Properties

Label 234.2.a.e.1.1
Level $234$
Weight $2$
Character 234.1
Self dual yes
Analytic conductor $1.868$
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] = [234,2,Mod(1,234)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(234, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("234.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 234 = 2 \cdot 3^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 234.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: \(1.86849940730\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 26)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 234.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{4} +3.00000 q^{5} -1.00000 q^{7} +1.00000 q^{8} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{4} +3.00000 q^{5} -1.00000 q^{7} +1.00000 q^{8} +3.00000 q^{10} -6.00000 q^{11} +1.00000 q^{13} -1.00000 q^{14} +1.00000 q^{16} +3.00000 q^{17} +2.00000 q^{19} +3.00000 q^{20} -6.00000 q^{22} +4.00000 q^{25} +1.00000 q^{26} -1.00000 q^{28} -6.00000 q^{29} -4.00000 q^{31} +1.00000 q^{32} +3.00000 q^{34} -3.00000 q^{35} -7.00000 q^{37} +2.00000 q^{38} +3.00000 q^{40} -1.00000 q^{43} -6.00000 q^{44} -3.00000 q^{47} -6.00000 q^{49} +4.00000 q^{50} +1.00000 q^{52} -18.0000 q^{55} -1.00000 q^{56} -6.00000 q^{58} +6.00000 q^{59} +8.00000 q^{61} -4.00000 q^{62} +1.00000 q^{64} +3.00000 q^{65} +14.0000 q^{67} +3.00000 q^{68} -3.00000 q^{70} +3.00000 q^{71} +2.00000 q^{73} -7.00000 q^{74} +2.00000 q^{76} +6.00000 q^{77} +8.00000 q^{79} +3.00000 q^{80} -12.0000 q^{83} +9.00000 q^{85} -1.00000 q^{86} -6.00000 q^{88} +6.00000 q^{89} -1.00000 q^{91} -3.00000 q^{94} +6.00000 q^{95} -10.0000 q^{97} -6.00000 q^{98} +O(q^{100})\)

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
\(5\) 3.00000 1.34164 0.670820 0.741620i \(-0.265942\pi\)
0.670820 + 0.741620i \(0.265942\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964 −0.188982 0.981981i \(-0.560519\pi\)
−0.188982 + 0.981981i \(0.560519\pi\)
\(8\) 1.00000 0.353553
\(9\) 0 0
\(10\) 3.00000 0.948683
\(11\) −6.00000 −1.80907 −0.904534 0.426401i \(-0.859781\pi\)
−0.904534 + 0.426401i \(0.859781\pi\)
\(12\) 0 0
\(13\) 1.00000 0.277350
\(14\) −1.00000 −0.267261
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 3.00000 0.727607 0.363803 0.931476i \(-0.381478\pi\)
0.363803 + 0.931476i \(0.381478\pi\)
\(18\) 0 0
\(19\) 2.00000 0.458831 0.229416 0.973329i \(-0.426318\pi\)
0.229416 + 0.973329i \(0.426318\pi\)
\(20\) 3.00000 0.670820
\(21\) 0 0
\(22\) −6.00000 −1.27920
\(23\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(24\) 0 0
\(25\) 4.00000 0.800000
\(26\) 1.00000 0.196116
\(27\) 0 0
\(28\) −1.00000 −0.188982
\(29\) −6.00000 −1.11417 −0.557086 0.830455i \(-0.688081\pi\)
−0.557086 + 0.830455i \(0.688081\pi\)
\(30\) 0 0
\(31\) −4.00000 −0.718421 −0.359211 0.933257i \(-0.616954\pi\)
−0.359211 + 0.933257i \(0.616954\pi\)
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) 3.00000 0.514496
\(35\) −3.00000 −0.507093
\(36\) 0 0
\(37\) −7.00000 −1.15079 −0.575396 0.817875i \(-0.695152\pi\)
−0.575396 + 0.817875i \(0.695152\pi\)
\(38\) 2.00000 0.324443
\(39\) 0 0
\(40\) 3.00000 0.474342
\(41\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(42\) 0 0
\(43\) −1.00000 −0.152499 −0.0762493 0.997089i \(-0.524294\pi\)
−0.0762493 + 0.997089i \(0.524294\pi\)
\(44\) −6.00000 −0.904534
\(45\) 0 0
\(46\) 0 0
\(47\) −3.00000 −0.437595 −0.218797 0.975770i \(-0.570213\pi\)
−0.218797 + 0.975770i \(0.570213\pi\)
\(48\) 0 0
\(49\) −6.00000 −0.857143
\(50\) 4.00000 0.565685
\(51\) 0 0
\(52\) 1.00000 0.138675
\(53\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(54\) 0 0
\(55\) −18.0000 −2.42712
\(56\) −1.00000 −0.133631
\(57\) 0 0
\(58\) −6.00000 −0.787839
\(59\) 6.00000 0.781133 0.390567 0.920575i \(-0.372279\pi\)
0.390567 + 0.920575i \(0.372279\pi\)
\(60\) 0 0
\(61\) 8.00000 1.02430 0.512148 0.858898i \(-0.328850\pi\)
0.512148 + 0.858898i \(0.328850\pi\)
\(62\) −4.00000 −0.508001
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 3.00000 0.372104
\(66\) 0 0
\(67\) 14.0000 1.71037 0.855186 0.518321i \(-0.173443\pi\)
0.855186 + 0.518321i \(0.173443\pi\)
\(68\) 3.00000 0.363803
\(69\) 0 0
\(70\) −3.00000 −0.358569
\(71\) 3.00000 0.356034 0.178017 0.984027i \(-0.443032\pi\)
0.178017 + 0.984027i \(0.443032\pi\)
\(72\) 0 0
\(73\) 2.00000 0.234082 0.117041 0.993127i \(-0.462659\pi\)
0.117041 + 0.993127i \(0.462659\pi\)
\(74\) −7.00000 −0.813733
\(75\) 0 0
\(76\) 2.00000 0.229416
\(77\) 6.00000 0.683763
\(78\) 0 0
\(79\) 8.00000 0.900070 0.450035 0.893011i \(-0.351411\pi\)
0.450035 + 0.893011i \(0.351411\pi\)
\(80\) 3.00000 0.335410
\(81\) 0 0
\(82\) 0 0
