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Properties

Label 2790.2.a.k.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)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} +2.00000 q^{7} -1.00000 q^{8} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} +2.00000 q^{7} -1.00000 q^{8} -1.00000 q^{10} -4.00000 q^{13} -2.00000 q^{14} +1.00000 q^{16} -6.00000 q^{17} +8.00000 q^{19} +1.00000 q^{20} +1.00000 q^{25} +4.00000 q^{26} +2.00000 q^{28} +1.00000 q^{31} -1.00000 q^{32} +6.00000 q^{34} +2.00000 q^{35} -4.00000 q^{37} -8.00000 q^{38} -1.00000 q^{40} +6.00000 q^{41} +8.00000 q^{43} +12.0000 q^{47} -3.00000 q^{49} -1.00000 q^{50} -4.00000 q^{52} +6.00000 q^{53} -2.00000 q^{56} +6.00000 q^{59} +2.00000 q^{61} -1.00000 q^{62} +1.00000 q^{64} -4.00000 q^{65} +2.00000 q^{67} -6.00000 q^{68} -2.00000 q^{70} +6.00000 q^{71} +8.00000 q^{73} +4.00000 q^{74} +8.00000 q^{76} +8.00000 q^{79} +1.00000 q^{80} -6.00000 q^{82} -12.0000 q^{83} -6.00000 q^{85} -8.00000 q^{86} -8.00000 q^{91} -12.0000 q^{94} +8.00000 q^{95} -10.0000 q^{97} +3.00000 q^{98} +O(q^{100})\)

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
\(7\) 2.00000 0.755929 0.377964 − 0.925820i \(-0.376624\pi\)
0.377964 + 0.925820i \(0.376624\pi\)
\(8\) −1.00000 −0.353553
\(9\) 0 0
\(10\) −1.00000 −0.316228
\(11\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) 0 0
\(13\) −4.00000 −1.10940 −0.554700 − 0.832050i \(-0.687167\pi\)
−0.554700 + 0.832050i \(0.687167\pi\)
\(14\) −2.00000 −0.534522
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −6.00000 −1.45521 −0.727607 − 0.685994i \(-0.759367\pi\)
−0.727607 + 0.685994i \(0.759367\pi\)
\(18\) 0 0
\(19\) 8.00000 1.83533 0.917663 − 0.397360i \(-0.130073\pi\)
0.917663 + 0.397360i \(0.130073\pi\)
\(20\) 1.00000 0.223607
\(21\) 0 0
\(22\) 0 0
\(23\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 4.00000 0.784465
\(27\) 0 0
\(28\) 2.00000 0.377964
\(29\) 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
\(34\) 6.00000 1.02899
\(35\) 2.00000 0.338062
\(36\) 0 0
\(37\) −4.00000 −0.657596 −0.328798 − 0.944400i \(-0.606644\pi\)
−0.328798 + 0.944400i \(0.606644\pi\)
\(38\) −8.00000 −1.29777
\(39\) 0 0
\(40\) −1.00000 −0.158114
\(41\) 6.00000 0.937043 0.468521 − 0.883452i \(-0.344787\pi\)
0.468521 + 0.883452i \(0.344787\pi\)
\(42\) 0 0
\(43\) 8.00000 1.21999 0.609994 − 0.792406i \(-0.291172\pi\)
0.609994 + 0.792406i \(0.291172\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 12.0000 1.75038 0.875190 − 0.483779i \(-0.160736\pi\)
0.875190 + 0.483779i \(0.160736\pi\)
\(48\) 0 0
\(49\) −3.00000 −0.428571
\(50\) −1.00000 −0.141421
\(51\) 0 0
\(52\) −4.00000 −0.554700
\(53\) 6.00000 0.824163 0.412082 − 0.911147i \(-0.364802\pi\)
0.412082 + 0.911147i \(0.364802\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −2.00000 −0.267261
\(57\) 0 0
\(58\) 0 0
\(59\) 6.00000 0.781133 0.390567 − 0.920575i \(-0.372279\pi\)
0.390567 + 0.920575i \(0.372279\pi\)
\(60\) 0 0
\(61\) 2.00000 0.256074 0.128037 − 0.991769i \(-0.459132\pi\)
0.128037 + 0.991769i \(0.459132\pi\)
\(62\) −1.00000 −0.127000
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) −4.00000 −0.496139
\(66\) 0 0
\(67\) 2.00000 0.244339 0.122169 − 0.992509i \(-0.461015\pi\)
0.122169 + 0.992509i \(0.461015\pi\)
\(68\) −6.00000 −0.727607
\(69\) 0 0
\(70\) −2.00000 −0.239046
\(71\) 6.00000 0.712069 0.356034 − 0.934473i \(-0.384129\pi\)
0.356034 + 0.934473i \(0.384129\pi\)
\(72\) 0 0
\(73\) 8.00000 0.936329 0.468165 − 0.883641i \(-0.344915\pi\)
0.468165 + 0.883641i \(0.344915\pi\)
\(74\) 4.00000 0.464991
\(75\) 0 0
\(76\) 8.00000 0.917663
\(77\) 0 0
\(78\) 0 0
\(79\) 8.00000 0.900070 0.450035 − 0.893011i \(-0.351411\pi\)
0.450035 + 0.893011i \(0.351411\pi\)
\(80\) 1.00000 0.111803
\(81\) 0 0
\(82\) −6.00000 −0.662589
\(83\) −12.0000 −1.31717 −0.658586 − 0.752506i \(-0.728845\pi\)
−0.658586 + 0.752506i \(0.728845\pi\)
\(84\) 0 0
\(85\) −6.00000 −0.650791
\(86\) −8.00000 −0.862662
\(87\) 0 0
\(88\) 0 0
\(89\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(90\) 0 0
\(91\) −8.00000 −0.838628
\(92\) 0 0
\(93\) 0 0
\(94\) −12.0000 −1.23771
\(95\) 8.00000 0.820783
\(96\) 0 0
