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Properties

Label 630.2.a.b.1.1
Level $630$
Weight $2$
Character 630.1
Self dual yes
Analytic conductor $5.031$
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] = [630,2,Mod(1,630)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(630, 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("630.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 630 = 2 \cdot 3^{2} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 630.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: \(5.03057532734\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 210)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 630.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{7} -1.00000 q^{8} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{10} -4.00000 q^{11} -2.00000 q^{13} -1.00000 q^{14} +1.00000 q^{16} -2.00000 q^{17} -4.00000 q^{19} -1.00000 q^{20} +4.00000 q^{22} +8.00000 q^{23} +1.00000 q^{25} +2.00000 q^{26} +1.00000 q^{28} -6.00000 q^{29} -8.00000 q^{31} -1.00000 q^{32} +2.00000 q^{34} -1.00000 q^{35} -2.00000 q^{37} +4.00000 q^{38} +1.00000 q^{40} -2.00000 q^{41} -12.0000 q^{43} -4.00000 q^{44} -8.00000 q^{46} +8.00000 q^{47} +1.00000 q^{49} -1.00000 q^{50} -2.00000 q^{52} -6.00000 q^{53} +4.00000 q^{55} -1.00000 q^{56} +6.00000 q^{58} -4.00000 q^{59} -2.00000 q^{61} +8.00000 q^{62} +1.00000 q^{64} +2.00000 q^{65} +12.0000 q^{67} -2.00000 q^{68} +1.00000 q^{70} -8.00000 q^{71} -14.0000 q^{73} +2.00000 q^{74} -4.00000 q^{76} -4.00000 q^{77} -1.00000 q^{80} +2.00000 q^{82} -12.0000 q^{83} +2.00000 q^{85} +12.0000 q^{86} +4.00000 q^{88} -2.00000 q^{89} -2.00000 q^{91} +8.00000 q^{92} -8.00000 q^{94} +4.00000 q^{95} +10.0000 q^{97} -1.00000 q^{98} +O(q^{100})\)

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
\(8\) −1.00000 −0.353553
\(9\) 0 0
\(10\) 1.00000 0.316228
\(11\) −4.00000 −1.20605 −0.603023 − 0.797724i \(-0.706037\pi\)
−0.603023 + 0.797724i \(0.706037\pi\)
\(12\) 0 0
\(13\) −2.00000 −0.554700 −0.277350 − 0.960769i \(-0.589456\pi\)
−0.277350 + 0.960769i \(0.589456\pi\)
\(14\) −1.00000 −0.267261
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −2.00000 −0.485071 −0.242536 − 0.970143i \(-0.577979\pi\)
−0.242536 + 0.970143i \(0.577979\pi\)
\(18\) 0 0
\(19\) −4.00000 −0.917663 −0.458831 − 0.888523i \(-0.651732\pi\)
−0.458831 + 0.888523i \(0.651732\pi\)
\(20\) −1.00000 −0.223607
\(21\) 0 0
\(22\) 4.00000 0.852803
\(23\) 8.00000 1.66812 0.834058 − 0.551677i \(-0.186012\pi\)
0.834058 + 0.551677i \(0.186012\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 2.00000 0.392232
\(27\) 0 0
\(28\) 1.00000 0.188982
\(29\) −6.00000 −1.11417 −0.557086 − 0.830455i \(-0.688081\pi\)
−0.557086 + 0.830455i \(0.688081\pi\)
\(30\) 0 0
\(31\) −8.00000 −1.43684 −0.718421 − 0.695608i \(-0.755135\pi\)
−0.718421 + 0.695608i \(0.755135\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0 0
\(34\) 2.00000 0.342997
\(35\) −1.00000 −0.169031
\(36\) 0 0
\(37\) −2.00000 −0.328798 −0.164399 − 0.986394i \(-0.552568\pi\)
−0.164399 + 0.986394i \(0.552568\pi\)
\(38\) 4.00000 0.648886
\(39\) 0 0
\(40\) 1.00000 0.158114
\(41\) −2.00000 −0.312348 −0.156174 − 0.987730i \(-0.549916\pi\)
−0.156174 + 0.987730i \(0.549916\pi\)
\(42\) 0 0
\(43\) −12.0000 −1.82998 −0.914991 − 0.403473i \(-0.867803\pi\)
−0.914991 + 0.403473i \(0.867803\pi\)
\(44\) −4.00000 −0.603023
\(45\) 0 0
\(46\) −8.00000 −1.17954
\(47\) 8.00000 1.16692 0.583460 − 0.812142i \(-0.301699\pi\)
0.583460 + 0.812142i \(0.301699\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) −1.00000 −0.141421
\(51\) 0 0
\(52\) −2.00000 −0.277350
\(53\) −6.00000 −0.824163 −0.412082 − 0.911147i \(-0.635198\pi\)
−0.412082 + 0.911147i \(0.635198\pi\)
\(54\) 0 0
\(55\) 4.00000 0.539360
\(56\) −1.00000 −0.133631
\(57\) 0 0
\(58\) 6.00000 0.787839
\(59\) −4.00000 −0.520756 −0.260378 − 0.965507i \(-0.583847\pi\)
−0.260378 + 0.965507i \(0.583847\pi\)
\(60\) 0 0
\(61\) −2.00000 −0.256074 −0.128037 − 0.991769i \(-0.540868\pi\)
−0.128037 + 0.991769i \(0.540868\pi\)
\(62\) 8.00000 1.01600
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 2.00000 0.248069
\(66\) 0 0
\(67\) 12.0000 1.46603 0.733017 − 0.680211i \(-0.238112\pi\)
0.733017 + 0.680211i \(0.238112\pi\)
\(68\) −2.00000 −0.242536
\(69\) 0 0
\(70\) 1.00000 0.119523
\(71\) −8.00000 −0.949425 −0.474713 − 0.880141i \(-0.657448\pi\)
−0.474713 + 0.880141i \(0.657448\pi\)
\(72\) 0 0
\(73\) −14.0000 −1.63858 −0.819288 − 0.573382i \(-0.805631\pi\)
−0.819288 + 0.573382i \(0.805631\pi\)