\(83\) −12.0000 −1.31717 −0.658586 0.752506i \(-0.728845\pi\)
−0.658586 + 0.752506i \(0.728845\pi\)
\(84\) 0 0
\(85\) 9.00000 0.976187
\(86\) −1.00000 −0.107833
\(87\) 0 0
\(88\) −6.00000 −0.639602
\(89\) 6.00000 0.635999 0.317999 0.948091i \(-0.396989\pi\)
0.317999 + 0.948091i \(0.396989\pi\)
\(90\) 0 0
\(91\) −1.00000 −0.104828
\(92\) 0 0
\(93\) 0 0
\(94\) −3.00000 −0.309426
\(95\) 6.00000 0.615587
\(96\) 0 0
\(97\) −10.0000 −1.01535 −0.507673 0.861550i \(-0.669494\pi\)
−0.507673 + 0.861550i \(0.669494\pi\)
\(98\) −6.00000 −0.606092
\(99\) 0 0
\(100\) 4.00000 0.400000
\(101\) 12.0000 1.19404 0.597022 0.802225i \(-0.296350\pi\)
0.597022 + 0.802225i \(0.296350\pi\)
\(102\) 0 0
\(103\) −4.00000 −0.394132 −0.197066 0.980390i \(-0.563141\pi\)
−0.197066 + 0.980390i \(0.563141\pi\)
\(104\) 1.00000 0.0980581
\(105\) 0 0
\(106\) 0 0
\(107\) −12.0000 −1.16008 −0.580042 0.814587i \(-0.696964\pi\)
−0.580042 + 0.814587i \(0.696964\pi\)
\(108\) 0 0
\(109\) −7.00000 −0.670478 −0.335239 0.942133i \(-0.608817\pi\)
−0.335239 + 0.942133i \(0.608817\pi\)
\(110\) −18.0000 −1.71623
\(111\) 0 0
\(112\) −1.00000 −0.0944911
\(113\) 6.00000 0.564433 0.282216 0.959351i \(-0.408930\pi\)
0.282216 + 0.959351i \(0.408930\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) −6.00000 −0.557086
\(117\) 0 0
\(118\) 6.00000 0.552345
\(119\) −3.00000 −0.275010
\(120\) 0 0
\(121\) 25.0000 2.27273
\(122\) 8.00000 0.724286
\(123\) 0 0
\(124\) −4.00000 −0.359211
\(125\) −3.00000 −0.268328
\(126\) 0 0
\(127\) 20.0000 1.77471 0.887357 0.461084i \(-0.152539\pi\)
0.887357 + 0.461084i \(0.152539\pi\)
\(128\) 1.00000 0.0883883
\(129\) 0 0
\(130\) 3.00000 0.263117
\(131\) 21.0000 1.83478 0.917389 0.397991i \(-0.130293\pi\)
0.917389 + 0.397991i \(0.130293\pi\)
\(132\) 0 0
\(133\) −2.00000 −0.173422
\(134\) 14.0000 1.20942
\(135\) 0 0
\(136\) 3.00000 0.257248
\(137\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(138\) 0 0
\(139\) −13.0000 −1.10265 −0.551323 0.834292i \(-0.685877\pi\)
−0.551323 + 0.834292i \(0.685877\pi\)
\(140\) −3.00000 −0.253546
\(141\) 0 0
\(142\) 3.00000 0.251754
\(143\) −6.00000 −0.501745
\(144\) 0 0
\(145\) −18.0000 −1.49482
\(146\) 2.00000 0.165521
\(147\) 0 0
\(148\) −7.00000 −0.575396
\(149\) 6.00000 0.491539 0.245770 0.969328i \(-0.420959\pi\)
0.245770 + 0.969328i \(0.420959\pi\)
\(150\) 0 0
\(151\) 17.0000 1.38344 0.691720 0.722166i \(-0.256853\pi\)
0.691720 + 0.722166i \(0.256853\pi\)
\(152\) 2.00000 0.162221
\(153\) 0 0
\(154\) 6.00000 0.483494
\(155\) −12.0000 −0.963863
\(156\) 0 0
\(157\) 14.0000 1.11732 0.558661 0.829396i \(-0.311315\pi\)
0.558661 + 0.829396i \(0.311315\pi\)
\(158\) 8.00000 0.636446
\(159\) 0 0
\(160\) 3.00000 0.237171
\(161\) 0 0
\(162\) 0 0
\(163\) −16.0000 −1.25322 −0.626608 0.779334i \(-0.715557\pi\)
−0.626608 + 0.779334i \(0.715557\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) −12.0000 −0.931381
\(167\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(168\) 0 0
\(169\) 1.00000 0.0769231
\(170\) 9.00000 0.690268
\(171\) 0 0
\(172\) −1.00000 −0.0762493
\(173\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(174\) 0 0
\(175\) −4.00000 −0.302372
\(176\) −6.00000 −0.452267
\(177\) 0 0
\(178\) 6.00000 0.449719
\(179\) −3.00000 −0.224231 −0.112115 0.993695i \(-0.535763\pi\)
−0.112115 + 0.993695i \(0.535763\pi\)
\(180\) 0 0
\(181\) 20.0000 1.48659 0.743294 0.668965i \(-0.233262\pi\)
0.743294 + 0.668965i \(0.233262\pi\)
\(182\) −1.00000 −0.0741249
\(183\) 0 0
\(184\) 0 0
\(185\) −21.0000 −1.54395
\(186\) 0 0
\(187\) −18.0000 −1.31629
\(188\) −3.00000 −0.218797
\(189\) 0 0
\(190\) 6.00000 0.435286
\(191\) 18.0000 1.30243 0.651217 0.758891i \(-0.274259\pi\)
0.651217 + 0.758891i \(0.274259\pi\)
\(192\) 0 0
\(193\) −4.00000 −0.287926 −0.143963 0.989583i \(-0.545985\pi\)
−0.143963 + 0.989583i \(0.545985\pi\)
\(194\) −10.0000 −0.717958
\(195\) 0 0
\(196\) −6.00000 −0.428571
\(197\) −3.00000 −0.213741 −0.106871 0.994273i \(-0.534083\pi\)
−0.106871 + 0.994273i \(0.534083\pi\)
\(198\) 0 0
\(199\) 2.00000 0.141776 0.0708881 0.997484i \(-0.477417\pi\)
0.0708881 + 0.997484i \(0.477417\pi\)
\(200\) 4.00000 0.282843
\(201\) 0 0
\(202\) 12.0000 0.844317
\(203\) 6.00000 0.421117
\(204\) 0 0
\(205\) 0 0
\(206\) −4.00000 −0.278693
\(207\) 0 0
\(208\) 1.00000 0.0693375
\(209\) −12.0000 −0.830057
\(210\) 0 0
\(211\) −13.0000 −0.894957 −0.447478 0.894295i \(-0.647678\pi\)
−0.447478 + 0.894295i \(0.647678\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) −12.0000 −0.820303
\(215\) −3.00000 −0.204598
\(216\) 0 0
\(217\) 4.00000 0.271538
\(218\) −7.00000 −0.474100
\(219\) 0 0
\(220\) −18.0000 −1.21356
\(221\) 3.00000 0.201802
\(222\) 0 0
