Properties

Label 1881.2.a.k.1.2
Level $1881$
Weight $2$
Character 1881.1
Self dual yes
Analytic conductor $15.020$
Analytic rank $0$
Dimension $5$
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] = [1881,2,Mod(1,1881)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1881, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1881.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1881 = 3^{2} \cdot 11 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1881.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: \(15.0198606202\)
Analytic rank: \(0\)
Dimension: \(5\)
Coefficient field: 5.5.246832.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{5} - 2x^{4} - 5x^{3} + 6x^{2} + 7x - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 209)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-1.51908\) of defining polynomial
Character \(\chi\) \(=\) 1881.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.82669 q^{2} +1.33679 q^{4} -2.34577 q^{5} +1.69239 q^{7} +1.21147 q^{8} +O(q^{10})\) \(q-1.82669 q^{2} +1.33679 q^{4} -2.34577 q^{5} +1.69239 q^{7} +1.21147 q^{8} +4.28499 q^{10} -1.00000 q^{11} +4.11168 q^{13} -3.09147 q^{14} -4.88657 q^{16} +6.16499 q^{17} -1.00000 q^{19} -3.13581 q^{20} +1.82669 q^{22} -3.52199 q^{23} +0.502638 q^{25} -7.51076 q^{26} +2.26238 q^{28} +8.10336 q^{29} -2.30144 q^{31} +6.50330 q^{32} -11.2615 q^{34} -3.96996 q^{35} -6.56016 q^{37} +1.82669 q^{38} -2.84184 q^{40} -7.75013 q^{41} +7.75102 q^{43} -1.33679 q^{44} +6.43359 q^{46} +10.8969 q^{47} -4.13581 q^{49} -0.918163 q^{50} +5.49647 q^{52} -7.93511 q^{53} +2.34577 q^{55} +2.05029 q^{56} -14.8023 q^{58} -10.9247 q^{59} -4.51162 q^{61} +4.20401 q^{62} -2.10636 q^{64} -9.64506 q^{65} +14.7201 q^{67} +8.24132 q^{68} +7.25189 q^{70} -3.12026 q^{71} +11.5827 q^{73} +11.9834 q^{74} -1.33679 q^{76} -1.69239 q^{77} +4.96184 q^{79} +11.4628 q^{80} +14.1571 q^{82} +1.82905 q^{83} -14.4617 q^{85} -14.1587 q^{86} -1.21147 q^{88} +9.37496 q^{89} +6.95858 q^{91} -4.70818 q^{92} -19.9053 q^{94} +2.34577 q^{95} -10.9937 q^{97} +7.55484 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q - 2 q^{2} + 6 q^{4} + 5 q^{5} + 6 q^{7} - 6 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 5 q - 2 q^{2} + 6 q^{4} + 5 q^{5} + 6 q^{7} - 6 q^{8} + 12 q^{10} - 5 q^{11} + 4 q^{13} + 14 q^{14} + 8 q^{16} + 4 q^{17} - 5 q^{19} + 8 q^{20} + 2 q^{22} - 3 q^{23} + 6 q^{25} + 6 q^{26} - 10 q^{28} - 10 q^{29} + 11 q^{31} - 14 q^{32} - 4 q^{34} + 8 q^{35} + q^{37} + 2 q^{38} - 16 q^{40} - 2 q^{41} + 20 q^{43} - 6 q^{44} - 4 q^{46} + 20 q^{47} + 3 q^{49} + 32 q^{50} + 6 q^{52} + 14 q^{53} - 5 q^{55} + 38 q^{56} - 6 q^{58} - 3 q^{59} - 10 q^{61} + 6 q^{62} + 9 q^{67} - 24 q^{68} + 50 q^{70} - 23 q^{71} - 8 q^{74} - 6 q^{76} - 6 q^{77} + 44 q^{79} + 18 q^{80} - 30 q^{82} + 14 q^{83} - 12 q^{85} - 52 q^{86} + 6 q^{88} + 27 q^{89} + 24 q^{91} - 58 q^{92} - 8 q^{94} - 5 q^{95} + 15 q^{97} + 10 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.82669 −1.29166 −0.645832 0.763479i \(-0.723489\pi\)
−0.645832 + 0.763479i \(0.723489\pi\)
\(3\) 0 0
\(4\) 1.33679 0.668396
\(5\) −2.34577 −1.04906 −0.524530 0.851392i \(-0.675759\pi\)
−0.524530 + 0.851392i \(0.675759\pi\)
\(6\) 0 0
\(7\) 1.69239 0.639664 0.319832 0.947474i \(-0.396374\pi\)
0.319832 + 0.947474i \(0.396374\pi\)
\(8\) 1.21147 0.428321
\(9\) 0 0
\(10\) 4.28499 1.35503
\(11\) −1.00000 −0.301511
\(12\) 0 0
\(13\) 4.11168 1.14038 0.570188 0.821514i \(-0.306870\pi\)
0.570188 + 0.821514i \(0.306870\pi\)
\(14\) −3.09147 −0.826231
\(15\) 0 0
\(16\) −4.88657 −1.22164
\(17\) 6.16499 1.49523 0.747615 0.664132i \(-0.231199\pi\)
0.747615 + 0.664132i \(0.231199\pi\)
\(18\) 0 0
\(19\) −1.00000 −0.229416
\(20\) −3.13581 −0.701188
\(21\) 0 0
\(22\) 1.82669 0.389451
\(23\) −3.52199 −0.734386 −0.367193 0.930145i \(-0.619681\pi\)
−0.367193 + 0.930145i \(0.619681\pi\)
\(24\) 0 0
\(25\) 0.502638 0.100528
\(26\) −7.51076 −1.47298
\(27\) 0 0
\(28\) 2.26238 0.427549
\(29\) 8.10336 1.50476 0.752378 0.658731i \(-0.228906\pi\)
0.752378 + 0.658731i \(0.228906\pi\)
\(30\) 0 0
\(31\) −2.30144 −0.413350 −0.206675 0.978410i \(-0.566264\pi\)
−0.206675 + 0.978410i \(0.566264\pi\)
\(32\) 6.50330 1.14963
\(33\) 0 0
\(34\) −11.2615 −1.93134
\(35\) −3.96996 −0.671046
\(36\) 0 0
\(37\) −6.56016 −1.07848 −0.539242 0.842151i \(-0.681289\pi\)
−0.539242 + 0.842151i \(0.681289\pi\)
\(38\) 1.82669 0.296328
\(39\) 0 0
\(40\) −2.84184 −0.449334
\(41\) −7.75013 −1.21037 −0.605183 0.796086i \(-0.706900\pi\)
−0.605183 + 0.796086i \(0.706900\pi\)
\(42\) 0 0
\(43\) 7.75102 1.18202 0.591010 0.806664i \(-0.298729\pi\)
0.591010 + 0.806664i \(0.298729\pi\)
