Properties

Label 1456.2.a.t.1.1
Level $1456$
Weight $2$
Character 1456.1
Self dual yes
Analytic conductor $11.626$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1456,2,Mod(1,1456)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1456, 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("1456.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1456 = 2^{4} \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1456.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: \(11.6262185343\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.316.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 4x + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 91)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(0.470683\) of defining polynomial
Character \(\chi\) \(=\) 1456.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.24914 q^{3} +0.529317 q^{5} +1.00000 q^{7} +2.05863 q^{9} +O(q^{10})\) \(q-2.24914 q^{3} +0.529317 q^{5} +1.00000 q^{7} +2.05863 q^{9} +2.24914 q^{11} +1.00000 q^{13} -1.19051 q^{15} -1.30777 q^{17} +1.47068 q^{19} -2.24914 q^{21} -5.83709 q^{23} -4.71982 q^{25} +2.11727 q^{27} +5.22154 q^{29} +7.02760 q^{31} -5.05863 q^{33} +0.529317 q^{35} -2.36641 q^{37} -2.24914 q^{39} +6.49828 q^{41} -11.3940 q^{43} +1.08967 q^{45} -8.58451 q^{47} +1.00000 q^{49} +2.94137 q^{51} +11.2767 q^{53} +1.19051 q^{55} -3.30777 q^{57} +12.1725 q^{59} -2.00000 q^{61} +2.05863 q^{63} +0.529317 q^{65} +15.9379 q^{67} +13.1284 q^{69} -1.19051 q^{71} +7.64315 q^{73} +10.6155 q^{75} +2.24914 q^{77} +1.33881 q^{79} -10.9379 q^{81} +16.3500 q^{83} -0.692226 q^{85} -11.7440 q^{87} +6.91033 q^{89} +1.00000 q^{91} -15.8061 q^{93} +0.778457 q^{95} -3.47068 q^{97} +4.63016 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 2 q^{3} + 2 q^{5} + 3 q^{7} + 7 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 2 q^{3} + 2 q^{5} + 3 q^{7} + 7 q^{9} - 2 q^{11} + 3 q^{13} + 6 q^{15} + 4 q^{17} + 4 q^{19} + 2 q^{21} - 10 q^{23} - 5 q^{25} + 8 q^{27} + 24 q^{29} + 4 q^{31} - 16 q^{33} + 2 q^{35} + 2 q^{39} + 2 q^{41} - 10 q^{43} + 22 q^{45} + 8 q^{47} + 3 q^{49} + 8 q^{51} + 8 q^{53} - 6 q^{55} - 2 q^{57} + 4 q^{59} - 6 q^{61} + 7 q^{63} + 2 q^{65} + 12 q^{67} - 6 q^{69} + 6 q^{71} - 10 q^{73} + 16 q^{75} - 2 q^{77} + 14 q^{79} + 3 q^{81} + 12 q^{83} - 10 q^{85} + 26 q^{87} + 2 q^{89} + 3 q^{91} - 22 q^{93} - 6 q^{95} - 10 q^{97} - 14 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −2.24914 −1.29854 −0.649271 0.760557i \(-0.724926\pi\)
−0.649271 + 0.760557i \(0.724926\pi\)
\(4\) 0 0
\(5\) 0.529317 0.236718 0.118359 0.992971i \(-0.462237\pi\)
0.118359 + 0.992971i \(0.462237\pi\)
\(6\) 0 0
\(7\) 1.00000 0.377964
\(8\) 0 0
\(9\) 2.05863 0.686211
\(10\) 0 0
\(11\) 2.24914 0.678141 0.339071 0.940761i \(-0.389887\pi\)
0.339071 + 0.940761i \(0.389887\pi\)
\(12\) 0 0
\(13\) 1.00000 0.277350
\(14\) 0 0
\(15\) −1.19051 −0.307388
\(16\) 0 0
\(17\) −1.30777 −0.317182 −0.158591 0.987344i \(-0.550695\pi\)
−0.158591 + 0.987344i \(0.550695\pi\)
\(18\) 0 0
\(19\) 1.47068 0.337398 0.168699 0.985668i \(-0.446043\pi\)
0.168699 + 0.985668i \(0.446043\pi\)
\(20\) 0 0
\(21\) −2.24914 −0.490803
\(22\) 0 0
\(23\) −5.83709 −1.21712 −0.608559 0.793509i \(-0.708252\pi\)
−0.608559 + 0.793509i \(0.708252\pi\)
\(24\) 0 0
\(25\) −4.71982 −0.943965
\(26\) 0 0
\(27\) 2.11727 0.407468
\(28\) 0 0
\(29\) 5.22154 0.969616 0.484808 0.874621i \(-0.338889\pi\)
0.484808 + 0.874621i \(0.338889\pi\)
\(30\) 0 0
\(31\) 7.02760 1.26219 0.631097 0.775704i \(-0.282605\pi\)
0.631097 + 0.775704i \(0.282605\pi\)
\(32\) 0 0
\(33\) −5.05863 −0.880595
\(34\) 0 0
\(35\) 0.529317 0.0894708
\(36\) 0 0
\(37\) −2.36641 −0.389035 −0.194517 0.980899i \(-0.562314\pi\)
−0.194517 + 0.980899i \(0.562314\pi\)
\(38\) 0 0
\(39\) −2.24914 −0.360151
\(40\) 0 0
\(41\) 6.49828 1.01486 0.507431 0.861693i \(-0.330595\pi\)
0.507431 + 0.861693i \(0.330595\pi\)
\(42\) 0 0
\(43\) −11.3940 −1.73757 −0.868785 0.495190i \(-0.835098\pi\)
−0.868785 + 0.495190i \(0.835098\pi\)
\(44\) 0 0
\(45\) 1.08967 0.162438
\(46\) 0 0
\(47\) −8.58451 −1.25218 −0.626090 0.779751i \(-0.715346\pi\)
−0.626090 + 0.779751i \(0.715346\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 2.94137 0.411874
\(52\) 0 0
\(53\) 11.2767 1.54898 0.774490 0.632587i \(-0.218007\pi\)
0.774490 + 0.632587i \(0.218007\pi\)
\(54\) 0 0
\(55\) 1.19051 0.160528
