Properties

Label 8048.2.a.o.1.5
Level $8048$
Weight $2$
Character 8048.1
Self dual yes
Analytic conductor $64.264$
Analytic rank $0$
Dimension $5$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [8048,2,Mod(1,8048)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8048, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("8048.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8048 = 2^{4} \cdot 503 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8048.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: \(64.2636035467\)
Analytic rank: \(0\)
Dimension: \(5\)
Coefficient field: 5.5.205225.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{5} - x^{4} - 6x^{3} + 3x^{2} + 7x - 3 \) 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 1006)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.5
Root \(-1.77799\) of defining polynomial
Character \(\chi\) \(=\) 8048.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+2.77799 q^{3} -2.86114 q^{5} -4.41712 q^{7} +4.71723 q^{9} +O(q^{10})\) \(q+2.77799 q^{3} -2.86114 q^{5} -4.41712 q^{7} +4.71723 q^{9} +5.26928 q^{11} -0.388086 q^{13} -7.94822 q^{15} +2.44041 q^{17} -5.10049 q^{19} -12.2707 q^{21} +5.55115 q^{23} +3.18613 q^{25} +4.77045 q^{27} -5.70472 q^{29} +7.16125 q^{31} +14.6380 q^{33} +12.6380 q^{35} -10.0188 q^{37} -1.07810 q^{39} +0.949119 q^{41} +7.06469 q^{43} -13.4967 q^{45} +8.52538 q^{47} +12.5110 q^{49} +6.77943 q^{51} -4.00665 q^{53} -15.0762 q^{55} -14.1691 q^{57} -9.25587 q^{59} -6.57332 q^{61} -20.8366 q^{63} +1.11037 q^{65} -13.6981 q^{67} +15.4211 q^{69} +14.7319 q^{71} +0.383259 q^{73} +8.85104 q^{75} -23.2751 q^{77} +5.00988 q^{79} -0.899429 q^{81} +10.3761 q^{83} -6.98235 q^{85} -15.8477 q^{87} +10.8825 q^{89} +1.71422 q^{91} +19.8939 q^{93} +14.5932 q^{95} -3.81018 q^{97} +24.8564 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q + 4 q^{3} - 3 q^{5} + 9 q^{7} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 5 q + 4 q^{3} - 3 q^{5} + 9 q^{7} + q^{9} + 15 q^{11} - 5 q^{13} + 4 q^{15} - 6 q^{17} - 10 q^{19} - 12 q^{21} + 12 q^{23} + 6 q^{25} + 7 q^{27} - 16 q^{29} + 33 q^{31} + 21 q^{33} + 11 q^{35} + 2 q^{37} - 4 q^{39} - 12 q^{41} + 17 q^{43} + 3 q^{45} + 7 q^{47} + 36 q^{49} - 14 q^{51} - 2 q^{55} - 6 q^{57} - 18 q^{59} + q^{61} - 31 q^{63} - 14 q^{65} - 6 q^{67} + 15 q^{69} + 26 q^{71} + 9 q^{73} + 33 q^{75} + 16 q^{77} + 21 q^{79} - 3 q^{81} - 28 q^{85} - 34 q^{87} + 15 q^{89} - 26 q^{91} + 19 q^{93} + 18 q^{95} + 26 q^{97} + 44 q^{99}+O(q^{100}) \) 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
\(3\) 2.77799 1.60387 0.801937 0.597409i \(-0.203803\pi\)
0.801937 + 0.597409i \(0.203803\pi\)
\(4\) 0 0
\(5\) −2.86114 −1.27954 −0.639771 0.768566i \(-0.720971\pi\)
−0.639771 + 0.768566i \(0.720971\pi\)
\(6\) 0 0
\(7\) −4.41712 −1.66952 −0.834758 0.550618i \(-0.814392\pi\)
−0.834758 + 0.550618i \(0.814392\pi\)
\(8\) 0 0
\(9\) 4.71723 1.57241
\(10\) 0 0
\(11\) 5.26928 1.58875 0.794374 0.607429i \(-0.207799\pi\)
0.794374 + 0.607429i \(0.207799\pi\)
\(12\) 0 0
\(13\) −0.388086 −0.107636 −0.0538178 0.998551i \(-0.517139\pi\)
−0.0538178 + 0.998551i \(0.517139\pi\)
\(14\) 0 0
\(15\) −7.94822 −2.05222
\(16\) 0 0
\(17\) 2.44041 0.591886 0.295943 0.955206i \(-0.404366\pi\)
0.295943 + 0.955206i \(0.404366\pi\)
\(18\) 0 0
\(19\) −5.10049 −1.17013 −0.585066 0.810985i \(-0.698932\pi\)
−0.585066 + 0.810985i \(0.698932\pi\)
\(20\) 0 0
\(21\) −12.2707 −2.67769
\(22\) 0 0
\(23\) 5.55115 1.15750 0.578748 0.815507i \(-0.303542\pi\)
0.578748 + 0.815507i \(0.303542\pi\)
\(24\) 0 0
\(25\) 3.18613 0.637226
\(26\) 0 0
\(27\) 4.77045 0.918073
\(28\) 0 0
\(29\) −5.70472 −1.05934 −0.529670 0.848204i \(-0.677684\pi\)
−0.529670 + 0.848204i \(0.677684\pi\)
\(30\) 0 0
\(31\) 7.16125 1.28620 0.643099 0.765783i \(-0.277648\pi\)
0.643099 + 0.765783i \(0.277648\pi\)
\(32\) 0 0
\(33\) 14.6380 2.54815
\(34\) 0 0
\(35\) 12.6380 2.13621
\(36\) 0 0
\(37\) −10.0188 −1.64708 −0.823539 0.567260i \(-0.808003\pi\)
−0.823539 + 0.567260i \(0.808003\pi\)
\(38\) 0 0
\(39\) −1.07810 −0.172634
\(40\) 0 0
\(41\) 0.949119 0.148227 0.0741137 0.997250i \(-0.476387\pi\)
0.0741137 + 0.997250i \(0.476387\pi\)
\(42\) 0 0
\(43\) 7.06469 1.07736 0.538678 0.842512i \(-0.318924\pi\)
0.538678 + 0.842512i \(0.318924\pi\)
