Properties

Label 3344.2.a.t.1.4
Level $3344$
Weight $2$
Character 3344.1
Self dual yes
Analytic conductor $26.702$
Analytic rank $1$
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] = [3344,2,Mod(1,3344)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3344, 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("3344.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3344 = 2^{4} \cdot 11 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3344.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: \(26.7019744359\)
Analytic rank: \(1\)
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: \(\Z[a_1, a_2, a_3]\)
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 1.4
Root \(-1.51908\) of defining polynomial
Character \(\chi\) \(=\) 3344.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+0.563416 q^{3} +2.34577 q^{5} -1.69239 q^{7} -2.68256 q^{9} +O(q^{10})\) \(q+0.563416 q^{3} +2.34577 q^{5} -1.69239 q^{7} -2.68256 q^{9} -1.00000 q^{11} +4.11168 q^{13} +1.32164 q^{15} -6.16499 q^{17} +1.00000 q^{19} -0.953520 q^{21} -3.52199 q^{23} +0.502638 q^{25} -3.20164 q^{27} -8.10336 q^{29} +2.30144 q^{31} -0.563416 q^{33} -3.96996 q^{35} -6.56016 q^{37} +2.31659 q^{39} +7.75013 q^{41} -7.75102 q^{43} -6.29268 q^{45} +10.8969 q^{47} -4.13581 q^{49} -3.47345 q^{51} +7.93511 q^{53} -2.34577 q^{55} +0.563416 q^{57} -10.9247 q^{59} -4.51162 q^{61} +4.53995 q^{63} +9.64506 q^{65} -14.7201 q^{67} -1.98435 q^{69} -3.12026 q^{71} +11.5827 q^{73} +0.283194 q^{75} +1.69239 q^{77} -4.96184 q^{79} +6.24383 q^{81} +1.82905 q^{83} -14.4617 q^{85} -4.56556 q^{87} -9.37496 q^{89} -6.95858 q^{91} +1.29666 q^{93} +2.34577 q^{95} -10.9937 q^{97} +2.68256 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q - q^{3} - 5 q^{5} - 6 q^{7} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 5 q - q^{3} - 5 q^{5} - 6 q^{7} + 4 q^{9} - 5 q^{11} + 4 q^{13} - 3 q^{15} - 4 q^{17} + 5 q^{19} + 10 q^{21} - 3 q^{23} + 6 q^{25} + 11 q^{27} + 10 q^{29} - 11 q^{31} + q^{33} + 8 q^{35} + q^{37} - 2 q^{39} + 2 q^{41} - 20 q^{43} - 28 q^{45} + 20 q^{47} + 3 q^{49} - 24 q^{51} - 14 q^{53} + 5 q^{55} - q^{57} - 3 q^{59} - 10 q^{61} - 24 q^{63} - 9 q^{67} - 5 q^{69} - 23 q^{71} + 18 q^{75} + 6 q^{77} - 44 q^{79} + q^{81} + 14 q^{83} - 12 q^{85} - 28 q^{87} - 27 q^{89} - 24 q^{91} - 27 q^{93} - 5 q^{95} + 15 q^{97} - 4 q^{99}+O(q^{100}) \) 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
\(3\) 0.563416 0.325288 0.162644 0.986685i \(-0.447998\pi\)
0.162644 + 0.986685i \(0.447998\pi\)
\(4\) 0 0
\(5\) 2.34577 1.04906 0.524530 0.851392i \(-0.324241\pi\)
0.524530 + 0.851392i \(0.324241\pi\)
\(6\) 0 0
\(7\) −1.69239 −0.639664 −0.319832 0.947474i \(-0.603626\pi\)
−0.319832 + 0.947474i \(0.603626\pi\)
\(8\) 0 0
\(9\) −2.68256 −0.894188
\(10\) 0 0
\(11\) −1.00000 −0.301511
\(12\) 0 0
\(13\) 4.11168 1.14038 0.570188 0.821514i \(-0.306870\pi\)
0.570188 + 0.821514i \(0.306870\pi\)
\(14\) 0 0
\(15\) 1.32164 0.341247
\(16\) 0 0
\(17\) −6.16499 −1.49523 −0.747615 0.664132i \(-0.768801\pi\)
−0.747615 + 0.664132i \(0.768801\pi\)
\(18\) 0 0
\(19\) 1.00000 0.229416
\(20\) 0 0
\(21\) −0.953520 −0.208075
\(22\) 0 0
\(23\) −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
\(26\) 0 0
\(27\) −3.20164 −0.616157
\(28\) 0 0
\(29\) −8.10336 −1.50476 −0.752378 0.658731i \(-0.771094\pi\)
−0.752378 + 0.658731i \(0.771094\pi\)
\(30\) 0 0
\(31\) 2.30144 0.413350 0.206675 0.978410i \(-0.433736\pi\)
0.206675 + 0.978410i \(0.433736\pi\)
\(32\) 0 0
\(33\) −0.563416 −0.0980781
\(34\) 0 0
\(35\) −3.96996 −0.671046
\(36\) 0 0
\(37\) −6.56016 −1.07848 −0.539242 0.842151i \(-0.681289\pi\)
−0.539242 + 0.842151i \(0.681289\pi\)
\(38\) 0 0
\(39\) 2.31659 0.370951
\(40\) 0 0
\(41\) 7.75013 1.21037 0.605183 0.796086i \(-0.293100\pi\)
