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Properties

Label 7350.2.a.p.1.1
Level $7350$
Weight $2$
Character 7350.1
Self dual yes
Analytic conductor $58.690$
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] = [7350,2,Mod(1,7350)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(7350, 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("7350.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 7350 = 2 \cdot 3 \cdot 5^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7350.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: \(58.6900454856\)
Analytic rank: \(0\)
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 \(\chi\) \(=\) 7350.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} +4.00000 q^{11} -1.00000 q^{12} -2.00000 q^{13} +1.00000 q^{16} +2.00000 q^{17} -1.00000 q^{18} +4.00000 q^{19} -4.00000 q^{22} +8.00000 q^{23} +1.00000 q^{24} +2.00000 q^{26} -1.00000 q^{27} +6.00000 q^{29} +8.00000 q^{31} -1.00000 q^{32} -4.00000 q^{33} -2.00000 q^{34} +1.00000 q^{36} +2.00000 q^{37} -4.00000 q^{38} +2.00000 q^{39} -2.00000 q^{41} +12.0000 q^{43} +4.00000 q^{44} -8.00000 q^{46} -8.00000 q^{47} -1.00000 q^{48} -2.00000 q^{51} -2.00000 q^{52} -6.00000 q^{53} +1.00000 q^{54} -4.00000 q^{57} -6.00000 q^{58} -4.00000 q^{59} +2.00000 q^{61} -8.00000 q^{62} +1.00000 q^{64} +4.00000 q^{66} -12.0000 q^{67} +2.00000 q^{68} -8.00000 q^{69} +8.00000 q^{71} -1.00000 q^{72} -14.0000 q^{73} -2.00000 q^{74} +4.00000 q^{76} -2.00000 q^{78} +1.00000 q^{81} +2.00000 q^{82} +12.0000 q^{83} -12.0000 q^{86} -6.00000 q^{87} -4.00000 q^{88} -2.00000 q^{89} +8.00000 q^{92} -8.00000 q^{93} +8.00000 q^{94} +1.00000 q^{96} +10.0000 q^{97} +4.00000 q^{99} +O(q^{100})\)

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
\(3\) −1.00000 −0.577350
\(4\) 1.00000 0.500000
\(5\) 0 0
\(6\) 1.00000 0.408248
\(7\) 0 0
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 4.00000 1.20605 0.603023 − 0.797724i \(-0.293963\pi\)
0.603023 + 0.797724i \(0.293963\pi\)
\(12\) −1.00000 −0.288675
\(13\) −2.00000 −0.554700 −0.277350 − 0.960769i \(-0.589456\pi\)
−0.277350 + 0.960769i \(0.589456\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 2.00000 0.485071 0.242536 − 0.970143i \(-0.422021\pi\)
0.242536 + 0.970143i \(0.422021\pi\)
\(18\) −1.00000 −0.235702
\(19\) 4.00000 0.917663 0.458831 − 0.888523i \(-0.348268\pi\)
0.458831 + 0.888523i \(0.348268\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) −4.00000 −0.852803
\(23\) 8.00000 1.66812 0.834058 − 0.551677i \(-0.186012\pi\)
0.834058 + 0.551677i \(0.186012\pi\)
\(24\) 1.00000 0.204124
\(25\) 0 0
\(26\) 2.00000 0.392232
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) 6.00000 1.11417 0.557086 − 0.830455i \(-0.311919\pi\)
0.557086 + 0.830455i \(0.311919\pi\)
\(30\) 0 0
\(31\) 8.00000 1.43684 0.718421 − 0.695608i \(-0.244865\pi\)
0.718421 + 0.695608i \(0.244865\pi\)
\(32\) −1.00000 −0.176777
\(33\) −4.00000 −0.696311
\(34\) −2.00000 −0.342997
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) 2.00000 0.328798 0.164399 − 0.986394i \(-0.447432\pi\)
0.164399 + 0.986394i \(0.447432\pi\)
\(38\) −4.00000 −0.648886
\(39\) 2.00000 0.320256
\(40\) 0 0
\(41\) −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 \(-0.132197\pi\)
0.914991 + 0.403473i \(0.132197\pi\)
\(44\) 4.00000 0.603023
\(45\) 0 0
\(46\) −8.00000 −1.17954
\(47\) −8.00000 −1.16692 −0.583460 − 0.812142i \(-0.698301\pi\)
−0.583460 + 0.812142i \(0.698301\pi\)
\(48\) −1.00000 −0.144338
\(49\) 0 0
\(50\) 0 0
\(51\) −2.00000 −0.280056
\(52\) −2.00000 −0.277350
\(53\) −6.00000 −0.824163 −0.412082 − 0.911147i \(-0.635198\pi\)
−0.412082 + 0.911147i \(0.635198\pi\)
\(54\) 1.00000 0.136083
\(55\) 0 0
\(56\) 0 0
\(57\) −4.00000 −0.529813
\(58\) −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 \(-0.459132\pi\)
0.128037 + 0.991769i \(0.459132\pi\)
\(62\) −8.00000 −1.01600
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 4.00000 0.492366
\(67\) −12.0000 −1.46603 −0.733017 − 0.680211i \(-0.761888\pi\)
−0.733017 + 0.680211i \(0.761888\pi\)