\(44\) −1.33679 −0.201529
\(45\) 0 0
\(46\) 6.43359 0.948581
\(47\) 10.8969 1.58948 0.794742 0.606948i \(-0.207606\pi\)
0.794742 + 0.606948i \(0.207606\pi\)
\(48\) 0 0
\(49\) −4.13581 −0.590830
\(50\) −0.918163 −0.129848
\(51\) 0 0
\(52\) 5.49647 0.762223
\(53\) −7.93511 −1.08997 −0.544986 0.838445i \(-0.683465\pi\)
−0.544986 + 0.838445i \(0.683465\pi\)
\(54\) 0 0
\(55\) 2.34577 0.316304
\(56\) 2.05029 0.273981
\(57\) 0 0
\(58\) −14.8023 −1.94364
\(59\) −10.9247 −1.42228 −0.711140 0.703051i \(-0.751821\pi\)
−0.711140 + 0.703051i \(0.751821\pi\)
\(60\) 0 0
\(61\) −4.51162 −0.577653 −0.288827 0.957381i \(-0.593265\pi\)
−0.288827 + 0.957381i \(0.593265\pi\)
\(62\) 4.20401 0.533910
\(63\) 0 0
\(64\) −2.10636 −0.263295
\(65\) −9.64506 −1.19632
\(66\) 0 0
\(67\) 14.7201 1.79835 0.899173 0.437594i \(-0.144169\pi\)
0.899173 + 0.437594i \(0.144169\pi\)
\(68\) 8.24132 0.999407
\(69\) 0 0
\(70\) 7.25189 0.866766
\(71\) −3.12026 −0.370307 −0.185153 0.982710i \(-0.559278\pi\)
−0.185153 + 0.982710i \(0.559278\pi\)
\(72\) 0 0
\(73\) 11.5827 1.35565 0.677824 0.735224i \(-0.262923\pi\)
0.677824 + 0.735224i \(0.262923\pi\)
\(74\) 11.9834 1.39304
\(75\) 0 0
\(76\) −1.33679 −0.153341
\(77\) −1.69239 −0.192866
\(78\) 0 0
\(79\) 4.96184 0.558250 0.279125 0.960255i \(-0.409956\pi\)
0.279125 + 0.960255i \(0.409956\pi\)
\(80\) 11.4628 1.28158
\(81\) 0 0
\(82\) 14.1571 1.56339
\(83\) 1.82905 0.200765 0.100382 0.994949i \(-0.467993\pi\)
0.100382 + 0.994949i \(0.467993\pi\)
\(84\) 0 0
\(85\) −14.4617 −1.56859
\(86\) −14.1587 −1.52677
\(87\) 0 0
\(88\) −1.21147 −0.129144
\(89\) 9.37496 0.993743 0.496872 0.867824i \(-0.334482\pi\)
0.496872 + 0.867824i \(0.334482\pi\)
\(90\) 0 0
\(91\) 6.95858 0.729457
\(92\) −4.70818 −0.490861
\(93\) 0 0
\(94\) −19.9053 −2.05308
\(95\) 2.34577 0.240671
\(96\) 0 0
\(97\) −10.9937 −1.11625 −0.558123 0.829758i \(-0.688478\pi\)
−0.558123 + 0.829758i \(0.688478\pi\)
\(98\) 7.55484 0.763154
\(99\) 0 0
\(100\) 0.671923 0.0671923
\(101\) −2.30621 −0.229476 −0.114738 0.993396i \(-0.536603\pi\)
−0.114738 + 0.993396i \(0.536603\pi\)
\(102\) 0 0
\(103\) 6.12000 0.603021 0.301511 0.953463i \(-0.402509\pi\)
0.301511 + 0.953463i \(0.402509\pi\)
\(104\) 4.98119 0.488446
\(105\) 0 0
\(106\) 14.4950 1.40788
\(107\) −0.422947 −0.0408878 −0.0204439 0.999791i \(-0.506508\pi\)
−0.0204439 + 0.999791i \(0.506508\pi\)
\(108\) 0 0
\(109\) 14.0180 1.34268 0.671338 0.741151i \(-0.265720\pi\)
0.671338 + 0.741151i \(0.265720\pi\)
\(110\) −4.28499 −0.408558
\(111\) 0 0
\(112\) −8.26999 −0.781441
\(113\) 2.53638 0.238602 0.119301 0.992858i \(-0.461935\pi\)
0.119301 + 0.992858i \(0.461935\pi\)
\(114\) 0 0
\(115\) 8.26179 0.770416
\(116\) 10.8325 1.00577
\(117\) 0 0
\(118\) 19.9561 1.83711
\(119\) 10.4336 0.956445
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 8.24132 0.746134
\(123\) 0 0
\(124\) −3.07654 −0.276282
\(125\) 10.5498 0.943601
\(126\) 0 0
\(127\) 1.07275 0.0951914 0.0475957 0.998867i \(-0.484844\pi\)
0.0475957 + 0.998867i \(0.484844\pi\)
\(128\) −9.15893 −0.809543
\(129\) 0 0
\(130\) 17.6185 1.54525
\(131\) 7.82709 0.683856 0.341928 0.939726i \(-0.388920\pi\)
0.341928 + 0.939726i \(0.388920\pi\)
\(132\) 0 0
\(133\) −1.69239 −0.146749
\(134\) −26.8890 −2.32286
\(135\) 0 0
\(136\) 7.46873 0.640438
\(137\) 11.2473 0.960919 0.480460 0.877017i \(-0.340470\pi\)
0.480460 + 0.877017i \(0.340470\pi\)
\(138\) 0 0
\(139\) −0.905926 −0.0768397 −0.0384198 0.999262i \(-0.512232\pi\)
−0.0384198 + 0.999262i \(0.512232\pi\)
\(140\) −5.30702 −0.448525
\(141\) 0 0
\(142\) 5.69975 0.478312
\(143\) −4.11168 −0.343836
\(144\) 0 0
\(145\) −19.0086 −1.57858
\(146\) −21.1579 −1.75104
\(147\) 0 0
\(148\) −8.76957 −0.720854
\(149\) −10.5174 −0.861622 −0.430811 0.902442i \(-0.641772\pi\)
−0.430811 + 0.902442i \(0.641772\pi\)
\(150\) 0 0
\(151\) 13.2436 1.07775 0.538873 0.842387i \(-0.318850\pi\)
0.538873 + 0.842387i \(0.318850\pi\)
\(152\) −1.21147 −0.0982635
\(153\) 0 0
\(154\) 3.09147 0.249118
\(155\) 5.39864 0.433629
\(156\) 0 0
\(157\) 1.99915 0.159549 0.0797747 0.996813i \(-0.474580\pi\)
0.0797747 + 0.996813i \(0.474580\pi\)
\(158\) −9.06373 −0.721072
\(159\) 0 0
\(160\) −15.2552 −1.20603
\(161\) −5.96059 −0.469761
\(162\) 0 0
\(163\) −18.7557 −1.46906 −0.734531 0.678575i \(-0.762598\pi\)
−0.734531 + 0.678575i \(0.762598\pi\)
\(164\) −10.3603 −0.809005
\(165\) 0 0
\(166\) −3.34111 −0.259320
\(167\) 6.31203 0.488440 0.244220 0.969720i \(-0.421468\pi\)
0.244220 + 0.969720i \(0.421468\pi\)
\(168\) 0 0
\(169\) 3.90593 0.300456
\(170\) 26.4170 2.02609
\(171\) 0 0
\(172\) 10.3615 0.790058
