Properties

Label 2320.2.a.s.1.1
Level $2320$
Weight $2$
Character 2320.1
Self dual yes
Analytic conductor $18.525$
Analytic rank $1$
Dimension $3$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2320,2,Mod(1,2320)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2320, 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("2320.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2320 = 2^{4} \cdot 5 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2320.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: \(18.5252932689\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.148.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 3x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 145)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(2.17009\) of defining polynomial
Character \(\chi\) \(=\) 2320.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.70928 q^{3} -1.00000 q^{5} -0.630898 q^{7} -0.0783777 q^{9} +O(q^{10})\) \(q-1.70928 q^{3} -1.00000 q^{5} -0.630898 q^{7} -0.0783777 q^{9} -0.290725 q^{11} -0.921622 q^{13} +1.70928 q^{15} +4.97107 q^{17} +6.04945 q^{19} +1.07838 q^{21} -2.29072 q^{23} +1.00000 q^{25} +5.26180 q^{27} +1.00000 q^{29} -10.0494 q^{31} +0.496928 q^{33} +0.630898 q^{35} +1.55252 q^{37} +1.57531 q^{39} +0.340173 q^{41} +5.70928 q^{43} +0.0783777 q^{45} +1.12783 q^{47} -6.60197 q^{49} -8.49693 q^{51} -0.340173 q^{53} +0.290725 q^{55} -10.3402 q^{57} -9.75872 q^{59} +3.07838 q^{61} +0.0494483 q^{63} +0.921622 q^{65} +5.70928 q^{67} +3.91548 q^{69} -9.07838 q^{71} -6.94441 q^{73} -1.70928 q^{75} +0.183417 q^{77} -12.3896 q^{79} -8.75872 q^{81} -2.78765 q^{83} -4.97107 q^{85} -1.70928 q^{87} +4.73820 q^{89} +0.581449 q^{91} +17.1773 q^{93} -6.04945 q^{95} -15.8927 q^{97} +0.0227863 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 2 q^{3} - 3 q^{5} + 2 q^{7} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 2 q^{3} - 3 q^{5} + 2 q^{7} + 3 q^{9} - 8 q^{11} - 6 q^{13} - 2 q^{15} - 14 q^{23} + 3 q^{25} + 8 q^{27} + 3 q^{29} - 12 q^{31} - 16 q^{33} - 2 q^{35} + 4 q^{37} - 16 q^{39} - 10 q^{41} + 10 q^{43} - 3 q^{45} - 18 q^{47} - q^{49} - 8 q^{51} + 10 q^{53} + 8 q^{55} - 20 q^{57} - 4 q^{59} + 6 q^{61} - 18 q^{63} + 6 q^{65} + 10 q^{67} - 20 q^{69} - 24 q^{71} - 4 q^{73} + 2 q^{75} - 4 q^{77} - 8 q^{79} - q^{81} + 2 q^{83} + 2 q^{87} + 22 q^{89} + 16 q^{91} + 12 q^{93} - 36 q^{97} - 20 q^{99}+O(q^{100}) \) 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
\(3\) −1.70928 −0.986851 −0.493425 0.869788i \(-0.664255\pi\)
−0.493425 + 0.869788i \(0.664255\pi\)
\(4\) 0 0
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) −0.630898 −0.238457 −0.119228 0.992867i \(-0.538042\pi\)
−0.119228 + 0.992867i \(0.538042\pi\)
\(8\) 0 0
\(9\) −0.0783777 −0.0261259
\(10\) 0 0
\(11\) −0.290725 −0.0876568 −0.0438284 0.999039i \(-0.513955\pi\)
−0.0438284 + 0.999039i \(0.513955\pi\)
\(12\) 0 0
\(13\) −0.921622 −0.255612 −0.127806 0.991799i \(-0.540793\pi\)
−0.127806 + 0.991799i \(0.540793\pi\)
\(14\) 0 0
\(15\) 1.70928 0.441333
\(16\) 0 0
\(17\) 4.97107 1.20566 0.602831 0.797869i \(-0.294039\pi\)
0.602831 + 0.797869i \(0.294039\pi\)
\(18\) 0 0
\(19\) 6.04945 1.38784 0.693919 0.720053i \(-0.255882\pi\)
0.693919 + 0.720053i \(0.255882\pi\)
\(20\) 0 0
\(21\) 1.07838 0.235321
\(22\) 0 0
\(23\) −2.29072 −0.477649 −0.238825 0.971063i \(-0.576762\pi\)
−0.238825 + 0.971063i \(0.576762\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 5.26180 1.01263
\(28\) 0 0
\(29\) 1.00000 0.185695
\(30\) 0 0
\(31\) −10.0494 −1.80493 −0.902467 0.430759i \(-0.858246\pi\)
−0.902467 + 0.430759i \(0.858246\pi\)
\(32\) 0 0
\(33\) 0.496928 0.0865041
\(34\) 0 0
\(35\) 0.630898 0.106641
\(36\) 0 0
\(37\) 1.55252 0.255233 0.127616 0.991824i \(-0.459267\pi\)
0.127616 + 0.991824i \(0.459267\pi\)
\(38\) 0 0
\(39\) 1.57531 0.252251
\(40\) 0 0
\(41\) 0.340173 0.0531261 0.0265630 0.999647i \(-0.491544\pi\)
0.0265630 + 0.999647i \(0.491544\pi\)
\(42\) 0 0
\(43\) 5.70928 0.870656 0.435328 0.900272i \(-0.356632\pi\)
0.435328 + 0.900272i \(0.356632\pi\)
\(44\) 0 0
\(45\) 0.0783777 0.0116839
\(46\) 0 0
\(47\) 1.12783 0.164510 0.0822552 0.996611i \(-0.473788\pi\)
0.0822552 + 0.996611i \(0.473788\pi\)
\(48\) 0 0
\(49\) −6.60197 −0.943138
\(50\) 0 0
\(51\) −8.49693 −1.18981
\(52\) 0 0
\(53\) −0.340173 −0.0467264 −0.0233632 0.999727i \(-0.507437\pi\)
