Properties

Label 5625.2.a.bc.1.8
Level $5625$
Weight $2$
Character 5625.1
Self dual yes
Analytic conductor $44.916$
Analytic rank $1$
Dimension $8$
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] = [5625,2,Mod(1,5625)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(5625, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("5625.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 5625 = 3^{2} \cdot 5^{4} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5625.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: \(44.9158511370\)
Analytic rank: \(1\)
Dimension: \(8\)
Coefficient field: 8.8.13366265625.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{7} - 12x^{6} + 10x^{5} + 41x^{4} - 20x^{3} - 48x^{2} + 8x + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1875)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.8
Root \(2.59716\) of defining polynomial
Character \(\chi\) \(=\) 5625.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+2.59716 q^{2} +4.74525 q^{4} -3.28414 q^{7} +7.12986 q^{8} +O(q^{10})\) \(q+2.59716 q^{2} +4.74525 q^{4} -3.28414 q^{7} +7.12986 q^{8} -4.30834 q^{11} +3.46120 q^{13} -8.52943 q^{14} +9.02691 q^{16} -5.44757 q^{17} -7.63427 q^{19} -11.1895 q^{22} -5.04986 q^{23} +8.98929 q^{26} -15.5841 q^{28} +3.12329 q^{29} -2.06658 q^{31} +9.18462 q^{32} -14.1482 q^{34} +1.89142 q^{37} -19.8274 q^{38} -3.89896 q^{41} +3.20815 q^{43} -20.4442 q^{44} -13.1153 q^{46} +6.28577 q^{47} +3.78555 q^{49} +16.4243 q^{52} -2.51480 q^{53} -23.4154 q^{56} +8.11170 q^{58} -7.72948 q^{59} -2.95661 q^{61} -5.36725 q^{62} +5.80013 q^{64} -12.9952 q^{67} -25.8501 q^{68} +4.72665 q^{71} +1.64933 q^{73} +4.91232 q^{74} -36.2265 q^{76} +14.1492 q^{77} +13.7349 q^{79} -10.1262 q^{82} -5.01687 q^{83} +8.33208 q^{86} -30.7179 q^{88} +9.00209 q^{89} -11.3670 q^{91} -23.9628 q^{92} +16.3252 q^{94} -2.57278 q^{97} +9.83169 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + q^{2} + 9 q^{4} - 12 q^{7} + 3 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + q^{2} + 9 q^{4} - 12 q^{7} + 3 q^{8} - 12 q^{11} - 14 q^{13} - 16 q^{14} + 15 q^{16} - q^{17} + 16 q^{19} - 18 q^{22} - 4 q^{23} + 34 q^{26} + 21 q^{28} - 2 q^{29} + 13 q^{31} - 18 q^{32} - 37 q^{34} + 8 q^{37} - 24 q^{38} + 12 q^{41} - 20 q^{43} - 47 q^{44} + 33 q^{46} - 15 q^{47} + 30 q^{49} + q^{52} - 4 q^{53} - 60 q^{56} - 2 q^{58} - 14 q^{59} + 10 q^{61} + 4 q^{62} + 41 q^{64} - 19 q^{67} - 33 q^{68} - 21 q^{71} + 19 q^{73} + 9 q^{74} - q^{76} - 11 q^{77} + 10 q^{79} - 24 q^{82} - 27 q^{83} - 42 q^{86} - 53 q^{88} + 9 q^{89} - 12 q^{91} - 63 q^{92} + 14 q^{94} - 24 q^{97} - 24 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.59716 1.83647 0.918235 0.396035i \(-0.129614\pi\)
0.918235 + 0.396035i \(0.129614\pi\)
\(3\) 0 0
\(4\) 4.74525 2.37263
\(5\) 0 0
\(6\) 0 0
\(7\) −3.28414 −1.24129 −0.620643 0.784093i \(-0.713129\pi\)
−0.620643 + 0.784093i \(0.713129\pi\)
\(8\) 7.12986 2.52079
\(9\) 0 0
\(10\) 0 0
\(11\) −4.30834 −1.29901 −0.649507 0.760356i \(-0.725025\pi\)
−0.649507 + 0.760356i \(0.725025\pi\)
\(12\) 0 0
\(13\) 3.46120 0.959964 0.479982 0.877278i \(-0.340643\pi\)
0.479982 + 0.877278i \(0.340643\pi\)
\(14\) −8.52943 −2.27959
\(15\) 0 0
\(16\) 9.02691 2.25673
\(17\) −5.44757 −1.32123 −0.660615 0.750725i \(-0.729705\pi\)
−0.660615 + 0.750725i \(0.729705\pi\)
\(18\) 0 0
\(19\) −7.63427 −1.75142 −0.875711 0.482835i \(-0.839607\pi\)
−0.875711 + 0.482835i \(0.839607\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) −11.1895 −2.38560
\(23\) −5.04986 −1.05297 −0.526484 0.850185i \(-0.676490\pi\)
−0.526484 + 0.850185i \(0.676490\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 8.98929 1.76295
\(27\) 0 0
\(28\) −15.5841 −2.94511
\(29\) 3.12329 0.579981 0.289990 0.957030i \(-0.406348\pi\)
0.289990 + 0.957030i \(0.406348\pi\)
\(30\) 0 0
\(31\) −2.06658 −0.371169 −0.185584 0.982628i \(-0.559418\pi\)
−0.185584 + 0.982628i \(0.559418\pi\)
\(32\) 9.18462 1.62363
\(33\) 0 0
\(34\) −14.1482 −2.42640
\(35\) 0 0
\(36\) 0 0
\(37\) 1.89142 0.310947 0.155474 0.987840i \(-0.450310\pi\)
0.155474 + 0.987840i \(0.450310\pi\)
\(38\) −19.8274 −3.21644
\(39\) 0 0
\(40\) 0 0
\(41\) −3.89896 −0.608915 −0.304457 0.952526i \(-0.598475\pi\)
−0.304457 + 0.952526i \(0.598475\pi\)
\(42\) 0 0
\(43\) 3.20815 0.489238 0.244619 0.969619i \(-0.421337\pi\)
0.244619 + 0.969619i \(0.421337\pi\)
\(44\) −20.4442 −3.08207
\(45\) 0 0
\(46\) −13.1153 −1.93375