0.605183 + 0.796086i \(0.293100\pi\)
\(42\) 0 0
\(43\) −7.75102 −1.18202 −0.591010 0.806664i \(-0.701271\pi\)
−0.591010 + 0.806664i \(0.701271\pi\)
\(44\) 0 0
\(45\) −6.29268 −0.938057
\(46\) 0 0
\(47\) 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 0
\(51\) −3.47345 −0.486381
\(52\) 0 0
\(53\) 7.93511 1.08997 0.544986 0.838445i \(-0.316535\pi\)
0.544986 + 0.838445i \(0.316535\pi\)
\(54\) 0 0
\(55\) −2.34577 −0.316304
\(56\) 0 0
\(57\) 0.563416 0.0746262
\(58\) 0 0
\(59\) −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 \(0.593265\pi\)
\(62\) 0 0
\(63\) 4.53995 0.571980
\(64\) 0 0
\(65\) 9.64506 1.19632
\(66\) 0 0
\(67\) −14.7201 −1.79835 −0.899173 0.437594i \(-0.855831\pi\)
−0.899173 + 0.437594i \(0.855831\pi\)
\(68\) 0 0
\(69\) −1.98435 −0.238887
\(70\) 0 0
\(71\) −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 \(0.262923\pi\)
\(74\) 0 0
\(75\) 0.283194 0.0327004
\(76\) 0 0
\(77\) 1.69239 0.192866
\(78\) 0 0
\(79\) −4.96184 −0.558250 −0.279125 0.960255i \(-0.590044\pi\)
−0.279125 + 0.960255i \(0.590044\pi\)
\(80\) 0 0
\(81\) 6.24383 0.693759
\(82\) 0 0
\(83\) 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
\(86\) 0 0
\(87\) −4.56556 −0.489480
\(88\) 0 0
\(89\) −9.37496 −0.993743 −0.496872 0.867824i \(-0.665518\pi\)
−0.496872 + 0.867824i \(0.665518\pi\)
\(90\) 0 0
\(91\) −6.95858 −0.729457
\(92\) 0 0
\(93\) 1.29666 0.134458
\(94\) 0 0
\(95\) 2.34577 0.240671
\(96\) 0 0
\(97\) −10.9937 −1.11625 −0.558123 0.829758i \(-0.688478\pi\)
−0.558123 + 0.829758i \(0.688478\pi\)
\(98\) 0 0
\(99\) 2.68256 0.269608
\(100\) 0 0
\(101\) 2.30621 0.229476 0.114738 0.993396i \(-0.463397\pi\)
0.114738 + 0.993396i \(0.463397\pi\)
\(102\) 0 0
\(103\) −6.12000 −0.603021 −0.301511 0.953463i \(-0.597491\pi\)
−0.301511 + 0.953463i \(0.597491\pi\)
\(104\) 0 0
\(105\) −2.23674 −0.218283
\(106\) 0 0
\(107\) −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 \(0.265720\pi\)
\(110\) 0 0
\(111\) −3.69609 −0.350818
\(112\) 0 0
\(113\) −2.53638 −0.238602 −0.119301 0.992858i \(-0.538065\pi\)
−0.119301 + 0.992858i \(0.538065\pi\)
\(114\) 0 0
\(115\) −8.26179 −0.770416
\(116\) 0 0
\(117\) −11.0298 −1.01971
\(118\) 0 0
\(119\) 10.4336 0.956445
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 0 0
\(123\) 4.36654 0.393718
\(124\) 0 0
\(125\) −10.5498 −0.943601
\(126\) 0 0
\(127\) −1.07275 −0.0951914 −0.0475957 0.998867i \(-0.515156\pi\)
−0.0475957 + 0.998867i \(0.515156\pi\)
\(128\) 0 0
\(129\) −4.36705 −0.384497
\(130\) 0 0
\(131\) 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
\(134\) 0 0
\(135\) −7.51032 −0.646386
\(136\) 0 0
\(137\) −11.2473 −0.960919 −0.480460 0.877017i \(-0.659530\pi\)
−0.480460 + 0.877017i \(0.659530\pi\)
\(138\) 0 0
\(139\) 0.905926 0.0768397 0.0384198 0.999262i \(-0.487768\pi\)
0.0384198 + 0.999262i \(0.487768\pi\)
\(140\) 0 0
\(141\) 6.13951 0.517040
\(142\) 0 0
\(143\) −4.11168 −0.343836
\(144\) 0 0
\(145\) −19.0086 −1.57858
\(146\) 0 0
\(147\) −2.33018 −0.192190
\(148\) 0 0
\(149\) 10.5174 0.861622 0.430811 0.902442i \(-0.358228\pi\)
0.430811 + 0.902442i \(0.358228\pi\)
\(150\) 0 0
\(151\) −13.2436 −1.07775 −0.538873 0.842387i \(-0.681150\pi\)
−0.538873 + 0.842387i \(0.681150\pi\)
\(152\) 0 0
\(153\) 16.5380 1.33702
\(154\) 0 0
\(155\) 5.39864 0.433629
\(156\) 0 0
\(157\) 1.99915 0.159549 0.0797747 0.996813i \(-0.474580\pi\)
0.0797747 + 0.996813i \(0.474580\pi\)
\(158\) 0 0
\(159\) 4.47076 0.354555
\(160\) 0 0
\(161\) 5.96059 0.469761
\(162\) 0 0
\(163\) 18.7557 1.46906 0.734531 0.678575i \(-0.237402\pi\)
0.734531 + 0.678575i \(0.237402\pi\)
\(164\) 0 0
\(165\) −1.32164 −0.102890
\(166\) 0 0
\(167\) 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
\(170\) 0 0
\(171\) −2.68256 −0.205141
\(172\) 0 0
\(173\) −21.5269 −1.63666 −0.818328 0.574751i \(-0.805099\pi\)
