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Properties

Label 2790.2.a.r.1.1
Level $2790$
Weight $2$
Character 2790.1
Self dual yes
Analytic conductor $22.278$
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] = [2790,2,Mod(1,2790)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2790, 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("2790.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2790 = 2 \cdot 3^{2} \cdot 5 \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2790.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: \(22.2782621639\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 930)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 2790.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{7} +1.00000 q^{8} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{7} +1.00000 q^{8} -1.00000 q^{10} +3.00000 q^{11} +2.00000 q^{13} -1.00000 q^{14} +1.00000 q^{16} +5.00000 q^{19} -1.00000 q^{20} +3.00000 q^{22} -9.00000 q^{23} +1.00000 q^{25} +2.00000 q^{26} -1.00000 q^{28} +1.00000 q^{31} +1.00000 q^{32} +1.00000 q^{35} +8.00000 q^{37} +5.00000 q^{38} -1.00000 q^{40} +11.0000 q^{43} +3.00000 q^{44} -9.00000 q^{46} +6.00000 q^{47} -6.00000 q^{49} +1.00000 q^{50} +2.00000 q^{52} +9.00000 q^{53} -3.00000 q^{55} -1.00000 q^{56} -6.00000 q^{59} -10.0000 q^{61} +1.00000 q^{62} +1.00000 q^{64} -2.00000 q^{65} +14.0000 q^{67} +1.00000 q^{70} -9.00000 q^{71} -1.00000 q^{73} +8.00000 q^{74} +5.00000 q^{76} -3.00000 q^{77} -1.00000 q^{79} -1.00000 q^{80} +12.0000 q^{83} +11.0000 q^{86} +3.00000 q^{88} +9.00000 q^{89} -2.00000 q^{91} -9.00000 q^{92} +6.00000 q^{94} -5.00000 q^{95} +14.0000 q^{97} -6.00000 q^{98} +O(q^{100})\)

Display 100 coefficients 10 coefficients

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) 0 0
\(4\) 1.00000 0.500000
\(5\) −1.00000 −0.447214
\(6\) 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
\(10\) −1.00000 −0.316228
\(11\) 3.00000 0.904534 0.452267 − 0.891883i \(-0.350615\pi\)
0.452267 + 0.891883i \(0.350615\pi\)
\(12\) 0 0
\(13\) 2.00000 0.554700 0.277350 − 0.960769i \(-0.410544\pi\)
0.277350 + 0.960769i \(0.410544\pi\)
\(14\) −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
\(19\) 5.00000 1.14708 0.573539 − 0.819178i \(-0.305570\pi\)
0.573539 + 0.819178i \(0.305570\pi\)
\(20\) −1.00000 −0.223607
\(21\) 0 0
\(22\) 3.00000 0.639602
\(23\) −9.00000 −1.87663 −0.938315 − 0.345782i \(-0.887614\pi\)
−0.938315 + 0.345782i \(0.887614\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 2.00000 0.392232
\(27\) 0 0
\(28\) −1.00000 −0.188982
\(29\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(30\) 0 0
\(31\) 1.00000 0.179605
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) 0 0
\(35\) 1.00000 0.169031
\(36\) 0 0
\(37\) 8.00000 1.31519 0.657596 − 0.753371i \(-0.271573\pi\)
0.657596 + 0.753371i \(0.271573\pi\)
\(38\) 5.00000 0.811107
\(39\) 0 0
\(40\) −1.00000 −0.158114
\(41\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(42\) 0 0
\(43\) 11.0000 1.67748 0.838742 − 0.544529i \(-0.183292\pi\)
0.838742 + 0.544529i \(0.183292\pi\)
\(44\) 3.00000 0.452267
\(45\) 0 0
\(46\) −9.00000 −1.32698
\(47\) 6.00000 0.875190 0.437595 − 0.899172i \(-0.355830\pi\)
0.437595 + 0.899172i \(0.355830\pi\)
\(48\) 0 0
\(49\) −6.00000 −0.857143
\(50\) 1.00000 0.141421
\(51\) 0 0
\(52\) 2.00000 0.277350
\(53\) 9.00000 1.23625 0.618123 − 0.786082i \(-0.287894\pi\)
0.618123 + 0.786082i \(0.287894\pi\)
\(54\) 0 0
\(55\) −3.00000 −0.404520
\(56\) −1.00000 −0.133631
\(57\) 0 0
\(58\) 0 0
\(59\) −6.00000 −0.781133 −0.390567 − 0.920575i \(-0.627721\pi\)
−0.390567 + 0.920575i \(0.627721\pi\)
\(60\) 0 0
\(61\) −10.0000 −1.28037 −0.640184 − 0.768221i \(-0.721142\pi\)
−0.640184 + 0.768221i \(0.721142\pi\)
\(62\) 1.00000 0.127000
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) −2.00000 −0.248069
\(66\) 0 0
\(67\) 14.0000 1.71037 0.855186 − 0.518321i \(-0.173443\pi\)
0.855186 + 0.518321i \(0.173443\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 1.00000 0.119523
\(71\) −9.00000 −1.06810 −0.534052 − 0.845452i \(-0.679331\pi\)
−0.534052 + 0.845452i \(0.679331\pi\)
\(72\) 0 0
\(73\) −1.00000 −0.117041 −0.0585206 − 0.998286i \(-0.518638\pi\)
−0.0585206 + 0.998286i \(0.518638\pi\)
\(74\) 8.00000 0.929981
\(75\) 0 0