\(68\) 2.00000 0.242536
\(69\) −8.00000 −0.963087
\(70\) 0 0
\(71\) 8.00000 0.949425 0.474713 − 0.880141i \(-0.342552\pi\)
0.474713 + 0.880141i \(0.342552\pi\)
\(72\) −1.00000 −0.117851
\(73\) −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
\(77\) 0 0
\(78\) −2.00000 −0.226455
\(79\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 2.00000 0.220863
\(83\) 12.0000 1.31717 0.658586 − 0.752506i \(-0.271155\pi\)
0.658586 + 0.752506i \(0.271155\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −12.0000 −1.29399
\(87\) −6.00000 −0.643268
\(88\) −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
\(91\) 0 0
\(92\) 8.00000 0.834058
\(93\) −8.00000 −0.829561
\(94\) 8.00000 0.825137
\(95\) 0 0
\(96\) 1.00000 0.102062
\(97\) 10.0000 1.01535 0.507673 − 0.861550i \(-0.330506\pi\)
0.507673 + 0.861550i \(0.330506\pi\)
\(98\) 0 0
\(99\) 4.00000 0.402015
\(100\) 0 0
\(101\) −6.00000 −0.597022 −0.298511 − 0.954406i \(-0.596490\pi\)
−0.298511 + 0.954406i \(0.596490\pi\)
\(102\) 2.00000 0.198030
\(103\) 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 \(0.0823304\pi\)
\(108\) −1.00000 −0.0962250
\(109\) −2.00000 −0.191565 −0.0957826 − 0.995402i \(-0.530535\pi\)
−0.0957826 + 0.995402i \(0.530535\pi\)
\(110\) 0 0
\(111\) −2.00000 −0.189832
\(112\) 0 0
\(113\) 14.0000 1.31701 0.658505 − 0.752577i \(-0.271189\pi\)
0.658505 + 0.752577i \(0.271189\pi\)
\(114\) 4.00000 0.374634
\(115\) 0 0
\(116\) 6.00000 0.557086
\(117\) −2.00000 −0.184900
\(118\) 4.00000 0.368230
\(119\) 0 0
\(120\) 0 0
\(121\) 5.00000 0.454545
\(122\) −2.00000 −0.181071
\(123\) 2.00000 0.180334
\(124\) 8.00000 0.718421
\(125\) 0 0
\(126\) 0 0
\(127\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(128\) −1.00000 −0.0883883
\(129\) −12.0000 −1.05654
\(130\) 0 0
\(131\) −12.0000 −1.04844 −0.524222 − 0.851581i \(-0.675644\pi\)
−0.524222 + 0.851581i \(0.675644\pi\)
\(132\) −4.00000 −0.348155
\(133\) 0 0
\(134\) 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 \(0.640493\pi\)
\(138\) 8.00000 0.681005
\(139\) −20.0000 −1.69638 −0.848189 − 0.529694i \(-0.822307\pi\)
−0.848189 + 0.529694i \(0.822307\pi\)
\(140\) 0 0
\(141\) 8.00000 0.673722
\(142\) −8.00000 −0.671345
\(143\) −8.00000 −0.668994
\(144\) 1.00000 0.0833333
\(145\) 0 0
\(146\) 14.0000 1.15865
\(147\) 0 0
\(148\) 2.00000 0.164399
\(149\) −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 \(0.394462\pi\)
\(152\) −4.00000 −0.324443
\(153\) 2.00000 0.161690
\(154\) 0 0
\(155\) 0 0
\(156\) 2.00000 0.160128
\(157\) 14.0000 1.11732 0.558661 − 0.829396i \(-0.311315\pi\)
0.558661 + 0.829396i \(0.311315\pi\)
\(158\) 0 0
\(159\) 6.00000 0.475831
\(160\) 0 0
\(161\) 0 0
\(162\) −1.00000 −0.0785674
\(163\) −12.0000 −0.939913 −0.469956 − 0.882690i \(-0.655730\pi\)
−0.469956 + 0.882690i \(0.655730\pi\)
\(164\) −2.00000 −0.156174
\(165\) 0 0
\(166\) −12.0000 −0.931381
\(167\) −16.0000 −1.23812 −0.619059 − 0.785345i \(-0.712486\pi\)
−0.619059 + 0.785345i \(0.712486\pi\)
\(168\) 0 0
\(169\) −9.00000 −0.692308
\(170\) 0 0
\(171\) 4.00000 0.305888
\(172\) 12.0000 0.914991
\(173\) 14.0000 1.06440 0.532200 − 0.846619i \(-0.321365\pi\)
0.532200 + 0.846619i \(0.321365\pi\)
\(174\) 6.00000 0.454859
\(175\) 0 0
\(176\) 4.00000 0.301511
\(177\) 4.00000 0.300658
\(178\) 2.00000 0.149906
\(179\) −20.0000 −1.49487 −0.747435 − 0.664335i \(-0.768715\pi\)
−0.747435 + 0.664335i \(0.768715\pi\)
\(180\) 0 0
\(181\) 10.0000 0.743294 0.371647 − 0.928374i \(-0.378793\pi\)
0.371647 + 0.928374i \(0.378793\pi\)
\(182\) 0 0
\(183\) −2.00000 −0.147844
\(184\) −8.00000 −0.589768
\(185\) 0 0
\(186\) 8.00000 0.586588
\(187\) 8.00000 0.585018
\(188\) −8.00000 −0.583460
\(189\) 0 0
\(190\) 0 0
\(191\) −16.0000 −1.15772 −0.578860 − 0.815427i \(-0.696502\pi\)
−0.578860 + 0.815427i \(0.696502\pi\)
\(192\) −1.00000 −0.0721688
\(193\) 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
\(196\) 0 0
\(197\) −6.00000 −0.427482 −0.213741 − 0.976890i \(-0.568565\pi\)
−0.213741 + 0.976890i \(0.568565\pi\)
\(198\) −4.00000 −0.284268
\(199\) 16.0000 1.13421 0.567105 − 0.823646i \(-0.308063\pi\)