\(223\) −19.0000 −1.27233 −0.636167 0.771551i \(-0.719481\pi\)
−0.636167 + 0.771551i \(0.719481\pi\)
\(224\) −1.00000 −0.0668153
\(225\) 0 0
\(226\) 6.00000 0.399114
\(227\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(228\) 0 0
\(229\) −13.0000 −0.859064 −0.429532 0.903052i \(-0.641321\pi\)
−0.429532 + 0.903052i \(0.641321\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) −6.00000 −0.393919
\(233\) 27.0000 1.76883 0.884414 0.466702i \(-0.154558\pi\)
0.884414 + 0.466702i \(0.154558\pi\)
\(234\) 0 0
\(235\) −9.00000 −0.587095
\(236\) 6.00000 0.390567
\(237\) 0 0
\(238\) −3.00000 −0.194461
\(239\) −15.0000 −0.970269 −0.485135 0.874439i \(-0.661229\pi\)
−0.485135 + 0.874439i \(0.661229\pi\)
\(240\) 0 0
\(241\) −10.0000 −0.644157 −0.322078 0.946713i \(-0.604381\pi\)
−0.322078 + 0.946713i \(0.604381\pi\)
\(242\) 25.0000 1.60706
\(243\) 0 0
\(244\) 8.00000 0.512148
\(245\) −18.0000 −1.14998
\(246\) 0 0
\(247\) 2.00000 0.127257
\(248\) −4.00000 −0.254000
\(249\) 0 0
\(250\) −3.00000 −0.189737
\(251\) −24.0000 −1.51487 −0.757433 0.652913i \(-0.773547\pi\)
−0.757433 + 0.652913i \(0.773547\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 20.0000 1.25491
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −9.00000 −0.561405 −0.280702 0.959795i \(-0.590567\pi\)
−0.280702 + 0.959795i \(0.590567\pi\)
\(258\) 0 0
\(259\) 7.00000 0.434959
\(260\) 3.00000 0.186052
\(261\) 0 0
\(262\) 21.0000 1.29738
\(263\) 12.0000 0.739952 0.369976 0.929041i \(-0.379366\pi\)
0.369976 + 0.929041i \(0.379366\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) −2.00000 −0.122628
\(267\) 0 0
\(268\) 14.0000 0.855186
\(269\) −24.0000 −1.46331 −0.731653 0.681677i \(-0.761251\pi\)
−0.731653 + 0.681677i \(0.761251\pi\)
\(270\) 0 0
\(271\) 11.0000 0.668202 0.334101 0.942537i \(-0.391567\pi\)
0.334101 + 0.942537i \(0.391567\pi\)
\(272\) 3.00000 0.181902
\(273\) 0 0
\(274\) 0 0
\(275\) −24.0000 −1.44725
\(276\) 0 0
\(277\) −28.0000 −1.68236 −0.841178 0.540758i \(-0.818138\pi\)
−0.841178 + 0.540758i \(0.818138\pi\)
\(278\) −13.0000 −0.779688
\(279\) 0 0
\(280\) −3.00000 −0.179284
\(281\) 6.00000 0.357930 0.178965 0.983855i \(-0.442725\pi\)
0.178965 + 0.983855i \(0.442725\pi\)
\(282\) 0 0
\(283\) −4.00000 −0.237775 −0.118888 0.992908i \(-0.537933\pi\)
−0.118888 + 0.992908i \(0.537933\pi\)
\(284\) 3.00000 0.178017
\(285\) 0 0
\(286\) −6.00000 −0.354787
\(287\) 0 0
\(288\) 0 0
\(289\) −8.00000 −0.470588
\(290\) −18.0000 −1.05700
\(291\) 0 0
\(292\) 2.00000 0.117041
\(293\) −21.0000 −1.22683 −0.613417 0.789760i \(-0.710205\pi\)
−0.613417 + 0.789760i \(0.710205\pi\)
\(294\) 0 0
\(295\) 18.0000 1.04800
\(296\) −7.00000 −0.406867
\(297\) 0 0
\(298\) 6.00000 0.347571
\(299\) 0 0
\(300\) 0 0
\(301\) 1.00000 0.0576390
\(302\) 17.0000 0.978240
\(303\) 0 0
\(304\) 2.00000 0.114708
\(305\) 24.0000 1.37424
\(306\) 0 0
\(307\) 2.00000 0.114146 0.0570730 0.998370i \(-0.481823\pi\)
0.0570730 + 0.998370i \(0.481823\pi\)
\(308\) 6.00000 0.341882
\(309\) 0 0
\(310\) −12.0000 −0.681554
\(311\) 30.0000 1.70114 0.850572 0.525859i \(-0.176256\pi\)
0.850572 + 0.525859i \(0.176256\pi\)
\(312\) 0 0
\(313\) −1.00000 −0.0565233 −0.0282617 0.999601i \(-0.508997\pi\)
−0.0282617 + 0.999601i \(0.508997\pi\)
\(314\) 14.0000 0.790066
\(315\) 0 0
\(316\) 8.00000 0.450035
\(317\) 6.00000 0.336994 0.168497 0.985702i \(-0.446109\pi\)
0.168497 + 0.985702i \(0.446109\pi\)
\(318\) 0 0
\(319\) 36.0000 2.01561
\(320\) 3.00000 0.167705
\(321\) 0 0
\(322\) 0 0
\(323\) 6.00000 0.333849
\(324\) 0 0
\(325\) 4.00000 0.221880
\(326\) −16.0000 −0.886158
\(327\) 0 0
\(328\) 0 0
\(329\) 3.00000 0.165395
\(330\) 0 0
\(331\) 8.00000 0.439720 0.219860 0.975531i \(-0.429440\pi\)
0.219860 + 0.975531i \(0.429440\pi\)
\(332\) −12.0000 −0.658586
\(333\) 0 0
\(334\) 0 0
\(335\) 42.0000 2.29471
\(336\) 0 0
\(337\) 23.0000 1.25289 0.626445 0.779466i \(-0.284509\pi\)
0.626445 + 0.779466i \(0.284509\pi\)
\(338\) 1.00000 0.0543928
\(339\) 0 0
\(340\) 9.00000 0.488094
\(341\) 24.0000 1.29967
\(342\) 0 0
\(343\) 13.0000 0.701934
\(344\) −1.00000 −0.0539164
\(345\) 0 0
\(346\) 0 0
\(347\) −3.00000 −0.161048 −0.0805242 0.996753i \(-0.525659\pi\)
−0.0805242 + 0.996753i \(0.525659\pi\)
\(348\) 0 0
\(349\) −19.0000 −1.01705 −0.508523 0.861048i \(-0.669808\pi\)
−0.508523 + 0.861048i \(0.669808\pi\)
\(350\) −4.00000 −0.213809
\(351\) 0 0
\(352\) −6.00000 −0.319801
\(353\) −24.0000 −1.27739 −0.638696 0.769460i \(-0.720526\pi\)
−0.638696 + 0.769460i \(0.720526\pi\)
\(354\) 0 0
\(355\) 9.00000 0.477670
\(356\) 6.00000 0.317999
\(357\) 0 0
\(358\) −3.00000 −0.158555
\(359\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(360\) 0 0
\(361\) −15.0000 −0.789474
