Properties

Label 8048.2.a.u.1.20
Level $8048$
Weight $2$
Character 8048.1
Self dual yes
Analytic conductor $64.264$
Analytic rank $0$
Dimension $26$
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: \(26\)
Twist minimal: no (minimal twist has level 503)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.20
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)
 
\(f(q)\) \(=\) \(q+1.65492 q^{3} +1.16845 q^{5} +5.22170 q^{7} -0.261242 q^{9} +O(q^{10})\) \(q+1.65492 q^{3} +1.16845 q^{5} +5.22170 q^{7} -0.261242 q^{9} +1.87602 q^{11} +3.59450 q^{13} +1.93370 q^{15} -4.39494 q^{17} +3.70486 q^{19} +8.64150 q^{21} -7.20094 q^{23} -3.63472 q^{25} -5.39709 q^{27} +5.84811 q^{29} +1.35487 q^{31} +3.10465 q^{33} +6.10132 q^{35} +3.71544 q^{37} +5.94860 q^{39} -2.17811 q^{41} -10.7599 q^{43} -0.305249 q^{45} -2.70411 q^{47} +20.2662 q^{49} -7.27328 q^{51} +13.6672 q^{53} +2.19204 q^{55} +6.13125 q^{57} +10.8525 q^{59} +13.7302 q^{61} -1.36413 q^{63} +4.20000 q^{65} -2.91833 q^{67} -11.9170 q^{69} +5.23876 q^{71} -6.15254 q^{73} -6.01516 q^{75} +9.79599 q^{77} +8.25148 q^{79} -8.14803 q^{81} +13.1873 q^{83} -5.13529 q^{85} +9.67814 q^{87} -12.9803 q^{89} +18.7694 q^{91} +2.24219 q^{93} +4.32896 q^{95} +8.59794 q^{97} -0.490093 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 26 q - 4 q^{3} + 9 q^{5} - 11 q^{7} + 42 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 26 q - 4 q^{3} + 9 q^{5} - 11 q^{7} + 42 q^{9} + 17 q^{11} + 14 q^{13} - 18 q^{15} + 17 q^{17} + 22 q^{19} - 16 q^{21} - 27 q^{23} + 93 q^{25} - 31 q^{27} + 13 q^{29} - 26 q^{31} + 6 q^{33} + 22 q^{35} + 55 q^{37} + 15 q^{39} + 24 q^{41} - 20 q^{43} - 8 q^{45} + 25 q^{47} + 65 q^{49} - 7 q^{51} + 30 q^{53} - 25 q^{55} + 9 q^{57} + 26 q^{59} + 15 q^{61} + 19 q^{63} + 20 q^{65} + 20 q^{67} - 27 q^{69} + 35 q^{71} + 38 q^{73} - 2 q^{75} - 6 q^{77} - 21 q^{79} + 70 q^{81} + 48 q^{83} + 6 q^{85} + 9 q^{87} - 5 q^{89} + 24 q^{91} - 8 q^{93} - 43 q^{95} + 142 q^{97} + 50 q^{99}+O(q^{100}) \) 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
\(3\) 1.65492 0.955468 0.477734 0.878504i \(-0.341458\pi\)
0.477734 + 0.878504i \(0.341458\pi\)
\(4\) 0 0
\(5\) 1.16845 0.522548 0.261274 0.965265i \(-0.415857\pi\)
0.261274 + 0.965265i \(0.415857\pi\)
\(6\) 0 0
\(7\) 5.22170 1.97362 0.986809 0.161889i \(-0.0517587\pi\)
0.986809 + 0.161889i \(0.0517587\pi\)
\(8\) 0 0
\(9\) −0.261242 −0.0870806
\(10\) 0 0
\(11\) 1.87602 0.565640 0.282820 0.959173i \(-0.408730\pi\)
0.282820 + 0.959173i \(0.408730\pi\)
\(12\) 0 0
\(13\) 3.59450 0.996934 0.498467 0.866909i \(-0.333896\pi\)
0.498467 + 0.866909i \(0.333896\pi\)
\(14\) 0 0
\(15\) 1.93370 0.499278
\(16\) 0 0
\(17\) −4.39494 −1.06593 −0.532965 0.846137i \(-0.678922\pi\)
−0.532965 + 0.846137i \(0.678922\pi\)
\(18\) 0 0
\(19\) 3.70486 0.849954 0.424977 0.905204i \(-0.360282\pi\)
0.424977 + 0.905204i \(0.360282\pi\)
\(20\) 0 0
\(21\) 8.64150 1.88573
\(22\) 0 0
\(23\) −7.20094 −1.50150 −0.750750 0.660587i \(-0.770308\pi\)
−0.750750 + 0.660587i \(0.770308\pi\)
\(24\) 0 0
\(25\) −3.63472 −0.726943
\(26\) 0 0
\(27\) −5.39709 −1.03867
\(28\) 0 0
\(29\) 5.84811 1.08597 0.542983 0.839744i \(-0.317295\pi\)
0.542983 + 0.839744i \(0.317295\pi\)
\(30\) 0 0
\(31\) 1.35487 0.243341 0.121671 0.992571i \(-0.461175\pi\)
0.121671 + 0.992571i \(0.461175\pi\)
\(32\) 0 0
\(33\) 3.10465 0.540451
\(34\) 0 0
\(35\) 6.10132 1.03131
\(36\) 0 0
\(37\) 3.71544 0.610814 0.305407 0.952222i \(-0.401207\pi\)
0.305407 + 0.952222i \(0.401207\pi\)
\(38\) 0 0
\(39\) 5.94860 0.952539
\(40\) 0 0
\(41\) −2.17811 −0.340163 −0.170082 0.985430i \(-0.554403\pi\)
−0.170082 + 0.985430i \(0.554403\pi\)
\(42\) 0 0
\(43\) −10.7599 −1.64087 −0.820435 0.571740i \(-0.806269\pi\)
−0.820435 + 0.571740i \(0.806269\pi\)
\(44\) 0 0
\(45\) −0.305249 −0.0455038
\(46\) 0 0
\(47\) −2.70411 −0.394435 −0.197218 0.980360i \(-0.563191\pi\)
−0.197218 + 0.980360i \(0.563191\pi\)
\(48\) 0 0
\(49\) 20.2662 2.89517
\(50\) 0 0
\(51\) −7.27328 −1.01846
\(52\) 0 0
\(53\) 13.6672 1.87734 0.938668 0.344823i \(-0.112061\pi\)
0.938668 + 0.344823i \(0.112061\pi\)
\(54\) 0 0
\(55\) 2.19204 0.295574
\(56\) 0 0
\(57\) 6.13125 0.812104
\(58\) 0 0
\(59\) 10.8525 1.41288 0.706439 0.707774i \(-0.250300\pi\)