\(76\) 5.00000 0.573539
\(77\) −3.00000 −0.341882
\(78\) 0 0
\(79\) −1.00000 −0.112509 −0.0562544 − 0.998416i \(-0.517916\pi\)
−0.0562544 + 0.998416i \(0.517916\pi\)
\(80\) −1.00000 −0.111803
\(81\) 0 0
\(82\) 0 0
\(83\) 12.0000 1.31717 0.658586 − 0.752506i \(-0.271155\pi\)
0.658586 + 0.752506i \(0.271155\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 11.0000 1.18616
\(87\) 0 0
\(88\) 3.00000 0.319801
\(89\) 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
\(94\) 6.00000 0.618853
\(95\) −5.00000 −0.512989
\(96\) 0 0
\(97\) 14.0000 1.42148 0.710742 − 0.703452i \(-0.248359\pi\)
0.710742 + 0.703452i \(0.248359\pi\)
\(98\) −6.00000 −0.606092
\(99\) 0 0
\(100\) 1.00000 0.100000
\(101\) 15.0000 1.49256 0.746278 − 0.665635i \(-0.231839\pi\)
0.746278 + 0.665635i \(0.231839\pi\)
\(102\) 0 0
\(103\) 8.00000 0.788263 0.394132 − 0.919054i \(-0.371045\pi\)
0.394132 + 0.919054i \(0.371045\pi\)
\(104\) 2.00000 0.196116
\(105\) 0 0
\(106\) 9.00000 0.874157
\(107\) 3.00000 0.290021 0.145010 − 0.989430i \(-0.453678\pi\)
0.145010 + 0.989430i \(0.453678\pi\)
\(108\) 0 0
\(109\) 8.00000 0.766261 0.383131 − 0.923694i \(-0.374846\pi\)
0.383131 + 0.923694i \(0.374846\pi\)
\(110\) −3.00000 −0.286039
\(111\) 0 0
\(112\) −1.00000 −0.0944911
\(113\) 3.00000 0.282216 0.141108 − 0.989994i \(-0.454933\pi\)
0.141108 + 0.989994i \(0.454933\pi\)
\(114\) 0 0
\(115\) 9.00000 0.839254
\(116\) 0 0
\(117\) 0 0
\(118\) −6.00000 −0.552345
\(119\) 0 0
\(120\) 0 0
\(121\) −2.00000 −0.181818
\(122\) −10.0000 −0.905357
\(123\) 0 0
\(124\) 1.00000 0.0898027
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) 8.00000 0.709885 0.354943 − 0.934888i \(-0.384500\pi\)
0.354943 + 0.934888i \(0.384500\pi\)
\(128\) 1.00000 0.0883883
\(129\) 0 0
\(130\) −2.00000 −0.175412
\(131\) 6.00000 0.524222 0.262111 − 0.965038i \(-0.415581\pi\)
0.262111 + 0.965038i \(0.415581\pi\)
\(132\) 0 0
\(133\) −5.00000 −0.433555
\(134\) 14.0000 1.20942
\(135\) 0 0
\(136\) 0 0
\(137\) −18.0000 −1.53784 −0.768922 − 0.639343i \(-0.779207\pi\)
−0.768922 + 0.639343i \(0.779207\pi\)
\(138\) 0 0
\(139\) 14.0000 1.18746 0.593732 − 0.804663i \(-0.297654\pi\)
0.593732 + 0.804663i \(0.297654\pi\)
\(140\) 1.00000 0.0845154
\(141\) 0 0
\(142\) −9.00000 −0.755263
\(143\) 6.00000 0.501745
\(144\) 0 0
\(145\) 0 0
\(146\) −1.00000 −0.0827606
\(147\) 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
\(151\) −16.0000 −1.30206 −0.651031 − 0.759051i \(-0.725663\pi\)
−0.651031 + 0.759051i \(0.725663\pi\)
\(152\) 5.00000 0.405554
\(153\) 0 0
\(154\) −3.00000 −0.241747
\(155\) −1.00000 −0.0803219
\(156\) 0 0
\(157\) −7.00000 −0.558661 −0.279330 − 0.960195i \(-0.590112\pi\)
−0.279330 + 0.960195i \(0.590112\pi\)
\(158\) −1.00000 −0.0795557
\(159\) 0 0
\(160\) −1.00000 −0.0790569
\(161\) 9.00000 0.709299
\(162\) 0 0
\(163\) −22.0000 −1.72317 −0.861586 − 0.507611i \(-0.830529\pi\)
−0.861586 + 0.507611i \(0.830529\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 12.0000 0.931381
\(167\) −9.00000 −0.696441 −0.348220 − 0.937413i \(-0.613214\pi\)
−0.348220 + 0.937413i \(0.613214\pi\)
\(168\) 0 0
\(169\) −9.00000 −0.692308
\(170\) 0 0
\(171\) 0 0
\(172\) 11.0000 0.838742
\(173\) −6.00000 −0.456172 −0.228086 − 0.973641i \(-0.573247\pi\)
−0.228086 + 0.973641i \(0.573247\pi\)
\(174\) 0 0
\(175\) −1.00000 −0.0755929
\(176\) 3.00000 0.226134
\(177\) 0 0
\(178\) 9.00000 0.674579
\(179\) 12.0000 0.896922 0.448461 − 0.893802i \(-0.351972\pi\)
0.448461 + 0.893802i \(0.351972\pi\)
\(180\) 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
\(185\) −8.00000 −0.588172
\(186\) 0 0
\(187\) 0 0
\(188\) 6.00000 0.437595
\(189\) 0 0
\(190\) −5.00000 −0.362738
\(191\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(192\) 0 0
\(193\) −10.0000 −0.719816 −0.359908 − 0.932988i \(-0.617192\pi\)
−0.359908 + 0.932988i \(0.617192\pi\)
\(194\) 14.0000 1.00514
\(195\) 0 0
\(196\) −6.00000 −0.428571
\(197\) 6.00000 0.427482 0.213741 − 0.976890i \(-0.431435\pi\)
0.213741 + 0.976890i \(0.431435\pi\)
\(198\) 0 0
\(199\) −25.0000 −1.77220 −0.886102 − 0.463491i \(-0.846597\pi\)
−0.886102 + 0.463491i \(0.846597\pi\)
\(200\) 1.00000 0.0707107
\(201\) 0 0
\(202\) 15.0000 1.05540
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 8.00000 0.557386
\(207\) 0 0
\(208\) 2.00000 0.138675