−0.818328 + 0.574751i \(0.805099\pi\)
\(174\) 0 0
\(175\) −0.850661 −0.0643039
\(176\) 0 0
\(177\) −6.15516 −0.462650
\(178\) 0 0
\(179\) 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 \(0.513840\pi\)
\(182\) 0 0
\(183\) −2.54191 −0.187904
\(184\) 0 0
\(185\) −15.3886 −1.13139
\(186\) 0 0
\(187\) 6.16499 0.450829
\(188\) 0 0
\(189\) 5.41844 0.394133
\(190\) 0 0
\(191\) 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 \(0.215133\pi\)
\(194\) 0 0
\(195\) 5.43418 0.389149
\(196\) 0 0
\(197\) 4.58794 0.326877 0.163439 0.986554i \(-0.447741\pi\)
0.163439 + 0.986554i \(0.447741\pi\)
\(198\) 0 0
\(199\) −13.0619 −0.925937 −0.462968 0.886375i \(-0.653216\pi\)
−0.462968 + 0.886375i \(0.653216\pi\)
\(200\) 0 0
\(201\) −8.29353 −0.584980
\(202\) 0 0
\(203\) 13.7141 0.962539
\(204\) 0 0
\(205\) 18.1800 1.26975
\(206\) 0 0
\(207\) 9.44797 0.656679
\(208\) 0 0
\(209\) −1.00000 −0.0691714
\(210\) 0 0
\(211\) −2.91188 −0.200462 −0.100231 0.994964i \(-0.531958\pi\)
−0.100231 + 0.994964i \(0.531958\pi\)
\(212\) 0 0
\(213\) −1.75800 −0.120456
\(214\) 0 0
\(215\) −18.1821 −1.24001
\(216\) 0 0
\(217\) −3.89493 −0.264405
\(218\) 0 0
\(219\) 6.52585 0.440976
\(220\) 0 0
\(221\) −25.3485 −1.70512
\(222\) 0 0
\(223\) −6.34161 −0.424665 −0.212333 0.977197i \(-0.568106\pi\)
−0.212333 + 0.977197i \(0.568106\pi\)
\(224\) 0 0
\(225\) −1.34836 −0.0898905
\(226\) 0 0
\(227\) 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 \(0.743935\pi\)
\(230\) 0 0
\(231\) 0.953520 0.0627370
\(232\) 0 0
\(233\) −17.6006 −1.15305 −0.576527 0.817078i \(-0.695593\pi\)
−0.576527 + 0.817078i \(0.695593\pi\)
\(234\) 0 0
\(235\) 25.5617 1.66746
\(236\) 0 0
\(237\) −2.79558 −0.181592
\(238\) 0 0
\(239\) −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 \(0.360968\pi\)
\(242\) 0 0
\(243\) 13.1228 0.841828
\(244\) 0 0
\(245\) −9.70166 −0.619816
\(246\) 0 0
\(247\) 4.11168 0.261620
\(248\) 0 0
\(249\) 1.03052 0.0653063
\(250\) 0 0
\(251\) 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
\(254\) 0 0
\(255\) −8.14792 −0.510243
\(256\) 0 0
\(257\) 12.6117 0.786697 0.393349 0.919389i \(-0.371317\pi\)
0.393349 + 0.919389i \(0.371317\pi\)
\(258\) 0 0
\(259\) 11.1024 0.689867
\(260\) 0 0
\(261\) 21.7378 1.34554
\(262\) 0 0
\(263\) 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
\(266\) 0 0
\(267\) −5.28200 −0.323253
\(268\) 0 0
\(269\) −20.1150 −1.22644 −0.613218 0.789914i \(-0.710125\pi\)
−0.613218 + 0.789914i \(0.710125\pi\)
\(270\) 0 0
\(271\) −15.5586 −0.945117 −0.472558 0.881299i \(-0.656669\pi\)
−0.472558 + 0.881299i \(0.656669\pi\)
\(272\) 0 0
\(273\) −3.92057 −0.237284
\(274\) 0 0
\(275\) −0.502638 −0.0303102
\(276\) 0 0
\(277\) 1.89211 0.113686 0.0568429 0.998383i \(-0.481897\pi\)
0.0568429 + 0.998383i \(0.481897\pi\)
\(278\) 0 0
\(279\) −6.17375 −0.369613
\(280\) 0 0
\(281\) 4.59299 0.273995 0.136997 0.990571i \(-0.456255\pi\)
0.136997 + 0.990571i \(0.456255\pi\)
\(282\) 0 0
\(283\) 1.17798 0.0700234 0.0350117 0.999387i \(-0.488853\pi\)
0.0350117 + 0.999387i \(0.488853\pi\)
\(284\) 0 0
\(285\) 1.32164 0.0782874
\(286\) 0 0
\(287\) −13.1163 −0.774228
\(288\) 0 0
\(289\) 21.0071 1.23571
\(290\) 0 0
\(291\) −6.19405 −0.363101
\(292\) 0 0
\(293\) −11.2606 −0.657851 −0.328925 0.944356i \(-0.606686\pi\)
−0.328925 + 0.944356i \(0.606686\pi\)
\(294\) 0 0
\(295\) −25.6269 −1.49206
\(296\) 0 0
\(297\) 3.20164 0.185778
\(298\) 0 0
\(299\) −14.4813 −0.837476
\(300\) 0 0
\(301\) 13.1178 0.756096
\(302\) 0 0
\(303\) 1.29935 0.0746459
\(304\) 0 0
\(305\) −10.5832 −0.605993
\(306\) 0 0
\(307\) −4.35245 −0.248407 −0.124204 0.992257i \(-0.539638\pi\)
−0.124204 + 0.992257i \(0.539638\pi\)
\(308\) 0 0
\(309\) −3.44810 −0.196156
\(310\) 0 0
\(311\) −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 \(0.665355\pi\)