\(362\) 20.0000 1.05118
\(363\) 0 0
\(364\) −1.00000 −0.0524142
\(365\) 6.00000 0.314054
\(366\) 0 0
\(367\) 26.0000 1.35719 0.678594 0.734513i \(-0.262589\pi\)
0.678594 + 0.734513i \(0.262589\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) −21.0000 −1.09174
\(371\) 0 0
\(372\) 0 0
\(373\) −4.00000 −0.207112 −0.103556 0.994624i \(-0.533022\pi\)
−0.103556 + 0.994624i \(0.533022\pi\)
\(374\) −18.0000 −0.930758
\(375\) 0 0
\(376\) −3.00000 −0.154713
\(377\) −6.00000 −0.309016
\(378\) 0 0
\(379\) 20.0000 1.02733 0.513665 0.857991i \(-0.328287\pi\)
0.513665 + 0.857991i \(0.328287\pi\)
\(380\) 6.00000 0.307794
\(381\) 0 0
\(382\) 18.0000 0.920960
\(383\) −21.0000 −1.07305 −0.536525 0.843884i \(-0.680263\pi\)
−0.536525 + 0.843884i \(0.680263\pi\)
\(384\) 0 0
\(385\) 18.0000 0.917365
\(386\) −4.00000 −0.203595
\(387\) 0 0
\(388\) −10.0000 −0.507673
\(389\) 6.00000 0.304212 0.152106 0.988364i \(-0.451394\pi\)
0.152106 + 0.988364i \(0.451394\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) −6.00000 −0.303046
\(393\) 0 0
\(394\) −3.00000 −0.151138
\(395\) 24.0000 1.20757
\(396\) 0 0
\(397\) −34.0000 −1.70641 −0.853206 0.521575i \(-0.825345\pi\)
−0.853206 + 0.521575i \(0.825345\pi\)
\(398\) 2.00000 0.100251
\(399\) 0 0
\(400\) 4.00000 0.200000
\(401\) −36.0000 −1.79775 −0.898877 0.438201i \(-0.855616\pi\)
−0.898877 + 0.438201i \(0.855616\pi\)
\(402\) 0 0
\(403\) −4.00000 −0.199254
\(404\) 12.0000 0.597022
\(405\) 0 0
\(406\) 6.00000 0.297775
\(407\) 42.0000 2.08186
\(408\) 0 0
\(409\) 32.0000 1.58230 0.791149 0.611623i \(-0.209483\pi\)
0.791149 + 0.611623i \(0.209483\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) −4.00000 −0.197066
\(413\) −6.00000 −0.295241
\(414\) 0 0
\(415\) −36.0000 −1.76717
\(416\) 1.00000 0.0490290
\(417\) 0 0
\(418\) −12.0000 −0.586939
\(419\) −9.00000 −0.439679 −0.219839 0.975536i \(-0.570553\pi\)
−0.219839 + 0.975536i \(0.570553\pi\)
\(420\) 0 0
\(421\) 17.0000 0.828529 0.414265 0.910156i \(-0.364039\pi\)
0.414265 + 0.910156i \(0.364039\pi\)
\(422\) −13.0000 −0.632830
\(423\) 0 0
\(424\) 0 0
\(425\) 12.0000 0.582086
\(426\) 0 0
\(427\) −8.00000 −0.387147
\(428\) −12.0000 −0.580042
\(429\) 0 0
\(430\) −3.00000 −0.144673
\(431\) 33.0000 1.58955 0.794777 0.606902i \(-0.207588\pi\)
0.794777 + 0.606902i \(0.207588\pi\)
\(432\) 0 0
\(433\) −25.0000 −1.20142 −0.600712 0.799466i \(-0.705116\pi\)
−0.600712 + 0.799466i \(0.705116\pi\)
\(434\) 4.00000 0.192006
\(435\) 0 0
\(436\) −7.00000 −0.335239
\(437\) 0 0
\(438\) 0 0
\(439\) 26.0000 1.24091 0.620456 0.784241i \(-0.286947\pi\)
0.620456 + 0.784241i \(0.286947\pi\)
\(440\) −18.0000 −0.858116
\(441\) 0 0
\(442\) 3.00000 0.142695
\(443\) −21.0000 −0.997740 −0.498870 0.866677i \(-0.666252\pi\)
−0.498870 + 0.866677i \(0.666252\pi\)
\(444\) 0 0
\(445\) 18.0000 0.853282
\(446\) −19.0000 −0.899676
\(447\) 0 0
\(448\) −1.00000 −0.0472456
\(449\) −6.00000 −0.283158 −0.141579 0.989927i \(-0.545218\pi\)
−0.141579 + 0.989927i \(0.545218\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 6.00000 0.282216
\(453\) 0 0
\(454\) 0 0
\(455\) −3.00000 −0.140642
\(456\) 0 0
\(457\) −10.0000 −0.467780 −0.233890 0.972263i \(-0.575146\pi\)
−0.233890 + 0.972263i \(0.575146\pi\)
\(458\) −13.0000 −0.607450
\(459\) 0 0
\(460\) 0 0
\(461\) −9.00000 −0.419172 −0.209586 0.977790i \(-0.567212\pi\)
−0.209586 + 0.977790i \(0.567212\pi\)
\(462\) 0 0
\(463\) −40.0000 −1.85896 −0.929479 0.368875i \(-0.879743\pi\)
−0.929479 + 0.368875i \(0.879743\pi\)
\(464\) −6.00000 −0.278543
\(465\) 0 0
\(466\) 27.0000 1.25075
\(467\) −36.0000 −1.66588 −0.832941 0.553362i \(-0.813345\pi\)
−0.832941 + 0.553362i \(0.813345\pi\)
\(468\) 0 0
\(469\) −14.0000 −0.646460
\(470\) −9.00000 −0.415139
\(471\) 0 0
\(472\) 6.00000 0.276172
\(473\) 6.00000 0.275880
\(474\) 0 0
\(475\) 8.00000 0.367065
\(476\) −3.00000 −0.137505
\(477\) 0 0
\(478\) −15.0000 −0.686084
\(479\) 21.0000 0.959514 0.479757 0.877401i \(-0.340725\pi\)
0.479757 + 0.877401i \(0.340725\pi\)
\(480\) 0 0
\(481\) −7.00000 −0.319173
\(482\) −10.0000 −0.455488
\(483\) 0 0
\(484\) 25.0000 1.13636
\(485\) −30.0000 −1.36223
\(486\) 0 0
\(487\) −16.0000 −0.725029 −0.362515 0.931978i \(-0.618082\pi\)
−0.362515 + 0.931978i \(0.618082\pi\)
\(488\) 8.00000 0.362143
\(489\) 0 0
\(490\) −18.0000 −0.813157
\(491\) 9.00000 0.406164 0.203082 0.979162i \(-0.434904\pi\)
0.203082 + 0.979162i \(0.434904\pi\)
\(492\) 0 0
\(493\) −18.0000 −0.810679
\(494\) 2.00000 0.0899843
\(495\) 0 0
\(496\) −4.00000 −0.179605
\(497\) −3.00000 −0.134568
\(498\) 0 0
\(499\) −40.0000 −1.79065 −0.895323 0.445418i \(-0.853055\pi\)
−0.895323 + 0.445418i \(0.853055\pi\)
\(500\) −3.00000 −0.134164
\(501\) 0 0
