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Properties

Label 4050.2.a.n.1.1
Level $4050$
Weight $2$
Character 4050.1
Self dual yes
Analytic conductor $32.339$
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] = [4050,2,Mod(1,4050)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4050, 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("4050.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4050 = 2 \cdot 3^{4} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4050.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: \(32.3394128186\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 90)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 4050.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{7} -1.00000 q^{8} +6.00000 q^{11} -2.00000 q^{13} -1.00000 q^{14} +1.00000 q^{16} -4.00000 q^{19} -6.00000 q^{22} -9.00000 q^{23} +2.00000 q^{26} +1.00000 q^{28} +3.00000 q^{29} -4.00000 q^{31} -1.00000 q^{32} -8.00000 q^{37} +4.00000 q^{38} -3.00000 q^{41} -8.00000 q^{43} +6.00000 q^{44} +9.00000 q^{46} +3.00000 q^{47} -6.00000 q^{49} -2.00000 q^{52} -6.00000 q^{53} -1.00000 q^{56} -3.00000 q^{58} +6.00000 q^{59} -13.0000 q^{61} +4.00000 q^{62} +1.00000 q^{64} +13.0000 q^{67} -6.00000 q^{71} +4.00000 q^{73} +8.00000 q^{74} -4.00000 q^{76} +6.00000 q^{77} -10.0000 q^{79} +3.00000 q^{82} +9.00000 q^{83} +8.00000 q^{86} -6.00000 q^{88} +9.00000 q^{89} -2.00000 q^{91} -9.00000 q^{92} -3.00000 q^{94} -2.00000 q^{97} +6.00000 q^{98} +O(q^{100})\)

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
\(5\) 0 0
\(6\) 0 0
\(7\) 1.00000 0.377964 0.188982 − 0.981981i \(-0.439481\pi\)
0.188982 + 0.981981i \(0.439481\pi\)
\(8\) −1.00000 −0.353553
\(9\) 0 0
\(10\) 0 0
\(11\) 6.00000 1.80907 0.904534 − 0.426401i \(-0.140219\pi\)
0.904534 + 0.426401i \(0.140219\pi\)
\(12\) 0 0
\(13\) −2.00000 −0.554700 −0.277350 − 0.960769i \(-0.589456\pi\)
−0.277350 + 0.960769i \(0.589456\pi\)
\(14\) −1.00000 −0.267261
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(18\) 0 0
\(19\) −4.00000 −0.917663 −0.458831 − 0.888523i \(-0.651732\pi\)
−0.458831 + 0.888523i \(0.651732\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) −6.00000 −1.27920
\(23\) −9.00000 −1.87663 −0.938315 − 0.345782i \(-0.887614\pi\)
−0.938315 + 0.345782i \(0.887614\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 2.00000 0.392232
\(27\) 0 0
\(28\) 1.00000 0.188982
\(29\) 3.00000 0.557086 0.278543 − 0.960424i \(-0.410149\pi\)
0.278543 + 0.960424i \(0.410149\pi\)
\(30\) 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
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −8.00000 −1.31519 −0.657596 − 0.753371i \(-0.728427\pi\)
−0.657596 + 0.753371i \(0.728427\pi\)
\(38\) 4.00000 0.648886
\(39\) 0 0
\(40\) 0 0
\(41\) −3.00000 −0.468521 −0.234261 − 0.972174i \(-0.575267\pi\)
−0.234261 + 0.972174i \(0.575267\pi\)
\(42\) 0 0
\(43\) −8.00000 −1.21999 −0.609994 − 0.792406i \(-0.708828\pi\)
−0.609994 + 0.792406i \(0.708828\pi\)
\(44\) 6.00000 0.904534
\(45\) 0 0
\(46\) 9.00000 1.32698
\(47\) 3.00000 0.437595 0.218797 − 0.975770i \(-0.429787\pi\)
0.218797 + 0.975770i \(0.429787\pi\)
\(48\) 0 0
\(49\) −6.00000 −0.857143
\(50\) 0 0
\(51\) 0 0
\(52\) −2.00000 −0.277350
\(53\) −6.00000 −0.824163 −0.412082 − 0.911147i \(-0.635198\pi\)
−0.412082 + 0.911147i \(0.635198\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −1.00000 −0.133631
\(57\) 0 0
\(58\) −3.00000 −0.393919
\(59\) 6.00000 0.781133 0.390567 − 0.920575i \(-0.372279\pi\)
0.390567 + 0.920575i \(0.372279\pi\)
\(60\) 0 0
\(61\) −13.0000 −1.66448 −0.832240 − 0.554416i \(-0.812942\pi\)
−0.832240 + 0.554416i \(0.812942\pi\)
\(62\) 4.00000 0.508001
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 0 0
\(67\) 13.0000 1.58820 0.794101 − 0.607785i \(-0.207942\pi\)
0.794101 + 0.607785i \(0.207942\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −6.00000 −0.712069 −0.356034 − 0.934473i \(-0.615871\pi\)
−0.356034 + 0.934473i \(0.615871\pi\)
\(72\) 0 0
\(73\) 4.00000 0.468165 0.234082 − 0.972217i \(-0.424791\pi\)
0.234082 + 0.972217i \(0.424791\pi\)
\(74\) 8.00000 0.929981
\(75\) 0 0
\(76\) −4.00000 −0.458831
\(77\) 6.00000 0.683763
\(78\) 0 0
\(79\) −10.0000 −1.12509 −0.562544 − 0.826767i \(-0.690177\pi\)
−0.562544 + 0.826767i \(0.690177\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 3.00000 0.331295