\(173\) 21.5269 1.63666 0.818328 0.574751i \(-0.194901\pi\)
0.818328 + 0.574751i \(0.194901\pi\)
\(174\) 0 0
\(175\) 0.850661 0.0643039
\(176\) 4.88657 0.368339
\(177\) 0 0
\(178\) −17.1251 −1.28358
\(179\) 7.97018 0.595719 0.297860 0.954610i \(-0.403727\pi\)
0.297860 + 0.954610i \(0.403727\pi\)
\(180\) 0 0
\(181\) −1.16955 −0.0869317 −0.0434659 0.999055i \(-0.513840\pi\)
−0.0434659 + 0.999055i \(0.513840\pi\)
\(182\) −12.7112 −0.942214
\(183\) 0 0
\(184\) −4.26680 −0.314553
\(185\) 15.3886 1.13139
\(186\) 0 0
\(187\) −6.16499 −0.450829
\(188\) 14.5670 1.06240
\(189\) 0 0
\(190\) −4.28499 −0.310866
\(191\) 15.6673 1.13365 0.566824 0.823839i \(-0.308172\pi\)
0.566824 + 0.823839i \(0.308172\pi\)
\(192\) 0 0
\(193\) 21.6769 1.56034 0.780169 0.625568i \(-0.215133\pi\)
0.780169 + 0.625568i \(0.215133\pi\)
\(194\) 20.0821 1.44181
\(195\) 0 0
\(196\) −5.52872 −0.394908
\(197\) −4.58794 −0.326877 −0.163439 0.986554i \(-0.552259\pi\)
−0.163439 + 0.986554i \(0.552259\pi\)
\(198\) 0 0
\(199\) 13.0619 0.925937 0.462968 0.886375i \(-0.346784\pi\)
0.462968 + 0.886375i \(0.346784\pi\)
\(200\) 0.608933 0.0430580
\(201\) 0 0
\(202\) 4.21272 0.296406
\(203\) 13.7141 0.962539
\(204\) 0 0
\(205\) 18.1800 1.26975
\(206\) −11.1793 −0.778901
\(207\) 0 0
\(208\) −20.0920 −1.39313
\(209\) 1.00000 0.0691714
\(210\) 0 0
\(211\) 2.91188 0.200462 0.100231 0.994964i \(-0.468042\pi\)
0.100231 + 0.994964i \(0.468042\pi\)
\(212\) −10.6076 −0.728533
\(213\) 0 0
\(214\) 0.772592 0.0528133
\(215\) −18.1821 −1.24001
\(216\) 0 0
\(217\) −3.89493 −0.264405
\(218\) −25.6064 −1.73429
\(219\) 0 0
\(220\) 3.13581 0.211416
\(221\) 25.3485 1.70512
\(222\) 0 0
\(223\) 6.34161 0.424665 0.212333 0.977197i \(-0.431894\pi\)
0.212333 + 0.977197i \(0.431894\pi\)
\(224\) 11.0061 0.735378
\(225\) 0 0
\(226\) −4.63317 −0.308194
\(227\) 2.00429 0.133029 0.0665147 0.997785i \(-0.478812\pi\)
0.0665147 + 0.997785i \(0.478812\pi\)
\(228\) 0 0
\(229\) −20.9893 −1.38701 −0.693507 0.720450i \(-0.743935\pi\)
−0.693507 + 0.720450i \(0.743935\pi\)
\(230\) −15.0917 −0.995118
\(231\) 0 0
\(232\) 9.81701 0.644518
\(233\) 17.6006 1.15305 0.576527 0.817078i \(-0.304407\pi\)
0.576527 + 0.817078i \(0.304407\pi\)
\(234\) 0 0
\(235\) −25.5617 −1.66746
\(236\) −14.6041 −0.950646
\(237\) 0 0
\(238\) −19.0589 −1.23541
\(239\) −26.6207 −1.72195 −0.860975 0.508647i \(-0.830146\pi\)
−0.860975 + 0.508647i \(0.830146\pi\)
\(240\) 0 0
\(241\) 13.1342 0.846049 0.423024 0.906118i \(-0.360968\pi\)
0.423024 + 0.906118i \(0.360968\pi\)
\(242\) −1.82669 −0.117424
\(243\) 0 0
\(244\) −6.03109 −0.386101
\(245\) 9.70166 0.619816
\(246\) 0 0
\(247\) −4.11168 −0.261620
\(248\) −2.78813 −0.177046
\(249\) 0 0
\(250\) −19.2712 −1.21882
\(251\) 26.1636 1.65143 0.825715 0.564087i \(-0.190772\pi\)
0.825715 + 0.564087i \(0.190772\pi\)
\(252\) 0 0
\(253\) 3.52199 0.221426
\(254\) −1.95959 −0.122955
\(255\) 0 0
\(256\) 20.9432 1.30895
\(257\) −12.6117 −0.786697 −0.393349 0.919389i \(-0.628683\pi\)
−0.393349 + 0.919389i \(0.628683\pi\)
\(258\) 0 0
\(259\) −11.1024 −0.689867
\(260\) −12.8934 −0.799618
\(261\) 0 0
\(262\) −14.2977 −0.883312
\(263\) 20.0550 1.23664 0.618322 0.785925i \(-0.287813\pi\)
0.618322 + 0.785925i \(0.287813\pi\)
\(264\) 0 0
\(265\) 18.6139 1.14345
\(266\) 3.09147 0.189550
\(267\) 0 0
\(268\) 19.6777 1.20201
\(269\) 20.1150 1.22644 0.613218 0.789914i \(-0.289875\pi\)
0.613218 + 0.789914i \(0.289875\pi\)
\(270\) 0 0
\(271\) 15.5586 0.945117 0.472558 0.881299i \(-0.343331\pi\)
0.472558 + 0.881299i \(0.343331\pi\)
\(272\) −30.1257 −1.82664
\(273\) 0 0
\(274\) −20.5453 −1.24119
\(275\) −0.502638 −0.0303102
\(276\) 0 0
\(277\) 1.89211 0.113686 0.0568429 0.998383i \(-0.481897\pi\)
0.0568429 + 0.998383i \(0.481897\pi\)
\(278\) 1.65485 0.0992510
\(279\) 0 0
\(280\) −4.80951 −0.287423
\(281\) −4.59299 −0.273995 −0.136997 0.990571i \(-0.543745\pi\)
−0.136997 + 0.990571i \(0.543745\pi\)
\(282\) 0 0
\(283\) −1.17798 −0.0700234 −0.0350117 0.999387i \(-0.511147\pi\)
−0.0350117 + 0.999387i \(0.511147\pi\)
\(284\) −4.17114 −0.247512
\(285\) 0 0
\(286\) 7.51076 0.444121
\(287\) −13.1163 −0.774228
\(288\) 0 0
\(289\) 21.0071 1.23571
\(290\) 34.7229 2.03900
\(291\) 0 0
\(292\) 15.4836 0.906110
\(293\) 11.2606 0.657851 0.328925 0.944356i \(-0.393314\pi\)
0.328925 + 0.944356i \(0.393314\pi\)
\(294\) 0 0
\(295\) 25.6269 1.49206
\(296\) −7.94745 −0.461936
\(297\) 0 0
\(298\) 19.2121 1.11293
\(299\) −14.4813 −0.837476
\(300\) 0 0
\(301\) 13.1178 0.756096
\(302\) −24.1919 −1.39209
\(303\) 0 0
\(304\) 4.88657 0.280264
\(305\) 10.5832 0.605993
\(306\) 0 0
\(307\) 4.35245 0.248407 0.124204 0.992257i \(-0.460362\pi\)
