Properties

Label 135.2.a.c.1.1
Level $135$
Weight $2$
Character 135.1
Self dual yes
Analytic conductor $1.078$
Analytic rank $0$
Dimension $2$
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] = [135,2,Mod(1,135)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(135, 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("135.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 135 = 3^{3} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 135.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: \(1.07798042729\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{13}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(2.30278\) of defining polynomial
Character \(\chi\) \(=\) 135.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.30278 q^{2} +3.30278 q^{4} +1.00000 q^{5} -2.60555 q^{7} -3.00000 q^{8} +O(q^{10})\) \(q-2.30278 q^{2} +3.30278 q^{4} +1.00000 q^{5} -2.60555 q^{7} -3.00000 q^{8} -2.30278 q^{10} +4.60555 q^{11} +6.60555 q^{13} +6.00000 q^{14} +0.302776 q^{16} +1.60555 q^{17} +3.60555 q^{19} +3.30278 q^{20} -10.6056 q^{22} -3.00000 q^{23} +1.00000 q^{25} -15.2111 q^{26} -8.60555 q^{28} +1.39445 q^{29} -5.60555 q^{31} +5.30278 q^{32} -3.69722 q^{34} -2.60555 q^{35} +2.00000 q^{37} -8.30278 q^{38} -3.00000 q^{40} -4.60555 q^{41} +0.605551 q^{43} +15.2111 q^{44} +6.90833 q^{46} -9.21110 q^{47} -0.211103 q^{49} -2.30278 q^{50} +21.8167 q^{52} +1.60555 q^{53} +4.60555 q^{55} +7.81665 q^{56} -3.21110 q^{58} -1.39445 q^{59} -4.21110 q^{61} +12.9083 q^{62} -12.8167 q^{64} +6.60555 q^{65} -0.788897 q^{67} +5.30278 q^{68} +6.00000 q^{70} +7.39445 q^{71} +12.6056 q^{73} -4.60555 q^{74} +11.9083 q^{76} -12.0000 q^{77} -11.6056 q^{79} +0.302776 q^{80} +10.6056 q^{82} +3.00000 q^{83} +1.60555 q^{85} -1.39445 q^{86} -13.8167 q^{88} -13.8167 q^{89} -17.2111 q^{91} -9.90833 q^{92} +21.2111 q^{94} +3.60555 q^{95} +8.00000 q^{97} +0.486122 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{2} + 3 q^{4} + 2 q^{5} + 2 q^{7} - 6 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - q^{2} + 3 q^{4} + 2 q^{5} + 2 q^{7} - 6 q^{8} - q^{10} + 2 q^{11} + 6 q^{13} + 12 q^{14} - 3 q^{16} - 4 q^{17} + 3 q^{20} - 14 q^{22} - 6 q^{23} + 2 q^{25} - 16 q^{26} - 10 q^{28} + 10 q^{29} - 4 q^{31} + 7 q^{32} - 11 q^{34} + 2 q^{35} + 4 q^{37} - 13 q^{38} - 6 q^{40} - 2 q^{41} - 6 q^{43} + 16 q^{44} + 3 q^{46} - 4 q^{47} + 14 q^{49} - q^{50} + 22 q^{52} - 4 q^{53} + 2 q^{55} - 6 q^{56} + 8 q^{58} - 10 q^{59} + 6 q^{61} + 15 q^{62} - 4 q^{64} + 6 q^{65} - 16 q^{67} + 7 q^{68} + 12 q^{70} + 22 q^{71} + 18 q^{73} - 2 q^{74} + 13 q^{76} - 24 q^{77} - 16 q^{79} - 3 q^{80} + 14 q^{82} + 6 q^{83} - 4 q^{85} - 10 q^{86} - 6 q^{88} - 6 q^{89} - 20 q^{91} - 9 q^{92} + 28 q^{94} + 16 q^{97} + 19 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.30278 −1.62831 −0.814154 0.580649i \(-0.802799\pi\)
−0.814154 + 0.580649i \(0.802799\pi\)
\(3\) 0 0
\(4\) 3.30278 1.65139
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) −2.60555 −0.984806 −0.492403 0.870367i \(-0.663881\pi\)
−0.492403 + 0.870367i \(0.663881\pi\)
\(8\) −3.00000 −1.06066
\(9\) 0 0
\(10\) −2.30278 −0.728202
\(11\) 4.60555 1.38863 0.694313 0.719673i \(-0.255708\pi\)
0.694313 + 0.719673i \(0.255708\pi\)
\(12\) 0 0
\(13\) 6.60555 1.83205 0.916025 0.401121i \(-0.131379\pi\)
0.916025 + 0.401121i \(0.131379\pi\)
\(14\) 6.00000 1.60357
\(15\) 0 0
\(16\) 0.302776 0.0756939
\(17\) 1.60555 0.389403 0.194702 0.980863i \(-0.437626\pi\)
0.194702 + 0.980863i \(0.437626\pi\)
\(18\) 0 0
\(19\) 3.60555 0.827170 0.413585 0.910465i \(-0.364276\pi\)
0.413585 + 0.910465i \(0.364276\pi\)
\(20\) 3.30278 0.738523
\(21\) 0 0
\(22\) −10.6056 −2.26111
\(23\) −3.00000 −0.625543 −0.312772 0.949828i \(-0.601257\pi\)
−0.312772 + 0.949828i \(0.601257\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) −15.2111 −2.98314
\(27\) 0 0
\(28\) −8.60555 −1.62630
\(29\) 1.39445 0.258943 0.129471 0.991583i \(-0.458672\pi\)
0.129471 + 0.991583i \(0.458672\pi\)
\(30\) 0 0
\(31\) −5.60555 −1.00679 −0.503393 0.864057i \(-0.667915\pi\)
−0.503393 + 0.864057i \(0.667915\pi\)
\(32\) 5.30278 0.937407
\(33\) 0 0
\(34\) −3.69722 −0.634069
\(35\) −2.60555 −0.440419
\(36\) 0 0
\(37\) 2.00000 0.328798 0.164399 0.986394i \(-0.447432\pi\)
0.164399 + 0.986394i \(0.447432\pi\)
\(38\) −8.30278 −1.34689
\(39\) 0 0
\(40\) −3.00000 −0.474342
\(41\) −4.60555 −0.719266 −0.359633 0.933094i \(-0.617098\pi\)
−0.359633 + 0.933094i \(0.617098\pi\)
\(42\) 0 0
\(43\) 0.605551 0.0923457 0.0461729 0.998933i \(-0.485297\pi\)
0.0461729 + 0.998933i \(0.485297\pi\)
\(44\) 15.2111 2.29316
\(45\) 0 0
\(46\) 6.90833 1.01858
\(47\) −9.21110 −1.34358 −0.671789 0.740743i \(-0.734474\pi\)
−0.671789 + 0.740743i \(0.734474\pi\)