\(74\) 2.00000 0.232495
\(75\) 0 0
\(76\) −4.00000 −0.458831
\(77\) −4.00000 −0.455842
\(78\) 0 0
\(79\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(80\) −1.00000 −0.111803
\(81\) 0 0
\(82\) 2.00000 0.220863
\(83\) −12.0000 −1.31717 −0.658586 − 0.752506i \(-0.728845\pi\)
−0.658586 + 0.752506i \(0.728845\pi\)
\(84\) 0 0
\(85\) 2.00000 0.216930
\(86\) 12.0000 1.29399
\(87\) 0 0
\(88\) 4.00000 0.426401
\(89\) −2.00000 −0.212000 −0.106000 − 0.994366i \(-0.533804\pi\)
−0.106000 + 0.994366i \(0.533804\pi\)
\(90\) 0 0
\(91\) −2.00000 −0.209657
\(92\) 8.00000 0.834058
\(93\) 0 0
\(94\) −8.00000 −0.825137
\(95\) 4.00000 0.410391
\(96\) 0 0
\(97\) 10.0000 1.01535 0.507673 − 0.861550i \(-0.330506\pi\)
0.507673 + 0.861550i \(0.330506\pi\)
\(98\) −1.00000 −0.101015
\(99\) 0 0
\(100\) 1.00000 0.100000
\(101\) −6.00000 −0.597022 −0.298511 − 0.954406i \(-0.596490\pi\)
−0.298511 + 0.954406i \(0.596490\pi\)
\(102\) 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
\(106\) 6.00000 0.582772
\(107\) 20.0000 1.93347 0.966736 − 0.255774i \(-0.0823304\pi\)
0.966736 + 0.255774i \(0.0823304\pi\)
\(108\) 0 0
\(109\) −2.00000 −0.191565 −0.0957826 − 0.995402i \(-0.530535\pi\)
−0.0957826 + 0.995402i \(0.530535\pi\)
\(110\) −4.00000 −0.381385
\(111\) 0 0
\(112\) 1.00000 0.0944911
\(113\) 14.0000 1.31701 0.658505 − 0.752577i \(-0.271189\pi\)
0.658505 + 0.752577i \(0.271189\pi\)
\(114\) 0 0
\(115\) −8.00000 −0.746004
\(116\) −6.00000 −0.557086
\(117\) 0 0
\(118\) 4.00000 0.368230
\(119\) −2.00000 −0.183340
\(120\) 0 0
\(121\) 5.00000 0.454545
\(122\) 2.00000 0.181071
\(123\) 0 0
\(124\) −8.00000 −0.718421
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 0 0
\(130\) −2.00000 −0.175412
\(131\) −12.0000 −1.04844 −0.524222 − 0.851581i \(-0.675644\pi\)
−0.524222 + 0.851581i \(0.675644\pi\)
\(132\) 0 0
\(133\) −4.00000 −0.346844
\(134\) −12.0000 −1.03664
\(135\) 0 0
\(136\) 2.00000 0.171499
\(137\) −10.0000 −0.854358 −0.427179 − 0.904167i \(-0.640493\pi\)
−0.427179 + 0.904167i \(0.640493\pi\)
\(138\) 0 0
\(139\) 20.0000 1.69638 0.848189 − 0.529694i \(-0.177693\pi\)
0.848189 + 0.529694i \(0.177693\pi\)
\(140\) −1.00000 −0.0845154
\(141\) 0 0
\(142\) 8.00000 0.671345
\(143\) 8.00000 0.668994
\(144\) 0 0
\(145\) 6.00000 0.498273
\(146\) 14.0000 1.15865
\(147\) 0 0
\(148\) −2.00000 −0.164399
\(149\) 18.0000 1.47462 0.737309 − 0.675556i \(-0.236096\pi\)
0.737309 + 0.675556i \(0.236096\pi\)
\(150\) 0 0
\(151\) 8.00000 0.651031 0.325515 − 0.945537i \(-0.394462\pi\)
0.325515 + 0.945537i \(0.394462\pi\)
\(152\) 4.00000 0.324443
\(153\) 0 0
\(154\) 4.00000 0.322329
\(155\) 8.00000 0.642575
\(156\) 0 0
\(157\) 14.0000 1.11732 0.558661 − 0.829396i \(-0.311315\pi\)
0.558661 + 0.829396i \(0.311315\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 1.00000 0.0790569
\(161\) 8.00000 0.630488
\(162\) 0 0
\(163\) 12.0000 0.939913 0.469956 − 0.882690i \(-0.344270\pi\)
0.469956 + 0.882690i \(0.344270\pi\)
\(164\) −2.00000 −0.156174
\(165\) 0 0
\(166\) 12.0000 0.931381
\(167\) 16.0000 1.23812 0.619059 − 0.785345i \(-0.287514\pi\)
0.619059 + 0.785345i \(0.287514\pi\)
\(168\) 0 0
\(169\) −9.00000 −0.692308
\(170\) −2.00000 −0.153393
\(171\) 0 0
\(172\) −12.0000 −0.914991
\(173\) −14.0000 −1.06440 −0.532200 − 0.846619i \(-0.678635\pi\)
−0.532200 + 0.846619i \(0.678635\pi\)
\(174\) 0 0
\(175\) 1.00000 0.0755929
\(176\) −4.00000 −0.301511
\(177\) 0 0
\(178\) 2.00000 0.149906
\(179\) 20.0000 1.49487 0.747435 − 0.664335i \(-0.231285\pi\)
0.747435 + 0.664335i \(0.231285\pi\)
\(180\) 0 0
\(181\) −10.0000 −0.743294 −0.371647 − 0.928374i \(-0.621207\pi\)
−0.371647 + 0.928374i \(0.621207\pi\)
\(182\) 2.00000 0.148250
\(183\) 0 0
\(184\) −8.00000 −0.589768
\(185\) 2.00000 0.147043
\(186\) 0 0
\(187\) 8.00000 0.585018
\(188\) 8.00000 0.583460
\(189\) 0 0
\(190\) −4.00000 −0.290191
\(191\) 16.0000 1.15772 0.578860 − 0.815427i \(-0.303498\pi\)
0.578860 + 0.815427i \(0.303498\pi\)
\(192\) 0 0
\(193\) −14.0000 −1.00774 −0.503871 − 0.863779i \(-0.668091\pi\)
−0.503871 + 0.863779i \(0.668091\pi\)
\(194\) −10.0000 −0.717958
\(195\) 0 0
\(196\) 1.00000 0.0714286
\(197\) −6.00000 −0.427482 −0.213741 − 0.976890i \(-0.568565\pi\)
−0.213741 + 0.976890i \(0.568565\pi\)
\(198\) 0 0
\(199\) −16.0000 −1.13421 −0.567105 − 0.823646i \(-0.691937\pi\)
−0.567105 + 0.823646i \(0.691937\pi\)
\(200\) −1.00000 −0.0707107
\(201\) 0 0
\(202\) 6.00000 0.422159
\(203\) −6.00000 −0.421117
\(204\) 0 0
\(205\) 2.00000 0.139686
\(206\) −8.00000 −0.557386
\(207\) 0 0