\(314\) 0 0
\(315\) 10.6497 0.600041
\(316\) 0 0
\(317\) −1.94192 −0.109069 −0.0545345 0.998512i \(-0.517367\pi\)
−0.0545345 + 0.998512i \(0.517367\pi\)
\(318\) 0 0
\(319\) 8.10336 0.453701
\(320\) 0 0
\(321\) −0.238295 −0.0133003
\(322\) 0 0
\(323\) −6.16499 −0.343029
\(324\) 0 0
\(325\) 2.06669 0.114639
\(326\) 0 0
\(327\) 7.89793 0.436757
\(328\) 0 0
\(329\) −18.4419 −1.01674
\(330\) 0 0
\(331\) −13.3988 −0.736462 −0.368231 0.929734i \(-0.620036\pi\)
−0.368231 + 0.929734i \(0.620036\pi\)
\(332\) 0 0
\(333\) 17.5980 0.964366
\(334\) 0 0
\(335\) −34.5300 −1.88657
\(336\) 0 0
\(337\) 33.1631 1.80651 0.903254 0.429107i \(-0.141172\pi\)
0.903254 + 0.429107i \(0.141172\pi\)
\(338\) 0 0
\(339\) −1.42903 −0.0776145
\(340\) 0 0
\(341\) −2.30144 −0.124630
\(342\) 0 0
\(343\) 18.8462 1.01760
\(344\) 0 0
\(345\) −4.65482 −0.250607
\(346\) 0 0
\(347\) −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 \(0.424195\pi\)
\(350\) 0 0
\(351\) −13.1641 −0.702650
\(352\) 0 0
\(353\) −1.59249 −0.0847599 −0.0423799 0.999102i \(-0.513494\pi\)
−0.0423799 + 0.999102i \(0.513494\pi\)
\(354\) 0 0
\(355\) −7.31942 −0.388474
\(356\) 0 0
\(357\) 5.87844 0.311120
\(358\) 0 0
\(359\) −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
\(362\) 0 0
\(363\) 0.563416 0.0295716
\(364\) 0 0
\(365\) 27.1703 1.42216
\(366\) 0 0
\(367\) 5.00276 0.261142 0.130571 0.991439i \(-0.458319\pi\)
0.130571 + 0.991439i \(0.458319\pi\)
\(368\) 0 0
\(369\) −20.7902 −1.08229
\(370\) 0 0
\(371\) −13.4293 −0.697216
\(372\) 0 0
\(373\) −34.1313 −1.76725 −0.883625 0.468195i \(-0.844904\pi\)
−0.883625 + 0.468195i \(0.844904\pi\)
\(374\) 0 0
\(375\) −5.94391 −0.306942
\(376\) 0 0
\(377\) −33.3185 −1.71599
\(378\) 0 0
\(379\) −15.7749 −0.810302 −0.405151 0.914250i \(-0.632781\pi\)
−0.405151 + 0.914250i \(0.632781\pi\)
\(380\) 0 0
\(381\) −0.604405 −0.0309646
\(382\) 0 0
\(383\) 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
\(386\) 0 0
\(387\) 20.7926 1.05695
\(388\) 0 0
\(389\) −18.3488 −0.930321 −0.465160 0.885226i \(-0.654003\pi\)
−0.465160 + 0.885226i \(0.654003\pi\)
\(390\) 0 0
\(391\) 21.7131 1.09808
\(392\) 0 0
\(393\) 4.40990 0.222450
\(394\) 0 0
\(395\) −11.6393 −0.585638
\(396\) 0 0
\(397\) −0.511483 −0.0256706 −0.0128353 0.999918i \(-0.504086\pi\)
−0.0128353 + 0.999918i \(0.504086\pi\)
\(398\) 0 0
\(399\) −0.953520 −0.0477357
\(400\) 0 0
\(401\) 18.9824 0.947935 0.473967 0.880542i \(-0.342821\pi\)
0.473967 + 0.880542i \(0.342821\pi\)
\(402\) 0 0
\(403\) 9.46277 0.471374
\(404\) 0 0
\(405\) 14.6466 0.727795
\(406\) 0 0
\(407\) 6.56016 0.325175
\(408\) 0 0
\(409\) 1.10089 0.0544355 0.0272177 0.999630i \(-0.491335\pi\)
0.0272177 + 0.999630i \(0.491335\pi\)
\(410\) 0 0
\(411\) −6.33689 −0.312576
\(412\) 0 0
\(413\) 18.4889 0.909781
\(414\) 0 0
\(415\) 4.29054 0.210614
\(416\) 0 0
\(417\) 0.510413 0.0249950
\(418\) 0 0
\(419\) 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 \(0.706191\pi\)
\(422\) 0 0
\(423\) −29.2318 −1.42130
\(424\) 0 0
\(425\) −3.09876 −0.150312
\(426\) 0 0
\(427\) 7.63542 0.369504
\(428\) 0 0
\(429\) −2.31659 −0.111846
\(430\) 0 0
\(431\) 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 \(0.619867\pi\)
\(434\) 0 0
\(435\) −10.7098 −0.513494
\(436\) 0 0
\(437\) −3.52199 −0.168480
\(438\) 0 0
\(439\) −26.1812 −1.24956 −0.624781 0.780800i \(-0.714812\pi\)
−0.624781 + 0.780800i \(0.714812\pi\)
\(440\) 0 0
\(441\) 11.0946 0.528313
\(442\) 0 0
\(443\) −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
\(446\) 0 0
\(447\) 5.92569 0.280275
\(448\) 0 0
\(449\) 11.7990 0.556829 0.278415 0.960461i \(-0.410191\pi\)
0.278415 + 0.960461i \(0.410191\pi\)
\(450\) 0 0
\(451\) −7.75013 −0.364939
\(452\) 0 0
\(453\) −7.46163 −0.350578
\(454\) 0 0
\(455\) −16.3232 −0.765245
\(456\) 0 0
