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Properties

Label 4410.2.a.r.1.1
Level $4410$
Weight $2$
Character 4410.1
Self dual yes
Analytic conductor $35.214$
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] = [4410,2,Mod(1,4410)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4410, 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("4410.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4410 = 2 \cdot 3^{2} \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4410.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: \(35.2140272914\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 70)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 4410.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{8} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{8} -1.00000 q^{10} +2.00000 q^{11} +1.00000 q^{16} +4.00000 q^{17} -6.00000 q^{19} +1.00000 q^{20} -2.00000 q^{22} -3.00000 q^{23} +1.00000 q^{25} -9.00000 q^{29} -4.00000 q^{31} -1.00000 q^{32} -4.00000 q^{34} -4.00000 q^{37} +6.00000 q^{38} -1.00000 q^{40} +7.00000 q^{41} -5.00000 q^{43} +2.00000 q^{44} +3.00000 q^{46} -8.00000 q^{47} -1.00000 q^{50} +2.00000 q^{53} +2.00000 q^{55} +9.00000 q^{58} -10.0000 q^{59} +1.00000 q^{61} +4.00000 q^{62} +1.00000 q^{64} -9.00000 q^{67} +4.00000 q^{68} -2.00000 q^{71} -4.00000 q^{73} +4.00000 q^{74} -6.00000 q^{76} +10.0000 q^{79} +1.00000 q^{80} -7.00000 q^{82} +7.00000 q^{83} +4.00000 q^{85} +5.00000 q^{86} -2.00000 q^{88} -1.00000 q^{89} -3.00000 q^{92} +8.00000 q^{94} -6.00000 q^{95} +14.0000 q^{97} +O(q^{100})\)

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
\(7\) 0 0
\(8\) −1.00000 −0.353553
\(9\) 0 0
\(10\) −1.00000 −0.316228
\(11\) 2.00000 0.603023 0.301511 − 0.953463i \(-0.402509\pi\)
0.301511 + 0.953463i \(0.402509\pi\)
\(12\) 0 0
\(13\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 4.00000 0.970143 0.485071 − 0.874475i \(-0.338794\pi\)
0.485071 + 0.874475i \(0.338794\pi\)
\(18\) 0 0
\(19\) −6.00000 −1.37649 −0.688247 − 0.725476i \(-0.741620\pi\)
−0.688247 + 0.725476i \(0.741620\pi\)
\(20\) 1.00000 0.223607
\(21\) 0 0
\(22\) −2.00000 −0.426401
\(23\) −3.00000 −0.625543 −0.312772 − 0.949828i \(-0.601257\pi\)
−0.312772 + 0.949828i \(0.601257\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −9.00000 −1.67126 −0.835629 − 0.549294i \(-0.814897\pi\)
−0.835629 + 0.549294i \(0.814897\pi\)
\(30\) 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
\(34\) −4.00000 −0.685994
\(35\) 0 0
\(36\) 0 0
\(37\) −4.00000 −0.657596 −0.328798 − 0.944400i \(-0.606644\pi\)
−0.328798 + 0.944400i \(0.606644\pi\)
\(38\) 6.00000 0.973329
\(39\) 0 0
\(40\) −1.00000 −0.158114
\(41\) 7.00000 1.09322 0.546608 − 0.837389i \(-0.315919\pi\)
0.546608 + 0.837389i \(0.315919\pi\)
\(42\) 0 0
\(43\) −5.00000 −0.762493 −0.381246 − 0.924473i \(-0.624505\pi\)
−0.381246 + 0.924473i \(0.624505\pi\)
\(44\) 2.00000 0.301511
\(45\) 0 0
\(46\) 3.00000 0.442326
\(47\) −8.00000 −1.16692 −0.583460 − 0.812142i \(-0.698301\pi\)
−0.583460 + 0.812142i \(0.698301\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) −1.00000 −0.141421
\(51\) 0 0
\(52\) 0 0
\(53\) 2.00000 0.274721 0.137361 − 0.990521i \(-0.456138\pi\)
0.137361 + 0.990521i \(0.456138\pi\)
\(54\) 0 0
\(55\) 2.00000 0.269680
\(56\) 0 0
\(57\) 0 0
\(58\) 9.00000 1.18176
\(59\) −10.0000 −1.30189 −0.650945 − 0.759125i \(-0.725627\pi\)
−0.650945 + 0.759125i \(0.725627\pi\)
\(60\) 0 0
\(61\) 1.00000 0.128037 0.0640184 − 0.997949i \(-0.479608\pi\)
0.0640184 + 0.997949i \(0.479608\pi\)
\(62\) 4.00000 0.508001
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 0 0
\(67\) −9.00000 −1.09952 −0.549762 − 0.835321i \(-0.685282\pi\)
−0.549762 + 0.835321i \(0.685282\pi\)
\(68\) 4.00000 0.485071
\(69\) 0 0
\(70\) 0 0
\(71\) −2.00000 −0.237356 −0.118678 − 0.992933i \(-0.537866\pi\)
−0.118678 + 0.992933i \(0.537866\pi\)
\(72\) 0 0
\(73\) −4.00000 −0.468165 −0.234082 − 0.972217i \(-0.575209\pi\)
−0.234082 + 0.972217i \(0.575209\pi\)
\(74\) 4.00000 0.464991
\(75\) 0 0
\(76\) −6.00000 −0.688247
\(77\) 0 0
\(78\) 0 0
\(79\) 10.0000 1.12509 0.562544 − 0.826767i \(-0.309823\pi\)
0.562544 + 0.826767i \(0.309823\pi\)
\(80\) 1.00000 0.111803
\(81\) 0 0
\(82\) −7.00000 −0.773021
\(83\) 7.00000 0.768350 0.384175 − 0.923260i \(-0.374486\pi\)
0.384175 + 0.923260i \(0.374486\pi\)
\(84\) 0 0
\(85\) 4.00000 0.433861