\(44\) 0 0
\(45\) −13.4967 −2.01196
\(46\) 0 0
\(47\) 8.52538 1.24355 0.621777 0.783194i \(-0.286411\pi\)
0.621777 + 0.783194i \(0.286411\pi\)
\(48\) 0 0
\(49\) 12.5110 1.78728
\(50\) 0 0
\(51\) 6.77943 0.949310
\(52\) 0 0
\(53\) −4.00665 −0.550355 −0.275178 0.961393i \(-0.588737\pi\)
−0.275178 + 0.961393i \(0.588737\pi\)
\(54\) 0 0
\(55\) −15.0762 −2.03287
\(56\) 0 0
\(57\) −14.1691 −1.87674
\(58\) 0 0
\(59\) −9.25587 −1.20501 −0.602506 0.798114i \(-0.705831\pi\)
−0.602506 + 0.798114i \(0.705831\pi\)
\(60\) 0 0
\(61\) −6.57332 −0.841627 −0.420814 0.907147i \(-0.638255\pi\)
−0.420814 + 0.907147i \(0.638255\pi\)
\(62\) 0 0
\(63\) −20.8366 −2.62516
\(64\) 0 0
\(65\) 1.11037 0.137724
\(66\) 0 0
\(67\) −13.6981 −1.67349 −0.836745 0.547593i \(-0.815544\pi\)
−0.836745 + 0.547593i \(0.815544\pi\)
\(68\) 0 0
\(69\) 15.4211 1.85648
\(70\) 0 0
\(71\) 14.7319 1.74835 0.874175 0.485612i \(-0.161403\pi\)
0.874175 + 0.485612i \(0.161403\pi\)
\(72\) 0 0
\(73\) 0.383259 0.0448571 0.0224286 0.999748i \(-0.492860\pi\)
0.0224286 + 0.999748i \(0.492860\pi\)
\(74\) 0 0
\(75\) 8.85104 1.02203
\(76\) 0 0
\(77\) −23.2751 −2.65244
\(78\) 0 0
\(79\) 5.00988 0.563655 0.281828 0.959465i \(-0.409059\pi\)
0.281828 + 0.959465i \(0.409059\pi\)
\(80\) 0 0
\(81\) −0.899429 −0.0999366
\(82\) 0 0
\(83\) 10.3761 1.13893 0.569464 0.822017i \(-0.307151\pi\)
0.569464 + 0.822017i \(0.307151\pi\)
\(84\) 0 0
\(85\) −6.98235 −0.757343
\(86\) 0 0
\(87\) −15.8477 −1.69905
\(88\) 0 0
\(89\) 10.8825 1.15354 0.576771 0.816906i \(-0.304313\pi\)
0.576771 + 0.816906i \(0.304313\pi\)
\(90\) 0 0
\(91\) 1.71422 0.179699
\(92\) 0 0
\(93\) 19.8939 2.06290
\(94\) 0 0
\(95\) 14.5932 1.49723
\(96\) 0 0
\(97\) −3.81018 −0.386865 −0.193433 0.981114i \(-0.561962\pi\)
−0.193433 + 0.981114i \(0.561962\pi\)
\(98\) 0 0
\(99\) 24.8564 2.49816
\(100\) 0 0
\(101\) 12.4917 1.24297 0.621485 0.783426i \(-0.286530\pi\)
0.621485 + 0.783426i \(0.286530\pi\)
\(102\) 0 0
\(103\) 14.6787 1.44634 0.723168 0.690672i \(-0.242685\pi\)
0.723168 + 0.690672i \(0.242685\pi\)
\(104\) 0 0
\(105\) 35.1083 3.42622
\(106\) 0 0
\(107\) 9.98246 0.965041 0.482521 0.875885i \(-0.339721\pi\)
0.482521 + 0.875885i \(0.339721\pi\)
\(108\) 0 0
\(109\) −0.875847 −0.0838909 −0.0419454 0.999120i \(-0.513356\pi\)
−0.0419454 + 0.999120i \(0.513356\pi\)
\(110\) 0 0
\(111\) −27.8321 −2.64170
\(112\) 0 0
\(113\) 8.23069 0.774278 0.387139 0.922021i \(-0.373463\pi\)
0.387139 + 0.922021i \(0.373463\pi\)
\(114\) 0 0
\(115\) −15.8826 −1.48106
\(116\) 0 0
\(117\) −1.83069 −0.169247
\(118\) 0 0
\(119\) −10.7796 −0.988163
\(120\) 0 0
\(121\) 16.7653 1.52412
\(122\) 0 0
\(123\) 2.63664 0.237738
\(124\) 0 0
\(125\) 5.18974 0.464184
\(126\) 0 0
\(127\) 4.99246 0.443009 0.221505 0.975159i \(-0.428903\pi\)
0.221505 + 0.975159i \(0.428903\pi\)
\(128\) 0 0
\(129\) 19.6256 1.72794
\(130\) 0 0
\(131\) 7.48542 0.654004 0.327002 0.945024i \(-0.393962\pi\)
0.327002 + 0.945024i \(0.393962\pi\)
\(132\) 0 0
\(133\) 22.5295 1.95355
\(134\) 0 0
\(135\) −13.6489 −1.17471
\(136\) 0 0
\(137\) 19.3953 1.65705 0.828525 0.559952i \(-0.189180\pi\)
0.828525 + 0.559952i \(0.189180\pi\)
\(138\) 0 0
\(139\) −1.29884 −0.110166 −0.0550829 0.998482i \(-0.517542\pi\)
−0.0550829 + 0.998482i \(0.517542\pi\)
\(140\) 0 0
\(141\) 23.6834 1.99450
\(142\) 0 0
\(143\) −2.04493 −0.171006
\(144\) 0 0
\(145\) 16.3220 1.35547
\(146\) 0 0
\(147\) 34.7553 2.86657
\(148\) 0 0
\(149\) −11.7129 −0.959560 −0.479780 0.877389i \(-0.659284\pi\)
−0.479780 + 0.877389i \(0.659284\pi\)
\(150\) 0 0
\(151\) −15.8591 −1.29060 −0.645299 0.763930i \(-0.723267\pi\)
−0.645299 + 0.763930i \(0.723267\pi\)
\(152\) 0 0
\(153\) 11.5120 0.930688
\(154\) 0 0
\(155\) −20.4893 −1.64574
\(156\) 0 0
\(157\) 13.1874 1.05247 0.526236 0.850339i \(-0.323603\pi\)
0.526236 + 0.850339i \(0.323603\pi\)
\(158\) 0 0
\(159\) −11.1304 −0.882700
\(160\) 0 0
\(161\) −24.5201 −1.93246
\(162\) 0 0
\(163\) −9.87472 −0.773448 −0.386724 0.922195i \(-0.626393\pi\)
−0.386724 + 0.922195i \(0.626393\pi\)
\(164\) 0 0
\(165\) −41.8814 −3.26046
\(166\) 0 0
\(167\) 3.78258 0.292705 0.146353 0.989232i \(-0.453247\pi\)
0.146353 + 0.989232i \(0.453247\pi\)
\(168\) 0 0
\(169\) −12.8494 −0.988415
\(170\) 0 0
\(171\) −24.0602 −1.83993