\(97\) −10.0000 −1.01535 −0.507673 − 0.861550i \(-0.669494\pi\)
−0.507673 + 0.861550i \(0.669494\pi\)
\(98\) 3.00000 0.303046
\(99\) 0 0
\(100\) 1.00000 0.100000
\(101\) 6.00000 0.597022 0.298511 − 0.954406i \(-0.403510\pi\)
0.298511 + 0.954406i \(0.403510\pi\)
\(102\) 0 0
\(103\) −10.0000 −0.985329 −0.492665 − 0.870219i \(-0.663977\pi\)
−0.492665 + 0.870219i \(0.663977\pi\)
\(104\) 4.00000 0.392232
\(105\) 0 0
\(106\) −6.00000 −0.582772
\(107\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(108\) 0 0
\(109\) 2.00000 0.191565 0.0957826 − 0.995402i \(-0.469465\pi\)
0.0957826 + 0.995402i \(0.469465\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 2.00000 0.188982
\(113\) 18.0000 1.69330 0.846649 − 0.532152i \(-0.178617\pi\)
0.846649 + 0.532152i \(0.178617\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 0 0
\(118\) −6.00000 −0.552345
\(119\) −12.0000 −1.10004
\(120\) 0 0
\(121\) −11.0000 −1.00000
\(122\) −2.00000 −0.181071
\(123\) 0 0
\(124\) 1.00000 0.0898027
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) −16.0000 −1.41977 −0.709885 − 0.704317i \(-0.751253\pi\)
−0.709885 + 0.704317i \(0.751253\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 0 0
\(130\) 4.00000 0.350823
\(131\) 6.00000 0.524222 0.262111 − 0.965038i \(-0.415581\pi\)
0.262111 + 0.965038i \(0.415581\pi\)
\(132\) 0 0
\(133\) 16.0000 1.38738
\(134\) −2.00000 −0.172774
\(135\) 0 0
\(136\) 6.00000 0.514496
\(137\) 6.00000 0.512615 0.256307 − 0.966595i \(-0.417494\pi\)
0.256307 + 0.966595i \(0.417494\pi\)
\(138\) 0 0
\(139\) −4.00000 −0.339276 −0.169638 − 0.985506i \(-0.554260\pi\)
−0.169638 + 0.985506i \(0.554260\pi\)
\(140\) 2.00000 0.169031
\(141\) 0 0
\(142\) −6.00000 −0.503509
\(143\) 0 0
\(144\) 0 0
\(145\) 0 0
\(146\) −8.00000 −0.662085
\(147\) 0 0
\(148\) −4.00000 −0.328798
\(149\) −18.0000 −1.47462 −0.737309 − 0.675556i \(-0.763904\pi\)
−0.737309 + 0.675556i \(0.763904\pi\)
\(150\) 0 0
\(151\) 8.00000 0.651031 0.325515 − 0.945537i \(-0.394462\pi\)
0.325515 + 0.945537i \(0.394462\pi\)
\(152\) −8.00000 −0.648886
\(153\) 0 0
\(154\) 0 0
\(155\) 1.00000 0.0803219
\(156\) 0 0
\(157\) 14.0000 1.11732 0.558661 − 0.829396i \(-0.311315\pi\)
0.558661 + 0.829396i \(0.311315\pi\)
\(158\) −8.00000 −0.636446
\(159\) 0 0
\(160\) −1.00000 −0.0790569
\(161\) 0 0
\(162\) 0 0
\(163\) 2.00000 0.156652 0.0783260 − 0.996928i \(-0.475042\pi\)
0.0783260 + 0.996928i \(0.475042\pi\)
\(164\) 6.00000 0.468521
\(165\) 0 0
\(166\) 12.0000 0.931381
\(167\) 24.0000 1.85718 0.928588 − 0.371113i \(-0.121024\pi\)
0.928588 + 0.371113i \(0.121024\pi\)
\(168\) 0 0
\(169\) 3.00000 0.230769
\(170\) 6.00000 0.460179
\(171\) 0 0
\(172\) 8.00000 0.609994
\(173\) 6.00000 0.456172 0.228086 − 0.973641i \(-0.426753\pi\)
0.228086 + 0.973641i \(0.426753\pi\)
\(174\) 0 0
\(175\) 2.00000 0.151186
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(180\) 0 0
\(181\) 2.00000 0.148659 0.0743294 − 0.997234i \(-0.476318\pi\)
0.0743294 + 0.997234i \(0.476318\pi\)
\(182\) 8.00000 0.592999
\(183\) 0 0
\(184\) 0 0
\(185\) −4.00000 −0.294086
\(186\) 0 0
\(187\) 0 0
\(188\) 12.0000 0.875190
\(189\) 0 0
\(190\) −8.00000 −0.580381
\(191\) 18.0000 1.30243 0.651217 − 0.758891i \(-0.274259\pi\)
0.651217 + 0.758891i \(0.274259\pi\)
\(192\) 0 0
\(193\) 14.0000 1.00774 0.503871 − 0.863779i \(-0.331909\pi\)
0.503871 + 0.863779i \(0.331909\pi\)
\(194\) 10.0000 0.717958
\(195\) 0 0
\(196\) −3.00000 −0.214286
\(197\) −6.00000 −0.427482 −0.213741 − 0.976890i \(-0.568565\pi\)
−0.213741 + 0.976890i \(0.568565\pi\)
\(198\) 0 0
\(199\) 8.00000 0.567105 0.283552 − 0.958957i \(-0.408487\pi\)
0.283552 + 0.958957i \(0.408487\pi\)
\(200\) −1.00000 −0.0707107
\(201\) 0 0
\(202\) −6.00000 −0.422159
\(203\) 0 0
\(204\) 0 0
\(205\) 6.00000 0.419058
\(206\) 10.0000 0.696733
\(207\) 0 0
\(208\) −4.00000 −0.277350
\(209\) 0 0
\(210\) 0 0
\(211\) 8.00000 0.550743 0.275371 − 0.961338i \(-0.411199\pi\)
0.275371 + 0.961338i \(0.411199\pi\)
\(212\) 6.00000 0.412082
\(213\) 0 0
\(214\) 0 0
\(215\) 8.00000 0.545595
\(216\) 0 0
\(217\) 2.00000 0.135769
\(218\) −2.00000 −0.135457
\(219\) 0 0
\(220\) 0 0
\(221\) 24.0000 1.61441
\(222\) 0 0