\(208\) −2.00000 −0.138675
\(209\) 16.0000 1.10674
\(210\) 0 0
\(211\) 20.0000 1.37686 0.688428 − 0.725304i \(-0.258301\pi\)
0.688428 + 0.725304i \(0.258301\pi\)
\(212\) −6.00000 −0.412082
\(213\) 0 0
\(214\) −20.0000 −1.36717
\(215\) 12.0000 0.818393
\(216\) 0 0
\(217\) −8.00000 −0.543075
\(218\) 2.00000 0.135457
\(219\) 0 0
\(220\) 4.00000 0.269680
\(221\) 4.00000 0.269069
\(222\) 0 0
\(223\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(224\) −1.00000 −0.0668153
\(225\) 0 0
\(226\) −14.0000 −0.931266
\(227\) −12.0000 −0.796468 −0.398234 − 0.917284i \(-0.630377\pi\)
−0.398234 + 0.917284i \(0.630377\pi\)
\(228\) 0 0
\(229\) −26.0000 −1.71813 −0.859064 − 0.511868i \(-0.828954\pi\)
−0.859064 + 0.511868i \(0.828954\pi\)
\(230\) 8.00000 0.527504
\(231\) 0 0
\(232\) 6.00000 0.393919
\(233\) −10.0000 −0.655122 −0.327561 − 0.944830i \(-0.606227\pi\)
−0.327561 + 0.944830i \(0.606227\pi\)
\(234\) 0 0
\(235\) −8.00000 −0.521862
\(236\) −4.00000 −0.260378
\(237\) 0 0
\(238\) 2.00000 0.129641
\(239\) 16.0000 1.03495 0.517477 − 0.855697i \(-0.326871\pi\)
0.517477 + 0.855697i \(0.326871\pi\)
\(240\) 0 0
\(241\) 18.0000 1.15948 0.579741 − 0.814801i \(-0.303154\pi\)
0.579741 + 0.814801i \(0.303154\pi\)
\(242\) −5.00000 −0.321412
\(243\) 0 0
\(244\) −2.00000 −0.128037
\(245\) −1.00000 −0.0638877
\(246\) 0 0
\(247\) 8.00000 0.509028
\(248\) 8.00000 0.508001
\(249\) 0 0
\(250\) 1.00000 0.0632456
\(251\) −4.00000 −0.252478 −0.126239 − 0.992000i \(-0.540291\pi\)
−0.126239 + 0.992000i \(0.540291\pi\)
\(252\) 0 0
\(253\) −32.0000 −2.01182
\(254\) 0 0
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −2.00000 −0.124757 −0.0623783 − 0.998053i \(-0.519869\pi\)
−0.0623783 + 0.998053i \(0.519869\pi\)
\(258\) 0 0
\(259\) −2.00000 −0.124274
\(260\) 2.00000 0.124035
\(261\) 0 0
\(262\) 12.0000 0.741362
\(263\) −24.0000 −1.47990 −0.739952 − 0.672660i \(-0.765152\pi\)
−0.739952 + 0.672660i \(0.765152\pi\)
\(264\) 0 0
\(265\) 6.00000 0.368577
\(266\) 4.00000 0.245256
\(267\) 0 0
\(268\) 12.0000 0.733017
\(269\) −30.0000 −1.82913 −0.914566 − 0.404436i \(-0.867468\pi\)
−0.914566 + 0.404436i \(0.867468\pi\)
\(270\) 0 0
\(271\) −24.0000 −1.45790 −0.728948 − 0.684569i \(-0.759990\pi\)
−0.728948 + 0.684569i \(0.759990\pi\)
\(272\) −2.00000 −0.121268
\(273\) 0 0
\(274\) 10.0000 0.604122
\(275\) −4.00000 −0.241209
\(276\) 0 0
\(277\) 14.0000 0.841178 0.420589 − 0.907251i \(-0.361823\pi\)
0.420589 + 0.907251i \(0.361823\pi\)
\(278\) −20.0000 −1.19952
\(279\) 0 0
\(280\) 1.00000 0.0597614
\(281\) −10.0000 −0.596550 −0.298275 − 0.954480i \(-0.596411\pi\)
−0.298275 + 0.954480i \(0.596411\pi\)
\(282\) 0 0
\(283\) 20.0000 1.18888 0.594438 − 0.804141i \(-0.297374\pi\)
0.594438 + 0.804141i \(0.297374\pi\)
\(284\) −8.00000 −0.474713
\(285\) 0 0
\(286\) −8.00000 −0.473050
\(287\) −2.00000 −0.118056
\(288\) 0 0
\(289\) −13.0000 −0.764706
\(290\) −6.00000 −0.352332
\(291\) 0 0
\(292\) −14.0000 −0.819288
\(293\) −6.00000 −0.350524 −0.175262 − 0.984522i \(-0.556077\pi\)
−0.175262 + 0.984522i \(0.556077\pi\)
\(294\) 0 0
\(295\) 4.00000 0.232889
\(296\) 2.00000 0.116248
\(297\) 0 0
\(298\) −18.0000 −1.04271
\(299\) −16.0000 −0.925304
\(300\) 0 0
\(301\) −12.0000 −0.691669
\(302\) −8.00000 −0.460348
\(303\) 0 0
\(304\) −4.00000 −0.229416
\(305\) 2.00000 0.114520
\(306\) 0 0
\(307\) −4.00000 −0.228292 −0.114146 − 0.993464i \(-0.536413\pi\)
−0.114146 + 0.993464i \(0.536413\pi\)
\(308\) −4.00000 −0.227921
\(309\) 0 0
\(310\) −8.00000 −0.454369
\(311\) −8.00000 −0.453638 −0.226819 − 0.973937i \(-0.572833\pi\)
−0.226819 + 0.973937i \(0.572833\pi\)
\(312\) 0 0
\(313\) 34.0000 1.92179 0.960897 − 0.276907i \(-0.0893093\pi\)
0.960897 + 0.276907i \(0.0893093\pi\)
\(314\) −14.0000 −0.790066
\(315\) 0 0
\(316\) 0 0
\(317\) −14.0000 −0.786318 −0.393159 − 0.919470i \(-0.628618\pi\)
−0.393159 + 0.919470i \(0.628618\pi\)
\(318\) 0 0
\(319\) 24.0000 1.34374
\(320\) −1.00000 −0.0559017
\(321\) 0 0
\(322\) −8.00000 −0.445823
\(323\) 8.00000 0.445132
\(324\) 0 0
\(325\) −2.00000 −0.110940
\(326\) −12.0000 −0.664619
\(327\) 0 0
\(328\) 2.00000 0.110432
\(329\) 8.00000 0.441054
\(330\) 0 0
\(331\) 28.0000 1.53902 0.769510 − 0.638635i \(-0.220501\pi\)
0.769510 + 0.638635i \(0.220501\pi\)
\(332\) −12.0000 −0.658586
\(333\) 0 0
\(334\) −16.0000 −0.875481
\(335\) −12.0000 −0.655630
\(336\) 0 0
\(337\) 2.00000 0.108947 0.0544735 − 0.998515i \(-0.482652\pi\)
0.0544735 + 0.998515i \(0.482652\pi\)
\(338\) 9.00000 0.489535
\(339\) 0 0
\(340\) 2.00000 0.108465
\(341\) 32.0000 1.73290