\(48\) 0 0
\(49\) −0.211103 −0.0301575
\(50\) −2.30278 −0.325662
\(51\) 0 0
\(52\) 21.8167 3.02543
\(53\) 1.60555 0.220539 0.110270 0.993902i \(-0.464829\pi\)
0.110270 + 0.993902i \(0.464829\pi\)
\(54\) 0 0
\(55\) 4.60555 0.621012
\(56\) 7.81665 1.04454
\(57\) 0 0
\(58\) −3.21110 −0.421638
\(59\) −1.39445 −0.181542 −0.0907709 0.995872i \(-0.528933\pi\)
−0.0907709 + 0.995872i \(0.528933\pi\)
\(60\) 0 0
\(61\) −4.21110 −0.539176 −0.269588 0.962976i \(-0.586888\pi\)
−0.269588 + 0.962976i \(0.586888\pi\)
\(62\) 12.9083 1.63936
\(63\) 0 0
\(64\) −12.8167 −1.60208
\(65\) 6.60555 0.819318
\(66\) 0 0
\(67\) −0.788897 −0.0963792 −0.0481896 0.998838i \(-0.515345\pi\)
−0.0481896 + 0.998838i \(0.515345\pi\)
\(68\) 5.30278 0.643056
\(69\) 0 0
\(70\) 6.00000 0.717137
\(71\) 7.39445 0.877560 0.438780 0.898595i \(-0.355411\pi\)
0.438780 + 0.898595i \(0.355411\pi\)
\(72\) 0 0
\(73\) 12.6056 1.47537 0.737684 0.675146i \(-0.235919\pi\)
0.737684 + 0.675146i \(0.235919\pi\)
\(74\) −4.60555 −0.535384
\(75\) 0 0
\(76\) 11.9083 1.36598
\(77\) −12.0000 −1.36753
\(78\) 0 0
\(79\) −11.6056 −1.30573 −0.652863 0.757476i \(-0.726432\pi\)
−0.652863 + 0.757476i \(0.726432\pi\)
\(80\) 0.302776 0.0338513
\(81\) 0 0
\(82\) 10.6056 1.17119
\(83\) 3.00000 0.329293 0.164646 0.986353i \(-0.447352\pi\)
0.164646 + 0.986353i \(0.447352\pi\)
\(84\) 0 0
\(85\) 1.60555 0.174146
\(86\) −1.39445 −0.150367
\(87\) 0 0
\(88\) −13.8167 −1.47286
\(89\) −13.8167 −1.46456 −0.732281 0.681002i \(-0.761544\pi\)
−0.732281 + 0.681002i \(0.761544\pi\)
\(90\) 0 0
\(91\) −17.2111 −1.80421
\(92\) −9.90833 −1.03301
\(93\) 0 0
\(94\) 21.2111 2.18776
\(95\) 3.60555 0.369922
\(96\) 0 0
\(97\) 8.00000 0.812277 0.406138 0.913812i \(-0.366875\pi\)
0.406138 + 0.913812i \(0.366875\pi\)
\(98\) 0.486122 0.0491057
\(99\) 0 0
\(100\) 3.30278 0.330278
\(101\) 12.0000 1.19404 0.597022 0.802225i \(-0.296350\pi\)
0.597022 + 0.802225i \(0.296350\pi\)
\(102\) 0 0
\(103\) −4.00000 −0.394132 −0.197066 0.980390i \(-0.563141\pi\)
−0.197066 + 0.980390i \(0.563141\pi\)
\(104\) −19.8167 −1.94318
\(105\) 0 0
\(106\) −3.69722 −0.359106
\(107\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(108\) 0 0
\(109\) −7.00000 −0.670478 −0.335239 0.942133i \(-0.608817\pi\)
−0.335239 + 0.942133i \(0.608817\pi\)
\(110\) −10.6056 −1.01120
\(111\) 0 0
\(112\) −0.788897 −0.0745438
\(113\) −15.2111 −1.43094 −0.715470 0.698643i \(-0.753787\pi\)
−0.715470 + 0.698643i \(0.753787\pi\)
\(114\) 0 0
\(115\) −3.00000 −0.279751
\(116\) 4.60555 0.427615
\(117\) 0 0
\(118\) 3.21110 0.295606
\(119\) −4.18335 −0.383487
\(120\) 0 0
\(121\) 10.2111 0.928282
\(122\) 9.69722 0.877945
\(123\) 0 0
\(124\) −18.5139 −1.66260
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) −19.2111 −1.70471 −0.852355 0.522964i \(-0.824826\pi\)
−0.852355 + 0.522964i \(0.824826\pi\)
\(128\) 18.9083 1.67128
\(129\) 0 0
\(130\) −15.2111 −1.33410
\(131\) 6.00000 0.524222 0.262111 0.965038i \(-0.415581\pi\)
0.262111 + 0.965038i \(0.415581\pi\)
\(132\) 0 0
\(133\) −9.39445 −0.814602
\(134\) 1.81665 0.156935
\(135\) 0 0
\(136\) −4.81665 −0.413025
\(137\) 16.8167 1.43674 0.718372 0.695659i \(-0.244888\pi\)
0.718372 + 0.695659i \(0.244888\pi\)
\(138\) 0 0
\(139\) −4.00000 −0.339276 −0.169638 0.985506i \(-0.554260\pi\)
−0.169638 + 0.985506i \(0.554260\pi\)
\(140\) −8.60555 −0.727302
\(141\) 0 0
\(142\) −17.0278 −1.42894
\(143\) 30.4222 2.54403
\(144\) 0 0
\(145\) 1.39445 0.115803
\(146\) −29.0278 −2.40235
\(147\) 0 0
\(148\) 6.60555 0.542973
\(149\) 23.0278 1.88651 0.943254 0.332073i \(-0.107748\pi\)
0.943254 + 0.332073i \(0.107748\pi\)
\(150\) 0 0
\(151\) 14.4222 1.17366 0.586831 0.809709i \(-0.300375\pi\)
0.586831 + 0.809709i \(0.300375\pi\)
\(152\) −10.8167 −0.877346
\(153\) 0 0
\(154\) 27.6333 2.22676
\(155\) −5.60555 −0.450249
\(156\) 0 0
\(157\) −17.8167 −1.42192 −0.710962 0.703231i \(-0.751740\pi\)
−0.710962 + 0.703231i \(0.751740\pi\)
\(158\) 26.7250 2.12613
\(159\) 0 0
\(160\) 5.30278 0.419221
\(161\) 7.81665 0.616039
\(162\) 0 0
\(163\) 2.00000 0.156652 0.0783260 0.996928i \(-0.475042\pi\)
0.0783260 + 0.996928i \(0.475042\pi\)
\(164\) −15.2111 −1.18779
\(165\) 0 0
\(166\) −6.90833 −0.536190
\(167\) −3.00000 −0.232147 −0.116073 0.993241i \(-0.537031\pi\)
−0.116073 + 0.993241i \(0.537031\pi\)
\(168\) 0 0
\(169\) 30.6333 2.35641
\(170\) −3.69722 −0.283564
\(171\) 0 0
\(172\) 2.00000 0.152499
\(173\) −10.8167 −0.822375 −0.411187 0.911551i \(-0.634886\pi\)
−0.411187 + 0.911551i \(0.634886\pi\)
\(174\) 0 0
\(175\) −2.60555 −0.196961
\(176\) 1.39445 0.105111
\(177\) 0 0
\(178\) 31.8167 2.38476
\(179\) −21.2111 −1.58539 −0.792696 0.609617i \(-0.791323\pi\)
−0.792696 + 0.609617i \(0.791323\pi\)
\(180\) 0 0