−0.0233632 + 0.999727i \(0.507437\pi\)
\(54\) 0 0
\(55\) 0.290725 0.0392013
\(56\) 0 0
\(57\) −10.3402 −1.36959
\(58\) 0 0
\(59\) −9.75872 −1.27048 −0.635239 0.772316i \(-0.719098\pi\)
−0.635239 + 0.772316i \(0.719098\pi\)
\(60\) 0 0
\(61\) 3.07838 0.394146 0.197073 0.980389i \(-0.436856\pi\)
0.197073 + 0.980389i \(0.436856\pi\)
\(62\) 0 0
\(63\) 0.0494483 0.00622990
\(64\) 0 0
\(65\) 0.921622 0.114313
\(66\) 0 0
\(67\) 5.70928 0.697499 0.348749 0.937216i \(-0.386606\pi\)
0.348749 + 0.937216i \(0.386606\pi\)
\(68\) 0 0
\(69\) 3.91548 0.471368
\(70\) 0 0
\(71\) −9.07838 −1.07741 −0.538703 0.842496i \(-0.681085\pi\)
−0.538703 + 0.842496i \(0.681085\pi\)
\(72\) 0 0
\(73\) −6.94441 −0.812782 −0.406391 0.913699i \(-0.633213\pi\)
−0.406391 + 0.913699i \(0.633213\pi\)
\(74\) 0 0
\(75\) −1.70928 −0.197370
\(76\) 0 0
\(77\) 0.183417 0.0209024
\(78\) 0 0
\(79\) −12.3896 −1.39394 −0.696971 0.717100i \(-0.745469\pi\)
−0.696971 + 0.717100i \(0.745469\pi\)
\(80\) 0 0
\(81\) −8.75872 −0.973192
\(82\) 0 0
\(83\) −2.78765 −0.305985 −0.152992 0.988227i \(-0.548891\pi\)
−0.152992 + 0.988227i \(0.548891\pi\)
\(84\) 0 0
\(85\) −4.97107 −0.539188
\(86\) 0 0
\(87\) −1.70928 −0.183254
\(88\) 0 0
\(89\) 4.73820 0.502249 0.251124 0.967955i \(-0.419200\pi\)
0.251124 + 0.967955i \(0.419200\pi\)
\(90\) 0 0
\(91\) 0.581449 0.0609524
\(92\) 0 0
\(93\) 17.1773 1.78120
\(94\) 0 0
\(95\) −6.04945 −0.620660
\(96\) 0 0
\(97\) −15.8927 −1.61366 −0.806829 0.590785i \(-0.798818\pi\)
−0.806829 + 0.590785i \(0.798818\pi\)
\(98\) 0 0
\(99\) 0.0227863 0.00229011
\(100\) 0 0
\(101\) −12.2557 −1.21948 −0.609741 0.792600i \(-0.708727\pi\)
−0.609741 + 0.792600i \(0.708727\pi\)
\(102\) 0 0
\(103\) 7.86603 0.775063 0.387532 0.921856i \(-0.373328\pi\)
0.387532 + 0.921856i \(0.373328\pi\)
\(104\) 0 0
\(105\) −1.07838 −0.105239
\(106\) 0 0
\(107\) −12.7298 −1.23064 −0.615318 0.788279i \(-0.710972\pi\)
−0.615318 + 0.788279i \(0.710972\pi\)
\(108\) 0 0
\(109\) 12.4391 1.19145 0.595723 0.803190i \(-0.296865\pi\)
0.595723 + 0.803190i \(0.296865\pi\)
\(110\) 0 0
\(111\) −2.65368 −0.251877
\(112\) 0 0
\(113\) 12.5730 1.18277 0.591386 0.806389i \(-0.298581\pi\)
0.591386 + 0.806389i \(0.298581\pi\)
\(114\) 0 0
\(115\) 2.29072 0.213611
\(116\) 0 0
\(117\) 0.0722347 0.00667810
\(118\) 0 0
\(119\) −3.13624 −0.287498
\(120\) 0 0
\(121\) −10.9155 −0.992316
\(122\) 0 0
\(123\) −0.581449 −0.0524275
\(124\) 0 0
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) 20.9132 1.85575 0.927874 0.372895i \(-0.121635\pi\)
0.927874 + 0.372895i \(0.121635\pi\)
\(128\) 0 0
\(129\) −9.75872 −0.859208
\(130\) 0 0
\(131\) 13.4680 1.17670 0.588352 0.808605i \(-0.299777\pi\)
0.588352 + 0.808605i \(0.299777\pi\)
\(132\) 0 0
\(133\) −3.81658 −0.330940
\(134\) 0 0
\(135\) −5.26180 −0.452863
\(136\) 0 0
\(137\) −13.5525 −1.15787 −0.578935 0.815374i \(-0.696531\pi\)
−0.578935 + 0.815374i \(0.696531\pi\)
\(138\) 0 0
\(139\) 4.89496 0.415185 0.207593 0.978215i \(-0.433437\pi\)
0.207593 + 0.978215i \(0.433437\pi\)
\(140\) 0 0
\(141\) −1.92777 −0.162347
\(142\) 0 0
\(143\) 0.267938 0.0224061
\(144\) 0 0
\(145\) −1.00000 −0.0830455
\(146\) 0 0
\(147\) 11.2846 0.930737
\(148\) 0 0
\(149\) −12.5236 −1.02597 −0.512986 0.858397i \(-0.671461\pi\)
−0.512986 + 0.858397i \(0.671461\pi\)
\(150\) 0 0
\(151\) −7.60197 −0.618639 −0.309320 0.950958i \(-0.600101\pi\)
−0.309320 + 0.950958i \(0.600101\pi\)
\(152\) 0 0
\(153\) −0.389621 −0.0314990
\(154\) 0 0
\(155\) 10.0494 0.807191
\(156\) 0 0
\(157\) −24.8865 −1.98616 −0.993081 0.117428i \(-0.962535\pi\)
−0.993081 + 0.117428i \(0.962535\pi\)
\(158\) 0 0
\(159\) 0.581449 0.0461119
\(160\) 0 0
\(161\) 1.44521 0.113899
\(162\) 0 0
\(163\) −0.447480 −0.0350493 −0.0175247 0.999846i \(-0.505579\pi\)
−0.0175247 + 0.999846i \(0.505579\pi\)
\(164\) 0 0
\(165\) −0.496928 −0.0386858
\(166\) 0 0
\(167\) −19.8660 −1.53728 −0.768640 0.639682i \(-0.779066\pi\)
−0.768640 + 0.639682i \(0.779066\pi\)
\(168\) 0 0
\(169\) −12.1506 −0.934662
\(170\) 0 0
\(171\) −0.474142 −0.0362586
\(172\) 0 0
\(173\) −25.4329 −1.93363 −0.966815 0.255478i \(-0.917767\pi\)
−0.966815 + 0.255478i \(0.917767\pi\)
\(174\) 0 0
\(175\) −0.630898 −0.0476914
\(176\) 0 0
\(177\) 16.6803 1.25377
\(178\) 0 0
\(179\) −14.8371 −1.10898 −0.554489 0.832191i \(-0.687086\pi\)
−0.554489 + 0.832191i \(0.687086\pi\)
\(180\) 0 0