\(83\) 9.00000 0.987878 0.493939 − 0.869496i \(-0.335557\pi\)
0.493939 + 0.869496i \(0.335557\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 8.00000 0.862662
\(87\) 0 0
\(88\) −6.00000 −0.639602
\(89\) 9.00000 0.953998 0.476999 − 0.878904i \(-0.341725\pi\)
0.476999 + 0.878904i \(0.341725\pi\)
\(90\) 0 0
\(91\) −2.00000 −0.209657
\(92\) −9.00000 −0.938315
\(93\) 0 0
\(94\) −3.00000 −0.309426
\(95\) 0 0
\(96\) 0 0
\(97\) −2.00000 −0.203069 −0.101535 − 0.994832i \(-0.532375\pi\)
−0.101535 + 0.994832i \(0.532375\pi\)
\(98\) 6.00000 0.606092
\(99\) 0 0
\(100\) 0 0
\(101\) 6.00000 0.597022 0.298511 − 0.954406i \(-0.403510\pi\)
0.298511 + 0.954406i \(0.403510\pi\)
\(102\) 0 0
\(103\) −8.00000 −0.788263 −0.394132 − 0.919054i \(-0.628955\pi\)
−0.394132 + 0.919054i \(0.628955\pi\)
\(104\) 2.00000 0.196116
\(105\) 0 0
\(106\) 6.00000 0.582772
\(107\) 3.00000 0.290021 0.145010 − 0.989430i \(-0.453678\pi\)
0.145010 + 0.989430i \(0.453678\pi\)
\(108\) 0 0
\(109\) −7.00000 −0.670478 −0.335239 − 0.942133i \(-0.608817\pi\)
−0.335239 + 0.942133i \(0.608817\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 1.00000 0.0944911
\(113\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 3.00000 0.278543
\(117\) 0 0
\(118\) −6.00000 −0.552345
\(119\) 0 0
\(120\) 0 0
\(121\) 25.0000 2.27273
\(122\) 13.0000 1.17696
\(123\) 0 0
\(124\) −4.00000 −0.359211
\(125\) 0 0
\(126\) 0 0
\(127\) 7.00000 0.621150 0.310575 − 0.950549i \(-0.399478\pi\)
0.310575 + 0.950549i \(0.399478\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 0 0
\(130\) 0 0
\(131\) −18.0000 −1.57267 −0.786334 − 0.617802i \(-0.788023\pi\)
−0.786334 + 0.617802i \(0.788023\pi\)
\(132\) 0 0
\(133\) −4.00000 −0.346844
\(134\) −13.0000 −1.12303
\(135\) 0 0
\(136\) 0 0
\(137\) 12.0000 1.02523 0.512615 − 0.858619i \(-0.328677\pi\)
0.512615 + 0.858619i \(0.328677\pi\)
\(138\) 0 0
\(139\) −16.0000 −1.35710 −0.678551 − 0.734553i \(-0.737392\pi\)
−0.678551 + 0.734553i \(0.737392\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 6.00000 0.503509
\(143\) −12.0000 −1.00349
\(144\) 0 0
\(145\) 0 0
\(146\) −4.00000 −0.331042
\(147\) 0 0
\(148\) −8.00000 −0.657596
\(149\) −3.00000 −0.245770 −0.122885 − 0.992421i \(-0.539215\pi\)
−0.122885 + 0.992421i \(0.539215\pi\)
\(150\) 0 0
\(151\) 14.0000 1.13930 0.569652 − 0.821886i \(-0.307078\pi\)
0.569652 + 0.821886i \(0.307078\pi\)
\(152\) 4.00000 0.324443
\(153\) 0 0
\(154\) −6.00000 −0.483494
\(155\) 0 0
\(156\) 0 0
\(157\) −14.0000 −1.11732 −0.558661 − 0.829396i \(-0.688685\pi\)
−0.558661 + 0.829396i \(0.688685\pi\)
\(158\) 10.0000 0.795557
\(159\) 0 0
\(160\) 0 0
\(161\) −9.00000 −0.709299
\(162\) 0 0
\(163\) 4.00000 0.313304 0.156652 − 0.987654i \(-0.449930\pi\)
0.156652 + 0.987654i \(0.449930\pi\)
\(164\) −3.00000 −0.234261
\(165\) 0 0
\(166\) −9.00000 −0.698535
\(167\) 3.00000 0.232147 0.116073 − 0.993241i \(-0.462969\pi\)
0.116073 + 0.993241i \(0.462969\pi\)
\(168\) 0 0
\(169\) −9.00000 −0.692308
\(170\) 0 0
\(171\) 0 0
\(172\) −8.00000 −0.609994
\(173\) −24.0000 −1.82469 −0.912343 − 0.409426i \(-0.865729\pi\)
−0.912343 + 0.409426i \(0.865729\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 6.00000 0.452267
\(177\) 0 0
\(178\) −9.00000 −0.674579
\(179\) −18.0000 −1.34538 −0.672692 − 0.739923i \(-0.734862\pi\)
−0.672692 + 0.739923i \(0.734862\pi\)
\(180\) 0 0
\(181\) 5.00000 0.371647 0.185824 − 0.982583i \(-0.440505\pi\)
0.185824 + 0.982583i \(0.440505\pi\)
\(182\) 2.00000 0.148250
\(183\) 0 0
\(184\) 9.00000 0.663489
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) 3.00000 0.218797
\(189\) 0 0
\(190\) 0 0
\(191\) −12.0000 −0.868290 −0.434145 − 0.900843i \(-0.642949\pi\)
−0.434145 + 0.900843i \(0.642949\pi\)
\(192\) 0 0
\(193\) −2.00000 −0.143963 −0.0719816 − 0.997406i \(-0.522932\pi\)
−0.0719816 + 0.997406i \(0.522932\pi\)
\(194\) 2.00000 0.143592
\(195\) 0 0
\(196\) −6.00000 −0.428571
\(197\) 12.0000 0.854965 0.427482 − 0.904024i \(-0.359401\pi\)
0.427482 + 0.904024i \(0.359401\pi\)
\(198\) 0 0
\(199\) 8.00000 0.567105 0.283552 − 0.958957i \(-0.408487\pi\)
0.283552 + 0.958957i \(0.408487\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) −6.00000 −0.422159
\(203\) 3.00000 0.210559
\(204\) 0 0
\(205\) 0 0
\(206\) 8.00000 0.557386
\(207\) 0 0
\(208\) −2.00000 −0.138675
\(209\) −24.0000 −1.66011
\(210\) 0 0
\(211\) 2.00000 0.137686 0.0688428 − 0.997628i \(-0.478069\pi\)