\(47\) 6.28577 0.916874 0.458437 0.888727i \(-0.348409\pi\)
0.458437 + 0.888727i \(0.348409\pi\)
\(48\) 0 0
\(49\) 3.78555 0.540793
\(50\) 0 0
\(51\) 0 0
\(52\) 16.4243 2.27763
\(53\) −2.51480 −0.345434 −0.172717 0.984971i \(-0.555255\pi\)
−0.172717 + 0.984971i \(0.555255\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −23.4154 −3.12902
\(57\) 0 0
\(58\) 8.11170 1.06512
\(59\) −7.72948 −1.00629 −0.503146 0.864201i \(-0.667824\pi\)
−0.503146 + 0.864201i \(0.667824\pi\)
\(60\) 0 0
\(61\) −2.95661 −0.378555 −0.189278 0.981924i \(-0.560615\pi\)
−0.189278 + 0.981924i \(0.560615\pi\)
\(62\) −5.36725 −0.681641
\(63\) 0 0
\(64\) 5.80013 0.725016
\(65\) 0 0
\(66\) 0 0
\(67\) −12.9952 −1.58762 −0.793811 0.608165i \(-0.791906\pi\)
−0.793811 + 0.608165i \(0.791906\pi\)
\(68\) −25.8501 −3.13479
\(69\) 0 0
\(70\) 0 0
\(71\) 4.72665 0.560950 0.280475 0.959861i \(-0.409508\pi\)
0.280475 + 0.959861i \(0.409508\pi\)
\(72\) 0 0
\(73\) 1.64933 0.193040 0.0965198 0.995331i \(-0.469229\pi\)
0.0965198 + 0.995331i \(0.469229\pi\)
\(74\) 4.91232 0.571046
\(75\) 0 0
\(76\) −36.2265 −4.15547
\(77\) 14.1492 1.61245
\(78\) 0 0
\(79\) 13.7349 1.54530 0.772648 0.634835i \(-0.218932\pi\)
0.772648 + 0.634835i \(0.218932\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) −10.1262 −1.11825
\(83\) −5.01687 −0.550673 −0.275336 0.961348i \(-0.588789\pi\)
−0.275336 + 0.961348i \(0.588789\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 8.33208 0.898472
\(87\) 0 0
\(88\) −30.7179 −3.27454
\(89\) 9.00209 0.954220 0.477110 0.878844i \(-0.341684\pi\)
0.477110 + 0.878844i \(0.341684\pi\)
\(90\) 0 0
\(91\) −11.3670 −1.19159
\(92\) −23.9628 −2.49830
\(93\) 0 0
\(94\) 16.3252 1.68381
\(95\) 0 0
\(96\) 0 0
\(97\) −2.57278 −0.261226 −0.130613 0.991433i \(-0.541695\pi\)
−0.130613 + 0.991433i \(0.541695\pi\)
\(98\) 9.83169 0.993150
\(99\) 0 0
\(100\) 0 0
\(101\) −7.87863 −0.783953 −0.391976 0.919975i \(-0.628209\pi\)
−0.391976 + 0.919975i \(0.628209\pi\)
\(102\) 0 0
\(103\) −12.4046 −1.22226 −0.611131 0.791530i \(-0.709285\pi\)
−0.611131 + 0.791530i \(0.709285\pi\)
\(104\) 24.6779 2.41986
\(105\) 0 0
\(106\) −6.53134 −0.634380
\(107\) 8.52094 0.823750 0.411875 0.911240i \(-0.364874\pi\)
0.411875 + 0.911240i \(0.364874\pi\)
\(108\) 0 0
\(109\) 17.5139 1.67752 0.838762 0.544499i \(-0.183280\pi\)
0.838762 + 0.544499i \(0.183280\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) −29.6456 −2.80125
\(113\) 4.17711 0.392949 0.196475 0.980509i \(-0.437051\pi\)
0.196475 + 0.980509i \(0.437051\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 14.8208 1.37608
\(117\) 0 0
\(118\) −20.0747 −1.84803
\(119\) 17.8906 1.64003
\(120\) 0 0
\(121\) 7.56181 0.687438
\(122\) −7.67880 −0.695206
\(123\) 0 0
\(124\) −9.80645 −0.880645
\(125\) 0 0
\(126\) 0 0
\(127\) −10.2112 −0.906097 −0.453049 0.891486i \(-0.649664\pi\)
−0.453049 + 0.891486i \(0.649664\pi\)
\(128\) −3.30537 −0.292156
\(129\) 0 0
\(130\) 0 0
\(131\) 4.91686 0.429588 0.214794 0.976659i \(-0.431092\pi\)
0.214794 + 0.976659i \(0.431092\pi\)
\(132\) 0 0
\(133\) 25.0720 2.17402
\(134\) −33.7508 −2.91562
\(135\) 0 0
\(136\) −38.8404 −3.33054
\(137\) −12.3817 −1.05784 −0.528919 0.848672i \(-0.677403\pi\)
−0.528919 + 0.848672i \(0.677403\pi\)
\(138\) 0 0
\(139\) 13.2420 1.12317 0.561584 0.827420i \(-0.310192\pi\)
0.561584 + 0.827420i \(0.310192\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 12.2759 1.03017
\(143\) −14.9120 −1.24701
\(144\) 0 0
\(145\) 0 0
\(146\) 4.28358 0.354512
\(147\) 0 0
\(148\) 8.97526 0.737762
\(149\) −5.20686 −0.426563 −0.213281 0.976991i \(-0.568415\pi\)
−0.213281 + 0.976991i \(0.568415\pi\)
\(150\) 0 0
\(151\) 7.78162 0.633259 0.316630 0.948549i \(-0.397449\pi\)
0.316630 + 0.948549i \(0.397449\pi\)
\(152\) −54.4313 −4.41496
\(153\) 0 0
\(154\) 36.7477 2.96122
\(155\) 0 0
\(156\) 0 0
\(157\) 16.4474 1.31264 0.656322 0.754481i \(-0.272111\pi\)
0.656322 + 0.754481i \(0.272111\pi\)
\(158\) 35.6717 2.83789
\(159\) 0 0
\(160\) 0 0
\(161\) 16.5844 1.30704
\(162\) 0 0
\(163\) −7.81342 −0.611994 −0.305997 0.952032i \(-0.598990\pi\)
−0.305997 + 0.952032i \(0.598990\pi\)
\(164\) −18.5015 −1.44473
\(165\) 0 0
\(166\) −13.0296 −1.01129
\(167\) −2.77030 −0.214372 −0.107186 0.994239i \(-0.534184\pi\)
−0.107186 + 0.994239i \(0.534184\pi\)
\(168\) 0 0
\(169\) −1.02011 −0.0784700
\(170\) 0 0
\(171\) 0 0
\(172\) 15.2235 1.16078
\(173\) −16.9573 −1.28924 −0.644620 0.764503i \(-0.722985\pi\)