\(223\) −28.0000 −1.87502 −0.937509 − 0.347960i \(-0.886874\pi\)
−0.937509 + 0.347960i \(0.886874\pi\)
\(224\) −2.00000 −0.133631
\(225\) 0 0
\(226\) −18.0000 −1.19734
\(227\) −12.0000 −0.796468 −0.398234 − 0.917284i \(-0.630377\pi\)
−0.398234 + 0.917284i \(0.630377\pi\)
\(228\) 0 0
\(229\) 14.0000 0.925146 0.462573 − 0.886581i \(-0.346926\pi\)
0.462573 + 0.886581i \(0.346926\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −18.0000 −1.17922 −0.589610 − 0.807688i \(-0.700718\pi\)
−0.589610 + 0.807688i \(0.700718\pi\)
\(234\) 0 0
\(235\) 12.0000 0.782794
\(236\) 6.00000 0.390567
\(237\) 0 0
\(238\) 12.0000 0.777844
\(239\) −12.0000 −0.776215 −0.388108 − 0.921614i \(-0.626871\pi\)
−0.388108 + 0.921614i \(0.626871\pi\)
\(240\) 0 0
\(241\) 2.00000 0.128831 0.0644157 − 0.997923i \(-0.479482\pi\)
0.0644157 + 0.997923i \(0.479482\pi\)
\(242\) 11.0000 0.707107
\(243\) 0 0
\(244\) 2.00000 0.128037
\(245\) −3.00000 −0.191663
\(246\) 0 0
\(247\) −32.0000 −2.03611
\(248\) −1.00000 −0.0635001
\(249\) 0 0
\(250\) −1.00000 −0.0632456
\(251\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 16.0000 1.00393
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 30.0000 1.87135 0.935674 − 0.352865i \(-0.114792\pi\)
0.935674 + 0.352865i \(0.114792\pi\)
\(258\) 0 0
\(259\) −8.00000 −0.497096
\(260\) −4.00000 −0.248069
\(261\) 0 0
\(262\) −6.00000 −0.370681
\(263\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(264\) 0 0
\(265\) 6.00000 0.368577
\(266\) −16.0000 −0.981023
\(267\) 0 0
\(268\) 2.00000 0.122169
\(269\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(270\) 0 0
\(271\) 32.0000 1.94386 0.971931 − 0.235267i \(-0.0755965\pi\)
0.971931 + 0.235267i \(0.0755965\pi\)
\(272\) −6.00000 −0.363803
\(273\) 0 0
\(274\) −6.00000 −0.362473
\(275\) 0 0
\(276\) 0 0
\(277\) 8.00000 0.480673 0.240337 − 0.970690i \(-0.422742\pi\)
0.240337 + 0.970690i \(0.422742\pi\)
\(278\) 4.00000 0.239904
\(279\) 0 0
\(280\) −2.00000 −0.119523
\(281\) 18.0000 1.07379 0.536895 − 0.843649i \(-0.319597\pi\)
0.536895 + 0.843649i \(0.319597\pi\)
\(282\) 0 0
\(283\) −22.0000 −1.30776 −0.653882 − 0.756596i \(-0.726861\pi\)
−0.653882 + 0.756596i \(0.726861\pi\)
\(284\) 6.00000 0.356034
\(285\) 0 0
\(286\) 0 0
\(287\) 12.0000 0.708338
\(288\) 0 0
\(289\) 19.0000 1.11765
\(290\) 0 0
\(291\) 0 0
\(292\) 8.00000 0.468165
\(293\) −6.00000 −0.350524 −0.175262 − 0.984522i \(-0.556077\pi\)
−0.175262 + 0.984522i \(0.556077\pi\)
\(294\) 0 0
\(295\) 6.00000 0.349334
\(296\) 4.00000 0.232495
\(297\) 0 0
\(298\) 18.0000 1.04271
\(299\) 0 0
\(300\) 0 0
\(301\) 16.0000 0.922225
\(302\) −8.00000 −0.460348
\(303\) 0 0
\(304\) 8.00000 0.458831
\(305\) 2.00000 0.114520
\(306\) 0 0
\(307\) −34.0000 −1.94048 −0.970241 − 0.242140i \(-0.922151\pi\)
−0.970241 + 0.242140i \(0.922151\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) −1.00000 −0.0567962
\(311\) 18.0000 1.02069 0.510343 − 0.859971i \(-0.329518\pi\)
0.510343 + 0.859971i \(0.329518\pi\)
\(312\) 0 0
\(313\) −16.0000 −0.904373 −0.452187 − 0.891923i \(-0.649356\pi\)
−0.452187 + 0.891923i \(0.649356\pi\)
\(314\) −14.0000 −0.790066
\(315\) 0 0
\(316\) 8.00000 0.450035
\(317\) −6.00000 −0.336994 −0.168497 − 0.985702i \(-0.553891\pi\)
−0.168497 + 0.985702i \(0.553891\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 1.00000 0.0559017
\(321\) 0 0
\(322\) 0 0
\(323\) −48.0000 −2.67079
\(324\) 0 0
\(325\) −4.00000 −0.221880
\(326\) −2.00000 −0.110770
\(327\) 0 0
\(328\) −6.00000 −0.331295
\(329\) 24.0000 1.32316
\(330\) 0 0
\(331\) −4.00000 −0.219860 −0.109930 − 0.993939i \(-0.535063\pi\)
−0.109930 + 0.993939i \(0.535063\pi\)
\(332\) −12.0000 −0.658586
\(333\) 0 0
\(334\) −24.0000 −1.31322
\(335\) 2.00000 0.109272
\(336\) 0 0
\(337\) 20.0000 1.08947 0.544735 − 0.838608i \(-0.316630\pi\)
0.544735 + 0.838608i \(0.316630\pi\)
\(338\) −3.00000 −0.163178
\(339\) 0 0
\(340\) −6.00000 −0.325396
\(341\) 0 0
\(342\) 0 0
\(343\) −20.0000 −1.07990
\(344\) −8.00000 −0.431331
\(345\) 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
\(349\) 2.00000 0.107058 0.0535288 − 0.998566i \(-0.482953\pi\)
0.0535288 + 0.998566i \(0.482953\pi\)
\(350\) −2.00000 −0.106904