0.0688428 + 0.997628i \(0.478069\pi\)
\(212\) −6.00000 −0.412082
\(213\) 0 0
\(214\) −3.00000 −0.205076
\(215\) 0 0
\(216\) 0 0
\(217\) −4.00000 −0.271538
\(218\) 7.00000 0.474100
\(219\) 0 0
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 1.00000 0.0669650 0.0334825 − 0.999439i \(-0.489340\pi\)
0.0334825 + 0.999439i \(0.489340\pi\)
\(224\) −1.00000 −0.0668153
\(225\) 0 0
\(226\) 0 0
\(227\) 12.0000 0.796468 0.398234 − 0.917284i \(-0.369623\pi\)
0.398234 + 0.917284i \(0.369623\pi\)
\(228\) 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
\(232\) −3.00000 −0.196960
\(233\) 12.0000 0.786146 0.393073 − 0.919507i \(-0.371412\pi\)
0.393073 + 0.919507i \(0.371412\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 6.00000 0.390567
\(237\) 0 0
\(238\) 0 0
\(239\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(240\) 0 0
\(241\) 29.0000 1.86805 0.934027 − 0.357202i \(-0.116269\pi\)
0.934027 + 0.357202i \(0.116269\pi\)
\(242\) −25.0000 −1.60706
\(243\) 0 0
\(244\) −13.0000 −0.832240
\(245\) 0 0
\(246\) 0 0
\(247\) 8.00000 0.509028
\(248\) 4.00000 0.254000
\(249\) 0 0
\(250\) 0 0
\(251\) 12.0000 0.757433 0.378717 − 0.925513i \(-0.376365\pi\)
0.378717 + 0.925513i \(0.376365\pi\)
\(252\) 0 0
\(253\) −54.0000 −3.39495
\(254\) −7.00000 −0.439219
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −18.0000 −1.12281 −0.561405 − 0.827541i \(-0.689739\pi\)
−0.561405 + 0.827541i \(0.689739\pi\)
\(258\) 0 0
\(259\) −8.00000 −0.497096
\(260\) 0 0
\(261\) 0 0
\(262\) 18.0000 1.11204
\(263\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 4.00000 0.245256
\(267\) 0 0
\(268\) 13.0000 0.794101
\(269\) 21.0000 1.28039 0.640196 − 0.768211i \(-0.278853\pi\)
0.640196 + 0.768211i \(0.278853\pi\)
\(270\) 0 0
\(271\) −4.00000 −0.242983 −0.121491 − 0.992592i \(-0.538768\pi\)
−0.121491 + 0.992592i \(0.538768\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) −12.0000 −0.724947
\(275\) 0 0
\(276\) 0 0
\(277\) −8.00000 −0.480673 −0.240337 − 0.970690i \(-0.577258\pi\)
−0.240337 + 0.970690i \(0.577258\pi\)
\(278\) 16.0000 0.959616
\(279\) 0 0
\(280\) 0 0
\(281\) −15.0000 −0.894825 −0.447412 − 0.894328i \(-0.647654\pi\)
−0.447412 + 0.894328i \(0.647654\pi\)
\(282\) 0 0
\(283\) 13.0000 0.772770 0.386385 − 0.922338i \(-0.373724\pi\)
0.386385 + 0.922338i \(0.373724\pi\)
\(284\) −6.00000 −0.356034
\(285\) 0 0
\(286\) 12.0000 0.709575
\(287\) −3.00000 −0.177084
\(288\) 0 0
\(289\) −17.0000 −1.00000
\(290\) 0 0
\(291\) 0 0
\(292\) 4.00000 0.234082
\(293\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 8.00000 0.464991
\(297\) 0 0
\(298\) 3.00000 0.173785
\(299\) 18.0000 1.04097
\(300\) 0 0
\(301\) −8.00000 −0.461112
\(302\) −14.0000 −0.805609
\(303\) 0 0
\(304\) −4.00000 −0.229416
\(305\) 0 0
\(306\) 0 0
\(307\) 7.00000 0.399511 0.199756 − 0.979846i \(-0.435985\pi\)
0.199756 + 0.979846i \(0.435985\pi\)
\(308\) 6.00000 0.341882
\(309\) 0 0
\(310\) 0 0
\(311\) 12.0000 0.680458 0.340229 − 0.940343i \(-0.389495\pi\)
0.340229 + 0.940343i \(0.389495\pi\)
\(312\) 0 0
\(313\) −2.00000 −0.113047 −0.0565233 − 0.998401i \(-0.518002\pi\)
−0.0565233 + 0.998401i \(0.518002\pi\)
\(314\) 14.0000 0.790066
\(315\) 0 0
\(316\) −10.0000 −0.562544
\(317\) 6.00000 0.336994 0.168497 − 0.985702i \(-0.446109\pi\)
0.168497 + 0.985702i \(0.446109\pi\)
\(318\) 0 0
\(319\) 18.0000 1.00781
\(320\) 0 0
\(321\) 0 0
\(322\) 9.00000 0.501550
\(323\) 0 0
\(324\) 0 0
\(325\) 0 0
\(326\) −4.00000 −0.221540
\(327\) 0 0
\(328\) 3.00000 0.165647
\(329\) 3.00000 0.165395
\(330\) 0 0
\(331\) −10.0000 −0.549650 −0.274825 − 0.961494i \(-0.588620\pi\)
−0.274825 + 0.961494i \(0.588620\pi\)
\(332\) 9.00000 0.493939
\(333\) 0 0
\(334\) −3.00000 −0.164153
\(335\) 0 0
\(336\) 0 0
\(337\) −8.00000 −0.435788 −0.217894 − 0.975972i \(-0.569919\pi\)
−0.217894 + 0.975972i \(0.569919\pi\)
\(338\) 9.00000 0.489535
\(339\) 0 0
\(340\) 0 0
\(341\) −24.0000 −1.29967
\(342\) 0 0
\(343\) −13.0000 −0.701934
\(344\) 8.00000 0.431331
\(345\) 0 0
\(346\) 24.0000 1.29025
\(347\) −12.0000 −0.644194 −0.322097 − 0.946707i \(-0.604388\pi\)
−0.322097 + 0.946707i \(0.604388\pi\)
\(348\) 0 0
\(349\) 23.0000 1.23116 0.615581 − 0.788074i \(-0.288921\pi\)
0.615581 + 0.788074i \(0.288921\pi\)
\(350\) 0 0
\(351\) 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
\(355\) 0 0
\(356\) 9.00000 0.476999
\(357\) 0 0
\(358\) 18.0000 0.951330