\(181\) −7.00000 −0.520306 −0.260153 0.965567i \(-0.583773\pi\)
−0.260153 + 0.965567i \(0.583773\pi\)
\(182\) 39.6333 2.93782
\(183\) 0 0
\(184\) 9.00000 0.663489
\(185\) 2.00000 0.147043
\(186\) 0 0
\(187\) 7.39445 0.540736
\(188\) −30.4222 −2.21877
\(189\) 0 0
\(190\) −8.30278 −0.602347
\(191\) −12.4222 −0.898839 −0.449420 0.893321i \(-0.648369\pi\)
−0.449420 + 0.893321i \(0.648369\pi\)
\(192\) 0 0
\(193\) 0.183346 0.0131975 0.00659877 0.999978i \(-0.497900\pi\)
0.00659877 + 0.999978i \(0.497900\pi\)
\(194\) −18.4222 −1.32264
\(195\) 0 0
\(196\) −0.697224 −0.0498017
\(197\) −22.8167 −1.62562 −0.812810 0.582529i \(-0.802063\pi\)
−0.812810 + 0.582529i \(0.802063\pi\)
\(198\) 0 0
\(199\) −1.21110 −0.0858528 −0.0429264 0.999078i \(-0.513668\pi\)
−0.0429264 + 0.999078i \(0.513668\pi\)
\(200\) −3.00000 −0.212132
\(201\) 0 0
\(202\) −27.6333 −1.94427
\(203\) −3.63331 −0.255008
\(204\) 0 0
\(205\) −4.60555 −0.321666
\(206\) 9.21110 0.641768
\(207\) 0 0
\(208\) 2.00000 0.138675
\(209\) 16.6056 1.14863
\(210\) 0 0
\(211\) −8.81665 −0.606963 −0.303482 0.952837i \(-0.598149\pi\)
−0.303482 + 0.952837i \(0.598149\pi\)
\(212\) 5.30278 0.364196
\(213\) 0 0
\(214\) 0 0
\(215\) 0.605551 0.0412983
\(216\) 0 0
\(217\) 14.6056 0.991489
\(218\) 16.1194 1.09175
\(219\) 0 0
\(220\) 15.2111 1.02553
\(221\) 10.6056 0.713407
\(222\) 0 0
\(223\) −10.0000 −0.669650 −0.334825 0.942280i \(-0.608677\pi\)
−0.334825 + 0.942280i \(0.608677\pi\)
\(224\) −13.8167 −0.923164
\(225\) 0 0
\(226\) 35.0278 2.33001
\(227\) −11.7889 −0.782457 −0.391228 0.920294i \(-0.627950\pi\)
−0.391228 + 0.920294i \(0.627950\pi\)
\(228\) 0 0
\(229\) 8.21110 0.542605 0.271302 0.962494i \(-0.412546\pi\)
0.271302 + 0.962494i \(0.412546\pi\)
\(230\) 6.90833 0.455522
\(231\) 0 0
\(232\) −4.18335 −0.274650
\(233\) −18.0000 −1.17922 −0.589610 0.807688i \(-0.700718\pi\)
−0.589610 + 0.807688i \(0.700718\pi\)
\(234\) 0 0
\(235\) −9.21110 −0.600866
\(236\) −4.60555 −0.299796
\(237\) 0 0
\(238\) 9.63331 0.624435
\(239\) −15.2111 −0.983924 −0.491962 0.870617i \(-0.663720\pi\)
−0.491962 + 0.870617i \(0.663720\pi\)
\(240\) 0 0
\(241\) 13.7889 0.888221 0.444110 0.895972i \(-0.353520\pi\)
0.444110 + 0.895972i \(0.353520\pi\)
\(242\) −23.5139 −1.51153
\(243\) 0 0
\(244\) −13.9083 −0.890389
\(245\) −0.211103 −0.0134868
\(246\) 0 0
\(247\) 23.8167 1.51542
\(248\) 16.8167 1.06786
\(249\) 0 0
\(250\) −2.30278 −0.145640
\(251\) 27.6333 1.74420 0.872099 0.489329i \(-0.162758\pi\)
0.872099 + 0.489329i \(0.162758\pi\)
\(252\) 0 0
\(253\) −13.8167 −0.868646
\(254\) 44.2389 2.77579
\(255\) 0 0
\(256\) −17.9083 −1.11927
\(257\) 1.18335 0.0738151 0.0369076 0.999319i \(-0.488249\pi\)
0.0369076 + 0.999319i \(0.488249\pi\)
\(258\) 0 0
\(259\) −5.21110 −0.323802
\(260\) 21.8167 1.35301
\(261\) 0 0
\(262\) −13.8167 −0.853596
\(263\) −2.78890 −0.171971 −0.0859854 0.996296i \(-0.527404\pi\)
−0.0859854 + 0.996296i \(0.527404\pi\)
\(264\) 0 0
\(265\) 1.60555 0.0986282
\(266\) 21.6333 1.32642
\(267\) 0 0
\(268\) −2.60555 −0.159159
\(269\) −3.21110 −0.195784 −0.0978922 0.995197i \(-0.531210\pi\)
−0.0978922 + 0.995197i \(0.531210\pi\)
\(270\) 0 0
\(271\) 31.2389 1.89763 0.948813 0.315839i \(-0.102286\pi\)
0.948813 + 0.315839i \(0.102286\pi\)
\(272\) 0.486122 0.0294755
\(273\) 0 0
\(274\) −38.7250 −2.33946
\(275\) 4.60555 0.277725
\(276\) 0 0
\(277\) 7.02776 0.422257 0.211128 0.977458i \(-0.432286\pi\)
0.211128 + 0.977458i \(0.432286\pi\)
\(278\) 9.21110 0.552445
\(279\) 0 0
\(280\) 7.81665 0.467134
\(281\) −19.8167 −1.18216 −0.591081 0.806612i \(-0.701299\pi\)
−0.591081 + 0.806612i \(0.701299\pi\)
\(282\) 0 0
\(283\) 3.39445 0.201779 0.100890 0.994898i \(-0.467831\pi\)
0.100890 + 0.994898i \(0.467831\pi\)
\(284\) 24.4222 1.44919
\(285\) 0 0
\(286\) −70.0555 −4.14247
\(287\) 12.0000 0.708338
\(288\) 0 0
\(289\) −14.4222 −0.848365
\(290\) −3.21110 −0.188562
\(291\) 0 0
\(292\) 41.6333 2.43641
\(293\) −7.18335 −0.419656 −0.209828 0.977738i \(-0.567290\pi\)
−0.209828 + 0.977738i \(0.567290\pi\)
\(294\) 0 0
\(295\) −1.39445 −0.0811879
\(296\) −6.00000 −0.348743
\(297\) 0 0
\(298\) −53.0278 −3.07182
\(299\) −19.8167 −1.14603
\(300\) 0 0
\(301\) −1.57779 −0.0909426
\(302\) −33.2111 −1.91108
\(303\) 0 0
\(304\) 1.09167 0.0626117
\(305\) −4.21110 −0.241127
\(306\) 0 0
\(307\) 8.42221 0.480681 0.240340 0.970689i \(-0.422741\pi\)
0.240340 + 0.970689i \(0.422741\pi\)
\(308\) −39.6333 −2.25832
\(309\) 0 0
\(310\) 12.9083 0.733144
\(311\) −7.81665 −0.443242 −0.221621 0.975133i \(-0.571135\pi\)
−0.221621 + 0.975133i \(0.571135\pi\)
\(312\) 0 0
\(313\) −19.6333 −1.10974 −0.554870 0.831937i \(-0.687232\pi\)
−0.554870 + 0.831937i \(0.687232\pi\)
\(314\) 41.0278 2.31533
\(315\) 0 0