\(181\) −5.91548 −0.439694 −0.219847 0.975534i \(-0.570556\pi\)
−0.219847 + 0.975534i \(0.570556\pi\)
\(182\) 0 0
\(183\) −5.26180 −0.388963
\(184\) 0 0
\(185\) −1.55252 −0.114144
\(186\) 0 0
\(187\) −1.44521 −0.105684
\(188\) 0 0
\(189\) −3.31965 −0.241469
\(190\) 0 0
\(191\) −7.02893 −0.508595 −0.254298 0.967126i \(-0.581844\pi\)
−0.254298 + 0.967126i \(0.581844\pi\)
\(192\) 0 0
\(193\) 17.8660 1.28603 0.643013 0.765856i \(-0.277684\pi\)
0.643013 + 0.765856i \(0.277684\pi\)
\(194\) 0 0
\(195\) −1.57531 −0.112810
\(196\) 0 0
\(197\) 6.09890 0.434528 0.217264 0.976113i \(-0.430287\pi\)
0.217264 + 0.976113i \(0.430287\pi\)
\(198\) 0 0
\(199\) 9.75872 0.691778 0.345889 0.938276i \(-0.387577\pi\)
0.345889 + 0.938276i \(0.387577\pi\)
\(200\) 0 0
\(201\) −9.75872 −0.688327
\(202\) 0 0
\(203\) −0.630898 −0.0442803
\(204\) 0 0
\(205\) −0.340173 −0.0237587
\(206\) 0 0
\(207\) 0.179542 0.0124790
\(208\) 0 0
\(209\) −1.75872 −0.121653
\(210\) 0 0
\(211\) −9.86603 −0.679206 −0.339603 0.940569i \(-0.610293\pi\)
−0.339603 + 0.940569i \(0.610293\pi\)
\(212\) 0 0
\(213\) 15.5174 1.06324
\(214\) 0 0
\(215\) −5.70928 −0.389369
\(216\) 0 0
\(217\) 6.34017 0.430399
\(218\) 0 0
\(219\) 11.8699 0.802094
\(220\) 0 0
\(221\) −4.58145 −0.308182
\(222\) 0 0
\(223\) −10.9711 −0.734677 −0.367339 0.930087i \(-0.619731\pi\)
−0.367339 + 0.930087i \(0.619731\pi\)
\(224\) 0 0
\(225\) −0.0783777 −0.00522518
\(226\) 0 0
\(227\) 12.5464 0.832732 0.416366 0.909197i \(-0.363303\pi\)
0.416366 + 0.909197i \(0.363303\pi\)
\(228\) 0 0
\(229\) 23.3607 1.54372 0.771859 0.635794i \(-0.219327\pi\)
0.771859 + 0.635794i \(0.219327\pi\)
\(230\) 0 0
\(231\) −0.313511 −0.0206275
\(232\) 0 0
\(233\) 12.4703 0.816954 0.408477 0.912769i \(-0.366060\pi\)
0.408477 + 0.912769i \(0.366060\pi\)
\(234\) 0 0
\(235\) −1.12783 −0.0735713
\(236\) 0 0
\(237\) 21.1773 1.37561
\(238\) 0 0
\(239\) −13.7587 −0.889978 −0.444989 0.895536i \(-0.646792\pi\)
−0.444989 + 0.895536i \(0.646792\pi\)
\(240\) 0 0
\(241\) −14.6803 −0.945644 −0.472822 0.881158i \(-0.656765\pi\)
−0.472822 + 0.881158i \(0.656765\pi\)
\(242\) 0 0
\(243\) −0.814315 −0.0522383
\(244\) 0 0
\(245\) 6.60197 0.421784
\(246\) 0 0
\(247\) −5.57531 −0.354748
\(248\) 0 0
\(249\) 4.76487 0.301961
\(250\) 0 0
\(251\) −15.4413 −0.974649 −0.487324 0.873221i \(-0.662027\pi\)
−0.487324 + 0.873221i \(0.662027\pi\)
\(252\) 0 0
\(253\) 0.665970 0.0418692
\(254\) 0 0
\(255\) 8.49693 0.532098
\(256\) 0 0
\(257\) −6.28231 −0.391880 −0.195940 0.980616i \(-0.562776\pi\)
−0.195940 + 0.980616i \(0.562776\pi\)
\(258\) 0 0
\(259\) −0.979481 −0.0608620
\(260\) 0 0
\(261\) −0.0783777 −0.00485146
\(262\) 0 0
\(263\) −10.0761 −0.621320 −0.310660 0.950521i \(-0.600550\pi\)
−0.310660 + 0.950521i \(0.600550\pi\)
\(264\) 0 0
\(265\) 0.340173 0.0208967
\(266\) 0 0
\(267\) −8.09890 −0.495644
\(268\) 0 0
\(269\) 28.1711 1.71762 0.858812 0.512291i \(-0.171203\pi\)
0.858812 + 0.512291i \(0.171203\pi\)
\(270\) 0 0
\(271\) −28.8020 −1.74960 −0.874799 0.484485i \(-0.839007\pi\)
−0.874799 + 0.484485i \(0.839007\pi\)
\(272\) 0 0
\(273\) −0.993857 −0.0601510
\(274\) 0 0
\(275\) −0.290725 −0.0175314
\(276\) 0 0
\(277\) 0.0266620 0.00160196 0.000800982 1.00000i \(-0.499745\pi\)
0.000800982 1.00000i \(0.499745\pi\)
\(278\) 0 0
\(279\) 0.787653 0.0471556
\(280\) 0 0
\(281\) −28.0722 −1.67465 −0.837325 0.546706i \(-0.815881\pi\)
−0.837325 + 0.546706i \(0.815881\pi\)
\(282\) 0 0
\(283\) 20.8143 1.23728 0.618641 0.785674i \(-0.287683\pi\)
0.618641 + 0.785674i \(0.287683\pi\)
\(284\) 0 0
\(285\) 10.3402 0.612499
\(286\) 0 0
\(287\) −0.214614 −0.0126683
\(288\) 0 0
\(289\) 7.71154 0.453620
\(290\) 0 0
\(291\) 27.1650 1.59244
\(292\) 0 0
\(293\) −15.4101 −0.900270 −0.450135 0.892961i \(-0.648624\pi\)
−0.450135 + 0.892961i \(0.648624\pi\)
\(294\) 0 0
\(295\) 9.75872 0.568175
\(296\) 0 0
\(297\) −1.52973 −0.0887641
\(298\) 0 0
\(299\) 2.11118 0.122093
\(300\) 0 0
\(301\) −3.60197 −0.207614
\(302\) 0 0
\(303\) 20.9483 1.20345
\(304\) 0 0
\(305\) −3.07838 −0.176267
\(306\) 0 0
\(307\) 28.4307 1.62262 0.811312 0.584614i \(-0.198754\pi\)
0.811312 + 0.584614i \(0.198754\pi\)
\(308\) 0 0
\(309\) −13.4452 −0.764871
\(310\) 0 0
\(311\) 19.6248 1.11282 0.556409 0.830909i \(-0.312179\pi\)
0.556409 + 0.830909i \(0.312179\pi\)
\(312\) 0 0
\(313\) 22.9093 1.29491 0.647456 0.762103i \(-0.275833\pi\)