\(86\) 5.00000 0.539164
\(87\) 0 0
\(88\) −2.00000 −0.213201
\(89\) −1.00000 −0.106000 −0.0529999 − 0.998595i \(-0.516878\pi\)
−0.0529999 + 0.998595i \(0.516878\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) −3.00000 −0.312772
\(93\) 0 0
\(94\) 8.00000 0.825137
\(95\) −6.00000 −0.615587
\(96\) 0 0
\(97\) 14.0000 1.42148 0.710742 − 0.703452i \(-0.248359\pi\)
0.710742 + 0.703452i \(0.248359\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 1.00000 0.100000
\(101\) −3.00000 −0.298511 −0.149256 − 0.988799i \(-0.547688\pi\)
−0.149256 + 0.988799i \(0.547688\pi\)
\(102\) 0 0
\(103\) 1.00000 0.0985329 0.0492665 − 0.998786i \(-0.484312\pi\)
0.0492665 + 0.998786i \(0.484312\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) −2.00000 −0.194257
\(107\) −3.00000 −0.290021 −0.145010 − 0.989430i \(-0.546322\pi\)
−0.145010 + 0.989430i \(0.546322\pi\)
\(108\) 0 0
\(109\) −9.00000 −0.862044 −0.431022 − 0.902342i \(-0.641847\pi\)
−0.431022 + 0.902342i \(0.641847\pi\)
\(110\) −2.00000 −0.190693
\(111\) 0 0
\(112\) 0 0
\(113\) −2.00000 −0.188144 −0.0940721 − 0.995565i \(-0.529988\pi\)
−0.0940721 + 0.995565i \(0.529988\pi\)
\(114\) 0 0
\(115\) −3.00000 −0.279751
\(116\) −9.00000 −0.835629
\(117\) 0 0
\(118\) 10.0000 0.920575
\(119\) 0 0
\(120\) 0 0
\(121\) −7.00000 −0.636364
\(122\) −1.00000 −0.0905357
\(123\) 0 0
\(124\) −4.00000 −0.359211
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) 16.0000 1.41977 0.709885 − 0.704317i \(-0.248747\pi\)
0.709885 + 0.704317i \(0.248747\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 0 0
\(130\) 0 0
\(131\) −8.00000 −0.698963 −0.349482 − 0.936943i \(-0.613642\pi\)
−0.349482 + 0.936943i \(0.613642\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 9.00000 0.777482
\(135\) 0 0
\(136\) −4.00000 −0.342997
\(137\) −12.0000 −1.02523 −0.512615 − 0.858619i \(-0.671323\pi\)
−0.512615 + 0.858619i \(0.671323\pi\)
\(138\) 0 0
\(139\) −14.0000 −1.18746 −0.593732 − 0.804663i \(-0.702346\pi\)
−0.593732 + 0.804663i \(0.702346\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 2.00000 0.167836
\(143\) 0 0
\(144\) 0 0
\(145\) −9.00000 −0.747409
\(146\) 4.00000 0.331042
\(147\) 0 0
\(148\) −4.00000 −0.328798
\(149\) −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 \(0.725663\pi\)
\(152\) 6.00000 0.486664
\(153\) 0 0
\(154\) 0 0
\(155\) −4.00000 −0.321288
\(156\) 0 0
\(157\) 10.0000 0.798087 0.399043 − 0.916932i \(-0.369342\pi\)
0.399043 + 0.916932i \(0.369342\pi\)
\(158\) −10.0000 −0.795557
\(159\) 0 0
\(160\) −1.00000 −0.0790569
\(161\) 0 0
\(162\) 0 0
\(163\) −4.00000 −0.313304 −0.156652 − 0.987654i \(-0.550070\pi\)
−0.156652 + 0.987654i \(0.550070\pi\)
\(164\) 7.00000 0.546608
\(165\) 0 0
\(166\) −7.00000 −0.543305
\(167\) 21.0000 1.62503 0.812514 − 0.582941i \(-0.198098\pi\)
0.812514 + 0.582941i \(0.198098\pi\)
\(168\) 0 0
\(169\) −13.0000 −1.00000
\(170\) −4.00000 −0.306786
\(171\) 0 0
\(172\) −5.00000 −0.381246
\(173\) −8.00000 −0.608229 −0.304114 − 0.952636i \(-0.598361\pi\)
−0.304114 + 0.952636i \(0.598361\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 2.00000 0.150756
\(177\) 0 0
\(178\) 1.00000 0.0749532
\(179\) −12.0000 −0.896922 −0.448461 − 0.893802i \(-0.648028\pi\)
−0.448461 + 0.893802i \(0.648028\pi\)
\(180\) 0 0
\(181\) 7.00000 0.520306 0.260153 − 0.965567i \(-0.416227\pi\)
0.260153 + 0.965567i \(0.416227\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 3.00000 0.221163
\(185\) −4.00000 −0.294086
\(186\) 0 0
\(187\) 8.00000 0.585018
\(188\) −8.00000 −0.583460
\(189\) 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
\(193\) 26.0000 1.87152 0.935760 − 0.352636i \(-0.114715\pi\)
0.935760 + 0.352636i \(0.114715\pi\)
\(194\) −14.0000 −1.00514
\(195\) 0 0
\(196\) 0 0
\(197\) −2.00000 −0.142494 −0.0712470 − 0.997459i \(-0.522698\pi\)
−0.0712470 + 0.997459i \(0.522698\pi\)
\(198\) 0 0
\(199\) −4.00000 −0.283552 −0.141776 − 0.989899i \(-0.545281\pi\)
−0.141776 + 0.989899i \(0.545281\pi\)
\(200\) −1.00000 −0.0707107
\(201\) 0 0
\(202\) 3.00000 0.211079
\(203\) 0 0
\(204\) 0 0
\(205\) 7.00000 0.488901
\(206\) −1.00000 −0.0696733
\(207\) 0 0
\(208\) 0 0
\(209\) −12.0000 −0.830057
\(210\) 0 0
\(211\) −26.0000 −1.78991 −0.894957 − 0.446153i \(-0.852794\pi\)
−0.894957 + 0.446153i \(0.852794\pi\)
\(212\) 2.00000 0.137361
\(213\) 0 0
\(214\) 3.00000 0.205076
\(215\) −5.00000 −0.340997
\(216\) 0 0
\(217\) 0 0
