Properties

Label 1470.2.a.q.1.1
Level $1470$
Weight $2$
Character 1470.1
Self dual yes
Analytic conductor $11.738$
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] = [1470,2,Mod(1,1470)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1470, 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("1470.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1470 = 2 \cdot 3 \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1470.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: \(11.7380090971\)
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 \(\chi\) \(=\) 1470.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} +4.00000 q^{11} +1.00000 q^{12} +2.00000 q^{13} -1.00000 q^{15} +1.00000 q^{16} -2.00000 q^{17} +1.00000 q^{18} +4.00000 q^{19} -1.00000 q^{20} +4.00000 q^{22} -8.00000 q^{23} +1.00000 q^{24} +1.00000 q^{25} +2.00000 q^{26} +1.00000 q^{27} +6.00000 q^{29} -1.00000 q^{30} +8.00000 q^{31} +1.00000 q^{32} +4.00000 q^{33} -2.00000 q^{34} +1.00000 q^{36} -2.00000 q^{37} +4.00000 q^{38} +2.00000 q^{39} -1.00000 q^{40} -2.00000 q^{41} -12.0000 q^{43} +4.00000 q^{44} -1.00000 q^{45} -8.00000 q^{46} +8.00000 q^{47} +1.00000 q^{48} +1.00000 q^{50} -2.00000 q^{51} +2.00000 q^{52} +6.00000 q^{53} +1.00000 q^{54} -4.00000 q^{55} +4.00000 q^{57} +6.00000 q^{58} -4.00000 q^{59} -1.00000 q^{60} +2.00000 q^{61} +8.00000 q^{62} +1.00000 q^{64} -2.00000 q^{65} +4.00000 q^{66} +12.0000 q^{67} -2.00000 q^{68} -8.00000 q^{69} +8.00000 q^{71} +1.00000 q^{72} +14.0000 q^{73} -2.00000 q^{74} +1.00000 q^{75} +4.00000 q^{76} +2.00000 q^{78} -1.00000 q^{80} +1.00000 q^{81} -2.00000 q^{82} -12.0000 q^{83} +2.00000 q^{85} -12.0000 q^{86} +6.00000 q^{87} +4.00000 q^{88} -2.00000 q^{89} -1.00000 q^{90} -8.00000 q^{92} +8.00000 q^{93} +8.00000 q^{94} -4.00000 q^{95} +1.00000 q^{96} -10.0000 q^{97} +4.00000 q^{99} +O(q^{100})\)

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
\(5\) −1.00000 −0.447214
\(6\) 1.00000 0.408248
\(7\) 0 0
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) −1.00000 −0.316228
\(11\) 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 \(-0.410544\pi\)
0.277350 + 0.960769i \(0.410544\pi\)
\(14\) 0 0
\(15\) −1.00000 −0.258199
\(16\) 1.00000 0.250000
\(17\) −2.00000 −0.485071 −0.242536 0.970143i \(-0.577979\pi\)
−0.242536 + 0.970143i \(0.577979\pi\)
\(18\) 1.00000 0.235702
\(19\) 4.00000 0.917663 0.458831 0.888523i \(-0.348268\pi\)
0.458831 + 0.888523i \(0.348268\pi\)
\(20\) −1.00000 −0.223607
\(21\) 0 0
\(22\) 4.00000 0.852803
\(23\) −8.00000 −1.66812 −0.834058 0.551677i \(-0.813988\pi\)
−0.834058 + 0.551677i \(0.813988\pi\)
\(24\) 1.00000 0.204124
\(25\) 1.00000 0.200000
\(26\) 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 \(0.311919\pi\)
\(30\) −1.00000 −0.182574
\(31\) 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 \(-0.552568\pi\)
−0.164399 + 0.986394i \(0.552568\pi\)
\(38\) 4.00000 0.648886
\(39\) 2.00000 0.320256
\(40\) −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
\(45\) −1.00000 −0.149071
\(46\) −8.00000 −1.17954
\(47\) 8.00000 1.16692 0.583460 0.812142i \(-0.301699\pi\)
0.583460 + 0.812142i \(0.301699\pi\)
\(48\) 1.00000 0.144338
\(49\) 0 0
\(50\) 1.00000 0.141421
\(51\) −2.00000 −0.280056
\(52\) 2.00000 0.277350
\(53\) 6.00000 0.824163 0.412082 0.911147i \(-0.364802\pi\)
0.412082 + 0.911147i \(0.364802\pi\)
\(54\) 1.00000 0.136083
\(55\) −4.00000 −0.539360
\(56\) 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 \(0.583847\pi\)
\(60\) −1.00000 −0.129099