\(457\) −15.6863 −0.733773 −0.366886 0.930266i \(-0.619576\pi\)
−0.366886 + 0.930266i \(0.619576\pi\)
\(458\) 0 0
\(459\) 19.7381 0.921296
\(460\) 0 0
\(461\) 27.2451 1.26893 0.634466 0.772951i \(-0.281220\pi\)
0.634466 + 0.772951i \(0.281220\pi\)
\(462\) 0 0
\(463\) 17.9364 0.833573 0.416787 0.909004i \(-0.363156\pi\)
0.416787 + 0.909004i \(0.363156\pi\)
\(464\) 0 0
\(465\) 3.04168 0.141054
\(466\) 0 0
\(467\) −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
\(470\) 0 0
\(471\) 1.12635 0.0518995
\(472\) 0 0
\(473\) 7.75102 0.356392
\(474\) 0 0
\(475\) 0.502638 0.0230626
\(476\) 0 0
\(477\) −21.2864 −0.974639
\(478\) 0 0
\(479\) −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
\(482\) 0 0
\(483\) 3.35829 0.152808
\(484\) 0 0
\(485\) −25.7888 −1.17101
\(486\) 0 0
\(487\) −6.38349 −0.289264 −0.144632 0.989486i \(-0.546200\pi\)
−0.144632 + 0.989486i \(0.546200\pi\)
\(488\) 0 0
\(489\) 10.5673 0.477869
\(490\) 0 0
\(491\) 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
\(494\) 0 0
\(495\) 6.29268 0.282835
\(496\) 0 0
\(497\) 5.28071 0.236872
\(498\) 0 0
\(499\) 31.9667 1.43103 0.715514 0.698599i \(-0.246193\pi\)
0.715514 + 0.698599i \(0.246193\pi\)
\(500\) 0 0
\(501\) 3.55630 0.158884
\(502\) 0 0
\(503\) 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
\(506\) 0 0
\(507\) 2.20066 0.0977347
\(508\) 0 0
\(509\) 25.7904 1.14314 0.571569 0.820554i \(-0.306335\pi\)
0.571569 + 0.820554i \(0.306335\pi\)
\(510\) 0 0
\(511\) −19.6024 −0.867160
\(512\) 0 0
\(513\) −3.20164 −0.141356
\(514\) 0 0
\(515\) −14.3561 −0.632606
\(516\) 0 0
\(517\) −10.8969 −0.479247
\(518\) 0 0
\(519\) −12.1286 −0.532385
\(520\) 0 0
\(521\) 31.3774 1.37467 0.687334 0.726342i \(-0.258781\pi\)
0.687334 + 0.726342i \(0.258781\pi\)
\(522\) 0 0
\(523\) 23.2388 1.01616 0.508082 0.861309i \(-0.330355\pi\)
0.508082 + 0.861309i \(0.330355\pi\)
\(524\) 0 0
\(525\) −0.479275 −0.0209173
\(526\) 0 0
\(527\) −14.1883 −0.618054
\(528\) 0 0
\(529\) −10.5956 −0.460677
\(530\) 0 0
\(531\) 29.3063 1.27178
\(532\) 0 0
\(533\) 31.8661 1.38027
\(534\) 0 0
\(535\) −0.992136 −0.0428938
\(536\) 0 0
\(537\) 4.49052 0.193780
\(538\) 0 0
\(539\) 4.13581 0.178142
\(540\) 0 0
\(541\) 36.0085 1.54812 0.774062 0.633109i \(-0.218222\pi\)
0.774062 + 0.633109i \(0.218222\pi\)
\(542\) 0 0
\(543\) −0.658941 −0.0282779
\(544\) 0 0
\(545\) 32.8829 1.40855
\(546\) 0 0
\(547\) 28.1855 1.20513 0.602563 0.798071i \(-0.294146\pi\)
0.602563 + 0.798071i \(0.294146\pi\)
\(548\) 0 0
\(549\) 12.1027 0.516530
\(550\) 0 0
\(551\) −8.10336 −0.345215
\(552\) 0 0
\(553\) 8.39738 0.357093
\(554\) 0 0
\(555\) −8.67019 −0.368029
\(556\) 0 0
\(557\) 21.6248 0.916274 0.458137 0.888882i \(-0.348517\pi\)
0.458137 + 0.888882i \(0.348517\pi\)
\(558\) 0 0
\(559\) −31.8697 −1.34795
\(560\) 0 0
\(561\) 3.47345 0.146649
\(562\) 0 0
\(563\) −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
\(566\) 0 0
\(567\) −10.5670 −0.443773
\(568\) 0 0
\(569\) 8.61143 0.361010 0.180505 0.983574i \(-0.442227\pi\)
0.180505 + 0.983574i \(0.442227\pi\)
\(570\) 0 0
\(571\) 17.6546 0.738821 0.369410 0.929266i \(-0.379560\pi\)
0.369410 + 0.929266i \(0.379560\pi\)
\(572\) 0 0
\(573\) 8.82722 0.368762
\(574\) 0 0
\(575\) −1.77029 −0.0738261
\(576\) 0 0
\(577\) −31.0907 −1.29432 −0.647162 0.762352i \(-0.724044\pi\)
−0.647162 + 0.762352i \(0.724044\pi\)
\(578\) 0 0
\(579\) 12.2131 0.507560
\(580\) 0 0
\(581\) −3.09547 −0.128422
\(582\) 0 0
\(583\) −7.93511 −0.328639
\(584\) 0 0
\(585\) −25.8735 −1.06974
\(586\) 0 0
\(587\) 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
\(590\) 0 0
\(591\) 2.58492 0.106329
\(592\) 0 0
\(593\) −24.2460 −0.995666 −0.497833 0.867273i \(-0.665871\pi\)
−0.497833 + 0.867273i \(0.665871\pi\)
\(594\) 0 0
\(595\) 24.4748 1.00337
\(596\) 0 0
\(597\) −7.35930 −0.301196
\(598\) 0 0
