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Properties

Label 2415.2.a.b.1.1
Level $2415$
Weight $2$
Character 2415.1
Self dual yes
Analytic conductor $19.284$
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] = [2415,2,Mod(1,2415)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2415, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("2415.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2415 = 3 \cdot 5 \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2415.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: \(19.2838720881\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 2415.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{3} -1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{6} +1.00000 q^{7} +3.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{3} -1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{6} +1.00000 q^{7} +3.00000 q^{8} +1.00000 q^{9} +1.00000 q^{10} +4.00000 q^{11} -1.00000 q^{12} +2.00000 q^{13} -1.00000 q^{14} -1.00000 q^{15} -1.00000 q^{16} -2.00000 q^{17} -1.00000 q^{18} +1.00000 q^{20} +1.00000 q^{21} -4.00000 q^{22} +1.00000 q^{23} +3.00000 q^{24} +1.00000 q^{25} -2.00000 q^{26} +1.00000 q^{27} -1.00000 q^{28} +6.00000 q^{29} +1.00000 q^{30} -8.00000 q^{31} -5.00000 q^{32} +4.00000 q^{33} +2.00000 q^{34} -1.00000 q^{35} -1.00000 q^{36} +10.0000 q^{37} +2.00000 q^{39} -3.00000 q^{40} -2.00000 q^{41} -1.00000 q^{42} +4.00000 q^{43} -4.00000 q^{44} -1.00000 q^{45} -1.00000 q^{46} -8.00000 q^{47} -1.00000 q^{48} +1.00000 q^{49} -1.00000 q^{50} -2.00000 q^{51} -2.00000 q^{52} +6.00000 q^{53} -1.00000 q^{54} -4.00000 q^{55} +3.00000 q^{56} -6.00000 q^{58} -4.00000 q^{59} +1.00000 q^{60} +10.0000 q^{61} +8.00000 q^{62} +1.00000 q^{63} +7.00000 q^{64} -2.00000 q^{65} -4.00000 q^{66} -12.0000 q^{67} +2.00000 q^{68} +1.00000 q^{69} +1.00000 q^{70} -12.0000 q^{71} +3.00000 q^{72} -10.0000 q^{73} -10.0000 q^{74} +1.00000 q^{75} +4.00000 q^{77} -2.00000 q^{78} +16.0000 q^{79} +1.00000 q^{80} +1.00000 q^{81} +2.00000 q^{82} -1.00000 q^{84} +2.00000 q^{85} -4.00000 q^{86} +6.00000 q^{87} +12.0000 q^{88} +6.00000 q^{89} +1.00000 q^{90} +2.00000 q^{91} -1.00000 q^{92} -8.00000 q^{93} +8.00000 q^{94} -5.00000 q^{96} -2.00000 q^{97} -1.00000 q^{98} +4.00000 q^{99} +O(q^{100})\)

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 −0.353553 − 0.935414i \(-0.615027\pi\)
−0.353553 + 0.935414i \(0.615027\pi\)
\(3\) 1.00000 0.577350
\(4\) −1.00000 −0.500000
\(5\) −1.00000 −0.447214
\(6\) −1.00000 −0.408248
\(7\) 1.00000 0.377964
\(8\) 3.00000 1.06066
\(9\) 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 \(0.410544\pi\)
\(14\) −1.00000 −0.267261
\(15\) −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
\(19\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(20\) 1.00000 0.223607
\(21\) 1.00000 0.218218
\(22\) −4.00000 −0.852803
\(23\) 1.00000 0.208514
\(24\) 3.00000 0.612372
\(25\) 1.00000 0.200000
\(26\) −2.00000 −0.392232
\(27\) 1.00000 0.192450
\(28\) −1.00000 −0.188982
\(29\) 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 \(-0.755135\pi\)
−0.718421 + 0.695608i \(0.755135\pi\)
\(32\) −5.00000 −0.883883
\(33\) 4.00000 0.696311
\(34\) 2.00000 0.342997
\(35\) −1.00000 −0.169031
\(36\) −1.00000 −0.166667
\(37\) 10.0000 1.64399 0.821995 − 0.569495i \(-0.192861\pi\)
0.821995 + 0.569495i \(0.192861\pi\)
\(38\) 0 0
\(39\) 2.00000 0.320256
\(40\) −3.00000 −0.474342
\(41\) −2.00000 −0.312348 −0.156174 − 0.987730i \(-0.549916\pi\)
−0.156174 + 0.987730i \(0.549916\pi\)
\(42\) −1.00000 −0.154303
\(43\) 4.00000 0.609994 0.304997 − 0.952353i \(-0.401344\pi\)
0.304997 + 0.952353i \(0.401344\pi\)
\(44\) −4.00000 −0.603023
\(45\) −1.00000 −0.149071
\(46\) −1.00000 −0.147442
\(47\) −8.00000 −1.16692 −0.583460 − 0.812142i \(-0.698301\pi\)
−0.583460 + 0.812142i \(0.698301\pi\)
\(48\) −1.00000 −0.144338
\(49\) 1.00000 0.142857
\(50\) −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
\(56\) 3.00000 0.400892
\(57\) 0 0
\(58\) −6.00000 −0.787839
\(59\) −4.00000 −0.520756 −0.260378 − 0.965507i \(-0.583847\pi\)
−0.260378 + 0.965507i \(0.583847\pi\)
\(60\) 1.00000 0.129099
\(61\) 10.0000 1.28037 0.640184 − 0.768221i \(-0.278858\pi\)
0.640184 + 0.768221i \(0.278858\pi\)
\(62\) 8.00000 1.01600
\(63\) 1.00000 0.125988
\(64\) 7.00000 0.875000
\(65\) −2.00000 −0.248069
\(66\) −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
\(69\) 1.00000 0.120386
\(70\) 1.00000 0.119523
\(71\) −12.0000 −1.42414 −0.712069 − 0.702109i \(-0.752242\pi\)
−0.712069 + 0.702109i \(0.752242\pi\)
\(72\) 3.00000 0.353553
\(73\) −10.0000 −1.17041 −0.585206 − 0.810885i \(-0.698986\pi\)
−0.585206 + 0.810885i \(0.698986\pi\)
\(74\) −10.0000 −1.16248
\(75\) 1.00000 0.115470
\(76\) 0 0
\(77\) 4.00000 0.455842
\(78\) −2.00000 −0.226455