\(351\) 0 0
\(352\) 0 0
\(353\) −30.0000 −1.59674 −0.798369 − 0.602168i \(-0.794304\pi\)
−0.798369 + 0.602168i \(0.794304\pi\)
\(354\) 0 0
\(355\) 6.00000 0.318447
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −30.0000 −1.58334 −0.791670 − 0.610949i \(-0.790788\pi\)
−0.791670 + 0.610949i \(0.790788\pi\)
\(360\) 0 0
\(361\) 45.0000 2.36842
\(362\) −2.00000 −0.105118
\(363\) 0 0
\(364\) −8.00000 −0.419314
\(365\) 8.00000 0.418739
\(366\) 0 0
\(367\) 20.0000 1.04399 0.521996 − 0.852948i \(-0.325188\pi\)
0.521996 + 0.852948i \(0.325188\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 4.00000 0.207950
\(371\) 12.0000 0.623009
\(372\) 0 0
\(373\) 38.0000 1.96757 0.983783 − 0.179364i \(-0.0574041\pi\)
0.983783 + 0.179364i \(0.0574041\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) −12.0000 −0.618853
\(377\) 0 0
\(378\) 0 0
\(379\) 32.0000 1.64373 0.821865 − 0.569683i \(-0.192934\pi\)
0.821865 + 0.569683i \(0.192934\pi\)
\(380\) 8.00000 0.410391
\(381\) 0 0
\(382\) −18.0000 −0.920960
\(383\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −14.0000 −0.712581
\(387\) 0 0
\(388\) −10.0000 −0.507673
\(389\) −24.0000 −1.21685 −0.608424 − 0.793612i \(-0.708198\pi\)
−0.608424 + 0.793612i \(0.708198\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 3.00000 0.151523
\(393\) 0 0
\(394\) 6.00000 0.302276
\(395\) 8.00000 0.402524
\(396\) 0 0
\(397\) 2.00000 0.100377 0.0501886 − 0.998740i \(-0.484018\pi\)
0.0501886 + 0.998740i \(0.484018\pi\)
\(398\) −8.00000 −0.401004
\(399\) 0 0
\(400\) 1.00000 0.0500000
\(401\) −36.0000 −1.79775 −0.898877 − 0.438201i \(-0.855616\pi\)
−0.898877 + 0.438201i \(0.855616\pi\)
\(402\) 0 0
\(403\) −4.00000 −0.199254
\(404\) 6.00000 0.298511
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 14.0000 0.692255 0.346128 − 0.938187i \(-0.387496\pi\)
0.346128 + 0.938187i \(0.387496\pi\)
\(410\) −6.00000 −0.296319
\(411\) 0 0
\(412\) −10.0000 −0.492665
\(413\) 12.0000 0.590481
\(414\) 0 0
\(415\) −12.0000 −0.589057
\(416\) 4.00000 0.196116
\(417\) 0 0
\(418\) 0 0
\(419\) −30.0000 −1.46560 −0.732798 − 0.680446i \(-0.761786\pi\)
−0.732798 + 0.680446i \(0.761786\pi\)
\(420\) 0 0
\(421\) 26.0000 1.26716 0.633581 − 0.773676i \(-0.281584\pi\)
0.633581 + 0.773676i \(0.281584\pi\)
\(422\) −8.00000 −0.389434
\(423\) 0 0
\(424\) −6.00000 −0.291386
\(425\) −6.00000 −0.291043
\(426\) 0 0
\(427\) 4.00000 0.193574
\(428\) 0 0
\(429\) 0 0
\(430\) −8.00000 −0.385794
\(431\) −6.00000 −0.289010 −0.144505 − 0.989504i \(-0.546159\pi\)
−0.144505 + 0.989504i \(0.546159\pi\)
\(432\) 0 0
\(433\) −16.0000 −0.768911 −0.384455 − 0.923144i \(-0.625611\pi\)
−0.384455 + 0.923144i \(0.625611\pi\)
\(434\) −2.00000 −0.0960031
\(435\) 0 0
\(436\) 2.00000 0.0957826
\(437\) 0 0
\(438\) 0 0
\(439\) 8.00000 0.381819 0.190910 − 0.981608i \(-0.438856\pi\)
0.190910 + 0.981608i \(0.438856\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) −24.0000 −1.14156
\(443\) 36.0000 1.71041 0.855206 − 0.518289i \(-0.173431\pi\)
0.855206 + 0.518289i \(0.173431\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 28.0000 1.32584
\(447\) 0 0
\(448\) 2.00000 0.0944911
\(449\) −12.0000 −0.566315 −0.283158 − 0.959073i \(-0.591382\pi\)
−0.283158 + 0.959073i \(0.591382\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 18.0000 0.846649
\(453\) 0 0
\(454\) 12.0000 0.563188
\(455\) −8.00000 −0.375046
\(456\) 0 0
\(457\) 8.00000 0.374224 0.187112 − 0.982339i \(-0.440087\pi\)
0.187112 + 0.982339i \(0.440087\pi\)
\(458\) −14.0000 −0.654177
\(459\) 0 0
\(460\) 0 0
\(461\) 36.0000 1.67669 0.838344 − 0.545142i \(-0.183524\pi\)
0.838344 + 0.545142i \(0.183524\pi\)
\(462\) 0 0
\(463\) −16.0000 −0.743583 −0.371792 − 0.928316i \(-0.621256\pi\)
−0.371792 + 0.928316i \(0.621256\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 18.0000 0.833834
\(467\) −24.0000 −1.11059 −0.555294 − 0.831654i \(-0.687394\pi\)
−0.555294 + 0.831654i \(0.687394\pi\)
\(468\) 0 0
\(469\) 4.00000 0.184703
\(470\) −12.0000 −0.553519
\(471\) 0 0
\(472\) −6.00000 −0.276172
\(473\) 0 0
\(474\) 0 0
\(475\) 8.00000 0.367065
\(476\) −12.0000 −0.550019
\(477\) 0 0
\(478\) 12.0000 0.548867
