−0.791670 + 0.610949i \(0.790788\pi\)
\(360\) −1.00000 −0.0527046
\(361\) −18.0000 −0.947368
\(362\) −2.00000 −0.105118
\(363\) 10.0000 0.524864
\(364\) 0 0
\(365\) −11.0000 −0.575766
\(366\) 11.0000 0.574979
\(367\) −8.00000 −0.417597 −0.208798 0.977959i \(-0.566955\pi\)
−0.208798 + 0.977959i \(0.566955\pi\)
\(368\) −3.00000 −0.156386
\(369\) 8.00000 0.416463
\(370\) −9.00000 −0.467888
\(371\) −10.0000 −0.519174
\(372\) 4.00000 0.207390
\(373\) −14.0000 −0.724893 −0.362446 0.932005i \(-0.618058\pi\)
−0.362446 + 0.932005i \(0.618058\pi\)
\(374\) −1.00000 −0.0517088
\(375\) 9.00000 0.464758
\(376\) 8.00000 0.412568
\(377\) 0 0
\(378\) 1.00000 0.0514344
\(379\) 16.0000 0.821865 0.410932 0.911666i \(-0.365203\pi\)
0.410932 + 0.911666i \(0.365203\pi\)
\(380\) −1.00000 −0.0512989
\(381\) 16.0000 0.819705
\(382\) 11.0000 0.562809
\(383\) 35.0000 1.78842 0.894208 0.447651i \(-0.147739\pi\)
0.894208 + 0.447651i \(0.147739\pi\)
\(384\) 1.00000 0.0510310
\(385\) −1.00000 −0.0509647
\(386\) −12.0000 −0.610784
\(387\) −7.00000 −0.355830
\(388\) −2.00000 −0.101535
\(389\) −2.00000 −0.101404 −0.0507020 0.998714i \(-0.516146\pi\)
−0.0507020 + 0.998714i \(0.516146\pi\)
\(390\) 0 0
\(391\) 3.00000 0.151717
\(392\) −1.00000 −0.0505076
\(393\) 3.00000 0.151330
\(394\) 2.00000 0.100759
\(395\) −12.0000 −0.603786
\(396\) −1.00000 −0.0502519
\(397\) −4.00000 −0.200754 −0.100377 0.994949i \(-0.532005\pi\)
−0.100377 + 0.994949i \(0.532005\pi\)
\(398\) 5.00000 0.250627
\(399\) 1.00000 0.0500626
\(400\) −4.00000 −0.200000
\(401\) −14.0000 −0.699127 −0.349563 0.936913i \(-0.613670\pi\)
−0.349563 + 0.936913i \(0.613670\pi\)
\(402\) −12.0000 −0.598506
\(403\) 0 0
\(404\) 2.00000 0.0995037
\(405\) 1.00000 0.0496904
\(406\) −9.00000 −0.446663
\(407\) −9.00000 −0.446113
\(408\) −1.00000 −0.0495074
\(409\) 27.0000 1.33506 0.667532 0.744581i \(-0.267351\pi\)
0.667532 + 0.744581i \(0.267351\pi\)
\(410\) −8.00000 −0.395092
\(411\) −15.0000 −0.739895
\(412\) 5.00000 0.246332
\(413\) 6.00000 0.295241
\(414\) 3.00000 0.147442
\(415\) 6.00000 0.294528
\(416\) 0 0
\(417\) 12.0000 0.587643
\(418\) −1.00000 −0.0489116
\(419\) 23.0000 1.12362 0.561812 0.827265i \(-0.310105\pi\)
0.561812 + 0.827265i \(0.310105\pi\)
\(420\) −1.00000 −0.0487950
\(421\) 22.0000 1.07221 0.536107 0.844150i \(-0.319894\pi\)
0.536107 + 0.844150i \(0.319894\pi\)
\(422\) 5.00000 0.243396
\(423\) −8.00000 −0.388973
\(424\) 10.0000 0.485643
\(425\) 4.00000 0.194029
\(426\) 6.00000 0.290701
\(427\) 11.0000 0.532327
\(428\) −2.00000 −0.0966736
\(429\) 0 0
\(430\) 7.00000 0.337570
\(431\) −18.0000 −0.867029 −0.433515 0.901146i \(-0.642727\pi\)
−0.433515 + 0.901146i \(0.642727\pi\)
\(432\) −1.00000 −0.0481125
\(433\) −4.00000 −0.192228 −0.0961139 0.995370i \(-0.530641\pi\)
−0.0961139 + 0.995370i \(0.530641\pi\)
\(434\) 4.00000 0.192006
\(435\) −9.00000 −0.431517
\(436\) 7.00000 0.335239
\(437\) 3.00000 0.143509
\(438\) −11.0000 −0.525600
\(439\) 33.0000 1.57500 0.787502 0.616312i \(-0.211374\pi\)
0.787502 + 0.616312i \(0.211374\pi\)