\(172\) 0 0
\(173\) 2.33969 0.177884 0.0889418 0.996037i \(-0.471651\pi\)
0.0889418 + 0.996037i \(0.471651\pi\)
\(174\) 0 0
\(175\) −14.0735 −1.06386
\(176\) 0 0
\(177\) −25.7127 −1.93269
\(178\) 0 0
\(179\) 21.8041 1.62971 0.814857 0.579662i \(-0.196815\pi\)
0.814857 + 0.579662i \(0.196815\pi\)
\(180\) 0 0
\(181\) 14.1966 1.05522 0.527611 0.849486i \(-0.323088\pi\)
0.527611 + 0.849486i \(0.323088\pi\)
\(182\) 0 0
\(183\) −18.2606 −1.34986
\(184\) 0 0
\(185\) 28.6651 2.10750
\(186\) 0 0
\(187\) 12.8592 0.940358
\(188\) 0 0
\(189\) −21.0717 −1.53274
\(190\) 0 0
\(191\) 19.8117 1.43352 0.716762 0.697318i \(-0.245623\pi\)
0.716762 + 0.697318i \(0.245623\pi\)
\(192\) 0 0
\(193\) 14.9356 1.07509 0.537545 0.843235i \(-0.319352\pi\)
0.537545 + 0.843235i \(0.319352\pi\)
\(194\) 0 0
\(195\) 3.08459 0.220892
\(196\) 0 0
\(197\) 9.14044 0.651230 0.325615 0.945502i \(-0.394429\pi\)
0.325615 + 0.945502i \(0.394429\pi\)
\(198\) 0 0
\(199\) 0.183873 0.0130344 0.00651721 0.999979i \(-0.497925\pi\)
0.00651721 + 0.999979i \(0.497925\pi\)
\(200\) 0 0
\(201\) −38.0532 −2.68406
\(202\) 0 0
\(203\) 25.1984 1.76858
\(204\) 0 0
\(205\) −2.71556 −0.189663
\(206\) 0 0
\(207\) 26.1861 1.82006
\(208\) 0 0
\(209\) −26.8759 −1.85905
\(210\) 0 0
\(211\) 4.70810 0.324119 0.162059 0.986781i \(-0.448186\pi\)
0.162059 + 0.986781i \(0.448186\pi\)
\(212\) 0 0
\(213\) 40.9249 2.80413
\(214\) 0 0
\(215\) −20.2131 −1.37852
\(216\) 0 0
\(217\) −31.6321 −2.14733
\(218\) 0 0
\(219\) 1.06469 0.0719451
\(220\) 0 0
\(221\) −0.947088 −0.0637080
\(222\) 0 0
\(223\) 5.74679 0.384833 0.192417 0.981313i \(-0.438368\pi\)
0.192417 + 0.981313i \(0.438368\pi\)
\(224\) 0 0
\(225\) 15.0297 1.00198
\(226\) 0 0
\(227\) −21.1278 −1.40230 −0.701151 0.713013i \(-0.747330\pi\)
−0.701151 + 0.713013i \(0.747330\pi\)
\(228\) 0 0
\(229\) −1.39276 −0.0920362 −0.0460181 0.998941i \(-0.514653\pi\)
−0.0460181 + 0.998941i \(0.514653\pi\)
\(230\) 0 0
\(231\) −64.6579 −4.25418
\(232\) 0 0
\(233\) 12.3430 0.808619 0.404310 0.914622i \(-0.367512\pi\)
0.404310 + 0.914622i \(0.367512\pi\)
\(234\) 0 0
\(235\) −24.3923 −1.59118
\(236\) 0 0
\(237\) 13.9174 0.904032
\(238\) 0 0
\(239\) 7.52471 0.486733 0.243367 0.969934i \(-0.421748\pi\)
0.243367 + 0.969934i \(0.421748\pi\)
\(240\) 0 0
\(241\) −30.0844 −1.93791 −0.968953 0.247246i \(-0.920474\pi\)
−0.968953 + 0.247246i \(0.920474\pi\)
\(242\) 0 0
\(243\) −16.8100 −1.07836
\(244\) 0 0
\(245\) −35.7956 −2.28690
\(246\) 0 0
\(247\) 1.97943 0.125948
\(248\) 0 0
\(249\) 28.8248 1.82669
\(250\) 0 0
\(251\) 12.7443 0.804413 0.402206 0.915549i \(-0.368243\pi\)
0.402206 + 0.915549i \(0.368243\pi\)
\(252\) 0 0
\(253\) 29.2506 1.83897
\(254\) 0 0
\(255\) −19.3969 −1.21468
\(256\) 0 0
\(257\) 6.35968 0.396706 0.198353 0.980131i \(-0.436441\pi\)
0.198353 + 0.980131i \(0.436441\pi\)
\(258\) 0 0
\(259\) 44.2542 2.74982
\(260\) 0 0
\(261\) −26.9105 −1.66572
\(262\) 0 0
\(263\) −5.38358 −0.331966 −0.165983 0.986129i \(-0.553080\pi\)
−0.165983 + 0.986129i \(0.553080\pi\)
\(264\) 0 0
\(265\) 11.4636 0.704202
\(266\) 0 0
\(267\) 30.2315 1.85014
\(268\) 0 0
\(269\) 17.7333 1.08122 0.540609 0.841274i \(-0.318194\pi\)
0.540609 + 0.841274i \(0.318194\pi\)
\(270\) 0 0
\(271\) 20.7268 1.25906 0.629530 0.776976i \(-0.283247\pi\)
0.629530 + 0.776976i \(0.283247\pi\)
\(272\) 0 0
\(273\) 4.76209 0.288215
\(274\) 0 0
\(275\) 16.7886 1.01239
\(276\) 0 0
\(277\) −22.7280 −1.36559 −0.682797 0.730608i \(-0.739237\pi\)
−0.682797 + 0.730608i \(0.739237\pi\)
\(278\) 0 0
\(279\) 33.7813 2.02243
\(280\) 0 0
\(281\) −16.5814 −0.989162 −0.494581 0.869131i \(-0.664679\pi\)
−0.494581 + 0.869131i \(0.664679\pi\)
\(282\) 0 0
\(283\) 17.6737 1.05059 0.525296 0.850920i \(-0.323955\pi\)
0.525296 + 0.850920i \(0.323955\pi\)
\(284\) 0 0
\(285\) 40.5398 2.40137
\(286\) 0 0
\(287\) −4.19237 −0.247468
\(288\) 0 0
\(289\) −11.0444 −0.649671
\(290\) 0 0
\(291\) −10.5846 −0.620483
\(292\) 0 0
\(293\) 6.55216 0.382781 0.191391 0.981514i \(-0.438700\pi\)
0.191391 + 0.981514i \(0.438700\pi\)
\(294\) 0 0
\(295\) 26.4824 1.54186
\(296\) 0 0
\(297\) 25.1368 1.45859
\(298\) 0 0
\(299\) −2.15432 −0.124588
\(300\) 0 0
\(301\) −31.2056 −1.79866
\(302\) 0 0
\(303\) 34.7018 1.99357
\(304\) 0 0
\(305\) 18.8072 1.07690
\(306\) 0 0