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) 12.0000 0.646997
\(345\) 0 0
\(346\) 14.0000 0.752645
\(347\) −12.0000 −0.644194 −0.322097 − 0.946707i \(-0.604388\pi\)
−0.322097 + 0.946707i \(0.604388\pi\)
\(348\) 0 0
\(349\) −34.0000 −1.81998 −0.909989 − 0.414632i \(-0.863910\pi\)
−0.909989 + 0.414632i \(0.863910\pi\)
\(350\) −1.00000 −0.0534522
\(351\) 0 0
\(352\) 4.00000 0.213201
\(353\) −18.0000 −0.958043 −0.479022 − 0.877803i \(-0.659008\pi\)
−0.479022 + 0.877803i \(0.659008\pi\)
\(354\) 0 0
\(355\) 8.00000 0.424596
\(356\) −2.00000 −0.106000
\(357\) 0 0
\(358\) −20.0000 −1.05703
\(359\) 8.00000 0.422224 0.211112 − 0.977462i \(-0.432292\pi\)
0.211112 + 0.977462i \(0.432292\pi\)
\(360\) 0 0
\(361\) −3.00000 −0.157895
\(362\) 10.0000 0.525588
\(363\) 0 0
\(364\) −2.00000 −0.104828
\(365\) 14.0000 0.732793
\(366\) 0 0
\(367\) −32.0000 −1.67039 −0.835193 − 0.549957i \(-0.814644\pi\)
−0.835193 + 0.549957i \(0.814644\pi\)
\(368\) 8.00000 0.417029
\(369\) 0 0
\(370\) −2.00000 −0.103975
\(371\) −6.00000 −0.311504
\(372\) 0 0
\(373\) 14.0000 0.724893 0.362446 − 0.932005i \(-0.381942\pi\)
0.362446 + 0.932005i \(0.381942\pi\)
\(374\) −8.00000 −0.413670
\(375\) 0 0
\(376\) −8.00000 −0.412568
\(377\) 12.0000 0.618031
\(378\) 0 0
\(379\) 28.0000 1.43826 0.719132 − 0.694874i \(-0.244540\pi\)
0.719132 + 0.694874i \(0.244540\pi\)
\(380\) 4.00000 0.205196
\(381\) 0 0
\(382\) −16.0000 −0.818631
\(383\) −24.0000 −1.22634 −0.613171 − 0.789950i \(-0.710106\pi\)
−0.613171 + 0.789950i \(0.710106\pi\)
\(384\) 0 0
\(385\) 4.00000 0.203859
\(386\) 14.0000 0.712581
\(387\) 0 0
\(388\) 10.0000 0.507673
\(389\) 18.0000 0.912636 0.456318 − 0.889817i \(-0.349168\pi\)
0.456318 + 0.889817i \(0.349168\pi\)
\(390\) 0 0
\(391\) −16.0000 −0.809155
\(392\) −1.00000 −0.0505076
\(393\) 0 0
\(394\) 6.00000 0.302276
\(395\) 0 0
\(396\) 0 0
\(397\) −2.00000 −0.100377 −0.0501886 − 0.998740i \(-0.515982\pi\)
−0.0501886 + 0.998740i \(0.515982\pi\)
\(398\) 16.0000 0.802008
\(399\) 0 0
\(400\) 1.00000 0.0500000
\(401\) −18.0000 −0.898877 −0.449439 − 0.893311i \(-0.648376\pi\)
−0.449439 + 0.893311i \(0.648376\pi\)
\(402\) 0 0
\(403\) 16.0000 0.797017
\(404\) −6.00000 −0.298511
\(405\) 0 0
\(406\) 6.00000 0.297775
\(407\) 8.00000 0.396545
\(408\) 0 0
\(409\) 10.0000 0.494468 0.247234 − 0.968956i \(-0.420478\pi\)
0.247234 + 0.968956i \(0.420478\pi\)
\(410\) −2.00000 −0.0987730
\(411\) 0 0
\(412\) 8.00000 0.394132
\(413\) −4.00000 −0.196827
\(414\) 0 0
\(415\) 12.0000 0.589057
\(416\) 2.00000 0.0980581
\(417\) 0 0
\(418\) −16.0000 −0.782586
\(419\) −12.0000 −0.586238 −0.293119 − 0.956076i \(-0.594693\pi\)
−0.293119 + 0.956076i \(0.594693\pi\)
\(420\) 0 0
\(421\) −26.0000 −1.26716 −0.633581 − 0.773676i \(-0.718416\pi\)
−0.633581 + 0.773676i \(0.718416\pi\)
\(422\) −20.0000 −0.973585
\(423\) 0 0
\(424\) 6.00000 0.291386
\(425\) −2.00000 −0.0970143
\(426\) 0 0
\(427\) −2.00000 −0.0967868
\(428\) 20.0000 0.966736
\(429\) 0 0
\(430\) −12.0000 −0.578691
\(431\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(432\) 0 0
\(433\) −6.00000 −0.288342 −0.144171 − 0.989553i \(-0.546051\pi\)
−0.144171 + 0.989553i \(0.546051\pi\)
\(434\) 8.00000 0.384012
\(435\) 0 0
\(436\) −2.00000 −0.0957826
\(437\) −32.0000 −1.53077
\(438\) 0 0
\(439\) 32.0000 1.52728 0.763638 − 0.645644i \(-0.223411\pi\)
0.763638 + 0.645644i \(0.223411\pi\)
\(440\) −4.00000 −0.190693
\(441\) 0 0
\(442\) −4.00000 −0.190261
\(443\) 4.00000 0.190046 0.0950229 − 0.995475i \(-0.469708\pi\)
0.0950229 + 0.995475i \(0.469708\pi\)
\(444\) 0 0
\(445\) 2.00000 0.0948091
\(446\) 0 0
\(447\) 0 0
\(448\) 1.00000 0.0472456
\(449\) 14.0000 0.660701 0.330350 − 0.943858i \(-0.392833\pi\)
0.330350 + 0.943858i \(0.392833\pi\)
\(450\) 0 0
\(451\) 8.00000 0.376705
\(452\) 14.0000 0.658505
\(453\) 0 0
\(454\) 12.0000 0.563188
\(455\) 2.00000 0.0937614
\(456\) 0 0
\(457\) −38.0000 −1.77757 −0.888783 − 0.458329i \(-0.848448\pi\)
−0.888783 + 0.458329i \(0.848448\pi\)
\(458\) 26.0000 1.21490
\(459\) 0 0
\(460\) −8.00000 −0.373002
\(461\) 34.0000 1.58354 0.791769 − 0.610821i \(-0.209160\pi\)
0.791769 + 0.610821i \(0.209160\pi\)
\(462\) 0 0
\(463\) −16.0000 −0.743583 −0.371792 − 0.928316i \(-0.621256\pi\)
−0.371792 + 0.928316i \(0.621256\pi\)
\(464\) −6.00000 −0.278543
\(465\) 0 0
\(466\) 10.0000 0.463241
\(467\) 20.0000 0.925490 0.462745 − 0.886492i \(-0.346865\pi\)
0.462745 + 0.886492i \(0.346865\pi\)
\(468\) 0 0
\(469\) 12.0000 0.554109
\(470\) 8.00000 0.369012
\(471\) 0 0
\(472\) 4.00000 0.184115