\(79\) 16.0000 1.80014 0.900070 − 0.435745i \(-0.143515\pi\)
0.900070 + 0.435745i \(0.143515\pi\)
\(80\) 1.00000 0.111803
\(81\) 1.00000 0.111111
\(82\) 2.00000 0.220863
\(83\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(84\) −1.00000 −0.109109
\(85\) 2.00000 0.216930
\(86\) −4.00000 −0.431331
\(87\) 6.00000 0.643268
\(88\) 12.0000 1.27920
\(89\) 6.00000 0.635999 0.317999 − 0.948091i \(-0.396989\pi\)
0.317999 + 0.948091i \(0.396989\pi\)
\(90\) 1.00000 0.105409
\(91\) 2.00000 0.209657
\(92\) −1.00000 −0.104257
\(93\) −8.00000 −0.829561
\(94\) 8.00000 0.825137
\(95\) 0 0
\(96\) −5.00000 −0.510310
\(97\) −2.00000 −0.203069 −0.101535 − 0.994832i \(-0.532375\pi\)
−0.101535 + 0.994832i \(0.532375\pi\)
\(98\) −1.00000 −0.101015
\(99\) 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 \(-0.371045\pi\)
0.394132 + 0.919054i \(0.371045\pi\)
\(104\) 6.00000 0.588348
\(105\) −1.00000 −0.0975900
\(106\) −6.00000 −0.582772
\(107\) 4.00000 0.386695 0.193347 − 0.981130i \(-0.438066\pi\)
0.193347 + 0.981130i \(0.438066\pi\)
\(108\) −1.00000 −0.0962250
\(109\) 14.0000 1.34096 0.670478 − 0.741929i \(-0.266089\pi\)
0.670478 + 0.741929i \(0.266089\pi\)
\(110\) 4.00000 0.381385
\(111\) 10.0000 0.949158
\(112\) −1.00000 −0.0944911
\(113\) 10.0000 0.940721 0.470360 − 0.882474i \(-0.344124\pi\)
0.470360 + 0.882474i \(0.344124\pi\)
\(114\) 0 0
\(115\) −1.00000 −0.0932505
\(116\) −6.00000 −0.557086
\(117\) 2.00000 0.184900
\(118\) 4.00000 0.368230
\(119\) −2.00000 −0.183340
\(120\) −3.00000 −0.273861
\(121\) 5.00000 0.454545
\(122\) −10.0000 −0.905357
\(123\) −2.00000 −0.180334
\(124\) 8.00000 0.718421
\(125\) −1.00000 −0.0894427
\(126\) −1.00000 −0.0890871
\(127\) 20.0000 1.77471 0.887357 − 0.461084i \(-0.152539\pi\)
0.887357 + 0.461084i \(0.152539\pi\)
\(128\) 3.00000 0.265165
\(129\) 4.00000 0.352180
\(130\) 2.00000 0.175412
\(131\) 12.0000 1.04844 0.524222 − 0.851581i \(-0.324356\pi\)
0.524222 + 0.851581i \(0.324356\pi\)
\(132\) −4.00000 −0.348155
\(133\) 0 0
\(134\) 12.0000 1.03664
\(135\) −1.00000 −0.0860663
\(136\) −6.00000 −0.514496
\(137\) 10.0000 0.854358 0.427179 − 0.904167i \(-0.359507\pi\)
0.427179 + 0.904167i \(0.359507\pi\)
\(138\) −1.00000 −0.0851257
\(139\) 20.0000 1.69638 0.848189 − 0.529694i \(-0.177693\pi\)
0.848189 + 0.529694i \(0.177693\pi\)
\(140\) 1.00000 0.0845154
\(141\) −8.00000 −0.673722
\(142\) 12.0000 1.00702
\(143\) 8.00000 0.668994
\(144\) −1.00000 −0.0833333
\(145\) −6.00000 −0.498273
\(146\) 10.0000 0.827606
\(147\) 1.00000 0.0824786
\(148\) −10.0000 −0.821995
\(149\) −22.0000 −1.80231 −0.901155 − 0.433497i \(-0.857280\pi\)
−0.901155 + 0.433497i \(0.857280\pi\)
\(150\) −1.00000 −0.0816497
\(151\) 16.0000 1.30206 0.651031 − 0.759051i \(-0.274337\pi\)
0.651031 + 0.759051i \(0.274337\pi\)
\(152\) 0 0
\(153\) −2.00000 −0.161690
\(154\) −4.00000 −0.322329
\(155\) 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 \(0.688685\pi\)
\(158\) −16.0000 −1.27289
\(159\) 6.00000 0.475831
\(160\) 5.00000 0.395285
\(161\) 1.00000 0.0788110
\(162\) −1.00000 −0.0785674
\(163\) 16.0000 1.25322 0.626608 − 0.779334i \(-0.284443\pi\)
0.626608 + 0.779334i \(0.284443\pi\)
\(164\) 2.00000 0.156174
\(165\) −4.00000 −0.311400
\(166\) 0 0
\(167\) −16.0000 −1.23812 −0.619059 − 0.785345i \(-0.712486\pi\)
−0.619059 + 0.785345i \(0.712486\pi\)
\(168\) 3.00000 0.231455
\(169\) −9.00000 −0.692308
\(170\) −2.00000 −0.153393
\(171\) 0 0
\(172\) −4.00000 −0.304997
\(173\) 14.0000 1.06440 0.532200 − 0.846619i \(-0.321365\pi\)
0.532200 + 0.846619i \(0.321365\pi\)
\(174\) −6.00000 −0.454859
\(175\) 1.00000 0.0755929
\(176\) −4.00000 −0.301511
\(177\) −4.00000 −0.300658
\(178\) −6.00000 −0.449719
\(179\) 16.0000 1.19590 0.597948 − 0.801535i \(-0.295983\pi\)
0.597948 + 0.801535i \(0.295983\pi\)
\(180\) 1.00000 0.0745356
\(181\) 10.0000 0.743294 0.371647 − 0.928374i \(-0.378793\pi\)
0.371647 + 0.928374i \(0.378793\pi\)
\(182\) −2.00000 −0.148250
\(183\) 10.0000 0.739221
\(184\) 3.00000 0.221163
\(185\) −10.0000 −0.735215
\(186\) 8.00000 0.586588
\(187\) −8.00000 −0.585018
\(188\) 8.00000 0.583460
\(189\) 1.00000 0.0727393
\(190\) 0 0
\(191\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(192\) 7.00000 0.505181
\(193\) 10.0000 0.719816 0.359908 − 0.932988i \(-0.382808\pi\)
0.359908 + 0.932988i \(0.382808\pi\)
\(194\) 2.00000 0.143592
\(195\) −2.00000 −0.143223
\(196\) −1.00000 −0.0714286
\(197\) 6.00000 0.427482 0.213741 − 0.976890i \(-0.431435\pi\)
0.213741 + 0.976890i \(0.431435\pi\)
\(198\) −4.00000 −0.284268
\(199\) −20.0000 −1.41776 −0.708881 − 0.705328i \(-0.750800\pi\)
−0.708881 + 0.705328i \(0.750800\pi\)
\(200\) 3.00000 0.212132
\(201\) −12.0000 −0.846415
\(202\) 6.00000 0.422159
\(203\) 6.00000 0.421117
\(204\) 2.00000 0.140028
\(205\) 2.00000 0.139686
\(206\) −8.00000 −0.557386
\(207\) 1.00000 0.0695048
