Properties

Label 8568.2.a.bj.1.3
Level $8568$
Weight $2$
Character 8568.1
Self dual yes
Analytic conductor $68.416$
Analytic rank $1$
Dimension $4$
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] = [8568,2,Mod(1,8568)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8568, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("8568.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8568 = 2^{3} \cdot 3^{2} \cdot 7 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8568.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: \(68.4158244518\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: 4.4.5225.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 8x^{2} + x + 11 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 952)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-1.88301\) of defining polynomial
Character \(\chi\) \(=\) 8568.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.78180 q^{5} -1.00000 q^{7} +O(q^{10})\) \(q+1.78180 q^{5} -1.00000 q^{7} -3.23607 q^{11} +2.47214 q^{13} -1.00000 q^{17} +7.76602 q^{19} +5.32962 q^{23} -1.82519 q^{25} -10.5657 q^{29} -9.64137 q^{31} -1.78180 q^{35} -1.62142 q^{37} -2.82519 q^{41} +0.587035 q^{43} -12.1871 q^{47} +1.00000 q^{49} -3.40321 q^{53} -5.76602 q^{55} +7.32962 q^{59} -3.82310 q^{61} +4.40485 q^{65} +0.394361 q^{67} +2.15137 q^{71} -14.3572 q^{73} +3.23607 q^{77} +4.18710 q^{79} -8.42108 q^{83} -1.78180 q^{85} +17.0446 q^{89} -2.47214 q^{91} +13.8375 q^{95} -7.72398 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + q^{5} - 4 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + q^{5} - 4 q^{7} - 4 q^{11} - 8 q^{13} - 4 q^{17} + 14 q^{19} - 8 q^{23} + 11 q^{25} - 4 q^{29} + 5 q^{31} - q^{35} - 4 q^{37} + 7 q^{41} + 19 q^{43} - 8 q^{47} + 4 q^{49} - 5 q^{53} - 6 q^{55} - 23 q^{61} + 8 q^{65} + 15 q^{67} - 2 q^{71} - 5 q^{73} + 4 q^{77} - 24 q^{79} - 10 q^{83} - q^{85} + 16 q^{89} + 8 q^{91} - 22 q^{95} - 15 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 0 0
\(4\) 0 0
\(5\) 1.78180 0.796845 0.398422 0.917202i \(-0.369558\pi\)
0.398422 + 0.917202i \(0.369558\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −3.23607 −0.975711 −0.487856 0.872924i \(-0.662221\pi\)
−0.487856 + 0.872924i \(0.662221\pi\)
\(12\) 0 0
\(13\) 2.47214 0.685647 0.342824 0.939400i \(-0.388617\pi\)
0.342824 + 0.939400i \(0.388617\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −1.00000 −0.242536
\(18\) 0 0
\(19\) 7.76602 1.78165 0.890824 0.454349i \(-0.150128\pi\)
0.890824 + 0.454349i \(0.150128\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 5.32962 1.11130 0.555651 0.831415i \(-0.312469\pi\)
0.555651 + 0.831415i \(0.312469\pi\)
\(24\) 0 0
\(25\) −1.82519 −0.365039
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −10.5657 −1.96200 −0.980999 0.194010i \(-0.937850\pi\)
−0.980999 + 0.194010i \(0.937850\pi\)
\(30\) 0 0
\(31\) −9.64137 −1.73164 −0.865821 0.500354i \(-0.833203\pi\)
−0.865821 + 0.500354i \(0.833203\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −1.78180 −0.301179
\(36\) 0 0
\(37\) −1.62142 −0.266559 −0.133280 0.991078i \(-0.542551\pi\)
−0.133280 + 0.991078i \(0.542551\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −2.82519 −0.441221 −0.220610 0.975362i \(-0.570805\pi\)
−0.220610 + 0.975362i \(0.570805\pi\)
\(42\) 0 0
\(43\) 0.587035 0.0895219 0.0447610 0.998998i \(-0.485747\pi\)
0.0447610 + 0.998998i \(0.485747\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −12.1871 −1.77767 −0.888836 0.458226i \(-0.848485\pi\)
−0.888836 + 0.458226i \(0.848485\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −3.40321 −0.467468 −0.233734 0.972301i \(-0.575094\pi\)
−0.233734 + 0.972301i \(0.575094\pi\)
\(54\) 0 0
\(55\) −5.76602 −0.777490
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 7.32962 0.954235 0.477118 0.878839i \(-0.341682\pi\)
0.477118 + 0.878839i \(0.341682\pi\)
\(60\) 0 0
\(61\) −3.82310 −0.489498 −0.244749 0.969586i \(-0.578706\pi\)
−0.244749 + 0.969586i \(0.578706\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 4.40485 0.546354
\(66\) 0 0