0.706439 + 0.707774i \(0.250300\pi\)
\(60\) 0 0
\(61\) 13.7302 1.75797 0.878985 0.476849i \(-0.158221\pi\)
0.878985 + 0.476849i \(0.158221\pi\)
\(62\) 0 0
\(63\) −1.36413 −0.171864
\(64\) 0 0
\(65\) 4.20000 0.520946
\(66\) 0 0
\(67\) −2.91833 −0.356531 −0.178265 0.983982i \(-0.557049\pi\)
−0.178265 + 0.983982i \(0.557049\pi\)
\(68\) 0 0
\(69\) −11.9170 −1.43463
\(70\) 0 0
\(71\) 5.23876 0.621727 0.310863 0.950455i \(-0.399382\pi\)
0.310863 + 0.950455i \(0.399382\pi\)
\(72\) 0 0
\(73\) −6.15254 −0.720100 −0.360050 0.932933i \(-0.617240\pi\)
−0.360050 + 0.932933i \(0.617240\pi\)
\(74\) 0 0
\(75\) −6.01516 −0.694571
\(76\) 0 0
\(77\) 9.79599 1.11636
\(78\) 0 0
\(79\) 8.25148 0.928364 0.464182 0.885740i \(-0.346348\pi\)
0.464182 + 0.885740i \(0.346348\pi\)
\(80\) 0 0
\(81\) −8.14803 −0.905336
\(82\) 0 0
\(83\) 13.1873 1.44750 0.723749 0.690063i \(-0.242417\pi\)
0.723749 + 0.690063i \(0.242417\pi\)
\(84\) 0 0
\(85\) −5.13529 −0.557000
\(86\) 0 0
\(87\) 9.67814 1.03761
\(88\) 0 0
\(89\) −12.9803 −1.37591 −0.687957 0.725752i \(-0.741492\pi\)
−0.687957 + 0.725752i \(0.741492\pi\)
\(90\) 0 0
\(91\) 18.7694 1.96757
\(92\) 0 0
\(93\) 2.24219 0.232505
\(94\) 0 0
\(95\) 4.32896 0.444142
\(96\) 0 0
\(97\) 8.59794 0.872988 0.436494 0.899707i \(-0.356220\pi\)
0.436494 + 0.899707i \(0.356220\pi\)
\(98\) 0 0
\(99\) −0.490093 −0.0492562
\(100\) 0 0
\(101\) 14.7250 1.46519 0.732596 0.680663i \(-0.238308\pi\)
0.732596 + 0.680663i \(0.238308\pi\)
\(102\) 0 0
\(103\) −17.8630 −1.76010 −0.880049 0.474883i \(-0.842490\pi\)
−0.880049 + 0.474883i \(0.842490\pi\)
\(104\) 0 0
\(105\) 10.0972 0.985385
\(106\) 0 0
\(107\) 4.24062 0.409956 0.204978 0.978767i \(-0.434288\pi\)
0.204978 + 0.978767i \(0.434288\pi\)
\(108\) 0 0
\(109\) 7.64169 0.731942 0.365971 0.930626i \(-0.380737\pi\)
0.365971 + 0.930626i \(0.380737\pi\)
\(110\) 0 0
\(111\) 6.14875 0.583613
\(112\) 0 0
\(113\) −6.24322 −0.587313 −0.293656 0.955911i \(-0.594872\pi\)
−0.293656 + 0.955911i \(0.594872\pi\)
\(114\) 0 0
\(115\) −8.41396 −0.784606
\(116\) 0 0
\(117\) −0.939033 −0.0868136
\(118\) 0 0
\(119\) −22.9491 −2.10374
\(120\) 0 0
\(121\) −7.48057 −0.680052
\(122\) 0 0
\(123\) −3.60459 −0.325015
\(124\) 0 0
\(125\) −10.0893 −0.902411
\(126\) 0 0
\(127\) 18.0174 1.59878 0.799392 0.600810i \(-0.205155\pi\)
0.799392 + 0.600810i \(0.205155\pi\)
\(128\) 0 0
\(129\) −17.8068 −1.56780
\(130\) 0 0
\(131\) 4.17633 0.364888 0.182444 0.983216i \(-0.441599\pi\)
0.182444 + 0.983216i \(0.441599\pi\)
\(132\) 0 0
\(133\) 19.3457 1.67748
\(134\) 0 0
\(135\) −6.30625 −0.542756
\(136\) 0 0
\(137\) 5.22839 0.446692 0.223346 0.974739i \(-0.428302\pi\)
0.223346 + 0.974739i \(0.428302\pi\)
\(138\) 0 0
\(139\) −12.6668 −1.07438 −0.537191 0.843460i \(-0.680515\pi\)
−0.537191 + 0.843460i \(0.680515\pi\)
\(140\) 0 0
\(141\) −4.47509 −0.376870
\(142\) 0 0
\(143\) 6.74333 0.563906
\(144\) 0 0
\(145\) 6.83324 0.567470
\(146\) 0 0
\(147\) 33.5389 2.76624
\(148\) 0 0
\(149\) −6.95650 −0.569898 −0.284949 0.958543i \(-0.591977\pi\)
−0.284949 + 0.958543i \(0.591977\pi\)
\(150\) 0 0
\(151\) 4.24542 0.345488 0.172744 0.984967i \(-0.444737\pi\)
0.172744 + 0.984967i \(0.444737\pi\)
\(152\) 0 0
\(153\) 1.14814 0.0928218
\(154\) 0 0
\(155\) 1.58310 0.127157
\(156\) 0 0
\(157\) 12.7081 1.01421 0.507107 0.861883i \(-0.330715\pi\)
0.507107 + 0.861883i \(0.330715\pi\)
\(158\) 0 0
\(159\) 22.6181 1.79373
\(160\) 0 0
\(161\) −37.6012 −2.96339
\(162\) 0 0
\(163\) −11.2202 −0.878831 −0.439416 0.898284i \(-0.644814\pi\)
−0.439416 + 0.898284i \(0.644814\pi\)
\(164\) 0 0
\(165\) 3.62764 0.282412
\(166\) 0 0
\(167\) −24.3997 −1.88811 −0.944054 0.329790i \(-0.893022\pi\)
−0.944054 + 0.329790i \(0.893022\pi\)
\(168\) 0 0
\(169\) −0.0795839 −0.00612184
\(170\) 0 0
\(171\) −0.967865 −0.0740145
\(172\) 0 0
\(173\) −15.7024 −1.19383 −0.596916 0.802304i \(-0.703607\pi\)
−0.596916 + 0.802304i \(0.703607\pi\)
\(174\) 0 0
\(175\) −18.9794 −1.43471
\(176\) 0 0
\(177\) 17.9601 1.34996
\(178\) 0 0
\(179\) −13.2973 −0.993889 −0.496944 0.867782i \(-0.665545\pi\)
−0.496944 + 0.867782i \(0.665545\pi\)
\(180\) 0 0
\(181\) 4.74072 0.352375 0.176187 0.984357i \(-0.443624\pi\)
0.176187 + 0.984357i \(0.443624\pi\)
\(182\) 0 0