\(479\) 6.00000 0.274147 0.137073 − 0.990561i \(-0.456230\pi\)
0.137073 + 0.990561i \(0.456230\pi\)
\(480\) 0 0
\(481\) 16.0000 0.729537
\(482\) −2.00000 −0.0910975
\(483\) 0 0
\(484\) −11.0000 −0.500000
\(485\) −10.0000 −0.454077
\(486\) 0 0
\(487\) −40.0000 −1.81257 −0.906287 − 0.422664i \(-0.861095\pi\)
−0.906287 + 0.422664i \(0.861095\pi\)
\(488\) −2.00000 −0.0905357
\(489\) 0 0
\(490\) 3.00000 0.135526
\(491\) −12.0000 −0.541552 −0.270776 − 0.962642i \(-0.587280\pi\)
−0.270776 + 0.962642i \(0.587280\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 32.0000 1.43975
\(495\) 0 0
\(496\) 1.00000 0.0449013
\(497\) 12.0000 0.538274
\(498\) 0 0
\(499\) −4.00000 −0.179065 −0.0895323 − 0.995984i \(-0.528537\pi\)
−0.0895323 + 0.995984i \(0.528537\pi\)
\(500\) 1.00000 0.0447214
\(501\) 0 0
\(502\) 0 0
\(503\) −36.0000 −1.60516 −0.802580 − 0.596544i \(-0.796540\pi\)
−0.802580 + 0.596544i \(0.796540\pi\)
\(504\) 0 0
\(505\) 6.00000 0.266996
\(506\) 0 0
\(507\) 0 0
\(508\) −16.0000 −0.709885
\(509\) −12.0000 −0.531891 −0.265945 − 0.963988i \(-0.585684\pi\)
−0.265945 + 0.963988i \(0.585684\pi\)
\(510\) 0 0
\(511\) 16.0000 0.707798
\(512\) −1.00000 −0.0441942
\(513\) 0 0
\(514\) −30.0000 −1.32324
\(515\) −10.0000 −0.440653
\(516\) 0 0
\(517\) 0 0
\(518\) 8.00000 0.351500
\(519\) 0 0
\(520\) 4.00000 0.175412
\(521\) 6.00000 0.262865 0.131432 − 0.991325i \(-0.458042\pi\)
0.131432 + 0.991325i \(0.458042\pi\)
\(522\) 0 0
\(523\) −40.0000 −1.74908 −0.874539 − 0.484955i \(-0.838836\pi\)
−0.874539 + 0.484955i \(0.838836\pi\)
\(524\) 6.00000 0.262111
\(525\) 0 0
\(526\) 0 0
\(527\) −6.00000 −0.261364
\(528\) 0 0
\(529\) −23.0000 −1.00000
\(530\) −6.00000 −0.260623
\(531\) 0 0
\(532\) 16.0000 0.693688
\(533\) −24.0000 −1.03956
\(534\) 0 0
\(535\) 0 0
\(536\) −2.00000 −0.0863868
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 2.00000 0.0859867 0.0429934 − 0.999075i \(-0.486311\pi\)
0.0429934 + 0.999075i \(0.486311\pi\)
\(542\) −32.0000 −1.37452
\(543\) 0 0
\(544\) 6.00000 0.257248
\(545\) 2.00000 0.0856706
\(546\) 0 0
\(547\) 2.00000 0.0855138 0.0427569 − 0.999086i \(-0.486386\pi\)
0.0427569 + 0.999086i \(0.486386\pi\)
\(548\) 6.00000 0.256307
\(549\) 0 0
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 16.0000 0.680389
\(554\) −8.00000 −0.339887
\(555\) 0 0
\(556\) −4.00000 −0.169638
\(557\) −42.0000 −1.77960 −0.889799 − 0.456354i \(-0.849155\pi\)
−0.889799 + 0.456354i \(0.849155\pi\)
\(558\) 0 0
\(559\) −32.0000 −1.35346
\(560\) 2.00000 0.0845154
\(561\) 0 0
\(562\) −18.0000 −0.759284
\(563\) 24.0000 1.01148 0.505740 − 0.862686i \(-0.331220\pi\)
0.505740 + 0.862686i \(0.331220\pi\)
\(564\) 0 0
\(565\) 18.0000 0.757266
\(566\) 22.0000 0.924729
\(567\) 0 0
\(568\) −6.00000 −0.251754
\(569\) −36.0000 −1.50920 −0.754599 − 0.656186i \(-0.772169\pi\)
−0.754599 + 0.656186i \(0.772169\pi\)
\(570\) 0 0
\(571\) 20.0000 0.836974 0.418487 − 0.908223i \(-0.362561\pi\)
0.418487 + 0.908223i \(0.362561\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) −12.0000 −0.500870
\(575\) 0 0
\(576\) 0 0
\(577\) −22.0000 −0.915872 −0.457936 − 0.888985i \(-0.651411\pi\)
−0.457936 + 0.888985i \(0.651411\pi\)
\(578\) −19.0000 −0.790296
\(579\) 0 0
\(580\) 0 0
\(581\) −24.0000 −0.995688
\(582\) 0 0
\(583\) 0 0
\(584\) −8.00000 −0.331042
\(585\) 0 0
\(586\) 6.00000 0.247858
\(587\) 12.0000 0.495293 0.247647 − 0.968850i \(-0.420343\pi\)
0.247647 + 0.968850i \(0.420343\pi\)
\(588\) 0 0
\(589\) 8.00000 0.329634
\(590\) −6.00000 −0.247016
\(591\) 0 0
\(592\) −4.00000 −0.164399
\(593\) −6.00000 −0.246390 −0.123195 − 0.992382i \(-0.539314\pi\)
−0.123195 + 0.992382i \(0.539314\pi\)
\(594\) 0 0
\(595\) −12.0000 −0.491952
\(596\) −18.0000 −0.737309
\(597\) 0 0
\(598\) 0 0
\(599\) −6.00000 −0.245153 −0.122577 − 0.992459i \(-0.539116\pi\)
−0.122577 + 0.992459i \(0.539116\pi\)
\(600\) 0 0
\(601\) −10.0000 −0.407909 −0.203954 − 0.978980i \(-0.565379\pi\)
−0.203954 + 0.978980i \(0.565379\pi\)
\(602\) −16.0000 −0.652111
\(603\) 0 0
\(604\) 8.00000 0.325515
\(605\) −11.0000 −0.447214
\(606\) 0 0
\(607\) −34.0000 −1.38002 −0.690009 − 0.723801i \(-0.742393\pi\)
−0.690009 + 0.723801i \(0.742393\pi\)
\(608\) −8.00000 −0.324443