\(440\) 1.00000 0.0476731
\(441\) 1.00000 0.0476190
\(442\) 0 0
\(443\) −6.00000 −0.285069 −0.142534 0.989790i \(-0.545525\pi\)
−0.142534 + 0.989790i \(0.545525\pi\)
\(444\) −9.00000 −0.427121
\(445\) −12.0000 −0.568855
\(446\) 8.00000 0.378811
\(447\) −6.00000 −0.283790
\(448\) 1.00000 0.0472456
\(449\) 15.0000 0.707894 0.353947 0.935266i \(-0.384839\pi\)
0.353947 + 0.935266i \(0.384839\pi\)
\(450\) 4.00000 0.188562
\(451\) −8.00000 −0.376705
\(452\) −2.00000 −0.0940721
\(453\) 1.00000 0.0469841
\(454\) 12.0000 0.563188
\(455\) 0 0
\(456\) −1.00000 −0.0468293
\(457\) 4.00000 0.187112 0.0935561 0.995614i \(-0.470177\pi\)
0.0935561 + 0.995614i \(0.470177\pi\)
\(458\) 2.00000 0.0934539
\(459\) 1.00000 0.0466760
\(460\) −3.00000 −0.139876
\(461\) 3.00000 0.139724 0.0698620 0.997557i \(-0.477744\pi\)
0.0698620 + 0.997557i \(0.477744\pi\)
\(462\) −1.00000 −0.0465242
\(463\) −29.0000 −1.34774 −0.673872 0.738848i \(-0.735370\pi\)
−0.673872 + 0.738848i \(0.735370\pi\)
\(464\) 9.00000 0.417815
\(465\) 4.00000 0.185496
\(466\) −18.0000 −0.833834
\(467\) 37.0000 1.71216 0.856078 0.516847i \(-0.172894\pi\)
0.856078 + 0.516847i \(0.172894\pi\)
\(468\) 0 0
\(469\) −12.0000 −0.554109
\(470\) 8.00000 0.369012
\(471\) 21.0000 0.967629
\(472\) −6.00000 −0.276172
\(473\) 7.00000 0.321860
\(474\) −12.0000 −0.551178
\(475\) 4.00000 0.183533
\(476\) −1.00000 −0.0458349
\(477\) −10.0000 −0.457869
\(478\) 8.00000 0.365911
\(479\) −39.0000 −1.78196 −0.890978 0.454047i \(-0.849980\pi\)
−0.890978 + 0.454047i \(0.849980\pi\)
\(480\) 1.00000 0.0456435
\(481\) 0 0
\(482\) 10.0000 0.455488
\(483\) 3.00000 0.136505
\(484\) −10.0000 −0.454545
\(485\) −2.00000 −0.0908153
\(486\) 1.00000 0.0453609
\(487\) 32.0000 1.45006 0.725029 0.688718i \(-0.241826\pi\)
0.725029 + 0.688718i \(0.241826\pi\)
\(488\) −11.0000 −0.497947
\(489\) −20.0000 −0.904431
\(490\) −1.00000 −0.0451754
\(491\) −12.0000 −0.541552 −0.270776 0.962642i \(-0.587280\pi\)
−0.270776 + 0.962642i \(0.587280\pi\)
\(492\) −8.00000 −0.360668
\(493\) −9.00000 −0.405340
\(494\) 0 0
\(495\) −1.00000 −0.0449467
\(496\) −4.00000 −0.179605
\(497\) 6.00000 0.269137
\(498\) 6.00000 0.268866
\(499\) −10.0000 −0.447661 −0.223831 0.974628i \(-0.571856\pi\)
−0.223831 + 0.974628i \(0.571856\pi\)
\(500\) −9.00000 −0.402492
\(501\) −11.0000 −0.491444
\(502\) 11.0000 0.490954
\(503\) −38.0000 −1.69434 −0.847168 0.531325i \(-0.821694\pi\)
−0.847168 + 0.531325i \(0.821694\pi\)
\(504\) −1.00000 −0.0445435
\(505\) 2.00000 0.0889988
\(506\) −3.00000 −0.133366
\(507\) 0 0
\(508\) −16.0000 −0.709885
\(509\) −37.0000 −1.64000 −0.819998 0.572366i \(-0.806026\pi\)
−0.819998 + 0.572366i \(0.806026\pi\)
\(510\) −1.00000 −0.0442807
\(511\) −11.0000 −0.486611
\(512\) −1.00000 −0.0441942
\(513\) 1.00000 0.0441511
\(514\) 14.0000 0.617514
\(515\) 5.00000 0.220326
\(516\) 7.00000 0.308158
\(517\) 8.00000 0.351840
\(518\) −9.00000 −0.395437
\(519\) 14.0000 0.614532
\(520\) 0 0
\(521\) −31.0000 −1.35813 −0.679067 0.734076i \(-0.737616\pi\)