\(183\) 22.7224 1.67968
\(184\) 0 0
\(185\) 4.34132 0.319180
\(186\) 0 0
\(187\) −8.24498 −0.602933
\(188\) 0 0
\(189\) −28.1820 −2.04994
\(190\) 0 0
\(191\) −12.0868 −0.874570 −0.437285 0.899323i \(-0.644060\pi\)
−0.437285 + 0.899323i \(0.644060\pi\)
\(192\) 0 0
\(193\) 16.6225 1.19652 0.598259 0.801303i \(-0.295860\pi\)
0.598259 + 0.801303i \(0.295860\pi\)
\(194\) 0 0
\(195\) 6.95067 0.497748
\(196\) 0 0
\(197\) 9.11878 0.649686 0.324843 0.945768i \(-0.394689\pi\)
0.324843 + 0.945768i \(0.394689\pi\)
\(198\) 0 0
\(199\) 5.97681 0.423685 0.211842 0.977304i \(-0.432054\pi\)
0.211842 + 0.977304i \(0.432054\pi\)
\(200\) 0 0
\(201\) −4.82960 −0.340654
\(202\) 0 0
\(203\) 30.5371 2.14328
\(204\) 0 0
\(205\) −2.54502 −0.177752
\(206\) 0 0
\(207\) 1.88119 0.130751
\(208\) 0 0
\(209\) 6.95038 0.480768
\(210\) 0 0
\(211\) −4.05588 −0.279218 −0.139609 0.990207i \(-0.544585\pi\)
−0.139609 + 0.990207i \(0.544585\pi\)
\(212\) 0 0
\(213\) 8.66973 0.594040
\(214\) 0 0
\(215\) −12.5724 −0.857434
\(216\) 0 0
\(217\) 7.07471 0.480262
\(218\) 0 0
\(219\) −10.1820 −0.688033
\(220\) 0 0
\(221\) −15.7976 −1.06266
\(222\) 0 0
\(223\) 2.35562 0.157744 0.0788719 0.996885i \(-0.474868\pi\)
0.0788719 + 0.996885i \(0.474868\pi\)
\(224\) 0 0
\(225\) 0.949540 0.0633026
\(226\) 0 0
\(227\) −6.87919 −0.456588 −0.228294 0.973592i \(-0.573315\pi\)
−0.228294 + 0.973592i \(0.573315\pi\)
\(228\) 0 0
\(229\) −1.96260 −0.129692 −0.0648462 0.997895i \(-0.520656\pi\)
−0.0648462 + 0.997895i \(0.520656\pi\)
\(230\) 0 0
\(231\) 16.2116 1.06664
\(232\) 0 0
\(233\) −16.7753 −1.09899 −0.549493 0.835499i \(-0.685179\pi\)
−0.549493 + 0.835499i \(0.685179\pi\)
\(234\) 0 0
\(235\) −3.15963 −0.206112
\(236\) 0 0
\(237\) 13.6555 0.887023
\(238\) 0 0
\(239\) −4.26878 −0.276124 −0.138062 0.990424i \(-0.544087\pi\)
−0.138062 + 0.990424i \(0.544087\pi\)
\(240\) 0 0
\(241\) −6.02897 −0.388360 −0.194180 0.980966i \(-0.562205\pi\)
−0.194180 + 0.980966i \(0.562205\pi\)
\(242\) 0 0
\(243\) 2.70695 0.173651
\(244\) 0 0
\(245\) 23.6801 1.51286
\(246\) 0 0
\(247\) 13.3171 0.847349
\(248\) 0 0
\(249\) 21.8240 1.38304
\(250\) 0 0
\(251\) −22.4540 −1.41728 −0.708642 0.705568i \(-0.750692\pi\)
−0.708642 + 0.705568i \(0.750692\pi\)
\(252\) 0 0
\(253\) −13.5091 −0.849308
\(254\) 0 0
\(255\) −8.49849 −0.532196
\(256\) 0 0
\(257\) 5.60541 0.349656 0.174828 0.984599i \(-0.444063\pi\)
0.174828 + 0.984599i \(0.444063\pi\)
\(258\) 0 0
\(259\) 19.4009 1.20551
\(260\) 0 0
\(261\) −1.52777 −0.0945665
\(262\) 0 0
\(263\) −1.68398 −0.103839 −0.0519194 0.998651i \(-0.516534\pi\)
−0.0519194 + 0.998651i \(0.516534\pi\)
\(264\) 0 0
\(265\) 15.9695 0.980998
\(266\) 0 0
\(267\) −21.4814 −1.31464
\(268\) 0 0
\(269\) −11.6951 −0.713063 −0.356532 0.934283i \(-0.616041\pi\)
−0.356532 + 0.934283i \(0.616041\pi\)
\(270\) 0 0
\(271\) −8.02617 −0.487555 −0.243778 0.969831i \(-0.578387\pi\)
−0.243778 + 0.969831i \(0.578387\pi\)
\(272\) 0 0
\(273\) 31.0618 1.87995
\(274\) 0 0
\(275\) −6.81878 −0.411188
\(276\) 0 0
\(277\) 24.9879 1.50138 0.750689 0.660656i \(-0.229722\pi\)
0.750689 + 0.660656i \(0.229722\pi\)
\(278\) 0 0
\(279\) −0.353948 −0.0211903
\(280\) 0 0
\(281\) 13.2415 0.789921 0.394961 0.918698i \(-0.370758\pi\)
0.394961 + 0.918698i \(0.370758\pi\)
\(282\) 0 0
\(283\) 20.7184 1.23158 0.615791 0.787910i \(-0.288837\pi\)
0.615791 + 0.787910i \(0.288837\pi\)
\(284\) 0 0
\(285\) 7.16408 0.424364
\(286\) 0 0
\(287\) −11.3734 −0.671352
\(288\) 0 0
\(289\) 2.31552 0.136207
\(290\) 0 0
\(291\) 14.2289 0.834113
\(292\) 0 0
\(293\) 6.18432 0.361292 0.180646 0.983548i \(-0.442181\pi\)
0.180646 + 0.983548i \(0.442181\pi\)
\(294\) 0 0
\(295\) 12.6807 0.738297
\(296\) 0 0
\(297\) −10.1250 −0.587514
\(298\) 0 0
\(299\) −25.8838 −1.49690
\(300\) 0 0
\(301\) −56.1850 −3.23845
\(302\) 0 0
\(303\) 24.3687 1.39994
\(304\) 0 0
\(305\) 16.0431 0.918624
\(306\) 0 0
\(307\) −6.11954 −0.349260 −0.174630 0.984634i \(-0.555873\pi\)
−0.174630 + 0.984634i \(0.555873\pi\)
\(308\) 0 0
\(309\) −29.5619 −1.68172
\(310\) 0 0
\(311\) −11.4142 −0.647241 −0.323621 0.946187i \(-0.604900\pi\)
−0.323621 + 0.946187i \(0.604900\pi\)
\(312\) 0 0
\(313\) 32.2666 1.82382 0.911908 0.410394i \(-0.134609\pi\)
0.911908 + 0.410394i \(0.134609\pi\)