\(502\) −24.0000 −1.07117
\(503\) 30.0000 1.33763 0.668817 0.743427i \(-0.266801\pi\)
0.668817 + 0.743427i \(0.266801\pi\)
\(504\) 0 0
\(505\) 36.0000 1.60198
\(506\) 0 0
\(507\) 0 0
\(508\) 20.0000 0.887357
\(509\) 18.0000 0.797836 0.398918 0.916987i \(-0.369386\pi\)
0.398918 + 0.916987i \(0.369386\pi\)
\(510\) 0 0
\(511\) −2.00000 −0.0884748
\(512\) 1.00000 0.0441942
\(513\) 0 0
\(514\) −9.00000 −0.396973
\(515\) −12.0000 −0.528783
\(516\) 0 0
\(517\) 18.0000 0.791639
\(518\) 7.00000 0.307562
\(519\) 0 0
\(520\) 3.00000 0.131559
\(521\) 9.00000 0.394297 0.197149 0.980374i \(-0.436832\pi\)
0.197149 + 0.980374i \(0.436832\pi\)
\(522\) 0 0
\(523\) 20.0000 0.874539 0.437269 0.899331i \(-0.355946\pi\)
0.437269 + 0.899331i \(0.355946\pi\)
\(524\) 21.0000 0.917389
\(525\) 0 0
\(526\) 12.0000 0.523225
\(527\) −12.0000 −0.522728
\(528\) 0 0
\(529\) −23.0000 −1.00000
\(530\) 0 0
\(531\) 0 0
\(532\) −2.00000 −0.0867110
\(533\) 0 0
\(534\) 0 0
\(535\) −36.0000 −1.55642
\(536\) 14.0000 0.604708
\(537\) 0 0
\(538\) −24.0000 −1.03471
\(539\) 36.0000 1.55063
\(540\) 0 0
\(541\) 11.0000 0.472927 0.236463 0.971640i \(-0.424012\pi\)
0.236463 + 0.971640i \(0.424012\pi\)
\(542\) 11.0000 0.472490
\(543\) 0 0
\(544\) 3.00000 0.128624
\(545\) −21.0000 −0.899541
\(546\) 0 0
\(547\) 17.0000 0.726868 0.363434 0.931620i \(-0.381604\pi\)
0.363434 + 0.931620i \(0.381604\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) −24.0000 −1.02336
\(551\) −12.0000 −0.511217
\(552\) 0 0
\(553\) −8.00000 −0.340195
\(554\) −28.0000 −1.18961
\(555\) 0 0
\(556\) −13.0000 −0.551323
\(557\) −3.00000 −0.127114 −0.0635570 0.997978i \(-0.520244\pi\)
−0.0635570 + 0.997978i \(0.520244\pi\)
\(558\) 0 0
\(559\) −1.00000 −0.0422955
\(560\) −3.00000 −0.126773
\(561\) 0 0
\(562\) 6.00000 0.253095
\(563\) −39.0000 −1.64365 −0.821827 0.569737i \(-0.807045\pi\)
−0.821827 + 0.569737i \(0.807045\pi\)
\(564\) 0 0
\(565\) 18.0000 0.757266
\(566\) −4.00000 −0.168133
\(567\) 0 0
\(568\) 3.00000 0.125877
\(569\) −15.0000 −0.628833 −0.314416 0.949285i \(-0.601809\pi\)
−0.314416 + 0.949285i \(0.601809\pi\)
\(570\) 0 0
\(571\) 5.00000 0.209243 0.104622 0.994512i \(-0.466637\pi\)
0.104622 + 0.994512i \(0.466637\pi\)
\(572\) −6.00000 −0.250873
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 38.0000 1.58196 0.790980 0.611842i \(-0.209571\pi\)
0.790980 + 0.611842i \(0.209571\pi\)
\(578\) −8.00000 −0.332756
\(579\) 0 0
\(580\) −18.0000 −0.747409
\(581\) 12.0000 0.497844
\(582\) 0 0
\(583\) 0 0
\(584\) 2.00000 0.0827606
\(585\) 0 0
\(586\) −21.0000 −0.867502
\(587\) −24.0000 −0.990586 −0.495293 0.868726i \(-0.664939\pi\)
−0.495293 + 0.868726i \(0.664939\pi\)
\(588\) 0 0
\(589\) −8.00000 −0.329634
\(590\) 18.0000 0.741048
\(591\) 0 0
\(592\) −7.00000 −0.287698
\(593\) −18.0000 −0.739171 −0.369586 0.929197i \(-0.620500\pi\)
−0.369586 + 0.929197i \(0.620500\pi\)
\(594\) 0 0
\(595\) −9.00000 −0.368964
\(596\) 6.00000 0.245770
\(597\) 0 0
\(598\) 0 0
\(599\) −6.00000 −0.245153 −0.122577 0.992459i \(-0.539116\pi\)
−0.122577 + 0.992459i \(0.539116\pi\)
\(600\) 0 0
\(601\) −19.0000 −0.775026 −0.387513 0.921864i \(-0.626666\pi\)
−0.387513 + 0.921864i \(0.626666\pi\)
\(602\) 1.00000 0.0407570
\(603\) 0 0
\(604\) 17.0000 0.691720
\(605\) 75.0000 3.04918
\(606\) 0 0
\(607\) 14.0000 0.568242 0.284121 0.958788i \(-0.408298\pi\)
0.284121 + 0.958788i \(0.408298\pi\)
\(608\) 2.00000 0.0811107
\(609\) 0 0
\(610\) 24.0000 0.971732
\(611\) −3.00000 −0.121367
\(612\) 0 0
\(613\) 38.0000 1.53481 0.767403 0.641165i \(-0.221549\pi\)
0.767403 + 0.641165i \(0.221549\pi\)
\(614\) 2.00000 0.0807134
\(615\) 0 0
\(616\) 6.00000 0.241747
\(617\) 24.0000 0.966204 0.483102 0.875564i \(-0.339510\pi\)
0.483102 + 0.875564i \(0.339510\pi\)
\(618\) 0 0
\(619\) −28.0000 −1.12542 −0.562708 0.826656i \(-0.690240\pi\)
−0.562708 + 0.826656i \(0.690240\pi\)
\(620\) −12.0000 −0.481932
\(621\) 0 0
\(622\) 30.0000 1.20289
\(623\) −6.00000 −0.240385
\(624\) 0 0
\(625\) −29.0000 −1.16000
\(626\) −1.00000 −0.0399680
\(627\) 0 0
\(628\) 14.0000 0.558661
\(629\) −21.0000 −0.837325
\(630\) 0 0
\(631\) 29.0000 1.15447 0.577236 0.816577i \(-0.304131\pi\)
0.577236 + 0.816577i \(0.304131\pi\)
\(632\) 8.00000 0.318223
\(633\) 0 0
\(634\) 6.00000 0.238290
\(635\) 60.0000 2.38103
\(636\) 0 0
\(637\) −6.00000 −0.237729
\(638\) 36.0000 1.42525
\(639\) 0 0
\(640\) 3.00000 0.118585
\(641\) 18.0000 0.710957 0.355479 0.934684i \(-0.384318\pi\)
0.355479 + 0.934684i \(0.384318\pi\)
\(642\) 0 0
\(643\) 14.0000 0.552106 0.276053 0.961142i \(-0.410973\pi\)
0.276053 + 0.961142i \(0.410973\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 6.00000 0.236067