\(609\) 0 0
\(610\) −2.00000 −0.0809776
\(611\) −48.0000 −1.94187
\(612\) 0 0
\(613\) −40.0000 −1.61558 −0.807792 − 0.589467i \(-0.799338\pi\)
−0.807792 + 0.589467i \(0.799338\pi\)
\(614\) 34.0000 1.37213
\(615\) 0 0
\(616\) 0 0
\(617\) −6.00000 −0.241551 −0.120775 − 0.992680i \(-0.538538\pi\)
−0.120775 + 0.992680i \(0.538538\pi\)
\(618\) 0 0
\(619\) −4.00000 −0.160774 −0.0803868 − 0.996764i \(-0.525616\pi\)
−0.0803868 + 0.996764i \(0.525616\pi\)
\(620\) 1.00000 0.0401610
\(621\) 0 0
\(622\) −18.0000 −0.721734
\(623\) 0 0
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 16.0000 0.639489
\(627\) 0 0
\(628\) 14.0000 0.558661
\(629\) 24.0000 0.956943
\(630\) 0 0
\(631\) 8.00000 0.318475 0.159237 − 0.987240i \(-0.449096\pi\)
0.159237 + 0.987240i \(0.449096\pi\)
\(632\) −8.00000 −0.318223
\(633\) 0 0
\(634\) 6.00000 0.238290
\(635\) −16.0000 −0.634941
\(636\) 0 0
\(637\) 12.0000 0.475457
\(638\) 0 0
\(639\) 0 0
\(640\) −1.00000 −0.0395285
\(641\) 12.0000 0.473972 0.236986 − 0.971513i \(-0.423841\pi\)
0.236986 + 0.971513i \(0.423841\pi\)
\(642\) 0 0
\(643\) 32.0000 1.26196 0.630978 − 0.775800i \(-0.282654\pi\)
0.630978 + 0.775800i \(0.282654\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 48.0000 1.88853
\(647\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 4.00000 0.156893
\(651\) 0 0
\(652\) 2.00000 0.0783260
\(653\) −6.00000 −0.234798 −0.117399 − 0.993085i \(-0.537456\pi\)
−0.117399 + 0.993085i \(0.537456\pi\)
\(654\) 0 0
\(655\) 6.00000 0.234439
\(656\) 6.00000 0.234261
\(657\) 0 0
\(658\) −24.0000 −0.935617
\(659\) −6.00000 −0.233727 −0.116863 − 0.993148i \(-0.537284\pi\)
−0.116863 + 0.993148i \(0.537284\pi\)
\(660\) 0 0
\(661\) −10.0000 −0.388955 −0.194477 − 0.980907i \(-0.562301\pi\)
−0.194477 + 0.980907i \(0.562301\pi\)
\(662\) 4.00000 0.155464
\(663\) 0 0
\(664\) 12.0000 0.465690
\(665\) 16.0000 0.620453
\(666\) 0 0
\(667\) 0 0
\(668\) 24.0000 0.928588
\(669\) 0 0
\(670\) −2.00000 −0.0772667
\(671\) 0 0
\(672\) 0 0
\(673\) 44.0000 1.69608 0.848038 − 0.529936i \(-0.177784\pi\)
0.848038 + 0.529936i \(0.177784\pi\)
\(674\) −20.0000 −0.770371
\(675\) 0 0
\(676\) 3.00000 0.115385
\(677\) −18.0000 −0.691796 −0.345898 − 0.938272i \(-0.612426\pi\)
−0.345898 + 0.938272i \(0.612426\pi\)
\(678\) 0 0
\(679\) −20.0000 −0.767530
\(680\) 6.00000 0.230089
\(681\) 0 0
\(682\) 0 0
\(683\) 36.0000 1.37750 0.688751 − 0.724998i \(-0.258159\pi\)
0.688751 + 0.724998i \(0.258159\pi\)
\(684\) 0 0
\(685\) 6.00000 0.229248
\(686\) 20.0000 0.763604
\(687\) 0 0
\(688\) 8.00000 0.304997
\(689\) −24.0000 −0.914327
\(690\) 0 0
\(691\) 44.0000 1.67384 0.836919 − 0.547326i \(-0.184354\pi\)
0.836919 + 0.547326i \(0.184354\pi\)
\(692\) 6.00000 0.228086
\(693\) 0 0
\(694\) −36.0000 −1.36654
\(695\) −4.00000 −0.151729
\(696\) 0 0
\(697\) −36.0000 −1.36360
\(698\) −2.00000 −0.0757011
\(699\) 0 0
\(700\) 2.00000 0.0755929
\(701\) −30.0000 −1.13308 −0.566542 − 0.824033i \(-0.691719\pi\)
−0.566542 + 0.824033i \(0.691719\pi\)
\(702\) 0 0
\(703\) −32.0000 −1.20690
\(704\) 0 0
\(705\) 0 0
\(706\) 30.0000 1.12906
\(707\) 12.0000 0.451306
\(708\) 0 0
\(709\) −10.0000 −0.375558 −0.187779 − 0.982211i \(-0.560129\pi\)
−0.187779 + 0.982211i \(0.560129\pi\)
\(710\) −6.00000 −0.225176
\(711\) 0 0
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 30.0000 1.11959
\(719\) 24.0000 0.895049 0.447524 − 0.894272i \(-0.352306\pi\)
0.447524 + 0.894272i \(0.352306\pi\)
\(720\) 0 0
\(721\) −20.0000 −0.744839
\(722\) −45.0000 −1.67473
\(723\) 0 0
\(724\) 2.00000 0.0743294
\(725\) 0 0
\(726\) 0 0
\(727\) 26.0000 0.964287 0.482143 − 0.876092i \(-0.339858\pi\)
0.482143 + 0.876092i \(0.339858\pi\)
\(728\) 8.00000 0.296500
\(729\) 0 0
\(730\) −8.00000 −0.296093
\(731\) −48.0000 −1.77534
\(732\) 0 0
\(733\) −22.0000 −0.812589 −0.406294 − 0.913742i \(-0.633179\pi\)
−0.406294 + 0.913742i \(0.633179\pi\)
\(734\) −20.0000 −0.738213
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) −52.0000 −1.91285 −0.956425 − 0.291977i \(-0.905687\pi\)
−0.956425 + 0.291977i \(0.905687\pi\)
\(740\) −4.00000 −0.147043
\(741\) 0 0
\(742\) −12.0000 −0.440534
\(743\) −24.0000 −0.880475 −0.440237 − 0.897881i \(-0.645106\pi\)