\(67\) 0.394361 0.0481788 0.0240894 0.999710i \(-0.492331\pi\)
0.0240894 + 0.999710i \(0.492331\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 2.15137 0.255321 0.127660 0.991818i \(-0.459253\pi\)
0.127660 + 0.991818i \(0.459253\pi\)
\(72\) 0 0
\(73\) −14.3572 −1.68039 −0.840194 0.542286i \(-0.817559\pi\)
−0.840194 + 0.542286i \(0.817559\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 3.23607 0.368784
\(78\) 0 0
\(79\) 4.18710 0.471086 0.235543 0.971864i \(-0.424313\pi\)
0.235543 + 0.971864i \(0.424313\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −8.42108 −0.924334 −0.462167 0.886793i \(-0.652928\pi\)
−0.462167 + 0.886793i \(0.652928\pi\)
\(84\) 0 0
\(85\) −1.78180 −0.193263
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 17.0446 1.80672 0.903361 0.428880i \(-0.141092\pi\)
0.903361 + 0.428880i \(0.141092\pi\)
\(90\) 0 0
\(91\) −2.47214 −0.259150
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 13.8375 1.41970
\(96\) 0 0
\(97\) −7.72398 −0.784251 −0.392126 0.919912i \(-0.628260\pi\)
−0.392126 + 0.919912i \(0.628260\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 13.6834 1.36155 0.680775 0.732492i \(-0.261643\pi\)
0.680775 + 0.732492i \(0.261643\pi\)
\(102\) 0 0
\(103\) 6.90854 0.680718 0.340359 0.940295i \(-0.389451\pi\)
0.340359 + 0.940295i \(0.389451\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 14.9868 1.44883 0.724413 0.689366i \(-0.242111\pi\)
0.724413 + 0.689366i \(0.242111\pi\)
\(108\) 0 0
\(109\) −19.3874 −1.85698 −0.928490 0.371358i \(-0.878892\pi\)
−0.928490 + 0.371358i \(0.878892\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −6.26971 −0.589805 −0.294902 0.955527i \(-0.595287\pi\)
−0.294902 + 0.955527i \(0.595287\pi\)
\(114\) 0 0
\(115\) 9.49631 0.885536
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 1.00000 0.0916698
\(120\) 0 0
\(121\) −0.527864 −0.0479876
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −12.1611 −1.08772
\(126\) 0 0
\(127\) 0.362807 0.0321939 0.0160970 0.999870i \(-0.494876\pi\)
0.0160970 + 0.999870i \(0.494876\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 1.94218 0.169689 0.0848446 0.996394i \(-0.472961\pi\)
0.0848446 + 0.996394i \(0.472961\pi\)
\(132\) 0 0
\(133\) −7.76602 −0.673400
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −5.17481 −0.442114 −0.221057 0.975261i \(-0.570951\pi\)
−0.221057 + 0.975261i \(0.570951\pi\)
\(138\) 0 0
\(139\) 4.13559 0.350776 0.175388 0.984499i \(-0.443882\pi\)
0.175388 + 0.984499i \(0.443882\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −8.00000 −0.668994
\(144\) 0 0
\(145\) −18.8259 −1.56341
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 2.17013 0.177784 0.0888921 0.996041i \(-0.471667\pi\)
0.0888921 + 0.996041i \(0.471667\pi\)
\(150\) 0 0
\(151\) 3.05917 0.248952 0.124476 0.992223i \(-0.460275\pi\)
0.124476 + 0.992223i \(0.460275\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −17.1790 −1.37985
\(156\) 0 0
\(157\) 9.95313 0.794346 0.397173 0.917744i \(-0.369991\pi\)
0.397173 + 0.917744i \(0.369991\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −5.32962 −0.420033
\(162\) 0 0
\(163\) 15.6613 1.22669 0.613345 0.789815i \(-0.289824\pi\)
0.613345 + 0.789815i \(0.289824\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 8.11908 0.628273 0.314137 0.949378i \(-0.398285\pi\)
0.314137 + 0.949378i \(0.398285\pi\)
\(168\) 0 0
\(169\) −6.88854 −0.529888
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −12.3640 −0.940018 −0.470009 0.882662i \(-0.655749\pi\)
−0.470009 + 0.882662i \(0.655749\pi\)
\(174\) 0 0
\(175\) 1.82519 0.137972
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 18.1233 1.35460 0.677298 0.735709i \(-0.263151\pi\)
0.677298 + 0.735709i \(0.263151\pi\)
\(180\) 0 0
\(181\) −18.4143 −1.36873 −0.684363 0.729142i \(-0.739919\pi\)
−0.684363 + 0.729142i \(0.739919\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −2.88904 −0.212406
\(186\) 0 0
\(187\) 3.23607 0.236645
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −10.7013 −0.774318 −0.387159 0.922013i \(-0.626543\pi\)