\(359\) −12.0000 −0.633336 −0.316668 − 0.948536i \(-0.602564\pi\)
−0.316668 + 0.948536i \(0.602564\pi\)
\(360\) 0 0
\(361\) −3.00000 −0.157895
\(362\) −5.00000 −0.262794
\(363\) 0 0
\(364\) −2.00000 −0.104828
\(365\) 0 0
\(366\) 0 0
\(367\) −8.00000 −0.417597 −0.208798 − 0.977959i \(-0.566955\pi\)
−0.208798 + 0.977959i \(0.566955\pi\)
\(368\) −9.00000 −0.469157
\(369\) 0 0
\(370\) 0 0
\(371\) −6.00000 −0.311504
\(372\) 0 0
\(373\) −26.0000 −1.34623 −0.673114 − 0.739538i \(-0.735044\pi\)
−0.673114 + 0.739538i \(0.735044\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) −3.00000 −0.154713
\(377\) −6.00000 −0.309016
\(378\) 0 0
\(379\) −22.0000 −1.13006 −0.565032 − 0.825069i \(-0.691136\pi\)
−0.565032 + 0.825069i \(0.691136\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 12.0000 0.613973
\(383\) 12.0000 0.613171 0.306586 − 0.951843i \(-0.400813\pi\)
0.306586 + 0.951843i \(0.400813\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 2.00000 0.101797
\(387\) 0 0
\(388\) −2.00000 −0.101535
\(389\) 21.0000 1.06474 0.532371 − 0.846511i \(-0.321301\pi\)
0.532371 + 0.846511i \(0.321301\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 6.00000 0.303046
\(393\) 0 0
\(394\) −12.0000 −0.604551
\(395\) 0 0
\(396\) 0 0
\(397\) −2.00000 −0.100377 −0.0501886 − 0.998740i \(-0.515982\pi\)
−0.0501886 + 0.998740i \(0.515982\pi\)
\(398\) −8.00000 −0.401004
\(399\) 0 0
\(400\) 0 0
\(401\) 6.00000 0.299626 0.149813 − 0.988714i \(-0.452133\pi\)
0.149813 + 0.988714i \(0.452133\pi\)
\(402\) 0 0
\(403\) 8.00000 0.398508
\(404\) 6.00000 0.298511
\(405\) 0 0
\(406\) −3.00000 −0.148888
\(407\) −48.0000 −2.37927
\(408\) 0 0
\(409\) −10.0000 −0.494468 −0.247234 − 0.968956i \(-0.579522\pi\)
−0.247234 + 0.968956i \(0.579522\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) −8.00000 −0.394132
\(413\) 6.00000 0.295241
\(414\) 0 0
\(415\) 0 0
\(416\) 2.00000 0.0980581
\(417\) 0 0
\(418\) 24.0000 1.17388
\(419\) −30.0000 −1.46560 −0.732798 − 0.680446i \(-0.761786\pi\)
−0.732798 + 0.680446i \(0.761786\pi\)
\(420\) 0 0
\(421\) −22.0000 −1.07221 −0.536107 − 0.844150i \(-0.680106\pi\)
−0.536107 + 0.844150i \(0.680106\pi\)
\(422\) −2.00000 −0.0973585
\(423\) 0 0
\(424\) 6.00000 0.291386
\(425\) 0 0
\(426\) 0 0
\(427\) −13.0000 −0.629114
\(428\) 3.00000 0.145010
\(429\) 0 0
\(430\) 0 0
\(431\) 6.00000 0.289010 0.144505 − 0.989504i \(-0.453841\pi\)
0.144505 + 0.989504i \(0.453841\pi\)
\(432\) 0 0
\(433\) 16.0000 0.768911 0.384455 − 0.923144i \(-0.374389\pi\)
0.384455 + 0.923144i \(0.374389\pi\)
\(434\) 4.00000 0.192006
\(435\) 0 0
\(436\) −7.00000 −0.335239
\(437\) 36.0000 1.72211
\(438\) 0 0
\(439\) −28.0000 −1.33637 −0.668184 − 0.743996i \(-0.732928\pi\)
−0.668184 + 0.743996i \(0.732928\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −9.00000 −0.427603 −0.213801 − 0.976877i \(-0.568585\pi\)
−0.213801 + 0.976877i \(0.568585\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) −1.00000 −0.0473514
\(447\) 0 0
\(448\) 1.00000 0.0472456
\(449\) 6.00000 0.283158 0.141579 − 0.989927i \(-0.454782\pi\)
0.141579 + 0.989927i \(0.454782\pi\)
\(450\) 0 0
\(451\) −18.0000 −0.847587
\(452\) 0 0
\(453\) 0 0
\(454\) −12.0000 −0.563188
\(455\) 0 0
\(456\) 0 0
\(457\) −8.00000 −0.374224 −0.187112 − 0.982339i \(-0.559913\pi\)
−0.187112 + 0.982339i \(0.559913\pi\)
\(458\) 13.0000 0.607450
\(459\) 0 0
\(460\) 0 0
\(461\) −27.0000 −1.25752 −0.628758 − 0.777601i \(-0.716436\pi\)
−0.628758 + 0.777601i \(0.716436\pi\)
\(462\) 0 0
\(463\) 4.00000 0.185896 0.0929479 − 0.995671i \(-0.470371\pi\)
0.0929479 + 0.995671i \(0.470371\pi\)
\(464\) 3.00000 0.139272
\(465\) 0 0
\(466\) −12.0000 −0.555889
\(467\) −36.0000 −1.66588 −0.832941 − 0.553362i \(-0.813345\pi\)
−0.832941 + 0.553362i \(0.813345\pi\)
\(468\) 0 0
\(469\) 13.0000 0.600284
\(470\) 0 0
\(471\) 0 0
\(472\) −6.00000 −0.276172
\(473\) −48.0000 −2.20704
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −30.0000 −1.37073 −0.685367 − 0.728197i \(-0.740358\pi\)
−0.685367 + 0.728197i \(0.740358\pi\)
\(480\) 0 0
\(481\) 16.0000 0.729537
\(482\) −29.0000 −1.32091
\(483\) 0 0
\(484\) 25.0000 1.13636
\(485\) 0 0
\(486\) 0 0
\(487\) −8.00000 −0.362515 −0.181257 − 0.983436i \(-0.558017\pi\)
−0.181257 + 0.983436i \(0.558017\pi\)
\(488\) 13.0000 0.588482
\(489\) 0 0
\(490\) 0 0
\(491\) 12.0000 0.541552 0.270776 − 0.962642i \(-0.412720\pi\)