\(208\) −2.00000 −0.138675
\(209\) 0 0
\(210\) 1.00000 0.0690066
\(211\) 20.0000 1.37686 0.688428 − 0.725304i \(-0.258301\pi\)
0.688428 + 0.725304i \(0.258301\pi\)
\(212\) −6.00000 −0.412082
\(213\) −12.0000 −0.822226
\(214\) −4.00000 −0.273434
\(215\) −4.00000 −0.272798
\(216\) 3.00000 0.204124
\(217\) −8.00000 −0.543075
\(218\) −14.0000 −0.948200
\(219\) −10.0000 −0.675737
\(220\) 4.00000 0.269680
\(221\) −4.00000 −0.269069
\(222\) −10.0000 −0.671156
\(223\) 16.0000 1.07144 0.535720 − 0.844396i \(-0.320040\pi\)
0.535720 + 0.844396i \(0.320040\pi\)
\(224\) −5.00000 −0.334077
\(225\) 1.00000 0.0666667
\(226\) −10.0000 −0.665190
\(227\) −8.00000 −0.530979 −0.265489 − 0.964114i \(-0.585534\pi\)
−0.265489 + 0.964114i \(0.585534\pi\)
\(228\) 0 0
\(229\) 10.0000 0.660819 0.330409 − 0.943838i \(-0.392813\pi\)
0.330409 + 0.943838i \(0.392813\pi\)
\(230\) 1.00000 0.0659380
\(231\) 4.00000 0.263181
\(232\) 18.0000 1.18176
\(233\) −6.00000 −0.393073 −0.196537 − 0.980497i \(-0.562969\pi\)
−0.196537 + 0.980497i \(0.562969\pi\)
\(234\) −2.00000 −0.130744
\(235\) 8.00000 0.521862
\(236\) 4.00000 0.260378
\(237\) 16.0000 1.03931
\(238\) 2.00000 0.129641
\(239\) −28.0000 −1.81117 −0.905585 − 0.424165i \(-0.860568\pi\)
−0.905585 + 0.424165i \(0.860568\pi\)
\(240\) 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
\(244\) −10.0000 −0.640184
\(245\) −1.00000 −0.0638877
\(246\) 2.00000 0.127515
\(247\) 0 0
\(248\) −24.0000 −1.52400
\(249\) 0 0
\(250\) 1.00000 0.0632456
\(251\) −20.0000 −1.26239 −0.631194 − 0.775625i \(-0.717435\pi\)
−0.631194 + 0.775625i \(0.717435\pi\)
\(252\) −1.00000 −0.0629941
\(253\) 4.00000 0.251478
\(254\) −20.0000 −1.25491
\(255\) 2.00000 0.125245
\(256\) −17.0000 −1.06250
\(257\) −6.00000 −0.374270 −0.187135 − 0.982334i \(-0.559920\pi\)
−0.187135 + 0.982334i \(0.559920\pi\)
\(258\) −4.00000 −0.249029
\(259\) 10.0000 0.621370
\(260\) 2.00000 0.124035
\(261\) 6.00000 0.371391
\(262\) −12.0000 −0.741362
\(263\) −16.0000 −0.986602 −0.493301 − 0.869859i \(-0.664210\pi\)
−0.493301 + 0.869859i \(0.664210\pi\)
\(264\) 12.0000 0.738549
\(265\) −6.00000 −0.368577
\(266\) 0 0
\(267\) 6.00000 0.367194
\(268\) 12.0000 0.733017
\(269\) 10.0000 0.609711 0.304855 − 0.952399i \(-0.401392\pi\)
0.304855 + 0.952399i \(0.401392\pi\)
\(270\) 1.00000 0.0608581
\(271\) −16.0000 −0.971931 −0.485965 − 0.873978i \(-0.661532\pi\)
−0.485965 + 0.873978i \(0.661532\pi\)
\(272\) 2.00000 0.121268
\(273\) 2.00000 0.121046
\(274\) −10.0000 −0.604122
\(275\) 4.00000 0.241209
\(276\) −1.00000 −0.0601929
\(277\) 22.0000 1.32185 0.660926 − 0.750451i \(-0.270164\pi\)
0.660926 + 0.750451i \(0.270164\pi\)
\(278\) −20.0000 −1.19952
\(279\) −8.00000 −0.478947
\(280\) −3.00000 −0.179284
\(281\) −18.0000 −1.07379 −0.536895 − 0.843649i \(-0.680403\pi\)
−0.536895 + 0.843649i \(0.680403\pi\)
\(282\) 8.00000 0.476393
\(283\) 28.0000 1.66443 0.832214 − 0.554455i \(-0.187073\pi\)
0.832214 + 0.554455i \(0.187073\pi\)
\(284\) 12.0000 0.712069
\(285\) 0 0
\(286\) −8.00000 −0.473050
\(287\) −2.00000 −0.118056
\(288\) −5.00000 −0.294628
\(289\) −13.0000 −0.764706
\(290\) 6.00000 0.352332
\(291\) −2.00000 −0.117242
\(292\) 10.0000 0.585206
\(293\) 26.0000 1.51894 0.759468 − 0.650545i \(-0.225459\pi\)
0.759468 + 0.650545i \(0.225459\pi\)
\(294\) −1.00000 −0.0583212
\(295\) 4.00000 0.232889
\(296\) 30.0000 1.74371
\(297\) 4.00000 0.232104
\(298\) 22.0000 1.27443
\(299\) 2.00000 0.115663
\(300\) −1.00000 −0.0577350
\(301\) 4.00000 0.230556
\(302\) −16.0000 −0.920697
\(303\) −6.00000 −0.344691
\(304\) 0 0
\(305\) −10.0000 −0.572598
\(306\) 2.00000 0.114332
\(307\) 28.0000 1.59804 0.799022 − 0.601302i \(-0.205351\pi\)
0.799022 + 0.601302i \(0.205351\pi\)
\(308\) −4.00000 −0.227921
\(309\) 8.00000 0.455104
\(310\) −8.00000 −0.454369
\(311\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(312\) 6.00000 0.339683
\(313\) 22.0000 1.24351 0.621757 − 0.783210i \(-0.286419\pi\)
0.621757 + 0.783210i \(0.286419\pi\)
\(314\) 14.0000 0.790066
\(315\) −1.00000 −0.0563436
\(316\) −16.0000 −0.900070
\(317\) −2.00000 −0.112331 −0.0561656 − 0.998421i \(-0.517887\pi\)
−0.0561656 + 0.998421i \(0.517887\pi\)
\(318\) −6.00000 −0.336463
\(319\) 24.0000 1.34374
\(320\) −7.00000 −0.391312
\(321\) 4.00000 0.223258
\(322\) −1.00000 −0.0557278
\(323\) 0 0
\(324\) −1.00000 −0.0555556
\(325\) 2.00000 0.110940
\(326\) −16.0000 −0.886158
\(327\) 14.0000 0.774202
\(328\) −6.00000 −0.331295
\(329\) −8.00000 −0.441054
\(330\) 4.00000 0.220193
\(331\) −28.0000 −1.53902 −0.769510 − 0.638635i \(-0.779499\pi\)
−0.769510 + 0.638635i \(0.779499\pi\)
\(332\) 0 0
\(333\) 10.0000 0.547997
\(334\) 16.0000 0.875481
\(335\) 12.0000 0.655630
\(336\) −1.00000 −0.0545545
\(337\) −18.0000 −0.980522 −0.490261 − 0.871576i \(-0.663099\pi\)
−0.490261 + 0.871576i \(0.663099\pi\)
\(338\) 9.00000 0.489535