−0.679067 + 0.734076i \(0.737616\pi\)
\(522\) −9.00000 −0.393919
\(523\) −36.0000 −1.57417 −0.787085 0.616844i \(-0.788411\pi\)
−0.787085 + 0.616844i \(0.788411\pi\)
\(524\) −3.00000 −0.131056
\(525\) 4.00000 0.174574
\(526\) −16.0000 −0.697633
\(527\) 4.00000 0.174243
\(528\) 1.00000 0.0435194
\(529\) −14.0000 −0.608696
\(530\) 10.0000 0.434372
\(531\) 6.00000 0.260378
\(532\) −1.00000 −0.0433555
\(533\) 0 0
\(534\) −12.0000 −0.519291
\(535\) −2.00000 −0.0864675
\(536\) 12.0000 0.518321
\(537\) −18.0000 −0.776757
\(538\) 20.0000 0.862261
\(539\) −1.00000 −0.0430730
\(540\) −1.00000 −0.0430331
\(541\) −41.0000 −1.76273 −0.881364 0.472438i \(-0.843374\pi\)
−0.881364 + 0.472438i \(0.843374\pi\)
\(542\) −4.00000 −0.171815
\(543\) −2.00000 −0.0858282
\(544\) 1.00000 0.0428746
\(545\) 7.00000 0.299847
\(546\) 0 0
\(547\) −28.0000 −1.19719 −0.598597 0.801050i \(-0.704275\pi\)
−0.598597 + 0.801050i \(0.704275\pi\)
\(548\) 15.0000 0.640768
\(549\) 11.0000 0.469469
\(550\) −4.00000 −0.170561
\(551\) −9.00000 −0.383413
\(552\) −3.00000 −0.127688
\(553\) −12.0000 −0.510292
\(554\) 16.0000 0.679775
\(555\) −9.00000 −0.382029
\(556\) −12.0000 −0.508913
\(557\) 2.00000 0.0847427 0.0423714 0.999102i \(-0.486509\pi\)
0.0423714 + 0.999102i \(0.486509\pi\)
\(558\) 4.00000 0.169334
\(559\) 0 0
\(560\) 1.00000 0.0422577
\(561\) −1.00000 −0.0422200
\(562\) −22.0000 −0.928014
\(563\) −11.0000 −0.463595 −0.231797 0.972764i \(-0.574461\pi\)
−0.231797 + 0.972764i \(0.574461\pi\)
\(564\) 8.00000 0.336861
\(565\) −2.00000 −0.0841406
\(566\) −14.0000 −0.588464
\(567\) 1.00000 0.0419961
\(568\) −6.00000 −0.251754
\(569\) 30.0000 1.25767 0.628833 0.777541i \(-0.283533\pi\)
0.628833 + 0.777541i \(0.283533\pi\)
\(570\) −1.00000 −0.0418854
\(571\) −32.0000 −1.33916 −0.669579 0.742741i \(-0.733526\pi\)
−0.669579 + 0.742741i \(0.733526\pi\)
\(572\) 0 0
\(573\) 11.0000 0.459532
\(574\) −8.00000 −0.333914
\(575\) 12.0000 0.500435
\(576\) 1.00000 0.0416667
\(577\) −18.0000 −0.749350 −0.374675 0.927156i \(-0.622246\pi\)
−0.374675 + 0.927156i \(0.622246\pi\)
\(578\) 16.0000 0.665512
\(579\) −12.0000 −0.498703
\(580\) 9.00000 0.373705
\(581\) 6.00000 0.248922
\(582\) −2.00000 −0.0829027
\(583\) 10.0000 0.414158
\(584\) 11.0000 0.455183
\(585\) 0 0
\(586\) −26.0000 −1.07405
\(587\) −16.0000 −0.660391 −0.330195 0.943913i \(-0.607115\pi\)
−0.330195 + 0.943913i \(0.607115\pi\)
\(588\) −1.00000 −0.0412393
\(589\) 4.00000 0.164817
\(590\) −6.00000 −0.247016
\(591\) 2.00000 0.0822690
\(592\) 9.00000 0.369898
\(593\) 28.0000 1.14982 0.574911 0.818216i \(-0.305037\pi\)
0.574911 + 0.818216i \(0.305037\pi\)
\(594\) −1.00000 −0.0410305
\(595\) −1.00000 −0.0409960
\(596\) 6.00000 0.245770
\(597\) 5.00000 0.204636
\(598\) 0 0
\(599\) 39.0000 1.59350 0.796748 0.604311i \(-0.206552\pi\)
0.796748 + 0.604311i \(0.206552\pi\)
\(600\) −4.00000 −0.163299
\(601\) −16.0000 −0.652654 −0.326327 0.945257i \(-0.605811\pi\)
−0.326327 + 0.945257i \(0.605811\pi\)
\(602\) 7.00000 0.285299
\(603\) −12.0000 −0.488678