\(61\) 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
\(65\) −2.00000 −0.248069
\(66\) 4.00000 0.492366
\(67\) 12.0000 1.46603 0.733017 0.680211i \(-0.238112\pi\)
0.733017 + 0.680211i \(0.238112\pi\)
\(68\) −2.00000 −0.242536
\(69\) −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 \(-0.194369\pi\)
0.819288 + 0.573382i \(0.194369\pi\)
\(74\) −2.00000 −0.232495
\(75\) 1.00000 0.115470
\(76\) 4.00000 0.458831
\(77\) 0 0
\(78\) 2.00000 0.226455
\(79\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(80\) −1.00000 −0.111803
\(81\) 1.00000 0.111111
\(82\) −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
\(87\) 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 \(0.533804\pi\)
\(90\) −1.00000 −0.105409
\(91\) 0 0
\(92\) −8.00000 −0.834058
\(93\) 8.00000 0.829561
\(94\) 8.00000 0.825137
\(95\) −4.00000 −0.410391
\(96\) 1.00000 0.102062
\(97\) −10.0000 −1.01535 −0.507673 0.861550i \(-0.669494\pi\)
−0.507673 + 0.861550i \(0.669494\pi\)
\(98\) 0 0
\(99\) 4.00000 0.402015
\(100\) 1.00000 0.100000
\(101\) −6.00000 −0.597022 −0.298511 0.954406i \(-0.596490\pi\)
−0.298511 + 0.954406i \(0.596490\pi\)
\(102\) −2.00000 −0.198030
\(103\) −8.00000 −0.788263 −0.394132 0.919054i \(-0.628955\pi\)
−0.394132 + 0.919054i \(0.628955\pi\)
\(104\) 2.00000 0.196116
\(105\) 0 0
\(106\) 6.00000 0.582772
\(107\) −20.0000 −1.93347 −0.966736 0.255774i \(-0.917670\pi\)
−0.966736 + 0.255774i \(0.917670\pi\)
\(108\) 1.00000 0.0962250
\(109\) −2.00000 −0.191565 −0.0957826 0.995402i \(-0.530535\pi\)
−0.0957826 + 0.995402i \(0.530535\pi\)
\(110\) −4.00000 −0.381385
\(111\) −2.00000 −0.189832
\(112\) 0 0
\(113\) −14.0000 −1.31701 −0.658505 0.752577i \(-0.728811\pi\)
−0.658505 + 0.752577i \(0.728811\pi\)
\(114\) 4.00000 0.374634
\(115\) 8.00000 0.746004
\(116\) 6.00000 0.557086
\(117\) 2.00000 0.184900
\(118\) −4.00000 −0.368230
\(119\) 0 0
\(120\) −1.00000 −0.0912871
\(121\) 5.00000 0.454545
\(122\) 2.00000 0.181071
\(123\) −2.00000 −0.180334
\(124\) 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
\(129\) −12.0000 −1.05654
\(130\) −2.00000 −0.175412
\(131\) −12.0000 −1.04844 −0.524222 0.851581i \(-0.675644\pi\)
−0.524222 + 0.851581i \(0.675644\pi\)
\(132\) 4.00000 0.348155
\(133\) 0 0
\(134\) 12.0000 1.03664
\(135\) −1.00000 −0.0860663
\(136\) −2.00000 −0.171499
\(137\) 10.0000 0.854358 0.427179 0.904167i \(-0.359507\pi\)
0.427179 + 0.904167i \(0.359507\pi\)
\(138\) −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
\(145\) −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 \(-0.763904\pi\)
−0.737309 + 0.675556i \(0.763904\pi\)
\(150\) 1.00000 0.0816497
\(151\) 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
\(155\) −8.00000 −0.642575
\(156\) 2.00000 0.160128
\(157\) −14.0000 −1.11732 −0.558661 0.829396i \(-0.688685\pi\)
−0.558661 + 0.829396i \(0.688685\pi\)
\(158\) 0 0
\(159\) 6.00000 0.475831
\(160\) −1.00000 −0.0790569
\(161\) 0 0
\(162\) 1.00000 0.0785674
\(163\) 12.0000 0.939913 0.469956 0.882690i \(-0.344270\pi\)
0.469956 + 0.882690i \(0.344270\pi\)
\(164\) −2.00000 −0.156174
\(165\) −4.00000 −0.311400
\(166\) −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
\(171\) 4.00000 0.305888
\(172\) −12.0000 −0.914991
\(173\) −14.0000 −1.06440 −0.532200 0.846619i \(-0.678635\pi\)
−0.532200 + 0.846619i \(0.678635\pi\)
\(174\) 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 \(0.768715\pi\)
\(180\) −1.00000 −0.0745356
\(181\) 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
\(185\) 2.00000 0.147043
\(186\) 8.00000 0.586588