\(473\) 48.0000 2.20704
\(474\) 0 0
\(475\) −4.00000 −0.183533
\(476\) −2.00000 −0.0916698
\(477\) 0 0
\(478\) −16.0000 −0.731823
\(479\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(480\) 0 0
\(481\) 4.00000 0.182384
\(482\) −18.0000 −0.819878
\(483\) 0 0
\(484\) 5.00000 0.227273
\(485\) −10.0000 −0.454077
\(486\) 0 0
\(487\) −24.0000 −1.08754 −0.543772 − 0.839233i \(-0.683004\pi\)
−0.543772 + 0.839233i \(0.683004\pi\)
\(488\) 2.00000 0.0905357
\(489\) 0 0
\(490\) 1.00000 0.0451754
\(491\) −36.0000 −1.62466 −0.812329 − 0.583200i \(-0.801800\pi\)
−0.812329 + 0.583200i \(0.801800\pi\)
\(492\) 0 0
\(493\) 12.0000 0.540453
\(494\) −8.00000 −0.359937
\(495\) 0 0
\(496\) −8.00000 −0.359211
\(497\) −8.00000 −0.358849
\(498\) 0 0
\(499\) 4.00000 0.179065 0.0895323 − 0.995984i \(-0.471463\pi\)
0.0895323 + 0.995984i \(0.471463\pi\)
\(500\) −1.00000 −0.0447214
\(501\) 0 0
\(502\) 4.00000 0.178529
\(503\) −16.0000 −0.713405 −0.356702 − 0.934218i \(-0.616099\pi\)
−0.356702 + 0.934218i \(0.616099\pi\)
\(504\) 0 0
\(505\) 6.00000 0.266996
\(506\) 32.0000 1.42257
\(507\) 0 0
\(508\) 0 0
\(509\) −14.0000 −0.620539 −0.310270 − 0.950649i \(-0.600419\pi\)
−0.310270 + 0.950649i \(0.600419\pi\)
\(510\) 0 0
\(511\) −14.0000 −0.619324
\(512\) −1.00000 −0.0441942
\(513\) 0 0
\(514\) 2.00000 0.0882162
\(515\) −8.00000 −0.352522
\(516\) 0 0
\(517\) −32.0000 −1.40736
\(518\) 2.00000 0.0878750
\(519\) 0 0
\(520\) −2.00000 −0.0877058
\(521\) 30.0000 1.31432 0.657162 − 0.753749i \(-0.271757\pi\)
0.657162 + 0.753749i \(0.271757\pi\)
\(522\) 0 0
\(523\) −28.0000 −1.22435 −0.612177 − 0.790721i \(-0.709706\pi\)
−0.612177 + 0.790721i \(0.709706\pi\)
\(524\) −12.0000 −0.524222
\(525\) 0 0
\(526\) 24.0000 1.04645
\(527\) 16.0000 0.696971
\(528\) 0 0
\(529\) 41.0000 1.78261
\(530\) −6.00000 −0.260623
\(531\) 0 0
\(532\) −4.00000 −0.173422
\(533\) 4.00000 0.173259
\(534\) 0 0
\(535\) −20.0000 −0.864675
\(536\) −12.0000 −0.518321
\(537\) 0 0
\(538\) 30.0000 1.29339
\(539\) −4.00000 −0.172292
\(540\) 0 0
\(541\) 14.0000 0.601907 0.300954 − 0.953639i \(-0.402695\pi\)
0.300954 + 0.953639i \(0.402695\pi\)
\(542\) 24.0000 1.03089
\(543\) 0 0
\(544\) 2.00000 0.0857493
\(545\) 2.00000 0.0856706
\(546\) 0 0
\(547\) 28.0000 1.19719 0.598597 − 0.801050i \(-0.295725\pi\)
0.598597 + 0.801050i \(0.295725\pi\)
\(548\) −10.0000 −0.427179
\(549\) 0 0
\(550\) 4.00000 0.170561
\(551\) 24.0000 1.02243
\(552\) 0 0
\(553\) 0 0
\(554\) −14.0000 −0.594803
\(555\) 0 0
\(556\) 20.0000 0.848189
\(557\) 2.00000 0.0847427 0.0423714 − 0.999102i \(-0.486509\pi\)
0.0423714 + 0.999102i \(0.486509\pi\)
\(558\) 0 0
\(559\) 24.0000 1.01509
\(560\) −1.00000 −0.0422577
\(561\) 0 0
\(562\) 10.0000 0.421825
\(563\) 36.0000 1.51722 0.758610 − 0.651546i \(-0.225879\pi\)
0.758610 + 0.651546i \(0.225879\pi\)
\(564\) 0 0
\(565\) −14.0000 −0.588984
\(566\) −20.0000 −0.840663
\(567\) 0 0
\(568\) 8.00000 0.335673
\(569\) 6.00000 0.251533 0.125767 − 0.992060i \(-0.459861\pi\)
0.125767 + 0.992060i \(0.459861\pi\)
\(570\) 0 0
\(571\) −20.0000 −0.836974 −0.418487 − 0.908223i \(-0.637439\pi\)
−0.418487 + 0.908223i \(0.637439\pi\)
\(572\) 8.00000 0.334497
\(573\) 0 0
\(574\) 2.00000 0.0834784
\(575\) 8.00000 0.333623
\(576\) 0 0
\(577\) 26.0000 1.08239 0.541197 − 0.840896i \(-0.317971\pi\)
0.541197 + 0.840896i \(0.317971\pi\)
\(578\) 13.0000 0.540729
\(579\) 0 0
\(580\) 6.00000 0.249136
\(581\) −12.0000 −0.497844
\(582\) 0 0
\(583\) 24.0000 0.993978
\(584\) 14.0000 0.579324
\(585\) 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
\(589\) 32.0000 1.31854
\(590\) −4.00000 −0.164677
\(591\) 0 0
\(592\) −2.00000 −0.0821995
\(593\) −18.0000 −0.739171 −0.369586 − 0.929197i \(-0.620500\pi\)
−0.369586 + 0.929197i \(0.620500\pi\)
\(594\) 0 0
\(595\) 2.00000 0.0819920
\(596\) 18.0000 0.737309
\(597\) 0 0
\(598\) 16.0000 0.654289
\(599\) −8.00000 −0.326871 −0.163436 − 0.986554i \(-0.552258\pi\)
−0.163436 + 0.986554i \(0.552258\pi\)
\(600\) 0 0
\(601\) −22.0000 −0.897399 −0.448699 − 0.893683i \(-0.648113\pi\)
−0.448699 + 0.893683i \(0.648113\pi\)
\(602\) 12.0000 0.489083
\(603\) 0 0
\(604\) 8.00000 0.325515
\(605\) −5.00000 −0.203279
\(606\) 0 0
\(607\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(608\) 4.00000 0.162221
\(609\) 0 0
\(610\) −2.00000 −0.0809776
\(611\) −16.0000 −0.647291
\(612\) 0 0
\(613\) −34.0000 −1.37325 −0.686624 − 0.727013i \(-0.740908\pi\)
−0.686624 + 0.727013i \(0.740908\pi\)
\(614\) 4.00000 0.161427
\(615\) 0 0
\(616\) 4.00000 0.161165