\(604\) −1.00000 −0.0406894
\(605\) −10.0000 −0.406558
\(606\) 2.00000 0.0812444
\(607\) 1.00000 0.0405887 0.0202944 0.999794i \(-0.493540\pi\)
0.0202944 + 0.999794i \(0.493540\pi\)
\(608\) 1.00000 0.0405554
\(609\) −9.00000 −0.364698
\(610\) −11.0000 −0.445377
\(611\) 0 0
\(612\) −1.00000 −0.0404226
\(613\) −17.0000 −0.686624 −0.343312 0.939222i \(-0.611549\pi\)
−0.343312 + 0.939222i \(0.611549\pi\)
\(614\) −8.00000 −0.322854
\(615\) −8.00000 −0.322591
\(616\) 1.00000 0.0402911
\(617\) −19.0000 −0.764911 −0.382456 0.923974i \(-0.624922\pi\)
−0.382456 + 0.923974i \(0.624922\pi\)
\(618\) 5.00000 0.201129
\(619\) −21.0000 −0.844061 −0.422031 0.906582i \(-0.638683\pi\)
−0.422031 + 0.906582i \(0.638683\pi\)
\(620\) −4.00000 −0.160644
\(621\) 3.00000 0.120386
\(622\) 8.00000 0.320771
\(623\) −12.0000 −0.480770
\(624\) 0 0
\(625\) 11.0000 0.440000
\(626\) 14.0000 0.559553
\(627\) −1.00000 −0.0399362
\(628\) −21.0000 −0.837991
\(629\) −9.00000 −0.358854
\(630\) −1.00000 −0.0398410
\(631\) −29.0000 −1.15447 −0.577236 0.816577i \(-0.695869\pi\)
−0.577236 + 0.816577i \(0.695869\pi\)
\(632\) 12.0000 0.477334
\(633\) 5.00000 0.198732
\(634\) 12.0000 0.476581
\(635\) −16.0000 −0.634941
\(636\) 10.0000 0.396526
\(637\) 0 0
\(638\) 9.00000 0.356313
\(639\) 6.00000 0.237356
\(640\) −1.00000 −0.0395285
\(641\) 44.0000 1.73790 0.868948 0.494904i \(-0.164797\pi\)
0.868948 + 0.494904i \(0.164797\pi\)
\(642\) −2.00000 −0.0789337
\(643\) −19.0000 −0.749287 −0.374643 0.927169i \(-0.622235\pi\)
−0.374643 + 0.927169i \(0.622235\pi\)
\(644\) −3.00000 −0.118217
\(645\) 7.00000 0.275625
\(646\) −1.00000 −0.0393445
\(647\) −42.0000 −1.65119 −0.825595 0.564263i \(-0.809160\pi\)
−0.825595 + 0.564263i \(0.809160\pi\)
\(648\) −1.00000 −0.0392837
\(649\) −6.00000 −0.235521
\(650\) 0 0
\(651\) 4.00000 0.156772
\(652\) 20.0000 0.783260
\(653\) −13.0000 −0.508729 −0.254365 0.967108i \(-0.581866\pi\)
−0.254365 + 0.967108i \(0.581866\pi\)
\(654\) 7.00000 0.273722
\(655\) −3.00000 −0.117220
\(656\) 8.00000 0.312348
\(657\) −11.0000 −0.429151
\(658\) 8.00000 0.311872
\(659\) −30.0000 −1.16863 −0.584317 0.811525i \(-0.698638\pi\)
−0.584317 + 0.811525i \(0.698638\pi\)
\(660\) 1.00000 0.0389249
\(661\) −26.0000 −1.01128 −0.505641 0.862744i \(-0.668744\pi\)
−0.505641 + 0.862744i \(0.668744\pi\)
\(662\) 8.00000 0.310929
\(663\) 0 0
\(664\) −6.00000 −0.232845
\(665\) −1.00000 −0.0387783
\(666\) −9.00000 −0.348743
\(667\) −27.0000 −1.04544
\(668\) 11.0000 0.425603
\(669\) 8.00000 0.309298
\(670\) 12.0000 0.463600
\(671\) −11.0000 −0.424650
\(672\) 1.00000 0.0385758
\(673\) −29.0000 −1.11787 −0.558934 0.829212i \(-0.688789\pi\)
−0.558934 + 0.829212i \(0.688789\pi\)
\(674\) 17.0000 0.654816
\(675\) 4.00000 0.153960
\(676\) 0 0
\(677\) 32.0000 1.22986 0.614930 0.788582i \(-0.289184\pi\)
0.614930 + 0.788582i \(0.289184\pi\)
\(678\) −2.00000 −0.0768095
\(679\) −2.00000 −0.0767530
\(680\) 1.00000 0.0383482
\(681\) 12.0000 0.459841
\(682\) −4.00000 −0.153168
