Properties

Label 7942.2.a.i.1.1
Level $7942$
Weight $2$
Character 7942.1
Self dual yes
Analytic conductor $63.417$
Analytic rank $2$
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] = [7942,2,Mod(1,7942)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(7942, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("7942.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 7942 = 2 \cdot 11 \cdot 19^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7942.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: \(63.4171892853\)
Analytic rank: \(2\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 418)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 7942.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -2.00000 q^{5} -1.00000 q^{6} -3.00000 q^{7} -1.00000 q^{8} -2.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -2.00000 q^{5} -1.00000 q^{6} -3.00000 q^{7} -1.00000 q^{8} -2.00000 q^{9} +2.00000 q^{10} -1.00000 q^{11} +1.00000 q^{12} -1.00000 q^{13} +3.00000 q^{14} -2.00000 q^{15} +1.00000 q^{16} -7.00000 q^{17} +2.00000 q^{18} -2.00000 q^{20} -3.00000 q^{21} +1.00000 q^{22} -5.00000 q^{23} -1.00000 q^{24} -1.00000 q^{25} +1.00000 q^{26} -5.00000 q^{27} -3.00000 q^{28} -1.00000 q^{29} +2.00000 q^{30} -10.0000 q^{31} -1.00000 q^{32} -1.00000 q^{33} +7.00000 q^{34} +6.00000 q^{35} -2.00000 q^{36} +6.00000 q^{37} -1.00000 q^{39} +2.00000 q^{40} -6.00000 q^{41} +3.00000 q^{42} -4.00000 q^{43} -1.00000 q^{44} +4.00000 q^{45} +5.00000 q^{46} +1.00000 q^{48} +2.00000 q^{49} +1.00000 q^{50} -7.00000 q^{51} -1.00000 q^{52} +1.00000 q^{53} +5.00000 q^{54} +2.00000 q^{55} +3.00000 q^{56} +1.00000 q^{58} -3.00000 q^{59} -2.00000 q^{60} -12.0000 q^{61} +10.0000 q^{62} +6.00000 q^{63} +1.00000 q^{64} +2.00000 q^{65} +1.00000 q^{66} -3.00000 q^{67} -7.00000 q^{68} -5.00000 q^{69} -6.00000 q^{70} +10.0000 q^{71} +2.00000 q^{72} +3.00000 q^{73} -6.00000 q^{74} -1.00000 q^{75} +3.00000 q^{77} +1.00000 q^{78} -8.00000 q^{79} -2.00000 q^{80} +1.00000 q^{81} +6.00000 q^{82} +8.00000 q^{83} -3.00000 q^{84} +14.0000 q^{85} +4.00000 q^{86} -1.00000 q^{87} +1.00000 q^{88} +8.00000 q^{89} -4.00000 q^{90} +3.00000 q^{91} -5.00000 q^{92} -10.0000 q^{93} -1.00000 q^{96} -8.00000 q^{97} -2.00000 q^{98} +2.00000 q^{99} +O(q^{100})\)

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 0.288675 0.957427i \(-0.406785\pi\)
0.288675 + 0.957427i \(0.406785\pi\)
\(4\) 1.00000 0.500000
\(5\) −2.00000 −0.894427 −0.447214 0.894427i \(-0.647584\pi\)
−0.447214 + 0.894427i \(0.647584\pi\)
\(6\) −1.00000 −0.408248
\(7\) −3.00000 −1.13389 −0.566947 0.823754i \(-0.691875\pi\)
−0.566947 + 0.823754i \(0.691875\pi\)
\(8\) −1.00000 −0.353553
\(9\) −2.00000 −0.666667
\(10\) 2.00000 0.632456
\(11\) −1.00000 −0.301511
\(12\) 1.00000 0.288675
\(13\) −1.00000 −0.277350 −0.138675 0.990338i \(-0.544284\pi\)
−0.138675 + 0.990338i \(0.544284\pi\)
\(14\) 3.00000 0.801784
\(15\) −2.00000 −0.516398
\(16\) 1.00000 0.250000
\(17\) −7.00000 −1.69775 −0.848875 0.528594i \(-0.822719\pi\)
−0.848875 + 0.528594i \(0.822719\pi\)
\(18\) 2.00000 0.471405
\(19\) 0 0
\(20\) −2.00000 −0.447214
\(21\) −3.00000 −0.654654
\(22\) 1.00000 0.213201
\(23\) −5.00000 −1.04257 −0.521286 0.853382i \(-0.674548\pi\)
−0.521286 + 0.853382i \(0.674548\pi\)
\(24\) −1.00000 −0.204124
\(25\) −1.00000 −0.200000
\(26\) 1.00000 0.196116
\(27\) −5.00000 −0.962250
\(28\) −3.00000 −0.566947
\(29\) −1.00000 −0.185695 −0.0928477 0.995680i \(-0.529597\pi\)
−0.0928477 + 0.995680i \(0.529597\pi\)
\(30\) 2.00000 0.365148
\(31\) −10.0000 −1.79605 −0.898027 0.439941i \(-0.854999\pi\)
−0.898027 + 0.439941i \(0.854999\pi\)
\(32\) −1.00000 −0.176777
\(33\) −1.00000 −0.174078
\(34\) 7.00000 1.20049
\(35\) 6.00000 1.01419
\(36\) −2.00000 −0.333333
\(37\) 6.00000 0.986394 0.493197 0.869918i \(-0.335828\pi\)
0.493197 + 0.869918i \(0.335828\pi\)
\(38\) 0 0
\(39\) −1.00000 −0.160128
\(40\) 2.00000 0.316228
\(41\) −6.00000 −0.937043 −0.468521 0.883452i \(-0.655213\pi\)
−0.468521 + 0.883452i \(0.655213\pi\)
\(42\) 3.00000 0.462910
\(43\) −4.00000 −0.609994 −0.304997 0.952353i \(-0.598656\pi\)
−0.304997 + 0.952353i \(0.598656\pi\)
\(44\) −1.00000 −0.150756
\(45\) 4.00000 0.596285
\(46\) 5.00000 0.737210
\(47\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 1.00000 0.144338
\(49\) 2.00000 0.285714
\(50\) 1.00000 0.141421
\(51\) −7.00000 −0.980196
\(52\) −1.00000 −0.138675
\(53\) 1.00000 0.137361 0.0686803 0.997639i \(-0.478121\pi\)
0.0686803 + 0.997639i \(0.478121\pi\)
\(54\) 5.00000 0.680414
\(55\) 2.00000 0.269680
\(56\) 3.00000 0.400892
\(57\) 0 0
\(58\) 1.00000 0.131306
\(59\) −3.00000 −0.390567 −0.195283 0.980747i \(-0.562563\pi\)
−0.195283 + 0.980747i \(0.562563\pi\)
\(60\) −2.00000 −0.258199
\(61\) −12.0000 −1.53644 −0.768221 0.640184i \(-0.778858\pi\)
−0.768221 + 0.640184i \(0.778858\pi\)
\(62\) 10.0000 1.27000
\(63\) 6.00000 0.755929
\(64\) 1.00000 0.125000
\(65\) 2.00000 0.248069
\(66\) 1.00000 0.123091