−0.644620 + 0.764503i \(0.722985\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −38.8910 −2.93152
\(177\) 0 0
\(178\) 23.3799 1.75240
\(179\) −1.56173 −0.116729 −0.0583645 0.998295i \(-0.518589\pi\)
−0.0583645 + 0.998295i \(0.518589\pi\)
\(180\) 0 0
\(181\) 1.55277 0.115416 0.0577081 0.998333i \(-0.481621\pi\)
0.0577081 + 0.998333i \(0.481621\pi\)
\(182\) −29.5221 −2.18832
\(183\) 0 0
\(184\) −36.0048 −2.65431
\(185\) 0 0
\(186\) 0 0
\(187\) 23.4700 1.71630
\(188\) 29.8276 2.17540
\(189\) 0 0
\(190\) 0 0
\(191\) −26.5640 −1.92211 −0.961054 0.276362i \(-0.910871\pi\)
−0.961054 + 0.276362i \(0.910871\pi\)
\(192\) 0 0
\(193\) 13.7642 0.990772 0.495386 0.868673i \(-0.335027\pi\)
0.495386 + 0.868673i \(0.335027\pi\)
\(194\) −6.68192 −0.479734
\(195\) 0 0
\(196\) 17.9634 1.28310
\(197\) 6.25851 0.445901 0.222950 0.974830i \(-0.428431\pi\)
0.222950 + 0.974830i \(0.428431\pi\)
\(198\) 0 0
\(199\) 3.35009 0.237482 0.118741 0.992925i \(-0.462114\pi\)
0.118741 + 0.992925i \(0.462114\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) −20.4621 −1.43971
\(203\) −10.2573 −0.719923
\(204\) 0 0
\(205\) 0 0
\(206\) −32.2168 −2.24465
\(207\) 0 0
\(208\) 31.2439 2.16638
\(209\) 32.8911 2.27512
\(210\) 0 0
\(211\) 24.0984 1.65900 0.829502 0.558504i \(-0.188624\pi\)
0.829502 + 0.558504i \(0.188624\pi\)
\(212\) −11.9333 −0.819586
\(213\) 0 0
\(214\) 22.1303 1.51279
\(215\) 0 0
\(216\) 0 0
\(217\) 6.78693 0.460727
\(218\) 45.4863 3.08072
\(219\) 0 0
\(220\) 0 0
\(221\) −18.8551 −1.26833
\(222\) 0 0
\(223\) 18.2697 1.22343 0.611714 0.791079i \(-0.290480\pi\)
0.611714 + 0.791079i \(0.290480\pi\)
\(224\) −30.1635 −2.01539
\(225\) 0 0
\(226\) 10.8486 0.721640
\(227\) −17.9602 −1.19206 −0.596031 0.802962i \(-0.703256\pi\)
−0.596031 + 0.802962i \(0.703256\pi\)
\(228\) 0 0
\(229\) 6.00658 0.396926 0.198463 0.980108i \(-0.436405\pi\)
0.198463 + 0.980108i \(0.436405\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 22.2687 1.46201
\(233\) −20.8526 −1.36610 −0.683049 0.730373i \(-0.739346\pi\)
−0.683049 + 0.730373i \(0.739346\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) −36.6783 −2.38755
\(237\) 0 0
\(238\) 46.4647 3.01186
\(239\) 27.8722 1.80290 0.901452 0.432880i \(-0.142503\pi\)
0.901452 + 0.432880i \(0.142503\pi\)
\(240\) 0 0
\(241\) −18.7159 −1.20560 −0.602798 0.797894i \(-0.705947\pi\)
−0.602798 + 0.797894i \(0.705947\pi\)
\(242\) 19.6393 1.26246
\(243\) 0 0
\(244\) −14.0299 −0.898170
\(245\) 0 0
\(246\) 0 0
\(247\) −26.4237 −1.68130
\(248\) −14.7344 −0.935638
\(249\) 0 0
\(250\) 0 0
\(251\) −11.0995 −0.700594 −0.350297 0.936639i \(-0.613919\pi\)
−0.350297 + 0.936639i \(0.613919\pi\)
\(252\) 0 0
\(253\) 21.7565 1.36782
\(254\) −26.5201 −1.66402
\(255\) 0 0
\(256\) −20.1848 −1.26155
\(257\) −17.8844 −1.11560 −0.557800 0.829975i \(-0.688354\pi\)
−0.557800 + 0.829975i \(0.688354\pi\)
\(258\) 0 0
\(259\) −6.21168 −0.385975
\(260\) 0 0
\(261\) 0 0
\(262\) 12.7699 0.788927
\(263\) −18.1341 −1.11819 −0.559097 0.829102i \(-0.688852\pi\)
−0.559097 + 0.829102i \(0.688852\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 65.1160 3.99252
\(267\) 0 0
\(268\) −61.6657 −3.76683
\(269\) 22.2963 1.35943 0.679714 0.733477i \(-0.262104\pi\)
0.679714 + 0.733477i \(0.262104\pi\)
\(270\) 0 0
\(271\) 16.8621 1.02430 0.512149 0.858896i \(-0.328849\pi\)
0.512149 + 0.858896i \(0.328849\pi\)
\(272\) −49.1747 −2.98166
\(273\) 0 0
\(274\) −32.1572 −1.94269
\(275\) 0 0
\(276\) 0 0
\(277\) 18.6929 1.12315 0.561575 0.827426i \(-0.310196\pi\)
0.561575 + 0.827426i \(0.310196\pi\)
\(278\) 34.3915 2.06266
\(279\) 0 0
\(280\) 0 0
\(281\) −25.9852 −1.55015 −0.775075 0.631870i \(-0.782288\pi\)
−0.775075 + 0.631870i \(0.782288\pi\)
\(282\) 0 0
\(283\) −16.9219 −1.00591 −0.502953 0.864314i \(-0.667753\pi\)
−0.502953 + 0.864314i \(0.667753\pi\)
\(284\) 22.4291 1.33092
\(285\) 0 0
\(286\) −38.7289 −2.29009
\(287\) 12.8047 0.755838
\(288\) 0 0
\(289\) 12.6760 0.745650
\(290\) 0 0
\(291\) 0 0
\(292\) 7.82649 0.458011
\(293\) 1.05979 0.0619134 0.0309567 0.999521i \(-0.490145\pi\)
0.0309567 + 0.999521i \(0.490145\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 13.4856 0.783832
\(297\) 0 0
\(298\) −13.5231 −0.783370
\(299\) −17.4786 −1.01081
\(300\) 0 0
\(301\) −10.5360 −0.607285
\(302\) 20.2101 1.16296
\(303\) 0 0
\(304\) −68.9139 −3.95248
\(305\) 0 0
\(306\) 0 0
\(307\) −29.7239 −1.69643 −0.848215 0.529652i \(-0.822323\pi\)
−0.848215 + 0.529652i \(0.822323\pi\)
\(308\) 67.1414 3.82574