\(683\) −19.0000 −0.727015 −0.363507 0.931591i \(-0.618421\pi\)
−0.363507 + 0.931591i \(0.618421\pi\)
\(684\) −1.00000 −0.0382360
\(685\) 15.0000 0.573121
\(686\) −1.00000 −0.0381802
\(687\) 2.00000 0.0763048
\(688\) −7.00000 −0.266872
\(689\) 0 0
\(690\) −3.00000 −0.114208
\(691\) −40.0000 −1.52167 −0.760836 0.648944i \(-0.775211\pi\)
−0.760836 + 0.648944i \(0.775211\pi\)
\(692\) −14.0000 −0.532200
\(693\) −1.00000 −0.0379869
\(694\) 0 0
\(695\) −12.0000 −0.455186
\(696\) 9.00000 0.341144
\(697\) −8.00000 −0.303022
\(698\) −4.00000 −0.151402
\(699\) −18.0000 −0.680823
\(700\) −4.00000 −0.151186
\(701\) 30.0000 1.13308 0.566542 0.824033i \(-0.308281\pi\)
0.566542 + 0.824033i \(0.308281\pi\)
\(702\) 0 0
\(703\) −9.00000 −0.339441
\(704\) −1.00000 −0.0376889
\(705\) 8.00000 0.301297
\(706\) 16.0000 0.602168
\(707\) 2.00000 0.0752177
\(708\) −6.00000 −0.225494
\(709\) 14.0000 0.525781 0.262891 0.964826i \(-0.415324\pi\)
0.262891 + 0.964826i \(0.415324\pi\)
\(710\) −6.00000 −0.225176
\(711\) −12.0000 −0.450035
\(712\) 12.0000 0.449719
\(713\) 12.0000 0.449404
\(714\) −1.00000 −0.0374241
\(715\) 0 0
\(716\) 18.0000 0.672692
\(717\) 8.00000 0.298765
\(718\) 30.0000 1.11959
\(719\) 8.00000 0.298350 0.149175 0.988811i \(-0.452338\pi\)
0.149175 + 0.988811i \(0.452338\pi\)
\(720\) 1.00000 0.0372678
\(721\) 5.00000 0.186210
\(722\) 18.0000 0.669891
\(723\) 10.0000 0.371904
\(724\) 2.00000 0.0743294
\(725\) −36.0000 −1.33701
\(726\) −10.0000 −0.371135
\(727\) −23.0000 −0.853023 −0.426511 0.904482i \(-0.640258\pi\)
−0.426511 + 0.904482i \(0.640258\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 11.0000 0.407128
\(731\) 7.00000 0.258904
\(732\) −11.0000 −0.406572
\(733\) −40.0000 −1.47743 −0.738717 0.674016i \(-0.764568\pi\)
−0.738717 + 0.674016i \(0.764568\pi\)
\(734\) 8.00000 0.295285
\(735\) −1.00000 −0.0368856
\(736\) 3.00000 0.110581
\(737\) 12.0000 0.442026
\(738\) −8.00000 −0.294484
\(739\) −8.00000 −0.294285 −0.147142 0.989115i \(-0.547008\pi\)
−0.147142 + 0.989115i \(0.547008\pi\)
\(740\) 9.00000 0.330847
\(741\) 0 0
\(742\) 10.0000 0.367112
\(743\) −36.0000 −1.32071 −0.660356 0.750953i \(-0.729595\pi\)
−0.660356 + 0.750953i \(0.729595\pi\)
\(744\) −4.00000 −0.146647
\(745\) 6.00000 0.219823
\(746\) 14.0000 0.512576
\(747\) 6.00000 0.219529
\(748\) 1.00000 0.0365636
\(749\) −2.00000 −0.0730784
\(750\) −9.00000 −0.328634
\(751\) −20.0000 −0.729810 −0.364905 0.931045i \(-0.618899\pi\)
−0.364905 + 0.931045i \(0.618899\pi\)
\(752\) −8.00000 −0.291730
\(753\) 11.0000 0.400862
\(754\) 0 0
\(755\) −1.00000 −0.0363937
\(756\) −1.00000 −0.0363696
\(757\) 38.0000 1.38113 0.690567 0.723269i \(-0.257361\pi\)
0.690567 + 0.723269i \(0.257361\pi\)
\(758\) −16.0000 −0.581146
\(759\) −3.00000 −0.108893
\(760\) 1.00000 0.0362738
\(761\) 42.0000 1.52250 0.761249 0.648459i \(-0.224586\pi\)
0.761249 + 0.648459i \(0.224586\pi\)
\(762\) −16.0000 −0.579619
\(763\) 7.00000 0.253417
\(764\) −11.0000 −0.397966
\(765\) −1.00000 −0.0361551
\(766\) −35.0000 −1.26460
\(767\) 0 0
\(768\) −1.00000 −0.0360844