0.124204 + 0.992257i \(0.460362\pi\)
\(308\) −2.26238 −0.128911
\(309\) 0 0
\(310\) −9.86164 −0.560103
\(311\) −7.61725 −0.431935 −0.215967 0.976401i \(-0.569291\pi\)
−0.215967 + 0.976401i \(0.569291\pi\)
\(312\) 0 0
\(313\) −17.5654 −0.992855 −0.496427 0.868078i \(-0.665355\pi\)
−0.496427 + 0.868078i \(0.665355\pi\)
\(314\) −3.65182 −0.206084
\(315\) 0 0
\(316\) 6.63295 0.373132
\(317\) 1.94192 0.109069 0.0545345 0.998512i \(-0.482633\pi\)
0.0545345 + 0.998512i \(0.482633\pi\)
\(318\) 0 0
\(319\) −8.10336 −0.453701
\(320\) 4.94104 0.276213
\(321\) 0 0
\(322\) 10.8882 0.606773
\(323\) −6.16499 −0.343029
\(324\) 0 0
\(325\) 2.06669 0.114639
\(326\) 34.2609 1.89754
\(327\) 0 0
\(328\) −9.38907 −0.518425
\(329\) 18.4419 1.01674
\(330\) 0 0
\(331\) 13.3988 0.736462 0.368231 0.929734i \(-0.379964\pi\)
0.368231 + 0.929734i \(0.379964\pi\)
\(332\) 2.44506 0.134190
\(333\) 0 0
\(334\) −11.5301 −0.630900
\(335\) −34.5300 −1.88657
\(336\) 0 0
\(337\) 33.1631 1.80651 0.903254 0.429107i \(-0.141172\pi\)
0.903254 + 0.429107i \(0.141172\pi\)
\(338\) −7.13491 −0.388088
\(339\) 0 0
\(340\) −19.3322 −1.04844
\(341\) 2.30144 0.124630
\(342\) 0 0
\(343\) −18.8462 −1.01760
\(344\) 9.39016 0.506283
\(345\) 0 0
\(346\) −39.3229 −2.11401
\(347\) −18.0268 −0.967730 −0.483865 0.875143i \(-0.660767\pi\)
−0.483865 + 0.875143i \(0.660767\pi\)
\(348\) 0 0
\(349\) 8.81411 0.471808 0.235904 0.971776i \(-0.424195\pi\)
0.235904 + 0.971776i \(0.424195\pi\)
\(350\) −1.55389 −0.0830590
\(351\) 0 0
\(352\) −6.50330 −0.346627
\(353\) 1.59249 0.0847599 0.0423799 0.999102i \(-0.486506\pi\)
0.0423799 + 0.999102i \(0.486506\pi\)
\(354\) 0 0
\(355\) 7.31942 0.388474
\(356\) 12.5324 0.664214
\(357\) 0 0
\(358\) −14.5590 −0.769469
\(359\) −36.3774 −1.91993 −0.959963 0.280126i \(-0.909624\pi\)
−0.959963 + 0.280126i \(0.909624\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 2.13640 0.112287
\(363\) 0 0
\(364\) 9.30218 0.487567
\(365\) −27.1703 −1.42216
\(366\) 0 0
\(367\) −5.00276 −0.261142 −0.130571 0.991439i \(-0.541681\pi\)
−0.130571 + 0.991439i \(0.541681\pi\)
\(368\) 17.2105 0.897158
\(369\) 0 0
\(370\) −28.1102 −1.46138
\(371\) −13.4293 −0.697216
\(372\) 0 0
\(373\) −34.1313 −1.76725 −0.883625 0.468195i \(-0.844904\pi\)
−0.883625 + 0.468195i \(0.844904\pi\)
\(374\) 11.2615 0.582320
\(375\) 0 0
\(376\) 13.2014 0.680808
\(377\) 33.3185 1.71599
\(378\) 0 0
\(379\) 15.7749 0.810302 0.405151 0.914250i \(-0.367219\pi\)
0.405151 + 0.914250i \(0.367219\pi\)
\(380\) 3.13581 0.160864
\(381\) 0 0
\(382\) −28.6193 −1.46429
\(383\) 22.4260 1.14592 0.572958 0.819585i \(-0.305796\pi\)
0.572958 + 0.819585i \(0.305796\pi\)
\(384\) 0 0
\(385\) 3.96996 0.202328
\(386\) −39.5970 −2.01543
\(387\) 0 0
\(388\) −14.6964 −0.746094
\(389\) 18.3488 0.930321 0.465160 0.885226i \(-0.345997\pi\)
0.465160 + 0.885226i \(0.345997\pi\)
\(390\) 0 0
\(391\) −21.7131 −1.09808
\(392\) −5.01042 −0.253065
\(393\) 0 0
\(394\) 8.38074 0.422216
\(395\) −11.6393 −0.585638
\(396\) 0 0
\(397\) −0.511483 −0.0256706 −0.0128353 0.999918i \(-0.504086\pi\)
−0.0128353 + 0.999918i \(0.504086\pi\)
\(398\) −23.8601 −1.19600
\(399\) 0 0
\(400\) −2.45618 −0.122809
\(401\) −18.9824 −0.947935 −0.473967 0.880542i \(-0.657179\pi\)
−0.473967 + 0.880542i \(0.657179\pi\)
\(402\) 0 0
\(403\) −9.46277 −0.471374
\(404\) −3.08292 −0.153381
\(405\) 0 0
\(406\) −25.0513 −1.24328
\(407\) 6.56016 0.325175
\(408\) 0 0
\(409\) 1.10089 0.0544355 0.0272177 0.999630i \(-0.491335\pi\)
0.0272177 + 0.999630i \(0.491335\pi\)
\(410\) −33.2092 −1.64009
\(411\) 0 0
\(412\) 8.18117 0.403057
\(413\) −18.4889 −0.909781
\(414\) 0 0
\(415\) −4.29054 −0.210614
\(416\) 26.7395 1.31101
\(417\) 0 0
\(418\) −1.82669 −0.0893463
\(419\) 18.7929 0.918094 0.459047 0.888412i \(-0.348191\pi\)
0.459047 + 0.888412i \(0.348191\pi\)
\(420\) 0 0
\(421\) −24.7618 −1.20682 −0.603408 0.797433i \(-0.706191\pi\)
−0.603408 + 0.797433i \(0.706191\pi\)
\(422\) −5.31910 −0.258930
\(423\) 0 0
\(424\) −9.61318 −0.466857
\(425\) 3.09876 0.150312
\(426\) 0 0
\(427\) −7.63542 −0.369504
\(428\) −0.565392 −0.0273293
\(429\) 0 0
\(430\) 33.2131 1.60168
\(431\) 32.1985 1.55095 0.775474 0.631380i \(-0.217511\pi\)
0.775474 + 0.631380i \(0.217511\pi\)
\(432\) 0 0
\(433\) −15.3042 −0.735475 −0.367737 0.929930i \(-0.619867\pi\)
−0.367737 + 0.929930i \(0.619867\pi\)
\(434\) 7.11483 0.341523
\(435\) 0 0
\(436\) 18.7391 0.897440
\(437\) 3.52199 0.168480
\(438\) 0 0
\(439\) 26.1812 1.24956 0.624781 0.780800i \(-0.285188\pi\)
0.624781 + 0.780800i \(0.285188\pi\)
\(440\) 2.84184 0.135479
\(441\) 0 0
\(442\) −46.3038 −2.20245
\(443\) −21.5579 −1.02425 −0.512123 0.858912i \(-0.671141\pi\)