\(187\) −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 \(-0.696502\pi\)
−0.578860 + 0.815427i \(0.696502\pi\)
\(192\) 1.00000 0.0721688
\(193\) −14.0000 −1.00774 −0.503871 0.863779i \(-0.668091\pi\)
−0.503871 + 0.863779i \(0.668091\pi\)
\(194\) −10.0000 −0.717958
\(195\) −2.00000 −0.143223
\(196\) 0 0
\(197\) 6.00000 0.427482 0.213741 0.976890i \(-0.431435\pi\)
0.213741 + 0.976890i \(0.431435\pi\)
\(198\) 4.00000 0.284268
\(199\) 16.0000 1.13421 0.567105 0.823646i \(-0.308063\pi\)
0.567105 + 0.823646i \(0.308063\pi\)
\(200\) 1.00000 0.0707107
\(201\) 12.0000 0.846415
\(202\) −6.00000 −0.422159
\(203\) 0 0
\(204\) −2.00000 −0.140028
\(205\) 2.00000 0.139686
\(206\) −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
\(215\) 12.0000 0.818393
\(216\) 1.00000 0.0680414
\(217\) 0 0
\(218\) −2.00000 −0.135457
\(219\) 14.0000 0.946032
\(220\) −4.00000 −0.269680
\(221\) −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
\(225\) 1.00000 0.0666667
\(226\) −14.0000 −0.931266
\(227\) −12.0000 −0.796468 −0.398234 0.917284i \(-0.630377\pi\)
−0.398234 + 0.917284i \(0.630377\pi\)
\(228\) 4.00000 0.264906
\(229\) 26.0000 1.71813 0.859064 0.511868i \(-0.171046\pi\)
0.859064 + 0.511868i \(0.171046\pi\)
\(230\) 8.00000 0.527504
\(231\) 0 0
\(232\) 6.00000 0.393919
\(233\) 10.0000 0.655122 0.327561 0.944830i \(-0.393773\pi\)
0.327561 + 0.944830i \(0.393773\pi\)
\(234\) 2.00000 0.130744
\(235\) −8.00000 −0.521862
\(236\) −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 \(0.673129\pi\)
\(240\) −1.00000 −0.0645497
\(241\) −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
\(250\) −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
\(255\) 2.00000 0.125245
\(256\) 1.00000 0.0625000
\(257\) −2.00000 −0.124757 −0.0623783 0.998053i \(-0.519869\pi\)
−0.0623783 + 0.998053i \(0.519869\pi\)
\(258\) −12.0000 −0.747087
\(259\) 0 0
\(260\) −2.00000 −0.124035
\(261\) 6.00000 0.371391
\(262\) −12.0000 −0.741362
\(263\) 24.0000 1.47990 0.739952 0.672660i \(-0.234848\pi\)
0.739952 + 0.672660i \(0.234848\pi\)
\(264\) 4.00000 0.246183
\(265\) −6.00000 −0.368577
\(266\) 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 \(0.867468\pi\)
\(270\) −1.00000 −0.0608581
\(271\) 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
\(275\) 4.00000 0.241209
\(276\) −8.00000 −0.481543
\(277\) 14.0000 0.841178 0.420589 0.907251i \(-0.361823\pi\)
0.420589 + 0.907251i \(0.361823\pi\)
\(278\) −20.0000 −1.19952
\(279\) 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 \(-0.702626\pi\)
−0.594438 + 0.804141i \(0.702626\pi\)
\(284\) 8.00000 0.474713
\(285\) −4.00000 −0.236940
\(286\) 8.00000 0.473050
\(287\) 0 0
\(288\) 1.00000 0.0589256
\(289\) −13.0000 −0.764706
\(290\) −6.00000 −0.352332
\(291\) −10.0000 −0.586210
\(292\) 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
\(297\) 4.00000 0.232104
\(298\) −18.0000 −1.04271
\(299\) −16.0000 −0.925304
\(300\) 1.00000 0.0577350
\(301\) 0 0
\(302\) 8.00000 0.460348
\(303\) −6.00000 −0.344691
\(304\) 4.00000 0.229416
\(305\) −2.00000 −0.114520
\(306\) −2.00000 −0.114332
\(307\) 4.00000 0.228292 0.114146 0.993464i \(-0.463587\pi\)
0.114146 + 0.993464i \(0.463587\pi\)
\(308\) 0 0
\(309\) −8.00000 −0.455104
\(310\) −8.00000 −0.454369
\(311\) −8.00000 −0.453638 −0.226819 0.973937i \(-0.572833\pi\)
−0.226819 + 0.973937i \(0.572833\pi\)
\(312\) 2.00000 0.113228
\(313\) −34.0000 −1.92179 −0.960897 0.276907i \(-0.910691\pi\)
−0.960897 + 0.276907i \(0.910691\pi\)
\(314\) −14.0000 −0.790066
\(315\) 0 0
\(316\) 0 0