0.270776 + 0.962642i \(0.412720\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) −8.00000 −0.359937
\(495\) 0 0
\(496\) −4.00000 −0.179605
\(497\) −6.00000 −0.269137
\(498\) 0 0
\(499\) 32.0000 1.43252 0.716258 − 0.697835i \(-0.245853\pi\)
0.716258 + 0.697835i \(0.245853\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) −12.0000 −0.535586
\(503\) −27.0000 −1.20387 −0.601935 − 0.798545i \(-0.705603\pi\)
−0.601935 + 0.798545i \(0.705603\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 54.0000 2.40059
\(507\) 0 0
\(508\) 7.00000 0.310575
\(509\) 15.0000 0.664863 0.332432 − 0.943127i \(-0.392131\pi\)
0.332432 + 0.943127i \(0.392131\pi\)
\(510\) 0 0
\(511\) 4.00000 0.176950
\(512\) −1.00000 −0.0441942
\(513\) 0 0
\(514\) 18.0000 0.793946
\(515\) 0 0
\(516\) 0 0
\(517\) 18.0000 0.791639
\(518\) 8.00000 0.351500
\(519\) 0 0
\(520\) 0 0
\(521\) 27.0000 1.18289 0.591446 − 0.806345i \(-0.298557\pi\)
0.591446 + 0.806345i \(0.298557\pi\)
\(522\) 0 0
\(523\) 19.0000 0.830812 0.415406 − 0.909636i \(-0.363640\pi\)
0.415406 + 0.909636i \(0.363640\pi\)
\(524\) −18.0000 −0.786334
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) 58.0000 2.52174
\(530\) 0 0
\(531\) 0 0
\(532\) −4.00000 −0.173422
\(533\) 6.00000 0.259889
\(534\) 0 0
\(535\) 0 0
\(536\) −13.0000 −0.561514
\(537\) 0 0
\(538\) −21.0000 −0.905374
\(539\) −36.0000 −1.55063
\(540\) 0 0
\(541\) 29.0000 1.24681 0.623404 − 0.781900i \(-0.285749\pi\)
0.623404 + 0.781900i \(0.285749\pi\)
\(542\) 4.00000 0.171815
\(543\) 0 0
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 43.0000 1.83855 0.919274 − 0.393619i \(-0.128777\pi\)
0.919274 + 0.393619i \(0.128777\pi\)
\(548\) 12.0000 0.512615
\(549\) 0 0
\(550\) 0 0
\(551\) −12.0000 −0.511217
\(552\) 0 0
\(553\) −10.0000 −0.425243
\(554\) 8.00000 0.339887
\(555\) 0 0
\(556\) −16.0000 −0.678551
\(557\) 6.00000 0.254228 0.127114 − 0.991888i \(-0.459429\pi\)
0.127114 + 0.991888i \(0.459429\pi\)
\(558\) 0 0
\(559\) 16.0000 0.676728
\(560\) 0 0
\(561\) 0 0
\(562\) 15.0000 0.632737
\(563\) 3.00000 0.126435 0.0632175 − 0.998000i \(-0.479864\pi\)
0.0632175 + 0.998000i \(0.479864\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) −13.0000 −0.546431
\(567\) 0 0
\(568\) 6.00000 0.251754
\(569\) −6.00000 −0.251533 −0.125767 − 0.992060i \(-0.540139\pi\)
−0.125767 + 0.992060i \(0.540139\pi\)
\(570\) 0 0
\(571\) −40.0000 −1.67395 −0.836974 − 0.547243i \(-0.815677\pi\)
−0.836974 + 0.547243i \(0.815677\pi\)
\(572\) −12.0000 −0.501745
\(573\) 0 0
\(574\) 3.00000 0.125218
\(575\) 0 0
\(576\) 0 0
\(577\) 10.0000 0.416305 0.208153 − 0.978096i \(-0.433255\pi\)
0.208153 + 0.978096i \(0.433255\pi\)
\(578\) 17.0000 0.707107
\(579\) 0 0
\(580\) 0 0
\(581\) 9.00000 0.373383
\(582\) 0 0
\(583\) −36.0000 −1.49097
\(584\) −4.00000 −0.165521
\(585\) 0 0
\(586\) 0 0
\(587\) 15.0000 0.619116 0.309558 − 0.950881i \(-0.399819\pi\)
0.309558 + 0.950881i \(0.399819\pi\)
\(588\) 0 0
\(589\) 16.0000 0.659269
\(590\) 0 0
\(591\) 0 0
\(592\) −8.00000 −0.328798
\(593\) −24.0000 −0.985562 −0.492781 − 0.870153i \(-0.664020\pi\)
−0.492781 + 0.870153i \(0.664020\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −3.00000 −0.122885
\(597\) 0 0
\(598\) −18.0000 −0.736075
\(599\) −6.00000 −0.245153 −0.122577 − 0.992459i \(-0.539116\pi\)
−0.122577 + 0.992459i \(0.539116\pi\)
\(600\) 0 0
\(601\) 26.0000 1.06056 0.530281 − 0.847822i \(-0.322086\pi\)
0.530281 + 0.847822i \(0.322086\pi\)
\(602\) 8.00000 0.326056
\(603\) 0 0
\(604\) 14.0000 0.569652
\(605\) 0 0
\(606\) 0 0
\(607\) −29.0000 −1.17707 −0.588537 − 0.808470i \(-0.700296\pi\)
−0.588537 + 0.808470i \(0.700296\pi\)
\(608\) 4.00000 0.162221
\(609\) 0 0
\(610\) 0 0
\(611\) −6.00000 −0.242734
\(612\) 0 0
\(613\) 40.0000 1.61558 0.807792 − 0.589467i \(-0.200662\pi\)
0.807792 + 0.589467i \(0.200662\pi\)
\(614\) −7.00000 −0.282497
\(615\) 0 0
\(616\) −6.00000 −0.241747
\(617\) −12.0000 −0.483102 −0.241551 − 0.970388i \(-0.577656\pi\)
−0.241551 + 0.970388i \(0.577656\pi\)
\(618\) 0 0
\(619\) −40.0000 −1.60774 −0.803868 − 0.594808i \(-0.797228\pi\)
−0.803868 + 0.594808i \(0.797228\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) −12.0000 −0.481156
\(623\) 9.00000 0.360577
\(624\) 0 0
\(625\) 0 0
\(626\) 2.00000 0.0799361
\(627\) 0 0
\(628\) −14.0000 −0.558661
\(629\) 0 0
\(630\) 0 0
\(631\) 32.0000 1.27390 0.636950 − 0.770905i \(-0.280196\pi\)