\(209\) 15.0000 1.03757
\(210\) 0 0
\(211\) −19.0000 −1.30801 −0.654007 − 0.756489i \(-0.726913\pi\)
−0.654007 + 0.756489i \(0.726913\pi\)
\(212\) 9.00000 0.618123
\(213\) 0 0
\(214\) 3.00000 0.205076
\(215\) −11.0000 −0.750194
\(216\) 0 0
\(217\) −1.00000 −0.0678844
\(218\) 8.00000 0.541828
\(219\) 0 0
\(220\) −3.00000 −0.202260
\(221\) 0 0
\(222\) 0 0
\(223\) −10.0000 −0.669650 −0.334825 − 0.942280i \(-0.608677\pi\)
−0.334825 + 0.942280i \(0.608677\pi\)
\(224\) −1.00000 −0.0668153
\(225\) 0 0
\(226\) 3.00000 0.199557
\(227\) 15.0000 0.995585 0.497792 − 0.867296i \(-0.334144\pi\)
0.497792 + 0.867296i \(0.334144\pi\)
\(228\) 0 0
\(229\) 5.00000 0.330409 0.165205 − 0.986259i \(-0.447172\pi\)
0.165205 + 0.986259i \(0.447172\pi\)
\(230\) 9.00000 0.593442
\(231\) 0 0
\(232\) 0 0
\(233\) −3.00000 −0.196537 −0.0982683 − 0.995160i \(-0.531330\pi\)
−0.0982683 + 0.995160i \(0.531330\pi\)
\(234\) 0 0
\(235\) −6.00000 −0.391397
\(236\) −6.00000 −0.390567
\(237\) 0 0
\(238\) 0 0
\(239\) −12.0000 −0.776215 −0.388108 − 0.921614i \(-0.626871\pi\)
−0.388108 + 0.921614i \(0.626871\pi\)
\(240\) 0 0
\(241\) −22.0000 −1.41714 −0.708572 − 0.705638i \(-0.750660\pi\)
−0.708572 + 0.705638i \(0.750660\pi\)
\(242\) −2.00000 −0.128565
\(243\) 0 0
\(244\) −10.0000 −0.640184
\(245\) 6.00000 0.383326
\(246\) 0 0
\(247\) 10.0000 0.636285
\(248\) 1.00000 0.0635001
\(249\) 0 0
\(250\) −1.00000 −0.0632456
\(251\) −12.0000 −0.757433 −0.378717 − 0.925513i \(-0.623635\pi\)
−0.378717 + 0.925513i \(0.623635\pi\)
\(252\) 0 0
\(253\) −27.0000 −1.69748
\(254\) 8.00000 0.501965
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 27.0000 1.68421 0.842107 − 0.539311i \(-0.181315\pi\)
0.842107 + 0.539311i \(0.181315\pi\)
\(258\) 0 0
\(259\) −8.00000 −0.497096
\(260\) −2.00000 −0.124035
\(261\) 0 0
\(262\) 6.00000 0.370681
\(263\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(264\) 0 0
\(265\) −9.00000 −0.552866
\(266\) −5.00000 −0.306570
\(267\) 0 0
\(268\) 14.0000 0.855186
\(269\) −6.00000 −0.365826 −0.182913 − 0.983129i \(-0.558553\pi\)
−0.182913 + 0.983129i \(0.558553\pi\)
\(270\) 0 0
\(271\) 5.00000 0.303728 0.151864 − 0.988401i \(-0.451472\pi\)
0.151864 + 0.988401i \(0.451472\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) −18.0000 −1.08742
\(275\) 3.00000 0.180907
\(276\) 0 0
\(277\) 8.00000 0.480673 0.240337 − 0.970690i \(-0.422742\pi\)
0.240337 + 0.970690i \(0.422742\pi\)
\(278\) 14.0000 0.839664
\(279\) 0 0
\(280\) 1.00000 0.0597614
\(281\) 24.0000 1.43172 0.715860 − 0.698244i \(-0.246035\pi\)
0.715860 + 0.698244i \(0.246035\pi\)
\(282\) 0 0
\(283\) −4.00000 −0.237775 −0.118888 − 0.992908i \(-0.537933\pi\)
−0.118888 + 0.992908i \(0.537933\pi\)
\(284\) −9.00000 −0.534052
\(285\) 0 0
\(286\) 6.00000 0.354787
\(287\) 0 0
\(288\) 0 0
\(289\) −17.0000 −1.00000
\(290\) 0 0
\(291\) 0 0
\(292\) −1.00000 −0.0585206
\(293\) −18.0000 −1.05157 −0.525786 − 0.850617i \(-0.676229\pi\)
−0.525786 + 0.850617i \(0.676229\pi\)
\(294\) 0 0
\(295\) 6.00000 0.349334
\(296\) 8.00000 0.464991
\(297\) 0 0
\(298\) −3.00000 −0.173785
\(299\) −18.0000 −1.04097
\(300\) 0 0
\(301\) −11.0000 −0.634029
\(302\) −16.0000 −0.920697
\(303\) 0 0
\(304\) 5.00000 0.286770
\(305\) 10.0000 0.572598
\(306\) 0 0
\(307\) 32.0000 1.82634 0.913168 − 0.407583i \(-0.133628\pi\)
0.913168 + 0.407583i \(0.133628\pi\)
\(308\) −3.00000 −0.170941
\(309\) 0 0
\(310\) −1.00000 −0.0567962
\(311\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(312\) 0 0
\(313\) −22.0000 −1.24351 −0.621757 − 0.783210i \(-0.713581\pi\)
−0.621757 + 0.783210i \(0.713581\pi\)
\(314\) −7.00000 −0.395033
\(315\) 0 0
\(316\) −1.00000 −0.0562544
\(317\) −24.0000 −1.34797 −0.673987 − 0.738743i \(-0.735420\pi\)
−0.673987 + 0.738743i \(0.735420\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) −1.00000 −0.0559017
\(321\) 0 0
\(322\) 9.00000 0.501550
\(323\) 0 0
\(324\) 0 0
\(325\) 2.00000 0.110940
\(326\) −22.0000 −1.21847
\(327\) 0 0
\(328\) 0 0
\(329\) −6.00000 −0.330791
\(330\) 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
\(334\) −9.00000 −0.492458
\(335\) −14.0000 −0.764902
\(336\) 0 0
\(337\) −22.0000 −1.19842 −0.599208 − 0.800593i \(-0.704518\pi\)
−0.599208 + 0.800593i \(0.704518\pi\)
\(338\) −9.00000 −0.489535
\(339\) 0 0
\(340\) 0 0
\(341\) 3.00000 0.162459