\(617\) −10.0000 −0.402585 −0.201292 − 0.979531i \(-0.564514\pi\)
−0.201292 + 0.979531i \(0.564514\pi\)
\(618\) 0 0
\(619\) −44.0000 −1.76851 −0.884255 − 0.467005i \(-0.845333\pi\)
−0.884255 + 0.467005i \(0.845333\pi\)
\(620\) 8.00000 0.321288
\(621\) 0 0
\(622\) 8.00000 0.320771
\(623\) −2.00000 −0.0801283
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) −34.0000 −1.35891
\(627\) 0 0
\(628\) 14.0000 0.558661
\(629\) 4.00000 0.159490
\(630\) 0 0
\(631\) 24.0000 0.955425 0.477712 − 0.878516i \(-0.341466\pi\)
0.477712 + 0.878516i \(0.341466\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 14.0000 0.556011
\(635\) 0 0
\(636\) 0 0
\(637\) −2.00000 −0.0792429
\(638\) −24.0000 −0.950169
\(639\) 0 0
\(640\) 1.00000 0.0395285
\(641\) 30.0000 1.18493 0.592464 − 0.805597i \(-0.298155\pi\)
0.592464 + 0.805597i \(0.298155\pi\)
\(642\) 0 0
\(643\) −20.0000 −0.788723 −0.394362 − 0.918955i \(-0.629034\pi\)
−0.394362 + 0.918955i \(0.629034\pi\)
\(644\) 8.00000 0.315244
\(645\) 0 0
\(646\) −8.00000 −0.314756
\(647\) 32.0000 1.25805 0.629025 − 0.777385i \(-0.283454\pi\)
0.629025 + 0.777385i \(0.283454\pi\)
\(648\) 0 0
\(649\) 16.0000 0.628055
\(650\) 2.00000 0.0784465
\(651\) 0 0
\(652\) 12.0000 0.469956
\(653\) 34.0000 1.33052 0.665261 − 0.746611i \(-0.268320\pi\)
0.665261 + 0.746611i \(0.268320\pi\)
\(654\) 0 0
\(655\) 12.0000 0.468879
\(656\) −2.00000 −0.0780869
\(657\) 0 0
\(658\) −8.00000 −0.311872
\(659\) −28.0000 −1.09073 −0.545363 − 0.838200i \(-0.683608\pi\)
−0.545363 + 0.838200i \(0.683608\pi\)
\(660\) 0 0
\(661\) −10.0000 −0.388955 −0.194477 − 0.980907i \(-0.562301\pi\)
−0.194477 + 0.980907i \(0.562301\pi\)
\(662\) −28.0000 −1.08825
\(663\) 0 0
\(664\) 12.0000 0.465690
\(665\) 4.00000 0.155113
\(666\) 0 0
\(667\) −48.0000 −1.85857
\(668\) 16.0000 0.619059
\(669\) 0 0
\(670\) 12.0000 0.463600
\(671\) 8.00000 0.308837
\(672\) 0 0
\(673\) 2.00000 0.0770943 0.0385472 − 0.999257i \(-0.487727\pi\)
0.0385472 + 0.999257i \(0.487727\pi\)
\(674\) −2.00000 −0.0770371
\(675\) 0 0
\(676\) −9.00000 −0.346154
\(677\) −6.00000 −0.230599 −0.115299 − 0.993331i \(-0.536783\pi\)
−0.115299 + 0.993331i \(0.536783\pi\)
\(678\) 0 0
\(679\) 10.0000 0.383765
\(680\) −2.00000 −0.0766965
\(681\) 0 0
\(682\) −32.0000 −1.22534
\(683\) 4.00000 0.153056 0.0765279 − 0.997067i \(-0.475617\pi\)
0.0765279 + 0.997067i \(0.475617\pi\)
\(684\) 0 0
\(685\) 10.0000 0.382080
\(686\) −1.00000 −0.0381802
\(687\) 0 0
\(688\) −12.0000 −0.457496
\(689\) 12.0000 0.457164
\(690\) 0 0
\(691\) −20.0000 −0.760836 −0.380418 − 0.924815i \(-0.624220\pi\)
−0.380418 + 0.924815i \(0.624220\pi\)
\(692\) −14.0000 −0.532200
\(693\) 0 0
\(694\) 12.0000 0.455514
\(695\) −20.0000 −0.758643
\(696\) 0 0
\(697\) 4.00000 0.151511
\(698\) 34.0000 1.28692
\(699\) 0 0
\(700\) 1.00000 0.0377964
\(701\) −6.00000 −0.226617 −0.113308 − 0.993560i \(-0.536145\pi\)
−0.113308 + 0.993560i \(0.536145\pi\)
\(702\) 0 0
\(703\) 8.00000 0.301726
\(704\) −4.00000 −0.150756
\(705\) 0 0
\(706\) 18.0000 0.677439
\(707\) −6.00000 −0.225653
\(708\) 0 0
\(709\) −26.0000 −0.976450 −0.488225 − 0.872718i \(-0.662356\pi\)
−0.488225 + 0.872718i \(0.662356\pi\)
\(710\) −8.00000 −0.300235
\(711\) 0 0
\(712\) 2.00000 0.0749532
\(713\) −64.0000 −2.39682
\(714\) 0 0
\(715\) −8.00000 −0.299183
\(716\) 20.0000 0.747435
\(717\) 0 0
\(718\) −8.00000 −0.298557
\(719\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(720\) 0 0
\(721\) 8.00000 0.297936
\(722\) 3.00000 0.111648
\(723\) 0 0
\(724\) −10.0000 −0.371647
\(725\) −6.00000 −0.222834
\(726\) 0 0
\(727\) −24.0000 −0.890111 −0.445055 − 0.895503i \(-0.646816\pi\)
−0.445055 + 0.895503i \(0.646816\pi\)
\(728\) 2.00000 0.0741249
\(729\) 0 0
\(730\) −14.0000 −0.518163
\(731\) 24.0000 0.887672
\(732\) 0 0
\(733\) 30.0000 1.10808 0.554038 − 0.832492i \(-0.313086\pi\)
0.554038 + 0.832492i \(0.313086\pi\)
\(734\) 32.0000 1.18114
\(735\) 0 0
\(736\) −8.00000 −0.294884
\(737\) −48.0000 −1.76810
\(738\) 0 0
\(739\) 36.0000 1.32428 0.662141 − 0.749380i \(-0.269648\pi\)
0.662141 + 0.749380i \(0.269648\pi\)
\(740\) 2.00000 0.0735215
\(741\) 0 0
\(742\) 6.00000 0.220267
\(743\) −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
\(746\) −14.0000 −0.512576
\(747\) 0 0
\(748\) 8.00000 0.292509
\(749\) 20.0000 0.730784
\(750\) 0 0
\(751\) −32.0000 −1.16770 −0.583848 − 0.811863i \(-0.698454\pi\)
−0.583848 + 0.811863i \(0.698454\pi\)
\(752\) 8.00000 0.291730
\(753\) 0 0
\(754\) −12.0000 −0.437014
\(755\) −8.00000 −0.291150
\(756\) 0 0
\(757\) −34.0000 −1.23575 −0.617876 − 0.786276i \(-0.712006\pi\)