−0.387159 + 0.922013i \(0.626543\pi\)
\(192\) 0 0
\(193\) −1.51254 −0.108875 −0.0544376 0.998517i \(-0.517337\pi\)
−0.0544376 + 0.998517i \(0.517337\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 7.27180 0.518094 0.259047 0.965865i \(-0.416591\pi\)
0.259047 + 0.965865i \(0.416591\pi\)
\(198\) 0 0
\(199\) −23.1832 −1.64341 −0.821706 0.569912i \(-0.806977\pi\)
−0.821706 + 0.569912i \(0.806977\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 10.5657 0.741566
\(204\) 0 0
\(205\) −5.03393 −0.351585
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −25.1314 −1.73837
\(210\) 0 0
\(211\) 11.1535 0.767836 0.383918 0.923367i \(-0.374575\pi\)
0.383918 + 0.923367i \(0.374575\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 1.04598 0.0713351
\(216\) 0 0
\(217\) 9.64137 0.654499
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −2.47214 −0.166294
\(222\) 0 0
\(223\) −15.5636 −1.04222 −0.521108 0.853491i \(-0.674481\pi\)
−0.521108 + 0.853491i \(0.674481\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −15.3649 −1.01980 −0.509902 0.860232i \(-0.670318\pi\)
−0.509902 + 0.860232i \(0.670318\pi\)
\(228\) 0 0
\(229\) −14.9085 −0.985184 −0.492592 0.870260i \(-0.663950\pi\)
−0.492592 + 0.870260i \(0.663950\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −12.9085 −0.845666 −0.422833 0.906208i \(-0.638964\pi\)
−0.422833 + 0.906208i \(0.638964\pi\)
\(234\) 0 0
\(235\) −21.7150 −1.41653
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −7.05917 −0.456620 −0.228310 0.973589i \(-0.573320\pi\)
−0.228310 + 0.973589i \(0.573320\pi\)
\(240\) 0 0
\(241\) −22.0309 −1.41914 −0.709568 0.704637i \(-0.751110\pi\)
−0.709568 + 0.704637i \(0.751110\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 1.78180 0.113835
\(246\) 0 0
\(247\) 19.1987 1.22158
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −28.1267 −1.77534 −0.887671 0.460479i \(-0.847678\pi\)
−0.887671 + 0.460479i \(0.847678\pi\)
\(252\) 0 0
\(253\) −17.2470 −1.08431
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −9.04459 −0.564186 −0.282093 0.959387i \(-0.591029\pi\)
−0.282093 + 0.959387i \(0.591029\pi\)
\(258\) 0 0
\(259\) 1.62142 0.100750
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −14.6592 −0.903927 −0.451964 0.892036i \(-0.649276\pi\)
−0.451964 + 0.892036i \(0.649276\pi\)
\(264\) 0 0
\(265\) −6.06384 −0.372499
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 9.75926 0.595032 0.297516 0.954717i \(-0.403842\pi\)
0.297516 + 0.954717i \(0.403842\pi\)
\(270\) 0 0
\(271\) 15.1602 0.920918 0.460459 0.887681i \(-0.347685\pi\)
0.460459 + 0.887681i \(0.347685\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 5.90645 0.356172
\(276\) 0 0
\(277\) 10.7324 0.644846 0.322423 0.946596i \(-0.395503\pi\)
0.322423 + 0.946596i \(0.395503\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 2.06245 0.123036 0.0615179 0.998106i \(-0.480406\pi\)
0.0615179 + 0.998106i \(0.480406\pi\)
\(282\) 0 0
\(283\) −15.4364 −0.917597 −0.458798 0.888540i \(-0.651720\pi\)
−0.458798 + 0.888540i \(0.651720\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 2.82519 0.166766
\(288\) 0 0
\(289\) 1.00000 0.0588235
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −2.28503 −0.133493 −0.0667465 0.997770i \(-0.521262\pi\)
−0.0667465 + 0.997770i \(0.521262\pi\)
\(294\) 0 0
\(295\) 13.0599 0.760377
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 13.1755 0.761961
\(300\) 0 0
\(301\) −0.587035 −0.0338361
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −6.81200 −0.390054
\(306\) 0 0
\(307\) −20.6277 −1.17728 −0.588642 0.808394i \(-0.700337\pi\)
−0.588642 + 0.808394i \(0.700337\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −18.2250 −1.03344 −0.516721 0.856154i \(-0.672848\pi\)
−0.516721 + 0.856154i \(0.672848\pi\)
\(312\) 0 0
\(313\) −15.6609 −0.885205 −0.442602 0.896718i \(-0.645945\pi\)
−0.442602 + 0.896718i \(0.645945\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −1.75508 −0.0985750 −0.0492875 0.998785i \(-0.515695\pi\)
−0.0492875 + 0.998785i \(0.515695\pi\)
\(318\) 0 0