\(342\) 0 0
\(343\) 13.0000 0.701934
\(344\) 11.0000 0.593080
\(345\) 0 0
\(346\) −6.00000 −0.322562
\(347\) 36.0000 1.93258 0.966291 − 0.257454i \(-0.0828835\pi\)
0.966291 + 0.257454i \(0.0828835\pi\)
\(348\) 0 0
\(349\) −4.00000 −0.214115 −0.107058 − 0.994253i \(-0.534143\pi\)
−0.107058 + 0.994253i \(0.534143\pi\)
\(350\) −1.00000 −0.0534522
\(351\) 0 0
\(352\) 3.00000 0.159901
\(353\) −6.00000 −0.319348 −0.159674 − 0.987170i \(-0.551044\pi\)
−0.159674 + 0.987170i \(0.551044\pi\)
\(354\) 0 0
\(355\) 9.00000 0.477670
\(356\) 9.00000 0.476999
\(357\) 0 0
\(358\) 12.0000 0.634220
\(359\) −9.00000 −0.475002 −0.237501 − 0.971387i \(-0.576328\pi\)
−0.237501 + 0.971387i \(0.576328\pi\)
\(360\) 0 0
\(361\) 6.00000 0.315789
\(362\) 5.00000 0.262794
\(363\) 0 0
\(364\) −2.00000 −0.104828
\(365\) 1.00000 0.0523424
\(366\) 0 0
\(367\) 26.0000 1.35719 0.678594 − 0.734513i \(-0.262589\pi\)
0.678594 + 0.734513i \(0.262589\pi\)
\(368\) −9.00000 −0.469157
\(369\) 0 0
\(370\) −8.00000 −0.415900
\(371\) −9.00000 −0.467257
\(372\) 0 0
\(373\) −19.0000 −0.983783 −0.491891 − 0.870657i \(-0.663694\pi\)
−0.491891 + 0.870657i \(0.663694\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 6.00000 0.309426
\(377\) 0 0
\(378\) 0 0
\(379\) −1.00000 −0.0513665 −0.0256833 − 0.999670i \(-0.508176\pi\)
−0.0256833 + 0.999670i \(0.508176\pi\)
\(380\) −5.00000 −0.256495
\(381\) 0 0
\(382\) 0 0
\(383\) −36.0000 −1.83951 −0.919757 − 0.392488i \(-0.871614\pi\)
−0.919757 + 0.392488i \(0.871614\pi\)
\(384\) 0 0
\(385\) 3.00000 0.152894
\(386\) −10.0000 −0.508987
\(387\) 0 0
\(388\) 14.0000 0.710742
\(389\) 30.0000 1.52106 0.760530 − 0.649303i \(-0.224939\pi\)
0.760530 + 0.649303i \(0.224939\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) −6.00000 −0.303046
\(393\) 0 0
\(394\) 6.00000 0.302276
\(395\) 1.00000 0.0503155
\(396\) 0 0
\(397\) −7.00000 −0.351320 −0.175660 − 0.984451i \(-0.556206\pi\)
−0.175660 + 0.984451i \(0.556206\pi\)
\(398\) −25.0000 −1.25314
\(399\) 0 0
\(400\) 1.00000 0.0500000
\(401\) −15.0000 −0.749064 −0.374532 − 0.927214i \(-0.622197\pi\)
−0.374532 + 0.927214i \(0.622197\pi\)
\(402\) 0 0
\(403\) 2.00000 0.0996271
\(404\) 15.0000 0.746278
\(405\) 0 0
\(406\) 0 0
\(407\) 24.0000 1.18964
\(408\) 0 0
\(409\) −28.0000 −1.38451 −0.692255 − 0.721653i \(-0.743383\pi\)
−0.692255 + 0.721653i \(0.743383\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 8.00000 0.394132
\(413\) 6.00000 0.295241
\(414\) 0 0
\(415\) −12.0000 −0.589057
\(416\) 2.00000 0.0980581
\(417\) 0 0
\(418\) 15.0000 0.733674
\(419\) 24.0000 1.17248 0.586238 − 0.810139i \(-0.300608\pi\)
0.586238 + 0.810139i \(0.300608\pi\)
\(420\) 0 0
\(421\) −10.0000 −0.487370 −0.243685 − 0.969854i \(-0.578356\pi\)
−0.243685 + 0.969854i \(0.578356\pi\)
\(422\) −19.0000 −0.924906
\(423\) 0 0
\(424\) 9.00000 0.437079
\(425\) 0 0
\(426\) 0 0
\(427\) 10.0000 0.483934
\(428\) 3.00000 0.145010
\(429\) 0 0
\(430\) −11.0000 −0.530467
\(431\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(432\) 0 0
\(433\) 5.00000 0.240285 0.120142 − 0.992757i \(-0.461665\pi\)
0.120142 + 0.992757i \(0.461665\pi\)
\(434\) −1.00000 −0.0480015
\(435\) 0 0
\(436\) 8.00000 0.383131
\(437\) −45.0000 −2.15264
\(438\) 0 0
\(439\) 14.0000 0.668184 0.334092 − 0.942541i \(-0.391570\pi\)
0.334092 + 0.942541i \(0.391570\pi\)
\(440\) −3.00000 −0.143019
\(441\) 0 0
\(442\) 0 0
\(443\) −27.0000 −1.28281 −0.641404 − 0.767203i \(-0.721648\pi\)
−0.641404 + 0.767203i \(0.721648\pi\)
\(444\) 0 0
\(445\) −9.00000 −0.426641
\(446\) −10.0000 −0.473514
\(447\) 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
\(452\) 3.00000 0.141108
\(453\) 0 0
\(454\) 15.0000 0.703985
\(455\) 2.00000 0.0937614
\(456\) 0 0
\(457\) 38.0000 1.77757 0.888783 − 0.458329i \(-0.151552\pi\)
0.888783 + 0.458329i \(0.151552\pi\)
\(458\) 5.00000 0.233635
\(459\) 0 0
\(460\) 9.00000 0.419627
\(461\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(462\) 0 0
\(463\) 32.0000 1.48717 0.743583 − 0.668644i \(-0.233125\pi\)
0.743583 + 0.668644i \(0.233125\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) −3.00000 −0.138972
\(467\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(468\) 0 0
\(469\) −14.0000 −0.646460
\(470\) −6.00000 −0.276759
\(471\) 0 0
\(472\) −6.00000 −0.276172
\(473\) 33.0000 1.51734