\(307\) −5.89018 −0.336170 −0.168085 0.985772i \(-0.553758\pi\)
−0.168085 + 0.985772i \(0.553758\pi\)
\(308\) 0 0
\(309\) 40.7773 2.31974
\(310\) 0 0
\(311\) −19.5605 −1.10917 −0.554586 0.832126i \(-0.687123\pi\)
−0.554586 + 0.832126i \(0.687123\pi\)
\(312\) 0 0
\(313\) 7.25925 0.410317 0.205159 0.978729i \(-0.434229\pi\)
0.205159 + 0.978729i \(0.434229\pi\)
\(314\) 0 0
\(315\) 59.6164 3.35900
\(316\) 0 0
\(317\) 22.4711 1.26210 0.631051 0.775741i \(-0.282624\pi\)
0.631051 + 0.775741i \(0.282624\pi\)
\(318\) 0 0
\(319\) −30.0598 −1.68302
\(320\) 0 0
\(321\) 27.7312 1.54780
\(322\) 0 0
\(323\) −12.4473 −0.692585
\(324\) 0 0
\(325\) −1.23649 −0.0685882
\(326\) 0 0
\(327\) −2.43309 −0.134550
\(328\) 0 0
\(329\) −37.6576 −2.07613
\(330\) 0 0
\(331\) 27.4449 1.50851 0.754254 0.656582i \(-0.227999\pi\)
0.754254 + 0.656582i \(0.227999\pi\)
\(332\) 0 0
\(333\) −47.2609 −2.58988
\(334\) 0 0
\(335\) 39.1922 2.14130
\(336\) 0 0
\(337\) 27.1727 1.48019 0.740096 0.672501i \(-0.234780\pi\)
0.740096 + 0.672501i \(0.234780\pi\)
\(338\) 0 0
\(339\) 22.8648 1.24184
\(340\) 0 0
\(341\) 37.7346 2.04344
\(342\) 0 0
\(343\) −24.3426 −1.31438
\(344\) 0 0
\(345\) −44.1218 −2.37544
\(346\) 0 0
\(347\) −9.69471 −0.520440 −0.260220 0.965549i \(-0.583795\pi\)
−0.260220 + 0.965549i \(0.583795\pi\)
\(348\) 0 0
\(349\) −33.5404 −1.79538 −0.897689 0.440630i \(-0.854755\pi\)
−0.897689 + 0.440630i \(0.854755\pi\)
\(350\) 0 0
\(351\) −1.85134 −0.0988174
\(352\) 0 0
\(353\) −12.0918 −0.643579 −0.321790 0.946811i \(-0.604284\pi\)
−0.321790 + 0.946811i \(0.604284\pi\)
\(354\) 0 0
\(355\) −42.1499 −2.23709
\(356\) 0 0
\(357\) −29.9456 −1.58489
\(358\) 0 0
\(359\) 4.38635 0.231503 0.115751 0.993278i \(-0.463072\pi\)
0.115751 + 0.993278i \(0.463072\pi\)
\(360\) 0 0
\(361\) 7.01499 0.369210
\(362\) 0 0
\(363\) 46.5739 2.44450
\(364\) 0 0
\(365\) −1.09656 −0.0573965
\(366\) 0 0
\(367\) −18.5437 −0.967974 −0.483987 0.875075i \(-0.660812\pi\)
−0.483987 + 0.875075i \(0.660812\pi\)
\(368\) 0 0
\(369\) 4.47721 0.233074
\(370\) 0 0
\(371\) 17.6978 0.918826
\(372\) 0 0
\(373\) −29.2488 −1.51444 −0.757222 0.653158i \(-0.773444\pi\)
−0.757222 + 0.653158i \(0.773444\pi\)
\(374\) 0 0
\(375\) 14.4170 0.744493
\(376\) 0 0
\(377\) 2.21392 0.114023
\(378\) 0 0
\(379\) −1.10977 −0.0570048 −0.0285024 0.999594i \(-0.509074\pi\)
−0.0285024 + 0.999594i \(0.509074\pi\)
\(380\) 0 0
\(381\) 13.8690 0.710531
\(382\) 0 0
\(383\) −7.22623 −0.369243 −0.184621 0.982810i \(-0.559106\pi\)
−0.184621 + 0.982810i \(0.559106\pi\)
\(384\) 0 0
\(385\) 66.5932 3.39390
\(386\) 0 0
\(387\) 33.3258 1.69404
\(388\) 0 0
\(389\) −17.4907 −0.886814 −0.443407 0.896320i \(-0.646230\pi\)
−0.443407 + 0.896320i \(0.646230\pi\)
\(390\) 0 0
\(391\) 13.5471 0.685106
\(392\) 0 0
\(393\) 20.7944 1.04894
\(394\) 0 0
\(395\) −14.3340 −0.721220
\(396\) 0 0
\(397\) −10.8378 −0.543933 −0.271966 0.962307i \(-0.587674\pi\)
−0.271966 + 0.962307i \(0.587674\pi\)
\(398\) 0 0
\(399\) 62.5867 3.13325
\(400\) 0 0
\(401\) 27.2985 1.36322 0.681611 0.731715i \(-0.261280\pi\)
0.681611 + 0.731715i \(0.261280\pi\)
\(402\) 0 0
\(403\) −2.77918 −0.138441
\(404\) 0 0
\(405\) 2.57339 0.127873
\(406\) 0 0
\(407\) −52.7918 −2.61679
\(408\) 0 0
\(409\) 20.5561 1.01643 0.508217 0.861229i \(-0.330305\pi\)
0.508217 + 0.861229i \(0.330305\pi\)
\(410\) 0 0
\(411\) 53.8799 2.65770
\(412\) 0 0
\(413\) 40.8843 2.01179
\(414\) 0 0
\(415\) −29.6875 −1.45730
\(416\) 0 0
\(417\) −3.60815 −0.176692
\(418\) 0 0
\(419\) −20.7529 −1.01384 −0.506922 0.861992i \(-0.669217\pi\)
−0.506922 + 0.861992i \(0.669217\pi\)
\(420\) 0 0
\(421\) 27.8137 1.35556 0.677780 0.735265i \(-0.262942\pi\)
0.677780 + 0.735265i \(0.262942\pi\)
\(422\) 0 0
\(423\) 40.2162 1.95538
\(424\) 0 0
\(425\) 7.77546 0.377165
\(426\) 0 0
\(427\) 29.0352 1.40511
\(428\) 0 0
\(429\) −5.68080 −0.274272
\(430\) 0 0
\(431\) 7.07019 0.340559 0.170280 0.985396i \(-0.445533\pi\)
0.170280 + 0.985396i \(0.445533\pi\)
\(432\) 0 0
\(433\) −28.2977 −1.35990 −0.679949 0.733259i \(-0.737998\pi\)
−0.679949 + 0.733259i \(0.737998\pi\)
\(434\) 0 0
\(435\) 45.3424 2.17400
\(436\) 0 0
\(437\) −28.3136 −1.35442
\(438\) 0 0
\(439\) 12.2043 0.582479 0.291240 0.956650i \(-0.405932\pi\)
0.291240 + 0.956650i \(0.405932\pi\)
\(440\) 0 0