−0.440237 + 0.897881i \(0.645106\pi\)
\(744\) 0 0
\(745\) −18.0000 −0.659469
\(746\) −38.0000 −1.39128
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) −16.0000 −0.583848 −0.291924 − 0.956441i \(-0.594295\pi\)
−0.291924 + 0.956441i \(0.594295\pi\)
\(752\) 12.0000 0.437595
\(753\) 0 0
\(754\) 0 0
\(755\) 8.00000 0.291150
\(756\) 0 0
\(757\) −4.00000 −0.145382 −0.0726912 − 0.997354i \(-0.523159\pi\)
−0.0726912 + 0.997354i \(0.523159\pi\)
\(758\) −32.0000 −1.16229
\(759\) 0 0
\(760\) −8.00000 −0.290191
\(761\) −12.0000 −0.435000 −0.217500 − 0.976060i \(-0.569790\pi\)
−0.217500 + 0.976060i \(0.569790\pi\)
\(762\) 0 0
\(763\) 4.00000 0.144810
\(764\) 18.0000 0.651217
\(765\) 0 0
\(766\) 0 0
\(767\) −24.0000 −0.866590
\(768\) 0 0
\(769\) −34.0000 −1.22607 −0.613036 − 0.790055i \(-0.710052\pi\)
−0.613036 + 0.790055i \(0.710052\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 14.0000 0.503871
\(773\) 54.0000 1.94225 0.971123 − 0.238581i \(-0.0766824\pi\)
0.971123 + 0.238581i \(0.0766824\pi\)
\(774\) 0 0
\(775\) 1.00000 0.0359211
\(776\) 10.0000 0.358979
\(777\) 0 0
\(778\) 24.0000 0.860442
\(779\) 48.0000 1.71978
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) −3.00000 −0.107143
\(785\) 14.0000 0.499681
\(786\) 0 0
\(787\) 20.0000 0.712923 0.356462 − 0.934310i \(-0.383983\pi\)
0.356462 + 0.934310i \(0.383983\pi\)
\(788\) −6.00000 −0.213741
\(789\) 0 0
\(790\) −8.00000 −0.284627
\(791\) 36.0000 1.28001
\(792\) 0 0
\(793\) −8.00000 −0.284088
\(794\) −2.00000 −0.0709773
\(795\) 0 0
\(796\) 8.00000 0.283552
\(797\) −6.00000 −0.212531 −0.106265 − 0.994338i \(-0.533889\pi\)
−0.106265 + 0.994338i \(0.533889\pi\)
\(798\) 0 0
\(799\) −72.0000 −2.54718
\(800\) −1.00000 −0.0353553
\(801\) 0 0
\(802\) 36.0000 1.27120
\(803\) 0 0
\(804\) 0 0
\(805\) 0 0
\(806\) 4.00000 0.140894
\(807\) 0 0
\(808\) −6.00000 −0.211079
\(809\) −36.0000 −1.26569 −0.632846 − 0.774277i \(-0.718114\pi\)
−0.632846 + 0.774277i \(0.718114\pi\)
\(810\) 0 0
\(811\) −16.0000 −0.561836 −0.280918 − 0.959732i \(-0.590639\pi\)
−0.280918 + 0.959732i \(0.590639\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 2.00000 0.0700569
\(816\) 0 0
\(817\) 64.0000 2.23908
\(818\) −14.0000 −0.489499
\(819\) 0 0
\(820\) 6.00000 0.209529
\(821\) 24.0000 0.837606 0.418803 − 0.908077i \(-0.362450\pi\)
0.418803 + 0.908077i \(0.362450\pi\)
\(822\) 0 0
\(823\) −16.0000 −0.557725 −0.278862 − 0.960331i \(-0.589957\pi\)
−0.278862 + 0.960331i \(0.589957\pi\)
\(824\) 10.0000 0.348367
\(825\) 0 0
\(826\) −12.0000 −0.417533
\(827\) 12.0000 0.417281 0.208640 − 0.977992i \(-0.433096\pi\)
0.208640 + 0.977992i \(0.433096\pi\)
\(828\) 0 0
\(829\) −22.0000 −0.764092 −0.382046 − 0.924143i \(-0.624780\pi\)
−0.382046 + 0.924143i \(0.624780\pi\)
\(830\) 12.0000 0.416526
\(831\) 0 0
\(832\) −4.00000 −0.138675
\(833\) 18.0000 0.623663
\(834\) 0 0
\(835\) 24.0000 0.830554
\(836\) 0 0
\(837\) 0 0
\(838\) 30.0000 1.03633
\(839\) 18.0000 0.621429 0.310715 − 0.950503i \(-0.399432\pi\)
0.310715 + 0.950503i \(0.399432\pi\)
\(840\) 0 0
\(841\) −29.0000 −1.00000
\(842\) −26.0000 −0.896019
\(843\) 0 0
\(844\) 8.00000 0.275371
\(845\) 3.00000 0.103203
\(846\) 0 0
\(847\) −22.0000 −0.755929
\(848\) 6.00000 0.206041
\(849\) 0 0
\(850\) 6.00000 0.205798
\(851\) 0 0
\(852\) 0 0
\(853\) 2.00000 0.0684787 0.0342393 − 0.999414i \(-0.489099\pi\)
0.0342393 + 0.999414i \(0.489099\pi\)
\(854\) −4.00000 −0.136877
\(855\) 0 0
\(856\) 0 0
\(857\) −54.0000 −1.84460 −0.922302 − 0.386469i \(-0.873695\pi\)
−0.922302 + 0.386469i \(0.873695\pi\)
\(858\) 0 0
\(859\) 20.0000 0.682391 0.341196 − 0.939992i \(-0.389168\pi\)
0.341196 + 0.939992i \(0.389168\pi\)
\(860\) 8.00000 0.272798
\(861\) 0 0
\(862\) 6.00000 0.204361
\(863\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(864\) 0 0
\(865\) 6.00000 0.204006
\(866\) 16.0000 0.543702
\(867\) 0 0
\(868\) 2.00000 0.0678844
\(869\) 0 0
\(870\) 0 0
\(871\) −8.00000 −0.271070
\(872\) −2.00000 −0.0677285
\(873\) 0 0
\(874\) 0 0
\(875\) 2.00000 0.0676123
\(876\) 0 0
\(877\) −22.0000 −0.742887 −0.371444 − 0.928456i \(-0.621137\pi\)
−0.371444 + 0.928456i \(0.621137\pi\)