\(309\) 0 0
\(310\) 0 0
\(311\) −5.08513 −0.288351 −0.144176 0.989552i \(-0.546053\pi\)
−0.144176 + 0.989552i \(0.546053\pi\)
\(312\) 0 0
\(313\) −25.9388 −1.46615 −0.733074 0.680149i \(-0.761915\pi\)
−0.733074 + 0.680149i \(0.761915\pi\)
\(314\) 42.7165 2.41063
\(315\) 0 0
\(316\) 65.1755 3.66641
\(317\) 19.7727 1.11054 0.555271 0.831669i \(-0.312614\pi\)
0.555271 + 0.831669i \(0.312614\pi\)
\(318\) 0 0
\(319\) −13.4562 −0.753403
\(320\) 0 0
\(321\) 0 0
\(322\) 43.0724 2.40033
\(323\) 41.5883 2.31403
\(324\) 0 0
\(325\) 0 0
\(326\) −20.2927 −1.12391
\(327\) 0 0
\(328\) −27.7990 −1.53494
\(329\) −20.6433 −1.13810
\(330\) 0 0
\(331\) −23.9589 −1.31690 −0.658449 0.752625i \(-0.728787\pi\)
−0.658449 + 0.752625i \(0.728787\pi\)
\(332\) −23.8063 −1.30654
\(333\) 0 0
\(334\) −7.19492 −0.393689
\(335\) 0 0
\(336\) 0 0
\(337\) 27.6779 1.50771 0.753857 0.657039i \(-0.228191\pi\)
0.753857 + 0.657039i \(0.228191\pi\)
\(338\) −2.64939 −0.144108
\(339\) 0 0
\(340\) 0 0
\(341\) 8.90354 0.482154
\(342\) 0 0
\(343\) 10.5567 0.570008
\(344\) 22.8737 1.23327
\(345\) 0 0
\(346\) −44.0409 −2.36765
\(347\) −13.4880 −0.724075 −0.362037 0.932164i \(-0.617919\pi\)
−0.362037 + 0.932164i \(0.617919\pi\)
\(348\) 0 0
\(349\) 19.1678 1.02603 0.513015 0.858379i \(-0.328528\pi\)
0.513015 + 0.858379i \(0.328528\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) −39.5705 −2.10911
\(353\) 7.61950 0.405545 0.202773 0.979226i \(-0.435005\pi\)
0.202773 + 0.979226i \(0.435005\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 42.7172 2.26401
\(357\) 0 0
\(358\) −4.05606 −0.214369
\(359\) −31.2943 −1.65165 −0.825826 0.563925i \(-0.809291\pi\)
−0.825826 + 0.563925i \(0.809291\pi\)
\(360\) 0 0
\(361\) 39.2821 2.06748
\(362\) 4.03279 0.211959
\(363\) 0 0
\(364\) −53.9395 −2.82720
\(365\) 0 0
\(366\) 0 0
\(367\) 6.35079 0.331508 0.165754 0.986167i \(-0.446994\pi\)
0.165754 + 0.986167i \(0.446994\pi\)
\(368\) −45.5846 −2.37626
\(369\) 0 0
\(370\) 0 0
\(371\) 8.25894 0.428783
\(372\) 0 0
\(373\) −11.1710 −0.578410 −0.289205 0.957267i \(-0.593391\pi\)
−0.289205 + 0.957267i \(0.593391\pi\)
\(374\) 60.9554 3.15193
\(375\) 0 0
\(376\) 44.8167 2.31124
\(377\) 10.8103 0.556761
\(378\) 0 0
\(379\) −20.7113 −1.06387 −0.531933 0.846787i \(-0.678534\pi\)
−0.531933 + 0.846787i \(0.678534\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) −68.9911 −3.52989
\(383\) −12.5869 −0.643158 −0.321579 0.946883i \(-0.604214\pi\)
−0.321579 + 0.946883i \(0.604214\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 35.7480 1.81952
\(387\) 0 0
\(388\) −12.2085 −0.619792
\(389\) −10.8301 −0.549110 −0.274555 0.961571i \(-0.588531\pi\)
−0.274555 + 0.961571i \(0.588531\pi\)
\(390\) 0 0
\(391\) 27.5095 1.39121
\(392\) 26.9905 1.36322
\(393\) 0 0
\(394\) 16.2544 0.818884
\(395\) 0 0
\(396\) 0 0
\(397\) 4.90704 0.246277 0.123139 0.992389i \(-0.460704\pi\)
0.123139 + 0.992389i \(0.460704\pi\)
\(398\) 8.70072 0.436128
\(399\) 0 0
\(400\) 0 0
\(401\) 27.5223 1.37440 0.687198 0.726470i \(-0.258840\pi\)
0.687198 + 0.726470i \(0.258840\pi\)
\(402\) 0 0
\(403\) −7.15285 −0.356309
\(404\) −37.3861 −1.86003
\(405\) 0 0
\(406\) −26.6399 −1.32212
\(407\) −8.14888 −0.403925
\(408\) 0 0
\(409\) 19.7665 0.977390 0.488695 0.872455i \(-0.337473\pi\)
0.488695 + 0.872455i \(0.337473\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) −58.8630 −2.89997
\(413\) 25.3847 1.24910
\(414\) 0 0
\(415\) 0 0
\(416\) 31.7898 1.55862
\(417\) 0 0
\(418\) 85.4234 4.17820
\(419\) 6.67948 0.326314 0.163157 0.986600i \(-0.447832\pi\)
0.163157 + 0.986600i \(0.447832\pi\)
\(420\) 0 0
\(421\) 5.65996 0.275849 0.137925 0.990443i \(-0.455957\pi\)
0.137925 + 0.990443i \(0.455957\pi\)
\(422\) 62.5875 3.04671
\(423\) 0 0
\(424\) −17.9302 −0.870766
\(425\) 0 0
\(426\) 0 0
\(427\) 9.70991 0.469896
\(428\) 40.4340 1.95445
\(429\) 0 0
\(430\) 0 0
\(431\) −6.41717 −0.309104 −0.154552 0.987985i \(-0.549393\pi\)
−0.154552 + 0.987985i \(0.549393\pi\)
\(432\) 0 0
\(433\) 1.10506 0.0531059 0.0265530 0.999647i \(-0.491547\pi\)
0.0265530 + 0.999647i \(0.491547\pi\)
\(434\) 17.6268 0.846112
\(435\) 0 0
\(436\) 83.1077 3.98014
\(437\) 38.5520 1.84419
\(438\) 0 0
\(439\) −5.11313 −0.244036 −0.122018 0.992528i \(-0.538937\pi\)
−0.122018 + 0.992528i \(0.538937\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) −48.9698 −2.32926
\(443\) −17.9079 −0.850829 −0.425414 0.904999i \(-0.639872\pi\)