\(647\) 6.00000 0.235884 0.117942 0.993020i \(-0.462370\pi\)
0.117942 + 0.993020i \(0.462370\pi\)
\(648\) 0 0
\(649\) −36.0000 −1.41312
\(650\) 4.00000 0.156893
\(651\) 0 0
\(652\) −16.0000 −0.626608
\(653\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(654\) 0 0
\(655\) 63.0000 2.46161
\(656\) 0 0
\(657\) 0 0
\(658\) 3.00000 0.116952
\(659\) 36.0000 1.40236 0.701180 0.712984i \(-0.252657\pi\)
0.701180 + 0.712984i \(0.252657\pi\)
\(660\) 0 0
\(661\) −22.0000 −0.855701 −0.427850 0.903850i \(-0.640729\pi\)
−0.427850 + 0.903850i \(0.640729\pi\)
\(662\) 8.00000 0.310929
\(663\) 0 0
\(664\) −12.0000 −0.465690
\(665\) −6.00000 −0.232670
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 0 0
\(670\) 42.0000 1.62260
\(671\) −48.0000 −1.85302
\(672\) 0 0
\(673\) −19.0000 −0.732396 −0.366198 0.930537i \(-0.619341\pi\)
−0.366198 + 0.930537i \(0.619341\pi\)
\(674\) 23.0000 0.885927
\(675\) 0 0
\(676\) 1.00000 0.0384615
\(677\) −48.0000 −1.84479 −0.922395 0.386248i \(-0.873771\pi\)
−0.922395 + 0.386248i \(0.873771\pi\)
\(678\) 0 0
\(679\) 10.0000 0.383765
\(680\) 9.00000 0.345134
\(681\) 0 0
\(682\) 24.0000 0.919007
\(683\) −24.0000 −0.918334 −0.459167 0.888350i \(-0.651852\pi\)
−0.459167 + 0.888350i \(0.651852\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 13.0000 0.496342
\(687\) 0 0
\(688\) −1.00000 −0.0381246
\(689\) 0 0
\(690\) 0 0
\(691\) 8.00000 0.304334 0.152167 0.988355i \(-0.451375\pi\)
0.152167 + 0.988355i \(0.451375\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) −3.00000 −0.113878
\(695\) −39.0000 −1.47935
\(696\) 0 0
\(697\) 0 0
\(698\) −19.0000 −0.719161
\(699\) 0 0
\(700\) −4.00000 −0.151186
\(701\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(702\) 0 0
\(703\) −14.0000 −0.528020
\(704\) −6.00000 −0.226134
\(705\) 0 0
\(706\) −24.0000 −0.903252
\(707\) −12.0000 −0.451306
\(708\) 0 0
\(709\) 26.0000 0.976450 0.488225 0.872718i \(-0.337644\pi\)
0.488225 + 0.872718i \(0.337644\pi\)
\(710\) 9.00000 0.337764
\(711\) 0 0
\(712\) 6.00000 0.224860
\(713\) 0 0
\(714\) 0 0
\(715\) −18.0000 −0.673162
\(716\) −3.00000 −0.112115
\(717\) 0 0
\(718\) 0 0
\(719\) −6.00000 −0.223762 −0.111881 0.993722i \(-0.535688\pi\)
−0.111881 + 0.993722i \(0.535688\pi\)
\(720\) 0 0
\(721\) 4.00000 0.148968
\(722\) −15.0000 −0.558242
\(723\) 0 0
\(724\) 20.0000 0.743294
\(725\) −24.0000 −0.891338
\(726\) 0 0
\(727\) −10.0000 −0.370879 −0.185440 0.982656i \(-0.559371\pi\)
−0.185440 + 0.982656i \(0.559371\pi\)
\(728\) −1.00000 −0.0370625
\(729\) 0 0
\(730\) 6.00000 0.222070
\(731\) −3.00000 −0.110959
\(732\) 0 0
\(733\) 23.0000 0.849524 0.424762 0.905305i \(-0.360358\pi\)
0.424762 + 0.905305i \(0.360358\pi\)
\(734\) 26.0000 0.959678
\(735\) 0 0
\(736\) 0 0
\(737\) −84.0000 −3.09418
\(738\) 0 0
\(739\) 20.0000 0.735712 0.367856 0.929883i \(-0.380092\pi\)
0.367856 + 0.929883i \(0.380092\pi\)
\(740\) −21.0000 −0.771975
\(741\) 0 0
\(742\) 0 0
\(743\) 9.00000 0.330178 0.165089 0.986279i \(-0.447209\pi\)
0.165089 + 0.986279i \(0.447209\pi\)
\(744\) 0 0
\(745\) 18.0000 0.659469
\(746\) −4.00000 −0.146450
\(747\) 0 0
\(748\) −18.0000 −0.658145
\(749\) 12.0000 0.438470
\(750\) 0 0
\(751\) −40.0000 −1.45962 −0.729810 0.683650i \(-0.760392\pi\)
−0.729810 + 0.683650i \(0.760392\pi\)
\(752\) −3.00000 −0.109399
\(753\) 0 0
\(754\) −6.00000 −0.218507
\(755\) 51.0000 1.85608
\(756\) 0 0
\(757\) −16.0000 −0.581530 −0.290765 0.956795i \(-0.593910\pi\)
−0.290765 + 0.956795i \(0.593910\pi\)
\(758\) 20.0000 0.726433
\(759\) 0 0
\(760\) 6.00000 0.217643
\(761\) 6.00000 0.217500 0.108750 0.994069i \(-0.465315\pi\)
0.108750 + 0.994069i \(0.465315\pi\)
\(762\) 0 0
\(763\) 7.00000 0.253417
\(764\) 18.0000 0.651217
\(765\) 0 0
\(766\) −21.0000 −0.758761
\(767\) 6.00000 0.216647
\(768\) 0 0
\(769\) 32.0000 1.15395 0.576975 0.816762i \(-0.304233\pi\)
0.576975 + 0.816762i \(0.304233\pi\)
\(770\) 18.0000 0.648675
\(771\) 0 0
\(772\) −4.00000 −0.143963
\(773\) 39.0000 1.40273 0.701366 0.712801i \(-0.252574\pi\)
0.701366 + 0.712801i \(0.252574\pi\)
\(774\) 0 0
\(775\) −16.0000 −0.574737
\(776\) −10.0000 −0.358979
\(777\) 0 0
\(778\) 6.00000 0.215110
\(779\) 0 0
\(780\) 0 0
\(781\) −18.0000 −0.644091
\(782\) 0 0
\(783\) 0 0
\(784\) −6.00000 −0.214286
\(785\) 42.0000 1.49904
\(786\) 0 0
\(787\) −40.0000 −1.42585 −0.712923 0.701242i \(-0.752629\pi\)
−0.712923 + 0.701242i \(0.752629\pi\)
\(788\) −3.00000 −0.106871
\(789\) 0 0
\(790\) 24.0000 0.853882
\(791\) −6.00000 −0.213335
\(792\) 0 0
\(793\) 8.00000 0.284088
\(794\) −34.0000 −1.20661
\(795\) 0 0
\(796\) 2.00000 0.0708881
\(797\) 42.0000 1.48772 0.743858 0.668338i \(-0.232994\pi\)
0.743858 + 0.668338i \(0.232994\pi\)
\(798\) 0 0
\(799\) −9.00000 −0.318397