\(218\) 9.00000 0.609557
\(219\) 0 0
\(220\) 2.00000 0.134840
\(221\) 0 0
\(222\) 0 0
\(223\) 28.0000 1.87502 0.937509 − 0.347960i \(-0.113126\pi\)
0.937509 + 0.347960i \(0.113126\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 2.00000 0.133038
\(227\) 4.00000 0.265489 0.132745 − 0.991150i \(-0.457621\pi\)
0.132745 + 0.991150i \(0.457621\pi\)
\(228\) 0 0
\(229\) 22.0000 1.45380 0.726900 − 0.686743i \(-0.240960\pi\)
0.726900 + 0.686743i \(0.240960\pi\)
\(230\) 3.00000 0.197814
\(231\) 0 0
\(232\) 9.00000 0.590879
\(233\) −24.0000 −1.57229 −0.786146 − 0.618041i \(-0.787927\pi\)
−0.786146 + 0.618041i \(0.787927\pi\)
\(234\) 0 0
\(235\) −8.00000 −0.521862
\(236\) −10.0000 −0.650945
\(237\) 0 0
\(238\) 0 0
\(239\) −16.0000 −1.03495 −0.517477 − 0.855697i \(-0.673129\pi\)
−0.517477 + 0.855697i \(0.673129\pi\)
\(240\) 0 0
\(241\) 10.0000 0.644157 0.322078 − 0.946713i \(-0.395619\pi\)
0.322078 + 0.946713i \(0.395619\pi\)
\(242\) 7.00000 0.449977
\(243\) 0 0
\(244\) 1.00000 0.0640184
\(245\) 0 0
\(246\) 0 0
\(247\) 0 0
\(248\) 4.00000 0.254000
\(249\) 0 0
\(250\) −1.00000 −0.0632456
\(251\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(252\) 0 0
\(253\) −6.00000 −0.377217
\(254\) −16.0000 −1.00393
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −8.00000 −0.499026 −0.249513 − 0.968371i \(-0.580271\pi\)
−0.249513 + 0.968371i \(0.580271\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) 8.00000 0.494242
\(263\) −5.00000 −0.308313 −0.154157 − 0.988046i \(-0.549266\pi\)
−0.154157 + 0.988046i \(0.549266\pi\)
\(264\) 0 0
\(265\) 2.00000 0.122859
\(266\) 0 0
\(267\) 0 0
\(268\) −9.00000 −0.549762
\(269\) −3.00000 −0.182913 −0.0914566 − 0.995809i \(-0.529152\pi\)
−0.0914566 + 0.995809i \(0.529152\pi\)
\(270\) 0 0
\(271\) −6.00000 −0.364474 −0.182237 − 0.983255i \(-0.558334\pi\)
−0.182237 + 0.983255i \(0.558334\pi\)
\(272\) 4.00000 0.242536
\(273\) 0 0
\(274\) 12.0000 0.724947
\(275\) 2.00000 0.120605
\(276\) 0 0
\(277\) 12.0000 0.721010 0.360505 − 0.932757i \(-0.382604\pi\)
0.360505 + 0.932757i \(0.382604\pi\)
\(278\) 14.0000 0.839664
\(279\) 0 0
\(280\) 0 0
\(281\) −2.00000 −0.119310 −0.0596550 − 0.998219i \(-0.519000\pi\)
−0.0596550 + 0.998219i \(0.519000\pi\)
\(282\) 0 0
\(283\) −4.00000 −0.237775 −0.118888 − 0.992908i \(-0.537933\pi\)
−0.118888 + 0.992908i \(0.537933\pi\)
\(284\) −2.00000 −0.118678
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −1.00000 −0.0588235
\(290\) 9.00000 0.528498
\(291\) 0 0
\(292\) −4.00000 −0.234082
\(293\) −28.0000 −1.63578 −0.817889 − 0.575376i \(-0.804856\pi\)
−0.817889 + 0.575376i \(0.804856\pi\)
\(294\) 0 0
\(295\) −10.0000 −0.582223
\(296\) 4.00000 0.232495
\(297\) 0 0
\(298\) 3.00000 0.173785
\(299\) 0 0
\(300\) 0 0
\(301\) 0 0
\(302\) 16.0000 0.920697
\(303\) 0 0
\(304\) −6.00000 −0.344124
\(305\) 1.00000 0.0572598
\(306\) 0 0
\(307\) 7.00000 0.399511 0.199756 − 0.979846i \(-0.435985\pi\)
0.199756 + 0.979846i \(0.435985\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 4.00000 0.227185
\(311\) 18.0000 1.02069 0.510343 − 0.859971i \(-0.329518\pi\)
0.510343 + 0.859971i \(0.329518\pi\)
\(312\) 0 0
\(313\) 8.00000 0.452187 0.226093 − 0.974106i \(-0.427405\pi\)
0.226093 + 0.974106i \(0.427405\pi\)
\(314\) −10.0000 −0.564333
\(315\) 0 0
\(316\) 10.0000 0.562544
\(317\) 32.0000 1.79730 0.898650 − 0.438667i \(-0.144549\pi\)
0.898650 + 0.438667i \(0.144549\pi\)
\(318\) 0 0
\(319\) −18.0000 −1.00781
\(320\) 1.00000 0.0559017
\(321\) 0 0
\(322\) 0 0
\(323\) −24.0000 −1.33540
\(324\) 0 0
\(325\) 0 0
\(326\) 4.00000 0.221540
\(327\) 0 0
\(328\) −7.00000 −0.386510
\(329\) 0 0
\(330\) 0 0
\(331\) −32.0000 −1.75888 −0.879440 − 0.476011i \(-0.842082\pi\)
−0.879440 + 0.476011i \(0.842082\pi\)
\(332\) 7.00000 0.384175
\(333\) 0 0
\(334\) −21.0000 −1.14907
\(335\) −9.00000 −0.491723
\(336\) 0 0
\(337\) −26.0000 −1.41631 −0.708155 − 0.706057i \(-0.750472\pi\)
−0.708155 + 0.706057i \(0.750472\pi\)
\(338\) 13.0000 0.707107
\(339\) 0 0
\(340\) 4.00000 0.216930
\(341\) −8.00000 −0.433224
\(342\) 0 0
\(343\) 0 0
\(344\) 5.00000 0.269582
\(345\) 0 0
\(346\) 8.00000 0.430083
\(347\) −19.0000 −1.01997 −0.509987 − 0.860182i \(-0.670350\pi\)
−0.509987 + 0.860182i \(0.670350\pi\)
\(348\) 0 0
\(349\) −35.0000 −1.87351 −0.936754 − 0.349990i \(-0.886185\pi\)
−0.936754 + 0.349990i \(0.886185\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) −2.00000 −0.106600