\(441\) 59.0171 2.81034
\(442\) 0 0
\(443\) 18.6611 0.886614 0.443307 0.896370i \(-0.353805\pi\)
0.443307 + 0.896370i \(0.353805\pi\)
\(444\) 0 0
\(445\) −31.1363 −1.47600
\(446\) 0 0
\(447\) −32.5384 −1.53901
\(448\) 0 0
\(449\) 31.3463 1.47932 0.739661 0.672979i \(-0.234986\pi\)
0.739661 + 0.672979i \(0.234986\pi\)
\(450\) 0 0
\(451\) 5.00117 0.235496
\(452\) 0 0
\(453\) −44.0565 −2.06995
\(454\) 0 0
\(455\) −4.90463 −0.229933
\(456\) 0 0
\(457\) 20.5025 0.959068 0.479534 0.877523i \(-0.340806\pi\)
0.479534 + 0.877523i \(0.340806\pi\)
\(458\) 0 0
\(459\) 11.6418 0.543395
\(460\) 0 0
\(461\) 28.9423 1.34798 0.673990 0.738741i \(-0.264579\pi\)
0.673990 + 0.738741i \(0.264579\pi\)
\(462\) 0 0
\(463\) −1.51527 −0.0704208 −0.0352104 0.999380i \(-0.511210\pi\)
−0.0352104 + 0.999380i \(0.511210\pi\)
\(464\) 0 0
\(465\) −56.9192 −2.63957
\(466\) 0 0
\(467\) −30.6493 −1.41828 −0.709139 0.705068i \(-0.750916\pi\)
−0.709139 + 0.705068i \(0.750916\pi\)
\(468\) 0 0
\(469\) 60.5062 2.79392
\(470\) 0 0
\(471\) 36.6345 1.68803
\(472\) 0 0
\(473\) 37.2258 1.71165
\(474\) 0 0
\(475\) −16.2508 −0.745639
\(476\) 0 0
\(477\) −18.9003 −0.865384
\(478\) 0 0
\(479\) 22.6034 1.03278 0.516388 0.856355i \(-0.327276\pi\)
0.516388 + 0.856355i \(0.327276\pi\)
\(480\) 0 0
\(481\) 3.88815 0.177284
\(482\) 0 0
\(483\) −68.1167 −3.09942
\(484\) 0 0
\(485\) 10.9015 0.495010
\(486\) 0 0
\(487\) 6.04914 0.274113 0.137056 0.990563i \(-0.456236\pi\)
0.137056 + 0.990563i \(0.456236\pi\)
\(488\) 0 0
\(489\) −27.4319 −1.24051
\(490\) 0 0
\(491\) 22.1868 1.00127 0.500637 0.865657i \(-0.333099\pi\)
0.500637 + 0.865657i \(0.333099\pi\)
\(492\) 0 0
\(493\) −13.9218 −0.627008
\(494\) 0 0
\(495\) −71.1177 −3.19650
\(496\) 0 0
\(497\) −65.0724 −2.91890
\(498\) 0 0
\(499\) −31.0030 −1.38789 −0.693943 0.720030i \(-0.744128\pi\)
−0.693943 + 0.720030i \(0.744128\pi\)
\(500\) 0 0
\(501\) 10.5080 0.469462
\(502\) 0 0
\(503\) −1.00000 −0.0445878
\(504\) 0 0
\(505\) −35.7405 −1.59043
\(506\) 0 0
\(507\) −35.6955 −1.58529
\(508\) 0 0
\(509\) −21.8647 −0.969137 −0.484569 0.874753i \(-0.661023\pi\)
−0.484569 + 0.874753i \(0.661023\pi\)
\(510\) 0 0
\(511\) −1.69290 −0.0748896
\(512\) 0 0
\(513\) −24.3316 −1.07427
\(514\) 0 0
\(515\) −41.9979 −1.85065
\(516\) 0 0
\(517\) 44.9226 1.97569
\(518\) 0 0
\(519\) 6.49965 0.285303
\(520\) 0 0
\(521\) −1.92786 −0.0844612 −0.0422306 0.999108i \(-0.513446\pi\)
−0.0422306 + 0.999108i \(0.513446\pi\)
\(522\) 0 0
\(523\) −19.0191 −0.831646 −0.415823 0.909446i \(-0.636506\pi\)
−0.415823 + 0.909446i \(0.636506\pi\)
\(524\) 0 0
\(525\) −39.0961 −1.70629
\(526\) 0 0
\(527\) 17.4764 0.761283
\(528\) 0 0
\(529\) 7.81531 0.339796
\(530\) 0 0
\(531\) −43.6621 −1.89477
\(532\) 0 0
\(533\) −0.368340 −0.0159546
\(534\) 0 0
\(535\) −28.5612 −1.23481
\(536\) 0 0
\(537\) 60.5715 2.61385
\(538\) 0 0
\(539\) 65.9238 2.83954
\(540\) 0 0
\(541\) −25.2620 −1.08610 −0.543049 0.839701i \(-0.682730\pi\)
−0.543049 + 0.839701i \(0.682730\pi\)
\(542\) 0 0
\(543\) 39.4379 1.69244
\(544\) 0 0
\(545\) 2.50592 0.107342
\(546\) 0 0
\(547\) 15.8303 0.676853 0.338427 0.940993i \(-0.390105\pi\)
0.338427 + 0.940993i \(0.390105\pi\)
\(548\) 0 0
\(549\) −31.0079 −1.32338
\(550\) 0 0
\(551\) 29.0969 1.23957
\(552\) 0 0
\(553\) −22.1292 −0.941031
\(554\) 0 0
\(555\) 79.6315 3.38017
\(556\) 0 0
\(557\) −22.1472 −0.938409 −0.469204 0.883090i \(-0.655459\pi\)
−0.469204 + 0.883090i \(0.655459\pi\)
\(558\) 0 0
\(559\) −2.74171 −0.115962
\(560\) 0 0
\(561\) 35.7227 1.50821
\(562\) 0 0
\(563\) −9.30504 −0.392161 −0.196080 0.980588i \(-0.562821\pi\)
−0.196080 + 0.980588i \(0.562821\pi\)
\(564\) 0 0
\(565\) −23.5492 −0.990720
\(566\) 0 0
\(567\) 3.97289 0.166846
\(568\) 0 0
\(569\) 4.92502 0.206467 0.103234 0.994657i \(-0.467081\pi\)
0.103234 + 0.994657i \(0.467081\pi\)
\(570\) 0 0
\(571\) −17.2749 −0.722932 −0.361466 0.932385i \(-0.617724\pi\)
−0.361466 + 0.932385i \(0.617724\pi\)
\(572\) 0 0
\(573\) 55.0367 2.29919
\(574\) 0 0
\(575\) 17.6867 0.737586
\(576\) 0 0
\(577\) 33.9915 1.41509 0.707543 0.706671i \(-0.249804\pi\)
0.707543 + 0.706671i \(0.249804\pi\)
\(578\) 0 0
\(579\) 41.4910 1.72431
\(580\) 0 0
\(581\) −45.8326 −1.90146
\(582\) 0 0
\(583\) −21.1121 −0.874375
\(584\) 0 0
\(585\) 5.23786 0.216559
\(586\) 0 0