\(769\) 31.0000 1.11789 0.558944 0.829205i \(-0.311207\pi\)
0.558944 + 0.829205i \(0.311207\pi\)
\(770\) 1.00000 0.0360375
\(771\) 14.0000 0.504198
\(772\) 12.0000 0.431889
\(773\) 19.0000 0.683383 0.341691 0.939812i \(-0.389000\pi\)
0.341691 + 0.939812i \(0.389000\pi\)
\(774\) 7.00000 0.251610
\(775\) 16.0000 0.574737
\(776\) 2.00000 0.0717958
\(777\) −9.00000 −0.322873
\(778\) 2.00000 0.0717035
\(779\) −8.00000 −0.286630
\(780\) 0 0
\(781\) −6.00000 −0.214697
\(782\) −3.00000 −0.107280
\(783\) −9.00000 −0.321634
\(784\) 1.00000 0.0357143
\(785\) −21.0000 −0.749522
\(786\) −3.00000 −0.107006
\(787\) −53.0000 −1.88925 −0.944623 0.328158i \(-0.893572\pi\)
−0.944623 + 0.328158i \(0.893572\pi\)
\(788\) −2.00000 −0.0712470
\(789\) −16.0000 −0.569615
\(790\) 12.0000 0.426941
\(791\) −2.00000 −0.0711118
\(792\) 1.00000 0.0355335
\(793\) 0 0
\(794\) 4.00000 0.141955
\(795\) 10.0000 0.354663
\(796\) −5.00000 −0.177220
\(797\) −2.00000 −0.0708436 −0.0354218 0.999372i \(-0.511277\pi\)
−0.0354218 + 0.999372i \(0.511277\pi\)
\(798\) −1.00000 −0.0353996
\(799\) 8.00000 0.283020
\(800\) 4.00000 0.141421
\(801\) −12.0000 −0.423999
\(802\) 14.0000 0.494357
\(803\) 11.0000 0.388182
\(804\) 12.0000 0.423207
\(805\) −3.00000 −0.105736
\(806\) 0 0
\(807\) 20.0000 0.704033
\(808\) −2.00000 −0.0703598
\(809\) −4.00000 −0.140633 −0.0703163 0.997525i \(-0.522401\pi\)
−0.0703163 + 0.997525i \(0.522401\pi\)
\(810\) −1.00000 −0.0351364
\(811\) −21.0000 −0.737410 −0.368705 0.929547i \(-0.620199\pi\)
−0.368705 + 0.929547i \(0.620199\pi\)
\(812\) 9.00000 0.315838
\(813\) −4.00000 −0.140286
\(814\) 9.00000 0.315450
\(815\) 20.0000 0.700569
\(816\) 1.00000 0.0350070
\(817\) 7.00000 0.244899
\(818\) −27.0000 −0.944033
\(819\) 0 0
\(820\) 8.00000 0.279372
\(821\) 36.0000 1.25641 0.628204 0.778048i \(-0.283790\pi\)
0.628204 + 0.778048i \(0.283790\pi\)
\(822\) 15.0000 0.523185
\(823\) 8.00000 0.278862 0.139431 0.990232i \(-0.455473\pi\)
0.139431 + 0.990232i \(0.455473\pi\)
\(824\) −5.00000 −0.174183
\(825\) −4.00000 −0.139262
\(826\) −6.00000 −0.208767
\(827\) 3.00000 0.104320 0.0521601 0.998639i \(-0.483389\pi\)
0.0521601 + 0.998639i \(0.483389\pi\)
\(828\) −3.00000 −0.104257
\(829\) −25.0000 −0.868286 −0.434143 0.900844i \(-0.642949\pi\)
−0.434143 + 0.900844i \(0.642949\pi\)
\(830\) −6.00000 −0.208263
\(831\) 16.0000 0.555034
\(832\) 0 0
\(833\) −1.00000 −0.0346479
\(834\) −12.0000 −0.415526
\(835\) 11.0000 0.380671
\(836\) 1.00000 0.0345857
\(837\) 4.00000 0.138260
\(838\) −23.0000 −0.794522
\(839\) −52.0000 −1.79524 −0.897620 0.440771i \(-0.854705\pi\)
−0.897620 + 0.440771i \(0.854705\pi\)
\(840\) 1.00000 0.0345033
\(841\) 52.0000 1.79310
\(842\) −22.0000 −0.758170
\(843\) −22.0000 −0.757720
\(844\) −5.00000 −0.172107
\(845\) 0 0
\(846\) 8.00000 0.275046
\(847\) −10.0000 −0.343604
\(848\) −10.0000 −0.343401
\(849\) −14.0000 −0.480479
\(850\) −4.00000 −0.137199
\(851\) −27.0000 −0.925548
\(852\) −6.00000 −0.205557
\(853\) 56.0000 1.91740 0.958702 0.284413i \(-0.0917988\pi\)