0.567105 + 0.823646i \(0.308063\pi\)
\(200\) 0 0
\(201\) 12.0000 0.846415
\(202\) 6.00000 0.422159
\(203\) 0 0
\(204\) −2.00000 −0.140028
\(205\) 0 0
\(206\) −8.00000 −0.557386
\(207\) 8.00000 0.556038
\(208\) −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
\(213\) −8.00000 −0.548151
\(214\) −20.0000 −1.36717
\(215\) 0 0
\(216\) 1.00000 0.0680414
\(217\) 0 0
\(218\) 2.00000 0.135457
\(219\) 14.0000 0.946032
\(220\) 0 0
\(221\) −4.00000 −0.269069
\(222\) 2.00000 0.134231
\(223\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) −14.0000 −0.931266
\(227\) 12.0000 0.796468 0.398234 − 0.917284i \(-0.369623\pi\)
0.398234 + 0.917284i \(0.369623\pi\)
\(228\) −4.00000 −0.264906
\(229\) 26.0000 1.71813 0.859064 − 0.511868i \(-0.171046\pi\)
0.859064 + 0.511868i \(0.171046\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) −6.00000 −0.393919
\(233\) −10.0000 −0.655122 −0.327561 − 0.944830i \(-0.606227\pi\)
−0.327561 + 0.944830i \(0.606227\pi\)
\(234\) 2.00000 0.130744
\(235\) 0 0
\(236\) −4.00000 −0.260378
\(237\) 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
\(241\) −18.0000 −1.15948 −0.579741 − 0.814801i \(-0.696846\pi\)
−0.579741 + 0.814801i \(0.696846\pi\)
\(242\) −5.00000 −0.321412
\(243\) −1.00000 −0.0641500
\(244\) 2.00000 0.128037
\(245\) 0 0
\(246\) −2.00000 −0.127515
\(247\) −8.00000 −0.509028
\(248\) −8.00000 −0.508001
\(249\) −12.0000 −0.760469
\(250\) 0 0
\(251\) −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 \(-0.480131\pi\)
0.0623783 + 0.998053i \(0.480131\pi\)
\(258\) 12.0000 0.747087
\(259\) 0 0
\(260\) 0 0
\(261\) 6.00000 0.371391
\(262\) 12.0000 0.741362
\(263\) −24.0000 −1.47990 −0.739952 − 0.672660i \(-0.765152\pi\)
−0.739952 + 0.672660i \(0.765152\pi\)
\(264\) 4.00000 0.246183
\(265\) 0 0
\(266\) 0 0
\(267\) 2.00000 0.122398
\(268\) −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 \(-0.240010\pi\)
0.728948 + 0.684569i \(0.240010\pi\)
\(272\) 2.00000 0.121268
\(273\) 0 0
\(274\) 10.0000 0.604122
\(275\) 0 0
\(276\) −8.00000 −0.481543
\(277\) −14.0000 −0.841178 −0.420589 − 0.907251i \(-0.638177\pi\)
−0.420589 + 0.907251i \(0.638177\pi\)
\(278\) 20.0000 1.19952
\(279\) 8.00000 0.478947
\(280\) 0 0
\(281\) 10.0000 0.596550 0.298275 − 0.954480i \(-0.403589\pi\)
0.298275 + 0.954480i \(0.403589\pi\)
\(282\) −8.00000 −0.476393
\(283\) 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
\(287\) 0 0
\(288\) −1.00000 −0.0589256
\(289\) −13.0000 −0.764706
\(290\) 0 0
\(291\) −10.0000 −0.586210
\(292\) −14.0000 −0.819288
\(293\) 6.00000 0.350524 0.175262 − 0.984522i \(-0.443923\pi\)
0.175262 + 0.984522i \(0.443923\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) −2.00000 −0.116248
\(297\) −4.00000 −0.232104
\(298\) 18.0000 1.04271
\(299\) −16.0000 −0.925304
\(300\) 0 0
\(301\) 0 0
\(302\) −8.00000 −0.460348
\(303\) 6.00000 0.344691
\(304\) 4.00000 0.229416
\(305\) 0 0
\(306\) −2.00000 −0.114332
\(307\) −4.00000 −0.228292 −0.114146 − 0.993464i \(-0.536413\pi\)
−0.114146 + 0.993464i \(0.536413\pi\)
\(308\) 0 0
\(309\) −8.00000 −0.455104
\(310\) 0 0
\(311\) −8.00000 −0.453638 −0.226819 − 0.973937i \(-0.572833\pi\)
−0.226819 + 0.973937i \(0.572833\pi\)
\(312\) −2.00000 −0.113228
\(313\) 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 \(0.628618\pi\)
\(318\) −6.00000 −0.336463
\(319\) 24.0000 1.34374
\(320\) 0 0
\(321\) −20.0000 −1.11629
\(322\) 0 0
\(323\) 8.00000 0.445132
\(324\) 1.00000 0.0555556
\(325\) 0 0
\(326\) 12.0000 0.664619
\(327\) 2.00000 0.110600
\(328\) 2.00000 0.110432
\(329\) 0 0
\(330\) 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
\(333\) 2.00000 0.109599
\(334\) 16.0000 0.875481
\(335\) 0 0
\(336\) 0 0
\(337\) −2.00000 −0.108947 −0.0544735 − 0.998515i \(-0.517348\pi\)
−0.0544735 + 0.998515i \(0.517348\pi\)
\(338\) 9.00000 0.489535
\(339\) −14.0000 −0.760376
\(340\) 0 0
\(341\) 32.0000 1.73290
\(342\) −4.00000 −0.216295
\(343\) 0 0
\(344\) −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 \(0.604388\pi\)
\(348\) −6.00000 −0.321634
\(349\) 34.0000 1.81998 0.909989 − 0.414632i \(-0.136090\pi\)