−0.512123 + 0.858912i \(0.671141\pi\)
\(444\) 0 0
\(445\) −21.9915 −1.04250
\(446\) −11.5841 −0.548525
\(447\) 0 0
\(448\) −3.56479 −0.168420
\(449\) −11.7990 −0.556829 −0.278415 0.960461i \(-0.589809\pi\)
−0.278415 + 0.960461i \(0.589809\pi\)
\(450\) 0 0
\(451\) 7.75013 0.364939
\(452\) 3.39061 0.159481
\(453\) 0 0
\(454\) −3.66122 −0.171829
\(455\) −16.3232 −0.765245
\(456\) 0 0
\(457\) −15.6863 −0.733773 −0.366886 0.930266i \(-0.619576\pi\)
−0.366886 + 0.930266i \(0.619576\pi\)
\(458\) 38.3410 1.79156
\(459\) 0 0
\(460\) 11.0443 0.514943
\(461\) −27.2451 −1.26893 −0.634466 0.772951i \(-0.718780\pi\)
−0.634466 + 0.772951i \(0.718780\pi\)
\(462\) 0 0
\(463\) −17.9364 −0.833573 −0.416787 0.909004i \(-0.636844\pi\)
−0.416787 + 0.909004i \(0.636844\pi\)
\(464\) −39.5977 −1.83828
\(465\) 0 0
\(466\) −32.1509 −1.48936
\(467\) −29.0100 −1.34242 −0.671211 0.741266i \(-0.734226\pi\)
−0.671211 + 0.741266i \(0.734226\pi\)
\(468\) 0 0
\(469\) 24.9122 1.15034
\(470\) 46.6933 2.15380
\(471\) 0 0
\(472\) −13.2350 −0.609191
\(473\) −7.75102 −0.356392
\(474\) 0 0
\(475\) −0.502638 −0.0230626
\(476\) 13.9475 0.639285
\(477\) 0 0
\(478\) 48.6277 2.22418
\(479\) −34.0896 −1.55759 −0.778797 0.627276i \(-0.784170\pi\)
−0.778797 + 0.627276i \(0.784170\pi\)
\(480\) 0 0
\(481\) −26.9733 −1.22988
\(482\) −23.9921 −1.09281
\(483\) 0 0
\(484\) 1.33679 0.0607633
\(485\) 25.7888 1.17101
\(486\) 0 0
\(487\) 6.38349 0.289264 0.144632 0.989486i \(-0.453800\pi\)
0.144632 + 0.989486i \(0.453800\pi\)
\(488\) −5.46570 −0.247421
\(489\) 0 0
\(490\) −17.7219 −0.800594
\(491\) 31.4552 1.41956 0.709778 0.704426i \(-0.248795\pi\)
0.709778 + 0.704426i \(0.248795\pi\)
\(492\) 0 0
\(493\) 49.9572 2.24996
\(494\) 7.51076 0.337925
\(495\) 0 0
\(496\) 11.2461 0.504966
\(497\) −5.28071 −0.236872
\(498\) 0 0
\(499\) −31.9667 −1.43103 −0.715514 0.698599i \(-0.753807\pi\)
−0.715514 + 0.698599i \(0.753807\pi\)
\(500\) 14.1029 0.630699
\(501\) 0 0
\(502\) −47.7927 −2.13309
\(503\) 34.0522 1.51831 0.759157 0.650907i \(-0.225611\pi\)
0.759157 + 0.650907i \(0.225611\pi\)
\(504\) 0 0
\(505\) 5.40983 0.240734
\(506\) −6.43359 −0.286008
\(507\) 0 0
\(508\) 1.43405 0.0636256
\(509\) −25.7904 −1.14314 −0.571569 0.820554i \(-0.693665\pi\)
−0.571569 + 0.820554i \(0.693665\pi\)
\(510\) 0 0
\(511\) 19.6024 0.867160
\(512\) −19.9389 −0.881184
\(513\) 0 0
\(514\) 23.0377 1.01615
\(515\) −14.3561 −0.632606
\(516\) 0 0
\(517\) −10.8969 −0.479247
\(518\) 20.2806 0.891076
\(519\) 0 0
\(520\) −11.6847 −0.512410
\(521\) −31.3774 −1.37467 −0.687334 0.726342i \(-0.741219\pi\)
−0.687334 + 0.726342i \(0.741219\pi\)
\(522\) 0 0
\(523\) −23.2388 −1.01616 −0.508082 0.861309i \(-0.669645\pi\)
−0.508082 + 0.861309i \(0.669645\pi\)
\(524\) 10.4632 0.457087
\(525\) 0 0
\(526\) −36.6343 −1.59733
\(527\) −14.1883 −0.618054
\(528\) 0 0
\(529\) −10.5956 −0.460677
\(530\) −34.0019 −1.47695
\(531\) 0 0
\(532\) −2.26238 −0.0980865
\(533\) −31.8661 −1.38027
\(534\) 0 0
\(535\) 0.992136 0.0428938
\(536\) 17.8330 0.770268
\(537\) 0 0
\(538\) −36.7439 −1.58414
\(539\) 4.13581 0.178142
\(540\) 0 0
\(541\) 36.0085 1.54812 0.774062 0.633109i \(-0.218222\pi\)
0.774062 + 0.633109i \(0.218222\pi\)
\(542\) −28.4207 −1.22077
\(543\) 0 0
\(544\) 40.0928 1.71896
\(545\) −32.8829 −1.40855
\(546\) 0 0
\(547\) −28.1855 −1.20513 −0.602563 0.798071i \(-0.705854\pi\)
−0.602563 + 0.798071i \(0.705854\pi\)
\(548\) 15.0353 0.642275
\(549\) 0 0
\(550\) 0.918163 0.0391506
\(551\) −8.10336 −0.345215
\(552\) 0 0
\(553\) 8.39738 0.357093
\(554\) −3.45629 −0.146844
\(555\) 0 0
\(556\) −1.21104 −0.0513594
\(557\) −21.6248 −0.916274 −0.458137 0.888882i \(-0.651483\pi\)
−0.458137 + 0.888882i \(0.651483\pi\)
\(558\) 0 0
\(559\) 31.8697 1.34795
\(560\) 19.3995 0.819779
\(561\) 0 0
\(562\) 8.38997 0.353909
\(563\) −13.1791 −0.555434 −0.277717 0.960663i \(-0.589578\pi\)
−0.277717 + 0.960663i \(0.589578\pi\)
\(564\) 0 0
\(565\) −5.94976 −0.250308
\(566\) 2.15179 0.0904467
\(567\) 0 0
\(568\) −3.78011 −0.158610
\(569\) −8.61143 −0.361010 −0.180505 0.983574i \(-0.557773\pi\)
−0.180505 + 0.983574i \(0.557773\pi\)
\(570\) 0 0
\(571\) −17.6546 −0.738821 −0.369410 0.929266i \(-0.620440\pi\)
−0.369410 + 0.929266i \(0.620440\pi\)
\(572\) −5.49647 −0.229819
\(573\) 0 0
\(574\) 23.9593 1.00004
\(575\) −1.77029 −0.0738261
\(576\) 0 0
\(577\) −31.0907 −1.29432 −0.647162 0.762352i \(-0.724044\pi\)
−0.647162 + 0.762352i \(0.724044\pi\)
\(578\) −38.3735 −1.59613
\(579\) 0 0
\(580\) −25.4106 −1.05512
\(581\) 3.09547 0.128422
\(582\) 0 0
\(583\) 7.93511 0.328639
\(584\) 14.0321 0.580652
\(585\) 0 0
\(586\) −20.5696 −0.849722