0.647456 + 0.762103i \(0.275833\pi\)
\(314\) 0 0
\(315\) −0.0494483 −0.00278610
\(316\) 0 0
\(317\) 22.8599 1.28394 0.641970 0.766730i \(-0.278118\pi\)
0.641970 + 0.766730i \(0.278118\pi\)
\(318\) 0 0
\(319\) −0.290725 −0.0162775
\(320\) 0 0
\(321\) 21.7587 1.21445
\(322\) 0 0
\(323\) 30.0722 1.67326
\(324\) 0 0
\(325\) −0.921622 −0.0511224
\(326\) 0 0
\(327\) −21.2618 −1.17578
\(328\) 0 0
\(329\) −0.711543 −0.0392286
\(330\) 0 0
\(331\) −24.0905 −1.32413 −0.662066 0.749445i \(-0.730320\pi\)
−0.662066 + 0.749445i \(0.730320\pi\)
\(332\) 0 0
\(333\) −0.121683 −0.00666819
\(334\) 0 0
\(335\) −5.70928 −0.311931
\(336\) 0 0
\(337\) 12.7877 0.696588 0.348294 0.937385i \(-0.386761\pi\)
0.348294 + 0.937385i \(0.386761\pi\)
\(338\) 0 0
\(339\) −21.4908 −1.16722
\(340\) 0 0
\(341\) 2.92162 0.158215
\(342\) 0 0
\(343\) 8.58145 0.463355
\(344\) 0 0
\(345\) −3.91548 −0.210802
\(346\) 0 0
\(347\) −8.41628 −0.451810 −0.225905 0.974149i \(-0.572534\pi\)
−0.225905 + 0.974149i \(0.572534\pi\)
\(348\) 0 0
\(349\) 22.1978 1.18822 0.594110 0.804384i \(-0.297504\pi\)
0.594110 + 0.804384i \(0.297504\pi\)
\(350\) 0 0
\(351\) −4.84939 −0.258841
\(352\) 0 0
\(353\) −6.18342 −0.329110 −0.164555 0.986368i \(-0.552619\pi\)
−0.164555 + 0.986368i \(0.552619\pi\)
\(354\) 0 0
\(355\) 9.07838 0.481830
\(356\) 0 0
\(357\) 5.36069 0.283718
\(358\) 0 0
\(359\) −5.05559 −0.266824 −0.133412 0.991061i \(-0.542593\pi\)
−0.133412 + 0.991061i \(0.542593\pi\)
\(360\) 0 0
\(361\) 17.5958 0.926096
\(362\) 0 0
\(363\) 18.6576 0.979268
\(364\) 0 0
\(365\) 6.94441 0.363487
\(366\) 0 0
\(367\) −29.5402 −1.54199 −0.770994 0.636843i \(-0.780240\pi\)
−0.770994 + 0.636843i \(0.780240\pi\)
\(368\) 0 0
\(369\) −0.0266620 −0.00138797
\(370\) 0 0
\(371\) 0.214614 0.0111422
\(372\) 0 0
\(373\) 14.4124 0.746246 0.373123 0.927782i \(-0.378287\pi\)
0.373123 + 0.927782i \(0.378287\pi\)
\(374\) 0 0
\(375\) 1.70928 0.0882666
\(376\) 0 0
\(377\) −0.921622 −0.0474660
\(378\) 0 0
\(379\) −14.1340 −0.726013 −0.363007 0.931787i \(-0.618250\pi\)
−0.363007 + 0.931787i \(0.618250\pi\)
\(380\) 0 0
\(381\) −35.7464 −1.83135
\(382\) 0 0
\(383\) 15.7815 0.806397 0.403199 0.915112i \(-0.367898\pi\)
0.403199 + 0.915112i \(0.367898\pi\)
\(384\) 0 0
\(385\) −0.183417 −0.00934782
\(386\) 0 0
\(387\) −0.447480 −0.0227467
\(388\) 0 0
\(389\) −13.8166 −0.700529 −0.350264 0.936651i \(-0.613908\pi\)
−0.350264 + 0.936651i \(0.613908\pi\)
\(390\) 0 0
\(391\) −11.3874 −0.575883
\(392\) 0 0
\(393\) −23.0205 −1.16123
\(394\) 0 0
\(395\) 12.3896 0.623390
\(396\) 0 0
\(397\) 9.05172 0.454293 0.227146 0.973861i \(-0.427060\pi\)
0.227146 + 0.973861i \(0.427060\pi\)
\(398\) 0 0
\(399\) 6.52359 0.326588
\(400\) 0 0
\(401\) 19.7587 0.986704 0.493352 0.869830i \(-0.335772\pi\)
0.493352 + 0.869830i \(0.335772\pi\)
\(402\) 0 0
\(403\) 9.26180 0.461363
\(404\) 0 0
\(405\) 8.75872 0.435224
\(406\) 0 0
\(407\) −0.451356 −0.0223729
\(408\) 0 0
\(409\) −1.71769 −0.0849341 −0.0424670 0.999098i \(-0.513522\pi\)
−0.0424670 + 0.999098i \(0.513522\pi\)
\(410\) 0 0
\(411\) 23.1650 1.14264
\(412\) 0 0
\(413\) 6.15676 0.302954
\(414\) 0 0
\(415\) 2.78765 0.136841
\(416\) 0 0
\(417\) −8.36683 −0.409726
\(418\) 0 0
\(419\) 35.5318 1.73584 0.867922 0.496701i \(-0.165456\pi\)
0.867922 + 0.496701i \(0.165456\pi\)
\(420\) 0 0
\(421\) −12.0722 −0.588365 −0.294182 0.955749i \(-0.595047\pi\)
−0.294182 + 0.955749i \(0.595047\pi\)
\(422\) 0 0
\(423\) −0.0883965 −0.00429798
\(424\) 0 0
\(425\) 4.97107 0.241132
\(426\) 0 0
\(427\) −1.94214 −0.0939868
\(428\) 0 0
\(429\) −0.457980 −0.0221115
\(430\) 0 0
\(431\) −19.8310 −0.955224 −0.477612 0.878571i \(-0.658497\pi\)
−0.477612 + 0.878571i \(0.658497\pi\)
\(432\) 0 0
\(433\) 14.8143 0.711931 0.355965 0.934499i \(-0.384152\pi\)
0.355965 + 0.934499i \(0.384152\pi\)
\(434\) 0 0
\(435\) 1.70928 0.0819535
\(436\) 0 0
\(437\) −13.8576 −0.662900
\(438\) 0 0
\(439\) 17.8576 0.852298 0.426149 0.904653i \(-0.359870\pi\)
0.426149 + 0.904653i \(0.359870\pi\)
\(440\) 0 0
\(441\) 0.517447 0.0246404
\(442\) 0 0
\(443\) −33.5936 −1.59608 −0.798039 0.602606i \(-0.794129\pi\)
−0.798039 + 0.602606i \(0.794129\pi\)
\(444\) 0 0
\(445\) −4.73820 −0.224612
\(446\) 0 0
\(447\) 21.4063 1.01248
\(448\) 0 0
\(449\) 7.07838 0.334049 0.167025 0.985953i \(-0.446584\pi\)
0.167025 + 0.985953i \(0.446584\pi\)
\(450\) 0 0
\(451\) −0.0988967 −0.00465686