\(317\) 14.0000 0.786318 0.393159 0.919470i \(-0.371382\pi\)
0.393159 + 0.919470i \(0.371382\pi\)
\(318\) 6.00000 0.336463
\(319\) 24.0000 1.34374
\(320\) −1.00000 −0.0559017
\(321\) −20.0000 −1.11629
\(322\) 0 0
\(323\) −8.00000 −0.445132
\(324\) 1.00000 0.0555556
\(325\) 2.00000 0.110940
\(326\) 12.0000 0.664619
\(327\) −2.00000 −0.110600
\(328\) −2.00000 −0.110432
\(329\) 0 0
\(330\) −4.00000 −0.220193
\(331\) 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
\(335\) −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
\(339\) −14.0000 −0.760376
\(340\) 2.00000 0.108465
\(341\) 32.0000 1.73290
\(342\) 4.00000 0.216295
\(343\) 0 0
\(344\) −12.0000 −0.646997
\(345\) 8.00000 0.430706
\(346\) −14.0000 −0.752645
\(347\) 12.0000 0.644194 0.322097 0.946707i \(-0.395612\pi\)
0.322097 + 0.946707i \(0.395612\pi\)
\(348\) 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 \(-0.659008\pi\)
−0.479022 + 0.877803i \(0.659008\pi\)
\(354\) −4.00000 −0.212598
\(355\) −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 \(-0.567708\pi\)
−0.211112 + 0.977462i \(0.567708\pi\)
\(360\) −1.00000 −0.0527046
\(361\) −3.00000 −0.157895
\(362\) 10.0000 0.525588
\(363\) 5.00000 0.262432
\(364\) 0 0
\(365\) −14.0000 −0.732793
\(366\) 2.00000 0.104542
\(367\) 32.0000 1.67039 0.835193 0.549957i \(-0.185356\pi\)
0.835193 + 0.549957i \(0.185356\pi\)
\(368\) −8.00000 −0.417029
\(369\) −2.00000 −0.104116
\(370\) 2.00000 0.103975
\(371\) 0 0
\(372\) 8.00000 0.414781
\(373\) 14.0000 0.724893 0.362446 0.932005i \(-0.381942\pi\)
0.362446 + 0.932005i \(0.381942\pi\)
\(374\) −8.00000 −0.413670
\(375\) −1.00000 −0.0516398
\(376\) 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 \(0.710106\pi\)
\(384\) 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 \(0.650832\pi\)
\(390\) −2.00000 −0.101274
\(391\) 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 \(-0.484018\pi\)
0.0501886 + 0.998740i \(0.484018\pi\)
\(398\) 16.0000 0.802008
\(399\) 0 0
\(400\) 1.00000 0.0500000
\(401\) 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
\(405\) −1.00000 −0.0496904
\(406\) 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 \(0.579522\pi\)
\(410\) 2.00000 0.0987730
\(411\) 10.0000 0.493264
\(412\) −8.00000 −0.394132
\(413\) 0 0
\(414\) −8.00000 −0.393179
\(415\) 12.0000 0.589057
\(416\) 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
\(425\) −2.00000 −0.0970143
\(426\) 8.00000 0.387601
\(427\) 0 0
\(428\) −20.0000 −0.966736
\(429\) 8.00000 0.386244
\(430\) 12.0000 0.578691
\(431\) 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 \(-0.453949\pi\)
0.144171 + 0.989553i \(0.453949\pi\)
\(434\) 0 0
\(435\) −6.00000 −0.287678
\(436\) −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 \(0.776589\pi\)
\(440\) −4.00000 −0.190693
\(441\) 0 0
\(442\) −4.00000 −0.190261
\(443\) −4.00000 −0.190046 −0.0950229 0.995475i \(-0.530292\pi\)
−0.0950229 + 0.995475i \(0.530292\pi\)
\(444\) −2.00000 −0.0949158
\(445\) 2.00000 0.0948091
\(446\) 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 \(0.607167\pi\)
\(450\) 1.00000 0.0471405
\(451\) −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 \(-0.848448\pi\)
−0.888783 + 0.458329i \(0.848448\pi\)
\(458\) 26.0000 1.21490
\(459\) −2.00000 −0.0933520
\(460\) 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
\(465\) −8.00000 −0.370991
\(466\) 10.0000 0.463241
\(467\) 20.0000 0.925490 0.462745 0.886492i \(-0.346865\pi\)
0.462745 + 0.886492i \(0.346865\pi\)
\(468\) 2.00000 0.0924500
\(469\) 0 0