\(599\) 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 \(0.217917\pi\)
\(602\) 0 0
\(603\) 39.4876 1.60806
\(604\) 0 0
\(605\) 2.34577 0.0953691
\(606\) 0 0
\(607\) 20.0130 0.812301 0.406151 0.913806i \(-0.366871\pi\)
0.406151 + 0.913806i \(0.366871\pi\)
\(608\) 0 0
\(609\) 7.72672 0.313103
\(610\) 0 0
\(611\) 44.8048 1.81261
\(612\) 0 0
\(613\) −42.1414 −1.70208 −0.851038 0.525105i \(-0.824026\pi\)
−0.851038 + 0.525105i \(0.824026\pi\)
\(614\) 0 0
\(615\) 10.2429 0.413034
\(616\) 0 0
\(617\) 16.9044 0.680545 0.340273 0.940327i \(-0.389481\pi\)
0.340273 + 0.940327i \(0.389481\pi\)
\(618\) 0 0
\(619\) −5.31177 −0.213498 −0.106749 0.994286i \(-0.534044\pi\)
−0.106749 + 0.994286i \(0.534044\pi\)
\(620\) 0 0
\(621\) 11.2762 0.452497
\(622\) 0 0
\(623\) 15.8661 0.635662
\(624\) 0 0
\(625\) −27.2605 −1.09042
\(626\) 0 0
\(627\) −0.563416 −0.0225006
\(628\) 0 0
\(629\) 40.4433 1.61258
\(630\) 0 0
\(631\) −34.0425 −1.35521 −0.677604 0.735427i \(-0.736982\pi\)
−0.677604 + 0.735427i \(0.736982\pi\)
\(632\) 0 0
\(633\) −1.64060 −0.0652079
\(634\) 0 0
\(635\) −2.51643 −0.0998615
\(636\) 0 0
\(637\) −17.0051 −0.673768
\(638\) 0 0
\(639\) 8.37030 0.331124
\(640\) 0 0
\(641\) −15.0405 −0.594063 −0.297031 0.954868i \(-0.595997\pi\)
−0.297031 + 0.954868i \(0.595997\pi\)
\(642\) 0 0
\(643\) −32.3618 −1.27622 −0.638112 0.769943i \(-0.720285\pi\)
−0.638112 + 0.769943i \(0.720285\pi\)
\(644\) 0 0
\(645\) −10.2441 −0.403361
\(646\) 0 0
\(647\) 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
\(650\) 0 0
\(651\) −2.19447 −0.0860079
\(652\) 0 0
\(653\) 31.1992 1.22092 0.610460 0.792047i \(-0.290984\pi\)
0.610460 + 0.792047i \(0.290984\pi\)
\(654\) 0 0
\(655\) 18.3605 0.717406
\(656\) 0 0
\(657\) −31.0712 −1.21220
\(658\) 0 0
\(659\) 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 \(0.271128\pi\)
\(662\) 0 0
\(663\) −14.2817 −0.554657
\(664\) 0 0
\(665\) −3.96996 −0.153949
\(666\) 0 0
\(667\) 28.5400 1.10507
\(668\) 0 0
\(669\) −3.57296 −0.138139
\(670\) 0 0
\(671\) 4.51162 0.174169
\(672\) 0 0
\(673\) 21.1648 0.815842 0.407921 0.913017i \(-0.366254\pi\)
0.407921 + 0.913017i \(0.366254\pi\)
\(674\) 0 0
\(675\) −1.60927 −0.0619408
\(676\) 0 0
\(677\) −34.5239 −1.32686 −0.663431 0.748238i \(-0.730900\pi\)
−0.663431 + 0.748238i \(0.730900\pi\)
\(678\) 0 0
\(679\) 18.6057 0.714022
\(680\) 0 0
\(681\) 1.12925 0.0432729
\(682\) 0 0
\(683\) −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
\(686\) 0 0
\(687\) −11.8257 −0.451179
\(688\) 0 0
\(689\) 32.6267 1.24298
\(690\) 0 0
\(691\) −39.5406 −1.50419 −0.752097 0.659052i \(-0.770958\pi\)
−0.752097 + 0.659052i \(0.770958\pi\)
\(692\) 0 0
\(693\) −4.53995 −0.172458
\(694\) 0 0
\(695\) 2.12510 0.0806094
\(696\) 0 0
\(697\) −47.7795 −1.80978
\(698\) 0 0
\(699\) −9.91646 −0.375075
\(700\) 0 0
\(701\) 22.4546 0.848097 0.424049 0.905639i \(-0.360609\pi\)
0.424049 + 0.905639i \(0.360609\pi\)
\(702\) 0 0
\(703\) −6.56016 −0.247421
\(704\) 0 0
\(705\) 14.4019 0.542406
\(706\) 0 0
\(707\) −3.90301 −0.146788
\(708\) 0 0
\(709\) −1.41244 −0.0530451 −0.0265226 0.999648i \(-0.508443\pi\)
−0.0265226 + 0.999648i \(0.508443\pi\)
\(710\) 0 0
\(711\) 13.3104 0.499181
\(712\) 0 0
\(713\) −8.10564 −0.303559
\(714\) 0 0
\(715\) −9.64506 −0.360705
\(716\) 0 0
\(717\) −14.9985 −0.560130
\(718\) 0 0
\(719\) 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
\(722\) 0 0
\(723\) 7.40002 0.275210
\(724\) 0 0
\(725\) −4.07306 −0.151270
\(726\) 0 0
\(727\) −9.42098 −0.349405 −0.174702 0.984621i \(-0.555896\pi\)
−0.174702 + 0.984621i \(0.555896\pi\)
\(728\) 0 0
\(729\) −11.3379 −0.419922
\(730\) 0 0
\(731\) 47.7850 1.76739
\(732\) 0 0
\(733\) 4.63361 0.171146 0.0855731 0.996332i \(-0.472728\pi\)
0.0855731 + 0.996332i \(0.472728\pi\)
\(734\) 0 0
\(735\) −5.46606 −0.201619