\(319\) 34.1913 1.91434
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −7.76602 −0.432113
\(324\) 0 0
\(325\) −4.51212 −0.250288
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 12.1871 0.671897
\(330\) 0 0
\(331\) −14.2829 −0.785059 −0.392530 0.919739i \(-0.628400\pi\)
−0.392530 + 0.919739i \(0.628400\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 0.702671 0.0383910
\(336\) 0 0
\(337\) 24.9290 1.35797 0.678983 0.734154i \(-0.262421\pi\)
0.678983 + 0.734154i \(0.262421\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 31.2001 1.68958
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −35.1934 −1.88928 −0.944640 0.328110i \(-0.893588\pi\)
−0.944640 + 0.328110i \(0.893588\pi\)
\(348\) 0 0
\(349\) 2.71029 0.145079 0.0725394 0.997366i \(-0.476890\pi\)
0.0725394 + 0.997366i \(0.476890\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 12.3580 0.657749 0.328874 0.944374i \(-0.393331\pi\)
0.328874 + 0.944374i \(0.393331\pi\)
\(354\) 0 0
\(355\) 3.83331 0.203451
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 5.04942 0.266498 0.133249 0.991083i \(-0.457459\pi\)
0.133249 + 0.991083i \(0.457459\pi\)
\(360\) 0 0
\(361\) 41.3111 2.17427
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −25.5817 −1.33901
\(366\) 0 0
\(367\) 25.9566 1.35492 0.677461 0.735559i \(-0.263080\pi\)
0.677461 + 0.735559i \(0.263080\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 3.40321 0.176686
\(372\) 0 0
\(373\) −8.23890 −0.426594 −0.213297 0.976987i \(-0.568420\pi\)
−0.213297 + 0.976987i \(0.568420\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −26.1198 −1.34524
\(378\) 0 0
\(379\) 11.7755 0.604866 0.302433 0.953171i \(-0.402201\pi\)
0.302433 + 0.953171i \(0.402201\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 23.2827 1.18969 0.594846 0.803839i \(-0.297213\pi\)
0.594846 + 0.803839i \(0.297213\pi\)
\(384\) 0 0
\(385\) 5.76602 0.293864
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 11.3717 0.576566 0.288283 0.957545i \(-0.406916\pi\)
0.288283 + 0.957545i \(0.406916\pi\)
\(390\) 0 0
\(391\) −5.32962 −0.269530
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 7.46058 0.375382
\(396\) 0 0
\(397\) 25.0645 1.25795 0.628977 0.777424i \(-0.283474\pi\)
0.628977 + 0.777424i \(0.283474\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −7.49631 −0.374348 −0.187174 0.982327i \(-0.559933\pi\)
−0.187174 + 0.982327i \(0.559933\pi\)
\(402\) 0 0
\(403\) −23.8348 −1.18730
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 5.24701 0.260085
\(408\) 0 0
\(409\) −11.4657 −0.566941 −0.283470 0.958981i \(-0.591486\pi\)
−0.283470 + 0.958981i \(0.591486\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −7.32962 −0.360667
\(414\) 0 0
\(415\) −15.0047 −0.736550
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −3.40998 −0.166588 −0.0832942 0.996525i \(-0.526544\pi\)
−0.0832942 + 0.996525i \(0.526544\pi\)
\(420\) 0 0
\(421\) 13.9524 0.679998 0.339999 0.940426i \(-0.389573\pi\)
0.339999 + 0.940426i \(0.389573\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 1.82519 0.0885349
\(426\) 0 0
\(427\) 3.82310 0.185013
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −27.7702 −1.33764 −0.668822 0.743423i \(-0.733201\pi\)
−0.668822 + 0.743423i \(0.733201\pi\)
\(432\) 0 0
\(433\) −16.1913 −0.778103 −0.389052 0.921216i \(-0.627197\pi\)
−0.389052 + 0.921216i \(0.627197\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 41.3899 1.97995
\(438\) 0 0
\(439\) −14.5266 −0.693318 −0.346659 0.937991i \(-0.612684\pi\)
−0.346659 + 0.937991i \(0.612684\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −33.5993 −1.59635 −0.798176 0.602424i \(-0.794202\pi\)
−0.798176 + 0.602424i \(0.794202\pi\)
\(444\) 0 0
\(445\) 30.3700 1.43968
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 13.7191 0.647447 0.323723 0.946152i \(-0.395065\pi\)
0.323723 + 0.946152i \(0.395065\pi\)
\(450\) 0 0
\(451\) 9.14252 0.430504
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −4.40485 −0.206503
\(456\) 0 0
\(457\) 9.18874 0.429831 0.214916 0.976633i \(-0.431052\pi\)