0.636950 + 0.770905i \(0.280196\pi\)
\(632\) 10.0000 0.397779
\(633\) 0 0
\(634\) −6.00000 −0.238290
\(635\) 0 0
\(636\) 0 0
\(637\) 12.0000 0.475457
\(638\) −18.0000 −0.712627
\(639\) 0 0
\(640\) 0 0
\(641\) 33.0000 1.30342 0.651711 − 0.758468i \(-0.274052\pi\)
0.651711 + 0.758468i \(0.274052\pi\)
\(642\) 0 0
\(643\) 31.0000 1.22252 0.611260 − 0.791430i \(-0.290663\pi\)
0.611260 + 0.791430i \(0.290663\pi\)
\(644\) −9.00000 −0.354650
\(645\) 0 0
\(646\) 0 0
\(647\) 3.00000 0.117942 0.0589711 − 0.998260i \(-0.481218\pi\)
0.0589711 + 0.998260i \(0.481218\pi\)
\(648\) 0 0
\(649\) 36.0000 1.41312
\(650\) 0 0
\(651\) 0 0
\(652\) 4.00000 0.156652
\(653\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) −3.00000 −0.117130
\(657\) 0 0
\(658\) −3.00000 −0.116952
\(659\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(660\) 0 0
\(661\) −46.0000 −1.78919 −0.894596 − 0.446875i \(-0.852537\pi\)
−0.894596 + 0.446875i \(0.852537\pi\)
\(662\) 10.0000 0.388661
\(663\) 0 0
\(664\) −9.00000 −0.349268
\(665\) 0 0
\(666\) 0 0
\(667\) −27.0000 −1.04544
\(668\) 3.00000 0.116073
\(669\) 0 0
\(670\) 0 0
\(671\) −78.0000 −3.01116
\(672\) 0 0
\(673\) 46.0000 1.77317 0.886585 − 0.462566i \(-0.153071\pi\)
0.886585 + 0.462566i \(0.153071\pi\)
\(674\) 8.00000 0.308148
\(675\) 0 0
\(676\) −9.00000 −0.346154
\(677\) −18.0000 −0.691796 −0.345898 − 0.938272i \(-0.612426\pi\)
−0.345898 + 0.938272i \(0.612426\pi\)
\(678\) 0 0
\(679\) −2.00000 −0.0767530
\(680\) 0 0
\(681\) 0 0
\(682\) 24.0000 0.919007
\(683\) −12.0000 −0.459167 −0.229584 − 0.973289i \(-0.573736\pi\)
−0.229584 + 0.973289i \(0.573736\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 13.0000 0.496342
\(687\) 0 0
\(688\) −8.00000 −0.304997
\(689\) 12.0000 0.457164
\(690\) 0 0
\(691\) −10.0000 −0.380418 −0.190209 − 0.981744i \(-0.560917\pi\)
−0.190209 + 0.981744i \(0.560917\pi\)
\(692\) −24.0000 −0.912343
\(693\) 0 0
\(694\) 12.0000 0.455514
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) −23.0000 −0.870563
\(699\) 0 0
\(700\) 0 0
\(701\) −45.0000 −1.69963 −0.849813 − 0.527084i \(-0.823285\pi\)
−0.849813 + 0.527084i \(0.823285\pi\)
\(702\) 0 0
\(703\) 32.0000 1.20690
\(704\) 6.00000 0.226134
\(705\) 0 0
\(706\) 24.0000 0.903252
\(707\) 6.00000 0.225653
\(708\) 0 0
\(709\) 11.0000 0.413114 0.206557 − 0.978435i \(-0.433774\pi\)
0.206557 + 0.978435i \(0.433774\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) −9.00000 −0.337289
\(713\) 36.0000 1.34821
\(714\) 0 0
\(715\) 0 0
\(716\) −18.0000 −0.672692
\(717\) 0 0
\(718\) 12.0000 0.447836
\(719\) −6.00000 −0.223762 −0.111881 − 0.993722i \(-0.535688\pi\)
−0.111881 + 0.993722i \(0.535688\pi\)
\(720\) 0 0
\(721\) −8.00000 −0.297936
\(722\) 3.00000 0.111648
\(723\) 0 0
\(724\) 5.00000 0.185824
\(725\) 0 0
\(726\) 0 0
\(727\) −53.0000 −1.96566 −0.982831 − 0.184510i \(-0.940930\pi\)
−0.982831 + 0.184510i \(0.940930\pi\)
\(728\) 2.00000 0.0741249
\(729\) 0 0
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) −14.0000 −0.517102 −0.258551 − 0.965998i \(-0.583245\pi\)
−0.258551 + 0.965998i \(0.583245\pi\)
\(734\) 8.00000 0.295285
\(735\) 0 0
\(736\) 9.00000 0.331744
\(737\) 78.0000 2.87317
\(738\) 0 0
\(739\) 2.00000 0.0735712 0.0367856 − 0.999323i \(-0.488288\pi\)
0.0367856 + 0.999323i \(0.488288\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 6.00000 0.220267
\(743\) 15.0000 0.550297 0.275148 − 0.961402i \(-0.411273\pi\)
0.275148 + 0.961402i \(0.411273\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 26.0000 0.951928
\(747\) 0 0
\(748\) 0 0
\(749\) 3.00000 0.109618
\(750\) 0 0
\(751\) 2.00000 0.0729810 0.0364905 − 0.999334i \(-0.488382\pi\)
0.0364905 + 0.999334i \(0.488382\pi\)
\(752\) 3.00000 0.109399
\(753\) 0 0
\(754\) 6.00000 0.218507
\(755\) 0 0
\(756\) 0 0
\(757\) 46.0000 1.67190 0.835949 − 0.548807i \(-0.184918\pi\)
0.835949 + 0.548807i \(0.184918\pi\)
\(758\) 22.0000 0.799076
\(759\) 0 0
\(760\) 0 0
\(761\) −33.0000 −1.19625 −0.598125 − 0.801403i \(-0.704087\pi\)
−0.598125 + 0.801403i \(0.704087\pi\)
\(762\) 0 0
\(763\) −7.00000 −0.253417
\(764\) −12.0000 −0.434145
\(765\) 0 0
\(766\) −12.0000 −0.433578
\(767\) −12.0000 −0.433295
\(768\) 0 0
\(769\) 29.0000 1.04577 0.522883 − 0.852404i \(-0.324856\pi\)
0.522883 + 0.852404i \(0.324856\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −2.00000 −0.0719816
\(773\) 48.0000 1.72644 0.863220 − 0.504828i \(-0.168444\pi\)