\(339\) 10.0000 0.543125
\(340\) −2.00000 −0.108465
\(341\) −32.0000 −1.73290
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) 12.0000 0.646997
\(345\) −1.00000 −0.0538382
\(346\) −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
\(349\) −2.00000 −0.107058 −0.0535288 − 0.998566i \(-0.517047\pi\)
−0.0535288 + 0.998566i \(0.517047\pi\)
\(350\) −1.00000 −0.0534522
\(351\) 2.00000 0.106752
\(352\) −20.0000 −1.06600
\(353\) 26.0000 1.38384 0.691920 − 0.721974i \(-0.256765\pi\)
0.691920 + 0.721974i \(0.256765\pi\)
\(354\) 4.00000 0.212598
\(355\) 12.0000 0.636894
\(356\) −6.00000 −0.317999
\(357\) −2.00000 −0.105851
\(358\) −16.0000 −0.845626
\(359\) −32.0000 −1.68890 −0.844448 − 0.535638i \(-0.820071\pi\)
−0.844448 + 0.535638i \(0.820071\pi\)
\(360\) −3.00000 −0.158114
\(361\) −19.0000 −1.00000
\(362\) −10.0000 −0.525588
\(363\) 5.00000 0.262432
\(364\) −2.00000 −0.104828
\(365\) 10.0000 0.523424
\(366\) −10.0000 −0.522708
\(367\) −16.0000 −0.835193 −0.417597 − 0.908633i \(-0.637127\pi\)
−0.417597 + 0.908633i \(0.637127\pi\)
\(368\) −1.00000 −0.0521286
\(369\) −2.00000 −0.104116
\(370\) 10.0000 0.519875
\(371\) 6.00000 0.311504
\(372\) 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
\(375\) −1.00000 −0.0516398
\(376\) −24.0000 −1.23771
\(377\) 12.0000 0.618031
\(378\) −1.00000 −0.0514344
\(379\) 20.0000 1.02733 0.513665 − 0.857991i \(-0.328287\pi\)
0.513665 + 0.857991i \(0.328287\pi\)
\(380\) 0 0
\(381\) 20.0000 1.02463
\(382\) 0 0
\(383\) 12.0000 0.613171 0.306586 − 0.951843i \(-0.400813\pi\)
0.306586 + 0.951843i \(0.400813\pi\)
\(384\) 3.00000 0.153093
\(385\) −4.00000 −0.203859
\(386\) −10.0000 −0.508987
\(387\) 4.00000 0.203331
\(388\) 2.00000 0.101535
\(389\) −30.0000 −1.52106 −0.760530 − 0.649303i \(-0.775061\pi\)
−0.760530 + 0.649303i \(0.775061\pi\)
\(390\) 2.00000 0.101274
\(391\) −2.00000 −0.101144
\(392\) 3.00000 0.151523
\(393\) 12.0000 0.605320
\(394\) −6.00000 −0.302276
\(395\) −16.0000 −0.805047
\(396\) −4.00000 −0.201008
\(397\) 18.0000 0.903394 0.451697 − 0.892171i \(-0.350819\pi\)
0.451697 + 0.892171i \(0.350819\pi\)
\(398\) 20.0000 1.00251
\(399\) 0 0
\(400\) −1.00000 −0.0500000
\(401\) 22.0000 1.09863 0.549314 − 0.835616i \(-0.314889\pi\)
0.549314 + 0.835616i \(0.314889\pi\)
\(402\) 12.0000 0.598506
\(403\) −16.0000 −0.797017
\(404\) 6.00000 0.298511
\(405\) −1.00000 −0.0496904
\(406\) −6.00000 −0.297775
\(407\) 40.0000 1.98273
\(408\) −6.00000 −0.297044
\(409\) 10.0000 0.494468 0.247234 − 0.968956i \(-0.420478\pi\)
0.247234 + 0.968956i \(0.420478\pi\)
\(410\) −2.00000 −0.0987730
\(411\) 10.0000 0.493264
\(412\) −8.00000 −0.394132
\(413\) −4.00000 −0.196827
\(414\) −1.00000 −0.0491473
\(415\) 0 0
\(416\) −10.0000 −0.490290
\(417\) 20.0000 0.979404
\(418\) 0 0
\(419\) 12.0000 0.586238 0.293119 − 0.956076i \(-0.405307\pi\)
0.293119 + 0.956076i \(0.405307\pi\)
\(420\) 1.00000 0.0487950
\(421\) −10.0000 −0.487370 −0.243685 − 0.969854i \(-0.578356\pi\)
−0.243685 + 0.969854i \(0.578356\pi\)
\(422\) −20.0000 −0.973585
\(423\) −8.00000 −0.388973
\(424\) 18.0000 0.874157
\(425\) −2.00000 −0.0970143
\(426\) 12.0000 0.581402
\(427\) 10.0000 0.483934
\(428\) −4.00000 −0.193347
\(429\) 8.00000 0.386244
\(430\) 4.00000 0.192897
\(431\) −32.0000 −1.54139 −0.770693 − 0.637207i \(-0.780090\pi\)
−0.770693 + 0.637207i \(0.780090\pi\)
\(432\) −1.00000 −0.0481125
\(433\) −34.0000 −1.63394 −0.816968 − 0.576683i \(-0.804347\pi\)
−0.816968 + 0.576683i \(0.804347\pi\)
\(434\) 8.00000 0.384012
\(435\) −6.00000 −0.287678
\(436\) −14.0000 −0.670478
\(437\) 0 0
\(438\) 10.0000 0.477818
\(439\) 16.0000 0.763638 0.381819 − 0.924237i \(-0.375298\pi\)
0.381819 + 0.924237i \(0.375298\pi\)
\(440\) −12.0000 −0.572078
\(441\) 1.00000 0.0476190
\(442\) 4.00000 0.190261
\(443\) −20.0000 −0.950229 −0.475114 − 0.879924i \(-0.657593\pi\)
−0.475114 + 0.879924i \(0.657593\pi\)
\(444\) −10.0000 −0.474579
\(445\) −6.00000 −0.284427
\(446\) −16.0000 −0.757622
\(447\) −22.0000 −1.04056
\(448\) 7.00000 0.330719
\(449\) 2.00000 0.0943858 0.0471929 − 0.998886i \(-0.484972\pi\)
0.0471929 + 0.998886i \(0.484972\pi\)
\(450\) −1.00000 −0.0471405
\(451\) −8.00000 −0.376705
\(452\) −10.0000 −0.470360
\(453\) 16.0000 0.751746
\(454\) 8.00000 0.375459
\(455\) −2.00000 −0.0937614
\(456\) 0 0
\(457\) 22.0000 1.02912 0.514558 − 0.857455i \(-0.327956\pi\)
0.514558 + 0.857455i \(0.327956\pi\)
\(458\) −10.0000 −0.467269
\(459\) −2.00000 −0.0933520
\(460\) 1.00000 0.0466252
\(461\) 18.0000 0.838344 0.419172 − 0.907907i \(-0.362320\pi\)
0.419172 + 0.907907i \(0.362320\pi\)
\(462\) −4.00000 −0.186097
\(463\) −20.0000 −0.929479 −0.464739 − 0.885448i \(-0.653852\pi\)
−0.464739 + 0.885448i \(0.653852\pi\)
\(464\) −6.00000 −0.278543
\(465\) 8.00000 0.370991
\(466\) 6.00000 0.277945
\(467\) 8.00000 0.370196 0.185098 − 0.982720i \(-0.440740\pi\)