\(67\) −3.00000 −0.366508 −0.183254 0.983066i \(-0.558663\pi\)
−0.183254 + 0.983066i \(0.558663\pi\)
\(68\) −7.00000 −0.848875
\(69\) −5.00000 −0.601929
\(70\) −6.00000 −0.717137
\(71\) 10.0000 1.18678 0.593391 0.804914i \(-0.297789\pi\)
0.593391 + 0.804914i \(0.297789\pi\)
\(72\) 2.00000 0.235702
\(73\) 3.00000 0.351123 0.175562 0.984468i \(-0.443826\pi\)
0.175562 + 0.984468i \(0.443826\pi\)
\(74\) −6.00000 −0.697486
\(75\) −1.00000 −0.115470
\(76\) 0 0
\(77\) 3.00000 0.341882
\(78\) 1.00000 0.113228
\(79\) −8.00000 −0.900070 −0.450035 0.893011i \(-0.648589\pi\)
−0.450035 + 0.893011i \(0.648589\pi\)
\(80\) −2.00000 −0.223607
\(81\) 1.00000 0.111111
\(82\) 6.00000 0.662589
\(83\) 8.00000 0.878114 0.439057 0.898459i \(-0.355313\pi\)
0.439057 + 0.898459i \(0.355313\pi\)
\(84\) −3.00000 −0.327327
\(85\) 14.0000 1.51851
\(86\) 4.00000 0.431331
\(87\) −1.00000 −0.107211
\(88\) 1.00000 0.106600
\(89\) 8.00000 0.847998 0.423999 0.905663i \(-0.360626\pi\)
0.423999 + 0.905663i \(0.360626\pi\)
\(90\) −4.00000 −0.421637
\(91\) 3.00000 0.314485
\(92\) −5.00000 −0.521286
\(93\) −10.0000 −1.03695
\(94\) 0 0
\(95\) 0 0
\(96\) −1.00000 −0.102062
\(97\) −8.00000 −0.812277 −0.406138 0.913812i \(-0.633125\pi\)
−0.406138 + 0.913812i \(0.633125\pi\)
\(98\) −2.00000 −0.202031
\(99\) 2.00000 0.201008
\(100\) −1.00000 −0.100000
\(101\) 6.00000 0.597022 0.298511 0.954406i \(-0.403510\pi\)
0.298511 + 0.954406i \(0.403510\pi\)
\(102\) 7.00000 0.693103
\(103\) −12.0000 −1.18240 −0.591198 0.806527i \(-0.701345\pi\)
−0.591198 + 0.806527i \(0.701345\pi\)
\(104\) 1.00000 0.0980581
\(105\) 6.00000 0.585540
\(106\) −1.00000 −0.0971286
\(107\) 13.0000 1.25676 0.628379 0.777908i \(-0.283719\pi\)
0.628379 + 0.777908i \(0.283719\pi\)
\(108\) −5.00000 −0.481125
\(109\) 13.0000 1.24517 0.622587 0.782551i \(-0.286082\pi\)
0.622587 + 0.782551i \(0.286082\pi\)
\(110\) −2.00000 −0.190693
\(111\) 6.00000 0.569495
\(112\) −3.00000 −0.283473
\(113\) 12.0000 1.12887 0.564433 0.825479i \(-0.309095\pi\)
0.564433 + 0.825479i \(0.309095\pi\)
\(114\) 0 0
\(115\) 10.0000 0.932505
\(116\) −1.00000 −0.0928477
\(117\) 2.00000 0.184900
\(118\) 3.00000 0.276172
\(119\) 21.0000 1.92507
\(120\) 2.00000 0.182574
\(121\) 1.00000 0.0909091
\(122\) 12.0000 1.08643
\(123\) −6.00000 −0.541002
\(124\) −10.0000 −0.898027
\(125\) 12.0000 1.07331
\(126\) −6.00000 −0.534522
\(127\) −16.0000 −1.41977 −0.709885 0.704317i \(-0.751253\pi\)
−0.709885 + 0.704317i \(0.751253\pi\)
\(128\) −1.00000 −0.0883883
\(129\) −4.00000 −0.352180
\(130\) −2.00000 −0.175412
\(131\) −22.0000 −1.92215 −0.961074 0.276289i \(-0.910895\pi\)
−0.961074 + 0.276289i \(0.910895\pi\)
\(132\) −1.00000 −0.0870388
\(133\) 0 0
\(134\) 3.00000 0.259161
\(135\) 10.0000 0.860663
\(136\) 7.00000 0.600245
\(137\) −17.0000 −1.45241 −0.726204 0.687479i \(-0.758717\pi\)
−0.726204 + 0.687479i \(0.758717\pi\)
\(138\) 5.00000 0.425628
\(139\) −14.0000 −1.18746 −0.593732 0.804663i \(-0.702346\pi\)
−0.593732 + 0.804663i \(0.702346\pi\)
\(140\) 6.00000 0.507093
\(141\) 0 0
\(142\) −10.0000 −0.839181
\(143\) 1.00000 0.0836242
\(144\) −2.00000 −0.166667
\(145\) 2.00000 0.166091
\(146\) −3.00000 −0.248282
\(147\) 2.00000 0.164957
\(148\) 6.00000 0.493197
\(149\) −18.0000 −1.47462 −0.737309 0.675556i \(-0.763904\pi\)
−0.737309 + 0.675556i \(0.763904\pi\)
\(150\) 1.00000 0.0816497
\(151\) −12.0000 −0.976546 −0.488273 0.872691i \(-0.662373\pi\)
−0.488273 + 0.872691i \(0.662373\pi\)
\(152\) 0 0
\(153\) 14.0000 1.13183
\(154\) −3.00000 −0.241747
\(155\) 20.0000 1.60644
\(156\) −1.00000 −0.0800641
\(157\) −22.0000 −1.75579 −0.877896 0.478852i \(-0.841053\pi\)
−0.877896 + 0.478852i \(0.841053\pi\)
\(158\) 8.00000 0.636446
\(159\) 1.00000 0.0793052
\(160\) 2.00000 0.158114
\(161\) 15.0000 1.18217
\(162\) −1.00000 −0.0785674
\(163\) −2.00000 −0.156652 −0.0783260 0.996928i \(-0.524958\pi\)
−0.0783260 + 0.996928i \(0.524958\pi\)
\(164\) −6.00000 −0.468521
\(165\) 2.00000 0.155700
\(166\) −8.00000 −0.620920
\(167\) 22.0000 1.70241 0.851206 0.524832i \(-0.175872\pi\)
0.851206 + 0.524832i \(0.175872\pi\)
\(168\) 3.00000 0.231455
\(169\) −12.0000 −0.923077
\(170\) −14.0000 −1.07375
\(171\) 0 0
\(172\) −4.00000 −0.304997
\(173\) −14.0000 −1.06440 −0.532200 0.846619i \(-0.678635\pi\)
−0.532200 + 0.846619i \(0.678635\pi\)
\(174\) 1.00000 0.0758098
\(175\) 3.00000 0.226779
\(176\) −1.00000 −0.0753778
\(177\) −3.00000 −0.225494
\(178\) −8.00000 −0.599625
\(179\) −4.00000 −0.298974 −0.149487 0.988764i \(-0.547762\pi\)
−0.149487 + 0.988764i \(0.547762\pi\)
\(180\) 4.00000 0.298142
\(181\) 2.00000 0.148659 0.0743294 0.997234i \(-0.476318\pi\)
0.0743294 + 0.997234i \(0.476318\pi\)
\(182\) −3.00000 −0.222375
\(183\) −12.0000 −0.887066
\(184\) 5.00000 0.368605
\(185\) −12.0000 −0.882258
\(186\) 10.0000 0.733236
\(187\) 7.00000 0.511891
\(188\) 0 0
\(189\) 15.0000 1.09109
\(190\) 0 0
\(191\) 3.00000 0.217072 0.108536 0.994092i \(-0.465384\pi\)
0.108536 + 0.994092i \(0.465384\pi\)
\(192\) 1.00000 0.0721688