\(587\) 26.7211 1.10290 0.551450 0.834208i \(-0.314075\pi\)
0.551450 + 0.834208i \(0.314075\pi\)
\(588\) 0 0
\(589\) 2.30144 0.0948290
\(590\) −46.8124 −1.92724
\(591\) 0 0
\(592\) 32.0567 1.31752
\(593\) 24.2460 0.995666 0.497833 0.867273i \(-0.334129\pi\)
0.497833 + 0.867273i \(0.334129\pi\)
\(594\) 0 0
\(595\) −24.4748 −1.00337
\(596\) −14.0596 −0.575905
\(597\) 0 0
\(598\) 26.4529 1.08174
\(599\) 18.2095 0.744019 0.372009 0.928229i \(-0.378669\pi\)
0.372009 + 0.928229i \(0.378669\pi\)
\(600\) 0 0
\(601\) 37.9824 1.54934 0.774668 0.632368i \(-0.217917\pi\)
0.774668 + 0.632368i \(0.217917\pi\)
\(602\) −23.9621 −0.976622
\(603\) 0 0
\(604\) 17.7039 0.720362
\(605\) −2.34577 −0.0953691
\(606\) 0 0
\(607\) −20.0130 −0.812301 −0.406151 0.913806i \(-0.633129\pi\)
−0.406151 + 0.913806i \(0.633129\pi\)
\(608\) −6.50330 −0.263744
\(609\) 0 0
\(610\) −19.3322 −0.782739
\(611\) 44.8048 1.81261
\(612\) 0 0
\(613\) −42.1414 −1.70208 −0.851038 0.525105i \(-0.824026\pi\)
−0.851038 + 0.525105i \(0.824026\pi\)
\(614\) −7.95057 −0.320859
\(615\) 0 0
\(616\) −2.05029 −0.0826085
\(617\) −16.9044 −0.680545 −0.340273 0.940327i \(-0.610519\pi\)
−0.340273 + 0.940327i \(0.610519\pi\)
\(618\) 0 0
\(619\) 5.31177 0.213498 0.106749 0.994286i \(-0.465956\pi\)
0.106749 + 0.994286i \(0.465956\pi\)
\(620\) 7.21686 0.289836
\(621\) 0 0
\(622\) 13.9144 0.557915
\(623\) 15.8661 0.635662
\(624\) 0 0
\(625\) −27.2605 −1.09042
\(626\) 32.0865 1.28243
\(627\) 0 0
\(628\) 2.67245 0.106642
\(629\) −40.4433 −1.61258
\(630\) 0 0
\(631\) 34.0425 1.35521 0.677604 0.735427i \(-0.263018\pi\)
0.677604 + 0.735427i \(0.263018\pi\)
\(632\) 6.01113 0.239110
\(633\) 0 0
\(634\) −3.54728 −0.140880
\(635\) −2.51643 −0.0998615
\(636\) 0 0
\(637\) −17.0051 −0.673768
\(638\) 14.8023 0.586030
\(639\) 0 0
\(640\) 21.4847 0.849259
\(641\) 15.0405 0.594063 0.297031 0.954868i \(-0.404003\pi\)
0.297031 + 0.954868i \(0.404003\pi\)
\(642\) 0 0
\(643\) 32.3618 1.27622 0.638112 0.769943i \(-0.279715\pi\)
0.638112 + 0.769943i \(0.279715\pi\)
\(644\) −7.96808 −0.313986
\(645\) 0 0
\(646\) 11.2615 0.443079
\(647\) 49.0042 1.92655 0.963276 0.268512i \(-0.0865319\pi\)
0.963276 + 0.268512i \(0.0865319\pi\)
\(648\) 0 0
\(649\) 10.9247 0.428833
\(650\) −3.77520 −0.148075
\(651\) 0 0
\(652\) −25.0725 −0.981916
\(653\) −31.1992 −1.22092 −0.610460 0.792047i \(-0.709016\pi\)
−0.610460 + 0.792047i \(0.709016\pi\)
\(654\) 0 0
\(655\) −18.3605 −0.717406
\(656\) 37.8715 1.47864
\(657\) 0 0
\(658\) −33.6876 −1.31328
\(659\) 35.7237 1.39160 0.695799 0.718237i \(-0.255050\pi\)
0.695799 + 0.718237i \(0.255050\pi\)
\(660\) 0 0
\(661\) 33.8677 1.31730 0.658651 0.752449i \(-0.271128\pi\)
0.658651 + 0.752449i \(0.271128\pi\)
\(662\) −24.4754 −0.951262
\(663\) 0 0
\(664\) 2.21585 0.0859916
\(665\) 3.96996 0.153949
\(666\) 0 0
\(667\) −28.5400 −1.10507
\(668\) 8.43788 0.326471
\(669\) 0 0
\(670\) 63.0755 2.43682
\(671\) 4.51162 0.174169
\(672\) 0 0
\(673\) 21.1648 0.815842 0.407921 0.913017i \(-0.366254\pi\)
0.407921 + 0.913017i \(0.366254\pi\)
\(674\) −60.5786 −2.33340
\(675\) 0 0
\(676\) 5.22141 0.200824
\(677\) 34.5239 1.32686 0.663431 0.748238i \(-0.269100\pi\)
0.663431 + 0.748238i \(0.269100\pi\)
\(678\) 0 0
\(679\) −18.6057 −0.714022
\(680\) −17.5199 −0.671858
\(681\) 0 0
\(682\) −4.20401 −0.160980
\(683\) −12.8022 −0.489862 −0.244931 0.969540i \(-0.578765\pi\)
−0.244931 + 0.969540i \(0.578765\pi\)
\(684\) 0 0
\(685\) −26.3835 −1.00806
\(686\) 34.4261 1.31439
\(687\) 0 0
\(688\) −37.8759 −1.44401
\(689\) −32.6267 −1.24298
\(690\) 0 0
\(691\) 39.5406 1.50419 0.752097 0.659052i \(-0.229042\pi\)
0.752097 + 0.659052i \(0.229042\pi\)
\(692\) 28.7769 1.09394
\(693\) 0 0
\(694\) 32.9294 1.24998
\(695\) 2.12510 0.0806094
\(696\) 0 0
\(697\) −47.7795 −1.80978
\(698\) −16.1006 −0.609418
\(699\) 0 0
\(700\) 1.13716 0.0429805
\(701\) −22.4546 −0.848097 −0.424049 0.905639i \(-0.639391\pi\)
−0.424049 + 0.905639i \(0.639391\pi\)
\(702\) 0 0
\(703\) 6.56016 0.247421
\(704\) 2.10636 0.0793865
\(705\) 0 0
\(706\) −2.90899 −0.109481
\(707\) −3.90301 −0.146788
\(708\) 0 0
\(709\) −1.41244 −0.0530451 −0.0265226 0.999648i \(-0.508443\pi\)
−0.0265226 + 0.999648i \(0.508443\pi\)
\(710\) −13.3703 −0.501778
\(711\) 0 0
\(712\) 11.3575 0.425641
\(713\) 8.10564 0.303559
\(714\) 0 0
\(715\) 9.64506 0.360705
\(716\) 10.6545 0.398177
\(717\) 0 0
\(718\) 66.4502 2.47990
\(719\) 39.0879 1.45773 0.728867 0.684655i \(-0.240047\pi\)
0.728867 + 0.684655i \(0.240047\pi\)
\(720\) 0 0
\(721\) 10.3574 0.385731
\(722\) −1.82669 −0.0679823
\(723\) 0 0
\(724\) −1.56344 −0.0581049
\(725\) 4.07306 0.151270
\(726\) 0 0