0.185098 + 0.982720i \(0.440740\pi\)
\(468\) −2.00000 −0.0924500
\(469\) −12.0000 −0.554109
\(470\) −8.00000 −0.369012
\(471\) −14.0000 −0.645086
\(472\) −12.0000 −0.552345
\(473\) 16.0000 0.735681
\(474\) −16.0000 −0.734904
\(475\) 0 0
\(476\) 2.00000 0.0916698
\(477\) 6.00000 0.274721
\(478\) 28.0000 1.28069
\(479\) −32.0000 −1.46212 −0.731059 − 0.682315i \(-0.760973\pi\)
−0.731059 + 0.682315i \(0.760973\pi\)
\(480\) 5.00000 0.228218
\(481\) 20.0000 0.911922
\(482\) 18.0000 0.819878
\(483\) 1.00000 0.0455016
\(484\) −5.00000 −0.227273
\(485\) 2.00000 0.0908153
\(486\) −1.00000 −0.0453609
\(487\) −20.0000 −0.906287 −0.453143 − 0.891438i \(-0.649697\pi\)
−0.453143 + 0.891438i \(0.649697\pi\)
\(488\) 30.0000 1.35804
\(489\) 16.0000 0.723545
\(490\) 1.00000 0.0451754
\(491\) 24.0000 1.08310 0.541552 − 0.840667i \(-0.317837\pi\)
0.541552 + 0.840667i \(0.317837\pi\)
\(492\) 2.00000 0.0901670
\(493\) −12.0000 −0.540453
\(494\) 0 0
\(495\) −4.00000 −0.179787
\(496\) 8.00000 0.359211
\(497\) −12.0000 −0.538274
\(498\) 0 0
\(499\) 12.0000 0.537194 0.268597 − 0.963253i \(-0.413440\pi\)
0.268597 + 0.963253i \(0.413440\pi\)
\(500\) 1.00000 0.0447214
\(501\) −16.0000 −0.714827
\(502\) 20.0000 0.892644
\(503\) −12.0000 −0.535054 −0.267527 − 0.963550i \(-0.586206\pi\)
−0.267527 + 0.963550i \(0.586206\pi\)
\(504\) 3.00000 0.133631
\(505\) 6.00000 0.266996
\(506\) −4.00000 −0.177822
\(507\) −9.00000 −0.399704
\(508\) −20.0000 −0.887357
\(509\) −30.0000 −1.32973 −0.664863 − 0.746965i \(-0.731510\pi\)
−0.664863 + 0.746965i \(0.731510\pi\)
\(510\) −2.00000 −0.0885615
\(511\) −10.0000 −0.442374
\(512\) 11.0000 0.486136
\(513\) 0 0
\(514\) 6.00000 0.264649
\(515\) −8.00000 −0.352522
\(516\) −4.00000 −0.176090
\(517\) −32.0000 −1.40736
\(518\) −10.0000 −0.439375
\(519\) 14.0000 0.614532
\(520\) −6.00000 −0.263117
\(521\) 22.0000 0.963837 0.481919 − 0.876216i \(-0.339940\pi\)
0.481919 + 0.876216i \(0.339940\pi\)
\(522\) −6.00000 −0.262613
\(523\) 44.0000 1.92399 0.961993 − 0.273075i \(-0.0880406\pi\)
0.961993 + 0.273075i \(0.0880406\pi\)
\(524\) −12.0000 −0.524222
\(525\) 1.00000 0.0436436
\(526\) 16.0000 0.697633
\(527\) 16.0000 0.696971
\(528\) −4.00000 −0.174078
\(529\) 1.00000 0.0434783
\(530\) 6.00000 0.260623
\(531\) −4.00000 −0.173585
\(532\) 0 0
\(533\) −4.00000 −0.173259
\(534\) −6.00000 −0.259645
\(535\) −4.00000 −0.172935
\(536\) −36.0000 −1.55496
\(537\) 16.0000 0.690451
\(538\) −10.0000 −0.431131
\(539\) 4.00000 0.172292
\(540\) 1.00000 0.0430331
\(541\) −34.0000 −1.46177 −0.730887 − 0.682498i \(-0.760893\pi\)
−0.730887 + 0.682498i \(0.760893\pi\)
\(542\) 16.0000 0.687259
\(543\) 10.0000 0.429141
\(544\) 10.0000 0.428746
\(545\) −14.0000 −0.599694
\(546\) −2.00000 −0.0855921
\(547\) −8.00000 −0.342055 −0.171028 − 0.985266i \(-0.554709\pi\)
−0.171028 + 0.985266i \(0.554709\pi\)
\(548\) −10.0000 −0.427179
\(549\) 10.0000 0.426790
\(550\) −4.00000 −0.170561
\(551\) 0 0
\(552\) 3.00000 0.127688
\(553\) 16.0000 0.680389
\(554\) −22.0000 −0.934690
\(555\) −10.0000 −0.424476
\(556\) −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
\(559\) 8.00000 0.338364
\(560\) 1.00000 0.0422577
\(561\) −8.00000 −0.337760
\(562\) 18.0000 0.759284
\(563\) −32.0000 −1.34864 −0.674320 − 0.738440i \(-0.735563\pi\)
−0.674320 + 0.738440i \(0.735563\pi\)
\(564\) 8.00000 0.336861
\(565\) −10.0000 −0.420703
\(566\) −28.0000 −1.17693
\(567\) 1.00000 0.0419961
\(568\) −36.0000 −1.51053
\(569\) 30.0000 1.25767 0.628833 − 0.777541i \(-0.283533\pi\)
0.628833 + 0.777541i \(0.283533\pi\)
\(570\) 0 0
\(571\) −44.0000 −1.84134 −0.920671 − 0.390339i \(-0.872358\pi\)
−0.920671 + 0.390339i \(0.872358\pi\)
\(572\) −8.00000 −0.334497
\(573\) 0 0
\(574\) 2.00000 0.0834784
\(575\) 1.00000 0.0417029
\(576\) 7.00000 0.291667
\(577\) −26.0000 −1.08239 −0.541197 − 0.840896i \(-0.682029\pi\)
−0.541197 + 0.840896i \(0.682029\pi\)
\(578\) 13.0000 0.540729
\(579\) 10.0000 0.415586
\(580\) 6.00000 0.249136
\(581\) 0 0
\(582\) 2.00000 0.0829027
\(583\) 24.0000 0.993978
\(584\) −30.0000 −1.24141
\(585\) −2.00000 −0.0826898
\(586\) −26.0000 −1.07405
\(587\) 20.0000 0.825488 0.412744 − 0.910847i \(-0.364570\pi\)
0.412744 + 0.910847i \(0.364570\pi\)
\(588\) −1.00000 −0.0412393
\(589\) 0 0
\(590\) −4.00000 −0.164677
\(591\) 6.00000 0.246807
\(592\) −10.0000 −0.410997
\(593\) −22.0000 −0.903432 −0.451716 − 0.892162i \(-0.649188\pi\)
−0.451716 + 0.892162i \(0.649188\pi\)
\(594\) −4.00000 −0.164122
\(595\) 2.00000 0.0819920
\(596\) 22.0000 0.901155
\(597\) −20.0000 −0.818546
\(598\) −2.00000 −0.0817861
\(599\) −36.0000 −1.47092 −0.735460 − 0.677568i \(-0.763034\pi\)
−0.735460 + 0.677568i \(0.763034\pi\)
\(600\) 3.00000 0.122474
\(601\) 42.0000 1.71322 0.856608 − 0.515968i \(-0.172568\pi\)
0.856608 + 0.515968i \(0.172568\pi\)