\(587\) −21.4528 −0.885451 −0.442726 0.896657i \(-0.645988\pi\)
−0.442726 + 0.896657i \(0.645988\pi\)
\(588\) 0 0
\(589\) −36.5259 −1.50502
\(590\) 0 0
\(591\) 25.3921 1.04449
\(592\) 0 0
\(593\) −10.9293 −0.448815 −0.224407 0.974495i \(-0.572045\pi\)
−0.224407 + 0.974495i \(0.572045\pi\)
\(594\) 0 0
\(595\) 30.8419 1.26440
\(596\) 0 0
\(597\) 0.510798 0.0209056
\(598\) 0 0
\(599\) −34.0120 −1.38969 −0.694847 0.719158i \(-0.744528\pi\)
−0.694847 + 0.719158i \(0.744528\pi\)
\(600\) 0 0
\(601\) 15.6512 0.638425 0.319213 0.947683i \(-0.396582\pi\)
0.319213 + 0.947683i \(0.396582\pi\)
\(602\) 0 0
\(603\) −64.6171 −2.63141
\(604\) 0 0
\(605\) −47.9679 −1.95017
\(606\) 0 0
\(607\) 5.42603 0.220236 0.110118 0.993919i \(-0.464877\pi\)
0.110118 + 0.993919i \(0.464877\pi\)
\(608\) 0 0
\(609\) 70.0010 2.83658
\(610\) 0 0
\(611\) −3.30858 −0.133851
\(612\) 0 0
\(613\) 11.8043 0.476771 0.238385 0.971171i \(-0.423382\pi\)
0.238385 + 0.971171i \(0.423382\pi\)
\(614\) 0 0
\(615\) −7.54381 −0.304196
\(616\) 0 0
\(617\) −22.0439 −0.887454 −0.443727 0.896162i \(-0.646344\pi\)
−0.443727 + 0.896162i \(0.646344\pi\)
\(618\) 0 0
\(619\) 3.88761 0.156256 0.0781282 0.996943i \(-0.475106\pi\)
0.0781282 + 0.996943i \(0.475106\pi\)
\(620\) 0 0
\(621\) 26.4815 1.06267
\(622\) 0 0
\(623\) −48.0693 −1.92586
\(624\) 0 0
\(625\) −30.7792 −1.23117
\(626\) 0 0
\(627\) −74.6610 −2.98167
\(628\) 0 0
\(629\) −24.4499 −0.974882
\(630\) 0 0
\(631\) −42.4500 −1.68991 −0.844953 0.534841i \(-0.820372\pi\)
−0.844953 + 0.534841i \(0.820372\pi\)
\(632\) 0 0
\(633\) 13.0790 0.519845
\(634\) 0 0
\(635\) −14.2841 −0.566848
\(636\) 0 0
\(637\) −4.85533 −0.192375
\(638\) 0 0
\(639\) 69.4935 2.74912
\(640\) 0 0
\(641\) −19.7521 −0.780161 −0.390080 0.920781i \(-0.627553\pi\)
−0.390080 + 0.920781i \(0.627553\pi\)
\(642\) 0 0
\(643\) −31.8338 −1.25540 −0.627701 0.778455i \(-0.716004\pi\)
−0.627701 + 0.778455i \(0.716004\pi\)
\(644\) 0 0
\(645\) −56.1517 −2.21097
\(646\) 0 0
\(647\) 2.56968 0.101024 0.0505122 0.998723i \(-0.483915\pi\)
0.0505122 + 0.998723i \(0.483915\pi\)
\(648\) 0 0
\(649\) −48.7718 −1.91446
\(650\) 0 0
\(651\) −87.8737 −3.44404
\(652\) 0 0
\(653\) 48.4354 1.89543 0.947713 0.319125i \(-0.103389\pi\)
0.947713 + 0.319125i \(0.103389\pi\)
\(654\) 0 0
\(655\) −21.4169 −0.836826
\(656\) 0 0
\(657\) 1.80792 0.0705338
\(658\) 0 0
\(659\) 19.2938 0.751580 0.375790 0.926705i \(-0.377371\pi\)
0.375790 + 0.926705i \(0.377371\pi\)
\(660\) 0 0
\(661\) 27.6053 1.07372 0.536861 0.843671i \(-0.319610\pi\)
0.536861 + 0.843671i \(0.319610\pi\)
\(662\) 0 0
\(663\) −2.63100 −0.102180
\(664\) 0 0
\(665\) −64.4600 −2.49965
\(666\) 0 0
\(667\) −31.6678 −1.22618
\(668\) 0 0
\(669\) 15.9645 0.617224
\(670\) 0 0
\(671\) −34.6367 −1.33713
\(672\) 0 0
\(673\) −12.1806 −0.469528 −0.234764 0.972052i \(-0.575432\pi\)
−0.234764 + 0.972052i \(0.575432\pi\)
\(674\) 0 0
\(675\) 15.1993 0.585020
\(676\) 0 0
\(677\) 30.3761 1.16745 0.583723 0.811953i \(-0.301595\pi\)
0.583723 + 0.811953i \(0.301595\pi\)
\(678\) 0 0
\(679\) 16.8300 0.645877
\(680\) 0 0
\(681\) −58.6928 −2.24911
\(682\) 0 0
\(683\) 8.65487 0.331169 0.165585 0.986196i \(-0.447049\pi\)
0.165585 + 0.986196i \(0.447049\pi\)
\(684\) 0 0
\(685\) −55.4926 −2.12026
\(686\) 0 0
\(687\) −3.86908 −0.147614
\(688\) 0 0
\(689\) 1.55492 0.0592378
\(690\) 0 0
\(691\) −19.6457 −0.747359 −0.373679 0.927558i \(-0.621904\pi\)
−0.373679 + 0.927558i \(0.621904\pi\)
\(692\) 0 0
\(693\) −109.794 −4.17072
\(694\) 0 0
\(695\) 3.71615 0.140962
\(696\) 0 0
\(697\) 2.31624 0.0877338
\(698\) 0 0
\(699\) 34.2888 1.29692
\(700\) 0 0
\(701\) −12.6055 −0.476104 −0.238052 0.971252i \(-0.576509\pi\)
−0.238052 + 0.971252i \(0.576509\pi\)
\(702\) 0 0
\(703\) 51.1007 1.92730
\(704\) 0 0
\(705\) −67.7616 −2.55205
\(706\) 0 0
\(707\) −55.1773 −2.07516
\(708\) 0 0
\(709\) 33.8692 1.27199 0.635993 0.771695i \(-0.280591\pi\)
0.635993 + 0.771695i \(0.280591\pi\)
\(710\) 0 0
\(711\) 23.6328 0.886297
\(712\) 0 0
\(713\) 39.7532 1.48877
\(714\) 0 0
\(715\) 5.85084 0.218809
\(716\) 0 0
\(717\) 20.9036 0.780659
\(718\) 0 0
\(719\) −19.9061 −0.742372 −0.371186 0.928559i \(-0.621049\pi\)
−0.371186 + 0.928559i \(0.621049\pi\)
\(720\) 0 0
\(721\) −64.8377 −2.41468
\(722\) 0 0
\(723\) −83.5741 −3.10815