\(193\) 4.00000 0.287926 0.143963 0.989583i \(-0.454015\pi\)
0.143963 + 0.989583i \(0.454015\pi\)
\(194\) 8.00000 0.574367
\(195\) 2.00000 0.143223
\(196\) 2.00000 0.142857
\(197\) −12.0000 −0.854965 −0.427482 0.904024i \(-0.640599\pi\)
−0.427482 + 0.904024i \(0.640599\pi\)
\(198\) −2.00000 −0.142134
\(199\) 23.0000 1.63043 0.815213 0.579161i \(-0.196620\pi\)
0.815213 + 0.579161i \(0.196620\pi\)
\(200\) 1.00000 0.0707107
\(201\) −3.00000 −0.211604
\(202\) −6.00000 −0.422159
\(203\) 3.00000 0.210559
\(204\) −7.00000 −0.490098
\(205\) 12.0000 0.838116
\(206\) 12.0000 0.836080
\(207\) 10.0000 0.695048
\(208\) −1.00000 −0.0693375
\(209\) 0 0
\(210\) −6.00000 −0.414039
\(211\) −1.00000 −0.0688428 −0.0344214 0.999407i \(-0.510959\pi\)
−0.0344214 + 0.999407i \(0.510959\pi\)
\(212\) 1.00000 0.0686803
\(213\) 10.0000 0.685189
\(214\) −13.0000 −0.888662
\(215\) 8.00000 0.545595
\(216\) 5.00000 0.340207
\(217\) 30.0000 2.03653
\(218\) −13.0000 −0.880471
\(219\) 3.00000 0.202721
\(220\) 2.00000 0.134840
\(221\) 7.00000 0.470871
\(222\) −6.00000 −0.402694
\(223\) −10.0000 −0.669650 −0.334825 0.942280i \(-0.608677\pi\)
−0.334825 + 0.942280i \(0.608677\pi\)
\(224\) 3.00000 0.200446
\(225\) 2.00000 0.133333
\(226\) −12.0000 −0.798228
\(227\) 15.0000 0.995585 0.497792 0.867296i \(-0.334144\pi\)
0.497792 + 0.867296i \(0.334144\pi\)
\(228\) 0 0
\(229\) 20.0000 1.32164 0.660819 0.750546i \(-0.270209\pi\)
0.660819 + 0.750546i \(0.270209\pi\)
\(230\) −10.0000 −0.659380
\(231\) 3.00000 0.197386
\(232\) 1.00000 0.0656532
\(233\) −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
\(236\) −3.00000 −0.195283
\(237\) −8.00000 −0.519656
\(238\) −21.0000 −1.36123
\(239\) 25.0000 1.61712 0.808558 0.588417i \(-0.200249\pi\)
0.808558 + 0.588417i \(0.200249\pi\)
\(240\) −2.00000 −0.129099
\(241\) 4.00000 0.257663 0.128831 0.991667i \(-0.458877\pi\)
0.128831 + 0.991667i \(0.458877\pi\)
\(242\) −1.00000 −0.0642824
\(243\) 16.0000 1.02640
\(244\) −12.0000 −0.768221
\(245\) −4.00000 −0.255551
\(246\) 6.00000 0.382546
\(247\) 0 0
\(248\) 10.0000 0.635001
\(249\) 8.00000 0.506979
\(250\) −12.0000 −0.758947
\(251\) −6.00000 −0.378717 −0.189358 0.981908i \(-0.560641\pi\)
−0.189358 + 0.981908i \(0.560641\pi\)
\(252\) 6.00000 0.377964
\(253\) 5.00000 0.314347
\(254\) 16.0000 1.00393
\(255\) 14.0000 0.876714
\(256\) 1.00000 0.0625000
\(257\) 22.0000 1.37232 0.686161 0.727450i \(-0.259294\pi\)
0.686161 + 0.727450i \(0.259294\pi\)
\(258\) 4.00000 0.249029
\(259\) −18.0000 −1.11847
\(260\) 2.00000 0.124035
\(261\) 2.00000 0.123797
\(262\) 22.0000 1.35916
\(263\) 4.00000 0.246651 0.123325 0.992366i \(-0.460644\pi\)
0.123325 + 0.992366i \(0.460644\pi\)
\(264\) 1.00000 0.0615457
\(265\) −2.00000 −0.122859
\(266\) 0 0
\(267\) 8.00000 0.489592
\(268\) −3.00000 −0.183254
\(269\) −18.0000 −1.09748 −0.548740 0.835993i \(-0.684892\pi\)
−0.548740 + 0.835993i \(0.684892\pi\)
\(270\) −10.0000 −0.608581
\(271\) −15.0000 −0.911185 −0.455593 0.890188i \(-0.650573\pi\)
−0.455593 + 0.890188i \(0.650573\pi\)
\(272\) −7.00000 −0.424437
\(273\) 3.00000 0.181568
\(274\) 17.0000 1.02701
\(275\) 1.00000 0.0603023
\(276\) −5.00000 −0.300965
\(277\) −16.0000 −0.961347 −0.480673 0.876900i \(-0.659608\pi\)
−0.480673 + 0.876900i \(0.659608\pi\)
\(278\) 14.0000 0.839664
\(279\) 20.0000 1.19737
\(280\) −6.00000 −0.358569
\(281\) −16.0000 −0.954480 −0.477240 0.878773i \(-0.658363\pi\)
−0.477240 + 0.878773i \(0.658363\pi\)
\(282\) 0 0
\(283\) −14.0000 −0.832214 −0.416107 0.909316i \(-0.636606\pi\)
−0.416107 + 0.909316i \(0.636606\pi\)
\(284\) 10.0000 0.593391
\(285\) 0 0
\(286\) −1.00000 −0.0591312
\(287\) 18.0000 1.06251
\(288\) 2.00000 0.117851
\(289\) 32.0000 1.88235
\(290\) −2.00000 −0.117444
\(291\) −8.00000 −0.468968
\(292\) 3.00000 0.175562
\(293\) 9.00000 0.525786 0.262893 0.964825i \(-0.415323\pi\)
0.262893 + 0.964825i \(0.415323\pi\)
\(294\) −2.00000 −0.116642
\(295\) 6.00000 0.349334
\(296\) −6.00000 −0.348743
\(297\) 5.00000 0.290129
\(298\) 18.0000 1.04271
\(299\) 5.00000 0.289157
\(300\) −1.00000 −0.0577350
\(301\) 12.0000 0.691669
\(302\) 12.0000 0.690522
\(303\) 6.00000 0.344691
\(304\) 0 0
\(305\) 24.0000 1.37424
\(306\) −14.0000 −0.800327
\(307\) 16.0000 0.913168 0.456584 0.889680i \(-0.349073\pi\)
0.456584 + 0.889680i \(0.349073\pi\)
\(308\) 3.00000 0.170941
\(309\) −12.0000 −0.682656
\(310\) −20.0000 −1.13592
\(311\) 3.00000 0.170114 0.0850572 0.996376i \(-0.472893\pi\)
0.0850572 + 0.996376i \(0.472893\pi\)
\(312\) 1.00000 0.0566139
\(313\) 21.0000 1.18699 0.593495 0.804838i \(-0.297748\pi\)
0.593495 + 0.804838i \(0.297748\pi\)
\(314\) 22.0000 1.24153
\(315\) −12.0000 −0.676123
\(316\) −8.00000 −0.450035
\(317\) −21.0000 −1.17948 −0.589739 0.807594i \(-0.700769\pi\)
−0.589739 + 0.807594i \(0.700769\pi\)
\(318\) −1.00000 −0.0560772
\(319\) 1.00000 0.0559893
\(320\) −2.00000 −0.111803
\(321\) 13.0000 0.725589
\(322\) −15.0000 −0.835917
\(323\) 0 0
\(324\) 1.00000 0.0555556
\(325\) 1.00000 0.0554700
\(326\) 2.00000 0.110770