\(602\) −4.00000 −0.163028
\(603\) −12.0000 −0.488678
\(604\) −16.0000 −0.651031
\(605\) −5.00000 −0.203279
\(606\) 6.00000 0.243733
\(607\) −24.0000 −0.974130 −0.487065 − 0.873366i \(-0.661933\pi\)
−0.487065 + 0.873366i \(0.661933\pi\)
\(608\) 0 0
\(609\) 6.00000 0.243132
\(610\) 10.0000 0.404888
\(611\) −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 \(0.259092\pi\)
\(614\) −28.0000 −1.12999
\(615\) 2.00000 0.0806478
\(616\) 12.0000 0.483494
\(617\) 34.0000 1.36879 0.684394 − 0.729112i \(-0.260067\pi\)
0.684394 + 0.729112i \(0.260067\pi\)
\(618\) −8.00000 −0.321807
\(619\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(620\) −8.00000 −0.321288
\(621\) 1.00000 0.0401286
\(622\) 0 0
\(623\) 6.00000 0.240385
\(624\) −2.00000 −0.0800641
\(625\) 1.00000 0.0400000
\(626\) −22.0000 −0.879297
\(627\) 0 0
\(628\) 14.0000 0.558661
\(629\) −20.0000 −0.797452
\(630\) 1.00000 0.0398410
\(631\) 16.0000 0.636950 0.318475 − 0.947931i \(-0.396829\pi\)
0.318475 + 0.947931i \(0.396829\pi\)
\(632\) 48.0000 1.90934
\(633\) 20.0000 0.794929
\(634\) 2.00000 0.0794301
\(635\) −20.0000 −0.793676
\(636\) −6.00000 −0.237915
\(637\) 2.00000 0.0792429
\(638\) −24.0000 −0.950169
\(639\) −12.0000 −0.474713
\(640\) −3.00000 −0.118585
\(641\) −18.0000 −0.710957 −0.355479 − 0.934684i \(-0.615682\pi\)
−0.355479 + 0.934684i \(0.615682\pi\)
\(642\) −4.00000 −0.157867
\(643\) −36.0000 −1.41970 −0.709851 − 0.704352i \(-0.751238\pi\)
−0.709851 + 0.704352i \(0.751238\pi\)
\(644\) −1.00000 −0.0394055
\(645\) −4.00000 −0.157500
\(646\) 0 0
\(647\) −16.0000 −0.629025 −0.314512 − 0.949253i \(-0.601841\pi\)
−0.314512 + 0.949253i \(0.601841\pi\)
\(648\) 3.00000 0.117851
\(649\) −16.0000 −0.628055
\(650\) −2.00000 −0.0784465
\(651\) −8.00000 −0.313545
\(652\) −16.0000 −0.626608
\(653\) −18.0000 −0.704394 −0.352197 − 0.935926i \(-0.614565\pi\)
−0.352197 + 0.935926i \(0.614565\pi\)
\(654\) −14.0000 −0.547443
\(655\) −12.0000 −0.468879
\(656\) 2.00000 0.0780869
\(657\) −10.0000 −0.390137
\(658\) 8.00000 0.311872
\(659\) 12.0000 0.467454 0.233727 − 0.972302i \(-0.424908\pi\)
0.233727 + 0.972302i \(0.424908\pi\)
\(660\) 4.00000 0.155700
\(661\) −14.0000 −0.544537 −0.272268 − 0.962221i \(-0.587774\pi\)
−0.272268 + 0.962221i \(0.587774\pi\)
\(662\) 28.0000 1.08825
\(663\) −4.00000 −0.155347
\(664\) 0 0
\(665\) 0 0
\(666\) −10.0000 −0.387492
\(667\) 6.00000 0.232321
\(668\) 16.0000 0.619059
\(669\) 16.0000 0.618596
\(670\) −12.0000 −0.463600
\(671\) 40.0000 1.54418
\(672\) −5.00000 −0.192879
\(673\) 26.0000 1.00223 0.501113 − 0.865382i \(-0.332924\pi\)
0.501113 + 0.865382i \(0.332924\pi\)
\(674\) 18.0000 0.693334
\(675\) 1.00000 0.0384900
\(676\) 9.00000 0.346154
\(677\) −14.0000 −0.538064 −0.269032 − 0.963131i \(-0.586704\pi\)
−0.269032 + 0.963131i \(0.586704\pi\)
\(678\) −10.0000 −0.384048
\(679\) −2.00000 −0.0767530
\(680\) 6.00000 0.230089
\(681\) −8.00000 −0.306561
\(682\) 32.0000 1.22534
\(683\) 20.0000 0.765279 0.382639 − 0.923898i \(-0.375015\pi\)
0.382639 + 0.923898i \(0.375015\pi\)
\(684\) 0 0
\(685\) −10.0000 −0.382080
\(686\) −1.00000 −0.0381802
\(687\) 10.0000 0.381524
\(688\) −4.00000 −0.152499
\(689\) 12.0000 0.457164
\(690\) 1.00000 0.0380693
\(691\) −28.0000 −1.06517 −0.532585 − 0.846376i \(-0.678779\pi\)
−0.532585 + 0.846376i \(0.678779\pi\)
\(692\) −14.0000 −0.532200
\(693\) 4.00000 0.151947
\(694\) −12.0000 −0.455514
\(695\) −20.0000 −0.758643
\(696\) 18.0000 0.682288
\(697\) 4.00000 0.151511
\(698\) 2.00000 0.0757011
\(699\) −6.00000 −0.226941
\(700\) −1.00000 −0.0377964
\(701\) −6.00000 −0.226617 −0.113308 − 0.993560i \(-0.536145\pi\)
−0.113308 + 0.993560i \(0.536145\pi\)
\(702\) −2.00000 −0.0754851
\(703\) 0 0
\(704\) 28.0000 1.05529
\(705\) 8.00000 0.301297
\(706\) −26.0000 −0.978523
\(707\) −6.00000 −0.225653
\(708\) 4.00000 0.150329
\(709\) −18.0000 −0.676004 −0.338002 − 0.941145i \(-0.609751\pi\)
−0.338002 + 0.941145i \(0.609751\pi\)
\(710\) −12.0000 −0.450352
\(711\) 16.0000 0.600047
\(712\) 18.0000 0.674579
\(713\) −8.00000 −0.299602
\(714\) 2.00000 0.0748481
\(715\) −8.00000 −0.299183
\(716\) −16.0000 −0.597948
\(717\) −28.0000 −1.04568
\(718\) 32.0000 1.19423
\(719\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(720\) 1.00000 0.0372678
\(721\) 8.00000 0.297936
\(722\) 19.0000 0.707107
\(723\) −18.0000 −0.669427
\(724\) −10.0000 −0.371647
\(725\) 6.00000 0.222834
\(726\) −5.00000 −0.185567
\(727\) 8.00000 0.296704 0.148352 − 0.988935i \(-0.452603\pi\)
0.148352 + 0.988935i \(0.452603\pi\)
\(728\) 6.00000 0.222375
\(729\) 1.00000 0.0370370
\(730\) −10.0000 −0.370117
\(731\) −8.00000 −0.295891
\(732\) −10.0000 −0.369611
\(733\) −30.0000 −1.10808 −0.554038 − 0.832492i \(-0.686914\pi\)
−0.554038 + 0.832492i \(0.686914\pi\)
\(734\) 16.0000 0.590571
\(735\) −1.00000 −0.0368856
\(736\) −5.00000 −0.184302