0.863220 + 0.504828i \(0.168444\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 2.00000 0.0717958
\(777\) 0 0
\(778\) −21.0000 −0.752886
\(779\) 12.0000 0.429945
\(780\) 0 0
\(781\) −36.0000 −1.28818
\(782\) 0 0
\(783\) 0 0
\(784\) −6.00000 −0.214286
\(785\) 0 0
\(786\) 0 0
\(787\) −20.0000 −0.712923 −0.356462 − 0.934310i \(-0.616017\pi\)
−0.356462 + 0.934310i \(0.616017\pi\)
\(788\) 12.0000 0.427482
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 26.0000 0.923287
\(794\) 2.00000 0.0709773
\(795\) 0 0
\(796\) 8.00000 0.283552
\(797\) 42.0000 1.48772 0.743858 − 0.668338i \(-0.232994\pi\)
0.743858 + 0.668338i \(0.232994\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 0 0
\(802\) −6.00000 −0.211867
\(803\) 24.0000 0.846942
\(804\) 0 0
\(805\) 0 0
\(806\) −8.00000 −0.281788
\(807\) 0 0
\(808\) −6.00000 −0.211079
\(809\) 18.0000 0.632846 0.316423 − 0.948618i \(-0.397518\pi\)
0.316423 + 0.948618i \(0.397518\pi\)
\(810\) 0 0
\(811\) 2.00000 0.0702295 0.0351147 − 0.999383i \(-0.488820\pi\)
0.0351147 + 0.999383i \(0.488820\pi\)
\(812\) 3.00000 0.105279
\(813\) 0 0
\(814\) 48.0000 1.68240
\(815\) 0 0
\(816\) 0 0
\(817\) 32.0000 1.11954
\(818\) 10.0000 0.349642
\(819\) 0 0
\(820\) 0 0
\(821\) 27.0000 0.942306 0.471153 − 0.882051i \(-0.343838\pi\)
0.471153 + 0.882051i \(0.343838\pi\)
\(822\) 0 0
\(823\) 25.0000 0.871445 0.435723 − 0.900081i \(-0.356493\pi\)
0.435723 + 0.900081i \(0.356493\pi\)
\(824\) 8.00000 0.278693
\(825\) 0 0
\(826\) −6.00000 −0.208767
\(827\) −3.00000 −0.104320 −0.0521601 − 0.998639i \(-0.516611\pi\)
−0.0521601 + 0.998639i \(0.516611\pi\)
\(828\) 0 0
\(829\) −7.00000 −0.243120 −0.121560 − 0.992584i \(-0.538790\pi\)
−0.121560 + 0.992584i \(0.538790\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) −2.00000 −0.0693375
\(833\) 0 0
\(834\) 0 0
\(835\) 0 0
\(836\) −24.0000 −0.830057
\(837\) 0 0
\(838\) 30.0000 1.03633
\(839\) 36.0000 1.24286 0.621429 − 0.783470i \(-0.286552\pi\)
0.621429 + 0.783470i \(0.286552\pi\)
\(840\) 0 0
\(841\) −20.0000 −0.689655
\(842\) 22.0000 0.758170
\(843\) 0 0
\(844\) 2.00000 0.0688428
\(845\) 0 0
\(846\) 0 0
\(847\) 25.0000 0.859010
\(848\) −6.00000 −0.206041
\(849\) 0 0
\(850\) 0 0
\(851\) 72.0000 2.46813
\(852\) 0 0
\(853\) 10.0000 0.342393 0.171197 − 0.985237i \(-0.445237\pi\)
0.171197 + 0.985237i \(0.445237\pi\)
\(854\) 13.0000 0.444851
\(855\) 0 0
\(856\) −3.00000 −0.102538
\(857\) 42.0000 1.43469 0.717346 − 0.696717i \(-0.245357\pi\)
0.717346 + 0.696717i \(0.245357\pi\)
\(858\) 0 0
\(859\) −34.0000 −1.16007 −0.580033 − 0.814593i \(-0.696960\pi\)
−0.580033 + 0.814593i \(0.696960\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) −6.00000 −0.204361
\(863\) −3.00000 −0.102121 −0.0510606 − 0.998696i \(-0.516260\pi\)
−0.0510606 + 0.998696i \(0.516260\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) −16.0000 −0.543702
\(867\) 0 0
\(868\) −4.00000 −0.135769
\(869\) −60.0000 −2.03536
\(870\) 0 0
\(871\) −26.0000 −0.880976
\(872\) 7.00000 0.237050
\(873\) 0 0
\(874\) −36.0000 −1.21772
\(875\) 0 0
\(876\) 0 0
\(877\) 22.0000 0.742887 0.371444 − 0.928456i \(-0.378863\pi\)
0.371444 + 0.928456i \(0.378863\pi\)
\(878\) 28.0000 0.944954
\(879\) 0 0
\(880\) 0 0
\(881\) 21.0000 0.707508 0.353754 − 0.935339i \(-0.384905\pi\)
0.353754 + 0.935339i \(0.384905\pi\)
\(882\) 0 0
\(883\) 31.0000 1.04323 0.521617 − 0.853180i \(-0.325329\pi\)
0.521617 + 0.853180i \(0.325329\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 9.00000 0.302361
\(887\) −36.0000 −1.20876 −0.604381 − 0.796696i \(-0.706579\pi\)
−0.604381 + 0.796696i \(0.706579\pi\)
\(888\) 0 0
\(889\) 7.00000 0.234772
\(890\) 0 0
\(891\) 0 0
\(892\) 1.00000 0.0334825
\(893\) −12.0000 −0.401565
\(894\) 0 0
\(895\) 0 0
\(896\) −1.00000 −0.0334077
\(897\) 0 0
\(898\) −6.00000 −0.200223
\(899\) −12.0000 −0.400222
\(900\) 0 0
\(901\) 0 0
\(902\) 18.0000 0.599334
\(903\) 0 0
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 37.0000 1.22856 0.614282 − 0.789086i \(-0.289446\pi\)
0.614282 + 0.789086i \(0.289446\pi\)
\(908\) 12.0000 0.398234
\(909\) 0 0
\(910\) 0 0
\(911\) −30.0000 −0.993944 −0.496972 − 0.867766i \(-0.665555\pi\)
−0.496972 + 0.867766i \(0.665555\pi\)
\(912\) 0 0
\(913\) 54.0000 1.78714
\(914\) 8.00000 0.264616
\(915\) 0 0
\(916\) −13.0000 −0.429532
\(917\) −18.0000 −0.594412
\(918\) 0 0
\(919\) 38.0000 1.25350 0.626752 − 0.779219i \(-0.284384\pi\)