\(452\) 0 0
\(453\) 12.9939 0.610505
\(454\) 0 0
\(455\) −0.581449 −0.0272588
\(456\) 0 0
\(457\) −5.81658 −0.272088 −0.136044 0.990703i \(-0.543439\pi\)
−0.136044 + 0.990703i \(0.543439\pi\)
\(458\) 0 0
\(459\) 26.1568 1.22089
\(460\) 0 0
\(461\) −32.3090 −1.50478 −0.752390 0.658718i \(-0.771099\pi\)
−0.752390 + 0.658718i \(0.771099\pi\)
\(462\) 0 0
\(463\) −1.44134 −0.0669846 −0.0334923 0.999439i \(-0.510663\pi\)
−0.0334923 + 0.999439i \(0.510663\pi\)
\(464\) 0 0
\(465\) −17.1773 −0.796577
\(466\) 0 0
\(467\) 11.7503 0.543740 0.271870 0.962334i \(-0.412358\pi\)
0.271870 + 0.962334i \(0.412358\pi\)
\(468\) 0 0
\(469\) −3.60197 −0.166323
\(470\) 0 0
\(471\) 42.5380 1.96005
\(472\) 0 0
\(473\) −1.65983 −0.0763189
\(474\) 0 0
\(475\) 6.04945 0.277568
\(476\) 0 0
\(477\) 0.0266620 0.00122077
\(478\) 0 0
\(479\) −17.1689 −0.784465 −0.392233 0.919866i \(-0.628297\pi\)
−0.392233 + 0.919866i \(0.628297\pi\)
\(480\) 0 0
\(481\) −1.43084 −0.0652405
\(482\) 0 0
\(483\) −2.47027 −0.112401
\(484\) 0 0
\(485\) 15.8927 0.721650
\(486\) 0 0
\(487\) 4.10277 0.185914 0.0929572 0.995670i \(-0.470368\pi\)
0.0929572 + 0.995670i \(0.470368\pi\)
\(488\) 0 0
\(489\) 0.764867 0.0345885
\(490\) 0 0
\(491\) −40.7708 −1.83996 −0.919981 0.391963i \(-0.871796\pi\)
−0.919981 + 0.391963i \(0.871796\pi\)
\(492\) 0 0
\(493\) 4.97107 0.223886
\(494\) 0 0
\(495\) −0.0227863 −0.00102417
\(496\) 0 0
\(497\) 5.72753 0.256915
\(498\) 0 0
\(499\) 18.4703 0.826843 0.413421 0.910540i \(-0.364334\pi\)
0.413421 + 0.910540i \(0.364334\pi\)
\(500\) 0 0
\(501\) 33.9565 1.51707
\(502\) 0 0
\(503\) 21.4947 0.958400 0.479200 0.877706i \(-0.340927\pi\)
0.479200 + 0.877706i \(0.340927\pi\)
\(504\) 0 0
\(505\) 12.2557 0.545369
\(506\) 0 0
\(507\) 20.7687 0.922372
\(508\) 0 0
\(509\) 3.75872 0.166602 0.0833012 0.996524i \(-0.473454\pi\)
0.0833012 + 0.996524i \(0.473454\pi\)
\(510\) 0 0
\(511\) 4.38121 0.193813
\(512\) 0 0
\(513\) 31.8310 1.40537
\(514\) 0 0
\(515\) −7.86603 −0.346619
\(516\) 0 0
\(517\) −0.327887 −0.0144204
\(518\) 0 0
\(519\) 43.4719 1.90820
\(520\) 0 0
\(521\) 12.8059 0.561037 0.280518 0.959849i \(-0.409494\pi\)
0.280518 + 0.959849i \(0.409494\pi\)
\(522\) 0 0
\(523\) 21.1278 0.923855 0.461928 0.886918i \(-0.347158\pi\)
0.461928 + 0.886918i \(0.347158\pi\)
\(524\) 0 0
\(525\) 1.07838 0.0470643
\(526\) 0 0
\(527\) −49.9565 −2.17614
\(528\) 0 0
\(529\) −17.7526 −0.771851
\(530\) 0 0
\(531\) 0.764867 0.0331924
\(532\) 0 0
\(533\) −0.313511 −0.0135797
\(534\) 0 0
\(535\) 12.7298 0.550357
\(536\) 0 0
\(537\) 25.3607 1.09439
\(538\) 0 0
\(539\) 1.91935 0.0826725
\(540\) 0 0
\(541\) 32.7382 1.40753 0.703763 0.710435i \(-0.251502\pi\)
0.703763 + 0.710435i \(0.251502\pi\)
\(542\) 0 0
\(543\) 10.1112 0.433912
\(544\) 0 0
\(545\) −12.4391 −0.532831
\(546\) 0 0
\(547\) 22.1073 0.945240 0.472620 0.881266i \(-0.343308\pi\)
0.472620 + 0.881266i \(0.343308\pi\)
\(548\) 0 0
\(549\) −0.241276 −0.0102974
\(550\) 0 0
\(551\) 6.04945 0.257715
\(552\) 0 0
\(553\) 7.81658 0.332395
\(554\) 0 0
\(555\) 2.65368 0.112643
\(556\) 0 0
\(557\) −39.8720 −1.68943 −0.844715 0.535216i \(-0.820230\pi\)
−0.844715 + 0.535216i \(0.820230\pi\)
\(558\) 0 0
\(559\) −5.26180 −0.222550
\(560\) 0 0
\(561\) 2.47027 0.104295
\(562\) 0 0
\(563\) −10.1217 −0.426578 −0.213289 0.976989i \(-0.568418\pi\)
−0.213289 + 0.976989i \(0.568418\pi\)
\(564\) 0 0
\(565\) −12.5730 −0.528952
\(566\) 0 0
\(567\) 5.52586 0.232064
\(568\) 0 0
\(569\) −24.4391 −1.02454 −0.512270 0.858825i \(-0.671195\pi\)
−0.512270 + 0.858825i \(0.671195\pi\)
\(570\) 0 0
\(571\) 28.2511 1.18227 0.591136 0.806572i \(-0.298680\pi\)
0.591136 + 0.806572i \(0.298680\pi\)
\(572\) 0 0
\(573\) 12.0144 0.501908
\(574\) 0 0
\(575\) −2.29072 −0.0955298
\(576\) 0 0
\(577\) 46.1171 1.91988 0.959941 0.280202i \(-0.0904015\pi\)
0.959941 + 0.280202i \(0.0904015\pi\)
\(578\) 0 0
\(579\) −30.5380 −1.26911
\(580\) 0 0
\(581\) 1.75872 0.0729642
\(582\) 0 0
\(583\) 0.0988967 0.00409588
\(584\) 0 0
\(585\) −0.0722347 −0.00298654
\(586\) 0 0
\(587\) 0.715418 0.0295285 0.0147642 0.999891i \(-0.495300\pi\)
0.0147642 + 0.999891i \(0.495300\pi\)
\(588\) 0 0
\(589\) −60.7936 −2.50496
\(590\) 0 0
\(591\) −10.4247 −0.428815
\(592\) 0 0
\(593\) 15.5441 0.638320 0.319160 0.947701i \(-0.396599\pi\)
0.319160 + 0.947701i \(0.396599\pi\)
\(594\) 0 0
\(595\) 3.13624 0.128573