0.214916 + 0.976633i \(0.431052\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 27.3338 1.27306 0.636531 0.771251i \(-0.280369\pi\)
0.636531 + 0.771251i \(0.280369\pi\)
\(462\) 0 0
\(463\) 7.36191 0.342137 0.171069 0.985259i \(-0.445278\pi\)
0.171069 + 0.985259i \(0.445278\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 28.4805 1.31792 0.658960 0.752178i \(-0.270997\pi\)
0.658960 + 0.752178i \(0.270997\pi\)
\(468\) 0 0
\(469\) −0.394361 −0.0182099
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −1.89968 −0.0873476
\(474\) 0 0
\(475\) −14.1745 −0.650370
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −7.72259 −0.352854 −0.176427 0.984314i \(-0.556454\pi\)
−0.176427 + 0.984314i \(0.556454\pi\)
\(480\) 0 0
\(481\) −4.00836 −0.182766
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −13.7626 −0.624927
\(486\) 0 0
\(487\) 29.0172 1.31490 0.657448 0.753500i \(-0.271636\pi\)
0.657448 + 0.753500i \(0.271636\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −14.2250 −0.641964 −0.320982 0.947085i \(-0.604013\pi\)
−0.320982 + 0.947085i \(0.604013\pi\)
\(492\) 0 0
\(493\) 10.5657 0.475855
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −2.15137 −0.0965021
\(498\) 0 0
\(499\) −21.6971 −0.971294 −0.485647 0.874155i \(-0.661416\pi\)
−0.485647 + 0.874155i \(0.661416\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 26.4700 1.18024 0.590120 0.807316i \(-0.299081\pi\)
0.590120 + 0.807316i \(0.299081\pi\)
\(504\) 0 0
\(505\) 24.3811 1.08494
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −37.3380 −1.65498 −0.827488 0.561483i \(-0.810231\pi\)
−0.827488 + 0.561483i \(0.810231\pi\)
\(510\) 0 0
\(511\) 14.3572 0.635127
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 12.3096 0.542427
\(516\) 0 0
\(517\) 39.4383 1.73449
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −40.7287 −1.78436 −0.892179 0.451681i \(-0.850824\pi\)
−0.892179 + 0.451681i \(0.850824\pi\)
\(522\) 0 0
\(523\) 29.7056 1.29894 0.649468 0.760389i \(-0.274992\pi\)
0.649468 + 0.760389i \(0.274992\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 9.64137 0.419985
\(528\) 0 0
\(529\) 5.40485 0.234993
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −6.98426 −0.302522
\(534\) 0 0
\(535\) 26.7034 1.15449
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −3.23607 −0.139387
\(540\) 0 0
\(541\) −45.4631 −1.95461 −0.977305 0.211835i \(-0.932056\pi\)
−0.977305 + 0.211835i \(0.932056\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −34.5445 −1.47972
\(546\) 0 0
\(547\) −10.3113 −0.440879 −0.220440 0.975401i \(-0.570749\pi\)
−0.220440 + 0.975401i \(0.570749\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −82.0534 −3.49559
\(552\) 0 0
\(553\) −4.18710 −0.178054
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −16.8770 −0.715101 −0.357550 0.933894i \(-0.616388\pi\)
−0.357550 + 0.933894i \(0.616388\pi\)
\(558\) 0 0
\(559\) 1.45123 0.0613805
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 35.6197 1.50119 0.750597 0.660761i \(-0.229766\pi\)
0.750597 + 0.660761i \(0.229766\pi\)
\(564\) 0 0
\(565\) −11.1714 −0.469983
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −32.2578 −1.35232 −0.676159 0.736755i \(-0.736357\pi\)
−0.676159 + 0.736755i \(0.736357\pi\)
\(570\) 0 0
\(571\) 35.4946 1.48540 0.742702 0.669622i \(-0.233544\pi\)
0.742702 + 0.669622i \(0.233544\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −9.72758 −0.405668
\(576\) 0 0
\(577\) 33.1356 1.37945 0.689726 0.724071i \(-0.257731\pi\)
0.689726 + 0.724071i \(0.257731\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 8.42108 0.349365
\(582\) 0 0
\(583\) 11.0130 0.456113
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 35.1397 1.45037 0.725186 0.688553i \(-0.241754\pi\)
0.725186 + 0.688553i \(0.241754\pi\)
\(588\) 0 0
\(589\) −74.8751 −3.08518
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 30.0729 1.23495 0.617474 0.786591i \(-0.288156\pi\)
0.617474 + 0.786591i \(0.288156\pi\)
\(594\) 0 0
\(595\) 1.78180 0.0730466
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −20.6464 −0.843591 −0.421796 0.906691i \(-0.638600\pi\)