0.626752 + 0.779219i \(0.284384\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 27.0000 0.889198
\(923\) 12.0000 0.394985
\(924\) 0 0
\(925\) 0 0
\(926\) −4.00000 −0.131448
\(927\) 0 0
\(928\) −3.00000 −0.0984798
\(929\) 6.00000 0.196854 0.0984268 − 0.995144i \(-0.468619\pi\)
0.0984268 + 0.995144i \(0.468619\pi\)
\(930\) 0 0
\(931\) 24.0000 0.786568
\(932\) 12.0000 0.393073
\(933\) 0 0
\(934\) 36.0000 1.17796
\(935\) 0 0
\(936\) 0 0
\(937\) −56.0000 −1.82944 −0.914720 − 0.404088i \(-0.867589\pi\)
−0.914720 + 0.404088i \(0.867589\pi\)
\(938\) −13.0000 −0.424465
\(939\) 0 0
\(940\) 0 0
\(941\) 21.0000 0.684580 0.342290 − 0.939594i \(-0.388797\pi\)
0.342290 + 0.939594i \(0.388797\pi\)
\(942\) 0 0
\(943\) 27.0000 0.879241
\(944\) 6.00000 0.195283
\(945\) 0 0
\(946\) 48.0000 1.56061
\(947\) −39.0000 −1.26733 −0.633665 − 0.773608i \(-0.718450\pi\)
−0.633665 + 0.773608i \(0.718450\pi\)
\(948\) 0 0
\(949\) −8.00000 −0.259691
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 6.00000 0.194359 0.0971795 − 0.995267i \(-0.469018\pi\)
0.0971795 + 0.995267i \(0.469018\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) 30.0000 0.969256
\(959\) 12.0000 0.387500
\(960\) 0 0
\(961\) −15.0000 −0.483871
\(962\) −16.0000 −0.515861
\(963\) 0 0
\(964\) 29.0000 0.934027
\(965\) 0 0
\(966\) 0 0
\(967\) 37.0000 1.18984 0.594920 − 0.803785i \(-0.297184\pi\)
0.594920 + 0.803785i \(0.297184\pi\)
\(968\) −25.0000 −0.803530
\(969\) 0 0
\(970\) 0 0
\(971\) −24.0000 −0.770197 −0.385098 − 0.922876i \(-0.625832\pi\)
−0.385098 + 0.922876i \(0.625832\pi\)
\(972\) 0 0
\(973\) −16.0000 −0.512936
\(974\) 8.00000 0.256337
\(975\) 0 0
\(976\) −13.0000 −0.416120
\(977\) 42.0000 1.34370 0.671850 − 0.740688i \(-0.265500\pi\)
0.671850 + 0.740688i \(0.265500\pi\)
\(978\) 0 0
\(979\) 54.0000 1.72585
\(980\) 0 0
\(981\) 0 0
\(982\) −12.0000 −0.382935
\(983\) −9.00000 −0.287055 −0.143528 − 0.989646i \(-0.545845\pi\)
−0.143528 + 0.989646i \(0.545845\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 8.00000 0.254514
\(989\) 72.0000 2.28947
\(990\) 0 0
\(991\) −10.0000 −0.317660 −0.158830 − 0.987306i \(-0.550772\pi\)
−0.158830 + 0.987306i \(0.550772\pi\)
\(992\) 4.00000 0.127000
\(993\) 0 0
\(994\) 6.00000 0.190308
\(995\) 0 0
\(996\) 0 0
\(997\) 10.0000 0.316703 0.158352 − 0.987383i \(-0.449382\pi\)
0.158352 + 0.987383i \(0.449382\pi\)
\(998\) −32.0000 −1.01294
\(999\) 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 4050.2.a.n.1.1 1
3.2 odd 2 4050.2.a.ba.1.1 1
5.2 odd 4 4050.2.c.t.649.1 2
5.3 odd 4 4050.2.c.t.649.2 2
5.4 even 2 810.2.a.g.1.1 1
9.2 odd 6 1350.2.e.b.901.1 2
9.4 even 3 450.2.e.e.151.1 2
9.5 odd 6 1350.2.e.b.451.1 2
9.7 even 3 450.2.e.e.301.1 2
15.2 even 4 4050.2.c.a.649.2 2
15.8 even 4 4050.2.c.a.649.1 2
15.14 odd 2 810.2.a.b.1.1 1
20.19 odd 2 6480.2.a.g.1.1 1
45.2 even 12 1350.2.j.e.199.2 4
45.4 even 6 90.2.e.a.61.1 yes 2
45.7 odd 12 450.2.j.c.49.1 4
45.13 odd 12 450.2.j.c.349.1 4
45.14 odd 6 270.2.e.b.181.1 2
45.22 odd 12 450.2.j.c.349.2 4
45.23 even 12 1350.2.j.e.1099.2 4
45.29 odd 6 270.2.e.b.91.1 2
45.32 even 12 1350.2.j.e.1099.1 4
45.34 even 6 90.2.e.a.31.1 ✓ 2
45.38 even 12 1350.2.j.e.199.1 4
45.43 odd 12 450.2.j.c.49.2 4
60.59 even 2 6480.2.a.v.1.1 1
180.59 even 6 2160.2.q.b.721.1 2
180.79 odd 6 720.2.q.b.481.1 2
180.119 even 6 2160.2.q.b.1441.1 2
180.139 odd 6 720.2.q.b.241.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
90.2.e.a.31.1 ✓ 2 45.34 even 6
90.2.e.a.61.1 yes 2 45.4 even 6
270.2.e.b.91.1 2 45.29 odd 6
270.2.e.b.181.1 2 45.14 odd 6
450.2.e.e.151.1 2 9.4 even 3
450.2.e.e.301.1 2 9.7 even 3
450.2.j.c.49.1 4 45.7 odd 12
450.2.j.c.49.2 4 45.43 odd 12
450.2.j.c.349.1 4 45.13 odd 12
450.2.j.c.349.2 4 45.22 odd 12
720.2.q.b.241.1 2 180.139 odd 6
720.2.q.b.481.1 2 180.79 odd 6
810.2.a.b.1.1 1 15.14 odd 2
810.2.a.g.1.1 1 5.4 even 2
1350.2.e.b.451.1 2 9.5 odd 6
1350.2.e.b.901.1 2 9.2 odd 6
1350.2.j.e.199.1 4 45.38 even 12
1350.2.j.e.199.2 4 45.2 even 12
1350.2.j.e.1099.1 4 45.32 even 12
1350.2.j.e.1099.2 4 45.23 even 12
2160.2.q.b.721.1 2 180.59 even 6
2160.2.q.b.1441.1 2 180.119 even 6
4050.2.a.n.1.1 1 1.1 even 1 trivial
4050.2.a.ba.1.1 1 3.2 odd 2
4050.2.c.a.649.1 2 15.8 even 4
4050.2.c.a.649.2 2 15.2 even 4
4050.2.c.t.649.1 2 5.2 odd 4
4050.2.c.t.649.2 2 5.3 odd 4
6480.2.a.g.1.1 1 20.19 odd 2
6480.2.a.v.1.1 1 60.59 even 2