0.909989 + 0.414632i \(0.136090\pi\)
\(350\) 0 0
\(351\) 2.00000 0.106752
\(352\) −4.00000 −0.213201
\(353\) 18.0000 0.958043 0.479022 − 0.877803i \(-0.340992\pi\)
0.479022 + 0.877803i \(0.340992\pi\)
\(354\) −4.00000 −0.212598
\(355\) 0 0
\(356\) −2.00000 −0.106000
\(357\) 0 0
\(358\) 20.0000 1.05703
\(359\) −8.00000 −0.422224 −0.211112 − 0.977462i \(-0.567708\pi\)
−0.211112 + 0.977462i \(0.567708\pi\)
\(360\) 0 0
\(361\) −3.00000 −0.157895
\(362\) −10.0000 −0.525588
\(363\) −5.00000 −0.262432
\(364\) 0 0
\(365\) 0 0
\(366\) 2.00000 0.104542
\(367\) −32.0000 −1.67039 −0.835193 − 0.549957i \(-0.814644\pi\)
−0.835193 + 0.549957i \(0.814644\pi\)
\(368\) 8.00000 0.417029
\(369\) −2.00000 −0.104116
\(370\) 0 0
\(371\) 0 0
\(372\) −8.00000 −0.414781
\(373\) −14.0000 −0.724893 −0.362446 − 0.932005i \(-0.618058\pi\)
−0.362446 + 0.932005i \(0.618058\pi\)
\(374\) −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 \(0.244540\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 16.0000 0.818631
\(383\) 24.0000 1.22634 0.613171 − 0.789950i \(-0.289894\pi\)
0.613171 + 0.789950i \(0.289894\pi\)
\(384\) 1.00000 0.0510310
\(385\) 0 0
\(386\) −14.0000 −0.712581
\(387\) 12.0000 0.609994
\(388\) 10.0000 0.507673
\(389\) −18.0000 −0.912636 −0.456318 − 0.889817i \(-0.650832\pi\)
−0.456318 + 0.889817i \(0.650832\pi\)
\(390\) 0 0
\(391\) 16.0000 0.809155
\(392\) 0 0
\(393\) 12.0000 0.605320
\(394\) 6.00000 0.302276
\(395\) 0 0
\(396\) 4.00000 0.201008
\(397\) −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
\(400\) 0 0
\(401\) 18.0000 0.898877 0.449439 − 0.893311i \(-0.351624\pi\)
0.449439 + 0.893311i \(0.351624\pi\)
\(402\) −12.0000 −0.598506
\(403\) −16.0000 −0.797017
\(404\) −6.00000 −0.298511
\(405\) 0 0
\(406\) 0 0
\(407\) 8.00000 0.396545
\(408\) 2.00000 0.0990148
\(409\) −10.0000 −0.494468 −0.247234 − 0.968956i \(-0.579522\pi\)
−0.247234 + 0.968956i \(0.579522\pi\)
\(410\) 0 0
\(411\) 10.0000 0.493264
\(412\) 8.00000 0.394132
\(413\) 0 0
\(414\) −8.00000 −0.393179
\(415\) 0 0
\(416\) 2.00000 0.0980581
\(417\) 20.0000 0.979404
\(418\) −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
\(423\) −8.00000 −0.388973
\(424\) 6.00000 0.291386
\(425\) 0 0
\(426\) 8.00000 0.387601
\(427\) 0 0
\(428\) 20.0000 0.966736
\(429\) 8.00000 0.386244
\(430\) 0 0
\(431\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(432\) −1.00000 −0.0481125
\(433\) −6.00000 −0.288342 −0.144171 − 0.989553i \(-0.546051\pi\)
−0.144171 + 0.989553i \(0.546051\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −2.00000 −0.0957826
\(437\) 32.0000 1.53077
\(438\) −14.0000 −0.668946
\(439\) −32.0000 −1.52728 −0.763638 − 0.645644i \(-0.776589\pi\)
−0.763638 + 0.645644i \(0.776589\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 4.00000 0.190261
\(443\) 4.00000 0.190046 0.0950229 − 0.995475i \(-0.469708\pi\)
0.0950229 + 0.995475i \(0.469708\pi\)
\(444\) −2.00000 −0.0949158
\(445\) 0 0
\(446\) 0 0
\(447\) 18.0000 0.851371
\(448\) 0 0
\(449\) −14.0000 −0.660701 −0.330350 − 0.943858i \(-0.607167\pi\)
−0.330350 + 0.943858i \(0.607167\pi\)
\(450\) 0 0
\(451\) −8.00000 −0.376705
\(452\) 14.0000 0.658505
\(453\) −8.00000 −0.375873
\(454\) −12.0000 −0.563188
\(455\) 0 0
\(456\) 4.00000 0.187317
\(457\) 38.0000 1.77757 0.888783 − 0.458329i \(-0.151552\pi\)
0.888783 + 0.458329i \(0.151552\pi\)
\(458\) −26.0000 −1.21490
\(459\) −2.00000 −0.0933520
\(460\) 0 0
\(461\) 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 \(-0.378744\pi\)
0.371792 + 0.928316i \(0.378744\pi\)
\(464\) 6.00000 0.278543
\(465\) 0 0
\(466\) 10.0000 0.463241
\(467\) −20.0000 −0.925490 −0.462745 − 0.886492i \(-0.653135\pi\)
−0.462745 + 0.886492i \(0.653135\pi\)
\(468\) −2.00000 −0.0924500
\(469\) 0 0
\(470\) 0 0
\(471\) −14.0000 −0.645086
\(472\) 4.00000 0.184115
\(473\) 48.0000 2.20704
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) −6.00000 −0.274721
\(478\) 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
\(485\) 0 0
\(486\) 1.00000 0.0453609
\(487\) 24.0000 1.08754 0.543772 − 0.839233i \(-0.316996\pi\)