\(474\) 0 0
\(475\) 5.00000 0.229416
\(476\) 0 0
\(477\) 0 0
\(478\) −12.0000 −0.548867
\(479\) −21.0000 −0.959514 −0.479757 − 0.877401i \(-0.659275\pi\)
−0.479757 + 0.877401i \(0.659275\pi\)
\(480\) 0 0
\(481\) 16.0000 0.729537
\(482\) −22.0000 −1.00207
\(483\) 0 0
\(484\) −2.00000 −0.0909091
\(485\) −14.0000 −0.635707
\(486\) 0 0
\(487\) −40.0000 −1.81257 −0.906287 − 0.422664i \(-0.861095\pi\)
−0.906287 + 0.422664i \(0.861095\pi\)
\(488\) −10.0000 −0.452679
\(489\) 0 0
\(490\) 6.00000 0.271052
\(491\) −21.0000 −0.947717 −0.473858 − 0.880601i \(-0.657139\pi\)
−0.473858 + 0.880601i \(0.657139\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 10.0000 0.449921
\(495\) 0 0
\(496\) 1.00000 0.0449013
\(497\) 9.00000 0.403705
\(498\) 0 0
\(499\) 32.0000 1.43252 0.716258 − 0.697835i \(-0.245853\pi\)
0.716258 + 0.697835i \(0.245853\pi\)
\(500\) −1.00000 −0.0447214
\(501\) 0 0
\(502\) −12.0000 −0.535586
\(503\) 24.0000 1.07011 0.535054 − 0.844818i \(-0.320291\pi\)
0.535054 + 0.844818i \(0.320291\pi\)
\(504\) 0 0
\(505\) −15.0000 −0.667491
\(506\) −27.0000 −1.20030
\(507\) 0 0
\(508\) 8.00000 0.354943
\(509\) −36.0000 −1.59567 −0.797836 − 0.602875i \(-0.794022\pi\)
−0.797836 + 0.602875i \(0.794022\pi\)
\(510\) 0 0
\(511\) 1.00000 0.0442374
\(512\) 1.00000 0.0441942
\(513\) 0 0
\(514\) 27.0000 1.19092
\(515\) −8.00000 −0.352522
\(516\) 0 0
\(517\) 18.0000 0.791639
\(518\) −8.00000 −0.351500
\(519\) 0 0
\(520\) −2.00000 −0.0877058
\(521\) −18.0000 −0.788594 −0.394297 − 0.918983i \(-0.629012\pi\)
−0.394297 + 0.918983i \(0.629012\pi\)
\(522\) 0 0
\(523\) −25.0000 −1.09317 −0.546587 − 0.837402i \(-0.684073\pi\)
−0.546587 + 0.837402i \(0.684073\pi\)
\(524\) 6.00000 0.262111
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) 58.0000 2.52174
\(530\) −9.00000 −0.390935
\(531\) 0 0
\(532\) −5.00000 −0.216777
\(533\) 0 0
\(534\) 0 0
\(535\) −3.00000 −0.129701
\(536\) 14.0000 0.604708
\(537\) 0 0
\(538\) −6.00000 −0.258678
\(539\) −18.0000 −0.775315
\(540\) 0 0
\(541\) −34.0000 −1.46177 −0.730887 − 0.682498i \(-0.760893\pi\)
−0.730887 + 0.682498i \(0.760893\pi\)
\(542\) 5.00000 0.214768
\(543\) 0 0
\(544\) 0 0
\(545\) −8.00000 −0.342682
\(546\) 0 0
\(547\) −34.0000 −1.45374 −0.726868 − 0.686778i \(-0.759025\pi\)
−0.726868 + 0.686778i \(0.759025\pi\)
\(548\) −18.0000 −0.768922
\(549\) 0 0
\(550\) 3.00000 0.127920
\(551\) 0 0
\(552\) 0 0
\(553\) 1.00000 0.0425243
\(554\) 8.00000 0.339887
\(555\) 0 0
\(556\) 14.0000 0.593732
\(557\) −3.00000 −0.127114 −0.0635570 − 0.997978i \(-0.520244\pi\)
−0.0635570 + 0.997978i \(0.520244\pi\)
\(558\) 0 0
\(559\) 22.0000 0.930501
\(560\) 1.00000 0.0422577
\(561\) 0 0
\(562\) 24.0000 1.01238
\(563\) 12.0000 0.505740 0.252870 − 0.967500i \(-0.418626\pi\)
0.252870 + 0.967500i \(0.418626\pi\)
\(564\) 0 0
\(565\) −3.00000 −0.126211
\(566\) −4.00000 −0.168133
\(567\) 0 0
\(568\) −9.00000 −0.377632
\(569\) −15.0000 −0.628833 −0.314416 − 0.949285i \(-0.601809\pi\)
−0.314416 + 0.949285i \(0.601809\pi\)
\(570\) 0 0
\(571\) 14.0000 0.585882 0.292941 − 0.956131i \(-0.405366\pi\)
0.292941 + 0.956131i \(0.405366\pi\)
\(572\) 6.00000 0.250873
\(573\) 0 0
\(574\) 0 0
\(575\) −9.00000 −0.375326
\(576\) 0 0
\(577\) −16.0000 −0.666089 −0.333044 − 0.942911i \(-0.608076\pi\)
−0.333044 + 0.942911i \(0.608076\pi\)
\(578\) −17.0000 −0.707107
\(579\) 0 0
\(580\) 0 0
\(581\) −12.0000 −0.497844
\(582\) 0 0
\(583\) 27.0000 1.11823
\(584\) −1.00000 −0.0413803
\(585\) 0 0
\(586\) −18.0000 −0.743573
\(587\) 24.0000 0.990586 0.495293 − 0.868726i \(-0.335061\pi\)
0.495293 + 0.868726i \(0.335061\pi\)
\(588\) 0 0
\(589\) 5.00000 0.206021
\(590\) 6.00000 0.247016
\(591\) 0 0
\(592\) 8.00000 0.328798
\(593\) −6.00000 −0.246390 −0.123195 − 0.992382i \(-0.539314\pi\)
−0.123195 + 0.992382i \(0.539314\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −3.00000 −0.122885
\(597\) 0 0
\(598\) −18.0000 −0.736075
\(599\) −27.0000 −1.10319 −0.551595 − 0.834112i \(-0.685981\pi\)
−0.551595 + 0.834112i \(0.685981\pi\)
\(600\) 0 0
\(601\) −10.0000 −0.407909 −0.203954 − 0.978980i \(-0.565379\pi\)
−0.203954 + 0.978980i \(0.565379\pi\)
\(602\) −11.0000 −0.448327
\(603\) 0 0
\(604\) −16.0000 −0.651031
\(605\) 2.00000 0.0813116
\(606\) 0 0
\(607\) −1.00000 −0.0405887 −0.0202944 − 0.999794i \(-0.506460\pi\)