0.958702 + 0.284413i \(0.0917988\pi\)
\(854\) −11.0000 −0.376412
\(855\) −1.00000 −0.0341993
\(856\) 2.00000 0.0683586
\(857\) 22.0000 0.751506 0.375753 0.926720i \(-0.377384\pi\)
0.375753 + 0.926720i \(0.377384\pi\)
\(858\) 0 0
\(859\) 10.0000 0.341196 0.170598 0.985341i \(-0.445430\pi\)
0.170598 + 0.985341i \(0.445430\pi\)
\(860\) −7.00000 −0.238698
\(861\) −8.00000 −0.272639
\(862\) 18.0000 0.613082
\(863\) 48.0000 1.63394 0.816970 0.576681i \(-0.195652\pi\)
0.816970 + 0.576681i \(0.195652\pi\)
\(864\) 1.00000 0.0340207
\(865\) −14.0000 −0.476014
\(866\) 4.00000 0.135926
\(867\) 16.0000 0.543388
\(868\) −4.00000 −0.135769
\(869\) 12.0000 0.407072
\(870\) 9.00000 0.305129
\(871\) 0 0
\(872\) −7.00000 −0.237050
\(873\) −2.00000 −0.0676897
\(874\) −3.00000 −0.101477
\(875\) −9.00000 −0.304256
\(876\) 11.0000 0.371656
\(877\) −22.0000 −0.742887 −0.371444 0.928456i \(-0.621137\pi\)
−0.371444 + 0.928456i \(0.621137\pi\)
\(878\) −33.0000 −1.11370
\(879\) −26.0000 −0.876958
\(880\) −1.00000 −0.0337100
\(881\) 59.0000 1.98776 0.993880 0.110463i \(-0.0352333\pi\)
0.993880 + 0.110463i \(0.0352333\pi\)
\(882\) −1.00000 −0.0336718
\(883\) −31.0000 −1.04323 −0.521617 0.853180i \(-0.674671\pi\)
−0.521617 + 0.853180i \(0.674671\pi\)
\(884\) 0 0
\(885\) −6.00000 −0.201688
\(886\) 6.00000 0.201574
\(887\) 2.00000 0.0671534 0.0335767 0.999436i \(-0.489310\pi\)
0.0335767 + 0.999436i \(0.489310\pi\)
\(888\) 9.00000 0.302020
\(889\) −16.0000 −0.536623
\(890\) 12.0000 0.402241
\(891\) −1.00000 −0.0335013
\(892\) −8.00000 −0.267860
\(893\) 8.00000 0.267710
\(894\) 6.00000 0.200670
\(895\) 18.0000 0.601674
\(896\) −1.00000 −0.0334077
\(897\) 0 0
\(898\) −15.0000 −0.500556
\(899\) −36.0000 −1.20067
\(900\) −4.00000 −0.133333
\(901\) 10.0000 0.333148
\(902\) 8.00000 0.266371
\(903\) 7.00000 0.232945
\(904\) 2.00000 0.0665190
\(905\) 2.00000 0.0664822
\(906\) −1.00000 −0.0332228
\(907\) 12.0000 0.398453 0.199227 0.979953i \(-0.436157\pi\)
0.199227 + 0.979953i \(0.436157\pi\)
\(908\) −12.0000 −0.398234
\(909\) 2.00000 0.0663358
\(910\) 0 0
\(911\) −29.0000 −0.960813 −0.480406 0.877046i \(-0.659511\pi\)
−0.480406 + 0.877046i \(0.659511\pi\)
\(912\) 1.00000 0.0331133
\(913\) −6.00000 −0.198571
\(914\) −4.00000 −0.132308
\(915\) −11.0000 −0.363649
\(916\) −2.00000 −0.0660819
\(917\) −3.00000 −0.0990687
\(918\) −1.00000 −0.0330049
\(919\) 34.0000 1.12156 0.560778 0.827966i \(-0.310502\pi\)
0.560778 + 0.827966i \(0.310502\pi\)
\(920\) 3.00000 0.0989071
\(921\) −8.00000 −0.263609
\(922\) −3.00000 −0.0987997
\(923\) 0 0
\(924\) 1.00000 0.0328976
\(925\) −36.0000 −1.18367
\(926\) 29.0000 0.952999
\(927\) 5.00000 0.164222
\(928\) −9.00000 −0.295439
\(929\) −26.0000 −0.853032 −0.426516 0.904480i \(-0.640259\pi\)
−0.426516 + 0.904480i \(0.640259\pi\)
\(930\) −4.00000 −0.131165
\(931\) −1.00000 −0.0327737
\(932\) 18.0000 0.589610
\(933\) 8.00000 0.261908
\(934\) −37.0000 −1.21068
\(935\) 1.00000 0.0327035
\(936\) 0 0
\(937\) −20.0000 −0.653372 −0.326686 0.945133i \(-0.605932\pi\)