\(596\) 0 0
\(597\) −16.6803 −0.682681
\(598\) 0 0
\(599\) 9.59809 0.392167 0.196084 0.980587i \(-0.437178\pi\)
0.196084 + 0.980587i \(0.437178\pi\)
\(600\) 0 0
\(601\) 6.81044 0.277804 0.138902 0.990306i \(-0.455643\pi\)
0.138902 + 0.990306i \(0.455643\pi\)
\(602\) 0 0
\(603\) −0.447480 −0.0182228
\(604\) 0 0
\(605\) 10.9155 0.443777
\(606\) 0 0
\(607\) −31.6970 −1.28654 −0.643271 0.765639i \(-0.722423\pi\)
−0.643271 + 0.765639i \(0.722423\pi\)
\(608\) 0 0
\(609\) 1.07838 0.0436981
\(610\) 0 0
\(611\) −1.03943 −0.0420508
\(612\) 0 0
\(613\) 1.20394 0.0486265 0.0243133 0.999704i \(-0.492260\pi\)
0.0243133 + 0.999704i \(0.492260\pi\)
\(614\) 0 0
\(615\) 0.581449 0.0234463
\(616\) 0 0
\(617\) −37.9337 −1.52715 −0.763577 0.645716i \(-0.776559\pi\)
−0.763577 + 0.645716i \(0.776559\pi\)
\(618\) 0 0
\(619\) 4.60424 0.185060 0.0925299 0.995710i \(-0.470505\pi\)
0.0925299 + 0.995710i \(0.470505\pi\)
\(620\) 0 0
\(621\) −12.0533 −0.483683
\(622\) 0 0
\(623\) −2.98932 −0.119765
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 3.00614 0.120054
\(628\) 0 0
\(629\) 7.71769 0.307724
\(630\) 0 0
\(631\) 8.41241 0.334893 0.167446 0.985881i \(-0.446448\pi\)
0.167446 + 0.985881i \(0.446448\pi\)
\(632\) 0 0
\(633\) 16.8638 0.670274
\(634\) 0 0
\(635\) −20.9132 −0.829915
\(636\) 0 0
\(637\) 6.08452 0.241077
\(638\) 0 0
\(639\) 0.711543 0.0281482
\(640\) 0 0
\(641\) 32.5380 1.28517 0.642586 0.766213i \(-0.277861\pi\)
0.642586 + 0.766213i \(0.277861\pi\)
\(642\) 0 0
\(643\) 2.09293 0.0825372 0.0412686 0.999148i \(-0.486860\pi\)
0.0412686 + 0.999148i \(0.486860\pi\)
\(644\) 0 0
\(645\) 9.75872 0.384249
\(646\) 0 0
\(647\) −45.1955 −1.77682 −0.888410 0.459051i \(-0.848189\pi\)
−0.888410 + 0.459051i \(0.848189\pi\)
\(648\) 0 0
\(649\) 2.83710 0.111366
\(650\) 0 0
\(651\) −10.8371 −0.424739
\(652\) 0 0
\(653\) −2.14834 −0.0840712 −0.0420356 0.999116i \(-0.513384\pi\)
−0.0420356 + 0.999116i \(0.513384\pi\)
\(654\) 0 0
\(655\) −13.4680 −0.526238
\(656\) 0 0
\(657\) 0.544287 0.0212347
\(658\) 0 0
\(659\) 45.0843 1.75624 0.878118 0.478444i \(-0.158799\pi\)
0.878118 + 0.478444i \(0.158799\pi\)
\(660\) 0 0
\(661\) −36.3234 −1.41281 −0.706407 0.707806i \(-0.749685\pi\)
−0.706407 + 0.707806i \(0.749685\pi\)
\(662\) 0 0
\(663\) 7.83096 0.304129
\(664\) 0 0
\(665\) 3.81658 0.148001
\(666\) 0 0
\(667\) −2.29072 −0.0886972
\(668\) 0 0
\(669\) 18.7526 0.725017
\(670\) 0 0
\(671\) −0.894960 −0.0345496
\(672\) 0 0
\(673\) −17.4719 −0.673491 −0.336746 0.941596i \(-0.609326\pi\)
−0.336746 + 0.941596i \(0.609326\pi\)
\(674\) 0 0
\(675\) 5.26180 0.202527
\(676\) 0 0
\(677\) −40.0372 −1.53875 −0.769377 0.638796i \(-0.779433\pi\)
−0.769377 + 0.638796i \(0.779433\pi\)
\(678\) 0 0
\(679\) 10.0267 0.384788
\(680\) 0 0
\(681\) −21.4452 −0.821782
\(682\) 0 0
\(683\) −2.07611 −0.0794402 −0.0397201 0.999211i \(-0.512647\pi\)
−0.0397201 + 0.999211i \(0.512647\pi\)
\(684\) 0 0
\(685\) 13.5525 0.517815
\(686\) 0 0
\(687\) −39.9299 −1.52342
\(688\) 0 0
\(689\) 0.313511 0.0119438
\(690\) 0 0
\(691\) −26.7070 −1.01598 −0.507991 0.861362i \(-0.669612\pi\)
−0.507991 + 0.861362i \(0.669612\pi\)
\(692\) 0 0
\(693\) −0.0143758 −0.000546093 0
\(694\) 0 0
\(695\) −4.89496 −0.185676
\(696\) 0 0
\(697\) 1.69102 0.0640521
\(698\) 0 0
\(699\) −21.3151 −0.806212
\(700\) 0 0
\(701\) −21.9155 −0.827736 −0.413868 0.910337i \(-0.635823\pi\)
−0.413868 + 0.910337i \(0.635823\pi\)
\(702\) 0 0
\(703\) 9.39189 0.354222
\(704\) 0 0
\(705\) 1.92777 0.0726038
\(706\) 0 0
\(707\) 7.73206 0.290794
\(708\) 0 0
\(709\) −4.60811 −0.173061 −0.0865306 0.996249i \(-0.527578\pi\)
−0.0865306 + 0.996249i \(0.527578\pi\)
\(710\) 0 0
\(711\) 0.971071 0.0364180
\(712\) 0 0
\(713\) 23.0205 0.862125
\(714\) 0 0
\(715\) −0.267938 −0.0100203
\(716\) 0 0
\(717\) 23.5174 0.878275
\(718\) 0 0
\(719\) −6.80590 −0.253817 −0.126909 0.991914i \(-0.540506\pi\)
−0.126909 + 0.991914i \(0.540506\pi\)
\(720\) 0 0
\(721\) −4.96266 −0.184819
\(722\) 0 0
\(723\) 25.0928 0.933210
\(724\) 0 0
\(725\) 1.00000 0.0371391
\(726\) 0 0
\(727\) 26.9711 1.00030 0.500151 0.865938i \(-0.333278\pi\)
0.500151 + 0.865938i \(0.333278\pi\)
\(728\) 0 0
\(729\) 27.6681 1.02474
\(730\) 0 0
\(731\) 28.3812 1.04972
\(732\) 0 0
\(733\) −30.0638 −1.11043 −0.555216 0.831706i \(-0.687365\pi\)
−0.555216 + 0.831706i \(0.687365\pi\)
\(734\) 0 0
\(735\) −11.2846 −0.416238