\(736\) 0 0
\(737\) 14.7201 0.542222
\(738\) 0 0
\(739\) 13.1654 0.484298 0.242149 0.970239i \(-0.422148\pi\)
0.242149 + 0.970239i \(0.422148\pi\)
\(740\) 0 0
\(741\) 2.31659 0.0851019
\(742\) 0 0
\(743\) 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
\(746\) 0 0
\(747\) −4.90655 −0.179521
\(748\) 0 0
\(749\) 0.715792 0.0261545
\(750\) 0 0
\(751\) −35.0415 −1.27868 −0.639341 0.768924i \(-0.720793\pi\)
−0.639341 + 0.768924i \(0.720793\pi\)
\(752\) 0 0
\(753\) 14.7410 0.537191
\(754\) 0 0
\(755\) −31.0664 −1.13062
\(756\) 0 0
\(757\) −23.8005 −0.865044 −0.432522 0.901623i \(-0.642376\pi\)
−0.432522 + 0.901623i \(0.642376\pi\)
\(758\) 0 0
\(759\) 1.98435 0.0720272
\(760\) 0 0
\(761\) −39.0282 −1.41477 −0.707386 0.706827i \(-0.750126\pi\)
−0.707386 + 0.706827i \(0.750126\pi\)
\(762\) 0 0
\(763\) −23.7239 −0.858862
\(764\) 0 0
\(765\) 38.7943 1.40261
\(766\) 0 0
\(767\) −44.9190 −1.62193
\(768\) 0 0
\(769\) 9.84757 0.355113 0.177556 0.984111i \(-0.443181\pi\)
0.177556 + 0.984111i \(0.443181\pi\)
\(770\) 0 0
\(771\) 7.10564 0.255903
\(772\) 0 0
\(773\) −42.9837 −1.54602 −0.773008 0.634396i \(-0.781249\pi\)
−0.773008 + 0.634396i \(0.781249\pi\)
\(774\) 0 0
\(775\) 1.15679 0.0415531
\(776\) 0 0
\(777\) 6.25524 0.224405
\(778\) 0 0
\(779\) 7.75013 0.277677
\(780\) 0 0
\(781\) 3.12026 0.111652
\(782\) 0 0
\(783\) 25.9441 0.927166
\(784\) 0 0
\(785\) 4.68954 0.167377
\(786\) 0 0
\(787\) 25.8709 0.922199 0.461099 0.887349i \(-0.347455\pi\)
0.461099 + 0.887349i \(0.347455\pi\)
\(788\) 0 0
\(789\) 11.2993 0.402266
\(790\) 0 0
\(791\) 4.29254 0.152625
\(792\) 0 0
\(793\) −18.5503 −0.658741
\(794\) 0 0
\(795\) 10.4874 0.371949
\(796\) 0 0
\(797\) 3.62989 0.128577 0.0642886 0.997931i \(-0.479522\pi\)
0.0642886 + 0.997931i \(0.479522\pi\)
\(798\) 0 0
\(799\) −67.1796 −2.37664
\(800\) 0 0
\(801\) 25.1489 0.888593
\(802\) 0 0
\(803\) −11.5827 −0.408743
\(804\) 0 0
\(805\) 13.9822 0.492807
\(806\) 0 0
\(807\) −11.3331 −0.398945
\(808\) 0 0
\(809\) −13.1327 −0.461720 −0.230860 0.972987i \(-0.574154\pi\)
−0.230860 + 0.972987i \(0.574154\pi\)
\(810\) 0 0
\(811\) −9.40230 −0.330160 −0.165080 0.986280i \(-0.552788\pi\)
−0.165080 + 0.986280i \(0.552788\pi\)
\(812\) 0 0
\(813\) −8.76595 −0.307435
\(814\) 0 0
\(815\) 43.9966 1.54114
\(816\) 0 0
\(817\) −7.75102 −0.271174
\(818\) 0 0
\(819\) 18.6668 0.652272
\(820\) 0 0
\(821\) −21.1701 −0.738843 −0.369422 0.929262i \(-0.620444\pi\)
−0.369422 + 0.929262i \(0.620444\pi\)
\(822\) 0 0
\(823\) −25.2277 −0.879384 −0.439692 0.898149i \(-0.644912\pi\)
−0.439692 + 0.898149i \(0.644912\pi\)
\(824\) 0 0
\(825\) −0.283194 −0.00985955
\(826\) 0 0
\(827\) 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 \(0.718574\pi\)
\(830\) 0 0
\(831\) 1.06604 0.0369806
\(832\) 0 0
\(833\) 25.4972 0.883427
\(834\) 0 0
\(835\) 14.8066 0.512403
\(836\) 0 0
\(837\) −7.36838 −0.254688
\(838\) 0 0
\(839\) −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
\(842\) 0 0
\(843\) 2.58776 0.0891273
\(844\) 0 0
\(845\) 9.16241 0.315196
\(846\) 0 0
\(847\) −1.69239 −0.0581513
\(848\) 0 0
\(849\) 0.663690 0.0227778
\(850\) 0 0
\(851\) 23.1048 0.792023
\(852\) 0 0
\(853\) 21.5088 0.736446 0.368223 0.929738i \(-0.379966\pi\)
0.368223 + 0.929738i \(0.379966\pi\)
\(854\) 0 0
\(855\) −6.29268 −0.215205
\(856\) 0 0
\(857\) −12.6370 −0.431670 −0.215835 0.976430i \(-0.569247\pi\)
−0.215835 + 0.976430i \(0.569247\pi\)
\(858\) 0 0
\(859\) 3.39720 0.115911 0.0579555 0.998319i \(-0.481542\pi\)
0.0579555 + 0.998319i \(0.481542\pi\)
\(860\) 0 0
\(861\) −7.38990 −0.251847
\(862\) 0 0
\(863\) 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
\(866\) 0 0
\(867\) 11.8358 0.401963
\(868\) 0 0
\(869\) 4.96184 0.168319
\(870\) 0 0
\(871\) −60.5243 −2.05079
\(872\) 0 0
\(873\) 29.4914 0.998133