\(353\) 18.0000 0.958043 0.479022 − 0.877803i \(-0.340992\pi\)
0.479022 + 0.877803i \(0.340992\pi\)
\(354\) 0 0
\(355\) −2.00000 −0.106149
\(356\) −1.00000 −0.0529999
\(357\) 0 0
\(358\) 12.0000 0.634220
\(359\) 4.00000 0.211112 0.105556 − 0.994413i \(-0.466338\pi\)
0.105556 + 0.994413i \(0.466338\pi\)
\(360\) 0 0
\(361\) 17.0000 0.894737
\(362\) −7.00000 −0.367912
\(363\) 0 0
\(364\) 0 0
\(365\) −4.00000 −0.209370
\(366\) 0 0
\(367\) −11.0000 −0.574195 −0.287098 − 0.957901i \(-0.592690\pi\)
−0.287098 + 0.957901i \(0.592690\pi\)
\(368\) −3.00000 −0.156386
\(369\) 0 0
\(370\) 4.00000 0.207950
\(371\) 0 0
\(372\) 0 0
\(373\) −4.00000 −0.207112 −0.103556 − 0.994624i \(-0.533022\pi\)
−0.103556 + 0.994624i \(0.533022\pi\)
\(374\) −8.00000 −0.413670
\(375\) 0 0
\(376\) 8.00000 0.412568
\(377\) 0 0
\(378\) 0 0
\(379\) 30.0000 1.54100 0.770498 − 0.637442i \(-0.220007\pi\)
0.770498 + 0.637442i \(0.220007\pi\)
\(380\) −6.00000 −0.307794
\(381\) 0 0
\(382\) −18.0000 −0.920960
\(383\) −15.0000 −0.766464 −0.383232 − 0.923652i \(-0.625189\pi\)
−0.383232 + 0.923652i \(0.625189\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −26.0000 −1.32337
\(387\) 0 0
\(388\) 14.0000 0.710742
\(389\) −26.0000 −1.31825 −0.659126 − 0.752032i \(-0.729074\pi\)
−0.659126 + 0.752032i \(0.729074\pi\)
\(390\) 0 0
\(391\) −12.0000 −0.606866
\(392\) 0 0
\(393\) 0 0
\(394\) 2.00000 0.100759
\(395\) 10.0000 0.503155
\(396\) 0 0
\(397\) 22.0000 1.10415 0.552074 − 0.833795i \(-0.313837\pi\)
0.552074 + 0.833795i \(0.313837\pi\)
\(398\) 4.00000 0.200502
\(399\) 0 0
\(400\) 1.00000 0.0500000
\(401\) −31.0000 −1.54807 −0.774033 − 0.633145i \(-0.781764\pi\)
−0.774033 + 0.633145i \(0.781764\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) −3.00000 −0.149256
\(405\) 0 0
\(406\) 0 0
\(407\) −8.00000 −0.396545
\(408\) 0 0
\(409\) 3.00000 0.148340 0.0741702 − 0.997246i \(-0.476369\pi\)
0.0741702 + 0.997246i \(0.476369\pi\)
\(410\) −7.00000 −0.345705
\(411\) 0 0
\(412\) 1.00000 0.0492665
\(413\) 0 0
\(414\) 0 0
\(415\) 7.00000 0.343616
\(416\) 0 0
\(417\) 0 0
\(418\) 12.0000 0.586939
\(419\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(420\) 0 0
\(421\) −19.0000 −0.926003 −0.463002 − 0.886357i \(-0.653228\pi\)
−0.463002 + 0.886357i \(0.653228\pi\)
\(422\) 26.0000 1.26566
\(423\) 0 0
\(424\) −2.00000 −0.0971286
\(425\) 4.00000 0.194029
\(426\) 0 0
\(427\) 0 0
\(428\) −3.00000 −0.145010
\(429\) 0 0
\(430\) 5.00000 0.241121
\(431\) 30.0000 1.44505 0.722525 − 0.691345i \(-0.242982\pi\)
0.722525 + 0.691345i \(0.242982\pi\)
\(432\) 0 0
\(433\) 14.0000 0.672797 0.336399 − 0.941720i \(-0.390791\pi\)
0.336399 + 0.941720i \(0.390791\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −9.00000 −0.431022
\(437\) 18.0000 0.861057
\(438\) 0 0
\(439\) −20.0000 −0.954548 −0.477274 − 0.878755i \(-0.658375\pi\)
−0.477274 + 0.878755i \(0.658375\pi\)
\(440\) −2.00000 −0.0953463
\(441\) 0 0
\(442\) 0 0
\(443\) −31.0000 −1.47285 −0.736427 − 0.676517i \(-0.763489\pi\)
−0.736427 + 0.676517i \(0.763489\pi\)
\(444\) 0 0
\(445\) −1.00000 −0.0474045
\(446\) −28.0000 −1.32584
\(447\) 0 0
\(448\) 0 0
\(449\) 33.0000 1.55737 0.778683 − 0.627417i \(-0.215888\pi\)
0.778683 + 0.627417i \(0.215888\pi\)
\(450\) 0 0
\(451\) 14.0000 0.659234
\(452\) −2.00000 −0.0940721
\(453\) 0 0
\(454\) −4.00000 −0.187729
\(455\) 0 0
\(456\) 0 0
\(457\) −32.0000 −1.49690 −0.748448 − 0.663193i \(-0.769201\pi\)
−0.748448 + 0.663193i \(0.769201\pi\)
\(458\) −22.0000 −1.02799
\(459\) 0 0
\(460\) −3.00000 −0.139876
\(461\) 14.0000 0.652045 0.326023 − 0.945362i \(-0.394291\pi\)
0.326023 + 0.945362i \(0.394291\pi\)
\(462\) 0 0
\(463\) −19.0000 −0.883005 −0.441502 − 0.897260i \(-0.645554\pi\)
−0.441502 + 0.897260i \(0.645554\pi\)
\(464\) −9.00000 −0.417815
\(465\) 0 0
\(466\) 24.0000 1.11178
\(467\) 13.0000 0.601568 0.300784 − 0.953692i \(-0.402752\pi\)
0.300784 + 0.953692i \(0.402752\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 8.00000 0.369012
\(471\) 0 0
\(472\) 10.0000 0.460287
\(473\) −10.0000 −0.459800
\(474\) 0 0
\(475\) −6.00000 −0.275299
\(476\) 0 0
\(477\) 0 0
\(478\) 16.0000 0.731823
\(479\) −24.0000 −1.09659 −0.548294 − 0.836286i \(-0.684723\pi\)
−0.548294 + 0.836286i \(0.684723\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) −10.0000 −0.455488
\(483\) 0 0
\(484\) −7.00000 −0.318182
\(485\) 14.0000 0.635707