\(56\) 0 0
\(57\) −3.30777 −0.438125
\(58\) 0 0
\(59\) 12.1725 1.58472 0.792360 0.610054i \(-0.208852\pi\)
0.792360 + 0.610054i \(0.208852\pi\)
\(60\) 0 0
\(61\) −2.00000 −0.256074 −0.128037 0.991769i \(-0.540868\pi\)
−0.128037 + 0.991769i \(0.540868\pi\)
\(62\) 0 0
\(63\) 2.05863 0.259363
\(64\) 0 0
\(65\) 0.529317 0.0656536
\(66\) 0 0
\(67\) 15.9379 1.94713 0.973564 0.228415i \(-0.0733542\pi\)
0.973564 + 0.228415i \(0.0733542\pi\)
\(68\) 0 0
\(69\) 13.1284 1.58048
\(70\) 0 0
\(71\) −1.19051 −0.141287 −0.0706436 0.997502i \(-0.522505\pi\)
−0.0706436 + 0.997502i \(0.522505\pi\)
\(72\) 0 0
\(73\) 7.64315 0.894562 0.447281 0.894393i \(-0.352392\pi\)
0.447281 + 0.894393i \(0.352392\pi\)
\(74\) 0 0
\(75\) 10.6155 1.22578
\(76\) 0 0
\(77\) 2.24914 0.256313
\(78\) 0 0
\(79\) 1.33881 0.150628 0.0753139 0.997160i \(-0.476004\pi\)
0.0753139 + 0.997160i \(0.476004\pi\)
\(80\) 0 0
\(81\) −10.9379 −1.21533
\(82\) 0 0
\(83\) 16.3500 1.79464 0.897322 0.441377i \(-0.145510\pi\)
0.897322 + 0.441377i \(0.145510\pi\)
\(84\) 0 0
\(85\) −0.692226 −0.0750825
\(86\) 0 0
\(87\) −11.7440 −1.25909
\(88\) 0 0
\(89\) 6.91033 0.732494 0.366247 0.930518i \(-0.380643\pi\)
0.366247 + 0.930518i \(0.380643\pi\)
\(90\) 0 0
\(91\) 1.00000 0.104828
\(92\) 0 0
\(93\) −15.8061 −1.63901
\(94\) 0 0
\(95\) 0.778457 0.0798680
\(96\) 0 0
\(97\) −3.47068 −0.352395 −0.176197 0.984355i \(-0.556380\pi\)
−0.176197 + 0.984355i \(0.556380\pi\)
\(98\) 0 0
\(99\) 4.63016 0.465348
\(100\) 0 0
\(101\) 7.75086 0.771239 0.385620 0.922658i \(-0.373988\pi\)
0.385620 + 0.922658i \(0.373988\pi\)
\(102\) 0 0
\(103\) 16.9966 1.67472 0.837361 0.546651i \(-0.184098\pi\)
0.837361 + 0.546651i \(0.184098\pi\)
\(104\) 0 0
\(105\) −1.19051 −0.116182
\(106\) 0 0
\(107\) 5.55691 0.537207 0.268604 0.963251i \(-0.413438\pi\)
0.268604 + 0.963251i \(0.413438\pi\)
\(108\) 0 0
\(109\) 7.92332 0.758917 0.379458 0.925209i \(-0.376110\pi\)
0.379458 + 0.925209i \(0.376110\pi\)
\(110\) 0 0
\(111\) 5.32238 0.505178
\(112\) 0 0
\(113\) −9.89229 −0.930588 −0.465294 0.885156i \(-0.654051\pi\)
−0.465294 + 0.885156i \(0.654051\pi\)
\(114\) 0 0
\(115\) −3.08967 −0.288113
\(116\) 0 0
\(117\) 2.05863 0.190321
\(118\) 0 0
\(119\) −1.30777 −0.119883
\(120\) 0 0
\(121\) −5.94137 −0.540124
\(122\) 0 0
\(123\) −14.6155 −1.31784
\(124\) 0 0
\(125\) −5.14486 −0.460171
\(126\) 0 0
\(127\) −0.824101 −0.0731271 −0.0365635 0.999331i \(-0.511641\pi\)
−0.0365635 + 0.999331i \(0.511641\pi\)
\(128\) 0 0
\(129\) 25.6267 2.25631
\(130\) 0 0
\(131\) −10.6155 −0.927485 −0.463742 0.885970i \(-0.653494\pi\)
−0.463742 + 0.885970i \(0.653494\pi\)
\(132\) 0 0
\(133\) 1.47068 0.127524
\(134\) 0 0
\(135\) 1.12070 0.0964549
\(136\) 0 0
\(137\) 11.3630 0.970804 0.485402 0.874291i \(-0.338673\pi\)
0.485402 + 0.874291i \(0.338673\pi\)
\(138\) 0 0
\(139\) 13.9233 1.18096 0.590480 0.807052i \(-0.298938\pi\)
0.590480 + 0.807052i \(0.298938\pi\)
\(140\) 0 0
\(141\) 19.3078 1.62601
\(142\) 0 0
\(143\) 2.24914 0.188083
\(144\) 0 0
\(145\) 2.76385 0.229525
\(146\) 0 0
\(147\) −2.24914 −0.185506
\(148\) 0 0
\(149\) 9.30777 0.762523 0.381261 0.924467i \(-0.375490\pi\)
0.381261 + 0.924467i \(0.375490\pi\)
\(150\) 0 0
\(151\) −7.07324 −0.575612 −0.287806 0.957689i \(-0.592926\pi\)
−0.287806 + 0.957689i \(0.592926\pi\)
\(152\) 0 0
\(153\) −2.69223 −0.217654
\(154\) 0 0
\(155\) 3.71982 0.298783
\(156\) 0 0
\(157\) −6.04059 −0.482091 −0.241046 0.970514i \(-0.577490\pi\)
−0.241046 + 0.970514i \(0.577490\pi\)
\(158\) 0 0
\(159\) −25.3630 −2.01141
\(160\) 0 0
\(161\) −5.83709 −0.460027
\(162\) 0 0
\(163\) −6.38101 −0.499800 −0.249900 0.968272i \(-0.580398\pi\)
−0.249900 + 0.968272i \(0.580398\pi\)
\(164\) 0 0
\(165\) −2.67762 −0.208452
\(166\) 0 0
\(167\) 16.5845 1.28335 0.641674 0.766977i \(-0.278240\pi\)
0.641674 + 0.766977i \(0.278240\pi\)
\(168\) 0 0
\(169\) 1.00000 0.0769231
\(170\) 0 0
\(171\) 3.02760 0.231526
\(172\) 0 0
\(173\) −23.3009 −1.77153 −0.885767 0.464130i \(-0.846367\pi\)
−0.885767 + 0.464130i \(0.846367\pi\)
\(174\) 0 0
\(175\) −4.71982 −0.356785
\(176\) 0 0
\(177\) −27.3776 −2.05782
\(178\) 0 0
\(179\) −21.0422 −1.57277 −0.786384 0.617738i \(-0.788049\pi\)
−0.786384 + 0.617738i \(0.788049\pi\)
\(180\) 0 0
\(181\) 16.7474 1.24483 0.622413 0.782689i \(-0.286152\pi\)
0.622413 + 0.782689i \(0.286152\pi\)
\(182\) 0 0
\(183\) 4.49828 0.332523
\(184\) 0 0