\(327\) 13.0000 0.718902
\(328\) 6.00000 0.331295
\(329\) 0 0
\(330\) −2.00000 −0.110096
\(331\) −29.0000 −1.59398 −0.796992 0.603990i \(-0.793577\pi\)
−0.796992 + 0.603990i \(0.793577\pi\)
\(332\) 8.00000 0.439057
\(333\) −12.0000 −0.657596
\(334\) −22.0000 −1.20379
\(335\) 6.00000 0.327815
\(336\) −3.00000 −0.163663
\(337\) 20.0000 1.08947 0.544735 0.838608i \(-0.316630\pi\)
0.544735 + 0.838608i \(0.316630\pi\)
\(338\) 12.0000 0.652714
\(339\) 12.0000 0.651751
\(340\) 14.0000 0.759257
\(341\) 10.0000 0.541530
\(342\) 0 0
\(343\) 15.0000 0.809924
\(344\) 4.00000 0.215666
\(345\) 10.0000 0.538382
\(346\) 14.0000 0.752645
\(347\) −26.0000 −1.39575 −0.697877 0.716218i \(-0.745872\pi\)
−0.697877 + 0.716218i \(0.745872\pi\)
\(348\) −1.00000 −0.0536056
\(349\) 10.0000 0.535288 0.267644 0.963518i \(-0.413755\pi\)
0.267644 + 0.963518i \(0.413755\pi\)
\(350\) −3.00000 −0.160357
\(351\) 5.00000 0.266880
\(352\) 1.00000 0.0533002
\(353\) 1.00000 0.0532246 0.0266123 0.999646i \(-0.491528\pi\)
0.0266123 + 0.999646i \(0.491528\pi\)
\(354\) 3.00000 0.159448
\(355\) −20.0000 −1.06149
\(356\) 8.00000 0.423999
\(357\) 21.0000 1.11144
\(358\) 4.00000 0.211407
\(359\) −17.0000 −0.897226 −0.448613 0.893726i \(-0.648082\pi\)
−0.448613 + 0.893726i \(0.648082\pi\)
\(360\) −4.00000 −0.210819
\(361\) 0 0
\(362\) −2.00000 −0.105118
\(363\) 1.00000 0.0524864
\(364\) 3.00000 0.157243
\(365\) −6.00000 −0.314054
\(366\) 12.0000 0.627250
\(367\) 16.0000 0.835193 0.417597 0.908633i \(-0.362873\pi\)
0.417597 + 0.908633i \(0.362873\pi\)
\(368\) −5.00000 −0.260643
\(369\) 12.0000 0.624695
\(370\) 12.0000 0.623850
\(371\) −3.00000 −0.155752
\(372\) −10.0000 −0.518476
\(373\) −11.0000 −0.569558 −0.284779 0.958593i \(-0.591920\pi\)
−0.284779 + 0.958593i \(0.591920\pi\)
\(374\) −7.00000 −0.361961
\(375\) 12.0000 0.619677
\(376\) 0 0
\(377\) 1.00000 0.0515026
\(378\) −15.0000 −0.771517
\(379\) 5.00000 0.256833 0.128416 0.991720i \(-0.459011\pi\)
0.128416 + 0.991720i \(0.459011\pi\)
\(380\) 0 0
\(381\) −16.0000 −0.819705
\(382\) −3.00000 −0.153493
\(383\) −2.00000 −0.102195 −0.0510976 0.998694i \(-0.516272\pi\)
−0.0510976 + 0.998694i \(0.516272\pi\)
\(384\) −1.00000 −0.0510310
\(385\) −6.00000 −0.305788
\(386\) −4.00000 −0.203595
\(387\) 8.00000 0.406663
\(388\) −8.00000 −0.406138
\(389\) 18.0000 0.912636 0.456318 0.889817i \(-0.349168\pi\)
0.456318 + 0.889817i \(0.349168\pi\)
\(390\) −2.00000 −0.101274
\(391\) 35.0000 1.77003
\(392\) −2.00000 −0.101015
\(393\) −22.0000 −1.10975
\(394\) 12.0000 0.604551
\(395\) 16.0000 0.805047
\(396\) 2.00000 0.100504
\(397\) 34.0000 1.70641 0.853206 0.521575i \(-0.174655\pi\)
0.853206 + 0.521575i \(0.174655\pi\)
\(398\) −23.0000 −1.15289
\(399\) 0 0
\(400\) −1.00000 −0.0500000
\(401\) 2.00000 0.0998752 0.0499376 0.998752i \(-0.484098\pi\)
0.0499376 + 0.998752i \(0.484098\pi\)
\(402\) 3.00000 0.149626
\(403\) 10.0000 0.498135
\(404\) 6.00000 0.298511
\(405\) −2.00000 −0.0993808
\(406\) −3.00000 −0.148888
\(407\) −6.00000 −0.297409
\(408\) 7.00000 0.346552
\(409\) −34.0000 −1.68119 −0.840596 0.541663i \(-0.817795\pi\)
−0.840596 + 0.541663i \(0.817795\pi\)
\(410\) −12.0000 −0.592638
\(411\) −17.0000 −0.838548
\(412\) −12.0000 −0.591198
\(413\) 9.00000 0.442861
\(414\) −10.0000 −0.491473
\(415\) −16.0000 −0.785409
\(416\) 1.00000 0.0490290
\(417\) −14.0000 −0.685583
\(418\) 0 0
\(419\) 10.0000 0.488532 0.244266 0.969708i \(-0.421453\pi\)
0.244266 + 0.969708i \(0.421453\pi\)
\(420\) 6.00000 0.292770
\(421\) −15.0000 −0.731055 −0.365528 0.930800i \(-0.619111\pi\)
−0.365528 + 0.930800i \(0.619111\pi\)
\(422\) 1.00000 0.0486792
\(423\) 0 0
\(424\) −1.00000 −0.0485643
\(425\) 7.00000 0.339550
\(426\) −10.0000 −0.484502
\(427\) 36.0000 1.74216
\(428\) 13.0000 0.628379
\(429\) 1.00000 0.0482805
\(430\) −8.00000 −0.385794
\(431\) −34.0000 −1.63772 −0.818861 0.573992i \(-0.805394\pi\)
−0.818861 + 0.573992i \(0.805394\pi\)
\(432\) −5.00000 −0.240563
\(433\) −24.0000 −1.15337 −0.576683 0.816968i \(-0.695653\pi\)
−0.576683 + 0.816968i \(0.695653\pi\)
\(434\) −30.0000 −1.44005
\(435\) 2.00000 0.0958927
\(436\) 13.0000 0.622587
\(437\) 0 0
\(438\) −3.00000 −0.143346
\(439\) 14.0000 0.668184 0.334092 0.942541i \(-0.391570\pi\)
0.334092 + 0.942541i \(0.391570\pi\)
\(440\) −2.00000 −0.0953463
\(441\) −4.00000 −0.190476
\(442\) −7.00000 −0.332956
\(443\) 18.0000 0.855206 0.427603 0.903967i \(-0.359358\pi\)
0.427603 + 0.903967i \(0.359358\pi\)
\(444\) 6.00000 0.284747
\(445\) −16.0000 −0.758473
\(446\) 10.0000 0.473514
\(447\) −18.0000 −0.851371
\(448\) −3.00000 −0.141737
\(449\) −24.0000 −1.13263 −0.566315 0.824189i \(-0.691631\pi\)
−0.566315 + 0.824189i \(0.691631\pi\)
\(450\) −2.00000 −0.0942809
\(451\) 6.00000 0.282529
\(452\) 12.0000 0.564433
\(453\) −12.0000 −0.563809
\(454\) −15.0000 −0.703985
\(455\) −6.00000 −0.281284
\(456\) 0 0
\(457\) −37.0000 −1.73079 −0.865393 0.501093i \(-0.832931\pi\)
−0.865393 + 0.501093i \(0.832931\pi\)
\(458\) −20.0000 −0.934539