−0.421796 + 0.906691i \(0.638600\pi\)
\(600\) 0 0
\(601\) −20.5961 −0.840134 −0.420067 0.907493i \(-0.637993\pi\)
−0.420067 + 0.907493i \(0.637993\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −0.940548 −0.0382387
\(606\) 0 0
\(607\) −14.9199 −0.605582 −0.302791 0.953057i \(-0.597918\pi\)
−0.302791 + 0.953057i \(0.597918\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −30.1282 −1.21886
\(612\) 0 0
\(613\) −24.9575 −1.00802 −0.504011 0.863697i \(-0.668143\pi\)
−0.504011 + 0.863697i \(0.668143\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 7.19596 0.289698 0.144849 0.989454i \(-0.453730\pi\)
0.144849 + 0.989454i \(0.453730\pi\)
\(618\) 0 0
\(619\) −17.7770 −0.714517 −0.357258 0.934006i \(-0.616288\pi\)
−0.357258 + 0.934006i \(0.616288\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −17.0446 −0.682877
\(624\) 0 0
\(625\) −12.5427 −0.501708
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 1.62142 0.0646501
\(630\) 0 0
\(631\) 35.3428 1.40698 0.703488 0.710708i \(-0.251625\pi\)
0.703488 + 0.710708i \(0.251625\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 0.646450 0.0256536
\(636\) 0 0
\(637\) 2.47214 0.0979496
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 3.66153 0.144622 0.0723108 0.997382i \(-0.476963\pi\)
0.0723108 + 0.997382i \(0.476963\pi\)
\(642\) 0 0
\(643\) 29.6783 1.17040 0.585199 0.810890i \(-0.301016\pi\)
0.585199 + 0.810890i \(0.301016\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −28.9758 −1.13916 −0.569579 0.821937i \(-0.692894\pi\)
−0.569579 + 0.821937i \(0.692894\pi\)
\(648\) 0 0
\(649\) −23.7191 −0.931058
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −19.2003 −0.751367 −0.375684 0.926748i \(-0.622592\pi\)
−0.375684 + 0.926748i \(0.622592\pi\)
\(654\) 0 0
\(655\) 3.46058 0.135216
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 5.13350 0.199973 0.0999865 0.994989i \(-0.468120\pi\)
0.0999865 + 0.994989i \(0.468120\pi\)
\(660\) 0 0
\(661\) −34.8839 −1.35683 −0.678413 0.734681i \(-0.737332\pi\)
−0.678413 + 0.734681i \(0.737332\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −13.8375 −0.536595
\(666\) 0 0
\(667\) −56.3111 −2.18037
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 12.3718 0.477609
\(672\) 0 0
\(673\) −46.2976 −1.78464 −0.892320 0.451403i \(-0.850924\pi\)
−0.892320 + 0.451403i \(0.850924\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 8.29450 0.318784 0.159392 0.987215i \(-0.449047\pi\)
0.159392 + 0.987215i \(0.449047\pi\)
\(678\) 0 0
\(679\) 7.72398 0.296419
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 43.0198 1.64611 0.823053 0.567964i \(-0.192269\pi\)
0.823053 + 0.567964i \(0.192269\pi\)
\(684\) 0 0
\(685\) −9.22047 −0.352296
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −8.41321 −0.320518
\(690\) 0 0
\(691\) −36.3817 −1.38403 −0.692013 0.721885i \(-0.743276\pi\)
−0.692013 + 0.721885i \(0.743276\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 7.36880 0.279514
\(696\) 0 0
\(697\) 2.82519 0.107012
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −18.5920 −0.702208 −0.351104 0.936336i \(-0.614194\pi\)
−0.351104 + 0.936336i \(0.614194\pi\)
\(702\) 0 0
\(703\) −12.5920 −0.474915
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −13.6834 −0.514618
\(708\) 0 0
\(709\) 11.0871 0.416384 0.208192 0.978088i \(-0.433242\pi\)
0.208192 + 0.978088i \(0.433242\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −51.3849 −1.92438
\(714\) 0 0
\(715\) −14.2544 −0.533084
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −0.357237 −0.0133227 −0.00666135 0.999978i \(-0.502120\pi\)
−0.00666135 + 0.999978i \(0.502120\pi\)
\(720\) 0 0
\(721\) −6.90854 −0.257287
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 19.2844 0.716205
\(726\) 0 0
\(727\) −3.07614 −0.114088 −0.0570439 0.998372i \(-0.518167\pi\)
−0.0570439 + 0.998372i \(0.518167\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −0.587035 −0.0217123
\(732\) 0 0
\(733\) 17.9930 0.664588 0.332294 0.943176i \(-0.392177\pi\)