\(737\) −48.0000 −1.76810
\(738\) 2.00000 0.0736210
\(739\) −12.0000 −0.441427 −0.220714 − 0.975339i \(-0.570839\pi\)
−0.220714 + 0.975339i \(0.570839\pi\)
\(740\) 10.0000 0.367607
\(741\) 0 0
\(742\) −6.00000 −0.220267
\(743\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(744\) −24.0000 −0.879883
\(745\) 22.0000 0.806018
\(746\) 14.0000 0.512576
\(747\) 0 0
\(748\) 8.00000 0.292509
\(749\) 4.00000 0.146157
\(750\) 1.00000 0.0365148
\(751\) −40.0000 −1.45962 −0.729810 − 0.683650i \(-0.760392\pi\)
−0.729810 + 0.683650i \(0.760392\pi\)
\(752\) 8.00000 0.291730
\(753\) −20.0000 −0.728841
\(754\) −12.0000 −0.437014
\(755\) −16.0000 −0.582300
\(756\) −1.00000 −0.0363696
\(757\) −6.00000 −0.218074 −0.109037 − 0.994038i \(-0.534777\pi\)
−0.109037 + 0.994038i \(0.534777\pi\)
\(758\) −20.0000 −0.726433
\(759\) 4.00000 0.145191
\(760\) 0 0
\(761\) −42.0000 −1.52250 −0.761249 − 0.648459i \(-0.775414\pi\)
−0.761249 + 0.648459i \(0.775414\pi\)
\(762\) −20.0000 −0.724524
\(763\) 14.0000 0.506834
\(764\) 0 0
\(765\) 2.00000 0.0723102
\(766\) −12.0000 −0.433578
\(767\) −8.00000 −0.288863
\(768\) −17.0000 −0.613435
\(769\) −2.00000 −0.0721218 −0.0360609 − 0.999350i \(-0.511481\pi\)
−0.0360609 + 0.999350i \(0.511481\pi\)
\(770\) 4.00000 0.144150
\(771\) −6.00000 −0.216085
\(772\) −10.0000 −0.359908
\(773\) 18.0000 0.647415 0.323708 − 0.946157i \(-0.395071\pi\)
0.323708 + 0.946157i \(0.395071\pi\)
\(774\) −4.00000 −0.143777
\(775\) −8.00000 −0.287368
\(776\) −6.00000 −0.215387
\(777\) 10.0000 0.358748
\(778\) 30.0000 1.07555
\(779\) 0 0
\(780\) 2.00000 0.0716115
\(781\) −48.0000 −1.71758
\(782\) 2.00000 0.0715199
\(783\) 6.00000 0.214423
\(784\) −1.00000 −0.0357143
\(785\) 14.0000 0.499681
\(786\) −12.0000 −0.428026
\(787\) 36.0000 1.28326 0.641631 − 0.767014i \(-0.278258\pi\)
0.641631 + 0.767014i \(0.278258\pi\)
\(788\) −6.00000 −0.213741
\(789\) −16.0000 −0.569615
\(790\) 16.0000 0.569254
\(791\) 10.0000 0.355559
\(792\) 12.0000 0.426401
\(793\) 20.0000 0.710221
\(794\) −18.0000 −0.638796
\(795\) −6.00000 −0.212798
\(796\) 20.0000 0.708881
\(797\) −14.0000 −0.495905 −0.247953 − 0.968772i \(-0.579758\pi\)
−0.247953 + 0.968772i \(0.579758\pi\)
\(798\) 0 0
\(799\) 16.0000 0.566039
\(800\) −5.00000 −0.176777
\(801\) 6.00000 0.212000
\(802\) −22.0000 −0.776847
\(803\) −40.0000 −1.41157
\(804\) 12.0000 0.423207
\(805\) −1.00000 −0.0352454
\(806\) 16.0000 0.563576
\(807\) 10.0000 0.352017
\(808\) −18.0000 −0.633238
\(809\) 18.0000 0.632846 0.316423 − 0.948618i \(-0.397518\pi\)
0.316423 + 0.948618i \(0.397518\pi\)
\(810\) 1.00000 0.0351364
\(811\) −12.0000 −0.421377 −0.210688 − 0.977553i \(-0.567571\pi\)
−0.210688 + 0.977553i \(0.567571\pi\)
\(812\) −6.00000 −0.210559
\(813\) −16.0000 −0.561144
\(814\) −40.0000 −1.40200
\(815\) −16.0000 −0.560456
\(816\) 2.00000 0.0700140
\(817\) 0 0
\(818\) −10.0000 −0.349642
\(819\) 2.00000 0.0698857
\(820\) −2.00000 −0.0698430
\(821\) 30.0000 1.04701 0.523504 − 0.852023i \(-0.324625\pi\)
0.523504 + 0.852023i \(0.324625\pi\)
\(822\) −10.0000 −0.348790
\(823\) 4.00000 0.139431 0.0697156 − 0.997567i \(-0.477791\pi\)
0.0697156 + 0.997567i \(0.477791\pi\)
\(824\) 24.0000 0.836080
\(825\) 4.00000 0.139262
\(826\) 4.00000 0.139178
\(827\) 12.0000 0.417281 0.208640 − 0.977992i \(-0.433096\pi\)
0.208640 + 0.977992i \(0.433096\pi\)
\(828\) −1.00000 −0.0347524
\(829\) −18.0000 −0.625166 −0.312583 − 0.949890i \(-0.601194\pi\)
−0.312583 + 0.949890i \(0.601194\pi\)
\(830\) 0 0
\(831\) 22.0000 0.763172
\(832\) 14.0000 0.485363
\(833\) −2.00000 −0.0692959
\(834\) −20.0000 −0.692543
\(835\) 16.0000 0.553703
\(836\) 0 0
\(837\) −8.00000 −0.276520
\(838\) −12.0000 −0.414533
\(839\) −40.0000 −1.38095 −0.690477 − 0.723355i \(-0.742599\pi\)
−0.690477 + 0.723355i \(0.742599\pi\)
\(840\) −3.00000 −0.103510
\(841\) 7.00000 0.241379
\(842\) 10.0000 0.344623
\(843\) −18.0000 −0.619953
\(844\) −20.0000 −0.688428
\(845\) 9.00000 0.309609
\(846\) 8.00000 0.275046
\(847\) 5.00000 0.171802
\(848\) −6.00000 −0.206041
\(849\) 28.0000 0.960958
\(850\) 2.00000 0.0685994
\(851\) 10.0000 0.342796
\(852\) 12.0000 0.411113
\(853\) 58.0000 1.98588 0.992941 − 0.118609i \(-0.0378434\pi\)
0.992941 + 0.118609i \(0.0378434\pi\)
\(854\) −10.0000 −0.342193
\(855\) 0 0
\(856\) 12.0000 0.410152
\(857\) −22.0000 −0.751506 −0.375753 − 0.926720i \(-0.622616\pi\)
−0.375753 + 0.926720i \(0.622616\pi\)
\(858\) −8.00000 −0.273115
\(859\) 44.0000 1.50126 0.750630 − 0.660722i \(-0.229750\pi\)
0.750630 + 0.660722i \(0.229750\pi\)
\(860\) 4.00000 0.136399
\(861\) −2.00000 −0.0681598
\(862\) 32.0000 1.08992
\(863\) −24.0000 −0.816970 −0.408485 − 0.912765i \(-0.633943\pi\)
−0.408485 + 0.912765i \(0.633943\pi\)
\(864\) −5.00000 −0.170103
\(865\) −14.0000 −0.476014
\(866\) 34.0000 1.15537
\(867\) −13.0000 −0.441503