\(459\) 35.0000 1.63366
\(460\) 10.0000 0.466252
\(461\) −8.00000 −0.372597 −0.186299 0.982493i \(-0.559649\pi\)
−0.186299 + 0.982493i \(0.559649\pi\)
\(462\) −3.00000 −0.139573
\(463\) 16.0000 0.743583 0.371792 0.928316i \(-0.378744\pi\)
0.371792 + 0.928316i \(0.378744\pi\)
\(464\) −1.00000 −0.0464238
\(465\) 20.0000 0.927478
\(466\) 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
\(469\) 9.00000 0.415581
\(470\) 0 0
\(471\) −22.0000 −1.01371
\(472\) 3.00000 0.138086
\(473\) 4.00000 0.183920
\(474\) 8.00000 0.367452
\(475\) 0 0
\(476\) 21.0000 0.962533
\(477\) −2.00000 −0.0915737
\(478\) −25.0000 −1.14347
\(479\) −12.0000 −0.548294 −0.274147 0.961688i \(-0.588395\pi\)
−0.274147 + 0.961688i \(0.588395\pi\)
\(480\) 2.00000 0.0912871
\(481\) −6.00000 −0.273576
\(482\) −4.00000 −0.182195
\(483\) 15.0000 0.682524
\(484\) 1.00000 0.0454545
\(485\) 16.0000 0.726523
\(486\) −16.0000 −0.725775
\(487\) 12.0000 0.543772 0.271886 0.962329i \(-0.412353\pi\)
0.271886 + 0.962329i \(0.412353\pi\)
\(488\) 12.0000 0.543214
\(489\) −2.00000 −0.0904431
\(490\) 4.00000 0.180702
\(491\) 10.0000 0.451294 0.225647 0.974209i \(-0.427550\pi\)
0.225647 + 0.974209i \(0.427550\pi\)
\(492\) −6.00000 −0.270501
\(493\) 7.00000 0.315264
\(494\) 0 0
\(495\) −4.00000 −0.179787
\(496\) −10.0000 −0.449013
\(497\) −30.0000 −1.34568
\(498\) −8.00000 −0.358489
\(499\) −28.0000 −1.25345 −0.626726 0.779240i \(-0.715605\pi\)
−0.626726 + 0.779240i \(0.715605\pi\)
\(500\) 12.0000 0.536656
\(501\) 22.0000 0.982888
\(502\) 6.00000 0.267793
\(503\) −19.0000 −0.847168 −0.423584 0.905857i \(-0.639228\pi\)
−0.423584 + 0.905857i \(0.639228\pi\)
\(504\) −6.00000 −0.267261
\(505\) −12.0000 −0.533993
\(506\) −5.00000 −0.222277
\(507\) −12.0000 −0.532939
\(508\) −16.0000 −0.709885
\(509\) 34.0000 1.50702 0.753512 0.657434i \(-0.228358\pi\)
0.753512 + 0.657434i \(0.228358\pi\)
\(510\) −14.0000 −0.619930
\(511\) −9.00000 −0.398137
\(512\) −1.00000 −0.0441942
\(513\) 0 0
\(514\) −22.0000 −0.970378
\(515\) 24.0000 1.05757
\(516\) −4.00000 −0.176090
\(517\) 0 0
\(518\) 18.0000 0.790875
\(519\) −14.0000 −0.614532
\(520\) −2.00000 −0.0877058
\(521\) 20.0000 0.876216 0.438108 0.898922i \(-0.355649\pi\)
0.438108 + 0.898922i \(0.355649\pi\)
\(522\) −2.00000 −0.0875376
\(523\) 21.0000 0.918266 0.459133 0.888368i \(-0.348160\pi\)
0.459133 + 0.888368i \(0.348160\pi\)
\(524\) −22.0000 −0.961074
\(525\) 3.00000 0.130931
\(526\) −4.00000 −0.174408
\(527\) 70.0000 3.04925
\(528\) −1.00000 −0.0435194
\(529\) 2.00000 0.0869565
\(530\) 2.00000 0.0868744
\(531\) 6.00000 0.260378
\(532\) 0 0
\(533\) 6.00000 0.259889
\(534\) −8.00000 −0.346194
\(535\) −26.0000 −1.12408
\(536\) 3.00000 0.129580
\(537\) −4.00000 −0.172613
\(538\) 18.0000 0.776035
\(539\) −2.00000 −0.0861461
\(540\) 10.0000 0.430331
\(541\) −16.0000 −0.687894 −0.343947 0.938989i \(-0.611764\pi\)
−0.343947 + 0.938989i \(0.611764\pi\)
\(542\) 15.0000 0.644305
\(543\) 2.00000 0.0858282
\(544\) 7.00000 0.300123
\(545\) −26.0000 −1.11372
\(546\) −3.00000 −0.128388
\(547\) −8.00000 −0.342055 −0.171028 0.985266i \(-0.554709\pi\)
−0.171028 + 0.985266i \(0.554709\pi\)
\(548\) −17.0000 −0.726204
\(549\) 24.0000 1.02430
\(550\) −1.00000 −0.0426401
\(551\) 0 0
\(552\) 5.00000 0.212814
\(553\) 24.0000 1.02058
\(554\) 16.0000 0.679775
\(555\) −12.0000 −0.509372
\(556\) −14.0000 −0.593732
\(557\) −6.00000 −0.254228 −0.127114 0.991888i \(-0.540571\pi\)
−0.127114 + 0.991888i \(0.540571\pi\)
\(558\) −20.0000 −0.846668
\(559\) 4.00000 0.169182
\(560\) 6.00000 0.253546
\(561\) 7.00000 0.295540
\(562\) 16.0000 0.674919
\(563\) −24.0000 −1.01148 −0.505740 0.862686i \(-0.668780\pi\)
−0.505740 + 0.862686i \(0.668780\pi\)
\(564\) 0 0
\(565\) −24.0000 −1.00969
\(566\) 14.0000 0.588464
\(567\) −3.00000 −0.125988
\(568\) −10.0000 −0.419591
\(569\) 12.0000 0.503066 0.251533 0.967849i \(-0.419065\pi\)
0.251533 + 0.967849i \(0.419065\pi\)
\(570\) 0 0
\(571\) −14.0000 −0.585882 −0.292941 0.956131i \(-0.594634\pi\)
−0.292941 + 0.956131i \(0.594634\pi\)
\(572\) 1.00000 0.0418121
\(573\) 3.00000 0.125327
\(574\) −18.0000 −0.751305
\(575\) 5.00000 0.208514
\(576\) −2.00000 −0.0833333
\(577\) 23.0000 0.957503 0.478751 0.877951i \(-0.341090\pi\)
0.478751 + 0.877951i \(0.341090\pi\)
\(578\) −32.0000 −1.33102
\(579\) 4.00000 0.166234
\(580\) 2.00000 0.0830455
\(581\) −24.0000 −0.995688
\(582\) 8.00000 0.331611
\(583\) −1.00000 −0.0414158
\(584\) −3.00000 −0.124141
\(585\) −4.00000 −0.165380
\(586\) −9.00000 −0.371787
\(587\) −12.0000 −0.495293 −0.247647 0.968850i \(-0.579657\pi\)
−0.247647 + 0.968850i \(0.579657\pi\)
\(588\) 2.00000 0.0824786
\(589\) 0 0
\(590\) −6.00000 −0.247016
\(591\) −12.0000 −0.493614
\(592\) 6.00000 0.246598
\(593\) 42.0000 1.72473 0.862367 0.506284i \(-0.168981\pi\)
0.862367 + 0.506284i \(0.168981\pi\)
\(594\) −5.00000 −0.205152
\(595\) −42.0000 −1.72183
\(596\) −18.0000 −0.737309
\(597\) 23.0000 0.941327
\(598\) −5.00000 −0.204465
\(599\) 4.00000 0.163436 0.0817178 0.996656i \(-0.473959\pi\)