−0.425414 + 0.904999i \(0.639872\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 47.4493 2.24679
\(447\) 0 0
\(448\) −19.0484 −0.899953
\(449\) −27.3710 −1.29172 −0.645859 0.763457i \(-0.723501\pi\)
−0.645859 + 0.763457i \(0.723501\pi\)
\(450\) 0 0
\(451\) 16.7980 0.790989
\(452\) 19.8214 0.932322
\(453\) 0 0
\(454\) −46.6456 −2.18919
\(455\) 0 0
\(456\) 0 0
\(457\) −8.92806 −0.417637 −0.208819 0.977954i \(-0.566962\pi\)
−0.208819 + 0.977954i \(0.566962\pi\)
\(458\) 15.6001 0.728943
\(459\) 0 0
\(460\) 0 0
\(461\) −1.81204 −0.0843952 −0.0421976 0.999109i \(-0.513436\pi\)
−0.0421976 + 0.999109i \(0.513436\pi\)
\(462\) 0 0
\(463\) −2.72192 −0.126498 −0.0632492 0.997998i \(-0.520146\pi\)
−0.0632492 + 0.997998i \(0.520146\pi\)
\(464\) 28.1937 1.30886
\(465\) 0 0
\(466\) −54.1575 −2.50880
\(467\) −40.2261 −1.86144 −0.930721 0.365729i \(-0.880820\pi\)
−0.930721 + 0.365729i \(0.880820\pi\)
\(468\) 0 0
\(469\) 42.6781 1.97069
\(470\) 0 0
\(471\) 0 0
\(472\) −55.1101 −2.53665
\(473\) −13.8218 −0.635527
\(474\) 0 0
\(475\) 0 0
\(476\) 84.8953 3.89117
\(477\) 0 0
\(478\) 72.3886 3.31098
\(479\) 27.9457 1.27687 0.638435 0.769676i \(-0.279582\pi\)
0.638435 + 0.769676i \(0.279582\pi\)
\(480\) 0 0
\(481\) 6.54657 0.298498
\(482\) −48.6082 −2.21404
\(483\) 0 0
\(484\) 35.8827 1.63103
\(485\) 0 0
\(486\) 0 0
\(487\) 25.8089 1.16951 0.584756 0.811209i \(-0.301190\pi\)
0.584756 + 0.811209i \(0.301190\pi\)
\(488\) −21.0802 −0.954257
\(489\) 0 0
\(490\) 0 0
\(491\) −23.7044 −1.06976 −0.534882 0.844927i \(-0.679644\pi\)
−0.534882 + 0.844927i \(0.679644\pi\)
\(492\) 0 0
\(493\) −17.0144 −0.766288
\(494\) −68.6267 −3.08766
\(495\) 0 0
\(496\) −18.6548 −0.837627
\(497\) −15.5230 −0.696300
\(498\) 0 0
\(499\) 8.95752 0.400994 0.200497 0.979694i \(-0.435744\pi\)
0.200497 + 0.979694i \(0.435744\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) −28.8272 −1.28662
\(503\) 10.1655 0.453256 0.226628 0.973981i \(-0.427230\pi\)
0.226628 + 0.973981i \(0.427230\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 56.5052 2.51196
\(507\) 0 0
\(508\) −48.4547 −2.14983
\(509\) 6.54473 0.290090 0.145045 0.989425i \(-0.453667\pi\)
0.145045 + 0.989425i \(0.453667\pi\)
\(510\) 0 0
\(511\) −5.41663 −0.239618
\(512\) −45.8125 −2.02465
\(513\) 0 0
\(514\) −46.4488 −2.04877
\(515\) 0 0
\(516\) 0 0
\(517\) −27.0813 −1.19103
\(518\) −16.1327 −0.708832
\(519\) 0 0
\(520\) 0 0
\(521\) 25.6022 1.12165 0.560827 0.827933i \(-0.310483\pi\)
0.560827 + 0.827933i \(0.310483\pi\)
\(522\) 0 0
\(523\) −5.02210 −0.219601 −0.109801 0.993954i \(-0.535021\pi\)
−0.109801 + 0.993954i \(0.535021\pi\)
\(524\) 23.3318 1.01925
\(525\) 0 0
\(526\) −47.0971 −2.05353
\(527\) 11.2579 0.490400
\(528\) 0 0
\(529\) 2.50105 0.108741
\(530\) 0 0
\(531\) 0 0
\(532\) 118.973 5.15813
\(533\) −13.4951 −0.584536
\(534\) 0 0
\(535\) 0 0
\(536\) −92.6543 −4.00206
\(537\) 0 0
\(538\) 57.9071 2.49655
\(539\) −16.3094 −0.702498
\(540\) 0 0
\(541\) −7.99099 −0.343560 −0.171780 0.985135i \(-0.554952\pi\)
−0.171780 + 0.985135i \(0.554952\pi\)
\(542\) 43.7936 1.88109
\(543\) 0 0
\(544\) −50.0339 −2.14518
\(545\) 0 0
\(546\) 0 0
\(547\) 21.9399 0.938084 0.469042 0.883176i \(-0.344599\pi\)
0.469042 + 0.883176i \(0.344599\pi\)
\(548\) −58.7542 −2.50986
\(549\) 0 0
\(550\) 0 0
\(551\) −23.8441 −1.01579
\(552\) 0 0
\(553\) −45.1072 −1.91815
\(554\) 48.5486 2.06263
\(555\) 0 0
\(556\) 62.8364 2.66486
\(557\) 32.6366 1.38286 0.691429 0.722444i \(-0.256981\pi\)
0.691429 + 0.722444i \(0.256981\pi\)
\(558\) 0 0
\(559\) 11.1040 0.469651
\(560\) 0 0
\(561\) 0 0
\(562\) −67.4879 −2.84680
\(563\) −5.25771 −0.221586 −0.110793 0.993843i \(-0.535339\pi\)
−0.110793 + 0.993843i \(0.535339\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) −43.9490 −1.84732
\(567\) 0 0
\(568\) 33.7004 1.41404
\(569\) −15.1931 −0.636926 −0.318463 0.947935i \(-0.603167\pi\)
−0.318463 + 0.947935i \(0.603167\pi\)
\(570\) 0 0
\(571\) −8.29434 −0.347107 −0.173554 0.984824i \(-0.555525\pi\)
−0.173554 + 0.984824i \(0.555525\pi\)
\(572\) −70.7613 −2.95868
\(573\) 0 0
\(574\) 33.2559 1.38807
\(575\) 0 0
\(576\) 0 0
\(577\) −2.69155 −0.112051 −0.0560253 0.998429i \(-0.517843\pi\)
−0.0560253 + 0.998429i \(0.517843\pi\)
\(578\) 32.9218 1.36936
\(579\) 0 0
\(580\) 0 0
\(581\) 16.4761 0.683543
\(582\) 0 0
\(583\) 10.8346 0.448724
\(584\) 11.7595 0.486612
\(585\) 0 0
\(586\) 2.75244 0.113702
\(587\) 27.2841 1.12613 0.563067 0.826411i \(-0.309621\pi\)
0.563067 + 0.826411i \(0.309621\pi\)