\(314\) 0 0
\(315\) −1.59392 −0.0898071
\(316\) 0 0
\(317\) −16.2640 −0.913476 −0.456738 0.889601i \(-0.650982\pi\)
−0.456738 + 0.889601i \(0.650982\pi\)
\(318\) 0 0
\(319\) 10.9711 0.614266
\(320\) 0 0
\(321\) 7.01788 0.391700
\(322\) 0 0
\(323\) −16.2827 −0.905992
\(324\) 0 0
\(325\) −13.0650 −0.724715
\(326\) 0 0
\(327\) 12.6464 0.699347
\(328\) 0 0
\(329\) −14.1201 −0.778465
\(330\) 0 0
\(331\) −17.1918 −0.944945 −0.472473 0.881345i \(-0.656638\pi\)
−0.472473 + 0.881345i \(0.656638\pi\)
\(332\) 0 0
\(333\) −0.970627 −0.0531900
\(334\) 0 0
\(335\) −3.40993 −0.186304
\(336\) 0 0
\(337\) −12.7167 −0.692724 −0.346362 0.938101i \(-0.612583\pi\)
−0.346362 + 0.938101i \(0.612583\pi\)
\(338\) 0 0
\(339\) −10.3320 −0.561158
\(340\) 0 0
\(341\) 2.54175 0.137643
\(342\) 0 0
\(343\) 69.2720 3.74034
\(344\) 0 0
\(345\) −13.9244 −0.749666
\(346\) 0 0
\(347\) −11.1578 −0.598982 −0.299491 0.954099i \(-0.596817\pi\)
−0.299491 + 0.954099i \(0.596817\pi\)
\(348\) 0 0
\(349\) −5.37351 −0.287638 −0.143819 0.989604i \(-0.545938\pi\)
−0.143819 + 0.989604i \(0.545938\pi\)
\(350\) 0 0
\(351\) −19.3998 −1.03549
\(352\) 0 0
\(353\) −13.5118 −0.719163 −0.359582 0.933114i \(-0.617081\pi\)
−0.359582 + 0.933114i \(0.617081\pi\)
\(354\) 0 0
\(355\) 6.12125 0.324882
\(356\) 0 0
\(357\) −37.9789 −2.01006
\(358\) 0 0
\(359\) −16.0870 −0.849039 −0.424519 0.905419i \(-0.639557\pi\)
−0.424519 + 0.905419i \(0.639557\pi\)
\(360\) 0 0
\(361\) −5.27398 −0.277578
\(362\) 0 0
\(363\) −12.3797 −0.649768
\(364\) 0 0
\(365\) −7.18896 −0.376287
\(366\) 0 0
\(367\) −17.6750 −0.922626 −0.461313 0.887237i \(-0.652622\pi\)
−0.461313 + 0.887237i \(0.652622\pi\)
\(368\) 0 0
\(369\) 0.569013 0.0296216
\(370\) 0 0
\(371\) 71.3661 3.70514
\(372\) 0 0
\(373\) 8.48505 0.439339 0.219670 0.975574i \(-0.429502\pi\)
0.219670 + 0.975574i \(0.429502\pi\)
\(374\) 0 0
\(375\) −16.6969 −0.862225
\(376\) 0 0
\(377\) 21.0210 1.08264
\(378\) 0 0
\(379\) 9.85389 0.506160 0.253080 0.967445i \(-0.418556\pi\)
0.253080 + 0.967445i \(0.418556\pi\)
\(380\) 0 0
\(381\) 29.8173 1.52759
\(382\) 0 0
\(383\) 10.4001 0.531420 0.265710 0.964053i \(-0.414394\pi\)
0.265710 + 0.964053i \(0.414394\pi\)
\(384\) 0 0
\(385\) 11.4462 0.583350
\(386\) 0 0
\(387\) 2.81094 0.142888
\(388\) 0 0
\(389\) −1.07680 −0.0545960 −0.0272980 0.999627i \(-0.508690\pi\)
−0.0272980 + 0.999627i \(0.508690\pi\)
\(390\) 0 0
\(391\) 31.6477 1.60049
\(392\) 0 0
\(393\) 6.91149 0.348639
\(394\) 0 0
\(395\) 9.64147 0.485115
\(396\) 0 0
\(397\) −22.0327 −1.10579 −0.552895 0.833251i \(-0.686477\pi\)
−0.552895 + 0.833251i \(0.686477\pi\)
\(398\) 0 0
\(399\) 32.0156 1.60278
\(400\) 0 0
\(401\) 1.11136 0.0554988 0.0277494 0.999615i \(-0.491166\pi\)
0.0277494 + 0.999615i \(0.491166\pi\)
\(402\) 0 0
\(403\) 4.87006 0.242595
\(404\) 0 0
\(405\) −9.52059 −0.473082
\(406\) 0 0
\(407\) 6.97022 0.345501
\(408\) 0 0
\(409\) −12.7644 −0.631158 −0.315579 0.948899i \(-0.602199\pi\)
−0.315579 + 0.948899i \(0.602199\pi\)
\(410\) 0 0
\(411\) 8.65257 0.426800
\(412\) 0 0
\(413\) 56.6686 2.78848
\(414\) 0 0
\(415\) 15.4088 0.756388
\(416\) 0 0
\(417\) −20.9625 −1.02654
\(418\) 0 0
\(419\) −2.81345 −0.137446 −0.0687230 0.997636i \(-0.521892\pi\)
−0.0687230 + 0.997636i \(0.521892\pi\)
\(420\) 0 0
\(421\) 0.340713 0.0166053 0.00830267 0.999966i \(-0.497357\pi\)
0.00830267 + 0.999966i \(0.497357\pi\)
\(422\) 0 0
\(423\) 0.706427 0.0343477
\(424\) 0 0
\(425\) 15.9744 0.774871
\(426\) 0 0
\(427\) 71.6949 3.46956
\(428\) 0 0
\(429\) 11.1597 0.538794
\(430\) 0 0
\(431\) −17.9632 −0.865257 −0.432629 0.901572i \(-0.642414\pi\)
−0.432629 + 0.901572i \(0.642414\pi\)
\(432\) 0 0
\(433\) 15.4246 0.741259 0.370629 0.928781i \(-0.379142\pi\)
0.370629 + 0.928781i \(0.379142\pi\)
\(434\) 0 0
\(435\) 11.3085 0.542199
\(436\) 0 0
\(437\) −26.6785 −1.27621
\(438\) 0 0
\(439\) −6.69721 −0.319640 −0.159820 0.987146i \(-0.551091\pi\)
−0.159820 + 0.987146i \(0.551091\pi\)
\(440\) 0 0
\(441\) −5.29437 −0.252113
\(442\) 0 0
\(443\) −33.6219 −1.59743 −0.798713 0.601712i \(-0.794486\pi\)
−0.798713 + 0.601712i \(0.794486\pi\)
\(444\) 0 0
\(445\) −15.1669 −0.718981
\(446\) 0 0
\(447\) −11.5124 −0.544520
\(448\) 0 0