−0.617876 + 0.786276i \(0.712006\pi\)
\(758\) −28.0000 −1.01701
\(759\) 0 0
\(760\) −4.00000 −0.145095
\(761\) −2.00000 −0.0724999 −0.0362500 − 0.999343i \(-0.511541\pi\)
−0.0362500 + 0.999343i \(0.511541\pi\)
\(762\) 0 0
\(763\) −2.00000 −0.0724049
\(764\) 16.0000 0.578860
\(765\) 0 0
\(766\) 24.0000 0.867155
\(767\) 8.00000 0.288863
\(768\) 0 0
\(769\) −14.0000 −0.504853 −0.252426 − 0.967616i \(-0.581229\pi\)
−0.252426 + 0.967616i \(0.581229\pi\)
\(770\) −4.00000 −0.144150
\(771\) 0 0
\(772\) −14.0000 −0.503871
\(773\) 26.0000 0.935155 0.467578 − 0.883952i \(-0.345127\pi\)
0.467578 + 0.883952i \(0.345127\pi\)
\(774\) 0 0
\(775\) −8.00000 −0.287368
\(776\) −10.0000 −0.358979
\(777\) 0 0
\(778\) −18.0000 −0.645331
\(779\) 8.00000 0.286630
\(780\) 0 0
\(781\) 32.0000 1.14505
\(782\) 16.0000 0.572159
\(783\) 0 0
\(784\) 1.00000 0.0357143
\(785\) −14.0000 −0.499681
\(786\) 0 0
\(787\) −4.00000 −0.142585 −0.0712923 − 0.997455i \(-0.522712\pi\)
−0.0712923 + 0.997455i \(0.522712\pi\)
\(788\) −6.00000 −0.213741
\(789\) 0 0
\(790\) 0 0
\(791\) 14.0000 0.497783
\(792\) 0 0
\(793\) 4.00000 0.142044
\(794\) 2.00000 0.0709773
\(795\) 0 0
\(796\) −16.0000 −0.567105
\(797\) −30.0000 −1.06265 −0.531327 − 0.847167i \(-0.678307\pi\)
−0.531327 + 0.847167i \(0.678307\pi\)
\(798\) 0 0
\(799\) −16.0000 −0.566039
\(800\) −1.00000 −0.0353553
\(801\) 0 0
\(802\) 18.0000 0.635602
\(803\) 56.0000 1.97620
\(804\) 0 0
\(805\) −8.00000 −0.281963
\(806\) −16.0000 −0.563576
\(807\) 0 0
\(808\) 6.00000 0.211079
\(809\) 38.0000 1.33601 0.668004 − 0.744157i \(-0.267149\pi\)
0.668004 + 0.744157i \(0.267149\pi\)
\(810\) 0 0
\(811\) 36.0000 1.26413 0.632065 − 0.774915i \(-0.282207\pi\)
0.632065 + 0.774915i \(0.282207\pi\)
\(812\) −6.00000 −0.210559
\(813\) 0 0
\(814\) −8.00000 −0.280400
\(815\) −12.0000 −0.420342
\(816\) 0 0
\(817\) 48.0000 1.67931
\(818\) −10.0000 −0.349642
\(819\) 0 0
\(820\) 2.00000 0.0698430
\(821\) −46.0000 −1.60541 −0.802706 − 0.596376i \(-0.796607\pi\)
−0.802706 + 0.596376i \(0.796607\pi\)
\(822\) 0 0
\(823\) −8.00000 −0.278862 −0.139431 − 0.990232i \(-0.544527\pi\)
−0.139431 + 0.990232i \(0.544527\pi\)
\(824\) −8.00000 −0.278693
\(825\) 0 0
\(826\) 4.00000 0.139178
\(827\) −12.0000 −0.417281 −0.208640 − 0.977992i \(-0.566904\pi\)
−0.208640 + 0.977992i \(0.566904\pi\)
\(828\) 0 0
\(829\) −34.0000 −1.18087 −0.590434 − 0.807086i \(-0.701044\pi\)
−0.590434 + 0.807086i \(0.701044\pi\)
\(830\) −12.0000 −0.416526
\(831\) 0 0
\(832\) −2.00000 −0.0693375
\(833\) −2.00000 −0.0692959
\(834\) 0 0
\(835\) −16.0000 −0.553703
\(836\) 16.0000 0.553372
\(837\) 0 0
\(838\) 12.0000 0.414533
\(839\) 24.0000 0.828572 0.414286 − 0.910147i \(-0.364031\pi\)
0.414286 + 0.910147i \(0.364031\pi\)
\(840\) 0 0
\(841\) 7.00000 0.241379
\(842\) 26.0000 0.896019
\(843\) 0 0
\(844\) 20.0000 0.688428
\(845\) 9.00000 0.309609
\(846\) 0 0
\(847\) 5.00000 0.171802
\(848\) −6.00000 −0.206041
\(849\) 0 0
\(850\) 2.00000 0.0685994
\(851\) −16.0000 −0.548473
\(852\) 0 0
\(853\) −26.0000 −0.890223 −0.445112 − 0.895475i \(-0.646836\pi\)
−0.445112 + 0.895475i \(0.646836\pi\)
\(854\) 2.00000 0.0684386
\(855\) 0 0
\(856\) −20.0000 −0.683586
\(857\) 22.0000 0.751506 0.375753 − 0.926720i \(-0.377384\pi\)
0.375753 + 0.926720i \(0.377384\pi\)
\(858\) 0 0
\(859\) −28.0000 −0.955348 −0.477674 − 0.878537i \(-0.658520\pi\)
−0.477674 + 0.878537i \(0.658520\pi\)
\(860\) 12.0000 0.409197
\(861\) 0 0
\(862\) 0 0
\(863\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(864\) 0 0
\(865\) 14.0000 0.476014
\(866\) 6.00000 0.203888
\(867\) 0 0
\(868\) −8.00000 −0.271538
\(869\) 0 0
\(870\) 0 0
\(871\) −24.0000 −0.813209
\(872\) 2.00000 0.0677285
\(873\) 0 0
\(874\) 32.0000 1.08242
\(875\) −1.00000 −0.0338062
\(876\) 0 0
\(877\) −10.0000 −0.337676 −0.168838 − 0.985644i \(-0.554001\pi\)
−0.168838 + 0.985644i \(0.554001\pi\)
\(878\) −32.0000 −1.07995
\(879\) 0 0
\(880\) 4.00000 0.134840
\(881\) −42.0000 −1.41502 −0.707508 − 0.706705i \(-0.750181\pi\)
−0.707508 + 0.706705i \(0.750181\pi\)
\(882\) 0 0
\(883\) 28.0000 0.942275 0.471138 − 0.882060i \(-0.343844\pi\)
0.471138 + 0.882060i \(0.343844\pi\)
\(884\) 4.00000 0.134535
\(885\) 0 0
\(886\) −4.00000 −0.134383
\(887\) −16.0000 −0.537227 −0.268614 − 0.963248i \(-0.586566\pi\)
−0.268614 + 0.963248i \(0.586566\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) −2.00000 −0.0670402
\(891\) 0 0
\(892\) 0 0
\(893\) −32.0000 −1.07084
\(894\) 0 0
\(895\) −20.0000 −0.668526
\(896\) −1.00000 −0.0334077
\(897\) 0 0