0.0817178 + 0.996656i \(0.473959\pi\)
\(600\) 1.00000 0.0408248
\(601\) −8.00000 −0.326327 −0.163163 0.986599i \(-0.552170\pi\)
−0.163163 + 0.986599i \(0.552170\pi\)
\(602\) −12.0000 −0.489083
\(603\) 6.00000 0.244339
\(604\) −12.0000 −0.488273
\(605\) −2.00000 −0.0813116
\(606\) −6.00000 −0.243733
\(607\) 22.0000 0.892952 0.446476 0.894795i \(-0.352679\pi\)
0.446476 + 0.894795i \(0.352679\pi\)
\(608\) 0 0
\(609\) 3.00000 0.121566
\(610\) −24.0000 −0.971732
\(611\) 0 0
\(612\) 14.0000 0.565916
\(613\) −28.0000 −1.13091 −0.565455 0.824779i \(-0.691299\pi\)
−0.565455 + 0.824779i \(0.691299\pi\)
\(614\) −16.0000 −0.645707
\(615\) 12.0000 0.483887
\(616\) −3.00000 −0.120873
\(617\) −30.0000 −1.20775 −0.603877 0.797077i \(-0.706378\pi\)
−0.603877 + 0.797077i \(0.706378\pi\)
\(618\) 12.0000 0.482711
\(619\) −20.0000 −0.803868 −0.401934 0.915669i \(-0.631662\pi\)
−0.401934 + 0.915669i \(0.631662\pi\)
\(620\) 20.0000 0.803219
\(621\) 25.0000 1.00322
\(622\) −3.00000 −0.120289
\(623\) −24.0000 −0.961540
\(624\) −1.00000 −0.0400320
\(625\) −19.0000 −0.760000
\(626\) −21.0000 −0.839329
\(627\) 0 0
\(628\) −22.0000 −0.877896
\(629\) −42.0000 −1.67465
\(630\) 12.0000 0.478091
\(631\) −8.00000 −0.318475 −0.159237 0.987240i \(-0.550904\pi\)
−0.159237 + 0.987240i \(0.550904\pi\)
\(632\) 8.00000 0.318223
\(633\) −1.00000 −0.0397464
\(634\) 21.0000 0.834017
\(635\) 32.0000 1.26988
\(636\) 1.00000 0.0396526
\(637\) −2.00000 −0.0792429
\(638\) −1.00000 −0.0395904
\(639\) −20.0000 −0.791188
\(640\) 2.00000 0.0790569
\(641\) −40.0000 −1.57991 −0.789953 0.613168i \(-0.789895\pi\)
−0.789953 + 0.613168i \(0.789895\pi\)
\(642\) −13.0000 −0.513069
\(643\) −26.0000 −1.02534 −0.512670 0.858586i \(-0.671344\pi\)
−0.512670 + 0.858586i \(0.671344\pi\)
\(644\) 15.0000 0.591083
\(645\) 8.00000 0.315000
\(646\) 0 0
\(647\) 3.00000 0.117942 0.0589711 0.998260i \(-0.481218\pi\)
0.0589711 + 0.998260i \(0.481218\pi\)
\(648\) −1.00000 −0.0392837
\(649\) 3.00000 0.117760
\(650\) −1.00000 −0.0392232
\(651\) 30.0000 1.17579
\(652\) −2.00000 −0.0783260
\(653\) 4.00000 0.156532 0.0782660 0.996933i \(-0.475062\pi\)
0.0782660 + 0.996933i \(0.475062\pi\)
\(654\) −13.0000 −0.508340
\(655\) 44.0000 1.71922
\(656\) −6.00000 −0.234261
\(657\) −6.00000 −0.234082
\(658\) 0 0
\(659\) −11.0000 −0.428499 −0.214250 0.976779i \(-0.568731\pi\)
−0.214250 + 0.976779i \(0.568731\pi\)
\(660\) 2.00000 0.0778499
\(661\) −29.0000 −1.12797 −0.563985 0.825785i \(-0.690732\pi\)
−0.563985 + 0.825785i \(0.690732\pi\)
\(662\) 29.0000 1.12712
\(663\) 7.00000 0.271857
\(664\) −8.00000 −0.310460
\(665\) 0 0
\(666\) 12.0000 0.464991
\(667\) 5.00000 0.193601
\(668\) 22.0000 0.851206
\(669\) −10.0000 −0.386622
\(670\) −6.00000 −0.231800
\(671\) 12.0000 0.463255
\(672\) 3.00000 0.115728
\(673\) 24.0000 0.925132 0.462566 0.886585i \(-0.346929\pi\)
0.462566 + 0.886585i \(0.346929\pi\)
\(674\) −20.0000 −0.770371
\(675\) 5.00000 0.192450
\(676\) −12.0000 −0.461538
\(677\) 29.0000 1.11456 0.557280 0.830324i \(-0.311845\pi\)
0.557280 + 0.830324i \(0.311845\pi\)
\(678\) −12.0000 −0.460857
\(679\) 24.0000 0.921035
\(680\) −14.0000 −0.536875
\(681\) 15.0000 0.574801
\(682\) −10.0000 −0.382920
\(683\) 8.00000 0.306111 0.153056 0.988218i \(-0.451089\pi\)
0.153056 + 0.988218i \(0.451089\pi\)
\(684\) 0 0
\(685\) 34.0000 1.29907
\(686\) −15.0000 −0.572703
\(687\) 20.0000 0.763048
\(688\) −4.00000 −0.152499
\(689\) −1.00000 −0.0380970
\(690\) −10.0000 −0.380693
\(691\) −14.0000 −0.532585 −0.266293 0.963892i \(-0.585799\pi\)
−0.266293 + 0.963892i \(0.585799\pi\)
\(692\) −14.0000 −0.532200
\(693\) −6.00000 −0.227921
\(694\) 26.0000 0.986947
\(695\) 28.0000 1.06210
\(696\) 1.00000 0.0379049
\(697\) 42.0000 1.59086
\(698\) −10.0000 −0.378506
\(699\) −10.0000 −0.378235
\(700\) 3.00000 0.113389
\(701\) 24.0000 0.906467 0.453234 0.891392i \(-0.350270\pi\)
0.453234 + 0.891392i \(0.350270\pi\)
\(702\) −5.00000 −0.188713
\(703\) 0 0
\(704\) −1.00000 −0.0376889
\(705\) 0 0
\(706\) −1.00000 −0.0376355
\(707\) −18.0000 −0.676960
\(708\) −3.00000 −0.112747
\(709\) −26.0000 −0.976450 −0.488225 0.872718i \(-0.662356\pi\)
−0.488225 + 0.872718i \(0.662356\pi\)
\(710\) 20.0000 0.750587
\(711\) 16.0000 0.600047
\(712\) −8.00000 −0.299813
\(713\) 50.0000 1.87251
\(714\) −21.0000 −0.785905
\(715\) −2.00000 −0.0747958
\(716\) −4.00000 −0.149487
\(717\) 25.0000 0.933642
\(718\) 17.0000 0.634434
\(719\) −33.0000 −1.23069 −0.615346 0.788257i \(-0.710984\pi\)
−0.615346 + 0.788257i \(0.710984\pi\)
\(720\) 4.00000 0.149071
\(721\) 36.0000 1.34071
\(722\) 0 0
\(723\) 4.00000 0.148762
\(724\) 2.00000 0.0743294
\(725\) 1.00000 0.0371391
\(726\) −1.00000 −0.0371135
\(727\) 47.0000 1.74313 0.871567 0.490277i \(-0.163104\pi\)
0.871567 + 0.490277i \(0.163104\pi\)
\(728\) −3.00000 −0.111187
\(729\) 13.0000 0.481481
\(730\) 6.00000 0.222070
\(731\) 28.0000 1.03562
\(732\) −12.0000 −0.443533
\(733\) −14.0000 −0.517102 −0.258551 0.965998i \(-0.583245\pi\)