0.543772 + 0.839233i \(0.316996\pi\)
\(488\) −2.00000 −0.0905357
\(489\) 12.0000 0.542659
\(490\) 0 0
\(491\) 36.0000 1.62466 0.812329 − 0.583200i \(-0.198200\pi\)
0.812329 + 0.583200i \(0.198200\pi\)
\(492\) 2.00000 0.0901670
\(493\) 12.0000 0.540453
\(494\) 8.00000 0.359937
\(495\) 0 0
\(496\) 8.00000 0.359211
\(497\) 0 0
\(498\) 12.0000 0.537733
\(499\) 4.00000 0.179065 0.0895323 − 0.995984i \(-0.471463\pi\)
0.0895323 + 0.995984i \(0.471463\pi\)
\(500\) 0 0
\(501\) 16.0000 0.714827
\(502\) 4.00000 0.178529
\(503\) 16.0000 0.713405 0.356702 − 0.934218i \(-0.383901\pi\)
0.356702 + 0.934218i \(0.383901\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) −32.0000 −1.42257
\(507\) 9.00000 0.399704
\(508\) 0 0
\(509\) −14.0000 −0.620539 −0.310270 − 0.950649i \(-0.600419\pi\)
−0.310270 + 0.950649i \(0.600419\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) −1.00000 −0.0441942
\(513\) −4.00000 −0.176604
\(514\) −2.00000 −0.0882162
\(515\) 0 0
\(516\) −12.0000 −0.528271
\(517\) −32.0000 −1.40736
\(518\) 0 0
\(519\) −14.0000 −0.614532
\(520\) 0 0
\(521\) 30.0000 1.31432 0.657162 − 0.753749i \(-0.271757\pi\)
0.657162 + 0.753749i \(0.271757\pi\)
\(522\) −6.00000 −0.262613
\(523\) −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
\(528\) −4.00000 −0.174078
\(529\) 41.0000 1.78261
\(530\) 0 0
\(531\) −4.00000 −0.173585
\(532\) 0 0
\(533\) 4.00000 0.173259
\(534\) −2.00000 −0.0865485
\(535\) 0 0
\(536\) 12.0000 0.518321
\(537\) 20.0000 0.863064
\(538\) 30.0000 1.29339
\(539\) 0 0
\(540\) 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
\(543\) −10.0000 −0.429141
\(544\) −2.00000 −0.0857493
\(545\) 0 0
\(546\) 0 0
\(547\) −28.0000 −1.19719 −0.598597 − 0.801050i \(-0.704275\pi\)
−0.598597 + 0.801050i \(0.704275\pi\)
\(548\) −10.0000 −0.427179
\(549\) 2.00000 0.0853579
\(550\) 0 0
\(551\) 24.0000 1.02243
\(552\) 8.00000 0.340503
\(553\) 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 \(0.486509\pi\)
\(558\) −8.00000 −0.338667
\(559\) −24.0000 −1.01509
\(560\) 0 0
\(561\) −8.00000 −0.337760
\(562\) −10.0000 −0.421825
\(563\) −36.0000 −1.51722 −0.758610 − 0.651546i \(-0.774121\pi\)
−0.758610 + 0.651546i \(0.774121\pi\)
\(564\) 8.00000 0.336861
\(565\) 0 0
\(566\) −20.0000 −0.840663
\(567\) 0 0
\(568\) −8.00000 −0.335673
\(569\) −6.00000 −0.251533 −0.125767 − 0.992060i \(-0.540139\pi\)
−0.125767 + 0.992060i \(0.540139\pi\)
\(570\) 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
\(573\) 16.0000 0.668410
\(574\) 0 0
\(575\) 0 0
\(576\) 1.00000 0.0416667
\(577\) 26.0000 1.08239 0.541197 − 0.840896i \(-0.317971\pi\)
0.541197 + 0.840896i \(0.317971\pi\)
\(578\) 13.0000 0.540729
\(579\) −14.0000 −0.581820
\(580\) 0 0
\(581\) 0 0
\(582\) 10.0000 0.414513
\(583\) −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 \(-0.579657\pi\)
−0.247647 + 0.968850i \(0.579657\pi\)
\(588\) 0 0
\(589\) 32.0000 1.31854
\(590\) 0 0
\(591\) 6.00000 0.246807
\(592\) 2.00000 0.0821995
\(593\) 18.0000 0.739171 0.369586 − 0.929197i \(-0.379500\pi\)
0.369586 + 0.929197i \(0.379500\pi\)
\(594\) 4.00000 0.164122
\(595\) 0 0
\(596\) −18.0000 −0.737309
\(597\) −16.0000 −0.654836
\(598\) 16.0000 0.654289
\(599\) 8.00000 0.326871 0.163436 − 0.986554i \(-0.447742\pi\)
0.163436 + 0.986554i \(0.447742\pi\)
\(600\) 0 0
\(601\) 22.0000 0.897399 0.448699 − 0.893683i \(-0.351887\pi\)
0.448699 + 0.893683i \(0.351887\pi\)
\(602\) 0 0
\(603\) −12.0000 −0.488678
\(604\) 8.00000 0.325515
\(605\) 0 0
\(606\) −6.00000 −0.243733
\(607\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(608\) −4.00000 −0.162221
\(609\) 0 0
\(610\) 0 0
\(611\) 16.0000 0.647291
\(612\) 2.00000 0.0808452
\(613\) 34.0000 1.37325 0.686624 − 0.727013i \(-0.259092\pi\)
0.686624 + 0.727013i \(0.259092\pi\)
\(614\) 4.00000 0.161427
\(615\) 0 0
\(616\) 0 0
\(617\) −10.0000 −0.402585 −0.201292 − 0.979531i \(-0.564514\pi\)
−0.201292 + 0.979531i \(0.564514\pi\)
\(618\) 8.00000 0.321807
\(619\) 44.0000 1.76851 0.884255 − 0.467005i \(-0.154667\pi\)
0.884255 + 0.467005i \(0.154667\pi\)
\(620\) 0 0
\(621\) −8.00000 −0.321029