\(470\) −8.00000 −0.369012
\(471\) −14.0000 −0.645086
\(472\) −4.00000 −0.184115
\(473\) −48.0000 −2.20704
\(474\) 0 0
\(475\) 4.00000 0.183533
\(476\) 0 0
\(477\) 6.00000 0.274721
\(478\) −16.0000 −0.731823
\(479\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(480\) −1.00000 −0.0456435
\(481\) −4.00000 −0.182384
\(482\) −18.0000 −0.819878
\(483\) 0 0
\(484\) 5.00000 0.227273
\(485\) 10.0000 0.454077
\(486\) 1.00000 0.0453609
\(487\) −24.0000 −1.08754 −0.543772 0.839233i \(-0.683004\pi\)
−0.543772 + 0.839233i \(0.683004\pi\)
\(488\) 2.00000 0.0905357
\(489\) 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
\(495\) −4.00000 −0.179787
\(496\) 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 \(0.471463\pi\)
\(500\) −1.00000 −0.0447214
\(501\) 16.0000 0.714827
\(502\) −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
\(507\) −9.00000 −0.399704
\(508\) 0 0
\(509\) −14.0000 −0.620539 −0.310270 0.950649i \(-0.600419\pi\)
−0.310270 + 0.950649i \(0.600419\pi\)
\(510\) 2.00000 0.0885615
\(511\) 0 0
\(512\) 1.00000 0.0441942
\(513\) 4.00000 0.176604
\(514\) −2.00000 −0.0882162
\(515\) 8.00000 0.352522
\(516\) −12.0000 −0.528271
\(517\) 32.0000 1.40736
\(518\) 0 0
\(519\) −14.0000 −0.614532
\(520\) −2.00000 −0.0877058
\(521\) 30.0000 1.31432 0.657162 0.753749i \(-0.271757\pi\)
0.657162 + 0.753749i \(0.271757\pi\)
\(522\) 6.00000 0.262613
\(523\) 28.0000 1.22435 0.612177 0.790721i \(-0.290294\pi\)
0.612177 + 0.790721i \(0.290294\pi\)
\(524\) −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
\(530\) −6.00000 −0.260623
\(531\) −4.00000 −0.173585
\(532\) 0 0
\(533\) −4.00000 −0.173259
\(534\) −2.00000 −0.0865485
\(535\) 20.0000 0.864675
\(536\) 12.0000 0.518321
\(537\) −20.0000 −0.863064
\(538\) −30.0000 −1.29339
\(539\) 0 0
\(540\) −1.00000 −0.0430331
\(541\) 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
\(545\) 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
\(549\) 2.00000 0.0853579
\(550\) 4.00000 0.170561
\(551\) 24.0000 1.02243
\(552\) −8.00000 −0.340503
\(553\) 0 0
\(554\) 14.0000 0.594803
\(555\) 2.00000 0.0848953
\(556\) −20.0000 −0.848189
\(557\) −2.00000 −0.0847427 −0.0423714 0.999102i \(-0.513491\pi\)
−0.0423714 + 0.999102i \(0.513491\pi\)
\(558\) 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 \(-0.225879\pi\)
0.758610 + 0.651546i \(0.225879\pi\)
\(564\) 8.00000 0.336861
\(565\) 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 \(-0.540139\pi\)
−0.125767 + 0.992060i \(0.540139\pi\)
\(570\) −4.00000 −0.167542
\(571\) −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
\(575\) −8.00000 −0.333623
\(576\) 1.00000 0.0416667
\(577\) −26.0000 −1.08239 −0.541197 0.840896i \(-0.682029\pi\)
−0.541197 + 0.840896i \(0.682029\pi\)
\(578\) −13.0000 −0.540729
\(579\) −14.0000 −0.581820
\(580\) −6.00000 −0.249136
\(581\) 0 0
\(582\) −10.0000 −0.414513
\(583\) 24.0000 0.993978
\(584\) 14.0000 0.579324
\(585\) −2.00000 −0.0826898
\(586\) −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
\(591\) 6.00000 0.246807
\(592\) −2.00000 −0.0821995
\(593\) −18.0000 −0.739171 −0.369586 0.929197i \(-0.620500\pi\)
−0.369586 + 0.929197i \(0.620500\pi\)
\(594\) 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 \(0.447742\pi\)
\(600\) 1.00000 0.0408248
\(601\) 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
\(605\) −5.00000 −0.203279
\(606\) −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
\(610\) −2.00000 −0.0809776
\(611\) 16.0000 0.647291
\(612\) −2.00000 −0.0808452
\(613\) −34.0000 −1.37325 −0.686624 0.727013i \(-0.740908\pi\)