\(724\) 0 0
\(725\) −18.1760 −0.675039
\(726\) 0 0
\(727\) 48.0664 1.78268 0.891342 0.453331i \(-0.149764\pi\)
0.891342 + 0.453331i \(0.149764\pi\)
\(728\) 0 0
\(729\) −43.9996 −1.62961
\(730\) 0 0
\(731\) 17.2407 0.637672
\(732\) 0 0
\(733\) −21.7845 −0.804629 −0.402314 0.915502i \(-0.631794\pi\)
−0.402314 + 0.915502i \(0.631794\pi\)
\(734\) 0 0
\(735\) −99.4399 −3.66790
\(736\) 0 0
\(737\) −72.1791 −2.65875
\(738\) 0 0
\(739\) −48.4145 −1.78096 −0.890479 0.455024i \(-0.849631\pi\)
−0.890479 + 0.455024i \(0.849631\pi\)
\(740\) 0 0
\(741\) 5.49883 0.202005
\(742\) 0 0
\(743\) −35.3436 −1.29663 −0.648316 0.761372i \(-0.724526\pi\)
−0.648316 + 0.761372i \(0.724526\pi\)
\(744\) 0 0
\(745\) 33.5123 1.22780
\(746\) 0 0
\(747\) 48.9465 1.79086
\(748\) 0 0
\(749\) −44.0938 −1.61115
\(750\) 0 0
\(751\) 4.51093 0.164606 0.0823031 0.996607i \(-0.473772\pi\)
0.0823031 + 0.996607i \(0.473772\pi\)
\(752\) 0 0
\(753\) 35.4035 1.29018
\(754\) 0 0
\(755\) 45.3752 1.65137
\(756\) 0 0
\(757\) 6.09638 0.221577 0.110788 0.993844i \(-0.464662\pi\)
0.110788 + 0.993844i \(0.464662\pi\)
\(758\) 0 0
\(759\) 81.2578 2.94947
\(760\) 0 0
\(761\) −10.2423 −0.371284 −0.185642 0.982617i \(-0.559436\pi\)
−0.185642 + 0.982617i \(0.559436\pi\)
\(762\) 0 0
\(763\) 3.86872 0.140057
\(764\) 0 0
\(765\) −32.9374 −1.19085
\(766\) 0 0
\(767\) 3.59207 0.129702
\(768\) 0 0
\(769\) −26.5975 −0.959129 −0.479565 0.877507i \(-0.659205\pi\)
−0.479565 + 0.877507i \(0.659205\pi\)
\(770\) 0 0
\(771\) 17.6671 0.636266
\(772\) 0 0
\(773\) −18.6278 −0.669995 −0.334997 0.942219i \(-0.608735\pi\)
−0.334997 + 0.942219i \(0.608735\pi\)
\(774\) 0 0
\(775\) 22.8167 0.819599
\(776\) 0 0
\(777\) 122.938 4.41036
\(778\) 0 0
\(779\) −4.84097 −0.173446
\(780\) 0 0
\(781\) 77.6263 2.77769
\(782\) 0 0
\(783\) −27.2141 −0.972551
\(784\) 0 0
\(785\) −37.7311 −1.34668
\(786\) 0 0
\(787\) 11.7361 0.418347 0.209174 0.977879i \(-0.432923\pi\)
0.209174 + 0.977879i \(0.432923\pi\)
\(788\) 0 0
\(789\) −14.9555 −0.532431
\(790\) 0 0
\(791\) −36.3559 −1.29267
\(792\) 0 0
\(793\) 2.55101 0.0905891
\(794\) 0 0
\(795\) 31.8457 1.12945
\(796\) 0 0
\(797\) −25.1954 −0.892468 −0.446234 0.894916i \(-0.647235\pi\)
−0.446234 + 0.894916i \(0.647235\pi\)
\(798\) 0 0
\(799\) 20.8054 0.736042
\(800\) 0 0
\(801\) 51.3352 1.81384
\(802\) 0 0
\(803\) 2.01950 0.0712666
\(804\) 0 0
\(805\) 70.1555 2.47266
\(806\) 0 0
\(807\) 49.2629 1.73414
\(808\) 0 0
\(809\) −33.8088 −1.18866 −0.594328 0.804223i \(-0.702582\pi\)
−0.594328 + 0.804223i \(0.702582\pi\)
\(810\) 0 0
\(811\) 48.7547 1.71201 0.856005 0.516967i \(-0.172939\pi\)
0.856005 + 0.516967i \(0.172939\pi\)
\(812\) 0 0
\(813\) 57.5787 2.01937
\(814\) 0 0
\(815\) 28.2530 0.989659
\(816\) 0 0
\(817\) −36.0334 −1.26065
\(818\) 0 0
\(819\) 8.08638 0.282561
\(820\) 0 0
\(821\) −56.3460 −1.96649 −0.983244 0.182295i \(-0.941647\pi\)
−0.983244 + 0.182295i \(0.941647\pi\)
\(822\) 0 0
\(823\) 33.9547 1.18359 0.591793 0.806090i \(-0.298420\pi\)
0.591793 + 0.806090i \(0.298420\pi\)
\(824\) 0 0
\(825\) 46.6386 1.62375
\(826\) 0 0
\(827\) −29.1900 −1.01504 −0.507518 0.861641i \(-0.669437\pi\)
−0.507518 + 0.861641i \(0.669437\pi\)
\(828\) 0 0
\(829\) 4.84344 0.168220 0.0841098 0.996456i \(-0.473195\pi\)
0.0841098 + 0.996456i \(0.473195\pi\)
\(830\) 0 0
\(831\) −63.1382 −2.19024
\(832\) 0 0
\(833\) 30.5319 1.05787
\(834\) 0 0
\(835\) −10.8225 −0.374528
\(836\) 0 0
\(837\) 34.1624 1.18082
\(838\) 0 0
\(839\) 51.8249 1.78919 0.894596 0.446875i \(-0.147463\pi\)
0.894596 + 0.446875i \(0.147463\pi\)
\(840\) 0 0
\(841\) 3.54381 0.122200
\(842\) 0 0
\(843\) −46.0629 −1.58649
\(844\) 0 0
\(845\) 36.7639 1.26472
\(846\) 0 0
\(847\) −74.0544 −2.54454
\(848\) 0 0
\(849\) 49.0973 1.68502
\(850\) 0 0
\(851\) −55.6158 −1.90648
\(852\) 0 0
\(853\) 29.3479 1.00485 0.502427 0.864620i \(-0.332441\pi\)
0.502427 + 0.864620i \(0.332441\pi\)
\(854\) 0 0
\(855\) 68.8396 2.35426
\(856\) 0 0
\(857\) 21.5484 0.736080 0.368040 0.929810i \(-0.380029\pi\)
0.368040 + 0.929810i \(0.380029\pi\)
\(858\) 0 0
\(859\) −20.3226 −0.693399 −0.346700 0.937976i \(-0.612698\pi\)
−0.346700 + 0.937976i \(0.612698\pi\)
\(860\) 0 0
\(861\) −11.6464 −0.396907
\(862\) 0 0
\(863\) 45.9037 1.56258 0.781291 0.624167i \(-0.214562\pi\)