\(449\) 27.8142 1.31263 0.656316 0.754486i \(-0.272114\pi\)
0.656316 + 0.754486i \(0.272114\pi\)
\(450\) 0 0
\(451\) −4.08616 −0.192410
\(452\) 0 0
\(453\) 7.02583 0.330102
\(454\) 0 0
\(455\) 21.9312 1.02815
\(456\) 0 0
\(457\) −2.39554 −0.112059 −0.0560294 0.998429i \(-0.517844\pi\)
−0.0560294 + 0.998429i \(0.517844\pi\)
\(458\) 0 0
\(459\) 23.7199 1.10715
\(460\) 0 0
\(461\) −24.6862 −1.14975 −0.574876 0.818241i \(-0.694950\pi\)
−0.574876 + 0.818241i \(0.694950\pi\)
\(462\) 0 0
\(463\) −12.5562 −0.583537 −0.291769 0.956489i \(-0.594244\pi\)
−0.291769 + 0.956489i \(0.594244\pi\)
\(464\) 0 0
\(465\) 2.61990 0.121495
\(466\) 0 0
\(467\) −3.68834 −0.170676 −0.0853380 0.996352i \(-0.527197\pi\)
−0.0853380 + 0.996352i \(0.527197\pi\)
\(468\) 0 0
\(469\) −15.2386 −0.703655
\(470\) 0 0
\(471\) 21.0308 0.969049
\(472\) 0 0
\(473\) −20.1857 −0.928141
\(474\) 0 0
\(475\) −13.4661 −0.617868
\(476\) 0 0
\(477\) −3.57044 −0.163479
\(478\) 0 0
\(479\) 8.22389 0.375759 0.187880 0.982192i \(-0.439839\pi\)
0.187880 + 0.982192i \(0.439839\pi\)
\(480\) 0 0
\(481\) 13.3551 0.608942
\(482\) 0 0
\(483\) −62.2269 −2.83142
\(484\) 0 0
\(485\) 10.0463 0.456179
\(486\) 0 0
\(487\) −15.8574 −0.718569 −0.359284 0.933228i \(-0.616979\pi\)
−0.359284 + 0.933228i \(0.616979\pi\)
\(488\) 0 0
\(489\) −18.5685 −0.839695
\(490\) 0 0
\(491\) 29.4948 1.33108 0.665541 0.746361i \(-0.268201\pi\)
0.665541 + 0.746361i \(0.268201\pi\)
\(492\) 0 0
\(493\) −25.7021 −1.15756
\(494\) 0 0
\(495\) −0.572651 −0.0257388
\(496\) 0 0
\(497\) 27.3553 1.22705
\(498\) 0 0
\(499\) −31.8227 −1.42458 −0.712291 0.701884i \(-0.752342\pi\)
−0.712291 + 0.701884i \(0.752342\pi\)
\(500\) 0 0
\(501\) −40.3796 −1.80403
\(502\) 0 0
\(503\) −1.00000 −0.0445878
\(504\) 0 0
\(505\) 17.2055 0.765634
\(506\) 0 0
\(507\) −0.131705 −0.00584923
\(508\) 0 0
\(509\) −11.5533 −0.512092 −0.256046 0.966665i \(-0.582420\pi\)
−0.256046 + 0.966665i \(0.582420\pi\)
\(510\) 0 0
\(511\) −32.1267 −1.42120
\(512\) 0 0
\(513\) −19.9955 −0.882823
\(514\) 0 0
\(515\) −20.8721 −0.919736
\(516\) 0 0
\(517\) −5.07296 −0.223108
\(518\) 0 0
\(519\) −25.9862 −1.14067
\(520\) 0 0
\(521\) −27.4131 −1.20099 −0.600494 0.799629i \(-0.705029\pi\)
−0.600494 + 0.799629i \(0.705029\pi\)
\(522\) 0 0
\(523\) 9.22449 0.403359 0.201679 0.979452i \(-0.435360\pi\)
0.201679 + 0.979452i \(0.435360\pi\)
\(524\) 0 0
\(525\) −31.4094 −1.37082
\(526\) 0 0
\(527\) −5.95456 −0.259385
\(528\) 0 0
\(529\) 28.8535 1.25450
\(530\) 0 0
\(531\) −2.83513 −0.123034
\(532\) 0 0
\(533\) −7.82921 −0.339121
\(534\) 0 0
\(535\) 4.95496 0.214222
\(536\) 0 0
\(537\) −22.0060 −0.949629
\(538\) 0 0
\(539\) 38.0197 1.63762
\(540\) 0 0
\(541\) 39.5153 1.69890 0.849449 0.527671i \(-0.176935\pi\)
0.849449 + 0.527671i \(0.176935\pi\)
\(542\) 0 0
\(543\) 7.84551 0.336683
\(544\) 0 0
\(545\) 8.92896 0.382475
\(546\) 0 0
\(547\) 18.3165 0.783157 0.391579 0.920145i \(-0.371929\pi\)
0.391579 + 0.920145i \(0.371929\pi\)
\(548\) 0 0
\(549\) −3.58690 −0.153085
\(550\) 0 0
\(551\) 21.6664 0.923021
\(552\) 0 0
\(553\) 43.0868 1.83224
\(554\) 0 0
\(555\) 7.18453 0.304966
\(556\) 0 0
\(557\) 1.84697 0.0782588 0.0391294 0.999234i \(-0.487542\pi\)
0.0391294 + 0.999234i \(0.487542\pi\)
\(558\) 0 0
\(559\) −38.6764 −1.63584
\(560\) 0 0
\(561\) −13.6448 −0.576083
\(562\) 0 0
\(563\) 18.6098 0.784312 0.392156 0.919899i \(-0.371729\pi\)
0.392156 + 0.919899i \(0.371729\pi\)
\(564\) 0 0
\(565\) −7.29491 −0.306899
\(566\) 0 0
\(567\) −42.5466 −1.78679
\(568\) 0 0
\(569\) −19.8106 −0.830503 −0.415251 0.909707i \(-0.636306\pi\)
−0.415251 + 0.909707i \(0.636306\pi\)
\(570\) 0 0
\(571\) −33.9800 −1.42202 −0.711008 0.703183i \(-0.751761\pi\)
−0.711008 + 0.703183i \(0.751761\pi\)
\(572\) 0 0
\(573\) −20.0027 −0.835624
\(574\) 0 0
\(575\) 26.1734 1.09150
\(576\) 0 0
\(577\) 12.7808 0.532073 0.266037 0.963963i \(-0.414286\pi\)
0.266037 + 0.963963i \(0.414286\pi\)
\(578\) 0 0
\(579\) 27.5090 1.14323
\(580\) 0 0
\(581\) 68.8604 2.85681
\(582\) 0 0
\(583\) 25.6399 1.06190
\(584\) 0 0
\(585\) −1.09722 −0.0453643
\(586\) 0 0
\(587\) 29.4463 1.21538 0.607690 0.794174i \(-0.292096\pi\)
0.607690 + 0.794174i \(0.292096\pi\)
\(588\) 0 0
\(589\) 5.01959 0.206829
\(590\) 0 0
\(591\) 15.0908 0.620754