\(736\) 0 0
\(737\) −1.65983 −0.0611405
\(738\) 0 0
\(739\) −51.1422 −1.88130 −0.940648 0.339383i \(-0.889782\pi\)
−0.940648 + 0.339383i \(0.889782\pi\)
\(740\) 0 0
\(741\) 9.52973 0.350084
\(742\) 0 0
\(743\) −11.1857 −0.410363 −0.205181 0.978724i \(-0.565778\pi\)
−0.205181 + 0.978724i \(0.565778\pi\)
\(744\) 0 0
\(745\) 12.5236 0.458829
\(746\) 0 0
\(747\) 0.218490 0.00799413
\(748\) 0 0
\(749\) 8.03120 0.293454
\(750\) 0 0
\(751\) 18.3630 0.670074 0.335037 0.942205i \(-0.391251\pi\)
0.335037 + 0.942205i \(0.391251\pi\)
\(752\) 0 0
\(753\) 26.3935 0.961832
\(754\) 0 0
\(755\) 7.60197 0.276664
\(756\) 0 0
\(757\) 15.8927 0.577630 0.288815 0.957385i \(-0.406739\pi\)
0.288815 + 0.957385i \(0.406739\pi\)
\(758\) 0 0
\(759\) −1.13833 −0.0413186
\(760\) 0 0
\(761\) 13.8843 0.503305 0.251652 0.967818i \(-0.419026\pi\)
0.251652 + 0.967818i \(0.419026\pi\)
\(762\) 0 0
\(763\) −7.84778 −0.284109
\(764\) 0 0
\(765\) 0.389621 0.0140868
\(766\) 0 0
\(767\) 8.99386 0.324749
\(768\) 0 0
\(769\) −35.4063 −1.27678 −0.638391 0.769712i \(-0.720400\pi\)
−0.638391 + 0.769712i \(0.720400\pi\)
\(770\) 0 0
\(771\) 10.7382 0.386727
\(772\) 0 0
\(773\) −0.488518 −0.0175708 −0.00878539 0.999961i \(-0.502797\pi\)
−0.00878539 + 0.999961i \(0.502797\pi\)
\(774\) 0 0
\(775\) −10.0494 −0.360987
\(776\) 0 0
\(777\) 1.67420 0.0600617
\(778\) 0 0
\(779\) 2.05786 0.0737304
\(780\) 0 0
\(781\) 2.63931 0.0944419
\(782\) 0 0
\(783\) 5.26180 0.188041
\(784\) 0 0
\(785\) 24.8865 0.888239
\(786\) 0 0
\(787\) 1.99159 0.0709925 0.0354962 0.999370i \(-0.488699\pi\)
0.0354962 + 0.999370i \(0.488699\pi\)
\(788\) 0 0
\(789\) 17.2228 0.613150
\(790\) 0 0
\(791\) −7.93230 −0.282040
\(792\) 0 0
\(793\) −2.83710 −0.100748
\(794\) 0 0
\(795\) −0.581449 −0.0206219
\(796\) 0 0
\(797\) −17.2702 −0.611742 −0.305871 0.952073i \(-0.598948\pi\)
−0.305871 + 0.952073i \(0.598948\pi\)
\(798\) 0 0
\(799\) 5.60650 0.198344
\(800\) 0 0
\(801\) −0.371370 −0.0131217
\(802\) 0 0
\(803\) 2.01891 0.0712458
\(804\) 0 0
\(805\) −1.44521 −0.0509371
\(806\) 0 0
\(807\) −48.1522 −1.69504
\(808\) 0 0
\(809\) −56.5068 −1.98667 −0.993336 0.115254i \(-0.963232\pi\)
−0.993336 + 0.115254i \(0.963232\pi\)
\(810\) 0 0
\(811\) 8.77924 0.308281 0.154140 0.988049i \(-0.450739\pi\)
0.154140 + 0.988049i \(0.450739\pi\)
\(812\) 0 0
\(813\) 49.2306 1.72659
\(814\) 0 0
\(815\) 0.447480 0.0156745
\(816\) 0 0
\(817\) 34.5380 1.20833
\(818\) 0 0
\(819\) −0.0455727 −0.00159244
\(820\) 0 0
\(821\) −28.1568 −0.982678 −0.491339 0.870969i \(-0.663492\pi\)
−0.491339 + 0.870969i \(0.663492\pi\)
\(822\) 0 0
\(823\) −20.7442 −0.723096 −0.361548 0.932353i \(-0.617752\pi\)
−0.361548 + 0.932353i \(0.617752\pi\)
\(824\) 0 0
\(825\) 0.496928 0.0173008
\(826\) 0 0
\(827\) 9.12783 0.317406 0.158703 0.987326i \(-0.449269\pi\)
0.158703 + 0.987326i \(0.449269\pi\)
\(828\) 0 0
\(829\) 31.8576 1.10646 0.553230 0.833028i \(-0.313395\pi\)
0.553230 + 0.833028i \(0.313395\pi\)
\(830\) 0 0
\(831\) −0.0455727 −0.00158090
\(832\) 0 0
\(833\) −32.8188 −1.13711
\(834\) 0 0
\(835\) 19.8660 0.687492
\(836\) 0 0
\(837\) −52.8781 −1.82774
\(838\) 0 0
\(839\) −27.4413 −0.947380 −0.473690 0.880692i \(-0.657078\pi\)
−0.473690 + 0.880692i \(0.657078\pi\)
\(840\) 0 0
\(841\) 1.00000 0.0344828
\(842\) 0 0
\(843\) 47.9832 1.65263
\(844\) 0 0
\(845\) 12.1506 0.417994
\(846\) 0 0
\(847\) 6.88655 0.236625
\(848\) 0 0
\(849\) −35.5774 −1.22101
\(850\) 0 0
\(851\) −3.55640 −0.121912
\(852\) 0 0
\(853\) −56.0515 −1.91917 −0.959584 0.281422i \(-0.909194\pi\)
−0.959584 + 0.281422i \(0.909194\pi\)
\(854\) 0 0
\(855\) 0.474142 0.0162153
\(856\) 0 0
\(857\) −6.08452 −0.207843 −0.103922 0.994585i \(-0.533139\pi\)
−0.103922 + 0.994585i \(0.533139\pi\)
\(858\) 0 0
\(859\) −35.5936 −1.21444 −0.607218 0.794535i \(-0.707715\pi\)
−0.607218 + 0.794535i \(0.707715\pi\)
\(860\) 0 0
\(861\) 0.366835 0.0125017
\(862\) 0 0
\(863\) 12.8287 0.436694 0.218347 0.975871i \(-0.429934\pi\)
0.218347 + 0.975871i \(0.429934\pi\)
\(864\) 0 0
\(865\) 25.4329 0.864745
\(866\) 0 0
\(867\) −13.1812 −0.447655
\(868\) 0 0
\(869\) 3.60197 0.122188
\(870\) 0 0
\(871\) −5.26180 −0.178289
\(872\) 0 0
\(873\) 1.24563 0.0421583
\(874\) 0 0
\(875\) 0.630898 0.0213282
\(876\) 0 0
\(877\) −1.50307 −0.0507551 −0.0253776 0.999678i \(-0.508079\pi\)
−0.0253776 + 0.999678i \(0.508079\pi\)
\(878\) 0 0
\(879\) 26.3402 0.888432