\(316\) −38.3305 −2.15626
\(317\) −7.60555 −0.427170 −0.213585 0.976924i \(-0.568514\pi\)
−0.213585 + 0.976924i \(0.568514\pi\)
\(318\) 0 0
\(319\) 6.42221 0.359574
\(320\) −12.8167 −0.716473
\(321\) 0 0
\(322\) −18.0000 −1.00310
\(323\) 5.78890 0.322103
\(324\) 0 0
\(325\) 6.60555 0.366410
\(326\) −4.60555 −0.255078
\(327\) 0 0
\(328\) 13.8167 0.762897
\(329\) 24.0000 1.32316
\(330\) 0 0
\(331\) 29.2111 1.60559 0.802794 0.596257i \(-0.203346\pi\)
0.802794 + 0.596257i \(0.203346\pi\)
\(332\) 9.90833 0.543790
\(333\) 0 0
\(334\) 6.90833 0.378007
\(335\) −0.788897 −0.0431021
\(336\) 0 0
\(337\) 6.60555 0.359827 0.179914 0.983682i \(-0.442418\pi\)
0.179914 + 0.983682i \(0.442418\pi\)
\(338\) −70.5416 −3.83696
\(339\) 0 0
\(340\) 5.30278 0.287583
\(341\) −25.8167 −1.39805
\(342\) 0 0
\(343\) 18.7889 1.01451
\(344\) −1.81665 −0.0979474
\(345\) 0 0
\(346\) 24.9083 1.33908
\(347\) 30.4222 1.63315 0.816575 0.577240i \(-0.195870\pi\)
0.816575 + 0.577240i \(0.195870\pi\)
\(348\) 0 0
\(349\) −31.8444 −1.70459 −0.852296 0.523060i \(-0.824791\pi\)
−0.852296 + 0.523060i \(0.824791\pi\)
\(350\) 6.00000 0.320713
\(351\) 0 0
\(352\) 24.4222 1.30171
\(353\) 21.6333 1.15142 0.575712 0.817652i \(-0.304725\pi\)
0.575712 + 0.817652i \(0.304725\pi\)
\(354\) 0 0
\(355\) 7.39445 0.392457
\(356\) −45.6333 −2.41856
\(357\) 0 0
\(358\) 48.8444 2.58151
\(359\) −9.63331 −0.508427 −0.254213 0.967148i \(-0.581817\pi\)
−0.254213 + 0.967148i \(0.581817\pi\)
\(360\) 0 0
\(361\) −6.00000 −0.315789
\(362\) 16.1194 0.847218
\(363\) 0 0
\(364\) −56.8444 −2.97946
\(365\) 12.6056 0.659805
\(366\) 0 0
\(367\) −2.60555 −0.136009 −0.0680043 0.997685i \(-0.521663\pi\)
−0.0680043 + 0.997685i \(0.521663\pi\)
\(368\) −0.908327 −0.0473498
\(369\) 0 0
\(370\) −4.60555 −0.239431
\(371\) −4.18335 −0.217189
\(372\) 0 0
\(373\) −25.2111 −1.30538 −0.652691 0.757624i \(-0.726360\pi\)
−0.652691 + 0.757624i \(0.726360\pi\)
\(374\) −17.0278 −0.880484
\(375\) 0 0
\(376\) 27.6333 1.42508
\(377\) 9.21110 0.474396
\(378\) 0 0
\(379\) 21.6056 1.10980 0.554901 0.831916i \(-0.312756\pi\)
0.554901 + 0.831916i \(0.312756\pi\)
\(380\) 11.9083 0.610884
\(381\) 0 0
\(382\) 28.6056 1.46359
\(383\) 24.6333 1.25870 0.629352 0.777121i \(-0.283321\pi\)
0.629352 + 0.777121i \(0.283321\pi\)
\(384\) 0 0
\(385\) −12.0000 −0.611577
\(386\) −0.422205 −0.0214897
\(387\) 0 0
\(388\) 26.4222 1.34138
\(389\) 25.8167 1.30896 0.654478 0.756081i \(-0.272888\pi\)
0.654478 + 0.756081i \(0.272888\pi\)
\(390\) 0 0
\(391\) −4.81665 −0.243589
\(392\) 0.633308 0.0319869
\(393\) 0 0
\(394\) 52.5416 2.64701
\(395\) −11.6056 −0.583939
\(396\) 0 0
\(397\) −5.39445 −0.270740 −0.135370 0.990795i \(-0.543222\pi\)
−0.135370 + 0.990795i \(0.543222\pi\)
\(398\) 2.78890 0.139795
\(399\) 0 0
\(400\) 0.302776 0.0151388
\(401\) −30.0000 −1.49813 −0.749064 0.662497i \(-0.769497\pi\)
−0.749064 + 0.662497i \(0.769497\pi\)
\(402\) 0 0
\(403\) −37.0278 −1.84448
\(404\) 39.6333 1.97183
\(405\) 0 0
\(406\) 8.36669 0.415232
\(407\) 9.21110 0.456577
\(408\) 0 0
\(409\) 5.00000 0.247234 0.123617 0.992330i \(-0.460551\pi\)
0.123617 + 0.992330i \(0.460551\pi\)
\(410\) 10.6056 0.523771
\(411\) 0 0
\(412\) −13.2111 −0.650864
\(413\) 3.63331 0.178783
\(414\) 0 0
\(415\) 3.00000 0.147264
\(416\) 35.0278 1.71738
\(417\) 0 0
\(418\) −38.2389 −1.87032
\(419\) 23.0278 1.12498 0.562490 0.826804i \(-0.309844\pi\)
0.562490 + 0.826804i \(0.309844\pi\)
\(420\) 0 0
\(421\) 5.42221 0.264262 0.132131 0.991232i \(-0.457818\pi\)
0.132131 + 0.991232i \(0.457818\pi\)
\(422\) 20.3028 0.988324
\(423\) 0 0
\(424\) −4.81665 −0.233917
\(425\) 1.60555 0.0778807
\(426\) 0 0
\(427\) 10.9722 0.530984
\(428\) 0 0
\(429\) 0 0
\(430\) −1.39445 −0.0672463
\(431\) 4.18335 0.201505 0.100752 0.994912i \(-0.467875\pi\)
0.100752 + 0.994912i \(0.467875\pi\)
\(432\) 0 0
\(433\) 22.2389 1.06873 0.534366 0.845253i \(-0.320551\pi\)
0.534366 + 0.845253i \(0.320551\pi\)
\(434\) −33.6333 −1.61445
\(435\) 0 0
\(436\) −23.1194 −1.10722
\(437\) −10.8167 −0.517431
\(438\) 0 0
\(439\) 27.6056 1.31754 0.658771 0.752344i \(-0.271077\pi\)
0.658771 + 0.752344i \(0.271077\pi\)
\(440\) −13.8167 −0.658683
\(441\) 0 0
\(442\) −24.4222 −1.16165
\(443\) −24.6333 −1.17036 −0.585182 0.810902i \(-0.698977\pi\)
−0.585182 + 0.810902i \(0.698977\pi\)
\(444\) 0 0
\(445\) −13.8167 −0.654972
\(446\) 23.0278 1.09040
\(447\) 0 0
\(448\) 33.3944 1.57774
\(449\) 38.2389 1.80460 0.902302 0.431105i \(-0.141876\pi\)
0.902302 + 0.431105i \(0.141876\pi\)
\(450\) 0 0
\(451\) −21.2111 −0.998792
\(452\) −50.2389 −2.36304
\(453\) 0 0
\(454\) 27.1472 1.27408
\(455\) −17.2111 −0.806869
\(456\) 0 0
\(457\) −13.2111 −0.617989 −0.308995 0.951064i \(-0.599993\pi\)
−0.308995 + 0.951064i \(0.599993\pi\)