\(486\) 0 0
\(487\) −16.0000 −0.725029 −0.362515 − 0.931978i \(-0.618082\pi\)
−0.362515 + 0.931978i \(0.618082\pi\)
\(488\) −1.00000 −0.0452679
\(489\) 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
\(493\) −36.0000 −1.62136
\(494\) 0 0
\(495\) 0 0
\(496\) −4.00000 −0.179605
\(497\) 0 0
\(498\) 0 0
\(499\) −18.0000 −0.805791 −0.402895 − 0.915246i \(-0.631996\pi\)
−0.402895 + 0.915246i \(0.631996\pi\)
\(500\) 1.00000 0.0447214
\(501\) 0 0
\(502\) 0 0
\(503\) 21.0000 0.936344 0.468172 − 0.883637i \(-0.344913\pi\)
0.468172 + 0.883637i \(0.344913\pi\)
\(504\) 0 0
\(505\) −3.00000 −0.133498
\(506\) 6.00000 0.266733
\(507\) 0 0
\(508\) 16.0000 0.709885
\(509\) −1.00000 −0.0443242 −0.0221621 − 0.999754i \(-0.507055\pi\)
−0.0221621 + 0.999754i \(0.507055\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) −1.00000 −0.0441942
\(513\) 0 0
\(514\) 8.00000 0.352865
\(515\) 1.00000 0.0440653
\(516\) 0 0
\(517\) −16.0000 −0.703679
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −38.0000 −1.66481 −0.832405 − 0.554168i \(-0.813037\pi\)
−0.832405 + 0.554168i \(0.813037\pi\)
\(522\) 0 0
\(523\) −20.0000 −0.874539 −0.437269 − 0.899331i \(-0.644054\pi\)
−0.437269 + 0.899331i \(0.644054\pi\)
\(524\) −8.00000 −0.349482
\(525\) 0 0
\(526\) 5.00000 0.218010
\(527\) −16.0000 −0.696971
\(528\) 0 0
\(529\) −14.0000 −0.608696
\(530\) −2.00000 −0.0868744
\(531\) 0 0
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) −3.00000 −0.129701
\(536\) 9.00000 0.388741
\(537\) 0 0
\(538\) 3.00000 0.129339
\(539\) 0 0
\(540\) 0 0
\(541\) 3.00000 0.128980 0.0644900 − 0.997918i \(-0.479458\pi\)
0.0644900 + 0.997918i \(0.479458\pi\)
\(542\) 6.00000 0.257722
\(543\) 0 0
\(544\) −4.00000 −0.171499
\(545\) −9.00000 −0.385518
\(546\) 0 0
\(547\) −33.0000 −1.41098 −0.705489 − 0.708721i \(-0.749273\pi\)
−0.705489 + 0.708721i \(0.749273\pi\)
\(548\) −12.0000 −0.512615
\(549\) 0 0
\(550\) −2.00000 −0.0852803
\(551\) 54.0000 2.30048
\(552\) 0 0
\(553\) 0 0
\(554\) −12.0000 −0.509831
\(555\) 0 0
\(556\) −14.0000 −0.593732
\(557\) 2.00000 0.0847427 0.0423714 − 0.999102i \(-0.486509\pi\)
0.0423714 + 0.999102i \(0.486509\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) 2.00000 0.0843649
\(563\) −17.0000 −0.716465 −0.358232 − 0.933632i \(-0.616620\pi\)
−0.358232 + 0.933632i \(0.616620\pi\)
\(564\) 0 0
\(565\) −2.00000 −0.0841406
\(566\) 4.00000 0.168133
\(567\) 0 0
\(568\) 2.00000 0.0839181
\(569\) 18.0000 0.754599 0.377300 − 0.926091i \(-0.376853\pi\)
0.377300 + 0.926091i \(0.376853\pi\)
\(570\) 0 0
\(571\) −30.0000 −1.25546 −0.627730 − 0.778431i \(-0.716016\pi\)
−0.627730 + 0.778431i \(0.716016\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −3.00000 −0.125109
\(576\) 0 0
\(577\) 10.0000 0.416305 0.208153 − 0.978096i \(-0.433255\pi\)
0.208153 + 0.978096i \(0.433255\pi\)
\(578\) 1.00000 0.0415945
\(579\) 0 0
\(580\) −9.00000 −0.373705
\(581\) 0 0
\(582\) 0 0
\(583\) 4.00000 0.165663
\(584\) 4.00000 0.165521
\(585\) 0 0
\(586\) 28.0000 1.15667
\(587\) 28.0000 1.15568 0.577842 − 0.816149i \(-0.303895\pi\)
0.577842 + 0.816149i \(0.303895\pi\)
\(588\) 0 0
\(589\) 24.0000 0.988903
\(590\) 10.0000 0.411693
\(591\) 0 0
\(592\) −4.00000 −0.164399
\(593\) 6.00000 0.246390 0.123195 − 0.992382i \(-0.460686\pi\)
0.123195 + 0.992382i \(0.460686\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −3.00000 −0.122885
\(597\) 0 0
\(598\) 0 0
\(599\) −12.0000 −0.490307 −0.245153 − 0.969484i \(-0.578838\pi\)
−0.245153 + 0.969484i \(0.578838\pi\)
\(600\) 0 0
\(601\) 42.0000 1.71322 0.856608 − 0.515968i \(-0.172568\pi\)
0.856608 + 0.515968i \(0.172568\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) −16.0000 −0.651031
\(605\) −7.00000 −0.284590
\(606\) 0 0
\(607\) 1.00000 0.0405887 0.0202944 − 0.999794i \(-0.493540\pi\)
0.0202944 + 0.999794i \(0.493540\pi\)
\(608\) 6.00000 0.243332
\(609\) 0 0
\(610\) −1.00000 −0.0404888
\(611\) 0 0
\(612\) 0 0
\(613\) 12.0000 0.484675 0.242338 − 0.970192i \(-0.422086\pi\)
0.242338 + 0.970192i \(0.422086\pi\)
\(614\) −7.00000 −0.282497
\(615\) 0 0
\(616\) 0 0
\(617\) −44.0000 −1.77137 −0.885687 − 0.464283i \(-0.846312\pi\)
−0.885687 + 0.464283i \(0.846312\pi\)
\(618\) 0 0
\(619\) −46.0000 −1.84890 −0.924448 − 0.381308i \(-0.875474\pi\)
−0.924448 + 0.381308i \(0.875474\pi\)
\(620\) −4.00000 −0.160644
\(621\) 0 0
\(622\) −18.0000 −0.721734
\(623\) 0 0