−0.258551 + 0.965998i \(0.583245\pi\)
\(734\) −16.0000 −0.590571
\(735\) −4.00000 −0.147542
\(736\) 5.00000 0.184302
\(737\) 3.00000 0.110506
\(738\) −12.0000 −0.441726
\(739\) −28.0000 −1.03000 −0.514998 0.857191i \(-0.672207\pi\)
−0.514998 + 0.857191i \(0.672207\pi\)
\(740\) −12.0000 −0.441129
\(741\) 0 0
\(742\) 3.00000 0.110133
\(743\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(744\) 10.0000 0.366618
\(745\) 36.0000 1.31894
\(746\) 11.0000 0.402739
\(747\) −16.0000 −0.585409
\(748\) 7.00000 0.255945
\(749\) −39.0000 −1.42503
\(750\) −12.0000 −0.438178
\(751\) 50.0000 1.82453 0.912263 0.409605i \(-0.134333\pi\)
0.912263 + 0.409605i \(0.134333\pi\)
\(752\) 0 0
\(753\) −6.00000 −0.218652
\(754\) −1.00000 −0.0364179
\(755\) 24.0000 0.873449
\(756\) 15.0000 0.545545
\(757\) 38.0000 1.38113 0.690567 0.723269i \(-0.257361\pi\)
0.690567 + 0.723269i \(0.257361\pi\)
\(758\) −5.00000 −0.181608
\(759\) 5.00000 0.181489
\(760\) 0 0
\(761\) −11.0000 −0.398750 −0.199375 0.979923i \(-0.563891\pi\)
−0.199375 + 0.979923i \(0.563891\pi\)
\(762\) 16.0000 0.579619
\(763\) −39.0000 −1.41189
\(764\) 3.00000 0.108536
\(765\) −28.0000 −1.01234
\(766\) 2.00000 0.0722629
\(767\) 3.00000 0.108324
\(768\) 1.00000 0.0360844
\(769\) −49.0000 −1.76699 −0.883493 0.468445i \(-0.844814\pi\)
−0.883493 + 0.468445i \(0.844814\pi\)
\(770\) 6.00000 0.216225
\(771\) 22.0000 0.792311
\(772\) 4.00000 0.143963
\(773\) 23.0000 0.827253 0.413626 0.910447i \(-0.364262\pi\)
0.413626 + 0.910447i \(0.364262\pi\)
\(774\) −8.00000 −0.287554
\(775\) 10.0000 0.359211
\(776\) 8.00000 0.287183
\(777\) −18.0000 −0.645746
\(778\) −18.0000 −0.645331
\(779\) 0 0
\(780\) 2.00000 0.0716115
\(781\) −10.0000 −0.357828
\(782\) −35.0000 −1.25160
\(783\) 5.00000 0.178685
\(784\) 2.00000 0.0714286
\(785\) 44.0000 1.57043
\(786\) 22.0000 0.784714
\(787\) −49.0000 −1.74666 −0.873331 0.487128i \(-0.838045\pi\)
−0.873331 + 0.487128i \(0.838045\pi\)
\(788\) −12.0000 −0.427482
\(789\) 4.00000 0.142404
\(790\) −16.0000 −0.569254
\(791\) −36.0000 −1.28001
\(792\) −2.00000 −0.0710669
\(793\) 12.0000 0.426132
\(794\) −34.0000 −1.20661
\(795\) −2.00000 −0.0709327
\(796\) 23.0000 0.815213
\(797\) −19.0000 −0.673015 −0.336507 0.941681i \(-0.609246\pi\)
−0.336507 + 0.941681i \(0.609246\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 1.00000 0.0353553
\(801\) −16.0000 −0.565332
\(802\) −2.00000 −0.0706225
\(803\) −3.00000 −0.105868
\(804\) −3.00000 −0.105802
\(805\) −30.0000 −1.05736
\(806\) −10.0000 −0.352235
\(807\) −18.0000 −0.633630
\(808\) −6.00000 −0.211079
\(809\) −13.0000 −0.457056 −0.228528 0.973537i \(-0.573391\pi\)
−0.228528 + 0.973537i \(0.573391\pi\)
\(810\) 2.00000 0.0702728
\(811\) 25.0000 0.877869 0.438934 0.898519i \(-0.355356\pi\)
0.438934 + 0.898519i \(0.355356\pi\)
\(812\) 3.00000 0.105279
\(813\) −15.0000 −0.526073
\(814\) 6.00000 0.210300
\(815\) 4.00000 0.140114
\(816\) −7.00000 −0.245049
\(817\) 0 0
\(818\) 34.0000 1.18878
\(819\) −6.00000 −0.209657
\(820\) 12.0000 0.419058
\(821\) −16.0000 −0.558404 −0.279202 0.960232i \(-0.590070\pi\)
−0.279202 + 0.960232i \(0.590070\pi\)
\(822\) 17.0000 0.592943
\(823\) −43.0000 −1.49889 −0.749443 0.662069i \(-0.769679\pi\)
−0.749443 + 0.662069i \(0.769679\pi\)
\(824\) 12.0000 0.418040
\(825\) 1.00000 0.0348155
\(826\) −9.00000 −0.313150
\(827\) −37.0000 −1.28662 −0.643308 0.765607i \(-0.722439\pi\)
−0.643308 + 0.765607i \(0.722439\pi\)
\(828\) 10.0000 0.347524
\(829\) −25.0000 −0.868286 −0.434143 0.900844i \(-0.642949\pi\)
−0.434143 + 0.900844i \(0.642949\pi\)
\(830\) 16.0000 0.555368
\(831\) −16.0000 −0.555034
\(832\) −1.00000 −0.0346688
\(833\) −14.0000 −0.485071
\(834\) 14.0000 0.484780
\(835\) −44.0000 −1.52268
\(836\) 0 0
\(837\) 50.0000 1.72825
\(838\) −10.0000 −0.345444
\(839\) −40.0000 −1.38095 −0.690477 0.723355i \(-0.742599\pi\)
−0.690477 + 0.723355i \(0.742599\pi\)
\(840\) −6.00000 −0.207020
\(841\) −28.0000 −0.965517
\(842\) 15.0000 0.516934
\(843\) −16.0000 −0.551069
\(844\) −1.00000 −0.0344214
\(845\) 24.0000 0.825625
\(846\) 0 0
\(847\) −3.00000 −0.103081
\(848\) 1.00000 0.0343401
\(849\) −14.0000 −0.480479
\(850\) −7.00000 −0.240098
\(851\) −30.0000 −1.02839
\(852\) 10.0000 0.342594
\(853\) 6.00000 0.205436 0.102718 0.994711i \(-0.467246\pi\)
0.102718 + 0.994711i \(0.467246\pi\)
\(854\) −36.0000 −1.23189
\(855\) 0 0
\(856\) −13.0000 −0.444331
\(857\) 30.0000 1.02478 0.512390 0.858753i \(-0.328760\pi\)
0.512390 + 0.858753i \(0.328760\pi\)
\(858\) −1.00000 −0.0341394
\(859\) −30.0000 −1.02359 −0.511793 0.859109i \(-0.671019\pi\)
−0.511793 + 0.859109i \(0.671019\pi\)
\(860\) 8.00000 0.272798
\(861\) 18.0000 0.613438
\(862\) 34.0000 1.15804
\(863\) 6.00000 0.204242 0.102121 0.994772i \(-0.467437\pi\)
0.102121 + 0.994772i \(0.467437\pi\)
\(864\) 5.00000 0.170103
\(865\) 28.0000 0.952029
\(866\) 24.0000 0.815553
\(867\) 32.0000 1.08678
\(868\) 30.0000 1.01827
\(869\) 8.00000 0.271381
\(870\) −2.00000 −0.0678064
\(871\) 3.00000 0.101651