\(622\) 8.00000 0.320771
\(623\) 0 0
\(624\) 2.00000 0.0800641
\(625\) 0 0
\(626\) −34.0000 −1.35891
\(627\) −16.0000 −0.638978
\(628\) 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
\(633\) −20.0000 −0.794929
\(634\) 14.0000 0.556011
\(635\) 0 0
\(636\) 6.00000 0.237915
\(637\) 0 0
\(638\) −24.0000 −0.950169
\(639\) 8.00000 0.316475
\(640\) 0 0
\(641\) −30.0000 −1.18493 −0.592464 − 0.805597i \(-0.701845\pi\)
−0.592464 + 0.805597i \(0.701845\pi\)
\(642\) 20.0000 0.789337
\(643\) −20.0000 −0.788723 −0.394362 − 0.918955i \(-0.629034\pi\)
−0.394362 + 0.918955i \(0.629034\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) −8.00000 −0.314756
\(647\) −32.0000 −1.25805 −0.629025 − 0.777385i \(-0.716546\pi\)
−0.629025 + 0.777385i \(0.716546\pi\)
\(648\) −1.00000 −0.0392837
\(649\) −16.0000 −0.628055
\(650\) 0 0
\(651\) 0 0
\(652\) −12.0000 −0.469956
\(653\) 34.0000 1.33052 0.665261 − 0.746611i \(-0.268320\pi\)
0.665261 + 0.746611i \(0.268320\pi\)
\(654\) −2.00000 −0.0782062
\(655\) 0 0
\(656\) −2.00000 −0.0780869
\(657\) −14.0000 −0.546192
\(658\) 0 0
\(659\) 28.0000 1.09073 0.545363 − 0.838200i \(-0.316392\pi\)
0.545363 + 0.838200i \(0.316392\pi\)
\(660\) 0 0
\(661\) 10.0000 0.388955 0.194477 − 0.980907i \(-0.437699\pi\)
0.194477 + 0.980907i \(0.437699\pi\)
\(662\) −28.0000 −1.08825
\(663\) 4.00000 0.155347
\(664\) −12.0000 −0.465690
\(665\) 0 0
\(666\) −2.00000 −0.0774984
\(667\) 48.0000 1.85857
\(668\) −16.0000 −0.619059
\(669\) 0 0
\(670\) 0 0
\(671\) 8.00000 0.308837
\(672\) 0 0
\(673\) −2.00000 −0.0770943 −0.0385472 − 0.999257i \(-0.512273\pi\)
−0.0385472 + 0.999257i \(0.512273\pi\)
\(674\) 2.00000 0.0770371
\(675\) 0 0
\(676\) −9.00000 −0.346154
\(677\) 6.00000 0.230599 0.115299 − 0.993331i \(-0.463217\pi\)
0.115299 + 0.993331i \(0.463217\pi\)
\(678\) 14.0000 0.537667
\(679\) 0 0
\(680\) 0 0
\(681\) −12.0000 −0.459841
\(682\) −32.0000 −1.22534
\(683\) 4.00000 0.153056 0.0765279 − 0.997067i \(-0.475617\pi\)
0.0765279 + 0.997067i \(0.475617\pi\)
\(684\) 4.00000 0.152944
\(685\) 0 0
\(686\) 0 0
\(687\) −26.0000 −0.991962
\(688\) 12.0000 0.457496
\(689\) 12.0000 0.457164
\(690\) 0 0
\(691\) 20.0000 0.760836 0.380418 − 0.924815i \(-0.375780\pi\)
0.380418 + 0.924815i \(0.375780\pi\)
\(692\) 14.0000 0.532200
\(693\) 0 0
\(694\) 12.0000 0.455514
\(695\) 0 0
\(696\) 6.00000 0.227429
\(697\) −4.00000 −0.151511
\(698\) −34.0000 −1.28692
\(699\) 10.0000 0.378235
\(700\) 0 0
\(701\) 6.00000 0.226617 0.113308 − 0.993560i \(-0.463855\pi\)
0.113308 + 0.993560i \(0.463855\pi\)
\(702\) −2.00000 −0.0754851
\(703\) 8.00000 0.301726
\(704\) 4.00000 0.150756
\(705\) 0 0
\(706\) −18.0000 −0.677439
\(707\) 0 0
\(708\) 4.00000 0.150329
\(709\) −26.0000 −0.976450 −0.488225 − 0.872718i \(-0.662356\pi\)
−0.488225 + 0.872718i \(0.662356\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 2.00000 0.0749532
\(713\) 64.0000 2.39682
\(714\) 0 0
\(715\) 0 0
\(716\) −20.0000 −0.747435
\(717\) 16.0000 0.597531
\(718\) 8.00000 0.298557
\(719\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 3.00000 0.111648
\(723\) 18.0000 0.669427
\(724\) 10.0000 0.371647
\(725\) 0 0
\(726\) 5.00000 0.185567
\(727\) −24.0000 −0.890111 −0.445055 − 0.895503i \(-0.646816\pi\)
−0.445055 + 0.895503i \(0.646816\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 24.0000 0.887672
\(732\) −2.00000 −0.0739221
\(733\) 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
\(738\) 2.00000 0.0736210
\(739\) 36.0000 1.32428 0.662141 − 0.749380i \(-0.269648\pi\)
0.662141 + 0.749380i \(0.269648\pi\)
\(740\) 0 0
\(741\) 8.00000 0.293887
\(742\) 0 0
\(743\) −24.0000 −0.880475 −0.440237 − 0.897881i \(-0.645106\pi\)
−0.440237 + 0.897881i \(0.645106\pi\)
\(744\) 8.00000 0.293294
\(745\) 0 0
\(746\) 14.0000 0.512576
\(747\) 12.0000 0.439057
\(748\) 8.00000 0.292509
\(749\) 0 0
\(750\) 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
\(753\) 4.00000 0.145768
\(754\) 12.0000 0.437014
\(755\) 0 0
\(756\) 0 0
\(757\) 34.0000 1.23575 0.617876 − 0.786276i \(-0.287994\pi\)
0.617876 + 0.786276i \(0.287994\pi\)