−0.0202944 + 0.999794i \(0.506460\pi\)
\(608\) 5.00000 0.202777
\(609\) 0 0
\(610\) 10.0000 0.404888
\(611\) 12.0000 0.485468
\(612\) 0 0
\(613\) −22.0000 −0.888572 −0.444286 − 0.895885i \(-0.646543\pi\)
−0.444286 + 0.895885i \(0.646543\pi\)
\(614\) 32.0000 1.29141
\(615\) 0 0
\(616\) −3.00000 −0.120873
\(617\) 15.0000 0.603877 0.301939 − 0.953327i \(-0.402366\pi\)
0.301939 + 0.953327i \(0.402366\pi\)
\(618\) 0 0
\(619\) 14.0000 0.562708 0.281354 − 0.959604i \(-0.409217\pi\)
0.281354 + 0.959604i \(0.409217\pi\)
\(620\) −1.00000 −0.0401610
\(621\) 0 0
\(622\) 0 0
\(623\) −9.00000 −0.360577
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) −22.0000 −0.879297
\(627\) 0 0
\(628\) −7.00000 −0.279330
\(629\) 0 0
\(630\) 0 0
\(631\) −7.00000 −0.278666 −0.139333 − 0.990246i \(-0.544496\pi\)
−0.139333 + 0.990246i \(0.544496\pi\)
\(632\) −1.00000 −0.0397779
\(633\) 0 0
\(634\) −24.0000 −0.953162
\(635\) −8.00000 −0.317470
\(636\) 0 0
\(637\) −12.0000 −0.475457
\(638\) 0 0
\(639\) 0 0
\(640\) −1.00000 −0.0395285
\(641\) −30.0000 −1.18493 −0.592464 − 0.805597i \(-0.701845\pi\)
−0.592464 + 0.805597i \(0.701845\pi\)
\(642\) 0 0
\(643\) −31.0000 −1.22252 −0.611260 − 0.791430i \(-0.709337\pi\)
−0.611260 + 0.791430i \(0.709337\pi\)
\(644\) 9.00000 0.354650
\(645\) 0 0
\(646\) 0 0
\(647\) 33.0000 1.29736 0.648682 − 0.761060i \(-0.275321\pi\)
0.648682 + 0.761060i \(0.275321\pi\)
\(648\) 0 0
\(649\) −18.0000 −0.706562
\(650\) 2.00000 0.0784465
\(651\) 0 0
\(652\) −22.0000 −0.861586
\(653\) −30.0000 −1.17399 −0.586995 − 0.809590i \(-0.699689\pi\)
−0.586995 + 0.809590i \(0.699689\pi\)
\(654\) 0 0
\(655\) −6.00000 −0.234439
\(656\) 0 0
\(657\) 0 0
\(658\) −6.00000 −0.233904
\(659\) 48.0000 1.86981 0.934907 − 0.354892i \(-0.115482\pi\)
0.934907 + 0.354892i \(0.115482\pi\)
\(660\) 0 0
\(661\) −46.0000 −1.78919 −0.894596 − 0.446875i \(-0.852537\pi\)
−0.894596 + 0.446875i \(0.852537\pi\)
\(662\) 8.00000 0.310929
\(663\) 0 0
\(664\) 12.0000 0.465690
\(665\) 5.00000 0.193892
\(666\) 0 0
\(667\) 0 0
\(668\) −9.00000 −0.348220
\(669\) 0 0
\(670\) −14.0000 −0.540867
\(671\) −30.0000 −1.15814
\(672\) 0 0
\(673\) −10.0000 −0.385472 −0.192736 − 0.981251i \(-0.561736\pi\)
−0.192736 + 0.981251i \(0.561736\pi\)
\(674\) −22.0000 −0.847408
\(675\) 0 0
\(676\) −9.00000 −0.346154
\(677\) 33.0000 1.26829 0.634147 − 0.773213i \(-0.281352\pi\)
0.634147 + 0.773213i \(0.281352\pi\)
\(678\) 0 0
\(679\) −14.0000 −0.537271
\(680\) 0 0
\(681\) 0 0
\(682\) 3.00000 0.114876
\(683\) 15.0000 0.573959 0.286980 − 0.957937i \(-0.407349\pi\)
0.286980 + 0.957937i \(0.407349\pi\)
\(684\) 0 0
\(685\) 18.0000 0.687745
\(686\) 13.0000 0.496342
\(687\) 0 0
\(688\) 11.0000 0.419371
\(689\) 18.0000 0.685745
\(690\) 0 0
\(691\) 17.0000 0.646710 0.323355 − 0.946278i \(-0.395189\pi\)
0.323355 + 0.946278i \(0.395189\pi\)
\(692\) −6.00000 −0.228086
\(693\) 0 0
\(694\) 36.0000 1.36654
\(695\) −14.0000 −0.531050
\(696\) 0 0
\(697\) 0 0
\(698\) −4.00000 −0.151402
\(699\) 0 0
\(700\) −1.00000 −0.0377964
\(701\) 3.00000 0.113308 0.0566542 − 0.998394i \(-0.481957\pi\)
0.0566542 + 0.998394i \(0.481957\pi\)
\(702\) 0 0
\(703\) 40.0000 1.50863
\(704\) 3.00000 0.113067
\(705\) 0 0
\(706\) −6.00000 −0.225813
\(707\) −15.0000 −0.564133
\(708\) 0 0
\(709\) 23.0000 0.863783 0.431892 − 0.901926i \(-0.357846\pi\)
0.431892 + 0.901926i \(0.357846\pi\)
\(710\) 9.00000 0.337764
\(711\) 0 0
\(712\) 9.00000 0.337289
\(713\) −9.00000 −0.337053
\(714\) 0 0
\(715\) −6.00000 −0.224387
\(716\) 12.0000 0.448461
\(717\) 0 0
\(718\) −9.00000 −0.335877
\(719\) −18.0000 −0.671287 −0.335643 − 0.941989i \(-0.608954\pi\)
−0.335643 + 0.941989i \(0.608954\pi\)
\(720\) 0 0
\(721\) −8.00000 −0.297936
\(722\) 6.00000 0.223297
\(723\) 0 0
\(724\) 5.00000 0.185824
\(725\) 0 0
\(726\) 0 0
\(727\) 23.0000 0.853023 0.426511 − 0.904482i \(-0.359742\pi\)
0.426511 + 0.904482i \(0.359742\pi\)
\(728\) −2.00000 −0.0741249
\(729\) 0 0
\(730\) 1.00000 0.0370117
\(731\) 0 0
\(732\) 0 0
\(733\) 26.0000 0.960332 0.480166 − 0.877178i \(-0.340576\pi\)
0.480166 + 0.877178i \(0.340576\pi\)
\(734\) 26.0000 0.959678
\(735\) 0 0
\(736\) −9.00000 −0.331744
\(737\) 42.0000 1.54709
\(738\) 0 0
\(739\) 32.0000 1.17714 0.588570 − 0.808447i \(-0.299691\pi\)
0.588570 + 0.808447i \(0.299691\pi\)