\(185\) −1.25258 −0.0920914
\(186\) 0 0
\(187\) −2.94137 −0.215094
\(188\) 0 0
\(189\) 2.11727 0.154008
\(190\) 0 0
\(191\) 7.43965 0.538314 0.269157 0.963096i \(-0.413255\pi\)
0.269157 + 0.963096i \(0.413255\pi\)
\(192\) 0 0
\(193\) −1.50172 −0.108096 −0.0540480 0.998538i \(-0.517212\pi\)
−0.0540480 + 0.998538i \(0.517212\pi\)
\(194\) 0 0
\(195\) −1.19051 −0.0852540
\(196\) 0 0
\(197\) 23.9931 1.70944 0.854720 0.519090i \(-0.173729\pi\)
0.854720 + 0.519090i \(0.173729\pi\)
\(198\) 0 0
\(199\) 2.01461 0.142812 0.0714059 0.997447i \(-0.477251\pi\)
0.0714059 + 0.997447i \(0.477251\pi\)
\(200\) 0 0
\(201\) −35.8466 −2.52843
\(202\) 0 0
\(203\) 5.22154 0.366480
\(204\) 0 0
\(205\) 3.43965 0.240235
\(206\) 0 0
\(207\) −12.0164 −0.835199
\(208\) 0 0
\(209\) 3.30777 0.228803
\(210\) 0 0
\(211\) −10.1008 −0.695370 −0.347685 0.937611i \(-0.613032\pi\)
−0.347685 + 0.937611i \(0.613032\pi\)
\(212\) 0 0
\(213\) 2.67762 0.183467
\(214\) 0 0
\(215\) −6.03104 −0.411313
\(216\) 0 0
\(217\) 7.02760 0.477064
\(218\) 0 0
\(219\) −17.1905 −1.16163
\(220\) 0 0
\(221\) −1.30777 −0.0879704
\(222\) 0 0
\(223\) −10.1414 −0.679120 −0.339560 0.940584i \(-0.610278\pi\)
−0.339560 + 0.940584i \(0.610278\pi\)
\(224\) 0 0
\(225\) −9.71639 −0.647759
\(226\) 0 0
\(227\) 5.38445 0.357379 0.178689 0.983906i \(-0.442814\pi\)
0.178689 + 0.983906i \(0.442814\pi\)
\(228\) 0 0
\(229\) 3.32238 0.219549 0.109775 0.993957i \(-0.464987\pi\)
0.109775 + 0.993957i \(0.464987\pi\)
\(230\) 0 0
\(231\) −5.05863 −0.332834
\(232\) 0 0
\(233\) 13.7198 0.898816 0.449408 0.893327i \(-0.351635\pi\)
0.449408 + 0.893327i \(0.351635\pi\)
\(234\) 0 0
\(235\) −4.54392 −0.296413
\(236\) 0 0
\(237\) −3.01117 −0.195597
\(238\) 0 0
\(239\) 3.50172 0.226507 0.113254 0.993566i \(-0.463873\pi\)
0.113254 + 0.993566i \(0.463873\pi\)
\(240\) 0 0
\(241\) 1.58795 0.102289 0.0511444 0.998691i \(-0.483713\pi\)
0.0511444 + 0.998691i \(0.483713\pi\)
\(242\) 0 0
\(243\) 18.2491 1.17068
\(244\) 0 0
\(245\) 0.529317 0.0338168
\(246\) 0 0
\(247\) 1.47068 0.0935773
\(248\) 0 0
\(249\) −36.7734 −2.33042
\(250\) 0 0
\(251\) 4.92676 0.310974 0.155487 0.987838i \(-0.450305\pi\)
0.155487 + 0.987838i \(0.450305\pi\)
\(252\) 0 0
\(253\) −13.1284 −0.825378
\(254\) 0 0
\(255\) 1.55691 0.0974978
\(256\) 0 0
\(257\) −8.01461 −0.499938 −0.249969 0.968254i \(-0.580420\pi\)
−0.249969 + 0.968254i \(0.580420\pi\)
\(258\) 0 0
\(259\) −2.36641 −0.147041
\(260\) 0 0
\(261\) 10.7492 0.665361
\(262\) 0 0
\(263\) −1.60256 −0.0988179 −0.0494090 0.998779i \(-0.515734\pi\)
−0.0494090 + 0.998779i \(0.515734\pi\)
\(264\) 0 0
\(265\) 5.96896 0.366671
\(266\) 0 0
\(267\) −15.5423 −0.951174
\(268\) 0 0
\(269\) −11.8207 −0.720719 −0.360359 0.932814i \(-0.617346\pi\)
−0.360359 + 0.932814i \(0.617346\pi\)
\(270\) 0 0
\(271\) −21.8827 −1.32928 −0.664641 0.747163i \(-0.731415\pi\)
−0.664641 + 0.747163i \(0.731415\pi\)
\(272\) 0 0
\(273\) −2.24914 −0.136124
\(274\) 0 0
\(275\) −10.6155 −0.640142
\(276\) 0 0
\(277\) −10.2181 −0.613946 −0.306973 0.951718i \(-0.599316\pi\)
−0.306973 + 0.951718i \(0.599316\pi\)
\(278\) 0 0
\(279\) 14.4672 0.866131
\(280\) 0 0
\(281\) 1.54231 0.0920063 0.0460031 0.998941i \(-0.485352\pi\)
0.0460031 + 0.998941i \(0.485352\pi\)
\(282\) 0 0
\(283\) −15.8466 −0.941985 −0.470993 0.882137i \(-0.656104\pi\)
−0.470993 + 0.882137i \(0.656104\pi\)
\(284\) 0 0
\(285\) −1.75086 −0.103712
\(286\) 0 0
\(287\) 6.49828 0.383581
\(288\) 0 0
\(289\) −15.2897 −0.899396
\(290\) 0 0
\(291\) 7.80605 0.457599
\(292\) 0 0
\(293\) 11.0828 0.647464 0.323732 0.946149i \(-0.395062\pi\)
0.323732 + 0.946149i \(0.395062\pi\)
\(294\) 0 0
\(295\) 6.44309 0.375131
\(296\) 0 0
\(297\) 4.76203 0.276321
\(298\) 0 0
\(299\) −5.83709 −0.337568
\(300\) 0 0
\(301\) −11.3940 −0.656740
\(302\) 0 0
\(303\) −17.4328 −1.00149
\(304\) 0 0
\(305\) −1.05863 −0.0606172
\(306\) 0 0
\(307\) −20.4121 −1.16498 −0.582489 0.812839i \(-0.697921\pi\)
−0.582489 + 0.812839i \(0.697921\pi\)
\(308\) 0 0
\(309\) −38.2277 −2.17470
\(310\) 0 0
\(311\) 1.92332 0.109062 0.0545308 0.998512i \(-0.482634\pi\)
0.0545308 + 0.998512i \(0.482634\pi\)
\(312\) 0 0
\(313\) −14.3664 −0.812037 −0.406019 0.913865i \(-0.633083\pi\)
−0.406019 + 0.913865i \(0.633083\pi\)
\(314\) 0 0
\(315\) 1.08967 0.0613959
\(316\) 0 0
\(317\) 15.5569 0.873763 0.436882 0.899519i \(-0.356083\pi\)
0.436882 + 0.899519i \(0.356083\pi\)