\(880\) 0 0
\(881\) 23.4908 0.791425 0.395712 0.918375i \(-0.370498\pi\)
0.395712 + 0.918375i \(0.370498\pi\)
\(882\) 0 0
\(883\) 29.0433 0.977385 0.488693 0.872456i \(-0.337474\pi\)
0.488693 + 0.872456i \(0.337474\pi\)
\(884\) 0 0
\(885\) −16.6803 −0.560704
\(886\) 0 0
\(887\) 19.0700 0.640307 0.320153 0.947366i \(-0.396266\pi\)
0.320153 + 0.947366i \(0.396266\pi\)
\(888\) 0 0
\(889\) −13.1941 −0.442516
\(890\) 0 0
\(891\) 2.54638 0.0853068
\(892\) 0 0
\(893\) 6.82273 0.228314
\(894\) 0 0
\(895\) 14.8371 0.495950
\(896\) 0 0
\(897\) −3.60859 −0.120487
\(898\) 0 0
\(899\) −10.0494 −0.335168
\(900\) 0 0
\(901\) −1.69102 −0.0563362
\(902\) 0 0
\(903\) 6.15676 0.204884
\(904\) 0 0
\(905\) 5.91548 0.196637
\(906\) 0 0
\(907\) −5.54023 −0.183960 −0.0919802 0.995761i \(-0.529320\pi\)
−0.0919802 + 0.995761i \(0.529320\pi\)
\(908\) 0 0
\(909\) 0.960570 0.0318601
\(910\) 0 0
\(911\) −53.2990 −1.76587 −0.882937 0.469492i \(-0.844437\pi\)
−0.882937 + 0.469492i \(0.844437\pi\)
\(912\) 0 0
\(913\) 0.810439 0.0268216
\(914\) 0 0
\(915\) 5.26180 0.173950
\(916\) 0 0
\(917\) −8.49693 −0.280593
\(918\) 0 0
\(919\) −37.5897 −1.23997 −0.619985 0.784614i \(-0.712861\pi\)
−0.619985 + 0.784614i \(0.712861\pi\)
\(920\) 0 0
\(921\) −48.5958 −1.60129
\(922\) 0 0
\(923\) 8.36683 0.275398
\(924\) 0 0
\(925\) 1.55252 0.0510465
\(926\) 0 0
\(927\) −0.616522 −0.0202492
\(928\) 0 0
\(929\) 37.3197 1.22442 0.612209 0.790696i \(-0.290281\pi\)
0.612209 + 0.790696i \(0.290281\pi\)
\(930\) 0 0
\(931\) −39.9383 −1.30892
\(932\) 0 0
\(933\) −33.5441 −1.09818
\(934\) 0 0
\(935\) 1.44521 0.0472635
\(936\) 0 0
\(937\) −22.8638 −0.746927 −0.373463 0.927645i \(-0.621830\pi\)
−0.373463 + 0.927645i \(0.621830\pi\)
\(938\) 0 0
\(939\) −39.1584 −1.27788
\(940\) 0 0
\(941\) 0.523590 0.0170686 0.00853428 0.999964i \(-0.497283\pi\)
0.00853428 + 0.999964i \(0.497283\pi\)
\(942\) 0 0
\(943\) −0.779243 −0.0253756
\(944\) 0 0
\(945\) 3.31965 0.107988
\(946\) 0 0
\(947\) 10.0228 0.325697 0.162848 0.986651i \(-0.447932\pi\)
0.162848 + 0.986651i \(0.447932\pi\)
\(948\) 0 0
\(949\) 6.40012 0.207757
\(950\) 0 0
\(951\) −39.0738 −1.26706
\(952\) 0 0
\(953\) −8.15676 −0.264223 −0.132112 0.991235i \(-0.542176\pi\)
−0.132112 + 0.991235i \(0.542176\pi\)
\(954\) 0 0
\(955\) 7.02893 0.227451
\(956\) 0 0
\(957\) 0.496928 0.0160634
\(958\) 0 0
\(959\) 8.55025 0.276102
\(960\) 0 0
\(961\) 69.9914 2.25779
\(962\) 0 0
\(963\) 0.997733 0.0321515
\(964\) 0 0
\(965\) −17.8660 −0.575128
\(966\) 0 0
\(967\) −15.7671 −0.507037 −0.253518 0.967331i \(-0.581588\pi\)
−0.253518 + 0.967331i \(0.581588\pi\)
\(968\) 0 0
\(969\) −51.4017 −1.65126
\(970\) 0 0
\(971\) 17.8804 0.573810 0.286905 0.957959i \(-0.407374\pi\)
0.286905 + 0.957959i \(0.407374\pi\)
\(972\) 0 0
\(973\) −3.08822 −0.0990037
\(974\) 0 0
\(975\) 1.57531 0.0504502
\(976\) 0 0
\(977\) −55.1071 −1.76303 −0.881517 0.472153i \(-0.843477\pi\)
−0.881517 + 0.472153i \(0.843477\pi\)
\(978\) 0 0
\(979\) −1.37751 −0.0440255
\(980\) 0 0
\(981\) −0.974946 −0.0311276
\(982\) 0 0
\(983\) 1.29687 0.0413637 0.0206818 0.999786i \(-0.493416\pi\)
0.0206818 + 0.999786i \(0.493416\pi\)
\(984\) 0 0
\(985\) −6.09890 −0.194327
\(986\) 0 0
\(987\) 1.21622 0.0387128
\(988\) 0 0
\(989\) −13.0784 −0.415868
\(990\) 0 0
\(991\) 3.11942 0.0990915 0.0495458 0.998772i \(-0.484223\pi\)
0.0495458 + 0.998772i \(0.484223\pi\)
\(992\) 0 0
\(993\) 41.1773 1.30672
\(994\) 0 0
\(995\) −9.75872 −0.309372
\(996\) 0 0
\(997\) 30.2472 0.957940 0.478970 0.877831i \(-0.341010\pi\)
0.478970 + 0.877831i \(0.341010\pi\)
\(998\) 0 0
\(999\) 8.16904 0.258457
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 2320.2.a.s.1.1 3
4.3 odd 2 145.2.a.d.1.2 3
8.3 odd 2 9280.2.a.bu.1.1 3
8.5 even 2 9280.2.a.bm.1.3 3
12.11 even 2 1305.2.a.o.1.2 3
20.3 even 4 725.2.b.d.349.2 6
20.7 even 4 725.2.b.d.349.5 6
20.19 odd 2 725.2.a.d.1.2 3
28.27 even 2 7105.2.a.p.1.2 3
60.59 even 2 6525.2.a.bh.1.2 3
116.115 odd 2 4205.2.a.e.1.2 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
145.2.a.d.1.2 3 4.3 odd 2
725.2.a.d.1.2 3 20.19 odd 2
725.2.b.d.349.2 6 20.3 even 4
725.2.b.d.349.5 6 20.7 even 4
1305.2.a.o.1.2 3 12.11 even 2
2320.2.a.s.1.1 3 1.1 even 1 trivial
4205.2.a.e.1.2 3 116.115 odd 2
6525.2.a.bh.1.2 3 60.59 even 2
7105.2.a.p.1.2 3 28.27 even 2
9280.2.a.bm.1.3 3 8.5 even 2
9280.2.a.bu.1.1 3 8.3 odd 2