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) −8.00000 −0.319744
\(627\) 0 0
\(628\) 10.0000 0.399043
\(629\) −16.0000 −0.637962
\(630\) 0 0
\(631\) 2.00000 0.0796187 0.0398094 − 0.999207i \(-0.487325\pi\)
0.0398094 + 0.999207i \(0.487325\pi\)
\(632\) −10.0000 −0.397779
\(633\) 0 0
\(634\) −32.0000 −1.27088
\(635\) 16.0000 0.634941
\(636\) 0 0
\(637\) 0 0
\(638\) 18.0000 0.712627
\(639\) 0 0
\(640\) −1.00000 −0.0395285
\(641\) −5.00000 −0.197488 −0.0987441 − 0.995113i \(-0.531483\pi\)
−0.0987441 + 0.995113i \(0.531483\pi\)
\(642\) 0 0
\(643\) 28.0000 1.10421 0.552106 − 0.833774i \(-0.313824\pi\)
0.552106 + 0.833774i \(0.313824\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 24.0000 0.944267
\(647\) 11.0000 0.432455 0.216227 − 0.976343i \(-0.430625\pi\)
0.216227 + 0.976343i \(0.430625\pi\)
\(648\) 0 0
\(649\) −20.0000 −0.785069
\(650\) 0 0
\(651\) 0 0
\(652\) −4.00000 −0.156652
\(653\) 4.00000 0.156532 0.0782660 − 0.996933i \(-0.475062\pi\)
0.0782660 + 0.996933i \(0.475062\pi\)
\(654\) 0 0
\(655\) −8.00000 −0.312586
\(656\) 7.00000 0.273304
\(657\) 0 0
\(658\) 0 0
\(659\) 26.0000 1.01282 0.506408 − 0.862294i \(-0.330973\pi\)
0.506408 + 0.862294i \(0.330973\pi\)
\(660\) 0 0
\(661\) −11.0000 −0.427850 −0.213925 − 0.976850i \(-0.568625\pi\)
−0.213925 + 0.976850i \(0.568625\pi\)
\(662\) 32.0000 1.24372
\(663\) 0 0
\(664\) −7.00000 −0.271653
\(665\) 0 0
\(666\) 0 0
\(667\) 27.0000 1.04544
\(668\) 21.0000 0.812514
\(669\) 0 0
\(670\) 9.00000 0.347700
\(671\) 2.00000 0.0772091
\(672\) 0 0
\(673\) 16.0000 0.616755 0.308377 − 0.951264i \(-0.400214\pi\)
0.308377 + 0.951264i \(0.400214\pi\)
\(674\) 26.0000 1.00148
\(675\) 0 0
\(676\) −13.0000 −0.500000
\(677\) 48.0000 1.84479 0.922395 − 0.386248i \(-0.126229\pi\)
0.922395 + 0.386248i \(0.126229\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) −4.00000 −0.153393
\(681\) 0 0
\(682\) 8.00000 0.306336
\(683\) 37.0000 1.41577 0.707883 − 0.706330i \(-0.249650\pi\)
0.707883 + 0.706330i \(0.249650\pi\)
\(684\) 0 0
\(685\) −12.0000 −0.458496
\(686\) 0 0
\(687\) 0 0
\(688\) −5.00000 −0.190623
\(689\) 0 0
\(690\) 0 0
\(691\) 22.0000 0.836919 0.418460 − 0.908235i \(-0.362570\pi\)
0.418460 + 0.908235i \(0.362570\pi\)
\(692\) −8.00000 −0.304114
\(693\) 0 0
\(694\) 19.0000 0.721230
\(695\) −14.0000 −0.531050
\(696\) 0 0
\(697\) 28.0000 1.06058
\(698\) 35.0000 1.32477
\(699\) 0 0
\(700\) 0 0
\(701\) 47.0000 1.77517 0.887583 − 0.460648i \(-0.152383\pi\)
0.887583 + 0.460648i \(0.152383\pi\)
\(702\) 0 0
\(703\) 24.0000 0.905177
\(704\) 2.00000 0.0753778
\(705\) 0 0
\(706\) −18.0000 −0.677439
\(707\) 0 0
\(708\) 0 0
\(709\) −11.0000 −0.413114 −0.206557 − 0.978435i \(-0.566226\pi\)
−0.206557 + 0.978435i \(0.566226\pi\)
\(710\) 2.00000 0.0750587
\(711\) 0 0
\(712\) 1.00000 0.0374766
\(713\) 12.0000 0.449404
\(714\) 0 0
\(715\) 0 0
\(716\) −12.0000 −0.448461
\(717\) 0 0
\(718\) −4.00000 −0.149279
\(719\) 6.00000 0.223762 0.111881 − 0.993722i \(-0.464312\pi\)
0.111881 + 0.993722i \(0.464312\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) −17.0000 −0.632674
\(723\) 0 0
\(724\) 7.00000 0.260153
\(725\) −9.00000 −0.334252
\(726\) 0 0
\(727\) −21.0000 −0.778847 −0.389423 − 0.921059i \(-0.627326\pi\)
−0.389423 + 0.921059i \(0.627326\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 4.00000 0.148047
\(731\) −20.0000 −0.739727
\(732\) 0 0
\(733\) 22.0000 0.812589 0.406294 − 0.913742i \(-0.366821\pi\)
0.406294 + 0.913742i \(0.366821\pi\)
\(734\) 11.0000 0.406017
\(735\) 0 0
\(736\) 3.00000 0.110581
\(737\) −18.0000 −0.663039
\(738\) 0 0
\(739\) −2.00000 −0.0735712 −0.0367856 − 0.999323i \(-0.511712\pi\)
−0.0367856 + 0.999323i \(0.511712\pi\)
\(740\) −4.00000 −0.147043
\(741\) 0 0
\(742\) 0 0
\(743\) −9.00000 −0.330178 −0.165089 − 0.986279i \(-0.552791\pi\)
−0.165089 + 0.986279i \(0.552791\pi\)
\(744\) 0 0
\(745\) −3.00000 −0.109911
\(746\) 4.00000 0.146450
\(747\) 0 0
\(748\) 8.00000 0.292509
\(749\) 0 0
\(750\) 0 0
\(751\) −4.00000 −0.145962 −0.0729810 − 0.997333i \(-0.523251\pi\)
−0.0729810 + 0.997333i \(0.523251\pi\)
\(752\) −8.00000 −0.291730
\(753\) 0 0
\(754\) 0 0
\(755\) −16.0000 −0.582300
\(756\) 0 0
\(757\) 16.0000 0.581530 0.290765 − 0.956795i \(-0.406090\pi\)
0.290765 + 0.956795i \(0.406090\pi\)
\(758\) −30.0000 −1.08965
\(759\) 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
\(763\) 0 0