\(318\) 0 0
\(319\) 11.7440 0.657537
\(320\) 0 0
\(321\) −12.4983 −0.697586
\(322\) 0 0
\(323\) −1.92332 −0.107016
\(324\) 0 0
\(325\) −4.71982 −0.261809
\(326\) 0 0
\(327\) −17.8207 −0.985485
\(328\) 0 0
\(329\) −8.58451 −0.473279
\(330\) 0 0
\(331\) −31.5500 −1.73415 −0.867073 0.498180i \(-0.834002\pi\)
−0.867073 + 0.498180i \(0.834002\pi\)
\(332\) 0 0
\(333\) −4.87156 −0.266960
\(334\) 0 0
\(335\) 8.43621 0.460919
\(336\) 0 0
\(337\) −8.42666 −0.459029 −0.229515 0.973305i \(-0.573714\pi\)
−0.229515 + 0.973305i \(0.573714\pi\)
\(338\) 0 0
\(339\) 22.2491 1.20841
\(340\) 0 0
\(341\) 15.8061 0.855946
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) 0 0
\(345\) 6.94910 0.374127
\(346\) 0 0
\(347\) −19.3484 −1.03867 −0.519337 0.854569i \(-0.673821\pi\)
−0.519337 + 0.854569i \(0.673821\pi\)
\(348\) 0 0
\(349\) 27.2553 1.45894 0.729470 0.684013i \(-0.239767\pi\)
0.729470 + 0.684013i \(0.239767\pi\)
\(350\) 0 0
\(351\) 2.11727 0.113011
\(352\) 0 0
\(353\) −25.6742 −1.36650 −0.683249 0.730185i \(-0.739434\pi\)
−0.683249 + 0.730185i \(0.739434\pi\)
\(354\) 0 0
\(355\) −0.630155 −0.0334452
\(356\) 0 0
\(357\) 2.94137 0.155674
\(358\) 0 0
\(359\) 23.4182 1.23596 0.617982 0.786192i \(-0.287951\pi\)
0.617982 + 0.786192i \(0.287951\pi\)
\(360\) 0 0
\(361\) −16.8371 −0.886163
\(362\) 0 0
\(363\) 13.3630 0.701374
\(364\) 0 0
\(365\) 4.04564 0.211759
\(366\) 0 0
\(367\) 14.6854 0.766569 0.383285 0.923630i \(-0.374793\pi\)
0.383285 + 0.923630i \(0.374793\pi\)
\(368\) 0 0
\(369\) 13.3776 0.696409
\(370\) 0 0
\(371\) 11.2767 0.585459
\(372\) 0 0
\(373\) −23.6673 −1.22545 −0.612723 0.790298i \(-0.709926\pi\)
−0.612723 + 0.790298i \(0.709926\pi\)
\(374\) 0 0
\(375\) 11.5715 0.597551
\(376\) 0 0
\(377\) 5.22154 0.268923
\(378\) 0 0
\(379\) 32.7405 1.68177 0.840884 0.541215i \(-0.182035\pi\)
0.840884 + 0.541215i \(0.182035\pi\)
\(380\) 0 0
\(381\) 1.85352 0.0949586
\(382\) 0 0
\(383\) 22.6155 1.15560 0.577800 0.816178i \(-0.303911\pi\)
0.577800 + 0.816178i \(0.303911\pi\)
\(384\) 0 0
\(385\) 1.19051 0.0606739
\(386\) 0 0
\(387\) −23.4561 −1.19234
\(388\) 0 0
\(389\) 38.0483 1.92913 0.964563 0.263852i \(-0.0849930\pi\)
0.964563 + 0.263852i \(0.0849930\pi\)
\(390\) 0 0
\(391\) 7.63359 0.386047
\(392\) 0 0
\(393\) 23.8759 1.20438
\(394\) 0 0
\(395\) 0.708654 0.0356562
\(396\) 0 0
\(397\) −11.7052 −0.587468 −0.293734 0.955887i \(-0.594898\pi\)
−0.293734 + 0.955887i \(0.594898\pi\)
\(398\) 0 0
\(399\) −3.30777 −0.165596
\(400\) 0 0
\(401\) 3.55691 0.177624 0.0888119 0.996048i \(-0.471693\pi\)
0.0888119 + 0.996048i \(0.471693\pi\)
\(402\) 0 0
\(403\) 7.02760 0.350070
\(404\) 0 0
\(405\) −5.78963 −0.287689
\(406\) 0 0
\(407\) −5.32238 −0.263821
\(408\) 0 0
\(409\) 5.26213 0.260196 0.130098 0.991501i \(-0.458471\pi\)
0.130098 + 0.991501i \(0.458471\pi\)
\(410\) 0 0
\(411\) −25.5569 −1.26063
\(412\) 0 0
\(413\) 12.1725 0.598968
\(414\) 0 0
\(415\) 8.65432 0.424824
\(416\) 0 0
\(417\) −31.3155 −1.53353
\(418\) 0 0
\(419\) −26.0337 −1.27183 −0.635915 0.771759i \(-0.719377\pi\)
−0.635915 + 0.771759i \(0.719377\pi\)
\(420\) 0 0
\(421\) 22.2423 1.08402 0.542011 0.840372i \(-0.317663\pi\)
0.542011 + 0.840372i \(0.317663\pi\)
\(422\) 0 0
\(423\) −17.6724 −0.859260
\(424\) 0 0
\(425\) 6.17246 0.299408
\(426\) 0 0
\(427\) −2.00000 −0.0967868
\(428\) 0 0
\(429\) −5.05863 −0.244233
\(430\) 0 0
\(431\) −27.6742 −1.33302 −0.666509 0.745497i \(-0.732212\pi\)
−0.666509 + 0.745497i \(0.732212\pi\)
\(432\) 0 0
\(433\) −12.7880 −0.614552 −0.307276 0.951620i \(-0.599418\pi\)
−0.307276 + 0.951620i \(0.599418\pi\)
\(434\) 0 0
\(435\) −6.21629 −0.298048
\(436\) 0 0
\(437\) −8.58451 −0.410653
\(438\) 0 0
\(439\) 18.1656 0.866996 0.433498 0.901155i \(-0.357279\pi\)
0.433498 + 0.901155i \(0.357279\pi\)
\(440\) 0 0
\(441\) 2.05863 0.0980302
\(442\) 0 0
\(443\) −0.107714 −0.00511767 −0.00255883 0.999997i \(-0.500815\pi\)
−0.00255883 + 0.999997i \(0.500815\pi\)
\(444\) 0 0
\(445\) 3.65775 0.173394
\(446\) 0 0
\(447\) −20.9345 −0.990167
\(448\) 0 0
\(449\) −22.1725 −1.04638 −0.523192 0.852215i \(-0.675259\pi\)
−0.523192 + 0.852215i \(0.675259\pi\)
\(450\) 0 0
\(451\) 14.6155 0.688219
\(452\) 0 0
\(453\) 15.9087 0.747457
\(454\) 0 0
\(455\) 0.529317 0.0248147
\(456\) 0 0
\(457\) 4.35953 0.203930 0.101965 0.994788i \(-0.467487\pi\)
0.101965 + 0.994788i \(0.467487\pi\)