\(898\) −14.0000 −0.467186
\(899\) 48.0000 1.60089
\(900\) 0 0
\(901\) 12.0000 0.399778
\(902\) −8.00000 −0.266371
\(903\) 0 0
\(904\) −14.0000 −0.465633
\(905\) 10.0000 0.332411
\(906\) 0 0
\(907\) −12.0000 −0.398453 −0.199227 − 0.979953i \(-0.563843\pi\)
−0.199227 + 0.979953i \(0.563843\pi\)
\(908\) −12.0000 −0.398234
\(909\) 0 0
\(910\) −2.00000 −0.0662994
\(911\) −48.0000 −1.59031 −0.795155 − 0.606406i \(-0.792611\pi\)
−0.795155 + 0.606406i \(0.792611\pi\)
\(912\) 0 0
\(913\) 48.0000 1.58857
\(914\) 38.0000 1.25693
\(915\) 0 0
\(916\) −26.0000 −0.859064
\(917\) −12.0000 −0.396275
\(918\) 0 0
\(919\) −8.00000 −0.263896 −0.131948 − 0.991257i \(-0.542123\pi\)
−0.131948 + 0.991257i \(0.542123\pi\)
\(920\) 8.00000 0.263752
\(921\) 0 0
\(922\) −34.0000 −1.11973
\(923\) 16.0000 0.526646
\(924\) 0 0
\(925\) −2.00000 −0.0657596
\(926\) 16.0000 0.525793
\(927\) 0 0
\(928\) 6.00000 0.196960
\(929\) −10.0000 −0.328089 −0.164045 − 0.986453i \(-0.552454\pi\)
−0.164045 + 0.986453i \(0.552454\pi\)
\(930\) 0 0
\(931\) −4.00000 −0.131095
\(932\) −10.0000 −0.327561
\(933\) 0 0
\(934\) −20.0000 −0.654420
\(935\) −8.00000 −0.261628
\(936\) 0 0
\(937\) 18.0000 0.588034 0.294017 − 0.955800i \(-0.405008\pi\)
0.294017 + 0.955800i \(0.405008\pi\)
\(938\) −12.0000 −0.391814
\(939\) 0 0
\(940\) −8.00000 −0.260931
\(941\) 18.0000 0.586783 0.293392 − 0.955992i \(-0.405216\pi\)
0.293392 + 0.955992i \(0.405216\pi\)
\(942\) 0 0
\(943\) −16.0000 −0.521032
\(944\) −4.00000 −0.130189
\(945\) 0 0
\(946\) −48.0000 −1.56061
\(947\) 12.0000 0.389948 0.194974 − 0.980808i \(-0.437538\pi\)
0.194974 + 0.980808i \(0.437538\pi\)
\(948\) 0 0
\(949\) 28.0000 0.908918
\(950\) 4.00000 0.129777
\(951\) 0 0
\(952\) 2.00000 0.0648204
\(953\) 6.00000 0.194359 0.0971795 − 0.995267i \(-0.469018\pi\)
0.0971795 + 0.995267i \(0.469018\pi\)
\(954\) 0 0
\(955\) −16.0000 −0.517748
\(956\) 16.0000 0.517477
\(957\) 0 0
\(958\) 0 0
\(959\) −10.0000 −0.322917
\(960\) 0 0
\(961\) 33.0000 1.06452
\(962\) −4.00000 −0.128965
\(963\) 0 0
\(964\) 18.0000 0.579741
\(965\) 14.0000 0.450676
\(966\) 0 0
\(967\) −8.00000 −0.257263 −0.128631 − 0.991692i \(-0.541058\pi\)
−0.128631 + 0.991692i \(0.541058\pi\)
\(968\) −5.00000 −0.160706
\(969\) 0 0
\(970\) 10.0000 0.321081
\(971\) −36.0000 −1.15529 −0.577647 − 0.816286i \(-0.696029\pi\)
−0.577647 + 0.816286i \(0.696029\pi\)
\(972\) 0 0
\(973\) 20.0000 0.641171
\(974\) 24.0000 0.769010
\(975\) 0 0
\(976\) −2.00000 −0.0640184
\(977\) −18.0000 −0.575871 −0.287936 − 0.957650i \(-0.592969\pi\)
−0.287936 + 0.957650i \(0.592969\pi\)
\(978\) 0 0
\(979\) 8.00000 0.255681
\(980\) −1.00000 −0.0319438
\(981\) 0 0
\(982\) 36.0000 1.14881
\(983\) −32.0000 −1.02064 −0.510321 − 0.859984i \(-0.670473\pi\)
−0.510321 + 0.859984i \(0.670473\pi\)
\(984\) 0 0
\(985\) 6.00000 0.191176
\(986\) −12.0000 −0.382158
\(987\) 0 0
\(988\) 8.00000 0.254514
\(989\) −96.0000 −3.05262
\(990\) 0 0
\(991\) −32.0000 −1.01651 −0.508257 − 0.861206i \(-0.669710\pi\)
−0.508257 + 0.861206i \(0.669710\pi\)
\(992\) 8.00000 0.254000
\(993\) 0 0
\(994\) 8.00000 0.253745
\(995\) 16.0000 0.507234
\(996\) 0 0
\(997\) 38.0000 1.20347 0.601736 − 0.798695i \(-0.294476\pi\)
0.601736 + 0.798695i \(0.294476\pi\)
\(998\) −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 630.2.a.b.1.1 1
3.2 odd 2 210.2.a.c.1.1 ✓ 1
4.3 odd 2 5040.2.a.i.1.1 1
5.2 odd 4 3150.2.g.e.2899.1 2
5.3 odd 4 3150.2.g.e.2899.2 2
5.4 even 2 3150.2.a.w.1.1 1
7.6 odd 2 4410.2.a.l.1.1 1
12.11 even 2 1680.2.a.q.1.1 1
15.2 even 4 1050.2.g.d.799.2 2
15.8 even 4 1050.2.g.d.799.1 2
15.14 odd 2 1050.2.a.h.1.1 1
21.2 odd 6 1470.2.i.f.361.1 2
21.5 even 6 1470.2.i.b.361.1 2
21.11 odd 6 1470.2.i.f.961.1 2
21.17 even 6 1470.2.i.b.961.1 2
21.20 even 2 1470.2.a.q.1.1 1
24.5 odd 2 6720.2.a.bp.1.1 1
24.11 even 2 6720.2.a.k.1.1 1
60.59 even 2 8400.2.a.p.1.1 1
105.104 even 2 7350.2.a.p.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
210.2.a.c.1.1 ✓ 1 3.2 odd 2
630.2.a.b.1.1 1 1.1 even 1 trivial
1050.2.a.h.1.1 1 15.14 odd 2
1050.2.g.d.799.1 2 15.8 even 4
1050.2.g.d.799.2 2 15.2 even 4
1470.2.a.q.1.1 1 21.20 even 2
1470.2.i.b.361.1 2 21.5 even 6
1470.2.i.b.961.1 2 21.17 even 6
1470.2.i.f.361.1 2 21.2 odd 6
1470.2.i.f.961.1 2 21.11 odd 6
1680.2.a.q.1.1 1 12.11 even 2
3150.2.a.w.1.1 1 5.4 even 2
3150.2.g.e.2899.1 2 5.2 odd 4
3150.2.g.e.2899.2 2 5.3 odd 4
4410.2.a.l.1.1 1 7.6 odd 2
5040.2.a.i.1.1 1 4.3 odd 2
6720.2.a.k.1.1 1 24.11 even 2
6720.2.a.bp.1.1 1 24.5 odd 2
7350.2.a.p.1.1 1 105.104 even 2
8400.2.a.p.1.1 1 60.59 even 2