\(758\) −28.0000 −1.01701
\(759\) −32.0000 −1.16153
\(760\) 0 0
\(761\) −2.00000 −0.0724999 −0.0362500 − 0.999343i \(-0.511541\pi\)
−0.0362500 + 0.999343i \(0.511541\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) −16.0000 −0.578860
\(765\) 0 0
\(766\) −24.0000 −0.867155
\(767\) 8.00000 0.288863
\(768\) −1.00000 −0.0360844
\(769\) 14.0000 0.504853 0.252426 − 0.967616i \(-0.418771\pi\)
0.252426 + 0.967616i \(0.418771\pi\)
\(770\) 0 0
\(771\) −2.00000 −0.0720282
\(772\) 14.0000 0.503871
\(773\) −26.0000 −0.935155 −0.467578 − 0.883952i \(-0.654873\pi\)
−0.467578 + 0.883952i \(0.654873\pi\)
\(774\) −12.0000 −0.431331
\(775\) 0 0
\(776\) −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
\(783\) −6.00000 −0.214423
\(784\) 0 0
\(785\) 0 0
\(786\) −12.0000 −0.428026
\(787\) −4.00000 −0.142585 −0.0712923 − 0.997455i \(-0.522712\pi\)
−0.0712923 + 0.997455i \(0.522712\pi\)
\(788\) −6.00000 −0.213741
\(789\) 24.0000 0.854423
\(790\) 0 0
\(791\) 0 0
\(792\) −4.00000 −0.142134
\(793\) −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 \(-0.321693\pi\)
0.531327 + 0.847167i \(0.321693\pi\)
\(798\) 0 0
\(799\) −16.0000 −0.566039
\(800\) 0 0
\(801\) −2.00000 −0.0706665
\(802\) −18.0000 −0.635602
\(803\) −56.0000 −1.97620
\(804\) 12.0000 0.423207
\(805\) 0 0
\(806\) 16.0000 0.563576
\(807\) 30.0000 1.05605
\(808\) 6.00000 0.211079
\(809\) −38.0000 −1.33601 −0.668004 − 0.744157i \(-0.732851\pi\)
−0.668004 + 0.744157i \(0.732851\pi\)
\(810\) 0 0
\(811\) −36.0000 −1.26413 −0.632065 − 0.774915i \(-0.717793\pi\)
−0.632065 + 0.774915i \(0.717793\pi\)
\(812\) 0 0
\(813\) −24.0000 −0.841717
\(814\) −8.00000 −0.280400
\(815\) 0 0
\(816\) −2.00000 −0.0700140
\(817\) 48.0000 1.67931
\(818\) 10.0000 0.349642
\(819\) 0 0
\(820\) 0 0
\(821\) 46.0000 1.60541 0.802706 − 0.596376i \(-0.203393\pi\)
0.802706 + 0.596376i \(0.203393\pi\)
\(822\) −10.0000 −0.348790
\(823\) 8.00000 0.278862 0.139431 − 0.990232i \(-0.455473\pi\)
0.139431 + 0.990232i \(0.455473\pi\)
\(824\) −8.00000 −0.278693
\(825\) 0 0
\(826\) 0 0
\(827\) −12.0000 −0.417281 −0.208640 − 0.977992i \(-0.566904\pi\)
−0.208640 + 0.977992i \(0.566904\pi\)
\(828\) 8.00000 0.278019
\(829\) 34.0000 1.18087 0.590434 − 0.807086i \(-0.298956\pi\)
0.590434 + 0.807086i \(0.298956\pi\)
\(830\) 0 0
\(831\) 14.0000 0.485655
\(832\) −2.00000 −0.0693375
\(833\) 0 0
\(834\) −20.0000 −0.692543
\(835\) 0 0
\(836\) 16.0000 0.553372
\(837\) −8.00000 −0.276520
\(838\) 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
\(843\) −10.0000 −0.344418
\(844\) 20.0000 0.688428
\(845\) 0 0
\(846\) 8.00000 0.275046
\(847\) 0 0
\(848\) −6.00000 −0.206041
\(849\) −20.0000 −0.686398
\(850\) 0 0
\(851\) 16.0000 0.548473
\(852\) −8.00000 −0.274075
\(853\) −26.0000 −0.890223 −0.445112 − 0.895475i \(-0.646836\pi\)
−0.445112 + 0.895475i \(0.646836\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) −20.0000 −0.683586
\(857\) −22.0000 −0.751506 −0.375753 − 0.926720i \(-0.622616\pi\)
−0.375753 + 0.926720i \(0.622616\pi\)
\(858\) −8.00000 −0.273115
\(859\) 28.0000 0.955348 0.477674 − 0.878537i \(-0.341480\pi\)
0.477674 + 0.878537i \(0.341480\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(864\) 1.00000 0.0340207
\(865\) 0 0
\(866\) 6.00000 0.203888
\(867\) 13.0000 0.441503
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 24.0000 0.813209
\(872\) 2.00000 0.0677285
\(873\) 10.0000 0.338449
\(874\) −32.0000 −1.08242
\(875\) 0 0
\(876\) 14.0000 0.473016
\(877\) 10.0000 0.337676 0.168838 − 0.985644i \(-0.445999\pi\)
0.168838 + 0.985644i \(0.445999\pi\)
\(878\) 32.0000 1.07995
\(879\) −6.00000 −0.202375
\(880\) 0 0
\(881\) −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 \(-0.656156\pi\)
−0.471138 + 0.882060i \(0.656156\pi\)
\(884\) −4.00000 −0.134535
\(885\) 0 0
\(886\) −4.00000 −0.134383
\(887\) 16.0000 0.537227 0.268614 − 0.963248i \(-0.413434\pi\)
0.268614 + 0.963248i \(0.413434\pi\)
\(888\) 2.00000 0.0671156
\(889\) 0 0
\(890\) 0 0
\(891\) 4.00000 0.134005
\(892\) 0 0
\(893\) −32.0000 −1.07084
\(894\) −18.0000 −0.602010
\(895\) 0 0