−0.686624 + 0.727013i \(0.740908\pi\)
\(614\) 4.00000 0.161427
\(615\) 2.00000 0.0806478
\(616\) 0 0
\(617\) 10.0000 0.402585 0.201292 0.979531i \(-0.435486\pi\)
0.201292 + 0.979531i \(0.435486\pi\)
\(618\) −8.00000 −0.321807
\(619\) 44.0000 1.76851 0.884255 0.467005i \(-0.154667\pi\)
0.884255 + 0.467005i \(0.154667\pi\)
\(620\) −8.00000 −0.321288
\(621\) −8.00000 −0.321029
\(622\) −8.00000 −0.320771
\(623\) 0 0
\(624\) 2.00000 0.0800641
\(625\) 1.00000 0.0400000
\(626\) −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
\(640\) −1.00000 −0.0395285
\(641\) −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 \(-0.370966\pi\)
0.394362 + 0.918955i \(0.370966\pi\)
\(644\) 0 0
\(645\) 12.0000 0.472500
\(646\) −8.00000 −0.314756
\(647\) 32.0000 1.25805 0.629025 0.777385i \(-0.283454\pi\)
0.629025 + 0.777385i \(0.283454\pi\)
\(648\) 1.00000 0.0392837
\(649\) −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 \(-0.731680\pi\)
−0.665261 + 0.746611i \(0.731680\pi\)
\(654\) −2.00000 −0.0782062
\(655\) 12.0000 0.468879
\(656\) −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 \(0.316392\pi\)
\(660\) −4.00000 −0.155700
\(661\) 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
\(670\) −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
\(675\) 1.00000 0.0384900
\(676\) −9.00000 −0.346154
\(677\) −6.00000 −0.230599 −0.115299 0.993331i \(-0.536783\pi\)
−0.115299 + 0.993331i \(0.536783\pi\)
\(678\) −14.0000 −0.537667
\(679\) 0 0
\(680\) 2.00000 0.0766965
\(681\) −12.0000 −0.459841
\(682\) 32.0000 1.22534
\(683\) −4.00000 −0.153056 −0.0765279 0.997067i \(-0.524383\pi\)
−0.0765279 + 0.997067i \(0.524383\pi\)
\(684\) 4.00000 0.152944
\(685\) −10.0000 −0.382080
\(686\) 0 0
\(687\) 26.0000 0.991962
\(688\) −12.0000 −0.457496
\(689\) 12.0000 0.457164
\(690\) 8.00000 0.304555
\(691\) 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
\(695\) 20.0000 0.758643
\(696\) 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
\(705\) −8.00000 −0.301297
\(706\) −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 \(0.662356\pi\)
\(710\) −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
\(717\) −16.0000 −0.597531
\(718\) −8.00000 −0.298557
\(719\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(720\) −1.00000 −0.0372678
\(721\) 0 0
\(722\) −3.00000 −0.111648
\(723\) −18.0000 −0.669427
\(724\) 10.0000 0.371647
\(725\) 6.00000 0.222834
\(726\) 5.00000 0.185567
\(727\) 24.0000 0.890111 0.445055 0.895503i \(-0.353184\pi\)
0.445055 + 0.895503i \(0.353184\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) −14.0000 −0.518163
\(731\) 24.0000 0.887672
\(732\) 2.00000 0.0739221
\(733\) −30.0000 −1.10808 −0.554038 0.832492i \(-0.686914\pi\)
−0.554038 + 0.832492i \(0.686914\pi\)
\(734\) 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 \(0.269648\pi\)
\(740\) 2.00000 0.0735215
\(741\) 8.00000 0.293887
\(742\) 0 0
\(743\) 24.0000 0.880475 0.440237 0.897881i \(-0.354894\pi\)
0.440237 + 0.897881i \(0.354894\pi\)
\(744\) 8.00000 0.293294
\(745\) 18.0000 0.659469
\(746\) 14.0000 0.512576
\(747\) −12.0000 −0.439057
\(748\) −8.00000 −0.292509
\(749\) 0 0
\(750\) −1.00000 −0.0365148
\(751\) −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
\(755\) −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
\(759\) −32.0000 −1.16153
\(760\) −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
\(763\) 0 0
\(764\) −16.0000 −0.578860
\(765\) 2.00000 0.0723102
\(766\) −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 \(-0.345127\pi\)