\(458\) 0 0
\(459\) −2.76891 −0.129241
\(460\) 0 0
\(461\) 32.3810 1.50813 0.754067 0.656797i \(-0.228089\pi\)
0.754067 + 0.656797i \(0.228089\pi\)
\(462\) 0 0
\(463\) −8.36641 −0.388820 −0.194410 0.980920i \(-0.562279\pi\)
−0.194410 + 0.980920i \(0.562279\pi\)
\(464\) 0 0
\(465\) −8.36641 −0.387983
\(466\) 0 0
\(467\) 19.5423 0.904310 0.452155 0.891939i \(-0.350655\pi\)
0.452155 + 0.891939i \(0.350655\pi\)
\(468\) 0 0
\(469\) 15.9379 0.735945
\(470\) 0 0
\(471\) 13.5861 0.626016
\(472\) 0 0
\(473\) −25.6267 −1.17832
\(474\) 0 0
\(475\) −6.94137 −0.318492
\(476\) 0 0
\(477\) 23.2147 1.06293
\(478\) 0 0
\(479\) 28.5224 1.30322 0.651612 0.758553i \(-0.274093\pi\)
0.651612 + 0.758553i \(0.274093\pi\)
\(480\) 0 0
\(481\) −2.36641 −0.107899
\(482\) 0 0
\(483\) 13.1284 0.597365
\(484\) 0 0
\(485\) −1.83709 −0.0834180
\(486\) 0 0
\(487\) −24.8241 −1.12489 −0.562444 0.826836i \(-0.690139\pi\)
−0.562444 + 0.826836i \(0.690139\pi\)
\(488\) 0 0
\(489\) 14.3518 0.649011
\(490\) 0 0
\(491\) −29.1690 −1.31638 −0.658190 0.752852i \(-0.728678\pi\)
−0.658190 + 0.752852i \(0.728678\pi\)
\(492\) 0 0
\(493\) −6.82860 −0.307545
\(494\) 0 0
\(495\) 2.45082 0.110156
\(496\) 0 0
\(497\) −1.19051 −0.0534016
\(498\) 0 0
\(499\) 33.3009 1.49075 0.745376 0.666644i \(-0.232270\pi\)
0.745376 + 0.666644i \(0.232270\pi\)
\(500\) 0 0
\(501\) −37.3009 −1.66648
\(502\) 0 0
\(503\) 12.3258 0.549581 0.274791 0.961504i \(-0.411391\pi\)
0.274791 + 0.961504i \(0.411391\pi\)
\(504\) 0 0
\(505\) 4.10266 0.182566
\(506\) 0 0
\(507\) −2.24914 −0.0998878
\(508\) 0 0
\(509\) 23.7052 1.05072 0.525358 0.850882i \(-0.323932\pi\)
0.525358 + 0.850882i \(0.323932\pi\)
\(510\) 0 0
\(511\) 7.64315 0.338113
\(512\) 0 0
\(513\) 3.11383 0.137479
\(514\) 0 0
\(515\) 8.99656 0.396436
\(516\) 0 0
\(517\) −19.3078 −0.849155
\(518\) 0 0
\(519\) 52.4070 2.30041
\(520\) 0 0
\(521\) 43.9018 1.92337 0.961687 0.274149i \(-0.0883962\pi\)
0.961687 + 0.274149i \(0.0883962\pi\)
\(522\) 0 0
\(523\) −37.4328 −1.63682 −0.818410 0.574634i \(-0.805144\pi\)
−0.818410 + 0.574634i \(0.805144\pi\)
\(524\) 0 0
\(525\) 10.6155 0.463300
\(526\) 0 0
\(527\) −9.19051 −0.400345
\(528\) 0 0
\(529\) 11.0716 0.481375
\(530\) 0 0
\(531\) 25.0586 1.08745
\(532\) 0 0
\(533\) 6.49828 0.281472
\(534\) 0 0
\(535\) 2.94137 0.127166
\(536\) 0 0
\(537\) 47.3269 2.04231
\(538\) 0 0
\(539\) 2.24914 0.0968773
\(540\) 0 0
\(541\) −34.9751 −1.50370 −0.751848 0.659336i \(-0.770837\pi\)
−0.751848 + 0.659336i \(0.770837\pi\)
\(542\) 0 0
\(543\) −37.6673 −1.61646
\(544\) 0 0
\(545\) 4.19395 0.179649
\(546\) 0 0
\(547\) −6.50783 −0.278255 −0.139127 0.990274i \(-0.544430\pi\)
−0.139127 + 0.990274i \(0.544430\pi\)
\(548\) 0 0
\(549\) −4.11727 −0.175721
\(550\) 0 0
\(551\) 7.67924 0.327146
\(552\) 0 0
\(553\) 1.33881 0.0569320
\(554\) 0 0
\(555\) 2.81722 0.119585
\(556\) 0 0
\(557\) −43.4328 −1.84031 −0.920153 0.391559i \(-0.871936\pi\)
−0.920153 + 0.391559i \(0.871936\pi\)
\(558\) 0 0
\(559\) −11.3940 −0.481915
\(560\) 0 0
\(561\) 6.61555 0.279309
\(562\) 0 0
\(563\) 33.8827 1.42799 0.713993 0.700152i \(-0.246885\pi\)
0.713993 + 0.700152i \(0.246885\pi\)
\(564\) 0 0
\(565\) −5.23615 −0.220287
\(566\) 0 0
\(567\) −10.9379 −0.459350
\(568\) 0 0
\(569\) 19.2147 0.805521 0.402760 0.915305i \(-0.368051\pi\)
0.402760 + 0.915305i \(0.368051\pi\)
\(570\) 0 0
\(571\) −20.8268 −0.871573 −0.435787 0.900050i \(-0.643530\pi\)
−0.435787 + 0.900050i \(0.643530\pi\)
\(572\) 0 0
\(573\) −16.7328 −0.699023
\(574\) 0 0
\(575\) 27.5500 1.14892
\(576\) 0 0
\(577\) 28.6448 1.19250 0.596249 0.802800i \(-0.296657\pi\)
0.596249 + 0.802800i \(0.296657\pi\)
\(578\) 0 0
\(579\) 3.37758 0.140367
\(580\) 0 0
\(581\) 16.3500 0.678311
\(582\) 0 0
\(583\) 25.3630 1.05043
\(584\) 0 0
\(585\) 1.08967 0.0450523
\(586\) 0 0
\(587\) 4.32076 0.178337 0.0891685 0.996017i \(-0.471579\pi\)
0.0891685 + 0.996017i \(0.471579\pi\)
\(588\) 0 0
\(589\) 10.3354 0.425862
\(590\) 0 0
\(591\) −53.9639 −2.21978
\(592\) 0 0
\(593\) −15.9690 −0.655767 −0.327883 0.944718i \(-0.606335\pi\)
−0.327883 + 0.944718i \(0.606335\pi\)
\(594\) 0 0
\(595\) −0.692226 −0.0283785
\(596\) 0 0
\(597\) −4.53114 −0.185447
\(598\) 0 0
\(599\) 16.8697 0.689279 0.344640 0.938735i \(-0.388001\pi\)
0.344640 + 0.938735i \(0.388001\pi\)
\(600\) 0 0
\(601\) −15.3415 −0.625792 −0.312896 0.949787i \(-0.601299\pi\)