\(800\) 4.00000 0.141421
\(801\) 0 0
\(802\) −36.0000 −1.27120
\(803\) −12.0000 −0.423471
\(804\) 0 0
\(805\) 0 0
\(806\) −4.00000 −0.140894
\(807\) 0 0
\(808\) 12.0000 0.422159
\(809\) 33.0000 1.16022 0.580109 0.814539i \(-0.303010\pi\)
0.580109 + 0.814539i \(0.303010\pi\)
\(810\) 0 0
\(811\) 20.0000 0.702295 0.351147 0.936320i \(-0.385792\pi\)
0.351147 + 0.936320i \(0.385792\pi\)
\(812\) 6.00000 0.210559
\(813\) 0 0
\(814\) 42.0000 1.47210
\(815\) −48.0000 −1.68137
\(816\) 0 0
\(817\) −2.00000 −0.0699711
\(818\) 32.0000 1.11885
\(819\) 0 0
\(820\) 0 0
\(821\) 3.00000 0.104701 0.0523504 0.998629i \(-0.483329\pi\)
0.0523504 + 0.998629i \(0.483329\pi\)
\(822\) 0 0
\(823\) 14.0000 0.488009 0.244005 0.969774i \(-0.421539\pi\)
0.244005 + 0.969774i \(0.421539\pi\)
\(824\) −4.00000 −0.139347
\(825\) 0 0
\(826\) −6.00000 −0.208767
\(827\) −18.0000 −0.625921 −0.312961 0.949766i \(-0.601321\pi\)
−0.312961 + 0.949766i \(0.601321\pi\)
\(828\) 0 0
\(829\) 38.0000 1.31979 0.659897 0.751356i \(-0.270600\pi\)
0.659897 + 0.751356i \(0.270600\pi\)
\(830\) −36.0000 −1.24958
\(831\) 0 0
\(832\) 1.00000 0.0346688
\(833\) −18.0000 −0.623663
\(834\) 0 0
\(835\) 0 0
\(836\) −12.0000 −0.415029
\(837\) 0 0
\(838\) −9.00000 −0.310900
\(839\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(840\) 0 0
\(841\) 7.00000 0.241379
\(842\) 17.0000 0.585859
\(843\) 0 0
\(844\) −13.0000 −0.447478
\(845\) 3.00000 0.103203
\(846\) 0 0
\(847\) −25.0000 −0.859010
\(848\) 0 0
\(849\) 0 0
\(850\) 12.0000 0.411597
\(851\) 0 0
\(852\) 0 0
\(853\) −37.0000 −1.26686 −0.633428 0.773802i \(-0.718353\pi\)
−0.633428 + 0.773802i \(0.718353\pi\)
\(854\) −8.00000 −0.273754
\(855\) 0 0
\(856\) −12.0000 −0.410152
\(857\) 42.0000 1.43469 0.717346 0.696717i \(-0.245357\pi\)
0.717346 + 0.696717i \(0.245357\pi\)
\(858\) 0 0
\(859\) −4.00000 −0.136478 −0.0682391 0.997669i \(-0.521738\pi\)
−0.0682391 + 0.997669i \(0.521738\pi\)
\(860\) −3.00000 −0.102299
\(861\) 0 0
\(862\) 33.0000 1.12398
\(863\) 45.0000 1.53182 0.765909 0.642949i \(-0.222289\pi\)
0.765909 + 0.642949i \(0.222289\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) −25.0000 −0.849535
\(867\) 0 0
\(868\) 4.00000 0.135769
\(869\) −48.0000 −1.62829
\(870\) 0 0
\(871\) 14.0000 0.474372
\(872\) −7.00000 −0.237050
\(873\) 0 0
\(874\) 0 0
\(875\) 3.00000 0.101419
\(876\) 0 0
\(877\) −13.0000 −0.438979 −0.219489 0.975615i \(-0.570439\pi\)
−0.219489 + 0.975615i \(0.570439\pi\)
\(878\) 26.0000 0.877457
\(879\) 0 0
\(880\) −18.0000 −0.606780
\(881\) −21.0000 −0.707508 −0.353754 0.935339i \(-0.615095\pi\)
−0.353754 + 0.935339i \(0.615095\pi\)
\(882\) 0 0
\(883\) 29.0000 0.975928 0.487964 0.872864i \(-0.337740\pi\)
0.487964 + 0.872864i \(0.337740\pi\)
\(884\) 3.00000 0.100901
\(885\) 0 0
\(886\) −21.0000 −0.705509
\(887\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(888\) 0 0
\(889\) −20.0000 −0.670778
\(890\) 18.0000 0.603361
\(891\) 0 0
\(892\) −19.0000 −0.636167
\(893\) −6.00000 −0.200782
\(894\) 0 0
\(895\) −9.00000 −0.300837
\(896\) −1.00000 −0.0334077
\(897\) 0 0
\(898\) −6.00000 −0.200223
\(899\) 24.0000 0.800445
\(900\) 0 0
\(901\) 0 0
\(902\) 0 0
\(903\) 0 0
\(904\) 6.00000 0.199557
\(905\) 60.0000 1.99447
\(906\) 0 0
\(907\) −37.0000 −1.22856 −0.614282 0.789086i \(-0.710554\pi\)
−0.614282 + 0.789086i \(0.710554\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) −3.00000 −0.0994490
\(911\) −30.0000 −0.993944 −0.496972 0.867766i \(-0.665555\pi\)
−0.496972 + 0.867766i \(0.665555\pi\)
\(912\) 0 0
\(913\) 72.0000 2.38285
\(914\) −10.0000 −0.330771
\(915\) 0 0
\(916\) −13.0000 −0.429532
\(917\) −21.0000 −0.693481
\(918\) 0 0
\(919\) −16.0000 −0.527791 −0.263896 0.964551i \(-0.585007\pi\)
−0.263896 + 0.964551i \(0.585007\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) −9.00000 −0.296399
\(923\) 3.00000 0.0987462
\(924\) 0 0
\(925\) −28.0000 −0.920634
\(926\) −40.0000 −1.31448
\(927\) 0 0
\(928\) −6.00000 −0.196960
\(929\) −36.0000 −1.18112 −0.590561 0.806993i \(-0.701093\pi\)
−0.590561 + 0.806993i \(0.701093\pi\)
\(930\) 0 0
\(931\) −12.0000 −0.393284
\(932\) 27.0000 0.884414
\(933\) 0 0
\(934\) −36.0000 −1.17796
\(935\) −54.0000 −1.76599
\(936\) 0 0
\(937\) −34.0000 −1.11073 −0.555366 0.831606i \(-0.687422\pi\)
−0.555366 + 0.831606i \(0.687422\pi\)
\(938\) −14.0000 −0.457116
\(939\) 0 0
\(940\) −9.00000 −0.293548
\(941\) 21.0000 0.684580 0.342290 0.939594i \(-0.388797\pi\)
0.342290 + 0.939594i \(0.388797\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 6.00000 0.195283
\(945\) 0 0
\(946\) 6.00000 0.195077
\(947\) −6.00000 −0.194974 −0.0974869 0.995237i \(-0.531080\pi\)
−0.0974869 + 0.995237i \(0.531080\pi\)