−0.326686 + 0.945133i \(0.605932\pi\)
\(938\) 12.0000 0.391814
\(939\) 14.0000 0.456873
\(940\) −8.00000 −0.260931
\(941\) −30.0000 −0.977972 −0.488986 0.872292i \(-0.662633\pi\)
−0.488986 + 0.872292i \(0.662633\pi\)
\(942\) −21.0000 −0.684217
\(943\) −24.0000 −0.781548
\(944\) 6.00000 0.195283
\(945\) −1.00000 −0.0325300
\(946\) −7.00000 −0.227590
\(947\) 35.0000 1.13735 0.568674 0.822563i \(-0.307457\pi\)
0.568674 + 0.822563i \(0.307457\pi\)
\(948\) 12.0000 0.389742
\(949\) 0 0
\(950\) −4.00000 −0.129777
\(951\) 12.0000 0.389127
\(952\) 1.00000 0.0324102
\(953\) 42.0000 1.36051 0.680257 0.732974i \(-0.261868\pi\)
0.680257 + 0.732974i \(0.261868\pi\)
\(954\) 10.0000 0.323762
\(955\) −11.0000 −0.355952
\(956\) −8.00000 −0.258738
\(957\) 9.00000 0.290929
\(958\) 39.0000 1.26003
\(959\) 15.0000 0.484375
\(960\) −1.00000 −0.0322749
\(961\) −15.0000 −0.483871
\(962\) 0 0
\(963\) −2.00000 −0.0644491
\(964\) −10.0000 −0.322078
\(965\) 12.0000 0.386294
\(966\) −3.00000 −0.0965234
\(967\) 9.00000 0.289420 0.144710 0.989474i \(-0.453775\pi\)
0.144710 + 0.989474i \(0.453775\pi\)
\(968\) 10.0000 0.321412
\(969\) −1.00000 −0.0321246
\(970\) 2.00000 0.0642161
\(971\) −4.00000 −0.128366 −0.0641831 0.997938i \(-0.520444\pi\)
−0.0641831 + 0.997938i \(0.520444\pi\)
\(972\) −1.00000 −0.0320750
\(973\) −12.0000 −0.384702
\(974\) −32.0000 −1.02535
\(975\) 0 0
\(976\) 11.0000 0.352101
\(977\) −39.0000 −1.24772 −0.623860 0.781536i \(-0.714437\pi\)
−0.623860 + 0.781536i \(0.714437\pi\)
\(978\) 20.0000 0.639529
\(979\) 12.0000 0.383522
\(980\) 1.00000 0.0319438
\(981\) 7.00000 0.223493
\(982\) 12.0000 0.382935
\(983\) 9.00000 0.287055 0.143528 0.989646i \(-0.454155\pi\)
0.143528 + 0.989646i \(0.454155\pi\)
\(984\) 8.00000 0.255031
\(985\) −2.00000 −0.0637253
\(986\) 9.00000 0.286618
\(987\) 8.00000 0.254643
\(988\) 0 0
\(989\) 21.0000 0.667761
\(990\) 1.00000 0.0317821
\(991\) −12.0000 −0.381193 −0.190596 0.981669i \(-0.561042\pi\)
−0.190596 + 0.981669i \(0.561042\pi\)
\(992\) 4.00000 0.127000
\(993\) 8.00000 0.253872
\(994\) −6.00000 −0.190308
\(995\) −5.00000 −0.158511
\(996\) −6.00000 −0.190117
\(997\) −14.0000 −0.443384 −0.221692 0.975117i \(-0.571158\pi\)
−0.221692 + 0.975117i \(0.571158\pi\)
\(998\) 10.0000 0.316544
\(999\) −9.00000 −0.284747
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 7098.2.a.d.1.1 1
13.5 odd 4 546.2.c.a.337.2 yes 2
13.8 odd 4 546.2.c.a.337.1 2
13.12 even 2 7098.2.a.s.1.1 1
39.5 even 4 1638.2.c.f.883.1 2
39.8 even 4 1638.2.c.f.883.2 2
52.31 even 4 4368.2.h.k.337.1 2
52.47 even 4 4368.2.h.k.337.2 2
91.34 even 4 3822.2.c.e.883.1 2
91.83 even 4 3822.2.c.e.883.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
546.2.c.a.337.1 2 13.8 odd 4
546.2.c.a.337.2 yes 2 13.5 odd 4
1638.2.c.f.883.1 2 39.5 even 4
1638.2.c.f.883.2 2 39.8 even 4
3822.2.c.e.883.1 2 91.34 even 4
3822.2.c.e.883.2 2 91.83 even 4
4368.2.h.k.337.1 2 52.31 even 4
4368.2.h.k.337.2 2 52.47 even 4
7098.2.a.d.1.1 1 1.1 even 1 trivial
7098.2.a.s.1.1 1 13.12 even 2