\(764\) 18.0000 0.651217
\(765\) 0 0
\(766\) 15.0000 0.541972
\(767\) 0 0
\(768\) 0 0
\(769\) −14.0000 −0.504853 −0.252426 − 0.967616i \(-0.581229\pi\)
−0.252426 + 0.967616i \(0.581229\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 26.0000 0.935760
\(773\) −24.0000 −0.863220 −0.431610 − 0.902060i \(-0.642054\pi\)
−0.431610 + 0.902060i \(0.642054\pi\)
\(774\) 0 0
\(775\) −4.00000 −0.143684
\(776\) −14.0000 −0.502571
\(777\) 0 0
\(778\) 26.0000 0.932145
\(779\) −42.0000 −1.50481
\(780\) 0 0
\(781\) −4.00000 −0.143131
\(782\) 12.0000 0.429119
\(783\) 0 0
\(784\) 0 0
\(785\) 10.0000 0.356915
\(786\) 0 0
\(787\) 31.0000 1.10503 0.552515 − 0.833503i \(-0.313668\pi\)
0.552515 + 0.833503i \(0.313668\pi\)
\(788\) −2.00000 −0.0712470
\(789\) 0 0
\(790\) −10.0000 −0.355784
\(791\) 0 0
\(792\) 0 0
\(793\) 0 0
\(794\) −22.0000 −0.780751
\(795\) 0 0
\(796\) −4.00000 −0.141776
\(797\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(798\) 0 0
\(799\) −32.0000 −1.13208
\(800\) −1.00000 −0.0353553
\(801\) 0 0
\(802\) 31.0000 1.09465
\(803\) −8.00000 −0.282314
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 3.00000 0.105540
\(809\) 51.0000 1.79306 0.896532 − 0.442978i \(-0.146078\pi\)
0.896532 + 0.442978i \(0.146078\pi\)
\(810\) 0 0
\(811\) −28.0000 −0.983213 −0.491606 − 0.870817i \(-0.663590\pi\)
−0.491606 + 0.870817i \(0.663590\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 8.00000 0.280400
\(815\) −4.00000 −0.140114
\(816\) 0 0
\(817\) 30.0000 1.04957
\(818\) −3.00000 −0.104893
\(819\) 0 0
\(820\) 7.00000 0.244451
\(821\) 18.0000 0.628204 0.314102 − 0.949389i \(-0.398297\pi\)
0.314102 + 0.949389i \(0.398297\pi\)
\(822\) 0 0
\(823\) 19.0000 0.662298 0.331149 − 0.943578i \(-0.392564\pi\)
0.331149 + 0.943578i \(0.392564\pi\)
\(824\) −1.00000 −0.0348367
\(825\) 0 0
\(826\) 0 0
\(827\) 19.0000 0.660695 0.330347 − 0.943859i \(-0.392834\pi\)
0.330347 + 0.943859i \(0.392834\pi\)
\(828\) 0 0
\(829\) −46.0000 −1.59765 −0.798823 − 0.601566i \(-0.794544\pi\)
−0.798823 + 0.601566i \(0.794544\pi\)
\(830\) −7.00000 −0.242974
\(831\) 0 0
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 21.0000 0.726735
\(836\) −12.0000 −0.415029
\(837\) 0 0
\(838\) 0 0
\(839\) −14.0000 −0.483334 −0.241667 − 0.970359i \(-0.577694\pi\)
−0.241667 + 0.970359i \(0.577694\pi\)
\(840\) 0 0
\(841\) 52.0000 1.79310
\(842\) 19.0000 0.654783
\(843\) 0 0
\(844\) −26.0000 −0.894957
\(845\) −13.0000 −0.447214
\(846\) 0 0
\(847\) 0 0
\(848\) 2.00000 0.0686803
\(849\) 0 0
\(850\) −4.00000 −0.137199
\(851\) 12.0000 0.411355
\(852\) 0 0
\(853\) 14.0000 0.479351 0.239675 − 0.970853i \(-0.422959\pi\)
0.239675 + 0.970853i \(0.422959\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 3.00000 0.102538
\(857\) 18.0000 0.614868 0.307434 − 0.951569i \(-0.400530\pi\)
0.307434 + 0.951569i \(0.400530\pi\)
\(858\) 0 0
\(859\) −48.0000 −1.63774 −0.818869 − 0.573980i \(-0.805399\pi\)
−0.818869 + 0.573980i \(0.805399\pi\)
\(860\) −5.00000 −0.170499
\(861\) 0 0
\(862\) −30.0000 −1.02180
\(863\) 11.0000 0.374444 0.187222 − 0.982318i \(-0.440052\pi\)
0.187222 + 0.982318i \(0.440052\pi\)
\(864\) 0 0
\(865\) −8.00000 −0.272008
\(866\) −14.0000 −0.475739
\(867\) 0 0
\(868\) 0 0
\(869\) 20.0000 0.678454
\(870\) 0 0
\(871\) 0 0
\(872\) 9.00000 0.304778
\(873\) 0 0
\(874\) −18.0000 −0.608859
\(875\) 0 0
\(876\) 0 0
\(877\) 38.0000 1.28317 0.641584 − 0.767052i \(-0.278277\pi\)
0.641584 + 0.767052i \(0.278277\pi\)
\(878\) 20.0000 0.674967
\(879\) 0 0
\(880\) 2.00000 0.0674200
\(881\) 7.00000 0.235836 0.117918 − 0.993023i \(-0.462378\pi\)
0.117918 + 0.993023i \(0.462378\pi\)
\(882\) 0 0
\(883\) −12.0000 −0.403832 −0.201916 − 0.979403i \(-0.564717\pi\)
−0.201916 + 0.979403i \(0.564717\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 31.0000 1.04147
\(887\) −29.0000 −0.973725 −0.486862 − 0.873479i \(-0.661859\pi\)
−0.486862 + 0.873479i \(0.661859\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 1.00000 0.0335201
\(891\) 0 0
\(892\) 28.0000 0.937509
\(893\) 48.0000 1.60626
\(894\) 0 0
\(895\) −12.0000 −0.401116
\(896\) 0 0
\(897\) 0 0
\(898\) −33.0000 −1.10122
\(899\) 36.0000 1.20067
\(900\) 0 0
\(901\) 8.00000 0.266519
\(902\) −14.0000 −0.466149
\(903\) 0 0
\(904\) 2.00000 0.0665190
\(905\) 7.00000 0.232688
\(906\) 0 0
\(907\) 5.00000 0.166022 0.0830111 − 0.996549i \(-0.473546\pi\)
0.0830111 + 0.996549i \(0.473546\pi\)