\(872\) −13.0000 −0.440236
\(873\) 16.0000 0.541518
\(874\) 0 0
\(875\) −36.0000 −1.21702
\(876\) 3.00000 0.101361
\(877\) −27.0000 −0.911725 −0.455863 0.890050i \(-0.650669\pi\)
−0.455863 + 0.890050i \(0.650669\pi\)
\(878\) −14.0000 −0.472477
\(879\) 9.00000 0.303562
\(880\) 2.00000 0.0674200
\(881\) 42.0000 1.41502 0.707508 0.706705i \(-0.249819\pi\)
0.707508 + 0.706705i \(0.249819\pi\)
\(882\) 4.00000 0.134687
\(883\) −44.0000 −1.48072 −0.740359 0.672212i \(-0.765344\pi\)
−0.740359 + 0.672212i \(0.765344\pi\)
\(884\) 7.00000 0.235435
\(885\) 6.00000 0.201688
\(886\) −18.0000 −0.604722
\(887\) 4.00000 0.134307 0.0671534 0.997743i \(-0.478608\pi\)
0.0671534 + 0.997743i \(0.478608\pi\)
\(888\) −6.00000 −0.201347
\(889\) 48.0000 1.60987
\(890\) 16.0000 0.536321
\(891\) −1.00000 −0.0335013
\(892\) −10.0000 −0.334825
\(893\) 0 0
\(894\) 18.0000 0.602010
\(895\) 8.00000 0.267411
\(896\) 3.00000 0.100223
\(897\) 5.00000 0.166945
\(898\) 24.0000 0.800890
\(899\) 10.0000 0.333519
\(900\) 2.00000 0.0666667
\(901\) −7.00000 −0.233204
\(902\) −6.00000 −0.199778
\(903\) 12.0000 0.399335
\(904\) −12.0000 −0.399114
\(905\) −4.00000 −0.132964
\(906\) 12.0000 0.398673
\(907\) −13.0000 −0.431658 −0.215829 0.976431i \(-0.569245\pi\)
−0.215829 + 0.976431i \(0.569245\pi\)
\(908\) 15.0000 0.497792
\(909\) −12.0000 −0.398015
\(910\) 6.00000 0.198898
\(911\) 6.00000 0.198789 0.0993944 0.995048i \(-0.468309\pi\)
0.0993944 + 0.995048i \(0.468309\pi\)
\(912\) 0 0
\(913\) −8.00000 −0.264761
\(914\) 37.0000 1.22385
\(915\) 24.0000 0.793416
\(916\) 20.0000 0.660819
\(917\) 66.0000 2.17951
\(918\) −35.0000 −1.15517
\(919\) 3.00000 0.0989609 0.0494804 0.998775i \(-0.484243\pi\)
0.0494804 + 0.998775i \(0.484243\pi\)
\(920\) −10.0000 −0.329690
\(921\) 16.0000 0.527218
\(922\) 8.00000 0.263466
\(923\) −10.0000 −0.329154
\(924\) 3.00000 0.0986928
\(925\) −6.00000 −0.197279
\(926\) −16.0000 −0.525793
\(927\) 24.0000 0.788263
\(928\) 1.00000 0.0328266
\(929\) 29.0000 0.951459 0.475730 0.879592i \(-0.342184\pi\)
0.475730 + 0.879592i \(0.342184\pi\)
\(930\) −20.0000 −0.655826
\(931\) 0 0
\(932\) −10.0000 −0.327561
\(933\) 3.00000 0.0982156
\(934\) −20.0000 −0.654420
\(935\) −14.0000 −0.457849
\(936\) −2.00000 −0.0653720
\(937\) −5.00000 −0.163343 −0.0816714 0.996659i \(-0.526026\pi\)
−0.0816714 + 0.996659i \(0.526026\pi\)
\(938\) −9.00000 −0.293860
\(939\) 21.0000 0.685309
\(940\) 0 0
\(941\) −33.0000 −1.07577 −0.537885 0.843018i \(-0.680776\pi\)
−0.537885 + 0.843018i \(0.680776\pi\)
\(942\) 22.0000 0.716799
\(943\) 30.0000 0.976934
\(944\) −3.00000 −0.0976417
\(945\) −30.0000 −0.975900
\(946\) −4.00000 −0.130051
\(947\) −48.0000 −1.55979 −0.779895 0.625910i \(-0.784728\pi\)
−0.779895 + 0.625910i \(0.784728\pi\)
\(948\) −8.00000 −0.259828
\(949\) −3.00000 −0.0973841
\(950\) 0 0
\(951\) −21.0000 −0.680972
\(952\) −21.0000 −0.680614
\(953\) −60.0000 −1.94359 −0.971795 0.235826i \(-0.924220\pi\)
−0.971795 + 0.235826i \(0.924220\pi\)
\(954\) 2.00000 0.0647524
\(955\) −6.00000 −0.194155
\(956\) 25.0000 0.808558
\(957\) 1.00000 0.0323254
\(958\) 12.0000 0.387702
\(959\) 51.0000 1.64688
\(960\) −2.00000 −0.0645497
\(961\) 69.0000 2.22581
\(962\) 6.00000 0.193448
\(963\) −26.0000 −0.837838
\(964\) 4.00000 0.128831
\(965\) −8.00000 −0.257529
\(966\) −15.0000 −0.482617
\(967\) 32.0000 1.02905 0.514525 0.857475i \(-0.327968\pi\)
0.514525 + 0.857475i \(0.327968\pi\)
\(968\) −1.00000 −0.0321412
\(969\) 0 0
\(970\) −16.0000 −0.513729
\(971\) 44.0000 1.41203 0.706014 0.708198i \(-0.250492\pi\)
0.706014 + 0.708198i \(0.250492\pi\)
\(972\) 16.0000 0.513200
\(973\) 42.0000 1.34646
\(974\) −12.0000 −0.384505
\(975\) 1.00000 0.0320256
\(976\) −12.0000 −0.384111
\(977\) 60.0000 1.91957 0.959785 0.280736i \(-0.0905785\pi\)
0.959785 + 0.280736i \(0.0905785\pi\)
\(978\) 2.00000 0.0639529
\(979\) −8.00000 −0.255681
\(980\) −4.00000 −0.127775
\(981\) −26.0000 −0.830116
\(982\) −10.0000 −0.319113
\(983\) 46.0000 1.46717 0.733586 0.679597i \(-0.237845\pi\)
0.733586 + 0.679597i \(0.237845\pi\)
\(984\) 6.00000 0.191273
\(985\) 24.0000 0.764704
\(986\) −7.00000 −0.222925
\(987\) 0 0
\(988\) 0 0
\(989\) 20.0000 0.635963
\(990\) 4.00000 0.127128
\(991\) −44.0000 −1.39771 −0.698853 0.715265i \(-0.746306\pi\)
−0.698853 + 0.715265i \(0.746306\pi\)
\(992\) 10.0000 0.317500
\(993\) −29.0000 −0.920287
\(994\) 30.0000 0.951542
\(995\) −46.0000 −1.45830
\(996\) 8.00000 0.253490
\(997\) −10.0000 −0.316703 −0.158352 0.987383i \(-0.550618\pi\)
−0.158352 + 0.987383i \(0.550618\pi\)
\(998\) 28.0000 0.886325
\(999\) −30.0000 −0.949158
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 7942.2.a.i.1.1 1
19.18 odd 2 418.2.a.a.1.1 1
57.56 even 2 3762.2.a.g.1.1 1
76.75 even 2 3344.2.a.h.1.1 1
209.208 even 2 4598.2.a.b.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
418.2.a.a.1.1 1 19.18 odd 2
3344.2.a.h.1.1 1 76.75 even 2
3762.2.a.g.1.1 1 57.56 even 2
4598.2.a.b.1.1 1 209.208 even 2
7942.2.a.i.1.1 1 1.1 even 1 trivial