−0.312896 + 0.949787i \(0.601299\pi\)
\(602\) 0 0
\(603\) 32.8103 1.33614
\(604\) 0 0
\(605\) −3.14486 −0.127857
\(606\) 0 0
\(607\) −35.8353 −1.45451 −0.727254 0.686368i \(-0.759204\pi\)
−0.727254 + 0.686368i \(0.759204\pi\)
\(608\) 0 0
\(609\) −11.7440 −0.475890
\(610\) 0 0
\(611\) −8.58451 −0.347292
\(612\) 0 0
\(613\) 19.6673 0.794355 0.397177 0.917742i \(-0.369990\pi\)
0.397177 + 0.917742i \(0.369990\pi\)
\(614\) 0 0
\(615\) −7.73625 −0.311956
\(616\) 0 0
\(617\) −41.4588 −1.66907 −0.834533 0.550958i \(-0.814263\pi\)
−0.834533 + 0.550958i \(0.814263\pi\)
\(618\) 0 0
\(619\) −10.8793 −0.437276 −0.218638 0.975806i \(-0.570161\pi\)
−0.218638 + 0.975806i \(0.570161\pi\)
\(620\) 0 0
\(621\) −12.3587 −0.495937
\(622\) 0 0
\(623\) 6.91033 0.276857
\(624\) 0 0
\(625\) 20.8759 0.835034
\(626\) 0 0
\(627\) −7.43965 −0.297111
\(628\) 0 0
\(629\) 3.09472 0.123395
\(630\) 0 0
\(631\) 31.4396 1.25159 0.625796 0.779987i \(-0.284774\pi\)
0.625796 + 0.779987i \(0.284774\pi\)
\(632\) 0 0
\(633\) 22.7182 0.902968
\(634\) 0 0
\(635\) −0.436210 −0.0173105
\(636\) 0 0
\(637\) 1.00000 0.0396214
\(638\) 0 0
\(639\) −2.45082 −0.0969529
\(640\) 0 0
\(641\) 3.04221 0.120160 0.0600799 0.998194i \(-0.480864\pi\)
0.0600799 + 0.998194i \(0.480864\pi\)
\(642\) 0 0
\(643\) −8.02922 −0.316641 −0.158321 0.987388i \(-0.550608\pi\)
−0.158321 + 0.987388i \(0.550608\pi\)
\(644\) 0 0
\(645\) 13.5646 0.534107
\(646\) 0 0
\(647\) −7.07324 −0.278078 −0.139039 0.990287i \(-0.544401\pi\)
−0.139039 + 0.990287i \(0.544401\pi\)
\(648\) 0 0
\(649\) 27.3776 1.07466
\(650\) 0 0
\(651\) −15.8061 −0.619488
\(652\) 0 0
\(653\) −15.7586 −0.616681 −0.308341 0.951276i \(-0.599774\pi\)
−0.308341 + 0.951276i \(0.599774\pi\)
\(654\) 0 0
\(655\) −5.61899 −0.219552
\(656\) 0 0
\(657\) 15.7344 0.613859
\(658\) 0 0
\(659\) 12.2181 0.475950 0.237975 0.971271i \(-0.423516\pi\)
0.237975 + 0.971271i \(0.423516\pi\)
\(660\) 0 0
\(661\) 3.73443 0.145253 0.0726263 0.997359i \(-0.476862\pi\)
0.0726263 + 0.997359i \(0.476862\pi\)
\(662\) 0 0
\(663\) 2.94137 0.114233
\(664\) 0 0
\(665\) 0.778457 0.0301873
\(666\) 0 0
\(667\) −30.4786 −1.18014
\(668\) 0 0
\(669\) 22.8095 0.881866
\(670\) 0 0
\(671\) −4.49828 −0.173654
\(672\) 0 0
\(673\) −5.65775 −0.218090 −0.109045 0.994037i \(-0.534779\pi\)
−0.109045 + 0.994037i \(0.534779\pi\)
\(674\) 0 0
\(675\) −9.99312 −0.384636
\(676\) 0 0
\(677\) 9.39906 0.361235 0.180618 0.983553i \(-0.442190\pi\)
0.180618 + 0.983553i \(0.442190\pi\)
\(678\) 0 0
\(679\) −3.47068 −0.133193
\(680\) 0 0
\(681\) −12.1104 −0.464071
\(682\) 0 0
\(683\) 20.7328 0.793319 0.396660 0.917966i \(-0.370169\pi\)
0.396660 + 0.917966i \(0.370169\pi\)
\(684\) 0 0
\(685\) 6.01461 0.229806
\(686\) 0 0
\(687\) −7.47250 −0.285094
\(688\) 0 0
\(689\) 11.2767 0.429610
\(690\) 0 0
\(691\) −16.0862 −0.611949 −0.305975 0.952040i \(-0.598982\pi\)
−0.305975 + 0.952040i \(0.598982\pi\)
\(692\) 0 0
\(693\) 4.63016 0.175885
\(694\) 0 0
\(695\) 7.36984 0.279554
\(696\) 0 0
\(697\) −8.49828 −0.321895
\(698\) 0 0
\(699\) −30.8578 −1.16715
\(700\) 0 0
\(701\) −6.98013 −0.263636 −0.131818 0.991274i \(-0.542081\pi\)
−0.131818 + 0.991274i \(0.542081\pi\)
\(702\) 0 0
\(703\) −3.48024 −0.131260
\(704\) 0 0
\(705\) 10.2199 0.384905
\(706\) 0 0
\(707\) 7.75086 0.291501
\(708\) 0 0
\(709\) 8.39239 0.315183 0.157591 0.987504i \(-0.449627\pi\)
0.157591 + 0.987504i \(0.449627\pi\)
\(710\) 0 0
\(711\) 2.75612 0.103362
\(712\) 0 0
\(713\) −41.0207 −1.53624
\(714\) 0 0
\(715\) 1.19051 0.0445225
\(716\) 0 0
\(717\) −7.87586 −0.294129
\(718\) 0 0
\(719\) 5.16129 0.192484 0.0962418 0.995358i \(-0.469318\pi\)
0.0962418 + 0.995358i \(0.469318\pi\)
\(720\) 0 0
\(721\) 16.9966 0.632985
\(722\) 0 0
\(723\) −3.57152 −0.132826
\(724\) 0 0
\(725\) −24.6448 −0.915284
\(726\) 0 0
\(727\) 40.4362 1.49970 0.749848 0.661610i \(-0.230127\pi\)
0.749848 + 0.661610i \(0.230127\pi\)
\(728\) 0 0
\(729\) −8.23109 −0.304855
\(730\) 0 0
\(731\) 14.9008 0.551125
\(732\) 0 0
\(733\) −39.1311 −1.44534 −0.722670 0.691193i \(-0.757085\pi\)
−0.722670 + 0.691193i \(0.757085\pi\)
\(734\) 0 0
\(735\) −1.19051 −0.0439125
\(736\) 0 0
\(737\) 35.8466 1.32043
\(738\) 0 0
\(739\) 7.13531 0.262477 0.131238 0.991351i \(-0.458105\pi\)
0.131238 + 0.991351i \(0.458105\pi\)
\(740\) 0 0
\(741\) −3.30777 −0.121514
\(742\) 0 0
\(743\) −13.8827 −0.509308 −0.254654 0.967032i \(-0.581962\pi\)