\(948\) 0 0
\(949\) 2.00000 0.0649227
\(950\) 8.00000 0.259554
\(951\) 0 0
\(952\) −3.00000 −0.0972306
\(953\) −15.0000 −0.485898 −0.242949 0.970039i \(-0.578115\pi\)
−0.242949 + 0.970039i \(0.578115\pi\)
\(954\) 0 0
\(955\) 54.0000 1.74740
\(956\) −15.0000 −0.485135
\(957\) 0 0
\(958\) 21.0000 0.678479
\(959\) 0 0
\(960\) 0 0
\(961\) −15.0000 −0.483871
\(962\) −7.00000 −0.225689
\(963\) 0 0
\(964\) −10.0000 −0.322078
\(965\) −12.0000 −0.386294
\(966\) 0 0
\(967\) −31.0000 −0.996893 −0.498446 0.866921i \(-0.666096\pi\)
−0.498446 + 0.866921i \(0.666096\pi\)
\(968\) 25.0000 0.803530
\(969\) 0 0
\(970\) −30.0000 −0.963242
\(971\) 3.00000 0.0962746 0.0481373 0.998841i \(-0.484672\pi\)
0.0481373 + 0.998841i \(0.484672\pi\)
\(972\) 0 0
\(973\) 13.0000 0.416761
\(974\) −16.0000 −0.512673
\(975\) 0 0
\(976\) 8.00000 0.256074
\(977\) 54.0000 1.72761 0.863807 0.503824i \(-0.168074\pi\)
0.863807 + 0.503824i \(0.168074\pi\)
\(978\) 0 0
\(979\) −36.0000 −1.15056
\(980\) −18.0000 −0.574989
\(981\) 0 0
\(982\) 9.00000 0.287202
\(983\) −39.0000 −1.24391 −0.621953 0.783054i \(-0.713661\pi\)
−0.621953 + 0.783054i \(0.713661\pi\)
\(984\) 0 0
\(985\) −9.00000 −0.286764
\(986\) −18.0000 −0.573237
\(987\) 0 0
\(988\) 2.00000 0.0636285
\(989\) 0 0
\(990\) 0 0
\(991\) 2.00000 0.0635321 0.0317660 0.999495i \(-0.489887\pi\)
0.0317660 + 0.999495i \(0.489887\pi\)
\(992\) −4.00000 −0.127000
\(993\) 0 0
\(994\) −3.00000 −0.0951542
\(995\) 6.00000 0.190213
\(996\) 0 0
\(997\) −46.0000 −1.45683 −0.728417 0.685134i \(-0.759744\pi\)
−0.728417 + 0.685134i \(0.759744\pi\)
\(998\) −40.0000 −1.26618
\(999\) 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 234.2.a.e.1.1 1
3.2 odd 2 26.2.a.a.1.1 1
4.3 odd 2 1872.2.a.q.1.1 1
5.2 odd 4 5850.2.e.a.5149.2 2
5.3 odd 4 5850.2.e.a.5149.1 2
5.4 even 2 5850.2.a.p.1.1 1
8.3 odd 2 7488.2.a.h.1.1 1
8.5 even 2 7488.2.a.g.1.1 1
9.2 odd 6 2106.2.e.ba.1405.1 2
9.4 even 3 2106.2.e.b.703.1 2
9.5 odd 6 2106.2.e.ba.703.1 2
9.7 even 3 2106.2.e.b.1405.1 2
12.11 even 2 208.2.a.a.1.1 1
13.5 odd 4 3042.2.b.a.1351.1 2
13.8 odd 4 3042.2.b.a.1351.2 2
13.12 even 2 3042.2.a.a.1.1 1
15.2 even 4 650.2.b.d.599.1 2
15.8 even 4 650.2.b.d.599.2 2
15.14 odd 2 650.2.a.j.1.1 1
21.2 odd 6 1274.2.f.p.1145.1 2
21.5 even 6 1274.2.f.r.1145.1 2
21.11 odd 6 1274.2.f.p.79.1 2
21.17 even 6 1274.2.f.r.79.1 2
21.20 even 2 1274.2.a.d.1.1 1
24.5 odd 2 832.2.a.d.1.1 1
24.11 even 2 832.2.a.i.1.1 1
33.32 even 2 3146.2.a.n.1.1 1
39.2 even 12 338.2.e.a.147.1 4
39.5 even 4 338.2.b.c.337.2 2
39.8 even 4 338.2.b.c.337.1 2
39.11 even 12 338.2.e.a.147.2 4
39.17 odd 6 338.2.c.a.315.1 2
39.20 even 12 338.2.e.a.23.1 4
39.23 odd 6 338.2.c.a.191.1 2
39.29 odd 6 338.2.c.d.191.1 2
39.32 even 12 338.2.e.a.23.2 4
39.35 odd 6 338.2.c.d.315.1 2
39.38 odd 2 338.2.a.f.1.1 1
48.5 odd 4 3328.2.b.m.1665.1 2
48.11 even 4 3328.2.b.j.1665.2 2
48.29 odd 4 3328.2.b.m.1665.2 2
48.35 even 4 3328.2.b.j.1665.1 2
51.50 odd 2 7514.2.a.c.1.1 1
57.56 even 2 9386.2.a.j.1.1 1
60.59 even 2 5200.2.a.x.1.1 1
156.47 odd 4 2704.2.f.d.337.1 2
156.83 odd 4 2704.2.f.d.337.2 2
156.155 even 2 2704.2.a.f.1.1 1
195.194 odd 2 8450.2.a.c.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
26.2.a.a.1.1 1 3.2 odd 2
208.2.a.a.1.1 1 12.11 even 2
234.2.a.e.1.1 1 1.1 even 1 trivial
338.2.a.f.1.1 1 39.38 odd 2
338.2.b.c.337.1 2 39.8 even 4
338.2.b.c.337.2 2 39.5 even 4
338.2.c.a.191.1 2 39.23 odd 6
338.2.c.a.315.1 2 39.17 odd 6
338.2.c.d.191.1 2 39.29 odd 6
338.2.c.d.315.1 2 39.35 odd 6
338.2.e.a.23.1 4 39.20 even 12
338.2.e.a.23.2 4 39.32 even 12
338.2.e.a.147.1 4 39.2 even 12
338.2.e.a.147.2 4 39.11 even 12
650.2.a.j.1.1 1 15.14 odd 2
650.2.b.d.599.1 2 15.2 even 4
650.2.b.d.599.2 2 15.8 even 4
832.2.a.d.1.1 1 24.5 odd 2
832.2.a.i.1.1 1 24.11 even 2
1274.2.a.d.1.1 1 21.20 even 2
1274.2.f.p.79.1 2 21.11 odd 6
1274.2.f.p.1145.1 2 21.2 odd 6
1274.2.f.r.79.1 2 21.17 even 6
1274.2.f.r.1145.1 2 21.5 even 6
1872.2.a.q.1.1 1 4.3 odd 2
2106.2.e.b.703.1 2 9.4 even 3
2106.2.e.b.1405.1 2 9.7 even 3
2106.2.e.ba.703.1 2 9.5 odd 6
2106.2.e.ba.1405.1 2 9.2 odd 6
2704.2.a.f.1.1 1 156.155 even 2
2704.2.f.d.337.1 2 156.47 odd 4
2704.2.f.d.337.2 2 156.83 odd 4
3042.2.a.a.1.1 1 13.12 even 2
3042.2.b.a.1351.1 2 13.5 odd 4
3042.2.b.a.1351.2 2 13.8 odd 4
3146.2.a.n.1.1 1 33.32 even 2
3328.2.b.j.1665.1 2 48.35 even 4
3328.2.b.j.1665.2 2 48.11 even 4
3328.2.b.m.1665.1 2 48.5 odd 4
3328.2.b.m.1665.2 2 48.29 odd 4
5200.2.a.x.1.1 1 60.59 even 2
5850.2.a.p.1.1 1 5.4 even 2
5850.2.e.a.5149.1 2 5.3 odd 4
5850.2.e.a.5149.2 2 5.2 odd 4
7488.2.a.g.1.1 1 8.5 even 2
7488.2.a.h.1.1 1 8.3 odd 2
7514.2.a.c.1.1 1 51.50 odd 2
8450.2.a.c.1.1 1 195.194 odd 2
9386.2.a.j.1.1 1 57.56 even 2