\(878\) −8.00000 −0.269987
\(879\) 0 0
\(880\) 0 0
\(881\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(882\) 0 0
\(883\) 44.0000 1.48072 0.740359 − 0.672212i \(-0.234656\pi\)
0.740359 + 0.672212i \(0.234656\pi\)
\(884\) 24.0000 0.807207
\(885\) 0 0
\(886\) −36.0000 −1.20944
\(887\) −24.0000 −0.805841 −0.402921 − 0.915235i \(-0.632005\pi\)
−0.402921 + 0.915235i \(0.632005\pi\)
\(888\) 0 0
\(889\) −32.0000 −1.07325
\(890\) 0 0
\(891\) 0 0
\(892\) −28.0000 −0.937509
\(893\) 96.0000 3.21252
\(894\) 0 0
\(895\) 0 0
\(896\) −2.00000 −0.0668153
\(897\) 0 0
\(898\) 12.0000 0.400445
\(899\) 0 0
\(900\) 0 0
\(901\) −36.0000 −1.19933
\(902\) 0 0
\(903\) 0 0
\(904\) −18.0000 −0.598671
\(905\) 2.00000 0.0664822
\(906\) 0 0
\(907\) −10.0000 −0.332045 −0.166022 − 0.986122i \(-0.553092\pi\)
−0.166022 + 0.986122i \(0.553092\pi\)
\(908\) −12.0000 −0.398234
\(909\) 0 0
\(910\) 8.00000 0.265197
\(911\) −24.0000 −0.795155 −0.397578 − 0.917568i \(-0.630149\pi\)
−0.397578 + 0.917568i \(0.630149\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) −8.00000 −0.264616
\(915\) 0 0
\(916\) 14.0000 0.462573
\(917\) 12.0000 0.396275
\(918\) 0 0
\(919\) −52.0000 −1.71532 −0.857661 − 0.514216i \(-0.828083\pi\)
−0.857661 + 0.514216i \(0.828083\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) −36.0000 −1.18560
\(923\) −24.0000 −0.789970
\(924\) 0 0
\(925\) −4.00000 −0.131519
\(926\) 16.0000 0.525793
\(927\) 0 0
\(928\) 0 0
\(929\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(930\) 0 0
\(931\) −24.0000 −0.786568
\(932\) −18.0000 −0.589610
\(933\) 0 0
\(934\) 24.0000 0.785304
\(935\) 0 0
\(936\) 0 0
\(937\) 38.0000 1.24141 0.620703 − 0.784046i \(-0.286847\pi\)
0.620703 + 0.784046i \(0.286847\pi\)
\(938\) −4.00000 −0.130605
\(939\) 0 0
\(940\) 12.0000 0.391397
\(941\) 48.0000 1.56476 0.782378 − 0.622804i \(-0.214007\pi\)
0.782378 + 0.622804i \(0.214007\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 6.00000 0.195283
\(945\) 0 0
\(946\) 0 0
\(947\) −12.0000 −0.389948 −0.194974 − 0.980808i \(-0.562462\pi\)
−0.194974 + 0.980808i \(0.562462\pi\)
\(948\) 0 0
\(949\) −32.0000 −1.03876
\(950\) −8.00000 −0.259554
\(951\) 0 0
\(952\) 12.0000 0.388922
\(953\) −18.0000 −0.583077 −0.291539 − 0.956559i \(-0.594167\pi\)
−0.291539 + 0.956559i \(0.594167\pi\)
\(954\) 0 0
\(955\) 18.0000 0.582466
\(956\) −12.0000 −0.388108
\(957\) 0 0
\(958\) −6.00000 −0.193851
\(959\) 12.0000 0.387500
\(960\) 0 0
\(961\) 1.00000 0.0322581
\(962\) −16.0000 −0.515861
\(963\) 0 0
\(964\) 2.00000 0.0644157
\(965\) 14.0000 0.450676
\(966\) 0 0
\(967\) −4.00000 −0.128631 −0.0643157 − 0.997930i \(-0.520486\pi\)
−0.0643157 + 0.997930i \(0.520486\pi\)
\(968\) 11.0000 0.353553
\(969\) 0 0
\(970\) 10.0000 0.321081
\(971\) −6.00000 −0.192549 −0.0962746 − 0.995355i \(-0.530693\pi\)
−0.0962746 + 0.995355i \(0.530693\pi\)
\(972\) 0 0
\(973\) −8.00000 −0.256468
\(974\) 40.0000 1.28168
\(975\) 0 0
\(976\) 2.00000 0.0640184
\(977\) 18.0000 0.575871 0.287936 − 0.957650i \(-0.407031\pi\)
0.287936 + 0.957650i \(0.407031\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) −3.00000 −0.0958315
\(981\) 0 0
\(982\) 12.0000 0.382935
\(983\) −24.0000 −0.765481 −0.382741 − 0.923856i \(-0.625020\pi\)
−0.382741 + 0.923856i \(0.625020\pi\)
\(984\) 0 0
\(985\) −6.00000 −0.191176
\(986\) 0 0
\(987\) 0 0
\(988\) −32.0000 −1.01806
\(989\) 0 0
\(990\) 0 0
\(991\) −16.0000 −0.508257 −0.254128 − 0.967170i \(-0.581789\pi\)
−0.254128 + 0.967170i \(0.581789\pi\)
\(992\) −1.00000 −0.0317500
\(993\) 0 0
\(994\) −12.0000 −0.380617
\(995\) 8.00000 0.253617
\(996\) 0 0
\(997\) −10.0000 −0.316703 −0.158352 − 0.987383i \(-0.550618\pi\)
−0.158352 + 0.987383i \(0.550618\pi\)
\(998\) 4.00000 0.126618
\(999\) 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.k.1.1 1
3.2 odd 2 930.2.a.n.1.1 ✓ 1
12.11 even 2 7440.2.a.b.1.1 1
15.2 even 4 4650.2.d.u.3349.2 2
15.8 even 4 4650.2.d.u.3349.1 2
15.14 odd 2 4650.2.a.d.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
930.2.a.n.1.1 ✓ 1 3.2 odd 2
2790.2.a.k.1.1 1 1.1 even 1 trivial
4650.2.a.d.1.1 1 15.14 odd 2
4650.2.d.u.3349.1 2 15.8 even 4
4650.2.d.u.3349.2 2 15.2 even 4
7440.2.a.b.1.1 1 12.11 even 2