\(874\) 0 0
\(875\) 17.8544 0.603588
\(876\) 0 0
\(877\) 31.7230 1.07121 0.535605 0.844468i \(-0.320083\pi\)
0.535605 + 0.844468i \(0.320083\pi\)
\(878\) 0 0
\(879\) −6.34439 −0.213991
\(880\) 0 0
\(881\) −6.45839 −0.217589 −0.108794 0.994064i \(-0.534699\pi\)
−0.108794 + 0.994064i \(0.534699\pi\)
\(882\) 0 0
\(883\) 48.5242 1.63297 0.816485 0.577367i \(-0.195920\pi\)
0.816485 + 0.577367i \(0.195920\pi\)
\(884\) 0 0
\(885\) −14.4386 −0.485348
\(886\) 0 0
\(887\) 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
\(890\) 0 0
\(891\) −6.24383 −0.209176
\(892\) 0 0
\(893\) 10.8969 0.364652
\(894\) 0 0
\(895\) 18.6962 0.624946
\(896\) 0 0
\(897\) −8.15900 −0.272421
\(898\) 0 0
\(899\) −18.6494 −0.621991
\(900\) 0 0
\(901\) −48.9199 −1.62976
\(902\) 0 0
\(903\) 7.39076 0.245949
\(904\) 0 0
\(905\) −2.74349 −0.0911966
\(906\) 0 0
\(907\) 2.05084 0.0680971 0.0340485 0.999420i \(-0.489160\pi\)
0.0340485 + 0.999420i \(0.489160\pi\)
\(908\) 0 0
\(909\) −6.18655 −0.205195
\(910\) 0 0
\(911\) 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
\(914\) 0 0
\(915\) −5.96275 −0.197122
\(916\) 0 0
\(917\) −13.2465 −0.437438
\(918\) 0 0
\(919\) −8.01221 −0.264299 −0.132149 0.991230i \(-0.542188\pi\)
−0.132149 + 0.991230i \(0.542188\pi\)
\(920\) 0 0
\(921\) −2.45224 −0.0808039
\(922\) 0 0
\(923\) −12.8295 −0.422289
\(924\) 0 0
\(925\) −3.29738 −0.108417
\(926\) 0 0
\(927\) 16.4173 0.539214
\(928\) 0 0
\(929\) 16.7897 0.550853 0.275426 0.961322i \(-0.411181\pi\)
0.275426 + 0.961322i \(0.411181\pi\)
\(930\) 0 0
\(931\) −4.13581 −0.135546
\(932\) 0 0
\(933\) −4.29168 −0.140503
\(934\) 0 0
\(935\) 14.4617 0.472947
\(936\) 0 0
\(937\) 21.8471 0.713713 0.356856 0.934159i \(-0.383849\pi\)
0.356856 + 0.934159i \(0.383849\pi\)
\(938\) 0 0
\(939\) −9.89661 −0.322964
\(940\) 0 0
\(941\) 17.8421 0.581635 0.290817 0.956779i \(-0.406073\pi\)
0.290817 + 0.956779i \(0.406073\pi\)
\(942\) 0 0
\(943\) −27.2959 −0.888877
\(944\) 0 0
\(945\) 12.7104 0.413470
\(946\) 0 0
\(947\) −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 0
\(951\) −1.09411 −0.0354788
\(952\) 0 0
\(953\) −8.06945 −0.261395 −0.130697 0.991422i \(-0.541722\pi\)
−0.130697 + 0.991422i \(0.541722\pi\)
\(954\) 0 0
\(955\) 36.7520 1.18927
\(956\) 0 0
\(957\) 4.56556 0.147584
\(958\) 0 0
\(959\) 19.0348 0.614666
\(960\) 0 0
\(961\) −25.7034 −0.829142
\(962\) 0 0
\(963\) 1.13458 0.0365614
\(964\) 0 0
\(965\) 50.8491 1.63689
\(966\) 0 0
\(967\) 3.86360 0.124245 0.0621225 0.998069i \(-0.480213\pi\)
0.0621225 + 0.998069i \(0.480213\pi\)
\(968\) 0 0
\(969\) −3.47345 −0.111583
\(970\) 0 0
\(971\) −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
\(974\) 0 0
\(975\) 1.16440 0.0372908
\(976\) 0 0
\(977\) 54.5146 1.74408 0.872039 0.489437i \(-0.162798\pi\)
0.872039 + 0.489437i \(0.162798\pi\)
\(978\) 0 0
\(979\) 9.37496 0.299625
\(980\) 0 0
\(981\) −37.6040 −1.20060
\(982\) 0 0
\(983\) −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
\(986\) 0 0
\(987\) −10.3905 −0.330732
\(988\) 0 0
\(989\) 27.2991 0.868060
\(990\) 0 0
\(991\) −31.7767 −1.00942 −0.504710 0.863289i \(-0.668400\pi\)
−0.504710 + 0.863289i \(0.668400\pi\)
\(992\) 0 0
\(993\) −7.54907 −0.239562
\(994\) 0 0
\(995\) −30.6403 −0.971363
\(996\) 0 0
\(997\) −47.5668 −1.50646 −0.753228 0.657759i \(-0.771504\pi\)
−0.753228 + 0.657759i \(0.771504\pi\)
\(998\) 0 0
\(999\) 21.0033 0.664515
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 3344.2.a.t.1.4 5
4.3 odd 2 209.2.a.c.1.4 5
12.11 even 2 1881.2.a.k.1.2 5
20.19 odd 2 5225.2.a.h.1.2 5
44.43 even 2 2299.2.a.n.1.2 5
76.75 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 4.3 odd 2
1881.2.a.k.1.2 5 12.11 even 2
2299.2.a.n.1.2 5 44.43 even 2
3344.2.a.t.1.4 5 1.1 even 1 trivial
3971.2.a.h.1.2 5 76.75 even 2
5225.2.a.h.1.2 5 20.19 odd 2