\(458\) −18.9083 −0.883528
\(459\) 0 0
\(460\) −9.90833 −0.461978
\(461\) 21.6333 1.00756 0.503782 0.863831i \(-0.331942\pi\)
0.503782 + 0.863831i \(0.331942\pi\)
\(462\) 0 0
\(463\) −0.788897 −0.0366632 −0.0183316 0.999832i \(-0.505835\pi\)
−0.0183316 + 0.999832i \(0.505835\pi\)
\(464\) 0.422205 0.0196004
\(465\) 0 0
\(466\) 41.4500 1.92013
\(467\) 12.2111 0.565062 0.282531 0.959258i \(-0.408826\pi\)
0.282531 + 0.959258i \(0.408826\pi\)
\(468\) 0 0
\(469\) 2.05551 0.0949148
\(470\) 21.2111 0.978395
\(471\) 0 0
\(472\) 4.18335 0.192554
\(473\) 2.78890 0.128234
\(474\) 0 0
\(475\) 3.60555 0.165434
\(476\) −13.8167 −0.633285
\(477\) 0 0
\(478\) 35.0278 1.60213
\(479\) −37.8167 −1.72789 −0.863944 0.503589i \(-0.832013\pi\)
−0.863944 + 0.503589i \(0.832013\pi\)
\(480\) 0 0
\(481\) 13.2111 0.602374
\(482\) −31.7527 −1.44630
\(483\) 0 0
\(484\) 33.7250 1.53295
\(485\) 8.00000 0.363261
\(486\) 0 0
\(487\) −29.8167 −1.35112 −0.675561 0.737304i \(-0.736098\pi\)
−0.675561 + 0.737304i \(0.736098\pi\)
\(488\) 12.6333 0.571883
\(489\) 0 0
\(490\) 0.486122 0.0219607
\(491\) 27.2111 1.22802 0.614010 0.789298i \(-0.289555\pi\)
0.614010 + 0.789298i \(0.289555\pi\)
\(492\) 0 0
\(493\) 2.23886 0.100833
\(494\) −54.8444 −2.46757
\(495\) 0 0
\(496\) −1.69722 −0.0762076
\(497\) −19.2666 −0.864226
\(498\) 0 0
\(499\) −20.3944 −0.912981 −0.456490 0.889728i \(-0.650894\pi\)
−0.456490 + 0.889728i \(0.650894\pi\)
\(500\) 3.30278 0.147705
\(501\) 0 0
\(502\) −63.6333 −2.84009
\(503\) 2.57779 0.114938 0.0574691 0.998347i \(-0.481697\pi\)
0.0574691 + 0.998347i \(0.481697\pi\)
\(504\) 0 0
\(505\) 12.0000 0.533993
\(506\) 31.8167 1.41442
\(507\) 0 0
\(508\) −63.4500 −2.81514
\(509\) −24.8444 −1.10121 −0.550605 0.834766i \(-0.685603\pi\)
−0.550605 + 0.834766i \(0.685603\pi\)
\(510\) 0 0
\(511\) −32.8444 −1.45295
\(512\) 3.42221 0.151242
\(513\) 0 0
\(514\) −2.72498 −0.120194
\(515\) −4.00000 −0.176261
\(516\) 0 0
\(517\) −42.4222 −1.86573
\(518\) 12.0000 0.527250
\(519\) 0 0
\(520\) −19.8167 −0.869018
\(521\) −28.6056 −1.25323 −0.626616 0.779328i \(-0.715561\pi\)
−0.626616 + 0.779328i \(0.715561\pi\)
\(522\) 0 0
\(523\) 30.6056 1.33829 0.669144 0.743133i \(-0.266661\pi\)
0.669144 + 0.743133i \(0.266661\pi\)
\(524\) 19.8167 0.865695
\(525\) 0 0
\(526\) 6.42221 0.280021
\(527\) −9.00000 −0.392046
\(528\) 0 0
\(529\) −14.0000 −0.608696
\(530\) −3.69722 −0.160597
\(531\) 0 0
\(532\) −31.0278 −1.34522
\(533\) −30.4222 −1.31773
\(534\) 0 0
\(535\) 0 0
\(536\) 2.36669 0.102226
\(537\) 0 0
\(538\) 7.39445 0.318797
\(539\) −0.972244 −0.0418775
\(540\) 0 0
\(541\) −40.4222 −1.73789 −0.868943 0.494912i \(-0.835200\pi\)
−0.868943 + 0.494912i \(0.835200\pi\)
\(542\) −71.9361 −3.08992
\(543\) 0 0
\(544\) 8.51388 0.365030
\(545\) −7.00000 −0.299847
\(546\) 0 0
\(547\) 0.605551 0.0258915 0.0129458 0.999916i \(-0.495879\pi\)
0.0129458 + 0.999916i \(0.495879\pi\)
\(548\) 55.5416 2.37262
\(549\) 0 0
\(550\) −10.6056 −0.452222
\(551\) 5.02776 0.214190
\(552\) 0 0
\(553\) 30.2389 1.28589
\(554\) −16.1833 −0.687564
\(555\) 0 0
\(556\) −13.2111 −0.560276
\(557\) 9.63331 0.408176 0.204088 0.978953i \(-0.434577\pi\)
0.204088 + 0.978953i \(0.434577\pi\)
\(558\) 0 0
\(559\) 4.00000 0.169182
\(560\) −0.788897 −0.0333370
\(561\) 0 0
\(562\) 45.6333 1.92492
\(563\) 24.0000 1.01148 0.505740 0.862686i \(-0.331220\pi\)
0.505740 + 0.862686i \(0.331220\pi\)
\(564\) 0 0
\(565\) −15.2111 −0.639936
\(566\) −7.81665 −0.328558
\(567\) 0 0
\(568\) −22.1833 −0.930793
\(569\) 16.1833 0.678441 0.339221 0.940707i \(-0.389837\pi\)
0.339221 + 0.940707i \(0.389837\pi\)
\(570\) 0 0
\(571\) 28.4500 1.19059 0.595297 0.803506i \(-0.297034\pi\)
0.595297 + 0.803506i \(0.297034\pi\)
\(572\) 100.478 4.20118
\(573\) 0 0
\(574\) −27.6333 −1.15339
\(575\) −3.00000 −0.125109
\(576\) 0 0
\(577\) 6.18335 0.257416 0.128708 0.991683i \(-0.458917\pi\)
0.128708 + 0.991683i \(0.458917\pi\)
\(578\) 33.2111 1.38140
\(579\) 0 0
\(580\) 4.60555 0.191235
\(581\) −7.81665 −0.324289
\(582\) 0 0
\(583\) 7.39445 0.306247
\(584\) −37.8167 −1.56486
\(585\) 0 0
\(586\) 16.5416 0.683329
\(587\) 21.0000 0.866763 0.433381 0.901211i \(-0.357320\pi\)
0.433381 + 0.901211i \(0.357320\pi\)
\(588\) 0 0
\(589\) −20.2111 −0.832784
\(590\) 3.21110 0.132199
\(591\) 0 0
\(592\) 0.605551 0.0248880
\(593\) −29.2389 −1.20070 −0.600348 0.799739i \(-0.704971\pi\)
−0.600348 + 0.799739i \(0.704971\pi\)
\(594\) 0 0
\(595\) −4.18335 −0.171500
\(596\) 76.0555 3.11536
\(597\) 0 0
\(598\) 45.6333 1.86608
\(599\) −22.6056 −0.923638 −0.461819 0.886974i \(-0.652803\pi\)
−0.461819 + 0.886974i \(0.652803\pi\)
\(600\) 0 0
\(601\) −10.6333 −0.433742 −0.216871 0.976200i \(-0.569585\pi\)
−0.216871 + 0.976200i \(0.569585\pi\)