0.467578 + 0.883952i \(0.345127\pi\)
\(774\) −12.0000 −0.431331
\(775\) 8.00000 0.287368
\(776\) −10.0000 −0.358979
\(777\) 0 0
\(778\) −18.0000 −0.645331
\(779\) −8.00000 −0.286630
\(780\) −2.00000 −0.0716115
\(781\) 32.0000 1.14505
\(782\) 16.0000 0.572159
\(783\) 6.00000 0.214423
\(784\) 0 0
\(785\) 14.0000 0.499681
\(786\) −12.0000 −0.428026
\(787\) 4.00000 0.142585 0.0712923 0.997455i \(-0.477288\pi\)
0.0712923 + 0.997455i \(0.477288\pi\)
\(788\) 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
\(795\) −6.00000 −0.212798
\(796\) 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
\(801\) −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 \(0.732851\pi\)
\(810\) −1.00000 −0.0351364
\(811\) −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
\(815\) −12.0000 −0.420342
\(816\) −2.00000 −0.0700140
\(817\) −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 \(-0.203393\pi\)
0.802706 + 0.596376i \(0.203393\pi\)
\(822\) 10.0000 0.348790
\(823\) −8.00000 −0.278862 −0.139431 0.990232i \(-0.544527\pi\)
−0.139431 + 0.990232i \(0.544527\pi\)
\(824\) −8.00000 −0.278693
\(825\) 4.00000 0.139262
\(826\) 0 0
\(827\) 12.0000 0.417281 0.208640 0.977992i \(-0.433096\pi\)
0.208640 + 0.977992i \(0.433096\pi\)
\(828\) −8.00000 −0.278019
\(829\) 34.0000 1.18087 0.590434 0.807086i \(-0.298956\pi\)
0.590434 + 0.807086i \(0.298956\pi\)
\(830\) 12.0000 0.416526
\(831\) 14.0000 0.485655
\(832\) 2.00000 0.0693375
\(833\) 0 0
\(834\) −20.0000 −0.692543
\(835\) −16.0000 −0.553703
\(836\) 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
\(845\) 9.00000 0.309609
\(846\) 8.00000 0.275046
\(847\) 0 0
\(848\) 6.00000 0.206041
\(849\) −20.0000 −0.686398
\(850\) −2.00000 −0.0685994
\(851\) 16.0000 0.548473
\(852\) 8.00000 0.274075
\(853\) 26.0000 0.890223 0.445112 0.895475i \(-0.353164\pi\)
0.445112 + 0.895475i \(0.353164\pi\)
\(854\) 0 0
\(855\) −4.00000 −0.136797
\(856\) −20.0000 −0.683586
\(857\) 22.0000 0.751506 0.375753 0.926720i \(-0.377384\pi\)
0.375753 + 0.926720i \(0.377384\pi\)
\(858\) 8.00000 0.273115
\(859\) 28.0000 0.955348 0.477674 0.878537i \(-0.341480\pi\)
0.477674 + 0.878537i \(0.341480\pi\)
\(860\) 12.0000 0.409197
\(861\) 0 0
\(862\) 0 0
\(863\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(864\) 1.00000 0.0340207
\(865\) 14.0000 0.476014
\(866\) 6.00000 0.203888
\(867\) −13.0000 −0.441503
\(868\) 0 0
\(869\) 0 0
\(870\) −6.00000 −0.203419
\(871\) 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 \(-0.554001\pi\)
−0.168838 + 0.985644i \(0.554001\pi\)
\(878\) −32.0000 −1.07995
\(879\) −6.00000 −0.202375
\(880\) −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
\(885\) 4.00000 0.134459
\(886\) −4.00000 −0.134383
\(887\) −16.0000 −0.537227 −0.268614 0.963248i \(-0.586566\pi\)
−0.268614 + 0.963248i \(0.586566\pi\)
\(888\) −2.00000 −0.0671156
\(889\) 0 0
\(890\) 2.00000 0.0670402
\(891\) 4.00000 0.134005
\(892\) 0 0
\(893\) 32.0000 1.07084
\(894\) −18.0000 −0.602010
\(895\) 20.0000 0.668526
\(896\) 0 0
\(897\) −16.0000 −0.534224
\(898\) −14.0000 −0.467186
\(899\) 48.0000 1.60089
\(900\) 1.00000 0.0333333
\(901\) −12.0000 −0.399778
\(902\) −8.00000 −0.266371
\(903\) 0 0
\(904\) −14.0000 −0.465633
\(905\) −10.0000 −0.332411
\(906\) 8.00000 0.265782
\(907\) −12.0000 −0.398453 −0.199227 0.979953i \(-0.563843\pi\)
−0.199227 + 0.979953i \(0.563843\pi\)
\(908\) −12.0000 −0.398234
\(909\) −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
\(915\) −2.00000 −0.0661180