\(740\) −8.00000 −0.294086
\(741\) 0 0
\(742\) −9.00000 −0.330400
\(743\) 9.00000 0.330178 0.165089 − 0.986279i \(-0.447209\pi\)
0.165089 + 0.986279i \(0.447209\pi\)
\(744\) 0 0
\(745\) 3.00000 0.109911
\(746\) −19.0000 −0.695639
\(747\) 0 0
\(748\) 0 0
\(749\) −3.00000 −0.109618
\(750\) 0 0
\(751\) 26.0000 0.948753 0.474377 − 0.880322i \(-0.342673\pi\)
0.474377 + 0.880322i \(0.342673\pi\)
\(752\) 6.00000 0.218797
\(753\) 0 0
\(754\) 0 0
\(755\) 16.0000 0.582300
\(756\) 0 0
\(757\) 2.00000 0.0726912 0.0363456 − 0.999339i \(-0.488428\pi\)
0.0363456 + 0.999339i \(0.488428\pi\)
\(758\) −1.00000 −0.0363216
\(759\) 0 0
\(760\) −5.00000 −0.181369
\(761\) 21.0000 0.761249 0.380625 − 0.924730i \(-0.375709\pi\)
0.380625 + 0.924730i \(0.375709\pi\)
\(762\) 0 0
\(763\) −8.00000 −0.289619
\(764\) 0 0
\(765\) 0 0
\(766\) −36.0000 −1.30073
\(767\) −12.0000 −0.433295
\(768\) 0 0
\(769\) −31.0000 −1.11789 −0.558944 − 0.829205i \(-0.688793\pi\)
−0.558944 + 0.829205i \(0.688793\pi\)
\(770\) 3.00000 0.108112
\(771\) 0 0
\(772\) −10.0000 −0.359908
\(773\) 27.0000 0.971123 0.485561 − 0.874203i \(-0.338615\pi\)
0.485561 + 0.874203i \(0.338615\pi\)
\(774\) 0 0
\(775\) 1.00000 0.0359211
\(776\) 14.0000 0.502571
\(777\) 0 0
\(778\) 30.0000 1.07555
\(779\) 0 0
\(780\) 0 0
\(781\) −27.0000 −0.966136
\(782\) 0 0
\(783\) 0 0
\(784\) −6.00000 −0.214286
\(785\) 7.00000 0.249841
\(786\) 0 0
\(787\) −25.0000 −0.891154 −0.445577 − 0.895244i \(-0.647001\pi\)
−0.445577 + 0.895244i \(0.647001\pi\)
\(788\) 6.00000 0.213741
\(789\) 0 0
\(790\) 1.00000 0.0355784
\(791\) −3.00000 −0.106668
\(792\) 0 0
\(793\) −20.0000 −0.710221
\(794\) −7.00000 −0.248421
\(795\) 0 0
\(796\) −25.0000 −0.886102
\(797\) −18.0000 −0.637593 −0.318796 − 0.947823i \(-0.603279\pi\)
−0.318796 + 0.947823i \(0.603279\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 1.00000 0.0353553
\(801\) 0 0
\(802\) −15.0000 −0.529668
\(803\) −3.00000 −0.105868
\(804\) 0 0
\(805\) −9.00000 −0.317208
\(806\) 2.00000 0.0704470
\(807\) 0 0
\(808\) 15.0000 0.527698
\(809\) −39.0000 −1.37117 −0.685583 − 0.727994i \(-0.740453\pi\)
−0.685583 + 0.727994i \(0.740453\pi\)
\(810\) 0 0
\(811\) −43.0000 −1.50993 −0.754967 − 0.655763i \(-0.772347\pi\)
−0.754967 + 0.655763i \(0.772347\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 24.0000 0.841200
\(815\) 22.0000 0.770626
\(816\) 0 0
\(817\) 55.0000 1.92421
\(818\) −28.0000 −0.978997
\(819\) 0 0
\(820\) 0 0
\(821\) −24.0000 −0.837606 −0.418803 − 0.908077i \(-0.637550\pi\)
−0.418803 + 0.908077i \(0.637550\pi\)
\(822\) 0 0
\(823\) −22.0000 −0.766872 −0.383436 − 0.923567i \(-0.625259\pi\)
−0.383436 + 0.923567i \(0.625259\pi\)
\(824\) 8.00000 0.278693
\(825\) 0 0
\(826\) 6.00000 0.208767
\(827\) 36.0000 1.25184 0.625921 − 0.779886i \(-0.284723\pi\)
0.625921 + 0.779886i \(0.284723\pi\)
\(828\) 0 0
\(829\) 29.0000 1.00721 0.503606 − 0.863934i \(-0.332006\pi\)
0.503606 + 0.863934i \(0.332006\pi\)
\(830\) −12.0000 −0.416526
\(831\) 0 0
\(832\) 2.00000 0.0693375
\(833\) 0 0
\(834\) 0 0
\(835\) 9.00000 0.311458
\(836\) 15.0000 0.518786
\(837\) 0 0
\(838\) 24.0000 0.829066
\(839\) −21.0000 −0.725001 −0.362500 − 0.931984i \(-0.618077\pi\)
−0.362500 + 0.931984i \(0.618077\pi\)
\(840\) 0 0
\(841\) −29.0000 −1.00000
\(842\) −10.0000 −0.344623
\(843\) 0 0
\(844\) −19.0000 −0.654007
\(845\) 9.00000 0.309609
\(846\) 0 0
\(847\) 2.00000 0.0687208
\(848\) 9.00000 0.309061
\(849\) 0 0
\(850\) 0 0
\(851\) −72.0000 −2.46813
\(852\) 0 0
\(853\) −19.0000 −0.650548 −0.325274 − 0.945620i \(-0.605456\pi\)
−0.325274 + 0.945620i \(0.605456\pi\)
\(854\) 10.0000 0.342193
\(855\) 0 0
\(856\) 3.00000 0.102538
\(857\) 6.00000 0.204956 0.102478 − 0.994735i \(-0.467323\pi\)
0.102478 + 0.994735i \(0.467323\pi\)
\(858\) 0 0
\(859\) 8.00000 0.272956 0.136478 − 0.990643i \(-0.456422\pi\)
0.136478 + 0.990643i \(0.456422\pi\)
\(860\) −11.0000 −0.375097
\(861\) 0 0
\(862\) 0 0
\(863\) −15.0000 −0.510606 −0.255303 − 0.966861i \(-0.582175\pi\)
−0.255303 + 0.966861i \(0.582175\pi\)
\(864\) 0 0
\(865\) 6.00000 0.204006
\(866\) 5.00000 0.169907
\(867\) 0 0
\(868\) −1.00000 −0.0339422
\(869\) −3.00000 −0.101768
\(870\) 0 0
\(871\) 28.0000 0.948744
\(872\) 8.00000 0.270914
\(873\) 0 0
\(874\) −45.0000 −1.52215
\(875\) 1.00000 0.0338062
\(876\) 0 0