\(602\) 3.63331 0.148083
\(603\) 0 0
\(604\) 47.6333 1.93817
\(605\) 10.2111 0.415140
\(606\) 0 0
\(607\) 24.6056 0.998709 0.499354 0.866398i \(-0.333571\pi\)
0.499354 + 0.866398i \(0.333571\pi\)
\(608\) 19.1194 0.775395
\(609\) 0 0
\(610\) 9.69722 0.392629
\(611\) −60.8444 −2.46150
\(612\) 0 0
\(613\) −28.8444 −1.16501 −0.582507 0.812825i \(-0.697928\pi\)
−0.582507 + 0.812825i \(0.697928\pi\)
\(614\) −19.3944 −0.782696
\(615\) 0 0
\(616\) 36.0000 1.45048
\(617\) 38.4500 1.54794 0.773969 0.633224i \(-0.218269\pi\)
0.773969 + 0.633224i \(0.218269\pi\)
\(618\) 0 0
\(619\) 35.6333 1.43222 0.716112 0.697986i \(-0.245920\pi\)
0.716112 + 0.697986i \(0.245920\pi\)
\(620\) −18.5139 −0.743535
\(621\) 0 0
\(622\) 18.0000 0.721734
\(623\) 36.0000 1.44231
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 45.2111 1.80700
\(627\) 0 0
\(628\) −58.8444 −2.34815
\(629\) 3.21110 0.128035
\(630\) 0 0
\(631\) −6.02776 −0.239961 −0.119981 0.992776i \(-0.538283\pi\)
−0.119981 + 0.992776i \(0.538283\pi\)
\(632\) 34.8167 1.38493
\(633\) 0 0
\(634\) 17.5139 0.695565
\(635\) −19.2111 −0.762369
\(636\) 0 0
\(637\) −1.39445 −0.0552501
\(638\) −14.7889 −0.585498
\(639\) 0 0
\(640\) 18.9083 0.747417
\(641\) −24.0000 −0.947943 −0.473972 0.880540i \(-0.657180\pi\)
−0.473972 + 0.880540i \(0.657180\pi\)
\(642\) 0 0
\(643\) 13.0278 0.513765 0.256882 0.966443i \(-0.417305\pi\)
0.256882 + 0.966443i \(0.417305\pi\)
\(644\) 25.8167 1.01732
\(645\) 0 0
\(646\) −13.3305 −0.524483
\(647\) 23.7889 0.935238 0.467619 0.883930i \(-0.345112\pi\)
0.467619 + 0.883930i \(0.345112\pi\)
\(648\) 0 0
\(649\) −6.42221 −0.252094
\(650\) −15.2111 −0.596629
\(651\) 0 0
\(652\) 6.60555 0.258693
\(653\) −23.2389 −0.909407 −0.454703 0.890643i \(-0.650255\pi\)
−0.454703 + 0.890643i \(0.650255\pi\)
\(654\) 0 0
\(655\) 6.00000 0.234439
\(656\) −1.39445 −0.0544441
\(657\) 0 0
\(658\) −55.2666 −2.15452
\(659\) −13.3944 −0.521774 −0.260887 0.965369i \(-0.584015\pi\)
−0.260887 + 0.965369i \(0.584015\pi\)
\(660\) 0 0
\(661\) −34.8444 −1.35529 −0.677645 0.735389i \(-0.736999\pi\)
−0.677645 + 0.735389i \(0.736999\pi\)
\(662\) −67.2666 −2.61439
\(663\) 0 0
\(664\) −9.00000 −0.349268
\(665\) −9.39445 −0.364301
\(666\) 0 0
\(667\) −4.18335 −0.161980
\(668\) −9.90833 −0.383365
\(669\) 0 0
\(670\) 1.81665 0.0701835
\(671\) −19.3944 −0.748714
\(672\) 0 0
\(673\) 25.0278 0.964749 0.482375 0.875965i \(-0.339774\pi\)
0.482375 + 0.875965i \(0.339774\pi\)
\(674\) −15.2111 −0.585910
\(675\) 0 0
\(676\) 101.175 3.89134
\(677\) 21.6333 0.831436 0.415718 0.909494i \(-0.363530\pi\)
0.415718 + 0.909494i \(0.363530\pi\)
\(678\) 0 0
\(679\) −20.8444 −0.799935
\(680\) −4.81665 −0.184710
\(681\) 0 0
\(682\) 59.4500 2.27646
\(683\) −9.00000 −0.344375 −0.172188 0.985064i \(-0.555084\pi\)
−0.172188 + 0.985064i \(0.555084\pi\)
\(684\) 0 0
\(685\) 16.8167 0.642531
\(686\) −43.2666 −1.65193
\(687\) 0 0
\(688\) 0.183346 0.00699001
\(689\) 10.6056 0.404039
\(690\) 0 0
\(691\) 9.60555 0.365412 0.182706 0.983168i \(-0.441514\pi\)
0.182706 + 0.983168i \(0.441514\pi\)
\(692\) −35.7250 −1.35806
\(693\) 0 0
\(694\) −70.0555 −2.65927
\(695\) −4.00000 −0.151729
\(696\) 0 0
\(697\) −7.39445 −0.280085
\(698\) 73.3305 2.77560
\(699\) 0 0
\(700\) −8.60555 −0.325259
\(701\) 11.5778 0.437287 0.218644 0.975805i \(-0.429837\pi\)
0.218644 + 0.975805i \(0.429837\pi\)
\(702\) 0 0
\(703\) 7.21110 0.271972
\(704\) −59.0278 −2.22469
\(705\) 0 0
\(706\) −49.8167 −1.87487
\(707\) −31.2666 −1.17590
\(708\) 0 0
\(709\) −22.8444 −0.857940 −0.428970 0.903319i \(-0.641123\pi\)
−0.428970 + 0.903319i \(0.641123\pi\)
\(710\) −17.0278 −0.639040
\(711\) 0 0
\(712\) 41.4500 1.55340
\(713\) 16.8167 0.629789
\(714\) 0 0
\(715\) 30.4222 1.13773
\(716\) −70.0555 −2.61810
\(717\) 0 0
\(718\) 22.1833 0.827875
\(719\) 37.2666 1.38981 0.694905 0.719101i \(-0.255446\pi\)
0.694905 + 0.719101i \(0.255446\pi\)
\(720\) 0 0
\(721\) 10.4222 0.388143
\(722\) 13.8167 0.514203
\(723\) 0 0
\(724\) −23.1194 −0.859227
\(725\) 1.39445 0.0517885
\(726\) 0 0
\(727\) 35.6333 1.32157 0.660783 0.750577i \(-0.270224\pi\)
0.660783 + 0.750577i \(0.270224\pi\)
\(728\) 51.6333 1.91366
\(729\) 0 0
\(730\) −29.0278 −1.07437
\(731\) 0.972244 0.0359597
\(732\) 0 0
\(733\) 32.0000 1.18195 0.590973 0.806691i \(-0.298744\pi\)
0.590973 + 0.806691i \(0.298744\pi\)
\(734\) 6.00000 0.221464
\(735\) 0 0
\(736\) −15.9083 −0.586389
\(737\) −3.63331 −0.133835
\(738\) 0 0
\(739\) −6.02776 −0.221735 −0.110867 0.993835i \(-0.535363\pi\)
−0.110867 + 0.993835i \(0.535363\pi\)
\(740\) 6.60555 0.242825
\(741\) 0 0
\(742\) 9.63331 0.353650
\(743\) −5.57779 −0.204629 −0.102315 0.994752i \(-0.532625\pi\)
−0.102315 + 0.994752i \(0.532625\pi\)