\(896\) 0 0
\(897\) 16.0000 0.534224
\(898\) 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
\(905\) 0 0
\(906\) 8.00000 0.265782
\(907\) 12.0000 0.398453 0.199227 − 0.979953i \(-0.436157\pi\)
0.199227 + 0.979953i \(0.436157\pi\)
\(908\) 12.0000 0.398234
\(909\) −6.00000 −0.199007
\(910\) 0 0
\(911\) 48.0000 1.59031 0.795155 − 0.606406i \(-0.207389\pi\)
0.795155 + 0.606406i \(0.207389\pi\)
\(912\) −4.00000 −0.132453
\(913\) 48.0000 1.58857
\(914\) −38.0000 −1.25693
\(915\) 0 0
\(916\) 26.0000 0.859064
\(917\) 0 0
\(918\) 2.00000 0.0660098
\(919\) −8.00000 −0.263896 −0.131948 − 0.991257i \(-0.542123\pi\)
−0.131948 + 0.991257i \(0.542123\pi\)
\(920\) 0 0
\(921\) 4.00000 0.131804
\(922\) −34.0000 −1.11973
\(923\) −16.0000 −0.526646
\(924\) 0 0
\(925\) 0 0
\(926\) −16.0000 −0.525793
\(927\) 8.00000 0.262754
\(928\) −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
\(931\) 0 0
\(932\) −10.0000 −0.327561
\(933\) 8.00000 0.261908
\(934\) 20.0000 0.654420
\(935\) 0 0
\(936\) 2.00000 0.0653720
\(937\) 18.0000 0.588034 0.294017 − 0.955800i \(-0.405008\pi\)
0.294017 + 0.955800i \(0.405008\pi\)
\(938\) 0 0
\(939\) −34.0000 −1.10955
\(940\) 0 0
\(941\) 18.0000 0.586783 0.293392 − 0.955992i \(-0.405216\pi\)
0.293392 + 0.955992i \(0.405216\pi\)
\(942\) 14.0000 0.456145
\(943\) −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
\(950\) 0 0
\(951\) 14.0000 0.453981
\(952\) 0 0
\(953\) 6.00000 0.194359 0.0971795 − 0.995267i \(-0.469018\pi\)
0.0971795 + 0.995267i \(0.469018\pi\)
\(954\) 6.00000 0.194257
\(955\) 0 0
\(956\) −16.0000 −0.517477
\(957\) −24.0000 −0.775810
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 33.0000 1.06452
\(962\) 4.00000 0.128965
\(963\) 20.0000 0.644491
\(964\) −18.0000 −0.579741
\(965\) 0 0
\(966\) 0 0
\(967\) 8.00000 0.257263 0.128631 − 0.991692i \(-0.458942\pi\)
0.128631 + 0.991692i \(0.458942\pi\)
\(968\) −5.00000 −0.160706
\(969\) −8.00000 −0.256997
\(970\) 0 0
\(971\) −36.0000 −1.15529 −0.577647 − 0.816286i \(-0.696029\pi\)
−0.577647 + 0.816286i \(0.696029\pi\)
\(972\) −1.00000 −0.0320750
\(973\) 0 0
\(974\) −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 \(0.592969\pi\)
\(978\) −12.0000 −0.383718
\(979\) −8.00000 −0.255681
\(980\) 0 0
\(981\) −2.00000 −0.0638551
\(982\) −36.0000 −1.14881
\(983\) 32.0000 1.02064 0.510321 − 0.859984i \(-0.329527\pi\)
0.510321 + 0.859984i \(0.329527\pi\)
\(984\) −2.00000 −0.0637577
\(985\) 0 0
\(986\) −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
\(993\) −28.0000 −0.888553
\(994\) 0 0
\(995\) 0 0
\(996\) −12.0000 −0.380235
\(997\) 38.0000 1.20347 0.601736 − 0.798695i \(-0.294476\pi\)
0.601736 + 0.798695i \(0.294476\pi\)
\(998\) −4.00000 −0.126618
\(999\) −2.00000 −0.0632772
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 7350.2.a.p.1.1 1
5.4 even 2 1470.2.a.q.1.1 1
7.6 odd 2 1050.2.a.h.1.1 1
15.14 odd 2 4410.2.a.l.1.1 1
21.20 even 2 3150.2.a.w.1.1 1
28.27 even 2 8400.2.a.p.1.1 1
35.4 even 6 1470.2.i.b.961.1 2
35.9 even 6 1470.2.i.b.361.1 2
35.13 even 4 1050.2.g.d.799.2 2
35.19 odd 6 1470.2.i.f.361.1 2
35.24 odd 6 1470.2.i.f.961.1 2
35.27 even 4 1050.2.g.d.799.1 2
35.34 odd 2 210.2.a.c.1.1 ✓ 1
105.62 odd 4 3150.2.g.e.2899.2 2
105.83 odd 4 3150.2.g.e.2899.1 2
105.104 even 2 630.2.a.b.1.1 1
140.139 even 2 1680.2.a.q.1.1 1
280.69 odd 2 6720.2.a.bp.1.1 1
280.139 even 2 6720.2.a.k.1.1 1
420.419 odd 2 5040.2.a.i.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
210.2.a.c.1.1 ✓ 1 35.34 odd 2
630.2.a.b.1.1 1 105.104 even 2
1050.2.a.h.1.1 1 7.6 odd 2
1050.2.g.d.799.1 2 35.27 even 4
1050.2.g.d.799.2 2 35.13 even 4
1470.2.a.q.1.1 1 5.4 even 2
1470.2.i.b.361.1 2 35.9 even 6
1470.2.i.b.961.1 2 35.4 even 6
1470.2.i.f.361.1 2 35.19 odd 6
1470.2.i.f.961.1 2 35.24 odd 6
1680.2.a.q.1.1 1 140.139 even 2
3150.2.a.w.1.1 1 21.20 even 2
3150.2.g.e.2899.1 2 105.83 odd 4
3150.2.g.e.2899.2 2 105.62 odd 4
4410.2.a.l.1.1 1 15.14 odd 2
5040.2.a.i.1.1 1 420.419 odd 2
6720.2.a.k.1.1 1 280.139 even 2
6720.2.a.bp.1.1 1 280.69 odd 2
7350.2.a.p.1.1 1 1.1 even 1 trivial
8400.2.a.p.1.1 1 28.27 even 2