\(588\) 0 0
\(589\) 15.7768 0.650074
\(590\) 0 0
\(591\) 0 0
\(592\) 17.0737 0.701723
\(593\) −3.04738 −0.125141 −0.0625704 0.998041i \(-0.519930\pi\)
−0.0625704 + 0.998041i \(0.519930\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −24.7079 −1.01207
\(597\) 0 0
\(598\) −45.3946 −1.85632
\(599\) 21.1104 0.862549 0.431274 0.902221i \(-0.358064\pi\)
0.431274 + 0.902221i \(0.358064\pi\)
\(600\) 0 0
\(601\) 1.78443 0.0727884 0.0363942 0.999338i \(-0.488413\pi\)
0.0363942 + 0.999338i \(0.488413\pi\)
\(602\) −27.3637 −1.11526
\(603\) 0 0
\(604\) 36.9258 1.50249
\(605\) 0 0
\(606\) 0 0
\(607\) −43.9927 −1.78561 −0.892805 0.450444i \(-0.851266\pi\)
−0.892805 + 0.450444i \(0.851266\pi\)
\(608\) −70.1179 −2.84366
\(609\) 0 0
\(610\) 0 0
\(611\) 21.7563 0.880166
\(612\) 0 0
\(613\) 37.9419 1.53246 0.766230 0.642567i \(-0.222130\pi\)
0.766230 + 0.642567i \(0.222130\pi\)
\(614\) −77.1977 −3.11544
\(615\) 0 0
\(616\) 100.882 4.06464
\(617\) −7.92668 −0.319116 −0.159558 0.987189i \(-0.551007\pi\)
−0.159558 + 0.987189i \(0.551007\pi\)
\(618\) 0 0
\(619\) −7.83411 −0.314880 −0.157440 0.987529i \(-0.550324\pi\)
−0.157440 + 0.987529i \(0.550324\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) −13.2069 −0.529548
\(623\) −29.5641 −1.18446
\(624\) 0 0
\(625\) 0 0
\(626\) −67.3673 −2.69254
\(627\) 0 0
\(628\) 78.0469 3.11441
\(629\) −10.3036 −0.410833
\(630\) 0 0
\(631\) 19.0248 0.757366 0.378683 0.925526i \(-0.376377\pi\)
0.378683 + 0.925526i \(0.376377\pi\)
\(632\) 97.9279 3.89536
\(633\) 0 0
\(634\) 51.3528 2.03948
\(635\) 0 0
\(636\) 0 0
\(637\) 13.1025 0.519141
\(638\) −34.9480 −1.38360
\(639\) 0 0
\(640\) 0 0
\(641\) 32.5783 1.28677 0.643383 0.765545i \(-0.277530\pi\)
0.643383 + 0.765545i \(0.277530\pi\)
\(642\) 0 0
\(643\) −40.0025 −1.57755 −0.788773 0.614684i \(-0.789283\pi\)
−0.788773 + 0.614684i \(0.789283\pi\)
\(644\) 78.6972 3.10111
\(645\) 0 0
\(646\) 108.011 4.24965
\(647\) −20.0199 −0.787063 −0.393532 0.919311i \(-0.628747\pi\)
−0.393532 + 0.919311i \(0.628747\pi\)
\(648\) 0 0
\(649\) 33.3012 1.30719
\(650\) 0 0
\(651\) 0 0
\(652\) −37.0766 −1.45203
\(653\) −10.3332 −0.404369 −0.202185 0.979347i \(-0.564804\pi\)
−0.202185 + 0.979347i \(0.564804\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) −35.1955 −1.37415
\(657\) 0 0
\(658\) −53.6141 −2.09009
\(659\) 47.2403 1.84022 0.920111 0.391657i \(-0.128098\pi\)
0.920111 + 0.391657i \(0.128098\pi\)
\(660\) 0 0
\(661\) 51.3528 1.99739 0.998696 0.0510604i \(-0.0162601\pi\)
0.998696 + 0.0510604i \(0.0162601\pi\)
\(662\) −62.2250 −2.41844
\(663\) 0 0
\(664\) −35.7696 −1.38813
\(665\) 0 0
\(666\) 0 0
\(667\) −15.7722 −0.610701
\(668\) −13.1458 −0.508625
\(669\) 0 0
\(670\) 0 0
\(671\) 12.7381 0.491749
\(672\) 0 0
\(673\) −40.5790 −1.56420 −0.782102 0.623150i \(-0.785853\pi\)
−0.782102 + 0.623150i \(0.785853\pi\)
\(674\) 71.8841 2.76887
\(675\) 0 0
\(676\) −4.84068 −0.186180
\(677\) 9.21482 0.354154 0.177077 0.984197i \(-0.443336\pi\)
0.177077 + 0.984197i \(0.443336\pi\)
\(678\) 0 0
\(679\) 8.44935 0.324256
\(680\) 0 0
\(681\) 0 0
\(682\) 23.1239 0.885461
\(683\) 13.3281 0.509987 0.254994 0.966943i \(-0.417927\pi\)
0.254994 + 0.966943i \(0.417927\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 27.4174 1.04680
\(687\) 0 0
\(688\) 28.9597 1.10408
\(689\) −8.70421 −0.331604
\(690\) 0 0
\(691\) −32.5862 −1.23964 −0.619819 0.784745i \(-0.712794\pi\)
−0.619819 + 0.784745i \(0.712794\pi\)
\(692\) −80.4667 −3.05889
\(693\) 0 0
\(694\) −35.0306 −1.32974
\(695\) 0 0
\(696\) 0 0
\(697\) 21.2398 0.804516
\(698\) 49.7820 1.88428
\(699\) 0 0
\(700\) 0 0
\(701\) 33.5547 1.26734 0.633672 0.773602i \(-0.281547\pi\)
0.633672 + 0.773602i \(0.281547\pi\)
\(702\) 0 0
\(703\) −14.4396 −0.544600
\(704\) −24.9889 −0.941806
\(705\) 0 0
\(706\) 19.7891 0.744772
\(707\) 25.8745 0.973110
\(708\) 0 0
\(709\) 12.4731 0.468436 0.234218 0.972184i \(-0.424747\pi\)
0.234218 + 0.972184i \(0.424747\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 64.1837 2.40539
\(713\) 10.4359 0.390829
\(714\) 0 0
\(715\) 0 0
\(716\) −7.41079 −0.276954
\(717\) 0 0
\(718\) −81.2765 −3.03321
\(719\) 7.53586 0.281040 0.140520 0.990078i \(-0.455123\pi\)
0.140520 + 0.990078i \(0.455123\pi\)
\(720\) 0 0
\(721\) 40.7384 1.51718
\(722\) 102.022 3.79687
\(723\) 0 0
\(724\) 7.36827 0.273839
\(725\) 0 0
\(726\) 0 0
\(727\) 42.8265 1.58835 0.794173 0.607691i \(-0.207904\pi\)
0.794173 + 0.607691i \(0.207904\pi\)