\(592\) 0 0
\(593\) −20.7161 −0.850706 −0.425353 0.905027i \(-0.639850\pi\)
−0.425353 + 0.905027i \(0.639850\pi\)
\(594\) 0 0
\(595\) −26.8149 −1.09931
\(596\) 0 0
\(597\) 9.89114 0.404817
\(598\) 0 0
\(599\) −33.4780 −1.36787 −0.683937 0.729541i \(-0.739734\pi\)
−0.683937 + 0.729541i \(0.739734\pi\)
\(600\) 0 0
\(601\) −25.6180 −1.04498 −0.522489 0.852646i \(-0.674997\pi\)
−0.522489 + 0.852646i \(0.674997\pi\)
\(602\) 0 0
\(603\) 0.762389 0.0310469
\(604\) 0 0
\(605\) −8.74069 −0.355360
\(606\) 0 0
\(607\) −31.2416 −1.26806 −0.634028 0.773310i \(-0.718600\pi\)
−0.634028 + 0.773310i \(0.718600\pi\)
\(608\) 0 0
\(609\) 50.5364 2.04784
\(610\) 0 0
\(611\) −9.71993 −0.393226
\(612\) 0 0
\(613\) 40.2405 1.62530 0.812650 0.582752i \(-0.198024\pi\)
0.812650 + 0.582752i \(0.198024\pi\)
\(614\) 0 0
\(615\) −4.21180 −0.169836
\(616\) 0 0
\(617\) 9.27736 0.373492 0.186746 0.982408i \(-0.440206\pi\)
0.186746 + 0.982408i \(0.440206\pi\)
\(618\) 0 0
\(619\) −7.92403 −0.318494 −0.159247 0.987239i \(-0.550907\pi\)
−0.159247 + 0.987239i \(0.550907\pi\)
\(620\) 0 0
\(621\) 38.8641 1.55956
\(622\) 0 0
\(623\) −67.7795 −2.71553
\(624\) 0 0
\(625\) 6.38474 0.255390
\(626\) 0 0
\(627\) 11.5023 0.459358
\(628\) 0 0
\(629\) −16.3291 −0.651085
\(630\) 0 0
\(631\) 9.56630 0.380828 0.190414 0.981704i \(-0.439017\pi\)
0.190414 + 0.981704i \(0.439017\pi\)
\(632\) 0 0
\(633\) −6.71216 −0.266784
\(634\) 0 0
\(635\) 21.0525 0.835442
\(636\) 0 0
\(637\) 72.8467 2.88629
\(638\) 0 0
\(639\) −1.36858 −0.0541403
\(640\) 0 0
\(641\) −28.1267 −1.11094 −0.555469 0.831538i \(-0.687461\pi\)
−0.555469 + 0.831538i \(0.687461\pi\)
\(642\) 0 0
\(643\) −3.46879 −0.136796 −0.0683978 0.997658i \(-0.521789\pi\)
−0.0683978 + 0.997658i \(0.521789\pi\)
\(644\) 0 0
\(645\) −20.8064 −0.819251
\(646\) 0 0
\(647\) −14.4099 −0.566512 −0.283256 0.959044i \(-0.591415\pi\)
−0.283256 + 0.959044i \(0.591415\pi\)
\(648\) 0 0
\(649\) 20.3595 0.799180
\(650\) 0 0
\(651\) 11.7081 0.458875
\(652\) 0 0
\(653\) −17.1316 −0.670411 −0.335205 0.942145i \(-0.608806\pi\)
−0.335205 + 0.942145i \(0.608806\pi\)
\(654\) 0 0
\(655\) 4.87985 0.190671
\(656\) 0 0
\(657\) 1.60730 0.0627067
\(658\) 0 0
\(659\) −3.66347 −0.142709 −0.0713543 0.997451i \(-0.522732\pi\)
−0.0713543 + 0.997451i \(0.522732\pi\)
\(660\) 0 0
\(661\) 39.6530 1.54232 0.771162 0.636639i \(-0.219676\pi\)
0.771162 + 0.636639i \(0.219676\pi\)
\(662\) 0 0
\(663\) −26.1438 −1.01534
\(664\) 0 0
\(665\) 22.6045 0.876567
\(666\) 0 0
\(667\) −42.1118 −1.63058
\(668\) 0 0
\(669\) 3.89836 0.150719
\(670\) 0 0
\(671\) 25.7580 0.994378
\(672\) 0 0
\(673\) 34.3247 1.32312 0.661560 0.749892i \(-0.269895\pi\)
0.661560 + 0.749892i \(0.269895\pi\)
\(674\) 0 0
\(675\) 19.6169 0.755055
\(676\) 0 0
\(677\) 20.2805 0.779442 0.389721 0.920933i \(-0.372572\pi\)
0.389721 + 0.920933i \(0.372572\pi\)
\(678\) 0 0
\(679\) 44.8959 1.72295
\(680\) 0 0
\(681\) −11.3845 −0.436255
\(682\) 0 0
\(683\) −24.3123 −0.930283 −0.465141 0.885236i \(-0.653997\pi\)
−0.465141 + 0.885236i \(0.653997\pi\)
\(684\) 0 0
\(685\) 6.10914 0.233418
\(686\) 0 0
\(687\) −3.24795 −0.123917
\(688\) 0 0
\(689\) 49.1267 1.87158
\(690\) 0 0
\(691\) −17.3936 −0.661684 −0.330842 0.943686i \(-0.607333\pi\)
−0.330842 + 0.943686i \(0.607333\pi\)
\(692\) 0 0
\(693\) −2.55912 −0.0972130
\(694\) 0 0
\(695\) −14.8006 −0.561417
\(696\) 0 0
\(697\) 9.57266 0.362590
\(698\) 0 0
\(699\) −27.7617 −1.05005
\(700\) 0 0
\(701\) −25.2331 −0.953042 −0.476521 0.879163i \(-0.658102\pi\)
−0.476521 + 0.879163i \(0.658102\pi\)
\(702\) 0 0
\(703\) 13.7652 0.519164
\(704\) 0 0
\(705\) −5.22893 −0.196933
\(706\) 0 0
\(707\) 76.8896 2.89173
\(708\) 0 0
\(709\) 26.0105 0.976845 0.488422 0.872607i \(-0.337573\pi\)
0.488422 + 0.872607i \(0.337573\pi\)
\(710\) 0 0
\(711\) −2.15563 −0.0808425
\(712\) 0 0
\(713\) −9.75631 −0.365377
\(714\) 0 0
\(715\) 7.87927 0.294668
\(716\) 0 0
\(717\) −7.06448 −0.263828
\(718\) 0 0
\(719\) 11.2045 0.417858 0.208929 0.977931i \(-0.433002\pi\)
0.208929 + 0.977931i \(0.433002\pi\)
\(720\) 0 0
\(721\) −93.2755 −3.47376
\(722\) 0 0
\(723\) −9.97745 −0.371065
\(724\) 0 0
\(725\) −21.2562 −0.789436
\(726\) 0 0
\(727\) 37.1106 1.37636 0.688179 0.725541i \(-0.258411\pi\)