\(744\) 0 0
\(745\) 23.0278 0.843672
\(746\) 58.0555 2.12556
\(747\) 0 0
\(748\) 24.4222 0.892964
\(749\) 0 0
\(750\) 0 0
\(751\) −30.0278 −1.09573 −0.547864 0.836567i \(-0.684559\pi\)
−0.547864 + 0.836567i \(0.684559\pi\)
\(752\) −2.78890 −0.101701
\(753\) 0 0
\(754\) −21.2111 −0.772463
\(755\) 14.4222 0.524878
\(756\) 0 0
\(757\) 16.7889 0.610203 0.305101 0.952320i \(-0.401310\pi\)
0.305101 + 0.952320i \(0.401310\pi\)
\(758\) −49.7527 −1.80710
\(759\) 0 0
\(760\) −10.8167 −0.392361
\(761\) 11.4500 0.415061 0.207530 0.978229i \(-0.433457\pi\)
0.207530 + 0.978229i \(0.433457\pi\)
\(762\) 0 0
\(763\) 18.2389 0.660291
\(764\) −41.0278 −1.48433
\(765\) 0 0
\(766\) −56.7250 −2.04956
\(767\) −9.21110 −0.332594
\(768\) 0 0
\(769\) 41.0000 1.47850 0.739249 0.673432i \(-0.235181\pi\)
0.739249 + 0.673432i \(0.235181\pi\)
\(770\) 27.6333 0.995835
\(771\) 0 0
\(772\) 0.605551 0.0217943
\(773\) −4.81665 −0.173243 −0.0866215 0.996241i \(-0.527607\pi\)
−0.0866215 + 0.996241i \(0.527607\pi\)
\(774\) 0 0
\(775\) −5.60555 −0.201357
\(776\) −24.0000 −0.861550
\(777\) 0 0
\(778\) −59.4500 −2.13138
\(779\) −16.6056 −0.594956
\(780\) 0 0
\(781\) 34.0555 1.21860
\(782\) 11.0917 0.396637
\(783\) 0 0
\(784\) −0.0639167 −0.00228274
\(785\) −17.8167 −0.635904
\(786\) 0 0
\(787\) −39.0278 −1.39119 −0.695595 0.718434i \(-0.744859\pi\)
−0.695595 + 0.718434i \(0.744859\pi\)
\(788\) −75.3583 −2.68453
\(789\) 0 0
\(790\) 26.7250 0.950832
\(791\) 39.6333 1.40920
\(792\) 0 0
\(793\) −27.8167 −0.987798
\(794\) 12.4222 0.440848
\(795\) 0 0
\(796\) −4.00000 −0.141776
\(797\) 41.6611 1.47571 0.737855 0.674959i \(-0.235839\pi\)
0.737855 + 0.674959i \(0.235839\pi\)
\(798\) 0 0
\(799\) −14.7889 −0.523194
\(800\) 5.30278 0.187481
\(801\) 0 0
\(802\) 69.0833 2.43942
\(803\) 58.0555 2.04873
\(804\) 0 0
\(805\) 7.81665 0.275501
\(806\) 85.2666 3.00339
\(807\) 0 0
\(808\) −36.0000 −1.26648
\(809\) 43.3944 1.52567 0.762834 0.646595i \(-0.223807\pi\)
0.762834 + 0.646595i \(0.223807\pi\)
\(810\) 0 0
\(811\) 13.5778 0.476781 0.238390 0.971169i \(-0.423380\pi\)
0.238390 + 0.971169i \(0.423380\pi\)
\(812\) −12.0000 −0.421117
\(813\) 0 0
\(814\) −21.2111 −0.743449
\(815\) 2.00000 0.0700569
\(816\) 0 0
\(817\) 2.18335 0.0763856
\(818\) −11.5139 −0.402573
\(819\) 0 0
\(820\) −15.2111 −0.531195
\(821\) 30.0000 1.04701 0.523504 0.852023i \(-0.324625\pi\)
0.523504 + 0.852023i \(0.324625\pi\)
\(822\) 0 0
\(823\) −17.8167 −0.621050 −0.310525 0.950565i \(-0.600505\pi\)
−0.310525 + 0.950565i \(0.600505\pi\)
\(824\) 12.0000 0.418040
\(825\) 0 0
\(826\) −8.36669 −0.291114
\(827\) −48.2111 −1.67646 −0.838232 0.545314i \(-0.816411\pi\)
−0.838232 + 0.545314i \(0.816411\pi\)
\(828\) 0 0
\(829\) −12.7889 −0.444177 −0.222088 0.975027i \(-0.571287\pi\)
−0.222088 + 0.975027i \(0.571287\pi\)
\(830\) −6.90833 −0.239792
\(831\) 0 0
\(832\) −84.6611 −2.93509
\(833\) −0.338936 −0.0117434
\(834\) 0 0
\(835\) −3.00000 −0.103819
\(836\) 54.8444 1.89683
\(837\) 0 0
\(838\) −53.0278 −1.83181
\(839\) 33.2111 1.14657 0.573287 0.819354i \(-0.305668\pi\)
0.573287 + 0.819354i \(0.305668\pi\)
\(840\) 0 0
\(841\) −27.0555 −0.932949
\(842\) −12.4861 −0.430300
\(843\) 0 0
\(844\) −29.1194 −1.00233
\(845\) 30.6333 1.05382
\(846\) 0 0
\(847\) −26.6056 −0.914178
\(848\) 0.486122 0.0166935
\(849\) 0 0
\(850\) −3.69722 −0.126814
\(851\) −6.00000 −0.205677
\(852\) 0 0
\(853\) 29.2111 1.00017 0.500085 0.865977i \(-0.333302\pi\)
0.500085 + 0.865977i \(0.333302\pi\)
\(854\) −25.2666 −0.864606
\(855\) 0 0
\(856\) 0 0
\(857\) 5.23886 0.178956 0.0894780 0.995989i \(-0.471480\pi\)
0.0894780 + 0.995989i \(0.471480\pi\)
\(858\) 0 0
\(859\) −30.0278 −1.02453 −0.512267 0.858826i \(-0.671194\pi\)
−0.512267 + 0.858826i \(0.671194\pi\)
\(860\) 2.00000 0.0681994
\(861\) 0 0
\(862\) −9.63331 −0.328112
\(863\) 30.2111 1.02840 0.514199 0.857671i \(-0.328089\pi\)
0.514199 + 0.857671i \(0.328089\pi\)
\(864\) 0 0
\(865\) −10.8167 −0.367777
\(866\) −51.2111 −1.74022
\(867\) 0 0
\(868\) 48.2389 1.63733
\(869\) −53.4500 −1.81317
\(870\) 0 0
\(871\) −5.21110 −0.176571
\(872\) 21.0000 0.711150
\(873\) 0 0
\(874\) 24.9083 0.842537
\(875\) −2.60555 −0.0880837
\(876\) 0 0
\(877\) 22.6611 0.765210 0.382605 0.923912i \(-0.375027\pi\)
0.382605 + 0.923912i \(0.375027\pi\)
\(878\) −63.5694 −2.14536
\(879\) 0 0
\(880\) 1.39445 0.0470069
\(881\) −25.3944 −0.855561 −0.427780 0.903883i \(-0.640704\pi\)
−0.427780 + 0.903883i \(0.640704\pi\)
\(882\) 0 0
\(883\) 15.8167 0.532273 0.266136 0.963935i \(-0.414253\pi\)
0.266136 + 0.963935i \(0.414253\pi\)
\(884\) 35.0278 1.17811
\(885\) 0 0
\(886\) 56.7250 1.90571
\(887\) −30.6333 −1.02857 −0.514283 0.857621i \(-0.671942\pi\)
−0.514283 + 0.857621i \(0.671942\pi\)
\(888\) 0 0