\(728\) −81.0455 −3.00375
\(729\) 0 0
\(730\) 0 0
\(731\) −17.4766 −0.646396
\(732\) 0 0
\(733\) −48.4170 −1.78832 −0.894161 0.447745i \(-0.852227\pi\)
−0.894161 + 0.447745i \(0.852227\pi\)
\(734\) 16.4940 0.608806
\(735\) 0 0
\(736\) −46.3810 −1.70963
\(737\) 55.9880 2.06234
\(738\) 0 0
\(739\) 31.1926 1.14744 0.573720 0.819052i \(-0.305500\pi\)
0.573720 + 0.819052i \(0.305500\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 21.4498 0.787447
\(743\) 45.9840 1.68699 0.843495 0.537137i \(-0.180494\pi\)
0.843495 + 0.537137i \(0.180494\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) −29.0128 −1.06223
\(747\) 0 0
\(748\) 111.371 4.07213
\(749\) −27.9839 −1.02251
\(750\) 0 0
\(751\) 2.51488 0.0917695 0.0458847 0.998947i \(-0.485389\pi\)
0.0458847 + 0.998947i \(0.485389\pi\)
\(752\) 56.7411 2.06913
\(753\) 0 0
\(754\) 28.0762 1.02247
\(755\) 0 0
\(756\) 0 0
\(757\) 39.6597 1.44146 0.720728 0.693218i \(-0.243808\pi\)
0.720728 + 0.693218i \(0.243808\pi\)
\(758\) −53.7905 −1.95376
\(759\) 0 0
\(760\) 0 0
\(761\) −17.2752 −0.626225 −0.313113 0.949716i \(-0.601372\pi\)
−0.313113 + 0.949716i \(0.601372\pi\)
\(762\) 0 0
\(763\) −57.5179 −2.08229
\(764\) −126.053 −4.56044
\(765\) 0 0
\(766\) −32.6901 −1.18114
\(767\) −26.7533 −0.966004
\(768\) 0 0
\(769\) −4.50943 −0.162614 −0.0813071 0.996689i \(-0.525909\pi\)
−0.0813071 + 0.996689i \(0.525909\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 65.3148 2.35073
\(773\) −33.6503 −1.21032 −0.605159 0.796105i \(-0.706890\pi\)
−0.605159 + 0.796105i \(0.706890\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) −18.3436 −0.658495
\(777\) 0 0
\(778\) −28.1276 −1.00842
\(779\) 29.7657 1.06647
\(780\) 0 0
\(781\) −20.3640 −0.728682
\(782\) 71.4465 2.55492
\(783\) 0 0
\(784\) 34.1718 1.22042
\(785\) 0 0
\(786\) 0 0
\(787\) −36.8366 −1.31308 −0.656542 0.754289i \(-0.727982\pi\)
−0.656542 + 0.754289i \(0.727982\pi\)
\(788\) 29.6982 1.05796
\(789\) 0 0
\(790\) 0 0
\(791\) −13.7182 −0.487763
\(792\) 0 0
\(793\) −10.2334 −0.363399
\(794\) 12.7444 0.452281
\(795\) 0 0
\(796\) 15.8970 0.563455
\(797\) −40.4962 −1.43445 −0.717224 0.696843i \(-0.754588\pi\)
−0.717224 + 0.696843i \(0.754588\pi\)
\(798\) 0 0
\(799\) −34.2422 −1.21140
\(800\) 0 0
\(801\) 0 0
\(802\) 71.4798 2.52404
\(803\) −7.10589 −0.250761
\(804\) 0 0
\(805\) 0 0
\(806\) −18.5771 −0.654350
\(807\) 0 0
\(808\) −56.1735 −1.97618
\(809\) 11.9110 0.418767 0.209384 0.977834i \(-0.432854\pi\)
0.209384 + 0.977834i \(0.432854\pi\)
\(810\) 0 0
\(811\) −42.2433 −1.48336 −0.741681 0.670752i \(-0.765971\pi\)
−0.741681 + 0.670752i \(0.765971\pi\)
\(812\) −48.6736 −1.70811
\(813\) 0 0
\(814\) −21.1640 −0.741796
\(815\) 0 0
\(816\) 0 0
\(817\) −24.4919 −0.856863
\(818\) 51.3368 1.79495
\(819\) 0 0
\(820\) 0 0
\(821\) −44.5710 −1.55554 −0.777769 0.628551i \(-0.783648\pi\)
−0.777769 + 0.628551i \(0.783648\pi\)
\(822\) 0 0
\(823\) −12.3125 −0.429186 −0.214593 0.976704i \(-0.568843\pi\)
−0.214593 + 0.976704i \(0.568843\pi\)
\(824\) −88.4431 −3.08106
\(825\) 0 0
\(826\) 65.9281 2.29393
\(827\) −24.7421 −0.860367 −0.430183 0.902742i \(-0.641551\pi\)
−0.430183 + 0.902742i \(0.641551\pi\)
\(828\) 0 0
\(829\) −53.7221 −1.86584 −0.932922 0.360077i \(-0.882750\pi\)
−0.932922 + 0.360077i \(0.882750\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 20.0754 0.695989
\(833\) −20.6221 −0.714512
\(834\) 0 0
\(835\) 0 0
\(836\) 156.076 5.39801
\(837\) 0 0
\(838\) 17.3477 0.599266
\(839\) 45.7865 1.58073 0.790363 0.612639i \(-0.209892\pi\)
0.790363 + 0.612639i \(0.209892\pi\)
\(840\) 0 0
\(841\) −19.2450 −0.663622
\(842\) 14.6998 0.506589
\(843\) 0 0
\(844\) 114.353 3.93620
\(845\) 0 0
\(846\) 0 0
\(847\) −24.8340 −0.853307
\(848\) −22.7009 −0.779550
\(849\) 0 0
\(850\) 0 0
\(851\) −9.55139 −0.327418
\(852\) 0 0
\(853\) 20.3074 0.695311 0.347655 0.937622i \(-0.386978\pi\)
0.347655 + 0.937622i \(0.386978\pi\)
\(854\) 25.2182 0.862950
\(855\) 0 0
\(856\) 60.7531 2.07650
\(857\) −47.4031 −1.61926 −0.809630 0.586941i \(-0.800332\pi\)
−0.809630 + 0.586941i \(0.800332\pi\)
\(858\) 0 0
\(859\) −5.37538 −0.183406 −0.0917028 0.995786i \(-0.529231\pi\)
−0.0917028 + 0.995786i \(0.529231\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) −16.6664 −0.567661
\(863\) −15.0679 −0.512916 −0.256458 0.966555i \(-0.582556\pi\)
−0.256458 + 0.966555i \(0.582556\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 2.87003 0.0975275
\(867\) 0 0
\(868\) 32.2057 1.09313