−0.254654 + 0.967032i \(0.581962\pi\)
\(744\) 0 0
\(745\) 4.92676 0.180502
\(746\) 0 0
\(747\) 33.6586 1.23150
\(748\) 0 0
\(749\) 5.55691 0.203045
\(750\) 0 0
\(751\) 37.3251 1.36201 0.681005 0.732278i \(-0.261543\pi\)
0.681005 + 0.732278i \(0.261543\pi\)
\(752\) 0 0
\(753\) −11.0810 −0.403813
\(754\) 0 0
\(755\) −3.74398 −0.136258
\(756\) 0 0
\(757\) −7.10428 −0.258209 −0.129105 0.991631i \(-0.541210\pi\)
−0.129105 + 0.991631i \(0.541210\pi\)
\(758\) 0 0
\(759\) 29.5277 1.07179
\(760\) 0 0
\(761\) −25.9621 −0.941125 −0.470562 0.882367i \(-0.655949\pi\)
−0.470562 + 0.882367i \(0.655949\pi\)
\(762\) 0 0
\(763\) 7.92332 0.286843
\(764\) 0 0
\(765\) −1.42504 −0.0515224
\(766\) 0 0
\(767\) 12.1725 0.439522
\(768\) 0 0
\(769\) −21.4638 −0.774005 −0.387002 0.922079i \(-0.626489\pi\)
−0.387002 + 0.922079i \(0.626489\pi\)
\(770\) 0 0
\(771\) 18.0260 0.649190
\(772\) 0 0
\(773\) −40.4914 −1.45637 −0.728187 0.685378i \(-0.759637\pi\)
−0.728187 + 0.685378i \(0.759637\pi\)
\(774\) 0 0
\(775\) −33.1690 −1.19147
\(776\) 0 0
\(777\) 5.32238 0.190939
\(778\) 0 0
\(779\) 9.55691 0.342412
\(780\) 0 0
\(781\) −2.67762 −0.0958127
\(782\) 0 0
\(783\) 11.0554 0.395088
\(784\) 0 0
\(785\) −3.19738 −0.114119
\(786\) 0 0
\(787\) 5.02072 0.178969 0.0894847 0.995988i \(-0.471478\pi\)
0.0894847 + 0.995988i \(0.471478\pi\)
\(788\) 0 0
\(789\) 3.60438 0.128319
\(790\) 0 0
\(791\) −9.89229 −0.351729
\(792\) 0 0
\(793\) −2.00000 −0.0710221
\(794\) 0 0
\(795\) −13.4250 −0.476137
\(796\) 0 0
\(797\) 19.7002 0.697815 0.348908 0.937157i \(-0.386553\pi\)
0.348908 + 0.937157i \(0.386553\pi\)
\(798\) 0 0
\(799\) 11.2266 0.397169
\(800\) 0 0
\(801\) 14.2258 0.502645
\(802\) 0 0
\(803\) 17.1905 0.606640
\(804\) 0 0
\(805\) −3.08967 −0.108897
\(806\) 0 0
\(807\) 26.5863 0.935883
\(808\) 0 0
\(809\) −2.57678 −0.0905947 −0.0452974 0.998974i \(-0.514424\pi\)
−0.0452974 + 0.998974i \(0.514424\pi\)
\(810\) 0 0
\(811\) −41.2311 −1.44782 −0.723910 0.689895i \(-0.757657\pi\)
−0.723910 + 0.689895i \(0.757657\pi\)
\(812\) 0 0
\(813\) 49.2173 1.72613
\(814\) 0 0
\(815\) −3.37758 −0.118311
\(816\) 0 0
\(817\) −16.7570 −0.586252
\(818\) 0 0
\(819\) 2.05863 0.0719345
\(820\) 0 0
\(821\) 5.26719 0.183826 0.0919130 0.995767i \(-0.470702\pi\)
0.0919130 + 0.995767i \(0.470702\pi\)
\(822\) 0 0
\(823\) −36.0191 −1.25555 −0.627774 0.778396i \(-0.716034\pi\)
−0.627774 + 0.778396i \(0.716034\pi\)
\(824\) 0 0
\(825\) 23.8759 0.831251
\(826\) 0 0
\(827\) 16.3157 0.567353 0.283676 0.958920i \(-0.408446\pi\)
0.283676 + 0.958920i \(0.408446\pi\)
\(828\) 0 0
\(829\) −30.3956 −1.05568 −0.527842 0.849343i \(-0.676999\pi\)
−0.527842 + 0.849343i \(0.676999\pi\)
\(830\) 0 0
\(831\) 22.9820 0.797235
\(832\) 0 0
\(833\) −1.30777 −0.0453117
\(834\) 0 0
\(835\) 8.77846 0.303791
\(836\) 0 0
\(837\) 14.8793 0.514304
\(838\) 0 0
\(839\) −29.8398 −1.03018 −0.515092 0.857135i \(-0.672242\pi\)
−0.515092 + 0.857135i \(0.672242\pi\)
\(840\) 0 0
\(841\) −1.73549 −0.0598445
\(842\) 0 0
\(843\) −3.46886 −0.119474
\(844\) 0 0
\(845\) 0.529317 0.0182090
\(846\) 0 0
\(847\) −5.94137 −0.204148
\(848\) 0 0
\(849\) 35.6413 1.22321
\(850\) 0 0
\(851\) 13.8129 0.473501
\(852\) 0 0
\(853\) −0.203497 −0.00696761 −0.00348380 0.999994i \(-0.501109\pi\)
−0.00348380 + 0.999994i \(0.501109\pi\)
\(854\) 0 0
\(855\) 1.60256 0.0548063
\(856\) 0 0
\(857\) −12.6155 −0.430939 −0.215469 0.976511i \(-0.569128\pi\)
−0.215469 + 0.976511i \(0.569128\pi\)
\(858\) 0 0
\(859\) 27.9671 0.954227 0.477113 0.878842i \(-0.341683\pi\)
0.477113 + 0.878842i \(0.341683\pi\)
\(860\) 0 0
\(861\) −14.6155 −0.498097
\(862\) 0 0
\(863\) 2.76891 0.0942546 0.0471273 0.998889i \(-0.484993\pi\)
0.0471273 + 0.998889i \(0.484993\pi\)
\(864\) 0 0
\(865\) −12.3336 −0.419353
\(866\) 0 0
\(867\) 34.3887 1.16790
\(868\) 0 0
\(869\) 3.01117 0.102147
\(870\) 0 0
\(871\) 15.9379 0.540036
\(872\) 0 0
\(873\) −7.14486 −0.241817
\(874\) 0 0
\(875\) −5.14486 −0.173928
\(876\) 0 0
\(877\) 6.71133 0.226626 0.113313 0.993559i \(-0.463854\pi\)
0.113313 + 0.993559i \(0.463854\pi\)
\(878\) 0 0
\(879\) −24.9268 −0.840759
\(880\) 0 0
\(881\) −18.3741 −0.619040 −0.309520 0.950893i \(-0.600168\pi\)
−0.309520 + 0.950893i \(0.600168\pi\)
\(882\) 0 0
\(883\) −9.93105 −0.334207 −0.167103 0.985939i \(-0.553441\pi\)
−0.167103 + 0.985939i \(0.553441\pi\)
\(884\) 0 0
\(885\) −14.4914 −0.487123