\(727\) 9.42098 0.349405 0.174702 0.984621i \(-0.444104\pi\)
0.174702 + 0.984621i \(0.444104\pi\)
\(728\) 8.43013 0.312442
\(729\) 0 0
\(730\) 49.6316 1.83695
\(731\) 47.7850 1.76739
\(732\) 0 0
\(733\) 4.63361 0.171146 0.0855731 0.996332i \(-0.472728\pi\)
0.0855731 + 0.996332i \(0.472728\pi\)
\(734\) 9.13849 0.337308
\(735\) 0 0
\(736\) −22.9046 −0.844274
\(737\) −14.7201 −0.542222
\(738\) 0 0
\(739\) −13.1654 −0.484298 −0.242149 0.970239i \(-0.577852\pi\)
−0.242149 + 0.970239i \(0.577852\pi\)
\(740\) 20.5714 0.756219
\(741\) 0 0
\(742\) 24.5312 0.900568
\(743\) 14.7689 0.541817 0.270909 0.962605i \(-0.412676\pi\)
0.270909 + 0.962605i \(0.412676\pi\)
\(744\) 0 0
\(745\) 24.6715 0.903894
\(746\) 62.3472 2.28269
\(747\) 0 0
\(748\) −8.24132 −0.301332
\(749\) −0.715792 −0.0261545
\(750\) 0 0
\(751\) 35.0415 1.27868 0.639341 0.768924i \(-0.279207\pi\)
0.639341 + 0.768924i \(0.279207\pi\)
\(752\) −53.2487 −1.94178
\(753\) 0 0
\(754\) −60.8625 −2.21648
\(755\) −31.0664 −1.13062
\(756\) 0 0
\(757\) −23.8005 −0.865044 −0.432522 0.901623i \(-0.642376\pi\)
−0.432522 + 0.901623i \(0.642376\pi\)
\(758\) −28.8158 −1.04664
\(759\) 0 0
\(760\) 2.84184 0.103084
\(761\) 39.0282 1.41477 0.707386 0.706827i \(-0.249874\pi\)
0.707386 + 0.706827i \(0.249874\pi\)
\(762\) 0 0
\(763\) 23.7239 0.858862
\(764\) 20.9440 0.757726
\(765\) 0 0
\(766\) −40.9654 −1.48014
\(767\) −44.9190 −1.62193
\(768\) 0 0
\(769\) 9.84757 0.355113 0.177556 0.984111i \(-0.443181\pi\)
0.177556 + 0.984111i \(0.443181\pi\)
\(770\) −7.25189 −0.261340
\(771\) 0 0
\(772\) 28.9776 1.04292
\(773\) 42.9837 1.54602 0.773008 0.634396i \(-0.218751\pi\)
0.773008 + 0.634396i \(0.218751\pi\)
\(774\) 0 0
\(775\) −1.15679 −0.0415531
\(776\) −13.3186 −0.478111
\(777\) 0 0
\(778\) −33.5175 −1.20166
\(779\) 7.75013 0.277677
\(780\) 0 0
\(781\) 3.12026 0.111652
\(782\) 39.6630 1.41835
\(783\) 0 0
\(784\) 20.2099 0.721783
\(785\) −4.68954 −0.167377
\(786\) 0 0
\(787\) −25.8709 −0.922199 −0.461099 0.887349i \(-0.652545\pi\)
−0.461099 + 0.887349i \(0.652545\pi\)
\(788\) −6.13313 −0.218484
\(789\) 0 0
\(790\) 21.2614 0.756448
\(791\) 4.29254 0.152625
\(792\) 0 0
\(793\) −18.5503 −0.658741
\(794\) 0.934320 0.0331578
\(795\) 0 0
\(796\) 17.4611 0.618893
\(797\) −3.62989 −0.128577 −0.0642886 0.997931i \(-0.520478\pi\)
−0.0642886 + 0.997931i \(0.520478\pi\)
\(798\) 0 0
\(799\) 67.1796 2.37664
\(800\) 3.26880 0.115570
\(801\) 0 0
\(802\) 34.6749 1.22441
\(803\) −11.5827 −0.408743
\(804\) 0 0
\(805\) 13.9822 0.492807
\(806\) 17.2855 0.608857
\(807\) 0 0
\(808\) −2.79391 −0.0982894
\(809\) 13.1327 0.461720 0.230860 0.972987i \(-0.425846\pi\)
0.230860 + 0.972987i \(0.425846\pi\)
\(810\) 0 0
\(811\) 9.40230 0.330160 0.165080 0.986280i \(-0.447212\pi\)
0.165080 + 0.986280i \(0.447212\pi\)
\(812\) 18.3329 0.643358
\(813\) 0 0
\(814\) −11.9834 −0.420017
\(815\) 43.9966 1.54114
\(816\) 0 0
\(817\) −7.75102 −0.271174
\(818\) −2.01098 −0.0703124
\(819\) 0 0
\(820\) 24.3029 0.848695
\(821\) 21.1701 0.738843 0.369422 0.929262i \(-0.379556\pi\)
0.369422 + 0.929262i \(0.379556\pi\)
\(822\) 0 0
\(823\) 25.2277 0.879384 0.439692 0.898149i \(-0.355088\pi\)
0.439692 + 0.898149i \(0.355088\pi\)
\(824\) 7.41422 0.258286
\(825\) 0 0
\(826\) 33.7735 1.17513
\(827\) 39.4460 1.37167 0.685836 0.727756i \(-0.259437\pi\)
0.685836 + 0.727756i \(0.259437\pi\)
\(828\) 0 0
\(829\) −36.5068 −1.26793 −0.633967 0.773360i \(-0.718574\pi\)
−0.633967 + 0.773360i \(0.718574\pi\)
\(830\) 7.83748 0.272043
\(831\) 0 0
\(832\) −8.66069 −0.300255
\(833\) −25.4972 −0.883427
\(834\) 0 0
\(835\) −14.8066 −0.512403
\(836\) 1.33679 0.0462339
\(837\) 0 0
\(838\) −34.3288 −1.18587
\(839\) −42.4670 −1.46612 −0.733061 0.680163i \(-0.761909\pi\)
−0.733061 + 0.680163i \(0.761909\pi\)
\(840\) 0 0
\(841\) 36.6645 1.26429
\(842\) 45.2321 1.55880
\(843\) 0 0
\(844\) 3.89258 0.133988
\(845\) −9.16241 −0.315196
\(846\) 0 0
\(847\) 1.69239 0.0581513
\(848\) 38.7755 1.33156
\(849\) 0 0
\(850\) −5.66047 −0.194153
\(851\) 23.1048 0.792023
\(852\) 0 0
\(853\) 21.5088 0.736446 0.368223 0.929738i \(-0.379966\pi\)
0.368223 + 0.929738i \(0.379966\pi\)
\(854\) 13.9475 0.477275
\(855\) 0 0
\(856\) −0.512389 −0.0175131
\(857\) 12.6370 0.431670 0.215835 0.976430i \(-0.430753\pi\)
0.215835 + 0.976430i \(0.430753\pi\)
\(858\) 0 0
\(859\) −3.39720 −0.115911 −0.0579555 0.998319i \(-0.518458\pi\)
−0.0579555 + 0.998319i \(0.518458\pi\)
\(860\) −24.3057 −0.828818
\(861\) 0 0
\(862\) −58.8167 −2.00330
\(863\) 10.9308 0.372088 0.186044 0.982541i \(-0.440433\pi\)
0.186044 + 0.982541i \(0.440433\pi\)
\(864\) 0 0
\(865\) −50.4971 −1.71695
\(866\) 27.9561 0.949987