\(869\) −59.1746 −2.00736
\(870\) 0 0
\(871\) −44.9791 −1.52406
\(872\) 124.871 4.22868
\(873\) 0 0
\(874\) 100.126 3.38680
\(875\) 0 0
\(876\) 0 0
\(877\) −6.43572 −0.217319 −0.108659 0.994079i \(-0.534656\pi\)
−0.108659 + 0.994079i \(0.534656\pi\)
\(878\) −13.2796 −0.448165
\(879\) 0 0
\(880\) 0 0
\(881\) −46.2411 −1.55790 −0.778950 0.627085i \(-0.784248\pi\)
−0.778950 + 0.627085i \(0.784248\pi\)
\(882\) 0 0
\(883\) 12.8545 0.432590 0.216295 0.976328i \(-0.430603\pi\)
0.216295 + 0.976328i \(0.430603\pi\)
\(884\) −89.4723 −3.00928
\(885\) 0 0
\(886\) −46.5097 −1.56252
\(887\) −33.5617 −1.12689 −0.563446 0.826153i \(-0.690525\pi\)
−0.563446 + 0.826153i \(0.690525\pi\)
\(888\) 0 0
\(889\) 33.5350 1.12473
\(890\) 0 0
\(891\) 0 0
\(892\) 86.6942 2.90274
\(893\) −47.9873 −1.60583
\(894\) 0 0
\(895\) 0 0
\(896\) 10.8553 0.362649
\(897\) 0 0
\(898\) −71.0870 −2.37220
\(899\) −6.45454 −0.215271
\(900\) 0 0
\(901\) 13.6995 0.456398
\(902\) 43.6272 1.45263
\(903\) 0 0
\(904\) 29.7822 0.990542
\(905\) 0 0
\(906\) 0 0
\(907\) −5.73678 −0.190487 −0.0952433 0.995454i \(-0.530363\pi\)
−0.0952433 + 0.995454i \(0.530363\pi\)
\(908\) −85.2257 −2.82831
\(909\) 0 0
\(910\) 0 0
\(911\) −4.08321 −0.135283 −0.0676413 0.997710i \(-0.521547\pi\)
−0.0676413 + 0.997710i \(0.521547\pi\)
\(912\) 0 0
\(913\) 21.6144 0.715332
\(914\) −23.1876 −0.766978
\(915\) 0 0
\(916\) 28.5027 0.941757
\(917\) −16.1476 −0.533242
\(918\) 0 0
\(919\) 34.1894 1.12780 0.563902 0.825842i \(-0.309300\pi\)
0.563902 + 0.825842i \(0.309300\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) −4.70617 −0.154989
\(923\) 16.3599 0.538492
\(924\) 0 0
\(925\) 0 0
\(926\) −7.06927 −0.232311
\(927\) 0 0
\(928\) 28.6863 0.941672
\(929\) 23.9398 0.785438 0.392719 0.919659i \(-0.371535\pi\)
0.392719 + 0.919659i \(0.371535\pi\)
\(930\) 0 0
\(931\) −28.8999 −0.947157
\(932\) −98.9507 −3.24124
\(933\) 0 0
\(934\) −104.474 −3.41849
\(935\) 0 0
\(936\) 0 0
\(937\) −21.4640 −0.701199 −0.350600 0.936525i \(-0.614022\pi\)
−0.350600 + 0.936525i \(0.614022\pi\)
\(938\) 110.842 3.61912
\(939\) 0 0
\(940\) 0 0
\(941\) 1.78646 0.0582371 0.0291185 0.999576i \(-0.490730\pi\)
0.0291185 + 0.999576i \(0.490730\pi\)
\(942\) 0 0
\(943\) 19.6892 0.641167
\(944\) −69.7733 −2.27093
\(945\) 0 0
\(946\) −35.8975 −1.16713
\(947\) −45.4097 −1.47562 −0.737809 0.675009i \(-0.764140\pi\)
−0.737809 + 0.675009i \(0.764140\pi\)
\(948\) 0 0
\(949\) 5.70866 0.185311
\(950\) 0 0
\(951\) 0 0
\(952\) 127.557 4.13416
\(953\) 1.00850 0.0326686 0.0163343 0.999867i \(-0.494800\pi\)
0.0163343 + 0.999867i \(0.494800\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 132.261 4.27761
\(957\) 0 0
\(958\) 72.5794 2.34493
\(959\) 40.6631 1.31308
\(960\) 0 0
\(961\) −26.7292 −0.862234
\(962\) 17.0025 0.548183
\(963\) 0 0
\(964\) −88.8115 −2.86043
\(965\) 0 0
\(966\) 0 0
\(967\) −21.4625 −0.690188 −0.345094 0.938568i \(-0.612153\pi\)
−0.345094 + 0.938568i \(0.612153\pi\)
\(968\) 53.9147 1.73288
\(969\) 0 0
\(970\) 0 0
\(971\) −47.3619 −1.51992 −0.759958 0.649972i \(-0.774781\pi\)
−0.759958 + 0.649972i \(0.774781\pi\)
\(972\) 0 0
\(973\) −43.4884 −1.39417
\(974\) 67.0298 2.14777
\(975\) 0 0
\(976\) −26.6891 −0.854296
\(977\) 6.29459 0.201382 0.100691 0.994918i \(-0.467895\pi\)
0.100691 + 0.994918i \(0.467895\pi\)
\(978\) 0 0
\(979\) −38.7841 −1.23955
\(980\) 0 0
\(981\) 0 0
\(982\) −61.5641 −1.96459
\(983\) −7.80790 −0.249033 −0.124517 0.992218i \(-0.539738\pi\)
−0.124517 + 0.992218i \(0.539738\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) −44.1891 −1.40727
\(987\) 0 0
\(988\) −125.387 −3.98910
\(989\) −16.2007 −0.515152
\(990\) 0 0
\(991\) −21.0569 −0.668896 −0.334448 0.942414i \(-0.608550\pi\)
−0.334448 + 0.942414i \(0.608550\pi\)
\(992\) −18.9808 −0.602640
\(993\) 0 0
\(994\) −40.3156 −1.27873
\(995\) 0 0
\(996\) 0 0
\(997\) −29.3042 −0.928072 −0.464036 0.885816i \(-0.653599\pi\)
−0.464036 + 0.885816i \(0.653599\pi\)
\(998\) 23.2641 0.736413
\(999\) 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 5625.2.a.bc.1.8 8
3.2 odd 2 1875.2.a.n.1.1 8
5.4 even 2 5625.2.a.u.1.1 8
15.2 even 4 1875.2.b.g.1249.2 16
15.8 even 4 1875.2.b.g.1249.15 16
15.14 odd 2 1875.2.a.o.1.8 yes 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1875.2.a.n.1.1 8 3.2 odd 2
1875.2.a.o.1.8 yes 8 15.14 odd 2
1875.2.b.g.1249.2 16 15.2 even 4
1875.2.b.g.1249.15 16 15.8 even 4
5625.2.a.u.1.1 8 5.4 even 2
5625.2.a.bc.1.8 8 1.1 even 1 trivial