\(908\) 4.00000 0.132745
\(909\) 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
\(913\) 14.0000 0.463332
\(914\) 32.0000 1.05847
\(915\) 0 0
\(916\) 22.0000 0.726900
\(917\) 0 0
\(918\) 0 0
\(919\) 38.0000 1.25350 0.626752 − 0.779219i \(-0.284384\pi\)
0.626752 + 0.779219i \(0.284384\pi\)
\(920\) 3.00000 0.0989071
\(921\) 0 0
\(922\) −14.0000 −0.461065
\(923\) 0 0
\(924\) 0 0
\(925\) −4.00000 −0.131519
\(926\) 19.0000 0.624379
\(927\) 0 0
\(928\) 9.00000 0.295439
\(929\) −43.0000 −1.41078 −0.705392 − 0.708817i \(-0.749229\pi\)
−0.705392 + 0.708817i \(0.749229\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) −24.0000 −0.786146
\(933\) 0 0
\(934\) −13.0000 −0.425373
\(935\) 8.00000 0.261628
\(936\) 0 0
\(937\) −28.0000 −0.914720 −0.457360 − 0.889282i \(-0.651205\pi\)
−0.457360 + 0.889282i \(0.651205\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) −8.00000 −0.260931
\(941\) 46.0000 1.49956 0.749779 − 0.661689i \(-0.230160\pi\)
0.749779 + 0.661689i \(0.230160\pi\)
\(942\) 0 0
\(943\) −21.0000 −0.683854
\(944\) −10.0000 −0.325472
\(945\) 0 0
\(946\) 10.0000 0.325128
\(947\) 25.0000 0.812391 0.406195 − 0.913786i \(-0.366855\pi\)
0.406195 + 0.913786i \(0.366855\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 6.00000 0.194666
\(951\) 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
\(955\) 18.0000 0.582466
\(956\) −16.0000 −0.517477
\(957\) 0 0
\(958\) 24.0000 0.775405
\(959\) 0 0
\(960\) 0 0
\(961\) −15.0000 −0.483871
\(962\) 0 0
\(963\) 0 0
\(964\) 10.0000 0.322078
\(965\) 26.0000 0.836970
\(966\) 0 0
\(967\) 37.0000 1.18984 0.594920 − 0.803785i \(-0.297184\pi\)
0.594920 + 0.803785i \(0.297184\pi\)
\(968\) 7.00000 0.224989
\(969\) 0 0
\(970\) −14.0000 −0.449513
\(971\) 48.0000 1.54039 0.770197 − 0.637806i \(-0.220158\pi\)
0.770197 + 0.637806i \(0.220158\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 16.0000 0.512673
\(975\) 0 0
\(976\) 1.00000 0.0320092
\(977\) 30.0000 0.959785 0.479893 − 0.877327i \(-0.340676\pi\)
0.479893 + 0.877327i \(0.340676\pi\)
\(978\) 0 0
\(979\) −2.00000 −0.0639203
\(980\) 0 0
\(981\) 0 0
\(982\) −12.0000 −0.382935
\(983\) −17.0000 −0.542216 −0.271108 − 0.962549i \(-0.587390\pi\)
−0.271108 + 0.962549i \(0.587390\pi\)
\(984\) 0 0
\(985\) −2.00000 −0.0637253
\(986\) 36.0000 1.14647
\(987\) 0 0
\(988\) 0 0
\(989\) 15.0000 0.476972
\(990\) 0 0
\(991\) 40.0000 1.27064 0.635321 − 0.772248i \(-0.280868\pi\)
0.635321 + 0.772248i \(0.280868\pi\)
\(992\) 4.00000 0.127000
\(993\) 0 0
\(994\) 0 0
\(995\) −4.00000 −0.126809
\(996\) 0 0
\(997\) −46.0000 −1.45683 −0.728417 − 0.685134i \(-0.759744\pi\)
−0.728417 + 0.685134i \(0.759744\pi\)
\(998\) 18.0000 0.569780
\(999\) 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 4410.2.a.r.1.1 1
3.2 odd 2 490.2.a.k.1.1 1
7.2 even 3 630.2.k.f.361.1 2
7.4 even 3 630.2.k.f.541.1 2
7.6 odd 2 4410.2.a.h.1.1 1
12.11 even 2 3920.2.a.b.1.1 1
15.2 even 4 2450.2.c.s.99.2 2
15.8 even 4 2450.2.c.s.99.1 2
15.14 odd 2 2450.2.a.b.1.1 1
21.2 odd 6 70.2.e.a.11.1 ✓ 2
21.5 even 6 490.2.e.f.361.1 2
21.11 odd 6 70.2.e.a.51.1 yes 2
21.17 even 6 490.2.e.f.471.1 2
21.20 even 2 490.2.a.e.1.1 1
84.11 even 6 560.2.q.i.401.1 2
84.23 even 6 560.2.q.i.81.1 2
84.83 odd 2 3920.2.a.bk.1.1 1
105.2 even 12 350.2.j.f.249.2 4
105.23 even 12 350.2.j.f.249.1 4
105.32 even 12 350.2.j.f.149.1 4
105.44 odd 6 350.2.e.l.151.1 2
105.53 even 12 350.2.j.f.149.2 4
105.62 odd 4 2450.2.c.a.99.2 2
105.74 odd 6 350.2.e.l.51.1 2
105.83 odd 4 2450.2.c.a.99.1 2
105.104 even 2 2450.2.a.q.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
70.2.e.a.11.1 ✓ 2 21.2 odd 6
70.2.e.a.51.1 yes 2 21.11 odd 6
350.2.e.l.51.1 2 105.74 odd 6
350.2.e.l.151.1 2 105.44 odd 6
350.2.j.f.149.1 4 105.32 even 12
350.2.j.f.149.2 4 105.53 even 12
350.2.j.f.249.1 4 105.23 even 12
350.2.j.f.249.2 4 105.2 even 12
490.2.a.e.1.1 1 21.20 even 2
490.2.a.k.1.1 1 3.2 odd 2
490.2.e.f.361.1 2 21.5 even 6
490.2.e.f.471.1 2 21.17 even 6
560.2.q.i.81.1 2 84.23 even 6
560.2.q.i.401.1 2 84.11 even 6
630.2.k.f.361.1 2 7.2 even 3
630.2.k.f.541.1 2 7.4 even 3
2450.2.a.b.1.1 1 15.14 odd 2
2450.2.a.q.1.1 1 105.104 even 2
2450.2.c.a.99.1 2 105.83 odd 4
2450.2.c.a.99.2 2 105.62 odd 4
2450.2.c.s.99.1 2 15.8 even 4
2450.2.c.s.99.2 2 15.2 even 4
3920.2.a.b.1.1 1 12.11 even 2
3920.2.a.bk.1.1 1 84.83 odd 2
4410.2.a.h.1.1 1 7.6 odd 2
4410.2.a.r.1.1 1 1.1 even 1 trivial