\(867\) 0 0
\(868\) −5.20672 −0.176727
\(869\) −4.96184 −0.168319
\(870\) 0 0
\(871\) 60.5243 2.05079
\(872\) 16.9824 0.575096
\(873\) 0 0
\(874\) −6.43359 −0.217619
\(875\) 17.8544 0.603588
\(876\) 0 0
\(877\) 31.7230 1.07121 0.535605 0.844468i \(-0.320083\pi\)
0.535605 + 0.844468i \(0.320083\pi\)
\(878\) −47.8250 −1.61402
\(879\) 0 0
\(880\) −11.4628 −0.386410
\(881\) 6.45839 0.217589 0.108794 0.994064i \(-0.465301\pi\)
0.108794 + 0.994064i \(0.465301\pi\)
\(882\) 0 0
\(883\) −48.5242 −1.63297 −0.816485 0.577367i \(-0.804080\pi\)
−0.816485 + 0.577367i \(0.804080\pi\)
\(884\) 33.8857 1.13970
\(885\) 0 0
\(886\) 39.3796 1.32298
\(887\) 44.0097 1.47770 0.738851 0.673869i \(-0.235369\pi\)
0.738851 + 0.673869i \(0.235369\pi\)
\(888\) 0 0
\(889\) 1.81552 0.0608905
\(890\) 40.1716 1.34656
\(891\) 0 0
\(892\) 8.47742 0.283845
\(893\) −10.8969 −0.364652
\(894\) 0 0
\(895\) −18.6962 −0.624946
\(896\) −15.5005 −0.517835
\(897\) 0 0
\(898\) 21.5531 0.719236
\(899\) −18.6494 −0.621991
\(900\) 0 0
\(901\) −48.9199 −1.62976
\(902\) −14.1571 −0.471379
\(903\) 0 0
\(904\) 3.07275 0.102198
\(905\) 2.74349 0.0911966
\(906\) 0 0
\(907\) −2.05084 −0.0680971 −0.0340485 0.999420i \(-0.510840\pi\)
−0.0340485 + 0.999420i \(0.510840\pi\)
\(908\) 2.67932 0.0889164
\(909\) 0 0
\(910\) 29.8175 0.988439
\(911\) 16.8417 0.557990 0.278995 0.960293i \(-0.409999\pi\)
0.278995 + 0.960293i \(0.409999\pi\)
\(912\) 0 0
\(913\) −1.82905 −0.0605328
\(914\) 28.6539 0.947788
\(915\) 0 0
\(916\) −28.0584 −0.927075
\(917\) 13.2465 0.437438
\(918\) 0 0
\(919\) 8.01221 0.264299 0.132149 0.991230i \(-0.457812\pi\)
0.132149 + 0.991230i \(0.457812\pi\)
\(920\) 10.0089 0.329985
\(921\) 0 0
\(922\) 49.7683 1.63903
\(923\) −12.8295 −0.422289
\(924\) 0 0
\(925\) −3.29738 −0.108417
\(926\) 32.7641 1.07670
\(927\) 0 0
\(928\) 52.6986 1.72992
\(929\) −16.7897 −0.550853 −0.275426 0.961322i \(-0.588819\pi\)
−0.275426 + 0.961322i \(0.588819\pi\)
\(930\) 0 0
\(931\) 4.13581 0.135546
\(932\) 23.5284 0.770698
\(933\) 0 0
\(934\) 52.9922 1.73396
\(935\) 14.4617 0.472947
\(936\) 0 0
\(937\) 21.8471 0.713713 0.356856 0.934159i \(-0.383849\pi\)
0.356856 + 0.934159i \(0.383849\pi\)
\(938\) −45.5068 −1.48585
\(939\) 0 0
\(940\) −34.1707 −1.11453
\(941\) −17.8421 −0.581635 −0.290817 0.956779i \(-0.593927\pi\)
−0.290817 + 0.956779i \(0.593927\pi\)
\(942\) 0 0
\(943\) 27.2959 0.888877
\(944\) 53.3845 1.73752
\(945\) 0 0
\(946\) 14.1587 0.460339
\(947\) −41.9736 −1.36396 −0.681980 0.731371i \(-0.738881\pi\)
−0.681980 + 0.731371i \(0.738881\pi\)
\(948\) 0 0
\(949\) 47.6242 1.54595
\(950\) 0.918163 0.0297892
\(951\) 0 0
\(952\) 12.6400 0.409665
\(953\) 8.06945 0.261395 0.130697 0.991422i \(-0.458278\pi\)
0.130697 + 0.991422i \(0.458278\pi\)
\(954\) 0 0
\(955\) −36.7520 −1.18927
\(956\) −35.5864 −1.15094
\(957\) 0 0
\(958\) 62.2711 2.01189
\(959\) 19.0348 0.614666
\(960\) 0 0
\(961\) −25.7034 −0.829142
\(962\) 49.2718 1.58859
\(963\) 0 0
\(964\) 17.5577 0.565496
\(965\) −50.8491 −1.63689
\(966\) 0 0
\(967\) −3.86360 −0.124245 −0.0621225 0.998069i \(-0.519787\pi\)
−0.0621225 + 0.998069i \(0.519787\pi\)
\(968\) 1.21147 0.0389382
\(969\) 0 0
\(970\) −47.1081 −1.51255
\(971\) −42.7787 −1.37283 −0.686417 0.727208i \(-0.740818\pi\)
−0.686417 + 0.727208i \(0.740818\pi\)
\(972\) 0 0
\(973\) −1.53318 −0.0491516
\(974\) −11.6607 −0.373631
\(975\) 0 0
\(976\) 22.0463 0.705686
\(977\) −54.5146 −1.74408 −0.872039 0.489437i \(-0.837202\pi\)
−0.872039 + 0.489437i \(0.837202\pi\)
\(978\) 0 0
\(979\) −9.37496 −0.299625
\(980\) 12.9691 0.414283
\(981\) 0 0
\(982\) −57.4590 −1.83359
\(983\) −33.5422 −1.06983 −0.534916 0.844905i \(-0.679657\pi\)
−0.534916 + 0.844905i \(0.679657\pi\)
\(984\) 0 0
\(985\) 10.7623 0.342914
\(986\) −91.2562 −2.90619
\(987\) 0 0
\(988\) −5.49647 −0.174866
\(989\) −27.2991 −0.868060
\(990\) 0 0
\(991\) 31.7767 1.00942 0.504710 0.863289i \(-0.331600\pi\)
0.504710 + 0.863289i \(0.331600\pi\)
\(992\) −14.9669 −0.475200
\(993\) 0 0
\(994\) 9.64621 0.305959
\(995\) −30.6403 −0.971363
\(996\) 0 0
\(997\) −47.5668 −1.50646 −0.753228 0.657759i \(-0.771504\pi\)
−0.753228 + 0.657759i \(0.771504\pi\)
\(998\) 58.3933 1.84841
\(999\) 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 1881.2.a.k.1.2 5
3.2 odd 2 209.2.a.c.1.4 5
12.11 even 2 3344.2.a.t.1.4 5
15.14 odd 2 5225.2.a.h.1.2 5
33.32 even 2 2299.2.a.n.1.2 5
57.56 even 2 3971.2.a.h.1.2 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
209.2.a.c.1.4 5 3.2 odd 2
1881.2.a.k.1.2 5 1.1 even 1 trivial
2299.2.a.n.1.2 5 33.32 even 2
3344.2.a.t.1.4 5 12.11 even 2
3971.2.a.h.1.2 5 57.56 even 2
5225.2.a.h.1.2 5 15.14 odd 2