\(916\) 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 \(0.542123\pi\)
\(920\) 8.00000 0.263752
\(921\) 4.00000 0.131804
\(922\) 34.0000 1.11973
\(923\) 16.0000 0.526646
\(924\) 0 0
\(925\) −2.00000 −0.0657596
\(926\) −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 \(0.552454\pi\)
\(930\) −8.00000 −0.262330
\(931\) 0 0
\(932\) 10.0000 0.327561
\(933\) −8.00000 −0.261908
\(934\) 20.0000 0.654420
\(935\) 8.00000 0.261628
\(936\) 2.00000 0.0653720
\(937\) −18.0000 −0.588034 −0.294017 0.955800i \(-0.594992\pi\)
−0.294017 + 0.955800i \(0.594992\pi\)
\(938\) 0 0
\(939\) −34.0000 −1.10955
\(940\) −8.00000 −0.260931
\(941\) 18.0000 0.586783 0.293392 0.955992i \(-0.405216\pi\)
0.293392 + 0.955992i \(0.405216\pi\)
\(942\) −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 \(-0.562462\pi\)
−0.194974 + 0.980808i \(0.562462\pi\)
\(948\) 0 0
\(949\) 28.0000 0.908918
\(950\) 4.00000 0.129777
\(951\) 14.0000 0.453981
\(952\) 0 0
\(953\) −6.00000 −0.194359 −0.0971795 0.995267i \(-0.530982\pi\)
−0.0971795 + 0.995267i \(0.530982\pi\)
\(954\) 6.00000 0.194257
\(955\) 16.0000 0.517748
\(956\) −16.0000 −0.517477
\(957\) 24.0000 0.775810
\(958\) 0 0
\(959\) 0 0
\(960\) −1.00000 −0.0322749
\(961\) 33.0000 1.06452
\(962\) −4.00000 −0.128965
\(963\) −20.0000 −0.644491
\(964\) −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
\(969\) −8.00000 −0.256997
\(970\) 10.0000 0.321081
\(971\) −36.0000 −1.15529 −0.577647 0.816286i \(-0.696029\pi\)
−0.577647 + 0.816286i \(0.696029\pi\)
\(972\) 1.00000 0.0320750
\(973\) 0 0
\(974\) −24.0000 −0.769010
\(975\) 2.00000 0.0640513
\(976\) 2.00000 0.0640184
\(977\) 18.0000 0.575871 0.287936 0.957650i \(-0.407031\pi\)
0.287936 + 0.957650i \(0.407031\pi\)
\(978\) 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 \(-0.670473\pi\)
−0.510321 + 0.859984i \(0.670473\pi\)
\(984\) −2.00000 −0.0637577
\(985\) −6.00000 −0.191176
\(986\) −12.0000 −0.382158
\(987\) 0 0
\(988\) 8.00000 0.254514
\(989\) 96.0000 3.05262
\(990\) −4.00000 −0.127128
\(991\) −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
\(995\) −16.0000 −0.507234
\(996\) −12.0000 −0.380235
\(997\) −38.0000 −1.20347 −0.601736 0.798695i \(-0.705524\pi\)
−0.601736 + 0.798695i \(0.705524\pi\)
\(998\) 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 1470.2.a.q.1.1 1
3.2 odd 2 4410.2.a.l.1.1 1
5.4 even 2 7350.2.a.p.1.1 1
7.2 even 3 1470.2.i.b.361.1 2
7.3 odd 6 1470.2.i.f.961.1 2
7.4 even 3 1470.2.i.b.961.1 2
7.5 odd 6 1470.2.i.f.361.1 2
7.6 odd 2 210.2.a.c.1.1 1
21.20 even 2 630.2.a.b.1.1 1
28.27 even 2 1680.2.a.q.1.1 1
35.13 even 4 1050.2.g.d.799.1 2
35.27 even 4 1050.2.g.d.799.2 2
35.34 odd 2 1050.2.a.h.1.1 1
56.13 odd 2 6720.2.a.bp.1.1 1
56.27 even 2 6720.2.a.k.1.1 1
84.83 odd 2 5040.2.a.i.1.1 1
105.62 odd 4 3150.2.g.e.2899.1 2
105.83 odd 4 3150.2.g.e.2899.2 2
105.104 even 2 3150.2.a.w.1.1 1
140.139 even 2 8400.2.a.p.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
210.2.a.c.1.1 1 7.6 odd 2
630.2.a.b.1.1 1 21.20 even 2
1050.2.a.h.1.1 1 35.34 odd 2
1050.2.g.d.799.1 2 35.13 even 4
1050.2.g.d.799.2 2 35.27 even 4
1470.2.a.q.1.1 1 1.1 even 1 trivial
1470.2.i.b.361.1 2 7.2 even 3
1470.2.i.b.961.1 2 7.4 even 3
1470.2.i.f.361.1 2 7.5 odd 6
1470.2.i.f.961.1 2 7.3 odd 6
1680.2.a.q.1.1 1 28.27 even 2
3150.2.a.w.1.1 1 105.104 even 2
3150.2.g.e.2899.1 2 105.62 odd 4
3150.2.g.e.2899.2 2 105.83 odd 4
4410.2.a.l.1.1 1 3.2 odd 2
5040.2.a.i.1.1 1 84.83 odd 2
6720.2.a.k.1.1 1 56.27 even 2
6720.2.a.bp.1.1 1 56.13 odd 2
7350.2.a.p.1.1 1 5.4 even 2
8400.2.a.p.1.1 1 140.139 even 2