0.688179 + 0.725541i \(0.258411\pi\)
\(728\) 0 0
\(729\) 28.9239 1.07125
\(730\) 0 0
\(731\) 47.2892 1.74905
\(732\) 0 0
\(733\) 32.5163 1.20102 0.600509 0.799618i \(-0.294965\pi\)
0.600509 + 0.799618i \(0.294965\pi\)
\(734\) 0 0
\(735\) 39.1886 1.44549
\(736\) 0 0
\(737\) −5.47483 −0.201668
\(738\) 0 0
\(739\) 25.9897 0.956046 0.478023 0.878347i \(-0.341353\pi\)
0.478023 + 0.878347i \(0.341353\pi\)
\(740\) 0 0
\(741\) 22.0388 0.809615
\(742\) 0 0
\(743\) 5.63810 0.206842 0.103421 0.994638i \(-0.467021\pi\)
0.103421 + 0.994638i \(0.467021\pi\)
\(744\) 0 0
\(745\) −8.12834 −0.297799
\(746\) 0 0
\(747\) −3.44508 −0.126049
\(748\) 0 0
\(749\) 22.1432 0.809096
\(750\) 0 0
\(751\) −48.6229 −1.77428 −0.887138 0.461504i \(-0.847310\pi\)
−0.887138 + 0.461504i \(0.847310\pi\)
\(752\) 0 0
\(753\) −37.1595 −1.35417
\(754\) 0 0
\(755\) 4.96058 0.180534
\(756\) 0 0
\(757\) −37.9095 −1.37784 −0.688922 0.724835i \(-0.741916\pi\)
−0.688922 + 0.724835i \(0.741916\pi\)
\(758\) 0 0
\(759\) −22.3564 −0.811487
\(760\) 0 0
\(761\) 32.9830 1.19563 0.597817 0.801633i \(-0.296035\pi\)
0.597817 + 0.801633i \(0.296035\pi\)
\(762\) 0 0
\(763\) 39.9026 1.44457
\(764\) 0 0
\(765\) 1.34155 0.0485039
\(766\) 0 0
\(767\) 39.0094 1.40855
\(768\) 0 0
\(769\) 15.9036 0.573499 0.286749 0.958006i \(-0.407425\pi\)
0.286749 + 0.958006i \(0.407425\pi\)
\(770\) 0 0
\(771\) 9.27651 0.334085
\(772\) 0 0
\(773\) −6.67230 −0.239986 −0.119993 0.992775i \(-0.538287\pi\)
−0.119993 + 0.992775i \(0.538287\pi\)
\(774\) 0 0
\(775\) −4.92455 −0.176895
\(776\) 0 0
\(777\) 32.1069 1.15183
\(778\) 0 0
\(779\) −8.06960 −0.289123
\(780\) 0 0
\(781\) 9.82800 0.351673
\(782\) 0 0
\(783\) −31.5628 −1.12796
\(784\) 0 0
\(785\) 14.8488 0.529976
\(786\) 0 0
\(787\) 17.7383 0.632302 0.316151 0.948709i \(-0.397609\pi\)
0.316151 + 0.948709i \(0.397609\pi\)
\(788\) 0 0
\(789\) −2.78685 −0.0992146
\(790\) 0 0
\(791\) −32.6002 −1.15913
\(792\) 0 0
\(793\) 49.3531 1.75258
\(794\) 0 0
\(795\) 26.4282 0.937313
\(796\) 0 0
\(797\) −22.1191 −0.783497 −0.391749 0.920072i \(-0.628130\pi\)
−0.391749 + 0.920072i \(0.628130\pi\)
\(798\) 0 0
\(799\) 11.8844 0.420441
\(800\) 0 0
\(801\) 3.39101 0.119815
\(802\) 0 0
\(803\) −11.5423 −0.407317
\(804\) 0 0
\(805\) −43.9352 −1.54851
\(806\) 0 0
\(807\) −19.3545 −0.681309
\(808\) 0 0
\(809\) 42.0410 1.47808 0.739042 0.673659i \(-0.235278\pi\)
0.739042 + 0.673659i \(0.235278\pi\)
\(810\) 0 0
\(811\) 18.8593 0.662238 0.331119 0.943589i \(-0.392574\pi\)
0.331119 + 0.943589i \(0.392574\pi\)
\(812\) 0 0
\(813\) −13.2827 −0.465843
\(814\) 0 0
\(815\) −13.1102 −0.459232
\(816\) 0 0
\(817\) −39.8640 −1.39466
\(818\) 0 0
\(819\) −4.90335 −0.171337
\(820\) 0 0
\(821\) −46.4269 −1.62031 −0.810155 0.586216i \(-0.800617\pi\)
−0.810155 + 0.586216i \(0.800617\pi\)
\(822\) 0 0
\(823\) −10.2034 −0.355669 −0.177835 0.984060i \(-0.556909\pi\)
−0.177835 + 0.984060i \(0.556909\pi\)
\(824\) 0 0
\(825\) −11.2845 −0.392877
\(826\) 0 0
\(827\) 25.0611 0.871461 0.435730 0.900077i \(-0.356490\pi\)
0.435730 + 0.900077i \(0.356490\pi\)
\(828\) 0 0
\(829\) −19.9198 −0.691844 −0.345922 0.938263i \(-0.612434\pi\)
−0.345922 + 0.938263i \(0.612434\pi\)
\(830\) 0 0
\(831\) 41.3530 1.43452
\(832\) 0 0
\(833\) −89.0687 −3.08605
\(834\) 0 0
\(835\) −28.5100 −0.986628
\(836\) 0 0
\(837\) −7.31234 −0.252751
\(838\) 0 0
\(839\) −52.0091 −1.79555 −0.897777 0.440451i \(-0.854818\pi\)
−0.897777 + 0.440451i \(0.854818\pi\)
\(840\) 0 0
\(841\) 5.20033 0.179322
\(842\) 0 0
\(843\) 21.9136 0.754744
\(844\) 0 0
\(845\) −0.0929901 −0.00319896
\(846\) 0 0
\(847\) −39.0613 −1.34216
\(848\) 0 0
\(849\) 34.2873 1.17674
\(850\) 0 0
\(851\) −26.7546 −0.917137
\(852\) 0 0
\(853\) 37.6780 1.29007 0.645036 0.764152i \(-0.276842\pi\)
0.645036 + 0.764152i \(0.276842\pi\)
\(854\) 0 0
\(855\) −1.13091 −0.0386762
\(856\) 0 0
\(857\) −34.6052 −1.18209 −0.591045 0.806638i \(-0.701285\pi\)
−0.591045 + 0.806638i \(0.701285\pi\)
\(858\) 0 0
\(859\) 27.9524 0.953723 0.476861 0.878978i \(-0.341774\pi\)
0.476861 + 0.878978i \(0.341774\pi\)
\(860\) 0 0
\(861\) −18.8221 −0.641456
\(862\) 0 0
\(863\) −35.2339 −1.19938 −0.599688 0.800234i \(-0.704709\pi\)
−0.599688 + 0.800234i \(0.704709\pi\)
\(864\) 0 0