\(889\) 50.0555 1.67881
\(890\) 31.8167 1.06650
\(891\) 0 0
\(892\) −33.0278 −1.10585
\(893\) −33.2111 −1.11137
\(894\) 0 0
\(895\) −21.2111 −0.709009
\(896\) −49.2666 −1.64588
\(897\) 0 0
\(898\) −88.0555 −2.93845
\(899\) −7.81665 −0.260700
\(900\) 0 0
\(901\) 2.57779 0.0858788
\(902\) 48.8444 1.62634
\(903\) 0 0
\(904\) 45.6333 1.51774
\(905\) −7.00000 −0.232688
\(906\) 0 0
\(907\) 1.57779 0.0523898 0.0261949 0.999657i \(-0.491661\pi\)
0.0261949 + 0.999657i \(0.491661\pi\)
\(908\) −38.9361 −1.29214
\(909\) 0 0
\(910\) 39.6333 1.31383
\(911\) −5.02776 −0.166577 −0.0832885 0.996525i \(-0.526542\pi\)
−0.0832885 + 0.996525i \(0.526542\pi\)
\(912\) 0 0
\(913\) 13.8167 0.457265
\(914\) 30.4222 1.00628
\(915\) 0 0
\(916\) 27.1194 0.896051
\(917\) −15.6333 −0.516257
\(918\) 0 0
\(919\) 26.4222 0.871588 0.435794 0.900046i \(-0.356468\pi\)
0.435794 + 0.900046i \(0.356468\pi\)
\(920\) 9.00000 0.296721
\(921\) 0 0
\(922\) −49.8167 −1.64062
\(923\) 48.8444 1.60773
\(924\) 0 0
\(925\) 2.00000 0.0657596
\(926\) 1.81665 0.0596989
\(927\) 0 0
\(928\) 7.39445 0.242735
\(929\) 19.3944 0.636311 0.318156 0.948039i \(-0.396937\pi\)
0.318156 + 0.948039i \(0.396937\pi\)
\(930\) 0 0
\(931\) −0.761141 −0.0249454
\(932\) −59.4500 −1.94735
\(933\) 0 0
\(934\) −28.1194 −0.920096
\(935\) 7.39445 0.241824
\(936\) 0 0
\(937\) 41.2111 1.34631 0.673154 0.739502i \(-0.264939\pi\)
0.673154 + 0.739502i \(0.264939\pi\)
\(938\) −4.73338 −0.154550
\(939\) 0 0
\(940\) −30.4222 −0.992263
\(941\) −0.422205 −0.0137635 −0.00688175 0.999976i \(-0.502191\pi\)
−0.00688175 + 0.999976i \(0.502191\pi\)
\(942\) 0 0
\(943\) 13.8167 0.449932
\(944\) −0.422205 −0.0137416
\(945\) 0 0
\(946\) −6.42221 −0.208804
\(947\) 39.0000 1.26733 0.633665 0.773608i \(-0.281550\pi\)
0.633665 + 0.773608i \(0.281550\pi\)
\(948\) 0 0
\(949\) 83.2666 2.70295
\(950\) −8.30278 −0.269378
\(951\) 0 0
\(952\) 12.5500 0.406749
\(953\) −30.8444 −0.999148 −0.499574 0.866271i \(-0.666510\pi\)
−0.499574 + 0.866271i \(0.666510\pi\)
\(954\) 0 0
\(955\) −12.4222 −0.401973
\(956\) −50.2389 −1.62484
\(957\) 0 0
\(958\) 87.0833 2.81353
\(959\) −43.8167 −1.41491
\(960\) 0 0
\(961\) 0.422205 0.0136195
\(962\) −30.4222 −0.980851
\(963\) 0 0
\(964\) 45.5416 1.46680
\(965\) 0.183346 0.00590212
\(966\) 0 0
\(967\) 50.0000 1.60789 0.803946 0.594703i \(-0.202730\pi\)
0.803946 + 0.594703i \(0.202730\pi\)
\(968\) −30.6333 −0.984592
\(969\) 0 0
\(970\) −18.4222 −0.591501
\(971\) −18.0000 −0.577647 −0.288824 0.957382i \(-0.593264\pi\)
−0.288824 + 0.957382i \(0.593264\pi\)
\(972\) 0 0
\(973\) 10.4222 0.334121
\(974\) 68.6611 2.20004
\(975\) 0 0
\(976\) −1.27502 −0.0408124
\(977\) −15.2111 −0.486646 −0.243323 0.969945i \(-0.578238\pi\)
−0.243323 + 0.969945i \(0.578238\pi\)
\(978\) 0 0
\(979\) −63.6333 −2.03373
\(980\) −0.697224 −0.0222720
\(981\) 0 0
\(982\) −62.6611 −1.99959
\(983\) −42.6333 −1.35979 −0.679896 0.733309i \(-0.737975\pi\)
−0.679896 + 0.733309i \(0.737975\pi\)
\(984\) 0 0
\(985\) −22.8167 −0.726999
\(986\) −5.15559 −0.164187
\(987\) 0 0
\(988\) 78.6611 2.50254
\(989\) −1.81665 −0.0577662
\(990\) 0 0
\(991\) −29.1833 −0.927040 −0.463520 0.886087i \(-0.653414\pi\)
−0.463520 + 0.886087i \(0.653414\pi\)
\(992\) −29.7250 −0.943769
\(993\) 0 0
\(994\) 44.3667 1.40723
\(995\) −1.21110 −0.0383945
\(996\) 0 0
\(997\) −39.8722 −1.26276 −0.631382 0.775472i \(-0.717512\pi\)
−0.631382 + 0.775472i \(0.717512\pi\)
\(998\) 46.9638 1.48661
\(999\) 0 0
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 135.2.a.c.1.1 2
3.2 odd 2 135.2.a.d.1.2 yes 2
4.3 odd 2 2160.2.a.ba.1.2 2
5.2 odd 4 675.2.b.i.649.1 4
5.3 odd 4 675.2.b.i.649.4 4
5.4 even 2 675.2.a.p.1.2 2
7.6 odd 2 6615.2.a.p.1.1 2
8.3 odd 2 8640.2.a.ck.1.2 2
8.5 even 2 8640.2.a.cr.1.1 2
9.2 odd 6 405.2.e.j.271.1 4
9.4 even 3 405.2.e.k.136.2 4
9.5 odd 6 405.2.e.j.136.1 4
9.7 even 3 405.2.e.k.271.2 4
12.11 even 2 2160.2.a.y.1.2 2
15.2 even 4 675.2.b.h.649.4 4
15.8 even 4 675.2.b.h.649.1 4
15.14 odd 2 675.2.a.k.1.1 2
21.20 even 2 6615.2.a.v.1.2 2
24.5 odd 2 8640.2.a.df.1.1 2
24.11 even 2 8640.2.a.cy.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
135.2.a.c.1.1 2 1.1 even 1 trivial
135.2.a.d.1.2 yes 2 3.2 odd 2
405.2.e.j.136.1 4 9.5 odd 6
405.2.e.j.271.1 4 9.2 odd 6
405.2.e.k.136.2 4 9.4 even 3
405.2.e.k.271.2 4 9.7 even 3
675.2.a.k.1.1 2 15.14 odd 2
675.2.a.p.1.2 2 5.4 even 2
675.2.b.h.649.1 4 15.8 even 4
675.2.b.h.649.4 4 15.2 even 4
675.2.b.i.649.1 4 5.2 odd 4
675.2.b.i.649.4 4 5.3 odd 4
2160.2.a.y.1.2 2 12.11 even 2
2160.2.a.ba.1.2 2 4.3 odd 2
6615.2.a.p.1.1 2 7.6 odd 2
6615.2.a.v.1.2 2 21.20 even 2
8640.2.a.ck.1.2 2 8.3 odd 2
8640.2.a.cr.1.1 2 8.5 even 2
8640.2.a.cy.1.2 2 24.11 even 2
8640.2.a.df.1.1 2 24.5 odd 2