\(886\) 0 0
\(887\) 51.3776 1.72509 0.862545 0.505980i \(-0.168869\pi\)
0.862545 + 0.505980i \(0.168869\pi\)
\(888\) 0 0
\(889\) −0.824101 −0.0276394
\(890\) 0 0
\(891\) −24.6009 −0.824162
\(892\) 0 0
\(893\) −12.6251 −0.422483
\(894\) 0 0
\(895\) −11.1380 −0.372302
\(896\) 0 0
\(897\) 13.1284 0.438346
\(898\) 0 0
\(899\) 36.6949 1.22384
\(900\) 0 0
\(901\) −14.7474 −0.491308
\(902\) 0 0
\(903\) 25.6267 0.852804
\(904\) 0 0
\(905\) 8.86469 0.294672
\(906\) 0 0
\(907\) −4.34225 −0.144182 −0.0720910 0.997398i \(-0.522967\pi\)
−0.0720910 + 0.997398i \(0.522967\pi\)
\(908\) 0 0
\(909\) 15.9562 0.529233
\(910\) 0 0
\(911\) −31.4853 −1.04315 −0.521577 0.853204i \(-0.674656\pi\)
−0.521577 + 0.853204i \(0.674656\pi\)
\(912\) 0 0
\(913\) 36.7734 1.21702
\(914\) 0 0
\(915\) 2.38101 0.0787139
\(916\) 0 0
\(917\) −10.6155 −0.350556
\(918\) 0 0
\(919\) −43.4588 −1.43357 −0.716786 0.697293i \(-0.754388\pi\)
−0.716786 + 0.697293i \(0.754388\pi\)
\(920\) 0 0
\(921\) 45.9096 1.51277
\(922\) 0 0
\(923\) −1.19051 −0.0391860
\(924\) 0 0
\(925\) 11.1690 0.367235
\(926\) 0 0
\(927\) 34.9897 1.14921
\(928\) 0 0
\(929\) 4.40517 0.144529 0.0722645 0.997385i \(-0.476977\pi\)
0.0722645 + 0.997385i \(0.476977\pi\)
\(930\) 0 0
\(931\) 1.47068 0.0481997
\(932\) 0 0
\(933\) −4.32582 −0.141621
\(934\) 0 0
\(935\) −1.55691 −0.0509165
\(936\) 0 0
\(937\) −34.0990 −1.11397 −0.556983 0.830524i \(-0.688041\pi\)
−0.556983 + 0.830524i \(0.688041\pi\)
\(938\) 0 0
\(939\) 32.3121 1.05446
\(940\) 0 0
\(941\) 44.4672 1.44959 0.724795 0.688964i \(-0.241934\pi\)
0.724795 + 0.688964i \(0.241934\pi\)
\(942\) 0 0
\(943\) −37.9311 −1.23521
\(944\) 0 0
\(945\) 1.12070 0.0364565
\(946\) 0 0
\(947\) −57.9311 −1.88251 −0.941253 0.337702i \(-0.890350\pi\)
−0.941253 + 0.337702i \(0.890350\pi\)
\(948\) 0 0
\(949\) 7.64315 0.248107
\(950\) 0 0
\(951\) −34.9897 −1.13462
\(952\) 0 0
\(953\) 35.3060 1.14367 0.571836 0.820368i \(-0.306231\pi\)
0.571836 + 0.820368i \(0.306231\pi\)
\(954\) 0 0
\(955\) 3.93793 0.127428
\(956\) 0 0
\(957\) −26.4139 −0.853839
\(958\) 0 0
\(959\) 11.3630 0.366929
\(960\) 0 0
\(961\) 18.3871 0.593133
\(962\) 0 0
\(963\) 11.4396 0.368638
\(964\) 0 0
\(965\) −0.794885 −0.0255882
\(966\) 0 0
\(967\) −23.7148 −0.762616 −0.381308 0.924448i \(-0.624526\pi\)
−0.381308 + 0.924448i \(0.624526\pi\)
\(968\) 0 0
\(969\) 4.32582 0.138965
\(970\) 0 0
\(971\) 23.9379 0.768205 0.384102 0.923291i \(-0.374511\pi\)
0.384102 + 0.923291i \(0.374511\pi\)
\(972\) 0 0
\(973\) 13.9233 0.446361
\(974\) 0 0
\(975\) 10.6155 0.339970
\(976\) 0 0
\(977\) −16.1871 −0.517870 −0.258935 0.965895i \(-0.583372\pi\)
−0.258935 + 0.965895i \(0.583372\pi\)
\(978\) 0 0
\(979\) 15.5423 0.496734
\(980\) 0 0
\(981\) 16.3112 0.520777
\(982\) 0 0
\(983\) −45.7243 −1.45838 −0.729190 0.684312i \(-0.760103\pi\)
−0.729190 + 0.684312i \(0.760103\pi\)
\(984\) 0 0
\(985\) 12.7000 0.404654
\(986\) 0 0
\(987\) 19.3078 0.614573
\(988\) 0 0
\(989\) 66.5078 2.11483
\(990\) 0 0
\(991\) 5.40356 0.171650 0.0858248 0.996310i \(-0.472647\pi\)
0.0858248 + 0.996310i \(0.472647\pi\)
\(992\) 0 0
\(993\) 70.9605 2.25186
\(994\) 0 0
\(995\) 1.06637 0.0338061
\(996\) 0 0
\(997\) 6.04832 0.191552 0.0957761 0.995403i \(-0.469467\pi\)
0.0957761 + 0.995403i \(0.469467\pi\)
\(998\) 0 0
\(999\) −5.01031 −0.158519
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 1456.2.a.t.1.1 3
4.3 odd 2 91.2.a.d.1.2 3
8.3 odd 2 5824.2.a.by.1.1 3
8.5 even 2 5824.2.a.bs.1.3 3
12.11 even 2 819.2.a.i.1.2 3
20.19 odd 2 2275.2.a.m.1.2 3
28.3 even 6 637.2.e.i.79.2 6
28.11 odd 6 637.2.e.j.79.2 6
28.19 even 6 637.2.e.i.508.2 6
28.23 odd 6 637.2.e.j.508.2 6
28.27 even 2 637.2.a.j.1.2 3
52.31 even 4 1183.2.c.f.337.3 6
52.47 even 4 1183.2.c.f.337.4 6
52.51 odd 2 1183.2.a.i.1.2 3
84.83 odd 2 5733.2.a.x.1.2 3
364.363 even 2 8281.2.a.bg.1.2 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
91.2.a.d.1.2 3 4.3 odd 2
637.2.a.j.1.2 3 28.27 even 2
637.2.e.i.79.2 6 28.3 even 6
637.2.e.i.508.2 6 28.19 even 6
637.2.e.j.79.2 6 28.11 odd 6
637.2.e.j.508.2 6 28.23 odd 6
819.2.a.i.1.2 3 12.11 even 2
1183.2.a.i.1.2 3 52.51 odd 2
1183.2.c.f.337.3 6 52.31 even 4
1183.2.c.f.337.4 6 52.47 even 4
1456.2.a.t.1.1 3 1.1 even 1 trivial
2275.2.a.m.1.2 3 20.19 odd 2
5733.2.a.x.1.2 3 84.83 odd 2
5824.2.a.bs.1.3 3 8.5 even 2
5824.2.a.by.1.1 3 8.3 odd 2
8281.2.a.bg.1.2 3 364.363 even 2