\(865\) −18.3475 −0.623835
\(866\) 0 0
\(867\) 3.83201 0.130142
\(868\) 0 0
\(869\) 15.4799 0.525120
\(870\) 0 0
\(871\) −10.4899 −0.355438
\(872\) 0 0
\(873\) −2.24614 −0.0760203
\(874\) 0 0
\(875\) −52.6831 −1.78102
\(876\) 0 0
\(877\) −29.6552 −1.00139 −0.500693 0.865625i \(-0.666921\pi\)
−0.500693 + 0.865625i \(0.666921\pi\)
\(878\) 0 0
\(879\) 10.2346 0.345203
\(880\) 0 0
\(881\) −24.3183 −0.819305 −0.409653 0.912242i \(-0.634350\pi\)
−0.409653 + 0.912242i \(0.634350\pi\)
\(882\) 0 0
\(883\) −25.5391 −0.859458 −0.429729 0.902958i \(-0.641391\pi\)
−0.429729 + 0.902958i \(0.641391\pi\)
\(884\) 0 0
\(885\) 20.9855 0.705419
\(886\) 0 0
\(887\) 10.9652 0.368177 0.184088 0.982910i \(-0.441067\pi\)
0.184088 + 0.982910i \(0.441067\pi\)
\(888\) 0 0
\(889\) 94.0814 3.15539
\(890\) 0 0
\(891\) −15.2858 −0.512094
\(892\) 0 0
\(893\) −10.0184 −0.335252
\(894\) 0 0
\(895\) −15.5373 −0.519355
\(896\) 0 0
\(897\) −42.8355 −1.43024
\(898\) 0 0
\(899\) 7.92340 0.264260
\(900\) 0 0
\(901\) −60.0666 −2.00111
\(902\) 0 0
\(903\) −92.9816 −3.09424
\(904\) 0 0
\(905\) 5.53931 0.184133
\(906\) 0 0
\(907\) −21.4093 −0.710883 −0.355441 0.934699i \(-0.615669\pi\)
−0.355441 + 0.934699i \(0.615669\pi\)
\(908\) 0 0
\(909\) −3.84679 −0.127590
\(910\) 0 0
\(911\) 2.28839 0.0758178 0.0379089 0.999281i \(-0.487930\pi\)
0.0379089 + 0.999281i \(0.487930\pi\)
\(912\) 0 0
\(913\) 24.7397 0.818763
\(914\) 0 0
\(915\) 26.5500 0.877716
\(916\) 0 0
\(917\) 21.8076 0.720149
\(918\) 0 0
\(919\) 4.90496 0.161800 0.0808999 0.996722i \(-0.474221\pi\)
0.0808999 + 0.996722i \(0.474221\pi\)
\(920\) 0 0
\(921\) −10.1273 −0.333707
\(922\) 0 0
\(923\) 18.8307 0.619821
\(924\) 0 0
\(925\) −13.5046 −0.444027
\(926\) 0 0
\(927\) 4.66657 0.153270
\(928\) 0 0
\(929\) 1.20150 0.0394200 0.0197100 0.999806i \(-0.493726\pi\)
0.0197100 + 0.999806i \(0.493726\pi\)
\(930\) 0 0
\(931\) 75.0834 2.46076
\(932\) 0 0
\(933\) −18.8896 −0.618419
\(934\) 0 0
\(935\) −9.63388 −0.315061
\(936\) 0 0
\(937\) −21.8422 −0.713554 −0.356777 0.934190i \(-0.616124\pi\)
−0.356777 + 0.934190i \(0.616124\pi\)
\(938\) 0 0
\(939\) 53.3986 1.74260
\(940\) 0 0
\(941\) −49.3887 −1.61003 −0.805013 0.593257i \(-0.797842\pi\)
−0.805013 + 0.593257i \(0.797842\pi\)
\(942\) 0 0
\(943\) 15.6844 0.510755
\(944\) 0 0
\(945\) −32.9294 −1.07119
\(946\) 0 0
\(947\) 5.09737 0.165642 0.0828211 0.996564i \(-0.473607\pi\)
0.0828211 + 0.996564i \(0.473607\pi\)
\(948\) 0 0
\(949\) −22.1153 −0.717893
\(950\) 0 0
\(951\) −26.9156 −0.872797
\(952\) 0 0
\(953\) 38.7086 1.25389 0.626947 0.779062i \(-0.284304\pi\)
0.626947 + 0.779062i \(0.284304\pi\)
\(954\) 0 0
\(955\) −14.1229 −0.457005
\(956\) 0 0
\(957\) 18.1563 0.586911
\(958\) 0 0
\(959\) 27.3011 0.881599
\(960\) 0 0
\(961\) −29.1643 −0.940785
\(962\) 0 0
\(963\) −1.10783 −0.0356992
\(964\) 0 0
\(965\) 19.4227 0.625238
\(966\) 0 0
\(967\) −17.8409 −0.573725 −0.286863 0.957972i \(-0.592612\pi\)
−0.286863 + 0.957972i \(0.592612\pi\)
\(968\) 0 0
\(969\) −26.9465 −0.865646
\(970\) 0 0
\(971\) −8.85285 −0.284101 −0.142051 0.989859i \(-0.545370\pi\)
−0.142051 + 0.989859i \(0.545370\pi\)
\(972\) 0 0
\(973\) −66.1422 −2.12042
\(974\) 0 0
\(975\) −21.6215 −0.692442
\(976\) 0 0
\(977\) 47.4775 1.51894 0.759471 0.650542i \(-0.225458\pi\)
0.759471 + 0.650542i \(0.225458\pi\)
\(978\) 0 0
\(979\) −24.3513 −0.778271
\(980\) 0 0
\(981\) −1.99633 −0.0637379
\(982\) 0 0
\(983\) −19.8880 −0.634328 −0.317164 0.948371i \(-0.602731\pi\)
−0.317164 + 0.948371i \(0.602731\pi\)
\(984\) 0 0
\(985\) 10.6549 0.339492
\(986\) 0 0
\(987\) −23.3676 −0.743798
\(988\) 0 0
\(989\) 77.4814 2.46376
\(990\) 0 0
\(991\) 10.7313 0.340890 0.170445 0.985367i \(-0.445479\pi\)
0.170445 + 0.985367i \(0.445479\pi\)
\(992\) 0 0
\(993\) −28.4510 −0.902865
\(994\) 0 0
\(995\) 6.98362 0.221396
\(996\) 0 0
\(997\) 58.1899 1.84289 0.921447 0.388504i \(-0.127008\pi\)
0.921447 + 0.388504i \(0.127008\pi\)
\(998\) 0 0
\(999\) −20.0526 −0.634435
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.u.1.20 26
4.3 odd 2 503.2.a.f.1.7 26
12.11 even 2 4527.2.a.o.1.20 26
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
503.2.a.f.1.7 26 4.3 odd 2
4527.2.a.o.1.20 26 12.11 even 2
8048.2.a.u.1.20 26 1.1 even 1 trivial