\(868\) 8.00000 0.271538
\(869\) 64.0000 2.17105
\(870\) 6.00000 0.203419
\(871\) −24.0000 −0.813209
\(872\) 42.0000 1.42230
\(873\) −2.00000 −0.0676897
\(874\) 0 0
\(875\) −1.00000 −0.0338062
\(876\) 10.0000 0.337869
\(877\) 22.0000 0.742887 0.371444 − 0.928456i \(-0.378863\pi\)
0.371444 + 0.928456i \(0.378863\pi\)
\(878\) −16.0000 −0.539974
\(879\) 26.0000 0.876958
\(880\) 4.00000 0.134840
\(881\) −18.0000 −0.606435 −0.303218 − 0.952921i \(-0.598061\pi\)
−0.303218 + 0.952921i \(0.598061\pi\)
\(882\) −1.00000 −0.0336718
\(883\) −24.0000 −0.807664 −0.403832 − 0.914833i \(-0.632322\pi\)
−0.403832 + 0.914833i \(0.632322\pi\)
\(884\) 4.00000 0.134535
\(885\) 4.00000 0.134459
\(886\) 20.0000 0.671913
\(887\) 32.0000 1.07445 0.537227 − 0.843437i \(-0.319472\pi\)
0.537227 + 0.843437i \(0.319472\pi\)
\(888\) 30.0000 1.00673
\(889\) 20.0000 0.670778
\(890\) 6.00000 0.201120
\(891\) 4.00000 0.134005
\(892\) −16.0000 −0.535720
\(893\) 0 0
\(894\) 22.0000 0.735790
\(895\) −16.0000 −0.534821
\(896\) 3.00000 0.100223
\(897\) 2.00000 0.0667781
\(898\) −2.00000 −0.0667409
\(899\) −48.0000 −1.60089
\(900\) −1.00000 −0.0333333
\(901\) −12.0000 −0.399778
\(902\) 8.00000 0.266371
\(903\) 4.00000 0.133112
\(904\) 30.0000 0.997785
\(905\) −10.0000 −0.332411
\(906\) −16.0000 −0.531564
\(907\) −12.0000 −0.398453 −0.199227 − 0.979953i \(-0.563843\pi\)
−0.199227 + 0.979953i \(0.563843\pi\)
\(908\) 8.00000 0.265489
\(909\) −6.00000 −0.199007
\(910\) 2.00000 0.0662994
\(911\) 40.0000 1.32526 0.662630 − 0.748947i \(-0.269440\pi\)
0.662630 + 0.748947i \(0.269440\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) −22.0000 −0.727695
\(915\) −10.0000 −0.330590
\(916\) −10.0000 −0.330409
\(917\) 12.0000 0.396275
\(918\) 2.00000 0.0660098
\(919\) −40.0000 −1.31948 −0.659739 − 0.751495i \(-0.729333\pi\)
−0.659739 + 0.751495i \(0.729333\pi\)
\(920\) −3.00000 −0.0989071
\(921\) 28.0000 0.922631
\(922\) −18.0000 −0.592798
\(923\) −24.0000 −0.789970
\(924\) −4.00000 −0.131590
\(925\) 10.0000 0.328798
\(926\) 20.0000 0.657241
\(927\) 8.00000 0.262754
\(928\) −30.0000 −0.984798
\(929\) −18.0000 −0.590561 −0.295280 − 0.955411i \(-0.595413\pi\)
−0.295280 + 0.955411i \(0.595413\pi\)
\(930\) −8.00000 −0.262330
\(931\) 0 0
\(932\) 6.00000 0.196537
\(933\) 0 0
\(934\) −8.00000 −0.261768
\(935\) 8.00000 0.261628
\(936\) 6.00000 0.196116
\(937\) −26.0000 −0.849383 −0.424691 − 0.905338i \(-0.639617\pi\)
−0.424691 + 0.905338i \(0.639617\pi\)
\(938\) 12.0000 0.391814
\(939\) 22.0000 0.717943
\(940\) −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
\(943\) −2.00000 −0.0651290
\(944\) 4.00000 0.130189
\(945\) −1.00000 −0.0325300
\(946\) −16.0000 −0.520205
\(947\) 20.0000 0.649913 0.324956 − 0.945729i \(-0.394650\pi\)
0.324956 + 0.945729i \(0.394650\pi\)
\(948\) −16.0000 −0.519656
\(949\) −20.0000 −0.649227
\(950\) 0 0
\(951\) −2.00000 −0.0648544
\(952\) −6.00000 −0.194461
\(953\) 34.0000 1.10137 0.550684 − 0.834714i \(-0.314367\pi\)
0.550684 + 0.834714i \(0.314367\pi\)
\(954\) −6.00000 −0.194257
\(955\) 0 0
\(956\) 28.0000 0.905585
\(957\) 24.0000 0.775810
\(958\) 32.0000 1.03387
\(959\) 10.0000 0.322917
\(960\) −7.00000 −0.225924
\(961\) 33.0000 1.06452
\(962\) −20.0000 −0.644826
\(963\) 4.00000 0.128898
\(964\) 18.0000 0.579741
\(965\) −10.0000 −0.321911
\(966\) −1.00000 −0.0321745
\(967\) −44.0000 −1.41494 −0.707472 − 0.706741i \(-0.750165\pi\)
−0.707472 + 0.706741i \(0.750165\pi\)
\(968\) 15.0000 0.482118
\(969\) 0 0
\(970\) −2.00000 −0.0642161
\(971\) −52.0000 −1.66876 −0.834380 − 0.551190i \(-0.814174\pi\)
−0.834380 + 0.551190i \(0.814174\pi\)
\(972\) −1.00000 −0.0320750
\(973\) 20.0000 0.641171
\(974\) 20.0000 0.640841
\(975\) 2.00000 0.0640513
\(976\) −10.0000 −0.320092
\(977\) 42.0000 1.34370 0.671850 − 0.740688i \(-0.265500\pi\)
0.671850 + 0.740688i \(0.265500\pi\)
\(978\) −16.0000 −0.511624
\(979\) 24.0000 0.767043
\(980\) 1.00000 0.0319438
\(981\) 14.0000 0.446986
\(982\) −24.0000 −0.765871
\(983\) −20.0000 −0.637901 −0.318950 − 0.947771i \(-0.603330\pi\)
−0.318950 + 0.947771i \(0.603330\pi\)
\(984\) −6.00000 −0.191273
\(985\) −6.00000 −0.191176
\(986\) 12.0000 0.382158
\(987\) −8.00000 −0.254643
\(988\) 0 0
\(989\) 4.00000 0.127193
\(990\) 4.00000 0.127128
\(991\) 40.0000 1.27064 0.635321 − 0.772248i \(-0.280868\pi\)
0.635321 + 0.772248i \(0.280868\pi\)
\(992\) 40.0000 1.27000
\(993\) −28.0000 −0.888553
\(994\) 12.0000 0.380617
\(995\) 20.0000 0.634043
\(996\) 0 0
\(997\) 42.0000 1.33015 0.665077 − 0.746775i \(-0.268399\pi\)
0.665077 + 0.746775i \(0.268399\pi\)
\(998\) −12.0000 −0.379853
\(999\) 10.0000 0.316386
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 2415.2.a.b.1.1 ✓ 1
3.2 odd 2 7245.2.a.s.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2415.2.a.b.1.1 ✓ 1 1.1 even 1 trivial
7245.2.a.s.1.1 1 3.2 odd 2