0.332294 + 0.943176i \(0.392177\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −1.27618 −0.0470086
\(738\) 0 0
\(739\) −8.47140 −0.311625 −0.155813 0.987787i \(-0.549800\pi\)
−0.155813 + 0.987787i \(0.549800\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −23.1587 −0.849612 −0.424806 0.905284i \(-0.639658\pi\)
−0.424806 + 0.905284i \(0.639658\pi\)
\(744\) 0 0
\(745\) 3.86674 0.141666
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −14.9868 −0.547605
\(750\) 0 0
\(751\) −19.3849 −0.707363 −0.353682 0.935366i \(-0.615070\pi\)
−0.353682 + 0.935366i \(0.615070\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 5.45083 0.198376
\(756\) 0 0
\(757\) 48.9669 1.77973 0.889866 0.456222i \(-0.150798\pi\)
0.889866 + 0.456222i \(0.150798\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 25.9354 0.940158 0.470079 0.882624i \(-0.344225\pi\)
0.470079 + 0.882624i \(0.344225\pi\)
\(762\) 0 0
\(763\) 19.3874 0.701872
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 18.1198 0.654269
\(768\) 0 0
\(769\) −5.37649 −0.193881 −0.0969407 0.995290i \(-0.530906\pi\)
−0.0969407 + 0.995290i \(0.530906\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 3.82642 0.137627 0.0688135 0.997630i \(-0.478079\pi\)
0.0688135 + 0.997630i \(0.478079\pi\)
\(774\) 0 0
\(775\) 17.5974 0.632116
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −21.9405 −0.786100
\(780\) 0 0
\(781\) −6.96198 −0.249119
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 17.7345 0.632970
\(786\) 0 0
\(787\) −1.50487 −0.0536427 −0.0268214 0.999640i \(-0.508539\pi\)
−0.0268214 + 0.999640i \(0.508539\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 6.26971 0.222925
\(792\) 0 0
\(793\) −9.45123 −0.335623
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −28.4448 −1.00757 −0.503783 0.863830i \(-0.668059\pi\)
−0.503783 + 0.863830i \(0.668059\pi\)
\(798\) 0 0
\(799\) 12.1871 0.431149
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 46.4610 1.63957
\(804\) 0 0
\(805\) −9.49631 −0.334701
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 29.5952 1.04051 0.520255 0.854011i \(-0.325837\pi\)
0.520255 + 0.854011i \(0.325837\pi\)
\(810\) 0 0
\(811\) 49.2818 1.73052 0.865259 0.501325i \(-0.167154\pi\)
0.865259 + 0.501325i \(0.167154\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 27.9053 0.977481
\(816\) 0 0
\(817\) 4.55892 0.159497
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 39.0114 1.36151 0.680753 0.732513i \(-0.261653\pi\)
0.680753 + 0.732513i \(0.261653\pi\)
\(822\) 0 0
\(823\) −36.4772 −1.27152 −0.635758 0.771888i \(-0.719312\pi\)
−0.635758 + 0.771888i \(0.719312\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 7.06846 0.245795 0.122897 0.992419i \(-0.460781\pi\)
0.122897 + 0.992419i \(0.460781\pi\)
\(828\) 0 0
\(829\) −19.5952 −0.680568 −0.340284 0.940323i \(-0.610523\pi\)
−0.340284 + 0.940323i \(0.610523\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −1.00000 −0.0346479
\(834\) 0 0
\(835\) 14.4666 0.500636
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −5.42576 −0.187318 −0.0936589 0.995604i \(-0.529856\pi\)
−0.0936589 + 0.995604i \(0.529856\pi\)
\(840\) 0 0
\(841\) 82.6338 2.84944
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) −12.2740 −0.422238
\(846\) 0 0
\(847\) 0.527864 0.0181376
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) −8.64153 −0.296228
\(852\) 0 0
\(853\) 8.52577 0.291917 0.145958 0.989291i \(-0.453373\pi\)
0.145958 + 0.989291i \(0.453373\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −28.9710 −0.989630 −0.494815 0.868998i \(-0.664764\pi\)
−0.494815 + 0.868998i \(0.664764\pi\)
\(858\) 0 0
\(859\) 9.67776 0.330201 0.165100 0.986277i \(-0.447205\pi\)
0.165100 + 0.986277i \(0.447205\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −9.67293 −0.329270 −0.164635 0.986355i \(-0.552645\pi\)
−0.164635 + 0.986355i \(0.552645\pi\)
\(864\) 0 0
\(865\) −22.0302 −0.749048
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −13.5498 −0.459644
\(870\) 0 0
\(871\) 0.974913 0.0330337
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 12.1611 0.411121