\(877\) −22.0000 −0.742887 −0.371444 − 0.928456i \(-0.621137\pi\)
−0.371444 + 0.928456i \(0.621137\pi\)
\(878\) 14.0000 0.472477
\(879\) 0 0
\(880\) −3.00000 −0.101130
\(881\) 18.0000 0.606435 0.303218 − 0.952921i \(-0.401939\pi\)
0.303218 + 0.952921i \(0.401939\pi\)
\(882\) 0 0
\(883\) −31.0000 −1.04323 −0.521617 − 0.853180i \(-0.674671\pi\)
−0.521617 + 0.853180i \(0.674671\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) −27.0000 −0.907083
\(887\) 18.0000 0.604381 0.302190 − 0.953248i \(-0.402282\pi\)
0.302190 + 0.953248i \(0.402282\pi\)
\(888\) 0 0
\(889\) −8.00000 −0.268311
\(890\) −9.00000 −0.301681
\(891\) 0 0
\(892\) −10.0000 −0.334825
\(893\) 30.0000 1.00391
\(894\) 0 0
\(895\) −12.0000 −0.401116
\(896\) −1.00000 −0.0334077
\(897\) 0 0
\(898\) −6.00000 −0.200223
\(899\) 0 0
\(900\) 0 0
\(901\) 0 0
\(902\) 0 0
\(903\) 0 0
\(904\) 3.00000 0.0997785
\(905\) −5.00000 −0.166206
\(906\) 0 0
\(907\) 8.00000 0.265636 0.132818 − 0.991140i \(-0.457597\pi\)
0.132818 + 0.991140i \(0.457597\pi\)
\(908\) 15.0000 0.497792
\(909\) 0 0
\(910\) 2.00000 0.0662994
\(911\) −12.0000 −0.397578 −0.198789 − 0.980042i \(-0.563701\pi\)
−0.198789 + 0.980042i \(0.563701\pi\)
\(912\) 0 0
\(913\) 36.0000 1.19143
\(914\) 38.0000 1.25693
\(915\) 0 0
\(916\) 5.00000 0.165205
\(917\) −6.00000 −0.198137
\(918\) 0 0
\(919\) 26.0000 0.857661 0.428830 − 0.903385i \(-0.358926\pi\)
0.428830 + 0.903385i \(0.358926\pi\)
\(920\) 9.00000 0.296721
\(921\) 0 0
\(922\) 0 0
\(923\) −18.0000 −0.592477
\(924\) 0 0
\(925\) 8.00000 0.263038
\(926\) 32.0000 1.05159
\(927\) 0 0
\(928\) 0 0
\(929\) 15.0000 0.492134 0.246067 − 0.969253i \(-0.420862\pi\)
0.246067 + 0.969253i \(0.420862\pi\)
\(930\) 0 0
\(931\) −30.0000 −0.983210
\(932\) −3.00000 −0.0982683
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 20.0000 0.653372 0.326686 − 0.945133i \(-0.394068\pi\)
0.326686 + 0.945133i \(0.394068\pi\)
\(938\) −14.0000 −0.457116
\(939\) 0 0
\(940\) −6.00000 −0.195698
\(941\) 12.0000 0.391189 0.195594 − 0.980685i \(-0.437336\pi\)
0.195594 + 0.980685i \(0.437336\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) −6.00000 −0.195283
\(945\) 0 0
\(946\) 33.0000 1.07292
\(947\) −54.0000 −1.75476 −0.877382 − 0.479792i \(-0.840712\pi\)
−0.877382 + 0.479792i \(0.840712\pi\)
\(948\) 0 0
\(949\) −2.00000 −0.0649227
\(950\) 5.00000 0.162221
\(951\) 0 0
\(952\) 0 0
\(953\) 12.0000 0.388718 0.194359 − 0.980930i \(-0.437737\pi\)
0.194359 + 0.980930i \(0.437737\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) −12.0000 −0.388108
\(957\) 0 0
\(958\) −21.0000 −0.678479
\(959\) 18.0000 0.581250
\(960\) 0 0
\(961\) 1.00000 0.0322581
\(962\) 16.0000 0.515861
\(963\) 0 0
\(964\) −22.0000 −0.708572
\(965\) 10.0000 0.321911
\(966\) 0 0
\(967\) −4.00000 −0.128631 −0.0643157 − 0.997930i \(-0.520486\pi\)
−0.0643157 + 0.997930i \(0.520486\pi\)
\(968\) −2.00000 −0.0642824
\(969\) 0 0
\(970\) −14.0000 −0.449513
\(971\) 42.0000 1.34784 0.673922 − 0.738802i \(-0.264608\pi\)
0.673922 + 0.738802i \(0.264608\pi\)
\(972\) 0 0
\(973\) −14.0000 −0.448819
\(974\) −40.0000 −1.28168
\(975\) 0 0
\(976\) −10.0000 −0.320092
\(977\) 18.0000 0.575871 0.287936 − 0.957650i \(-0.407031\pi\)
0.287936 + 0.957650i \(0.407031\pi\)
\(978\) 0 0
\(979\) 27.0000 0.862924
\(980\) 6.00000 0.191663
\(981\) 0 0
\(982\) −21.0000 −0.670137
\(983\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(984\) 0 0
\(985\) −6.00000 −0.191176
\(986\) 0 0
\(987\) 0 0
\(988\) 10.0000 0.318142
\(989\) −99.0000 −3.14802
\(990\) 0 0
\(991\) 23.0000 0.730619 0.365310 − 0.930886i \(-0.380963\pi\)
0.365310 + 0.930886i \(0.380963\pi\)
\(992\) 1.00000 0.0317500
\(993\) 0 0
\(994\) 9.00000 0.285463
\(995\) 25.0000 0.792553
\(996\) 0 0
\(997\) 26.0000 0.823428 0.411714 − 0.911313i \(-0.364930\pi\)
0.411714 + 0.911313i \(0.364930\pi\)
\(998\) 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 2790.2.a.r.1.1 1
3.2 odd 2 930.2.a.i.1.1 ✓ 1
12.11 even 2 7440.2.a.k.1.1 1
15.2 even 4 4650.2.d.d.3349.1 2
15.8 even 4 4650.2.d.d.3349.2 2
15.14 odd 2 4650.2.a.bd.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
930.2.a.i.1.1 ✓ 1 3.2 odd 2
2790.2.a.r.1.1 1 1.1 even 1 trivial
4650.2.a.bd.1.1 1 15.14 odd 2
4650.2.d.d.3349.1 2 15.2 even 4
4650.2.d.d.3349.2 2 15.8 even 4
7440.2.a.k.1.1 1 12.11 even 2