0.781291 + 0.624167i \(0.214562\pi\)
\(864\) 0 0
\(865\) −6.69419 −0.227609
\(866\) 0 0
\(867\) −30.6813 −1.04199
\(868\) 0 0
\(869\) 26.3985 0.895506
\(870\) 0 0
\(871\) 5.31604 0.180127
\(872\) 0 0
\(873\) −17.9735 −0.608311
\(874\) 0 0
\(875\) −22.9237 −0.774963
\(876\) 0 0
\(877\) 47.3822 1.59998 0.799992 0.600011i \(-0.204837\pi\)
0.799992 + 0.600011i \(0.204837\pi\)
\(878\) 0 0
\(879\) 18.2018 0.613933
\(880\) 0 0
\(881\) 35.1715 1.18496 0.592479 0.805586i \(-0.298149\pi\)
0.592479 + 0.805586i \(0.298149\pi\)
\(882\) 0 0
\(883\) 18.8095 0.632990 0.316495 0.948594i \(-0.397494\pi\)
0.316495 + 0.948594i \(0.397494\pi\)
\(884\) 0 0
\(885\) 73.5677 2.47295
\(886\) 0 0
\(887\) −42.2215 −1.41766 −0.708830 0.705380i \(-0.750776\pi\)
−0.708830 + 0.705380i \(0.750776\pi\)
\(888\) 0 0
\(889\) −22.0523 −0.739610
\(890\) 0 0
\(891\) −4.73934 −0.158774
\(892\) 0 0
\(893\) −43.4836 −1.45512
\(894\) 0 0
\(895\) −62.3846 −2.08529
\(896\) 0 0
\(897\) −5.98469 −0.199823
\(898\) 0 0
\(899\) −40.8529 −1.36252
\(900\) 0 0
\(901\) −9.77785 −0.325747
\(902\) 0 0
\(903\) −86.6889 −2.88482
\(904\) 0 0
\(905\) −40.6183 −1.35020
\(906\) 0 0
\(907\) 8.13790 0.270214 0.135107 0.990831i \(-0.456862\pi\)
0.135107 + 0.990831i \(0.456862\pi\)
\(908\) 0 0
\(909\) 58.9262 1.95446
\(910\) 0 0
\(911\) −0.335989 −0.0111318 −0.00556590 0.999985i \(-0.501772\pi\)
−0.00556590 + 0.999985i \(0.501772\pi\)
\(912\) 0 0
\(913\) 54.6747 1.80947
\(914\) 0 0
\(915\) 52.2462 1.72721
\(916\) 0 0
\(917\) −33.0640 −1.09187
\(918\) 0 0
\(919\) −40.3906 −1.33236 −0.666182 0.745789i \(-0.732073\pi\)
−0.666182 + 0.745789i \(0.732073\pi\)
\(920\) 0 0
\(921\) −16.3629 −0.539174
\(922\) 0 0
\(923\) −5.71722 −0.188185
\(924\) 0 0
\(925\) −31.9211 −1.04956
\(926\) 0 0
\(927\) 69.2429 2.27423
\(928\) 0 0
\(929\) −26.6253 −0.873548 −0.436774 0.899571i \(-0.643879\pi\)
−0.436774 + 0.899571i \(0.643879\pi\)
\(930\) 0 0
\(931\) −63.8121 −2.09136
\(932\) 0 0
\(933\) −54.3388 −1.77897
\(934\) 0 0
\(935\) −36.7920 −1.20323
\(936\) 0 0
\(937\) 10.7576 0.351436 0.175718 0.984441i \(-0.443775\pi\)
0.175718 + 0.984441i \(0.443775\pi\)
\(938\) 0 0
\(939\) 20.1661 0.658097
\(940\) 0 0
\(941\) −46.3237 −1.51011 −0.755055 0.655662i \(-0.772390\pi\)
−0.755055 + 0.655662i \(0.772390\pi\)
\(942\) 0 0
\(943\) 5.26870 0.171573
\(944\) 0 0
\(945\) 60.2890 1.96120
\(946\) 0 0
\(947\) −25.0484 −0.813962 −0.406981 0.913437i \(-0.633419\pi\)
−0.406981 + 0.913437i \(0.633419\pi\)
\(948\) 0 0
\(949\) −0.148738 −0.00482822
\(950\) 0 0
\(951\) 62.4245 2.02425
\(952\) 0 0
\(953\) −20.0611 −0.649843 −0.324921 0.945741i \(-0.605338\pi\)
−0.324921 + 0.945741i \(0.605338\pi\)
\(954\) 0 0
\(955\) −56.6841 −1.83425
\(956\) 0 0
\(957\) −83.5057 −2.69936
\(958\) 0 0
\(959\) −85.6713 −2.76647
\(960\) 0 0
\(961\) 20.2835 0.654306
\(962\) 0 0
\(963\) 47.0896 1.51744
\(964\) 0 0
\(965\) −42.7329 −1.37562
\(966\) 0 0
\(967\) 22.1889 0.713547 0.356773 0.934191i \(-0.383877\pi\)
0.356773 + 0.934191i \(0.383877\pi\)
\(968\) 0 0
\(969\) −34.5784 −1.11082
\(970\) 0 0
\(971\) 29.7101 0.953443 0.476721 0.879054i \(-0.341825\pi\)
0.476721 + 0.879054i \(0.341825\pi\)
\(972\) 0 0
\(973\) 5.73711 0.183923
\(974\) 0 0
\(975\) −3.43496 −0.110007
\(976\) 0 0
\(977\) 15.7076 0.502530 0.251265 0.967918i \(-0.419153\pi\)
0.251265 + 0.967918i \(0.419153\pi\)
\(978\) 0 0
\(979\) 57.3429 1.83269
\(980\) 0 0
\(981\) −4.13157 −0.131911
\(982\) 0 0
\(983\) −15.1215 −0.482301 −0.241150 0.970488i \(-0.577525\pi\)
−0.241150 + 0.970488i \(0.577525\pi\)
\(984\) 0 0
\(985\) −26.1521 −0.833275
\(986\) 0 0
\(987\) −104.613 −3.32985
\(988\) 0 0
\(989\) 39.2172 1.24703
\(990\) 0 0
\(991\) −58.7879 −1.86746 −0.933730 0.357979i \(-0.883466\pi\)
−0.933730 + 0.357979i \(0.883466\pi\)
\(992\) 0 0
\(993\) 76.2417 2.41946
\(994\) 0 0
\(995\) −0.526087 −0.0166781
\(996\) 0 0
\(997\) 25.4219 0.805120 0.402560 0.915394i \(-0.368120\pi\)
0.402560 + 0.915394i \(0.368120\pi\)
\(998\) 0 0
\(999\) −47.7941 −1.51214
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 8048.2.a.o.1.5 5
4.3 odd 2 1006.2.a.h.1.1 5
12.11 even 2 9054.2.a.ba.1.5 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1006.2.a.h.1.1 5 4.3 odd 2
8048.2.a.o.1.5 5 1.1 even 1 trivial
9054.2.a.ba.1.5 5 12.11 even 2