\(876\) 0 0
\(877\) −40.8080 −1.37799 −0.688995 0.724766i \(-0.741948\pi\)
−0.688995 + 0.724766i \(0.741948\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −0.544255 −0.0183364 −0.00916821 0.999958i \(-0.502918\pi\)
−0.00916821 + 0.999958i \(0.502918\pi\)
\(882\) 0 0
\(883\) 7.31126 0.246043 0.123022 0.992404i \(-0.460742\pi\)
0.123022 + 0.992404i \(0.460742\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −13.8600 −0.465374 −0.232687 0.972552i \(-0.574752\pi\)
−0.232687 + 0.972552i \(0.574752\pi\)
\(888\) 0 0
\(889\) −0.362807 −0.0121682
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −94.6453 −3.16718
\(894\) 0 0
\(895\) 32.2920 1.07940
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 101.868 3.39748
\(900\) 0 0
\(901\) 3.40321 0.113378
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −32.8106 −1.09066
\(906\) 0 0
\(907\) 50.8411 1.68815 0.844075 0.536225i \(-0.180150\pi\)
0.844075 + 0.536225i \(0.180150\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −25.8459 −0.856311 −0.428156 0.903705i \(-0.640836\pi\)
−0.428156 + 0.903705i \(0.640836\pi\)
\(912\) 0 0
\(913\) 27.2512 0.901883
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −1.94218 −0.0641365
\(918\) 0 0
\(919\) 44.8471 1.47937 0.739684 0.672954i \(-0.234975\pi\)
0.739684 + 0.672954i \(0.234975\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 5.31848 0.175060
\(924\) 0 0
\(925\) 2.95940 0.0973044
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 44.2402 1.45147 0.725737 0.687972i \(-0.241499\pi\)
0.725737 + 0.687972i \(0.241499\pi\)
\(930\) 0 0
\(931\) 7.76602 0.254521
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 5.76602 0.188569
\(936\) 0 0
\(937\) −41.2813 −1.34860 −0.674300 0.738457i \(-0.735555\pi\)
−0.674300 + 0.738457i \(0.735555\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 11.1816 0.364509 0.182254 0.983251i \(-0.441661\pi\)
0.182254 + 0.983251i \(0.441661\pi\)
\(942\) 0 0
\(943\) −15.0572 −0.490330
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −18.5810 −0.603802 −0.301901 0.953339i \(-0.597621\pi\)
−0.301901 + 0.953339i \(0.597621\pi\)
\(948\) 0 0
\(949\) −35.4930 −1.15215
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −13.3359 −0.431993 −0.215997 0.976394i \(-0.569300\pi\)
−0.215997 + 0.976394i \(0.569300\pi\)
\(954\) 0 0
\(955\) −19.0675 −0.617011
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 5.17481 0.167103
\(960\) 0 0
\(961\) 61.9561 1.99858
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) −2.69505 −0.0867567
\(966\) 0 0
\(967\) 10.9877 0.353341 0.176670 0.984270i \(-0.443467\pi\)
0.176670 + 0.984270i \(0.443467\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −47.3831 −1.52059 −0.760297 0.649575i \(-0.774947\pi\)
−0.760297 + 0.649575i \(0.774947\pi\)
\(972\) 0 0
\(973\) −4.13559 −0.132581
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −1.49148 −0.0477166 −0.0238583 0.999715i \(-0.507595\pi\)
−0.0238583 + 0.999715i \(0.507595\pi\)
\(978\) 0 0
\(979\) −55.1574 −1.76284
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 53.9441 1.72055 0.860275 0.509830i \(-0.170292\pi\)
0.860275 + 0.509830i \(0.170292\pi\)
\(984\) 0 0
\(985\) 12.9569 0.412841
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 3.12867 0.0994860
\(990\) 0 0
\(991\) −51.2401 −1.62769 −0.813847 0.581079i \(-0.802631\pi\)
−0.813847 + 0.581079i \(0.802631\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −41.3077 −1.30954
\(996\) 0 0
\(997\) −12.0544 −0.381766 −0.190883 0.981613i \(-0.561135\pi\)
−0.190883 + 0.981613i \(0.561135\pi\)
\(998\) 0 0
\(999\) 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 8568.2.a.bj.1.3 4
3.2 odd 2 952.2.a.g.1.4 4
12.11 even 2 1904.2.a.q.1.1 4
21.20 even 2 6664.2.a.o.1.1 4
24.5 odd 2 7616.2.a.bj.1.1 4
24.11 even 2 7616.2.a.bp.1.4 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
952.2.a.g.1.4 4 3.2 odd 2
1904.2.a.q.1.1 4 12.11 even 2
6664.2.a.o.1.1 4 21.20 even 2
7616.2.a.bj.1.1 4 24.5 odd 2
7616.2.a.bp.1.4 4 24.11 even 2
8568.2.a.bj.1.3 4 1.1 even 1 trivial